Properties

Label 3136.1.d
Level 3136
Weight 1
Character orbit d
Rep. character \(\chi_{3136}(1471,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 4
Sturm bound 448
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3136.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(448\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3136, [\chi])\).

Total New Old
Modular forms 74 12 62
Cusp forms 26 7 19
Eisenstein series 48 5 43

The following table gives the dimensions of subspaces with specified projective image type.

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 3 4 0 0

Trace form

\( 7q + 3q^{9} + O(q^{10}) \) \( 7q + 3q^{9} + q^{25} + 2q^{29} + 6q^{37} + 6q^{53} - 4q^{57} - 4q^{65} - q^{81} - 4q^{93} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3136, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3136.1.d.a \(1\) \(1.565\) \(\Q\) \(D_{2}\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{7}) \) \(0\) \(0\) \(0\) \(0\) \(q+q^{9}-q^{25}+2q^{29}+2q^{37}-2q^{53}+\cdots\)
3136.1.d.b \(2\) \(1.565\) \(\Q(\sqrt{-1}) \) \(A_{4}\) None None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{3}-q^{5}-iq^{11}-iq^{15}+q^{17}+\cdots\)
3136.1.d.c \(2\) \(1.565\) \(\Q(\sqrt{2}) \) \(D_{4}\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{5}+q^{9}+\beta q^{13}-\beta q^{17}+q^{25}+\cdots\)
3136.1.d.d \(2\) \(1.565\) \(\Q(\sqrt{-1}) \) \(A_{4}\) None None \(0\) \(0\) \(2\) \(0\) \(q-iq^{3}+q^{5}-iq^{11}-iq^{15}-q^{17}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(3136, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(3136, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(1568, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$5$ (\( 1 + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 - T + T^{2} )^{2} \))
$7$ 1
$11$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$13$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))
$17$ (\( 1 + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T + T^{2} )^{2} \))
$19$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$23$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$29$ (\( ( 1 - T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$31$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$37$ (\( ( 1 - T )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$41$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))
$43$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$47$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$53$ (\( ( 1 + T )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T + T^{2} )^{2} \))
$59$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$61$ (\( 1 + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T + T^{2} )^{2} \))
$67$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$71$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$73$ (\( 1 + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 - T + T^{2} )^{2} \))
$79$ (\( ( 1 - T )( 1 + T ) \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( 1 - T^{2} + T^{4} \))
$83$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$89$ (\( 1 + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 - T + T^{2} )^{2} \))
$97$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))
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