LMFDB - Space of modular forms of level 8015, weight 2, and Trivial character

Defining parameters

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 15 $
Sturm bound:			$1840$
Trace bound:			$3$
Distinguishing $T_p$:			$2$, $3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(8015))$.

	Total	New	Old
Modular forms	924	455	469
Cusp forms	917	455	462
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$5$	$7$	$229$	Fricke	Dim.
$+$	$+$	$+$	$+$	$52$
$+$	$+$	$-$	$-$	$63$
$+$	$-$	$+$	$-$	$68$
$+$	$-$	$-$	$+$	$45$
$-$	$+$	$+$	$-$	$67$
$-$	$+$	$-$	$+$	$44$
$-$	$-$	$+$	$+$	$41$
$-$	$-$	$-$	$-$	$75$
Plus space			$+$	$182$
Minus space			$-$	$273$

Trace form

$ 455q + 5q^{2} + 4q^{3} + 453q^{4} - q^{5} + 20q^{6} + 3q^{7} + 33q^{8} + 471q^{9} + O(q^{10}) $

\( 455q + 5q^{2} + 4q^{3} + 453q^{4} - q^{5} + 20q^{6} + 3q^{7} + 33q^{8} + 471q^{9} - 3q^{10} + 8q^{11} + 20q^{12} - 14q^{13} + q^{14} + 453q^{16} + 14q^{17} + 25q^{18} + 12q^{19} - 7q^{20} + 20q^{22} + 32q^{23} + 12q^{24} + 455q^{25} - 2q^{26} + 64q^{27} + 21q^{28} + 22q^{29} - 4q^{30} + 16q^{31} + 81q^{32} + 72q^{33} + 26q^{34} + 7q^{35} + 481q^{36} + 2q^{37} + 4q^{38} + 68q^{39} - 15q^{40} + 38q^{41} - 12q^{42} + 12q^{43} + 20q^{44} + 3q^{45} - 32q^{46} + 8q^{47} + 4q^{48} + 455q^{49} + 5q^{50} - 4q^{51} + 14q^{52} - 30q^{53} - 16q^{54} + 4q^{55} - 3q^{56} + 24q^{57} + 22q^{58} + 28q^{59} - 12q^{60} - 30q^{61} + 16q^{62} + 7q^{63} + 509q^{64} + 14q^{65} + 24q^{66} + 28q^{67} + 114q^{68} - 8q^{69} - 3q^{70} + 16q^{71} + 149q^{72} + 38q^{73} + 46q^{74} + 4q^{75} + 76q^{76} + 20q^{77} + 72q^{78} + 12q^{79} + q^{80} + 559q^{81} + 98q^{82} - 12q^{83} - 12q^{84} - 6q^{85} + 68q^{86} - 40q^{87} + 108q^{88} - 2q^{89} - 7q^{90} - 2q^{91} + 32q^{92} + 16q^{93} + 8q^{94} - 4q^{95} + 108q^{96} + 78q^{97} + 5q^{98} - 20q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(8015))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	5	7	229	$q$-expansion
8015.2.a.a	$1$	$64.000$	$\Q$	None	$-2$	$-3$	$1$	$1$	$-$	$-$	$-$	$q-2q^{2}-3q^{3}+2q^{4}+q^{5}+6q^{6}+\cdots$
8015.2.a.b	$1$	$64.000$	$\Q$	None	$-1$	$-2$	$1$	$1$	$-$	$-$	$+$	$q-q^{2}-2q^{3}-q^{4}+q^{5}+2q^{6}+q^{7}+\cdots$
8015.2.a.c	$1$	$64.000$	$\Q$	None	$-1$	$1$	$1$	$1$	$-$	$-$	$+$	$q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+q^{7}+\cdots$
8015.2.a.d	$1$	$64.000$	$\Q$	None	$1$	$0$	$1$	$1$	$-$	$-$	$-$	$q+q^{2}-q^{4}+q^{5}+q^{7}-3q^{8}-3q^{9}+\cdots$
8015.2.a.e	$1$	$64.000$	$\Q$	None	$1$	$2$	$-1$	$-1$	$+$	$+$	$-$	$q+q^{2}+2q^{3}-q^{4}-q^{5}+2q^{6}-q^{7}+\cdots$
8015.2.a.f	$1$	$64.000$	$\Q$	None	$2$	$1$	$1$	$1$	$-$	$-$	$+$	$q+2q^{2}+q^{3}+2q^{4}+q^{5}+2q^{6}+q^{7}+\cdots$
8015.2.a.g	$3$	$64.000$	3.3.148.1	None	$1$	$-1$	$-3$	$-3$	$+$	$+$	$+$	$q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$
8015.2.a.h	$38$	$64.000$		None	$-6$	$-9$	$38$	$38$	$-$	$-$	$+$
8015.2.a.i	$44$	$64.000$		None	$-2$	$0$	$44$	$-44$	$-$	$+$	$-$
8015.2.a.j	$45$	$64.000$		None	$-6$	$0$	$-45$	$45$	$+$	$-$	$-$
8015.2.a.k	$49$	$64.000$		None	$-3$	$-10$	$-49$	$-49$	$+$	$+$	$+$
8015.2.a.l	$62$	$64.000$		None	$2$	$11$	$-62$	$-62$	$+$	$+$	$-$
8015.2.a.m	$67$	$64.000$		None	$3$	$0$	$67$	$-67$	$-$	$+$	$+$
8015.2.a.n	$68$	$64.000$		None	$9$	$0$	$-68$	$68$	$+$	$-$	$+$
8015.2.a.o	$73$	$64.000$		None	$7$	$14$	$73$	$73$	$-$	$-$	$-$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(8015))$ into lower level spaces

