Properties

Label 4140.3.d.c.2161.18
Level $4140$
Weight $3$
Character 4140.2161
Analytic conductor $112.807$
Analytic rank $0$
Dimension $32$
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] = [4140,3,Mod(2161,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.2161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4140.d (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: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2161.18
Character \(\chi\) \(=\) 4140.2161
Dual form 4140.3.d.c.2161.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607i q^{5} -11.5587i q^{7} +O(q^{10})\) \(q+2.23607i q^{5} -11.5587i q^{7} -6.61510i q^{11} -4.35365 q^{13} -15.5392i q^{17} +32.8419i q^{19} +(3.09068 - 22.7914i) q^{23} -5.00000 q^{25} -36.6564 q^{29} +41.7912 q^{31} +25.8461 q^{35} +12.5508i q^{37} -40.5335 q^{41} -83.7540i q^{43} +69.6716 q^{47} -84.6040 q^{49} -23.7139i q^{53} +14.7918 q^{55} -45.4954 q^{59} -99.3377i q^{61} -9.73505i q^{65} +77.0336i q^{67} -52.2590 q^{71} +91.3817 q^{73} -76.4621 q^{77} +11.5142i q^{79} -29.1026i q^{83} +34.7466 q^{85} +31.3082i q^{89} +50.3226i q^{91} -73.4368 q^{95} -110.693i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 11.5587i 1.65125i −0.564222 0.825623i \(-0.690824\pi\)
0.564222 0.825623i \(-0.309176\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.61510i 0.601373i −0.953723 0.300686i \(-0.902784\pi\)
0.953723 0.300686i \(-0.0972157\pi\)
\(12\) 0 0
\(13\) −4.35365 −0.334896 −0.167448 0.985881i \(-0.553553\pi\)
−0.167448 + 0.985881i \(0.553553\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.5392i 0.914068i −0.889449 0.457034i \(-0.848912\pi\)
0.889449 0.457034i \(-0.151088\pi\)
\(18\) 0 0
\(19\) 32.8419i 1.72852i 0.503043 + 0.864262i \(0.332214\pi\)
−0.503043 + 0.864262i \(0.667786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.09068 22.7914i 0.134377 0.990930i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −36.6564 −1.26401 −0.632007 0.774963i \(-0.717769\pi\)
−0.632007 + 0.774963i \(0.717769\pi\)
\(30\) 0 0
\(31\) 41.7912 1.34810 0.674052 0.738684i \(-0.264552\pi\)
0.674052 + 0.738684i \(0.264552\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 25.8461 0.738460
\(36\) 0 0
\(37\) 12.5508i 0.339210i 0.985512 + 0.169605i \(0.0542491\pi\)
−0.985512 + 0.169605i \(0.945751\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −40.5335 −0.988621 −0.494311 0.869285i \(-0.664579\pi\)
−0.494311 + 0.869285i \(0.664579\pi\)
\(42\) 0 0
\(43\) 83.7540i 1.94777i −0.227046 0.973884i \(-0.572907\pi\)
0.227046 0.973884i \(-0.427093\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 69.6716 1.48237 0.741187 0.671299i \(-0.234263\pi\)
0.741187 + 0.671299i \(0.234263\pi\)
\(48\) 0 0
\(49\) −84.6040 −1.72661
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 23.7139i 0.447431i −0.974654 0.223716i \(-0.928181\pi\)
0.974654 0.223716i \(-0.0718187\pi\)
\(54\) 0 0
\(55\) 14.7918 0.268942
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −45.4954 −0.771108 −0.385554 0.922685i \(-0.625990\pi\)
−0.385554 + 0.922685i \(0.625990\pi\)
\(60\) 0 0
\(61\) 99.3377i 1.62849i −0.580524 0.814243i \(-0.697152\pi\)
0.580524 0.814243i \(-0.302848\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.73505i 0.149770i
\(66\) 0 0
\(67\) 77.0336i 1.14975i 0.818240 + 0.574877i \(0.194950\pi\)
−0.818240 + 0.574877i \(0.805050\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −52.2590 −0.736042 −0.368021 0.929818i \(-0.619965\pi\)
−0.368021 + 0.929818i \(0.619965\pi\)
\(72\) 0 0
\(73\) 91.3817 1.25180 0.625902 0.779902i \(-0.284731\pi\)
0.625902 + 0.779902i \(0.284731\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −76.4621 −0.993014
\(78\) 0 0
\(79\) 11.5142i 0.145749i 0.997341 + 0.0728747i \(0.0232173\pi\)
−0.997341 + 0.0728747i \(0.976783\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 29.1026i 0.350633i −0.984512 0.175317i \(-0.943905\pi\)
0.984512 0.175317i \(-0.0560949\pi\)
\(84\) 0 0
\(85\) 34.7466 0.408784
