Properties

Label 7623.2.a.w.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.85410 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.85410 q^{5} -1.00000 q^{7} +3.00000 q^{8} -2.85410 q^{10} +1.23607 q^{13} +1.00000 q^{14} -1.00000 q^{16} +7.85410 q^{17} -2.61803 q^{19} -2.85410 q^{20} +3.09017 q^{23} +3.14590 q^{25} -1.23607 q^{26} +1.00000 q^{28} -2.00000 q^{29} -7.32624 q^{31} -5.00000 q^{32} -7.85410 q^{34} -2.85410 q^{35} -12.0902 q^{37} +2.61803 q^{38} +8.56231 q^{40} -10.8541 q^{41} +1.23607 q^{43} -3.09017 q^{46} -2.00000 q^{47} +1.00000 q^{49} -3.14590 q^{50} -1.23607 q^{52} -12.1803 q^{53} -3.00000 q^{56} +2.00000 q^{58} -6.76393 q^{59} -8.94427 q^{61} +7.32624 q^{62} +7.00000 q^{64} +3.52786 q^{65} +8.00000 q^{67} -7.85410 q^{68} +2.85410 q^{70} -8.94427 q^{71} +5.23607 q^{73} +12.0902 q^{74} +2.61803 q^{76} +14.0000 q^{79} -2.85410 q^{80} +10.8541 q^{82} +2.94427 q^{83} +22.4164 q^{85} -1.23607 q^{86} -1.09017 q^{89} -1.23607 q^{91} -3.09017 q^{92} +2.00000 q^{94} -7.47214 q^{95} -3.52786 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{4} - q^{5} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{4} - q^{5} - 2q^{7} + 6q^{8} + q^{10} - 2q^{13} + 2q^{14} - 2q^{16} + 9q^{17} - 3q^{19} + q^{20} - 5q^{23} + 13q^{25} + 2q^{26} + 2q^{28} - 4q^{29} + q^{31} - 10q^{32} - 9q^{34} + q^{35} - 13q^{37} + 3q^{38} - 3q^{40} - 15q^{41} - 2q^{43} + 5q^{46} - 4q^{47} + 2q^{49} - 13q^{50} + 2q^{52} - 2q^{53} - 6q^{56} + 4q^{58} - 18q^{59} - q^{62} + 14q^{64} + 16q^{65} + 16q^{67} - 9q^{68} - q^{70} + 6q^{73} + 13q^{74} + 3q^{76} + 28q^{79} + q^{80} + 15q^{82} - 12q^{83} + 18q^{85} + 2q^{86} + 9q^{89} + 2q^{91} + 5q^{92} + 4q^{94} - 6q^{95} - 16q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.85410 1.27639 0.638197 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −2.85410 −0.902546
\(11\) 0 0
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 7.85410 1.90490 0.952450 0.304696i \(-0.0985548\pi\)
0.952450 + 0.304696i \(0.0985548\pi\)
\(18\) 0 0
\(19\) −2.61803 −0.600618 −0.300309 0.953842i \(-0.597090\pi\)
−0.300309 + 0.953842i \(0.597090\pi\)
\(20\) −2.85410 −0.638197
\(21\) 0 0
\(22\) 0 0
\(23\) 3.09017 0.644345 0.322172 0.946681i \(-0.395587\pi\)
0.322172 + 0.946681i \(0.395587\pi\)
\(24\) 0 0
\(25\) 3.14590 0.629180
\(26\) −1.23607 −0.242413
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −7.32624 −1.31583 −0.657916 0.753092i \(-0.728562\pi\)
−0.657916 + 0.753092i \(0.728562\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −7.85410 −1.34697
\(35\) −2.85410 −0.482431
\(36\) 0 0
\(37\) −12.0902 −1.98761 −0.993806 0.111130i \(-0.964553\pi\)
−0.993806 + 0.111130i \(0.964553\pi\)
\(38\) 2.61803 0.424701
\(39\) 0 0
\(40\) 8.56231 1.35382
\(41\) −10.8541 −1.69513 −0.847563 0.530695i \(-0.821931\pi\)
−0.847563 + 0.530695i \(0.821931\pi\)
\(42\) 0 0
\(43\) 1.23607 0.188499 0.0942493 0.995549i \(-0.469955\pi\)
0.0942493 + 0.995549i \(0.469955\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.09017 −0.455621
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.14590 −0.444897
\(51\) 0 0
\(52\) −1.23607 −0.171412
\(53\) −12.1803 −1.67310 −0.836549 0.547892i \(-0.815431\pi\)
−0.836549 + 0.547892i \(0.815431\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −6.76393 −0.880589 −0.440294 0.897853i \(-0.645126\pi\)
−0.440294 + 0.897853i \(0.645126\pi\)
\(60\) 0 0
\(61\) −8.94427 −1.14520 −0.572598 0.819836i \(-0.694065\pi\)
−0.572598 + 0.819836i \(0.694065\pi\)
\(62\) 7.32624 0.930433
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 3.52786 0.437578
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −7.85410 −0.952450
\(69\) 0 0
\(70\) 2.85410 0.341130
\(71\) −8.94427 −1.06149 −0.530745 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) 5.23607 0.612835 0.306418 0.951897i \(-0.400870\pi\)
0.306418 + 0.951897i \(0.400870\pi\)
\(74\) 12.0902 1.40545
\(75\) 0 0
\(76\) 2.61803 0.300309
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) −2.85410 −0.319098
\(81\) 0 0
\(82\) 10.8541 1.19864
\(83\) 2.94427 0.323176 0.161588 0.986858i \(-0.448338\pi\)
0.161588 + 0.986858i \(0.448338\pi\)
\(84\) 0 0
\(85\) 22.4164 2.43140
\(86\) −1.23607 −0.133289
\(87\) 0 0
\(88\) 0 0
\(89\) −1.09017 −0.115558 −0.0577789 0.998329i \(-0.518402\pi\)
