Properties

Label 9075.2.a.bx.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +3.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +3.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{12} +5.23607 q^{13} -2.00000 q^{14} +1.85410 q^{16} -5.47214 q^{17} -0.618034 q^{18} -6.47214 q^{19} +3.23607 q^{21} +4.70820 q^{23} +2.23607 q^{24} -3.23607 q^{26} +1.00000 q^{27} -5.23607 q^{28} +1.23607 q^{29} -6.70820 q^{31} -5.61803 q^{32} +3.38197 q^{34} -1.61803 q^{36} -0.763932 q^{37} +4.00000 q^{38} +5.23607 q^{39} -3.52786 q^{41} -2.00000 q^{42} -5.23607 q^{43} -2.90983 q^{46} -8.70820 q^{47} +1.85410 q^{48} +3.47214 q^{49} -5.47214 q^{51} -8.47214 q^{52} -9.94427 q^{53} -0.618034 q^{54} +7.23607 q^{56} -6.47214 q^{57} -0.763932 q^{58} +11.7082 q^{59} +1.47214 q^{61} +4.14590 q^{62} +3.23607 q^{63} -0.236068 q^{64} -11.2361 q^{67} +8.85410 q^{68} +4.70820 q^{69} -14.4721 q^{71} +2.23607 q^{72} -10.4721 q^{73} +0.472136 q^{74} +10.4721 q^{76} -3.23607 q^{78} -12.7082 q^{79} +1.00000 q^{81} +2.18034 q^{82} -4.00000 q^{83} -5.23607 q^{84} +3.23607 q^{86} +1.23607 q^{87} +4.76393 q^{89} +16.9443 q^{91} -7.61803 q^{92} -6.70820 q^{93} +5.38197 q^{94} -5.61803 q^{96} -12.7639 q^{97} -2.14590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} + 2q^{9} - q^{12} + 6q^{13} - 4q^{14} - 3q^{16} - 2q^{17} + q^{18} - 4q^{19} + 2q^{21} - 4q^{23} - 2q^{26} + 2q^{27} - 6q^{28} - 2q^{29} - 9q^{32} + 9q^{34} - q^{36} - 6q^{37} + 8q^{38} + 6q^{39} - 16q^{41} - 4q^{42} - 6q^{43} - 17q^{46} - 4q^{47} - 3q^{48} - 2q^{49} - 2q^{51} - 8q^{52} - 2q^{53} + q^{54} + 10q^{56} - 4q^{57} - 6q^{58} + 10q^{59} - 6q^{61} + 15q^{62} + 2q^{63} + 4q^{64} - 18q^{67} + 11q^{68} - 4q^{69} - 20q^{71} - 12q^{73} - 8q^{74} + 12q^{76} - 2q^{78} - 12q^{79} + 2q^{81} - 18q^{82} - 8q^{83} - 6q^{84} + 2q^{86} - 2q^{87} + 14q^{89} + 16q^{91} - 13q^{92} + 13q^{94} - 9q^{96} - 30q^{97} - 11q^{98} + O(q^{100}) \)

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 \(0.570119\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) 5.23607 1.45222 0.726112 0.687576i \(-0.241325\pi\)
0.726112 + 0.687576i \(0.241325\pi\)
\(14\) −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 \(0.730970\pi\)
\(18\) −0.618034 −0.145672
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 0 0
\(21\) 3.23607 0.706168
\(22\) 0 0
\(23\) 4.70820 0.981728 0.490864 0.871236i \(-0.336681\pi\)
0.490864 + 0.871236i \(0.336681\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −3.23607 −0.634645
\(27\) 1.00000 0.192450
\(28\) −5.23607 −0.989524
\(29\) 1.23607 0.229532 0.114766 0.993393i \(-0.463388\pi\)
0.114766 + 0.993393i \(0.463388\pi\)
\(30\) 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
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) −0.763932 −0.125590 −0.0627948 0.998026i \(-0.520001\pi\)
−0.0627948 + 0.998026i \(0.520001\pi\)
\(38\) 4.00000 0.648886
\(39\) 5.23607 0.838442
\(40\) 0 0
\(41\) −3.52786 −0.550960 −0.275480 0.961307i \(-0.588837\pi\)
−0.275480 + 0.961307i \(0.588837\pi\)
\(42\) −2.00000 −0.308607
\(43\) −5.23607 −0.798493 −0.399246 0.916844i \(-0.630728\pi\)
−0.399246 + 0.916844i \(0.630728\pi\)
\(44\) 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 \(0.719046\pi\)
\(48\) 1.85410 0.267617
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) −5.47214 −0.766252
\(52\) −8.47214 −1.17487
\(53\) −9.94427 −1.36595 −0.682975 0.730441i \(-0.739314\pi\)
−0.682975 + 0.730441i \(0.739314\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) 7.23607 0.966960
\(57\) −6.47214 −0.857255
\(58\) −0.763932 −0.100309
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 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
\(63\) 3.23607 0.407706
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) −11.2361 −1.37270 −0.686352 0.727269i \(-0.740789\pi\)
−0.686352 + 0.727269i \(0.740789\pi\)
\(68\) 8.85410 1.07372
\(69\) 4.70820 0.566801
\(70\) 0 0
