Properties

Label 825.2.c.e.199.4
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.e.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -0.828427i q^{7} -4.41421i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -0.828427i q^{7} -4.41421i q^{8} -1.00000 q^{9} -1.00000 q^{11} +3.82843i q^{12} -5.65685i q^{13} +2.00000 q^{14} +3.00000 q^{16} +1.17157i q^{17} -2.41421i q^{18} +6.82843 q^{19} -0.828427 q^{21} -2.41421i q^{22} -4.00000i q^{23} -4.41421 q^{24} +13.6569 q^{26} +1.00000i q^{27} +3.17157i q^{28} +4.82843 q^{29} -1.58579i q^{32} +1.00000i q^{33} -2.82843 q^{34} +3.82843 q^{36} -11.6569i q^{37} +16.4853i q^{38} -5.65685 q^{39} +4.82843 q^{41} -2.00000i q^{42} -8.82843i q^{43} +3.82843 q^{44} +9.65685 q^{46} +4.00000i q^{47} -3.00000i q^{48} +6.31371 q^{49} +1.17157 q^{51} +21.6569i q^{52} +9.31371i q^{53} -2.41421 q^{54} -3.65685 q^{56} -6.82843i q^{57} +11.6569i q^{58} +4.00000 q^{59} -11.6569 q^{61} +0.828427i q^{63} +9.82843 q^{64} -2.41421 q^{66} +5.65685i q^{67} -4.48528i q^{68} -4.00000 q^{69} +2.34315 q^{71} +4.41421i q^{72} +11.3137i q^{73} +28.1421 q^{74} -26.1421 q^{76} +0.828427i q^{77} -13.6569i q^{78} -8.48528 q^{79} +1.00000 q^{81} +11.6569i q^{82} -10.0000i q^{83} +3.17157 q^{84} +21.3137 q^{86} -4.82843i q^{87} +4.41421i q^{88} -3.65685 q^{89} -4.68629 q^{91} +15.3137i q^{92} -9.65685 q^{94} -1.58579 q^{96} -11.6569i q^{97} +15.2426i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} - 4q^{11} + 8q^{14} + 12q^{16} + 16q^{19} + 8q^{21} - 12q^{24} + 32q^{26} + 8q^{29} + 4q^{36} + 8q^{41} + 4q^{44} + 16q^{46} - 20q^{49} + 16q^{51} - 4q^{54} + 8q^{56} + 16q^{59} - 24q^{61} + 28q^{64} - 4q^{66} - 16q^{69} + 32q^{71} + 56q^{74} - 48q^{76} + 4q^{81} + 24q^{84} + 40q^{86} + 8q^{89} - 64q^{91} - 16q^{94} - 12q^{96} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421i 1.70711i 0.521005 + 0.853553i \(0.325557\pi\)
−0.521005 + 0.853553i \(0.674443\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) 2.41421 0.985599
\(7\) − 0.828427i − 0.313116i −0.987669 0.156558i \(-0.949960\pi\)
0.987669 0.156558i \(-0.0500398\pi\)
\(8\) − 4.41421i − 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 3.82843i 1.10517i
\(13\) − 5.65685i − 1.56893i −0.620174 0.784465i \(-0.712938\pi\)
0.620174 0.784465i \(-0.287062\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 1.17157i 0.284148i 0.989856 + 0.142074i \(0.0453771\pi\)
−0.989856 + 0.142074i \(0.954623\pi\)
\(18\) − 2.41421i − 0.569036i
\(19\) 6.82843 1.56655 0.783274 0.621676i \(-0.213548\pi\)
0.783274 + 0.621676i \(0.213548\pi\)
\(20\) 0 0
\(21\) −0.828427 −0.180778
\(22\) − 2.41421i − 0.514712i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −4.41421 −0.901048
\(25\) 0 0
\(26\) 13.6569 2.67833
\(27\) 1.00000i 0.192450i
\(28\) 3.17157i 0.599371i
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.58579i − 0.280330i
\(33\) 1.00000i 0.174078i
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) − 11.6569i − 1.91638i −0.286141 0.958188i \(-0.592373\pi\)
0.286141 0.958188i \(-0.407627\pi\)
\(38\) 16.4853i 2.67427i
\(39\) −5.65685 −0.905822
\(40\) 0 0
\(41\) 4.82843 0.754074 0.377037 0.926198i \(-0.376943\pi\)
0.377037 + 0.926198i \(0.376943\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 8.82843i − 1.34632i −0.739496 0.673161i \(-0.764936\pi\)
0.739496 0.673161i \(-0.235064\pi\)
\(44\) 3.82843 0.577157
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) 6.31371 0.901958
\(50\) 0 0
\(51\) 1.17157 0.164053
\(52\) 21.6569i 3.00327i
\(53\) 9.31371i 1.27934i 0.768651 + 0.639668i \(0.220928\pi\)
−0.768651 + 0.639668i \(0.779072\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) −3.65685 −0.488668
\(57\) − 6.82843i − 0.904447i
\(58\) 11.6569i 1.53062i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −11.6569 −1.49251 −0.746254 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0 0
\(63\) 0.828427i 0.104372i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) −2.41421 −0.297169
\(67\) 5.65685i 0.691095i 0.938401 + 0.345547i \(0.112307\pi\)
−0.938401 + 0.345547i \(0.887693\pi\)
\(68\) − 4.48528i − 0.543920i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 2.34315 0.278080 0.139040 0.990287i \(-0.455598\pi\)
0.139040 + 0.990287i \(0.455598\pi\)
\(72\) 4.41421i 0.520220i
\(73\) 11.3137i 1.32417i 0.749429 + 0.662085i \(0.230328\pi\)
−0.749429 + 0.662085i \(0.769672\pi\)
\(74\) 28.1421 3.27146
\(75\) 0 0
\(76\) −26.1421 −2.99871
\(77\) 0.828427i 0.0944080i
\(78\) − 13.6569i − 1.54633i
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 11.6569i 1.28728i
\(83\) − 10.0000i − 1.09764i −0.835940 0.548821i \(-0.815077\pi\)