−0.0577789 + 0.998329i \(0.518402\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) −3.09017 −0.322172
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) −7.47214 −0.766625
\(96\) 0 0
\(97\) −3.52786 −0.358200 −0.179100 0.983831i \(-0.557319\pi\)
−0.179100 + 0.983831i \(0.557319\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −3.14590 −0.314590
\(101\) −2.09017 −0.207980 −0.103990 0.994578i \(-0.533161\pi\)
−0.103990 + 0.994578i \(0.533161\pi\)
\(102\) 0 0
\(103\) −10.3262 −1.01747 −0.508737 0.860922i \(-0.669888\pi\)
−0.508737 + 0.860922i \(0.669888\pi\)
\(104\) 3.70820 0.363619
\(105\) 0 0
\(106\) 12.1803 1.18306
\(107\) 4.90983 0.474651 0.237326 0.971430i \(-0.423729\pi\)
0.237326 + 0.971430i \(0.423729\pi\)
\(108\) 0 0
\(109\) −10.5623 −1.01169 −0.505843 0.862626i \(-0.668818\pi\)
−0.505843 + 0.862626i \(0.668818\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −0.763932 −0.0718647 −0.0359323 0.999354i \(-0.511440\pi\)
−0.0359323 + 0.999354i \(0.511440\pi\)
\(114\) 0 0
\(115\) 8.81966 0.822438
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 6.76393 0.622670
\(119\) −7.85410 −0.719984
\(120\) 0 0
\(121\) 0 0
\(122\) 8.94427 0.809776
\(123\) 0 0
\(124\) 7.32624 0.657916
\(125\) −5.29180 −0.473313
\(126\) 0 0
\(127\) 3.23607 0.287155 0.143577 0.989639i \(-0.454139\pi\)
0.143577 + 0.989639i \(0.454139\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −3.52786 −0.309414
\(131\) 2.29180 0.200235 0.100118 0.994976i \(-0.468078\pi\)
0.100118 + 0.994976i \(0.468078\pi\)
\(132\) 0 0
\(133\) 2.61803 0.227012
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 23.5623 2.02045
\(137\) −6.76393 −0.577882 −0.288941 0.957347i \(-0.593303\pi\)
−0.288941 + 0.957347i \(0.593303\pi\)
\(138\) 0 0
\(139\) −4.79837 −0.406993 −0.203496 0.979076i \(-0.565231\pi\)
−0.203496 + 0.979076i \(0.565231\pi\)
\(140\) 2.85410 0.241216
\(141\) 0 0
\(142\) 8.94427 0.750587
\(143\) 0 0
\(144\) 0 0
\(145\) −5.70820 −0.474041
\(146\) −5.23607 −0.433340
\(147\) 0 0
\(148\) 12.0902 0.993806
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −7.85410 −0.637052
\(153\) 0 0
\(154\) 0 0
\(155\) −20.9098 −1.67952
\(156\) 0 0
\(157\) 9.70820 0.774799 0.387400 0.921912i \(-0.373373\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) −14.2705 −1.12818
\(161\) −3.09017 −0.243540
\(162\) 0 0
\(163\) −5.41641 −0.424246 −0.212123 0.977243i \(-0.568038\pi\)
−0.212123 + 0.977243i \(0.568038\pi\)
\(164\) 10.8541 0.847563
\(165\) 0 0
\(166\) −2.94427 −0.228520
\(167\) 8.18034 0.633014 0.316507 0.948590i \(-0.397490\pi\)
0.316507 + 0.948590i \(0.397490\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) −22.4164 −1.71926
\(171\) 0 0
\(172\) −1.23607 −0.0942493
\(173\) 19.8541 1.50948 0.754740 0.656024i \(-0.227763\pi\)
0.754740 + 0.656024i \(0.227763\pi\)
\(174\) 0 0
\(175\) −3.14590 −0.237808
\(176\) 0 0
\(177\) 0 0
\(178\) 1.09017 0.0817117
\(179\) −8.38197 −0.626498 −0.313249 0.949671i \(-0.601417\pi\)
−0.313249 + 0.949671i \(0.601417\pi\)
\(180\) 0 0
\(181\) −18.9443 −1.40812 −0.704058 0.710142i \(-0.748631\pi\)
−0.704058 + 0.710142i \(0.748631\pi\)
\(182\) 1.23607 0.0916235
\(183\) 0 0
\(184\) 9.27051 0.683431
\(185\) −34.5066 −2.53697
\(186\) 0 0
\(187\) 0 0
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 7.47214 0.542086
\(191\) 18.2705 1.32201 0.661004 0.750383i \(-0.270131\pi\)
0.661004 + 0.750383i \(0.270131\pi\)
\(192\) 0 0
\(193\) −18.3820 −1.32316 −0.661581 0.749873i \(-0.730114\pi\)
−0.661581 + 0.749873i \(0.730114\pi\)
\(194\) 3.52786 0.253286
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) −2.61803 −0.185588 −0.0927938 0.995685i \(-0.529580\pi\)
−0.0927938 + 0.995685i \(0.529580\pi\)
\(200\) 9.43769 0.667346
\(201\) 0 0
\(202\) 2.09017 0.147064
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −30.9787 −2.16365
\(206\) 10.3262 0.719463
\(207\) 0 0
\(208\) −1.23607 −0.0857059
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 12.1803 0.836549
\(213\) 0 0
\(214\) −4.90983 −0.335629
\(215\) 3.52786 0.240598
\(216\) 0 0
\(217\) 7.32624 0.497337
\(218\) 10.5623 0.715370
\(219\) 0 0
\(220\) 0 0
\(221\) 9.70820 0.653044
\(222\) 0 0
\(223\) 20.7984 1.39276 0.696381 0.717672i \(-0.254792\pi\)
0.696381 + 0.717672i \(0.254792\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 0.763932 0.0508160
