Properties

Label 2760.2.a.v.1.3
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.36007\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +0.512641 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +0.512641 q^{7} +1.00000 q^{9} -1.76485 q^{11} -4.95530 q^{13} +1.00000 q^{15} +7.97235 q^{17} +1.93002 q^{19} -0.512641 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.16280 q^{29} +4.20750 q^{31} +1.76485 q^{33} -0.512641 q^{35} +6.37267 q^{37} +4.95530 q^{39} +4.27749 q^{41} -12.8853 q^{43} -1.00000 q^{45} -1.93002 q^{47} -6.73720 q^{49} -7.97235 q^{51} +4.60782 q^{53} +1.76485 q^{55} -1.93002 q^{57} -11.1628 q^{59} +8.65016 q^{61} +0.512641 q^{63} +4.95530 q^{65} -11.1822 q^{67} -1.00000 q^{69} +2.27749 q^{71} -8.95530 q^{73} -1.00000 q^{75} -0.904733 q^{77} +8.58018 q^{79} +1.00000 q^{81} -6.91296 q^{83} -7.97235 q^{85} +9.16280 q^{87} -1.69486 q^{89} -2.54029 q^{91} -4.20750 q^{93} -1.93002 q^{95} -14.4150 q^{97} -1.76485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{15} + 2 q^{17} - 6 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} - 4 q^{37} + 2 q^{39} + 8 q^{41} - 20 q^{43} - 4 q^{45} + 6 q^{47} + 10 q^{49} - 2 q^{51} - 4 q^{53} + 6 q^{57} - 4 q^{59} - 4 q^{61} + 2 q^{65} - 26 q^{67} - 4 q^{69} - 18 q^{73} - 4 q^{75} + 6 q^{77} - 18 q^{79} + 4 q^{81} - 26 q^{83} - 2 q^{85} - 4 q^{87} + 14 q^{89} - 38 q^{91} + 6 q^{93} + 6 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 0.512641 0.193760 0.0968801 0.995296i \(-0.469114\pi\)
0.0968801 + 0.995296i \(0.469114\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.76485 −0.532122 −0.266061 0.963956i \(-0.585722\pi\)
−0.266061 + 0.963956i \(0.585722\pi\)
\(12\) 0 0
\(13\) −4.95530 −1.37435 −0.687176 0.726491i \(-0.741150\pi\)
−0.687176 + 0.726491i \(0.741150\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 7.97235 1.93358 0.966790 0.255573i \(-0.0822643\pi\)
0.966790 + 0.255573i \(0.0822643\pi\)
\(18\) 0 0
\(19\) 1.93002 0.442776 0.221388 0.975186i \(-0.428941\pi\)
0.221388 + 0.975186i \(0.428941\pi\)
\(20\) 0 0
\(21\) −0.512641 −0.111867
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.16280 −1.70149 −0.850745 0.525579i \(-0.823849\pi\)
−0.850745 + 0.525579i \(0.823849\pi\)
\(30\) 0 0
\(31\) 4.20750 0.755690 0.377845 0.925869i \(-0.376665\pi\)
0.377845 + 0.925869i \(0.376665\pi\)
\(32\) 0 0
\(33\) 1.76485 0.307221
\(34\) 0 0
\(35\) −0.512641 −0.0866522
\(36\) 0 0
\(37\) 6.37267 1.04766 0.523830 0.851823i \(-0.324503\pi\)
0.523830 + 0.851823i \(0.324503\pi\)
\(38\) 0 0
\(39\) 4.95530 0.793483
\(40\) 0 0
\(41\) 4.27749 0.668031 0.334016 0.942567i \(-0.391596\pi\)
0.334016 + 0.942567i \(0.391596\pi\)
\(42\) 0 0
\(43\) −12.8853 −1.96499 −0.982496 0.186284i \(-0.940356\pi\)
−0.982496 + 0.186284i \(0.940356\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −1.93002 −0.281522 −0.140761 0.990044i \(-0.544955\pi\)
−0.140761 + 0.990044i \(0.544955\pi\)
\(48\) 0 0
\(49\) −6.73720 −0.962457
\(50\) 0 0
\(51\) −7.97235 −1.11635
\(52\) 0 0
\(53\) 4.60782 0.632933 0.316467 0.948604i \(-0.397503\pi\)
0.316467 + 0.948604i \(0.397503\pi\)
\(54\) 0 0
\(55\) 1.76485 0.237972
\(56\) 0 0
\(57\) −1.93002 −0.255637
\(58\) 0 0
\(59\) −11.1628 −1.45327 −0.726637 0.687022i \(-0.758918\pi\)
−0.726637 + 0.687022i \(0.758918\pi\)
\(60\) 0 0
\(61\) 8.65016 1.10754 0.553770 0.832670i \(-0.313189\pi\)
0.553770 + 0.832670i \(0.313189\pi\)
\(62\) 0 0
\(63\) 0.512641 0.0645867
\(64\) 0 0
\(65\) 4.95530 0.614629
\(66\) 0 0
\(67\) −11.1822 −1.36613 −0.683063 0.730360i \(-0.739353\pi\)
−0.683063 + 0.730360i \(0.739353\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.27749 0.270288 0.135144 0.990826i \(-0.456850\pi\)
0.135144 + 0.990826i \(0.456850\pi\)
\(72\) 0 0
\(73\) −8.95530 −1.04814 −0.524069 0.851676i \(-0.675587\pi\)
−0.524069 + 0.851676i \(0.675587\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −0.904733 −0.103104
\(78\) 0 0
\(79\) 8.58018 0.965345 0.482673 0.875801i \(-0.339666\pi\)
0.482673 + 0.875801i \(0.339666\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.91296 −0.758796 −0.379398 0.925234i \(-0.623869\pi\)
