Properties

Label 7623.2.a.cp.1.5
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.7674048.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.82356\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.823556 q^{2} -1.32176 q^{4} -2.98565 q^{5} +1.00000 q^{7} -2.73565 q^{8} +O(q^{10})\) \(q+0.823556 q^{2} -1.32176 q^{4} -2.98565 q^{5} +1.00000 q^{7} -2.73565 q^{8} -2.45885 q^{10} +2.20144 q^{13} +0.823556 q^{14} +0.390549 q^{16} -4.40669 q^{17} -1.72563 q^{19} +3.94630 q^{20} +8.39774 q^{23} +3.91410 q^{25} +1.81301 q^{26} -1.32176 q^{28} -3.29295 q^{29} +7.47573 q^{31} +5.79294 q^{32} -3.62916 q^{34} -2.98565 q^{35} -8.78071 q^{37} -1.42115 q^{38} +8.16769 q^{40} -5.39351 q^{41} +9.44629 q^{43} +6.91601 q^{46} +5.39667 q^{47} +1.00000 q^{49} +3.22348 q^{50} -2.90977 q^{52} -9.39774 q^{53} -2.73565 q^{56} -2.71193 q^{58} -3.47462 q^{59} +12.9942 q^{61} +6.15668 q^{62} +3.98971 q^{64} -6.57274 q^{65} +4.32138 q^{67} +5.82457 q^{68} -2.45885 q^{70} -4.40046 q^{71} +14.7509 q^{73} -7.23140 q^{74} +2.28086 q^{76} -7.18768 q^{79} -1.16604 q^{80} -4.44186 q^{82} -7.63445 q^{83} +13.1568 q^{85} +7.77955 q^{86} -10.8428 q^{89} +2.20144 q^{91} -11.0998 q^{92} +4.44446 q^{94} +5.15211 q^{95} +2.85498 q^{97} +0.823556 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} - 8q^{10} + 4q^{13} - 4q^{14} + 8q^{16} - 22q^{17} + 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} + 4q^{28} - 12q^{29} - 2q^{31} - 8q^{32} + 24q^{34} + 4q^{35} + 14q^{37} + 22q^{38} + 18q^{40} - 26q^{41} - 4q^{43} + 12q^{46} + 16q^{47} + 6q^{49} + 4q^{50} + 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} - 8q^{61} - 20q^{62} + 26q^{64} - 24q^{65} + 6q^{67} - 12q^{68} - 8q^{70} - 22q^{71} + 14q^{73} - 44q^{74} - 30q^{76} - 28q^{79} + 4q^{80} - 4q^{82} - 22q^{83} - 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} - 38q^{94} + 24q^{95} - 4q^{97} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.823556 0.582342 0.291171 0.956671i \(-0.405955\pi\)
0.291171 + 0.956671i \(0.405955\pi\)
\(3\) 0 0
\(4\) −1.32176 −0.660878
\(5\) −2.98565 −1.33522 −0.667611 0.744510i \(-0.732683\pi\)
−0.667611 + 0.744510i \(0.732683\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.73565 −0.967199
\(9\) 0 0
\(10\) −2.45885 −0.777556
\(11\) 0 0
\(12\) 0 0
\(13\) 2.20144 0.610571 0.305285 0.952261i \(-0.401248\pi\)
0.305285 + 0.952261i \(0.401248\pi\)
\(14\) 0.823556 0.220105
\(15\) 0 0
\(16\) 0.390549 0.0976372
\(17\) −4.40669 −1.06878 −0.534390 0.845238i \(-0.679459\pi\)
−0.534390 + 0.845238i \(0.679459\pi\)
\(18\) 0 0
\(19\) −1.72563 −0.395886 −0.197943 0.980214i \(-0.563426\pi\)
−0.197943 + 0.980214i \(0.563426\pi\)
\(20\) 3.94630 0.882419
\(21\) 0 0
\(22\) 0 0
\(23\) 8.39774 1.75105 0.875525 0.483173i \(-0.160516\pi\)
0.875525 + 0.483173i \(0.160516\pi\)
\(24\) 0 0
\(25\) 3.91410 0.782819
\(26\) 1.81301 0.355561
\(27\) 0 0
\(28\) −1.32176 −0.249788
\(29\) −3.29295 −0.611485 −0.305743 0.952114i \(-0.598905\pi\)
−0.305743 + 0.952114i \(0.598905\pi\)
\(30\) 0 0
\(31\) 7.47573 1.34268 0.671341 0.741149i \(-0.265719\pi\)
0.671341 + 0.741149i \(0.265719\pi\)
\(32\) 5.79294 1.02406
\(33\) 0 0
\(34\) −3.62916 −0.622396
\(35\) −2.98565 −0.504667
\(36\) 0 0
\(37\) −8.78071 −1.44354 −0.721770 0.692133i \(-0.756671\pi\)
−0.721770 + 0.692133i \(0.756671\pi\)
\(38\) −1.42115 −0.230541
\(39\) 0 0
\(40\) 8.16769 1.29143
\(41\) −5.39351 −0.842325 −0.421163 0.906985i \(-0.638378\pi\)
−0.421163 + 0.906985i \(0.638378\pi\)
\(42\) 0 0
\(43\) 9.44629 1.44055 0.720273 0.693691i \(-0.244017\pi\)
0.720273 + 0.693691i \(0.244017\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.91601 1.01971
\(47\) 5.39667 0.787185 0.393593 0.919285i \(-0.371232\pi\)
0.393593 + 0.919285i \(0.371232\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.22348 0.455869
\(51\) 0 0
\(52\) −2.90977 −0.403513
\(53\) −9.39774 −1.29088 −0.645439 0.763812i \(-0.723326\pi\)
−0.645439 + 0.763812i \(0.723326\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.73565 −0.365567
\(57\) 0 0
\(58\) −2.71193 −0.356094
\(59\) −3.47462 −0.452357 −0.226178 0.974086i \(-0.572623\pi\)
−0.226178 + 0.974086i \(0.572623\pi\)
\(60\) 0 0
\(61\) 12.9942 1.66374 0.831870 0.554970i \(-0.187270\pi\)
0.831870 + 0.554970i \(0.187270\pi\)
\(62\) 6.15668 0.781900
\(63\) 0 0
\(64\) 3.98971 0.498714
\(65\) −6.57274 −0.815248
\(66\) 0 0
\(67\) 4.32138 0.527940 0.263970 0.964531i \(-0.414968\pi\)
0.263970 + 0.964531i \(0.414968\pi\)
\(68\) 5.82457 0.706333
\(69\) 0 0
\(70\) −2.45885 −0.293889
