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Properties

Label 1764.2.l
Level $1764$
Weight $2$
Character orbit 1764.l
Rep. character $\chi_{1764}(949,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $10$
Sturm bound $672$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.l (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(672\)
Trace bound: \(3\)
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

\( 80 q - 8 q^{5} - 6 q^{9} + O(q^{10}) \) \( 80 q - 8 q^{5} - 6 q^{9} - 8 q^{11} + q^{13} - 17 q^{15} + 5 q^{17} - 2 q^{19} + 22 q^{23} + 80 q^{25} - 9 q^{27} - 6 q^{29} - 2 q^{31} - 5 q^{33} + q^{37} + 41 q^{39} + 24 q^{41} - 2 q^{43} + 7 q^{45} + 6 q^{47} + 5 q^{51} + 22 q^{53} + 12 q^{55} + 15 q^{57} + 7 q^{59} + 13 q^{61} - 11 q^{65} + 7 q^{67} + 43 q^{69} + 38 q^{71} - 14 q^{73} - q^{75} + 7 q^{79} - 58 q^{81} + 26 q^{83} + 12 q^{85} + 5 q^{87} + 21 q^{89} - 11 q^{93} + 54 q^{95} + q^{97} - 23 q^{99} + O(q^{100}) \)

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.l.a 1764.l 63.g $2$ $14.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\zeta_{6})q^{3}-3q^{5}+3\zeta_{6}q^{9}+3q^{11}+\cdots\)
1764.2.l.b 1764.l 63.g $2$ $14.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-2\zeta_{6})q^{3}-2q^{5}-3q^{9}+4q^{11}+\cdots\)
1764.2.l.c 1764.l 63.g $2$ $14.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\zeta_{6})q^{3}+3q^{5}+3\zeta_{6}q^{9}+3q^{11}+\cdots\)
1764.2.l.d 1764.l 63.g $6$ $14.086$ 6.0.309123.1 None \(0\) \(-4\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{4})q^{3}+(-1-\beta _{2}-\beta _{4}-\beta _{5})q^{5}+\cdots\)
1764.2.l.e 1764.l 63.g $6$ $14.086$ 6.0.309123.1 None \(0\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{2}+\beta _{5})q^{3}+(-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.l.f 1764.l 63.g $6$ $14.086$ 6.0.309123.1 None \(0\) \(2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{3}+\beta _{4}+\beta _{5})q^{3}+(\beta _{2}+\beta _{3})q^{5}+\cdots\)
1764.2.l.g 1764.l 63.g $6$ $14.086$ 6.0.309123.1 None \(0\) \(4\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{4})q^{3}+(1+\beta _{2}+\beta _{4}+\beta _{5})q^{5}+\cdots\)
1764.2.l.h 1764.l 63.g $12$ $14.086$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+\beta _{9})q^{3}+\beta _{11}q^{5}+(1-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
1764.2.l.i 1764.l 63.g $14$ $14.086$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{3}+\beta _{4})q^{3}+(\beta _{5}-\beta _{10})q^{5}+(1+\cdots)q^{9}+\cdots\)
1764.2.l.j 1764.l 63.g $24$ $14.086$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$

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

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