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 31.3082i 0.351777i 0.984410 + 0.175889i \(0.0562798\pi\)
−0.984410 + 0.175889i \(0.943720\pi\)
\(90\) 0 0
\(91\) 50.3226i 0.552996i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −73.4368 −0.773019
\(96\) 0 0
\(97\) 110.693i 1.14116i −0.821241 0.570582i \(-0.806718\pi\)
0.821241 0.570582i \(-0.193282\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −165.111 −1.63476 −0.817379 0.576100i \(-0.804574\pi\)
−0.817379 + 0.576100i \(0.804574\pi\)
\(102\) 0 0
\(103\) 74.1229i 0.719640i 0.933022 + 0.359820i \(0.117162\pi\)
−0.933022 + 0.359820i \(0.882838\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 43.9973i 0.411190i 0.978637 + 0.205595i \(0.0659129\pi\)
−0.978637 + 0.205595i \(0.934087\pi\)
\(108\) 0 0
\(109\) 82.2989i 0.755036i 0.926002 + 0.377518i \(0.123222\pi\)
−0.926002 + 0.377518i \(0.876778\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 112.935i 0.999429i 0.866190 + 0.499715i \(0.166562\pi\)
−0.866190 + 0.499715i \(0.833438\pi\)
\(114\) 0 0
\(115\) 50.9631 + 6.91096i 0.443157 + 0.0600953i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −179.613 −1.50935
\(120\) 0 0
\(121\) 77.2404 0.638351
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −172.032 −1.35458 −0.677291 0.735715i \(-0.736846\pi\)
−0.677291 + 0.735715i \(0.736846\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 49.6869 0.379289 0.189645 0.981853i \(-0.439266\pi\)
0.189645 + 0.981853i \(0.439266\pi\)
\(132\) 0 0
\(133\) 379.611 2.85422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 184.096i 1.34377i 0.740657 + 0.671884i \(0.234515\pi\)
−0.740657 + 0.671884i \(0.765485\pi\)
\(138\) 0 0
\(139\) 61.9715 0.445838 0.222919 0.974837i \(-0.428441\pi\)
0.222919 + 0.974837i \(0.428441\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 28.7998i 0.201397i
\(144\) 0 0
\(145\) 81.9662i 0.565284i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 252.750i 1.69631i −0.529750 0.848154i \(-0.677714\pi\)
0.529750 0.848154i \(-0.322286\pi\)
\(150\) 0 0
\(151\) −167.183 −1.10718 −0.553588 0.832791i \(-0.686742\pi\)
−0.553588 + 0.832791i \(0.686742\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 93.4480i 0.602890i
\(156\) 0 0
\(157\) 238.657i 1.52011i 0.649859 + 0.760055i \(0.274828\pi\)
−0.649859 + 0.760055i \(0.725172\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −263.439 35.7243i −1.63627 0.221890i
\(162\) 0 0
\(163\) 109.935 0.674448 0.337224 0.941424i \(-0.390512\pi\)
0.337224 + 0.941424i \(0.390512\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −193.817 −1.16058 −0.580289 0.814410i \(-0.697061\pi\)
−0.580289 + 0.814410i \(0.697061\pi\)
\(168\) 0 0
\(169\) −150.046 −0.887845
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 182.435 1.05454 0.527268 0.849699i \(-0.323216\pi\)
0.527268 + 0.849699i \(0.323216\pi\)
\(174\) 0 0
\(175\) 57.7936i 0.330249i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −143.275 −0.800419 −0.400209 0.916424i \(-0.631063\pi\)
−0.400209 + 0.916424i \(0.631063\pi\)
\(180\) 0 0
\(181\) 184.195i 1.01765i 0.860869 + 0.508826i \(0.169920\pi\)
−0.860869 + 0.508826i \(0.830080\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −28.0643 −0.151699
\(186\) 0 0
\(187\) −102.793 −0.549696
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 69.5741i 0.364262i 0.983274 + 0.182131i \(0.0582995\pi\)
−0.983274 + 0.182131i \(0.941700\pi\)
\(192\) 0 0
\(193\) 246.167 1.27548 0.637739 0.770252i \(-0.279870\pi\)
0.637739 + 0.770252i \(0.279870\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −329.331 −1.67173 −0.835865 0.548935i \(-0.815034\pi\)
−0.835865 + 0.548935i \(0.815034\pi\)
\(198\) 0 0
\(199\) 63.0077i 0.316622i 0.987389 + 0.158311i \(0.0506048\pi\)
−0.987389 + 0.158311i \(0.949395\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 423.701i 2.08720i
\(204\) 0 0
\(205\) 90.6356i 0.442125i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 217.253 1.03949
\(210\) 0 0
\(211\) 38.9809 0.184744 0.0923719 0.995725i \(-0.470555\pi\)
