Properties

Label 1089.3.b.g.485.6
Level $1089$
Weight $3$
Character 1089.485
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.65306824704.6
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 485.6
Root \(1.13623 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1089.485
Dual form 1089.3.b.g.485.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.60688i q^{2} +1.41795 q^{4} +6.21249i q^{5} -11.1468 q^{7} +8.70597i q^{8} +O(q^{10})\) \(q+1.60688i q^{2} +1.41795 q^{4} +6.21249i q^{5} -11.1468 q^{7} +8.70597i q^{8} -9.98270 q^{10} +17.7871 q^{13} -17.9115i q^{14} -8.31762 q^{16} +8.53964i q^{17} -16.4632 q^{19} +8.80900i q^{20} +25.0834i q^{23} -13.5950 q^{25} +28.5817i q^{26} -15.8056 q^{28} +9.46540i q^{29} -46.1952 q^{31} +21.4585i q^{32} -13.7221 q^{34} -69.2494i q^{35} -37.4758 q^{37} -26.4543i q^{38} -54.0858 q^{40} -51.8375i q^{41} +55.6297 q^{43} -40.3058 q^{46} -37.8600i q^{47} +75.2512 q^{49} -21.8455i q^{50} +25.2212 q^{52} -58.5761i q^{53} -97.0437i q^{56} -15.2097 q^{58} +44.6296i q^{59} +10.1633 q^{61} -74.2300i q^{62} -67.7516 q^{64} +110.502i q^{65} -58.3755 q^{67} +12.1088i q^{68} +111.275 q^{70} -75.1766i q^{71} -112.274 q^{73} -60.2190i q^{74} -23.3440 q^{76} -52.2849 q^{79} -51.6731i q^{80} +83.2963 q^{82} +18.0976i q^{83} -53.0524 q^{85} +89.3901i q^{86} +136.779i q^{89} -198.269 q^{91} +35.5669i q^{92} +60.8364 q^{94} -102.278i q^{95} +127.754 q^{97} +120.919i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 16 q^{7} + 48 q^{10} + 8 q^{13} + 104 q^{16} - 40 q^{19} - 112 q^{25} + 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} - 432 q^{40} + 104 q^{43} - 24 q^{46} - 96 q^{49} - 280 q^{52} - 432 q^{58} + 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} - 448 q^{73} + 344 q^{76} - 448 q^{79} + 504 q^{82} - 48 q^{85} - 544 q^{91} - 360 q^{94} - 152 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.60688i 0.803438i 0.915763 + 0.401719i \(0.131587\pi\)
−0.915763 + 0.401719i \(0.868413\pi\)
\(3\) 0 0
\(4\) 1.41795 0.354488
\(5\) 6.21249i 1.24250i 0.783613 + 0.621249i \(0.213374\pi\)
−0.783613 + 0.621249i \(0.786626\pi\)
\(6\) 0 0
\(7\) −11.1468 −1.59240 −0.796200 0.605033i \(-0.793160\pi\)
−0.796200 + 0.605033i \(0.793160\pi\)
\(8\) 8.70597i 1.08825i
\(9\) 0 0
\(10\) −9.98270 −0.998270
\(11\) 0 0
\(12\) 0 0
\(13\) 17.7871 1.36824 0.684119 0.729370i \(-0.260187\pi\)
0.684119 + 0.729370i \(0.260187\pi\)
\(14\) − 17.9115i − 1.27939i
\(15\) 0 0
\(16\) −8.31762 −0.519851
\(17\) 8.53964i 0.502332i 0.967944 + 0.251166i \(0.0808139\pi\)
−0.967944 + 0.251166i \(0.919186\pi\)
\(18\) 0 0
\(19\) −16.4632 −0.866485 −0.433242 0.901277i \(-0.642631\pi\)
−0.433242 + 0.901277i \(0.642631\pi\)
\(20\) 8.80900i 0.440450i
\(21\) 0 0
\(22\) 0 0
\(23\) 25.0834i 1.09058i 0.838247 + 0.545290i \(0.183581\pi\)
−0.838247 + 0.545290i \(0.816419\pi\)
\(24\) 0 0
\(25\) −13.5950 −0.543802
\(26\) 28.5817i 1.09929i
\(27\) 0 0
\(28\) −15.8056 −0.564486
\(29\) 9.46540i 0.326393i 0.986594 + 0.163197i \(0.0521805\pi\)
−0.986594 + 0.163197i \(0.947820\pi\)
\(30\) 0 0
\(31\) −46.1952 −1.49017 −0.745084 0.666970i \(-0.767591\pi\)
−0.745084 + 0.666970i \(0.767591\pi\)
\(32\) 21.4585i 0.670579i
\(33\) 0 0
\(34\) −13.7221 −0.403592
\(35\) − 69.2494i − 1.97855i
\(36\) 0 0
\(37\) −37.4758 −1.01286 −0.506430 0.862281i \(-0.669035\pi\)
−0.506430 + 0.862281i \(0.669035\pi\)
\(38\) − 26.4543i − 0.696167i
\(39\) 0 0
\(40\) −54.0858 −1.35214
\(41\) − 51.8375i − 1.26433i −0.774835 0.632164i \(-0.782167\pi\)
0.774835 0.632164i \(-0.217833\pi\)
\(42\) 0 0
\(43\) 55.6297 1.29371 0.646857 0.762611i \(-0.276083\pi\)
0.646857 + 0.762611i \(0.276083\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −40.3058 −0.876214
\(47\) − 37.8600i − 0.805533i −0.915303 0.402766i \(-0.868049\pi\)
0.915303 0.402766i \(-0.131951\pi\)
\(48\) 0 0
\(49\) 75.2512 1.53574
\(50\) − 21.8455i − 0.436911i
\(51\) 0 0
\(52\) 25.2212 0.485024
\(53\) − 58.5761i − 1.10521i −0.833444 0.552604i \(-0.813634\pi\)
0.833444 0.552604i \(-0.186366\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 97.0437i − 1.73292i
\(57\) 0 0
\(58\) −15.2097 −0.262237
\(59\) 44.6296i 0.756435i 0.925717 + 0.378217i \(0.123463\pi\)
−0.925717 + 0.378217i \(0.876537\pi\)
\(60\) 0 0
\(61\) 10.1633 0.166612 0.0833060 0.996524i \(-0.473452\pi\)
0.0833060 + 0.996524i \(0.473452\pi\)
\(62\) − 74.2300i − 1.19726i
\(63\) 0 0
\(64\) −67.7516 −1.05862
\(65\) 110.502i 1.70003i
\(66\) 0 0
\(67\) −58.3755 −0.871276 −0.435638 0.900122i \(-0.643477\pi\)
−0.435638 + 0.900122i \(0.643477\pi\)
\(68\) 12.1088i 0.178070i
