Properties

Label 684.2.k
Level $684$
Weight $2$
Character orbit 684.k
Rep. character $\chi_{684}(505,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $18$
Newform subspaces $7$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(684, [\chi])\).

Total New Old
Modular forms 264 18 246
Cusp forms 216 18 198
Eisenstein series 48 0 48

Trace form

\( 18 q + q^{5} + O(q^{10}) \) \( 18 q + q^{5} - 4 q^{11} - 7 q^{13} + 7 q^{17} - 4 q^{19} - 13 q^{23} - 16 q^{25} + 11 q^{29} + 6 q^{35} - 4 q^{37} + 3 q^{41} + 9 q^{43} + 5 q^{47} + 34 q^{49} + 9 q^{53} - 16 q^{55} + 13 q^{59} + 9 q^{61} + 50 q^{65} - 17 q^{67} + 3 q^{71} + 21 q^{73} - 20 q^{77} + q^{79} + 8 q^{83} + 3 q^{85} + 13 q^{89} + 20 q^{91} - 61 q^{95} - 11 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
684.2.k.a 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{5}-3q^{7}+4q^{11}-5\zeta_{6}q^{13}+\cdots\)
684.2.k.b 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+4q^{11}+\zeta_{6}q^{13}+\cdots\)
684.2.k.c 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-2\) $\mathrm{U}(1)[D_{3}]$ \(q-q^{7}+7\zeta_{6}q^{13}+(5-2\zeta_{6})q^{19}+5\zeta_{6}q^{25}+\cdots\)
684.2.k.d 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(10\) $\mathrm{U}(1)[D_{3}]$ \(q+5q^{7}-5\zeta_{6}q^{13}+(3+2\zeta_{6})q^{19}+\cdots\)
684.2.k.e 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{5}-3q^{7}-4q^{11}-5\zeta_{6}q^{13}+\cdots\)
684.2.k.f 684.k 19.c $4$ $5.462$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{5}+(-1+\beta _{3})q^{7}-\beta _{3}q^{11}+\cdots\)
684.2.k.g 684.k 19.c $4$ $5.462$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(2\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{5}+(2-\beta _{3})q^{7}+(-3+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

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