Properties

Label 3640.2.a.x.1.2
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.487928\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.512072 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.73778 q^{9} +O(q^{10})\) \(q+0.512072 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.73778 q^{9} -3.81141 q^{11} +1.00000 q^{13} -0.512072 q^{15} -0.761926 q^{17} -1.04948 q^{19} -0.512072 q^{21} +5.81141 q^{23} +1.00000 q^{25} -2.93816 q^{27} +4.32348 q^{29} +4.61103 q^{31} -1.95171 q^{33} +1.00000 q^{35} -5.92273 q^{37} +0.512072 q^{39} -7.43600 q^{41} -2.78726 q^{43} +2.73778 q^{45} -3.12311 q^{47} +1.00000 q^{49} -0.390161 q^{51} +7.12311 q^{53} +3.81141 q^{55} -0.537410 q^{57} +14.0866 q^{59} +8.24621 q^{61} +2.73778 q^{63} -1.00000 q^{65} +2.21393 q^{67} +2.97586 q^{69} +4.58205 q^{71} -3.07482 q^{73} +0.512072 q^{75} +3.81141 q^{77} +9.11439 q^{79} +6.70880 q^{81} -11.3210 q^{83} +0.761926 q^{85} +2.21393 q^{87} +17.3500 q^{89} -1.00000 q^{91} +2.36118 q^{93} +1.04948 q^{95} +11.7972 q^{97} +10.4348 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 4 q^{5} - 4 q^{7} + 3 q^{9} + 6 q^{11} + 4 q^{13} - 3 q^{15} + 9 q^{17} + 5 q^{19} - 3 q^{21} + 2 q^{23} + 4 q^{25} + 6 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 4 q^{35} - 11 q^{37} + 3 q^{39} - 5 q^{41} + 12 q^{43} - 3 q^{45} + 4 q^{47} + 4 q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 8 q^{57} + 11 q^{59} - 3 q^{63} - 4 q^{65} + 19 q^{67} + 10 q^{69} - 8 q^{71} + 8 q^{73} + 3 q^{75} - 6 q^{77} + 13 q^{79} + 4 q^{81} + 8 q^{83} - 9 q^{85} + 19 q^{87} + 25 q^{89} - 4 q^{91} + 5 q^{93} - 5 q^{95} + 18 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.512072 0.295645 0.147822 0.989014i \(-0.452774\pi\)
0.147822 + 0.989014i \(0.452774\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.73778 −0.912594
\(10\) 0 0
\(11\) −3.81141 −1.14918 −0.574591 0.818441i \(-0.694839\pi\)
−0.574591 + 0.818441i \(0.694839\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.512072 −0.132216
\(16\) 0 0
\(17\) −0.761926 −0.184794 −0.0923971 0.995722i \(-0.529453\pi\)
−0.0923971 + 0.995722i \(0.529453\pi\)
\(18\) 0 0
\(19\) −1.04948 −0.240767 −0.120384 0.992727i \(-0.538413\pi\)
−0.120384 + 0.992727i \(0.538413\pi\)
\(20\) 0 0
\(21\) −0.512072 −0.111743
\(22\) 0 0
\(23\) 5.81141 1.21176 0.605881 0.795555i \(-0.292821\pi\)
0.605881 + 0.795555i \(0.292821\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.93816 −0.565448
\(28\) 0 0
\(29\) 4.32348 0.802850 0.401425 0.915892i \(-0.368515\pi\)
0.401425 + 0.915892i \(0.368515\pi\)
\(30\) 0 0
\(31\) 4.61103 0.828166 0.414083 0.910239i \(-0.364102\pi\)
0.414083 + 0.910239i \(0.364102\pi\)
\(32\) 0 0
\(33\) −1.95171 −0.339750
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −5.92273 −0.973691 −0.486846 0.873488i \(-0.661853\pi\)
−0.486846 + 0.873488i \(0.661853\pi\)
\(38\) 0 0
\(39\) 0.512072 0.0819971
\(40\) 0 0
\(41\) −7.43600 −1.16131 −0.580654 0.814150i \(-0.697203\pi\)
−0.580654 + 0.814150i \(0.697203\pi\)
\(42\) 0 0
\(43\) −2.78726 −0.425054 −0.212527 0.977155i \(-0.568169\pi\)
−0.212527 + 0.977155i \(0.568169\pi\)
\(44\) 0 0
\(45\) 2.73778 0.408125
\(46\) 0 0
\(47\) −3.12311 −0.455552 −0.227776 0.973714i \(-0.573145\pi\)
−0.227776 + 0.973714i \(0.573145\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.390161 −0.0546334
\(52\) 0 0
\(53\) 7.12311 0.978434 0.489217 0.872162i \(-0.337283\pi\)
0.489217 + 0.872162i \(0.337283\pi\)
\(54\) 0 0
\(55\) 3.81141 0.513930
\(56\) 0 0
\(57\) −0.537410 −0.0711816
\(58\) 0 0
\(59\) 14.0866 1.83392 0.916960 0.398980i \(-0.130636\pi\)
0.916960 + 0.398980i \(0.130636\pi\)
\(60\) 0 0
\(61\) 8.24621 1.05582 0.527910 0.849301i \(-0.322976\pi\)
0.527910 + 0.849301i \(0.322976\pi\)
\(62\) 0 0
\(63\) 2.73778 0.344928
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.21393 0.270475 0.135237 0.990813i \(-0.456820\pi\)
0.135237 + 0.990813i \(0.456820\pi\)
\(68\) 0 0
\(69\) 2.97586 0.358251
\(70\) 0 0
\(71\) 4.58205 0.543790 0.271895 0.962327i \(-0.412350\pi\)
0.271895 + 0.962327i \(0.412350\pi\)
\(72\) 0 0
\(73\) −3.07482 −0.359880 −0.179940 0.983678i \(-0.557590\pi\)
