Properties

Label 108.4.b
Level 108
Weight 4
Character orbit b
Rep. character \(\chi_{108}(107,\cdot)\)
Character field \(\Q\)
Dimension 24
Newform subspaces 2
Sturm bound 72
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(72\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(108, [\chi])\).

Total New Old
Modular forms 60 24 36
Cusp forms 48 24 24
Eisenstein series 12 0 12

Trace form

\( 24q - 6q^{4} + O(q^{10}) \) \( 24q - 6q^{4} + 66q^{10} - 36q^{13} + 138q^{16} + 186q^{22} - 516q^{25} + 138q^{28} - 684q^{34} + 276q^{37} - 378q^{40} + 1056q^{46} - 432q^{49} + 1128q^{52} + 1884q^{58} - 828q^{61} - 570q^{64} - 5406q^{70} + 816q^{73} - 4428q^{76} + 1944q^{82} + 888q^{85} + 8190q^{88} + 4164q^{94} + 3048q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
108.4.b.a \(12\) \(6.372\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{2}+(-1-\beta _{4})q^{4}-\beta _{9}q^{5}+(\beta _{2}+\cdots)q^{7}+\cdots\)
108.4.b.b \(12\) \(6.372\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}-\beta _{6}q^{4}+\beta _{5}q^{5}-\beta _{4}q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(108, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(108, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 6 T^{2} + 12 T^{4} - 416 T^{6} + 768 T^{8} + 24576 T^{10} + 262144 T^{12} \))(\( 1 - 3 T^{2} - 24 T^{4} + 592 T^{6} - 1536 T^{8} - 12288 T^{10} + 262144 T^{12} \))
$3$ 1
$5$ (\( ( 1 - 342 T^{2} + 73575 T^{4} - 11181908 T^{6} + 1149609375 T^{8} - 83496093750 T^{10} + 3814697265625 T^{12} )^{2} \))(\( ( 1 - 279 T^{2} + 14742 T^{4} + 1160125 T^{6} + 230343750 T^{8} - 68115234375 T^{10} + 3814697265625 T^{12} )^{2} \))
$7$ (\( ( 1 - 849 T^{2} + 387966 T^{4} - 141880421 T^{6} + 45643811934 T^{8} - 11751252833649 T^{10} + 1628413597910449 T^{12} )^{2} \))(\( ( 1 - 1101 T^{2} + 652854 T^{4} - 259162985 T^{6} + 76807620246 T^{8} - 15239257208301 T^{10} + 1628413597910449 T^{12} )^{2} \))
$11$ (\( ( 1 + 4074 T^{2} + 8337351 T^{4} + 12032309932 T^{6} + 14770125874911 T^{8} + 12785957206761354 T^{10} + 5559917313492231481 T^{12} )^{2} \))(\( ( 1 + 3057 T^{2} + 6556998 T^{4} + 9226981429 T^{6} + 11616121933878 T^{8} + 9594175547636097 T^{10} + 5559917313492231481 T^{12} )^{2} \))
$13$ (\( ( 1 - 9 T + 1962 T^{2} + 66427 T^{3} + 4310514 T^{4} - 43441281 T^{5} + 10604499373 T^{6} )^{4} \))(\( ( 1 + 18 T + 3915 T^{2} + 8332 T^{3} + 8601255 T^{4} + 86882562 T^{5} + 10604499373 T^{6} )^{4} \))
$17$ (\( ( 1 - 13134 T^{2} + 78982671 T^{4} - 357919940036 T^{6} + 1906449671066799 T^{8} - 7652160463775680974 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 - 11946 T^{2} + 72737391 T^{4} - 332667352460 T^{6} + 1755703794142479 T^{8} - 6960005245946724906 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))
$19$ (\( ( 1 - 18825 T^{2} + 202841862 T^{4} - 1656627349277 T^{6} + 9542874101470422 T^{8} - 41665653351420480825 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 8322 T^{2} + 146058135 T^{4} - 772095813500 T^{6} + 6871433638291935 T^{8} - 18419206756468591842 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \))
$23$ (\( ( 1 + 6306 T^{2} + 110681535 T^{4} + 2149854095644 T^{6} + 16384839429609615 T^{8} + \)\(13\!\cdots\!26\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \))(\( ( 1 + 52314 T^{2} + 1249837599 T^{4} + 18511268917228 T^{6} + 185020820073590511 T^{8} + \)\(11\!\cdots\!94\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \))
$29$ (\( ( 1 - 43326 T^{2} + 1075658103 T^{4} - 29470174509188 T^{6} + 639826525087020063 T^{8} - \)\(15\!\cdots\!66\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 117954 T^{2} + 6353615223 T^{4} - 198355607069564 T^{6} + 3779278507301015583 T^{8} - \)\(41\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$31$ (\( ( 1 - 153750 T^{2} + 10409070351 T^{4} - 400107422263412 T^{6} + 9238088252300462031 T^{8} - \)\(12\!\cdots\!50\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 45741 T^{2} + 1957054854 T^{4} - 51110094297449 T^{6} + 1736893386843917574 T^{8} - \)\(36\!\cdots\!01\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))
$37$ (\( ( 1 - 129 T + 39378 T^{2} + 7527427 T^{3} + 1994613834 T^{4} - 330978706761 T^{5} + 129961739795077 T^{6} )^{4} \))(\( ( 1 + 60 T + 83559 T^{2} - 2089640 T^{3} + 4232514027 T^{4} + 153943584540 T^{5} + 129961739795077 T^{6} )^{4} \))
