Properties

Label 441.4.c.a.440.1
Level $441$
Weight $4$
Character 441.440
Analytic conductor $26.020$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 440.1
Root \(-4.21355 + 2.43270i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.4.c.a.440.15

$q$-expansion

\(f(q)\) \(=\) \(q-4.86539i q^{2} -15.6720 q^{4} -12.7643 q^{5} +37.3274i q^{8} +O(q^{10})\) \(q-4.86539i q^{2} -15.6720 q^{4} -12.7643 q^{5} +37.3274i q^{8} +62.1035i q^{10} +54.1131i q^{11} +8.85528i q^{13} +56.2362 q^{16} +68.9173 q^{17} -163.849i q^{19} +200.043 q^{20} +263.281 q^{22} +93.9704i q^{23} +37.9283 q^{25} +43.0844 q^{26} +119.620i q^{29} -98.8783i q^{31} +25.0082i q^{32} -335.310i q^{34} +94.1898 q^{37} -797.190 q^{38} -476.460i q^{40} +259.347 q^{41} +5.01418 q^{43} -848.062i q^{44} +457.203 q^{46} -57.3493 q^{47} -184.536i q^{50} -138.780i q^{52} -470.030i q^{53} -690.718i q^{55} +581.998 q^{58} +225.958 q^{59} +427.990i q^{61} -481.082 q^{62} +571.564 q^{64} -113.032i q^{65} +163.853 q^{67} -1080.07 q^{68} +79.8529i q^{71} -769.547i q^{73} -458.270i q^{74} +2567.85i q^{76} +534.817 q^{79} -717.818 q^{80} -1261.82i q^{82} +438.520 q^{83} -879.684 q^{85} -24.3959i q^{86} -2019.90 q^{88} -25.6483 q^{89} -1472.71i q^{92} +279.027i q^{94} +2091.43i q^{95} +1381.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 64q^{4} + O(q^{10}) \) \( 16q - 64q^{4} + 376q^{16} + 528q^{22} + 40q^{25} + 2392q^{37} + 328q^{43} + 2784q^{46} + 6744q^{58} + 5432q^{64} - 616q^{67} + 4352q^{79} - 4608q^{85} - 1416q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.86539i − 1.72018i −0.510146 0.860088i \(-0.670409\pi\)
0.510146 0.860088i \(-0.329591\pi\)
\(3\) 0 0
\(4\) −15.6720 −1.95900
\(5\) −12.7643 −1.14168 −0.570839 0.821062i \(-0.693382\pi\)
−0.570839 + 0.821062i \(0.693382\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 37.3274i 1.64965i
\(9\) 0 0
\(10\) 62.1035i 1.96388i
\(11\) 54.1131i 1.48325i 0.670817 + 0.741623i \(0.265944\pi\)
−0.670817 + 0.741623i \(0.734056\pi\)
\(12\) 0 0
\(13\) 8.85528i 0.188924i 0.995528 + 0.0944620i \(0.0301131\pi\)
−0.995528 + 0.0944620i \(0.969887\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 56.2362 0.878691
\(17\) 68.9173 0.983230 0.491615 0.870813i \(-0.336407\pi\)
0.491615 + 0.870813i \(0.336407\pi\)
\(18\) 0 0
\(19\) − 163.849i − 1.97840i −0.146578 0.989199i \(-0.546826\pi\)
0.146578 0.989199i \(-0.453174\pi\)
\(20\) 200.043 2.23655
\(21\) 0 0
\(22\) 263.281 2.55144
\(23\) 93.9704i 0.851921i 0.904742 + 0.425960i \(0.140064\pi\)
−0.904742 + 0.425960i \(0.859936\pi\)
\(24\) 0 0
\(25\) 37.9283 0.303427
\(26\) 43.0844 0.324982
\(27\) 0 0
\(28\) 0 0
\(29\) 119.620i 0.765961i 0.923756 + 0.382981i \(0.125102\pi\)
−0.923756 + 0.382981i \(0.874898\pi\)
\(30\) 0 0
\(31\) − 98.8783i − 0.572873i −0.958099 0.286437i \(-0.907529\pi\)
0.958099 0.286437i \(-0.0924708\pi\)
\(32\) 25.0082i 0.138152i
\(33\) 0 0
\(34\) − 335.310i − 1.69133i
\(35\) 0 0
\(36\) 0 0
\(37\) 94.1898 0.418506 0.209253 0.977862i \(-0.432897\pi\)
0.209253 + 0.977862i \(0.432897\pi\)
\(38\) −797.190 −3.40319
\(39\) 0 0
\(40\) − 476.460i − 1.88337i
\(41\) 259.347 0.987883 0.493941 0.869495i \(-0.335556\pi\)
0.493941 + 0.869495i \(0.335556\pi\)
\(42\) 0 0
\(43\) 5.01418 0.0177827 0.00889133 0.999960i \(-0.497170\pi\)
0.00889133 + 0.999960i \(0.497170\pi\)
\(44\) − 848.062i − 2.90568i
\(45\) 0 0
\(46\) 457.203 1.46545
\(47\) −57.3493 −0.177984 −0.0889921 0.996032i \(-0.528365\pi\)
−0.0889921 + 0.996032i \(0.528365\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 184.536i − 0.521947i
\(51\) 0 0
\(52\) − 138.780i − 0.370103i
\(53\) − 470.030i − 1.21818i −0.793101 0.609090i \(-0.791535\pi\)
0.793101 0.609090i \(-0.208465\pi\)
\(54\) 0 0
\(55\) − 690.718i − 1.69339i
\(56\) 0 0
\(57\) 0 0
\(58\) 581.998 1.31759
\(59\) 225.958 0.498597 0.249299 0.968427i \(-0.419800\pi\)
0.249299 + 0.968427i \(0.419800\pi\)
\(60\) 0 0
\(61\) 427.990i 0.898336i 0.893447 + 0.449168i \(0.148280\pi\)
−0.893447 + 0.449168i \(0.851720\pi\)
\(62\) −481.082 −0.985442
\(63\) 0 0
\(64\) 571.564 1.11634
\(65\) − 113.032i − 0.215690i
\(66\) 0 0
\(67\) 163.853 0.298774 0.149387 0.988779i \(-0.452270\pi\)
0.149387 + 0.988779i \(0.452270\pi\)
\(68\) −1080.07 −1.92615
\(69\) 0 0
\(70\) 0 0
\(71\) 79.8529i 0.133476i 0.997771 + 0.0667380i \(0.0212592\pi\)
−0.997771 + 0.0667380i \(0.978741\pi\)
\(72\) 0 0
\(73\) − 769.547i − 1.23382i −0.787035 0.616909i \(-0.788385\pi\)
0.787035 0.616909i \(-0.211615\pi\)
\(74\) − 458.270i − 0.719903i
\(75\) 0 0
\(76\) 2567.85i 3.87569i
\(77\) 0 0