−0.179940 + 0.983678i \(0.557590\pi\)
\(74\) 0 0
\(75\) 0.512072 0.0591289
\(76\) 0 0
\(77\) 3.81141 0.434350
\(78\) 0 0
\(79\) 9.11439 1.02545 0.512724 0.858553i \(-0.328636\pi\)
0.512724 + 0.858553i \(0.328636\pi\)
\(80\) 0 0
\(81\) 6.70880 0.745422
\(82\) 0 0
\(83\) −11.3210 −1.24264 −0.621322 0.783555i \(-0.713404\pi\)
−0.621322 + 0.783555i \(0.713404\pi\)
\(84\) 0 0
\(85\) 0.761926 0.0826425
\(86\) 0 0
\(87\) 2.21393 0.237358
\(88\) 0 0
\(89\) 17.3500 1.83910 0.919549 0.392976i \(-0.128554\pi\)
0.919549 + 0.392976i \(0.128554\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 2.36118 0.244843
\(94\) 0 0
\(95\) 1.04948 0.107674
\(96\) 0 0
\(97\) 11.7972 1.19782 0.598911 0.800816i \(-0.295600\pi\)
0.598911 + 0.800816i \(0.295600\pi\)
\(98\) 0 0
\(99\) 10.4348 1.04874
\(100\) 0 0
\(101\) 4.25350 0.423239 0.211619 0.977352i \(-0.432126\pi\)
0.211619 + 0.977352i \(0.432126\pi\)
\(102\) 0 0
\(103\) 12.1243 1.19464 0.597321 0.802002i \(-0.296232\pi\)
0.597321 + 0.802002i \(0.296232\pi\)
\(104\) 0 0
\(105\) 0.512072 0.0499731
\(106\) 0 0
\(107\) 15.2870 1.47785 0.738924 0.673789i \(-0.235334\pi\)
0.738924 + 0.673789i \(0.235334\pi\)
\(108\) 0 0
\(109\) −17.2727 −1.65443 −0.827214 0.561886i \(-0.810076\pi\)
−0.827214 + 0.561886i \(0.810076\pi\)
\(110\) 0 0
\(111\) −3.03286 −0.287867
\(112\) 0 0
\(113\) −0.598671 −0.0563182 −0.0281591 0.999603i \(-0.508965\pi\)
−0.0281591 + 0.999603i \(0.508965\pi\)
\(114\) 0 0
\(115\) −5.81141 −0.541917
\(116\) 0 0
\(117\) −2.73778 −0.253108
\(118\) 0 0
\(119\) 0.761926 0.0698456
\(120\) 0 0
\(121\) 3.52682 0.320620
\(122\) 0 0
\(123\) −3.80776 −0.343335
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.0093 −0.888185 −0.444092 0.895981i \(-0.646474\pi\)
−0.444092 + 0.895981i \(0.646474\pi\)
\(128\) 0 0
\(129\) −1.42728 −0.125665
\(130\) 0 0
\(131\) 0.876894 0.0766146 0.0383073 0.999266i \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) 0 0
\(133\) 1.04948 0.0910016
\(134\) 0 0
\(135\) 2.93816 0.252876
\(136\) 0 0
\(137\) 7.29934 0.623624 0.311812 0.950144i \(-0.399064\pi\)
0.311812 + 0.950144i \(0.399064\pi\)
\(138\) 0 0
\(139\) −15.6735 −1.32941 −0.664704 0.747107i \(-0.731442\pi\)
−0.664704 + 0.747107i \(0.731442\pi\)
\(140\) 0 0
\(141\) −1.59925 −0.134681
\(142\) 0 0
\(143\) −3.81141 −0.318726
\(144\) 0 0
\(145\) −4.32348 −0.359045
\(146\) 0 0
\(147\) 0.512072 0.0422350
\(148\) 0 0
\(149\) −7.12311 −0.583548 −0.291774 0.956487i \(-0.594245\pi\)
−0.291774 + 0.956487i \(0.594245\pi\)
\(150\) 0 0
\(151\) 3.61587 0.294255 0.147128 0.989118i \(-0.452997\pi\)
0.147128 + 0.989118i \(0.452997\pi\)
\(152\) 0 0
\(153\) 2.08599 0.168642
\(154\) 0 0
\(155\) −4.61103 −0.370367
\(156\) 0 0
\(157\) −1.82616 −0.145743 −0.0728717 0.997341i \(-0.523216\pi\)
−0.0728717 + 0.997341i \(0.523216\pi\)
\(158\) 0 0
\(159\) 3.64754 0.289269
\(160\) 0 0
\(161\) −5.81141 −0.458003
\(162\) 0 0
\(163\) 22.5721 1.76798 0.883991 0.467504i \(-0.154847\pi\)
0.883991 + 0.467504i \(0.154847\pi\)
\(164\) 0 0
\(165\) 1.95171 0.151941
\(166\) 0 0
\(167\) −1.82564 −0.141272 −0.0706360 0.997502i \(-0.522503\pi\)
−0.0706360 + 0.997502i \(0.522503\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.87325 0.219723
\(172\) 0 0
\(173\) −8.22691 −0.625480 −0.312740 0.949839i \(-0.601247\pi\)
−0.312740 + 0.949839i \(0.601247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 7.21335 0.542188
\(178\) 0 0
\(179\) 26.1312 1.95314 0.976570 0.215198i \(-0.0690398\pi\)
0.976570 + 0.215198i \(0.0690398\pi\)
\(180\) 0 0
\(181\) −8.40075 −0.624423 −0.312211 0.950013i \(-0.601070\pi\)
−0.312211 + 0.950013i \(0.601070\pi\)
\(182\) 0 0
\(183\) 4.22265 0.312147
\(184\) 0 0
\(185\) 5.92273 0.435448
\(186\) 0 0
\(187\) 2.90401 0.212362
\(188\) 0 0
\(189\) 2.93816 0.213719
\(190\) 0 0
\(191\) −6.17623 −0.446896 −0.223448 0.974716i \(-0.571731\pi\)
−0.223448 + 0.974716i \(0.571731\pi\)
\(192\) 0 0
\(193\) 3.26163 0.234778 0.117389 0.993086i \(-0.462548\pi\)
0.117389 + 0.993086i \(0.462548\pi\)
\(194\) 0 0
\(195\) −0.512072 −0.0366702
\(196\) 0 0
\(197\) 21.1684 1.50818 0.754092 0.656769i \(-0.228077\pi\)
0.754092 + 0.656769i \(0.228077\pi\)
