Properties

Label 400.1.b
Level 400
Weight 1
Character orbit b
Rep. character \(\chi_{400}(351,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 60
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 400.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(60\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(400, [\chi])\).

Total New Old
Modular forms 19 1 18
Cusp forms 1 1 0
Eisenstein series 18 0 18

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 1 0 0 0

Trace form

\( q + q^{9} + O(q^{10}) \) \( q + q^{9} - 2q^{29} - 2q^{41} + q^{49} - 2q^{61} + q^{81} - 2q^{89} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
400.1.b.a \(1\) \(0.200\) \(\Q\) \(D_{2}\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \) \(\Q(\sqrt{5}) \) \(0\) \(0\) \(0\) \(0\) \(q+q^{9}-2q^{29}-2q^{41}+q^{49}-2q^{61}+\cdots\)

Hecke characteristic polynomials

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