Properties

Label 1089.3.b.a.485.2
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.a.485.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} +2.00000 q^{4} -5.65685i q^{5} -1.00000 q^{7} +8.48528i q^{8} +8.00000 q^{10} -16.0000 q^{13} -1.41421i q^{14} -4.00000 q^{16} -11.3137i q^{17} -33.0000 q^{19} -11.3137i q^{20} -24.0416i q^{23} -7.00000 q^{25} -22.6274i q^{26} -2.00000 q^{28} +12.7279i q^{29} +31.0000 q^{31} +28.2843i q^{32} +16.0000 q^{34} +5.65685i q^{35} -57.0000 q^{37} -46.6690i q^{38} +48.0000 q^{40} -15.5563i q^{41} +34.0000 q^{46} +60.8112i q^{47} -48.0000 q^{49} -9.89949i q^{50} -32.0000 q^{52} -89.0955i q^{53} -8.48528i q^{56} -18.0000 q^{58} +69.2965i q^{59} -105.000 q^{61} +43.8406i q^{62} -56.0000 q^{64} +90.5097i q^{65} -103.000 q^{67} -22.6274i q^{68} -8.00000 q^{70} -118.794i q^{71} +47.0000 q^{73} -80.6102i q^{74} -66.0000 q^{76} -23.0000 q^{79} +22.6274i q^{80} +22.0000 q^{82} -106.066i q^{83} -64.0000 q^{85} +50.9117i q^{89} +16.0000 q^{91} -48.0833i q^{92} -86.0000 q^{94} +186.676i q^{95} -25.0000 q^{97} -67.8823i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 2 q^{7} + 16 q^{10} - 32 q^{13} - 8 q^{16} - 66 q^{19} - 14 q^{25} - 4 q^{28} + 62 q^{31} + 32 q^{34} - 114 q^{37} + 96 q^{40} + 68 q^{46} - 96 q^{49} - 64 q^{52} - 36 q^{58} - 210 q^{61} - 112 q^{64} - 206 q^{67} - 16 q^{70} + 94 q^{73} - 132 q^{76} - 46 q^{79} + 44 q^{82} - 128 q^{85} + 32 q^{91} - 172 q^{94} - 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) − 5.65685i − 1.13137i −0.824621 0.565685i \(-0.808612\pi\)
0.824621 0.565685i \(-0.191388\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.142857 −0.0714286 0.997446i \(-0.522756\pi\)
−0.0714286 + 0.997446i \(0.522756\pi\)
\(8\) 8.48528i 1.06066i
\(9\) 0 0
\(10\) 8.00000 0.800000
\(11\) 0 0
\(12\) 0 0
\(13\) −16.0000 −1.23077 −0.615385 0.788227i \(-0.710999\pi\)
−0.615385 + 0.788227i \(0.710999\pi\)
\(14\) − 1.41421i − 0.101015i
\(15\) 0 0
\(16\) −4.00000 −0.250000
\(17\) − 11.3137i − 0.665512i −0.943013 0.332756i \(-0.892021\pi\)
0.943013 0.332756i \(-0.107979\pi\)
\(18\) 0 0
\(19\) −33.0000 −1.73684 −0.868421 0.495827i \(-0.834865\pi\)
−0.868421 + 0.495827i \(0.834865\pi\)
\(20\) − 11.3137i − 0.565685i
\(21\) 0 0
\(22\) 0 0
\(23\) − 24.0416i − 1.04529i −0.852551 0.522644i \(-0.824946\pi\)
0.852551 0.522644i \(-0.175054\pi\)
\(24\) 0 0
\(25\) −7.00000 −0.280000
\(26\) − 22.6274i − 0.870285i
\(27\) 0 0
\(28\) −2.00000 −0.0714286
\(29\) 12.7279i 0.438894i 0.975624 + 0.219447i \(0.0704253\pi\)
−0.975624 + 0.219447i \(0.929575\pi\)
\(30\) 0 0
\(31\) 31.0000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 28.2843i 0.883883i
\(33\) 0 0
\(34\) 16.0000 0.470588
\(35\) 5.65685i 0.161624i
\(36\) 0 0
\(37\) −57.0000 −1.54054 −0.770270 0.637718i \(-0.779879\pi\)
−0.770270 + 0.637718i \(0.779879\pi\)
\(38\) − 46.6690i − 1.22813i
\(39\) 0 0
\(40\) 48.0000 1.20000
\(41\) − 15.5563i − 0.379423i −0.981840 0.189712i \(-0.939245\pi\)
0.981840 0.189712i \(-0.0607553\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 34.0000 0.739130
\(47\) 60.8112i 1.29385i 0.762552 + 0.646927i \(0.223946\pi\)
−0.762552 + 0.646927i \(0.776054\pi\)
\(48\) 0 0
\(49\) −48.0000 −0.979592
\(50\) − 9.89949i − 0.197990i
\(51\) 0 0
\(52\) −32.0000 −0.615385
\(53\) − 89.0955i − 1.68105i −0.541776 0.840523i \(-0.682248\pi\)
0.541776 0.840523i \(-0.317752\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 8.48528i − 0.151523i
\(57\) 0 0
\(58\) −18.0000 −0.310345
\(59\) 69.2965i 1.17452i 0.809400 + 0.587258i \(0.199793\pi\)
−0.809400 + 0.587258i \(0.800207\pi\)
\(60\) 0 0
\(61\) −105.000 −1.72131 −0.860656 0.509187i \(-0.829946\pi\)
−0.860656 + 0.509187i \(0.829946\pi\)
\(62\) 43.8406i 0.707107i
\(63\) 0 0
\(64\) −56.0000 −0.875000
\(65\) 90.5097i 1.39246i
\(66\) 0 0
\(67\) −103.000 −1.53731 −0.768657 0.639662i \(-0.779075\pi\)
−0.768657 + 0.639662i \(0.779075\pi\)
\(68\) − 22.6274i − 0.332756i
\(69\) 0 0
\(70\) −8.00000 −0.114286
\(71\) − 118.794i − 1.67315i −0.547849 0.836577i \(-0.684553\pi\)
0.547849 0.836577i \(-0.315447\pi\)
\(72\) 0 0
\(73\) 47.0000 0.643836 0.321918 0.946768i \(-0.395673\pi\)
0.321918 + 0.946768i \(0.395673\pi\)
\(74\) − 80.6102i − 1.08933i
\(75\) 0 0
\(76\) −66.0000 −0.868421
\(77\) 0 0
\(78\) 0 0
\(79\) −23.0000 −0.291139 −0.145570 0.989348i \(-0.546501\pi\)
−0.145570 + 0.989348i \(0.546501\pi\)
\(80\) 22.6274i 0.282843i
\(81\) 0 0
\(82\) 22.0000 0.268293
