Properties

Label 2205.4.a.w.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.31662 q^{2} -2.63325 q^{4} -5.00000 q^{5} -24.6332 q^{8} +O(q^{10})\) \(q+2.31662 q^{2} -2.63325 q^{4} -5.00000 q^{5} -24.6332 q^{8} -11.5831 q^{10} -46.2665 q^{11} -61.3325 q^{13} -36.0000 q^{16} -101.332 q^{17} +3.66750 q^{19} +13.1662 q^{20} -107.182 q^{22} -84.8655 q^{23} +25.0000 q^{25} -142.084 q^{26} -30.1980 q^{29} +188.997 q^{31} +113.668 q^{32} -234.749 q^{34} +18.0685 q^{37} +8.49623 q^{38} +123.166 q^{40} -481.662 q^{41} -97.7995 q^{43} +121.831 q^{44} -196.602 q^{46} -117.665 q^{47} +57.9156 q^{50} +161.504 q^{52} -667.995 q^{53} +231.332 q^{55} -69.9574 q^{58} -57.3350 q^{59} +738.997 q^{61} +437.836 q^{62} +551.325 q^{64} +306.662 q^{65} +552.396 q^{67} +266.834 q^{68} +740.264 q^{71} +233.325 q^{73} +41.8580 q^{74} -9.65745 q^{76} -1075.19 q^{79} +180.000 q^{80} -1115.83 q^{82} -683.325 q^{83} +506.662 q^{85} -226.565 q^{86} +1139.69 q^{88} +1380.32 q^{89} +223.472 q^{92} -272.586 q^{94} -18.3375 q^{95} +218.008 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 8 q^{4} - 10 q^{5} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 8 q^{4} - 10 q^{5} - 36 q^{8} + 10 q^{10} - 66 q^{11} + 10 q^{13} - 72 q^{16} - 70 q^{17} + 140 q^{19} - 40 q^{20} - 22 q^{22} + 16 q^{23} + 50 q^{25} - 450 q^{26} + 258 q^{29} - 20 q^{31} + 360 q^{32} - 370 q^{34} + 328 q^{37} - 580 q^{38} + 180 q^{40} - 300 q^{41} - 116 q^{43} - 88 q^{44} - 632 q^{46} + 30 q^{47} - 50 q^{50} + 920 q^{52} - 540 q^{53} + 330 q^{55} - 1314 q^{58} - 380 q^{59} + 1080 q^{61} + 1340 q^{62} - 224 q^{64} - 50 q^{65} + 468 q^{67} + 600 q^{68} + 1056 q^{71} - 860 q^{73} - 1296 q^{74} + 1440 q^{76} + 158 q^{79} + 360 q^{80} - 1900 q^{82} - 40 q^{83} + 350 q^{85} - 148 q^{86} + 1364 q^{88} + 240 q^{89} + 1296 q^{92} - 910 q^{94} - 700 q^{95} + 1630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.31662 0.819051 0.409525 0.912299i \(-0.365694\pi\)
0.409525 + 0.912299i \(0.365694\pi\)
\(3\) 0 0
\(4\) −2.63325 −0.329156
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −24.6332 −1.08865
\(9\) 0 0
\(10\) −11.5831 −0.366291
\(11\) −46.2665 −1.26817 −0.634085 0.773263i \(-0.718623\pi\)
−0.634085 + 0.773263i \(0.718623\pi\)
\(12\) 0 0
\(13\) −61.3325 −1.30851 −0.654253 0.756276i \(-0.727017\pi\)
−0.654253 + 0.756276i \(0.727017\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −36.0000 −0.562500
\(17\) −101.332 −1.44569 −0.722845 0.691010i \(-0.757166\pi\)
−0.722845 + 0.691010i \(0.757166\pi\)
\(18\) 0 0
\(19\) 3.66750 0.0442833 0.0221417 0.999755i \(-0.492952\pi\)
0.0221417 + 0.999755i \(0.492952\pi\)
\(20\) 13.1662 0.147203
\(21\) 0 0
\(22\) −107.182 −1.03870
\(23\) −84.8655 −0.769377 −0.384689 0.923046i \(-0.625691\pi\)
−0.384689 + 0.923046i \(0.625691\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −142.084 −1.07173
\(27\) 0 0
\(28\) 0 0
\(29\) −30.1980 −0.193366 −0.0966832 0.995315i \(-0.530823\pi\)
−0.0966832 + 0.995315i \(0.530823\pi\)
\(30\) 0 0
\(31\) 188.997 1.09500 0.547499 0.836806i \(-0.315580\pi\)
0.547499 + 0.836806i \(0.315580\pi\)
\(32\) 113.668 0.627930
\(33\) 0 0
\(34\) −234.749 −1.18409
\(35\) 0 0
\(36\) 0 0
\(37\) 18.0685 0.0802823 0.0401411 0.999194i \(-0.487219\pi\)
0.0401411 + 0.999194i \(0.487219\pi\)
\(38\) 8.49623 0.0362703
\(39\) 0 0
\(40\) 123.166 0.486857
\(41\) −481.662 −1.83471 −0.917354 0.398072i \(-0.869679\pi\)
−0.917354 + 0.398072i \(0.869679\pi\)
\(42\) 0 0
\(43\) −97.7995 −0.346844 −0.173422 0.984848i \(-0.555482\pi\)
−0.173422 + 0.984848i \(0.555482\pi\)
\(44\) 121.831 0.417426
\(45\) 0 0
\(46\) −196.602 −0.630159
\(47\) −117.665 −0.365175 −0.182587 0.983190i \(-0.558447\pi\)
−0.182587 + 0.983190i \(0.558447\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 57.9156 0.163810
\(51\) 0 0
\(52\) 161.504 0.430703
\(53\) −667.995 −1.73125 −0.865624 0.500694i \(-0.833078\pi\)
−0.865624 + 0.500694i \(0.833078\pi\)
\(54\) 0 0
\(55\) 231.332 0.567143
\(56\) 0 0
\(57\) 0 0
\(58\) −69.9574 −0.158377
\(59\) −57.3350 −0.126515 −0.0632575 0.997997i \(-0.520149\pi\)
−0.0632575 + 0.997997i \(0.520149\pi\)
\(60\) 0 0
\(61\) 738.997 1.55113 0.775565 0.631268i \(-0.217465\pi\)
0.775565 + 0.631268i \(0.217465\pi\)
\(62\) 437.836 0.896859
\(63\) 0 0
\(64\) 551.325 1.07681
\(65\) 306.662 0.585182
\(66\) 0 0
\(67\) 552.396 1.00725 0.503626 0.863922i \(-0.331999\pi\)
0.503626 + 0.863922i \(0.331999\pi\)
\(68\) 266.834 0.475858
