Properties

Label 668.2.a.b.1.2
Level $668$
Weight $2$
Character 668.1
Self dual yes
Analytic conductor $5.334$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.826865.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.69135\) of defining polynomial
Character \(\chi\) \(=\) 668.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.55202 q^{3} +2.00000 q^{5} +4.48567 q^{7} -0.591244 q^{9} +O(q^{10})\) \(q-1.55202 q^{3} +2.00000 q^{5} +4.48567 q^{7} -0.591244 q^{9} -1.62500 q^{11} +3.38270 q^{13} -3.10403 q^{15} -5.30425 q^{17} +6.97394 q^{19} -6.96184 q^{21} +0.132692 q^{23} -1.00000 q^{25} +5.57367 q^{27} -7.86837 q^{29} +10.5868 q^{31} +2.52203 q^{33} +8.97134 q^{35} +1.72133 q^{37} -5.25001 q^{39} +8.40828 q^{41} -6.76540 q^{43} -1.18249 q^{45} +0.208543 q^{47} +13.1212 q^{49} +8.23228 q^{51} +8.68695 q^{53} -3.25001 q^{55} -10.8237 q^{57} +13.8152 q^{59} -3.06741 q^{61} -2.65212 q^{63} +6.76540 q^{65} +7.51019 q^{67} -0.205940 q^{69} +2.06752 q^{71} +5.43694 q^{73} +1.55202 q^{75} -7.28924 q^{77} -17.5256 q^{79} -6.87670 q^{81} -12.1481 q^{83} -10.6085 q^{85} +12.2118 q^{87} -10.8020 q^{89} +15.1737 q^{91} -16.4310 q^{93} +13.9479 q^{95} -4.21809 q^{97} +0.960773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + 5q^{11} - 4q^{13} + 6q^{15} - 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} - 5q^{25} + 12q^{27} - 5q^{29} + 9q^{31} - 8q^{33} + 18q^{35} + 8q^{37} - 4q^{41} + 8q^{43} + 12q^{45} + 13q^{47} + 14q^{49} - 2q^{53} + 10q^{55} - 12q^{57} + 4q^{59} + 11q^{61} - q^{63} - 8q^{65} + 28q^{67} - 16q^{69} + 2q^{71} + 8q^{73} - 3q^{75} - 12q^{77} - 10q^{79} - 15q^{81} + 2q^{83} - 4q^{85} + 4q^{87} - 17q^{89} - 12q^{91} - 12q^{93} + 10q^{95} - 27q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) −1.55202 −0.896057 −0.448029 0.894019i \(-0.647874\pi\)
−0.448029 + 0.894019i \(0.647874\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 4.48567 1.69542 0.847712 0.530457i \(-0.177979\pi\)
0.847712 + 0.530457i \(0.177979\pi\)
\(8\) 0 0
\(9\) −0.591244 −0.197081
\(10\) 0 0
\(11\) −1.62500 −0.489957 −0.244979 0.969528i \(-0.578781\pi\)
−0.244979 + 0.969528i \(0.578781\pi\)
\(12\) 0 0
\(13\) 3.38270 0.938192 0.469096 0.883147i \(-0.344580\pi\)
0.469096 + 0.883147i \(0.344580\pi\)
\(14\) 0 0
\(15\) −3.10403 −0.801458
\(16\) 0 0
\(17\) −5.30425 −1.28647 −0.643235 0.765669i \(-0.722408\pi\)
−0.643235 + 0.765669i \(0.722408\pi\)
\(18\) 0 0
\(19\) 6.97394 1.59993 0.799966 0.600045i \(-0.204851\pi\)
0.799966 + 0.600045i \(0.204851\pi\)
\(20\) 0 0
\(21\) −6.96184 −1.51920
\(22\) 0 0
\(23\) 0.132692 0.0276682 0.0138341 0.999904i \(-0.495596\pi\)
0.0138341 + 0.999904i \(0.495596\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.57367 1.07265
\(28\) 0 0
\(29\) −7.86837 −1.46112 −0.730560 0.682849i \(-0.760741\pi\)
−0.730560 + 0.682849i \(0.760741\pi\)
\(30\) 0 0
\(31\) 10.5868 1.90145 0.950726 0.310031i \(-0.100339\pi\)
0.950726 + 0.310031i \(0.100339\pi\)
\(32\) 0 0
\(33\) 2.52203 0.439030
\(34\) 0 0
\(35\) 8.97134 1.51643
\(36\) 0 0
\(37\) 1.72133 0.282985 0.141493 0.989939i \(-0.454810\pi\)
0.141493 + 0.989939i \(0.454810\pi\)
\(38\) 0 0
\(39\) −5.25001 −0.840674
\(40\) 0 0
\(41\) 8.40828 1.31315 0.656576 0.754259i \(-0.272004\pi\)
0.656576 + 0.754259i \(0.272004\pi\)
\(42\) 0 0
\(43\) −6.76540 −1.03171 −0.515857 0.856675i \(-0.672526\pi\)
−0.515857 + 0.856675i \(0.672526\pi\)
\(44\) 0 0
\(45\) −1.18249 −0.176275
\(46\) 0 0
\(47\) 0.208543 0.0304191 0.0152095 0.999884i \(-0.495158\pi\)
0.0152095 + 0.999884i \(0.495158\pi\)
\(48\) 0 0
\(49\) 13.1212 1.87446
\(50\) 0 0
\(51\) 8.23228 1.15275
\(52\) 0 0
\(53\) 8.68695 1.19324 0.596622 0.802522i \(-0.296509\pi\)
0.596622 + 0.802522i \(0.296509\pi\)
\(54\) 0 0
\(55\) −3.25001 −0.438231
\(56\) 0 0
\(57\) −10.8237 −1.43363
\(58\) 0 0
\(59\) 13.8152 1.79859 0.899293 0.437347i \(-0.144082\pi\)
0.899293 + 0.437347i \(0.144082\pi\)
\(60\) 0 0
\(61\) −3.06741 −0.392742 −0.196371 0.980530i \(-0.562916\pi\)
−0.196371 + 0.980530i \(0.562916\pi\)
\(62\) 0 0
\(63\) −2.65212 −0.334136
\(64\) 0 0
\(65\) 6.76540 0.839145
\(66\) 0 0
\(67\) 7.51019 0.917515 0.458758 0.888561i \(-0.348295\pi\)