\(71\) −4.40046 −0.522238 −0.261119 0.965307i \(-0.584092\pi\)
−0.261119 + 0.965307i \(0.584092\pi\)
\(72\) 0 0
\(73\) 14.7509 1.72647 0.863233 0.504805i \(-0.168436\pi\)
0.863233 + 0.504805i \(0.168436\pi\)
\(74\) −7.23140 −0.840634
\(75\) 0 0
\(76\) 2.28086 0.261632
\(77\) 0 0
\(78\) 0 0
\(79\) −7.18768 −0.808677 −0.404338 0.914609i \(-0.632498\pi\)
−0.404338 + 0.914609i \(0.632498\pi\)
\(80\) −1.16604 −0.130367
\(81\) 0 0
\(82\) −4.44186 −0.490521
\(83\) −7.63445 −0.837990 −0.418995 0.907989i \(-0.637617\pi\)
−0.418995 + 0.907989i \(0.637617\pi\)
\(84\) 0 0
\(85\) 13.1568 1.42706
\(86\) 7.77955 0.838890
\(87\) 0 0
\(88\) 0 0
\(89\) −10.8428 −1.14934 −0.574668 0.818386i \(-0.694869\pi\)
−0.574668 + 0.818386i \(0.694869\pi\)
\(90\) 0 0
\(91\) 2.20144 0.230774
\(92\) −11.0998 −1.15723
\(93\) 0 0
\(94\) 4.44446 0.458411
\(95\) 5.15211 0.528596
\(96\) 0 0
\(97\) 2.85498 0.289879 0.144940 0.989440i \(-0.453701\pi\)
0.144940 + 0.989440i \(0.453701\pi\)
\(98\) 0.823556 0.0831917
\(99\) 0 0
\(100\) −5.17348 −0.517348
\(101\) 14.6011 1.45287 0.726434 0.687236i \(-0.241176\pi\)
0.726434 + 0.687236i \(0.241176\pi\)
\(102\) 0 0
\(103\) −0.107767 −0.0106186 −0.00530932 0.999986i \(-0.501690\pi\)
−0.00530932 + 0.999986i \(0.501690\pi\)
\(104\) −6.02238 −0.590543
\(105\) 0 0
\(106\) −7.73956 −0.751733
\(107\) 4.64902 0.449438 0.224719 0.974424i \(-0.427854\pi\)
0.224719 + 0.974424i \(0.427854\pi\)
\(108\) 0 0
\(109\) 3.23140 0.309512 0.154756 0.987953i \(-0.450541\pi\)
0.154756 + 0.987953i \(0.450541\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.390549 0.0369034
\(113\) 11.3194 1.06484 0.532420 0.846480i \(-0.321283\pi\)
0.532420 + 0.846480i \(0.321283\pi\)
\(114\) 0 0
\(115\) −25.0727 −2.33804
\(116\) 4.35247 0.404117
\(117\) 0 0
\(118\) −2.86154 −0.263426
\(119\) −4.40669 −0.403961
\(120\) 0 0
\(121\) 0 0
\(122\) 10.7015 0.968866
\(123\) 0 0
\(124\) −9.88109 −0.887348
\(125\) 3.24213 0.289985
\(126\) 0 0
\(127\) −19.9802 −1.77295 −0.886476 0.462774i \(-0.846854\pi\)
−0.886476 + 0.462774i \(0.846854\pi\)
\(128\) −8.30013 −0.733635
\(129\) 0 0
\(130\) −5.41302 −0.474753
\(131\) −8.45523 −0.738737 −0.369368 0.929283i \(-0.620426\pi\)
−0.369368 + 0.929283i \(0.620426\pi\)
\(132\) 0 0
\(133\) −1.72563 −0.149631
\(134\) 3.55890 0.307442
\(135\) 0 0
\(136\) 12.0552 1.03372
\(137\) −8.35456 −0.713779 −0.356889 0.934147i \(-0.616163\pi\)
−0.356889 + 0.934147i \(0.616163\pi\)
\(138\) 0 0
\(139\) −14.7134 −1.24797 −0.623986 0.781436i \(-0.714488\pi\)
−0.623986 + 0.781436i \(0.714488\pi\)
\(140\) 3.94630 0.333523
\(141\) 0 0
\(142\) −3.62402 −0.304121
\(143\) 0 0
\(144\) 0 0
\(145\) 9.83159 0.816469
\(146\) 12.1482 1.00539
\(147\) 0 0
\(148\) 11.6059 0.954003
\(149\) −8.84271 −0.724423 −0.362212 0.932096i \(-0.617978\pi\)
−0.362212 + 0.932096i \(0.617978\pi\)
\(150\) 0 0
\(151\) −19.9699 −1.62513 −0.812563 0.582873i \(-0.801929\pi\)
−0.812563 + 0.582873i \(0.801929\pi\)
\(152\) 4.72071 0.382900
\(153\) 0 0
\(154\) 0 0
\(155\) −22.3199 −1.79278
\(156\) 0 0
\(157\) 13.0517 1.04164 0.520818 0.853668i \(-0.325627\pi\)
0.520818 + 0.853668i \(0.325627\pi\)
\(158\) −5.91945 −0.470926
\(159\) 0 0
\(160\) −17.2957 −1.36734
\(161\) 8.39774 0.661834
\(162\) 0 0
\(163\) 5.96447 0.467174 0.233587 0.972336i \(-0.424954\pi\)
0.233587 + 0.972336i \(0.424954\pi\)
\(164\) 7.12891 0.556674
\(165\) 0 0
\(166\) −6.28740 −0.487997
\(167\) −13.2472 −1.02510 −0.512548 0.858659i \(-0.671298\pi\)
−0.512548 + 0.858659i \(0.671298\pi\)
\(168\) 0 0
\(169\) −8.15365 −0.627204
\(170\) 10.8354 0.831037
\(171\) 0 0
\(172\) −12.4857 −0.952025
\(173\) 0.517716 0.0393612 0.0196806 0.999806i \(-0.493735\pi\)
0.0196806 + 0.999806i \(0.493735\pi\)
\(174\) 0 0
\(175\) 3.91410 0.295878
\(176\) 0 0
\(177\) 0 0
\(178\) −8.92967 −0.669307
\(179\) −1.69615 −0.126776 −0.0633881 0.997989i \(-0.520191\pi\)
−0.0633881 + 0.997989i \(0.520191\pi\)
\(180\) 0 0
\(181\) −9.88599 −0.734820 −0.367410 0.930059i \(-0.619755\pi\)
−0.367410 + 0.930059i \(0.619755\pi\)
\(182\) 1.81301 0.134389
\(183\) 0 0
\(184\) −22.9733 −1.69361
\(185\) 26.2161 1.92745
\(186\) 0 0
\(187\) 0 0
\(188\) −7.13308 −0.520233
\(189\) 0 0
\(190\) 4.24305 0.307823
\(191\) −23.1329 −1.67384 −0.836918 0.547328i \(-0.815645\pi\)
−0.836918 + 0.547328i \(0.815645\pi\)
\(192\) 0 0
\(193\) −23.0888 −1.66197 −0.830983 0.556297i \(-0.812222\pi\)
−0.830983 + 0.556297i \(0.812222\pi\)
\(194\) 2.35124 0.168809
\(195\) 0 0
\(196\) −1.32176 −0.0944111
\(197\) −6.68989 −0.476635 −0.238318 0.971187i \(-0.576596\pi\)
