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Properties

Label 416.2.u
Level $416$
Weight $2$
Character orbit 416.u
Rep. character $\chi_{416}(47,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $24$
Newform subspaces $2$
Sturm bound $112$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.u (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 104 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(112\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(416, [\chi])\).

Total New Old
Modular forms 128 32 96
Cusp forms 96 24 72
Eisenstein series 32 8 24

Trace form

\( 24 q + 8 q^{3} + 8 q^{9} + 4 q^{11} + 4 q^{19} + 32 q^{27} - 16 q^{33} + 8 q^{35} - 52 q^{59} - 8 q^{65} + 4 q^{67} - 40 q^{81} - 36 q^{83} - 24 q^{89} + 20 q^{91} - 16 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(416, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
416.2.u.a 416.u 104.m $4$ $3.322$ \(\Q(i, \sqrt{26})\) None 104.2.m.a \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{3}+\beta _{1}q^{5}-\beta _{3}q^{7}-2q^{9}+(-1+\cdots)q^{11}+\cdots\)
416.2.u.b 416.u 104.m $20$ $3.322$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 104.2.m.b \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{3}+\beta _{15}q^{5}+(\beta _{4}-\beta _{9})q^{7}+(1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(416, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(416, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 3}\)