Properties

Label 693.2.i
Level $693$
Weight $2$
Character orbit 693.i
Rep. character $\chi_{693}(100,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $68$
Newform subspaces $12$
Sturm bound $192$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(192\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(693, [\chi])\).

Total New Old
Modular forms 208 68 140
Cusp forms 176 68 108
Eisenstein series 32 0 32

Trace form

\( 68q - 36q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 68q - 36q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + 6q^{10} + 8q^{13} + 4q^{14} - 36q^{16} - 6q^{17} - 10q^{19} - 8q^{20} + 4q^{23} - 42q^{25} + 14q^{26} - 64q^{28} + 24q^{29} + 6q^{31} + 32q^{32} + 48q^{34} - 42q^{35} - 4q^{38} + 24q^{40} + 24q^{41} - 52q^{43} + 4q^{44} + 10q^{46} + 22q^{47} - 4q^{49} - 88q^{50} + 54q^{52} - 10q^{53} + 16q^{55} - 30q^{56} - 2q^{58} + 40q^{59} + 28q^{61} + 68q^{62} + 44q^{64} - 10q^{65} + 28q^{67} - 52q^{68} - 6q^{70} - 12q^{71} - 46q^{73} - 8q^{74} + 32q^{76} - 2q^{77} - 20q^{79} - 2q^{80} - 30q^{82} + 84q^{83} + 80q^{85} + 6q^{86} + 12q^{88} + 6q^{89} - 26q^{91} - 124q^{92} - 22q^{94} - 24q^{95} - 84q^{97} + 18q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(693, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
693.2.i.a \(2\) \(5.534\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(0\) \(5\) \(q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots\)
693.2.i.b \(2\) \(5.534\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(4\) \(-4\) \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots\)
693.2.i.c \(2\) \(5.534\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(-5\) \(q+(2-2\zeta_{6})q^{4}-4\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
693.2.i.d \(2\) \(5.534\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(-5\) \(q+(2-2\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
693.2.i.e \(2\) \(5.534\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-4\) \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+(-1-2\zeta_{6})q^{7}+\cdots\)
693.2.i.f \(4\) \(5.534\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(-4\) \(4\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
693.2.i.g \(6\) \(5.534\) 6.0.1783323.2 None \(0\) \(0\) \(-2\) \(2\) \(q+(\beta _{3}-\beta _{5})q^{2}+(-1-\beta _{1}+\beta _{4}+\beta _{5})q^{4}+\cdots\)
693.2.i.h \(6\) \(5.534\) \(\Q(\zeta_{18})\) None \(0\) \(0\) \(6\) \(0\) \(q+(\zeta_{18}-\zeta_{18}^{2}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
693.2.i.i \(8\) \(5.534\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(2\) \(0\) \(4\) \(2\) \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{6})q^{4}+\cdots\)
693.2.i.j \(10\) \(5.534\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(2\) \(0\) \(-4\) \(-1\) \(q+\beta _{3}q^{2}+(-2-2\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
693.2.i.k \(12\) \(5.534\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(-4\) \(6\) \(q+\beta _{6}q^{2}+(-2+2\beta _{4}-\beta _{8})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
693.2.i.l \(12\) \(5.534\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(4\) \(6\) \(q-\beta _{6}q^{2}+(-2+2\beta _{4}-\beta _{8})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(693, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(693, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 2}\)