−0.238318 + 0.971187i \(0.576596\pi\)
\(198\) 0 0
\(199\) −15.8233 −1.12169 −0.560844 0.827922i \(-0.689523\pi\)
−0.560844 + 0.827922i \(0.689523\pi\)
\(200\) −10.7076 −0.757142
\(201\) 0 0
\(202\) 12.0249 0.846066
\(203\) −3.29295 −0.231120
\(204\) 0 0
\(205\) 16.1031 1.12469
\(206\) −0.0887525 −0.00618368
\(207\) 0 0
\(208\) 0.859771 0.0596144
\(209\) 0 0
\(210\) 0 0
\(211\) 11.3525 0.781538 0.390769 0.920489i \(-0.372209\pi\)
0.390769 + 0.920489i \(0.372209\pi\)
\(212\) 12.4215 0.853113
\(213\) 0 0
\(214\) 3.82873 0.261727
\(215\) −28.2033 −1.92345
\(216\) 0 0
\(217\) 7.47573 0.507486
\(218\) 2.66124 0.180242
\(219\) 0 0
\(220\) 0 0
\(221\) −9.70109 −0.652566
\(222\) 0 0
\(223\) −10.6899 −0.715846 −0.357923 0.933751i \(-0.616515\pi\)
−0.357923 + 0.933751i \(0.616515\pi\)
\(224\) 5.79294 0.387057
\(225\) 0 0
\(226\) 9.32217 0.620102
\(227\) 3.73783 0.248089 0.124044 0.992277i \(-0.460414\pi\)
0.124044 + 0.992277i \(0.460414\pi\)
\(228\) 0 0
\(229\) −23.6585 −1.56340 −0.781699 0.623656i \(-0.785647\pi\)
−0.781699 + 0.623656i \(0.785647\pi\)
\(230\) −20.6488 −1.36154
\(231\) 0 0
\(232\) 9.00836 0.591428
\(233\) −3.86675 −0.253319 −0.126660 0.991946i \(-0.540426\pi\)
−0.126660 + 0.991946i \(0.540426\pi\)
\(234\) 0 0
\(235\) −16.1126 −1.05107
\(236\) 4.59259 0.298952
\(237\) 0 0
\(238\) −3.62916 −0.235243
\(239\) 10.0717 0.651482 0.325741 0.945459i \(-0.394386\pi\)
0.325741 + 0.945459i \(0.394386\pi\)
\(240\) 0 0
\(241\) 13.4265 0.864878 0.432439 0.901663i \(-0.357653\pi\)
0.432439 + 0.901663i \(0.357653\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −17.1752 −1.09953
\(245\) −2.98565 −0.190746
\(246\) 0 0
\(247\) −3.79887 −0.241716
\(248\) −20.4510 −1.29864
\(249\) 0 0
\(250\) 2.67007 0.168870
\(251\) −0.0764471 −0.00482530 −0.00241265 0.999997i \(-0.500768\pi\)
−0.00241265 + 0.999997i \(0.500768\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −16.4548 −1.03246
\(255\) 0 0
\(256\) −14.8151 −0.925941
\(257\) −9.51194 −0.593338 −0.296669 0.954980i \(-0.595876\pi\)
−0.296669 + 0.954980i \(0.595876\pi\)
\(258\) 0 0
\(259\) −8.78071 −0.545607
\(260\) 8.68755 0.538779
\(261\) 0 0
\(262\) −6.96335 −0.430197
\(263\) −14.7919 −0.912107 −0.456053 0.889952i \(-0.650737\pi\)
−0.456053 + 0.889952i \(0.650737\pi\)
\(264\) 0 0
\(265\) 28.0583 1.72361
\(266\) −1.42115 −0.0871363
\(267\) 0 0
\(268\) −5.71181 −0.348904
\(269\) 30.0556 1.83252 0.916262 0.400580i \(-0.131191\pi\)
0.916262 + 0.400580i \(0.131191\pi\)
\(270\) 0 0
\(271\) −10.3918 −0.631260 −0.315630 0.948882i \(-0.602216\pi\)
−0.315630 + 0.948882i \(0.602216\pi\)
\(272\) −1.72103 −0.104353
\(273\) 0 0
\(274\) −6.88045 −0.415663
\(275\) 0 0
\(276\) 0 0
\(277\) 28.1138 1.68919 0.844597 0.535403i \(-0.179840\pi\)
0.844597 + 0.535403i \(0.179840\pi\)
\(278\) −12.1173 −0.726747
\(279\) 0 0
\(280\) 8.16769 0.488113
\(281\) −9.57010 −0.570904 −0.285452 0.958393i \(-0.592144\pi\)
−0.285452 + 0.958393i \(0.592144\pi\)
\(282\) 0 0
\(283\) −1.65526 −0.0983947 −0.0491974 0.998789i \(-0.515666\pi\)
−0.0491974 + 0.998789i \(0.515666\pi\)
\(284\) 5.81633 0.345136
\(285\) 0 0
\(286\) 0 0
\(287\) −5.39351 −0.318369
\(288\) 0 0
\(289\) 2.41896 0.142292
\(290\) 8.09686 0.475464
\(291\) 0 0
\(292\) −19.4971 −1.14098
\(293\) −4.68188 −0.273519 −0.136759 0.990604i \(-0.543669\pi\)
−0.136759 + 0.990604i \(0.543669\pi\)
\(294\) 0 0
\(295\) 10.3740 0.603997
\(296\) 24.0210 1.39619
\(297\) 0 0
\(298\) −7.28247 −0.421862
\(299\) 18.4871 1.06914
\(300\) 0 0
\(301\) 9.44629 0.544475
\(302\) −16.4463 −0.946379
\(303\) 0 0
\(304\) −0.673941 −0.0386532
\(305\) −38.7962 −2.22146
\(306\) 0 0
\(307\) −16.9829 −0.969266 −0.484633 0.874718i \(-0.661047\pi\)
−0.484633 + 0.874718i \(0.661047\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −18.3817 −1.04401
\(311\) 21.9257 1.24329 0.621645 0.783299i \(-0.286465\pi\)
0.621645 + 0.783299i \(0.286465\pi\)
\(312\) 0 0
\(313\) 10.0555 0.568368 0.284184 0.958770i \(-0.408277\pi\)
0.284184 + 0.958770i \(0.408277\pi\)
\(314\) 10.7488 0.606589
\(315\) 0 0
\(316\) 9.50035 0.534436
\(317\) −23.5602 −1.32327 −0.661635 0.749826i \(-0.730137\pi\)
−0.661635 + 0.749826i \(0.730137\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −11.9119 −0.665895
\(321\) 0 0
\(322\) 6.91601 0.385414
\(323\) 7.60431 0.423115
\(324\) 0 0
\(325\) 8.61666 0.477966
\(326\) 4.91208 0.272055
\(327\) 0 0
\(328\) 14.7548 0.814696
\(329\) 5.39667 0.297528
\(330\) 0 0
\(331\) −20.8607 −1.14661 −0.573304 0.819343i \(-0.694339\pi\)