−0.379398 + 0.925234i \(0.623869\pi\)
\(84\) 0 0
\(85\) −7.97235 −0.864723
\(86\) 0 0
\(87\) 9.16280 0.982355
\(88\) 0 0
\(89\) −1.69486 −0.179655 −0.0898276 0.995957i \(-0.528632\pi\)
−0.0898276 + 0.995957i \(0.528632\pi\)
\(90\) 0 0
\(91\) −2.54029 −0.266295
\(92\) 0 0
\(93\) −4.20750 −0.436298
\(94\) 0 0
\(95\) −1.93002 −0.198015
\(96\) 0 0
\(97\) −14.4150 −1.46362 −0.731811 0.681507i \(-0.761325\pi\)
−0.731811 + 0.681507i \(0.761325\pi\)
\(98\) 0 0
\(99\) −1.76485 −0.177374
\(100\) 0 0
\(101\) 9.63311 0.958530 0.479265 0.877670i \(-0.340903\pi\)
0.479265 + 0.877670i \(0.340903\pi\)
\(102\) 0 0
\(103\) −7.83483 −0.771989 −0.385994 0.922501i \(-0.626142\pi\)
−0.385994 + 0.922501i \(0.626142\pi\)
\(104\) 0 0
\(105\) 0.512641 0.0500287
\(106\) 0 0
\(107\) 2.44266 0.236141 0.118070 0.993005i \(-0.462329\pi\)
0.118070 + 0.993005i \(0.462329\pi\)
\(108\) 0 0
\(109\) 3.76485 0.360607 0.180303 0.983611i \(-0.442292\pi\)
0.180303 + 0.983611i \(0.442292\pi\)
\(110\) 0 0
\(111\) −6.37267 −0.604867
\(112\) 0 0
\(113\) −2.53206 −0.238196 −0.119098 0.992882i \(-0.538000\pi\)
−0.119098 + 0.992882i \(0.538000\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −4.95530 −0.458117
\(118\) 0 0
\(119\) 4.08696 0.374651
\(120\) 0 0
\(121\) −7.88531 −0.716847
\(122\) 0 0
\(123\) −4.27749 −0.385688
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.4344 −1.10338 −0.551689 0.834050i \(-0.686016\pi\)
−0.551689 + 0.834050i \(0.686016\pi\)
\(128\) 0 0
\(129\) 12.8853 1.13449
\(130\) 0 0
\(131\) −1.94944 −0.170323 −0.0851615 0.996367i \(-0.527141\pi\)
−0.0851615 + 0.996367i \(0.527141\pi\)
\(132\) 0 0
\(133\) 0.989405 0.0857923
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 9.22456 0.788107 0.394054 0.919087i \(-0.371072\pi\)
0.394054 + 0.919087i \(0.371072\pi\)
\(138\) 0 0
\(139\) −17.5631 −1.48968 −0.744842 0.667241i \(-0.767475\pi\)
−0.744842 + 0.667241i \(0.767475\pi\)
\(140\) 0 0
\(141\) 1.93002 0.162537
\(142\) 0 0
\(143\) 8.74534 0.731322
\(144\) 0 0
\(145\) 9.16280 0.760929
\(146\) 0 0
\(147\) 6.73720 0.555675
\(148\) 0 0
\(149\) 14.3956 1.17933 0.589666 0.807647i \(-0.299259\pi\)
0.589666 + 0.807647i \(0.299259\pi\)
\(150\) 0 0
\(151\) −14.9359 −1.21546 −0.607732 0.794142i \(-0.707921\pi\)
−0.607732 + 0.794142i \(0.707921\pi\)
\(152\) 0 0
\(153\) 7.97235 0.644526
\(154\) 0 0
\(155\) −4.20750 −0.337955
\(156\) 0 0
\(157\) −18.4573 −1.47306 −0.736528 0.676407i \(-0.763536\pi\)
−0.736528 + 0.676407i \(0.763536\pi\)
\(158\) 0 0
\(159\) −4.60782 −0.365424
\(160\) 0 0
\(161\) 0.512641 0.0404018
\(162\) 0 0
\(163\) −3.58499 −0.280798 −0.140399 0.990095i \(-0.544839\pi\)
−0.140399 + 0.990095i \(0.544839\pi\)
\(164\) 0 0
\(165\) −1.76485 −0.137393
\(166\) 0 0
\(167\) −0.120549 −0.00932835 −0.00466418 0.999989i \(-0.501485\pi\)
−0.00466418 + 0.999989i \(0.501485\pi\)
\(168\) 0 0
\(169\) 11.5550 0.888844
\(170\) 0 0
\(171\) 1.93002 0.147592
\(172\) 0 0
\(173\) −11.0253 −0.838237 −0.419118 0.907932i \(-0.637661\pi\)
−0.419118 + 0.907932i \(0.637661\pi\)
\(174\) 0 0
\(175\) 0.512641 0.0387520
\(176\) 0 0
\(177\) 11.1628 0.839048
\(178\) 0 0
\(179\) 16.2750 1.21645 0.608227 0.793763i \(-0.291881\pi\)
0.608227 + 0.793763i \(0.291881\pi\)
\(180\) 0 0
\(181\) 12.7543 0.948016 0.474008 0.880520i \(-0.342807\pi\)
0.474008 + 0.880520i \(0.342807\pi\)
\(182\) 0 0
\(183\) −8.65016 −0.639438
\(184\) 0 0
\(185\) −6.37267 −0.468528
\(186\) 0 0
\(187\) −14.0700 −1.02890
\(188\) 0 0
\(189\) −0.512641 −0.0372892
\(190\) 0 0
\(191\) −14.9000 −1.07813 −0.539063 0.842265i \(-0.681222\pi\)
−0.539063 + 0.842265i \(0.681222\pi\)
\(192\) 0 0
\(193\) −17.9447 −1.29169 −0.645844 0.763469i \(-0.723494\pi\)
−0.645844 + 0.763469i \(0.723494\pi\)
\(194\) 0 0
\(195\) −4.95530 −0.354856
\(196\) 0 0
\(197\) −6.96999 −0.496591 −0.248295 0.968684i \(-0.579870\pi\)
−0.248295 + 0.968684i \(0.579870\pi\)
\(198\) 0 0
\(199\) −21.4655 −1.52165 −0.760824 0.648958i \(-0.775205\pi\)
−0.760824 + 0.648958i \(0.775205\pi\)
\(200\) 0 0
\(201\) 11.1822 0.788733
\(202\) 0 0
\(203\) −4.69723 −0.329681