\(69\) 0 0
\(70\) 111.275 1.58965
\(71\) − 75.1766i − 1.05882i −0.848365 0.529412i \(-0.822412\pi\)
0.848365 0.529412i \(-0.177588\pi\)
\(72\) 0 0
\(73\) −112.274 −1.53800 −0.769000 0.639249i \(-0.779245\pi\)
−0.769000 + 0.639249i \(0.779245\pi\)
\(74\) − 60.2190i − 0.813771i
\(75\) 0 0
\(76\) −23.3440 −0.307158
\(77\) 0 0
\(78\) 0 0
\(79\) −52.2849 −0.661834 −0.330917 0.943660i \(-0.607358\pi\)
−0.330917 + 0.943660i \(0.607358\pi\)
\(80\) − 51.6731i − 0.645914i
\(81\) 0 0
\(82\) 83.2963 1.01581
\(83\) 18.0976i 0.218043i 0.994039 + 0.109022i \(0.0347717\pi\)
−0.994039 + 0.109022i \(0.965228\pi\)
\(84\) 0 0
\(85\) −53.0524 −0.624146
\(86\) 89.3901i 1.03942i
\(87\) 0 0
\(88\) 0 0
\(89\) 136.779i 1.53684i 0.639943 + 0.768422i \(0.278958\pi\)
−0.639943 + 0.768422i \(0.721042\pi\)
\(90\) 0 0
\(91\) −198.269 −2.17878
\(92\) 35.5669i 0.386597i
\(93\) 0 0
\(94\) 60.8364 0.647196
\(95\) − 102.278i − 1.07661i
\(96\) 0 0
\(97\) 127.754 1.31705 0.658525 0.752559i \(-0.271181\pi\)
0.658525 + 0.752559i \(0.271181\pi\)
\(98\) 120.919i 1.23387i
\(99\) 0 0
\(100\) −19.2771 −0.192771
\(101\) 5.72551i 0.0566882i 0.999598 + 0.0283441i \(0.00902342\pi\)
−0.999598 + 0.0283441i \(0.990977\pi\)
\(102\) 0 0
\(103\) 27.6215 0.268170 0.134085 0.990970i \(-0.457191\pi\)
0.134085 + 0.990970i \(0.457191\pi\)
\(104\) 154.854i 1.48898i
\(105\) 0 0
\(106\) 94.1245 0.887967
\(107\) − 47.9169i − 0.447821i −0.974610 0.223911i \(-0.928118\pi\)
0.974610 0.223911i \(-0.0718824\pi\)
\(108\) 0 0
\(109\) 79.6706 0.730923 0.365461 0.930826i \(-0.380911\pi\)
0.365461 + 0.930826i \(0.380911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 92.7148 0.827811
\(113\) − 214.762i − 1.90055i −0.311415 0.950274i \(-0.600803\pi\)
0.311415 0.950274i \(-0.399197\pi\)
\(114\) 0 0
\(115\) −155.830 −1.35504
\(116\) 13.4215i 0.115702i
\(117\) 0 0
\(118\) −71.7143 −0.607748
\(119\) − 95.1896i − 0.799913i
\(120\) 0 0
\(121\) 0 0
\(122\) 16.3312i 0.133862i
\(123\) 0 0
\(124\) −65.5025 −0.528246
\(125\) 70.8532i 0.566826i
\(126\) 0 0
\(127\) 15.6836 0.123493 0.0617465 0.998092i \(-0.480333\pi\)
0.0617465 + 0.998092i \(0.480333\pi\)
\(128\) − 23.0344i − 0.179956i
\(129\) 0 0
\(130\) −177.563 −1.36587
\(131\) − 78.9245i − 0.602477i −0.953549 0.301239i \(-0.902600\pi\)
0.953549 0.301239i \(-0.0974001\pi\)
\(132\) 0 0
\(133\) 183.512 1.37979
\(134\) − 93.8022i − 0.700016i
\(135\) 0 0
\(136\) −74.3458 −0.546661
\(137\) 135.084i 0.986017i 0.870025 + 0.493008i \(0.164103\pi\)
−0.870025 + 0.493008i \(0.835897\pi\)
\(138\) 0 0
\(139\) −181.554 −1.30615 −0.653073 0.757295i \(-0.726520\pi\)
−0.653073 + 0.757295i \(0.726520\pi\)
\(140\) − 98.1922i − 0.701373i
\(141\) 0 0
\(142\) 120.799 0.850700
\(143\) 0 0
\(144\) 0 0
\(145\) −58.8037 −0.405543
\(146\) − 180.410i − 1.23569i
\(147\) 0 0
\(148\) −53.1389 −0.359047
\(149\) 123.629i 0.829725i 0.909884 + 0.414862i \(0.136170\pi\)
−0.909884 + 0.414862i \(0.863830\pi\)
\(150\) 0 0
\(151\) −67.4578 −0.446740 −0.223370 0.974734i \(-0.571706\pi\)
−0.223370 + 0.974734i \(0.571706\pi\)
\(152\) − 143.328i − 0.942949i
\(153\) 0 0
\(154\) 0 0
\(155\) − 286.987i − 1.85153i
\(156\) 0 0
\(157\) 168.395 1.07258 0.536291 0.844033i \(-0.319825\pi\)
0.536291 + 0.844033i \(0.319825\pi\)
\(158\) − 84.0154i − 0.531743i
\(159\) 0 0
\(160\) −133.311 −0.833193
\(161\) − 279.599i − 1.73664i
\(162\) 0 0
\(163\) 128.845 0.790458 0.395229 0.918583i \(-0.370665\pi\)
0.395229 + 0.918583i \(0.370665\pi\)
\(164\) − 73.5029i − 0.448189i
\(165\) 0 0
\(166\) −29.0806 −0.175184
\(167\) − 51.8678i − 0.310586i −0.987868 0.155293i \(-0.950368\pi\)
0.987868 0.155293i \(-0.0496322\pi\)
\(168\) 0 0
\(169\) 147.381 0.872077
\(170\) − 85.2486i − 0.501463i
\(171\) 0 0
\(172\) 78.8802 0.458606
\(173\) 231.087i 1.33576i 0.744268 + 0.667881i \(0.232799\pi\)
−0.744268 + 0.667881i \(0.767201\pi\)
\(174\) 0 0
\(175\) 151.541 0.865950
\(176\) 0 0
\(177\) 0 0
\(178\) −219.787 −1.23476
\(179\) 294.651i 1.64609i 0.567973 + 0.823047i \(0.307728\pi\)
−0.567973 + 0.823047i \(0.692272\pi\)
\(180\) 0 0
\(181\) −157.848 −0.872090 −0.436045 0.899925i \(-0.643621\pi\)
−0.436045 + 0.899925i \(0.643621\pi\)
\(182\) − 318.594i − 1.75052i
\(183\) 0 0
\(184\) −218.375 −1.18682
\(185\) − 232.818i − 1.25848i
\(186\) 0 0
\(187\) 0 0
\(188\) − 53.6837i − 0.285551i
\(189\) 0 0
\(190\) 164.347 0.864986
\(191\) 215.226i 1.12684i 0.826172 + 0.563419i \(0.190514\pi\)
−0.826172 + 0.563419i \(0.809486\pi\)
\(192\) 0 0
\(193\) −239.813 −1.24255 −0.621277 0.783591i \(-0.713386\pi\)
−0.621277 + 0.783591i \(0.713386\pi\)
