Properties

Label 385.2.a
Level $385$
Weight $2$
Character orbit 385.a
Rep. character $\chi_{385}(1,\cdot)$
Character field $\Q$
Dimension $19$
Newform subspaces $8$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(385))\).

Total New Old
Modular forms 52 19 33
Cusp forms 45 19 26
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.

\(5\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(8\)
Minus space\(-\)\(11\)

Trace form

\( 19q - 3q^{2} - 4q^{3} + 21q^{4} - q^{5} + 20q^{6} - q^{7} + 9q^{8} + 7q^{9} + O(q^{10}) \) \( 19q - 3q^{2} - 4q^{3} + 21q^{4} - q^{5} + 20q^{6} - q^{7} + 9q^{8} + 7q^{9} + 5q^{10} - q^{11} - 4q^{12} - 14q^{13} - 3q^{14} + 4q^{15} + 13q^{16} - 10q^{17} - 15q^{18} + 4q^{19} - 7q^{20} - 4q^{21} + q^{22} + 16q^{23} + 12q^{24} + 19q^{25} - 10q^{26} + 8q^{27} - 7q^{28} - 22q^{29} - 12q^{30} - 8q^{31} - 15q^{32} - 4q^{33} - 14q^{34} - q^{35} - 15q^{36} - 22q^{37} - 36q^{38} + 8q^{39} + 33q^{40} - 2q^{41} - 4q^{42} - 20q^{43} - 7q^{44} + 3q^{45} - 24q^{47} - 52q^{48} + 19q^{49} - 3q^{50} - 8q^{51} - 26q^{52} + 10q^{53} + 32q^{54} - q^{55} - 15q^{56} - 16q^{57} + 14q^{58} + 36q^{59} - 4q^{60} - 14q^{61} + 56q^{62} + 3q^{63} + 93q^{64} - 6q^{65} - 12q^{66} - 4q^{67} - 46q^{68} + 16q^{69} + q^{70} + 8q^{71} - 19q^{72} - 34q^{73} + 46q^{74} - 4q^{75} - 4q^{76} + 7q^{77} + 8q^{78} + q^{80} - 21q^{81} + 26q^{82} + 4q^{83} + 28q^{84} + 14q^{85} - 52q^{86} + 8q^{87} - 3q^{88} - 2q^{89} - 7q^{90} + 2q^{91} + 40q^{92} + 48q^{93} + 56q^{94} + 4q^{95} - 76q^{96} - 18q^{97} - 3q^{98} - 13q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(385))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7 11
385.2.a.a \(1\) \(3.074\) \(\Q\) None \(-1\) \(-2\) \(1\) \(1\) \(-\) \(-\) \(+\) \(q-q^{2}-2q^{3}-q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
385.2.a.b \(1\) \(3.074\) \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) \(-\) \(+\) \(-\) \(q-q^{2}-q^{4}+q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
385.2.a.c \(2\) \(3.074\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(2\) \(+\) \(-\) \(+\) \(q+\beta q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(3+\beta )q^{6}+\cdots\)
385.2.a.d \(2\) \(3.074\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}+\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
385.2.a.e \(3\) \(3.074\) 3.3.148.1 None \(-3\) \(-4\) \(-3\) \(3\) \(+\) \(-\) \(-\) \(q+(-1-\beta _{2})q^{2}+(-1-\beta _{1})q^{3}+(2+\cdots)q^{4}+\cdots\)
385.2.a.f \(3\) \(3.074\) 3.3.148.1 None \(-3\) \(-2\) \(-3\) \(-3\) \(+\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{2}+(-1+\beta _{1})q^{3}+(2+\cdots)q^{4}+\cdots\)
385.2.a.g \(3\) \(3.074\) 3.3.148.1 None \(1\) \(0\) \(3\) \(3\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
385.2.a.h \(4\) \(3.074\) 4.4.11348.1 None \(2\) \(2\) \(4\) \(-4\) \(-\) \(+\) \(+\) \(q+(1+\beta _{2})q^{2}+(1+\beta _{3})q^{3}+(2+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(385))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(385)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)