Properties

Label 3920.2.a.br.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
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: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\) \(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +0.171573 q^{11} +4.41421 q^{13} +0.414214 q^{15} +3.24264 q^{17} -6.00000 q^{19} -7.41421 q^{23} +1.00000 q^{25} -2.41421 q^{27} -8.65685 q^{29} -10.2426 q^{31} +0.0710678 q^{33} +2.24264 q^{37} +1.82843 q^{39} -6.24264 q^{41} -2.00000 q^{43} -2.82843 q^{45} +7.24264 q^{47} +1.34315 q^{51} +4.24264 q^{53} +0.171573 q^{55} -2.48528 q^{57} +2.24264 q^{59} -2.82843 q^{61} +4.41421 q^{65} +8.24264 q^{67} -3.07107 q^{69} +3.17157 q^{71} -8.48528 q^{73} +0.414214 q^{75} -1.48528 q^{79} +7.48528 q^{81} +3.24264 q^{85} -3.58579 q^{87} -8.00000 q^{89} -4.24264 q^{93} -6.00000 q^{95} +13.2426 q^{97} -0.485281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} + 6q^{11} + 6q^{13} - 2q^{15} - 2q^{17} - 12q^{19} - 12q^{23} + 2q^{25} - 2q^{27} - 6q^{29} - 12q^{31} - 14q^{33} - 4q^{37} - 2q^{39} - 4q^{41} - 4q^{43} + 6q^{47} + 14q^{51} + 6q^{55} + 12q^{57} - 4q^{59} + 6q^{65} + 8q^{67} + 8q^{69} + 12q^{71} - 2q^{75} + 14q^{79} - 2q^{81} - 2q^{85} - 10q^{87} - 16q^{89} - 12q^{95} + 18q^{97} + 16q^{99} + O(q^{100}) \)

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
\(11\) 0.171573 0.0517312 0.0258656 0.999665i \(-0.491766\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) 4.41421 1.22428 0.612141 0.790748i \(-0.290308\pi\)
0.612141 + 0.790748i \(0.290308\pi\)
\(14\) 0 0
\(15\) 0.414214 0.106949
\(16\) 0 0
\(17\) 3.24264 0.786456 0.393228 0.919441i \(-0.371358\pi\)
0.393228 + 0.919441i \(0.371358\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.41421 −1.54597 −0.772985 0.634424i \(-0.781237\pi\)
−0.772985 + 0.634424i \(0.781237\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0 0
\(31\) −10.2426 −1.83963 −0.919816 0.392349i \(-0.871662\pi\)
−0.919816 + 0.392349i \(0.871662\pi\)
\(32\) 0 0
\(33\) 0.0710678 0.0123713
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.24264 0.368688 0.184344 0.982862i \(-0.440984\pi\)
0.184344 + 0.982862i \(0.440984\pi\)
\(38\) 0 0
\(39\) 1.82843 0.292783
\(40\) 0 0
\(41\) −6.24264 −0.974937 −0.487468 0.873141i \(-0.662080\pi\)
−0.487468 + 0.873141i \(0.662080\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 7.24264 1.05645 0.528224 0.849105i \(-0.322858\pi\)
0.528224 + 0.849105i \(0.322858\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.34315 0.188078
\(52\) 0 0
\(53\) 4.24264 0.582772 0.291386 0.956606i \(-0.405884\pi\)
0.291386 + 0.956606i \(0.405884\pi\)
\(54\) 0 0
\(55\) 0.171573 0.0231349
\(56\) 0 0
\(57\) −2.48528 −0.329184
\(58\) 0 0
\(59\) 2.24264 0.291967 0.145983 0.989287i \(-0.453365\pi\)
0.145983 + 0.989287i \(0.453365\pi\)
\(60\) 0 0
\(61\) −2.82843 −0.362143 −0.181071 0.983470i \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.41421 0.547516
\(66\) 0 0
\(67\) 8.24264 1.00700 0.503499 0.863996i \(-0.332046\pi\)
0.503499 + 0.863996i \(0.332046\pi\)
\(68\) 0 0
\(69\) −3.07107 −0.369713
\(70\) 0 0
\(71\) 3.17157 0.376396 0.188198 0.982131i \(-0.439735\pi\)
0.188198 + 0.982131i \(0.439735\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 0 0
\(75\) 0.414214 0.0478293
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.48528 −0.167107 −0.0835536 0.996503i \(-0.526627\pi\)
−0.0835536 + 0.996503i \(0.526627\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.24264 0.351714
\(86\) 0 0
\(87\) −3.58579 −0.384437
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.24264 −0.439941
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 13.2426 1.34459 0.672293 0.740285i \(-0.265309\pi\)
0.672293 + 0.740285i \(0.265309\pi\)
\(98\) 0 0
\(99\) −0.485281 −0.0487726
\(100\) 0 0
\(101\) 2.48528 0.247295 0.123647 0.992326i \(-0.460541\pi\)
