Properties

Label 4028.2.c.a.3497.5
Level $4028$
Weight $2$
Character 4028.3497
Analytic conductor $32.164$
Analytic rank $0$
Dimension $82$
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] = [4028,2,Mod(3497,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4028.3497");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3497.5
Character \(\chi\) \(=\) 4028.3497
Dual form 4028.2.c.a.3497.78

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.07917i q^{3} +0.328536i q^{5} -0.0673750 q^{7} -6.48127 q^{9} +O(q^{10})\) \(q-3.07917i q^{3} +0.328536i q^{5} -0.0673750 q^{7} -6.48127 q^{9} +3.92700 q^{11} -3.28260 q^{13} +1.01162 q^{15} +6.87583 q^{17} -1.00000i q^{19} +0.207459i q^{21} -2.75415i q^{23} +4.89206 q^{25} +10.7194i q^{27} +6.62262 q^{29} -8.36320i q^{31} -12.0919i q^{33} -0.0221352i q^{35} +6.84219 q^{37} +10.1077i q^{39} +7.14612i q^{41} -10.6184 q^{43} -2.12933i q^{45} +6.02840 q^{47} -6.99546 q^{49} -21.1718i q^{51} +(5.06972 + 5.22475i) q^{53} +1.29016i q^{55} -3.07917 q^{57} +10.4122 q^{59} -9.76505i q^{61} +0.436676 q^{63} -1.07845i q^{65} +2.57441i q^{67} -8.48049 q^{69} +3.31248i q^{71} +2.82088i q^{73} -15.0635i q^{75} -0.264582 q^{77} -4.59891i q^{79} +13.5631 q^{81} -14.3011i q^{83} +2.25896i q^{85} -20.3921i q^{87} -7.42589 q^{89} +0.221166 q^{91} -25.7517 q^{93} +0.328536 q^{95} -8.36020 q^{97} -25.4519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times\).

\(n\) \(2015\) \(2281\) \(2757\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.07917i 1.77776i −0.458142 0.888879i \(-0.651485\pi\)
0.458142 0.888879i \(-0.348515\pi\)
\(4\) 0 0
\(5\) 0.328536i 0.146926i 0.997298 + 0.0734630i \(0.0234051\pi\)
−0.997298 + 0.0734630i \(0.976595\pi\)
\(6\) 0 0
\(7\) −0.0673750 −0.0254654 −0.0127327 0.999919i \(-0.504053\pi\)
−0.0127327 + 0.999919i \(0.504053\pi\)
\(8\) 0 0
\(9\) −6.48127 −2.16042
\(10\) 0 0
\(11\) 3.92700 1.18403 0.592017 0.805925i \(-0.298332\pi\)
0.592017 + 0.805925i \(0.298332\pi\)
\(12\) 0 0
\(13\) −3.28260 −0.910431 −0.455215 0.890381i \(-0.650438\pi\)
−0.455215 + 0.890381i \(0.650438\pi\)
\(14\) 0 0
\(15\) 1.01162 0.261199
\(16\) 0 0
\(17\) 6.87583 1.66763 0.833817 0.552040i \(-0.186151\pi\)
0.833817 + 0.552040i \(0.186151\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 0.207459i 0.0452713i
\(22\) 0 0
\(23\) 2.75415i 0.574280i −0.957889 0.287140i \(-0.907296\pi\)
0.957889 0.287140i \(-0.0927045\pi\)
\(24\) 0 0
\(25\) 4.89206 0.978413
\(26\) 0 0
\(27\) 10.7194i 2.06295i
\(28\) 0 0
\(29\) 6.62262 1.22979 0.614894 0.788609i \(-0.289199\pi\)
0.614894 + 0.788609i \(0.289199\pi\)
\(30\) 0 0
\(31\) 8.36320i 1.50207i −0.660260 0.751037i \(-0.729554\pi\)
0.660260 0.751037i \(-0.270446\pi\)
\(32\) 0 0
\(33\) 12.0919i 2.10493i
\(34\) 0 0
\(35\) 0.0221352i 0.00374152i
\(36\) 0 0
\(37\) 6.84219 1.12485 0.562424 0.826849i \(-0.309869\pi\)
0.562424 + 0.826849i \(0.309869\pi\)
\(38\) 0 0
\(39\) 10.1077i 1.61853i
\(40\) 0 0
\(41\) 7.14612i 1.11604i 0.829829 + 0.558018i \(0.188438\pi\)
−0.829829 + 0.558018i \(0.811562\pi\)
\(42\) 0 0
\(43\) −10.6184 −1.61929 −0.809647 0.586918i \(-0.800341\pi\)
−0.809647 + 0.586918i \(0.800341\pi\)
\(44\) 0 0
\(45\) 2.12933i 0.317422i
\(46\) 0 0
\(47\) 6.02840 0.879332 0.439666 0.898161i \(-0.355097\pi\)
0.439666 + 0.898161i \(0.355097\pi\)
\(48\) 0 0
\(49\) −6.99546 −0.999352
\(50\) 0 0
\(51\) 21.1718i 2.96465i
\(52\) 0 0
\(53\) 5.06972 + 5.22475i 0.696379 + 0.717674i
\(54\) 0 0
\(55\) 1.29016i 0.173965i
\(56\) 0 0
\(57\) −3.07917 −0.407846
\(58\) 0 0
\(59\) 10.4122 1.35556 0.677779 0.735266i \(-0.262943\pi\)
0.677779 + 0.735266i \(0.262943\pi\)
\(60\) 0 0
\(61\) 9.76505i 1.25029i −0.780510 0.625144i \(-0.785040\pi\)
0.780510 0.625144i \(-0.214960\pi\)
\(62\) 0 0
\(63\) 0.436676 0.0550160
\(64\) 0 0
\(65\) 1.07845i 0.133766i
\(66\) 0 0
\(67\) 2.57441i 0.314514i 0.987558 + 0.157257i \(0.0502652\pi\)
−0.987558 + 0.157257i \(0.949735\pi\)
\(68\) 0 0
\(69\) −8.48049 −1.02093
\(70\) 0 0
\(71\) 3.31248i 0.393119i 0.980492 + 0.196559i \(0.0629769\pi\)
−0.980492 + 0.196559i \(0.937023\pi\)
\(72\) 0 0
\(73\) 2.82088i 0.330159i 0.986280 + 0.165079i \(0.0527881\pi\)
−0.986280 + 0.165079i \(0.947212\pi\)
\(74\) 0 0