\(78\) 0 0
\(79\) 534.817 0.761666 0.380833 0.924644i \(-0.375637\pi\)
0.380833 + 0.924644i \(0.375637\pi\)
\(80\) −717.818 −1.00318
\(81\) 0 0
\(82\) − 1261.82i − 1.69933i
\(83\) 438.520 0.579926 0.289963 0.957038i \(-0.406357\pi\)
0.289963 + 0.957038i \(0.406357\pi\)
\(84\) 0 0
\(85\) −879.684 −1.12253
\(86\) − 24.3959i − 0.0305893i
\(87\) 0 0
\(88\) −2019.90 −2.44684
\(89\) −25.6483 −0.0305474 −0.0152737 0.999883i \(-0.504862\pi\)
−0.0152737 + 0.999883i \(0.504862\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 1472.71i − 1.66892i
\(93\) 0 0
\(94\) 279.027i 0.306164i
\(95\) 2091.43i 2.25869i
\(96\) 0 0
\(97\) 1381.00i 1.44555i 0.691081 + 0.722777i \(0.257135\pi\)
−0.691081 + 0.722777i \(0.742865\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −594.414 −0.594414
\(101\) 713.616 0.703044 0.351522 0.936180i \(-0.385664\pi\)
0.351522 + 0.936180i \(0.385664\pi\)
\(102\) 0 0
\(103\) 1792.58i 1.71483i 0.514624 + 0.857416i \(0.327932\pi\)
−0.514624 + 0.857416i \(0.672068\pi\)
\(104\) −330.544 −0.311659
\(105\) 0 0
\(106\) −2286.88 −2.09548
\(107\) − 22.5589i − 0.0203818i −0.999948 0.0101909i \(-0.996756\pi\)
0.999948 0.0101909i \(-0.00324391\pi\)
\(108\) 0 0
\(109\) 952.420 0.836930 0.418465 0.908233i \(-0.362568\pi\)
0.418465 + 0.908233i \(0.362568\pi\)
\(110\) −3360.61 −2.91292
\(111\) 0 0
\(112\) 0 0
\(113\) − 120.145i − 0.100020i −0.998749 0.0500102i \(-0.984075\pi\)
0.998749 0.0500102i \(-0.0159254\pi\)
\(114\) 0 0
\(115\) − 1199.47i − 0.972619i
\(116\) − 1874.69i − 1.50052i
\(117\) 0 0
\(118\) − 1099.37i − 0.857674i
\(119\) 0 0
\(120\) 0 0
\(121\) −1597.22 −1.20002
\(122\) 2082.34 1.54530
\(123\) 0 0
\(124\) 1549.62i 1.12226i
\(125\) 1111.41 0.795262
\(126\) 0 0
\(127\) 884.302 0.617867 0.308934 0.951084i \(-0.400028\pi\)
0.308934 + 0.951084i \(0.400028\pi\)
\(128\) − 2580.82i − 1.78214i
\(129\) 0 0
\(130\) −549.944 −0.371025
\(131\) 1606.88 1.07171 0.535853 0.844311i \(-0.319990\pi\)
0.535853 + 0.844311i \(0.319990\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 797.210i − 0.513944i
\(135\) 0 0
\(136\) 2572.50i 1.62199i
\(137\) 710.483i 0.443071i 0.975152 + 0.221535i \(0.0711068\pi\)
−0.975152 + 0.221535i \(0.928893\pi\)
\(138\) 0 0
\(139\) − 1531.91i − 0.934782i −0.884051 0.467391i \(-0.845194\pi\)
0.884051 0.467391i \(-0.154806\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 388.515 0.229602
\(143\) −479.186 −0.280221
\(144\) 0 0
\(145\) − 1526.87i − 0.874480i
\(146\) −3744.15 −2.12238
\(147\) 0 0
\(148\) −1476.15 −0.819854
\(149\) 2401.07i 1.32016i 0.751197 + 0.660079i \(0.229477\pi\)
−0.751197 + 0.660079i \(0.770523\pi\)
\(150\) 0 0
\(151\) 2467.97 1.33007 0.665035 0.746812i \(-0.268416\pi\)
0.665035 + 0.746812i \(0.268416\pi\)
\(152\) 6116.06 3.26367
\(153\) 0 0
\(154\) 0 0
\(155\) 1262.12i 0.654036i
\(156\) 0 0
\(157\) − 2436.12i − 1.23837i −0.785246 0.619184i \(-0.787463\pi\)
0.785246 0.619184i \(-0.212537\pi\)
\(158\) − 2602.09i − 1.31020i
\(159\) 0 0
\(160\) − 319.213i − 0.157725i
\(161\) 0 0
\(162\) 0 0
\(163\) 3277.01 1.57469 0.787347 0.616510i \(-0.211454\pi\)
0.787347 + 0.616510i \(0.211454\pi\)
\(164\) −4064.49 −1.93527
\(165\) 0 0
\(166\) − 2133.57i − 0.997574i
\(167\) −365.585 −0.169400 −0.0847000 0.996406i \(-0.526993\pi\)
−0.0847000 + 0.996406i \(0.526993\pi\)
\(168\) 0 0
\(169\) 2118.58 0.964308
\(170\) 4280.01i 1.93095i
\(171\) 0 0
\(172\) −78.5823 −0.0348363
\(173\) −2092.93 −0.919785 −0.459892 0.887975i \(-0.652112\pi\)
−0.459892 + 0.887975i \(0.652112\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3043.11i 1.30331i
\(177\) 0 0
\(178\) 124.789i 0.0525469i
\(179\) 1759.78i 0.734817i 0.930060 + 0.367408i \(0.119755\pi\)
−0.930060 + 0.367408i \(0.880245\pi\)
\(180\) 0 0
\(181\) − 3197.54i − 1.31310i −0.754282 0.656551i \(-0.772015\pi\)
0.754282 0.656551i \(-0.227985\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3507.67 −1.40537
\(185\) −1202.27 −0.477798
\(186\) 0 0
\(187\) 3729.33i 1.45837i
\(188\) 898.780 0.348672
\(189\) 0 0
\(190\) 10175.6 3.88535
\(191\) − 549.374i − 0.208122i −0.994571 0.104061i \(-0.966816\pi\)
0.994571 0.104061i \(-0.0331837\pi\)
\(192\) 0 0
\(193\) −704.476 −0.262742 −0.131371 0.991333i \(-0.541938\pi\)
−0.131371 + 0.991333i \(0.541938\pi\)
\(194\) 6719.08 2.48661
\(195\) 0 0
\(196\) 0 0
\(197\) 5317.81i 1.92324i 0.274384 + 0.961620i \(0.411526\pi\)
−0.274384 + 0.961620i \(0.588474\pi\)
\(198\) 0 0
\(199\) 2489.30i 0.886743i 0.896338 + 0.443371i \(0.146218\pi\)
−0.896338 + 0.443371i \(0.853782\pi\)
\(200\) 1415.77i 0.500549i
