Properties

Label 5445.2.a.x.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -3.23607 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -3.23607 q^{7} +2.23607 q^{8} -0.618034 q^{10} -5.23607 q^{13} +2.00000 q^{14} +1.85410 q^{16} -5.47214 q^{17} -6.47214 q^{19} -1.61803 q^{20} +4.70820 q^{23} +1.00000 q^{25} +3.23607 q^{26} +5.23607 q^{28} -1.23607 q^{29} -6.70820 q^{31} -5.61803 q^{32} +3.38197 q^{34} -3.23607 q^{35} +0.763932 q^{37} +4.00000 q^{38} +2.23607 q^{40} +3.52786 q^{41} +5.23607 q^{43} -2.90983 q^{46} -8.70820 q^{47} +3.47214 q^{49} -0.618034 q^{50} +8.47214 q^{52} -9.94427 q^{53} -7.23607 q^{56} +0.763932 q^{58} -11.7082 q^{59} +1.47214 q^{61} +4.14590 q^{62} -0.236068 q^{64} -5.23607 q^{65} +11.2361 q^{67} +8.85410 q^{68} +2.00000 q^{70} +14.4721 q^{71} +10.4721 q^{73} -0.472136 q^{74} +10.4721 q^{76} -12.7082 q^{79} +1.85410 q^{80} -2.18034 q^{82} -4.00000 q^{83} -5.47214 q^{85} -3.23607 q^{86} -4.76393 q^{89} +16.9443 q^{91} -7.61803 q^{92} +5.38197 q^{94} -6.47214 q^{95} +12.7639 q^{97} -2.14590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 2q^{5} - 2q^{7} + q^{10} - 6q^{13} + 4q^{14} - 3q^{16} - 2q^{17} - 4q^{19} - q^{20} - 4q^{23} + 2q^{25} + 2q^{26} + 6q^{28} + 2q^{29} - 9q^{32} + 9q^{34} - 2q^{35} + 6q^{37} + 8q^{38} + 16q^{41} + 6q^{43} - 17q^{46} - 4q^{47} - 2q^{49} + q^{50} + 8q^{52} - 2q^{53} - 10q^{56} + 6q^{58} - 10q^{59} - 6q^{61} + 15q^{62} + 4q^{64} - 6q^{65} + 18q^{67} + 11q^{68} + 4q^{70} + 20q^{71} + 12q^{73} + 8q^{74} + 12q^{76} - 12q^{79} - 3q^{80} + 18q^{82} - 8q^{83} - 2q^{85} - 2q^{86} - 14q^{89} + 16q^{91} - 13q^{92} + 13q^{94} - 4q^{95} + 30q^{97} - 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) −0.618034 −0.195440
\(11\) 0 0
\(12\) 0 0
\(13\) −5.23607 −1.45222 −0.726112 0.687576i \(-0.758675\pi\)
−0.726112 + 0.687576i \(0.758675\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −5.47214 −1.32719 −0.663594 0.748093i \(-0.730970\pi\)
−0.663594 + 0.748093i \(0.730970\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) −1.61803 −0.361803
\(21\) 0 0
\(22\) 0 0
\(23\) 4.70820 0.981728 0.490864 0.871236i \(-0.336681\pi\)
0.490864 + 0.871236i \(0.336681\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.23607 0.634645
\(27\) 0 0
\(28\) 5.23607 0.989524
\(29\) −1.23607 −0.229532 −0.114766 0.993393i \(-0.536612\pi\)
−0.114766 + 0.993393i \(0.536612\pi\)
\(30\) 0 0
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) 3.38197 0.580002
\(35\) −3.23607 −0.546995
\(36\) 0 0
\(37\) 0.763932 0.125590 0.0627948 0.998026i \(-0.479999\pi\)
0.0627948 + 0.998026i \(0.479999\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.23607 0.353553
\(41\) 3.52786 0.550960 0.275480 0.961307i \(-0.411163\pi\)
0.275480 + 0.961307i \(0.411163\pi\)
\(42\) 0 0
\(43\) 5.23607 0.798493 0.399246 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.90983 −0.429031
\(47\) −8.70820 −1.27022 −0.635111 0.772421i \(-0.719046\pi\)
−0.635111 + 0.772421i \(0.719046\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) −0.618034 −0.0874032
\(51\) 0 0
\(52\) 8.47214 1.17487
\(53\) −9.94427 −1.36595 −0.682975 0.730441i \(-0.739314\pi\)
−0.682975 + 0.730441i \(0.739314\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.23607 −0.966960
\(57\) 0 0
\(58\) 0.763932 0.100309
\(59\) −11.7082 −1.52428 −0.762139 0.647413i \(-0.775851\pi\)
−0.762139 + 0.647413i \(0.775851\pi\)
\(60\) 0 0
\(61\) 1.47214 0.188488 0.0942438 0.995549i \(-0.469957\pi\)
0.0942438 + 0.995549i \(0.469957\pi\)
\(62\) 4.14590 0.526530
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) −5.23607 −0.649454
\(66\) 0 0
\(67\) 11.2361 1.37270 0.686352 0.727269i \(-0.259211\pi\)
0.686352 + 0.727269i \(0.259211\pi\)
\(68\) 8.85410 1.07372
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 14.4721 1.71753 0.858763 0.512373i \(-0.171233\pi\)
0.858763 + 0.512373i \(0.171233\pi\)
\(72\) 0 0
\(73\) 10.4721 1.22567 0.612835 0.790211i \(-0.290029\pi\)
0.612835 + 0.790211i \(0.290029\pi\)
\(74\) −0.472136 −0.0548847
\(75\) 0 0
\(76\) 10.4721 1.20124
\(77\) 0 0
\(78\) 0 0
\(79\) −12.7082 −1.42978 −0.714892 0.699235i \(-0.753524\pi\)
−0.714892 + 0.699235i \(0.753524\pi\)
\(80\) 1.85410 0.207295
\(81\) 0 0
