Properties

Label 12.7
Level 12
Weight 7
Dimension 8
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 56
Trace bound 1

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Defining parameters

Level: \( N \) = \( 12\( 12 = 2^{2} \cdot 3 \) \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(12))\).

Total New Old
Modular forms 29 8 21
Cusp forms 19 8 11
Eisenstein series 10 0 10

Trace form

\( 8q - 10q^{2} - 6q^{3} + 156q^{4} - 44q^{5} - 162q^{6} + 484q^{7} + 1136q^{8} - 2880q^{9} + O(q^{10}) \) \( 8q - 10q^{2} - 6q^{3} + 156q^{4} - 44q^{5} - 162q^{6} + 484q^{7} + 1136q^{8} - 2880q^{9} + 84q^{10} - 972q^{12} + 1888q^{13} + 4776q^{14} - 8640q^{15} - 9744q^{16} + 12220q^{17} + 2430q^{18} + 11572q^{19} + 17608q^{20} - 11172q^{21} - 13512q^{22} - 7776q^{24} + 35828q^{25} - 59252q^{26} + 12906q^{27} - 17808q^{28} - 84860q^{29} + 57348q^{30} - 40892q^{31} + 61280q^{32} + 99576q^{33} + 109404q^{34} - 37908q^{36} - 72752q^{37} - 128088q^{38} - 15708q^{39} - 195552q^{40} - 65252q^{41} + 210600q^{42} + 137236q^{43} + 445008q^{44} + 62532q^{45} + 81120q^{46} - 276048q^{48} - 229716q^{49} - 743118q^{50} - 380160q^{51} - 179592q^{52} + 470308q^{53} + 39366q^{54} + 570240q^{55} + 793728q^{56} - 238836q^{57} + 529860q^{58} - 723816q^{60} + 62512q^{61} - 513384q^{62} - 344124q^{63} - 642432q^{64} + 321512q^{65} + 771768q^{66} - 168716q^{67} + 690328q^{68} + 1042848q^{69} + 938928q^{70} - 276048q^{72} - 1511024q^{73} - 1522916q^{74} + 61770q^{75} + 67824q^{76} - 1487328q^{77} + 396252q^{78} - 319484q^{79} + 1272352q^{80} + 1313496q^{81} + 240444q^{82} - 143856q^{84} - 1466040q^{85} - 1154568q^{86} + 665280q^{87} + 489600q^{88} + 730924q^{89} - 20412q^{90} + 1267112q^{91} - 1338816q^{92} + 39084q^{93} - 390288q^{94} + 624672q^{96} + 5048464q^{97} + 1604918q^{98} - 570240q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(12))\)

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
12.7.c \(\chi_{12}(5, \cdot)\) 12.7.c.a 2 1
12.7.d \(\chi_{12}(7, \cdot)\) 12.7.d.a 6 1

