Properties

Label 684.3.e.a.305.3
Level $684$
Weight $3$
Character 684.305
Analytic conductor $18.638$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 156 x^{10} + 8721 x^{8} + 208784 x^{6} + 2024760 x^{4} + 7117056 x^{2} + 6533136\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 305.3
Root \(-5.71859i\) of defining polynomial
Character \(\chi\) \(=\) 684.305
Dual form 684.3.e.a.305.10

$q$-expansion

\(f(q)\) \(=\) \(q-5.71859i q^{5} +5.91816 q^{7} +O(q^{10})\) \(q-5.71859i q^{5} +5.91816 q^{7} +2.41452i q^{11} +17.1103 q^{13} +16.1412i q^{17} +4.35890 q^{19} -15.9801i q^{23} -7.70232 q^{25} -31.2245i q^{29} -13.5110 q^{31} -33.8436i q^{35} +45.7135 q^{37} -30.3783i q^{41} -21.0735 q^{43} -43.6245i q^{47} -13.9754 q^{49} +43.5224i q^{53} +13.8077 q^{55} -86.2708i q^{59} +112.834 q^{61} -97.8468i q^{65} -41.3253 q^{67} -13.6014i q^{71} -21.7775 q^{73} +14.2895i q^{77} +71.2807 q^{79} +19.9149i q^{83} +92.3052 q^{85} -20.3030i q^{89} +101.261 q^{91} -24.9268i q^{95} +20.1043 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{7} + O(q^{10}) \) \( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 5.71859i − 1.14372i −0.820352 0.571859i \(-0.806222\pi\)
0.820352 0.571859i \(-0.193778\pi\)
\(6\) 0 0
\(7\) 5.91816 0.845452 0.422726 0.906258i \(-0.361073\pi\)
0.422726 + 0.906258i \(0.361073\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.41452i 0.219502i 0.993959 + 0.109751i \(0.0350053\pi\)
−0.993959 + 0.109751i \(0.964995\pi\)
\(12\) 0 0
\(13\) 17.1103 1.31618 0.658088 0.752941i \(-0.271365\pi\)
0.658088 + 0.752941i \(0.271365\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.1412i 0.949485i 0.880125 + 0.474742i \(0.157459\pi\)
−0.880125 + 0.474742i \(0.842541\pi\)
\(18\) 0 0
\(19\) 4.35890 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 15.9801i − 0.694786i −0.937720 0.347393i \(-0.887067\pi\)
0.937720 0.347393i \(-0.112933\pi\)
\(24\) 0 0
\(25\) −7.70232 −0.308093
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 31.2245i − 1.07671i −0.842719 0.538354i \(-0.819046\pi\)
0.842719 0.538354i \(-0.180954\pi\)
\(30\) 0 0
\(31\) −13.5110 −0.435839 −0.217920 0.975967i \(-0.569927\pi\)
−0.217920 + 0.975967i \(0.569927\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 33.8436i − 0.966959i
\(36\) 0 0
\(37\) 45.7135 1.23550 0.617750 0.786375i \(-0.288044\pi\)
0.617750 + 0.786375i \(0.288044\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 30.3783i − 0.740935i −0.928845 0.370467i \(-0.879197\pi\)
0.928845 0.370467i \(-0.120803\pi\)
\(42\) 0 0
\(43\) −21.0735 −0.490081 −0.245041 0.969513i \(-0.578801\pi\)
−0.245041 + 0.969513i \(0.578801\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 43.6245i − 0.928182i −0.885788 0.464091i \(-0.846381\pi\)
0.885788 0.464091i \(-0.153619\pi\)
\(48\) 0 0
\(49\) −13.9754 −0.285212
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 43.5224i 0.821177i 0.911821 + 0.410589i \(0.134677\pi\)
−0.911821 + 0.410589i \(0.865323\pi\)
\(54\) 0 0
\(55\) 13.8077 0.251048
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 86.2708i − 1.46222i −0.682261 0.731109i \(-0.739003\pi\)
0.682261 0.731109i \(-0.260997\pi\)
\(60\) 0 0
\(61\) 112.834 1.84974 0.924870 0.380284i \(-0.124174\pi\)
0.924870 + 0.380284i \(0.124174\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 97.8468i − 1.50534i
\(66\) 0 0
\(67\) −41.3253 −0.616796 −0.308398 0.951257i \(-0.599793\pi\)
−0.308398 + 0.951257i \(0.599793\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 13.6014i − 0.191569i −0.995402 0.0957846i \(-0.969464\pi\)
0.995402 0.0957846i \(-0.0305360\pi\)
\(72\) 0 0
\(73\) −21.7775 −0.298322 −0.149161 0.988813i \(-0.547657\pi\)
−0.149161 + 0.988813i \(0.547657\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.2895i 0.185578i
\(78\) 0 0
\(79\) 71.2807 0.902287 0.451143 0.892451i \(-0.351016\pi\)
0.451143 + 0.892451i \(0.351016\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 19.9149i 0.239939i 0.992778 + 0.119969i \(0.0382796\pi\)