\(194\) 205.284i 1.05817i
\(195\) 0 0
\(196\) 106.702 0.544400
\(197\) 298.396i 1.51470i 0.653010 + 0.757349i \(0.273506\pi\)
−0.653010 + 0.757349i \(0.726494\pi\)
\(198\) 0 0
\(199\) −66.9651 −0.336508 −0.168254 0.985744i \(-0.553813\pi\)
−0.168254 + 0.985744i \(0.553813\pi\)
\(200\) − 118.358i − 0.591790i
\(201\) 0 0
\(202\) −9.20019 −0.0455455
\(203\) − 105.509i − 0.519748i
\(204\) 0 0
\(205\) 322.040 1.57093
\(206\) 44.3843i 0.215458i
\(207\) 0 0
\(208\) −147.946 −0.711280
\(209\) 0 0
\(210\) 0 0
\(211\) −58.8876 −0.279088 −0.139544 0.990216i \(-0.544564\pi\)
−0.139544 + 0.990216i \(0.544564\pi\)
\(212\) − 83.0580i − 0.391783i
\(213\) 0 0
\(214\) 76.9965 0.359797
\(215\) 345.599i 1.60744i
\(216\) 0 0
\(217\) 514.929 2.37295
\(218\) 128.021i 0.587251i
\(219\) 0 0
\(220\) 0 0
\(221\) 151.895i 0.687310i
\(222\) 0 0
\(223\) −216.873 −0.972525 −0.486263 0.873813i \(-0.661640\pi\)
−0.486263 + 0.873813i \(0.661640\pi\)
\(224\) − 239.194i − 1.06783i
\(225\) 0 0
\(226\) 345.096 1.52697
\(227\) − 126.476i − 0.557163i −0.960413 0.278582i \(-0.910136\pi\)
0.960413 0.278582i \(-0.0898643\pi\)
\(228\) 0 0
\(229\) 29.4781 0.128726 0.0643628 0.997927i \(-0.479499\pi\)
0.0643628 + 0.997927i \(0.479499\pi\)
\(230\) − 250.400i − 1.08869i
\(231\) 0 0
\(232\) −82.4055 −0.355196
\(233\) 236.199i 1.01373i 0.862026 + 0.506864i \(0.169196\pi\)
−0.862026 + 0.506864i \(0.830804\pi\)
\(234\) 0 0
\(235\) 235.205 1.00087
\(236\) 63.2826i 0.268147i
\(237\) 0 0
\(238\) 152.958 0.642680
\(239\) − 119.069i − 0.498197i −0.968478 0.249098i \(-0.919866\pi\)
0.968478 0.249098i \(-0.0801342\pi\)
\(240\) 0 0
\(241\) −102.143 −0.423828 −0.211914 0.977288i \(-0.567970\pi\)
−0.211914 + 0.977288i \(0.567970\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 14.4111 0.0590619
\(245\) 467.497i 1.90815i
\(246\) 0 0
\(247\) −292.833 −1.18556
\(248\) − 402.174i − 1.62167i
\(249\) 0 0
\(250\) −113.852 −0.455409
\(251\) 264.723i 1.05467i 0.849656 + 0.527337i \(0.176810\pi\)
−0.849656 + 0.527337i \(0.823190\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 25.2016i 0.0992190i
\(255\) 0 0
\(256\) −233.993 −0.914036
\(257\) − 299.407i − 1.16501i −0.812828 0.582504i \(-0.802073\pi\)
0.812828 0.582504i \(-0.197927\pi\)
\(258\) 0 0
\(259\) 417.736 1.61288
\(260\) 156.687i 0.602641i
\(261\) 0 0
\(262\) 126.822 0.484053
\(263\) − 345.564i − 1.31393i −0.753921 0.656966i \(-0.771840\pi\)
0.753921 0.656966i \(-0.228160\pi\)
\(264\) 0 0
\(265\) 363.903 1.37322
\(266\) 294.881i 1.10858i
\(267\) 0 0
\(268\) −82.7736 −0.308857
\(269\) 158.607i 0.589618i 0.955556 + 0.294809i \(0.0952560\pi\)
−0.955556 + 0.294809i \(0.904744\pi\)
\(270\) 0 0
\(271\) −121.065 −0.446734 −0.223367 0.974734i \(-0.571705\pi\)
−0.223367 + 0.974734i \(0.571705\pi\)
\(272\) − 71.0294i − 0.261138i
\(273\) 0 0
\(274\) −217.064 −0.792203
\(275\) 0 0
\(276\) 0 0
\(277\) 221.505 0.799656 0.399828 0.916590i \(-0.369070\pi\)
0.399828 + 0.916590i \(0.369070\pi\)
\(278\) − 291.735i − 1.04941i
\(279\) 0 0
\(280\) 602.883 2.15315
\(281\) − 26.5964i − 0.0946489i −0.998880 0.0473245i \(-0.984931\pi\)
0.998880 0.0473245i \(-0.0150695\pi\)
\(282\) 0 0
\(283\) 216.323 0.764393 0.382197 0.924081i \(-0.375168\pi\)
0.382197 + 0.924081i \(0.375168\pi\)
\(284\) − 106.597i − 0.375340i
\(285\) 0 0
\(286\) 0 0
\(287\) 577.822i 2.01332i
\(288\) 0 0
\(289\) 216.075 0.747663
\(290\) − 94.4902i − 0.325828i
\(291\) 0 0
\(292\) −159.199 −0.545202
\(293\) 249.421i 0.851266i 0.904896 + 0.425633i \(0.139949\pi\)
−0.904896 + 0.425633i \(0.860051\pi\)
\(294\) 0 0
\(295\) −277.261 −0.939869
\(296\) − 326.264i − 1.10224i
\(297\) 0 0
\(298\) −198.656 −0.666632
\(299\) 446.160i 1.49217i
\(300\) 0 0
\(301\) −620.093 −2.06011
\(302\) − 108.396i − 0.358928i
\(303\) 0 0
\(304\) 136.935 0.450443
\(305\) 63.1396i 0.207015i
\(306\) 0 0
\(307\) −211.978 −0.690483 −0.345242 0.938514i \(-0.612203\pi\)
−0.345242 + 0.938514i \(0.612203\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 461.153 1.48759
\(311\) 278.236i 0.894649i 0.894372 + 0.447324i \(0.147623\pi\)
−0.894372 + 0.447324i \(0.852377\pi\)
\(312\) 0 0
\(313\) 378.259 1.20850 0.604248 0.796796i \(-0.293474\pi\)
0.604248 + 0.796796i \(0.293474\pi\)
\(314\) 270.590i 0.861753i
\(315\) 0 0
\(316\) −74.1374 −0.234612
\(317\) 446.608i 1.40886i 0.709774 + 0.704429i \(0.248797\pi\)
−0.709774 + 0.704429i \(0.751203\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 420.906i − 1.31533i
\(321\) 0 0
\(322\) 449.281 1.39528
\(323\) − 140.590i − 0.435263i
\(324\) 0 0
\(325\) −241.816 −0.744051