\(71\) −14.4721 −1.71753 −0.858763 0.512373i \(-0.828767\pi\)
−0.858763 + 0.512373i \(0.828767\pi\)
\(72\) 2.23607 0.263523
\(73\) −10.4721 −1.22567 −0.612835 0.790211i \(-0.709971\pi\)
−0.612835 + 0.790211i \(0.709971\pi\)
\(74\) 0.472136 0.0548847
\(75\) 0 0
\(76\) 10.4721 1.20124
\(77\) 0 0
\(78\) −3.23607 −0.366413
\(79\) −12.7082 −1.42978 −0.714892 0.699235i \(-0.753524\pi\)
−0.714892 + 0.699235i \(0.753524\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.18034 0.240778
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −5.23607 −0.571302
\(85\) 0 0
\(86\) 3.23607 0.348954
\(87\) 1.23607 0.132520
\(88\) 0 0
\(89\) 4.76393 0.504976 0.252488 0.967600i \(-0.418751\pi\)
0.252488 + 0.967600i \(0.418751\pi\)
\(90\) 0 0
\(91\) 16.9443 1.77624
\(92\) −7.61803 −0.794235
\(93\) −6.70820 −0.695608
\(94\) 5.38197 0.555107
\(95\) 0 0
\(96\) −5.61803 −0.573388
\(97\) −12.7639 −1.29598 −0.647990 0.761648i \(-0.724390\pi\)
−0.647990 + 0.761648i \(0.724390\pi\)
\(98\) −2.14590 −0.216768
\(99\) 0 0
\(100\) 0 0
\(101\) 5.52786 0.550043 0.275022 0.961438i \(-0.411315\pi\)
0.275022 + 0.961438i \(0.411315\pi\)
\(102\) 3.38197 0.334865
\(103\) 0.944272 0.0930419 0.0465209 0.998917i \(-0.485187\pi\)
0.0465209 + 0.998917i \(0.485187\pi\)
\(104\) 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 \(0.258433\pi\)
\(108\) −1.61803 −0.155695
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) −0.763932 −0.0725092
\(112\) 6.00000 0.566947
\(113\) 12.4164 1.16804 0.584019 0.811740i \(-0.301479\pi\)
0.584019 + 0.811740i \(0.301479\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 5.23607 0.484075
\(118\) −7.23607 −0.666134
\(119\) −17.7082 −1.62331
\(120\) 0 0
\(121\) 0 0
\(122\) −0.909830 −0.0823721
\(123\) −3.52786 −0.318097
\(124\) 10.8541 0.974727
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 3.70820 0.329050 0.164525 0.986373i \(-0.447391\pi\)
0.164525 + 0.986373i \(0.447391\pi\)
\(128\) 11.3820 1.00603
\(129\) −5.23607 −0.461010
\(130\) 0 0
\(131\) 0.472136 0.0412507 0.0206254 0.999787i \(-0.493434\pi\)
0.0206254 + 0.999787i \(0.493434\pi\)
\(132\) 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 \(0.768206\pi\)
\(138\) −2.90983 −0.247701
\(139\) 7.29180 0.618482 0.309241 0.950984i \(-0.399925\pi\)
0.309241 + 0.950984i \(0.399925\pi\)
\(140\) 0 0
\(141\) −8.70820 −0.733363
\(142\) 8.94427 0.750587
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) 6.47214 0.535638
\(147\) 3.47214 0.286377
\(148\) 1.23607 0.101604
\(149\) 4.29180 0.351598 0.175799 0.984426i \(-0.443749\pi\)
0.175799 + 0.984426i \(0.443749\pi\)
\(150\) 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
\(153\) −5.47214 −0.442396
\(154\) 0 0
\(155\) 0 0
\(156\) −8.47214 −0.678314
\(157\) −2.29180 −0.182905 −0.0914526 0.995809i \(-0.529151\pi\)
−0.0914526 + 0.995809i \(0.529151\pi\)
\(158\) 7.85410 0.624839
\(159\) −9.94427 −0.788632
\(160\) 0 0
\(161\) 15.2361 1.20077
\(162\) −0.618034 −0.0485573
\(163\) −6.18034 −0.484082 −0.242041 0.970266i \(-0.577817\pi\)
−0.242041 + 0.970266i \(0.577817\pi\)
\(164\) 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 \(0.460732\pi\)
\(168\) 7.23607 0.558275
\(169\) 14.4164 1.10895
\(170\) 0 0
\(171\) −6.47214 −0.494937
\(172\) 8.47214 0.645994
\(173\) 2.94427 0.223849 0.111924 0.993717i \(-0.464299\pi\)
0.111924 + 0.993717i \(0.464299\pi\)
\(174\) −0.763932 −0.0579135
\(175\) 0 0
\(176\) 0 0
\(177\) 11.7082 0.880042
\(178\) −2.94427 −0.220683
\(179\) −20.1803 −1.50835 −0.754175 0.656674i \(-0.771963\pi\)
−0.754175 + 0.656674i \(0.771963\pi\)
\(180\) 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
\(183\) 1.47214 0.108823
\(184\) 10.5279 0.776124
\(185\) 0 0
\(186\) 4.14590 0.303992
\(187\) 0 0
\(188\) 14.0902 1.02763
\(189\) 3.23607 0.235389
\(190\) 0 0
\(191\) 27.5967 1.99683 0.998415 0.0562752i \(-0.0179224\pi\)
0.998415 + 0.0562752i \(0.0179224\pi\)
\(192\) −0.236068 −0.0170367