\(83\) − 106.066i − 1.27790i −0.769247 0.638952i \(-0.779368\pi\)
0.769247 0.638952i \(-0.220632\pi\)
\(84\) 0 0
\(85\) −64.0000 −0.752941
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 50.9117i 0.572041i 0.958223 + 0.286021i \(0.0923326\pi\)
−0.958223 + 0.286021i \(0.907667\pi\)
\(90\) 0 0
\(91\) 16.0000 0.175824
\(92\) − 48.0833i − 0.522644i
\(93\) 0 0
\(94\) −86.0000 −0.914894
\(95\) 186.676i 1.96501i
\(96\) 0 0
\(97\) −25.0000 −0.257732 −0.128866 0.991662i \(-0.541134\pi\)
−0.128866 + 0.991662i \(0.541134\pi\)
\(98\) − 67.8823i − 0.692676i
\(99\) 0 0
\(100\) −14.0000 −0.140000
\(101\) 151.321i 1.49823i 0.662442 + 0.749113i \(0.269520\pi\)
−0.662442 + 0.749113i \(0.730480\pi\)
\(102\) 0 0
\(103\) −25.0000 −0.242718 −0.121359 0.992609i \(-0.538725\pi\)
−0.121359 + 0.992609i \(0.538725\pi\)
\(104\) − 135.765i − 1.30543i
\(105\) 0 0
\(106\) 126.000 1.18868
\(107\) − 131.522i − 1.22918i −0.788848 0.614588i \(-0.789322\pi\)
0.788848 0.614588i \(-0.210678\pi\)
\(108\) 0 0
\(109\) 119.000 1.09174 0.545872 0.837869i \(-0.316199\pi\)
0.545872 + 0.837869i \(0.316199\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.0357143
\(113\) 12.7279i 0.112636i 0.998413 + 0.0563182i \(0.0179361\pi\)
−0.998413 + 0.0563182i \(0.982064\pi\)
\(114\) 0 0
\(115\) −136.000 −1.18261
\(116\) 25.4558i 0.219447i
\(117\) 0 0
\(118\) −98.0000 −0.830508
\(119\) 11.3137i 0.0950732i
\(120\) 0 0
\(121\) 0 0
\(122\) − 148.492i − 1.21715i
\(123\) 0 0
\(124\) 62.0000 0.500000
\(125\) − 101.823i − 0.814587i
\(126\) 0 0
\(127\) 79.0000 0.622047 0.311024 0.950402i \(-0.399328\pi\)
0.311024 + 0.950402i \(0.399328\pi\)
\(128\) 33.9411i 0.265165i
\(129\) 0 0
\(130\) −128.000 −0.984615
\(131\) 114.551i 0.874437i 0.899355 + 0.437219i \(0.144036\pi\)
−0.899355 + 0.437219i \(0.855964\pi\)
\(132\) 0 0
\(133\) 33.0000 0.248120
\(134\) − 145.664i − 1.08704i
\(135\) 0 0
\(136\) 96.0000 0.705882
\(137\) 90.5097i 0.660655i 0.943866 + 0.330327i \(0.107159\pi\)
−0.943866 + 0.330327i \(0.892841\pi\)
\(138\) 0 0
\(139\) −58.0000 −0.417266 −0.208633 0.977994i \(-0.566901\pi\)
−0.208633 + 0.977994i \(0.566901\pi\)
\(140\) 11.3137i 0.0808122i
\(141\) 0 0
\(142\) 168.000 1.18310
\(143\) 0 0
\(144\) 0 0
\(145\) 72.0000 0.496552
\(146\) 66.4680i 0.455261i
\(147\) 0 0
\(148\) −114.000 −0.770270
\(149\) − 260.215i − 1.74641i −0.487352 0.873206i \(-0.662037\pi\)
0.487352 0.873206i \(-0.337963\pi\)
\(150\) 0 0
\(151\) 64.0000 0.423841 0.211921 0.977287i \(-0.432028\pi\)
0.211921 + 0.977287i \(0.432028\pi\)
\(152\) − 280.014i − 1.84220i
\(153\) 0 0
\(154\) 0 0
\(155\) − 175.362i − 1.13137i
\(156\) 0 0
\(157\) 257.000 1.63694 0.818471 0.574547i \(-0.194822\pi\)
0.818471 + 0.574547i \(0.194822\pi\)
\(158\) − 32.5269i − 0.205867i
\(159\) 0 0
\(160\) 160.000 1.00000
\(161\) 24.0416i 0.149327i
\(162\) 0 0
\(163\) 79.0000 0.484663 0.242331 0.970194i \(-0.422088\pi\)
0.242331 + 0.970194i \(0.422088\pi\)
\(164\) − 31.1127i − 0.189712i
\(165\) 0 0
\(166\) 150.000 0.903614
\(167\) 29.6985i 0.177835i 0.996039 + 0.0889176i \(0.0283408\pi\)
−0.996039 + 0.0889176i \(0.971659\pi\)
\(168\) 0 0
\(169\) 87.0000 0.514793
\(170\) − 90.5097i − 0.532410i
\(171\) 0 0
\(172\) 0 0
\(173\) − 67.8823i − 0.392383i −0.980566 0.196191i \(-0.937143\pi\)
0.980566 0.196191i \(-0.0628574\pi\)
\(174\) 0 0
\(175\) 7.00000 0.0400000
\(176\) 0 0
\(177\) 0 0
\(178\) −72.0000 −0.404494
\(179\) − 199.404i − 1.11399i −0.830516 0.556995i \(-0.811954\pi\)
0.830516 0.556995i \(-0.188046\pi\)
\(180\) 0 0
\(181\) −287.000 −1.58564 −0.792818 0.609459i \(-0.791387\pi\)
−0.792818 + 0.609459i \(0.791387\pi\)
\(182\) 22.6274i 0.124326i
\(183\) 0 0
\(184\) 204.000 1.10870
\(185\) 322.441i 1.74292i
\(186\) 0 0
\(187\) 0 0
\(188\) 121.622i 0.646927i
\(189\) 0 0
\(190\) −264.000 −1.38947
\(191\) − 164.049i − 0.858894i −0.903092 0.429447i \(-0.858708\pi\)
0.903092 0.429447i \(-0.141292\pi\)
\(192\) 0 0
\(193\) −145.000 −0.751295 −0.375648 0.926763i \(-0.622580\pi\)
−0.375648 + 0.926763i \(0.622580\pi\)
\(194\) − 35.3553i − 0.182244i
\(195\) 0 0
\(196\) −96.0000 −0.489796
\(197\) − 45.2548i − 0.229720i −0.993382 0.114860i \(-0.963358\pi\)
0.993382 0.114860i \(-0.0366419\pi\)
\(198\) 0 0
\(199\) −57.0000 −0.286432 −0.143216 0.989691i \(-0.545744\pi\)
−0.143216 + 0.989691i \(0.545744\pi\)
\(200\) − 59.3970i − 0.296985i
\(201\) 0 0
\(202\) −214.000 −1.05941
\(203\) − 12.7279i − 0.0626991i
\(204\) 0 0
\(205\) −88.0000 −0.429268
