Properties

Label 2205.4.a.x.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
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: \(1\)
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 \(-0.272983\pi\)
0.654253 + 0.756276i \(0.272983\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −36.0000 −0.562500
\(17\) 101.332 1.44569 0.722845 0.691010i \(-0.242834\pi\)
0.722845 + 0.691010i \(0.242834\pi\)
\(18\) 0 0
\(19\) −3.66750 −0.0442833 −0.0221417 0.999755i \(-0.507048\pi\)
−0.0221417 + 0.999755i \(0.507048\pi\)
\(20\) −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 \(-0.684420\pi\)
−0.547499 + 0.836806i \(0.684420\pi\)
\(32\) 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 \(-0.130321\pi\)
0.917354 + 0.398072i \(0.130321\pi\)
\(42\) 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 \(-0.441553\pi\)
0.182587 + 0.983190i \(0.441553\pi\)
\(48\) 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 \(-0.479851\pi\)
0.0632575 + 0.997997i \(0.479851\pi\)
\(60\) 0 0
\(61\) −738.997 −1.55113 −0.775565 0.631268i \(-0.782535\pi\)
−0.775565 + 0.631268i \(0.782535\pi\)
\(62\) −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 \(-0.559891\pi\)
−0.187045 + 0.982351i \(0.559891\pi\)
\(74\) 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 \(-0.350769\pi\)
0.451835 + 0.892101i \(0.350769\pi\)
\(84\) 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 \(-0.807133\pi\)
−0.821985 + 0.569509i \(0.807133\pi\)
\(90\) 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 \(-0.536398\pi\)
−0.114100 + 0.993469i \(0.536398\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −65.8312 −0.0658312
\(101\) −1474.33 −1.45249 −0.726243 0.687438i \(-0.758735\pi\)
−0.726243 + 0.687438i \(0.758735\pi\)
\(102\) 0 0
\(103\) 810.990 0.775818 0.387909 0.921698i \(-0.373198\pi\)
0.387909 + 0.921698i \(0.373198\pi\)
\(104\) −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 \(-0.652259\pi\)
−0.460301 + 0.887763i \(0.652259\pi\)
\(132\) 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 \(-0.826008\pi\)
−0.854291 + 0.519795i \(0.826008\pi\)
\(140\) 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 \(-0.503290\pi\)
−0.0103345 + 0.999947i \(0.503290\pi\)
\(158\) −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 \(-0.213606\pi\)
0.783161 + 0.621819i \(0.213606\pi\)
\(168\) 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 \(-0.251118\pi\)
0.704619 + 0.709586i \(0.251118\pi\)
\(174\) 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 \(-0.440260\pi\)
0.186579 + 0.982440i \(0.440260\pi\)
\(182\) 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 \(-0.757681\pi\)
−0.723962 + 0.689840i \(0.757681\pi\)
\(200\) −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 \(-0.452849\pi\)
0.147589 + 0.989049i \(0.452849\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2229.68 −0.656266
\(227\) −1660.96 −0.485648 −0.242824 0.970070i \(-0.578074\pi\)
−0.242824 + 0.970070i \(0.578074\pi\)
\(228\) 0 0
\(229\) 574.327 0.165732 0.0828660 0.996561i \(-0.473593\pi\)
0.0828660 + 0.996561i \(0.473593\pi\)
\(230\) −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 \(-0.606015\pi\)
−0.326934 + 0.945047i \(0.606015\pi\)
\(242\) 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 \(-0.619205\pi\)
−0.365802 + 0.930693i \(0.619205\pi\)
\(252\) 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 \(-0.493483\pi\)
0.0204708 + 0.999790i \(0.493483\pi\)
\(258\) 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 \(-0.604636\pi\)
−0.322836 + 0.946455i \(0.604636\pi\)
\(270\) 0 0
\(271\) 2850.98 0.639057 0.319529 0.947577i \(-0.396475\pi\)
0.319529 + 0.947577i \(0.396475\pi\)
