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Properties

Label 228.3.j
Level $228$
Weight $3$
Character orbit 228.j
Rep. character $\chi_{228}(7,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $80$
Newform subspaces $4$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 76 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(228, [\chi])\).

Total New Old
Modular forms 168 80 88
Cusp forms 152 80 72
Eisenstein series 16 0 16

Trace form

\( 80 q + 2 q^{4} + 6 q^{6} - 12 q^{8} + 120 q^{9} + O(q^{10}) \) \( 80 q + 2 q^{4} + 6 q^{6} - 12 q^{8} + 120 q^{9} + 8 q^{10} - 8 q^{13} - 30 q^{14} - 14 q^{16} + 124 q^{20} + 24 q^{21} - 60 q^{22} - 36 q^{24} - 200 q^{25} + 120 q^{26} + 114 q^{28} - 40 q^{32} - 16 q^{34} - 6 q^{36} + 80 q^{37} + 294 q^{38} + 28 q^{40} + 16 q^{41} - 210 q^{44} - 320 q^{46} - 24 q^{48} - 512 q^{49} - 728 q^{50} + 130 q^{52} - 16 q^{53} - 18 q^{54} + 192 q^{56} + 72 q^{57} + 8 q^{58} + 42 q^{60} + 168 q^{61} - 78 q^{62} - 124 q^{64} + 320 q^{65} - 144 q^{66} - 224 q^{68} + 384 q^{69} + 36 q^{70} - 18 q^{72} + 120 q^{73} + 160 q^{74} - 342 q^{76} + 64 q^{77} - 78 q^{78} + 78 q^{80} - 360 q^{81} + 460 q^{82} + 432 q^{84} - 320 q^{85} - 170 q^{86} + 1120 q^{88} - 96 q^{89} - 24 q^{90} + 162 q^{92} - 168 q^{93} + 1040 q^{94} - 108 q^{96} + 288 q^{97} - 580 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(228, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
228.3.j.a 228.j 76.g $2$ $6.213$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-3\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+2\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
228.3.j.b 228.j 76.g $2$ $6.213$ \(\Q(\sqrt{-3}) \) None \(4\) \(3\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+2q^{2}+(2-\zeta_{6})q^{3}+4q^{4}-4\zeta_{6}q^{5}+\cdots\)
228.3.j.c 228.j 76.g $38$ $6.213$ None \(-1\) \(-57\) \(4\) \(0\) $\mathrm{SU}(2)[C_{6}]$
228.3.j.d 228.j 76.g $38$ $6.213$ None \(-1\) \(57\) \(4\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(228, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(228, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 2}\)