Properties

Label 12.8
Level 12
Weight 8
Dimension 14
Nonzero newspaces 2
Newform subspaces 4
Sturm bound 64
Trace bound 1

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Defining parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(12))\).

Total New Old
Modular forms 33 18 15
Cusp forms 23 14 9
Eisenstein series 10 4 6

Trace form

\( 14q + 24q^{4} - 108q^{5} - 216q^{6} + 280q^{7} + 2574q^{9} + O(q^{10}) \) \( 14q + 24q^{4} - 108q^{5} - 216q^{6} + 280q^{7} + 2574q^{9} - 1200q^{10} - 8208q^{11} + 792q^{12} + 13828q^{13} + 17496q^{15} - 14304q^{16} - 58860q^{17} - 11376q^{18} + 35440q^{19} + 58032q^{21} + 58224q^{22} - 92880q^{23} + 99360q^{24} + 7274q^{25} - 169392q^{28} - 108q^{29} - 342000q^{30} - 120392q^{31} - 150552q^{33} + 698688q^{34} + 614736q^{35} + 950328q^{36} - 92636q^{37} - 524880q^{39} - 1909440q^{40} + 418500q^{41} - 2233872q^{42} - 518960q^{43} + 125748q^{45} + 3504480q^{46} + 1328400q^{47} + 4119840q^{48} + 540054q^{49} - 384912q^{51} - 5965680q^{52} - 2043900q^{53} - 7675560q^{54} - 606528q^{55} + 1372464q^{57} + 10151088q^{58} - 433728q^{59} + 12640320q^{60} + 2129716q^{61} + 204120q^{63} - 15471744q^{64} - 6854760q^{65} - 16368336q^{66} - 311360q^{67} + 4585248q^{69} + 21838560q^{70} + 1643760q^{71} + 22404672q^{72} + 3111148q^{73} - 1889568q^{75} - 27292752q^{76} - 4298400q^{77} - 29031696q^{78} + 1171864q^{79} - 939762q^{81} + 30103200q^{82} + 10620720q^{83} + 31202640q^{84} - 4923384q^{85} - 8485560q^{87} - 33590592q^{88} + 3245940q^{89} - 35812080q^{90} - 17453680q^{91} + 6418080q^{93} + 31652544q^{94} + 20131200q^{95} + 34992000q^{96} + 25180924q^{97} - 5983632q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)

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
12.8.a \(\chi_{12}(1, \cdot)\) 12.8.a.a 1 1
12.8.a.b 1
12.8.b \(\chi_{12}(11, \cdot)\) 12.8.b.a 4 1
12.8.b.b 8