\(193\) 16.1803 1.16469 0.582343 0.812943i \(-0.302136\pi\)
0.582343 + 0.812943i \(0.302136\pi\)
\(194\) 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 \(0.304818\pi\)
\(200\) 0 0
\(201\) −11.2361 −0.792531
\(202\) −3.41641 −0.240378
\(203\) 4.00000 0.280745
\(204\) 8.85410 0.619911
\(205\) 0 0
\(206\) −0.583592 −0.0406608
\(207\) 4.70820 0.327243
\(208\) 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
\(213\) −14.4721 −0.991614
\(214\) −8.79837 −0.601444
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) −21.7082 −1.47365
\(218\) 9.23607 0.625545
\(219\) −10.4721 −0.707641
\(220\) 0 0
\(221\) −28.6525 −1.92737
\(222\) 0.472136 0.0316877
\(223\) −9.05573 −0.606416 −0.303208 0.952924i \(-0.598058\pi\)
−0.303208 + 0.952924i \(0.598058\pi\)
\(224\) −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 \(0.312917\pi\)
\(228\) 10.4721 0.693534
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.76393 0.181461
\(233\) 10.8885 0.713332 0.356666 0.934232i \(-0.383913\pi\)
0.356666 + 0.934232i \(0.383913\pi\)
\(234\) −3.23607 −0.211548
\(235\) 0 0
\(236\) −18.9443 −1.23317
\(237\) −12.7082 −0.825487
\(238\) 10.9443 0.709412
\(239\) 24.6525 1.59464 0.797318 0.603559i \(-0.206251\pi\)
0.797318 + 0.603559i \(0.206251\pi\)
\(240\) 0 0
\(241\) 17.9443 1.15589 0.577946 0.816075i \(-0.303854\pi\)
0.577946 + 0.816075i \(0.303854\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.38197 −0.152490
\(245\) 0 0
\(246\) 2.18034 0.139013
\(247\) −33.8885 −2.15628
\(248\) −15.0000 −0.952501
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −23.7082 −1.49645 −0.748224 0.663446i \(-0.769093\pi\)
−0.748224 + 0.663446i \(0.769093\pi\)
\(252\) −5.23607 −0.329841
\(253\) 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 \(0.857172\pi\)
\(258\) 3.23607 0.201469
\(259\) −2.47214 −0.153611
\(260\) 0 0
\(261\) 1.23607 0.0765107
\(262\) −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
\(265\) 0 0
\(266\) 12.9443 0.793664
\(267\) 4.76393 0.291548
\(268\) 18.1803 1.11054
\(269\) 7.05573 0.430195 0.215098 0.976593i \(-0.430993\pi\)
0.215098 + 0.976593i \(0.430993\pi\)
\(270\) 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
\(273\) 16.9443 1.02551
\(274\) 10.7984 0.652354
\(275\) 0 0
\(276\) −7.61803 −0.458552
\(277\) −4.94427 −0.297073 −0.148536 0.988907i \(-0.547456\pi\)
−0.148536 + 0.988907i \(0.547456\pi\)
\(278\) −4.50658 −0.270287
\(279\) −6.70820 −0.401610
\(280\) 0 0
\(281\) −15.2361 −0.908908 −0.454454 0.890770i \(-0.650166\pi\)
−0.454454 + 0.890770i \(0.650166\pi\)
\(282\) 5.38197 0.320491
\(283\) 11.8885 0.706701 0.353350 0.935491i \(-0.385042\pi\)
0.353350 + 0.935491i \(0.385042\pi\)
\(284\) 23.4164 1.38951
\(285\) 0 0
\(286\) 0 0
\(287\) −11.4164 −0.673889
\(288\) −5.61803 −0.331046
\(289\) 12.9443 0.761428
\(290\) 0 0
\(291\) −12.7639 −0.748235
\(292\) 16.9443 0.991589
\(293\) 21.4721 1.25442 0.627208 0.778852i \(-0.284198\pi\)
0.627208 + 0.778852i \(0.284198\pi\)
\(294\) −2.14590 −0.125151
\(295\) 0 0
\(296\) −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
\(303\) 5.52786 0.317567
\(304\) −12.0000 −0.688247
\(305\) 0 0
\(306\) 3.38197 0.193334
\(307\) −11.8885 −0.678515 −0.339258 0.940694i \(-0.610176\pi\)
−0.339258 + 0.940694i \(0.610176\pi\)
\(308\) 0 0
\(309\) 0.944272 0.0537178
\(310\) 0 0
\(311\) −3.23607 −0.183501 −0.0917503 0.995782i \(-0.529246\pi\)
−0.0917503 + 0.995782i \(0.529246\pi\)
\(312\) 11.7082 0.662847
\(313\) −22.7639 −1.28669 −0.643347 0.765575i \(-0.722455\pi\)
−0.643347 + 0.765575i \(0.722455\pi\)
\(314\) 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 \(0.464670\pi\)
\(318\) 6.14590 0.344645
\(319\) 0 0
\(320\) 0 0
\(321\) 14.2361 0.794580
\(322\) −9.41641 −0.524756
\(323\) 35.4164 1.97062
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) 3.81966 0.211551
\(327\) −14.9443 −0.826420
\(328\) −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
\(333\) −0.763932 −0.0418632