0.835940 0.548821i \(-0.184923\pi\)
\(84\) 3.17157 0.346047
\(85\) 0 0
\(86\) 21.3137 2.29832
\(87\) − 4.82843i − 0.517662i
\(88\) 4.41421i 0.470557i
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) −4.68629 −0.491257
\(92\) 15.3137i 1.59656i
\(93\) 0 0
\(94\) −9.65685 −0.996028
\(95\) 0 0
\(96\) −1.58579 −0.161849
\(97\) − 11.6569i − 1.18357i −0.806094 0.591787i \(-0.798423\pi\)
0.806094 0.591787i \(-0.201577\pi\)
\(98\) 15.2426i 1.53974i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −0.828427 −0.0824316 −0.0412158 0.999150i \(-0.513123\pi\)
−0.0412158 + 0.999150i \(0.513123\pi\)
\(102\) 2.82843i 0.280056i
\(103\) − 3.31371i − 0.326509i −0.986584 0.163255i \(-0.947801\pi\)
0.986584 0.163255i \(-0.0521992\pi\)
\(104\) −24.9706 −2.44857
\(105\) 0 0
\(106\) −22.4853 −2.18396
\(107\) − 17.3137i − 1.67378i −0.547372 0.836890i \(-0.684372\pi\)
0.547372 0.836890i \(-0.315628\pi\)
\(108\) − 3.82843i − 0.368391i
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) 0 0
\(111\) −11.6569 −1.10642
\(112\) − 2.48528i − 0.234837i
\(113\) − 18.9706i − 1.78460i −0.451442 0.892300i \(-0.649090\pi\)
0.451442 0.892300i \(-0.350910\pi\)
\(114\) 16.4853 1.54399
\(115\) 0 0
\(116\) −18.4853 −1.71632
\(117\) 5.65685i 0.522976i
\(118\) 9.65685i 0.888985i
\(119\) 0.970563 0.0889713
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 28.1421i − 2.54787i
\(123\) − 4.82843i − 0.435365i
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 14.4853i 1.28536i 0.766134 + 0.642680i \(0.222178\pi\)
−0.766134 + 0.642680i \(0.777822\pi\)
\(128\) 20.5563i 1.81694i
\(129\) −8.82843 −0.777300
\(130\) 0 0
\(131\) 3.31371 0.289520 0.144760 0.989467i \(-0.453759\pi\)
0.144760 + 0.989467i \(0.453759\pi\)
\(132\) − 3.82843i − 0.333222i
\(133\) − 5.65685i − 0.490511i
\(134\) −13.6569 −1.17977
\(135\) 0 0
\(136\) 5.17157 0.443459
\(137\) 13.3137i 1.13747i 0.822522 + 0.568733i \(0.192566\pi\)
−0.822522 + 0.568733i \(0.807434\pi\)
\(138\) − 9.65685i − 0.822046i
\(139\) −0.485281 −0.0411610 −0.0205805 0.999788i \(-0.506551\pi\)
−0.0205805 + 0.999788i \(0.506551\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 5.65685i 0.474713i
\(143\) 5.65685i 0.473050i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −27.3137 −2.26050
\(147\) − 6.31371i − 0.520746i
\(148\) 44.6274i 3.66835i
\(149\) −1.51472 −0.124091 −0.0620453 0.998073i \(-0.519762\pi\)
−0.0620453 + 0.998073i \(0.519762\pi\)
\(150\) 0 0
\(151\) 16.4853 1.34155 0.670777 0.741659i \(-0.265961\pi\)
0.670777 + 0.741659i \(0.265961\pi\)
\(152\) − 30.1421i − 2.44485i
\(153\) − 1.17157i − 0.0947161i
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 21.6569 1.73394
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) − 20.4853i − 1.62972i
\(159\) 9.31371 0.738625
\(160\) 0 0
\(161\) −3.31371 −0.261157
\(162\) 2.41421i 0.189679i
\(163\) − 7.31371i − 0.572854i −0.958102 0.286427i \(-0.907532\pi\)
0.958102 0.286427i \(-0.0924676\pi\)
\(164\) −18.4853 −1.44346
\(165\) 0 0
\(166\) 24.1421 1.87379
\(167\) 13.3137i 1.03025i 0.857116 + 0.515123i \(0.172254\pi\)
−0.857116 + 0.515123i \(0.827746\pi\)
\(168\) 3.65685i 0.282132i
\(169\) −19.0000 −1.46154
\(170\) 0 0
\(171\) −6.82843 −0.522183
\(172\) 33.7990i 2.57715i
\(173\) 2.82843i 0.215041i 0.994203 + 0.107521i \(0.0342912\pi\)
−0.994203 + 0.107521i \(0.965709\pi\)
\(174\) 11.6569 0.883704
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) − 4.00000i − 0.300658i
\(178\) − 8.82843i − 0.661719i
\(179\) 17.6569 1.31974 0.659868 0.751382i \(-0.270612\pi\)
0.659868 + 0.751382i \(0.270612\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) − 11.3137i − 0.838628i
\(183\) 11.6569i 0.861699i
\(184\) −17.6569 −1.30168
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.17157i − 0.0856739i
\(188\) − 15.3137i − 1.11687i
\(189\) 0.828427 0.0602592
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) − 9.82843i − 0.709306i
\(193\) 13.6569i 0.983042i 0.870866 + 0.491521i \(0.163559\pi\)
−0.870866 + 0.491521i \(0.836441\pi\)
\(194\) 28.1421 2.02049
\(195\) 0 0
\(196\) −24.1716 −1.72654
\(197\) 8.48528i 0.604551i 0.953221 + 0.302276i \(0.0977463\pi\)
−0.953221 + 0.302276i \(0.902254\pi\)
\(198\) 2.41421i 0.171571i
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) − 2.00000i − 0.140720i
\(203\) − 4.00000i − 0.280745i
\(204\) −4.48528 −0.314033
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000i 0.278019i
\(208\) − 16.9706i − 1.17670i
\(209\) −6.82843 −0.472332
\(210\) 0 0
\(211\) 1.17157 0.0806544 0.0403272 0.999187i \(-0.487160\pi\)
0.0403272 + 0.999187i \(0.487160\pi\)
\(212\) − 35.6569i − 2.44892i
