Properties

Label 4100.2.d.c.1149.7
Level $4100$
Weight $2$
Character 4100.1149
Analytic conductor $32.739$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 14 x^{4} - 14 x^{3} + 8 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 164)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1149.7
Root \(-0.187509 - 0.187509i\) of defining polynomial
Character \(\chi\) \(=\) 4100.1149
Dual form 4100.2.d.c.1149.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.92968i q^{3} -2.77840i q^{7} -5.58303 q^{9} +O(q^{10})\) \(q+2.92968i q^{3} -2.77840i q^{7} -5.58303 q^{9} +2.02837 q^{11} -3.10932i q^{13} +7.91609i q^{17} -2.17964 q^{19} +8.13983 q^{21} -4.75004i q^{23} -7.56744i q^{27} -3.85936 q^{29} -10.6661 q^{31} +5.94246i q^{33} -6.58303i q^{37} +9.10932 q^{39} -1.00000 q^{41} +4.80677i q^{43} -4.03900i q^{47} -0.719526 q^{49} -23.1916 q^{51} -1.94327i q^{53} -6.38566i q^{57} -4.75004 q^{59} +2.89068 q^{61} +15.5119i q^{63} -5.97163i q^{67} +13.9161 q^{69} +14.4026 q^{71} -9.24916i q^{73} -5.63562i q^{77} +0.320282 q^{79} +5.42111 q^{81} -3.69745i q^{83} -11.3067i q^{87} -8.96868 q^{89} -8.63895 q^{91} -31.2484i q^{93} -17.8322i q^{97} -11.3244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 8q^{11} - 12q^{19} + 8q^{29} - 16q^{31} + 48q^{39} - 8q^{41} - 32q^{49} - 8q^{51} - 24q^{59} + 48q^{61} + 56q^{69} - 4q^{71} + 36q^{79} + 56q^{81} - 8q^{89} + 72q^{91} - 116q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 2.92968i 1.69145i 0.533618 + 0.845726i \(0.320832\pi\)
−0.533618 + 0.845726i \(0.679168\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.77840i − 1.05014i −0.851060 0.525069i \(-0.824040\pi\)
0.851060 0.525069i \(-0.175960\pi\)
\(8\) 0 0
\(9\) −5.58303 −1.86101
\(10\) 0 0
\(11\) 2.02837 0.611575 0.305788 0.952100i \(-0.401080\pi\)
0.305788 + 0.952100i \(0.401080\pi\)
\(12\) 0 0
\(13\) − 3.10932i − 0.862371i −0.902263 0.431186i \(-0.858095\pi\)
0.902263 0.431186i \(-0.141905\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.91609i 1.91993i 0.280112 + 0.959967i \(0.409628\pi\)
−0.280112 + 0.959967i \(0.590372\pi\)
\(18\) 0 0
\(19\) −2.17964 −0.500044 −0.250022 0.968240i \(-0.580438\pi\)
−0.250022 + 0.968240i \(0.580438\pi\)
\(20\) 0 0
\(21\) 8.13983 1.77626
\(22\) 0 0
\(23\) − 4.75004i − 0.990451i −0.868764 0.495226i \(-0.835085\pi\)
0.868764 0.495226i \(-0.164915\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 7.56744i − 1.45636i
\(28\) 0 0
\(29\) −3.85936 −0.716665 −0.358333 0.933594i \(-0.616655\pi\)
−0.358333 + 0.933594i \(0.616655\pi\)
\(30\) 0 0
\(31\) −10.6661 −1.91569 −0.957847 0.287280i \(-0.907249\pi\)
−0.957847 + 0.287280i \(0.907249\pi\)
\(32\) 0 0
\(33\) 5.94246i 1.03445i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.58303i − 1.08224i −0.840944 0.541122i \(-0.818000\pi\)
0.840944 0.541122i \(-0.182000\pi\)
\(38\) 0 0
\(39\) 9.10932 1.45866
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 4.80677i 0.733025i 0.930413 + 0.366513i \(0.119448\pi\)
−0.930413 + 0.366513i \(0.880552\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.03900i − 0.589149i −0.955628 0.294575i \(-0.904822\pi\)
0.955628 0.294575i \(-0.0951779\pi\)
\(48\) 0 0
\(49\) −0.719526 −0.102789
\(50\) 0 0
\(51\) −23.1916 −3.24748
\(52\) 0 0
\(53\) − 1.94327i − 0.266928i −0.991054 0.133464i \(-0.957390\pi\)
0.991054 0.133464i \(-0.0426101\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.38566i − 0.845801i
\(58\) 0 0
\(59\) −4.75004 −0.618402 −0.309201 0.950997i \(-0.600062\pi\)
−0.309201 + 0.950997i \(0.600062\pi\)
\(60\) 0 0
\(61\) 2.89068 0.370113 0.185057 0.982728i \(-0.440753\pi\)
0.185057 + 0.982728i \(0.440753\pi\)
\(62\) 0 0
\(63\) 15.5119i 1.95432i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.97163i − 0.729551i −0.931095 0.364776i \(-0.881146\pi\)
0.931095 0.364776i \(-0.118854\pi\)
\(68\) 0 0
\(69\) 13.9161 1.67530
\(70\) 0 0
\(71\) 14.4026 1.70927 0.854636 0.519228i \(-0.173780\pi\)
0.854636 + 0.519228i \(0.173780\pi\)
\(72\) 0 0
\(73\) − 9.24916i − 1.08253i −0.840851 0.541266i \(-0.817945\pi\)
0.840851 0.541266i \(-0.182055\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.63562i − 0.642238i