\(272\) −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 \(-0.710809\pi\)
−0.614913 + 0.788595i \(0.710809\pi\)
\(284\) −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 \(-0.671081\pi\)
−0.511962 + 0.859008i \(0.671081\pi\)
\(294\) 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 \(-0.562625\pi\)
−0.195477 + 0.980708i \(0.562625\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2189.18 −0.401088
\(311\) −5764.30 −1.05101 −0.525504 0.850791i \(-0.676123\pi\)
−0.525504 + 0.850791i \(0.676123\pi\)
\(312\) 0 0
\(313\) 1360.01 0.245599 0.122799 0.992432i \(-0.460813\pi\)
0.122799 + 0.992432i \(0.460813\pi\)
\(314\) −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 \(-0.541346\pi\)
−0.129527 + 0.991576i \(0.541346\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5259.00 −0.796322
\(353\) −3921.36 −0.591254 −0.295627 0.955303i \(-0.595529\pi\)
−0.295627 + 0.955303i \(0.595529\pi\)
\(354\) 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 \(-0.804947\pi\)
−0.818054 + 0.575142i \(0.804947\pi\)
\(368\) 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 \(-0.434047\pi\)
0.205719 + 0.978611i \(0.434047\pi\)
\(384\) 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 \(-0.221779\pi\)
0.766939 + 0.641720i \(0.221779\pi\)
\(398\) −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 \(-0.578131\pi\)
−0.242998 + 0.970027i \(0.578131\pi\)
\(410\) 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 \(-0.454619\pi\)
0.142087 + 0.989854i \(0.454619\pi\)
\(420\) 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 \(-0.279984\pi\)
0.637462 + 0.770481i \(0.279984\pi\)
\(434\) 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 \(-0.334334\pi\)
0.497274 + 0.867594i \(0.334334\pi\)
\(440\) 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 \(-0.763019\pi\)
−0.735429 + 0.677602i \(0.763019\pi\)
\(462\) 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 \(-0.404228\pi\)
0.296357 + 0.955077i \(0.404228\pi\)
\(468\) 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 \(-0.314705\pi\)
0.549796 + 0.835299i \(0.314705\pi\)
\(480\) 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 \(-0.489393\pi\)
0.0333156 + 0.999445i \(0.489393\pi\)
\(504\) 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 \(-0.319539\pi\)
0.537049 + 0.843551i \(0.319539\pi\)
\(510\) 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 \(-0.476740\pi\)
0.0730082 + 0.997331i \(0.476740\pi\)
\(522\) 0 0
\(523\) −1421.42 −0.118842 −0.0594210 0.998233i \(-0.518925\pi\)
−0.0594210 + 0.998233i \(0.518925\pi\)
\(524\) 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 \(-0.136888\pi\)
0.908946 + 0.416913i \(0.136888\pi\)
\(564\) 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 \(-0.204682\pi\)
0.800285 + 0.599620i \(0.204682\pi\)
\(578\) 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 \(-0.378380\pi\)
0.372852 + 0.927891i \(0.378380\pi\)
\(588\) 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 \(-0.430259\pi\)
0.217349 + 0.976094i \(0.430259\pi\)
\(594\) 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 \(-0.185194\pi\)
0.835472 + 0.549533i \(0.185194\pi\)
\(602\) 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 \(-0.622780\pi\)
−0.376230 + 0.926526i \(0.622780\pi\)
\(608\) −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 \(-0.224267\pi\)
0.761900 + 0.647694i \(0.224267\pi\)
\(620\) 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 \(-0.530257\pi\)
−0.0949113 + 0.995486i \(0.530257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −860.944 −0.0524356
\(647\) 9178.63 0.557727 0.278863 0.960331i \(-0.410042\pi\)
0.278863 + 0.960331i \(0.410042\pi\)
