Properties

Label 980.2.a.j.1.2
Level $980$
Weight $2$
Character 980.1
Self dual yes
Analytic conductor $7.825$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.82533939809\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 980.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\) \(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} -3.82843 q^{11} -3.58579 q^{13} +0.414214 q^{15} -6.41421 q^{17} -3.65685 q^{19} +0.585786 q^{23} +1.00000 q^{25} -2.41421 q^{27} +6.65685 q^{29} -4.58579 q^{31} -1.58579 q^{33} -3.41421 q^{37} -1.48528 q^{39} -0.585786 q^{41} +11.6569 q^{43} -2.82843 q^{45} +8.89949 q^{47} -2.65685 q^{51} -3.75736 q^{53} -3.82843 q^{55} -1.51472 q^{57} -3.41421 q^{59} -5.17157 q^{61} -3.58579 q^{65} -11.0711 q^{67} +0.242641 q^{69} +6.48528 q^{71} -5.17157 q^{73} +0.414214 q^{75} +13.1421 q^{79} +7.48528 q^{81} +8.00000 q^{83} -6.41421 q^{85} +2.75736 q^{87} -16.9706 q^{89} -1.89949 q^{93} -3.65685 q^{95} -15.7279 q^{97} +10.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 2q^{11} - 10q^{13} - 2q^{15} - 10q^{17} + 4q^{19} + 4q^{23} + 2q^{25} - 2q^{27} + 2q^{29} - 12q^{31} - 6q^{33} - 4q^{37} + 14q^{39} - 4q^{41} + 12q^{43} - 2q^{47} + 6q^{51} - 16q^{53} - 2q^{55} - 20q^{57} - 4q^{59} - 16q^{61} - 10q^{65} - 8q^{67} - 8q^{69} - 4q^{71} - 16q^{73} - 2q^{75} - 2q^{79} - 2q^{81} + 16q^{83} - 10q^{85} + 14q^{87} + 16q^{93} + 4q^{95} - 6q^{97} + 16q^{99} + O(q^{100}) \)

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
\(3\) 0.414214 0.239146 0.119573 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −3.82843 −1.15431 −0.577157 0.816633i \(-0.695838\pi\)
−0.577157 + 0.816633i \(0.695838\pi\)
\(12\) 0 0
\(13\) −3.58579 −0.994518 −0.497259 0.867602i \(-0.665660\pi\)
−0.497259 + 0.867602i \(0.665660\pi\)
\(14\) 0 0
\(15\) 0.414214 0.106949
\(16\) 0 0
\(17\) −6.41421 −1.55568 −0.777838 0.628465i \(-0.783683\pi\)
−0.777838 + 0.628465i \(0.783683\pi\)
\(18\) 0 0
\(19\) −3.65685 −0.838940 −0.419470 0.907769i \(-0.637784\pi\)
−0.419470 + 0.907769i \(0.637784\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.585786 0.122145 0.0610725 0.998133i \(-0.480548\pi\)
0.0610725 + 0.998133i \(0.480548\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) 6.65685 1.23615 0.618073 0.786120i \(-0.287913\pi\)
0.618073 + 0.786120i \(0.287913\pi\)
\(30\) 0 0
\(31\) −4.58579 −0.823632 −0.411816 0.911267i \(-0.635105\pi\)
−0.411816 + 0.911267i \(0.635105\pi\)
\(32\) 0 0
\(33\) −1.58579 −0.276050
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.41421 −0.561293 −0.280647 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) 0 0
\(39\) −1.48528 −0.237835
\(40\) 0 0
\(41\) −0.585786 −0.0914845 −0.0457422 0.998953i \(-0.514565\pi\)
−0.0457422 + 0.998953i \(0.514565\pi\)
\(42\) 0 0
\(43\) 11.6569 1.77765 0.888827 0.458243i \(-0.151521\pi\)
0.888827 + 0.458243i \(0.151521\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 8.89949 1.29812 0.649062 0.760735i \(-0.275161\pi\)
0.649062 + 0.760735i \(0.275161\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.65685 −0.372034
\(52\) 0 0
\(53\) −3.75736 −0.516113 −0.258056 0.966130i \(-0.583082\pi\)
−0.258056 + 0.966130i \(0.583082\pi\)
\(54\) 0 0
\(55\) −3.82843 −0.516225
\(56\) 0 0
\(57\) −1.51472 −0.200629
\(58\) 0 0
\(59\) −3.41421 −0.444493 −0.222246 0.974991i \(-0.571339\pi\)
−0.222246 + 0.974991i \(0.571339\pi\)
\(60\) 0 0
\(61\) −5.17157 −0.662152 −0.331076 0.943604i \(-0.607412\pi\)
−0.331076 + 0.943604i \(0.607412\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.58579 −0.444762
\(66\) 0 0
\(67\) −11.0711 −1.35255 −0.676273 0.736651i \(-0.736406\pi\)
−0.676273 + 0.736651i \(0.736406\pi\)
\(68\) 0 0
\(69\) 0.242641 0.0292105
\(70\) 0 0
\(71\) 6.48528 0.769661 0.384831 0.922987i \(-0.374260\pi\)
0.384831 + 0.922987i \(0.374260\pi\)
\(72\) 0 0
\(73\) −5.17157 −0.605287 −0.302643 0.953104i \(-0.597869\pi\)
−0.302643 + 0.953104i \(0.597869\pi\)
\(74\) 0 0
\(75\) 0.414214 0.0478293
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.1421 1.47861 0.739303 0.673373i \(-0.235155\pi\)