0.458758 + 0.888561i \(0.348295\pi\)
\(68\) 0 0
\(69\) −0.205940 −0.0247923
\(70\) 0 0
\(71\) 2.06752 0.245370 0.122685 0.992446i \(-0.460850\pi\)
0.122685 + 0.992446i \(0.460850\pi\)
\(72\) 0 0
\(73\) 5.43694 0.636346 0.318173 0.948033i \(-0.396931\pi\)
0.318173 + 0.948033i \(0.396931\pi\)
\(74\) 0 0
\(75\) 1.55202 0.179211
\(76\) 0 0
\(77\) −7.28924 −0.830686
\(78\) 0 0
\(79\) −17.5256 −1.97178 −0.985892 0.167383i \(-0.946468\pi\)
−0.985892 + 0.167383i \(0.946468\pi\)
\(80\) 0 0
\(81\) −6.87670 −0.764078
\(82\) 0 0
\(83\) −12.1481 −1.33343 −0.666714 0.745314i \(-0.732300\pi\)
−0.666714 + 0.745314i \(0.732300\pi\)
\(84\) 0 0
\(85\) −10.6085 −1.15065
\(86\) 0 0
\(87\) 12.2118 1.30925
\(88\) 0 0
\(89\) −10.8020 −1.14501 −0.572506 0.819900i \(-0.694029\pi\)
−0.572506 + 0.819900i \(0.694029\pi\)
\(90\) 0 0
\(91\) 15.1737 1.59063
\(92\) 0 0
\(93\) −16.4310 −1.70381
\(94\) 0 0
\(95\) 13.9479 1.43102
\(96\) 0 0
\(97\) −4.21809 −0.428282 −0.214141 0.976803i \(-0.568695\pi\)
−0.214141 + 0.976803i \(0.568695\pi\)
\(98\) 0 0
\(99\) 0.960773 0.0965614
\(100\) 0 0
\(101\) −2.33291 −0.232133 −0.116066 0.993241i \(-0.537029\pi\)
−0.116066 + 0.993241i \(0.537029\pi\)
\(102\) 0 0
\(103\) −11.3121 −1.11461 −0.557307 0.830306i \(-0.688165\pi\)
−0.557307 + 0.830306i \(0.688165\pi\)
\(104\) 0 0
\(105\) −13.9237 −1.35881
\(106\) 0 0
\(107\) 11.0942 1.07251 0.536257 0.844055i \(-0.319838\pi\)
0.536257 + 0.844055i \(0.319838\pi\)
\(108\) 0 0
\(109\) −2.77633 −0.265924 −0.132962 0.991121i \(-0.542449\pi\)
−0.132962 + 0.991121i \(0.542449\pi\)
\(110\) 0 0
\(111\) −2.67154 −0.253571
\(112\) 0 0
\(113\) −17.2579 −1.62348 −0.811742 0.584017i \(-0.801480\pi\)
−0.811742 + 0.584017i \(0.801480\pi\)
\(114\) 0 0
\(115\) 0.265384 0.0247472
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −23.7931 −2.18111
\(120\) 0 0
\(121\) −8.35936 −0.759942
\(122\) 0 0
\(123\) −13.0498 −1.17666
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.31768 −0.738075 −0.369037 0.929415i \(-0.620313\pi\)
−0.369037 + 0.929415i \(0.620313\pi\)
\(128\) 0 0
\(129\) 10.5000 0.924475
\(130\) 0 0
\(131\) −5.84309 −0.510513 −0.255257 0.966873i \(-0.582160\pi\)
−0.255257 + 0.966873i \(0.582160\pi\)
\(132\) 0 0
\(133\) 31.2828 2.71256
\(134\) 0 0
\(135\) 11.1473 0.959410
\(136\) 0 0
\(137\) 13.6762 1.16843 0.584217 0.811598i \(-0.301402\pi\)
0.584217 + 0.811598i \(0.301402\pi\)
\(138\) 0 0
\(139\) 10.2998 0.873618 0.436809 0.899554i \(-0.356109\pi\)
0.436809 + 0.899554i \(0.356109\pi\)
\(140\) 0 0
\(141\) −0.323662 −0.0272572
\(142\) 0 0
\(143\) −5.49690 −0.459674
\(144\) 0 0
\(145\) −15.7367 −1.30687
\(146\) 0 0
\(147\) −20.3644 −1.67963
\(148\) 0 0
\(149\) −7.84309 −0.642531 −0.321266 0.946989i \(-0.604108\pi\)
−0.321266 + 0.946989i \(0.604108\pi\)
\(150\) 0 0
\(151\) −15.5232 −1.26326 −0.631632 0.775268i \(-0.717615\pi\)
−0.631632 + 0.775268i \(0.717615\pi\)
\(152\) 0 0
\(153\) 3.13610 0.253539
\(154\) 0 0
\(155\) 21.1737 1.70071
\(156\) 0 0
\(157\) 17.4174 1.39006 0.695030 0.718980i \(-0.255391\pi\)
0.695030 + 0.718980i \(0.255391\pi\)
\(158\) 0 0
\(159\) −13.4823 −1.06922
\(160\) 0 0
\(161\) 0.595212 0.0469093
\(162\) 0 0
\(163\) 23.9912 1.87914 0.939568 0.342363i \(-0.111227\pi\)
0.939568 + 0.342363i \(0.111227\pi\)
\(164\) 0 0
\(165\) 5.04407 0.392680
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −1.55733 −0.119795
\(170\) 0 0
\(171\) −4.12330 −0.315317
\(172\) 0 0
\(173\) −1.68232 −0.127904 −0.0639522 0.997953i \(-0.520371\pi\)
−0.0639522 + 0.997953i \(0.520371\pi\)
\(174\) 0 0
\(175\) −4.48567 −0.339085
\(176\) 0 0
\(177\) −21.4414 −1.61164
\(178\) 0 0
\(179\) −0.760672 −0.0568553 −0.0284276 0.999596i \(-0.509050\pi\)
−0.0284276 + 0.999596i \(0.509050\pi\)
\(180\) 0 0
\(181\) −6.35665 −0.472486 −0.236243 0.971694i \(-0.575916\pi\)
−0.236243 + 0.971694i \(0.575916\pi\)
\(182\) 0 0
\(183\) 4.76067 0.351919
\(184\) 0 0
\(185\) 3.44267 0.253110
\(186\) 0 0
\(187\) 8.61943 0.630315
\(188\) 0 0
\(189\) 25.0017 1.81860
\(190\) 0 0
\(191\) 11.1989 0.810327 0.405163 0.914244i \(-0.367215\pi\)
0.405163 + 0.914244i \(0.367215\pi\)
\(192\) 0 0
\(193\) −14.9471 −1.07592 −0.537959 0.842971i \(-0.680804\pi\)
