Properties

Label 441.4.c.a.440.13
Level $441$
Weight $4$
Character 441.440
Analytic conductor $26.020$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 440.13
Root \(3.91663 - 2.26127i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.4.c.a.440.3

$q$-expansion

\(f(q)\) \(=\) \(q+4.52254i q^{2} -12.4534 q^{4} -1.26570 q^{5} -20.1405i q^{8} +O(q^{10})\) \(q+4.52254i q^{2} -12.4534 q^{4} -1.26570 q^{5} -20.1405i q^{8} -5.72419i q^{10} +41.5978i q^{11} +85.7355i q^{13} -8.54055 q^{16} +77.7859 q^{17} +48.6865i q^{19} +15.7623 q^{20} -188.128 q^{22} -90.9487i q^{23} -123.398 q^{25} -387.742 q^{26} -151.196i q^{29} -88.1799i q^{31} -199.749i q^{32} +351.790i q^{34} +90.5828 q^{37} -220.187 q^{38} +25.4919i q^{40} -383.530 q^{41} -227.894 q^{43} -518.033i q^{44} +411.319 q^{46} +139.106 q^{47} -558.072i q^{50} -1067.70i q^{52} -334.573i q^{53} -52.6505i q^{55} +683.789 q^{58} -880.100 q^{59} +13.1571i q^{61} +398.797 q^{62} +835.050 q^{64} -108.516i q^{65} -442.424 q^{67} -968.696 q^{68} +341.552i q^{71} +921.703i q^{73} +409.664i q^{74} -606.311i q^{76} -413.129 q^{79} +10.8098 q^{80} -1734.53i q^{82} +954.307 q^{83} -98.4538 q^{85} -1030.66i q^{86} +837.803 q^{88} -29.6981 q^{89} +1132.62i q^{92} +629.111i q^{94} -61.6227i q^{95} +1199.63i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 64q^{4} + O(q^{10}) \) \( 16q - 64q^{4} + 376q^{16} + 528q^{22} + 40q^{25} + 2392q^{37} + 328q^{43} + 2784q^{46} + 6744q^{58} + 5432q^{64} - 616q^{67} + 4352q^{79} - 4608q^{85} - 1416q^{88} + O(q^{100}) \)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.52254i 1.59896i 0.600693 + 0.799480i \(0.294891\pi\)
−0.600693 + 0.799480i \(0.705109\pi\)
\(3\) 0 0
\(4\) −12.4534 −1.55667
\(5\) −1.26570 −0.113208 −0.0566040 0.998397i \(-0.518027\pi\)
−0.0566040 + 0.998397i \(0.518027\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 20.1405i − 0.890094i
\(9\) 0 0
\(10\) − 5.72419i − 0.181015i
\(11\) 41.5978i 1.14020i 0.821575 + 0.570101i \(0.193096\pi\)
−0.821575 + 0.570101i \(0.806904\pi\)
\(12\) 0 0
\(13\) 85.7355i 1.82914i 0.404433 + 0.914568i \(0.367469\pi\)
−0.404433 + 0.914568i \(0.632531\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.54055 −0.133446
\(17\) 77.7859 1.10976 0.554878 0.831932i \(-0.312765\pi\)
0.554878 + 0.831932i \(0.312765\pi\)
\(18\) 0 0
\(19\) 48.6865i 0.587866i 0.955826 + 0.293933i \(0.0949642\pi\)
−0.955826 + 0.293933i \(0.905036\pi\)
\(20\) 15.7623 0.176227
\(21\) 0 0
\(22\) −188.128 −1.82314
\(23\) − 90.9487i − 0.824527i −0.911065 0.412263i \(-0.864738\pi\)
0.911065 0.412263i \(-0.135262\pi\)
\(24\) 0 0
\(25\) −123.398 −0.987184
\(26\) −387.742 −2.92471
\(27\) 0 0
\(28\) 0 0
\(29\) − 151.196i − 0.968151i −0.875026 0.484075i \(-0.839156\pi\)
0.875026 0.484075i \(-0.160844\pi\)
\(30\) 0 0
\(31\) − 88.1799i − 0.510890i −0.966824 0.255445i \(-0.917778\pi\)
0.966824 0.255445i \(-0.0822219\pi\)
\(32\) − 199.749i − 1.10347i
\(33\) 0 0
\(34\) 351.790i 1.77445i
\(35\) 0 0
\(36\) 0 0
\(37\) 90.5828 0.402479 0.201239 0.979542i \(-0.435503\pi\)
0.201239 + 0.979542i \(0.435503\pi\)
\(38\) −220.187 −0.939974
\(39\) 0 0
\(40\) 25.4919i 0.100766i
\(41\) −383.530 −1.46091 −0.730455 0.682961i \(-0.760692\pi\)
−0.730455 + 0.682961i \(0.760692\pi\)
\(42\) 0 0
\(43\) −227.894 −0.808222 −0.404111 0.914710i \(-0.632419\pi\)
−0.404111 + 0.914710i \(0.632419\pi\)
\(44\) − 518.033i − 1.77492i
\(45\) 0 0
\(46\) 411.319 1.31838
\(47\) 139.106 0.431716 0.215858 0.976425i \(-0.430745\pi\)
0.215858 + 0.976425i \(0.430745\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 558.072i − 1.57847i
\(51\) 0 0
\(52\) − 1067.70i − 2.84736i
\(53\) − 334.573i − 0.867116i −0.901126 0.433558i \(-0.857258\pi\)
0.901126 0.433558i \(-0.142742\pi\)
\(54\) 0 0
\(55\) − 52.6505i − 0.129080i
\(56\) 0 0
\(57\) 0 0
\(58\) 683.789 1.54803
\(59\) −880.100 −1.94202 −0.971010 0.239037i \(-0.923168\pi\)
−0.971010 + 0.239037i \(0.923168\pi\)
\(60\) 0 0
\(61\) 13.1571i 0.0276162i 0.999905 + 0.0138081i \(0.00439540\pi\)
−0.999905 + 0.0138081i \(0.995605\pi\)
\(62\) 398.797 0.816892
\(63\) 0 0
\(64\) 835.050 1.63096
\(65\) − 108.516i − 0.207073i
\(66\) 0 0
\(67\) −442.424 −0.806728 −0.403364 0.915040i \(-0.632159\pi\)
−0.403364 + 0.915040i \(0.632159\pi\)
\(68\) −968.696 −1.72752
\(69\) 0 0
\(70\) 0 0