\(227\) 21.4164 1.42146 0.710728 0.703466i \(-0.248365\pi\)
0.710728 + 0.703466i \(0.248365\pi\)
\(228\) 0 0
\(229\) 21.2361 1.40332 0.701659 0.712512i \(-0.252443\pi\)
0.701659 + 0.712512i \(0.252443\pi\)
\(230\) −8.81966 −0.581551
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −5.70820 −0.373957 −0.186978 0.982364i \(-0.559869\pi\)
−0.186978 + 0.982364i \(0.559869\pi\)
\(234\) 0 0
\(235\) −5.70820 −0.372362
\(236\) 6.76393 0.440294
\(237\) 0 0
\(238\) 7.85410 0.509106
\(239\) −1.56231 −0.101057 −0.0505286 0.998723i \(-0.516091\pi\)
−0.0505286 + 0.998723i \(0.516091\pi\)
\(240\) 0 0
\(241\) −4.94427 −0.318489 −0.159244 0.987239i \(-0.550906\pi\)
−0.159244 + 0.987239i \(0.550906\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 8.94427 0.572598
\(245\) 2.85410 0.182342
\(246\) 0 0
\(247\) −3.23607 −0.205906
\(248\) −21.9787 −1.39565
\(249\) 0 0
\(250\) 5.29180 0.334683
\(251\) −24.1803 −1.52625 −0.763125 0.646251i \(-0.776336\pi\)
−0.763125 + 0.646251i \(0.776336\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3.23607 −0.203049
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −2.61803 −0.163308 −0.0816542 0.996661i \(-0.526020\pi\)
−0.0816542 + 0.996661i \(0.526020\pi\)
\(258\) 0 0
\(259\) 12.0902 0.751247
\(260\) −3.52786 −0.218789
\(261\) 0 0
\(262\) −2.29180 −0.141588
\(263\) 2.43769 0.150315 0.0751573 0.997172i \(-0.476054\pi\)
0.0751573 + 0.997172i \(0.476054\pi\)
\(264\) 0 0
\(265\) −34.7639 −2.13553
\(266\) −2.61803 −0.160522
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −27.8885 −1.70039 −0.850197 0.526464i \(-0.823517\pi\)
−0.850197 + 0.526464i \(0.823517\pi\)
\(270\) 0 0
\(271\) −24.0902 −1.46337 −0.731687 0.681641i \(-0.761267\pi\)
−0.731687 + 0.681641i \(0.761267\pi\)
\(272\) −7.85410 −0.476225
\(273\) 0 0
\(274\) 6.76393 0.408624
\(275\) 0 0
\(276\) 0 0
\(277\) 10.4377 0.627140 0.313570 0.949565i \(-0.398475\pi\)
0.313570 + 0.949565i \(0.398475\pi\)
\(278\) 4.79837 0.287787
\(279\) 0 0
\(280\) −8.56231 −0.511696
\(281\) 1.05573 0.0629795 0.0314897 0.999504i \(-0.489975\pi\)
0.0314897 + 0.999504i \(0.489975\pi\)
\(282\) 0 0
\(283\) −13.4377 −0.798788 −0.399394 0.916779i \(-0.630779\pi\)
−0.399394 + 0.916779i \(0.630779\pi\)
\(284\) 8.94427 0.530745
\(285\) 0 0
\(286\) 0 0
\(287\) 10.8541 0.640697
\(288\) 0 0
\(289\) 44.6869 2.62864
\(290\) 5.70820 0.335197
\(291\) 0 0
\(292\) −5.23607 −0.306418
\(293\) 19.7984 1.15663 0.578317 0.815812i \(-0.303710\pi\)
0.578317 + 0.815812i \(0.303710\pi\)
\(294\) 0 0
\(295\) −19.3050 −1.12398
\(296\) −36.2705 −2.10818
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) 3.81966 0.220897
\(300\) 0 0
\(301\) −1.23607 −0.0712458
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 2.61803 0.150155
\(305\) −25.5279 −1.46172
\(306\) 0 0
\(307\) 24.2705 1.38519 0.692596 0.721326i \(-0.256467\pi\)
0.692596 + 0.721326i \(0.256467\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 20.9098 1.18760
\(311\) 0.763932 0.0433186 0.0216593 0.999765i \(-0.493105\pi\)
0.0216593 + 0.999765i \(0.493105\pi\)
\(312\) 0 0
\(313\) −9.81966 −0.555040 −0.277520 0.960720i \(-0.589512\pi\)
−0.277520 + 0.960720i \(0.589512\pi\)
\(314\) −9.70820 −0.547866
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) 27.5967 1.54999 0.774994 0.631969i \(-0.217753\pi\)
0.774994 + 0.631969i \(0.217753\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 19.9787 1.11684
\(321\) 0 0
\(322\) 3.09017 0.172208
\(323\) −20.5623 −1.14412
\(324\) 0 0
\(325\) 3.88854 0.215698
\(326\) 5.41641 0.299987
\(327\) 0 0
\(328\) −32.5623 −1.79795
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 30.3607 1.66877 0.834387 0.551179i \(-0.185822\pi\)
0.834387 + 0.551179i \(0.185822\pi\)
\(332\) −2.94427 −0.161588
\(333\) 0 0
\(334\) −8.18034 −0.447608
\(335\) 22.8328 1.24749
\(336\) 0 0
\(337\) 27.1459 1.47873 0.739366 0.673304i \(-0.235126\pi\)
0.739366 + 0.673304i \(0.235126\pi\)
\(338\) 11.4721 0.624002
\(339\) 0 0
\(340\) −22.4164 −1.21570
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 3.70820 0.199933
\(345\) 0 0
\(346\) −19.8541 −1.06736
\(347\) −12.3820 −0.664699 −0.332349 0.943156i \(-0.607841\pi\)
−0.332349 + 0.943156i \(0.607841\pi\)
\(348\) 0 0
\(349\) −22.7639 −1.21853 −0.609263 0.792968i \(-0.708534\pi\)
