Properties

Label 684.3.e.a.305.9
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 305.9
Root \(3.29597i\) of defining polynomial
Character \(\chi\) \(=\) 684.305
Dual form 684.3.e.a.305.4

$q$-expansion

\(f(q)\) \(=\) \(q+3.29597i q^{5} -2.46307 q^{7} +O(q^{10})\) \(q+3.29597i q^{5} -2.46307 q^{7} -10.4157i q^{11} +2.93243 q^{13} +27.8783i q^{17} -4.35890 q^{19} +24.3461i q^{23} +14.1366 q^{25} +7.80708i q^{29} -17.4419 q^{31} -8.11820i q^{35} -48.3596 q^{37} +51.2260i q^{41} -82.7891 q^{43} -22.2146i q^{47} -42.9333 q^{49} +16.6275i q^{53} +34.3300 q^{55} +49.5384i q^{59} +16.9475 q^{61} +9.66522i q^{65} -1.05167 q^{67} +79.0160i q^{71} -53.4664 q^{73} +25.6547i q^{77} +137.382 q^{79} -4.56292i q^{83} -91.8862 q^{85} +120.627i q^{89} -7.22278 q^{91} -14.3668i q^{95} +13.4947 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{7} + O(q^{10}) \) \( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97} + O(q^{100}) \)

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
\(5\) 3.29597i 0.659195i 0.944122 + 0.329597i \(0.106913\pi\)
−0.944122 + 0.329597i \(0.893087\pi\)
\(6\) 0 0
\(7\) −2.46307 −0.351867 −0.175933 0.984402i \(-0.556294\pi\)
−0.175933 + 0.984402i \(0.556294\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 10.4157i − 0.946886i −0.880825 0.473443i \(-0.843011\pi\)
0.880825 0.473443i \(-0.156989\pi\)
\(12\) 0 0
\(13\) 2.93243 0.225572 0.112786 0.993619i \(-0.464023\pi\)
0.112786 + 0.993619i \(0.464023\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 27.8783i 1.63990i 0.572433 + 0.819951i \(0.306000\pi\)
−0.572433 + 0.819951i \(0.694000\pi\)
\(18\) 0 0
\(19\) −4.35890 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 24.3461i 1.05853i 0.848458 + 0.529263i \(0.177532\pi\)
−0.848458 + 0.529263i \(0.822468\pi\)
\(24\) 0 0
\(25\) 14.1366 0.565463
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.80708i 0.269210i 0.990899 + 0.134605i \(0.0429765\pi\)
−0.990899 + 0.134605i \(0.957024\pi\)
\(30\) 0 0
\(31\) −17.4419 −0.562642 −0.281321 0.959614i \(-0.590773\pi\)
−0.281321 + 0.959614i \(0.590773\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 8.11820i − 0.231949i
\(36\) 0 0
\(37\) −48.3596 −1.30702 −0.653508 0.756919i \(-0.726704\pi\)
−0.653508 + 0.756919i \(0.726704\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 51.2260i 1.24942i 0.780859 + 0.624708i \(0.214782\pi\)
−0.780859 + 0.624708i \(0.785218\pi\)
\(42\) 0 0
\(43\) −82.7891 −1.92533 −0.962664 0.270700i \(-0.912745\pi\)
−0.962664 + 0.270700i \(0.912745\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 22.2146i − 0.472651i −0.971674 0.236325i \(-0.924057\pi\)
0.971674 0.236325i \(-0.0759432\pi\)
\(48\) 0 0
\(49\) −42.9333 −0.876190
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 16.6275i 0.313726i 0.987620 + 0.156863i \(0.0501382\pi\)
−0.987620 + 0.156863i \(0.949862\pi\)
\(54\) 0 0
\(55\) 34.3300 0.624182
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 49.5384i 0.839634i 0.907609 + 0.419817i \(0.137906\pi\)
−0.907609 + 0.419817i \(0.862094\pi\)
\(60\) 0 0
\(61\) 16.9475 0.277828 0.138914 0.990304i \(-0.455639\pi\)
0.138914 + 0.990304i \(0.455639\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.66522i 0.148696i
\(66\) 0 0
\(67\) −1.05167 −0.0156965 −0.00784826 0.999969i \(-0.502498\pi\)
−0.00784826 + 0.999969i \(0.502498\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 79.0160i 1.11290i 0.830881 + 0.556451i \(0.187837\pi\)
−0.830881 + 0.556451i \(0.812163\pi\)
\(72\) 0 0
\(73\) −53.4664 −0.732417 −0.366208 0.930533i \(-0.619344\pi\)
−0.366208 + 0.930533i \(0.619344\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 25.6547i 0.333178i
\(78\) 0 0
\(79\) 137.382 1.73902 0.869508 0.493919i \(-0.164436\pi\)
0.869508 + 0.493919i \(0.164436\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 4.56292i − 0.0549750i −0.999622 0.0274875i \(-0.991249\pi\)
0.999622 0.0274875i \(-0.00875064\pi\)
\(84\) 0 0
\(85\) −91.8862 −1.08101
