Properties

Label 668.2.a.b.1.5
Level $668$
Weight $2$
Character 668.1
Self dual yes
Analytic conductor $5.334$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.5
Root \(0.147687\) of defining polynomial
Character \(\chi\) \(=\) 668.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.12588 q^{3} +2.00000 q^{5} -0.516539 q^{7} +6.77110 q^{9} +O(q^{10})\) \(q+3.12588 q^{3} +2.00000 q^{5} -0.516539 q^{7} +6.77110 q^{9} +0.538350 q^{11} -0.295373 q^{13} +6.25175 q^{15} -6.99507 q^{17} -4.06647 q^{19} -1.61464 q^{21} +0.781327 q^{23} -1.00000 q^{25} +11.7880 q^{27} +0.811912 q^{29} +3.07629 q^{31} +1.68282 q^{33} -1.03308 q^{35} -3.95638 q^{37} -0.923300 q^{39} +0.743318 q^{41} +0.590746 q^{43} +13.5422 q^{45} -3.47572 q^{47} -6.73319 q^{49} -21.8657 q^{51} +6.69970 q^{53} +1.07670 q^{55} -12.7113 q^{57} -8.91427 q^{59} +4.63992 q^{61} -3.49753 q^{63} -0.590746 q^{65} +6.55274 q^{67} +2.44233 q^{69} +12.4655 q^{71} +7.77640 q^{73} -3.12588 q^{75} -0.278079 q^{77} -4.88529 q^{79} +16.5345 q^{81} -1.11388 q^{83} -13.9901 q^{85} +2.53794 q^{87} -1.79742 q^{89} +0.152572 q^{91} +9.61609 q^{93} -8.13294 q^{95} -17.1192 q^{97} +3.64522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + 5q^{11} - 4q^{13} + 6q^{15} - 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} - 5q^{25} + 12q^{27} - 5q^{29} + 9q^{31} - 8q^{33} + 18q^{35} + 8q^{37} - 4q^{41} + 8q^{43} + 12q^{45} + 13q^{47} + 14q^{49} - 2q^{53} + 10q^{55} - 12q^{57} + 4q^{59} + 11q^{61} - q^{63} - 8q^{65} + 28q^{67} - 16q^{69} + 2q^{71} + 8q^{73} - 3q^{75} - 12q^{77} - 10q^{79} - 15q^{81} + 2q^{83} - 4q^{85} + 4q^{87} - 17q^{89} - 12q^{91} - 12q^{93} + 10q^{95} - 27q^{97} + 3q^{99} + O(q^{100}) \)

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
\(3\) 3.12588 1.80472 0.902362 0.430978i \(-0.141831\pi\)
0.902362 + 0.430978i \(0.141831\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −0.516539 −0.195233 −0.0976167 0.995224i \(-0.531122\pi\)
−0.0976167 + 0.995224i \(0.531122\pi\)
\(8\) 0 0
\(9\) 6.77110 2.25703
\(10\) 0 0
\(11\) 0.538350 0.162319 0.0811593 0.996701i \(-0.474138\pi\)
0.0811593 + 0.996701i \(0.474138\pi\)
\(12\) 0 0
\(13\) −0.295373 −0.0819218 −0.0409609 0.999161i \(-0.513042\pi\)
−0.0409609 + 0.999161i \(0.513042\pi\)
\(14\) 0 0
\(15\) 6.25175 1.61420
\(16\) 0 0
\(17\) −6.99507 −1.69655 −0.848277 0.529553i \(-0.822360\pi\)
−0.848277 + 0.529553i \(0.822360\pi\)
\(18\) 0 0
\(19\) −4.06647 −0.932912 −0.466456 0.884544i \(-0.654469\pi\)
−0.466456 + 0.884544i \(0.654469\pi\)
\(20\) 0 0
\(21\) −1.61464 −0.352342
\(22\) 0 0
\(23\) 0.781327 0.162918 0.0814590 0.996677i \(-0.474042\pi\)
0.0814590 + 0.996677i \(0.474042\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 11.7880 2.26860
\(28\) 0 0
\(29\) 0.811912 0.150768 0.0753841 0.997155i \(-0.475982\pi\)
0.0753841 + 0.997155i \(0.475982\pi\)
\(30\) 0 0
\(31\) 3.07629 0.552517 0.276259 0.961083i \(-0.410905\pi\)
0.276259 + 0.961083i \(0.410905\pi\)
\(32\) 0 0
\(33\) 1.68282 0.292941
\(34\) 0 0
\(35\) −1.03308 −0.174622
\(36\) 0 0
\(37\) −3.95638 −0.650424 −0.325212 0.945641i \(-0.605436\pi\)
−0.325212 + 0.945641i \(0.605436\pi\)
\(38\) 0 0
\(39\) −0.923300 −0.147846
\(40\) 0 0
\(41\) 0.743318 0.116087 0.0580434 0.998314i \(-0.481514\pi\)
0.0580434 + 0.998314i \(0.481514\pi\)
\(42\) 0 0
\(43\) 0.590746 0.0900880 0.0450440 0.998985i \(-0.485657\pi\)
0.0450440 + 0.998985i \(0.485657\pi\)
\(44\) 0 0
\(45\) 13.5422 2.01875
\(46\) 0 0
\(47\) −3.47572 −0.506986 −0.253493 0.967337i \(-0.581580\pi\)
−0.253493 + 0.967337i \(0.581580\pi\)
\(48\) 0 0
\(49\) −6.73319 −0.961884
\(50\) 0 0
\(51\) −21.8657 −3.06181
\(52\) 0 0
\(53\) 6.69970 0.920274 0.460137 0.887848i \(-0.347800\pi\)
0.460137 + 0.887848i \(0.347800\pi\)
\(54\) 0 0
\(55\) 1.07670 0.145182
\(56\) 0 0
\(57\) −12.7113 −1.68365
\(58\) 0 0
\(59\) −8.91427 −1.16054 −0.580269 0.814425i \(-0.697053\pi\)
−0.580269 + 0.814425i \(0.697053\pi\)
\(60\) 0 0
\(61\) 4.63992 0.594081 0.297041 0.954865i \(-0.404000\pi\)
0.297041 + 0.954865i \(0.404000\pi\)
\(62\) 0 0
\(63\) −3.49753 −0.440648
\(64\) 0 0
\(65\) −0.590746 −0.0732731
\(66\) 0 0
\(67\) 6.55274 0.800544 0.400272 0.916396i \(-0.368916\pi\)