\(82\) −2.18034 −0.240778
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −5.47214 −0.593536
\(86\) −3.23607 −0.348954
\(87\) 0 0
\(88\) 0 0
\(89\) −4.76393 −0.504976 −0.252488 0.967600i \(-0.581249\pi\)
−0.252488 + 0.967600i \(0.581249\pi\)
\(90\) 0 0
\(91\) 16.9443 1.77624
\(92\) −7.61803 −0.794235
\(93\) 0 0
\(94\) 5.38197 0.555107
\(95\) −6.47214 −0.664027
\(96\) 0 0
\(97\) 12.7639 1.29598 0.647990 0.761648i \(-0.275610\pi\)
0.647990 + 0.761648i \(0.275610\pi\)
\(98\) −2.14590 −0.216768
\(99\) 0 0
\(100\) −1.61803 −0.161803
\(101\) −5.52786 −0.550043 −0.275022 0.961438i \(-0.588685\pi\)
−0.275022 + 0.961438i \(0.588685\pi\)
\(102\) 0 0
\(103\) −0.944272 −0.0930419 −0.0465209 0.998917i \(-0.514813\pi\)
−0.0465209 + 0.998917i \(0.514813\pi\)
\(104\) −11.7082 −1.14808
\(105\) 0 0
\(106\) 6.14590 0.596942
\(107\) 14.2361 1.37625 0.688126 0.725591i \(-0.258433\pi\)
0.688126 + 0.725591i \(0.258433\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −6.00000 −0.566947
\(113\) 12.4164 1.16804 0.584019 0.811740i \(-0.301479\pi\)
0.584019 + 0.811740i \(0.301479\pi\)
\(114\) 0 0
\(115\) 4.70820 0.439042
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 7.23607 0.666134
\(119\) 17.7082 1.62331
\(120\) 0 0
\(121\) 0 0
\(122\) −0.909830 −0.0823721
\(123\) 0 0
\(124\) 10.8541 0.974727
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −3.70820 −0.329050 −0.164525 0.986373i \(-0.552609\pi\)
−0.164525 + 0.986373i \(0.552609\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) 3.23607 0.283822
\(131\) −0.472136 −0.0412507 −0.0206254 0.999787i \(-0.506566\pi\)
−0.0206254 + 0.999787i \(0.506566\pi\)
\(132\) 0 0
\(133\) 20.9443 1.81610
\(134\) −6.94427 −0.599894
\(135\) 0 0
\(136\) −12.2361 −1.04923
\(137\) −17.4721 −1.49275 −0.746373 0.665528i \(-0.768206\pi\)
−0.746373 + 0.665528i \(0.768206\pi\)
\(138\) 0 0
\(139\) 7.29180 0.618482 0.309241 0.950984i \(-0.399925\pi\)
0.309241 + 0.950984i \(0.399925\pi\)
\(140\) 5.23607 0.442529
\(141\) 0 0
\(142\) −8.94427 −0.750587
\(143\) 0 0
\(144\) 0 0
\(145\) −1.23607 −0.102650
\(146\) −6.47214 −0.535638
\(147\) 0 0
\(148\) −1.23607 −0.101604
\(149\) −4.29180 −0.351598 −0.175799 0.984426i \(-0.556251\pi\)
−0.175799 + 0.984426i \(0.556251\pi\)
\(150\) 0 0
\(151\) −7.29180 −0.593398 −0.296699 0.954971i \(-0.595886\pi\)
−0.296699 + 0.954971i \(0.595886\pi\)
\(152\) −14.4721 −1.17385
\(153\) 0 0
\(154\) 0 0
\(155\) −6.70820 −0.538816
\(156\) 0 0
\(157\) 2.29180 0.182905 0.0914526 0.995809i \(-0.470849\pi\)
0.0914526 + 0.995809i \(0.470849\pi\)
\(158\) 7.85410 0.624839
\(159\) 0 0
\(160\) −5.61803 −0.444145
\(161\) −15.2361 −1.20077
\(162\) 0 0
\(163\) 6.18034 0.484082 0.242041 0.970266i \(-0.422183\pi\)
0.242041 + 0.970266i \(0.422183\pi\)
\(164\) −5.70820 −0.445736
\(165\) 0 0
\(166\) 2.47214 0.191875
\(167\) 3.18034 0.246102 0.123051 0.992400i \(-0.460732\pi\)
0.123051 + 0.992400i \(0.460732\pi\)
\(168\) 0 0
\(169\) 14.4164 1.10895
\(170\) 3.38197 0.259385
\(171\) 0 0
\(172\) −8.47214 −0.645994
\(173\) 2.94427 0.223849 0.111924 0.993717i \(-0.464299\pi\)
0.111924 + 0.993717i \(0.464299\pi\)
\(174\) 0 0
\(175\) −3.23607 −0.244624
\(176\) 0 0
\(177\) 0 0
\(178\) 2.94427 0.220683
\(179\) 20.1803 1.50835 0.754175 0.656674i \(-0.228037\pi\)
0.754175 + 0.656674i \(0.228037\pi\)
\(180\) 0 0
\(181\) −21.4164 −1.59187 −0.795935 0.605383i \(-0.793020\pi\)
−0.795935 + 0.605383i \(0.793020\pi\)
\(182\) −10.4721 −0.776246
\(183\) 0 0
\(184\) 10.5279 0.776124
\(185\) 0.763932 0.0561654
\(186\) 0 0
\(187\) 0 0
\(188\) 14.0902 1.02763
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −27.5967 −1.99683 −0.998415 0.0562752i \(-0.982078\pi\)
−0.998415 + 0.0562752i \(0.982078\pi\)
\(192\) 0 0
\(193\) −16.1803 −1.16469 −0.582343 0.812943i \(-0.697864\pi\)
−0.582343 + 0.812943i \(0.697864\pi\)
\(194\) −7.88854 −0.566364
\(195\) 0 0
\(196\) −5.61803 −0.401288
\(197\) −1.41641 −0.100915 −0.0504574 0.998726i \(-0.516068\pi\)
−0.0504574 + 0.998726i \(0.516068\pi\)
\(198\) 0 0
\(199\) 16.2361 1.15094 0.575472 0.817821i \(-0.304818\pi\)
0.575472 + 0.817821i \(0.304818\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 3.41641 0.240378
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 3.52786 0.246397
