Properties

Label 825.2.a.g.1.2
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} -0.828427 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +2.41421 q^{6} -0.828427 q^{7} +4.41421 q^{8} +1.00000 q^{9} -1.00000 q^{11} +3.82843 q^{12} +5.65685 q^{13} -2.00000 q^{14} +3.00000 q^{16} +1.17157 q^{17} +2.41421 q^{18} -6.82843 q^{19} -0.828427 q^{21} -2.41421 q^{22} +4.00000 q^{23} +4.41421 q^{24} +13.6569 q^{26} +1.00000 q^{27} -3.17157 q^{28} -4.82843 q^{29} -1.58579 q^{32} -1.00000 q^{33} +2.82843 q^{34} +3.82843 q^{36} -11.6569 q^{37} -16.4853 q^{38} +5.65685 q^{39} +4.82843 q^{41} -2.00000 q^{42} +8.82843 q^{43} -3.82843 q^{44} +9.65685 q^{46} +4.00000 q^{47} +3.00000 q^{48} -6.31371 q^{49} +1.17157 q^{51} +21.6569 q^{52} -9.31371 q^{53} +2.41421 q^{54} -3.65685 q^{56} -6.82843 q^{57} -11.6569 q^{58} -4.00000 q^{59} -11.6569 q^{61} -0.828427 q^{63} -9.82843 q^{64} -2.41421 q^{66} +5.65685 q^{67} +4.48528 q^{68} +4.00000 q^{69} +2.34315 q^{71} +4.41421 q^{72} -11.3137 q^{73} -28.1421 q^{74} -26.1421 q^{76} +0.828427 q^{77} +13.6569 q^{78} +8.48528 q^{79} +1.00000 q^{81} +11.6569 q^{82} +10.0000 q^{83} -3.17157 q^{84} +21.3137 q^{86} -4.82843 q^{87} -4.41421 q^{88} +3.65685 q^{89} -4.68629 q^{91} +15.3137 q^{92} +9.65685 q^{94} -1.58579 q^{96} -11.6569 q^{97} -15.2426 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} - 2q^{11} + 2q^{12} - 4q^{14} + 6q^{16} + 8q^{17} + 2q^{18} - 8q^{19} + 4q^{21} - 2q^{22} + 8q^{23} + 6q^{24} + 16q^{26} + 2q^{27} - 12q^{28} - 4q^{29} - 6q^{32} - 2q^{33} + 2q^{36} - 12q^{37} - 16q^{38} + 4q^{41} - 4q^{42} + 12q^{43} - 2q^{44} + 8q^{46} + 8q^{47} + 6q^{48} + 10q^{49} + 8q^{51} + 32q^{52} + 4q^{53} + 2q^{54} + 4q^{56} - 8q^{57} - 12q^{58} - 8q^{59} - 12q^{61} + 4q^{63} - 14q^{64} - 2q^{66} - 8q^{68} + 8q^{69} + 16q^{71} + 6q^{72} - 28q^{74} - 24q^{76} - 4q^{77} + 16q^{78} + 2q^{81} + 12q^{82} + 20q^{83} - 12q^{84} + 20q^{86} - 4q^{87} - 6q^{88} - 4q^{89} - 32q^{91} + 8q^{92} + 8q^{94} - 6q^{96} - 12q^{97} - 22q^{98} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 2.41421 0.985599
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 3.82843 1.10517
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 2.41421 0.569036
\(19\) −6.82843 −1.56655 −0.783274 0.621676i \(-0.786452\pi\)
−0.783274 + 0.621676i \(0.786452\pi\)
\(20\) 0 0
\(21\) −0.828427 −0.180778
\(22\) −2.41421 −0.514712
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 4.41421 0.901048
\(25\) 0 0
\(26\) 13.6569 2.67833
\(27\) 1.00000 0.192450
\(28\) −3.17157 −0.599371
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.58579 −0.280330
\(33\) −1.00000 −0.174078
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) −11.6569 −1.91638 −0.958188 0.286141i \(-0.907627\pi\)
−0.958188 + 0.286141i \(0.907627\pi\)
\(38\) −16.4853 −2.67427
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 4.82843 0.754074 0.377037 0.926198i \(-0.376943\pi\)
0.377037 + 0.926198i \(0.376943\pi\)
\(42\) −2.00000 −0.308607
\(43\) 8.82843 1.34632 0.673161 0.739496i \(-0.264936\pi\)
0.673161 + 0.739496i \(0.264936\pi\)
\(44\) −3.82843 −0.577157
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 3.00000 0.433013
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) 1.17157 0.164053
\(52\) 21.6569 3.00327
\(53\) −9.31371 −1.27934 −0.639668 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) −3.65685 −0.488668
\(57\) −6.82843 −0.904447
\(58\) −11.6569 −1.53062
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −11.6569 −1.49251 −0.746254 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0 0
\(63\) −0.828427 −0.104372
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) −2.41421 −0.297169
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 4.48528 0.543920
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.34315 0.278080 0.139040 0.990287i \(-0.455598\pi\)
0.139040 + 0.990287i \(0.455598\pi\)
\(72\) 4.41421 0.520220
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) −28.1421 −3.27146
\(75\) 0 0
\(76\) −26.1421 −2.99871
\(77\) 0.828427 0.0944080
\(78\) 13.6569 1.54633
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 11.6569 1.28728
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) −3.17157 −0.346047
\(85\) 0 0
\(86\) 21.3137 2.29832
\(87\) −4.82843 −0.517662
\(88\) −4.41421 −0.470557
\(89\) 3.65685 0.387626 0.193813 0.981039i \(-0.437915\pi\)
