Properties

Label 315.4.a
Level 315
Weight 4
Character orbit a
Rep. character \(\chi_{315}(1,\cdot)\)
Character field \(\Q\)
Dimension 30
Newform subspaces 16
Sturm bound 192
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(192\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(315))\).

Total New Old
Modular forms 152 30 122
Cusp forms 136 30 106
Eisenstein series 16 0 16

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

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(17\)
Minus space\(-\)\(13\)

Trace form

\( 30q - 2q^{2} + 98q^{4} + 14q^{7} - 54q^{8} + O(q^{10}) \) \( 30q - 2q^{2} + 98q^{4} + 14q^{7} - 54q^{8} + 60q^{10} + 116q^{11} + 16q^{13} - 70q^{14} + 466q^{16} + 260q^{17} + 196q^{19} + 36q^{22} - 400q^{23} + 750q^{25} + 468q^{26} + 70q^{28} - 16q^{29} - 432q^{31} - 54q^{32} + 608q^{34} - 140q^{35} - 116q^{37} + 748q^{38} + 600q^{40} - 356q^{41} - 872q^{43} + 1944q^{44} - 1064q^{46} + 272q^{47} + 1470q^{49} - 50q^{50} + 648q^{52} - 1900q^{53} + 360q^{55} - 546q^{56} - 928q^{58} + 2276q^{59} - 1024q^{61} + 1680q^{62} + 3178q^{64} - 40q^{65} - 1456q^{67} + 604q^{68} + 140q^{70} - 40q^{71} + 1548q^{73} - 2772q^{74} + 2228q^{76} - 1568q^{77} + 564q^{79} + 3200q^{80} - 6516q^{82} + 484q^{83} + 940q^{85} - 1728q^{86} - 1024q^{88} + 4236q^{89} + 1092q^{91} - 4896q^{92} - 2356q^{94} - 500q^{95} + 124q^{97} - 98q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
315.4.a.a \(1\) \(18.586\) \(\Q\) None \(-5\) \(0\) \(-5\) \(7\) \(-\) \(+\) \(-\) \(q-5q^{2}+17q^{4}-5q^{5}+7q^{7}-45q^{8}+\cdots\)
315.4.a.b \(1\) \(18.586\) \(\Q\) None \(-3\) \(0\) \(-5\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{2}+q^{4}-5q^{5}+7q^{7}+21q^{8}+\cdots\)
315.4.a.c \(1\) \(18.586\) \(\Q\) None \(-1\) \(0\) \(5\) \(7\) \(-\) \(-\) \(-\) \(q-q^{2}-7q^{4}+5q^{5}+7q^{7}+15q^{8}+\cdots\)
315.4.a.d \(1\) \(18.586\) \(\Q\) None \(0\) \(0\) \(-5\) \(7\) \(-\) \(+\) \(-\) \(q-8q^{4}-5q^{5}+7q^{7}-42q^{11}+20q^{13}+\cdots\)
315.4.a.e \(1\) \(18.586\) \(\Q\) None \(3\) \(0\) \(5\) \(7\) \(+\) \(-\) \(-\) \(q+3q^{2}+q^{4}+5q^{5}+7q^{7}-21q^{8}+\cdots\)
315.4.a.f \(2\) \(18.586\) \(\Q(\sqrt{2}) \) None \(-8\) \(0\) \(10\) \(-14\) \(-\) \(-\) \(+\) \(q+(-4+\beta )q^{2}+(10-8\beta )q^{4}+5q^{5}+\cdots\)
315.4.a.g \(2\) \(18.586\) \(\Q(\sqrt{41}) \) None \(-3\) \(0\) \(10\) \(14\) \(-\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+(3+3\beta )q^{4}+5q^{5}+\cdots\)
315.4.a.h \(2\) \(18.586\) \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(-10\) \(-14\) \(+\) \(+\) \(+\) \(q-\beta q^{2}+(-4+\beta )q^{4}-5q^{5}-7q^{7}+\cdots\)
315.4.a.i \(2\) \(18.586\) \(\Q(\sqrt{65}) \) None \(-1\) \(0\) \(-10\) \(-14\) \(-\) \(+\) \(+\) \(q-\beta q^{2}+(8+\beta )q^{4}-5q^{5}-7q^{7}+(-2^{4}+\cdots)q^{8}+\cdots\)
315.4.a.j \(2\) \(18.586\) \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(10\) \(-14\) \(+\) \(-\) \(+\) \(q+\beta q^{2}+(-4+\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.k \(2\) \(18.586\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-10\) \(-14\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-5q^{5}-7q^{7}+\cdots\)
315.4.a.l \(2\) \(18.586\) \(\Q(\sqrt{5}) \) None \(4\) \(0\) \(10\) \(-14\) \(-\) \(-\) \(+\) \(q+(2+\beta )q^{2}+(1+4\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.m \(2\) \(18.586\) \(\Q(\sqrt{17}) \) None \(7\) \(0\) \(10\) \(-14\) \(-\) \(-\) \(+\) \(q+(4-\beta )q^{2}+(12-7\beta )q^{4}+5q^{5}-7q^{7}+\cdots\)
