Properties

Label 5445.2.a.o.1.2
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
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 \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) −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 \(-0.241325\pi\)
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 \(-0.269030\pi\)
0.663594 + 0.748093i \(0.269030\pi\)
\(18\) 0 0
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) −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 \(-0.463388\pi\)
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
\(35\) 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 \(-0.588837\pi\)
−0.275480 + 0.961307i \(0.588837\pi\)
\(42\) 0 0
\(43\) −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 \(0.719046\pi\)
\(48\) 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 \(-0.530043\pi\)
−0.0942438 + 0.995549i \(0.530043\pi\)
\(62\) −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 \(-0.709971\pi\)
−0.612835 + 0.790211i \(0.709971\pi\)
\(74\) 0.472136 0.0548847
\(75\) 0 0
\(76\) −10.4721 −1.20124
\(77\) 0 0
\(78\) 0 0
\(79\) 12.7082 1.42978 0.714892 0.699235i \(-0.246476\pi\)
0.714892 + 0.699235i \(0.246476\pi\)
\(80\) 1.85410 0.207295
\(81\) 0 0
\(82\) −2.18034 −0.240778
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 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 \(-0.411315\pi\)
0.275022 + 0.961438i \(0.411315\pi\)
\(102\) 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 \(-0.741567\pi\)
−0.688126 + 0.725591i \(0.741567\pi\)
\(108\) 0 0
\(109\) 14.9443 1.43140 0.715701 0.698407i \(-0.246107\pi\)
0.715701 + 0.698407i \(0.246107\pi\)
\(110\) 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 \(-0.447391\pi\)
0.164525 + 0.986373i \(0.447391\pi\)
\(128\) −11.3820 −1.00603
\(129\) 0 0
\(130\) 3.23607 0.283822
\(131\) 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 \(0.768206\pi\)
\(138\) 0 0
\(139\) −7.29180 −0.618482 −0.309241 0.950984i \(-0.600075\pi\)
−0.309241 + 0.950984i \(0.600075\pi\)
\(140\) −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 \(-0.443749\pi\)
0.175799 + 0.984426i \(0.443749\pi\)
\(150\) 0 0
\(151\) 7.29180 0.593398 0.296699 0.954971i \(-0.404114\pi\)
0.296699 + 0.954971i \(0.404114\pi\)
\(152\) −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 \(-0.539268\pi\)
−0.123051 + 0.992400i \(0.539268\pi\)
\(168\) 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 \(-0.535701\pi\)
−0.111924 + 0.993717i \(0.535701\pi\)
\(174\) 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 \(-0.302136\pi\)
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 \(-0.483932\pi\)
0.0504574 + 0.998726i \(0.483932\pi\)
\(198\) 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 \(-0.480661\pi\)
0.0607170 + 0.998155i \(0.480661\pi\)
\(212\) 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 \(-0.687083\pi\)
−0.554481 + 0.832196i \(0.687083\pi\)
\(228\) 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 \(-0.616087\pi\)
−0.356666 + 0.934232i \(0.616087\pi\)
\(234\) 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 \(-0.206251\pi\)
0.797318 + 0.603559i \(0.206251\pi\)
\(240\) 0 0
\(241\) −17.9443 −1.15589 −0.577946 0.816075i \(-0.696146\pi\)
−0.577946 + 0.816075i \(0.696146\pi\)
\(242\) 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 \(-0.713057\pi\)
−0.620468 + 0.784231i \(0.713057\pi\)
\(264\) 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 \(-0.763343\pi\)
−0.736118 + 0.676853i \(0.763343\pi\)
\(272\) 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 \(-0.547456\pi\)
−0.148536 + 0.988907i \(0.547456\pi\)
\(278\) −4.50658 −0.270287
\(279\) 0 0
\(280\) −7.23607 −0.432438
\(281\) −15.2361 −0.908908 −0.454454 0.890770i \(-0.650166\pi\)
−0.454454 + 0.890770i \(0.650166\pi\)
\(282\) 0 0
\(283\) 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
\(288\) 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 \(-0.715802\pi\)
−0.627208 + 0.778852i \(0.715802\pi\)
\(294\) 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 \(-0.610176\pi\)
−0.339258 + 0.940694i \(0.610176\pi\)
\(308\) 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 \(-0.568929\pi\)
−0.214858 + 0.976645i \(0.568929\pi\)
\(338\) 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 \(-0.375195\pi\)
0.382116 + 0.924114i \(0.375195\pi\)
\(348\) 0 0
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 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 \(-0.562702\pi\)
−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
\(365\) −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 \(-0.218361\pi\)
0.773785 + 0.633448i \(0.218361\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 19.4721 1.00420
\(377\) 6.47214 0.333332
\(378\) 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 \(-0.136652\pi\)
0.909255 + 0.416240i \(0.136652\pi\)
\(410\) −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 \(-0.455238\pi\)
0.140162 + 0.990129i \(0.455238\pi\)
\(432\) 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 \(-0.562108\pi\)
−0.193883 + 0.981025i \(0.562108\pi\)
\(440\) 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 \(-0.797118\pi\)
