\documentclass[12pt]{article} \usepackage[centertags, sumlimits, namelimits]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{newlfont} \usepackage{longtable} \textwidth= 6.25in \textheight= 9.0in \topmargin = -10pt \evensidemargin=10pt \oddsidemargin=10pt \headsep=25pt \parskip=10pt \font\smallit=cmti10 \font\smalltt=cmtt10 \font\smallrm=cmr9 \tolerance=1000 \hfuzz30pt \newlength{\defbaselineskip} \setlength{\defbaselineskip}{\baselineskip} \newcommand{\setlinespacing}[1] {\setlength{\baselineskip}{#1 \defbaselineskip}} \newcommand{\doublespacing}{\setlength{\baselineskip} {2.0 \defbaselineskip}} \newcommand{\singlespacing}{\setlength{\baselineskip}{\defbaselineskip}} \DeclareMathOperator{\ord}{ord} \newtheorem{theorem}{Theorem} \newtheorem{cor}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition}[section] \theoremstyle{remark} \newtheorem{rem}{Remark} \newcommand{\Sum}[1]{\ensuremath{\sum_{#1=0}^{\infty}}} \newcommand{\Prod}[1]{\ensuremath{\prod_{#1=1}^{\infty}}} \allowdisplaybreaks \begin{document} \begin{center} {\bf A MODULAR IDENTITY FOR THE RAMANUJAN IDENTITY MODULO 35} \vskip 20pt {\bf Adrian Dan Stanger}\\ {\smallit Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA}\\ {\tt adrian@math.byu.edu} \end{center} \vskip 30pt \centerline{\smallit Received: 3/20/02, Revised: 6/27/02, Accepted: 6/27/02, Published: 6/28/02} \vskip 30pt \centerline{\bf Abstract} \noindent In Rademacher's proof \cite{Rad} of the Ramanujan identities modulo 5 and 7, the function $$ L_{k} (z) = \eta(kz) \sum_{\mu=0}^{k-1} \left\{\frac{1}{\eta \left(\frac{z+24\mu}{k} \right) } \right\}$$ appears. This is a modular function on $ \Gamma_0(k)$, provided that $(k,6) = 1$. The basic identities \begin{gather*} p(5n + 4) \equiv 0 \bmod 5, \\ p(7n + 5) \equiv 0 \bmod 7, \end{gather*} (as well as some generalizations to higher powers of 5 and 7) are proved by examining the coefficients of the expansions of $L_5$ and $L_7$ at infinity. The two identities above immediately imply the following identity: $$ p(35n + 19) \equiv 0 \bmod 35. $$ Rademacher asked if there was a corresponding modular identity. Such an identity is derived in this paper. Also, another identity is obtained from the expansion of $L_{35}$ at 0. \pagestyle{myheadings} \markright{\smalltt INTEGERS: \smallrm ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 1 (2002), \#A09\hfill} \thispagestyle{empty} \baselineskip=15pt \vskip 30pt \section*{\normalsize 1. Generalities} \subsection*{\normalsize 1.1 Basics} Throughout, $ p(n) $ refers to the unrestricted partition function. Euler proved the identity $$ \sum_{n=0}^{\infty} p(n) q^n = \prod_{m=1}^{\infty} \frac{1}{1-q^m}. $$ We will see that this ties the partition function intimately to certain types of modular functions (and forms). Let $ \Gamma(1) = \mathrm{PSL}_{2}(\mathbb{Z}) $, which is the special linear group modulo its center $(\pm I).$ $\Gamma(1)$ is generated by the two transformations $$ S = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \ \ \ T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}.