\(75\) 15.0635i 1.73938i
\(76\) 0 0
\(77\) −0.264582 −0.0301519
\(78\) 0 0
\(79\) 4.59891i 0.517418i −0.965955 0.258709i \(-0.916703\pi\)
0.965955 0.258709i \(-0.0832971\pi\)
\(80\) 0 0
\(81\) 13.5631 1.50701
\(82\) 0 0
\(83\) 14.3011i 1.56976i −0.619651 0.784878i \(-0.712726\pi\)
0.619651 0.784878i \(-0.287274\pi\)
\(84\) 0 0
\(85\) 2.25896i 0.245019i
\(86\) 0 0
\(87\) 20.3921i 2.18627i
\(88\) 0 0
\(89\) −7.42589 −0.787143 −0.393571 0.919294i \(-0.628761\pi\)
−0.393571 + 0.919294i \(0.628761\pi\)
\(90\) 0 0
\(91\) 0.221166 0.0231845
\(92\) 0 0
\(93\) −25.7517 −2.67033
\(94\) 0 0
\(95\) 0.328536 0.0337071
\(96\) 0 0
\(97\) −8.36020 −0.848849 −0.424425 0.905463i \(-0.639524\pi\)
−0.424425 + 0.905463i \(0.639524\pi\)
\(98\) 0 0
\(99\) −25.4519 −2.55802
\(100\) 0 0
\(101\) 6.72655i 0.669316i −0.942340 0.334658i \(-0.891379\pi\)
0.942340 0.334658i \(-0.108621\pi\)
\(102\) 0 0
\(103\) 12.1248i 1.19469i −0.801985 0.597344i \(-0.796223\pi\)
0.801985 0.597344i \(-0.203777\pi\)
\(104\) 0 0
\(105\) −0.0681578 −0.00665152
\(106\) 0 0
\(107\) −8.10962 −0.783987 −0.391993 0.919968i \(-0.628214\pi\)
−0.391993 + 0.919968i \(0.628214\pi\)
\(108\) 0 0
\(109\) 12.3685i 1.18469i 0.805684 + 0.592346i \(0.201798\pi\)
−0.805684 + 0.592346i \(0.798202\pi\)
\(110\) 0 0
\(111\) 21.0682i 1.99971i
\(112\) 0 0
\(113\) 7.22026 0.679225 0.339613 0.940565i \(-0.389704\pi\)
0.339613 + 0.940565i \(0.389704\pi\)
\(114\) 0 0
\(115\) 0.904838 0.0843766
\(116\) 0 0
\(117\) 21.2754 1.96692
\(118\) 0 0
\(119\) −0.463260 −0.0424669
\(120\) 0 0
\(121\) 4.42131 0.401937
\(122\) 0 0
\(123\) 22.0041 1.98404
\(124\) 0 0
\(125\) 3.24990i 0.290680i
\(126\) 0 0
\(127\) 11.2119i 0.994893i −0.867495 0.497447i \(-0.834271\pi\)
0.867495 0.497447i \(-0.165729\pi\)
\(128\) 0 0
\(129\) 32.6959i 2.87871i
\(130\) 0 0
\(131\) −6.97641 −0.609532 −0.304766 0.952427i \(-0.598578\pi\)
−0.304766 + 0.952427i \(0.598578\pi\)
\(132\) 0 0
\(133\) 0.0673750i 0.00584216i
\(134\) 0 0
\(135\) −3.52172 −0.303101
\(136\) 0 0
\(137\) 3.29776i 0.281747i −0.990028 0.140873i \(-0.955009\pi\)
0.990028 0.140873i \(-0.0449911\pi\)
\(138\) 0 0
\(139\) 5.00576i 0.424583i −0.977206 0.212291i \(-0.931907\pi\)
0.977206 0.212291i \(-0.0680926\pi\)
\(140\) 0 0
\(141\) 18.5624i 1.56324i
\(142\) 0 0
\(143\) −12.8908 −1.07798
\(144\) 0 0
\(145\) 2.17577i 0.180688i
\(146\) 0 0
\(147\) 21.5402i 1.77661i
\(148\) 0 0
\(149\) −0.891002 −0.0729937 −0.0364968 0.999334i \(-0.511620\pi\)
−0.0364968 + 0.999334i \(0.511620\pi\)
\(150\) 0 0
\(151\) 6.03360i 0.491007i −0.969396 0.245504i \(-0.921047\pi\)
0.969396 0.245504i \(-0.0789534\pi\)
\(152\) 0 0
\(153\) −44.5641 −3.60280
\(154\) 0 0
\(155\) 2.74761 0.220694
\(156\) 0 0
\(157\) 7.68739i 0.613520i −0.951787 0.306760i \(-0.900755\pi\)
0.951787 0.306760i \(-0.0992450\pi\)
\(158\) 0 0
\(159\) 16.0879 15.6105i 1.27585 1.23799i
\(160\) 0 0
\(161\) 0.185561i 0.0146242i
\(162\) 0 0
\(163\) −16.0388 −1.25626 −0.628129 0.778109i \(-0.716179\pi\)
−0.628129 + 0.778109i \(0.716179\pi\)
\(164\) 0 0
\(165\) 3.97262 0.309268
\(166\) 0 0
\(167\) 3.21312i 0.248638i 0.992242 + 0.124319i \(0.0396747\pi\)
−0.992242 + 0.124319i \(0.960325\pi\)
\(168\) 0 0
\(169\) −2.22451 −0.171116
\(170\) 0 0
\(171\) 6.48127i 0.495635i
\(172\) 0 0
\(173\) 13.3330i 1.01369i 0.862038 + 0.506844i \(0.169188\pi\)
−0.862038 + 0.506844i \(0.830812\pi\)
\(174\) 0 0
\(175\) −0.329603 −0.0249156
\(176\) 0 0
\(177\) 32.0610i 2.40985i
\(178\) 0 0
\(179\) 17.7858i 1.32937i −0.747124 0.664685i \(-0.768566\pi\)
0.747124 0.664685i \(-0.231434\pi\)
\(180\) 0 0
\(181\) 15.8658i 1.17929i 0.807661 + 0.589647i \(0.200733\pi\)
−0.807661 + 0.589647i \(0.799267\pi\)
\(182\) 0 0
\(183\) −30.0682 −2.22271
\(184\) 0 0
\(185\) 2.24791i 0.165269i
\(186\) 0 0
\(187\) 27.0014 1.97454
\(188\) 0 0
\(189\) 0.722221i 0.0525338i
\(190\) 0 0
\(191\) 22.8865i 1.65601i 0.560719 + 0.828006i \(0.310525\pi\)
−0.560719 + 0.828006i \(0.689475\pi\)
\(192\) 0 0
\(193\) 5.74851i 0.413787i 0.978364 + 0.206893i \(0.0663353\pi\)
−0.978364 + 0.206893i \(0.933665\pi\)
\(194\) 0 0
\(195\) −3.32074 −0.237803
\(196\) 0 0
\(197\) −1.31373 −0.0935997 −0.0467999 0.998904i \(-0.514902\pi\)
−0.0467999 + 0.998904i \(0.514902\pi\)