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 120.627i 1.35536i 0.735358 + 0.677679i \(0.237014\pi\)
−0.735358 + 0.677679i \(0.762986\pi\)
\(90\) 0 0
\(91\) −7.22278 −0.0793712
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 14.3668i − 0.151230i
\(96\) 0 0
\(97\) 13.4947 0.139121 0.0695604 0.997578i \(-0.477840\pi\)
0.0695604 + 0.997578i \(0.477840\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 101.454i 1.00449i 0.864724 + 0.502247i \(0.167493\pi\)
−0.864724 + 0.502247i \(0.832507\pi\)
\(102\) 0 0
\(103\) −141.381 −1.37263 −0.686313 0.727306i \(-0.740772\pi\)
−0.686313 + 0.727306i \(0.740772\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 21.6171i − 0.202029i −0.994885 0.101014i \(-0.967791\pi\)
0.994885 0.101014i \(-0.0322088\pi\)
\(108\) 0 0
\(109\) 30.9232 0.283699 0.141850 0.989888i \(-0.454695\pi\)
0.141850 + 0.989888i \(0.454695\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 154.363i − 1.36605i −0.730397 0.683023i \(-0.760665\pi\)
0.730397 0.683023i \(-0.239335\pi\)
\(114\) 0 0
\(115\) −80.2441 −0.697775
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 68.6662i − 0.577027i
\(120\) 0 0
\(121\) 12.5123 0.103407
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 128.993i 1.03194i
\(126\) 0 0
\(127\) 71.0621 0.559544 0.279772 0.960066i \(-0.409741\pi\)
0.279772 + 0.960066i \(0.409741\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 61.8603i − 0.472216i −0.971727 0.236108i \(-0.924128\pi\)
0.971727 0.236108i \(-0.0758719\pi\)
\(132\) 0 0
\(133\) 10.7363 0.0807238
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 86.2875i 0.629836i 0.949119 + 0.314918i \(0.101977\pi\)
−0.949119 + 0.314918i \(0.898023\pi\)
\(138\) 0 0
\(139\) 173.550 1.24856 0.624280 0.781201i \(-0.285393\pi\)
0.624280 + 0.781201i \(0.285393\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 30.5435i − 0.213591i
\(144\) 0 0
\(145\) −25.7319 −0.177461
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 46.9468i − 0.315079i −0.987513 0.157540i \(-0.949644\pi\)
0.987513 0.157540i \(-0.0503562\pi\)
\(150\) 0 0
\(151\) −1.01794 −0.00674136 −0.00337068 0.999994i \(-0.501073\pi\)
−0.00337068 + 0.999994i \(0.501073\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 57.4880i − 0.370890i
\(156\) 0 0
\(157\) 6.17959 0.0393604 0.0196802 0.999806i \(-0.493735\pi\)
0.0196802 + 0.999806i \(0.493735\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 59.9661i − 0.372460i
\(162\) 0 0
\(163\) 37.5244 0.230211 0.115106 0.993353i \(-0.463279\pi\)
0.115106 + 0.993353i \(0.463279\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 146.277i − 0.875907i −0.898998 0.437954i \(-0.855703\pi\)
0.898998 0.437954i \(-0.144297\pi\)
\(168\) 0 0
\(169\) −160.401 −0.949117
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 43.4528i − 0.251172i −0.992083 0.125586i \(-0.959919\pi\)
0.992083 0.125586i \(-0.0400811\pi\)
\(174\) 0 0
\(175\) −34.8193 −0.198968
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 99.6202i − 0.556537i −0.960503 0.278269i \(-0.910239\pi\)
0.960503 0.278269i \(-0.0897606\pi\)
\(180\) 0 0
\(181\) 294.049 1.62458 0.812291 0.583252i \(-0.198220\pi\)
0.812291 + 0.583252i \(0.198220\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 159.392i − 0.861578i
\(186\) 0 0
\(187\) 290.374 1.55280
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 149.910i − 0.784869i −0.919780 0.392435i \(-0.871633\pi\)
0.919780 0.392435i \(-0.128367\pi\)
\(192\) 0 0
\(193\) −376.836 −1.95252 −0.976259 0.216608i \(-0.930501\pi\)
−0.976259 + 0.216608i \(0.930501\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 100.556i − 0.510439i −0.966883 0.255219i \(-0.917852\pi\)
0.966883 0.255219i \(-0.0821477\pi\)
\(198\) 0 0
\(199\) 150.210 0.754822 0.377411 0.926046i \(-0.376814\pi\)
0.377411 + 0.926046i \(0.376814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 19.2294i − 0.0947259i
\(204\) 0 0
\(205\) −168.840 −0.823608
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 45.4012i 0.217231i
\(210\) 0 0