315.4.a.n \(3\) \(18.586\) 3.3.22952.1 None \(-2\) \(0\) \(-15\) \(21\) \(+\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
315.4.a.o \(3\) \(18.586\) 3.3.22952.1 None \(2\) \(0\) \(15\) \(21\) \(+\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
315.4.a.p \(3\) \(18.586\) 3.3.14360.1 None \(3\) \(0\) \(-15\) \(21\) \(-\) \(+\) \(-\) \(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(315)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 5 T + 8 T^{2} \))(\( 1 + 3 T + 8 T^{2} \))(\( 1 + T + 8 T^{2} \))(\( 1 + 8 T^{2} \))(\( 1 - 3 T + 8 T^{2} \))(\( 1 + 8 T + 30 T^{2} + 64 T^{3} + 64 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 24 T^{3} + 64 T^{4} \))(\( 1 + T + 12 T^{2} + 8 T^{3} + 64 T^{4} \))(\( 1 + T + 8 T^{3} + 64 T^{4} \))(\( 1 - T + 12 T^{2} - 8 T^{3} + 64 T^{4} \))(\( 1 - 2 T + 9 T^{2} - 16 T^{3} + 64 T^{4} \))(\( 1 - 4 T + 15 T^{2} - 32 T^{3} + 64 T^{4} \))(\( 1 - 7 T + 24 T^{2} - 56 T^{3} + 64 T^{4} \))(\( 1 + 2 T + 7 T^{2} + 28 T^{3} + 56 T^{4} + 128 T^{5} + 512 T^{6} \))(\( 1 - 2 T + 7 T^{2} - 28 T^{3} + 56 T^{4} - 128 T^{5} + 512 T^{6} \))(\( 1 - 3 T + 10 T^{2} - 18 T^{3} + 80 T^{4} - 192 T^{5} + 512 T^{6} \))
$3$ 1
$5$ (\( 1 + 5 T \))(\( 1 + 5 T \))(\( 1 - 5 T \))(\( 1 + 5 T \))(\( 1 - 5 T \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 + 5 T )^{2} \))(\( ( 1 + 5 T )^{2} \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 + 5 T )^{2} \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 + 5 T )^{3} \))(\( ( 1 - 5 T )^{3} \))(\( ( 1 + 5 T )^{3} \))
$7$ (\( 1 - 7 T \))(\( 1 - 7 T \))(\( 1 - 7 T \))(\( 1 - 7 T \))(\( 1 - 7 T \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{3} \))(\( ( 1 - 7 T )^{3} \))(\( ( 1 - 7 T )^{3} \))
$11$ (\( 1 + 12 T + 1331 T^{2} \))(\( 1 + 60 T + 1331 T^{2} \))(\( 1 + 12 T + 1331 T^{2} \))(\( 1 + 42 T + 1331 T^{2} \))(\( 1 - 60 T + 1331 T^{2} \))(\( 1 - 14 T + 663 T^{2} - 18634 T^{3} + 1771561 T^{4} \))(\( 1 + 62 T + 3582 T^{2} + 82522 T^{3} + 1771561 T^{4} \))(\( 1 - 4 T + 2598 T^{2} - 5324 T^{3} + 1771561 T^{4} \))(\( 1 - 22 T + 2718 T^{2} - 29282 T^{3} + 1771561 T^{4} \))(\( 1 + 4 T + 2598 T^{2} + 5324 T^{3} + 1771561 T^{4} \))(\( 1 - 16 T - 474 T^{2} - 21296 T^{3} + 1771561 T^{4} \))(\( 1 - 92 T + 4758 T^{2} - 122452 T^{3} + 1771561 T^{4} \))(\( 1 - 26 T + 2406 T^{2} - 34606 T^{3} + 1771561 T^{4} \))(\( 1 - 20 T + 1781 T^{2} - 21080 T^{3} + 2370511 T^{4} - 35431220 T^{5} + 2357947691 T^{6} \))(\( 1 + 20 T + 1781 T^{2} + 21080 T^{3} + 2370511 T^{4} + 35431220 T^{5} + 2357947691 T^{6} \))(\( 1 - 74 T + 5570 T^{2} - 204680 T^{3} + 7413670 T^{4} - 131095514 T^{5} + 2357947691 T^{6} \))
$13$ (\( 1 - 30 T + 2197 T^{2} \))(\( 1 - 38 T + 2197 T^{2} \))(\( 1 + 78 T + 2197 T^{2} \))(\( 1 - 20 T + 2197 T^{2} \))(\( 1 - 38 T + 2197 T^{2} \))(\( 1 - 50 T + 4987 T^{2} - 109850 T^{3} + 4826809 T^{4} \))(\( 1 + 6 T + 3378 T^{2} + 13182 T^{3} + 4826809 T^{4} \))(\( 1 + 22 T + 2458 T^{2} + 48334 T^{3} + 4826809 T^{4} \))(\( 1 + 22 T + 4450 T^{2} + 48334 T^{3} + 4826809 T^{4} \))(\( 1 + 22 T + 2458 T^{2} + 48334 T^{3} + 4826809 T^{4} \))(\( 1 + 76 T + 5806 T^{2} + 166972 T^{3} + 4826809 T^{4} \))(\( 1 - 8 T - 2810 T^{2} - 17576 T^{3} + 4826809 T^{4} \))(\( 1 - 14 T + 2386 T^{2} - 30758 T^{3} + 4826809 T^{4} \))(\( 1 + 4843 T^{2} - 17328 T^{3} + 10640071 T^{4} + 10604499373 T^{6} \))(\( 1 + 4843 T^{2} - 17328 T^{3} + 10640071 T^{4} + 10604499373 T^{6} \))(\( 1 - 44 T + 3100 T^{2} - 238206 T^{3} + 6810700 T^{4} - 212379596 T^{5} + 10604499373 T^{6} \))
