Properties

Label 6030.2.a.bu.1.4
Level $6030$
Weight $2$
Character 6030.1
Self dual yes
Analytic conductor $48.150$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 10 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2010)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.88727\) of defining polynomial
Character \(\chi\) \(=\) 6030.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.33630 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.33630 q^{7} +1.00000 q^{8} +1.00000 q^{10} +4.74553 q^{11} +0.409227 q^{13} +4.33630 q^{14} +1.00000 q^{16} -5.77453 q^{17} -5.77453 q^{19} +1.00000 q^{20} +4.74553 q^{22} +9.36530 q^{23} +1.00000 q^{25} +0.409227 q^{26} +4.33630 q^{28} +1.30730 q^{29} +2.40923 q^{31} +1.00000 q^{32} -5.77453 q^{34} +4.33630 q^{35} +3.02900 q^{37} -5.77453 q^{38} +1.00000 q^{40} -1.71653 q^{41} -9.49106 q^{43} +4.74553 q^{44} +9.36530 q^{46} +1.51115 q^{47} +11.8035 q^{49} +1.00000 q^{50} +0.409227 q^{52} +10.6726 q^{53} +4.74553 q^{55} +4.33630 q^{56} +1.30730 q^{58} -1.71653 q^{59} +6.33630 q^{61} +2.40923 q^{62} +1.00000 q^{64} +0.409227 q^{65} -1.00000 q^{67} -5.77453 q^{68} +4.33630 q^{70} -8.11084 q^{71} -8.05800 q^{73} +3.02900 q^{74} -5.77453 q^{76} +20.5781 q^{77} +7.64878 q^{79} +1.00000 q^{80} -1.71653 q^{82} -8.57806 q^{83} -5.77453 q^{85} -9.49106 q^{86} +4.74553 q^{88} +9.08700 q^{89} +1.77453 q^{91} +9.36530 q^{92} +1.51115 q^{94} -5.77453 q^{95} -4.74553 q^{97} +11.8035 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + 4q^{10} + 3q^{11} + 2q^{13} + q^{14} + 4q^{16} + 2q^{17} + 2q^{19} + 4q^{20} + 3q^{22} + 12q^{23} + 4q^{25} + 2q^{26} + q^{28} - 2q^{29} + 10q^{31} + 4q^{32} + 2q^{34} + q^{35} + 3q^{37} + 2q^{38} + 4q^{40} - 6q^{43} + 3q^{44} + 12q^{46} + 14q^{47} + 13q^{49} + 4q^{50} + 2q^{52} + 10q^{53} + 3q^{55} + q^{56} - 2q^{58} + 9q^{61} + 10q^{62} + 4q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + q^{70} + 9q^{71} - 14q^{73} + 3q^{74} + 2q^{76} + 23q^{77} + 12q^{79} + 4q^{80} + 25q^{83} + 2q^{85} - 6q^{86} + 3q^{88} + 9q^{89} - 18q^{91} + 12q^{92} + 14q^{94} + 2q^{95} - 3q^{97} + 13q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.33630 1.63897 0.819484 0.573101i \(-0.194260\pi\)
0.819484 + 0.573101i \(0.194260\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 4.74553 1.43083 0.715416 0.698699i \(-0.246237\pi\)
0.715416 + 0.698699i \(0.246237\pi\)
\(12\) 0 0
\(13\) 0.409227 0.113499 0.0567495 0.998388i \(-0.481926\pi\)
0.0567495 + 0.998388i \(0.481926\pi\)
\(14\) 4.33630 1.15893
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.77453 −1.40053 −0.700265 0.713883i \(-0.746935\pi\)
−0.700265 + 0.713883i \(0.746935\pi\)
\(18\) 0 0
\(19\) −5.77453 −1.32477 −0.662384 0.749164i \(-0.730455\pi\)
−0.662384 + 0.749164i \(0.730455\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.74553 1.01175
\(23\) 9.36530 1.95280 0.976401 0.215968i \(-0.0692906\pi\)
0.976401 + 0.215968i \(0.0692906\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.409227 0.0802560
\(27\) 0 0
\(28\) 4.33630 0.819484
\(29\) 1.30730 0.242760 0.121380 0.992606i \(-0.461268\pi\)
0.121380 + 0.992606i \(0.461268\pi\)
\(30\) 0 0
\(31\) 2.40923 0.432710 0.216355 0.976315i \(-0.430583\pi\)
0.216355 + 0.976315i \(0.430583\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.77453 −0.990324
\(35\) 4.33630 0.732969
\(36\) 0 0
\(37\) 3.02900 0.497965 0.248982 0.968508i \(-0.419904\pi\)
0.248982 + 0.968508i \(0.419904\pi\)
\(38\) −5.77453 −0.936753
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −1.71653 −0.268077 −0.134038 0.990976i \(-0.542795\pi\)
−0.134038 + 0.990976i \(0.542795\pi\)
\(42\) 0 0
\(43\) −9.49106 −1.44737 −0.723687 0.690129i \(-0.757554\pi\)
−0.723687 + 0.690129i \(0.757554\pi\)
\(44\) 4.74553 0.715416
\(45\) 0 0
\(46\) 9.36530 1.38084
\(47\) 1.51115 0.220424 0.110212 0.993908i \(-0.464847\pi\)
0.110212 + 0.993908i \(0.464847\pi\)
\(48\) 0 0
\(49\) 11.8035 1.68622
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0.409227 0.0567495
\(53\) 10.6726 1.46600 0.732998 0.680231i \(-0.238121\pi\)
0.732998 + 0.680231i \(0.238121\pi\)
\(54\) 0 0
\(55\) 4.74553 0.639887
\(56\) 4.33630 0.579463
\(57\) 0 0
\(58\) 1.30730 0.171657
\(59\) −1.71653 −0.223473 −0.111737 0.993738i \(-0.535641\pi\)
−0.111737 + 0.993738i \(0.535641\pi\)
\(60\) 0 0
\(61\) 6.33630 0.811281 0.405640 0.914033i \(-0.367049\pi\)
0.405640 + 0.914033i \(0.367049\pi\)
\(62\) 2.40923 0.305972
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.409227 0.0507583