\(198\) 0 0
\(199\) 25.9270 1.83792 0.918958 0.394356i \(-0.129032\pi\)
0.918958 + 0.394356i \(0.129032\pi\)
\(200\) 0 0
\(201\) 1.13369 0.0799644
\(202\) 0 0
\(203\) −4.32348 −0.303449
\(204\) 0 0
\(205\) 7.43600 0.519353
\(206\) 0 0
\(207\) −15.9104 −1.10585
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −18.6098 −1.28115 −0.640575 0.767896i \(-0.721304\pi\)
−0.640575 + 0.767896i \(0.721304\pi\)
\(212\) 0 0
\(213\) 2.34634 0.160769
\(214\) 0 0
\(215\) 2.78726 0.190090
\(216\) 0 0
\(217\) −4.61103 −0.313017
\(218\) 0 0
\(219\) −1.57453 −0.106397
\(220\) 0 0
\(221\) −0.761926 −0.0512527
\(222\) 0 0
\(223\) 1.29747 0.0868850 0.0434425 0.999056i \(-0.486167\pi\)
0.0434425 + 0.999056i \(0.486167\pi\)
\(224\) 0 0
\(225\) −2.73778 −0.182519
\(226\) 0 0
\(227\) 3.63010 0.240938 0.120469 0.992717i \(-0.461560\pi\)
0.120469 + 0.992717i \(0.461560\pi\)
\(228\) 0 0
\(229\) −27.4089 −1.81123 −0.905615 0.424101i \(-0.860590\pi\)
−0.905615 + 0.424101i \(0.860590\pi\)
\(230\) 0 0
\(231\) 1.95171 0.128413
\(232\) 0 0
\(233\) −17.3210 −1.13474 −0.567369 0.823464i \(-0.692039\pi\)
−0.567369 + 0.823464i \(0.692039\pi\)
\(234\) 0 0
\(235\) 3.12311 0.203729
\(236\) 0 0
\(237\) 4.66722 0.303168
\(238\) 0 0
\(239\) 0.236879 0.0153225 0.00766123 0.999971i \(-0.497561\pi\)
0.00766123 + 0.999971i \(0.497561\pi\)
\(240\) 0 0
\(241\) −12.6654 −0.815847 −0.407924 0.913016i \(-0.633747\pi\)
−0.407924 + 0.913016i \(0.633747\pi\)
\(242\) 0 0
\(243\) 12.2499 0.785829
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −1.04948 −0.0667769
\(248\) 0 0
\(249\) −5.79718 −0.367381
\(250\) 0 0
\(251\) 17.3234 1.09344 0.546722 0.837314i \(-0.315875\pi\)
0.546722 + 0.837314i \(0.315875\pi\)
\(252\) 0 0
\(253\) −22.1496 −1.39254
\(254\) 0 0
\(255\) 0.390161 0.0244328
\(256\) 0 0
\(257\) −11.6155 −0.724557 −0.362278 0.932070i \(-0.618001\pi\)
−0.362278 + 0.932070i \(0.618001\pi\)
\(258\) 0 0
\(259\) 5.92273 0.368021
\(260\) 0 0
\(261\) −11.8367 −0.732676
\(262\) 0 0
\(263\) 24.1874 1.49146 0.745730 0.666248i \(-0.232101\pi\)
0.745730 + 0.666248i \(0.232101\pi\)
\(264\) 0 0
\(265\) −7.12311 −0.437569
\(266\) 0 0
\(267\) 8.88445 0.543719
\(268\) 0 0
\(269\) 14.4997 0.884063 0.442031 0.897000i \(-0.354258\pi\)
0.442031 + 0.897000i \(0.354258\pi\)
\(270\) 0 0
\(271\) −22.8591 −1.38859 −0.694296 0.719690i \(-0.744284\pi\)
−0.694296 + 0.719690i \(0.744284\pi\)
\(272\) 0 0
\(273\) −0.512072 −0.0309920
\(274\) 0 0
\(275\) −3.81141 −0.229836
\(276\) 0 0
\(277\) 8.10135 0.486763 0.243382 0.969931i \(-0.421743\pi\)
0.243382 + 0.969931i \(0.421743\pi\)
\(278\) 0 0
\(279\) −12.6240 −0.755780
\(280\) 0 0
\(281\) 13.8714 0.827499 0.413750 0.910391i \(-0.364219\pi\)
0.413750 + 0.910391i \(0.364219\pi\)
\(282\) 0 0
\(283\) 27.5728 1.63903 0.819515 0.573058i \(-0.194243\pi\)
0.819515 + 0.573058i \(0.194243\pi\)
\(284\) 0 0
\(285\) 0.537410 0.0318334
\(286\) 0 0
\(287\) 7.43600 0.438933
\(288\) 0 0
\(289\) −16.4195 −0.965851
\(290\) 0 0
\(291\) 6.04100 0.354130
\(292\) 0 0
\(293\) −28.1461 −1.64431 −0.822156 0.569262i \(-0.807229\pi\)
−0.822156 + 0.569262i \(0.807229\pi\)
\(294\) 0 0
\(295\) −14.0866 −0.820154
\(296\) 0 0
\(297\) 11.1985 0.649803
\(298\) 0 0
\(299\) 5.81141 0.336082
\(300\) 0 0
\(301\) 2.78726 0.160655
\(302\) 0 0
\(303\) 2.17810 0.125128
\(304\) 0 0
\(305\) −8.24621 −0.472177
\(306\) 0 0
\(307\) 10.4514 0.596494 0.298247 0.954489i \(-0.403598\pi\)
0.298247 + 0.954489i \(0.403598\pi\)
\(308\) 0 0
\(309\) 6.20851 0.353190
\(310\) 0 0
\(311\) −17.2148 −0.976161 −0.488080 0.872799i \(-0.662303\pi\)
−0.488080 + 0.872799i \(0.662303\pi\)
\(312\) 0 0
\(313\) 18.4460 1.04263 0.521315 0.853364i \(-0.325442\pi\)
0.521315 + 0.853364i \(0.325442\pi\)
\(314\) 0 0
\(315\) −2.73778 −0.154257
\(316\) 0 0
\(317\) 31.5932 1.77445 0.887225 0.461337i \(-0.152630\pi\)
0.887225 + 0.461337i \(0.152630\pi\)
\(318\) 0 0
\(319\) −16.4785 −0.922621
\(320\) 0 0
\(321\) 7.82802 0.436918
\(322\) 0 0
\(323\) 0.799627 0.0444924
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −8.84488 −0.489123
\(328\) 0 0
\(329\) 3.12311 0.172182
\(330\) 0 0