$ S_{2}^{\mathrm{old}}(\Gamma_0(8015)) \cong $ $S_{2}^{\mathrm{new}}(\Gamma_0(35))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(229))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1145))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1603))$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 + 2 T + 2 T^{2} $)($ 1 + T + 2 T^{2} $)($ 1 + T + 2 T^{2} $)($ 1 - T + 2 T^{2} $)($ 1 - T + 2 T^{2} $)($ 1 - 2 T + 2 T^{2} $)($ 1 - T + 3 T^{2} - 3 T^{3} + 6 T^{4} - 4 T^{5} + 8 T^{6} $)
$3$	($ 1 + 3 T + 3 T^{2} $)($ 1 + 2 T + 3 T^{2} $)($ 1 - T + 3 T^{2} $)($ 1 + 3 T^{2} $)($ 1 - 2 T + 3 T^{2} $)($ 1 - T + 3 T^{2} $)($ 1 + T - 7 T^{3} + 9 T^{5} + 27 T^{6} $)
$5$	($ 1 - T $)($ 1 - T $)($ 1 - T $)($ 1 - T $)($ 1 + T $)($ 1 - T $)($ ( 1 + T )^{3} $)
$7$	($ 1 - T $)($ 1 - T $)($ 1 - T $)($ 1 - T $)($ 1 + T $)($ 1 - T $)($ ( 1 + T )^{3} $)
$11$	($ 1 - 5 T + 11 T^{2} $)($ 1 - 3 T + 11 T^{2} $)($ 1 + 11 T^{2} $)($ 1 + 4 T + 11 T^{2} $)($ 1 + T + 11 T^{2} $)($ 1 + 3 T + 11 T^{2} $)($ 1 + 4 T + 29 T^{2} + 68 T^{3} + 319 T^{4} + 484 T^{5} + 1331 T^{6} $)
$13$	($ 1 + T + 13 T^{2} $)($ 1 + 7 T + 13 T^{2} $)($ 1 - 5 T + 13 T^{2} $)($ 1 - 2 T + 13 T^{2} $)($ 1 + 5 T + 13 T^{2} $)($ 1 - 3 T + 13 T^{2} $)($ 1 + 3 T + 8 T^{2} - 17 T^{3} + 104 T^{4} + 507 T^{5} + 2197 T^{6} $)
$17$	($ 1 - 3 T + 17 T^{2} $)($ 1 - 6 T + 17 T^{2} $)($ 1 + 17 T^{2} $)($ 1 + 6 T + 17 T^{2} $)($ 1 - 2 T + 17 T^{2} $)($ 1 + T + 17 T^{2} $)($ 1 + 4 T + 43 T^{2} + 120 T^{3} + 731 T^{4} + 1156 T^{5} + 4913 T^{6} $)
$19$	($ 1 - 2 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 + 8 T + 19 T^{2} $)($ 1 + 4 T + 19 T^{2} $)($ 1 - 4 T + 19 T^{2} $)($ 1 + 6 T + 19 T^{2} $)($ ( 1 + 19 T^{2} )^{3} $)
$23$	($ 1 - 6 T + 23 T^{2} $)($ 1 + 23 T^{2} $)($ 1 - 6 T + 23 T^{2} $)($ 1 + 23 T^{2} $)($ 1 + 4 T + 23 T^{2} $)($ 1 + 2 T + 23 T^{2} $)($ 1 + 4 T + 29 T^{2} + 116 T^{3} + 667 T^{4} + 2116 T^{5} + 12167 T^{6} $)
