Properties

Label 224.3.r.c.95.2
Level $224$
Weight $3$
Character 224.95
Analytic conductor $6.104$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(95,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.95");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.r (of order \(6\), degree \(2\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 28 x^{9} - 100 x^{8} + 140 x^{7} + 392 x^{6} + 1400 x^{5} + 8040 x^{4} + 11256 x^{3} + 9800 x^{2} + 13720 x + 9604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 95.2
Root \(3.45178 + 0.924901i\) of defining polynomial
Character \(\chi\) \(=\) 224.95
Dual form 224.3.r.c.191.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.51065 - 2.02688i) q^{3} +(1.21646 + 2.10698i) q^{5} +(2.97300 - 6.33729i) q^{7} +(3.71646 + 6.43710i) q^{9} +O(q^{10})\) \(q+(-3.51065 - 2.02688i) q^{3} +(1.21646 + 2.10698i) q^{5} +(2.97300 - 6.33729i) q^{7} +(3.71646 + 6.43710i) q^{9} +(-6.39055 - 3.68959i) q^{11} -8.56707 q^{13} -9.86249i q^{15} +(-7.45812 + 12.9178i) q^{17} +(-22.9151 + 13.2300i) q^{19} +(-23.2821 + 16.2221i) q^{21} +(-12.9208 + 7.45980i) q^{23} +(9.54044 - 16.5245i) q^{25} +6.35253i q^{27} -52.0780 q^{29} +(-18.4688 - 10.6629i) q^{31} +(14.9567 + 25.9057i) q^{33} +(16.9691 - 1.44504i) q^{35} +(10.9063 + 18.8902i) q^{37} +(30.0760 + 17.3644i) q^{39} +58.5138 q^{41} -58.6983i q^{43} +(-9.04188 + 15.6610i) q^{45} +(-57.4110 + 33.1463i) q^{47} +(-31.3225 - 37.6816i) q^{49} +(52.3658 - 30.2334i) q^{51} +(36.0642 - 62.4650i) q^{53} -17.9530i q^{55} +107.263 q^{57} +(15.3096 + 8.83897i) q^{59} +(-44.1746 - 76.5126i) q^{61} +(51.8428 - 4.41480i) q^{63} +(-10.4215 - 18.0506i) q^{65} +(80.0413 + 46.2119i) q^{67} +60.4804 q^{69} -11.8459i q^{71} +(-33.2555 + 57.6001i) q^{73} +(-66.9863 + 38.6746i) q^{75} +(-42.3811 + 29.5296i) q^{77} +(-15.9202 + 9.19151i) q^{79} +(46.3240 - 80.2355i) q^{81} +101.779i q^{83} -36.2901 q^{85} +(182.828 + 105.556i) q^{87} +(-23.2850 - 40.3308i) q^{89} +(-25.4699 + 54.2920i) q^{91} +(43.2250 + 74.8678i) q^{93} +(-55.7507 - 32.1877i) q^{95} -135.924 q^{97} -54.8488i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{5} + 32 q^{9} - 128 q^{13} - 6 q^{17} - 62 q^{21} - 32 q^{25} + 128 q^{29} - 134 q^{33} - 66 q^{37} + 384 q^{41} - 192 q^{45} - 12 q^{49} - 2 q^{53} + 468 q^{57} - 434 q^{61} + 160 q^{65} + 124 q^{69} - 10 q^{73} - 370 q^{77} + 422 q^{81} + 1020 q^{85} - 522 q^{89} + 306 q^{93} - 768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.51065 2.02688i −1.17022 0.675626i −0.216487 0.976286i \(-0.569460\pi\)
−0.953731 + 0.300660i \(0.902793\pi\)
\(4\) 0 0
\(5\) 1.21646 + 2.10698i 0.243293 + 0.421395i 0.961650 0.274279i \(-0.0884391\pi\)
−0.718358 + 0.695674i \(0.755106\pi\)
\(6\) 0 0
\(7\) 2.97300 6.33729i 0.424714 0.905327i
\(8\) 0 0
\(9\) 3.71646 + 6.43710i 0.412940 + 0.715234i
\(10\) 0 0
\(11\) −6.39055 3.68959i −0.580959 0.335417i 0.180555 0.983565i \(-0.442211\pi\)
−0.761514 + 0.648148i \(0.775544\pi\)
\(12\) 0 0
\(13\) −8.56707 −0.659006 −0.329503 0.944155i \(-0.606881\pi\)
−0.329503 + 0.944155i \(0.606881\pi\)
\(14\) 0 0
\(15\) 9.86249i 0.657499i
\(16\) 0 0
\(17\) −7.45812 + 12.9178i −0.438713 + 0.759873i −0.997591 0.0693771i \(-0.977899\pi\)
0.558878 + 0.829250i \(0.311232\pi\)
\(18\) 0 0
\(19\) −22.9151 + 13.2300i −1.20606 + 0.696317i −0.961895 0.273418i \(-0.911846\pi\)
−0.244161 + 0.969735i \(0.578512\pi\)
\(20\) 0 0
\(21\) −23.2821 + 16.2221i −1.10867 + 0.772482i
\(22\) 0 0
\(23\) −12.9208 + 7.45980i −0.561772 + 0.324339i −0.753856 0.657039i \(-0.771809\pi\)
0.192084 + 0.981378i \(0.438475\pi\)
\(24\) 0 0
\(25\) 9.54044 16.5245i 0.381617 0.660981i
\(26\) 0 0
\(27\) 6.35253i 0.235279i
\(28\) 0 0
\(29\) −52.0780 −1.79579 −0.897896 0.440207i \(-0.854905\pi\)
−0.897896 + 0.440207i \(0.854905\pi\)
\(30\) 0 0
\(31\) −18.4688 10.6629i −0.595767 0.343966i 0.171608 0.985165i \(-0.445104\pi\)
−0.767374 + 0.641199i \(0.778437\pi\)
\(32\) 0 0
\(33\) 14.9567 + 25.9057i 0.453233 + 0.785022i
\(34\) 0 0
\(35\) 16.9691 1.44504i 0.484830 0.0412868i
\(36\) 0 0
\(37\) 10.9063 + 18.8902i 0.294765 + 0.510547i 0.974930 0.222511i \(-0.0714255\pi\)
−0.680166 + 0.733059i \(0.738092\pi\)
\(38\) 0 0
\(39\) 30.0760 + 17.3644i 0.771180 + 0.445241i
\(40\) 0 0
\(41\) 58.5138 1.42717 0.713583 0.700571i \(-0.247071\pi\)
0.713583 + 0.700571i \(0.247071\pi\)
\(42\) 0 0
\(43\) 58.6983i 1.36508i −0.730849 0.682539i \(-0.760876\pi\)
0.730849 0.682539i \(-0.239124\pi\)
\(44\) 0 0
\(45\) −9.04188 + 15.6610i −0.200931 + 0.348022i
\(46\) 0 0
\(47\) −57.4110 + 33.1463i −1.22151 + 0.705240i −0.965240 0.261365i \(-0.915827\pi\)
−0.256271 + 0.966605i \(0.582494\pi\)
\(48\) 0 0
\(49\) −31.3225 37.6816i −0.639235 0.769011i
\(50\) 0 0
\(51\) 52.3658 30.2334i 1.02678 0.592812i
\(52\) 0 0
\(53\) 36.0642 62.4650i 0.680456 1.17858i −0.294385 0.955687i \(-0.595115\pi\)
0.974842 0.222898i \(-0.0715517\pi\)
\(54\) 0 0
\(55\) 17.9530i 0.326418i
\(56\) 0 0
\(57\) 107.263 1.88180
\(58\) 0 0
\(59\) 15.3096 + 8.83897i 0.259484 + 0.149813i 0.624099 0.781345i \(-0.285466\pi\)
−0.364615 + 0.931158i \(0.618799\pi\)
\(60\) 0 0
\(61\) −44.1746 76.5126i −0.724173 1.25431i −0.959313 0.282344i \(-0.908888\pi\)
0.235140 0.971962i \(-0.424445\pi\)
\(62\) 0 0
\(63\) 51.8428 4.41480i 0.822902 0.0700761i
\(64\) 0 0
\(65\) −10.4215 18.0506i −0.160331 0.277702i
