Properties

Label 4560.2.a.bu.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.27307\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +0.166860 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +0.166860 q^{7} +1.00000 q^{9} -4.71301 q^{11} +1.83314 q^{13} +1.00000 q^{15} -4.54615 q^{17} -1.00000 q^{19} +0.166860 q^{21} +4.54615 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.71301 q^{29} +2.54615 q^{31} -4.71301 q^{33} +0.166860 q^{35} +2.16686 q^{37} +1.83314 q^{39} +10.7130 q^{41} -9.25915 q^{43} +1.00000 q^{45} -0.546146 q^{47} -6.97216 q^{49} -4.54615 q^{51} +10.5461 q^{53} -4.71301 q^{55} -1.00000 q^{57} +9.42601 q^{59} +12.8799 q^{61} +0.166860 q^{63} +1.83314 q^{65} +4.54615 q^{69} +5.42601 q^{71} +15.0923 q^{73} +1.00000 q^{75} -0.786413 q^{77} -13.7597 q^{79} +1.00000 q^{81} +13.9722 q^{83} -4.54615 q^{85} +6.71301 q^{87} +6.04557 q^{89} +0.305878 q^{91} +2.54615 q^{93} -1.00000 q^{95} -2.16686 q^{97} -4.71301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99} + O(q^{100}) \)

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.166860 0.0630672 0.0315336 0.999503i \(-0.489961\pi\)
0.0315336 + 0.999503i \(0.489961\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.71301 −1.42102 −0.710512 0.703685i \(-0.751537\pi\)
−0.710512 + 0.703685i \(0.751537\pi\)
\(12\) 0 0
\(13\) 1.83314 0.508421 0.254211 0.967149i \(-0.418184\pi\)
0.254211 + 0.967149i \(0.418184\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.54615 −1.10260 −0.551301 0.834306i \(-0.685868\pi\)
−0.551301 + 0.834306i \(0.685868\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.166860 0.0364119
\(22\) 0 0
\(23\) 4.54615 0.947937 0.473968 0.880542i \(-0.342821\pi\)
0.473968 + 0.880542i \(0.342821\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.71301 1.24657 0.623287 0.781993i \(-0.285797\pi\)
0.623287 + 0.781993i \(0.285797\pi\)
\(30\) 0 0
\(31\) 2.54615 0.457301 0.228651 0.973509i \(-0.426569\pi\)
0.228651 + 0.973509i \(0.426569\pi\)
\(32\) 0 0
\(33\) −4.71301 −0.820429
\(34\) 0 0
\(35\) 0.166860 0.0282045
\(36\) 0 0
\(37\) 2.16686 0.356230 0.178115 0.984010i \(-0.443000\pi\)
0.178115 + 0.984010i \(0.443000\pi\)
\(38\) 0 0
\(39\) 1.83314 0.293537
\(40\) 0 0
\(41\) 10.7130 1.67309 0.836545 0.547898i \(-0.184572\pi\)
0.836545 + 0.547898i \(0.184572\pi\)
\(42\) 0 0
\(43\) −9.25915 −1.41201 −0.706004 0.708208i \(-0.749504\pi\)
−0.706004 + 0.708208i \(0.749504\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −0.546146 −0.0796635 −0.0398318 0.999206i \(-0.512682\pi\)
−0.0398318 + 0.999206i \(0.512682\pi\)
\(48\) 0 0
\(49\) −6.97216 −0.996023
\(50\) 0 0
\(51\) −4.54615 −0.636588
\(52\) 0 0
\(53\) 10.5461 1.44862 0.724312 0.689472i \(-0.242157\pi\)
0.724312 + 0.689472i \(0.242157\pi\)
\(54\) 0 0
\(55\) −4.71301 −0.635502
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 9.42601 1.22716 0.613581 0.789632i \(-0.289728\pi\)
0.613581 + 0.789632i \(0.289728\pi\)
\(60\) 0 0
\(61\) 12.8799 1.64910 0.824549 0.565791i \(-0.191429\pi\)
0.824549 + 0.565791i \(0.191429\pi\)
\(62\) 0 0
\(63\) 0.166860 0.0210224
\(64\) 0 0
\(65\) 1.83314 0.227373
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 4.54615 0.547292
\(70\) 0 0
\(71\) 5.42601 0.643949 0.321975 0.946748i \(-0.395653\pi\)
0.321975 + 0.946748i \(0.395653\pi\)
\(72\) 0 0
\(73\) 15.0923 1.76642 0.883210 0.468979i \(-0.155378\pi\)
0.883210 + 0.468979i \(0.155378\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −0.786413 −0.0896201
\(78\) 0 0
\(79\) −13.7597 −1.54809 −0.774045 0.633130i \(-0.781770\pi\)
−0.774045 + 0.633130i \(0.781770\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.9722 1.53364 0.766822 0.641860i \(-0.221837\pi\)
0.766822 + 0.641860i \(0.221837\pi\)
\(84\) 0 0
\(85\) −4.54615 −0.493099
\(86\) 0 0
\(87\) 6.71301 0.719710
\(88\) 0 0
\(89\) 6.04557 0.640829 0.320414 0.947278i \(-0.396178\pi\)
0.320414 + 0.947278i \(0.396178\pi\)
\(90\) 0 0
\(91\) 0.305878 0.0320647
\(92\) 0 0
\(93\) 2.54615 0.264023
