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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
735.2.i.a 735.i 7.c $2$ $5.869$ \(\Q(\sqrt{-3}) \) None 105.2.a.a \(-1\) \(-1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.b 735.i 7.c $2$ $5.869$ \(\Q(\sqrt{-3}) \) None 105.2.a.a \(-1\) \(1\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.c 735.i 7.c $2$ $5.869$ \(\Q(\sqrt{-3}) \) None 105.2.i.a \(0\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots\)
735.2.i.d 735.i 7.c $2$ $5.869$ \(\Q(\sqrt{-3}) \) None 15.2.a.a \(1\) \(-1\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.e 735.i 7.c $2$ $5.869$ \(\Q(\sqrt{-3}) \) None 15.2.a.a \(1\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
735.2.i.f 735.i 7.c $2$ $5.869$ \(\Q(\sqrt{-3}) \) None 105.2.i.b \(2\) \(1\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
735.2.i.g 735.i 7.c $4$ $5.869$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 735.2.a.l \(-2\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.h 735.i 7.c $4$ $5.869$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 735.2.a.l \(-2\) \(2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.i 735.i 7.c $4$ $5.869$ \(\Q(\sqrt{-3}, \sqrt{5})\) None 105.2.a.b \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{2}+\beta _{1}q^{3}+3\beta _{1}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
735.2.i.j 735.i 7.c $4$ $5.869$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 105.2.i.c \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+\beta _{3}q^{6}+\cdots\)
735.2.i.k 735.i 7.c $4$ $5.869$ \(\Q(\sqrt{-3}, \sqrt{5})\) None 105.2.a.b \(0\) \(2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{2}-\beta _{1}q^{3}+3\beta _{1}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
735.2.i.l 735.i 7.c $4$ $5.869$ \(\Q(\zeta_{12})\) None 105.2.i.d \(2\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta_{2}+\beta_1)q^{2}+(\beta_1-1)q^{3}+\cdots\)
735.2.i.m 735.i 7.c $8$ $5.869$ 8.0.\(\cdots\).10 None 735.2.a.n \(-4\) \(-4\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{5})q^{2}+\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
735.2.i.n 735.i 7.c $8$ $5.869$ 8.0.\(\cdots\).10 None 735.2.a.n \(-4\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{5})q^{2}-\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)

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

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