\(66\) 0 0
\(67\) 80.0413 + 46.2119i 1.19465 + 0.689730i 0.959357 0.282196i \(-0.0910627\pi\)
0.235290 + 0.971925i \(0.424396\pi\)
\(68\) 0 0
\(69\) 60.4804 0.876528
\(70\) 0 0
\(71\) 11.8459i 0.166844i −0.996514 0.0834222i \(-0.973415\pi\)
0.996514 0.0834222i \(-0.0265850\pi\)
\(72\) 0 0
\(73\) −33.2555 + 57.6001i −0.455554 + 0.789043i −0.998720 0.0505826i \(-0.983892\pi\)
0.543166 + 0.839626i \(0.317226\pi\)
\(74\) 0 0
\(75\) −66.9863 + 38.6746i −0.893151 + 0.515661i
\(76\) 0 0
\(77\) −42.3811 + 29.5296i −0.550404 + 0.383502i
\(78\) 0 0
\(79\) −15.9202 + 9.19151i −0.201521 + 0.116348i −0.597365 0.801970i \(-0.703786\pi\)
0.395844 + 0.918318i \(0.370452\pi\)
\(80\) 0 0
\(81\) 46.3240 80.2355i 0.571901 0.990561i
\(82\) 0 0
\(83\) 101.779i 1.22626i 0.789984 + 0.613128i \(0.210089\pi\)
−0.789984 + 0.613128i \(0.789911\pi\)
\(84\) 0 0
\(85\) −36.2901 −0.426942
\(86\) 0 0
\(87\) 182.828 + 105.556i 2.10147 + 1.21328i
\(88\) 0 0
\(89\) −23.2850 40.3308i −0.261629 0.453155i 0.705046 0.709162i \(-0.250926\pi\)
−0.966675 + 0.256007i \(0.917593\pi\)
\(90\) 0 0
\(91\) −25.4699 + 54.2920i −0.279889 + 0.596616i
\(92\) 0 0
\(93\) 43.2250 + 74.8678i 0.464785 + 0.805031i
\(94\) 0 0
\(95\) −55.7507 32.1877i −0.586849 0.338818i
\(96\) 0 0
\(97\) −135.924 −1.40128 −0.700639 0.713516i \(-0.747102\pi\)
−0.700639 + 0.713516i \(0.747102\pi\)
\(98\) 0 0
\(99\) 54.8488i 0.554029i
\(100\) 0 0
\(101\) 44.7274 77.4701i 0.442845 0.767030i −0.555054 0.831814i \(-0.687302\pi\)
0.997899 + 0.0647838i \(0.0206358\pi\)
\(102\) 0 0
\(103\) −4.54400 + 2.62348i −0.0441165 + 0.0254707i −0.521896 0.853009i \(-0.674775\pi\)
0.477780 + 0.878480i \(0.341442\pi\)
\(104\) 0 0
\(105\) −62.5014 29.3212i −0.595252 0.279249i
\(106\) 0 0
\(107\) 13.0278 7.52159i 0.121755 0.0702952i −0.437886 0.899031i \(-0.644273\pi\)
0.559641 + 0.828735i \(0.310939\pi\)
\(108\) 0 0
\(109\) 3.67603 6.36707i 0.0337250 0.0584135i −0.848670 0.528922i \(-0.822596\pi\)
0.882395 + 0.470509i \(0.155930\pi\)
\(110\) 0 0
\(111\) 88.4228i 0.796602i
\(112\) 0 0
\(113\) 170.683 1.51047 0.755236 0.655454i \(-0.227522\pi\)
0.755236 + 0.655454i \(0.227522\pi\)
\(114\) 0 0
\(115\) −31.4353 18.1492i −0.273350 0.157819i
\(116\) 0 0
\(117\) −31.8392 55.1471i −0.272130 0.471343i
\(118\) 0 0
\(119\) 59.6911 + 85.6690i 0.501606 + 0.719908i
\(120\) 0 0
\(121\) −33.2739 57.6321i −0.274991 0.476298i
\(122\) 0 0
\(123\) −205.422 118.600i −1.67010 0.964230i
\(124\) 0 0
\(125\) 107.246 0.857964
\(126\) 0 0
\(127\) 194.746i 1.53343i −0.641987 0.766716i \(-0.721890\pi\)
0.641987 0.766716i \(-0.278110\pi\)
\(128\) 0 0
\(129\) −118.974 + 206.070i −0.922282 + 1.59744i
\(130\) 0 0
\(131\) 25.9409 14.9770i 0.198022 0.114328i −0.397711 0.917511i \(-0.630195\pi\)
0.595732 + 0.803183i \(0.296862\pi\)
\(132\) 0 0
\(133\) 15.7160 + 184.552i 0.118165 + 1.38761i
\(134\) 0 0
\(135\) −13.3846 + 7.72762i −0.0991454 + 0.0572416i
\(136\) 0 0
\(137\) −88.9157 + 154.007i −0.649020 + 1.12414i 0.334337 + 0.942454i \(0.391488\pi\)
−0.983357 + 0.181682i \(0.941846\pi\)
\(138\) 0 0
\(139\) 120.142i 0.864334i −0.901793 0.432167i \(-0.857749\pi\)
0.901793 0.432167i \(-0.142251\pi\)
\(140\) 0 0
\(141\) 268.734 1.90591
\(142\) 0 0
\(143\) 54.7483 + 31.6090i 0.382855 + 0.221042i
\(144\) 0 0
\(145\) −63.3509 109.727i −0.436903 0.756738i
\(146\) 0 0
\(147\) 33.5867 + 195.774i 0.228481 + 1.33179i
\(148\) 0 0
\(149\) 41.2640 + 71.4713i 0.276939 + 0.479673i 0.970623 0.240607i \(-0.0773466\pi\)
−0.693683 + 0.720280i \(0.744013\pi\)
\(150\) 0 0
\(151\) −246.068 142.067i −1.62959 0.940843i −0.984215 0.176975i \(-0.943369\pi\)
−0.645373 0.763868i \(-0.723298\pi\)
\(152\) 0 0
\(153\) −110.871 −0.724649
\(154\) 0 0
\(155\) 51.8843i 0.334738i
\(156\) 0 0
\(157\) −28.1271 + 48.7175i −0.179153 + 0.310303i −0.941591 0.336759i \(-0.890669\pi\)
0.762437 + 0.647062i \(0.224003\pi\)
\(158\) 0 0
\(159\) −253.218 + 146.195i −1.59256 + 0.919468i
\(160\) 0 0
\(161\) 8.86152 + 104.061i 0.0550405 + 0.646339i
\(162\) 0 0
\(163\) 166.695 96.2414i 1.02267 0.590438i 0.107793 0.994173i \(-0.465622\pi\)
0.914876 + 0.403735i \(0.132288\pi\)
\(164\) 0 0
\(165\) −36.3885 + 63.0267i −0.220536 + 0.381980i
\(166\) 0 0
\(167\) 132.185i 0.791530i −0.918352 0.395765i \(-0.870480\pi\)
0.918352 0.395765i \(-0.129520\pi\)
\(168\) 0 0
\(169\) −95.6052 −0.565712
\(170\) 0 0
\(171\) −170.326 98.3378i −0.996059 0.575075i
\(172\) 0 0
\(173\) −102.145 176.920i −0.590434 1.02266i −0.994174 0.107788i \(-0.965623\pi\)
0.403740 0.914874i \(-0.367710\pi\)
\(174\) 0 0
\(175\) −76.3570 109.588i −0.436326 0.626217i
\(176\) 0 0
\(177\) −35.8310 62.0612i −0.202435 0.350628i
\(178\) 0 0
\(179\) 106.903 + 61.7205i 0.597224 + 0.344808i 0.767949 0.640511i \(-0.221278\pi\)
−0.170725 + 0.985319i \(0.554611\pi\)
\(180\) 0 0
\(181\) −131.612 −0.727136 −0.363568 0.931568i \(-0.618442\pi\)
−0.363568 + 0.931568i \(0.618442\pi\)
\(182\) 0 0
\(183\) 358.146i 1.95708i
\(184\) 0 0
\(185\) −26.5342 + 45.9586i −0.143428 + 0.248425i
\(186\) 0 0
\(187\) 95.3230 55.0347i 0.509749 0.294303i
\(188\) 0 0
\(189\) 40.2579 + 18.8861i 0.213005 + 0.0999264i
\(190\) 0 0
\(191\) −149.814 + 86.4954i −0.784369 + 0.452856i −0.837976 0.545706i \(-0.816261\pi\)
0.0536074 + 0.998562i \(0.482928\pi\)
\(192\) 0 0