\(78\) 0 0
\(79\) 0.320282 0.0360345 0.0180173 0.999838i \(-0.494265\pi\)
0.0180173 + 0.999838i \(0.494265\pi\)
\(80\) 0 0
\(81\) 5.42111 0.602346
\(82\) 0 0
\(83\) − 3.69745i − 0.405847i −0.979195 0.202924i \(-0.934956\pi\)
0.979195 0.202924i \(-0.0650443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 11.3067i − 1.21220i
\(88\) 0 0
\(89\) −8.96868 −0.950679 −0.475339 0.879803i \(-0.657675\pi\)
−0.475339 + 0.879803i \(0.657675\pi\)
\(90\) 0 0
\(91\) −8.63895 −0.905608
\(92\) 0 0
\(93\) − 31.2484i − 3.24030i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 17.8322i − 1.81058i −0.424790 0.905292i \(-0.639652\pi\)
0.424790 0.905292i \(-0.360348\pi\)
\(98\) 0 0
\(99\) −11.3244 −1.13815
\(100\) 0 0
\(101\) −13.0254 −1.29608 −0.648039 0.761607i \(-0.724410\pi\)
−0.648039 + 0.761607i \(0.724410\pi\)
\(102\) 0 0
\(103\) − 8.05673i − 0.793853i −0.917850 0.396927i \(-0.870077\pi\)
0.917850 0.396927i \(-0.129923\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.41602i 0.426912i 0.976953 + 0.213456i \(0.0684721\pi\)
−0.976953 + 0.213456i \(0.931528\pi\)
\(108\) 0 0
\(109\) −18.5255 −1.77442 −0.887210 0.461366i \(-0.847360\pi\)
−0.887210 + 0.461366i \(0.847360\pi\)
\(110\) 0 0
\(111\) 19.2862 1.83056
\(112\) 0 0
\(113\) − 8.08310i − 0.760394i −0.924905 0.380197i \(-0.875856\pi\)
0.924905 0.380197i \(-0.124144\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 17.3594i 1.60488i
\(118\) 0 0
\(119\) 21.9941 2.01620
\(120\) 0 0
\(121\) −6.88573 −0.625976
\(122\) 0 0
\(123\) − 2.92968i − 0.264160i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.44748i − 0.394650i −0.980338 0.197325i \(-0.936775\pi\)
0.980338 0.197325i \(-0.0632255\pi\)
\(128\) 0 0
\(129\) −14.0823 −1.23988
\(130\) 0 0
\(131\) 9.85936 0.861416 0.430708 0.902491i \(-0.358264\pi\)
0.430708 + 0.902491i \(0.358264\pi\)
\(132\) 0 0
\(133\) 6.05593i 0.525115i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.30255i 0.196720i 0.995151 + 0.0983602i \(0.0313597\pi\)
−0.995151 + 0.0983602i \(0.968640\pi\)
\(138\) 0 0
\(139\) −5.16605 −0.438179 −0.219090 0.975705i \(-0.570309\pi\)
−0.219090 + 0.975705i \(0.570309\pi\)
\(140\) 0 0
\(141\) 11.8330 0.996517
\(142\) 0 0
\(143\) − 6.30684i − 0.527405i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.10798i − 0.173863i
\(148\) 0 0
\(149\) −3.85936 −0.316171 −0.158086 0.987425i \(-0.550532\pi\)
−0.158086 + 0.987425i \(0.550532\pi\)
\(150\) 0 0
\(151\) −4.71103 −0.383379 −0.191689 0.981456i \(-0.561397\pi\)
−0.191689 + 0.981456i \(0.561397\pi\)
\(152\) 0 0
\(153\) − 44.1958i − 3.57302i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 8.30684i − 0.662958i −0.943463 0.331479i \(-0.892452\pi\)
0.943463 0.331479i \(-0.107548\pi\)
\(158\) 0 0
\(159\) 5.69316 0.451497
\(160\) 0 0
\(161\) −13.1975 −1.04011
\(162\) 0 0
\(163\) − 13.6348i − 1.06796i −0.845497 0.533981i \(-0.820696\pi\)
0.845497 0.533981i \(-0.179304\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 9.97163i − 0.771628i −0.922577 0.385814i \(-0.873921\pi\)
0.922577 0.385814i \(-0.126079\pi\)
\(168\) 0 0
\(169\) 3.33211 0.256316
\(170\) 0 0
\(171\) 12.1690 0.930587
\(172\) 0 0
\(173\) 9.11361i 0.692895i 0.938069 + 0.346448i \(0.112612\pi\)
−0.938069 + 0.346448i \(0.887388\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 13.9161i − 1.04600i
\(178\) 0 0
\(179\) 12.9403 0.967205 0.483602 0.875288i \(-0.339328\pi\)
0.483602 + 0.875288i \(0.339328\pi\)
\(180\) 0 0
\(181\) 8.16191 0.606670 0.303335 0.952884i \(-0.401900\pi\)
0.303335 + 0.952884i \(0.401900\pi\)
\(182\) 0 0
\(183\) 8.46876i 0.626029i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.0567i 1.17418i
\(188\) 0 0
\(189\) −21.0254 −1.52937
\(190\) 0 0
\(191\) 4.52829 0.327656 0.163828 0.986489i \(-0.447616\pi\)
0.163828 + 0.986489i \(0.447616\pi\)
\(192\) 0 0
\(193\) − 14.1662i − 1.01971i −0.860262 0.509853i \(-0.829700\pi\)
0.860262 0.509853i \(-0.170300\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.5042i 0.890888i 0.895310 + 0.445444i \(0.146954\pi\)
−0.895310 + 0.445444i \(0.853046\pi\)
\(198\) 0 0
\(199\) 14.5853 1.03393 0.516963 0.856008i \(-0.327062\pi\)
0.516963 + 0.856008i \(0.327062\pi\)
