Properties

Label 441.4.a.n.1.2
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -7.82843 q^{4} -0.100505 q^{5} -6.55635 q^{8} +O(q^{10})\) \(q+0.414214 q^{2} -7.82843 q^{4} -0.100505 q^{5} -6.55635 q^{8} -0.0416306 q^{10} +43.9411 q^{11} +16.6447 q^{13} +59.9117 q^{16} -121.640 q^{17} +127.113 q^{19} +0.786797 q^{20} +18.2010 q^{22} -53.5980 q^{23} -124.990 q^{25} +6.89444 q^{26} -235.681 q^{29} +18.7107 q^{31} +77.2670 q^{32} -50.3848 q^{34} -191.882 q^{37} +52.6518 q^{38} +0.658946 q^{40} -319.713 q^{41} -218.579 q^{43} -343.990 q^{44} -22.2010 q^{46} +401.553 q^{47} -51.7725 q^{50} -130.302 q^{52} -643.117 q^{53} -4.41631 q^{55} -97.6224 q^{58} -11.6123 q^{59} +12.2426 q^{61} +7.75022 q^{62} -447.288 q^{64} -1.67287 q^{65} +669.048 q^{67} +952.247 q^{68} -822.098 q^{71} -515.100 q^{73} -79.4802 q^{74} -995.092 q^{76} -805.754 q^{79} -6.02143 q^{80} -132.429 q^{82} -394.863 q^{83} +12.2254 q^{85} -90.5382 q^{86} -288.093 q^{88} +673.418 q^{89} +419.588 q^{92} +166.329 q^{94} -12.7755 q^{95} +1091.11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 10q^{4} - 20q^{5} + 18q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 10q^{4} - 20q^{5} + 18q^{8} + 48q^{10} + 20q^{11} + 104q^{13} + 18q^{16} - 116q^{17} + 192q^{19} + 44q^{20} + 76q^{22} - 28q^{23} + 146q^{25} - 204q^{26} - 296q^{29} - 104q^{31} - 18q^{32} - 64q^{34} - 248q^{37} - 104q^{38} - 488q^{40} - 20q^{41} - 720q^{43} - 292q^{44} - 84q^{46} + 96q^{47} - 706q^{50} - 320q^{52} - 268q^{53} + 472q^{55} + 48q^{58} + 616q^{59} + 16q^{61} + 304q^{62} + 118q^{64} - 1740q^{65} - 144q^{67} + 940q^{68} - 988q^{71} + 104q^{73} + 56q^{74} - 1136q^{76} - 944q^{79} + 828q^{80} - 856q^{82} - 1016q^{83} - 100q^{85} + 1120q^{86} - 876q^{88} + 388q^{89} + 364q^{92} + 904q^{94} - 1304q^{95} + 488q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.146447 0.0732233 0.997316i \(-0.476671\pi\)
0.0732233 + 0.997316i \(0.476671\pi\)
\(3\) 0 0
\(4\) −7.82843 −0.978553
\(5\) −0.100505 −0.00898945 −0.00449472 0.999990i \(-0.501431\pi\)
−0.00449472 + 0.999990i \(0.501431\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −6.55635 −0.289752
\(9\) 0 0
\(10\) −0.0416306 −0.00131647
\(11\) 43.9411 1.20443 0.602216 0.798333i \(-0.294285\pi\)
0.602216 + 0.798333i \(0.294285\pi\)
\(12\) 0 0
\(13\) 16.6447 0.355108 0.177554 0.984111i \(-0.443182\pi\)
0.177554 + 0.984111i \(0.443182\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 59.9117 0.936120
\(17\) −121.640 −1.73541 −0.867704 0.497081i \(-0.834405\pi\)
−0.867704 + 0.497081i \(0.834405\pi\)
\(18\) 0 0
\(19\) 127.113 1.53482 0.767412 0.641154i \(-0.221544\pi\)
0.767412 + 0.641154i \(0.221544\pi\)
\(20\) 0.786797 0.00879665
\(21\) 0 0
\(22\) 18.2010 0.176385
\(23\) −53.5980 −0.485911 −0.242955 0.970037i \(-0.578117\pi\)
−0.242955 + 0.970037i \(0.578117\pi\)
\(24\) 0 0
\(25\) −124.990 −0.999919
\(26\) 6.89444 0.0520043
\(27\) 0 0
\(28\) 0 0
\(29\) −235.681 −1.50913 −0.754567 0.656223i \(-0.772153\pi\)
−0.754567 + 0.656223i \(0.772153\pi\)
\(30\) 0 0
\(31\) 18.7107 0.108404 0.0542022 0.998530i \(-0.482738\pi\)
0.0542022 + 0.998530i \(0.482738\pi\)
\(32\) 77.2670 0.426844
\(33\) 0 0
\(34\) −50.3848 −0.254145
\(35\) 0 0
\(36\) 0 0
\(37\) −191.882 −0.852574 −0.426287 0.904588i \(-0.640179\pi\)
−0.426287 + 0.904588i \(0.640179\pi\)
\(38\) 52.6518 0.224770
\(39\) 0 0
\(40\) 0.658946 0.00260471
\(41\) −319.713 −1.21782 −0.608912 0.793238i \(-0.708394\pi\)
−0.608912 + 0.793238i \(0.708394\pi\)
\(42\) 0 0
\(43\) −218.579 −0.775184 −0.387592 0.921831i \(-0.626693\pi\)
−0.387592 + 0.921831i \(0.626693\pi\)
\(44\) −343.990 −1.17860
\(45\) 0 0
\(46\) −22.2010 −0.0711600
\(47\) 401.553 1.24623 0.623113 0.782132i \(-0.285868\pi\)
0.623113 + 0.782132i \(0.285868\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −51.7725 −0.146435
\(51\) 0 0
\(52\) −130.302 −0.347492
\(53\) −643.117 −1.66677 −0.833386 0.552692i \(-0.813601\pi\)
−0.833386 + 0.552692i \(0.813601\pi\)
\(54\) 0 0
\(55\) −4.41631 −0.0108272
\(56\) 0 0
\(57\) 0 0
\(58\) −97.6224 −0.221008
\(59\) −11.6123 −0.0256235 −0.0128118 0.999918i \(-0.504078\pi\)
−0.0128118 + 0.999918i \(0.504078\pi\)
\(60\) 0 0
\(61\) 12.2426 0.0256969 0.0128484 0.999917i \(-0.495910\pi\)
0.0128484 + 0.999917i \(0.495910\pi\)
\(62\) 7.75022 0.0158755
\(63\) 0 0
\(64\) −447.288 −0.873610
\(65\) −1.67287 −0.00319222
\(66\) 0 0
\(67\) 669.048 1.21996 0.609979 0.792417i \(-0.291178\pi\)
0.609979 + 0.792417i \(0.291178\pi\)
\(68\) 952.247 1.69819
\(69\) 0 0