\(201\) 0 0
\(202\) − 3472.02i − 1.20936i
\(203\) 0 0
\(204\) 0 0
\(205\) −3310.39 −1.12784
\(206\) 8721.58 2.94981
\(207\) 0 0
\(208\) 497.987i 0.166006i
\(209\) 8866.38 2.93445
\(210\) 0 0
\(211\) −3454.31 −1.12704 −0.563519 0.826103i \(-0.690553\pi\)
−0.563519 + 0.826103i \(0.690553\pi\)
\(212\) 7366.32i 2.38642i
\(213\) 0 0
\(214\) −109.758 −0.0350602
\(215\) −64.0026 −0.0203021
\(216\) 0 0
\(217\) 0 0
\(218\) − 4633.90i − 1.43967i
\(219\) 0 0
\(220\) 10824.9i 3.31735i
\(221\) 610.282i 0.185756i
\(222\) 0 0
\(223\) 3896.38i 1.17005i 0.811016 + 0.585024i \(0.198915\pi\)
−0.811016 + 0.585024i \(0.801085\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −584.553 −0.172053
\(227\) 605.495 0.177040 0.0885201 0.996074i \(-0.471786\pi\)
0.0885201 + 0.996074i \(0.471786\pi\)
\(228\) 0 0
\(229\) − 2208.92i − 0.637422i −0.947852 0.318711i \(-0.896750\pi\)
0.947852 0.318711i \(-0.103250\pi\)
\(230\) −5835.89 −1.67307
\(231\) 0 0
\(232\) −4465.10 −1.26357
\(233\) − 2385.35i − 0.670683i −0.942097 0.335342i \(-0.891148\pi\)
0.942097 0.335342i \(-0.108852\pi\)
\(234\) 0 0
\(235\) 732.026 0.203201
\(236\) −3541.22 −0.976753
\(237\) 0 0
\(238\) 0 0
\(239\) 3017.95i 0.816798i 0.912803 + 0.408399i \(0.133913\pi\)
−0.912803 + 0.408399i \(0.866087\pi\)
\(240\) 0 0
\(241\) 2515.49i 0.672353i 0.941799 + 0.336176i \(0.109134\pi\)
−0.941799 + 0.336176i \(0.890866\pi\)
\(242\) 7771.12i 2.06424i
\(243\) 0 0
\(244\) − 6707.47i − 1.75984i
\(245\) 0 0
\(246\) 0 0
\(247\) 1450.93 0.373767
\(248\) 3690.87 0.945042
\(249\) 0 0
\(250\) − 5407.45i − 1.36799i
\(251\) −1306.11 −0.328451 −0.164226 0.986423i \(-0.552512\pi\)
−0.164226 + 0.986423i \(0.552512\pi\)
\(252\) 0 0
\(253\) −5085.03 −1.26361
\(254\) − 4302.48i − 1.06284i
\(255\) 0 0
\(256\) −7984.17 −1.94926
\(257\) 7471.83 1.81354 0.906770 0.421626i \(-0.138541\pi\)
0.906770 + 0.421626i \(0.138541\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1771.44i 0.422538i
\(261\) 0 0
\(262\) − 7818.09i − 1.84352i
\(263\) − 1536.64i − 0.360278i −0.983641 0.180139i \(-0.942345\pi\)
0.983641 0.180139i \(-0.0576547\pi\)
\(264\) 0 0
\(265\) 5999.62i 1.39077i
\(266\) 0 0
\(267\) 0 0
\(268\) −2567.91 −0.585300
\(269\) 3916.18 0.887636 0.443818 0.896117i \(-0.353624\pi\)
0.443818 + 0.896117i \(0.353624\pi\)
\(270\) 0 0
\(271\) 3599.68i 0.806882i 0.915006 + 0.403441i \(0.132186\pi\)
−0.915006 + 0.403441i \(0.867814\pi\)
\(272\) 3875.65 0.863955
\(273\) 0 0
\(274\) 3456.78 0.762159
\(275\) 2052.42i 0.450056i
\(276\) 0 0
\(277\) 284.081 0.0616200 0.0308100 0.999525i \(-0.490191\pi\)
0.0308100 + 0.999525i \(0.490191\pi\)
\(278\) −7453.33 −1.60799
\(279\) 0 0
\(280\) 0 0
\(281\) − 321.256i − 0.0682011i −0.999418 0.0341006i \(-0.989143\pi\)
0.999418 0.0341006i \(-0.0108566\pi\)
\(282\) 0 0
\(283\) − 6802.80i − 1.42892i −0.699676 0.714460i \(-0.746672\pi\)
0.699676 0.714460i \(-0.253328\pi\)
\(284\) − 1251.46i − 0.261480i
\(285\) 0 0
\(286\) 2331.43i 0.482029i
\(287\) 0 0
\(288\) 0 0
\(289\) −163.403 −0.0332594
\(290\) −7428.82 −1.50426
\(291\) 0 0
\(292\) 12060.4i 2.41705i
\(293\) −3180.05 −0.634063 −0.317031 0.948415i \(-0.602686\pi\)
−0.317031 + 0.948415i \(0.602686\pi\)
\(294\) 0 0
\(295\) −2884.20 −0.569237
\(296\) 3515.86i 0.690390i
\(297\) 0 0
\(298\) 11682.1 2.27090
\(299\) −832.134 −0.160948
\(300\) 0 0
\(301\) 0 0
\(302\) − 12007.6i − 2.28795i
\(303\) 0 0
\(304\) − 9214.25i − 1.73840i
\(305\) − 5463.01i − 1.02561i
\(306\) 0 0
\(307\) 2976.39i 0.553328i 0.960967 + 0.276664i \(0.0892289\pi\)
−0.960967 + 0.276664i \(0.910771\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 6140.69 1.12506
\(311\) −4681.79 −0.853633 −0.426817 0.904338i \(-0.640365\pi\)
−0.426817 + 0.904338i \(0.640365\pi\)
\(312\) 0 0
\(313\) − 982.047i − 0.177344i −0.996061 0.0886718i \(-0.971738\pi\)
0.996061 0.0886718i \(-0.0282622\pi\)
\(314\) −11852.7 −2.13021
\(315\) 0 0
\(316\) −8381.66 −1.49211
\(317\) − 4930.34i − 0.873550i −0.899571 0.436775i \(-0.856121\pi\)
0.899571 0.436775i \(-0.143879\pi\)
\(318\) 0 0
\(319\) −6473.01 −1.13611
\(320\) −7295.64 −1.27450
\(321\) 0 0
\(322\) 0 0
\(323\) − 11292.0i − 1.94522i
\(324\) 0 0
\(325\) 335.866i 0.0573246i
\(326\) − 15943.9i − 2.70875i
\(327\) 0 0
\(328\) 9680.75i 1.62966i
\(329\) 0 0
\(330\) 0 0
\(331\) −4034.51 −0.669959 −0.334980 0.942225i \(-0.608729\pi\)
−0.334980 + 0.942225i \(0.608729\pi\)
\(332\) −6872.50 −1.13608
\(333\) 0 0
\(334\) 1778.71i 0.291398i
\(335\) −2091.48 −0.341104
\(336\) 0 0
\(337\) 2771.62 0.448011 0.224006 0.974588i \(-0.428087\pi\)
