Properties

Label 1856.2.a.r.1.1
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{3} +1.00000 q^{5} -2.82843 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} +1.00000 q^{5} -2.82843 q^{7} +2.82843 q^{9} +0.414214 q^{11} +3.82843 q^{13} -2.41421 q^{15} +0.828427 q^{17} -6.00000 q^{19} +6.82843 q^{21} +3.65685 q^{23} -4.00000 q^{25} +0.414214 q^{27} -1.00000 q^{29} +10.0711 q^{31} -1.00000 q^{33} -2.82843 q^{35} +4.00000 q^{37} -9.24264 q^{39} -4.48528 q^{41} -3.58579 q^{43} +2.82843 q^{45} -3.24264 q^{47} +1.00000 q^{49} -2.00000 q^{51} -9.48528 q^{53} +0.414214 q^{55} +14.4853 q^{57} +3.65685 q^{59} +4.82843 q^{61} -8.00000 q^{63} +3.82843 q^{65} -5.65685 q^{67} -8.82843 q^{69} -8.82843 q^{71} +4.00000 q^{73} +9.65685 q^{75} -1.17157 q^{77} -2.41421 q^{79} -9.48528 q^{81} -7.65685 q^{83} +0.828427 q^{85} +2.41421 q^{87} -12.4853 q^{89} -10.8284 q^{91} -24.3137 q^{93} -6.00000 q^{95} +4.48528 q^{97} +1.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 4 q^{17} - 12 q^{19} + 8 q^{21} - 4 q^{23} - 8 q^{25} - 2 q^{27} - 2 q^{29} + 6 q^{31} - 2 q^{33} + 8 q^{37} - 10 q^{39} + 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 4 q^{51} - 2 q^{53} - 2 q^{55} + 12 q^{57} - 4 q^{59} + 4 q^{61} - 16 q^{63} + 2 q^{65} - 12 q^{69} - 12 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 2 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} + 2 q^{87} - 8 q^{89} - 16 q^{91} - 26 q^{93} - 12 q^{95} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) 0.414214 0.124890 0.0624450 0.998048i \(-0.480110\pi\)
0.0624450 + 0.998048i \(0.480110\pi\)
\(12\) 0 0
\(13\) 3.82843 1.06181 0.530907 0.847430i \(-0.321851\pi\)
0.530907 + 0.847430i \(0.321851\pi\)
\(14\) 0 0
\(15\) −2.41421 −0.623347
\(16\) 0 0
\(17\) 0.828427 0.200923 0.100462 0.994941i \(-0.467968\pi\)
0.100462 + 0.994941i \(0.467968\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 6.82843 1.49008
\(22\) 0 0
\(23\) 3.65685 0.762507 0.381253 0.924471i \(-0.375493\pi\)
0.381253 + 0.924471i \(0.375493\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 10.0711 1.80882 0.904409 0.426667i \(-0.140313\pi\)
0.904409 + 0.426667i \(0.140313\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −9.24264 −1.48001
\(40\) 0 0
\(41\) −4.48528 −0.700483 −0.350242 0.936659i \(-0.613901\pi\)
−0.350242 + 0.936659i \(0.613901\pi\)
\(42\) 0 0
\(43\) −3.58579 −0.546827 −0.273414 0.961897i \(-0.588153\pi\)
−0.273414 + 0.961897i \(0.588153\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) −3.24264 −0.472988 −0.236494 0.971633i \(-0.575998\pi\)
−0.236494 + 0.971633i \(0.575998\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −9.48528 −1.30290 −0.651452 0.758690i \(-0.725840\pi\)
−0.651452 + 0.758690i \(0.725840\pi\)
\(54\) 0 0
\(55\) 0.414214 0.0558525
\(56\) 0 0
\(57\) 14.4853 1.91862
\(58\) 0 0
\(59\) 3.65685 0.476082 0.238041 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) 0 0
\(61\) 4.82843 0.618217 0.309108 0.951027i \(-0.399969\pi\)
0.309108 + 0.951027i \(0.399969\pi\)
\(62\) 0 0
\(63\) −8.00000 −1.00791
\(64\) 0 0
\(65\) 3.82843 0.474858
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) −8.82843 −1.06282
\(70\) 0 0
\(71\) −8.82843 −1.04774 −0.523871 0.851798i \(-0.675513\pi\)
−0.523871 + 0.851798i \(0.675513\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 9.65685 1.11508
\(76\) 0 0
\(77\) −1.17157 −0.133513
\(78\) 0 0
\(79\) −2.41421 −0.271620 −0.135810 0.990735i \(-0.543364\pi\)
−0.135810 + 0.990735i \(0.543364\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) −7.65685 −0.840449 −0.420224 0.907420i \(-0.638049\pi\)
−0.420224 + 0.907420i \(0.638049\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) 0 0
\(87\) 2.41421 0.258831
\(88\) 0 0
\(89\) −12.4853 −1.32344 −0.661719 0.749752i \(-0.730173\pi\)
−0.661719 + 0.749752i \(0.730173\pi\)
\(90\) 0 0
\(91\) −10.8284 −1.13513
\(92\) 0 0
\(93\) −24.3137 −2.52121
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 4.48528 0.455411 0.227706 0.973730i \(-0.426878\pi\)
0.227706 + 0.973730i \(0.426878\pi\)
\(98\) 0 0
\(99\) 1.17157 0.117748
\(100\) 0 0
\(101\) 2.34315 0.233152 0.116576 0.993182i \(-0.462808\pi\)
0.116576 + 0.993182i \(0.462808\pi\)
\(102\) 0 0