0.123647 + 0.992326i \(0.460541\pi\)
\(102\) 0 0
\(103\) −19.2426 −1.89603 −0.948017 0.318220i \(-0.896915\pi\)
−0.948017 + 0.318220i \(0.896915\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.48528 0.240261 0.120131 0.992758i \(-0.461669\pi\)
0.120131 + 0.992758i \(0.461669\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 0.928932 0.0881703
\(112\) 0 0
\(113\) −13.0711 −1.22962 −0.614811 0.788674i \(-0.710768\pi\)
−0.614811 + 0.788674i \(0.710768\pi\)
\(114\) 0 0
\(115\) −7.41421 −0.691379
\(116\) 0 0
\(117\) −12.4853 −1.15426
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9706 −0.997324
\(122\) 0 0
\(123\) −2.58579 −0.233153
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.24264 −0.731416 −0.365708 0.930730i \(-0.619173\pi\)
−0.365708 + 0.930730i \(0.619173\pi\)
\(128\) 0 0
\(129\) −0.828427 −0.0729389
\(130\) 0 0
\(131\) −12.2426 −1.06964 −0.534822 0.844965i \(-0.679621\pi\)
−0.534822 + 0.844965i \(0.679621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.41421 −0.207782
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −7.75736 −0.657971 −0.328985 0.944335i \(-0.606707\pi\)
−0.328985 + 0.944335i \(0.606707\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 0.757359 0.0633336
\(144\) 0 0
\(145\) −8.65685 −0.718913
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.17157 −0.751365 −0.375682 0.926749i \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 0 0
\(151\) 7.48528 0.609144 0.304572 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(152\) 0 0
\(153\) −9.17157 −0.741478
\(154\) 0 0
\(155\) −10.2426 −0.822709
\(156\) 0 0
\(157\) −15.1716 −1.21082 −0.605412 0.795913i \(-0.706992\pi\)
−0.605412 + 0.795913i \(0.706992\pi\)
\(158\) 0 0
\(159\) 1.75736 0.139368
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.2426 0.802266 0.401133 0.916020i \(-0.368617\pi\)
0.401133 + 0.916020i \(0.368617\pi\)
\(164\) 0 0
\(165\) 0.0710678 0.00553262
\(166\) 0 0
\(167\) −0.757359 −0.0586062 −0.0293031 0.999571i \(-0.509329\pi\)
−0.0293031 + 0.999571i \(0.509329\pi\)
\(168\) 0 0
\(169\) 6.48528 0.498868
\(170\) 0 0
\(171\) 16.9706 1.29777
\(172\) 0 0
\(173\) −7.24264 −0.550648 −0.275324 0.961352i \(-0.588785\pi\)
−0.275324 + 0.961352i \(0.588785\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.928932 0.0698228
\(178\) 0 0
\(179\) 14.4853 1.08268 0.541340 0.840804i \(-0.317917\pi\)
0.541340 + 0.840804i \(0.317917\pi\)
\(180\) 0 0
\(181\) −18.7279 −1.39204 −0.696018 0.718025i \(-0.745047\pi\)
−0.696018 + 0.718025i \(0.745047\pi\)
\(182\) 0 0
\(183\) −1.17157 −0.0866052
\(184\) 0 0
\(185\) 2.24264 0.164882
\(186\) 0 0
\(187\) 0.556349 0.0406843
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.9706 −1.01087 −0.505437 0.862863i \(-0.668669\pi\)
−0.505437 + 0.862863i \(0.668669\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 1.82843 0.130936
\(196\) 0 0
\(197\) 13.4142 0.955723 0.477862 0.878435i \(-0.341412\pi\)
0.477862 + 0.878435i \(0.341412\pi\)
\(198\) 0 0
\(199\) 7.41421 0.525580 0.262790 0.964853i \(-0.415357\pi\)
0.262790 + 0.964853i \(0.415357\pi\)
\(200\) 0 0
\(201\) 3.41421 0.240820
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.24264 −0.436005
\(206\) 0 0
\(207\) 20.9706 1.45755
\(208\) 0 0
\(209\) −1.02944 −0.0712077
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 0 0
\(213\) 1.31371 0.0900138
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.51472 −0.237503
\(220\) 0 0
\(221\) 14.3137 0.962844
\(222\) 0 0
\(223\) 24.2132 1.62144 0.810718 0.585437i \(-0.199077\pi\)
0.810718 + 0.585437i \(0.199077\pi\)
\(224\) 0 0
\(225\) −2.82843 −0.188562
\(226\) 0 0
\(227\) −15.7279 −1.04390 −0.521949 0.852976i \(-0.674795\pi\)
−0.521949 + 0.852976i \(0.674795\pi\)
\(228\) 0 0
\(229\) 18.0416 1.19222 0.596112 0.802901i \(-0.296711\pi\)
0.596112 + 0.802901i \(0.296711\pi\)