0.224006 + 0.974588i \(0.428087\pi\)
\(338\) − 10307.7i − 1.65878i
\(339\) 0 0
\(340\) 13786.4 2.19904
\(341\) 5350.61 0.849712
\(342\) 0 0
\(343\) 0 0
\(344\) 187.166i 0.0293352i
\(345\) 0 0
\(346\) 10182.9i 1.58219i
\(347\) − 4322.21i − 0.668670i −0.942454 0.334335i \(-0.891488\pi\)
0.942454 0.334335i \(-0.108512\pi\)
\(348\) 0 0
\(349\) − 1331.65i − 0.204245i −0.994772 0.102122i \(-0.967437\pi\)
0.994772 0.102122i \(-0.0325634\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1353.27 −0.204913
\(353\) −11348.5 −1.71111 −0.855553 0.517715i \(-0.826783\pi\)
−0.855553 + 0.517715i \(0.826783\pi\)
\(354\) 0 0
\(355\) − 1019.27i − 0.152386i
\(356\) 401.962 0.0598425
\(357\) 0 0
\(358\) 8562.02 1.26401
\(359\) 8368.68i 1.23031i 0.788406 + 0.615156i \(0.210907\pi\)
−0.788406 + 0.615156i \(0.789093\pi\)
\(360\) 0 0
\(361\) −19987.5 −2.91406
\(362\) −15557.3 −2.25876
\(363\) 0 0
\(364\) 0 0
\(365\) 9822.76i 1.40862i
\(366\) 0 0
\(367\) − 2715.06i − 0.386171i −0.981182 0.193086i \(-0.938150\pi\)
0.981182 0.193086i \(-0.0618495\pi\)
\(368\) 5284.54i 0.748575i
\(369\) 0 0
\(370\) 5849.52i 0.821897i
\(371\) 0 0
\(372\) 0 0
\(373\) 6097.11 0.846371 0.423186 0.906043i \(-0.360912\pi\)
0.423186 + 0.906043i \(0.360912\pi\)
\(374\) 18144.6 2.50865
\(375\) 0 0
\(376\) − 2140.70i − 0.293612i
\(377\) −1059.27 −0.144708
\(378\) 0 0
\(379\) −9922.24 −1.34478 −0.672389 0.740198i \(-0.734732\pi\)
−0.672389 + 0.740198i \(0.734732\pi\)
\(380\) − 32776.9i − 4.42479i
\(381\) 0 0
\(382\) −2672.92 −0.358006
\(383\) −1219.06 −0.162640 −0.0813202 0.996688i \(-0.525914\pi\)
−0.0813202 + 0.996688i \(0.525914\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3427.55i 0.451963i
\(387\) 0 0
\(388\) − 21643.0i − 2.83185i
\(389\) 13134.3i 1.71192i 0.517045 + 0.855958i \(0.327032\pi\)
−0.517045 + 0.855958i \(0.672968\pi\)
\(390\) 0 0
\(391\) 6476.19i 0.837634i
\(392\) 0 0
\(393\) 0 0
\(394\) 25873.2 3.30831
\(395\) −6826.58 −0.869576
\(396\) 0 0
\(397\) 3115.02i 0.393799i 0.980424 + 0.196900i \(0.0630873\pi\)
−0.980424 + 0.196900i \(0.936913\pi\)
\(398\) 12111.4 1.52535
\(399\) 0 0
\(400\) 2132.94 0.266618
\(401\) 11228.8i 1.39836i 0.714946 + 0.699179i \(0.246451\pi\)
−0.714946 + 0.699179i \(0.753549\pi\)
\(402\) 0 0
\(403\) 875.595 0.108229
\(404\) −11183.8 −1.37727
\(405\) 0 0
\(406\) 0 0
\(407\) 5096.90i 0.620747i
\(408\) 0 0
\(409\) 12400.8i 1.49922i 0.661880 + 0.749610i \(0.269759\pi\)
−0.661880 + 0.749610i \(0.730241\pi\)
\(410\) 16106.4i 1.94009i
\(411\) 0 0
\(412\) − 28093.3i − 3.35936i
\(413\) 0 0
\(414\) 0 0
\(415\) −5597.42 −0.662088
\(416\) −221.454 −0.0261002
\(417\) 0 0
\(418\) − 43138.4i − 5.04777i
\(419\) 8260.19 0.963095 0.481547 0.876420i \(-0.340075\pi\)
0.481547 + 0.876420i \(0.340075\pi\)
\(420\) 0 0
\(421\) 5571.81 0.645020 0.322510 0.946566i \(-0.395473\pi\)
0.322510 + 0.946566i \(0.395473\pi\)
\(422\) 16806.6i 1.93870i
\(423\) 0 0
\(424\) 17545.0 2.00957
\(425\) 2613.92 0.298338
\(426\) 0 0
\(427\) 0 0
\(428\) 353.543i 0.0399279i
\(429\) 0 0
\(430\) 311.398i 0.0349231i
\(431\) − 7037.76i − 0.786535i −0.919424 0.393268i \(-0.871345\pi\)
0.919424 0.393268i \(-0.128655\pi\)
\(432\) 0 0
\(433\) 9212.26i 1.02243i 0.859452 + 0.511216i \(0.170805\pi\)
−0.859452 + 0.511216i \(0.829195\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14926.4 −1.63955
\(437\) 15397.0 1.68544
\(438\) 0 0
\(439\) − 8481.62i − 0.922109i −0.887372 0.461054i \(-0.847471\pi\)
0.887372 0.461054i \(-0.152529\pi\)
\(440\) 25782.7 2.79350
\(441\) 0 0
\(442\) 2969.26 0.319532
\(443\) − 13055.0i − 1.40014i −0.714076 0.700068i \(-0.753153\pi\)
0.714076 0.700068i \(-0.246847\pi\)
\(444\) 0 0
\(445\) 327.384 0.0348753
\(446\) 18957.4 2.01269
\(447\) 0 0
\(448\) 0 0
\(449\) − 14114.0i − 1.48348i −0.670689 0.741738i \(-0.734002\pi\)
0.670689 0.741738i \(-0.265998\pi\)
\(450\) 0 0
\(451\) 14034.1i 1.46527i
\(452\) 1882.92i 0.195940i
\(453\) 0 0
\(454\) − 2945.97i − 0.304540i
\(455\) 0 0
\(456\) 0 0
\(457\) −8973.74 −0.918543 −0.459271 0.888296i \(-0.651889\pi\)
−0.459271 + 0.888296i \(0.651889\pi\)
\(458\) −10747.3 −1.09648
\(459\) 0 0
\(460\) 18798.1i 1.90536i
\(461\) −955.010 −0.0964842 −0.0482421 0.998836i \(-0.515362\pi\)
−0.0482421 + 0.998836i \(0.515362\pi\)
\(462\) 0 0
\(463\) 12004.5 1.20496 0.602479 0.798135i \(-0.294180\pi\)
0.602479 + 0.798135i \(0.294180\pi\)
\(464\) 6726.97i 0.673043i
\(465\) 0 0
\(466\) −11605.6 −1.15369
\(467\) 5064.91 0.501876 0.250938 0.968003i \(-0.419261\pi\)
0.250938 + 0.968003i \(0.419261\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 3561.59i − 0.349540i
