Properties

Label 210.3.h
Level 210
Weight 3
Character orbit h
Rep. character \(\chi_{210}(139,\cdot)\)
Character field \(\Q\)
Dimension 16
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.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
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 16 88
Cusp forms 88 16 72
Eisenstein series 16 0 16

Trace form

\( 16q - 32q^{4} + 48q^{9} + O(q^{10}) \) \( 16q - 32q^{4} + 48q^{9} + 96q^{11} + 16q^{14} - 24q^{15} + 64q^{16} + 24q^{21} + 24q^{25} + 64q^{29} + 24q^{30} - 8q^{35} - 96q^{36} - 144q^{39} - 192q^{44} - 176q^{46} + 224q^{49} - 96q^{50} - 48q^{51} - 32q^{56} + 48q^{60} - 128q^{64} + 368q^{65} - 56q^{70} - 384q^{71} + 224q^{74} - 608q^{79} + 144q^{81} - 48q^{84} - 440q^{85} + 416q^{86} + 224q^{91} - 560q^{95} + 288q^{99} + O(q^{100}) \)

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.h.a \(16\) \(5.722\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{5}q^{2}-\beta _{1}q^{3}-2q^{4}+\beta _{8}q^{5}-\beta _{4}q^{6}+\cdots\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{8} \)
$3$ \( ( 1 - 3 T^{2} )^{8} \)
$5$ \( 1 - 12 T^{2} - 376 T^{4} - 3300 T^{6} + 588750 T^{8} - 2062500 T^{10} - 146875000 T^{12} - 2929687500 T^{14} + 152587890625 T^{16} \)
$7$ \( 1 - 112 T^{2} + 3932 T^{4} - 24720 T^{6} - 1183546 T^{8} - 59352720 T^{10} + 22667197532 T^{12} - 1550224166512 T^{14} + 33232930569601 T^{16} \)
$11$ \( ( 1 - 24 T + 512 T^{2} - 6936 T^{3} + 89778 T^{4} - 839256 T^{5} + 7496192 T^{6} - 42517464 T^{7} + 214358881 T^{8} )^{4} \)
$13$ \( ( 1 + 308 T^{2} + 54824 T^{4} + 3243420 T^{6} - 106725490 T^{8} + 92635318620 T^{10} + 44721621048104 T^{12} + 7175810217724148 T^{14} + 665416609183179841 T^{16} )^{2} \)
$17$ \( ( 1 + 1316 T^{2} + 881960 T^{4} + 398466540 T^{6} + 132906424526 T^{8} + 33280323887340 T^{10} + 6152339032664360 T^{12} + 766730864194365476 T^{14} + 48661191875666868481 T^{16} )^{2} \)
$19$ \( ( 1 - 1092 T^{2} + 576680 T^{4} - 222586380 T^{6} + 81331407246 T^{8} - 29007679627980 T^{10} + 9794081134483880 T^{12} - 2416939891620247812 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \)
$23$ \( ( 1 - 1300 T^{2} + 1274456 T^{4} - 947494620 T^{6} + 528617435630 T^{8} - 265147841955420 T^{10} + 99803905057282136 T^{12} - 28489011761626417300 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \)
$29$ \( ( 1 - 16 T + 1276 T^{2} - 55408 T^{3} + 751270 T^{4} - 46598128 T^{5} + 902490556 T^{6} - 9517173136 T^{7} + 500246412961 T^{8} )^{4} \)
$31$ \( ( 1 - 3948 T^{2} + 5903432 T^{4} - 3942646212 T^{6} + 1976573418510 T^{8} - 3641116572352452 T^{10} + 5034984242942397512 T^{12} - \)\(31\!\cdots\!28\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( ( 1 - 2416 T^{2} + 5835932 T^{4} - 9959371920 T^{6} + 14384225705606 T^{8} - 18665466436959120 T^{10} + 20498591244480089372 T^{12} - \)\(15\!\cdots\!96\)\( T^{14} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \)
$41$ \( ( 1 - 12516 T^{2} + 70001768 T^{4} - 228356671020 T^{6} + 475054496880078 T^{8} - 645281375058146220 T^{10} + \)\(55\!\cdots\!28\)\( T^{12} - \)\(28\!\cdots\!96\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \)
$43$ \( ( 1 - 1912 T^{2} + 2635964 T^{4} - 3280241736 T^{6} + 4054697588678 T^{8} - 11214493727278536 T^{10} + 30809675156546242364 T^{12} - \)\(76\!\cdots\!12\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( ( 1 + 6768 T^{2} + 28283420 T^{4} + 81813152400 T^{6} + 198252999971526 T^{8} + 399222085316384400 T^{10} + \)\(67\!\cdots\!20\)\( T^{12} + \)\(78\!\cdots\!88\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} )^{2} \)
$53$ \( ( 1 - 6868 T^{2} + 39779480 T^{4} - 162177493980 T^{6} + 501674308076846 T^{8} - 1279658434876804380 T^{10} + \)\(24\!\cdots\!80\)\( T^{12} - \)\(33\!\cdots\!88\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} )^{2} \)
$59$ \( ( 1 - 11736 T^{2} + 79787516 T^{4} - 412369170408 T^{6} + 1633889740528710 T^{8} - 4996826103104253288 T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(20\!\cdots\!16\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \)
$61$ \( ( 1 - 15944 T^{2} + 96425692 T^{4} - 269693317496 T^{6} + 596407807846918 T^{8} - 3734130792812134136 T^{10} + \)\(18\!\cdots\!52\)\( T^{12} - \)\(42\!\cdots\!24\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \)
$67$ \( ( 1 - 18688 T^{2} + 141351740 T^{4} - 579001624320 T^{6} + 2070725817126086 T^{8} - 11667531790868862720 T^{10} + \)\(57\!\cdots\!40\)\( T^{12} - \)\(15\!\cdots\!68\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} )^{2} \)
$71$ \( ( 1 + 96 T + 14048 T^{2} + 646080 T^{3} + 72093138 T^{4} + 3256889280 T^{5} + 356983294688 T^{6} + 12297627256416 T^{7} + 645753531245761 T^{8} )^{4} \)
$73$ \( ( 1 + 18612 T^{2} + 177429992 T^{4} + 1152928560348 T^{6} + 6336699932034510 T^{8} + 32741143112545547868 T^{10} + \)\(14\!\cdots\!52\)\( T^{12} + \)\(42\!\cdots\!52\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \)
$79$ \( ( 1 + 152 T + 28916 T^{2} + 2541000 T^{3} + 270417254 T^{4} + 15858381000 T^{5} + 1126280542196 T^{6} + 36949293239192 T^{7} + 1517108809906561 T^{8} )^{4} \)
$83$ \( ( 1 + 39632 T^{2} + 756012572 T^{4} + 9074582537520 T^{6} + 74694570513655814 T^{8} + \)\(43\!\cdots\!20\)\( T^{10} + \)\(17\!\cdots\!52\)\( T^{12} + \)\(42\!\cdots\!52\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \)
$89$ \( ( 1 - 38508 T^{2} + 778838024 T^{4} - 10318651040964 T^{6} + 96273467949842958 T^{8} - \)\(64\!\cdots\!24\)\( T^{10} + \)\(30\!\cdots\!44\)\( T^{12} - \)\(95\!\cdots\!68\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \)
$97$ \( ( 1 + 54804 T^{2} + 1439825768 T^{4} + 23691050976252 T^{6} + 266516143722806094 T^{8} + \)\(20\!\cdots\!12\)\( T^{10} + \)\(11\!\cdots\!48\)\( T^{12} + \)\(38\!\cdots\!64\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \)
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