\(103\) −4.82843 −0.475759 −0.237880 0.971295i \(-0.576452\pi\)
−0.237880 + 0.971295i \(0.576452\pi\)
\(104\) 0 0
\(105\) 6.82843 0.666386
\(106\) 0 0
\(107\) 14.8284 1.43352 0.716759 0.697321i \(-0.245625\pi\)
0.716759 + 0.697321i \(0.245625\pi\)
\(108\) 0 0
\(109\) −12.6569 −1.21231 −0.606153 0.795348i \(-0.707288\pi\)
−0.606153 + 0.795348i \(0.707288\pi\)
\(110\) 0 0
\(111\) −9.65685 −0.916588
\(112\) 0 0
\(113\) −13.3137 −1.25245 −0.626224 0.779643i \(-0.715401\pi\)
−0.626224 + 0.779643i \(0.715401\pi\)
\(114\) 0 0
\(115\) 3.65685 0.341003
\(116\) 0 0
\(117\) 10.8284 1.00109
\(118\) 0 0
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) −10.8284 −0.984402
\(122\) 0 0
\(123\) 10.8284 0.976366
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −4.34315 −0.385392 −0.192696 0.981259i \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(128\) 0 0
\(129\) 8.65685 0.762194
\(130\) 0 0
\(131\) −21.3137 −1.86219 −0.931094 0.364780i \(-0.881144\pi\)
−0.931094 + 0.364780i \(0.881144\pi\)
\(132\) 0 0
\(133\) 16.9706 1.47153
\(134\) 0 0
\(135\) 0.414214 0.0356498
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 7.82843 0.659272
\(142\) 0 0
\(143\) 1.58579 0.132610
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) −2.41421 −0.199121
\(148\) 0 0
\(149\) 7.82843 0.641330 0.320665 0.947193i \(-0.396094\pi\)
0.320665 + 0.947193i \(0.396094\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 0 0
\(153\) 2.34315 0.189432
\(154\) 0 0
\(155\) 10.0711 0.808928
\(156\) 0 0
\(157\) −8.48528 −0.677199 −0.338600 0.940931i \(-0.609953\pi\)
−0.338600 + 0.940931i \(0.609953\pi\)
\(158\) 0 0
\(159\) 22.8995 1.81605
\(160\) 0 0
\(161\) −10.3431 −0.815154
\(162\) 0 0
\(163\) −3.92893 −0.307738 −0.153869 0.988091i \(-0.549173\pi\)
−0.153869 + 0.988091i \(0.549173\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −3.17157 −0.245424 −0.122712 0.992442i \(-0.539159\pi\)
−0.122712 + 0.992442i \(0.539159\pi\)
\(168\) 0 0
\(169\) 1.65685 0.127450
\(170\) 0 0
\(171\) −16.9706 −1.29777
\(172\) 0 0
\(173\) −12.3431 −0.938432 −0.469216 0.883083i \(-0.655463\pi\)
−0.469216 + 0.883083i \(0.655463\pi\)
\(174\) 0 0
\(175\) 11.3137 0.855236
\(176\) 0 0
\(177\) −8.82843 −0.663585
\(178\) 0 0
\(179\) 6.48528 0.484733 0.242366 0.970185i \(-0.422076\pi\)
0.242366 + 0.970185i \(0.422076\pi\)
\(180\) 0 0
\(181\) −8.31371 −0.617953 −0.308977 0.951070i \(-0.599986\pi\)
−0.308977 + 0.951070i \(0.599986\pi\)
\(182\) 0 0
\(183\) −11.6569 −0.861699
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0.343146 0.0250933
\(188\) 0 0
\(189\) −1.17157 −0.0852194
\(190\) 0 0
\(191\) 25.3137 1.83164 0.915818 0.401594i \(-0.131544\pi\)
0.915818 + 0.401594i \(0.131544\pi\)
\(192\) 0 0
\(193\) −5.17157 −0.372258 −0.186129 0.982525i \(-0.559594\pi\)
−0.186129 + 0.982525i \(0.559594\pi\)
\(194\) 0 0
\(195\) −9.24264 −0.661879
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −0.485281 −0.0344007 −0.0172003 0.999852i \(-0.505475\pi\)
−0.0172003 + 0.999852i \(0.505475\pi\)
\(200\) 0 0
\(201\) 13.6569 0.963280
\(202\) 0 0
\(203\) 2.82843 0.198517
\(204\) 0 0
\(205\) −4.48528 −0.313266
\(206\) 0 0
\(207\) 10.3431 0.718898
\(208\) 0 0
\(209\) −2.48528 −0.171911
\(210\) 0 0
\(211\) 19.3848 1.33450 0.667252 0.744832i \(-0.267471\pi\)
0.667252 + 0.744832i \(0.267471\pi\)
\(212\) 0 0
\(213\) 21.3137 1.46039
\(214\) 0 0
\(215\) −3.58579 −0.244549
\(216\) 0 0
\(217\) −28.4853 −1.93371
\(218\) 0 0
\(219\) −9.65685 −0.652550
\(220\) 0 0
\(221\) 3.17157 0.213343
\(222\) 0 0
\(223\) −3.17157 −0.212384 −0.106192 0.994346i \(-0.533866\pi\)
−0.106192 + 0.994346i \(0.533866\pi\)
\(224\) 0 0
\(225\) −11.3137 −0.754247
\(226\) 0 0
\(227\) 8.14214 0.540413 0.270206 0.962802i \(-0.412908\pi\)
0.270206 + 0.962802i \(0.412908\pi\)
\(228\) 0 0
\(229\) 3.51472 0.232259 0.116130 0.993234i \(-0.462951\pi\)
0.116130 + 0.993234i \(0.462951\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 18.3137 1.19977 0.599885 0.800086i \(-0.295213\pi\)
0.599885 + 0.800086i \(0.295213\pi\)
\(234\) 0 0
\(235\) −3.24264 −0.211527
\(236\) 0 0
\(237\) 5.82843 0.378597
\(238\) 0 0
\(239\) −19.6569 −1.27150 −0.635748 0.771897i \(-0.719308\pi\)
−0.635748 + 0.771897i \(0.719308\pi\)
\(240\) 0 0