$17$ (\( 1 - 134 T + 4913 T^{2} \))(\( 1 - 84 T + 4913 T^{2} \))(\( 1 - 94 T + 4913 T^{2} \))(\( 1 + 66 T + 4913 T^{2} \))(\( 1 + 84 T + 4913 T^{2} \))(\( 1 - 50 T + 6579 T^{2} - 245650 T^{3} + 24137569 T^{4} \))(\( 1 + 40 T + 8750 T^{2} + 196520 T^{3} + 24137569 T^{4} \))(\( 1 - 58 T + 7794 T^{2} - 284954 T^{3} + 24137569 T^{4} \))(\( 1 + 116 T + 9030 T^{2} + 569908 T^{3} + 24137569 T^{4} \))(\( 1 + 58 T + 7794 T^{2} + 284954 T^{3} + 24137569 T^{4} \))(\( 1 - 124 T + 13638 T^{2} - 609212 T^{3} + 24137569 T^{4} \))(\( 1 - 44 T + 630 T^{2} - 216172 T^{3} + 24137569 T^{4} \))(\( 1 + 16 T + 6558 T^{2} + 78608 T^{3} + 24137569 T^{4} \))(\( 1 + 234 T + 30351 T^{2} + 2581612 T^{3} + 149114463 T^{4} + 5648191146 T^{5} + 118587876497 T^{6} \))(\( 1 - 234 T + 30351 T^{2} - 2581612 T^{3} + 149114463 T^{4} - 5648191146 T^{5} + 118587876497 T^{6} \))(\( 1 - 52 T + 2992 T^{2} - 454246 T^{3} + 14699696 T^{4} - 1255153588 T^{5} + 118587876497 T^{6} \))
$19$ (\( 1 + 92 T + 6859 T^{2} \))(\( 1 - 110 T + 6859 T^{2} \))(\( 1 - 40 T + 6859 T^{2} \))(\( 1 - 38 T + 6859 T^{2} \))(\( 1 - 110 T + 6859 T^{2} \))(\( 1 - 36 T + 10170 T^{2} - 246924 T^{3} + 47045881 T^{4} \))(\( 1 + 122 T + 17398 T^{2} + 836798 T^{3} + 47045881 T^{4} \))(\( 1 + 5490 T^{2} + 47045881 T^{4} \))(\( 1 - 102 T + 13134 T^{2} - 699618 T^{3} + 47045881 T^{4} \))(\( 1 + 5490 T^{2} + 47045881 T^{4} \))(\( 1 + 96 T + 13974 T^{2} + 658464 T^{3} + 47045881 T^{4} \))(\( 1 + 108 T + 13254 T^{2} + 740772 T^{3} + 47045881 T^{4} \))(\( 1 - 174 T + 18414 T^{2} - 1193466 T^{3} + 47045881 T^{4} \))(\( 1 + 82 T + 15513 T^{2} + 1140156 T^{3} + 106403667 T^{4} + 3857762242 T^{5} + 322687697779 T^{6} \))(\( 1 + 82 T + 15513 T^{2} + 1140156 T^{3} + 106403667 T^{4} + 3857762242 T^{5} + 322687697779 T^{6} \))(\( 1 - 168 T + 26197 T^{2} - 2333344 T^{3} + 179685223 T^{4} - 7903708008 T^{5} + 322687697779 T^{6} \))
$23$ (\( 1 + 112 T + 12167 T^{2} \))(\( 1 + 120 T + 12167 T^{2} \))(\( 1 + 32 T + 12167 T^{2} \))(\( 1 + 12 T + 12167 T^{2} \))(\( 1 - 120 T + 12167 T^{2} \))(\( 1 + 244 T + 29970 T^{2} + 2968748 T^{3} + 148035889 T^{4} \))(\( 1 + 16 T + 782 T^{2} + 194672 T^{3} + 148035889 T^{4} \))(\( 1 - 82 T + 25182 T^{2} - 997694 T^{3} + 148035889 T^{4} \))(\( 1 + 260 T + 34734 T^{2} + 3163420 T^{3} + 148035889 T^{4} \))(\( 1 + 82 T + 25182 T^{2} + 997694 T^{3} + 148035889 T^{4} \))(\( 1 - 16 T + 15150 T^{2} - 194672 T^{3} + 148035889 T^{4} \))(\( 1 - 320 T + 47934 T^{2} - 3893440 T^{3} + 148035889 T^{4} \))(\( 1 + 184 T + 19470 T^{2} + 2238728 T^{3} + 148035889 T^{4} \))(\( 1 + 30 T + 19233 T^{2} + 1108300 T^{3} + 234007911 T^{4} + 4441076670 T^{5} + 1801152661463 T^{6} \))(\( 1 - 30 T + 19233 T^{2} - 1108300 T^{3} + 234007911 T^{4} - 4441076670 T^{5} + 1801152661463 T^{6} \))(\( 1 - 124 T + 38233 T^{2} - 2923048 T^{3} + 465180911 T^{4} - 18356450236 T^{5} + 1801152661463 T^{6} \))