\(213\) − 2.34315i − 0.160550i
\(214\) 41.7990 2.85732
\(215\) 0 0
\(216\) 4.41421 0.300349
\(217\) 0 0
\(218\) − 41.7990i − 2.83098i
\(219\) 11.3137 0.764510
\(220\) 0 0
\(221\) 6.62742 0.445808
\(222\) − 28.1421i − 1.88878i
\(223\) − 6.34315i − 0.424768i −0.977186 0.212384i \(-0.931877\pi\)
0.977186 0.212384i \(-0.0681228\pi\)
\(224\) −1.31371 −0.0877758
\(225\) 0 0
\(226\) 45.7990 3.04650
\(227\) 14.0000i 0.929213i 0.885517 + 0.464606i \(0.153804\pi\)
−0.885517 + 0.464606i \(0.846196\pi\)
\(228\) 26.1421i 1.73131i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0.828427 0.0545065
\(232\) − 21.3137i − 1.39931i
\(233\) − 18.8284i − 1.23349i −0.787163 0.616746i \(-0.788451\pi\)
0.787163 0.616746i \(-0.211549\pi\)
\(234\) −13.6569 −0.892776
\(235\) 0 0
\(236\) −15.3137 −0.996838
\(237\) 8.48528i 0.551178i
\(238\) 2.34315i 0.151884i
\(239\) −17.6569 −1.14213 −0.571063 0.820906i \(-0.693469\pi\)
−0.571063 + 0.820906i \(0.693469\pi\)
\(240\) 0 0
\(241\) −12.3431 −0.795092 −0.397546 0.917582i \(-0.630138\pi\)
−0.397546 + 0.917582i \(0.630138\pi\)
\(242\) 2.41421i 0.155192i
\(243\) − 1.00000i − 0.0641500i
\(244\) 44.6274 2.85698
\(245\) 0 0
\(246\) 11.6569 0.743214
\(247\) − 38.6274i − 2.45780i
\(248\) 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 \(0.269782\pi\)
\(252\) − 3.17157i − 0.199790i
\(253\) 4.00000i 0.251478i
\(254\) −34.9706 −2.19425
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 16.3431i 1.01946i 0.860335 + 0.509729i \(0.170254\pi\)
−0.860335 + 0.509729i \(0.829746\pi\)
\(258\) − 21.3137i − 1.32693i
\(259\) −9.65685 −0.600048
\(260\) 0 0
\(261\) −4.82843 −0.298872
\(262\) 8.00000i 0.494242i
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 4.41421 0.271676
\(265\) 0 0
\(266\) 13.6569 0.837355
\(267\) 3.65685i 0.223796i
\(268\) − 21.6569i − 1.32290i
\(269\) −20.6274 −1.25768 −0.628838 0.777536i \(-0.716469\pi\)
−0.628838 + 0.777536i \(0.716469\pi\)
\(270\) 0 0
\(271\) −11.7990 −0.716738 −0.358369 0.933580i \(-0.616667\pi\)
−0.358369 + 0.933580i \(0.616667\pi\)
\(272\) 3.51472i 0.213111i
\(273\) 4.68629i 0.283627i
\(274\) −32.1421 −1.94178
\(275\) 0 0
\(276\) 15.3137 0.921777
\(277\) 2.34315i 0.140786i 0.997519 + 0.0703930i \(0.0224253\pi\)
−0.997519 + 0.0703930i \(0.977575\pi\)
\(278\) − 1.17157i − 0.0702663i
\(279\) 0 0
\(280\) 0 0
\(281\) −11.1716 −0.666440 −0.333220 0.942849i \(-0.608135\pi\)
−0.333220 + 0.942849i \(0.608135\pi\)
\(282\) 9.65685i 0.575057i
\(283\) − 8.82843i − 0.524796i −0.964960 0.262398i \(-0.915487\pi\)
0.964960 0.262398i \(-0.0845132\pi\)
\(284\) −8.97056 −0.532305
\(285\) 0 0
\(286\) −13.6569 −0.807547
\(287\) − 4.00000i − 0.236113i
\(288\) 1.58579i 0.0934434i
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) −11.6569 −0.683337
\(292\) − 43.3137i − 2.53474i
\(293\) − 6.82843i − 0.398921i −0.979906 0.199460i \(-0.936081\pi\)
0.979906 0.199460i \(-0.0639190\pi\)
\(294\) 15.2426 0.888969
\(295\) 0 0
\(296\) −51.4558 −2.99081
\(297\) − 1.00000i − 0.0580259i
\(298\) − 3.65685i − 0.211836i
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) −7.31371 −0.421555
\(302\) 39.7990i 2.29017i
\(303\) 0.828427i 0.0475919i
\(304\) 20.4853 1.17491
\(305\) 0 0
\(306\) 2.82843 0.161690
\(307\) 3.17157i 0.181011i 0.995896 + 0.0905056i \(0.0288483\pi\)
−0.995896 + 0.0905056i \(0.971152\pi\)
\(308\) − 3.17157i − 0.180717i
\(309\) −3.31371 −0.188510
\(310\) 0 0
\(311\) −3.31371 −0.187903 −0.0939516 0.995577i \(-0.529950\pi\)
−0.0939516 + 0.995577i \(0.529950\pi\)
\(312\) 24.9706i 1.41368i
\(313\) 15.6569i 0.884978i 0.896774 + 0.442489i \(0.145904\pi\)
−0.896774 + 0.442489i \(0.854096\pi\)
\(314\) 43.4558 2.45236
\(315\) 0 0
\(316\) 32.4853 1.82744
\(317\) 26.2843i 1.47627i 0.674652 + 0.738136i \(0.264294\pi\)
−0.674652 + 0.738136i \(0.735706\pi\)
\(318\) 22.4853i 1.26091i
\(319\) −4.82843 −0.270340
\(320\) 0 0
\(321\) −17.3137 −0.966357
\(322\) − 8.00000i − 0.445823i
\(323\) 8.00000i 0.445132i
\(324\) −3.82843 −0.212690
\(325\) 0 0
\(326\) 17.6569 0.977923
\(327\) 17.3137i 0.957450i
\(328\) − 21.3137i − 1.17685i
\(329\) 3.31371 0.182691
\(330\) 0 0
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) 38.2843i 2.10112i
\(333\) 11.6569i 0.638792i
\(334\) −32.1421 −1.75874
\(335\) 0 0
\(336\) −2.48528 −0.135583
\(337\) − 3.31371i − 0.180509i −0.995919 0.0902546i \(-0.971232\pi\)
0.995919 0.0902546i \(-0.0287681\pi\)
\(338\) − 45.8701i − 2.49500i
\(339\) −18.9706 −1.03034
\(340\) 0 0
\(341\) 0 0
\(342\) − 16.4853i − 0.891422i
\(343\) − 11.0294i − 0.595534i
\(344\) −38.9706 −2.10115
\(345\) 0 0
\(346\) −6.82843 −0.367099
\(347\) 29.3137i 1.57364i 0.617181 + 0.786821i \(0.288275\pi\)
