Properties

Label 5520.2.be.c.1471.20
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.20
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.19

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -2.01693 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -2.01693 q^{7} -1.00000 q^{9} -0.697775 q^{11} +0.195235 q^{13} -1.00000 q^{15} +0.430024i q^{17} +7.71006 q^{19} -2.01693i q^{21} +(-4.23049 + 2.25897i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -9.64181 q^{29} -1.05072i q^{31} -0.697775i q^{33} -2.01693i q^{35} +4.89822i q^{37} +0.195235i q^{39} -3.71938 q^{41} +5.85669 q^{43} -1.00000i q^{45} -10.6764i q^{47} -2.93201 q^{49} -0.430024 q^{51} +0.464177i q^{53} -0.697775i q^{55} +7.71006i q^{57} +7.64910i q^{59} -4.18983i q^{61} +2.01693 q^{63} +0.195235i q^{65} -3.70657 q^{67} +(-2.25897 - 4.23049i) q^{69} -5.69569i q^{71} -3.51188 q^{73} -1.00000i q^{75} +1.40736 q^{77} -9.68233 q^{79} +1.00000 q^{81} +0.982199 q^{83} -0.430024 q^{85} -9.64181i q^{87} -5.48101i q^{89} -0.393776 q^{91} +1.05072 q^{93} +7.71006i q^{95} +13.5553i q^{97} +0.697775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −2.01693 −0.762327 −0.381163 0.924508i \(-0.624476\pi\)
−0.381163 + 0.924508i \(0.624476\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −0.697775 −0.210387 −0.105194 0.994452i \(-0.533546\pi\)
−0.105194 + 0.994452i \(0.533546\pi\)
\(12\) 0 0
\(13\) 0.195235 0.0541486 0.0270743 0.999633i \(-0.491381\pi\)
0.0270743 + 0.999633i \(0.491381\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0.430024i 0.104296i 0.998639 + 0.0521480i \(0.0166068\pi\)
−0.998639 + 0.0521480i \(0.983393\pi\)
\(18\) 0 0
\(19\) 7.71006 1.76881 0.884405 0.466720i \(-0.154564\pi\)
0.884405 + 0.466720i \(0.154564\pi\)
\(20\) 0 0
\(21\) 2.01693i 0.440130i
\(22\) 0 0
\(23\) −4.23049 + 2.25897i −0.882119 + 0.471027i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −9.64181 −1.79044 −0.895220 0.445625i \(-0.852982\pi\)
−0.895220 + 0.445625i \(0.852982\pi\)
\(30\) 0 0
\(31\) 1.05072i 0.188714i −0.995538 0.0943570i \(-0.969920\pi\)
0.995538 0.0943570i \(-0.0300795\pi\)
\(32\) 0 0
\(33\) 0.697775i 0.121467i
\(34\) 0 0
\(35\) 2.01693i 0.340923i
\(36\) 0 0
\(37\) 4.89822i 0.805262i 0.915362 + 0.402631i \(0.131904\pi\)
−0.915362 + 0.402631i \(0.868096\pi\)
\(38\) 0 0
\(39\) 0.195235i 0.0312627i
\(40\) 0 0
\(41\) −3.71938 −0.580870 −0.290435 0.956895i \(-0.593800\pi\)
−0.290435 + 0.956895i \(0.593800\pi\)
\(42\) 0 0
\(43\) 5.85669 0.893136 0.446568 0.894750i \(-0.352646\pi\)
0.446568 + 0.894750i \(0.352646\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 10.6764i 1.55731i −0.627452 0.778655i \(-0.715902\pi\)
0.627452 0.778655i \(-0.284098\pi\)
\(48\) 0 0
\(49\) −2.93201 −0.418858
\(50\) 0 0
\(51\) −0.430024 −0.0602154
\(52\) 0 0
\(53\) 0.464177i 0.0637597i 0.999492 + 0.0318798i \(0.0101494\pi\)
−0.999492 + 0.0318798i \(0.989851\pi\)
\(54\) 0 0
\(55\) 0.697775i 0.0940880i
\(56\) 0 0
\(57\) 7.71006i 1.02122i
\(58\) 0 0
\(59\) 7.64910i 0.995828i 0.867226 + 0.497914i \(0.165901\pi\)
−0.867226 + 0.497914i \(0.834099\pi\)
\(60\) 0 0
\(61\) 4.18983i 0.536452i −0.963356 0.268226i \(-0.913563\pi\)
0.963356 0.268226i \(-0.0864375\pi\)
\(62\) 0 0
\(63\) 2.01693 0.254109
\(64\) 0 0
\(65\) 0.195235i 0.0242160i
\(66\) 0 0
\(67\) −3.70657 −0.452830 −0.226415 0.974031i \(-0.572700\pi\)
−0.226415 + 0.974031i \(0.572700\pi\)