\(71\) 341.552i 0.570912i 0.958392 + 0.285456i \(0.0921450\pi\)
−0.958392 + 0.285456i \(0.907855\pi\)
\(72\) 0 0
\(73\) 921.703i 1.47777i 0.673832 + 0.738885i \(0.264647\pi\)
−0.673832 + 0.738885i \(0.735353\pi\)
\(74\) 409.664i 0.643547i
\(75\) 0 0
\(76\) − 606.311i − 0.915114i
\(77\) 0 0
\(78\) 0 0
\(79\) −413.129 −0.588362 −0.294181 0.955750i \(-0.595047\pi\)
−0.294181 + 0.955750i \(0.595047\pi\)
\(80\) 10.8098 0.0151072
\(81\) 0 0
\(82\) − 1734.53i − 2.33594i
\(83\) 954.307 1.26203 0.631017 0.775769i \(-0.282638\pi\)
0.631017 + 0.775769i \(0.282638\pi\)
\(84\) 0 0
\(85\) −98.4538 −0.125633
\(86\) − 1030.66i − 1.29231i
\(87\) 0 0
\(88\) 837.803 1.01489
\(89\) −29.6981 −0.0353707 −0.0176853 0.999844i \(-0.505630\pi\)
−0.0176853 + 0.999844i \(0.505630\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1132.62i 1.28352i
\(93\) 0 0
\(94\) 629.111i 0.690297i
\(95\) − 61.6227i − 0.0665511i
\(96\) 0 0
\(97\) 1199.63i 1.25572i 0.778328 + 0.627858i \(0.216068\pi\)
−0.778328 + 0.627858i \(0.783932\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1536.72 1.53672
\(101\) 654.844 0.645143 0.322571 0.946545i \(-0.395453\pi\)
0.322571 + 0.946545i \(0.395453\pi\)
\(102\) 0 0
\(103\) − 1369.49i − 1.31009i −0.755589 0.655047i \(-0.772649\pi\)
0.755589 0.655047i \(-0.227351\pi\)
\(104\) 1726.76 1.62810
\(105\) 0 0
\(106\) 1513.12 1.38648
\(107\) 428.753i 0.387375i 0.981063 + 0.193688i \(0.0620448\pi\)
−0.981063 + 0.193688i \(0.937955\pi\)
\(108\) 0 0
\(109\) −668.521 −0.587456 −0.293728 0.955889i \(-0.594896\pi\)
−0.293728 + 0.955889i \(0.594896\pi\)
\(110\) 238.114 0.206393
\(111\) 0 0
\(112\) 0 0
\(113\) 914.837i 0.761598i 0.924658 + 0.380799i \(0.124351\pi\)
−0.924658 + 0.380799i \(0.875649\pi\)
\(114\) 0 0
\(115\) 115.114i 0.0933429i
\(116\) 1882.90i 1.50709i
\(117\) 0 0
\(118\) − 3980.29i − 3.10521i
\(119\) 0 0
\(120\) 0 0
\(121\) −399.379 −0.300060
\(122\) −59.5034 −0.0441572
\(123\) 0 0
\(124\) 1098.14i 0.795287i
\(125\) 314.398 0.224965
\(126\) 0 0
\(127\) 1260.95 0.881034 0.440517 0.897744i \(-0.354795\pi\)
0.440517 + 0.897744i \(0.354795\pi\)
\(128\) 2178.55i 1.50436i
\(129\) 0 0
\(130\) 490.767 0.331101
\(131\) 1367.20 0.911853 0.455926 0.890017i \(-0.349308\pi\)
0.455926 + 0.890017i \(0.349308\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 2000.88i − 1.28992i
\(135\) 0 0
\(136\) − 1566.65i − 0.987787i
\(137\) − 1101.16i − 0.686703i −0.939207 0.343351i \(-0.888438\pi\)
0.939207 0.343351i \(-0.111562\pi\)
\(138\) 0 0
\(139\) 2306.56i 1.40748i 0.710458 + 0.703739i \(0.248488\pi\)
−0.710458 + 0.703739i \(0.751512\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1544.68 −0.912865
\(143\) −3566.41 −2.08558
\(144\) 0 0
\(145\) 191.369i 0.109602i
\(146\) −4168.44 −2.36289
\(147\) 0 0
\(148\) −1128.06 −0.626527
\(149\) − 1755.80i − 0.965376i −0.875793 0.482688i \(-0.839661\pi\)
0.875793 0.482688i \(-0.160339\pi\)
\(150\) 0 0
\(151\) −524.981 −0.282930 −0.141465 0.989943i \(-0.545181\pi\)
−0.141465 + 0.989943i \(0.545181\pi\)
\(152\) 980.573 0.523256
\(153\) 0 0
\(154\) 0 0
\(155\) 111.610i 0.0578368i
\(156\) 0 0
\(157\) 1318.02i 0.669995i 0.942219 + 0.334998i \(0.108736\pi\)
−0.942219 + 0.334998i \(0.891264\pi\)
\(158\) − 1868.39i − 0.940767i
\(159\) 0 0
\(160\) 252.823i 0.124921i
\(161\) 0 0
\(162\) 0 0
\(163\) −447.832 −0.215196 −0.107598 0.994195i \(-0.534316\pi\)
−0.107598 + 0.994195i \(0.534316\pi\)
\(164\) 4776.24 2.27416
\(165\) 0 0
\(166\) 4315.89i 2.01794i
\(167\) −811.124 −0.375848 −0.187924 0.982184i \(-0.560176\pi\)
−0.187924 + 0.982184i \(0.560176\pi\)
\(168\) 0 0
\(169\) −5153.58 −2.34574
\(170\) − 445.261i − 0.200882i
\(171\) 0 0
\(172\) 2838.05 1.25814
\(173\) 2242.47 0.985503 0.492751 0.870170i \(-0.335991\pi\)
0.492751 + 0.870170i \(0.335991\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 355.269i − 0.152156i
\(177\) 0 0
\(178\) − 134.311i − 0.0565563i
\(179\) − 2923.43i − 1.22071i −0.792127 0.610357i \(-0.791026\pi\)
0.792127 0.610357i \(-0.208974\pi\)
\(180\) 0 0
\(181\) − 282.859i − 0.116159i −0.998312 0.0580794i \(-0.981502\pi\)
0.998312 0.0580794i \(-0.0184977\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1831.76 −0.733907
\(185\) −114.651 −0.0455638
\(186\) 0 0
\(187\) 3235.72i 1.26534i
\(188\) −1732.34 −0.672040
\(189\) 0 0
\(190\) 278.691 0.106412