\(193\) −99.7579 + 172.786i −0.516880 + 0.895263i 0.482928 + 0.875660i \(0.339573\pi\)
−0.999808 + 0.0196024i \(0.993760\pi\)
\(194\) 0 0
\(195\) 84.4926i 0.433296i
\(196\) 0 0
\(197\) −9.40658 −0.0477492 −0.0238746 0.999715i \(-0.507600\pi\)
−0.0238746 + 0.999715i \(0.507600\pi\)
\(198\) 0 0
\(199\) 266.746 + 154.006i 1.34043 + 0.773899i 0.986871 0.161513i \(-0.0516372\pi\)
0.353561 + 0.935411i \(0.384971\pi\)
\(200\) 0 0
\(201\) −187.332 324.468i −0.931998 1.61427i
\(202\) 0 0
\(203\) −154.828 + 330.033i −0.762699 + 1.62578i
\(204\) 0 0
\(205\) 71.1799 + 123.287i 0.347219 + 0.601401i
\(206\) 0 0
\(207\) −96.0390 55.4482i −0.463957 0.267866i
\(208\) 0 0
\(209\) 195.253 0.934226
\(210\) 0 0
\(211\) 274.103i 1.29906i −0.760334 0.649532i \(-0.774965\pi\)
0.760334 0.649532i \(-0.225035\pi\)
\(212\) 0 0
\(213\) −24.0103 + 41.5870i −0.112724 + 0.195244i
\(214\) 0 0
\(215\) 123.676 71.4044i 0.575237 0.332113i
\(216\) 0 0
\(217\) −122.482 + 85.3410i −0.564433 + 0.393276i
\(218\) 0 0
\(219\) 233.497 134.809i 1.06620 0.615568i
\(220\) 0 0
\(221\) 63.8943 110.668i 0.289114 0.500761i
\(222\) 0 0
\(223\) 93.9923i 0.421490i −0.977541 0.210745i \(-0.932411\pi\)
0.977541 0.210745i \(-0.0675890\pi\)
\(224\) 0 0
\(225\) 141.827 0.630341
\(226\) 0 0
\(227\) 136.226 + 78.6500i 0.600113 + 0.346476i 0.769086 0.639145i \(-0.220712\pi\)
−0.168973 + 0.985621i \(0.554045\pi\)
\(228\) 0 0
\(229\) −41.1365 71.2506i −0.179636 0.311138i 0.762120 0.647436i \(-0.224159\pi\)
−0.941756 + 0.336298i \(0.890825\pi\)
\(230\) 0 0
\(231\) 208.638 17.7671i 0.903196 0.0769137i
\(232\) 0 0
\(233\) 166.177 + 287.828i 0.713207 + 1.23531i 0.963647 + 0.267179i \(0.0860915\pi\)
−0.250440 + 0.968132i \(0.580575\pi\)
\(234\) 0 0
\(235\) −139.677 80.6424i −0.594369 0.343159i
\(236\) 0 0
\(237\) 74.5202 0.314431
\(238\) 0 0
\(239\) 36.6386i 0.153300i 0.997058 + 0.0766499i \(0.0244224\pi\)
−0.997058 + 0.0766499i \(0.975578\pi\)
\(240\) 0 0
\(241\) 230.374 399.020i 0.955910 1.65568i 0.223637 0.974673i \(-0.428207\pi\)
0.732273 0.681011i \(-0.238460\pi\)
\(242\) 0 0
\(243\) −275.742 + 159.200i −1.13474 + 0.655142i
\(244\) 0 0
\(245\) 41.2914 111.834i 0.168536 0.456465i
\(246\) 0 0
\(247\) 196.315 113.343i 0.794798 0.458877i
\(248\) 0 0
\(249\) 206.294 357.312i 0.828490 1.43499i
\(250\) 0 0
\(251\) 23.1867i 0.0923775i −0.998933 0.0461887i \(-0.985292\pi\)
0.998933 0.0461887i \(-0.0147076\pi\)
\(252\) 0 0
\(253\) 110.094 0.435155
\(254\) 0 0
\(255\) 127.402 + 73.5556i 0.499616 + 0.288453i
\(256\) 0 0
\(257\) 101.126 + 175.155i 0.393486 + 0.681538i 0.992907 0.118896i \(-0.0379356\pi\)
−0.599420 + 0.800434i \(0.704602\pi\)
\(258\) 0 0
\(259\) 152.137 12.9556i 0.587403 0.0500216i
\(260\) 0 0
\(261\) −193.546 335.231i −0.741555 1.28441i
\(262\) 0 0
\(263\) 27.1209 + 15.6583i 0.103121 + 0.0595372i 0.550674 0.834721i \(-0.314371\pi\)
−0.447552 + 0.894258i \(0.647704\pi\)
\(264\) 0 0
\(265\) 175.483 0.662200
\(266\) 0 0
\(267\) 188.783i 0.707053i
\(268\) 0 0
\(269\) −44.6656 + 77.3631i −0.166043 + 0.287595i −0.937025 0.349262i \(-0.886432\pi\)
0.770982 + 0.636857i \(0.219766\pi\)
\(270\) 0 0
\(271\) 46.3103 26.7372i 0.170887 0.0986614i −0.412117 0.911131i \(-0.635211\pi\)
0.583004 + 0.812469i \(0.301877\pi\)
\(272\) 0 0
\(273\) 199.459 138.976i 0.730621 0.509070i
\(274\) 0 0
\(275\) −121.937 + 70.4005i −0.443408 + 0.256002i
\(276\) 0 0
\(277\) −20.3184 + 35.1924i −0.0733515 + 0.127048i −0.900368 0.435129i \(-0.856703\pi\)
0.827017 + 0.562177i \(0.190036\pi\)
\(278\) 0 0
\(279\) 158.514i 0.568150i
\(280\) 0 0
\(281\) 5.49585 0.0195582 0.00977910 0.999952i \(-0.496887\pi\)
0.00977910 + 0.999952i \(0.496887\pi\)
\(282\) 0 0
\(283\) −255.061 147.260i −0.901276 0.520352i −0.0236618 0.999720i \(-0.507533\pi\)
−0.877614 + 0.479368i \(0.840866\pi\)
\(284\) 0 0
\(285\) 130.481 + 226.000i 0.457828 + 0.792981i
\(286\) 0 0
\(287\) 173.962 370.819i 0.606138 1.29205i
\(288\) 0 0
\(289\) 33.2529 + 57.5957i 0.115062 + 0.199293i
\(290\) 0 0
\(291\) 477.182 + 275.501i 1.63980 + 0.946739i
\(292\) 0 0
\(293\) −163.827 −0.559135 −0.279568 0.960126i \(-0.590191\pi\)
−0.279568 + 0.960126i \(0.590191\pi\)
\(294\) 0 0
\(295\) 43.0091i 0.145794i
\(296\) 0 0
\(297\) 23.4382 40.5962i 0.0789165 0.136687i
\(298\) 0 0
\(299\) 110.693 63.9087i 0.370211 0.213741i
\(300\) 0 0
\(301\) −371.988 174.510i −1.23584 0.579768i
\(302\) 0 0
\(303\) −314.045 + 181.314i −1.03645 + 0.598395i
\(304\) 0 0
\(305\) 107.473 186.150i 0.352372 0.610326i
\(306\) 0 0
\(307\) 559.026i 1.82093i 0.413586 + 0.910465i \(0.364276\pi\)
−0.413586 + 0.910465i \(0.635724\pi\)
\(308\) 0 0
\(309\) 21.2699 0.0688346
\(310\) 0 0
\(311\) 280.938 + 162.199i 0.903336 + 0.521541i 0.878281 0.478145i \(-0.158691\pi\)
0.0250551 + 0.999686i \(0.492024\pi\)
\(312\) 0 0
\(313\) 50.9596 + 88.2646i 0.162810 + 0.281995i 0.935875 0.352331i \(-0.114611\pi\)
−0.773065 + 0.634326i \(0.781278\pi\)
\(314\) 0 0
\(315\) 72.3668 + 103.861i 0.229736 + 0.329718i
\(316\) 0 0
\(317\) −274.099 474.754i −0.864667 1.49765i −0.867377 0.497651i \(-0.834196\pi\)
0.00271028 0.999996i \(-0.499137\pi\)
\(318\) 0 0
\(319\) 332.807 + 192.146i 1.04328 + 0.602339i
\(320\) 0 0
\(321\) −60.9813 −0.189973
\(322\) 0 0
\(323\) 394.684i 1.22193i
\(324\) 0 0
\(325\) −81.7336 + 141.567i −0.251488 + 0.435590i
\(326\) 0 0
\(327\) −25.8105 + 14.9017i −0.0789313 + 0.0455710i
