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Properties

Label 684.3.g
Level $684$
Weight $3$
Character orbit 684.g
Rep. character $\chi_{684}(343,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $4$
Sturm bound $360$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(360\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(684, [\chi])\).

Total New Old
Modular forms 248 90 158
Cusp forms 232 90 142
Eisenstein series 16 0 16

Trace form

\( 90 q - 2 q^{2} - 6 q^{4} - 4 q^{5} + 28 q^{8} + O(q^{10}) \) \( 90 q - 2 q^{2} - 6 q^{4} - 4 q^{5} + 28 q^{8} + 8 q^{10} + 12 q^{13} - 24 q^{14} - 62 q^{16} + 20 q^{17} + 40 q^{20} - 72 q^{22} + 374 q^{25} + 86 q^{26} - 90 q^{28} - 100 q^{29} - 52 q^{32} - 76 q^{34} + 92 q^{37} + 180 q^{40} + 100 q^{41} - 120 q^{44} + 164 q^{46} - 710 q^{49} + 14 q^{50} + 132 q^{52} - 276 q^{53} + 180 q^{56} - 262 q^{58} - 20 q^{61} - 60 q^{62} - 174 q^{64} - 88 q^{65} + 46 q^{68} + 12 q^{70} - 44 q^{73} + 304 q^{74} + 104 q^{77} - 388 q^{80} + 564 q^{82} + 64 q^{85} - 400 q^{86} + 80 q^{88} + 500 q^{89} + 342 q^{92} - 224 q^{94} + 100 q^{97} + 578 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
684.3.g.a 684.g 4.b $4$ $18.638$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(4\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{1})q^{2}+(-2-2\beta _{1})q^{4}+(2-2\beta _{2}+\cdots)q^{5}+\cdots\)
684.3.g.b 684.g 4.b $14$ $18.638$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{10}q^{5}+(\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)
684.3.g.c 684.g 4.b $36$ $18.638$ None \(-4\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{2}]$
684.3.g.d 684.g 4.b $36$ $18.638$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)