\(331\) 5.56520 0.305891 0.152945 0.988235i \(-0.451124\pi\)
0.152945 + 0.988235i \(0.451124\pi\)
\(332\) 0 0
\(333\) 16.2152 0.888585
\(334\) 0 0
\(335\) −2.21393 −0.120960
\(336\) 0 0
\(337\) −8.84978 −0.482078 −0.241039 0.970515i \(-0.577488\pi\)
−0.241039 + 0.970515i \(0.577488\pi\)
\(338\) 0 0
\(339\) −0.306562 −0.0166502
\(340\) 0 0
\(341\) −17.5745 −0.951714
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.97586 −0.160215
\(346\) 0 0
\(347\) 0.982799 0.0527594 0.0263797 0.999652i \(-0.491602\pi\)
0.0263797 + 0.999652i \(0.491602\pi\)
\(348\) 0 0
\(349\) −17.7248 −0.948787 −0.474394 0.880313i \(-0.657333\pi\)
−0.474394 + 0.880313i \(0.657333\pi\)
\(350\) 0 0
\(351\) −2.93816 −0.156827
\(352\) 0 0
\(353\) 12.0483 0.641266 0.320633 0.947204i \(-0.396104\pi\)
0.320633 + 0.947204i \(0.396104\pi\)
\(354\) 0 0
\(355\) −4.58205 −0.243190
\(356\) 0 0
\(357\) 0.390161 0.0206495
\(358\) 0 0
\(359\) 23.4107 1.23557 0.617784 0.786348i \(-0.288031\pi\)
0.617784 + 0.786348i \(0.288031\pi\)
\(360\) 0 0
\(361\) −17.8986 −0.942031
\(362\) 0 0
\(363\) 1.80599 0.0947897
\(364\) 0 0
\(365\) 3.07482 0.160943
\(366\) 0 0
\(367\) 31.4058 1.63937 0.819684 0.572816i \(-0.194149\pi\)
0.819684 + 0.572816i \(0.194149\pi\)
\(368\) 0 0
\(369\) 20.3581 1.05980
\(370\) 0 0
\(371\) −7.12311 −0.369813
\(372\) 0 0
\(373\) −19.7701 −1.02366 −0.511828 0.859088i \(-0.671031\pi\)
−0.511828 + 0.859088i \(0.671031\pi\)
\(374\) 0 0
\(375\) −0.512072 −0.0264433
\(376\) 0 0
\(377\) 4.32348 0.222670
\(378\) 0 0
\(379\) 16.1132 0.827679 0.413840 0.910350i \(-0.364187\pi\)
0.413840 + 0.910350i \(0.364187\pi\)
\(380\) 0 0
\(381\) −5.12549 −0.262587
\(382\) 0 0
\(383\) 22.2003 1.13438 0.567192 0.823586i \(-0.308030\pi\)
0.567192 + 0.823586i \(0.308030\pi\)
\(384\) 0 0
\(385\) −3.81141 −0.194247
\(386\) 0 0
\(387\) 7.63092 0.387902
\(388\) 0 0
\(389\) −29.6377 −1.50269 −0.751344 0.659910i \(-0.770594\pi\)
−0.751344 + 0.659910i \(0.770594\pi\)
\(390\) 0 0
\(391\) −4.42786 −0.223927
\(392\) 0 0
\(393\) 0.449033 0.0226507
\(394\) 0 0
\(395\) −9.11439 −0.458595
\(396\) 0 0
\(397\) 1.75618 0.0881400 0.0440700 0.999028i \(-0.485968\pi\)
0.0440700 + 0.999028i \(0.485968\pi\)
\(398\) 0 0
\(399\) 0.537410 0.0269041
\(400\) 0 0
\(401\) −26.5716 −1.32692 −0.663460 0.748212i \(-0.730913\pi\)
−0.663460 + 0.748212i \(0.730913\pi\)
\(402\) 0 0
\(403\) 4.61103 0.229692
\(404\) 0 0
\(405\) −6.70880 −0.333363
\(406\) 0 0
\(407\) 22.5739 1.11895
\(408\) 0 0
\(409\) 36.2239 1.79116 0.895578 0.444905i \(-0.146763\pi\)
0.895578 + 0.444905i \(0.146763\pi\)
\(410\) 0 0
\(411\) 3.73778 0.184371
\(412\) 0 0
\(413\) −14.0866 −0.693156
\(414\) 0 0
\(415\) 11.3210 0.555728
\(416\) 0 0
\(417\) −8.02595 −0.393032
\(418\) 0 0
\(419\) 20.9203 1.02202 0.511011 0.859574i \(-0.329271\pi\)
0.511011 + 0.859574i \(0.329271\pi\)
\(420\) 0 0
\(421\) −29.6638 −1.44573 −0.722863 0.690991i \(-0.757174\pi\)
−0.722863 + 0.690991i \(0.757174\pi\)
\(422\) 0 0
\(423\) 8.55038 0.415734
\(424\) 0 0
\(425\) −0.761926 −0.0369588
\(426\) 0 0
\(427\) −8.24621 −0.399062
\(428\) 0 0
\(429\) −1.95171 −0.0942296
\(430\) 0 0
\(431\) −26.0812 −1.25629 −0.628143 0.778098i \(-0.716185\pi\)
−0.628143 + 0.778098i \(0.716185\pi\)
\(432\) 0 0
\(433\) 10.9907 0.528179 0.264090 0.964498i \(-0.414929\pi\)
0.264090 + 0.964498i \(0.414929\pi\)
\(434\) 0 0
\(435\) −2.21393 −0.106150
\(436\) 0 0
\(437\) −6.09896 −0.291753
\(438\) 0 0
\(439\) −11.8207 −0.564173 −0.282087 0.959389i \(-0.591027\pi\)
−0.282087 + 0.959389i \(0.591027\pi\)
\(440\) 0 0
\(441\) −2.73778 −0.130371
\(442\) 0 0
\(443\) −19.2290 −0.913598 −0.456799 0.889570i \(-0.651004\pi\)
−0.456799 + 0.889570i \(0.651004\pi\)
\(444\) 0 0
\(445\) −17.3500 −0.822469
\(446\) 0 0
\(447\) −3.64754 −0.172523
\(448\) 0 0
\(449\) −17.8455 −0.842180 −0.421090 0.907019i \(-0.638352\pi\)
−0.421090 + 0.907019i \(0.638352\pi\)
\(450\) 0 0
\(451\) 28.3416 1.33455
\(452\) 0 0
\(453\) 1.85159 0.0869951
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) 16.3953 0.766941 0.383470 0.923553i \(-0.374729\pi\)
0.383470 + 0.923553i \(0.374729\pi\)
\(458\) 0 0
\(459\) 2.23866 0.104492
\(460\) 0 0