\(66\) 0 0
\(67\) −1.00000 −0.122169
\(68\) −5.77453 −0.700265
\(69\) 0 0
\(70\) 4.33630 0.518287
\(71\) −8.11084 −0.962579 −0.481290 0.876562i \(-0.659831\pi\)
−0.481290 + 0.876562i \(0.659831\pi\)
\(72\) 0 0
\(73\) −8.05800 −0.943118 −0.471559 0.881835i \(-0.656309\pi\)
−0.471559 + 0.881835i \(0.656309\pi\)
\(74\) 3.02900 0.352114
\(75\) 0 0
\(76\) −5.77453 −0.662384
\(77\) 20.5781 2.34509
\(78\) 0 0
\(79\) 7.64878 0.860554 0.430277 0.902697i \(-0.358416\pi\)
0.430277 + 0.902697i \(0.358416\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −1.71653 −0.189559
\(83\) −8.57806 −0.941565 −0.470782 0.882249i \(-0.656028\pi\)
−0.470782 + 0.882249i \(0.656028\pi\)
\(84\) 0 0
\(85\) −5.77453 −0.626336
\(86\) −9.49106 −1.02345
\(87\) 0 0
\(88\) 4.74553 0.505875
\(89\) 9.08700 0.963220 0.481610 0.876386i \(-0.340052\pi\)
0.481610 + 0.876386i \(0.340052\pi\)
\(90\) 0 0
\(91\) 1.77453 0.186021
\(92\) 9.36530 0.976401
\(93\) 0 0
\(94\) 1.51115 0.155863
\(95\) −5.77453 −0.592454
\(96\) 0 0
\(97\) −4.74553 −0.481836 −0.240918 0.970546i \(-0.577448\pi\)
−0.240918 + 0.970546i \(0.577448\pi\)
\(98\) 11.8035 1.19234
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.52006 −0.847778 −0.423889 0.905714i \(-0.639335\pi\)
−0.423889 + 0.905714i \(0.639335\pi\)
\(102\) 0 0
\(103\) −13.4911 −1.32931 −0.664657 0.747149i \(-0.731422\pi\)
−0.664657 + 0.747149i \(0.731422\pi\)
\(104\) 0.409227 0.0401280
\(105\) 0 0
\(106\) 10.6726 1.03662
\(107\) 8.81845 0.852512 0.426256 0.904603i \(-0.359832\pi\)
0.426256 + 0.904603i \(0.359832\pi\)
\(108\) 0 0
\(109\) −17.8854 −1.71311 −0.856554 0.516058i \(-0.827399\pi\)
−0.856554 + 0.516058i \(0.827399\pi\)
\(110\) 4.74553 0.452469
\(111\) 0 0
\(112\) 4.33630 0.409742
\(113\) 3.86908 0.363972 0.181986 0.983301i \(-0.441747\pi\)
0.181986 + 0.983301i \(0.441747\pi\)
\(114\) 0 0
\(115\) 9.36530 0.873319
\(116\) 1.30730 0.121380
\(117\) 0 0
\(118\) −1.71653 −0.158019
\(119\) −25.0401 −2.29542
\(120\) 0 0
\(121\) 11.5201 1.04728
\(122\) 6.33630 0.573662
\(123\) 0 0
\(124\) 2.40923 0.216355
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.23955 −0.819877 −0.409939 0.912113i \(-0.634450\pi\)
−0.409939 + 0.912113i \(0.634450\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0.409227 0.0358916
\(131\) 18.9561 1.65620 0.828100 0.560580i \(-0.189422\pi\)
0.828100 + 0.560580i \(0.189422\pi\)
\(132\) 0 0
\(133\) −25.0401 −2.17125
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) −5.77453 −0.495162
\(137\) 0.970999 0.0829581 0.0414790 0.999139i \(-0.486793\pi\)
0.0414790 + 0.999139i \(0.486793\pi\)
\(138\) 0 0
\(139\) 9.41144 0.798268 0.399134 0.916893i \(-0.369311\pi\)
0.399134 + 0.916893i \(0.369311\pi\)
\(140\) 4.33630 0.366485
\(141\) 0 0
\(142\) −8.11084 −0.680646
\(143\) 1.94200 0.162398
\(144\) 0 0
\(145\) 1.30730 0.108566
\(146\) −8.05800 −0.666885
\(147\) 0 0
\(148\) 3.02900 0.248982
\(149\) 1.30730 0.107098 0.0535492 0.998565i \(-0.482947\pi\)
0.0535492 + 0.998565i \(0.482947\pi\)
\(150\) 0 0
\(151\) −15.3385 −1.24823 −0.624115 0.781332i \(-0.714540\pi\)
−0.624115 + 0.781332i \(0.714540\pi\)
\(152\) −5.77453 −0.468376
\(153\) 0 0
\(154\) 20.5781 1.65823
\(155\) 2.40923 0.193514
\(156\) 0 0
\(157\) −7.79615 −0.622201 −0.311100 0.950377i \(-0.600698\pi\)
−0.311100 + 0.950377i \(0.600698\pi\)
\(158\) 7.64878 0.608504
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 40.6108 3.20058
\(162\) 0 0
\(163\) −6.97100 −0.546011 −0.273005 0.962013i \(-0.588018\pi\)
−0.273005 + 0.962013i \(0.588018\pi\)
\(164\) −1.71653 −0.134038
\(165\) 0 0
\(166\) −8.57806 −0.665787
\(167\) −0.546851 −0.0423166 −0.0211583 0.999776i \(-0.506735\pi\)
−0.0211583 + 0.999776i \(0.506735\pi\)
\(168\) 0 0
\(169\) −12.8325 −0.987118
\(170\) −5.77453 −0.442886
\(171\) 0 0
\(172\) −9.49106 −0.723687
\(173\) 16.3147 1.24038 0.620191 0.784451i \(-0.287055\pi\)
0.620191 + 0.784451i \(0.287055\pi\)
\(174\) 0 0
\(175\) 4.33630 0.327794
\(176\) 4.74553 0.357708
\(177\) 0 0
\(178\) 9.08700 0.681100
\(179\) 1.92038 0.143536 0.0717679 0.997421i \(-0.477136\pi\)
0.0717679 + 0.997421i \(0.477136\pi\)
\(180\) 0 0
\(181\) 17.4032 1.29357 0.646785 0.762672i \(-0.276113\pi\)
0.646785 + 0.762672i \(0.276113\pi\)
\(182\) 1.77453 0.131537
\(183\) 0 0
\(184\) 9.36530 0.690419
\(185\) 3.02900 0.222697
\(186\) 0 0
\(187\) −27.4032 −2.00392
\(188\) 1.51115 0.110212
\(189\) 0 0
\(190\) −5.77453 −0.418929
\(191\) 2.50894 0.181540 0.0907702 0.995872i \(-0.471067\pi\)