\(334\) −1.96556 −0.107551
\(335\) 0 0
\(336\) 6.00000 0.327327
\(337\) −7.88854 −0.429716 −0.214858 0.976645i \(-0.568929\pi\)
−0.214858 + 0.976645i \(0.568929\pi\)
\(338\) −8.90983 −0.484631
\(339\) 12.4164 0.674367
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −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 \(0.624805\pi\)
\(348\) −2.00000 −0.107211
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) 5.23607 0.279481
\(352\) 0 0
\(353\) 7.00000 0.372572 0.186286 0.982496i \(-0.440355\pi\)
0.186286 + 0.982496i \(0.440355\pi\)
\(354\) −7.23607 −0.384593
\(355\) 0 0
\(356\) −7.70820 −0.408534
\(357\) −17.7082 −0.937218
\(358\) 12.4721 0.659173
\(359\) −7.41641 −0.391423 −0.195712 0.980662i \(-0.562702\pi\)
−0.195712 + 0.980662i \(0.562702\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 13.2361 0.695672
\(363\) 0 0
\(364\) −27.4164 −1.43701
\(365\) 0 0
\(366\) −0.909830 −0.0475576
\(367\) −4.76393 −0.248675 −0.124338 0.992240i \(-0.539681\pi\)
−0.124338 + 0.992240i \(0.539681\pi\)
\(368\) 8.72949 0.455056
\(369\) −3.52786 −0.183653
\(370\) 0 0
\(371\) −32.1803 −1.67072
\(372\) 10.8541 0.562759
\(373\) 29.8885 1.54757 0.773785 0.633448i \(-0.218361\pi\)
0.773785 + 0.633448i \(0.218361\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −19.4721 −1.00420
\(377\) 6.47214 0.333332
\(378\) −2.00000 −0.102869
\(379\) 30.5967 1.57165 0.785825 0.618449i \(-0.212239\pi\)
0.785825 + 0.618449i \(0.212239\pi\)
\(380\) 0 0
\(381\) 3.70820 0.189977
\(382\) −17.0557 −0.872647
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −5.23607 −0.266164
\(388\) 20.6525 1.04847
\(389\) −21.5967 −1.09500 −0.547499 0.836806i \(-0.684420\pi\)
−0.547499 + 0.836806i \(0.684420\pi\)
\(390\) 0 0
\(391\) −25.7639 −1.30294
\(392\) 7.76393 0.392138
\(393\) 0.472136 0.0238161
\(394\) 0.875388 0.0441014
\(395\) 0 0
\(396\) 0 0
\(397\) 17.4164 0.874104 0.437052 0.899436i \(-0.356022\pi\)
0.437052 + 0.899436i \(0.356022\pi\)
\(398\) −10.0344 −0.502981
\(399\) −20.9443 −1.04853
\(400\) 0 0
\(401\) 20.9443 1.04591 0.522954 0.852361i \(-0.324830\pi\)
0.522954 + 0.852361i \(0.324830\pi\)
\(402\) 6.94427 0.346349
\(403\) −35.1246 −1.74968
\(404\) −8.94427 −0.444994
\(405\) 0 0
\(406\) −2.47214 −0.122690
\(407\) 0 0
\(408\) −12.2361 −0.605776
\(409\) −36.7771 −1.81851 −0.909255 0.416240i \(-0.863348\pi\)
−0.909255 + 0.416240i \(0.863348\pi\)
\(410\) 0 0
\(411\) −17.4721 −0.861837
\(412\) −1.52786 −0.0752725
\(413\) 37.8885 1.86437
\(414\) −2.90983 −0.143010
\(415\) 0 0
\(416\) −29.4164 −1.44226
\(417\) 7.29180 0.357081
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 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
\(423\) −8.70820 −0.423407
\(424\) −22.2361 −1.07988
\(425\) 0 0
\(426\) 8.94427 0.433351
\(427\) 4.76393 0.230543
\(428\) −23.0344 −1.11341
\(429\) 0 0
\(430\) 0 0
\(431\) 5.81966 0.280323 0.140162 0.990129i \(-0.455238\pi\)
0.140162 + 0.990129i \(0.455238\pi\)
\(432\) 1.85410 0.0892055
\(433\) −25.2361 −1.21277 −0.606384 0.795172i \(-0.707381\pi\)
−0.606384 + 0.795172i \(0.707381\pi\)
\(434\) 13.4164 0.644008
\(435\) 0 0
\(436\) 24.1803 1.15803
\(437\) −30.4721 −1.45768
\(438\) 6.47214 0.309251
\(439\) 8.12461 0.387767 0.193883 0.981025i \(-0.437892\pi\)
0.193883 + 0.981025i \(0.437892\pi\)
\(440\) 0 0
\(441\) 3.47214 0.165340
\(442\) 17.7082 0.842293
\(443\) −7.41641 −0.352364 −0.176182 0.984358i \(-0.556375\pi\)
−0.176182 + 0.984358i \(0.556375\pi\)
\(444\) 1.23607 0.0586612
\(445\) 0 0
\(446\) 5.59675 0.265014
\(447\) 4.29180 0.202995
\(448\) −0.763932 −0.0360924
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −20.0902 −0.944962
\(453\) −7.29180 −0.342598
\(454\) −10.3262 −0.484634
\(455\) 0 0
\(456\) −14.4721 −0.677720
\(457\) −34.3607 −1.60732 −0.803662 0.595085i \(-0.797118\pi\)
−0.803662 + 0.595085i \(0.797118\pi\)
\(458\) −4.32624 −0.202152
\(459\) −5.47214 −0.255417
\(460\) 0 0