0.400272 + 0.916396i \(0.368916\pi\)
\(68\) 0 0
\(69\) 2.44233 0.294022
\(70\) 0 0
\(71\) 12.4655 1.47938 0.739691 0.672947i \(-0.234972\pi\)
0.739691 + 0.672947i \(0.234972\pi\)
\(72\) 0 0
\(73\) 7.77640 0.910158 0.455079 0.890451i \(-0.349611\pi\)
0.455079 + 0.890451i \(0.349611\pi\)
\(74\) 0 0
\(75\) −3.12588 −0.360945
\(76\) 0 0
\(77\) −0.278079 −0.0316900
\(78\) 0 0
\(79\) −4.88529 −0.549638 −0.274819 0.961496i \(-0.588618\pi\)
−0.274819 + 0.961496i \(0.588618\pi\)
\(80\) 0 0
\(81\) 16.5345 1.83716
\(82\) 0 0
\(83\) −1.11388 −0.122264 −0.0611321 0.998130i \(-0.519471\pi\)
−0.0611321 + 0.998130i \(0.519471\pi\)
\(84\) 0 0
\(85\) −13.9901 −1.51744
\(86\) 0 0
\(87\) 2.53794 0.272095
\(88\) 0 0
\(89\) −1.79742 −0.190527 −0.0952633 0.995452i \(-0.530369\pi\)
−0.0952633 + 0.995452i \(0.530369\pi\)
\(90\) 0 0
\(91\) 0.152572 0.0159939
\(92\) 0 0
\(93\) 9.61609 0.997142
\(94\) 0 0
\(95\) −8.13294 −0.834422
\(96\) 0 0
\(97\) −17.1192 −1.73819 −0.869097 0.494641i \(-0.835300\pi\)
−0.869097 + 0.494641i \(0.835300\pi\)
\(98\) 0 0
\(99\) 3.64522 0.366358
\(100\) 0 0
\(101\) −14.0281 −1.39585 −0.697926 0.716170i \(-0.745894\pi\)
−0.697926 + 0.716170i \(0.745894\pi\)
\(102\) 0 0
\(103\) 16.7553 1.65094 0.825472 0.564443i \(-0.190909\pi\)
0.825472 + 0.564443i \(0.190909\pi\)
\(104\) 0 0
\(105\) −3.22927 −0.315145
\(106\) 0 0
\(107\) 9.47360 0.915847 0.457924 0.888992i \(-0.348593\pi\)
0.457924 + 0.888992i \(0.348593\pi\)
\(108\) 0 0
\(109\) 20.3467 1.94886 0.974429 0.224695i \(-0.0721384\pi\)
0.974429 + 0.224695i \(0.0721384\pi\)
\(110\) 0 0
\(111\) −12.3671 −1.17384
\(112\) 0 0
\(113\) 16.8270 1.58295 0.791476 0.611200i \(-0.209313\pi\)
0.791476 + 0.611200i \(0.209313\pi\)
\(114\) 0 0
\(115\) 1.56265 0.145718
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 3.61322 0.331224
\(120\) 0 0
\(121\) −10.7102 −0.973653
\(122\) 0 0
\(123\) 2.32352 0.209505
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 9.52781 0.845456 0.422728 0.906257i \(-0.361073\pi\)
0.422728 + 0.906257i \(0.361073\pi\)
\(128\) 0 0
\(129\) 1.84660 0.162584
\(130\) 0 0
\(131\) −16.5809 −1.44868 −0.724339 0.689444i \(-0.757855\pi\)
−0.724339 + 0.689444i \(0.757855\pi\)
\(132\) 0 0
\(133\) 2.10049 0.182136
\(134\) 0 0
\(135\) 23.5760 2.02909
\(136\) 0 0
\(137\) 3.53903 0.302360 0.151180 0.988506i \(-0.451693\pi\)
0.151180 + 0.988506i \(0.451693\pi\)
\(138\) 0 0
\(139\) −9.40022 −0.797316 −0.398658 0.917100i \(-0.630524\pi\)
−0.398658 + 0.917100i \(0.630524\pi\)
\(140\) 0 0
\(141\) −10.8647 −0.914971
\(142\) 0 0
\(143\) −0.159014 −0.0132974
\(144\) 0 0
\(145\) 1.62382 0.134851
\(146\) 0 0
\(147\) −21.0471 −1.73594
\(148\) 0 0
\(149\) −18.5809 −1.52221 −0.761103 0.648631i \(-0.775342\pi\)
−0.761103 + 0.648631i \(0.775342\pi\)
\(150\) 0 0
\(151\) 17.2644 1.40495 0.702477 0.711706i \(-0.252077\pi\)
0.702477 + 0.711706i \(0.252077\pi\)
\(152\) 0 0
\(153\) −47.3643 −3.82918
\(154\) 0 0
\(155\) 6.15257 0.494186
\(156\) 0 0
\(157\) −7.19863 −0.574513 −0.287256 0.957854i \(-0.592743\pi\)
−0.287256 + 0.957854i \(0.592743\pi\)
\(158\) 0 0
\(159\) 20.9424 1.66084
\(160\) 0 0
\(161\) −0.403586 −0.0318070
\(162\) 0 0
\(163\) 23.6948 1.85592 0.927959 0.372683i \(-0.121562\pi\)
0.927959 + 0.372683i \(0.121562\pi\)
\(164\) 0 0
\(165\) 3.36563 0.262014
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −12.9128 −0.993289
\(170\) 0 0
\(171\) −27.5345 −2.10561
\(172\) 0 0
\(173\) −19.5278 −1.48467 −0.742336 0.670028i \(-0.766282\pi\)
−0.742336 + 0.670028i \(0.766282\pi\)
\(174\) 0 0
\(175\) 0.516539 0.0390467
\(176\) 0 0
\(177\) −27.8649 −2.09445
\(178\) 0 0
\(179\) −10.5038 −0.785092 −0.392546 0.919732i \(-0.628406\pi\)
−0.392546 + 0.919732i \(0.628406\pi\)
\(180\) 0 0
\(181\) 8.36184 0.621531 0.310765 0.950487i \(-0.399415\pi\)
0.310765 + 0.950487i \(0.399415\pi\)
\(182\) 0 0
\(183\) 14.5038 1.07215
\(184\) 0 0
\(185\) −7.91275 −0.581757
\(186\) 0 0
\(187\) −3.76580 −0.275382
\(188\) 0 0
\(189\) −6.08895 −0.442906
\(190\) 0 0
\(191\) 9.43844 0.682942 0.341471 0.939892i \(-0.389075\pi\)
0.341471 + 0.939892i \(0.389075\pi\)
\(192\) 0 0
\(193\) −16.3291 −1.17540 −0.587698 0.809080i \(-0.699966\pi\)