\(211\) 23.1583 0.109755 0.0548774 0.998493i \(-0.482523\pi\)
0.0548774 + 0.998493i \(0.482523\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 272.871i − 1.26917i
\(216\) 0 0
\(217\) 42.9606 0.197975
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 81.7514i 0.369916i
\(222\) 0 0
\(223\) 438.633 1.96696 0.983481 0.181010i \(-0.0579365\pi\)
0.983481 + 0.181010i \(0.0579365\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 123.454i − 0.543850i −0.962318 0.271925i \(-0.912340\pi\)
0.962318 0.271925i \(-0.0876603\pi\)
\(228\) 0 0
\(229\) 159.655 0.697182 0.348591 0.937275i \(-0.386660\pi\)
0.348591 + 0.937275i \(0.386660\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 257.960i 1.10712i 0.832808 + 0.553561i \(0.186732\pi\)
−0.832808 + 0.553561i \(0.813268\pi\)
\(234\) 0 0
\(235\) 73.2187 0.311569
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.7567i 0.0659276i 0.999457 + 0.0329638i \(0.0104946\pi\)
−0.999457 + 0.0329638i \(0.989505\pi\)
\(240\) 0 0
\(241\) −29.0324 −0.120466 −0.0602331 0.998184i \(-0.519184\pi\)
−0.0602331 + 0.998184i \(0.519184\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 141.507i − 0.577579i
\(246\) 0 0
\(247\) −12.7822 −0.0517497
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 72.1205i − 0.287333i −0.989626 0.143666i \(-0.954111\pi\)
0.989626 0.143666i \(-0.0458892\pi\)
\(252\) 0 0
\(253\) 253.583 1.00230
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 252.576i 0.982787i 0.870938 + 0.491393i \(0.163512\pi\)
−0.870938 + 0.491393i \(0.836488\pi\)
\(258\) 0 0
\(259\) 119.113 0.459896
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 47.7320i 0.181491i 0.995874 + 0.0907453i \(0.0289249\pi\)
−0.995874 + 0.0907453i \(0.971075\pi\)
\(264\) 0 0
\(265\) −54.8038 −0.206807
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 443.061i − 1.64707i −0.567266 0.823534i \(-0.691999\pi\)
0.567266 0.823534i \(-0.308001\pi\)
\(270\) 0 0
\(271\) 101.973 0.376286 0.188143 0.982142i \(-0.439753\pi\)
0.188143 + 0.982142i \(0.439753\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 147.243i − 0.535429i
\(276\) 0 0
\(277\) 193.897 0.699989 0.349995 0.936752i \(-0.386183\pi\)
0.349995 + 0.936752i \(0.386183\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 287.031i − 1.02146i −0.859741 0.510731i \(-0.829375\pi\)
0.859741 0.510731i \(-0.170625\pi\)
\(282\) 0 0
\(283\) −253.702 −0.896474 −0.448237 0.893915i \(-0.647948\pi\)
−0.448237 + 0.893915i \(0.647948\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 126.173i − 0.439628i
\(288\) 0 0
\(289\) −488.202 −1.68928
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 563.387i 1.92282i 0.275115 + 0.961411i \(0.411284\pi\)
−0.275115 + 0.961411i \(0.588716\pi\)
\(294\) 0 0
\(295\) −163.277 −0.553482
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 71.3934i 0.238774i
\(300\) 0 0
\(301\) 203.915 0.677459
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 55.8586i 0.183143i
\(306\) 0 0
\(307\) 224.823 0.732321 0.366161 0.930552i \(-0.380672\pi\)
0.366161 + 0.930552i \(0.380672\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 495.020i − 1.59171i −0.605490 0.795853i \(-0.707023\pi\)
0.605490 0.795853i \(-0.292977\pi\)
\(312\) 0 0
\(313\) 317.555 1.01455 0.507276 0.861784i \(-0.330653\pi\)
0.507276 + 0.861784i \(0.330653\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 290.705i 0.917051i 0.888681 + 0.458525i \(0.151622\pi\)
−0.888681 + 0.458525i \(0.848378\pi\)
\(318\) 0 0
\(319\) 81.3165 0.254911
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 121.519i − 0.376219i
\(324\) 0 0
\(325\) 41.4545 0.127552
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 54.7160i 0.166310i
\(330\) 0 0
\(331\) 3.55596 0.0107431 0.00537154 0.999986i \(-0.498290\pi\)
0.00537154 + 0.999986i \(0.498290\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 3.46626i − 0.0103471i
\(336\) 0 0
\(337\) −447.418 −1.32765 −0.663825 0.747888i \(-0.731068\pi\)
−0.663825 + 0.747888i \(0.731068\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 181.670i 0.532757i
