Properties

Label 416.1.t
Level $416$
Weight $1$
Character orbit 416.t
Rep. character $\chi_{416}(161,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $2$
Newform subspaces $1$
Sturm bound $56$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 416.t (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(56\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 18 2 16
Cusp forms 2 2 0
Eisenstein series 16 0 16

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

\( 2 q + 2 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{9} + 2 q^{37} - 2 q^{41} - 2 q^{45} - 4 q^{53} - 4 q^{61} - 2 q^{65} + 2 q^{73} + 2 q^{81} + 4 q^{85} - 2 q^{89} + 2 q^{97} + O(q^{100}) \)

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.t.a 416.t 13.d $2$ $0.208$ \(\Q(\sqrt{-1}) \) $D_{4}$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+(1+i)q^{5}-q^{9}+iq^{13}-iq^{17}+\cdots\)