LMFDB - Space of modular forms of level 24 and weight 4

Defining parameters

Level:	$ N $	=	$ 24\( 24 = 2^{3} \cdot 3 $ \)
Weight:	$ k $	=	$ 4 $
Nonzero newspaces:			$ 3 $
Newform subspaces:			$ 4 $
Sturm bound:			$128$
Trace bound:			$1$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(\Gamma_1(24))$.

	Total	New	Old
Modular forms	60	21	39
Cusp forms	36	17	19
Eisenstein series	24	4	20

Trace form

$ 17q + 2q^{2} + q^{3} + 20q^{4} + 14q^{5} - 14q^{6} + 4q^{7} - 76q^{8} - 47q^{9} + O(q^{10}) $

\( 17q + 2q^{2} + q^{3} + 20q^{4} + 14q^{5} - 14q^{6} + 4q^{7} - 76q^{8} - 47q^{9} + 36q^{10} - 28q^{11} - 56q^{12} - 74q^{13} - 100q^{14} - 18q^{15} - 96q^{16} + 134q^{17} + 166q^{18} + 64q^{19} + 56q^{20} - 72q^{21} + 448q^{22} + 336q^{23} + 532q^{24} + 11q^{25} + 56q^{26} - 107q^{27} + 176q^{28} - 138q^{29} - 252q^{30} - 556q^{31} - 248q^{32} - 148q^{33} - 1332q^{34} - 336q^{35} - 1028q^{36} + 30q^{37} - 776q^{38} + 90q^{39} - 1016q^{40} + 518q^{41} + 756q^{42} + 432q^{43} + 1152q^{44} + 126q^{45} + 1768q^{46} - 168q^{47} + 2296q^{48} + 621q^{49} + 1970q^{50} + 998q^{51} + 1744q^{52} - 130q^{53} - 2114q^{54} + 632q^{55} - 1864q^{56} + 224q^{57} - 2476q^{58} + 596q^{59} - 3792q^{60} - 218q^{61} - 2108q^{62} - 468q^{63} - 1552q^{64} - 2780q^{65} + 1352q^{66} - 2072q^{67} + 2976q^{68} + 24q^{69} + 5048q^{70} - 848q^{71} + 3964q^{72} + 170q^{73} + 1568q^{74} - 1745q^{75} + 1864q^{76} + 672q^{77} - 2088q^{78} - 76q^{79} - 2112q^{80} + 721q^{81} - 5372q^{82} - 1508q^{83} - 6216q^{84} + 1148q^{85} - 760q^{86} + 630q^{87} - 2576q^{88} - 466q^{89} + 3564q^{90} + 4944q^{91} + 1728q^{92} + 240q^{93} + 6888q^{94} + 6392q^{95} + 6424q^{96} - 1630q^{97} + 3354q^{98} + 3860q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{4}^{\mathrm{new}}(\Gamma_1(24))$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $ S_k^{\mathrm{new}}(N, \chi) $ we list the newforms together with their dimension.

Label	$\chi$	Newforms	Dimension	$\chi$ degree
24.4.a	$\chi_{24}(1, \cdot)$	24.4.a.a	1	1
24.4.c	$\chi_{24}(23, \cdot)$	None	0	1
24.4.d	$\chi_{24}(13, \cdot)$	24.4.d.a	6	1
24.4.f	$\chi_{24}(11, \cdot)$	24.4.f.a	2	1
24.4.f	$\chi_{24}(11, \cdot)$	24.4.f.b	8	1