\(68\) 0 0
\(69\) −2.25897 4.23049i −0.271948 0.509292i
\(70\) 0 0
\(71\) 5.69569i 0.675954i −0.941154 0.337977i \(-0.890257\pi\)
0.941154 0.337977i \(-0.109743\pi\)
\(72\) 0 0
\(73\) −3.51188 −0.411034 −0.205517 0.978654i \(-0.565888\pi\)
−0.205517 + 0.978654i \(0.565888\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 1.40736 0.160384
\(78\) 0 0
\(79\) −9.68233 −1.08935 −0.544674 0.838648i \(-0.683346\pi\)
−0.544674 + 0.838648i \(0.683346\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.982199 0.107810 0.0539052 0.998546i \(-0.482833\pi\)
0.0539052 + 0.998546i \(0.482833\pi\)
\(84\) 0 0
\(85\) −0.430024 −0.0466426
\(86\) 0 0
\(87\) 9.64181i 1.03371i
\(88\) 0 0
\(89\) 5.48101i 0.580986i −0.956877 0.290493i \(-0.906181\pi\)
0.956877 0.290493i \(-0.0938192\pi\)
\(90\) 0 0
\(91\) −0.393776 −0.0412789
\(92\) 0 0
\(93\) 1.05072 0.108954
\(94\) 0 0
\(95\) 7.71006i 0.791036i
\(96\) 0 0
\(97\) 13.5553i 1.37634i 0.725552 + 0.688168i \(0.241585\pi\)
−0.725552 + 0.688168i \(0.758415\pi\)
\(98\) 0 0
\(99\) 0.697775 0.0701291
\(100\) 0 0
\(101\) −9.87390 −0.982490 −0.491245 0.871021i \(-0.663458\pi\)
−0.491245 + 0.871021i \(0.663458\pi\)
\(102\) 0 0
\(103\) 7.65129 0.753904 0.376952 0.926233i \(-0.376972\pi\)
0.376952 + 0.926233i \(0.376972\pi\)
\(104\) 0 0
\(105\) 2.01693 0.196832
\(106\) 0 0
\(107\) 15.7693 1.52448 0.762238 0.647296i \(-0.224100\pi\)
0.762238 + 0.647296i \(0.224100\pi\)
\(108\) 0 0
\(109\) 7.29036i 0.698289i −0.937069 0.349145i \(-0.886472\pi\)
0.937069 0.349145i \(-0.113528\pi\)
\(110\) 0 0
\(111\) −4.89822 −0.464918
\(112\) 0 0
\(113\) 11.6243i 1.09352i −0.837288 0.546762i \(-0.815860\pi\)
0.837288 0.546762i \(-0.184140\pi\)
\(114\) 0 0
\(115\) −2.25897 4.23049i −0.210650 0.394496i
\(116\) 0 0
\(117\) −0.195235 −0.0180495
\(118\) 0 0
\(119\) 0.867327i 0.0795077i
\(120\) 0 0
\(121\) −10.5131 −0.955737
\(122\) 0 0
\(123\) 3.71938i 0.335365i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 5.20841i 0.462172i 0.972933 + 0.231086i \(0.0742279\pi\)
−0.972933 + 0.231086i \(0.925772\pi\)
\(128\) 0 0
\(129\) 5.85669i 0.515653i
\(130\) 0 0
\(131\) 2.04355i 0.178546i 0.996007 + 0.0892728i \(0.0284543\pi\)
−0.996007 + 0.0892728i \(0.971546\pi\)
\(132\) 0 0
\(133\) −15.5506 −1.34841
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 21.0004i 1.79419i −0.441841 0.897093i \(-0.645675\pi\)
0.441841 0.897093i \(-0.354325\pi\)
\(138\) 0 0
\(139\) 2.56106i 0.217227i −0.994084 0.108613i \(-0.965359\pi\)
0.994084 0.108613i \(-0.0346410\pi\)
\(140\) 0 0
\(141\) 10.6764 0.899113
\(142\) 0 0
\(143\) −0.136231 −0.0113922
\(144\) 0 0
\(145\) 9.64181i 0.800709i
\(146\) 0 0
\(147\) 2.93201i 0.241828i
\(148\) 0 0
\(149\) 9.82277i 0.804712i −0.915483 0.402356i \(-0.868191\pi\)
0.915483 0.402356i \(-0.131809\pi\)
\(150\) 0 0
\(151\) 14.8419i 1.20782i −0.797054 0.603909i \(-0.793609\pi\)
0.797054 0.603909i \(-0.206391\pi\)
\(152\) 0 0
\(153\) 0.430024i 0.0347654i
\(154\) 0 0
\(155\) 1.05072 0.0843955
\(156\) 0 0
\(157\) 1.12666i 0.0899172i −0.998989 0.0449586i \(-0.985684\pi\)
0.998989 0.0449586i \(-0.0143156\pi\)
\(158\) 0 0
\(159\) −0.464177 −0.0368117
\(160\) 0 0
\(161\) 8.53260 4.55617i 0.672463 0.359077i
\(162\) 0 0
\(163\) 17.1655i 1.34451i −0.740321 0.672254i \(-0.765326\pi\)
0.740321 0.672254i \(-0.234674\pi\)
\(164\) 0 0
\(165\) 0.697775 0.0543217
\(166\) 0 0