\(200\) 0 0
\(201\) 17.4950 1.23400
\(202\) 0 0
\(203\) 10.7229i 0.752597i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 26.5196i 1.84324i
\(208\) 0 0
\(209\) −4.42111 −0.305815
\(210\) 0 0
\(211\) −26.3860 −1.81649 −0.908245 0.418439i \(-0.862577\pi\)
−0.908245 + 0.418439i \(0.862577\pi\)
\(212\) 0 0
\(213\) 42.1950i 2.89115i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 29.6348i 2.01174i
\(218\) 0 0
\(219\) 27.0971 1.83105
\(220\) 0 0
\(221\) 24.6137 1.65570
\(222\) 0 0
\(223\) − 1.89482i − 0.126886i −0.997985 0.0634432i \(-0.979792\pi\)
0.997985 0.0634432i \(-0.0202082\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.9805i 1.65801i 0.559238 + 0.829007i \(0.311094\pi\)
−0.559238 + 0.829007i \(0.688906\pi\)
\(228\) 0 0
\(229\) 2.52549 0.166889 0.0834446 0.996512i \(-0.473408\pi\)
0.0834446 + 0.996512i \(0.473408\pi\)
\(230\) 0 0
\(231\) 16.5106 1.08632
\(232\) 0 0
\(233\) 9.24835i 0.605880i 0.953010 + 0.302940i \(0.0979681\pi\)
−0.953010 + 0.302940i \(0.902032\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.938323i 0.0609507i
\(238\) 0 0
\(239\) −2.34746 −0.151844 −0.0759222 0.997114i \(-0.524190\pi\)
−0.0759222 + 0.997114i \(0.524190\pi\)
\(240\) 0 0
\(241\) 2.89068 0.186205 0.0931024 0.995657i \(-0.470322\pi\)
0.0931024 + 0.995657i \(0.470322\pi\)
\(242\) 0 0
\(243\) − 6.82021i − 0.437516i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.77721i 0.431224i
\(248\) 0 0
\(249\) 10.8323 0.686471
\(250\) 0 0
\(251\) 24.9730 1.57628 0.788140 0.615496i \(-0.211044\pi\)
0.788140 + 0.615496i \(0.211044\pi\)
\(252\) 0 0
\(253\) − 9.63481i − 0.605736i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.7500i − 0.670569i −0.942117 0.335284i \(-0.891168\pi\)
0.942117 0.335284i \(-0.108832\pi\)
\(258\) 0 0
\(259\) −18.2903 −1.13650
\(260\) 0 0
\(261\) 21.5469 1.33372
\(262\) 0 0
\(263\) − 4.90250i − 0.302301i −0.988511 0.151151i \(-0.951702\pi\)
0.988511 0.151151i \(-0.0482979\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 26.2754i − 1.60803i
\(268\) 0 0
\(269\) −3.32797 −0.202910 −0.101455 0.994840i \(-0.532350\pi\)
−0.101455 + 0.994840i \(0.532350\pi\)
\(270\) 0 0
\(271\) 4.58222 0.278350 0.139175 0.990268i \(-0.455555\pi\)
0.139175 + 0.990268i \(0.455555\pi\)
\(272\) 0 0
\(273\) − 25.3094i − 1.53179i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 24.2178i 1.45511i 0.686050 + 0.727555i \(0.259343\pi\)
−0.686050 + 0.727555i \(0.740657\pi\)
\(278\) 0 0
\(279\) 59.5493 3.56512
\(280\) 0 0
\(281\) 6.61369 0.394540 0.197270 0.980349i \(-0.436792\pi\)
0.197270 + 0.980349i \(0.436792\pi\)
\(282\) 0 0
\(283\) 5.16605i 0.307090i 0.988142 + 0.153545i \(0.0490690\pi\)
−0.988142 + 0.153545i \(0.950931\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.77840i 0.164004i
\(288\) 0 0
\(289\) −45.6645 −2.68615
\(290\) 0 0
\(291\) 52.2426 3.06252
\(292\) 0 0
\(293\) − 10.2458i − 0.598567i −0.954164 0.299284i \(-0.903252\pi\)
0.954164 0.299284i \(-0.0967477\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 15.3495i − 0.890671i
\(298\) 0 0
\(299\) −14.7694 −0.854137
\(300\) 0 0
\(301\) 13.3551 0.769778
\(302\) 0 0
\(303\) − 38.1603i − 2.19225i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 18.2754i 1.04303i 0.853242 + 0.521515i \(0.174633\pi\)
−0.853242 + 0.521515i \(0.825367\pi\)
\(308\) 0 0
\(309\) 23.6036 1.34276
\(310\) 0 0
\(311\) 4.50303 0.255343 0.127672 0.991816i \(-0.459250\pi\)
0.127672 + 0.991816i \(0.459250\pi\)
\(312\) 0 0
\(313\) 6.56095i 0.370847i 0.982659 + 0.185423i \(0.0593656\pi\)
−0.982659 + 0.185423i \(0.940634\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 12.8363i − 0.720960i −0.932767 0.360480i \(-0.882613\pi\)
0.932767 0.360480i \(-0.117387\pi\)
\(318\) 0 0
\(319\) −7.82820 −0.438295
\(320\) 0 0
\(321\) −12.9375 −0.722102
\(322\) 0 0
\(323\) − 17.2543i − 0.960052i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 54.2738i − 3.00135i
\(328\) 0 0
\(329\) −11.2220 −0.618688
\(330\) 0 0
\(331\) −8.99705 −0.494523 −0.247261 0.968949i \(-0.579531\pi\)
−0.247261 + 0.968949i \(0.579531\pi\)
\(332\) 0 0
\(333\) 36.7532i 2.01406i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 35.0502i 1.90930i 0.297723 + 0.954652i \(0.403773\pi\)