0.193813 + 0.981039i \(0.437915\pi\)
\(90\) 0 0
\(91\) −4.68629 −0.491257
\(92\) 15.3137 1.59656
\(93\) 0 0
\(94\) 9.65685 0.996028
\(95\) 0 0
\(96\) −1.58579 −0.161849
\(97\) −11.6569 −1.18357 −0.591787 0.806094i \(-0.701577\pi\)
−0.591787 + 0.806094i \(0.701577\pi\)
\(98\) −15.2426 −1.53974
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −0.828427 −0.0824316 −0.0412158 0.999150i \(-0.513123\pi\)
−0.0412158 + 0.999150i \(0.513123\pi\)
\(102\) 2.82843 0.280056
\(103\) 3.31371 0.326509 0.163255 0.986584i \(-0.447801\pi\)
0.163255 + 0.986584i \(0.447801\pi\)
\(104\) 24.9706 2.44857
\(105\) 0 0
\(106\) −22.4853 −2.18396
\(107\) −17.3137 −1.67378 −0.836890 0.547372i \(-0.815628\pi\)
−0.836890 + 0.547372i \(0.815628\pi\)
\(108\) 3.82843 0.368391
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 0 0
\(111\) −11.6569 −1.10642
\(112\) −2.48528 −0.234837
\(113\) 18.9706 1.78460 0.892300 0.451442i \(-0.149090\pi\)
0.892300 + 0.451442i \(0.149090\pi\)
\(114\) −16.4853 −1.54399
\(115\) 0 0
\(116\) −18.4853 −1.71632
\(117\) 5.65685 0.522976
\(118\) −9.65685 −0.888985
\(119\) −0.970563 −0.0889713
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −28.1421 −2.54787
\(123\) 4.82843 0.435365
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 14.4853 1.28536 0.642680 0.766134i \(-0.277822\pi\)
0.642680 + 0.766134i \(0.277822\pi\)
\(128\) −20.5563 −1.81694
\(129\) 8.82843 0.777300
\(130\) 0 0
\(131\) 3.31371 0.289520 0.144760 0.989467i \(-0.453759\pi\)
0.144760 + 0.989467i \(0.453759\pi\)
\(132\) −3.82843 −0.333222
\(133\) 5.65685 0.490511
\(134\) 13.6569 1.17977
\(135\) 0 0
\(136\) 5.17157 0.443459
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 9.65685 0.822046
\(139\) 0.485281 0.0411610 0.0205805 0.999788i \(-0.493449\pi\)
0.0205805 + 0.999788i \(0.493449\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 5.65685 0.474713
\(143\) −5.65685 −0.473050
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −27.3137 −2.26050
\(147\) −6.31371 −0.520746
\(148\) −44.6274 −3.66835
\(149\) 1.51472 0.124091 0.0620453 0.998073i \(-0.480238\pi\)
0.0620453 + 0.998073i \(0.480238\pi\)
\(150\) 0 0
\(151\) 16.4853 1.34155 0.670777 0.741659i \(-0.265961\pi\)
0.670777 + 0.741659i \(0.265961\pi\)
\(152\) −30.1421 −2.44485
\(153\) 1.17157 0.0947161
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 21.6569 1.73394
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 20.4853 1.62972
\(159\) −9.31371 −0.738625
\(160\) 0 0
\(161\) −3.31371 −0.261157
\(162\) 2.41421 0.189679
\(163\) 7.31371 0.572854 0.286427 0.958102i \(-0.407532\pi\)
0.286427 + 0.958102i \(0.407532\pi\)
\(164\) 18.4853 1.44346
\(165\) 0 0
\(166\) 24.1421 1.87379
\(167\) 13.3137 1.03025 0.515123 0.857116i \(-0.327746\pi\)
0.515123 + 0.857116i \(0.327746\pi\)
\(168\) −3.65685 −0.282132
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) −6.82843 −0.522183
\(172\) 33.7990 2.57715
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) −11.6569 −0.883704
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −4.00000 −0.300658
\(178\) 8.82843 0.661719
\(179\) −17.6569 −1.31974 −0.659868 0.751382i \(-0.729388\pi\)
−0.659868 + 0.751382i \(0.729388\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −11.3137 −0.838628
\(183\) −11.6569 −0.861699
\(184\) 17.6569 1.30168
\(185\) 0 0
\(186\) 0 0
\(187\) −1.17157 −0.0856739
\(188\) 15.3137 1.11687
\(189\) −0.828427 −0.0602592
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) −9.82843 −0.709306
\(193\) −13.6569 −0.983042 −0.491521 0.870866i \(-0.663559\pi\)
−0.491521 + 0.870866i \(0.663559\pi\)
\(194\) −28.1421 −2.02049
\(195\) 0 0
\(196\) −24.1716 −1.72654
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) −2.41421 −0.171571
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) −2.00000 −0.140720
\(203\) 4.00000 0.280745
\(204\) 4.48528 0.314033
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) 16.9706 1.17670
\(209\) 6.82843 0.472332
\(210\) 0 0
\(211\) 1.17157 0.0806544 0.0403272 0.999187i \(-0.487160\pi\)
0.0403272 + 0.999187i \(0.487160\pi\)
\(212\) −35.6569 −2.44892
\(213\) 2.34315 0.160550
\(214\) −41.7990 −2.85732
\(215\) 0 0
\(216\) 4.41421 0.300349
\(217\) 0 0
\(218\) 41.7990 2.83098
\(219\) −11.3137 −0.764510
\(220\) 0 0
\(221\) 6.62742 0.445808
\(222\) −28.1421 −1.88878
\(223\) 6.34315 0.424768 0.212384 0.977186i \(-0.431877\pi\)
