Properties

Label 3920.2.a.ca.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
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 490)
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

\(f(q)\) \(=\) \(q+3.41421 q^{3} -1.00000 q^{5} +8.65685 q^{9} +O(q^{10})\) \(q+3.41421 q^{3} -1.00000 q^{5} +8.65685 q^{9} +0.828427 q^{11} +4.82843 q^{13} -3.41421 q^{15} -2.58579 q^{17} +0.585786 q^{19} +1.17157 q^{23} +1.00000 q^{25} +19.3137 q^{27} -4.82843 q^{29} -2.82843 q^{31} +2.82843 q^{33} -7.65685 q^{37} +16.4853 q^{39} +3.07107 q^{41} +8.82843 q^{43} -8.65685 q^{45} +5.17157 q^{47} -8.82843 q^{51} +6.48528 q^{53} -0.828427 q^{55} +2.00000 q^{57} +8.58579 q^{59} -9.31371 q^{61} -4.82843 q^{65} -1.65685 q^{67} +4.00000 q^{69} +4.48528 q^{71} +9.41421 q^{73} +3.41421 q^{75} +6.82843 q^{79} +39.9706 q^{81} -2.24264 q^{83} +2.58579 q^{85} -16.4853 q^{87} -12.7279 q^{89} -9.65685 q^{93} -0.585786 q^{95} +7.75736 q^{97} +7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 2q^{5} + 6q^{9} - 4q^{11} + 4q^{13} - 4q^{15} - 8q^{17} + 4q^{19} + 8q^{23} + 2q^{25} + 16q^{27} - 4q^{29} - 4q^{37} + 16q^{39} - 8q^{41} + 12q^{43} - 6q^{45} + 16q^{47} - 12q^{51} - 4q^{53} + 4q^{55} + 4q^{57} + 20q^{59} + 4q^{61} - 4q^{65} + 8q^{67} + 8q^{69} - 8q^{71} + 16q^{73} + 4q^{75} + 8q^{79} + 46q^{81} + 4q^{83} + 8q^{85} - 16q^{87} - 8q^{93} - 4q^{95} + 24q^{97} + 20q^{99} + O(q^{100}) \)

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
\(3\) 3.41421 1.97120 0.985599 0.169102i \(-0.0540867\pi\)
0.985599 + 0.169102i \(0.0540867\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 8.65685 2.88562
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) −2.58579 −0.627145 −0.313573 0.949564i \(-0.601526\pi\)
−0.313573 + 0.949564i \(0.601526\pi\)
\(18\) 0 0
\(19\) 0.585786 0.134389 0.0671943 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.17157 0.244290 0.122145 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 19.3137 3.71692
\(28\) 0 0
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 16.4853 2.63976
\(40\) 0 0
\(41\) 3.07107 0.479620 0.239810 0.970820i \(-0.422915\pi\)
0.239810 + 0.970820i \(0.422915\pi\)
\(42\) 0 0
\(43\) 8.82843 1.34632 0.673161 0.739496i \(-0.264936\pi\)
0.673161 + 0.739496i \(0.264936\pi\)
\(44\) 0 0
\(45\) −8.65685 −1.29049
\(46\) 0 0
\(47\) 5.17157 0.754351 0.377176 0.926142i \(-0.376895\pi\)
0.377176 + 0.926142i \(0.376895\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.82843 −1.23623
\(52\) 0 0
\(53\) 6.48528 0.890822 0.445411 0.895326i \(-0.353058\pi\)
0.445411 + 0.895326i \(0.353058\pi\)
\(54\) 0 0
\(55\) −0.828427 −0.111705
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 8.58579 1.11777 0.558887 0.829244i \(-0.311229\pi\)
0.558887 + 0.829244i \(0.311229\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.82843 −0.598893
\(66\) 0 0
\(67\) −1.65685 −0.202417 −0.101208 0.994865i \(-0.532271\pi\)
−0.101208 + 0.994865i \(0.532271\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) 9.41421 1.10185 0.550925 0.834555i \(-0.314275\pi\)
0.550925 + 0.834555i \(0.314275\pi\)
\(74\) 0 0
\(75\) 3.41421 0.394239
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.82843 0.768258 0.384129 0.923279i \(-0.374502\pi\)
0.384129 + 0.923279i \(0.374502\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 0 0
\(83\) −2.24264 −0.246162 −0.123081 0.992397i \(-0.539277\pi\)
−0.123081 + 0.992397i \(0.539277\pi\)
\(84\) 0 0
\(85\) 2.58579 0.280468
\(86\) 0 0
\(87\) −16.4853 −1.76741
\(88\) 0 0
\(89\) −12.7279 −1.34916 −0.674579 0.738203i \(-0.735675\pi\)
−0.674579 + 0.738203i \(0.735675\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −9.65685 −1.00137
\(94\) 0 0
\(95\) −0.585786 −0.0601004
\(96\) 0 0
\(97\) 7.75736 0.787641 0.393820 0.919187i \(-0.371153\pi\)
