Properties

Label 4100.2.d.c.1149.1
Level $4100$
Weight $2$
Character 4100.1149
Analytic conductor $32.739$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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 1149.1
Root \(1.76639 + 1.76639i\) of defining polynomial
Character \(\chi\) \(=\) 4100.1149
Dual form 4100.2.d.c.1149.8

$q$-expansion

\(f(q)\) \(=\) \(q-3.24028i q^{3} +0.858626i q^{7} -7.49944 q^{9} +O(q^{10})\) \(q-3.24028i q^{3} +0.858626i q^{7} -7.49944 q^{9} +6.20694 q^{11} +1.41500i q^{13} +3.93332i q^{17} -3.82529 q^{19} +2.78219 q^{21} +3.06557i q^{23} +14.5795i q^{27} +8.48057 q^{29} +1.13225 q^{31} -20.1123i q^{33} -8.49944i q^{37} +4.58500 q^{39} -1.00000 q^{41} +5.34832i q^{43} +6.65528i q^{47} +6.26276 q^{49} +12.7451 q^{51} +6.41389i q^{53} +12.3950i q^{57} +3.06557 q^{59} +7.41500 q^{61} -6.43922i q^{63} -1.79306i q^{67} +9.93332 q^{69} -3.02422 q^{71} +0.632807i q^{73} +5.32944i q^{77} +14.3059 q^{79} +24.7433 q^{81} -8.76332i q^{83} -27.4795i q^{87} +7.89557 q^{89} -1.21496 q^{91} -3.66882i q^{93} -9.86664i q^{97} -46.5486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 8q^{11} - 12q^{19} + 8q^{29} - 16q^{31} + 48q^{39} - 8q^{41} - 32q^{49} - 8q^{51} - 24q^{59} + 48q^{61} + 56q^{69} - 4q^{71} + 36q^{79} + 56q^{81} - 8q^{89} + 72q^{91} - 116q^{99} + O(q^{100}) \)

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
\(3\) − 3.24028i − 1.87078i −0.353619 0.935390i \(-0.615049\pi\)
0.353619 0.935390i \(-0.384951\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.858626i 0.324530i 0.986747 + 0.162265i \(0.0518800\pi\)
−0.986747 + 0.162265i \(0.948120\pi\)
\(8\) 0 0
\(9\) −7.49944 −2.49981
\(10\) 0 0
\(11\) 6.20694 1.87146 0.935732 0.352712i \(-0.114740\pi\)
0.935732 + 0.352712i \(0.114740\pi\)
\(12\) 0 0
\(13\) 1.41500i 0.392450i 0.980559 + 0.196225i \(0.0628683\pi\)
−0.980559 + 0.196225i \(0.937132\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.93332i 0.953970i 0.878911 + 0.476985i \(0.158270\pi\)
−0.878911 + 0.476985i \(0.841730\pi\)
\(18\) 0 0
\(19\) −3.82529 −0.877581 −0.438790 0.898589i \(-0.644593\pi\)
−0.438790 + 0.898589i \(0.644593\pi\)
\(20\) 0 0
\(21\) 2.78219 0.607124
\(22\) 0 0
\(23\) 3.06557i 0.639215i 0.947550 + 0.319608i \(0.103551\pi\)
−0.947550 + 0.319608i \(0.896449\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.5795i 2.80582i
\(28\) 0 0
\(29\) 8.48057 1.57480 0.787401 0.616441i \(-0.211426\pi\)
0.787401 + 0.616441i \(0.211426\pi\)
\(30\) 0 0
\(31\) 1.13225 0.203358 0.101679 0.994817i \(-0.467578\pi\)
0.101679 + 0.994817i \(0.467578\pi\)
\(32\) 0 0
\(33\) − 20.1123i − 3.50110i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.49944i − 1.39730i −0.715464 0.698650i \(-0.753784\pi\)
0.715464 0.698650i \(-0.246216\pi\)
\(38\) 0 0
\(39\) 4.58500 0.734188
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 5.34832i 0.815611i 0.913069 + 0.407805i \(0.133706\pi\)
−0.913069 + 0.407805i \(0.866294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.65528i 0.970773i 0.874300 + 0.485386i \(0.161321\pi\)
−0.874300 + 0.485386i \(0.838679\pi\)
\(48\) 0 0
\(49\) 6.26276 0.894680
\(50\) 0 0
\(51\) 12.7451 1.78467
\(52\) 0 0
\(53\) 6.41389i 0.881015i 0.897749 + 0.440508i \(0.145202\pi\)
−0.897749 + 0.440508i \(0.854798\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.3950i 1.64176i
\(58\) 0 0
\(59\) 3.06557 0.399103 0.199552 0.979887i \(-0.436051\pi\)
0.199552 + 0.979887i \(0.436051\pi\)
\(60\) 0 0
\(61\) 7.41500 0.949393 0.474697 0.880149i \(-0.342558\pi\)
0.474697 + 0.880149i \(0.342558\pi\)
\(62\) 0 0
\(63\) − 6.43922i − 0.811265i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.79306i − 0.219057i −0.993984 0.109528i \(-0.965066\pi\)
0.993984 0.109528i \(-0.0349340\pi\)
\(68\) 0 0
\(69\) 9.93332 1.19583
\(70\) 0 0
\(71\) −3.02422 −0.358909 −0.179454 0.983766i \(-0.557433\pi\)
−0.179454 + 0.983766i \(0.557433\pi\)
\(72\) 0 0
\(73\) 0.632807i 0.0740645i 0.999314 + 0.0370322i \(0.0117904\pi\)
−0.999314 + 0.0370322i \(0.988210\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.32944i 0.607346i
\(78\) 0 0
\(79\) 14.3059 1.60953 0.804767 0.593591i \(-0.202290\pi\)
