Properties

Label 378.2.g
Level 378
Weight 2
Character orbit g
Rep. character \(\chi_{378}(109,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 20
Newforms 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: \(\Q(\zeta_{3})\)
Newforms: \( 8 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(378, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
378.2.g.a \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
378.2.g.b \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
378.2.g.c \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
378.2.g.d \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
378.2.g.e \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
378.2.g.f \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(3\) \(-4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
378.2.g.g \(4\) \(3.018\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(2\) \(0\) \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
378.2.g.h \(4\) \(3.018\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(0\) \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

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

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