\(230\) 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
\(235\) 7.24264 0.472458
\(236\) 0 0
\(237\) −0.615224 −0.0399631
\(238\) 0 0
\(239\) 17.4853 1.13103 0.565514 0.824738i \(-0.308678\pi\)
0.565514 + 0.824738i \(0.308678\pi\)
\(240\) 0 0
\(241\) −0.727922 −0.0468896 −0.0234448 0.999725i \(-0.507463\pi\)
−0.0234448 + 0.999725i \(0.507463\pi\)
\(242\) 0 0
\(243\) 10.3431 0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −26.4853 −1.68522
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −17.2132 −1.08649 −0.543244 0.839575i \(-0.682804\pi\)
−0.543244 + 0.839575i \(0.682804\pi\)
\(252\) 0 0
\(253\) −1.27208 −0.0799749
\(254\) 0 0
\(255\) 1.34315 0.0841110
\(256\) 0 0
\(257\) −9.51472 −0.593512 −0.296756 0.954953i \(-0.595905\pi\)
−0.296756 + 0.954953i \(0.595905\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 24.4853 1.51560
\(262\) 0 0
\(263\) 16.6274 1.02529 0.512645 0.858601i \(-0.328666\pi\)
0.512645 + 0.858601i \(0.328666\pi\)
\(264\) 0 0
\(265\) 4.24264 0.260623
\(266\) 0 0
\(267\) −3.31371 −0.202796
\(268\) 0 0
\(269\) −16.2426 −0.990331 −0.495166 0.868799i \(-0.664893\pi\)
−0.495166 + 0.868799i \(0.664893\pi\)
\(270\) 0 0
\(271\) 0.686292 0.0416892 0.0208446 0.999783i \(-0.493364\pi\)
0.0208446 + 0.999783i \(0.493364\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.171573 0.0103462
\(276\) 0 0
\(277\) 15.2132 0.914073 0.457036 0.889448i \(-0.348911\pi\)
0.457036 + 0.889448i \(0.348911\pi\)
\(278\) 0 0
\(279\) 28.9706 1.73442
\(280\) 0 0
\(281\) 2.31371 0.138024 0.0690121 0.997616i \(-0.478015\pi\)
0.0690121 + 0.997616i \(0.478015\pi\)
\(282\) 0 0
\(283\) −24.5563 −1.45972 −0.729862 0.683595i \(-0.760416\pi\)
−0.729862 + 0.683595i \(0.760416\pi\)
\(284\) 0 0
\(285\) −2.48528 −0.147215
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.48528 −0.381487
\(290\) 0 0
\(291\) 5.48528 0.321553
\(292\) 0 0
\(293\) 25.7279 1.50304 0.751521 0.659710i \(-0.229321\pi\)
0.751521 + 0.659710i \(0.229321\pi\)
\(294\) 0 0
\(295\) 2.24264 0.130572
\(296\) 0 0
\(297\) −0.414214 −0.0240351
\(298\) 0 0
\(299\) −32.7279 −1.89270
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.02944 0.0591396
\(304\) 0 0
\(305\) −2.82843 −0.161955
\(306\) 0 0
\(307\) −30.8995 −1.76353 −0.881764 0.471691i \(-0.843644\pi\)
−0.881764 + 0.471691i \(0.843644\pi\)
\(308\) 0 0
\(309\) −7.97056 −0.453429
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) −18.2132 −1.02947 −0.514736 0.857349i \(-0.672110\pi\)
−0.514736 + 0.857349i \(0.672110\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.343146 −0.0192730 −0.00963649 0.999954i \(-0.503067\pi\)
−0.00963649 + 0.999954i \(0.503067\pi\)
\(318\) 0 0
\(319\) −1.48528 −0.0831598
\(320\) 0 0
\(321\) 1.02944 0.0574576
\(322\) 0 0
\(323\) −19.4558 −1.08255
\(324\) 0 0
\(325\) 4.41421 0.244857
\(326\) 0 0
\(327\) 2.07107 0.114530
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 27.4558 1.50911 0.754555 0.656237i \(-0.227853\pi\)
0.754555 + 0.656237i \(0.227853\pi\)
\(332\) 0 0
\(333\) −6.34315 −0.347602
\(334\) 0 0
\(335\) 8.24264 0.450344
\(336\) 0 0
\(337\) 22.2426 1.21163 0.605817 0.795604i \(-0.292846\pi\)
0.605817 + 0.795604i \(0.292846\pi\)
\(338\) 0 0
\(339\) −5.41421 −0.294060
\(340\) 0 0
\(341\) −1.75736 −0.0951663
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.07107 −0.165341
\(346\) 0 0
\(347\) −13.0711 −0.701692 −0.350846 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(348\) 0 0
\(349\) −10.9706 −0.587241 −0.293620 0.955922i \(-0.594860\pi\)
−0.293620 + 0.955922i \(0.594860\pi\)
\(350\) 0 0
\(351\) −10.6569 −0.568821
\(352\) 0 0
\(353\) −6.21320 −0.330695 −0.165348 0.986235i \(-0.552875\pi\)
−0.165348 + 0.986235i \(0.552875\pi\)
\(354\) 0 0
\(355\) 3.17157 0.168330
\(356\) 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
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −4.54416 −0.238506
