Properties

Label 1764.2.k
Level $1764$
Weight $2$
Character orbit 1764.k
Rep. character $\chi_{1764}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $34$
Newform subspaces $13$
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.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 13 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 768 34 734
Cusp forms 576 34 542
Eisenstein series 192 0 192

Trace form

\( 34q - q^{5} + O(q^{10}) \) \( 34q - q^{5} + 3q^{11} - 8q^{13} - 11q^{17} - 3q^{19} - 17q^{23} - 28q^{25} - 4q^{29} + 7q^{31} - 9q^{37} + 24q^{41} + 52q^{43} + 3q^{47} - 9q^{53} + 26q^{55} - 13q^{59} + 7q^{61} - 40q^{65} - 11q^{67} - 20q^{71} + 5q^{73} - 17q^{79} + 28q^{83} + 90q^{85} - 15q^{89} + 7q^{95} - 28q^{97} + O(q^{100}) \)

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.k.a \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+6q^{13}+\cdots\)
1764.2.k.b \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(0\) \(q-3\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{11}-2q^{13}+\cdots\)
1764.2.k.c \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+4q^{13}+\cdots\)
1764.2.k.d \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-6+6\zeta_{6})q^{11}-2q^{13}-4\zeta_{6}q^{19}+\cdots\)
1764.2.k.e \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-6+6\zeta_{6})q^{11}+2q^{13}+4\zeta_{6}q^{19}+\cdots\)
1764.2.k.f \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-5q^{13}-\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.g \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-2q^{13}+8\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.h \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+2q^{13}-8\zeta_{6}q^{19}+(5-5\zeta_{6})q^{25}+\cdots\)
1764.2.k.i \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-4q^{13}+\cdots\)
1764.2.k.j \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) \(q+2\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}+3q^{13}+\cdots\)
1764.2.k.k \(2\) \(14.086\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(0\) \(q+4\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-6q^{13}+\cdots\)
1764.2.k.l \(4\) \(14.086\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}-4\beta _{2}q^{11}-3\beta _{3}q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
1764.2.k.m \(8\) \(14.086\) 8.0.\(\cdots\).4 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{7}q^{5}-\beta _{5}q^{11}-3\beta _{6}q^{13}+\beta _{3}q^{17}+\cdots\)

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

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