Properties

Label 400.1.x
Level $400$
Weight $1$
Character orbit 400.x
Rep. character $\chi_{400}(79,\cdot)$
Character field $\Q(\zeta_{10})$
Dimension $4$
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.x (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 100 \)
Character field: \(\Q(\zeta_{10})\)
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 28 4 24
Cusp forms 4 4 0
Eisenstein series 24 0 24

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

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

Trace form

\( 4 q + q^{5} + q^{9} + O(q^{10}) \) \( 4 q + q^{5} + q^{9} - q^{25} - 2 q^{29} - 5 q^{37} + 2 q^{41} - q^{45} - 4 q^{49} - 5 q^{53} + 2 q^{61} - 5 q^{65} - q^{81} + 5 q^{85} + 3 q^{89} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field Image CM RM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.1.x.a 400.x 100.h $4$ $0.200$ \(\Q(\zeta_{10})\) $D_{10}$ \(\Q(\sqrt{-1}) \) None 400.1.x.a \(0\) \(0\) \(1\) \(0\) \(q+\zeta_{10}^{3}q^{5}-\zeta_{10}^{4}q^{9}+(\zeta_{10}+\zeta_{10}^{2}+\cdots)q^{13}+\cdots\)