\(70\) 0 0
\(71\) −822.098 −1.37416 −0.687078 0.726584i \(-0.741107\pi\)
−0.687078 + 0.726584i \(0.741107\pi\)
\(72\) 0 0
\(73\) −515.100 −0.825861 −0.412930 0.910763i \(-0.635495\pi\)
−0.412930 + 0.910763i \(0.635495\pi\)
\(74\) −79.4802 −0.124857
\(75\) 0 0
\(76\) −995.092 −1.50191
\(77\) 0 0
\(78\) 0 0
\(79\) −805.754 −1.14752 −0.573762 0.819022i \(-0.694517\pi\)
−0.573762 + 0.819022i \(0.694517\pi\)
\(80\) −6.02143 −0.00841520
\(81\) 0 0
\(82\) −132.429 −0.178346
\(83\) −394.863 −0.522191 −0.261095 0.965313i \(-0.584084\pi\)
−0.261095 + 0.965313i \(0.584084\pi\)
\(84\) 0 0
\(85\) 12.2254 0.0156004
\(86\) −90.5382 −0.113523
\(87\) 0 0
\(88\) −288.093 −0.348987
\(89\) 673.418 0.802047 0.401024 0.916068i \(-0.368655\pi\)
0.401024 + 0.916068i \(0.368655\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 419.588 0.475490
\(93\) 0 0
\(94\) 166.329 0.182505
\(95\) −12.7755 −0.0137972
\(96\) 0 0
\(97\) 1091.11 1.14212 0.571061 0.820908i \(-0.306532\pi\)
0.571061 + 0.820908i \(0.306532\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 978.474 0.978474
\(101\) −1370.79 −1.35048 −0.675242 0.737597i \(-0.735961\pi\)
−0.675242 + 0.737597i \(0.735961\pi\)
\(102\) 0 0
\(103\) 1413.96 1.35263 0.676316 0.736611i \(-0.263575\pi\)
0.676316 + 0.736611i \(0.263575\pi\)
\(104\) −109.128 −0.102893
\(105\) 0 0
\(106\) −266.388 −0.244093
\(107\) 343.539 0.310385 0.155192 0.987884i \(-0.450400\pi\)
0.155192 + 0.987884i \(0.450400\pi\)
\(108\) 0 0
\(109\) −317.657 −0.279138 −0.139569 0.990212i \(-0.544572\pi\)
−0.139569 + 0.990212i \(0.544572\pi\)
\(110\) −1.82929 −0.00158560
\(111\) 0 0
\(112\) 0 0
\(113\) −798.373 −0.664643 −0.332321 0.943166i \(-0.607832\pi\)
−0.332321 + 0.943166i \(0.607832\pi\)
\(114\) 0 0
\(115\) 5.38687 0.00436807
\(116\) 1845.01 1.47677
\(117\) 0 0
\(118\) −4.80996 −0.00375248
\(119\) 0 0
\(120\) 0 0
\(121\) 599.823 0.450656
\(122\) 5.07107 0.00376322
\(123\) 0 0
\(124\) −146.475 −0.106080
\(125\) 25.1253 0.0179782
\(126\) 0 0
\(127\) 1071.40 0.748593 0.374297 0.927309i \(-0.377884\pi\)
0.374297 + 0.927309i \(0.377884\pi\)
\(128\) −803.409 −0.554781
\(129\) 0 0
\(130\) −0.692927 −0.000467490 0
\(131\) −2515.02 −1.67739 −0.838695 0.544601i \(-0.816681\pi\)
−0.838695 + 0.544601i \(0.816681\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 277.129 0.178659
\(135\) 0 0
\(136\) 797.512 0.502839
\(137\) −251.064 −0.156568 −0.0782841 0.996931i \(-0.524944\pi\)
−0.0782841 + 0.996931i \(0.524944\pi\)
\(138\) 0 0
\(139\) −886.067 −0.540685 −0.270343 0.962764i \(-0.587137\pi\)
−0.270343 + 0.962764i \(0.587137\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −340.524 −0.201240
\(143\) 731.385 0.427703
\(144\) 0 0
\(145\) 23.6872 0.0135663
\(146\) −213.361 −0.120945
\(147\) 0 0
\(148\) 1502.14 0.834289
\(149\) −582.626 −0.320339 −0.160170 0.987090i \(-0.551204\pi\)
−0.160170 + 0.987090i \(0.551204\pi\)
\(150\) 0 0
\(151\) −2811.76 −1.51535 −0.757676 0.652631i \(-0.773665\pi\)
−0.757676 + 0.652631i \(0.773665\pi\)
\(152\) −833.395 −0.444719
\(153\) 0 0
\(154\) 0 0
\(155\) −1.88052 −0.000974496 0
\(156\) 0 0
\(157\) 1691.34 0.859770 0.429885 0.902884i \(-0.358554\pi\)
0.429885 + 0.902884i \(0.358554\pi\)
\(158\) −333.754 −0.168051
\(159\) 0 0
\(160\) −7.76573 −0.00383709
\(161\) 0 0
\(162\) 0 0
\(163\) −40.7232 −0.0195686 −0.00978432 0.999952i \(-0.503114\pi\)
−0.00978432 + 0.999952i \(0.503114\pi\)
\(164\) 2502.85 1.19170
\(165\) 0 0
\(166\) −163.558 −0.0764731
\(167\) 2900.47 1.34398 0.671990 0.740560i \(-0.265440\pi\)
0.671990 + 0.740560i \(0.265440\pi\)
\(168\) 0 0
\(169\) −1919.96 −0.873899
\(170\) 5.06393 0.00228462
\(171\) 0 0
\(172\) 1711.13 0.758559
\(173\) −2146.15 −0.943171 −0.471585 0.881820i \(-0.656318\pi\)
−0.471585 + 0.881820i \(0.656318\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2632.59 1.12749
\(177\) 0 0
\(178\) 278.939 0.117457
\(179\) −1203.54 −0.502552 −0.251276 0.967916i \(-0.580850\pi\)
−0.251276 + 0.967916i \(0.580850\pi\)
\(180\) 0 0
\(181\) 2990.47 1.22807 0.614033 0.789280i \(-0.289546\pi\)
0.614033 + 0.789280i \(0.289546\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 351.407 0.140794
\(185\) 19.2851 0.00766417
\(186\) 0 0
\(187\) −5344.98 −2.09018
\(188\) −3143.53 −1.21950
\(189\) 0 0
\(190\) −5.29177 −0.00202056
\(191\) 2807.41 1.06354 0.531772 0.846887i \(-0.321526\pi\)
0.531772 + 0.846887i \(0.321526\pi\)
\(192\) 0 0
\(193\) 3336.37 1.24434 0.622169 0.782883i \(-0.286252\pi\)
0.622169 + 0.782883i \(0.286252\pi\)