0.212384 + 0.977186i \(0.431877\pi\)
\(224\) 1.31371 0.0877758
\(225\) 0 0
\(226\) 45.7990 3.04650
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) −26.1421 −1.73131
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0.828427 0.0545065
\(232\) −21.3137 −1.39931
\(233\) 18.8284 1.23349 0.616746 0.787163i \(-0.288451\pi\)
0.616746 + 0.787163i \(0.288451\pi\)
\(234\) 13.6569 0.892776
\(235\) 0 0
\(236\) −15.3137 −0.996838
\(237\) 8.48528 0.551178
\(238\) −2.34315 −0.151884
\(239\) 17.6569 1.14213 0.571063 0.820906i \(-0.306531\pi\)
0.571063 + 0.820906i \(0.306531\pi\)
\(240\) 0 0
\(241\) −12.3431 −0.795092 −0.397546 0.917582i \(-0.630138\pi\)
−0.397546 + 0.917582i \(0.630138\pi\)
\(242\) 2.41421 0.155192
\(243\) 1.00000 0.0641500
\(244\) −44.6274 −2.85698
\(245\) 0 0
\(246\) 11.6569 0.743214
\(247\) −38.6274 −2.45780
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) 20.9706 1.32365 0.661825 0.749658i \(-0.269782\pi\)
0.661825 + 0.749658i \(0.269782\pi\)
\(252\) −3.17157 −0.199790
\(253\) −4.00000 −0.251478
\(254\) 34.9706 2.19425
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 16.3431 1.01946 0.509729 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(258\) 21.3137 1.32693
\(259\) 9.65685 0.600048
\(260\) 0 0
\(261\) −4.82843 −0.298872
\(262\) 8.00000 0.494242
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) −4.41421 −0.271676
\(265\) 0 0
\(266\) 13.6569 0.837355
\(267\) 3.65685 0.223796
\(268\) 21.6569 1.32290
\(269\) 20.6274 1.25768 0.628838 0.777536i \(-0.283531\pi\)
0.628838 + 0.777536i \(0.283531\pi\)
\(270\) 0 0
\(271\) −11.7990 −0.716738 −0.358369 0.933580i \(-0.616667\pi\)
−0.358369 + 0.933580i \(0.616667\pi\)
\(272\) 3.51472 0.213111
\(273\) −4.68629 −0.283627
\(274\) 32.1421 1.94178
\(275\) 0 0
\(276\) 15.3137 0.921777
\(277\) 2.34315 0.140786 0.0703930 0.997519i \(-0.477575\pi\)
0.0703930 + 0.997519i \(0.477575\pi\)
\(278\) 1.17157 0.0702663
\(279\) 0 0
\(280\) 0 0
\(281\) −11.1716 −0.666440 −0.333220 0.942849i \(-0.608135\pi\)
−0.333220 + 0.942849i \(0.608135\pi\)
\(282\) 9.65685 0.575057
\(283\) 8.82843 0.524796 0.262398 0.964960i \(-0.415487\pi\)
0.262398 + 0.964960i \(0.415487\pi\)
\(284\) 8.97056 0.532305
\(285\) 0 0
\(286\) −13.6569 −0.807547
\(287\) −4.00000 −0.236113
\(288\) −1.58579 −0.0934434
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) −11.6569 −0.683337
\(292\) −43.3137 −2.53474
\(293\) 6.82843 0.398921 0.199460 0.979906i \(-0.436081\pi\)
0.199460 + 0.979906i \(0.436081\pi\)
\(294\) −15.2426 −0.888969
\(295\) 0 0
\(296\) −51.4558 −2.99081
\(297\) −1.00000 −0.0580259
\(298\) 3.65685 0.211836
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) −7.31371 −0.421555
\(302\) 39.7990 2.29017
\(303\) −0.828427 −0.0475919
\(304\) −20.4853 −1.17491
\(305\) 0 0
\(306\) 2.82843 0.161690
\(307\) 3.17157 0.181011 0.0905056 0.995896i \(-0.471152\pi\)
0.0905056 + 0.995896i \(0.471152\pi\)
\(308\) 3.17157 0.180717
\(309\) 3.31371 0.188510
\(310\) 0 0
\(311\) −3.31371 −0.187903 −0.0939516 0.995577i \(-0.529950\pi\)
−0.0939516 + 0.995577i \(0.529950\pi\)
\(312\) 24.9706 1.41368
\(313\) −15.6569 −0.884978 −0.442489 0.896774i \(-0.645904\pi\)
−0.442489 + 0.896774i \(0.645904\pi\)
\(314\) −43.4558 −2.45236
\(315\) 0 0
\(316\) 32.4853 1.82744
\(317\) 26.2843 1.47627 0.738136 0.674652i \(-0.235706\pi\)
0.738136 + 0.674652i \(0.235706\pi\)
\(318\) −22.4853 −1.26091
\(319\) 4.82843 0.270340
\(320\) 0 0
\(321\) −17.3137 −0.966357
\(322\) −8.00000 −0.445823
\(323\) −8.00000 −0.445132
\(324\) 3.82843 0.212690
\(325\) 0 0
\(326\) 17.6569 0.977923
\(327\) 17.3137 0.957450
\(328\) 21.3137 1.17685
\(329\) −3.31371 −0.182691
\(330\) 0 0
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) 38.2843 2.10112
\(333\) −11.6569 −0.638792
\(334\) 32.1421 1.75874
\(335\) 0 0
\(336\) −2.48528 −0.135583
\(337\) −3.31371 −0.180509 −0.0902546 0.995919i \(-0.528768\pi\)
−0.0902546 + 0.995919i \(0.528768\pi\)
\(338\) 45.8701 2.49500
\(339\) 18.9706 1.03034
\(340\) 0 0
\(341\) 0 0
\(342\) −16.4853 −0.891422
\(343\) 11.0294 0.595534
\(344\) 38.9706 2.10115
\(345\) 0 0
\(346\) −6.82843 −0.367099
\(347\) 29.3137 1.57364 0.786821 0.617181i \(-0.211725\pi\)
0.786821 + 0.617181i \(0.211725\pi\)
\(348\) −18.4853 −0.990915
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 1.58579 0.0845227
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) −9.65685 −0.513256