\(471\) 0 0
\(472\) 8434.43i 0.822513i
\(473\) 271.333i 0.0263761i
\(474\) 0 0
\(475\) − 6214.52i − 0.600299i
\(476\) 0 0
\(477\) 0 0
\(478\) 14683.5 1.40504
\(479\) −15213.7 −1.45121 −0.725607 0.688109i \(-0.758441\pi\)
−0.725607 + 0.688109i \(0.758441\pi\)
\(480\) 0 0
\(481\) 834.077i 0.0790658i
\(482\) 12238.8 1.15656
\(483\) 0 0
\(484\) 25031.7 2.35084
\(485\) − 17627.5i − 1.65036i
\(486\) 0 0
\(487\) 15811.8 1.47126 0.735629 0.677384i \(-0.236887\pi\)
0.735629 + 0.677384i \(0.236887\pi\)
\(488\) −15975.8 −1.48194
\(489\) 0 0
\(490\) 0 0
\(491\) 18064.2i 1.66034i 0.557512 + 0.830169i \(0.311756\pi\)
−0.557512 + 0.830169i \(0.688244\pi\)
\(492\) 0 0
\(493\) 8243.89i 0.753116i
\(494\) − 7059.34i − 0.642945i
\(495\) 0 0
\(496\) − 5560.54i − 0.503378i
\(497\) 0 0
\(498\) 0 0
\(499\) −10524.7 −0.944185 −0.472092 0.881549i \(-0.656501\pi\)
−0.472092 + 0.881549i \(0.656501\pi\)
\(500\) −17418.1 −1.55792
\(501\) 0 0
\(502\) 6354.76i 0.564994i
\(503\) −7790.82 −0.690607 −0.345304 0.938491i \(-0.612224\pi\)
−0.345304 + 0.938491i \(0.612224\pi\)
\(504\) 0 0
\(505\) −9108.83 −0.802649
\(506\) 24740.6i 2.17363i
\(507\) 0 0
\(508\) −13858.8 −1.21040
\(509\) −10196.5 −0.887919 −0.443960 0.896047i \(-0.646427\pi\)
−0.443960 + 0.896047i \(0.646427\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 18199.6i 1.57093i
\(513\) 0 0
\(514\) − 36353.4i − 3.11961i
\(515\) − 22881.0i − 1.95778i
\(516\) 0 0
\(517\) − 3103.35i − 0.263994i
\(518\) 0 0
\(519\) 0 0
\(520\) 4219.18 0.355814
\(521\) −1927.58 −0.162090 −0.0810449 0.996710i \(-0.525826\pi\)
−0.0810449 + 0.996710i \(0.525826\pi\)
\(522\) 0 0
\(523\) − 7755.77i − 0.648444i −0.945981 0.324222i \(-0.894897\pi\)
0.945981 0.324222i \(-0.105103\pi\)
\(524\) −25183.0 −2.09948
\(525\) 0 0
\(526\) −7476.33 −0.619741
\(527\) − 6814.43i − 0.563266i
\(528\) 0 0
\(529\) 3336.56 0.274231
\(530\) 29190.5 2.39236
\(531\) 0 0
\(532\) 0 0
\(533\) 2296.59i 0.186635i
\(534\) 0 0
\(535\) 287.949i 0.0232694i
\(536\) 6116.22i 0.492874i
\(537\) 0 0
\(538\) − 19053.8i − 1.52689i
\(539\) 0 0
\(540\) 0 0
\(541\) 16760.8 1.33199 0.665993 0.745958i \(-0.268008\pi\)
0.665993 + 0.745958i \(0.268008\pi\)
\(542\) 17513.9 1.38798
\(543\) 0 0
\(544\) 1723.50i 0.135835i
\(545\) −12157.0 −0.955503
\(546\) 0 0
\(547\) 5869.79 0.458819 0.229410 0.973330i \(-0.426320\pi\)
0.229410 + 0.973330i \(0.426320\pi\)
\(548\) − 11134.7i − 0.867977i
\(549\) 0 0
\(550\) 9985.82 0.774176
\(551\) 19599.6 1.51538
\(552\) 0 0
\(553\) 0 0
\(554\) − 1382.16i − 0.105997i
\(555\) 0 0
\(556\) 24008.1i 1.83124i
\(557\) 21658.1i 1.64755i 0.566918 + 0.823774i \(0.308136\pi\)
−0.566918 + 0.823774i \(0.691864\pi\)
\(558\) 0 0
\(559\) 44.4019i 0.00335957i
\(560\) 0 0
\(561\) 0 0
\(562\) −1563.03 −0.117318
\(563\) −9590.71 −0.717940 −0.358970 0.933349i \(-0.616872\pi\)
−0.358970 + 0.933349i \(0.616872\pi\)
\(564\) 0 0
\(565\) 1533.57i 0.114191i
\(566\) −33098.3 −2.45799
\(567\) 0 0
\(568\) −2980.70 −0.220189
\(569\) 16634.0i 1.22554i 0.790260 + 0.612772i \(0.209946\pi\)
−0.790260 + 0.612772i \(0.790054\pi\)
\(570\) 0 0
\(571\) 6331.03 0.464002 0.232001 0.972716i \(-0.425473\pi\)
0.232001 + 0.972716i \(0.425473\pi\)
\(572\) 7509.82 0.548953
\(573\) 0 0
\(574\) 0 0
\(575\) 3564.14i 0.258495i
\(576\) 0 0
\(577\) − 9736.36i − 0.702479i −0.936286 0.351239i \(-0.885760\pi\)
0.936286 0.351239i \(-0.114240\pi\)
\(578\) 795.021i 0.0572120i
\(579\) 0 0
\(580\) 23929.1i 1.71311i
\(581\) 0 0
\(582\) 0 0
\(583\) 25434.7 1.80686
\(584\) 28725.2 2.03537
\(585\) 0 0
\(586\) 15472.2i 1.09070i
\(587\) 10940.3 0.769255 0.384628 0.923072i \(-0.374330\pi\)
0.384628 + 0.923072i \(0.374330\pi\)
\(588\) 0 0
\(589\) −16201.1 −1.13337
\(590\) 14032.8i 0.979187i
\(591\) 0 0
\(592\) 5296.88 0.367737
\(593\) −24860.7 −1.72160 −0.860799 0.508946i \(-0.830035\pi\)
−0.860799 + 0.508946i \(0.830035\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 37629.6i − 2.58619i
\(597\) 0 0
\(598\) 4048.66i 0.276859i
\(599\) − 23333.9i − 1.59165i −0.605526 0.795825i \(-0.707037\pi\)
0.605526 0.795825i \(-0.292963\pi\)
\(600\) 0 0
\(601\) 13012.4i 0.883175i 0.897218 + 0.441587i \(0.145584\pi\)
−0.897218 + 0.441587i \(0.854416\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −38678.1 −2.60561
\(605\) 20387.5 1.37003
\(606\) 0 0
\(607\) 8493.62i 0.567950i 0.958832 + 0.283975i \(0.0916532\pi\)
−0.958832 + 0.283975i \(0.908347\pi\)
\(608\) 4097.57 0.273320
\(609\) 0 0
\(610\) −26579.7 −1.76423
\(611\) − 507.844i − 0.0336255i
\(612\) 0 0
