Properties

Label 24.8.f
Level 24
Weight 8
Character orbit f
Rep. character \(\chi_{24}(11,\cdot)\)
Character field \(\Q\)
Dimension 26
Newform subspaces 3
Sturm bound 32
Trace bound 1

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Defining parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 24.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(24, [\chi])\).

Total New Old
Modular forms 30 30 0
Cusp forms 26 26 0
Eisenstein series 4 4 0

Trace form

\( 26q - 2q^{3} - 28q^{4} + 40q^{6} - 2q^{9} + O(q^{10}) \) \( 26q - 2q^{3} - 28q^{4} + 40q^{6} - 2q^{9} - 5304q^{10} - 12812q^{12} - 14872q^{16} - 9800q^{18} + 60580q^{19} - 82496q^{22} - 81080q^{24} + 281246q^{25} + 238042q^{27} - 50352q^{28} - 24624q^{30} - 45136q^{33} - 332816q^{34} - 148244q^{36} + 115872q^{40} + 331272q^{42} - 752852q^{43} + 378144q^{46} + 566200q^{48} - 2158138q^{49} - 1112848q^{51} + 1542144q^{52} + 26104q^{54} - 347548q^{57} + 2501160q^{58} + 414000q^{60} - 728272q^{64} - 91288q^{66} + 1552540q^{67} + 5522736q^{70} + 3503824q^{72} + 1267060q^{73} + 573562q^{75} - 213032q^{76} - 10346544q^{78} + 6421738q^{81} - 17006144q^{82} - 12156480q^{84} - 17283152q^{88} + 14281704q^{90} + 4997664q^{91} - 2342208q^{94} + 7955728q^{96} + 13155724q^{97} + 14430800q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(24, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
24.8.f.a \(2\) \(7.497\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(-86\) \(0\) \(0\) \(q+8\beta q^{2}+(-43+13\beta )q^{3}-2^{7}q^{4}+\cdots\)
24.8.f.b \(4\) \(7.497\) \(\Q(\sqrt{6}, \sqrt{-26})\) None \(0\) \(-36\) \(0\) \(0\) \(q+(2\beta _{1}+\beta _{2})q^{2}+(-9-9\beta _{1})q^{3}+(-80+\cdots)q^{4}+\cdots\)
24.8.f.c \(20\) \(7.497\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(120\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(6+\beta _{1}-\beta _{4})q^{3}+(3^{3}+\beta _{3}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 128 T^{2} \))(\( 1 + 160 T^{2} + 16384 T^{4} \))(\( 1 - 274 T^{2} + 45864 T^{4} - 5031360 T^{6} + 776389632 T^{8} - 102828146688 T^{10} + 12720367730688 T^{12} - 1350595415900160 T^{14} + 201712005185273856 T^{16} - 19743780766392254464 T^{18} + \)\(11\!\cdots\!24\)\( T^{20} \))
$3$ (\( 1 + 86 T + 2187 T^{2} \))(\( ( 1 + 18 T + 2187 T^{2} )^{2} \))(\( ( 1 - 60 T + 531 T^{2} + 25920 T^{3} + 766422 T^{4} - 160875720 T^{5} + 1676164914 T^{6} + 123974556480 T^{7} + 5554447550793 T^{8} - 1372607547297660 T^{9} + 50031545098999707 T^{10} )^{2} \))
$5$ (\( ( 1 + 78125 T^{2} )^{2} \))(\( ( 1 + 146650 T^{2} + 6103515625 T^{4} )^{2} \))(\( ( 1 + 212726 T^{2} + 34462341957 T^{4} + 4087236092863080 T^{6} + \)\(39\!\cdots\!50\)\( T^{8} + \)\(34\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!50\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(78\!\cdots\!25\)\( T^{16} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(84\!\cdots\!25\)\( T^{20} )^{2} \))
$7$ (\( ( 1 - 823543 T^{2} )^{2} \))(\( ( 1 + 751570 T^{2} + 678223072849 T^{4} )^{2} \))(\( ( 1 - 4741522 T^{2} + 10982550245901 T^{4} - 16959553029024120120 T^{6} + \)\(19\!\cdots\!94\)\( T^{8} - \)\(18\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!06\)\( T^{12} - \)\(78\!\cdots\!20\)\( T^{14} + \)\(34\!\cdots\!49\)\( T^{16} - \)\(10\!\cdots\!22\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \))