0.739303 + 0.673373i \(0.235155\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −6.41421 −0.695719
\(86\) 0 0
\(87\) 2.75736 0.295620
\(88\) 0 0
\(89\) −16.9706 −1.79888 −0.899438 0.437048i \(-0.856024\pi\)
−0.899438 + 0.437048i \(0.856024\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.89949 −0.196968
\(94\) 0 0
\(95\) −3.65685 −0.375185
\(96\) 0 0
\(97\) −15.7279 −1.59693 −0.798464 0.602042i \(-0.794354\pi\)
−0.798464 + 0.602042i \(0.794354\pi\)
\(98\) 0 0
\(99\) 10.8284 1.08830
\(100\) 0 0
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) 0 0
\(103\) −1.58579 −0.156252 −0.0781261 0.996943i \(-0.524894\pi\)
−0.0781261 + 0.996943i \(0.524894\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.8284 −1.62687 −0.813433 0.581659i \(-0.802404\pi\)
−0.813433 + 0.581659i \(0.802404\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) −1.41421 −0.134231
\(112\) 0 0
\(113\) −5.07107 −0.477046 −0.238523 0.971137i \(-0.576663\pi\)
−0.238523 + 0.971137i \(0.576663\pi\)
\(114\) 0 0
\(115\) 0.585786 0.0546249
\(116\) 0 0
\(117\) 10.1421 0.937641
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.65685 0.332441
\(122\) 0 0
\(123\) −0.242641 −0.0218782
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −21.8995 −1.94327 −0.971633 0.236494i \(-0.924002\pi\)
−0.971633 + 0.236494i \(0.924002\pi\)
\(128\) 0 0
\(129\) 4.82843 0.425119
\(130\) 0 0
\(131\) 11.7574 1.02725 0.513623 0.858016i \(-0.328303\pi\)
0.513623 + 0.858016i \(0.328303\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.41421 −0.207782
\(136\) 0 0
\(137\) 12.9706 1.10815 0.554075 0.832467i \(-0.313072\pi\)
0.554075 + 0.832467i \(0.313072\pi\)
\(138\) 0 0
\(139\) 13.8995 1.17894 0.589470 0.807790i \(-0.299337\pi\)
0.589470 + 0.807790i \(0.299337\pi\)
\(140\) 0 0
\(141\) 3.68629 0.310442
\(142\) 0 0
\(143\) 13.7279 1.14799
\(144\) 0 0
\(145\) 6.65685 0.552822
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.51472 0.287937 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(150\) 0 0
\(151\) −11.8284 −0.962584 −0.481292 0.876560i \(-0.659832\pi\)
−0.481292 + 0.876560i \(0.659832\pi\)
\(152\) 0 0
\(153\) 18.1421 1.46670
\(154\) 0 0
\(155\) −4.58579 −0.368339
\(156\) 0 0
\(157\) −10.4853 −0.836817 −0.418408 0.908259i \(-0.637412\pi\)
−0.418408 + 0.908259i \(0.637412\pi\)
\(158\) 0 0
\(159\) −1.55635 −0.123427
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 21.5563 1.68842 0.844212 0.536010i \(-0.180069\pi\)
0.844212 + 0.536010i \(0.180069\pi\)
\(164\) 0 0
\(165\) −1.58579 −0.123453
\(166\) 0 0
\(167\) −2.41421 −0.186817 −0.0934087 0.995628i \(-0.529776\pi\)
−0.0934087 + 0.995628i \(0.529776\pi\)
\(168\) 0 0
\(169\) −0.142136 −0.0109335
\(170\) 0 0
\(171\) 10.3431 0.790960
\(172\) 0 0
\(173\) −18.5563 −1.41081 −0.705407 0.708803i \(-0.749236\pi\)
−0.705407 + 0.708803i \(0.749236\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.41421 −0.106299
\(178\) 0 0
\(179\) 6.48528 0.484733 0.242366 0.970185i \(-0.422076\pi\)
0.242366 + 0.970185i \(0.422076\pi\)
\(180\) 0 0
\(181\) −1.75736 −0.130623 −0.0653117 0.997865i \(-0.520804\pi\)
−0.0653117 + 0.997865i \(0.520804\pi\)
\(182\) 0 0
\(183\) −2.14214 −0.158351
\(184\) 0 0
\(185\) −3.41421 −0.251018
\(186\) 0 0
\(187\) 24.5563 1.79574
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.6569 1.20525 0.602624 0.798025i \(-0.294122\pi\)
0.602624 + 0.798025i \(0.294122\pi\)
\(192\) 0 0
\(193\) 5.65685 0.407189 0.203595 0.979055i \(-0.434738\pi\)
0.203595 + 0.979055i \(0.434738\pi\)
\(194\) 0 0
\(195\) −1.48528 −0.106363
\(196\) 0 0
\(197\) −27.5563 −1.96331 −0.981654 0.190669i \(-0.938934\pi\)
−0.981654 + 0.190669i \(0.938934\pi\)
\(198\) 0 0
\(199\) 26.7279 1.89469 0.947346 0.320212i \(-0.103754\pi\)
0.947346 + 0.320212i \(0.103754\pi\)
\(200\) 0 0
\(201\) −4.58579 −0.323456
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.585786 −0.0409131
\(206\) 0 0
\(207\) −1.65685 −0.115159
\(208\) 0 0
\(209\) 14.0000 0.968400
