Properties

Label 756.2.bi
Level 756
Weight 2
Character orbit bi
Rep. character \(\chi_{756}(307,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 88
Newform subspaces 3
Sturm bound 288
Trace bound 5

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Defining parameters

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

\( 88q + 2q^{2} - 2q^{4} + 8q^{8} + O(q^{10}) \) \( 88q + 2q^{2} - 2q^{4} + 8q^{8} - 2q^{16} - 10q^{22} + 24q^{25} - 12q^{28} + 12q^{29} + 32q^{32} - 16q^{37} + 68q^{44} - 16q^{46} - 2q^{49} + 4q^{50} + 16q^{53} - 12q^{56} + 6q^{58} - 8q^{64} + 52q^{65} - 16q^{70} - 64q^{74} + 22q^{77} + 16q^{85} + 56q^{86} + 14q^{88} + 10q^{92} - 64q^{98} + O(q^{100}) \)

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.bi.a \(4\) \(6.037\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-6\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}-2\zeta_{12}q^{4}+\cdots\)
756.2.bi.b \(4\) \(6.037\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(6\) \(0\) \(q+(-1+\zeta_{12}+\zeta_{12}^{2})q^{2}+(-2\zeta_{12}+\cdots)q^{4}+\cdots\)
756.2.bi.c \(80\) \(6.037\) None \(6\) \(0\) \(0\) \(0\)

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

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))
$3$ 1
$5$ (\( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))
$11$ (\( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \))(\( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2} \))
$17$ (\( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \))(\( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} ) \))(\( ( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} ) \))
$37$ (\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 37 T^{2} )^{4} \))
$41$ (\( ( 1 - 15 T + 116 T^{2} - 615 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 15 T + 116 T^{2} + 615 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8} \))(\( 1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8} \))(\( 1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 + 4 T + 53 T^{2} )^{4} \))(\( ( 1 + 4 T + 53 T^{2} )^{4} \))
$59$ (\( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} \))(\( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 - 9 T + 88 T^{2} - 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 91 T^{2} + 3792 T^{4} - 408499 T^{6} + 20151121 T^{8} \))(\( 1 - 91 T^{2} + 3792 T^{4} - 408499 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 126 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 126 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8} \))(\( 1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8} \))(\( 1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 + 122 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 122 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 5 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))
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