−0.803662 + 0.595085i \(0.797118\pi\)
\(458\) 4.32624 0.202152
\(459\) 0 0
\(460\) −7.61803 −0.355193
\(461\) −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 \(-0.455476\pi\)
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 \(-0.497878\pi\)
0.00666625 + 0.999978i \(0.497878\pi\)
\(480\) 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 \(-0.165334\pi\)
0.868112 + 0.496369i \(0.165334\pi\)
\(492\) 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 \(-0.327130\pi\)
0.516780 + 0.856118i \(0.327130\pi\)
\(504\) 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 \(-0.680273\pi\)
−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
\(530\) −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 \(-0.880986\pi\)
−0.930913 + 0.365242i \(0.880986\pi\)
\(542\) −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 \(-0.565773\pi\)
−0.205164 + 0.978728i \(0.565773\pi\)
\(548\) 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 \(-0.442174\pi\)
0.180669 + 0.983544i \(0.442174\pi\)
\(558\) 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 \(-0.493666\pi\)
0.0198982 + 0.999802i \(0.493666\pi\)
\(564\) 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 \(-0.364444\pi\)
0.413105 + 0.910683i \(0.364444\pi\)
\(570\) 0 0
\(571\) 0.124612 0.00521484 0.00260742 0.999997i \(-0.499170\pi\)
0.00260742 + 0.999997i \(0.499170\pi\)
\(572\) 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 \(-0.245289\pi\)
0.717495 + 0.696564i \(0.245289\pi\)
\(594\) 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 \(-0.302778\pi\)
0.580701 + 0.814117i \(0.302778\pi\)
\(602\) −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 \(-0.459244\pi\)
0.127688 + 0.991814i \(0.459244\pi\)
\(608\) 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 \(-0.596835\pi\)
−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 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 \(-0.743855\pi\)
−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
\(663\) 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 \(-0.678517\pi\)
−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 \(-0.533191\pi\)
−0.104085 + 0.994568i \(0.533191\pi\)
\(678\) 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 \(-0.739437\pi\)
−0.683256 + 0.730179i \(0.739437\pi\)
\(702\) 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 \(-0.154567\pi\)
0.884402 + 0.466726i \(0.154567\pi\)
\(734\) 2.94427 0.108675
\(735\) 0 0
\(736\) 26.4508 0.974991
\(737\) 0 0
\(738\) 0 0
\(739\) 1.65248 0.0607873 0.0303937 0.999538i \(-0.490324\pi\)
0.0303937 + 0.999538i \(0.490324\pi\)
\(740\) −1.23607 −0.0454388
\(741\) 0 0
\(742\) −19.8885 −0.730131
\(743\) −2.23607 −0.0820334 −0.0410167 0.999158i \(-0.513060\pi\)
−0.0410167 + 0.999158i \(0.513060\pi\)
\(744\) 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 \(-0.199833\pi\)
0.809325 + 0.587360i \(0.199833\pi\)
\(762\) 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 \(-0.645819\pi\)
−0.442249 + 0.896892i \(0.645819\pi\)
\(770\) 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 \(-0.660168\pi\)
−0.482216 + 0.876052i \(0.660168\pi\)
\(788\) −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 \(-0.455086\pi\)
0.140633 + 0.990062i \(0.455086\pi\)
\(810\) 0 0
\(811\) 7.87539 0.276542 0.138271 0.990394i \(-0.455845\pi\)
0.138271 + 0.990394i \(0.455845\pi\)
\(812\) −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 \(-0.299359\pi\)
0.589414 + 0.807831i \(0.299359\pi\)
\(822\) 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 \(-0.549703\pi\)
−0.155511 + 0.987834i \(0.549703\pi\)
\(828\) 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 \(-0.683820\pi\)
−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 \(-0.505437\pi\)
−0.0170797 + 0.999854i \(0.505437\pi\)
\(858\) 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 \(-0.438869\pi\)
0.190871 + 0.981615i \(0.438869\pi\)
\(878\) −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 \(-0.591816\pi\)
−0.284466 + 0.958686i \(0.591816\pi\)
\(888\) 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 \(-0.482054\pi\)
0.0563484 + 0.998411i \(0.482054\pi\)
\(920\) −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 \(-0.318021\pi\)
0.541067 + 0.840980i \(0.318021\pi\)
\(938\) 22.4721 0.733741
\(939\) 0 0
\(940\) 14.0902 0.459571
\(941\) 45.2361 1.47465 0.737327 0.675536i \(-0.236088\pi\)
0.737327 + 0.675536i \(0.236088\pi\)
\(942\) 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 \(-0.672760\pi\)
−0.516486 + 0.856296i \(0.672760\pi\)
\(954\) 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 \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 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 \(-0.289627\pi\)
0.613833 + 0.789436i \(0.289627\pi\)
\(998\) −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.o.1.2 2
3.2 odd 2 1815.2.a.j.1.1 yes 2
11.10 odd 2 5445.2.a.x.1.1 2
15.14 odd 2 9075.2.a.bd.1.2 2
33.32 even 2 1815.2.a.f.1.2 2
165.164 even 2 9075.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.2 2 33.32 even 2
1815.2.a.j.1.1 yes 2 3.2 odd 2
5445.2.a.o.1.2 2 1.1 even 1 trivial
5445.2.a.x.1.1 2 11.10 odd 2
9075.2.a.bd.1.2 2 15.14 odd 2
9075.2.a.bx.1.1 2 165.164 even 2