\(210\) 0 0
\(211\) −24.3137 −1.67382 −0.836912 0.547337i \(-0.815642\pi\)
−0.836912 + 0.547337i \(0.815642\pi\)
\(212\) 0 0
\(213\) 2.68629 0.184062
\(214\) 0 0
\(215\) 11.6569 0.794991
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.14214 −0.144752
\(220\) 0 0
\(221\) 23.0000 1.54715
\(222\) 0 0
\(223\) −24.0711 −1.61192 −0.805959 0.591971i \(-0.798350\pi\)
−0.805959 + 0.591971i \(0.798350\pi\)
\(224\) 0 0
\(225\) −2.82843 −0.188562
\(226\) 0 0
\(227\) −7.72792 −0.512920 −0.256460 0.966555i \(-0.582556\pi\)
−0.256460 + 0.966555i \(0.582556\pi\)
\(228\) 0 0
\(229\) −23.8995 −1.57932 −0.789662 0.613543i \(-0.789744\pi\)
−0.789662 + 0.613543i \(0.789744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.17157 −0.600850 −0.300425 0.953805i \(-0.597128\pi\)
−0.300425 + 0.953805i \(0.597128\pi\)
\(234\) 0 0
\(235\) 8.89949 0.580539
\(236\) 0 0
\(237\) 5.44365 0.353603
\(238\) 0 0
\(239\) 14.1716 0.916683 0.458341 0.888776i \(-0.348444\pi\)
0.458341 + 0.888776i \(0.348444\pi\)
\(240\) 0 0
\(241\) 3.55635 0.229085 0.114542 0.993418i \(-0.463460\pi\)
0.114542 + 0.993418i \(0.463460\pi\)
\(242\) 0 0
\(243\) 10.3431 0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.1127 0.834341
\(248\) 0 0
\(249\) 3.31371 0.209998
\(250\) 0 0
\(251\) 27.0711 1.70871 0.854355 0.519689i \(-0.173952\pi\)
0.854355 + 0.519689i \(0.173952\pi\)
\(252\) 0 0
\(253\) −2.24264 −0.140994
\(254\) 0 0
\(255\) −2.65685 −0.166379
\(256\) 0 0
\(257\) 25.7990 1.60930 0.804648 0.593752i \(-0.202354\pi\)
0.804648 + 0.593752i \(0.202354\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −18.8284 −1.16545
\(262\) 0 0
\(263\) 10.0000 0.616626 0.308313 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) 0 0
\(265\) −3.75736 −0.230813
\(266\) 0 0
\(267\) −7.02944 −0.430195
\(268\) 0 0
\(269\) 16.7279 1.01992 0.509960 0.860198i \(-0.329660\pi\)
0.509960 + 0.860198i \(0.329660\pi\)
\(270\) 0 0
\(271\) −28.9706 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.82843 −0.230863
\(276\) 0 0
\(277\) −7.41421 −0.445477 −0.222738 0.974878i \(-0.571500\pi\)
−0.222738 + 0.974878i \(0.571500\pi\)
\(278\) 0 0
\(279\) 12.9706 0.776527
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 18.7574 1.11501 0.557505 0.830174i \(-0.311759\pi\)
0.557505 + 0.830174i \(0.311759\pi\)
\(284\) 0 0
\(285\) −1.51472 −0.0897242
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 24.1421 1.42013
\(290\) 0 0
\(291\) −6.51472 −0.381900
\(292\) 0 0
\(293\) 14.4142 0.842087 0.421044 0.907040i \(-0.361664\pi\)
0.421044 + 0.907040i \(0.361664\pi\)
\(294\) 0 0
\(295\) −3.41421 −0.198783
\(296\) 0 0
\(297\) 9.24264 0.536312
\(298\) 0 0
\(299\) −2.10051 −0.121475
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −5.17157 −0.296123
\(306\) 0 0
\(307\) −3.58579 −0.204652 −0.102326 0.994751i \(-0.532628\pi\)
−0.102326 + 0.994751i \(0.532628\pi\)
\(308\) 0 0
\(309\) −0.656854 −0.0373671
\(310\) 0 0
\(311\) −6.97056 −0.395264 −0.197632 0.980276i \(-0.563325\pi\)
−0.197632 + 0.980276i \(0.563325\pi\)
\(312\) 0 0
\(313\) −16.5563 −0.935820 −0.467910 0.883776i \(-0.654993\pi\)
−0.467910 + 0.883776i \(0.654993\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.9706 −0.840831 −0.420415 0.907332i \(-0.638116\pi\)
−0.420415 + 0.907332i \(0.638116\pi\)
\(318\) 0 0
\(319\) −25.4853 −1.42690
\(320\) 0 0
\(321\) −6.97056 −0.389059
\(322\) 0 0
\(323\) 23.4558 1.30512
\(324\) 0 0
\(325\) −3.58579 −0.198904
\(326\) 0 0
\(327\) 3.72792 0.206155
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.82843 0.265394 0.132697 0.991157i \(-0.457636\pi\)
0.132697 + 0.991157i \(0.457636\pi\)
\(332\) 0 0
\(333\) 9.65685 0.529192
\(334\) 0 0
\(335\) −11.0711 −0.604877
\(336\) 0 0
\(337\) −18.7279 −1.02017 −0.510087 0.860123i \(-0.670387\pi\)
−0.510087 + 0.860123i \(0.670387\pi\)
\(338\) 0 0
\(339\) −2.10051 −0.114084
\(340\) 0 0
\(341\) 17.5563 0.950730
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0.242641 0.0130633
\(346\) 0 0
\(347\) −7.41421 −0.398016 −0.199008 0.979998i \(-0.563772\pi\)