0.0923719 + 0.995725i \(0.470555\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 187.280 0.871068
\(216\) 0 0
\(217\) 483.053i 2.22605i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 67.6520i 0.306118i
\(222\) 0 0
\(223\) −414.767 −1.85994 −0.929971 0.367632i \(-0.880169\pi\)
−0.929971 + 0.367632i \(0.880169\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 190.504i 0.839224i 0.907703 + 0.419612i \(0.137834\pi\)
−0.907703 + 0.419612i \(0.862166\pi\)
\(228\) 0 0
\(229\) 126.193i 0.551063i 0.961292 + 0.275532i \(0.0888538\pi\)
−0.961292 + 0.275532i \(0.911146\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −374.063 −1.60542 −0.802711 0.596368i \(-0.796610\pi\)
−0.802711 + 0.596368i \(0.796610\pi\)
\(234\) 0 0
\(235\) 155.790i 0.662938i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −112.894 −0.472360 −0.236180 0.971709i \(-0.575896\pi\)
−0.236180 + 0.971709i \(0.575896\pi\)
\(240\) 0 0
\(241\) 280.944i 1.16574i −0.812565 0.582870i \(-0.801929\pi\)
0.812565 0.582870i \(-0.198071\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 189.180i 0.772165i
\(246\) 0 0
\(247\) 142.982i 0.578875i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 389.074i 1.55009i 0.631904 + 0.775047i \(0.282274\pi\)
−0.631904 + 0.775047i \(0.717726\pi\)
\(252\) 0 0
\(253\) −150.767 20.4451i −0.595918 0.0808108i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −129.931 −0.505568 −0.252784 0.967523i \(-0.581346\pi\)
−0.252784 + 0.967523i \(0.581346\pi\)
\(258\) 0 0
\(259\) 145.071 0.560118
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 439.611i 1.67152i −0.549091 0.835762i \(-0.685026\pi\)
0.549091 0.835762i \(-0.314974\pi\)
\(264\) 0 0
\(265\) 53.0258 0.200097
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −291.536 −1.08378 −0.541888 0.840450i \(-0.682290\pi\)
−0.541888 + 0.840450i \(0.682290\pi\)
\(270\) 0 0
\(271\) 95.0531 0.350749 0.175375 0.984502i \(-0.443886\pi\)
0.175375 + 0.984502i \(0.443886\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.0755i 0.120275i
\(276\) 0 0
\(277\) 173.565 0.626587 0.313294 0.949656i \(-0.398568\pi\)
0.313294 + 0.949656i \(0.398568\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 254.496i 0.905681i −0.891591 0.452841i \(-0.850411\pi\)
0.891591 0.452841i \(-0.149589\pi\)
\(282\) 0 0
\(283\) 49.3160i 0.174261i −0.996197 0.0871307i \(-0.972230\pi\)
0.996197 0.0871307i \(-0.0277698\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 468.515i 1.63246i
\(288\) 0 0
\(289\) 47.5347 0.164480
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 425.822i 1.45332i −0.686999 0.726659i \(-0.741072\pi\)
0.686999 0.726659i \(-0.258928\pi\)
\(294\) 0 0
\(295\) 101.731i 0.344850i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −13.4557 + 99.2257i −0.0450024 + 0.331858i
\(300\) 0 0
\(301\) −968.089 −3.21624
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 222.126 0.728281
\(306\) 0 0
\(307\) −308.764 −1.00575 −0.502873 0.864360i \(-0.667724\pi\)
−0.502873 + 0.864360i \(0.667724\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 551.432 1.77309 0.886546 0.462640i \(-0.153098\pi\)
0.886546 + 0.462640i \(0.153098\pi\)
\(312\) 0 0
\(313\) 193.853i 0.619340i −0.950844 0.309670i \(-0.899781\pi\)
0.950844 0.309670i \(-0.100219\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −260.610 −0.822113 −0.411057 0.911610i \(-0.634840\pi\)
−0.411057 + 0.911610i \(0.634840\pi\)
\(318\) 0 0
\(319\) 242.486i 0.760143i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 510.336 1.57999
\(324\) 0 0
\(325\) 21.7682 0.0669792
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 805.314i 2.44776i
\(330\) 0 0
\(331\) −263.090 −0.794834 −0.397417 0.917638i \(-0.630093\pi\)
−0.397417 + 0.917638i \(0.630093\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −172.252 −0.514186
\(336\) 0 0
\(337\) 348.620i 1.03448i 0.855840 + 0.517241i \(0.173041\pi\)
−0.855840 + 0.517241i \(0.826959\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 276.453i 0.810713i