−0.992778 + 0.119969i \(0.961720\pi\)
\(84\) 0 0
\(85\) 92.3052 1.08594
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 20.3030i − 0.228124i −0.993474 0.114062i \(-0.963614\pi\)
0.993474 0.114062i \(-0.0363862\pi\)
\(90\) 0 0
\(91\) 101.261 1.11276
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 24.9268i − 0.262387i
\(96\) 0 0
\(97\) 20.1043 0.207261 0.103631 0.994616i \(-0.466954\pi\)
0.103631 + 0.994616i \(0.466954\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 109.949i − 1.08860i −0.838890 0.544300i \(-0.816795\pi\)
0.838890 0.544300i \(-0.183205\pi\)
\(102\) 0 0
\(103\) 96.5263 0.937148 0.468574 0.883424i \(-0.344768\pi\)
0.468574 + 0.883424i \(0.344768\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 158.697i 1.48315i 0.670872 + 0.741574i \(0.265920\pi\)
−0.670872 + 0.741574i \(0.734080\pi\)
\(108\) 0 0
\(109\) −79.4341 −0.728753 −0.364376 0.931252i \(-0.618718\pi\)
−0.364376 + 0.931252i \(0.618718\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 27.4358i 0.242795i 0.992604 + 0.121397i \(0.0387375\pi\)
−0.992604 + 0.121397i \(0.961262\pi\)
\(114\) 0 0
\(115\) −91.3836 −0.794640
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 95.5265i 0.802743i
\(120\) 0 0
\(121\) 115.170 0.951819
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 98.9184i − 0.791347i
\(126\) 0 0
\(127\) −192.044 −1.51216 −0.756079 0.654480i \(-0.772888\pi\)
−0.756079 + 0.654480i \(0.772888\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 210.592i 1.60757i 0.594920 + 0.803785i \(0.297184\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(132\) 0 0
\(133\) 25.7967 0.193960
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 136.988i − 0.999915i −0.866050 0.499957i \(-0.833349\pi\)
0.866050 0.499957i \(-0.166651\pi\)
\(138\) 0 0
\(139\) −203.487 −1.46394 −0.731969 0.681338i \(-0.761399\pi\)
−0.731969 + 0.681338i \(0.761399\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 41.3131i 0.288903i
\(144\) 0 0
\(145\) −178.560 −1.23145
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 100.061i − 0.671552i −0.941942 0.335776i \(-0.891001\pi\)
0.941942 0.335776i \(-0.108999\pi\)
\(150\) 0 0
\(151\) −21.6662 −0.143485 −0.0717425 0.997423i \(-0.522856\pi\)
−0.0717425 + 0.997423i \(0.522856\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 77.2640i 0.498478i
\(156\) 0 0
\(157\) 150.489 0.958530 0.479265 0.877670i \(-0.340903\pi\)
0.479265 + 0.877670i \(0.340903\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 94.5727i − 0.587408i
\(162\) 0 0
\(163\) −17.2416 −0.105777 −0.0528884 0.998600i \(-0.516843\pi\)
−0.0528884 + 0.998600i \(0.516843\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 27.7925i 0.166422i 0.996532 + 0.0832110i \(0.0265176\pi\)
−0.996532 + 0.0832110i \(0.973482\pi\)
\(168\) 0 0
\(169\) 123.762 0.732319
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 30.8921i 0.178567i 0.996006 + 0.0892836i \(0.0284577\pi\)
−0.996006 + 0.0892836i \(0.971542\pi\)
\(174\) 0 0
\(175\) −45.5836 −0.260478
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 200.232i 1.11862i 0.828960 + 0.559308i \(0.188933\pi\)
−0.828960 + 0.559308i \(0.811067\pi\)
\(180\) 0 0
\(181\) −226.588 −1.25187 −0.625933 0.779877i \(-0.715282\pi\)
−0.625933 + 0.779877i \(0.715282\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 261.417i − 1.41306i
\(186\) 0 0
\(187\) −38.9733 −0.208414
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.67013i 0.0506290i 0.999680 + 0.0253145i \(0.00805871\pi\)
−0.999680 + 0.0253145i \(0.991941\pi\)
\(192\) 0 0
\(193\) 22.9769 0.119051 0.0595256 0.998227i \(-0.481041\pi\)
0.0595256 + 0.998227i \(0.481041\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 336.959i 1.71045i 0.518257 + 0.855225i \(0.326581\pi\)
−0.518257 + 0.855225i \(0.673419\pi\)
\(198\) 0 0
\(199\) 101.008 0.507580 0.253790 0.967259i \(-0.418323\pi\)
0.253790 + 0.967259i \(0.418323\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 184.792i − 0.910304i
\(204\) 0 0
\(205\) −173.721 −0.847421