\(204\) 0 0
\(205\) −4.27749 −0.298753
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −3.40618 −0.235611
\(210\) 0 0
\(211\) −25.3175 −1.74293 −0.871463 0.490462i \(-0.836828\pi\)
−0.871463 + 0.490462i \(0.836828\pi\)
\(212\) 0 0
\(213\) −2.27749 −0.156051
\(214\) 0 0
\(215\) 12.8853 0.878771
\(216\) 0 0
\(217\) 2.15694 0.146423
\(218\) 0 0
\(219\) 8.95530 0.605143
\(220\) 0 0
\(221\) −39.5054 −2.65742
\(222\) 0 0
\(223\) −14.3644 −0.961914 −0.480957 0.876744i \(-0.659711\pi\)
−0.480957 + 0.876744i \(0.659711\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −16.2498 −1.07854 −0.539270 0.842133i \(-0.681300\pi\)
−0.539270 + 0.842133i \(0.681300\pi\)
\(228\) 0 0
\(229\) 14.8048 0.978330 0.489165 0.872191i \(-0.337302\pi\)
0.489165 + 0.872191i \(0.337302\pi\)
\(230\) 0 0
\(231\) 0.904733 0.0595271
\(232\) 0 0
\(233\) −29.8047 −1.95257 −0.976287 0.216482i \(-0.930542\pi\)
−0.976287 + 0.216482i \(0.930542\pi\)
\(234\) 0 0
\(235\) 1.93002 0.125900
\(236\) 0 0
\(237\) −8.58018 −0.557342
\(238\) 0 0
\(239\) −11.1075 −0.718485 −0.359242 0.933244i \(-0.616965\pi\)
−0.359242 + 0.933244i \(0.616965\pi\)
\(240\) 0 0
\(241\) 20.3062 1.30804 0.654018 0.756479i \(-0.273082\pi\)
0.654018 + 0.756479i \(0.273082\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.73720 0.430424
\(246\) 0 0
\(247\) −9.56380 −0.608530
\(248\) 0 0
\(249\) 6.91296 0.438091
\(250\) 0 0
\(251\) 2.21573 0.139856 0.0699279 0.997552i \(-0.477723\pi\)
0.0699279 + 0.997552i \(0.477723\pi\)
\(252\) 0 0
\(253\) −1.76485 −0.110955
\(254\) 0 0
\(255\) 7.97235 0.499248
\(256\) 0 0
\(257\) 2.90473 0.181192 0.0905961 0.995888i \(-0.471123\pi\)
0.0905961 + 0.995888i \(0.471123\pi\)
\(258\) 0 0
\(259\) 3.26689 0.202995
\(260\) 0 0
\(261\) −9.16280 −0.567163
\(262\) 0 0
\(263\) 3.75662 0.231643 0.115822 0.993270i \(-0.463050\pi\)
0.115822 + 0.993270i \(0.463050\pi\)
\(264\) 0 0
\(265\) −4.60782 −0.283056
\(266\) 0 0
\(267\) 1.69486 0.103724
\(268\) 0 0
\(269\) 7.58254 0.462316 0.231158 0.972916i \(-0.425749\pi\)
0.231158 + 0.972916i \(0.425749\pi\)
\(270\) 0 0
\(271\) −15.0423 −0.913752 −0.456876 0.889530i \(-0.651032\pi\)
−0.456876 + 0.889530i \(0.651032\pi\)
\(272\) 0 0
\(273\) 2.54029 0.153745
\(274\) 0 0
\(275\) −1.76485 −0.106424
\(276\) 0 0
\(277\) 8.05056 0.483712 0.241856 0.970312i \(-0.422244\pi\)
0.241856 + 0.970312i \(0.422244\pi\)
\(278\) 0 0
\(279\) 4.20750 0.251897
\(280\) 0 0
\(281\) −14.3704 −0.857266 −0.428633 0.903479i \(-0.641005\pi\)
−0.428633 + 0.903479i \(0.641005\pi\)
\(282\) 0 0
\(283\) 24.0287 1.42836 0.714179 0.699963i \(-0.246800\pi\)
0.714179 + 0.699963i \(0.246800\pi\)
\(284\) 0 0
\(285\) 1.93002 0.114324
\(286\) 0 0
\(287\) 2.19282 0.129438
\(288\) 0 0
\(289\) 46.5584 2.73873
\(290\) 0 0
\(291\) 14.4150 0.845023
\(292\) 0 0
\(293\) 8.46786 0.494697 0.247349 0.968927i \(-0.420441\pi\)
0.247349 + 0.968927i \(0.420441\pi\)
\(294\) 0 0
\(295\) 11.1628 0.649923
\(296\) 0 0
\(297\) 1.76485 0.102407
\(298\) 0 0
\(299\) −4.95530 −0.286572
\(300\) 0 0
\(301\) −6.60554 −0.380737
\(302\) 0 0
\(303\) −9.63311 −0.553408
\(304\) 0 0
\(305\) −8.65016 −0.495307
\(306\) 0 0
\(307\) 17.9594 1.02500 0.512498 0.858688i \(-0.328720\pi\)
0.512498 + 0.858688i \(0.328720\pi\)
\(308\) 0 0
\(309\) 7.83483 0.445708
\(310\) 0 0
\(311\) 23.8894 1.35464 0.677322 0.735687i \(-0.263140\pi\)
0.677322 + 0.735687i \(0.263140\pi\)
\(312\) 0 0
\(313\) 25.4184 1.43673 0.718367 0.695664i \(-0.244890\pi\)
0.718367 + 0.695664i \(0.244890\pi\)
\(314\) 0 0
\(315\) −0.512641 −0.0288841
\(316\) 0 0
\(317\) 24.0147 1.34880 0.674400 0.738367i \(-0.264403\pi\)
0.674400 + 0.738367i \(0.264403\pi\)
\(318\) 0 0
\(319\) 16.1709 0.905399
\(320\) 0 0
\(321\) −2.44266 −0.136336
\(322\) 0 0
\(323\) 15.3868 0.856142
\(324\) 0 0
\(325\) −4.95530 −0.274870
\(326\) 0 0
\(327\) −3.76485 −0.208197
\(328\) 0 0
\(329\) −0.989405 −0.0545477
\(330\) 0 0
\(331\) −12.7625 −0.701489 −0.350745 0.936471i \(-0.614072\pi\)
−0.350745 + 0.936471i \(0.614072\pi\)
\(332\) 0 0
\(333\) 6.37267 0.349220
\(334\) 0 0
\(335\) 11.1822 0.610950
\(336\) 0 0