0.804767 + 0.593591i \(0.202290\pi\)
\(80\) 0 0
\(81\) 24.7433 2.74926
\(82\) 0 0
\(83\) − 8.76332i − 0.961899i −0.876748 0.480950i \(-0.840292\pi\)
0.876748 0.480950i \(-0.159708\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 27.4795i − 2.94611i
\(88\) 0 0
\(89\) 7.89557 0.836929 0.418464 0.908233i \(-0.362569\pi\)
0.418464 + 0.908233i \(0.362569\pi\)
\(90\) 0 0
\(91\) −1.21496 −0.127362
\(92\) 0 0
\(93\) − 3.66882i − 0.380439i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 9.86664i − 1.00181i −0.865504 0.500903i \(-0.833001\pi\)
0.865504 0.500903i \(-0.166999\pi\)
\(98\) 0 0
\(99\) −46.5486 −4.67831
\(100\) 0 0
\(101\) −4.51832 −0.449590 −0.224795 0.974406i \(-0.572171\pi\)
−0.224795 + 0.974406i \(0.572171\pi\)
\(102\) 0 0
\(103\) − 16.4139i − 1.61731i −0.588284 0.808654i \(-0.700196\pi\)
0.588284 0.808654i \(-0.299804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.0645i 1.55301i 0.630111 + 0.776505i \(0.283009\pi\)
−0.630111 + 0.776505i \(0.716991\pi\)
\(108\) 0 0
\(109\) 5.61282 0.537611 0.268805 0.963195i \(-0.413371\pi\)
0.268805 + 0.963195i \(0.413371\pi\)
\(110\) 0 0
\(111\) −27.5406 −2.61404
\(112\) 0 0
\(113\) 5.63170i 0.529785i 0.964278 + 0.264893i \(0.0853365\pi\)
−0.964278 + 0.264893i \(0.914663\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 10.6117i − 0.981053i
\(118\) 0 0
\(119\) −3.37725 −0.309592
\(120\) 0 0
\(121\) 27.5262 2.50238
\(122\) 0 0
\(123\) 3.24028i 0.292167i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1.69775i − 0.150651i −0.997159 0.0753254i \(-0.976000\pi\)
0.997159 0.0753254i \(-0.0239995\pi\)
\(128\) 0 0
\(129\) 17.3301 1.52583
\(130\) 0 0
\(131\) −2.48057 −0.216728 −0.108364 0.994111i \(-0.534561\pi\)
−0.108364 + 0.994111i \(0.534561\pi\)
\(132\) 0 0
\(133\) − 3.28449i − 0.284801i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.76332i − 0.236086i −0.993008 0.118043i \(-0.962338\pi\)
0.993008 0.118043i \(-0.0376621\pi\)
\(138\) 0 0
\(139\) −8.99889 −0.763276 −0.381638 0.924312i \(-0.624640\pi\)
−0.381638 + 0.924312i \(0.624640\pi\)
\(140\) 0 0
\(141\) 21.5650 1.81610
\(142\) 0 0
\(143\) 8.78282i 0.734456i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 20.2931i − 1.67375i
\(148\) 0 0
\(149\) 8.48057 0.694755 0.347378 0.937725i \(-0.387072\pi\)
0.347378 + 0.937725i \(0.387072\pi\)
\(150\) 0 0
\(151\) −7.58971 −0.617642 −0.308821 0.951120i \(-0.599934\pi\)
−0.308821 + 0.951120i \(0.599934\pi\)
\(152\) 0 0
\(153\) − 29.4977i − 2.38475i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.78282i 0.541328i 0.962674 + 0.270664i \(0.0872433\pi\)
−0.962674 + 0.270664i \(0.912757\pi\)
\(158\) 0 0
\(159\) 20.7828 1.64818
\(160\) 0 0
\(161\) −2.63218 −0.207445
\(162\) 0 0
\(163\) 15.0278i 1.17707i 0.808472 + 0.588535i \(0.200295\pi\)
−0.808472 + 0.588535i \(0.799705\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 5.79306i − 0.448280i −0.974557 0.224140i \(-0.928043\pi\)
0.974557 0.224140i \(-0.0719573\pi\)
\(168\) 0 0
\(169\) 10.9978 0.845983
\(170\) 0 0
\(171\) 28.6875 2.19379
\(172\) 0 0
\(173\) − 5.43450i − 0.413178i −0.978428 0.206589i \(-0.933764\pi\)
0.978428 0.206589i \(-0.0662362\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 9.93332i − 0.746634i
\(178\) 0 0
\(179\) −8.10251 −0.605610 −0.302805 0.953052i \(-0.597923\pi\)
−0.302805 + 0.953052i \(0.597923\pi\)
\(180\) 0 0
\(181\) −9.24389 −0.687093 −0.343546 0.939136i \(-0.611628\pi\)
−0.343546 + 0.939136i \(0.611628\pi\)
\(182\) 0 0
\(183\) − 24.0267i − 1.77611i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.4139i 1.78532i
\(188\) 0 0
\(189\) −12.5183 −0.910574
\(190\) 0 0
\(191\) 24.3381 1.76104 0.880521 0.474007i \(-0.157193\pi\)
0.880521 + 0.474007i \(0.157193\pi\)
\(192\) 0 0
\(193\) 13.2634i 0.954720i 0.878708 + 0.477360i \(0.158406\pi\)
−0.878708 + 0.477360i \(0.841594\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.1116i 1.29040i 0.764013 + 0.645200i \(0.223226\pi\)
−0.764013 + 0.645200i \(0.776774\pi\)
\(198\) 0 0
\(199\) −19.7726 −1.40164 −0.700821 0.713337i \(-0.747183\pi\)
−0.700821 + 0.713337i \(0.747183\pi\)
\(200\) 0 0
\(201\) −5.81001 −0.409807
\(202\) 0 0