\(364\) 0 0
\(365\) −8.48528 −0.444140
\(366\) 0 0
\(367\) 23.8701 1.24601 0.623003 0.782219i \(-0.285912\pi\)
0.623003 + 0.782219i \(0.285912\pi\)
\(368\) 0 0
\(369\) 17.6569 0.919179
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −16.4853 −0.853576 −0.426788 0.904352i \(-0.640355\pi\)
−0.426788 + 0.904352i \(0.640355\pi\)
\(374\) 0 0
\(375\) 0.414214 0.0213899
\(376\) 0 0
\(377\) −38.2132 −1.96808
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) −3.41421 −0.174915
\(382\) 0 0
\(383\) −4.48528 −0.229187 −0.114594 0.993412i \(-0.536557\pi\)
−0.114594 + 0.993412i \(0.536557\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.65685 0.287554
\(388\) 0 0
\(389\) −6.85786 −0.347708 −0.173854 0.984771i \(-0.555622\pi\)
−0.173854 + 0.984771i \(0.555622\pi\)
\(390\) 0 0
\(391\) −24.0416 −1.21584
\(392\) 0 0
\(393\) −5.07107 −0.255802
\(394\) 0 0
\(395\) −1.48528 −0.0747326
\(396\) 0 0
\(397\) 1.58579 0.0795883 0.0397942 0.999208i \(-0.487330\pi\)
0.0397942 + 0.999208i \(0.487330\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.8284 0.590683 0.295342 0.955392i \(-0.404566\pi\)
0.295342 + 0.955392i \(0.404566\pi\)
\(402\) 0 0
\(403\) −45.2132 −2.25223
\(404\) 0 0
\(405\) 7.48528 0.371947
\(406\) 0 0
\(407\) 0.384776 0.0190727
\(408\) 0 0
\(409\) 2.48528 0.122889 0.0614446 0.998110i \(-0.480429\pi\)
0.0614446 + 0.998110i \(0.480429\pi\)
\(410\) 0 0
\(411\) 4.97056 0.245180
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.21320 −0.157351
\(418\) 0 0
\(419\) −18.7279 −0.914919 −0.457459 0.889230i \(-0.651241\pi\)
−0.457459 + 0.889230i \(0.651241\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) −20.4853 −0.996028
\(424\) 0 0
\(425\) 3.24264 0.157291
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.313708 0.0151460
\(430\) 0 0
\(431\) 28.7990 1.38720 0.693599 0.720361i \(-0.256024\pi\)
0.693599 + 0.720361i \(0.256024\pi\)
\(432\) 0 0
\(433\) 10.9706 0.527212 0.263606 0.964630i \(-0.415088\pi\)
0.263606 + 0.964630i \(0.415088\pi\)
\(434\) 0 0
\(435\) −3.58579 −0.171925
\(436\) 0 0
\(437\) 44.4853 2.12802
\(438\) 0 0
\(439\) 30.3848 1.45019 0.725093 0.688651i \(-0.241797\pi\)
0.725093 + 0.688651i \(0.241797\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.1716 −0.720823 −0.360412 0.932793i \(-0.617364\pi\)
−0.360412 + 0.932793i \(0.617364\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 0 0
\(447\) −3.79899 −0.179686
\(448\) 0 0
\(449\) −24.1716 −1.14073 −0.570364 0.821392i \(-0.693198\pi\)
−0.570364 + 0.821392i \(0.693198\pi\)
\(450\) 0 0
\(451\) −1.07107 −0.0504346
\(452\) 0 0
\(453\) 3.10051 0.145674
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.7574 −0.549986 −0.274993 0.961446i \(-0.588676\pi\)
−0.274993 + 0.961446i \(0.588676\pi\)
\(458\) 0 0
\(459\) −7.82843 −0.365400
\(460\) 0 0
\(461\) 3.02944 0.141095 0.0705475 0.997508i \(-0.477525\pi\)
0.0705475 + 0.997508i \(0.477525\pi\)
\(462\) 0 0
\(463\) 21.4558 0.997138 0.498569 0.866850i \(-0.333859\pi\)
0.498569 + 0.866850i \(0.333859\pi\)
\(464\) 0 0
\(465\) −4.24264 −0.196748
\(466\) 0 0
\(467\) 5.72792 0.265057 0.132528 0.991179i \(-0.457690\pi\)
0.132528 + 0.991179i \(0.457690\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.28427 −0.289564
\(472\) 0 0
\(473\) −0.343146 −0.0157779
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) −11.7574 −0.537207 −0.268604 0.963251i \(-0.586562\pi\)
−0.268604 + 0.963251i \(0.586562\pi\)
\(480\) 0 0
\(481\) 9.89949 0.451378
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.2426 0.601317
\(486\) 0 0
\(487\) −27.6985 −1.25514 −0.627569 0.778561i \(-0.715950\pi\)
−0.627569 + 0.778561i \(0.715950\pi\)
\(488\) 0 0
\(489\) 4.24264 0.191859
\(490\) 0 0
\(491\) 37.2843 1.68262 0.841308 0.540556i \(-0.181786\pi\)
0.841308 + 0.540556i \(0.181786\pi\)
\(492\) 0 0
\(493\) −28.0711 −1.26426