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.5246i 0.0503572i
\(210\) 0 0
\(211\) 16.5805 0.0785805 0.0392902 0.999228i \(-0.487490\pi\)
0.0392902 + 0.999228i \(0.487490\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 120.511i 0.560515i
\(216\) 0 0
\(217\) −79.9604 −0.368481
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 276.181i 1.24969i
\(222\) 0 0
\(223\) −292.950 −1.31368 −0.656839 0.754031i \(-0.728107\pi\)
−0.656839 + 0.754031i \(0.728107\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 395.334i 1.74156i 0.491675 + 0.870779i \(0.336385\pi\)
−0.491675 + 0.870779i \(0.663615\pi\)
\(228\) 0 0
\(229\) 271.347 1.18492 0.592460 0.805600i \(-0.298157\pi\)
0.592460 + 0.805600i \(0.298157\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 78.1541i 0.335425i 0.985836 + 0.167713i \(0.0536381\pi\)
−0.985836 + 0.167713i \(0.946362\pi\)
\(234\) 0 0
\(235\) −249.471 −1.06158
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 376.765i 1.57642i 0.615405 + 0.788211i \(0.288992\pi\)
−0.615405 + 0.788211i \(0.711008\pi\)
\(240\) 0 0
\(241\) −428.672 −1.77872 −0.889361 0.457206i \(-0.848850\pi\)
−0.889361 + 0.457206i \(0.848850\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 79.9194i 0.326202i
\(246\) 0 0
\(247\) 74.5820 0.301951
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 329.477i 1.31266i 0.754475 + 0.656329i \(0.227892\pi\)
−0.754475 + 0.656329i \(0.772108\pi\)
\(252\) 0 0
\(253\) 38.5842 0.152507
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.5478i 0.0527150i 0.999653 + 0.0263575i \(0.00839083\pi\)
−0.999653 + 0.0263575i \(0.991609\pi\)
\(258\) 0 0
\(259\) 270.540 1.04456
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 89.8023i 0.341454i 0.985318 + 0.170727i \(0.0546115\pi\)
−0.985318 + 0.170727i \(0.945388\pi\)
\(264\) 0 0
\(265\) 248.887 0.939196
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 179.454i − 0.667114i −0.942730 0.333557i \(-0.891751\pi\)
0.942730 0.333557i \(-0.108249\pi\)
\(270\) 0 0
\(271\) 279.633 1.03186 0.515928 0.856632i \(-0.327447\pi\)
0.515928 + 0.856632i \(0.327447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 18.5974i − 0.0676270i
\(276\) 0 0
\(277\) −180.273 −0.650804 −0.325402 0.945576i \(-0.605500\pi\)
−0.325402 + 0.945576i \(0.605500\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 34.2781i 0.121986i 0.998138 + 0.0609931i \(0.0194268\pi\)
−0.998138 + 0.0609931i \(0.980573\pi\)
\(282\) 0 0
\(283\) 234.711 0.829368 0.414684 0.909965i \(-0.363892\pi\)
0.414684 + 0.909965i \(0.363892\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 179.784i − 0.626425i
\(288\) 0 0
\(289\) 28.4603 0.0984787
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 502.963i 1.71660i 0.513150 + 0.858299i \(0.328478\pi\)
−0.513150 + 0.858299i \(0.671522\pi\)
\(294\) 0 0
\(295\) −493.348 −1.67237
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 273.424i − 0.914461i
\(300\) 0 0
\(301\) −124.716 −0.414340
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 645.253i − 2.11558i
\(306\) 0 0
\(307\) −140.853 −0.458805 −0.229402 0.973332i \(-0.573677\pi\)
−0.229402 + 0.973332i \(0.573677\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 200.164i − 0.643615i −0.946805 0.321808i \(-0.895710\pi\)
0.946805 0.321808i \(-0.104290\pi\)
\(312\) 0 0
\(313\) 136.384 0.435731 0.217865 0.975979i \(-0.430091\pi\)
0.217865 + 0.975979i \(0.430091\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 136.215i 0.429700i 0.976647 + 0.214850i \(0.0689263\pi\)
−0.976647 + 0.214850i \(0.931074\pi\)
\(318\) 0 0
\(319\) 75.3923 0.236339
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 70.3580i 0.217827i
\(324\) 0 0
\(325\) −131.789 −0.405505
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 258.177i − 0.784733i
\(330\) 0 0
\(331\) −572.123 −1.72847 −0.864235 0.503089i \(-0.832197\pi\)
−0.864235 + 0.503089i \(0.832197\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 236.323i 0.705441i
\(336\) 0 0
\(337\) 466.379 1.38391 0.691957 0.721938i \(-0.256749\pi\)
