Properties

Label 693.2.m
Level $693$
Weight $2$
Character orbit 693.m
Rep. character $\chi_{693}(64,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $120$
Newform subspaces $11$
Sturm bound $192$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.m (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 11 \)
Sturm bound: \(192\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 416 120 296
Cusp forms 352 120 232
Eisenstein series 64 0 64

Trace form

\( 120 q - 32 q^{4} + 2 q^{5} - 2 q^{7} + 10 q^{8} + O(q^{10}) \) \( 120 q - 32 q^{4} + 2 q^{5} - 2 q^{7} + 10 q^{8} + 16 q^{10} + 10 q^{11} + 14 q^{13} + 3 q^{14} - 50 q^{16} + 8 q^{17} + 26 q^{19} + 38 q^{20} - 6 q^{22} - 60 q^{25} + 58 q^{26} - q^{28} - 10 q^{29} - 10 q^{31} - 100 q^{32} - 16 q^{34} + 8 q^{35} + 2 q^{37} - 34 q^{38} + 70 q^{40} + 24 q^{41} + 24 q^{43} - 9 q^{44} + 27 q^{46} + 24 q^{47} - 30 q^{49} + 46 q^{50} - 26 q^{52} + 46 q^{53} - 42 q^{55} - 18 q^{56} - 51 q^{58} - 38 q^{59} - 106 q^{62} - 72 q^{64} - 56 q^{65} + 44 q^{67} - 94 q^{68} + 20 q^{70} + 24 q^{71} - 64 q^{73} + 66 q^{74} - 112 q^{76} - 8 q^{77} - 66 q^{79} - 56 q^{80} - 86 q^{82} + 60 q^{83} - 12 q^{85} - 33 q^{86} + 84 q^{88} + 24 q^{89} + 18 q^{91} + 79 q^{92} + 10 q^{94} + 72 q^{95} + 58 q^{97} - 10 q^{98} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
693.2.m.a 693.m 11.c $4$ $5.534$ \(\Q(\zeta_{10})\) None \(-5\) \(0\) \(3\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
693.2.m.b 693.m 11.c $4$ $5.534$ \(\Q(\zeta_{10})\) None \(-2\) \(0\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\cdots)q^{4}+\cdots\)
693.2.m.c 693.m 11.c $4$ $5.534$ \(\Q(\zeta_{10})\) None \(-1\) \(0\) \(-7\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
693.2.m.d 693.m 11.c $4$ $5.534$ \(\Q(\zeta_{10})\) None \(2\) \(0\) \(-3\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+(-1+\zeta_{10}+\zeta_{10}^{3})q^{4}+\cdots\)
693.2.m.e 693.m 11.c $4$ $5.534$ \(\Q(\zeta_{10})\) None \(4\) \(0\) \(3\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+(3+\cdots)q^{4}+\cdots\)
693.2.m.f 693.m 11.c $8$ $5.534$ 8.0.13140625.1 None \(-2\) \(0\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{3}-\beta _{5}-\beta _{6})q^{2}+(1+\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots\)
693.2.m.g 693.m 11.c $8$ $5.534$ 8.0.159390625.1 None \(1\) \(0\) \(-3\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{4}q^{2}+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
693.2.m.h 693.m 11.c $16$ $5.534$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{1}q^{2}-\beta _{7}q^{4}+(\beta _{9}-\beta _{13})q^{5}-\beta _{4}q^{7}+\cdots\)
693.2.m.i 693.m 11.c $16$ $5.534$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(0\) \(5\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{5}+\beta _{6})q^{2}+(\beta _{2}+\beta _{7}+\beta _{11})q^{4}+\cdots\)
693.2.m.j 693.m 11.c $20$ $5.534$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(5\) \(5\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{7}q^{2}+(\beta _{3}-\beta _{6}-\beta _{8}-2\beta _{9}-\beta _{10}+\cdots)q^{4}+\cdots\)
693.2.m.k 693.m 11.c $32$ $5.534$ None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{5}]$

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

\( S_{2}^{\mathrm{old}}(693, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 2}\)