−0.537959 + 0.842971i \(0.680804\pi\)
\(194\) 0 0
\(195\) −10.5000 −0.751922
\(196\) 0 0
\(197\) 18.0775 1.28797 0.643984 0.765039i \(-0.277280\pi\)
0.643984 + 0.765039i \(0.277280\pi\)
\(198\) 0 0
\(199\) −3.67617 −0.260596 −0.130298 0.991475i \(-0.541593\pi\)
−0.130298 + 0.991475i \(0.541593\pi\)
\(200\) 0 0
\(201\) −11.6559 −0.822147
\(202\) 0 0
\(203\) −35.2949 −2.47722
\(204\) 0 0
\(205\) 16.8166 1.17452
\(206\) 0 0
\(207\) −0.0784532 −0.00545288
\(208\) 0 0
\(209\) −11.3327 −0.783899
\(210\) 0 0
\(211\) −10.2297 −0.704243 −0.352122 0.935954i \(-0.614540\pi\)
−0.352122 + 0.935954i \(0.614540\pi\)
\(212\) 0 0
\(213\) −3.20883 −0.219865
\(214\) 0 0
\(215\) −13.5308 −0.922793
\(216\) 0 0
\(217\) 47.4891 3.22377
\(218\) 0 0
\(219\) −8.43822 −0.570202
\(220\) 0 0
\(221\) −17.9427 −1.20696
\(222\) 0 0
\(223\) −22.1032 −1.48014 −0.740070 0.672530i \(-0.765208\pi\)
−0.740070 + 0.672530i \(0.765208\pi\)
\(224\) 0 0
\(225\) 0.591244 0.0394162
\(226\) 0 0
\(227\) 22.2050 1.47380 0.736898 0.676003i \(-0.236290\pi\)
0.736898 + 0.676003i \(0.236290\pi\)
\(228\) 0 0
\(229\) 1.66030 0.109716 0.0548580 0.998494i \(-0.482529\pi\)
0.0548580 + 0.998494i \(0.482529\pi\)
\(230\) 0 0
\(231\) 11.3130 0.744342
\(232\) 0 0
\(233\) −19.0524 −1.24817 −0.624083 0.781358i \(-0.714527\pi\)
−0.624083 + 0.781358i \(0.714527\pi\)
\(234\) 0 0
\(235\) 0.417085 0.0272076
\(236\) 0 0
\(237\) 27.2000 1.76683
\(238\) 0 0
\(239\) 5.68706 0.367865 0.183933 0.982939i \(-0.441117\pi\)
0.183933 + 0.982939i \(0.441117\pi\)
\(240\) 0 0
\(241\) −15.9721 −1.02885 −0.514427 0.857534i \(-0.671995\pi\)
−0.514427 + 0.857534i \(0.671995\pi\)
\(242\) 0 0
\(243\) −6.04826 −0.387996
\(244\) 0 0
\(245\) 26.2425 1.67657
\(246\) 0 0
\(247\) 23.5908 1.50104
\(248\) 0 0
\(249\) 18.8541 1.19483
\(250\) 0 0
\(251\) −30.9878 −1.95593 −0.977967 0.208760i \(-0.933057\pi\)
−0.977967 + 0.208760i \(0.933057\pi\)
\(252\) 0 0
\(253\) −0.215625 −0.0135562
\(254\) 0 0
\(255\) 16.4646 1.03105
\(256\) 0 0
\(257\) −3.32611 −0.207477 −0.103739 0.994605i \(-0.533081\pi\)
−0.103739 + 0.994605i \(0.533081\pi\)
\(258\) 0 0
\(259\) 7.72133 0.479780
\(260\) 0 0
\(261\) 4.65212 0.287959
\(262\) 0 0
\(263\) 8.47433 0.522549 0.261275 0.965265i \(-0.415857\pi\)
0.261275 + 0.965265i \(0.415857\pi\)
\(264\) 0 0
\(265\) 17.3739 1.06727
\(266\) 0 0
\(267\) 16.7649 1.02600
\(268\) 0 0
\(269\) −2.94140 −0.179340 −0.0896702 0.995972i \(-0.528581\pi\)
−0.0896702 + 0.995972i \(0.528581\pi\)
\(270\) 0 0
\(271\) 12.7654 0.775443 0.387721 0.921777i \(-0.373262\pi\)
0.387721 + 0.921777i \(0.373262\pi\)
\(272\) 0 0
\(273\) −23.5498 −1.42530
\(274\) 0 0
\(275\) 1.62500 0.0979915
\(276\) 0 0
\(277\) −6.54097 −0.393009 −0.196505 0.980503i \(-0.562959\pi\)
−0.196505 + 0.980503i \(0.562959\pi\)
\(278\) 0 0
\(279\) −6.25940 −0.374741
\(280\) 0 0
\(281\) 5.83281 0.347956 0.173978 0.984750i \(-0.444338\pi\)
0.173978 + 0.984750i \(0.444338\pi\)
\(282\) 0 0
\(283\) 6.70270 0.398434 0.199217 0.979955i \(-0.436160\pi\)
0.199217 + 0.979955i \(0.436160\pi\)
\(284\) 0 0
\(285\) −21.6474 −1.28228
\(286\) 0 0
\(287\) 37.7168 2.22635
\(288\) 0 0
\(289\) 11.1350 0.655003
\(290\) 0 0
\(291\) 6.54655 0.383765
\(292\) 0 0
\(293\) −0.113390 −0.00662433 −0.00331217 0.999995i \(-0.501054\pi\)
−0.00331217 + 0.999995i \(0.501054\pi\)
\(294\) 0 0
\(295\) 27.6304 1.60870
\(296\) 0 0
\(297\) −9.05724 −0.525554
\(298\) 0 0
\(299\) 0.448857 0.0259581
\(300\) 0 0
\(301\) −30.3474 −1.74919
\(302\) 0 0
\(303\) 3.62071 0.208004
\(304\) 0 0
\(305\) −6.13482 −0.351279
\(306\) 0 0
\(307\) −19.7204 −1.12550 −0.562751 0.826627i \(-0.690257\pi\)
−0.562751 + 0.826627i \(0.690257\pi\)
\(308\) 0 0
\(309\) 17.5566 0.998759
\(310\) 0 0
\(311\) −28.2858 −1.60394 −0.801970 0.597364i \(-0.796215\pi\)
−0.801970 + 0.597364i \(0.796215\pi\)
\(312\) 0 0
\(313\) −2.95285 −0.166905 −0.0834526 0.996512i \(-0.526595\pi\)
−0.0834526 + 0.996512i \(0.526595\pi\)
\(314\) 0 0
\(315\) −5.30425 −0.298861
\(316\) 0 0
\(317\) −23.5101 −1.32046 −0.660230 0.751064i \(-0.729541\pi\)
−0.660230 + 0.751064i \(0.729541\pi\)
\(318\) 0 0
\(319\) 12.7861 0.715886
\(320\) 0 0
\(321\) −17.2183 −0.961034