−0.587698 + 0.809080i \(0.699966\pi\)
\(194\) 0 0
\(195\) −1.84660 −0.132238
\(196\) 0 0
\(197\) −17.3460 −1.23585 −0.617926 0.786237i \(-0.712027\pi\)
−0.617926 + 0.786237i \(0.712027\pi\)
\(198\) 0 0
\(199\) 6.46097 0.458006 0.229003 0.973426i \(-0.426453\pi\)
0.229003 + 0.973426i \(0.426453\pi\)
\(200\) 0 0
\(201\) 20.4830 1.44476
\(202\) 0 0
\(203\) −0.419384 −0.0294350
\(204\) 0 0
\(205\) 1.48664 0.103831
\(206\) 0 0
\(207\) 5.29044 0.367711
\(208\) 0 0
\(209\) −2.18918 −0.151429
\(210\) 0 0
\(211\) −2.41035 −0.165935 −0.0829677 0.996552i \(-0.526440\pi\)
−0.0829677 + 0.996552i \(0.526440\pi\)
\(212\) 0 0
\(213\) 38.9656 2.66988
\(214\) 0 0
\(215\) 1.18149 0.0805771
\(216\) 0 0
\(217\) −1.58902 −0.107870
\(218\) 0 0
\(219\) 24.3080 1.64259
\(220\) 0 0
\(221\) 2.06616 0.138985
\(222\) 0 0
\(223\) −0.798106 −0.0534452 −0.0267226 0.999643i \(-0.508507\pi\)
−0.0267226 + 0.999643i \(0.508507\pi\)
\(224\) 0 0
\(225\) −6.77110 −0.451406
\(226\) 0 0
\(227\) −10.4979 −0.696769 −0.348385 0.937352i \(-0.613270\pi\)
−0.348385 + 0.937352i \(0.613270\pi\)
\(228\) 0 0
\(229\) 11.6916 0.772602 0.386301 0.922373i \(-0.373753\pi\)
0.386301 + 0.922373i \(0.373753\pi\)
\(230\) 0 0
\(231\) −0.869240 −0.0571918
\(232\) 0 0
\(233\) −23.8858 −1.56481 −0.782404 0.622771i \(-0.786007\pi\)
−0.782404 + 0.622771i \(0.786007\pi\)
\(234\) 0 0
\(235\) −6.95145 −0.453462
\(236\) 0 0
\(237\) −15.2708 −0.991946
\(238\) 0 0
\(239\) 21.8051 1.41046 0.705228 0.708981i \(-0.250845\pi\)
0.705228 + 0.708981i \(0.250845\pi\)
\(240\) 0 0
\(241\) 17.4951 1.12696 0.563481 0.826129i \(-0.309462\pi\)
0.563481 + 0.826129i \(0.309462\pi\)
\(242\) 0 0
\(243\) 16.3207 1.04697
\(244\) 0 0
\(245\) −13.4664 −0.860335
\(246\) 0 0
\(247\) 1.20113 0.0764258
\(248\) 0 0
\(249\) −3.48185 −0.220653
\(250\) 0 0
\(251\) −1.26889 −0.0800917 −0.0400458 0.999198i \(-0.512750\pi\)
−0.0400458 + 0.999198i \(0.512750\pi\)
\(252\) 0 0
\(253\) 0.420628 0.0264446
\(254\) 0 0
\(255\) −43.7314 −2.73857
\(256\) 0 0
\(257\) 26.5168 1.65407 0.827036 0.562149i \(-0.190025\pi\)
0.827036 + 0.562149i \(0.190025\pi\)
\(258\) 0 0
\(259\) 2.04362 0.126985
\(260\) 0 0
\(261\) 5.49753 0.340289
\(262\) 0 0
\(263\) −14.2975 −0.881622 −0.440811 0.897600i \(-0.645309\pi\)
−0.440811 + 0.897600i \(0.645309\pi\)
\(264\) 0 0
\(265\) 13.3994 0.823118
\(266\) 0 0
\(267\) −5.61852 −0.343848
\(268\) 0 0
\(269\) −18.0183 −1.09859 −0.549297 0.835627i \(-0.685104\pi\)
−0.549297 + 0.835627i \(0.685104\pi\)
\(270\) 0 0
\(271\) 5.40925 0.328589 0.164294 0.986411i \(-0.447465\pi\)
0.164294 + 0.986411i \(0.447465\pi\)
\(272\) 0 0
\(273\) 0.476920 0.0288645
\(274\) 0 0
\(275\) −0.538350 −0.0324637
\(276\) 0 0
\(277\) 0.475355 0.0285613 0.0142806 0.999898i \(-0.495454\pi\)
0.0142806 + 0.999898i \(0.495454\pi\)
\(278\) 0 0
\(279\) 20.8298 1.24705
\(280\) 0 0
\(281\) −9.23067 −0.550655 −0.275328 0.961350i \(-0.588786\pi\)
−0.275328 + 0.961350i \(0.588786\pi\)
\(282\) 0 0
\(283\) 22.6333 1.34541 0.672704 0.739911i \(-0.265133\pi\)
0.672704 + 0.739911i \(0.265133\pi\)
\(284\) 0 0
\(285\) −25.4226 −1.50590
\(286\) 0 0
\(287\) −0.383953 −0.0226640
\(288\) 0 0
\(289\) 31.9310 1.87829
\(290\) 0 0
\(291\) −53.5126 −3.13696
\(292\) 0 0
\(293\) −10.0171 −0.585207 −0.292604 0.956234i \(-0.594522\pi\)
−0.292604 + 0.956234i \(0.594522\pi\)
\(294\) 0 0
\(295\) −17.8285 −1.03802
\(296\) 0 0
\(297\) 6.34606 0.368236
\(298\) 0 0
\(299\) −0.230783 −0.0133465
\(300\) 0 0
\(301\) −0.305143 −0.0175882
\(302\) 0 0
\(303\) −43.8502 −2.51913
\(304\) 0 0
\(305\) 9.27984 0.531362
\(306\) 0 0
\(307\) 16.0119 0.913849 0.456925 0.889505i \(-0.348951\pi\)
0.456925 + 0.889505i \(0.348951\pi\)
\(308\) 0 0
\(309\) 52.3748 2.97950
\(310\) 0 0
\(311\) −10.3613 −0.587537 −0.293768 0.955877i \(-0.594909\pi\)
−0.293768 + 0.955877i \(0.594909\pi\)
\(312\) 0 0
\(313\) 9.36002 0.529059 0.264530 0.964378i \(-0.414783\pi\)
0.264530 + 0.964378i \(0.414783\pi\)
\(314\) 0 0
\(315\) −6.99507 −0.394127
\(316\) 0 0
\(317\) 12.8595 0.722260 0.361130 0.932515i \(-0.382391\pi\)
0.361130 + 0.932515i \(0.382391\pi\)
\(318\) 0 0
\(319\) 0.437093 0.0244725
\(320\) 0 0
\(321\) 29.6133 1.65285
\(322\) 0 0