\(461\) 16.8739 0.785894 0.392947 0.919561i \(-0.371456\pi\)
0.392947 + 0.919561i \(0.371456\pi\)
\(462\) 0 0
\(463\) −8.51082 −0.395531 −0.197766 0.980249i \(-0.563368\pi\)
−0.197766 + 0.980249i \(0.563368\pi\)
\(464\) 0 0
\(465\) −2.36118 −0.109497
\(466\) 0 0
\(467\) −7.09777 −0.328446 −0.164223 0.986423i \(-0.552512\pi\)
−0.164223 + 0.986423i \(0.552512\pi\)
\(468\) 0 0
\(469\) −2.21393 −0.102230
\(470\) 0 0
\(471\) −0.935124 −0.0430883
\(472\) 0 0
\(473\) 10.6234 0.488464
\(474\) 0 0
\(475\) −1.04948 −0.0481535
\(476\) 0 0
\(477\) −19.5015 −0.892913
\(478\) 0 0
\(479\) 7.46081 0.340893 0.170447 0.985367i \(-0.445479\pi\)
0.170447 + 0.985367i \(0.445479\pi\)
\(480\) 0 0
\(481\) −5.92273 −0.270053
\(482\) 0 0
\(483\) −2.97586 −0.135406
\(484\) 0 0
\(485\) −11.7972 −0.535682
\(486\) 0 0
\(487\) −7.20037 −0.326280 −0.163140 0.986603i \(-0.552162\pi\)
−0.163140 + 0.986603i \(0.552162\pi\)
\(488\) 0 0
\(489\) 11.5585 0.522694
\(490\) 0 0
\(491\) 12.2945 0.554843 0.277421 0.960748i \(-0.410520\pi\)
0.277421 + 0.960748i \(0.410520\pi\)
\(492\) 0 0
\(493\) −3.29417 −0.148362
\(494\) 0 0
\(495\) −10.4348 −0.469010
\(496\) 0 0
\(497\) −4.58205 −0.205533
\(498\) 0 0
\(499\) −5.66655 −0.253670 −0.126835 0.991924i \(-0.540482\pi\)
−0.126835 + 0.991924i \(0.540482\pi\)
\(500\) 0 0
\(501\) −0.934856 −0.0417663
\(502\) 0 0
\(503\) 11.2363 0.501002 0.250501 0.968116i \(-0.419405\pi\)
0.250501 + 0.968116i \(0.419405\pi\)
\(504\) 0 0
\(505\) −4.25350 −0.189278
\(506\) 0 0
\(507\) 0.512072 0.0227419
\(508\) 0 0
\(509\) 21.9864 0.974529 0.487265 0.873254i \(-0.337995\pi\)
0.487265 + 0.873254i \(0.337995\pi\)
\(510\) 0 0
\(511\) 3.07482 0.136022
\(512\) 0 0
\(513\) 3.08354 0.136142
\(514\) 0 0
\(515\) −12.1243 −0.534261
\(516\) 0 0
\(517\) 11.9034 0.523512
\(518\) 0 0
\(519\) −4.21277 −0.184920
\(520\) 0 0
\(521\) 39.2727 1.72057 0.860285 0.509813i \(-0.170286\pi\)
0.860285 + 0.509813i \(0.170286\pi\)
\(522\) 0 0
\(523\) 14.1132 0.617127 0.308563 0.951204i \(-0.400152\pi\)
0.308563 + 0.951204i \(0.400152\pi\)
\(524\) 0 0
\(525\) −0.512072 −0.0223486
\(526\) 0 0
\(527\) −3.51327 −0.153040
\(528\) 0 0
\(529\) 10.7725 0.468367
\(530\) 0 0
\(531\) −38.5660 −1.67362
\(532\) 0 0
\(533\) −7.43600 −0.322089
\(534\) 0 0
\(535\) −15.2870 −0.660913
\(536\) 0 0
\(537\) 13.3811 0.577436
\(538\) 0 0
\(539\) −3.81141 −0.164169
\(540\) 0 0
\(541\) −31.4967 −1.35415 −0.677075 0.735914i \(-0.736753\pi\)
−0.677075 + 0.735914i \(0.736753\pi\)
\(542\) 0 0
\(543\) −4.30178 −0.184607
\(544\) 0 0
\(545\) 17.2727 0.739883
\(546\) 0 0
\(547\) −19.5096 −0.834171 −0.417086 0.908867i \(-0.636948\pi\)
−0.417086 + 0.908867i \(0.636948\pi\)
\(548\) 0 0
\(549\) −22.5763 −0.963534
\(550\) 0 0
\(551\) −4.53741 −0.193300
\(552\) 0 0
\(553\) −9.11439 −0.387583
\(554\) 0 0
\(555\) 3.03286 0.128738
\(556\) 0 0
\(557\) −34.7559 −1.47265 −0.736327 0.676626i \(-0.763441\pi\)
−0.736327 + 0.676626i \(0.763441\pi\)
\(558\) 0 0
\(559\) −2.78726 −0.117889
\(560\) 0 0
\(561\) 1.48706 0.0627838
\(562\) 0 0
\(563\) −21.7508 −0.916689 −0.458344 0.888775i \(-0.651557\pi\)
−0.458344 + 0.888775i \(0.651557\pi\)
\(564\) 0 0
\(565\) 0.598671 0.0251863
\(566\) 0 0
\(567\) −6.70880 −0.281743
\(568\) 0 0
\(569\) −0.898589 −0.0376708 −0.0188354 0.999823i \(-0.505996\pi\)
−0.0188354 + 0.999823i \(0.505996\pi\)
\(570\) 0 0
\(571\) −13.9536 −0.583939 −0.291970 0.956428i \(-0.594311\pi\)
−0.291970 + 0.956428i \(0.594311\pi\)
\(572\) 0 0
\(573\) −3.16267 −0.132123
\(574\) 0 0
\(575\) 5.81141 0.242352
\(576\) 0 0
\(577\) 1.43218 0.0596222 0.0298111 0.999556i \(-0.490509\pi\)
0.0298111 + 0.999556i \(0.490509\pi\)
\(578\) 0 0
\(579\) 1.67019 0.0694107
\(580\) 0 0
\(581\) 11.3210 0.469675
\(582\) 0 0
\(583\) −27.1491 −1.12440
\(584\) 0 0
\(585\) 2.73778 0.113193
\(586\) 0 0
\(587\) −7.82429 −0.322943 −0.161472 0.986877i \(-0.551624\pi\)
−0.161472 + 0.986877i \(0.551624\pi\)
\(588\) 0 0
\(589\) −4.83919 −0.199395
\(590\) 0 0
\(591\) 10.8397 0.445886
\(592\) 0 0
\(593\) 22.0259 0.904497 0.452249 0.891892i \(-0.350622\pi\)
0.452249 + 0.891892i \(0.350622\pi\)
\(594\) 0 0
\(595\) −0.761926 −0.0312359
\(596\) 0 0
\(597\) 13.2765 0.543370