−0.609263 + 0.792968i \(0.708534\pi\)
\(350\) 3.14590 0.168155
\(351\) 0 0
\(352\) 0 0
\(353\) −12.4721 −0.663825 −0.331912 0.943310i \(-0.607694\pi\)
−0.331912 + 0.943310i \(0.607694\pi\)
\(354\) 0 0
\(355\) −25.5279 −1.35488
\(356\) 1.09017 0.0577789
\(357\) 0 0
\(358\) 8.38197 0.443001
\(359\) 7.03444 0.371264 0.185632 0.982619i \(-0.440567\pi\)
0.185632 + 0.982619i \(0.440567\pi\)
\(360\) 0 0
\(361\) −12.1459 −0.639258
\(362\) 18.9443 0.995689
\(363\) 0 0
\(364\) 1.23607 0.0647876
\(365\) 14.9443 0.782219
\(366\) 0 0
\(367\) 16.2705 0.849314 0.424657 0.905354i \(-0.360395\pi\)
0.424657 + 0.905354i \(0.360395\pi\)
\(368\) −3.09017 −0.161086
\(369\) 0 0
\(370\) 34.5066 1.79391
\(371\) 12.1803 0.632372
\(372\) 0 0
\(373\) 23.3262 1.20779 0.603893 0.797065i \(-0.293615\pi\)
0.603893 + 0.797065i \(0.293615\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −2.47214 −0.127321
\(378\) 0 0
\(379\) 2.47214 0.126985 0.0634925 0.997982i \(-0.479776\pi\)
0.0634925 + 0.997982i \(0.479776\pi\)
\(380\) 7.47214 0.383312
\(381\) 0 0
\(382\) −18.2705 −0.934801
\(383\) 13.0557 0.667117 0.333558 0.942729i \(-0.391751\pi\)
0.333558 + 0.942729i \(0.391751\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.3820 0.935617
\(387\) 0 0
\(388\) 3.52786 0.179100
\(389\) −22.1803 −1.12459 −0.562294 0.826937i \(-0.690081\pi\)
−0.562294 + 0.826937i \(0.690081\pi\)
\(390\) 0 0
\(391\) 24.2705 1.22741
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) 39.9574 2.01048
\(396\) 0 0
\(397\) 14.6525 0.735387 0.367693 0.929947i \(-0.380148\pi\)
0.367693 + 0.929947i \(0.380148\pi\)
\(398\) 2.61803 0.131230
\(399\) 0 0
\(400\) −3.14590 −0.157295
\(401\) −18.4721 −0.922454 −0.461227 0.887282i \(-0.652591\pi\)
−0.461227 + 0.887282i \(0.652591\pi\)
\(402\) 0 0
\(403\) −9.05573 −0.451098
\(404\) 2.09017 0.103990
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) −28.7639 −1.42228 −0.711142 0.703048i \(-0.751822\pi\)
−0.711142 + 0.703048i \(0.751822\pi\)
\(410\) 30.9787 1.52993
\(411\) 0 0
\(412\) 10.3262 0.508737
\(413\) 6.76393 0.332831
\(414\) 0 0
\(415\) 8.40325 0.412499
\(416\) −6.18034 −0.303016
\(417\) 0 0
\(418\) 0 0
\(419\) −14.7639 −0.721265 −0.360633 0.932708i \(-0.617439\pi\)
−0.360633 + 0.932708i \(0.617439\pi\)
\(420\) 0 0
\(421\) −4.32624 −0.210848 −0.105424 0.994427i \(-0.533620\pi\)
−0.105424 + 0.994427i \(0.533620\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 0 0
\(424\) −36.5410 −1.77459
\(425\) 24.7082 1.19852
\(426\) 0 0
\(427\) 8.94427 0.432844
\(428\) −4.90983 −0.237326
\(429\) 0 0
\(430\) −3.52786 −0.170129
\(431\) −2.09017 −0.100680 −0.0503400 0.998732i \(-0.516030\pi\)
−0.0503400 + 0.998732i \(0.516030\pi\)
\(432\) 0 0
\(433\) −17.2361 −0.828313 −0.414156 0.910206i \(-0.635923\pi\)
−0.414156 + 0.910206i \(0.635923\pi\)
\(434\) −7.32624 −0.351671
\(435\) 0 0
\(436\) 10.5623 0.505843
\(437\) −8.09017 −0.387005
\(438\) 0 0
\(439\) 21.7984 1.04038 0.520190 0.854051i \(-0.325861\pi\)
0.520190 + 0.854051i \(0.325861\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −9.70820 −0.461772
\(443\) −4.90983 −0.233273 −0.116637 0.993175i \(-0.537211\pi\)
−0.116637 + 0.993175i \(0.537211\pi\)
\(444\) 0 0
\(445\) −3.11146 −0.147497
\(446\) −20.7984 −0.984832
\(447\) 0 0
\(448\) −7.00000 −0.330719
\(449\) −41.8885 −1.97684 −0.988421 0.151734i \(-0.951514\pi\)
−0.988421 + 0.151734i \(0.951514\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.763932 0.0359323
\(453\) 0 0
\(454\) −21.4164 −1.00512
\(455\) −3.52786 −0.165389
\(456\) 0 0
\(457\) −12.4721 −0.583422 −0.291711 0.956507i \(-0.594225\pi\)
−0.291711 + 0.956507i \(0.594225\pi\)
\(458\) −21.2361 −0.992296
\(459\) 0 0
\(460\) −8.81966 −0.411219
\(461\) −6.36068 −0.296246 −0.148123 0.988969i \(-0.547323\pi\)
−0.148123 + 0.988969i \(0.547323\pi\)
\(462\) 0 0
\(463\) 11.4164 0.530565 0.265283 0.964171i \(-0.414535\pi\)
0.265283 + 0.964171i \(0.414535\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 5.70820 0.264427
\(467\) −19.2361 −0.890139 −0.445070 0.895496i \(-0.646821\pi\)
−0.445070 + 0.895496i \(0.646821\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 5.70820 0.263300
\(471\) 0 0
\(472\) −20.2918 −0.934006
\(473\) 0 0
\(474\) 0 0
\(475\) −8.23607 −0.377897
\(476\) 7.85410 0.359992
\(477\) 0 0
\(478\) 1.56231 0.0714582