\(167\) 0.809244i 0.0626212i 0.999510 + 0.0313106i \(0.00996809\pi\)
−0.999510 + 0.0313106i \(0.990032\pi\)
\(168\) 0 0
\(169\) −12.9619 −0.997068
\(170\) 0 0
\(171\) −7.71006 −0.589603
\(172\) 0 0
\(173\) 0.192946 0.0146694 0.00733472 0.999973i \(-0.497665\pi\)
0.00733472 + 0.999973i \(0.497665\pi\)
\(174\) 0 0
\(175\) 2.01693 0.152465
\(176\) 0 0
\(177\) −7.64910 −0.574942
\(178\) 0 0
\(179\) 23.5203i 1.75799i −0.476832 0.878994i \(-0.658215\pi\)
0.476832 0.878994i \(-0.341785\pi\)
\(180\) 0 0
\(181\) 9.77478i 0.726554i −0.931681 0.363277i \(-0.881658\pi\)
0.931681 0.363277i \(-0.118342\pi\)
\(182\) 0 0
\(183\) 4.18983 0.309721
\(184\) 0 0
\(185\) −4.89822 −0.360124
\(186\) 0 0
\(187\) 0.300060i 0.0219426i
\(188\) 0 0
\(189\) 2.01693i 0.146710i
\(190\) 0 0
\(191\) 7.21224 0.521859 0.260930 0.965358i \(-0.415971\pi\)
0.260930 + 0.965358i \(0.415971\pi\)
\(192\) 0 0
\(193\) 24.9860 1.79853 0.899266 0.437402i \(-0.144101\pi\)
0.899266 + 0.437402i \(0.144101\pi\)
\(194\) 0 0
\(195\) −0.195235 −0.0139811
\(196\) 0 0
\(197\) −6.96165 −0.495997 −0.247998 0.968760i \(-0.579773\pi\)
−0.247998 + 0.968760i \(0.579773\pi\)
\(198\) 0 0
\(199\) 2.94085 0.208471 0.104236 0.994553i \(-0.466760\pi\)
0.104236 + 0.994553i \(0.466760\pi\)
\(200\) 0 0
\(201\) 3.70657i 0.261441i
\(202\) 0 0
\(203\) 19.4468 1.36490
\(204\) 0 0
\(205\) 3.71938i 0.259773i
\(206\) 0 0
\(207\) 4.23049 2.25897i 0.294040 0.157009i
\(208\) 0 0
\(209\) −5.37989 −0.372135
\(210\) 0 0
\(211\) 28.0259i 1.92938i −0.263381 0.964692i \(-0.584838\pi\)
0.263381 0.964692i \(-0.415162\pi\)
\(212\) 0 0
\(213\) 5.69569 0.390262
\(214\) 0 0
\(215\) 5.85669i 0.399423i
\(216\) 0 0
\(217\) 2.11922i 0.143862i
\(218\) 0 0
\(219\) 3.51188i 0.237311i
\(220\) 0 0
\(221\) 0.0839559i 0.00564748i
\(222\) 0 0
\(223\) 1.81830i 0.121763i 0.998145 + 0.0608813i \(0.0193911\pi\)
−0.998145 + 0.0608813i \(0.980609\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −15.7824 −1.04752 −0.523758 0.851867i \(-0.675470\pi\)
−0.523758 + 0.851867i \(0.675470\pi\)
\(228\) 0 0
\(229\) 7.05803i 0.466408i 0.972428 + 0.233204i \(0.0749210\pi\)
−0.972428 + 0.233204i \(0.925079\pi\)
\(230\) 0 0
\(231\) 1.40736i 0.0925976i
\(232\) 0 0
\(233\) −5.71879 −0.374650 −0.187325 0.982298i \(-0.559982\pi\)
−0.187325 + 0.982298i \(0.559982\pi\)
\(234\) 0 0
\(235\) 10.6764 0.696450
\(236\) 0 0
\(237\) 9.68233i 0.628935i
\(238\) 0 0
\(239\) 20.6125i 1.33331i 0.745365 + 0.666657i \(0.232275\pi\)
−0.745365 + 0.666657i \(0.767725\pi\)
\(240\) 0 0
\(241\) 23.5987i 1.52013i 0.649849 + 0.760063i \(0.274832\pi\)
−0.649849 + 0.760063i \(0.725168\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.93201i 0.187319i
\(246\) 0 0
\(247\) 1.50528 0.0957785
\(248\) 0 0
\(249\) 0.982199i 0.0622443i
\(250\) 0 0
\(251\) 1.45913 0.0920994 0.0460497 0.998939i \(-0.485337\pi\)
0.0460497 + 0.998939i \(0.485337\pi\)
\(252\) 0 0
\(253\) 2.95193 1.57625i 0.185587 0.0990981i
\(254\) 0 0
\(255\) 0.430024i 0.0269291i
\(256\) 0 0
\(257\) −20.3730 −1.27083 −0.635417 0.772169i \(-0.719172\pi\)
−0.635417 + 0.772169i \(0.719172\pi\)
\(258\) 0 0
\(259\) 9.87935i 0.613873i
\(260\) 0 0
\(261\) 9.64181 0.596813
\(262\) 0 0
\(263\) 6.84599 0.422142 0.211071 0.977471i \(-0.432305\pi\)
0.211071 + 0.977471i \(0.432305\pi\)
\(264\) 0 0
\(265\) −0.464177 −0.0285142
\(266\) 0 0
\(267\) 5.48101 0.335432
\(268\) 0 0
\(269\) −15.7765 −0.961912 −0.480956 0.876745i \(-0.659710\pi\)
