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Properties

Label 57.2.e.b.49.2
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 49.2
Root \(-1.62241 + 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 57.49
Dual form 57.2.e.b.7.2

$q$-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+(0.285997 - 0.495361i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.836412 + 1.44871i) q^{4} +(1.33641 - 2.31473i) q^{5} +(0.285997 + 0.495361i) q^{6} -3.67282 q^{7} +2.10083 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.764419 - 1.32401i) q^{10} -3.81681 q^{11} -1.67282 q^{12} +(-0.0719933 - 0.124696i) q^{13} +(-1.05042 + 1.81937i) q^{14} +(1.33641 + 2.31473i) q^{15} +(-1.07199 + 1.85675i) q^{16} -0.571993 q^{18} +(4.24482 + 0.990721i) q^{19} +4.47116 q^{20} +(1.83641 - 3.18076i) q^{21} +(-1.09159 + 1.89070i) q^{22} +(-3.76442 - 6.52016i) q^{23} +(-1.05042 + 1.81937i) q^{24} +(-1.07199 - 1.85675i) q^{25} -0.0823593 q^{26} +1.00000 q^{27} +(-3.07199 - 5.32085i) q^{28} +(2.67282 + 4.62947i) q^{29} +1.52884 q^{30} +8.81681 q^{31} +(2.71400 + 4.70079i) q^{32} +(1.90841 - 3.30545i) q^{33} +(-4.90841 + 8.50161i) q^{35} +(0.836412 - 1.44871i) q^{36} -1.00000 q^{37} +(1.70477 - 1.81937i) q^{38} +0.143987 q^{39} +(2.80757 - 4.86286i) q^{40} +(-2.67282 + 4.62947i) q^{41} +(-1.05042 - 1.81937i) q^{42} +(-1.40841 + 2.43943i) q^{43} +(-3.19243 - 5.52944i) q^{44} -2.67282 q^{45} -4.30644 q^{46} +(3.00000 + 5.19615i) q^{47} +(-1.07199 - 1.85675i) q^{48} +6.48963 q^{49} -1.22635 q^{50} +(0.120432 - 0.208594i) q^{52} +(-4.00924 - 6.94420i) q^{53} +(0.285997 - 0.495361i) q^{54} +(-5.10083 + 8.83490i) q^{55} -7.71598 q^{56} +(-2.98040 + 3.18076i) q^{57} +3.05767 q^{58} +(-1.90841 + 3.30545i) q^{59} +(-2.23558 + 3.87214i) q^{60} +(-5.74482 - 9.95031i) q^{61} +(2.52158 - 4.36750i) q^{62} +(1.83641 + 3.18076i) q^{63} -1.18319 q^{64} -0.384851 q^{65} +(-1.09159 - 1.89070i) q^{66} +(-2.69243 - 4.66342i) q^{67} +7.52884 q^{69} +(2.80757 + 4.86286i) q^{70} +(6.81681 - 11.8071i) q^{71} +(-1.05042 - 1.81937i) q^{72} +(0.172824 - 0.299339i) q^{73} +(-0.285997 + 0.495361i) q^{74} +2.14399 q^{75} +(2.11515 + 6.97815i) q^{76} +14.0185 q^{77} +(0.0411797 - 0.0713253i) q^{78} +(-3.26442 + 5.65414i) q^{79} +(2.86525 + 4.96276i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(1.52884 + 2.64802i) q^{82} -2.28797 q^{83} +6.14399 q^{84} +(0.805598 + 1.39534i) q^{86} -5.34565 q^{87} -8.01847 q^{88} +(4.33641 + 7.51089i) q^{89} +(-0.764419 + 1.32401i) q^{90} +(0.264419 + 0.457986i) q^{91} +(6.29721 - 10.9071i) q^{92} +(-4.40841 + 7.63558i) q^{93} +3.43196 q^{94} +(7.96608 - 8.50161i) q^{95} -5.42801 q^{96} +(2.95684 - 5.12140i) q^{97} +(1.85601 - 3.21471i) q^{98} +(1.90841 + 3.30545i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.285997 − 0.495361i 0.202230 − 0.350273i −0.747017 − 0.664805i \(-0.768514\pi\)
0.949247 + 0.314533i \(0.101848\pi\)
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0.836412 + 1.44871i 0.418206 + 0.724354i
\(5\) 1.33641 − 2.31473i 0.597662 − 1.03518i −0.395504 − 0.918464i \(-0.629430\pi\)