\(337\) −5.38973 −0.293597 −0.146799 0.989166i \(-0.546897\pi\)
−0.146799 + 0.989166i \(0.546897\pi\)
\(338\) 0 0
\(339\) 2.53206 0.137523
\(340\) 0 0
\(341\) −7.42560 −0.402119
\(342\) 0 0
\(343\) −7.04225 −0.380246
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 7.52969 0.404215 0.202108 0.979363i \(-0.435221\pi\)
0.202108 + 0.979363i \(0.435221\pi\)
\(348\) 0 0
\(349\) −5.59723 −0.299613 −0.149806 0.988715i \(-0.547865\pi\)
−0.149806 + 0.988715i \(0.547865\pi\)
\(350\) 0 0
\(351\) 4.95530 0.264494
\(352\) 0 0
\(353\) −37.3150 −1.98608 −0.993039 0.117788i \(-0.962420\pi\)
−0.993039 + 0.117788i \(0.962420\pi\)
\(354\) 0 0
\(355\) −2.27749 −0.120877
\(356\) 0 0
\(357\) −4.08696 −0.216305
\(358\) 0 0
\(359\) 15.1799 0.801167 0.400583 0.916260i \(-0.368808\pi\)
0.400583 + 0.916260i \(0.368808\pi\)
\(360\) 0 0
\(361\) −15.2750 −0.803949
\(362\) 0 0
\(363\) 7.88531 0.413872
\(364\) 0 0
\(365\) 8.95530 0.468742
\(366\) 0 0
\(367\) −35.6389 −1.86033 −0.930167 0.367136i \(-0.880338\pi\)
−0.930167 + 0.367136i \(0.880338\pi\)
\(368\) 0 0
\(369\) 4.27749 0.222677
\(370\) 0 0
\(371\) 2.36216 0.122637
\(372\) 0 0
\(373\) −0.864846 −0.0447800 −0.0223900 0.999749i \(-0.507128\pi\)
−0.0223900 + 0.999749i \(0.507128\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 45.4044 2.33845
\(378\) 0 0
\(379\) −16.8300 −0.864500 −0.432250 0.901754i \(-0.642280\pi\)
−0.432250 + 0.901754i \(0.642280\pi\)
\(380\) 0 0
\(381\) 12.4344 0.637035
\(382\) 0 0
\(383\) 18.3622 0.938263 0.469131 0.883128i \(-0.344567\pi\)
0.469131 + 0.883128i \(0.344567\pi\)
\(384\) 0 0
\(385\) 0.904733 0.0461095
\(386\) 0 0
\(387\) −12.8853 −0.654997
\(388\) 0 0
\(389\) 22.0294 1.11693 0.558467 0.829527i \(-0.311390\pi\)
0.558467 + 0.829527i \(0.311390\pi\)
\(390\) 0 0
\(391\) 7.97235 0.403179
\(392\) 0 0
\(393\) 1.94944 0.0983360
\(394\) 0 0
\(395\) −8.58018 −0.431716
\(396\) 0 0
\(397\) 23.8553 1.19726 0.598632 0.801024i \(-0.295711\pi\)
0.598632 + 0.801024i \(0.295711\pi\)
\(398\) 0 0
\(399\) −0.989405 −0.0495322
\(400\) 0 0
\(401\) 18.5413 0.925910 0.462955 0.886382i \(-0.346789\pi\)
0.462955 + 0.886382i \(0.346789\pi\)
\(402\) 0 0
\(403\) −20.8494 −1.03858
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −11.2468 −0.557483
\(408\) 0 0
\(409\) 24.5884 1.21582 0.607909 0.794007i \(-0.292008\pi\)
0.607909 + 0.794007i \(0.292008\pi\)
\(410\) 0 0
\(411\) −9.22456 −0.455014
\(412\) 0 0
\(413\) −5.72251 −0.281586
\(414\) 0 0
\(415\) 6.91296 0.339344
\(416\) 0 0
\(417\) 17.5631 0.860070
\(418\) 0 0
\(419\) 5.86598 0.286572 0.143286 0.989681i \(-0.454233\pi\)
0.143286 + 0.989681i \(0.454233\pi\)
\(420\) 0 0
\(421\) −13.3955 −0.652857 −0.326429 0.945222i \(-0.605845\pi\)
−0.326429 + 0.945222i \(0.605845\pi\)
\(422\) 0 0
\(423\) −1.93002 −0.0938406
\(424\) 0 0
\(425\) 7.97235 0.386716
\(426\) 0 0
\(427\) 4.43443 0.214597
\(428\) 0 0
\(429\) −8.74534 −0.422229
\(430\) 0 0
\(431\) 39.3509 1.89547 0.947733 0.319065i \(-0.103369\pi\)
0.947733 + 0.319065i \(0.103369\pi\)
\(432\) 0 0
\(433\) −37.0034 −1.77827 −0.889135 0.457644i \(-0.848693\pi\)
−0.889135 + 0.457644i \(0.848693\pi\)
\(434\) 0 0
\(435\) −9.16280 −0.439323
\(436\) 0 0
\(437\) 1.93002 0.0923252
\(438\) 0 0
\(439\) −7.75434 −0.370094 −0.185047 0.982730i \(-0.559244\pi\)
−0.185047 + 0.982730i \(0.559244\pi\)
\(440\) 0 0
\(441\) −6.73720 −0.320819
\(442\) 0 0
\(443\) −28.6706 −1.36218 −0.681091 0.732198i \(-0.738494\pi\)
−0.681091 + 0.732198i \(0.738494\pi\)
\(444\) 0 0
\(445\) 1.69486 0.0803442
\(446\) 0 0
\(447\) −14.3956 −0.680888
\(448\) 0 0
\(449\) −17.5943 −0.830325 −0.415162 0.909747i \(-0.636275\pi\)
−0.415162 + 0.909747i \(0.636275\pi\)
\(450\) 0 0
\(451\) −7.54911 −0.355474
\(452\) 0 0
\(453\) 14.9359 0.701749
\(454\) 0 0
\(455\) 2.54029 0.119091
\(456\) 0 0
\(457\) 20.1569 0.942902 0.471451 0.881892i \(-0.343730\pi\)
0.471451 + 0.881892i \(0.343730\pi\)
\(458\) 0 0
\(459\) −7.97235 −0.372118
\(460\) 0 0
\(461\) 27.6307 1.28689 0.643444 0.765493i \(-0.277505\pi\)
0.643444 + 0.765493i \(0.277505\pi\)
\(462\) 0 0
\(463\) 25.0398 1.16370 0.581849 0.813297i \(-0.302329\pi\)