−0.480956 + 0.876745i \(0.659710\pi\)
\(270\) 0 0
\(271\) 15.4121i 0.936216i −0.883671 0.468108i \(-0.844936\pi\)
0.883671 0.468108i \(-0.155064\pi\)
\(272\) 0 0
\(273\) 0.393776i 0.0238324i
\(274\) 0 0
\(275\) 0.697775 0.0420774
\(276\) 0 0
\(277\) 7.24637 0.435392 0.217696 0.976017i \(-0.430146\pi\)
0.217696 + 0.976017i \(0.430146\pi\)
\(278\) 0 0
\(279\) 1.05072i 0.0629047i
\(280\) 0 0
\(281\) 0.321739i 0.0191933i −0.999954 0.00959666i \(-0.996945\pi\)
0.999954 0.00959666i \(-0.00305476\pi\)
\(282\) 0 0
\(283\) 16.0317 0.952988 0.476494 0.879178i \(-0.341907\pi\)
0.476494 + 0.879178i \(0.341907\pi\)
\(284\) 0 0
\(285\) −7.71006 −0.456705
\(286\) 0 0
\(287\) 7.50172 0.442812
\(288\) 0 0
\(289\) 16.8151 0.989122
\(290\) 0 0
\(291\) −13.5553 −0.794628
\(292\) 0 0
\(293\) 6.96765i 0.407055i −0.979069 0.203527i \(-0.934759\pi\)
0.979069 0.203527i \(-0.0652406\pi\)
\(294\) 0 0
\(295\) −7.64910 −0.445348
\(296\) 0 0
\(297\) 0.697775i 0.0404890i
\(298\) 0 0
\(299\) −0.825942 + 0.441030i −0.0477655 + 0.0255054i
\(300\) 0 0
\(301\) −11.8125 −0.680862
\(302\) 0 0
\(303\) 9.87390i 0.567241i
\(304\) 0 0
\(305\) 4.18983 0.239909
\(306\) 0 0
\(307\) 23.8378i 1.36049i 0.732983 + 0.680246i \(0.238127\pi\)
−0.732983 + 0.680246i \(0.761873\pi\)
\(308\) 0 0
\(309\) 7.65129i 0.435266i
\(310\) 0 0
\(311\) 2.02594i 0.114880i 0.998349 + 0.0574401i \(0.0182938\pi\)
−0.998349 + 0.0574401i \(0.981706\pi\)
\(312\) 0 0
\(313\) 16.4162i 0.927900i 0.885861 + 0.463950i \(0.153568\pi\)
−0.885861 + 0.463950i \(0.846432\pi\)
\(314\) 0 0
\(315\) 2.01693i 0.113641i
\(316\) 0 0
\(317\) −21.2081 −1.19117 −0.595583 0.803294i \(-0.703079\pi\)
−0.595583 + 0.803294i \(0.703079\pi\)
\(318\) 0 0
\(319\) 6.72782 0.376686
\(320\) 0 0
\(321\) 15.7693i 0.880157i
\(322\) 0 0
\(323\) 3.31551i 0.184480i
\(324\) 0 0
\(325\) −0.195235 −0.0108297
\(326\) 0 0
\(327\) 7.29036 0.403158
\(328\) 0 0
\(329\) 21.5335i 1.18718i
\(330\) 0 0
\(331\) 2.15797i 0.118613i 0.998240 + 0.0593063i \(0.0188889\pi\)
−0.998240 + 0.0593063i \(0.981111\pi\)
\(332\) 0 0
\(333\) 4.89822i 0.268421i
\(334\) 0 0
\(335\) 3.70657i 0.202512i
\(336\) 0 0
\(337\) 3.24431i 0.176729i 0.996088 + 0.0883645i \(0.0281640\pi\)
−0.996088 + 0.0883645i \(0.971836\pi\)
\(338\) 0 0
\(339\) 11.6243 0.631346
\(340\) 0 0
\(341\) 0.733163i 0.0397030i
\(342\) 0 0
\(343\) 20.0321 1.08163
\(344\) 0 0
\(345\) 4.23049 2.25897i 0.227762 0.121619i
\(346\) 0 0
\(347\) 24.8303i 1.33296i −0.745524 0.666479i \(-0.767800\pi\)
0.745524 0.666479i \(-0.232200\pi\)
\(348\) 0 0
\(349\) 2.69841 0.144442 0.0722212 0.997389i \(-0.476991\pi\)
0.0722212 + 0.997389i \(0.476991\pi\)
\(350\) 0 0
\(351\) 0.195235i 0.0104209i
\(352\) 0 0
\(353\) 6.10284 0.324821 0.162411 0.986723i \(-0.448073\pi\)
0.162411 + 0.986723i \(0.448073\pi\)
\(354\) 0 0
\(355\) 5.69569 0.302296
\(356\) 0 0
\(357\) 0.867327 0.0459038
\(358\) 0 0
\(359\) 13.7445 0.725406 0.362703 0.931905i \(-0.381854\pi\)
0.362703 + 0.931905i \(0.381854\pi\)
\(360\) 0 0
\(361\) 40.4451 2.12869
\(362\) 0 0
\(363\) 10.5131i 0.551795i
\(364\) 0 0
\(365\) 3.51188i 0.183820i
\(366\) 0 0
\(367\) −2.83223 −0.147841 −0.0739206 0.997264i \(-0.523551\pi\)
−0.0739206 + 0.997264i \(0.523551\pi\)
\(368\) 0 0
\(369\) 3.71938 0.193623
\(370\) 0 0
\(371\) 0.936212i 0.0486057i
\(372\) 0 0
\(373\) 13.7736i 0.713168i −0.934263 0.356584i \(-0.883941\pi\)