\(241\) −18.3137 −1.17969 −0.589845 0.807517i \(-0.700811\pi\)
−0.589845 + 0.807517i \(0.700811\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −22.9706 −1.46158
\(248\) 0 0
\(249\) 18.4853 1.17146
\(250\) 0 0
\(251\) −20.0711 −1.26687 −0.633437 0.773794i \(-0.718357\pi\)
−0.633437 + 0.773794i \(0.718357\pi\)
\(252\) 0 0
\(253\) 1.51472 0.0952295
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) −18.1716 −1.13351 −0.566756 0.823886i \(-0.691802\pi\)
−0.566756 + 0.823886i \(0.691802\pi\)
\(258\) 0 0
\(259\) −11.3137 −0.703000
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) 2.75736 0.170026 0.0850130 0.996380i \(-0.472907\pi\)
0.0850130 + 0.996380i \(0.472907\pi\)
\(264\) 0 0
\(265\) −9.48528 −0.582676
\(266\) 0 0
\(267\) 30.1421 1.84467
\(268\) 0 0
\(269\) −31.4558 −1.91790 −0.958948 0.283581i \(-0.908478\pi\)
−0.958948 + 0.283581i \(0.908478\pi\)
\(270\) 0 0
\(271\) 16.5563 1.00573 0.502863 0.864366i \(-0.332280\pi\)
0.502863 + 0.864366i \(0.332280\pi\)
\(272\) 0 0
\(273\) 26.1421 1.58219
\(274\) 0 0
\(275\) −1.65685 −0.0999121
\(276\) 0 0
\(277\) 17.3137 1.04028 0.520140 0.854081i \(-0.325880\pi\)
0.520140 + 0.854081i \(0.325880\pi\)
\(278\) 0 0
\(279\) 28.4853 1.70537
\(280\) 0 0
\(281\) 31.9706 1.90720 0.953602 0.301070i \(-0.0973439\pi\)
0.953602 + 0.301070i \(0.0973439\pi\)
\(282\) 0 0
\(283\) −11.6569 −0.692928 −0.346464 0.938063i \(-0.612618\pi\)
−0.346464 + 0.938063i \(0.612618\pi\)
\(284\) 0 0
\(285\) 14.4853 0.858034
\(286\) 0 0
\(287\) 12.6863 0.748848
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) −10.8284 −0.634774
\(292\) 0 0
\(293\) −7.65685 −0.447318 −0.223659 0.974667i \(-0.571800\pi\)
−0.223659 + 0.974667i \(0.571800\pi\)
\(294\) 0 0
\(295\) 3.65685 0.212910
\(296\) 0 0
\(297\) 0.171573 0.00995567
\(298\) 0 0
\(299\) 14.0000 0.809641
\(300\) 0 0
\(301\) 10.1421 0.584583
\(302\) 0 0
\(303\) −5.65685 −0.324978
\(304\) 0 0
\(305\) 4.82843 0.276475
\(306\) 0 0
\(307\) −2.89949 −0.165483 −0.0827415 0.996571i \(-0.526368\pi\)
−0.0827415 + 0.996571i \(0.526368\pi\)
\(308\) 0 0
\(309\) 11.6569 0.663135
\(310\) 0 0
\(311\) 2.68629 0.152326 0.0761628 0.997095i \(-0.475733\pi\)
0.0761628 + 0.997095i \(0.475733\pi\)
\(312\) 0 0
\(313\) 9.82843 0.555536 0.277768 0.960648i \(-0.410405\pi\)
0.277768 + 0.960648i \(0.410405\pi\)
\(314\) 0 0
\(315\) −8.00000 −0.450749
\(316\) 0 0
\(317\) 31.4558 1.76674 0.883368 0.468680i \(-0.155270\pi\)
0.883368 + 0.468680i \(0.155270\pi\)
\(318\) 0 0
\(319\) −0.414214 −0.0231915
\(320\) 0 0
\(321\) −35.7990 −1.99810
\(322\) 0 0
\(323\) −4.97056 −0.276570
\(324\) 0 0
\(325\) −15.3137 −0.849452
\(326\) 0 0
\(327\) 30.5563 1.68977
\(328\) 0 0
\(329\) 9.17157 0.505645
\(330\) 0 0
\(331\) 2.41421 0.132697 0.0663486 0.997797i \(-0.478865\pi\)
0.0663486 + 0.997797i \(0.478865\pi\)
\(332\) 0 0
\(333\) 11.3137 0.619987
\(334\) 0 0
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) 21.7990 1.18747 0.593733 0.804662i \(-0.297653\pi\)
0.593733 + 0.804662i \(0.297653\pi\)
\(338\) 0 0
\(339\) 32.1421 1.74572
\(340\) 0 0
\(341\) 4.17157 0.225903
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −8.82843 −0.475307
\(346\) 0 0
\(347\) −2.48528 −0.133417 −0.0667084 0.997773i \(-0.521250\pi\)
−0.0667084 + 0.997773i \(0.521250\pi\)
\(348\) 0 0
\(349\) 5.14214 0.275252 0.137626 0.990484i \(-0.456053\pi\)
0.137626 + 0.990484i \(0.456053\pi\)
\(350\) 0 0
\(351\) 1.58579 0.0846430
\(352\) 0 0
\(353\) 26.9706 1.43550 0.717749 0.696302i \(-0.245172\pi\)
0.717749 + 0.696302i \(0.245172\pi\)
\(354\) 0 0
\(355\) −8.82843 −0.468564
\(356\) 0 0
\(357\) 5.65685 0.299392
\(358\) 0 0
\(359\) 3.92893 0.207361 0.103681 0.994611i \(-0.466938\pi\)
0.103681 + 0.994611i \(0.466938\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 26.1421 1.37211
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 0 0
\(369\) −12.6863 −0.660422
\(370\) 0 0
\(371\) 26.8284 1.39286
\(372\) 0 0
\(373\) 26.3137 1.36247 0.681236 0.732064i \(-0.261443\pi\)
0.681236 + 0.732064i \(0.261443\pi\)
\(374\) 0 0
\(375\) 21.7279 1.12203
\(376\) 0 0
\(377\) −3.82843 −0.197174
\(378\) 0 0
\(379\) 6.97056 0.358054 0.179027 0.983844i \(-0.442705\pi\)
0.179027 + 0.983844i \(0.442705\pi\)
\(380\) 0 0
\(381\) 10.4853 0.537177
