Properties

Label 735.2.i
Level $735$
Weight $2$
Character orbit 735.i
Rep. character $\chi_{735}(226,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $52$
Newform subspaces $14$
Sturm bound $224$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(735, [\chi])\).

Total New Old
Modular forms 256 52 204
Cusp forms 192 52 140
Eisenstein series 64 0 64

Trace form

\( 52q - 8q^{2} - 2q^{3} - 32q^{4} + 48q^{8} - 26q^{9} + O(q^{10}) \) \( 52q - 8q^{2} - 2q^{3} - 32q^{4} + 48q^{8} - 26q^{9} - 4q^{10} - 12q^{11} - 4q^{12} + 4q^{13} - 36q^{16} + 8q^{17} - 8q^{18} + 6q^{19} - 16q^{20} - 8q^{22} - 16q^{23} + 12q^{24} - 26q^{25} + 8q^{26} + 4q^{27} + 40q^{29} + 6q^{31} - 36q^{32} - 8q^{33} + 16q^{34} + 64q^{36} + 2q^{37} - 32q^{38} + 22q^{39} - 12q^{40} - 8q^{41} + 20q^{43} - 28q^{44} - 8q^{46} + 8q^{47} + 16q^{48} + 16q^{50} - 12q^{51} + 24q^{55} - 68q^{57} + 4q^{58} - 4q^{59} + 4q^{61} - 24q^{62} + 56q^{64} + 12q^{65} + 12q^{66} - 2q^{67} - 12q^{68} - 24q^{69} - 24q^{72} + 10q^{73} - 12q^{74} - 2q^{75} - 40q^{76} + 8q^{78} + 6q^{79} + 16q^{80} - 26q^{81} + 28q^{82} - 8q^{83} - 8q^{85} + 132q^{86} - 24q^{87} + 136q^{88} - 16q^{89} + 8q^{90} - 200q^{92} + 34q^{93} + 24q^{94} - 8q^{95} + 24q^{96} + 32q^{97} + 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(735, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
735.2.i.a \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(-1\) \(-1\) \(-1\) \(0\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.b \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(1\) \(0\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.c \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(0\) \(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots\)
735.2.i.d \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(1\) \(0\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.e \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(-1\) \(0\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.f \(2\) \(5.869\) \(\Q(\sqrt{-3}) \) None \(2\) \(1\) \(1\) \(0\) \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
735.2.i.g \(4\) \(5.869\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(-2\) \(-2\) \(0\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.h \(4\) \(5.869\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(2\) \(2\) \(0\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.i \(4\) \(5.869\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(-2\) \(-2\) \(0\) \(q-\beta _{2}q^{2}+\beta _{1}q^{3}+3\beta _{1}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
735.2.i.j \(4\) \(5.869\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(-2\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+\beta _{3}q^{6}+\cdots\)
735.2.i.k \(4\) \(5.869\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(2\) \(2\) \(0\) \(q-\beta _{2}q^{2}-\beta _{1}q^{3}+3\beta _{1}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
735.2.i.l \(4\) \(5.869\) \(\Q(\zeta_{12})\) None \(2\) \(-2\) \(2\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-1+\zeta_{12})q^{3}+\cdots\)
735.2.i.m \(8\) \(5.869\) 8.0.\(\cdots\).10 None \(-4\) \(-4\) \(4\) \(0\) \(q+(-1-\beta _{1}-\beta _{5})q^{2}+\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.n \(8\) \(5.869\) 8.0.\(\cdots\).10 None \(-4\) \(4\) \(-4\) \(0\) \(q+(-1-\beta _{1}-\beta _{5})q^{2}-\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(735, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(735, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)