Properties

Label 342.2.a
Level $342$
Weight $2$
Character orbit 342.a
Rep. character $\chi_{342}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $7$
Sturm bound $120$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(120\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(342))\).

Total New Old
Modular forms 68 7 61
Cusp forms 53 7 46
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.

\(2\)\(3\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7 q - q^{2} + 7 q^{4} + 2 q^{5} + 2 q^{7} - q^{8} + O(q^{10}) \) \( 7 q - q^{2} + 7 q^{4} + 2 q^{5} + 2 q^{7} - q^{8} + 2 q^{10} + 4 q^{11} - 6 q^{13} + 4 q^{14} + 7 q^{16} - 4 q^{17} - q^{19} + 2 q^{20} + 4 q^{22} + 10 q^{23} - 7 q^{25} + 8 q^{26} + 2 q^{28} - 2 q^{29} - 4 q^{31} - q^{32} + 2 q^{34} + 12 q^{35} - 18 q^{37} + 3 q^{38} + 2 q^{40} - 18 q^{41} + 8 q^{43} + 4 q^{44} + 4 q^{46} - 20 q^{47} - 7 q^{49} - 15 q^{50} - 6 q^{52} + 6 q^{53} - 8 q^{55} + 4 q^{56} - 8 q^{58} - 28 q^{59} - 10 q^{61} + 4 q^{62} + 7 q^{64} - 8 q^{65} + 4 q^{67} - 4 q^{68} - 12 q^{70} + 12 q^{71} + 20 q^{73} - 10 q^{74} - q^{76} - 28 q^{77} + 4 q^{79} + 2 q^{80} - 6 q^{82} + 28 q^{83} - 24 q^{85} - 8 q^{86} + 4 q^{88} + 26 q^{89} + 8 q^{91} + 10 q^{92} - 8 q^{94} - 2 q^{95} - 2 q^{97} - q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(342))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 19
342.2.a.a 342.a 1.a $1$ $2.731$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-2q^{11}+\cdots\)
342.2.a.b 342.a 1.a $1$ $2.731$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}+4q^{11}+\cdots\)
342.2.a.c 342.a 1.a $1$ $2.731$ \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{13}+4q^{14}+\cdots\)
342.2.a.d 342.a 1.a $1$ $2.731$ \(\Q\) None \(-1\) \(0\) \(4\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+4q^{5}+3q^{7}-q^{8}-4q^{10}+\cdots\)
342.2.a.e 342.a 1.a $1$ $2.731$ \(\Q\) None \(1\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{7}+q^{8}+6q^{11}+5q^{13}+\cdots\)
342.2.a.f 342.a 1.a $1$ $2.731$ \(\Q\) None \(1\) \(0\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+4q^{7}+q^{8}-4q^{11}+4q^{14}+\cdots\)
342.2.a.g 342.a 1.a $1$ $2.731$ \(\Q\) None \(1\) \(0\) \(2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}+q^{8}+2q^{10}+2q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(342))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(342)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)