Properties

Label 668.2.a.b.1.4
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 1.4
Root \(1.66287\) of defining polynomial
Character \(\chi\) \(=\) 668.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.89773 q^{3} +2.00000 q^{5} +4.92082 q^{7} +0.601369 q^{9} +O(q^{10})\) \(q+1.89773 q^{3} +2.00000 q^{5} +4.92082 q^{7} +0.601369 q^{9} -2.15568 q^{11} -3.32575 q^{13} +3.79545 q^{15} +5.91846 q^{17} -0.927114 q^{19} +9.33838 q^{21} -7.63710 q^{23} -1.00000 q^{25} -4.55195 q^{27} -1.59508 q^{29} -5.18270 q^{31} -4.09089 q^{33} +9.84165 q^{35} +1.53029 q^{37} -6.31136 q^{39} -9.71392 q^{41} +6.65149 q^{43} +1.20274 q^{45} +5.72438 q^{47} +17.2145 q^{49} +11.2316 q^{51} -9.24421 q^{53} -4.31136 q^{55} -1.75941 q^{57} +5.78287 q^{59} +14.8606 q^{61} +2.95923 q^{63} -6.65149 q^{65} +10.5747 q^{67} -14.4931 q^{69} +5.51410 q^{71} -13.5556 q^{73} -1.89773 q^{75} -10.6077 q^{77} -8.23454 q^{79} -10.4425 q^{81} +7.97724 q^{83} +11.8369 q^{85} -3.02702 q^{87} -8.41363 q^{89} -16.3654 q^{91} -9.83536 q^{93} -1.85423 q^{95} +5.34111 q^{97} -1.29636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + 5q^{11} - 4q^{13} + 6q^{15} - 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} - 5q^{25} + 12q^{27} - 5q^{29} + 9q^{31} - 8q^{33} + 18q^{35} + 8q^{37} - 4q^{41} + 8q^{43} + 12q^{45} + 13q^{47} + 14q^{49} - 2q^{53} + 10q^{55} - 12q^{57} + 4q^{59} + 11q^{61} - q^{63} - 8q^{65} + 28q^{67} - 16q^{69} + 2q^{71} + 8q^{73} - 3q^{75} - 12q^{77} - 10q^{79} - 15q^{81} + 2q^{83} - 4q^{85} + 4q^{87} - 17q^{89} - 12q^{91} - 12q^{93} + 10q^{95} - 27q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) 1.89773 1.09565 0.547827 0.836592i \(-0.315455\pi\)
0.547827 + 0.836592i \(0.315455\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 4.92082 1.85990 0.929948 0.367690i \(-0.119851\pi\)
0.929948 + 0.367690i \(0.119851\pi\)
\(8\) 0 0
\(9\) 0.601369 0.200456
\(10\) 0 0
\(11\) −2.15568 −0.649962 −0.324981 0.945721i \(-0.605358\pi\)
−0.324981 + 0.945721i \(0.605358\pi\)
\(12\) 0 0
\(13\) −3.32575 −0.922396 −0.461198 0.887297i \(-0.652580\pi\)
−0.461198 + 0.887297i \(0.652580\pi\)
\(14\) 0 0
\(15\) 3.79545 0.979982
\(16\) 0 0
\(17\) 5.91846 1.43544 0.717719 0.696333i \(-0.245186\pi\)
0.717719 + 0.696333i \(0.245186\pi\)
\(18\) 0 0
\(19\) −0.927114 −0.212695 −0.106347 0.994329i \(-0.533916\pi\)
−0.106347 + 0.994329i \(0.533916\pi\)
\(20\) 0 0
\(21\) 9.33838 2.03780
\(22\) 0 0
\(23\) −7.63710 −1.59245 −0.796223 0.605003i \(-0.793172\pi\)
−0.796223 + 0.605003i \(0.793172\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −4.55195 −0.876023
\(28\) 0 0
\(29\) −1.59508 −0.296199 −0.148099 0.988972i \(-0.547316\pi\)
−0.148099 + 0.988972i \(0.547316\pi\)
\(30\) 0 0
\(31\) −5.18270 −0.930841 −0.465421 0.885090i \(-0.654097\pi\)
−0.465421 + 0.885090i \(0.654097\pi\)
\(32\) 0 0
\(33\) −4.09089 −0.712133
\(34\) 0 0
\(35\) 9.84165 1.66354
\(36\) 0 0
\(37\) 1.53029 0.251578 0.125789 0.992057i \(-0.459854\pi\)
0.125789 + 0.992057i \(0.459854\pi\)
\(38\) 0 0
\(39\) −6.31136 −1.01063
\(40\) 0 0
\(41\) −9.71392 −1.51706 −0.758529 0.651639i \(-0.774082\pi\)
−0.758529 + 0.651639i \(0.774082\pi\)
\(42\) 0 0
\(43\) 6.65149 1.01434 0.507171 0.861845i \(-0.330691\pi\)
0.507171 + 0.861845i \(0.330691\pi\)
\(44\) 0 0
\(45\) 1.20274 0.179294
\(46\) 0 0
\(47\) 5.72438 0.834986 0.417493 0.908680i \(-0.362909\pi\)
0.417493 + 0.908680i \(0.362909\pi\)
\(48\) 0 0
\(49\) 17.2145 2.45922
\(50\) 0 0
\(51\) 11.2316 1.57274
\(52\) 0 0
\(53\) −9.24421 −1.26979 −0.634895 0.772599i \(-0.718957\pi\)
−0.634895 + 0.772599i \(0.718957\pi\)
\(54\) 0 0
\(55\) −4.31136 −0.581343
\(56\) 0 0
\(57\) −1.75941 −0.233040
\(58\) 0 0
\(59\) 5.78287 0.752866 0.376433 0.926444i \(-0.377150\pi\)
0.376433 + 0.926444i \(0.377150\pi\)
\(60\) 0 0
\(61\) 14.8606 1.90270 0.951351 0.308110i \(-0.0996964\pi\)
0.951351 + 0.308110i \(0.0996964\pi\)
\(62\) 0 0
\(63\) 2.95923 0.372828
\(64\) 0 0
\(65\) −6.65149 −0.825016
\(66\) 0 0
\(67\) 10.5747 1.29190 0.645951 0.763379i \(-0.276461\pi\)
0.645951 + 0.763379i \(0.276461\pi\)
\(68\) 0 0
\(69\) −14.4931 −1.74477
\(70\) 0 0