\(206\) − 35.3553i − 0.171628i
\(207\) 0 0
\(208\) 64.0000 0.307692
\(209\) 0 0
\(210\) 0 0
\(211\) 105.000 0.497630 0.248815 0.968551i \(-0.419959\pi\)
0.248815 + 0.968551i \(0.419959\pi\)
\(212\) − 178.191i − 0.840523i
\(213\) 0 0
\(214\) 186.000 0.869159
\(215\) 0 0
\(216\) 0 0
\(217\) −31.0000 −0.142857
\(218\) 168.291i 0.771979i
\(219\) 0 0
\(220\) 0 0
\(221\) 181.019i 0.819092i
\(222\) 0 0
\(223\) 199.000 0.892377 0.446188 0.894939i \(-0.352781\pi\)
0.446188 + 0.894939i \(0.352781\pi\)
\(224\) − 28.2843i − 0.126269i
\(225\) 0 0
\(226\) −18.0000 −0.0796460
\(227\) − 186.676i − 0.822362i −0.911554 0.411181i \(-0.865116\pi\)
0.911554 0.411181i \(-0.134884\pi\)
\(228\) 0 0
\(229\) −72.0000 −0.314410 −0.157205 0.987566i \(-0.550248\pi\)
−0.157205 + 0.987566i \(0.550248\pi\)
\(230\) − 192.333i − 0.836231i
\(231\) 0 0
\(232\) −108.000 −0.465517
\(233\) − 131.522i − 0.564472i −0.959345 0.282236i \(-0.908924\pi\)
0.959345 0.282236i \(-0.0910760\pi\)
\(234\) 0 0
\(235\) 344.000 1.46383
\(236\) 138.593i 0.587258i
\(237\) 0 0
\(238\) −16.0000 −0.0672269
\(239\) − 4.24264i − 0.0177516i −0.999961 0.00887582i \(-0.997175\pi\)
0.999961 0.00887582i \(-0.00282530\pi\)
\(240\) 0 0
\(241\) −160.000 −0.663900 −0.331950 0.943297i \(-0.607707\pi\)
−0.331950 + 0.943297i \(0.607707\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −210.000 −0.860656
\(245\) 271.529i 1.10828i
\(246\) 0 0
\(247\) 528.000 2.13765
\(248\) 263.044i 1.06066i
\(249\) 0 0
\(250\) 144.000 0.576000
\(251\) 108.894i 0.433842i 0.976189 + 0.216921i \(0.0696015\pi\)
−0.976189 + 0.216921i \(0.930399\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 111.723i 0.439854i
\(255\) 0 0
\(256\) −272.000 −1.06250
\(257\) 261.630i 1.01801i 0.860762 + 0.509007i \(0.169987\pi\)
−0.860762 + 0.509007i \(0.830013\pi\)
\(258\) 0 0
\(259\) 57.0000 0.220077
\(260\) 181.019i 0.696228i
\(261\) 0 0
\(262\) −162.000 −0.618321
\(263\) 439.820i 1.67232i 0.548485 + 0.836160i \(0.315205\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(264\) 0 0
\(265\) −504.000 −1.90189
\(266\) 46.6690i 0.175448i
\(267\) 0 0
\(268\) −206.000 −0.768657
\(269\) 244.659i 0.909513i 0.890616 + 0.454756i \(0.150274\pi\)
−0.890616 + 0.454756i \(0.849726\pi\)
\(270\) 0 0
\(271\) −344.000 −1.26937 −0.634686 0.772770i \(-0.718871\pi\)
−0.634686 + 0.772770i \(0.718871\pi\)
\(272\) 45.2548i 0.166378i
\(273\) 0 0
\(274\) −128.000 −0.467153
\(275\) 0 0
\(276\) 0 0
\(277\) 95.0000 0.342960 0.171480 0.985188i \(-0.445145\pi\)
0.171480 + 0.985188i \(0.445145\pi\)
\(278\) − 82.0244i − 0.295052i
\(279\) 0 0
\(280\) −48.0000 −0.171429
\(281\) 390.323i 1.38905i 0.719469 + 0.694525i \(0.244385\pi\)
−0.719469 + 0.694525i \(0.755615\pi\)
\(282\) 0 0
\(283\) 335.000 1.18375 0.591873 0.806031i \(-0.298389\pi\)
0.591873 + 0.806031i \(0.298389\pi\)
\(284\) − 237.588i − 0.836577i
\(285\) 0 0
\(286\) 0 0
\(287\) 15.5563i 0.0542033i
\(288\) 0 0
\(289\) 161.000 0.557093
\(290\) 101.823i 0.351115i
\(291\) 0 0
\(292\) 94.0000 0.321918
\(293\) − 560.029i − 1.91136i −0.294407 0.955680i \(-0.595122\pi\)
0.294407 0.955680i \(-0.404878\pi\)
\(294\) 0 0
\(295\) 392.000 1.32881
\(296\) − 483.661i − 1.63399i
\(297\) 0 0
\(298\) 368.000 1.23490
\(299\) 384.666i 1.28651i
\(300\) 0 0
\(301\) 0 0
\(302\) 90.5097i 0.299701i
\(303\) 0 0
\(304\) 132.000 0.434211
\(305\) 593.970i 1.94744i
\(306\) 0 0
\(307\) −311.000 −1.01303 −0.506515 0.862231i \(-0.669066\pi\)
−0.506515 + 0.862231i \(0.669066\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 248.000 0.800000
\(311\) 16.9706i 0.0545677i 0.999628 + 0.0272839i \(0.00868580\pi\)
−0.999628 + 0.0272839i \(0.991314\pi\)
\(312\) 0 0
\(313\) 200.000 0.638978 0.319489 0.947590i \(-0.396489\pi\)
0.319489 + 0.947590i \(0.396489\pi\)
\(314\) 363.453i 1.15749i
\(315\) 0 0
\(316\) −46.0000 −0.145570
\(317\) 282.843i 0.892248i 0.894971 + 0.446124i \(0.147196\pi\)
−0.894971 + 0.446124i \(0.852804\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 316.784i 0.989949i
\(321\) 0 0
\(322\) −34.0000 −0.105590
\(323\) 373.352i 1.15589i
\(324\) 0 0
\(325\) 112.000 0.344615
\(326\) 111.723i 0.342708i
\(327\) 0 0
\(328\) 132.000 0.402439
\(329\) − 60.8112i − 0.184836i
\(330\) 0 0
\(331\) 577.000 1.74320 0.871601 0.490216i \(-0.163082\pi\)
0.871601 + 0.490216i \(0.163082\pi\)
\(332\) − 212.132i − 0.638952i
\(333\) 0 0
\(334\) −42.0000 −0.125749
\(335\) 582.656i 1.73927i
\(336\) 0 0
\(337\) 289.000 0.857567 0.428783 0.903407i \(-0.358942\pi\)
