Properties

Label 756.2.f
Level 756
Weight 2
Character orbit f
Rep. character \(\chi_{756}(377,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 4
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.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
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 162 10 152
Cusp forms 126 10 116
Eisenstein series 36 0 36

Trace form

\( 10q - q^{7} + O(q^{10}) \) \( 10q - q^{7} - 2q^{25} + 10q^{37} + 52q^{43} - 5q^{49} - 10q^{67} + 50q^{79} - 60q^{85} - 3q^{91} + O(q^{100}) \)

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.f.a \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(4\) \(q-3q^{5}+(2-\zeta_{6})q^{7}+3\zeta_{6}q^{11}-2\zeta_{6}q^{13}+\cdots\)
756.2.f.b \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(q+(-2-\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+(5+\cdots)q^{19}+\cdots\)
756.2.f.c \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(4\) \(q+3q^{5}+(2-\zeta_{6})q^{7}-3\zeta_{6}q^{11}-2\zeta_{6}q^{13}+\cdots\)
756.2.f.d \(4\) \(6.037\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{2}q^{5}+(-1+\beta _{1})q^{7}+\beta _{3}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))
$11$ (\( 1 + 5 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 121 T^{4} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 19 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 19 T^{2} + 529 T^{4} \))(\( ( 1 + 26 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 + 50 T^{2} + 841 T^{4} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 - 35 T^{2} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 35 T^{2} + 961 T^{4} \))(\( ( 1 - 8 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 - T + 37 T^{2} )^{4} \))
$41$ (\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 7 T + 43 T^{2} )^{4} \))
$47$ (\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 6 T + 47 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{4} \))
$59$ (\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( ( 1 - 116 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 - 2 T + 67 T^{2} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T + 67 T^{2} )^{2} \))(\( ( 1 + 10 T + 67 T^{2} )^{4} \))
$71$ (\( 1 - 115 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 115 T^{2} + 5041 T^{4} \))(\( ( 1 - 70 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 134 T^{2} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 134 T^{2} + 5329 T^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))
$79$ (\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 - 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 5 T + 79 T^{2} )^{4} \))
$83$ (\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 89 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 188 T^{2} + 9409 T^{4} )^{2} \))
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