0.393820 + 0.919187i \(0.371153\pi\)
\(98\) 0 0
\(99\) 7.17157 0.720770
\(100\) 0 0
\(101\) −13.3137 −1.32476 −0.662382 0.749166i \(-0.730454\pi\)
−0.662382 + 0.749166i \(0.730454\pi\)
\(102\) 0 0
\(103\) −14.8284 −1.46109 −0.730544 0.682865i \(-0.760734\pi\)
−0.730544 + 0.682865i \(0.760734\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.65685 −0.933563 −0.466782 0.884373i \(-0.654587\pi\)
−0.466782 + 0.884373i \(0.654587\pi\)
\(108\) 0 0
\(109\) 2.48528 0.238047 0.119023 0.992891i \(-0.462024\pi\)
0.119023 + 0.992891i \(0.462024\pi\)
\(110\) 0 0
\(111\) −26.1421 −2.48130
\(112\) 0 0
\(113\) −15.3137 −1.44059 −0.720296 0.693667i \(-0.755994\pi\)
−0.720296 + 0.693667i \(0.755994\pi\)
\(114\) 0 0
\(115\) −1.17157 −0.109250
\(116\) 0 0
\(117\) 41.7990 3.86432
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) 10.4853 0.945426
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) 0 0
\(129\) 30.1421 2.65387
\(130\) 0 0
\(131\) −6.24264 −0.545422 −0.272711 0.962096i \(-0.587920\pi\)
−0.272711 + 0.962096i \(0.587920\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −19.3137 −1.66226
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) 19.8995 1.68785 0.843927 0.536459i \(-0.180238\pi\)
0.843927 + 0.536459i \(0.180238\pi\)
\(140\) 0 0
\(141\) 17.6569 1.48698
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 4.82843 0.400979
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 11.3137 0.920697 0.460348 0.887738i \(-0.347725\pi\)
0.460348 + 0.887738i \(0.347725\pi\)
\(152\) 0 0
\(153\) −22.3848 −1.80970
\(154\) 0 0
\(155\) 2.82843 0.227185
\(156\) 0 0
\(157\) −6.48528 −0.517582 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) 0 0
\(159\) 22.1421 1.75599
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.1421 1.57765 0.788827 0.614615i \(-0.210689\pi\)
0.788827 + 0.614615i \(0.210689\pi\)
\(164\) 0 0
\(165\) −2.82843 −0.220193
\(166\) 0 0
\(167\) 15.7990 1.22256 0.611281 0.791413i \(-0.290654\pi\)
0.611281 + 0.791413i \(0.290654\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) 5.07107 0.387794
\(172\) 0 0
\(173\) 8.82843 0.671213 0.335606 0.942002i \(-0.391059\pi\)
0.335606 + 0.942002i \(0.391059\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 29.3137 2.20335
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 2.48528 0.184730 0.0923648 0.995725i \(-0.470557\pi\)
0.0923648 + 0.995725i \(0.470557\pi\)
\(182\) 0 0
\(183\) −31.7990 −2.35065
\(184\) 0 0
\(185\) 7.65685 0.562943
\(186\) 0 0
\(187\) −2.14214 −0.156648
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.1421 −0.733859 −0.366930 0.930249i \(-0.619591\pi\)
−0.366930 + 0.930249i \(0.619591\pi\)
\(192\) 0 0
\(193\) 5.65685 0.407189 0.203595 0.979055i \(-0.434738\pi\)
0.203595 + 0.979055i \(0.434738\pi\)
\(194\) 0 0
\(195\) −16.4853 −1.18054
\(196\) 0 0
\(197\) −25.7990 −1.83810 −0.919051 0.394139i \(-0.871043\pi\)
−0.919051 + 0.394139i \(0.871043\pi\)
\(198\) 0 0
\(199\) 16.4853 1.16861 0.584305 0.811534i \(-0.301367\pi\)
0.584305 + 0.811534i \(0.301367\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.07107 −0.214493
\(206\) 0 0
\(207\) 10.1421 0.704927
\(208\) 0 0
\(209\) 0.485281 0.0335676
\(210\) 0 0
\(211\) −18.6274 −1.28236 −0.641182 0.767389i \(-0.721556\pi\)
−0.641182 + 0.767389i \(0.721556\pi\)
\(212\) 0 0
\(213\) 15.3137 1.04928
\(214\) 0 0
\(215\) −8.82843 −0.602094
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 32.1421 2.17196
\(220\) 0 0
\(221\) −12.4853 −0.839851
\(222\) 0 0
\(223\) −7.31371 −0.489762 −0.244881 0.969553i \(-0.578749\pi\)
−0.244881 + 0.969553i \(0.578749\pi\)
\(224\) 0 0
\(225\) 8.65685 0.577124
\(226\) 0 0
\(227\) 18.2426 1.21081 0.605403 0.795919i \(-0.293012\pi\)
0.605403 + 0.795919i \(0.293012\pi\)
\(228\) 0 0
\(229\) 16.1421 1.06670 0.533351 0.845894i \(-0.320932\pi\)