\(382\) 0 0
\(383\) −3.51472 −0.179594 −0.0897969 0.995960i \(-0.528622\pi\)
−0.0897969 + 0.995960i \(0.528622\pi\)
\(384\) 0 0
\(385\) −1.17157 −0.0597089
\(386\) 0 0
\(387\) −10.1421 −0.515554
\(388\) 0 0
\(389\) −3.02944 −0.153599 −0.0767993 0.997047i \(-0.524470\pi\)
−0.0767993 + 0.997047i \(0.524470\pi\)
\(390\) 0 0
\(391\) 3.02944 0.153205
\(392\) 0 0
\(393\) 51.4558 2.59560
\(394\) 0 0
\(395\) −2.41421 −0.121472
\(396\) 0 0
\(397\) −19.3431 −0.970805 −0.485402 0.874291i \(-0.661327\pi\)
−0.485402 + 0.874291i \(0.661327\pi\)
\(398\) 0 0
\(399\) −40.9706 −2.05109
\(400\) 0 0
\(401\) −18.6569 −0.931679 −0.465839 0.884869i \(-0.654248\pi\)
−0.465839 + 0.884869i \(0.654248\pi\)
\(402\) 0 0
\(403\) 38.5563 1.92063
\(404\) 0 0
\(405\) −9.48528 −0.471327
\(406\) 0 0
\(407\) 1.65685 0.0821272
\(408\) 0 0
\(409\) −18.9706 −0.938034 −0.469017 0.883189i \(-0.655392\pi\)
−0.469017 + 0.883189i \(0.655392\pi\)
\(410\) 0 0
\(411\) −28.9706 −1.42901
\(412\) 0 0
\(413\) −10.3431 −0.508953
\(414\) 0 0
\(415\) −7.65685 −0.375860
\(416\) 0 0
\(417\) 33.7990 1.65514
\(418\) 0 0
\(419\) 9.51472 0.464824 0.232412 0.972617i \(-0.425338\pi\)
0.232412 + 0.972617i \(0.425338\pi\)
\(420\) 0 0
\(421\) −37.1127 −1.80876 −0.904381 0.426726i \(-0.859667\pi\)
−0.904381 + 0.426726i \(0.859667\pi\)
\(422\) 0 0
\(423\) −9.17157 −0.445937
\(424\) 0 0
\(425\) −3.31371 −0.160738
\(426\) 0 0
\(427\) −13.6569 −0.660901
\(428\) 0 0
\(429\) −3.82843 −0.184838
\(430\) 0 0
\(431\) 19.6569 0.946837 0.473419 0.880838i \(-0.343020\pi\)
0.473419 + 0.880838i \(0.343020\pi\)
\(432\) 0 0
\(433\) 30.6274 1.47186 0.735930 0.677058i \(-0.236745\pi\)
0.735930 + 0.677058i \(0.236745\pi\)
\(434\) 0 0
\(435\) 2.41421 0.115753
\(436\) 0 0
\(437\) −21.9411 −1.04959
\(438\) 0 0
\(439\) −0.343146 −0.0163775 −0.00818873 0.999966i \(-0.502607\pi\)
−0.00818873 + 0.999966i \(0.502607\pi\)
\(440\) 0 0
\(441\) 2.82843 0.134687
\(442\) 0 0
\(443\) 24.3431 1.15658 0.578289 0.815832i \(-0.303721\pi\)
0.578289 + 0.815832i \(0.303721\pi\)
\(444\) 0 0
\(445\) −12.4853 −0.591859
\(446\) 0 0
\(447\) −18.8995 −0.893915
\(448\) 0 0
\(449\) −34.9706 −1.65036 −0.825181 0.564868i \(-0.808927\pi\)
−0.825181 + 0.564868i \(0.808927\pi\)
\(450\) 0 0
\(451\) −1.85786 −0.0874834
\(452\) 0 0
\(453\) 34.1421 1.60414
\(454\) 0 0
\(455\) −10.8284 −0.507644
\(456\) 0 0
\(457\) 1.02944 0.0481550 0.0240775 0.999710i \(-0.492335\pi\)
0.0240775 + 0.999710i \(0.492335\pi\)
\(458\) 0 0
\(459\) 0.343146 0.0160167
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 0 0
\(465\) −24.3137 −1.12752
\(466\) 0 0
\(467\) 38.3553 1.77487 0.887437 0.460930i \(-0.152484\pi\)
0.887437 + 0.460930i \(0.152484\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 20.4853 0.943912
\(472\) 0 0
\(473\) −1.48528 −0.0682933
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) −26.8284 −1.22839
\(478\) 0 0
\(479\) 6.89949 0.315246 0.157623 0.987499i \(-0.449617\pi\)
0.157623 + 0.987499i \(0.449617\pi\)
\(480\) 0 0
\(481\) 15.3137 0.698245
\(482\) 0 0
\(483\) 24.9706 1.13620
\(484\) 0 0
\(485\) 4.48528 0.203666
\(486\) 0 0
\(487\) −11.5147 −0.521782 −0.260891 0.965368i \(-0.584016\pi\)
−0.260891 + 0.965368i \(0.584016\pi\)
\(488\) 0 0
\(489\) 9.48528 0.428939
\(490\) 0 0
\(491\) 21.2426 0.958667 0.479333 0.877633i \(-0.340878\pi\)
0.479333 + 0.877633i \(0.340878\pi\)
\(492\) 0 0
\(493\) −0.828427 −0.0373105
\(494\) 0 0
\(495\) 1.17157 0.0526583
\(496\) 0 0
\(497\) 24.9706 1.12008
\(498\) 0 0
\(499\) −18.9706 −0.849239 −0.424620 0.905372i \(-0.639592\pi\)
−0.424620 + 0.905372i \(0.639592\pi\)
\(500\) 0 0
\(501\) 7.65685 0.342083
\(502\) 0 0
\(503\) 0.272078 0.0121314 0.00606568 0.999982i \(-0.498069\pi\)
0.00606568 + 0.999982i \(0.498069\pi\)
\(504\) 0 0
\(505\) 2.34315 0.104269
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 0 0
\(509\) 10.5147 0.466057 0.233028 0.972470i \(-0.425137\pi\)
0.233028 + 0.972470i \(0.425137\pi\)
\(510\) 0 0
\(511\) −11.3137 −0.500489
\(512\) 0 0
\(513\) −2.48528 −0.109728
\(514\) 0 0
\(515\) −4.82843 −0.212766
\(516\) 0 0
\(517\) −1.34315 −0.0590715
\(518\) 0 0
\(519\) 29.7990 1.30803
\(520\) 0 0
\(521\) −29.1421 −1.27674 −0.638370 0.769730i \(-0.720391\pi\)
−0.638370 + 0.769730i \(0.720391\pi\)