\(203\) 7.28164i 0.511071i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 22.9901i − 1.59792i
\(208\) 0 0
\(209\) −23.7433 −1.64236
\(210\) 0 0
\(211\) 1.53924 0.105966 0.0529828 0.998595i \(-0.483127\pi\)
0.0529828 + 0.998595i \(0.483127\pi\)
\(212\) 0 0
\(213\) 9.79933i 0.671439i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.972180i 0.0659959i
\(218\) 0 0
\(219\) 2.05047 0.138558
\(220\) 0 0
\(221\) −5.56564 −0.374386
\(222\) 0 0
\(223\) − 27.6578i − 1.85210i −0.377399 0.926051i \(-0.623181\pi\)
0.377399 0.926051i \(-0.376819\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.79635i 0.119228i 0.998221 + 0.0596141i \(0.0189870\pi\)
−0.998221 + 0.0596141i \(0.981013\pi\)
\(228\) 0 0
\(229\) −21.6128 −1.42822 −0.714108 0.700036i \(-0.753167\pi\)
−0.714108 + 0.700036i \(0.753167\pi\)
\(230\) 0 0
\(231\) 17.2689 1.13621
\(232\) 0 0
\(233\) − 18.3312i − 1.20092i −0.799656 0.600458i \(-0.794985\pi\)
0.799656 0.600458i \(-0.205015\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 46.3550i − 3.01108i
\(238\) 0 0
\(239\) −11.9586 −0.773541 −0.386770 0.922176i \(-0.626409\pi\)
−0.386770 + 0.922176i \(0.626409\pi\)
\(240\) 0 0
\(241\) 7.41500 0.477642 0.238821 0.971064i \(-0.423239\pi\)
0.238821 + 0.971064i \(0.423239\pi\)
\(242\) 0 0
\(243\) − 36.4370i − 2.33743i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.41278i − 0.344407i
\(248\) 0 0
\(249\) −28.3956 −1.79950
\(250\) 0 0
\(251\) −1.91507 −0.120878 −0.0604392 0.998172i \(-0.519250\pi\)
−0.0604392 + 0.998172i \(0.519250\pi\)
\(252\) 0 0
\(253\) 19.0278i 1.19627i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.93443i − 0.183045i −0.995803 0.0915224i \(-0.970827\pi\)
0.995803 0.0915224i \(-0.0291733\pi\)
\(258\) 0 0
\(259\) 7.29784 0.453466
\(260\) 0 0
\(261\) −63.5996 −3.93671
\(262\) 0 0
\(263\) − 3.10692i − 0.191581i −0.995402 0.0957905i \(-0.969462\pi\)
0.995402 0.0957905i \(-0.0305379\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 25.5839i − 1.56571i
\(268\) 0 0
\(269\) 10.2450 0.624649 0.312324 0.949976i \(-0.398892\pi\)
0.312324 + 0.949976i \(0.398892\pi\)
\(270\) 0 0
\(271\) −11.1989 −0.680287 −0.340143 0.940374i \(-0.610476\pi\)
−0.340143 + 0.940374i \(0.610476\pi\)
\(272\) 0 0
\(273\) 3.93680i 0.238266i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.52838i − 0.151915i −0.997111 0.0759577i \(-0.975799\pi\)
0.997111 0.0759577i \(-0.0242014\pi\)
\(278\) 0 0
\(279\) −8.49125 −0.508358
\(280\) 0 0
\(281\) −23.5656 −1.40581 −0.702904 0.711285i \(-0.748114\pi\)
−0.702904 + 0.711285i \(0.748114\pi\)
\(282\) 0 0
\(283\) 8.99889i 0.534928i 0.963568 + 0.267464i \(0.0861857\pi\)
−0.963568 + 0.267464i \(0.913814\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 0.858626i − 0.0506831i
\(288\) 0 0
\(289\) 1.52900 0.0899415
\(290\) 0 0
\(291\) −31.9707 −1.87416
\(292\) 0 0
\(293\) 3.17721i 0.185614i 0.995684 + 0.0928072i \(0.0295840\pi\)
−0.995684 + 0.0928072i \(0.970416\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 90.4940i 5.25100i
\(298\) 0 0
\(299\) −4.33778 −0.250860
\(300\) 0 0
\(301\) −4.59220 −0.264690
\(302\) 0 0
\(303\) 14.6406i 0.841083i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.5839i 1.00357i 0.864994 + 0.501783i \(0.167322\pi\)
−0.864994 + 0.501783i \(0.832678\pi\)
\(308\) 0 0
\(309\) −53.1857 −3.02563
\(310\) 0 0
\(311\) 1.55749 0.0883169 0.0441584 0.999025i \(-0.485939\pi\)
0.0441584 + 0.999025i \(0.485939\pi\)
\(312\) 0 0
\(313\) 20.5255i 1.16017i 0.814556 + 0.580086i \(0.196981\pi\)
−0.814556 + 0.580086i \(0.803019\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 26.1094i − 1.46645i −0.679986 0.733225i \(-0.738014\pi\)
0.679986 0.733225i \(-0.261986\pi\)
\(318\) 0 0
\(319\) 52.6384 2.94719
\(320\) 0 0
\(321\) 52.0534 2.90534
\(322\) 0 0
\(323\) − 15.0461i − 0.837185i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 18.1871i − 1.00575i
\(328\) 0 0
\(329\) −5.71440 −0.315045
\(330\) 0 0
\(331\) 3.68862 0.202745 0.101373 0.994849i \(-0.467677\pi\)
0.101373 + 0.994849i \(0.467677\pi\)
\(332\) 0 0
\(333\) 63.7411i 3.49299i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 30.9240i − 1.68454i −0.539057 0.842269i \(-0.681219\pi\)