\(598\) 0 0
\(599\) 28.9028 1.18094 0.590469 0.807060i \(-0.298943\pi\)
0.590469 + 0.807060i \(0.298943\pi\)
\(600\) 0 0
\(601\) −13.5510 −0.552755 −0.276378 0.961049i \(-0.589134\pi\)
−0.276378 + 0.961049i \(0.589134\pi\)
\(602\) 0 0
\(603\) −6.06126 −0.246834
\(604\) 0 0
\(605\) −3.52682 −0.143386
\(606\) 0 0
\(607\) 33.7504 1.36989 0.684944 0.728596i \(-0.259827\pi\)
0.684944 + 0.728596i \(0.259827\pi\)
\(608\) 0 0
\(609\) −2.21393 −0.0897130
\(610\) 0 0
\(611\) −3.12311 −0.126347
\(612\) 0 0
\(613\) −6.95230 −0.280801 −0.140400 0.990095i \(-0.544839\pi\)
−0.140400 + 0.990095i \(0.544839\pi\)
\(614\) 0 0
\(615\) 3.80776 0.153544
\(616\) 0 0
\(617\) −22.0003 −0.885697 −0.442849 0.896596i \(-0.646032\pi\)
−0.442849 + 0.896596i \(0.646032\pi\)
\(618\) 0 0
\(619\) −6.11610 −0.245827 −0.122913 0.992417i \(-0.539224\pi\)
−0.122913 + 0.992417i \(0.539224\pi\)
\(620\) 0 0
\(621\) −17.0748 −0.685189
\(622\) 0 0
\(623\) −17.3500 −0.695114
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.04829 0.0818007
\(628\) 0 0
\(629\) 4.51268 0.179932
\(630\) 0 0
\(631\) 34.6278 1.37851 0.689256 0.724518i \(-0.257937\pi\)
0.689256 + 0.724518i \(0.257937\pi\)
\(632\) 0 0
\(633\) −9.52954 −0.378765
\(634\) 0 0
\(635\) 10.0093 0.397208
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −12.5447 −0.496259
\(640\) 0 0
\(641\) −25.8153 −1.01964 −0.509822 0.860280i \(-0.670289\pi\)
−0.509822 + 0.860280i \(0.670289\pi\)
\(642\) 0 0
\(643\) −23.0737 −0.909936 −0.454968 0.890508i \(-0.650349\pi\)
−0.454968 + 0.890508i \(0.650349\pi\)
\(644\) 0 0
\(645\) 1.42728 0.0561990
\(646\) 0 0
\(647\) 7.48554 0.294287 0.147143 0.989115i \(-0.452992\pi\)
0.147143 + 0.989115i \(0.452992\pi\)
\(648\) 0 0
\(649\) −53.6898 −2.10751
\(650\) 0 0
\(651\) −2.36118 −0.0925419
\(652\) 0 0
\(653\) 21.7972 0.852990 0.426495 0.904490i \(-0.359748\pi\)
0.426495 + 0.904490i \(0.359748\pi\)
\(654\) 0 0
\(655\) −0.876894 −0.0342631
\(656\) 0 0
\(657\) 8.41819 0.328425
\(658\) 0 0
\(659\) 11.0329 0.429779 0.214890 0.976638i \(-0.431061\pi\)
0.214890 + 0.976638i \(0.431061\pi\)
\(660\) 0 0
\(661\) 33.5633 1.30546 0.652731 0.757590i \(-0.273623\pi\)
0.652731 + 0.757590i \(0.273623\pi\)
\(662\) 0 0
\(663\) −0.390161 −0.0151526
\(664\) 0 0
\(665\) −1.04948 −0.0406971
\(666\) 0 0
\(667\) 25.1255 0.972863
\(668\) 0 0
\(669\) 0.664398 0.0256871
\(670\) 0 0
\(671\) −31.4297 −1.21333
\(672\) 0 0
\(673\) 38.8129 1.49613 0.748063 0.663628i \(-0.230984\pi\)
0.748063 + 0.663628i \(0.230984\pi\)
\(674\) 0 0
\(675\) −2.93816 −0.113090
\(676\) 0 0
\(677\) 27.9371 1.07371 0.536856 0.843674i \(-0.319612\pi\)
0.536856 + 0.843674i \(0.319612\pi\)
\(678\) 0 0
\(679\) −11.7972 −0.452734
\(680\) 0 0
\(681\) 1.85887 0.0712321
\(682\) 0 0
\(683\) −43.7564 −1.67429 −0.837147 0.546978i \(-0.815778\pi\)
−0.837147 + 0.546978i \(0.815778\pi\)
\(684\) 0 0
\(685\) −7.29934 −0.278893
\(686\) 0 0
\(687\) −14.0353 −0.535481
\(688\) 0 0
\(689\) 7.12311 0.271369
\(690\) 0 0
\(691\) 19.0281 0.723861 0.361931 0.932205i \(-0.382118\pi\)
0.361931 + 0.932205i \(0.382118\pi\)
\(692\) 0 0
\(693\) −10.4348 −0.396385
\(694\) 0 0
\(695\) 15.6735 0.594529
\(696\) 0 0
\(697\) 5.66568 0.214603
\(698\) 0 0
\(699\) −8.86961 −0.335479
\(700\) 0 0
\(701\) −42.7087 −1.61309 −0.806543 0.591175i \(-0.798664\pi\)
−0.806543 + 0.591175i \(0.798664\pi\)
\(702\) 0 0
\(703\) 6.21580 0.234433
\(704\) 0 0
\(705\) 1.59925 0.0602314
\(706\) 0 0
\(707\) −4.25350 −0.159969
\(708\) 0 0
\(709\) −45.9988 −1.72752 −0.863761 0.503901i \(-0.831898\pi\)
−0.863761 + 0.503901i \(0.831898\pi\)
\(710\) 0 0
\(711\) −24.9532 −0.935818
\(712\) 0 0
\(713\) 26.7966 1.00354
\(714\) 0 0
\(715\) 3.81141 0.142539
\(716\) 0 0
\(717\) 0.121299 0.00453000
\(718\) 0 0
\(719\) −0.524435 −0.0195581 −0.00977906 0.999952i \(-0.503113\pi\)
−0.00977906 + 0.999952i \(0.503113\pi\)
\(720\) 0 0
\(721\) −12.1243 −0.451533
\(722\) 0 0
\(723\) −6.48557 −0.241201
\(724\) 0 0
\(725\) 4.32348 0.160570
\(726\) 0 0
\(727\) 2.78913 0.103443 0.0517215 0.998662i \(-0.483529\pi\)
0.0517215 + 0.998662i \(0.483529\pi\)
\(728\) 0 0
\(729\) −13.8536 −0.513096
\(730\) 0 0