\(206\) 0.583592 0.0406608
\(207\) 0 0
\(208\) −9.70820 −0.673143
\(209\) 0 0
\(210\) 0 0
\(211\) −1.76393 −0.121434 −0.0607170 0.998155i \(-0.519339\pi\)
−0.0607170 + 0.998155i \(0.519339\pi\)
\(212\) 16.0902 1.10508
\(213\) 0 0
\(214\) −8.79837 −0.601444
\(215\) 5.23607 0.357097
\(216\) 0 0
\(217\) 21.7082 1.47365
\(218\) 9.23607 0.625545
\(219\) 0 0
\(220\) 0 0
\(221\) 28.6525 1.92737
\(222\) 0 0
\(223\) 9.05573 0.606416 0.303208 0.952924i \(-0.401942\pi\)
0.303208 + 0.952924i \(0.401942\pi\)
\(224\) 18.1803 1.21473
\(225\) 0 0
\(226\) −7.67376 −0.510451
\(227\) 16.7082 1.10896 0.554481 0.832196i \(-0.312917\pi\)
0.554481 + 0.832196i \(0.312917\pi\)
\(228\) 0 0
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) −2.90983 −0.191869
\(231\) 0 0
\(232\) −2.76393 −0.181461
\(233\) 10.8885 0.713332 0.356666 0.934232i \(-0.383913\pi\)
0.356666 + 0.934232i \(0.383913\pi\)
\(234\) 0 0
\(235\) −8.70820 −0.568061
\(236\) 18.9443 1.23317
\(237\) 0 0
\(238\) −10.9443 −0.709412
\(239\) −24.6525 −1.59464 −0.797318 0.603559i \(-0.793749\pi\)
−0.797318 + 0.603559i \(0.793749\pi\)
\(240\) 0 0
\(241\) 17.9443 1.15589 0.577946 0.816075i \(-0.303854\pi\)
0.577946 + 0.816075i \(0.303854\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.38197 −0.152490
\(245\) 3.47214 0.221827
\(246\) 0 0
\(247\) 33.8885 2.15628
\(248\) −15.0000 −0.952501
\(249\) 0 0
\(250\) −0.618034 −0.0390879
\(251\) 23.7082 1.49645 0.748224 0.663446i \(-0.230907\pi\)
0.748224 + 0.663446i \(0.230907\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.29180 0.143800
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −28.8885 −1.80202 −0.901009 0.433801i \(-0.857172\pi\)
−0.901009 + 0.433801i \(0.857172\pi\)
\(258\) 0 0
\(259\) −2.47214 −0.153611
\(260\) 8.47214 0.525420
\(261\) 0 0
\(262\) 0.291796 0.0180272
\(263\) 20.1246 1.24094 0.620468 0.784231i \(-0.286943\pi\)
0.620468 + 0.784231i \(0.286943\pi\)
\(264\) 0 0
\(265\) −9.94427 −0.610872
\(266\) −12.9443 −0.793664
\(267\) 0 0
\(268\) −18.1803 −1.11054
\(269\) −7.05573 −0.430195 −0.215098 0.976593i \(-0.569007\pi\)
−0.215098 + 0.976593i \(0.569007\pi\)
\(270\) 0 0
\(271\) 24.2361 1.47224 0.736118 0.676853i \(-0.236657\pi\)
0.736118 + 0.676853i \(0.236657\pi\)
\(272\) −10.1459 −0.615185
\(273\) 0 0
\(274\) 10.7984 0.652354
\(275\) 0 0
\(276\) 0 0
\(277\) 4.94427 0.297073 0.148536 0.988907i \(-0.452544\pi\)
0.148536 + 0.988907i \(0.452544\pi\)
\(278\) −4.50658 −0.270287
\(279\) 0 0
\(280\) −7.23607 −0.432438
\(281\) 15.2361 0.908908 0.454454 0.890770i \(-0.349834\pi\)
0.454454 + 0.890770i \(0.349834\pi\)
\(282\) 0 0
\(283\) −11.8885 −0.706701 −0.353350 0.935491i \(-0.614958\pi\)
−0.353350 + 0.935491i \(0.614958\pi\)
\(284\) −23.4164 −1.38951
\(285\) 0 0
\(286\) 0 0
\(287\) −11.4164 −0.673889
\(288\) 0 0
\(289\) 12.9443 0.761428
\(290\) 0.763932 0.0448596
\(291\) 0 0
\(292\) −16.9443 −0.991589
\(293\) 21.4721 1.25442 0.627208 0.778852i \(-0.284198\pi\)
0.627208 + 0.778852i \(0.284198\pi\)
\(294\) 0 0
\(295\) −11.7082 −0.681678
\(296\) 1.70820 0.0992873
\(297\) 0 0
\(298\) 2.65248 0.153654
\(299\) −24.6525 −1.42569
\(300\) 0 0
\(301\) −16.9443 −0.976652
\(302\) 4.50658 0.259324
\(303\) 0 0
\(304\) −12.0000 −0.688247
\(305\) 1.47214 0.0842943
\(306\) 0 0
\(307\) 11.8885 0.678515 0.339258 0.940694i \(-0.389824\pi\)
0.339258 + 0.940694i \(0.389824\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.14590 0.235471
\(311\) 3.23607 0.183501 0.0917503 0.995782i \(-0.470754\pi\)
0.0917503 + 0.995782i \(0.470754\pi\)
\(312\) 0 0
\(313\) 22.7639 1.28669 0.643347 0.765575i \(-0.277545\pi\)
0.643347 + 0.765575i \(0.277545\pi\)
\(314\) −1.41641 −0.0799325
\(315\) 0 0
\(316\) 20.5623 1.15672
\(317\) 3.94427 0.221532 0.110766 0.993846i \(-0.464670\pi\)
0.110766 + 0.993846i \(0.464670\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.236068 −0.0131966
\(321\) 0 0
\(322\) 9.41641 0.524756
\(323\) 35.4164 1.97062
\(324\) 0 0
\(325\) −5.23607 −0.290445
\(326\) −3.81966 −0.211551
\(327\) 0 0
\(328\) 7.88854 0.435572
\(329\) 28.1803 1.55363
\(330\) 0 0
\(331\) −17.1803 −0.944317 −0.472158 0.881514i \(-0.656525\pi\)
−0.472158 + 0.881514i \(0.656525\pi\)
\(332\) 6.47214 0.355205
\(333\) 0 0
