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Properties

Label 57.2.e
Level $57$
Weight $2$
Character orbit 57.e
Rep. character $\chi_{57}(7,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $2$
Sturm bound $13$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(57, [\chi])\).

Total New Old
Modular forms 16 8 8
Cusp forms 8 8 0
Eisenstein series 8 0 8

Trace form

\( 8 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{6} - 12 q^{8} - 4 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{3} - 4 q^{4} - 2 q^{5} + 2 q^{6} - 12 q^{8} - 4 q^{9} + 4 q^{10} - 4 q^{11} + 12 q^{12} - 4 q^{13} + 2 q^{14} - 2 q^{15} - 4 q^{16} + 4 q^{17} - 4 q^{19} + 44 q^{20} + 2 q^{21} - 16 q^{22} - 10 q^{23} - 32 q^{26} + 4 q^{27} - 16 q^{28} + 4 q^{29} - 8 q^{30} + 24 q^{31} + 12 q^{32} - 2 q^{33} + 4 q^{34} - 18 q^{35} - 4 q^{36} + 42 q^{38} - 12 q^{39} + 24 q^{40} + 16 q^{41} + 4 q^{42} + 4 q^{43} - 14 q^{44} + 4 q^{45} + 32 q^{46} + 24 q^{47} - 6 q^{48} - 16 q^{49} - 56 q^{50} - 4 q^{51} + 2 q^{53} + 2 q^{54} - 12 q^{55} - 48 q^{56} - 6 q^{57} - 32 q^{58} - 10 q^{59} - 22 q^{60} + 26 q^{62} - 16 q^{64} + 12 q^{65} - 20 q^{66} - 20 q^{67} + 8 q^{68} + 36 q^{69} + 24 q^{70} + 12 q^{71} + 6 q^{72} - 8 q^{73} - 4 q^{74} + 20 q^{75} + 20 q^{76} + 20 q^{77} + 26 q^{78} - 12 q^{79} - 10 q^{80} - 4 q^{81} + 4 q^{82} - 8 q^{83} + 36 q^{84} + 18 q^{86} + 24 q^{87} + 24 q^{88} + 22 q^{89} + 4 q^{90} - 12 q^{91} - 2 q^{92} - 18 q^{93} + 2 q^{95} - 44 q^{96} + 20 q^{98} + 2 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(57, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
57.2.e.a 57.e 19.c $2$ $0.455$ \(\Q(\sqrt{-3}) \) None 57.2.e.a \(-1\) \(1\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
57.2.e.b 57.e 19.c $6$ $0.455$ 6.0.954288.1 None 57.2.e.b \(1\) \(-3\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{4}+\beta _{5})q^{2}-\beta _{3}q^{3}+(-2-\beta _{1}+\cdots)q^{4}+\cdots\)