\(326\) 207.037i 0.635084i
\(327\) 0 0
\(328\) 451.295 1.37590
\(329\) 422.018i 1.28273i
\(330\) 0 0
\(331\) 80.3709 0.242812 0.121406 0.992603i \(-0.461260\pi\)
0.121406 + 0.992603i \(0.461260\pi\)
\(332\) 25.6615i 0.0772935i
\(333\) 0 0
\(334\) 83.3452 0.249536
\(335\) − 362.657i − 1.08256i
\(336\) 0 0
\(337\) −171.695 −0.509482 −0.254741 0.967009i \(-0.581990\pi\)
−0.254741 + 0.967009i \(0.581990\pi\)
\(338\) 236.823i 0.700660i
\(339\) 0 0
\(340\) −75.2257 −0.221252
\(341\) 0 0
\(342\) 0 0
\(343\) −292.616 −0.853109
\(344\) 484.311i 1.40788i
\(345\) 0 0
\(346\) −371.328 −1.07320
\(347\) 466.748i 1.34510i 0.740053 + 0.672548i \(0.234800\pi\)
−0.740053 + 0.672548i \(0.765200\pi\)
\(348\) 0 0
\(349\) 298.940 0.856561 0.428280 0.903646i \(-0.359120\pi\)
0.428280 + 0.903646i \(0.359120\pi\)
\(350\) 243.508i 0.695737i
\(351\) 0 0
\(352\) 0 0
\(353\) 43.9061i 0.124380i 0.998064 + 0.0621899i \(0.0198084\pi\)
−0.998064 + 0.0621899i \(0.980192\pi\)
\(354\) 0 0
\(355\) 467.034 1.31559
\(356\) 193.946i 0.544792i
\(357\) 0 0
\(358\) −473.467 −1.32253
\(359\) 657.031i 1.83017i 0.403261 + 0.915085i \(0.367877\pi\)
−0.403261 + 0.915085i \(0.632123\pi\)
\(360\) 0 0
\(361\) −89.9627 −0.249204
\(362\) − 253.642i − 0.700670i
\(363\) 0 0
\(364\) −281.136 −0.772352
\(365\) − 697.501i − 1.91096i
\(366\) 0 0
\(367\) −397.449 −1.08297 −0.541484 0.840711i \(-0.682137\pi\)
−0.541484 + 0.840711i \(0.682137\pi\)
\(368\) − 208.634i − 0.566939i
\(369\) 0 0
\(370\) 374.110 1.01111
\(371\) 652.936i 1.75993i
\(372\) 0 0
\(373\) 11.2157 0.0300688 0.0150344 0.999887i \(-0.495214\pi\)
0.0150344 + 0.999887i \(0.495214\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 329.609 0.876618
\(377\) 168.362i 0.446584i
\(378\) 0 0
\(379\) −99.5442 −0.262650 −0.131325 0.991339i \(-0.541923\pi\)
−0.131325 + 0.991339i \(0.541923\pi\)
\(380\) − 145.024i − 0.381643i
\(381\) 0 0
\(382\) −345.841 −0.905344
\(383\) 164.100i 0.428461i 0.976783 + 0.214230i \(0.0687243\pi\)
−0.976783 + 0.214230i \(0.931276\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 385.349i − 0.998315i
\(387\) 0 0
\(388\) 181.148 0.466878
\(389\) 553.313i 1.42240i 0.702991 + 0.711199i \(0.251848\pi\)
−0.702991 + 0.711199i \(0.748152\pi\)
\(390\) 0 0
\(391\) −214.203 −0.547833
\(392\) 655.135i 1.67126i
\(393\) 0 0
\(394\) −479.485 −1.21697
\(395\) − 324.820i − 0.822328i
\(396\) 0 0
\(397\) 18.8569 0.0474984 0.0237492 0.999718i \(-0.492440\pi\)
0.0237492 + 0.999718i \(0.492440\pi\)
\(398\) − 107.605i − 0.270363i
\(399\) 0 0
\(400\) 113.078 0.282696
\(401\) − 254.177i − 0.633858i −0.948449 0.316929i \(-0.897348\pi\)
0.948449 0.316929i \(-0.102652\pi\)
\(402\) 0 0
\(403\) −821.680 −2.03891
\(404\) 8.11849i 0.0200953i
\(405\) 0 0
\(406\) 169.540 0.417585
\(407\) 0 0
\(408\) 0 0
\(409\) 533.620 1.30469 0.652347 0.757920i \(-0.273784\pi\)
0.652347 + 0.757920i \(0.273784\pi\)
\(410\) 517.478i 1.26214i
\(411\) 0 0
\(412\) 39.1659 0.0950628
\(413\) − 497.478i − 1.20455i
\(414\) 0 0
\(415\) −112.431 −0.270918
\(416\) 381.685i 0.917512i
\(417\) 0 0
\(418\) 0 0
\(419\) 345.851i 0.825420i 0.910862 + 0.412710i \(0.135418\pi\)
−0.910862 + 0.412710i \(0.864582\pi\)
\(420\) 0 0
\(421\) 410.003 0.973880 0.486940 0.873436i \(-0.338113\pi\)
0.486940 + 0.873436i \(0.338113\pi\)
\(422\) − 94.6250i − 0.224230i
\(423\) 0 0
\(424\) 509.962 1.20274
\(425\) − 116.097i − 0.273169i
\(426\) 0 0
\(427\) −113.289 −0.265313
\(428\) − 67.9438i − 0.158747i
\(429\) 0 0
\(430\) −555.335 −1.29148
\(431\) − 101.979i − 0.236610i −0.992977 0.118305i \(-0.962254\pi\)
0.992977 0.118305i \(-0.0377461\pi\)
\(432\) 0 0
\(433\) −378.091 −0.873189 −0.436595 0.899658i \(-0.643816\pi\)
−0.436595 + 0.899658i \(0.643816\pi\)
\(434\) 827.427i 1.90651i
\(435\) 0 0
\(436\) 112.969 0.259103
\(437\) − 412.953i − 0.944971i
\(438\) 0 0
\(439\) −594.777 −1.35484 −0.677422 0.735594i \(-0.736903\pi\)
−0.677422 + 0.735594i \(0.736903\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −244.077 −0.552210
\(443\) 47.2023i 0.106551i 0.998580 + 0.0532757i \(0.0169662\pi\)
−0.998580 + 0.0532757i \(0.983034\pi\)
\(444\) 0 0
\(445\) −849.739 −1.90953
\(446\) − 348.488i − 0.781364i
\(447\) 0 0
\(448\) 755.214 1.68575
\(449\) − 608.886i − 1.35609i −0.735019 0.678047i \(-0.762827\pi\)
0.735019 0.678047i \(-0.237173\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 304.522i − 0.673721i
\(453\) 0 0
\(454\) 203.231 0.447646
\(455\) − 1231.75i − 2.70713i
\(456\) 0 0
\(457\) 479.540 1.04932 0.524661 0.851311i \(-0.324192\pi\)
0.524661 + 0.851311i \(0.324192\pi\)
\(458\) 47.3677i 0.103423i