\(334\) −1.96556 −0.107551
\(335\) 11.2361 0.613892
\(336\) 0 0
\(337\) 7.88854 0.429716 0.214858 0.976645i \(-0.431071\pi\)
0.214858 + 0.976645i \(0.431071\pi\)
\(338\) −8.90983 −0.484631
\(339\) 0 0
\(340\) 8.85410 0.480181
\(341\) 0 0
\(342\) 0 0
\(343\) 11.4164 0.616428
\(344\) 11.7082 0.631264
\(345\) 0 0
\(346\) −1.81966 −0.0978255
\(347\) −14.2361 −0.764232 −0.382116 0.924114i \(-0.624805\pi\)
−0.382116 + 0.924114i \(0.624805\pi\)
\(348\) 0 0
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) 7.00000 0.372572 0.186286 0.982496i \(-0.440355\pi\)
0.186286 + 0.982496i \(0.440355\pi\)
\(354\) 0 0
\(355\) 14.4721 0.768101
\(356\) 7.70820 0.408534
\(357\) 0 0
\(358\) −12.4721 −0.659173
\(359\) 7.41641 0.391423 0.195712 0.980662i \(-0.437298\pi\)
0.195712 + 0.980662i \(0.437298\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 13.2361 0.695672
\(363\) 0 0
\(364\) −27.4164 −1.43701
\(365\) 10.4721 0.548137
\(366\) 0 0
\(367\) 4.76393 0.248675 0.124338 0.992240i \(-0.460319\pi\)
0.124338 + 0.992240i \(0.460319\pi\)
\(368\) 8.72949 0.455056
\(369\) 0 0
\(370\) −0.472136 −0.0245452
\(371\) 32.1803 1.67072
\(372\) 0 0
\(373\) −29.8885 −1.54757 −0.773785 0.633448i \(-0.781639\pi\)
−0.773785 + 0.633448i \(0.781639\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −19.4721 −1.00420
\(377\) 6.47214 0.333332
\(378\) 0 0
\(379\) 30.5967 1.57165 0.785825 0.618449i \(-0.212239\pi\)
0.785825 + 0.618449i \(0.212239\pi\)
\(380\) 10.4721 0.537209
\(381\) 0 0
\(382\) 17.0557 0.872647
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) −20.6525 −1.04847
\(389\) 21.5967 1.09500 0.547499 0.836806i \(-0.315580\pi\)
0.547499 + 0.836806i \(0.315580\pi\)
\(390\) 0 0
\(391\) −25.7639 −1.30294
\(392\) 7.76393 0.392138
\(393\) 0 0
\(394\) 0.875388 0.0441014
\(395\) −12.7082 −0.639419
\(396\) 0 0
\(397\) −17.4164 −0.874104 −0.437052 0.899436i \(-0.643978\pi\)
−0.437052 + 0.899436i \(0.643978\pi\)
\(398\) −10.0344 −0.502981
\(399\) 0 0
\(400\) 1.85410 0.0927051
\(401\) −20.9443 −1.04591 −0.522954 0.852361i \(-0.675170\pi\)
−0.522954 + 0.852361i \(0.675170\pi\)
\(402\) 0 0
\(403\) 35.1246 1.74968
\(404\) 8.94427 0.444994
\(405\) 0 0
\(406\) −2.47214 −0.122690
\(407\) 0 0
\(408\) 0 0
\(409\) −36.7771 −1.81851 −0.909255 0.416240i \(-0.863348\pi\)
−0.909255 + 0.416240i \(0.863348\pi\)
\(410\) −2.18034 −0.107679
\(411\) 0 0
\(412\) 1.52786 0.0752725
\(413\) 37.8885 1.86437
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 29.4164 1.44226
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 29.8328 1.45396 0.726981 0.686657i \(-0.240923\pi\)
0.726981 + 0.686657i \(0.240923\pi\)
\(422\) 1.09017 0.0530686
\(423\) 0 0
\(424\) −22.2361 −1.07988
\(425\) −5.47214 −0.265438
\(426\) 0 0
\(427\) −4.76393 −0.230543
\(428\) −23.0344 −1.11341
\(429\) 0 0
\(430\) −3.23607 −0.156057
\(431\) −5.81966 −0.280323 −0.140162 0.990129i \(-0.544762\pi\)
−0.140162 + 0.990129i \(0.544762\pi\)
\(432\) 0 0
\(433\) 25.2361 1.21277 0.606384 0.795172i \(-0.292619\pi\)
0.606384 + 0.795172i \(0.292619\pi\)
\(434\) −13.4164 −0.644008
\(435\) 0 0
\(436\) 24.1803 1.15803
\(437\) −30.4721 −1.45768
\(438\) 0 0
\(439\) 8.12461 0.387767 0.193883 0.981025i \(-0.437892\pi\)
0.193883 + 0.981025i \(0.437892\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −17.7082 −0.842293
\(443\) −7.41641 −0.352364 −0.176182 0.984358i \(-0.556375\pi\)
−0.176182 + 0.984358i \(0.556375\pi\)
\(444\) 0 0
\(445\) −4.76393 −0.225832
\(446\) −5.59675 −0.265014
\(447\) 0 0
\(448\) 0.763932 0.0360924
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −20.0902 −0.944962
\(453\) 0 0
\(454\) −10.3262 −0.484634
\(455\) 16.9443 0.794360
\(456\) 0 0
\(457\) 34.3607 1.60732 0.803662 0.595085i \(-0.202882\pi\)
0.803662 + 0.595085i \(0.202882\pi\)
\(458\) −4.32624 −0.202152
\(459\) 0 0
\(460\) −7.61803 −0.355193
\(461\) 30.1803 1.40564 0.702819 0.711368i \(-0.251924\pi\)
0.702819 + 0.711368i \(0.251924\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −2.29180 −0.106394
\(465\) 0 0
\(466\) −6.72949 −0.311738
\(467\) 3.76393 0.174174 0.0870870 0.996201i \(-0.472244\pi\)
0.0870870 + 0.996201i \(0.472244\pi\)
\(468\) 0 0
\(469\) −36.3607 −1.67898
\(470\) 5.38197 0.248252
\(471\) 0 0