−0.199008 + 0.979998i \(0.563772\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 8.65685 0.462069
\(352\) 0 0
\(353\) 2.07107 0.110232 0.0551159 0.998480i \(-0.482447\pi\)
0.0551159 + 0.998480i \(0.482447\pi\)
\(354\) 0 0
\(355\) 6.48528 0.344203
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) −5.62742 −0.296180
\(362\) 0 0
\(363\) 1.51472 0.0795021
\(364\) 0 0
\(365\) −5.17157 −0.270692
\(366\) 0 0
\(367\) −19.7279 −1.02979 −0.514895 0.857254i \(-0.672169\pi\)
−0.514895 + 0.857254i \(0.672169\pi\)
\(368\) 0 0
\(369\) 1.65685 0.0862524
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.4853 0.853576 0.426788 0.904352i \(-0.359645\pi\)
0.426788 + 0.904352i \(0.359645\pi\)
\(374\) 0 0
\(375\) 0.414214 0.0213899
\(376\) 0 0
\(377\) −23.8701 −1.22937
\(378\) 0 0
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) −9.07107 −0.464725
\(382\) 0 0
\(383\) 3.51472 0.179594 0.0897969 0.995960i \(-0.471378\pi\)
0.0897969 + 0.995960i \(0.471378\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −32.9706 −1.67599
\(388\) 0 0
\(389\) 13.1421 0.666333 0.333166 0.942868i \(-0.391883\pi\)
0.333166 + 0.942868i \(0.391883\pi\)
\(390\) 0 0
\(391\) −3.75736 −0.190018
\(392\) 0 0
\(393\) 4.87006 0.245662
\(394\) 0 0
\(395\) 13.1421 0.661253
\(396\) 0 0
\(397\) −14.4142 −0.723429 −0.361714 0.932289i \(-0.617808\pi\)
−0.361714 + 0.932289i \(0.617808\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.514719 0.0257038 0.0128519 0.999917i \(-0.495909\pi\)
0.0128519 + 0.999917i \(0.495909\pi\)
\(402\) 0 0
\(403\) 16.4437 0.819117
\(404\) 0 0
\(405\) 7.48528 0.371947
\(406\) 0 0
\(407\) 13.0711 0.647909
\(408\) 0 0
\(409\) −32.8284 −1.62326 −0.811631 0.584171i \(-0.801420\pi\)
−0.811631 + 0.584171i \(0.801420\pi\)
\(410\) 0 0
\(411\) 5.37258 0.265010
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 5.75736 0.281939
\(418\) 0 0
\(419\) 9.55635 0.466858 0.233429 0.972374i \(-0.425005\pi\)
0.233429 + 0.972374i \(0.425005\pi\)
\(420\) 0 0
\(421\) 20.3137 0.990030 0.495015 0.868885i \(-0.335163\pi\)
0.495015 + 0.868885i \(0.335163\pi\)
\(422\) 0 0
\(423\) −25.1716 −1.22388
\(424\) 0 0
\(425\) −6.41421 −0.311135
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5.68629 0.274537
\(430\) 0 0
\(431\) −9.82843 −0.473419 −0.236709 0.971581i \(-0.576069\pi\)
−0.236709 + 0.971581i \(0.576069\pi\)
\(432\) 0 0
\(433\) 22.2843 1.07091 0.535457 0.844563i \(-0.320139\pi\)
0.535457 + 0.844563i \(0.320139\pi\)
\(434\) 0 0
\(435\) 2.75736 0.132205
\(436\) 0 0
\(437\) −2.14214 −0.102472
\(438\) 0 0
\(439\) −24.2426 −1.15704 −0.578519 0.815669i \(-0.696369\pi\)
−0.578519 + 0.815669i \(0.696369\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −35.4558 −1.68456 −0.842279 0.539042i \(-0.818786\pi\)
−0.842279 + 0.539042i \(0.818786\pi\)
\(444\) 0 0
\(445\) −16.9706 −0.804482
\(446\) 0 0
\(447\) 1.45584 0.0688591
\(448\) 0 0
\(449\) 23.8284 1.12453 0.562267 0.826956i \(-0.309930\pi\)
0.562267 + 0.826956i \(0.309930\pi\)
\(450\) 0 0
\(451\) 2.24264 0.105602
\(452\) 0 0
\(453\) −4.89949 −0.230198
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0711 −0.704995 −0.352497 0.935813i \(-0.614667\pi\)
−0.352497 + 0.935813i \(0.614667\pi\)
\(458\) 0 0
\(459\) 15.4853 0.722791
\(460\) 0 0
\(461\) −22.3431 −1.04062 −0.520312 0.853976i \(-0.674184\pi\)
−0.520312 + 0.853976i \(0.674184\pi\)
\(462\) 0 0
\(463\) −13.4558 −0.625346 −0.312673 0.949861i \(-0.601224\pi\)
−0.312673 + 0.949861i \(0.601224\pi\)
\(464\) 0 0
\(465\) −1.89949 −0.0880870
\(466\) 0 0
\(467\) −0.899495 −0.0416237 −0.0208118 0.999783i \(-0.506625\pi\)
−0.0208118 + 0.999783i \(0.506625\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.34315 −0.200122
\(472\) 0 0
\(473\) −44.6274 −2.05197
\(474\) 0 0
\(475\) −3.65685 −0.167788
\(476\) 0 0
\(477\) 10.6274 0.486596
\(478\) 0 0
\(479\) −9.41421 −0.430146 −0.215073 0.976598i \(-0.568999\pi\)
−0.215073 + 0.976598i \(0.568999\pi\)
\(480\) 0 0
\(481\) 12.2426 0.558216
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.7279 −0.714168
