Properties

Label 770.2.a
Level $770$
Weight $2$
Character orbit 770.a
Rep. character $\chi_{770}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $13$
Sturm bound $288$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(770))\).

Total New Old
Modular forms 152 21 131
Cusp forms 137 21 116
Eisenstein series 15 0 15

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

\(2\)\(5\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
Plus space\(+\)\(3\)
Minus space\(-\)\(18\)

Trace form

\( 21q + q^{2} + 4q^{3} + 21q^{4} + q^{5} + 4q^{6} + q^{7} + q^{8} + 33q^{9} + O(q^{10}) \) \( 21q + q^{2} + 4q^{3} + 21q^{4} + q^{5} + 4q^{6} + q^{7} + q^{8} + 33q^{9} + q^{10} + q^{11} + 4q^{12} + 14q^{13} + q^{14} + 4q^{15} + 21q^{16} + 2q^{17} + 13q^{18} + 4q^{19} + q^{20} + 4q^{21} - 3q^{22} + 4q^{24} + 21q^{25} + 6q^{26} + 40q^{27} + q^{28} + 14q^{29} + 4q^{30} + 24q^{31} + q^{32} + 4q^{33} + 18q^{34} + q^{35} + 33q^{36} + 14q^{37} - 4q^{38} + 24q^{39} + q^{40} - 22q^{41} - 4q^{42} + 12q^{43} + q^{44} - 19q^{45} - 8q^{46} + 8q^{47} + 4q^{48} + 21q^{49} + q^{50} - 8q^{51} + 14q^{52} - 34q^{53} - 8q^{54} + q^{55} + q^{56} - 2q^{58} - 20q^{59} + 4q^{60} + 30q^{61} - 48q^{62} - 19q^{63} + 21q^{64} - 2q^{65} + 4q^{66} - 4q^{67} + 2q^{68} - 16q^{69} - 3q^{70} - 24q^{71} + 13q^{72} + 10q^{73} + 14q^{74} + 4q^{75} + 4q^{76} - 7q^{77} - 40q^{78} - 8q^{79} + q^{80} - 3q^{81} + 10q^{82} - 28q^{83} + 4q^{84} + 18q^{85} - 4q^{86} - 8q^{87} - 3q^{88} + 2q^{89} + 13q^{90} - 2q^{91} - 32q^{94} - 20q^{95} + 4q^{96} + 10q^{97} + q^{98} + 13q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(770))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7 11
770.2.a.a \(1\) \(6.148\) \(\Q\) None \(-1\) \(-2\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+q^{7}+\cdots\)
770.2.a.b \(1\) \(6.148\) \(\Q\) None \(-1\) \(-2\) \(1\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots\)
770.2.a.c \(1\) \(6.148\) \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
770.2.a.d \(1\) \(6.148\) \(\Q\) None \(-1\) \(0\) \(1\) \(1\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
770.2.a.e \(1\) \(6.148\) \(\Q\) None \(-1\) \(2\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
770.2.a.f \(1\) \(6.148\) \(\Q\) None \(1\) \(-2\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
770.2.a.g \(1\) \(6.148\) \(\Q\) None \(1\) \(-2\) \(1\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\cdots\)
770.2.a.h \(2\) \(6.148\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(2\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
770.2.a.i \(2\) \(6.148\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}-q^{7}+\cdots\)
770.2.a.j \(2\) \(6.148\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(-2\) \(2\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(1+\beta )q^{6}+\cdots\)
770.2.a.k \(2\) \(6.148\) \(\Q(\sqrt{33}) \) None \(2\) \(4\) \(2\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+q^{7}+\cdots\)
770.2.a.l \(3\) \(6.148\) 3.3.892.1 None \(-3\) \(0\) \(-3\) \(-3\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
770.2.a.m \(3\) \(6.148\) 3.3.316.1 None \(3\) \(2\) \(3\) \(-3\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+q^{5}+(1+\beta _{2})q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(770))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(770)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(385))\)\(^{\oplus 2}\)