\(472\) −26.1803 −1.20505
\(473\) 0 0
\(474\) 0 0
\(475\) −6.47214 −0.296962
\(476\) −28.6525 −1.31328
\(477\) 0 0
\(478\) 15.2361 0.696882
\(479\) −0.291796 −0.0133325 −0.00666625 0.999978i \(-0.502122\pi\)
−0.00666625 + 0.999978i \(0.502122\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −11.0902 −0.505143
\(483\) 0 0
\(484\) 0 0
\(485\) 12.7639 0.579580
\(486\) 0 0
\(487\) −11.7082 −0.530549 −0.265275 0.964173i \(-0.585463\pi\)
−0.265275 + 0.964173i \(0.585463\pi\)
\(488\) 3.29180 0.149013
\(489\) 0 0
\(490\) −2.14590 −0.0969418
\(491\) −38.4721 −1.73622 −0.868112 0.496369i \(-0.834666\pi\)
−0.868112 + 0.496369i \(0.834666\pi\)
\(492\) 0 0
\(493\) 6.76393 0.304632
\(494\) −20.9443 −0.942327
\(495\) 0 0
\(496\) −12.4377 −0.558469
\(497\) −46.8328 −2.10074
\(498\) 0 0
\(499\) −0.944272 −0.0422714 −0.0211357 0.999777i \(-0.506728\pi\)
−0.0211357 + 0.999777i \(0.506728\pi\)
\(500\) −1.61803 −0.0723607
\(501\) 0 0
\(502\) −14.6525 −0.653972
\(503\) −23.1803 −1.03356 −0.516780 0.856118i \(-0.672870\pi\)
−0.516780 + 0.856118i \(0.672870\pi\)
\(504\) 0 0
\(505\) −5.52786 −0.245987
\(506\) 0 0
\(507\) 0 0
\(508\) 6.00000 0.266207
\(509\) −15.5967 −0.691314 −0.345657 0.938361i \(-0.612344\pi\)
−0.345657 + 0.938361i \(0.612344\pi\)
\(510\) 0 0
\(511\) −33.8885 −1.49914
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) 17.8541 0.787511
\(515\) −0.944272 −0.0416096
\(516\) 0 0
\(517\) 0 0
\(518\) 1.52786 0.0671305
\(519\) 0 0
\(520\) −11.7082 −0.513439
\(521\) −17.8197 −0.780693 −0.390347 0.920668i \(-0.627645\pi\)
−0.390347 + 0.920668i \(0.627645\pi\)
\(522\) 0 0
\(523\) 24.5410 1.07310 0.536552 0.843867i \(-0.319727\pi\)
0.536552 + 0.843867i \(0.319727\pi\)
\(524\) 0.763932 0.0333725
\(525\) 0 0
\(526\) −12.4377 −0.542309
\(527\) 36.7082 1.59903
\(528\) 0 0
\(529\) −0.832816 −0.0362094
\(530\) 6.14590 0.266961
\(531\) 0 0
\(532\) −33.8885 −1.46925
\(533\) −18.4721 −0.800117
\(534\) 0 0
\(535\) 14.2361 0.615479
\(536\) 25.1246 1.08522
\(537\) 0 0
\(538\) 4.36068 0.188002
\(539\) 0 0
\(540\) 0 0
\(541\) 43.3050 1.86183 0.930913 0.365242i \(-0.119014\pi\)
0.930913 + 0.365242i \(0.119014\pi\)
\(542\) −14.9787 −0.643391
\(543\) 0 0
\(544\) 30.7426 1.31808
\(545\) −14.9443 −0.640142
\(546\) 0 0
\(547\) 9.59675 0.410327 0.205164 0.978728i \(-0.434227\pi\)
0.205164 + 0.978728i \(0.434227\pi\)
\(548\) 28.2705 1.20766
\(549\) 0 0
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 41.1246 1.74880
\(554\) −3.05573 −0.129825
\(555\) 0 0
\(556\) −11.7984 −0.500363
\(557\) −8.52786 −0.361337 −0.180669 0.983544i \(-0.557826\pi\)
−0.180669 + 0.983544i \(0.557826\pi\)
\(558\) 0 0
\(559\) −27.4164 −1.15959
\(560\) −6.00000 −0.253546
\(561\) 0 0
\(562\) −9.41641 −0.397207
\(563\) −0.944272 −0.0397963 −0.0198982 0.999802i \(-0.506334\pi\)
−0.0198982 + 0.999802i \(0.506334\pi\)
\(564\) 0 0
\(565\) 12.4164 0.522362
\(566\) 7.34752 0.308839
\(567\) 0 0
\(568\) 32.3607 1.35782
\(569\) −19.7082 −0.826211 −0.413105 0.910683i \(-0.635556\pi\)
−0.413105 + 0.910683i \(0.635556\pi\)
\(570\) 0 0
\(571\) −0.124612 −0.00521484 −0.00260742 0.999997i \(-0.500830\pi\)
−0.00260742 + 0.999997i \(0.500830\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7.05573 0.294500
\(575\) 4.70820 0.196346
\(576\) 0 0
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 12.9443 0.537019
\(582\) 0 0
\(583\) 0 0
\(584\) 23.4164 0.968978
\(585\) 0 0
\(586\) −13.2705 −0.548200
\(587\) 11.6525 0.480949 0.240475 0.970655i \(-0.422697\pi\)
0.240475 + 0.970655i \(0.422697\pi\)
\(588\) 0 0
\(589\) 43.4164 1.78894
\(590\) 7.23607 0.297904
\(591\) 0 0
\(592\) 1.41641 0.0582140
\(593\) −34.9443 −1.43499 −0.717495 0.696564i \(-0.754711\pi\)
−0.717495 + 0.696564i \(0.754711\pi\)
\(594\) 0 0
\(595\) 17.7082 0.725966
\(596\) 6.94427 0.284448
\(597\) 0 0
\(598\) 15.2361 0.623049
\(599\) 31.0132 1.26716 0.633582 0.773676i \(-0.281584\pi\)
0.633582 + 0.773676i \(0.281584\pi\)
\(600\) 0 0
\(601\) −28.4721 −1.16140 −0.580701 0.814117i \(-0.697222\pi\)
−0.580701 + 0.814117i \(0.697222\pi\)
\(602\) 10.4721 0.426812
\(603\) 0 0
\(604\) 11.7984 0.480069
\(605\) 0 0
\(606\) 0 0
\(607\) −6.29180 −0.255376 −0.127688 0.991814i \(-0.540756\pi\)
