↑

Properties

Label 378.4.g
Level $378$
Weight $4$
Character orbit 378.g
Rep. character $\chi_{378}(109,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $64$
Newform subspaces $8$
Sturm bound $288$
Trace bound $5$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.g (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(378, [\chi])\).

Total New Old
Modular forms 456 64 392
Cusp forms 408 64 344
Eisenstein series 48 0 48

Trace form

\( 64 q - 128 q^{4} + 52 q^{7} + O(q^{10}) \) \( 64 q - 128 q^{4} + 52 q^{7} + 12 q^{10} + 236 q^{13} - 512 q^{16} + 266 q^{19} - 312 q^{22} - 806 q^{25} - 104 q^{28} + 110 q^{31} + 240 q^{34} - 262 q^{37} + 48 q^{40} - 3280 q^{43} - 96 q^{46} - 3104 q^{49} - 472 q^{52} + 744 q^{55} + 816 q^{58} + 836 q^{61} + 4096 q^{64} - 790 q^{67} + 780 q^{70} - 2578 q^{73} - 2128 q^{76} - 2254 q^{79} - 1056 q^{82} + 8448 q^{85} + 624 q^{88} - 2692 q^{91} + 2544 q^{94} + 1124 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
378.4.g.a 378.g 7.c $6$ $22.303$ 6.0.\(\cdots\).2 None \(-6\) \(0\) \(-5\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)
378.4.g.b 378.g 7.c $6$ $22.303$ 6.0.\(\cdots\).2 None \(6\) \(0\) \(5\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(2\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
378.4.g.c 378.g 7.c $8$ $22.303$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(2\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{1})q^{2}-4\beta _{1}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
378.4.g.d 378.g 7.c $8$ $22.303$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(0\) \(4\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-4-4\beta _{1})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
378.4.g.e 378.g 7.c $8$ $22.303$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(-4\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(-4-4\beta _{1})q^{4}+(\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
378.4.g.f 378.g 7.c $8$ $22.303$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(-2\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
378.4.g.g 378.g 7.c $10$ $22.303$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(0\) \(2\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{2})q^{2}+4\beta _{2}q^{4}+\beta _{5}q^{5}+\cdots\)
378.4.g.h 378.g 7.c $10$ $22.303$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(0\) \(-2\) \(-13\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{2})q^{2}+4\beta _{2}q^{4}-\beta _{5}q^{5}+(3\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)