Decomposition of $S_{4}^{\mathrm{old}}(\Gamma_1(24))$ into lower level spaces

$ S_{4}^{\mathrm{old}}(\Gamma_1(24)) \cong $ $S_{4}^{\mathrm{new}}(\Gamma_1(6))$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$^{\oplus 2}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	($ 1 - 2 T - 6 T^{2} + 40 T^{3} - 48 T^{4} - 128 T^{5} + 512 T^{6} $)($ 1 + 8 T^{2} $)($ 1 - 10 T^{2} + 120 T^{4} - 640 T^{6} + 4096 T^{8} $)
$3$	($ 1 - 3 T $)($ ( 1 + 9 T^{2} )^{3} $)($ 1 - 10 T + 27 T^{2} $)($ ( 1 + 6 T + 30 T^{2} + 162 T^{3} + 729 T^{4} )^{2} $)
$5$	($ 1 - 14 T + 125 T^{2} $)($ 1 - 322 T^{2} + 49351 T^{4} - 6170684 T^{6} + 771109375 T^{8} - 78613281250 T^{10} + 3814697265625 T^{12} $)($ ( 1 + 125 T^{2} )^{2} $)($ ( 1 + 176 T^{2} + 38862 T^{4} + 2750000 T^{6} + 244140625 T^{8} )^{2} $)
$7$	($ 1 + 24 T + 343 T^{2} $)($ ( 1 - 14 T + 449 T^{2} - 3788 T^{3} + 154007 T^{4} - 1647086 T^{5} + 40353607 T^{6} )^{2} $)($ ( 1 - 343 T^{2} )^{2} $)($ ( 1 - 448 T^{2} + 215646 T^{4} - 52706752 T^{6} + 13841287201 T^{8} )^{2} $)
$11$	($ 1 + 28 T + 1331 T^{2} $)($ 1 - 2354 T^{2} + 4518503 T^{4} - 5987722076 T^{6} + 8004803693183 T^{8} - 7387860398801234 T^{10} + 5559917313492231481 T^{12} $)($ ( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} ) $)($ ( 1 - 4840 T^{2} + 9341310 T^{4} - 8574355240 T^{6} + 3138428376721 T^{8} )^{2} $)
$13$	($ 1 + 74 T + 2197 T^{2} $)($ 1 - 8270 T^{2} + 36264983 T^{4} - 97600232804 T^{6} + 175044146329247 T^{8} - 192675163962917870 T^{10} + $$11\!\cdots\!29$$ T^{12} $)($ ( 1 - 2197 T^{2} )^{2} $)($ ( 1 - 2980 T^{2} + 4014966 T^{4} - 14383890820 T^{6} + 23298085122481 T^{8} )^{2} $)
$17$	($ 1 - 82 T + 4913 T^{2} $)($ ( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 17760495 T^{4} - 627576794 T^{5} + 118587876497 T^{6} )^{2} $)($ ( 1 - 90 T + 4913 T^{2} )( 1 + 90 T + 4913 T^{2} ) $)($ ( 1 - 17188 T^{2} + 120704262 T^{4} - 414876535972 T^{6} + 582622237229761 T^{8} )^{2} $)
$19$	($ 1 - 92 T + 6859 T^{2} $)($ 1 - 18194 T^{2} + 183315287 T^{4} - 1372700323292 T^{6} + 8624229177682847 T^{8} - 40269051637489733234 T^{10} + $$10\!\cdots\!41$$ T^{12} $)($ ( 1 + 106 T + 6859 T^{2} )^{2} $)($ ( 1 - 46 T + 10254 T^{2} - 315514 T^{3} + 47045881 T^{4} )^{4} $)
$23$	($ 1 - 8 T + 12167 T^{2} $)($ ( 1 - 164 T + 42885 T^{2} - 4036280 T^{3} + 521781795 T^{4} - 24277885796 T^{5} + 1801152661463 T^{6} )^{2} $)($ ( 1 + 12167 T^{2} )^{2} $)($ ( 1 + 31196 T^{2} + 464722854 T^{4} + 4618127593244 T^{6} + 21914624432020321 T^{8} )^{2} $)
