Properties

Label 756.2.be
Level 756
Weight 2
Character orbit be
Rep. character \(\chi_{756}(107,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 128
Newform subspaces 5
Sturm bound 288
Trace bound 7

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.be (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 84 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(19\)

Dimensions

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

Total New Old
Modular forms 312 128 184
Cusp forms 264 128 136
Eisenstein series 48 0 48

Trace form

\( 128q + O(q^{10}) \) \( 128q - 8q^{13} + 16q^{16} - 20q^{22} + 60q^{25} + 22q^{28} + 8q^{34} - 4q^{37} + 26q^{40} - 6q^{46} + 8q^{49} + 26q^{52} - 12q^{58} + 8q^{61} - 60q^{64} + 78q^{70} + 4q^{73} - 144q^{76} + 74q^{82} + 16q^{85} - 40q^{88} - 30q^{94} + 88q^{97} + O(q^{100}) \)

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.be.a \(4\) \(6.037\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{1}q^{5}+(-1+\cdots)q^{7}+\cdots\)
756.2.be.b \(4\) \(6.037\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(8\) \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{1}q^{5}+(1+2\beta _{2}+\cdots)q^{7}+\cdots\)
756.2.be.c \(28\) \(6.037\) None \(0\) \(0\) \(0\) \(-2\)
756.2.be.d \(28\) \(6.037\) None \(0\) \(0\) \(0\) \(2\)
756.2.be.e \(64\) \(6.037\) None \(0\) \(0\) \(0\) \(0\)

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

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - 2 T^{2} + 4 T^{4} \))
$3$ 1
$5$ (\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))
$7$ (\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))
$11$ (\( 1 - 16 T^{2} + 135 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( 1 - 16 T^{2} + 135 T^{4} - 1936 T^{6} + 14641 T^{8} \))
$13$ (\( ( 1 + T + 13 T^{2} )^{4} \))(\( ( 1 + T + 13 T^{2} )^{4} \))
$17$ (\( 1 + 32 T^{2} + 735 T^{4} + 9248 T^{6} + 83521 T^{8} \))(\( 1 + 32 T^{2} + 735 T^{4} + 9248 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \))(\( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 32 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 11 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} \))(\( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 + 8 T^{2} - 2745 T^{4} + 22472 T^{6} + 7890481 T^{8} \))(\( 1 + 8 T^{2} - 2745 T^{4} + 22472 T^{6} + 7890481 T^{8} \))
$59$ (\( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} \))(\( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 9 T + 94 T^{2} + 603 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 - 74 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 + 27 T + 322 T^{2} + 2133 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 27 T + 322 T^{2} - 2133 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 - 22 T^{2} - 7437 T^{4} - 174262 T^{6} + 62742241 T^{8} \))(\( 1 - 22 T^{2} - 7437 T^{4} - 174262 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 - 5 T + 97 T^{2} )^{4} \))(\( ( 1 - 5 T + 97 T^{2} )^{4} \))
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