0.428783 + 0.903407i \(0.358942\pi\)
\(338\) 123.037i 0.364014i
\(339\) 0 0
\(340\) −128.000 −0.376471
\(341\) 0 0
\(342\) 0 0
\(343\) 97.0000 0.282799
\(344\) 0 0
\(345\) 0 0
\(346\) 96.0000 0.277457
\(347\) − 541.644i − 1.56093i −0.625198 0.780467i \(-0.714982\pi\)
0.625198 0.780467i \(-0.285018\pi\)
\(348\) 0 0
\(349\) −513.000 −1.46991 −0.734957 0.678114i \(-0.762798\pi\)
−0.734957 + 0.678114i \(0.762798\pi\)
\(350\) 9.89949i 0.0282843i
\(351\) 0 0
\(352\) 0 0
\(353\) − 254.558i − 0.721129i −0.932734 0.360564i \(-0.882584\pi\)
0.932734 0.360564i \(-0.117416\pi\)
\(354\) 0 0
\(355\) −672.000 −1.89296
\(356\) 101.823i 0.286021i
\(357\) 0 0
\(358\) 282.000 0.787709
\(359\) − 452.548i − 1.26058i −0.776360 0.630290i \(-0.782936\pi\)
0.776360 0.630290i \(-0.217064\pi\)
\(360\) 0 0
\(361\) 728.000 2.01662
\(362\) − 405.879i − 1.12121i
\(363\) 0 0
\(364\) 32.0000 0.0879121
\(365\) − 265.872i − 0.728417i
\(366\) 0 0
\(367\) 160.000 0.435967 0.217984 0.975952i \(-0.430052\pi\)
0.217984 + 0.975952i \(0.430052\pi\)
\(368\) 96.1665i 0.261322i
\(369\) 0 0
\(370\) −456.000 −1.23243
\(371\) 89.0955i 0.240149i
\(372\) 0 0
\(373\) −127.000 −0.340483 −0.170241 0.985402i \(-0.554455\pi\)
−0.170241 + 0.985402i \(0.554455\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −516.000 −1.37234
\(377\) − 203.647i − 0.540177i
\(378\) 0 0
\(379\) 454.000 1.19789 0.598945 0.800790i \(-0.295587\pi\)
0.598945 + 0.800790i \(0.295587\pi\)
\(380\) 373.352i 0.982506i
\(381\) 0 0
\(382\) 232.000 0.607330
\(383\) − 74.9533i − 0.195701i −0.995201 0.0978503i \(-0.968803\pi\)
0.995201 0.0978503i \(-0.0311966\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 205.061i − 0.531246i
\(387\) 0 0
\(388\) −50.0000 −0.128866
\(389\) 4.24264i 0.0109065i 0.999985 + 0.00545327i \(0.00173584\pi\)
−0.999985 + 0.00545327i \(0.998264\pi\)
\(390\) 0 0
\(391\) −272.000 −0.695652
\(392\) − 407.294i − 1.03901i
\(393\) 0 0
\(394\) 64.0000 0.162437
\(395\) 130.108i 0.329386i
\(396\) 0 0
\(397\) 481.000 1.21159 0.605793 0.795622i \(-0.292856\pi\)
0.605793 + 0.795622i \(0.292856\pi\)
\(398\) − 80.6102i − 0.202538i
\(399\) 0 0
\(400\) 28.0000 0.0700000
\(401\) 429.921i 1.07212i 0.844179 + 0.536061i \(0.180088\pi\)
−0.844179 + 0.536061i \(0.819912\pi\)
\(402\) 0 0
\(403\) −496.000 −1.23077
\(404\) 302.642i 0.749113i
\(405\) 0 0
\(406\) 18.0000 0.0443350
\(407\) 0 0
\(408\) 0 0
\(409\) −177.000 −0.432763 −0.216381 0.976309i \(-0.569425\pi\)
−0.216381 + 0.976309i \(0.569425\pi\)
\(410\) − 124.451i − 0.303539i
\(411\) 0 0
\(412\) −50.0000 −0.121359
\(413\) − 69.2965i − 0.167788i
\(414\) 0 0
\(415\) −600.000 −1.44578
\(416\) − 452.548i − 1.08786i
\(417\) 0 0
\(418\) 0 0
\(419\) 306.884i 0.732421i 0.930532 + 0.366210i \(0.119345\pi\)
−0.930532 + 0.366210i \(0.880655\pi\)
\(420\) 0 0
\(421\) 160.000 0.380048 0.190024 0.981779i \(-0.439143\pi\)
0.190024 + 0.981779i \(0.439143\pi\)
\(422\) 148.492i 0.351878i
\(423\) 0 0
\(424\) 756.000 1.78302
\(425\) 79.1960i 0.186343i
\(426\) 0 0
\(427\) 105.000 0.245902
\(428\) − 263.044i − 0.614588i
\(429\) 0 0
\(430\) 0 0
\(431\) 691.550i 1.60453i 0.596971 + 0.802263i \(0.296371\pi\)
−0.596971 + 0.802263i \(0.703629\pi\)
\(432\) 0 0
\(433\) −143.000 −0.330254 −0.165127 0.986272i \(-0.552803\pi\)
−0.165127 + 0.986272i \(0.552803\pi\)
\(434\) − 43.8406i − 0.101015i
\(435\) 0 0
\(436\) 238.000 0.545872
\(437\) 793.374i 1.81550i
\(438\) 0 0
\(439\) 335.000 0.763098 0.381549 0.924349i \(-0.375391\pi\)
0.381549 + 0.924349i \(0.375391\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −256.000 −0.579186
\(443\) 130.108i 0.293697i 0.989159 + 0.146848i \(0.0469129\pi\)
−0.989159 + 0.146848i \(0.953087\pi\)
\(444\) 0 0
\(445\) 288.000 0.647191
\(446\) 281.428i 0.631006i
\(447\) 0 0
\(448\) 56.0000 0.125000
\(449\) 854.185i 1.90242i 0.308550 + 0.951208i \(0.400156\pi\)
−0.308550 + 0.951208i \(0.599844\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 25.4558i 0.0563182i
\(453\) 0 0
\(454\) 264.000 0.581498
\(455\) − 90.5097i − 0.198922i
\(456\) 0 0
\(457\) −264.000 −0.577681 −0.288840 0.957377i \(-0.593270\pi\)
−0.288840 + 0.957377i \(0.593270\pi\)
\(458\) − 101.823i − 0.222322i
\(459\) 0 0
\(460\) −272.000 −0.591304
\(461\) 123.037i 0.266891i 0.991056 + 0.133445i \(0.0426041\pi\)
−0.991056 + 0.133445i \(0.957396\pi\)
\(462\) 0 0
\(463\) −608.000 −1.31317 −0.656587 0.754250i \(-0.728001\pi\)
−0.656587 + 0.754250i \(0.728001\pi\)
