Properties

Label 416.1.h
Level $416$
Weight $1$
Character orbit 416.h
Rep. character $\chi_{416}(207,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $56$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 416.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 104 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(416, [\chi])\).

Total New Old
Modular forms 16 4 12
Cusp forms 8 2 6
Eisenstein series 8 2 6

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 2 0 0 0

Trace form

\( 2 q + 2 q^{3} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{17} - 2 q^{27} - 2 q^{35} + 2 q^{43} - 2 q^{51} - 2 q^{65} - 2 q^{81} + 2 q^{91} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(416, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
416.1.h.a 416.h 104.h $1$ $0.208$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-26}) \) None \(0\) \(1\) \(-1\) \(1\) \(q+q^{3}-q^{5}+q^{7}+q^{13}-q^{15}-q^{17}+\cdots\)
416.1.h.b 416.h 104.h $1$ $0.208$ \(\Q\) $D_{3}$ \(\Q(\sqrt{-26}) \) None \(0\) \(1\) \(1\) \(-1\) \(q+q^{3}+q^{5}-q^{7}-q^{13}+q^{15}-q^{17}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(416, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(416, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 3}\)