\(191\) 4617.22i 1.74916i 0.484878 + 0.874582i \(0.338864\pi\)
−0.484878 + 0.874582i \(0.661136\pi\)
\(192\) 0 0
\(193\) −4155.45 −1.54982 −0.774912 0.632070i \(-0.782206\pi\)
−0.774912 + 0.632070i \(0.782206\pi\)
\(194\) −5425.39 −2.00784
\(195\) 0 0
\(196\) 0 0
\(197\) 1626.36i 0.588190i 0.955776 + 0.294095i \(0.0950183\pi\)
−0.955776 + 0.294095i \(0.904982\pi\)
\(198\) 0 0
\(199\) − 174.199i − 0.0620535i −0.999519 0.0310267i \(-0.990122\pi\)
0.999519 0.0310267i \(-0.00987771\pi\)
\(200\) 2485.30i 0.878687i
\(201\) 0 0
\(202\) 2961.56i 1.03156i
\(203\) 0 0
\(204\) 0 0
\(205\) 485.435 0.165387
\(206\) 6193.56 2.09479
\(207\) 0 0
\(208\) − 732.229i − 0.244091i
\(209\) −2025.25 −0.670286
\(210\) 0 0
\(211\) −2942.35 −0.959999 −0.479999 0.877269i \(-0.659363\pi\)
−0.479999 + 0.877269i \(0.659363\pi\)
\(212\) 4166.56i 1.34981i
\(213\) 0 0
\(214\) −1939.05 −0.619397
\(215\) 288.446 0.0914972
\(216\) 0 0
\(217\) 0 0
\(218\) − 3023.42i − 0.939319i
\(219\) 0 0
\(220\) 655.676i 0.200935i
\(221\) 6669.01i 2.02989i
\(222\) 0 0
\(223\) 3374.75i 1.01341i 0.862120 + 0.506704i \(0.169136\pi\)
−0.862120 + 0.506704i \(0.830864\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4137.39 −1.21776
\(227\) 3030.85 0.886188 0.443094 0.896475i \(-0.353881\pi\)
0.443094 + 0.896475i \(0.353881\pi\)
\(228\) 0 0
\(229\) 1108.55i 0.319890i 0.987126 + 0.159945i \(0.0511317\pi\)
−0.987126 + 0.159945i \(0.948868\pi\)
\(230\) −520.608 −0.149252
\(231\) 0 0
\(232\) −3045.17 −0.861746
\(233\) − 3099.87i − 0.871585i −0.900047 0.435793i \(-0.856468\pi\)
0.900047 0.435793i \(-0.143532\pi\)
\(234\) 0 0
\(235\) −176.067 −0.0488737
\(236\) 10960.2 3.02309
\(237\) 0 0
\(238\) 0 0
\(239\) − 1735.25i − 0.469640i −0.972039 0.234820i \(-0.924550\pi\)
0.972039 0.234820i \(-0.0754501\pi\)
\(240\) 0 0
\(241\) 1200.04i 0.320752i 0.987056 + 0.160376i \(0.0512707\pi\)
−0.987056 + 0.160376i \(0.948729\pi\)
\(242\) − 1806.21i − 0.479783i
\(243\) 0 0
\(244\) − 163.850i − 0.0429894i
\(245\) 0 0
\(246\) 0 0
\(247\) −4174.16 −1.07529
\(248\) −1775.99 −0.454740
\(249\) 0 0
\(250\) 1421.88i 0.359710i
\(251\) 3712.56 0.933603 0.466802 0.884362i \(-0.345406\pi\)
0.466802 + 0.884362i \(0.345406\pi\)
\(252\) 0 0
\(253\) 3783.27 0.940126
\(254\) 5702.70i 1.40874i
\(255\) 0 0
\(256\) −3172.19 −0.774460
\(257\) −779.148 −0.189113 −0.0945563 0.995520i \(-0.530143\pi\)
−0.0945563 + 0.995520i \(0.530143\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1351.39i 0.322344i
\(261\) 0 0
\(262\) 6183.21i 1.45802i
\(263\) 1965.38i 0.460801i 0.973096 + 0.230400i \(0.0740035\pi\)
−0.973096 + 0.230400i \(0.925996\pi\)
\(264\) 0 0
\(265\) 423.470i 0.0981644i
\(266\) 0 0
\(267\) 0 0
\(268\) 5509.68 1.25581
\(269\) −2472.82 −0.560486 −0.280243 0.959929i \(-0.590415\pi\)
−0.280243 + 0.959929i \(0.590415\pi\)
\(270\) 0 0
\(271\) 4729.42i 1.06012i 0.847961 + 0.530058i \(0.177830\pi\)
−0.847961 + 0.530058i \(0.822170\pi\)
\(272\) −664.334 −0.148093
\(273\) 0 0
\(274\) 4980.03 1.09801
\(275\) − 5133.09i − 1.12559i
\(276\) 0 0
\(277\) 1173.16 0.254470 0.127235 0.991873i \(-0.459390\pi\)
0.127235 + 0.991873i \(0.459390\pi\)
\(278\) −10431.5 −2.25050
\(279\) 0 0
\(280\) 0 0
\(281\) 8195.18i 1.73980i 0.493229 + 0.869899i \(0.335816\pi\)
−0.493229 + 0.869899i \(0.664184\pi\)
\(282\) 0 0
\(283\) − 2589.34i − 0.543889i −0.962313 0.271944i \(-0.912333\pi\)
0.962313 0.271944i \(-0.0876666\pi\)
\(284\) − 4253.47i − 0.888722i
\(285\) 0 0
\(286\) − 16129.2i − 3.33476i
\(287\) 0 0
\(288\) 0 0
\(289\) 1137.64 0.231557
\(290\) −865.474 −0.175250
\(291\) 0 0
\(292\) − 11478.3i − 2.30040i
\(293\) 8871.16 1.76880 0.884400 0.466729i \(-0.154568\pi\)
0.884400 + 0.466729i \(0.154568\pi\)
\(294\) 0 0
\(295\) 1113.94 0.219852
\(296\) − 1824.39i − 0.358244i
\(297\) 0 0
\(298\) 7940.69 1.54360
\(299\) 7797.53 1.50817
\(300\) 0 0
\(301\) 0 0
\(302\) − 2374.25i − 0.452393i
\(303\) 0 0
\(304\) − 415.810i − 0.0784484i
\(305\) − 16.6529i − 0.00312638i
\(306\) 0 0
\(307\) − 2707.52i − 0.503344i −0.967813 0.251672i \(-0.919020\pi\)
0.967813 0.251672i \(-0.0809804\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −504.759 −0.0924786
\(311\) −2161.09 −0.394033 −0.197017 0.980400i \(-0.563125\pi\)
−0.197017 + 0.980400i \(0.563125\pi\)
\(312\) 0 0
\(313\) − 8429.61i − 1.52227i −0.648595 0.761133i \(-0.724643\pi\)