\(464\) − 50.9117i − 0.109723i
\(465\) 0 0
\(466\) 186.000 0.399142
\(467\) − 424.264i − 0.908488i −0.890877 0.454244i \(-0.849909\pi\)
0.890877 0.454244i \(-0.150091\pi\)
\(468\) 0 0
\(469\) 103.000 0.219616
\(470\) 486.489i 1.03508i
\(471\) 0 0
\(472\) −588.000 −1.24576
\(473\) 0 0
\(474\) 0 0
\(475\) 231.000 0.486316
\(476\) 22.6274i 0.0475366i
\(477\) 0 0
\(478\) 6.00000 0.0125523
\(479\) 830.143i 1.73308i 0.499111 + 0.866538i \(0.333660\pi\)
−0.499111 + 0.866538i \(0.666340\pi\)
\(480\) 0 0
\(481\) 912.000 1.89605
\(482\) − 226.274i − 0.469448i
\(483\) 0 0
\(484\) 0 0
\(485\) 141.421i 0.291590i
\(486\) 0 0
\(487\) 56.0000 0.114990 0.0574949 0.998346i \(-0.481689\pi\)
0.0574949 + 0.998346i \(0.481689\pi\)
\(488\) − 890.955i − 1.82573i
\(489\) 0 0
\(490\) −384.000 −0.783673
\(491\) − 278.600i − 0.567414i −0.958911 0.283707i \(-0.908436\pi\)
0.958911 0.283707i \(-0.0915642\pi\)
\(492\) 0 0
\(493\) 144.000 0.292089
\(494\) 746.705i 1.51155i
\(495\) 0 0
\(496\) −124.000 −0.250000
\(497\) 118.794i 0.239022i
\(498\) 0 0
\(499\) −439.000 −0.879760 −0.439880 0.898057i \(-0.644979\pi\)
−0.439880 + 0.898057i \(0.644979\pi\)
\(500\) − 203.647i − 0.407294i
\(501\) 0 0
\(502\) −154.000 −0.306773
\(503\) − 605.283i − 1.20335i −0.798742 0.601673i \(-0.794501\pi\)
0.798742 0.601673i \(-0.205499\pi\)
\(504\) 0 0
\(505\) 856.000 1.69505
\(506\) 0 0
\(507\) 0 0
\(508\) 158.000 0.311024
\(509\) 352.139i 0.691825i 0.938267 + 0.345913i \(0.112431\pi\)
−0.938267 + 0.345913i \(0.887569\pi\)
\(510\) 0 0
\(511\) −47.0000 −0.0919765
\(512\) − 248.902i − 0.486136i
\(513\) 0 0
\(514\) −370.000 −0.719844
\(515\) 141.421i 0.274605i
\(516\) 0 0
\(517\) 0 0
\(518\) 80.6102i 0.155618i
\(519\) 0 0
\(520\) −768.000 −1.47692
\(521\) 581.242i 1.11563i 0.829966 + 0.557814i \(0.188360\pi\)
−0.829966 + 0.557814i \(0.811640\pi\)
\(522\) 0 0
\(523\) 385.000 0.736138 0.368069 0.929799i \(-0.380019\pi\)
0.368069 + 0.929799i \(0.380019\pi\)
\(524\) 229.103i 0.437219i
\(525\) 0 0
\(526\) −622.000 −1.18251
\(527\) − 350.725i − 0.665512i
\(528\) 0 0
\(529\) −49.0000 −0.0926276
\(530\) − 712.764i − 1.34484i
\(531\) 0 0
\(532\) 66.0000 0.124060
\(533\) 248.902i 0.466982i
\(534\) 0 0
\(535\) −744.000 −1.39065
\(536\) − 873.984i − 1.63057i
\(537\) 0 0
\(538\) −346.000 −0.643123
\(539\) 0 0
\(540\) 0 0
\(541\) −88.0000 −0.162662 −0.0813309 0.996687i \(-0.525917\pi\)
−0.0813309 + 0.996687i \(0.525917\pi\)
\(542\) − 486.489i − 0.897582i
\(543\) 0 0
\(544\) 320.000 0.588235
\(545\) − 673.166i − 1.23517i
\(546\) 0 0
\(547\) 424.000 0.775137 0.387569 0.921841i \(-0.373315\pi\)
0.387569 + 0.921841i \(0.373315\pi\)
\(548\) 181.019i 0.330327i
\(549\) 0 0
\(550\) 0 0
\(551\) − 420.021i − 0.762289i
\(552\) 0 0
\(553\) 23.0000 0.0415913
\(554\) 134.350i 0.242510i
\(555\) 0 0
\(556\) −116.000 −0.208633
\(557\) 209.304i 0.375769i 0.982191 + 0.187885i \(0.0601631\pi\)
−0.982191 + 0.187885i \(0.939837\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) − 22.6274i − 0.0404061i
\(561\) 0 0
\(562\) −552.000 −0.982206
\(563\) − 469.519i − 0.833959i −0.908916 0.416979i \(-0.863089\pi\)
0.908916 0.416979i \(-0.136911\pi\)
\(564\) 0 0
\(565\) 72.0000 0.127434
\(566\) 473.762i 0.837035i
\(567\) 0 0
\(568\) 1008.00 1.77465
\(569\) − 831.558i − 1.46144i −0.682679 0.730718i \(-0.739185\pi\)
0.682679 0.730718i \(-0.260815\pi\)
\(570\) 0 0
\(571\) −503.000 −0.880911 −0.440455 0.897775i \(-0.645183\pi\)
−0.440455 + 0.897775i \(0.645183\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −22.0000 −0.0383275
\(575\) 168.291i 0.292681i
\(576\) 0 0
\(577\) −615.000 −1.06586 −0.532929 0.846160i \(-0.678909\pi\)
−0.532929 + 0.846160i \(0.678909\pi\)
\(578\) 227.688i 0.393925i
\(579\) 0 0
\(580\) 144.000 0.248276
\(581\) 106.066i 0.182558i
\(582\) 0 0
\(583\) 0 0
\(584\) 398.808i 0.682891i
\(585\) 0 0
\(586\) 792.000 1.35154
\(587\) − 5.65685i − 0.00963689i −0.999988 0.00481844i \(-0.998466\pi\)
0.999988 0.00481844i \(-0.00153376\pi\)
\(588\) 0 0
\(589\) −1023.00 −1.73684
\(590\) 554.372i 0.939613i
\(591\) 0 0
\(592\) 228.000 0.385135
\(593\) 287.085i 0.484124i 0.970261 + 0.242062i \(0.0778237\pi\)
−0.970261 + 0.242062i \(0.922176\pi\)
\(594\) 0 0
\(595\) 64.0000 0.107563
\(596\) − 520.431i − 0.873206i
\(597\) 0 0
\(598\) −544.000 −0.909699
\(599\) − 956.008i − 1.59601i −0.602653 0.798004i \(-0.705890\pi\)
0.602653 0.798004i \(-0.294110\pi\)
\(600\) 0 0