\(459\) 0 0
\(460\) −220.959 −0.480346
\(461\) 372.082i 0.807119i 0.914953 + 0.403559i \(0.132227\pi\)
−0.914953 + 0.403559i \(0.867773\pi\)
\(462\) 0 0
\(463\) 418.247 0.903342 0.451671 0.892185i \(-0.350828\pi\)
0.451671 + 0.892185i \(0.350828\pi\)
\(464\) − 78.7296i − 0.169676i
\(465\) 0 0
\(466\) −379.542 −0.814468
\(467\) − 118.057i − 0.252799i −0.991979 0.126400i \(-0.959658\pi\)
0.991979 0.126400i \(-0.0403422\pi\)
\(468\) 0 0
\(469\) 650.700 1.38742
\(470\) 377.946i 0.804139i
\(471\) 0 0
\(472\) −388.544 −0.823187
\(473\) 0 0
\(474\) 0 0
\(475\) 223.818 0.471196
\(476\) − 134.974i − 0.283559i
\(477\) 0 0
\(478\) 191.329 0.400270
\(479\) 692.647i 1.44603i 0.690833 + 0.723014i \(0.257244\pi\)
−0.690833 + 0.723014i \(0.742756\pi\)
\(480\) 0 0
\(481\) −666.587 −1.38584
\(482\) − 164.131i − 0.340520i
\(483\) 0 0
\(484\) 0 0
\(485\) 793.669i 1.63643i
\(486\) 0 0
\(487\) −813.256 −1.66993 −0.834965 0.550303i \(-0.814512\pi\)
−0.834965 + 0.550303i \(0.814512\pi\)
\(488\) 88.4817i 0.181315i
\(489\) 0 0
\(490\) −751.210 −1.53308
\(491\) − 265.228i − 0.540179i −0.962835 0.270089i \(-0.912947\pi\)
0.962835 0.270089i \(-0.0870532\pi\)
\(492\) 0 0
\(493\) −80.8311 −0.163958
\(494\) − 470.546i − 0.952522i
\(495\) 0 0
\(496\) 384.234 0.774666
\(497\) 837.978i 1.68607i
\(498\) 0 0
\(499\) 492.715 0.987406 0.493703 0.869631i \(-0.335643\pi\)
0.493703 + 0.869631i \(0.335643\pi\)
\(500\) 100.466i 0.200933i
\(501\) 0 0
\(502\) −425.377 −0.847365
\(503\) 88.4886i 0.175922i 0.996124 + 0.0879608i \(0.0280350\pi\)
−0.996124 + 0.0879608i \(0.971965\pi\)
\(504\) 0 0
\(505\) −35.5697 −0.0704350
\(506\) 0 0
\(507\) 0 0
\(508\) 22.2386 0.0437767
\(509\) − 261.110i − 0.512986i −0.966546 0.256493i \(-0.917433\pi\)
0.966546 0.256493i \(-0.0825671\pi\)
\(510\) 0 0
\(511\) 1251.50 2.44911
\(512\) − 468.135i − 0.914327i
\(513\) 0 0
\(514\) 481.110 0.936012
\(515\) 171.598i 0.333200i
\(516\) 0 0
\(517\) 0 0
\(518\) 671.250i 1.29585i
\(519\) 0 0
\(520\) −962.029 −1.85006
\(521\) 221.574i 0.425287i 0.977130 + 0.212643i \(0.0682072\pi\)
−0.977130 + 0.212643i \(0.931793\pi\)
\(522\) 0 0
\(523\) 444.427 0.849765 0.424882 0.905249i \(-0.360315\pi\)
0.424882 + 0.905249i \(0.360315\pi\)
\(524\) − 111.911i − 0.213571i
\(525\) 0 0
\(526\) 555.278 1.05566
\(527\) − 394.491i − 0.748559i
\(528\) 0 0
\(529\) −100.174 −0.189366
\(530\) 584.747i 1.10330i
\(531\) 0 0
\(532\) 260.211 0.489119
\(533\) − 922.038i − 1.72990i
\(534\) 0 0
\(535\) 297.683 0.556417
\(536\) − 508.216i − 0.948163i
\(537\) 0 0
\(538\) −254.862 −0.473721
\(539\) 0 0
\(540\) 0 0
\(541\) −508.057 −0.939107 −0.469553 0.882904i \(-0.655585\pi\)
−0.469553 + 0.882904i \(0.655585\pi\)
\(542\) − 194.536i − 0.358923i
\(543\) 0 0
\(544\) −183.248 −0.336853
\(545\) 494.953i 0.908170i
\(546\) 0 0
\(547\) −44.8213 −0.0819403 −0.0409701 0.999160i \(-0.513045\pi\)
−0.0409701 + 0.999160i \(0.513045\pi\)
\(548\) 191.543i 0.349531i
\(549\) 0 0
\(550\) 0 0
\(551\) − 155.831i − 0.282815i
\(552\) 0 0
\(553\) 582.810 1.05391
\(554\) 355.931i 0.642474i
\(555\) 0 0
\(556\) −257.435 −0.463012
\(557\) − 89.6474i − 0.160947i −0.996757 0.0804734i \(-0.974357\pi\)
0.996757 0.0804734i \(-0.0256432\pi\)
\(558\) 0 0
\(559\) 989.492 1.77011
\(560\) 575.990i 1.02855i
\(561\) 0 0
\(562\) 42.7370 0.0760445
\(563\) − 312.733i − 0.555476i −0.960657 0.277738i \(-0.910415\pi\)
0.960657 0.277738i \(-0.0895847\pi\)
\(564\) 0 0
\(565\) 1334.21 2.36143
\(566\) 347.605i 0.614142i
\(567\) 0 0
\(568\) 654.485 1.15226
\(569\) 484.269i 0.851088i 0.904938 + 0.425544i \(0.139917\pi\)
−0.904938 + 0.425544i \(0.860083\pi\)
\(570\) 0 0
\(571\) 1049.98 1.83884 0.919422 0.393272i \(-0.128657\pi\)
0.919422 + 0.393272i \(0.128657\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −928.488 −1.61757
\(575\) − 341.009i − 0.593060i
\(576\) 0 0
\(577\) 309.817 0.536945 0.268473 0.963287i \(-0.413481\pi\)
0.268473 + 0.963287i \(0.413481\pi\)
\(578\) 347.205i 0.600701i
\(579\) 0 0
\(580\) −83.3807 −0.143760
\(581\) − 201.730i − 0.347212i
\(582\) 0 0
\(583\) 0 0
\(584\) − 977.454i − 1.67372i
\(585\) 0 0
\(586\) −400.789 −0.683940
\(587\) 258.506i 0.440385i 0.975456 + 0.220192i \(0.0706685\pi\)
−0.975456 + 0.220192i \(0.929332\pi\)
\(588\) 0 0
\(589\) 760.522 1.29121
\(590\) − 445.524i − 0.755126i
\(591\) 0 0
\(592\) 311.710 0.526537
\(593\) − 601.720i − 1.01470i −0.861739 0.507352i \(-0.830624\pi\)
0.861739 0.507352i \(-0.169376\pi\)
\(594\) 0 0
\(595\) 591.365 0.993890
\(596\) 175.300i 0.294127i
\(597\) 0 0
\(598\) −716.924 −1.19887