\(198\) 0 0
\(199\) 0.356112 0.0252441 0.0126220 0.999920i \(-0.495982\pi\)
0.0126220 + 0.999920i \(0.495982\pi\)
\(200\) 0 0
\(201\) 7.92704 0.559130
\(202\) 0 0
\(203\) −0.446199 −0.0313170
\(204\) 0 0
\(205\) −2.34776 −0.163975
\(206\) 0 0
\(207\) 17.8504i 1.24069i
\(208\) 0 0
\(209\) 3.92700i 0.271636i
\(210\) 0 0
\(211\) 22.6585 1.55987 0.779936 0.625859i \(-0.215252\pi\)
0.779936 + 0.625859i \(0.215252\pi\)
\(212\) 0 0
\(213\) 10.1997 0.698870
\(214\) 0 0
\(215\) 3.48854i 0.237916i
\(216\) 0 0
\(217\) 0.563471i 0.0382509i
\(218\) 0 0
\(219\) 8.68596 0.586943
\(220\) 0 0
\(221\) −22.5706 −1.51827
\(222\) 0 0
\(223\) 0.394431 0.0264131 0.0132065 0.999913i \(-0.495796\pi\)
0.0132065 + 0.999913i \(0.495796\pi\)
\(224\) 0 0
\(225\) −31.7068 −2.11379
\(226\) 0 0
\(227\) 24.4605 1.62350 0.811751 0.584003i \(-0.198515\pi\)
0.811751 + 0.584003i \(0.198515\pi\)
\(228\) 0 0
\(229\) 2.12203 0.140228 0.0701139 0.997539i \(-0.477664\pi\)
0.0701139 + 0.997539i \(0.477664\pi\)
\(230\) 0 0
\(231\) 0.814691i 0.0536027i
\(232\) 0 0
\(233\) 16.5633i 1.08510i 0.840024 + 0.542549i \(0.182541\pi\)
−0.840024 + 0.542549i \(0.817459\pi\)
\(234\) 0 0
\(235\) 1.98055i 0.129197i
\(236\) 0 0
\(237\) −14.1608 −0.919844
\(238\) 0 0
\(239\) 6.57821i 0.425509i −0.977106 0.212754i \(-0.931757\pi\)
0.977106 0.212754i \(-0.0682434\pi\)
\(240\) 0 0
\(241\) 10.0210 0.645511 0.322756 0.946482i \(-0.395391\pi\)
0.322756 + 0.946482i \(0.395391\pi\)
\(242\) 0 0
\(243\) 9.60468i 0.616141i
\(244\) 0 0
\(245\) 2.29826i 0.146831i
\(246\) 0 0
\(247\) 3.28260i 0.208867i
\(248\) 0 0
\(249\) −44.0356 −2.79064
\(250\) 0 0
\(251\) 8.17074i 0.515732i 0.966181 + 0.257866i \(0.0830194\pi\)
−0.966181 + 0.257866i \(0.916981\pi\)
\(252\) 0 0
\(253\) 10.8155i 0.679967i
\(254\) 0 0
\(255\) 6.95572 0.435584
\(256\) 0 0
\(257\) 30.1648i 1.88163i −0.338926 0.940813i \(-0.610064\pi\)
0.338926 0.940813i \(-0.389936\pi\)
\(258\) 0 0
\(259\) −0.460993 −0.0286447
\(260\) 0 0
\(261\) −42.9230 −2.65686
\(262\) 0 0
\(263\) 25.2691i 1.55816i 0.626926 + 0.779079i \(0.284313\pi\)
−0.626926 + 0.779079i \(0.715687\pi\)
\(264\) 0 0
\(265\) −1.71652 + 1.66559i −0.105445 + 0.102316i
\(266\) 0 0
\(267\) 22.8656i 1.39935i
\(268\) 0 0
\(269\) −1.13598 −0.0692621 −0.0346310 0.999400i \(-0.511026\pi\)
−0.0346310 + 0.999400i \(0.511026\pi\)
\(270\) 0 0
\(271\) −4.38846 −0.266580 −0.133290 0.991077i \(-0.542554\pi\)
−0.133290 + 0.991077i \(0.542554\pi\)
\(272\) 0 0
\(273\) 0.681006i 0.0412163i
\(274\) 0 0
\(275\) 19.2111 1.15847
\(276\) 0 0
\(277\) 8.36304i 0.502486i −0.967924 0.251243i \(-0.919161\pi\)
0.967924 0.251243i \(-0.0808394\pi\)
\(278\) 0 0
\(279\) 54.2042i 3.24512i
\(280\) 0 0
\(281\) 2.20514 0.131548 0.0657738 0.997835i \(-0.479048\pi\)
0.0657738 + 0.997835i \(0.479048\pi\)
\(282\) 0 0
\(283\) 2.42637i 0.144233i 0.997396 + 0.0721165i \(0.0229753\pi\)
−0.997396 + 0.0721165i \(0.977025\pi\)
\(284\) 0 0
\(285\) 1.01162i 0.0599231i
\(286\) 0 0
\(287\) 0.481470i 0.0284203i
\(288\) 0 0
\(289\) 30.2771 1.78101
\(290\) 0 0
\(291\) 25.7424i 1.50905i
\(292\) 0 0
\(293\) 2.49769 0.145917 0.0729584 0.997335i \(-0.476756\pi\)
0.0729584 + 0.997335i \(0.476756\pi\)
\(294\) 0 0
\(295\) 3.42080i 0.199167i
\(296\) 0 0
\(297\) 42.0951i 2.44261i
\(298\) 0 0
\(299\) 9.04078i 0.522842i
\(300\) 0 0
\(301\) 0.715416 0.0412359
\(302\) 0 0
\(303\) −20.7122 −1.18988
\(304\) 0 0
\(305\) 3.20818 0.183700
\(306\) 0 0
\(307\) −21.8905 −1.24935 −0.624677 0.780883i \(-0.714770\pi\)
−0.624677 + 0.780883i \(0.714770\pi\)
\(308\) 0 0
\(309\) −37.3342 −2.12387
\(310\) 0 0
\(311\) −12.0040 −0.680682 −0.340341 0.940302i \(-0.610543\pi\)
−0.340341 + 0.940302i \(0.610543\pi\)
\(312\) 0 0
\(313\) 19.1611i 1.08305i −0.840684 0.541525i \(-0.817847\pi\)
0.840684 0.541525i \(-0.182153\pi\)
\(314\) 0 0
\(315\) 0.143464i 0.00808328i
\(316\) 0 0
\(317\) 26.5137 1.48916 0.744580 0.667533i \(-0.232650\pi\)
0.744580 + 0.667533i \(0.232650\pi\)
\(318\) 0 0
\(319\) 26.0070 1.45611
\(320\) 0 0
\(321\) 24.9709i 1.39374i
\(322\) 0 0
\(323\) 6.87583i 0.382582i
\(324\) 0 0
\(325\) −16.0587 −0.890777
\(326\) 0 0
\(327\) 38.0848 2.10609
\(328\) 0 0
\(329\) −0.406164 −0.0223925
\(330\) 0 0