0.533351 + 0.845894i \(0.320932\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.3137 1.52733 0.763666 0.645612i \(-0.223397\pi\)
0.763666 + 0.645612i \(0.223397\pi\)
\(234\) 0 0
\(235\) −5.17157 −0.337356
\(236\) 0 0
\(237\) 23.3137 1.51439
\(238\) 0 0
\(239\) −1.65685 −0.107173 −0.0535865 0.998563i \(-0.517065\pi\)
−0.0535865 + 0.998563i \(0.517065\pi\)
\(240\) 0 0
\(241\) 13.4142 0.864085 0.432043 0.901853i \(-0.357793\pi\)
0.432043 + 0.901853i \(0.357793\pi\)
\(242\) 0 0
\(243\) 78.5269 5.03750
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.82843 0.179969
\(248\) 0 0
\(249\) −7.65685 −0.485233
\(250\) 0 0
\(251\) 0.585786 0.0369745 0.0184873 0.999829i \(-0.494115\pi\)
0.0184873 + 0.999829i \(0.494115\pi\)
\(252\) 0 0
\(253\) 0.970563 0.0610188
\(254\) 0 0
\(255\) 8.82843 0.552858
\(256\) 0 0
\(257\) 9.89949 0.617514 0.308757 0.951141i \(-0.400087\pi\)
0.308757 + 0.951141i \(0.400087\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −41.7990 −2.58729
\(262\) 0 0
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) −6.48528 −0.398388
\(266\) 0 0
\(267\) −43.4558 −2.65945
\(268\) 0 0
\(269\) −18.4853 −1.12707 −0.563534 0.826093i \(-0.690559\pi\)
−0.563534 + 0.826093i \(0.690559\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.828427 0.0499560
\(276\) 0 0
\(277\) −8.14214 −0.489214 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(278\) 0 0
\(279\) −24.4853 −1.46590
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) −2.24264 −0.133311 −0.0666556 0.997776i \(-0.521233\pi\)
−0.0666556 + 0.997776i \(0.521233\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.3137 −0.606689
\(290\) 0 0
\(291\) 26.4853 1.55259
\(292\) 0 0
\(293\) 8.34315 0.487412 0.243706 0.969849i \(-0.421637\pi\)
0.243706 + 0.969849i \(0.421637\pi\)
\(294\) 0 0
\(295\) −8.58579 −0.499884
\(296\) 0 0
\(297\) 16.0000 0.928414
\(298\) 0 0
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −45.4558 −2.61137
\(304\) 0 0
\(305\) 9.31371 0.533301
\(306\) 0 0
\(307\) 14.9289 0.852039 0.426020 0.904714i \(-0.359915\pi\)
0.426020 + 0.904714i \(0.359915\pi\)
\(308\) 0 0
\(309\) −50.6274 −2.88009
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −14.3848 −0.813076 −0.406538 0.913634i \(-0.633264\pi\)
−0.406538 + 0.913634i \(0.633264\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.4853 0.588912 0.294456 0.955665i \(-0.404862\pi\)
0.294456 + 0.955665i \(0.404862\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) −32.9706 −1.84024
\(322\) 0 0
\(323\) −1.51472 −0.0842812
\(324\) 0 0
\(325\) 4.82843 0.267833
\(326\) 0 0
\(327\) 8.48528 0.469237
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −33.7990 −1.85776 −0.928880 0.370380i \(-0.879227\pi\)
−0.928880 + 0.370380i \(0.879227\pi\)
\(332\) 0 0
\(333\) −66.2843 −3.63236
\(334\) 0 0
\(335\) 1.65685 0.0905236
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) −52.2843 −2.83969
\(340\) 0 0
\(341\) −2.34315 −0.126888
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 3.17157 0.170259 0.0851295 0.996370i \(-0.472870\pi\)
0.0851295 + 0.996370i \(0.472870\pi\)
\(348\) 0 0
\(349\) −2.48528 −0.133034 −0.0665170 0.997785i \(-0.521189\pi\)
−0.0665170 + 0.997785i \(0.521189\pi\)
\(350\) 0 0
\(351\) 93.2548 4.97757
\(352\) 0 0
\(353\) 2.38478 0.126929 0.0634644 0.997984i \(-0.479785\pi\)
0.0634644 + 0.997984i \(0.479785\pi\)
\(354\) 0 0
\(355\) −4.48528 −0.238054
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) 0 0
\(363\) −35.2132 −1.84821
\(364\) 0 0
\(365\) −9.41421 −0.492762
\(366\) 0 0
\(367\) 24.9706 1.30345 0.651726 0.758454i \(-0.274045\pi\)
0.651726 + 0.758454i \(0.274045\pi\)
\(368\) 0 0
\(369\) 26.5858 1.38400
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −30.4853 −1.57847 −0.789234 0.614093i \(-0.789522\pi\)
