↑

Properties

Label 57.2.i
Level $57$
Weight $2$
Character orbit 57.i
Rep. character $\chi_{57}(4,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $18$
Newform subspaces $2$
Sturm bound $13$
Trace bound $1$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.i (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{9})\)
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 54 18 36
Cusp forms 30 18 12
Eisenstein series 24 0 24

Trace form

\( 18 q - 6 q^{4} - 6 q^{6} - 6 q^{7} + O(q^{10}) \) \( 18 q - 6 q^{4} - 6 q^{6} - 6 q^{7} - 24 q^{10} - 6 q^{11} - 6 q^{12} - 9 q^{13} + 6 q^{14} + 12 q^{15} + 18 q^{16} + 9 q^{19} - 12 q^{20} + 3 q^{21} - 24 q^{22} + 24 q^{24} - 12 q^{25} + 12 q^{26} - 3 q^{27} + 30 q^{28} + 12 q^{29} - 24 q^{31} - 6 q^{32} - 6 q^{33} + 42 q^{34} + 24 q^{35} - 6 q^{36} + 24 q^{38} + 6 q^{40} - 12 q^{41} - 12 q^{42} + 33 q^{43} + 24 q^{44} - 6 q^{45} + 18 q^{46} - 42 q^{47} - 24 q^{48} + 3 q^{49} + 18 q^{50} - 12 q^{51} - 18 q^{52} - 54 q^{53} - 6 q^{54} - 18 q^{55} - 72 q^{56} - 12 q^{57} - 48 q^{58} - 30 q^{59} - 12 q^{60} - 48 q^{62} + 6 q^{63} - 18 q^{64} + 36 q^{65} + 48 q^{66} + 33 q^{67} - 18 q^{68} + 12 q^{70} - 12 q^{71} + 36 q^{72} - 51 q^{73} + 60 q^{74} + 42 q^{75} - 60 q^{76} + 96 q^{77} + 36 q^{79} + 66 q^{80} - 18 q^{82} + 18 q^{83} + 18 q^{85} + 48 q^{86} + 24 q^{87} + 48 q^{88} + 48 q^{89} - 6 q^{90} + 12 q^{91} + 72 q^{92} + 12 q^{94} - 66 q^{95} - 24 q^{96} - 6 q^{97} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
57.2.i.a 57.i 19.e $6$ $0.455$ \(\Q(\zeta_{18})\) None \(3\) \(0\) \(-6\) \(3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{5})q^{2}-\zeta_{18}^{4}q^{3}+\cdots\)
57.2.i.b 57.i 19.e $12$ $0.455$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(0\) \(6\) \(-9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1-\beta _{1}-\beta _{2}+\beta _{5}-\beta _{9})q^{2}-\beta _{6}q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(57, [\chi])\) into lower level spaces

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