0.691957 + 0.721938i \(0.256749\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 32.6226i − 0.0956675i
\(342\) 0 0
\(343\) −372.698 −1.08658
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 365.970i − 1.05467i −0.849658 0.527334i \(-0.823192\pi\)
0.849658 0.527334i \(-0.176808\pi\)
\(348\) 0 0
\(349\) 498.969 1.42971 0.714855 0.699273i \(-0.246493\pi\)
0.714855 + 0.699273i \(0.246493\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 230.666i − 0.653444i −0.945120 0.326722i \(-0.894056\pi\)
0.945120 0.326722i \(-0.105944\pi\)
\(354\) 0 0
\(355\) −77.7809 −0.219101
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 80.5301i − 0.224318i −0.993690 0.112159i \(-0.964223\pi\)
0.993690 0.112159i \(-0.0357766\pi\)
\(360\) 0 0
\(361\) 19.0000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 124.537i 0.341197i
\(366\) 0 0
\(367\) −225.261 −0.613790 −0.306895 0.951743i \(-0.599290\pi\)
−0.306895 + 0.951743i \(0.599290\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 257.573i 0.694266i
\(372\) 0 0
\(373\) −11.0567 −0.0296427 −0.0148214 0.999890i \(-0.504718\pi\)
−0.0148214 + 0.999890i \(0.504718\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 534.261i − 1.41714i
\(378\) 0 0
\(379\) −124.413 −0.328267 −0.164133 0.986438i \(-0.552483\pi\)
−0.164133 + 0.986438i \(0.552483\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 181.006i 0.472601i 0.971680 + 0.236301i \(0.0759350\pi\)
−0.971680 + 0.236301i \(0.924065\pi\)
\(384\) 0 0
\(385\) 81.7160 0.212249
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 118.699i − 0.305139i −0.988293 0.152569i \(-0.951245\pi\)
0.988293 0.152569i \(-0.0487548\pi\)
\(390\) 0 0
\(391\) 257.938 0.659689
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 407.625i − 1.03196i
\(396\) 0 0
\(397\) 307.471 0.774486 0.387243 0.921978i \(-0.373427\pi\)
0.387243 + 0.921978i \(0.373427\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 284.383i 0.709185i 0.935021 + 0.354593i \(0.115380\pi\)
−0.935021 + 0.354593i \(0.884620\pi\)
\(402\) 0 0
\(403\) −231.177 −0.573641
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 110.376i 0.271194i
\(408\) 0 0
\(409\) 172.237 0.421117 0.210559 0.977581i \(-0.432472\pi\)
0.210559 + 0.977581i \(0.432472\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 510.565i − 1.23623i
\(414\) 0 0
\(415\) 113.885 0.274422
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 149.429i 0.356633i 0.983973 + 0.178317i \(0.0570651\pi\)
−0.983973 + 0.178317i \(0.942935\pi\)
\(420\) 0 0
\(421\) 42.9576 0.102037 0.0510185 0.998698i \(-0.483753\pi\)
0.0510185 + 0.998698i \(0.483753\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 124.325i − 0.292530i
\(426\) 0 0
\(427\) 667.771 1.56387
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 400.606i 0.929480i 0.885447 + 0.464740i \(0.153852\pi\)
−0.885447 + 0.464740i \(0.846148\pi\)
\(432\) 0 0
\(433\) 428.936 0.990614 0.495307 0.868718i \(-0.335056\pi\)
0.495307 + 0.868718i \(0.335056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 69.6556i − 0.159395i
\(438\) 0 0
\(439\) −101.286 −0.230720 −0.115360 0.993324i \(-0.536802\pi\)
−0.115360 + 0.993324i \(0.536802\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 762.277i − 1.72071i −0.509691 0.860357i \(-0.670240\pi\)
0.509691 0.860357i \(-0.329760\pi\)
\(444\) 0 0
\(445\) −116.105 −0.260909
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 447.630i − 0.996949i −0.866904 0.498474i \(-0.833894\pi\)
0.866904 0.498474i \(-0.166106\pi\)
\(450\) 0 0
\(451\) 73.3491 0.162637
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 579.073i − 1.27269i
\(456\) 0 0
\(457\) 433.951 0.949565 0.474782 0.880103i \(-0.342527\pi\)
0.474782 + 0.880103i \(0.342527\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 517.149i 1.12180i 0.827884 + 0.560899i \(0.189544\pi\)
−0.827884 + 0.560899i \(0.810456\pi\)
\(462\) 0 0
\(463\) 620.085 1.33928 0.669638 0.742688i \(-0.266449\pi\)