\(328\) 0 0
\(329\) 39.3745 + 462.374i 0.119679 + 1.40539i
\(330\) 0 0
\(331\) 140.905 81.3518i 0.425696 0.245776i −0.271815 0.962349i \(-0.587624\pi\)
0.697511 + 0.716574i \(0.254291\pi\)
\(332\) 0 0
\(333\) −81.0656 + 140.410i −0.243440 + 0.421651i
\(334\) 0 0
\(335\) 224.860i 0.671225i
\(336\) 0 0
\(337\) −598.848 −1.77700 −0.888499 0.458879i \(-0.848251\pi\)
−0.888499 + 0.458879i \(0.848251\pi\)
\(338\) 0 0
\(339\) −599.210 345.954i −1.76758 1.02051i
\(340\) 0 0
\(341\) 78.6837 + 136.284i 0.230744 + 0.399660i
\(342\) 0 0
\(343\) −331.921 + 86.4727i −0.967699 + 0.252107i
\(344\) 0 0
\(345\) 73.5722 + 127.431i 0.213253 + 0.369365i
\(346\) 0 0
\(347\) 94.3313 + 54.4622i 0.271848 + 0.156952i 0.629727 0.776816i \(-0.283167\pi\)
−0.357879 + 0.933768i \(0.616500\pi\)
\(348\) 0 0
\(349\) −277.893 −0.796255 −0.398128 0.917330i \(-0.630340\pi\)
−0.398128 + 0.917330i \(0.630340\pi\)
\(350\) 0 0
\(351\) 54.4226i 0.155050i
\(352\) 0 0
\(353\) 52.0900 90.2225i 0.147564 0.255588i −0.782763 0.622320i \(-0.786190\pi\)
0.930327 + 0.366732i \(0.119524\pi\)
\(354\) 0 0
\(355\) 24.9591 14.4102i 0.0703074 0.0405920i
\(356\) 0 0
\(357\) −35.9143 421.741i −0.100600 1.18135i
\(358\) 0 0
\(359\) −581.956 + 335.992i −1.62105 + 0.935911i −0.634405 + 0.773001i \(0.718755\pi\)
−0.986641 + 0.162911i \(0.947912\pi\)
\(360\) 0 0
\(361\) 169.567 293.699i 0.469714 0.813569i
\(362\) 0 0
\(363\) 269.769i 0.743164i
\(364\) 0 0
\(365\) −161.816 −0.443332
\(366\) 0 0
\(367\) −86.1727 49.7518i −0.234803 0.135564i 0.377983 0.925813i \(-0.376618\pi\)
−0.612786 + 0.790249i \(0.709951\pi\)
\(368\) 0 0
\(369\) 217.464 + 376.659i 0.589334 + 1.02076i
\(370\) 0 0
\(371\) −288.640 414.258i −0.778006 1.11660i
\(372\) 0 0
\(373\) 308.281 + 533.959i 0.826491 + 1.43152i 0.900775 + 0.434287i \(0.143000\pi\)
−0.0742836 + 0.997237i \(0.523667\pi\)
\(374\) 0 0
\(375\) −376.502 217.373i −1.00401 0.579663i
\(376\) 0 0
\(377\) 446.156 1.18344
\(378\) 0 0
\(379\) 212.094i 0.559616i −0.960056 0.279808i \(-0.909729\pi\)
0.960056 0.279808i \(-0.0902707\pi\)
\(380\) 0 0
\(381\) −394.726 + 683.685i −1.03603 + 1.79445i
\(382\) 0 0
\(383\) −388.045 + 224.038i −1.01317 + 0.584955i −0.912119 0.409926i \(-0.865555\pi\)
−0.101053 + 0.994881i \(0.532221\pi\)
\(384\) 0 0
\(385\) −113.773 53.3742i −0.295515 0.138634i
\(386\) 0 0
\(387\) 377.847 218.150i 0.976349 0.563696i
\(388\) 0 0
\(389\) 15.0297 26.0322i 0.0386368 0.0669208i −0.846060 0.533087i \(-0.821032\pi\)
0.884697 + 0.466166i \(0.154365\pi\)
\(390\) 0 0
\(391\) 222.544i 0.569167i
\(392\) 0 0
\(393\) −121.426 −0.308972
\(394\) 0 0
\(395\) −38.7326 22.3623i −0.0980571 0.0566133i
\(396\) 0 0
\(397\) 150.913 + 261.389i 0.380133 + 0.658410i 0.991081 0.133261i \(-0.0425448\pi\)
−0.610948 + 0.791671i \(0.709211\pi\)
\(398\) 0 0
\(399\) 318.892 679.754i 0.799227 1.70364i
\(400\) 0 0
\(401\) −27.1956 47.1042i −0.0678195 0.117467i 0.830122 0.557582i \(-0.188271\pi\)
−0.897941 + 0.440115i \(0.854938\pi\)
\(402\) 0 0
\(403\) 158.223 + 91.3503i 0.392614 + 0.226676i
\(404\) 0 0
\(405\) 225.406 0.556557
\(406\) 0 0
\(407\) 160.959i 0.395476i
\(408\) 0 0
\(409\) 77.7710 134.703i 0.190149 0.329348i −0.755150 0.655552i \(-0.772436\pi\)
0.945300 + 0.326204i \(0.105769\pi\)
\(410\) 0 0
\(411\) 624.305 360.443i 1.51899 0.876989i
\(412\) 0 0
\(413\) 101.530 70.7428i 0.245837 0.171290i
\(414\) 0 0
\(415\) −214.446 + 123.811i −0.516738 + 0.298339i
\(416\) 0 0
\(417\) −243.514 + 421.779i −0.583967 + 1.01146i
\(418\) 0 0
\(419\) 384.000i 0.916468i −0.888832 0.458234i \(-0.848482\pi\)
0.888832 0.458234i \(-0.151518\pi\)
\(420\) 0 0
\(421\) −295.132 −0.701026 −0.350513 0.936558i \(-0.613993\pi\)
−0.350513 + 0.936558i \(0.613993\pi\)
\(422\) 0 0
\(423\) −426.732 246.374i −1.00882 0.582444i
\(424\) 0 0
\(425\) 142.307 + 246.484i 0.334841 + 0.579962i
\(426\) 0 0
\(427\) −616.214 + 52.4751i −1.44312 + 0.122892i
\(428\) 0 0
\(429\) −128.135 221.936i −0.298683 0.517334i
\(430\) 0 0
\(431\) 417.311 + 240.935i 0.968239 + 0.559013i 0.898699 0.438566i \(-0.144514\pi\)
0.0695400 + 0.997579i \(0.477847\pi\)
\(432\) 0 0
\(433\) 9.18001 0.0212009 0.0106005 0.999944i \(-0.496626\pi\)
0.0106005 + 0.999944i \(0.496626\pi\)
\(434\) 0 0
\(435\) 513.618i 1.18073i
\(436\) 0 0
\(437\) 197.387 341.884i 0.451686 0.782343i
\(438\) 0 0
\(439\) 169.939 98.1143i 0.387105 0.223495i −0.293800 0.955867i \(-0.594920\pi\)
0.680905 + 0.732372i \(0.261587\pi\)
\(440\) 0 0
\(441\) 126.151 341.668i 0.286057 0.774758i
\(442\) 0 0
\(443\) −55.9819 + 32.3212i −0.126370 + 0.0729598i −0.561852 0.827237i \(-0.689911\pi\)
0.435482 + 0.900197i \(0.356578\pi\)
\(444\) 0 0
\(445\) 56.6506 98.1218i 0.127305 0.220498i
\(446\) 0 0
\(447\) 334.548i 0.748429i
\(448\) 0 0
\(449\) −834.833 −1.85932 −0.929658 0.368424i \(-0.879898\pi\)
−0.929658 + 0.368424i \(0.879898\pi\)
\(450\) 0 0
\(451\) −373.935 215.892i −0.829125 0.478696i
\(452\) 0 0
\(453\) 575.906 + 997.498i 1.27132 + 2.20198i
\(454\) 0 0
\(455\) −145.375 + 12.3798i −0.319506 + 0.0272083i
\(456\) 0 0
\(457\) −25.0154 43.3280i −0.0547383 0.0948095i 0.837358 0.546655i \(-0.184099\pi\)
−0.892096 + 0.451846i \(0.850766\pi\)
\(458\) 0 0
\(459\) −82.0610 47.3780i −0.178782 0.103220i
\(460\) 0 0
\(461\) −439.355 −0.953048 −0.476524 0.879161i \(-0.658104\pi\)
−0.476524 + 0.879161i \(0.658104\pi\)