−0.789234 + 0.614093i \(0.789522\pi\)
\(374\) 0 0
\(375\) −3.41421 −0.176309
\(376\) 0 0
\(377\) −23.3137 −1.20072
\(378\) 0 0
\(379\) −34.4853 −1.77139 −0.885695 0.464268i \(-0.846318\pi\)
−0.885695 + 0.464268i \(0.846318\pi\)
\(380\) 0 0
\(381\) −9.65685 −0.494736
\(382\) 0 0
\(383\) −32.4853 −1.65992 −0.829960 0.557823i \(-0.811637\pi\)
−0.829960 + 0.557823i \(0.811637\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 76.4264 3.88497
\(388\) 0 0
\(389\) 28.1421 1.42686 0.713431 0.700725i \(-0.247140\pi\)
0.713431 + 0.700725i \(0.247140\pi\)
\(390\) 0 0
\(391\) −3.02944 −0.153205
\(392\) 0 0
\(393\) −21.3137 −1.07513
\(394\) 0 0
\(395\) −6.82843 −0.343575
\(396\) 0 0
\(397\) −33.7990 −1.69632 −0.848161 0.529738i \(-0.822290\pi\)
−0.848161 + 0.529738i \(0.822290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −13.6569 −0.680296
\(404\) 0 0
\(405\) −39.9706 −1.98615
\(406\) 0 0
\(407\) −6.34315 −0.314418
\(408\) 0 0
\(409\) 10.5858 0.523433 0.261717 0.965145i \(-0.415711\pi\)
0.261717 + 0.965145i \(0.415711\pi\)
\(410\) 0 0
\(411\) −54.6274 −2.69457
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2.24264 0.110087
\(416\) 0 0
\(417\) 67.9411 3.32709
\(418\) 0 0
\(419\) −20.8701 −1.01957 −0.509785 0.860302i \(-0.670275\pi\)
−0.509785 + 0.860302i \(0.670275\pi\)
\(420\) 0 0
\(421\) 17.3137 0.843819 0.421909 0.906638i \(-0.361360\pi\)
0.421909 + 0.906638i \(0.361360\pi\)
\(422\) 0 0
\(423\) 44.7696 2.17677
\(424\) 0 0
\(425\) −2.58579 −0.125429
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 13.6569 0.659359
\(430\) 0 0
\(431\) 22.3431 1.07623 0.538116 0.842871i \(-0.319136\pi\)
0.538116 + 0.842871i \(0.319136\pi\)
\(432\) 0 0
\(433\) −10.5858 −0.508720 −0.254360 0.967110i \(-0.581865\pi\)
−0.254360 + 0.967110i \(0.581865\pi\)
\(434\) 0 0
\(435\) 16.4853 0.790409
\(436\) 0 0
\(437\) 0.686292 0.0328298
\(438\) 0 0
\(439\) −24.9706 −1.19178 −0.595890 0.803066i \(-0.703201\pi\)
−0.595890 + 0.803066i \(0.703201\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.02944 −0.143933 −0.0719665 0.997407i \(-0.522927\pi\)
−0.0719665 + 0.997407i \(0.522927\pi\)
\(444\) 0 0
\(445\) 12.7279 0.603361
\(446\) 0 0
\(447\) −20.4853 −0.968921
\(448\) 0 0
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) 2.54416 0.119800
\(452\) 0 0
\(453\) 38.6274 1.81487
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 21.6569 1.01306 0.506532 0.862221i \(-0.330927\pi\)
0.506532 + 0.862221i \(0.330927\pi\)
\(458\) 0 0
\(459\) −49.9411 −2.33105
\(460\) 0 0
\(461\) 12.8284 0.597479 0.298740 0.954335i \(-0.403434\pi\)
0.298740 + 0.954335i \(0.403434\pi\)
\(462\) 0 0
\(463\) 16.9706 0.788689 0.394344 0.918963i \(-0.370972\pi\)
0.394344 + 0.918963i \(0.370972\pi\)
\(464\) 0 0
\(465\) 9.65685 0.447826
\(466\) 0 0
\(467\) −15.8995 −0.735741 −0.367870 0.929877i \(-0.619913\pi\)
−0.367870 + 0.929877i \(0.619913\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −22.1421 −1.02026
\(472\) 0 0
\(473\) 7.31371 0.336285
\(474\) 0 0
\(475\) 0.585786 0.0268777
\(476\) 0 0
\(477\) 56.1421 2.57057
\(478\) 0 0
\(479\) −17.1716 −0.784589 −0.392295 0.919840i \(-0.628319\pi\)
−0.392295 + 0.919840i \(0.628319\pi\)
\(480\) 0 0
\(481\) −36.9706 −1.68571
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.75736 −0.352244
\(486\) 0 0
\(487\) −31.7990 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(488\) 0 0
\(489\) 68.7696 3.10987
\(490\) 0 0
\(491\) −32.2843 −1.45697 −0.728484 0.685062i \(-0.759775\pi\)
−0.728484 + 0.685062i \(0.759775\pi\)
\(492\) 0 0
\(493\) 12.4853 0.562309
\(494\) 0 0
\(495\) −7.17157 −0.322338
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −30.3431 −1.35835 −0.679173 0.733978i \(-0.737661\pi\)
−0.679173 + 0.733978i \(0.737661\pi\)
\(500\) 0 0
\(501\) 53.9411 2.40991
\(502\) 0 0