\(323\) 28.4452 1.58273
\(324\) 0 0
\(325\) 0.295373 0.0163844
\(326\) 0 0
\(327\) 63.6012 3.51715
\(328\) 0 0
\(329\) 1.79535 0.0989806
\(330\) 0 0
\(331\) −2.81289 −0.154611 −0.0773053 0.997007i \(-0.524632\pi\)
−0.0773053 + 0.997007i \(0.524632\pi\)
\(332\) 0 0
\(333\) −26.7890 −1.46803
\(334\) 0 0
\(335\) 13.1055 0.716029
\(336\) 0 0
\(337\) 19.3021 1.05145 0.525725 0.850654i \(-0.323794\pi\)
0.525725 + 0.850654i \(0.323794\pi\)
\(338\) 0 0
\(339\) 52.5992 2.85679
\(340\) 0 0
\(341\) 1.65612 0.0896839
\(342\) 0 0
\(343\) 7.09373 0.383025
\(344\) 0 0
\(345\) 4.88466 0.262981
\(346\) 0 0
\(347\) 28.4120 1.52524 0.762618 0.646849i \(-0.223914\pi\)
0.762618 + 0.646849i \(0.223914\pi\)
\(348\) 0 0
\(349\) 7.74264 0.414454 0.207227 0.978293i \(-0.433556\pi\)
0.207227 + 0.978293i \(0.433556\pi\)
\(350\) 0 0
\(351\) −3.48185 −0.185847
\(352\) 0 0
\(353\) 19.3204 1.02832 0.514161 0.857694i \(-0.328104\pi\)
0.514161 + 0.857694i \(0.328104\pi\)
\(354\) 0 0
\(355\) 24.9310 1.32320
\(356\) 0 0
\(357\) 11.2945 0.597768
\(358\) 0 0
\(359\) −2.57278 −0.135786 −0.0678932 0.997693i \(-0.521628\pi\)
−0.0678932 + 0.997693i \(0.521628\pi\)
\(360\) 0 0
\(361\) −2.46383 −0.129675
\(362\) 0 0
\(363\) −33.4787 −1.75718
\(364\) 0 0
\(365\) 15.5528 0.814070
\(366\) 0 0
\(367\) −23.1615 −1.20902 −0.604509 0.796598i \(-0.706631\pi\)
−0.604509 + 0.796598i \(0.706631\pi\)
\(368\) 0 0
\(369\) 5.03308 0.262012
\(370\) 0 0
\(371\) −3.46065 −0.179668
\(372\) 0 0
\(373\) 20.9233 1.08337 0.541684 0.840582i \(-0.317787\pi\)
0.541684 + 0.840582i \(0.317787\pi\)
\(374\) 0 0
\(375\) −37.5105 −1.93703
\(376\) 0 0
\(377\) −0.239817 −0.0123512
\(378\) 0 0
\(379\) 30.4945 1.56640 0.783198 0.621773i \(-0.213587\pi\)
0.783198 + 0.621773i \(0.213587\pi\)
\(380\) 0 0
\(381\) 29.7827 1.52582
\(382\) 0 0
\(383\) 27.8997 1.42561 0.712804 0.701363i \(-0.247425\pi\)
0.712804 + 0.701363i \(0.247425\pi\)
\(384\) 0 0
\(385\) −0.556158 −0.0283444
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 18.9087 0.958707 0.479353 0.877622i \(-0.340871\pi\)
0.479353 + 0.877622i \(0.340871\pi\)
\(390\) 0 0
\(391\) −5.46544 −0.276399
\(392\) 0 0
\(393\) −51.8298 −2.61447
\(394\) 0 0
\(395\) −9.77058 −0.491611
\(396\) 0 0
\(397\) −23.2771 −1.16824 −0.584122 0.811666i \(-0.698561\pi\)
−0.584122 + 0.811666i \(0.698561\pi\)
\(398\) 0 0
\(399\) 6.56587 0.328705
\(400\) 0 0
\(401\) −23.3692 −1.16700 −0.583500 0.812113i \(-0.698317\pi\)
−0.583500 + 0.812113i \(0.698317\pi\)
\(402\) 0 0
\(403\) −0.908652 −0.0452632
\(404\) 0 0
\(405\) 33.0689 1.64321
\(406\) 0 0
\(407\) −2.12992 −0.105576
\(408\) 0 0
\(409\) −13.5940 −0.672178 −0.336089 0.941830i \(-0.609104\pi\)
−0.336089 + 0.941830i \(0.609104\pi\)
\(410\) 0 0
\(411\) 11.0626 0.545676
\(412\) 0 0
\(413\) 4.60456 0.226576
\(414\) 0 0
\(415\) −2.22776 −0.109356
\(416\) 0 0
\(417\) −29.3839 −1.43894
\(418\) 0 0
\(419\) 20.6740 1.00999 0.504996 0.863121i \(-0.331494\pi\)
0.504996 + 0.863121i \(0.331494\pi\)
\(420\) 0 0
\(421\) 33.9625 1.65523 0.827614 0.561297i \(-0.189698\pi\)
0.827614 + 0.561297i \(0.189698\pi\)
\(422\) 0 0
\(423\) −23.5345 −1.14428
\(424\) 0 0
\(425\) 6.99507 0.339311
\(426\) 0 0
\(427\) −2.39670 −0.115984
\(428\) 0 0
\(429\) −0.497059 −0.0239982
\(430\) 0 0
\(431\) −4.35940 −0.209985 −0.104992 0.994473i \(-0.533482\pi\)
−0.104992 + 0.994473i \(0.533482\pi\)
\(432\) 0 0
\(433\) −8.95196 −0.430204 −0.215102 0.976592i \(-0.569008\pi\)
−0.215102 + 0.976592i \(0.569008\pi\)
\(434\) 0 0
\(435\) 5.07587 0.243369
\(436\) 0 0
\(437\) −3.17724 −0.151988
\(438\) 0 0
\(439\) 30.0473 1.43408 0.717039 0.697033i \(-0.245497\pi\)
0.717039 + 0.697033i \(0.245497\pi\)
\(440\) 0 0
\(441\) −45.5911 −2.17100
\(442\) 0 0
\(443\) −14.2252 −0.675858 −0.337929 0.941172i \(-0.609726\pi\)
−0.337929 + 0.941172i \(0.609726\pi\)
\(444\) 0 0
\(445\) −3.59485 −0.170412
\(446\) 0 0
\(447\) −58.0815 −2.74716
\(448\) 0 0
\(449\) −5.54334 −0.261606 −0.130803 0.991408i \(-0.541756\pi\)
−0.130803 + 0.991408i \(0.541756\pi\)
\(450\) 0 0
\(451\) 0.400165 0.0188431
\(452\) 0 0
\(453\) 53.9663 2.53556
\(454\) 0 0
\(455\) 0.305143 0.0143053
\(456\) 0 0
\(457\) −11.2195 −0.524826 −0.262413 0.964956i \(-0.584518\pi\)