$29$	($ 1 + 138 T + 24389 T^{2} $)($ 1 - 123986 T^{2} + 6779237687 T^{4} - 212188653261788 T^{6} + 4032448674829698527 T^{8} - $$43\!\cdots\!26$$ T^{10} + $$21\!\cdots\!61$$ T^{12} $)($ ( 1 + 24389 T^{2} )^{2} $)($ ( 1 + 53936 T^{2} + 1889351598 T^{4} + 32082390641456 T^{6} + 353814783205469041 T^{8} )^{2} $)
$31$	($ 1 - 80 T + 29791 T^{2} $)($ ( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 2795855559 T^{4} + 282226170558 T^{5} + 26439622160671 T^{6} )^{2} $)($ ( 1 - 29791 T^{2} )^{2} $)($ ( 1 - 98176 T^{2} + 4116020094 T^{4} - 87131561385856 T^{6} + 787662783788549761 T^{8} )^{2} $)
$37$	($ 1 - 30 T + 50653 T^{2} $)($ 1 - 124142 T^{2} + 4233590471 T^{4} - 51775900405988 T^{6} + 10862234876335448639 T^{8} - $$81\!\cdots\!02$$ T^{10} + $$16\!\cdots\!29$$ T^{12} $)($ ( 1 - 50653 T^{2} )^{2} $)($ ( 1 - 124996 T^{2} + 9025572822 T^{4} - 320705538219364 T^{6} + 6582952005840035281 T^{8} )^{2} $)
$41$	($ 1 - 282 T + 68921 T^{2} $)($ ( 1 - 118 T + 89463 T^{2} + 3720620 T^{3} + 6165879423 T^{4} - 560512300438 T^{5} + 327381934393961 T^{6} )^{2} $)($ ( 1 - 522 T + 68921 T^{2} )( 1 + 522 T + 68921 T^{2} ) $)($ ( 1 - 82084 T^{2} + 4965540774 T^{4} - 389907556518244 T^{6} + 22563490300366186081 T^{8} )^{2} $)
$43$	($ 1 - 4 T + 79507 T^{2} $)($ 1 - 247490 T^{2} + 32846327015 T^{4} - 3025278566260412 T^{6} + $$20\!\cdots\!35$$ T^{8} - $$98\!\cdots\!90$$ T^{10} + $$25\!\cdots\!49$$ T^{12} $)($ ( 1 - 290 T + 79507 T^{2} )^{2} $)($ ( 1 + 38 T + 109182 T^{2} + 3021266 T^{3} + 6321363049 T^{4} )^{4} $)
$47$	($ 1 - 240 T + 103823 T^{2} $)($ ( 1 + 204 T + 283677 T^{2} + 40395048 T^{3} + 29452197171 T^{4} + 2198959927116 T^{5} + 1119130473102767 T^{6} )^{2} $)($ ( 1 + 103823 T^{2} )^{2} $)($ ( 1 + 93500 T^{2} - 1715710650 T^{4} + 1007856633261500 T^{6} + $$11\!\cdots\!41$$ T^{8} )^{2} $)
$53$	($ 1 + 130 T + 148877 T^{2} $)($ 1 - 292802 T^{2} + 13793539751 T^{4} + 2699981198933572 T^{6} + $$30\!\cdots\!79$$ T^{8} - $$14\!\cdots\!82$$ T^{10} + $$10\!\cdots\!89$$ T^{12} $)($ ( 1 + 148877 T^{2} )^{2} $)($ ( 1 + 418352 T^{2} + 87270813582 T^{4} + 9272504807039408 T^{6} + $$49\!\cdots\!41$$ T^{8} )^{2} $)
$59$	($ 1 - 596 T + 205379 T^{2} $)($ 1 - 1093858 T^{2} + 524838290887 T^{4} - 140555838506313212 T^{6} + $$22\!\cdots\!67$$ T^{8} - $$19\!\cdots\!98$$ T^{10} + $$75\!\cdots\!21$$ T^{12} $)($ ( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} ) $)($ ( 1 - 799912 T^{2} + 244209482046 T^{4} - 33740715025839592 T^{6} + $$17\!\cdots\!81$$ T^{8} )^{2} $)
$61$	($ 1 + 218 T + 226981 T^{2} $)($ 1 - 459870 T^{2} + 162687674679 T^{4} - 38865284671151684 T^{6} + $$83\!\cdots\!19$$ T^{8} - $$12\!\cdots\!70$$ T^{10} + $$13\!\cdots\!81$$ T^{12} $)($ ( 1 - 226981 T^{2} )^{2} $)($ ( 1 - 357220 T^{2} + 77943572022 T^{4} - 18404108129236420 T^{6} + $$26\!\cdots\!21$$ T^{8} )^{2} $)