0.539057 0.842269i \(-0.318781\pi\)
\(338\) 0 0
\(339\) 18.2483 0.991111
\(340\) 0 0
\(341\) 7.02782 0.380578
\(342\) 0 0
\(343\) 11.3878i 0.614881i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.1264i − 1.45622i −0.685459 0.728111i \(-0.740398\pi\)
0.685459 0.728111i \(-0.259602\pi\)
\(348\) 0 0
\(349\) 9.30051 0.497845 0.248922 0.968523i \(-0.419924\pi\)
0.248922 + 0.968523i \(0.419924\pi\)
\(350\) 0 0
\(351\) −20.6300 −1.10115
\(352\) 0 0
\(353\) 13.3961i 0.713004i 0.934295 + 0.356502i \(0.116031\pi\)
−0.934295 + 0.356502i \(0.883969\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.9432i 0.579178i
\(358\) 0 0
\(359\) 28.9611 1.52851 0.764255 0.644914i \(-0.223107\pi\)
0.764255 + 0.644914i \(0.223107\pi\)
\(360\) 0 0
\(361\) −4.36719 −0.229852
\(362\) 0 0
\(363\) − 89.1926i − 4.68140i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 31.4795i 1.64321i 0.570054 + 0.821607i \(0.306922\pi\)
−0.570054 + 0.821607i \(0.693078\pi\)
\(368\) 0 0
\(369\) 7.49944 0.390405
\(370\) 0 0
\(371\) −5.50713 −0.285916
\(372\) 0 0
\(373\) − 15.5266i − 0.803939i −0.915653 0.401969i \(-0.868326\pi\)
0.915653 0.401969i \(-0.131674\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −1.97107 −0.101247 −0.0506235 0.998718i \(-0.516121\pi\)
−0.0506235 + 0.998718i \(0.516121\pi\)
\(380\) 0 0
\(381\) −5.50119 −0.281834
\(382\) 0 0
\(383\) − 12.8404i − 0.656113i −0.944658 0.328056i \(-0.893606\pi\)
0.944658 0.328056i \(-0.106394\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 40.1094i − 2.03888i
\(388\) 0 0
\(389\) −1.03886 −0.0526724 −0.0263362 0.999653i \(-0.508384\pi\)
−0.0263362 + 0.999653i \(0.508384\pi\)
\(390\) 0 0
\(391\) −12.0579 −0.609792
\(392\) 0 0
\(393\) 8.03775i 0.405451i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.6406i 1.23668i 0.785911 + 0.618339i \(0.212194\pi\)
−0.785911 + 0.618339i \(0.787806\pi\)
\(398\) 0 0
\(399\) −10.6427 −0.532800
\(400\) 0 0
\(401\) 0.397236 0.0198370 0.00991851 0.999951i \(-0.496843\pi\)
0.00991851 + 0.999951i \(0.496843\pi\)
\(402\) 0 0
\(403\) 1.60213i 0.0798080i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 52.7556i − 2.61500i
\(408\) 0 0
\(409\) 2.76395 0.136668 0.0683342 0.997662i \(-0.478232\pi\)
0.0683342 + 0.997662i \(0.478232\pi\)
\(410\) 0 0
\(411\) −8.95393 −0.441665
\(412\) 0 0
\(413\) 2.63218i 0.129521i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 29.1590i 1.42792i
\(418\) 0 0
\(419\) −1.48168 −0.0723848 −0.0361924 0.999345i \(-0.511523\pi\)
−0.0361924 + 0.999345i \(0.511523\pi\)
\(420\) 0 0
\(421\) −18.9039 −0.921319 −0.460659 0.887577i \(-0.652387\pi\)
−0.460659 + 0.887577i \(0.652387\pi\)
\(422\) 0 0
\(423\) − 49.9109i − 2.42675i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.36671i 0.308107i
\(428\) 0 0
\(429\) 28.4588 1.37401
\(430\) 0 0
\(431\) −7.28164 −0.350744 −0.175372 0.984502i \(-0.556113\pi\)
−0.175372 + 0.984502i \(0.556113\pi\)
\(432\) 0 0
\(433\) − 0.397865i − 0.0191202i −0.999954 0.00956009i \(-0.996957\pi\)
0.999954 0.00956009i \(-0.00304312\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 11.7267i − 0.560963i
\(438\) 0 0
\(439\) −33.9798 −1.62177 −0.810885 0.585206i \(-0.801014\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(440\) 0 0
\(441\) −46.9672 −2.23653
\(442\) 0 0
\(443\) 16.2050i 0.769924i 0.922932 + 0.384962i \(0.125785\pi\)
−0.922932 + 0.384962i \(0.874215\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 27.4795i − 1.29973i
\(448\) 0 0
\(449\) 23.1118 1.09071 0.545356 0.838204i \(-0.316394\pi\)
0.545356 + 0.838204i \(0.316394\pi\)
\(450\) 0 0
\(451\) −6.20694 −0.292274
\(452\) 0 0
\(453\) 24.5928i 1.15547i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 35.6301i − 1.66671i −0.552741 0.833353i \(-0.686418\pi\)
0.552741 0.833353i \(-0.313582\pi\)
\(458\) 0 0
\(459\) −57.3457 −2.67667
\(460\) 0 0
\(461\) 6.63058 0.308817 0.154409 0.988007i \(-0.450653\pi\)
0.154409 + 0.988007i \(0.450653\pi\)
\(462\) 0 0
\(463\) − 21.0220i − 0.976975i −0.872571 0.488487i \(-0.837549\pi\)
0.872571 0.488487i \(-0.162451\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 15.8933i − 0.735456i −0.929933 0.367728i \(-0.880136\pi\)