−0.297723 + 0.954652i \(0.596227\pi\)
\(338\) 0 0
\(339\) 23.6809 1.28617
\(340\) 0 0
\(341\) −21.6348 −1.17159
\(342\) 0 0
\(343\) − 17.4497i − 0.942195i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.8982i − 1.01451i −0.861796 0.507255i \(-0.830660\pi\)
0.861796 0.507255i \(-0.169340\pi\)
\(348\) 0 0
\(349\) 23.1653 1.24001 0.620004 0.784599i \(-0.287131\pi\)
0.620004 + 0.784599i \(0.287131\pi\)
\(350\) 0 0
\(351\) −23.5296 −1.25592
\(352\) 0 0
\(353\) − 1.55171i − 0.0825893i −0.999147 0.0412946i \(-0.986852\pi\)
0.999147 0.0412946i \(-0.0131482\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 64.4357i 3.41030i
\(358\) 0 0
\(359\) 4.28128 0.225957 0.112979 0.993597i \(-0.463961\pi\)
0.112979 + 0.993597i \(0.463961\pi\)
\(360\) 0 0
\(361\) −14.2492 −0.749956
\(362\) 0 0
\(363\) − 20.1730i − 1.05881i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 15.3067i 0.799003i 0.916733 + 0.399501i \(0.130817\pi\)
−0.916733 + 0.399501i \(0.869183\pi\)
\(368\) 0 0
\(369\) 5.58303 0.290641
\(370\) 0 0
\(371\) −5.39918 −0.280312
\(372\) 0 0
\(373\) − 5.39489i − 0.279337i −0.990198 0.139668i \(-0.955396\pi\)
0.990198 0.139668i \(-0.0446037\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −26.8009 −1.37667 −0.688334 0.725394i \(-0.741658\pi\)
−0.688334 + 0.725394i \(0.741658\pi\)
\(380\) 0 0
\(381\) 13.0297 0.667532
\(382\) 0 0
\(383\) 21.6675i 1.10716i 0.832797 + 0.553578i \(0.186738\pi\)
−0.832797 + 0.553578i \(0.813262\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 26.8363i − 1.36417i
\(388\) 0 0
\(389\) −25.7187 −1.30399 −0.651995 0.758223i \(-0.726068\pi\)
−0.651995 + 0.758223i \(0.726068\pi\)
\(390\) 0 0
\(391\) 37.6017 1.90160
\(392\) 0 0
\(393\) 28.8848i 1.45704i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 28.1603i − 1.41333i −0.707551 0.706663i \(-0.750200\pi\)
0.707551 0.706663i \(-0.249800\pi\)
\(398\) 0 0
\(399\) −17.7419 −0.888208
\(400\) 0 0
\(401\) −10.7178 −0.535220 −0.267610 0.963527i \(-0.586234\pi\)
−0.267610 + 0.963527i \(0.586234\pi\)
\(402\) 0 0
\(403\) 33.1644i 1.65204i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 13.3528i − 0.661873i
\(408\) 0 0
\(409\) −22.7492 −1.12488 −0.562439 0.826839i \(-0.690137\pi\)
−0.562439 + 0.826839i \(0.690137\pi\)
\(410\) 0 0
\(411\) −6.74575 −0.332743
\(412\) 0 0
\(413\) 13.1975i 0.649408i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 15.1349i − 0.741159i
\(418\) 0 0
\(419\) 7.02542 0.343214 0.171607 0.985165i \(-0.445104\pi\)
0.171607 + 0.985165i \(0.445104\pi\)
\(420\) 0 0
\(421\) 16.5992 0.808996 0.404498 0.914539i \(-0.367446\pi\)
0.404498 + 0.914539i \(0.367446\pi\)
\(422\) 0 0
\(423\) 22.5499i 1.09641i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.03147i − 0.388670i
\(428\) 0 0
\(429\) 18.4770 0.892080
\(430\) 0 0
\(431\) −10.7229 −0.516502 −0.258251 0.966078i \(-0.583146\pi\)
−0.258251 + 0.966078i \(0.583146\pi\)
\(432\) 0 0
\(433\) 31.1644i 1.49767i 0.662758 + 0.748834i \(0.269386\pi\)
−0.662758 + 0.748834i \(0.730614\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.3534i 0.495270i
\(438\) 0 0
\(439\) 34.7981 1.66082 0.830411 0.557152i \(-0.188106\pi\)
0.830411 + 0.557152i \(0.188106\pi\)
\(440\) 0 0
\(441\) 4.01714 0.191292
\(442\) 0 0
\(443\) − 25.8806i − 1.22963i −0.788673 0.614813i \(-0.789231\pi\)
0.788673 0.614813i \(-0.210769\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 11.3067i − 0.534788i
\(448\) 0 0
\(449\) −27.2231 −1.28474 −0.642368 0.766396i \(-0.722048\pi\)
−0.642368 + 0.766396i \(0.722048\pi\)
\(450\) 0 0
\(451\) −2.02837 −0.0955120
\(452\) 0 0
\(453\) − 13.8018i − 0.648466i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.19767i 0.289915i 0.989438 + 0.144957i \(0.0463045\pi\)
−0.989438 + 0.144957i \(0.953695\pi\)
\(458\) 0 0
\(459\) 59.9046 2.79611
\(460\) 0 0
\(461\) −10.9170 −0.508458 −0.254229 0.967144i \(-0.581822\pi\)
−0.254229 + 0.967144i \(0.581822\pi\)
\(462\) 0 0
\(463\) 4.07047i 0.189171i 0.995517 + 0.0945854i \(0.0301525\pi\)
−0.995517 + 0.0945854i \(0.969847\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.63657i 0.399653i 0.979831 + 0.199826i \(0.0640379\pi\)