$67$	($ 1 + 436 T + 300763 T^{2} $)($ 1 - 750066 T^{2} + 226548162807 T^{4} - 53949461413257884 T^{6} + $$20\!\cdots\!83$$ T^{8} - $$61\!\cdots\!26$$ T^{10} + $$74\!\cdots\!09$$ T^{12} $)($ ( 1 + 70 T + 300763 T^{2} )^{2} $)($ ( 1 + 374 T + 633822 T^{2} + 112485362 T^{3} + 90458382169 T^{4} )^{4} $)
$71$	($ 1 - 856 T + 357911 T^{2} $)($ ( 1 + 852 T + 1006773 T^{2} + 524795352 T^{3} + 360335131203 T^{4} + 109141441900692 T^{5} + 45848500718449031 T^{6} )^{2} $)($ ( 1 + 357911 T^{2} )^{2} $)($ ( 1 + 776732 T^{2} + 403415373030 T^{4} + 99499589730526172 T^{6} + $$16\!\cdots\!41$$ T^{8} )^{2} $)
$73$	($ 1 + 998 T + 389017 T^{2} $)($ ( 1 - 478 T + 911095 T^{2} - 251066948 T^{3} + 354431443615 T^{4} - 72337760166142 T^{5} + 58871586708267913 T^{6} )^{2} $)($ ( 1 + 430 T + 389017 T^{2} )^{2} $)($ ( 1 - 268 T + 73158 T^{2} - 104256556 T^{3} + 151334226289 T^{4} )^{4} $)
$79$	($ 1 + 32 T + 493039 T^{2} $)($ ( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 693998245127 T^{4} + 5347924021462 T^{5} + 119851595982618319 T^{6} )^{2} $)($ ( 1 - 493039 T^{2} )^{2} $)($ ( 1 - 943744 T^{2} + 544083847614 T^{4} - 229412327623210624 T^{6} + $$59\!\cdots\!41$$ T^{8} )^{2} $)
$83$	($ 1 + 1508 T + 571787 T^{2} $)($ 1 - 2910274 T^{2} + 3775777045015 T^{4} - 2787348361783974908 T^{6} + $$12\!\cdots\!35$$ T^{8} - $$31\!\cdots\!14$$ T^{10} + $$34\!\cdots\!09$$ T^{12} $)($ ( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} ) $)($ ( 1 - 1890664 T^{2} + 1510361878110 T^{4} - 618134394075327016 T^{6} + $$10\!\cdots\!61$$ T^{8} )^{2} $)
$89$	($ 1 + 246 T + 704969 T^{2} $)($ ( 1 + 110 T + 2073543 T^{2} + 156516836 T^{3} + 1461783535167 T^{4} + 54667942005710 T^{5} + 350356403707485209 T^{6} )^{2} $)($ ( 1 - 1026 T + 704969 T^{2} )( 1 + 1026 T + 704969 T^{2} ) $)($ ( 1 - 175300 T^{2} - 599506777050 T^{4} - 87120820305463300 T^{6} + $$24\!\cdots\!21$$ T^{8} )^{2} $)
$97$	($ 1 - 866 T + 912673 T^{2} $)($ ( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2728599301967 T^{4} + 1017891790023238 T^{5} + 760231058654565217 T^{6} )^{2} $)($ ( 1 - 1910 T + 912673 T^{2} )^{2} $)($ ( 1 + 968 T + 1792302 T^{2} + 883467464 T^{3} + 832972004929 T^{4} )^{4} $)
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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:	\( N \)	=	\( 24\( 24 = 2^{3} \cdot 3 \) \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 3 \)
Newform subspaces:			\( 4 \)
Sturm bound:			\(128\)
Trace bound:			\(1\)