0.929933 0.367728i \(-0.119864\pi\)
\(468\) 0 0
\(469\) 1.53956 0.0710905
\(470\) 0 0
\(471\) 21.9783 1.01271
\(472\) 0 0
\(473\) 33.1967i 1.52639i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 48.1006i − 2.20237i
\(478\) 0 0
\(479\) 34.8564 1.59263 0.796315 0.604882i \(-0.206780\pi\)
0.796315 + 0.604882i \(0.206780\pi\)
\(480\) 0 0
\(481\) 12.0267 0.548371
\(482\) 0 0
\(483\) 8.52900i 0.388083i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.44393i 0.427945i 0.976840 + 0.213973i \(0.0686404\pi\)
−0.976840 + 0.213973i \(0.931360\pi\)
\(488\) 0 0
\(489\) 48.6944 2.20204
\(490\) 0 0
\(491\) 33.1577 1.49639 0.748193 0.663481i \(-0.230922\pi\)
0.748193 + 0.663481i \(0.230922\pi\)
\(492\) 0 0
\(493\) 33.3568i 1.50231i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.59667i − 0.116477i
\(498\) 0 0
\(499\) −12.7597 −0.571203 −0.285602 0.958348i \(-0.592193\pi\)
−0.285602 + 0.958348i \(0.592193\pi\)
\(500\) 0 0
\(501\) −18.7712 −0.838633
\(502\) 0 0
\(503\) − 17.0025i − 0.758104i −0.925375 0.379052i \(-0.876250\pi\)
0.925375 0.379052i \(-0.123750\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 35.6359i − 1.58265i
\(508\) 0 0
\(509\) 2.50950 0.111232 0.0556158 0.998452i \(-0.482288\pi\)
0.0556158 + 0.998452i \(0.482288\pi\)
\(510\) 0 0
\(511\) −0.543345 −0.0240361
\(512\) 0 0
\(513\) − 55.7707i − 2.46234i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 41.3090i 1.81677i
\(518\) 0 0
\(519\) −17.6093 −0.772964
\(520\) 0 0
\(521\) 13.1300 0.575237 0.287618 0.957745i \(-0.407136\pi\)
0.287618 + 0.957745i \(0.407136\pi\)
\(522\) 0 0
\(523\) 23.8301i 1.04202i 0.853551 + 0.521010i \(0.174444\pi\)
−0.853551 + 0.521010i \(0.825556\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.45350i 0.193998i
\(528\) 0 0
\(529\) 13.6023 0.591404
\(530\) 0 0
\(531\) −22.9901 −0.997684
\(532\) 0 0
\(533\) − 1.41500i − 0.0612904i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 26.2544i 1.13296i
\(538\) 0 0
\(539\) 38.8726 1.67436
\(540\) 0 0
\(541\) 14.6995 0.631980 0.315990 0.948762i \(-0.397663\pi\)
0.315990 + 0.948762i \(0.397663\pi\)
\(542\) 0 0
\(543\) 29.9528i 1.28540i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.2209i − 0.864584i −0.901734 0.432292i \(-0.857705\pi\)
0.901734 0.432292i \(-0.142295\pi\)
\(548\) 0 0
\(549\) −55.6084 −2.37331
\(550\) 0 0
\(551\) −32.4406 −1.38202
\(552\) 0 0
\(553\) 12.2834i 0.522342i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 40.6784i 1.72360i 0.507249 + 0.861799i \(0.330662\pi\)
−0.507249 + 0.861799i \(0.669338\pi\)
\(558\) 0 0
\(559\) −7.56787 −0.320087
\(560\) 0 0
\(561\) 79.1079 3.33994
\(562\) 0 0
\(563\) − 19.4486i − 0.819661i −0.912162 0.409831i \(-0.865588\pi\)
0.912162 0.409831i \(-0.134412\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 21.2453i 0.892217i
\(568\) 0 0
\(569\) 31.1850 1.30734 0.653672 0.756778i \(-0.273227\pi\)
0.653672 + 0.756778i \(0.273227\pi\)
\(570\) 0 0
\(571\) −0.190921 −0.00798981 −0.00399491 0.999992i \(-0.501272\pi\)
−0.00399491 + 0.999992i \(0.501272\pi\)
\(572\) 0 0
\(573\) − 78.8623i − 3.29452i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.0278i 1.20844i 0.796816 + 0.604222i \(0.206516\pi\)
−0.796816 + 0.604222i \(0.793484\pi\)
\(578\) 0 0
\(579\) 42.9772 1.78607
\(580\) 0 0
\(581\) 7.52441 0.312165
\(582\) 0 0
\(583\) 39.8106i 1.64879i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.5735i 0.477690i 0.971058 + 0.238845i \(0.0767687\pi\)
−0.971058 + 0.238845i \(0.923231\pi\)
\(588\) 0 0
\(589\) −4.33118 −0.178463
\(590\) 0 0
\(591\) 58.6869 2.41405
\(592\) 0 0
\(593\) 34.5557i 1.41903i 0.704689 + 0.709517i \(0.251087\pi\)
−0.704689 + 0.709517i \(0.748913\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 64.0688i 2.62216i
\(598\) 0 0
\(599\) −26.9967 −1.10305 −0.551527 0.834157i \(-0.685955\pi\)
−0.551527 + 0.834157i \(0.685955\pi\)
\(600\) 0 0
\(601\) 47.8534 1.95198 0.975990 0.217816i \(-0.0698932\pi\)
0.975990 + 0.217816i \(0.0698932\pi\)
\(602\) 0 0
\(603\) 13.4469i 0.547601i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 12.3761i − 0.502332i −0.967944 0.251166i \(-0.919186\pi\)