\(601\) 601.000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 128.000 0.211921
\(605\) 0 0
\(606\) 0 0
\(607\) 1024.00 1.68699 0.843493 0.537141i \(-0.180496\pi\)
0.843493 + 0.537141i \(0.180496\pi\)
\(608\) − 933.381i − 1.53517i
\(609\) 0 0
\(610\) −840.000 −1.37705
\(611\) − 972.979i − 1.59244i
\(612\) 0 0
\(613\) 721.000 1.17618 0.588091 0.808795i \(-0.299880\pi\)
0.588091 + 0.808795i \(0.299880\pi\)
\(614\) − 439.820i − 0.716320i
\(615\) 0 0
\(616\) 0 0
\(617\) 50.9117i 0.0825149i 0.999149 + 0.0412574i \(0.0131364\pi\)
−0.999149 + 0.0412574i \(0.986864\pi\)
\(618\) 0 0
\(619\) 152.000 0.245557 0.122779 0.992434i \(-0.460819\pi\)
0.122779 + 0.992434i \(0.460819\pi\)
\(620\) − 350.725i − 0.565685i
\(621\) 0 0
\(622\) −24.0000 −0.0385852
\(623\) − 50.9117i − 0.0817202i
\(624\) 0 0
\(625\) −751.000 −1.20160
\(626\) 282.843i 0.451825i
\(627\) 0 0
\(628\) 514.000 0.818471
\(629\) 644.881i 1.02525i
\(630\) 0 0
\(631\) 544.000 0.862124 0.431062 0.902322i \(-0.358139\pi\)
0.431062 + 0.902322i \(0.358139\pi\)
\(632\) − 195.161i − 0.308800i
\(633\) 0 0
\(634\) −400.000 −0.630915
\(635\) − 446.891i − 0.703766i
\(636\) 0 0
\(637\) 768.000 1.20565
\(638\) 0 0
\(639\) 0 0
\(640\) 192.000 0.300000
\(641\) 814.587i 1.27081i 0.772180 + 0.635403i \(0.219166\pi\)
−0.772180 + 0.635403i \(0.780834\pi\)
\(642\) 0 0
\(643\) −1231.00 −1.91446 −0.957232 0.289322i \(-0.906570\pi\)
−0.957232 + 0.289322i \(0.906570\pi\)
\(644\) 48.0833i 0.0746634i
\(645\) 0 0
\(646\) −528.000 −0.817337
\(647\) 158.392i 0.244810i 0.992480 + 0.122405i \(0.0390606\pi\)
−0.992480 + 0.122405i \(0.960939\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 158.392i 0.243680i
\(651\) 0 0
\(652\) 158.000 0.242331
\(653\) − 264.458i − 0.404989i −0.979283 0.202495i \(-0.935095\pi\)
0.979283 0.202495i \(-0.0649049\pi\)
\(654\) 0 0
\(655\) 648.000 0.989313
\(656\) 62.2254i 0.0948558i
\(657\) 0 0
\(658\) 86.0000 0.130699
\(659\) − 490.732i − 0.744662i −0.928100 0.372331i \(-0.878559\pi\)
0.928100 0.372331i \(-0.121441\pi\)
\(660\) 0 0
\(661\) −521.000 −0.788200 −0.394100 0.919068i \(-0.628944\pi\)
−0.394100 + 0.919068i \(0.628944\pi\)
\(662\) 816.001i 1.23263i
\(663\) 0 0
\(664\) 900.000 1.35542
\(665\) − 186.676i − 0.280716i
\(666\) 0 0
\(667\) 306.000 0.458771
\(668\) 59.3970i 0.0889176i
\(669\) 0 0
\(670\) −824.000 −1.22985
\(671\) 0 0
\(672\) 0 0
\(673\) −239.000 −0.355126 −0.177563 0.984109i \(-0.556821\pi\)
−0.177563 + 0.984109i \(0.556821\pi\)
\(674\) 408.708i 0.606391i
\(675\) 0 0
\(676\) 174.000 0.257396
\(677\) − 305.470i − 0.451211i −0.974219 0.225606i \(-0.927564\pi\)
0.974219 0.225606i \(-0.0724361\pi\)
\(678\) 0 0
\(679\) 25.0000 0.0368189
\(680\) − 543.058i − 0.798615i
\(681\) 0 0
\(682\) 0 0
\(683\) − 1101.67i − 1.61299i −0.591241 0.806495i \(-0.701362\pi\)
0.591241 0.806495i \(-0.298638\pi\)
\(684\) 0 0
\(685\) 512.000 0.747445
\(686\) 137.179i 0.199969i
\(687\) 0 0
\(688\) 0 0
\(689\) 1425.53i 2.06898i
\(690\) 0 0
\(691\) −919.000 −1.32996 −0.664978 0.746863i \(-0.731559\pi\)
−0.664978 + 0.746863i \(0.731559\pi\)
\(692\) − 135.765i − 0.196191i
\(693\) 0 0
\(694\) 766.000 1.10375
\(695\) 328.098i 0.472083i
\(696\) 0 0
\(697\) −176.000 −0.252511
\(698\) − 725.492i − 1.03939i
\(699\) 0 0
\(700\) 14.0000 0.0200000
\(701\) − 646.296i − 0.921962i −0.887410 0.460981i \(-0.847498\pi\)
0.887410 0.460981i \(-0.152502\pi\)
\(702\) 0 0
\(703\) 1881.00 2.67568
\(704\) 0 0
\(705\) 0 0
\(706\) 360.000 0.509915
\(707\) − 151.321i − 0.214032i
\(708\) 0 0
\(709\) 518.000 0.730606 0.365303 0.930889i \(-0.380965\pi\)
0.365303 + 0.930889i \(0.380965\pi\)
\(710\) − 950.352i − 1.33852i
\(711\) 0 0
\(712\) −432.000 −0.606742
\(713\) − 745.291i − 1.04529i
\(714\) 0 0
\(715\) 0 0
\(716\) − 398.808i − 0.556995i
\(717\) 0 0
\(718\) 640.000 0.891365
\(719\) − 182.434i − 0.253732i −0.991920 0.126866i \(-0.959508\pi\)
0.991920 0.126866i \(-0.0404919\pi\)
\(720\) 0 0
\(721\) 25.0000 0.0346741
\(722\) 1029.55i 1.42597i
\(723\) 0 0
\(724\) −574.000 −0.792818
\(725\) − 89.0955i − 0.122890i
\(726\) 0 0
\(727\) −114.000 −0.156809 −0.0784044 0.996922i \(-0.524983\pi\)
−0.0784044 + 0.996922i \(0.524983\pi\)
\(728\) 135.765i 0.186490i
\(729\) 0 0
\(730\) 376.000 0.515068
\(731\) 0 0
\(732\) 0 0
\(733\) 234.000 0.319236 0.159618 0.987179i \(-0.448974\pi\)
0.159618 + 0.987179i \(0.448974\pi\)
\(734\) 226.274i 0.308275i
\(735\) 0 0
\(736\) 680.000 0.923913
\(737\) 0 0
\(738\) 0 0
\(739\) −1241.00 −1.67930 −0.839648 0.543131i \(-0.817239\pi\)