−0.979831 + 0.199826i \(0.935962\pi\)
\(468\) 0 0
\(469\) −16.5916 −0.766129
\(470\) 0 0
\(471\) 24.3364 1.12136
\(472\) 0 0
\(473\) 9.74989i 0.448300i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.8493i 0.496756i
\(478\) 0 0
\(479\) 23.5537 1.07620 0.538098 0.842882i \(-0.319143\pi\)
0.538098 + 0.842882i \(0.319143\pi\)
\(480\) 0 0
\(481\) −20.4688 −0.933295
\(482\) 0 0
\(483\) − 38.6645i − 1.75930i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 19.9102i − 0.902217i −0.892469 0.451108i \(-0.851029\pi\)
0.892469 0.451108i \(-0.148971\pi\)
\(488\) 0 0
\(489\) 39.9456 1.80640
\(490\) 0 0
\(491\) 29.7585 1.34298 0.671490 0.741013i \(-0.265654\pi\)
0.671490 + 0.741013i \(0.265654\pi\)
\(492\) 0 0
\(493\) − 30.5511i − 1.37595i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 40.0162i − 1.79497i
\(498\) 0 0
\(499\) −18.9297 −0.847409 −0.423704 0.905801i \(-0.639270\pi\)
−0.423704 + 0.905801i \(0.639270\pi\)
\(500\) 0 0
\(501\) 29.2137 1.30517
\(502\) 0 0
\(503\) − 1.93382i − 0.0862248i −0.999070 0.0431124i \(-0.986273\pi\)
0.999070 0.0431124i \(-0.0137274\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.76202i 0.433546i
\(508\) 0 0
\(509\) −34.6602 −1.53629 −0.768144 0.640277i \(-0.778819\pi\)
−0.768144 + 0.640277i \(0.778819\pi\)
\(510\) 0 0
\(511\) −25.6979 −1.13681
\(512\) 0 0
\(513\) 16.4943i 0.728242i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.19258i − 0.360309i
\(518\) 0 0
\(519\) −26.7000 −1.17200
\(520\) 0 0
\(521\) −6.33402 −0.277498 −0.138749 0.990328i \(-0.544308\pi\)
−0.138749 + 0.990328i \(0.544308\pi\)
\(522\) 0 0
\(523\) − 29.9460i − 1.30944i −0.755869 0.654722i \(-0.772785\pi\)
0.755869 0.654722i \(-0.227215\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 84.4341i − 3.67801i
\(528\) 0 0
\(529\) 0.437142 0.0190062
\(530\) 0 0
\(531\) 26.5196 1.15085
\(532\) 0 0
\(533\) 3.10932i 0.134680i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 37.9110i 1.63598i
\(538\) 0 0
\(539\) −1.45946 −0.0628635
\(540\) 0 0
\(541\) 0.834750 0.0358887 0.0179443 0.999839i \(-0.494288\pi\)
0.0179443 + 0.999839i \(0.494288\pi\)
\(542\) 0 0
\(543\) 23.9118i 1.02615i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.6527i 0.883045i 0.897250 + 0.441523i \(0.145562\pi\)
−0.897250 + 0.441523i \(0.854438\pi\)
\(548\) 0 0
\(549\) −16.1387 −0.688784
\(550\) 0 0
\(551\) 8.41203 0.358364
\(552\) 0 0
\(553\) − 0.889872i − 0.0378412i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.72447i 0.369668i 0.982770 + 0.184834i \(0.0591747\pi\)
−0.982770 + 0.184834i \(0.940825\pi\)
\(558\) 0 0
\(559\) 14.9458 0.632140
\(560\) 0 0
\(561\) −47.0411 −1.98608
\(562\) 0 0
\(563\) 9.80144i 0.413081i 0.978438 + 0.206541i \(0.0662206\pi\)
−0.978438 + 0.206541i \(0.933779\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 15.0620i − 0.632546i
\(568\) 0 0
\(569\) −25.1570 −1.05464 −0.527318 0.849668i \(-0.676802\pi\)
−0.527318 + 0.849668i \(0.676802\pi\)
\(570\) 0 0
\(571\) 27.1928 1.13798 0.568992 0.822343i \(-0.307334\pi\)
0.568992 + 0.822343i \(0.307334\pi\)
\(572\) 0 0
\(573\) 13.2664i 0.554214i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.365186i 0.0152029i 0.999971 + 0.00760144i \(0.00241964\pi\)
−0.999971 + 0.00760144i \(0.997580\pi\)
\(578\) 0 0
\(579\) 41.5025 1.72478
\(580\) 0 0
\(581\) −10.2730 −0.426196
\(582\) 0 0
\(583\) − 3.94166i − 0.163247i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.7242i 1.55704i 0.627617 + 0.778522i \(0.284030\pi\)
−0.627617 + 0.778522i \(0.715970\pi\)
\(588\) 0 0
\(589\) 23.2484 0.957932
\(590\) 0 0
\(591\) −36.6334 −1.50689
\(592\) 0 0
\(593\) − 45.1333i − 1.85340i −0.375800 0.926701i \(-0.622632\pi\)
0.375800 0.926701i \(-0.377368\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 42.7303i 1.74884i
\(598\) 0 0
\(599\) −15.4982 −0.633238 −0.316619 0.948553i \(-0.602548\pi\)
−0.316619 + 0.948553i \(0.602548\pi\)
\(600\) 0 0
\(601\) −5.18924 −0.211674 −0.105837 0.994384i \(-0.533752\pi\)
−0.105837 + 0.994384i \(0.533752\pi\)
\(602\) 0 0
\(603\) 33.3398i 1.35770i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.8280i 0.683029i 0.939876 + 0.341515i \(0.110940\pi\)