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −2.16686 −0.220011 −0.110006 0.993931i \(-0.535087\pi\)
−0.110006 + 0.993931i \(0.535087\pi\)
\(98\) 0 0
\(99\) −4.71301 −0.473675
\(100\) 0 0
\(101\) −16.5183 −1.64363 −0.821816 0.569753i \(-0.807039\pi\)
−0.821816 + 0.569753i \(0.807039\pi\)
\(102\) 0 0
\(103\) 5.09229 0.501758 0.250879 0.968018i \(-0.419280\pi\)
0.250879 + 0.968018i \(0.419280\pi\)
\(104\) 0 0
\(105\) 0.166860 0.0162839
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) 15.4260 1.47754 0.738772 0.673955i \(-0.235406\pi\)
0.738772 + 0.673955i \(0.235406\pi\)
\(110\) 0 0
\(111\) 2.16686 0.205669
\(112\) 0 0
\(113\) 14.8799 1.39978 0.699890 0.714251i \(-0.253232\pi\)
0.699890 + 0.714251i \(0.253232\pi\)
\(114\) 0 0
\(115\) 4.54615 0.423930
\(116\) 0 0
\(117\) 1.83314 0.169474
\(118\) 0 0
\(119\) −0.758571 −0.0695381
\(120\) 0 0
\(121\) 11.2124 1.01931
\(122\) 0 0
\(123\) 10.7130 0.965959
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) −9.25915 −0.815223
\(130\) 0 0
\(131\) −14.1390 −1.23533 −0.617666 0.786441i \(-0.711922\pi\)
−0.617666 + 0.786441i \(0.711922\pi\)
\(132\) 0 0
\(133\) −0.166860 −0.0144686
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −9.42601 −0.799504 −0.399752 0.916623i \(-0.630904\pi\)
−0.399752 + 0.916623i \(0.630904\pi\)
\(140\) 0 0
\(141\) −0.546146 −0.0459938
\(142\) 0 0
\(143\) −8.63960 −0.722480
\(144\) 0 0
\(145\) 6.71301 0.557485
\(146\) 0 0
\(147\) −6.97216 −0.575054
\(148\) 0 0
\(149\) −15.0923 −1.23641 −0.618204 0.786017i \(-0.712140\pi\)
−0.618204 + 0.786017i \(0.712140\pi\)
\(150\) 0 0
\(151\) −11.2136 −0.912549 −0.456274 0.889839i \(-0.650816\pi\)
−0.456274 + 0.889839i \(0.650816\pi\)
\(152\) 0 0
\(153\) −4.54615 −0.367534
\(154\) 0 0
\(155\) 2.54615 0.204511
\(156\) 0 0
\(157\) −14.7586 −1.17786 −0.588931 0.808183i \(-0.700451\pi\)
−0.588931 + 0.808183i \(0.700451\pi\)
\(158\) 0 0
\(159\) 10.5461 0.836364
\(160\) 0 0
\(161\) 0.758571 0.0597838
\(162\) 0 0
\(163\) 3.49942 0.274096 0.137048 0.990564i \(-0.456239\pi\)
0.137048 + 0.990564i \(0.456239\pi\)
\(164\) 0 0
\(165\) −4.71301 −0.366907
\(166\) 0 0
\(167\) −6.21243 −0.480732 −0.240366 0.970682i \(-0.577267\pi\)
−0.240366 + 0.970682i \(0.577267\pi\)
\(168\) 0 0
\(169\) −9.63960 −0.741508
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −1.12013 −0.0851622 −0.0425811 0.999093i \(-0.513558\pi\)
−0.0425811 + 0.999093i \(0.513558\pi\)
\(174\) 0 0
\(175\) 0.166860 0.0126134
\(176\) 0 0
\(177\) 9.42601 0.708502
\(178\) 0 0
\(179\) 11.6663 0.871979 0.435989 0.899952i \(-0.356399\pi\)
0.435989 + 0.899952i \(0.356399\pi\)
\(180\) 0 0
\(181\) 9.66628 0.718489 0.359244 0.933243i \(-0.383034\pi\)
0.359244 + 0.933243i \(0.383034\pi\)
\(182\) 0 0
\(183\) 12.8799 0.952107
\(184\) 0 0
\(185\) 2.16686 0.159311
\(186\) 0 0
\(187\) 21.4260 1.56683
\(188\) 0 0
\(189\) 0.166860 0.0121373
\(190\) 0 0
\(191\) −1.04673 −0.0757385 −0.0378692 0.999283i \(-0.512057\pi\)
−0.0378692 + 0.999283i \(0.512057\pi\)
\(192\) 0 0
\(193\) −10.9254 −0.786430 −0.393215 0.919447i \(-0.628637\pi\)
−0.393215 + 0.919447i \(0.628637\pi\)
\(194\) 0 0
\(195\) 1.83314 0.131274
\(196\) 0 0
\(197\) −11.4539 −0.816053 −0.408027 0.912970i \(-0.633783\pi\)
−0.408027 + 0.912970i \(0.633783\pi\)
\(198\) 0 0
\(199\) 24.7586 1.75509 0.877544 0.479496i \(-0.159180\pi\)
0.877544 + 0.479496i \(0.159180\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.12013 0.0786180
\(204\) 0 0
\(205\) 10.7130 0.748229
\(206\) 0 0
\(207\) 4.54615 0.315979
\(208\) 0 0
\(209\) 4.71301 0.326005
\(210\) 0 0
\(211\) −16.4249 −1.13073 −0.565367 0.824840i \(-0.691265\pi\)
−0.565367 + 0.824840i \(0.691265\pi\)
\(212\) 0 0
\(213\) 5.42601 0.371784
\(214\) 0 0
\(215\) −9.25915 −0.631469
\(216\) 0 0
\(217\) 0.424850 0.0288407
\(218\) 0 0
\(219\) 15.0923 1.01984
\(220\) 0 0
\(221\) −8.33372 −0.560587
\(222\) 0 0
\(223\) 5.09229 0.341005 0.170503 0.985357i \(-0.445461\pi\)
