Properties

Label 770.2.n
Level $770$
Weight $2$
Character orbit 770.n
Rep. character $\chi_{770}(71,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $96$
Newform subspaces $11$
Sturm bound $288$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.n (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 11 \)
Sturm bound: \(288\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(770, [\chi])\).

Total New Old
Modular forms 608 96 512
Cusp forms 544 96 448
Eisenstein series 64 0 64

Trace form

\( 96q - 4q^{2} - 8q^{3} - 24q^{4} + 12q^{6} - 4q^{8} - 20q^{9} + O(q^{10}) \) \( 96q - 4q^{2} - 8q^{3} - 24q^{4} + 12q^{6} - 4q^{8} - 20q^{9} + 24q^{11} - 8q^{12} + 24q^{13} + 12q^{15} - 24q^{16} - 24q^{17} + 36q^{19} - 8q^{21} - 12q^{22} + 16q^{23} + 12q^{24} - 24q^{25} - 8q^{26} - 20q^{27} + 64q^{29} - 8q^{30} + 8q^{31} + 16q^{32} - 28q^{33} - 40q^{34} - 4q^{35} - 20q^{36} + 48q^{37} - 16q^{38} + 60q^{39} - 24q^{41} - 24q^{43} + 4q^{44} + 24q^{46} - 32q^{47} - 8q^{48} - 24q^{49} - 4q^{50} - 24q^{51} - 16q^{52} - 16q^{53} - 32q^{54} + 32q^{55} - 60q^{57} - 40q^{58} - 4q^{59} - 8q^{60} + 16q^{61} - 24q^{64} + 56q^{65} - 8q^{66} - 40q^{67} - 24q^{68} - 16q^{71} - 20q^{72} - 8q^{73} - 32q^{74} + 12q^{75} - 24q^{76} + 64q^{78} - 16q^{79} - 56q^{81} + 4q^{82} + 68q^{83} + 12q^{84} - 32q^{85} - 28q^{86} - 112q^{87} - 12q^{88} - 72q^{89} - 32q^{90} + 36q^{91} - 24q^{92} + 72q^{93} - 32q^{94} - 8q^{96} + 148q^{97} + 16q^{98} - 144q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(770, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
770.2.n.a \(4\) \(6.148\) \(\Q(\zeta_{10})\) None \(-1\) \(-3\) \(-1\) \(-1\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
770.2.n.b \(4\) \(6.148\) \(\Q(\zeta_{10})\) None \(1\) \(-4\) \(1\) \(1\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\cdots)q^{3}+\cdots\)
770.2.n.c \(4\) \(6.148\) \(\Q(\zeta_{10})\) None \(1\) \(-2\) \(1\) \(-1\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\cdots)q^{3}+\cdots\)
770.2.n.d \(4\) \(6.148\) \(\Q(\zeta_{10})\) None \(1\) \(2\) \(1\) \(-1\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(\zeta_{10}+\cdots)q^{3}+\cdots\)
770.2.n.e \(8\) \(6.148\) 8.0.682515625.5 None \(2\) \(-1\) \(2\) \(2\) \(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{4}-\beta _{6})q^{3}+\beta _{3}q^{4}+\cdots\)
770.2.n.f \(8\) \(6.148\) 8.0.484000000.9 None \(2\) \(2\) \(-2\) \(2\) \(q+\beta _{5}q^{2}+(-\beta _{3}-\beta _{4}-\beta _{7})q^{3}+(-1+\cdots)q^{4}+\cdots\)
770.2.n.g \(12\) \(6.148\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(-2\) \(-3\) \(3\) \(q+\beta _{8}q^{2}+(-\beta _{4}-\beta _{5})q^{3}+\beta _{6}q^{4}+\cdots\)
770.2.n.h \(12\) \(6.148\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(0\) \(3\) \(-3\) \(q+(-1-\beta _{2}-\beta _{3}-\beta _{10})q^{2}+(-\beta _{5}+\cdots)q^{3}+\cdots\)
770.2.n.i \(12\) \(6.148\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(2\) \(-3\) \(-3\) \(q-\beta _{8}q^{2}+(1-\beta _{2}+\beta _{6}+\beta _{7}-\beta _{8}+\cdots)q^{3}+\cdots\)
770.2.n.j \(12\) \(6.148\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(3\) \(3\) \(-3\) \(-3\) \(q+\beta _{6}q^{2}+\beta _{4}q^{3}-\beta _{8}q^{4}+\beta _{7}q^{5}+\cdots\)
770.2.n.k \(16\) \(6.148\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-4\) \(-5\) \(4\) \(4\) \(q+(-1-\beta _{3}+\beta _{8}+\beta _{9})q^{2}-\beta _{6}q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(770, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(770, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(22, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(154, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(385, [\chi])\)\(^{\oplus 2}\)