\(486\) 0 0
\(487\) −7.41421 −0.335970 −0.167985 0.985790i \(-0.553726\pi\)
−0.167985 + 0.985790i \(0.553726\pi\)
\(488\) 0 0
\(489\) 8.92893 0.403780
\(490\) 0 0
\(491\) 17.2843 0.780028 0.390014 0.920809i \(-0.372470\pi\)
0.390014 + 0.920809i \(0.372470\pi\)
\(492\) 0 0
\(493\) −42.6985 −1.92304
\(494\) 0 0
\(495\) 10.8284 0.486702
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −13.6274 −0.610047 −0.305023 0.952345i \(-0.598664\pi\)
−0.305023 + 0.952345i \(0.598664\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −39.0416 −1.74078 −0.870390 0.492363i \(-0.836133\pi\)
−0.870390 + 0.492363i \(0.836133\pi\)
\(504\) 0 0
\(505\) 4.82843 0.214862
\(506\) 0 0
\(507\) −0.0588745 −0.00261471
\(508\) 0 0
\(509\) 35.0711 1.55450 0.777249 0.629193i \(-0.216615\pi\)
0.777249 + 0.629193i \(0.216615\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.82843 0.389785
\(514\) 0 0
\(515\) −1.58579 −0.0698781
\(516\) 0 0
\(517\) −34.0711 −1.49844
\(518\) 0 0
\(519\) −7.68629 −0.337391
\(520\) 0 0
\(521\) −6.68629 −0.292932 −0.146466 0.989216i \(-0.546790\pi\)
−0.146466 + 0.989216i \(0.546790\pi\)
\(522\) 0 0
\(523\) −1.85786 −0.0812387 −0.0406194 0.999175i \(-0.512933\pi\)
−0.0406194 + 0.999175i \(0.512933\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.4142 1.28130
\(528\) 0 0
\(529\) −22.6569 −0.985081
\(530\) 0 0
\(531\) 9.65685 0.419072
\(532\) 0 0
\(533\) 2.10051 0.0909830
\(534\) 0 0
\(535\) −16.8284 −0.727556
\(536\) 0 0
\(537\) 2.68629 0.115922
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15.9706 0.686628 0.343314 0.939221i \(-0.388450\pi\)
0.343314 + 0.939221i \(0.388450\pi\)
\(542\) 0 0
\(543\) −0.727922 −0.0312381
\(544\) 0 0
\(545\) 9.00000 0.385518
\(546\) 0 0
\(547\) 7.51472 0.321306 0.160653 0.987011i \(-0.448640\pi\)
0.160653 + 0.987011i \(0.448640\pi\)
\(548\) 0 0
\(549\) 14.6274 0.624283
\(550\) 0 0
\(551\) −24.3431 −1.03705
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.41421 −0.0600300
\(556\) 0 0
\(557\) −7.79899 −0.330454 −0.165227 0.986256i \(-0.552836\pi\)
−0.165227 + 0.986256i \(0.552836\pi\)
\(558\) 0 0
\(559\) −41.7990 −1.76791
\(560\) 0 0
\(561\) 10.1716 0.429444
\(562\) 0 0
\(563\) 20.6274 0.869342 0.434671 0.900589i \(-0.356865\pi\)
0.434671 + 0.900589i \(0.356865\pi\)
\(564\) 0 0
\(565\) −5.07107 −0.213341
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.7990 −1.33308 −0.666542 0.745468i \(-0.732226\pi\)
−0.666542 + 0.745468i \(0.732226\pi\)
\(570\) 0 0
\(571\) 4.82843 0.202063 0.101032 0.994883i \(-0.467786\pi\)
0.101032 + 0.994883i \(0.467786\pi\)
\(572\) 0 0
\(573\) 6.89949 0.288231
\(574\) 0 0
\(575\) 0.585786 0.0244290
\(576\) 0 0
\(577\) 14.0711 0.585786 0.292893 0.956145i \(-0.405382\pi\)
0.292893 + 0.956145i \(0.405382\pi\)
\(578\) 0 0
\(579\) 2.34315 0.0973778
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 14.3848 0.595757
\(584\) 0 0
\(585\) 10.1421 0.419326
\(586\) 0 0
\(587\) 30.8284 1.27243 0.636213 0.771514i \(-0.280500\pi\)
0.636213 + 0.771514i \(0.280500\pi\)
\(588\) 0 0
\(589\) 16.7696 0.690977
\(590\) 0 0
\(591\) −11.4142 −0.469518
\(592\) 0 0
\(593\) −17.7279 −0.727999 −0.363999 0.931399i \(-0.618589\pi\)
−0.363999 + 0.931399i \(0.618589\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0711 0.453109
\(598\) 0 0
\(599\) −12.7990 −0.522953 −0.261476 0.965210i \(-0.584209\pi\)
−0.261476 + 0.965210i \(0.584209\pi\)
\(600\) 0 0
\(601\) −39.6569 −1.61764 −0.808818 0.588058i \(-0.799892\pi\)
−0.808818 + 0.588058i \(0.799892\pi\)
\(602\) 0 0
\(603\) 31.3137 1.27519
\(604\) 0 0
\(605\) 3.65685 0.148672
\(606\) 0 0
\(607\) −13.9289 −0.565358 −0.282679 0.959215i \(-0.591223\pi\)
−0.282679 + 0.959215i \(0.591223\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.9117 −1.29101
\(612\) 0 0
\(613\) −41.3137 −1.66864 −0.834322 0.551277i \(-0.814141\pi\)
−0.834322 + 0.551277i \(0.814141\pi\)
\(614\) 0 0
\(615\) −0.242641 −0.00978422
\(616\) 0 0
\(617\) 40.8701 1.64537 0.822683 0.568500i \(-0.192476\pi\)