$41$ (\( ( 1 - 232950 T^{2} + 25214641887 T^{4} - 1908694392304628 T^{6} + \)\(11\!\cdots\!67\)\( T^{8} - \)\(52\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 189606 T^{2} + 21529296543 T^{4} - 1820253257865236 T^{6} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(42\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))
$43$ (\( ( 1 - 262254 T^{2} + 37283382183 T^{4} - 3613899564944228 T^{6} + \)\(23\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 + 27042 T^{2} + 15896713575 T^{4} + 267144985943548 T^{6} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \))
$47$ (\( ( 1 + 179538 T^{2} + 22658226447 T^{4} + 3067316968586236 T^{6} + \)\(24\!\cdots\!63\)\( T^{8} + \)\(20\!\cdots\!58\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 + 377142 T^{2} + 74876518767 T^{4} + 9593174285133748 T^{6} + \)\(80\!\cdots\!43\)\( T^{8} + \)\(43\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \))
$53$ (\( ( 1 - 768750 T^{2} + 259538463591 T^{4} - 49798991602056548 T^{6} + \)\(57\!\cdots\!39\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 - 398679 T^{2} + 82394238966 T^{4} - 12639200632897475 T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!39\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))
$59$ (\( ( 1 + 523338 T^{2} + 180986988135 T^{4} + 40857650874211948 T^{6} + \)\(76\!\cdots\!35\)\( T^{8} + \)\(93\!\cdots\!78\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 + 275982 T^{2} + 30860919687 T^{4} - 3157264383153500 T^{6} + \)\(13\!\cdots\!67\)\( T^{8} + \)\(49\!\cdots\!42\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \))
$61$ (\( ( 1 + 243 T + 648738 T^{2} + 106648063 T^{3} + 147251199978 T^{4} + 12519450969723 T^{5} + 11694146092834141 T^{6} )^{4} \))(\( ( 1 - 36 T + 351135 T^{2} - 84569384 T^{3} + 79700973435 T^{4} - 1854733476996 T^{5} + 11694146092834141 T^{6} )^{4} \))
$67$ (\( ( 1 - 1637409 T^{2} + 1163514522486 T^{4} - 457988207385885221 T^{6} + \)\(10\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!49\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 - 1341390 T^{2} + 846750752247 T^{4} - 319903491280410596 T^{6} + \)\(76\!\cdots\!43\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \))
$71$ (\( ( 1 + 377994 T^{2} + 13348121343 T^{4} - 1180461843095924 T^{6} + \)\(17\!\cdots\!03\)\( T^{8} + \)\(62\!\cdots\!54\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 + 1203654 T^{2} + 624403431231 T^{4} + 229814346307125076 T^{6} + \)\(79\!\cdots\!51\)\( T^{8} + \)\(19\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$73$ (\( ( 1 - 165 T + 834942 T^{2} - 185551841 T^{3} + 324806632014 T^{4} - 24970147337685 T^{5} + 58871586708267913 T^{6} )^{4} \))(\( ( 1 - 39 T + 955110 T^{2} - 15631571 T^{3} + 371554026870 T^{4} - 5902034825271 T^{5} + 58871586708267913 T^{6} )^{4} \))
$79$ (\( ( 1 - 2108721 T^{2} + 2206739657838 T^{4} - 1369032710710548773 T^{6} + \)\(53\!\cdots\!98\)\( T^{8} - \)\(12\!\cdots\!61\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 1716006 T^{2} + 1350264931311 T^{4} - 731914440695115860 T^{6} + \)\(32\!\cdots\!31\)\( T^{8} - \)\(10\!\cdots\!46\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))
$83$ (\( ( 1 + 290850 T^{2} + 143373074583 T^{4} + 347024073827678524 T^{6} + \)\(46\!\cdots\!27\)\( T^{8} + \)\(31\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 + 1600737 T^{2} + 1256006483670 T^{4} + 724491266389094437 T^{6} + \)\(41\!\cdots\!30\)\( T^{8} + \)\(17\!\cdots\!57\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \))
$89$ (\( ( 1 - 1956702 T^{2} + 2321980636575 T^{4} - 1921108256399844452 T^{6} + \)\(11\!\cdots\!75\)\( T^{8} - \)\(48\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 + 123846 T^{2} + 872233472127 T^{4} - 77031279330160556 T^{6} + \)\(43\!\cdots\!47\)\( T^{8} + \)\(30\!\cdots\!66\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))
$97$ (\( ( 1 - 633 T + 2697438 T^{2} - 1126776701 T^{3} + 2461878831774 T^{4} - 527271279120057 T^{5} + 760231058654565217 T^{6} )^{4} \))(\( ( 1 - 129 T + 1814286 T^{2} - 533624789 T^{3} + 1655849846478 T^{4} - 107453388635841 T^{5} + 760231058654565217 T^{6} )^{4} \))
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