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Properties

Label 57.2.e.b.49.1
Level $57$
Weight $2$
Character 57.49
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [57,2,Mod(7,57)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("57.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
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
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.1
Root \(1.71903 + 0.211943i\) of defining polynomial
Character \(\chi\) \(=\) 57.49
Dual form 57.2.e.b.7.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.04307 + 1.80664i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-1.17597 - 2.03684i) q^{4} +(-0.675970 + 1.17081i) q^{5} +(-1.04307 - 1.80664i) q^{6} +0.351939 q^{7} +0.734191 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.41016 - 2.44247i) q^{10} +5.52420 q^{11} +2.35194 q^{12} +(2.58613 + 4.47931i) q^{13} +(-0.367095 + 0.635828i) q^{14} +(-0.675970 - 1.17081i) q^{15} +(1.58613 - 2.74726i) q^{16} +2.08613 q^{18} +(-2.43807 - 3.61328i) q^{19} +3.17968 q^{20} +(-0.175970 + 0.304788i) q^{21} +(-5.76210 + 9.98025i) q^{22} +(-4.41016 - 7.63862i) q^{23} +(-0.367095 + 0.635828i) q^{24} +(1.58613 + 2.74726i) q^{25} -10.7900 q^{26} +1.00000 q^{27} +(-0.413870 - 0.716844i) q^{28} +(-1.35194 - 2.34163i) q^{29} +2.82032 q^{30} -0.524200 q^{31} +(4.04307 + 7.00279i) q^{32} +(-2.76210 + 4.78410i) q^{33} +(-0.237900 + 0.412055i) q^{35} +(-1.17597 + 2.03684i) q^{36} -1.00000 q^{37} +(9.07097 - 0.635828i) q^{38} -5.17226 q^{39} +(-0.496291 + 0.859601i) q^{40} +(1.35194 - 2.34163i) q^{41} +(-0.367095 - 0.635828i) q^{42} +(3.26210 - 5.65012i) q^{43} +(-6.49629 - 11.2519i) q^{44} +1.35194 q^{45} +18.4003 q^{46} +(3.00000 + 5.19615i) q^{47} +(1.58613 + 2.74726i) q^{48} -6.87614 q^{49} -6.61775 q^{50} +(6.08242 - 10.5351i) q^{52} +(2.02791 + 3.51244i) q^{53} +(-1.04307 + 1.80664i) q^{54} +(-3.73419 + 6.46781i) q^{55} +0.258391 q^{56} +(4.34823 - 0.304788i) q^{57} +5.64064 q^{58} +(2.76210 - 4.78410i) q^{59} +(-1.58984 + 2.75368i) q^{60} +(0.938069 + 1.62478i) q^{61} +(0.546774 - 0.947041i) q^{62} +(-0.175970 - 0.304788i) q^{63} -10.5242 q^{64} -6.99258 q^{65} +(-5.76210 - 9.98025i) q^{66} +(-5.99629 - 10.3859i) q^{67} +8.82032 q^{69} +(-0.496291 - 0.859601i) q^{70} +(-2.52420 + 4.37204i) q^{71} +(-0.367095 - 0.635828i) q^{72} +(-3.85194 + 6.67175i) q^{73} +(1.04307 - 1.80664i) q^{74} -3.17226 q^{75} +(-4.49258 + 9.21507i) q^{76} +1.94418 q^{77} +(5.39500 - 9.34442i) q^{78} +(-3.91016 + 6.77260i) q^{79} +(2.14435 + 3.71413i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(2.82032 + 4.88494i) q^{82} +8.34452 q^{83} +0.827740 q^{84} +(6.80516 + 11.7869i) q^{86} +2.70388 q^{87} +4.05582 q^{88} +(2.32403 + 4.02534i) q^{89} +(-1.41016 + 2.44247i) q^{90} +(0.910161 + 1.57644i) q^{91} +(-10.3724 + 17.9656i) q^{92} +(0.262100 - 0.453970i) q^{93} -12.5168 q^{94} +(5.87854 - 0.412055i) q^{95} -8.08613 q^{96} +(6.90645 - 11.9623i) q^{97} +(7.17226 - 12.4227i) q^{98} +(-2.76210 - 4.78410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 3 q^{3} - 5 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - 3 q^{9} + 4 q^{10} + 10 q^{12} + q^{13} + 3 q^{14} - 2 q^{15} - 5 q^{16} - 2 q^{18} + 4 q^{19} + 44 q^{20} + q^{21} - 18 q^{22}+ \cdots + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.04307 + 1.80664i −0.737558 + 1.27749i 0.216033 + 0.976386i \(0.430688\pi\)
