Properties

Label 414.2.a
Level $414$
Weight $2$
Character orbit 414.a
Rep. character $\chi_{414}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $7$
Sturm bound $144$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(414))\).

Total New Old
Modular forms 80 10 70
Cusp forms 65 10 55
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(23\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(8\)

Trace form

\( 10 q + 10 q^{4} - 2 q^{5} + 4 q^{7} + O(q^{10}) \) \( 10 q + 10 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{10} + 10 q^{11} - 4 q^{13} - 4 q^{14} + 10 q^{16} + 8 q^{17} + 2 q^{19} - 2 q^{20} - 6 q^{22} + 18 q^{25} + 4 q^{28} + 16 q^{31} - 12 q^{34} + 32 q^{35} + 2 q^{37} + 10 q^{38} + 2 q^{40} - 16 q^{41} + 6 q^{43} + 10 q^{44} - 2 q^{46} - 8 q^{47} + 10 q^{49} + 4 q^{50} - 4 q^{52} - 18 q^{53} - 16 q^{55} - 4 q^{56} - 20 q^{58} - 8 q^{59} + 6 q^{61} + 8 q^{62} + 10 q^{64} - 12 q^{65} - 22 q^{67} + 8 q^{68} - 16 q^{73} - 34 q^{74} + 2 q^{76} - 24 q^{77} - 28 q^{79} - 2 q^{80} - 28 q^{82} - 34 q^{83} - 60 q^{85} + 6 q^{86} - 6 q^{88} - 12 q^{89} - 24 q^{91} - 8 q^{95} + 8 q^{97} - 16 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(414))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 23
414.2.a.a $1$ $3.306$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $-$ $-$ \(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-2q^{13}+\cdots\)
414.2.a.b $1$ $3.306$ \(\Q\) None \(1\) \(0\) \(-4\) \(-4\) $-$ $-$ $+$ \(q+q^{2}+q^{4}-4q^{5}-4q^{7}+q^{8}-4q^{10}+\cdots\)
414.2.a.c $1$ $3.306$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $-$ $-$ $-$ \(q+q^{2}+q^{4}+2q^{7}+q^{8}+2q^{13}+2q^{14}+\cdots\)
414.2.a.d $1$ $3.306$ \(\Q\) None \(1\) \(0\) \(2\) \(-2\) $-$ $-$ $-$ \(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
414.2.a.e $2$ $3.306$ \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(-2\) \(4\) $+$ $+$ $-$ \(q-q^{2}+q^{4}+(-1+\beta )q^{5}+2q^{7}-q^{8}+\cdots\)
414.2.a.f $2$ $3.306$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(2\) \(0\) $+$ $-$ $+$ \(q-q^{2}+q^{4}+(1+\beta )q^{5}+2\beta q^{7}-q^{8}+\cdots\)
414.2.a.g $2$ $3.306$ \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(2\) \(4\) $-$ $+$ $+$ \(q+q^{2}+q^{4}+(1+\beta )q^{5}+2q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(414))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(414)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 2}\)