Properties

Label 57.2.e.b
Level 57
Weight 2
Character orbit 57.e
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.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} - \nu^{3} + 9 \nu^{2} - 21 \nu - 9 \)\()/27\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} - \nu^{3} - 18 \nu^{2} + 33 \nu - 9 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 36 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} + 7 \nu^{3} + 9 \nu^{2} + 12 \nu + 9 \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} + 3 \nu - 72 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{5} + 7 \beta_{4} - 11 \beta_{3} + \beta_{2} - \beta_{1} + 7\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 20\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(7 \beta_{5} - 2 \beta_{4} - 20 \beta_{3} + \beta_{2} - 10 \beta_{1} + 43\)\()/3\)

Character Values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.71903 0.211943i
−1.62241 0.606458i
0.403374 + 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 1.68443i
−1.04307 1.80664i −0.500000 0.866025i −1.17597 + 2.03684i −0.675970 1.17081i −1.04307 + 1.80664i 0.351939 0.734191 −0.500000 + 0.866025i −1.41016 + 2.44247i
7.2 0.285997 + 0.495361i −0.500000 0.866025i 0.836412 1.44871i 1.33641 + 2.31473i 0.285997 0.495361i −3.67282 2.10083 −0.500000 + 0.866025i −0.764419 + 1.32401i
7.3 1.25707 + 2.17731i −0.500000 0.866025i −2.16044 + 3.74200i −1.66044 2.87597i 1.25707 2.17731i 2.32088 −5.83502 −0.500000 + 0.866025i 4.17458 7.23058i
49.1 −1.04307 + 1.80664i −0.500000 + 0.866025i −1.17597 2.03684i −0.675970 + 1.17081i −1.04307 1.80664i 0.351939 0.734191 −0.500000 0.866025i −1.41016 2.44247i
49.2 0.285997 0.495361i −0.500000 + 0.866025i 0.836412 + 1.44871i 1.33641 2.31473i 0.285997 + 0.495361i −3.67282 2.10083 −0.500000 0.866025i −0.764419 1.32401i
49.3 1.25707 2.17731i −0.500000 + 0.866025i −2.16044 3.74200i −1.66044 + 2.87597i 1.25707 + 2.17731i 2.32088 −5.83502 −0.500000 0.866025i 4.17458 + 7.23058i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 6 T_{2}^{4} - T_{2}^{3} + 28 T_{2}^{2} - 15 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).