Properties

Label 756.2.n
Level 756
Weight 2
Character orbit n
Rep. character \(\chi_{756}(19,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 88
Newform subspaces 2
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.n (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 252 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(756, [\chi])\).

Total New Old
Modular forms 312 104 208
Cusp forms 264 88 176
Eisenstein series 48 16 32

Trace form

\( 88q - q^{2} + q^{4} + 8q^{8} + O(q^{10}) \) \( 88q - q^{2} + q^{4} + 8q^{8} - 6q^{10} + 15q^{14} + q^{16} + 12q^{17} - 24q^{20} + 2q^{22} - 60q^{25} + 12q^{26} - q^{32} + 6q^{34} - 4q^{37} + 5q^{44} + 2q^{46} - 2q^{49} + 31q^{50} + 4q^{53} + 48q^{56} + 6q^{58} - 6q^{61} - 8q^{64} - 14q^{65} + 8q^{70} - 12q^{73} - 34q^{74} - 12q^{76} - 62q^{77} - 51q^{80} - 12q^{82} - 14q^{85} + 26q^{86} + 14q^{88} + 72q^{89} - 44q^{92} - 3q^{94} + 5q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
756.2.n.a \(4\) \(6.037\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(0\) \(0\) \(q+(-1+\zeta_{12}+\zeta_{12}^{2})q^{2}+(-2\zeta_{12}+\cdots)q^{4}+\cdots\)
756.2.n.b \(84\) \(6.037\) None \(1\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))
$3$ 1
$5$ (\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 - 18 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 - 2 T + 13 T^{2} )^{2} \))
$17$ (\( ( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 - 35 T^{2} + 264 T^{4} - 33635 T^{6} + 923521 T^{8} \))
$37$ (\( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 + 3 T + 44 T^{2} + 123 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 - 91 T^{2} + 6072 T^{4} - 201019 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 + 15 T + 164 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} )^{2} \))
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