$29$ (\( 1 - 58 T + 24389 T^{2} \))(\( 1 + 162 T + 24389 T^{2} \))(\( 1 - 50 T + 24389 T^{2} \))(\( 1 - 258 T + 24389 T^{2} \))(\( 1 - 162 T + 24389 T^{2} \))(\( 1 - 26 T + 47795 T^{2} - 634114 T^{3} + 594823321 T^{4} \))(\( 1 + 352 T + 78278 T^{2} + 8584928 T^{3} + 594823321 T^{4} \))(\( 1 - 334 T + 72842 T^{2} - 8145926 T^{3} + 594823321 T^{4} \))(\( 1 - 196 T + 20942 T^{2} - 4780244 T^{3} + 594823321 T^{4} \))(\( 1 + 334 T + 72842 T^{2} + 8145926 T^{3} + 594823321 T^{4} \))(\( 1 + 188 T + 34286 T^{2} + 4585132 T^{3} + 594823321 T^{4} \))(\( 1 - 236 T + 61982 T^{2} - 5755804 T^{3} + 594823321 T^{4} \))(\( 1 - 32 T + 19046 T^{2} - 780448 T^{3} + 594823321 T^{4} \))(\( 1 - 32 T + 28367 T^{2} - 1970496 T^{3} + 691842763 T^{4} - 19034346272 T^{5} + 14507145975869 T^{6} \))(\( 1 + 32 T + 28367 T^{2} + 1970496 T^{3} + 691842763 T^{4} + 19034346272 T^{5} + 14507145975869 T^{6} \))(\( 1 + 332 T + 80572 T^{2} + 13628846 T^{3} + 1965070508 T^{4} + 197481342572 T^{5} + 14507145975869 T^{6} \))
$31$ (\( 1 + 224 T + 29791 T^{2} \))(\( 1 - 236 T + 29791 T^{2} \))(\( 1 + 248 T + 29791 T^{2} \))(\( 1 - 146 T + 29791 T^{2} \))(\( 1 - 236 T + 29791 T^{2} \))(\( 1 + 120 T - 1618 T^{2} + 3574920 T^{3} + 887503681 T^{4} \))(\( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} \))(\( 1 + 210 T + 69230 T^{2} + 6256110 T^{3} + 887503681 T^{4} \))(\( 1 - 150 T + 36542 T^{2} - 4468650 T^{3} + 887503681 T^{4} \))(\( 1 + 210 T + 69230 T^{2} + 6256110 T^{3} + 887503681 T^{4} \))(\( 1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4} \))(\( 1 + 60 T + 8462 T^{2} + 1787460 T^{3} + 887503681 T^{4} \))(\( 1 - 330 T + 85430 T^{2} - 9831030 T^{3} + 887503681 T^{4} \))(\( 1 + 362 T + 131221 T^{2} + 23112348 T^{3} + 3909204811 T^{4} + 321276332522 T^{5} + 26439622160671 T^{6} \))(\( 1 + 362 T + 131221 T^{2} + 23112348 T^{3} + 3909204811 T^{4} + 321276332522 T^{5} + 26439622160671 T^{6} \))(\( 1 - 320 T + 113341 T^{2} - 19016064 T^{3} + 3376541731 T^{4} - 284001177920 T^{5} + 26439622160671 T^{6} \))
$37$ (\( 1 + 146 T + 50653 T^{2} \))(\( 1 + 376 T + 50653 T^{2} \))(\( 1 + 434 T + 50653 T^{2} \))(\( 1 - 434 T + 50653 T^{2} \))(\( 1 + 376 T + 50653 T^{2} \))(\( 1 - 564 T + 173630 T^{2} - 28568292 T^{3} + 2565726409 T^{4} \))(\( 1 + 188 T + 44542 T^{2} + 9522764 T^{3} + 2565726409 T^{4} \))(\( 1 - 6 T + 97490 T^{2} - 303918 T^{3} + 2565726409 T^{4} \))(\( 1 + 96 T + 82550 T^{2} + 4862688 T^{3} + 2565726409 T^{4} \))(\( 1 - 6 T + 97490 T^{2} - 303918 T^{3} + 2565726409 T^{4} \))(\( 1 + 132 T + 70814 T^{2} + 6686196 T^{3} + 2565726409 T^{4} \))(\( 1 - 204 T + 108830 T^{2} - 10333212 T^{3} + 2565726409 T^{4} \))(\( 1 + 132 T + 103214 T^{2} + 6686196 T^{3} + 2565726409 T^{4} \))(\( 1 - 302 T + 153027 T^{2} - 29168916 T^{3} + 7751276631 T^{4} - 774849375518 T^{5} + 129961739795077 T^{6} \))(\( 1 - 302 T + 153027 T^{2} - 29168916 T^{3} + 7751276631 T^{4} - 774849375518 T^{5} + 129961739795077 T^{6} \))(\( 1 + 54 T + 139843 T^{2} + 5496260 T^{3} + 7083467479 T^{4} + 138549226086 T^{5} + 129961739795077 T^{6} \))
$41$ (\( 1 + 18 T + 68921 T^{2} \))(\( 1 - 126 T + 68921 T^{2} \))(\( 1 + 402 T + 68921 T^{2} \))(\( 1 - 282 T + 68921 T^{2} \))(\( 1 + 126 T + 68921 T^{2} \))(\( 1 - 328 T + 133986 T^{2} - 22606088 T^{3} + 4750104241 T^{4} \))(\( 1 + 16 T + 18350 T^{2} + 1102736 T^{3} + 4750104241 T^{4} \))(\( 1 - 176 T + 134094 T^{2} - 12130096 T^{3} + 4750104241 T^{4} \))(\( 1 - 176 T - 16914 T^{2} - 12130096 T^{3} + 4750104241 T^{4} \))(\( 1 + 176 T + 134094 T^{2} + 12130096 T^{3} + 4750104241 T^{4} \))(\( 1 + 100 T + 89142 T^{2} + 6892100 T^{3} + 4750104241 T^{4} \))(\( 1 + 44 T + 106326 T^{2} + 3032524 T^{3} + 4750104241 T^{4} \))(\( 1 + 200 T + 64542 T^{2} + 13784200 T^{3} + 4750104241 T^{4} \))(\( 1 + 626 T + 246775 T^{2} + 70209884 T^{3} + 17007979775 T^{4} + 2973565254866 T^{5} + 327381934393961 T^{6} \))(\( 1 - 626 T + 246775 T^{2} - 70209884 T^{3} + 17007979775 T^{4} - 2973565254866 T^{5} + 327381934393961 T^{6} \))(\( 1 + 362 T + 248299 T^{2} + 51434996 T^{3} + 17113015379 T^{4} + 1719537735242 T^{5} + 327381934393961 T^{6} \))