0.170503 + 0.985357i \(0.445461\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 24.6396 1.63539 0.817694 0.575653i \(-0.195252\pi\)
0.817694 + 0.575653i \(0.195252\pi\)
\(228\) 0 0
\(229\) −7.78757 −0.514617 −0.257309 0.966329i \(-0.582836\pi\)
−0.257309 + 0.966329i \(0.582836\pi\)
\(230\) 0 0
\(231\) −0.786413 −0.0517422
\(232\) 0 0
\(233\) −25.9443 −1.69967 −0.849834 0.527050i \(-0.823298\pi\)
−0.849834 + 0.527050i \(0.823298\pi\)
\(234\) 0 0
\(235\) −0.546146 −0.0356266
\(236\) 0 0
\(237\) −13.7597 −0.893791
\(238\) 0 0
\(239\) 16.3793 1.05949 0.529744 0.848158i \(-0.322288\pi\)
0.529744 + 0.848158i \(0.322288\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.97216 −0.445435
\(246\) 0 0
\(247\) −1.83314 −0.116640
\(248\) 0 0
\(249\) 13.9722 0.885450
\(250\) 0 0
\(251\) 21.4716 1.35527 0.677637 0.735397i \(-0.263004\pi\)
0.677637 + 0.735397i \(0.263004\pi\)
\(252\) 0 0
\(253\) −21.4260 −1.34704
\(254\) 0 0
\(255\) −4.54615 −0.284691
\(256\) 0 0
\(257\) −15.9722 −0.996316 −0.498158 0.867086i \(-0.665990\pi\)
−0.498158 + 0.867086i \(0.665990\pi\)
\(258\) 0 0
\(259\) 0.361563 0.0224664
\(260\) 0 0
\(261\) 6.71301 0.415525
\(262\) 0 0
\(263\) −17.3047 −1.06705 −0.533527 0.845783i \(-0.679134\pi\)
−0.533527 + 0.845783i \(0.679134\pi\)
\(264\) 0 0
\(265\) 10.5461 0.647845
\(266\) 0 0
\(267\) 6.04557 0.369983
\(268\) 0 0
\(269\) −7.04673 −0.429646 −0.214823 0.976653i \(-0.568918\pi\)
−0.214823 + 0.976653i \(0.568918\pi\)
\(270\) 0 0
\(271\) −11.6663 −0.708676 −0.354338 0.935117i \(-0.615294\pi\)
−0.354338 + 0.935117i \(0.615294\pi\)
\(272\) 0 0
\(273\) 0.305878 0.0185126
\(274\) 0 0
\(275\) −4.71301 −0.284205
\(276\) 0 0
\(277\) 29.9443 1.79918 0.899590 0.436736i \(-0.143866\pi\)
0.899590 + 0.436736i \(0.143866\pi\)
\(278\) 0 0
\(279\) 2.54615 0.152434
\(280\) 0 0
\(281\) 24.1390 1.44001 0.720007 0.693967i \(-0.244139\pi\)
0.720007 + 0.693967i \(0.244139\pi\)
\(282\) 0 0
\(283\) −4.92543 −0.292786 −0.146393 0.989226i \(-0.546766\pi\)
−0.146393 + 0.989226i \(0.546766\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 1.78757 0.105517
\(288\) 0 0
\(289\) 3.66744 0.215732
\(290\) 0 0
\(291\) −2.16686 −0.127024
\(292\) 0 0
\(293\) 9.12013 0.532804 0.266402 0.963862i \(-0.414165\pi\)
0.266402 + 0.963862i \(0.414165\pi\)
\(294\) 0 0
\(295\) 9.42601 0.548804
\(296\) 0 0
\(297\) −4.71301 −0.273476
\(298\) 0 0
\(299\) 8.33372 0.481951
\(300\) 0 0
\(301\) −1.54498 −0.0890514
\(302\) 0 0
\(303\) −16.5183 −0.948952
\(304\) 0 0
\(305\) 12.8799 0.737499
\(306\) 0 0
\(307\) 8.75857 0.499878 0.249939 0.968262i \(-0.419589\pi\)
0.249939 + 0.968262i \(0.419589\pi\)
\(308\) 0 0
\(309\) 5.09229 0.289690
\(310\) 0 0
\(311\) −1.04673 −0.0593544 −0.0296772 0.999560i \(-0.509448\pi\)
−0.0296772 + 0.999560i \(0.509448\pi\)
\(312\) 0 0
\(313\) −21.6663 −1.22465 −0.612325 0.790606i \(-0.709766\pi\)
−0.612325 + 0.790606i \(0.709766\pi\)
\(314\) 0 0
\(315\) 0.166860 0.00940151
\(316\) 0 0
\(317\) −14.5461 −0.816993 −0.408496 0.912760i \(-0.633947\pi\)
−0.408496 + 0.912760i \(0.633947\pi\)
\(318\) 0 0
\(319\) −31.6384 −1.77141
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 4.54615 0.252954
\(324\) 0 0
\(325\) 1.83314 0.101684
\(326\) 0 0
\(327\) 15.4260 0.853060
\(328\) 0 0
\(329\) −0.0911300 −0.00502416
\(330\) 0 0
\(331\) −7.21359 −0.396495 −0.198247 0.980152i \(-0.563525\pi\)
−0.198247 + 0.980152i \(0.563525\pi\)
\(332\) 0 0
\(333\) 2.16686 0.118743
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.59171 −0.141179 −0.0705897 0.997505i \(-0.522488\pi\)
−0.0705897 + 0.997505i \(0.522488\pi\)
\(338\) 0 0
\(339\) 14.8799 0.808163
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −2.33140 −0.125884
\(344\) 0 0
\(345\) 4.54615 0.244756
\(346\) 0 0
\(347\) 3.54498 0.190305 0.0951524 0.995463i \(-0.469666\pi\)
0.0951524 + 0.995463i \(0.469666\pi\)
\(348\) 0 0
\(349\) −28.8520 −1.54441 −0.772207 0.635371i \(-0.780847\pi\)
−0.772207 + 0.635371i \(0.780847\pi\)
\(350\) 0 0
\(351\) 1.83314 0.0978458
\(352\) 0 0
\(353\) 20.8520 1.10984 0.554921 0.831903i \(-0.312749\pi\)
