Properties

Label 370.2.a
Level $370$
Weight $2$
Character orbit 370.a
Rep. character $\chi_{370}(1,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $7$
Sturm bound $114$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(114\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(370))\).

Total New Old
Modular forms 60 11 49
Cusp forms 53 11 42
Eisenstein series 7 0 7

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

\(2\)\(5\)\(37\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(3\)
Minus space\(-\)\(8\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 37
370.2.a.a \(1\) \(2.954\) \(\Q\) None \(-1\) \(-2\) \(-1\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
370.2.a.b \(1\) \(2.954\) \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{5}-q^{8}-3q^{9}+q^{10}+\cdots\)
370.2.a.c \(1\) \(2.954\) \(\Q\) None \(-1\) \(2\) \(1\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\cdots\)
370.2.a.d \(1\) \(2.954\) \(\Q\) None \(1\) \(-2\) \(1\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+2q^{7}+\cdots\)
370.2.a.e \(2\) \(2.954\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(-6\) \(+\) \(-\) \(-\) \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
370.2.a.f \(2\) \(2.954\) \(\Q(\sqrt{33}) \) None \(2\) \(4\) \(2\) \(-3\) \(-\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+(-1+\cdots)q^{7}+\cdots\)
370.2.a.g \(3\) \(2.954\) 3.3.892.1 None \(3\) \(0\) \(-3\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}-\beta _{2}q^{3}+q^{4}-q^{5}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(370))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(370)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 2}\)