\(69\) 0 0
\(70\) 0 0
\(71\) 740.264 1.23737 0.618684 0.785640i \(-0.287666\pi\)
0.618684 + 0.785640i \(0.287666\pi\)
\(72\) 0 0
\(73\) 233.325 0.374091 0.187045 0.982351i \(-0.440109\pi\)
0.187045 + 0.982351i \(0.440109\pi\)
\(74\) 41.8580 0.0657553
\(75\) 0 0
\(76\) −9.65745 −0.0145761
\(77\) 0 0
\(78\) 0 0
\(79\) −1075.19 −1.53124 −0.765619 0.643294i \(-0.777567\pi\)
−0.765619 + 0.643294i \(0.777567\pi\)
\(80\) 180.000 0.251558
\(81\) 0 0
\(82\) −1115.83 −1.50272
\(83\) −683.325 −0.903671 −0.451835 0.892101i \(-0.649231\pi\)
−0.451835 + 0.892101i \(0.649231\pi\)
\(84\) 0 0
\(85\) 506.662 0.646532
\(86\) −226.565 −0.284083
\(87\) 0 0
\(88\) 1139.69 1.38059
\(89\) 1380.32 1.64397 0.821985 0.569509i \(-0.192867\pi\)
0.821985 + 0.569509i \(0.192867\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 223.472 0.253245
\(93\) 0 0
\(94\) −272.586 −0.299096
\(95\) −18.3375 −0.0198041
\(96\) 0 0
\(97\) 218.008 0.228199 0.114100 0.993469i \(-0.463602\pi\)
0.114100 + 0.993469i \(0.463602\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −65.8312 −0.0658312
\(101\) 1474.33 1.45249 0.726243 0.687438i \(-0.241265\pi\)
0.726243 + 0.687438i \(0.241265\pi\)
\(102\) 0 0
\(103\) −810.990 −0.775818 −0.387909 0.921698i \(-0.626802\pi\)
−0.387909 + 0.921698i \(0.626802\pi\)
\(104\) 1510.82 1.42450
\(105\) 0 0
\(106\) −1547.49 −1.41798
\(107\) −440.660 −0.398133 −0.199066 0.979986i \(-0.563791\pi\)
−0.199066 + 0.979986i \(0.563791\pi\)
\(108\) 0 0
\(109\) −1906.19 −1.67504 −0.837522 0.546404i \(-0.815996\pi\)
−0.837522 + 0.546404i \(0.815996\pi\)
\(110\) 535.911 0.464519
\(111\) 0 0
\(112\) 0 0
\(113\) −962.470 −0.801252 −0.400626 0.916242i \(-0.631207\pi\)
−0.400626 + 0.916242i \(0.631207\pi\)
\(114\) 0 0
\(115\) 424.327 0.344076
\(116\) 79.5188 0.0636478
\(117\) 0 0
\(118\) −132.824 −0.103622
\(119\) 0 0
\(120\) 0 0
\(121\) 809.589 0.608256
\(122\) 1711.98 1.27045
\(123\) 0 0
\(124\) −497.678 −0.360426
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1621.74 −1.13312 −0.566558 0.824022i \(-0.691725\pi\)
−0.566558 + 0.824022i \(0.691725\pi\)
\(128\) 367.873 0.254029
\(129\) 0 0
\(130\) 710.422 0.479293
\(131\) 1380.32 0.920602 0.460301 0.887763i \(-0.347741\pi\)
0.460301 + 0.887763i \(0.347741\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1279.69 0.824991
\(135\) 0 0
\(136\) 2496.15 1.57385
\(137\) 1949.66 1.21584 0.607921 0.793997i \(-0.292004\pi\)
0.607921 + 0.793997i \(0.292004\pi\)
\(138\) 0 0
\(139\) 2800.00 1.70858 0.854291 0.519795i \(-0.173992\pi\)
0.854291 + 0.519795i \(0.173992\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1714.91 1.01347
\(143\) 2837.64 1.65941
\(144\) 0 0
\(145\) 150.990 0.0864761
\(146\) 540.526 0.306399
\(147\) 0 0
\(148\) −47.5789 −0.0264254
\(149\) 1434.12 0.788506 0.394253 0.919002i \(-0.371003\pi\)
0.394253 + 0.919002i \(0.371003\pi\)
\(150\) 0 0
\(151\) −1985.58 −1.07009 −0.535047 0.844822i \(-0.679706\pi\)
−0.535047 + 0.844822i \(0.679706\pi\)
\(152\) −90.3425 −0.0482089
\(153\) 0 0
\(154\) 0 0
\(155\) −944.987 −0.489698
\(156\) 0 0
\(157\) 40.6600 0.0206689 0.0103345 0.999947i \(-0.496710\pi\)
0.0103345 + 0.999947i \(0.496710\pi\)
\(158\) −2490.80 −1.25416
\(159\) 0 0
\(160\) −568.338 −0.280819
\(161\) 0 0
\(162\) 0 0
\(163\) −3953.98 −1.90000 −0.950000 0.312250i \(-0.898917\pi\)
−0.950000 + 0.312250i \(0.898917\pi\)
\(164\) 1268.34 0.603906
\(165\) 0 0
\(166\) −1583.01 −0.740152
\(167\) −3380.30 −1.56632 −0.783161 0.621819i \(-0.786394\pi\)
−0.783161 + 0.621819i \(0.786394\pi\)
\(168\) 0 0
\(169\) 1564.68 0.712187
\(170\) 1173.75 0.529543
\(171\) 0 0
\(172\) 257.530 0.114166
\(173\) −3206.66 −1.40924 −0.704619 0.709586i \(-0.748882\pi\)
−0.704619 + 0.709586i \(0.748882\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1665.59 0.713346
\(177\) 0 0
\(178\) 3197.68 1.34649
\(179\) −1442.65 −0.602395 −0.301198 0.953562i \(-0.597386\pi\)
−0.301198 + 0.953562i \(0.597386\pi\)
\(180\) 0 0
\(181\) −908.680 −0.373158 −0.186579 0.982440i \(-0.559740\pi\)
−0.186579 + 0.982440i \(0.559740\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2090.51 0.837580
\(185\) −90.3425 −0.0359033
\(186\) 0 0
\(187\) 4688.30 1.83338
\(188\) 309.841 0.120199
\(189\) 0 0
\(190\) −42.4812 −0.0162206
\(191\) −2474.64 −0.937479 −0.468739 0.883336i \(-0.655292\pi\)
−0.468739 + 0.883336i \(0.655292\pi\)
\(192\) 0 0
\(193\) 3533.52 1.31787 0.658934 0.752201i \(-0.271008\pi\)
