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Properties

Label 585.2.bu
Level $585$
Weight $2$
Character orbit 585.bu
Rep. character $\chi_{585}(316,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $48$
Newform subspaces $5$
Sturm bound $168$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bu (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(168\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(585, [\chi])\).

Total New Old
Modular forms 184 48 136
Cusp forms 152 48 104
Eisenstein series 32 0 32

Trace form

\( 48 q + 26 q^{4} + 6 q^{7} + O(q^{10}) \) \( 48 q + 26 q^{4} + 6 q^{7} - 2 q^{10} - 12 q^{13} + 4 q^{14} - 34 q^{16} + 14 q^{17} + 12 q^{19} + 12 q^{20} + 12 q^{22} + 14 q^{23} - 48 q^{25} + 22 q^{26} + 78 q^{28} - 66 q^{32} - 6 q^{35} - 18 q^{37} + 16 q^{38} - 12 q^{40} - 12 q^{41} + 2 q^{43} - 18 q^{46} + 4 q^{49} + 58 q^{52} + 56 q^{53} + 16 q^{55} - 28 q^{56} - 60 q^{58} + 12 q^{59} - 8 q^{61} + 48 q^{62} - 200 q^{64} - 12 q^{65} - 54 q^{67} - 10 q^{68} - 48 q^{71} - 6 q^{74} + 30 q^{76} - 4 q^{77} - 24 q^{79} - 52 q^{82} + 18 q^{85} + 38 q^{88} - 48 q^{89} + 16 q^{91} - 52 q^{92} + 24 q^{94} + 24 q^{95} + 78 q^{97} + 24 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(585, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
585.2.bu.a 585.bu 13.e $4$ $4.671$ \(\Q(\zeta_{12})\) None \(6\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(2+\cdots)q^{4}+\cdots\)
585.2.bu.b 585.bu 13.e $8$ $4.671$ 8.0.56070144.2 None \(-6\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{2}+\beta _{5}+\beta _{6})q^{2}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
585.2.bu.c 585.bu 13.e $8$ $4.671$ 8.0.22581504.2 None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{3}+\beta _{5}+\beta _{7})q^{2}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
585.2.bu.d 585.bu 13.e $8$ $4.671$ 8.0.191102976.5 None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2\beta _{1}+\beta _{3}-\beta _{5}-\beta _{7})q^{2}+(-2\beta _{2}+\cdots)q^{4}+\cdots\)
585.2.bu.e 585.bu 13.e $20$ $4.671$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(\beta _{2}+\beta _{5}-\beta _{7})q^{4}+(\beta _{6}-\beta _{16}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(585, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(585, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 2}\)