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Properties

Label 684.2.bt
Level $684$
Weight $2$
Character orbit 684.bt
Rep. character $\chi_{684}(67,\cdot)$
Character field $\Q(\zeta_{18})$
Dimension $696$
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.bt (of order \(18\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 684 \)
Character field: \(\Q(\zeta_{18})\)
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 744 744 0
Cusp forms 696 696 0
Eisenstein series 48 48 0

Trace form

\( 696 q - 3 q^{2} - 3 q^{4} - 6 q^{5} - 6 q^{6} - 18 q^{8} - 18 q^{9} + O(q^{10}) \) \( 696 q - 3 q^{2} - 3 q^{4} - 6 q^{5} - 6 q^{6} - 18 q^{8} - 18 q^{9} - 12 q^{10} - 9 q^{12} - 6 q^{13} + 21 q^{14} - 3 q^{16} - 24 q^{17} - 6 q^{20} + 24 q^{21} - 15 q^{22} - 39 q^{24} - 6 q^{25} - 30 q^{26} - 6 q^{29} + 42 q^{30} - 3 q^{32} - 36 q^{33} - 21 q^{34} + 54 q^{36} - 3 q^{38} + 12 q^{40} - 36 q^{41} + 66 q^{42} - 33 q^{44} - 6 q^{45} - 18 q^{46} + 6 q^{48} - 564 q^{49} - 9 q^{50} - 3 q^{52} - 24 q^{53} + 66 q^{54} + 63 q^{56} - 12 q^{57} - 6 q^{58} - 9 q^{60} - 6 q^{61} - 18 q^{62} - 24 q^{64} - 18 q^{65} + 6 q^{66} - 6 q^{68} - 18 q^{69} + 18 q^{70} + 12 q^{72} - 60 q^{73} - 60 q^{74} - 3 q^{76} + 72 q^{77} + 9 q^{78} - 12 q^{80} + 18 q^{81} - 12 q^{82} - 9 q^{84} - 6 q^{85} + 105 q^{86} - 24 q^{89} - 141 q^{90} - 171 q^{92} + 6 q^{93} + 15 q^{96} - 24 q^{97} - 63 q^{98} + O(q^{100}) \)

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.bt.a 684.bt 684.at $696$ $5.462$ None \(-3\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{18}]$