0.581849 + 0.813297i \(0.302329\pi\)
\(464\) 0 0
\(465\) 4.20750 0.195118
\(466\) 0 0
\(467\) 8.06176 0.373054 0.186527 0.982450i \(-0.440277\pi\)
0.186527 + 0.982450i \(0.440277\pi\)
\(468\) 0 0
\(469\) −5.73247 −0.264701
\(470\) 0 0
\(471\) 18.4573 0.850470
\(472\) 0 0
\(473\) 22.7406 1.04561
\(474\) 0 0
\(475\) 1.93002 0.0885552
\(476\) 0 0
\(477\) 4.60782 0.210978
\(478\) 0 0
\(479\) 4.92119 0.224855 0.112427 0.993660i \(-0.464137\pi\)
0.112427 + 0.993660i \(0.464137\pi\)
\(480\) 0 0
\(481\) −31.5785 −1.43986
\(482\) 0 0
\(483\) −0.512641 −0.0233260
\(484\) 0 0
\(485\) 14.4150 0.654552
\(486\) 0 0
\(487\) 7.96412 0.360889 0.180444 0.983585i \(-0.442246\pi\)
0.180444 + 0.983585i \(0.442246\pi\)
\(488\) 0 0
\(489\) 3.58499 0.162119
\(490\) 0 0
\(491\) 37.9587 1.71305 0.856526 0.516103i \(-0.172618\pi\)
0.856526 + 0.516103i \(0.172618\pi\)
\(492\) 0 0
\(493\) −73.0491 −3.28997
\(494\) 0 0
\(495\) 1.76485 0.0793240
\(496\) 0 0
\(497\) 1.16753 0.0523711
\(498\) 0 0
\(499\) −31.8887 −1.42754 −0.713768 0.700382i \(-0.753013\pi\)
−0.713768 + 0.700382i \(0.753013\pi\)
\(500\) 0 0
\(501\) 0.120549 0.00538573
\(502\) 0 0
\(503\) −35.3990 −1.57836 −0.789182 0.614160i \(-0.789495\pi\)
−0.789182 + 0.614160i \(0.789495\pi\)
\(504\) 0 0
\(505\) −9.63311 −0.428668
\(506\) 0 0
\(507\) −11.5550 −0.513175
\(508\) 0 0
\(509\) −16.0341 −0.710699 −0.355350 0.934733i \(-0.615638\pi\)
−0.355350 + 0.934733i \(0.615638\pi\)
\(510\) 0 0
\(511\) −4.59085 −0.203087
\(512\) 0 0
\(513\) −1.93002 −0.0852123
\(514\) 0 0
\(515\) 7.83483 0.345244
\(516\) 0 0
\(517\) 3.40618 0.149804
\(518\) 0 0
\(519\) 11.0253 0.483956
\(520\) 0 0
\(521\) 16.0399 0.702720 0.351360 0.936240i \(-0.385719\pi\)
0.351360 + 0.936240i \(0.385719\pi\)
\(522\) 0 0
\(523\) 31.1098 1.36034 0.680168 0.733056i \(-0.261907\pi\)
0.680168 + 0.733056i \(0.261907\pi\)
\(524\) 0 0
\(525\) −0.512641 −0.0223735
\(526\) 0 0
\(527\) 33.5437 1.46119
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.1628 −0.484424
\(532\) 0 0
\(533\) −21.1962 −0.918111
\(534\) 0 0
\(535\) −2.44266 −0.105605
\(536\) 0 0
\(537\) −16.2750 −0.702320
\(538\) 0 0
\(539\) 11.8901 0.512144
\(540\) 0 0
\(541\) 24.5160 1.05402 0.527012 0.849858i \(-0.323312\pi\)
0.527012 + 0.849858i \(0.323312\pi\)
\(542\) 0 0
\(543\) −12.7543 −0.547337
\(544\) 0 0
\(545\) −3.76485 −0.161268
\(546\) 0 0
\(547\) −2.22937 −0.0953211 −0.0476606 0.998864i \(-0.515177\pi\)
−0.0476606 + 0.998864i \(0.515177\pi\)
\(548\) 0 0
\(549\) 8.65016 0.369180
\(550\) 0 0
\(551\) −17.6844 −0.753379
\(552\) 0 0
\(553\) 4.39855 0.187045
\(554\) 0 0
\(555\) 6.37267 0.270505
\(556\) 0 0
\(557\) −12.5525 −0.531868 −0.265934 0.963991i \(-0.585680\pi\)
−0.265934 + 0.963991i \(0.585680\pi\)
\(558\) 0 0
\(559\) 63.8506 2.70059
\(560\) 0 0
\(561\) 14.0700 0.594035
\(562\) 0 0
\(563\) −27.8829 −1.17513 −0.587563 0.809178i \(-0.699913\pi\)
−0.587563 + 0.809178i \(0.699913\pi\)
\(564\) 0 0
\(565\) 2.53206 0.106525
\(566\) 0 0
\(567\) 0.512641 0.0215289
\(568\) 0 0
\(569\) −22.8717 −0.958830 −0.479415 0.877588i \(-0.659151\pi\)
−0.479415 + 0.877588i \(0.659151\pi\)
\(570\) 0 0
\(571\) 30.9847 1.29667 0.648334 0.761356i \(-0.275466\pi\)
0.648334 + 0.761356i \(0.275466\pi\)
\(572\) 0 0
\(573\) 14.9000 0.622456
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −29.2274 −1.21675 −0.608376 0.793649i \(-0.708179\pi\)
−0.608376 + 0.793649i \(0.708179\pi\)
\(578\) 0 0
\(579\) 17.9447 0.745756
\(580\) 0 0
\(581\) −3.54387 −0.147024
\(582\) 0 0
\(583\) −8.13211 −0.336798
\(584\) 0 0
\(585\) 4.95530 0.204876
\(586\) 0 0
\(587\) −4.36461 −0.180147 −0.0900734 0.995935i \(-0.528710\pi\)
−0.0900734 + 0.995935i \(0.528710\pi\)
\(588\) 0 0
\(589\) 8.12055 0.334601
\(590\) 0 0
\(591\) 6.96999 0.286707
\(592\) 0 0
\(593\) −28.7765 −1.18171 −0.590854 0.806778i \(-0.701209\pi\)
−0.590854 + 0.806778i \(0.701209\pi\)
\(594\) 0 0
\(595\) −4.08696 −0.167549
\(596\) 0 0
\(597\) 21.4655 0.878524
\(598\) 0 0
\(599\) 15.9612 0.652155 0.326078 0.945343i \(-0.394273\pi\)
0.326078 + 0.945343i \(0.394273\pi\)
\(600\) 0 0