\(342\) 0 0
\(343\) 411.537i 1.19982i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −415.018 −1.19602 −0.598009 0.801489i \(-0.704041\pi\)
−0.598009 + 0.801489i \(0.704041\pi\)
\(348\) 0 0
\(349\) 336.999 0.965612 0.482806 0.875727i \(-0.339618\pi\)
0.482806 + 0.875727i \(0.339618\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −435.958 −1.23501 −0.617505 0.786567i \(-0.711856\pi\)
−0.617505 + 0.786567i \(0.711856\pi\)
\(354\) 0 0
\(355\) 116.855i 0.329168i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 241.813i 0.673575i −0.941581 0.336788i \(-0.890660\pi\)
0.941581 0.336788i \(-0.109340\pi\)
\(360\) 0 0
\(361\) −717.593 −1.98779
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 204.336i 0.559824i
\(366\) 0 0
\(367\) 429.459i 1.17019i −0.810966 0.585094i \(-0.801058\pi\)
0.810966 0.585094i \(-0.198942\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −274.102 −0.738819
\(372\) 0 0
\(373\) 422.643i 1.13309i −0.824030 0.566546i \(-0.808280\pi\)
0.824030 0.566546i \(-0.191720\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 159.589 0.423313
\(378\) 0 0
\(379\) 678.289i 1.78968i 0.446386 + 0.894840i \(0.352711\pi\)
−0.446386 + 0.894840i \(0.647289\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 189.732i 0.495385i −0.968839 0.247692i \(-0.920328\pi\)
0.968839 0.247692i \(-0.0796722\pi\)
\(384\) 0 0
\(385\) 170.974i 0.444090i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 150.076i 0.385800i 0.981218 + 0.192900i \(0.0617894\pi\)
−0.981218 + 0.192900i \(0.938211\pi\)
\(390\) 0 0
\(391\) −354.159 48.0265i −0.905777 0.122830i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −25.7465 −0.0651811
\(396\) 0 0
\(397\) 654.883 1.64958 0.824789 0.565440i \(-0.191294\pi\)
0.824789 + 0.565440i \(0.191294\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 248.712i 0.620230i 0.950699 + 0.310115i \(0.100367\pi\)
−0.950699 + 0.310115i \(0.899633\pi\)
\(402\) 0 0
\(403\) −181.944 −0.451475
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 83.0245 0.203991
\(408\) 0 0
\(409\) −492.502 −1.20416 −0.602081 0.798435i \(-0.705662\pi\)
−0.602081 + 0.798435i \(0.705662\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 525.869i 1.27329i
\(414\) 0 0
\(415\) 65.0753 0.156808
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 131.015i 0.312685i 0.987703 + 0.156343i \(0.0499704\pi\)
−0.987703 + 0.156343i \(0.950030\pi\)
\(420\) 0 0
\(421\) 119.454i 0.283738i 0.989885 + 0.141869i \(0.0453112\pi\)
−0.989885 + 0.141869i \(0.954689\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 77.6958i 0.182814i
\(426\) 0 0
\(427\) −1148.22 −2.68903
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 462.137i 1.07224i −0.844141 0.536121i \(-0.819889\pi\)
0.844141 0.536121i \(-0.180111\pi\)
\(432\) 0 0
\(433\) 88.9988i 0.205540i 0.994705 + 0.102770i \(0.0327706\pi\)
−0.994705 + 0.102770i \(0.967229\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 748.514 + 101.504i 1.71285 + 0.232274i
\(438\) 0 0
\(439\) −690.443 −1.57276 −0.786382 0.617741i \(-0.788048\pi\)
−0.786382 + 0.617741i \(0.788048\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −499.195 −1.12685 −0.563426 0.826167i \(-0.690517\pi\)
−0.563426 + 0.826167i \(0.690517\pi\)
\(444\) 0 0
\(445\) −70.0072 −0.157319
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 219.842 0.489627 0.244813 0.969570i \(-0.421273\pi\)
0.244813 + 0.969570i \(0.421273\pi\)
\(450\) 0 0
\(451\) 268.133i 0.594530i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −112.525 −0.247307
\(456\) 0 0
\(457\) 28.0141i 0.0613001i −0.999530 0.0306501i \(-0.990242\pi\)
0.999530 0.0306501i \(-0.00975774\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −527.172 −1.14354 −0.571771 0.820414i \(-0.693743\pi\)
−0.571771 + 0.820414i \(0.693743\pi\)
\(462\) 0 0
\(463\) −66.6980 −0.144056 −0.0720281 0.997403i \(-0.522947\pi\)
−0.0720281 + 0.997403i \(0.522947\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 249.178i 0.533571i 0.963756 + 0.266785i \(0.0859615\pi\)