\(522\) 0 0
\(523\) −4.68629 −0.204917 −0.102459 0.994737i \(-0.532671\pi\)
−0.102459 + 0.994737i \(0.532671\pi\)
\(524\) 0 0
\(525\) −27.3137 −1.19207
\(526\) 0 0
\(527\) 8.34315 0.363433
\(528\) 0 0
\(529\) −9.62742 −0.418583
\(530\) 0 0
\(531\) 10.3431 0.448854
\(532\) 0 0
\(533\) −17.1716 −0.743783
\(534\) 0 0
\(535\) 14.8284 0.641089
\(536\) 0 0
\(537\) −15.6569 −0.675643
\(538\) 0 0
\(539\) 0.414214 0.0178414
\(540\) 0 0
\(541\) 10.3431 0.444687 0.222343 0.974968i \(-0.428629\pi\)
0.222343 + 0.974968i \(0.428629\pi\)
\(542\) 0 0
\(543\) 20.0711 0.861332
\(544\) 0 0
\(545\) −12.6569 −0.542160
\(546\) 0 0
\(547\) −35.7990 −1.53065 −0.765327 0.643641i \(-0.777423\pi\)
−0.765327 + 0.643641i \(0.777423\pi\)
\(548\) 0 0
\(549\) 13.6569 0.582860
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 6.82843 0.290374
\(554\) 0 0
\(555\) −9.65685 −0.409911
\(556\) 0 0
\(557\) 17.3137 0.733605 0.366803 0.930299i \(-0.380452\pi\)
0.366803 + 0.930299i \(0.380452\pi\)
\(558\) 0 0
\(559\) −13.7279 −0.580629
\(560\) 0 0
\(561\) −0.828427 −0.0349762
\(562\) 0 0
\(563\) 0.757359 0.0319189 0.0159594 0.999873i \(-0.494920\pi\)
0.0159594 + 0.999873i \(0.494920\pi\)
\(564\) 0 0
\(565\) −13.3137 −0.560112
\(566\) 0 0
\(567\) 26.8284 1.12669
\(568\) 0 0
\(569\) −39.6569 −1.66250 −0.831251 0.555897i \(-0.812375\pi\)
−0.831251 + 0.555897i \(0.812375\pi\)
\(570\) 0 0
\(571\) −14.6274 −0.612138 −0.306069 0.952009i \(-0.599014\pi\)
−0.306069 + 0.952009i \(0.599014\pi\)
\(572\) 0 0
\(573\) −61.1127 −2.55302
\(574\) 0 0
\(575\) −14.6274 −0.610005
\(576\) 0 0
\(577\) −29.7990 −1.24055 −0.620274 0.784385i \(-0.712979\pi\)
−0.620274 + 0.784385i \(0.712979\pi\)
\(578\) 0 0
\(579\) 12.4853 0.518871
\(580\) 0 0
\(581\) 21.6569 0.898478
\(582\) 0 0
\(583\) −3.92893 −0.162720
\(584\) 0 0
\(585\) 10.8284 0.447700
\(586\) 0 0
\(587\) −7.65685 −0.316032 −0.158016 0.987437i \(-0.550510\pi\)
−0.158016 + 0.987437i \(0.550510\pi\)
\(588\) 0 0
\(589\) −60.4264 −2.48983
\(590\) 0 0
\(591\) 4.82843 0.198615
\(592\) 0 0
\(593\) −19.4853 −0.800165 −0.400082 0.916479i \(-0.631018\pi\)
−0.400082 + 0.916479i \(0.631018\pi\)
\(594\) 0 0
\(595\) −2.34315 −0.0960596
\(596\) 0 0
\(597\) 1.17157 0.0479493
\(598\) 0 0
\(599\) 9.87006 0.403280 0.201640 0.979460i \(-0.435373\pi\)
0.201640 + 0.979460i \(0.435373\pi\)
\(600\) 0 0
\(601\) −17.1716 −0.700443 −0.350222 0.936667i \(-0.613894\pi\)
−0.350222 + 0.936667i \(0.613894\pi\)
\(602\) 0 0
\(603\) −16.0000 −0.651570
\(604\) 0 0
\(605\) −10.8284 −0.440238
\(606\) 0 0
\(607\) −7.72792 −0.313667 −0.156833 0.987625i \(-0.550129\pi\)
−0.156833 + 0.987625i \(0.550129\pi\)
\(608\) 0 0
\(609\) −6.82843 −0.276702
\(610\) 0 0
\(611\) −12.4142 −0.502225
\(612\) 0 0
\(613\) 9.00000 0.363507 0.181753 0.983344i \(-0.441823\pi\)
0.181753 + 0.983344i \(0.441823\pi\)
\(614\) 0 0
\(615\) 10.8284 0.436644
\(616\) 0 0
\(617\) 0.686292 0.0276291 0.0138145 0.999905i \(-0.495603\pi\)
0.0138145 + 0.999905i \(0.495603\pi\)
\(618\) 0 0
\(619\) −33.5858 −1.34993 −0.674963 0.737851i \(-0.735841\pi\)
−0.674963 + 0.737851i \(0.735841\pi\)
\(620\) 0 0
\(621\) 1.51472 0.0607836
\(622\) 0 0
\(623\) 35.3137 1.41481
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) 3.31371 0.132126
\(630\) 0 0
\(631\) −36.8284 −1.46612 −0.733058 0.680166i \(-0.761908\pi\)
−0.733058 + 0.680166i \(0.761908\pi\)
\(632\) 0 0
\(633\) −46.7990 −1.86009
\(634\) 0 0
\(635\) −4.34315 −0.172352
\(636\) 0 0
\(637\) 3.82843 0.151688
\(638\) 0 0
\(639\) −24.9706 −0.987820
\(640\) 0 0
\(641\) 17.7990 0.703018 0.351509 0.936185i \(-0.385669\pi\)
0.351509 + 0.936185i \(0.385669\pi\)
\(642\) 0 0
\(643\) −32.4853 −1.28109 −0.640547 0.767919i \(-0.721292\pi\)
−0.640547 + 0.767919i \(0.721292\pi\)
\(644\) 0 0
\(645\) 8.65685 0.340863
\(646\) 0 0
\(647\) 39.6569 1.55907 0.779536 0.626358i \(-0.215455\pi\)
0.779536 + 0.626358i \(0.215455\pi\)
\(648\) 0 0
\(649\) 1.51472 0.0594579
\(650\) 0 0
\(651\) 68.7696 2.69529
\(652\) 0 0
\(653\) 30.1421 1.17955 0.589776 0.807567i \(-0.299216\pi\)
0.589776 + 0.807567i \(0.299216\pi\)
\(654\) 0 0
\(655\) −21.3137 −0.832796
\(656\) 0 0
\(657\) 11.3137 0.441390
\(658\) 0 0
\(659\) −14.4142 −0.561498 −0.280749 0.959781i \(-0.590583\pi\)