$11$ (\( ( 1 - 8814 T + 19487171 T^{2} )( 1 + 8814 T + 19487171 T^{2} ) \))(\( ( 1 - 35789342 T^{2} + 379749833583241 T^{4} )^{2} \))(\( ( 1 - 64937722 T^{2} + 3455038618503141 T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!62\)\( T^{8} - \)\(69\!\cdots\!64\)\( T^{10} + \)\(12\!\cdots\!42\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!61\)\( T^{16} - \)\(13\!\cdots\!42\)\( T^{18} + \)\(78\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( ( 1 - 62748517 T^{2} )^{2} \))(\( ( 1 - 22639370 T^{2} + 3937376385699289 T^{4} )^{2} \))(\( ( 1 - 352407394 T^{2} + 62269551776806821 T^{4} - \)\(72\!\cdots\!68\)\( T^{6} + \)\(63\!\cdots\!46\)\( T^{8} - \)\(44\!\cdots\!72\)\( T^{10} + \)\(25\!\cdots\!94\)\( T^{12} - \)\(11\!\cdots\!28\)\( T^{14} + \)\(38\!\cdots\!49\)\( T^{16} - \)\(84\!\cdots\!54\)\( T^{18} + \)\(94\!\cdots\!49\)\( T^{20} )^{2} \))
$17$ (\( ( 1 - 22182 T + 410338673 T^{2} )( 1 + 22182 T + 410338673 T^{2} ) \))(\( ( 1 - 61393730 T^{2} + 168377826559400929 T^{4} )^{2} \))(\( ( 1 - 2730456490 T^{2} + 3432956136635397261 T^{4} - \)\(26\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(68\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!26\)\( T^{12} - \)\(76\!\cdots\!92\)\( T^{14} + \)\(16\!\cdots\!29\)\( T^{16} - \)\(21\!\cdots\!90\)\( T^{18} + \)\(13\!\cdots\!49\)\( T^{20} )^{2} \))
$19$ (\( ( 1 + 59722 T + 893871739 T^{2} )^{2} \))(\( ( 1 - 11570 T + 893871739 T^{2} )^{4} \))(\( ( 1 - 33436 T + 3303470211 T^{2} - 96285443658240 T^{3} + 5143998080498378646 T^{4} - \)\(12\!\cdots\!92\)\( T^{5} + \)\(45\!\cdots\!94\)\( T^{6} - \)\(76\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!76\)\( T^{9} + \)\(57\!\cdots\!99\)\( T^{10} )^{4} \))
$23$ (\( ( 1 + 3404825447 T^{2} )^{2} \))(\( ( 1 + 3725533390 T^{2} + 11592836324538749809 T^{4} )^{2} \))(\( ( 1 + 14309806406 T^{2} + 96167780623641456861 T^{4} + \)\(42\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!66\)\( T^{8} + \)\(48\!\cdots\!52\)\( T^{10} + \)\(17\!\cdots\!94\)\( T^{12} + \)\(57\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!69\)\( T^{16} + \)\(25\!\cdots\!66\)\( T^{18} + \)\(20\!\cdots\!49\)\( T^{20} )^{2} \))
$29$ (\( ( 1 + 17249876309 T^{2} )^{2} \))(\( ( 1 + 31499935018 T^{2} + \)\(29\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 73882100390 T^{2} + \)\(28\!\cdots\!49\)\( T^{4} + \)\(79\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!74\)\( T^{8} + \)\(33\!\cdots\!40\)\( T^{10} + \)\(53\!\cdots\!94\)\( T^{12} + \)\(70\!\cdots\!56\)\( T^{14} + \)\(74\!\cdots\!09\)\( T^{16} + \)\(57\!\cdots\!90\)\( T^{18} + \)\(23\!\cdots\!01\)\( T^{20} )^{2} \))
$31$ (\( ( 1 - 27512614111 T^{2} )^{2} \))(\( ( 1 - 49885402622 T^{2} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 102708614914 T^{2} + \)\(77\!\cdots\!29\)\( T^{4} - \)\(38\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!74\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!54\)\( T^{12} - \)\(21\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!69\)\( T^{16} - \)\(33\!\cdots\!34\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( ( 1 - 94931877133 T^{2} )^{2} \))(\( ( 1 - 109874639450 T^{2} + \)\(90\!\cdots\!89\)\( T^{4} )^{2} \))(\( ( 1 - 483705831538 T^{2} + \)\(12\!\cdots\!41\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!66\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!74\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!29\)\( T^{16} - \)\(31\!\cdots\!58\)\( T^{18} + \)\(59\!\cdots\!49\)\( T^{20} )^{2} \))