0.669638 + 0.742688i \(0.266449\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 551.719i 1.18141i 0.806887 + 0.590705i \(0.201150\pi\)
−0.806887 + 0.590705i \(0.798850\pi\)
\(468\) 0 0
\(469\) −244.570 −0.521471
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 50.8824i − 0.107574i
\(474\) 0 0
\(475\) −33.5737 −0.0706814
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 728.217i 1.52029i 0.649755 + 0.760143i \(0.274871\pi\)
−0.649755 + 0.760143i \(0.725129\pi\)
\(480\) 0 0
\(481\) 782.171 1.62613
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 114.968i − 0.237048i
\(486\) 0 0
\(487\) −588.424 −1.20826 −0.604131 0.796885i \(-0.706480\pi\)
−0.604131 + 0.796885i \(0.706480\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 181.398i − 0.369446i −0.982791 0.184723i \(-0.940861\pi\)
0.982791 0.184723i \(-0.0591387\pi\)
\(492\) 0 0
\(493\) 504.003 1.02232
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 80.4953i − 0.161962i
\(498\) 0 0
\(499\) −360.725 −0.722895 −0.361448 0.932392i \(-0.617717\pi\)
−0.361448 + 0.932392i \(0.617717\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 875.415i − 1.74039i −0.492709 0.870194i \(-0.663993\pi\)
0.492709 0.870194i \(-0.336007\pi\)
\(504\) 0 0
\(505\) −628.752 −1.24505
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 99.8530i 0.196175i 0.995178 + 0.0980874i \(0.0312725\pi\)
−0.995178 + 0.0980874i \(0.968728\pi\)
\(510\) 0 0
\(511\) −128.883 −0.252217
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 551.995i − 1.07183i
\(516\) 0 0
\(517\) 105.332 0.203738
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 46.6446i 0.0895290i 0.998998 + 0.0447645i \(0.0142538\pi\)
−0.998998 + 0.0447645i \(0.985746\pi\)
\(522\) 0 0
\(523\) 677.022 1.29450 0.647248 0.762279i \(-0.275920\pi\)
0.647248 + 0.762279i \(0.275920\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 218.085i − 0.413823i
\(528\) 0 0
\(529\) 273.637 0.517272
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 519.782i − 0.975201i
\(534\) 0 0
\(535\) 907.522 1.69630
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 33.7438i − 0.0626045i
\(540\) 0 0
\(541\) −1007.63 −1.86252 −0.931262 0.364349i \(-0.881292\pi\)
−0.931262 + 0.364349i \(0.881292\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 454.251i 0.833488i
\(546\) 0 0
\(547\) 575.298 1.05173 0.525867 0.850567i \(-0.323741\pi\)
0.525867 + 0.850567i \(0.323741\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 136.105i − 0.247014i
\(552\) 0 0
\(553\) 421.850 0.762840
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 140.156i − 0.251626i −0.992054 0.125813i \(-0.959846\pi\)
0.992054 0.125813i \(-0.0401539\pi\)
\(558\) 0 0
\(559\) −360.574 −0.645033
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 157.113i 0.279064i 0.990218 + 0.139532i \(0.0445598\pi\)
−0.990218 + 0.139532i \(0.955440\pi\)
\(564\) 0 0
\(565\) 156.894 0.277689
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 661.449i − 1.16248i −0.813733 0.581238i \(-0.802568\pi\)
0.813733 0.581238i \(-0.197432\pi\)
\(570\) 0 0
\(571\) 496.494 0.869517 0.434759 0.900547i \(-0.356834\pi\)
0.434759 + 0.900547i \(0.356834\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 123.084i 0.214059i
\(576\) 0 0
\(577\) −700.475 −1.21399 −0.606997 0.794704i \(-0.707626\pi\)
−0.606997 + 0.794704i \(0.707626\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 117.860i 0.202857i
\(582\) 0 0
\(583\) −105.086 −0.180250
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 318.368i − 0.542364i −0.962528 0.271182i \(-0.912585\pi\)
0.962528 0.271182i \(-0.0874145\pi\)
\(588\) 0 0
\(589\) −58.8931 −0.0999884
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 340.951i 0.574959i 0.957787 + 0.287479i \(0.0928173\pi\)
−0.957787 + 0.287479i \(0.907183\pi\)
\(594\) 0 0
\(595\) 546.277 0.918113
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 993.451i − 1.65852i −0.558866 0.829258i \(-0.688764\pi\)
0.558866 0.829258i \(-0.311236\pi\)
\(600\) 0 0
\(601\) −265.068 −0.441044 −0.220522 0.975382i \(-0.570776\pi\)