\(731\) 2.12369 0.0785475
\(732\) 0 0
\(733\) −24.8546 −0.918024 −0.459012 0.888430i \(-0.651797\pi\)
−0.459012 + 0.888430i \(0.651797\pi\)
\(734\) 0 0
\(735\) −0.512072 −0.0188880
\(736\) 0 0
\(737\) −8.43819 −0.310825
\(738\) 0 0
\(739\) 45.6817 1.68043 0.840213 0.542256i \(-0.182430\pi\)
0.840213 + 0.542256i \(0.182430\pi\)
\(740\) 0 0
\(741\) −0.537410 −0.0197422
\(742\) 0 0
\(743\) −15.0376 −0.551678 −0.275839 0.961204i \(-0.588956\pi\)
−0.275839 + 0.961204i \(0.588956\pi\)
\(744\) 0 0
\(745\) 7.12311 0.260970
\(746\) 0 0
\(747\) 30.9945 1.13403
\(748\) 0 0
\(749\) −15.2870 −0.558574
\(750\) 0 0
\(751\) 7.45626 0.272083 0.136041 0.990703i \(-0.456562\pi\)
0.136041 + 0.990703i \(0.456562\pi\)
\(752\) 0 0
\(753\) 8.87083 0.323271
\(754\) 0 0
\(755\) −3.61587 −0.131595
\(756\) 0 0
\(757\) −10.6778 −0.388091 −0.194046 0.980992i \(-0.562161\pi\)
−0.194046 + 0.980992i \(0.562161\pi\)
\(758\) 0 0
\(759\) −11.3422 −0.411696
\(760\) 0 0
\(761\) 8.29636 0.300743 0.150371 0.988630i \(-0.451953\pi\)
0.150371 + 0.988630i \(0.451953\pi\)
\(762\) 0 0
\(763\) 17.2727 0.625315
\(764\) 0 0
\(765\) −2.08599 −0.0754190
\(766\) 0 0
\(767\) 14.0866 0.508638
\(768\) 0 0
\(769\) 3.22446 0.116277 0.0581385 0.998309i \(-0.481484\pi\)
0.0581385 + 0.998309i \(0.481484\pi\)
\(770\) 0 0
\(771\) −5.94798 −0.214211
\(772\) 0 0
\(773\) −3.57453 −0.128567 −0.0642834 0.997932i \(-0.520476\pi\)
−0.0642834 + 0.997932i \(0.520476\pi\)
\(774\) 0 0
\(775\) 4.61103 0.165633
\(776\) 0 0
\(777\) 3.03286 0.108803
\(778\) 0 0
\(779\) 7.80394 0.279605
\(780\) 0 0
\(781\) −17.4641 −0.624914
\(782\) 0 0
\(783\) −12.7031 −0.453970
\(784\) 0 0
\(785\) 1.82616 0.0651784
\(786\) 0 0
\(787\) −21.1641 −0.754419 −0.377209 0.926128i \(-0.623116\pi\)
−0.377209 + 0.926128i \(0.623116\pi\)
\(788\) 0 0
\(789\) 12.3857 0.440942
\(790\) 0 0
\(791\) 0.598671 0.0212863
\(792\) 0 0
\(793\) 8.24621 0.292832
\(794\) 0 0
\(795\) −3.64754 −0.129365
\(796\) 0 0
\(797\) 14.6856 0.520191 0.260096 0.965583i \(-0.416246\pi\)
0.260096 + 0.965583i \(0.416246\pi\)
\(798\) 0 0
\(799\) 2.37958 0.0841833
\(800\) 0 0
\(801\) −47.5006 −1.67835
\(802\) 0 0
\(803\) 11.7194 0.413568
\(804\) 0 0
\(805\) 5.81141 0.204825
\(806\) 0 0
\(807\) 7.42489 0.261368
\(808\) 0 0
\(809\) 16.0160 0.563093 0.281546 0.959548i \(-0.409153\pi\)
0.281546 + 0.959548i \(0.409153\pi\)
\(810\) 0 0
\(811\) 30.6243 1.07536 0.537682 0.843148i \(-0.319300\pi\)
0.537682 + 0.843148i \(0.319300\pi\)
\(812\) 0 0
\(813\) −11.7055 −0.410530
\(814\) 0 0
\(815\) −22.5721 −0.790665
\(816\) 0 0
\(817\) 2.92518 0.102339
\(818\) 0 0
\(819\) 2.73778 0.0956659
\(820\) 0 0
\(821\) 12.8449 0.448289 0.224145 0.974556i \(-0.428041\pi\)
0.224145 + 0.974556i \(0.428041\pi\)
\(822\) 0 0
\(823\) −2.30634 −0.0803939 −0.0401969 0.999192i \(-0.512799\pi\)
−0.0401969 + 0.999192i \(0.512799\pi\)
\(824\) 0 0
\(825\) −1.95171 −0.0679499
\(826\) 0 0
\(827\) 44.6409 1.55232 0.776158 0.630538i \(-0.217166\pi\)
0.776158 + 0.630538i \(0.217166\pi\)
\(828\) 0 0
\(829\) 31.9583 1.10996 0.554979 0.831864i \(-0.312726\pi\)
0.554979 + 0.831864i \(0.312726\pi\)
\(830\) 0 0
\(831\) 4.14847 0.143909
\(832\) 0 0
\(833\) −0.761926 −0.0263992
\(834\) 0 0
\(835\) 1.82564 0.0631787
\(836\) 0 0
\(837\) −13.5479 −0.468285
\(838\) 0 0
\(839\) 9.15955 0.316223 0.158111 0.987421i \(-0.449459\pi\)
0.158111 + 0.987421i \(0.449459\pi\)
\(840\) 0 0
\(841\) −10.3075 −0.355432
\(842\) 0 0
\(843\) 7.10316 0.244646
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −3.52682 −0.121183
\(848\) 0 0
\(849\) 14.1192 0.484570
\(850\) 0 0
\(851\) −34.4194 −1.17988
\(852\) 0 0
\(853\) −51.8950 −1.77685 −0.888425 0.459022i \(-0.848200\pi\)
−0.888425 + 0.459022i \(0.848200\pi\)
\(854\) 0 0
\(855\) −2.87325 −0.0982631
\(856\) 0 0
\(857\) 25.5479 0.872701 0.436350 0.899777i \(-0.356271\pi\)
0.436350 + 0.899777i \(0.356271\pi\)
\(858\) 0 0
\(859\) 30.7399 1.04883 0.524415 0.851463i \(-0.324284\pi\)
0.524415 + 0.851463i \(0.324284\pi\)
\(860\) 0 0
\(861\) 3.80776 0.129768
\(862\) 0 0
\(863\) 33.3171 1.13413 0.567063 0.823674i \(-0.308079\pi\)
0.567063 + 0.823674i \(0.308079\pi\)
\(864\) 0 0
\(865\) 8.22691 0.279723