0.658934 + 0.752201i \(0.271008\pi\)
\(194\) 505.042 0.186907
\(195\) 0 0
\(196\) 0 0
\(197\) 1952.57 0.706165 0.353083 0.935592i \(-0.385133\pi\)
0.353083 + 0.935592i \(0.385133\pi\)
\(198\) 0 0
\(199\) 4064.67 1.44792 0.723962 0.689840i \(-0.242319\pi\)
0.723962 + 0.689840i \(0.242319\pi\)
\(200\) −615.831 −0.217729
\(201\) 0 0
\(202\) 3415.46 1.18966
\(203\) 0 0
\(204\) 0 0
\(205\) 2408.31 0.820507
\(206\) −1878.76 −0.635434
\(207\) 0 0
\(208\) 2207.97 0.736034
\(209\) −169.683 −0.0561588
\(210\) 0 0
\(211\) −4325.34 −1.41123 −0.705613 0.708598i \(-0.749328\pi\)
−0.705613 + 0.708598i \(0.749328\pi\)
\(212\) 1759.00 0.569851
\(213\) 0 0
\(214\) −1020.84 −0.326091
\(215\) 488.997 0.155113
\(216\) 0 0
\(217\) 0 0
\(218\) −4415.92 −1.37194
\(219\) 0 0
\(220\) −609.156 −0.186679
\(221\) 6214.97 1.89169
\(222\) 0 0
\(223\) −982.970 −0.295177 −0.147589 0.989049i \(-0.547151\pi\)
−0.147589 + 0.989049i \(0.547151\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2229.68 −0.656266
\(227\) 1660.96 0.485648 0.242824 0.970070i \(-0.421926\pi\)
0.242824 + 0.970070i \(0.421926\pi\)
\(228\) 0 0
\(229\) −574.327 −0.165732 −0.0828660 0.996561i \(-0.526407\pi\)
−0.0828660 + 0.996561i \(0.526407\pi\)
\(230\) 983.008 0.281816
\(231\) 0 0
\(232\) 743.875 0.210508
\(233\) 2316.48 0.651320 0.325660 0.945487i \(-0.394414\pi\)
0.325660 + 0.945487i \(0.394414\pi\)
\(234\) 0 0
\(235\) 588.325 0.163311
\(236\) 150.977 0.0416432
\(237\) 0 0
\(238\) 0 0
\(239\) 3659.31 0.990382 0.495191 0.868784i \(-0.335098\pi\)
0.495191 + 0.868784i \(0.335098\pi\)
\(240\) 0 0
\(241\) 2446.33 0.653868 0.326934 0.945047i \(-0.393985\pi\)
0.326934 + 0.945047i \(0.393985\pi\)
\(242\) 1875.51 0.498193
\(243\) 0 0
\(244\) −1945.96 −0.510564
\(245\) 0 0
\(246\) 0 0
\(247\) −224.937 −0.0579450
\(248\) −4655.62 −1.19207
\(249\) 0 0
\(250\) −289.578 −0.0732581
\(251\) 2909.29 0.731605 0.365802 0.930693i \(-0.380795\pi\)
0.365802 + 0.930693i \(0.380795\pi\)
\(252\) 0 0
\(253\) 3926.43 0.975702
\(254\) −3756.95 −0.928080
\(255\) 0 0
\(256\) −3558.38 −0.868744
\(257\) −168.680 −0.0409415 −0.0204708 0.999790i \(-0.506517\pi\)
−0.0204708 + 0.999790i \(0.506517\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −807.519 −0.192616
\(261\) 0 0
\(262\) 3197.68 0.754020
\(263\) 3244.47 0.760695 0.380347 0.924844i \(-0.375804\pi\)
0.380347 + 0.924844i \(0.375804\pi\)
\(264\) 0 0
\(265\) 3339.97 0.774238
\(266\) 0 0
\(267\) 0 0
\(268\) −1454.60 −0.331543
\(269\) 2848.65 0.645671 0.322836 0.946455i \(-0.395364\pi\)
0.322836 + 0.946455i \(0.395364\pi\)
\(270\) 0 0
\(271\) −2850.98 −0.639057 −0.319529 0.947577i \(-0.603525\pi\)
−0.319529 + 0.947577i \(0.603525\pi\)
\(272\) 3647.97 0.813201
\(273\) 0 0
\(274\) 4516.62 0.995837
\(275\) −1156.66 −0.253634
\(276\) 0 0
\(277\) 2298.63 0.498597 0.249298 0.968427i \(-0.419800\pi\)
0.249298 + 0.968427i \(0.419800\pi\)
\(278\) 6486.55 1.39942
\(279\) 0 0
\(280\) 0 0
\(281\) −6109.20 −1.29695 −0.648477 0.761234i \(-0.724594\pi\)
−0.648477 + 0.761234i \(0.724594\pi\)
\(282\) 0 0
\(283\) 5854.95 1.22983 0.614913 0.788595i \(-0.289191\pi\)
0.614913 + 0.788595i \(0.289191\pi\)
\(284\) −1949.30 −0.407288
\(285\) 0 0
\(286\) 6573.75 1.35914
\(287\) 0 0
\(288\) 0 0
\(289\) 5355.27 1.09002
\(290\) 349.787 0.0708283
\(291\) 0 0
\(292\) −614.403 −0.123134
\(293\) 5135.34 1.02392 0.511962 0.859008i \(-0.328919\pi\)
0.511962 + 0.859008i \(0.328919\pi\)
\(294\) 0 0
\(295\) 286.675 0.0565792
\(296\) −445.086 −0.0873990
\(297\) 0 0
\(298\) 3322.31 0.645827
\(299\) 5205.01 1.00673
\(300\) 0 0
\(301\) 0 0
\(302\) −4599.85 −0.876462
\(303\) 0 0
\(304\) −132.030 −0.0249094
\(305\) −3694.99 −0.693686
\(306\) 0 0
\(307\) 2102.97 0.390954 0.195477 0.980708i \(-0.437375\pi\)
0.195477 + 0.980708i \(0.437375\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2189.18 −0.401088
\(311\) 5764.30 1.05101 0.525504 0.850791i \(-0.323877\pi\)
0.525504 + 0.850791i \(0.323877\pi\)
\(312\) 0 0
\(313\) −1360.01 −0.245599 −0.122799 0.992432i \(-0.539187\pi\)
−0.122799 + 0.992432i \(0.539187\pi\)
\(314\) 94.1939 0.0169289
\(315\) 0 0
\(316\) 2831.23 0.504017
\(317\) −5138.95 −0.910511 −0.455256 0.890361i \(-0.650452\pi\)
−0.455256 + 0.890361i \(0.650452\pi\)
\(318\) 0 0
\(319\) 1397.16 0.245222
\(320\) −2756.62 −0.481563
\(321\) 0 0
\(322\) 0 0
\(323\) −371.637 −0.0640200
\(324\) 0 0
\(325\) −1533.31 −0.261701
\(326\) −9159.90 −1.55620
\(327\) 0 0
