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Properties

Label 378.3.r
Level $378$
Weight $3$
Character orbit 378.r
Rep. character $\chi_{378}(233,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $32$
Newform subspaces $1$
Sturm bound $216$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(216\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(378, [\chi])\).

Total New Old
Modular forms 312 32 280
Cusp forms 264 32 232
Eisenstein series 48 0 48

Trace form

\( 32 q - 64 q^{4} + 2 q^{7} + O(q^{10}) \) \( 32 q - 64 q^{4} + 2 q^{7} + 36 q^{11} + 10 q^{13} - 36 q^{14} + 128 q^{16} + 54 q^{17} + 28 q^{19} + 126 q^{23} + 80 q^{25} + 72 q^{26} - 4 q^{28} - 36 q^{29} + 16 q^{31} + 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} - 72 q^{44} - 12 q^{46} + 2 q^{49} + 288 q^{50} - 20 q^{52} + 72 q^{53} - 24 q^{55} + 72 q^{56} - 24 q^{58} + 124 q^{61} - 256 q^{64} - 140 q^{67} - 108 q^{68} + 72 q^{70} + 196 q^{73} - 216 q^{74} - 56 q^{76} - 486 q^{77} + 76 q^{79} + 60 q^{85} + 144 q^{86} + 486 q^{89} - 122 q^{91} - 252 q^{92} - 336 q^{94} - 38 q^{97} - 288 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
378.3.r.a 378.r 63.j $32$ $10.300$ None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)