\(503\) 17.6569 0.787280 0.393640 0.919265i \(-0.371216\pi\)
0.393640 + 0.919265i \(0.371216\pi\)
\(504\) 0 0
\(505\) 13.3137 0.592452
\(506\) 0 0
\(507\) 35.2132 1.56387
\(508\) 0 0
\(509\) −5.79899 −0.257036 −0.128518 0.991707i \(-0.541022\pi\)
−0.128518 + 0.991707i \(0.541022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 11.3137 0.499512
\(514\) 0 0
\(515\) 14.8284 0.653419
\(516\) 0 0
\(517\) 4.28427 0.188422
\(518\) 0 0
\(519\) 30.1421 1.32309
\(520\) 0 0
\(521\) −19.0711 −0.835519 −0.417759 0.908558i \(-0.637184\pi\)
−0.417759 + 0.908558i \(0.637184\pi\)
\(522\) 0 0
\(523\) 23.8995 1.04505 0.522526 0.852623i \(-0.324990\pi\)
0.522526 + 0.852623i \(0.324990\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.31371 0.318590
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) 74.3259 3.22547
\(532\) 0 0
\(533\) 14.8284 0.642290
\(534\) 0 0
\(535\) 9.65685 0.417502
\(536\) 0 0
\(537\) −13.6569 −0.589337
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) 0 0
\(543\) 8.48528 0.364138
\(544\) 0 0
\(545\) −2.48528 −0.106458
\(546\) 0 0
\(547\) 10.4853 0.448318 0.224159 0.974553i \(-0.428036\pi\)
0.224159 + 0.974553i \(0.428036\pi\)
\(548\) 0 0
\(549\) −80.6274 −3.44109
\(550\) 0 0
\(551\) −2.82843 −0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 26.1421 1.10967
\(556\) 0 0
\(557\) 15.1716 0.642840 0.321420 0.946937i \(-0.395840\pi\)
0.321420 + 0.946937i \(0.395840\pi\)
\(558\) 0 0
\(559\) 42.6274 1.80295
\(560\) 0 0
\(561\) −7.31371 −0.308785
\(562\) 0 0
\(563\) −36.5858 −1.54191 −0.770954 0.636891i \(-0.780220\pi\)
−0.770954 + 0.636891i \(0.780220\pi\)
\(564\) 0 0
\(565\) 15.3137 0.644253
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −29.3137 −1.22889 −0.614447 0.788958i \(-0.710621\pi\)
−0.614447 + 0.788958i \(0.710621\pi\)
\(570\) 0 0
\(571\) 2.20101 0.0921094 0.0460547 0.998939i \(-0.485335\pi\)
0.0460547 + 0.998939i \(0.485335\pi\)
\(572\) 0 0
\(573\) −34.6274 −1.44658
\(574\) 0 0
\(575\) 1.17157 0.0488580
\(576\) 0 0
\(577\) −6.10051 −0.253967 −0.126984 0.991905i \(-0.540530\pi\)
−0.126984 + 0.991905i \(0.540530\pi\)
\(578\) 0 0
\(579\) 19.3137 0.802650
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 5.37258 0.222510
\(584\) 0 0
\(585\) −41.7990 −1.72818
\(586\) 0 0
\(587\) 17.0711 0.704598 0.352299 0.935887i \(-0.385400\pi\)
0.352299 + 0.935887i \(0.385400\pi\)
\(588\) 0 0
\(589\) −1.65685 −0.0682695
\(590\) 0 0
\(591\) −88.0833 −3.62326
\(592\) 0 0
\(593\) −3.27208 −0.134368 −0.0671841 0.997741i \(-0.521401\pi\)
−0.0671841 + 0.997741i \(0.521401\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 56.2843 2.30356
\(598\) 0 0
\(599\) 10.8284 0.442438 0.221219 0.975224i \(-0.428997\pi\)
0.221219 + 0.975224i \(0.428997\pi\)
\(600\) 0 0
\(601\) −6.58579 −0.268640 −0.134320 0.990938i \(-0.542885\pi\)
−0.134320 + 0.990938i \(0.542885\pi\)
\(602\) 0 0
\(603\) −14.3431 −0.584098
\(604\) 0 0
\(605\) 10.3137 0.419312
\(606\) 0 0
\(607\) 16.2843 0.660958 0.330479 0.943813i \(-0.392790\pi\)
0.330479 + 0.943813i \(0.392790\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.9706 1.01020
\(612\) 0 0
\(613\) −12.3431 −0.498535 −0.249267 0.968435i \(-0.580190\pi\)
−0.249267 + 0.968435i \(0.580190\pi\)
\(614\) 0 0
\(615\) −10.4853 −0.422807
\(616\) 0 0
\(617\) −33.3137 −1.34116 −0.670580 0.741837i \(-0.733955\pi\)
−0.670580 + 0.741837i \(0.733955\pi\)
\(618\) 0 0
\(619\) 29.0711 1.16846 0.584232 0.811586i \(-0.301396\pi\)
0.584232 + 0.811586i \(0.301396\pi\)
\(620\) 0 0
\(621\) 22.6274 0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.65685 0.0661684
\(628\) 0 0
\(629\) 19.7990 0.789437
\(630\) 0 0
\(631\) 12.4853 0.497031 0.248516 0.968628i \(-0.420057\pi\)
0.248516 + 0.968628i \(0.420057\pi\)
\(632\) 0 0
\(633\) −63.5980 −2.52779
\(634\) 0 0
\(635\) 2.82843 0.112243