0.554921 + 0.831903i \(0.312749\pi\)
\(354\) 0 0
\(355\) 5.42601 0.287983
\(356\) 0 0
\(357\) −0.758571 −0.0401478
\(358\) 0 0
\(359\) 10.4727 0.552730 0.276365 0.961053i \(-0.410870\pi\)
0.276365 + 0.961053i \(0.410870\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 11.2124 0.588500
\(364\) 0 0
\(365\) 15.0923 0.789967
\(366\) 0 0
\(367\) −2.40713 −0.125651 −0.0628255 0.998025i \(-0.520011\pi\)
−0.0628255 + 0.998025i \(0.520011\pi\)
\(368\) 0 0
\(369\) 10.7130 0.557697
\(370\) 0 0
\(371\) 1.75973 0.0913608
\(372\) 0 0
\(373\) −31.2592 −1.61854 −0.809269 0.587439i \(-0.800136\pi\)
−0.809269 + 0.587439i \(0.800136\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 12.3059 0.633785
\(378\) 0 0
\(379\) 11.6384 0.597826 0.298913 0.954280i \(-0.403376\pi\)
0.298913 + 0.954280i \(0.403376\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −25.0923 −1.28216 −0.641078 0.767476i \(-0.721513\pi\)
−0.641078 + 0.767476i \(0.721513\pi\)
\(384\) 0 0
\(385\) −0.786413 −0.0400793
\(386\) 0 0
\(387\) −9.25915 −0.470669
\(388\) 0 0
\(389\) 11.7597 0.596242 0.298121 0.954528i \(-0.403640\pi\)
0.298121 + 0.954528i \(0.403640\pi\)
\(390\) 0 0
\(391\) −20.6674 −1.04520
\(392\) 0 0
\(393\) −14.1390 −0.713219
\(394\) 0 0
\(395\) −13.7597 −0.692327
\(396\) 0 0
\(397\) 25.5195 1.28079 0.640393 0.768048i \(-0.278772\pi\)
0.640393 + 0.768048i \(0.278772\pi\)
\(398\) 0 0
\(399\) −0.166860 −0.00835346
\(400\) 0 0
\(401\) 0.194703 0.00972298 0.00486149 0.999988i \(-0.498453\pi\)
0.00486149 + 0.999988i \(0.498453\pi\)
\(402\) 0 0
\(403\) 4.66744 0.232502
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −10.2124 −0.506211
\(408\) 0 0
\(409\) 29.6106 1.46415 0.732075 0.681224i \(-0.238552\pi\)
0.732075 + 0.681224i \(0.238552\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) 1.57283 0.0773937
\(414\) 0 0
\(415\) 13.9722 0.685866
\(416\) 0 0
\(417\) −9.42601 −0.461594
\(418\) 0 0
\(419\) 3.95443 0.193187 0.0965934 0.995324i \(-0.469205\pi\)
0.0965934 + 0.995324i \(0.469205\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −0.546146 −0.0265545
\(424\) 0 0
\(425\) −4.54615 −0.220520
\(426\) 0 0
\(427\) 2.14914 0.104004
\(428\) 0 0
\(429\) −8.63960 −0.417124
\(430\) 0 0
\(431\) 29.8509 1.43787 0.718933 0.695080i \(-0.244631\pi\)
0.718933 + 0.695080i \(0.244631\pi\)
\(432\) 0 0
\(433\) 5.07457 0.243868 0.121934 0.992538i \(-0.461090\pi\)
0.121934 + 0.992538i \(0.461090\pi\)
\(434\) 0 0
\(435\) 6.71301 0.321864
\(436\) 0 0
\(437\) −4.54615 −0.217472
\(438\) 0 0
\(439\) 34.8520 1.66340 0.831698 0.555228i \(-0.187369\pi\)
0.831698 + 0.555228i \(0.187369\pi\)
\(440\) 0 0
\(441\) −6.97216 −0.332008
\(442\) 0 0
\(443\) −15.8787 −0.754420 −0.377210 0.926128i \(-0.623117\pi\)
−0.377210 + 0.926128i \(0.623117\pi\)
\(444\) 0 0
\(445\) 6.04557 0.286587
\(446\) 0 0
\(447\) −15.0923 −0.713841
\(448\) 0 0
\(449\) 11.8053 0.557126 0.278563 0.960418i \(-0.410142\pi\)
0.278563 + 0.960418i \(0.410142\pi\)
\(450\) 0 0
\(451\) −50.4905 −2.37750
\(452\) 0 0
\(453\) −11.2136 −0.526860
\(454\) 0 0
\(455\) 0.305878 0.0143398
\(456\) 0 0
\(457\) −35.4260 −1.65716 −0.828579 0.559871i \(-0.810851\pi\)
−0.828579 + 0.559871i \(0.810851\pi\)
\(458\) 0 0
\(459\) −4.54615 −0.212196
\(460\) 0 0
\(461\) −10.6674 −0.496832 −0.248416 0.968653i \(-0.579910\pi\)
−0.248416 + 0.968653i \(0.579910\pi\)
\(462\) 0 0
\(463\) −13.9266 −0.647224 −0.323612 0.946190i \(-0.604897\pi\)
−0.323612 + 0.946190i \(0.604897\pi\)
\(464\) 0 0
\(465\) 2.54615 0.118075
\(466\) 0 0
\(467\) −2.30588 −0.106703 −0.0533517 0.998576i \(-0.516990\pi\)
−0.0533517 + 0.998576i \(0.516990\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.7586 −0.680039
\(472\) 0 0
\(473\) 43.6384 2.00650
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 10.5461 0.482875
\(478\) 0 0
\(479\) −2.13902 −0.0977342 −0.0488671 0.998805i \(-0.515561\pi\)
−0.0488671 + 0.998805i \(0.515561\pi\)
\(480\) 0 0
\(481\) 3.97216 0.181115
\(482\) 0 0