\(494\) 0 0
\(495\) −0.485281 −0.0218118
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3.00000 −0.134298 −0.0671492 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(500\) 0 0
\(501\) −0.313708 −0.0140155
\(502\) 0 0
\(503\) 41.2426 1.83892 0.919459 0.393185i \(-0.128627\pi\)
0.919459 + 0.393185i \(0.128627\pi\)
\(504\) 0 0
\(505\) 2.48528 0.110594
\(506\) 0 0
\(507\) 2.68629 0.119302
\(508\) 0 0
\(509\) −25.2132 −1.11756 −0.558778 0.829317i \(-0.688730\pi\)
−0.558778 + 0.829317i \(0.688730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 14.4853 0.639541
\(514\) 0 0
\(515\) −19.2426 −0.847932
\(516\) 0 0
\(517\) 1.24264 0.0546513
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) 14.9706 0.655872 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(522\) 0 0
\(523\) −32.4853 −1.42048 −0.710241 0.703959i \(-0.751414\pi\)
−0.710241 + 0.703959i \(0.751414\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −33.2132 −1.44679
\(528\) 0 0
\(529\) 31.9706 1.39002
\(530\) 0 0
\(531\) −6.34315 −0.275269
\(532\) 0 0
\(533\) −27.5563 −1.19360
\(534\) 0 0
\(535\) 2.48528 0.107448
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −21.9706 −0.944588 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(542\) 0 0
\(543\) −7.75736 −0.332900
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) 0 0
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 51.9411 2.21277
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.928932 0.0394310
\(556\) 0 0
\(557\) −7.79899 −0.330454 −0.165227 0.986256i \(-0.552836\pi\)
−0.165227 + 0.986256i \(0.552836\pi\)
\(558\) 0 0
\(559\) −8.82843 −0.373403
\(560\) 0 0
\(561\) 0.230447 0.00972950
\(562\) 0 0
\(563\) 31.9411 1.34616 0.673079 0.739571i \(-0.264971\pi\)
0.673079 + 0.739571i \(0.264971\pi\)
\(564\) 0 0
\(565\) −13.0711 −0.549904
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.1421 1.09594 0.547968 0.836500i \(-0.315402\pi\)
0.547968 + 0.836500i \(0.315402\pi\)
\(570\) 0 0
\(571\) 17.5147 0.732968 0.366484 0.930424i \(-0.380561\pi\)
0.366484 + 0.930424i \(0.380561\pi\)
\(572\) 0 0
\(573\) −5.78680 −0.241747
\(574\) 0 0
\(575\) −7.41421 −0.309194
\(576\) 0 0
\(577\) 15.7279 0.654762 0.327381 0.944892i \(-0.393834\pi\)
0.327381 + 0.944892i \(0.393834\pi\)
\(578\) 0 0
\(579\) −6.62742 −0.275426
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.727922 0.0301475
\(584\) 0 0
\(585\) −12.4853 −0.516203
\(586\) 0 0
\(587\) 37.4558 1.54597 0.772984 0.634425i \(-0.218763\pi\)
0.772984 + 0.634425i \(0.218763\pi\)
\(588\) 0 0
\(589\) 61.4558 2.53224
\(590\) 0 0
\(591\) 5.55635 0.228558
\(592\) 0 0
\(593\) 19.2426 0.790201 0.395100 0.918638i \(-0.370710\pi\)
0.395100 + 0.918638i \(0.370710\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.07107 0.125690
\(598\) 0 0
\(599\) 17.8284 0.728450 0.364225 0.931311i \(-0.381334\pi\)
0.364225 + 0.931311i \(0.381334\pi\)
\(600\) 0 0
\(601\) −10.9706 −0.447499 −0.223749 0.974647i \(-0.571830\pi\)
−0.223749 + 0.974647i \(0.571830\pi\)
\(602\) 0 0
\(603\) −23.3137 −0.949408
\(604\) 0 0
\(605\) −10.9706 −0.446017
\(606\) 0 0
\(607\) 5.10051 0.207023 0.103512 0.994628i \(-0.466992\pi\)
0.103512 + 0.994628i \(0.466992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 31.9706 1.29339
\(612\) 0 0
\(613\) −23.9411 −0.966973 −0.483486 0.875352i \(-0.660630\pi\)
−0.483486 + 0.875352i \(0.660630\pi\)
\(614\) 0 0
\(615\) −2.58579 −0.104269
\(616\) 0 0
\(617\) 4.58579 0.184617 0.0923084 0.995730i \(-0.470575\pi\)
0.0923084 + 0.995730i \(0.470575\pi\)
\(618\) 0 0
\(619\) −16.9289 −0.680431 −0.340216 0.940347i \(-0.610500\pi\)
−0.340216 + 0.940347i \(0.610500\pi\)
\(620\) 0 0
\(621\) 17.8995 0.718282
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.426407 −0.0170291
\(628\) 0 0
\(629\) 7.27208 0.289957
\(630\) 0 0
\(631\) 42.4558 1.69014 0.845070 0.534655i \(-0.179559\pi\)