\(648\) 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 \(-0.594829\pi\)
−0.293526 + 0.955951i \(0.594829\pi\)
\(662\) 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 \(-0.470965\pi\)
0.0910907 + 0.995843i \(0.470965\pi\)
\(678\) 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 \(-0.369826\pi\)
0.397649 + 0.917538i \(0.369826\pi\)
\(692\) −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 \(-0.838532\pi\)
−0.874076 + 0.485790i \(0.838532\pi\)
\(720\) 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 \(-0.219111\pi\)
0.772290 + 0.635270i \(0.219111\pi\)
\(728\) 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 \(-0.662228\pi\)
−0.487874 + 0.872914i \(0.662228\pi\)
\(734\) −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 \(-0.336969\pi\)
0.490075 + 0.871680i \(0.336969\pi\)
\(762\) 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 \(-0.477097\pi\)
0.0718907 + 0.997413i \(0.477097\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9304.64 −0.433784
\(773\) −19387.0 −0.902074 −0.451037 0.892505i \(-0.648946\pi\)
−0.451037 + 0.892505i \(0.648946\pi\)
\(774\) 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 \(-0.939588\pi\)
−0.982044 + 0.188651i \(0.939588\pi\)
\(788\) −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 \(-0.375676\pi\)
0.380720 + 0.924691i \(0.375676\pi\)
\(798\) 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 \(-0.639441\pi\)
−0.424189 + 0.905574i \(0.639441\pi\)
\(812\) 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 \(-0.295836\pi\)
0.598318 + 0.801259i \(0.295836\pi\)
\(830\) 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 \(-0.278940\pi\)
0.639987 + 0.768386i \(0.278940\pi\)
\(840\) 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 \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10854.9 0.433426
\(857\) −6459.44 −0.257468 −0.128734 0.991679i \(-0.541091\pi\)
−0.128734 + 0.991679i \(0.541091\pi\)
\(858\) 0 0
\(859\) −48214.4 −1.91508 −0.957541 0.288298i \(-0.906910\pi\)
−0.957541 + 0.288298i \(0.906910\pi\)
\(860\) 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 \(-0.429615\pi\)
0.219323 + 0.975652i \(0.429615\pi\)
\(882\) 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 \(-0.595318\pi\)
−0.294994 + 0.955499i \(0.595318\pi\)
\(888\) 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 \(-0.463385\pi\)
0.114776 + 0.993391i \(0.463385\pi\)
\(930\) 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 \(-0.442705\pi\)
0.179028 + 0.983844i \(0.442705\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1549.21 −0.0537548
\(941\) 34396.2 1.19159 0.595794 0.803137i \(-0.296837\pi\)
0.595794 + 0.803137i \(0.296837\pi\)
\(942\) 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 \(-0.713528\pi\)
−0.621626 + 0.783314i \(0.713528\pi\)
\(972\) 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 \(-0.620351\pi\)
−0.369149 + 0.929370i \(0.620351\pi\)
\(984\) 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 \(-0.422481\pi\)
0.241132 + 0.970492i \(0.422481\pi\)
\(998\) −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.x.1.2 2
3.2 odd 2 245.4.a.i.1.1 2
7.6 odd 2 2205.4.a.w.1.2 2
15.14 odd 2 1225.4.a.q.1.2 2
21.2 odd 6 245.4.e.k.116.2 4
21.5 even 6 245.4.e.j.116.2 4
21.11 odd 6 245.4.e.k.226.2 4
21.17 even 6 245.4.e.j.226.2 4
21.20 even 2 245.4.a.j.1.1 yes 2
105.104 even 2 1225.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.1 2 3.2 odd 2
245.4.a.j.1.1 yes 2 21.20 even 2
245.4.e.j.116.2 4 21.5 even 6
245.4.e.j.226.2 4 21.17 even 6
245.4.e.k.116.2 4 21.2 odd 6
245.4.e.k.226.2 4 21.11 odd 6
1225.4.a.p.1.2 2 105.104 even 2
1225.4.a.q.1.2 2 15.14 odd 2
2205.4.a.w.1.2 2 7.6 odd 2
2205.4.a.x.1.2 2 1.1 even 1 trivial