\(483\) 0.758571 0.0345162
\(484\) 0 0
\(485\) −2.16686 −0.0983921
\(486\) 0 0
\(487\) 8.09113 0.366644 0.183322 0.983053i \(-0.441315\pi\)
0.183322 + 0.983053i \(0.441315\pi\)
\(488\) 0 0
\(489\) 3.49942 0.158249
\(490\) 0 0
\(491\) −4.37929 −0.197634 −0.0988172 0.995106i \(-0.531506\pi\)
−0.0988172 + 0.995106i \(0.531506\pi\)
\(492\) 0 0
\(493\) −30.5183 −1.37448
\(494\) 0 0
\(495\) −4.71301 −0.211834
\(496\) 0 0
\(497\) 0.905386 0.0406121
\(498\) 0 0
\(499\) 10.0935 0.451845 0.225923 0.974145i \(-0.427460\pi\)
0.225923 + 0.974145i \(0.427460\pi\)
\(500\) 0 0
\(501\) −6.21243 −0.277551
\(502\) 0 0
\(503\) 8.21243 0.366174 0.183087 0.983097i \(-0.441391\pi\)
0.183087 + 0.983097i \(0.441391\pi\)
\(504\) 0 0
\(505\) −16.5183 −0.735055
\(506\) 0 0
\(507\) −9.63960 −0.428110
\(508\) 0 0
\(509\) 32.8065 1.45412 0.727060 0.686574i \(-0.240886\pi\)
0.727060 + 0.686574i \(0.240886\pi\)
\(510\) 0 0
\(511\) 2.51830 0.111403
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 5.09229 0.224393
\(516\) 0 0
\(517\) 2.57399 0.113204
\(518\) 0 0
\(519\) −1.12013 −0.0491684
\(520\) 0 0
\(521\) −9.62071 −0.421491 −0.210746 0.977541i \(-0.567589\pi\)
−0.210746 + 0.977541i \(0.567589\pi\)
\(522\) 0 0
\(523\) 31.6106 1.38223 0.691117 0.722743i \(-0.257119\pi\)
0.691117 + 0.722743i \(0.257119\pi\)
\(524\) 0 0
\(525\) 0.166860 0.00728238
\(526\) 0 0
\(527\) −11.5751 −0.504221
\(528\) 0 0
\(529\) −2.33256 −0.101416
\(530\) 0 0
\(531\) 9.42601 0.409054
\(532\) 0 0
\(533\) 19.6384 0.850635
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) 11.6663 0.503437
\(538\) 0 0
\(539\) 32.8598 1.41537
\(540\) 0 0
\(541\) −32.1846 −1.38372 −0.691862 0.722030i \(-0.743209\pi\)
−0.691862 + 0.722030i \(0.743209\pi\)
\(542\) 0 0
\(543\) 9.66628 0.414820
\(544\) 0 0
\(545\) 15.4260 0.660778
\(546\) 0 0
\(547\) −33.4260 −1.42919 −0.714597 0.699537i \(-0.753390\pi\)
−0.714597 + 0.699537i \(0.753390\pi\)
\(548\) 0 0
\(549\) 12.8799 0.549699
\(550\) 0 0
\(551\) −6.71301 −0.285984
\(552\) 0 0
\(553\) −2.29595 −0.0976338
\(554\) 0 0
\(555\) 2.16686 0.0919781
\(556\) 0 0
\(557\) 25.2769 1.07102 0.535508 0.844530i \(-0.320120\pi\)
0.535508 + 0.844530i \(0.320120\pi\)
\(558\) 0 0
\(559\) −16.9733 −0.717895
\(560\) 0 0
\(561\) 21.4260 0.904607
\(562\) 0 0
\(563\) 32.6396 1.37560 0.687798 0.725903i \(-0.258578\pi\)
0.687798 + 0.725903i \(0.258578\pi\)
\(564\) 0 0
\(565\) 14.8799 0.626001
\(566\) 0 0
\(567\) 0.166860 0.00700747
\(568\) 0 0
\(569\) 25.6207 1.07408 0.537038 0.843558i \(-0.319543\pi\)
0.537038 + 0.843558i \(0.319543\pi\)
\(570\) 0 0
\(571\) 27.1857 1.13769 0.568844 0.822445i \(-0.307391\pi\)
0.568844 + 0.822445i \(0.307391\pi\)
\(572\) 0 0
\(573\) −1.04673 −0.0437276
\(574\) 0 0
\(575\) 4.54615 0.189587
\(576\) 0 0
\(577\) 4.51830 0.188099 0.0940497 0.995568i \(-0.470019\pi\)
0.0940497 + 0.995568i \(0.470019\pi\)
\(578\) 0 0
\(579\) −10.9254 −0.454045
\(580\) 0 0
\(581\) 2.33140 0.0967227
\(582\) 0 0
\(583\) −49.7040 −2.05853
\(584\) 0 0
\(585\) 1.83314 0.0757910
\(586\) 0 0
\(587\) −26.9733 −1.11331 −0.556654 0.830744i \(-0.687915\pi\)
−0.556654 + 0.830744i \(0.687915\pi\)
\(588\) 0 0
\(589\) −2.54615 −0.104912
\(590\) 0 0
\(591\) −11.4539 −0.471149
\(592\) 0 0
\(593\) −7.09229 −0.291246 −0.145623 0.989340i \(-0.546519\pi\)
−0.145623 + 0.989340i \(0.546519\pi\)
\(594\) 0 0
\(595\) −0.758571 −0.0310984
\(596\) 0 0
\(597\) 24.7586 1.01330
\(598\) 0 0
\(599\) 6.85202 0.279966 0.139983 0.990154i \(-0.455295\pi\)
0.139983 + 0.990154i \(0.455295\pi\)
\(600\) 0 0
\(601\) −31.8509 −1.29922 −0.649612 0.760266i \(-0.725069\pi\)
−0.649612 + 0.760266i \(0.725069\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.2124 0.455850
\(606\) 0 0
\(607\) −41.3703 −1.67917 −0.839585 0.543229i \(-0.817202\pi\)
−0.839585 + 0.543229i \(0.817202\pi\)
\(608\) 0 0
\(609\) 1.12013 0.0453901
\(610\) 0 0
\(611\) −1.00116 −0.0405027
\(612\) 0 0
\(613\) 5.66628 0.228859 0.114429 0.993431i \(-0.463496\pi\)
0.114429 + 0.993431i \(0.463496\pi\)
