Properties

Label 637.2.c.f.246.6
Level $637$
Weight $2$
Character 637.246
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(246,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.246");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 31x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 246.6
Root \(1.07305i\) of defining polynomial
Character \(\chi\) \(=\) 637.246
Dual form 637.2.c.f.246.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.07305i q^{2} -2.43140 q^{3} +0.848553 q^{4} -0.625432i q^{5} -2.60903i q^{6} +3.05665i q^{8} +2.91173 q^{9} +O(q^{10})\) \(q+1.07305i q^{2} -2.43140 q^{3} +0.848553 q^{4} -0.625432i q^{5} -2.60903i q^{6} +3.05665i q^{8} +2.91173 q^{9} +0.671123 q^{10} -0.708521i q^{11} -2.06318 q^{12} +(0.848553 - 3.50428i) q^{13} +1.52068i q^{15} -1.58285 q^{16} +3.34313 q^{17} +3.12445i q^{18} +5.20276i q^{19} -0.530712i q^{20} +0.760282 q^{22} +4.43140 q^{23} -7.43196i q^{24} +4.60883 q^{25} +(3.76028 + 0.910544i) q^{26} +0.214623 q^{27} -6.59711 q^{29} -1.63177 q^{30} +4.39061i q^{31} +4.41482i q^{32} +1.72270i q^{33} +3.58737i q^{34} +2.47076 q^{36} +0.423409i q^{37} -5.58285 q^{38} +(-2.06318 + 8.52032i) q^{39} +1.91173 q^{40} +5.01604i q^{41} +11.2059 q^{43} -0.601218i q^{44} -1.82109i q^{45} +4.75514i q^{46} +8.07269i q^{47} +3.84855 q^{48} +4.94553i q^{50} -8.12851 q^{51} +(0.720042 - 2.97356i) q^{52} -0.697106 q^{53} +0.230302i q^{54} -0.443132 q^{55} -12.6500i q^{57} -7.07906i q^{58} +9.86319i q^{59} +1.29038i q^{60} +4.69711 q^{61} -4.71136 q^{62} -7.90305 q^{64} +(-2.19169 - 0.530712i) q^{65} -1.84855 q^{66} -10.4208i q^{67} +2.83683 q^{68} -10.7745 q^{69} +14.0876i q^{71} +8.90015i q^{72} -5.08383i q^{73} -0.454341 q^{74} -11.2059 q^{75} +4.41482i q^{76} +(-9.14277 - 2.21390i) q^{78} -3.91173 q^{79} +0.989966i q^{80} -9.25702 q^{81} -5.38249 q^{82} -10.2035i q^{83} -2.09090i q^{85} +12.0246i q^{86} +16.0402 q^{87} +2.16570 q^{88} -13.3791i q^{89} +1.95413 q^{90} +3.76028 q^{92} -10.6753i q^{93} -8.66244 q^{94} +3.25397 q^{95} -10.7342i q^{96} +0.202023i q^{97} -2.06302i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 6 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 6 q^{4} + 12 q^{9} + 6 q^{10} - 18 q^{12} - 6 q^{13} - 2 q^{16} - 8 q^{17} - 18 q^{22} + 12 q^{23} + 6 q^{26} + 16 q^{27} - 8 q^{29} - 38 q^{30} - 28 q^{36} - 34 q^{38} - 18 q^{39} + 4 q^{40} + 8 q^{43} + 18 q^{48} - 16 q^{51} + 42 q^{52} + 20 q^{53} + 12 q^{55} + 12 q^{61} + 22 q^{62} + 44 q^{64} + 30 q^{65} - 2 q^{66} + 2 q^{68} - 28 q^{69} - 42 q^{74} - 8 q^{75} + 10 q^{78} - 20 q^{79} + 24 q^{81} + 16 q^{82} + 68 q^{87} - 4 q^{88} - 108 q^{90} + 6 q^{92} + 26 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07305i 0.758764i 0.925240 + 0.379382i \(0.123863\pi\)
−0.925240 + 0.379382i \(0.876137\pi\)
\(3\) −2.43140 −1.40377 −0.701886 0.712289i \(-0.747658\pi\)
−0.701886 + 0.712289i \(0.747658\pi\)
\(4\) 0.848553 0.424277
\(5\) 0.625432i 0.279702i −0.990173 0.139851i \(-0.955338\pi\)
0.990173 0.139851i \(-0.0446623\pi\)
\(6\) 2.60903i 1.06513i
\(7\) 0 0
\(8\) 3.05665i 1.08069i
\(9\) 2.91173 0.970576
\(10\) 0.671123 0.212228
\(11\) 0.708521i 0.213627i −0.994279 0.106814i \(-0.965935\pi\)
0.994279 0.106814i \(-0.0340648\pi\)
\(12\) −2.06318 −0.595588
\(13\) 0.848553 3.50428i 0.235346 0.971912i
\(14\) 0 0
\(15\) 1.52068i 0.392637i
\(16\) −1.58285 −0.395713
\(17\) 3.34313 0.810829 0.405414 0.914133i \(-0.367127\pi\)
0.405414 + 0.914133i \(0.367127\pi\)
\(18\) 3.12445i 0.736439i
\(19\) 5.20276i 1.19360i 0.802392 + 0.596798i \(0.203561\pi\)
−0.802392 + 0.596798i \(0.796439\pi\)
\(20\) 0.530712i 0.118671i
\(21\) 0 0
\(22\) 0.760282 0.162093
\(23\) 4.43140 0.924012 0.462006 0.886877i \(-0.347130\pi\)
0.462006 + 0.886877i \(0.347130\pi\)
\(24\) 7.43196i 1.51704i
\(25\) 4.60883 0.921767
\(26\) 3.76028 + 0.910544i 0.737452 + 0.178572i
\(27\) 0.214623 0.0413042
\(28\) 0 0
\(29\) −6.59711 −1.22505 −0.612526 0.790450i \(-0.709847\pi\)
−0.612526 + 0.790450i \(0.709847\pi\)
\(30\) −1.63177 −0.297919
\(31\) 4.39061i 0.788576i 0.918987 + 0.394288i \(0.129009\pi\)
−0.918987 + 0.394288i \(0.870991\pi\)
\(32\) 4.41482i 0.780437i
\(33\) 1.72270i 0.299884i
\(34\) 3.58737i 0.615228i
\(35\) 0 0
\(36\) 2.47076 0.411793
\(37\) 0.423409i 0.0696080i 0.999394 + 0.0348040i \(0.0110807\pi\)
−0.999394 + 0.0348040i \(0.988919\pi\)
\(38\) −5.58285 −0.905658
\(39\) −2.06318 + 8.52032i −0.330373 + 1.36434i
\(40\) 1.91173 0.302271
\(41\) 5.01604i 0.783374i 0.920099 + 0.391687i \(0.128108\pi\)
−0.920099 + 0.391687i \(0.871892\pi\)
\(42\) 0 0
\(43\) 11.2059 1.70889 0.854445 0.519542i \(-0.173897\pi\)
0.854445 + 0.519542i \(0.173897\pi\)
\(44\) 0.601218i 0.0906370i
\(45\) 1.82109i 0.271472i
\(46\) 4.75514i 0.701107i
\(47\) 8.07269i 1.17752i 0.808307 + 0.588762i \(0.200384\pi\)
−0.808307 + 0.588762i \(0.799616\pi\)
\(48\) 3.84855 0.555491
\(49\) 0 0
\(50\) 4.94553i 0.699404i
\(51\) −8.12851 −1.13822
\(52\) 0.720042 2.97356i 0.0998519 0.412359i
\(53\) −0.697106 −0.0957549 −0.0478774 0.998853i \(-0.515246\pi\)
−0.0478774 + 0.998853i \(0.515246\pi\)
\(54\) 0.230302i 0.0313401i
\(55\) −0.443132 −0.0597519
\(56\) 0 0
\(57\) 12.6500i 1.67554i
\(58\) 7.07906i 0.929526i
\(59\) 9.86319i 1.28408i 0.766672 + 0.642039i \(0.221911\pi\)
−0.766672 + 0.642039i \(0.778089\pi\)
\(60\) 1.29038i 0.166587i