0.0907702 + 0.995872i \(0.471067\pi\)
\(192\) 0 0
\(193\) 25.9382 1.86707 0.933536 0.358483i \(-0.116706\pi\)
0.933536 + 0.358483i \(0.116706\pi\)
\(194\) −4.74553 −0.340709
\(195\) 0 0
\(196\) 11.8035 0.843109
\(197\) 1.18155 0.0841817 0.0420909 0.999114i \(-0.486598\pi\)
0.0420909 + 0.999114i \(0.486598\pi\)
\(198\) 0 0
\(199\) −16.5781 −1.17519 −0.587594 0.809156i \(-0.699925\pi\)
−0.587594 + 0.809156i \(0.699925\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −8.52006 −0.599470
\(203\) 5.66886 0.397876
\(204\) 0 0
\(205\) −1.71653 −0.119888
\(206\) −13.4911 −0.939967
\(207\) 0 0
\(208\) 0.409227 0.0283748
\(209\) −27.4032 −1.89552
\(210\) 0 0
\(211\) −25.0617 −1.72532 −0.862661 0.505783i \(-0.831204\pi\)
−0.862661 + 0.505783i \(0.831204\pi\)
\(212\) 10.6726 0.732998
\(213\) 0 0
\(214\) 8.81845 0.602817
\(215\) −9.49106 −0.647285
\(216\) 0 0
\(217\) 10.4471 0.709198
\(218\) −17.8854 −1.21135
\(219\) 0 0
\(220\) 4.74553 0.319944
\(221\) −2.36309 −0.158959
\(222\) 0 0
\(223\) −25.8586 −1.73162 −0.865809 0.500374i \(-0.833196\pi\)
−0.865809 + 0.500374i \(0.833196\pi\)
\(224\) 4.33630 0.289731
\(225\) 0 0
\(226\) 3.86908 0.257367
\(227\) −17.7127 −1.17564 −0.587818 0.808993i \(-0.700013\pi\)
−0.587818 + 0.808993i \(0.700013\pi\)
\(228\) 0 0
\(229\) −15.8274 −1.04590 −0.522951 0.852363i \(-0.675169\pi\)
−0.522951 + 0.852363i \(0.675169\pi\)
\(230\) 9.36530 0.617530
\(231\) 0 0
\(232\) 1.30730 0.0858287
\(233\) 22.4322 1.46958 0.734792 0.678293i \(-0.237280\pi\)
0.734792 + 0.678293i \(0.237280\pi\)
\(234\) 0 0
\(235\) 1.51115 0.0985766
\(236\) −1.71653 −0.111737
\(237\) 0 0
\(238\) −25.0401 −1.62311
\(239\) −9.23955 −0.597657 −0.298828 0.954307i \(-0.596596\pi\)
−0.298828 + 0.954307i \(0.596596\pi\)
\(240\) 0 0
\(241\) 6.17859 0.397998 0.198999 0.980000i \(-0.436231\pi\)
0.198999 + 0.980000i \(0.436231\pi\)
\(242\) 11.5201 0.740538
\(243\) 0 0
\(244\) 6.33630 0.405640
\(245\) 11.8035 0.754100
\(246\) 0 0
\(247\) −2.36309 −0.150360
\(248\) 2.40923 0.152986
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 6.97100 0.440006 0.220003 0.975499i \(-0.429393\pi\)
0.220003 + 0.975499i \(0.429393\pi\)
\(252\) 0 0
\(253\) 44.4433 2.79413
\(254\) −9.23955 −0.579741
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.7090 −0.917522 −0.458761 0.888560i \(-0.651707\pi\)
−0.458761 + 0.888560i \(0.651707\pi\)
\(258\) 0 0
\(259\) 13.1347 0.816149
\(260\) 0.409227 0.0253792
\(261\) 0 0
\(262\) 18.9561 1.17111
\(263\) −8.79836 −0.542530 −0.271265 0.962505i \(-0.587442\pi\)
−0.271265 + 0.962505i \(0.587442\pi\)
\(264\) 0 0
\(265\) 10.6726 0.655613
\(266\) −25.0401 −1.53531
\(267\) 0 0
\(268\) −1.00000 −0.0610847
\(269\) −28.8564 −1.75940 −0.879702 0.475526i \(-0.842258\pi\)
−0.879702 + 0.475526i \(0.842258\pi\)
\(270\) 0 0
\(271\) 13.8124 0.839046 0.419523 0.907745i \(-0.362197\pi\)
0.419523 + 0.907745i \(0.362197\pi\)
\(272\) −5.77453 −0.350132
\(273\) 0 0
\(274\) 0.970999 0.0586602
\(275\) 4.74553 0.286166
\(276\) 0 0
\(277\) −3.28051 −0.197107 −0.0985535 0.995132i \(-0.531422\pi\)
−0.0985535 + 0.995132i \(0.531422\pi\)
\(278\) 9.41144 0.564461
\(279\) 0 0
\(280\) 4.33630 0.259144
\(281\) −0.363093 −0.0216603 −0.0108302 0.999941i \(-0.503447\pi\)
−0.0108302 + 0.999941i \(0.503447\pi\)
\(282\) 0 0
\(283\) 12.1570 0.722657 0.361328 0.932439i \(-0.382323\pi\)
0.361328 + 0.932439i \(0.382323\pi\)
\(284\) −8.11084 −0.481290
\(285\) 0 0
\(286\) 1.94200 0.114833
\(287\) −7.44340 −0.439370
\(288\) 0 0
\(289\) 16.3452 0.961483
\(290\) 1.30730 0.0767675
\(291\) 0 0
\(292\) −8.05800 −0.471559
\(293\) 0.871287 0.0509012 0.0254506 0.999676i \(-0.491898\pi\)
0.0254506 + 0.999676i \(0.491898\pi\)
\(294\) 0 0
\(295\) −1.71653 −0.0999402
\(296\) 3.02900 0.176057
\(297\) 0 0
\(298\) 1.30730 0.0757300
\(299\) 3.83253 0.221641
\(300\) 0 0
\(301\) −41.1561 −2.37220
\(302\) −15.3385 −0.882632
\(303\) 0 0
\(304\) −5.77453 −0.331192
\(305\) 6.33630 0.362816
\(306\) 0 0
\(307\) 26.4583 1.51005 0.755026 0.655694i \(-0.227624\pi\)
0.755026 + 0.655694i \(0.227624\pi\)
\(308\) 20.5781 1.17254
\(309\) 0 0
\(310\) 2.40923 0.136835
\(311\) 26.1637 1.48361 0.741803 0.670618i \(-0.233971\pi\)
0.741803 + 0.670618i \(0.233971\pi\)
\(312\) 0 0
\(313\) 12.4404 0.703175 0.351588 0.936155i \(-0.385642\pi\)
0.351588 + 0.936155i \(0.385642\pi\)
\(314\) −7.79615 −0.439962
\(315\) 0 0
\(316\) 7.64878 0.430277
\(317\) 31.2558 1.75550 0.877751 0.479116i \(-0.159043\pi\)
0.877751 + 0.479116i \(0.159043\pi\)
\(318\) 0 0
\(319\) 6.20385 0.347349
