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Properties

Label 378.3.b
Level $378$
Weight $3$
Character orbit 378.b
Rep. character $\chi_{378}(323,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $3$
Sturm bound $216$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(216\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 156 16 140
Cusp forms 132 16 116
Eisenstein series 24 0 24

Trace form

\( 16 q - 32 q^{4} + O(q^{10}) \) \( 16 q - 32 q^{4} + 32 q^{10} - 56 q^{13} + 64 q^{16} - 40 q^{19} - 8 q^{22} + 120 q^{25} - 8 q^{31} - 128 q^{34} - 40 q^{37} - 64 q^{40} - 40 q^{43} - 8 q^{46} + 112 q^{49} + 112 q^{52} + 232 q^{55} + 176 q^{58} - 296 q^{61} - 128 q^{64} + 96 q^{67} - 56 q^{70} - 240 q^{73} + 80 q^{76} - 152 q^{79} + 144 q^{82} + 16 q^{85} + 16 q^{88} - 56 q^{91} + 48 q^{94} + 216 q^{97} + O(q^{100}) \)

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.b.a 378.b 3.b $4$ $10.300$ \(\Q(\sqrt{-2}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-2q^{4}+\beta _{1}q^{5}+\beta _{3}q^{7}+2\beta _{2}q^{8}+\cdots\)
378.3.b.b 378.b 3.b $4$ $10.300$ \(\Q(\sqrt{-2}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-2q^{4}+(\beta _{1}+3\beta _{2})q^{5}+\beta _{3}q^{7}+\cdots\)
378.3.b.c 378.b 3.b $8$ $10.300$ 8.0.157351936.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{2}-2q^{4}+(-\beta _{1}+\beta _{4}-\beta _{6}+\cdots)q^{5}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)