0.822683 + 0.568500i \(0.192476\pi\)
\(618\) 0 0
\(619\) −10.8701 −0.436905 −0.218452 0.975848i \(-0.570101\pi\)
−0.218452 + 0.975848i \(0.570101\pi\)
\(620\) 0 0
\(621\) −1.41421 −0.0567504
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.79899 0.231589
\(628\) 0 0
\(629\) 21.8995 0.873190
\(630\) 0 0
\(631\) −39.4853 −1.57188 −0.785942 0.618300i \(-0.787822\pi\)
−0.785942 + 0.618300i \(0.787822\pi\)
\(632\) 0 0
\(633\) −10.0711 −0.400289
\(634\) 0 0
\(635\) −21.8995 −0.869055
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −18.3431 −0.725644
\(640\) 0 0
\(641\) 39.3137 1.55280 0.776399 0.630242i \(-0.217044\pi\)
0.776399 + 0.630242i \(0.217044\pi\)
\(642\) 0 0
\(643\) −4.21320 −0.166153 −0.0830763 0.996543i \(-0.526475\pi\)
−0.0830763 + 0.996543i \(0.526475\pi\)
\(644\) 0 0
\(645\) 4.82843 0.190119
\(646\) 0 0
\(647\) −35.1127 −1.38042 −0.690211 0.723608i \(-0.742482\pi\)
−0.690211 + 0.723608i \(0.742482\pi\)
\(648\) 0 0
\(649\) 13.0711 0.513084
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −26.6863 −1.04432 −0.522158 0.852849i \(-0.674873\pi\)
−0.522158 + 0.852849i \(0.674873\pi\)
\(654\) 0 0
\(655\) 11.7574 0.459398
\(656\) 0 0
\(657\) 14.6274 0.570670
\(658\) 0 0
\(659\) −20.6569 −0.804677 −0.402338 0.915491i \(-0.631802\pi\)
−0.402338 + 0.915491i \(0.631802\pi\)
\(660\) 0 0
\(661\) 34.1421 1.32798 0.663988 0.747744i \(-0.268863\pi\)
0.663988 + 0.747744i \(0.268863\pi\)
\(662\) 0 0
\(663\) 9.52691 0.369995
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.89949 0.150989
\(668\) 0 0
\(669\) −9.97056 −0.385484
\(670\) 0 0
\(671\) 19.7990 0.764332
\(672\) 0 0
\(673\) −27.5147 −1.06061 −0.530307 0.847806i \(-0.677923\pi\)
−0.530307 + 0.847806i \(0.677923\pi\)
\(674\) 0 0
\(675\) −2.41421 −0.0929231
\(676\) 0 0
\(677\) 10.2721 0.394788 0.197394 0.980324i \(-0.436752\pi\)
0.197394 + 0.980324i \(0.436752\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.20101 −0.122663
\(682\) 0 0
\(683\) 35.1127 1.34355 0.671775 0.740755i \(-0.265532\pi\)
0.671775 + 0.740755i \(0.265532\pi\)
\(684\) 0 0
\(685\) 12.9706 0.495580
\(686\) 0 0
\(687\) −9.89949 −0.377689
\(688\) 0 0
\(689\) 13.4731 0.513284
\(690\) 0 0
\(691\) −3.51472 −0.133706 −0.0668531 0.997763i \(-0.521296\pi\)
−0.0668531 + 0.997763i \(0.521296\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.8995 0.527238
\(696\) 0 0
\(697\) 3.75736 0.142320
\(698\) 0 0
\(699\) −3.79899 −0.143691
\(700\) 0 0
\(701\) 0.514719 0.0194407 0.00972033 0.999953i \(-0.496906\pi\)
0.00972033 + 0.999953i \(0.496906\pi\)
\(702\) 0 0
\(703\) 12.4853 0.470891
\(704\) 0 0
\(705\) 3.68629 0.138834
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.62742 0.136231 0.0681153 0.997677i \(-0.478301\pi\)
0.0681153 + 0.997677i \(0.478301\pi\)
\(710\) 0 0
\(711\) −37.1716 −1.39404
\(712\) 0 0
\(713\) −2.68629 −0.100602
\(714\) 0 0
\(715\) 13.7279 0.513395
\(716\) 0 0
\(717\) 5.87006 0.219221
\(718\) 0 0
\(719\) −15.7574 −0.587650 −0.293825 0.955859i \(-0.594928\pi\)
−0.293825 + 0.955859i \(0.594928\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.47309 0.0547847
\(724\) 0 0
\(725\) 6.65685 0.247229
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) −74.7696 −2.76545
\(732\) 0 0
\(733\) 30.6985 1.13387 0.566937 0.823761i \(-0.308128\pi\)
0.566937 + 0.823761i \(0.308128\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.3848 1.56126
\(738\) 0 0
\(739\) 20.6569 0.759875 0.379937 0.925012i \(-0.375946\pi\)
0.379937 + 0.925012i \(0.375946\pi\)
\(740\) 0 0
\(741\) 5.43146 0.199530
\(742\) 0 0
\(743\) −8.92893 −0.327571 −0.163785 0.986496i \(-0.552370\pi\)
−0.163785 + 0.986496i \(0.552370\pi\)
\(744\) 0 0
\(745\) 3.51472 0.128769
\(746\) 0 0
\(747\) −22.6274 −0.827894
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.14214 0.0416771 0.0208386 0.999783i \(-0.493366\pi\)
0.0208386 + 0.999783i \(0.493366\pi\)
\(752\) 0 0
\(753\) 11.2132 0.408632
\(754\) 0 0
\(755\) −11.8284 −0.430481
\(756\) 0 0
\(757\) 37.1716 1.35102 0.675512 0.737349i \(-0.263923\pi\)