0.967944 0.251166i \(-0.0808140\pi\)
\(608\) 0 0
\(609\) 23.5946 0.956100
\(610\) 0 0
\(611\) −9.41722 −0.380980
\(612\) 0 0
\(613\) 12.9061i 0.521274i 0.965437 + 0.260637i \(0.0839325\pi\)
−0.965437 + 0.260637i \(0.916067\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.5268i 1.63155i 0.578372 + 0.815773i \(0.303688\pi\)
−0.578372 + 0.815773i \(0.696312\pi\)
\(618\) 0 0
\(619\) 2.65889 0.106870 0.0534348 0.998571i \(-0.482983\pi\)
0.0534348 + 0.998571i \(0.482983\pi\)
\(620\) 0 0
\(621\) −44.6944 −1.79353
\(622\) 0 0
\(623\) 6.77934i 0.271609i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 76.9351i 3.07249i
\(628\) 0 0
\(629\) 33.4310 1.33298
\(630\) 0 0
\(631\) −0.139958 −0.00557165 −0.00278583 0.999996i \(-0.500887\pi\)
−0.00278583 + 0.999996i \(0.500887\pi\)
\(632\) 0 0
\(633\) − 4.98757i − 0.198238i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.86180i 0.351117i
\(638\) 0 0
\(639\) 22.6800 0.897205
\(640\) 0 0
\(641\) 9.39675 0.371149 0.185575 0.982630i \(-0.440585\pi\)
0.185575 + 0.982630i \(0.440585\pi\)
\(642\) 0 0
\(643\) 6.02863i 0.237746i 0.992909 + 0.118873i \(0.0379281\pi\)
−0.992909 + 0.118873i \(0.962072\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 17.3861i − 0.683517i −0.939788 0.341758i \(-0.888978\pi\)
0.939788 0.341758i \(-0.111022\pi\)
\(648\) 0 0
\(649\) 19.0278 0.746907
\(650\) 0 0
\(651\) 3.15014 0.123464
\(652\) 0 0
\(653\) − 8.02832i − 0.314173i −0.987585 0.157086i \(-0.949790\pi\)
0.987585 0.157086i \(-0.0502101\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 4.74570i − 0.185147i
\(658\) 0 0
\(659\) 33.8809 1.31981 0.659907 0.751348i \(-0.270596\pi\)
0.659907 + 0.751348i \(0.270596\pi\)
\(660\) 0 0
\(661\) 34.3840 1.33738 0.668691 0.743541i \(-0.266855\pi\)
0.668691 + 0.743541i \(0.266855\pi\)
\(662\) 0 0
\(663\) 18.0343i 0.700393i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 25.9978i 1.00664i
\(668\) 0 0
\(669\) −89.6191 −3.46487
\(670\) 0 0
\(671\) 46.0245 1.77676
\(672\) 0 0
\(673\) 10.3689i 0.399693i 0.979827 + 0.199847i \(0.0640444\pi\)
−0.979827 + 0.199847i \(0.935956\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 28.0616i − 1.07850i −0.842147 0.539248i \(-0.818709\pi\)
0.842147 0.539248i \(-0.181291\pi\)
\(678\) 0 0
\(679\) 8.47175 0.325116
\(680\) 0 0
\(681\) 5.82070 0.223050
\(682\) 0 0
\(683\) 12.2209i 0.467621i 0.972282 + 0.233810i \(0.0751195\pi\)
−0.972282 + 0.233810i \(0.924881\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 70.0317i 2.67188i
\(688\) 0 0
\(689\) −9.07565 −0.345755
\(690\) 0 0
\(691\) 46.1797 1.75676 0.878379 0.477964i \(-0.158625\pi\)
0.878379 + 0.477964i \(0.158625\pi\)
\(692\) 0 0
\(693\) − 39.9679i − 1.51825i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 3.93332i − 0.148985i
\(698\) 0 0
\(699\) −59.3983 −2.24665
\(700\) 0 0
\(701\) −46.8161 −1.76822 −0.884110 0.467279i \(-0.845234\pi\)
−0.884110 + 0.467279i \(0.845234\pi\)
\(702\) 0 0
\(703\) 32.5128i 1.22624i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 3.87955i − 0.145905i
\(708\) 0 0
\(709\) −42.0840 −1.58050 −0.790248 0.612787i \(-0.790048\pi\)
−0.790248 + 0.612787i \(0.790048\pi\)
\(710\) 0 0
\(711\) −107.286 −4.02354
\(712\) 0 0
\(713\) 3.47100i 0.129990i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 38.7494i 1.44712i
\(718\) 0 0
\(719\) −1.77685 −0.0662653 −0.0331327 0.999451i \(-0.510548\pi\)
−0.0331327 + 0.999451i \(0.510548\pi\)
\(720\) 0 0
\(721\) 14.0934 0.524865
\(722\) 0 0
\(723\) − 24.0267i − 0.893563i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.4346i 0.646614i 0.946294 + 0.323307i \(0.104795\pi\)
−0.946294 + 0.323307i \(0.895205\pi\)
\(728\) 0 0
\(729\) −43.8362 −1.62356
\(730\) 0 0
\(731\) −21.0366 −0.778068
\(732\) 0 0
\(733\) − 8.31827i − 0.307242i −0.988130 0.153621i \(-0.950906\pi\)
0.988130 0.153621i \(-0.0490936\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 11.1294i − 0.409957i
\(738\) 0 0
\(739\) 0.139958 0.00514845 0.00257422 0.999997i \(-0.499181\pi\)
0.00257422 + 0.999997i \(0.499181\pi\)
\(740\) 0 0
\(741\) −17.5389 −0.644309
\(742\) 0 0
\(743\) − 3.92389i − 0.143954i −0.997406 0.0719768i \(-0.977069\pi\)