−0.617181 + 0.786821i \(0.711725\pi\)
\(348\) 18.4853i 0.990915i
\(349\) −10.9706 −0.587241 −0.293620 0.955922i \(-0.594860\pi\)
−0.293620 + 0.955922i \(0.594860\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 1.58579i 0.0845227i
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) 9.65685 0.513256
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 0.970563i − 0.0513676i
\(358\) 42.6274i 2.25293i
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) − 33.7990i − 1.77644i
\(363\) − 1.00000i − 0.0524864i
\(364\) 17.9411 0.940370
\(365\) 0 0
\(366\) −28.1421 −1.47101
\(367\) − 9.65685i − 0.504084i −0.967716 0.252042i \(-0.918898\pi\)
0.967716 0.252042i \(-0.0811021\pi\)
\(368\) − 12.0000i − 0.625543i
\(369\) −4.82843 −0.251358
\(370\) 0 0
\(371\) 7.71573 0.400581
\(372\) 0 0
\(373\) 10.6274i 0.550267i 0.961406 + 0.275133i \(0.0887220\pi\)
−0.961406 + 0.275133i \(0.911278\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) 17.6569 0.910583
\(377\) − 27.3137i − 1.40673i
\(378\) 2.00000i 0.102869i
\(379\) 23.3137 1.19754 0.598772 0.800919i \(-0.295655\pi\)
0.598772 + 0.800919i \(0.295655\pi\)
\(380\) 0 0
\(381\) 14.4853 0.742103
\(382\) 13.6569i 0.698745i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) 20.5563 1.04901
\(385\) 0 0
\(386\) −32.9706 −1.67816
\(387\) 8.82843i 0.448774i
\(388\) 44.6274i 2.26561i
\(389\) 23.6569 1.19945 0.599725 0.800206i \(-0.295277\pi\)
0.599725 + 0.800206i \(0.295277\pi\)
\(390\) 0 0
\(391\) 4.68629 0.236996
\(392\) − 27.8701i − 1.40765i
\(393\) − 3.31371i − 0.167154i
\(394\) −20.4853 −1.03203
\(395\) 0 0
\(396\) −3.82843 −0.192386
\(397\) 14.9706i 0.751351i 0.926751 + 0.375676i \(0.122589\pi\)
−0.926751 + 0.375676i \(0.877411\pi\)
\(398\) 52.2843i 2.62077i
\(399\) −5.65685 −0.283197
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 13.6569i 0.681142i
\(403\) 0 0
\(404\) 3.17157 0.157792
\(405\) 0 0
\(406\) 9.65685 0.479262
\(407\) 11.6569i 0.577809i
\(408\) − 5.17157i − 0.256031i
\(409\) 19.6569 0.971969 0.485984 0.873967i \(-0.338461\pi\)
0.485984 + 0.873967i \(0.338461\pi\)
\(410\) 0 0
\(411\) 13.3137 0.656717
\(412\) 12.6863i 0.625009i
\(413\) − 3.31371i − 0.163057i
\(414\) −9.65685 −0.474608
\(415\) 0 0
\(416\) −8.97056 −0.439818
\(417\) 0.485281i 0.0237643i
\(418\) − 16.4853i − 0.806321i
\(419\) 36.9706 1.80613 0.903065 0.429504i \(-0.141311\pi\)
0.903065 + 0.429504i \(0.141311\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.82843i 0.137686i
\(423\) − 4.00000i − 0.194487i
\(424\) 41.1127 1.99661
\(425\) 0 0
\(426\) 5.65685 0.274075
\(427\) 9.65685i 0.467328i
\(428\) 66.2843i 3.20397i
\(429\) 5.65685 0.273115
\(430\) 0 0
\(431\) 21.6569 1.04317 0.521587 0.853198i \(-0.325340\pi\)
0.521587 + 0.853198i \(0.325340\pi\)
\(432\) 3.00000i 0.144338i
\(433\) − 15.6569i − 0.752420i −0.926534 0.376210i \(-0.877227\pi\)
0.926534 0.376210i \(-0.122773\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 66.2843 3.17444
\(437\) − 27.3137i − 1.30659i
\(438\) 27.3137i 1.30510i
\(439\) 20.4853 0.977709 0.488855 0.872365i \(-0.337415\pi\)
0.488855 + 0.872365i \(0.337415\pi\)
\(440\) 0 0
\(441\) −6.31371 −0.300653
\(442\) 16.0000i 0.761042i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 44.6274 2.11792
\(445\) 0 0
\(446\) 15.3137 0.725125
\(447\) 1.51472i 0.0716437i
\(448\) − 8.14214i − 0.384680i
\(449\) −30.9706 −1.46159 −0.730796 0.682596i \(-0.760851\pi\)
−0.730796 + 0.682596i \(0.760851\pi\)
\(450\) 0 0
\(451\) −4.82843 −0.227362
\(452\) 72.6274i 3.41611i
\(453\) − 16.4853i − 0.774546i
\(454\) −33.7990 −1.58627
\(455\) 0 0
\(456\) −30.1421 −1.41153
\(457\) 23.3137i 1.09057i 0.838251 + 0.545285i \(0.183578\pi\)
−0.838251 + 0.545285i \(0.816422\pi\)
\(458\) 4.82843i 0.225618i
\(459\) −1.17157 −0.0546843
\(460\) 0 0
\(461\) 0.142136 0.00661992 0.00330996 0.999995i \(-0.498946\pi\)
0.00330996 + 0.999995i \(0.498946\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) 4.97056i 0.231002i 0.993307 + 0.115501i \(0.0368473\pi\)
−0.993307 + 0.115501i \(0.963153\pi\)
\(464\) 14.4853 0.672462
\(465\) 0 0
\(466\) 45.4558 2.10570
\(467\) 22.6274i 1.04707i 0.852004 + 0.523536i \(0.175387\pi\)
−0.852004 + 0.523536i \(0.824613\pi\)
\(468\) − 21.6569i − 1.00109i
\(469\) 4.68629 0.216393
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 17.6569i − 0.812723i
\(473\) 8.82843i 0.405932i
\(474\) −20.4853 −0.940920
\(475\) 0 0
\(476\) −3.71573 −0.170310
\(477\) − 9.31371i − 0.426445i
\(478\) − 42.6274i − 1.94973i
\(479\) −36.9706 −1.68923 −0.844614 0.535376i \(-0.820170\pi\)
−0.844614 + 0.535376i \(0.820170\pi\)
\(480\) 0 0
