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Properties

Label 57.3.g
Level $57$
Weight $3$
Character orbit 57.g
Rep. character $\chi_{57}(31,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $12$
Newform subspaces $2$
Sturm bound $20$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(57, [\chi])\).

Total New Old
Modular forms 32 12 20
Cusp forms 24 12 12
Eisenstein series 8 0 8

Trace form

\( 12 q + 10 q^{4} + 2 q^{5} - 6 q^{6} + 4 q^{7} + 18 q^{9} + O(q^{10}) \) \( 12 q + 10 q^{4} + 2 q^{5} - 6 q^{6} + 4 q^{7} + 18 q^{9} + 84 q^{10} - 36 q^{11} - 18 q^{13} - 138 q^{14} - 18 q^{15} - 22 q^{16} + 28 q^{17} - 56 q^{19} - 92 q^{20} + 72 q^{21} - 48 q^{22} + 30 q^{23} + 60 q^{24} - 6 q^{25} + 176 q^{26} - 82 q^{28} - 36 q^{29} - 48 q^{30} - 12 q^{32} + 54 q^{33} - 60 q^{34} - 30 q^{35} - 30 q^{36} + 202 q^{38} - 24 q^{39} + 192 q^{40} + 120 q^{41} - 24 q^{42} + 118 q^{43} + 82 q^{44} + 12 q^{45} - 76 q^{47} + 72 q^{48} + 48 q^{49} - 144 q^{51} + 78 q^{52} + 90 q^{53} + 18 q^{54} - 28 q^{55} - 144 q^{57} - 304 q^{58} + 162 q^{59} - 234 q^{60} - 114 q^{61} - 230 q^{62} + 6 q^{63} + 212 q^{64} - 120 q^{66} - 102 q^{67} - 176 q^{68} - 96 q^{70} - 288 q^{71} - 54 q^{72} + 26 q^{73} - 96 q^{74} - 2 q^{76} + 244 q^{77} - 90 q^{78} - 114 q^{79} + 318 q^{80} - 54 q^{81} + 76 q^{82} - 600 q^{83} + 184 q^{85} + 210 q^{86} + 336 q^{87} + 570 q^{89} + 252 q^{90} - 306 q^{91} - 14 q^{92} + 108 q^{93} - 110 q^{95} + 228 q^{96} - 240 q^{97} + 276 q^{98} - 54 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(57, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
57.3.g.a 57.g 19.d $6$ $1.553$ 6.0.6967728.1 None \(-3\) \(9\) \(-2\) \(26\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(2-\beta _{1})q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
57.3.g.b 57.g 19.d $6$ $1.553$ 6.0.92607408.1 None \(3\) \(-9\) \(4\) \(-22\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{4}q^{2}+(-2-\beta _{3})q^{3}+(1+\beta _{1}+2\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(57, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(57, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 2}\)