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 38.8284 1.53603
\(640\) 0 0
\(641\) −24.6274 −0.972724 −0.486362 0.873757i \(-0.661676\pi\)
−0.486362 + 0.873757i \(0.661676\pi\)
\(642\) 0 0
\(643\) 4.78680 0.188773 0.0943864 0.995536i \(-0.469911\pi\)
0.0943864 + 0.995536i \(0.469911\pi\)
\(644\) 0 0
\(645\) −30.1421 −1.18685
\(646\) 0 0
\(647\) −23.1127 −0.908654 −0.454327 0.890835i \(-0.650120\pi\)
−0.454327 + 0.890835i \(0.650120\pi\)
\(648\) 0 0
\(649\) 7.11270 0.279198
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.34315 0.169960 0.0849802 0.996383i \(-0.472917\pi\)
0.0849802 + 0.996383i \(0.472917\pi\)
\(654\) 0 0
\(655\) 6.24264 0.243920
\(656\) 0 0
\(657\) 81.4975 3.17952
\(658\) 0 0
\(659\) 27.1716 1.05845 0.529227 0.848480i \(-0.322482\pi\)
0.529227 + 0.848480i \(0.322482\pi\)
\(660\) 0 0
\(661\) 38.2843 1.48909 0.744543 0.667575i \(-0.232668\pi\)
0.744543 + 0.667575i \(0.232668\pi\)
\(662\) 0 0
\(663\) −42.6274 −1.65551
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.65685 −0.219034
\(668\) 0 0
\(669\) −24.9706 −0.965418
\(670\) 0 0
\(671\) −7.71573 −0.297862
\(672\) 0 0
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) 0 0
\(675\) 19.3137 0.743385
\(676\) 0 0
\(677\) −39.4558 −1.51641 −0.758206 0.652015i \(-0.773924\pi\)
−0.758206 + 0.652015i \(0.773924\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 62.2843 2.38674
\(682\) 0 0
\(683\) −33.6569 −1.28784 −0.643922 0.765091i \(-0.722694\pi\)
−0.643922 + 0.765091i \(0.722694\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 55.1127 2.10268
\(688\) 0 0
\(689\) 31.3137 1.19296
\(690\) 0 0
\(691\) 1.75736 0.0668531 0.0334265 0.999441i \(-0.489358\pi\)
0.0334265 + 0.999441i \(0.489358\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −19.8995 −0.754831
\(696\) 0 0
\(697\) −7.94113 −0.300792
\(698\) 0 0
\(699\) 79.5980 3.01067
\(700\) 0 0
\(701\) 2.48528 0.0938678 0.0469339 0.998898i \(-0.485055\pi\)
0.0469339 + 0.998898i \(0.485055\pi\)
\(702\) 0 0
\(703\) −4.48528 −0.169166
\(704\) 0 0
\(705\) −17.6569 −0.664996
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 45.1127 1.69424 0.847121 0.531399i \(-0.178334\pi\)
0.847121 + 0.531399i \(0.178334\pi\)
\(710\) 0 0
\(711\) 59.1127 2.21690
\(712\) 0 0
\(713\) −3.31371 −0.124099
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) −5.65685 −0.211259
\(718\) 0 0
\(719\) 41.4558 1.54604 0.773021 0.634380i \(-0.218745\pi\)
0.773021 + 0.634380i \(0.218745\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 45.7990 1.70328
\(724\) 0 0
\(725\) −4.82843 −0.179323
\(726\) 0 0
\(727\) −3.51472 −0.130354 −0.0651768 0.997874i \(-0.520761\pi\)
−0.0651768 + 0.997874i \(0.520761\pi\)
\(728\) 0 0
\(729\) 148.196 5.48874
\(730\) 0 0
\(731\) −22.8284 −0.844340
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.37258 −0.0505597
\(738\) 0 0
\(739\) −3.17157 −0.116668 −0.0583341 0.998297i \(-0.518579\pi\)
−0.0583341 + 0.998297i \(0.518579\pi\)
\(740\) 0 0
\(741\) 9.65685 0.354753
\(742\) 0 0
\(743\) −51.7990 −1.90032 −0.950160 0.311762i \(-0.899081\pi\)
−0.950160 + 0.311762i \(0.899081\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −19.4142 −0.710329
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 39.3137 1.43458 0.717289 0.696776i \(-0.245383\pi\)
0.717289 + 0.696776i \(0.245383\pi\)
\(752\) 0 0
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) −11.3137 −0.411748
\(756\) 0 0
\(757\) 3.65685 0.132911 0.0664553 0.997789i \(-0.478831\pi\)
0.0664553 + 0.997789i \(0.478831\pi\)
\(758\) 0 0
\(759\) 3.31371 0.120280
\(760\) 0 0
\(761\) −22.3848 −0.811448 −0.405724 0.913996i \(-0.632980\pi\)
−0.405724 + 0.913996i \(0.632980\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 22.3848 0.809323
\(766\) 0 0
\(767\) 41.4558 1.49688
\(768\) 0 0