−0.573304 + 0.819343i \(0.694339\pi\)
\(332\) 10.0909 0.553809
\(333\) 0 0
\(334\) −10.9098 −0.596956
\(335\) −12.9021 −0.704918
\(336\) 0 0
\(337\) 13.3544 0.727458 0.363729 0.931505i \(-0.381503\pi\)
0.363729 + 0.931505i \(0.381503\pi\)
\(338\) −6.71498 −0.365247
\(339\) 0 0
\(340\) −17.3901 −0.943112
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −25.8418 −1.39329
\(345\) 0 0
\(346\) 0.426368 0.0229217
\(347\) 1.59256 0.0854933 0.0427466 0.999086i \(-0.486389\pi\)
0.0427466 + 0.999086i \(0.486389\pi\)
\(348\) 0 0
\(349\) 6.63475 0.355150 0.177575 0.984107i \(-0.443175\pi\)
0.177575 + 0.984107i \(0.443175\pi\)
\(350\) 3.22348 0.172302
\(351\) 0 0
\(352\) 0 0
\(353\) 25.4141 1.35265 0.676327 0.736601i \(-0.263570\pi\)
0.676327 + 0.736601i \(0.263570\pi\)
\(354\) 0 0
\(355\) 13.1382 0.697304
\(356\) 14.3316 0.759571
\(357\) 0 0
\(358\) −1.39688 −0.0738271
\(359\) −16.0792 −0.848629 −0.424314 0.905515i \(-0.639485\pi\)
−0.424314 + 0.905515i \(0.639485\pi\)
\(360\) 0 0
\(361\) −16.0222 −0.843274
\(362\) −8.14167 −0.427916
\(363\) 0 0
\(364\) −2.90977 −0.152513
\(365\) −44.0411 −2.30522
\(366\) 0 0
\(367\) −3.02606 −0.157959 −0.0789796 0.996876i \(-0.525166\pi\)
−0.0789796 + 0.996876i \(0.525166\pi\)
\(368\) 3.27973 0.170968
\(369\) 0 0
\(370\) 21.5904 1.12243
\(371\) −9.39774 −0.487906
\(372\) 0 0
\(373\) 1.73856 0.0900192 0.0450096 0.998987i \(-0.485668\pi\)
0.0450096 + 0.998987i \(0.485668\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −14.7634 −0.761365
\(377\) −7.24924 −0.373355
\(378\) 0 0
\(379\) 19.5728 1.00538 0.502692 0.864465i \(-0.332343\pi\)
0.502692 + 0.864465i \(0.332343\pi\)
\(380\) −6.80983 −0.349337
\(381\) 0 0
\(382\) −19.0512 −0.974746
\(383\) −4.76472 −0.243466 −0.121733 0.992563i \(-0.538845\pi\)
−0.121733 + 0.992563i \(0.538845\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −19.0149 −0.967833
\(387\) 0 0
\(388\) −3.77359 −0.191575
\(389\) 1.42208 0.0721024 0.0360512 0.999350i \(-0.488522\pi\)
0.0360512 + 0.999350i \(0.488522\pi\)
\(390\) 0 0
\(391\) −37.0063 −1.87149
\(392\) −2.73565 −0.138171
\(393\) 0 0
\(394\) −5.50950 −0.277565
\(395\) 21.4599 1.07976
\(396\) 0 0
\(397\) −18.3969 −0.923314 −0.461657 0.887059i \(-0.652745\pi\)
−0.461657 + 0.887059i \(0.652745\pi\)
\(398\) −13.0314 −0.653206
\(399\) 0 0
\(400\) 1.52865 0.0764323
\(401\) 1.65268 0.0825308 0.0412654 0.999148i \(-0.486861\pi\)
0.0412654 + 0.999148i \(0.486861\pi\)
\(402\) 0 0
\(403\) 16.4574 0.819802
\(404\) −19.2991 −0.960168
\(405\) 0 0
\(406\) −2.71193 −0.134591
\(407\) 0 0
\(408\) 0 0
\(409\) −36.6858 −1.81400 −0.906999 0.421133i \(-0.861632\pi\)
−0.906999 + 0.421133i \(0.861632\pi\)
\(410\) 13.2618 0.654955
\(411\) 0 0
\(412\) 0.142442 0.00701762
\(413\) −3.47462 −0.170975
\(414\) 0 0
\(415\) 22.7938 1.11890
\(416\) 12.7528 0.625259
\(417\) 0 0
\(418\) 0 0
\(419\) 17.3452 0.847366 0.423683 0.905810i \(-0.360737\pi\)
0.423683 + 0.905810i \(0.360737\pi\)
\(420\) 0 0
\(421\) 5.32683 0.259614 0.129807 0.991539i \(-0.458564\pi\)
0.129807 + 0.991539i \(0.458564\pi\)
\(422\) 9.34941 0.455122
\(423\) 0 0
\(424\) 25.7089 1.24854
\(425\) −17.2482 −0.836662
\(426\) 0 0
\(427\) 12.9942 0.628835
\(428\) −6.14487 −0.297024
\(429\) 0 0
\(430\) −23.2270 −1.12011
\(431\) 28.2081 1.35873 0.679367 0.733798i \(-0.262254\pi\)
0.679367 + 0.733798i \(0.262254\pi\)
\(432\) 0 0
\(433\) 17.7060 0.850895 0.425447 0.904983i \(-0.360117\pi\)
0.425447 + 0.904983i \(0.360117\pi\)
\(434\) 6.15668 0.295530
\(435\) 0 0
\(436\) −4.27113 −0.204550
\(437\) −14.4914 −0.693216
\(438\) 0 0
\(439\) −25.1189 −1.19886 −0.599430 0.800427i \(-0.704606\pi\)
−0.599430 + 0.800427i \(0.704606\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.98939 −0.380017
\(443\) 26.9567 1.28075 0.640376 0.768061i \(-0.278778\pi\)
0.640376 + 0.768061i \(0.278778\pi\)
\(444\) 0 0
\(445\) 32.3728 1.53462
\(446\) −8.80369 −0.416867
\(447\) 0 0
\(448\) 3.98971 0.188496
\(449\) 21.0964 0.995599 0.497799 0.867292i \(-0.334142\pi\)
0.497799 + 0.867292i \(0.334142\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −14.9615 −0.703730
\(453\) 0 0
\(454\) 3.07831 0.144472
\(455\) −6.57274 −0.308135
\(456\) 0 0
\(457\) −1.90356 −0.0890448 −0.0445224 0.999008i \(-0.514177\pi\)
−0.0445224 + 0.999008i \(0.514177\pi\)
\(458\) −19.4841 −0.910432
\(459\) 0 0
\(460\) 33.1400 1.54516
\(461\) −25.0440 −1.16641 −0.583207 0.812324i \(-0.698202\pi\)
−0.583207 + 0.812324i \(0.698202\pi\)
\(462\) 0 0
\(463\) −21.1721 −0.983952 −0.491976 0.870609i \(-0.663725\pi\)