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −0.970563 −0.0513676
\(358\) −42.6274 −2.25293
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) −33.7990 −1.77644
\(363\) 1.00000 0.0524864
\(364\) −17.9411 −0.940370
\(365\) 0 0
\(366\) −28.1421 −1.47101
\(367\) −9.65685 −0.504084 −0.252042 0.967716i \(-0.581102\pi\)
−0.252042 + 0.967716i \(0.581102\pi\)
\(368\) 12.0000 0.625543
\(369\) 4.82843 0.251358
\(370\) 0 0
\(371\) 7.71573 0.400581
\(372\) 0 0
\(373\) −10.6274 −0.550267 −0.275133 0.961406i \(-0.588722\pi\)
−0.275133 + 0.961406i \(0.588722\pi\)
\(374\) −2.82843 −0.146254
\(375\) 0 0
\(376\) 17.6569 0.910583
\(377\) −27.3137 −1.40673
\(378\) −2.00000 −0.102869
\(379\) −23.3137 −1.19754 −0.598772 0.800919i \(-0.704345\pi\)
−0.598772 + 0.800919i \(0.704345\pi\)
\(380\) 0 0
\(381\) 14.4853 0.742103
\(382\) 13.6569 0.698745
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −20.5563 −1.04901
\(385\) 0 0
\(386\) −32.9706 −1.67816
\(387\) 8.82843 0.448774
\(388\) −44.6274 −2.26561
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) 0 0
\(391\) 4.68629 0.236996
\(392\) −27.8701 −1.40765
\(393\) 3.31371 0.167154
\(394\) 20.4853 1.03203
\(395\) 0 0
\(396\) −3.82843 −0.192386
\(397\) 14.9706 0.751351 0.375676 0.926751i \(-0.377411\pi\)
0.375676 + 0.926751i \(0.377411\pi\)
\(398\) −52.2843 −2.62077
\(399\) 5.65685 0.283197
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 13.6569 0.681142
\(403\) 0 0
\(404\) −3.17157 −0.157792
\(405\) 0 0
\(406\) 9.65685 0.479262
\(407\) 11.6569 0.577809
\(408\) 5.17157 0.256031
\(409\) −19.6569 −0.971969 −0.485984 0.873967i \(-0.661539\pi\)
−0.485984 + 0.873967i \(0.661539\pi\)
\(410\) 0 0
\(411\) 13.3137 0.656717
\(412\) 12.6863 0.625009
\(413\) 3.31371 0.163057
\(414\) 9.65685 0.474608
\(415\) 0 0
\(416\) −8.97056 −0.439818
\(417\) 0.485281 0.0237643
\(418\) 16.4853 0.806321
\(419\) −36.9706 −1.80613 −0.903065 0.429504i \(-0.858689\pi\)
−0.903065 + 0.429504i \(0.858689\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.82843 0.137686
\(423\) 4.00000 0.194487
\(424\) −41.1127 −1.99661
\(425\) 0 0
\(426\) 5.65685 0.274075
\(427\) 9.65685 0.467328
\(428\) −66.2843 −3.20397
\(429\) −5.65685 −0.273115
\(430\) 0 0
\(431\) 21.6569 1.04317 0.521587 0.853198i \(-0.325340\pi\)
0.521587 + 0.853198i \(0.325340\pi\)
\(432\) 3.00000 0.144338
\(433\) 15.6569 0.752420 0.376210 0.926534i \(-0.377227\pi\)
0.376210 + 0.926534i \(0.377227\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 66.2843 3.17444
\(437\) −27.3137 −1.30659
\(438\) −27.3137 −1.30510
\(439\) −20.4853 −0.977709 −0.488855 0.872365i \(-0.662585\pi\)
−0.488855 + 0.872365i \(0.662585\pi\)
\(440\) 0 0
\(441\) −6.31371 −0.300653
\(442\) 16.0000 0.761042
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −44.6274 −2.11792
\(445\) 0 0
\(446\) 15.3137 0.725125
\(447\) 1.51472 0.0716437
\(448\) 8.14214 0.384680
\(449\) 30.9706 1.46159 0.730796 0.682596i \(-0.239149\pi\)
0.730796 + 0.682596i \(0.239149\pi\)
\(450\) 0 0
\(451\) −4.82843 −0.227362
\(452\) 72.6274 3.41611
\(453\) 16.4853 0.774546
\(454\) 33.7990 1.58627
\(455\) 0 0
\(456\) −30.1421 −1.41153
\(457\) 23.3137 1.09057 0.545285 0.838251i \(-0.316422\pi\)
0.545285 + 0.838251i \(0.316422\pi\)
\(458\) −4.82843 −0.225618
\(459\) 1.17157 0.0546843
\(460\) 0 0
\(461\) 0.142136 0.00661992 0.00330996 0.999995i \(-0.498946\pi\)
0.00330996 + 0.999995i \(0.498946\pi\)
\(462\) 2.00000 0.0930484
\(463\) −4.97056 −0.231002 −0.115501 0.993307i \(-0.536847\pi\)
−0.115501 + 0.993307i \(0.536847\pi\)
\(464\) −14.4853 −0.672462
\(465\) 0 0
\(466\) 45.4558 2.10570
\(467\) 22.6274 1.04707 0.523536 0.852004i \(-0.324613\pi\)
0.523536 + 0.852004i \(0.324613\pi\)
\(468\) 21.6569 1.00109
\(469\) −4.68629 −0.216393
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) −17.6569 −0.812723
\(473\) −8.82843 −0.405932
\(474\) 20.4853 0.940920
\(475\) 0 0
\(476\) −3.71573 −0.170310
\(477\) −9.31371 −0.426445
\(478\) 42.6274 1.94973
\(479\) 36.9706 1.68923 0.844614 0.535376i \(-0.179830\pi\)
0.844614 + 0.535376i \(0.179830\pi\)
\(480\) 0 0
\(481\) −65.9411 −3.00666
\(482\) −29.7990 −1.35731
\(483\) −3.31371 −0.150779
\(484\) 3.82843 0.174019
\(485\) 0 0
\(486\) 2.41421 0.109511
\(487\) 12.9706 0.587752 0.293876 0.955844i \(-0.405055\pi\)
0.293876 + 0.955844i \(0.405055\pi\)
\(488\) −51.4558 −2.32930