−0.953592 + 0.301103i \(0.902645\pi\)
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) −1.17597 − 2.03684i −0.587985 − 1.01842i
\(5\) −0.675970 + 1.17081i −0.302303 + 0.523604i −0.976657 − 0.214804i \(-0.931089\pi\)
0.674354 + 0.738408i \(0.264422\pi\)
\(6\) −1.04307 − 1.80664i −0.425830 − 0.737558i
\(7\) 0.351939 0.133021 0.0665103 − 0.997786i \(-0.478813\pi\)
0.0665103 + 0.997786i \(0.478813\pi\)
\(8\) 0.734191 0.259576
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −1.41016 − 2.44247i −0.445932 − 0.772377i
\(11\) 5.52420 1.66561 0.832804 − 0.553567i \(-0.186734\pi\)
0.832804 + 0.553567i \(0.186734\pi\)
\(12\) 2.35194 0.678946
\(13\) 2.58613 + 4.47931i 0.717263 + 1.24234i 0.962080 + 0.272767i \(0.0879389\pi\)
−0.244817 + 0.969569i \(0.578728\pi\)
\(14\) −0.367095 + 0.635828i −0.0981104 + 0.169932i
\(15\) −0.675970 − 1.17081i −0.174535 − 0.302303i
\(16\) 1.58613 − 2.74726i 0.396533 − 0.686815i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 2.08613 0.491706
\(19\) −2.43807 − 3.61328i −0.559331 − 0.828944i
\(20\) 3.17968 0.710998
\(21\) −0.175970 + 0.304788i −0.0383997 + 0.0665103i
\(22\) −5.76210 + 9.98025i −1.22848 + 2.12780i
\(23\) −4.41016 − 7.63862i −0.919582 − 1.59276i −0.800051 − 0.599932i \(-0.795194\pi\)
−0.119531 − 0.992830i \(-0.538139\pi\)
\(24\) −0.367095 + 0.635828i −0.0749331 + 0.129788i
\(25\) 1.58613 + 2.74726i 0.317226 + 0.549452i
\(26\) −10.7900 −2.11609
\(27\) 1.00000 0.192450
\(28\) −0.413870 − 0.716844i −0.0782140 − 0.135471i
\(29\) −1.35194 − 2.34163i −0.251049 − 0.434829i 0.712766 − 0.701402i \(-0.247442\pi\)
−0.963815 + 0.266573i \(0.914109\pi\)
\(30\) 2.82032 0.514918
\(31\) −0.524200 −0.0941490 −0.0470745 − 0.998891i \(-0.514990\pi\)
−0.0470745 + 0.998891i \(0.514990\pi\)
\(32\) 4.04307 + 7.00279i 0.714720 + 1.23793i
\(33\) −2.76210 + 4.78410i −0.480820 + 0.832804i
\(34\) 0 0
\(35\) −0.237900 + 0.412055i −0.0402125 + 0.0696500i
\(36\) −1.17597 + 2.03684i −0.195995 + 0.339473i
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 9.07097 − 0.635828i 1.47151 − 0.103145i
\(39\) −5.17226 −0.828225
\(40\) −0.496291 + 0.859601i −0.0784705 + 0.135915i
\(41\) 1.35194 − 2.34163i 0.211137 − 0.365701i −0.740933 − 0.671579i \(-0.765617\pi\)
0.952071 + 0.305878i \(0.0989499\pi\)
\(42\) −0.367095 − 0.635828i −0.0566441 − 0.0981104i
\(43\) 3.26210 − 5.65012i 0.497466 − 0.861636i −0.502530 − 0.864560i \(-0.667597\pi\)
0.999996 + 0.00292406i \(0.000930757\pi\)
\(44\) −6.49629 − 11.2519i −0.979353 − 1.69629i
\(45\) 1.35194 0.201535
\(46\) 18.4003 2.71298
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 − 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 1.58613 + 2.74726i 0.228938 + 0.396533i
\(49\) −6.87614 −0.982306
\(50\) −6.61775 −0.935891
\(51\) 0 0
\(52\) 6.08242 − 10.5351i 0.843480 − 1.46095i
\(53\) 2.02791 + 3.51244i 0.278555 + 0.482471i 0.971026 − 0.238975i \(-0.0768113\pi\)
−0.692471 + 0.721446i \(0.743478\pi\)
\(54\) −1.04307 + 1.80664i −0.141943 + 0.245853i
\(55\) −3.73419 + 6.46781i −0.503518 + 0.872119i
\(56\) 0.258391 0.0345289