\(342\) 0 0
\(343\) 226.438 0.660169
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 154.029i − 0.443889i −0.975059 0.221944i \(-0.928760\pi\)
0.975059 0.221944i \(-0.0712403\pi\)
\(348\) 0 0
\(349\) −356.174 −1.02056 −0.510278 0.860009i \(-0.670458\pi\)
−0.510278 + 0.860009i \(0.670458\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 251.734i 0.713129i 0.934271 + 0.356564i \(0.116052\pi\)
−0.934271 + 0.356564i \(0.883948\pi\)
\(354\) 0 0
\(355\) −260.435 −0.733618
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 488.041i − 1.35944i −0.733470 0.679722i \(-0.762100\pi\)
0.733470 0.679722i \(-0.237900\pi\)
\(360\) 0 0
\(361\) 19.0000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 176.224i − 0.482805i
\(366\) 0 0
\(367\) −281.123 −0.766002 −0.383001 0.923748i \(-0.625109\pi\)
−0.383001 + 0.923748i \(0.625109\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 40.9547i − 0.110390i
\(372\) 0 0
\(373\) −343.161 −0.920001 −0.460001 0.887919i \(-0.652151\pi\)
−0.460001 + 0.887919i \(0.652151\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.8937i 0.0607261i
\(378\) 0 0
\(379\) 410.802 1.08391 0.541955 0.840408i \(-0.317684\pi\)
0.541955 + 0.840408i \(0.317684\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 370.444i 0.967217i 0.875285 + 0.483608i \(0.160674\pi\)
−0.875285 + 0.483608i \(0.839326\pi\)
\(384\) 0 0
\(385\) −84.5571 −0.219629
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 389.355i − 1.00091i −0.865762 0.500456i \(-0.833166\pi\)
0.865762 0.500456i \(-0.166834\pi\)
\(390\) 0 0
\(391\) −678.729 −1.73588
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 452.808i 1.14635i
\(396\) 0 0
\(397\) −25.0875 −0.0631928 −0.0315964 0.999501i \(-0.510059\pi\)
−0.0315964 + 0.999501i \(0.510059\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 602.238i 1.50184i 0.660393 + 0.750920i \(0.270390\pi\)
−0.660393 + 0.750920i \(0.729610\pi\)
\(402\) 0 0
\(403\) −51.1472 −0.126916
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 503.701i 1.23760i
\(408\) 0 0
\(409\) −47.6877 −0.116596 −0.0582979 0.998299i \(-0.518567\pi\)
−0.0582979 + 0.998299i \(0.518567\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 122.016i − 0.295439i
\(414\) 0 0
\(415\) 15.0393 0.0362392
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 19.3011i − 0.0460647i −0.999735 0.0230323i \(-0.992668\pi\)
0.999735 0.0230323i \(-0.00733207\pi\)
\(420\) 0 0
\(421\) 81.7466 0.194172 0.0970862 0.995276i \(-0.469048\pi\)
0.0970862 + 0.995276i \(0.469048\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 394.104i 0.927304i
\(426\) 0 0
\(427\) −41.7429 −0.0977586
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 675.971i 1.56838i 0.620522 + 0.784189i \(0.286921\pi\)
−0.620522 + 0.784189i \(0.713079\pi\)
\(432\) 0 0
\(433\) 391.222 0.903515 0.451758 0.892141i \(-0.350797\pi\)
0.451758 + 0.892141i \(0.350797\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 106.122i − 0.242843i
\(438\) 0 0
\(439\) 120.944 0.275498 0.137749 0.990467i \(-0.456013\pi\)
0.137749 + 0.990467i \(0.456013\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 65.2480i 0.147287i 0.997285 + 0.0736433i \(0.0234626\pi\)
−0.997285 + 0.0736433i \(0.976537\pi\)
\(444\) 0 0
\(445\) −397.583 −0.893444
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 486.760i 1.08410i 0.840347 + 0.542049i \(0.182351\pi\)
−0.840347 + 0.542049i \(0.817649\pi\)
\(450\) 0 0
\(451\) 533.557 1.18305
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 23.8061i − 0.0523211i
\(456\) 0 0
\(457\) 523.679 1.14591 0.572953 0.819588i \(-0.305798\pi\)
0.572953 + 0.819588i \(0.305798\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 142.495i − 0.309101i −0.987985 0.154550i \(-0.950607\pi\)
0.987985 0.154550i \(-0.0493929\pi\)
\(462\) 0 0
\(463\) −240.182 −0.518752 −0.259376 0.965776i \(-0.583517\pi\)
−0.259376 + 0.965776i \(0.583517\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 264.429i 0.566230i 0.959086 + 0.283115i \(0.0913678\pi\)