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 40.6108 2.26315
\(323\) 33.3452 1.85538
\(324\) 0 0
\(325\) 0.409227 0.0226998
\(326\) −6.97100 −0.386088
\(327\) 0 0
\(328\) −1.71653 −0.0947795
\(329\) 6.55281 0.361268
\(330\) 0 0
\(331\) 28.6041 1.57222 0.786112 0.618084i \(-0.212091\pi\)
0.786112 + 0.618084i \(0.212091\pi\)
\(332\) −8.57806 −0.470782
\(333\) 0 0
\(334\) −0.546851 −0.0299224
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) −17.5751 −0.957377 −0.478689 0.877985i \(-0.658888\pi\)
−0.478689 + 0.877985i \(0.658888\pi\)
\(338\) −12.8325 −0.697998
\(339\) 0 0
\(340\) −5.77453 −0.313168
\(341\) 11.4331 0.619135
\(342\) 0 0
\(343\) 20.8296 1.12469
\(344\) −9.49106 −0.511724
\(345\) 0 0
\(346\) 16.3147 0.877083
\(347\) 34.1122 1.83124 0.915620 0.402046i \(-0.131701\pi\)
0.915620 + 0.402046i \(0.131701\pi\)
\(348\) 0 0
\(349\) −12.9821 −0.694917 −0.347459 0.937695i \(-0.612955\pi\)
−0.347459 + 0.937695i \(0.612955\pi\)
\(350\) 4.33630 0.231785
\(351\) 0 0
\(352\) 4.74553 0.252938
\(353\) −3.55286 −0.189100 −0.0945498 0.995520i \(-0.530141\pi\)
−0.0945498 + 0.995520i \(0.530141\pi\)
\(354\) 0 0
\(355\) −8.11084 −0.430478
\(356\) 9.08700 0.481610
\(357\) 0 0
\(358\) 1.92038 0.101495
\(359\) 22.0230 1.16233 0.581165 0.813786i \(-0.302597\pi\)
0.581165 + 0.813786i \(0.302597\pi\)
\(360\) 0 0
\(361\) 14.3452 0.755011
\(362\) 17.4032 0.914693
\(363\) 0 0
\(364\) 1.77453 0.0930107
\(365\) −8.05800 −0.421775
\(366\) 0 0
\(367\) −2.34300 −0.122304 −0.0611519 0.998128i \(-0.519477\pi\)
−0.0611519 + 0.998128i \(0.519477\pi\)
\(368\) 9.36530 0.488200
\(369\) 0 0
\(370\) 3.02900 0.157470
\(371\) 46.2797 2.40272
\(372\) 0 0
\(373\) 16.9360 0.876912 0.438456 0.898753i \(-0.355525\pi\)
0.438456 + 0.898753i \(0.355525\pi\)
\(374\) −27.4032 −1.41699
\(375\) 0 0
\(376\) 1.51115 0.0779316
\(377\) 0.534983 0.0275530
\(378\) 0 0
\(379\) 7.93378 0.407531 0.203765 0.979020i \(-0.434682\pi\)
0.203765 + 0.979020i \(0.434682\pi\)
\(380\) −5.77453 −0.296227
\(381\) 0 0
\(382\) 2.50894 0.128368
\(383\) −26.3742 −1.34766 −0.673830 0.738887i \(-0.735352\pi\)
−0.673830 + 0.738887i \(0.735352\pi\)
\(384\) 0 0
\(385\) 20.5781 1.04876
\(386\) 25.9382 1.32022
\(387\) 0 0
\(388\) −4.74553 −0.240918
\(389\) −39.4277 −1.99907 −0.999533 0.0305693i \(-0.990268\pi\)
−0.999533 + 0.0305693i \(0.990268\pi\)
\(390\) 0 0
\(391\) −54.0802 −2.73496
\(392\) 11.8035 0.596168
\(393\) 0 0
\(394\) 1.18155 0.0595255
\(395\) 7.64878 0.384852
\(396\) 0 0
\(397\) −23.8073 −1.19485 −0.597426 0.801924i \(-0.703810\pi\)
−0.597426 + 0.801924i \(0.703810\pi\)
\(398\) −16.5781 −0.830983
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 17.8109 0.889435 0.444717 0.895671i \(-0.353304\pi\)
0.444717 + 0.895671i \(0.353304\pi\)
\(402\) 0 0
\(403\) 0.985920 0.0491122
\(404\) −8.52006 −0.423889
\(405\) 0 0
\(406\) 5.66886 0.281341
\(407\) 14.3742 0.712503
\(408\) 0 0
\(409\) 9.18155 0.453998 0.226999 0.973895i \(-0.427109\pi\)
0.226999 + 0.973895i \(0.427109\pi\)
\(410\) −1.71653 −0.0847734
\(411\) 0 0
\(412\) −13.4911 −0.664657
\(413\) −7.44340 −0.366266
\(414\) 0 0
\(415\) −8.57806 −0.421081
\(416\) 0.409227 0.0200640
\(417\) 0 0
\(418\) −27.4032 −1.34034
\(419\) 12.3013 0.600958 0.300479 0.953789i \(-0.402854\pi\)
0.300479 + 0.953789i \(0.402854\pi\)
\(420\) 0 0
\(421\) 32.8363 1.60034 0.800171 0.599772i \(-0.204742\pi\)
0.800171 + 0.599772i \(0.204742\pi\)
\(422\) −25.0617 −1.21999
\(423\) 0 0
\(424\) 10.6726 0.518308
\(425\) −5.77453 −0.280106
\(426\) 0 0
\(427\) 27.4761 1.32966
\(428\) 8.81845 0.426256
\(429\) 0 0
\(430\) −9.49106 −0.457700
\(431\) 41.0051 1.97515 0.987573 0.157158i \(-0.0502332\pi\)
0.987573 + 0.157158i \(0.0502332\pi\)
\(432\) 0 0
\(433\) 23.5274 1.13066 0.565328 0.824866i \(-0.308749\pi\)
0.565328 + 0.824866i \(0.308749\pi\)
\(434\) 10.4471 0.501479
\(435\) 0 0
\(436\) −17.8854 −0.856554
\(437\) −54.0802 −2.58701
\(438\) 0 0
\(439\) −22.0691 −1.05330 −0.526651 0.850082i \(-0.676553\pi\)
−0.526651 + 0.850082i \(0.676553\pi\)
\(440\) 4.74553 0.226234
\(441\) 0 0
\(442\) −2.36309 −0.112401
\(443\) −7.54906 −0.358667 −0.179333 0.983788i \(-0.557394\pi\)
−0.179333 + 0.983788i \(0.557394\pi\)
\(444\) 0 0
\(445\) 9.08700 0.430765
\(446\) −25.8586 −1.22444
\(447\) 0 0
\(448\) 4.33630 0.204871
\(449\) −27.4398 −1.29496 −0.647481 0.762081i \(-0.724178\pi\)
−0.647481 + 0.762081i \(0.724178\pi\)
\(450\) 0 0
\(451\) −8.14585 −0.383573
\(452\) 3.86908 0.181986
\(453\) 0 0
\(454\) −17.7127 −0.831300