−0.127688 + 0.991814i \(0.540756\pi\)
\(608\) 36.3607 1.47462
\(609\) 0 0
\(610\) −0.909830 −0.0368379
\(611\) 45.5967 1.84465
\(612\) 0 0
\(613\) 14.8328 0.599092 0.299546 0.954082i \(-0.403165\pi\)
0.299546 + 0.954082i \(0.403165\pi\)
\(614\) −7.34752 −0.296522
\(615\) 0 0
\(616\) 0 0
\(617\) −41.7771 −1.68188 −0.840941 0.541127i \(-0.817998\pi\)
−0.840941 + 0.541127i \(0.817998\pi\)
\(618\) 0 0
\(619\) −9.52786 −0.382957 −0.191479 0.981497i \(-0.561328\pi\)
−0.191479 + 0.981497i \(0.561328\pi\)
\(620\) 10.8541 0.435911
\(621\) 0 0
\(622\) −2.00000 −0.0801927
\(623\) 15.4164 0.617645
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −14.0689 −0.562306
\(627\) 0 0
\(628\) −3.70820 −0.147973
\(629\) −4.18034 −0.166681
\(630\) 0 0
\(631\) 9.18034 0.365464 0.182732 0.983163i \(-0.441506\pi\)
0.182732 + 0.983163i \(0.441506\pi\)
\(632\) −28.4164 −1.13034
\(633\) 0 0
\(634\) −2.43769 −0.0968132
\(635\) −3.70820 −0.147156
\(636\) 0 0
\(637\) −18.1803 −0.720331
\(638\) 0 0
\(639\) 0 0
\(640\) 11.3820 0.449912
\(641\) 9.12461 0.360400 0.180200 0.983630i \(-0.442325\pi\)
0.180200 + 0.983630i \(0.442325\pi\)
\(642\) 0 0
\(643\) −28.8328 −1.13706 −0.568528 0.822664i \(-0.692487\pi\)
−0.568528 + 0.822664i \(0.692487\pi\)
\(644\) 24.6525 0.971444
\(645\) 0 0
\(646\) −21.8885 −0.861193
\(647\) 16.2361 0.638306 0.319153 0.947703i \(-0.396602\pi\)
0.319153 + 0.947703i \(0.396602\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 3.23607 0.126929
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) −16.8328 −0.658719 −0.329359 0.944205i \(-0.606833\pi\)
−0.329359 + 0.944205i \(0.606833\pi\)
\(654\) 0 0
\(655\) −0.472136 −0.0184479
\(656\) 6.54102 0.255384
\(657\) 0 0
\(658\) −17.4164 −0.678962
\(659\) 35.5967 1.38665 0.693326 0.720624i \(-0.256145\pi\)
0.693326 + 0.720624i \(0.256145\pi\)
\(660\) 0 0
\(661\) 45.7771 1.78052 0.890261 0.455450i \(-0.150522\pi\)
0.890261 + 0.455450i \(0.150522\pi\)
\(662\) 10.6180 0.412682
\(663\) 0 0
\(664\) −8.94427 −0.347105
\(665\) 20.9443 0.812184
\(666\) 0 0
\(667\) −5.81966 −0.225338
\(668\) −5.14590 −0.199101
\(669\) 0 0
\(670\) −6.94427 −0.268281
\(671\) 0 0
\(672\) 0 0
\(673\) 27.5967 1.06378 0.531888 0.846815i \(-0.321483\pi\)
0.531888 + 0.846815i \(0.321483\pi\)
\(674\) −4.87539 −0.187793
\(675\) 0 0
\(676\) −23.3262 −0.897163
\(677\) 5.41641 0.208169 0.104085 0.994568i \(-0.466809\pi\)
0.104085 + 0.994568i \(0.466809\pi\)
\(678\) 0 0
\(679\) −41.3050 −1.58514
\(680\) −12.2361 −0.469232
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −17.4721 −0.667576
\(686\) −7.05573 −0.269389
\(687\) 0 0
\(688\) 9.70820 0.370122
\(689\) 52.0689 1.98367
\(690\) 0 0
\(691\) −37.5410 −1.42813 −0.714064 0.700081i \(-0.753147\pi\)
−0.714064 + 0.700081i \(0.753147\pi\)
\(692\) −4.76393 −0.181098
\(693\) 0 0
\(694\) 8.79837 0.333982
\(695\) 7.29180 0.276594
\(696\) 0 0
\(697\) −19.3050 −0.731227
\(698\) −9.27051 −0.350894
\(699\) 0 0
\(700\) 5.23607 0.197905
\(701\) 36.1803 1.36651 0.683256 0.730179i \(-0.260563\pi\)
0.683256 + 0.730179i \(0.260563\pi\)
\(702\) 0 0
\(703\) −4.94427 −0.186477
\(704\) 0 0
\(705\) 0 0
\(706\) −4.32624 −0.162820
\(707\) 17.8885 0.672768
\(708\) 0 0
\(709\) −17.9443 −0.673911 −0.336956 0.941521i \(-0.609397\pi\)
−0.336956 + 0.941521i \(0.609397\pi\)
\(710\) −8.94427 −0.335673
\(711\) 0 0
\(712\) −10.6525 −0.399218
\(713\) −31.5836 −1.18281
\(714\) 0 0
\(715\) 0 0
\(716\) −32.6525 −1.22028
\(717\) 0 0
\(718\) −4.58359 −0.171058
\(719\) −26.3607 −0.983087 −0.491544 0.870853i \(-0.663567\pi\)
−0.491544 + 0.870853i \(0.663567\pi\)
\(720\) 0 0
\(721\) 3.05573 0.113801
\(722\) −14.1459 −0.526456
\(723\) 0 0
\(724\) 34.6525 1.28785
\(725\) −1.23607 −0.0459064
\(726\) 0 0
\(727\) 44.8328 1.66276 0.831379 0.555706i \(-0.187552\pi\)
0.831379 + 0.555706i \(0.187552\pi\)
\(728\) 37.8885 1.40424
\(729\) 0 0
\(730\) −6.47214 −0.239544
\(731\) −28.6525 −1.05975
\(732\) 0 0
\(733\) −47.8885 −1.76880 −0.884402 0.466726i \(-0.845433\pi\)
−0.884402 + 0.466726i \(0.845433\pi\)
\(734\) −2.94427 −0.108675
\(735\) 0 0
\(736\) −26.4508 −0.974991
\(737\) 0 0
\(738\) 0 0
\(739\) −1.65248 −0.0607873 −0.0303937 0.999538i \(-0.509676\pi\)
−0.0303937 + 0.999538i \(0.509676\pi\)
\(740\) −1.23607 −0.0454388