0.845070 + 0.534655i \(0.179559\pi\)
\(632\) 0 0
\(633\) −3.72792 −0.148172
\(634\) 0 0
\(635\) −8.24264 −0.327099
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.97056 −0.354870
\(640\) 0 0
\(641\) −23.3137 −0.920836 −0.460418 0.887702i \(-0.652300\pi\)
−0.460418 + 0.887702i \(0.652300\pi\)
\(642\) 0 0
\(643\) −2.27208 −0.0896020 −0.0448010 0.998996i \(-0.514265\pi\)
−0.0448010 + 0.998996i \(0.514265\pi\)
\(644\) 0 0
\(645\) −0.828427 −0.0326193
\(646\) 0 0
\(647\) 11.5147 0.452690 0.226345 0.974047i \(-0.427322\pi\)
0.226345 + 0.974047i \(0.427322\pi\)
\(648\) 0 0
\(649\) 0.384776 0.0151038
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.9706 1.36850 0.684252 0.729246i \(-0.260129\pi\)
0.684252 + 0.729246i \(0.260129\pi\)
\(654\) 0 0
\(655\) −12.2426 −0.478360
\(656\) 0 0
\(657\) 24.0000 0.936329
\(658\) 0 0
\(659\) −19.9706 −0.777943 −0.388971 0.921250i \(-0.627169\pi\)
−0.388971 + 0.921250i \(0.627169\pi\)
\(660\) 0 0
\(661\) 13.4558 0.523372 0.261686 0.965153i \(-0.415722\pi\)
0.261686 + 0.965153i \(0.415722\pi\)
\(662\) 0 0
\(663\) 5.92893 0.230261
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 64.1838 2.48521
\(668\) 0 0
\(669\) 10.0294 0.387760
\(670\) 0 0
\(671\) −0.485281 −0.0187341
\(672\) 0 0
\(673\) 3.51472 0.135482 0.0677412 0.997703i \(-0.478421\pi\)
0.0677412 + 0.997703i \(0.478421\pi\)
\(674\) 0 0
\(675\) −2.41421 −0.0929231
\(676\) 0 0
\(677\) 44.2132 1.69925 0.849626 0.527386i \(-0.176828\pi\)
0.849626 + 0.527386i \(0.176828\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.51472 −0.249645
\(682\) 0 0
\(683\) −31.7990 −1.21675 −0.608377 0.793648i \(-0.708179\pi\)
−0.608377 + 0.793648i \(0.708179\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 7.47309 0.285116
\(688\) 0 0
\(689\) 18.7279 0.713477
\(690\) 0 0
\(691\) −14.8284 −0.564100 −0.282050 0.959400i \(-0.591014\pi\)
−0.282050 + 0.959400i \(0.591014\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.75736 −0.294253
\(696\) 0 0
\(697\) −20.2426 −0.766745
\(698\) 0 0
\(699\) −3.79899 −0.143691
\(700\) 0 0
\(701\) 46.4558 1.75461 0.877307 0.479931i \(-0.159338\pi\)
0.877307 + 0.479931i \(0.159338\pi\)
\(702\) 0 0
\(703\) −13.4558 −0.507497
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 4.20101 0.157550
\(712\) 0 0
\(713\) 75.9411 2.84402
\(714\) 0 0
\(715\) 0.757359 0.0283236
\(716\) 0 0
\(717\) 7.24264 0.270481
\(718\) 0 0
\(719\) 25.2132 0.940294 0.470147 0.882588i \(-0.344201\pi\)
0.470147 + 0.882588i \(0.344201\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −0.301515 −0.0112135
\(724\) 0 0
\(725\) −8.65685 −0.321507
\(726\) 0 0
\(727\) 6.68629 0.247981 0.123990 0.992283i \(-0.460431\pi\)
0.123990 + 0.992283i \(0.460431\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) −6.48528 −0.239867
\(732\) 0 0
\(733\) 14.6985 0.542901 0.271450 0.962452i \(-0.412497\pi\)
0.271450 + 0.962452i \(0.412497\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41421 0.0520932
\(738\) 0 0
\(739\) −29.9706 −1.10248 −0.551242 0.834345i \(-0.685846\pi\)
−0.551242 + 0.834345i \(0.685846\pi\)
\(740\) 0 0
\(741\) −10.9706 −0.403014
\(742\) 0 0
\(743\) −11.2721 −0.413532 −0.206766 0.978390i \(-0.566294\pi\)
−0.206766 + 0.978390i \(0.566294\pi\)
\(744\) 0 0
\(745\) −9.17157 −0.336020
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29.4853 −1.07593 −0.537967 0.842966i \(-0.680807\pi\)
−0.537967 + 0.842966i \(0.680807\pi\)
\(752\) 0 0
\(753\) −7.12994 −0.259830
\(754\) 0 0
\(755\) 7.48528 0.272417
\(756\) 0 0
\(757\) 0.485281 0.0176379 0.00881893 0.999961i \(-0.497193\pi\)
0.00881893 + 0.999961i \(0.497193\pi\)
\(758\) 0 0
\(759\) −0.526912 −0.0191257
\(760\) 0 0
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −9.17157 −0.331599
\(766\) 0 0
\(767\) 9.89949 0.357450
\(768\) 0 0
\(769\) −8.82843 −0.318361 −0.159181 0.987249i \(-0.550885\pi\)