\(462\) 0 0
\(463\) 244.469i 0.528010i 0.964521 + 0.264005i \(0.0850435\pi\)
−0.964521 + 0.264005i \(0.914956\pi\)
\(464\) 0 0
\(465\) −105.163 + 182.148i −0.226157 + 0.391716i
\(466\) 0 0
\(467\) −385.416 + 222.520i −0.825301 + 0.476488i −0.852241 0.523149i \(-0.824757\pi\)
0.0269400 + 0.999637i \(0.491424\pi\)
\(468\) 0 0
\(469\) 530.821 369.857i 1.13181 0.788608i
\(470\) 0 0
\(471\) 197.489 114.020i 0.419297 0.242081i
\(472\) 0 0
\(473\) −216.573 + 375.115i −0.457870 + 0.793054i
\(474\) 0 0
\(475\) 504.881i 1.06291i
\(476\) 0 0
\(477\) 536.125 1.12395
\(478\) 0 0
\(479\) 230.123 + 132.862i 0.480424 + 0.277373i 0.720593 0.693358i \(-0.243869\pi\)
−0.240169 + 0.970731i \(0.577203\pi\)
\(480\) 0 0
\(481\) −93.4350 161.834i −0.194251 0.336453i
\(482\) 0 0
\(483\) 179.808 383.282i 0.372274 0.793545i
\(484\) 0 0
\(485\) −165.346 286.388i −0.340920 0.590492i
\(486\) 0 0
\(487\) 21.3034 + 12.2995i 0.0437441 + 0.0252557i 0.521713 0.853121i \(-0.325293\pi\)
−0.477968 + 0.878377i \(0.658627\pi\)
\(488\) 0 0
\(489\) −780.278 −1.59566
\(490\) 0 0
\(491\) 532.126i 1.08376i 0.840456 + 0.541880i \(0.182287\pi\)
−0.840456 + 0.541880i \(0.817713\pi\)
\(492\) 0 0
\(493\) 388.404 672.735i 0.787837 1.36457i
\(494\) 0 0
\(495\) 115.565 66.7216i 0.233465 0.134791i
\(496\) 0 0
\(497\) −75.0712 35.2180i −0.151049 0.0708612i
\(498\) 0 0
\(499\) −628.228 + 362.708i −1.25897 + 0.726869i −0.972875 0.231330i \(-0.925692\pi\)
−0.286100 + 0.958200i \(0.592359\pi\)
\(500\) 0 0
\(501\) −267.924 + 464.058i −0.534778 + 0.926263i
\(502\) 0 0
\(503\) 46.8524i 0.0931459i −0.998915 0.0465729i \(-0.985170\pi\)
0.998915 0.0465729i \(-0.0148300\pi\)
\(504\) 0 0
\(505\) 217.637 0.430964
\(506\) 0 0
\(507\) 335.637 + 193.780i 0.662006 + 0.382209i
\(508\) 0 0
\(509\) −431.957 748.172i −0.848639 1.46989i −0.882423 0.470457i \(-0.844089\pi\)
0.0337834 0.999429i \(-0.489244\pi\)
\(510\) 0 0
\(511\) 266.160 + 381.995i 0.520862 + 0.747544i
\(512\) 0 0
\(513\) −84.0441 145.569i −0.163829 0.283760i
\(514\) 0 0
\(515\) −11.0552 6.38273i −0.0214664 0.0123937i
\(516\) 0 0
\(517\) 489.184 0.946197
\(518\) 0 0
\(519\) 828.142i 1.59565i
\(520\) 0 0
\(521\) 169.288 293.215i 0.324928 0.562792i −0.656570 0.754265i \(-0.727993\pi\)
0.981498 + 0.191473i \(0.0613266\pi\)
\(522\) 0 0
\(523\) 323.498 186.772i 0.618544 0.357116i −0.157758 0.987478i \(-0.550427\pi\)
0.776302 + 0.630361i \(0.217093\pi\)
\(524\) 0 0
\(525\) 45.9416 + 539.492i 0.0875079 + 1.02760i
\(526\) 0 0
\(527\) 275.485 159.051i 0.522741 0.301805i
\(528\) 0 0
\(529\) −153.203 + 265.355i −0.289608 + 0.501616i
\(530\) 0 0
\(531\) 131.399i 0.247456i
\(532\) 0 0
\(533\) −501.292 −0.940510
\(534\) 0 0
\(535\) 31.6956 + 18.2995i 0.0592441 + 0.0342046i
\(536\) 0 0
\(537\) −250.200 433.359i −0.465922 0.807000i
\(538\) 0 0
\(539\) 61.1389 + 356.373i 0.113430 + 0.661174i
\(540\) 0 0
\(541\) 294.267 + 509.685i 0.543931 + 0.942116i 0.998673 + 0.0514928i \(0.0163979\pi\)
−0.454743 + 0.890623i \(0.650269\pi\)
\(542\) 0 0
\(543\) 462.043 + 266.761i 0.850908 + 0.491272i
\(544\) 0 0
\(545\) 17.8870 0.0328202
\(546\) 0 0
\(547\) 206.784i 0.378032i −0.981974 0.189016i \(-0.939470\pi\)
0.981974 0.189016i \(-0.0605298\pi\)
\(548\) 0 0
\(549\) 328.346 568.713i 0.598081 1.03591i
\(550\) 0 0
\(551\) 1193.37 688.993i 2.16583 1.25044i
\(552\) 0 0
\(553\) 10.9186 + 128.217i 0.0197443 + 0.231857i
\(554\) 0 0
\(555\) 186.305 107.563i 0.335684 0.193807i
\(556\) 0 0
\(557\) 238.327 412.794i 0.427876 0.741103i −0.568808 0.822470i \(-0.692595\pi\)
0.996684 + 0.0813674i \(0.0259287\pi\)
\(558\) 0 0
\(559\) 502.873i 0.899594i
\(560\) 0 0
\(561\) −446.195 −0.795356
\(562\) 0 0
\(563\) −405.901 234.347i −0.720961 0.416247i 0.0941451 0.995558i \(-0.469988\pi\)
−0.815106 + 0.579311i \(0.803322\pi\)
\(564\) 0 0
\(565\) 207.630 + 359.625i 0.367486 + 0.636505i
\(566\) 0 0
\(567\) −370.754 532.109i −0.653888 0.938463i
\(568\) 0 0
\(569\) −179.141 310.282i −0.314835 0.545311i 0.664567 0.747228i \(-0.268616\pi\)
−0.979402 + 0.201918i \(0.935283\pi\)
\(570\) 0 0
\(571\) 321.513 + 185.626i 0.563070 + 0.325089i 0.754377 0.656442i \(-0.227939\pi\)
−0.191307 + 0.981530i \(0.561273\pi\)
\(572\) 0 0
\(573\) 701.262 1.22384
\(574\) 0 0
\(575\) 284.679i 0.495094i
\(576\) 0 0
\(577\) 535.691 927.844i 0.928408 1.60805i 0.142421 0.989806i \(-0.454511\pi\)
0.785987 0.618243i \(-0.212155\pi\)
\(578\) 0 0
\(579\) 700.431 404.394i 1.20972 0.698435i
\(580\) 0 0
\(581\) 645.004 + 302.590i 1.11016 + 0.520808i
\(582\) 0 0
\(583\) −460.940 + 266.124i −0.790635 + 0.456473i
\(584\) 0 0
\(585\) 77.4625 134.169i 0.132414 0.229349i
\(586\) 0 0
\(587\) 283.490i 0.482948i −0.970407 0.241474i \(-0.922369\pi\)
0.970407 0.241474i \(-0.0776308\pi\)
\(588\) 0 0
\(589\) 564.284 0.958037
\(590\) 0 0
\(591\) 33.0233 + 19.0660i 0.0558769 + 0.0322606i
\(592\) 0 0
\(593\) 310.680 + 538.113i 0.523912 + 0.907442i 0.999613 + 0.0278350i \(0.00886130\pi\)
−0.475700 + 0.879607i \(0.657805\pi\)
\(594\) 0 0
\(595\) −107.891 + 229.981i −0.181329 + 0.386523i
\(596\) 0 0
\(597\) −624.302 1081.32i −1.04573 1.81126i
\(598\) 0 0
\(599\) −910.469 525.659i −1.51998 0.877562i −0.999723 0.0235529i \(-0.992502\pi\)
−0.520259 0.854009i \(-0.674164\pi\)
\(600\) 0 0
\(601\) −427.201 −0.710817 −0.355409 0.934711i \(-0.615658\pi\)