\(599\) 168.322i 0.281005i 0.990080 + 0.140502i \(0.0448718\pi\)
−0.990080 + 0.140502i \(0.955128\pi\)
\(600\) 0 0
\(601\) 263.203 0.437942 0.218971 0.975731i \(-0.429730\pi\)
0.218971 + 0.975731i \(0.429730\pi\)
\(602\) − 996.413i − 1.65517i
\(603\) 0 0
\(604\) −95.6518 −0.158364
\(605\) 0 0
\(606\) 0 0
\(607\) −139.029 −0.229044 −0.114522 0.993421i \(-0.536534\pi\)
−0.114522 + 0.993421i \(0.536534\pi\)
\(608\) − 353.276i − 0.581046i
\(609\) 0 0
\(610\) −101.458 −0.166324
\(611\) − 673.421i − 1.10216i
\(612\) 0 0
\(613\) 724.937 1.18260 0.591302 0.806450i \(-0.298614\pi\)
0.591302 + 0.806450i \(0.298614\pi\)
\(614\) − 340.623i − 0.554760i
\(615\) 0 0
\(616\) 0 0
\(617\) − 323.005i − 0.523509i −0.965135 0.261754i \(-0.915699\pi\)
0.965135 0.261754i \(-0.0843011\pi\)
\(618\) 0 0
\(619\) 448.155 0.723999 0.362000 0.932178i \(-0.382094\pi\)
0.362000 + 0.932178i \(0.382094\pi\)
\(620\) − 406.934i − 0.656345i
\(621\) 0 0
\(622\) −447.090 −0.718795
\(623\) − 1524.65i − 2.44727i
\(624\) 0 0
\(625\) −780.051 −1.24808
\(626\) 607.816i 0.970952i
\(627\) 0 0
\(628\) 238.776 0.380217
\(629\) − 320.030i − 0.508792i
\(630\) 0 0
\(631\) 12.9974 0.0205982 0.0102991 0.999947i \(-0.496722\pi\)
0.0102991 + 0.999947i \(0.496722\pi\)
\(632\) − 455.191i − 0.720239i
\(633\) 0 0
\(634\) −717.644 −1.13193
\(635\) 97.4343i 0.153440i
\(636\) 0 0
\(637\) 1338.50 2.10126
\(638\) 0 0
\(639\) 0 0
\(640\) 143.101 0.223595
\(641\) 132.212i 0.206260i 0.994668 + 0.103130i \(0.0328857\pi\)
−0.994668 + 0.103130i \(0.967114\pi\)
\(642\) 0 0
\(643\) −1129.29 −1.75628 −0.878139 0.478405i \(-0.841215\pi\)
−0.878139 + 0.478405i \(0.841215\pi\)
\(644\) − 396.458i − 0.615617i
\(645\) 0 0
\(646\) 225.910 0.349707
\(647\) − 351.154i − 0.542741i −0.962475 0.271371i \(-0.912523\pi\)
0.962475 0.271371i \(-0.0874769\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 388.569i − 0.597798i
\(651\) 0 0
\(652\) 182.695 0.280207
\(653\) 1091.70i 1.67182i 0.548866 + 0.835910i \(0.315060\pi\)
−0.548866 + 0.835910i \(0.684940\pi\)
\(654\) 0 0
\(655\) 490.318 0.748577
\(656\) 431.164i 0.657262i
\(657\) 0 0
\(658\) −678.131 −1.03059
\(659\) 1111.39i 1.68648i 0.537534 + 0.843242i \(0.319356\pi\)
−0.537534 + 0.843242i \(0.680644\pi\)
\(660\) 0 0
\(661\) −496.270 −0.750786 −0.375393 0.926866i \(-0.622492\pi\)
−0.375393 + 0.926866i \(0.622492\pi\)
\(662\) 129.146i 0.195085i
\(663\) 0 0
\(664\) −157.557 −0.237285
\(665\) 1140.07i 1.71439i
\(666\) 0 0
\(667\) −237.424 −0.355958
\(668\) − 73.5460i − 0.110099i
\(669\) 0 0
\(670\) 582.745 0.869769
\(671\) 0 0
\(672\) 0 0
\(673\) 27.3883 0.0406959 0.0203479 0.999793i \(-0.493523\pi\)
0.0203479 + 0.999793i \(0.493523\pi\)
\(674\) − 275.893i − 0.409337i
\(675\) 0 0
\(676\) 208.979 0.309141
\(677\) − 381.717i − 0.563836i −0.959439 0.281918i \(-0.909029\pi\)
0.959439 0.281918i \(-0.0909706\pi\)
\(678\) 0 0
\(679\) −1424.05 −2.09727
\(680\) − 461.873i − 0.679225i
\(681\) 0 0
\(682\) 0 0
\(683\) − 867.261i − 1.26978i −0.772602 0.634891i \(-0.781045\pi\)
0.772602 0.634891i \(-0.218955\pi\)
\(684\) 0 0
\(685\) −839.210 −1.22512
\(686\) − 470.198i − 0.685420i
\(687\) 0 0
\(688\) −462.707 −0.672539
\(689\) − 1041.90i − 1.51219i
\(690\) 0 0
\(691\) 463.252 0.670408 0.335204 0.942146i \(-0.391195\pi\)
0.335204 + 0.942146i \(0.391195\pi\)
\(692\) 327.670i 0.473511i
\(693\) 0 0
\(694\) −750.007 −1.08070
\(695\) − 1127.90i − 1.62288i
\(696\) 0 0
\(697\) 442.673 0.635112
\(698\) 480.359i 0.688193i
\(699\) 0 0
\(700\) 214.878 0.306968
\(701\) − 577.756i − 0.824189i −0.911141 0.412094i \(-0.864797\pi\)
0.911141 0.412094i \(-0.135203\pi\)
\(702\) 0 0
\(703\) 616.973 0.877628
\(704\) 0 0
\(705\) 0 0
\(706\) −70.5516 −0.0999315
\(707\) − 63.8211i − 0.0902703i
\(708\) 0 0
\(709\) −870.594 −1.22792 −0.613959 0.789338i \(-0.710424\pi\)
−0.613959 + 0.789338i \(0.710424\pi\)
\(710\) 750.465i 1.05699i
\(711\) 0 0
\(712\) −1190.80 −1.67247
\(713\) − 1158.73i − 1.62515i
\(714\) 0 0
\(715\) 0 0
\(716\) 417.800i 0.583520i
\(717\) 0 0
\(718\) −1055.77 −1.47043
\(719\) − 533.694i − 0.742272i −0.928579 0.371136i \(-0.878968\pi\)
0.928579 0.371136i \(-0.121032\pi\)
\(720\) 0 0
\(721\) −307.891 −0.427033
\(722\) − 144.559i − 0.200220i
\(723\) 0 0
\(724\) −223.821 −0.309145
\(725\) − 128.682i − 0.177493i
\(726\) 0 0
\(727\) −749.734 −1.03127 −0.515636 0.856808i \(-0.672444\pi\)
−0.515636 + 0.856808i \(0.672444\pi\)
\(728\) − 1726.13i − 2.37105i
\(729\) 0 0
\(730\) 1120.80 1.53534
\(731\) 475.058i 0.649874i
\(732\) 0 0
\(733\) −461.131 −0.629101 −0.314550 0.949241i \(-0.601854\pi\)
