Properties

Label 57.2.f.a
Level 57
Weight 2
Character orbit 57.f
Analytic conductor 0.455
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
−0.965926 + 1.67303i 1.57313 0.724745i −0.866025 1.50000i 1.22474 + 0.707107i −0.307007 + 3.33195i −3.73205 −0.517638 1.94949 2.28024i −2.36603 + 1.36603i
8.2 −0.258819 + 0.448288i −1.57313 + 0.724745i 0.866025 + 1.50000i 1.22474 + 0.707107i 0.0822623 0.892794i −0.267949 −1.93185 1.94949 2.28024i −0.633975 + 0.366025i
8.3 0.258819 0.448288i −0.158919 1.72474i 0.866025 + 1.50000i −1.22474 0.707107i −0.814313 0.375156i −0.267949 1.93185 −2.94949 + 0.548188i −0.633975 + 0.366025i
8.4 0.965926 1.67303i 0.158919 + 1.72474i −0.866025 1.50000i −1.22474 0.707107i 3.03906 + 1.40010i −3.73205 0.517638 −2.94949 + 0.548188i −2.36603 + 1.36603i
50.1 −0.965926 1.67303i 1.57313 + 0.724745i −0.866025 + 1.50000i 1.22474 0.707107i −0.307007 3.33195i −3.73205 −0.517638 1.94949 + 2.28024i −2.36603 1.36603i
50.2 −0.258819 0.448288i −1.57313 0.724745i 0.866025 1.50000i 1.22474 0.707107i 0.0822623 + 0.892794i −0.267949 −1.93185 1.94949 + 2.28024i −0.633975 0.366025i
50.3 0.258819 + 0.448288i −0.158919 + 1.72474i 0.866025 1.50000i −1.22474 + 0.707107i −0.814313 + 0.375156i −0.267949 1.93185 −2.94949 0.548188i −0.633975 0.366025i
50.4 0.965926 + 1.67303i 0.158919 1.72474i −0.866025 + 1.50000i −1.22474 + 0.707107i 3.03906 1.40010i −3.73205 0.517638 −2.94949 0.548188i −2.36603 1.36603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
19.d Odd 1 yes
57.f Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(57, [\chi])\).