−0.839648 + 0.543131i \(0.817239\pi\)
\(740\) 644.881i 0.871461i
\(741\) 0 0
\(742\) −126.000 −0.169811
\(743\) 141.421i 0.190338i 0.995461 + 0.0951691i \(0.0303392\pi\)
−0.995461 + 0.0951691i \(0.969661\pi\)
\(744\) 0 0
\(745\) −1472.00 −1.97584
\(746\) − 179.605i − 0.240758i
\(747\) 0 0
\(748\) 0 0
\(749\) 131.522i 0.175597i
\(750\) 0 0
\(751\) −113.000 −0.150466 −0.0752330 0.997166i \(-0.523970\pi\)
−0.0752330 + 0.997166i \(0.523970\pi\)
\(752\) − 243.245i − 0.323464i
\(753\) 0 0
\(754\) 288.000 0.381963
\(755\) − 362.039i − 0.479521i
\(756\) 0 0
\(757\) −55.0000 −0.0726552 −0.0363276 0.999340i \(-0.511566\pi\)
−0.0363276 + 0.999340i \(0.511566\pi\)
\(758\) 642.053i 0.847036i
\(759\) 0 0
\(760\) −1584.00 −2.08421
\(761\) − 882.469i − 1.15962i −0.814752 0.579809i \(-0.803127\pi\)
0.814752 0.579809i \(-0.196873\pi\)
\(762\) 0 0
\(763\) −119.000 −0.155963
\(764\) − 328.098i − 0.429447i
\(765\) 0 0
\(766\) 106.000 0.138381
\(767\) − 1108.74i − 1.44556i
\(768\) 0 0
\(769\) −41.0000 −0.0533160 −0.0266580 0.999645i \(-0.508487\pi\)
−0.0266580 + 0.999645i \(0.508487\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −290.000 −0.375648
\(773\) 339.411i 0.439083i 0.975603 + 0.219542i \(0.0704562\pi\)
−0.975603 + 0.219542i \(0.929544\pi\)
\(774\) 0 0
\(775\) −217.000 −0.280000
\(776\) − 212.132i − 0.273366i
\(777\) 0 0
\(778\) −6.00000 −0.00771208
\(779\) 513.360i 0.658998i
\(780\) 0 0
\(781\) 0 0
\(782\) − 384.666i − 0.491900i
\(783\) 0 0
\(784\) 192.000 0.244898
\(785\) − 1453.81i − 1.85199i
\(786\) 0 0
\(787\) −1024.00 −1.30114 −0.650572 0.759445i \(-0.725471\pi\)
−0.650572 + 0.759445i \(0.725471\pi\)
\(788\) − 90.5097i − 0.114860i
\(789\) 0 0
\(790\) −184.000 −0.232911
\(791\) − 12.7279i − 0.0160909i
\(792\) 0 0
\(793\) 1680.00 2.11854
\(794\) 680.237i 0.856721i
\(795\) 0 0
\(796\) −114.000 −0.143216
\(797\) − 7.07107i − 0.00887211i −0.999990 0.00443605i \(-0.998588\pi\)
0.999990 0.00443605i \(-0.00141204\pi\)
\(798\) 0 0
\(799\) 688.000 0.861076
\(800\) − 197.990i − 0.247487i
\(801\) 0 0
\(802\) −608.000 −0.758105
\(803\) 0 0
\(804\) 0 0
\(805\) 136.000 0.168944
\(806\) − 701.450i − 0.870285i
\(807\) 0 0
\(808\) −1284.00 −1.58911
\(809\) − 933.381i − 1.15375i −0.816834 0.576873i \(-0.804273\pi\)
0.816834 0.576873i \(-0.195727\pi\)
\(810\) 0 0
\(811\) −1513.00 −1.86560 −0.932799 0.360397i \(-0.882641\pi\)
−0.932799 + 0.360397i \(0.882641\pi\)
\(812\) − 25.4558i − 0.0313496i
\(813\) 0 0
\(814\) 0 0
\(815\) − 446.891i − 0.548333i
\(816\) 0 0
\(817\) 0 0
\(818\) − 250.316i − 0.306010i
\(819\) 0 0
\(820\) −176.000 −0.214634
\(821\) 926.310i 1.12827i 0.825682 + 0.564135i \(0.190790\pi\)
−0.825682 + 0.564135i \(0.809210\pi\)
\(822\) 0 0
\(823\) 615.000 0.747266 0.373633 0.927577i \(-0.378112\pi\)
0.373633 + 0.927577i \(0.378112\pi\)
\(824\) − 212.132i − 0.257442i
\(825\) 0 0
\(826\) 98.0000 0.118644
\(827\) 1084.70i 1.31161i 0.754930 + 0.655805i \(0.227671\pi\)
−0.754930 + 0.655805i \(0.772329\pi\)
\(828\) 0 0
\(829\) 97.0000 0.117008 0.0585042 0.998287i \(-0.481367\pi\)
0.0585042 + 0.998287i \(0.481367\pi\)
\(830\) − 848.528i − 1.02232i
\(831\) 0 0
\(832\) 896.000 1.07692
\(833\) 543.058i 0.651930i
\(834\) 0 0
\(835\) 168.000 0.201198
\(836\) 0 0
\(837\) 0 0
\(838\) −434.000 −0.517900
\(839\) − 1363.30i − 1.62491i −0.583022 0.812456i \(-0.698130\pi\)
0.583022 0.812456i \(-0.301870\pi\)
\(840\) 0 0
\(841\) 679.000 0.807372
\(842\) 226.274i 0.268734i
\(843\) 0 0
\(844\) 210.000 0.248815
\(845\) − 492.146i − 0.582422i
\(846\) 0 0
\(847\) 0 0
\(848\) 356.382i 0.420262i
\(849\) 0 0
\(850\) −112.000 −0.131765
\(851\) 1370.37i 1.61031i
\(852\) 0 0
\(853\) −41.0000 −0.0480657 −0.0240328 0.999711i \(-0.507651\pi\)
−0.0240328 + 0.999711i \(0.507651\pi\)
\(854\) 148.492i 0.173879i
\(855\) 0 0
\(856\) 1116.00 1.30374
\(857\) 401.637i 0.468654i 0.972158 + 0.234327i \(0.0752887\pi\)
−0.972158 + 0.234327i \(0.924711\pi\)
\(858\) 0 0
\(859\) −1425.00 −1.65891 −0.829453 0.558577i \(-0.811348\pi\)
−0.829453 + 0.558577i \(0.811348\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −978.000 −1.13457
\(863\) − 1107.33i − 1.28312i −0.767074 0.641558i \(-0.778288\pi\)
0.767074 0.641558i \(-0.221712\pi\)
\(864\) 0 0
\(865\) −384.000 −0.443931
\(866\) − 202.233i − 0.233525i
\(867\) 0 0
\(868\) −62.0000 −0.0714286
\(869\) 0 0
\(870\) 0 0
\(871\) 1648.00 1.89208
\(872\) 1009.75i 1.15797i
\(873\) 0 0
\(874\) −1122.00 −1.28375