\(331\) −24.1473 −1.32726 −0.663628 0.748063i \(-0.730984\pi\)
−0.663628 + 0.748063i \(0.730984\pi\)
\(332\) 0 0
\(333\) −44.3461 −2.43015
\(334\) 0 0
\(335\) −0.845788 −0.0462103
\(336\) 0 0
\(337\) 8.69142i 0.473452i −0.971576 0.236726i \(-0.923926\pi\)
0.971576 0.236726i \(-0.0760743\pi\)
\(338\) 0 0
\(339\) 22.2324i 1.20750i
\(340\) 0 0
\(341\) 32.8423i 1.77851i
\(342\) 0 0
\(343\) 0.942945 0.0509142
\(344\) 0 0
\(345\) 2.78615i 0.150001i
\(346\) 0 0
\(347\) −21.6153 −1.16037 −0.580185 0.814485i \(-0.697020\pi\)
−0.580185 + 0.814485i \(0.697020\pi\)
\(348\) 0 0
\(349\) 18.6583i 0.998758i 0.866384 + 0.499379i \(0.166438\pi\)
−0.866384 + 0.499379i \(0.833562\pi\)
\(350\) 0 0
\(351\) 35.1876i 1.87817i
\(352\) 0 0
\(353\) 16.7132i 0.889554i −0.895641 0.444777i \(-0.853283\pi\)
0.895641 0.444777i \(-0.146717\pi\)
\(354\) 0 0
\(355\) −1.08827 −0.0577594
\(356\) 0 0
\(357\) 1.42645i 0.0754959i
\(358\) 0 0
\(359\) 14.7261i 0.777216i −0.921403 0.388608i \(-0.872956\pi\)
0.921403 0.388608i \(-0.127044\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 13.6139i 0.714547i
\(364\) 0 0
\(365\) −0.926761 −0.0485089
\(366\) 0 0
\(367\) 1.62402 0.0847731 0.0423865 0.999101i \(-0.486504\pi\)
0.0423865 + 0.999101i \(0.486504\pi\)
\(368\) 0 0
\(369\) 46.3160i 2.41111i
\(370\) 0 0
\(371\) −0.341572 0.352018i −0.0177335 0.0182758i
\(372\) 0 0
\(373\) 35.9917i 1.86358i −0.363001 0.931789i \(-0.618248\pi\)
0.363001 0.931789i \(-0.381752\pi\)
\(374\) 0 0
\(375\) 10.0070 0.516759
\(376\) 0 0
\(377\) −21.7394 −1.11964
\(378\) 0 0
\(379\) 31.5284i 1.61950i 0.586772 + 0.809752i \(0.300399\pi\)
−0.586772 + 0.809752i \(0.699601\pi\)
\(380\) 0 0
\(381\) −34.5232 −1.76868
\(382\) 0 0
\(383\) 5.61481i 0.286903i 0.989657 + 0.143452i \(0.0458202\pi\)
−0.989657 + 0.143452i \(0.954180\pi\)
\(384\) 0 0
\(385\) 0.0869247i 0.00443009i
\(386\) 0 0
\(387\) 68.8208 3.49836
\(388\) 0 0
\(389\) 32.5209i 1.64887i −0.565954 0.824437i \(-0.691492\pi\)
0.565954 0.824437i \(-0.308508\pi\)
\(390\) 0 0
\(391\) 18.9371i 0.957689i
\(392\) 0 0
\(393\) 21.4815i 1.08360i
\(394\) 0 0
\(395\) 1.51091 0.0760221
\(396\) 0 0
\(397\) 4.56480i 0.229101i 0.993417 + 0.114550i \(0.0365427\pi\)
−0.993417 + 0.114550i \(0.963457\pi\)
\(398\) 0 0
\(399\) 0.207459 0.0103859
\(400\) 0 0
\(401\) 1.92403i 0.0960813i 0.998845 + 0.0480407i \(0.0152977\pi\)
−0.998845 + 0.0480407i \(0.984702\pi\)
\(402\) 0 0
\(403\) 27.4531i 1.36753i
\(404\) 0 0
\(405\) 4.45596i 0.221418i
\(406\) 0 0
\(407\) 26.8692 1.33186
\(408\) 0 0
\(409\) −36.7067 −1.81503 −0.907514 0.420022i \(-0.862022\pi\)
−0.907514 + 0.420022i \(0.862022\pi\)
\(410\) 0 0
\(411\) −10.1544 −0.500878
\(412\) 0 0
\(413\) −0.701525 −0.0345198
\(414\) 0 0
\(415\) 4.69845 0.230638
\(416\) 0 0
\(417\) −15.4136 −0.754805
\(418\) 0 0
\(419\) 8.25974i 0.403515i −0.979436 0.201757i \(-0.935335\pi\)
0.979436 0.201757i \(-0.0646652\pi\)
\(420\) 0 0
\(421\) 1.85203i 0.0902624i −0.998981 0.0451312i \(-0.985629\pi\)
0.998981 0.0451312i \(-0.0143706\pi\)
\(422\) 0 0
\(423\) −39.0717 −1.89973
\(424\) 0 0
\(425\) 33.6370 1.63163
\(426\) 0 0
\(427\) 0.657921i 0.0318390i
\(428\) 0 0
\(429\) 39.6929i 1.91639i
\(430\) 0 0
\(431\) 25.0919 1.20863 0.604316 0.796745i \(-0.293446\pi\)
0.604316 + 0.796745i \(0.293446\pi\)
\(432\) 0 0
\(433\) −23.0086 −1.10572 −0.552862 0.833273i \(-0.686464\pi\)
−0.552862 + 0.833273i \(0.686464\pi\)
\(434\) 0 0
\(435\) 6.69956 0.321219
\(436\) 0 0
\(437\) −2.75415 −0.131749
\(438\) 0 0
\(439\) 6.79021 0.324079 0.162040 0.986784i \(-0.448193\pi\)
0.162040 + 0.986784i \(0.448193\pi\)
\(440\) 0 0
\(441\) 45.3395 2.15902
\(442\) 0 0
\(443\) 29.5124i 1.40218i 0.713074 + 0.701088i \(0.247302\pi\)
−0.713074 + 0.701088i \(0.752698\pi\)
\(444\) 0 0
\(445\) 2.43967i 0.115652i
\(446\) 0 0
\(447\) 2.74354i 0.129765i
\(448\) 0 0
\(449\) 3.80185 0.179421 0.0897103 0.995968i \(-0.471406\pi\)
0.0897103 + 0.995968i \(0.471406\pi\)
\(450\) 0 0
\(451\) 28.0628i 1.32143i
\(452\) 0 0
\(453\) −18.5785 −0.872892
\(454\) 0 0
\(455\) 0.0726609i 0.00340640i
\(456\) 0 0
\(457\) 23.5786i 1.10296i 0.834188 + 0.551480i \(0.185937\pi\)
−0.834188 + 0.551480i \(0.814063\pi\)
\(458\) 0 0
\(459\) 73.7049i 3.44025i
\(460\) 0 0
