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

Related objects

Downloads

Learn more about

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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{6} - 16q^{7} - 4q^{9} + O(q^{10}) \) \( 8q + 4q^{6} - 16q^{7} - 4q^{9} - 12q^{10} + 12q^{13} + 4q^{16} - 12q^{21} + 12q^{22} + 4q^{24} - 12q^{25} + 12q^{28} - 8q^{30} + 24q^{33} + 24q^{34} + 16q^{39} - 12q^{40} - 20q^{42} - 16q^{43} + 32q^{45} + 24q^{48} - 48q^{51} - 24q^{52} - 4q^{54} + 16q^{55} - 28q^{57} - 64q^{58} - 24q^{60} + 28q^{61} + 8q^{63} - 32q^{64} - 4q^{66} + 36q^{70} - 24q^{72} + 12q^{73} + 60q^{76} + 36q^{78} + 24q^{79} + 28q^{81} + 16q^{82} + 32q^{87} + 12q^{90} - 48q^{91} - 4q^{93} + 72q^{96} + 24q^{97} - 32q^{99} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(57, [\chi])\).