$29$	($ 1 - 5 T + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 + 29 T^{2} $)($ 1 + 10 T + 29 T^{2} $)($ 1 + 4 T + 29 T^{2} $)($ 1 + 3 T + 29 T^{2} $)($ 1 - 10 T + 111 T^{2} - 576 T^{3} + 3219 T^{4} - 8410 T^{5} + 24389 T^{6} $)
$31$	($ 1 - 2 T + 31 T^{2} $)($ 1 + 9 T + 31 T^{2} $)($ 1 + 31 T^{2} $)($ 1 - 8 T + 31 T^{2} $)($ 1 - 9 T + 31 T^{2} $)($ 1 + 2 T + 31 T^{2} $)($ 1 - 6 T + 21 T^{2} + 56 T^{3} + 651 T^{4} - 5766 T^{5} + 29791 T^{6} $)
$37$	($ 1 + 8 T + 37 T^{2} $)($ 1 + T + 37 T^{2} $)($ 1 - 2 T + 37 T^{2} $)($ 1 + 2 T + 37 T^{2} $)($ 1 - 7 T + 37 T^{2} $)($ 1 + 4 T + 37 T^{2} $)($ 1 + 6 T + 75 T^{2} + 336 T^{3} + 2775 T^{4} + 8214 T^{5} + 50653 T^{6} $)
$41$	($ 1 - 6 T + 41 T^{2} $)($ 1 + 5 T + 41 T^{2} $)($ 1 + 8 T + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 - T + 41 T^{2} $)($ 1 + 6 T + 41 T^{2} $)($ 1 + 26 T + 339 T^{2} + 2712 T^{3} + 13899 T^{4} + 43706 T^{5} + 68921 T^{6} $)
$43$	($ 1 + 4 T + 43 T^{2} $)($ 1 - T + 43 T^{2} $)($ 1 + 8 T + 43 T^{2} $)($ 1 - 8 T + 43 T^{2} $)($ 1 + 3 T + 43 T^{2} $)($ 1 + 8 T + 43 T^{2} $)($ 1 - 16 T + 193 T^{2} - 1408 T^{3} + 8299 T^{4} - 29584 T^{5} + 79507 T^{6} $)
$47$	($ 1 + T + 47 T^{2} $)($ 1 - 5 T + 47 T^{2} $)($ 1 + 10 T + 47 T^{2} $)($ 1 - 8 T + 47 T^{2} $)($ 1 - 3 T + 47 T^{2} $)($ 1 - 3 T + 47 T^{2} $)($ 1 + 8 T + 157 T^{2} + 756 T^{3} + 7379 T^{4} + 17672 T^{5} + 103823 T^{6} $)
$53$	($ 1 - 6 T + 53 T^{2} $)($ 1 + 5 T + 53 T^{2} $)($ 1 + 2 T + 53 T^{2} $)($ 1 - 6 T + 53 T^{2} $)($ 1 + 5 T + 53 T^{2} $)($ 1 - 2 T + 53 T^{2} $)($ 1 - 2 T + 155 T^{2} - 208 T^{3} + 8215 T^{4} - 5618 T^{5} + 148877 T^{6} $)
$59$	($ 1 + 59 T^{2} $)($ 1 + 12 T + 59 T^{2} $)($ 1 - 3 T + 59 T^{2} $)($ 1 - 12 T + 59 T^{2} $)($ 1 - 4 T + 59 T^{2} $)($ 1 + 8 T + 59 T^{2} $)($ 1 + 15 T + 182 T^{2} + 1459 T^{3} + 10738 T^{4} + 52215 T^{5} + 205379 T^{6} $)
$61$	($ 1 - 10 T + 61 T^{2} $)($ 1 - 14 T + 61 T^{2} $)($ 1 - 5 T + 61 T^{2} $)($ 1 + 2 T + 61 T^{2} $)($ 1 - 2 T + 61 T^{2} $)($ 1 + 10 T + 61 T^{2} $)($ 1 - T + 86 T^{2} - 285 T^{3} + 5246 T^{4} - 3721 T^{5} + 226981 T^{6} $)