\(489\) 7.31371 0.330737
\(490\) 0 0
\(491\) 14.3431 0.647297 0.323649 0.946177i \(-0.395090\pi\)
0.323649 + 0.946177i \(0.395090\pi\)
\(492\) 18.4853 0.833381
\(493\) −5.65685 −0.254772
\(494\) −93.2548 −4.19573
\(495\) 0 0
\(496\) 0 0
\(497\) −1.94113 −0.0870714
\(498\) 24.1421 1.08183
\(499\) −22.3431 −1.00022 −0.500108 0.865963i \(-0.666706\pi\)
−0.500108 + 0.865963i \(0.666706\pi\)
\(500\) 0 0
\(501\) 13.3137 0.594813
\(502\) 50.6274 2.25961
\(503\) −17.3137 −0.771980 −0.385990 0.922503i \(-0.626140\pi\)
−0.385990 + 0.922503i \(0.626140\pi\)
\(504\) −3.65685 −0.162889
\(505\) 0 0
\(506\) −9.65685 −0.429300
\(507\) 19.0000 0.843820
\(508\) 55.4558 2.46046
\(509\) 18.6863 0.828255 0.414128 0.910219i \(-0.364087\pi\)
0.414128 + 0.910219i \(0.364087\pi\)
\(510\) 0 0
\(511\) 9.37258 0.414619
\(512\) −31.2426 −1.38074
\(513\) −6.82843 −0.301482
\(514\) 39.4558 1.74032
\(515\) 0 0
\(516\) 33.7990 1.48792
\(517\) −4.00000 −0.175920
\(518\) 23.3137 1.02435
\(519\) −2.82843 −0.124154
\(520\) 0 0
\(521\) −32.6274 −1.42943 −0.714717 0.699414i \(-0.753444\pi\)
−0.714717 + 0.699414i \(0.753444\pi\)
\(522\) −11.6569 −0.510207
\(523\) 9.51472 0.416050 0.208025 0.978124i \(-0.433297\pi\)
0.208025 + 0.978124i \(0.433297\pi\)
\(524\) 12.6863 0.554203
\(525\) 0 0
\(526\) −43.4558 −1.89476
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 21.6569 0.938944
\(533\) 27.3137 1.18309
\(534\) 8.82843 0.382043
\(535\) 0 0
\(536\) 24.9706 1.07856
\(537\) −17.6569 −0.761950
\(538\) 49.7990 2.14699
\(539\) 6.31371 0.271951
\(540\) 0 0
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) −28.4853 −1.22355
\(543\) −14.0000 −0.600798
\(544\) −1.85786 −0.0796553
\(545\) 0 0
\(546\) −11.3137 −0.484182
\(547\) 8.14214 0.348133 0.174066 0.984734i \(-0.444309\pi\)
0.174066 + 0.984734i \(0.444309\pi\)
\(548\) 50.9706 2.17735
\(549\) −11.6569 −0.497502
\(550\) 0 0
\(551\) 32.9706 1.40459
\(552\) 17.6569 0.751526
\(553\) −7.02944 −0.298922
\(554\) 5.65685 0.240337
\(555\) 0 0
\(556\) 1.85786 0.0787910
\(557\) −5.17157 −0.219127 −0.109563 0.993980i \(-0.534945\pi\)
−0.109563 + 0.993980i \(0.534945\pi\)
\(558\) 0 0
\(559\) 49.9411 2.11228
\(560\) 0 0
\(561\) −1.17157 −0.0494638
\(562\) −26.9706 −1.13768
\(563\) −31.6569 −1.33418 −0.667089 0.744978i \(-0.732460\pi\)
−0.667089 + 0.744978i \(0.732460\pi\)
\(564\) 15.3137 0.644823
\(565\) 0 0
\(566\) 21.3137 0.895882
\(567\) −0.828427 −0.0347907
\(568\) 10.3431 0.433989
\(569\) −35.4558 −1.48639 −0.743193 0.669077i \(-0.766690\pi\)
−0.743193 + 0.669077i \(0.766690\pi\)
\(570\) 0 0
\(571\) −16.4853 −0.689888 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(572\) −21.6569 −0.905519
\(573\) 5.65685 0.236318
\(574\) −9.65685 −0.403069
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −37.7279 −1.56927
\(579\) −13.6569 −0.567559
\(580\) 0 0
\(581\) −8.28427 −0.343689
\(582\) −28.1421 −1.16653
\(583\) 9.31371 0.385734
\(584\) −49.9411 −2.06658
\(585\) 0 0
\(586\) 16.4853 0.681001
\(587\) −14.6274 −0.603738 −0.301869 0.953349i \(-0.597611\pi\)
−0.301869 + 0.953349i \(0.597611\pi\)
\(588\) −24.1716 −0.996819
\(589\) 0 0
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) −34.9706 −1.43728
\(593\) 22.8284 0.937451 0.468726 0.883344i \(-0.344713\pi\)
0.468726 + 0.883344i \(0.344713\pi\)
\(594\) −2.41421 −0.0990564
\(595\) 0 0
\(596\) 5.79899 0.237536
\(597\) −21.6569 −0.886356
\(598\) 54.6274 2.23388
\(599\) 27.3137 1.11601 0.558004 0.829838i \(-0.311567\pi\)
0.558004 + 0.829838i \(0.311567\pi\)
\(600\) 0 0
\(601\) −5.31371 −0.216751 −0.108375 0.994110i \(-0.534565\pi\)
−0.108375 + 0.994110i \(0.534565\pi\)
\(602\) −17.6569 −0.719640
\(603\) 5.65685 0.230365
\(604\) 63.1127 2.56802
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) −1.51472 −0.0614805 −0.0307403 0.999527i \(-0.509786\pi\)
−0.0307403 + 0.999527i \(0.509786\pi\)
\(608\) 10.8284 0.439151
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 4.48528 0.181307
\(613\) 45.9411 1.85554 0.927772 0.373147i \(-0.121721\pi\)
0.927772 + 0.373147i \(0.121721\pi\)
\(614\) 7.65685 0.309005
\(615\) 0 0
\(616\) 3.65685 0.147339
\(617\) −0.343146 −0.0138145 −0.00690726 0.999976i \(-0.502199\pi\)
−0.00690726 + 0.999976i \(0.502199\pi\)
\(618\) 8.00000 0.321807
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −8.00000 −0.320771
\(623\) −3.02944 −0.121372
\(624\) 16.9706 0.679366
\(625\) 0 0