\(614\) 0 0
\(615\) 10.7130 0.431990
\(616\) 0 0
\(617\) −30.7307 −1.23717 −0.618586 0.785717i \(-0.712294\pi\)
−0.618586 + 0.785717i \(0.712294\pi\)
\(618\) 0 0
\(619\) −2.99884 −0.120533 −0.0602667 0.998182i \(-0.519195\pi\)
−0.0602667 + 0.998182i \(0.519195\pi\)
\(620\) 0 0
\(621\) 4.54615 0.182431
\(622\) 0 0
\(623\) 1.00876 0.0404153
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.71301 0.188219
\(628\) 0 0
\(629\) −9.85086 −0.392780
\(630\) 0 0
\(631\) −25.5171 −1.01582 −0.507911 0.861410i \(-0.669582\pi\)
−0.507911 + 0.861410i \(0.669582\pi\)
\(632\) 0 0
\(633\) −16.4249 −0.652829
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) −12.7809 −0.506399
\(638\) 0 0
\(639\) 5.42601 0.214650
\(640\) 0 0
\(641\) −4.13902 −0.163481 −0.0817407 0.996654i \(-0.526048\pi\)
−0.0817407 + 0.996654i \(0.526048\pi\)
\(642\) 0 0
\(643\) −26.2603 −1.03561 −0.517803 0.855500i \(-0.673250\pi\)
−0.517803 + 0.855500i \(0.673250\pi\)
\(644\) 0 0
\(645\) −9.25915 −0.364579
\(646\) 0 0
\(647\) 22.3970 0.880517 0.440259 0.897871i \(-0.354887\pi\)
0.440259 + 0.897871i \(0.354887\pi\)
\(648\) 0 0
\(649\) −44.4249 −1.74383
\(650\) 0 0
\(651\) 0.424850 0.0166512
\(652\) 0 0
\(653\) 13.6384 0.533713 0.266857 0.963736i \(-0.414015\pi\)
0.266857 + 0.963736i \(0.414015\pi\)
\(654\) 0 0
\(655\) −14.1390 −0.552457
\(656\) 0 0
\(657\) 15.0923 0.588806
\(658\) 0 0
\(659\) 13.0923 0.510003 0.255002 0.966941i \(-0.417924\pi\)
0.255002 + 0.966941i \(0.417924\pi\)
\(660\) 0 0
\(661\) −20.5183 −0.798070 −0.399035 0.916936i \(-0.630655\pi\)
−0.399035 + 0.916936i \(0.630655\pi\)
\(662\) 0 0
\(663\) −8.33372 −0.323655
\(664\) 0 0
\(665\) −0.166860 −0.00647056
\(666\) 0 0
\(667\) 30.5183 1.18167
\(668\) 0 0
\(669\) 5.09229 0.196879
\(670\) 0 0
\(671\) −60.7029 −2.34341
\(672\) 0 0
\(673\) −2.16686 −0.0835263 −0.0417632 0.999128i \(-0.513297\pi\)
−0.0417632 + 0.999128i \(0.513297\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −22.3036 −0.857195 −0.428598 0.903495i \(-0.640992\pi\)
−0.428598 + 0.903495i \(0.640992\pi\)
\(678\) 0 0
\(679\) −0.361563 −0.0138755
\(680\) 0 0
\(681\) 24.6396 0.944191
\(682\) 0 0
\(683\) −7.33256 −0.280573 −0.140286 0.990111i \(-0.544802\pi\)
−0.140286 + 0.990111i \(0.544802\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −7.78757 −0.297115
\(688\) 0 0
\(689\) 19.3326 0.736512
\(690\) 0 0
\(691\) −0.333720 −0.0126953 −0.00634766 0.999980i \(-0.502021\pi\)
−0.00634766 + 0.999980i \(0.502021\pi\)
\(692\) 0 0
\(693\) −0.786413 −0.0298734
\(694\) 0 0
\(695\) −9.42601 −0.357549
\(696\) 0 0
\(697\) −48.7029 −1.84475
\(698\) 0 0
\(699\) −25.9443 −0.981304
\(700\) 0 0
\(701\) 19.7597 0.746315 0.373157 0.927768i \(-0.378275\pi\)
0.373157 + 0.927768i \(0.378275\pi\)
\(702\) 0 0
\(703\) −2.16686 −0.0817247
\(704\) 0 0
\(705\) −0.546146 −0.0205690
\(706\) 0 0
\(707\) −2.75625 −0.103659
\(708\) 0 0
\(709\) 30.3970 1.14158 0.570792 0.821095i \(-0.306636\pi\)
0.570792 + 0.821095i \(0.306636\pi\)
\(710\) 0 0
\(711\) −13.7597 −0.516030
\(712\) 0 0
\(713\) 11.5751 0.433493
\(714\) 0 0
\(715\) −8.63960 −0.323103
\(716\) 0 0
\(717\) 16.3793 0.611696
\(718\) 0 0
\(719\) 34.1390 1.27317 0.636585 0.771206i \(-0.280346\pi\)
0.636585 + 0.771206i \(0.280346\pi\)
\(720\) 0 0
\(721\) 0.849701 0.0316445
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) 6.71301 0.249315
\(726\) 0 0
\(727\) −41.2592 −1.53022 −0.765109 0.643901i \(-0.777315\pi\)
−0.765109 + 0.643901i \(0.777315\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.0935 1.55688
\(732\) 0 0
\(733\) 3.09229 0.114216 0.0571082 0.998368i \(-0.481812\pi\)
0.0571082 + 0.998368i \(0.481812\pi\)
\(734\) 0 0
\(735\) −6.97216 −0.257172
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 15.5195 0.570893 0.285446 0.958395i \(-0.407858\pi\)
0.285446 + 0.958395i \(0.407858\pi\)
\(740\) 0 0
\(741\) −1.83314 −0.0673421
\(742\) 0 0
\(743\) −26.4272 −0.969519 −0.484759 0.874647i \(-0.661093\pi\)
−0.484759 + 0.874647i \(0.661093\pi\)
\(744\) 0 0
