Properties

Label 756.2.bj
Level 756
Weight 2
Character orbit bj
Rep. character \(\chi_{756}(451,\cdot)\)
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.bj (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

\( 88q + 2q^{2} - 2q^{4} + 6q^{5} + 8q^{8} + O(q^{10}) \) \( 88q + 2q^{2} - 2q^{4} + 6q^{5} + 8q^{8} - 6q^{10} - 12q^{14} - 2q^{16} + 12q^{17} + 24q^{20} + 2q^{22} + 30q^{25} + 12q^{26} + 2q^{32} - 6q^{34} - 4q^{37} - 27q^{38} - 18q^{40} + 5q^{44} + 2q^{46} - 2q^{49} + 31q^{50} - 27q^{52} + 4q^{53} - 54q^{56} - 3q^{58} - 8q^{64} + 28q^{65} + 18q^{68} + 5q^{70} - 12q^{73} + 17q^{74} + 12q^{76} + 22q^{77} - 51q^{80} - 12q^{82} - 14q^{85} - 13q^{86} - 7q^{88} + 72q^{89} - 44q^{92} + 5q^{98} + O(q^{100}) \)

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.bj.a \(4\) \(6.037\) \(\Q(\zeta_{12})\) None \(4\) \(0\) \(12\) \(0\) \(q+(1+\zeta_{12}^{3})q^{2}+2\zeta_{12}^{3}q^{4}+(4-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
756.2.bj.b \(84\) \(6.037\) None \(-2\) \(0\) \(-6\) \(0\)

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

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} )^{2} \))
$3$ 1
$5$ (\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 - 13 T^{2} + 49 T^{4} \))
$11$ (\( 1 + 18 T^{2} + 203 T^{4} + 2178 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))
$17$ (\( ( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8} \))
$23$ (\( 1 + 30 T^{2} + 371 T^{4} + 15870 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 35 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 35 T^{2} - 624 T^{4} - 64715 T^{6} + 3418801 T^{8} \))
$47$ (\( ( 1 + 91 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 115 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 95 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 - 53 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 - 138 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 149 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 + 15 T + 164 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} )^{2} \))
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