0.934263 0.356584i \(-0.116059\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −1.88242 −0.0969498
\(378\) 0 0
\(379\) 12.0709 0.620040 0.310020 0.950730i \(-0.399664\pi\)
0.310020 + 0.950730i \(0.399664\pi\)
\(380\) 0 0
\(381\) −5.20841 −0.266835
\(382\) 0 0
\(383\) −10.1553 −0.518910 −0.259455 0.965755i \(-0.583543\pi\)
−0.259455 + 0.965755i \(0.583543\pi\)
\(384\) 0 0
\(385\) 1.40736i 0.0717258i
\(386\) 0 0
\(387\) −5.85669 −0.297712
\(388\) 0 0
\(389\) 12.2653i 0.621877i −0.950430 0.310939i \(-0.899357\pi\)
0.950430 0.310939i \(-0.100643\pi\)
\(390\) 0 0
\(391\) −0.971409 1.81921i −0.0491263 0.0920015i
\(392\) 0 0
\(393\) −2.04355 −0.103083
\(394\) 0 0
\(395\) 9.68233i 0.487171i
\(396\) 0 0
\(397\) −10.7383 −0.538942 −0.269471 0.963008i \(-0.586849\pi\)
−0.269471 + 0.963008i \(0.586849\pi\)
\(398\) 0 0
\(399\) 15.5506i 0.778506i
\(400\) 0 0
\(401\) 18.0035i 0.899049i −0.893268 0.449525i \(-0.851593\pi\)
0.893268 0.449525i \(-0.148407\pi\)
\(402\) 0 0
\(403\) 0.205137i 0.0102186i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 3.41786i 0.169417i
\(408\) 0 0
\(409\) −19.4255 −0.960527 −0.480264 0.877124i \(-0.659459\pi\)
−0.480264 + 0.877124i \(0.659459\pi\)
\(410\) 0 0
\(411\) 21.0004 1.03587
\(412\) 0 0
\(413\) 15.4277i 0.759147i
\(414\) 0 0
\(415\) 0.982199i 0.0482143i
\(416\) 0 0
\(417\) 2.56106 0.125416
\(418\) 0 0
\(419\) −19.8285 −0.968685 −0.484343 0.874878i \(-0.660941\pi\)
−0.484343 + 0.874878i \(0.660941\pi\)
\(420\) 0 0
\(421\) 14.9197i 0.727140i −0.931567 0.363570i \(-0.881558\pi\)
0.931567 0.363570i \(-0.118442\pi\)
\(422\) 0 0
\(423\) 10.6764i 0.519103i
\(424\) 0 0
\(425\) 0.430024i 0.0208592i
\(426\) 0 0
\(427\) 8.45058i 0.408952i
\(428\) 0 0
\(429\) 0.136231i 0.00657727i
\(430\) 0 0
\(431\) −37.3682 −1.79996 −0.899982 0.435926i \(-0.856421\pi\)
−0.899982 + 0.435926i \(0.856421\pi\)
\(432\) 0 0
\(433\) 4.24734i 0.204114i 0.994779 + 0.102057i \(0.0325425\pi\)
−0.994779 + 0.102057i \(0.967458\pi\)
\(434\) 0 0
\(435\) 9.64181 0.462290
\(436\) 0 0
\(437\) −32.6174 + 17.4168i −1.56030 + 0.833157i
\(438\) 0 0
\(439\) 36.0964i 1.72278i −0.507940 0.861392i \(-0.669593\pi\)
0.507940 0.861392i \(-0.330407\pi\)
\(440\) 0 0
\(441\) 2.93201 0.139619
\(442\) 0 0
\(443\) 8.68329i 0.412556i −0.978493 0.206278i \(-0.933865\pi\)
0.978493 0.206278i \(-0.0661350\pi\)
\(444\) 0 0
\(445\) 5.48101 0.259825
\(446\) 0 0
\(447\) 9.82277 0.464601
\(448\) 0 0
\(449\) 13.7648 0.649599 0.324799 0.945783i \(-0.394703\pi\)
0.324799 + 0.945783i \(0.394703\pi\)
\(450\) 0 0
\(451\) 2.59529 0.122208
\(452\) 0 0
\(453\) 14.8419 0.697334
\(454\) 0 0
\(455\) 0.393776i 0.0184605i
\(456\) 0 0
\(457\) 13.9155i 0.650939i 0.945553 + 0.325469i \(0.105522\pi\)
−0.945553 + 0.325469i \(0.894478\pi\)
\(458\) 0 0
\(459\) 0.430024 0.0200718
\(460\) 0 0
\(461\) −23.7166 −1.10459 −0.552296 0.833648i \(-0.686248\pi\)
−0.552296 + 0.833648i \(0.686248\pi\)
\(462\) 0 0
\(463\) 9.78160i 0.454589i 0.973826 + 0.227295i \(0.0729881\pi\)
−0.973826 + 0.227295i \(0.927012\pi\)
\(464\) 0 0
\(465\) 1.05072i 0.0487258i
\(466\) 0 0
\(467\) −37.7537 −1.74703 −0.873516 0.486795i \(-0.838166\pi\)
−0.873516 + 0.486795i \(0.838166\pi\)
\(468\) 0 0
\(469\) 7.47588 0.345204
\(470\) 0 0
\(471\) 1.12666 0.0519137
\(472\) 0 0
\(473\) −4.08665 −0.187904
\(474\) 0 0
\(475\) −7.71006 −0.353762
\(476\) 0 0
\(477\) 0.464177i 0.0212532i
\(478\) 0 0