\(479\) −11.0557 −0.505149 −0.252575 0.967577i \(-0.581277\pi\)
−0.252575 + 0.967577i \(0.581277\pi\)
\(480\) 0 0
\(481\) −14.9443 −0.681400
\(482\) 4.94427 0.225205
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0689 −0.457204
\(486\) 0 0
\(487\) −39.2361 −1.77796 −0.888978 0.457950i \(-0.848584\pi\)
−0.888978 + 0.457950i \(0.848584\pi\)
\(488\) −26.8328 −1.21466
\(489\) 0 0
\(490\) −2.85410 −0.128935
\(491\) −0.326238 −0.0147229 −0.00736146 0.999973i \(-0.502343\pi\)
−0.00736146 + 0.999973i \(0.502343\pi\)
\(492\) 0 0
\(493\) −15.7082 −0.707462
\(494\) 3.23607 0.145598
\(495\) 0 0
\(496\) 7.32624 0.328958
\(497\) 8.94427 0.401205
\(498\) 0 0
\(499\) 26.1803 1.17199 0.585996 0.810314i \(-0.300703\pi\)
0.585996 + 0.810314i \(0.300703\pi\)
\(500\) 5.29180 0.236656
\(501\) 0 0
\(502\) 24.1803 1.07922
\(503\) −0.583592 −0.0260211 −0.0130105 0.999915i \(-0.504142\pi\)
−0.0130105 + 0.999915i \(0.504142\pi\)
\(504\) 0 0
\(505\) −5.96556 −0.265464
\(506\) 0 0
\(507\) 0 0
\(508\) −3.23607 −0.143577
\(509\) 5.90983 0.261949 0.130974 0.991386i \(-0.458189\pi\)
0.130974 + 0.991386i \(0.458189\pi\)
\(510\) 0 0
\(511\) −5.23607 −0.231630
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 2.61803 0.115477
\(515\) −29.4721 −1.29870
\(516\) 0 0
\(517\) 0 0
\(518\) −12.0902 −0.531212
\(519\) 0 0
\(520\) 10.5836 0.464121
\(521\) 11.6180 0.508995 0.254498 0.967073i \(-0.418090\pi\)
0.254498 + 0.967073i \(0.418090\pi\)
\(522\) 0 0
\(523\) −2.90983 −0.127238 −0.0636190 0.997974i \(-0.520264\pi\)
−0.0636190 + 0.997974i \(0.520264\pi\)
\(524\) −2.29180 −0.100118
\(525\) 0 0
\(526\) −2.43769 −0.106289
\(527\) −57.5410 −2.50653
\(528\) 0 0
\(529\) −13.4508 −0.584820
\(530\) 34.7639 1.51005
\(531\) 0 0
\(532\) −2.61803 −0.113506
\(533\) −13.4164 −0.581129
\(534\) 0 0
\(535\) 14.0132 0.605842
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) 27.8885 1.20236
\(539\) 0 0
\(540\) 0 0
\(541\) −16.8541 −0.724614 −0.362307 0.932059i \(-0.618011\pi\)
−0.362307 + 0.932059i \(0.618011\pi\)
\(542\) 24.0902 1.03476
\(543\) 0 0
\(544\) −39.2705 −1.68371
\(545\) −30.1459 −1.29131
\(546\) 0 0
\(547\) 36.4721 1.55944 0.779718 0.626131i \(-0.215362\pi\)
0.779718 + 0.626131i \(0.215362\pi\)
\(548\) 6.76393 0.288941
\(549\) 0 0
\(550\) 0 0
\(551\) 5.23607 0.223064
\(552\) 0 0
\(553\) −14.0000 −0.595341
\(554\) −10.4377 −0.443455
\(555\) 0 0
\(556\) 4.79837 0.203496
\(557\) −21.4164 −0.907442 −0.453721 0.891144i \(-0.649904\pi\)
−0.453721 + 0.891144i \(0.649904\pi\)
\(558\) 0 0
\(559\) 1.52786 0.0646218
\(560\) 2.85410 0.120608
\(561\) 0 0
\(562\) −1.05573 −0.0445332
\(563\) 40.3607 1.70100 0.850500 0.525975i \(-0.176300\pi\)
0.850500 + 0.525975i \(0.176300\pi\)
\(564\) 0 0
\(565\) −2.18034 −0.0917276
\(566\) 13.4377 0.564828
\(567\) 0 0
\(568\) −26.8328 −1.12588
\(569\) −18.1803 −0.762159 −0.381080 0.924542i \(-0.624448\pi\)
−0.381080 + 0.924542i \(0.624448\pi\)
\(570\) 0 0
\(571\) −28.5410 −1.19440 −0.597202 0.802091i \(-0.703721\pi\)
−0.597202 + 0.802091i \(0.703721\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −10.8541 −0.453041
\(575\) 9.72136 0.405409
\(576\) 0 0
\(577\) −14.6525 −0.609991 −0.304995 0.952354i \(-0.598655\pi\)
−0.304995 + 0.952354i \(0.598655\pi\)
\(578\) −44.6869 −1.85873
\(579\) 0 0
\(580\) 5.70820 0.237020
\(581\) −2.94427 −0.122149
\(582\) 0 0
\(583\) 0 0
\(584\) 15.7082 0.650010
\(585\) 0 0
\(586\) −19.7984 −0.817863
\(587\) −3.88854 −0.160497 −0.0802487 0.996775i \(-0.525571\pi\)
−0.0802487 + 0.996775i \(0.525571\pi\)
\(588\) 0 0
\(589\) 19.1803 0.790312
\(590\) 19.3050 0.794772
\(591\) 0 0
\(592\) 12.0902 0.496903
\(593\) 3.43769 0.141169 0.0705846 0.997506i \(-0.477514\pi\)
0.0705846 + 0.997506i \(0.477514\pi\)
\(594\) 0 0
\(595\) −22.4164 −0.918983
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) −3.81966 −0.156198
\(599\) 1.56231 0.0638341 0.0319170 0.999491i \(-0.489839\pi\)
0.0319170 + 0.999491i \(0.489839\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 1.23607 0.0503784
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 10.3262 0.419129 0.209565 0.977795i \(-0.432795\pi\)
0.209565 + 0.977795i \(0.432795\pi\)
\(608\) 13.0902 0.530876
\(609\) 0 0
\(610\) 25.5279 1.03359
\(611\) −2.47214 −0.100012
\(612\) 0 0
\(613\) 5.85410 0.236445 0.118222 0.992987i \(-0.462280\pi\)