\(328\) 11864.9 1.99735
\(329\) 0 0
\(330\) 0 0
\(331\) 1825.70 0.303171 0.151585 0.988444i \(-0.451562\pi\)
0.151585 + 0.988444i \(0.451562\pi\)
\(332\) 1799.37 0.297449
\(333\) 0 0
\(334\) −7830.90 −1.28290
\(335\) −2761.98 −0.450457
\(336\) 0 0
\(337\) 153.985 0.0248905 0.0124452 0.999923i \(-0.496038\pi\)
0.0124452 + 0.999923i \(0.496038\pi\)
\(338\) 3624.76 0.583317
\(339\) 0 0
\(340\) −1334.17 −0.212810
\(341\) −8744.25 −1.38864
\(342\) 0 0
\(343\) 0 0
\(344\) 2409.12 0.377590
\(345\) 0 0
\(346\) −7428.63 −1.15424
\(347\) −4359.39 −0.674421 −0.337211 0.941429i \(-0.609483\pi\)
−0.337211 + 0.941429i \(0.609483\pi\)
\(348\) 0 0
\(349\) 1689.00 0.259054 0.129527 0.991576i \(-0.458654\pi\)
0.129527 + 0.991576i \(0.458654\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5259.00 −0.796322
\(353\) 3921.36 0.591254 0.295627 0.955303i \(-0.404471\pi\)
0.295627 + 0.955303i \(0.404471\pi\)
\(354\) 0 0
\(355\) −3701.32 −0.553368
\(356\) −3634.72 −0.541123
\(357\) 0 0
\(358\) −3342.08 −0.493392
\(359\) 2867.86 0.421616 0.210808 0.977528i \(-0.432391\pi\)
0.210808 + 0.977528i \(0.432391\pi\)
\(360\) 0 0
\(361\) −6845.55 −0.998039
\(362\) −2105.07 −0.305636
\(363\) 0 0
\(364\) 0 0
\(365\) −1166.62 −0.167298
\(366\) 0 0
\(367\) 11503.0 1.63611 0.818054 0.575142i \(-0.195053\pi\)
0.818054 + 0.575142i \(0.195053\pi\)
\(368\) 3055.16 0.432775
\(369\) 0 0
\(370\) −209.290 −0.0294066
\(371\) 0 0
\(372\) 0 0
\(373\) 5086.43 0.706073 0.353037 0.935610i \(-0.385149\pi\)
0.353037 + 0.935610i \(0.385149\pi\)
\(374\) 10861.0 1.50163
\(375\) 0 0
\(376\) 2898.47 0.397546
\(377\) 1852.12 0.253021
\(378\) 0 0
\(379\) 954.827 0.129409 0.0647047 0.997904i \(-0.479389\pi\)
0.0647047 + 0.997904i \(0.479389\pi\)
\(380\) 48.2873 0.00651864
\(381\) 0 0
\(382\) −5732.81 −0.767843
\(383\) −3083.91 −0.411437 −0.205719 0.978611i \(-0.565953\pi\)
−0.205719 + 0.978611i \(0.565953\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8185.84 1.07940
\(387\) 0 0
\(388\) −574.068 −0.0751131
\(389\) 6331.15 0.825198 0.412599 0.910913i \(-0.364621\pi\)
0.412599 + 0.910913i \(0.364621\pi\)
\(390\) 0 0
\(391\) 8599.63 1.11228
\(392\) 0 0
\(393\) 0 0
\(394\) 4523.36 0.578385
\(395\) 5375.93 0.684791
\(396\) 0 0
\(397\) −12133.2 −1.53388 −0.766939 0.641720i \(-0.778221\pi\)
−0.766939 + 0.641720i \(0.778221\pi\)
\(398\) 9416.32 1.18592
\(399\) 0 0
\(400\) −900.000 −0.112500
\(401\) 270.669 0.0337072 0.0168536 0.999858i \(-0.494635\pi\)
0.0168536 + 0.999858i \(0.494635\pi\)
\(402\) 0 0
\(403\) −11591.7 −1.43281
\(404\) −3882.27 −0.478095
\(405\) 0 0
\(406\) 0 0
\(407\) −835.967 −0.101812
\(408\) 0 0
\(409\) 4019.92 0.485996 0.242998 0.970027i \(-0.421869\pi\)
0.242998 + 0.970027i \(0.421869\pi\)
\(410\) 5579.16 0.672036
\(411\) 0 0
\(412\) 2135.54 0.255365
\(413\) 0 0
\(414\) 0 0
\(415\) 3416.62 0.404134
\(416\) −6971.51 −0.821650
\(417\) 0 0
\(418\) −393.091 −0.0459969
\(419\) −2437.28 −0.284175 −0.142087 0.989854i \(-0.545381\pi\)
−0.142087 + 0.989854i \(0.545381\pi\)
\(420\) 0 0
\(421\) −4751.36 −0.550041 −0.275020 0.961438i \(-0.588685\pi\)
−0.275020 + 0.961438i \(0.588685\pi\)
\(422\) −10020.2 −1.15586
\(423\) 0 0
\(424\) 16454.9 1.88472
\(425\) −2533.31 −0.289138
\(426\) 0 0
\(427\) 0 0
\(428\) 1160.37 0.131048
\(429\) 0 0
\(430\) 1132.82 0.127046
\(431\) −7925.19 −0.885714 −0.442857 0.896592i \(-0.646035\pi\)
−0.442857 + 0.896592i \(0.646035\pi\)
\(432\) 0 0
\(433\) −11487.3 −1.27492 −0.637462 0.770481i \(-0.720016\pi\)
−0.637462 + 0.770481i \(0.720016\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5019.47 0.551351
\(437\) −311.245 −0.0340706
\(438\) 0 0
\(439\) −9147.92 −0.994548 −0.497274 0.867594i \(-0.665666\pi\)
−0.497274 + 0.867594i \(0.665666\pi\)
\(440\) −5698.47 −0.617418
\(441\) 0 0
\(442\) 14397.8 1.54939
\(443\) −1864.35 −0.199950 −0.0999752 0.994990i \(-0.531876\pi\)
−0.0999752 + 0.994990i \(0.531876\pi\)
\(444\) 0 0
\(445\) −6901.59 −0.735206
\(446\) −2277.17 −0.241765
\(447\) 0 0
\(448\) 0 0
\(449\) −4490.88 −0.472022 −0.236011 0.971750i \(-0.575840\pi\)
−0.236011 + 0.971750i \(0.575840\pi\)
\(450\) 0 0
\(451\) 22284.8 2.32672
\(452\) 2534.42 0.263737
\(453\) 0 0
\(454\) 3847.83 0.397770
\(455\) 0 0
\(456\) 0 0
\(457\) −14343.8 −1.46822 −0.734109 0.679032i \(-0.762400\pi\)
−0.734109 + 0.679032i \(0.762400\pi\)
\(458\) −1330.50 −0.135743
\(459\) 0 0
\(460\) −1117.36 −0.113255
\(461\) 14558.7 1.47086 0.735429 0.677602i \(-0.236981\pi\)