−0.959086 + 0.283115i \(0.908632\pi\)
\(468\) 0 0
\(469\) 2.59033 0.00552308
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 862.310i 1.82307i
\(474\) 0 0
\(475\) −61.6199 −0.129726
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 723.453i 1.51034i 0.655529 + 0.755170i \(0.272446\pi\)
−0.655529 + 0.755170i \(0.727554\pi\)
\(480\) 0 0
\(481\) −141.811 −0.294826
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 44.4782i 0.0917077i
\(486\) 0 0
\(487\) −728.561 −1.49602 −0.748009 0.663688i \(-0.768990\pi\)
−0.748009 + 0.663688i \(0.768990\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 888.765i − 1.81011i −0.425293 0.905056i \(-0.639829\pi\)
0.425293 0.905056i \(-0.360171\pi\)
\(492\) 0 0
\(493\) −217.648 −0.441477
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 194.622i − 0.391593i
\(498\) 0 0
\(499\) −740.968 −1.48491 −0.742453 0.669899i \(-0.766338\pi\)
−0.742453 + 0.669899i \(0.766338\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 195.451i − 0.388571i −0.980945 0.194285i \(-0.937761\pi\)
0.980945 0.194285i \(-0.0622388\pi\)
\(504\) 0 0
\(505\) −334.389 −0.662156
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 232.031i 0.455857i 0.973678 + 0.227929i \(0.0731953\pi\)
−0.973678 + 0.227929i \(0.926805\pi\)
\(510\) 0 0
\(511\) 131.691 0.257713
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 465.986i − 0.904828i
\(516\) 0 0
\(517\) −231.382 −0.447546
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 313.989i 0.602666i 0.953519 + 0.301333i \(0.0974316\pi\)
−0.953519 + 0.301333i \(0.902568\pi\)
\(522\) 0 0
\(523\) −414.025 −0.791635 −0.395817 0.918329i \(-0.629539\pi\)
−0.395817 + 0.918329i \(0.629539\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 486.251i − 0.922678i
\(528\) 0 0
\(529\) −63.7334 −0.120479
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 150.217i 0.281833i
\(534\) 0 0
\(535\) 71.2493 0.133176
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 447.182i 0.829652i
\(540\) 0 0
\(541\) 569.365 1.05243 0.526215 0.850351i \(-0.323611\pi\)
0.526215 + 0.850351i \(0.323611\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 101.922i 0.187013i
\(546\) 0 0
\(547\) 527.255 0.963902 0.481951 0.876198i \(-0.339928\pi\)
0.481951 + 0.876198i \(0.339928\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 34.0303i − 0.0617609i
\(552\) 0 0
\(553\) −338.382 −0.611902
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1092.85i − 1.96203i −0.193930 0.981015i \(-0.562123\pi\)
0.193930 0.981015i \(-0.437877\pi\)
\(558\) 0 0
\(559\) −242.774 −0.434300
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 662.414i 1.17658i 0.808650 + 0.588290i \(0.200199\pi\)
−0.808650 + 0.588290i \(0.799801\pi\)
\(564\) 0 0
\(565\) 508.777 0.900490
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 23.5128i − 0.0413230i −0.999787 0.0206615i \(-0.993423\pi\)
0.999787 0.0206615i \(-0.00657723\pi\)
\(570\) 0 0
\(571\) −531.954 −0.931618 −0.465809 0.884885i \(-0.654237\pi\)
−0.465809 + 0.884885i \(0.654237\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 344.170i 0.598557i
\(576\) 0 0
\(577\) 441.929 0.765909 0.382954 0.923767i \(-0.374907\pi\)
0.382954 + 0.923767i \(0.374907\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 11.2388i 0.0193439i
\(582\) 0 0
\(583\) 173.188 0.297063
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 716.074i 1.21989i 0.792445 + 0.609944i \(0.208808\pi\)
−0.792445 + 0.609944i \(0.791192\pi\)
\(588\) 0 0
\(589\) 76.0275 0.129079
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 598.295i 1.00893i 0.863432 + 0.504465i \(0.168310\pi\)
−0.863432 + 0.504465i \(0.831690\pi\)
\(594\) 0 0
\(595\) 226.322 0.380373
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 600.674i 1.00279i 0.865217 + 0.501397i \(0.167181\pi\)
−0.865217 + 0.501397i \(0.832819\pi\)
\(600\) 0 0
\(601\) −663.305 −1.10367 −0.551834 0.833954i \(-0.686072\pi\)