−0.939876 + 0.341515i \(0.889060\pi\)
\(608\) 0 0
\(609\) −31.4146 −1.27298
\(610\) 0 0
\(611\) −12.5586 −0.508065
\(612\) 0 0
\(613\) 25.1042i 1.01395i 0.861961 + 0.506975i \(0.169236\pi\)
−0.861961 + 0.506975i \(0.830764\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 14.3324i − 0.577001i −0.957480 0.288501i \(-0.906843\pi\)
0.957480 0.288501i \(-0.0931567\pi\)
\(618\) 0 0
\(619\) −19.2712 −0.774576 −0.387288 0.921959i \(-0.626588\pi\)
−0.387288 + 0.921959i \(0.626588\pi\)
\(620\) 0 0
\(621\) −35.9456 −1.44245
\(622\) 0 0
\(623\) 24.9186i 0.998344i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 12.9524i − 0.517271i
\(628\) 0 0
\(629\) 52.1119 2.07784
\(630\) 0 0
\(631\) −30.1856 −1.20167 −0.600834 0.799374i \(-0.705165\pi\)
−0.600834 + 0.799374i \(0.705165\pi\)
\(632\) 0 0
\(633\) − 77.3027i − 3.07251i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.23724i 0.0886427i
\(638\) 0 0
\(639\) −80.4100 −3.18097
\(640\) 0 0
\(641\) −25.9984 −1.02687 −0.513437 0.858127i \(-0.671628\pi\)
−0.513437 + 0.858127i \(0.671628\pi\)
\(642\) 0 0
\(643\) 11.4402i 0.451159i 0.974225 + 0.225580i \(0.0724276\pi\)
−0.974225 + 0.225580i \(0.927572\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 37.6915i − 1.48181i −0.671611 0.740904i \(-0.734397\pi\)
0.671611 0.740904i \(-0.265603\pi\)
\(648\) 0 0
\(649\) −9.63481 −0.378200
\(650\) 0 0
\(651\) −86.8205 −3.40277
\(652\) 0 0
\(653\) − 43.6814i − 1.70938i −0.519136 0.854692i \(-0.673746\pi\)
0.519136 0.854692i \(-0.326254\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 51.6383i 2.01460i
\(658\) 0 0
\(659\) −25.0900 −0.977367 −0.488684 0.872461i \(-0.662523\pi\)
−0.488684 + 0.872461i \(0.662523\pi\)
\(660\) 0 0
\(661\) −37.7392 −1.46788 −0.733942 0.679212i \(-0.762322\pi\)
−0.733942 + 0.679212i \(0.762322\pi\)
\(662\) 0 0
\(663\) 72.1102i 2.80053i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.3321i 0.709822i
\(668\) 0 0
\(669\) 5.55121 0.214622
\(670\) 0 0
\(671\) 5.86335 0.226352
\(672\) 0 0
\(673\) 3.63642i 0.140174i 0.997541 + 0.0700869i \(0.0223277\pi\)
−0.997541 + 0.0700869i \(0.977672\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 29.1948i − 1.12205i −0.827800 0.561024i \(-0.810408\pi\)
0.827800 0.561024i \(-0.189592\pi\)
\(678\) 0 0
\(679\) −49.5450 −1.90136
\(680\) 0 0
\(681\) −73.1849 −2.80445
\(682\) 0 0
\(683\) − 28.6527i − 1.09636i −0.836359 0.548182i \(-0.815320\pi\)
0.836359 0.548182i \(-0.184680\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.39888i 0.282285i
\(688\) 0 0
\(689\) −6.04225 −0.230191
\(690\) 0 0
\(691\) 17.6880 0.672883 0.336442 0.941704i \(-0.390777\pi\)
0.336442 + 0.941704i \(0.390777\pi\)
\(692\) 0 0
\(693\) 31.4638i 1.19521i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 7.91609i − 0.299843i
\(698\) 0 0
\(699\) −27.0947 −1.02482
\(700\) 0 0
\(701\) 2.79339 0.105505 0.0527525 0.998608i \(-0.483201\pi\)
0.0527525 + 0.998608i \(0.483201\pi\)
\(702\) 0 0
\(703\) 14.3486i 0.541169i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.1899i 1.36106i
\(708\) 0 0
\(709\) −20.4117 −0.766578 −0.383289 0.923628i \(-0.625209\pi\)
−0.383289 + 0.923628i \(0.625209\pi\)
\(710\) 0 0
\(711\) −1.78814 −0.0670606
\(712\) 0 0
\(713\) 50.6645i 1.89740i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6.87730i − 0.256838i
\(718\) 0 0
\(719\) −34.9848 −1.30471 −0.652356 0.757912i \(-0.726219\pi\)
−0.652356 + 0.757912i \(0.726219\pi\)
\(720\) 0 0
\(721\) −22.3849 −0.833655
\(722\) 0 0
\(723\) 8.46876i 0.314957i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 24.8796i 0.922734i 0.887209 + 0.461367i \(0.152641\pi\)
−0.887209 + 0.461367i \(0.847359\pi\)
\(728\) 0 0
\(729\) 36.2444 1.34238
\(730\) 0 0
\(731\) −38.0508 −1.40736
\(732\) 0 0
\(733\) − 28.7737i − 1.06278i −0.847127 0.531390i \(-0.821670\pi\)
0.847127 0.531390i \(-0.178330\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 12.1127i − 0.446176i
\(738\) 0 0
\(739\) 30.1856 1.11039 0.555197 0.831719i \(-0.312643\pi\)
0.555197 + 0.831719i \(0.312643\pi\)
\(740\) 0 0
\(741\) −19.8551 −0.729394
\(742\) 0 0
\(743\) − 22.7127i − 0.833247i −0.909079 0.416624i \(-0.863213\pi\)