−0.963756 + 0.266785i \(0.914039\pi\)
\(468\) 0 0
\(469\) 890.410 1.89853
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −554.041 −1.17133
\(474\) 0 0
\(475\) 164.210i 0.345705i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 669.591i 1.39789i −0.715174 0.698946i \(-0.753653\pi\)
0.715174 0.698946i \(-0.246347\pi\)
\(480\) 0 0
\(481\) 54.6416i 0.113600i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 247.517 0.510344
\(486\) 0 0
\(487\) 42.3713 0.0870047 0.0435024 0.999053i \(-0.486148\pi\)
0.0435024 + 0.999053i \(0.486148\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 65.4718 0.133344 0.0666719 0.997775i \(-0.478762\pi\)
0.0666719 + 0.997775i \(0.478762\pi\)
\(492\) 0 0
\(493\) 569.609i 1.15539i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 604.047i 1.21539i
\(498\) 0 0
\(499\) 490.380 0.982725 0.491363 0.870955i \(-0.336499\pi\)
0.491363 + 0.870955i \(0.336499\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.7682i 0.0611693i 0.999532 + 0.0305847i \(0.00973692\pi\)
−0.999532 + 0.0305847i \(0.990263\pi\)
\(504\) 0 0
\(505\) 369.198i 0.731086i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 64.9076 0.127520 0.0637599 0.997965i \(-0.479691\pi\)
0.0637599 + 0.997965i \(0.479691\pi\)
\(510\) 0 0
\(511\) 1056.26i 2.06704i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −165.744 −0.321833
\(516\) 0 0
\(517\) 460.885i 0.891459i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 468.510i 0.899252i −0.893217 0.449626i \(-0.851557\pi\)
0.893217 0.449626i \(-0.148443\pi\)
\(522\) 0 0
\(523\) 957.668i 1.83111i 0.402198 + 0.915553i \(0.368246\pi\)
−0.402198 + 0.915553i \(0.631754\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 649.400i 1.23226i
\(528\) 0 0
\(529\) −509.895 140.882i −0.963886 0.266317i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 176.468 0.331085
\(534\) 0 0
\(535\) −98.3810 −0.183890
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 559.664i 1.03834i
\(540\) 0 0
\(541\) 822.318 1.52000 0.759998 0.649925i \(-0.225200\pi\)
0.759998 + 0.649925i \(0.225200\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −184.026 −0.337662
\(546\) 0 0
\(547\) −440.483 −0.805270 −0.402635 0.915361i \(-0.631906\pi\)
−0.402635 + 0.915361i \(0.631906\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1203.87i 2.18488i
\(552\) 0 0
\(553\) 133.089 0.240668
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 665.131i 1.19413i 0.802193 + 0.597065i \(0.203667\pi\)
−0.802193 + 0.597065i \(0.796333\pi\)
\(558\) 0 0
\(559\) 364.635i 0.652300i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 517.477i 0.919142i −0.888141 0.459571i \(-0.848003\pi\)
0.888141 0.459571i \(-0.151997\pi\)
\(564\) 0 0
\(565\) −252.531 −0.446958
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 440.146i 0.773542i 0.922176 + 0.386771i \(0.126410\pi\)
−0.922176 + 0.386771i \(0.873590\pi\)
\(570\) 0 0
\(571\) 507.292i 0.888427i −0.895921 0.444214i \(-0.853483\pi\)
0.895921 0.444214i \(-0.146517\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15.4534 + 113.957i −0.0268754 + 0.198186i
\(576\) 0 0
\(577\) −1094.87 −1.89751 −0.948757 0.316005i \(-0.897658\pi\)
−0.948757 + 0.316005i \(0.897658\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −336.388 −0.578982
\(582\) 0 0
\(583\) −156.870 −0.269073
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 811.775 1.38292 0.691461 0.722414i \(-0.256967\pi\)
0.691461 + 0.722414i \(0.256967\pi\)
\(588\) 0 0
\(589\) 1372.50i 2.33023i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 384.656 0.648661 0.324331 0.945944i \(-0.394861\pi\)
0.324331 + 0.945944i \(0.394861\pi\)
\(594\) 0 0
\(595\) 401.626i 0.675002i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −503.582 −0.840704 −0.420352 0.907361i \(-0.638093\pi\)
−0.420352 + 0.907361i \(0.638093\pi\)
\(600\) 0 0
\(601\) 629.639 1.04765 0.523826 0.851825i \(-0.324504\pi\)