\(613\) −9139.58 −0.602193 −0.301097 0.953594i \(-0.597353\pi\)
−0.301097 + 0.953594i \(0.597353\pi\)
\(614\) 14481.3 0.951822
\(615\) 0 0
\(616\) 0 0
\(617\) − 7360.91i − 0.480290i −0.970737 0.240145i \(-0.922805\pi\)
0.970737 0.240145i \(-0.0771950\pi\)
\(618\) 0 0
\(619\) − 22953.7i − 1.49045i −0.666813 0.745225i \(-0.732342\pi\)
0.666813 0.745225i \(-0.267658\pi\)
\(620\) − 19779.9i − 1.28126i
\(621\) 0 0
\(622\) 22778.7i 1.46840i
\(623\) 0 0
\(624\) 0 0
\(625\) −18927.5 −1.21136
\(626\) −4778.04 −0.305062
\(627\) 0 0
\(628\) 38179.0i 2.42597i
\(629\) 6491.31 0.411487
\(630\) 0 0
\(631\) 21126.0 1.33282 0.666412 0.745584i \(-0.267829\pi\)
0.666412 + 0.745584i \(0.267829\pi\)
\(632\) 19963.3i 1.25648i
\(633\) 0 0
\(634\) −23988.0 −1.50266
\(635\) −11287.5 −0.705405
\(636\) 0 0
\(637\) 0 0
\(638\) 31493.7i 1.95431i
\(639\) 0 0
\(640\) 32942.4i 2.03463i
\(641\) 17837.2i 1.09911i 0.835458 + 0.549554i \(0.185202\pi\)
−0.835458 + 0.549554i \(0.814798\pi\)
\(642\) 0 0
\(643\) − 25449.0i − 1.56082i −0.625266 0.780412i \(-0.715009\pi\)
0.625266 0.780412i \(-0.284991\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −54940.2 −3.34612
\(647\) 26298.4 1.59799 0.798993 0.601341i \(-0.205367\pi\)
0.798993 + 0.601341i \(0.205367\pi\)
\(648\) 0 0
\(649\) 12227.3i 0.739542i
\(650\) 1634.12 0.0986083
\(651\) 0 0
\(652\) −51357.4 −3.08483
\(653\) 9472.92i 0.567694i 0.958870 + 0.283847i \(0.0916107\pi\)
−0.958870 + 0.283847i \(0.908389\pi\)
\(654\) 0 0
\(655\) −20510.7 −1.22354
\(656\) 14584.7 0.868043
\(657\) 0 0
\(658\) 0 0
\(659\) − 22384.5i − 1.32318i −0.749865 0.661591i \(-0.769882\pi\)
0.749865 0.661591i \(-0.230118\pi\)
\(660\) 0 0
\(661\) 22929.3i 1.34924i 0.738167 + 0.674618i \(0.235692\pi\)
−0.738167 + 0.674618i \(0.764308\pi\)
\(662\) 19629.5i 1.15245i
\(663\) 0 0
\(664\) 16368.8i 0.956677i
\(665\) 0 0
\(666\) 0 0
\(667\) −11240.7 −0.652538
\(668\) 5729.45 0.331855
\(669\) 0 0
\(670\) 10175.9i 0.586758i
\(671\) −23159.9 −1.33245
\(672\) 0 0
\(673\) −4873.86 −0.279158 −0.139579 0.990211i \(-0.544575\pi\)
−0.139579 + 0.990211i \(0.544575\pi\)
\(674\) − 13485.0i − 0.770658i
\(675\) 0 0
\(676\) −33202.5 −1.88908
\(677\) −16247.4 −0.922363 −0.461181 0.887306i \(-0.652574\pi\)
−0.461181 + 0.887306i \(0.652574\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 32836.3i − 1.85179i
\(681\) 0 0
\(682\) − 26032.8i − 1.46165i
\(683\) 21692.4i 1.21528i 0.794212 + 0.607641i \(0.207884\pi\)
−0.794212 + 0.607641i \(0.792116\pi\)
\(684\) 0 0
\(685\) − 9068.85i − 0.505844i
\(686\) 0 0
\(687\) 0 0
\(688\) 281.978 0.0156255
\(689\) 4162.24 0.230143
\(690\) 0 0
\(691\) − 22516.5i − 1.23961i −0.784757 0.619803i \(-0.787212\pi\)
0.784757 0.619803i \(-0.212788\pi\)
\(692\) 32800.5 1.80186
\(693\) 0 0
\(694\) −21029.2 −1.15023
\(695\) 19553.8i 1.06722i
\(696\) 0 0
\(697\) 17873.5 0.971316
\(698\) −6478.99 −0.351337
\(699\) 0 0
\(700\) 0 0
\(701\) − 33929.0i − 1.82807i −0.405631 0.914037i \(-0.632948\pi\)
0.405631 0.914037i \(-0.367052\pi\)
\(702\) 0 0
\(703\) − 15432.9i − 0.827971i
\(704\) 30929.1i 1.65580i
\(705\) 0 0
\(706\) 55214.9i 2.94340i
\(707\) 0 0
\(708\) 0 0
\(709\) 19187.2 1.01635 0.508175 0.861254i \(-0.330320\pi\)
0.508175 + 0.861254i \(0.330320\pi\)
\(710\) −4959.14 −0.262131
\(711\) 0 0
\(712\) − 957.386i − 0.0503927i
\(713\) 9291.63 0.488043
\(714\) 0 0
\(715\) 6116.49 0.319922
\(716\) − 27579.3i − 1.43951i
\(717\) 0 0
\(718\) 40716.9 2.11635
\(719\) −13766.9 −0.714071 −0.357036 0.934091i \(-0.616212\pi\)
−0.357036 + 0.934091i \(0.616212\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 97247.2i 5.01269i
\(723\) 0 0
\(724\) 50111.9i 2.57237i
\(725\) 4536.99i 0.232413i
\(726\) 0 0
\(727\) − 12226.1i − 0.623717i −0.950129 0.311858i \(-0.899049\pi\)
0.950129 0.311858i \(-0.100951\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 47791.6 2.42308
\(731\) 345.564 0.0174844
\(732\) 0 0
\(733\) 2605.89i 0.131311i 0.997842 + 0.0656553i \(0.0209138\pi\)
−0.997842 + 0.0656553i \(0.979086\pi\)
\(734\) −13209.8 −0.664282
\(735\) 0 0
\(736\) −2350.03 −0.117695
\(737\) 8866.61i 0.443156i
\(738\) 0 0
\(739\) 33743.4 1.67967 0.839833 0.542845i \(-0.182653\pi\)
0.839833 + 0.542845i \(0.182653\pi\)
\(740\) 18842.0 0.936009
\(741\) 0 0
\(742\) 0 0
\(743\) − 14586.9i − 0.720244i −0.932905 0.360122i \(-0.882735\pi\)
0.932905 0.360122i \(-0.117265\pi\)
\(744\) 0 0
\(745\) − 30648.1i − 1.50719i
\(746\) − 29664.8i − 1.45591i
\(747\) 0 0
\(748\) − 58446.1i − 2.85695i
\(749\) 0 0
\(750\) 0 0
\(751\) −7515.24 −0.365160 −0.182580 0.983191i \(-0.558445\pi\)