0.648595 0.761133i \(-0.275357\pi\)
\(314\) −5960.79 −1.07130
\(315\) 0 0
\(316\) 5144.84 0.915886
\(317\) 9595.64i 1.70014i 0.526669 + 0.850071i \(0.323441\pi\)
−0.526669 + 0.850071i \(0.676559\pi\)
\(318\) 0 0
\(319\) 6289.42 1.10389
\(320\) −1056.93 −0.184637
\(321\) 0 0
\(322\) 0 0
\(323\) 3787.12i 0.652387i
\(324\) 0 0
\(325\) − 10579.6i − 1.80569i
\(326\) − 2025.34i − 0.344089i
\(327\) 0 0
\(328\) 7724.50i 1.30035i
\(329\) 0 0
\(330\) 0 0
\(331\) 8543.91 1.41878 0.709390 0.704817i \(-0.248971\pi\)
0.709390 + 0.704817i \(0.248971\pi\)
\(332\) −11884.3 −1.96457
\(333\) 0 0
\(334\) − 3668.34i − 0.600966i
\(335\) 559.978 0.0913280
\(336\) 0 0
\(337\) 598.875 0.0968036 0.0484018 0.998828i \(-0.484587\pi\)
0.0484018 + 0.998828i \(0.484587\pi\)
\(338\) − 23307.3i − 3.75074i
\(339\) 0 0
\(340\) 1226.08 0.195569
\(341\) 3668.09 0.582517
\(342\) 0 0
\(343\) 0 0
\(344\) 4589.91i 0.719394i
\(345\) 0 0
\(346\) 10141.7i 1.57578i
\(347\) − 7100.97i − 1.09856i −0.835638 0.549280i \(-0.814902\pi\)
0.835638 0.549280i \(-0.185098\pi\)
\(348\) 0 0
\(349\) − 3620.71i − 0.555336i −0.960677 0.277668i \(-0.910438\pi\)
0.960677 0.277668i \(-0.0895616\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 8309.14 1.25818
\(353\) 2329.77 0.351278 0.175639 0.984455i \(-0.443801\pi\)
0.175639 + 0.984455i \(0.443801\pi\)
\(354\) 0 0
\(355\) − 432.303i − 0.0646317i
\(356\) 369.841 0.0550605
\(357\) 0 0
\(358\) 13221.3 1.95187
\(359\) 1757.95i 0.258443i 0.991616 + 0.129222i \(0.0412478\pi\)
−0.991616 + 0.129222i \(0.958752\pi\)
\(360\) 0 0
\(361\) 4488.62 0.654414
\(362\) 1279.24 0.185733
\(363\) 0 0
\(364\) 0 0
\(365\) − 1166.60i − 0.167295i
\(366\) 0 0
\(367\) − 1684.58i − 0.239603i −0.992798 0.119802i \(-0.961774\pi\)
0.992798 0.119802i \(-0.0382259\pi\)
\(368\) 776.752i 0.110030i
\(369\) 0 0
\(370\) − 518.513i − 0.0728547i
\(371\) 0 0
\(372\) 0 0
\(373\) −297.292 −0.0412686 −0.0206343 0.999787i \(-0.506569\pi\)
−0.0206343 + 0.999787i \(0.506569\pi\)
\(374\) −14633.7 −2.02323
\(375\) 0 0
\(376\) − 2801.66i − 0.384268i
\(377\) 12962.9 1.77088
\(378\) 0 0
\(379\) −7402.78 −1.00331 −0.501656 0.865067i \(-0.667276\pi\)
−0.501656 + 0.865067i \(0.667276\pi\)
\(380\) 767.410i 0.103598i
\(381\) 0 0
\(382\) −20881.6 −2.79684
\(383\) −12527.5 −1.67135 −0.835676 0.549223i \(-0.814924\pi\)
−0.835676 + 0.549223i \(0.814924\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 18793.2i − 2.47810i
\(387\) 0 0
\(388\) − 14939.5i − 1.95474i
\(389\) 7972.22i 1.03909i 0.854442 + 0.519547i \(0.173899\pi\)
−0.854442 + 0.519547i \(0.826101\pi\)
\(390\) 0 0
\(391\) − 7074.52i − 0.915023i
\(392\) 0 0
\(393\) 0 0
\(394\) −7355.29 −0.940493
\(395\) 522.898 0.0666072
\(396\) 0 0
\(397\) 12508.3i 1.58130i 0.612271 + 0.790648i \(0.290256\pi\)
−0.612271 + 0.790648i \(0.709744\pi\)
\(398\) 787.822 0.0992210
\(399\) 0 0
\(400\) 1053.89 0.131736
\(401\) 8017.72i 0.998469i 0.866467 + 0.499234i \(0.166385\pi\)
−0.866467 + 0.499234i \(0.833615\pi\)
\(402\) 0 0
\(403\) 7560.15 0.934487
\(404\) −8155.01 −1.00427
\(405\) 0 0
\(406\) 0 0
\(407\) 3768.05i 0.458907i
\(408\) 0 0
\(409\) 8736.51i 1.05622i 0.849177 + 0.528109i \(0.177099\pi\)
−0.849177 + 0.528109i \(0.822901\pi\)
\(410\) 2195.40i 0.264446i
\(411\) 0 0
\(412\) 17054.7i 2.03938i
\(413\) 0 0
\(414\) 0 0
\(415\) −1207.87 −0.142872
\(416\) 17125.6 2.01840
\(417\) 0 0
\(418\) − 9159.29i − 1.07176i
\(419\) −3926.67 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(420\) 0 0
\(421\) 1443.44 0.167100 0.0835499 0.996504i \(-0.473374\pi\)
0.0835499 + 0.996504i \(0.473374\pi\)
\(422\) − 13306.9i − 1.53500i
\(423\) 0 0
\(424\) −6738.48 −0.771815
\(425\) −9598.62 −1.09553
\(426\) 0 0
\(427\) 0 0
\(428\) − 5339.42i − 0.603016i
\(429\) 0 0
\(430\) 1304.51i 0.146300i
\(431\) 14803.8i 1.65447i 0.561857 + 0.827234i \(0.310087\pi\)
−0.561857 + 0.827234i \(0.689913\pi\)
\(432\) 0 0
\(433\) 15872.1i 1.76158i 0.473508 + 0.880790i \(0.342988\pi\)
−0.473508 + 0.880790i \(0.657012\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8325.34 0.914476
\(437\) 4427.97 0.484711
\(438\) 0 0
\(439\) − 3032.91i − 0.329733i −0.986316 0.164867i \(-0.947281\pi\)
0.986316 0.164867i \(-0.0527193\pi\)
\(440\) −1060.41 −0.114893
\(441\) 0 0
\(442\) −30160.9 −3.24572
\(443\) − 12848.2i − 1.37796i −0.724782 0.688978i \(-0.758060\pi\)