\(322\) 0 0
\(323\) −36.9915 −2.05826
\(324\) 0 0
\(325\) −3.38270 −0.187638
\(326\) 0 0
\(327\) 4.30892 0.238284
\(328\) 0 0
\(329\) 0.935453 0.0515732
\(330\) 0 0
\(331\) −3.53388 −0.194240 −0.0971198 0.995273i \(-0.530963\pi\)
−0.0971198 + 0.995273i \(0.530963\pi\)
\(332\) 0 0
\(333\) −1.01773 −0.0557711
\(334\) 0 0
\(335\) 15.0204 0.820651
\(336\) 0 0
\(337\) −30.7676 −1.67602 −0.838008 0.545657i \(-0.816280\pi\)
−0.838008 + 0.545657i \(0.816280\pi\)
\(338\) 0 0
\(339\) 26.7845 1.45473
\(340\) 0 0
\(341\) −17.2037 −0.931631
\(342\) 0 0
\(343\) 27.4579 1.48259
\(344\) 0 0
\(345\) −0.411880 −0.0221749
\(346\) 0 0
\(347\) 8.95468 0.480713 0.240356 0.970685i \(-0.422736\pi\)
0.240356 + 0.970685i \(0.422736\pi\)
\(348\) 0 0
\(349\) 3.10711 0.166320 0.0831599 0.996536i \(-0.473499\pi\)
0.0831599 + 0.996536i \(0.473499\pi\)
\(350\) 0 0
\(351\) 18.8541 1.00636
\(352\) 0 0
\(353\) −12.0170 −0.639600 −0.319800 0.947485i \(-0.603616\pi\)
−0.319800 + 0.947485i \(0.603616\pi\)
\(354\) 0 0
\(355\) 4.13504 0.219465
\(356\) 0 0
\(357\) 36.9273 1.95440
\(358\) 0 0
\(359\) −28.7949 −1.51974 −0.759869 0.650077i \(-0.774737\pi\)
−0.759869 + 0.650077i \(0.774737\pi\)
\(360\) 0 0
\(361\) 29.6359 1.55978
\(362\) 0 0
\(363\) 12.9739 0.680951
\(364\) 0 0
\(365\) 10.8739 0.569165
\(366\) 0 0
\(367\) 7.28255 0.380146 0.190073 0.981770i \(-0.439128\pi\)
0.190073 + 0.981770i \(0.439128\pi\)
\(368\) 0 0
\(369\) −4.97134 −0.258798
\(370\) 0 0
\(371\) 38.9668 2.02306
\(372\) 0 0
\(373\) 25.2500 1.30740 0.653698 0.756756i \(-0.273217\pi\)
0.653698 + 0.756756i \(0.273217\pi\)
\(374\) 0 0
\(375\) 18.6242 0.961750
\(376\) 0 0
\(377\) −26.6163 −1.37081
\(378\) 0 0
\(379\) −15.2733 −0.784535 −0.392268 0.919851i \(-0.628309\pi\)
−0.392268 + 0.919851i \(0.628309\pi\)
\(380\) 0 0
\(381\) 12.9092 0.661357
\(382\) 0 0
\(383\) 2.83494 0.144859 0.0724293 0.997374i \(-0.476925\pi\)
0.0724293 + 0.997374i \(0.476925\pi\)
\(384\) 0 0
\(385\) −14.5785 −0.742988
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) −17.8121 −0.903110 −0.451555 0.892243i \(-0.649131\pi\)
−0.451555 + 0.892243i \(0.649131\pi\)
\(390\) 0 0
\(391\) −0.703831 −0.0355942
\(392\) 0 0
\(393\) 9.06858 0.457449
\(394\) 0 0
\(395\) −35.0512 −1.76362
\(396\) 0 0
\(397\) 18.5585 0.931425 0.465712 0.884936i \(-0.345798\pi\)
0.465712 + 0.884936i \(0.345798\pi\)
\(398\) 0 0
\(399\) −48.5515 −2.43061
\(400\) 0 0
\(401\) 19.8040 0.988967 0.494483 0.869187i \(-0.335357\pi\)
0.494483 + 0.869187i \(0.335357\pi\)
\(402\) 0 0
\(403\) 35.8121 1.78393
\(404\) 0 0
\(405\) −13.7534 −0.683412
\(406\) 0 0
\(407\) −2.79717 −0.138651
\(408\) 0 0
\(409\) 24.2604 1.19960 0.599799 0.800151i \(-0.295247\pi\)
0.599799 + 0.800151i \(0.295247\pi\)
\(410\) 0 0
\(411\) −21.2256 −1.04698
\(412\) 0 0
\(413\) 61.9704 3.04937
\(414\) 0 0
\(415\) −24.2962 −1.19265
\(416\) 0 0
\(417\) −15.9855 −0.782812
\(418\) 0 0
\(419\) −16.5947 −0.810704 −0.405352 0.914161i \(-0.632851\pi\)
−0.405352 + 0.914161i \(0.632851\pi\)
\(420\) 0 0
\(421\) −24.1760 −1.17827 −0.589134 0.808035i \(-0.700531\pi\)
−0.589134 + 0.808035i \(0.700531\pi\)
\(422\) 0 0
\(423\) −0.123299 −0.00599502
\(424\) 0 0
\(425\) 5.30425 0.257294
\(426\) 0 0
\(427\) −13.7594 −0.665864
\(428\) 0 0
\(429\) 8.53129 0.411894
\(430\) 0 0
\(431\) 0.233761 0.0112598 0.00562992 0.999984i \(-0.498208\pi\)
0.00562992 + 0.999984i \(0.498208\pi\)
\(432\) 0 0
\(433\) 31.0595 1.49263 0.746313 0.665595i \(-0.231822\pi\)
0.746313 + 0.665595i \(0.231822\pi\)
\(434\) 0 0
\(435\) 24.4237 1.17103
\(436\) 0 0
\(437\) 0.925386 0.0442672
\(438\) 0 0
\(439\) −20.3994 −0.973610 −0.486805 0.873511i \(-0.661838\pi\)
−0.486805 + 0.873511i \(0.661838\pi\)
\(440\) 0 0
\(441\) −7.75785 −0.369421
\(442\) 0 0
\(443\) 0.445779 0.0211796 0.0105898 0.999944i \(-0.496629\pi\)
0.0105898 + 0.999944i \(0.496629\pi\)
\(444\) 0 0
\(445\) −21.6041 −1.02413
\(446\) 0 0
\(447\) 12.1726 0.575745
\(448\) 0 0
\(449\) 23.8874 1.12732 0.563659 0.826007i \(-0.309393\pi\)
0.563659 + 0.826007i \(0.309393\pi\)
\(450\) 0 0
\(451\) −13.6635 −0.643389
\(452\) 0 0
\(453\) 24.0923 1.13196
\(454\) 0 0
\(455\) 30.3474 1.42271
\(456\) 0 0