\(626\) −37.7990 −1.51075
\(627\) 6.82843 0.272701
\(628\) −68.9117 −2.74988
\(629\) −13.6569 −0.544534
\(630\) 0 0
\(631\) 45.6569 1.81757 0.908785 0.417264i \(-0.137011\pi\)
0.908785 + 0.417264i \(0.137011\pi\)
\(632\) 37.4558 1.48991
\(633\) 1.17157 0.0465658
\(634\) 63.4558 2.52015
\(635\) 0 0
\(636\) −35.6569 −1.41389
\(637\) −35.7157 −1.41511
\(638\) 11.6569 0.461499
\(639\) 2.34315 0.0926934
\(640\) 0 0
\(641\) 6.97056 0.275321 0.137660 0.990479i \(-0.456042\pi\)
0.137660 + 0.990479i \(0.456042\pi\)
\(642\) −41.7990 −1.64967
\(643\) −37.9411 −1.49625 −0.748126 0.663557i \(-0.769046\pi\)
−0.748126 + 0.663557i \(0.769046\pi\)
\(644\) −12.6863 −0.499910
\(645\) 0 0
\(646\) −19.3137 −0.759888
\(647\) −4.68629 −0.184237 −0.0921186 0.995748i \(-0.529364\pi\)
−0.0921186 + 0.995748i \(0.529364\pi\)
\(648\) 4.41421 0.173407
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 28.0000 1.09656
\(653\) −6.97056 −0.272779 −0.136390 0.990655i \(-0.543550\pi\)
−0.136390 + 0.990655i \(0.543550\pi\)
\(654\) 41.7990 1.63447
\(655\) 0 0
\(656\) 14.4853 0.565555
\(657\) −11.3137 −0.441390
\(658\) −8.00000 −0.311872
\(659\) −15.3137 −0.596537 −0.298269 0.954482i \(-0.596409\pi\)
−0.298269 + 0.954482i \(0.596409\pi\)
\(660\) 0 0
\(661\) 9.31371 0.362261 0.181131 0.983459i \(-0.442024\pi\)
0.181131 + 0.983459i \(0.442024\pi\)
\(662\) 15.3137 0.595184
\(663\) 6.62742 0.257388
\(664\) 44.1421 1.71305
\(665\) 0 0
\(666\) −28.1421 −1.09049
\(667\) −19.3137 −0.747830
\(668\) 50.9706 1.97211
\(669\) 6.34315 0.245240
\(670\) 0 0
\(671\) 11.6569 0.450008
\(672\) 1.31371 0.0506774
\(673\) −18.3431 −0.707076 −0.353538 0.935420i \(-0.615022\pi\)
−0.353538 + 0.935420i \(0.615022\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 72.7401 2.79770
\(677\) −29.4558 −1.13208 −0.566040 0.824378i \(-0.691525\pi\)
−0.566040 + 0.824378i \(0.691525\pi\)
\(678\) 45.7990 1.75890
\(679\) 9.65685 0.370596
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −26.1421 −0.999570
\(685\) 0 0
\(686\) 26.6274 1.01664
\(687\) −2.00000 −0.0763048
\(688\) 26.4853 1.00974
\(689\) −52.6863 −2.00719
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −10.8284 −0.411635
\(693\) 0.828427 0.0314693
\(694\) 70.7696 2.68638
\(695\) 0 0
\(696\) −21.3137 −0.807894
\(697\) 5.65685 0.214269
\(698\) 26.4853 1.00248
\(699\) 18.8284 0.712157
\(700\) 0 0
\(701\) 36.1421 1.36507 0.682535 0.730853i \(-0.260878\pi\)
0.682535 + 0.730853i \(0.260878\pi\)
\(702\) 13.6569 0.515445
\(703\) 79.5980 3.00209
\(704\) 9.82843 0.370423
\(705\) 0 0
\(706\) −62.7696 −2.36236
\(707\) 0.686292 0.0258106
\(708\) −15.3137 −0.575524
\(709\) 6.68629 0.251109 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 16.1421 0.604952
\(713\) 0 0
\(714\) −2.34315 −0.0876900
\(715\) 0 0
\(716\) −67.5980 −2.52626
\(717\) 17.6569 0.659407
\(718\) 28.9706 1.08117
\(719\) −47.5980 −1.77511 −0.887553 0.460706i \(-0.847596\pi\)
−0.887553 + 0.460706i \(0.847596\pi\)
\(720\) 0 0
\(721\) −2.74517 −0.102235
\(722\) 66.6985 2.48226
\(723\) −12.3431 −0.459047
\(724\) −53.5980 −1.99195
\(725\) 0 0
\(726\) 2.41421 0.0895999
\(727\) −33.9411 −1.25881 −0.629403 0.777079i \(-0.716701\pi\)
−0.629403 + 0.777079i \(0.716701\pi\)
\(728\) −20.6863 −0.766685
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.3431 0.382555
\(732\) −44.6274 −1.64948
\(733\) 6.34315 0.234289 0.117145 0.993115i \(-0.462626\pi\)
0.117145 + 0.993115i \(0.462626\pi\)
\(734\) −23.3137 −0.860525
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) −5.65685 −0.208373
\(738\) 11.6569 0.429095
\(739\) −15.1127 −0.555930 −0.277965 0.960591i \(-0.589660\pi\)
−0.277965 + 0.960591i \(0.589660\pi\)
\(740\) 0 0
\(741\) −38.6274 −1.41901
\(742\) 18.6274 0.683834
\(743\) −36.3431 −1.33330 −0.666650 0.745371i \(-0.732273\pi\)
−0.666650 + 0.745371i \(0.732273\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −25.6569 −0.939364
\(747\) 10.0000 0.365881
\(748\) −4.48528 −0.163998
\(749\) 14.3431 0.524087
\(750\) 0 0
\(751\) 20.2843 0.740184 0.370092 0.928995i \(-0.379326\pi\)
0.370092 + 0.928995i \(0.379326\pi\)
\(752\) 12.0000 0.437595
\(753\) 20.9706 0.764210
\(754\) −65.9411 −2.40143
\(755\) 0 0
\(756\) −3.17157 −0.115349
\(757\) 36.6274 1.33125 0.665623 0.746288i \(-0.268166\pi\)
0.665623 + 0.746288i \(0.268166\pi\)
\(758\) −56.2843 −2.04434
