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 \(3\), degree \(2\), minimal)

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: \( 3 \)
Twist minimal: yes
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 Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.e.b 6
3.b odd 2 1 171.2.f.b 6
4.b odd 2 1 912.2.q.l 6
12.b even 2 1 2736.2.s.z 6
19.c even 3 1 inner 57.2.e.b 6
19.c even 3 1 1083.2.a.l 3
19.d odd 6 1 1083.2.a.o 3
57.f even 6 1 3249.2.a.t 3
57.h odd 6 1 171.2.f.b 6
57.h odd 6 1 3249.2.a.y 3
76.g odd 6 1 912.2.q.l 6
228.m even 6 1 2736.2.s.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 1.a even 1 1 trivial
57.2.e.b 6 19.c even 3 1 inner
171.2.f.b 6 3.b odd 2 1
171.2.f.b 6 57.h odd 6 1
912.2.q.l 6 4.b odd 2 1
912.2.q.l 6 76.g odd 6 1
1083.2.a.l 3 19.c even 3 1
1083.2.a.o 3 19.d odd 6 1
2736.2.s.z 6 12.b even 2 1
2736.2.s.z 6 228.m even 6 1
3249.2.a.t 3 57.f even 6 1
3249.2.a.y 3 57.h odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{3} - 2 T^{4} - T^{5} + 11 T^{6} - 2 T^{7} - 8 T^{8} + 8 T^{9} - 32 T^{11} + 64 T^{12} \)
$3$ \( ( 1 + T + T^{2} )^{3} \)
$5$ \( 1 + 2 T - 3 T^{2} - 2 T^{3} - 2 T^{4} - 34 T^{5} - 31 T^{6} - 170 T^{7} - 50 T^{8} - 250 T^{9} - 1875 T^{10} + 6250 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 + T + 12 T^{2} + 17 T^{3} + 84 T^{4} + 49 T^{5} + 343 T^{6} )^{2} \)
$11$ \( ( 1 + 9 T^{2} - 36 T^{3} + 99 T^{4} + 1331 T^{6} )^{2} \)
$13$ \( 1 - T - 17 T^{2} + 40 T^{3} + 61 T^{4} - 223 T^{5} + 854 T^{6} - 2899 T^{7} + 10309 T^{8} + 87880 T^{9} - 485537 T^{10} - 371293 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 - 17 T^{2} + 289 T^{4} )^{3} \)
$19$ \( 1 - 4 T + 17 T^{2} - 136 T^{3} + 323 T^{4} - 1444 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 14 T + 99 T^{2} + 382 T^{3} + 346 T^{4} - 7726 T^{5} - 57613 T^{6} - 177698 T^{7} + 183034 T^{8} + 4647794 T^{9} + 27704259 T^{10} + 90108802 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 4 T - 39 T^{2} - 52 T^{3} + 886 T^{4} - 1916 T^{5} - 33907 T^{6} - 55564 T^{7} + 745126 T^{8} - 1268228 T^{9} - 27583959 T^{10} + 82044596 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 - 15 T + 144 T^{2} - 899 T^{3} + 4464 T^{4} - 14415 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{6} \)
$41$ \( 1 - 4 T - 75 T^{2} + 100 T^{3} + 3622 T^{4} + 2012 T^{5} - 174751 T^{6} + 82492 T^{7} + 6088582 T^{8} + 6892100 T^{9} - 211932075 T^{10} - 463424804 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 3 T - 99 T^{2} + 218 T^{3} + 6207 T^{4} - 6951 T^{5} - 285738 T^{6} - 298893 T^{7} + 11476743 T^{8} + 17332526 T^{9} - 338461299 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( ( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{3} \)
$53$ \( 1 - 6 T - 51 T^{2} + 102 T^{3} + 1086 T^{4} + 11334 T^{5} - 107183 T^{6} + 600702 T^{7} + 3050574 T^{8} + 15185454 T^{9} - 402414531 T^{10} - 2509172958 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 153 T^{2} + 72 T^{3} + 14382 T^{4} - 5508 T^{5} - 969077 T^{6} - 324972 T^{7} + 50063742 T^{8} + 14787288 T^{9} - 1853956233 T^{10} + 42180533641 T^{12} \)
$61$ \( 1 + 13 T - 25 T^{2} - 504 T^{3} + 6133 T^{4} + 34211 T^{5} - 181514 T^{6} + 2086871 T^{7} + 22820893 T^{8} - 114398424 T^{9} - 346146025 T^{10} + 10979751913 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 9 T - 39 T^{2} - 250 T^{3} + 375 T^{4} - 24519 T^{5} - 284658 T^{6} - 1642773 T^{7} + 1683375 T^{8} - 75190750 T^{9} - 785893719 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 18 T + 99 T^{2} - 234 T^{3} + 306 T^{4} + 55062 T^{5} - 871373 T^{6} + 3909402 T^{7} + 1542546 T^{8} - 83751174 T^{9} + 2515756419 T^{10} - 32476128318 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 19 T + 59 T^{2} + 252 T^{3} + 17041 T^{4} + 91889 T^{5} - 279578 T^{6} + 6707897 T^{7} + 90811489 T^{8} + 98032284 T^{9} + 1675496219 T^{10} + 39388360267 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 11 T - 119 T^{2} - 494 T^{3} + 21403 T^{4} + 27611 T^{5} - 1863994 T^{6} + 2181269 T^{7} + 133576123 T^{8} - 243561266 T^{9} - 4635059639 T^{10} + 33847620389 T^{11} + 243087455521 T^{12} \)
$83$ \( ( 1 + 4 T + 169 T^{2} + 472 T^{3} + 14027 T^{4} + 27556 T^{5} + 571787 T^{6} )^{2} \)
$89$ \( 1 - 16 T - 87 T^{2} + 424 T^{3} + 39826 T^{4} - 163780 T^{5} - 2350663 T^{6} - 14576420 T^{7} + 315461746 T^{8} + 298906856 T^{9} - 5458574967 T^{10} - 89344951184 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 2 T - 19 T^{2} - 2166 T^{3} + 418 T^{4} + 14486 T^{5} + 2859997 T^{6} + 1405142 T^{7} + 3932962 T^{8} - 1976849718 T^{9} - 1682056339 T^{10} - 17174680514 T^{11} + 832972004929 T^{12} \)
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