\(194\) 451.954 0.167260
\(195\) 0 0
\(196\) 0 0
\(197\) 4226.65 1.52861 0.764305 0.644855i \(-0.223082\pi\)
0.764305 + 0.644855i \(0.223082\pi\)
\(198\) 0 0
\(199\) −4385.69 −1.56228 −0.781140 0.624356i \(-0.785361\pi\)
−0.781140 + 0.624356i \(0.785361\pi\)
\(200\) 819.477 0.289729
\(201\) 0 0
\(202\) −567.800 −0.197774
\(203\) 0 0
\(204\) 0 0
\(205\) 32.1328 0.0109476
\(206\) 585.680 0.198088
\(207\) 0 0
\(208\) 997.210 0.332423
\(209\) 5585.48 1.84859
\(210\) 0 0
\(211\) 2291.56 0.747665 0.373833 0.927496i \(-0.378043\pi\)
0.373833 + 0.927496i \(0.378043\pi\)
\(212\) 5034.59 1.63103
\(213\) 0 0
\(214\) 142.299 0.0454548
\(215\) 21.9683 0.00696848
\(216\) 0 0
\(217\) 0 0
\(218\) −131.578 −0.0408788
\(219\) 0 0
\(220\) 34.5727 0.0105950
\(221\) −2024.65 −0.616257
\(222\) 0 0
\(223\) 217.970 0.0654544 0.0327272 0.999464i \(-0.489581\pi\)
0.0327272 + 0.999464i \(0.489581\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −330.697 −0.0973347
\(227\) 1835.36 0.536639 0.268320 0.963330i \(-0.413532\pi\)
0.268320 + 0.963330i \(0.413532\pi\)
\(228\) 0 0
\(229\) −2774.01 −0.800488 −0.400244 0.916409i \(-0.631075\pi\)
−0.400244 + 0.916409i \(0.631075\pi\)
\(230\) 2.23131 0.000639689 0
\(231\) 0 0
\(232\) 1545.21 0.437275
\(233\) 988.712 0.277994 0.138997 0.990293i \(-0.455612\pi\)
0.138997 + 0.990293i \(0.455612\pi\)
\(234\) 0 0
\(235\) −40.3581 −0.0112029
\(236\) 90.9058 0.0250740
\(237\) 0 0
\(238\) 0 0
\(239\) 837.928 0.226783 0.113391 0.993550i \(-0.463829\pi\)
0.113391 + 0.993550i \(0.463829\pi\)
\(240\) 0 0
\(241\) 3454.99 0.923466 0.461733 0.887019i \(-0.347228\pi\)
0.461733 + 0.887019i \(0.347228\pi\)
\(242\) 248.455 0.0659970
\(243\) 0 0
\(244\) −95.8406 −0.0251458
\(245\) 0 0
\(246\) 0 0
\(247\) 2115.75 0.545028
\(248\) −122.674 −0.0314104
\(249\) 0 0
\(250\) 10.4072 0.00263284
\(251\) 5635.01 1.41705 0.708523 0.705688i \(-0.249362\pi\)
0.708523 + 0.705688i \(0.249362\pi\)
\(252\) 0 0
\(253\) −2355.16 −0.585246
\(254\) 443.788 0.109629
\(255\) 0 0
\(256\) 3245.52 0.792364
\(257\) −2271.16 −0.551248 −0.275624 0.961265i \(-0.588885\pi\)
−0.275624 + 0.961265i \(0.588885\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 13.0960 0.00312376
\(261\) 0 0
\(262\) −1041.75 −0.245648
\(263\) −163.867 −0.0384201 −0.0192101 0.999815i \(-0.506115\pi\)
−0.0192101 + 0.999815i \(0.506115\pi\)
\(264\) 0 0
\(265\) 64.6365 0.0149834
\(266\) 0 0
\(267\) 0 0
\(268\) −5237.59 −1.19379
\(269\) −5167.10 −1.17116 −0.585582 0.810613i \(-0.699134\pi\)
−0.585582 + 0.810613i \(0.699134\pi\)
\(270\) 0 0
\(271\) 1622.27 0.363638 0.181819 0.983332i \(-0.441801\pi\)
0.181819 + 0.983332i \(0.441801\pi\)
\(272\) −7287.63 −1.62455
\(273\) 0 0
\(274\) −103.994 −0.0229289
\(275\) −5492.20 −1.20433
\(276\) 0 0
\(277\) −4612.37 −1.00047 −0.500235 0.865890i \(-0.666753\pi\)
−0.500235 + 0.865890i \(0.666753\pi\)
\(278\) −367.021 −0.0791815
\(279\) 0 0
\(280\) 0 0
\(281\) 2125.22 0.451174 0.225587 0.974223i \(-0.427570\pi\)
0.225587 + 0.974223i \(0.427570\pi\)
\(282\) 0 0
\(283\) −2571.54 −0.540149 −0.270075 0.962839i \(-0.587048\pi\)
−0.270075 + 0.962839i \(0.587048\pi\)
\(284\) 6435.73 1.34468
\(285\) 0 0
\(286\) 302.950 0.0626356
\(287\) 0 0
\(288\) 0 0
\(289\) 9883.19 2.01164
\(290\) 9.81154 0.00198674
\(291\) 0 0
\(292\) 4032.42 0.808149
\(293\) −3324.96 −0.662957 −0.331478 0.943463i \(-0.607547\pi\)
−0.331478 + 0.943463i \(0.607547\pi\)
\(294\) 0 0
\(295\) 1.16709 0.000230341 0
\(296\) 1258.05 0.247035
\(297\) 0 0
\(298\) −241.331 −0.0469126
\(299\) −892.120 −0.172551
\(300\) 0 0
\(301\) 0 0
\(302\) −1164.67 −0.221918
\(303\) 0 0
\(304\) 7615.54 1.43678
\(305\) −1.23045 −0.000231001 0
\(306\) 0 0
\(307\) −887.096 −0.164916 −0.0824580 0.996595i \(-0.526277\pi\)
−0.0824580 + 0.996595i \(0.526277\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.778936 −0.000142712 0
\(311\) 4510.82 0.822460 0.411230 0.911532i \(-0.365099\pi\)
0.411230 + 0.911532i \(0.365099\pi\)
\(312\) 0 0
\(313\) 3715.78 0.671018 0.335509 0.942037i \(-0.391092\pi\)
0.335509 + 0.942037i \(0.391092\pi\)
\(314\) 700.577 0.125910
\(315\) 0 0
\(316\) 6307.79 1.12291
\(317\) −6954.52 −1.23219 −0.616096 0.787671i \(-0.711287\pi\)
−0.616096 + 0.787671i \(0.711287\pi\)
\(318\) 0 0
\(319\) −10356.1 −1.81765
\(320\) 44.9548 0.00785327
\(321\) 0 0
\(322\) 0 0
\(323\) −15461.9 −2.66355
\(324\) 0 0
\(325\) −2080.41 −0.355079
\(326\) −16.8681 −0.00286576
\(327\) 0 0
\(328\) 2096.15 0.352867
\(329\) 0 0
\(330\) 0 0