0.735429 + 0.677602i \(0.236981\pi\)
\(462\) 0 0
\(463\) −1809.56 −0.181636 −0.0908178 0.995868i \(-0.528948\pi\)
−0.0908178 + 0.995868i \(0.528948\pi\)
\(464\) 1087.13 0.108769
\(465\) 0 0
\(466\) 5366.41 0.533464
\(467\) −5981.65 −0.592715 −0.296357 0.955077i \(-0.595772\pi\)
−0.296357 + 0.955077i \(0.595772\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1362.93 0.133760
\(471\) 0 0
\(472\) 1412.35 0.137730
\(473\) 4524.84 0.439857
\(474\) 0 0
\(475\) 91.6876 0.00885666
\(476\) 0 0
\(477\) 0 0
\(478\) 8477.25 0.811173
\(479\) −11527.5 −1.09959 −0.549796 0.835299i \(-0.685295\pi\)
−0.549796 + 0.835299i \(0.685295\pi\)
\(480\) 0 0
\(481\) −1108.19 −0.105050
\(482\) 5667.23 0.535551
\(483\) 0 0
\(484\) −2131.85 −0.200211
\(485\) −1090.04 −0.102054
\(486\) 0 0
\(487\) 15791.5 1.46936 0.734682 0.678411i \(-0.237331\pi\)
0.734682 + 0.678411i \(0.237331\pi\)
\(488\) −18203.9 −1.68863
\(489\) 0 0
\(490\) 0 0
\(491\) −13064.9 −1.20083 −0.600417 0.799687i \(-0.704999\pi\)
−0.600417 + 0.799687i \(0.704999\pi\)
\(492\) 0 0
\(493\) 3060.04 0.279548
\(494\) −521.095 −0.0474599
\(495\) 0 0
\(496\) −6803.91 −0.615937
\(497\) 0 0
\(498\) 0 0
\(499\) −20135.8 −1.80642 −0.903209 0.429201i \(-0.858795\pi\)
−0.903209 + 0.429201i \(0.858795\pi\)
\(500\) 329.156 0.0294406
\(501\) 0 0
\(502\) 6739.73 0.599221
\(503\) −751.675 −0.0666313 −0.0333156 0.999445i \(-0.510607\pi\)
−0.0333156 + 0.999445i \(0.510607\pi\)
\(504\) 0 0
\(505\) −7371.64 −0.649571
\(506\) 9096.06 0.799149
\(507\) 0 0
\(508\) 4270.44 0.372972
\(509\) −12334.5 −1.07410 −0.537049 0.843551i \(-0.680461\pi\)
−0.537049 + 0.843551i \(0.680461\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11186.4 −0.965574
\(513\) 0 0
\(514\) −390.768 −0.0335332
\(515\) 4054.95 0.346956
\(516\) 0 0
\(517\) 5443.95 0.463104
\(518\) 0 0
\(519\) 0 0
\(520\) −7554.09 −0.637056
\(521\) −1736.43 −0.146016 −0.0730082 0.997331i \(-0.523260\pi\)
−0.0730082 + 0.997331i \(0.523260\pi\)
\(522\) 0 0
\(523\) 1421.42 0.118842 0.0594210 0.998233i \(-0.481075\pi\)
0.0594210 + 0.998233i \(0.481075\pi\)
\(524\) −3634.72 −0.303022
\(525\) 0 0
\(526\) 7516.23 0.623048
\(527\) −19151.6 −1.58303
\(528\) 0 0
\(529\) −4964.85 −0.408059
\(530\) 7737.47 0.634140
\(531\) 0 0
\(532\) 0 0
\(533\) 29541.6 2.40073
\(534\) 0 0
\(535\) 2203.30 0.178050
\(536\) −13607.3 −1.09654
\(537\) 0 0
\(538\) 6599.26 0.528837
\(539\) 0 0
\(540\) 0 0
\(541\) 5773.27 0.458802 0.229401 0.973332i \(-0.426323\pi\)
0.229401 + 0.973332i \(0.426323\pi\)
\(542\) −6604.64 −0.523420
\(543\) 0 0
\(544\) −11518.2 −0.907793
\(545\) 9530.94 0.749102
\(546\) 0 0
\(547\) −3941.30 −0.308076 −0.154038 0.988065i \(-0.549228\pi\)
−0.154038 + 0.988065i \(0.549228\pi\)
\(548\) −5133.93 −0.400202
\(549\) 0 0
\(550\) −2679.55 −0.207739
\(551\) −110.751 −0.00856291
\(552\) 0 0
\(553\) 0 0
\(554\) 5325.06 0.408376
\(555\) 0 0
\(556\) −7373.10 −0.562390
\(557\) 6951.74 0.528823 0.264412 0.964410i \(-0.414822\pi\)
0.264412 + 0.964410i \(0.414822\pi\)
\(558\) 0 0
\(559\) 5998.29 0.453847
\(560\) 0 0
\(561\) 0 0
\(562\) −14152.7 −1.06227
\(563\) −24284.6 −1.81789 −0.908946 0.416913i \(-0.863112\pi\)
−0.908946 + 0.416913i \(0.863112\pi\)
\(564\) 0 0
\(565\) 4812.35 0.358331
\(566\) 13563.7 1.00729
\(567\) 0 0
\(568\) −18235.1 −1.34706
\(569\) 21563.4 1.58873 0.794363 0.607443i \(-0.207805\pi\)
0.794363 + 0.607443i \(0.207805\pi\)
\(570\) 0 0
\(571\) −3689.56 −0.270409 −0.135204 0.990818i \(-0.543169\pi\)
−0.135204 + 0.990818i \(0.543169\pi\)
\(572\) −7472.21 −0.546204
\(573\) 0 0
\(574\) 0 0
\(575\) −2121.64 −0.153875
\(576\) 0 0
\(577\) −22183.9 −1.60057 −0.800285 0.599620i \(-0.795318\pi\)
−0.800285 + 0.599620i \(0.795318\pi\)
\(578\) 12406.2 0.892783
\(579\) 0 0
\(580\) −397.594 −0.0284641
\(581\) 0 0
\(582\) 0 0
\(583\) 30905.8 2.19552
\(584\) −5747.55 −0.407252
\(585\) 0 0
\(586\) 11896.7 0.838646
\(587\) −10605.3 −0.745705 −0.372852 0.927891i \(-0.621620\pi\)
−0.372852 + 0.927891i \(0.621620\pi\)
\(588\) 0 0
\(589\) 693.149 0.0484902
\(590\) 664.119 0.0463412
\(591\) 0 0
\(592\) −650.466 −0.0451588
\(593\) −6277.25 −0.434698 −0.217349 0.976094i \(-0.569741\pi\)
−0.217349 + 0.976094i \(0.569741\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3776.39 −0.259542
\(597\) 0 0
\(598\) 12058.1 0.824567
\(599\) 9970.73 0.680122 0.340061 0.940403i \(-0.389552\pi\)
0.340061 + 0.940403i \(0.389552\pi\)
\(600\) 0 0
