Properties

Label 57.2.i.a
Level 57
Weight 2
Character orbit 57.i
Analytic conductor 0.455
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.i (of order \(9\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{2} \) \( -\zeta_{18}^{4} q^{3} \) \( + ( -1 + \zeta_{18} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{4} \) \( + ( -1 - \zeta_{18} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{5} \) \( + ( \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{6} \) \( + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} \) \( + ( -\zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{8} \) \( + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{2} \) \( -\zeta_{18}^{4} q^{3} \) \( + ( -1 + \zeta_{18} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{4} \) \( + ( -1 - \zeta_{18} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{5} \) \( + ( \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{6} \) \( + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} \) \( + ( -\zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{8} \) \( + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{9} \) \( + ( -1 - \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{10} \) \( + ( -\zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} + 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} \) \( + ( \zeta_{18} - \zeta_{18}^{2} ) q^{12} \) \( + ( 2 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{13} \) \( + ( 1 - 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{14} \) \( + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{15} \) \( + ( -2 + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{16} \) \( + ( -1 + 3 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{17} \) \( + ( -\zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{18} \) \( + ( 4 - 2 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{19} \) \( + ( 1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{20} \) \( + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{21} \) \( + ( -2 + 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{22} \) \( + ( -3 + 3 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{23} \) \( + ( -1 - \zeta_{18} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{24} \) \( + ( 1 + \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{25} \) \( + ( 4 + 2 \zeta_{18} - \zeta_{18}^{2} - 4 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{26} \) \( + \zeta_{18}^{3} q^{27} \) \( + ( -1 - \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{28} \) \( + ( -3 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{29} \) \( + ( \zeta_{18}^{2} + 2 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{30} \) \( + ( -8 - \zeta_{18} + 8 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{31} \) \( + ( 1 + 2 \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{32} \) \( + ( 2 + \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{33} \) \( + ( 5 - 3 \zeta_{18} - 5 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{34} \) \( + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{35} \) \( + ( -1 + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{36} \) \( + ( 2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} \) \( + ( \zeta_{18} + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{38} \) \( + ( -1 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} ) q^{39} \) \( + ( 1 + 3 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{40} \) \( + ( 3 - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{41} \) \( + ( -2 + \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{42} \) \( + ( 4 - 7 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 7 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{43} \) \( + ( 2 - \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{44} \) \( + ( 1 + \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{45} \) \( + ( 5 \zeta_{18} - 3 \zeta_{18}^{3} + 5 \zeta_{18}^{5} ) q^{46} \) \( + ( -4 - 4 \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 7 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{47} \) \( + ( -1 - \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{48} \) \( + ( 3 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} - \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{49} \) \( + ( 3 + 2 \zeta_{18} - 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{50} \) \( + ( -1 - 2 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{51} \) \( + ( 2 - 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{52} \) \( + ( -5 + 2 \zeta_{18} - 6 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 5 \zeta_{18}^{4} ) q^{53} \) \( + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{54} \) \( + ( -5 - 2 \zeta_{18} - 6 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{55} \) \( + ( 1 + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{56} \) \( + ( -1 - 2 \zeta_{18} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{57} \) \( + ( -4 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{58} \) \( + ( 1 + 5 \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{59} \) \( + ( \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{60} \) \( + ( -1 - 5 \zeta_{18} - 3 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{61} \) \( + ( -9 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{62} \) \( + ( \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{63} \) \( + ( 1 - 3 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{64} \) \( + ( \zeta_{18} + 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{65} \) \( + ( 4 + 4 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{66} \) \( + ( 4 + 4 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{67} \) \( + ( -6 \zeta_{18} + 4 \zeta_{18}^{2} + \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{68} \) \( + ( 2 + 3 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} ) q^{69} \) \( + ( 2 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{70} \) \( + ( 4 - 3 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{71} \) \( + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{72} \) \( + ( -5 + 5 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{73} \) \( + ( 2 + \zeta_{18} + 5 