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Properties

Label 1764.2.x
Level $1764$
Weight $2$
Character orbit 1764.x
Rep. character $\chi_{1764}(293,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $80$
Newform subspaces $3$
Sturm bound $672$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.x (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1764, [\chi])\).

Total New Old
Modular forms 720 80 640
Cusp forms 624 80 544
Eisenstein series 96 0 96

Trace form

\( 80 q + 4 q^{9} + O(q^{10}) \) \( 80 q + 4 q^{9} - 12 q^{11} + 18 q^{15} + 6 q^{23} - 40 q^{25} + 12 q^{29} - 4 q^{37} - 28 q^{39} - 4 q^{43} + 12 q^{51} + 46 q^{57} + 6 q^{65} - 14 q^{67} - 26 q^{79} + 60 q^{81} - 12 q^{85} + 84 q^{95} - 58 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1764.2.x.a 1764.x 63.o $16$ $14.086$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{3}-\beta _{11}q^{5}+(\beta _{2}-\beta _{3}+\beta _{14}+\cdots)q^{9}+\cdots\)
1764.2.x.b 1764.x 63.o $16$ $14.086$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{3}+\beta _{7}q^{5}+\beta _{5}q^{9}+(1-\beta _{3}+\cdots)q^{11}+\cdots\)
1764.2.x.c 1764.x 63.o $48$ $14.086$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)