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 $3$

Related objects

Downloads

Learn more

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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\) 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 37
370.2.a.a 370.a 1.a $1$ $2.954$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
370.2.a.b 370.a 1.a $1$ $2.954$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}-3q^{9}+q^{10}+\cdots\)
370.2.a.c 370.a 1.a $1$ $2.954$ \(\Q\) None \(-1\) \(2\) \(1\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\cdots\)
370.2.a.d 370.a 1.a $1$ $2.954$ \(\Q\) None \(1\) \(-2\) \(1\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+2q^{7}+\cdots\)
370.2.a.e 370.a 1.a $2$ $2.954$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(-6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
370.2.a.f 370.a 1.a $2$ $2.954$ \(\Q(\sqrt{33}) \) None \(2\) \(4\) \(2\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+(-1+\cdots)q^{7}+\cdots\)
370.2.a.g 370.a 1.a $3$ $2.954$ 3.3.892.1 None \(3\) \(0\) \(-3\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{2}q^{3}+q^{4}-q^{5}-\beta _{2}q^{6}+\cdots\)

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

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