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Properties

Label 353.2.a
Level $353$
Weight $2$
Character orbit 353.a
Rep. character $\chi_{353}(1,\cdot)$
Character field $\Q$
Dimension $29$
Newform subspaces $4$
Sturm bound $59$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 353.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(59\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(353))\).

Total New Old
Modular forms 30 30 0
Cusp forms 29 29 0
Eisenstein series 1 1 0

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

\(353\)Dim
\(+\)\(11\)
\(-\)\(18\)

Trace form

\( 29 q - q^{2} - 2 q^{3} + 27 q^{4} - 4 q^{5} - 6 q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} + O(q^{10}) \) \( 29 q - q^{2} - 2 q^{3} + 27 q^{4} - 4 q^{5} - 6 q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} - 6 q^{10} - 4 q^{11} - 22 q^{12} + 2 q^{13} + 8 q^{14} - 2 q^{15} + 27 q^{16} - 6 q^{17} - 7 q^{18} + 2 q^{20} + 2 q^{21} + 6 q^{22} - 10 q^{23} + 19 q^{25} + 12 q^{26} - 8 q^{27} + 18 q^{28} + 6 q^{29} + 2 q^{30} + 2 q^{31} + 3 q^{32} - 18 q^{33} - 26 q^{34} - 4 q^{35} + 27 q^{36} + 6 q^{37} + 12 q^{38} - 30 q^{39} - 36 q^{40} - 10 q^{41} - 20 q^{42} - 14 q^{43} - 10 q^{44} - 44 q^{45} - 2 q^{46} - 10 q^{47} - 60 q^{48} + 59 q^{49} - 3 q^{50} - 22 q^{51} - 2 q^{52} - 8 q^{53} - 40 q^{54} + 4 q^{55} + 58 q^{56} + 28 q^{57} - 16 q^{58} - 2 q^{59} + 16 q^{61} + 14 q^{62} + 24 q^{63} + 5 q^{64} - 8 q^{65} + 24 q^{66} + 24 q^{67} - 20 q^{68} - 10 q^{69} - 24 q^{70} + 10 q^{71} + 21 q^{72} + 10 q^{73} - 2 q^{74} + 46 q^{75} - 12 q^{76} - 32 q^{77} + 6 q^{78} + 12 q^{79} - 12 q^{80} + 21 q^{81} - 18 q^{82} - 8 q^{83} - 44 q^{84} - 52 q^{85} - 16 q^{86} + 24 q^{87} - 22 q^{88} - 2 q^{89} - 40 q^{90} + 46 q^{91} + 34 q^{92} - 20 q^{93} - 16 q^{94} - 12 q^{95} - 18 q^{96} - 8 q^{97} - 81 q^{98} - 38 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(353))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 353
353.2.a.a 353.a 1.a $1$ $2.819$ \(\Q\) None \(-1\) \(2\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}+2q^{5}-2q^{6}-2q^{7}+\cdots\)
353.2.a.b 353.a 1.a $3$ $2.819$ 3.3.229.1 None \(1\) \(3\) \(2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+(\beta _{1}-\beta _{2})q^{2}+(1+\beta _{1})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
353.2.a.c 353.a 1.a $11$ $2.819$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-5\) \(-5\) \(-4\) \(-25\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{2}-\beta _{5}-\beta _{8}+\cdots)q^{3}+\cdots\)
353.2.a.d 353.a 1.a $14$ $2.819$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(4\) \(-2\) \(-4\) \(29\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+\beta _{12}q^{5}+\cdots\)