Properties

Label 403.2.a
Level $403$
Weight $2$
Character orbit 403.a
Rep. character $\chi_{403}(1,\cdot)$
Character field $\Q$
Dimension $31$
Newform subspaces $5$
Sturm bound $74$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(74\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(403))\).

Total New Old
Modular forms 38 31 7
Cusp forms 35 31 4
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.

\(13\)\(31\)FrickeDim
\(+\)\(+\)$+$\(8\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(10\)
\(-\)\(-\)$+$\(6\)
Plus space\(+\)\(14\)
Minus space\(-\)\(17\)

Trace form

\( 31 q - 3 q^{2} + 33 q^{4} - 2 q^{5} - 9 q^{8} + 23 q^{9} + O(q^{10}) \) \( 31 q - 3 q^{2} + 33 q^{4} - 2 q^{5} - 9 q^{8} + 23 q^{9} + 4 q^{10} - 4 q^{11} - 12 q^{12} + q^{13} - 14 q^{14} + 12 q^{15} + 25 q^{16} - 14 q^{17} - 39 q^{18} - 4 q^{19} - 20 q^{20} - 12 q^{21} - 8 q^{22} - 4 q^{23} - 24 q^{24} + 57 q^{25} + 3 q^{26} + 12 q^{27} - 26 q^{28} - 14 q^{29} - 4 q^{30} - 5 q^{31} - 11 q^{32} - 6 q^{34} + 4 q^{35} + 29 q^{36} + 2 q^{37} + 2 q^{38} - 4 q^{39} + 22 q^{40} - 14 q^{41} + 8 q^{42} - 16 q^{43} + 8 q^{44} - 22 q^{45} - 4 q^{47} - 40 q^{48} + 11 q^{49} + 13 q^{50} - 8 q^{51} - q^{52} + 26 q^{53} + 28 q^{54} - 4 q^{55} - 12 q^{56} - 40 q^{57} + 18 q^{58} - 24 q^{59} + 92 q^{60} + 6 q^{61} + 3 q^{62} - 44 q^{63} + 3 q^{64} + 6 q^{65} + 4 q^{66} - 36 q^{67} - 2 q^{68} + 12 q^{69} + 2 q^{70} - 12 q^{71} - 21 q^{72} + 6 q^{73} + 38 q^{74} - 56 q^{75} - 10 q^{76} - 52 q^{77} - 12 q^{78} - 12 q^{79} + 4 q^{80} - 9 q^{81} + 12 q^{82} - 8 q^{83} - 4 q^{84} + 32 q^{85} - 20 q^{86} - 4 q^{88} - 2 q^{89} + 36 q^{90} - 76 q^{92} + 24 q^{94} + 12 q^{95} - 4 q^{96} - 22 q^{97} + 99 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 13 31
403.2.a.a 403.a 1.a $2$ $3.218$ \(\Q(\sqrt{5}) \) None \(3\) \(-4\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-2q^{3}+3\beta q^{4}+(-1+2\beta )q^{5}+\cdots\)
403.2.a.b 403.a 1.a $6$ $3.218$ 6.6.5748973.1 None \(-2\) \(-5\) \(-9\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{2}+(-1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
403.2.a.c 403.a 1.a $7$ $3.218$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(2\) \(5\) \(11\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+(1-\beta _{1})q^{3}+(\beta _{1}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
403.2.a.d 403.a 1.a $8$ $3.218$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-5\) \(-3\) \(-15\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{6}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
403.2.a.e 403.a 1.a $8$ $3.218$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(7\) \(11\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(403))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(403)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 2}\)