$43$ (\( 1 - 340 T + 79507 T^{2} \))(\( 1 + 34 T + 79507 T^{2} \))(\( 1 + 68 T + 79507 T^{2} \))(\( 1 - 20 T + 79507 T^{2} \))(\( 1 + 34 T + 79507 T^{2} \))(\( 1 + 260 T + 166666 T^{2} + 20671820 T^{3} + 6321363049 T^{4} \))(\( 1 + 396 T + 95718 T^{2} + 31484772 T^{3} + 6321363049 T^{4} \))(\( 1 - 46 T + 42430 T^{2} - 3657322 T^{3} + 6321363049 T^{4} \))(\( 1 + 344 T + 171958 T^{2} + 27350408 T^{3} + 6321363049 T^{4} \))(\( 1 - 46 T + 42430 T^{2} - 3657322 T^{3} + 6321363049 T^{4} \))(\( 1 + 536 T + 200086 T^{2} + 42615752 T^{3} + 6321363049 T^{4} \))(\( 1 - 136 T + 81718 T^{2} - 10812952 T^{3} + 6321363049 T^{4} \))(\( 1 - 364 T + 172486 T^{2} - 28940548 T^{3} + 6321363049 T^{4} \))(\( 1 + 68 T + 56521 T^{2} + 70488 T^{3} + 4493815147 T^{4} + 429852687332 T^{5} + 502592611936843 T^{6} \))(\( 1 + 68 T + 56521 T^{2} + 70488 T^{3} + 4493815147 T^{4} + 429852687332 T^{5} + 502592611936843 T^{6} \))(\( 1 + 16 T + 149005 T^{2} + 1019664 T^{3} + 11846940535 T^{4} + 101141808784 T^{5} + 502592611936843 T^{6} \))
$47$ (\( 1 + 208 T + 103823 T^{2} \))(\( 1 - 6 T + 103823 T^{2} \))(\( 1 + 536 T + 103823 T^{2} \))(\( 1 - 72 T + 103823 T^{2} \))(\( 1 + 6 T + 103823 T^{2} \))(\( 1 - 350 T + 203423 T^{2} - 36338050 T^{3} + 10779215329 T^{4} \))(\( 1 - 188 T + 15582 T^{2} - 19518724 T^{3} + 10779215329 T^{4} \))(\( 1 - 514 T + 269870 T^{2} - 53365022 T^{3} + 10779215329 T^{4} \))(\( 1 + 560 T + 248606 T^{2} + 58140880 T^{3} + 10779215329 T^{4} \))(\( 1 + 514 T + 269870 T^{2} + 53365022 T^{3} + 10779215329 T^{4} \))(\( 1 - 928 T + 408830 T^{2} - 96347744 T^{3} + 10779215329 T^{4} \))(\( 1 + 400 T + 106526 T^{2} + 41529200 T^{3} + 10779215329 T^{4} \))(\( 1 + 292 T + 179390 T^{2} + 30316316 T^{3} + 10779215329 T^{4} \))(\( 1 + 936 T + 600637 T^{2} + 223876784 T^{3} + 62359935251 T^{4} + 10089345547944 T^{5} + 1119130473102767 T^{6} \))(\( 1 - 936 T + 600637 T^{2} - 223876784 T^{3} + 62359935251 T^{4} - 10089345547944 T^{5} + 1119130473102767 T^{6} \))(\( 1 - 730 T + 427526 T^{2} - 156550492 T^{3} + 44387031898 T^{4} - 7868827190170 T^{5} + 1119130473102767 T^{6} \))
$53$ (\( 1 - 754 T + 148877 T^{2} \))(\( 1 + 582 T + 148877 T^{2} \))(\( 1 + 22 T + 148877 T^{2} \))(\( 1 + 336 T + 148877 T^{2} \))(\( 1 - 582 T + 148877 T^{2} \))(\( 1 - 56 T + 265770 T^{2} - 8337112 T^{3} + 22164361129 T^{4} \))(\( 1 + 982 T + 504354 T^{2} + 146197214 T^{3} + 22164361129 T^{4} \))(\( 1 - 808 T + 449478 T^{2} - 120292616 T^{3} + 22164361129 T^{4} \))(\( 1 + 326 T + 204138 T^{2} + 48533902 T^{3} + 22164361129 T^{4} \))(\( 1 + 808 T + 449478 T^{2} + 120292616 T^{3} + 22164361129 T^{4} \))(\( 1 + 884 T + 460350 T^{2} + 131607268 T^{3} + 22164361129 T^{4} \))(\( 1 + 16 T + 260838 T^{2} + 2382032 T^{3} + 22164361129 T^{4} \))(\( 1 + 34 T + 103410 T^{2} + 5061818 T^{3} + 22164361129 T^{4} \))(\( 1 + 298 T + 380319 T^{2} + 69347636 T^{3} + 56620751763 T^{4} + 6604979616442 T^{5} + 3299763591802133 T^{6} \))(\( 1 - 298 T + 380319 T^{2} - 69347636 T^{3} + 56620751763 T^{4} - 6604979616442 T^{5} + 3299763591802133 T^{6} \))(\( 1 + 110 T + 105959 T^{2} - 57565396 T^{3} + 15774858043 T^{4} + 2438079724190 T^{5} + 3299763591802133 T^{6} \))