\(461\) −30.1803 −1.40564 −0.702819 0.711368i \(-0.748076\pi\)
−0.702819 + 0.711368i \(0.748076\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 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 \(0.472244\pi\)
\(468\) −8.47214 −0.391625
\(469\) −36.3607 −1.67898
\(470\) 0 0
\(471\) −2.29180 −0.105600
\(472\) 26.1803 1.20505
\(473\) 0 0
\(474\) 7.85410 0.360751
\(475\) 0 0
\(476\) 28.6525 1.31328
\(477\) −9.94427 −0.455317
\(478\) −15.2361 −0.696882
\(479\) 0.291796 0.0133325 0.00666625 0.999978i \(-0.497878\pi\)
0.00666625 + 0.999978i \(0.497878\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −11.0902 −0.505143
\(483\) 15.2361 0.693265
\(484\) 0 0
\(485\) 0 0
\(486\) −0.618034 −0.0280346
\(487\) 11.7082 0.530549 0.265275 0.964173i \(-0.414537\pi\)
0.265275 + 0.964173i \(0.414537\pi\)
\(488\) 3.29180 0.149013
\(489\) −6.18034 −0.279485
\(490\) 0 0
\(491\) 38.4721 1.73622 0.868112 0.496369i \(-0.165334\pi\)
0.868112 + 0.496369i \(0.165334\pi\)
\(492\) 5.70820 0.257346
\(493\) −6.76393 −0.304632
\(494\) 20.9443 0.942327
\(495\) 0 0
\(496\) −12.4377 −0.558469
\(497\) −46.8328 −2.10074
\(498\) 2.47214 0.110779
\(499\) −0.944272 −0.0422714 −0.0211357 0.999777i \(-0.506728\pi\)
−0.0211357 + 0.999777i \(0.506728\pi\)
\(500\) 0 0
\(501\) 3.18034 0.142087
\(502\) 14.6525 0.653972
\(503\) −23.1803 −1.03356 −0.516780 0.856118i \(-0.672870\pi\)
−0.516780 + 0.856118i \(0.672870\pi\)
\(504\) 7.23607 0.322320
\(505\) 0 0
\(506\) 0 0
\(507\) 14.4164 0.640255
\(508\) −6.00000 −0.266207
\(509\) 15.5967 0.691314 0.345657 0.938361i \(-0.387656\pi\)
0.345657 + 0.938361i \(0.387656\pi\)
\(510\) 0 0
\(511\) −33.8885 −1.49914
\(512\) −18.7082 −0.826794
\(513\) −6.47214 −0.285752
\(514\) 17.8541 0.787511
\(515\) 0 0
\(516\) 8.47214 0.372965
\(517\) 0 0
\(518\) 1.52786 0.0671305
\(519\) 2.94427 0.129239
\(520\) 0 0
\(521\) 17.8197 0.780693 0.390347 0.920668i \(-0.372355\pi\)
0.390347 + 0.920668i \(0.372355\pi\)
\(522\) −0.763932 −0.0334364
\(523\) −24.5410 −1.07310 −0.536552 0.843867i \(-0.680273\pi\)
−0.536552 + 0.843867i \(0.680273\pi\)
\(524\) −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
\(530\) 0 0
\(531\) 11.7082 0.508093
\(532\) 33.8885 1.46925
\(533\) −18.4721 −0.800117
\(534\) −2.94427 −0.127411
\(535\) 0 0
\(536\) −25.1246 −1.08522
\(537\) −20.1803 −0.870846
\(538\) −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
\(543\) −21.4164 −0.919066
\(544\) 30.7426 1.31808
\(545\) 0 0
\(546\) −10.4721 −0.448166
\(547\) −9.59675 −0.410327 −0.205164 0.978728i \(-0.565773\pi\)
−0.205164 + 0.978728i \(0.565773\pi\)
\(548\) 28.2705 1.20766
\(549\) 1.47214 0.0628292
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 10.5279 0.448096
\(553\) −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 \(0.557826\pi\)
\(558\) 4.14590 0.175510
\(559\) −27.4164 −1.15959
\(560\) 0 0
\(561\) 0 0
\(562\) 9.41641 0.397207
\(563\) −0.944272 −0.0397963 −0.0198982 0.999802i \(-0.506334\pi\)
−0.0198982 + 0.999802i \(0.506334\pi\)
\(564\) 14.0902 0.593303
\(565\) 0 0
\(566\) −7.34752 −0.308839
\(567\) 3.23607 0.135902
\(568\) −32.3607 −1.35782
\(569\) 19.7082 0.826211 0.413105 0.910683i \(-0.364444\pi\)
0.413105 + 0.910683i \(0.364444\pi\)
\(570\) 0 0
\(571\) −0.124612 −0.00521484 −0.00260742 0.999997i \(-0.500830\pi\)
−0.00260742 + 0.999997i \(0.500830\pi\)
\(572\) 0 0
\(573\) 27.5967 1.15287
\(574\) 7.05573 0.294500
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) −8.00000 −0.332756
\(579\) 16.1803 0.672432
\(580\) 0 0
\(581\) −12.9443 −0.537019
\(582\) 7.88854 0.326991
\(583\) 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 \(0.422697\pi\)
\(588\) −5.61803 −0.231684
\(589\) 43.4164 1.78894
\(590\) 0 0
\(591\) −1.41641 −0.0582632
\(592\) −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
\(595\) 0 0
\(596\) −6.94427 −0.284448
\(597\) 16.2361 0.664498
\(598\) −15.2361 −0.623049
\(599\) −31.0132 −1.26716 −0.633582 0.773676i \(-0.718416\pi\)