\(455\) 1.77453 0.0831913
\(456\) 0 0
\(457\) −14.3891 −0.673095 −0.336548 0.941666i \(-0.609259\pi\)
−0.336548 + 0.941666i \(0.609259\pi\)
\(458\) −15.8274 −0.739564
\(459\) 0 0
\(460\) 9.36530 0.436660
\(461\) 14.5365 0.677033 0.338517 0.940960i \(-0.390075\pi\)
0.338517 + 0.940960i \(0.390075\pi\)
\(462\) 0 0
\(463\) 11.6124 0.539674 0.269837 0.962906i \(-0.413030\pi\)
0.269837 + 0.962906i \(0.413030\pi\)
\(464\) 1.30730 0.0606900
\(465\) 0 0
\(466\) 22.4322 1.03915
\(467\) −8.88316 −0.411063 −0.205532 0.978650i \(-0.565892\pi\)
−0.205532 + 0.978650i \(0.565892\pi\)
\(468\) 0 0
\(469\) −4.33630 −0.200232
\(470\) 1.51115 0.0697042
\(471\) 0 0
\(472\) −1.71653 −0.0790097
\(473\) −45.0401 −2.07095
\(474\) 0 0
\(475\) −5.77453 −0.264954
\(476\) −25.0401 −1.14771
\(477\) 0 0
\(478\) −9.23955 −0.422607
\(479\) −24.5319 −1.12089 −0.560446 0.828191i \(-0.689370\pi\)
−0.560446 + 0.828191i \(0.689370\pi\)
\(480\) 0 0
\(481\) 1.23955 0.0565185
\(482\) 6.17859 0.281427
\(483\) 0 0
\(484\) 11.5201 0.523639
\(485\) −4.74553 −0.215483
\(486\) 0 0
\(487\) −12.9553 −0.587062 −0.293531 0.955950i \(-0.594830\pi\)
−0.293531 + 0.955950i \(0.594830\pi\)
\(488\) 6.33630 0.286831
\(489\) 0 0
\(490\) 11.8035 0.533229
\(491\) −27.6911 −1.24968 −0.624841 0.780752i \(-0.714836\pi\)
−0.624841 + 0.780752i \(0.714836\pi\)
\(492\) 0 0
\(493\) −7.54906 −0.339993
\(494\) −2.36309 −0.106321
\(495\) 0 0
\(496\) 2.40923 0.108177
\(497\) −35.1710 −1.57764
\(498\) 0 0
\(499\) 32.7737 1.46715 0.733576 0.679607i \(-0.237850\pi\)
0.733576 + 0.679607i \(0.237850\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 6.97100 0.311131
\(503\) −20.5677 −0.917070 −0.458535 0.888676i \(-0.651626\pi\)
−0.458535 + 0.888676i \(0.651626\pi\)
\(504\) 0 0
\(505\) −8.52006 −0.379138
\(506\) 44.4433 1.97575
\(507\) 0 0
\(508\) −9.23955 −0.409939
\(509\) 1.97991 0.0877580 0.0438790 0.999037i \(-0.486028\pi\)
0.0438790 + 0.999037i \(0.486028\pi\)
\(510\) 0 0
\(511\) −34.9419 −1.54574
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.7090 −0.648786
\(515\) −13.4911 −0.594487
\(516\) 0 0
\(517\) 7.17121 0.315389
\(518\) 13.1347 0.577104
\(519\) 0 0
\(520\) 0.409227 0.0179458
\(521\) −22.6407 −0.991905 −0.495952 0.868350i \(-0.665181\pi\)
−0.495952 + 0.868350i \(0.665181\pi\)
\(522\) 0 0
\(523\) −23.8519 −1.04297 −0.521485 0.853261i \(-0.674622\pi\)
−0.521485 + 0.853261i \(0.674622\pi\)
\(524\) 18.9561 0.828100
\(525\) 0 0
\(526\) −8.79836 −0.383627
\(527\) −13.9122 −0.606023
\(528\) 0 0
\(529\) 64.7089 2.81343
\(530\) 10.6726 0.463588
\(531\) 0 0
\(532\) −25.0401 −1.08563
\(533\) −0.702450 −0.0304265
\(534\) 0 0
\(535\) 8.81845 0.381255
\(536\) −1.00000 −0.0431934
\(537\) 0 0
\(538\) −28.8564 −1.24409
\(539\) 56.0140 2.41269
\(540\) 0 0
\(541\) −4.49919 −0.193435 −0.0967175 0.995312i \(-0.530834\pi\)
−0.0967175 + 0.995312i \(0.530834\pi\)
\(542\) 13.8124 0.593295
\(543\) 0 0
\(544\) −5.77453 −0.247581
\(545\) −17.8854 −0.766125
\(546\) 0 0
\(547\) 5.68015 0.242866 0.121433 0.992600i \(-0.461251\pi\)
0.121433 + 0.992600i \(0.461251\pi\)
\(548\) 0.970999 0.0414790
\(549\) 0 0
\(550\) 4.74553 0.202350
\(551\) −7.54906 −0.321601
\(552\) 0 0
\(553\) 33.1674 1.41042
\(554\) −3.28051 −0.139376
\(555\) 0 0
\(556\) 9.41144 0.399134
\(557\) 20.6412 0.874597 0.437299 0.899316i \(-0.355935\pi\)
0.437299 + 0.899316i \(0.355935\pi\)
\(558\) 0 0
\(559\) −3.88400 −0.164276
\(560\) 4.33630 0.183242
\(561\) 0 0
\(562\) −0.363093 −0.0153162
\(563\) 2.50894 0.105739 0.0528696 0.998601i \(-0.483163\pi\)
0.0528696 + 0.998601i \(0.483163\pi\)
\(564\) 0 0
\(565\) 3.86908 0.162773
\(566\) 12.1570 0.510996
\(567\) 0 0
\(568\) −8.11084 −0.340323
\(569\) −41.6138 −1.74454 −0.872270 0.489025i \(-0.837353\pi\)
−0.872270 + 0.489025i \(0.837353\pi\)
\(570\) 0 0
\(571\) 11.5491 0.483313 0.241657 0.970362i \(-0.422309\pi\)
0.241657 + 0.970362i \(0.422309\pi\)
\(572\) 1.94200 0.0811990
\(573\) 0 0
\(574\) −7.44340 −0.310681
\(575\) 9.36530 0.390560
\(576\) 0 0
\(577\) −40.6211 −1.69108 −0.845540 0.533912i \(-0.820721\pi\)
−0.845540 + 0.533912i \(0.820721\pi\)
\(578\) 16.3452 0.679871
\(579\) 0 0
\(580\) 1.30730 0.0542828
\(581\) −37.1971 −1.54320
\(582\) 0 0
\(583\) 50.6472 2.09759
\(584\) −8.05800 −0.333442
\(585\) 0 0
\(586\) 0.871287 0.0359926
\(587\) −7.82220 −0.322857 −0.161428 0.986884i \(-0.551610\pi\)
−0.161428 + 0.986884i \(0.551610\pi\)
\(588\) 0 0
\(589\) −13.9122 −0.573240
\(590\) −1.71653 −0.0706684
\(591\) 0 0
\(592\) 3.02900 0.124491
