Properties

Label 210.3.f
Level 210
Weight 3
Character orbit f
Rep. character \(\chi_{210}(181,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 144
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(144\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(210, [\chi])\).

Total New Old
Modular forms 104 8 96
Cusp forms 88 8 80
Eisenstein series 16 0 16

Trace form

\( 8q + 16q^{4} - 24q^{9} + O(q^{10}) \) \( 8q + 16q^{4} - 24q^{9} - 16q^{11} + 32q^{14} + 32q^{16} + 96q^{22} - 40q^{25} - 144q^{29} - 80q^{35} - 48q^{36} - 48q^{37} - 48q^{39} - 48q^{42} - 64q^{43} - 32q^{44} + 128q^{46} - 24q^{49} + 128q^{53} + 64q^{56} + 144q^{57} + 224q^{58} + 64q^{64} + 80q^{65} - 192q^{67} + 176q^{71} - 160q^{74} + 192q^{77} - 96q^{78} - 288q^{79} + 72q^{81} - 240q^{85} - 64q^{86} + 192q^{88} + 64q^{91} + 336q^{93} + 80q^{95} + 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
210.3.f.a \(8\) \(5.722\) 8.0.3317760000.3 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+\beta _{5}q^{3}+2q^{4}+\beta _{6}q^{5}+\beta _{3}q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{4} \)
$3$ \( ( 1 + 3 T^{2} )^{4} \)
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( 1 + 12 T^{2} - 2842 T^{4} + 28812 T^{6} + 5764801 T^{8} \)
$11$ \( ( 1 + 8 T + 304 T^{2} + 2120 T^{3} + 45250 T^{4} + 256520 T^{5} + 4450864 T^{6} + 14172488 T^{7} + 214358881 T^{8} )^{2} \)
$13$ \( 1 - 1088 T^{2} + 557124 T^{4} - 173160640 T^{6} + 35590649030 T^{8} - 4945641039040 T^{10} + 454463162206404 T^{12} - 25348316613259328 T^{14} + 665416609183179841 T^{16} \)
$17$ \( ( 1 - 716 T^{2} + 266406 T^{4} - 59801036 T^{6} + 6975757441 T^{8} )^{2} \)
$19$ \( 1 - 1040 T^{2} + 715428 T^{4} - 373461040 T^{6} + 149925694598 T^{8} - 48669816193840 T^{10} + 12150516539296548 T^{12} - 2301847515828807440 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( ( 1 + 780 T^{2} + 15360 T^{3} + 320102 T^{4} + 8125440 T^{5} + 218275980 T^{6} + 78310985281 T^{8} )^{2} \)
$29$ \( ( 1 + 72 T + 4044 T^{2} + 159480 T^{3} + 5105990 T^{4} + 134122680 T^{5} + 2860244364 T^{6} + 42827279112 T^{7} + 500246412961 T^{8} )^{2} \)
$31$ \( 1 - 5072 T^{2} + 12940644 T^{4} - 21077526640 T^{6} + 23985269180870 T^{8} - 19465538480099440 T^{10} + 11036959286314652004 T^{12} - \)\(39\!\cdots\!92\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( ( 1 + 24 T + 2412 T^{2} - 16728 T^{3} + 1972550 T^{4} - 22900632 T^{5} + 4520476332 T^{6} + 61577433816 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( 1 - 11432 T^{2} + 60126684 T^{4} - 189254916760 T^{6} + 389111411703110 T^{8} - 534789162838654360 T^{10} + \)\(48\!\cdots\!64\)\( T^{12} - \)\(25\!\cdots\!92\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( ( 1 + 32 T + 3396 T^{2} + 140128 T^{3} + 8375270 T^{4} + 259096672 T^{5} + 11610248196 T^{6} + 202283617568 T^{7} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 - 9656 T^{2} + 49532316 T^{4} - 174004173832 T^{6} + 446973571058246 T^{8} - 849084860968707592 T^{10} + \)\(11\!\cdots\!76\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 - 64 T + 6168 T^{2} - 270464 T^{3} + 22592978 T^{4} - 759733376 T^{5} + 48668486808 T^{6} - 1418519112256 T^{7} + 62259690411361 T^{8} )^{2} \)
$59$ \( 1 - 13672 T^{2} + 87440284 T^{4} - 360721973080 T^{6} + 1260995942842630 T^{8} - 4370998368442641880 T^{10} + \)\(12\!\cdots\!64\)\( T^{12} - \)\(24\!\cdots\!32\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( 1 - 5096 T^{2} + 58282524 T^{4} - 205728528856 T^{6} + 1227606280747910 T^{8} - 2848484499704087896 T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( ( 1 + 96 T + 14188 T^{2} + 794016 T^{3} + 79706118 T^{4} + 3564337824 T^{5} + 285904104748 T^{6} + 8684004688224 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 - 88 T + 8304 T^{2} - 522200 T^{3} + 58604450 T^{4} - 2632410200 T^{5} + 211018599024 T^{6} - 11272824985048 T^{7} + 645753531245761 T^{8} )^{2} \)
$73$ \( 1 - 18048 T^{2} + 179971204 T^{4} - 1281534583680 T^{6} + 7286987478858630 T^{8} - 36393327957179306880 T^{10} + \)\(14\!\cdots\!24\)\( T^{12} - \)\(41\!\cdots\!08\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( ( 1 + 144 T + 27876 T^{2} + 2532528 T^{3} + 267406406 T^{4} + 15805507248 T^{5} + 1085772457956 T^{6} + 35004593595024 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( 1 - 37880 T^{2} + 719278428 T^{4} - 8629223884360 T^{6} + 71063717963313158 T^{8} - \)\(40\!\cdots\!60\)\( T^{10} + \)\(16\!\cdots\!48\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( 1 - 20024 T^{2} + 352120860 T^{4} - 3746723082376 T^{6} + 35814387700257734 T^{8} - \)\(23\!\cdots\!16\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{12} - \)\(49\!\cdots\!04\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( 1 - 40128 T^{2} + 785762308 T^{4} - 10615096489536 T^{6} + 111881094946850310 T^{8} - \)\(93\!\cdots\!16\)\( T^{10} + \)\(61\!\cdots\!88\)\( T^{12} - \)\(27\!\cdots\!48\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \)
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