\(741\) 0 0
\(742\) −19.8885 −0.730131
\(743\) 2.23607 0.0820334 0.0410167 0.999158i \(-0.486940\pi\)
0.0410167 + 0.999158i \(0.486940\pi\)
\(744\) 0 0
\(745\) −4.29180 −0.157239
\(746\) 18.4721 0.676313
\(747\) 0 0
\(748\) 0 0
\(749\) −46.0689 −1.68332
\(750\) 0 0
\(751\) −34.0132 −1.24116 −0.620579 0.784144i \(-0.713102\pi\)
−0.620579 + 0.784144i \(0.713102\pi\)
\(752\) −16.1459 −0.588780
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) −7.29180 −0.265376
\(756\) 0 0
\(757\) 18.9443 0.688541 0.344271 0.938870i \(-0.388126\pi\)
0.344271 + 0.938870i \(0.388126\pi\)
\(758\) −18.9098 −0.686836
\(759\) 0 0
\(760\) −14.4721 −0.524960
\(761\) −44.6525 −1.61865 −0.809325 0.587360i \(-0.800167\pi\)
−0.809325 + 0.587360i \(0.800167\pi\)
\(762\) 0 0
\(763\) 48.3607 1.75077
\(764\) 44.6525 1.61547
\(765\) 0 0
\(766\) 9.88854 0.357288
\(767\) 61.3050 2.21359
\(768\) 0 0
\(769\) 24.5279 0.884497 0.442249 0.896892i \(-0.354181\pi\)
0.442249 + 0.896892i \(0.354181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.1803 0.942251
\(773\) 7.94427 0.285736 0.142868 0.989742i \(-0.454368\pi\)
0.142868 + 0.989742i \(0.454368\pi\)
\(774\) 0 0
\(775\) −6.70820 −0.240966
\(776\) 28.5410 1.02456
\(777\) 0 0
\(778\) −13.3475 −0.478532
\(779\) −22.8328 −0.818071
\(780\) 0 0
\(781\) 0 0
\(782\) 15.9230 0.569405
\(783\) 0 0
\(784\) 6.43769 0.229918
\(785\) 2.29180 0.0817977
\(786\) 0 0
\(787\) 27.0557 0.964433 0.482216 0.876052i \(-0.339832\pi\)
0.482216 + 0.876052i \(0.339832\pi\)
\(788\) 2.29180 0.0816419
\(789\) 0 0
\(790\) 7.85410 0.279436
\(791\) −40.1803 −1.42865
\(792\) 0 0
\(793\) −7.70820 −0.273726
\(794\) 10.7639 0.381998
\(795\) 0 0
\(796\) −26.2705 −0.931134
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 47.6525 1.68582
\(800\) −5.61803 −0.198627
\(801\) 0 0
\(802\) 12.9443 0.457078
\(803\) 0 0
\(804\) 0 0
\(805\) −15.2361 −0.537001
\(806\) −21.7082 −0.764639
\(807\) 0 0
\(808\) −12.3607 −0.434847
\(809\) −8.00000 −0.281265 −0.140633 0.990062i \(-0.544914\pi\)
−0.140633 + 0.990062i \(0.544914\pi\)
\(810\) 0 0
\(811\) −7.87539 −0.276542 −0.138271 0.990394i \(-0.544155\pi\)
−0.138271 + 0.990394i \(0.544155\pi\)
\(812\) −6.47214 −0.227127
\(813\) 0 0
\(814\) 0 0
\(815\) 6.18034 0.216488
\(816\) 0 0
\(817\) −33.8885 −1.18561
\(818\) 22.7295 0.794718
\(819\) 0 0
\(820\) −5.70820 −0.199339
\(821\) −33.7771 −1.17883 −0.589414 0.807831i \(-0.700641\pi\)
−0.589414 + 0.807831i \(0.700641\pi\)
\(822\) 0 0
\(823\) 38.2492 1.33328 0.666642 0.745378i \(-0.267731\pi\)
0.666642 + 0.745378i \(0.267731\pi\)
\(824\) −2.11146 −0.0735561
\(825\) 0 0
\(826\) −23.4164 −0.814761
\(827\) 8.94427 0.311023 0.155511 0.987834i \(-0.450297\pi\)
0.155511 + 0.987834i \(0.450297\pi\)
\(828\) 0 0
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 2.47214 0.0858091
\(831\) 0 0
\(832\) 1.23607 0.0428529
\(833\) −19.0000 −0.658311
\(834\) 0 0
\(835\) 3.18034 0.110060
\(836\) 0 0
\(837\) 0 0
\(838\) −8.65248 −0.298895
\(839\) −28.3607 −0.979119 −0.489560 0.871970i \(-0.662842\pi\)
−0.489560 + 0.871970i \(0.662842\pi\)
\(840\) 0 0
\(841\) −27.4721 −0.947315
\(842\) −18.4377 −0.635405
\(843\) 0 0
\(844\) 2.85410 0.0982422
\(845\) 14.4164 0.495940
\(846\) 0 0
\(847\) 0 0
\(848\) −18.4377 −0.633153
\(849\) 0 0
\(850\) 3.38197 0.116000
\(851\) 3.59675 0.123295
\(852\) 0 0
\(853\) 31.8885 1.09184 0.545921 0.837836i \(-0.316180\pi\)
0.545921 + 0.837836i \(0.316180\pi\)
\(854\) 2.94427 0.100751
\(855\) 0 0
\(856\) 31.8328 1.08802
\(857\) 1.00000 0.0341593 0.0170797 0.999854i \(-0.494563\pi\)
0.0170797 + 0.999854i \(0.494563\pi\)
\(858\) 0 0
\(859\) −26.8328 −0.915524 −0.457762 0.889075i \(-0.651349\pi\)
−0.457762 + 0.889075i \(0.651349\pi\)
\(860\) −8.47214 −0.288897
\(861\) 0 0
\(862\) 3.59675 0.122506
\(863\) 37.8885 1.28974 0.644871 0.764292i \(-0.276911\pi\)
0.644871 + 0.764292i \(0.276911\pi\)
\(864\) 0 0
\(865\) 2.94427 0.100108
\(866\) −15.5967 −0.529999
\(867\) 0 0
\(868\) −35.1246 −1.19221
\(869\) 0 0
\(870\) 0 0
\(871\) −58.8328 −1.99347
\(872\) −33.4164 −1.13162
\(873\) 0 0
\(874\) 18.8328 0.637029
\(875\) −3.23607 −0.109399
\(876\) 0 0
\(877\) −11.3050 −0.381741 −0.190871 0.981615i \(-0.561131\pi\)
−0.190871 + 0.981615i \(0.561131\pi\)