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(12))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(12)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 10 T - 28 T^{2} - 992 T^{3} - 1792 T^{4} + 40960 T^{5} + 262144 T^{6} \))
$3$ (\( 1 + 6 T + 729 T^{2} \))(\( ( 1 + 243 T^{2} )^{3} \))
$5$ (\( 1 - 5330 T^{2} + 244140625 T^{4} \))(\( ( 1 + 22 T + 9575 T^{2} + 1350100 T^{3} + 149609375 T^{4} + 5371093750 T^{5} + 3814697265625 T^{6} )^{2} \))
$7$ (\( ( 1 - 242 T + 117649 T^{2} )^{2} \))(\( 1 - 297174 T^{2} + 49120386735 T^{4} - 6352070622707828 T^{6} + \)\(67\!\cdots\!35\)\( T^{8} - \)\(56\!\cdots\!74\)\( T^{10} + \)\(26\!\cdots\!01\)\( T^{12} \))
$11$ (\( 1 - 406802 T^{2} + 3138428376721 T^{4} \))(\( 1 - 2481510 T^{2} + 6024225785151 T^{4} - 16527738499614302996 T^{6} + \)\(18\!\cdots\!71\)\( T^{8} - \)\(24\!\cdots\!10\)\( T^{10} + \)\(30\!\cdots\!61\)\( T^{12} \))
$13$ (\( ( 1 - 2618 T + 4826809 T^{2} )^{2} \))(\( ( 1 + 1674 T + 11116791 T^{2} + 16065685420 T^{3} + 53658626849919 T^{4} + 39000994495033194 T^{5} + \)\(11\!\cdots\!29\)\( T^{6} )^{2} \))
$17$ (\( 1 + 1905982 T^{2} + 582622237229761 T^{4} \))(\( ( 1 - 6110 T + 66889295 T^{2} - 292605918500 T^{3} + 1614544973423855 T^{4} - 3559821869473839710 T^{5} + \)\(14\!\cdots\!09\)\( T^{6} )^{2} \))
$19$ (\( ( 1 - 5786 T + 47045881 T^{2} )^{2} \))(\( 1 - 146059206 T^{2} + 11584792041958815 T^{4} - \)\(63\!\cdots\!32\)\( T^{6} + \)\(25\!\cdots\!15\)\( T^{8} - \)\(71\!\cdots\!26\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} \))
$23$ (\( 1 - 208876898 T^{2} + 21914624432020321 T^{4} \))(\( 1 - 607352358 T^{2} + 180562295442724527 T^{4} - \)\(33\!\cdots\!16\)\( T^{6} + \)\(39\!\cdots\!67\)\( T^{8} - \)\(29\!\cdots\!78\)\( T^{10} + \)\(10\!\cdots\!61\)\( T^{12} \))
$29$ (\( 1 - 1035966962 T^{2} + 353814783205469041 T^{4} \))(\( ( 1 + 42430 T + 2077407863 T^{2} + 50868097395076 T^{3} + 1235690644141173023 T^{4} + \)\(15\!\cdots\!30\)\( T^{5} + \)\(21\!\cdots\!61\)\( T^{6} )^{2} \))
$31$ (\( ( 1 + 20446 T + 887503681 T^{2} )^{2} \))(\( 1 - 3163788534 T^{2} + 5235513742747890255 T^{4} - \)\(55\!\cdots\!16\)\( T^{6} + \)\(41\!\cdots\!55\)\( T^{8} - \)\(19\!\cdots\!14\)\( T^{10} + \)\(48\!\cdots\!81\)\( T^{12} \))
$37$ (\( ( 1 + 46774 T + 2565726409 T^{2} )^{2} \))(\( ( 1 - 10398 T + 1806728583 T^{2} - 208245913091588 T^{3} + 4635571239298248447 T^{4} - \)\(68\!\cdots\!38\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} )^{2} \))
$41$ (\( 1 - 9487663202 T^{2} + 22563490300366186081 T^{4} \))(\( ( 1 + 32626 T + 13292008703 T^{2} + 308481662881276 T^{3} + 63138426911529209423 T^{4} + \)\(73\!\cdots\!06\)\( T^{5} + \)\(10\!\cdots\!21\)\( T^{6} )^{2} \))
$43$ (\( ( 1 - 68618 T + 6321363049 T^{2} )^{2} \))(\( 1 - 29142871014 T^{2} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(31\!\cdots\!92\)\( T^{6} + \)\(15\!\cdots\!55\)\( T^{8} - \)\(46\!\cdots\!14\)\( T^{10} + \)\(63\!\cdots\!01\)\( T^{12} \))
$47$ (\( 1 - 21106800578 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - 39654949638 T^{2} + \)\(75\!\cdots\!55\)\( T^{4} - \)\(94\!\cdots\!52\)\( T^{6} + \)\(87\!\cdots\!55\)\( T^{8} - \)\(53\!\cdots\!78\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \))
$53$ (\( 1 - 14819087378 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 235154 T + 80653582439 T^{2} - 10625112229848476 T^{3} + \)\(17\!\cdots\!31\)\( T^{4} - \)\(11\!\cdots\!14\)\( T^{5} + \)\(10\!\cdots\!89\)\( T^{6} )^{2} \))
$59$ (\( 1 - 61991044562 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 38958254886 T^{2} + \)\(25\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!32\)\( T^{6} + \)\(45\!\cdots\!75\)\( T^{8} - \)\(12\!\cdots\!46\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \))
$61$ (\( ( 1 - 24794 T + 51520374361 T^{2} )^{2} \))(\( ( 1 - 6462 T + 83167803063 T^{2} - 6651331196354948 T^{3} + \)\(42\!\cdots\!43\)\( T^{4} - \)\(17\!\cdots\!02\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{2} \))
$67$ (\( ( 1 + 84358 T + 90458382169 T^{2} )^{2} \))(\( 1 - 296399142534 T^{2} + \)\(37\!\cdots\!95\)\( T^{4} - \)\(34\!\cdots\!68\)\( T^{6} + \)\(30\!\cdots\!95\)\( T^{8} - \)\(19\!\cdots\!14\)\( T^{10} + \)\(54\!\cdots\!81\)\( T^{12} \))
$71$ (\( 1 - 151063967522 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \))(\( 1 - 415188323430 T^{2} + \)\(92\!\cdots\!47\)\( T^{4} - \)\(13\!\cdots\!36\)\( T^{6} + \)\(15\!\cdots\!27\)\( T^{8} - \)\(11\!\cdots\!30\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))
$73$ (\( ( 1 + 113806 T + 151334226289 T^{2} )^{2} \))(\( ( 1 + 641706 T + 530324505759 T^{2} + 188743068109697740 T^{3} + \)\(80\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!26\)\( T^{5} + \)\(34\!\cdots\!69\)\( T^{6} )^{2} \))
$79$ (\( ( 1 + 159742 T + 243087455521 T^{2} )^{2} \))(\( 1 - 234724993590 T^{2} + \)\(48\!\cdots\!95\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!95\)\( T^{8} - \)\(81\!\cdots\!90\)\( T^{10} + \)\(20\!\cdots\!21\)\( T^{12} \))
$83$ (\( 1 - 388294032818 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \))(\( 1 - 954563652678 T^{2} + \)\(53\!\cdots\!43\)\( T^{4} - \)\(21\!\cdots\!84\)\( T^{6} + \)\(57\!\cdots\!23\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \))
$89$ (\( 1 + 583819025758 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \))(\( ( 1 - 365462 T + 1298555750591 T^{2} - 314016079540511540 T^{3} + \)\(64\!\cdots\!51\)\( T^{4} - \)\(90\!\cdots\!02\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} )^{2} \))
$97$ (\( ( 1 - 899522 T + 832972004929 T^{2} )^{2} \))(\( ( 1 - 1624710 T + 3125499884751 T^{2} - 2772177769643198996 T^{3} + \)\(26\!\cdots\!79\)\( T^{4} - \)\(11\!\cdots\!10\)\( T^{5} + \)\(57\!\cdots\!89\)\( T^{6} )^{2} \))
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