\(57\) 4.34823 − 0.304788i 0.575937 − 0.0403702i
\(58\) 5.64064 0.740653
\(59\) 2.76210 − 4.78410i 0.359595 − 0.622836i −0.628298 − 0.777972i \(-0.716248\pi\)
0.987893 + 0.155136i \(0.0495816\pi\)
\(60\) −1.58984 + 2.75368i −0.205247 + 0.355499i
\(61\) 0.938069 + 1.62478i 0.120107 + 0.208032i 0.919810 − 0.392364i \(-0.128343\pi\)
−0.799702 + 0.600397i \(0.795009\pi\)
\(62\) 0.546774 − 0.947041i 0.0694404 − 0.120274i
\(63\) −0.175970 − 0.304788i −0.0221701 − 0.0383997i
\(64\) −10.5242 −1.31552
\(65\) −6.99258 −0.867323
\(66\) −5.76210 − 9.98025i −0.709265 − 1.22848i
\(67\) −5.99629 − 10.3859i −0.732564 − 1.26884i −0.955784 − 0.294069i \(-0.904990\pi\)
0.223221 − 0.974768i \(-0.428343\pi\)
\(68\) 0 0
\(69\) 8.82032 1.06184
\(70\) −0.496291 − 0.859601i −0.0593181 − 0.102742i
\(71\) −2.52420 + 4.37204i −0.299567 + 0.518866i −0.976037 − 0.217605i \(-0.930176\pi\)
0.676470 + 0.736471i \(0.263509\pi\)
\(72\) −0.367095 − 0.635828i −0.0432626 − 0.0749331i
\(73\) −3.85194 + 6.67175i −0.450835 + 0.780870i −0.998438 − 0.0558687i \(-0.982207\pi\)
0.547603 + 0.836738i \(0.315541\pi\)
\(74\) 1.04307 − 1.80664i 0.121254 − 0.210018i
\(75\) −3.17226 −0.366301
\(76\) −4.49258 + 9.21507i −0.515334 + 1.05704i
\(77\) 1.94418 0.221560
\(78\) 5.39500 − 9.34442i 0.610864 − 1.05805i
\(79\) −3.91016 + 6.77260i −0.439927 + 0.761977i −0.997683 − 0.0680283i \(-0.978329\pi\)
0.557756 + 0.830005i \(0.311663\pi\)
\(80\) 2.14435 + 3.71413i 0.239746 + 0.415252i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 2.82032 + 4.88494i 0.311452 + 0.539451i
\(83\) 8.34452 0.915930 0.457965 − 0.888970i \(-0.348578\pi\)
0.457965 + 0.888970i \(0.348578\pi\)
\(84\) 0.827740 0.0903138
\(85\) 0 0
\(86\) 6.80516 + 11.7869i 0.733820 + 1.27101i
\(87\) 2.70388 0.289886
\(88\) 4.05582 0.432352
\(89\) 2.32403 + 4.02534i 0.246347 + 0.426685i 0.962509 − 0.271248i \(-0.0874365\pi\)
−0.716163 + 0.697933i \(0.754103\pi\)
\(90\) −1.41016 + 2.44247i −0.148644 + 0.257459i
\(91\) 0.910161 + 1.57644i 0.0954108 + 0.165256i
\(92\) −10.3724 + 17.9656i −1.08140 + 1.87304i
\(93\) 0.262100 − 0.453970i 0.0271785 − 0.0470745i
\(94\) −12.5168 −1.29101
\(95\) 5.87854 − 0.412055i 0.603126 − 0.0422760i
\(96\) −8.08613 −0.825287
\(97\) 6.90645 − 11.9623i 0.701244 − 1.21459i −0.266786 − 0.963756i \(-0.585962\pi\)
0.968030 − 0.250834i \(-0.0807049\pi\)
\(98\) 7.17226 − 12.4227i 0.724508 − 1.25488i
\(99\) −2.76210 − 4.78410i −0.277601 − 0.480820i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 57.2.e.b.49.1 yes 6
3.2 odd 2 171.2.f.b.163.3 6
4.3 odd 2 912.2.q.l.49.2 6
12.11 even 2 2736.2.s.z.1873.2 6
19.7 even 3 inner 57.2.e.b.7.1 ✓ 6
19.8 odd 6 1083.2.a.o.1.1 3
19.11 even 3 1083.2.a.l.1.3 3
57.8 even 6 3249.2.a.t.1.3 3
57.11 odd 6 3249.2.a.y.1.1 3
57.26 odd 6 171.2.f.b.64.3 6
76.7 odd 6 912.2.q.l.577.2 6
228.83 even 6 2736.2.s.z.577.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.1 ✓ 6 19.7 even 3 inner
57.2.e.b.49.1 yes 6 1.1 even 1 trivial
171.2.f.b.64.3 6 57.26 odd 6
171.2.f.b.163.3 6 3.2 odd 2
912.2.q.l.49.2 6 4.3 odd 2
912.2.q.l.577.2 6 76.7 odd 6
1083.2.a.l.1.3 3 19.11 even 3
1083.2.a.o.1.1 3 19.8 odd 6
2736.2.s.z.577.2 6 228.83 even 6
2736.2.s.z.1873.2 6 12.11 even 2
3249.2.a.t.1.3 3 57.8 even 6
3249.2.a.y.1.1 3 57.11 odd 6