−0.633582 + 0.773676i \(0.718416\pi\)
\(600\) 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
\(603\) −11.2361 −0.457568
\(604\) 11.7984 0.480069
\(605\) 0 0
\(606\) −3.41641 −0.138782
\(607\) 6.29180 0.255376 0.127688 0.991814i \(-0.459244\pi\)
0.127688 + 0.991814i \(0.459244\pi\)
\(608\) 36.3607 1.47462
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) −45.5967 −1.84465
\(612\) 8.85410 0.357906
\(613\) −14.8328 −0.599092 −0.299546 0.954082i \(-0.596835\pi\)
−0.299546 + 0.954082i \(0.596835\pi\)
\(614\) 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 \(0.817998\pi\)
\(618\) −0.583592 −0.0234755
\(619\) −9.52786 −0.382957 −0.191479 0.981497i \(-0.561328\pi\)
−0.191479 + 0.981497i \(0.561328\pi\)
\(620\) 0 0
\(621\) 4.70820 0.188934
\(622\) 2.00000 0.0801927
\(623\) 15.4164 0.617645
\(624\) 9.70820 0.388639
\(625\) 0 0
\(626\) 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
\(633\) −1.76393 −0.0701100
\(634\) −2.43769 −0.0968132
\(635\) 0 0
\(636\) 16.0902 0.638017
\(637\) 18.1803 0.720331
\(638\) 0 0
\(639\) −14.4721 −0.572509
\(640\) 0 0
\(641\) −9.12461 −0.360400 −0.180200 0.983630i \(-0.557675\pi\)
−0.180200 + 0.983630i \(0.557675\pi\)
\(642\) −8.79837 −0.347244
\(643\) 28.8328 1.13706 0.568528 0.822664i \(-0.307513\pi\)
0.568528 + 0.822664i \(0.307513\pi\)
\(644\) −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 \(0.396602\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) −21.7082 −0.850812
\(652\) 10.0000 0.391630
\(653\) −16.8328 −0.658719 −0.329359 0.944205i \(-0.606833\pi\)
−0.329359 + 0.944205i \(0.606833\pi\)
\(654\) 9.23607 0.361159
\(655\) 0 0
\(656\) −6.54102 −0.255384
\(657\) −10.4721 −0.408557
\(658\) 17.4164 0.678962
\(659\) −35.5967 −1.38665 −0.693326 0.720624i \(-0.743855\pi\)
−0.693326 + 0.720624i \(0.743855\pi\)
\(660\) 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
\(663\) −28.6525 −1.11277
\(664\) −8.94427 −0.347105
\(665\) 0 0
\(666\) 0.472136 0.0182949
\(667\) 5.81966 0.225338
\(668\) −5.14590 −0.199101
\(669\) −9.05573 −0.350115
\(670\) 0 0
\(671\) 0 0
\(672\) −18.1803 −0.701322
\(673\) −27.5967 −1.06378 −0.531888 0.846815i \(-0.678517\pi\)
−0.531888 + 0.846815i \(0.678517\pi\)
\(674\) 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 \(0.466809\pi\)
\(678\) −7.67376 −0.294709
\(679\) −41.3050 −1.58514
\(680\) 0 0
\(681\) 16.7082 0.640260
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 10.4721 0.400412
\(685\) 0 0
\(686\) 7.05573 0.269389
\(687\) 7.00000 0.267067
\(688\) −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
\(695\) 0 0
\(696\) 2.76393 0.104767
\(697\) 19.3050 0.731227
\(698\) −9.27051 −0.350894
\(699\) 10.8885 0.411843
\(700\) 0 0
\(701\) −36.1803 −1.36651 −0.683256 0.730179i \(-0.739437\pi\)
−0.683256 + 0.730179i \(0.739437\pi\)
\(702\) −3.23607 −0.122138
\(703\) 4.94427 0.186477
\(704\) 0 0
\(705\) 0 0
\(706\) −4.32624 −0.162820
\(707\) 17.8885 0.672768
\(708\) −18.9443 −0.711969
\(709\) −17.9443 −0.673911 −0.336956 0.941521i \(-0.609397\pi\)
−0.336956 + 0.941521i \(0.609397\pi\)
\(710\) 0 0
\(711\) −12.7082 −0.476595
\(712\) 10.6525 0.399218
\(713\) −31.5836 −1.18281
\(714\) 10.9443 0.409579
\(715\) 0 0
\(716\) 32.6525 1.22028
\(717\) 24.6525 0.920664
\(718\) 4.58359 0.171058
\(719\) 26.3607 0.983087 0.491544 0.870853i \(-0.336433\pi\)
0.491544 + 0.870853i \(0.336433\pi\)
\(720\) 0 0
\(721\) 3.05573 0.113801
\(722\) −14.1459 −0.526456
\(723\) 17.9443 0.667355
\(724\) 34.6525 1.28785
\(725\) 0 0
\(726\) 0 0
\(727\) −44.8328 −1.66276 −0.831379 0.555706i \(-0.812448\pi\)
−0.831379 + 0.555706i \(0.812448\pi\)
\(728\) 37.8885 1.40424
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.6525 1.05975
\(732\) −2.38197 −0.0880400
\(733\) 47.8885 1.76880 0.884402 0.466726i \(-0.154567\pi\)
0.884402 + 0.466726i \(0.154567\pi\)
\(734\) 2.94427 0.108675
\(735\) 0 0
\(736\) −26.4508 −0.974991
\(737\) 0 0
\(738\) 2.18034 0.0802594
