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Properties

Label 378.2.g
Level $378$
Weight $2$
Character orbit 378.g
Rep. character $\chi_{378}(109,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $20$
Newform subspaces $8$
Sturm bound $144$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.g (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(378, [\chi])\).

Total New Old
Modular forms 168 20 148
Cusp forms 120 20 100
Eisenstein series 48 0 48

Trace form

\( 20 q - 10 q^{4} - 14 q^{7} + O(q^{10}) \) \( 20 q - 10 q^{4} - 14 q^{7} + 2 q^{10} + 12 q^{13} - 10 q^{16} + 16 q^{19} + 4 q^{22} - 8 q^{25} + 4 q^{28} - 12 q^{31} - 8 q^{34} - 14 q^{37} + 2 q^{40} + 100 q^{43} - 8 q^{46} + 38 q^{49} - 6 q^{52} - 104 q^{55} - 16 q^{58} - 18 q^{61} + 20 q^{64} - 10 q^{67} - 50 q^{70} - 16 q^{73} - 32 q^{76} - 18 q^{79} + 40 q^{85} - 2 q^{88} + 60 q^{91} - 24 q^{94} + 32 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
378.2.g.a 378.g 7.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
378.2.g.b 378.g 7.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
378.2.g.c 378.g 7.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
378.2.g.d 378.g 7.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
378.2.g.e 378.g 7.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
378.2.g.f 378.g 7.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
378.2.g.g 378.g 7.c $4$ $3.018$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
378.2.g.h 378.g 7.c $4$ $3.018$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

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