−0.551834 + 0.833954i \(0.686072\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 41.2401i 0.0681655i
\(606\) 0 0
\(607\) 89.6567 0.147705 0.0738523 0.997269i \(-0.476471\pi\)
0.0738523 + 0.997269i \(0.476471\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 65.1428i − 0.106617i
\(612\) 0 0
\(613\) −129.918 −0.211938 −0.105969 0.994369i \(-0.533794\pi\)
−0.105969 + 0.994369i \(0.533794\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 262.827i − 0.425975i −0.977055 0.212988i \(-0.931681\pi\)
0.977055 0.212988i \(-0.0683194\pi\)
\(618\) 0 0
\(619\) −536.855 −0.867293 −0.433647 0.901083i \(-0.642773\pi\)
−0.433647 + 0.901083i \(0.642773\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 297.112i − 0.476905i
\(624\) 0 0
\(625\) −71.7434 −0.114789
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1348.19i − 2.14338i
\(630\) 0 0
\(631\) 1013.65 1.60641 0.803207 0.595700i \(-0.203125\pi\)
0.803207 + 0.595700i \(0.203125\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 234.219i 0.368848i
\(636\) 0 0
\(637\) −125.899 −0.197644
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 801.720i − 1.25073i −0.780331 0.625367i \(-0.784949\pi\)
0.780331 0.625367i \(-0.215051\pi\)
\(642\) 0 0
\(643\) −309.838 −0.481863 −0.240931 0.970542i \(-0.577453\pi\)
−0.240931 + 0.970542i \(0.577453\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 829.721i − 1.28241i −0.767369 0.641206i \(-0.778434\pi\)
0.767369 0.641206i \(-0.221566\pi\)
\(648\) 0 0
\(649\) 515.979 0.795037
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 785.167i − 1.20240i −0.799099 0.601200i \(-0.794689\pi\)
0.799099 0.601200i \(-0.205311\pi\)
\(654\) 0 0
\(655\) 203.890 0.311282
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1098.52i 1.66695i 0.552558 + 0.833474i \(0.313652\pi\)
−0.552558 + 0.833474i \(0.686348\pi\)
\(660\) 0 0
\(661\) −294.514 −0.445558 −0.222779 0.974869i \(-0.571513\pi\)
−0.222779 + 0.974869i \(0.571513\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 35.3864i 0.0532127i
\(666\) 0 0
\(667\) −190.072 −0.284966
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 176.521i − 0.263072i
\(672\) 0 0
\(673\) −924.048 −1.37303 −0.686514 0.727117i \(-0.740860\pi\)
−0.686514 + 0.727117i \(0.740860\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 615.975i 0.909860i 0.890527 + 0.454930i \(0.150336\pi\)
−0.890527 + 0.454930i \(0.849664\pi\)
\(678\) 0 0
\(679\) −33.2384 −0.0489520
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 49.5710i 0.0725783i 0.999341 + 0.0362891i \(0.0115537\pi\)
−0.999341 + 0.0362891i \(0.988446\pi\)
\(684\) 0 0
\(685\) −284.401 −0.415184
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 48.7590i 0.0707678i
\(690\) 0 0
\(691\) 1306.23 1.89035 0.945175 0.326564i \(-0.105891\pi\)
0.945175 + 0.326564i \(0.105891\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 572.015i 0.823044i
\(696\) 0 0
\(697\) −1428.10 −2.04892
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 814.490i − 1.16190i −0.813940 0.580949i \(-0.802682\pi\)
0.813940 0.580949i \(-0.197318\pi\)
\(702\) 0 0
\(703\) 210.795 0.299850
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 249.888i − 0.353448i
\(708\) 0 0
\(709\) 873.275 1.23170 0.615849 0.787864i \(-0.288813\pi\)
0.615849 + 0.787864i \(0.288813\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 424.642i − 0.595571i
\(714\) 0 0
\(715\) 100.670 0.140798
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 448.892i − 0.624328i −0.950028 0.312164i \(-0.898946\pi\)
0.950028 0.312164i \(-0.101054\pi\)
\(720\) 0 0
\(721\) 348.230 0.482982
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 110.365i 0.152228i
\(726\) 0 0
\(727\) 772.702 1.06286 0.531432 0.847101i \(-0.321654\pi\)
0.531432 + 0.847101i \(0.321654\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 2308.02i − 3.15735i
\(732\) 0 0
\(733\) 544.850 0.743316 0.371658 0.928370i \(-0.378789\pi\)
0.371658 + 0.928370i \(0.378789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.9539i 0.0148628i