\(457\) 11.8064 0.552280 0.276140 0.961117i \(-0.410945\pi\)
0.276140 + 0.961117i \(0.410945\pi\)
\(458\) 0 0
\(459\) −29.5641 −1.37994
\(460\) 0 0
\(461\) 14.0913 0.656295 0.328148 0.944626i \(-0.393576\pi\)
0.328148 + 0.944626i \(0.393576\pi\)
\(462\) 0 0
\(463\) 14.6663 0.681602 0.340801 0.940135i \(-0.389302\pi\)
0.340801 + 0.940135i \(0.389302\pi\)
\(464\) 0 0
\(465\) −32.8619 −1.52393
\(466\) 0 0
\(467\) 30.8445 1.42731 0.713656 0.700496i \(-0.247038\pi\)
0.713656 + 0.700496i \(0.247038\pi\)
\(468\) 0 0
\(469\) 33.6882 1.55558
\(470\) 0 0
\(471\) −27.0321 −1.24557
\(472\) 0 0
\(473\) 10.9938 0.505496
\(474\) 0 0
\(475\) −6.97394 −0.319987
\(476\) 0 0
\(477\) −5.13610 −0.235166
\(478\) 0 0
\(479\) 4.40615 0.201322 0.100661 0.994921i \(-0.467904\pi\)
0.100661 + 0.994921i \(0.467904\pi\)
\(480\) 0 0
\(481\) 5.82275 0.265495
\(482\) 0 0
\(483\) −0.923779 −0.0420334
\(484\) 0 0
\(485\) −8.43618 −0.383067
\(486\) 0 0
\(487\) 24.2128 1.09719 0.548594 0.836089i \(-0.315163\pi\)
0.548594 + 0.836089i \(0.315163\pi\)
\(488\) 0 0
\(489\) −37.2347 −1.68381
\(490\) 0 0
\(491\) −32.9516 −1.48709 −0.743543 0.668689i \(-0.766856\pi\)
−0.743543 + 0.668689i \(0.766856\pi\)
\(492\) 0 0
\(493\) 41.7358 1.87969
\(494\) 0 0
\(495\) 1.92155 0.0863671
\(496\) 0 0
\(497\) 9.27422 0.416006
\(498\) 0 0
\(499\) 9.79619 0.438538 0.219269 0.975664i \(-0.429633\pi\)
0.219269 + 0.975664i \(0.429633\pi\)
\(500\) 0 0
\(501\) 1.55202 0.0693390
\(502\) 0 0
\(503\) 14.0414 0.626075 0.313038 0.949741i \(-0.398653\pi\)
0.313038 + 0.949741i \(0.398653\pi\)
\(504\) 0 0
\(505\) −4.66581 −0.207626
\(506\) 0 0
\(507\) 2.41701 0.107343
\(508\) 0 0
\(509\) −30.9992 −1.37402 −0.687008 0.726650i \(-0.741076\pi\)
−0.687008 + 0.726650i \(0.741076\pi\)
\(510\) 0 0
\(511\) 24.3883 1.07888
\(512\) 0 0
\(513\) 38.8705 1.71617
\(514\) 0 0
\(515\) −22.6242 −0.996941
\(516\) 0 0
\(517\) −0.338883 −0.0149040
\(518\) 0 0
\(519\) 2.61099 0.114610
\(520\) 0 0
\(521\) 23.3877 1.02463 0.512317 0.858796i \(-0.328787\pi\)
0.512317 + 0.858796i \(0.328787\pi\)
\(522\) 0 0
\(523\) −14.2130 −0.621490 −0.310745 0.950493i \(-0.600579\pi\)
−0.310745 + 0.950493i \(0.600579\pi\)
\(524\) 0 0
\(525\) 6.96184 0.303839
\(526\) 0 0
\(527\) −56.1552 −2.44616
\(528\) 0 0
\(529\) −22.9824 −0.999234
\(530\) 0 0
\(531\) −8.16815 −0.354467
\(532\) 0 0
\(533\) 28.4427 1.23199
\(534\) 0 0
\(535\) 22.1883 0.959285
\(536\) 0 0
\(537\) 1.18058 0.0509456
\(538\) 0 0
\(539\) −21.3221 −0.918407
\(540\) 0 0
\(541\) −3.61867 −0.155579 −0.0777893 0.996970i \(-0.524786\pi\)
−0.0777893 + 0.996970i \(0.524786\pi\)
\(542\) 0 0
\(543\) 9.86562 0.423374
\(544\) 0 0
\(545\) −5.55267 −0.237850
\(546\) 0 0
\(547\) 16.1945 0.692426 0.346213 0.938156i \(-0.387467\pi\)
0.346213 + 0.938156i \(0.387467\pi\)
\(548\) 0 0
\(549\) 1.81359 0.0774020
\(550\) 0 0
\(551\) −54.8736 −2.33769
\(552\) 0 0
\(553\) −78.6141 −3.34301
\(554\) 0 0
\(555\) −5.34308 −0.226801
\(556\) 0 0
\(557\) −5.39689 −0.228674 −0.114337 0.993442i \(-0.536474\pi\)
−0.114337 + 0.993442i \(0.536474\pi\)
\(558\) 0 0
\(559\) −22.8853 −0.967946
\(560\) 0 0
\(561\) −13.3775 −0.564798
\(562\) 0 0
\(563\) 38.3990 1.61833 0.809163 0.587585i \(-0.199921\pi\)
0.809163 + 0.587585i \(0.199921\pi\)
\(564\) 0 0
\(565\) −34.5157 −1.45209
\(566\) 0 0
\(567\) −30.8466 −1.29544
\(568\) 0 0
\(569\) −7.14674 −0.299607 −0.149803 0.988716i \(-0.547864\pi\)
−0.149803 + 0.988716i \(0.547864\pi\)
\(570\) 0 0
\(571\) 9.09003 0.380406 0.190203 0.981745i \(-0.439085\pi\)
0.190203 + 0.981745i \(0.439085\pi\)
\(572\) 0 0
\(573\) −17.3809 −0.726099
\(574\) 0 0
\(575\) −0.132692 −0.00553363
\(576\) 0 0
\(577\) −5.92103 −0.246496 −0.123248 0.992376i \(-0.539331\pi\)
−0.123248 + 0.992376i \(0.539331\pi\)
\(578\) 0 0
\(579\) 23.1982 0.964084
\(580\) 0 0
\(581\) −54.4924 −2.26073
\(582\) 0 0
\(583\) −14.1163 −0.584639
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 27.2003 1.12267 0.561337 0.827587i \(-0.310287\pi\)
0.561337 + 0.827587i \(0.310287\pi\)
\(588\) 0 0
\(589\) 73.8320 3.04220
\(590\) 0 0
\(591\) −28.0566 −1.15409
\(592\) 0 0
\(593\) 16.3314 0.670648 0.335324 0.942103i \(-0.391154\pi\)