\(769\) −19.5563 −0.705220 −0.352610 0.935770i \(-0.614706\pi\)
−0.352610 + 0.935770i \(0.614706\pi\)
\(770\) 0 0
\(771\) 33.7990 1.21724
\(772\) 0 0
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) −2.82843 −0.101600
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.79899 0.0644555
\(780\) 0 0
\(781\) 3.71573 0.132959
\(782\) 0 0
\(783\) −93.2548 −3.33266
\(784\) 0 0
\(785\) 6.48528 0.231470
\(786\) 0 0
\(787\) −1.27208 −0.0453447 −0.0226723 0.999743i \(-0.507217\pi\)
−0.0226723 + 0.999743i \(0.507217\pi\)
\(788\) 0 0
\(789\) −95.5980 −3.40338
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −44.9706 −1.59695
\(794\) 0 0
\(795\) −22.1421 −0.785301
\(796\) 0 0
\(797\) 41.7990 1.48060 0.740298 0.672279i \(-0.234684\pi\)
0.740298 + 0.672279i \(0.234684\pi\)
\(798\) 0 0
\(799\) −13.3726 −0.473088
\(800\) 0 0
\(801\) −110.184 −3.89315
\(802\) 0 0
\(803\) 7.79899 0.275220
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −63.1127 −2.22167
\(808\) 0 0
\(809\) 3.02944 0.106509 0.0532547 0.998581i \(-0.483040\pi\)
0.0532547 + 0.998581i \(0.483040\pi\)
\(810\) 0 0
\(811\) 32.5858 1.14424 0.572121 0.820169i \(-0.306121\pi\)
0.572121 + 0.820169i \(0.306121\pi\)
\(812\) 0 0
\(813\) 40.9706 1.43690
\(814\) 0 0
\(815\) −20.1421 −0.705548
\(816\) 0 0
\(817\) 5.17157 0.180930
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17.3137 0.604253 0.302126 0.953268i \(-0.402304\pi\)
0.302126 + 0.953268i \(0.402304\pi\)
\(822\) 0 0
\(823\) −20.2843 −0.707065 −0.353533 0.935422i \(-0.615020\pi\)
−0.353533 + 0.935422i \(0.615020\pi\)
\(824\) 0 0
\(825\) 2.82843 0.0984732
\(826\) 0 0
\(827\) 5.37258 0.186823 0.0934115 0.995628i \(-0.470223\pi\)
0.0934115 + 0.995628i \(0.470223\pi\)
\(828\) 0 0
\(829\) 5.02944 0.174680 0.0873398 0.996179i \(-0.472163\pi\)
0.0873398 + 0.996179i \(0.472163\pi\)
\(830\) 0 0
\(831\) −27.7990 −0.964336
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −15.7990 −0.546747
\(836\) 0 0
\(837\) −54.6274 −1.88820
\(838\) 0 0
\(839\) −42.1421 −1.45491 −0.727454 0.686156i \(-0.759297\pi\)
−0.727454 + 0.686156i \(0.759297\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 0 0
\(843\) 27.3137 0.940734
\(844\) 0 0
\(845\) −10.3137 −0.354802
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7.65685 −0.262783
\(850\) 0 0
\(851\) −8.97056 −0.307507
\(852\) 0 0
\(853\) 43.1716 1.47817 0.739083 0.673614i \(-0.235259\pi\)
0.739083 + 0.673614i \(0.235259\pi\)
\(854\) 0 0
\(855\) −5.07107 −0.173427
\(856\) 0 0
\(857\) 4.92893 0.168369 0.0841846 0.996450i \(-0.473171\pi\)
0.0841846 + 0.996450i \(0.473171\pi\)
\(858\) 0 0
\(859\) −7.21320 −0.246111 −0.123056 0.992400i \(-0.539269\pi\)
−0.123056 + 0.992400i \(0.539269\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.97056 0.169200 0.0846000 0.996415i \(-0.473039\pi\)
0.0846000 + 0.996415i \(0.473039\pi\)
\(864\) 0 0
\(865\) −8.82843 −0.300176
\(866\) 0 0
\(867\) −35.2132 −1.19590
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 67.1543 2.27283
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.2843 −1.02263 −0.511314 0.859394i \(-0.670841\pi\)
−0.511314 + 0.859394i \(0.670841\pi\)
\(878\) 0 0
\(879\) 28.4853 0.960785
\(880\) 0 0
\(881\) 2.38478 0.0803452 0.0401726 0.999193i \(-0.487209\pi\)
0.0401726 + 0.999193i \(0.487209\pi\)
\(882\) 0 0
\(883\) 41.6569 1.40186 0.700932 0.713228i \(-0.252767\pi\)
0.700932 + 0.713228i \(0.252767\pi\)
\(884\) 0 0
\(885\) −29.3137 −0.985370
\(886\) 0 0
\(887\) −55.1127 −1.85050 −0.925252 0.379354i \(-0.876146\pi\)
−0.925252 + 0.379354i \(0.876146\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 33.1127 1.10932
\(892\) 0 0
\(893\) 3.02944 0.101376
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 19.3137 0.644866
\(898\) 0 0
\(899\) 13.6569 0.455482
\(900\) 0 0
\(901\) −16.7696 −0.558675