\(479\) 13.8613 0.633337 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(480\) 0 0
\(481\) 0.956306i 0.0436038i
\(482\) 0 0
\(483\) 4.55617 + 8.53260i 0.207313 + 0.388247i
\(484\) 0 0
\(485\) −13.5553 −0.615516
\(486\) 0 0
\(487\) 20.7362i 0.939646i −0.882761 0.469823i \(-0.844318\pi\)
0.882761 0.469823i \(-0.155682\pi\)
\(488\) 0 0
\(489\) 17.1655 0.776252
\(490\) 0 0
\(491\) 32.8875i 1.48419i −0.670295 0.742095i \(-0.733832\pi\)
0.670295 0.742095i \(-0.266168\pi\)
\(492\) 0 0
\(493\) 4.14621i 0.186736i
\(494\) 0 0
\(495\) 0.697775i 0.0313627i
\(496\) 0 0
\(497\) 11.4878i 0.515298i
\(498\) 0 0
\(499\) 0.150831i 0.00675210i −0.999994 0.00337605i \(-0.998925\pi\)
0.999994 0.00337605i \(-0.00107463\pi\)
\(500\) 0 0
\(501\) −0.809244 −0.0361543
\(502\) 0 0
\(503\) 6.93423 0.309182 0.154591 0.987979i \(-0.450594\pi\)
0.154591 + 0.987979i \(0.450594\pi\)
\(504\) 0 0
\(505\) 9.87390i 0.439383i
\(506\) 0 0
\(507\) 12.9619i 0.575657i
\(508\) 0 0
\(509\) −10.6312 −0.471220 −0.235610 0.971848i \(-0.575709\pi\)
−0.235610 + 0.971848i \(0.575709\pi\)
\(510\) 0 0
\(511\) 7.08320 0.313343
\(512\) 0 0
\(513\) 7.71006i 0.340408i
\(514\) 0 0
\(515\) 7.65129i 0.337156i
\(516\) 0 0
\(517\) 7.44971i 0.327638i
\(518\) 0 0
\(519\) 0.192946i 0.00846941i
\(520\) 0 0
\(521\) 12.6607i 0.554673i 0.960773 + 0.277337i \(0.0894517\pi\)
−0.960773 + 0.277337i \(0.910548\pi\)
\(522\) 0 0
\(523\) 35.9210 1.57072 0.785358 0.619042i \(-0.212479\pi\)
0.785358 + 0.619042i \(0.212479\pi\)
\(524\) 0 0
\(525\) 2.01693i 0.0880259i
\(526\) 0 0
\(527\) 0.451833 0.0196821
\(528\) 0 0
\(529\) 12.7941 19.1131i 0.556267 0.831004i
\(530\) 0 0
\(531\) 7.64910i 0.331943i
\(532\) 0 0
\(533\) −0.726155 −0.0314533
\(534\) 0 0
\(535\) 15.7693i 0.681767i
\(536\) 0 0
\(537\) 23.5203 1.01498
\(538\) 0 0
\(539\) 2.04588 0.0881224
\(540\) 0 0
\(541\) −15.5417 −0.668191 −0.334095 0.942539i \(-0.608431\pi\)
−0.334095 + 0.942539i \(0.608431\pi\)
\(542\) 0 0
\(543\) 9.77478 0.419476
\(544\) 0 0
\(545\) 7.29036 0.312285
\(546\) 0 0
\(547\) 12.4468i 0.532187i 0.963947 + 0.266093i \(0.0857330\pi\)
−0.963947 + 0.266093i \(0.914267\pi\)
\(548\) 0 0
\(549\) 4.18983i 0.178817i
\(550\) 0 0
\(551\) −74.3390 −3.16695
\(552\) 0 0
\(553\) 19.5286 0.830438
\(554\) 0 0
\(555\) 4.89822i 0.207918i
\(556\) 0 0
\(557\) 24.8342i 1.05226i 0.850404 + 0.526130i \(0.176357\pi\)
−0.850404 + 0.526130i \(0.823643\pi\)
\(558\) 0 0
\(559\) 1.14343 0.0483621
\(560\) 0 0
\(561\) 0.300060 0.0126685
\(562\) 0 0
\(563\) 41.0729 1.73101 0.865507 0.500896i \(-0.166996\pi\)
0.865507 + 0.500896i \(0.166996\pi\)
\(564\) 0 0
\(565\) 11.6243 0.489038
\(566\) 0 0
\(567\) −2.01693 −0.0847030
\(568\) 0 0
\(569\) 32.8724i 1.37808i 0.724722 + 0.689041i \(0.241968\pi\)
−0.724722 + 0.689041i \(0.758032\pi\)
\(570\) 0 0
\(571\) −38.5411 −1.61290 −0.806448 0.591305i \(-0.798613\pi\)
−0.806448 + 0.591305i \(0.798613\pi\)
\(572\) 0 0
\(573\) 7.21224i 0.301296i
\(574\) 0 0
\(575\) 4.23049 2.25897i 0.176424 0.0942054i
\(576\) 0 0
\(577\) 33.2323 1.38348 0.691740 0.722147i \(-0.256844\pi\)
0.691740 + 0.722147i \(0.256844\pi\)
\(578\) 0 0
\(579\) 24.9860i 1.03838i
\(580\) 0 0
\(581\) −1.98102 −0.0821867
\(582\) 0 0
\(583\) 0.323892i 0.0134142i
\(584\) 0 0
\(585\) 0.195235i 0.00807199i
\(586\) 0 0
\(587\) 10.1033i 0.417008i 0.978021 + 0.208504i \(0.0668595\pi\)
−0.978021 + 0.208504i \(0.933141\pi\)