\(61\) 4.69711 0.601403 0.300701 0.953718i \(-0.402779\pi\)
0.300701 + 0.953718i \(0.402779\pi\)
\(62\) −4.71136 −0.598344
\(63\) 0 0
\(64\) −7.90305 −0.987881
\(65\) −2.19169 0.530712i −0.271845 0.0658267i
\(66\) −1.84855 −0.227541
\(67\) 10.4208i 1.27311i −0.771233 0.636553i \(-0.780360\pi\)
0.771233 0.636553i \(-0.219640\pi\)
\(68\) 2.83683 0.344016
\(69\) −10.7745 −1.29710
\(70\) 0 0
\(71\) 14.0876i 1.67189i 0.548812 + 0.835946i \(0.315080\pi\)
−0.548812 + 0.835946i \(0.684920\pi\)
\(72\) 8.90015i 1.04889i
\(73\) 5.08383i 0.595017i −0.954719 0.297509i \(-0.903844\pi\)
0.954719 0.297509i \(-0.0961557\pi\)
\(74\) −0.454341 −0.0528161
\(75\) −11.2059 −1.29395
\(76\) 4.41482i 0.506415i
\(77\) 0 0
\(78\) −9.14277 2.21390i −1.03521 0.250675i
\(79\) −3.91173 −0.440104 −0.220052 0.975488i \(-0.570623\pi\)
−0.220052 + 0.975488i \(0.570623\pi\)
\(80\) 0.989966i 0.110682i
\(81\) −9.25702 −1.02856
\(82\) −5.38249 −0.594396
\(83\) 10.2035i 1.11998i −0.828499 0.559990i \(-0.810805\pi\)
0.828499 0.559990i \(-0.189195\pi\)
\(84\) 0 0
\(85\) 2.09090i 0.226790i
\(86\) 12.0246i 1.29665i
\(87\) 16.0402 1.71969
\(88\) 2.16570 0.230865
\(89\) 13.3791i 1.41818i −0.705117 0.709090i \(-0.749106\pi\)
0.705117 0.709090i \(-0.250894\pi\)
\(90\) 1.95413 0.205983
\(91\) 0 0
\(92\) 3.76028 0.392036
\(93\) 10.6753i 1.10698i
\(94\) −8.66244 −0.893463
\(95\) 3.25397 0.333851
\(96\) 10.7342i 1.09556i
\(97\) 0.202023i 0.0205123i 0.999947 + 0.0102562i \(0.00326470\pi\)
−0.999947 + 0.0102562i \(0.996735\pi\)
\(98\) 0 0
\(99\) 2.06302i 0.207341i
\(100\) 3.91084 0.391084
\(101\) 17.3345 1.72484 0.862421 0.506191i \(-0.168947\pi\)
0.862421 + 0.506191i \(0.168947\pi\)
\(102\) 8.72234i 0.863640i
\(103\) 10.8148 1.06561 0.532806 0.846237i \(-0.321138\pi\)
0.532806 + 0.846237i \(0.321138\pi\)
\(104\) 10.7114 + 2.59373i 1.05034 + 0.254336i
\(105\) 0 0
\(106\) 0.748033i 0.0726554i
\(107\) −6.11678 −0.591332 −0.295666 0.955291i \(-0.595542\pi\)
−0.295666 + 0.955291i \(0.595542\pi\)
\(108\) 0.182119 0.0175244
\(109\) 11.3992i 1.09184i −0.837837 0.545921i \(-0.816180\pi\)
0.837837 0.545921i \(-0.183820\pi\)
\(110\) 0.475505i 0.0453376i
\(111\) 1.02948i 0.0977137i
\(112\) 0 0
\(113\) −0.923456 −0.0868714 −0.0434357 0.999056i \(-0.513830\pi\)
−0.0434357 + 0.999056i \(0.513830\pi\)
\(114\) 13.5742 1.27134
\(115\) 2.77154i 0.258448i
\(116\) −5.59800 −0.519761
\(117\) 2.47076 10.2035i 0.228422 0.943314i
\(118\) −10.5837 −0.974312
\(119\) 0 0
\(120\) −4.64819 −0.424319
\(121\) 10.4980 0.954363
\(122\) 5.04025i 0.456323i
\(123\) 12.1960i 1.09968i
\(124\) 3.72566i 0.334574i
\(125\) 6.00967i 0.537521i
\(126\) 0 0
\(127\) 8.50972 0.755116 0.377558 0.925986i \(-0.376764\pi\)
0.377558 + 0.925986i \(0.376764\pi\)
\(128\) 0.349236i 0.0308684i
\(129\) −27.2462 −2.39889
\(130\) 0.569483 2.35180i 0.0499470 0.206267i
\(131\) −7.00305 −0.611859 −0.305930 0.952054i \(-0.598967\pi\)
−0.305930 + 0.952054i \(0.598967\pi\)
\(132\) 1.46180i 0.127234i
\(133\) 0 0
\(134\) 11.1821 0.965988
\(135\) 0.134232i 0.0115528i
\(136\) 10.2188i 0.876255i
\(137\) 6.21694i 0.531149i −0.964090 0.265575i \(-0.914438\pi\)
0.964090 0.265575i \(-0.0855617\pi\)
\(138\) 11.5617i 0.984195i
\(139\) 6.53140 0.553986 0.276993 0.960872i \(-0.410662\pi\)
0.276993 + 0.960872i \(0.410662\pi\)
\(140\) 0 0
\(141\) 19.6280i 1.65297i
\(142\) −15.1168 −1.26857
\(143\) −2.48285 0.601218i −0.207627 0.0502763i
\(144\) −4.60883 −0.384070
\(145\) 4.12604i 0.342649i
\(146\) 5.45523 0.451478
\(147\) 0 0
\(148\) 0.359285i 0.0295330i
\(149\) 3.69738i 0.302901i 0.988465 + 0.151451i \(0.0483945\pi\)
−0.988465 + 0.151451i \(0.951606\pi\)
\(150\) 12.0246i 0.981804i
\(151\) 4.87774i 0.396945i −0.980106 0.198473i \(-0.936402\pi\)
0.980106 0.198473i \(-0.0635981\pi\)
\(152\) −15.9030 −1.28991
\(153\) 9.73430 0.786971
\(154\) 0 0
\(155\) 2.74603 0.220566
\(156\) −1.75071 + 7.22994i −0.140169 + 0.578858i
\(157\) −9.51968 −0.759753 −0.379876 0.925037i \(-0.624033\pi\)
−0.379876 + 0.925037i \(0.624033\pi\)
\(158\) 4.19750i 0.333935i
\(159\) 1.69495 0.134418
\(160\) 2.76117 0.218290
\(161\) 0 0
\(162\) 9.93329i 0.780433i
\(163\) 23.7089i 1.85702i −0.371305 0.928511i \(-0.621090\pi\)
0.371305 0.928511i \(-0.378910\pi\)
\(164\) 4.25637i 0.332367i
\(165\) 1.07743 0.0838780
\(166\) 10.9489 0.849801
\(167\) 1.13193i 0.0875914i 0.999041 + 0.0437957i \(0.0139451\pi\)
−0.999041 + 0.0437957i \(0.986055\pi\)
\(168\) 0 0
\(169\) −11.5599 5.94713i −0.889224 0.457472i
\(170\) 2.24365 0.172080
\(171\) 15.1490i 1.15848i
\(172\) 9.50884 0.725042
\(173\) −11.9892 −0.911519 −0.455760 0.890103i \(-0.650632\pi\)
−0.455760 + 0.890103i \(0.650632\pi\)
\(174\) 17.2121i 1.30484i
\(175\) 0 0
\(176\) 1.12148i 0.0845350i
\(177\) 23.9814i 1.80255i
\(178\) 14.3565 1.07607
\(179\) −9.47076 −0.707878 −0.353939 0.935269i \(-0.615158\pi\)
−0.353939 + 0.935269i \(0.615158\pi\)
\(180\) 1.54529i 0.115179i
\(181\) −11.4314 −0.849690 −0.424845 0.905266i \(-0.639671\pi\)
−0.424845 + 0.905266i \(0.639671\pi\)
\(182\) 0 0
\(183\) −11.4206 −0.844233
\(184\) 13.5453i 0.998571i
\(185\) 0.264813 0.0194695
\(186\) 11.4552 0.839938
\(187\) 2.36868i 0.173215i
\(188\) 6.85011i 0.499595i
\(189\) 0 0
\(190\) 3.49169i 0.253314i
\(191\) 15.6875 1.13511 0.567555 0.823335i \(-0.307889\pi\)
0.567555 + 0.823335i \(0.307889\pi\)
\(192\) 19.2155 1.38676
\(193\) 23.0071i 1.65609i 0.560662 + 0.828045i \(0.310547\pi\)
