Properties

Label 338.2.a
Level $338$
Weight $2$
Character orbit 338.a
Rep. character $\chi_{338}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $8$
Sturm bound $91$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(91\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(338))\).

Total New Old
Modular forms 59 12 47
Cusp forms 32 12 20
Eisenstein series 27 0 27

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

\(2\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(9\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
338.2.a.a 338.a 1.a $1$ $2.699$ \(\Q\) None 26.2.a.b \(-1\) \(-3\) \(1\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
338.2.a.b 338.a 1.a $1$ $2.699$ \(\Q\) None 26.2.b.a \(-1\) \(-1\) \(3\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
338.2.a.c 338.a 1.a $1$ $2.699$ \(\Q\) None 26.2.c.a \(-1\) \(0\) \(1\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
338.2.a.d 338.a 1.a $1$ $2.699$ \(\Q\) None 26.2.b.a \(1\) \(-1\) \(-3\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
338.2.a.e 338.a 1.a $1$ $2.699$ \(\Q\) None 26.2.c.a \(1\) \(0\) \(-1\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots\)
338.2.a.f 338.a 1.a $1$ $2.699$ \(\Q\) None 26.2.a.a \(1\) \(1\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
338.2.a.g 338.a 1.a $3$ $2.699$ \(\Q(\zeta_{14})^+\) None 338.2.a.g \(-3\) \(3\) \(-2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots\)
338.2.a.h 338.a 1.a $3$ $2.699$ \(\Q(\zeta_{14})^+\) None 338.2.a.g \(3\) \(3\) \(2\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(338)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)