\(331\) −9863.18 −1.63785 −0.818926 0.573899i \(-0.805430\pi\)
−0.818926 + 0.573899i \(0.805430\pi\)
\(332\) 3091.16 0.510992
\(333\) 0 0
\(334\) 1201.41 0.196821
\(335\) −67.2427 −0.0109668
\(336\) 0 0
\(337\) −5945.06 −0.960974 −0.480487 0.877002i \(-0.659540\pi\)
−0.480487 + 0.877002i \(0.659540\pi\)
\(338\) −795.272 −0.127979
\(339\) 0 0
\(340\) −95.7056 −0.0152658
\(341\) 822.168 0.130566
\(342\) 0 0
\(343\) 0 0
\(344\) 1433.08 0.224612
\(345\) 0 0
\(346\) −888.963 −0.138124
\(347\) −1169.57 −0.180939 −0.0904697 0.995899i \(-0.528837\pi\)
−0.0904697 + 0.995899i \(0.528837\pi\)
\(348\) 0 0
\(349\) −9176.66 −1.40749 −0.703747 0.710451i \(-0.748491\pi\)
−0.703747 + 0.710451i \(0.748491\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3395.20 0.514104
\(353\) −10587.2 −1.59632 −0.798158 0.602448i \(-0.794192\pi\)
−0.798158 + 0.602448i \(0.794192\pi\)
\(354\) 0 0
\(355\) 82.6250 0.0123529
\(356\) −5271.81 −0.784846
\(357\) 0 0
\(358\) −498.522 −0.0735970
\(359\) −8615.21 −1.26656 −0.633278 0.773924i \(-0.718291\pi\)
−0.633278 + 0.773924i \(0.718291\pi\)
\(360\) 0 0
\(361\) 9298.64 1.35568
\(362\) 1238.69 0.179846
\(363\) 0 0
\(364\) 0 0
\(365\) 51.7701 0.00742403
\(366\) 0 0
\(367\) 8297.79 1.18022 0.590110 0.807323i \(-0.299084\pi\)
0.590110 + 0.807323i \(0.299084\pi\)
\(368\) −3211.15 −0.454871
\(369\) 0 0
\(370\) 7.98817 0.00112239
\(371\) 0 0
\(372\) 0 0
\(373\) −5123.86 −0.711269 −0.355634 0.934625i \(-0.615735\pi\)
−0.355634 + 0.934625i \(0.615735\pi\)
\(374\) −2213.96 −0.306100
\(375\) 0 0
\(376\) −2632.72 −0.361097
\(377\) −3922.83 −0.535905
\(378\) 0 0
\(379\) −1502.49 −0.203635 −0.101817 0.994803i \(-0.532466\pi\)
−0.101817 + 0.994803i \(0.532466\pi\)
\(380\) 100.012 0.0135013
\(381\) 0 0
\(382\) 1162.87 0.155753
\(383\) 10872.9 1.45060 0.725301 0.688431i \(-0.241700\pi\)
0.725301 + 0.688431i \(0.241700\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1381.97 0.182229
\(387\) 0 0
\(388\) −8541.71 −1.11763
\(389\) −4618.99 −0.602036 −0.301018 0.953618i \(-0.597326\pi\)
−0.301018 + 0.953618i \(0.597326\pi\)
\(390\) 0 0
\(391\) 6519.64 0.843254
\(392\) 0 0
\(393\) 0 0
\(394\) 1750.73 0.223860
\(395\) 80.9824 0.0103156
\(396\) 0 0
\(397\) 9606.95 1.21451 0.607253 0.794508i \(-0.292271\pi\)
0.607253 + 0.794508i \(0.292271\pi\)
\(398\) −1816.61 −0.228791
\(399\) 0 0
\(400\) −7488.36 −0.936044
\(401\) 10501.0 1.30772 0.653862 0.756614i \(-0.273148\pi\)
0.653862 + 0.756614i \(0.273148\pi\)
\(402\) 0 0
\(403\) 311.433 0.0384952
\(404\) 10731.1 1.32152
\(405\) 0 0
\(406\) 0 0
\(407\) −8431.52 −1.02687
\(408\) 0 0
\(409\) −12066.9 −1.45885 −0.729427 0.684059i \(-0.760213\pi\)
−0.729427 + 0.684059i \(0.760213\pi\)
\(410\) 13.3098 0.00160323
\(411\) 0 0
\(412\) −11069.0 −1.32362
\(413\) 0 0
\(414\) 0 0
\(415\) 39.6857 0.00469421
\(416\) 1286.08 0.151576
\(417\) 0 0
\(418\) 2313.58 0.270720
\(419\) 6366.31 0.742278 0.371139 0.928577i \(-0.378967\pi\)
0.371139 + 0.928577i \(0.378967\pi\)
\(420\) 0 0
\(421\) −4731.84 −0.547781 −0.273890 0.961761i \(-0.588311\pi\)
−0.273890 + 0.961761i \(0.588311\pi\)
\(422\) 949.194 0.109493
\(423\) 0 0
\(424\) 4216.50 0.482951
\(425\) 15203.7 1.73527
\(426\) 0 0
\(427\) 0 0
\(428\) −2689.37 −0.303728
\(429\) 0 0
\(430\) 9.09955 0.00102051
\(431\) −3752.78 −0.419409 −0.209704 0.977765i \(-0.567250\pi\)
−0.209704 + 0.977765i \(0.567250\pi\)
\(432\) 0 0
\(433\) 11709.2 1.29956 0.649780 0.760122i \(-0.274861\pi\)
0.649780 + 0.760122i \(0.274861\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2486.75 0.273151
\(437\) −6812.98 −0.745788
\(438\) 0 0
\(439\) −14924.7 −1.62259 −0.811296 0.584635i \(-0.801238\pi\)
−0.811296 + 0.584635i \(0.801238\pi\)
\(440\) 28.9548 0.00313720
\(441\) 0 0
\(442\) −838.638 −0.0902487
\(443\) 8517.66 0.913513 0.456756 0.889592i \(-0.349011\pi\)
0.456756 + 0.889592i \(0.349011\pi\)
\(444\) 0 0
\(445\) −67.6820 −0.00720996
\(446\) 90.2860 0.00958557
\(447\) 0 0
\(448\) 0 0
\(449\) 5965.73 0.627038 0.313519 0.949582i \(-0.398492\pi\)
0.313519 + 0.949582i \(0.398492\pi\)
\(450\) 0 0
\(451\) −14048.5 −1.46678
\(452\) 6250.01 0.650389
\(453\) 0 0
\(454\) 760.231 0.0785890
\(455\) 0 0
\(456\) 0 0
\(457\) −13860.6 −1.41875 −0.709376 0.704830i \(-0.751023\pi\)
−0.709376 + 0.704830i \(0.751023\pi\)
\(458\) −1149.03 −0.117229
\(459\) 0 0
\(460\) −42.1707 −0.00427439
\(461\) 149.312 0.0150850 0.00754249 0.999972i \(-0.497599\pi\)
0.00754249 + 0.999972i \(0.497599\pi\)
\(462\) 0 0
\(463\) 5403.95 0.542425 0.271213 0.962519i \(-0.412575\pi\)