0.993165 − 0.116716i \(-0.0372367\pi\)
\(6\) 0.285997 + 0.495361i 0.116758 + 0.202230i
\(7\) −3.67282 −1.38820 −0.694098 − 0.719880i \(-0.744197\pi\)
−0.694098 + 0.719880i \(0.744197\pi\)
\(8\) 2.10083 0.742756
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −0.764419 − 1.32401i −0.241730 − 0.418689i
\(11\) −3.81681 −1.15081 −0.575406 − 0.817868i \(-0.695156\pi\)
−0.575406 + 0.817868i \(0.695156\pi\)
\(12\) −1.67282 −0.482903
\(13\) −0.0719933 − 0.124696i −0.0199673 − 0.0345844i 0.855869 − 0.517193i \(-0.173023\pi\)
−0.875836 + 0.482608i \(0.839690\pi\)
\(14\) −1.05042 + 1.81937i −0.280735 + 0.486248i
\(15\) 1.33641 + 2.31473i 0.345060 + 0.597662i
\(16\) −1.07199 + 1.85675i −0.267998 + 0.464187i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) −0.571993 −0.134820
\(19\) 4.24482 + 0.990721i 0.973828 + 0.227287i
\(20\) 4.47116 0.999782
\(21\) 1.83641 − 3.18076i 0.400738 − 0.694098i
\(22\) −1.09159 + 1.89070i −0.232729 + 0.403098i
\(23\) −3.76442 − 6.52016i −0.784936 − 1.35955i −0.929038 − 0.369985i \(-0.879363\pi\)
0.144102 − 0.989563i \(-0.453971\pi\)
\(24\) −1.05042 + 1.81937i −0.214415 + 0.371378i
\(25\) −1.07199 − 1.85675i −0.214399 − 0.371349i
\(26\) −0.0823593 −0.0161520
\(27\) 1.00000 0.192450
\(28\) −3.07199 − 5.32085i −0.580552 − 1.00555i
\(29\) 2.67282 + 4.62947i 0.496331 + 0.859670i 0.999991 − 0.00423154i \(-0.00134695\pi\)
−0.503660 + 0.863902i \(0.668014\pi\)
\(30\) 1.52884 0.279126
\(31\) 8.81681 1.58355 0.791773 − 0.610816i \(-0.209158\pi\)
0.791773 + 0.610816i \(0.209158\pi\)
\(32\) 2.71400 + 4.70079i 0.479773 + 0.830990i
\(33\) 1.90841 − 3.30545i 0.332211 − 0.575406i
\(34\) 0 0
\(35\) −4.90841 + 8.50161i −0.829672 + 1.43703i
\(36\) 0.836412 − 1.44871i 0.139402 − 0.241451i
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 1.70477 − 1.81937i 0.276550 − 0.295141i
\(39\) 0.143987 0.0230563
\(40\) 2.80757 − 4.86286i 0.443917 − 0.768886i
\(41\) −2.67282 + 4.62947i −0.417425 + 0.723001i −0.995680 − 0.0928551i \(-0.970401\pi\)
0.578255 + 0.815856i \(0.303734\pi\)
\(42\) −1.05042 − 1.81937i −0.162083 − 0.280735i
\(43\) −1.40841 + 2.43943i −0.214780 + 0.372009i −0.953204 − 0.302327i \(-0.902237\pi\)
0.738425 + 0.674336i \(0.235570\pi\)
\(44\) −3.19243 − 5.52944i −0.481276 − 0.833595i
\(45\) −2.67282 −0.398441
\(46\) −4.30644 −0.634951
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 − 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) −1.07199 − 1.85675i −0.154729 − 0.267998i
\(49\) 6.48963 0.927091
\(50\) −1.22635 −0.173431
\(51\) 0 0
\(52\) 0.120432 − 0.208594i 0.0167009 − 0.0289268i
\(53\) −4.00924 − 6.94420i −0.550711 − 0.953859i −0.998223 − 0.0595815i \(-0.981023\pi\)
0.447513 − 0.894278i \(-0.352310\pi\)
\(54\) 0.285997 − 0.495361i 0.0389192 − 0.0674101i
\(55\) −5.10083 + 8.83490i −0.687796 + 1.19130i
\(56\) −7.71598 −1.03109
\(57\) −2.98040 + 3.18076i −0.394763 + 0.421302i
\(58\) 3.05767 0.401492
\(59\) −1.90841 + 3.30545i −0.248453 + 0.430334i −0.963097 − 0.269155i \(-0.913256\pi\)
0.714644 + 0.699489i \(0.246589\pi\)