−0.560662 + 0.828045i \(0.689453\pi\)
\(194\) −0.216782 −0.0155640
\(195\) 5.32888 + 1.29038i 0.381609 + 0.0924057i
\(196\) 0 0
\(197\) 10.2035i 0.726970i 0.931600 + 0.363485i \(0.118413\pi\)
−0.931600 + 0.363485i \(0.881587\pi\)
\(198\) 2.21373 0.157323
\(199\) −11.9235 −0.845231 −0.422616 0.906309i \(-0.638888\pi\)
−0.422616 + 0.906309i \(0.638888\pi\)
\(200\) 14.0876i 0.996145i
\(201\) 25.3372i 1.78715i
\(202\) 18.6008i 1.30875i
\(203\) 0 0
\(204\) −6.89747 −0.482920
\(205\) 3.13719 0.219111
\(206\) 11.6049i 0.808548i
\(207\) 12.9030 0.896824
\(208\) −1.34313 + 5.54675i −0.0931296 + 0.384598i
\(209\) 3.68627 0.254984
\(210\) 0 0
\(211\) −15.5893 −1.07321 −0.536606 0.843833i \(-0.680294\pi\)
−0.536606 + 0.843833i \(0.680294\pi\)
\(212\) −0.591531 −0.0406265
\(213\) 34.2527i 2.34696i
\(214\) 6.56365i 0.448682i
\(215\) 7.00855i 0.477979i
\(216\) 0.656028i 0.0446370i
\(217\) 0 0
\(218\) 12.2319 0.828451
\(219\) 12.3608i 0.835269i
\(220\) −0.376021 −0.0253513
\(221\) 2.83683 11.7153i 0.190826 0.788054i
\(222\) 1.10469 0.0741417
\(223\) 6.76662i 0.453126i −0.973996 0.226563i \(-0.927251\pi\)
0.973996 0.226563i \(-0.0727490\pi\)
\(224\) 0 0
\(225\) 13.4197 0.894645
\(226\) 0.990919i 0.0659149i
\(227\) 16.8245i 1.11668i 0.829612 + 0.558340i \(0.188562\pi\)
−0.829612 + 0.558340i \(0.811438\pi\)
\(228\) 10.7342i 0.710891i
\(229\) 11.0257i 0.728599i −0.931282 0.364300i \(-0.881308\pi\)
0.931282 0.364300i \(-0.118692\pi\)
\(230\) 2.97402 0.196101
\(231\) 0 0
\(232\) 20.1651i 1.32390i
\(233\) −17.3549 −1.13695 −0.568477 0.822699i \(-0.692467\pi\)
−0.568477 + 0.822699i \(0.692467\pi\)
\(234\) 10.9489 + 2.65126i 0.715753 + 0.173318i
\(235\) 5.04892 0.329355
\(236\) 8.36944i 0.544804i
\(237\) 9.51100 0.617806
\(238\) 0 0
\(239\) 19.7223i 1.27573i 0.770148 + 0.637865i \(0.220182\pi\)
−0.770148 + 0.637865i \(0.779818\pi\)
\(240\) 2.40701i 0.155372i
\(241\) 2.78413i 0.179341i −0.995971 0.0896706i \(-0.971419\pi\)
0.995971 0.0896706i \(-0.0285814\pi\)
\(242\) 11.2649i 0.724137i
\(243\) 21.8637 1.40256
\(244\) 3.98574 0.255161
\(245\) 0 0
\(246\) 13.0870 0.834397
\(247\) 18.2319 + 4.41482i 1.16007 + 0.280908i
\(248\) −13.4206 −0.852207
\(249\) 24.8088i 1.57220i
\(250\) 6.44871 0.407852
\(251\) −23.5608 −1.48714 −0.743572 0.668655i \(-0.766870\pi\)
−0.743572 + 0.668655i \(0.766870\pi\)
\(252\) 0 0
\(253\) 3.13974i 0.197394i
\(254\) 9.13140i 0.572955i
\(255\) 5.08383i 0.318362i
\(256\) −16.1808 −1.01130
\(257\) 3.43229 0.214101 0.107050 0.994254i \(-0.465859\pi\)
0.107050 + 0.994254i \(0.465859\pi\)
\(258\) 29.2367i 1.82019i
\(259\) 0 0
\(260\) −1.85976 0.450337i −0.115338 0.0279287i
\(261\) −19.2090 −1.18901
\(262\) 7.51465i 0.464257i
\(263\) −21.4491 −1.32261 −0.661303 0.750119i \(-0.729996\pi\)
−0.661303 + 0.750119i \(0.729996\pi\)
\(264\) −5.26570 −0.324082
\(265\) 0.435992i 0.0267828i
\(266\) 0 0
\(267\) 32.5300i 1.99080i
\(268\) 8.84262i 0.540149i
\(269\) 14.6569 0.893645 0.446822 0.894623i \(-0.352556\pi\)
0.446822 + 0.894623i \(0.352556\pi\)
\(270\) 0.144038 0.00876589
\(271\) 2.04366i 0.124143i 0.998072 + 0.0620717i \(0.0197707\pi\)
−0.998072 + 0.0620717i \(0.980229\pi\)
\(272\) −5.29168 −0.320856
\(273\) 0 0
\(274\) 6.67112 0.403017
\(275\) 3.26546i 0.196914i
\(276\) −9.14277 −0.550330
\(277\) 5.43356 0.326471 0.163236 0.986587i \(-0.447807\pi\)
0.163236 + 0.986587i \(0.447807\pi\)
\(278\) 7.00855i 0.420345i
\(279\) 12.7843i 0.765373i
\(280\) 0 0
\(281\) 20.2356i 1.20715i −0.797305 0.603577i \(-0.793742\pi\)
0.797305 0.603577i \(-0.206258\pi\)
\(282\) 21.0619 1.25422
\(283\) −1.73519 −0.103146 −0.0515731 0.998669i \(-0.516424\pi\)
−0.0515731 + 0.998669i \(0.516424\pi\)
\(284\) 11.9541i 0.709345i
\(285\) −7.91173 −0.468650
\(286\) 0.645140 2.66424i 0.0381479 0.157540i
\(287\) 0 0
\(288\) 12.8548i 0.757474i
\(289\) −5.82346 −0.342556
\(290\) −4.42747 −0.259990
\(291\) 0.491200i 0.0287947i
\(292\) 4.31390i 0.252452i
\(293\) 27.2441i 1.59162i −0.605547 0.795810i \(-0.707046\pi\)
0.605547 0.795810i \(-0.292954\pi\)
\(294\) 0 0
\(295\) 6.16875 0.359159
\(296\) −1.29421 −0.0752247
\(297\) 0.152065i 0.00882369i
\(298\) −3.96750 −0.229831
\(299\) 3.76028 15.5289i 0.217463 0.898058i
\(300\) −9.50884 −0.548993
\(301\) 0 0
\(302\) 5.23409 0.301188
\(303\) −42.1471 −2.42129
\(304\) 8.23520i 0.472321i
\(305\) 2.93772i 0.168213i
\(306\) 10.4454i 0.597126i
\(307\) 12.7138i 0.725612i −0.931865 0.362806i \(-0.881819\pi\)
0.931865 0.362806i \(-0.118181\pi\)
\(308\) 0 0
\(309\) −26.2951 −1.49588
\(310\) 2.94664i 0.167358i
\(311\) −9.61879 −0.545431 −0.272716 0.962095i \(-0.587922\pi\)
−0.272716 + 0.962095i \(0.587922\pi\)
\(312\) −26.0437 6.30641i −1.47443 0.357030i
\(313\) −9.02547 −0.510149 −0.255075 0.966921i \(-0.582100\pi\)
−0.255075 + 0.966921i \(0.582100\pi\)
\(314\) 10.2151i 0.576473i
\(315\) 0 0
\(316\) −3.31931 −0.186726
\(317\) 24.6262i 1.38314i 0.722307 + 0.691572i \(0.243082\pi\)
−0.722307 + 0.691572i \(0.756918\pi\)
\(318\) 1.81877i 0.101992i
\(319\) 4.67419i 0.261704i
\(320\) 4.94282i 0.276312i
\(321\) 14.8724 0.830095
\(322\) 0 0
\(323\) 17.3935i 0.967802i
\(324\) −7.85507 −0.436393
\(325\) 3.91084 16.1506i 0.216934 0.895876i
\(326\) 25.4409 1.40904
\(327\) 27.7160i 1.53270i
\(328\) −15.3323 −0.846584
\(329\) 0 0
\(330\) 1.15614i 0.0636436i
\(331\) 13.1718i 0.723989i 0.932180 + 0.361994i \(0.117904\pi\)