0.997406 0.0719768i \(-0.0229308\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 65.7200i 2.40457i
\(748\) 0 0
\(749\) −13.7934 −0.503998
\(750\) 0 0
\(751\) 38.2786 1.39681 0.698403 0.715705i \(-0.253894\pi\)
0.698403 + 0.715705i \(0.253894\pi\)
\(752\) 0 0
\(753\) 6.20538i 0.226137i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 31.0640i 1.12904i 0.825420 + 0.564519i \(0.190938\pi\)
−0.825420 + 0.564519i \(0.809062\pi\)
\(758\) 0 0
\(759\) 61.6556 2.23795
\(760\) 0 0
\(761\) −2.83945 −0.102930 −0.0514649 0.998675i \(-0.516389\pi\)
−0.0514649 + 0.998675i \(0.516389\pi\)
\(762\) 0 0
\(763\) 4.81931i 0.174471i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.33778i 0.156628i
\(768\) 0 0
\(769\) −38.7888 −1.39876 −0.699379 0.714751i \(-0.746540\pi\)
−0.699379 + 0.714751i \(0.746540\pi\)
\(770\) 0 0
\(771\) −9.50839 −0.342436
\(772\) 0 0
\(773\) 25.7150i 0.924905i 0.886644 + 0.462453i \(0.153030\pi\)
−0.886644 + 0.462453i \(0.846970\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 23.6471i − 0.848335i
\(778\) 0 0
\(779\) 3.82529 0.137055
\(780\) 0 0
\(781\) −18.7712 −0.671685
\(782\) 0 0
\(783\) 123.642i 4.41861i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 41.7817i − 1.48936i −0.667423 0.744679i \(-0.732603\pi\)
0.667423 0.744679i \(-0.267397\pi\)
\(788\) 0 0
\(789\) −10.0673 −0.358406
\(790\) 0 0
\(791\) −4.83552 −0.171931
\(792\) 0 0
\(793\) 10.4922i 0.372590i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 36.2818i − 1.28517i −0.766216 0.642583i \(-0.777863\pi\)
0.766216 0.642583i \(-0.222137\pi\)
\(798\) 0 0
\(799\) −26.1774 −0.926088
\(800\) 0 0
\(801\) −59.2124 −2.09217
\(802\) 0 0
\(803\) 3.92780i 0.138609i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 33.1967i − 1.16858i
\(808\) 0 0
\(809\) 37.8556 1.33093 0.665466 0.746428i \(-0.268233\pi\)
0.665466 + 0.746428i \(0.268233\pi\)
\(810\) 0 0
\(811\) −15.5751 −0.546915 −0.273457 0.961884i \(-0.588167\pi\)
−0.273457 + 0.961884i \(0.588167\pi\)
\(812\) 0 0
\(813\) 36.2877i 1.27267i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 20.4588i − 0.715764i
\(818\) 0 0
\(819\) 9.11149 0.318381
\(820\) 0 0
\(821\) −9.42283 −0.328859 −0.164430 0.986389i \(-0.552578\pi\)
−0.164430 + 0.986389i \(0.552578\pi\)
\(822\) 0 0
\(823\) − 13.9636i − 0.486741i −0.969933 0.243371i \(-0.921747\pi\)
0.969933 0.243371i \(-0.0782532\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 19.6886i − 0.684641i −0.939583 0.342320i \(-0.888787\pi\)
0.939583 0.342320i \(-0.111213\pi\)
\(828\) 0 0
\(829\) −52.9595 −1.83936 −0.919681 0.392668i \(-0.871552\pi\)
−0.919681 + 0.392668i \(0.871552\pi\)
\(830\) 0 0
\(831\) −8.19266 −0.284200
\(832\) 0 0
\(833\) 24.6334i 0.853498i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 16.5076i 0.570587i
\(838\) 0 0
\(839\) 17.1392 0.591709 0.295855 0.955233i \(-0.404396\pi\)
0.295855 + 0.955233i \(0.404396\pi\)
\(840\) 0 0
\(841\) 42.9201 1.48000
\(842\) 0 0
\(843\) 76.3594i 2.62996i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 23.6347i 0.812097i
\(848\) 0 0
\(849\) 29.1590 1.00073
\(850\) 0 0
\(851\) 26.0556 0.893176
\(852\) 0 0
\(853\) − 24.8256i − 0.850011i −0.905191 0.425005i \(-0.860272\pi\)
0.905191 0.425005i \(-0.139728\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 25.6584i − 0.876474i −0.898859 0.438237i \(-0.855603\pi\)
0.898859 0.438237i \(-0.144397\pi\)
\(858\) 0 0
\(859\) −11.9818 −0.408812 −0.204406 0.978886i \(-0.565526\pi\)
−0.204406 + 0.978886i \(0.565526\pi\)
\(860\) 0 0
\(861\) −2.78219 −0.0948169
\(862\) 0 0
\(863\) − 33.9138i − 1.15444i −0.816589 0.577220i \(-0.804138\pi\)
0.816589 0.577220i \(-0.195862\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 4.95441i − 0.168261i
\(868\) 0 0
\(869\) 88.7956 3.01219
\(870\) 0 0
\(871\) 2.53717 0.0859688
\(872\) 0 0
\(873\) 73.9943i 2.50433i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.1871i 0.479066i 0.970888 + 0.239533i \(0.0769943\pi\)
−0.970888 + 0.239533i \(0.923006\pi\)
\(878\) 0 0
\(879\) 10.2950 0.347243
\(880\) 0 0
\(881\) 42.3201 1.42580 0.712901 0.701265i \(-0.247381\pi\)
0.712901 + 0.701265i \(0.247381\pi\)
\(882\) 0 0