$67$	($ 1 - 8 T + 67 T^{2} $)($ 1 + 67 T^{2} $)($ 1 + 6 T + 67 T^{2} $)($ 1 + 4 T + 67 T^{2} $)($ 1 - 12 T + 67 T^{2} $)($ 1 - 8 T + 67 T^{2} $)($ 1 + 12 T + 201 T^{2} + 1460 T^{3} + 13467 T^{4} + 53868 T^{5} + 300763 T^{6} $)
$71$	($ 1 - 12 T + 71 T^{2} $)($ 1 + 3 T + 71 T^{2} $)($ 1 + 6 T + 71 T^{2} $)($ 1 + 71 T^{2} $)($ 1 - 9 T + 71 T^{2} $)($ 1 - 12 T + 71 T^{2} $)($ 1 - 18 T + 281 T^{2} - 2456 T^{3} + 19951 T^{4} - 90738 T^{5} + 357911 T^{6} $)
$73$	($ 1 + 10 T + 73 T^{2} $)($ 1 + 2 T + 73 T^{2} $)($ 1 + 11 T + 73 T^{2} $)($ 1 - 14 T + 73 T^{2} $)($ 1 + 14 T + 73 T^{2} $)($ 1 + 2 T + 73 T^{2} $)($ 1 + 15 T + 224 T^{2} + 1879 T^{3} + 16352 T^{4} + 79935 T^{5} + 389017 T^{6} $)
$79$	($ 1 - 9 T + 79 T^{2} $)($ 1 + 12 T + 79 T^{2} $)($ 1 - 3 T + 79 T^{2} $)($ 1 + 79 T^{2} $)($ 1 - 12 T + 79 T^{2} $)($ 1 - 9 T + 79 T^{2} $)($ 1 - 5 T + 180 T^{2} - 551 T^{3} + 14220 T^{4} - 31205 T^{5} + 493039 T^{6} $)
$83$	($ 1 - 4 T + 83 T^{2} $)($ 1 + 2 T + 83 T^{2} $)($ 1 - 7 T + 83 T^{2} $)($ 1 - 16 T + 83 T^{2} $)($ 1 - 6 T + 83 T^{2} $)($ 1 + 4 T + 83 T^{2} $)($ 1 + 7 T + 260 T^{2} + 1163 T^{3} + 21580 T^{4} + 48223 T^{5} + 571787 T^{6} $)
$89$	($ 1 - 12 T + 89 T^{2} $)($ 1 + 5 T + 89 T^{2} $)($ 1 - 10 T + 89 T^{2} $)($ 1 + 6 T + 89 T^{2} $)($ 1 - 9 T + 89 T^{2} $)($ 1 - 12 T + 89 T^{2} $)($ 1 + 6 T - T^{2} - 172 T^{3} - 89 T^{4} + 47526 T^{5} + 704969 T^{6} $)
$97$	($ 1 - 11 T + 97 T^{2} $)($ 1 + 2 T + 97 T^{2} $)($ 1 + 8 T + 97 T^{2} $)($ 1 - 2 T + 97 T^{2} $)($ 1 + 10 T + 97 T^{2} $)($ 1 - 7 T + 97 T^{2} $)($ 1 + 24 T + 419 T^{2} + 4528 T^{3} + 40643 T^{4} + 225816 T^{5} + 912673 T^{6} $)
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Overview	Features
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(1840\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8015))\) into lower level spaces