−0.220522 + 0.975382i \(0.570776\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 658.611i − 1.08861i
\(606\) 0 0
\(607\) −363.853 −0.599429 −0.299714 0.954029i \(-0.596891\pi\)
−0.299714 + 0.954029i \(0.596891\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 746.428i − 1.22165i
\(612\) 0 0
\(613\) −871.617 −1.42189 −0.710944 0.703249i \(-0.751732\pi\)
−0.710944 + 0.703249i \(0.751732\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 580.947i − 0.941567i −0.882249 0.470784i \(-0.843971\pi\)
0.882249 0.470784i \(-0.156029\pi\)
\(618\) 0 0
\(619\) −766.583 −1.23842 −0.619211 0.785225i \(-0.712548\pi\)
−0.619211 + 0.785225i \(0.712548\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 120.156i − 0.192868i
\(624\) 0 0
\(625\) −758.232 −1.21317
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 737.872i 1.17309i
\(630\) 0 0
\(631\) −828.521 −1.31303 −0.656514 0.754314i \(-0.727970\pi\)
−0.656514 + 0.754314i \(0.727970\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1098.22i 1.72948i
\(636\) 0 0
\(637\) −239.123 −0.375389
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 892.171i 1.39184i 0.718118 + 0.695921i \(0.245004\pi\)
−0.718118 + 0.695921i \(0.754996\pi\)
\(642\) 0 0
\(643\) 1204.02 1.87251 0.936254 0.351323i \(-0.114268\pi\)
0.936254 + 0.351323i \(0.114268\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1181.91i 1.82675i 0.407117 + 0.913376i \(0.366534\pi\)
−0.407117 + 0.913376i \(0.633466\pi\)
\(648\) 0 0
\(649\) 208.303 0.320959
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 891.748i − 1.36562i −0.730598 0.682808i \(-0.760758\pi\)
0.730598 0.682808i \(-0.239242\pi\)
\(654\) 0 0
\(655\) 1204.29 1.83861
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1111.12i 1.68608i 0.537854 + 0.843038i \(0.319235\pi\)
−0.537854 + 0.843038i \(0.680765\pi\)
\(660\) 0 0
\(661\) −939.722 −1.42167 −0.710834 0.703360i \(-0.751682\pi\)
−0.710834 + 0.703360i \(0.751682\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 147.521i − 0.221836i
\(666\) 0 0
\(667\) −498.971 −0.748082
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 272.440i 0.406021i
\(672\) 0 0
\(673\) 57.4396 0.0853485 0.0426743 0.999089i \(-0.486412\pi\)
0.0426743 + 0.999089i \(0.486412\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 975.447i 1.44084i 0.693539 + 0.720419i \(0.256051\pi\)
−0.693539 + 0.720419i \(0.743949\pi\)
\(678\) 0 0
\(679\) 118.981 0.175229
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 933.409i − 1.36663i −0.730123 0.683316i \(-0.760537\pi\)
0.730123 0.683316i \(-0.239463\pi\)
\(684\) 0 0
\(685\) −783.381 −1.14362
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 744.681i 1.08081i
\(690\) 0 0
\(691\) 341.846 0.494712 0.247356 0.968925i \(-0.420438\pi\)
0.247356 + 0.968925i \(0.420438\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1163.66i 1.67433i
\(696\) 0 0
\(697\) 490.344 0.703506
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 796.464i 1.13618i 0.822965 + 0.568092i \(0.192318\pi\)
−0.822965 + 0.568092i \(0.807682\pi\)
\(702\) 0 0
\(703\) 199.260 0.283443
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 650.694i − 0.920359i
\(708\) 0 0
\(709\) −84.7579 −0.119546 −0.0597728 0.998212i \(-0.519038\pi\)
−0.0597728 + 0.998212i \(0.519038\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 215.907i 0.302815i
\(714\) 0 0
\(715\) 236.253 0.330424
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 55.6567i − 0.0774084i −0.999251 0.0387042i \(-0.987677\pi\)
0.999251 0.0387042i \(-0.0123230\pi\)
\(720\) 0 0
\(721\) 571.258 0.792313
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 240.501i 0.331726i
\(726\) 0 0
\(727\) −1255.51 −1.72698 −0.863489 0.504368i \(-0.831725\pi\)
−0.863489 + 0.504368i \(0.831725\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 340.152i − 0.465325i
\(732\) 0 0
\(733\) −590.908 −0.806150 −0.403075 0.915167i \(-0.632059\pi\)
−0.403075 + 0.915167i \(0.632059\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 99.7809i − 0.135388i