−0.932180 + 0.361994i \(0.882096\pi\)
\(332\) 8.65821i 0.475181i
\(333\) 1.23285i 0.0675599i
\(334\) −1.21462 −0.0664612
\(335\) −6.51752 −0.356090
\(336\) 0 0
\(337\) 17.0307 0.927720 0.463860 0.885909i \(-0.346464\pi\)
0.463860 + 0.885909i \(0.346464\pi\)
\(338\) 6.38160 12.4044i 0.347113 0.674712i
\(339\) 2.24529 0.121948
\(340\) 1.77424i 0.0962218i
\(341\) 3.11084 0.168461
\(342\) −16.2558 −0.879010
\(343\) 0 0
\(344\) 34.2527i 1.84678i
\(345\) 6.73874i 0.362802i
\(346\) 12.8650i 0.691628i
\(347\) 0.459917 0.0246897 0.0123448 0.999924i \(-0.496070\pi\)
0.0123448 + 0.999924i \(0.496070\pi\)
\(348\) 13.6110 0.729626
\(349\) 6.87822i 0.368183i 0.982909 + 0.184091i \(0.0589342\pi\)
−0.982909 + 0.184091i \(0.941066\pi\)
\(350\) 0 0
\(351\) 0.182119 0.752098i 0.00972078 0.0401440i
\(352\) 3.12799 0.166723
\(353\) 1.53326i 0.0816073i 0.999167 + 0.0408036i \(0.0129918\pi\)
−0.999167 + 0.0408036i \(0.987008\pi\)
\(354\) 25.7334 1.36771
\(355\) 8.81084 0.467631
\(356\) 11.3529i 0.601701i
\(357\) 0 0
\(358\) 10.1626i 0.537112i
\(359\) 27.2068i 1.43592i 0.696085 + 0.717959i \(0.254923\pi\)
−0.696085 + 0.717959i \(0.745077\pi\)
\(360\) 5.56644 0.293377
\(361\) −8.06875 −0.424671
\(362\) 12.2665i 0.644714i
\(363\) −25.5249 −1.33971
\(364\) 0 0
\(365\) −3.17959 −0.166427
\(366\) 12.2549i 0.640574i
\(367\) 26.9814 1.40842 0.704208 0.709994i \(-0.251302\pi\)
0.704208 + 0.709994i \(0.251302\pi\)
\(368\) −7.01426 −0.365643
\(369\) 14.6053i 0.760324i
\(370\) 0.284159i 0.0147727i
\(371\) 0 0
\(372\) 9.05859i 0.469666i
\(373\) 3.97238 0.205682 0.102841 0.994698i \(-0.467207\pi\)
0.102841 + 0.994698i \(0.467207\pi\)
\(374\) 2.54172 0.131429
\(375\) 14.6119i 0.754558i
\(376\) −24.6754 −1.27254
\(377\) −5.59800 + 23.1181i −0.288311 + 1.19064i
\(378\) 0 0
\(379\) 11.4059i 0.585884i −0.956130 0.292942i \(-0.905366\pi\)
0.956130 0.292942i \(-0.0946343\pi\)
\(380\) 2.76117 0.141645
\(381\) −20.6906 −1.06001
\(382\) 16.8336i 0.861281i
\(383\) 23.7920i 1.21571i −0.794047 0.607856i \(-0.792030\pi\)
0.794047 0.607856i \(-0.207970\pi\)
\(384\) 0.849134i 0.0433322i
\(385\) 0 0
\(386\) −24.6879 −1.25658
\(387\) 32.6287 1.65861
\(388\) 0.171427i 0.00870290i
\(389\) 28.4110 1.44049 0.720247 0.693717i \(-0.244028\pi\)
0.720247 + 0.693717i \(0.244028\pi\)
\(390\) −1.38464 + 5.71818i −0.0701142 + 0.289551i
\(391\) 14.8148 0.749216
\(392\) 0 0
\(393\) 17.0272 0.858911
\(394\) −10.9489 −0.551599
\(395\) 2.44652i 0.123098i
\(396\) 1.75058i 0.0879701i
\(397\) 9.85912i 0.494815i 0.968912 + 0.247408i \(0.0795786\pi\)
−0.968912 + 0.247408i \(0.920421\pi\)
\(398\) 12.7945i 0.641332i
\(399\) 0 0
\(400\) −7.29510 −0.364755
\(401\) 12.6194i 0.630184i −0.949061 0.315092i \(-0.897965\pi\)
0.949061 0.315092i \(-0.102035\pi\)
\(402\) −27.1883 −1.35603
\(403\) 15.3859 + 3.72566i 0.766426 + 0.185588i
\(404\) 14.7092 0.731810
\(405\) 5.78964i 0.287689i
\(406\) 0 0
\(407\) 0.299994 0.0148701
\(408\) 24.8460i 1.23006i
\(409\) 18.0573i 0.892878i −0.894814 0.446439i \(-0.852692\pi\)
0.894814 0.446439i \(-0.147308\pi\)
\(410\) 3.36638i 0.166254i
\(411\) 15.1159i 0.745613i
\(412\) 9.17691 0.452114
\(413\) 0 0
\(414\) 13.8457i 0.680478i
\(415\) −6.38160 −0.313260
\(416\) 15.4708 + 3.74621i 0.758516 + 0.183673i
\(417\) −15.8805 −0.777671
\(418\) 3.95557i 0.193473i
\(419\) 14.2805 0.697647 0.348823 0.937188i \(-0.386581\pi\)
0.348823 + 0.937188i \(0.386581\pi\)
\(420\) 0 0
\(421\) 4.27439i 0.208321i 0.994561 + 0.104160i \(0.0332155\pi\)
−0.994561 + 0.104160i \(0.966784\pi\)
\(422\) 16.7282i 0.814316i
\(423\) 23.5055i 1.14288i
\(424\) 2.13081i 0.103481i
\(425\) 15.4080 0.747395
\(426\) 36.7550 1.78079
\(427\) 0 0
\(428\) −5.19042 −0.250888
\(429\) 6.03682 + 1.46180i 0.291461 + 0.0705765i
\(430\) 7.52056 0.362674
\(431\) 14.6067i 0.703580i 0.936079 + 0.351790i \(0.114427\pi\)
−0.936079 + 0.351790i \(0.885573\pi\)
\(432\) −0.339716 −0.0163446
\(433\) 28.0099 1.34607 0.673035 0.739611i \(-0.264991\pi\)
0.673035 + 0.739611i \(0.264991\pi\)
\(434\) 0 0
\(435\) 10.0321i 0.481001i
\(436\) 9.67279i 0.463243i
\(437\) 23.0556i 1.10290i
\(438\) −13.2639 −0.633772
\(439\) −17.0774 −0.815061 −0.407531 0.913192i \(-0.633610\pi\)
−0.407531 + 0.913192i \(0.633610\pi\)
\(440\) 1.35450i 0.0645733i
\(441\) 0 0
\(442\) 12.5711 + 3.04407i 0.597947 + 0.144792i
\(443\) 13.8157 0.656402 0.328201 0.944608i \(-0.393558\pi\)
0.328201 + 0.944608i \(0.393558\pi\)
\(444\) 0.873567i 0.0414576i
\(445\) −8.36771 −0.396668
\(446\) 7.26096 0.343816
\(447\) 8.98984i 0.425205i
\(448\) 0 0
\(449\) 32.6410i 1.54042i −0.637789 0.770211i \(-0.720151\pi\)
0.637789 0.770211i \(-0.279849\pi\)
\(450\) 14.4001i 0.678825i
\(451\) 3.55397 0.167350
\(452\) −0.783601 −0.0368575
\(453\) 11.8598i 0.557220i
\(454\) −18.0536 −0.847298
\(455\) 0 0
\(456\) 38.6667 1.81074
\(457\) 3.17034i 0.148302i −0.997247 0.0741511i \(-0.976375\pi\)
0.997247 0.0741511i \(-0.0236247\pi\)
\(458\) 11.8312 0.552835
\(459\) 0.717513 0.0334906
\(460\) 2.35180i 0.109653i
\(461\) 0.202023i 0.00940915i −0.999989 0.00470458i \(-0.998502\pi\)
0.999989 0.00470458i \(-0.00149752\pi\)
\(462\) 0 0
\(463\) 17.2121i 0.799912i 0.916534 + 0.399956i \(0.130975\pi\)
−0.916534 + 0.399956i \(0.869025\pi\)
\(464\) 10.4422 0.484769
\(465\) −6.67670 −0.309625
\(466\) 18.6227i 0.862681i
\(467\) 0.191169 0.00884625 0.00442312 0.999990i \(-0.498592\pi\)