0.271213 + 0.962519i \(0.412575\pi\)
\(464\) −14120.1 −1.41273
\(465\) 0 0
\(466\) 409.538 0.0407113
\(467\) −3704.66 −0.367090 −0.183545 0.983011i \(-0.558757\pi\)
−0.183545 + 0.983011i \(0.558757\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −16.7169 −0.00164062
\(471\) 0 0
\(472\) 76.1341 0.00742448
\(473\) −9604.59 −0.933657
\(474\) 0 0
\(475\) −15887.8 −1.53470
\(476\) 0 0
\(477\) 0 0
\(478\) 347.081 0.0332115
\(479\) 10671.8 1.01796 0.508982 0.860777i \(-0.330022\pi\)
0.508982 + 0.860777i \(0.330022\pi\)
\(480\) 0 0
\(481\) −3193.82 −0.302756
\(482\) 1431.10 0.135238
\(483\) 0 0
\(484\) −4695.67 −0.440990
\(485\) −109.662 −0.0102670
\(486\) 0 0
\(487\) 5853.92 0.544695 0.272348 0.962199i \(-0.412200\pi\)
0.272348 + 0.962199i \(0.412200\pi\)
\(488\) −80.2670 −0.00744573
\(489\) 0 0
\(490\) 0 0
\(491\) −4065.31 −0.373656 −0.186828 0.982393i \(-0.559821\pi\)
−0.186828 + 0.982393i \(0.559821\pi\)
\(492\) 0 0
\(493\) 28668.2 2.61896
\(494\) 876.371 0.0798174
\(495\) 0 0
\(496\) 1120.99 0.101480
\(497\) 0 0
\(498\) 0 0
\(499\) 4811.19 0.431620 0.215810 0.976435i \(-0.430761\pi\)
0.215810 + 0.976435i \(0.430761\pi\)
\(500\) −196.691 −0.0175926
\(501\) 0 0
\(502\) 2334.10 0.207522
\(503\) −17001.2 −1.50705 −0.753526 0.657418i \(-0.771649\pi\)
−0.753526 + 0.657418i \(0.771649\pi\)
\(504\) 0 0
\(505\) 137.771 0.0121401
\(506\) −975.537 −0.0857074
\(507\) 0 0
\(508\) −8387.37 −0.732538
\(509\) −13797.2 −1.20148 −0.600738 0.799446i \(-0.705126\pi\)
−0.600738 + 0.799446i \(0.705126\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 7771.61 0.670820
\(513\) 0 0
\(514\) −940.744 −0.0807284
\(515\) −142.110 −0.0121594
\(516\) 0 0
\(517\) 17644.7 1.50099
\(518\) 0 0
\(519\) 0 0
\(520\) 10.9679 0.000924954 0
\(521\) −3936.61 −0.331029 −0.165515 0.986207i \(-0.552928\pi\)
−0.165515 + 0.986207i \(0.552928\pi\)
\(522\) 0 0
\(523\) 17459.3 1.45973 0.729866 0.683590i \(-0.239582\pi\)
0.729866 + 0.683590i \(0.239582\pi\)
\(524\) 19688.6 1.64142
\(525\) 0 0
\(526\) −67.8760 −0.00562649
\(527\) −2275.96 −0.188126
\(528\) 0 0
\(529\) −9294.26 −0.763891
\(530\) 26.7733 0.00219426
\(531\) 0 0
\(532\) 0 0
\(533\) −5321.51 −0.432458
\(534\) 0 0
\(535\) −34.5274 −0.00279019
\(536\) −4386.51 −0.353486
\(537\) 0 0
\(538\) −2140.28 −0.171513
\(539\) 0 0
\(540\) 0 0
\(541\) 19101.3 1.51798 0.758992 0.651100i \(-0.225692\pi\)
0.758992 + 0.651100i \(0.225692\pi\)
\(542\) 671.967 0.0532536
\(543\) 0 0
\(544\) −9398.73 −0.740749
\(545\) 31.9261 0.00250929
\(546\) 0 0
\(547\) 15413.5 1.20481 0.602407 0.798189i \(-0.294208\pi\)
0.602407 + 0.798189i \(0.294208\pi\)
\(548\) 1965.44 0.153210
\(549\) 0 0
\(550\) −2274.94 −0.176371
\(551\) −29958.1 −2.31626
\(552\) 0 0
\(553\) 0 0
\(554\) −1910.51 −0.146516
\(555\) 0 0
\(556\) 6936.51 0.529089
\(557\) 20492.9 1.55891 0.779453 0.626460i \(-0.215497\pi\)
0.779453 + 0.626460i \(0.215497\pi\)
\(558\) 0 0
\(559\) −3638.17 −0.275274
\(560\) 0 0
\(561\) 0 0
\(562\) 880.294 0.0660729
\(563\) 7142.49 0.534671 0.267336 0.963603i \(-0.413857\pi\)
0.267336 + 0.963603i \(0.413857\pi\)
\(564\) 0 0
\(565\) 80.2406 0.00597477
\(566\) −1065.17 −0.0791030
\(567\) 0 0
\(568\) 5389.96 0.398165
\(569\) −4097.91 −0.301922 −0.150961 0.988540i \(-0.548237\pi\)
−0.150961 + 0.988540i \(0.548237\pi\)
\(570\) 0 0
\(571\) −2838.48 −0.208033 −0.104016 0.994576i \(-0.533169\pi\)
−0.104016 + 0.994576i \(0.533169\pi\)
\(572\) −5725.60 −0.418530
\(573\) 0 0
\(574\) 0 0
\(575\) 6699.21 0.485872
\(576\) 0 0
\(577\) −15464.5 −1.11576 −0.557881 0.829921i \(-0.688385\pi\)
−0.557881 + 0.829921i \(0.688385\pi\)
\(578\) 4093.75 0.294598
\(579\) 0 0
\(580\) −185.433 −0.0132753
\(581\) 0 0
\(582\) 0 0
\(583\) −28259.3 −2.00751
\(584\) 3377.17 0.239295
\(585\) 0 0
\(586\) −1377.24 −0.0970878
\(587\) −14003.6 −0.984652 −0.492326 0.870411i \(-0.663853\pi\)
−0.492326 + 0.870411i \(0.663853\pi\)
\(588\) 0 0
\(589\) 2378.36 0.166382
\(590\) 0.483425 3.37327e−5 0
\(591\) 0 0
\(592\) −11496.0 −0.798112
\(593\) −6504.50 −0.450435 −0.225217 0.974309i \(-0.572309\pi\)
−0.225217 + 0.974309i \(0.572309\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4561.04 0.313469
\(597\) 0 0
\(598\) −369.528 −0.0252695
\(599\) 12616.1 0.860567 0.430284 0.902694i \(-0.358414\pi\)
0.430284 + 0.902694i \(0.358414\pi\)
\(600\) 0 0
\(601\) 8270.87 0.561358 0.280679 0.959802i \(-0.409440\pi\)
0.280679 + 0.959802i \(0.409440\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 22011.7 1.48285
\(605\) −60.2852 −0.00405114