$$ We'll have need also for the product $TST,$ which we will call $W$. Then in $\Gamma(1)$, we have: $$ W = \begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix}. $$ $\Gamma(1) $ acts faithfully on the upper half plane $ \mathcal{H}=\{z \in \mathbb{C}: \Im(z)>0 \}$ via linear fractional transformation. Throughout, we will identify a linear fractional transformation and its matrix. We let $ q = e ^{2 \pi i z}$ throughout. In what follows, Dedekind's function $\eta(z)$ will be used, so we remind the reader of some known facts. Recall $ \eta (z) = e^{\frac{\pi i z}{12}}\prod_{n=1}^{\infty} \left(1-e^{2\pi i n z}\right)$, and note that $\eta(z)$ is periodic of period 24. It is pole-free and zero-free throughout $ \mathcal{H}.$ It is a modular form of weight one half, and has a multiplier which is a $24^{th}$ root of unity in its transformation equation. The root of unity $\varepsilon(a,b,c,d)$ is determined by the following formulas (provided $ c \ne 0$): \begin{multline} \label{root} \varepsilon(a,b,c,d) = \exp\left\{\pi i\left(\frac{a+d}{12c} + s(-d,c)\right) \right\}, \\ s(h,k) = \sum_{r=1}^{k-1} \frac{r}{k} \left(\frac{hr}{k}-\left[\frac{hr}{k}\right]-\frac{1}{2}\right), \end{multline} where [ ] is the greatest integer function. For properties of the Dedekind $s(h,k)$ sum see \cite{RWded, RGded}. We will often write just $\varepsilon$ in place of the more cumbersome $\varepsilon(a,b,c,d)$. Thus, for any $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma(1) $, $\eta (Az) = \varepsilon (cz + d)^{1/2} \eta (z).$ There is a relationship between $ \eta $ and $ p $. Namely, $$ \sum_{n=0}^{\infty} p(n) q^n = \frac{q^{\frac{1}{24}}}{\eta(z)}. $$ Also, $ L_k$ has $q$-expansion given by: \begin{multline}\label{q-expansion} L_k (z) = kq^{1 + \left[\frac{k}{24}\right ]}\prod_{m=1}^{\infty} (1-q^{km}) \sum_{n=0}^{\infty} p(kn + \Delta) q^n,\ \\ \Delta \ \mathrm{least \ positive \ solution \ of} \ 24n \equiv 1 \bmod k. \end{multline} It is clear that $L_k$ has all its poles located at the cusps of $ \Gamma_0 (k). $ Note that when $ k = 35 \ (\mathrm{resp.} \ 5, 7)$, $\Delta = 19 \ (\mathrm{resp.} \ 4, 5)$. This explains the form for the identities listed earlier. That $L_k(z)$ is a modular function on $\Gamma_0(k)$ may be seen as follows. One can show that $\frac{\eta(kz)}{\eta \left(\frac{z}{k}\right)}$ is a modular function on $\Gamma_0^0(k)$ using known facts about Dedekind sums (see particularly Theorems 17 and 18 of \cite{RWded}). The matrices $\begin{pmatrix} 1 & 24 \mu \\ 0 & 1 \end{pmatrix}, (0 \le \mu \le k-1)$ form a set of coset representatives for $\Gamma_0^0(k)$ in $\Gamma_0(k).$ Therefore, when we sum over this set of coset representatives to obtain $L_k(z),$ we obtain a modular function on $\Gamma_0(k).$ Let $\omega \in \mathbb{Q}\bigcup\{i\infty\}$ be a cusp for $\Gamma_0(k).$ We will let $H_{\omega} $ be the set of modular functions on $\Gamma_0(k)$ having poles only at $\omega$. It is with this space of functions that we will work. In particular we will consider functions in $H_{\omega}$, where $\omega=0$ or $i\infty.$ We mention also the Fricke involution $S_k(z) = \frac{-1}{kz}.$ Let $\mathcal{F}$ denote the set of modular functions on $\Gamma_0(k),$ and $f \in \mathcal{F}.$ $S_k(z)$ is an operator from $\mathcal{F}$ to itself. Moreover, it interchanges the polar orders of $f$ at $i \infty$ and $0$ in the appropriate uniformizing variables (i.e, the expansion of $f(S_k(z))$ at $i\infty$ has the same order in the variable $q$, as the expansion of $f(z)$ at zero has in the variable $q^{\frac{1}{k}}).