\(601\) −0.571948 −0.0233302 −0.0116651 0.999932i \(-0.503713\pi\)
−0.0116651 + 0.999932i \(0.503713\pi\)
\(602\) 0 0
\(603\) −11.1822 −0.455375
\(604\) 0 0
\(605\) 7.88531 0.320584
\(606\) 0 0
\(607\) 24.5356 0.995868 0.497934 0.867215i \(-0.334092\pi\)
0.497934 + 0.867215i \(0.334092\pi\)
\(608\) 0 0
\(609\) 4.69723 0.190341
\(610\) 0 0
\(611\) 9.56380 0.386910
\(612\) 0 0
\(613\) −11.8689 −0.479382 −0.239691 0.970849i \(-0.577046\pi\)
−0.239691 + 0.970849i \(0.577046\pi\)
\(614\) 0 0
\(615\) 4.27749 0.172485
\(616\) 0 0
\(617\) −30.5614 −1.23036 −0.615179 0.788388i \(-0.710916\pi\)
−0.615179 + 0.788388i \(0.710916\pi\)
\(618\) 0 0
\(619\) 10.9583 0.440450 0.220225 0.975449i \(-0.429321\pi\)
0.220225 + 0.975449i \(0.429321\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −0.868857 −0.0348100
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.40618 0.136030
\(628\) 0 0
\(629\) 50.8052 2.02574
\(630\) 0 0
\(631\) −8.21396 −0.326993 −0.163496 0.986544i \(-0.552277\pi\)
−0.163496 + 0.986544i \(0.552277\pi\)
\(632\) 0 0
\(633\) 25.3175 1.00628
\(634\) 0 0
\(635\) 12.4344 0.493445
\(636\) 0 0
\(637\) 33.3848 1.32276
\(638\) 0 0
\(639\) 2.27749 0.0900961
\(640\) 0 0
\(641\) 18.0401 0.712539 0.356270 0.934383i \(-0.384048\pi\)
0.356270 + 0.934383i \(0.384048\pi\)
\(642\) 0 0
\(643\) 8.39787 0.331180 0.165590 0.986195i \(-0.447047\pi\)
0.165590 + 0.986195i \(0.447047\pi\)
\(644\) 0 0
\(645\) −12.8853 −0.507359
\(646\) 0 0
\(647\) 8.57440 0.337094 0.168547 0.985694i \(-0.446092\pi\)
0.168547 + 0.985694i \(0.446092\pi\)
\(648\) 0 0
\(649\) 19.7006 0.773318
\(650\) 0 0
\(651\) −2.15694 −0.0845371
\(652\) 0 0
\(653\) 43.9206 1.71874 0.859372 0.511351i \(-0.170855\pi\)
0.859372 + 0.511351i \(0.170855\pi\)
\(654\) 0 0
\(655\) 1.94944 0.0761707
\(656\) 0 0
\(657\) −8.95530 −0.349379
\(658\) 0 0
\(659\) 42.7170 1.66402 0.832009 0.554762i \(-0.187191\pi\)
0.832009 + 0.554762i \(0.187191\pi\)
\(660\) 0 0
\(661\) −47.7017 −1.85538 −0.927690 0.373351i \(-0.878209\pi\)
−0.927690 + 0.373351i \(0.878209\pi\)
\(662\) 0 0
\(663\) 39.5054 1.53426
\(664\) 0 0
\(665\) −0.989405 −0.0383675
\(666\) 0 0
\(667\) −9.16280 −0.354785
\(668\) 0 0
\(669\) 14.3644 0.555361
\(670\) 0 0
\(671\) −15.2662 −0.589346
\(672\) 0 0
\(673\) 19.8912 0.766748 0.383374 0.923593i \(-0.374762\pi\)
0.383374 + 0.923593i \(0.374762\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −12.3457 −0.474484 −0.237242 0.971451i \(-0.576243\pi\)
−0.237242 + 0.971451i \(0.576243\pi\)
\(678\) 0 0
\(679\) −7.38973 −0.283592
\(680\) 0 0
\(681\) 16.2498 0.622695
\(682\) 0 0
\(683\) 42.9504 1.64345 0.821726 0.569883i \(-0.193012\pi\)
0.821726 + 0.569883i \(0.193012\pi\)
\(684\) 0 0
\(685\) −9.22456 −0.352452
\(686\) 0 0
\(687\) −14.8048 −0.564839
\(688\) 0 0
\(689\) −22.8331 −0.869874
\(690\) 0 0
\(691\) −33.2156 −1.26358 −0.631791 0.775138i \(-0.717680\pi\)
−0.631791 + 0.775138i \(0.717680\pi\)
\(692\) 0 0
\(693\) −0.904733 −0.0343680
\(694\) 0 0
\(695\) 17.5631 0.666207
\(696\) 0 0
\(697\) 34.1016 1.29169
\(698\) 0 0
\(699\) 29.8047 1.12732
\(700\) 0 0
\(701\) −6.62480 −0.250215 −0.125108 0.992143i \(-0.539928\pi\)
−0.125108 + 0.992143i \(0.539928\pi\)
\(702\) 0 0
\(703\) 12.2994 0.463879
\(704\) 0 0
\(705\) −1.93002 −0.0726886
\(706\) 0 0
\(707\) 4.93833 0.185725
\(708\) 0 0
\(709\) 3.88836 0.146030 0.0730152 0.997331i \(-0.476738\pi\)
0.0730152 + 0.997331i \(0.476738\pi\)
\(710\) 0 0
\(711\) 8.58018 0.321782
\(712\) 0 0
\(713\) 4.20750 0.157572
\(714\) 0 0
\(715\) −8.74534 −0.327057
\(716\) 0 0
\(717\) 11.1075 0.414817
\(718\) 0 0
\(719\) 29.1969 1.08886 0.544430 0.838806i \(-0.316746\pi\)
0.544430 + 0.838806i \(0.316746\pi\)
\(720\) 0 0
\(721\) −4.01646 −0.149581
\(722\) 0 0
\(723\) −20.3062 −0.755195
\(724\) 0 0
\(725\) −9.16280 −0.340298
\(726\) 0 0
\(727\) −1.16175 −0.0430871 −0.0215435 0.999768i \(-0.506858\pi\)
−0.0215435 + 0.999768i \(0.506858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −102.726 −3.79947
\(732\) 0 0
\(733\) 3.29226 0.121602 0.0608012 0.998150i \(-0.480634\pi\)
0.0608012 + 0.998150i \(0.480634\pi\)