\(481\) −65.9411 −3.00666
\(482\) − 29.7990i − 1.35731i
\(483\) 3.31371i 0.150779i
\(484\) −3.82843 −0.174019
\(485\) 0 0
\(486\) 2.41421 0.109511
\(487\) 12.9706i 0.587752i 0.955844 + 0.293876i \(0.0949453\pi\)
−0.955844 + 0.293876i \(0.905055\pi\)
\(488\) 51.4558i 2.32930i
\(489\) −7.31371 −0.330737
\(490\) 0 0
\(491\) 14.3431 0.647297 0.323649 0.946177i \(-0.395090\pi\)
0.323649 + 0.946177i \(0.395090\pi\)
\(492\) 18.4853i 0.833381i
\(493\) 5.65685i 0.254772i
\(494\) 93.2548 4.19573
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.94113i − 0.0870714i
\(498\) − 24.1421i − 1.08183i
\(499\) 22.3431 1.00022 0.500108 0.865963i \(-0.333294\pi\)
0.500108 + 0.865963i \(0.333294\pi\)
\(500\) 0 0
\(501\) 13.3137 0.594813
\(502\) 50.6274i 2.25961i
\(503\) 17.3137i 0.771980i 0.922503 + 0.385990i \(0.126140\pi\)
−0.922503 + 0.385990i \(0.873860\pi\)
\(504\) 3.65685 0.162889
\(505\) 0 0
\(506\) −9.65685 −0.429300
\(507\) 19.0000i 0.843820i
\(508\) − 55.4558i − 2.46046i
\(509\) −18.6863 −0.828255 −0.414128 0.910219i \(-0.635913\pi\)
−0.414128 + 0.910219i \(0.635913\pi\)
\(510\) 0 0
\(511\) 9.37258 0.414619
\(512\) − 31.2426i − 1.38074i
\(513\) 6.82843i 0.301482i
\(514\) −39.4558 −1.74032
\(515\) 0 0
\(516\) 33.7990 1.48792
\(517\) − 4.00000i − 0.175920i
\(518\) − 23.3137i − 1.02435i
\(519\) 2.82843 0.124154
\(520\) 0 0
\(521\) −32.6274 −1.42943 −0.714717 0.699414i \(-0.753444\pi\)
−0.714717 + 0.699414i \(0.753444\pi\)
\(522\) − 11.6569i − 0.510207i
\(523\) − 9.51472i − 0.416050i −0.978124 0.208025i \(-0.933297\pi\)
0.978124 0.208025i \(-0.0667035\pi\)
\(524\) −12.6863 −0.554203
\(525\) 0 0
\(526\) −43.4558 −1.89476
\(527\) 0 0
\(528\) 3.00000i 0.130558i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 21.6569i 0.938944i
\(533\) − 27.3137i − 1.18309i
\(534\) −8.82843 −0.382043
\(535\) 0 0
\(536\) 24.9706 1.07856
\(537\) − 17.6569i − 0.761950i
\(538\) − 49.7990i − 2.14699i
\(539\) −6.31371 −0.271951
\(540\) 0 0
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) − 28.4853i − 1.22355i
\(543\) 14.0000i 0.600798i
\(544\) 1.85786 0.0796553
\(545\) 0 0
\(546\) −11.3137 −0.484182
\(547\) 8.14214i 0.348133i 0.984734 + 0.174066i \(0.0556907\pi\)
−0.984734 + 0.174066i \(0.944309\pi\)
\(548\) − 50.9706i − 2.17735i
\(549\) 11.6569 0.497502
\(550\) 0 0
\(551\) 32.9706 1.40459
\(552\) 17.6569i 0.751526i
\(553\) 7.02944i 0.298922i
\(554\) −5.65685 −0.240337
\(555\) 0 0
\(556\) 1.85786 0.0787910
\(557\) − 5.17157i − 0.219127i −0.993980 0.109563i \(-0.965055\pi\)
0.993980 0.109563i \(-0.0349452\pi\)
\(558\) 0 0
\(559\) −49.9411 −2.11228
\(560\) 0 0
\(561\) −1.17157 −0.0494638
\(562\) − 26.9706i − 1.13768i
\(563\) 31.6569i 1.33418i 0.744978 + 0.667089i \(0.232460\pi\)
−0.744978 + 0.667089i \(0.767540\pi\)
\(564\) −15.3137 −0.644823
\(565\) 0 0
\(566\) 21.3137 0.895882
\(567\) − 0.828427i − 0.0347907i
\(568\) − 10.3431i − 0.433989i
\(569\) 35.4558 1.48639 0.743193 0.669077i \(-0.233310\pi\)
0.743193 + 0.669077i \(0.233310\pi\)
\(570\) 0 0
\(571\) −16.4853 −0.689888 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(572\) − 21.6569i − 0.905519i
\(573\) − 5.65685i − 0.236318i
\(574\) 9.65685 0.403069
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 37.7279i 1.56927i
\(579\) 13.6569 0.567559
\(580\) 0 0
\(581\) −8.28427 −0.343689
\(582\) − 28.1421i − 1.16653i
\(583\) − 9.31371i − 0.385734i
\(584\) 49.9411 2.06658
\(585\) 0 0
\(586\) 16.4853 0.681001
\(587\) − 14.6274i − 0.603738i −0.953349 0.301869i \(-0.902389\pi\)
0.953349 0.301869i \(-0.0976105\pi\)
\(588\) 24.1716i 0.996819i
\(589\) 0 0
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) − 34.9706i − 1.43728i
\(593\) − 22.8284i − 0.937451i −0.883344 0.468726i \(-0.844713\pi\)
0.883344 0.468726i \(-0.155287\pi\)
\(594\) 2.41421 0.0990564
\(595\) 0 0
\(596\) 5.79899 0.237536
\(597\) − 21.6569i − 0.886356i
\(598\) − 54.6274i − 2.23388i
\(599\) −27.3137 −1.11601 −0.558004 0.829838i \(-0.688433\pi\)
−0.558004 + 0.829838i \(0.688433\pi\)
\(600\) 0 0
\(601\) −5.31371 −0.216751 −0.108375 0.994110i \(-0.534565\pi\)
−0.108375 + 0.994110i \(0.534565\pi\)
\(602\) − 17.6569i − 0.719640i
\(603\) − 5.65685i − 0.230365i
\(604\) −63.1127 −2.56802
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) − 1.51472i − 0.0614805i −0.999527 0.0307403i \(-0.990214\pi\)
0.999527 0.0307403i \(-0.00978647\pi\)
\(608\) − 10.8284i − 0.439151i
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 4.48528i 0.181307i
\(613\) − 45.9411i − 1.85554i −0.373147 0.927772i \(-0.621721\pi\)
0.373147 0.927772i \(-0.378279\pi\)
\(614\) −7.65685 −0.309005
\(615\) 0 0
\(616\) 3.65685 0.147339