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{74} \) \( + ( 2 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{75} \) \( + ( -1 + 4 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{76} \) \( + ( -2 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{77} \) \( + ( -2 - 4 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{78} \) \( + ( 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{79} \) \( + ( 2 + 4 \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{80} \) \( + ( \zeta_{18} - \zeta_{18}^{4} ) q^{81} \) \( + ( 4 - \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{82} \) \( + ( 5 + \zeta_{18} + 3 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{83} \) \( + ( 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{84} \) \( + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{85} \) \( + ( -5 - 5 \zeta_{18} + 6 \zeta_{18}^{2} + 9 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{86} \) \( + ( 4 \zeta_{18} - 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{87} \) \( + ( 2 + 5 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{88} \) \( + ( 5 - 10 \zeta_{18} + 5 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{89} \) \( + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{90} \) \( + ( -3 + 2 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{91} \) \( + ( 2 - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 6 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{92} \) \( + ( 1 + 8 \zeta_{18} + \zeta_{18}^{2} ) q^{93} \) \( + ( -1 - 6 \zeta_{18} - 6 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{94} \) \( + ( -3 + \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{95} \) \( + ( 3 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{96} \) \( + ( 5 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{97} \) \( + ( -1 + \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{98} \) \( + ( 1 - 3 \zeta_{18} - \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 3q^{6} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut -\mathstrut 6q^{10} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut 15q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 18q^{19} \) \(\mathstrut +\mathstrut 6q^{20} \) \(\mathstrut +\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 9q^{23} \) \(\mathstrut -\mathstrut 6q^{24} \) \(\mathstrut +\mathstrut 12q^{26} \) \(\mathstrut +\mathstrut 3q^{27} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 24q^{31} \) \(\mathstrut -\mathstrut 3q^{32} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 21q^{34} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 3q^{36} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 6q^{38} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 9q^{40} \) \(\mathstrut +\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 9q^{42} \) \(\mathstrut +\mathstrut 18q^{43} \) \(\mathstrut +\mathstrut 9q^{44} \) \(\mathstrut +\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 9q^{46} \) \(\mathstrut -\mathstrut 33q^{47} \) \(\mathstrut -\mathstrut 12q^{48} \) \(\mathstrut +\mathstrut 12q^{49} \) \(\mathstrut +\mathstrut 9q^{50} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut +\mathstrut 9q^{52} \) \(\mathstrut -\mathstrut 24q^{53} \) \(\mathstrut -\mathstrut 3q^{54} \) \(\mathstrut -\mathstrut 33q^{55} \) \(\mathstrut +\mathstrut 6q^{56} \) \(\mathstrut -\mathstrut 9q^{57} \) \(\mathstrut -\mathstrut 24q^{58} \) \(\mathstrut +\mathstrut 3q^{60} \) \(\mathstrut -\mathstrut 21q^{61} \) \(\mathstrut -\mathstrut 51q^{62} \) \(\mathstrut -\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 15q^{65} \) \(\mathstrut +\mathstrut 18q^{66} \) \(\mathstrut +\mathstrut 30q^{67} \) \(\mathstrut +\mathstrut 3q^{68} \) \(\mathstrut +\mathstrut 6q^{69} \) \(\mathstrut -\mathstrut 6q^{70} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut +\mathstrut 6q^{72} \) \(\mathstrut -\mathstrut 27q^{73} \) \(\mathstrut +\mathstrut 21q^{74} \) \(\mathstrut +\mathstrut 12q^{75} \) \(\mathstrut -\mathstrut 9q^{78} \) \(\mathstrut +\mathstrut 12q^{79} \) \(\mathstrut +\mathstrut 24q^{80} \) \(\mathstrut +\mathstrut 15q^{82} \) \(\mathstrut +\mathstrut 15q^{83} \) \(\mathstrut -\mathstrut 3q^{84} \) \(\mathstrut +\mathstrut 15q^{85} \) \(\mathstrut -\mathstrut 3q^{86} \) \(\mathstrut +\mathstrut 9q^{87} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut +\mathstrut 45q^{89} \) \(\mathstrut +\mathstrut 3q^{90} \) \(\mathstrut -\mathstrut 12q^{91} \) \(\mathstrut +\mathstrut 3q^{92} \) \(\mathstrut +\mathstrut 6q^{93} \) \(\mathstrut -\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 18q^{96} \) \(\mathstrut +\mathstrut 21q^{97} \) \(\mathstrut +\mathstrut 3q^{98} \) \(\mathstrut -\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 0.342020i
−0.173648 + 0.984808i
1.43969 + 0.524005i −0.173648 0.984808i 0.266044 + 0.223238i −1.93969 + 1.62760i 0.266044 1.50881i −0.266044 + 0.460802i −1.26604 2.19285i −0.939693 + 0.342020i −3.64543 + 1.32683i
16.1 −0.266044 0.223238i 0.939693 0.342020i −0.326352 1.85083i −0.233956 + 1.32683i −0.326352 0.118782i 0.326352 + 0.565258i −0.673648 + 1.16679i 0.766044 0.642788i 0.358441 0.300767i
25.1 −0.266044 + 0.223238i 0.939693 + 0.342020i −0.326352 + 1.85083i −0.233956 1.32683i −0.326352 + 0.118782i 0.326352 0.565258i −0.673648 1.16679i 0.766044 + 0.642788i 0.358441 + 0.300767i
28.1 0.326352 + 1.85083i −0.766044 + 0.642788i −1.43969 + 0.524005i −0.826352 0.300767i −1.43969 1.20805i 1.43969 2.49362i 0.439693 + 0.761570i 0.173648 0.984808i 0.286989 1.62760i
43.1 1.43969 0.524005i −0.173648 + 0.984808i 0.266044 0.223238i −1.93969 1.62760i 0.266044 + 1.50881i −0.266044 0.460802i −1.26604 + 2.19285i −0.939693 0.342020i −3.64543 1.32683i
55.1 0.326352 1.85083i −0.766044 0.642788i −1.43969 0.524005i −0.826352 + 0.300767i −1.43969 + 1.20805i 1.43969 + 2.49362i 0.439693 0.761570i 0.173648 + 0.984808i 0.286989 + 1.62760i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
19.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 3 T_{2}^{5} \) \(\mathstrut +\mathstrut 6 T_{2}^{4} \) \(\mathstrut -\mathstrut 8 T_{2}^{3} \) \(\mathstrut +\mathstrut 3 T_{2}^{2} \) \(\mathstrut +\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).