−0.280749 + 0.959781i \(0.590583\pi\)
\(660\) 0 0
\(661\) −33.3137 −1.29575 −0.647877 0.761745i \(-0.724343\pi\)
−0.647877 + 0.761745i \(0.724343\pi\)
\(662\) 0 0
\(663\) −7.65685 −0.297368
\(664\) 0 0
\(665\) 16.9706 0.658090
\(666\) 0 0
\(667\) −3.65685 −0.141594
\(668\) 0 0
\(669\) 7.65685 0.296031
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −21.6274 −0.833676 −0.416838 0.908981i \(-0.636862\pi\)
−0.416838 + 0.908981i \(0.636862\pi\)
\(674\) 0 0
\(675\) −1.65685 −0.0637723
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −12.6863 −0.486855
\(680\) 0 0
\(681\) −19.6569 −0.753252
\(682\) 0 0
\(683\) −20.9706 −0.802416 −0.401208 0.915987i \(-0.631410\pi\)
−0.401208 + 0.915987i \(0.631410\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −8.48528 −0.323734
\(688\) 0 0
\(689\) −36.3137 −1.38344
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) 0 0
\(693\) −3.31371 −0.125877
\(694\) 0 0
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) −3.71573 −0.140743
\(698\) 0 0
\(699\) −44.2132 −1.67230
\(700\) 0 0
\(701\) 40.1127 1.51504 0.757518 0.652814i \(-0.226412\pi\)
0.757518 + 0.652814i \(0.226412\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 7.82843 0.294836
\(706\) 0 0
\(707\) −6.62742 −0.249250
\(708\) 0 0
\(709\) −29.1421 −1.09446 −0.547228 0.836984i \(-0.684317\pi\)
−0.547228 + 0.836984i \(0.684317\pi\)
\(710\) 0 0
\(711\) −6.82843 −0.256086
\(712\) 0 0
\(713\) 36.8284 1.37924
\(714\) 0 0
\(715\) 1.58579 0.0593051
\(716\) 0 0
\(717\) 47.4558 1.77227
\(718\) 0 0
\(719\) −20.1421 −0.751175 −0.375587 0.926787i \(-0.622559\pi\)
−0.375587 + 0.926787i \(0.622559\pi\)
\(720\) 0 0
\(721\) 13.6569 0.508608
\(722\) 0 0
\(723\) 44.2132 1.64431
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 1.31371 0.0487228 0.0243614 0.999703i \(-0.492245\pi\)
0.0243614 + 0.999703i \(0.492245\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −2.97056 −0.109870
\(732\) 0 0
\(733\) 41.2548 1.52378 0.761891 0.647705i \(-0.224271\pi\)
0.761891 + 0.647705i \(0.224271\pi\)
\(734\) 0 0
\(735\) −2.41421 −0.0890496
\(736\) 0 0
\(737\) −2.34315 −0.0863109
\(738\) 0 0
\(739\) −4.07107 −0.149757 −0.0748783 0.997193i \(-0.523857\pi\)
−0.0748783 + 0.997193i \(0.523857\pi\)
\(740\) 0 0
\(741\) 55.4558 2.03722
\(742\) 0 0
\(743\) 23.6569 0.867886 0.433943 0.900940i \(-0.357122\pi\)
0.433943 + 0.900940i \(0.357122\pi\)
\(744\) 0 0
\(745\) 7.82843 0.286811
\(746\) 0 0
\(747\) −21.6569 −0.792383
\(748\) 0 0
\(749\) −41.9411 −1.53250
\(750\) 0 0
\(751\) 25.3137 0.923710 0.461855 0.886955i \(-0.347184\pi\)
0.461855 + 0.886955i \(0.347184\pi\)
\(752\) 0 0
\(753\) 48.4558 1.76583
\(754\) 0 0
\(755\) −14.1421 −0.514685
\(756\) 0 0
\(757\) −25.5147 −0.927348 −0.463674 0.886006i \(-0.653469\pi\)
−0.463674 + 0.886006i \(0.653469\pi\)
\(758\) 0 0
\(759\) −3.65685 −0.132735
\(760\) 0 0
\(761\) 45.5980 1.65293 0.826463 0.562991i \(-0.190350\pi\)
0.826463 + 0.562991i \(0.190350\pi\)
\(762\) 0 0
\(763\) 35.7990 1.29601
\(764\) 0 0
\(765\) 2.34315 0.0847166
\(766\) 0 0
\(767\) 14.0000 0.505511
\(768\) 0 0
\(769\) −49.1127 −1.77105 −0.885525 0.464592i \(-0.846201\pi\)
−0.885525 + 0.464592i \(0.846201\pi\)
\(770\) 0 0
\(771\) 43.8701 1.57994
\(772\) 0 0
\(773\) 19.5147 0.701896 0.350948 0.936395i \(-0.385859\pi\)
0.350948 + 0.936395i \(0.385859\pi\)
\(774\) 0 0
\(775\) −40.2843 −1.44705
\(776\) 0 0
\(777\) 27.3137 0.979874
\(778\) 0 0
\(779\) 26.9117 0.964211
\(780\) 0 0
\(781\) −3.65685 −0.130853
\(782\) 0 0
\(783\) −0.414214 −0.0148028
\(784\) 0 0
\(785\) −8.48528 −0.302853
\(786\) 0 0
\(787\) 54.0833 1.92786 0.963930 0.266156i \(-0.0857536\pi\)
0.963930 + 0.266156i \(0.0857536\pi\)
\(788\) 0 0
\(789\) −6.65685 −0.236990
\(790\) 0 0
\(791\) 37.6569 1.33892
\(792\) 0 0
\(793\) 18.4853 0.656432
\(794\) 0 0
\(795\) 22.8995 0.812161
\(796\) 0 0
\(797\) −51.7401 −1.83273 −0.916364 0.400345i \(-0.868890\pi\)
−0.916364 + 0.400345i \(0.868890\pi\)
\(798\) 0 0
\(799\) −2.68629 −0.0950342
\(800\) 0 0
\(801\) −35.3137 −1.24775
\(802\) 0 0
\(803\) 1.65685 0.0584691
\(804\) 0 0
\(805\) −10.3431 −0.364548
\(806\) 0 0
\(807\) 75.9411 2.67325
\(808\) 0 0
\(809\) 36.2843 1.27569 0.637844 0.770166i \(-0.279827\pi\)
0.637844 + 0.770166i \(0.279827\pi\)
