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Properties

Label 684.6.k
Level $684$
Weight $6$
Character orbit 684.k
Rep. character $\chi_{684}(505,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $82$
Newform subspaces $7$
Sturm bound $720$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 684.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(720\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(684, [\chi])\).

Total New Old
Modular forms 1224 82 1142
Cusp forms 1176 82 1094
Eisenstein series 48 0 48

Trace form

\( 82 q + 11 q^{5} + 16 q^{7} + O(q^{10}) \) \( 82 q + 11 q^{5} + 16 q^{7} + 596 q^{11} - 541 q^{13} - 1359 q^{17} + 1536 q^{19} + 1847 q^{23} - 22098 q^{25} + 3577 q^{29} - 5328 q^{31} + 11574 q^{35} - 9420 q^{37} + 12007 q^{41} - 6327 q^{43} - 17575 q^{47} + 191890 q^{49} + 25043 q^{53} - 496 q^{55} - 25023 q^{59} - 685 q^{61} - 245402 q^{65} + 6475 q^{67} - 52225 q^{71} + 65677 q^{73} - 244500 q^{77} - 46219 q^{79} + 265152 q^{83} + 124653 q^{85} + 257955 q^{89} - 71068 q^{91} + 25783 q^{95} - 100647 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
684.6.k.a 684.k 19.c $2$ $109.703$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-422\) $\mathrm{U}(1)[D_{3}]$ \(q-211q^{7}+427\zeta_{6}q^{13}+(-1525+\cdots)q^{19}+\cdots\)
684.6.k.b 684.k 19.c $2$ $109.703$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-50\) $\mathrm{U}(1)[D_{3}]$ \(q-5^{2}q^{7}+775\zeta_{6}q^{13}+(93-1618\zeta_{6})q^{19}+\cdots\)
684.6.k.c 684.k 19.c $4$ $109.703$ \(\Q(\sqrt{-3}, \sqrt{1729})\) None \(0\) \(0\) \(0\) \(108\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{5}+3^{3}q^{7}-\beta _{3}q^{11}+(385-385\beta _{1}+\cdots)q^{13}+\cdots\)
684.6.k.d 684.k 19.c $16$ $109.703$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(96\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{5}+(6+\beta _{4}+\beta _{10})q^{7}+(-17+\cdots)q^{11}+\cdots\)
684.6.k.e 684.k 19.c $16$ $109.703$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(22\) \(-492\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+3\beta _{8}-\beta _{10})q^{5}+(-31+\beta _{1}+\cdots)q^{7}+\cdots\)
684.6.k.f 684.k 19.c $18$ $109.703$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(0\) \(0\) \(-11\) \(336\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{2}-\beta _{4})q^{5}+(19+\beta _{3}-\beta _{9}+\cdots)q^{7}+\cdots\)
684.6.k.g 684.k 19.c $24$ $109.703$ None \(0\) \(0\) \(0\) \(440\) $\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{6}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)