Properties

Label 224.3.d
Level 224
Weight 3
Character orbit d
Rep. character \(\chi_{224}(127,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 2
Sturm bound 96
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(224, [\chi])\).

Total New Old
Modular forms 72 12 60
Cusp forms 56 12 44
Eisenstein series 16 0 16

Trace form

\( 12q - 8q^{5} - 20q^{9} + O(q^{10}) \) \( 12q - 8q^{5} - 20q^{9} + 24q^{13} - 72q^{17} + 132q^{25} - 40q^{29} + 32q^{33} - 40q^{37} + 24q^{41} + 216q^{45} - 84q^{49} - 232q^{53} - 320q^{57} - 200q^{61} + 48q^{65} + 480q^{69} + 120q^{73} + 460q^{81} + 176q^{85} + 56q^{89} - 576q^{93} - 72q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(224, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
224.3.d.a \(4\) \(6.104\) \(\Q(i, \sqrt{7})\) None \(0\) \(0\) \(-8\) \(0\) \(q+\beta _{1}q^{3}+(-2+\beta _{2})q^{5}-\beta _{3}q^{7}+5q^{9}+\cdots\)
224.3.d.b \(8\) \(6.104\) 8.0.1997017344.2 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-\beta _{4}q^{5}-\beta _{3}q^{7}+(-5+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(224, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(224, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 - 14 T^{2} + 81 T^{4} )^{2} \))(\( 1 - 16 T^{2} + 92 T^{4} + 272 T^{6} - 8570 T^{8} + 22032 T^{10} + 603612 T^{12} - 8503056 T^{14} + 43046721 T^{16} \))
$5$ (\( ( 1 + 4 T + 26 T^{2} + 100 T^{3} + 625 T^{4} )^{2} \))(\( ( 1 + 24 T^{2} - 224 T^{3} + 78 T^{4} - 5600 T^{5} + 15000 T^{6} + 390625 T^{8} )^{2} \))
$7$ (\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{4} \))
$11$ (\( 1 - 228 T^{2} + 35110 T^{4} - 3338148 T^{6} + 214358881 T^{8} \))(\( 1 - 296 T^{2} + 72860 T^{4} - 12907416 T^{6} + 1704788486 T^{8} - 188977477656 T^{10} + 15618188069660 T^{12} - 928974799509416 T^{14} + 45949729863572161 T^{16} \))
$13$ (\( ( 1 + 4 T + 90 T^{2} + 676 T^{3} + 28561 T^{4} )^{2} \))(\( ( 1 - 16 T + 696 T^{2} - 7536 T^{3} + 176398 T^{4} - 1273584 T^{5} + 19878456 T^{6} - 77228944 T^{7} + 815730721 T^{8} )^{2} \))
$17$ (\( ( 1 + 28 T + 662 T^{2} + 8092 T^{3} + 83521 T^{4} )^{2} \))(\( ( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 409224 T^{5} + 35746988 T^{6} + 193100552 T^{7} + 6975757441 T^{8} )^{2} \))
$19$ (\( 1 - 348 T^{2} + 111718 T^{4} - 45351708 T^{6} + 16983563041 T^{8} \))(\( 1 - 1936 T^{2} + 1795420 T^{4} - 1065665392 T^{6} + 449725089670 T^{8} - 138878579550832 T^{10} + 30492628755072220 T^{12} - 4284977683312087696 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))
$23$ (\( 1 - 1604 T^{2} + 1138374 T^{4} - 448864964 T^{6} + 78310985281 T^{8} \))(\( 1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 536484029577912 T^{10} + 185352391597952156 T^{12} - 42251395904935178888 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \))
$29$ (\( ( 1 - 20 T + 774 T^{2} - 16820 T^{3} + 707281 T^{4} )^{2} \))(\( ( 1 + 40 T + 3660 T^{2} + 100632 T^{3} + 4741126 T^{4} + 84631512 T^{5} + 2588648460 T^{6} + 23792932840 T^{7} + 500246412961 T^{8} )^{2} \))
$31$ (\( ( 1 - 1906 T^{2} + 923521 T^{4} )^{2} \))(\( 1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5172694104774 T^{8} - 5454285613703192 T^{10} + 5077686791404993500 T^{12} - \)\(28\!\cdots\!64\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \))
$37$ (\( ( 1 - 68 T + 3782 T^{2} - 93092 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 + 88 T + 3692 T^{2} + 66024 T^{3} + 510982 T^{4} + 90386856 T^{5} + 6919402412 T^{6} + 225783923992 T^{7} + 3512479453921 T^{8} )^{2} \))