\(601\) −24619.2 −1.67094 −0.835472 0.549533i \(-0.814806\pi\)
−0.835472 + 0.549533i \(0.814806\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5228.53 0.352228
\(605\) −4047.94 −0.272020
\(606\) 0 0
\(607\) 11252.9 0.752460 0.376230 0.926526i \(-0.377220\pi\)
0.376230 + 0.926526i \(0.377220\pi\)
\(608\) 416.876 0.0278068
\(609\) 0 0
\(610\) −8559.90 −0.568164
\(611\) 7216.69 0.477833
\(612\) 0 0
\(613\) −15293.2 −1.00765 −0.503824 0.863807i \(-0.668074\pi\)
−0.503824 + 0.863807i \(0.668074\pi\)
\(614\) 4871.79 0.320211
\(615\) 0 0
\(616\) 0 0
\(617\) 17589.4 1.14769 0.573843 0.818966i \(-0.305452\pi\)
0.573843 + 0.818966i \(0.305452\pi\)
\(618\) 0 0
\(619\) −23467.4 −1.52380 −0.761900 0.647694i \(-0.775733\pi\)
−0.761900 + 0.647694i \(0.775733\pi\)
\(620\) 2488.39 0.161187
\(621\) 0 0
\(622\) 13353.7 0.860829
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −3150.64 −0.201158
\(627\) 0 0
\(628\) −107.068 −0.00680330
\(629\) −1830.93 −0.116063
\(630\) 0 0
\(631\) −6040.86 −0.381114 −0.190557 0.981676i \(-0.561029\pi\)
−0.190557 + 0.981676i \(0.561029\pi\)
\(632\) 26485.3 1.66698
\(633\) 0 0
\(634\) −11905.0 −0.745755
\(635\) 8108.68 0.506745
\(636\) 0 0
\(637\) 0 0
\(638\) 3236.68 0.200849
\(639\) 0 0
\(640\) −1839.37 −0.113605
\(641\) −25111.6 −1.54735 −0.773673 0.633586i \(-0.781582\pi\)
−0.773673 + 0.633586i \(0.781582\pi\)
\(642\) 0 0
\(643\) 3095.03 0.189823 0.0949113 0.995486i \(-0.469743\pi\)
0.0949113 + 0.995486i \(0.469743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −860.944 −0.0524356
\(647\) −9178.63 −0.557727 −0.278863 0.960331i \(-0.589958\pi\)
−0.278863 + 0.960331i \(0.589958\pi\)
\(648\) 0 0
\(649\) 2652.69 0.160443
\(650\) −3552.11 −0.214346
\(651\) 0 0
\(652\) 10411.8 0.625397
\(653\) 14438.4 0.865265 0.432632 0.901570i \(-0.357585\pi\)
0.432632 + 0.901570i \(0.357585\pi\)
\(654\) 0 0
\(655\) −6901.59 −0.411706
\(656\) 17339.8 1.03202
\(657\) 0 0
\(658\) 0 0
\(659\) 2900.64 0.171461 0.0857305 0.996318i \(-0.472678\pi\)
0.0857305 + 0.996318i \(0.472678\pi\)
\(660\) 0 0
\(661\) 9976.52 0.587053 0.293526 0.955951i \(-0.405171\pi\)
0.293526 + 0.955951i \(0.405171\pi\)
\(662\) 4229.46 0.248312
\(663\) 0 0
\(664\) 16832.5 0.983777
\(665\) 0 0
\(666\) 0 0
\(667\) 2562.77 0.148772
\(668\) 8901.19 0.515565
\(669\) 0 0
\(670\) −6398.47 −0.368947
\(671\) −34190.8 −1.96710
\(672\) 0 0
\(673\) 20760.8 1.18911 0.594554 0.804055i \(-0.297329\pi\)
0.594554 + 0.804055i \(0.297329\pi\)
\(674\) 356.725 0.0203866
\(675\) 0 0
\(676\) −4120.18 −0.234421
\(677\) −3209.13 −0.182181 −0.0910907 0.995843i \(-0.529035\pi\)
−0.0910907 + 0.995843i \(0.529035\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −12480.7 −0.703845
\(681\) 0 0
\(682\) −20257.2 −1.13737
\(683\) −4333.57 −0.242781 −0.121391 0.992605i \(-0.538735\pi\)
−0.121391 + 0.992605i \(0.538735\pi\)
\(684\) 0 0
\(685\) −9748.29 −0.543741
\(686\) 0 0
\(687\) 0 0
\(688\) 3520.78 0.195100
\(689\) 40969.8 2.26535
\(690\) 0 0
\(691\) −14446.0 −0.795297 −0.397649 0.917538i \(-0.630174\pi\)
−0.397649 + 0.917538i \(0.630174\pi\)
\(692\) 8443.94 0.463859
\(693\) 0 0
\(694\) −10099.1 −0.552385
\(695\) −14000.0 −0.764101
\(696\) 0 0
\(697\) 48808.1 2.65242
\(698\) 3912.77 0.212179
\(699\) 0 0
\(700\) 0 0
\(701\) 859.801 0.0463256 0.0231628 0.999732i \(-0.492626\pi\)
0.0231628 + 0.999732i \(0.492626\pi\)
\(702\) 0 0
\(703\) 66.2663 0.00355517
\(704\) −25507.9 −1.36557
\(705\) 0 0
\(706\) 9084.31 0.484267
\(707\) 0 0
\(708\) 0 0
\(709\) −7979.13 −0.422655 −0.211327 0.977415i \(-0.567779\pi\)
−0.211327 + 0.977415i \(0.567779\pi\)
\(710\) −8574.57 −0.453236
\(711\) 0 0
\(712\) −34001.7 −1.78970
\(713\) −16039.4 −0.842467
\(714\) 0 0
\(715\) −14188.2 −0.742110
\(716\) 3798.86 0.198282
\(717\) 0 0
\(718\) 6643.76 0.345325
\(719\) 33703.3 1.74815 0.874076 0.485790i \(-0.161468\pi\)
0.874076 + 0.485790i \(0.161468\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15858.6 −0.817444
\(723\) 0 0
\(724\) 2392.78 0.122827
\(725\) −754.950 −0.0386733
\(726\) 0 0
\(727\) −30277.0 −1.54458 −0.772290 0.635270i \(-0.780889\pi\)
−0.772290 + 0.635270i \(0.780889\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2702.63 −0.137026
\(731\) 9910.27 0.501429
\(732\) 0 0
\(733\) 19363.9 0.975749 0.487874 0.872914i \(-0.337772\pi\)
0.487874 + 0.872914i \(0.337772\pi\)
\(734\) 26648.1 1.34005
\(735\) 0 0
\(736\) −9646.45 −0.483115
\(737\) −25557.4 −1.27737