\(588\) 0 0
\(589\) 8.10108i 0.333799i
\(590\) 0 0
\(591\) 6.96165i 0.286364i
\(592\) 0 0
\(593\) −0.289921 −0.0119056 −0.00595281 0.999982i \(-0.501895\pi\)
−0.00595281 + 0.999982i \(0.501895\pi\)
\(594\) 0 0
\(595\) 0.867327 0.0355569
\(596\) 0 0
\(597\) 2.94085i 0.120361i
\(598\) 0 0
\(599\) 37.0346i 1.51319i 0.653883 + 0.756596i \(0.273139\pi\)
−0.653883 + 0.756596i \(0.726861\pi\)
\(600\) 0 0
\(601\) −38.1805 −1.55742 −0.778708 0.627387i \(-0.784125\pi\)
−0.778708 + 0.627387i \(0.784125\pi\)
\(602\) 0 0
\(603\) 3.70657 0.150943
\(604\) 0 0
\(605\) 10.5131i 0.427419i
\(606\) 0 0
\(607\) 39.4410i 1.60086i −0.599426 0.800430i \(-0.704605\pi\)
0.599426 0.800430i \(-0.295395\pi\)
\(608\) 0 0
\(609\) 19.4468i 0.788025i
\(610\) 0 0
\(611\) 2.08441i 0.0843261i
\(612\) 0 0
\(613\) 12.2889i 0.496345i 0.968716 + 0.248173i \(0.0798300\pi\)
−0.968716 + 0.248173i \(0.920170\pi\)
\(614\) 0 0
\(615\) 3.71938 0.149980
\(616\) 0 0
\(617\) 10.2034i 0.410773i −0.978681 0.205386i \(-0.934155\pi\)
0.978681 0.205386i \(-0.0658451\pi\)
\(618\) 0 0
\(619\) −31.5655 −1.26873 −0.634363 0.773036i \(-0.718737\pi\)
−0.634363 + 0.773036i \(0.718737\pi\)
\(620\) 0 0
\(621\) 2.25897 + 4.23049i 0.0906492 + 0.169764i
\(622\) 0 0
\(623\) 11.0548i 0.442901i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.37989i 0.214852i
\(628\) 0 0
\(629\) −2.10635 −0.0839857
\(630\) 0 0
\(631\) 18.4861 0.735919 0.367960 0.929842i \(-0.380057\pi\)
0.367960 + 0.929842i \(0.380057\pi\)
\(632\) 0 0
\(633\) 28.0259 1.11393
\(634\) 0 0
\(635\) −5.20841 −0.206690
\(636\) 0 0
\(637\) −0.572431 −0.0226806
\(638\) 0 0
\(639\) 5.69569i 0.225318i
\(640\) 0 0
\(641\) 31.0234i 1.22535i −0.790335 0.612674i \(-0.790094\pi\)
0.790335 0.612674i \(-0.209906\pi\)
\(642\) 0 0
\(643\) 4.62461 0.182377 0.0911884 0.995834i \(-0.470933\pi\)
0.0911884 + 0.995834i \(0.470933\pi\)
\(644\) 0 0
\(645\) −5.85669 −0.230607
\(646\) 0 0
\(647\) 20.6906i 0.813433i 0.913554 + 0.406716i \(0.133326\pi\)
−0.913554 + 0.406716i \(0.866674\pi\)
\(648\) 0 0
\(649\) 5.33736i 0.209510i
\(650\) 0 0
\(651\) −2.11922 −0.0830586
\(652\) 0 0
\(653\) −44.7001 −1.74925 −0.874626 0.484799i \(-0.838893\pi\)
−0.874626 + 0.484799i \(0.838893\pi\)
\(654\) 0 0
\(655\) −2.04355 −0.0798480
\(656\) 0 0
\(657\) 3.51188 0.137011
\(658\) 0 0
\(659\) −4.80658 −0.187238 −0.0936190 0.995608i \(-0.529844\pi\)
−0.0936190 + 0.995608i \(0.529844\pi\)
\(660\) 0 0
\(661\) 37.6532i 1.46454i 0.681014 + 0.732270i \(0.261539\pi\)
−0.681014 + 0.732270i \(0.738461\pi\)
\(662\) 0 0
\(663\) −0.0839559 −0.00326058
\(664\) 0 0
\(665\) 15.5506i 0.603028i
\(666\) 0 0
\(667\) 40.7896 21.7805i 1.57938 0.843346i
\(668\) 0 0
\(669\) −1.81830 −0.0702997
\(670\) 0 0
\(671\) 2.92356i 0.112863i
\(672\) 0 0
\(673\) −29.9838 −1.15579 −0.577895 0.816111i \(-0.696126\pi\)
−0.577895 + 0.816111i \(0.696126\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 26.9203i 1.03463i 0.855794 + 0.517316i \(0.173069\pi\)
−0.855794 + 0.517316i \(0.826931\pi\)
\(678\) 0 0
\(679\) 27.3401i 1.04922i
\(680\) 0 0
\(681\) 15.7824i 0.604783i
\(682\) 0 0
\(683\) 34.8703i 1.33427i 0.744935 + 0.667137i \(0.232480\pi\)
−0.744935 + 0.667137i \(0.767520\pi\)
\(684\) 0 0
\(685\) 21.0004 0.802385
\(686\) 0 0
\(687\) −7.05803 −0.269281
\(688\) 0 0
\(689\) 0.0906239i 0.00345250i
\(690\) 0 0
\(691\) 5.22615i 0.198812i 0.995047 + 0.0994061i \(0.0316943\pi\)