0.00442312 + 0.999990i \(0.498592\pi\)
\(468\) 2.09657 8.65821i 0.0969139 0.400226i
\(469\) 0 0
\(470\) 5.41777i 0.249903i
\(471\) 23.1462 1.06652
\(472\) −30.1483 −1.38769
\(473\) 7.93965i 0.365065i
\(474\) 10.2058i 0.468769i
\(475\) 23.9787i 1.10022i
\(476\) 0 0
\(477\) −2.02978 −0.0929374
\(478\) −21.1631 −0.967978
\(479\) 21.4785i 0.981377i −0.871335 0.490688i \(-0.836745\pi\)
0.871335 0.490688i \(-0.163255\pi\)
\(480\) −6.71352 −0.306429
\(481\) 1.48374 + 0.359285i 0.0676528 + 0.0163820i
\(482\) 2.98752 0.136078
\(483\) 0 0
\(484\) 8.90811 0.404914
\(485\) 0.126352 0.00573733
\(486\) 23.4609i 1.06421i
\(487\) 19.0484i 0.863167i 0.902073 + 0.431584i \(0.142045\pi\)
−0.902073 + 0.431584i \(0.857955\pi\)
\(488\) 14.3574i 0.649930i
\(489\) 57.6458i 2.60684i
\(490\) 0 0
\(491\) −35.7559 −1.61364 −0.806821 0.590796i \(-0.798814\pi\)
−0.806821 + 0.590796i \(0.798814\pi\)
\(492\) 10.3490i 0.466568i
\(493\) −22.0550 −0.993308
\(494\) −4.73735 + 19.5639i −0.213143 + 0.880220i
\(495\) −1.29028 −0.0579937
\(496\) 6.94968i 0.312050i
\(497\) 0 0
\(498\) −26.6213 −1.19293
\(499\) 17.6891i 0.791875i 0.918277 + 0.395937i \(0.129580\pi\)
−0.918277 + 0.395937i \(0.870420\pi\)
\(500\) 5.09953i 0.228058i
\(501\) 2.75218i 0.122958i
\(502\) 25.2820i 1.12839i
\(503\) 11.3305 0.505203 0.252601 0.967570i \(-0.418714\pi\)
0.252601 + 0.967570i \(0.418714\pi\)
\(504\) 0 0
\(505\) 10.8415i 0.482441i
\(506\) 3.36912 0.149776
\(507\) 28.1068 + 14.4599i 1.24827 + 0.642186i
\(508\) 7.22095 0.320378
\(509\) 19.3514i 0.857735i −0.903367 0.428868i \(-0.858913\pi\)
0.903367 0.428868i \(-0.141087\pi\)
\(510\) −5.45523 −0.241562
\(511\) 0 0
\(512\) 16.6645i 0.736472i
\(513\) 1.11663i 0.0493005i
\(514\) 3.68304i 0.162452i
\(515\) 6.76391i 0.298053i
\(516\) −23.1198 −1.01779
\(517\) 5.71967 0.251551
\(518\) 0 0
\(519\) 29.1505 1.27957
\(520\) 1.62220 6.69923i 0.0711383 0.293781i
\(521\) −7.71099 −0.337825 −0.168912 0.985631i \(-0.554025\pi\)
−0.168912 + 0.985631i \(0.554025\pi\)
\(522\) 20.6123i 0.902176i
\(523\) −35.0501 −1.53263 −0.766317 0.642462i \(-0.777913\pi\)
−0.766317 + 0.642462i \(0.777913\pi\)
\(524\) −5.94246 −0.259597
\(525\) 0 0
\(526\) 23.0160i 1.00355i
\(527\) 14.6784i 0.639401i
\(528\) 2.72678i 0.118668i
\(529\) −3.36265 −0.146202
\(530\) −0.467844 −0.0203218
\(531\) 28.7189i 1.24630i
\(532\) 0 0
\(533\) 17.5776 + 4.25637i 0.761370 + 0.184364i
\(534\) −34.9065 −1.51055
\(535\) 3.82563i 0.165396i
\(536\) 31.8529 1.37583
\(537\) 23.0272 0.993699
\(538\) 15.7276i 0.678066i
\(539\) 0 0
\(540\) 0.113903i 0.00490160i
\(541\) 12.1335i 0.521659i −0.965385 0.260829i \(-0.916004\pi\)
0.965385 0.260829i \(-0.0839960\pi\)
\(542\) −2.19296 −0.0941956
\(543\) 27.7944 1.19277
\(544\) 14.7593i 0.632801i
\(545\) −7.12940 −0.305390
\(546\) 0 0
\(547\) −5.12546 −0.219149 −0.109575 0.993979i \(-0.534949\pi\)
−0.109575 + 0.993979i \(0.534949\pi\)
\(548\) 5.27541i 0.225354i
\(549\) 13.6767 0.583707
\(550\) 3.50401 0.149412
\(551\) 34.3232i 1.46222i
\(552\) 32.9340i 1.40177i
\(553\) 0 0
\(554\) 5.83051i 0.247715i
\(555\) −0.643868 −0.0273307
\(556\) 5.54224 0.235043
\(557\) 37.5586i 1.59141i −0.605685 0.795705i \(-0.707101\pi\)
0.605685 0.795705i \(-0.292899\pi\)
\(558\) −13.7182 −0.580738
\(559\) 9.50884 39.2687i 0.402181 1.66089i
\(560\) 0 0
\(561\) 5.75922i 0.243154i
\(562\) 21.7139 0.915945
\(563\) −28.7009 −1.20960 −0.604799 0.796378i \(-0.706747\pi\)
−0.604799 + 0.796378i \(0.706747\pi\)
\(564\) 16.6554i 0.701318i
\(565\) 0.577559i 0.0242981i
\(566\) 1.86195i 0.0782636i
\(567\) 0 0
\(568\) −43.0610 −1.80680
\(569\) −17.9483 −0.752434 −0.376217 0.926532i \(-0.622775\pi\)
−0.376217 + 0.926532i \(0.622775\pi\)
\(570\) 8.48972i 0.355595i
\(571\) −17.8274 −0.746053 −0.373027 0.927821i \(-0.621680\pi\)
−0.373027 + 0.927821i \(0.621680\pi\)
\(572\) −2.10683 0.510165i −0.0880911 0.0213311i
\(573\) −38.1428 −1.59344
\(574\) 0 0
\(575\) 20.4236 0.851724
\(576\) −23.0115 −0.958814
\(577\) 33.0570i 1.37618i 0.725624 + 0.688091i \(0.241551\pi\)
−0.725624 + 0.688091i \(0.758449\pi\)
\(578\) 6.24889i 0.259920i
\(579\) 55.9396i 2.32477i
\(580\) 3.50117i 0.145378i
\(581\) 0 0
\(582\) 0.527085 0.0218484
\(583\) 0.493914i 0.0204558i
\(584\) 15.5395 0.643029
\(585\) −6.38160 1.54529i −0.263847 0.0638899i
\(586\) 29.2345 1.20766
\(587\) 14.7295i 0.607953i 0.952680 + 0.303976i \(0.0983144\pi\)
−0.952680 + 0.303976i \(0.901686\pi\)
\(588\) 0 0
\(589\) −22.8433 −0.941241
\(590\) 6.61941i 0.272517i
\(591\) 24.8088i 1.02050i
\(592\) 0.670193i 0.0275448i
\(593\) 9.20987i 0.378204i 0.981957 + 0.189102i \(0.0605577\pi\)
−0.981957 + 0.189102i \(0.939442\pi\)
\(594\) 0.163174 0.00669510
\(595\) 0 0
\(596\) 3.13743i 0.128514i
\(597\) 28.9907 1.18651
\(598\) 16.6633 + 4.03499i 0.681414 + 0.165003i
\(599\) −10.5745 −0.432064 −0.216032 0.976386i \(-0.569312\pi\)
−0.216032 + 0.976386i \(0.569312\pi\)
\(600\) 34.2527i 1.39836i
\(601\) 4.08916 0.166800 0.0834001 0.996516i \(-0.473422\pi\)
0.0834001 + 0.996516i \(0.473422\pi\)
\(602\) 0 0
\(603\) 30.3426i 1.23565i
\(604\) 4.13902i 0.168414i
\(605\) 6.56578i 0.266937i
\(606\) 45.2261i 1.83719i
\(607\) 3.60706 0.146406 0.0732030 0.997317i \(-0.476678\pi\)
0.0732030 + 0.997317i \(0.476678\pi\)
\(608\) −22.9693 −0.931527
\(609\) 0 0
\(610\) 3.15233 0.127634
\(611\) 28.2890 + 6.85011i 1.14445 + 0.277126i