−0.355409 + 0.934711i \(0.615658\pi\)
\(602\) 0 0
\(603\) 686.979i 1.13927i
\(604\) 0 0
\(605\) 80.9530 140.215i 0.133807 0.231760i
\(606\) 0 0
\(607\) 621.252 358.680i 1.02348 0.590906i 0.108369 0.994111i \(-0.465437\pi\)
0.915110 + 0.403205i \(0.132104\pi\)
\(608\) 0 0
\(609\) 1212.48 844.816i 1.99094 1.38722i
\(610\) 0 0
\(611\) 491.844 283.966i 0.804983 0.464757i
\(612\) 0 0
\(613\) 4.87305 8.44037i 0.00794951 0.0137690i −0.862023 0.506869i \(-0.830803\pi\)
0.869973 + 0.493100i \(0.164136\pi\)
\(614\) 0 0
\(615\) 577.091i 0.938360i
\(616\) 0 0
\(617\) 505.878 0.819900 0.409950 0.912108i \(-0.365546\pi\)
0.409950 + 0.912108i \(0.365546\pi\)
\(618\) 0 0
\(619\) 272.815 + 157.510i 0.440735 + 0.254458i 0.703909 0.710290i \(-0.251436\pi\)
−0.263175 + 0.964748i \(0.584770\pi\)
\(620\) 0 0
\(621\) −47.3886 82.0795i −0.0763102 0.132173i
\(622\) 0 0
\(623\) −324.814 + 27.6603i −0.521371 + 0.0443985i
\(624\) 0 0
\(625\) −108.051 187.149i −0.172881 0.299439i
\(626\) 0 0
\(627\) −685.466 395.754i −1.09325 0.631187i
\(628\) 0 0
\(629\) −325.362 −0.517268
\(630\) 0 0
\(631\) 381.905i 0.605238i 0.953112 + 0.302619i \(0.0978609\pi\)
−0.953112 + 0.302619i \(0.902139\pi\)
\(632\) 0 0
\(633\) −555.573 + 962.280i −0.877682 + 1.52019i
\(634\) 0 0
\(635\) 410.325 236.901i 0.646181 0.373073i
\(636\) 0 0
\(637\) 268.342 + 322.821i 0.421260 + 0.506783i
\(638\) 0 0
\(639\) 76.2536 44.0250i 0.119333 0.0688968i
\(640\) 0 0
\(641\) −68.4839 + 118.618i −0.106839 + 0.185051i −0.914488 0.404613i \(-0.867406\pi\)
0.807649 + 0.589664i \(0.200740\pi\)
\(642\) 0 0
\(643\) 260.130i 0.404556i 0.979328 + 0.202278i \(0.0648345\pi\)
−0.979328 + 0.202278i \(0.935166\pi\)
\(644\) 0 0
\(645\) −578.911 −0.897537
\(646\) 0 0
\(647\) −378.735 218.663i −0.585371 0.337964i 0.177894 0.984050i \(-0.443071\pi\)
−0.763265 + 0.646086i \(0.776405\pi\)
\(648\) 0 0
\(649\) −65.2243 112.972i −0.100500 0.174071i
\(650\) 0 0
\(651\) 602.967 51.3470i 0.926217 0.0788741i
\(652\) 0 0
\(653\) 590.447 + 1022.68i 0.904206 + 1.56613i 0.821979 + 0.569518i \(0.192870\pi\)
0.0822270 + 0.996614i \(0.473797\pi\)
\(654\) 0 0
\(655\) 63.1122 + 36.4379i 0.0963545 + 0.0556303i
\(656\) 0 0
\(657\) −494.371 −0.752467
\(658\) 0 0
\(659\) 129.818i 0.196992i 0.995137 + 0.0984960i \(0.0314032\pi\)
−0.995137 + 0.0984960i \(0.968597\pi\)
\(660\) 0 0
\(661\) 316.779 548.677i 0.479241 0.830070i −0.520475 0.853877i \(-0.674245\pi\)
0.999717 + 0.0238063i \(0.00757850\pi\)
\(662\) 0 0
\(663\) −448.621 + 259.012i −0.676654 + 0.390666i
\(664\) 0 0
\(665\) −369.729 + 257.614i −0.555984 + 0.387390i
\(666\) 0 0
\(667\) 672.887 388.491i 1.00883 0.582446i
\(668\) 0 0
\(669\) −190.511 + 329.975i −0.284770 + 0.493235i
\(670\) 0 0
\(671\) 651.944i 0.971600i
\(672\) 0 0
\(673\) 1074.74 1.59694 0.798471 0.602033i \(-0.205642\pi\)
0.798471 + 0.602033i \(0.205642\pi\)
\(674\) 0 0
\(675\) 104.973 + 60.6059i 0.155515 + 0.0897866i
\(676\) 0 0
\(677\) −277.711 481.010i −0.410208 0.710502i 0.584704 0.811247i \(-0.301211\pi\)
−0.994912 + 0.100745i \(0.967877\pi\)
\(678\) 0 0
\(679\) −404.102 + 861.390i −0.595143 + 1.26861i
\(680\) 0 0
\(681\) −318.828 552.226i −0.468176 0.810904i
\(682\) 0 0
\(683\) −128.117 73.9682i −0.187579 0.108299i 0.403270 0.915081i \(-0.367874\pi\)
−0.590849 + 0.806782i \(0.701207\pi\)
\(684\) 0 0
\(685\) −432.651 −0.631607
\(686\) 0 0
\(687\) 333.515i 0.485466i
\(688\) 0 0
\(689\) −308.965 + 535.142i −0.448425 + 0.776694i
\(690\) 0 0
\(691\) −506.502 + 292.429i −0.732999 + 0.423197i −0.819518 0.573053i \(-0.805759\pi\)
0.0865192 + 0.996250i \(0.472426\pi\)
\(692\) 0 0
\(693\) −347.593 163.066i −0.501577 0.235304i
\(694\) 0 0
\(695\) 253.137 146.149i 0.364226 0.210286i
\(696\) 0 0
\(697\) −436.403 + 755.872i −0.626116 + 1.08446i
\(698\) 0 0
\(699\) 1347.28i 1.92745i
\(700\) 0 0
\(701\) −1138.04 −1.62345 −0.811727 0.584037i \(-0.801472\pi\)
−0.811727 + 0.584037i \(0.801472\pi\)
\(702\) 0 0
\(703\) −499.837 288.581i −0.711005 0.410499i
\(704\) 0 0
\(705\) 326.905 + 566.215i 0.463694 + 0.803142i
\(706\) 0 0
\(707\) −357.976 513.769i −0.506331 0.726689i
\(708\) 0 0
\(709\) 421.341 + 729.785i 0.594276 + 1.02932i 0.993649 + 0.112527i \(0.0358945\pi\)
−0.399373 + 0.916789i \(0.630772\pi\)
\(710\) 0 0
\(711\) −118.333 68.3198i −0.166432 0.0960897i
\(712\) 0 0
\(713\) 318.174 0.446247
\(714\) 0 0
\(715\) 153.804i 0.215111i
\(716\) 0 0
\(717\) 74.2620 128.626i 0.103573 0.179394i
\(718\) 0 0
\(719\) −48.8258 + 28.1896i −0.0679079 + 0.0392066i −0.533570 0.845756i \(-0.679150\pi\)
0.465662 + 0.884963i \(0.345816\pi\)
\(720\) 0 0
\(721\) 3.11644 + 36.5963i 0.00432238 + 0.0507576i
\(722\) 0 0
\(723\) −1617.53 + 933.880i −2.23725 + 1.29167i
\(724\) 0 0
\(725\) −496.847 + 860.564i −0.685306 + 1.18698i
\(726\) 0 0
\(727\) 929.204i 1.27813i 0.769151 + 0.639067i \(0.220680\pi\)
−0.769151 + 0.639067i \(0.779320\pi\)
\(728\) 0 0
\(729\) 456.881 0.626723
\(730\) 0 0
\(731\) 758.256 + 437.779i 1.03729 + 0.598877i
\(732\) 0 0
\(733\) −273.917 474.437i −0.373692 0.647254i 0.616438 0.787403i \(-0.288575\pi\)
−0.990130 + 0.140149i \(0.955242\pi\)
\(734\) 0 0
\(735\) −371.634 + 308.918i −0.505624 + 0.420297i
\(736\) 0 0
\(737\) −341.005 590.639i −0.462694 0.801409i
\(738\) 0 0
\(739\) 473.319 + 273.271i 0.640485 + 0.369784i 0.784801 0.619747i \(-0.212765\pi\)