0.724782 0.688978i \(-0.241940\pi\)
\(444\) 0 0
\(445\) 37.5890 0.00400424
\(446\) −15262.4 −1.62040
\(447\) 0 0
\(448\) 0 0
\(449\) 107.668i 0.0113166i 0.999984 + 0.00565831i \(0.00180111\pi\)
−0.999984 + 0.00565831i \(0.998199\pi\)
\(450\) 0 0
\(451\) − 15954.0i − 1.66573i
\(452\) − 11392.8i − 1.18556i
\(453\) 0 0
\(454\) 13707.1i 1.41698i
\(455\) 0 0
\(456\) 0 0
\(457\) −9777.06 −1.00077 −0.500385 0.865803i \(-0.666808\pi\)
−0.500385 + 0.865803i \(0.666808\pi\)
\(458\) −5013.45 −0.511492
\(459\) 0 0
\(460\) − 1433.56i − 0.145304i
\(461\) 638.874 0.0645452 0.0322726 0.999479i \(-0.489726\pi\)
0.0322726 + 0.999479i \(0.489726\pi\)
\(462\) 0 0
\(463\) −5602.26 −0.562331 −0.281165 0.959659i \(-0.590721\pi\)
−0.281165 + 0.959659i \(0.590721\pi\)
\(464\) 1291.30i 0.129196i
\(465\) 0 0
\(466\) 14019.3 1.39363
\(467\) 5518.88 0.546859 0.273430 0.961892i \(-0.411842\pi\)
0.273430 + 0.961892i \(0.411842\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 796.268i − 0.0781470i
\(471\) 0 0
\(472\) 17725.7i 1.72858i
\(473\) − 9479.91i − 0.921536i
\(474\) 0 0
\(475\) − 6007.82i − 0.580332i
\(476\) 0 0
\(477\) 0 0
\(478\) 7847.74 0.750935
\(479\) 5468.79 0.521661 0.260830 0.965385i \(-0.416004\pi\)
0.260830 + 0.965385i \(0.416004\pi\)
\(480\) 0 0
\(481\) 7766.16i 0.736188i
\(482\) −5427.22 −0.512869
\(483\) 0 0
\(484\) 4973.62 0.467094
\(485\) − 1518.38i − 0.142157i
\(486\) 0 0
\(487\) −11733.4 −1.09177 −0.545886 0.837860i \(-0.683807\pi\)
−0.545886 + 0.837860i \(0.683807\pi\)
\(488\) 264.990 0.0245811
\(489\) 0 0
\(490\) 0 0
\(491\) 3514.92i 0.323068i 0.986867 + 0.161534i \(0.0516441\pi\)
−0.986867 + 0.161534i \(0.948356\pi\)
\(492\) 0 0
\(493\) − 11760.9i − 1.07441i
\(494\) − 18877.8i − 1.71934i
\(495\) 0 0
\(496\) 753.106i 0.0681763i
\(497\) 0 0
\(498\) 0 0
\(499\) 9888.98 0.887158 0.443579 0.896235i \(-0.353709\pi\)
0.443579 + 0.896235i \(0.353709\pi\)
\(500\) −3915.32 −0.350196
\(501\) 0 0
\(502\) 16790.2i 1.49279i
\(503\) −10172.2 −0.901698 −0.450849 0.892600i \(-0.648879\pi\)
−0.450849 + 0.892600i \(0.648879\pi\)
\(504\) 0 0
\(505\) −828.838 −0.0730353
\(506\) 17110.0i 1.50322i
\(507\) 0 0
\(508\) −15703.1 −1.37148
\(509\) 4299.12 0.374372 0.187186 0.982325i \(-0.440063\pi\)
0.187186 + 0.982325i \(0.440063\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3082.06i 0.266033i
\(513\) 0 0
\(514\) − 3523.73i − 0.302383i
\(515\) 1733.36i 0.148313i
\(516\) 0 0
\(517\) 5786.50i 0.492243i
\(518\) 0 0
\(519\) 0 0
\(520\) −2185.57 −0.184314
\(521\) 10993.0 0.924396 0.462198 0.886777i \(-0.347061\pi\)
0.462198 + 0.886777i \(0.347061\pi\)
\(522\) 0 0
\(523\) 8529.54i 0.713137i 0.934269 + 0.356569i \(0.116053\pi\)
−0.934269 + 0.356569i \(0.883947\pi\)
\(524\) −17026.2 −1.41946
\(525\) 0 0
\(526\) −8888.51 −0.736801
\(527\) − 6859.15i − 0.566963i
\(528\) 0 0
\(529\) 3895.34 0.320156
\(530\) −1915.16 −0.156961
\(531\) 0 0
\(532\) 0 0
\(533\) − 32882.2i − 2.67220i
\(534\) 0 0
\(535\) − 542.674i − 0.0438539i
\(536\) 8910.67i 0.718064i
\(537\) 0 0
\(538\) − 11183.4i − 0.896195i
\(539\) 0 0
\(540\) 0 0
\(541\) −8705.86 −0.691856 −0.345928 0.938261i \(-0.612436\pi\)
−0.345928 + 0.938261i \(0.612436\pi\)
\(542\) −21389.0 −1.69508
\(543\) 0 0
\(544\) − 15537.7i − 1.22458i
\(545\) 846.150 0.0665047
\(546\) 0 0
\(547\) 17183.8 1.34319 0.671596 0.740917i \(-0.265609\pi\)
0.671596 + 0.740917i \(0.265609\pi\)
\(548\) 13713.1i 1.06897i
\(549\) 0 0
\(550\) 23214.6 1.79977
\(551\) 7361.20 0.569143
\(552\) 0 0
\(553\) 0 0
\(554\) 5305.66i 0.406888i
\(555\) 0 0
\(556\) − 28724.4i − 2.19098i
\(557\) 10380.5i 0.789652i 0.918756 + 0.394826i \(0.129195\pi\)
−0.918756 + 0.394826i \(0.870805\pi\)
\(558\) 0 0
\(559\) − 19538.6i − 1.47835i
\(560\) 0 0
\(561\) 0 0
\(562\) −37063.0 −2.78187
\(563\) −18496.4 −1.38460 −0.692302 0.721608i \(-0.743403\pi\)
−0.692302 + 0.721608i \(0.743403\pi\)
\(564\) 0 0
\(565\) − 1157.91i − 0.0862189i
\(566\) 11710.4 0.869656
\(567\) 0 0
\(568\) 6879.04 0.508166
\(569\) − 4033.88i − 0.297204i −0.988897 0.148602i \(-0.952523\pi\)
0.988897 0.148602i \(-0.0474774\pi\)
\(570\) 0 0
\(571\) 12860.0 0.942513 0.471257 0.881996i \(-0.343801\pi\)
0.471257 + 0.881996i \(0.343801\pi\)
\(572\) 44413.8 3.24657
\(573\) 0 0
\(574\) 0 0
\(575\) 11222.9i 0.813959i
\(576\) 0 0
\(577\) 20402.7i 1.47205i 0.676953 + 0.736026i \(0.263300\pi\)
