↑

Properties

Label 378.2.e
Level $378$
Weight $2$
Character orbit 378.e
Rep. character $\chi_{378}(37,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $16$
Newform subspaces $4$
Sturm bound $144$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(2\)
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 16 152
Cusp forms 120 16 104
Eisenstein series 48 0 48

Trace form

\( 16 q + 16 q^{4} + 4 q^{5} - 2 q^{7} + O(q^{10}) \) \( 16 q + 16 q^{4} + 4 q^{5} - 2 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 14 q^{17} - 4 q^{19} + 4 q^{20} + 2 q^{23} - 8 q^{25} + 16 q^{26} - 2 q^{28} + 10 q^{29} - 4 q^{31} + 14 q^{35} + 2 q^{37} + 12 q^{38} + 6 q^{41} + 2 q^{43} - 4 q^{44} - 6 q^{46} - 12 q^{47} + 4 q^{49} + 4 q^{50} + 2 q^{52} - 24 q^{53} - 12 q^{55} - 2 q^{56} + 6 q^{58} - 44 q^{59} - 16 q^{61} - 44 q^{62} + 16 q^{64} - 12 q^{65} - 28 q^{67} + 14 q^{68} - 18 q^{70} - 52 q^{71} - 28 q^{73} + 6 q^{74} - 4 q^{76} - 50 q^{77} - 40 q^{79} + 4 q^{80} - 16 q^{83} + 12 q^{85} - 12 q^{86} + 36 q^{89} - 16 q^{91} + 2 q^{92} - 24 q^{94} + 68 q^{95} + 2 q^{97} + 24 q^{98} + O(q^{100}) \)

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.e.a 378.e 63.h $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}+q^{4}+(-3+3\zeta_{6})q^{5}+(-3+\cdots)q^{7}+\cdots\)
378.2.e.b 378.e 63.h $2$ $3.018$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+q^{2}+q^{4}+(3-3\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
378.2.e.c 378.e 63.h $6$ $3.018$ 6.0.309123.1 None \(-6\) \(0\) \(5\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}+q^{4}+(2-2\beta _{4}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots\)
378.2.e.d 378.e 63.h $6$ $3.018$ 6.0.309123.1 None \(6\) \(0\) \(-1\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+q^{2}+q^{4}-\beta _{2}q^{5}+(\beta _{3}+\beta _{4}-\beta _{5})q^{7}+\cdots\)

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

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