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Properties

Label 684.2.j
Level $684$
Weight $2$
Character orbit 684.j
Rep. character $\chi_{684}(49,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $1$
Sturm bound $240$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 171 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(240\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 252 40 212
Cusp forms 228 40 188
Eisenstein series 24 0 24

Trace form

\( 40 q - 2 q^{3} - q^{7} - 2 q^{9} + O(q^{10}) \) \( 40 q - 2 q^{3} - q^{7} - 2 q^{9} - q^{11} + 2 q^{13} + q^{15} + 5 q^{17} + q^{19} + 6 q^{21} + 8 q^{23} - 20 q^{25} + 7 q^{27} - 9 q^{29} + 2 q^{31} - q^{33} - 6 q^{35} + 2 q^{37} + 3 q^{39} - 25 q^{41} - 16 q^{43} + 44 q^{45} - 10 q^{47} - 21 q^{49} + 19 q^{51} - 8 q^{53} - 22 q^{59} - 7 q^{61} + 10 q^{63} - 6 q^{65} + 20 q^{67} + 38 q^{69} - 5 q^{71} - 10 q^{73} - 10 q^{77} + 2 q^{79} + 22 q^{81} + q^{83} - 52 q^{87} - 34 q^{89} + 16 q^{91} - 8 q^{93} + 36 q^{95} + 2 q^{97} + 28 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.j.a 684.j 171.h $40$ $5.462$ None \(0\) \(-2\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$

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

\( S_{2}^{\mathrm{old}}(684, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)