−0.314550 + 0.949241i \(0.601854\pi\)
\(734\) − 638.651i − 0.870097i
\(735\) 0 0
\(736\) −538.251 −0.731320
\(737\) 0 0
\(738\) 0 0
\(739\) −730.982 −0.989150 −0.494575 0.869135i \(-0.664676\pi\)
−0.494575 + 0.869135i \(0.664676\pi\)
\(740\) − 330.125i − 0.446115i
\(741\) 0 0
\(742\) −1049.19 −1.41400
\(743\) 147.952i 0.199128i 0.995031 + 0.0995642i \(0.0317449\pi\)
−0.995031 + 0.0995642i \(0.968255\pi\)
\(744\) 0 0
\(745\) −768.044 −1.03093
\(746\) 18.0222i 0.0241585i
\(747\) 0 0
\(748\) 0 0
\(749\) 534.120i 0.713111i
\(750\) 0 0
\(751\) 1020.61 1.35901 0.679503 0.733673i \(-0.262195\pi\)
0.679503 + 0.733673i \(0.262195\pi\)
\(752\) 314.905i 0.418757i
\(753\) 0 0
\(754\) −270.537 −0.358802
\(755\) − 419.081i − 0.555074i
\(756\) 0 0
\(757\) 157.707 0.208332 0.104166 0.994560i \(-0.466783\pi\)
0.104166 + 0.994560i \(0.466783\pi\)
\(758\) − 159.955i − 0.211023i
\(759\) 0 0
\(760\) 890.426 1.17161
\(761\) 727.440i 0.955900i 0.878387 + 0.477950i \(0.158620\pi\)
−0.878387 + 0.477950i \(0.841380\pi\)
\(762\) 0 0
\(763\) −888.072 −1.16392
\(764\) 305.180i 0.399450i
\(765\) 0 0
\(766\) −263.689 −0.344242
\(767\) 793.832i 1.03498i
\(768\) 0 0
\(769\) −1249.12 −1.62434 −0.812172 0.583418i \(-0.801715\pi\)
−0.812172 + 0.583418i \(0.801715\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −340.043 −0.440470
\(773\) − 1360.02i − 1.75941i −0.475521 0.879704i \(-0.657741\pi\)
0.475521 0.879704i \(-0.342259\pi\)
\(774\) 0 0
\(775\) 628.026 0.810356
\(776\) 1112.22i 1.43327i
\(777\) 0 0
\(778\) −889.105 −1.14281
\(779\) 853.411i 1.09552i
\(780\) 0 0
\(781\) 0 0
\(782\) − 344.197i − 0.440150i
\(783\) 0 0
\(784\) −625.910 −0.798355
\(785\) 1046.15i 1.33268i
\(786\) 0 0
\(787\) 20.7010 0.0263037 0.0131518 0.999914i \(-0.495814\pi\)
0.0131518 + 0.999914i \(0.495814\pi\)
\(788\) 423.110i 0.536942i
\(789\) 0 0
\(790\) 521.945 0.660689
\(791\) 2393.91i 3.02643i
\(792\) 0 0
\(793\) 180.776 0.227965
\(794\) 30.3006i 0.0381620i
\(795\) 0 0
\(796\) −94.9532 −0.119288
\(797\) − 514.624i − 0.645701i −0.946450 0.322851i \(-0.895359\pi\)
0.946450 0.322851i \(-0.104641\pi\)
\(798\) 0 0
\(799\) 323.311 0.404645
\(800\) − 291.729i − 0.364662i
\(801\) 0 0
\(802\) 408.431 0.509265
\(803\) 0 0
\(804\) 0 0
\(805\) 1737.01 2.15777
\(806\) − 1320.34i − 1.63814i
\(807\) 0 0
\(808\) −49.8462 −0.0616908
\(809\) − 983.864i − 1.21615i −0.793880 0.608074i \(-0.791942\pi\)
0.793880 0.608074i \(-0.208058\pi\)
\(810\) 0 0
\(811\) 1460.35 1.80068 0.900340 0.435186i \(-0.143317\pi\)
0.900340 + 0.435186i \(0.143317\pi\)
\(812\) − 149.606i − 0.184244i
\(813\) 0 0
\(814\) 0 0
\(815\) 800.446i 0.982142i
\(816\) 0 0
\(817\) −915.844 −1.12098
\(818\) 857.461i 1.04824i
\(819\) 0 0
\(820\) 456.636 0.556873
\(821\) 477.870i 0.582058i 0.956714 + 0.291029i \(0.0939977\pi\)
−0.956714 + 0.291029i \(0.906002\pi\)
\(822\) 0 0
\(823\) −987.820 −1.20027 −0.600134 0.799900i \(-0.704886\pi\)
−0.600134 + 0.799900i \(0.704886\pi\)
\(824\) 240.472i 0.291835i
\(825\) 0 0
\(826\) 799.385 0.967778
\(827\) 109.527i 0.132439i 0.997805 + 0.0662193i \(0.0210937\pi\)
−0.997805 + 0.0662193i \(0.978906\pi\)
\(828\) 0 0
\(829\) 1512.47 1.82445 0.912225 0.409689i \(-0.134363\pi\)
0.912225 + 0.409689i \(0.134363\pi\)
\(830\) − 180.663i − 0.217666i
\(831\) 0 0
\(832\) −1205.11 −1.44844
\(833\) 642.618i 0.771450i
\(834\) 0 0
\(835\) 322.229 0.385902
\(836\) 0 0
\(837\) 0 0
\(838\) −555.740 −0.663174
\(839\) − 497.928i − 0.593478i −0.954959 0.296739i \(-0.904101\pi\)
0.954959 0.296739i \(-0.0958992\pi\)
\(840\) 0 0
\(841\) 751.406 0.893468
\(842\) 658.824i 0.782452i
\(843\) 0 0
\(844\) −83.4997 −0.0989332
\(845\) 915.604i 1.08355i
\(846\) 0 0
\(847\) 0 0
\(848\) 487.213i 0.574544i
\(849\) 0 0
\(850\) 186.553 0.219474
\(851\) − 940.020i − 1.10461i
\(852\) 0 0
\(853\) 234.494 0.274905 0.137452 0.990508i \(-0.456109\pi\)
0.137452 + 0.990508i \(0.456109\pi\)
\(854\) − 182.041i − 0.213163i
\(855\) 0 0
\(856\) 417.163 0.487340
\(857\) − 473.810i − 0.552870i −0.961033 0.276435i \(-0.910847\pi\)
0.961033 0.276435i \(-0.0891531\pi\)
\(858\) 0 0
\(859\) −82.9051 −0.0965135 −0.0482568 0.998835i \(-0.515367\pi\)
−0.0482568 + 0.998835i \(0.515367\pi\)
\(860\) 490.042i 0.569817i
\(861\) 0 0
\(862\) 163.868 0.190102
\(863\) − 232.504i − 0.269414i −0.990886 0.134707i \(-0.956991\pi\)
0.990886 0.134707i \(-0.0430092\pi\)
\(864\) 0 0
\(865\) −1435.63 −1.65968
\(866\) − 607.545i − 0.701553i
\(867\) 0 0
\(868\) 730.144 0.841180
\(869\) 0 0
\(870\) 0 0
\(871\) −1038.33 −1.19211
\(872\) 693.610i 0.795424i
\(873\) 0 0
\(874\) 663.563 0.759226