0.335324 + 0.942103i \(0.391154\pi\)
\(594\) 0 0
\(595\) −47.5862 −1.95084
\(596\) 0 0
\(597\) 5.70547 0.233509
\(598\) 0 0
\(599\) −14.6823 −0.599902 −0.299951 0.953955i \(-0.596970\pi\)
−0.299951 + 0.953955i \(0.596970\pi\)
\(600\) 0 0
\(601\) −20.3773 −0.831206 −0.415603 0.909546i \(-0.636430\pi\)
−0.415603 + 0.909546i \(0.636430\pi\)
\(602\) 0 0
\(603\) −4.44035 −0.180825
\(604\) 0 0
\(605\) −16.7187 −0.679713
\(606\) 0 0
\(607\) −1.72706 −0.0700992 −0.0350496 0.999386i \(-0.511159\pi\)
−0.0350496 + 0.999386i \(0.511159\pi\)
\(608\) 0 0
\(609\) 54.7783 2.21973
\(610\) 0 0
\(611\) 0.705437 0.0285389
\(612\) 0 0
\(613\) −9.43357 −0.381018 −0.190509 0.981685i \(-0.561014\pi\)
−0.190509 + 0.981685i \(0.561014\pi\)
\(614\) 0 0
\(615\) −26.0996 −1.05244
\(616\) 0 0
\(617\) −19.2934 −0.776724 −0.388362 0.921507i \(-0.626959\pi\)
−0.388362 + 0.921507i \(0.626959\pi\)
\(618\) 0 0
\(619\) −11.7698 −0.473067 −0.236533 0.971623i \(-0.576011\pi\)
−0.236533 + 0.971623i \(0.576011\pi\)
\(620\) 0 0
\(621\) 0.739581 0.0296784
\(622\) 0 0
\(623\) −48.4543 −1.94128
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 17.5885 0.702418
\(628\) 0 0
\(629\) −9.13038 −0.364052
\(630\) 0 0
\(631\) 35.7072 1.42148 0.710742 0.703453i \(-0.248360\pi\)
0.710742 + 0.703453i \(0.248360\pi\)
\(632\) 0 0
\(633\) 15.8767 0.631042
\(634\) 0 0
\(635\) −16.6354 −0.660154
\(636\) 0 0
\(637\) 44.3852 1.75861
\(638\) 0 0
\(639\) −1.22241 −0.0483578
\(640\) 0 0
\(641\) −32.4396 −1.28129 −0.640644 0.767838i \(-0.721333\pi\)
−0.640644 + 0.767838i \(0.721333\pi\)
\(642\) 0 0
\(643\) 24.8629 0.980496 0.490248 0.871583i \(-0.336906\pi\)
0.490248 + 0.871583i \(0.336906\pi\)
\(644\) 0 0
\(645\) 21.0000 0.826876
\(646\) 0 0
\(647\) −22.5888 −0.888056 −0.444028 0.896013i \(-0.646451\pi\)
−0.444028 + 0.896013i \(0.646451\pi\)
\(648\) 0 0
\(649\) −22.4498 −0.881230
\(650\) 0 0
\(651\) −73.7039 −2.88868
\(652\) 0 0
\(653\) 22.9597 0.898481 0.449241 0.893411i \(-0.351695\pi\)
0.449241 + 0.893411i \(0.351695\pi\)
\(654\) 0 0
\(655\) −11.6862 −0.456617
\(656\) 0 0
\(657\) −3.21456 −0.125412
\(658\) 0 0
\(659\) −1.23673 −0.0481760 −0.0240880 0.999710i \(-0.507668\pi\)
−0.0240880 + 0.999710i \(0.507668\pi\)
\(660\) 0 0
\(661\) 12.5706 0.488940 0.244470 0.969657i \(-0.421386\pi\)
0.244470 + 0.969657i \(0.421386\pi\)
\(662\) 0 0
\(663\) 27.8473 1.08150
\(664\) 0 0
\(665\) 62.5656 2.42619
\(666\) 0 0
\(667\) −1.04407 −0.0404265
\(668\) 0 0
\(669\) 34.3045 1.32629
\(670\) 0 0
\(671\) 4.98455 0.192427
\(672\) 0 0
\(673\) −48.5661 −1.87208 −0.936042 0.351888i \(-0.885540\pi\)
−0.936042 + 0.351888i \(0.885540\pi\)
\(674\) 0 0
\(675\) −5.57367 −0.214531
\(676\) 0 0
\(677\) 3.90041 0.149905 0.0749525 0.997187i \(-0.476119\pi\)
0.0749525 + 0.997187i \(0.476119\pi\)
\(678\) 0 0
\(679\) −18.9210 −0.726120
\(680\) 0 0
\(681\) −34.4625 −1.32061
\(682\) 0 0
\(683\) −17.2389 −0.659626 −0.329813 0.944046i \(-0.606986\pi\)
−0.329813 + 0.944046i \(0.606986\pi\)
\(684\) 0 0
\(685\) 27.3523 1.04508
\(686\) 0 0
\(687\) −2.57682 −0.0983118
\(688\) 0 0
\(689\) 29.3853 1.11949
\(690\) 0 0
\(691\) −26.1641 −0.995330 −0.497665 0.867369i \(-0.665809\pi\)
−0.497665 + 0.867369i \(0.665809\pi\)
\(692\) 0 0
\(693\) 4.30971 0.163712
\(694\) 0 0
\(695\) 20.5996 0.781388
\(696\) 0 0
\(697\) −44.5996 −1.68933
\(698\) 0 0
\(699\) 29.5697 1.11843
\(700\) 0 0
\(701\) 31.0982 1.17456 0.587282 0.809382i \(-0.300198\pi\)
0.587282 + 0.809382i \(0.300198\pi\)
\(702\) 0 0
\(703\) 12.0045 0.452758
\(704\) 0 0
\(705\) −0.647323 −0.0243796
\(706\) 0 0
\(707\) −10.4646 −0.393564
\(708\) 0 0
\(709\) −9.23408 −0.346793 −0.173397 0.984852i \(-0.555474\pi\)
−0.173397 + 0.984852i \(0.555474\pi\)
\(710\) 0 0
\(711\) 10.3619 0.388601
\(712\) 0 0
\(713\) 1.40479 0.0526097
\(714\) 0 0
\(715\) −10.9938 −0.411145
\(716\) 0 0
\(717\) −8.82641 −0.329629
\(718\) 0 0
\(719\) 33.3348 1.24318 0.621590 0.783343i \(-0.286487\pi\)
0.621590 + 0.783343i \(0.286487\pi\)
\(720\) 0 0
\(721\) −50.7424 −1.88974
\(722\) 0 0
\(723\) 24.7890 0.921912
\(724\) 0 0
\(725\) 7.86837 0.292224
\(726\) 0 0
\(727\) 2.10327 0.0780061 0.0390030 0.999239i \(-0.487582\pi\)
