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Properties

Label 378.4.f
Level $378$
Weight $4$
Character orbit 378.f
Rep. character $\chi_{378}(127,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $36$
Newform subspaces $4$
Sturm bound $288$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(288\)
Trace bound: \(1\)
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 36 420
Cusp forms 408 36 372
Eisenstein series 48 0 48

Trace form

\( 36 q + 4 q^{2} - 72 q^{4} + 16 q^{5} - 32 q^{8} + O(q^{10}) \) \( 36 q + 4 q^{2} - 72 q^{4} + 16 q^{5} - 32 q^{8} + 86 q^{11} + 56 q^{14} - 288 q^{16} - 76 q^{17} - 180 q^{19} + 64 q^{20} + 36 q^{22} - 412 q^{23} - 162 q^{25} - 308 q^{29} + 72 q^{31} + 64 q^{32} - 180 q^{34} - 280 q^{35} + 288 q^{37} + 188 q^{38} - 522 q^{41} - 342 q^{43} - 688 q^{44} - 1188 q^{47} - 882 q^{49} + 1076 q^{50} - 432 q^{53} + 1584 q^{55} + 224 q^{56} - 362 q^{59} + 624 q^{62} + 2304 q^{64} + 1872 q^{65} - 1998 q^{67} + 152 q^{68} + 2216 q^{71} - 828 q^{73} + 872 q^{74} + 360 q^{76} + 616 q^{77} + 936 q^{79} - 512 q^{80} + 5832 q^{82} - 1384 q^{83} - 936 q^{85} + 2132 q^{86} + 144 q^{88} + 3120 q^{89} - 1008 q^{91} - 1648 q^{92} - 2308 q^{95} - 3114 q^{97} - 392 q^{98} + O(q^{100}) \)

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.f.a 378.f 9.c $6$ $22.303$ 6.0.309123.1 None \(-6\) \(0\) \(9\) \(-21\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{3}q^{2}+(-4+4\beta _{3})q^{4}+(3-2\beta _{1}+\cdots)q^{5}+\cdots\)
378.4.f.b 378.f 9.c $8$ $22.303$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(-1\) \(28\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{2})q^{2}-4\beta _{2}q^{4}+(\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
378.4.f.c 378.f 9.c $10$ $22.303$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(0\) \(-1\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-4-4\beta _{1})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
378.4.f.d 378.f 9.c $12$ $22.303$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(0\) \(9\) \(-42\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(-4-4\beta _{1})q^{4}+(2+\beta _{1}+\cdots)q^{5}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)