Introduction

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Modular forms

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Representations

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(24))\) 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	24.4

Level	24
Weight	4
Dimension	17
Nonzero newspaces	3
Newform subspaces	4
Sturm bound	128
Trace bound	1

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
24.4.a	\(\chi_{24}(1, \cdot)\)	24.4.a.a	1	1
24.4.c	\(\chi_{24}(23, \cdot)\)	None	0	1
24.4.d	\(\chi_{24}(13, \cdot)\)	24.4.d.a	6	1
24.4.f	\(\chi_{24}(11, \cdot)\)	24.4.f.a	2	1
24.4.f	\(\chi_{24}(11, \cdot)\)	24.4.f.b	8	1

$p$	$F_p(T)$
$2$	(\( 1 - 2 T - 6 T^{2} + 40 T^{3} - 48 T^{4} - 128 T^{5} + 512 T^{6} \))(\( 1 + 8 T^{2} \))(\( 1 - 10 T^{2} + 120 T^{4} - 640 T^{6} + 4096 T^{8} \))
$3$	(\( 1 - 3 T \))(\( ( 1 + 9 T^{2} )^{3} \))(\( 1 - 10 T + 27 T^{2} \))(\( ( 1 + 6 T + 30 T^{2} + 162 T^{3} + 729 T^{4} )^{2} \))
$5$	(\( 1 - 14 T + 125 T^{2} \))(\( 1 - 322 T^{2} + 49351 T^{4} - 6170684 T^{6} + 771109375 T^{8} - 78613281250 T^{10} + 3814697265625 T^{12} \))(\( ( 1 + 125 T^{2} )^{2} \))(\( ( 1 + 176 T^{2} + 38862 T^{4} + 2750000 T^{6} + 244140625 T^{8} )^{2} \))
$7$	(\( 1 + 24 T + 343 T^{2} \))(\( ( 1 - 14 T + 449 T^{2} - 3788 T^{3} + 154007 T^{4} - 1647086 T^{5} + 40353607 T^{6} )^{2} \))(\( ( 1 - 343 T^{2} )^{2} \))(\( ( 1 - 448 T^{2} + 215646 T^{4} - 52706752 T^{6} + 13841287201 T^{8} )^{2} \))
$11$	(\( 1 + 28 T + 1331 T^{2} \))(\( 1 - 2354 T^{2} + 4518503 T^{4} - 5987722076 T^{6} + 8004803693183 T^{8} - 7387860398801234 T^{10} + 5559917313492231481 T^{12} \))(\( ( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} ) \))(\( ( 1 - 4840 T^{2} + 9341310 T^{4} - 8574355240 T^{6} + 3138428376721 T^{8} )^{2} \))
$13$	(\( 1 + 74 T + 2197 T^{2} \))(\( 1 - 8270 T^{2} + 36264983 T^{4} - 97600232804 T^{6} + 175044146329247 T^{8} - 192675163962917870 T^{10} + \)\(11\!\cdots\!29\)\( T^{12} \))(\( ( 1 - 2197 T^{2} )^{2} \))(\( ( 1 - 2980 T^{2} + 4014966 T^{4} - 14383890820 T^{6} + 23298085122481 T^{8} )^{2} \))
$17$	(\( 1 - 82 T + 4913 T^{2} \))(\( ( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 17760495 T^{4} - 627576794 T^{5} + 118587876497 T^{6} )^{2} \))(\( ( 1 - 90 T + 4913 T^{2} )( 1 + 90 T + 4913 T^{2} ) \))(\( ( 1 - 17188 T^{2} + 120704262 T^{4} - 414876535972 T^{6} + 582622237229761 T^{8} )^{2} \))
$19$	(\( 1 - 92 T + 6859 T^{2} \))(\( 1 - 18194 T^{2} + 183315287 T^{4} - 1372700323292 T^{6} + 8624229177682847 T^{8} - 40269051637489733234 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} \))(\( ( 1 + 106 T + 6859 T^{2} )^{2} \))(\( ( 1 - 46 T + 10254 T^{2} - 315514 T^{3} + 47045881 T^{4} )^{4} \))