\(612\) 8.26007 0.333893
\(613\) 38.4845i 1.55437i −0.629271 0.777186i \(-0.716646\pi\)
0.629271 0.777186i \(-0.283354\pi\)
\(614\) 13.6426 0.550569
\(615\) −7.62778 −0.307582
\(616\) 0 0
\(617\) 3.09503i 0.124601i −0.998057 0.0623007i \(-0.980156\pi\)
0.998057 0.0623007i \(-0.0198438\pi\)
\(618\) 28.2161i 1.13502i
\(619\) 12.2692i 0.493142i −0.969125 0.246571i \(-0.920696\pi\)
0.969125 0.246571i \(-0.0793039\pi\)
\(620\) 2.33015 0.0935810
\(621\) 0.951081 0.0381655
\(622\) 10.3215i 0.413854i
\(623\) 0 0
\(624\) 3.26570 13.4864i 0.130733 0.539888i
\(625\) 19.2855 0.771421
\(626\) 9.68482i 0.387083i
\(627\) −8.96281 −0.357940
\(628\) −8.07795 −0.322345
\(629\) 1.41551i 0.0564402i
\(630\) 0 0
\(631\) 5.31780i 0.211698i 0.994382 + 0.105849i \(0.0337561\pi\)
−0.994382 + 0.105849i \(0.966244\pi\)
\(632\) 11.9568i 0.475616i
\(633\) 37.9039 1.50655
\(634\) −26.4253 −1.04948
\(635\) 5.32225i 0.211207i
\(636\) 1.43825 0.0570304
\(637\) 0 0
\(638\) −5.01566 −0.198572
\(639\) 41.0193i 1.62270i
\(640\) 0.218423 0.00863394
\(641\) −12.1904 −0.481493 −0.240746 0.970588i \(-0.577392\pi\)
−0.240746 + 0.970588i \(0.577392\pi\)
\(642\) 15.9589i 0.629847i
\(643\) 18.9733i 0.748235i −0.927381 0.374117i \(-0.877946\pi\)
0.927381 0.374117i \(-0.122054\pi\)
\(644\) 0 0
\(645\) 17.0406i 0.670974i
\(646\) −18.6642 −0.734334
\(647\) −19.7117 −0.774948 −0.387474 0.921881i \(-0.626652\pi\)
−0.387474 + 0.921881i \(0.626652\pi\)
\(648\) 28.2955i 1.11155i
\(649\) 6.98827 0.274314
\(650\) 17.3305 + 4.19655i 0.679759 + 0.164602i
\(651\) 0 0
\(652\) 20.1182i 0.787891i
\(653\) −20.3973 −0.798206 −0.399103 0.916906i \(-0.630678\pi\)
−0.399103 + 0.916906i \(0.630678\pi\)
\(654\) −29.7408 −1.16296
\(655\) 4.37993i 0.171138i
\(656\) 7.93965i 0.309991i
\(657\) 14.8027i 0.577510i
\(658\) 0 0
\(659\) 32.6628 1.27236 0.636181 0.771540i \(-0.280513\pi\)
0.636181 + 0.771540i \(0.280513\pi\)
\(660\) 0.914258 0.0355875
\(661\) 9.73692i 0.378722i 0.981908 + 0.189361i \(0.0606417\pi\)
−0.981908 + 0.189361i \(0.939358\pi\)
\(662\) −14.1341 −0.549337
\(663\) −6.89747 + 28.4846i −0.267876 + 1.10625i
\(664\) 31.1886 1.21035
\(665\) 0 0
\(666\) −1.32292 −0.0512620
\(667\) −29.2345 −1.13196
\(668\) 0.960502i 0.0371630i
\(669\) 16.4524i 0.636086i
\(670\) 6.99365i 0.270188i
\(671\) 3.32800i 0.128476i
\(672\) 0 0
\(673\) −39.4512 −1.52073 −0.760367 0.649494i \(-0.774981\pi\)
−0.760367 + 0.649494i \(0.774981\pi\)
\(674\) 18.2748i 0.703921i
\(675\) 0.989161 0.0380728
\(676\) −9.80920 5.04646i −0.377277 0.194094i
\(677\) −48.6339 −1.86915 −0.934576 0.355764i \(-0.884221\pi\)
−0.934576 + 0.355764i \(0.884221\pi\)
\(678\) 2.40932i 0.0925296i
\(679\) 0 0
\(680\) 6.39117 0.245090
\(681\) 40.9072i 1.56757i
\(682\) 3.33810i 0.127822i
\(683\) 5.70773i 0.218400i −0.994020 0.109200i \(-0.965171\pi\)
0.994020 0.109200i \(-0.0348289\pi\)
\(684\) 12.8548i 0.491514i
\(685\) −3.88828 −0.148563
\(686\) 0 0
\(687\) 26.8080i 1.02279i
\(688\) −17.7373 −0.676230
\(689\) −0.591531 + 2.44285i −0.0225356 + 0.0930653i
\(690\) −7.23104 −0.275281
\(691\) 41.7732i 1.58913i −0.607182 0.794563i \(-0.707700\pi\)
0.607182 0.794563i \(-0.292300\pi\)
\(692\) −10.1734 −0.386736
\(693\) 0 0
\(694\) 0.493517i 0.0187336i
\(695\) 4.08495i 0.154951i
\(696\) 49.0295i 1.85846i
\(697\) 16.7693i 0.635182i
\(698\) −7.38071 −0.279364
\(699\) 42.1967 1.59603
\(700\) 0 0
\(701\) 22.4361 0.847399 0.423700 0.905803i \(-0.360731\pi\)
0.423700 + 0.905803i \(0.360731\pi\)
\(702\) 0.807042 + 0.195424i 0.0304598 + 0.00737579i
\(703\) −2.20290 −0.0830838
\(704\) 5.59948i 0.211038i
\(705\) −12.2760 −0.462340
\(706\) −1.64527 −0.0619207
\(707\) 0 0
\(708\) 20.3495i 0.764780i
\(709\) 28.7468i 1.07961i 0.841790 + 0.539804i \(0.181502\pi\)
−0.841790 + 0.539804i \(0.818498\pi\)
\(710\) 9.45452i 0.354822i
\(711\) −11.3899 −0.427154
\(712\) 40.8953 1.53261
\(713\) 19.4566i 0.728654i
\(714\) 0 0
\(715\) −0.376021 + 1.55286i −0.0140624 + 0.0580735i
\(716\) −8.03644 −0.300336
\(717\) 47.9529i 1.79083i
\(718\) −29.1944 −1.08952
\(719\) 4.20899 0.156969 0.0784844 0.996915i \(-0.474992\pi\)
0.0784844 + 0.996915i \(0.474992\pi\)
\(720\) 2.88251i 0.107425i
\(721\) 0 0
\(722\) 8.65821i 0.322225i
\(723\) 6.76934i 0.251754i
\(724\) −9.70015 −0.360503
\(725\) −30.4050 −1.12921
\(726\) 27.3896i 1.01652i
\(727\) −43.4680 −1.61214 −0.806070 0.591820i \(-0.798409\pi\)
−0.806070 + 0.591820i \(0.798409\pi\)
\(728\) 0 0
\(729\) −25.3884 −0.940312
\(730\) 3.41187i 0.126279i
\(731\) 37.4630 1.38562
\(732\) −9.69096 −0.358188
\(733\) 9.09421i 0.335902i −0.985795 0.167951i \(-0.946285\pi\)
0.985795 0.167951i \(-0.0537151\pi\)
\(734\) 28.9525i 1.06866i
\(735\) 0 0
\(736\) 19.5639i 0.721133i
\(737\) −7.38337 −0.271970
\(738\) −15.6723 −0.576907
\(739\) 9.60867i 0.353461i 0.984259 + 0.176730i \(0.0565520\pi\)
−0.984259 + 0.176730i \(0.943448\pi\)
\(740\) 0.224708 0.00826044
\(741\) −44.3292 10.7342i −1.62847 0.394331i
\(742\) 0 0
\(743\) 32.1771i 1.18046i −0.807234 0.590231i \(-0.799036\pi\)
0.807234 0.590231i \(-0.200964\pi\)
\(744\) 32.6308 1.19630
\(745\) 2.31246 0.0847220
\(746\) 4.26258i 0.156064i
\(747\) 29.7098i 1.08703i
\(748\) 2.00995i 0.0734911i
\(749\) 0 0
\(750\) −15.6794 −0.572531
\(751\) −7.79784 −0.284547 −0.142274 0.989827i \(-0.545441\pi\)
−0.142274 + 0.989827i \(0.545441\pi\)
\(752\) 12.7779i 0.465961i
\(753\) 57.2858 2.08761