$41$ (\( ( 1 - 236886 T + 194754273881 T^{2} )( 1 + 236886 T + 194754273881 T^{2} ) \))(\( ( 1 - 226584602162 T^{2} + \)\(37\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 1090625824570 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} - \)\(19\!\cdots\!36\)\( T^{6} + \)\(47\!\cdots\!34\)\( T^{8} - \)\(98\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!74\)\( T^{12} - \)\(27\!\cdots\!56\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} - \)\(22\!\cdots\!70\)\( T^{18} + \)\(78\!\cdots\!01\)\( T^{20} )^{2} \))
$43$ (\( ( 1 + 220510 T + 271818611107 T^{2} )^{2} \))(\( ( 1 - 495062 T + 271818611107 T^{2} )^{4} \))(\( ( 1 + 573020 T + 765614977707 T^{2} + 388334673397088640 T^{3} + \)\(29\!\cdots\!86\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(80\!\cdots\!02\)\( T^{6} + \)\(28\!\cdots\!60\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} + \)\(31\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!07\)\( T^{10} )^{4} \))
$47$ (\( ( 1 + 506623120463 T^{2} )^{2} \))(\( ( 1 - 277104744290 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} )^{2} \))(\( ( 1 + 2220997832726 T^{2} + \)\(28\!\cdots\!69\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!22\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{10} + \)\(49\!\cdots\!18\)\( T^{12} + \)\(17\!\cdots\!20\)\( T^{14} + \)\(48\!\cdots\!21\)\( T^{16} + \)\(96\!\cdots\!46\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} )^{2} \))
$53$ (\( ( 1 + 1174711139837 T^{2} )^{2} \))(\( ( 1 + 2021761791610 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} )^{2} \))(\( ( 1 + 3461178859478 T^{2} + \)\(63\!\cdots\!13\)\( T^{4} + \)\(75\!\cdots\!12\)\( T^{6} + \)\(64\!\cdots\!46\)\( T^{8} + \)\(53\!\cdots\!64\)\( T^{10} + \)\(88\!\cdots\!74\)\( T^{12} + \)\(14\!\cdots\!32\)\( T^{14} + \)\(16\!\cdots\!17\)\( T^{16} + \)\(12\!\cdots\!38\)\( T^{18} + \)\(50\!\cdots\!49\)\( T^{20} )^{2} \))
$59$ (\( ( 1 - 1030926 T + 2488651484819 T^{2} )( 1 + 1030926 T + 2488651484819 T^{2} ) \))(\( ( 1 - 2902252328702 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 19732197121882 T^{2} + \)\(18\!\cdots\!29\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(12\!\cdots\!44\)\( T^{10} + \)\(26\!\cdots\!66\)\( T^{12} - \)\(41\!\cdots\!80\)\( T^{14} + \)\(43\!\cdots\!49\)\( T^{16} - \)\(29\!\cdots\!62\)\( T^{18} + \)\(91\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( ( 1 - 3142742836021 T^{2} )^{2} \))(\( ( 1 - 3094549289642 T^{2} + \)\(98\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 8130200545282 T^{2} + \)\(45\!\cdots\!25\)\( T^{4} - \)\(16\!\cdots\!48\)\( T^{6} + \)\(61\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!52\)\( T^{10} + \)\(60\!\cdots\!70\)\( T^{12} - \)\(16\!\cdots\!88\)\( T^{14} + \)\(43\!\cdots\!25\)\( T^{16} - \)\(77\!\cdots\!02\)\( T^{18} + \)\(93\!\cdots\!01\)\( T^{20} )^{2} \))