\(617\) − 0.343146i − 0.0138145i −0.999976 0.00690726i \(-0.997801\pi\)
0.999976 0.00690726i \(-0.00219867\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) 14.3431 0.576500 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 8.00000i − 0.320771i
\(623\) 3.02944i 0.121372i
\(624\) −16.9706 −0.679366
\(625\) 0 0
\(626\) −37.7990 −1.51075
\(627\) 6.82843i 0.272701i
\(628\) 68.9117i 2.74988i
\(629\) 13.6569 0.544534
\(630\) 0 0
\(631\) 45.6569 1.81757 0.908785 0.417264i \(-0.137011\pi\)
0.908785 + 0.417264i \(0.137011\pi\)
\(632\) 37.4558i 1.48991i
\(633\) − 1.17157i − 0.0465658i
\(634\) −63.4558 −2.52015
\(635\) 0 0
\(636\) −35.6569 −1.41389
\(637\) − 35.7157i − 1.41511i
\(638\) − 11.6569i − 0.461499i
\(639\) −2.34315 −0.0926934
\(640\) 0 0
\(641\) 6.97056 0.275321 0.137660 0.990479i \(-0.456042\pi\)
0.137660 + 0.990479i \(0.456042\pi\)
\(642\) − 41.7990i − 1.64967i
\(643\) 37.9411i 1.49625i 0.663557 + 0.748126i \(0.269046\pi\)
−0.663557 + 0.748126i \(0.730954\pi\)
\(644\) 12.6863 0.499910
\(645\) 0 0
\(646\) −19.3137 −0.759888
\(647\) − 4.68629i − 0.184237i −0.995748 0.0921186i \(-0.970636\pi\)
0.995748 0.0921186i \(-0.0293639\pi\)
\(648\) − 4.41421i − 0.173407i
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 28.0000i 1.09656i
\(653\) 6.97056i 0.272779i 0.990655 + 0.136390i \(0.0435499\pi\)
−0.990655 + 0.136390i \(0.956450\pi\)
\(654\) −41.7990 −1.63447
\(655\) 0 0
\(656\) 14.4853 0.565555
\(657\) − 11.3137i − 0.441390i
\(658\) 8.00000i 0.311872i
\(659\) 15.3137 0.596537 0.298269 0.954482i \(-0.403591\pi\)
0.298269 + 0.954482i \(0.403591\pi\)
\(660\) 0 0
\(661\) 9.31371 0.362261 0.181131 0.983459i \(-0.442024\pi\)
0.181131 + 0.983459i \(0.442024\pi\)
\(662\) 15.3137i 0.595184i
\(663\) − 6.62742i − 0.257388i
\(664\) −44.1421 −1.71305
\(665\) 0 0
\(666\) −28.1421 −1.09049
\(667\) − 19.3137i − 0.747830i
\(668\) − 50.9706i − 1.97211i
\(669\) −6.34315 −0.245240
\(670\) 0 0
\(671\) 11.6569 0.450008
\(672\) 1.31371i 0.0506774i
\(673\) 18.3431i 0.707076i 0.935420 + 0.353538i \(0.115022\pi\)
−0.935420 + 0.353538i \(0.884978\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) 72.7401 2.79770
\(677\) − 29.4558i − 1.13208i −0.824378 0.566040i \(-0.808475\pi\)
0.824378 0.566040i \(-0.191525\pi\)
\(678\) − 45.7990i − 1.75890i
\(679\) −9.65685 −0.370596
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 26.1421 0.999570
\(685\) 0 0
\(686\) 26.6274 1.01664
\(687\) − 2.00000i − 0.0763048i
\(688\) − 26.4853i − 1.00974i
\(689\) 52.6863 2.00719
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 10.8284i − 0.411635i
\(693\) − 0.828427i − 0.0314693i
\(694\) −70.7696 −2.68638
\(695\) 0 0
\(696\) −21.3137 −0.807894
\(697\) 5.65685i 0.214269i
\(698\) − 26.4853i − 1.00248i
\(699\) −18.8284 −0.712157
\(700\) 0 0
\(701\) 36.1421 1.36507 0.682535 0.730853i \(-0.260878\pi\)
0.682535 + 0.730853i \(0.260878\pi\)
\(702\) 13.6569i 0.515445i
\(703\) − 79.5980i − 3.00209i
\(704\) −9.82843 −0.370423
\(705\) 0 0
\(706\) −62.7696 −2.36236
\(707\) 0.686292i 0.0258106i
\(708\) 15.3137i 0.575524i
\(709\) −6.68629 −0.251109 −0.125554 0.992087i \(-0.540071\pi\)
−0.125554 + 0.992087i \(0.540071\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 16.1421i 0.604952i
\(713\) 0 0
\(714\) 2.34315 0.0876900
\(715\) 0 0
\(716\) −67.5980 −2.52626
\(717\) 17.6569i 0.659407i
\(718\) − 28.9706i − 1.08117i
\(719\) 47.5980 1.77511 0.887553 0.460706i \(-0.152404\pi\)
0.887553 + 0.460706i \(0.152404\pi\)
\(720\) 0 0
\(721\) −2.74517 −0.102235
\(722\) 66.6985i 2.48226i
\(723\) 12.3431i 0.459047i
\(724\) 53.5980 1.99195
\(725\) 0 0
\(726\) 2.41421 0.0895999
\(727\) − 33.9411i − 1.25881i −0.777079 0.629403i \(-0.783299\pi\)
0.777079 0.629403i \(-0.216701\pi\)
\(728\) 20.6863i 0.766685i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 10.3431 0.382555
\(732\) − 44.6274i − 1.64948i
\(733\) − 6.34315i − 0.234289i −0.993115 0.117145i \(-0.962626\pi\)
0.993115 0.117145i \(-0.0373741\pi\)
\(734\) 23.3137 0.860525
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) − 5.65685i − 0.208373i
\(738\) − 11.6569i − 0.429095i
\(739\) 15.1127 0.555930 0.277965 0.960591i \(-0.410340\pi\)
0.277965 + 0.960591i \(0.410340\pi\)
\(740\) 0 0
\(741\) −38.6274 −1.41901
\(742\) 18.6274i 0.683834i
\(743\) 36.3431i 1.33330i 0.745371 + 0.666650i \(0.232273\pi\)
−0.745371 + 0.666650i \(0.767727\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −25.6569 −0.939364
\(747\) 10.0000i 0.365881i
\(748\) 4.48528i 0.163998i
\(749\) −14.3431 −0.524087
\(750\) 0 0
\(751\) 20.2843 0.740184 0.370092 0.928995i \(-0.379326\pi\)
0.370092 + 0.928995i \(0.379326\pi\)