−0.159181 + 0.987249i \(0.550885\pi\)
\(770\) 0 0
\(771\) −3.94113 −0.141936
\(772\) 0 0
\(773\) −6.21320 −0.223473 −0.111737 0.993738i \(-0.535641\pi\)
−0.111737 + 0.993738i \(0.535641\pi\)
\(774\) 0 0
\(775\) −10.2426 −0.367927
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 37.4558 1.34199
\(780\) 0 0
\(781\) 0.544156 0.0194714
\(782\) 0 0
\(783\) 20.8995 0.746887
\(784\) 0 0
\(785\) −15.1716 −0.541497
\(786\) 0 0
\(787\) −20.2721 −0.722622 −0.361311 0.932445i \(-0.617671\pi\)
−0.361311 + 0.932445i \(0.617671\pi\)
\(788\) 0 0
\(789\) 6.88730 0.245194
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.4853 −0.443365
\(794\) 0 0
\(795\) 1.75736 0.0623271
\(796\) 0 0
\(797\) −21.1838 −0.750367 −0.375184 0.926950i \(-0.622420\pi\)
−0.375184 + 0.926950i \(0.622420\pi\)
\(798\) 0 0
\(799\) 23.4853 0.830850
\(800\) 0 0
\(801\) 22.6274 0.799500
\(802\) 0 0
\(803\) −1.45584 −0.0513756
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.72792 −0.236834
\(808\) 0 0
\(809\) −48.5980 −1.70861 −0.854307 0.519769i \(-0.826018\pi\)
−0.854307 + 0.519769i \(0.826018\pi\)
\(810\) 0 0
\(811\) −47.3553 −1.66287 −0.831435 0.555621i \(-0.812480\pi\)
−0.831435 + 0.555621i \(0.812480\pi\)
\(812\) 0 0
\(813\) 0.284271 0.00996983
\(814\) 0 0
\(815\) 10.2426 0.358784
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.4853 −0.819642 −0.409821 0.912166i \(-0.634409\pi\)
−0.409821 + 0.912166i \(0.634409\pi\)
\(822\) 0 0
\(823\) −16.7279 −0.583099 −0.291549 0.956556i \(-0.594171\pi\)
−0.291549 + 0.956556i \(0.594171\pi\)
\(824\) 0 0
\(825\) 0.0710678 0.00247426
\(826\) 0 0
\(827\) −42.0416 −1.46193 −0.730965 0.682415i \(-0.760930\pi\)
−0.730965 + 0.682415i \(0.760930\pi\)
\(828\) 0 0
\(829\) 5.95837 0.206943 0.103471 0.994632i \(-0.467005\pi\)
0.103471 + 0.994632i \(0.467005\pi\)
\(830\) 0 0
\(831\) 6.30152 0.218597
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.757359 −0.0262095
\(836\) 0 0
\(837\) 24.7279 0.854722
\(838\) 0 0
\(839\) 23.2721 0.803441 0.401721 0.915762i \(-0.368412\pi\)
0.401721 + 0.915762i \(0.368412\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) 0 0
\(843\) 0.958369 0.0330080
\(844\) 0 0
\(845\) 6.48528 0.223100
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −10.1716 −0.349087
\(850\) 0 0
\(851\) −16.6274 −0.569981
\(852\) 0 0
\(853\) 10.9706 0.375625 0.187812 0.982205i \(-0.439860\pi\)
0.187812 + 0.982205i \(0.439860\pi\)
\(854\) 0 0
\(855\) 16.9706 0.580381
\(856\) 0 0
\(857\) −22.4853 −0.768083 −0.384041 0.923316i \(-0.625468\pi\)
−0.384041 + 0.923316i \(0.625468\pi\)
\(858\) 0 0
\(859\) 24.7696 0.845126 0.422563 0.906334i \(-0.361130\pi\)
0.422563 + 0.906334i \(0.361130\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −18.3848 −0.625825 −0.312913 0.949782i \(-0.601305\pi\)
−0.312913 + 0.949782i \(0.601305\pi\)
\(864\) 0 0
\(865\) −7.24264 −0.246257
\(866\) 0 0
\(867\) −2.68629 −0.0912312
\(868\) 0 0
\(869\) −0.254834 −0.00864465
\(870\) 0 0
\(871\) 36.3848 1.23285
\(872\) 0 0
\(873\) −37.4558 −1.26769
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0294 0.439973 0.219986 0.975503i \(-0.429399\pi\)
0.219986 + 0.975503i \(0.429399\pi\)
\(878\) 0 0
\(879\) 10.6569 0.359447
\(880\) 0 0
\(881\) −52.9706 −1.78462 −0.892312 0.451420i \(-0.850918\pi\)
−0.892312 + 0.451420i \(0.850918\pi\)
\(882\) 0 0
\(883\) −31.5147 −1.06055 −0.530277 0.847824i \(-0.677912\pi\)
−0.530277 + 0.847824i \(0.677912\pi\)
\(884\) 0 0
\(885\) 0.928932 0.0312257
\(886\) 0 0
\(887\) −2.97056 −0.0997417 −0.0498709 0.998756i \(-0.515881\pi\)
−0.0498709 + 0.998756i \(0.515881\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.28427 0.0430247
\(892\) 0 0
\(893\) −43.4558 −1.45419
\(894\) 0 0
\(895\) 14.4853 0.484190
\(896\) 0 0
\(897\) −13.5563 −0.452633
\(898\) 0 0
\(899\) 88.6690 2.95728
\(900\) 0 0