\(461\) −33.2696 −1.54952 −0.774759 0.632256i \(-0.782129\pi\)
−0.774759 + 0.632256i \(0.782129\pi\)
\(462\) 0 0
\(463\) 27.4818i 1.27719i −0.769545 0.638593i \(-0.779517\pi\)
0.769545 0.638593i \(-0.220483\pi\)
\(464\) 0 0
\(465\) 8.46037i 0.392340i
\(466\) 0 0
\(467\) −38.9537 −1.80256 −0.901282 0.433234i \(-0.857372\pi\)
−0.901282 + 0.433234i \(0.857372\pi\)
\(468\) 0 0
\(469\) 0.173451i 0.00800922i
\(470\) 0 0
\(471\) −23.6708 −1.09069
\(472\) 0 0
\(473\) −41.6985 −1.91730
\(474\) 0 0
\(475\) 4.89206i 0.224463i
\(476\) 0 0
\(477\) −32.8582 33.8630i −1.50447 1.55048i
\(478\) 0 0
\(479\) 37.7395i 1.72436i −0.506601 0.862181i \(-0.669098\pi\)
0.506601 0.862181i \(-0.330902\pi\)
\(480\) 0 0
\(481\) −22.4602 −1.02410
\(482\) 0 0
\(483\) 0.571373 0.0259984
\(484\) 0 0
\(485\) 2.74663i 0.124718i
\(486\) 0 0
\(487\) 37.8215 1.71386 0.856928 0.515436i \(-0.172370\pi\)
0.856928 + 0.515436i \(0.172370\pi\)
\(488\) 0 0
\(489\) 49.3862i 2.23332i
\(490\) 0 0
\(491\) 13.2598i 0.598407i −0.954189 0.299204i \(-0.903279\pi\)
0.954189 0.299204i \(-0.0967210\pi\)
\(492\) 0 0
\(493\) 45.5360 2.05084
\(494\) 0 0
\(495\) 8.36189i 0.375839i
\(496\) 0 0
\(497\) 0.223178i 0.0100109i
\(498\) 0 0
\(499\) 12.5189i 0.560423i 0.959938 + 0.280212i \(0.0904047\pi\)
−0.959938 + 0.280212i \(0.909595\pi\)
\(500\) 0 0
\(501\) 9.89372 0.442019
\(502\) 0 0
\(503\) 32.5534i 1.45149i −0.687966 0.725743i \(-0.741496\pi\)
0.687966 0.725743i \(-0.258504\pi\)
\(504\) 0 0
\(505\) 2.20991 0.0983399
\(506\) 0 0
\(507\) 6.84964i 0.304203i
\(508\) 0 0
\(509\) 41.4168i 1.83577i 0.396847 + 0.917885i \(0.370104\pi\)
−0.396847 + 0.917885i \(0.629896\pi\)
\(510\) 0 0
\(511\) 0.190057i 0.00840762i
\(512\) 0 0
\(513\) 10.7194 0.473274
\(514\) 0 0
\(515\) 3.98343 0.175531
\(516\) 0 0
\(517\) 23.6735 1.04116
\(518\) 0 0
\(519\) 41.0545 1.80209
\(520\) 0 0
\(521\) 1.40801 0.0616860 0.0308430 0.999524i \(-0.490181\pi\)
0.0308430 + 0.999524i \(0.490181\pi\)
\(522\) 0 0
\(523\) 10.4483 0.456872 0.228436 0.973559i \(-0.426639\pi\)
0.228436 + 0.973559i \(0.426639\pi\)
\(524\) 0 0
\(525\) 1.01490i 0.0442940i
\(526\) 0 0
\(527\) 57.5040i 2.50491i
\(528\) 0 0
\(529\) 15.4147 0.670203
\(530\) 0 0
\(531\) −67.4845 −2.92858
\(532\) 0 0
\(533\) 23.4579i 1.01607i
\(534\) 0 0
\(535\) 2.66431i 0.115188i
\(536\) 0 0
\(537\) −54.7653 −2.36330
\(538\) 0 0
\(539\) −27.4712 −1.18327
\(540\) 0 0
\(541\) −4.09242 −0.175947 −0.0879734 0.996123i \(-0.528039\pi\)
−0.0879734 + 0.996123i \(0.528039\pi\)
\(542\) 0 0
\(543\) 48.8534 2.09650
\(544\) 0 0
\(545\) −4.06351 −0.174062
\(546\) 0 0
\(547\) 17.2792 0.738805 0.369402 0.929270i \(-0.379562\pi\)
0.369402 + 0.929270i \(0.379562\pi\)
\(548\) 0 0
\(549\) 63.2900i 2.70115i
\(550\) 0 0
\(551\) 6.62262i 0.282133i
\(552\) 0 0
\(553\) 0.309852i 0.0131762i
\(554\) 0 0
\(555\) 6.92168 0.293809
\(556\) 0 0
\(557\) 28.2663i 1.19768i −0.800868 0.598841i \(-0.795628\pi\)
0.800868 0.598841i \(-0.204372\pi\)
\(558\) 0 0
\(559\) 34.8561 1.47425
\(560\) 0 0
\(561\) 83.1418i 3.51025i
\(562\) 0 0
\(563\) 12.0007i 0.505768i 0.967497 + 0.252884i \(0.0813791\pi\)
−0.967497 + 0.252884i \(0.918621\pi\)
\(564\) 0 0
\(565\) 2.37212i 0.0997958i
\(566\) 0 0
\(567\) −0.913812 −0.0383765
\(568\) 0 0
\(569\) 27.0926i 1.13578i 0.823105 + 0.567889i \(0.192240\pi\)
−0.823105 + 0.567889i \(0.807760\pi\)
\(570\) 0 0
\(571\) 40.6812i 1.70245i 0.524797 + 0.851227i \(0.324141\pi\)
−0.524797 + 0.851227i \(0.675859\pi\)
\(572\) 0 0
\(573\) 70.4715 2.94399
\(574\) 0 0
\(575\) 13.4735i 0.561883i
\(576\) 0 0
\(577\) 29.8641 1.24326 0.621629 0.783312i \(-0.286471\pi\)
0.621629 + 0.783312i \(0.286471\pi\)
\(578\) 0 0
\(579\) 17.7006 0.735612
\(580\) 0 0
\(581\) 0.963540i 0.0399744i
\(582\) 0 0
\(583\) 19.9088 + 20.5176i 0.824537 + 0.849751i
\(584\) 0 0
\(585\) 6.98976i 0.288991i
\(586\) 0 0
\(587\) 22.7113 0.937397 0.468698 0.883358i \(-0.344723\pi\)
0.468698 + 0.883358i \(0.344723\pi\)
\(588\) 0 0
\(589\) −8.36320 −0.344600
\(590\) 0 0
\(591\) 4.04521i 0.166398i
\(592\) 0 0
\(593\) 17.4158 0.715183 0.357591 0.933878i \(-0.383598\pi\)
0.357591 + 0.933878i \(0.383598\pi\)
\(594\) 0 0
\(595\) 0.152198i 0.00623949i
\(596\) 0 0
\(597\) 1.09653i 0.0448779i
\(598\) 0 0
