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Properties

Label 340.2.a
Level $340$
Weight $2$
Character orbit 340.a
Rep. character $\chi_{340}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $2$
Sturm bound $108$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(108\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(340))\).

Total New Old
Modular forms 60 4 56
Cusp forms 49 4 45
Eisenstein series 11 0 11

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

\(2\)\(5\)\(17\)FrickeDim
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q + 2 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{5} - 4 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{17} + 16 q^{21} + 8 q^{23} + 4 q^{25} - 12 q^{27} - 8 q^{29} + 16 q^{31} - 4 q^{33} + 4 q^{35} - 8 q^{37} - 20 q^{39} + 12 q^{41} - 4 q^{43} + 10 q^{45} - 12 q^{47} + 4 q^{49} + 8 q^{53} - 32 q^{57} - 16 q^{59} + 4 q^{61} + 8 q^{65} - 28 q^{67} - 8 q^{69} + 20 q^{71} - 4 q^{73} - 12 q^{77} - 8 q^{79} + 20 q^{81} - 12 q^{83} + 2 q^{85} - 8 q^{87} + 8 q^{89} + 4 q^{91} - 28 q^{93} - 16 q^{97} - 32 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(340))\) 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 5 17
340.2.a.a 340.a 1.a $1$ $2.715$ \(\Q\) None \(0\) \(0\) \(-1\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{7}-3q^{9}+2q^{11}-6q^{13}+\cdots\)
340.2.a.b 340.a 1.a $3$ $2.715$ 3.3.404.1 None \(0\) \(0\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+q^{5}-\beta _{2}q^{7}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(340))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(340)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 2}\)