\(738\) 0 0
\(739\) 24952.4 1.24207 0.621035 0.783783i \(-0.286713\pi\)
0.621035 + 0.783783i \(0.286713\pi\)
\(740\) 237.894 0.0118178
\(741\) 0 0
\(742\) 0 0
\(743\) 8154.54 0.402640 0.201320 0.979526i \(-0.435477\pi\)
0.201320 + 0.979526i \(0.435477\pi\)
\(744\) 0 0
\(745\) −7170.58 −0.352631
\(746\) 11783.3 0.578310
\(747\) 0 0
\(748\) −12345.5 −0.603469
\(749\) 0 0
\(750\) 0 0
\(751\) −4311.26 −0.209481 −0.104740 0.994500i \(-0.533401\pi\)
−0.104740 + 0.994500i \(0.533401\pi\)
\(752\) 4235.94 0.205411
\(753\) 0 0
\(754\) 4290.66 0.207237
\(755\) 9927.91 0.478561
\(756\) 0 0
\(757\) 3624.79 0.174036 0.0870179 0.996207i \(-0.472266\pi\)
0.0870179 + 0.996207i \(0.472266\pi\)
\(758\) 2211.98 0.105993
\(759\) 0 0
\(760\) 451.713 0.0215597
\(761\) −20576.4 −0.980150 −0.490075 0.871680i \(-0.663031\pi\)
−0.490075 + 0.871680i \(0.663031\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 6516.34 0.308577
\(765\) 0 0
\(766\) −7144.26 −0.336988
\(767\) 3516.50 0.165546
\(768\) 0 0
\(769\) −3066.14 −0.143781 −0.0718907 0.997413i \(-0.522903\pi\)
−0.0718907 + 0.997413i \(0.522903\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9304.64 −0.433784
\(773\) 19387.0 0.902074 0.451037 0.892505i \(-0.351054\pi\)
0.451037 + 0.892505i \(0.351054\pi\)
\(774\) 0 0
\(775\) 4724.94 0.219000
\(776\) −5370.23 −0.248428
\(777\) 0 0
\(778\) 14666.9 0.675879
\(779\) −1766.50 −0.0812470
\(780\) 0 0
\(781\) −34249.4 −1.56919
\(782\) 19922.1 0.911015
\(783\) 0 0
\(784\) 0 0
\(785\) −203.300 −0.00924342
\(786\) 0 0
\(787\) 43363.4 1.96409 0.982044 0.188651i \(-0.0604115\pi\)
0.982044 + 0.188651i \(0.0604115\pi\)
\(788\) −5141.59 −0.232439
\(789\) 0 0
\(790\) 12454.0 0.560878
\(791\) 0 0
\(792\) 0 0
\(793\) −45324.6 −2.02966
\(794\) −28108.2 −1.25632
\(795\) 0 0
\(796\) −10703.3 −0.476593
\(797\) −17132.6 −0.761439 −0.380720 0.924691i \(-0.624324\pi\)
−0.380720 + 0.924691i \(0.624324\pi\)
\(798\) 0 0
\(799\) 11923.3 0.527929
\(800\) 2841.69 0.125586
\(801\) 0 0
\(802\) 627.039 0.0276079
\(803\) −10795.1 −0.474411
\(804\) 0 0
\(805\) 0 0
\(806\) −26853.6 −1.17355
\(807\) 0 0
\(808\) −36317.5 −1.58124
\(809\) 1080.49 0.0469566 0.0234783 0.999724i \(-0.492526\pi\)
0.0234783 + 0.999724i \(0.492526\pi\)
\(810\) 0 0
\(811\) 19593.9 0.848378 0.424189 0.905574i \(-0.360559\pi\)
0.424189 + 0.905574i \(0.360559\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1936.62 −0.0833889
\(815\) 19769.9 0.849706
\(816\) 0 0
\(817\) −358.680 −0.0153594
\(818\) 9312.66 0.398056
\(819\) 0 0
\(820\) −6341.69 −0.270075
\(821\) −5123.80 −0.217810 −0.108905 0.994052i \(-0.534734\pi\)
−0.108905 + 0.994052i \(0.534734\pi\)
\(822\) 0 0
\(823\) −13184.1 −0.558405 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(824\) 19977.3 0.844591
\(825\) 0 0
\(826\) 0 0
\(827\) −24658.7 −1.03684 −0.518421 0.855126i \(-0.673480\pi\)
−0.518421 + 0.855126i \(0.673480\pi\)
\(828\) 0 0
\(829\) −28562.3 −1.19664 −0.598318 0.801259i \(-0.704164\pi\)
−0.598318 + 0.801259i \(0.704164\pi\)
\(830\) 7915.04 0.331006
\(831\) 0 0
\(832\) −33814.1 −1.40901
\(833\) 0 0
\(834\) 0 0
\(835\) 16901.5 0.700481
\(836\) 446.817 0.0184850
\(837\) 0 0
\(838\) −5646.27 −0.232753
\(839\) −31106.0 −1.27997 −0.639987 0.768386i \(-0.721060\pi\)
−0.639987 + 0.768386i \(0.721060\pi\)
\(840\) 0 0
\(841\) −23477.1 −0.962609
\(842\) −11007.1 −0.450511
\(843\) 0 0
\(844\) 11389.7 0.464514
\(845\) −7823.38 −0.318500
\(846\) 0 0
\(847\) 0 0
\(848\) 24047.8 0.973827
\(849\) 0 0
\(850\) −5868.73 −0.236819
\(851\) −1533.39 −0.0617674
\(852\) 0 0
\(853\) −20567.9 −0.825596 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10854.9 0.433426
\(857\) 6459.44 0.257468 0.128734 0.991679i \(-0.458909\pi\)
0.128734 + 0.991679i \(0.458909\pi\)
\(858\) 0 0
\(859\) 48214.4 1.91508 0.957541 0.288298i \(-0.0930895\pi\)
0.957541 + 0.288298i \(0.0930895\pi\)
\(860\) −1287.65 −0.0510565
\(861\) 0 0
\(862\) −18359.7 −0.725445
\(863\) 31709.5 1.25076 0.625378 0.780322i \(-0.284945\pi\)
0.625378 + 0.780322i \(0.284945\pi\)
\(864\) 0 0
\(865\) 16033.3 0.630230
\(866\) −26611.7 −1.04423
\(867\) 0 0
\(868\) 0 0
\(869\) 49745.1 1.94187
\(870\) 0 0
\(871\) −33879.8 −1.31800
\(872\) 46955.6 1.82353
\(873\) 0 0
\(874\) −721.037 −0.0279055
\(875\) 0 0
\(876\) 0 0
\(877\) 25654.8 0.987799 0.493900 0.869519i \(-0.335571\pi\)
0.493900 + 0.869519i \(0.335571\pi\)
\(878\) −21192.3 −0.814585
\(879\) 0 0
\(880\) −8327.97 −0.319018
