↑

Properties

Label 378.2.d
Level $378$
Weight $2$
Character orbit 378.d
Rep. character $\chi_{378}(377,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $3$
Sturm bound $144$
Trace bound $22$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(144\)
Trace bound: \(22\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

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

Total New Old
Modular forms 84 12 72
Cusp forms 60 12 48
Eisenstein series 24 0 24

Trace form

\( 12 q - 12 q^{4} + 6 q^{7} + O(q^{10}) \) \( 12 q - 12 q^{4} + 6 q^{7} + 12 q^{16} + 12 q^{25} - 6 q^{28} + 12 q^{37} - 60 q^{43} + 30 q^{49} - 36 q^{58} - 12 q^{64} - 12 q^{67} - 18 q^{70} + 24 q^{79} + 72 q^{85} - 36 q^{91} + O(q^{100}) \)

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.d.a 378.d 21.c $4$ $3.018$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}^{3}q^{2}-q^{4}+(-2\zeta_{12}+\zeta_{12}^{3})q^{5}+\cdots\)
378.2.d.b 378.d 21.c $4$ $3.018$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}q^{2}-q^{4}-\zeta_{12}^{3}q^{5}+(2-\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
378.2.d.c 378.d 21.c $4$ $3.018$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{2}-q^{4}-2\zeta_{12}^{3}q^{5}+(2-\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)

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

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