−0.676953 + 0.736026i \(0.736700\pi\)
\(578\) 5145.02i 0.370250i
\(579\) 0 0
\(580\) − 2383.19i − 0.170615i
\(581\) 0 0
\(582\) 0 0
\(583\) 13917.5 0.988687
\(584\) 18563.6 1.31535
\(585\) 0 0
\(586\) 40120.2i 2.82824i
\(587\) −16279.7 −1.14470 −0.572348 0.820011i \(-0.693967\pi\)
−0.572348 + 0.820011i \(0.693967\pi\)
\(588\) 0 0
\(589\) 4293.17 0.300335
\(590\) 5037.86i 0.351535i
\(591\) 0 0
\(592\) −773.627 −0.0537093
\(593\) −2684.31 −0.185888 −0.0929440 0.995671i \(-0.529628\pi\)
−0.0929440 + 0.995671i \(0.529628\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 21865.7i 1.50277i
\(597\) 0 0
\(598\) 35264.7i 2.41150i
\(599\) − 14115.8i − 0.962864i −0.876483 0.481432i \(-0.840117\pi\)
0.876483 0.481432i \(-0.159883\pi\)
\(600\) 0 0
\(601\) − 11096.1i − 0.753109i −0.926394 0.376555i \(-0.877109\pi\)
0.926394 0.376555i \(-0.122891\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 6537.79 0.440428
\(605\) 505.495 0.0339691
\(606\) 0 0
\(607\) − 11076.7i − 0.740674i −0.928897 0.370337i \(-0.879242\pi\)
0.928897 0.370337i \(-0.120758\pi\)
\(608\) 9725.10 0.648692
\(609\) 0 0
\(610\) 75.3136 0.00499895
\(611\) 11926.3i 0.789667i
\(612\) 0 0
\(613\) −7602.68 −0.500929 −0.250464 0.968126i \(-0.580583\pi\)
−0.250464 + 0.968126i \(0.580583\pi\)
\(614\) 12244.9 0.804827
\(615\) 0 0
\(616\) 0 0
\(617\) 11325.9i 0.738998i 0.929231 + 0.369499i \(0.120471\pi\)
−0.929231 + 0.369499i \(0.879529\pi\)
\(618\) 0 0
\(619\) 19162.4i 1.24427i 0.782910 + 0.622136i \(0.213735\pi\)
−0.782910 + 0.622136i \(0.786265\pi\)
\(620\) − 1389.92i − 0.0900328i
\(621\) 0 0
\(622\) − 9773.64i − 0.630044i
\(623\) 0 0
\(624\) 0 0
\(625\) 15026.8 0.961716
\(626\) 38123.2 2.43404
\(627\) 0 0
\(628\) − 16413.8i − 1.04296i
\(629\) 7046.06 0.446653
\(630\) 0 0
\(631\) 10140.7 0.639768 0.319884 0.947457i \(-0.396356\pi\)
0.319884 + 0.947457i \(0.396356\pi\)
\(632\) 8320.63i 0.523698i
\(633\) 0 0
\(634\) −43396.7 −2.71846
\(635\) −1595.99 −0.0997400
\(636\) 0 0
\(637\) 0 0
\(638\) 28444.2i 1.76507i
\(639\) 0 0
\(640\) − 2757.40i − 0.170306i
\(641\) 3407.13i 0.209943i 0.994475 + 0.104972i \(0.0334751\pi\)
−0.994475 + 0.104972i \(0.966525\pi\)
\(642\) 0 0
\(643\) − 659.110i − 0.0404242i −0.999796 0.0202121i \(-0.993566\pi\)
0.999796 0.0202121i \(-0.00643415\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −17127.4 −1.04314
\(647\) −6606.94 −0.401462 −0.200731 0.979646i \(-0.564332\pi\)
−0.200731 + 0.979646i \(0.564332\pi\)
\(648\) 0 0
\(649\) − 36610.2i − 2.21429i
\(650\) 47846.6 2.88723
\(651\) 0 0
\(652\) 5577.01 0.334989
\(653\) 25885.1i 1.55124i 0.631198 + 0.775622i \(0.282564\pi\)
−0.631198 + 0.775622i \(0.717436\pi\)
\(654\) 0 0
\(655\) −1730.47 −0.103229
\(656\) 3275.56 0.194953
\(657\) 0 0
\(658\) 0 0
\(659\) − 7468.86i − 0.441495i −0.975331 0.220748i \(-0.929150\pi\)
0.975331 0.220748i \(-0.0708497\pi\)
\(660\) 0 0
\(661\) 6353.11i 0.373839i 0.982375 + 0.186919i \(0.0598503\pi\)
−0.982375 + 0.186919i \(0.940150\pi\)
\(662\) 38640.2i 2.26857i
\(663\) 0 0
\(664\) − 19220.3i − 1.12333i
\(665\) 0 0
\(666\) 0 0
\(667\) −13751.1 −0.798266
\(668\) 10101.2 0.585072
\(669\) 0 0
\(670\) 2532.52i 0.146030i
\(671\) −547.306 −0.0314881
\(672\) 0 0
\(673\) −20238.2 −1.15918 −0.579589 0.814909i \(-0.696787\pi\)
−0.579589 + 0.814909i \(0.696787\pi\)
\(674\) 2708.44i 0.154785i
\(675\) 0 0
\(676\) 64179.5 3.65154
\(677\) −11316.7 −0.642449 −0.321224 0.947003i \(-0.604094\pi\)
−0.321224 + 0.947003i \(0.604094\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1982.91i 0.111825i
\(681\) 0 0
\(682\) 16589.1i 0.931422i
\(683\) − 17169.7i − 0.961906i −0.876746 0.480953i \(-0.840291\pi\)
0.876746 0.480953i \(-0.159709\pi\)
\(684\) 0 0
\(685\) 1393.74i 0.0777402i
\(686\) 0 0
\(687\) 0 0
\(688\) 1946.34 0.107854
\(689\) 28684.8 1.58607
\(690\) 0 0
\(691\) − 8358.40i − 0.460157i −0.973172 0.230079i \(-0.926102\pi\)
0.973172 0.230079i \(-0.0738983\pi\)
\(692\) −27926.3 −1.53410
\(693\) 0 0
\(694\) 32114.4 1.75655
\(695\) − 2919.41i − 0.159338i
\(696\) 0 0
\(697\) −29833.2 −1.62125
\(698\) 16374.8 0.887960
\(699\) 0 0
\(700\) 0 0
\(701\) 19235.8i 1.03641i 0.855256 + 0.518206i \(0.173400\pi\)
−0.855256 + 0.518206i \(0.826600\pi\)
\(702\) 0 0
\(703\) 4410.16i 0.236604i
\(704\) 34736.3i 1.85962i
\(705\) 0 0
