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Properties

Label 735.3.h
Level $735$
Weight $3$
Character orbit 735.h
Rep. character $\chi_{735}(391,\cdot)$
Character field $\Q$
Dimension $52$
Newform subspaces $3$
Sturm bound $336$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 735.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(336\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(735, [\chi])\).

Total New Old
Modular forms 240 52 188
Cusp forms 208 52 156
Eisenstein series 32 0 32

Trace form

\( 52 q + 8 q^{2} + 72 q^{4} - 8 q^{8} - 156 q^{9} + O(q^{10}) \) \( 52 q + 8 q^{2} + 72 q^{4} - 8 q^{8} - 156 q^{9} + 48 q^{11} + 120 q^{16} - 24 q^{18} + 104 q^{22} + 128 q^{23} - 260 q^{25} - 72 q^{29} + 256 q^{32} - 216 q^{36} - 156 q^{37} + 108 q^{39} - 92 q^{43} - 112 q^{44} - 136 q^{46} - 40 q^{50} - 72 q^{51} - 336 q^{53} - 12 q^{57} + 472 q^{58} + 392 q^{64} + 240 q^{65} - 524 q^{67} + 112 q^{71} + 24 q^{72} - 176 q^{74} - 552 q^{78} + 260 q^{79} + 468 q^{81} - 120 q^{85} - 552 q^{86} + 1024 q^{88} + 808 q^{92} - 468 q^{93} - 80 q^{95} - 144 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(735, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
735.3.h.a 735.h 7.b $8$ $20.027$ 8.0.\(\cdots\).16 None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{5}q^{3}+(1-\beta _{1}+\beta _{3})q^{4}+\cdots\)
735.3.h.b 735.h 7.b $12$ $20.027$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}+\beta _{7}q^{3}+(4-\beta _{1}-\beta _{2})q^{4}+\cdots\)
735.3.h.c 735.h 7.b $32$ $20.027$ None \(16\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(735, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(735, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)