0.118222 + 0.992987i \(0.462280\pi\)
\(614\) −24.2705 −0.979478
\(615\) 0 0
\(616\) 0 0
\(617\) −44.6525 −1.79764 −0.898820 0.438317i \(-0.855575\pi\)
−0.898820 + 0.438317i \(0.855575\pi\)
\(618\) 0 0
\(619\) −11.5066 −0.462488 −0.231244 0.972896i \(-0.574280\pi\)
−0.231244 + 0.972896i \(0.574280\pi\)
\(620\) 20.9098 0.839759
\(621\) 0 0
\(622\) −0.763932 −0.0306309
\(623\) 1.09017 0.0436767
\(624\) 0 0
\(625\) −30.8328 −1.23331
\(626\) 9.81966 0.392473
\(627\) 0 0
\(628\) −9.70820 −0.387400
\(629\) −94.9574 −3.78620
\(630\) 0 0
\(631\) 29.8885 1.18984 0.594922 0.803783i \(-0.297183\pi\)
0.594922 + 0.803783i \(0.297183\pi\)
\(632\) 42.0000 1.67067
\(633\) 0 0
\(634\) −27.5967 −1.09601
\(635\) 9.23607 0.366522
\(636\) 0 0
\(637\) 1.23607 0.0489748
\(638\) 0 0
\(639\) 0 0
\(640\) 8.56231 0.338455
\(641\) −8.65248 −0.341752 −0.170876 0.985293i \(-0.554660\pi\)
−0.170876 + 0.985293i \(0.554660\pi\)
\(642\) 0 0
\(643\) 47.9230 1.88990 0.944949 0.327218i \(-0.106111\pi\)
0.944949 + 0.327218i \(0.106111\pi\)
\(644\) 3.09017 0.121770
\(645\) 0 0
\(646\) 20.5623 0.809013
\(647\) 30.3607 1.19360 0.596801 0.802389i \(-0.296438\pi\)
0.596801 + 0.802389i \(0.296438\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −3.88854 −0.152521
\(651\) 0 0
\(652\) 5.41641 0.212123
\(653\) 25.8885 1.01310 0.506549 0.862211i \(-0.330921\pi\)
0.506549 + 0.862211i \(0.330921\pi\)
\(654\) 0 0
\(655\) 6.54102 0.255579
\(656\) 10.8541 0.423781
\(657\) 0 0
\(658\) −2.00000 −0.0779681
\(659\) −35.0344 −1.36475 −0.682374 0.731003i \(-0.739052\pi\)
−0.682374 + 0.731003i \(0.739052\pi\)
\(660\) 0 0
\(661\) 8.29180 0.322513 0.161257 0.986912i \(-0.448445\pi\)
0.161257 + 0.986912i \(0.448445\pi\)
\(662\) −30.3607 −1.18000
\(663\) 0 0
\(664\) 8.83282 0.342780
\(665\) 7.47214 0.289757
\(666\) 0 0
\(667\) −6.18034 −0.239304
\(668\) −8.18034 −0.316507
\(669\) 0 0
\(670\) −22.8328 −0.882109
\(671\) 0 0
\(672\) 0 0
\(673\) −35.8885 −1.38340 −0.691701 0.722184i \(-0.743138\pi\)
−0.691701 + 0.722184i \(0.743138\pi\)
\(674\) −27.1459 −1.04562
\(675\) 0 0
\(676\) 11.4721 0.441236
\(677\) 23.8885 0.918111 0.459056 0.888408i \(-0.348188\pi\)
0.459056 + 0.888408i \(0.348188\pi\)
\(678\) 0 0
\(679\) 3.52786 0.135387
\(680\) 67.2492 2.57889
\(681\) 0 0
\(682\) 0 0
\(683\) −40.9230 −1.56587 −0.782937 0.622101i \(-0.786279\pi\)
−0.782937 + 0.622101i \(0.786279\pi\)
\(684\) 0 0
\(685\) −19.3050 −0.737604
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −1.23607 −0.0471246
\(689\) −15.0557 −0.573578
\(690\) 0 0
\(691\) 39.9230 1.51874 0.759371 0.650658i \(-0.225507\pi\)
0.759371 + 0.650658i \(0.225507\pi\)
\(692\) −19.8541 −0.754740
\(693\) 0 0
\(694\) 12.3820 0.470013
\(695\) −13.6950 −0.519483
\(696\) 0 0
\(697\) −85.2492 −3.22904
\(698\) 22.7639 0.861628
\(699\) 0 0
\(700\) 3.14590 0.118904
\(701\) 24.0689 0.909069 0.454535 0.890729i \(-0.349806\pi\)
0.454535 + 0.890729i \(0.349806\pi\)
\(702\) 0 0
\(703\) 31.6525 1.19380
\(704\) 0 0
\(705\) 0 0
\(706\) 12.4721 0.469395
\(707\) 2.09017 0.0786089
\(708\) 0 0
\(709\) −19.6869 −0.739358 −0.369679 0.929160i \(-0.620532\pi\)
−0.369679 + 0.929160i \(0.620532\pi\)
\(710\) 25.5279 0.958044
\(711\) 0 0
\(712\) −3.27051 −0.122568
\(713\) −22.6393 −0.847849
\(714\) 0 0
\(715\) 0 0
\(716\) 8.38197 0.313249
\(717\) 0 0
\(718\) −7.03444 −0.262523
\(719\) −4.87539 −0.181821 −0.0909106 0.995859i \(-0.528978\pi\)
−0.0909106 + 0.995859i \(0.528978\pi\)
\(720\) 0 0
\(721\) 10.3262 0.384569
\(722\) 12.1459 0.452024
\(723\) 0 0
\(724\) 18.9443 0.704058
\(725\) −6.29180 −0.233671
\(726\) 0 0
\(727\) 25.7426 0.954742 0.477371 0.878702i \(-0.341590\pi\)
0.477371 + 0.878702i \(0.341590\pi\)
\(728\) −3.70820 −0.137435
\(729\) 0 0
\(730\) −14.9443 −0.553112
\(731\) 9.70820 0.359071
\(732\) 0 0
\(733\) −31.1246 −1.14961 −0.574807 0.818289i \(-0.694923\pi\)
−0.574807 + 0.818289i \(0.694923\pi\)
\(734\) −16.2705 −0.600555
\(735\) 0 0
\(736\) −15.4508 −0.569526
\(737\) 0 0
\(738\) 0 0
\(739\) −49.0132 −1.80298 −0.901489 0.432802i \(-0.857525\pi\)
−0.901489 + 0.432802i \(0.857525\pi\)
\(740\) 34.5066 1.26849
\(741\) 0 0
\(742\) −12.1803 −0.447154
\(743\) 17.3262 0.635638 0.317819 0.948151i \(-0.397050\pi\)
0.317819 + 0.948151i \(0.397050\pi\)
\(744\) 0 0
\(745\) −57.0820 −2.09132