$59$ (\( 1 + 380 T + 205379 T^{2} \))(\( 1 + 492 T + 205379 T^{2} \))(\( 1 - 560 T + 205379 T^{2} \))(\( 1 - 360 T + 205379 T^{2} \))(\( 1 - 492 T + 205379 T^{2} \))(\( ( 1 - 616 T + 205379 T^{2} )^{2} \))(\( 1 + 516 T + 418118 T^{2} + 105975564 T^{3} + 42180533641 T^{4} \))(\( ( 1 - 284 T + 205379 T^{2} )^{2} \))(\( 1 - 844 T + 474182 T^{2} - 173339876 T^{3} + 42180533641 T^{4} \))(\( ( 1 + 284 T + 205379 T^{2} )^{2} \))(\( 1 + 104 T + 80534 T^{2} + 21359416 T^{3} + 42180533641 T^{4} \))(\( 1 - 464 T + 458102 T^{2} - 95295856 T^{3} + 42180533641 T^{4} \))(\( 1 + 364 T + 438374 T^{2} + 74757956 T^{3} + 42180533641 T^{4} \))(\( 1 + 428 T + 132409 T^{2} - 53779640 T^{3} + 27194028011 T^{4} + 18053268398348 T^{5} + 8662995818654939 T^{6} \))(\( 1 - 428 T + 132409 T^{2} + 53779640 T^{3} + 27194028011 T^{4} - 18053268398348 T^{5} + 8662995818654939 T^{6} \))(\( 1 - 180 T + 3577 T^{2} + 128522760 T^{3} + 734640683 T^{4} - 7592496055380 T^{5} + 8662995818654939 T^{6} \))
$61$ (\( 1 - 718 T + 226981 T^{2} \))(\( 1 + 880 T + 226981 T^{2} \))(\( 1 + 278 T + 226981 T^{2} \))(\( 1 + 682 T + 226981 T^{2} \))(\( 1 + 880 T + 226981 T^{2} \))(\( 1 - 336 T + 458858 T^{2} - 76265616 T^{3} + 51520374361 T^{4} \))(\( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} \))(\( 1 + 618 T + 515018 T^{2} + 140274258 T^{3} + 51520374361 T^{4} \))(\( 1 + 204 T + 455006 T^{2} + 46304124 T^{3} + 51520374361 T^{4} \))(\( 1 + 618 T + 515018 T^{2} + 140274258 T^{3} + 51520374361 T^{4} \))(\( 1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4} \))(\( 1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4} \))(\( 1 - 792 T + 340886 T^{2} - 179768952 T^{3} + 51520374361 T^{4} \))(\( 1 - 1050 T + 784443 T^{2} - 413439100 T^{3} + 178053656583 T^{4} - 54096393079050 T^{5} + 11694146092834141 T^{6} \))(\( 1 - 1050 T + 784443 T^{2} - 413439100 T^{3} + 178053656583 T^{4} - 54096393079050 T^{5} + 11694146092834141 T^{6} \))(\( 1 - 1222 T + 1103759 T^{2} - 593135356 T^{3} + 250532321579 T^{4} - 62957897469142 T^{5} + 11694146092834141 T^{6} \))
$67$ (\( 1 - 412 T + 300763 T^{2} \))(\( 1 + 826 T + 300763 T^{2} \))(\( 1 + 164 T + 300763 T^{2} \))(\( 1 - 812 T + 300763 T^{2} \))(\( 1 + 826 T + 300763 T^{2} \))(\( 1 + 152 T + 599110 T^{2} + 45715976 T^{3} + 90458382169 T^{4} \))(\( 1 + 356 T + 100374 T^{2} + 107071628 T^{3} + 90458382169 T^{4} \))(\( 1 - 694 T + 681118 T^{2} - 208729522 T^{3} + 90458382169 T^{4} \))(\( 1 + 104 T + 537670 T^{2} + 31279352 T^{3} + 90458382169 T^{4} \))(\( 1 - 694 T + 681118 T^{2} - 208729522 T^{3} + 90458382169 T^{4} \))(\( 1 + 1688 T + 1302310 T^{2} + 507687944 T^{3} + 90458382169 T^{4} \))(\( 1 - 736 T + 602470 T^{2} - 221361568 T^{3} + 90458382169 T^{4} \))(\( 1 + 788 T + 753430 T^{2} + 237001244 T^{3} + 90458382169 T^{4} \))(\( 1 + 52 T + 81089 T^{2} - 61774664 T^{3} + 24388570907 T^{4} + 4703835872788 T^{5} + 27206534396294947 T^{6} \))(\( 1 + 52 T + 81089 T^{2} - 61774664 T^{3} + 24388570907 T^{4} + 4703835872788 T^{5} + 27206534396294947 T^{6} \))(\( 1 - 204 T + 92049 T^{2} + 202232824 T^{3} + 27684933387 T^{4} - 18453509962476 T^{5} + 27206534396294947 T^{6} \))
