Properties

Label 1764.2.b
Level $1764$
Weight $2$
Character orbit 1764.b
Rep. character $\chi_{1764}(1567,\cdot)$
Character field $\Q$
Dimension $96$
Newform subspaces $14$
Sturm bound $672$
Trace bound $50$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(672\)
Trace bound: \(50\)
Distinguishing \(T_p\): \(5\), \(11\), \(19\), \(29\), \(53\)

Dimensions

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

Total New Old
Modular forms 368 104 264
Cusp forms 304 96 208
Eisenstein series 64 8 56

Trace form

\( 96q - 4q^{2} + 4q^{4} - 4q^{8} + O(q^{10}) \) \( 96q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{16} - 20q^{22} - 68q^{25} - 16q^{29} - 4q^{32} - 4q^{37} + 24q^{44} + 28q^{46} + 56q^{50} + 20q^{53} + 64q^{58} + 52q^{64} + 8q^{65} + 12q^{85} - 36q^{86} - 64q^{88} - 8q^{92} + O(q^{100}) \)

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.b.a \(4\) \(14.086\) \(\Q(\zeta_{12})\) None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\zeta_{12})q^{2}-2\zeta_{12}q^{4}-\zeta_{12}^{2}q^{5}+\cdots\)
1764.2.b.b \(4\) \(14.086\) \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(-2\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+(-2-\beta _{2})q^{4}+\beta _{3}q^{5}+(2+\cdots)q^{8}+\cdots\)
1764.2.b.c \(4\) \(14.086\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}+(1-\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
1764.2.b.d \(4\) \(14.086\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(1-\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)
1764.2.b.e \(4\) \(14.086\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+2q^{4}+(\beta _{1}+2\beta _{3})q^{5}+2\beta _{2}q^{8}+\cdots\)
1764.2.b.f \(4\) \(14.086\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+2q^{4}+\beta _{1}q^{5}+2\beta _{2}q^{8}+\cdots\)
1764.2.b.g \(4\) \(14.086\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+2q^{4}+(\beta _{1}+2\beta _{3})q^{5}-2\beta _{2}q^{8}+\cdots\)
1764.2.b.h \(4\) \(14.086\) \(\Q(\sqrt{-3}, \sqrt{-7})\) None \(2\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-2-\beta _{2})q^{4}+\beta _{3}q^{5}+(-2+\cdots)q^{8}+\cdots\)
1764.2.b.i \(8\) \(14.086\) 8.0.562828176.1 None \(2\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{2}-\beta _{5})q^{5}+(\beta _{3}+\cdots)q^{8}+\cdots\)
1764.2.b.j \(8\) \(14.086\) 8.0.562828176.1 None \(2\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{2}+\beta _{6}q^{4}+(\beta _{2}-\beta _{6})q^{5}+(-1+\cdots)q^{8}+\cdots\)
1764.2.b.k \(8\) \(14.086\) 8.0.\(\cdots\).10 None \(4\) \(0\) \(0\) \(0\) \(q+(1+\beta _{3})q^{2}+(1+\beta _{3}+\beta _{5}-\beta _{6})q^{4}+\cdots\)
1764.2.b.l \(12\) \(14.086\) 12.0.\(\cdots\).1 None \(-4\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+(\beta _{5}+\beta _{6})q^{4}+(1-\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
1764.2.b.m \(12\) \(14.086\) 12.0.\(\cdots\).1 None \(-4\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{2}+(-\beta _{2}-\beta _{11})q^{4}+(-1+\beta _{3}+\cdots)q^{5}+\cdots\)
1764.2.b.n \(16\) \(14.086\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{7}q^{2}+(-1+\beta _{1}-\beta _{10})q^{4}+\beta _{12}q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)