\(606\) 0 0
\(607\) 3811.84 0.254889 0.127445 0.991846i \(-0.459323\pi\)
0.127445 + 0.991846i \(0.459323\pi\)
\(608\) 9821.62 0.655130
\(609\) 0 0
\(610\) −0.509668 −3.38293e−5 0
\(611\) 6683.72 0.442544
\(612\) 0 0
\(613\) 11359.3 0.748445 0.374222 0.927339i \(-0.377910\pi\)
0.374222 + 0.927339i \(0.377910\pi\)
\(614\) −367.447 −0.0241514
\(615\) 0 0
\(616\) 0 0
\(617\) 18272.2 1.19224 0.596118 0.802896i \(-0.296709\pi\)
0.596118 + 0.802896i \(0.296709\pi\)
\(618\) 0 0
\(619\) 29600.2 1.92203 0.961013 0.276503i \(-0.0891756\pi\)
0.961013 + 0.276503i \(0.0891756\pi\)
\(620\) 14.7215 0.000953596 0
\(621\) 0 0
\(622\) 1868.44 0.120447
\(623\) 0 0
\(624\) 0 0
\(625\) 15621.2 0.999758
\(626\) 1539.13 0.0982682
\(627\) 0 0
\(628\) −13240.6 −0.841331
\(629\) 23340.5 1.47956
\(630\) 0 0
\(631\) 7185.41 0.453322 0.226661 0.973974i \(-0.427219\pi\)
0.226661 + 0.973974i \(0.427219\pi\)
\(632\) 5282.81 0.332498
\(633\) 0 0
\(634\) −2880.66 −0.180450
\(635\) −107.681 −0.00672944
\(636\) 0 0
\(637\) 0 0
\(638\) −4289.64 −0.266189
\(639\) 0 0
\(640\) 80.7467 0.00498718
\(641\) 232.982 0.0143561 0.00717803 0.999974i \(-0.497715\pi\)
0.00717803 + 0.999974i \(0.497715\pi\)
\(642\) 0 0
\(643\) −1837.96 −0.112725 −0.0563624 0.998410i \(-0.517950\pi\)
−0.0563624 + 0.998410i \(0.517950\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6404.54 −0.390067
\(647\) 18594.7 1.12988 0.564941 0.825131i \(-0.308899\pi\)
0.564941 + 0.825131i \(0.308899\pi\)
\(648\) 0 0
\(649\) −510.256 −0.0308618
\(650\) −861.736 −0.0520001
\(651\) 0 0
\(652\) 318.799 0.0191490
\(653\) 28864.7 1.72980 0.864902 0.501940i \(-0.167380\pi\)
0.864902 + 0.501940i \(0.167380\pi\)
\(654\) 0 0
\(655\) 252.772 0.0150788
\(656\) −19154.5 −1.14003
\(657\) 0 0
\(658\) 0 0
\(659\) 29066.3 1.71815 0.859076 0.511847i \(-0.171039\pi\)
0.859076 + 0.511847i \(0.171039\pi\)
\(660\) 0 0
\(661\) −3979.51 −0.234168 −0.117084 0.993122i \(-0.537355\pi\)
−0.117084 + 0.993122i \(0.537355\pi\)
\(662\) −4085.46 −0.239858
\(663\) 0 0
\(664\) 2588.86 0.151306
\(665\) 0 0
\(666\) 0 0
\(667\) 12632.0 0.733305
\(668\) −22706.1 −1.31516
\(669\) 0 0
\(670\) −27.8528 −0.00160604
\(671\) 537.955 0.0309501
\(672\) 0 0
\(673\) −184.229 −0.0105520 −0.00527601 0.999986i \(-0.501679\pi\)
−0.00527601 + 0.999986i \(0.501679\pi\)
\(674\) −2462.52 −0.140731
\(675\) 0 0
\(676\) 15030.2 0.855156
\(677\) 16683.5 0.947116 0.473558 0.880763i \(-0.342969\pi\)
0.473558 + 0.880763i \(0.342969\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −80.1540 −0.00452024
\(681\) 0 0
\(682\) 340.553 0.0191209
\(683\) −17808.2 −0.997676 −0.498838 0.866695i \(-0.666240\pi\)
−0.498838 + 0.866695i \(0.666240\pi\)
\(684\) 0 0
\(685\) 25.2332 0.00140746
\(686\) 0 0
\(687\) 0 0
\(688\) −13095.4 −0.725666
\(689\) −10704.5 −0.591883
\(690\) 0 0
\(691\) −20145.7 −1.10908 −0.554542 0.832156i \(-0.687106\pi\)
−0.554542 + 0.832156i \(0.687106\pi\)
\(692\) 16801.0 0.922943
\(693\) 0 0
\(694\) −484.453 −0.0264980
\(695\) 89.0542 0.00486046
\(696\) 0 0
\(697\) 38889.7 2.11342
\(698\) −3801.10 −0.206123
\(699\) 0 0
\(700\) 0 0
\(701\) 2719.67 0.146534 0.0732672 0.997312i \(-0.476657\pi\)
0.0732672 + 0.997312i \(0.476657\pi\)
\(702\) 0 0
\(703\) −24390.7 −1.30855
\(704\) −19654.4 −1.05220
\(705\) 0 0
\(706\) −4385.36 −0.233775
\(707\) 0 0
\(708\) 0 0
\(709\) −625.708 −0.0331438 −0.0165719 0.999863i \(-0.505275\pi\)
−0.0165719 + 0.999863i \(0.505275\pi\)
\(710\) 34.2244 0.00180904
\(711\) 0 0
\(712\) −4415.17 −0.232395
\(713\) −1002.85 −0.0526749
\(714\) 0 0
\(715\) −73.5079 −0.00384481
\(716\) 9421.82 0.491774
\(717\) 0 0
\(718\) −3568.54 −0.185483
\(719\) 9577.54 0.496776 0.248388 0.968661i \(-0.420099\pi\)
0.248388 + 0.968661i \(0.420099\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3851.62 0.198535
\(723\) 0 0
\(724\) −23410.7 −1.20173
\(725\) 29457.8 1.50901
\(726\) 0 0
\(727\) −16741.2 −0.854053 −0.427027 0.904239i \(-0.640439\pi\)
−0.427027 + 0.904239i \(0.640439\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 21.4439 0.00108722
\(731\) 26587.8 1.34526
\(732\) 0 0
\(733\) −6496.04 −0.327335 −0.163668 0.986516i \(-0.552332\pi\)
−0.163668 + 0.986516i \(0.552332\pi\)
\(734\) 3437.06 0.172839
\(735\) 0 0
\(736\) −4141.36 −0.207408
\(737\) 29398.7 1.46936
\(738\) 0 0
\(739\) 499.280 0.0248529 0.0124265 0.999923i \(-0.496044\pi\)
0.0124265 + 0.999923i \(0.496044\pi\)
\(740\) −150.972 −0.00749980
\(741\) 0 0
\(742\) 0 0
\(743\) 6367.30 0.314393 0.157196 0.987567i \(-0.449754\pi\)
