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Properties

Label 378.3.c
Level $378$
Weight $3$
Character orbit 378.c
Rep. character $\chi_{378}(55,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $3$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(216\)
Trace bound: \(7\)
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 20 136
Cusp forms 132 20 112
Eisenstein series 24 0 24

Trace form

\( 20 q + 40 q^{4} - 2 q^{7} + O(q^{10}) \) \( 20 q + 40 q^{4} - 2 q^{7} + 80 q^{16} - 24 q^{22} - 52 q^{25} - 4 q^{28} + 140 q^{37} + 128 q^{43} - 24 q^{46} + 302 q^{49} + 240 q^{58} + 160 q^{64} - 44 q^{67} - 48 q^{70} - 836 q^{79} - 912 q^{85} - 48 q^{88} - 498 q^{91} + O(q^{100}) \)

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.c.a 378.c 7.b $4$ $10.300$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-28\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+2q^{4}+\beta _{2}q^{5}-7q^{7}+2\beta _{1}q^{8}+\cdots\)
378.3.c.b 378.c 7.b $4$ $10.300$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(20\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+2q^{4}+\beta _{2}q^{5}+(5+2\beta _{3})q^{7}+\cdots\)
378.3.c.c 378.c 7.b $12$ $10.300$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{2}+2q^{4}-\beta _{8}q^{5}+(1+\beta _{2})q^{7}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)