\(738\) 0 0
\(739\) 840.323 1.13711 0.568554 0.822646i \(-0.307503\pi\)
0.568554 + 0.822646i \(0.307503\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1361.90i 1.83297i 0.400070 + 0.916485i \(0.368986\pi\)
−0.400070 + 0.916485i \(0.631014\pi\)
\(744\) 0 0
\(745\) 154.735 0.207699
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 53.2443i 0.0710872i
\(750\) 0 0
\(751\) −850.123 −1.13199 −0.565994 0.824409i \(-0.691507\pi\)
−0.565994 + 0.824409i \(0.691507\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 3.35512i − 0.00444387i
\(756\) 0 0
\(757\) 1375.84 1.81750 0.908748 0.417345i \(-0.137039\pi\)
0.908748 + 0.417345i \(0.137039\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 562.980i − 0.739790i −0.929074 0.369895i \(-0.879394\pi\)
0.929074 0.369895i \(-0.120606\pi\)
\(762\) 0 0
\(763\) −76.1659 −0.0998243
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 145.268i 0.189398i
\(768\) 0 0
\(769\) 1034.71 1.34552 0.672760 0.739861i \(-0.265109\pi\)
0.672760 + 0.739861i \(0.265109\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 144.532i − 0.186975i −0.995620 0.0934875i \(-0.970198\pi\)
0.995620 0.0934875i \(-0.0298015\pi\)
\(774\) 0 0
\(775\) −246.568 −0.318153
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 223.289i − 0.286636i
\(780\) 0 0
\(781\) 823.010 1.05379
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20.3677i 0.0259462i
\(786\) 0 0
\(787\) −589.006 −0.748419 −0.374210 0.927344i \(-0.622086\pi\)
−0.374210 + 0.927344i \(0.622086\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 380.207i 0.480666i
\(792\) 0 0
\(793\) 49.6975 0.0626703
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1241.83i − 1.55813i −0.626944 0.779064i \(-0.715695\pi\)
0.626944 0.779064i \(-0.284305\pi\)
\(798\) 0 0
\(799\) 619.306 0.775101
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 556.893i 0.693515i
\(804\) 0 0
\(805\) 197.647 0.245524
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1350.09i 1.66883i 0.551135 + 0.834416i \(0.314195\pi\)
−0.551135 + 0.834416i \(0.685805\pi\)
\(810\) 0 0
\(811\) −1180.46 −1.45555 −0.727777 0.685813i \(-0.759447\pi\)
−0.727777 + 0.685813i \(0.759447\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 123.680i 0.151754i
\(816\) 0 0
\(817\) 360.869 0.441700
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.71685i 0.00209117i 0.999999 + 0.00104558i \(0.000332820\pi\)
−0.999999 + 0.00104558i \(0.999667\pi\)
\(822\) 0 0
\(823\) −243.690 −0.296099 −0.148050 0.988980i \(-0.547300\pi\)
−0.148050 + 0.988980i \(0.547300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 586.109i 0.708717i 0.935110 + 0.354359i \(0.115301\pi\)
−0.935110 + 0.354359i \(0.884699\pi\)
\(828\) 0 0
\(829\) 1097.09 1.32339 0.661697 0.749772i \(-0.269837\pi\)
0.661697 + 0.749772i \(0.269837\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1196.91i − 1.43687i
\(834\) 0 0
\(835\) 482.123 0.577393
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1628.39i − 1.94087i −0.241357 0.970436i \(-0.577593\pi\)
0.241357 0.970436i \(-0.422407\pi\)
\(840\) 0 0
\(841\) 780.050 0.927526
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 528.677i − 0.625653i
\(846\) 0 0
\(847\) −30.8186 −0.0363856
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1177.37i − 1.38351i
\(852\) 0 0
\(853\) −379.377 −0.444757 −0.222378 0.974960i \(-0.571382\pi\)
−0.222378 + 0.974960i \(0.571382\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1094.67i − 1.27733i −0.769485 0.638665i \(-0.779487\pi\)
0.769485 0.638665i \(-0.220513\pi\)
\(858\) 0 0
\(859\) −1482.81 −1.72621 −0.863103 0.505028i \(-0.831482\pi\)
−0.863103 + 0.505028i \(0.831482\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1083.41i 1.25540i 0.778457 + 0.627698i \(0.216003\pi\)
−0.778457 + 0.627698i \(0.783997\pi\)
\(864\) 0 0
\(865\) 143.219 0.165571
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1430.94i − 1.64665i
\(870\) 0 0
\(871\) −3.08394 −0.00354069
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 317.719i − 0.363107i