\(706\) 10536.5i 0.561680i
\(707\) 0 0
\(708\) 0 0
\(709\) −10320.3 −0.546669 −0.273334 0.961919i \(-0.588127\pi\)
−0.273334 + 0.961919i \(0.588127\pi\)
\(710\) 1955.11 0.103344
\(711\) 0 0
\(712\) 598.135i 0.0314833i
\(713\) −8019.85 −0.421242
\(714\) 0 0
\(715\) 4514.02 0.236105
\(716\) 36406.6i 1.90025i
\(717\) 0 0
\(718\) −7950.40 −0.413240
\(719\) 23359.5 1.21163 0.605815 0.795605i \(-0.292847\pi\)
0.605815 + 0.795605i \(0.292847\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 20300.0i 1.04638i
\(723\) 0 0
\(724\) 3522.55i 0.180821i
\(725\) 18657.3i 0.955743i
\(726\) 0 0
\(727\) 22260.4i 1.13561i 0.823162 + 0.567807i \(0.192208\pi\)
−0.823162 + 0.567807i \(0.807792\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 5276.00 0.267498
\(731\) −17727.0 −0.896929
\(732\) 0 0
\(733\) 10447.5i 0.526451i 0.964734 + 0.263226i \(0.0847864\pi\)
−0.964734 + 0.263226i \(0.915214\pi\)
\(734\) 7618.58 0.383116
\(735\) 0 0
\(736\) −18166.9 −0.909840
\(737\) − 18403.9i − 0.919832i
\(738\) 0 0
\(739\) 13190.9 0.656609 0.328304 0.944572i \(-0.393523\pi\)
0.328304 + 0.944572i \(0.393523\pi\)
\(740\) 1427.79 0.0709278
\(741\) 0 0
\(742\) 0 0
\(743\) − 35382.2i − 1.74703i −0.486795 0.873517i \(-0.661834\pi\)
0.486795 0.873517i \(-0.338166\pi\)
\(744\) 0 0
\(745\) 2222.32i 0.109288i
\(746\) − 1344.52i − 0.0659869i
\(747\) 0 0
\(748\) − 40295.6i − 1.96973i
\(749\) 0 0
\(750\) 0 0
\(751\) −29384.6 −1.42778 −0.713888 0.700259i \(-0.753068\pi\)
−0.713888 + 0.700259i \(0.753068\pi\)
\(752\) −1188.04 −0.0576109
\(753\) 0 0
\(754\) 58625.1i 2.83156i
\(755\) 664.470 0.0320299
\(756\) 0 0
\(757\) 11329.1 0.543939 0.271969 0.962306i \(-0.412325\pi\)
0.271969 + 0.962306i \(0.412325\pi\)
\(758\) − 33479.3i − 1.60425i
\(759\) 0 0
\(760\) −1241.11 −0.0592367
\(761\) −25393.0 −1.20958 −0.604792 0.796383i \(-0.706744\pi\)
−0.604792 + 0.796383i \(0.706744\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 57499.9i − 2.72287i
\(765\) 0 0
\(766\) − 56656.3i − 2.67242i
\(767\) − 75455.8i − 3.55222i
\(768\) 0 0
\(769\) 18120.8i 0.849744i 0.905253 + 0.424872i \(0.139681\pi\)
−0.905253 + 0.424872i \(0.860319\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 51749.4 2.41256
\(773\) −17346.1 −0.807110 −0.403555 0.914955i \(-0.632226\pi\)
−0.403555 + 0.914955i \(0.632226\pi\)
\(774\) 0 0
\(775\) 10881.2i 0.504342i
\(776\) 24161.3 1.11771
\(777\) 0 0
\(778\) −36054.7 −1.66147
\(779\) − 18672.7i − 0.858819i
\(780\) 0 0
\(781\) −14207.8 −0.650955
\(782\) 31994.8 1.46308
\(783\) 0 0
\(784\) 0 0
\(785\) − 1668.22i − 0.0758488i
\(786\) 0 0
\(787\) − 2080.22i − 0.0942209i −0.998890 0.0471104i \(-0.984999\pi\)
0.998890 0.0471104i \(-0.0150013\pi\)
\(788\) − 20253.7i − 0.915619i
\(789\) 0 0
\(790\) 2364.83i 0.106502i
\(791\) 0 0
\(792\) 0 0
\(793\) −1128.03 −0.0505138
\(794\) −56569.4 −2.52843
\(795\) 0 0
\(796\) 2169.37i 0.0965969i
\(797\) −39737.3 −1.76608 −0.883041 0.469297i \(-0.844507\pi\)
−0.883041 + 0.469297i \(0.844507\pi\)
\(798\) 0 0
\(799\) 10820.5 0.479099
\(800\) 24648.7i 1.08933i
\(801\) 0 0
\(802\) −36260.5 −1.59651
\(803\) −38340.8 −1.68495
\(804\) 0 0
\(805\) 0 0
\(806\) 34191.1i 1.49421i
\(807\) 0 0
\(808\) − 13188.9i − 0.574238i
\(809\) 11216.0i 0.487435i 0.969846 + 0.243717i \(0.0783669\pi\)
−0.969846 + 0.243717i \(0.921633\pi\)
\(810\) 0 0
\(811\) − 14792.0i − 0.640465i −0.947339 0.320232i \(-0.896239\pi\)
0.947339 0.320232i \(-0.103761\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −17041.1 −0.733774
\(815\) 566.822 0.0243618
\(816\) 0 0
\(817\) − 11095.4i − 0.475126i
\(818\) −39511.2 −1.68885
\(819\) 0 0
\(820\) −6045.30 −0.257453
\(821\) − 33062.1i − 1.40545i −0.711460 0.702726i \(-0.751966\pi\)
0.711460 0.702726i \(-0.248034\pi\)
\(822\) 0 0
\(823\) 20090.7 0.850933 0.425467 0.904974i \(-0.360110\pi\)
0.425467 + 0.904974i \(0.360110\pi\)
\(824\) −27582.2 −1.16611
\(825\) 0 0
\(826\) 0 0
\(827\) − 36882.5i − 1.55082i −0.631458 0.775410i \(-0.717543\pi\)
0.631458 0.775410i \(-0.282457\pi\)
\(828\) 0 0
\(829\) 37755.2i 1.58178i 0.611961 + 0.790888i \(0.290381\pi\)
−0.611961 + 0.790888i \(0.709619\pi\)
\(830\) − 5462.64i − 0.228447i
\(831\) 0 0
\(832\) 71593.5i 2.98324i
\(833\) 0 0
\(834\) 0 0
\(835\) 1026.64 0.0425490
\(836\) 25221.2 1.04341
\(837\) 0 0
\(838\) − 17758.5i − 0.732049i