\(878\) −5.02129 −0.169460
\(879\) 0 0
\(880\) 0 0
\(881\) −38.8328 −1.30831 −0.654155 0.756360i \(-0.726976\pi\)
−0.654155 + 0.756360i \(0.726976\pi\)
\(882\) 0 0
\(883\) 10.1803 0.342596 0.171298 0.985219i \(-0.445204\pi\)
0.171298 + 0.985219i \(0.445204\pi\)
\(884\) −46.3607 −1.55928
\(885\) 0 0
\(886\) 4.58359 0.153989
\(887\) 16.9443 0.568933 0.284466 0.958686i \(-0.408184\pi\)
0.284466 + 0.958686i \(0.408184\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 2.94427 0.0986922
\(891\) 0 0
\(892\) −14.6525 −0.490601
\(893\) 56.3607 1.88604
\(894\) 0 0
\(895\) 20.1803 0.674554
\(896\) −36.8328 −1.23050
\(897\) 0 0
\(898\) −6.18034 −0.206241
\(899\) 8.29180 0.276547
\(900\) 0 0
\(901\) 54.4164 1.81287
\(902\) 0 0
\(903\) 0 0
\(904\) 27.7639 0.923415
\(905\) −21.4164 −0.711905
\(906\) 0 0
\(907\) −42.0000 −1.39459 −0.697294 0.716786i \(-0.745613\pi\)
−0.697294 + 0.716786i \(0.745613\pi\)
\(908\) −27.0344 −0.897169
\(909\) 0 0
\(910\) −10.4721 −0.347148
\(911\) −47.9574 −1.58890 −0.794450 0.607329i \(-0.792241\pi\)
−0.794450 + 0.607329i \(0.792241\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −21.2361 −0.702427
\(915\) 0 0
\(916\) −11.3262 −0.374229
\(917\) 1.52786 0.0504545
\(918\) 0 0
\(919\) −3.41641 −0.112697 −0.0563484 0.998411i \(-0.517946\pi\)
−0.0563484 + 0.998411i \(0.517946\pi\)
\(920\) 10.5279 0.347093
\(921\) 0 0
\(922\) −18.6525 −0.614287
\(923\) −75.7771 −2.49423
\(924\) 0 0
\(925\) 0.763932 0.0251179
\(926\) −3.70820 −0.121859
\(927\) 0 0
\(928\) 6.94427 0.227957
\(929\) −50.9443 −1.67143 −0.835714 0.549165i \(-0.814946\pi\)
−0.835714 + 0.549165i \(0.814946\pi\)
\(930\) 0 0
\(931\) −22.4721 −0.736495
\(932\) −17.6180 −0.577098
\(933\) 0 0
\(934\) −2.32624 −0.0761168
\(935\) 0 0
\(936\) 0 0
\(937\) −33.1246 −1.08213 −0.541067 0.840980i \(-0.681979\pi\)
−0.541067 + 0.840980i \(0.681979\pi\)
\(938\) 22.4721 0.733741
\(939\) 0 0
\(940\) 14.0902 0.459571
\(941\) −45.2361 −1.47465 −0.737327 0.675536i \(-0.763912\pi\)
−0.737327 + 0.675536i \(0.763912\pi\)
\(942\) 0 0
\(943\) 16.6099 0.540893
\(944\) −21.7082 −0.706542
\(945\) 0 0
\(946\) 0 0
\(947\) 18.2361 0.592593 0.296296 0.955096i \(-0.404248\pi\)
0.296296 + 0.955096i \(0.404248\pi\)
\(948\) 0 0
\(949\) −54.8328 −1.77995
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 39.5967 1.28334
\(953\) 31.8885 1.03297 0.516486 0.856296i \(-0.327240\pi\)
0.516486 + 0.856296i \(0.327240\pi\)
\(954\) 0 0
\(955\) −27.5967 −0.893010
\(956\) 39.8885 1.29009
\(957\) 0 0
\(958\) 0.180340 0.00582652
\(959\) 56.5410 1.82580
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 2.47214 0.0797049
\(963\) 0 0
\(964\) −29.0344 −0.935136
\(965\) −16.1803 −0.520864
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −7.88854 −0.253286
\(971\) −29.2361 −0.938230 −0.469115 0.883137i \(-0.655427\pi\)
−0.469115 + 0.883137i \(0.655427\pi\)
\(972\) 0 0
\(973\) −23.5967 −0.756477
\(974\) 7.23607 0.231859
\(975\) 0 0
\(976\) 2.72949 0.0873689
\(977\) 15.9443 0.510102 0.255051 0.966928i \(-0.417908\pi\)
0.255051 + 0.966928i \(0.417908\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.61803 −0.179462
\(981\) 0 0
\(982\) 23.7771 0.758757
\(983\) 55.1803 1.75998 0.879990 0.474993i \(-0.157549\pi\)
0.879990 + 0.474993i \(0.157549\pi\)
\(984\) 0 0
\(985\) −1.41641 −0.0451305
\(986\) −4.18034 −0.133129
\(987\) 0 0
\(988\) −54.8328 −1.74446
\(989\) 24.6525 0.783903
\(990\) 0 0
\(991\) 6.23607 0.198095 0.0990476 0.995083i \(-0.468420\pi\)
0.0990476 + 0.995083i \(0.468420\pi\)
\(992\) 37.6869 1.19656
\(993\) 0 0
\(994\) 28.9443 0.918057
\(995\) 16.2361 0.514718
\(996\) 0 0
\(997\) −38.7639 −1.22767 −0.613833 0.789436i \(-0.710373\pi\)
−0.613833 + 0.789436i \(0.710373\pi\)
\(998\) 0.583592 0.0184733
\(999\) 0 0
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 5445.2.a.x.1.1 2
3.2 odd 2 1815.2.a.f.1.2 2
11.10 odd 2 5445.2.a.o.1.2 2
15.14 odd 2 9075.2.a.bx.1.1 2
33.32 even 2 1815.2.a.j.1.1 yes 2
165.164 even 2 9075.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.2 2 3.2 odd 2
1815.2.a.j.1.1 yes 2 33.32 even 2
5445.2.a.o.1.2 2 11.10 odd 2
5445.2.a.x.1.1 2 1.1 even 1 trivial
9075.2.a.bd.1.2 2 165.164 even 2
9075.2.a.bx.1.1 2 15.14 odd 2