$ For the rest of section 1, let $k=pQ,$ where $p$ and $Q$ are two distinct primes strictly greater than 3. Then $S_k(z)$ interchanges the polar orders of $f$ at the cusps $\frac{1}{p}$ and $\frac{1}{Q}$ as well. \subsection*{\normalsize 1.2 Evaluation at $0$} Pursuant to our task, we need to determine the order of $L_k$ at each of the cusps. We determine the expansion about $0$. To do so we subject $L_k$ to the transformation $S_k$ and then compute the expansion as usual. $$L_k(S_k z) = \eta \left[ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} z \right ] \sum_{\mu = 0}^{k-1} \left\{\eta \left[\begin{pmatrix} 24\mu k & -1 \\ k^2 & 0 \end{pmatrix} z \right] \right\}^{-1}. $$ We need to factor the inner matrix as the product of a unimodular matrix and an upper triangular matrix. Upon factoring and applying the transformation identities we obtain: $$ L_k(S_k z) = \frac{1}{\sqrt{k}} \, \eta\left(z\right) \sum_{\mu=0}^{k-1} \frac{1}{\varepsilon(\frac{24\mu}{\delta}, b, \frac{k}{\delta}, d) \, {\delta}^{1/2} \, \eta\left(\delta^2 z + \frac{\delta y}{k}\right)}, $$ where $\delta = (\mu, k)$, and the matrix inside factors as: $$ \begin{pmatrix} 24\mu / \delta & b\\ k/ \delta & d \end{pmatrix} \begin{pmatrix} \delta k & y \\ 0 & k/ \delta \end{pmatrix},$$ for some integers $b,d$ and $y$ such that \begin{gather*} \frac{24\mu}{\delta}y + \frac{bk}{\delta} = -1, \\ \frac{24\mu}{\delta}d - \frac{bk}{\delta} = 1. \end{gather*} Together, the two equations yield that $y=-d$ (in the case $\mu = 0$ this is clear). Thus we obtain the following expression: \begin{equation} \label{at zero} L_k(S_k z) = \frac{1}{\sqrt{k}} \prod_{m=1}^{\infty} \left(1-q^m\right) \sum_{\mu=0}^{k-1} \left\{ \varepsilon \,\delta^{\frac{1}{2}} \, e^{\frac{\pi i y \delta}{12k}} \, q^{\frac{\delta^2 - 1}{24}} \prod_{n=1}^{\infty} \left(1-e^{2 \pi i \delta yn} q^{\delta^2n} \right)\right\}^{-1}. \end{equation} Now, since $\mu = 0$ is the only term of the sum in which $\delta=k $, and this yields the largest negative power of $q$, we see that $\ord_0(L_k) \ge (1-k^2)/24$. We will return to equation~\ref{at zero} later. \subsection*{\normalsize 1.3 Evaluation at all other cusps} Since k is square-free, all the other cusps of $\Gamma_0(k)$ occur at the points $\{1/l | \ 1< l < k, \ l|k\,\}$, in other words, at $\frac{1}{p}$ and $\frac{1}{Q}.$ To determine the order of $L_k$ at $1/l$, we subject it to the transformation $W^{-l}$ and then evaluate. Thus, we obtain \begin{gather*} L_k \left(W^{-l} z\right) = \eta \left(k W^{-l} z \right) \sum_{\mu = 0}^{k-1} \left\{\eta \left(\frac{W^{-l} z + 24\mu}{k}\right)\right\}^{-1} \\ \quad = \eta \left[\begin{pmatrix} k/l & 1 - k/l \\ -1 & 1 \end{pmatrix} \begin{pmatrix} l & k/l - 1 \\ 0 & k/l \end{pmatrix} z \right] \sum_{\mu = 0}^{k-1} \left\{\eta \left[ \begin{pmatrix} 1 - 24l\mu & 24\mu \\ -k l & k \end{pmatrix} z \right] \right\}^{-1}. \end{gather*} Now, let $\delta = (1 - 24\mu l , k/l)$. Then the matrix inside factors as $$ \begin{pmatrix} (1 - 24\mu l)/\delta & b \\ (-k l)/\delta & d \end{pmatrix} \begin{pmatrix} \delta & y \\ 0 & k/\delta \end{pmatrix}, $$ for integers $b,d$ and $y$ such that \begin{gather*} \frac{1 - 24\mu l }{\delta}\,d + \frac{bk}{\delta}\,l = 1, \\ \frac{1 - 24\mu l }{\delta}\, y + \frac{bk}{\delta} = 24\mu, \\ \frac{-kl}{\delta}\,y + \frac{kd}{\delta} = k. \end{gather*} The final equation above yields $ -l y + d =\delta$. Using this fact we see that \begin{gather*} L_k(W^{-l}(z)) = e^{-\frac{k+l}{12l} \pi