−0.144316 + 0.989532i \(0.546098\pi\)
\(740\) 0 0
\(741\) −918.926 −1.24012
\(742\) 0 0
\(743\) 904.617i 1.21752i −0.793354 0.608760i \(-0.791667\pi\)
0.793354 0.608760i \(-0.208333\pi\)
\(744\) 0 0
\(745\) −100.392 + 173.884i −0.134755 + 0.233402i
\(746\) 0 0
\(747\) −655.163 + 378.259i −0.877059 + 0.506370i
\(748\) 0 0
\(749\) −8.93491 104.922i −0.0119291 0.140083i
\(750\) 0 0
\(751\) 792.217 457.386i 1.05488 0.609037i 0.130870 0.991400i \(-0.458223\pi\)
0.924012 + 0.382363i \(0.124890\pi\)
\(752\) 0 0
\(753\) −46.9967 + 81.4006i −0.0624126 + 0.108102i
\(754\) 0 0
\(755\) 691.278i 0.915601i
\(756\) 0 0
\(757\) −780.766 −1.03140 −0.515698 0.856771i \(-0.672467\pi\)
−0.515698 + 0.856771i \(0.672467\pi\)
\(758\) 0 0
\(759\) −386.503 223.148i −0.509227 0.294002i
\(760\) 0 0
\(761\) −77.5839 134.379i −0.101950 0.176583i 0.810538 0.585686i \(-0.199175\pi\)
−0.912488 + 0.409104i \(0.865841\pi\)
\(762\) 0 0
\(763\) −29.4211 42.2254i −0.0385598 0.0553412i
\(764\) 0 0
\(765\) −134.871 233.603i −0.176302 0.305364i
\(766\) 0 0
\(767\) −131.158 75.7242i −0.171001 0.0987277i
\(768\) 0 0
\(769\) 354.951 0.461575 0.230788 0.973004i \(-0.425870\pi\)
0.230788 + 0.973004i \(0.425870\pi\)
\(770\) 0 0
\(771\) 819.880i 1.06340i
\(772\) 0 0
\(773\) −433.669 + 751.136i −0.561020 + 0.971715i 0.436388 + 0.899759i \(0.356258\pi\)
−0.997408 + 0.0719566i \(0.977076\pi\)
\(774\) 0 0
\(775\) −352.400 + 203.458i −0.454710 + 0.262527i
\(776\) 0 0
\(777\) −560.361 262.881i −0.721186 0.338328i
\(778\) 0 0
\(779\) −1340.85 + 774.139i −1.72124 + 0.993760i
\(780\) 0 0
\(781\) −43.7066 + 75.7021i −0.0559624 + 0.0969297i
\(782\) 0 0
\(783\) 330.827i 0.422512i
\(784\) 0 0
\(785\) −136.862 −0.174347
\(786\) 0 0
\(787\) −81.7643 47.2066i −0.103894 0.0599830i 0.447153 0.894458i \(-0.352438\pi\)
−0.551046 + 0.834475i \(0.685771\pi\)
\(788\) 0 0
\(789\) −63.4748 109.942i −0.0804497 0.139343i
\(790\) 0 0
\(791\) 507.441 1081.67i 0.641519 1.36747i
\(792\) 0 0
\(793\) 378.447 + 655.489i 0.477234 + 0.826594i
\(794\) 0 0
\(795\) −616.060 355.682i −0.774918 0.447399i
\(796\) 0 0
\(797\) 1065.20 1.33651 0.668256 0.743931i \(-0.267041\pi\)
0.668256 + 0.743931i \(0.267041\pi\)
\(798\) 0 0
\(799\) 988.835i 1.23759i
\(800\) 0 0
\(801\) 173.076 299.776i 0.216074 0.374252i
\(802\) 0 0
\(803\) 425.041 245.398i 0.529317 0.305601i
\(804\) 0 0
\(805\) −208.473 + 145.257i −0.258973 + 0.180443i
\(806\) 0 0
\(807\) 313.611 181.063i 0.388613 0.224366i
\(808\) 0 0
\(809\) 254.373 440.587i 0.314429 0.544607i −0.664887 0.746944i \(-0.731520\pi\)
0.979316 + 0.202337i \(0.0648537\pi\)
\(810\) 0 0
\(811\) 652.893i 0.805047i 0.915410 + 0.402524i \(0.131867\pi\)
−0.915410 + 0.402524i \(0.868133\pi\)
\(812\) 0 0
\(813\) −216.772 −0.266633
\(814\) 0 0
\(815\) 405.557 + 234.148i 0.497616 + 0.287298i
\(816\) 0 0
\(817\) 776.580 + 1345.08i 0.950527 + 1.64636i
\(818\) 0 0
\(819\) −444.141 + 37.8219i −0.542297 + 0.0461806i
\(820\) 0 0
\(821\) 87.7543 + 151.995i 0.106887 + 0.185134i 0.914508 0.404569i \(-0.132578\pi\)
−0.807621 + 0.589702i \(0.799245\pi\)
\(822\) 0 0
\(823\) 753.427 + 434.991i 0.915464 + 0.528544i 0.882185 0.470902i \(-0.156072\pi\)
0.0332791 + 0.999446i \(0.489405\pi\)
\(824\) 0 0
\(825\) 570.773 0.691846
\(826\) 0 0
\(827\) 269.724i 0.326148i 0.986614 + 0.163074i \(0.0521409\pi\)
−0.986614 + 0.163074i \(0.947859\pi\)
\(828\) 0 0
\(829\) −72.2989 + 125.225i −0.0872122 + 0.151056i −0.906332 0.422567i \(-0.861129\pi\)
0.819120 + 0.573623i \(0.194462\pi\)
\(830\) 0 0
\(831\) 142.661 82.3656i 0.171674 0.0991163i
\(832\) 0 0
\(833\) 720.372 123.586i 0.864792 0.148362i
\(834\) 0 0
\(835\) 278.512 160.799i 0.333547 0.192573i
\(836\) 0 0
\(837\) 67.7367 117.323i 0.0809280 0.140171i
\(838\) 0 0
\(839\) 920.554i 1.09720i −0.836084 0.548602i \(-0.815160\pi\)
0.836084 0.548602i \(-0.184840\pi\)
\(840\) 0 0
\(841\) 1871.12 2.22487
\(842\) 0 0
\(843\) −19.2940 11.1394i −0.0228874 0.0132140i
\(844\) 0 0
\(845\) −116.300 201.438i −0.137633 0.238388i
\(846\) 0 0
\(847\) −464.155 + 39.5262i −0.547999 + 0.0466661i
\(848\) 0 0
\(849\) 596.954 + 1033.95i 0.703126 + 1.21785i
\(850\) 0 0
\(851\) −281.835 162.718i −0.331181 0.191207i
\(852\) 0 0
\(853\) −206.715 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(854\) 0 0
\(855\) 478.497i 0.559646i
\(856\) 0 0
\(857\) −560.474 + 970.769i −0.653995 + 1.13275i 0.328150 + 0.944626i \(0.393575\pi\)
−0.982145 + 0.188127i \(0.939758\pi\)
\(858\) 0 0
\(859\) −23.2497 + 13.4232i −0.0270660 + 0.0156265i −0.513472 0.858106i \(-0.671641\pi\)
0.486406 + 0.873733i \(0.338308\pi\)
\(860\) 0 0
\(861\) −1362.32 + 949.219i −1.58226 + 1.10246i
\(862\) 0 0
\(863\) −338.354 + 195.349i −0.392067 + 0.226360i −0.683055 0.730367i \(-0.739349\pi\)
0.290988 + 0.956727i \(0.406016\pi\)
\(864\) 0 0
\(865\) 248.511 430.434i 0.287296 0.497612i
\(866\) 0 0
\(867\) 269.598i 0.310955i
\(868\) 0 0
\(869\) 135.651 0.156101
\(870\) 0 0
\(871\) −685.720 395.901i −0.787279 0.454536i
\(872\) 0 0
\(873\) −505.156 874.956i −0.578644 1.00224i
\(874\) 0 0
\(875\) 318.841 679.646i 0.364390 0.776738i
\(876\) 0 0
\(877\) −72.5325 125.630i −0.0827053 0.143250i 0.821706 0.569912i \(-0.193023\pi\)
−0.904411 + 0.426662i \(0.859689\pi\)
\(878\) 0 0
\(879\) 575.139 + 332.057i 0.654311 + 0.377766i
\(880\) 0 0
\(881\) −360.233 −0.408891 −0.204445 0.978878i \(-0.565539\pi\)