\(883\) 51.7258i 1.74071i 0.492422 + 0.870356i \(0.336112\pi\)
−0.492422 + 0.870356i \(0.663888\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 30.1270i − 1.01157i −0.862661 0.505783i \(-0.831204\pi\)
0.862661 0.505783i \(-0.168796\pi\)
\(888\) 0 0
\(889\) 1.45773 0.0488907
\(890\) 0 0
\(891\) 153.580 5.14514
\(892\) 0 0
\(893\) − 25.4584i − 0.851932i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 14.0556i 0.469304i
\(898\) 0 0
\(899\) 9.60213 0.320249
\(900\) 0 0
\(901\) −25.2279 −0.840462
\(902\) 0 0
\(903\) 14.8800i 0.495177i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.24848i 0.141068i 0.997509 + 0.0705342i \(0.0224704\pi\)
−0.997509 + 0.0705342i \(0.977530\pi\)
\(908\) 0 0
\(909\) 33.8849 1.12389
\(910\) 0 0
\(911\) −49.1395 −1.62806 −0.814031 0.580821i \(-0.802732\pi\)
−0.814031 + 0.580821i \(0.802732\pi\)
\(912\) 0 0
\(913\) − 54.3934i − 1.80016i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.12988i − 0.0703349i
\(918\) 0 0
\(919\) −6.59412 −0.217520 −0.108760 0.994068i \(-0.534688\pi\)
−0.108760 + 0.994068i \(0.534688\pi\)
\(920\) 0 0
\(921\) 56.9768 1.87745
\(922\) 0 0
\(923\) − 4.27927i − 0.140854i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 123.095i 4.04297i
\(928\) 0 0
\(929\) −54.3833 −1.78426 −0.892130 0.451779i \(-0.850789\pi\)
−0.892130 + 0.451779i \(0.850789\pi\)
\(930\) 0 0
\(931\) −23.9568 −0.785154
\(932\) 0 0
\(933\) − 5.04669i − 0.165221i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 43.0888i 1.40765i 0.710374 + 0.703825i \(0.248526\pi\)
−0.710374 + 0.703825i \(0.751474\pi\)
\(938\) 0 0
\(939\) 66.5085 2.17042
\(940\) 0 0
\(941\) 12.6966 0.413899 0.206949 0.978352i \(-0.433646\pi\)
0.206949 + 0.978352i \(0.433646\pi\)
\(942\) 0 0
\(943\) − 3.06557i − 0.0998287i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 29.6867i − 0.964688i −0.875982 0.482344i \(-0.839785\pi\)
0.875982 0.482344i \(-0.160215\pi\)
\(948\) 0 0
\(949\) −0.895422 −0.0290666
\(950\) 0 0
\(951\) −84.6019 −2.74341
\(952\) 0 0
\(953\) 7.62947i 0.247143i 0.992336 + 0.123571i \(0.0394348\pi\)
−0.992336 + 0.123571i \(0.960565\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 170.563i − 5.51353i
\(958\) 0 0
\(959\) 2.37266 0.0766171
\(960\) 0 0
\(961\) −29.7180 −0.958645
\(962\) 0 0
\(963\) − 120.475i − 3.88224i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.0187i 1.19044i 0.803562 + 0.595221i \(0.202935\pi\)
−0.803562 + 0.595221i \(0.797065\pi\)
\(968\) 0 0
\(969\) −48.7535 −1.56619
\(970\) 0 0
\(971\) −17.2425 −0.553338 −0.276669 0.960965i \(-0.589231\pi\)
−0.276669 + 0.960965i \(0.589231\pi\)
\(972\) 0 0
\(973\) − 7.72668i − 0.247706i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31.6112i 1.01133i 0.862729 + 0.505666i \(0.168753\pi\)
−0.862729 + 0.505666i \(0.831247\pi\)
\(978\) 0 0
\(979\) 49.0074 1.56628
\(980\) 0 0
\(981\) −42.0930 −1.34393
\(982\) 0 0
\(983\) 51.8543i 1.65390i 0.562278 + 0.826948i \(0.309925\pi\)
−0.562278 + 0.826948i \(0.690075\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.5163i 0.589380i
\(988\) 0 0
\(989\) −16.3956 −0.521351
\(990\) 0 0
\(991\) −37.7209 −1.19824 −0.599121 0.800658i \(-0.704483\pi\)
−0.599121 + 0.800658i \(0.704483\pi\)
\(992\) 0 0
\(993\) − 11.9522i − 0.379291i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.1882i 0.607698i 0.952720 + 0.303849i \(0.0982718\pi\)
−0.952720 + 0.303849i \(0.901728\pi\)
\(998\) 0 0
\(999\) 123.917 3.92058
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.1149.1 8
5.2 odd 4 164.2.a.a.1.1 4
5.3 odd 4 4100.2.a.c.1.4 4
5.4 even 2 inner 4100.2.d.c.1149.8 8
15.2 even 4 1476.2.a.g.1.2 4
20.7 even 4 656.2.a.i.1.4 4
35.27 even 4 8036.2.a.i.1.4 4
40.27 even 4 2624.2.a.y.1.1 4
40.37 odd 4 2624.2.a.v.1.4 4
60.47 odd 4 5904.2.a.bp.1.2 4
205.122 odd 4 6724.2.a.c.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.a.a.1.1 4 5.2 odd 4
656.2.a.i.1.4 4 20.7 even 4
1476.2.a.g.1.2 4 15.2 even 4
2624.2.a.v.1.4 4 40.37 odd 4
2624.2.a.y.1.1 4 40.27 even 4
4100.2.a.c.1.4 4 5.3 odd 4
4100.2.d.c.1149.1 8 1.1 even 1 trivial
4100.2.d.c.1149.8 8 5.4 even 2 inner
5904.2.a.bp.1.2 4 60.47 odd 4
6724.2.a.c.1.4 4 205.122 odd 4
8036.2.a.i.1.4 4 35.27 even 4