\(71\) 5.51410 0.654403 0.327201 0.944955i \(-0.393894\pi\)
0.327201 + 0.944955i \(0.393894\pi\)
\(72\) 0 0
\(73\) −13.5556 −1.58656 −0.793279 0.608858i \(-0.791628\pi\)
−0.793279 + 0.608858i \(0.791628\pi\)
\(74\) 0 0
\(75\) −1.89773 −0.219131
\(76\) 0 0
\(77\) −10.6077 −1.20886
\(78\) 0 0
\(79\) −8.23454 −0.926459 −0.463229 0.886238i \(-0.653309\pi\)
−0.463229 + 0.886238i \(0.653309\pi\)
\(80\) 0 0
\(81\) −10.4425 −1.16027
\(82\) 0 0
\(83\) 7.97724 0.875615 0.437808 0.899069i \(-0.355755\pi\)
0.437808 + 0.899069i \(0.355755\pi\)
\(84\) 0 0
\(85\) 11.8369 1.28389
\(86\) 0 0
\(87\) −3.02702 −0.324531
\(88\) 0 0
\(89\) −8.41363 −0.891843 −0.445922 0.895072i \(-0.647124\pi\)
−0.445922 + 0.895072i \(0.647124\pi\)
\(90\) 0 0
\(91\) −16.3654 −1.71556
\(92\) 0 0
\(93\) −9.83536 −1.01988
\(94\) 0 0
\(95\) −1.85423 −0.190240
\(96\) 0 0
\(97\) 5.34111 0.542308 0.271154 0.962536i \(-0.412595\pi\)
0.271154 + 0.962536i \(0.412595\pi\)
\(98\) 0 0
\(99\) −1.29636 −0.130289
\(100\) 0 0
\(101\) 9.76011 0.971167 0.485584 0.874190i \(-0.338607\pi\)
0.485584 + 0.874190i \(0.338607\pi\)
\(102\) 0 0
\(103\) 9.38636 0.924866 0.462433 0.886654i \(-0.346977\pi\)
0.462433 + 0.886654i \(0.346977\pi\)
\(104\) 0 0
\(105\) 18.6768 1.82267
\(106\) 0 0
\(107\) −10.9161 −1.05530 −0.527650 0.849462i \(-0.676927\pi\)
−0.527650 + 0.849462i \(0.676927\pi\)
\(108\) 0 0
\(109\) 9.57287 0.916915 0.458457 0.888716i \(-0.348402\pi\)
0.458457 + 0.888716i \(0.348402\pi\)
\(110\) 0 0
\(111\) 2.90407 0.275643
\(112\) 0 0
\(113\) −8.84346 −0.831922 −0.415961 0.909382i \(-0.636555\pi\)
−0.415961 + 0.909382i \(0.636555\pi\)
\(114\) 0 0
\(115\) −15.2742 −1.42433
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 29.1237 2.66977
\(120\) 0 0
\(121\) −6.35305 −0.577550
\(122\) 0 0
\(123\) −18.4344 −1.66217
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −9.52762 −0.845439 −0.422720 0.906260i \(-0.638925\pi\)
−0.422720 + 0.906260i \(0.638925\pi\)
\(128\) 0 0
\(129\) 12.6227 1.11137
\(130\) 0 0
\(131\) 3.18543 0.278313 0.139156 0.990270i \(-0.455561\pi\)
0.139156 + 0.990270i \(0.455561\pi\)
\(132\) 0 0
\(133\) −4.56217 −0.395590
\(134\) 0 0
\(135\) −9.10390 −0.783539
\(136\) 0 0
\(137\) −8.62066 −0.736513 −0.368257 0.929724i \(-0.620045\pi\)
−0.368257 + 0.929724i \(0.620045\pi\)
\(138\) 0 0
\(139\) 16.7457 1.42035 0.710177 0.704023i \(-0.248615\pi\)
0.710177 + 0.704023i \(0.248615\pi\)
\(140\) 0 0
\(141\) 10.8633 0.914855
\(142\) 0 0
\(143\) 7.16924 0.599522
\(144\) 0 0
\(145\) −3.19016 −0.264928
\(146\) 0 0
\(147\) 32.6684 2.69445
\(148\) 0 0
\(149\) 1.18543 0.0971145 0.0485572 0.998820i \(-0.484538\pi\)
0.0485572 + 0.998820i \(0.484538\pi\)
\(150\) 0 0
\(151\) 8.43075 0.686085 0.343042 0.939320i \(-0.388543\pi\)
0.343042 + 0.939320i \(0.388543\pi\)
\(152\) 0 0
\(153\) 3.55918 0.287743
\(154\) 0 0
\(155\) −10.3654 −0.832570
\(156\) 0 0
\(157\) −22.9854 −1.83443 −0.917217 0.398388i \(-0.869570\pi\)
−0.917217 + 0.398388i \(0.869570\pi\)
\(158\) 0 0
\(159\) −17.5430 −1.39125
\(160\) 0 0
\(161\) −37.5808 −2.96178
\(162\) 0 0
\(163\) −5.16267 −0.404371 −0.202186 0.979347i \(-0.564804\pi\)
−0.202186 + 0.979347i \(0.564804\pi\)
\(164\) 0 0
\(165\) −8.18178 −0.636951
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −1.93942 −0.149186
\(170\) 0 0
\(171\) −0.557538 −0.0426360
\(172\) 0 0
\(173\) −0.472383 −0.0359146 −0.0179573 0.999839i \(-0.505716\pi\)
−0.0179573 + 0.999839i \(0.505716\pi\)
\(174\) 0 0
\(175\) −4.92082 −0.371979
\(176\) 0 0
\(177\) 10.9743 0.824880
\(178\) 0 0
\(179\) −24.2013 −1.80889 −0.904446 0.426589i \(-0.859715\pi\)
−0.904446 + 0.426589i \(0.859715\pi\)
\(180\) 0 0
\(181\) 8.25286 0.613430 0.306715 0.951801i \(-0.400770\pi\)
0.306715 + 0.951801i \(0.400770\pi\)
\(182\) 0 0
\(183\) 28.2013 2.08470
\(184\) 0 0
\(185\) 3.06058 0.225018
\(186\) 0 0
\(187\) −12.7583 −0.932979
\(188\) 0 0
\(189\) −22.3993 −1.62931
\(190\) 0 0
\(191\) 19.6806 1.42404 0.712018 0.702161i \(-0.247781\pi\)
0.712018 + 0.702161i \(0.247781\pi\)
\(192\) 0 0
\(193\) 0.980888 0.0706059 0.0353029 0.999377i \(-0.488760\pi\)
0.0353029 + 0.999377i \(0.488760\pi\)
\(194\) 0 0
\(195\) −12.6227 −0.903931