\(599\) 47.3331 1.93398 0.966989 0.254819i \(-0.0820158\pi\)
0.966989 + 0.254819i \(0.0820158\pi\)
\(600\) 0 0
\(601\) 8.11541i 0.331035i −0.986207 0.165517i \(-0.947071\pi\)
0.986207 0.165517i \(-0.0529294\pi\)
\(602\) 0 0
\(603\) 16.6855i 0.679484i
\(604\) 0 0
\(605\) 1.45256i 0.0590550i
\(606\) 0 0
\(607\) 16.8574 0.684219 0.342109 0.939660i \(-0.388859\pi\)
0.342109 + 0.939660i \(0.388859\pi\)
\(608\) 0 0
\(609\) 1.37392i 0.0556741i
\(610\) 0 0
\(611\) −19.7888 −0.800571
\(612\) 0 0
\(613\) 22.4407i 0.906371i −0.891416 0.453186i \(-0.850287\pi\)
0.891416 0.453186i \(-0.149713\pi\)
\(614\) 0 0
\(615\) 7.22915i 0.291507i
\(616\) 0 0
\(617\) 31.7241i 1.27716i −0.769554 0.638582i \(-0.779521\pi\)
0.769554 0.638582i \(-0.220479\pi\)
\(618\) 0 0
\(619\) 13.2692 0.533334 0.266667 0.963789i \(-0.414078\pi\)
0.266667 + 0.963789i \(0.414078\pi\)
\(620\) 0 0
\(621\) 29.5229 1.18471
\(622\) 0 0
\(623\) 0.500320 0.0200449
\(624\) 0 0
\(625\) 23.3926 0.935704
\(626\) 0 0
\(627\) −12.0919 −0.482903
\(628\) 0 0
\(629\) 47.0457 1.87584
\(630\) 0 0
\(631\) 1.02862i 0.0409487i −0.999790 0.0204744i \(-0.993482\pi\)
0.999790 0.0204744i \(-0.00651765\pi\)
\(632\) 0 0
\(633\) 69.7692i 2.77308i
\(634\) 0 0
\(635\) 3.68351 0.146176
\(636\) 0 0
\(637\) 22.9633 0.909840
\(638\) 0 0
\(639\) 21.4691i 0.849303i
\(640\) 0 0
\(641\) 20.3773i 0.804857i 0.915452 + 0.402428i \(0.131834\pi\)
−0.915452 + 0.402428i \(0.868166\pi\)
\(642\) 0 0
\(643\) 0.0328109 0.00129393 0.000646967 1.00000i \(-0.499794\pi\)
0.000646967 1.00000i \(0.499794\pi\)
\(644\) 0 0
\(645\) −10.7418 −0.422957
\(646\) 0 0
\(647\) 31.7936 1.24994 0.624968 0.780650i \(-0.285112\pi\)
0.624968 + 0.780650i \(0.285112\pi\)
\(648\) 0 0
\(649\) 40.8888 1.60503
\(650\) 0 0
\(651\) 1.73502 0.0680008
\(652\) 0 0
\(653\) −32.7316 −1.28089 −0.640443 0.768006i \(-0.721249\pi\)
−0.640443 + 0.768006i \(0.721249\pi\)
\(654\) 0 0
\(655\) 2.29201i 0.0895561i
\(656\) 0 0
\(657\) 18.2829i 0.713283i
\(658\) 0 0
\(659\) 26.4997i 1.03228i −0.856503 0.516142i \(-0.827368\pi\)
0.856503 0.516142i \(-0.172632\pi\)
\(660\) 0 0
\(661\) −16.7653 −0.652093 −0.326047 0.945354i \(-0.605717\pi\)
−0.326047 + 0.945354i \(0.605717\pi\)
\(662\) 0 0
\(663\) 69.4988i 2.69911i
\(664\) 0 0
\(665\) −0.0221352 −0.000858364
\(666\) 0 0
\(667\) 18.2397i 0.706243i
\(668\) 0 0
\(669\) 1.21452i 0.0469560i
\(670\) 0 0
\(671\) 38.3473i 1.48038i
\(672\) 0 0
\(673\) 9.66250 0.372462 0.186231 0.982506i \(-0.440373\pi\)
0.186231 + 0.982506i \(0.440373\pi\)
\(674\) 0 0
\(675\) 52.4401i 2.01842i
\(676\) 0 0
\(677\) 29.5959i 1.13746i 0.822524 + 0.568731i \(0.192565\pi\)
−0.822524 + 0.568731i \(0.807435\pi\)
\(678\) 0 0
\(679\) 0.563269 0.0216163
\(680\) 0 0
\(681\) 75.3181i 2.88619i
\(682\) 0 0
\(683\) −23.6867 −0.906346 −0.453173 0.891422i \(-0.649708\pi\)
−0.453173 + 0.891422i \(0.649708\pi\)
\(684\) 0 0
\(685\) 1.08344 0.0413959
\(686\) 0 0
\(687\) 6.53409i 0.249291i
\(688\) 0 0
\(689\) −16.6419 17.1508i −0.634005 0.653393i
\(690\) 0 0
\(691\) 26.5901i 1.01153i 0.862670 + 0.505767i \(0.168790\pi\)
−0.862670 + 0.505767i \(0.831210\pi\)
\(692\) 0 0
\(693\) 1.71482 0.0651408
\(694\) 0 0
\(695\) 1.64457 0.0623822
\(696\) 0 0
\(697\) 49.1356i 1.86114i
\(698\) 0 0
\(699\) 51.0012 1.92904
\(700\) 0 0
\(701\) 33.1360i 1.25153i 0.780012 + 0.625764i \(0.215213\pi\)
−0.780012 + 0.625764i \(0.784787\pi\)
\(702\) 0 0
\(703\) 6.84219i 0.258058i
\(704\) 0 0
\(705\) 6.09844 0.229681
\(706\) 0 0
\(707\) 0.453201i 0.0170444i
\(708\) 0 0
\(709\) 51.5212i 1.93492i 0.253024 + 0.967460i \(0.418575\pi\)
−0.253024 + 0.967460i \(0.581425\pi\)
\(710\) 0 0
\(711\) 29.8068i 1.11784i
\(712\) 0 0
\(713\) −23.0335 −0.862611
\(714\) 0 0
\(715\) 4.23509i 0.158383i
\(716\) 0 0
\(717\) −20.2554 −0.756452
\(718\) 0 0
\(719\) 19.4058i 0.723713i −0.932234 0.361857i \(-0.882143\pi\)
0.932234 0.361857i \(-0.117857\pi\)
\(720\) 0 0
\(721\) 0.816907i 0.0304232i
\(722\) 0 0
\(723\) 30.8564i 1.14756i
\(724\) 0 0
\(725\) 32.3983 1.20324
\(726\) 0 0
\(727\) 39.2039 1.45399 0.726996 0.686642i \(-0.240916\pi\)
0.726996 + 0.686642i \(0.240916\pi\)
\(728\) 0 0
\(729\) 11.1148 0.411658
\(730\) 0 0
\(731\) −73.0105 −2.70039
\(732\) 0 0