0.0390030 + 0.999239i \(0.487582\pi\)
\(728\) 0 0
\(729\) 30.0171 1.11174
\(730\) 0 0
\(731\) 35.8854 1.32727
\(732\) 0 0
\(733\) −34.0965 −1.25938 −0.629691 0.776846i \(-0.716818\pi\)
−0.629691 + 0.776846i \(0.716818\pi\)
\(734\) 0 0
\(735\) −40.7288 −1.50230
\(736\) 0 0
\(737\) −12.2041 −0.449543
\(738\) 0 0
\(739\) −11.1479 −0.410081 −0.205040 0.978754i \(-0.565733\pi\)
−0.205040 + 0.978754i \(0.565733\pi\)
\(740\) 0 0
\(741\) −36.6133 −1.34502
\(742\) 0 0
\(743\) −23.4849 −0.861579 −0.430789 0.902452i \(-0.641765\pi\)
−0.430789 + 0.902452i \(0.641765\pi\)
\(744\) 0 0
\(745\) −15.6862 −0.574697
\(746\) 0 0
\(747\) 7.18249 0.262793
\(748\) 0 0
\(749\) 49.7648 1.81837
\(750\) 0 0
\(751\) 31.1954 1.13834 0.569169 0.822221i \(-0.307265\pi\)
0.569169 + 0.822221i \(0.307265\pi\)
\(752\) 0 0
\(753\) 48.0936 1.75263
\(754\) 0 0
\(755\) −31.0465 −1.12990
\(756\) 0 0
\(757\) −19.3753 −0.704208 −0.352104 0.935961i \(-0.614534\pi\)
−0.352104 + 0.935961i \(0.614534\pi\)
\(758\) 0 0
\(759\) 0.334653 0.0121472
\(760\) 0 0
\(761\) 34.7008 1.25790 0.628951 0.777445i \(-0.283485\pi\)
0.628951 + 0.777445i \(0.283485\pi\)
\(762\) 0 0
\(763\) −12.4537 −0.450855
\(764\) 0 0
\(765\) 6.27220 0.226772
\(766\) 0 0
\(767\) 46.7327 1.68742
\(768\) 0 0
\(769\) −25.6173 −0.923785 −0.461892 0.886936i \(-0.652829\pi\)
−0.461892 + 0.886936i \(0.652829\pi\)
\(770\) 0 0
\(771\) 5.16218 0.185911
\(772\) 0 0
\(773\) −25.4065 −0.913808 −0.456904 0.889516i \(-0.651042\pi\)
−0.456904 + 0.889516i \(0.651042\pi\)
\(774\) 0 0
\(775\) −10.5868 −0.380291
\(776\) 0 0
\(777\) −11.9836 −0.429911
\(778\) 0 0
\(779\) 58.6389 2.10096
\(780\) 0 0
\(781\) −3.35973 −0.120221
\(782\) 0 0
\(783\) −43.8557 −1.56728
\(784\) 0 0
\(785\) 34.8348 1.24331
\(786\) 0 0
\(787\) 34.5483 1.23151 0.615757 0.787936i \(-0.288850\pi\)
0.615757 + 0.787936i \(0.288850\pi\)
\(788\) 0 0
\(789\) −13.1523 −0.468234
\(790\) 0 0
\(791\) −77.4131 −2.75249
\(792\) 0 0
\(793\) −10.3761 −0.368467
\(794\) 0 0
\(795\) −26.9646 −0.956335
\(796\) 0 0
\(797\) 43.1106 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(798\) 0 0
\(799\) −1.10616 −0.0391332
\(800\) 0 0
\(801\) 6.38663 0.225660
\(802\) 0 0
\(803\) −8.83505 −0.311782
\(804\) 0 0
\(805\) 1.19042 0.0419569
\(806\) 0 0
\(807\) 4.56510 0.160699
\(808\) 0 0
\(809\) 24.4150 0.858386 0.429193 0.903213i \(-0.358798\pi\)
0.429193 + 0.903213i \(0.358798\pi\)
\(810\) 0 0
\(811\) −39.1406 −1.37441 −0.687206 0.726463i \(-0.741163\pi\)
−0.687206 + 0.726463i \(0.741163\pi\)
\(812\) 0 0
\(813\) −19.8121 −0.694841
\(814\) 0 0
\(815\) 47.9824 1.68075
\(816\) 0 0
\(817\) −47.1815 −1.65067
\(818\) 0 0
\(819\) −8.97134 −0.313484
\(820\) 0 0
\(821\) 26.3557 0.919822 0.459911 0.887965i \(-0.347881\pi\)
0.459911 + 0.887965i \(0.347881\pi\)
\(822\) 0 0
\(823\) 11.0365 0.384709 0.192354 0.981326i \(-0.438388\pi\)
0.192354 + 0.981326i \(0.438388\pi\)
\(824\) 0 0
\(825\) −2.52203 −0.0878060
\(826\) 0 0
\(827\) −11.5066 −0.400123 −0.200062 0.979783i \(-0.564114\pi\)
−0.200062 + 0.979783i \(0.564114\pi\)
\(828\) 0 0
\(829\) −17.9151 −0.622217 −0.311108 0.950374i \(-0.600700\pi\)
−0.311108 + 0.950374i \(0.600700\pi\)
\(830\) 0 0
\(831\) 10.1517 0.352159
\(832\) 0 0
\(833\) −69.5983 −2.41144
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 59.0076 2.03960
\(838\) 0 0
\(839\) 46.8888 1.61878 0.809390 0.587271i \(-0.199798\pi\)
0.809390 + 0.587271i \(0.199798\pi\)
\(840\) 0 0
\(841\) 32.9113 1.13487
\(842\) 0 0
\(843\) −9.05262 −0.311789
\(844\) 0 0
\(845\) −3.11467 −0.107148
\(846\) 0 0
\(847\) −37.4973 −1.28842
\(848\) 0 0
\(849\) −10.4027 −0.357020
\(850\) 0 0
\(851\) 0.228407 0.00782969
\(852\) 0 0
\(853\) −7.97457 −0.273044 −0.136522 0.990637i \(-0.543592\pi\)
−0.136522 + 0.990637i \(0.543592\pi\)
\(854\) 0 0
\(855\) −8.24660 −0.282028
\(856\) 0 0
\(857\) 38.3453 1.30985 0.654925 0.755694i \(-0.272700\pi\)
0.654925 + 0.755694i \(0.272700\pi\)
\(858\) 0 0
\(859\) 55.7156 1.90099 0.950497 0.310735i \(-0.100575\pi\)
0.950497 + 0.310735i \(0.100575\pi\)
\(860\) 0 0
\(861\) −58.5371 −1.99494
\(862\) 0 0
\(863\) 35.5096 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(864\) 0 0