\(752\) 12.0000i 0.437595i
\(753\) − 20.9706i − 0.764210i
\(754\) 65.9411 2.40143
\(755\) 0 0
\(756\) −3.17157 −0.115349
\(757\) 36.6274i 1.33125i 0.746288 + 0.665623i \(0.231834\pi\)
−0.746288 + 0.665623i \(0.768166\pi\)
\(758\) 56.2843i 2.04434i
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −28.8284 −1.04503 −0.522515 0.852630i \(-0.675006\pi\)
−0.522515 + 0.852630i \(0.675006\pi\)
\(762\) 34.9706i 1.26685i
\(763\) 14.3431i 0.519257i
\(764\) −21.6569 −0.783517
\(765\) 0 0
\(766\) 19.3137 0.697833
\(767\) − 22.6274i − 0.817029i
\(768\) 29.9706i 1.08147i
\(769\) −10.6863 −0.385358 −0.192679 0.981262i \(-0.561718\pi\)
−0.192679 + 0.981262i \(0.561718\pi\)
\(770\) 0 0
\(771\) 16.3431 0.588584
\(772\) − 52.2843i − 1.88175i
\(773\) 3.65685i 0.131528i 0.997835 + 0.0657640i \(0.0209484\pi\)
−0.997835 + 0.0657640i \(0.979052\pi\)
\(774\) −21.3137 −0.766105
\(775\) 0 0
\(776\) −51.4558 −1.84716
\(777\) 9.65685i 0.346438i
\(778\) 57.1127i 2.04759i
\(779\) 32.9706 1.18129
\(780\) 0 0
\(781\) −2.34315 −0.0838443
\(782\) 11.3137i 0.404577i
\(783\) 4.82843i 0.172554i
\(784\) 18.9411 0.676469
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) 20.1421i 0.717990i 0.933340 + 0.358995i \(0.116880\pi\)
−0.933340 + 0.358995i \(0.883120\pi\)
\(788\) − 32.4853i − 1.15724i
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −15.7157 −0.558787
\(792\) − 4.41421i − 0.156852i
\(793\) 65.9411i 2.34164i
\(794\) −36.1421 −1.28264
\(795\) 0 0
\(796\) −82.9117 −2.93873
\(797\) − 34.9706i − 1.23872i −0.785107 0.619360i \(-0.787392\pi\)
0.785107 0.619360i \(-0.212608\pi\)
\(798\) − 13.6569i − 0.483447i
\(799\) −4.68629 −0.165789
\(800\) 0 0
\(801\) 3.65685 0.129209
\(802\) − 16.1421i − 0.569999i
\(803\) − 11.3137i − 0.399252i
\(804\) −21.6569 −0.763778
\(805\) 0 0
\(806\) 0 0
\(807\) 20.6274i 0.726119i
\(808\) 3.65685i 0.128648i
\(809\) −28.4264 −0.999419 −0.499710 0.866193i \(-0.666560\pi\)
−0.499710 + 0.866193i \(0.666560\pi\)
\(810\) 0 0
\(811\) 0.485281 0.0170405 0.00852027 0.999964i \(-0.497288\pi\)
0.00852027 + 0.999964i \(0.497288\pi\)
\(812\) 15.3137i 0.537406i
\(813\) 11.7990i 0.413809i
\(814\) −28.1421 −0.986381
\(815\) 0 0
\(816\) 3.51472 0.123040
\(817\) − 60.2843i − 2.10908i
\(818\) 47.4558i 1.65925i
\(819\) 4.68629 0.163752
\(820\) 0 0
\(821\) −12.8284 −0.447715 −0.223858 0.974622i \(-0.571865\pi\)
−0.223858 + 0.974622i \(0.571865\pi\)
\(822\) 32.1421i 1.12109i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) −14.6274 −0.509570
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) − 41.3137i − 1.43662i −0.695724 0.718309i \(-0.744916\pi\)
0.695724 0.718309i \(-0.255084\pi\)
\(828\) − 15.3137i − 0.532188i
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 2.34315 0.0812828
\(832\) − 55.5980i − 1.92751i
\(833\) 7.39697i 0.256290i
\(834\) −1.17157 −0.0405683
\(835\) 0 0
\(836\) 26.1421 0.904145
\(837\) 0 0
\(838\) 89.2548i 3.08326i
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) − 14.4853i − 0.499196i
\(843\) 11.1716i 0.384769i
\(844\) −4.48528 −0.154390
\(845\) 0 0
\(846\) 9.65685 0.332009
\(847\) − 0.828427i − 0.0284651i
\(848\) 27.9411i 0.959502i
\(849\) −8.82843 −0.302991
\(850\) 0 0
\(851\) −46.6274 −1.59837
\(852\) 8.97056i 0.307326i
\(853\) − 8.68629i − 0.297413i −0.988881 0.148706i \(-0.952489\pi\)
0.988881 0.148706i \(-0.0475110\pi\)
\(854\) −23.3137 −0.797779
\(855\) 0 0
\(856\) −76.4264 −2.61220
\(857\) 28.4853i 0.973039i 0.873670 + 0.486519i \(0.161734\pi\)
−0.873670 + 0.486519i \(0.838266\pi\)
\(858\) 13.6569i 0.466237i
\(859\) 52.9706 1.80733 0.903666 0.428238i \(-0.140865\pi\)
0.903666 + 0.428238i \(0.140865\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 52.2843i 1.78081i
\(863\) − 20.6863i − 0.704170i −0.935968 0.352085i \(-0.885473\pi\)
0.935968 0.352085i \(-0.114527\pi\)
\(864\) 1.58579 0.0539496
\(865\) 0 0
\(866\) 37.7990 1.28446
\(867\) − 15.6274i − 0.530735i
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 76.4264i 2.58812i
\(873\) 11.6569i 0.394525i
\(874\) 65.9411 2.23049
\(875\) 0 0
\(876\) −43.3137 −1.46343
\(877\) − 2.62742i − 0.0887216i −0.999016 0.0443608i \(-0.985875\pi\)
0.999016 0.0443608i \(-0.0141251\pi\)
\(878\) 49.4558i 1.66905i
\(879\) −6.82843 −0.230317
\(880\) 0 0
\(881\) −46.9706 −1.58248 −0.791239 0.611507i \(-0.790564\pi\)
−0.791239 + 0.611507i \(0.790564\pi\)
\(882\) − 15.2426i − 0.513246i
\(883\) − 5.37258i − 0.180802i −0.995905 0.0904009i \(-0.971185\pi\)
0.995905 0.0904009i \(-0.0288148\pi\)
\(884\) −25.3726 −0.853372
\(885\) 0 0
\(886\) −28.9706 −0.973285
\(887\) 15.6569i 0.525706i 0.964836 + 0.262853i \(0.0846634\pi\)
−0.964836 + 0.262853i \(0.915337\pi\)
