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Properties

Label 378.2.u
Level $378$
Weight $2$
Character orbit 378.u
Rep. character $\chi_{378}(43,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $108$
Newform subspaces $5$
Sturm bound $144$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.u (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 27 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 5 \)
Sturm bound: \(144\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 456 108 348
Cusp forms 408 108 300
Eisenstein series 48 0 48

Trace form

\( 108 q + 12 q^{5} + 6 q^{8} + 12 q^{9} + O(q^{10}) \) \( 108 q + 12 q^{5} + 6 q^{8} + 12 q^{9} + 18 q^{11} + 6 q^{12} + 24 q^{15} - 12 q^{18} - 24 q^{20} + 18 q^{22} + 12 q^{23} + 36 q^{25} - 54 q^{27} - 72 q^{29} + 36 q^{31} + 18 q^{33} + 18 q^{34} + 12 q^{35} - 24 q^{36} + 24 q^{38} + 12 q^{39} + 54 q^{41} + 18 q^{43} + 12 q^{44} - 12 q^{45} + 72 q^{47} + 12 q^{48} + 12 q^{50} - 36 q^{51} - 72 q^{53} + 36 q^{54} - 42 q^{57} - 72 q^{59} - 54 q^{64} - 84 q^{65} - 54 q^{67} - 12 q^{68} - 96 q^{69} - 24 q^{71} - 24 q^{72} - 96 q^{74} - 120 q^{75} - 36 q^{76} - 36 q^{78} - 72 q^{79} - 24 q^{81} + 12 q^{84} - 72 q^{85} - 6 q^{86} + 60 q^{87} - 36 q^{88} - 66 q^{89} - 36 q^{90} - 24 q^{92} - 12 q^{93} + 12 q^{95} - 12 q^{96} - 54 q^{97} + 6 q^{98} + 72 q^{99} + O(q^{100}) \)

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.u.a 378.u 27.e $6$ $3.018$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q-\zeta_{18}q^{2}+(\zeta_{18}-2\zeta_{18}^{4})q^{3}+\zeta_{18}^{2}q^{4}+\cdots\)
378.2.u.b 378.u 27.e $12$ $3.018$ 12.0.\(\cdots\).1 None \(0\) \(3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q+\beta _{7}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{5}-\beta _{10}+\cdots)q^{3}+\cdots\)
378.2.u.c 378.u 27.e $24$ $3.018$ None \(0\) \(-3\) \(3\) \(0\) $\mathrm{SU}(2)[C_{9}]$
378.2.u.d 378.u 27.e $30$ $3.018$ None \(0\) \(-3\) \(3\) \(0\) $\mathrm{SU}(2)[C_{9}]$
378.2.u.e 378.u 27.e $36$ $3.018$ None \(0\) \(3\) \(3\) \(0\) $\mathrm{SU}(2)[C_{9}]$

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

\( S_{2}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)