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Properties

Label 407.2.a
Level $407$
Weight $2$
Character orbit 407.a
Rep. character $\chi_{407}(1,\cdot)$
Character field $\Q$
Dimension $31$
Newform subspaces $4$
Sturm bound $76$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(76\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(407))\).

Total New Old
Modular forms 40 31 9
Cusp forms 37 31 6
Eisenstein series 3 0 3

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

\(11\)\(37\)FrickeDim
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(11\)
\(-\)\(+\)$-$\(12\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(23\)

Trace form

\( 31 q + 3 q^{2} + 4 q^{3} + 29 q^{4} + 2 q^{5} - 16 q^{6} + 8 q^{7} - 9 q^{8} + 35 q^{9} + O(q^{10}) \) \( 31 q + 3 q^{2} + 4 q^{3} + 29 q^{4} + 2 q^{5} - 16 q^{6} + 8 q^{7} - 9 q^{8} + 35 q^{9} - 2 q^{10} + q^{11} + 16 q^{12} + 14 q^{13} - 12 q^{14} - 12 q^{15} + 29 q^{16} + 6 q^{17} + 11 q^{18} + 4 q^{19} + 10 q^{20} - 12 q^{21} + q^{22} - 16 q^{23} - 12 q^{24} + 53 q^{25} - 2 q^{26} + 4 q^{27} + 8 q^{28} - 2 q^{29} - 8 q^{30} - 45 q^{32} + 4 q^{33} - 10 q^{34} + 4 q^{35} - 7 q^{36} - q^{37} + 16 q^{39} + 2 q^{40} + 26 q^{41} + 28 q^{42} + 16 q^{43} - q^{44} + 10 q^{45} - 32 q^{46} - 8 q^{47} + 59 q^{49} - 63 q^{50} + 4 q^{51} + 26 q^{52} + 18 q^{53} - 32 q^{54} - 2 q^{55} - 52 q^{56} - 16 q^{57} + 18 q^{58} - 16 q^{59} - 44 q^{60} - 6 q^{61} + 44 q^{62} - 4 q^{63} + 13 q^{64} - 28 q^{65} - 16 q^{66} + 6 q^{68} - 20 q^{69} - 48 q^{70} + 12 q^{71} - 13 q^{72} + 14 q^{73} + 3 q^{74} - 4 q^{75} + 8 q^{76} + 4 q^{77} - 56 q^{78} + 24 q^{79} - 26 q^{80} + 7 q^{81} - 34 q^{82} + 24 q^{83} - 68 q^{84} - 8 q^{85} - 40 q^{86} + 32 q^{87} + 9 q^{88} - 42 q^{89} - 106 q^{90} - 8 q^{91} - 24 q^{92} - 36 q^{93} - 20 q^{94} - 36 q^{95} - 100 q^{96} + 26 q^{97} - 85 q^{98} + 13 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(407))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 11 37
407.2.a.a 407.a 1.a $4$ $3.250$ 4.4.1957.1 None \(-1\) \(0\) \(1\) \(-7\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(-\beta _{1}-2\beta _{2}-\beta _{3})q^{4}+\cdots\)
407.2.a.b 407.a 1.a $4$ $3.250$ 4.4.1957.1 None \(1\) \(-4\) \(-5\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{2}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
407.2.a.c 407.a 1.a $11$ $3.250$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(2\) \(0\) \(1\) \(9\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{2})q^{4}+\beta _{9}q^{5}+\cdots\)
407.2.a.d 407.a 1.a $12$ $3.250$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(1\) \(8\) \(5\) \(7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{10})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(407))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(407)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)