Properties

Label 12.7.d
Level 12
Weight 7
Character orbit d
Rep. character \(\chi_{12}(7,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 1
Sturm bound 14
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 12.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(14\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(12, [\chi])\).

Total New Old
Modular forms 14 6 8
Cusp forms 10 6 4
Eisenstein series 4 0 4

Trace form

\( 6q - 10q^{2} + 156q^{4} - 44q^{5} - 162q^{6} + 1136q^{8} - 1458q^{9} + O(q^{10}) \) \( 6q - 10q^{2} + 156q^{4} - 44q^{5} - 162q^{6} + 1136q^{8} - 1458q^{9} + 84q^{10} - 972q^{12} - 3348q^{13} + 4776q^{14} - 9744q^{16} + 12220q^{17} + 2430q^{18} + 17608q^{20} - 9720q^{21} - 13512q^{22} - 7776q^{24} + 56418q^{25} - 59252q^{26} - 17808q^{28} - 84860q^{29} + 57348q^{30} + 61280q^{32} + 4536q^{33} + 109404q^{34} - 37908q^{36} + 20796q^{37} - 128088q^{38} - 195552q^{40} - 65252q^{41} + 210600q^{42} + 445008q^{44} + 10692q^{45} + 81120q^{46} - 276048q^{48} - 111546q^{49} - 743118q^{50} - 179592q^{52} + 470308q^{53} + 39366q^{54} + 793728q^{56} - 204120q^{57} + 529860q^{58} - 723816q^{60} + 12924q^{61} - 513384q^{62} - 642432q^{64} + 321512q^{65} + 771768q^{66} + 690328q^{68} + 541728q^{69} + 938928q^{70} - 276048q^{72} - 1283412q^{73} - 1522916q^{74} + 67824q^{76} - 1487328q^{77} + 396252q^{78} + 1272352q^{80} + 354294q^{81} + 240444q^{82} - 143856q^{84} + 814920q^{85} - 1154568q^{86} + 489600q^{88} + 730924q^{89} - 20412q^{90} - 1338816q^{92} - 83592q^{93} - 390288q^{94} + 624672q^{96} + 3249420q^{97} + 1604918q^{98} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(12, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
12.7.d.a \(6\) \(2.761\) 6.0.50898483.1 None \(-10\) \(0\) \(-44\) \(0\) \(q+(-2-\beta _{1})q^{2}-\beta _{2}q^{3}+(3^{3}+2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(12, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(12, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T - 28 T^{2} - 992 T^{3} - 1792 T^{4} + 40960 T^{5} + 262144 T^{6} \)
$3$ \( ( 1 + 243 T^{2} )^{3} \)
$5$ \( ( 1 + 22 T + 9575 T^{2} + 1350100 T^{3} + 149609375 T^{4} + 5371093750 T^{5} + 3814697265625 T^{6} )^{2} \)
$7$ \( 1 - 297174 T^{2} + 49120386735 T^{4} - 6352070622707828 T^{6} + \)\(67\!\cdots\!35\)\( T^{8} - \)\(56\!\cdots\!74\)\( T^{10} + \)\(26\!\cdots\!01\)\( T^{12} \)
$11$ \( 1 - 2481510 T^{2} + 6024225785151 T^{4} - 16527738499614302996 T^{6} + \)\(18\!\cdots\!71\)\( T^{8} - \)\(24\!\cdots\!10\)\( T^{10} + \)\(30\!\cdots\!61\)\( T^{12} \)
$13$ \( ( 1 + 1674 T + 11116791 T^{2} + 16065685420 T^{3} + 53658626849919 T^{4} + 39000994495033194 T^{5} + \)\(11\!\cdots\!29\)\( T^{6} )^{2} \)
$17$ \( ( 1 - 6110 T + 66889295 T^{2} - 292605918500 T^{3} + 1614544973423855 T^{4} - 3559821869473839710 T^{5} + \)\(14\!\cdots\!09\)\( T^{6} )^{2} \)
$19$ \( 1 - 146059206 T^{2} + 11584792041958815 T^{4} - \)\(63\!\cdots\!32\)\( T^{6} + \)\(25\!\cdots\!15\)\( T^{8} - \)\(71\!\cdots\!26\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} \)
$23$ \( 1 - 607352358 T^{2} + 180562295442724527 T^{4} - \)\(33\!\cdots\!16\)\( T^{6} + \)\(39\!\cdots\!67\)\( T^{8} - \)\(29\!\cdots\!78\)\( T^{10} + \)\(10\!\cdots\!61\)\( T^{12} \)