\(733\) 12.0741 0.445967 0.222984 0.974822i \(-0.428420\pi\)
0.222984 + 0.974822i \(0.428420\pi\)
\(734\) 0 0
\(735\) −7.07674 −0.261029
\(736\) 0 0
\(737\) 10.1097i 0.372396i
\(738\) 0 0
\(739\) 7.42296i 0.273058i 0.990636 + 0.136529i \(0.0435947\pi\)
−0.990636 + 0.136529i \(0.956405\pi\)
\(740\) 0 0
\(741\) 10.1077 0.371315
\(742\) 0 0
\(743\) −17.8249 −0.653931 −0.326965 0.945036i \(-0.606026\pi\)
−0.326965 + 0.945036i \(0.606026\pi\)
\(744\) 0 0
\(745\) 0.292726i 0.0107247i
\(746\) 0 0
\(747\) 92.6896i 3.39134i
\(748\) 0 0
\(749\) 0.546386 0.0199645
\(750\) 0 0
\(751\) −48.9104 −1.78476 −0.892382 0.451280i \(-0.850968\pi\)
−0.892382 + 0.451280i \(0.850968\pi\)
\(752\) 0 0
\(753\) 25.1591 0.916847
\(754\) 0 0
\(755\) 1.98226 0.0721417
\(756\) 0 0
\(757\) 40.5789 1.47487 0.737433 0.675420i \(-0.236038\pi\)
0.737433 + 0.675420i \(0.236038\pi\)
\(758\) 0 0
\(759\) −33.3028 −1.20882
\(760\) 0 0
\(761\) 6.68772i 0.242430i 0.992626 + 0.121215i \(0.0386790\pi\)
−0.992626 + 0.121215i \(0.961321\pi\)
\(762\) 0 0
\(763\) 0.833331i 0.0301686i
\(764\) 0 0
\(765\) 14.6409i 0.529344i
\(766\) 0 0
\(767\) −34.1792 −1.23414
\(768\) 0 0
\(769\) 5.62446i 0.202823i −0.994845 0.101412i \(-0.967664\pi\)
0.994845 0.101412i \(-0.0323359\pi\)
\(770\) 0 0
\(771\) −92.8823 −3.34508
\(772\) 0 0
\(773\) 14.6890i 0.528328i −0.964478 0.264164i \(-0.914904\pi\)
0.964478 0.264164i \(-0.0850961\pi\)
\(774\) 0 0
\(775\) 40.9133i 1.46965i
\(776\) 0 0
\(777\) 1.41947i 0.0509233i
\(778\) 0 0
\(779\) 7.14612 0.256036
\(780\) 0 0
\(781\) 13.0081i 0.465466i
\(782\) 0 0
\(783\) 70.9906i 2.53700i
\(784\) 0 0
\(785\) 2.52559 0.0901421
\(786\) 0 0
\(787\) 38.7538i 1.38142i 0.723130 + 0.690712i \(0.242703\pi\)
−0.723130 + 0.690712i \(0.757297\pi\)
\(788\) 0 0
\(789\) 77.8077 2.77003
\(790\) 0 0
\(791\) −0.486466 −0.0172967
\(792\) 0 0
\(793\) 32.0548i 1.13830i
\(794\) 0 0
\(795\) 5.12862 + 5.28545i 0.181893 + 0.187456i
\(796\) 0 0
\(797\) 26.2079i 0.928332i −0.885748 0.464166i \(-0.846354\pi\)
0.885748 0.464166i \(-0.153646\pi\)
\(798\) 0 0
\(799\) 41.4503 1.46641
\(800\) 0 0
\(801\) 48.1292 1.70056
\(802\) 0 0
\(803\) 11.0776i 0.390919i
\(804\) 0 0
\(805\) −0.0609635 −0.00214868
\(806\) 0 0
\(807\) 3.49788i 0.123131i
\(808\) 0 0
\(809\) 31.0360i 1.09117i −0.838056 0.545584i \(-0.816308\pi\)
0.838056 0.545584i \(-0.183692\pi\)
\(810\) 0 0
\(811\) 22.1708 0.778523 0.389262 0.921127i \(-0.372730\pi\)
0.389262 + 0.921127i \(0.372730\pi\)
\(812\) 0 0
\(813\) 13.5128i 0.473914i
\(814\) 0 0
\(815\) 5.26934i 0.184577i
\(816\) 0 0
\(817\) 10.6184i 0.371491i
\(818\) 0 0
\(819\) −1.43343 −0.0500882
\(820\) 0 0
\(821\) 5.77096i 0.201408i 0.994916 + 0.100704i \(0.0321095\pi\)
−0.994916 + 0.100704i \(0.967891\pi\)
\(822\) 0 0
\(823\) −16.4344 −0.572869 −0.286434 0.958100i \(-0.592470\pi\)
−0.286434 + 0.958100i \(0.592470\pi\)
\(824\) 0 0
\(825\) 59.1543i 2.05949i
\(826\) 0 0
\(827\) 22.9599i 0.798395i 0.916865 + 0.399198i \(0.130711\pi\)
−0.916865 + 0.399198i \(0.869289\pi\)
\(828\) 0 0
\(829\) 9.23901i 0.320884i 0.987045 + 0.160442i \(0.0512920\pi\)
−0.987045 + 0.160442i \(0.948708\pi\)
\(830\) 0 0
\(831\) −25.7512 −0.893299
\(832\) 0 0
\(833\) −48.0996 −1.66655
\(834\) 0 0
\(835\) −1.05563 −0.0365314
\(836\) 0 0
\(837\) 89.6486 3.09871
\(838\) 0 0
\(839\) 41.8408 1.44451 0.722253 0.691629i \(-0.243107\pi\)
0.722253 + 0.691629i \(0.243107\pi\)
\(840\) 0 0
\(841\) 14.8590 0.512381
\(842\) 0 0
\(843\) 6.78999i 0.233860i
\(844\) 0 0
\(845\) 0.730833i 0.0251414i
\(846\) 0 0
\(847\) −0.297886 −0.0102355
\(848\) 0 0
\(849\) 7.47121 0.256411
\(850\) 0 0
\(851\) 18.8444i 0.645978i
\(852\) 0 0
\(853\) 8.39209i 0.287340i −0.989626 0.143670i \(-0.954110\pi\)
0.989626 0.143670i \(-0.0458903\pi\)
\(854\) 0 0
\(855\) −2.12933 −0.0728217
\(856\) 0 0
\(857\) 27.0489 0.923974 0.461987 0.886887i \(-0.347137\pi\)
0.461987 + 0.886887i \(0.347137\pi\)
\(858\) 0 0
\(859\) −33.4374 −1.14087 −0.570434 0.821343i \(-0.693225\pi\)
−0.570434 + 0.821343i \(0.693225\pi\)
\(860\) 0 0
\(861\) −1.48253 −0.0505244
\(862\) 0 0
\(863\) 34.5370 1.17565 0.587826 0.808987i \(-0.299984\pi\)
0.587826 + 0.808987i \(0.299984\pi\)
\(864\) 0 0
\(865\) −4.38037 −0.148937
\(866\) 0 0