0.157196 + 0.987567i \(0.449754\pi\)
\(744\) 0 0
\(745\) 58.5568 0.00287967
\(746\) −2122.37 −0.104163
\(747\) 0 0
\(748\) 41842.8 2.04535
\(749\) 0 0
\(750\) 0 0
\(751\) 496.098 0.0241050 0.0120525 0.999927i \(-0.496163\pi\)
0.0120525 + 0.999927i \(0.496163\pi\)
\(752\) 24057.7 1.16662
\(753\) 0 0
\(754\) −1624.89 −0.0784815
\(755\) 282.597 0.0136222
\(756\) 0 0
\(757\) 13025.9 0.625408 0.312704 0.949851i \(-0.398765\pi\)
0.312704 + 0.949851i \(0.398765\pi\)
\(758\) −622.350 −0.0298216
\(759\) 0 0
\(760\) 83.7604 0.00399778
\(761\) 25474.6 1.21347 0.606736 0.794904i \(-0.292479\pi\)
0.606736 + 0.794904i \(0.292479\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −21977.6 −1.04074
\(765\) 0 0
\(766\) 4503.72 0.212436
\(767\) −193.282 −0.00909911
\(768\) 0 0
\(769\) −29054.0 −1.36244 −0.681218 0.732080i \(-0.738550\pi\)
−0.681218 + 0.732080i \(0.738550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −26118.5 −1.21765
\(773\) −1897.35 −0.0882834 −0.0441417 0.999025i \(-0.514055\pi\)
−0.0441417 + 0.999025i \(0.514055\pi\)
\(774\) 0 0
\(775\) −2338.65 −0.108396
\(776\) −7153.72 −0.330933
\(777\) 0 0
\(778\) −1913.25 −0.0881662
\(779\) −40639.6 −1.86914
\(780\) 0 0
\(781\) −36123.9 −1.65508
\(782\) 2700.52 0.123492
\(783\) 0 0
\(784\) 0 0
\(785\) −169.989 −0.00772886
\(786\) 0 0
\(787\) 42650.3 1.93179 0.965895 0.258936i \(-0.0833718\pi\)
0.965895 + 0.258936i \(0.0833718\pi\)
\(788\) −33088.0 −1.49583
\(789\) 0 0
\(790\) 33.5440 0.00151069
\(791\) 0 0
\(792\) 0 0
\(793\) 203.775 0.00912516
\(794\) 3979.33 0.177860
\(795\) 0 0
\(796\) 34333.1 1.52877
\(797\) −36822.8 −1.63655 −0.818275 0.574828i \(-0.805069\pi\)
−0.818275 + 0.574828i \(0.805069\pi\)
\(798\) 0 0
\(799\) −48844.8 −2.16271
\(800\) −9657.60 −0.426810
\(801\) 0 0
\(802\) 4349.68 0.191512
\(803\) −22634.1 −0.994693
\(804\) 0 0
\(805\) 0 0
\(806\) 129.000 0.00563750
\(807\) 0 0
\(808\) 8987.38 0.391306
\(809\) −5081.94 −0.220855 −0.110427 0.993884i \(-0.535222\pi\)
−0.110427 + 0.993884i \(0.535222\pi\)
\(810\) 0 0
\(811\) 11873.4 0.514097 0.257048 0.966399i \(-0.417250\pi\)
0.257048 + 0.966399i \(0.417250\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3492.45 −0.150381
\(815\) 4.09289 0.000175911 0
\(816\) 0 0
\(817\) −27784.1 −1.18977
\(818\) −4998.29 −0.213644
\(819\) 0 0
\(820\) −251.549 −0.0107128
\(821\) −16969.4 −0.721361 −0.360681 0.932689i \(-0.617456\pi\)
−0.360681 + 0.932689i \(0.617456\pi\)
\(822\) 0 0
\(823\) 3995.94 0.169246 0.0846231 0.996413i \(-0.473031\pi\)
0.0846231 + 0.996413i \(0.473031\pi\)
\(824\) −9270.39 −0.391929
\(825\) 0 0
\(826\) 0 0
\(827\) 13589.7 0.571417 0.285708 0.958317i \(-0.407771\pi\)
0.285708 + 0.958317i \(0.407771\pi\)
\(828\) 0 0
\(829\) −30646.0 −1.28393 −0.641966 0.766733i \(-0.721881\pi\)
−0.641966 + 0.766733i \(0.721881\pi\)
\(830\) 16.4384 0.000687451 0
\(831\) 0 0
\(832\) −7444.96 −0.310226
\(833\) 0 0
\(834\) 0 0
\(835\) −291.511 −0.0120816
\(836\) −43725.5 −1.80894
\(837\) 0 0
\(838\) 2637.01 0.108704
\(839\) 7497.57 0.308516 0.154258 0.988031i \(-0.450701\pi\)
0.154258 + 0.988031i \(0.450701\pi\)
\(840\) 0 0
\(841\) 31156.6 1.27749
\(842\) −1959.99 −0.0802207
\(843\) 0 0
\(844\) −17939.3 −0.731630
\(845\) 192.965 0.00785586
\(846\) 0 0
\(847\) 0 0
\(848\) −38530.2 −1.56030
\(849\) 0 0
\(850\) 6297.59 0.254124
\(851\) 10284.5 0.414275
\(852\) 0 0
\(853\) −10347.6 −0.415352 −0.207676 0.978198i \(-0.566590\pi\)
−0.207676 + 0.978198i \(0.566590\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2252.36 −0.0899348
\(857\) −1550.00 −0.0617817 −0.0308909 0.999523i \(-0.509834\pi\)
−0.0308909 + 0.999523i \(0.509834\pi\)
\(858\) 0 0
\(859\) 15187.5 0.603250 0.301625 0.953427i \(-0.402471\pi\)
0.301625 + 0.953427i \(0.402471\pi\)
\(860\) −171.977 −0.00681903
\(861\) 0 0
\(862\) −1554.45 −0.0614210
\(863\) −41550.2 −1.63892 −0.819459 0.573138i \(-0.805726\pi\)
−0.819459 + 0.573138i \(0.805726\pi\)
\(864\) 0 0
\(865\) 215.699 0.00847858
\(866\) 4850.12 0.190316
\(867\) 0 0
\(868\) 0 0
\(869\) −35405.8 −1.38212
\(870\) 0 0
\(871\) 11136.1 0.433216
\(872\) 2082.67 0.0808808
\(873\) 0 0
\(874\) −2822.03 −0.109218
\(875\) 0 0
\(876\) 0 0
\(877\) 27837.4 1.07184 0.535919 0.844269i \(-0.319965\pi\)
0.535919 + 0.844269i \(0.319965\pi\)
\(878\) −6182.02 −0.237623
\(879\) 0 0
\(880\) −264.588 −0.0101355
\(881\) 2587.85 0.0989635 0.0494817 0.998775i \(-0.484243\pi\)
0.0494817 + 0.998775i \(0.484243\pi\)
\(882\) 0 0
\(883\) −16382.0 −0.624346 −0.312173 0.950025i \(-0.601057\pi\)
