Properties

Label 1764.2.j
Level $1764$
Weight $2$
Character orbit 1764.j
Rep. character $\chi_{1764}(589,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $82$
Newform subspaces $9$
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.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 9 \)
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 82 638
Cusp forms 624 82 542
Eisenstein series 96 0 96

Trace form

\( 82q + q^{5} + 6q^{9} + O(q^{10}) \) \( 82q + q^{5} + 6q^{9} + q^{11} - q^{13} - 17q^{15} - 16q^{17} - 4q^{19} + q^{23} - 44q^{25} - 15q^{29} - q^{31} + q^{33} + 8q^{37} + 17q^{39} - 3q^{41} + 5q^{43} + q^{45} - 3q^{47} + 20q^{51} + 28q^{53} - 6q^{55} + 42q^{57} + 25q^{59} - 13q^{61} + 13q^{65} - 7q^{67} - 23q^{69} + 44q^{71} + 8q^{73} - 34q^{75} + 11q^{79} + 14q^{81} - 7q^{83} - 6q^{85} + 77q^{87} - 48q^{89} - 29q^{93} - 13q^{97} - 17q^{99} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1764.2.j.a \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(2\) \(0\) \(q+(-1-\zeta_{6})q^{3}+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+\cdots\)
1764.2.j.b \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(0\) \(q+(1-2\zeta_{6})q^{3}+3\zeta_{6}q^{5}-3q^{9}+(-3+\cdots)q^{11}+\cdots\)
1764.2.j.c \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-2\) \(0\) \(q+(1+\zeta_{6})q^{3}-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots\)
1764.2.j.d \(6\) \(14.086\) 6.0.309123.1 None \(0\) \(-2\) \(-3\) \(0\) \(q+(-1+\beta _{2}+\beta _{3}+\beta _{4})q^{3}+(\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.j.e \(6\) \(14.086\) 6.0.309123.1 None \(0\) \(2\) \(1\) \(0\) \(q+(1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.j.f \(12\) \(14.086\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{3}+(-\beta _{8}+\beta _{11})q^{5}+(\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
1764.2.j.g \(14\) \(14.086\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-3\) \(-2\) \(0\) \(q-\beta _{3}q^{3}+\beta _{5}q^{5}+(-\beta _{1}-\beta _{2}+\beta _{11}+\cdots)q^{9}+\cdots\)
1764.2.j.h \(14\) \(14.086\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(3\) \(2\) \(0\) \(q+\beta _{3}q^{3}-\beta _{5}q^{5}+(-\beta _{1}-\beta _{2}+\beta _{11}+\cdots)q^{9}+\cdots\)
1764.2.j.i \(24\) \(14.086\) None \(0\) \(0\) \(0\) \(0\)

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

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)