−0.995047 + 0.0994061i \(0.968306\pi\)
\(692\) 0 0
\(693\) −1.40736 −0.0534613
\(694\) 0 0
\(695\) 2.56106 0.0971467
\(696\) 0 0
\(697\) 1.59942i 0.0605824i
\(698\) 0 0
\(699\) 5.71879i 0.216305i
\(700\) 0 0
\(701\) 15.5942i 0.588985i −0.955654 0.294493i \(-0.904849\pi\)
0.955654 0.294493i \(-0.0951507\pi\)
\(702\) 0 0
\(703\) 37.7656i 1.42436i
\(704\) 0 0
\(705\) 10.6764i 0.402096i
\(706\) 0 0
\(707\) 19.9149 0.748978
\(708\) 0 0
\(709\) 31.2212i 1.17254i −0.810117 0.586268i \(-0.800596\pi\)
0.810117 0.586268i \(-0.199404\pi\)
\(710\) 0 0
\(711\) 9.68233 0.363116
\(712\) 0 0
\(713\) 2.37353 + 4.44504i 0.0888894 + 0.166468i
\(714\) 0 0
\(715\) 0.136231i 0.00509473i
\(716\) 0 0
\(717\) −20.6125 −0.769789
\(718\) 0 0
\(719\) 2.74815i 0.102489i −0.998686 0.0512443i \(-0.983681\pi\)
0.998686 0.0512443i \(-0.0163187\pi\)
\(720\) 0 0
\(721\) −15.4321 −0.574721
\(722\) 0 0
\(723\) −23.5987 −0.877646
\(724\) 0 0
\(725\) 9.64181 0.358088
\(726\) 0 0
\(727\) −28.4656 −1.05573 −0.527865 0.849328i \(-0.677007\pi\)
−0.527865 + 0.849328i \(0.677007\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 2.51852i 0.0931506i
\(732\) 0 0
\(733\) 31.8684i 1.17709i −0.808466 0.588543i \(-0.799702\pi\)
0.808466 0.588543i \(-0.200298\pi\)
\(734\) 0 0
\(735\) 2.93201 0.108149
\(736\) 0 0
\(737\) 2.58635 0.0952695
\(738\) 0 0
\(739\) 19.4238i 0.714515i −0.934006 0.357258i \(-0.883712\pi\)
0.934006 0.357258i \(-0.116288\pi\)
\(740\) 0 0
\(741\) 1.50528i 0.0552978i
\(742\) 0 0
\(743\) −18.2199 −0.668425 −0.334212 0.942498i \(-0.608470\pi\)
−0.334212 + 0.942498i \(0.608470\pi\)
\(744\) 0 0
\(745\) 9.82277 0.359878
\(746\) 0 0
\(747\) −0.982199 −0.0359368
\(748\) 0 0
\(749\) −31.8055 −1.16215
\(750\) 0 0
\(751\) −50.1935 −1.83159 −0.915794 0.401649i \(-0.868437\pi\)
−0.915794 + 0.401649i \(0.868437\pi\)
\(752\) 0 0
\(753\) 1.45913i 0.0531736i
\(754\) 0 0
\(755\) 14.8419 0.540152
\(756\) 0 0
\(757\) 24.7967i 0.901250i 0.892713 + 0.450625i \(0.148799\pi\)
−0.892713 + 0.450625i \(0.851201\pi\)
\(758\) 0 0
\(759\) 1.57625 + 2.95193i 0.0572143 + 0.107148i
\(760\) 0 0
\(761\) −30.6359 −1.11055 −0.555275 0.831667i \(-0.687387\pi\)
−0.555275 + 0.831667i \(0.687387\pi\)
\(762\) 0 0
\(763\) 14.7041i 0.532325i
\(764\) 0 0
\(765\) 0.430024 0.0155475
\(766\) 0 0
\(767\) 1.49338i 0.0539227i
\(768\) 0 0
\(769\) 35.2375i 1.27070i 0.772226 + 0.635348i \(0.219143\pi\)
−0.772226 + 0.635348i \(0.780857\pi\)
\(770\) 0 0
\(771\) 20.3730i 0.733716i
\(772\) 0 0
\(773\) 51.1400i 1.83938i −0.392650 0.919688i \(-0.628441\pi\)
0.392650 0.919688i \(-0.371559\pi\)
\(774\) 0 0
\(775\) 1.05072i 0.0377428i
\(776\) 0 0
\(777\) 9.87935 0.354420
\(778\) 0 0
\(779\) −28.6767 −1.02745
\(780\) 0 0
\(781\) 3.97431i 0.142212i
\(782\) 0 0
\(783\) 9.64181i 0.344570i
\(784\) 0 0
\(785\) 1.12666 0.0402122
\(786\) 0 0
\(787\) −36.8424 −1.31329 −0.656645 0.754200i \(-0.728025\pi\)
−0.656645 + 0.754200i \(0.728025\pi\)
\(788\) 0 0
\(789\) 6.84599i 0.243724i
\(790\) 0 0
\(791\) 23.4454i 0.833622i
\(792\) 0 0
\(793\) 0.818003i 0.0290481i
\(794\) 0 0
\(795\) 0.464177i 0.0164627i
\(796\) 0 0
\(797\) 24.8874i 0.881556i −0.897616 0.440778i \(-0.854703\pi\)
0.897616 0.440778i \(-0.145297\pi\)
\(798\) 0 0
\(799\) 4.59110 0.162421
\(800\) 0 0
\(801\) 5.48101i 0.193662i
\(802\) 0 0
\(803\) 2.45050 0.0864764
\(804\) 0 0
\(805\) 4.55617 + 8.53260i 0.160584 + 0.300734i
\(806\) 0