0.675512 + 0.737349i \(0.263923\pi\)
\(758\) 0 0
\(759\) −0.928932 −0.0337181
\(760\) 0 0
\(761\) 18.8701 0.684039 0.342020 0.939693i \(-0.388889\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 18.1421 0.655930
\(766\) 0 0
\(767\) 12.2426 0.442056
\(768\) 0 0
\(769\) −28.1421 −1.01483 −0.507416 0.861701i \(-0.669399\pi\)
−0.507416 + 0.861701i \(0.669399\pi\)
\(770\) 0 0
\(771\) 10.6863 0.384857
\(772\) 0 0
\(773\) 43.7279 1.57278 0.786392 0.617728i \(-0.211947\pi\)
0.786392 + 0.617728i \(0.211947\pi\)
\(774\) 0 0
\(775\) −4.58579 −0.164726
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.14214 0.0767500
\(780\) 0 0
\(781\) −24.8284 −0.888431
\(782\) 0 0
\(783\) −16.0711 −0.574333
\(784\) 0 0
\(785\) −10.4853 −0.374236
\(786\) 0 0
\(787\) 43.7279 1.55873 0.779366 0.626569i \(-0.215541\pi\)
0.779366 + 0.626569i \(0.215541\pi\)
\(788\) 0 0
\(789\) 4.14214 0.147464
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18.5442 0.658522
\(794\) 0 0
\(795\) −1.55635 −0.0551980
\(796\) 0 0
\(797\) 35.3848 1.25339 0.626697 0.779263i \(-0.284407\pi\)
0.626697 + 0.779263i \(0.284407\pi\)
\(798\) 0 0
\(799\) −57.0833 −2.01946
\(800\) 0 0
\(801\) 48.0000 1.69600
\(802\) 0 0
\(803\) 19.7990 0.698691
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.92893 0.243910
\(808\) 0 0
\(809\) 6.02944 0.211984 0.105992 0.994367i \(-0.466198\pi\)
0.105992 + 0.994367i \(0.466198\pi\)
\(810\) 0 0
\(811\) 19.5563 0.686716 0.343358 0.939205i \(-0.388436\pi\)
0.343358 + 0.939205i \(0.388436\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) 21.5563 0.755086
\(816\) 0 0
\(817\) −42.6274 −1.49134
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 47.8284 1.66922 0.834612 0.550839i \(-0.185692\pi\)
0.834612 + 0.550839i \(0.185692\pi\)
\(822\) 0 0
\(823\) 36.5269 1.27325 0.636624 0.771174i \(-0.280330\pi\)
0.636624 + 0.771174i \(0.280330\pi\)
\(824\) 0 0
\(825\) −1.58579 −0.0552100
\(826\) 0 0
\(827\) −10.0416 −0.349182 −0.174591 0.984641i \(-0.555860\pi\)
−0.174591 + 0.984641i \(0.555860\pi\)
\(828\) 0 0
\(829\) −22.7279 −0.789373 −0.394687 0.918816i \(-0.629147\pi\)
−0.394687 + 0.918816i \(0.629147\pi\)
\(830\) 0 0
\(831\) −3.07107 −0.106534
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −2.41421 −0.0835473
\(836\) 0 0
\(837\) 11.0711 0.382672
\(838\) 0 0
\(839\) −22.3848 −0.772808 −0.386404 0.922330i \(-0.626283\pi\)
−0.386404 + 0.922330i \(0.626283\pi\)
\(840\) 0 0
\(841\) 15.3137 0.528059
\(842\) 0 0
\(843\) −0.414214 −0.0142663
\(844\) 0 0
\(845\) −0.142136 −0.00488961
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.76955 0.266650
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 30.2843 1.03691 0.518457 0.855104i \(-0.326507\pi\)
0.518457 + 0.855104i \(0.326507\pi\)
\(854\) 0 0
\(855\) 10.3431 0.353728
\(856\) 0 0
\(857\) 20.8284 0.711486 0.355743 0.934584i \(-0.384228\pi\)
0.355743 + 0.934584i \(0.384228\pi\)
\(858\) 0 0
\(859\) 45.4558 1.55093 0.775467 0.631388i \(-0.217515\pi\)
0.775467 + 0.631388i \(0.217515\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.7574 −0.672548 −0.336274 0.941764i \(-0.609167\pi\)
−0.336274 + 0.941764i \(0.609167\pi\)
\(864\) 0 0
\(865\) −18.5563 −0.630935
\(866\) 0 0
\(867\) 10.0000 0.339618
\(868\) 0 0
\(869\) −50.3137 −1.70678
\(870\) 0 0
\(871\) 39.6985 1.34513
\(872\) 0 0
\(873\) 44.4853 1.50560
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.6863 0.360850 0.180425 0.983589i \(-0.442253\pi\)
0.180425 + 0.983589i \(0.442253\pi\)
\(878\) 0 0
\(879\) 5.97056 0.201382
\(880\) 0 0
\(881\) 56.2843 1.89627 0.948133 0.317875i \(-0.102969\pi\)
0.948133 + 0.317875i \(0.102969\pi\)
\(882\) 0 0
\(883\) −7.11270 −0.239361 −0.119681 0.992812i \(-0.538187\pi\)
−0.119681 + 0.992812i \(0.538187\pi\)
\(884\) 0 0
\(885\) −1.41421 −0.0475383
\(886\) 0 0
\(887\) −57.5980 −1.93395 −0.966975 0.254870i \(-0.917967\pi\)
−0.966975 + 0.254870i \(0.917967\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −28.6569 −0.960041
\(892\) 0 0
\(893\) −32.5442 −1.08905
\(894\) 0 0