\(196\) 0 0
\(197\) −16.0379 −1.14265 −0.571325 0.820724i \(-0.693570\pi\)
−0.571325 + 0.820724i \(0.693570\pi\)
\(198\) 0 0
\(199\) 18.6207 1.31998 0.659992 0.751273i \(-0.270560\pi\)
0.659992 + 0.751273i \(0.270560\pi\)
\(200\) 0 0
\(201\) 20.0679 1.41548
\(202\) 0 0
\(203\) −7.84910 −0.550899
\(204\) 0 0
\(205\) −19.4278 −1.35690
\(206\) 0 0
\(207\) −4.59272 −0.319216
\(208\) 0 0
\(209\) 1.99856 0.138243
\(210\) 0 0
\(211\) 10.2451 0.705304 0.352652 0.935755i \(-0.385280\pi\)
0.352652 + 0.935755i \(0.385280\pi\)
\(212\) 0 0
\(213\) 10.4642 0.716999
\(214\) 0 0
\(215\) 13.3030 0.907256
\(216\) 0 0
\(217\) −25.5032 −1.73127
\(218\) 0 0
\(219\) −25.7248 −1.73832
\(220\) 0 0
\(221\) −19.6833 −1.32404
\(222\) 0 0
\(223\) 2.05135 0.137369 0.0686844 0.997638i \(-0.478120\pi\)
0.0686844 + 0.997638i \(0.478120\pi\)
\(224\) 0 0
\(225\) −0.601369 −0.0400913
\(226\) 0 0
\(227\) −2.13743 −0.141866 −0.0709332 0.997481i \(-0.522598\pi\)
−0.0709332 + 0.997481i \(0.522598\pi\)
\(228\) 0 0
\(229\) 9.18599 0.607027 0.303514 0.952827i \(-0.401840\pi\)
0.303514 + 0.952827i \(0.401840\pi\)
\(230\) 0 0
\(231\) −20.1306 −1.32449
\(232\) 0 0
\(233\) 16.5799 1.08619 0.543093 0.839672i \(-0.317253\pi\)
0.543093 + 0.839672i \(0.317253\pi\)
\(234\) 0 0
\(235\) 11.4488 0.746834
\(236\) 0 0
\(237\) −15.6269 −1.01508
\(238\) 0 0
\(239\) 9.13046 0.590601 0.295300 0.955404i \(-0.404580\pi\)
0.295300 + 0.955404i \(0.404580\pi\)
\(240\) 0 0
\(241\) −16.9683 −1.09302 −0.546512 0.837451i \(-0.684045\pi\)
−0.546512 + 0.837451i \(0.684045\pi\)
\(242\) 0 0
\(243\) −6.16110 −0.395235
\(244\) 0 0
\(245\) 34.4290 2.19959
\(246\) 0 0
\(247\) 3.08335 0.196189
\(248\) 0 0
\(249\) 15.1386 0.959371
\(250\) 0 0
\(251\) −5.45949 −0.344600 −0.172300 0.985045i \(-0.555120\pi\)
−0.172300 + 0.985045i \(0.555120\pi\)
\(252\) 0 0
\(253\) 16.4631 1.03503
\(254\) 0 0
\(255\) 22.4633 1.40670
\(256\) 0 0
\(257\) 5.76122 0.359375 0.179687 0.983724i \(-0.442491\pi\)
0.179687 + 0.983724i \(0.442491\pi\)
\(258\) 0 0
\(259\) 7.53029 0.467909
\(260\) 0 0
\(261\) −0.959231 −0.0593749
\(262\) 0 0
\(263\) 0.636216 0.0392307 0.0196154 0.999808i \(-0.493756\pi\)
0.0196154 + 0.999808i \(0.493756\pi\)
\(264\) 0 0
\(265\) −18.4884 −1.13573
\(266\) 0 0
\(267\) −15.9668 −0.977151
\(268\) 0 0
\(269\) 31.5970 1.92651 0.963253 0.268597i \(-0.0865601\pi\)
0.963253 + 0.268597i \(0.0865601\pi\)
\(270\) 0 0
\(271\) −0.651490 −0.0395752 −0.0197876 0.999804i \(-0.506299\pi\)
−0.0197876 + 0.999804i \(0.506299\pi\)
\(272\) 0 0
\(273\) −31.0571 −1.87966
\(274\) 0 0
\(275\) 2.15568 0.129992
\(276\) 0 0
\(277\) 19.3510 1.16269 0.581345 0.813657i \(-0.302527\pi\)
0.581345 + 0.813657i \(0.302527\pi\)
\(278\) 0 0
\(279\) −3.11672 −0.186593
\(280\) 0 0
\(281\) −25.5121 −1.52192 −0.760961 0.648797i \(-0.775272\pi\)
−0.760961 + 0.648797i \(0.775272\pi\)
\(282\) 0 0
\(283\) 11.6217 0.690840 0.345420 0.938448i \(-0.387736\pi\)
0.345420 + 0.938448i \(0.387736\pi\)
\(284\) 0 0
\(285\) −3.51882 −0.208437
\(286\) 0 0
\(287\) −47.8005 −2.82157
\(288\) 0 0
\(289\) 18.0282 1.06048
\(290\) 0 0
\(291\) 10.1360 0.594181
\(292\) 0 0
\(293\) −13.6824 −0.799335 −0.399668 0.916660i \(-0.630874\pi\)
−0.399668 + 0.916660i \(0.630874\pi\)
\(294\) 0 0
\(295\) 11.5657 0.673384
\(296\) 0 0
\(297\) 9.81254 0.569381
\(298\) 0 0
\(299\) 25.3991 1.46887
\(300\) 0 0
\(301\) 32.7308 1.88657
\(302\) 0 0
\(303\) 18.5220 1.06406
\(304\) 0 0
\(305\) 29.7212 1.70183
\(306\) 0 0
\(307\) 19.1003 1.09011 0.545055 0.838400i \(-0.316509\pi\)
0.545055 + 0.838400i \(0.316509\pi\)
\(308\) 0 0
\(309\) 17.8128 1.01333
\(310\) 0 0
\(311\) −23.1206 −1.31105 −0.655524 0.755174i \(-0.727552\pi\)
−0.655524 + 0.755174i \(0.727552\pi\)
\(312\) 0 0
\(313\) −5.63526 −0.318524 −0.159262 0.987236i \(-0.550911\pi\)
−0.159262 + 0.987236i \(0.550911\pi\)
\(314\) 0 0
\(315\) 5.91846 0.333468
\(316\) 0 0
\(317\) −9.44544 −0.530509 −0.265254 0.964178i \(-0.585456\pi\)
−0.265254 + 0.964178i \(0.585456\pi\)
\(318\) 0 0
\(319\) 3.43848 0.192518
\(320\) 0 0
\(321\) −20.7158 −1.15624
\(322\) 0 0
\(323\) −5.48709 −0.305310
\(324\) 0 0
\(325\) 3.32575 0.184479
\(326\) 0 0