\(593\) 12.4041 0.509374 0.254687 0.967024i \(-0.418028\pi\)
0.254687 + 0.967024i \(0.418028\pi\)
\(594\) 0 0
\(595\) −25.0401 −1.02655
\(596\) 1.30730 0.0535492
\(597\) 0 0
\(598\) 3.83253 0.156724
\(599\) −20.5134 −0.838153 −0.419077 0.907951i \(-0.637646\pi\)
−0.419077 + 0.907951i \(0.637646\pi\)
\(600\) 0 0
\(601\) 20.4984 0.836149 0.418074 0.908413i \(-0.362705\pi\)
0.418074 + 0.908413i \(0.362705\pi\)
\(602\) −41.1561 −1.67740
\(603\) 0 0
\(604\) −15.3385 −0.624115
\(605\) 11.5201 0.468357
\(606\) 0 0
\(607\) 8.96430 0.363850 0.181925 0.983312i \(-0.441767\pi\)
0.181925 + 0.983312i \(0.441767\pi\)
\(608\) −5.77453 −0.234188
\(609\) 0 0
\(610\) 6.33630 0.256549
\(611\) 0.618403 0.0250179
\(612\) 0 0
\(613\) −28.6904 −1.15880 −0.579398 0.815045i \(-0.696712\pi\)
−0.579398 + 0.815045i \(0.696712\pi\)
\(614\) 26.4583 1.06777
\(615\) 0 0
\(616\) 20.5781 0.829114
\(617\) 2.85483 0.114931 0.0574656 0.998347i \(-0.481698\pi\)
0.0574656 + 0.998347i \(0.481698\pi\)
\(618\) 0 0
\(619\) −25.7745 −1.03597 −0.517983 0.855391i \(-0.673317\pi\)
−0.517983 + 0.855391i \(0.673317\pi\)
\(620\) 2.40923 0.0967569
\(621\) 0 0
\(622\) 26.1637 1.04907
\(623\) 39.4040 1.57869
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 12.4404 0.497220
\(627\) 0 0
\(628\) −7.79615 −0.311100
\(629\) −17.4911 −0.697414
\(630\) 0 0
\(631\) −29.1398 −1.16004 −0.580019 0.814603i \(-0.696955\pi\)
−0.580019 + 0.814603i \(0.696955\pi\)
\(632\) 7.64878 0.304252
\(633\) 0 0
\(634\) 31.2558 1.24133
\(635\) −9.23955 −0.366660
\(636\) 0 0
\(637\) 4.83032 0.191384
\(638\) 6.20385 0.245613
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −31.0535 −1.22654 −0.613270 0.789873i \(-0.710146\pi\)
−0.613270 + 0.789873i \(0.710146\pi\)
\(642\) 0 0
\(643\) −28.1273 −1.10923 −0.554616 0.832106i \(-0.687135\pi\)
−0.554616 + 0.832106i \(0.687135\pi\)
\(644\) 40.6108 1.60029
\(645\) 0 0
\(646\) 33.3452 1.31195
\(647\) −37.2641 −1.46500 −0.732501 0.680766i \(-0.761647\pi\)
−0.732501 + 0.680766i \(0.761647\pi\)
\(648\) 0 0
\(649\) −8.14585 −0.319752
\(650\) 0.409227 0.0160512
\(651\) 0 0
\(652\) −6.97100 −0.273005
\(653\) 11.1963 0.438145 0.219073 0.975709i \(-0.429697\pi\)
0.219073 + 0.975709i \(0.429697\pi\)
\(654\) 0 0
\(655\) 18.9561 0.740675
\(656\) −1.71653 −0.0670192
\(657\) 0 0
\(658\) 6.55281 0.255455
\(659\) 18.8727 0.735174 0.367587 0.929989i \(-0.380184\pi\)
0.367587 + 0.929989i \(0.380184\pi\)
\(660\) 0 0
\(661\) 13.5290 0.526216 0.263108 0.964766i \(-0.415252\pi\)
0.263108 + 0.964766i \(0.415252\pi\)
\(662\) 28.6041 1.11173
\(663\) 0 0
\(664\) −8.57806 −0.332893
\(665\) −25.0401 −0.971014
\(666\) 0 0
\(667\) 12.2433 0.474062
\(668\) −0.546851 −0.0211583
\(669\) 0 0
\(670\) −1.00000 −0.0386334
\(671\) 30.0691 1.16081
\(672\) 0 0
\(673\) −43.2358 −1.66662 −0.833308 0.552809i \(-0.813556\pi\)
−0.833308 + 0.552809i \(0.813556\pi\)
\(674\) −17.5751 −0.676968
\(675\) 0 0
\(676\) −12.8325 −0.493559
\(677\) −3.63691 −0.139778 −0.0698888 0.997555i \(-0.522264\pi\)
−0.0698888 + 0.997555i \(0.522264\pi\)
\(678\) 0 0
\(679\) −20.5781 −0.789714
\(680\) −5.77453 −0.221443
\(681\) 0 0
\(682\) 11.4331 0.437794
\(683\) −42.9523 −1.64352 −0.821762 0.569831i \(-0.807009\pi\)
−0.821762 + 0.569831i \(0.807009\pi\)
\(684\) 0 0
\(685\) 0.970999 0.0371000
\(686\) 20.8296 0.795277
\(687\) 0 0
\(688\) −9.49106 −0.361843
\(689\) 4.36752 0.166389
\(690\) 0 0
\(691\) 43.0683 1.63839 0.819197 0.573512i \(-0.194419\pi\)
0.819197 + 0.573512i \(0.194419\pi\)
\(692\) 16.3147 0.620191
\(693\) 0 0
\(694\) 34.1122 1.29488
\(695\) 9.41144 0.356996
\(696\) 0 0
\(697\) 9.91216 0.375450
\(698\) −12.9821 −0.491381
\(699\) 0 0
\(700\) 4.33630 0.163897
\(701\) −42.5379 −1.60663 −0.803317 0.595552i \(-0.796933\pi\)
−0.803317 + 0.595552i \(0.796933\pi\)
\(702\) 0 0
\(703\) −17.4911 −0.659688
\(704\) 4.74553 0.178854
\(705\) 0 0
\(706\) −3.55286 −0.133714
\(707\) −36.9456 −1.38948
\(708\) 0 0
\(709\) −22.8319 −0.857468 −0.428734 0.903431i \(-0.641040\pi\)
−0.428734 + 0.903431i \(0.641040\pi\)
\(710\) −8.11084 −0.304394
\(711\) 0 0
\(712\) 9.08700 0.340550
\(713\) 22.5631 0.844996
\(714\) 0 0
\(715\) 1.94200 0.0726266
\(716\) 1.92038 0.0717679
\(717\) 0 0
\(718\) 22.0230 0.821891
\(719\) 11.3869 0.424661 0.212330 0.977198i \(-0.431895\pi\)
0.212330 + 0.977198i \(0.431895\pi\)
\(720\) 0 0
\(721\) −58.5013 −2.17870
\(722\) 14.3452 0.533874
\(723\) 0 0
\(724\) 17.4032 0.646785
\(725\) 1.30730 0.0485520
\(726\) 0 0