0.909079 0.416624i \(-0.136787\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 20.6429i 0.755286i
\(748\) 0 0
\(749\) 12.2695 0.448317
\(750\) 0 0
\(751\) −0.0200854 −0.000732927 0 −0.000366463 1.00000i \(-0.500117\pi\)
−0.000366463 1.00000i \(0.500117\pi\)
\(752\) 0 0
\(753\) 73.1628i 2.66620i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 44.9002i − 1.63192i −0.578106 0.815962i \(-0.696208\pi\)
0.578106 0.815962i \(-0.303792\pi\)
\(758\) 0 0
\(759\) 28.2269 1.02457
\(760\) 0 0
\(761\) −19.0203 −0.689486 −0.344743 0.938697i \(-0.612034\pi\)
−0.344743 + 0.938697i \(0.612034\pi\)
\(762\) 0 0
\(763\) 51.4713i 1.86339i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.7694i 0.533292i
\(768\) 0 0
\(769\) −42.1220 −1.51896 −0.759480 0.650531i \(-0.774546\pi\)
−0.759480 + 0.650531i \(0.774546\pi\)
\(770\) 0 0
\(771\) 31.4942 1.13424
\(772\) 0 0
\(773\) 10.7753i 0.387561i 0.981045 + 0.193780i \(0.0620749\pi\)
−0.981045 + 0.193780i \(0.937925\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 53.5848i − 1.92234i
\(778\) 0 0
\(779\) 2.17964 0.0780938
\(780\) 0 0
\(781\) 29.2137 1.04535
\(782\) 0 0
\(783\) 29.2055i 1.04372i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.8592i − 0.814843i −0.913240 0.407421i \(-0.866428\pi\)
0.913240 0.407421i \(-0.133572\pi\)
\(788\) 0 0
\(789\) 14.3628 0.511328
\(790\) 0 0
\(791\) −22.4581 −0.798519
\(792\) 0 0
\(793\) − 8.98805i − 0.319175i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.00444i 0.177266i 0.996064 + 0.0886332i \(0.0282499\pi\)
−0.996064 + 0.0886332i \(0.971750\pi\)
\(798\) 0 0
\(799\) 31.9731 1.13113
\(800\) 0 0
\(801\) 50.0724 1.76922
\(802\) 0 0
\(803\) − 18.7607i − 0.662050i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 9.74989i − 0.343212i
\(808\) 0 0
\(809\) −7.52135 −0.264437 −0.132218 0.991221i \(-0.542210\pi\)
−0.132218 + 0.991221i \(0.542210\pi\)
\(810\) 0 0
\(811\) 29.4103 1.03273 0.516367 0.856367i \(-0.327284\pi\)
0.516367 + 0.856367i \(0.327284\pi\)
\(812\) 0 0
\(813\) 13.4244i 0.470816i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 10.4770i − 0.366545i
\(818\) 0 0
\(819\) 48.2315 1.68535
\(820\) 0 0
\(821\) 38.0205 1.32692 0.663462 0.748210i \(-0.269087\pi\)
0.663462 + 0.748210i \(0.269087\pi\)
\(822\) 0 0
\(823\) 25.7849i 0.898805i 0.893329 + 0.449403i \(0.148363\pi\)
−0.893329 + 0.449403i \(0.851637\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 7.00295i − 0.243516i −0.992560 0.121758i \(-0.961147\pi\)
0.992560 0.121758i \(-0.0388533\pi\)
\(828\) 0 0
\(829\) −0.166709 −0.00579003 −0.00289501 0.999996i \(-0.500922\pi\)
−0.00289501 + 0.999996i \(0.500922\pi\)
\(830\) 0 0
\(831\) −70.9505 −2.46125
\(832\) 0 0
\(833\) − 5.69584i − 0.197349i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 80.7154i 2.78993i
\(838\) 0 0
\(839\) 13.1105 0.452625 0.226313 0.974055i \(-0.427333\pi\)
0.226313 + 0.974055i \(0.427333\pi\)
\(840\) 0 0
\(841\) −14.1053 −0.486391
\(842\) 0 0
\(843\) 19.3760i 0.667345i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 19.1313i 0.657361i
\(848\) 0 0
\(849\) −15.1349 −0.519428
\(850\) 0 0
\(851\) −31.2696 −1.07191
\(852\) 0 0
\(853\) − 0.445573i − 0.0152561i −0.999971 0.00762806i \(-0.997572\pi\)
0.999971 0.00762806i \(-0.00242811\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.5519i 0.702038i 0.936368 + 0.351019i \(0.114165\pi\)
−0.936368 + 0.351019i \(0.885835\pi\)
\(858\) 0 0
\(859\) 18.8891 0.644487 0.322243 0.946657i \(-0.395563\pi\)
0.322243 + 0.946657i \(0.395563\pi\)
\(860\) 0 0
\(861\) −8.13983 −0.277405
\(862\) 0 0
\(863\) − 47.9204i − 1.63123i −0.578596 0.815614i \(-0.696399\pi\)
0.578596 0.815614i \(-0.303601\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 133.782i − 4.54349i
\(868\) 0 0
\(869\) 0.649649 0.0220378
\(870\) 0 0
\(871\) −18.5677 −0.629144
\(872\) 0 0
\(873\) 99.5576i 3.36951i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 50.2738i 1.69762i 0.528694 + 0.848812i \(0.322682\pi\)
−0.528694 + 0.848812i \(0.677318\pi\)
\(878\) 0 0
\(879\) 30.0170 1.01245
\(880\) 0 0
\(881\) −38.6019 −1.30053 −0.650265 0.759707i \(-0.725342\pi\)
−0.650265 + 0.759707i \(0.725342\pi\)
\(882\) 0 0