\(754\) −24.8070 6.00696i −0.903417 0.218760i
\(755\) −3.05070 −0.111026
\(756\) 0 0
\(757\) 17.9970 0.654110 0.327055 0.945005i \(-0.393944\pi\)
0.327055 + 0.945005i \(0.393944\pi\)
\(758\) 12.2392 0.444548
\(759\) 7.63399i 0.277096i
\(760\) 9.94627i 0.360789i
\(761\) 40.6790i 1.47461i 0.675559 + 0.737306i \(0.263902\pi\)
−0.675559 + 0.737306i \(0.736098\pi\)
\(762\) 22.2021i 0.804299i
\(763\) 0 0
\(764\) 13.3117 0.481601
\(765\) 6.08814i 0.220117i
\(766\) 25.5301 0.922439
\(767\) 34.5633 + 8.36944i 1.24801 + 0.302203i
\(768\) 39.3422 1.41964
\(769\) 39.3098i 1.41755i 0.705435 + 0.708774i \(0.250752\pi\)
−0.705435 + 0.708774i \(0.749248\pi\)
\(770\) 0 0
\(771\) −8.34529 −0.300548
\(772\) 19.5228i 0.702640i
\(773\) 13.4736i 0.484611i 0.970200 + 0.242306i \(0.0779036\pi\)
−0.970200 + 0.242306i \(0.922096\pi\)
\(774\) 35.0124i 1.25849i
\(775\) 20.2356i 0.726884i
\(776\) −0.617515 −0.0221675
\(777\) 0 0
\(778\) 30.4866i 1.09300i
\(779\) −26.0973 −0.935032
\(780\) 4.52183 + 1.09495i 0.161908 + 0.0392056i
\(781\) 9.98137 0.357161
\(782\) 15.8971i 0.568478i
\(783\) −1.41589 −0.0505998
\(784\) 0 0
\(785\) 5.95391i 0.212504i
\(786\) 18.2712i 0.651711i
\(787\) 39.8291i 1.41975i −0.704326 0.709877i \(-0.748751\pi\)
0.704326 0.709877i \(-0.251249\pi\)
\(788\) 8.65821i 0.308436i
\(789\) 52.1514 1.85664
\(790\) −2.62525 −0.0934022
\(791\) 0 0
\(792\) 6.30594 0.224072
\(793\) 3.98574 16.4600i 0.141538 0.584510i
\(794\) −10.5794 −0.375448
\(795\) 1.06007i 0.0375969i
\(796\) −10.1177 −0.358612
\(797\) 21.2530 0.752821 0.376410 0.926453i \(-0.377158\pi\)
0.376410 + 0.926453i \(0.377158\pi\)
\(798\) 0 0
\(799\) 26.9881i 0.954770i
\(800\) 20.3472i 0.719382i
\(801\) 38.9563i 1.37645i
\(802\) 13.5413 0.478161
\(803\) −3.60200 −0.127112
\(804\) 21.5000i 0.758246i
\(805\) 0 0
\(806\) −3.99784 + 16.5099i −0.140818 + 0.581537i
\(807\) −35.6368 −1.25447
\(808\) 52.9854i 1.86402i
\(809\) 21.4175 0.753000 0.376500 0.926417i \(-0.377128\pi\)
0.376500 + 0.926417i \(0.377128\pi\)
\(810\) −6.21260 −0.218288
\(811\) 11.0116i 0.386669i −0.981133 0.193335i \(-0.938070\pi\)
0.981133 0.193335i \(-0.0619303\pi\)
\(812\) 0 0
\(813\) 4.96896i 0.174269i
\(814\) 0.321910i 0.0112829i
\(815\) −14.8283 −0.519412
\(816\) 12.8662 0.450408
\(817\) 58.3019i 2.03972i
\(818\) 19.3765 0.677484
\(819\) 0 0
\(820\) 2.66207 0.0929636
\(821\) 38.6685i 1.34954i −0.738029 0.674769i \(-0.764243\pi\)
0.738029 0.674769i \(-0.235757\pi\)
\(822\) −16.2202 −0.565744
\(823\) 20.4566 0.713073 0.356537 0.934281i \(-0.383957\pi\)
0.356537 + 0.934281i \(0.383957\pi\)
\(824\) 33.0570i 1.15160i
\(825\) 7.93965i 0.276423i
\(826\) 0 0
\(827\) 27.3451i 0.950881i −0.879748 0.475440i \(-0.842289\pi\)
0.879748 0.475440i \(-0.157711\pi\)
\(828\) 10.9489 0.380501
\(829\) 25.0086 0.868585 0.434292 0.900772i \(-0.356998\pi\)
0.434292 + 0.900772i \(0.356998\pi\)
\(830\) 6.84780i 0.237691i
\(831\) −13.2112 −0.458291
\(832\) −6.70616 + 27.6945i −0.232494 + 0.960133i
\(833\) 0 0
\(834\) 17.0406i 0.590069i
\(835\) 0.707945 0.0244994
\(836\) 3.12799 0.108184
\(837\) 0.942324i 0.0325715i
\(838\) 15.3237i 0.529350i
\(839\) 8.76981i 0.302768i −0.988475 0.151384i \(-0.951627\pi\)
0.988475 0.151384i \(-0.0483729\pi\)
\(840\) 0 0
\(841\) 14.5218 0.500753
\(842\) −4.58665 −0.158066
\(843\) 49.2009i 1.69457i
\(844\) −13.2284 −0.455339
\(845\) −3.71952 + 7.22994i −0.127956 + 0.248717i
\(846\) −25.2227 −0.867174
\(847\) 0 0
\(848\) 1.10342 0.0378914
\(849\) 4.21894 0.144794
\(850\) 16.5336i 0.567097i
\(851\) 1.87630i 0.0643186i
\(852\) 29.0652i 0.995758i
\(853\) 19.8232i 0.678734i −0.940654 0.339367i \(-0.889787\pi\)
0.940654 0.339367i \(-0.110213\pi\)
\(854\) 0 0
\(855\) 9.47469 0.324028
\(856\) 18.6969i 0.639047i
\(857\) 2.67037 0.0912181 0.0456090 0.998959i \(-0.485477\pi\)
0.0456090 + 0.998959i \(0.485477\pi\)
\(858\) −1.56860 + 6.47784i −0.0535510 + 0.221150i
\(859\) −38.9597 −1.32929 −0.664644 0.747160i \(-0.731417\pi\)
−0.664644 + 0.747160i \(0.731417\pi\)
\(860\) 5.94713i 0.202795i
\(861\) 0 0
\(862\) −15.6738 −0.533851
\(863\) 24.7976i 0.844121i −0.906568 0.422060i \(-0.861307\pi\)
0.906568 0.422060i \(-0.138693\pi\)
\(864\) 0.947521i 0.0322353i
\(865\) 7.49840i 0.254953i
\(866\) 30.0561i 1.02135i
\(867\) 14.1592 0.480871
\(868\) 0 0
\(869\) 2.77154i 0.0940181i
\(870\) 10.7650 0.364967
\(871\) −36.5175 8.84262i −1.23735 0.299621i
\(872\) 34.8433 1.17994
\(873\) 0.588237i 0.0199088i
\(874\) −24.7399 −0.836839
\(875\) 0 0
\(876\) 10.4888i 0.354385i
\(877\) 1.97840i 0.0668059i 0.999442 + 0.0334029i \(0.0106345\pi\)
−0.999442 + 0.0334029i \(0.989366\pi\)
\(878\) 18.3250i 0.618440i
\(879\) 66.2415i 2.23427i
\(880\) 0.701412 0.0236446
\(881\) −17.1466 −0.577683 −0.288841 0.957377i \(-0.593270\pi\)
−0.288841 + 0.957377i \(0.593270\pi\)
\(882\) 0 0
\(883\) 10.2168 0.343822 0.171911 0.985112i \(-0.445006\pi\)
0.171911 + 0.985112i \(0.445006\pi\)
\(884\) 2.40720 9.94102i 0.0809628 0.334353i
\(885\) −14.9987 −0.504177
\(886\) 14.8250i 0.498055i
\(887\) 50.9931 1.71218 0.856090 0.516826i \(-0.172887\pi\)
0.856090 + 0.516826i \(0.172887\pi\)
\(888\) 3.14676 0.105598
\(889\) 0 0
\(890\) 8.97901i 0.300977i
\(891\) 6.55879i 0.219728i
\(892\) 5.74184i 0.192251i
\(893\) −42.0003 −1.40549
\(894\) 9.64659 0.322630
\(895\) 5.92331i 0.197995i
\(896\) 0 0
\(897\) −9.14277 + 37.7570i −0.305268 + 1.26067i