\(739\) −1.65248 −0.0607873 −0.0303937 0.999538i \(-0.509676\pi\)
−0.0303937 + 0.999538i \(0.509676\pi\)
\(740\) 0 0
\(741\) −33.8885 −1.24493
\(742\) 19.8885 0.730131
\(743\) 2.23607 0.0820334 0.0410167 0.999158i \(-0.486940\pi\)
0.0410167 + 0.999158i \(0.486940\pi\)
\(744\) −15.0000 −0.549927
\(745\) 0 0
\(746\) −18.4721 −0.676313
\(747\) −4.00000 −0.146352
\(748\) 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
\(753\) −23.7082 −0.863975
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −5.23607 −0.190434
\(757\) −18.9443 −0.688541 −0.344271 0.938870i \(-0.611874\pi\)
−0.344271 + 0.938870i \(0.611874\pi\)
\(758\) −18.9098 −0.686836
\(759\) 0 0
\(760\) 0 0
\(761\) 44.6525 1.61865 0.809325 0.587360i \(-0.199833\pi\)
0.809325 + 0.587360i \(0.199833\pi\)
\(762\) −2.29180 −0.0830230
\(763\) −48.3607 −1.75077
\(764\) −44.6525 −1.61547
\(765\) 0 0
\(766\) 9.88854 0.357288
\(767\) 61.3050 2.21359
\(768\) −6.56231 −0.236797
\(769\) 24.5279 0.884497 0.442249 0.896892i \(-0.354181\pi\)
0.442249 + 0.896892i \(0.354181\pi\)
\(770\) 0 0
\(771\) −28.8885 −1.04040
\(772\) −26.1803 −0.942251
\(773\) 7.94427 0.285736 0.142868 0.989742i \(-0.454368\pi\)
0.142868 + 0.989742i \(0.454368\pi\)
\(774\) 3.23607 0.116318
\(775\) 0 0
\(776\) −28.5410 −1.02456
\(777\) −2.47214 −0.0886874
\(778\) 13.3475 0.478532
\(779\) 22.8328 0.818071
\(780\) 0 0
\(781\) 0 0
\(782\) 15.9230 0.569405
\(783\) 1.23607 0.0441735
\(784\) 6.43769 0.229918
\(785\) 0 0
\(786\) −0.291796 −0.0104080
\(787\) −27.0557 −0.964433 −0.482216 0.876052i \(-0.660168\pi\)
−0.482216 + 0.876052i \(0.660168\pi\)
\(788\) 2.29180 0.0816419
\(789\) 20.1246 0.716455
\(790\) 0 0
\(791\) 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 \(0.678307\pi\)
\(798\) 12.9443 0.458222
\(799\) 47.6525 1.68582
\(800\) 0 0
\(801\) 4.76393 0.168325
\(802\) −12.9443 −0.457078
\(803\) 0 0
\(804\) 18.1803 0.641171
\(805\) 0 0
\(806\) 21.7082 0.764639
\(807\) 7.05573 0.248373
\(808\) 12.3607 0.434847
\(809\) 8.00000 0.281265 0.140633 0.990062i \(-0.455086\pi\)
0.140633 + 0.990062i \(0.455086\pi\)
\(810\) 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
\(813\) 24.2361 0.849996
\(814\) 0 0
\(815\) 0 0
\(816\) −10.1459 −0.355177
\(817\) 33.8885 1.18561
\(818\) 22.7295 0.794718
\(819\) 16.9443 0.592081
\(820\) 0 0
\(821\) 33.7771 1.17883 0.589414 0.807831i \(-0.299359\pi\)
0.589414 + 0.807831i \(0.299359\pi\)
\(822\) 10.7984 0.376637
\(823\) −38.2492 −1.33328 −0.666642 0.745378i \(-0.732269\pi\)
−0.666642 + 0.745378i \(0.732269\pi\)
\(824\) 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 \(0.450297\pi\)
\(828\) −7.61803 −0.264745
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) −4.94427 −0.171515
\(832\) −1.23607 −0.0428529
\(833\) −19.0000 −0.658311
\(834\) −4.50658 −0.156050
\(835\) 0 0
\(836\) 0 0
\(837\) −6.70820 −0.231869
\(838\) 8.65248 0.298895
\(839\) 28.3607 0.979119 0.489560 0.871970i \(-0.337158\pi\)
0.489560 + 0.871970i \(0.337158\pi\)
\(840\) 0 0
\(841\) −27.4721 −0.947315
\(842\) −18.4377 −0.635405
\(843\) −15.2361 −0.524758
\(844\) 2.85410 0.0982422
\(845\) 0 0
\(846\) 5.38197 0.185036
\(847\) 0 0
\(848\) −18.4377 −0.633153
\(849\) 11.8885 0.408014
\(850\) 0 0
\(851\) −3.59675 −0.123295
\(852\) 23.4164 0.802233
\(853\) −31.8885 −1.09184 −0.545921 0.837836i \(-0.683820\pi\)
−0.545921 + 0.837836i \(0.683820\pi\)
\(854\) −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 \(0.651349\pi\)
\(860\) 0 0
\(861\) −11.4164 −0.389070
\(862\) −3.59675 −0.122506
\(863\) 37.8885 1.28974 0.644871 0.764292i \(-0.276911\pi\)
0.644871 + 0.764292i \(0.276911\pi\)
\(864\) −5.61803 −0.191129
\(865\) 0 0
\(866\) 15.5967 0.529999
\(867\) 12.9443 0.439611
\(868\) 35.1246 1.19221
\(869\) 0 0
\(870\) 0 0
\(871\) −58.8328 −1.99347
\(872\) −33.4164 −1.13162
\(873\) −12.7639 −0.431994
\(874\) 18.8328 0.637029
\(875\) 0 0
\(876\) 16.9443 0.572494
\(877\) 11.3050 0.381741 0.190871 0.981615i \(-0.438869\pi\)