\(810\) 0 0
\(811\) −10.8284 −0.380238 −0.190119 0.981761i \(-0.560887\pi\)
−0.190119 + 0.981761i \(0.560887\pi\)
\(812\) 0 0
\(813\) −39.9706 −1.40183
\(814\) 0 0
\(815\) −3.92893 −0.137624
\(816\) 0 0
\(817\) 21.5147 0.752705
\(818\) 0 0
\(819\) −30.6274 −1.07021
\(820\) 0 0
\(821\) 1.48528 0.0518367 0.0259183 0.999664i \(-0.491749\pi\)
0.0259183 + 0.999664i \(0.491749\pi\)
\(822\) 0 0
\(823\) −54.2843 −1.89223 −0.946115 0.323830i \(-0.895029\pi\)
−0.946115 + 0.323830i \(0.895029\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −32.8995 −1.14403 −0.572014 0.820244i \(-0.693838\pi\)
−0.572014 + 0.820244i \(0.693838\pi\)
\(828\) 0 0
\(829\) 29.7990 1.03496 0.517481 0.855695i \(-0.326870\pi\)
0.517481 + 0.855695i \(0.326870\pi\)
\(830\) 0 0
\(831\) −41.7990 −1.44999
\(832\) 0 0
\(833\) 0.828427 0.0287033
\(834\) 0 0
\(835\) −3.17157 −0.109757
\(836\) 0 0
\(837\) 4.17157 0.144191
\(838\) 0 0
\(839\) −7.92893 −0.273737 −0.136869 0.990589i \(-0.543704\pi\)
−0.136869 + 0.990589i \(0.543704\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −77.1838 −2.65835
\(844\) 0 0
\(845\) 1.65685 0.0569975
\(846\) 0 0
\(847\) 30.6274 1.05237
\(848\) 0 0
\(849\) 28.1421 0.965836
\(850\) 0 0
\(851\) 14.6274 0.501421
\(852\) 0 0
\(853\) 22.9706 0.786497 0.393249 0.919432i \(-0.371351\pi\)
0.393249 + 0.919432i \(0.371351\pi\)
\(854\) 0 0
\(855\) −16.9706 −0.580381
\(856\) 0 0
\(857\) −6.17157 −0.210817 −0.105408 0.994429i \(-0.533615\pi\)
−0.105408 + 0.994429i \(0.533615\pi\)
\(858\) 0 0
\(859\) −19.7279 −0.673108 −0.336554 0.941664i \(-0.609261\pi\)
−0.336554 + 0.941664i \(0.609261\pi\)
\(860\) 0 0
\(861\) −30.6274 −1.04378
\(862\) 0 0
\(863\) 17.1127 0.582523 0.291262 0.956643i \(-0.405925\pi\)
0.291262 + 0.956643i \(0.405925\pi\)
\(864\) 0 0
\(865\) −12.3431 −0.419680
\(866\) 0 0
\(867\) 39.3848 1.33758
\(868\) 0 0
\(869\) −1.00000 −0.0339227
\(870\) 0 0
\(871\) −21.6569 −0.733815
\(872\) 0 0
\(873\) 12.6863 0.429366
\(874\) 0 0
\(875\) 25.4558 0.860565
\(876\) 0 0
\(877\) 37.1421 1.25420 0.627100 0.778938i \(-0.284242\pi\)
0.627100 + 0.778938i \(0.284242\pi\)
\(878\) 0 0
\(879\) 18.4853 0.623493
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −38.4264 −1.29315 −0.646576 0.762850i \(-0.723800\pi\)
−0.646576 + 0.762850i \(0.723800\pi\)
\(884\) 0 0
\(885\) −8.82843 −0.296764
\(886\) 0 0
\(887\) 17.1005 0.574179 0.287089 0.957904i \(-0.407312\pi\)
0.287089 + 0.957904i \(0.407312\pi\)
\(888\) 0 0
\(889\) 12.2843 0.412001
\(890\) 0 0
\(891\) −3.92893 −0.131624
\(892\) 0 0
\(893\) 19.4558 0.651065
\(894\) 0 0
\(895\) 6.48528 0.216779
\(896\) 0 0
\(897\) −33.7990 −1.12852
\(898\) 0 0
\(899\) −10.0711 −0.335889
\(900\) 0 0
\(901\) −7.85786 −0.261783
\(902\) 0 0
\(903\) −24.4853 −0.814819
\(904\) 0 0
\(905\) −8.31371 −0.276357
\(906\) 0 0
\(907\) −22.2843 −0.739937 −0.369969 0.929044i \(-0.620632\pi\)
−0.369969 + 0.929044i \(0.620632\pi\)
\(908\) 0 0
\(909\) 6.62742 0.219818
\(910\) 0 0
\(911\) −15.4437 −0.511671 −0.255835 0.966720i \(-0.582351\pi\)
−0.255835 + 0.966720i \(0.582351\pi\)
\(912\) 0 0
\(913\) −3.17157 −0.104964
\(914\) 0 0
\(915\) −11.6569 −0.385364
\(916\) 0 0
\(917\) 60.2843 1.99076
\(918\) 0 0
\(919\) 8.14214 0.268584 0.134292 0.990942i \(-0.457124\pi\)
0.134292 + 0.990942i \(0.457124\pi\)
\(920\) 0 0
\(921\) 7.00000 0.230658
\(922\) 0 0
\(923\) −33.7990 −1.11251
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 0 0
\(927\) −13.6569 −0.448550
\(928\) 0 0
\(929\) 18.6863 0.613077 0.306539 0.951858i \(-0.400829\pi\)
0.306539 + 0.951858i \(0.400829\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) −6.48528 −0.212319
\(934\) 0 0
\(935\) 0.343146 0.0112221
\(936\) 0 0
\(937\) −16.6274 −0.543194 −0.271597 0.962411i \(-0.587552\pi\)
−0.271597 + 0.962411i \(0.587552\pi\)
\(938\) 0 0
\(939\) −23.7279 −0.774331
\(940\) 0 0
\(941\) 56.5980 1.84504 0.922521 0.385948i \(-0.126125\pi\)
0.922521 + 0.385948i \(0.126125\pi\)
\(942\) 0 0
\(943\) −16.4020 −0.534123
\(944\) 0 0
\(945\) −1.17157 −0.0381113
\(946\) 0 0
\(947\) −2.61522 −0.0849834 −0.0424917 0.999097i \(-0.513530\pi\)
−0.0424917 + 0.999097i \(0.513530\pi\)
\(948\) 0 0
\(949\) 15.3137 0.497104
\(950\) 0 0
\(951\) −75.9411 −2.46256
\(952\) 0 0