\(876\) 0 0
\(877\) 1265.49 1.44297 0.721487 0.692428i \(-0.243459\pi\)
0.721487 + 0.692428i \(0.243459\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 78.8518i − 0.0895027i −0.998998 0.0447513i \(-0.985750\pi\)
0.998998 0.0447513i \(-0.0142495\pi\)
\(882\) 0 0
\(883\) 86.6337 0.0981129 0.0490565 0.998796i \(-0.484379\pi\)
0.0490565 + 0.998796i \(0.484379\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 449.343i − 0.506587i −0.967389 0.253293i \(-0.918486\pi\)
0.967389 0.253293i \(-0.0815138\pi\)
\(888\) 0 0
\(889\) −175.031 −0.196885
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 96.8312i 0.108434i
\(894\) 0 0
\(895\) 328.345 0.366866
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 136.170i − 0.151469i
\(900\) 0 0
\(901\) −463.547 −0.514481
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 969.179i 1.07092i
\(906\) 0 0
\(907\) −1271.12 −1.40145 −0.700727 0.713429i \(-0.747141\pi\)
−0.700727 + 0.713429i \(0.747141\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 428.669i − 0.470548i −0.971929 0.235274i \(-0.924401\pi\)
0.971929 0.235274i \(-0.0755987\pi\)
\(912\) 0 0
\(913\) −47.5263 −0.0520550
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 152.366i 0.166157i
\(918\) 0 0
\(919\) −302.820 −0.329511 −0.164755 0.986334i \(-0.552683\pi\)
−0.164755 + 0.986334i \(0.552683\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 231.709i 0.251039i
\(924\) 0 0
\(925\) −683.639 −0.739069
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1191.45i 1.28250i 0.767331 + 0.641252i \(0.221585\pi\)
−0.767331 + 0.641252i \(0.778415\pi\)
\(930\) 0 0
\(931\) 187.142 0.201012
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 957.064i 1.02360i
\(936\) 0 0
\(937\) 438.222 0.467686 0.233843 0.972274i \(-0.424870\pi\)
0.233843 + 0.972274i \(0.424870\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 796.939i − 0.846907i −0.905918 0.423453i \(-0.860818\pi\)
0.905918 0.423453i \(-0.139182\pi\)
\(942\) 0 0
\(943\) −1247.15 −1.32254
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 744.367i − 0.786026i −0.919533 0.393013i \(-0.871433\pi\)
0.919533 0.393013i \(-0.128567\pi\)
\(948\) 0 0
\(949\) −156.787 −0.165213
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 370.138i 0.388392i 0.980963 + 0.194196i \(0.0622098\pi\)
−0.980963 + 0.194196i \(0.937790\pi\)
\(954\) 0 0
\(955\) 494.099 0.517382
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 212.532i − 0.221618i
\(960\) 0 0
\(961\) −656.780 −0.683434
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1242.04i − 1.28709i
\(966\) 0 0
\(967\) 509.275 0.526654 0.263327 0.964707i \(-0.415180\pi\)
0.263327 + 0.964707i \(0.415180\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 987.400i 1.01689i 0.861095 + 0.508445i \(0.169779\pi\)
−0.861095 + 0.508445i \(0.830221\pi\)
\(972\) 0 0
\(973\) −427.465 −0.439327
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1679.85i 1.71939i 0.510804 + 0.859697i \(0.329348\pi\)
−0.510804 + 0.859697i \(0.670652\pi\)
\(978\) 0 0
\(979\) 1256.42 1.28337
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 575.062i − 0.585007i −0.956264 0.292503i \(-0.905512\pi\)
0.956264 0.292503i \(-0.0944883\pi\)
\(984\) 0 0
\(985\) 331.431 0.336478
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 2015.59i − 2.03801i
\(990\) 0 0
\(991\) −544.246 −0.549188 −0.274594 0.961560i \(-0.588544\pi\)
−0.274594 + 0.961560i \(0.588544\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 495.087i 0.497575i
\(996\) 0 0
\(997\) 1569.52 1.57424 0.787120 0.616800i \(-0.211571\pi\)
0.787120 + 0.616800i \(0.211571\pi\)
\(998\) 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.305.9 yes 12
3.2 odd 2 inner 684.3.e.a.305.4 12
4.3 odd 2 2736.3.h.c.305.9 12
12.11 even 2 2736.3.h.c.305.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.e.a.305.4 12 3.2 odd 2 inner
684.3.e.a.305.9 yes 12 1.1 even 1 trivial
2736.3.h.c.305.4 12 12.11 even 2
2736.3.h.c.305.9 12 4.3 odd 2