Properties

Label 1764.3.z
Level $1764$
Weight $3$
Character orbit 1764.z
Rep. character $\chi_{1764}(325,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $66$
Newform subspaces $14$
Sturm bound $1008$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 14 \)
Sturm bound: \(1008\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\), \(19\)

Dimensions

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

Total New Old
Modular forms 1440 66 1374
Cusp forms 1248 66 1182
Eisenstein series 192 0 192

Trace form

\( 66q - 3q^{5} + O(q^{10}) \) \( 66q - 3q^{5} - 3q^{11} + 3q^{17} - 45q^{19} + 43q^{23} + 122q^{25} - 164q^{29} - 105q^{31} - 81q^{37} + 48q^{43} + 141q^{47} - 9q^{53} + 231q^{59} - 39q^{61} + 6q^{65} - 25q^{67} - 680q^{71} + 99q^{73} + 7q^{79} + 198q^{85} + 423q^{89} - 61q^{95} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1764.3.z.a \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-12\) \(0\) \(q+(-8+4\zeta_{6})q^{5}+(18-18\zeta_{6})q^{11}+\cdots\)
1764.3.z.b \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(8-2^{4}\zeta_{6})q^{13}+(-2^{5}+2^{4}\zeta_{6})q^{19}+\cdots\)
1764.3.z.c \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(15-30\zeta_{6})q^{13}+(10-5\zeta_{6})q^{19}+\cdots\)
1764.3.z.d \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-8+2^{4}\zeta_{6})q^{13}+(2^{5}-2^{4}\zeta_{6})q^{19}+\cdots\)
1764.3.z.e \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(0\) \(q+(2-\zeta_{6})q^{5}+(-3+3\zeta_{6})q^{11}+(4+\cdots)q^{13}+\cdots\)
1764.3.z.f \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(0\) \(q+(2-\zeta_{6})q^{5}+(15-15\zeta_{6})q^{11}+(-8+\cdots)q^{13}+\cdots\)
1764.3.z.g \(2\) \(48.066\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(12\) \(0\) \(q+(8-4\zeta_{6})q^{5}+(18-18\zeta_{6})q^{11}+(12+\cdots)q^{13}+\cdots\)
1764.3.z.h \(4\) \(48.066\) \(\Q(\sqrt{-3}, \sqrt{65})\) None \(0\) \(0\) \(-9\) \(0\) \(q+(-1-2\beta _{1}-\beta _{3})q^{5}+(-7\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1764.3.z.i \(4\) \(48.066\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{5}+(-6-6\beta _{1})q^{11}-\beta _{2}q^{13}+\cdots\)
1764.3.z.j \(4\) \(48.066\) \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{5}+(2\beta _{2}-\beta _{3})q^{11}+(-6-12\beta _{1}+\cdots)q^{13}+\cdots\)
1764.3.z.k \(8\) \(48.066\) 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+\beta _{2}-2\beta _{6})q^{5}+(-6\beta _{4}+\beta _{7})q^{11}+\cdots\)
1764.3.z.l \(8\) \(48.066\) 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) \(q+(-2\beta _{2}-\beta _{5}+\beta _{6})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1764.3.z.m \(8\) \(48.066\) 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) \(q+(2\beta _{2}+\beta _{5}+\beta _{6})q^{5}+(\beta _{1}-\beta _{2}-\beta _{5}+\cdots)q^{11}+\cdots\)
1764.3.z.n \(16\) \(48.066\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{4}-\beta _{12})q^{5}+\beta _{11}q^{11}+\beta _{13}q^{13}+\cdots\)

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

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