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(12))\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( 1 - 176 T^{2} + 16384 T^{4} \))(\( 1 + 164 T^{2} + 16128 T^{4} + 2686976 T^{6} + 268435456 T^{8} \))
$3$ (\( 1 + 27 T \))(\( 1 - 27 T \))(\( 1 + 486 T^{2} + 4782969 T^{4} \))(\( 1 - 1044 T^{2} - 3619242 T^{4} - 4993419636 T^{6} + 22876792454961 T^{8} \))
$5$ (\( 1 + 378 T + 78125 T^{2} \))(\( 1 - 270 T + 78125 T^{2} \))(\( ( 1 - 113930 T^{2} + 6103515625 T^{4} )^{2} \))(\( ( 1 - 107380 T^{2} + 8772021750 T^{4} - 655395507812500 T^{6} + 37252902984619140625 T^{8} )^{2} \))
$7$ (\( 1 + 832 T + 823543 T^{2} \))(\( 1 - 1112 T + 823543 T^{2} \))(\( ( 1 - 648626 T^{2} + 678223072849 T^{4} )^{2} \))(\( ( 1 - 1886596 T^{2} + 2237391652518 T^{4} - 1279532936344632004 T^{6} + \)\(45\!\cdots\!01\)\( T^{8} )^{2} \))
$11$ (\( 1 + 2484 T + 19487171 T^{2} \))(\( 1 + 5724 T + 19487171 T^{2} \))(\( ( 1 + 35075542 T^{2} + 379749833583241 T^{4} )^{2} \))(\( ( 1 + 15051980 T^{2} + 46104248605878 T^{4} + \)\(57\!\cdots\!80\)\( T^{6} + \)\(14\!\cdots\!81\)\( T^{8} )^{2} \))
$13$ (\( 1 - 14870 T + 62748517 T^{2} \))(\( 1 + 4570 T + 62748517 T^{2} \))(\( ( 1 + 12730 T + 62748517 T^{2} )^{4} \))(\( ( 1 - 13612 T + 170323374 T^{2} - 854132813404 T^{3} + 3937376385699289 T^{4} )^{4} \))
$17$ (\( 1 + 22302 T + 410338673 T^{2} \))(\( 1 + 36558 T + 410338673 T^{2} \))(\( ( 1 - 465184226 T^{2} + 168377826559400929 T^{4} )^{2} \))(\( ( 1 - 1564780996 T^{2} + 948082382592036678 T^{4} - \)\(26\!\cdots\!84\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} )^{2} \))
$19$ (\( 1 + 16300 T + 893871739 T^{2} \))(\( 1 - 51740 T + 893871739 T^{2} \))(\( ( 1 - 1215969338 T^{2} + 799006685782884121 T^{4} )^{2} \))(\( ( 1 - 2363096020 T^{2} + 2897086659724142358 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \))
$23$ (\( 1 + 115128 T + 3404825447 T^{2} \))(\( 1 - 22248 T + 3404825447 T^{2} \))(\( ( 1 + 1571942926 T^{2} + 11592836324538749809 T^{4} )^{2} \))(\( ( 1 + 10655298716 T^{2} + 50911417411485435558 T^{4} + \)\(12\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} )^{2} \))
$29$ (\( 1 - 157086 T + 17249876309 T^{2} \))(\( 1 + 157194 T + 17249876309 T^{2} \))(\( ( 1 - 23241291098 T^{2} + \)\(29\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 15119114260 T^{2} + \)\(42\!\cdots\!38\)\( T^{4} - \)\(44\!\cdots\!60\)\( T^{6} + \)\(88\!\cdots\!61\)\( T^{8} )^{2} \))
$31$ (\( 1 + 16456 T + 27512614111 T^{2} \))(\( 1 + 103936 T + 27512614111 T^{2} \))(\( ( 1 - 46858094882 T^{2} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 92139890404 T^{2} + \)\(35\!\cdots\!46\)\( T^{4} - \)\(69\!\cdots\!84\)\( T^{6} + \)\(57\!\cdots\!41\)\( T^{8} )^{2} \))
$37$ (\( 1 + 149266 T + 94931877133 T^{2} \))(\( 1 + 94834 T + 94931877133 T^{2} \))(\( ( 1 - 43310 T + 94931877133 T^{2} )^{4} \))(\( ( 1 + 5444 T + 86443034526 T^{2} + 516809139112052 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} )^{4} \))
$41$ (\( 1 + 241110 T + 194754273881 T^{2} \))(\( 1 - 659610 T + 194754273881 T^{2} \))(\( ( 1 + 227957477518 T^{2} + \)\(37\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 670723359844 T^{2} + \)\(18\!\cdots\!06\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(14\!\cdots\!21\)\( T^{8} )^{2} \))
$43$ (\( 1 + 443188 T + 271818611107 T^{2} \))(\( 1 + 75772 T + 271818611107 T^{2} \))(\( ( 1 - 538623749354 T^{2} + \)\(73\!\cdots\!49\)\( T^{4} )^{2} \))(\( ( 1 - 281458444084 T^{2} + \)\(11\!\cdots\!18\)\( T^{4} - \)\(20\!\cdots\!16\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} )^{2} \))
$47$ (\( 1 - 922752 T + 506623120463 T^{2} \))(\( 1 - 405648 T + 506623120463 T^{2} \))(\( ( 1 + 695418926494 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} )^{2} \))(\( ( 1 + 1594540776764 T^{2} + \)\(11\!\cdots\!18\)\( T^{4} + \)\(40\!\cdots\!16\)\( T^{6} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))
$53$ (\( 1 + 697626 T + 1174711139837 T^{2} \))(\( 1 + 1346274 T + 1174711139837 T^{2} \))(\( ( 1 - 2227661146154 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} )^{2} \))(\( ( 1 - 3369637018804 T^{2} + \)\(51\!\cdots\!38\)\( T^{4} - \)\(46\!\cdots\!76\)\( T^{6} + \)\(19\!\cdots\!61\)\( T^{8} )^{2} \))
$59$ (\( 1 - 870156 T + 2488651484819 T^{2} \))(\( 1 + 1303884 T + 2488651484819 T^{2} \))(\( ( 1 + 4402559099638 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 + 5549519857676 T^{2} + \)\(17\!\cdots\!66\)\( T^{4} + \)\(34\!\cdots\!36\)\( T^{6} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))
$61$ (\( 1 - 2067062 T + 3142742836021 T^{2} \))(\( 1 - 1833782 T + 3142742836021 T^{2} \))(\( ( 1 - 314198 T + 3142742836021 T^{2} )^{4} \))(\( ( 1 + 756980 T + 3358166464398 T^{2} + 2378993472011176580 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} )^{4} \))
$67$ (\( 1 + 1680748 T + 6060711605323 T^{2} \))(\( 1 - 1369388 T + 6060711605323 T^{2} \))(\( ( 1 - 11463132309146 T^{2} + \)\(36\!\cdots\!29\)\( T^{4} )^{2} \))(\( ( 1 - 1556381570836 T^{2} + \)\(48\!\cdots\!98\)\( T^{4} - \)\(57\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!41\)\( T^{8} )^{2} \))
$71$ (\( 1 + 1070280 T + 9095120158391 T^{2} \))(\( 1 - 2714040 T + 9095120158391 T^{2} \))(\( ( 1 + 18120013316782 T^{2} + \)\(82\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 11818476110300 T^{2} + \)\(50\!\cdots\!38\)\( T^{4} + \)\(97\!\cdots\!00\)\( T^{6} + \)\(68\!\cdots\!61\)\( T^{8} )^{2} \))
$73$ (\( 1 + 2403334 T + 11047398519097 T^{2} \))(\( 1 - 2868794 T + 11047398519097 T^{2} \))(\( ( 1 + 259270 T + 11047398519097 T^{2} )^{4} \))(\( ( 1 - 920692 T + 21663235778454 T^{2} - 10171251437344455124 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} )^{4} \))
$79$ (\( 1 - 2301512 T + 19203908986159 T^{2} \))(\( 1 + 1129648 T + 19203908986159 T^{2} \))(\( ( 1 - 11004425314178 T^{2} + \)\(36\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 27183610132900 T^{2} + \)\(70\!\cdots\!78\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!61\)\( T^{8} )^{2} \))
$83$ (\( 1 - 4708692 T + 27136050989627 T^{2} \))(\( 1 - 5912028 T + 27136050989627 T^{2} \))(\( ( 1 - 47771708513594 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} )^{2} \))(\( ( 1 + 92589213551276 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} + \)\(68\!\cdots\!04\)\( T^{6} + \)\(54\!\cdots\!41\)\( T^{8} )^{2} \))
$89$ (\( 1 - 4143690 T + 44231334895529 T^{2} \))(\( 1 + 897750 T + 44231334895529 T^{2} \))(\( ( 1 - 73372933660178 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 152646663101860 T^{2} + \)\(96\!\cdots\!78\)\( T^{4} - \)\(29\!\cdots\!60\)\( T^{6} + \)\(38\!\cdots\!81\)\( T^{8} )^{2} \))
$97$ (\( 1 + 1622974 T + 80798284478113 T^{2} \))(\( 1 - 13719074 T + 80798284478113 T^{2} \))(\( ( 1 - 7243010 T + 80798284478113 T^{2} )^{4} \))(\( ( 1 + 3971804 T + 152677108318086 T^{2} + \)\(32\!\cdots\!52\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} )^{4} \))
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