\(727\) 8.38761 0.311079 0.155540 0.987830i \(-0.450288\pi\)
0.155540 + 0.987830i \(0.450288\pi\)
\(728\) 1.77453 0.0657685
\(729\) 0 0
\(730\) −8.05800 −0.298240
\(731\) 54.8064 2.02709
\(732\) 0 0
\(733\) −18.8244 −0.695295 −0.347648 0.937625i \(-0.613019\pi\)
−0.347648 + 0.937625i \(0.613019\pi\)
\(734\) −2.34300 −0.0864819
\(735\) 0 0
\(736\) 9.36530 0.345210
\(737\) −4.74553 −0.174804
\(738\) 0 0
\(739\) −1.86238 −0.0685086 −0.0342543 0.999413i \(-0.510906\pi\)
−0.0342543 + 0.999413i \(0.510906\pi\)
\(740\) 3.02900 0.111348
\(741\) 0 0
\(742\) 46.2797 1.69898
\(743\) −3.21504 −0.117948 −0.0589741 0.998260i \(-0.518783\pi\)
−0.0589741 + 0.998260i \(0.518783\pi\)
\(744\) 0 0
\(745\) 1.30730 0.0478959
\(746\) 16.9360 0.620071
\(747\) 0 0
\(748\) −27.4032 −1.00196
\(749\) 38.2395 1.39724
\(750\) 0 0
\(751\) 14.9821 0.546705 0.273353 0.961914i \(-0.411867\pi\)
0.273353 + 0.961914i \(0.411867\pi\)
\(752\) 1.51115 0.0551060
\(753\) 0 0
\(754\) 0.534983 0.0194829
\(755\) −15.3385 −0.558226
\(756\) 0 0
\(757\) 47.8749 1.74004 0.870021 0.493015i \(-0.164105\pi\)
0.870021 + 0.493015i \(0.164105\pi\)
\(758\) 7.93378 0.288168
\(759\) 0 0
\(760\) −5.77453 −0.209464
\(761\) −10.3266 −0.374337 −0.187169 0.982328i \(-0.559931\pi\)
−0.187169 + 0.982328i \(0.559931\pi\)
\(762\) 0 0
\(763\) −77.5564 −2.80773
\(764\) 2.50894 0.0907702
\(765\) 0 0
\(766\) −26.3742 −0.952939
\(767\) −0.702450 −0.0253640
\(768\) 0 0
\(769\) −9.60264 −0.346280 −0.173140 0.984897i \(-0.555391\pi\)
−0.173140 + 0.984897i \(0.555391\pi\)
\(770\) 20.5781 0.741582
\(771\) 0 0
\(772\) 25.9382 0.933536
\(773\) 32.7151 1.17668 0.588340 0.808613i \(-0.299782\pi\)
0.588340 + 0.808613i \(0.299782\pi\)
\(774\) 0 0
\(775\) 2.40923 0.0865420
\(776\) −4.74553 −0.170355
\(777\) 0 0
\(778\) −39.4277 −1.41355
\(779\) 9.91216 0.355140
\(780\) 0 0
\(781\) −38.4902 −1.37729
\(782\) −54.0802 −1.93391
\(783\) 0 0
\(784\) 11.8035 0.421555
\(785\) −7.79615 −0.278257
\(786\) 0 0
\(787\) −15.3795 −0.548219 −0.274110 0.961698i \(-0.588383\pi\)
−0.274110 + 0.961698i \(0.588383\pi\)
\(788\) 1.18155 0.0420909
\(789\) 0 0
\(790\) 7.64878 0.272131
\(791\) 16.7775 0.596539
\(792\) 0 0
\(793\) 2.59299 0.0920796
\(794\) −23.8073 −0.844889
\(795\) 0 0
\(796\) −16.5781 −0.587594
\(797\) 24.5260 0.868756 0.434378 0.900731i \(-0.356968\pi\)
0.434378 + 0.900731i \(0.356968\pi\)
\(798\) 0 0
\(799\) −8.72619 −0.308710
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 17.8109 0.628925
\(803\) −38.2395 −1.34944
\(804\) 0 0
\(805\) 40.6108 1.43134
\(806\) 0.985920 0.0347276
\(807\) 0 0
\(808\) −8.52006 −0.299735
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −36.9329 −1.29689 −0.648445 0.761261i \(-0.724581\pi\)
−0.648445 + 0.761261i \(0.724581\pi\)
\(812\) 5.66886 0.198938
\(813\) 0 0
\(814\) 14.3742 0.503816
\(815\) −6.97100 −0.244183
\(816\) 0 0
\(817\) 54.8064 1.91743
\(818\) 9.18155 0.321025
\(819\) 0 0
\(820\) −1.71653 −0.0599438
\(821\) −45.4991 −1.58793 −0.793965 0.607963i \(-0.791987\pi\)
−0.793965 + 0.607963i \(0.791987\pi\)
\(822\) 0 0
\(823\) 28.9241 1.00823 0.504116 0.863636i \(-0.331818\pi\)
0.504116 + 0.863636i \(0.331818\pi\)
\(824\) −13.4911 −0.469983
\(825\) 0 0
\(826\) −7.44340 −0.258989
\(827\) 31.0767 1.08064 0.540321 0.841459i \(-0.318303\pi\)
0.540321 + 0.841459i \(0.318303\pi\)
\(828\) 0 0
\(829\) 10.2217 0.355013 0.177507 0.984120i \(-0.443197\pi\)
0.177507 + 0.984120i \(0.443197\pi\)
\(830\) −8.57806 −0.297749
\(831\) 0 0
\(832\) 0.409227 0.0141874
\(833\) −68.1599 −2.36160
\(834\) 0 0
\(835\) −0.546851 −0.0189246
\(836\) −27.4032 −0.947760
\(837\) 0 0
\(838\) 12.3013 0.424941
\(839\) 55.7870 1.92598 0.962991 0.269533i \(-0.0868695\pi\)
0.962991 + 0.269533i \(0.0868695\pi\)
\(840\) 0 0
\(841\) −27.2910 −0.941068
\(842\) 32.8363 1.13161
\(843\) 0 0
\(844\) −25.0617 −0.862661
\(845\) −12.8325 −0.441453
\(846\) 0 0
\(847\) 49.9545 1.71646
\(848\) 10.6726 0.366499
\(849\) 0 0
\(850\) −5.77453 −0.198065
\(851\) 28.3675 0.972426
\(852\) 0 0
\(853\) 31.9880 1.09525 0.547624 0.836725i \(-0.315533\pi\)
0.547624 + 0.836725i \(0.315533\pi\)
\(854\) 27.4761 0.940214
\(855\) 0 0
\(856\) 8.81845 0.301409
\(857\) −29.4329 −1.00541 −0.502704 0.864458i \(-0.667662\pi\)
−0.502704 + 0.864458i \(0.667662\pi\)
\(858\) 0 0
\(859\) −14.5306 −0.495776 −0.247888 0.968789i \(-0.579737\pi\)
−0.247888 + 0.968789i \(0.579737\pi\)
\(860\) −9.49106 −0.323643
\(861\) 0 0
\(862\) 41.0051 1.39664
\(863\) 20.9442 0.712949 0.356475 0.934305i \(-0.383979\pi\)