0.523826 + 0.851825i \(0.324504\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 172.715i 0.285479i
\(606\) 0 0
\(607\) 528.591 0.870826 0.435413 0.900231i \(-0.356602\pi\)
0.435413 + 0.900231i \(0.356602\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −303.325 −0.496441
\(612\) 0 0
\(613\) 9.16043i 0.0149436i −0.999972 0.00747180i \(-0.997622\pi\)
0.999972 0.00747180i \(-0.00237837\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 332.958i 0.539641i 0.962911 + 0.269820i \(0.0869643\pi\)
−0.962911 + 0.269820i \(0.913036\pi\)
\(618\) 0 0
\(619\) 367.054i 0.592979i −0.955036 0.296489i \(-0.904184\pi\)
0.955036 0.296489i \(-0.0958159\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 361.882 0.580870
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 195.028 0.310061
\(630\) 0 0
\(631\) 818.463i 1.29709i 0.761177 + 0.648544i \(0.224622\pi\)
−0.761177 + 0.648544i \(0.775378\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 384.675i 0.605787i
\(636\) 0 0
\(637\) 368.336 0.578236
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 506.746i 0.790555i 0.918562 + 0.395277i \(0.129352\pi\)
−0.918562 + 0.395277i \(0.870648\pi\)
\(642\) 0 0
\(643\) 67.4223i 0.104856i −0.998625 0.0524279i \(-0.983304\pi\)
0.998625 0.0524279i \(-0.0166960\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −300.396 −0.464290 −0.232145 0.972681i \(-0.574574\pi\)
−0.232145 + 0.972681i \(0.574574\pi\)
\(648\) 0 0
\(649\) 300.957i 0.463724i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 129.994 0.199072 0.0995360 0.995034i \(-0.468264\pi\)
0.0995360 + 0.995034i \(0.468264\pi\)
\(654\) 0 0
\(655\) 111.103i 0.169623i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 375.123i 0.569231i 0.958642 + 0.284615i \(0.0918659\pi\)
−0.958642 + 0.284615i \(0.908134\pi\)
\(660\) 0 0
\(661\) 601.794i 0.910429i −0.890382 0.455215i \(-0.849563\pi\)
0.890382 0.455215i \(-0.150437\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 848.836i 1.27644i
\(666\) 0 0
\(667\) −113.293 + 835.450i −0.169855 + 1.25255i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −657.129 −0.979328
\(672\) 0 0
\(673\) −442.603 −0.657657 −0.328829 0.944390i \(-0.606654\pi\)
−0.328829 + 0.944390i \(0.606654\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 303.051i 0.447638i −0.974631 0.223819i \(-0.928148\pi\)
0.974631 0.223819i \(-0.0718524\pi\)
\(678\) 0 0
\(679\) −1279.47 −1.88434
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −242.465 −0.355000 −0.177500 0.984121i \(-0.556801\pi\)
−0.177500 + 0.984121i \(0.556801\pi\)
\(684\) 0 0
\(685\) −411.652 −0.600951
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 103.242i 0.149843i
\(690\) 0 0
\(691\) 409.395 0.592468 0.296234 0.955115i \(-0.404269\pi\)
0.296234 + 0.955115i \(0.404269\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 138.573i 0.199385i
\(696\) 0 0
\(697\) 629.856i 0.903667i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1359.99i 1.94007i 0.242963 + 0.970036i \(0.421881\pi\)
−0.242963 + 0.970036i \(0.578119\pi\)
\(702\) 0 0
\(703\) −412.191 −0.586332
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1908.47i 2.69939i
\(708\) 0 0
\(709\) 442.186i 0.623676i 0.950135 + 0.311838i \(0.100945\pi\)
−0.950135 + 0.311838i \(0.899055\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 129.163 952.480i 0.181154 1.33588i
\(714\) 0 0
\(715\) −64.3983 −0.0900676
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −415.632 −0.578070 −0.289035 0.957319i \(-0.593334\pi\)
−0.289035 + 0.957319i \(0.593334\pi\)
\(720\) 0 0
\(721\) 856.766 1.18830
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 183.282 0.252803
\(726\) 0 0
\(727\) 161.744i 0.222482i −0.993793 0.111241i \(-0.964517\pi\)
0.993793 0.111241i \(-0.0354826\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1301.47 −1.78039
\(732\) 0 0
\(733\) 272.499i 0.371758i 0.982573 + 0.185879i \(0.0595133\pi\)
−0.982573 + 0.185879i \(0.940487\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 509.585 0.691431
\(738\) 0 0