\(881\) −11470.4 −0.438647 −0.219323 0.975652i \(-0.570385\pi\)
−0.219323 + 0.975652i \(0.570385\pi\)
\(882\) 0 0
\(883\) 39124.0 1.49108 0.745542 0.666459i \(-0.232191\pi\)
0.745542 + 0.666459i \(0.232191\pi\)
\(884\) −16365.6 −0.622663
\(885\) 0 0
\(886\) −4319.01 −0.163770
\(887\) 15585.8 0.589987 0.294994 0.955499i \(-0.404682\pi\)
0.294994 + 0.955499i \(0.404682\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −15988.4 −0.602171
\(891\) 0 0
\(892\) 2588.40 0.0971594
\(893\) −431.537 −0.0161711
\(894\) 0 0
\(895\) 7213.25 0.269399
\(896\) 0 0
\(897\) 0 0
\(898\) −10403.7 −0.386610
\(899\) −5707.34 −0.211736
\(900\) 0 0
\(901\) 67689.6 2.50285
\(902\) 51625.6 1.90570
\(903\) 0 0
\(904\) 23708.8 0.872280
\(905\) 4543.40 0.166881
\(906\) 0 0
\(907\) 27596.1 1.01027 0.505134 0.863041i \(-0.331443\pi\)
0.505134 + 0.863041i \(0.331443\pi\)
\(908\) −4373.73 −0.159854
\(909\) 0 0
\(910\) 0 0
\(911\) 14396.2 0.523565 0.261782 0.965127i \(-0.415690\pi\)
0.261782 + 0.965127i \(0.415690\pi\)
\(912\) 0 0
\(913\) 31615.1 1.14601
\(914\) −33229.2 −1.20254
\(915\) 0 0
\(916\) 1512.35 0.0545517
\(917\) 0 0
\(918\) 0 0
\(919\) 10279.6 0.368978 0.184489 0.982835i \(-0.440937\pi\)
0.184489 + 0.982835i \(0.440937\pi\)
\(920\) −10452.6 −0.374577
\(921\) 0 0
\(922\) 33727.0 1.20471
\(923\) −45402.2 −1.61910
\(924\) 0 0
\(925\) 451.713 0.0160565
\(926\) −4192.06 −0.148769
\(927\) 0 0
\(928\) −3432.53 −0.121421
\(929\) −6499.87 −0.229552 −0.114776 0.993391i \(-0.536615\pi\)
−0.114776 + 0.993391i \(0.536615\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6099.86 −0.214386
\(933\) 0 0
\(934\) −13857.2 −0.485463
\(935\) −23441.5 −0.819913
\(936\) 0 0
\(937\) −10269.8 −0.358056 −0.179028 0.983844i \(-0.557295\pi\)
−0.179028 + 0.983844i \(0.557295\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1549.21 −0.0537548
\(941\) −34396.2 −1.19159 −0.595794 0.803137i \(-0.703163\pi\)
−0.595794 + 0.803137i \(0.703163\pi\)
\(942\) 0 0
\(943\) 40876.5 1.41158
\(944\) 2064.06 0.0711647
\(945\) 0 0
\(946\) 10482.4 0.360265
\(947\) −27192.1 −0.933078 −0.466539 0.884501i \(-0.654499\pi\)
−0.466539 + 0.884501i \(0.654499\pi\)
\(948\) 0 0
\(949\) −14310.4 −0.489500
\(950\) 212.406 0.00725406
\(951\) 0 0
\(952\) 0 0
\(953\) −49965.2 −1.69836 −0.849178 0.528107i \(-0.822902\pi\)
−0.849178 + 0.528107i \(0.822902\pi\)
\(954\) 0 0
\(955\) 12373.2 0.419253
\(956\) −9635.88 −0.325990
\(957\) 0 0
\(958\) −26704.9 −0.900622
\(959\) 0 0
\(960\) 0 0
\(961\) 5929.05 0.199022
\(962\) −2567.25 −0.0860411
\(963\) 0 0
\(964\) −6441.80 −0.215225
\(965\) −17667.6 −0.589368
\(966\) 0 0
\(967\) −16755.5 −0.557208 −0.278604 0.960406i \(-0.589872\pi\)
−0.278604 + 0.960406i \(0.589872\pi\)
\(968\) −19942.8 −0.662176
\(969\) 0 0
\(970\) −2525.21 −0.0835872
\(971\) 37617.4 1.24325 0.621626 0.783314i \(-0.286472\pi\)
0.621626 + 0.783314i \(0.286472\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 36583.0 1.20348
\(975\) 0 0
\(976\) −26603.9 −0.872511
\(977\) 27690.9 0.906767 0.453384 0.891315i \(-0.350217\pi\)
0.453384 + 0.891315i \(0.350217\pi\)
\(978\) 0 0
\(979\) −63862.5 −2.08483
\(980\) 0 0
\(981\) 0 0
\(982\) −30266.4 −0.983544
\(983\) 22754.2 0.738299 0.369149 0.929370i \(-0.379649\pi\)
0.369149 + 0.929370i \(0.379649\pi\)
\(984\) 0 0
\(985\) −9762.83 −0.315807
\(986\) 7088.96 0.228964
\(987\) 0 0
\(988\) 592.316 0.0190729
\(989\) 8299.80 0.266854
\(990\) 0 0
\(991\) −55470.9 −1.77809 −0.889047 0.457816i \(-0.848632\pi\)
−0.889047 + 0.457816i \(0.848632\pi\)
\(992\) 21482.9 0.687583
\(993\) 0 0
\(994\) 0 0
\(995\) −20323.4 −0.647531
\(996\) 0 0
\(997\) −15181.9 −0.482264 −0.241132 0.970492i \(-0.577519\pi\)
−0.241132 + 0.970492i \(0.577519\pi\)
\(998\) −46647.1 −1.47955
\(999\) 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 2205.4.a.w.1.2 2
3.2 odd 2 245.4.a.j.1.1 yes 2
7.6 odd 2 2205.4.a.x.1.2 2
15.14 odd 2 1225.4.a.p.1.2 2
21.2 odd 6 245.4.e.j.116.2 4
21.5 even 6 245.4.e.k.116.2 4
21.11 odd 6 245.4.e.j.226.2 4
21.17 even 6 245.4.e.k.226.2 4
21.20 even 2 245.4.a.i.1.1 2
105.104 even 2 1225.4.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.1 2 21.20 even 2
245.4.a.j.1.1 yes 2 3.2 odd 2
245.4.e.j.116.2 4 21.2 odd 6
245.4.e.j.226.2 4 21.11 odd 6
245.4.e.k.116.2 4 21.5 even 6
245.4.e.k.226.2 4 21.17 even 6
1225.4.a.p.1.2 2 15.14 odd 2
1225.4.a.q.1.2 2 105.104 even 2
2205.4.a.w.1.2 2 1.1 even 1 trivial
2205.4.a.x.1.2 2 7.6 odd 2