\(327\) 18.1667 1.00462
\(328\) 0 0
\(329\) 28.1686 1.55299
\(330\) 0 0
\(331\) 12.7565 0.701158 0.350579 0.936533i \(-0.385985\pi\)
0.350579 + 0.936533i \(0.385985\pi\)
\(332\) 0 0
\(333\) 0.920269 0.0504304
\(334\) 0 0
\(335\) 21.1494 1.15551
\(336\) 0 0
\(337\) −6.10285 −0.332443 −0.166222 0.986088i \(-0.553157\pi\)
−0.166222 + 0.986088i \(0.553157\pi\)
\(338\) 0 0
\(339\) −16.7825 −0.911498
\(340\) 0 0
\(341\) 11.1722 0.605011
\(342\) 0 0
\(343\) 50.2638 2.71399
\(344\) 0 0
\(345\) −28.9863 −1.56057
\(346\) 0 0
\(347\) −9.85312 −0.528943 −0.264472 0.964393i \(-0.585198\pi\)
−0.264472 + 0.964393i \(0.585198\pi\)
\(348\) 0 0
\(349\) 6.75107 0.361376 0.180688 0.983540i \(-0.442168\pi\)
0.180688 + 0.983540i \(0.442168\pi\)
\(350\) 0 0
\(351\) 15.1386 0.808040
\(352\) 0 0
\(353\) −21.7715 −1.15878 −0.579390 0.815050i \(-0.696709\pi\)
−0.579390 + 0.815050i \(0.696709\pi\)
\(354\) 0 0
\(355\) 11.0282 0.585316
\(356\) 0 0
\(357\) 55.2689 2.92514
\(358\) 0 0
\(359\) 0.773613 0.0408297 0.0204149 0.999792i \(-0.493501\pi\)
0.0204149 + 0.999792i \(0.493501\pi\)
\(360\) 0 0
\(361\) −18.1405 −0.954761
\(362\) 0 0
\(363\) −12.0564 −0.632795
\(364\) 0 0
\(365\) −27.1111 −1.41906
\(366\) 0 0
\(367\) 34.9813 1.82601 0.913004 0.407951i \(-0.133757\pi\)
0.913004 + 0.407951i \(0.133757\pi\)
\(368\) 0 0
\(369\) −5.84165 −0.304104
\(370\) 0 0
\(371\) −45.4891 −2.36168
\(372\) 0 0
\(373\) 26.3114 1.36235 0.681175 0.732120i \(-0.261469\pi\)
0.681175 + 0.732120i \(0.261469\pi\)
\(374\) 0 0
\(375\) −22.7727 −1.17598
\(376\) 0 0
\(377\) 5.30483 0.273212
\(378\) 0 0
\(379\) 5.49668 0.282345 0.141173 0.989985i \(-0.454913\pi\)
0.141173 + 0.989985i \(0.454913\pi\)
\(380\) 0 0
\(381\) −18.0808 −0.926308
\(382\) 0 0
\(383\) 17.4512 0.891712 0.445856 0.895105i \(-0.352899\pi\)
0.445856 + 0.895105i \(0.352899\pi\)
\(384\) 0 0
\(385\) −21.2154 −1.08124
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 0.763649 0.0387185 0.0193593 0.999813i \(-0.493837\pi\)
0.0193593 + 0.999813i \(0.493837\pi\)
\(390\) 0 0
\(391\) −45.1999 −2.28586
\(392\) 0 0
\(393\) 6.04508 0.304934
\(394\) 0 0
\(395\) −16.4691 −0.828650
\(396\) 0 0
\(397\) 33.8451 1.69864 0.849318 0.527882i \(-0.177014\pi\)
0.849318 + 0.527882i \(0.177014\pi\)
\(398\) 0 0
\(399\) −8.65775 −0.433429
\(400\) 0 0
\(401\) −32.0451 −1.60026 −0.800128 0.599830i \(-0.795235\pi\)
−0.800128 + 0.599830i \(0.795235\pi\)
\(402\) 0 0
\(403\) 17.2364 0.858604
\(404\) 0 0
\(405\) −20.8849 −1.03778
\(406\) 0 0
\(407\) −3.29881 −0.163516
\(408\) 0 0
\(409\) 27.2109 1.34549 0.672747 0.739873i \(-0.265114\pi\)
0.672747 + 0.739873i \(0.265114\pi\)
\(410\) 0 0
\(411\) −16.3597 −0.806963
\(412\) 0 0
\(413\) 28.4565 1.40025
\(414\) 0 0
\(415\) 15.9545 0.783174
\(416\) 0 0
\(417\) 31.7788 1.55622
\(418\) 0 0
\(419\) 8.65061 0.422610 0.211305 0.977420i \(-0.432229\pi\)
0.211305 + 0.977420i \(0.432229\pi\)
\(420\) 0 0
\(421\) −14.8729 −0.724861 −0.362430 0.932011i \(-0.618053\pi\)
−0.362430 + 0.932011i \(0.618053\pi\)
\(422\) 0 0
\(423\) 3.44246 0.167378
\(424\) 0 0
\(425\) −5.91846 −0.287088
\(426\) 0 0
\(427\) 73.1263 3.53883
\(428\) 0 0
\(429\) 13.6053 0.656868
\(430\) 0 0
\(431\) −3.60441 −0.173618 −0.0868092 0.996225i \(-0.527667\pi\)
−0.0868092 + 0.996225i \(0.527667\pi\)
\(432\) 0 0
\(433\) 22.3165 1.07246 0.536231 0.844072i \(-0.319848\pi\)
0.536231 + 0.844072i \(0.319848\pi\)
\(434\) 0 0
\(435\) −6.05405 −0.290269
\(436\) 0 0
\(437\) 7.08047 0.338705
\(438\) 0 0
\(439\) −37.6145 −1.79524 −0.897620 0.440770i \(-0.854705\pi\)
−0.897620 + 0.440770i \(0.854705\pi\)
\(440\) 0 0
\(441\) 10.3523 0.492965
\(442\) 0 0
\(443\) 14.8525 0.705665 0.352833 0.935686i \(-0.385219\pi\)
0.352833 + 0.935686i \(0.385219\pi\)
\(444\) 0 0
\(445\) −16.8273 −0.797689
\(446\) 0 0
\(447\) 2.24963 0.106404
\(448\) 0 0
\(449\) −17.5558 −0.828511 −0.414256 0.910161i \(-0.635958\pi\)
−0.414256 + 0.910161i \(0.635958\pi\)
\(450\) 0 0
\(451\) 20.9401 0.986030
\(452\) 0 0
\(453\) 15.9993 0.751711
\(454\) 0 0
\(455\) −32.7308 −1.53444
\(456\) 0 0
\(457\) −25.3798 −1.18722 −0.593608 0.804754i \(-0.702297\pi\)
−0.593608 + 0.804754i \(0.702297\pi\)