\(746\) −23.3262 −0.854034
\(747\) 0 0
\(748\) 0 0
\(749\) −4.90983 −0.179401
\(750\) 0 0
\(751\) −10.4721 −0.382134 −0.191067 0.981577i \(-0.561195\pi\)
−0.191067 + 0.981577i \(0.561195\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) 2.47214 0.0900299
\(755\) 5.70820 0.207743
\(756\) 0 0
\(757\) 0.326238 0.0118573 0.00592866 0.999982i \(-0.498113\pi\)
0.00592866 + 0.999982i \(0.498113\pi\)
\(758\) −2.47214 −0.0897920
\(759\) 0 0
\(760\) −22.4164 −0.813129
\(761\) 45.7771 1.65942 0.829709 0.558196i \(-0.188506\pi\)
0.829709 + 0.558196i \(0.188506\pi\)
\(762\) 0 0
\(763\) 10.5623 0.382381
\(764\) −18.2705 −0.661004
\(765\) 0 0
\(766\) −13.0557 −0.471723
\(767\) −8.36068 −0.301887
\(768\) 0 0
\(769\) −14.5836 −0.525898 −0.262949 0.964810i \(-0.584695\pi\)
−0.262949 + 0.964810i \(0.584695\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.3820 0.661581
\(773\) 38.9443 1.40073 0.700364 0.713786i \(-0.253021\pi\)
0.700364 + 0.713786i \(0.253021\pi\)
\(774\) 0 0
\(775\) −23.0476 −0.827894
\(776\) −10.5836 −0.379929
\(777\) 0 0
\(778\) 22.1803 0.795204
\(779\) 28.4164 1.01812
\(780\) 0 0
\(781\) 0 0
\(782\) −24.2705 −0.867912
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 27.7082 0.988948
\(786\) 0 0
\(787\) −12.2705 −0.437396 −0.218698 0.975793i \(-0.570181\pi\)
−0.218698 + 0.975793i \(0.570181\pi\)
\(788\) −4.00000 −0.142494
\(789\) 0 0
\(790\) −39.9574 −1.42162
\(791\) 0.763932 0.0271623
\(792\) 0 0
\(793\) −11.0557 −0.392600
\(794\) −14.6525 −0.519997
\(795\) 0 0
\(796\) 2.61803 0.0927938
\(797\) −10.2705 −0.363800 −0.181900 0.983317i \(-0.558225\pi\)
−0.181900 + 0.983317i \(0.558225\pi\)
\(798\) 0 0
\(799\) −15.7082 −0.555716
\(800\) −15.7295 −0.556121
\(801\) 0 0
\(802\) 18.4721 0.652274
\(803\) 0 0
\(804\) 0 0
\(805\) −8.81966 −0.310852
\(806\) 9.05573 0.318974
\(807\) 0 0
\(808\) −6.27051 −0.220596
\(809\) −28.6525 −1.00737 −0.503684 0.863888i \(-0.668022\pi\)
−0.503684 + 0.863888i \(0.668022\pi\)
\(810\) 0 0
\(811\) 16.3607 0.574501 0.287251 0.957855i \(-0.407259\pi\)
0.287251 + 0.957855i \(0.407259\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) −15.4590 −0.541504
\(816\) 0 0
\(817\) −3.23607 −0.113216
\(818\) 28.7639 1.00571
\(819\) 0 0
\(820\) 30.9787 1.08182
\(821\) 30.3607 1.05960 0.529798 0.848124i \(-0.322268\pi\)
0.529798 + 0.848124i \(0.322268\pi\)
\(822\) 0 0
\(823\) 24.4721 0.853045 0.426523 0.904477i \(-0.359738\pi\)
0.426523 + 0.904477i \(0.359738\pi\)
\(824\) −30.9787 −1.07919
\(825\) 0 0
\(826\) −6.76393 −0.235347
\(827\) −4.32624 −0.150438 −0.0752190 0.997167i \(-0.523966\pi\)
−0.0752190 + 0.997167i \(0.523966\pi\)
\(828\) 0 0
\(829\) 28.9443 1.00528 0.502638 0.864497i \(-0.332363\pi\)
0.502638 + 0.864497i \(0.332363\pi\)
\(830\) −8.40325 −0.291681
\(831\) 0 0
\(832\) 8.65248 0.299971
\(833\) 7.85410 0.272129
\(834\) 0 0
\(835\) 23.3475 0.807974
\(836\) 0 0
\(837\) 0 0
\(838\) 14.7639 0.510012
\(839\) −33.4164 −1.15366 −0.576831 0.816863i \(-0.695711\pi\)
−0.576831 + 0.816863i \(0.695711\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 4.32624 0.149092
\(843\) 0 0
\(844\) −2.00000 −0.0688428
\(845\) −32.7426 −1.12638
\(846\) 0 0
\(847\) 0 0
\(848\) 12.1803 0.418275
\(849\) 0 0
\(850\) −24.7082 −0.847484
\(851\) −37.3607 −1.28071
\(852\) 0 0
\(853\) 15.7082 0.537839 0.268919 0.963163i \(-0.413333\pi\)
0.268919 + 0.963163i \(0.413333\pi\)
\(854\) −8.94427 −0.306067
\(855\) 0 0
\(856\) 14.7295 0.503444
\(857\) −18.9443 −0.647124 −0.323562 0.946207i \(-0.604880\pi\)
−0.323562 + 0.946207i \(0.604880\pi\)
\(858\) 0 0
\(859\) −5.88854 −0.200915 −0.100457 0.994941i \(-0.532031\pi\)
−0.100457 + 0.994941i \(0.532031\pi\)
\(860\) −3.52786 −0.120299
\(861\) 0 0
\(862\) 2.09017 0.0711915
\(863\) −37.6869 −1.28288 −0.641439 0.767174i \(-0.721662\pi\)
−0.641439 + 0.767174i \(0.721662\pi\)
\(864\) 0 0
\(865\) 56.6656 1.92669
\(866\) 17.2361 0.585705
\(867\) 0 0
\(868\) −7.32624 −0.248669
\(869\) 0 0
\(870\) 0 0
\(871\) 9.88854 0.335061
\(872\) −31.6869 −1.07305
\(873\) 0 0
\(874\) 8.09017 0.273654
\(875\) 5.29180 0.178895
\(876\) 0 0
\(877\) 46.3607 1.56549 0.782744 0.622343i \(-0.213819\pi\)
0.782744 + 0.622343i \(0.213819\pi\)
\(878\) −21.7984 −0.735659
\(879\) 0 0
\(880\) 0 0
\(881\) 33.4508 1.12699 0.563494 0.826120i \(-0.309457\pi\)