$67$ (\( ( 1 + 3851302 T + 6060711605323 T^{2} )^{2} \))(\( ( 1 - 1400126 T + 6060711605323 T^{2} )^{4} \))(\( ( 1 - 913660 T + 20744505994131 T^{2} - 20152579863740122560 T^{3} + \)\(20\!\cdots\!10\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(12\!\cdots\!30\)\( T^{6} - \)\(74\!\cdots\!40\)\( T^{7} + \)\(46\!\cdots\!77\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{9} + \)\(81\!\cdots\!43\)\( T^{10} )^{4} \))
$71$ (\( ( 1 + 9095120158391 T^{2} )^{2} \))(\( ( 1 + 5449089705838 T^{2} + \)\(82\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 78176481465446 T^{2} + \)\(28\!\cdots\!33\)\( T^{4} + \)\(63\!\cdots\!56\)\( T^{6} + \)\(95\!\cdots\!42\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(78\!\cdots\!02\)\( T^{12} + \)\(43\!\cdots\!16\)\( T^{14} + \)\(16\!\cdots\!53\)\( T^{16} + \)\(36\!\cdots\!66\)\( T^{18} + \)\(38\!\cdots\!01\)\( T^{20} )^{2} \))
$73$ (\( ( 1 + 4865614 T + 11047398519097 T^{2} )^{2} \))(\( ( 1 + 2223598 T + 11047398519097 T^{2} )^{4} \))(\( ( 1 - 4973170 T + 56350864331589 T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!70\)\( T^{6} - \)\(25\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!97\)\( T^{8} - \)\(74\!\cdots\!70\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{4} \))
$79$ (\( ( 1 - 19203908986159 T^{2} )^{2} \))(\( ( 1 - 7153320805022 T^{2} + \)\(36\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 126095156012386 T^{2} + \)\(68\!\cdots\!69\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!54\)\( T^{8} - \)\(77\!\cdots\!60\)\( T^{10} + \)\(15\!\cdots\!74\)\( T^{12} - \)\(28\!\cdots\!20\)\( T^{14} + \)\(34\!\cdots\!29\)\( T^{16} - \)\(23\!\cdots\!06\)\( T^{18} + \)\(68\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( ( 1 - 4808934 T + 27136050989627 T^{2} )( 1 + 4808934 T + 27136050989627 T^{2} ) \))(\( ( 1 - 45191963757710 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} )^{2} \))(\( ( 1 - 179430499509514 T^{2} + \)\(15\!\cdots\!93\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!46\)\( T^{8} - \)\(93\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!34\)\( T^{12} - \)\(43\!\cdots\!80\)\( T^{14} + \)\(60\!\cdots\!77\)\( T^{16} - \)\(52\!\cdots\!34\)\( T^{18} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \))
$89$ (\( ( 1 - 7073118 T + 44231334895529 T^{2} )( 1 + 7073118 T + 44231334895529 T^{2} ) \))(\( ( 1 - 54294758858162 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 287003084492698 T^{2} + \)\(42\!\cdots\!01\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!22\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{10} + \)\(53\!\cdots\!02\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!21\)\( T^{16} - \)\(42\!\cdots\!78\)\( T^{18} + \)\(28\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( ( 1 + 9938890 T + 80798284478113 T^{2} )^{2} \))(\( ( 1 - 6867926 T + 80798284478113 T^{2} )^{4} \))(\( ( 1 - 1390450 T + 285922062427293 T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!78\)\( T^{4} + \)\(61\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!14\)\( T^{6} + \)\(17\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} - \)\(59\!\cdots\!50\)\( T^{9} + \)\(34\!\cdots\!93\)\( T^{10} )^{4} \))
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