−0.262413 + 0.964956i \(0.584518\pi\)
\(458\) 0 0
\(459\) −82.4577 −3.84880
\(460\) 0 0
\(461\) −1.92450 −0.0896328 −0.0448164 0.998995i \(-0.514270\pi\)
−0.0448164 + 0.998995i \(0.514270\pi\)
\(462\) 0 0
\(463\) 26.4339 1.22849 0.614244 0.789116i \(-0.289461\pi\)
0.614244 + 0.789116i \(0.289461\pi\)
\(464\) 0 0
\(465\) 19.2322 0.891871
\(466\) 0 0
\(467\) −31.5897 −1.46180 −0.730898 0.682486i \(-0.760899\pi\)
−0.730898 + 0.682486i \(0.760899\pi\)
\(468\) 0 0
\(469\) −3.38474 −0.156293
\(470\) 0 0
\(471\) −22.5020 −1.03684
\(472\) 0 0
\(473\) 0.318028 0.0146230
\(474\) 0 0
\(475\) 4.06647 0.186582
\(476\) 0 0
\(477\) 45.3643 2.07709
\(478\) 0 0
\(479\) 12.8045 0.585052 0.292526 0.956258i \(-0.405504\pi\)
0.292526 + 0.956258i \(0.405504\pi\)
\(480\) 0 0
\(481\) 1.16861 0.0532839
\(482\) 0 0
\(483\) −1.26156 −0.0574029
\(484\) 0 0
\(485\) −34.2385 −1.55469
\(486\) 0 0
\(487\) −38.2482 −1.73319 −0.866596 0.499011i \(-0.833697\pi\)
−0.866596 + 0.499011i \(0.833697\pi\)
\(488\) 0 0
\(489\) 74.0669 3.34942
\(490\) 0 0
\(491\) −38.4176 −1.73376 −0.866881 0.498514i \(-0.833879\pi\)
−0.866881 + 0.498514i \(0.833879\pi\)
\(492\) 0 0
\(493\) −5.67938 −0.255786
\(494\) 0 0
\(495\) 7.29044 0.327681
\(496\) 0 0
\(497\) −6.43891 −0.288825
\(498\) 0 0
\(499\) −3.61884 −0.162001 −0.0810007 0.996714i \(-0.525812\pi\)
−0.0810007 + 0.996714i \(0.525812\pi\)
\(500\) 0 0
\(501\) −3.12588 −0.139654
\(502\) 0 0
\(503\) −31.3565 −1.39812 −0.699059 0.715064i \(-0.746398\pi\)
−0.699059 + 0.715064i \(0.746398\pi\)
\(504\) 0 0
\(505\) −28.0563 −1.24849
\(506\) 0 0
\(507\) −40.3637 −1.79261
\(508\) 0 0
\(509\) 22.9722 1.01822 0.509111 0.860701i \(-0.329974\pi\)
0.509111 + 0.860701i \(0.329974\pi\)
\(510\) 0 0
\(511\) −4.01681 −0.177693
\(512\) 0 0
\(513\) −47.9354 −2.11640
\(514\) 0 0
\(515\) 33.5105 1.47665
\(516\) 0 0
\(517\) −1.87116 −0.0822934
\(518\) 0 0
\(519\) −61.0415 −2.67942
\(520\) 0 0
\(521\) 12.1707 0.533210 0.266605 0.963806i \(-0.414098\pi\)
0.266605 + 0.963806i \(0.414098\pi\)
\(522\) 0 0
\(523\) 6.62991 0.289906 0.144953 0.989439i \(-0.453697\pi\)
0.144953 + 0.989439i \(0.453697\pi\)
\(524\) 0 0
\(525\) 1.61464 0.0704685
\(526\) 0 0
\(527\) −21.5188 −0.937375
\(528\) 0 0
\(529\) −22.3895 −0.973458
\(530\) 0 0
\(531\) −60.3593 −2.61937
\(532\) 0 0
\(533\) −0.219556 −0.00951003
\(534\) 0 0
\(535\) 18.9472 0.819159
\(536\) 0 0
\(537\) −32.8336 −1.41688
\(538\) 0 0
\(539\) −3.62481 −0.156132
\(540\) 0 0
\(541\) −14.6963 −0.631842 −0.315921 0.948785i \(-0.602313\pi\)
−0.315921 + 0.948785i \(0.602313\pi\)
\(542\) 0 0
\(543\) 26.1381 1.12169
\(544\) 0 0
\(545\) 40.6934 1.74311
\(546\) 0 0
\(547\) 40.9360 1.75030 0.875148 0.483856i \(-0.160764\pi\)
0.875148 + 0.483856i \(0.160764\pi\)
\(548\) 0 0
\(549\) 31.4174 1.34086
\(550\) 0 0
\(551\) −3.30162 −0.140654
\(552\) 0 0
\(553\) 2.52344 0.107308
\(554\) 0 0
\(555\) −24.7343 −1.04991
\(556\) 0 0
\(557\) 41.2803 1.74910 0.874552 0.484932i \(-0.161156\pi\)
0.874552 + 0.484932i \(0.161156\pi\)
\(558\) 0 0
\(559\) −0.174491 −0.00738017
\(560\) 0 0
\(561\) −11.7714 −0.496989
\(562\) 0 0
\(563\) 15.4103 0.649467 0.324733 0.945806i \(-0.394725\pi\)
0.324733 + 0.945806i \(0.394725\pi\)
\(564\) 0 0
\(565\) 33.6540 1.41584
\(566\) 0 0
\(567\) −8.54069 −0.358675
\(568\) 0 0
\(569\) 11.2870 0.473177 0.236588 0.971610i \(-0.423971\pi\)
0.236588 + 0.971610i \(0.423971\pi\)
\(570\) 0 0
\(571\) 1.50980 0.0631830 0.0315915 0.999501i \(-0.489942\pi\)
0.0315915 + 0.999501i \(0.489942\pi\)
\(572\) 0 0
\(573\) 29.5034 1.23252
\(574\) 0 0
\(575\) −0.781327 −0.0325836
\(576\) 0 0
\(577\) 24.9800 1.03993 0.519966 0.854187i \(-0.325945\pi\)
0.519966 + 0.854187i \(0.325945\pi\)
\(578\) 0 0
\(579\) −51.0428 −2.12127
\(580\) 0 0
\(581\) 0.575363 0.0238701
\(582\) 0 0
\(583\) 3.60678 0.149378
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 20.9400 0.864287 0.432144 0.901805i \(-0.357757\pi\)
0.432144 + 0.901805i \(0.357757\pi\)
\(588\) 0 0
\(589\) −12.5096 −0.515450
\(590\) 0 0
\(591\) −54.2214 −2.23037
\(592\) 0 0
\(593\) −32.8904 −1.35065 −0.675323 0.737522i \(-0.735996\pi\)
−0.675323 + 0.737522i \(0.735996\pi\)