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −28.8284 −1.04503 −0.522515 0.852630i \(-0.675006\pi\)
−0.522515 + 0.852630i \(0.675006\pi\)
\(762\) 34.9706 1.26685
\(763\) −14.3431 −0.519257
\(764\) 21.6569 0.783517
\(765\) 0 0
\(766\) 19.3137 0.697833
\(767\) −22.6274 −0.817029
\(768\) −29.9706 −1.08147
\(769\) 10.6863 0.385358 0.192679 0.981262i \(-0.438282\pi\)
0.192679 + 0.981262i \(0.438282\pi\)
\(770\) 0 0
\(771\) 16.3431 0.588584
\(772\) −52.2843 −1.88175
\(773\) −3.65685 −0.131528 −0.0657640 0.997835i \(-0.520948\pi\)
−0.0657640 + 0.997835i \(0.520948\pi\)
\(774\) 21.3137 0.766105
\(775\) 0 0
\(776\) −51.4558 −1.84716
\(777\) 9.65685 0.346438
\(778\) −57.1127 −2.04759
\(779\) −32.9706 −1.18129
\(780\) 0 0
\(781\) −2.34315 −0.0838443
\(782\) 11.3137 0.404577
\(783\) −4.82843 −0.172554
\(784\) −18.9411 −0.676469
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) 20.1421 0.717990 0.358995 0.933340i \(-0.383120\pi\)
0.358995 + 0.933340i \(0.383120\pi\)
\(788\) 32.4853 1.15724
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) −15.7157 −0.558787
\(792\) −4.41421 −0.156852
\(793\) −65.9411 −2.34164
\(794\) 36.1421 1.28264
\(795\) 0 0
\(796\) −82.9117 −2.93873
\(797\) −34.9706 −1.23872 −0.619360 0.785107i \(-0.712608\pi\)
−0.619360 + 0.785107i \(0.712608\pi\)
\(798\) 13.6569 0.483447
\(799\) 4.68629 0.165789
\(800\) 0 0
\(801\) 3.65685 0.129209
\(802\) −16.1421 −0.569999
\(803\) 11.3137 0.399252
\(804\) 21.6569 0.763778
\(805\) 0 0
\(806\) 0 0
\(807\) 20.6274 0.726119
\(808\) −3.65685 −0.128648
\(809\) 28.4264 0.999419 0.499710 0.866193i \(-0.333440\pi\)
0.499710 + 0.866193i \(0.333440\pi\)
\(810\) 0 0
\(811\) 0.485281 0.0170405 0.00852027 0.999964i \(-0.497288\pi\)
0.00852027 + 0.999964i \(0.497288\pi\)
\(812\) 15.3137 0.537406
\(813\) −11.7990 −0.413809
\(814\) 28.1421 0.986381
\(815\) 0 0
\(816\) 3.51472 0.123040
\(817\) −60.2843 −2.10908
\(818\) −47.4558 −1.65925
\(819\) −4.68629 −0.163752
\(820\) 0 0
\(821\) −12.8284 −0.447715 −0.223858 0.974622i \(-0.571865\pi\)
−0.223858 + 0.974622i \(0.571865\pi\)
\(822\) 32.1421 1.12109
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 14.6274 0.509570
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −41.3137 −1.43662 −0.718309 0.695724i \(-0.755084\pi\)
−0.718309 + 0.695724i \(0.755084\pi\)
\(828\) 15.3137 0.532188
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 2.34315 0.0812828
\(832\) −55.5980 −1.92751
\(833\) −7.39697 −0.256290
\(834\) 1.17157 0.0405683
\(835\) 0 0
\(836\) 26.1421 0.904145
\(837\) 0 0
\(838\) −89.2548 −3.08326
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) −14.4853 −0.499196
\(843\) −11.1716 −0.384769
\(844\) 4.48528 0.154390
\(845\) 0 0
\(846\) 9.65685 0.332009
\(847\) −0.828427 −0.0284651
\(848\) −27.9411 −0.959502
\(849\) 8.82843 0.302991
\(850\) 0 0
\(851\) −46.6274 −1.59837
\(852\) 8.97056 0.307326
\(853\) 8.68629 0.297413 0.148706 0.988881i \(-0.452489\pi\)
0.148706 + 0.988881i \(0.452489\pi\)
\(854\) 23.3137 0.797779
\(855\) 0 0
\(856\) −76.4264 −2.61220
\(857\) 28.4853 0.973039 0.486519 0.873670i \(-0.338266\pi\)
0.486519 + 0.873670i \(0.338266\pi\)
\(858\) −13.6569 −0.466237
\(859\) −52.9706 −1.80733 −0.903666 0.428238i \(-0.859135\pi\)
−0.903666 + 0.428238i \(0.859135\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 52.2843 1.78081
\(863\) 20.6863 0.704170 0.352085 0.935968i \(-0.385473\pi\)
0.352085 + 0.935968i \(0.385473\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 0 0
\(866\) 37.7990 1.28446
\(867\) −15.6274 −0.530735
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 76.4264 2.58812
\(873\) −11.6569 −0.394525
\(874\) −65.9411 −2.23049
\(875\) 0 0
\(876\) −43.3137 −1.46343
\(877\) −2.62742 −0.0887216 −0.0443608 0.999016i \(-0.514125\pi\)
−0.0443608 + 0.999016i \(0.514125\pi\)
\(878\) −49.4558 −1.66905
\(879\) 6.82843 0.230317
\(880\) 0 0
\(881\) −46.9706 −1.58248 −0.791239 0.611507i \(-0.790564\pi\)
−0.791239 + 0.611507i \(0.790564\pi\)
\(882\) −15.2426 −0.513246
\(883\) 5.37258 0.180802 0.0904009 0.995905i \(-0.471185\pi\)
0.0904009 + 0.995905i \(0.471185\pi\)
\(884\) 25.3726 0.853372
\(885\) 0 0
\(886\) −28.9706 −0.973285
\(887\) 15.6569 0.525706 0.262853 0.964836i \(-0.415337\pi\)
0.262853 + 0.964836i \(0.415337\pi\)
\(888\) −51.4558 −1.72675
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 24.2843 0.813098
\(893\) −27.3137 −0.914018
\(894\) 3.65685 0.122304