\(867\) 93.2282i 3.16620i
\(868\) 0 0
\(869\) 18.0599i 0.612641i
\(870\) 0 0
\(871\) 8.45077i 0.286343i
\(872\) 0 0
\(873\) 54.1847 1.83387
\(874\) 0 0
\(875\) 0.218962i 0.00740228i
\(876\) 0 0
\(877\) −36.4836 −1.23196 −0.615981 0.787761i \(-0.711240\pi\)
−0.615981 + 0.787761i \(0.711240\pi\)
\(878\) 0 0
\(879\) 7.69081i 0.259405i
\(880\) 0 0
\(881\) 14.4788i 0.487803i 0.969800 + 0.243902i \(0.0784274\pi\)
−0.969800 + 0.243902i \(0.921573\pi\)
\(882\) 0 0
\(883\) 7.25178i 0.244042i −0.992528 0.122021i \(-0.961062\pi\)
0.992528 0.122021i \(-0.0389375\pi\)
\(884\) 0 0
\(885\) 10.5332 0.354070
\(886\) 0 0
\(887\) 26.4542i 0.888244i 0.895966 + 0.444122i \(0.146484\pi\)
−0.895966 + 0.444122i \(0.853516\pi\)
\(888\) 0 0
\(889\) 0.755401i 0.0253353i
\(890\) 0 0
\(891\) 53.2621 1.78435
\(892\) 0 0
\(893\) 6.02840i 0.201733i
\(894\) 0 0
\(895\) 5.84327 0.195319
\(896\) 0 0
\(897\) 27.8381 0.929486
\(898\) 0 0
\(899\) 55.3862i 1.84723i
\(900\) 0 0
\(901\) 34.8585 + 35.9245i 1.16131 + 1.19682i
\(902\) 0 0
\(903\) 2.20289i 0.0733075i
\(904\) 0 0
\(905\) −5.21249 −0.173269
\(906\) 0 0
\(907\) 29.6638 0.984972 0.492486 0.870320i \(-0.336088\pi\)
0.492486 + 0.870320i \(0.336088\pi\)
\(908\) 0 0
\(909\) 43.5966i 1.44601i
\(910\) 0 0
\(911\) 33.5901 1.11289 0.556444 0.830885i \(-0.312165\pi\)
0.556444 + 0.830885i \(0.312165\pi\)
\(912\) 0 0
\(913\) 56.1606i 1.85864i
\(914\) 0 0
\(915\) 9.87851i 0.326573i
\(916\) 0 0
\(917\) 0.470036 0.0155220
\(918\) 0 0
\(919\) 56.2845i 1.85665i −0.371763 0.928327i \(-0.621247\pi\)
0.371763 0.928327i \(-0.378753\pi\)
\(920\) 0 0
\(921\) 67.4044i 2.22105i
\(922\) 0 0
\(923\) 10.8736i 0.357907i
\(924\) 0 0
\(925\) 33.4724 1.10057
\(926\) 0 0
\(927\) 78.5839i 2.58103i
\(928\) 0 0
\(929\) −20.8349 −0.683571 −0.341785 0.939778i \(-0.611032\pi\)
−0.341785 + 0.939778i \(0.611032\pi\)
\(930\) 0 0
\(931\) 6.99546i 0.229267i
\(932\) 0 0
\(933\) 36.9622i 1.21009i
\(934\) 0 0
\(935\) 8.87094i 0.290111i
\(936\) 0 0
\(937\) 0.666786 0.0217829 0.0108915 0.999941i \(-0.496533\pi\)
0.0108915 + 0.999941i \(0.496533\pi\)
\(938\) 0 0
\(939\) −59.0003 −1.92540
\(940\) 0 0
\(941\) 3.15647 0.102898 0.0514491 0.998676i \(-0.483616\pi\)
0.0514491 + 0.998676i \(0.483616\pi\)
\(942\) 0 0
\(943\) 19.6815 0.640917
\(944\) 0 0
\(945\) 0.237276 0.00771858
\(946\) 0 0
\(947\) −37.4996 −1.21857 −0.609286 0.792951i \(-0.708544\pi\)
−0.609286 + 0.792951i \(0.708544\pi\)
\(948\) 0 0
\(949\) 9.25983i 0.300587i
\(950\) 0 0
\(951\) 81.6402i 2.64737i
\(952\) 0 0
\(953\) 1.98085 0.0641661 0.0320830 0.999485i \(-0.489786\pi\)
0.0320830 + 0.999485i \(0.489786\pi\)
\(954\) 0 0
\(955\) −7.51906 −0.243311
\(956\) 0 0
\(957\) 80.0799i 2.58861i
\(958\) 0 0
\(959\) 0.222187i 0.00717479i
\(960\) 0 0
\(961\) −38.9431 −1.25623
\(962\) 0 0
\(963\) 52.5607 1.69374
\(964\) 0 0
\(965\) −1.88859 −0.0607960
\(966\) 0 0
\(967\) 16.3735 0.526538 0.263269 0.964723i \(-0.415199\pi\)
0.263269 + 0.964723i \(0.415199\pi\)
\(968\) 0 0
\(969\) −21.1718 −0.680138
\(970\) 0 0
\(971\) 52.8833 1.69710 0.848552 0.529111i \(-0.177475\pi\)
0.848552 + 0.529111i \(0.177475\pi\)
\(972\) 0 0
\(973\) 0.337263i 0.0108122i
\(974\) 0 0
\(975\) 49.4474i 1.58359i
\(976\) 0 0
\(977\) 49.3272i 1.57812i 0.614318 + 0.789058i \(0.289431\pi\)
−0.614318 + 0.789058i \(0.710569\pi\)
\(978\) 0 0
\(979\) −29.1614 −0.932004
\(980\) 0 0
\(981\) 80.1638i 2.55943i
\(982\) 0 0
\(983\) 43.4940 1.38724 0.693622 0.720339i \(-0.256014\pi\)
0.693622 + 0.720339i \(0.256014\pi\)
\(984\) 0 0
\(985\) 0.431610i 0.0137522i
\(986\) 0 0
\(987\) 1.25065i 0.0398085i
\(988\) 0 0
\(989\) 29.2447i 0.929927i
\(990\) 0 0
\(991\) 22.7711 0.723348 0.361674 0.932305i \(-0.382205\pi\)
0.361674 + 0.932305i \(0.382205\pi\)
\(992\) 0 0
\(993\) 74.3536i 2.35954i
\(994\) 0 0
\(995\) 0.116996i 0.00370901i
\(996\) 0 0
\(997\) −36.6175 −1.15969 −0.579844 0.814728i \(-0.696886\pi\)
−0.579844 + 0.814728i \(0.696886\pi\)
\(998\) 0 0
\(999\) 73.3442i 2.32051i
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 4028.2.c.a.3497.5 82
53.52 even 2 inner 4028.2.c.a.3497.78 yes 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.c.a.3497.5 82 1.1 even 1 trivial
4028.2.c.a.3497.78 yes 82 53.52 even 2 inner