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Properties

Label 378.2.f
Level $378$
Weight $2$
Character orbit 378.f
Rep. character $\chi_{378}(127,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $4$
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.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 168 12 156
Cusp forms 120 12 108
Eisenstein series 48 0 48

Trace form

\( 12 q - 2 q^{2} - 6 q^{4} + 4 q^{5} + 4 q^{8} + O(q^{10}) \) \( 12 q - 2 q^{2} - 6 q^{4} + 4 q^{5} + 4 q^{8} + 2 q^{11} - 4 q^{14} - 6 q^{16} - 4 q^{17} + 12 q^{19} + 4 q^{20} - 6 q^{22} - 4 q^{23} - 18 q^{25} - 8 q^{29} - 12 q^{31} - 2 q^{32} - 6 q^{34} + 8 q^{35} + 24 q^{37} + 14 q^{38} + 18 q^{41} - 6 q^{43} - 4 q^{44} - 6 q^{49} - 10 q^{50} + 24 q^{55} - 4 q^{56} - 14 q^{59} - 24 q^{62} + 12 q^{64} - 36 q^{65} - 18 q^{67} + 2 q^{68} - 16 q^{71} + 12 q^{73} - 4 q^{74} - 6 q^{76} - 8 q^{77} - 12 q^{79} - 8 q^{80} - 36 q^{82} + 44 q^{83} + 12 q^{85} + 2 q^{86} - 6 q^{88} + 24 q^{89} + 24 q^{91} - 4 q^{92} + 44 q^{95} + 6 q^{97} + 4 q^{98} + O(q^{100}) \)

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.f.a 378.f 9.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-3\zeta_{6}q^{5}+\cdots\)
378.2.f.b 378.f 9.c $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+2\zeta_{6}q^{5}+\cdots\)
378.2.f.c 378.f 9.c $4$ $3.018$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-2\) \(0\) \(3\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
378.2.f.d 378.f 9.c $4$ $3.018$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(2\) \(0\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)

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

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