−0.491976 + 0.870609i \(0.663725\pi\)
\(464\) −1.28606 −0.0597037
\(465\) 0 0
\(466\) −3.18449 −0.147519
\(467\) −16.6484 −0.770398 −0.385199 0.922834i \(-0.625867\pi\)
−0.385199 + 0.922834i \(0.625867\pi\)
\(468\) 0 0
\(469\) 4.32138 0.199543
\(470\) −13.2696 −0.612081
\(471\) 0 0
\(472\) 9.50534 0.437519
\(473\) 0 0
\(474\) 0 0
\(475\) −6.75427 −0.309907
\(476\) 5.82457 0.266969
\(477\) 0 0
\(478\) 8.29459 0.379386
\(479\) −21.6613 −0.989730 −0.494865 0.868970i \(-0.664783\pi\)
−0.494865 + 0.868970i \(0.664783\pi\)
\(480\) 0 0
\(481\) −19.3302 −0.881383
\(482\) 11.0575 0.503655
\(483\) 0 0
\(484\) 0 0
\(485\) −8.52397 −0.387053
\(486\) 0 0
\(487\) −2.81335 −0.127485 −0.0637426 0.997966i \(-0.520304\pi\)
−0.0637426 + 0.997966i \(0.520304\pi\)
\(488\) −35.5477 −1.60917
\(489\) 0 0
\(490\) −2.45885 −0.111079
\(491\) −3.20580 −0.144676 −0.0723379 0.997380i \(-0.523046\pi\)
−0.0723379 + 0.997380i \(0.523046\pi\)
\(492\) 0 0
\(493\) 14.5110 0.653543
\(494\) −3.12858 −0.140762
\(495\) 0 0
\(496\) 2.91964 0.131096
\(497\) −4.40046 −0.197388
\(498\) 0 0
\(499\) −20.0512 −0.897616 −0.448808 0.893628i \(-0.648151\pi\)
−0.448808 + 0.893628i \(0.648151\pi\)
\(500\) −4.28530 −0.191644
\(501\) 0 0
\(502\) −0.0629585 −0.00280997
\(503\) −35.5089 −1.58326 −0.791631 0.610999i \(-0.790768\pi\)
−0.791631 + 0.610999i \(0.790768\pi\)
\(504\) 0 0
\(505\) −43.5939 −1.93990
\(506\) 0 0
\(507\) 0 0
\(508\) 26.4089 1.17170
\(509\) −9.18980 −0.407331 −0.203665 0.979041i \(-0.565285\pi\)
−0.203665 + 0.979041i \(0.565285\pi\)
\(510\) 0 0
\(511\) 14.7509 0.652543
\(512\) 4.39924 0.194421
\(513\) 0 0
\(514\) −7.83361 −0.345526
\(515\) 0.321756 0.0141782
\(516\) 0 0
\(517\) 0 0
\(518\) −7.23140 −0.317730
\(519\) 0 0
\(520\) 17.9807 0.788507
\(521\) −12.3998 −0.543246 −0.271623 0.962404i \(-0.587560\pi\)
−0.271623 + 0.962404i \(0.587560\pi\)
\(522\) 0 0
\(523\) 22.7105 0.993062 0.496531 0.868019i \(-0.334607\pi\)
0.496531 + 0.868019i \(0.334607\pi\)
\(524\) 11.1757 0.488215
\(525\) 0 0
\(526\) −12.1819 −0.531158
\(527\) −32.9433 −1.43503
\(528\) 0 0
\(529\) 47.5220 2.06617
\(530\) 23.1076 1.00373
\(531\) 0 0
\(532\) 2.28086 0.0988877
\(533\) −11.8735 −0.514299
\(534\) 0 0
\(535\) −13.8803 −0.600100
\(536\) −11.8218 −0.510623
\(537\) 0 0
\(538\) 24.7525 1.06716
\(539\) 0 0
\(540\) 0 0
\(541\) 17.2748 0.742703 0.371351 0.928492i \(-0.378895\pi\)
0.371351 + 0.928492i \(0.378895\pi\)
\(542\) −8.55827 −0.367609
\(543\) 0 0
\(544\) −25.5277 −1.09449
\(545\) −9.64784 −0.413268
\(546\) 0 0
\(547\) −13.1740 −0.563278 −0.281639 0.959520i \(-0.590878\pi\)
−0.281639 + 0.959520i \(0.590878\pi\)
\(548\) 11.0427 0.471720
\(549\) 0 0
\(550\) 0 0
\(551\) 5.68240 0.242078
\(552\) 0 0
\(553\) −7.18768 −0.305651
\(554\) 23.1533 0.983689
\(555\) 0 0
\(556\) 19.4475 0.824757
\(557\) 12.6216 0.534793 0.267396 0.963587i \(-0.413837\pi\)
0.267396 + 0.963587i \(0.413837\pi\)
\(558\) 0 0
\(559\) 20.7955 0.879555
\(560\) −1.16604 −0.0492743
\(561\) 0 0
\(562\) −7.88151 −0.332462
\(563\) −9.81924 −0.413832 −0.206916 0.978359i \(-0.566343\pi\)
−0.206916 + 0.978359i \(0.566343\pi\)
\(564\) 0 0
\(565\) −33.7958 −1.42180
\(566\) −1.36320 −0.0572994
\(567\) 0 0
\(568\) 12.0381 0.505108
\(569\) −20.3544 −0.853301 −0.426650 0.904417i \(-0.640307\pi\)
−0.426650 + 0.904417i \(0.640307\pi\)
\(570\) 0 0
\(571\) −32.3174 −1.35244 −0.676221 0.736699i \(-0.736384\pi\)
−0.676221 + 0.736699i \(0.736384\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.44186 −0.185400
\(575\) 32.8696 1.37076
\(576\) 0 0
\(577\) −11.3181 −0.471179 −0.235590 0.971853i \(-0.575702\pi\)
−0.235590 + 0.971853i \(0.575702\pi\)
\(578\) 1.99215 0.0828623
\(579\) 0 0
\(580\) −12.9950 −0.539586
\(581\) −7.63445 −0.316730
\(582\) 0 0
\(583\) 0 0
\(584\) −40.3534 −1.66984
\(585\) 0 0
\(586\) −3.85579 −0.159281
\(587\) 38.2388 1.57828 0.789142 0.614211i \(-0.210526\pi\)
0.789142 + 0.614211i \(0.210526\pi\)
\(588\) 0 0
\(589\) −12.9003 −0.531548
\(590\) 8.54356 0.351733
\(591\) 0 0
\(592\) −3.42930 −0.140943
\(593\) 26.4263 1.08520 0.542598 0.839992i \(-0.317441\pi\)
0.542598 + 0.839992i \(0.317441\pi\)
\(594\) 0 0
\(595\) 13.1568 0.539378
\(596\) 11.6879 0.478755
\(597\) 0 0
\(598\) 15.2252 0.622605
\(599\) 20.2994 0.829413 0.414706 0.909955i \(-0.363884\pi\)
0.414706 + 0.909955i \(0.363884\pi\)
\(600\) 0 0
\(601\) −5.09410 −0.207793 −0.103896 0.994588i \(-0.533131\pi\)
−0.103896 + 0.994588i \(0.533131\pi\)
\(602\) 7.77955 0.317071
\(603\) 0 0
\(604\) 26.3953 1.07401
\(605\) 0 0