\(898\) 35.0255 1.16882
\(899\) 28.9653i 0.966047i
\(900\) 11.3873 0.379577
\(901\) −2.33052 −0.0776408
\(902\) 3.81360i 0.126979i
\(903\) 0 0
\(904\) 2.82268i 0.0938811i
\(905\) 7.14957i 0.237660i
\(906\) −12.7262 −0.422799
\(907\) 5.78538 0.192100 0.0960501 0.995376i \(-0.469379\pi\)
0.0960501 + 0.995376i \(0.469379\pi\)
\(908\) 14.2765i 0.473782i
\(909\) 50.4732 1.67409
\(910\) 0 0
\(911\) −1.70706 −0.0565573 −0.0282787 0.999600i \(-0.509003\pi\)
−0.0282787 + 0.999600i \(0.509003\pi\)
\(912\) 20.0231i 0.663032i
\(913\) −7.22940 −0.239258
\(914\) 3.40195 0.112526
\(915\) 7.14279i 0.236133i
\(916\) 9.35590i 0.309128i
\(917\) 0 0
\(918\) 0.769931i 0.0254115i
\(919\) 37.2050 1.22728 0.613640 0.789586i \(-0.289705\pi\)
0.613640 + 0.789586i \(0.289705\pi\)
\(920\) 8.47164 0.279302
\(921\) 30.9123i 1.01859i
\(922\) 0.216782 0.00713933
\(923\) 49.3669 + 11.9541i 1.62493 + 0.393474i
\(924\) 0 0
\(925\) 1.95142i 0.0641623i
\(926\) −18.4695 −0.606945
\(927\) 31.4897 1.03426
\(928\) 29.1251i 0.956077i
\(929\) 19.9046i 0.653048i −0.945189 0.326524i \(-0.894123\pi\)
0.945189 0.326524i \(-0.105877\pi\)
\(930\) 7.16447i 0.234932i
\(931\) 0 0
\(932\) −14.7265 −0.482383
\(933\) 23.3872 0.765661
\(934\) 0.205135i 0.00671222i
\(935\) −1.48145 −0.0484485
\(936\) 31.1886 + 7.55225i 1.01943 + 0.246853i
\(937\) 7.16949 0.234217 0.117109 0.993119i \(-0.462637\pi\)
0.117109 + 0.993119i \(0.462637\pi\)
\(938\) 0 0
\(939\) 21.9446 0.716134
\(940\) 4.28428 0.139738
\(941\) 3.06072i 0.0997766i −0.998755 0.0498883i \(-0.984113\pi\)
0.998755 0.0498883i \(-0.0158865\pi\)
\(942\) 24.8371i 0.809237i
\(943\) 22.2281i 0.723846i
\(944\) 15.6120i 0.508126i
\(945\) 0 0
\(946\) 8.51968 0.276999
\(947\) 44.2056i 1.43649i −0.695791 0.718244i \(-0.744946\pi\)
0.695791 0.718244i \(-0.255054\pi\)
\(948\) 8.07058 0.262120
\(949\) −17.8151 4.31390i −0.578304 0.140035i
\(950\) −25.7304 −0.834806
\(951\) 59.8762i 1.94162i
\(952\) 0 0
\(953\) 13.7002 0.443791 0.221896 0.975070i \(-0.428776\pi\)
0.221896 + 0.975070i \(0.428776\pi\)
\(954\) 2.17807i 0.0705176i
\(955\) 9.81149i 0.317492i
\(956\) 16.7354i 0.541262i
\(957\) 11.3648i 0.367373i
\(958\) 23.0476 0.744634
\(959\) 0 0
\(960\) 12.0180i 0.387879i
\(961\) 11.7226 0.378147
\(962\) −0.385532 + 1.59214i −0.0124301 + 0.0513325i
\(963\) −17.8104 −0.573933
\(964\) 2.36248i 0.0760903i
\(965\) 14.3894 0.463211
\(966\) 0 0
\(967\) 43.9429i 1.41311i 0.707659 + 0.706554i \(0.249751\pi\)
−0.707659 + 0.706554i \(0.750249\pi\)
\(968\) 32.0887i 1.03137i
\(969\) 42.2907i 1.35857i
\(970\) 0.135582i 0.00435329i
\(971\) −21.3171 −0.684098 −0.342049 0.939682i \(-0.611121\pi\)
−0.342049 + 0.939682i \(0.611121\pi\)
\(972\) 18.5525 0.595072
\(973\) 0 0
\(974\) −20.4400 −0.654941
\(975\) −9.50884 + 39.2687i −0.304527 + 1.25761i
\(976\) −7.43482 −0.237983
\(977\) 16.4466i 0.526173i 0.964772 + 0.263087i \(0.0847405\pi\)
−0.964772 + 0.263087i \(0.915260\pi\)
\(978\) −61.8572 −1.97797
\(979\) −9.47937 −0.302962
\(980\) 0 0
\(981\) 33.1913i 1.05972i
\(982\) 38.3681i 1.22437i
\(983\) 11.7104i 0.373504i 0.982407 + 0.186752i \(0.0597961\pi\)
−0.982407 + 0.186752i \(0.940204\pi\)
\(984\) 37.2790 1.18841
\(985\) 6.38160 0.203335
\(986\) 23.6662i 0.753687i
\(987\) 0 0
\(988\) 15.4708 + 3.74621i 0.492190 + 0.119183i
\(989\) 49.6581 1.57903
\(990\) 1.38454i 0.0440036i
\(991\) −31.3747 −0.996650 −0.498325 0.866990i \(-0.666051\pi\)
−0.498325 + 0.866990i \(0.666051\pi\)
\(992\) −19.3837 −0.615434
\(993\) 32.0260i 1.01632i
\(994\) 0 0
\(995\) 7.45731i 0.236413i
\(996\) 21.0516i 0.667046i
\(997\) −4.69711 −0.148759 −0.0743794 0.997230i \(-0.523698\pi\)
−0.0743794 + 0.997230i \(0.523698\pi\)
\(998\) −18.9814 −0.600847
\(999\) 0.0908732i 0.00287510i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.c.f.246.6 8
7.2 even 3 91.2.r.a.25.6 yes 16
7.3 odd 6 637.2.r.f.324.3 16
7.4 even 3 91.2.r.a.51.3 yes 16
7.5 odd 6 637.2.r.f.116.6 16
7.6 odd 2 637.2.c.e.246.6 8
13.5 odd 4 8281.2.a.ck.1.6 8
13.8 odd 4 8281.2.a.ck.1.3 8
13.12 even 2 inner 637.2.c.f.246.3 8
21.2 odd 6 819.2.dl.e.298.3 16
21.11 odd 6 819.2.dl.e.415.6 16
91.12 odd 6 637.2.r.f.116.3 16
91.18 odd 12 1183.2.e.i.170.3 16
91.25 even 6 91.2.r.a.51.6 yes 16
91.34 even 4 8281.2.a.cj.1.3 8
91.38 odd 6 637.2.r.f.324.6 16
91.44 odd 12 1183.2.e.i.508.3 16
91.51 even 6 91.2.r.a.25.3 16
91.60 odd 12 1183.2.e.i.170.6 16
91.83 even 4 8281.2.a.cj.1.6 8
91.86 odd 12 1183.2.e.i.508.6 16
91.90 odd 2 637.2.c.e.246.3 8
273.116 odd 6 819.2.dl.e.415.3 16
273.233 odd 6 819.2.dl.e.298.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.r.a.25.3 16 91.51 even 6
91.2.r.a.25.6 yes 16 7.2 even 3
91.2.r.a.51.3 yes 16 7.4 even 3
91.2.r.a.51.6 yes 16 91.25 even 6
637.2.c.e.246.3 8 91.90 odd 2
637.2.c.e.246.6 8 7.6 odd 2
637.2.c.f.246.3 8 13.12 even 2 inner
637.2.c.f.246.6 8 1.1 even 1 trivial
637.2.r.f.116.3 16 91.12 odd 6
637.2.r.f.116.6 16 7.5 odd 6
637.2.r.f.324.3 16 7.3 odd 6
637.2.r.f.324.6 16 91.38 odd 6
819.2.dl.e.298.3 16 21.2 odd 6
819.2.dl.e.298.6 16 273.233 odd 6
819.2.dl.e.415.3 16 273.116 odd 6
819.2.dl.e.415.6 16 21.11 odd 6
1183.2.e.i.170.3 16 91.18 odd 12
1183.2.e.i.170.6 16 91.60 odd 12
1183.2.e.i.508.3 16 91.44 odd 12
1183.2.e.i.508.6 16 91.86 odd 12
8281.2.a.cj.1.3 8 91.34 even 4
8281.2.a.cj.1.6 8 91.83 even 4
8281.2.a.ck.1.3 8 13.8 odd 4
8281.2.a.ck.1.6 8 13.5 odd 4