\(745\) −15.0923 −0.552939
\(746\) 0 0
\(747\) 13.9722 0.511215
\(748\) 0 0
\(749\) 2.66976 0.0975510
\(750\) 0 0
\(751\) −40.3059 −1.47078 −0.735391 0.677643i \(-0.763002\pi\)
−0.735391 + 0.677643i \(0.763002\pi\)
\(752\) 0 0
\(753\) 21.4716 0.782468
\(754\) 0 0
\(755\) −11.2136 −0.408104
\(756\) 0 0
\(757\) 0.573988 0.0208620 0.0104310 0.999946i \(-0.496680\pi\)
0.0104310 + 0.999946i \(0.496680\pi\)
\(758\) 0 0
\(759\) −21.4260 −0.777715
\(760\) 0 0
\(761\) −43.7597 −1.58629 −0.793145 0.609033i \(-0.791558\pi\)
−0.793145 + 0.609033i \(0.791558\pi\)
\(762\) 0 0
\(763\) 2.57399 0.0931846
\(764\) 0 0
\(765\) −4.54615 −0.164366
\(766\) 0 0
\(767\) 17.2792 0.623916
\(768\) 0 0
\(769\) 13.3326 0.480784 0.240392 0.970676i \(-0.422724\pi\)
0.240392 + 0.970676i \(0.422724\pi\)
\(770\) 0 0
\(771\) −15.9722 −0.575223
\(772\) 0 0
\(773\) −31.2136 −1.12267 −0.561337 0.827587i \(-0.689713\pi\)
−0.561337 + 0.827587i \(0.689713\pi\)
\(774\) 0 0
\(775\) 2.54615 0.0914603
\(776\) 0 0
\(777\) 0.361563 0.0129710
\(778\) 0 0
\(779\) −10.7130 −0.383833
\(780\) 0 0
\(781\) −25.5728 −0.915068
\(782\) 0 0
\(783\) 6.71301 0.239903
\(784\) 0 0
\(785\) −14.7586 −0.526756
\(786\) 0 0
\(787\) 48.2780 1.72093 0.860463 0.509513i \(-0.170174\pi\)
0.860463 + 0.509513i \(0.170174\pi\)
\(788\) 0 0
\(789\) −17.3047 −0.616064
\(790\) 0 0
\(791\) 2.48286 0.0882803
\(792\) 0 0
\(793\) 23.6106 0.838437
\(794\) 0 0
\(795\) 10.5461 0.374033
\(796\) 0 0
\(797\) 17.0644 0.604454 0.302227 0.953236i \(-0.402270\pi\)
0.302227 + 0.953236i \(0.402270\pi\)
\(798\) 0 0
\(799\) 2.48286 0.0878372
\(800\) 0 0
\(801\) 6.04557 0.213610
\(802\) 0 0
\(803\) −71.1301 −2.51013
\(804\) 0 0
\(805\) 0.758571 0.0267361
\(806\) 0 0
\(807\) −7.04673 −0.248057
\(808\) 0 0
\(809\) −3.75973 −0.132185 −0.0660926 0.997813i \(-0.521053\pi\)
−0.0660926 + 0.997813i \(0.521053\pi\)
\(810\) 0 0
\(811\) 5.33488 0.187333 0.0936665 0.995604i \(-0.470141\pi\)
0.0936665 + 0.995604i \(0.470141\pi\)
\(812\) 0 0
\(813\) −11.6663 −0.409154
\(814\) 0 0
\(815\) 3.49942 0.122579
\(816\) 0 0
\(817\) 9.25915 0.323937
\(818\) 0 0
\(819\) 0.305878 0.0106882
\(820\) 0 0
\(821\) 7.42601 0.259170 0.129585 0.991568i \(-0.458636\pi\)
0.129585 + 0.991568i \(0.458636\pi\)
\(822\) 0 0
\(823\) 30.6852 1.06962 0.534809 0.844973i \(-0.320384\pi\)
0.534809 + 0.844973i \(0.320384\pi\)
\(824\) 0 0
\(825\) −4.71301 −0.164086
\(826\) 0 0
\(827\) −30.8242 −1.07186 −0.535931 0.844262i \(-0.680039\pi\)
−0.535931 + 0.844262i \(0.680039\pi\)
\(828\) 0 0
\(829\) 20.5183 0.712630 0.356315 0.934366i \(-0.384033\pi\)
0.356315 + 0.934366i \(0.384033\pi\)
\(830\) 0 0
\(831\) 29.9443 1.03876
\(832\) 0 0
\(833\) 31.6964 1.09822
\(834\) 0 0
\(835\) −6.21243 −0.214990
\(836\) 0 0
\(837\) 2.54615 0.0880077
\(838\) 0 0
\(839\) −30.9432 −1.06828 −0.534138 0.845397i \(-0.679364\pi\)
−0.534138 + 0.845397i \(0.679364\pi\)
\(840\) 0 0
\(841\) 16.0644 0.553947
\(842\) 0 0
\(843\) 24.1390 0.831392
\(844\) 0 0
\(845\) −9.63960 −0.331612
\(846\) 0 0
\(847\) 1.87091 0.0642852
\(848\) 0 0
\(849\) −4.92543 −0.169040
\(850\) 0 0
\(851\) 9.85086 0.337683
\(852\) 0 0
\(853\) −31.4260 −1.07601 −0.538003 0.842943i \(-0.680821\pi\)
−0.538003 + 0.842943i \(0.680821\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) −11.6941 −0.399464 −0.199732 0.979851i \(-0.564007\pi\)
−0.199732 + 0.979851i \(0.564007\pi\)
\(858\) 0 0
\(859\) 38.1289 1.30094 0.650471 0.759531i \(-0.274572\pi\)
0.650471 + 0.759531i \(0.274572\pi\)
\(860\) 0 0
\(861\) 1.78757 0.0609204
\(862\) 0 0
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) −1.12013 −0.0380857
\(866\) 0 0
\(867\) 3.66744 0.124553
\(868\) 0 0
\(869\) 64.8497 2.19988
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.16686 −0.0733371
\(874\) 0 0
\(875\) 0.166860 0.00564091
\(876\) 0 0
\(877\) 22.8697 0.772256 0.386128 0.922445i \(-0.373812\pi\)
0.386128 + 0.922445i \(0.373812\pi\)
\(878\) 0 0
\(879\) 9.12013 0.307614
\(880\) 0 0
\(881\) −58.2780 −1.96344 −0.981718 0.190339i \(-0.939041\pi\)
