↑

Properties

Label 684.2.r
Level $684$
Weight $2$
Character orbit 684.r
Rep. character $\chi_{684}(487,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $96$
Newform subspaces $4$
Sturm bound $240$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 76 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(240\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\), \(23\)

Dimensions

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

Total New Old
Modular forms 256 104 152
Cusp forms 224 96 128
Eisenstein series 32 8 24

Trace form

\( 96 q + 3 q^{2} + q^{4} + 2 q^{5} + O(q^{10}) \) \( 96 q + 3 q^{2} + q^{4} + 2 q^{5} - 18 q^{10} + 6 q^{13} - 12 q^{14} + q^{16} - 2 q^{17} + 24 q^{20} + 3 q^{22} - 42 q^{25} + 24 q^{26} - 4 q^{28} + 6 q^{29} + 33 q^{32} + 24 q^{34} + 10 q^{38} + 48 q^{40} + 24 q^{41} + 7 q^{44} - 112 q^{49} - 42 q^{52} + 18 q^{53} + 24 q^{58} + 30 q^{61} - 6 q^{62} - 2 q^{64} - 44 q^{68} + 90 q^{70} - 8 q^{73} - 46 q^{74} - 9 q^{76} - 16 q^{77} - 32 q^{80} - 17 q^{82} - 2 q^{85} + 66 q^{86} + 30 q^{89} + 34 q^{92} - 12 q^{97} - 39 q^{98} + O(q^{100}) \)

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.r.a 684.r 76.f $16$ $5.462$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{12}q^{2}+(-1-\beta _{1}+\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots\)
684.2.r.b 684.r 76.f $20$ $5.462$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{4})q^{2}+(-\beta _{2}-\beta _{9})q^{4}+\cdots\)
684.2.r.c 684.r 76.f $20$ $5.462$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{16}q^{2}+(\beta _{14}-\beta _{17})q^{4}-\beta _{6}q^{5}+\cdots\)
684.2.r.d 684.r 76.f $40$ $5.462$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)