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.48528 −0.0826135
\(906\) 0 0
\(907\) −0.284271 −0.00943907 −0.00471954 0.999989i \(-0.501502\pi\)
−0.00471954 + 0.999989i \(0.501502\pi\)
\(908\) 0 0
\(909\) −115.255 −3.82276
\(910\) 0 0
\(911\) −36.2843 −1.20215 −0.601076 0.799192i \(-0.705261\pi\)
−0.601076 + 0.799192i \(0.705261\pi\)
\(912\) 0 0
\(913\) −1.85786 −0.0614863
\(914\) 0 0
\(915\) 31.7990 1.05124
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −15.5147 −0.511783 −0.255892 0.966705i \(-0.582369\pi\)
−0.255892 + 0.966705i \(0.582369\pi\)
\(920\) 0 0
\(921\) 50.9706 1.67954
\(922\) 0 0
\(923\) 21.6569 0.712844
\(924\) 0 0
\(925\) −7.65685 −0.251756
\(926\) 0 0
\(927\) −128.368 −4.21614
\(928\) 0 0
\(929\) 17.2132 0.564747 0.282373 0.959305i \(-0.408878\pi\)
0.282373 + 0.959305i \(0.408878\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13.6569 0.447105
\(934\) 0 0
\(935\) 2.14214 0.0700553
\(936\) 0 0
\(937\) −20.2426 −0.661298 −0.330649 0.943754i \(-0.607268\pi\)
−0.330649 + 0.943754i \(0.607268\pi\)
\(938\) 0 0
\(939\) −49.1127 −1.60273
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) 3.59798 0.117166
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.82843 0.156903 0.0784514 0.996918i \(-0.475002\pi\)
0.0784514 + 0.996918i \(0.475002\pi\)
\(948\) 0 0
\(949\) 45.4558 1.47556
\(950\) 0 0
\(951\) 35.7990 1.16086
\(952\) 0 0
\(953\) 0.343146 0.0111156 0.00555779 0.999985i \(-0.498231\pi\)
0.00555779 + 0.999985i \(0.498231\pi\)
\(954\) 0 0
\(955\) 10.1421 0.328192
\(956\) 0 0
\(957\) −13.6569 −0.441463
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) −83.5980 −2.69391
\(964\) 0 0
\(965\) −5.65685 −0.182101
\(966\) 0 0
\(967\) 37.4558 1.20450 0.602249 0.798308i \(-0.294271\pi\)
0.602249 + 0.798308i \(0.294271\pi\)
\(968\) 0 0
\(969\) −5.17157 −0.166135
\(970\) 0 0
\(971\) −33.3553 −1.07042 −0.535212 0.844718i \(-0.679768\pi\)
−0.535212 + 0.844718i \(0.679768\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 16.4853 0.527952
\(976\) 0 0
\(977\) −12.6863 −0.405870 −0.202935 0.979192i \(-0.565048\pi\)
−0.202935 + 0.979192i \(0.565048\pi\)
\(978\) 0 0
\(979\) −10.5442 −0.336993
\(980\) 0 0
\(981\) 21.5147 0.686912
\(982\) 0 0
\(983\) −12.2010 −0.389152 −0.194576 0.980887i \(-0.562333\pi\)
−0.194576 + 0.980887i \(0.562333\pi\)
\(984\) 0 0
\(985\) 25.7990 0.822024
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.3431 0.328893
\(990\) 0 0
\(991\) −44.7696 −1.42215 −0.711076 0.703115i \(-0.751792\pi\)
−0.711076 + 0.703115i \(0.751792\pi\)
\(992\) 0 0
\(993\) −115.397 −3.66201
\(994\) 0 0
\(995\) −16.4853 −0.522619
\(996\) 0 0
\(997\) 18.2843 0.579069 0.289534 0.957168i \(-0.406500\pi\)
0.289534 + 0.957168i \(0.406500\pi\)
\(998\) 0 0
\(999\) −147.882 −4.67879
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.ca.1.2 2
4.3 odd 2 490.2.a.l.1.1 2
7.6 odd 2 3920.2.a.bm.1.1 2
12.11 even 2 4410.2.a.by.1.2 2
20.3 even 4 2450.2.c.w.99.3 4
20.7 even 4 2450.2.c.w.99.2 4
20.19 odd 2 2450.2.a.bs.1.2 2
28.3 even 6 490.2.e.i.471.1 4
28.11 odd 6 490.2.e.j.471.2 4
28.19 even 6 490.2.e.i.361.1 4
28.23 odd 6 490.2.e.j.361.2 4
28.27 even 2 490.2.a.m.1.2 yes 2
84.83 odd 2 4410.2.a.bt.1.2 2
140.27 odd 4 2450.2.c.t.99.1 4
140.83 odd 4 2450.2.c.t.99.4 4
140.139 even 2 2450.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 4.3 odd 2
490.2.a.m.1.2 yes 2 28.27 even 2
490.2.e.i.361.1 4 28.19 even 6
490.2.e.i.471.1 4 28.3 even 6
490.2.e.j.361.2 4 28.23 odd 6
490.2.e.j.471.2 4 28.11 odd 6
2450.2.a.bn.1.1 2 140.139 even 2
2450.2.a.bs.1.2 2 20.19 odd 2
2450.2.c.t.99.1 4 140.27 odd 4
2450.2.c.t.99.4 4 140.83 odd 4
2450.2.c.w.99.2 4 20.7 even 4
2450.2.c.w.99.3 4 20.3 even 4
3920.2.a.bm.1.1 2 7.6 odd 2
3920.2.a.ca.1.2 2 1.1 even 1 trivial
4410.2.a.bt.1.2 2 84.83 odd 2
4410.2.a.by.1.2 2 12.11 even 2