\(895\) 0 0
\(896\) 17.0294 0.568914
\(897\) 22.6274 0.755507
\(898\) 74.7696 2.49509
\(899\) 0 0
\(900\) 0 0
\(901\) −10.9117 −0.363521
\(902\) −11.6569 −0.388131
\(903\) −7.31371 −0.243385
\(904\) 83.7401 2.78515
\(905\) 0 0
\(906\) 39.7990 1.32223
\(907\) −40.9706 −1.36041 −0.680203 0.733024i \(-0.738108\pi\)
−0.680203 + 0.733024i \(0.738108\pi\)
\(908\) 53.5980 1.77871
\(909\) −0.828427 −0.0274772
\(910\) 0 0
\(911\) −48.9706 −1.62247 −0.811234 0.584722i \(-0.801203\pi\)
−0.811234 + 0.584722i \(0.801203\pi\)
\(912\) −20.4853 −0.678335
\(913\) −10.0000 −0.330952
\(914\) 56.2843 1.86172
\(915\) 0 0
\(916\) −7.65685 −0.252990
\(917\) −2.74517 −0.0906534
\(918\) 2.82843 0.0933520
\(919\) 11.5147 0.379836 0.189918 0.981800i \(-0.439178\pi\)
0.189918 + 0.981800i \(0.439178\pi\)
\(920\) 0 0
\(921\) 3.17157 0.104507
\(922\) 0.343146 0.0113009
\(923\) 13.2548 0.436288
\(924\) 3.17157 0.104337
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 3.31371 0.108836
\(928\) 7.65685 0.251349
\(929\) −45.5980 −1.49602 −0.748011 0.663687i \(-0.768991\pi\)
−0.748011 + 0.663687i \(0.768991\pi\)
\(930\) 0 0
\(931\) 43.1127 1.41296
\(932\) 72.0833 2.36117
\(933\) −3.31371 −0.108486
\(934\) 54.6274 1.78746
\(935\) 0 0
\(936\) 24.9706 0.816188
\(937\) −11.0294 −0.360316 −0.180158 0.983638i \(-0.557661\pi\)
−0.180158 + 0.983638i \(0.557661\pi\)
\(938\) −11.3137 −0.369406
\(939\) −15.6569 −0.510942
\(940\) 0 0
\(941\) −34.7696 −1.13346 −0.566728 0.823905i \(-0.691791\pi\)
−0.566728 + 0.823905i \(0.691791\pi\)
\(942\) −43.4558 −1.41587
\(943\) 19.3137 0.628941
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −21.3137 −0.692968
\(947\) −6.62742 −0.215362 −0.107681 0.994185i \(-0.534343\pi\)
−0.107681 + 0.994185i \(0.534343\pi\)
\(948\) 32.4853 1.05507
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 26.2843 0.852326
\(952\) −4.28427 −0.138854
\(953\) 11.7990 0.382207 0.191103 0.981570i \(-0.438793\pi\)
0.191103 + 0.981570i \(0.438793\pi\)
\(954\) −22.4853 −0.727988
\(955\) 0 0
\(956\) 67.5980 2.18627
\(957\) 4.82843 0.156081
\(958\) 89.2548 2.88369
\(959\) −11.0294 −0.356159
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −159.196 −5.13268
\(963\) −17.3137 −0.557926
\(964\) −47.2548 −1.52198
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) −11.4558 −0.368395 −0.184198 0.982889i \(-0.558969\pi\)
−0.184198 + 0.982889i \(0.558969\pi\)
\(968\) 4.41421 0.141878
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −34.6274 −1.11125 −0.555623 0.831434i \(-0.687520\pi\)
−0.555623 + 0.831434i \(0.687520\pi\)
\(972\) 3.82843 0.122797
\(973\) −0.402020 −0.0128882
\(974\) 31.3137 1.00336
\(975\) 0 0
\(976\) −34.9706 −1.11938
\(977\) −2.68629 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(978\) 17.6569 0.564604
\(979\) −3.65685 −0.116874
\(980\) 0 0
\(981\) 17.3137 0.552784
\(982\) 34.6274 1.10501
\(983\) 30.6274 0.976863 0.488431 0.872602i \(-0.337569\pi\)
0.488431 + 0.872602i \(0.337569\pi\)
\(984\) 21.3137 0.679456
\(985\) 0 0
\(986\) −13.6569 −0.434923
\(987\) −3.31371 −0.105477
\(988\) −147.882 −4.70476
\(989\) 35.3137 1.12291
\(990\) 0 0
\(991\) −30.6274 −0.972912 −0.486456 0.873705i \(-0.661711\pi\)
−0.486456 + 0.873705i \(0.661711\pi\)
\(992\) 0 0
\(993\) 6.34315 0.201294
\(994\) −4.68629 −0.148640
\(995\) 0 0
\(996\) 38.2843 1.21308
\(997\) 39.3137 1.24508 0.622539 0.782589i \(-0.286101\pi\)
0.622539 + 0.782589i \(0.286101\pi\)
\(998\) −53.9411 −1.70748
\(999\) −11.6569 −0.368807
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 825.2.a.g.1.2 2
3.2 odd 2 2475.2.a.m.1.1 2
5.2 odd 4 825.2.c.e.199.4 4
5.3 odd 4 825.2.c.e.199.1 4
5.4 even 2 165.2.a.a.1.1 2
11.10 odd 2 9075.2.a.v.1.1 2
15.2 even 4 2475.2.c.m.199.1 4
15.8 even 4 2475.2.c.m.199.4 4
15.14 odd 2 495.2.a.d.1.2 2
20.19 odd 2 2640.2.a.bb.1.1 2
35.34 odd 2 8085.2.a.ba.1.1 2
55.54 odd 2 1815.2.a.k.1.2 2
60.59 even 2 7920.2.a.cg.1.1 2
165.164 even 2 5445.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 5.4 even 2
495.2.a.d.1.2 2 15.14 odd 2
825.2.a.g.1.2 2 1.1 even 1 trivial
825.2.c.e.199.1 4 5.3 odd 4
825.2.c.e.199.4 4 5.2 odd 4
1815.2.a.k.1.2 2 55.54 odd 2
2475.2.a.m.1.1 2 3.2 odd 2
2475.2.c.m.199.1 4 15.2 even 4
2475.2.c.m.199.4 4 15.8 even 4
2640.2.a.bb.1.1 2 20.19 odd 2
5445.2.a.m.1.1 2 165.164 even 2
7920.2.a.cg.1.1 2 60.59 even 2
8085.2.a.ba.1.1 2 35.34 odd 2
9075.2.a.v.1.1 2 11.10 odd 2