0.563494 + 0.826120i \(0.309457\pi\)
\(882\) 0 0
\(883\) 7.70820 0.259402 0.129701 0.991553i \(-0.458598\pi\)
0.129701 + 0.991553i \(0.458598\pi\)
\(884\) −9.70820 −0.326522
\(885\) 0 0
\(886\) 4.90983 0.164949
\(887\) −46.5410 −1.56269 −0.781347 0.624097i \(-0.785467\pi\)
−0.781347 + 0.624097i \(0.785467\pi\)
\(888\) 0 0
\(889\) −3.23607 −0.108534
\(890\) 3.11146 0.104296
\(891\) 0 0
\(892\) −20.7984 −0.696381
\(893\) 5.23607 0.175218
\(894\) 0 0
\(895\) −23.9230 −0.799657
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 41.8885 1.39784
\(899\) 14.6525 0.488687
\(900\) 0 0
\(901\) −95.6656 −3.18708
\(902\) 0 0
\(903\) 0 0
\(904\) −2.29180 −0.0762240
\(905\) −54.0689 −1.79731
\(906\) 0 0
\(907\) −47.2361 −1.56845 −0.784224 0.620478i \(-0.786939\pi\)
−0.784224 + 0.620478i \(0.786939\pi\)
\(908\) −21.4164 −0.710728
\(909\) 0 0
\(910\) 3.52786 0.116948
\(911\) 11.0557 0.366293 0.183146 0.983086i \(-0.441372\pi\)
0.183146 + 0.983086i \(0.441372\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 12.4721 0.412542
\(915\) 0 0
\(916\) −21.2361 −0.701659
\(917\) −2.29180 −0.0756818
\(918\) 0 0
\(919\) −7.41641 −0.244645 −0.122322 0.992490i \(-0.539034\pi\)
−0.122322 + 0.992490i \(0.539034\pi\)
\(920\) 26.4590 0.872327
\(921\) 0 0
\(922\) 6.36068 0.209478
\(923\) −11.0557 −0.363904
\(924\) 0 0
\(925\) −38.0344 −1.25056
\(926\) −11.4164 −0.375166
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) 28.7984 0.944844 0.472422 0.881372i \(-0.343380\pi\)
0.472422 + 0.881372i \(0.343380\pi\)
\(930\) 0 0
\(931\) −2.61803 −0.0858026
\(932\) 5.70820 0.186978
\(933\) 0 0
\(934\) 19.2361 0.629423
\(935\) 0 0
\(936\) 0 0
\(937\) −11.7082 −0.382490 −0.191245 0.981542i \(-0.561253\pi\)
−0.191245 + 0.981542i \(0.561253\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 5.70820 0.186181
\(941\) −11.8541 −0.386433 −0.193216 0.981156i \(-0.561892\pi\)
−0.193216 + 0.981156i \(0.561892\pi\)
\(942\) 0 0
\(943\) −33.5410 −1.09225
\(944\) 6.76393 0.220147
\(945\) 0 0
\(946\) 0 0
\(947\) 23.4377 0.761623 0.380811 0.924653i \(-0.375645\pi\)
0.380811 + 0.924653i \(0.375645\pi\)
\(948\) 0 0
\(949\) 6.47214 0.210094
\(950\) 8.23607 0.267213
\(951\) 0 0
\(952\) −23.5623 −0.763659
\(953\) 4.83282 0.156550 0.0782751 0.996932i \(-0.475059\pi\)
0.0782751 + 0.996932i \(0.475059\pi\)
\(954\) 0 0
\(955\) 52.1459 1.68740
\(956\) 1.56231 0.0505286
\(957\) 0 0
\(958\) 11.0557 0.357194
\(959\) 6.76393 0.218419
\(960\) 0 0
\(961\) 22.6738 0.731412
\(962\) 14.9443 0.481823
\(963\) 0 0
\(964\) 4.94427 0.159244
\(965\) −52.4640 −1.68888
\(966\) 0 0
\(967\) −49.3050 −1.58554 −0.792770 0.609521i \(-0.791362\pi\)
−0.792770 + 0.609521i \(0.791362\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 10.0689 0.323292
\(971\) −42.0689 −1.35005 −0.675027 0.737793i \(-0.735868\pi\)
−0.675027 + 0.737793i \(0.735868\pi\)
\(972\) 0 0
\(973\) 4.79837 0.153829
\(974\) 39.2361 1.25720
\(975\) 0 0
\(976\) 8.94427 0.286299
\(977\) 26.5836 0.850484 0.425242 0.905080i \(-0.360189\pi\)
0.425242 + 0.905080i \(0.360189\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.85410 −0.0911709
\(981\) 0 0
\(982\) 0.326238 0.0104107
\(983\) −48.4721 −1.54602 −0.773011 0.634393i \(-0.781250\pi\)
−0.773011 + 0.634393i \(0.781250\pi\)
\(984\) 0 0
\(985\) 11.4164 0.363757
\(986\) 15.7082 0.500251
\(987\) 0 0
\(988\) 3.23607 0.102953
\(989\) 3.81966 0.121458
\(990\) 0 0
\(991\) 18.6525 0.592515 0.296258 0.955108i \(-0.404261\pi\)
0.296258 + 0.955108i \(0.404261\pi\)
\(992\) 36.6312 1.16304
\(993\) 0 0
\(994\) −8.94427 −0.283695
\(995\) −7.47214 −0.236883
\(996\) 0 0
\(997\) 52.5410 1.66399 0.831995 0.554783i \(-0.187199\pi\)
0.831995 + 0.554783i \(0.187199\pi\)
\(998\) −26.1803 −0.828724
\(999\) 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 7623.2.a.w.1.2 2
3.2 odd 2 2541.2.a.bd.1.1 2
11.2 odd 10 693.2.m.c.631.1 4
11.6 odd 10 693.2.m.c.190.1 4
11.10 odd 2 7623.2.a.bu.1.2 2
33.2 even 10 231.2.j.c.169.1 4
33.17 even 10 231.2.j.c.190.1 yes 4
33.32 even 2 2541.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.c.169.1 4 33.2 even 10
231.2.j.c.190.1 yes 4 33.17 even 10
693.2.m.c.190.1 4 11.6 odd 10
693.2.m.c.631.1 4 11.2 odd 10
2541.2.a.n.1.1 2 33.32 even 2
2541.2.a.bd.1.1 2 3.2 odd 2
7623.2.a.w.1.2 2 1.1 even 1 trivial
7623.2.a.bu.1.2 2 11.10 odd 2