\(458\) 0 0
\(459\) −26.9405 −1.25748
\(460\) 0 0
\(461\) 15.0213 0.699614 0.349807 0.936822i \(-0.386247\pi\)
0.349807 + 0.936822i \(0.386247\pi\)
\(462\) 0 0
\(463\) 26.6335 1.23776 0.618881 0.785485i \(-0.287586\pi\)
0.618881 + 0.785485i \(0.287586\pi\)
\(464\) 0 0
\(465\) −19.6707 −0.912208
\(466\) 0 0
\(467\) 5.42230 0.250914 0.125457 0.992099i \(-0.459960\pi\)
0.125457 + 0.992099i \(0.459960\pi\)
\(468\) 0 0
\(469\) 52.0361 2.40280
\(470\) 0 0
\(471\) −43.6200 −2.00990
\(472\) 0 0
\(473\) −14.3385 −0.659284
\(474\) 0 0
\(475\) 0.927114 0.0425389
\(476\) 0 0
\(477\) −5.55918 −0.254537
\(478\) 0 0
\(479\) 14.3701 0.656588 0.328294 0.944576i \(-0.393526\pi\)
0.328294 + 0.944576i \(0.393526\pi\)
\(480\) 0 0
\(481\) −5.08936 −0.232055
\(482\) 0 0
\(483\) −71.3182 −3.24509
\(484\) 0 0
\(485\) 10.6822 0.485055
\(486\) 0 0
\(487\) −9.60533 −0.435259 −0.217630 0.976031i \(-0.569832\pi\)
−0.217630 + 0.976031i \(0.569832\pi\)
\(488\) 0 0
\(489\) −9.79734 −0.443051
\(490\) 0 0
\(491\) −3.60036 −0.162482 −0.0812409 0.996694i \(-0.525888\pi\)
−0.0812409 + 0.996694i \(0.525888\pi\)
\(492\) 0 0
\(493\) −9.44041 −0.425175
\(494\) 0 0
\(495\) −2.59272 −0.116534
\(496\) 0 0
\(497\) 27.1339 1.21712
\(498\) 0 0
\(499\) −32.5772 −1.45836 −0.729178 0.684325i \(-0.760097\pi\)
−0.729178 + 0.684325i \(0.760097\pi\)
\(500\) 0 0
\(501\) −1.89773 −0.0847842
\(502\) 0 0
\(503\) 5.05070 0.225199 0.112600 0.993640i \(-0.464082\pi\)
0.112600 + 0.993640i \(0.464082\pi\)
\(504\) 0 0
\(505\) 19.5202 0.868638
\(506\) 0 0
\(507\) −3.68049 −0.163456
\(508\) 0 0
\(509\) 10.9284 0.484393 0.242197 0.970227i \(-0.422132\pi\)
0.242197 + 0.970227i \(0.422132\pi\)
\(510\) 0 0
\(511\) −66.7045 −2.95084
\(512\) 0 0
\(513\) 4.22018 0.186325
\(514\) 0 0
\(515\) 18.7727 0.827225
\(516\) 0 0
\(517\) −12.3399 −0.542709
\(518\) 0 0
\(519\) −0.896453 −0.0393499
\(520\) 0 0
\(521\) 39.4092 1.72655 0.863274 0.504736i \(-0.168410\pi\)
0.863274 + 0.504736i \(0.168410\pi\)
\(522\) 0 0
\(523\) −38.5511 −1.68572 −0.842862 0.538130i \(-0.819131\pi\)
−0.842862 + 0.538130i \(0.819131\pi\)
\(524\) 0 0
\(525\) −9.33838 −0.407560
\(526\) 0 0
\(527\) −30.6736 −1.33616
\(528\) 0 0
\(529\) 35.3253 1.53588
\(530\) 0 0
\(531\) 3.47764 0.150917
\(532\) 0 0
\(533\) 32.3060 1.39933
\(534\) 0 0
\(535\) −21.8322 −0.943888
\(536\) 0 0
\(537\) −45.9275 −1.98192
\(538\) 0 0
\(539\) −37.1090 −1.59840
\(540\) 0 0
\(541\) 17.8850 0.768935 0.384467 0.923139i \(-0.374385\pi\)
0.384467 + 0.923139i \(0.374385\pi\)
\(542\) 0 0
\(543\) 15.6617 0.672107
\(544\) 0 0
\(545\) 19.1457 0.820113
\(546\) 0 0
\(547\) −6.73915 −0.288145 −0.144073 0.989567i \(-0.546020\pi\)
−0.144073 + 0.989567i \(0.546020\pi\)
\(548\) 0 0
\(549\) 8.93669 0.381409
\(550\) 0 0
\(551\) 1.47882 0.0629999
\(552\) 0 0
\(553\) −40.5207 −1.72312
\(554\) 0 0
\(555\) 5.80815 0.246542
\(556\) 0 0
\(557\) −11.2470 −0.476550 −0.238275 0.971198i \(-0.576582\pi\)
−0.238275 + 0.971198i \(0.576582\pi\)
\(558\) 0 0
\(559\) −22.1212 −0.935625
\(560\) 0 0
\(561\) −24.2118 −1.02222
\(562\) 0 0
\(563\) −28.1919 −1.18815 −0.594073 0.804411i \(-0.702481\pi\)
−0.594073 + 0.804411i \(0.702481\pi\)
\(564\) 0 0
\(565\) −17.6869 −0.744094
\(566\) 0 0
\(567\) −51.3855 −2.15799
\(568\) 0 0
\(569\) −15.2335 −0.638620 −0.319310 0.947650i \(-0.603451\pi\)
−0.319310 + 0.947650i \(0.603451\pi\)
\(570\) 0 0
\(571\) −9.42060 −0.394240 −0.197120 0.980379i \(-0.563159\pi\)
−0.197120 + 0.980379i \(0.563159\pi\)
\(572\) 0 0
\(573\) 37.3483 1.56025
\(574\) 0 0
\(575\) 7.63710 0.318489
\(576\) 0 0
\(577\) −14.3375 −0.596879 −0.298439 0.954429i \(-0.596466\pi\)
−0.298439 + 0.954429i \(0.596466\pi\)
\(578\) 0 0
\(579\) 1.86146 0.0773596
\(580\) 0 0
\(581\) 39.2546 1.62855
\(582\) 0 0
\(583\) 19.9275 0.825314
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 27.1224 1.11946 0.559731 0.828674i \(-0.310904\pi\)
0.559731 + 0.828674i \(0.310904\pi\)
\(588\) 0 0
\(589\) 4.80496 0.197985
\(590\) 0 0
\(591\) −30.4355 −1.25195
\(592\) 0 0
\(593\) −6.05331 −0.248580 −0.124290 0.992246i \(-0.539665\pi\)
−0.124290 + 0.992246i \(0.539665\pi\)
\(594\) 0 0
\(595\) 58.2474 2.38791