\(606\) 0 0
\(607\) 30.6407 1.24367 0.621833 0.783150i \(-0.286388\pi\)
0.621833 + 0.783150i \(0.286388\pi\)
\(608\) −9.99645 −0.405410
\(609\) 0 0
\(610\) −31.9508 −1.29365
\(611\) 11.8805 0.480632
\(612\) 0 0
\(613\) −24.8391 −1.00324 −0.501620 0.865088i \(-0.667263\pi\)
−0.501620 + 0.865088i \(0.667263\pi\)
\(614\) −13.9864 −0.564444
\(615\) 0 0
\(616\) 0 0
\(617\) −0.521714 −0.0210034 −0.0105017 0.999945i \(-0.503343\pi\)
−0.0105017 + 0.999945i \(0.503343\pi\)
\(618\) 0 0
\(619\) 31.4859 1.26552 0.632762 0.774346i \(-0.281921\pi\)
0.632762 + 0.774346i \(0.281921\pi\)
\(620\) 29.5015 1.18481
\(621\) 0 0
\(622\) 18.0570 0.724020
\(623\) −10.8428 −0.434408
\(624\) 0 0
\(625\) −29.2503 −1.17001
\(626\) 8.28124 0.330985
\(627\) 0 0
\(628\) −17.2511 −0.688395
\(629\) 38.6939 1.54283
\(630\) 0 0
\(631\) −0.114230 −0.00454742 −0.00227371 0.999997i \(-0.500724\pi\)
−0.00227371 + 0.999997i \(0.500724\pi\)
\(632\) 19.6630 0.782151
\(633\) 0 0
\(634\) −19.4031 −0.770596
\(635\) 59.6537 2.36729
\(636\) 0 0
\(637\) 2.20144 0.0872244
\(638\) 0 0
\(639\) 0 0
\(640\) 24.7813 0.979566
\(641\) −13.2141 −0.521927 −0.260963 0.965349i \(-0.584040\pi\)
−0.260963 + 0.965349i \(0.584040\pi\)
\(642\) 0 0
\(643\) 2.45599 0.0968550 0.0484275 0.998827i \(-0.484579\pi\)
0.0484275 + 0.998827i \(0.484579\pi\)
\(644\) −11.0998 −0.437392
\(645\) 0 0
\(646\) 6.26257 0.246398
\(647\) 18.6116 0.731698 0.365849 0.930674i \(-0.380779\pi\)
0.365849 + 0.930674i \(0.380779\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 7.09630 0.278340
\(651\) 0 0
\(652\) −7.88357 −0.308745
\(653\) −19.4481 −0.761065 −0.380532 0.924768i \(-0.624259\pi\)
−0.380532 + 0.924768i \(0.624259\pi\)
\(654\) 0 0
\(655\) 25.2443 0.986378
\(656\) −2.10643 −0.0822423
\(657\) 0 0
\(658\) 4.44446 0.173263
\(659\) −4.71629 −0.183721 −0.0918603 0.995772i \(-0.529281\pi\)
−0.0918603 + 0.995772i \(0.529281\pi\)
\(660\) 0 0
\(661\) 24.9330 0.969782 0.484891 0.874575i \(-0.338859\pi\)
0.484891 + 0.874575i \(0.338859\pi\)
\(662\) −17.1800 −0.667718
\(663\) 0 0
\(664\) 20.8852 0.810503
\(665\) 5.15211 0.199790
\(666\) 0 0
\(667\) −27.6533 −1.07074
\(668\) 17.5095 0.677463
\(669\) 0 0
\(670\) −10.6256 −0.410503
\(671\) 0 0
\(672\) 0 0
\(673\) −5.12972 −0.197736 −0.0988681 0.995101i \(-0.531522\pi\)
−0.0988681 + 0.995101i \(0.531522\pi\)
\(674\) 10.9981 0.423629
\(675\) 0 0
\(676\) 10.7771 0.414505
\(677\) 51.2610 1.97012 0.985060 0.172213i \(-0.0550917\pi\)
0.985060 + 0.172213i \(0.0550917\pi\)
\(678\) 0 0
\(679\) 2.85498 0.109564
\(680\) −35.9925 −1.38025
\(681\) 0 0
\(682\) 0 0
\(683\) 24.9448 0.954485 0.477242 0.878772i \(-0.341636\pi\)
0.477242 + 0.878772i \(0.341636\pi\)
\(684\) 0 0
\(685\) 24.9438 0.953053
\(686\) 0.823556 0.0314435
\(687\) 0 0
\(688\) 3.68924 0.140651
\(689\) −20.6886 −0.788172
\(690\) 0 0
\(691\) 11.7847 0.448312 0.224156 0.974553i \(-0.428038\pi\)
0.224156 + 0.974553i \(0.428038\pi\)
\(692\) −0.684294 −0.0260130
\(693\) 0 0
\(694\) 1.31156 0.0497863
\(695\) 43.9290 1.66632
\(696\) 0 0
\(697\) 23.7676 0.900261
\(698\) 5.46409 0.206819
\(699\) 0 0
\(700\) −5.17348 −0.195539
\(701\) 31.3172 1.18283 0.591417 0.806366i \(-0.298569\pi\)
0.591417 + 0.806366i \(0.298569\pi\)
\(702\) 0 0
\(703\) 15.1522 0.571477
\(704\) 0 0
\(705\) 0 0
\(706\) 20.9299 0.787708
\(707\) 14.6011 0.549133
\(708\) 0 0
\(709\) −41.4751 −1.55763 −0.778815 0.627254i \(-0.784179\pi\)
−0.778815 + 0.627254i \(0.784179\pi\)
\(710\) 10.8201 0.406070
\(711\) 0 0
\(712\) 29.6622 1.11164
\(713\) 62.7792 2.35110
\(714\) 0 0
\(715\) 0 0
\(716\) 2.24190 0.0837836
\(717\) 0 0
\(718\) −13.2421 −0.494192
\(719\) 7.89027 0.294258 0.147129 0.989117i \(-0.452997\pi\)
0.147129 + 0.989117i \(0.452997\pi\)
\(720\) 0 0
\(721\) −0.107767 −0.00401347
\(722\) −13.1952 −0.491074
\(723\) 0 0
\(724\) 13.0669 0.485626
\(725\) −12.8889 −0.478682
\(726\) 0 0
\(727\) 5.12729 0.190161 0.0950803 0.995470i \(-0.469689\pi\)
0.0950803 + 0.995470i \(0.469689\pi\)
\(728\) −6.02238 −0.223204
\(729\) 0 0
\(730\) −36.2703 −1.34242
\(731\) −41.6269 −1.53963
\(732\) 0 0
\(733\) −33.7736 −1.24746 −0.623728 0.781642i \(-0.714383\pi\)
−0.623728 + 0.781642i \(0.714383\pi\)
\(734\) −2.49213 −0.0919863
\(735\) 0 0
\(736\) 48.6476 1.79317
\(737\) 0 0
\(738\) 0 0
\(739\) −42.7431 −1.57233 −0.786164 0.618017i \(-0.787936\pi\)
−0.786164 + 0.618017i \(0.787936\pi\)
\(740\) −34.6513 −1.27381
\(741\) 0 0
\(742\) −7.73956 −0.284128
\(743\) −22.9565 −0.842192 −0.421096 0.907016i \(-0.638354\pi\)
−0.421096 + 0.907016i \(0.638354\pi\)