$23$	(\( 1 - 8 T + 12167 T^{2} \))(\( ( 1 - 164 T + 42885 T^{2} - 4036280 T^{3} + 521781795 T^{4} - 24277885796 T^{5} + 1801152661463 T^{6} )^{2} \))(\( ( 1 + 12167 T^{2} )^{2} \))(\( ( 1 + 31196 T^{2} + 464722854 T^{4} + 4618127593244 T^{6} + 21914624432020321 T^{8} )^{2} \))
$29$	(\( 1 + 138 T + 24389 T^{2} \))(\( 1 - 123986 T^{2} + 6779237687 T^{4} - 212188653261788 T^{6} + 4032448674829698527 T^{8} - \)\(43\!\cdots\!26\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} \))(\( ( 1 + 24389 T^{2} )^{2} \))(\( ( 1 + 53936 T^{2} + 1889351598 T^{4} + 32082390641456 T^{6} + 353814783205469041 T^{8} )^{2} \))
$31$	(\( 1 - 80 T + 29791 T^{2} \))(\( ( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 2795855559 T^{4} + 282226170558 T^{5} + 26439622160671 T^{6} )^{2} \))(\( ( 1 - 29791 T^{2} )^{2} \))(\( ( 1 - 98176 T^{2} + 4116020094 T^{4} - 87131561385856 T^{6} + 787662783788549761 T^{8} )^{2} \))
$37$	(\( 1 - 30 T + 50653 T^{2} \))(\( 1 - 124142 T^{2} + 4233590471 T^{4} - 51775900405988 T^{6} + 10862234876335448639 T^{8} - \)\(81\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!29\)\( T^{12} \))(\( ( 1 - 50653 T^{2} )^{2} \))(\( ( 1 - 124996 T^{2} + 9025572822 T^{4} - 320705538219364 T^{6} + 6582952005840035281 T^{8} )^{2} \))
$41$	(\( 1 - 282 T + 68921 T^{2} \))(\( ( 1 - 118 T + 89463 T^{2} + 3720620 T^{3} + 6165879423 T^{4} - 560512300438 T^{5} + 327381934393961 T^{6} )^{2} \))(\( ( 1 - 522 T + 68921 T^{2} )( 1 + 522 T + 68921 T^{2} ) \))(\( ( 1 - 82084 T^{2} + 4965540774 T^{4} - 389907556518244 T^{6} + 22563490300366186081 T^{8} )^{2} \))
$43$	(\( 1 - 4 T + 79507 T^{2} \))(\( 1 - 247490 T^{2} + 32846327015 T^{4} - 3025278566260412 T^{6} + \)\(20\!\cdots\!35\)\( T^{8} - \)\(98\!\cdots\!90\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} \))(\( ( 1 - 290 T + 79507 T^{2} )^{2} \))(\( ( 1 + 38 T + 109182 T^{2} + 3021266 T^{3} + 6321363049 T^{4} )^{4} \))
$47$	(\( 1 - 240 T + 103823 T^{2} \))(\( ( 1 + 204 T + 283677 T^{2} + 40395048 T^{3} + 29452197171 T^{4} + 2198959927116 T^{5} + 1119130473102767 T^{6} )^{2} \))(\( ( 1 + 103823 T^{2} )^{2} \))(\( ( 1 + 93500 T^{2} - 1715710650 T^{4} + 1007856633261500 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \))
$53$	(\( 1 + 130 T + 148877 T^{2} \))(\( 1 - 292802 T^{2} + 13793539751 T^{4} + 2699981198933572 T^{6} + \)\(30\!\cdots\!79\)\( T^{8} - \)\(14\!\cdots\!82\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} \))(\( ( 1 + 148877 T^{2} )^{2} \))(\( ( 1 + 418352 T^{2} + 87270813582 T^{4} + 9272504807039408 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} )^{2} \))
$59$	(\( 1 - 596 T + 205379 T^{2} \))(\( 1 - 1093858 T^{2} + 524838290887 T^{4} - 140555838506313212 T^{6} + \)\(22\!\cdots\!67\)\( T^{8} - \)\(19\!\cdots\!98\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} ) \))(\( ( 1 - 799912 T^{2} + 244209482046 T^{4} - 33740715025839592 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))
