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.

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

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

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

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 \(T_{2}^{6} \) \(\mathstrut -\mathstrut T_{2}^{5} \) \(\mathstrut +\mathstrut 6 T_{2}^{4} \) \(\mathstrut -\mathstrut T_{2}^{3} \) \(\mathstrut +\mathstrut 28 T_{2}^{2} \) \(\mathstrut -\mathstrut 15 T_{2} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).