\(738\) 0 0
\(739\) −429.402 −0.581058 −0.290529 0.956866i \(-0.593831\pi\)
−0.290529 + 0.956866i \(0.593831\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 212.103i 0.285468i 0.989761 + 0.142734i \(0.0455893\pi\)
−0.989761 + 0.142734i \(0.954411\pi\)
\(744\) 0 0
\(745\) −572.210 −0.768067
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 939.193i 1.25393i
\(750\) 0 0
\(751\) 828.843 1.10365 0.551826 0.833959i \(-0.313931\pi\)
0.551826 + 0.833959i \(0.313931\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 123.900i 0.164106i
\(756\) 0 0
\(757\) 430.874 0.569186 0.284593 0.958648i \(-0.408142\pi\)
0.284593 + 0.958648i \(0.408142\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 192.402i − 0.252828i −0.991978 0.126414i \(-0.959653\pi\)
0.991978 0.126414i \(-0.0403468\pi\)
\(762\) 0 0
\(763\) −470.104 −0.616125
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1476.12i − 1.92454i
\(768\) 0 0
\(769\) 592.450 0.770416 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1372.66i 1.77575i 0.460081 + 0.887877i \(0.347820\pi\)
−0.460081 + 0.887877i \(0.652180\pi\)
\(774\) 0 0
\(775\) 104.066 0.134279
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 132.416i − 0.169982i
\(780\) 0 0
\(781\) 32.8409 0.0420498
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 860.587i − 1.09629i
\(786\) 0 0
\(787\) −766.155 −0.973514 −0.486757 0.873537i \(-0.661820\pi\)
−0.486757 + 0.873537i \(0.661820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 162.369i 0.205271i
\(792\) 0 0
\(793\) 1930.62 2.43458
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 340.025i 0.426631i 0.976983 + 0.213315i \(0.0684262\pi\)
−0.976983 + 0.213315i \(0.931574\pi\)
\(798\) 0 0
\(799\) 704.154 0.881294
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 52.5823i − 0.0654823i
\(804\) 0 0
\(805\) −540.823 −0.671830
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1264.36i 1.56287i 0.623984 + 0.781437i \(0.285513\pi\)
−0.623984 + 0.781437i \(0.714487\pi\)
\(810\) 0 0
\(811\) −906.142 −1.11731 −0.558657 0.829399i \(-0.688683\pi\)
−0.558657 + 0.829399i \(0.688683\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 98.5979i 0.120979i
\(816\) 0 0
\(817\) −91.8572 −0.112432
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1093.41i 1.33180i 0.746041 + 0.665900i \(0.231952\pi\)
−0.746041 + 0.665900i \(0.768048\pi\)
\(822\) 0 0
\(823\) 1356.71 1.64849 0.824245 0.566233i \(-0.191600\pi\)
0.824245 + 0.566233i \(0.191600\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1606.81i 1.94294i 0.237166 + 0.971469i \(0.423782\pi\)
−0.237166 + 0.971469i \(0.576218\pi\)
\(828\) 0 0
\(829\) −1595.51 −1.92462 −0.962312 0.271946i \(-0.912333\pi\)
−0.962312 + 0.271946i \(0.912333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 225.580i − 0.270804i
\(834\) 0 0
\(835\) 158.934 0.190340
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 614.374i − 0.732270i −0.930562 0.366135i \(-0.880681\pi\)
0.930562 0.366135i \(-0.119319\pi\)
\(840\) 0 0
\(841\) −133.971 −0.159300
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 707.744i − 0.837567i
\(846\) 0 0
\(847\) 681.595 0.804717
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 730.505i − 0.858408i
\(852\) 0 0
\(853\) −1047.21 −1.22768 −0.613839 0.789431i \(-0.710376\pi\)
−0.613839 + 0.789431i \(0.710376\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 164.267i 0.191677i 0.995397 + 0.0958385i \(0.0305532\pi\)
−0.995397 + 0.0958385i \(0.969447\pi\)
\(858\) 0 0
\(859\) −620.596 −0.722464 −0.361232 0.932476i \(-0.617644\pi\)
−0.361232 + 0.932476i \(0.617644\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 86.1854i − 0.0998672i −0.998753 0.0499336i \(-0.984099\pi\)
0.998753 0.0499336i \(-0.0159010\pi\)
\(864\) 0 0
\(865\) 176.660 0.204231
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 172.109i 0.198054i
\(870\) 0 0
\(871\) −707.088 −0.811812
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 585.415i − 0.669046i