\(739\) −303.297 −0.410416 −0.205208 0.978718i \(-0.565787\pi\)
−0.205208 + 0.978718i \(0.565787\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 706.910i 0.951426i 0.879601 + 0.475713i \(0.157810\pi\)
−0.879601 + 0.475713i \(0.842190\pi\)
\(744\) 0 0
\(745\) 565.166 0.758612
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 508.553 0.678976
\(750\) 0 0
\(751\) 518.357i 0.690222i 0.938562 + 0.345111i \(0.112159\pi\)
−0.938562 + 0.345111i \(0.887841\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 373.834i 0.495144i
\(756\) 0 0
\(757\) 773.832i 1.02224i −0.859511 0.511118i \(-0.829232\pi\)
0.859511 0.511118i \(-0.170768\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 141.335 0.185723 0.0928615 0.995679i \(-0.470399\pi\)
0.0928615 + 0.995679i \(0.470399\pi\)
\(762\) 0 0
\(763\) 951.270 1.24675
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 198.071 0.258241
\(768\) 0 0
\(769\) 192.492i 0.250314i 0.992137 + 0.125157i \(0.0399435\pi\)
−0.992137 + 0.125157i \(0.960057\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1394.17i 1.80359i −0.432169 0.901793i \(-0.642252\pi\)
0.432169 0.901793i \(-0.357748\pi\)
\(774\) 0 0
\(775\) −208.956 −0.269621
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1331.20i 1.70885i
\(780\) 0 0
\(781\) 345.698i 0.442635i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −533.654 −0.679814
\(786\) 0 0
\(787\) 966.770i 1.22842i −0.789141 0.614212i \(-0.789474\pi\)
0.789141 0.614212i \(-0.210526\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1305.39 1.65030
\(792\) 0 0
\(793\) 432.481i 0.545374i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 422.069i 0.529572i 0.964307 + 0.264786i \(0.0853014\pi\)
−0.964307 + 0.264786i \(0.914699\pi\)
\(798\) 0 0
\(799\) 1082.64i 1.35499i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 604.499i 0.752801i
\(804\) 0 0
\(805\) 79.8819 589.068i 0.0992322 0.731762i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1002.82 −1.23958 −0.619791 0.784767i \(-0.712783\pi\)
−0.619791 + 0.784767i \(0.712783\pi\)
\(810\) 0 0
\(811\) −879.037 −1.08389 −0.541946 0.840413i \(-0.682312\pi\)
−0.541946 + 0.840413i \(0.682312\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 245.822i 0.301622i
\(816\) 0 0
\(817\) 2750.64 3.36676
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −147.227 −0.179327 −0.0896633 0.995972i \(-0.528579\pi\)
−0.0896633 + 0.995972i \(0.528579\pi\)
\(822\) 0 0
\(823\) 893.347 1.08548 0.542738 0.839902i \(-0.317388\pi\)
0.542738 + 0.839902i \(0.317388\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1040.81i 1.25854i 0.777186 + 0.629270i \(0.216646\pi\)
−0.777186 + 0.629270i \(0.783354\pi\)
\(828\) 0 0
\(829\) −284.524 −0.343214 −0.171607 0.985166i \(-0.554896\pi\)
−0.171607 + 0.985166i \(0.554896\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1314.68i 1.57824i
\(834\) 0 0
\(835\) 433.387i 0.519027i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 369.349i 0.440225i 0.975474 + 0.220113i \(0.0706425\pi\)
−0.975474 + 0.220113i \(0.929358\pi\)
\(840\) 0 0
\(841\) 502.691 0.597730
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 335.513i 0.397056i
\(846\) 0 0
\(847\) 892.801i 1.05407i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 286.049 + 38.7903i 0.336133 + 0.0455820i
\(852\) 0 0
\(853\) −1288.20 −1.51020 −0.755100 0.655610i \(-0.772412\pi\)
−0.755100 + 0.655610i \(0.772412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1442.40 1.68308 0.841538 0.540198i \(-0.181651\pi\)
0.841538 + 0.540198i \(0.181651\pi\)
\(858\) 0 0
\(859\) 997.689 1.16145 0.580727 0.814098i \(-0.302768\pi\)
0.580727 + 0.814098i \(0.302768\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −92.2907 −0.106942 −0.0534709 0.998569i \(-0.517028\pi\)
−0.0534709 + 0.998569i \(0.517028\pi\)
\(864\) 0 0
\(865\) 407.937i 0.471603i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 76.1676 0.0876497
\(870\) 0 0
\(871\) 335.377i 0.385048i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −129.230 −0.147692
