Properties

Label 70.2.i
Level 70
Weight 2
Character orbit i
Rep. character \(\chi_{70}(9,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 8
Newform subspaces 2
Sturm bound 24
Trace bound 5

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Defining parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.i (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

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

\( 8q + 4q^{4} + 2q^{5} - 12q^{6} + 6q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 2q^{5} - 12q^{6} + 6q^{9} + 2q^{10} - 6q^{11} - 2q^{14} - 24q^{15} - 4q^{16} - 14q^{19} + 4q^{20} + 6q^{21} - 6q^{24} + 14q^{26} + 20q^{29} + 6q^{30} - 16q^{31} + 16q^{34} + 26q^{35} + 12q^{36} + 12q^{39} - 2q^{40} + 6q^{44} + 24q^{45} + 16q^{46} - 26q^{49} - 32q^{50} - 12q^{51} - 18q^{54} - 24q^{55} + 2q^{56} - 4q^{59} - 12q^{60} + 6q^{61} - 8q^{64} - 2q^{65} - 12q^{69} - 2q^{70} - 14q^{74} + 24q^{75} - 28q^{76} + 8q^{79} + 2q^{80} - 36q^{81} - 24q^{84} + 8q^{85} - 14q^{86} - 10q^{89} + 60q^{90} + 20q^{91} + 2q^{94} + 16q^{95} + 6q^{96} + 36q^{99} + O(q^{100}) \)

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.i.a \(4\) \(0.559\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) \(q+\zeta_{12}q^{2}+(-3\zeta_{12}+3\zeta_{12}^{3})q^{3}+\cdots\)
70.2.i.b \(4\) \(0.559\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(2-\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)

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

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))
$3$ (\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))
$5$ (\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))(\( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \))
$19$ (\( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \))(\( 1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - T + 29 T^{2} )^{4} \))(\( ( 1 - 4 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} ) \))
$41$ (\( ( 1 + 3 T + 41 T^{2} )^{4} \))(\( ( 1 - 3 T + 41 T^{2} )^{4} \))
$43$ (\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \))(\( 1 + 45 T^{2} - 184 T^{4} + 99405 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} \))(\( 1 + 130 T^{2} + 12411 T^{4} + 583570 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 6 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \))(\( ( 1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4} )( 1 + 6 T - 37 T^{2} + 438 T^{3} + 5329 T^{4} ) \))
$79$ (\( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 - 85 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 50 T^{2} + 9409 T^{4} )^{2} \))
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