\(901\) 13.7574 0.458324
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.7279 −0.622537
\(906\) 0 0
\(907\) 44.1838 1.46710 0.733549 0.679637i \(-0.237863\pi\)
0.733549 + 0.679637i \(0.237863\pi\)
\(908\) 0 0
\(909\) −7.02944 −0.233152
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −1.17157 −0.0387310
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −12.4558 −0.410880 −0.205440 0.978670i \(-0.565863\pi\)
−0.205440 + 0.978670i \(0.565863\pi\)
\(920\) 0 0
\(921\) −12.7990 −0.421741
\(922\) 0 0
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) 2.24264 0.0737376
\(926\) 0 0
\(927\) 54.4264 1.78760
\(928\) 0 0
\(929\) 29.2721 0.960386 0.480193 0.877163i \(-0.340567\pi\)
0.480193 + 0.877163i \(0.340567\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 4.14214 0.135607
\(934\) 0 0
\(935\) 0.556349 0.0181946
\(936\) 0 0
\(937\) −48.5563 −1.58627 −0.793133 0.609048i \(-0.791552\pi\)
−0.793133 + 0.609048i \(0.791552\pi\)
\(938\) 0 0
\(939\) −7.54416 −0.246194
\(940\) 0 0
\(941\) 28.9706 0.944413 0.472207 0.881488i \(-0.343458\pi\)
0.472207 + 0.881488i \(0.343458\pi\)
\(942\) 0 0
\(943\) 46.2843 1.50722
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 52.2426 1.69766 0.848829 0.528668i \(-0.177308\pi\)
0.848829 + 0.528668i \(0.177308\pi\)
\(948\) 0 0
\(949\) −37.4558 −1.21587
\(950\) 0 0
\(951\) −0.142136 −0.00460906
\(952\) 0 0
\(953\) 53.0122 1.71723 0.858617 0.512618i \(-0.171324\pi\)
0.858617 + 0.512618i \(0.171324\pi\)
\(954\) 0 0
\(955\) −13.9706 −0.452077
\(956\) 0 0
\(957\) −0.615224 −0.0198874
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 73.9117 2.38425
\(962\) 0 0
\(963\) −7.02944 −0.226520
\(964\) 0 0
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) 60.4264 1.94318 0.971591 0.236666i \(-0.0760546\pi\)
0.971591 + 0.236666i \(0.0760546\pi\)
\(968\) 0 0
\(969\) −8.05887 −0.258888
\(970\) 0 0
\(971\) 17.2721 0.554287 0.277144 0.960828i \(-0.410612\pi\)
0.277144 + 0.960828i \(0.410612\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.82843 0.0585565
\(976\) 0 0
\(977\) 42.7696 1.36832 0.684160 0.729332i \(-0.260169\pi\)
0.684160 + 0.729332i \(0.260169\pi\)
\(978\) 0 0
\(979\) −1.37258 −0.0438679
\(980\) 0 0
\(981\) −14.1421 −0.451524
\(982\) 0 0
\(983\) −0.213203 −0.00680013 −0.00340007 0.999994i \(-0.501082\pi\)
−0.00340007 + 0.999994i \(0.501082\pi\)
\(984\) 0 0
\(985\) 13.4142 0.427412
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.8284 0.471517
\(990\) 0 0
\(991\) −19.9411 −0.633451 −0.316725 0.948517i \(-0.602583\pi\)
−0.316725 + 0.948517i \(0.602583\pi\)
\(992\) 0 0
\(993\) 11.3726 0.360898
\(994\) 0 0
\(995\) 7.41421 0.235046
\(996\) 0 0
\(997\) −21.7279 −0.688130 −0.344065 0.938946i \(-0.611804\pi\)
−0.344065 + 0.938946i \(0.611804\pi\)
\(998\) 0 0
\(999\) −5.41421 −0.171298
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 3920.2.a.br.1.2 2
4.3 odd 2 245.2.a.f.1.1 yes 2
7.6 odd 2 3920.2.a.bw.1.1 2
12.11 even 2 2205.2.a.t.1.2 2
20.3 even 4 1225.2.b.j.99.4 4
20.7 even 4 1225.2.b.j.99.1 4
20.19 odd 2 1225.2.a.p.1.2 2
28.3 even 6 245.2.e.g.226.2 4
28.11 odd 6 245.2.e.f.226.2 4
28.19 even 6 245.2.e.g.116.2 4
28.23 odd 6 245.2.e.f.116.2 4
28.27 even 2 245.2.a.e.1.1 2
84.83 odd 2 2205.2.a.v.1.2 2
140.27 odd 4 1225.2.b.i.99.2 4
140.83 odd 4 1225.2.b.i.99.3 4
140.139 even 2 1225.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.1 2 28.27 even 2
245.2.a.f.1.1 yes 2 4.3 odd 2
245.2.e.f.116.2 4 28.23 odd 6
245.2.e.f.226.2 4 28.11 odd 6
245.2.e.g.116.2 4 28.19 even 6
245.2.e.g.226.2 4 28.3 even 6
1225.2.a.p.1.2 2 20.19 odd 2
1225.2.a.r.1.2 2 140.139 even 2
1225.2.b.i.99.2 4 140.27 odd 4
1225.2.b.i.99.3 4 140.83 odd 4
1225.2.b.j.99.1 4 20.7 even 4
1225.2.b.j.99.4 4 20.3 even 4
2205.2.a.t.1.2 2 12.11 even 2
2205.2.a.v.1.2 2 84.83 odd 2
3920.2.a.br.1.2 2 1.1 even 1 trivial
3920.2.a.bw.1.1 2 7.6 odd 2