\(594\) 0 0
\(595\) 7.22645 0.296256
\(596\) 0 0
\(597\) 20.1962 0.826575
\(598\) 0 0
\(599\) −18.3583 −0.750098 −0.375049 0.927005i \(-0.622374\pi\)
−0.375049 + 0.927005i \(0.622374\pi\)
\(600\) 0 0
\(601\) −45.8253 −1.86925 −0.934626 0.355633i \(-0.884265\pi\)
−0.934626 + 0.355633i \(0.884265\pi\)
\(602\) 0 0
\(603\) 44.3692 1.80685
\(604\) 0 0
\(605\) −21.4204 −0.870861
\(606\) 0 0
\(607\) 17.6455 0.716210 0.358105 0.933681i \(-0.383423\pi\)
0.358105 + 0.933681i \(0.383423\pi\)
\(608\) 0 0
\(609\) −1.31094 −0.0531221
\(610\) 0 0
\(611\) 1.02664 0.0415332
\(612\) 0 0
\(613\) 2.27761 0.0919919 0.0459960 0.998942i \(-0.485354\pi\)
0.0459960 + 0.998942i \(0.485354\pi\)
\(614\) 0 0
\(615\) 4.64704 0.187387
\(616\) 0 0
\(617\) 1.33514 0.0537507 0.0268753 0.999639i \(-0.491444\pi\)
0.0268753 + 0.999639i \(0.491444\pi\)
\(618\) 0 0
\(619\) 16.6808 0.670458 0.335229 0.942137i \(-0.391186\pi\)
0.335229 + 0.942137i \(0.391186\pi\)
\(620\) 0 0
\(621\) 9.21027 0.369595
\(622\) 0 0
\(623\) 0.928440 0.0371971
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −6.84312 −0.273288
\(628\) 0 0
\(629\) 27.6751 1.10348
\(630\) 0 0
\(631\) −3.60704 −0.143594 −0.0717971 0.997419i \(-0.522873\pi\)
−0.0717971 + 0.997419i \(0.522873\pi\)
\(632\) 0 0
\(633\) −7.53445 −0.299468
\(634\) 0 0
\(635\) 19.0556 0.756199
\(636\) 0 0
\(637\) 1.98880 0.0787992
\(638\) 0 0
\(639\) 84.4050 3.33901
\(640\) 0 0
\(641\) 10.2139 0.403426 0.201713 0.979445i \(-0.435349\pi\)
0.201713 + 0.979445i \(0.435349\pi\)
\(642\) 0 0
\(643\) 2.82338 0.111343 0.0556717 0.998449i \(-0.482270\pi\)
0.0556717 + 0.998449i \(0.482270\pi\)
\(644\) 0 0
\(645\) 3.69320 0.145420
\(646\) 0 0
\(647\) −41.4408 −1.62921 −0.814604 0.580018i \(-0.803046\pi\)
−0.814604 + 0.580018i \(0.803046\pi\)
\(648\) 0 0
\(649\) −4.79900 −0.188377
\(650\) 0 0
\(651\) −4.96708 −0.194675
\(652\) 0 0
\(653\) −43.7584 −1.71240 −0.856200 0.516645i \(-0.827181\pi\)
−0.856200 + 0.516645i \(0.827181\pi\)
\(654\) 0 0
\(655\) −33.1618 −1.29574
\(656\) 0 0
\(657\) 52.6547 2.05426
\(658\) 0 0
\(659\) 7.47042 0.291006 0.145503 0.989358i \(-0.453520\pi\)
0.145503 + 0.989358i \(0.453520\pi\)
\(660\) 0 0
\(661\) 28.3065 1.10099 0.550497 0.834837i \(-0.314438\pi\)
0.550497 + 0.834837i \(0.314438\pi\)
\(662\) 0 0
\(663\) 6.45854 0.250829
\(664\) 0 0
\(665\) 4.20098 0.162907
\(666\) 0 0
\(667\) 0.634369 0.0245629
\(668\) 0 0
\(669\) −2.49478 −0.0964538
\(670\) 0 0
\(671\) 2.49790 0.0964305
\(672\) 0 0
\(673\) −31.8789 −1.22884 −0.614421 0.788979i \(-0.710610\pi\)
−0.614421 + 0.788979i \(0.710610\pi\)
\(674\) 0 0
\(675\) −11.7880 −0.453719
\(676\) 0 0
\(677\) 34.6470 1.33159 0.665797 0.746133i \(-0.268092\pi\)
0.665797 + 0.746133i \(0.268092\pi\)
\(678\) 0 0
\(679\) 8.84275 0.339354
\(680\) 0 0
\(681\) −32.8151 −1.25748
\(682\) 0 0
\(683\) 7.53159 0.288188 0.144094 0.989564i \(-0.453973\pi\)
0.144094 + 0.989564i \(0.453973\pi\)
\(684\) 0 0
\(685\) 7.07806 0.270439
\(686\) 0 0
\(687\) 36.5464 1.39433
\(688\) 0 0
\(689\) −1.97891 −0.0753905
\(690\) 0 0
\(691\) −34.3094 −1.30519 −0.652596 0.757706i \(-0.726320\pi\)
−0.652596 + 0.757706i \(0.726320\pi\)
\(692\) 0 0
\(693\) −1.88290 −0.0715254
\(694\) 0 0
\(695\) −18.8004 −0.713141
\(696\) 0 0
\(697\) −5.19956 −0.196947
\(698\) 0 0
\(699\) −74.6639 −2.82405
\(700\) 0 0
\(701\) 0.0251855 0.000951242 0 0.000475621 1.00000i \(-0.499849\pi\)
0.000475621 1.00000i \(0.499849\pi\)
\(702\) 0 0
\(703\) 16.0885 0.606789
\(704\) 0 0
\(705\) −21.7294 −0.818375
\(706\) 0 0
\(707\) 7.24608 0.272517
\(708\) 0 0
\(709\) −28.2131 −1.05957 −0.529783 0.848133i \(-0.677727\pi\)
−0.529783 + 0.848133i \(0.677727\pi\)
\(710\) 0 0
\(711\) −33.0788 −1.24055
\(712\) 0 0
\(713\) 2.40359 0.0900150
\(714\) 0 0
\(715\) −0.318028 −0.0118936
\(716\) 0 0
\(717\) 68.1600 2.54548
\(718\) 0 0
\(719\) −24.5507 −0.915585 −0.457792 0.889059i \(-0.651360\pi\)
−0.457792 + 0.889059i \(0.651360\pi\)
\(720\) 0 0
\(721\) −8.65474 −0.322319
\(722\) 0 0
\(723\) 54.6877 2.03386
\(724\) 0 0
\(725\) −0.811912 −0.0301537
\(726\) 0 0
\(727\) 16.2103 0.601208 0.300604 0.953749i \(-0.402812\pi\)
0.300604 + 0.953749i \(0.402812\pi\)
\(728\) 0 0
\(729\) 1.41315 0.0523389