\(744\) 0 0
\(745\) 26.4012 0.967266
\(746\) 1.43180 0.0524220
\(747\) 0 0
\(748\) 0 0
\(749\) 4.64902 0.169872
\(750\) 0 0
\(751\) −0.930719 −0.0339624 −0.0169812 0.999856i \(-0.505406\pi\)
−0.0169812 + 0.999856i \(0.505406\pi\)
\(752\) 2.10766 0.0768586
\(753\) 0 0
\(754\) −5.97016 −0.217420
\(755\) 59.6231 2.16991
\(756\) 0 0
\(757\) −11.4824 −0.417334 −0.208667 0.977987i \(-0.566913\pi\)
−0.208667 + 0.977987i \(0.566913\pi\)
\(758\) 16.1193 0.585478
\(759\) 0 0
\(760\) −14.0944 −0.511257
\(761\) −38.8640 −1.40882 −0.704410 0.709794i \(-0.748788\pi\)
−0.704410 + 0.709794i \(0.748788\pi\)
\(762\) 0 0
\(763\) 3.23140 0.116985
\(764\) 30.5760 1.10620
\(765\) 0 0
\(766\) −3.92402 −0.141780
\(767\) −7.64917 −0.276196
\(768\) 0 0
\(769\) −35.6509 −1.28560 −0.642802 0.766032i \(-0.722228\pi\)
−0.642802 + 0.766032i \(0.722228\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 30.5177 1.09836
\(773\) 1.77045 0.0636788 0.0318394 0.999493i \(-0.489863\pi\)
0.0318394 + 0.999493i \(0.489863\pi\)
\(774\) 0 0
\(775\) 29.2607 1.05108
\(776\) −7.81023 −0.280371
\(777\) 0 0
\(778\) 1.17116 0.0419883
\(779\) 9.30719 0.333465
\(780\) 0 0
\(781\) 0 0
\(782\) −30.4767 −1.08985
\(783\) 0 0
\(784\) 0.390549 0.0139482
\(785\) −38.9677 −1.39082
\(786\) 0 0
\(787\) 3.35315 0.119527 0.0597635 0.998213i \(-0.480965\pi\)
0.0597635 + 0.998213i \(0.480965\pi\)
\(788\) 8.84240 0.314998
\(789\) 0 0
\(790\) 17.6734 0.628792
\(791\) 11.3194 0.402472
\(792\) 0 0
\(793\) 28.6061 1.01583
\(794\) −15.1509 −0.537684
\(795\) 0 0
\(796\) 20.9146 0.741298
\(797\) −4.92542 −0.174467 −0.0872337 0.996188i \(-0.527803\pi\)
−0.0872337 + 0.996188i \(0.527803\pi\)
\(798\) 0 0
\(799\) −23.7815 −0.841328
\(800\) 22.6741 0.801652
\(801\) 0 0
\(802\) 1.36107 0.0480612
\(803\) 0 0
\(804\) 0 0
\(805\) −25.0727 −0.883696
\(806\) 13.5536 0.477405
\(807\) 0 0
\(808\) −39.9436 −1.40521
\(809\) 6.97322 0.245165 0.122583 0.992458i \(-0.460882\pi\)
0.122583 + 0.992458i \(0.460882\pi\)
\(810\) 0 0
\(811\) 5.93377 0.208363 0.104181 0.994558i \(-0.466778\pi\)
0.104181 + 0.994558i \(0.466778\pi\)
\(812\) 4.35247 0.152742
\(813\) 0 0
\(814\) 0 0
\(815\) −17.8078 −0.623781
\(816\) 0 0
\(817\) −16.3008 −0.570292
\(818\) −30.2128 −1.05637
\(819\) 0 0
\(820\) −21.2844 −0.743284
\(821\) −41.7468 −1.45697 −0.728487 0.685060i \(-0.759776\pi\)
−0.728487 + 0.685060i \(0.759776\pi\)
\(822\) 0 0
\(823\) −23.2004 −0.808714 −0.404357 0.914601i \(-0.632505\pi\)
−0.404357 + 0.914601i \(0.632505\pi\)
\(824\) 0.294814 0.0102703
\(825\) 0 0
\(826\) −2.86154 −0.0995658
\(827\) 13.1779 0.458240 0.229120 0.973398i \(-0.426415\pi\)
0.229120 + 0.973398i \(0.426415\pi\)
\(828\) 0 0
\(829\) 4.29090 0.149029 0.0745146 0.997220i \(-0.476259\pi\)
0.0745146 + 0.997220i \(0.476259\pi\)
\(830\) 18.7720 0.651584
\(831\) 0 0
\(832\) 8.78313 0.304500
\(833\) −4.40669 −0.152683
\(834\) 0 0
\(835\) 39.5513 1.36873
\(836\) 0 0
\(837\) 0 0
\(838\) 14.2847 0.493457
\(839\) 15.7453 0.543588 0.271794 0.962355i \(-0.412383\pi\)
0.271794 + 0.962355i \(0.412383\pi\)
\(840\) 0 0
\(841\) −18.1565 −0.626086
\(842\) 4.38694 0.151184
\(843\) 0 0
\(844\) −15.0052 −0.516501
\(845\) 24.3439 0.837456
\(846\) 0 0
\(847\) 0 0
\(848\) −3.67028 −0.126038
\(849\) 0 0
\(850\) −14.2049 −0.487223
\(851\) −73.7381 −2.52771
\(852\) 0 0
\(853\) 6.14015 0.210235 0.105117 0.994460i \(-0.466478\pi\)
0.105117 + 0.994460i \(0.466478\pi\)
\(854\) 10.7015 0.366197
\(855\) 0 0
\(856\) −12.7181 −0.434696
\(857\) −34.4740 −1.17761 −0.588804 0.808276i \(-0.700401\pi\)
−0.588804 + 0.808276i \(0.700401\pi\)
\(858\) 0 0
\(859\) 8.21553 0.280310 0.140155 0.990130i \(-0.455240\pi\)
0.140155 + 0.990130i \(0.455240\pi\)
\(860\) 37.2779 1.27116
\(861\) 0 0
\(862\) 23.2309 0.791248
\(863\) −13.0390 −0.443852 −0.221926 0.975063i \(-0.571234\pi\)
−0.221926 + 0.975063i \(0.571234\pi\)
\(864\) 0 0
\(865\) −1.54572 −0.0525560
\(866\) 14.5819 0.495512
\(867\) 0 0
\(868\) −9.88109 −0.335386
\(869\) 0 0
\(870\) 0 0
\(871\) 9.51327 0.322345
\(872\) −8.84000 −0.299360
\(873\) 0 0
\(874\) −11.9344 −0.403689
\(875\) 3.24213 0.109604
\(876\) 0 0
\(877\) 17.9259 0.605316 0.302658 0.953099i \(-0.402126\pi\)
0.302658 + 0.953099i \(0.402126\pi\)
\(878\) −20.6868 −0.698147
\(879\) 0 0
\(880\) 0 0
\(881\) 17.3276 0.583780 0.291890 0.956452i \(-0.405716\pi\)
0.291890 + 0.956452i \(0.405716\pi\)
\(882\) 0 0
\(883\) 21.8740 0.736120 0.368060 0.929802i \(-0.380022\pi\)
0.368060 + 0.929802i \(0.380022\pi\)
\(884\) 12.8225 0.431266
\(885\) 0 0