\(875\) 101.823i 0.116370i
\(876\) 0 0
\(877\) 1593.00 1.81642 0.908210 0.418515i \(-0.137449\pi\)
0.908210 + 0.418515i \(0.137449\pi\)
\(878\) 473.762i 0.539592i
\(879\) 0 0
\(880\) 0 0
\(881\) 646.296i 0.733593i 0.930301 + 0.366797i \(0.119546\pi\)
−0.930301 + 0.366797i \(0.880454\pi\)
\(882\) 0 0
\(883\) −1145.00 −1.29672 −0.648358 0.761336i \(-0.724544\pi\)
−0.648358 + 0.761336i \(0.724544\pi\)
\(884\) 362.039i 0.409546i
\(885\) 0 0
\(886\) −184.000 −0.207675
\(887\) 439.820i 0.495852i 0.968779 + 0.247926i \(0.0797489\pi\)
−0.968779 + 0.247926i \(0.920251\pi\)
\(888\) 0 0
\(889\) −79.0000 −0.0888639
\(890\) 407.294i 0.457633i
\(891\) 0 0
\(892\) 398.000 0.446188
\(893\) − 2006.77i − 2.24722i
\(894\) 0 0
\(895\) −1128.00 −1.26034
\(896\) − 33.9411i − 0.0378807i
\(897\) 0 0
\(898\) −1208.00 −1.34521
\(899\) 394.566i 0.438894i
\(900\) 0 0
\(901\) −1008.00 −1.11876
\(902\) 0 0
\(903\) 0 0
\(904\) −108.000 −0.119469
\(905\) 1623.52i 1.79394i
\(906\) 0 0
\(907\) −87.0000 −0.0959206 −0.0479603 0.998849i \(-0.515272\pi\)
−0.0479603 + 0.998849i \(0.515272\pi\)
\(908\) − 373.352i − 0.411181i
\(909\) 0 0
\(910\) 128.000 0.140659
\(911\) − 1493.41i − 1.63931i −0.572859 0.819654i \(-0.694166\pi\)
0.572859 0.819654i \(-0.305834\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 373.352i − 0.408482i
\(915\) 0 0
\(916\) −144.000 −0.157205
\(917\) − 114.551i − 0.124920i
\(918\) 0 0
\(919\) −865.000 −0.941240 −0.470620 0.882336i \(-0.655970\pi\)
−0.470620 + 0.882336i \(0.655970\pi\)
\(920\) − 1154.00i − 1.25435i
\(921\) 0 0
\(922\) −174.000 −0.188720
\(923\) 1900.70i 2.05927i
\(924\) 0 0
\(925\) 399.000 0.431351
\(926\) − 859.842i − 0.928555i
\(927\) 0 0
\(928\) −360.000 −0.387931
\(929\) − 1192.18i − 1.28330i −0.766999 0.641648i \(-0.778251\pi\)
0.766999 0.641648i \(-0.221749\pi\)
\(930\) 0 0
\(931\) 1584.00 1.70140
\(932\) − 263.044i − 0.282236i
\(933\) 0 0
\(934\) 600.000 0.642398
\(935\) 0 0
\(936\) 0 0
\(937\) −473.000 −0.504803 −0.252401 0.967623i \(-0.581220\pi\)
−0.252401 + 0.967623i \(0.581220\pi\)
\(938\) 145.664i 0.155292i
\(939\) 0 0
\(940\) 688.000 0.731915
\(941\) 114.551i 0.121734i 0.998146 + 0.0608668i \(0.0193865\pi\)
−0.998146 + 0.0608668i \(0.980614\pi\)
\(942\) 0 0
\(943\) −374.000 −0.396607
\(944\) − 277.186i − 0.293629i
\(945\) 0 0
\(946\) 0 0
\(947\) − 818.830i − 0.864656i −0.901716 0.432328i \(-0.857692\pi\)
0.901716 0.432328i \(-0.142308\pi\)
\(948\) 0 0
\(949\) −752.000 −0.792413
\(950\) 326.683i 0.343877i
\(951\) 0 0
\(952\) −96.0000 −0.100840
\(953\) − 1042.28i − 1.09368i −0.837238 0.546839i \(-0.815831\pi\)
0.837238 0.546839i \(-0.184169\pi\)
\(954\) 0 0
\(955\) −928.000 −0.971728
\(956\) − 8.48528i − 0.00887582i
\(957\) 0 0
\(958\) −1174.00 −1.22547
\(959\) − 90.5097i − 0.0943792i
\(960\) 0 0
\(961\) 0 0
\(962\) 1289.76i 1.34071i
\(963\) 0 0
\(964\) −320.000 −0.331950
\(965\) 820.244i 0.849994i
\(966\) 0 0
\(967\) −937.000 −0.968976 −0.484488 0.874798i \(-0.660994\pi\)
−0.484488 + 0.874798i \(0.660994\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −200.000 −0.206186
\(971\) − 694.379i − 0.715117i −0.933891 0.357559i \(-0.883609\pi\)
0.933891 0.357559i \(-0.116391\pi\)
\(972\) 0 0
\(973\) 58.0000 0.0596095
\(974\) 79.1960i 0.0813100i
\(975\) 0 0
\(976\) 420.000 0.430328
\(977\) − 708.521i − 0.725201i −0.931945 0.362600i \(-0.881889\pi\)
0.931945 0.362600i \(-0.118111\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 543.058i 0.554141i
\(981\) 0 0
\(982\) 394.000 0.401222
\(983\) 547.301i 0.556766i 0.960470 + 0.278383i \(0.0897984\pi\)
−0.960470 + 0.278383i \(0.910202\pi\)
\(984\) 0 0
\(985\) −256.000 −0.259898
\(986\) 203.647i 0.206538i
\(987\) 0 0
\(988\) 1056.00 1.06883
\(989\) 0 0
\(990\) 0 0
\(991\) 286.000 0.288597 0.144299 0.989534i \(-0.453907\pi\)
0.144299 + 0.989534i \(0.453907\pi\)
\(992\) 876.812i 0.883883i
\(993\) 0 0
\(994\) −168.000 −0.169014
\(995\) 322.441i 0.324061i
\(996\) 0 0
\(997\) 265.000 0.265797 0.132899 0.991130i \(-0.457572\pi\)
0.132899 + 0.991130i \(0.457572\pi\)
\(998\) − 620.840i − 0.622084i
\(999\) 0 0
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 1089.3.b.a.485.2 yes 2
3.2 odd 2 inner 1089.3.b.a.485.1 2
11.10 odd 2 1089.3.b.b.485.1 yes 2
33.32 even 2 1089.3.b.b.485.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.a.485.1 2 3.2 odd 2 inner
1089.3.b.a.485.2 yes 2 1.1 even 1 trivial
1089.3.b.b.485.1 yes 2 11.10 odd 2
1089.3.b.b.485.2 yes 2 33.32 even 2