−0.312173 + 0.950025i \(0.601057\pi\)
\(884\) 15849.8 0.603040
\(885\) 0 0
\(886\) 3528.13 0.133781
\(887\) 22980.2 0.869896 0.434948 0.900456i \(-0.356767\pi\)
0.434948 + 0.900456i \(0.356767\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −28.0348 −0.00105587
\(891\) 0 0
\(892\) −1706.36 −0.0640506
\(893\) 51042.5 1.91274
\(894\) 0 0
\(895\) 120.962 0.00451766
\(896\) 0 0
\(897\) 0 0
\(898\) 2471.09 0.0918276
\(899\) −4409.76 −0.163597
\(900\) 0 0
\(901\) 78228.5 2.89253
\(902\) −5819.10 −0.214806
\(903\) 0 0
\(904\) 5234.42 0.192582
\(905\) −300.557 −0.0110396
\(906\) 0 0
\(907\) −22715.4 −0.831590 −0.415795 0.909458i \(-0.636497\pi\)
−0.415795 + 0.909458i \(0.636497\pi\)
\(908\) −14368.0 −0.525130
\(909\) 0 0
\(910\) 0 0
\(911\) −34922.3 −1.27006 −0.635031 0.772487i \(-0.719013\pi\)
−0.635031 + 0.772487i \(0.719013\pi\)
\(912\) 0 0
\(913\) −17350.7 −0.628943
\(914\) −5741.24 −0.207772
\(915\) 0 0
\(916\) 21716.1 0.783320
\(917\) 0 0
\(918\) 0 0
\(919\) 11702.6 0.420059 0.210030 0.977695i \(-0.432644\pi\)
0.210030 + 0.977695i \(0.432644\pi\)
\(920\) −35.3182 −0.00126566
\(921\) 0 0
\(922\) 61.8473 0.00220914
\(923\) −13683.5 −0.487973
\(924\) 0 0
\(925\) 23983.3 0.852505
\(926\) 2238.39 0.0794364
\(927\) 0 0
\(928\) −18210.4 −0.644165
\(929\) 53096.5 1.87518 0.937588 0.347747i \(-0.113053\pi\)
0.937588 + 0.347747i \(0.113053\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −7740.06 −0.272032
\(933\) 0 0
\(934\) −1534.52 −0.0537591
\(935\) 537.198 0.0187896
\(936\) 0 0
\(937\) −39020.6 −1.36046 −0.680229 0.733000i \(-0.738120\pi\)
−0.680229 + 0.733000i \(0.738120\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 315.941 0.0109626
\(941\) −30743.8 −1.06506 −0.532528 0.846412i \(-0.678758\pi\)
−0.532528 + 0.846412i \(0.678758\pi\)
\(942\) 0 0
\(943\) 17136.0 0.591754
\(944\) −695.710 −0.0239867
\(945\) 0 0
\(946\) −3978.35 −0.136731
\(947\) −16300.3 −0.559332 −0.279666 0.960097i \(-0.590224\pi\)
−0.279666 + 0.960097i \(0.590224\pi\)
\(948\) 0 0
\(949\) −8573.66 −0.293269
\(950\) −6580.94 −0.224752
\(951\) 0 0
\(952\) 0 0
\(953\) 11512.7 0.391325 0.195663 0.980671i \(-0.437314\pi\)
0.195663 + 0.980671i \(0.437314\pi\)
\(954\) 0 0
\(955\) −282.159 −0.00956068
\(956\) −6559.65 −0.221919
\(957\) 0 0
\(958\) 4420.39 0.149078
\(959\) 0 0
\(960\) 0 0
\(961\) −29440.9 −0.988248
\(962\) −1322.92 −0.0443375
\(963\) 0 0
\(964\) −27047.1 −0.903661
\(965\) −335.322 −0.0111859
\(966\) 0 0
\(967\) −18178.4 −0.604528 −0.302264 0.953224i \(-0.597742\pi\)
−0.302264 + 0.953224i \(0.597742\pi\)
\(968\) −3932.65 −0.130579
\(969\) 0 0
\(970\) −45.4237 −0.00150357
\(971\) 28276.7 0.934543 0.467271 0.884114i \(-0.345237\pi\)
0.467271 + 0.884114i \(0.345237\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2424.77 0.0797688
\(975\) 0 0
\(976\) 733.477 0.0240554
\(977\) 13947.1 0.456710 0.228355 0.973578i \(-0.426665\pi\)
0.228355 + 0.973578i \(0.426665\pi\)
\(978\) 0 0
\(979\) 29590.8 0.966011
\(980\) 0 0
\(981\) 0 0
\(982\) −1683.91 −0.0547206
\(983\) −26576.8 −0.862327 −0.431164 0.902274i \(-0.641897\pi\)
−0.431164 + 0.902274i \(0.641897\pi\)
\(984\) 0 0
\(985\) −424.799 −0.0137414
\(986\) 11874.7 0.383539
\(987\) 0 0
\(988\) −16563.0 −0.533339
\(989\) 11715.4 0.376671
\(990\) 0 0
\(991\) 16249.9 0.520884 0.260442 0.965490i \(-0.416132\pi\)
0.260442 + 0.965490i \(0.416132\pi\)
\(992\) 1445.72 0.0462718
\(993\) 0 0
\(994\) 0 0
\(995\) 440.784 0.0140440
\(996\) 0 0
\(997\) −18814.1 −0.597642 −0.298821 0.954309i \(-0.596593\pi\)
−0.298821 + 0.954309i \(0.596593\pi\)
\(998\) 1992.86 0.0632092
\(999\) 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.a.n.1.2 2
3.2 odd 2 147.4.a.j.1.1 2
7.2 even 3 441.4.e.v.361.1 4
7.3 odd 6 441.4.e.u.226.1 4
7.4 even 3 441.4.e.v.226.1 4
7.5 odd 6 441.4.e.u.361.1 4
7.6 odd 2 441.4.a.o.1.2 2
12.11 even 2 2352.4.a.cf.1.1 2
21.2 odd 6 147.4.e.k.67.2 4
21.5 even 6 147.4.e.j.67.2 4
21.11 odd 6 147.4.e.k.79.2 4
21.17 even 6 147.4.e.j.79.2 4
21.20 even 2 147.4.a.k.1.1 yes 2
84.83 odd 2 2352.4.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.1 2 3.2 odd 2
147.4.a.k.1.1 yes 2 21.20 even 2
147.4.e.j.67.2 4 21.5 even 6
147.4.e.j.79.2 4 21.17 even 6
147.4.e.k.67.2 4 21.2 odd 6
147.4.e.k.79.2 4 21.11 odd 6
441.4.a.n.1.2 2 1.1 even 1 trivial
441.4.a.o.1.2 2 7.6 odd 2
441.4.e.u.226.1 4 7.3 odd 6
441.4.e.u.361.1 4 7.5 odd 6
441.4.e.v.226.1 4 7.4 even 3
441.4.e.v.361.1 4 7.2 even 3
2352.4.a.bl.1.2 2 84.83 odd 2
2352.4.a.cf.1.1 2 12.11 even 2