0.190871 + 0.981615i \(0.438869\pi\)
\(878\) −5.02129 −0.169460
\(879\) 21.4721 0.724237
\(880\) 0 0
\(881\) 38.8328 1.30831 0.654155 0.756360i \(-0.273024\pi\)
0.654155 + 0.756360i \(0.273024\pi\)
\(882\) −2.14590 −0.0722561
\(883\) −10.1803 −0.342596 −0.171298 0.985219i \(-0.554796\pi\)
−0.171298 + 0.985219i \(0.554796\pi\)
\(884\) 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 \(0.408184\pi\)
\(888\) −1.70820 −0.0573236
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 14.6525 0.490601
\(893\) 56.3607 1.88604
\(894\) −2.65248 −0.0887121
\(895\) 0 0
\(896\) 36.8328 1.23050
\(897\) 24.6525 0.823122
\(898\) 6.18034 0.206241
\(899\) −8.29180 −0.276547
\(900\) 0 0
\(901\) 54.4164 1.81287
\(902\) 0 0
\(903\) −16.9443 −0.563870
\(904\) 27.7639 0.923415
\(905\) 0 0
\(906\) 4.50658 0.149721
\(907\) 42.0000 1.39459 0.697294 0.716786i \(-0.254387\pi\)
0.697294 + 0.716786i \(0.254387\pi\)
\(908\) −27.0344 −0.897169
\(909\) 5.52786 0.183348
\(910\) 0 0
\(911\) 47.9574 1.58890 0.794450 0.607329i \(-0.207759\pi\)
0.794450 + 0.607329i \(0.207759\pi\)
\(912\) −12.0000 −0.397360
\(913\) 0 0
\(914\) 21.2361 0.702427
\(915\) 0 0
\(916\) −11.3262 −0.374229
\(917\) 1.52786 0.0504545
\(918\) 3.38197 0.111622
\(919\) −3.41641 −0.112697 −0.0563484 0.998411i \(-0.517946\pi\)
−0.0563484 + 0.998411i \(0.517946\pi\)
\(920\) 0 0
\(921\) −11.8885 −0.391741
\(922\) 18.6525 0.614287
\(923\) −75.7771 −2.49423
\(924\) 0 0
\(925\) 0 0
\(926\) 3.70820 0.121859
\(927\) 0.944272 0.0310140
\(928\) −6.94427 −0.227957
\(929\) 50.9443 1.67143 0.835714 0.549165i \(-0.185054\pi\)
0.835714 + 0.549165i \(0.185054\pi\)
\(930\) 0 0
\(931\) −22.4721 −0.736495
\(932\) −17.6180 −0.577098
\(933\) −3.23607 −0.105944
\(934\) −2.32624 −0.0761168
\(935\) 0 0
\(936\) 11.7082 0.382695
\(937\) 33.1246 1.08213 0.541067 0.840980i \(-0.318021\pi\)
0.541067 + 0.840980i \(0.318021\pi\)
\(938\) 22.4721 0.733741
\(939\) −22.7639 −0.742873
\(940\) 0 0
\(941\) 45.2361 1.47465 0.737327 0.675536i \(-0.236088\pi\)
0.737327 + 0.675536i \(0.236088\pi\)
\(942\) 1.41641 0.0461491
\(943\) −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 \(0.404248\pi\)
\(948\) 20.5623 0.667833
\(949\) −54.8328 −1.77995
\(950\) 0 0
\(951\) 3.94427 0.127902
\(952\) −39.5967 −1.28334
\(953\) 31.8885 1.03297 0.516486 0.856296i \(-0.327240\pi\)
0.516486 + 0.856296i \(0.327240\pi\)
\(954\) 6.14590 0.198981
\(955\) 0 0
\(956\) −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
\(963\) 14.2361 0.458751
\(964\) −29.0344 −0.935136
\(965\) 0 0
\(966\) −9.41641 −0.302968
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 35.4164 1.13774
\(970\) 0 0
\(971\) 29.2361 0.938230 0.469115 0.883137i \(-0.344573\pi\)
0.469115 + 0.883137i \(0.344573\pi\)
\(972\) −1.61803 −0.0518985
\(973\) 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 \(0.417908\pi\)
\(978\) 3.81966 0.122139
\(979\) 0 0
\(980\) 0 0
\(981\) −14.9443 −0.477134
\(982\) −23.7771 −0.758757
\(983\) 55.1803 1.75998 0.879990 0.474993i \(-0.157549\pi\)
0.879990 + 0.474993i \(0.157549\pi\)
\(984\) −7.88854 −0.251478
\(985\) 0 0
\(986\) 4.18034 0.133129
\(987\) −28.1803 −0.896990
\(988\) 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
\(993\) −17.1803 −0.545202
\(994\) 28.9443 0.918057
\(995\) 0 0
\(996\) 6.47214 0.205077
\(997\) 38.7639 1.22767 0.613833 0.789436i \(-0.289627\pi\)
0.613833 + 0.789436i \(0.289627\pi\)
\(998\) 0.583592 0.0184733
\(999\) −0.763932 −0.0241697
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 9075.2.a.bx.1.1 2
5.4 even 2 1815.2.a.f.1.2 2
11.10 odd 2 9075.2.a.bd.1.2 2
15.14 odd 2 5445.2.a.x.1.1 2
55.54 odd 2 1815.2.a.j.1.1 yes 2
165.164 even 2 5445.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.2 2 5.4 even 2
1815.2.a.j.1.1 yes 2 55.54 odd 2
5445.2.a.o.1.2 2 165.164 even 2
5445.2.a.x.1.1 2 15.14 odd 2
9075.2.a.bd.1.2 2 11.10 odd 2
9075.2.a.bx.1.1 2 1.1 even 1 trivial