\(876\) 0 0
\(877\) −1408.22 −1.60572 −0.802860 0.596167i \(-0.796689\pi\)
−0.802860 + 0.596167i \(0.796689\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 675.980i 0.767287i 0.923481 + 0.383643i \(0.125331\pi\)
−0.923481 + 0.383643i \(0.874669\pi\)
\(882\) 0 0
\(883\) 533.689 0.604404 0.302202 0.953244i \(-0.402278\pi\)
0.302202 + 0.953244i \(0.402278\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1403.05i 1.58179i 0.611954 + 0.790894i \(0.290384\pi\)
−0.611954 + 0.790894i \(0.709616\pi\)
\(888\) 0 0
\(889\) −1136.55 −1.27846
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 190.155i − 0.212939i
\(894\) 0 0
\(895\) 1145.05 1.27938
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 421.875i 0.469271i
\(900\) 0 0
\(901\) −702.505 −0.779695
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1295.76i 1.43178i
\(906\) 0 0
\(907\) −1071.75 −1.18165 −0.590823 0.806801i \(-0.701197\pi\)
−0.590823 + 0.806801i \(0.701197\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 706.802i − 0.775853i −0.921690 0.387926i \(-0.873191\pi\)
0.921690 0.387926i \(-0.126809\pi\)
\(912\) 0 0
\(913\) −48.0849 −0.0526670
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1246.32i 1.35912i
\(918\) 0 0
\(919\) −1120.97 −1.21978 −0.609888 0.792487i \(-0.708786\pi\)
−0.609888 + 0.792487i \(0.708786\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 232.724i − 0.252139i
\(924\) 0 0
\(925\) −352.100 −0.380649
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1470.71i − 1.58311i −0.611099 0.791554i \(-0.709272\pi\)
0.611099 0.791554i \(-0.290728\pi\)
\(930\) 0 0
\(931\) −60.9172 −0.0654320
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 222.873i 0.238367i
\(936\) 0 0
\(937\) 877.866 0.936890 0.468445 0.883493i \(-0.344814\pi\)
0.468445 + 0.883493i \(0.344814\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1367.92i 1.45369i 0.686804 + 0.726843i \(0.259013\pi\)
−0.686804 + 0.726843i \(0.740987\pi\)
\(942\) 0 0
\(943\) −485.448 −0.514791
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 254.710i 0.268965i 0.990916 + 0.134483i \(0.0429372\pi\)
−0.990916 + 0.134483i \(0.957063\pi\)
\(948\) 0 0
\(949\) −372.620 −0.392645
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1347.59i − 1.41405i −0.707188 0.707025i \(-0.750037\pi\)
0.707188 0.707025i \(-0.249963\pi\)
\(954\) 0 0
\(955\) 55.2996 0.0579053
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 810.719i − 0.845380i
\(960\) 0 0
\(961\) −778.452 −0.810044
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 131.395i − 0.136161i
\(966\) 0 0
\(967\) −1086.76 −1.12385 −0.561924 0.827189i \(-0.689939\pi\)
−0.561924 + 0.827189i \(0.689939\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1004.02i − 1.03401i −0.855983 0.517005i \(-0.827047\pi\)
0.855983 0.517005i \(-0.172953\pi\)
\(972\) 0 0
\(973\) −1204.27 −1.23769
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 765.973i 0.784005i 0.919964 + 0.392002i \(0.128218\pi\)
−0.919964 + 0.392002i \(0.871782\pi\)
\(978\) 0 0
\(979\) 49.0220 0.0500736
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1779.64i − 1.81042i −0.424965 0.905210i \(-0.639714\pi\)
0.424965 0.905210i \(-0.360286\pi\)
\(984\) 0 0
\(985\) 1926.93 1.95627
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 336.756i 0.340502i
\(990\) 0 0
\(991\) 126.056 0.127201 0.0636004 0.997975i \(-0.479742\pi\)
0.0636004 + 0.997975i \(0.479742\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 577.626i − 0.580529i
\(996\) 0 0
\(997\) 587.955 0.589724 0.294862 0.955540i \(-0.404726\pi\)
0.294862 + 0.955540i \(0.404726\pi\)
\(998\) 0 0
\(999\) 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 684.3.e.a.305.3 12
3.2 odd 2 inner 684.3.e.a.305.10 yes 12
4.3 odd 2 2736.3.h.c.305.3 12
12.11 even 2 2736.3.h.c.305.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.e.a.305.3 12 1.1 even 1 trivial
684.3.e.a.305.10 yes 12 3.2 odd 2 inner
2736.3.h.c.305.3 12 4.3 odd 2
2736.3.h.c.305.10 12 12.11 even 2