Properties

Label 70.2.g
Level 70
Weight 2
Character orbit g
Rep. character \(\chi_{70}(13,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 1
Sturm bound 24
Trace bound 0

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Defining parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(70, [\chi])\).

Total New Old
Modular forms 32 8 24
Cusp forms 16 8 8
Eisenstein series 16 0 16

Trace form

\( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} - 16q^{15} - 8q^{16} + 8q^{18} + 16q^{21} - 16q^{22} - 8q^{23} + 8q^{28} + 16q^{30} - 8q^{35} + 8q^{36} + 32q^{37} + 32q^{43} - 16q^{46} + 32q^{50} - 32q^{51} - 32q^{53} - 8q^{56} - 8q^{57} - 16q^{58} - 8q^{60} + 8q^{65} + 16q^{67} - 24q^{70} - 16q^{71} + 8q^{72} - 16q^{77} - 40q^{78} + 24q^{81} - 48q^{85} + 16q^{88} + 16q^{91} - 8q^{92} - 16q^{93} + 64q^{95} + 32q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(70, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
70.2.g.a \(8\) \(0.559\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(-8\) \(q-\zeta_{16}^{6}q^{2}+(\zeta_{16}-\zeta_{16}^{3})q^{3}-\zeta_{16}^{4}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(70, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 72 T^{6} + 81 T^{8} ) \)
$5$ \( 1 - 48 T^{4} + 625 T^{8} \)
$7$ \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 616 T^{5} + 1568 T^{6} + 2744 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + 14 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 240 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16} \)
$19$ \( ( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 4 T + 8 T^{2} + 84 T^{3} + 878 T^{4} + 1932 T^{5} + 4232 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 38480 T^{5} + 175232 T^{6} - 810448 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 8 T + 32 T^{2} - 344 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( 1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 16 T + 128 T^{2} + 784 T^{3} + 4786 T^{4} + 41552 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 8 T + 32 T^{2} + 552 T^{3} - 8974 T^{4} + 36984 T^{5} + 143648 T^{6} - 2406104 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( 1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( 1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16} \)
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