\(883\) 3.07860i 0.103603i 0.998657 + 0.0518016i \(0.0164963\pi\)
−0.998657 + 0.0518016i \(0.983504\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.0770i 1.04346i 0.853110 + 0.521731i \(0.174714\pi\)
−0.853110 + 0.521731i \(0.825286\pi\)
\(888\) 0 0
\(889\) −12.3569 −0.414437
\(890\) 0 0
\(891\) 10.9960 0.368380
\(892\) 0 0
\(893\) 8.80358i 0.294601i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 43.2696i − 1.44473i
\(898\) 0 0
\(899\) 41.1644 1.37291
\(900\) 0 0
\(901\) 15.3831 0.512485
\(902\) 0 0
\(903\) 39.1263i 1.30204i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 42.5534i − 1.41296i −0.707731 0.706482i \(-0.750281\pi\)
0.707731 0.706482i \(-0.249719\pi\)
\(908\) 0 0
\(909\) 72.7213 2.41201
\(910\) 0 0
\(911\) −14.8694 −0.492645 −0.246323 0.969188i \(-0.579222\pi\)
−0.246323 + 0.969188i \(0.579222\pi\)
\(912\) 0 0
\(913\) − 7.49977i − 0.248206i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 27.3933i − 0.904606i
\(918\) 0 0
\(919\) −26.5539 −0.875931 −0.437965 0.898992i \(-0.644301\pi\)
−0.437965 + 0.898992i \(0.644301\pi\)
\(920\) 0 0
\(921\) −53.5410 −1.76424
\(922\) 0 0
\(923\) − 44.7823i − 1.47403i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 44.9810i 1.47737i
\(928\) 0 0
\(929\) −2.70749 −0.0888298 −0.0444149 0.999013i \(-0.514142\pi\)
−0.0444149 + 0.999013i \(0.514142\pi\)
\(930\) 0 0
\(931\) 1.56831 0.0513993
\(932\) 0 0
\(933\) 13.1924i 0.431901i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 36.0067i 1.17629i 0.808757 + 0.588143i \(0.200141\pi\)
−0.808757 + 0.588143i \(0.799859\pi\)
\(938\) 0 0
\(939\) −19.2215 −0.627269
\(940\) 0 0
\(941\) 11.6135 0.378591 0.189295 0.981920i \(-0.439380\pi\)
0.189295 + 0.981920i \(0.439380\pi\)
\(942\) 0 0
\(943\) 4.75004i 0.154683i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.9061i 0.679355i 0.940542 + 0.339678i \(0.110318\pi\)
−0.940542 + 0.339678i \(0.889682\pi\)
\(948\) 0 0
\(949\) −28.7586 −0.933544
\(950\) 0 0
\(951\) 37.6063 1.21947
\(952\) 0 0
\(953\) − 13.7510i − 0.445438i −0.974883 0.222719i \(-0.928507\pi\)
0.974883 0.222719i \(-0.0714933\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 22.9341i − 0.741355i
\(958\) 0 0
\(959\) 6.39742 0.206584
\(960\) 0 0
\(961\) 82.7663 2.66988
\(962\) 0 0
\(963\) − 24.6547i − 0.794488i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 7.07935i − 0.227656i −0.993500 0.113828i \(-0.963689\pi\)
0.993500 0.113828i \(-0.0363114\pi\)
\(968\) 0 0
\(969\) 50.5494 1.62388
\(970\) 0 0
\(971\) −18.7382 −0.601338 −0.300669 0.953729i \(-0.597210\pi\)
−0.300669 + 0.953729i \(0.597210\pi\)
\(972\) 0 0
\(973\) 14.3534i 0.460148i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 60.6756i − 1.94118i −0.240729 0.970592i \(-0.577387\pi\)
0.240729 0.970592i \(-0.422613\pi\)
\(978\) 0 0
\(979\) −18.1918 −0.581412
\(980\) 0 0
\(981\) 103.428 3.30221
\(982\) 0 0
\(983\) 47.3720i 1.51093i 0.655188 + 0.755466i \(0.272590\pi\)
−0.655188 + 0.755466i \(0.727410\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 32.8768i − 1.04648i
\(988\) 0 0
\(989\) 22.8323 0.726026
\(990\) 0 0
\(991\) −19.2110 −0.610256 −0.305128 0.952311i \(-0.598699\pi\)
−0.305128 + 0.952311i \(0.598699\pi\)
\(992\) 0 0
\(993\) − 26.3585i − 0.836461i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 59.1077i 1.87196i 0.352053 + 0.935980i \(0.385484\pi\)
−0.352053 + 0.935980i \(0.614516\pi\)
\(998\) 0 0
\(999\) −49.8167 −1.57613
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 4100.2.d.c.1149.7 8
5.2 odd 4 164.2.a.a.1.4 4
5.3 odd 4 4100.2.a.c.1.1 4
5.4 even 2 inner 4100.2.d.c.1149.2 8
15.2 even 4 1476.2.a.g.1.4 4
20.7 even 4 656.2.a.i.1.1 4
35.27 even 4 8036.2.a.i.1.1 4
40.27 even 4 2624.2.a.y.1.4 4
40.37 odd 4 2624.2.a.v.1.1 4
60.47 odd 4 5904.2.a.bp.1.4 4
205.122 odd 4 6724.2.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.4 4 5.2 odd 4
656.2.a.i.1.1 4 20.7 even 4
1476.2.a.g.1.4 4 15.2 even 4
2624.2.a.v.1.1 4 40.37 odd 4
2624.2.a.y.1.4 4 40.27 even 4
4100.2.a.c.1.1 4 5.3 odd 4
4100.2.d.c.1149.2 8 5.4 even 2 inner
4100.2.d.c.1149.7 8 1.1 even 1 trivial
5904.2.a.bp.1.4 4 60.47 odd 4
6724.2.a.c.1.1 4 205.122 odd 4
8036.2.a.i.1.1 4 35.27 even 4