\(60\) −2.23558 + 3.87214i −0.288612 + 0.499891i
\(61\) −5.74482 − 9.95031i −0.735548 − 1.27401i −0.954482 − 0.298268i \(-0.903591\pi\)
0.218934 − 0.975740i \(-0.429742\pi\)
\(62\) 2.52158 − 4.36750i 0.320241 − 0.554673i
\(63\) 1.83641 + 3.18076i 0.231366 + 0.400738i
\(64\) −1.18319 −0.147899
\(65\) −0.384851 −0.0477348
\(66\) −1.09159 − 1.89070i −0.134366 − 0.232729i
\(67\) −2.69243 − 4.66342i −0.328932 − 0.569727i 0.653368 − 0.757040i \(-0.273355\pi\)
−0.982300 + 0.187313i \(0.940022\pi\)
\(68\) 0 0
\(69\) 7.52884 0.906365
\(70\) 2.80757 + 4.86286i 0.335569 + 0.581223i
\(71\) 6.81681 − 11.8071i 0.809007 − 1.40124i −0.104546 − 0.994520i \(-0.533339\pi\)
0.913553 − 0.406720i \(-0.133328\pi\)
\(72\) −1.05042 − 1.81937i −0.123793 − 0.214415i
\(73\) 0.172824 − 0.299339i 0.0202275 − 0.0350350i −0.855734 − 0.517415i \(-0.826894\pi\)
0.875962 + 0.482380i \(0.160228\pi\)
\(74\) −0.285997 + 0.495361i −0.0332464 + 0.0575845i
\(75\) 2.14399 0.247566
\(76\) 2.11515 + 6.97815i 0.242624 + 0.800449i
\(77\) 14.0185 1.59755
\(78\) 0.0411797 − 0.0713253i 0.00466268 − 0.00807600i
\(79\) −3.26442 + 5.65414i −0.367276 + 0.636140i −0.989139 − 0.146986i \(-0.953043\pi\)
0.621863 + 0.783126i \(0.286376\pi\)
\(80\) 2.86525 + 4.96276i 0.320345 + 0.554853i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 1.52884 + 2.64802i 0.168832 + 0.292425i
\(83\) −2.28797 −0.251138 −0.125569 − 0.992085i \(-0.540076\pi\)
−0.125569 + 0.992085i \(0.540076\pi\)
\(84\) 6.14399 0.670364
\(85\) 0 0
\(86\) 0.805598 + 1.39534i 0.0868699 + 0.150463i
\(87\) −5.34565 −0.573114
\(88\) −8.01847 −0.854772
\(89\) 4.33641 + 7.51089i 0.459659 + 0.796152i 0.998943 − 0.0459717i \(-0.0146384\pi\)
−0.539284 + 0.842124i \(0.681305\pi\)
\(90\) −0.764419 + 1.32401i −0.0805768 + 0.139563i
\(91\) 0.264419 + 0.457986i 0.0277186 + 0.0480100i
\(92\) 6.29721 − 10.9071i 0.656529 − 1.13714i
\(93\) −4.40841 + 7.63558i −0.457130 + 0.791773i
\(94\) 3.43196 0.353980
\(95\) 7.96608 − 8.50161i 0.817303 − 0.872246i
\(96\) −5.42801 −0.553994
\(97\) 2.95684 − 5.12140i 0.300222 − 0.520000i −0.675964 − 0.736935i \(-0.736273\pi\)
0.976186 + 0.216935i \(0.0696059\pi\)
\(98\) 1.85601 − 3.21471i 0.187486 − 0.324735i
\(99\) 1.90841 + 3.30545i 0.191802 + 0.332211i
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.49.2 yes 6
3.2 odd 2 171.2.f.b.163.2 6
4.3 odd 2 912.2.q.l.49.3 6
12.11 even 2 2736.2.s.z.1873.1 6
19.7 even 3 inner 57.2.e.b.7.2 ✓ 6
19.8 odd 6 1083.2.a.o.1.2 3
19.11 even 3 1083.2.a.l.1.2 3
57.8 even 6 3249.2.a.t.1.2 3
57.11 odd 6 3249.2.a.y.1.2 3
57.26 odd 6 171.2.f.b.64.2 6
76.7 odd 6 912.2.q.l.577.3 6
228.83 even 6 2736.2.s.z.577.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.2 ✓ 6 19.7 even 3 inner
57.2.e.b.49.2 yes 6 1.1 even 1 trivial
171.2.f.b.64.2 6 57.26 odd 6
171.2.f.b.163.2 6 3.2 odd 2
912.2.q.l.49.3 6 4.3 odd 2
912.2.q.l.577.3 6 76.7 odd 6
1083.2.a.l.1.2 3 19.11 even 3
1083.2.a.o.1.2 3 19.8 odd 6
2736.2.s.z.577.1 6 228.83 even 6
2736.2.s.z.1873.1 6 12.11 even 2
3249.2.a.t.1.2 3 57.8 even 6
3249.2.a.y.1.2 3 57.11 odd 6