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Properties

Label 806.2.e
Level $806$
Weight $2$
Character orbit 806.e
Rep. character $\chi_{806}(521,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $64$
Newform subspaces $5$
Sturm bound $224$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 31 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(224\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(806, [\chi])\).

Total New Old
Modular forms 232 64 168
Cusp forms 216 64 152
Eisenstein series 16 0 16

Trace form

\( 64 q + 8 q^{3} + 64 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} - 24 q^{9} + O(q^{10}) \) \( 64 q + 8 q^{3} + 64 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} - 24 q^{9} + 8 q^{10} + 4 q^{11} + 8 q^{12} + 2 q^{13} - 2 q^{14} + 64 q^{16} + 4 q^{17} - 4 q^{20} - 4 q^{21} - 14 q^{22} + 40 q^{23} + 4 q^{24} - 36 q^{25} + 6 q^{26} - 28 q^{27} - 4 q^{28} - 20 q^{29} - 8 q^{30} + 8 q^{31} - 16 q^{33} + 8 q^{34} - 20 q^{35} - 24 q^{36} + 10 q^{38} + 8 q^{40} - 12 q^{41} - 10 q^{42} - 20 q^{43} + 4 q^{44} - 20 q^{45} - 16 q^{46} + 24 q^{47} + 8 q^{48} - 22 q^{49} - 16 q^{50} + 42 q^{51} + 2 q^{52} - 18 q^{53} - 32 q^{54} - 40 q^{55} - 2 q^{56} - 16 q^{57} + 40 q^{58} - 24 q^{59} - 12 q^{61} - 12 q^{62} + 16 q^{63} + 64 q^{64} + 16 q^{65} + 32 q^{66} - 16 q^{67} + 4 q^{68} - 8 q^{69} + 72 q^{70} - 52 q^{71} + 4 q^{73} + 4 q^{74} + 58 q^{75} - 36 q^{77} + 12 q^{79} - 4 q^{80} - 8 q^{81} - 24 q^{82} - 16 q^{83} - 4 q^{84} - 56 q^{85} + 8 q^{86} - 14 q^{88} - 26 q^{90} - 8 q^{91} + 40 q^{92} - 28 q^{93} - 24 q^{94} - 32 q^{95} + 4 q^{96} + 16 q^{97} + 8 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(806, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
806.2.e.a 806.e 31.c $2$ $6.436$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-2\) \(-2\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}+(-2+2\zeta_{6})q^{3}+q^{4}-2\zeta_{6}q^{5}+\cdots\)
806.2.e.b 806.e 31.c $12$ $6.436$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(3\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}-\beta _{9}q^{3}+q^{4}+(-1+\beta _{2}+\beta _{5}+\cdots)q^{5}+\cdots\)
806.2.e.c 806.e 31.c $12$ $6.436$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(3\) \(6\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{4}-\beta _{9}+\cdots)q^{5}+\cdots\)
806.2.e.d 806.e 31.c $18$ $6.436$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-18\) \(1\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(-\beta _{8}+\beta _{12}+\cdots)q^{5}+\cdots\)
806.2.e.e 806.e 31.c $20$ $6.436$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(20\) \(3\) \(-4\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+q^{2}+(-\beta _{1}-\beta _{2})q^{3}+q^{4}-\beta _{13}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(806, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(806, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(31, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(62, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(403, [\chi])\)\(^{\oplus 2}\)