$29$ \( ( 1 + 42430 T + 2077407863 T^{2} + 50868097395076 T^{3} + 1235690644141173023 T^{4} + \)\(15\!\cdots\!30\)\( T^{5} + \)\(21\!\cdots\!61\)\( T^{6} )^{2} \)
$31$ \( 1 - 3163788534 T^{2} + 5235513742747890255 T^{4} - \)\(55\!\cdots\!16\)\( T^{6} + \)\(41\!\cdots\!55\)\( T^{8} - \)\(19\!\cdots\!14\)\( T^{10} + \)\(48\!\cdots\!81\)\( T^{12} \)
$37$ \( ( 1 - 10398 T + 1806728583 T^{2} - 208245913091588 T^{3} + 4635571239298248447 T^{4} - \)\(68\!\cdots\!38\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} )^{2} \)
$41$ \( ( 1 + 32626 T + 13292008703 T^{2} + 308481662881276 T^{3} + 63138426911529209423 T^{4} + \)\(73\!\cdots\!06\)\( T^{5} + \)\(10\!\cdots\!21\)\( T^{6} )^{2} \)
$43$ \( 1 - 29142871014 T^{2} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(31\!\cdots\!92\)\( T^{6} + \)\(15\!\cdots\!55\)\( T^{8} - \)\(46\!\cdots\!14\)\( T^{10} + \)\(63\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 - 39654949638 T^{2} + \)\(75\!\cdots\!55\)\( T^{4} - \)\(94\!\cdots\!52\)\( T^{6} + \)\(87\!\cdots\!55\)\( T^{8} - \)\(53\!\cdots\!78\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \)
$53$ \( ( 1 - 235154 T + 80653582439 T^{2} - 10625112229848476 T^{3} + \)\(17\!\cdots\!31\)\( T^{4} - \)\(11\!\cdots\!14\)\( T^{5} + \)\(10\!\cdots\!89\)\( T^{6} )^{2} \)
$59$ \( 1 - 38958254886 T^{2} + \)\(25\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!32\)\( T^{6} + \)\(45\!\cdots\!75\)\( T^{8} - \)\(12\!\cdots\!46\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \)
$61$ \( ( 1 - 6462 T + 83167803063 T^{2} - 6651331196354948 T^{3} + \)\(42\!\cdots\!43\)\( T^{4} - \)\(17\!\cdots\!02\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{2} \)
$67$ \( 1 - 296399142534 T^{2} + \)\(37\!\cdots\!95\)\( T^{4} - \)\(34\!\cdots\!68\)\( T^{6} + \)\(30\!\cdots\!95\)\( T^{8} - \)\(19\!\cdots\!14\)\( T^{10} + \)\(54\!\cdots\!81\)\( T^{12} \)
$71$ \( 1 - 415188323430 T^{2} + \)\(92\!\cdots\!47\)\( T^{4} - \)\(13\!\cdots\!36\)\( T^{6} + \)\(15\!\cdots\!27\)\( T^{8} - \)\(11\!\cdots\!30\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \)
$73$ \( ( 1 + 641706 T + 530324505759 T^{2} + 188743068109697740 T^{3} + \)\(80\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!26\)\( T^{5} + \)\(34\!\cdots\!69\)\( T^{6} )^{2} \)
$79$ \( 1 - 234724993590 T^{2} + \)\(48\!\cdots\!95\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!95\)\( T^{8} - \)\(81\!\cdots\!90\)\( T^{10} + \)\(20\!\cdots\!21\)\( T^{12} \)
$83$ \( 1 - 954563652678 T^{2} + \)\(53\!\cdots\!43\)\( T^{4} - \)\(21\!\cdots\!84\)\( T^{6} + \)\(57\!\cdots\!23\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \)
$89$ \( ( 1 - 365462 T + 1298555750591 T^{2} - 314016079540511540 T^{3} + \)\(64\!\cdots\!51\)\( T^{4} - \)\(90\!\cdots\!02\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} )^{2} \)
$97$ \( ( 1 - 1624710 T + 3125499884751 T^{2} - 2772177769643198996 T^{3} + \)\(26\!\cdots\!79\)\( T^{4} - \)\(11\!\cdots\!10\)\( T^{5} + \)\(57\!\cdots\!89\)\( T^{6} )^{2} \)
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