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8015.2.a

Level	8015
Weight	2
Character orbit	a
Rep. character	\(\chi_{8015}(1,\cdot)\)
Character field	\(\Q\)
Dimension	455
Newform subspaces	15
Sturm bound	1840
Trace bound	3

\(5\)	\(7\)	\(229\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(52\)
\(+\)	\(+\)	\(-\)	\(-\)	\(63\)
\(+\)	\(-\)	\(+\)	\(-\)	\(68\)
\(+\)	\(-\)	\(-\)	\(+\)	\(45\)
\(-\)	\(+\)	\(+\)	\(-\)	\(67\)
\(-\)	\(+\)	\(-\)	\(+\)	\(44\)
\(-\)	\(-\)	\(+\)	\(+\)	\(41\)
\(-\)	\(-\)	\(-\)	\(-\)	\(75\)
Plus space			\(+\)	\(182\)
Minus space			\(-\)	\(273\)

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs			$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	5	7	229	$q$-expansion
8015.2.a.a	\(1\)	\(64.000\)	\(\Q\)	None	\(-2\)	\(-3\)	\(1\)	\(1\)	\(-\)	\(-\)	\(-\)	\(q-2q^{2}-3q^{3}+2q^{4}+q^{5}+6q^{6}+\cdots\)
8015.2.a.b	\(1\)	\(64.000\)	\(\Q\)	None	\(-1\)	\(-2\)	\(1\)	\(1\)	\(-\)	\(-\)	\(+\)	\(q-q^{2}-2q^{3}-q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
8015.2.a.c	\(1\)	\(64.000\)	\(\Q\)	None	\(-1\)	\(1\)	\(1\)	\(1\)	\(-\)	\(-\)	\(+\)	\(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
8015.2.a.d	\(1\)	\(64.000\)	\(\Q\)	None	\(1\)	\(0\)	\(1\)	\(1\)	\(-\)	\(-\)	\(-\)	\(q+q^{2}-q^{4}+q^{5}+q^{7}-3q^{8}-3q^{9}+\cdots\)
8015.2.a.e	\(1\)	\(64.000\)	\(\Q\)	None	\(1\)	\(2\)	\(-1\)	\(-1\)	\(+\)	\(+\)	\(-\)	\(q+q^{2}+2q^{3}-q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
8015.2.a.f	\(1\)	\(64.000\)	\(\Q\)	None	\(2\)	\(1\)	\(1\)	\(1\)	\(-\)	\(-\)	\(+\)	\(q+2q^{2}+q^{3}+2q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
8015.2.a.g	\(3\)	\(64.000\)	3.3.148.1	None	\(1\)	\(-1\)	\(-3\)	\(-3\)	\(+\)	\(+\)	\(+\)	\(q+\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
8015.2.a.h	\(38\)	\(64.000\)		None	\(-6\)	\(-9\)	\(38\)	\(38\)	\(-\)	\(-\)	\(+\)
8015.2.a.i	\(44\)	\(64.000\)		None	\(-2\)	\(0\)	\(44\)	\(-44\)	\(-\)	\(+\)	\(-\)
8015.2.a.j	\(45\)	\(64.000\)		None	\(-6\)	\(0\)	\(-45\)	\(45\)	\(+\)	\(-\)	\(-\)
8015.2.a.k	\(49\)	\(64.000\)		None	\(-3\)	\(-10\)	\(-49\)	\(-49\)	\(+\)	\(+\)	\(+\)
8015.2.a.l	\(62\)	\(64.000\)		None	\(2\)	\(11\)	\(-62\)	\(-62\)	\(+\)	\(+\)	\(-\)
8015.2.a.m	\(67\)	\(64.000\)		None	\(3\)	\(0\)	\(67\)	\(-67\)	\(-\)	\(+\)	\(+\)
8015.2.a.n	\(68\)	\(64.000\)		None	\(9\)	\(0\)	\(-68\)	\(68\)	\(+\)	\(-\)	\(+\)
8015.2.a.o	\(73\)	\(64.000\)		None	\(7\)	\(14\)	\(73\)	\(73\)	\(-\)	\(-\)	\(-\)

