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Properties

Label 1764.2.i
Level $1764$
Weight $2$
Character orbit 1764.i
Rep. character $\chi_{1764}(373,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $10$
Sturm bound $672$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.i (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: \(5\)
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

\( 80 q + 4 q^{5} + O(q^{10}) \) \( 80 q + 4 q^{5} + 4 q^{11} + q^{13} - 17 q^{15} + 5 q^{17} - 2 q^{19} - 11 q^{23} - 40 q^{25} - 9 q^{27} - 6 q^{29} + 4 q^{31} - 8 q^{33} + q^{37} - 13 q^{39} + 24 q^{41} - 2 q^{43} + 4 q^{45} - 12 q^{47} + 11 q^{51} + 22 q^{53} + 12 q^{55} + 15 q^{57} - 14 q^{59} - 26 q^{61} + 22 q^{65} - 14 q^{67} + 43 q^{69} + 38 q^{71} - 14 q^{73} + 47 q^{75} - 14 q^{79} + 56 q^{81} + 26 q^{83} + 12 q^{85} - 28 q^{87} + 21 q^{89} - 35 q^{93} - 108 q^{95} + q^{97} - 23 q^{99} + O(q^{100}) \)

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.i.a 1764.i 63.h $2$ $14.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots\)
1764.2.i.b 1764.i 63.h $2$ $14.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}+(3-3\zeta_{6})q^{9}+\cdots\)
1764.2.i.c 1764.i 63.h $2$ $14.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots\)
1764.2.i.d 1764.i 63.h $6$ $14.086$ 6.0.309123.1 None \(0\) \(-4\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+\beta _{3}-\beta _{5})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.i.e 1764.i 63.h $6$ $14.086$ 6.0.309123.1 None \(0\) \(-2\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}+\beta _{3})q^{3}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
1764.2.i.f 1764.i 63.h $6$ $14.086$ 6.0.309123.1 None \(0\) \(2\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}-\beta _{3})q^{3}+(1+\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.i.g 1764.i 63.h $6$ $14.086$ 6.0.309123.1 None \(0\) \(4\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.i.h 1764.i 63.h $12$ $14.086$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{10})q^{3}+(-\beta _{8}+\beta _{11})q^{5}+\cdots\)
1764.2.i.i 1764.i 63.h $14$ $14.086$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-3\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{4})q^{3}-\beta _{8}q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
1764.2.i.j 1764.i 63.h $24$ $14.086$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$

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

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \)