Properties

Label 819.2.o
Level $819$
Weight $2$
Character orbit 819.o
Rep. character $\chi_{819}(568,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $72$
Newform subspaces $10$
Sturm bound $224$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.o (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(224\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(819, [\chi])\).

Total New Old
Modular forms 240 72 168
Cusp forms 208 72 136
Eisenstein series 32 0 32

Trace form

\( 72 q - 2 q^{2} - 38 q^{4} - 8 q^{5} + 24 q^{8} + O(q^{10}) \) \( 72 q - 2 q^{2} - 38 q^{4} - 8 q^{5} + 24 q^{8} + 10 q^{10} + 6 q^{11} - 14 q^{13} - 8 q^{14} - 34 q^{16} + 10 q^{17} + 8 q^{19} - 10 q^{20} + 14 q^{22} + 12 q^{23} + 76 q^{25} + 32 q^{26} + 10 q^{29} + 28 q^{31} - 6 q^{32} - 64 q^{34} + 6 q^{35} + 4 q^{37} - 4 q^{38} - 80 q^{40} - 2 q^{43} + 24 q^{44} + 20 q^{46} + 20 q^{47} - 36 q^{49} + 50 q^{50} + 22 q^{52} - 32 q^{53} + 6 q^{55} + 12 q^{56} - 50 q^{58} + 22 q^{59} + 12 q^{61} - 40 q^{62} + 80 q^{64} + 10 q^{65} - 20 q^{67} - 38 q^{68} - 48 q^{70} - 8 q^{71} - 8 q^{73} + 12 q^{74} - 26 q^{76} - 24 q^{77} + 8 q^{79} + 30 q^{80} + 16 q^{82} - 124 q^{83} + 14 q^{85} - 128 q^{86} + 18 q^{88} - 16 q^{89} - 14 q^{91} - 16 q^{92} - 36 q^{94} + 22 q^{95} - 26 q^{97} - 2 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(819, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
819.2.o.a 819.o 13.c $4$ $6.540$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+\zeta_{12}^{3}q^{5}+\cdots\)
819.2.o.b 819.o 13.c $4$ $6.540$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+\zeta_{12}^{3}q^{5}+\cdots\)
819.2.o.c 819.o 13.c $4$ $6.540$ \(\Q(\sqrt{-3}, \sqrt{5})\) None \(3\) \(0\) \(-6\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{3})q^{2}+(3\beta _{1}+3\beta _{2})q^{4}+\cdots\)
819.2.o.d 819.o 13.c $6$ $6.540$ 6.0.771147.1 None \(-2\) \(0\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{2}-\beta _{3}+\beta _{4}+\beta _{5})q^{2}+(-2+\cdots)q^{4}+\cdots\)
819.2.o.e 819.o 13.c $6$ $6.540$ 6.0.64827.1 None \(-2\) \(0\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{4}+\beta _{5})q^{2}+(-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
819.2.o.f 819.o 13.c $6$ $6.540$ 6.0.6040683.1 None \(0\) \(0\) \(-4\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{2})q^{2}+(\beta _{3}-\beta _{4}-\beta _{5})q^{4}+\cdots\)
819.2.o.g 819.o 13.c $6$ $6.540$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(-12\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{18}+\zeta_{18}^{2}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots\)
819.2.o.h 819.o 13.c $8$ $6.540$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-1\) \(0\) \(14\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
819.2.o.i 819.o 13.c $12$ $6.540$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{7}q^{2}+(-1-\beta _{3}+\beta _{4})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
819.2.o.j 819.o 13.c $16$ $6.540$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{9})q^{2}+(-1-\beta _{3}-\beta _{8}-\beta _{11}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(819, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(819, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(273, [\chi])\)\(^{\oplus 2}\)