\(596\) 0 0
\(597\) 35.3369 1.44624
\(598\) 0 0
\(599\) 35.9024 1.46693 0.733466 0.679726i \(-0.237902\pi\)
0.733466 + 0.679726i \(0.237902\pi\)
\(600\) 0 0
\(601\) 19.1106 0.779537 0.389768 0.920913i \(-0.372555\pi\)
0.389768 + 0.920913i \(0.372555\pi\)
\(602\) 0 0
\(603\) 6.35928 0.258970
\(604\) 0 0
\(605\) −12.7061 −0.516576
\(606\) 0 0
\(607\) −20.1464 −0.817719 −0.408859 0.912597i \(-0.634073\pi\)
−0.408859 + 0.912597i \(0.634073\pi\)
\(608\) 0 0
\(609\) −14.8955 −0.603594
\(610\) 0 0
\(611\) −19.0378 −0.770188
\(612\) 0 0
\(613\) −35.5775 −1.43696 −0.718481 0.695547i \(-0.755162\pi\)
−0.718481 + 0.695547i \(0.755162\pi\)
\(614\) 0 0
\(615\) −36.8687 −1.48669
\(616\) 0 0
\(617\) 45.6697 1.83859 0.919296 0.393567i \(-0.128759\pi\)
0.919296 + 0.393567i \(0.128759\pi\)
\(618\) 0 0
\(619\) −45.1754 −1.81575 −0.907876 0.419240i \(-0.862297\pi\)
−0.907876 + 0.419240i \(0.862297\pi\)
\(620\) 0 0
\(621\) 34.7637 1.39502
\(622\) 0 0
\(623\) −41.4020 −1.65874
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 3.79272 0.151467
\(628\) 0 0
\(629\) 9.05697 0.361125
\(630\) 0 0
\(631\) 6.77591 0.269745 0.134872 0.990863i \(-0.456938\pi\)
0.134872 + 0.990863i \(0.456938\pi\)
\(632\) 0 0
\(633\) 19.4425 0.772768
\(634\) 0 0
\(635\) −19.0552 −0.756184
\(636\) 0 0
\(637\) −57.2511 −2.26837
\(638\) 0 0
\(639\) 3.31601 0.131179
\(640\) 0 0
\(641\) −25.7595 −1.01744 −0.508719 0.860932i \(-0.669881\pi\)
−0.508719 + 0.860932i \(0.669881\pi\)
\(642\) 0 0
\(643\) 50.3013 1.98369 0.991845 0.127453i \(-0.0406803\pi\)
0.991845 + 0.127453i \(0.0406803\pi\)
\(644\) 0 0
\(645\) 25.2454 0.994038
\(646\) 0 0
\(647\) 30.0782 1.18250 0.591248 0.806490i \(-0.298635\pi\)
0.591248 + 0.806490i \(0.298635\pi\)
\(648\) 0 0
\(649\) −12.4660 −0.489334
\(650\) 0 0
\(651\) −48.3981 −1.89687
\(652\) 0 0
\(653\) −7.05608 −0.276126 −0.138063 0.990423i \(-0.544088\pi\)
−0.138063 + 0.990423i \(0.544088\pi\)
\(654\) 0 0
\(655\) 6.37087 0.248930
\(656\) 0 0
\(657\) −8.15190 −0.318036
\(658\) 0 0
\(659\) 13.4326 0.523258 0.261629 0.965168i \(-0.415740\pi\)
0.261629 + 0.965168i \(0.415740\pi\)
\(660\) 0 0
\(661\) 28.6833 1.11565 0.557826 0.829958i \(-0.311636\pi\)
0.557826 + 0.829958i \(0.311636\pi\)
\(662\) 0 0
\(663\) −37.3535 −1.45069
\(664\) 0 0
\(665\) −9.12433 −0.353826
\(666\) 0 0
\(667\) 12.1818 0.471680
\(668\) 0 0
\(669\) 3.89291 0.150509
\(670\) 0 0
\(671\) −32.0346 −1.23668
\(672\) 0 0
\(673\) 20.6474 0.795900 0.397950 0.917407i \(-0.369722\pi\)
0.397950 + 0.917407i \(0.369722\pi\)
\(674\) 0 0
\(675\) 4.55195 0.175205
\(676\) 0 0
\(677\) −6.86873 −0.263987 −0.131993 0.991251i \(-0.542138\pi\)
−0.131993 + 0.991251i \(0.542138\pi\)
\(678\) 0 0
\(679\) 26.2827 1.00864
\(680\) 0 0
\(681\) −4.05626 −0.155436
\(682\) 0 0
\(683\) 25.5166 0.976366 0.488183 0.872741i \(-0.337660\pi\)
0.488183 + 0.872741i \(0.337660\pi\)
\(684\) 0 0
\(685\) −17.2413 −0.658757
\(686\) 0 0
\(687\) 17.4325 0.665091
\(688\) 0 0
\(689\) 30.7439 1.17125
\(690\) 0 0
\(691\) 34.6547 1.31833 0.659164 0.751999i \(-0.270910\pi\)
0.659164 + 0.751999i \(0.270910\pi\)
\(692\) 0 0
\(693\) −6.37915 −0.242324
\(694\) 0 0
\(695\) 33.4914 1.27040
\(696\) 0 0
\(697\) −57.4914 −2.17764
\(698\) 0 0
\(699\) 31.4642 1.19008
\(700\) 0 0
\(701\) −11.5408 −0.435892 −0.217946 0.975961i \(-0.569936\pi\)
−0.217946 + 0.975961i \(0.569936\pi\)
\(702\) 0 0
\(703\) −1.41875 −0.0535093
\(704\) 0 0
\(705\) 21.7266 0.818271
\(706\) 0 0
\(707\) 48.0278 1.80627
\(708\) 0 0
\(709\) 13.5022 0.507085 0.253543 0.967324i \(-0.418404\pi\)
0.253543 + 0.967324i \(0.418404\pi\)
\(710\) 0 0
\(711\) −4.95200 −0.185714
\(712\) 0 0
\(713\) 39.5808 1.48231
\(714\) 0 0
\(715\) 14.3385 0.536229
\(716\) 0 0
\(717\) 17.3271 0.647094
\(718\) 0 0
\(719\) −45.3481 −1.69120 −0.845599 0.533819i \(-0.820756\pi\)
−0.845599 + 0.533819i \(0.820756\pi\)
\(720\) 0 0
\(721\) 46.1886 1.72016
\(722\) 0 0
\(723\) −32.2012 −1.19758
\(724\) 0 0
\(725\) 1.59508 0.0592397
\(726\) 0 0
\(727\) −4.92211 −0.182551 −0.0912756 0.995826i \(-0.529094\pi\)
−0.0912756 + 0.995826i \(0.529094\pi\)
\(728\) 0 0
\(729\) 19.6353 0.727233
\(730\) 0 0
