Properties

Label 12.5
Level 12
Weight 5
Dimension 5
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 40
Trace bound 1

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Defining parameters

Level: \( N \) = \( 12\( 12 = 2^{2} \cdot 3 \) \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(12))\).

Total New Old
Modular forms 21 5 16
Cusp forms 11 5 6
Eisenstein series 10 0 10

Trace form

\( 5q + 6q^{2} + 9q^{3} - 20q^{4} + 24q^{5} - 18q^{6} - 94q^{7} - 27q^{9} + O(q^{10}) \) \( 5q + 6q^{2} + 9q^{3} - 20q^{4} + 24q^{5} - 18q^{6} - 94q^{7} - 27q^{9} - 172q^{10} + 180q^{12} + 442q^{13} + 600q^{14} + 112q^{16} - 600q^{17} - 162q^{18} - 46q^{19} - 1368q^{20} - 990q^{21} - 1128q^{22} + 1296q^{24} + 1597q^{25} + 1692q^{26} + 729q^{27} + 1488q^{28} + 888q^{29} - 1980q^{30} + 194q^{31} - 2784q^{32} + 720q^{33} - 484q^{34} + 540q^{36} - 6470q^{37} + 4680q^{38} + 1314q^{39} + 1664q^{40} + 552q^{41} - 2088q^{42} - 3214q^{43} - 3696q^{44} - 648q^{45} - 384q^{46} + 1008q^{48} + 5863q^{49} - 1038q^{50} + 6008q^{52} + 5112q^{53} + 486q^{54} + 1728q^{56} + 5202q^{57} - 124q^{58} - 2664q^{60} + 2266q^{61} - 7224q^{62} - 7614q^{63} - 14720q^{64} - 18192q^{65} + 4824q^{66} + 5906q^{67} + 5496q^{68} - 9792q^{69} + 6096q^{70} + 298q^{73} - 4116q^{74} + 5625q^{75} - 1872q^{76} + 20928q^{77} + 9900q^{78} + 7682q^{79} + 25632q^{80} + 9477q^{81} + 3740q^{82} - 10512q^{84} - 10256q^{85} - 19560q^{86} - 8640q^{88} - 25080q^{89} + 4644q^{90} - 13724q^{91} + 18816q^{92} - 15390q^{93} - 5232q^{94} - 8352q^{96} + 4234q^{97} - 5850q^{98} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(12))\)

We only show spaces with odd 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
12.5.c \(\chi_{12}(5, \cdot)\) 12.5.c.a 1 1
12.5.d \(\chi_{12}(7, \cdot)\) 12.5.d.a 4 1

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(12))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(12)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 6 T + 28 T^{2} - 96 T^{3} + 256 T^{4} \))
$3$ (\( 1 - 9 T \))(\( ( 1 + 27 T^{2} )^{2} \))
$5$ (\( ( 1 - 25 T )( 1 + 25 T ) \))(\( ( 1 - 12 T + 454 T^{2} - 7500 T^{3} + 390625 T^{4} )^{2} \))
$7$ (\( 1 + 94 T + 2401 T^{2} \))(\( 1 - 4516 T^{2} + 16148934 T^{4} - 26033841316 T^{6} + 33232930569601 T^{8} \))
$11$ (\( ( 1 - 121 T )( 1 + 121 T ) \))(\( 1 - 36196 T^{2} + 708332166 T^{4} - 7758934056676 T^{6} + 45949729863572161 T^{8} \))
$13$ (\( 1 - 146 T + 28561 T^{2} \))(\( ( 1 - 148 T + 32646 T^{2} - 4227028 T^{3} + 815730721 T^{4} )^{2} \))
$17$ (\( ( 1 - 289 T )( 1 + 289 T ) \))(\( ( 1 + 300 T + 186214 T^{2} + 25056300 T^{3} + 6975757441 T^{4} )^{2} \))
$19$ (\( 1 + 46 T + 130321 T^{2} \))(\( 1 - 195556 T^{2} + 17286835974 T^{4} - 3321237654045796 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} \))
$23$ (\( ( 1 - 529 T )( 1 + 529 T ) \))(\( 1 - 655492 T^{2} + 255175536006 T^{4} - 51332224363813252 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} \))
$29$ (\( ( 1 - 841 T )( 1 + 841 T ) \))(\( ( 1 - 444 T + 1423078 T^{2} - 314032764 T^{3} + 500246412961 T^{4} )^{2} \))
$31$ (\( 1 - 194 T + 923521 T^{2} \))(\( 1 - 2090020 T^{2} + 2465294161734 T^{4} - 1782559326072438820 T^{6} + \)\(72\!\cdots\!81\)\( T^{8} \))
$37$ (\( 1 + 2062 T + 1874161 T^{2} \))(\( ( 1 + 2204 T + 4842918 T^{2} + 4130650844 T^{3} + 3512479453921 T^{4} )^{2} \))
$41$ (\( ( 1 - 1681 T )( 1 + 1681 T ) \))(\( ( 1 - 276 T + 5507494 T^{2} - 779910036 T^{3} + 7984925229121 T^{4} )^{2} \))
$43$ (\( 1 + 3214 T + 3418801 T^{2} \))(\( 1 - 3593572 T^{2} + 5893643026566 T^{4} - 42002389247979180772 T^{6} + \)\(13\!\cdots\!01\)\( T^{8} \))
$47$ (\( ( 1 - 2209 T )( 1 + 2209 T ) \))(\( 1 - 16853380 T^{2} + 118578854811654 T^{4} - \)\(40\!\cdots\!80\)\( T^{6} + \)\(56\!\cdots\!21\)\( T^{8} \))
$53$ (\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( ( 1 - 2556 T + 17173798 T^{2} - 20168069436 T^{3} + 62259690411361 T^{4} )^{2} \))
$59$ (\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( 1 - 32722276 T^{2} + 559903452302214 T^{4} - \)\(48\!\cdots\!96\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} \))
$61$ (\( 1 + 1966 T + 13845841 T^{2} \))(\( ( 1 - 2116 T + 16830246 T^{2} - 29297799556 T^{3} + 191707312997281 T^{4} )^{2} \))
$67$ (\( 1 - 5906 T + 20151121 T^{2} \))(\( 1 - 53256676 T^{2} + 1338152481850374 T^{4} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} \))
$71$ (\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( 1 - 79127428 T^{2} + 2740885222137990 T^{4} - \)\(51\!\cdots\!08\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} \))
$73$ (\( 1 + 8542 T + 28398241 T^{2} \))(\( ( 1 - 4420 T + 3693510 T^{2} - 125520225220 T^{3} + 806460091894081 T^{4} )^{2} \))
$79$ (\( 1 - 7682 T + 38950081 T^{2} \))(\( 1 - 86136100 T^{2} + 4459690111983174 T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!21\)\( T^{8} \))
$83$ (\( ( 1 - 6889 T )( 1 + 6889 T ) \))(\( 1 - 855268 T^{2} + 4298268871421190 T^{4} - \)\(19\!\cdots\!88\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} \))
$89$ (\( ( 1 - 7921 T )( 1 + 7921 T ) \))(\( ( 1 + 12540 T + 132835270 T^{2} + 786787702140 T^{3} + 3936588805702081 T^{4} )^{2} \))
$97$ (\( 1 + 18814 T + 88529281 T^{2} \))(\( ( 1 - 11524 T + 146880774 T^{2} - 1020211434244 T^{3} + 7837433594376961 T^{4} )^{2} \))
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