\(876\) 0 0
\(877\) 270.272 0.308178 0.154089 0.988057i \(-0.450756\pi\)
0.154089 + 0.988057i \(0.450756\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 950.026i 1.07835i −0.842194 0.539175i \(-0.818736\pi\)
0.842194 0.539175i \(-0.181264\pi\)
\(882\) 0 0
\(883\) 101.560 0.115017 0.0575085 0.998345i \(-0.481684\pi\)
0.0575085 + 0.998345i \(0.481684\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1486.55 −1.67593 −0.837964 0.545725i \(-0.816254\pi\)
−0.837964 + 0.545725i \(0.816254\pi\)
\(888\) 0 0
\(889\) 1988.47i 2.23675i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2288.15i 2.56232i
\(894\) 0 0
\(895\) 320.373i 0.357958i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1531.92 −1.70402
\(900\) 0 0
\(901\) −368.493 −0.408983
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −411.873 −0.455108
\(906\) 0 0
\(907\) 104.880i 0.115634i −0.998327 0.0578169i \(-0.981586\pi\)
0.998327 0.0578169i \(-0.0184140\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 435.963i 0.478555i −0.970951 0.239277i \(-0.923089\pi\)
0.970951 0.239277i \(-0.0769105\pi\)
\(912\) 0 0
\(913\) −192.516 −0.210861
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 574.317i 0.626300i
\(918\) 0 0
\(919\) 224.737i 0.244546i −0.992497 0.122273i \(-0.960982\pi\)
0.992497 0.122273i \(-0.0390183\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 227.517 0.246497
\(924\) 0 0
\(925\) 62.7538i 0.0678419i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1701.11 1.83112 0.915559 0.402183i \(-0.131748\pi\)
0.915559 + 0.402183i \(0.131748\pi\)
\(930\) 0 0
\(931\) 2778.56i 2.98449i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 229.852i 0.245831i
\(936\) 0 0
\(937\) 775.110i 0.827225i 0.910453 + 0.413612i \(0.135733\pi\)
−0.910453 + 0.413612i \(0.864267\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1193.83i 1.26868i −0.773052 0.634342i \(-0.781271\pi\)
0.773052 0.634342i \(-0.218729\pi\)
\(942\) 0 0
\(943\) −125.276 + 923.815i −0.132848 + 0.979655i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 430.109 0.454180 0.227090 0.973874i \(-0.427079\pi\)
0.227090 + 0.973874i \(0.427079\pi\)
\(948\) 0 0
\(949\) −397.844 −0.419224
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1390.03i 1.45859i −0.684200 0.729294i \(-0.739849\pi\)
0.684200 0.729294i \(-0.260151\pi\)
\(954\) 0 0
\(955\) −155.572 −0.162903
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2127.92 2.21889
\(960\) 0 0
\(961\) 785.507 0.817385
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 550.447i 0.570411i
\(966\) 0 0
\(967\) −824.460 −0.852595 −0.426298 0.904583i \(-0.640182\pi\)
−0.426298 + 0.904583i \(0.640182\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 614.100i 0.632441i 0.948686 + 0.316220i \(0.102414\pi\)
−0.948686 + 0.316220i \(0.897586\pi\)
\(972\) 0 0
\(973\) 716.311i 0.736188i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1170.50i 1.19806i −0.800728 0.599028i \(-0.795554\pi\)
0.800728 0.599028i \(-0.204446\pi\)
\(978\) 0 0
\(979\) 207.107 0.211549
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1798.20i 1.82930i −0.404244 0.914651i \(-0.632465\pi\)
0.404244 0.914651i \(-0.367535\pi\)
\(984\) 0 0
\(985\) 736.406i 0.747621i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1908.87 258.857i −1.93010 0.261736i
\(990\) 0 0
\(991\) 656.246 0.662206 0.331103 0.943595i \(-0.392579\pi\)
0.331103 + 0.943595i \(0.392579\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −140.890 −0.141598
\(996\) 0 0
\(997\) 1483.74 1.48821 0.744103 0.668065i \(-0.232877\pi\)
0.744103 + 0.668065i \(0.232877\pi\)
\(998\) 0 0
\(999\) 0 0
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 4140.3.d.c.2161.18 32
3.2 odd 2 1380.3.d.a.781.1 32
23.22 odd 2 inner 4140.3.d.c.2161.15 32
69.68 even 2 1380.3.d.a.781.16 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.3.d.a.781.1 32 3.2 odd 2
1380.3.d.a.781.16 yes 32 69.68 even 2
4140.3.d.c.2161.15 32 23.22 odd 2 inner
4140.3.d.c.2161.18 32 1.1 even 1 trivial