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Properties

Label 1764.3.d
Level $1764$
Weight $3$
Character orbit 1764.d
Rep. character $\chi_{1764}(685,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $8$
Sturm bound $1008$
Trace bound $23$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(1008\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1764, [\chi])\).

Total New Old
Modular forms 720 34 686
Cusp forms 624 34 590
Eisenstein series 96 0 96

Trace form

\( 34 q + O(q^{10}) \) \( 34 q + 30 q^{11} - 46 q^{23} - 196 q^{25} - 4 q^{29} + 58 q^{37} - 32 q^{43} + 330 q^{53} - 156 q^{65} - 54 q^{67} + 8 q^{71} + 10 q^{79} - 162 q^{85} + 658 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1764.3.d.a 1764.d 7.b $2$ $48.066$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{5}-15q^{11}-8\zeta_{6}q^{13}+17\zeta_{6}q^{17}+\cdots\)
1764.3.d.b 1764.d 7.b $2$ $48.066$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-15\zeta_{6}q^{13}+5\zeta_{6}q^{19}+5^{2}q^{25}+\cdots\)
1764.3.d.c 1764.d 7.b $2$ $48.066$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{5}+3q^{11}+4\zeta_{6}q^{13}-10\zeta_{6}q^{17}+\cdots\)
1764.3.d.d 1764.d 7.b $4$ $48.066$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{5}+\beta _{2}q^{11}-6\beta _{1}q^{13}+4\beta _{3}q^{17}+\cdots\)
1764.3.d.e 1764.d 7.b $4$ $48.066$ 4.0.2048.2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-2\beta _{2})q^{5}+(6-\beta _{3})q^{11}+(3\beta _{1}+\cdots)q^{13}+\cdots\)
1764.3.d.f 1764.d 7.b $4$ $48.066$ \(\Q(\sqrt{-3}, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2\beta _{1}+\beta _{2})q^{5}+(8-\beta _{3})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)
1764.3.d.g 1764.d 7.b $8$ $48.066$ 8.0.3288334336.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{5}-\beta _{3}q^{11}+\beta _{6}q^{13}+3\beta _{1}q^{17}+\cdots\)
1764.3.d.h 1764.d 7.b $8$ $48.066$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{2}+\beta _{3})q^{5}+(-\beta _{1}+\beta _{4})q^{11}+(\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)