\(886\) 22.2004 0.745836
\(887\) −25.6098 −0.859893 −0.429946 0.902854i \(-0.641468\pi\)
−0.429946 + 0.902854i \(0.641468\pi\)
\(888\) 0 0
\(889\) −19.9802 −0.670113
\(890\) 26.6608 0.893674
\(891\) 0 0
\(892\) 14.1294 0.473087
\(893\) −9.31263 −0.311635
\(894\) 0 0
\(895\) 5.06411 0.169275
\(896\) −8.30013 −0.277288
\(897\) 0 0
\(898\) 17.3740 0.579779
\(899\) −24.6172 −0.821030
\(900\) 0 0
\(901\) 41.4130 1.37967
\(902\) 0 0
\(903\) 0 0
\(904\) −30.9660 −1.02991
\(905\) 29.5161 0.981148
\(906\) 0 0
\(907\) −14.4388 −0.479431 −0.239716 0.970843i \(-0.577054\pi\)
−0.239716 + 0.970843i \(0.577054\pi\)
\(908\) −4.94050 −0.163956
\(909\) 0 0
\(910\) −5.41302 −0.179440
\(911\) −37.1858 −1.23202 −0.616010 0.787739i \(-0.711252\pi\)
−0.616010 + 0.787739i \(0.711252\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.56769 −0.0518545
\(915\) 0 0
\(916\) 31.2708 1.03321
\(917\) −8.45523 −0.279216
\(918\) 0 0
\(919\) −16.0496 −0.529426 −0.264713 0.964327i \(-0.585277\pi\)
−0.264713 + 0.964327i \(0.585277\pi\)
\(920\) 68.5901 2.26135
\(921\) 0 0
\(922\) −20.6251 −0.679252
\(923\) −9.68736 −0.318863
\(924\) 0 0
\(925\) −34.3685 −1.13003
\(926\) −17.4364 −0.572997
\(927\) 0 0
\(928\) −19.0759 −0.626196
\(929\) −33.8914 −1.11194 −0.555970 0.831202i \(-0.687653\pi\)
−0.555970 + 0.831202i \(0.687653\pi\)
\(930\) 0 0
\(931\) −1.72563 −0.0565551
\(932\) 5.11090 0.167413
\(933\) 0 0
\(934\) −13.7109 −0.448635
\(935\) 0 0
\(936\) 0 0
\(937\) −9.22624 −0.301408 −0.150704 0.988579i \(-0.548154\pi\)
−0.150704 + 0.988579i \(0.548154\pi\)
\(938\) 3.55890 0.116202
\(939\) 0 0
\(940\) 21.2969 0.694627
\(941\) 53.3626 1.73957 0.869786 0.493430i \(-0.164257\pi\)
0.869786 + 0.493430i \(0.164257\pi\)
\(942\) 0 0
\(943\) −45.2933 −1.47495
\(944\) −1.35701 −0.0441668
\(945\) 0 0
\(946\) 0 0
\(947\) −31.0986 −1.01057 −0.505285 0.862953i \(-0.668612\pi\)
−0.505285 + 0.862953i \(0.668612\pi\)
\(948\) 0 0
\(949\) 32.4734 1.05413
\(950\) −5.56252 −0.180472
\(951\) 0 0
\(952\) 12.0552 0.390711
\(953\) −34.9286 −1.13145 −0.565724 0.824595i \(-0.691403\pi\)
−0.565724 + 0.824595i \(0.691403\pi\)
\(954\) 0 0
\(955\) 69.0667 2.23494
\(956\) −13.3123 −0.430550
\(957\) 0 0
\(958\) −17.8393 −0.576361
\(959\) −8.35456 −0.269783
\(960\) 0 0
\(961\) 24.8866 0.802793
\(962\) −15.9195 −0.513266
\(963\) 0 0
\(964\) −17.7466 −0.571579
\(965\) 68.9350 2.21910
\(966\) 0 0
\(967\) −20.8029 −0.668975 −0.334488 0.942400i \(-0.608563\pi\)
−0.334488 + 0.942400i \(0.608563\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −7.01996 −0.225397
\(971\) 40.0308 1.28465 0.642325 0.766433i \(-0.277970\pi\)
0.642325 + 0.766433i \(0.277970\pi\)
\(972\) 0 0
\(973\) −14.7134 −0.471689
\(974\) −2.31695 −0.0742400
\(975\) 0 0
\(976\) 5.07488 0.162443
\(977\) −6.48609 −0.207509 −0.103754 0.994603i \(-0.533086\pi\)
−0.103754 + 0.994603i \(0.533086\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.94630 0.126060
\(981\) 0 0
\(982\) −2.64016 −0.0842508
\(983\) −10.9782 −0.350150 −0.175075 0.984555i \(-0.556017\pi\)
−0.175075 + 0.984555i \(0.556017\pi\)
\(984\) 0 0
\(985\) 19.9737 0.636414
\(986\) 11.9506 0.380586
\(987\) 0 0
\(988\) 5.02118 0.159745
\(989\) 79.3275 2.52247
\(990\) 0 0
\(991\) −59.2666 −1.88267 −0.941333 0.337479i \(-0.890426\pi\)
−0.941333 + 0.337479i \(0.890426\pi\)
\(992\) 43.3065 1.37498
\(993\) 0 0
\(994\) −3.62402 −0.114947
\(995\) 47.2430 1.49770
\(996\) 0 0
\(997\) −43.3517 −1.37296 −0.686481 0.727148i \(-0.740845\pi\)
−0.686481 + 0.727148i \(0.740845\pi\)
\(998\) −16.5133 −0.522720
\(999\) 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.cp.1.5 6
3.2 odd 2 847.2.a.n.1.2 yes 6
11.10 odd 2 7623.2.a.cs.1.2 6
21.20 even 2 5929.2.a.bm.1.2 6
33.2 even 10 847.2.f.z.323.2 24
33.5 odd 10 847.2.f.y.729.5 24
33.8 even 10 847.2.f.z.372.5 24
33.14 odd 10 847.2.f.y.372.2 24
33.17 even 10 847.2.f.z.729.2 24
33.20 odd 10 847.2.f.y.323.5 24
33.26 odd 10 847.2.f.y.148.2 24
33.29 even 10 847.2.f.z.148.5 24
33.32 even 2 847.2.a.m.1.5 6
231.230 odd 2 5929.2.a.bj.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.m.1.5 6 33.32 even 2
847.2.a.n.1.2 yes 6 3.2 odd 2
847.2.f.y.148.2 24 33.26 odd 10
847.2.f.y.323.5 24 33.20 odd 10
847.2.f.y.372.2 24 33.14 odd 10
847.2.f.y.729.5 24 33.5 odd 10
847.2.f.z.148.5 24 33.29 even 10
847.2.f.z.323.2 24 33.2 even 10
847.2.f.z.372.5 24 33.8 even 10
847.2.f.z.729.2 24 33.17 even 10
5929.2.a.bj.1.5 6 231.230 odd 2
5929.2.a.bm.1.2 6 21.20 even 2
7623.2.a.cp.1.5 6 1.1 even 1 trivial
7623.2.a.cs.1.2 6 11.10 odd 2