i} \sqrt{l/k} \,\, \eta \left(\frac{l^2 z - l + k}{k}\right) \sum_{\mu=0}^{k-1} \left\{\varepsilon \,\delta^{1/2} \,\eta \left(\frac{\delta^2 z + \delta y}{k}\right)\right\}^{-1} \ \\ = e^{-\frac{k + l}{12l} \pi i} \sqrt{l/k} \prod_{m=1}^{\infty} \left(1 - \rho^{(k-l)n} q^{\frac{l^2}{k} n}\right) \times \\ \sum_{\mu=0}^{k-1} \left\{\varepsilon \, \delta^{1/2} \rho^{\frac{dy+k-l}{24}} q^{\frac{\delta^2-l^2}{24k}} \prod_{n=1}^{\infty}\left(1-\rho^{\delta y n} q^{\frac{\delta^2}{k} n} \right) \right\}^{-1}, \ \ \left(\rho = e^{\frac{2 \pi i}{k}}\right). \end{gather*} \vskip 30pt \section*{\normalsize 2. Specialization to k = 35} \subsection*{\normalsize 2.1 Obtaining an explicit polynomial identity at $i\infty$} Now, we take $k=35$ for the remainder of this paper. We see from the results above that $L_{35} (z)$ has a zero of order $2$ at $i\infty$ in the variable $q$, a pole of order $51$ at $0$ in the variable $q^{\frac{1}{35}}$ , is holomorphic at $1/5$ in $q^{\frac{1}{7}},$ and has a zero of order at least $1$ at $1/7$ in the variable $q^{\frac{1}{5}}.$ We refer the reader to \cite{NewClass} for an explicit construction of the basis functions with which we will work, and for an account of the functions $A_9$ and $B_8$ (as well as other similar functions used in deriving the basis of functions at infinity). In summary, the basis functions $B_4, B_5, B_6, $ and $B_7$ all have poles only at $ i \infty$ in the same order as the value of their respective subscript. Similarly, $A_9$ and $B_8$ have poles of order 9 and 8, respectively, at $i \infty$, and $A_9$ has zeroes of orders 5, 1, and 3 at $0, 1/5$ and $1/7$ respectively, while $B_8$ has zeros of orders 6 and 2 at 0 and $1/7$ respectively. Suffice it to say that $G(z) = A_9 (z) B_8^8 (z) L_{35} (z)$ is pole-free in $\mathcal{H}$ and has order $-71$ at $i\infty$, $2$ at $0$, $0$ (or possibly greater) at $1/5$, and $20$ at $1/7$. So we have: \begin{theorem} $G(z) \in \mathbb{C}[B_4, B_5, B_6, B_7].$ \end{theorem} The proof follows from the work in the previous section and the fact that the $B_i$ do form a polynomial basis for the set of modular functions in $H_{i\infty}$. Since the order at $i\infty$ is known, the polynomial is determined, using Mathematica to calculate the coefficients. Among the possible ways of choosing an appropriate form, it was determined to use $B_7$ to keep the total degree (as a polynomial in the $B_i$) as small as possible. For instance the terms matching $q^{-65}, \ q^{-64}, \ q^{-63} $ are $B_7^8 B_5 B_4, \ B_7^8B_4^2, \ B_7^9$ respectively. Call the polynomial that we obtain $P_{\infty}$. Then $L_{35} (z) = A_9^{-1}B_8^{-8} P_{\infty}$. \begin{theorem} \label{polynomial} Let $c_i$ be the coefficient on the term of $P_{\infty}$ corresponding to $q^i$. Then the polynomial $P_{\infty}$ is determined by the coefficients in the appendix in table 1, where $c_i$ is the given $i^{th}$ coefficient multiplied by $1225$ $(={35}^2)$. \end{theorem} \begin{cor} $p(35n+19)\equiv 0 \bmod 35. $ \end{cor} \begin{rem} This identity also shows how the genus dramatically affects a polynomial identity associated with such an identity for the partition function. For the cases $k=5$ and $k=7$, where the genus is $0$, Rademacher obtains a linear polynomial and a quadratic polynomial, respectively, in the respective basis function. Here, the genus is 3 and a polynomial of degree $11$ in the \emph{four} basis functions is obtained. In a similar manner, modular identities for the congruences for 55 and 77 may surely be constructed. It is expected that more basis functions and a higher-degree polynomial will be necessary. \end{rem} \subsection*{\normalsize 2.2 An Identity at $0$} We return to equation~\ref{at zero}. Recalling the connection with the partition function $p$, and separating into cases depending on $\delta$, we have : $$ L_k(S_k(z)) = \sum_{\delta | 35} \frac{q^{\frac{1-\delta^2}{24}}}{\sqrt{k}} \sum_{\mu = 0}^{34} \varepsilon^{-1} \delta ^{-\frac{1}{2}} \sum_{n=0}^{\infty} e^{\frac{(24 n -1)\delta y \pi i}{12k}} p(n) q ^{\delta^2 n} \prod_{m=1}^{\infty} (1-q^m). $$ When we group the outer sum into terms by $\delta$, we obtain Gauss sums for the Jacobi-Legendre symbol of the appropriate modulus. We have the following terms: \begin{gather*} \delta = 35: \ \ \ \ \frac{1}{35q^{51}} \prod_{m=1}^{\infty} (1-q^m) \sum_{n=0}^{\infty} p(n)q^{1225n}, \\ \delta = 7: \ \ \ \ - \frac{1}{7q^2} \prod_{m=1}^{\infty} (1-q^m) \sum_{n=0}^{\infty} \left ( \frac{n-4}{5} \right ) p(n)q^{49n}, \\ \delta = 5: \ \ \ \ - \frac{1}{5q} \prod_{m=1}^{\infty} (1-q^m) \sum_{n=0}^{\infty} \left ( \frac{n-5}{7} \right ) p(n)q^{25n}, \\ \delta = 1: \ \ \ \ \prod_{m=1}^{\infty} (1-q^m) \sum_{n=0}^{\infty} \left ( \frac{n-19}{35} \right ) p(n)q^n, \end{gather*} where (-) indicates the Jacobi-Legendre symbol. Now, $L_{35}(S_{35} z)$ has order $-51$ (resp. $ 2, 1, -1$) at $ i \infty$ (resp. $ 0, 1/5, 1/7$). For the last two points we really only know that the order at each of these points is greater than or equal to the given value. Nevertheless, this is sufficient to determine that the function $ h(z) = B_8 (z) L_{35} (S_{35} z) $ has its only pole at $ i \infty $, and the order of the pole there is $ 59.$ Thus, $ h(z) \in \mathbb{C}[B_4, B_5, B_6, B_7].$ To clear fractions, multiply by $ 35 $ and then note $35 L_{35}(S_{35} z) = B_8^{-1} P_0 $, where $P_0$ is a polynomial to be determined in the four basis functions. So $$ q^{-51} -5 q^{-2} -7 q^{-1} + 35 + ... = P_0 B_8^{-1}\Sum{n} p(n)q^n. $$ This yields the following: \begin{theorem} $ 35 L_{35} (S_{35} z)$ is a polynomial in the basis functions $ B_4, B_5, B_6, B_7 $, with integral coefficients given in table 2 of the appendix. \end{theorem} \begin{cor} \begin{gather*} \Sum{n}\left\{ p(n)q^{1225n}-5\left(\frac{n-4}{5}\right)p(n)q^{49(n+1)} \right. \\ \left. - 7\left(\frac{n-5}{7}\right)p(n)q^{25(n+2)}+35\left(\frac{n-19}{35}\right)p(n)q^{n+51} \right\} \\ = \ q^{51}P_0B_8^{-1}\Sum{n}p(n)q^n. \end{gather*} \end{cor} I would like to thank my doctoral advisor, Morris Newman, who suggested this problem to me, and Bruce Berndt and the referee for their helpful comments. \vskip 30pt \section*{\normalsize 3. Appendix: tables} \begin{longtable}{|c|r|c|r|} \caption[]{Coefficients for $P_{\infty}$} \\ \hline i & \multicolumn{1}{c|}{coefficient} & i & \multicolumn{1}{c|}{coefficient} \\ \hline \endfirsthead \caption[]{\emph continued}\\ \hline i & \multicolumn{1}{c|}{coefficient} & i & \multicolumn{1}{c|}{coefficient} \\ \hline \endhead \hline \multicolumn{4}{r} {\emph {continued on next page}} \endfoot \hline \endlastfoot -69 & 14 & -33 & -10569223238228513913405 \\ -68 & 10487 & -34 & -29246449555394872646896 \\ -67 & 1050000 & -33 & -30752043774402282243297 \\ -66 & 37305717 & -32 & -10500761656485195685244 \\ -65 & 689742738 & -31 & -63994584518030323125 \\ -64 & 6710739396 & -30 & 747396610368371686739 \\ -63& 66723377634 & -29 & 41866861538739825786329 \\ -62 & 306529775888 & -28 & -154825275874629072148111 \\ -61& 944992220254 & -27 & -330675045299039974754087 \\ -60 & 8032835276603 & -26 & -330675045299039974754087 \\ -59 & 41786228768706 & -25 & -195434688782712582767044 \\ -58 & 163674979606784 & -24 & 106519373950966952221 \\ -57 & 201541582793635 & -23 & -4489388038461810926594 \\ -56 & 1192164476376773 & -22 & 93815803283069768407781 \\ -55 & 2133885512377322 & -21 & -868861692593019875600320 \\ -54 & -224422822302317 & -20 & -780240652273282530768564 \\ -53 & 9311212319487067 & -19 & -903341606561540466301311 \\ -52 & 53104246090438488 & -18 & -575916410654649291605538 \\ -51 & 174875061214811942 & -17 & -72387387334271714225 \\ -50 & 59790513579741686 & -16 & 5279005749531844180716 \\ -49 & 418363890733605649 & -15 & -329709784193546752041430 \\ -48 & 1123804424169272601 & -14 & -754715959977583205060136 \\ -47 & -1849756259047888932 & -13 & 2673409023878484316383343 \\ -46 & -3961615256484350591 & -12 & 3038512747510492176975342 \\ -45 & 1372328949965814202 & -11 & 1778681601887609961535840 \\ -44 & 19742808525931603996 & -10 & 16760790778989818745 \\ -43 & 2177933533113105157 & -9 & -1894749705906830722098 \\ -42 & 23516820211582241283 & -8 & 189938163107999640927156 \\ -41 & 123390318981474451231 & -7 & 4789820626890790384933368 \\-40 & -271841378745461882544 & -6 & -1532454919226226734377599 \\ -39 & -777621797568622505476 & -5 & -1735248899286586237586336 \\ -38 & -673143178730309518022 & -4 & -997109474512591275870705 \\ -37 & 268511875093390021788 & -3 & 0 \\ -36 & 10323721732692166606 & -2 & 0 \\ -35 & 334287613007887666292 & -1 & 0\\ -34 & 3963341600197861903691 & 0 & -3000539263195134467759624 \\ \hline \end{longtable} \begin{center} \begin{longtable}{|c|r|c|r|} \caption[]{Coefficients for $P_0$}\\ \hline i & \multicolumn{1}{c|}{coefficient} & i & \multicolumn{1}{c|}{coefficient} \\ \hline \endfirsthead \caption[]{\emph continued} \\ \hline i & \multicolumn{1}{c|}{coefficient} & i & \multicolumn{1}{c|}{coefficient} \\ \hline \endhead \hline \multicolumn{4}{r} {\emph {continued on next page}} \endfoot \hline \endlastfoot -59 & 1 & -29 & 101856\\ -58 & -5 & -28 & -1088550 \\ -57 & 13 & -27 & -779536 \\ -56 & -24 & -26 & -868518\\ -55 & -124 & -25 & -638720 \\ -54 & -167 & -24 & 183785 \\ -53 & -100 & -23 & 310027 \\ -52 & 186 & -22 & -1009582 \\ -51 & 82 & -21 & -5533 \\ -50 & -784 & -20 & 8917982 \\ -49 & 1899 & -19 & 10010374 \\ -48 & 7839 & -18 & 6789719 \\ -47 & 9103 & -17 & -21976 \\ -46 & 5400 & -16 & 113545 \\ -45 & -531 & -15& 35220 \\ -44 & 2196 & -14 & 21729982 \\ -43 & -786 & -13 & -703140 \\ -42 & 13705 & -12 & -920456 \\ -41 & -1006 & -11 & -843856 \\ -40 & -4690 & -10 & -236844 \\ -39 & 1882 & -9 & -376949 \\ -38 & -6611 & -8 & 1365595 \\ -37 & -16039 & -7 & -4046123 \\ -36 & 44380 & -6 & -12312578 \\ -35 & -84846 & -5 & -13942378 \\ -34 & -393717 & -4 & -9425016 \\ -33 & -427964 & -3 & 0 \\ -32 & -282747 & -2 & 0 \\ -31 & -17208 & -1 & 0 \\ -30 & -53917 & 0 & -30151150 \\ \hline \end{longtable} \end{center} \nocite{Ap41, ApIntro} \bibliographystyle{alpha} \begin{thebibliography}{New57} \bibitem[Apo95]{ApIntro} T.~M. 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