\(730\) 0 0
\(731\) −4.13231 −0.152839
\(732\) 0 0
\(733\) −16.0087 −0.591296 −0.295648 0.955297i \(-0.595535\pi\)
−0.295648 + 0.955297i \(0.595535\pi\)
\(734\) 0 0
\(735\) −42.0942 −1.55267
\(736\) 0 0
\(737\) 3.52767 0.129943
\(738\) 0 0
\(739\) −11.9536 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(740\) 0 0
\(741\) 3.75457 0.137928
\(742\) 0 0
\(743\) 38.6259 1.41705 0.708524 0.705686i \(-0.249361\pi\)
0.708524 + 0.705686i \(0.249361\pi\)
\(744\) 0 0
\(745\) −37.1618 −1.36150
\(746\) 0 0
\(747\) −7.54219 −0.275954
\(748\) 0 0
\(749\) −4.89348 −0.178804
\(750\) 0 0
\(751\) 21.6589 0.790346 0.395173 0.918607i \(-0.370685\pi\)
0.395173 + 0.918607i \(0.370685\pi\)
\(752\) 0 0
\(753\) −3.96639 −0.144543
\(754\) 0 0
\(755\) 34.5287 1.25663
\(756\) 0 0
\(757\) 43.6016 1.58473 0.792364 0.610048i \(-0.208850\pi\)
0.792364 + 0.610048i \(0.208850\pi\)
\(758\) 0 0
\(759\) 1.31483 0.0477253
\(760\) 0 0
\(761\) 32.5442 1.17973 0.589863 0.807504i \(-0.299182\pi\)
0.589863 + 0.807504i \(0.299182\pi\)
\(762\) 0 0
\(763\) −10.5099 −0.380482
\(764\) 0 0
\(765\) −94.7286 −3.42492
\(766\) 0 0
\(767\) 2.63303 0.0950734
\(768\) 0 0
\(769\) −11.9886 −0.432319 −0.216159 0.976358i \(-0.569353\pi\)
−0.216159 + 0.976358i \(0.569353\pi\)
\(770\) 0 0
\(771\) 82.8882 2.98515
\(772\) 0 0
\(773\) 31.3355 1.12706 0.563530 0.826096i \(-0.309443\pi\)
0.563530 + 0.826096i \(0.309443\pi\)
\(774\) 0 0
\(775\) −3.07629 −0.110503
\(776\) 0 0
\(777\) 6.38811 0.229172
\(778\) 0 0
\(779\) −3.02268 −0.108299
\(780\) 0 0
\(781\) 6.71080 0.240131
\(782\) 0 0
\(783\) 9.57080 0.342032
\(784\) 0 0
\(785\) −14.3973 −0.513860
\(786\) 0 0
\(787\) 9.39669 0.334956 0.167478 0.985876i \(-0.446438\pi\)
0.167478 + 0.985876i \(0.446438\pi\)
\(788\) 0 0
\(789\) −44.6922 −1.59109
\(790\) 0 0
\(791\) −8.69181 −0.309045
\(792\) 0 0
\(793\) −1.37051 −0.0486682
\(794\) 0 0
\(795\) 41.8848 1.48550
\(796\) 0 0
\(797\) 21.7756 0.771330 0.385665 0.922639i \(-0.373972\pi\)
0.385665 + 0.922639i \(0.373972\pi\)
\(798\) 0 0
\(799\) 24.3129 0.860129
\(800\) 0 0
\(801\) −12.1705 −0.430025
\(802\) 0 0
\(803\) 4.18642 0.147736
\(804\) 0 0
\(805\) −0.807172 −0.0284491
\(806\) 0 0
\(807\) −56.3229 −1.98266
\(808\) 0 0
\(809\) −1.83916 −0.0646614 −0.0323307 0.999477i \(-0.510293\pi\)
−0.0323307 + 0.999477i \(0.510293\pi\)
\(810\) 0 0
\(811\) 7.27579 0.255488 0.127744 0.991807i \(-0.459226\pi\)
0.127744 + 0.991807i \(0.459226\pi\)
\(812\) 0 0
\(813\) 16.9087 0.593012
\(814\) 0 0
\(815\) 47.3895 1.65998
\(816\) 0 0
\(817\) −2.40225 −0.0840441
\(818\) 0 0
\(819\) 1.03308 0.0360987
\(820\) 0 0
\(821\) −45.0675 −1.57287 −0.786434 0.617674i \(-0.788075\pi\)
−0.786434 + 0.617674i \(0.788075\pi\)
\(822\) 0 0
\(823\) −8.71724 −0.303864 −0.151932 0.988391i \(-0.548549\pi\)
−0.151932 + 0.988391i \(0.548549\pi\)
\(824\) 0 0
\(825\) −1.68282 −0.0585881
\(826\) 0 0
\(827\) −8.18072 −0.284471 −0.142236 0.989833i \(-0.545429\pi\)
−0.142236 + 0.989833i \(0.545429\pi\)
\(828\) 0 0
\(829\) −43.1349 −1.49814 −0.749068 0.662493i \(-0.769498\pi\)
−0.749068 + 0.662493i \(0.769498\pi\)
\(830\) 0 0
\(831\) 1.48590 0.0515453
\(832\) 0 0
\(833\) 47.0991 1.63189
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 36.2632 1.25344
\(838\) 0 0
\(839\) −37.2048 −1.28445 −0.642226 0.766515i \(-0.721989\pi\)
−0.642226 + 0.766515i \(0.721989\pi\)
\(840\) 0 0
\(841\) −28.3408 −0.977269
\(842\) 0 0
\(843\) −28.8539 −0.993782
\(844\) 0 0
\(845\) −25.8255 −0.888425
\(846\) 0 0
\(847\) 5.53222 0.190089
\(848\) 0 0
\(849\) 70.7488 2.42809
\(850\) 0 0
\(851\) −3.09123 −0.105966
\(852\) 0 0
\(853\) 11.9553 0.409341 0.204671 0.978831i \(-0.434388\pi\)
0.204671 + 0.978831i \(0.434388\pi\)
\(854\) 0 0
\(855\) −55.0689 −1.88332
\(856\) 0 0
\(857\) −2.28383 −0.0780142 −0.0390071 0.999239i \(-0.512420\pi\)
−0.0390071 + 0.999239i \(0.512420\pi\)
\(858\) 0 0
\(859\) −16.8387 −0.574529 −0.287265 0.957851i \(-0.592746\pi\)
−0.287265 + 0.957851i \(0.592746\pi\)
\(860\) 0 0
\(861\) −1.20019 −0.0409023
\(862\) 0 0
\(863\) −31.2591 −1.06407 −0.532036 0.846721i \(-0.678573\pi\)
−0.532036 + 0.846721i \(0.678573\pi\)
\(864\) 0 0
\(865\) −39.0556 −1.32793
\(866\) 0 0