\(731\) 39.3666 1.45603
\(732\) 0 0
\(733\) −17.4838 −0.645780 −0.322890 0.946437i \(-0.604654\pi\)
−0.322890 + 0.946437i \(0.604654\pi\)
\(734\) 0 0
\(735\) 65.3369 2.40999
\(736\) 0 0
\(737\) −22.7956 −0.839687
\(738\) 0 0
\(739\) −27.0498 −0.995042 −0.497521 0.867452i \(-0.665756\pi\)
−0.497521 + 0.867452i \(0.665756\pi\)
\(740\) 0 0
\(741\) 5.85135 0.214955
\(742\) 0 0
\(743\) −35.6126 −1.30650 −0.653250 0.757142i \(-0.726595\pi\)
−0.653250 + 0.757142i \(0.726595\pi\)
\(744\) 0 0
\(745\) 2.37087 0.0868618
\(746\) 0 0
\(747\) 4.79726 0.175523
\(748\) 0 0
\(749\) −53.7162 −1.96275
\(750\) 0 0
\(751\) −22.4268 −0.818364 −0.409182 0.912453i \(-0.634186\pi\)
−0.409182 + 0.912453i \(0.634186\pi\)
\(752\) 0 0
\(753\) −10.3606 −0.377562
\(754\) 0 0
\(755\) 16.8615 0.613653
\(756\) 0 0
\(757\) −16.4941 −0.599488 −0.299744 0.954020i \(-0.596901\pi\)
−0.299744 + 0.954020i \(0.596901\pi\)
\(758\) 0 0
\(759\) 31.2426 1.13403
\(760\) 0 0
\(761\) −12.9669 −0.470050 −0.235025 0.971989i \(-0.575517\pi\)
−0.235025 + 0.971989i \(0.575517\pi\)
\(762\) 0 0
\(763\) 47.1064 1.70537
\(764\) 0 0
\(765\) 7.11836 0.257365
\(766\) 0 0
\(767\) −19.2324 −0.694441
\(768\) 0 0
\(769\) −36.5712 −1.31879 −0.659395 0.751797i \(-0.729188\pi\)
−0.659395 + 0.751797i \(0.729188\pi\)
\(770\) 0 0
\(771\) 10.9332 0.393750
\(772\) 0 0
\(773\) −33.0199 −1.18764 −0.593821 0.804597i \(-0.702381\pi\)
−0.593821 + 0.804597i \(0.702381\pi\)
\(774\) 0 0
\(775\) 5.18270 0.186168
\(776\) 0 0
\(777\) 14.2904 0.512667
\(778\) 0 0
\(779\) 9.00591 0.322670
\(780\) 0 0
\(781\) −11.8866 −0.425337
\(782\) 0 0
\(783\) 7.26072 0.259477
\(784\) 0 0
\(785\) −45.9708 −1.64077
\(786\) 0 0
\(787\) −36.9726 −1.31793 −0.658965 0.752173i \(-0.729006\pi\)
−0.658965 + 0.752173i \(0.729006\pi\)
\(788\) 0 0
\(789\) 1.20736 0.0429833
\(790\) 0 0
\(791\) −43.5171 −1.54729
\(792\) 0 0
\(793\) −49.4225 −1.75504
\(794\) 0 0
\(795\) −35.0860 −1.24437
\(796\) 0 0
\(797\) −5.29826 −0.187674 −0.0938369 0.995588i \(-0.529913\pi\)
−0.0938369 + 0.995588i \(0.529913\pi\)
\(798\) 0 0
\(799\) 33.8795 1.19857
\(800\) 0 0
\(801\) −5.05970 −0.178776
\(802\) 0 0
\(803\) 29.2214 1.03120
\(804\) 0 0
\(805\) −75.1617 −2.64910
\(806\) 0 0
\(807\) 59.9626 2.11078
\(808\) 0 0
\(809\) −42.7600 −1.50336 −0.751680 0.659527i \(-0.770756\pi\)
−0.751680 + 0.659527i \(0.770756\pi\)
\(810\) 0 0
\(811\) −26.1404 −0.917914 −0.458957 0.888458i \(-0.651777\pi\)
−0.458957 + 0.888458i \(0.651777\pi\)
\(812\) 0 0
\(813\) −1.23635 −0.0433607
\(814\) 0 0
\(815\) −10.3253 −0.361681
\(816\) 0 0
\(817\) −6.16669 −0.215745
\(818\) 0 0
\(819\) −9.84165 −0.343895
\(820\) 0 0
\(821\) −10.0450 −0.350575 −0.175287 0.984517i \(-0.556085\pi\)
−0.175287 + 0.984517i \(0.556085\pi\)
\(822\) 0 0
\(823\) 0.690450 0.0240676 0.0120338 0.999928i \(-0.496169\pi\)
0.0120338 + 0.999928i \(0.496169\pi\)
\(824\) 0 0
\(825\) 4.09089 0.142427
\(826\) 0 0
\(827\) 32.1255 1.11711 0.558557 0.829466i \(-0.311355\pi\)
0.558557 + 0.829466i \(0.311355\pi\)
\(828\) 0 0
\(829\) 1.09009 0.0378605 0.0189302 0.999821i \(-0.493974\pi\)
0.0189302 + 0.999821i \(0.493974\pi\)
\(830\) 0 0
\(831\) 36.7230 1.27391
\(832\) 0 0
\(833\) 101.883 3.53005
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 23.5914 0.815438
\(838\) 0 0
\(839\) −23.0574 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(840\) 0 0
\(841\) −26.4557 −0.912266
\(842\) 0 0
\(843\) −48.4149 −1.66750
\(844\) 0 0
\(845\) −3.87884 −0.133436
\(846\) 0 0
\(847\) −31.2622 −1.07418
\(848\) 0 0
\(849\) 22.0549 0.756922
\(850\) 0 0
\(851\) −11.6870 −0.400625
\(852\) 0 0
\(853\) −16.0903 −0.550923 −0.275461 0.961312i \(-0.588831\pi\)
−0.275461 + 0.961312i \(0.588831\pi\)
\(854\) 0 0
\(855\) −1.11508 −0.0381348
\(856\) 0 0
\(857\) 20.1916 0.689733 0.344866 0.938652i \(-0.387924\pi\)
0.344866 + 0.938652i \(0.387924\pi\)
\(858\) 0 0
\(859\) 48.7326 1.66273 0.831367 0.555723i \(-0.187559\pi\)
0.831367 + 0.555723i \(0.187559\pi\)
\(860\) 0 0
\(861\) −90.7123 −3.09147
\(862\) 0 0
\(863\) −39.3807 −1.34054 −0.670268 0.742119i \(-0.733821\pi\)
−0.670268 + 0.742119i \(0.733821\pi\)
\(864\) 0 0
\(865\) −0.944765 −0.0321230
