Properties

Label 70.3.b
Level 70
Weight 3
Character orbit b
Rep. character \(\chi_{70}(41,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 36
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 70.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(36\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(70, [\chi])\).

Total New Old
Modular forms 28 8 20
Cusp forms 20 8 12
Eisenstein series 8 0 8

Trace form

\( 8q + 16q^{4} - 4q^{7} - 36q^{9} + O(q^{10}) \) \( 8q + 16q^{4} - 4q^{7} - 36q^{9} + 20q^{11} - 4q^{14} + 20q^{15} + 32q^{16} + 32q^{18} - 64q^{21} - 48q^{22} - 72q^{23} - 40q^{25} - 8q^{28} + 12q^{29} - 40q^{30} + 40q^{35} - 72q^{36} + 136q^{37} + 36q^{39} + 80q^{42} + 128q^{43} + 40q^{44} - 56q^{46} + 4q^{49} + 268q^{51} - 208q^{53} - 8q^{56} - 24q^{57} - 224q^{58} + 40q^{60} + 380q^{63} + 64q^{64} - 100q^{65} - 112q^{67} + 60q^{70} - 376q^{71} + 64q^{72} - 208q^{74} - 408q^{77} + 320q^{78} + 188q^{79} + 344q^{81} - 128q^{84} + 180q^{85} + 200q^{86} - 96q^{88} - 244q^{91} - 144q^{92} - 824q^{93} - 40q^{95} + 48q^{98} - 576q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(70, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
70.3.b.a \(8\) \(1.907\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{6}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+2q^{4}-\beta _{2}q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(70, [\chi])\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{4} \)
$3$ \( 1 - 18 T^{2} + 157 T^{4} - 702 T^{6} + 648 T^{8} - 56862 T^{10} + 1030077 T^{12} - 9565938 T^{14} + 43046721 T^{16} \)
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( 1 + 4 T + 6 T^{2} + 220 T^{3} + 4970 T^{4} + 10780 T^{5} + 14406 T^{6} + 470596 T^{7} + 5764801 T^{8} \)
$11$ \( ( 1 - 10 T + 173 T^{2} + 570 T^{3} + 1208 T^{4} + 68970 T^{5} + 2532893 T^{6} - 17715610 T^{7} + 214358881 T^{8} )^{2} \)
$13$ \( 1 - 470 T^{2} + 96425 T^{4} - 7707750 T^{6} + 323093332 T^{8} - 220141047750 T^{10} + 78656834772425 T^{12} - 10950100007566070 T^{14} + 665416609183179841 T^{16} \)
$17$ \( 1 - 774 T^{2} + 378041 T^{4} - 152875638 T^{6} + 49129851156 T^{8} - 12768326161398 T^{10} + 2637122318753081 T^{12} - 450949611615835014 T^{14} + 48661191875666868481 T^{16} \)
$19$ \( 1 - 1712 T^{2} + 1528412 T^{4} - 900114768 T^{6} + 380400506758 T^{8} - 117303856680528 T^{10} + 25957881554620892 T^{12} - 3789195141441267632 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( ( 1 + 36 T + 878 T^{2} + 19356 T^{3} + 572138 T^{4} + 10239324 T^{5} + 245700398 T^{6} + 5329292004 T^{7} + 78310985281 T^{8} )^{2} \)
$29$ \( ( 1 - 6 T + 2321 T^{2} - 22062 T^{3} + 2551316 T^{4} - 18554142 T^{5} + 1641599201 T^{6} - 3568939926 T^{7} + 500246412961 T^{8} )^{2} \)
$31$ \( 1 - 2336 T^{2} + 3598076 T^{4} - 4780408032 T^{6} + 5102420377606 T^{8} - 4414807206120672 T^{10} + 3068766772431563516 T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( ( 1 - 68 T + 6144 T^{2} - 261644 T^{3} + 12851966 T^{4} - 358190636 T^{5} + 11514845184 T^{6} - 174469395812 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( 1 - 7016 T^{2} + 25625276 T^{4} - 64434523992 T^{6} + 122819632739206 T^{8} - 182076564950157912 T^{10} + \)\(20\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!96\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( ( 1 - 64 T + 7682 T^{2} - 352392 T^{3} + 21563770 T^{4} - 651572808 T^{5} + 26263229282 T^{6} - 404567235136 T^{7} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 - 5618 T^{2} + 16643837 T^{4} - 38947889982 T^{6} + 83393318217448 T^{8} - 190053278735255742 T^{10} + \)\(39\!\cdots\!57\)\( T^{12} - \)\(65\!\cdots\!38\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 + 104 T + 12156 T^{2} + 784920 T^{3} + 53277670 T^{4} + 2204840280 T^{5} + 95916687036 T^{6} + 2305093557416 T^{7} + 62259690411361 T^{8} )^{2} \)
$59$ \( 1 - 16128 T^{2} + 123340988 T^{4} - 622901719296 T^{6} + 2414919503628870 T^{8} - 7547925000230297856 T^{10} + \)\(18\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!68\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( 1 - 27152 T^{2} + 331146332 T^{4} - 2369254902768 T^{6} + 10876351870920838 T^{8} - 32804326672196187888 T^{10} + \)\(63\!\cdots\!92\)\( T^{12} - \)\(72\!\cdots\!92\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( ( 1 + 56 T + 12026 T^{2} + 624480 T^{3} + 76112570 T^{4} + 2803290720 T^{5} + 242337381146 T^{6} + 5065669401464 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 + 188 T + 25704 T^{2} + 2576820 T^{3} + 208896910 T^{4} + 12989749620 T^{5} + 653181848424 T^{6} + 24082853377148 T^{7} + 645753531245761 T^{8} )^{2} \)
$73$ \( 1 - 34552 T^{2} + 545494172 T^{4} - 5215338412488 T^{6} + 33447987891134918 T^{8} - \)\(14\!\cdots\!08\)\( T^{10} + \)\(43\!\cdots\!32\)\( T^{12} - \)\(79\!\cdots\!92\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( ( 1 - 94 T + 13965 T^{2} - 1353586 T^{3} + 93524384 T^{4} - 8447730226 T^{5} + 543937881165 T^{6} - 22850220818974 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( 1 - 28196 T^{2} + 281031704 T^{4} - 911252649996 T^{6} - 186386635165010 T^{8} - 43246520775610816716 T^{10} + \)\(63\!\cdots\!64\)\( T^{12} - \)\(30\!\cdots\!56\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( 1 - 26360 T^{2} + 419977340 T^{4} - 4862799095880 T^{6} + 44341547037657862 T^{8} - \)\(30\!\cdots\!80\)\( T^{10} + \)\(16\!\cdots\!40\)\( T^{12} - \)\(65\!\cdots\!60\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( 1 - 33190 T^{2} + 746913305 T^{4} - 10749643947030 T^{6} + 119807243476794452 T^{8} - \)\(95\!\cdots\!30\)\( T^{10} + \)\(58\!\cdots\!05\)\( T^{12} - \)\(23\!\cdots\!90\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \)
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