\(888\) 51.4558i 1.72675i
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 24.2843i 0.813098i
\(893\) 27.3137i 0.914018i
\(894\) −3.65685 −0.122304
\(895\) 0 0
\(896\) 17.0294 0.568914
\(897\) 22.6274i 0.755507i
\(898\) − 74.7696i − 2.49509i
\(899\) 0 0
\(900\) 0 0
\(901\) −10.9117 −0.363521
\(902\) − 11.6569i − 0.388131i
\(903\) 7.31371i 0.243385i
\(904\) −83.7401 −2.78515
\(905\) 0 0
\(906\) 39.7990 1.32223
\(907\) − 40.9706i − 1.36041i −0.733024 0.680203i \(-0.761892\pi\)
0.733024 0.680203i \(-0.238108\pi\)
\(908\) − 53.5980i − 1.77871i
\(909\) 0.828427 0.0274772
\(910\) 0 0
\(911\) −48.9706 −1.62247 −0.811234 0.584722i \(-0.801203\pi\)
−0.811234 + 0.584722i \(0.801203\pi\)
\(912\) − 20.4853i − 0.678335i
\(913\) 10.0000i 0.330952i
\(914\) −56.2843 −1.86172
\(915\) 0 0
\(916\) −7.65685 −0.252990
\(917\) − 2.74517i − 0.0906534i
\(918\) − 2.82843i − 0.0933520i
\(919\) −11.5147 −0.379836 −0.189918 0.981800i \(-0.560822\pi\)
−0.189918 + 0.981800i \(0.560822\pi\)
\(920\) 0 0
\(921\) 3.17157 0.104507
\(922\) 0.343146i 0.0113009i
\(923\) − 13.2548i − 0.436288i
\(924\) −3.17157 −0.104337
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 3.31371i 0.108836i
\(928\) − 7.65685i − 0.251349i
\(929\) 45.5980 1.49602 0.748011 0.663687i \(-0.231009\pi\)
0.748011 + 0.663687i \(0.231009\pi\)
\(930\) 0 0
\(931\) 43.1127 1.41296
\(932\) 72.0833i 2.36117i
\(933\) 3.31371i 0.108486i
\(934\) −54.6274 −1.78746
\(935\) 0 0
\(936\) 24.9706 0.816188
\(937\) − 11.0294i − 0.360316i −0.983638 0.180158i \(-0.942339\pi\)
0.983638 0.180158i \(-0.0576609\pi\)
\(938\) 11.3137i 0.369406i
\(939\) 15.6569 0.510942
\(940\) 0 0
\(941\) −34.7696 −1.13346 −0.566728 0.823905i \(-0.691791\pi\)
−0.566728 + 0.823905i \(0.691791\pi\)
\(942\) − 43.4558i − 1.41587i
\(943\) − 19.3137i − 0.628941i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −21.3137 −0.692968
\(947\) − 6.62742i − 0.215362i −0.994185 0.107681i \(-0.965657\pi\)
0.994185 0.107681i \(-0.0343425\pi\)
\(948\) − 32.4853i − 1.05507i
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) 26.2843 0.852326
\(952\) − 4.28427i − 0.138854i
\(953\) − 11.7990i − 0.382207i −0.981570 0.191103i \(-0.938793\pi\)
0.981570 0.191103i \(-0.0612066\pi\)
\(954\) 22.4853 0.727988
\(955\) 0 0
\(956\) 67.5980 2.18627
\(957\) 4.82843i 0.156081i
\(958\) − 89.2548i − 2.88369i
\(959\) 11.0294 0.356159
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 159.196i − 5.13268i
\(963\) 17.3137i 0.557926i
\(964\) 47.2548 1.52198
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) − 11.4558i − 0.368395i −0.982889 0.184198i \(-0.941031\pi\)
0.982889 0.184198i \(-0.0589686\pi\)
\(968\) − 4.41421i − 0.141878i
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −34.6274 −1.11125 −0.555623 0.831434i \(-0.687520\pi\)
−0.555623 + 0.831434i \(0.687520\pi\)
\(972\) 3.82843i 0.122797i
\(973\) 0.402020i 0.0128882i
\(974\) −31.3137 −1.00336
\(975\) 0 0
\(976\) −34.9706 −1.11938
\(977\) − 2.68629i − 0.0859421i −0.999076 0.0429710i \(-0.986318\pi\)
0.999076 0.0429710i \(-0.0136823\pi\)
\(978\) − 17.6569i − 0.564604i
\(979\) 3.65685 0.116874
\(980\) 0 0
\(981\) 17.3137 0.552784
\(982\) 34.6274i 1.10501i
\(983\) − 30.6274i − 0.976863i −0.872602 0.488431i \(-0.837569\pi\)
0.872602 0.488431i \(-0.162431\pi\)
\(984\) −21.3137 −0.679456
\(985\) 0 0
\(986\) −13.6569 −0.434923
\(987\) − 3.31371i − 0.105477i
\(988\) 147.882i 4.70476i
\(989\) −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
\(993\) − 6.34315i − 0.201294i
\(994\) 4.68629 0.148640
\(995\) 0 0
\(996\) 38.2843 1.21308
\(997\) 39.3137i 1.24508i 0.782589 + 0.622539i \(0.213899\pi\)
−0.782589 + 0.622539i \(0.786101\pi\)
\(998\) 53.9411i 1.70748i
\(999\) 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.c.e.199.4 4
3.2 odd 2 2475.2.c.m.199.1 4
5.2 odd 4 165.2.a.a.1.1 2
5.3 odd 4 825.2.a.g.1.2 2
5.4 even 2 inner 825.2.c.e.199.1 4
15.2 even 4 495.2.a.d.1.2 2
15.8 even 4 2475.2.a.m.1.1 2
15.14 odd 2 2475.2.c.m.199.4 4
20.7 even 4 2640.2.a.bb.1.1 2
35.27 even 4 8085.2.a.ba.1.1 2
55.32 even 4 1815.2.a.k.1.2 2
55.43 even 4 9075.2.a.v.1.1 2
60.47 odd 4 7920.2.a.cg.1.1 2
165.32 odd 4 5445.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 5.2 odd 4
495.2.a.d.1.2 2 15.2 even 4
825.2.a.g.1.2 2 5.3 odd 4
825.2.c.e.199.1 4 5.4 even 2 inner
825.2.c.e.199.4 4 1.1 even 1 trivial
1815.2.a.k.1.2 2 55.32 even 4
2475.2.a.m.1.1 2 15.8 even 4
2475.2.c.m.199.1 4 3.2 odd 2
2475.2.c.m.199.4 4 15.14 odd 2
2640.2.a.bb.1.1 2 20.7 even 4
5445.2.a.m.1.1 2 165.32 odd 4
7920.2.a.cg.1.1 2 60.47 odd 4
8085.2.a.ba.1.1 2 35.27 even 4
9075.2.a.v.1.1 2 55.43 even 4