\(865\) −3.36464 −0.114401
\(866\) 0 0
\(867\) −17.2818 −0.586920
\(868\) 0 0
\(869\) 28.4792 0.966090
\(870\) 0 0
\(871\) 25.4047 0.860806
\(872\) 0 0
\(873\) 2.49392 0.0844063
\(874\) 0 0
\(875\) −53.8281 −1.81972
\(876\) 0 0
\(877\) 1.96237 0.0662646 0.0331323 0.999451i \(-0.489452\pi\)
0.0331323 + 0.999451i \(0.489452\pi\)
\(878\) 0 0
\(879\) 0.175984 0.00593578
\(880\) 0 0
\(881\) −25.9608 −0.874642 −0.437321 0.899305i \(-0.644073\pi\)
−0.437321 + 0.899305i \(0.644073\pi\)
\(882\) 0 0
\(883\) 32.1139 1.08072 0.540360 0.841434i \(-0.318288\pi\)
0.540360 + 0.841434i \(0.318288\pi\)
\(884\) 0 0
\(885\) −42.8828 −1.44149
\(886\) 0 0
\(887\) 6.96553 0.233880 0.116940 0.993139i \(-0.462692\pi\)
0.116940 + 0.993139i \(0.462692\pi\)
\(888\) 0 0
\(889\) −37.3104 −1.25135
\(890\) 0 0
\(891\) 11.1747 0.374366
\(892\) 0 0
\(893\) 1.45436 0.0486684
\(894\) 0 0
\(895\) −1.52134 −0.0508529
\(896\) 0 0
\(897\) −0.696634 −0.0232599
\(898\) 0 0
\(899\) −83.3012 −2.77825
\(900\) 0 0
\(901\) −46.0777 −1.53507
\(902\) 0 0
\(903\) 47.0996 1.56738
\(904\) 0 0
\(905\) −12.7133 −0.422604
\(906\) 0 0
\(907\) 34.2026 1.13568 0.567839 0.823140i \(-0.307780\pi\)
0.567839 + 0.823140i \(0.307780\pi\)
\(908\) 0 0
\(909\) 1.37932 0.0457490
\(910\) 0 0
\(911\) −33.5867 −1.11278 −0.556388 0.830922i \(-0.687813\pi\)
−0.556388 + 0.830922i \(0.687813\pi\)
\(912\) 0 0
\(913\) 19.7407 0.653323
\(914\) 0 0
\(915\) 9.52134 0.314766
\(916\) 0 0
\(917\) −26.2102 −0.865537
\(918\) 0 0
\(919\) 44.8672 1.48003 0.740017 0.672588i \(-0.234817\pi\)
0.740017 + 0.672588i \(0.234817\pi\)
\(920\) 0 0
\(921\) 30.6064 1.00851
\(922\) 0 0
\(923\) 6.99381 0.230204
\(924\) 0 0
\(925\) −1.72133 −0.0565971
\(926\) 0 0
\(927\) 6.68821 0.219670
\(928\) 0 0
\(929\) −12.6760 −0.415887 −0.207943 0.978141i \(-0.566677\pi\)
−0.207943 + 0.978141i \(0.566677\pi\)
\(930\) 0 0
\(931\) 91.5068 2.99902
\(932\) 0 0
\(933\) 43.9000 1.43722
\(934\) 0 0
\(935\) 17.2389 0.563771
\(936\) 0 0
\(937\) 3.52060 0.115013 0.0575065 0.998345i \(-0.481685\pi\)
0.0575065 + 0.998345i \(0.481685\pi\)
\(938\) 0 0
\(939\) 4.58288 0.149557
\(940\) 0 0
\(941\) 54.2981 1.77007 0.885034 0.465526i \(-0.154135\pi\)
0.885034 + 0.465526i \(0.154135\pi\)
\(942\) 0 0
\(943\) 1.11571 0.0363325
\(944\) 0 0
\(945\) 50.0033 1.62661
\(946\) 0 0
\(947\) 35.3015 1.14715 0.573573 0.819155i \(-0.305557\pi\)
0.573573 + 0.819155i \(0.305557\pi\)
\(948\) 0 0
\(949\) 18.3915 0.597015
\(950\) 0 0
\(951\) 36.4881 1.18321
\(952\) 0 0
\(953\) 41.2048 1.33475 0.667377 0.744720i \(-0.267417\pi\)
0.667377 + 0.744720i \(0.267417\pi\)
\(954\) 0 0
\(955\) 22.3979 0.724778
\(956\) 0 0
\(957\) −19.8443 −0.641475
\(958\) 0 0
\(959\) 61.3468 1.98099
\(960\) 0 0
\(961\) 81.0812 2.61552
\(962\) 0 0
\(963\) −6.55935 −0.211372
\(964\) 0 0
\(965\) −29.8943 −0.962330
\(966\) 0 0
\(967\) −16.5718 −0.532913 −0.266457 0.963847i \(-0.585853\pi\)
−0.266457 + 0.963847i \(0.585853\pi\)
\(968\) 0 0
\(969\) 57.4115 1.84432
\(970\) 0 0
\(971\) −41.8901 −1.34432 −0.672158 0.740408i \(-0.734632\pi\)
−0.672158 + 0.740408i \(0.734632\pi\)
\(972\) 0 0
\(973\) 46.2015 1.48115
\(974\) 0 0
\(975\) 5.25001 0.168135
\(976\) 0 0
\(977\) 54.2586 1.73589 0.867944 0.496663i \(-0.165441\pi\)
0.867944 + 0.496663i \(0.165441\pi\)
\(978\) 0 0
\(979\) 17.5533 0.561007
\(980\) 0 0
\(981\) 1.64149 0.0524087
\(982\) 0 0
\(983\) −40.0351 −1.27692 −0.638460 0.769655i \(-0.720428\pi\)
−0.638460 + 0.769655i \(0.720428\pi\)
\(984\) 0 0
\(985\) 36.1550 1.15199
\(986\) 0 0
\(987\) −1.45184 −0.0462126
\(988\) 0 0
\(989\) −0.897714 −0.0285456
\(990\) 0 0
\(991\) 6.86359 0.218029 0.109015 0.994040i \(-0.465230\pi\)
0.109015 + 0.994040i \(0.465230\pi\)
\(992\) 0 0
\(993\) 5.48464 0.174050
\(994\) 0 0
\(995\) −7.35233 −0.233085
\(996\) 0 0
\(997\) 26.6578 0.844260 0.422130 0.906535i \(-0.361283\pi\)
0.422130 + 0.906535i \(0.361283\pi\)
\(998\) 0 0
\(999\) 9.59414 0.303545
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 668.2.a.b.1.2 5
3.2 odd 2 6012.2.a.d.1.4 5
4.3 odd 2 2672.2.a.j.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.2 5 1.1 even 1 trivial
2672.2.a.j.1.4 5 4.3 odd 2
6012.2.a.d.1.4 5 3.2 odd 2