$71$ (\( 1 - 960 T + 357911 T^{2} \))(\( 1 - 666 T + 357911 T^{2} \))(\( 1 + 672 T + 357911 T^{2} \))(\( 1 + 810 T + 357911 T^{2} \))(\( 1 + 666 T + 357911 T^{2} \))(\( ( 1 - 952 T + 357911 T^{2} )^{2} \))(\( 1 + 310 T + 664038 T^{2} + 110952410 T^{3} + 128100283921 T^{4} \))(\( 1 - 814 T + 769934 T^{2} - 291339554 T^{3} + 128100283921 T^{4} \))(\( 1 + 1670 T + 1384382 T^{2} + 597711370 T^{3} + 128100283921 T^{4} \))(\( 1 + 814 T + 769934 T^{2} + 291339554 T^{3} + 128100283921 T^{4} \))(\( 1 - 136 T + 540446 T^{2} - 48675896 T^{3} + 128100283921 T^{4} \))(\( 1 - 740 T + 757502 T^{2} - 264854140 T^{3} + 128100283921 T^{4} \))(\( 1 + 454 T + 304526 T^{2} + 162491594 T^{3} + 128100283921 T^{4} \))(\( 1 + 416 T + 1050065 T^{2} + 283436624 T^{3} + 375829814215 T^{4} + 53289718111136 T^{5} + 45848500718449031 T^{6} \))(\( 1 - 416 T + 1050065 T^{2} - 283436624 T^{3} + 375829814215 T^{4} - 53289718111136 T^{5} + 45848500718449031 T^{6} \))(\( 1 - 136 T + 900677 T^{2} - 112926832 T^{3} + 322362205747 T^{4} - 17421638613256 T^{5} + 45848500718449031 T^{6} \))
$73$ (\( 1 - 1066 T + 389017 T^{2} \))(\( 1 + 826 T + 389017 T^{2} \))(\( 1 - 82 T + 389017 T^{2} \))(\( 1 + 124 T + 389017 T^{2} \))(\( 1 + 826 T + 389017 T^{2} \))(\( 1 - 676 T + 655606 T^{2} - 262975492 T^{3} + 151334226289 T^{4} \))(\( 1 - 326 T + 789802 T^{2} - 126819542 T^{3} + 151334226289 T^{4} \))(\( 1 - 82 T + 422290 T^{2} - 31899394 T^{3} + 151334226289 T^{4} \))(\( 1 + 386 T + 152218 T^{2} + 150160562 T^{3} + 151334226289 T^{4} \))(\( 1 - 82 T + 422290 T^{2} - 31899394 T^{3} + 151334226289 T^{4} \))(\( 1 - 508 T + 13078 T^{2} - 197620636 T^{3} + 151334226289 T^{4} \))(\( 1 - 424 T + 748558 T^{2} - 164943208 T^{3} + 151334226289 T^{4} \))(\( 1 - 778 T + 794698 T^{2} - 302655226 T^{3} + 151334226289 T^{4} \))(\( 1 + 312 T + 987991 T^{2} + 198811072 T^{3} + 384345294847 T^{4} + 47216278602168 T^{5} + 58871586708267913 T^{6} \))(\( 1 + 312 T + 987991 T^{2} + 198811072 T^{3} + 384345294847 T^{4} + 47216278602168 T^{5} + 58871586708267913 T^{6} \))(\( 1 - 310 T + 450279 T^{2} - 192471924 T^{3} + 175166185743 T^{4} - 46913610149590 T^{5} + 58871586708267913 T^{6} \))
$79$ (\( 1 - 896 T + 493039 T^{2} \))(\( 1 + 592 T + 493039 T^{2} \))(\( 1 + 1000 T + 493039 T^{2} \))(\( 1 - 1136 T + 493039 T^{2} \))(\( 1 + 592 T + 493039 T^{2} \))(\( 1 - 1014 T + 1120119 T^{2} - 499941546 T^{3} + 243087455521 T^{4} \))(\( 1 - 1832 T + 1824478 T^{2} - 903247448 T^{3} + 243087455521 T^{4} \))(\( 1 - 600 T + 618846 T^{2} - 295823400 T^{3} + 243087455521 T^{4} \))(\( 1 + 888 T + 1007454 T^{2} + 437818632 T^{3} + 243087455521 T^{4} \))(\( 1 - 600 T + 618846 T^{2} - 295823400 T^{3} + 243087455521 T^{4} \))(\( 1 + 432 T + 602142 T^{2} + 212992848 T^{3} + 243087455521 T^{4} \))(\( 1 + 408 T - 143586 T^{2} + 201159912 T^{3} + 243087455521 T^{4} \))(\( 1 - 408 T + 994782 T^{2} - 201159912 T^{3} + 243087455521 T^{4} \))(\( 1 + 488 T + 50733 T^{2} + 198311472 T^{3} + 25013347587 T^{4} + 118626678294248 T^{5} + 119851595982618319 T^{6} \))(\( 1 + 488 T + 50733 T^{2} + 198311472 T^{3} + 25013347587 T^{4} + 118626678294248 T^{5} + 119851595982618319 T^{6} \))(\( 1 + 1034 T + 1094262 T^{2} + 675989052 T^{3} + 539513842218 T^{4} + 251352429008714 T^{5} + 119851595982618319 T^{6} \))