$61$	(\( 1 + 218 T + 226981 T^{2} \))(\( 1 - 459870 T^{2} + 162687674679 T^{4} - 38865284671151684 T^{6} + \)\(83\!\cdots\!19\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{10} + \)\(13\!\cdots\!81\)\( T^{12} \))(\( ( 1 - 226981 T^{2} )^{2} \))(\( ( 1 - 357220 T^{2} + 77943572022 T^{4} - 18404108129236420 T^{6} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \))
$67$	(\( 1 + 436 T + 300763 T^{2} \))(\( 1 - 750066 T^{2} + 226548162807 T^{4} - 53949461413257884 T^{6} + \)\(20\!\cdots\!83\)\( T^{8} - \)\(61\!\cdots\!26\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} \))(\( ( 1 + 70 T + 300763 T^{2} )^{2} \))(\( ( 1 + 374 T + 633822 T^{2} + 112485362 T^{3} + 90458382169 T^{4} )^{4} \))
$71$	(\( 1 - 856 T + 357911 T^{2} \))(\( ( 1 + 852 T + 1006773 T^{2} + 524795352 T^{3} + 360335131203 T^{4} + 109141441900692 T^{5} + 45848500718449031 T^{6} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 + 776732 T^{2} + 403415373030 T^{4} + 99499589730526172 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))
$73$	(\( 1 + 998 T + 389017 T^{2} \))(\( ( 1 - 478 T + 911095 T^{2} - 251066948 T^{3} + 354431443615 T^{4} - 72337760166142 T^{5} + 58871586708267913 T^{6} )^{2} \))(\( ( 1 + 430 T + 389017 T^{2} )^{2} \))(\( ( 1 - 268 T + 73158 T^{2} - 104256556 T^{3} + 151334226289 T^{4} )^{4} \))
$79$	(\( 1 + 32 T + 493039 T^{2} \))(\( ( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 693998245127 T^{4} + 5347924021462 T^{5} + 119851595982618319 T^{6} )^{2} \))(\( ( 1 - 493039 T^{2} )^{2} \))(\( ( 1 - 943744 T^{2} + 544083847614 T^{4} - 229412327623210624 T^{6} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \))
$83$	(\( 1 + 1508 T + 571787 T^{2} \))(\( 1 - 2910274 T^{2} + 3775777045015 T^{4} - 2787348361783974908 T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(31\!\cdots\!14\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} \))(\( ( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} ) \))(\( ( 1 - 1890664 T^{2} + 1510361878110 T^{4} - 618134394075327016 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} )^{2} \))
$89$	(\( 1 + 246 T + 704969 T^{2} \))(\( ( 1 + 110 T + 2073543 T^{2} + 156516836 T^{3} + 1461783535167 T^{4} + 54667942005710 T^{5} + 350356403707485209 T^{6} )^{2} \))(\( ( 1 - 1026 T + 704969 T^{2} )( 1 + 1026 T + 704969 T^{2} ) \))(\( ( 1 - 175300 T^{2} - 599506777050 T^{4} - 87120820305463300 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$	(\( 1 - 866 T + 912673 T^{2} \))(\( ( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2728599301967 T^{4} + 1017891790023238 T^{5} + 760231058654565217 T^{6} )^{2} \))(\( ( 1 - 1910 T + 912673 T^{2} )^{2} \))(\( ( 1 + 968 T + 1792302 T^{2} + 883467464 T^{3} + 832972004929 T^{4} )^{4} \))
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