\(875\) − 789.786i − 0.902613i
\(876\) 0 0
\(877\) −594.018 −0.677329 −0.338665 0.940907i \(-0.609975\pi\)
−0.338665 + 0.940907i \(0.609975\pi\)
\(878\) − 955.732i − 1.08853i
\(879\) 0 0
\(880\) 0 0
\(881\) − 976.976i − 1.10894i −0.832204 0.554470i \(-0.812921\pi\)
0.832204 0.554470i \(-0.187079\pi\)
\(882\) 0 0
\(883\) 250.334 0.283503 0.141752 0.989902i \(-0.454727\pi\)
0.141752 + 0.989902i \(0.454727\pi\)
\(884\) 215.380i 0.243643i
\(885\) 0 0
\(886\) −75.8482 −0.0856075
\(887\) − 766.566i − 0.864223i −0.901820 0.432112i \(-0.857769\pi\)
0.901820 0.432112i \(-0.142231\pi\)
\(888\) 0 0
\(889\) −174.822 −0.196650
\(890\) − 1365.43i − 1.53419i
\(891\) 0 0
\(892\) −307.515 −0.344748
\(893\) 623.298i 0.697982i
\(894\) 0 0
\(895\) −1830.52 −2.04527
\(896\) 256.760i 0.286562i
\(897\) 0 0
\(898\) 978.404 1.08954
\(899\) − 437.256i − 0.486381i
\(900\) 0 0
\(901\) 500.218 0.555181
\(902\) 0 0
\(903\) 0 0
\(904\) 1869.71 2.06827
\(905\) − 980.631i − 1.08357i
\(906\) 0 0
\(907\) 1053.93 1.16199 0.580996 0.813907i \(-0.302663\pi\)
0.580996 + 0.813907i \(0.302663\pi\)
\(908\) − 179.337i − 0.197508i
\(909\) 0 0
\(910\) 1979.26 2.17501
\(911\) 643.333i 0.706183i 0.935589 + 0.353092i \(0.114870\pi\)
−0.935589 + 0.353092i \(0.885130\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 770.561i 0.843065i
\(915\) 0 0
\(916\) 41.7985 0.0456316
\(917\) 879.756i 0.959385i
\(918\) 0 0
\(919\) 33.6201 0.0365834 0.0182917 0.999833i \(-0.494177\pi\)
0.0182917 + 0.999833i \(0.494177\pi\)
\(920\) − 1356.65i − 1.47462i
\(921\) 0 0
\(922\) −597.889 −0.648470
\(923\) − 1337.17i − 1.44873i
\(924\) 0 0
\(925\) 509.486 0.550795
\(926\) 672.071i 0.725779i
\(927\) 0 0
\(928\) −203.113 −0.218872
\(929\) − 576.473i − 0.620530i −0.950650 0.310265i \(-0.899582\pi\)
0.950650 0.310265i \(-0.100418\pi\)
\(930\) 0 0
\(931\) −1238.88 −1.33069
\(932\) 334.918i 0.359354i
\(933\) 0 0
\(934\) 189.703 0.203108
\(935\) 0 0
\(936\) 0 0
\(937\) −46.1378 −0.0492399 −0.0246200 0.999697i \(-0.507838\pi\)
−0.0246200 + 0.999697i \(0.507838\pi\)
\(938\) 1045.59i 1.11471i
\(939\) 0 0
\(940\) 333.509 0.354797
\(941\) 863.034i 0.917146i 0.888657 + 0.458573i \(0.151639\pi\)
−0.888657 + 0.458573i \(0.848361\pi\)
\(942\) 0 0
\(943\) 1300.26 1.37885
\(944\) − 371.212i − 0.393233i
\(945\) 0 0
\(946\) 0 0
\(947\) − 676.033i − 0.713868i −0.934130 0.356934i \(-0.883822\pi\)
0.934130 0.356934i \(-0.116178\pi\)
\(948\) 0 0
\(949\) −1997.03 −2.10435
\(950\) 359.648i 0.378577i
\(951\) 0 0
\(952\) 828.718 0.870502
\(953\) 972.843i 1.02082i 0.859931 + 0.510411i \(0.170507\pi\)
−0.859931 + 0.510411i \(0.829493\pi\)
\(954\) 0 0
\(955\) −1337.09 −1.40009
\(956\) − 168.834i − 0.176605i
\(957\) 0 0
\(958\) −1113.00 −1.16179
\(959\) − 1505.76i − 1.57013i
\(960\) 0 0
\(961\) 1173.00 1.22060
\(962\) − 1071.12i − 1.11343i
\(963\) 0 0
\(964\) −144.833 −0.150242
\(965\) − 1489.84i − 1.54387i
\(966\) 0 0
\(967\) −1715.71 −1.77426 −0.887128 0.461523i \(-0.847303\pi\)
−0.887128 + 0.461523i \(0.847303\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1275.33 −1.31477
\(971\) 729.785i 0.751580i 0.926705 + 0.375790i \(0.122629\pi\)
−0.926705 + 0.375790i \(0.877371\pi\)
\(972\) 0 0
\(973\) 2023.75 2.07991
\(974\) − 1306.80i − 1.34169i
\(975\) 0 0
\(976\) −84.5347 −0.0866134
\(977\) 584.651i 0.598414i 0.954188 + 0.299207i \(0.0967222\pi\)
−0.954188 + 0.299207i \(0.903278\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 662.888i 0.676416i
\(981\) 0 0
\(982\) 426.188 0.434000
\(983\) 76.3328i 0.0776529i 0.999246 + 0.0388265i \(0.0123620\pi\)
−0.999246 + 0.0388265i \(0.987638\pi\)
\(984\) 0 0
\(985\) −1853.78 −1.88201
\(986\) − 129.885i − 0.131730i
\(987\) 0 0
\(988\) −415.222 −0.420266
\(989\) 1395.38i 1.41090i
\(990\) 0 0
\(991\) −1492.66 −1.50622 −0.753109 0.657895i \(-0.771447\pi\)
−0.753109 + 0.657895i \(0.771447\pi\)
\(992\) − 991.281i − 0.999275i
\(993\) 0 0
\(994\) −1346.53 −1.35465
\(995\) − 416.020i − 0.418111i
\(996\) 0 0
\(997\) 1492.32 1.49681 0.748405 0.663242i \(-0.230820\pi\)
0.748405 + 0.663242i \(0.230820\pi\)
\(998\) 791.732i 0.793319i
\(999\) 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.g.485.6 8
3.2 odd 2 inner 1089.3.b.g.485.3 8
11.10 odd 2 99.3.b.a.89.3 8
33.32 even 2 99.3.b.a.89.6 yes 8
44.43 even 2 1584.3.i.b.881.7 8
132.131 odd 2 1584.3.i.b.881.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.b.a.89.3 8 11.10 odd 2
99.3.b.a.89.6 yes 8 33.32 even 2
1089.3.b.g.485.3 8 3.2 odd 2 inner
1089.3.b.g.485.6 8 1.1 even 1 trivial
1584.3.i.b.881.2 8 132.131 odd 2
1584.3.i.b.881.7 8 44.43 even 2