\(867\) 99.8123 3.38980
\(868\) 0 0
\(869\) −2.63000 −0.0892165
\(870\) 0 0
\(871\) −1.93550 −0.0655820
\(872\) 0 0
\(873\) −115.916 −3.92316
\(874\) 0 0
\(875\) 6.19847 0.209546
\(876\) 0 0
\(877\) −6.86671 −0.231872 −0.115936 0.993257i \(-0.536987\pi\)
−0.115936 + 0.993257i \(0.536987\pi\)
\(878\) 0 0
\(879\) −31.3123 −1.05614
\(880\) 0 0
\(881\) −9.06820 −0.305515 −0.152758 0.988264i \(-0.548815\pi\)
−0.152758 + 0.988264i \(0.548815\pi\)
\(882\) 0 0
\(883\) −2.28390 −0.0768594 −0.0384297 0.999261i \(-0.512236\pi\)
−0.0384297 + 0.999261i \(0.512236\pi\)
\(884\) 0 0
\(885\) −55.7298 −1.87334
\(886\) 0 0
\(887\) −31.8293 −1.06872 −0.534361 0.845256i \(-0.679448\pi\)
−0.534361 + 0.845256i \(0.679448\pi\)
\(888\) 0 0
\(889\) −4.92148 −0.165061
\(890\) 0 0
\(891\) 8.90133 0.298206
\(892\) 0 0
\(893\) 14.1339 0.472974
\(894\) 0 0
\(895\) −21.0076 −0.702208
\(896\) 0 0
\(897\) −0.721399 −0.0240868
\(898\) 0 0
\(899\) 2.49767 0.0833021
\(900\) 0 0
\(901\) −46.8648 −1.56129
\(902\) 0 0
\(903\) −0.953840 −0.0317418
\(904\) 0 0
\(905\) 16.7237 0.555914
\(906\) 0 0
\(907\) −54.0598 −1.79502 −0.897512 0.440989i \(-0.854628\pi\)
−0.897512 + 0.440989i \(0.854628\pi\)
\(908\) 0 0
\(909\) −94.9859 −3.15048
\(910\) 0 0
\(911\) 6.19867 0.205371 0.102686 0.994714i \(-0.467256\pi\)
0.102686 + 0.994714i \(0.467256\pi\)
\(912\) 0 0
\(913\) −0.599658 −0.0198458
\(914\) 0 0
\(915\) 29.0076 0.958963
\(916\) 0 0
\(917\) 8.56467 0.282830
\(918\) 0 0
\(919\) 6.53624 0.215611 0.107805 0.994172i \(-0.465618\pi\)
0.107805 + 0.994172i \(0.465618\pi\)
\(920\) 0 0
\(921\) 50.0513 1.64925
\(922\) 0 0
\(923\) −3.68197 −0.121194
\(924\) 0 0
\(925\) 3.95638 0.130085
\(926\) 0 0
\(927\) 113.451 3.72623
\(928\) 0 0
\(929\) 21.4372 0.703331 0.351666 0.936126i \(-0.385615\pi\)
0.351666 + 0.936126i \(0.385615\pi\)
\(930\) 0 0
\(931\) 27.3803 0.897353
\(932\) 0 0
\(933\) −32.3882 −1.06034
\(934\) 0 0
\(935\) −7.53159 −0.246309
\(936\) 0 0
\(937\) −1.58083 −0.0516434 −0.0258217 0.999667i \(-0.508220\pi\)
−0.0258217 + 0.999667i \(0.508220\pi\)
\(938\) 0 0
\(939\) 29.2582 0.954807
\(940\) 0 0
\(941\) 10.6995 0.348794 0.174397 0.984675i \(-0.444202\pi\)
0.174397 + 0.984675i \(0.444202\pi\)
\(942\) 0 0
\(943\) 0.580775 0.0189126
\(944\) 0 0
\(945\) −12.1779 −0.396147
\(946\) 0 0
\(947\) 29.2664 0.951030 0.475515 0.879708i \(-0.342262\pi\)
0.475515 + 0.879708i \(0.342262\pi\)
\(948\) 0 0
\(949\) −2.29694 −0.0745618
\(950\) 0 0
\(951\) 40.1971 1.30348
\(952\) 0 0
\(953\) −45.0155 −1.45820 −0.729098 0.684410i \(-0.760060\pi\)
−0.729098 + 0.684410i \(0.760060\pi\)
\(954\) 0 0
\(955\) 18.8769 0.610842
\(956\) 0 0
\(957\) 1.36630 0.0441661
\(958\) 0 0
\(959\) −1.82805 −0.0590307
\(960\) 0 0
\(961\) −21.5365 −0.694725
\(962\) 0 0
\(963\) 64.1466 2.06710
\(964\) 0 0
\(965\) −32.6583 −1.05131
\(966\) 0 0
\(967\) 61.7031 1.98424 0.992118 0.125304i \(-0.0399907\pi\)
0.992118 + 0.125304i \(0.0399907\pi\)
\(968\) 0 0
\(969\) 88.9162 2.85640
\(970\) 0 0
\(971\) −11.4233 −0.366590 −0.183295 0.983058i \(-0.558676\pi\)
−0.183295 + 0.983058i \(0.558676\pi\)
\(972\) 0 0
\(973\) 4.85558 0.155663
\(974\) 0 0
\(975\) 0.923300 0.0295693
\(976\) 0 0
\(977\) −3.28909 −0.105227 −0.0526137 0.998615i \(-0.516755\pi\)
−0.0526137 + 0.998615i \(0.516755\pi\)
\(978\) 0 0
\(979\) −0.967644 −0.0309260
\(980\) 0 0
\(981\) 137.769 4.39864
\(982\) 0 0
\(983\) 15.4572 0.493009 0.246505 0.969142i \(-0.420718\pi\)
0.246505 + 0.969142i \(0.420718\pi\)
\(984\) 0 0
\(985\) −34.6920 −1.10538
\(986\) 0 0
\(987\) 5.61203 0.178633
\(988\) 0 0
\(989\) 0.461566 0.0146770
\(990\) 0 0
\(991\) −21.3319 −0.677630 −0.338815 0.940853i \(-0.610026\pi\)
−0.338815 + 0.940853i \(0.610026\pi\)
\(992\) 0 0
\(993\) −8.79276 −0.279030
\(994\) 0 0
\(995\) 12.9219 0.409653
\(996\) 0 0
\(997\) −36.9757 −1.17103 −0.585516 0.810661i \(-0.699108\pi\)
−0.585516 + 0.810661i \(0.699108\pi\)
\(998\) 0 0
\(999\) −46.6377 −1.47555
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.1.5 5
3.2 odd 2 6012.2.a.d.1.2 5
4.3 odd 2 2672.2.a.j.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.5 5 1.1 even 1 trivial
2672.2.a.j.1.1 5 4.3 odd 2
6012.2.a.d.1.2 5 3.2 odd 2