0.356475 + 0.934305i \(0.383979\pi\)
\(864\) 0 0
\(865\) 16.3147 0.554716
\(866\) 23.5274 0.799495
\(867\) 0 0
\(868\) 10.4471 0.354599
\(869\) 36.2975 1.23131
\(870\) 0 0
\(871\) −0.409227 −0.0138661
\(872\) −17.8854 −0.605675
\(873\) 0 0
\(874\) −54.0802 −1.82929
\(875\) 4.33630 0.146594
\(876\) 0 0
\(877\) −42.5789 −1.43779 −0.718893 0.695121i \(-0.755351\pi\)
−0.718893 + 0.695121i \(0.755351\pi\)
\(878\) −22.0691 −0.744797
\(879\) 0 0
\(880\) 4.74553 0.159972
\(881\) 40.1851 1.35387 0.676936 0.736042i \(-0.263308\pi\)
0.676936 + 0.736042i \(0.263308\pi\)
\(882\) 0 0
\(883\) −38.1637 −1.28431 −0.642155 0.766575i \(-0.721959\pi\)
−0.642155 + 0.766575i \(0.721959\pi\)
\(884\) −2.36309 −0.0794794
\(885\) 0 0
\(886\) −7.54906 −0.253616
\(887\) −22.4353 −0.753303 −0.376651 0.926355i \(-0.622925\pi\)
−0.376651 + 0.926355i \(0.622925\pi\)
\(888\) 0 0
\(889\) −40.0655 −1.34375
\(890\) 9.08700 0.304597
\(891\) 0 0
\(892\) −25.8586 −0.865809
\(893\) −8.72619 −0.292011
\(894\) 0 0
\(895\) 1.92038 0.0641911
\(896\) 4.33630 0.144866
\(897\) 0 0
\(898\) −27.4398 −0.915677
\(899\) 3.14959 0.105045
\(900\) 0 0
\(901\) −61.6293 −2.05317
\(902\) −8.14585 −0.271227
\(903\) 0 0
\(904\) 3.86908 0.128684
\(905\) 17.4032 0.578503
\(906\) 0 0
\(907\) 45.2945 1.50398 0.751990 0.659174i \(-0.229094\pi\)
0.751990 + 0.659174i \(0.229094\pi\)
\(908\) −17.7127 −0.587818
\(909\) 0 0
\(910\) 1.77453 0.0588252
\(911\) 6.75528 0.223813 0.111906 0.993719i \(-0.464304\pi\)
0.111906 + 0.993719i \(0.464304\pi\)
\(912\) 0 0
\(913\) −40.7075 −1.34722
\(914\) −14.3891 −0.475950
\(915\) 0 0
\(916\) −15.8274 −0.522951
\(917\) 82.1993 2.71446
\(918\) 0 0
\(919\) −1.48511 −0.0489891 −0.0244946 0.999700i \(-0.507798\pi\)
−0.0244946 + 0.999700i \(0.507798\pi\)
\(920\) 9.36530 0.308765
\(921\) 0 0
\(922\) 14.5365 0.478735
\(923\) −3.31917 −0.109252
\(924\) 0 0
\(925\) 3.02900 0.0995929
\(926\) 11.6124 0.381607
\(927\) 0 0
\(928\) 1.30730 0.0429143
\(929\) −30.3490 −0.995717 −0.497859 0.867258i \(-0.665880\pi\)
−0.497859 + 0.867258i \(0.665880\pi\)
\(930\) 0 0
\(931\) −68.1599 −2.23385
\(932\) 22.4322 0.734792
\(933\) 0 0
\(934\) −8.88316 −0.290666
\(935\) −27.4032 −0.896181
\(936\) 0 0
\(937\) −53.1992 −1.73794 −0.868971 0.494863i \(-0.835218\pi\)
−0.868971 + 0.494863i \(0.835218\pi\)
\(938\) −4.33630 −0.141585
\(939\) 0 0
\(940\) 1.51115 0.0492883
\(941\) −32.2395 −1.05098 −0.525489 0.850801i \(-0.676118\pi\)
−0.525489 + 0.850801i \(0.676118\pi\)
\(942\) 0 0
\(943\) −16.0758 −0.523501
\(944\) −1.71653 −0.0558683
\(945\) 0 0
\(946\) −45.0401 −1.46438
\(947\) 27.7060 0.900325 0.450163 0.892947i \(-0.351366\pi\)
0.450163 + 0.892947i \(0.351366\pi\)
\(948\) 0 0
\(949\) −3.29755 −0.107043
\(950\) −5.77453 −0.187351
\(951\) 0 0
\(952\) −25.0401 −0.811555
\(953\) 50.3771 1.63187 0.815937 0.578140i \(-0.196221\pi\)
0.815937 + 0.578140i \(0.196221\pi\)
\(954\) 0 0
\(955\) 2.50894 0.0811873
\(956\) −9.23955 −0.298828
\(957\) 0 0
\(958\) −24.5319 −0.792591
\(959\) 4.21055 0.135966
\(960\) 0 0
\(961\) −25.1956 −0.812762
\(962\) 1.23955 0.0399646
\(963\) 0 0
\(964\) 6.17859 0.198999
\(965\) 25.9382 0.834980
\(966\) 0 0
\(967\) 2.23675 0.0719291 0.0359646 0.999353i \(-0.488550\pi\)
0.0359646 + 0.999353i \(0.488550\pi\)
\(968\) 11.5201 0.370269
\(969\) 0 0
\(970\) −4.74553 −0.152370
\(971\) −37.2878 −1.19662 −0.598312 0.801263i \(-0.704162\pi\)
−0.598312 + 0.801263i \(0.704162\pi\)
\(972\) 0 0
\(973\) 40.8109 1.30834
\(974\) −12.9553 −0.415116
\(975\) 0 0
\(976\) 6.33630 0.202820
\(977\) −13.5261 −0.432738 −0.216369 0.976312i \(-0.569421\pi\)
−0.216369 + 0.976312i \(0.569421\pi\)
\(978\) 0 0
\(979\) 43.1227 1.37821
\(980\) 11.8035 0.377050
\(981\) 0 0
\(982\) −27.6911 −0.883659
\(983\) 59.9924 1.91346 0.956730 0.290976i \(-0.0939800\pi\)
0.956730 + 0.290976i \(0.0939800\pi\)
\(984\) 0 0
\(985\) 1.18155 0.0376472
\(986\) −7.54906 −0.240411
\(987\) 0 0
\(988\) −2.36309 −0.0751800
\(989\) −88.8867 −2.82643
\(990\) 0 0
\(991\) 3.37353 0.107164 0.0535818 0.998563i \(-0.482936\pi\)
0.0535818 + 0.998563i \(0.482936\pi\)
\(992\) 2.40923 0.0764930
\(993\) 0 0
\(994\) −35.1710 −1.11556
\(995\) −16.5781 −0.525560
\(996\) 0 0
\(997\) −10.0647 −0.318752 −0.159376 0.987218i \(-0.550948\pi\)
−0.159376 + 0.987218i \(0.550948\pi\)
\(998\) 32.7737 1.03743
\(999\) 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 6030.2.a.bu.1.4 4
3.2 odd 2 2010.2.a.r.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.4 4 3.2 odd 2
6030.2.a.bu.1.4 4 1.1 even 1 trivial