−0.182580 + 0.983191i \(0.558445\pi\)
\(752\) −3225.11 −0.156393
\(753\) 0 0
\(754\) 5153.75i 0.248924i
\(755\) −31502.0 −1.51851
\(756\) 0 0
\(757\) 23917.4 1.14834 0.574169 0.818737i \(-0.305325\pi\)
0.574169 + 0.818737i \(0.305325\pi\)
\(758\) 48275.6i 2.31326i
\(759\) 0 0
\(760\) −78067.5 −3.72606
\(761\) 12198.5 0.581073 0.290537 0.956864i \(-0.406166\pi\)
0.290537 + 0.956864i \(0.406166\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 8609.80i 0.407712i
\(765\) 0 0
\(766\) 5931.22i 0.279770i
\(767\) 2000.92i 0.0941969i
\(768\) 0 0
\(769\) − 2013.08i − 0.0943999i −0.998885 0.0471999i \(-0.984970\pi\)
0.998885 0.0471999i \(-0.0150298\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 11040.6 0.514713
\(773\) 30278.2 1.40884 0.704418 0.709785i \(-0.251208\pi\)
0.704418 + 0.709785i \(0.251208\pi\)
\(774\) 0 0
\(775\) − 3750.29i − 0.173825i
\(776\) −51549.0 −2.38467
\(777\) 0 0
\(778\) 63903.5 2.94480
\(779\) − 42493.8i − 1.95443i
\(780\) 0 0
\(781\) −4321.08 −0.197978
\(782\) 31509.2 1.44088
\(783\) 0 0
\(784\) 0 0
\(785\) 31095.5i 1.41382i
\(786\) 0 0
\(787\) − 28682.0i − 1.29911i −0.760313 0.649557i \(-0.774954\pi\)
0.760313 0.649557i \(-0.225046\pi\)
\(788\) − 83340.9i − 3.76763i
\(789\) 0 0
\(790\) 33214.0i 1.49582i
\(791\) 0 0
\(792\) 0 0
\(793\) −3789.97 −0.169717
\(794\) 15155.8 0.677404
\(795\) 0 0
\(796\) − 39012.4i − 1.73713i
\(797\) 27108.9 1.20483 0.602413 0.798184i \(-0.294206\pi\)
0.602413 + 0.798184i \(0.294206\pi\)
\(798\) 0 0
\(799\) −3952.36 −0.174999
\(800\) 948.519i 0.0419190i
\(801\) 0 0
\(802\) 54632.7 2.40542
\(803\) 41642.6 1.83006
\(804\) 0 0
\(805\) 0 0
\(806\) − 4260.11i − 0.186174i
\(807\) 0 0
\(808\) 26637.4i 1.15978i
\(809\) − 19778.1i − 0.859530i −0.902941 0.429765i \(-0.858596\pi\)
0.902941 0.429765i \(-0.141404\pi\)
\(810\) 0 0
\(811\) − 44456.6i − 1.92488i −0.271487 0.962442i \(-0.587516\pi\)
0.271487 0.962442i \(-0.412484\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 24798.4 1.06779
\(815\) −41828.8 −1.79779
\(816\) 0 0
\(817\) − 821.568i − 0.0351812i
\(818\) 60334.8 2.57892
\(819\) 0 0
\(820\) 51880.6 2.20945
\(821\) − 11499.2i − 0.488823i −0.969672 0.244411i \(-0.921405\pi\)
0.969672 0.244411i \(-0.0785948\pi\)
\(822\) 0 0
\(823\) 1193.10 0.0505332 0.0252666 0.999681i \(-0.491957\pi\)
0.0252666 + 0.999681i \(0.491957\pi\)
\(824\) −66912.2 −2.82888
\(825\) 0 0
\(826\) 0 0
\(827\) − 29552.9i − 1.24263i −0.783561 0.621315i \(-0.786599\pi\)
0.783561 0.621315i \(-0.213401\pi\)
\(828\) 0 0
\(829\) 13582.4i 0.569043i 0.958670 + 0.284521i \(0.0918347\pi\)
−0.958670 + 0.284521i \(0.908165\pi\)
\(830\) 27233.6i 1.13891i
\(831\) 0 0
\(832\) 5061.36i 0.210903i
\(833\) 0 0
\(834\) 0 0
\(835\) 4666.45 0.193400
\(836\) −138954. −5.74860
\(837\) 0 0
\(838\) − 40189.1i − 1.65669i
\(839\) −11623.6 −0.478297 −0.239149 0.970983i \(-0.576868\pi\)
−0.239149 + 0.970983i \(0.576868\pi\)
\(840\) 0 0
\(841\) 10080.1 0.413303
\(842\) − 27109.0i − 1.10955i
\(843\) 0 0
\(844\) 54136.1 2.20787
\(845\) −27042.3 −1.10093
\(846\) 0 0
\(847\) 0 0
\(848\) − 26432.7i − 1.07040i
\(849\) 0 0
\(850\) − 12717.7i − 0.513194i
\(851\) 8851.06i 0.356534i
\(852\) 0 0
\(853\) 1019.82i 0.0409354i 0.999791 + 0.0204677i \(0.00651553\pi\)
−0.999791 + 0.0204677i \(0.993484\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 842.064 0.0336228
\(857\) 8598.86 0.342744 0.171372 0.985206i \(-0.445180\pi\)
0.171372 + 0.985206i \(0.445180\pi\)
\(858\) 0 0
\(859\) 37160.7i 1.47603i 0.674786 + 0.738013i \(0.264236\pi\)
−0.674786 + 0.738013i \(0.735764\pi\)
\(860\) 1003.05 0.0397718
\(861\) 0 0
\(862\) −34241.4 −1.35298
\(863\) − 12279.4i − 0.484350i −0.970233 0.242175i \(-0.922139\pi\)
0.970233 0.242175i \(-0.0778608\pi\)
\(864\) 0 0
\(865\) 26714.9 1.05010
\(866\) 44821.2 1.75876
\(867\) 0 0
\(868\) 0 0
\(869\) 28940.6i 1.12974i
\(870\) 0 0
\(871\) 1450.97i 0.0564456i
\(872\) 35551.4i 1.38064i
\(873\) 0 0
\(874\) − 74912.3i − 2.89925i
\(875\) 0 0
\(876\) 0 0
\(877\) 31871.5 1.22717 0.613583 0.789630i \(-0.289727\pi\)
0.613583 + 0.789630i \(0.289727\pi\)
\(878\) −41266.4 −1.58619
\(879\) 0 0
\(880\) − 38843.3i − 1.48796i
\(881\) 29427.6 1.12536 0.562679 0.826676i \(-0.309771\pi\)
0.562679 + 0.826676i \(0.309771\pi\)
\(882\) 0 0
\(883\) 846.860 0.0322753 0.0161377 0.999870i \(-0.494863\pi\)
0.0161377 + 0.999870i \(0.494863\pi\)
\(884\) − 9564.35i − 0.363896i
\(885\) 0 0
\(886\) −63517.5 −2.40848
\(887\) −7880.13 −0.298296 −0.149148 0.988815i \(-0.547653\pi\)
−0.149148 + 0.988815i \(0.547653\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 1592.85i − 0.0599916i