$p$	$F_p(T)$
$2$	(\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - T + 3 T^{2} - 3 T^{3} + 6 T^{4} - 4 T^{5} + 8 T^{6} \))
$3$	(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + T - 7 T^{3} + 9 T^{5} + 27 T^{6} \))
$5$	(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))
$7$	(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))
$11$	(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 4 T + 29 T^{2} + 68 T^{3} + 319 T^{4} + 484 T^{5} + 1331 T^{6} \))
$13$	(\( 1 + T + 13 T^{2} \))(\( 1 + 7 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + 3 T + 8 T^{2} - 17 T^{3} + 104 T^{4} + 507 T^{5} + 2197 T^{6} \))
$17$	(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 + 4 T + 43 T^{2} + 120 T^{3} + 731 T^{4} + 1156 T^{5} + 4913 T^{6} \))
$19$	(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{3} \))
$23$	(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 4 T + 29 T^{2} + 116 T^{3} + 667 T^{4} + 2116 T^{5} + 12167 T^{6} \))
$29$	(\( 1 - 5 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - 10 T + 111 T^{2} - 576 T^{3} + 3219 T^{4} - 8410 T^{5} + 24389 T^{6} \))
$31$	(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 - 6 T + 21 T^{2} + 56 T^{3} + 651 T^{4} - 5766 T^{5} + 29791 T^{6} \))
$37$	(\( 1 + 8 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 6 T + 75 T^{2} + 336 T^{3} + 2775 T^{4} + 8214 T^{5} + 50653 T^{6} \))
$41$	(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 26 T + 339 T^{2} + 2712 T^{3} + 13899 T^{4} + 43706 T^{5} + 68921 T^{6} \))
$43$	(\( 1 + 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 3 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 16 T + 193 T^{2} - 1408 T^{3} + 8299 T^{4} - 29584 T^{5} + 79507 T^{6} \))
$47$	(\( 1 + T + 47 T^{2} \))(\( 1 - 5 T + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 8 T + 157 T^{2} + 756 T^{3} + 7379 T^{4} + 17672 T^{5} + 103823 T^{6} \))
$53$	(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 - 2 T + 155 T^{2} - 208 T^{3} + 8215 T^{4} - 5618 T^{5} + 148877 T^{6} \))
$59$	(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 + 15 T + 182 T^{2} + 1459 T^{3} + 10738 T^{4} + 52215 T^{5} + 205379 T^{6} \))
$61$	(\( 1 - 10 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - T + 86 T^{2} - 285 T^{3} + 5246 T^{4} - 3721 T^{5} + 226981 T^{6} \))
$67$	(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 6 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 12 T + 201 T^{2} + 1460 T^{3} + 13467 T^{4} + 53868 T^{5} + 300763 T^{6} \))
$71$	(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 - 18 T + 281 T^{2} - 2456 T^{3} + 19951 T^{4} - 90738 T^{5} + 357911 T^{6} \))
$73$	(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 15 T + 224 T^{2} + 1879 T^{3} + 16352 T^{4} + 79935 T^{5} + 389017 T^{6} \))
$79$	(\( 1 - 9 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 - 5 T + 180 T^{2} - 551 T^{3} + 14220 T^{4} - 31205 T^{5} + 493039 T^{6} \))
$83$	(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 7 T + 83 T^{2} \))(\( 1 - 16 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 7 T + 260 T^{2} + 1163 T^{3} + 21580 T^{4} + 48223 T^{5} + 571787 T^{6} \))
$89$	(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 6 T - T^{2} - 172 T^{3} - 89 T^{4} + 47526 T^{5} + 704969 T^{6} \))
$97$	(\( 1 - 11 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 24 T + 419 T^{2} + 4528 T^{3} + 40643 T^{4} + 225816 T^{5} + 912673 T^{6} \))
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