−0.981718 + 0.190339i \(0.939041\pi\)
\(882\) 0 0
\(883\) 1.31484 0.0442478 0.0221239 0.999755i \(-0.492957\pi\)
0.0221239 + 0.999755i \(0.492957\pi\)
\(884\) 0 0
\(885\) 9.42601 0.316852
\(886\) 0 0
\(887\) −10.1846 −0.341965 −0.170982 0.985274i \(-0.554694\pi\)
−0.170982 + 0.985274i \(0.554694\pi\)
\(888\) 0 0
\(889\) −0.667441 −0.0223853
\(890\) 0 0
\(891\) −4.71301 −0.157892
\(892\) 0 0
\(893\) 0.546146 0.0182761
\(894\) 0 0
\(895\) 11.6663 0.389961
\(896\) 0 0
\(897\) 8.33372 0.278255
\(898\) 0 0
\(899\) 17.0923 0.570060
\(900\) 0 0
\(901\) −47.9443 −1.59726
\(902\) 0 0
\(903\) −1.54498 −0.0514139
\(904\) 0 0
\(905\) 9.66628 0.321318
\(906\) 0 0
\(907\) 44.0354 1.46217 0.731086 0.682285i \(-0.239014\pi\)
0.731086 + 0.682285i \(0.239014\pi\)
\(908\) 0 0
\(909\) −16.5183 −0.547878
\(910\) 0 0
\(911\) 36.4249 1.20681 0.603405 0.797435i \(-0.293810\pi\)
0.603405 + 0.797435i \(0.293810\pi\)
\(912\) 0 0
\(913\) −65.8509 −2.17935
\(914\) 0 0
\(915\) 12.8799 0.425795
\(916\) 0 0
\(917\) −2.35924 −0.0779090
\(918\) 0 0
\(919\) 5.75973 0.189996 0.0949980 0.995477i \(-0.469716\pi\)
0.0949980 + 0.995477i \(0.469716\pi\)
\(920\) 0 0
\(921\) 8.75857 0.288605
\(922\) 0 0
\(923\) 9.94664 0.327398
\(924\) 0 0
\(925\) 2.16686 0.0712459
\(926\) 0 0
\(927\) 5.09229 0.167253
\(928\) 0 0
\(929\) 50.7586 1.66533 0.832667 0.553774i \(-0.186813\pi\)
0.832667 + 0.553774i \(0.186813\pi\)
\(930\) 0 0
\(931\) 6.97216 0.228503
\(932\) 0 0
\(933\) −1.04673 −0.0342683
\(934\) 0 0
\(935\) 21.4260 0.700706
\(936\) 0 0
\(937\) −38.2780 −1.25049 −0.625244 0.780429i \(-0.715001\pi\)
−0.625244 + 0.780429i \(0.715001\pi\)
\(938\) 0 0
\(939\) −21.6663 −0.707052
\(940\) 0 0
\(941\) 30.6573 0.999400 0.499700 0.866199i \(-0.333444\pi\)
0.499700 + 0.866199i \(0.333444\pi\)
\(942\) 0 0
\(943\) 48.7029 1.58598
\(944\) 0 0
\(945\) 0.166860 0.00542796
\(946\) 0 0
\(947\) −11.1201 −0.361356 −0.180678 0.983542i \(-0.557829\pi\)
−0.180678 + 0.983542i \(0.557829\pi\)
\(948\) 0 0
\(949\) 27.6663 0.898085
\(950\) 0 0
\(951\) −14.5461 −0.471691
\(952\) 0 0
\(953\) −30.6373 −0.992439 −0.496219 0.868197i \(-0.665279\pi\)
−0.496219 + 0.868197i \(0.665279\pi\)
\(954\) 0 0
\(955\) −1.04673 −0.0338713
\(956\) 0 0
\(957\) −31.6384 −1.02273
\(958\) 0 0
\(959\) −1.00116 −0.0323292
\(960\) 0 0
\(961\) −24.5171 −0.790876
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) −10.9254 −0.351702
\(966\) 0 0
\(967\) −11.6863 −0.375807 −0.187903 0.982188i \(-0.560169\pi\)
−0.187903 + 0.982188i \(0.560169\pi\)
\(968\) 0 0
\(969\) 4.54615 0.146043
\(970\) 0 0
\(971\) −15.2769 −0.490258 −0.245129 0.969490i \(-0.578830\pi\)
−0.245129 + 0.969490i \(0.578830\pi\)
\(972\) 0 0
\(973\) −1.57283 −0.0504225
\(974\) 0 0
\(975\) 1.83314 0.0587075
\(976\) 0 0
\(977\) −4.97332 −0.159111 −0.0795553 0.996830i \(-0.525350\pi\)
−0.0795553 + 0.996830i \(0.525350\pi\)
\(978\) 0 0
\(979\) −28.4928 −0.910633
\(980\) 0 0
\(981\) 15.4260 0.492515
\(982\) 0 0
\(983\) −27.9722 −0.892173 −0.446087 0.894990i \(-0.647183\pi\)
−0.446087 + 0.894990i \(0.647183\pi\)
\(984\) 0 0
\(985\) −11.4539 −0.364950
\(986\) 0 0
\(987\) −0.0911300 −0.00290070
\(988\) 0 0
\(989\) −42.0935 −1.33849
\(990\) 0 0
\(991\) 34.8520 1.10711 0.553556 0.832812i \(-0.313271\pi\)
0.553556 + 0.832812i \(0.313271\pi\)
\(992\) 0 0
\(993\) −7.21359 −0.228916
\(994\) 0 0
\(995\) 24.7586 0.784899
\(996\) 0 0
\(997\) 37.6106 1.19114 0.595570 0.803304i \(-0.296926\pi\)
0.595570 + 0.803304i \(0.296926\pi\)
\(998\) 0 0
\(999\) 2.16686 0.0685564
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 4560.2.a.bu.1.2 3
4.3 odd 2 1140.2.a.f.1.2 3
12.11 even 2 3420.2.a.k.1.2 3
20.3 even 4 5700.2.f.p.3649.2 6
20.7 even 4 5700.2.f.p.3649.5 6
20.19 odd 2 5700.2.a.z.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.f.1.2 3 4.3 odd 2
3420.2.a.k.1.2 3 12.11 even 2
4560.2.a.bu.1.2 3 1.1 even 1 trivial
5700.2.a.z.1.2 3 20.19 odd 2
5700.2.f.p.3649.2 6 20.3 even 4
5700.2.f.p.3649.5 6 20.7 even 4