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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 13
338.2.a.a \(1\) \(2.699\) \(\Q\) None \(-1\) \(-3\) \(1\) \(-1\) \(+\) \(+\) \(q-q^{2}-3q^{3}+q^{4}+q^{5}+3q^{6}-q^{7}+\cdots\)
338.2.a.b \(1\) \(2.699\) \(\Q\) None \(-1\) \(-1\) \(3\) \(3\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
338.2.a.c \(1\) \(2.699\) \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-3q^{9}+\cdots\)
338.2.a.d \(1\) \(2.699\) \(\Q\) None \(1\) \(-1\) \(-3\) \(-3\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
338.2.a.e \(1\) \(2.699\) \(\Q\) None \(1\) \(0\) \(-1\) \(4\) \(-\) \(+\) \(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-3q^{9}+\cdots\)
338.2.a.f \(1\) \(2.699\) \(\Q\) None \(1\) \(1\) \(3\) \(1\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
338.2.a.g \(3\) \(2.699\) \(\Q(\zeta_{14})^+\) None \(-3\) \(3\) \(-2\) \(4\) \(+\) \(-\) \(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots\)
338.2.a.h \(3\) \(2.699\) \(\Q(\zeta_{14})^+\) None \(3\) \(3\) \(2\) \(-4\) \(-\) \(+\) \(q+q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+2\beta _{1}q^{5}+\cdots\)

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

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