$83$ (\( 1 + 436 T + 571787 T^{2} \))(\( 1 + 792 T + 571787 T^{2} \))(\( 1 - 448 T + 571787 T^{2} \))(\( 1 + 156 T + 571787 T^{2} \))(\( 1 - 792 T + 571787 T^{2} \))(\( 1 - 376 T + 458918 T^{2} - 214991912 T^{3} + 326940373369 T^{4} \))(\( 1 - 680 T + 744870 T^{2} - 388815160 T^{3} + 326940373369 T^{4} \))(\( 1 + 268 T + 779030 T^{2} + 153238916 T^{3} + 326940373369 T^{4} \))(\( 1 + 928 T + 600710 T^{2} + 530618336 T^{3} + 326940373369 T^{4} \))(\( 1 - 268 T + 779030 T^{2} - 153238916 T^{3} + 326940373369 T^{4} \))(\( 1 - 584 T + 1172390 T^{2} - 333923608 T^{3} + 326940373369 T^{4} \))(\( 1 + 608 T + 1200710 T^{2} + 347646496 T^{3} + 326940373369 T^{4} \))(\( 1 + 1136 T + 1169990 T^{2} + 649550032 T^{3} + 326940373369 T^{4} \))(\( 1 + 948 T + 1105441 T^{2} + 652600952 T^{3} + 632076793067 T^{4} + 309939473953812 T^{5} + 186940255267540403 T^{6} \))(\( 1 - 948 T + 1105441 T^{2} - 652600952 T^{3} + 632076793067 T^{4} - 309939473953812 T^{5} + 186940255267540403 T^{6} \))(\( 1 - 1660 T + 2204369 T^{2} - 1855605736 T^{3} + 1260429537403 T^{4} - 542721019792540 T^{5} + 186940255267540403 T^{6} \))
$89$ (\( 1 - 1038 T + 704969 T^{2} \))(\( 1 + 1002 T + 704969 T^{2} \))(\( 1 - 870 T + 704969 T^{2} \))(\( 1 - 1038 T + 704969 T^{2} \))(\( 1 - 1002 T + 704969 T^{2} \))(\( 1 - 216 T + 1417730 T^{2} - 152273304 T^{3} + 496981290961 T^{4} \))(\( 1 + 796 T + 411158 T^{2} + 561155324 T^{3} + 496981290961 T^{4} \))(\( 1 + 72 T + 448286 T^{2} + 50757768 T^{3} + 496981290961 T^{4} \))(\( 1 + 588 T + 1495334 T^{2} + 414521772 T^{3} + 496981290961 T^{4} \))(\( 1 - 72 T + 448286 T^{2} - 50757768 T^{3} + 496981290961 T^{4} \))(\( 1 - 1404 T + 1802390 T^{2} - 989776476 T^{3} + 496981290961 T^{4} \))(\( 1 - 1332 T + 1790774 T^{2} - 939018708 T^{3} + 496981290961 T^{4} \))(\( 1 + 36 T - 842170 T^{2} + 25378884 T^{3} + 496981290961 T^{4} \))(\( 1 - 142 T + 1874695 T^{2} - 219867172 T^{3} + 1321601859455 T^{4} - 70571343316462 T^{5} + 350356403707485209 T^{6} \))(\( 1 + 142 T + 1874695 T^{2} + 219867172 T^{3} + 1321601859455 T^{4} + 70571343316462 T^{5} + 350356403707485209 T^{6} \))(\( 1 + 242 T + 1427227 T^{2} + 347564516 T^{3} + 1006150790963 T^{4} + 120269472412562 T^{5} + 350356403707485209 T^{6} \))
$97$ (\( 1 + 702 T + 912673 T^{2} \))(\( 1 - 1442 T + 912673 T^{2} \))(\( 1 - 1026 T + 912673 T^{2} \))(\( 1 - 1208 T + 912673 T^{2} \))(\( 1 - 1442 T + 912673 T^{2} \))(\( 1 - 2742 T + 3608187 T^{2} - 2502549366 T^{3} + 832972004929 T^{4} \))(\( 1 + 670 T + 1145410 T^{2} + 611490910 T^{3} + 832972004929 T^{4} \))(\( 1 - 1626 T + 1498938 T^{2} - 1484006298 T^{3} + 832972004929 T^{4} \))(\( 1 - 522 T + 291282 T^{2} - 476415306 T^{3} + 832972004929 T^{4} \))(\( 1 - 1626 T + 1498938 T^{2} - 1484006298 T^{3} + 832972004929 T^{4} \))(\( 1 + 1188 T + 2161254 T^{2} + 1084255524 T^{3} + 832972004929 T^{4} \))(\( 1 + 2448 T + 3286542 T^{2} + 2234223504 T^{3} + 832972004929 T^{4} \))(\( 1 + 498 T + 1796754 T^{2} + 454511154 T^{3} + 832972004929 T^{4} \))(\( 1 + 3052 T + 5048255 T^{2} + 5775117464 T^{3} + 4607406035615 T^{4} + 2542230559043308 T^{5} + 760231058654565217 T^{6} \))(\( 1 + 3052 T + 5048255 T^{2} + 5775117464 T^{3} + 4607406035615 T^{4} + 2542230559043308 T^{5} + 760231058654565217 T^{6} \))(\( 1 - 100 T + 2266056 T^{2} - 184512618 T^{3} + 2068168127688 T^{4} - 83297200492900 T^{5} + 760231058654565217 T^{6} \))
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