\(953\) −35.6274 −1.15409 −0.577043 0.816714i \(-0.695793\pi\)
−0.577043 + 0.816714i \(0.695793\pi\)
\(954\) 0 0
\(955\) 25.3137 0.819132
\(956\) 0 0
\(957\) 1.00000 0.0323254
\(958\) 0 0
\(959\) −33.9411 −1.09602
\(960\) 0 0
\(961\) 70.4264 2.27182
\(962\) 0 0
\(963\) 41.9411 1.35153
\(964\) 0 0
\(965\) −5.17157 −0.166479
\(966\) 0 0
\(967\) −35.2426 −1.13333 −0.566663 0.823949i \(-0.691766\pi\)
−0.566663 + 0.823949i \(0.691766\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 15.6569 0.502452 0.251226 0.967928i \(-0.419166\pi\)
0.251226 + 0.967928i \(0.419166\pi\)
\(972\) 0 0
\(973\) 39.5980 1.26945
\(974\) 0 0
\(975\) 36.9706 1.18401
\(976\) 0 0
\(977\) 36.1716 1.15723 0.578616 0.815600i \(-0.303593\pi\)
0.578616 + 0.815600i \(0.303593\pi\)
\(978\) 0 0
\(979\) −5.17157 −0.165284
\(980\) 0 0
\(981\) −35.7990 −1.14297
\(982\) 0 0
\(983\) −21.8701 −0.697547 −0.348773 0.937207i \(-0.613402\pi\)
−0.348773 + 0.937207i \(0.613402\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) −22.1421 −0.704792
\(988\) 0 0
\(989\) −13.1127 −0.416960
\(990\) 0 0
\(991\) −12.8284 −0.407508 −0.203754 0.979022i \(-0.565314\pi\)
−0.203754 + 0.979022i \(0.565314\pi\)
\(992\) 0 0
\(993\) −5.82843 −0.184960
\(994\) 0 0
\(995\) −0.485281 −0.0153845
\(996\) 0 0
\(997\) −28.2843 −0.895772 −0.447886 0.894091i \(-0.647823\pi\)
−0.447886 + 0.894091i \(0.647823\pi\)
\(998\) 0 0
\(999\) 1.65685 0.0524205
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 1856.2.a.r.1.1 2
4.3 odd 2 1856.2.a.w.1.2 2
8.3 odd 2 464.2.a.h.1.1 2
8.5 even 2 29.2.a.a.1.1 2
24.5 odd 2 261.2.a.d.1.2 2
24.11 even 2 4176.2.a.bq.1.2 2
40.13 odd 4 725.2.b.b.349.4 4
40.29 even 2 725.2.a.b.1.2 2
40.37 odd 4 725.2.b.b.349.1 4
56.13 odd 2 1421.2.a.j.1.1 2
88.21 odd 2 3509.2.a.j.1.2 2
104.77 even 2 4901.2.a.g.1.2 2
120.29 odd 2 6525.2.a.o.1.1 2
136.101 even 2 8381.2.a.e.1.1 2
232.5 even 14 841.2.d.f.605.1 12
232.13 even 14 841.2.d.f.778.2 12
232.21 odd 28 841.2.e.k.267.4 24
232.37 odd 28 841.2.e.k.267.1 24
232.45 even 14 841.2.d.j.778.1 12
232.53 even 14 841.2.d.j.605.2 12
232.61 odd 28 841.2.e.k.270.1 24
232.69 odd 28 841.2.e.k.63.1 24
232.77 odd 28 841.2.e.k.651.4 24
232.85 odd 28 841.2.e.k.236.4 24
232.93 even 14 841.2.d.f.645.1 12
232.101 odd 28 841.2.e.k.196.4 24
232.109 even 14 841.2.d.f.571.1 12
232.125 even 14 841.2.d.f.574.2 12
232.133 odd 4 841.2.b.a.840.4 4
232.141 even 14 841.2.d.j.190.2 12
232.149 even 14 841.2.d.f.190.1 12
232.157 odd 4 841.2.b.a.840.1 4
232.165 even 14 841.2.d.j.574.1 12
232.173 even 2 841.2.a.d.1.2 2
232.181 even 14 841.2.d.j.571.2 12
232.189 odd 28 841.2.e.k.196.1 24
232.197 even 14 841.2.d.j.645.2 12
232.205 odd 28 841.2.e.k.236.1 24
232.213 odd 28 841.2.e.k.651.1 24
232.221 odd 28 841.2.e.k.63.4 24
232.229 odd 28 841.2.e.k.270.4 24
696.173 odd 2 7569.2.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.a.a.1.1 2 8.5 even 2
261.2.a.d.1.2 2 24.5 odd 2
464.2.a.h.1.1 2 8.3 odd 2
725.2.a.b.1.2 2 40.29 even 2
725.2.b.b.349.1 4 40.37 odd 4
725.2.b.b.349.4 4 40.13 odd 4
841.2.a.d.1.2 2 232.173 even 2
841.2.b.a.840.1 4 232.157 odd 4
841.2.b.a.840.4 4 232.133 odd 4
841.2.d.f.190.1 12 232.149 even 14
841.2.d.f.571.1 12 232.109 even 14
841.2.d.f.574.2 12 232.125 even 14
841.2.d.f.605.1 12 232.5 even 14
841.2.d.f.645.1 12 232.93 even 14
841.2.d.f.778.2 12 232.13 even 14
841.2.d.j.190.2 12 232.141 even 14
841.2.d.j.571.2 12 232.181 even 14
841.2.d.j.574.1 12 232.165 even 14
841.2.d.j.605.2 12 232.53 even 14
841.2.d.j.645.2 12 232.197 even 14
841.2.d.j.778.1 12 232.45 even 14
841.2.e.k.63.1 24 232.69 odd 28
841.2.e.k.63.4 24 232.221 odd 28
841.2.e.k.196.1 24 232.189 odd 28
841.2.e.k.196.4 24 232.101 odd 28
841.2.e.k.236.1 24 232.205 odd 28
841.2.e.k.236.4 24 232.85 odd 28
841.2.e.k.267.1 24 232.37 odd 28
841.2.e.k.267.4 24 232.21 odd 28
841.2.e.k.270.1 24 232.61 odd 28
841.2.e.k.270.4 24 232.229 odd 28
841.2.e.k.651.1 24 232.213 odd 28
841.2.e.k.651.4 24 232.77 odd 28
1421.2.a.j.1.1 2 56.13 odd 2
1856.2.a.r.1.1 2 1.1 even 1 trivial
1856.2.a.w.1.2 2 4.3 odd 2
3509.2.a.j.1.2 2 88.21 odd 2
4176.2.a.bq.1.2 2 24.11 even 2
4901.2.a.g.1.2 2 104.77 even 2
6525.2.a.o.1.1 2 120.29 odd 2
7569.2.a.c.1.1 2 696.173 odd 2
8381.2.a.e.1.1 2 136.101 even 2