\(734\) 0 0
\(735\) −6.73720 −0.248505
\(736\) 0 0
\(737\) 19.7349 0.726945
\(738\) 0 0
\(739\) −13.4232 −0.493779 −0.246889 0.969044i \(-0.579408\pi\)
−0.246889 + 0.969044i \(0.579408\pi\)
\(740\) 0 0
\(741\) 9.56380 0.351335
\(742\) 0 0
\(743\) 10.6279 0.389901 0.194950 0.980813i \(-0.437545\pi\)
0.194950 + 0.980813i \(0.437545\pi\)
\(744\) 0 0
\(745\) −14.3956 −0.527414
\(746\) 0 0
\(747\) −6.91296 −0.252932
\(748\) 0 0
\(749\) 1.25221 0.0457546
\(750\) 0 0
\(751\) −4.92993 −0.179896 −0.0899479 0.995946i \(-0.528670\pi\)
−0.0899479 + 0.995946i \(0.528670\pi\)
\(752\) 0 0
\(753\) −2.21573 −0.0807458
\(754\) 0 0
\(755\) 14.9359 0.543572
\(756\) 0 0
\(757\) −31.2580 −1.13609 −0.568045 0.822997i \(-0.692300\pi\)
−0.568045 + 0.822997i \(0.692300\pi\)
\(758\) 0 0
\(759\) 1.76485 0.0640599
\(760\) 0 0
\(761\) −10.6972 −0.387774 −0.193887 0.981024i \(-0.562110\pi\)
−0.193887 + 0.981024i \(0.562110\pi\)
\(762\) 0 0
\(763\) 1.93002 0.0698713
\(764\) 0 0
\(765\) −7.97235 −0.288241
\(766\) 0 0
\(767\) 55.3150 1.99731
\(768\) 0 0
\(769\) 40.2897 1.45288 0.726442 0.687227i \(-0.241172\pi\)
0.726442 + 0.687227i \(0.241172\pi\)
\(770\) 0 0
\(771\) −2.90473 −0.104611
\(772\) 0 0
\(773\) −9.24976 −0.332691 −0.166345 0.986068i \(-0.553197\pi\)
−0.166345 + 0.986068i \(0.553197\pi\)
\(774\) 0 0
\(775\) 4.20750 0.151138
\(776\) 0 0
\(777\) −3.26689 −0.117199
\(778\) 0 0
\(779\) 8.25562 0.295788
\(780\) 0 0
\(781\) −4.01942 −0.143826
\(782\) 0 0
\(783\) 9.16280 0.327452
\(784\) 0 0
\(785\) 18.4573 0.658771
\(786\) 0 0
\(787\) −48.8129 −1.73999 −0.869996 0.493059i \(-0.835879\pi\)
−0.869996 + 0.493059i \(0.835879\pi\)
\(788\) 0 0
\(789\) −3.75662 −0.133739
\(790\) 0 0
\(791\) −1.29804 −0.0461529
\(792\) 0 0
\(793\) −42.8641 −1.52215
\(794\) 0 0
\(795\) 4.60782 0.163423
\(796\) 0 0
\(797\) −28.7594 −1.01871 −0.509354 0.860557i \(-0.670116\pi\)
−0.509354 + 0.860557i \(0.670116\pi\)
\(798\) 0 0
\(799\) −15.3868 −0.544345
\(800\) 0 0
\(801\) −1.69486 −0.0598850
\(802\) 0 0
\(803\) 15.8047 0.557737
\(804\) 0 0
\(805\) −0.512641 −0.0180682
\(806\) 0 0
\(807\) −7.58254 −0.266918
\(808\) 0 0
\(809\) 18.1034 0.636482 0.318241 0.948010i \(-0.396908\pi\)
0.318241 + 0.948010i \(0.396908\pi\)
\(810\) 0 0
\(811\) −9.59740 −0.337010 −0.168505 0.985701i \(-0.553894\pi\)
−0.168505 + 0.985701i \(0.553894\pi\)
\(812\) 0 0
\(813\) 15.0423 0.527555
\(814\) 0 0
\(815\) 3.58499 0.125577
\(816\) 0 0
\(817\) −24.8689 −0.870051
\(818\) 0 0
\(819\) −2.54029 −0.0887649
\(820\) 0 0
\(821\) −34.3044 −1.19723 −0.598616 0.801036i \(-0.704283\pi\)
−0.598616 + 0.801036i \(0.704283\pi\)
\(822\) 0 0
\(823\) 24.0504 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(824\) 0 0
\(825\) 1.76485 0.0614441
\(826\) 0 0
\(827\) −29.3621 −1.02102 −0.510510 0.859872i \(-0.670543\pi\)
−0.510510 + 0.859872i \(0.670543\pi\)
\(828\) 0 0
\(829\) −26.8472 −0.932440 −0.466220 0.884669i \(-0.654385\pi\)
−0.466220 + 0.884669i \(0.654385\pi\)
\(830\) 0 0
\(831\) −8.05056 −0.279271
\(832\) 0 0
\(833\) −53.7113 −1.86099
\(834\) 0 0
\(835\) 0.120549 0.00417177
\(836\) 0 0
\(837\) −4.20750 −0.145433
\(838\) 0 0
\(839\) −16.7453 −0.578114 −0.289057 0.957312i \(-0.593342\pi\)
−0.289057 + 0.957312i \(0.593342\pi\)
\(840\) 0 0
\(841\) 54.9569 1.89507
\(842\) 0 0
\(843\) 14.3704 0.494942
\(844\) 0 0
\(845\) −11.5550 −0.397503
\(846\) 0 0
\(847\) −4.04234 −0.138896
\(848\) 0 0
\(849\) −24.0287 −0.824663
\(850\) 0 0
\(851\) 6.37267 0.218452
\(852\) 0 0
\(853\) 24.3421 0.833456 0.416728 0.909031i \(-0.363177\pi\)
0.416728 + 0.909031i \(0.363177\pi\)
\(854\) 0 0
\(855\) −1.93002 −0.0660051
\(856\) 0 0
\(857\) 23.0921 0.788812 0.394406 0.918936i \(-0.370950\pi\)
0.394406 + 0.918936i \(0.370950\pi\)
\(858\) 0 0
\(859\) 16.4532 0.561375 0.280687 0.959799i \(-0.409438\pi\)
0.280687 + 0.959799i \(0.409438\pi\)
\(860\) 0 0
\(861\) −2.19282 −0.0747310
\(862\) 0 0
\(863\) −19.6144 −0.667681 −0.333840 0.942630i \(-0.608345\pi\)
−0.333840 + 0.942630i \(0.608345\pi\)
\(864\) 0 0
\(865\) 11.0253 0.374871
\(866\) 0 0
\(867\) −46.5584 −1.58121
\(868\) 0 0
\(869\) −15.1427 −0.513681
