Properties

Label 416.2.e
Level $416$
Weight $2$
Character orbit 416.e
Rep. character $\chi_{416}(337,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $3$
Sturm bound $112$
Trace bound $5$

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 64 16 48
Cusp forms 48 12 36
Eisenstein series 16 4 12

Trace form

\( 12 q - 12 q^{9} + O(q^{10}) \) \( 12 q - 12 q^{9} - 8 q^{17} - 8 q^{23} - 4 q^{25} - 16 q^{39} + 12 q^{49} + 56 q^{55} - 16 q^{79} + 4 q^{81} - 8 q^{87} + 16 q^{95} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
416.2.e.a 416.e 104.e $2$ $3.322$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-q^{5}-3iq^{7}+2q^{9}+2q^{11}+\cdots\)
416.2.e.b 416.e 104.e $2$ $3.322$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+q^{5}+3iq^{7}+2q^{9}-2q^{11}+\cdots\)
416.2.e.c 416.e 104.e $8$ $3.322$ 8.0.4521217600.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+\beta _{4}q^{5}+\beta _{3}q^{7}+(-3-\beta _{5}+\cdots)q^{9}+\cdots\)

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

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