\(895\) 6.48528 0.216779
\(896\) 0 0
\(897\) −0.870058 −0.0290504
\(898\) 0 0
\(899\) −30.5269 −1.01813
\(900\) 0 0
\(901\) 24.1005 0.802904
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.75736 −0.0584166
\(906\) 0 0
\(907\) 22.5269 0.747994 0.373997 0.927430i \(-0.377987\pi\)
0.373997 + 0.927430i \(0.377987\pi\)
\(908\) 0 0
\(909\) −13.6569 −0.452969
\(910\) 0 0
\(911\) −55.5980 −1.84204 −0.921022 0.389511i \(-0.872644\pi\)
−0.921022 + 0.389511i \(0.872644\pi\)
\(912\) 0 0
\(913\) −30.6274 −1.01362
\(914\) 0 0
\(915\) −2.14214 −0.0708168
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −18.5147 −0.610744 −0.305372 0.952233i \(-0.598781\pi\)
−0.305372 + 0.952233i \(0.598781\pi\)
\(920\) 0 0
\(921\) −1.48528 −0.0489417
\(922\) 0 0
\(923\) −23.2548 −0.765442
\(924\) 0 0
\(925\) −3.41421 −0.112259
\(926\) 0 0
\(927\) 4.48528 0.147316
\(928\) 0 0
\(929\) 21.2721 0.697914 0.348957 0.937139i \(-0.386536\pi\)
0.348957 + 0.937139i \(0.386536\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.88730 −0.0945260
\(934\) 0 0
\(935\) 24.5563 0.803078
\(936\) 0 0
\(937\) 7.72792 0.252460 0.126230 0.992001i \(-0.459712\pi\)
0.126230 + 0.992001i \(0.459712\pi\)
\(938\) 0 0
\(939\) −6.85786 −0.223798
\(940\) 0 0
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 0 0
\(943\) −0.343146 −0.0111744
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −32.0416 −1.04121 −0.520607 0.853797i \(-0.674294\pi\)
−0.520607 + 0.853797i \(0.674294\pi\)
\(948\) 0 0
\(949\) 18.5442 0.601969
\(950\) 0 0
\(951\) −6.20101 −0.201082
\(952\) 0 0
\(953\) −38.8701 −1.25912 −0.629562 0.776950i \(-0.716766\pi\)
−0.629562 + 0.776950i \(0.716766\pi\)
\(954\) 0 0
\(955\) 16.6569 0.539003
\(956\) 0 0
\(957\) −10.5563 −0.341238
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.97056 −0.321631
\(962\) 0 0
\(963\) 47.5980 1.53382
\(964\) 0 0
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) −7.45584 −0.239764 −0.119882 0.992788i \(-0.538252\pi\)
−0.119882 + 0.992788i \(0.538252\pi\)
\(968\) 0 0
\(969\) 9.71573 0.312114
\(970\) 0 0
\(971\) 22.5269 0.722923 0.361462 0.932387i \(-0.382278\pi\)
0.361462 + 0.932387i \(0.382278\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.48528 −0.0475671
\(976\) 0 0
\(977\) −8.14214 −0.260490 −0.130245 0.991482i \(-0.541576\pi\)
−0.130245 + 0.991482i \(0.541576\pi\)
\(978\) 0 0
\(979\) 64.9706 2.07647
\(980\) 0 0
\(981\) −25.4558 −0.812743
\(982\) 0 0
\(983\) 42.0122 1.33998 0.669990 0.742370i \(-0.266298\pi\)
0.669990 + 0.742370i \(0.266298\pi\)
\(984\) 0 0
\(985\) −27.5563 −0.878018
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.82843 0.217131
\(990\) 0 0
\(991\) 25.3137 0.804116 0.402058 0.915614i \(-0.368295\pi\)
0.402058 + 0.915614i \(0.368295\pi\)
\(992\) 0 0
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) 26.7279 0.847332
\(996\) 0 0
\(997\) 50.8406 1.61014 0.805069 0.593181i \(-0.202128\pi\)
0.805069 + 0.593181i \(0.202128\pi\)
\(998\) 0 0
\(999\) 8.24264 0.260786
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 980.2.a.j.1.2 2
3.2 odd 2 8820.2.a.bg.1.2 2
4.3 odd 2 3920.2.a.bx.1.1 2
5.2 odd 4 4900.2.e.q.2549.2 4
5.3 odd 4 4900.2.e.q.2549.3 4
5.4 even 2 4900.2.a.z.1.1 2
7.2 even 3 980.2.i.l.361.1 4
7.3 odd 6 980.2.i.k.961.2 4
7.4 even 3 980.2.i.l.961.1 4
7.5 odd 6 980.2.i.k.361.2 4
7.6 odd 2 980.2.a.k.1.1 yes 2
21.20 even 2 8820.2.a.bl.1.2 2
28.27 even 2 3920.2.a.bo.1.2 2
35.13 even 4 4900.2.e.r.2549.2 4
35.27 even 4 4900.2.e.r.2549.3 4
35.34 odd 2 4900.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.j.1.2 2 1.1 even 1 trivial
980.2.a.k.1.1 yes 2 7.6 odd 2
980.2.i.k.361.2 4 7.5 odd 6
980.2.i.k.961.2 4 7.3 odd 6
980.2.i.l.361.1 4 7.2 even 3
980.2.i.l.961.1 4 7.4 even 3
3920.2.a.bo.1.2 2 28.27 even 2
3920.2.a.bx.1.1 2 4.3 odd 2
4900.2.a.x.1.2 2 35.34 odd 2
4900.2.a.z.1.1 2 5.4 even 2
4900.2.e.q.2549.2 4 5.2 odd 4
4900.2.e.q.2549.3 4 5.3 odd 4
4900.2.e.r.2549.2 4 35.13 even 4
4900.2.e.r.2549.3 4 35.27 even 4
8820.2.a.bg.1.2 2 3.2 odd 2
8820.2.a.bl.1.2 2 21.20 even 2