\(870\) 0 0
\(871\) 55.4112 1.87754
\(872\) 0 0
\(873\) −14.4150 −0.487874
\(874\) 0 0
\(875\) −0.512641 −0.0173304
\(876\) 0 0
\(877\) 38.1305 1.28758 0.643788 0.765204i \(-0.277362\pi\)
0.643788 + 0.765204i \(0.277362\pi\)
\(878\) 0 0
\(879\) −8.46786 −0.285614
\(880\) 0 0
\(881\) 13.2051 0.444892 0.222446 0.974945i \(-0.428596\pi\)
0.222446 + 0.974945i \(0.428596\pi\)
\(882\) 0 0
\(883\) 33.3362 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(884\) 0 0
\(885\) −11.1628 −0.375233
\(886\) 0 0
\(887\) −17.2839 −0.580335 −0.290168 0.956976i \(-0.593711\pi\)
−0.290168 + 0.956976i \(0.593711\pi\)
\(888\) 0 0
\(889\) −6.37440 −0.213790
\(890\) 0 0
\(891\) −1.76485 −0.0591246
\(892\) 0 0
\(893\) −3.72496 −0.124651
\(894\) 0 0
\(895\) −16.2750 −0.544015
\(896\) 0 0
\(897\) 4.95530 0.165453
\(898\) 0 0
\(899\) −38.5525 −1.28580
\(900\) 0 0
\(901\) 36.7352 1.22383
\(902\) 0 0
\(903\) 6.60554 0.219819
\(904\) 0 0
\(905\) −12.7543 −0.423966
\(906\) 0 0
\(907\) −38.6731 −1.28412 −0.642059 0.766655i \(-0.721920\pi\)
−0.642059 + 0.766655i \(0.721920\pi\)
\(908\) 0 0
\(909\) 9.63311 0.319510
\(910\) 0 0
\(911\) 0.0503989 0.00166979 0.000834895 1.00000i \(-0.499734\pi\)
0.000834895 1.00000i \(0.499734\pi\)
\(912\) 0 0
\(913\) 12.2003 0.403772
\(914\) 0 0
\(915\) 8.65016 0.285966
\(916\) 0 0
\(917\) −0.999361 −0.0330018
\(918\) 0 0
\(919\) −6.47921 −0.213730 −0.106865 0.994274i \(-0.534081\pi\)
−0.106865 + 0.994274i \(0.534081\pi\)
\(920\) 0 0
\(921\) −17.9594 −0.591782
\(922\) 0 0
\(923\) −11.2856 −0.371471
\(924\) 0 0
\(925\) 6.37267 0.209532
\(926\) 0 0
\(927\) −7.83483 −0.257330
\(928\) 0 0
\(929\) 30.2563 0.992677 0.496338 0.868129i \(-0.334677\pi\)
0.496338 + 0.868129i \(0.334677\pi\)
\(930\) 0 0
\(931\) −13.0029 −0.426153
\(932\) 0 0
\(933\) −23.8894 −0.782104
\(934\) 0 0
\(935\) 14.0700 0.460138
\(936\) 0 0
\(937\) −21.2274 −0.693468 −0.346734 0.937963i \(-0.612709\pi\)
−0.346734 + 0.937963i \(0.612709\pi\)
\(938\) 0 0
\(939\) −25.4184 −0.829499
\(940\) 0 0
\(941\) 9.59495 0.312786 0.156393 0.987695i \(-0.450013\pi\)
0.156393 + 0.987695i \(0.450013\pi\)
\(942\) 0 0
\(943\) 4.27749 0.139294
\(944\) 0 0
\(945\) 0.512641 0.0166762
\(946\) 0 0
\(947\) 35.2403 1.14516 0.572578 0.819850i \(-0.305943\pi\)
0.572578 + 0.819850i \(0.305943\pi\)
\(948\) 0 0
\(949\) 44.3762 1.44051
\(950\) 0 0
\(951\) −24.0147 −0.778730
\(952\) 0 0
\(953\) 48.6717 1.57663 0.788315 0.615272i \(-0.210954\pi\)
0.788315 + 0.615272i \(0.210954\pi\)
\(954\) 0 0
\(955\) 14.9000 0.482153
\(956\) 0 0
\(957\) −16.1709 −0.522733
\(958\) 0 0
\(959\) 4.72889 0.152704
\(960\) 0 0
\(961\) −13.2969 −0.428933
\(962\) 0 0
\(963\) 2.44266 0.0787135
\(964\) 0 0
\(965\) 17.9447 0.577660
\(966\) 0 0
\(967\) 22.7765 0.732443 0.366221 0.930528i \(-0.380651\pi\)
0.366221 + 0.930528i \(0.380651\pi\)
\(968\) 0 0
\(969\) −15.3868 −0.494294
\(970\) 0 0
\(971\) −50.3120 −1.61459 −0.807294 0.590150i \(-0.799069\pi\)
−0.807294 + 0.590150i \(0.799069\pi\)
\(972\) 0 0
\(973\) −9.00358 −0.288641
\(974\) 0 0
\(975\) 4.95530 0.158697
\(976\) 0 0
\(977\) −38.6836 −1.23760 −0.618799 0.785549i \(-0.712380\pi\)
−0.618799 + 0.785549i \(0.712380\pi\)
\(978\) 0 0
\(979\) 2.99117 0.0955984
\(980\) 0 0
\(981\) 3.76485 0.120202
\(982\) 0 0
\(983\) 12.6584 0.403740 0.201870 0.979412i \(-0.435298\pi\)
0.201870 + 0.979412i \(0.435298\pi\)
\(984\) 0 0
\(985\) 6.96999 0.222082
\(986\) 0 0
\(987\) 0.989405 0.0314931
\(988\) 0 0
\(989\) −12.8853 −0.409729
\(990\) 0 0
\(991\) 11.7148 0.372133 0.186067 0.982537i \(-0.440426\pi\)
0.186067 + 0.982537i \(0.440426\pi\)
\(992\) 0 0
\(993\) 12.7625 0.405005
\(994\) 0 0
\(995\) 21.4655 0.680502
\(996\) 0 0
\(997\) −1.34389 −0.0425615 −0.0212808 0.999774i \(-0.506774\pi\)
−0.0212808 + 0.999774i \(0.506774\pi\)
\(998\) 0 0
\(999\) −6.37267 −0.201622
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 2760.2.a.v.1.3 4
3.2 odd 2 8280.2.a.bq.1.3 4
4.3 odd 2 5520.2.a.cb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.v.1.3 4 1.1 even 1 trivial
5520.2.a.cb.1.2 4 4.3 odd 2
8280.2.a.bq.1.3 4 3.2 odd 2