\(891\) 0 0
\(892\) − 61064.2i − 2.29213i
\(893\) 9396.64i 0.352124i
\(894\) 0 0
\(895\) − 22462.4i − 0.838923i
\(896\) 0 0
\(897\) 0 0
\(898\) −68670.1 −2.55184
\(899\) 11827.8 0.438799
\(900\) 0 0
\(901\) − 32393.2i − 1.19775i
\(902\) 68281.2 2.52053
\(903\) 0 0
\(904\) 4484.71 0.164999
\(905\) 40814.5i 1.49914i
\(906\) 0 0
\(907\) 21293.2 0.779524 0.389762 0.920916i \(-0.372557\pi\)
0.389762 + 0.920916i \(0.372557\pi\)
\(908\) −9489.33 −0.346822
\(909\) 0 0
\(910\) 0 0
\(911\) 14634.3i 0.532225i 0.963942 + 0.266112i \(0.0857393\pi\)
−0.963942 + 0.266112i \(0.914261\pi\)
\(912\) 0 0
\(913\) 23729.7i 0.860173i
\(914\) 43660.8i 1.58005i
\(915\) 0 0
\(916\) 34618.2i 1.24871i
\(917\) 0 0
\(918\) 0 0
\(919\) 12197.2 0.437812 0.218906 0.975746i \(-0.429751\pi\)
0.218906 + 0.975746i \(0.429751\pi\)
\(920\) 44773.1 1.60448
\(921\) 0 0
\(922\) 4646.49i 0.165970i
\(923\) −707.119 −0.0252168
\(924\) 0 0
\(925\) 3572.46 0.126986
\(926\) − 58406.4i − 2.07274i
\(927\) 0 0
\(928\) −2991.48 −0.105819
\(929\) −24696.2 −0.872181 −0.436091 0.899903i \(-0.643637\pi\)
−0.436091 + 0.899903i \(0.643637\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 37383.2i 1.31387i
\(933\) 0 0
\(934\) − 24642.8i − 0.863315i
\(935\) − 47602.4i − 1.66499i
\(936\) 0 0
\(937\) − 14448.0i − 0.503730i −0.967762 0.251865i \(-0.918956\pi\)
0.967762 0.251865i \(-0.0810439\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −11472.3 −0.398070
\(941\) 31160.8 1.07950 0.539752 0.841824i \(-0.318518\pi\)
0.539752 + 0.841824i \(0.318518\pi\)
\(942\) 0 0
\(943\) 24370.9i 0.841598i
\(944\) 12707.0 0.438113
\(945\) 0 0
\(946\) 1320.14 0.0453715
\(947\) 10262.4i 0.352146i 0.984377 + 0.176073i \(0.0563395\pi\)
−0.984377 + 0.176073i \(0.943660\pi\)
\(948\) 0 0
\(949\) 6814.55 0.233098
\(950\) −30236.1 −1.03262
\(951\) 0 0
\(952\) 0 0
\(953\) − 12686.9i − 0.431236i −0.976478 0.215618i \(-0.930823\pi\)
0.976478 0.215618i \(-0.0691766\pi\)
\(954\) 0 0
\(955\) 7012.40i 0.237608i
\(956\) − 47297.3i − 1.60011i
\(957\) 0 0
\(958\) 74020.6i 2.49634i
\(959\) 0 0
\(960\) 0 0
\(961\) 20014.1 0.671816
\(962\) 4058.11 0.136007
\(963\) 0 0
\(964\) − 39422.8i − 1.31714i
\(965\) 8992.17 0.299967
\(966\) 0 0
\(967\) 16129.8 0.536401 0.268200 0.963363i \(-0.413571\pi\)
0.268200 + 0.963363i \(0.413571\pi\)
\(968\) − 59620.3i − 1.97962i
\(969\) 0 0
\(970\) −85764.6 −2.83890
\(971\) −52764.8 −1.74387 −0.871937 0.489618i \(-0.837136\pi\)
−0.871937 + 0.489618i \(0.837136\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 76930.8i − 2.53082i
\(975\) 0 0
\(976\) 24068.5i 0.789360i
\(977\) 53993.2i 1.76806i 0.467430 + 0.884030i \(0.345180\pi\)
−0.467430 + 0.884030i \(0.654820\pi\)
\(978\) 0 0
\(979\) − 1387.91i − 0.0453093i
\(980\) 0 0
\(981\) 0 0
\(982\) 87889.4 2.85607
\(983\) 52238.4 1.69496 0.847480 0.530827i \(-0.178119\pi\)
0.847480 + 0.530827i \(0.178119\pi\)
\(984\) 0 0
\(985\) − 67878.3i − 2.19572i
\(986\) 40109.7 1.29549
\(987\) 0 0
\(988\) −22739.0 −0.732211
\(989\) 471.184i 0.0151494i
\(990\) 0 0
\(991\) −9624.42 −0.308506 −0.154253 0.988031i \(-0.549297\pi\)
−0.154253 + 0.988031i \(0.549297\pi\)
\(992\) 2472.77 0.0791436
\(993\) 0 0
\(994\) 0 0
\(995\) − 31774.3i − 1.01237i
\(996\) 0 0
\(997\) 28894.1i 0.917838i 0.888478 + 0.458919i \(0.151763\pi\)
−0.888478 + 0.458919i \(0.848237\pi\)
\(998\) 51206.6i 1.62416i
\(999\) 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 441.4.c.a.440.1 16
3.2 odd 2 inner 441.4.c.a.440.16 16
7.2 even 3 63.4.p.a.17.1 16
7.3 odd 6 63.4.p.a.26.8 yes 16
7.4 even 3 441.4.p.c.215.8 16
7.5 odd 6 441.4.p.c.80.1 16
7.6 odd 2 inner 441.4.c.a.440.2 16
21.2 odd 6 63.4.p.a.17.8 yes 16
21.5 even 6 441.4.p.c.80.8 16
21.11 odd 6 441.4.p.c.215.1 16
21.17 even 6 63.4.p.a.26.1 yes 16
21.20 even 2 inner 441.4.c.a.440.15 16
28.3 even 6 1008.4.bt.a.593.2 16
28.23 odd 6 1008.4.bt.a.17.7 16
84.23 even 6 1008.4.bt.a.17.2 16
84.59 odd 6 1008.4.bt.a.593.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.p.a.17.1 16 7.2 even 3
63.4.p.a.17.8 yes 16 21.2 odd 6
63.4.p.a.26.1 yes 16 21.17 even 6
63.4.p.a.26.8 yes 16 7.3 odd 6
441.4.c.a.440.1 16 1.1 even 1 trivial
441.4.c.a.440.2 16 7.6 odd 2 inner
441.4.c.a.440.15 16 21.20 even 2 inner
441.4.c.a.440.16 16 3.2 odd 2 inner
441.4.p.c.80.1 16 7.5 odd 6
441.4.p.c.80.8 16 21.5 even 6
441.4.p.c.215.1 16 21.11 odd 6
441.4.p.c.215.8 16 7.4 even 3
1008.4.bt.a.17.2 16 84.23 even 6
1008.4.bt.a.17.7 16 28.23 odd 6
1008.4.bt.a.593.2 16 28.3 even 6
1008.4.bt.a.593.7 16 84.59 odd 6