\(866\) 0 0
\(867\) 34.2126 1.16192
\(868\) 0 0
\(869\) 17.7510 0.602162
\(870\) 0 0
\(871\) −35.1687 −1.19165
\(872\) 0 0
\(873\) 3.21198 0.108709
\(874\) 0 0
\(875\) −59.0499 −1.99625
\(876\) 0 0
\(877\) 17.0862 0.576960 0.288480 0.957486i \(-0.406850\pi\)
0.288480 + 0.957486i \(0.406850\pi\)
\(878\) 0 0
\(879\) −25.9655 −0.875794
\(880\) 0 0
\(881\) 41.0783 1.38396 0.691981 0.721916i \(-0.256738\pi\)
0.691981 + 0.721916i \(0.256738\pi\)
\(882\) 0 0
\(883\) 44.5706 1.49992 0.749960 0.661483i \(-0.230073\pi\)
0.749960 + 0.661483i \(0.230073\pi\)
\(884\) 0 0
\(885\) 21.9486 0.737795
\(886\) 0 0
\(887\) 53.7165 1.80362 0.901812 0.432128i \(-0.142237\pi\)
0.901812 + 0.432128i \(0.142237\pi\)
\(888\) 0 0
\(889\) −46.8837 −1.57243
\(890\) 0 0
\(891\) 22.5106 0.754133
\(892\) 0 0
\(893\) −5.30715 −0.177597
\(894\) 0 0
\(895\) −48.4026 −1.61792
\(896\) 0 0
\(897\) 48.2005 1.60937
\(898\) 0 0
\(899\) 8.26682 0.275714
\(900\) 0 0
\(901\) −54.7115 −1.82270
\(902\) 0 0
\(903\) 62.1142 2.06703
\(904\) 0 0
\(905\) 16.5057 0.548669
\(906\) 0 0
\(907\) −21.9320 −0.728239 −0.364120 0.931352i \(-0.618630\pi\)
−0.364120 + 0.931352i \(0.618630\pi\)
\(908\) 0 0
\(909\) 5.86943 0.194677
\(910\) 0 0
\(911\) 52.9321 1.75372 0.876860 0.480746i \(-0.159634\pi\)
0.876860 + 0.480746i \(0.159634\pi\)
\(912\) 0 0
\(913\) −17.1964 −0.569116
\(914\) 0 0
\(915\) 56.4026 1.86461
\(916\) 0 0
\(917\) 15.6750 0.517633
\(918\) 0 0
\(919\) −33.5195 −1.10570 −0.552852 0.833279i \(-0.686461\pi\)
−0.552852 + 0.833279i \(0.686461\pi\)
\(920\) 0 0
\(921\) 36.2471 1.19438
\(922\) 0 0
\(923\) −18.3385 −0.603618
\(924\) 0 0
\(925\) −1.53029 −0.0503156
\(926\) 0 0
\(927\) 5.64467 0.185395
\(928\) 0 0
\(929\) −42.7611 −1.40295 −0.701473 0.712696i \(-0.747474\pi\)
−0.701473 + 0.712696i \(0.747474\pi\)
\(930\) 0 0
\(931\) −15.9598 −0.523062
\(932\) 0 0
\(933\) −43.8766 −1.43645
\(934\) 0 0
\(935\) −25.5166 −0.834482
\(936\) 0 0
\(937\) −28.5004 −0.931067 −0.465533 0.885030i \(-0.654137\pi\)
−0.465533 + 0.885030i \(0.654137\pi\)
\(938\) 0 0
\(939\) −10.6942 −0.348991
\(940\) 0 0
\(941\) 26.2339 0.855201 0.427601 0.903968i \(-0.359359\pi\)
0.427601 + 0.903968i \(0.359359\pi\)
\(942\) 0 0
\(943\) 74.1862 2.41583
\(944\) 0 0
\(945\) −44.7987 −1.45730
\(946\) 0 0
\(947\) −1.11732 −0.0363082 −0.0181541 0.999835i \(-0.505779\pi\)
−0.0181541 + 0.999835i \(0.505779\pi\)
\(948\) 0 0
\(949\) 45.0824 1.46344
\(950\) 0 0
\(951\) −17.9249 −0.581253
\(952\) 0 0
\(953\) −21.6413 −0.701032 −0.350516 0.936557i \(-0.613994\pi\)
−0.350516 + 0.936557i \(0.613994\pi\)
\(954\) 0 0
\(955\) 39.3611 1.27370
\(956\) 0 0
\(957\) 6.52529 0.210933
\(958\) 0 0
\(959\) −42.4208 −1.36984
\(960\) 0 0
\(961\) −4.13958 −0.133535
\(962\) 0 0
\(963\) −6.56460 −0.211541
\(964\) 0 0
\(965\) 1.96178 0.0631518
\(966\) 0 0
\(967\) −15.0526 −0.484060 −0.242030 0.970269i \(-0.577813\pi\)
−0.242030 + 0.970269i \(0.577813\pi\)
\(968\) 0 0
\(969\) −10.4130 −0.334514
\(970\) 0 0
\(971\) 8.32460 0.267149 0.133575 0.991039i \(-0.457354\pi\)
0.133575 + 0.991039i \(0.457354\pi\)
\(972\) 0 0
\(973\) 82.4028 2.64171
\(974\) 0 0
\(975\) 6.31136 0.202125
\(976\) 0 0
\(977\) 45.9701 1.47071 0.735357 0.677680i \(-0.237014\pi\)
0.735357 + 0.677680i \(0.237014\pi\)
\(978\) 0 0
\(979\) 18.1371 0.579664
\(980\) 0 0
\(981\) 5.75683 0.183801
\(982\) 0 0
\(983\) 40.8484 1.30286 0.651430 0.758709i \(-0.274169\pi\)
0.651430 + 0.758709i \(0.274169\pi\)
\(984\) 0 0
\(985\) −32.0757 −1.02202
\(986\) 0 0
\(987\) 53.4564 1.70154
\(988\) 0 0
\(989\) −50.7981 −1.61529
\(990\) 0 0
\(991\) 28.1825 0.895247 0.447624 0.894222i \(-0.352270\pi\)
0.447624 + 0.894222i \(0.352270\pi\)
\(992\) 0 0
\(993\) 24.2083 0.768226
\(994\) 0 0
\(995\) 37.2413 1.18063
\(996\) 0 0
\(997\) −18.9832 −0.601205 −0.300602 0.953750i \(-0.597188\pi\)
−0.300602 + 0.953750i \(0.597188\pi\)
\(998\) 0 0
\(999\) −6.96580 −0.220388
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.1.4 5
3.2 odd 2 6012.2.a.d.1.5 5
4.3 odd 2 2672.2.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.4 5 1.1 even 1 trivial
2672.2.a.j.1.2 5 4.3 odd 2
6012.2.a.d.1.5 5 3.2 odd 2