−0.204445 + 0.978878i \(0.565539\pi\)
\(882\) 0 0
\(883\) 145.555i 0.164841i 0.996598 + 0.0824205i \(0.0262650\pi\)
−0.996598 + 0.0824205i \(0.973735\pi\)
\(884\) 0 0
\(885\) 87.1743 150.990i 0.0985020 0.170610i
\(886\) 0 0
\(887\) −1149.57 + 663.707i −1.29603 + 0.748261i −0.979715 0.200395i \(-0.935777\pi\)
−0.316310 + 0.948656i \(0.602444\pi\)
\(888\) 0 0
\(889\) −1234.16 578.980i −1.38826 0.651271i
\(890\) 0 0
\(891\) −592.071 + 341.833i −0.664502 + 0.383650i
\(892\) 0 0
\(893\) 877.051 1519.10i 0.982140 1.70112i
\(894\) 0 0
\(895\) 300.323i 0.335556i
\(896\) 0 0
\(897\) −518.140 −0.577637
\(898\) 0 0
\(899\) 961.816 + 555.305i 1.06987 + 0.617692i
\(900\) 0 0
\(901\) 537.942 + 931.743i 0.597050 + 1.03412i
\(902\) 0 0
\(903\) 952.212 + 1366.62i 1.05450 + 1.51342i
\(904\) 0 0
\(905\) −160.101 277.303i −0.176907 0.306412i
\(906\) 0 0
\(907\) 392.834 + 226.803i 0.433114 + 0.250058i 0.700672 0.713483i \(-0.252884\pi\)
−0.267558 + 0.963542i \(0.586217\pi\)
\(908\) 0 0
\(909\) 664.910 0.731475
\(910\) 0 0
\(911\) 891.300i 0.978376i −0.872178 0.489188i \(-0.837293\pi\)
0.872178 0.489188i \(-0.162707\pi\)
\(912\) 0 0
\(913\) 375.523 650.425i 0.411307 0.712404i
\(914\) 0 0
\(915\) −754.605 + 435.671i −0.824704 + 0.476143i
\(916\) 0 0
\(917\) −17.7912 208.921i −0.0194015 0.227831i
\(918\) 0 0
\(919\) 1405.23 811.309i 1.52909 0.882818i 0.529685 0.848195i \(-0.322310\pi\)
0.999400 0.0346230i \(-0.0110230\pi\)
\(920\) 0 0
\(921\) 1133.08 1962.55i 1.23027 2.13089i
\(922\) 0 0
\(923\) 101.485i 0.109951i
\(924\) 0 0
\(925\) 416.203 0.449949
\(926\) 0 0
\(927\) −33.7752 19.5001i −0.0364350 0.0210357i
\(928\) 0 0
\(929\) −255.412 442.387i −0.274932 0.476197i 0.695186 0.718830i \(-0.255322\pi\)
−0.970118 + 0.242634i \(0.921989\pi\)
\(930\) 0 0
\(931\) 1216.29 + 449.078i 1.30643 + 0.482361i
\(932\) 0 0
\(933\) −657.516 1138.85i −0.704734 1.22063i
\(934\) 0 0
\(935\) 231.914 + 133.895i 0.248036 + 0.143204i
\(936\) 0 0
\(937\) 898.415 0.958820 0.479410 0.877591i \(-0.340851\pi\)
0.479410 + 0.877591i \(0.340851\pi\)
\(938\) 0 0
\(939\) 413.155i 0.439995i
\(940\) 0 0
\(941\) −853.746 + 1478.73i −0.907276 + 1.57145i −0.0894429 + 0.995992i \(0.528509\pi\)
−0.817833 + 0.575456i \(0.804825\pi\)
\(942\) 0 0
\(943\) −756.043 + 436.501i −0.801742 + 0.462886i
\(944\) 0 0
\(945\) 9.17966 + 107.797i 0.00971393 + 0.114070i
\(946\) 0 0
\(947\) 1211.13 699.247i 1.27891 0.738381i 0.302265 0.953224i \(-0.402257\pi\)
0.976649 + 0.214843i \(0.0689238\pi\)
\(948\) 0 0
\(949\) 284.902 493.465i 0.300213 0.519984i
\(950\) 0 0
\(951\) 2222.26i 2.33677i
\(952\) 0 0
\(953\) −99.5631 −0.104473 −0.0522367 0.998635i \(-0.516635\pi\)
−0.0522367 + 0.998635i \(0.516635\pi\)
\(954\) 0 0
\(955\) −364.488 210.437i −0.381662 0.220353i
\(956\) 0 0
\(957\) −778.913 1349.12i −0.813912 1.40974i
\(958\) 0 0
\(959\) 711.638 + 1021.35i 0.742063 + 1.06501i
\(960\) 0 0
\(961\) −253.103 438.387i −0.263375 0.456178i
\(962\) 0 0
\(963\) 96.8345 + 55.9074i 0.100555 + 0.0580555i
\(964\) 0 0
\(965\) −485.407 −0.503012
\(966\) 0 0
\(967\) 272.789i 0.282099i −0.990003 0.141049i \(-0.954952\pi\)
0.990003 0.141049i \(-0.0450476\pi\)
\(968\) 0 0
\(969\) −799.977 + 1385.60i −0.825569 + 1.42993i
\(970\) 0 0
\(971\) −1039.50 + 600.155i −1.07055 + 0.618080i −0.928330 0.371756i \(-0.878756\pi\)
−0.142215 + 0.989836i \(0.545422\pi\)
\(972\) 0 0
\(973\) −761.378 357.184i −0.782506 0.367095i
\(974\) 0 0
\(975\) 573.877 331.328i 0.588592 0.339824i
\(976\) 0 0
\(977\) −269.972 + 467.606i −0.276328 + 0.478614i −0.970469 0.241225i \(-0.922451\pi\)
0.694141 + 0.719839i \(0.255784\pi\)
\(978\) 0 0
\(979\) 343.648i 0.351019i
\(980\) 0 0
\(981\) 54.6473 0.0557057
\(982\) 0 0
\(983\) 487.338 + 281.365i 0.495766 + 0.286231i 0.726963 0.686676i \(-0.240931\pi\)
−0.231197 + 0.972907i \(0.574264\pi\)
\(984\) 0 0
\(985\) −11.4428 19.8194i −0.0116170 0.0201213i
\(986\) 0 0
\(987\) 798.945 1703.04i 0.809468 1.72547i
\(988\) 0 0
\(989\) 437.878 + 758.427i 0.442748 + 0.766862i
\(990\) 0 0
\(991\) −424.335 244.990i −0.428189 0.247215i 0.270386 0.962752i \(-0.412849\pi\)
−0.698575 + 0.715537i \(0.746182\pi\)
\(992\) 0 0
\(993\) −659.561 −0.664210
\(994\) 0 0
\(995\) 749.370i 0.753135i
\(996\) 0 0
\(997\) 832.723 1442.32i 0.835229 1.44666i −0.0586148 0.998281i \(-0.518668\pi\)
0.893844 0.448378i \(-0.147998\pi\)
\(998\) 0 0
\(999\) −120.001 + 69.2826i −0.120121 + 0.0693519i
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 224.3.r.c.95.2 12
4.3 odd 2 inner 224.3.r.c.95.5 yes 12
7.2 even 3 inner 224.3.r.c.191.5 yes 12
7.3 odd 6 1568.3.d.k.1471.2 6
7.4 even 3 1568.3.d.j.1471.5 6
8.3 odd 2 448.3.r.g.319.2 12
8.5 even 2 448.3.r.g.319.5 12
28.3 even 6 1568.3.d.k.1471.5 6
28.11 odd 6 1568.3.d.j.1471.2 6
28.23 odd 6 inner 224.3.r.c.191.2 yes 12
56.37 even 6 448.3.r.g.191.2 12
56.51 odd 6 448.3.r.g.191.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.r.c.95.2 12 1.1 even 1 trivial
224.3.r.c.95.5 yes 12 4.3 odd 2 inner
224.3.r.c.191.2 yes 12 28.23 odd 6 inner
224.3.r.c.191.5 yes 12 7.2 even 3 inner
448.3.r.g.191.2 12 56.37 even 6
448.3.r.g.191.5 12 56.51 odd 6
448.3.r.g.319.2 12 8.3 odd 2
448.3.r.g.319.5 12 8.5 even 2
1568.3.d.j.1471.2 6 28.11 odd 6
1568.3.d.j.1471.5 6 7.4 even 3
1568.3.d.k.1471.2 6 7.3 odd 6
1568.3.d.k.1471.5 6 28.3 even 6