\(866\) 0 0
\(867\) −8.40794 −0.285549
\(868\) 0 0
\(869\) −34.7386 −1.17843
\(870\) 0 0
\(871\) 2.21393 0.0750162
\(872\) 0 0
\(873\) −32.2981 −1.09313
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 21.1627 0.714613 0.357306 0.933987i \(-0.383695\pi\)
0.357306 + 0.933987i \(0.383695\pi\)
\(878\) 0 0
\(879\) −14.4128 −0.486132
\(880\) 0 0
\(881\) 29.5364 0.995106 0.497553 0.867433i \(-0.334232\pi\)
0.497553 + 0.867433i \(0.334232\pi\)
\(882\) 0 0
\(883\) −1.09144 −0.0367298 −0.0183649 0.999831i \(-0.505846\pi\)
−0.0183649 + 0.999831i \(0.505846\pi\)
\(884\) 0 0
\(885\) −7.21335 −0.242474
\(886\) 0 0
\(887\) −22.7793 −0.764855 −0.382428 0.923985i \(-0.624912\pi\)
−0.382428 + 0.923985i \(0.624912\pi\)
\(888\) 0 0
\(889\) 10.0093 0.335702
\(890\) 0 0
\(891\) −25.5700 −0.856626
\(892\) 0 0
\(893\) 3.27764 0.109682
\(894\) 0 0
\(895\) −26.1312 −0.873471
\(896\) 0 0
\(897\) 2.97586 0.0993610
\(898\) 0 0
\(899\) 19.9357 0.664893
\(900\) 0 0
\(901\) −5.42728 −0.180809
\(902\) 0 0
\(903\) 1.42728 0.0474969
\(904\) 0 0
\(905\) 8.40075 0.279250
\(906\) 0 0
\(907\) −32.8156 −1.08962 −0.544812 0.838558i \(-0.683399\pi\)
−0.544812 + 0.838558i \(0.683399\pi\)
\(908\) 0 0
\(909\) −11.6452 −0.386245
\(910\) 0 0
\(911\) −46.6062 −1.54413 −0.772067 0.635542i \(-0.780777\pi\)
−0.772067 + 0.635542i \(0.780777\pi\)
\(912\) 0 0
\(913\) 43.1491 1.42803
\(914\) 0 0
\(915\) −4.22265 −0.139597
\(916\) 0 0
\(917\) −0.876894 −0.0289576
\(918\) 0 0
\(919\) 30.7266 1.01358 0.506789 0.862070i \(-0.330832\pi\)
0.506789 + 0.862070i \(0.330832\pi\)
\(920\) 0 0
\(921\) 5.35188 0.176350
\(922\) 0 0
\(923\) 4.58205 0.150820
\(924\) 0 0
\(925\) −5.92273 −0.194738
\(926\) 0 0
\(927\) −33.1937 −1.09022
\(928\) 0 0
\(929\) −47.3451 −1.55334 −0.776671 0.629906i \(-0.783093\pi\)
−0.776671 + 0.629906i \(0.783093\pi\)
\(930\) 0 0
\(931\) −1.04948 −0.0343954
\(932\) 0 0
\(933\) −8.81520 −0.288597
\(934\) 0 0
\(935\) −2.90401 −0.0949713
\(936\) 0 0
\(937\) 32.0396 1.04669 0.523344 0.852122i \(-0.324684\pi\)
0.523344 + 0.852122i \(0.324684\pi\)
\(938\) 0 0
\(939\) 9.44567 0.308248
\(940\) 0 0
\(941\) −17.8676 −0.582467 −0.291233 0.956652i \(-0.594066\pi\)
−0.291233 + 0.956652i \(0.594066\pi\)
\(942\) 0 0
\(943\) −43.2136 −1.40723
\(944\) 0 0
\(945\) −2.93816 −0.0955782
\(946\) 0 0
\(947\) −32.3426 −1.05099 −0.525496 0.850796i \(-0.676120\pi\)
−0.525496 + 0.850796i \(0.676120\pi\)
\(948\) 0 0
\(949\) −3.07482 −0.0998129
\(950\) 0 0
\(951\) 16.1780 0.524607
\(952\) 0 0
\(953\) 16.5195 0.535120 0.267560 0.963541i \(-0.413783\pi\)
0.267560 + 0.963541i \(0.413783\pi\)
\(954\) 0 0
\(955\) 6.17623 0.199858
\(956\) 0 0
\(957\) −8.43819 −0.272768
\(958\) 0 0
\(959\) −7.29934 −0.235708
\(960\) 0 0
\(961\) −9.73837 −0.314141
\(962\) 0 0
\(963\) −41.8524 −1.34867
\(964\) 0 0
\(965\) −3.26163 −0.104996
\(966\) 0 0
\(967\) −37.9970 −1.22190 −0.610950 0.791669i \(-0.709212\pi\)
−0.610950 + 0.791669i \(0.709212\pi\)
\(968\) 0 0
\(969\) 0.409466 0.0131540
\(970\) 0 0
\(971\) −16.1695 −0.518903 −0.259451 0.965756i \(-0.583542\pi\)
−0.259451 + 0.965756i \(0.583542\pi\)
\(972\) 0 0
\(973\) 15.6735 0.502469
\(974\) 0 0
\(975\) 0.512072 0.0163994
\(976\) 0 0
\(977\) 31.1318 0.995995 0.497998 0.867178i \(-0.334069\pi\)
0.497998 + 0.867178i \(0.334069\pi\)
\(978\) 0 0
\(979\) −66.1280 −2.11346
\(980\) 0 0
\(981\) 47.2890 1.50982
\(982\) 0 0
\(983\) 51.5871 1.64537 0.822686 0.568496i \(-0.192475\pi\)
0.822686 + 0.568496i \(0.192475\pi\)
\(984\) 0 0
\(985\) −21.1684 −0.674480
\(986\) 0 0
\(987\) 1.59925 0.0509048
\(988\) 0 0
\(989\) −16.1979 −0.515064
\(990\) 0 0
\(991\) −35.0341 −1.11289 −0.556447 0.830883i \(-0.687836\pi\)
−0.556447 + 0.830883i \(0.687836\pi\)
\(992\) 0 0
\(993\) 2.84978 0.0904350
\(994\) 0 0
\(995\) −25.9270 −0.821941
\(996\) 0 0
\(997\) 25.1675 0.797061 0.398531 0.917155i \(-0.369520\pi\)
0.398531 + 0.917155i \(0.369520\pi\)
\(998\) 0 0
\(999\) 17.4019 0.550572
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 3640.2.a.x.1.2 4
4.3 odd 2 7280.2.a.br.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.x.1.2 4 1.1 even 1 trivial
7280.2.a.br.1.3 4 4.3 odd 2