$41$ (\( ( 1 + 60 T + 4150 T^{2} + 100860 T^{3} + 2825761 T^{4} )^{2} \))(\( ( 1 - 72 T + 4364 T^{2} - 180408 T^{3} + 8880422 T^{4} - 303265848 T^{5} + 12331621004 T^{6} - 342007505352 T^{7} + 7984925229121 T^{8} )^{2} \))
$43$ (\( 1 - 1508 T^{2} + 5793318 T^{4} - 5155551908 T^{6} + 11688200277601 T^{8} \))(\( 1 - 4392 T^{2} + 15655580 T^{4} - 32786290584 T^{6} + 70340912004102 T^{8} - 112089803034869784 T^{10} + \)\(18\!\cdots\!80\)\( T^{12} - \)\(17\!\cdots\!92\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 5348 T^{2} + 14587206 T^{4} - 26096533988 T^{6} + 23811286661761 T^{8} \))(\( 1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 531772641369485336 T^{10} + \)\(81\!\cdots\!04\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))
$53$ (\( ( 1 + 140 T + 10070 T^{2} + 393260 T^{3} + 7890481 T^{4} )^{2} \))(\( ( 1 - 24 T + 6108 T^{2} - 242920 T^{3} + 18930726 T^{4} - 682362280 T^{5} + 48195057948 T^{6} - 531944667096 T^{7} + 62259690411361 T^{8} )^{2} \))
$59$ (\( 1 - 9692 T^{2} + 45396006 T^{4} - 117441462812 T^{6} + 146830437604321 T^{8} \))(\( 1 - 21392 T^{2} + 213726428 T^{4} - 1316320178544 T^{6} + 5495654048256518 T^{8} - 15950326795002102384 T^{10} + \)\(31\!\cdots\!88\)\( T^{12} - \)\(38\!\cdots\!52\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))
$61$ (\( ( 1 + 4 T + 7194 T^{2} + 14884 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 + 96 T + 12728 T^{2} + 895872 T^{3} + 70065870 T^{4} + 3333539712 T^{5} + 176229864248 T^{6} + 4945955938656 T^{7} + 191707312997281 T^{8} )^{2} \))
$67$ (\( 1 - 11204 T^{2} + 65233446 T^{4} - 225773159684 T^{6} + 406067677556641 T^{8} \))(\( 1 - 16200 T^{2} + 120433884 T^{4} - 596093100152 T^{6} + 2634880977729030 T^{8} - 12011944188428070392 T^{10} + \)\(48\!\cdots\!44\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \))
$71$ (\( 1 - 8388 T^{2} + 39052870 T^{4} - 213153180228 T^{6} + 645753531245761 T^{8} \))(\( 1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 50600555550540147768 T^{10} + \)\(15\!\cdots\!80\)\( T^{12} - \)\(34\!\cdots\!00\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \))
$73$ (\( ( 1 + 76 T + 4934 T^{2} + 405004 T^{3} + 28398241 T^{4} )^{2} \))(\( ( 1 - 136 T + 21660 T^{2} - 1811512 T^{3} + 170528390 T^{4} - 9653547448 T^{5} + 615105900060 T^{6} - 20581454775304 T^{7} + 806460091894081 T^{8} )^{2} \))
$79$ (\( ( 1 - 1282 T^{2} + 38950081 T^{4} )^{2} \))(\( 1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 72117759397043305912 T^{10} + \)\(43\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!68\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))
$83$ (\( 1 - 11804 T^{2} + 67754214 T^{4} - 560198021084 T^{6} + 2252292232139041 T^{8} \))(\( 1 - 19280 T^{2} + 168996572 T^{4} - 1078781742000 T^{6} + 6945567152070790 T^{8} - 51197170200775182000 T^{10} + \)\(38\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!80\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))
$89$ (\( ( 1 - 68 T + 15206 T^{2} - 538628 T^{3} + 62742241 T^{4} )^{2} \))(\( ( 1 + 40 T + 25916 T^{2} + 837912 T^{3} + 285914566 T^{4} + 6637100952 T^{5} + 1626027917756 T^{6} + 19879251638440 T^{7} + 3936588805702081 T^{8} )^{2} \))
$97$ (\( ( 1 + 300 T + 41206 T^{2} + 2822700 T^{3} + 88529281 T^{4} )^{2} \))(\( ( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 68309565816 T^{5} + 4635747270284 T^{6} - 219904609301256 T^{7} + 7837433594376961 T^{8} )^{2} \))
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