\(839\) −44250.4 −1.82085 −0.910426 0.413672i \(-0.864246\pi\)
−0.910426 + 0.413672i \(0.864246\pi\)
\(840\) 0 0
\(841\) 1528.81 0.0626844
\(842\) 6528.02i 0.267186i
\(843\) 0 0
\(844\) 36642.2 1.49440
\(845\) 6522.91 0.265556
\(846\) 0 0
\(847\) 0 0
\(848\) 2857.44i 0.115713i
\(849\) 0 0
\(850\) − 43410.1i − 1.75171i
\(851\) − 8238.39i − 0.331855i
\(852\) 0 0
\(853\) 14952.2i 0.600179i 0.953911 + 0.300089i \(0.0970165\pi\)
−0.953911 + 0.300089i \(0.902983\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8635.32 0.344801
\(857\) 2449.53 0.0976363 0.0488181 0.998808i \(-0.484455\pi\)
0.0488181 + 0.998808i \(0.484455\pi\)
\(858\) 0 0
\(859\) − 6641.80i − 0.263813i −0.991262 0.131906i \(-0.957890\pi\)
0.991262 0.131906i \(-0.0421098\pi\)
\(860\) −3592.13 −0.142431
\(861\) 0 0
\(862\) −66951.0 −2.64543
\(863\) 18866.9i 0.744191i 0.928194 + 0.372096i \(0.121361\pi\)
−0.928194 + 0.372096i \(0.878639\pi\)
\(864\) 0 0
\(865\) −2838.30 −0.111567
\(866\) −71782.2 −2.81669
\(867\) 0 0
\(868\) 0 0
\(869\) − 17185.3i − 0.670851i
\(870\) 0 0
\(871\) − 37931.5i − 1.47561i
\(872\) 13464.4i 0.522892i
\(873\) 0 0
\(874\) 20025.7i 0.775033i
\(875\) 0 0
\(876\) 0 0
\(877\) 11979.7 0.461260 0.230630 0.973042i \(-0.425921\pi\)
0.230630 + 0.973042i \(0.425921\pi\)
\(878\) 13716.5 0.527230
\(879\) 0 0
\(880\) 449.664i 0.0172252i
\(881\) 34504.0 1.31949 0.659743 0.751491i \(-0.270665\pi\)
0.659743 + 0.751491i \(0.270665\pi\)
\(882\) 0 0
\(883\) −8148.85 −0.310567 −0.155283 0.987870i \(-0.549629\pi\)
−0.155283 + 0.987870i \(0.549629\pi\)
\(884\) − 83051.7i − 3.15988i
\(885\) 0 0
\(886\) 58106.3 2.20330
\(887\) 34955.2 1.32320 0.661602 0.749856i \(-0.269877\pi\)
0.661602 + 0.749856i \(0.269877\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 169.998i 0.00640262i
\(891\) 0 0
\(892\) − 42027.0i − 1.57754i
\(893\) 6772.57i 0.253791i
\(894\) 0 0
\(895\) 3700.20i 0.138194i
\(896\) 0 0
\(897\) 0 0
\(898\) −486.932 −0.0180948
\(899\) −13332.4 −0.494618
\(900\) 0 0
\(901\) − 26025.1i − 0.962287i
\(902\) 72152.7 2.66344
\(903\) 0 0
\(904\) 18425.3 0.677894
\(905\) 358.016i 0.0131501i
\(906\) 0 0
\(907\) −24989.0 −0.914824 −0.457412 0.889255i \(-0.651224\pi\)
−0.457412 + 0.889255i \(0.651224\pi\)
\(908\) −37744.3 −1.37950
\(909\) 0 0
\(910\) 0 0
\(911\) − 41609.1i − 1.51325i −0.653849 0.756625i \(-0.726847\pi\)
0.653849 0.756625i \(-0.273153\pi\)
\(912\) 0 0
\(913\) 39697.1i 1.43897i
\(914\) − 44217.1i − 1.60019i
\(915\) 0 0
\(916\) − 13805.1i − 0.497964i
\(917\) 0 0
\(918\) 0 0
\(919\) 50992.0 1.83033 0.915164 0.403082i \(-0.132061\pi\)
0.915164 + 0.403082i \(0.132061\pi\)
\(920\) 2318.46 0.0830840
\(921\) 0 0
\(922\) 2889.33i 0.103205i
\(923\) −29283.1 −1.04428
\(924\) 0 0
\(925\) −11177.7 −0.397321
\(926\) − 25336.5i − 0.899144i
\(927\) 0 0
\(928\) −30201.3 −1.06832
\(929\) −47493.0 −1.67728 −0.838642 0.544683i \(-0.816650\pi\)
−0.838642 + 0.544683i \(0.816650\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 38603.9i 1.35677i
\(933\) 0 0
\(934\) 24959.4i 0.874406i
\(935\) − 4095.46i − 0.143247i
\(936\) 0 0
\(937\) − 6811.98i − 0.237500i −0.992924 0.118750i \(-0.962111\pi\)
0.992924 0.118750i \(-0.0378887\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2192.62 0.0760802
\(941\) 32542.0 1.12735 0.563677 0.825995i \(-0.309386\pi\)
0.563677 + 0.825995i \(0.309386\pi\)
\(942\) 0 0
\(943\) 34881.5i 1.20456i
\(944\) 7516.54 0.259155
\(945\) 0 0
\(946\) 42873.3 1.47350
\(947\) 11685.4i 0.400976i 0.979696 + 0.200488i \(0.0642528\pi\)
−0.979696 + 0.200488i \(0.935747\pi\)
\(948\) 0 0
\(949\) −79022.7 −2.70304
\(950\) 27170.6 0.927927
\(951\) 0 0
\(952\) 0 0
\(953\) 46457.5i 1.57912i 0.613671 + 0.789562i \(0.289692\pi\)
−0.613671 + 0.789562i \(0.710308\pi\)
\(954\) 0 0
\(955\) − 5844.03i − 0.198019i
\(956\) 21609.7i 0.731075i
\(957\) 0 0
\(958\) 24732.8i 0.834114i
\(959\) 0 0
\(960\) 0 0
\(961\) 22015.3 0.738992
\(962\) −35122.8 −1.17714
\(963\) 0 0
\(964\) − 14944.5i − 0.499305i
\(965\) 5259.57 0.175452
\(966\) 0 0
\(967\) −27949.1 −0.929455 −0.464728 0.885454i \(-0.653848\pi\)
−0.464728 + 0.885454i \(0.653848\pi\)
\(968\) 8043.71i 0.267081i
\(969\) 0 0
\(970\) 6866.94 0.227303
\(971\) −3219.69 −0.106411 −0.0532053 0.998584i \(-0.516944\pi\)
−0.0532053 + 0.998584i \(0.516944\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 53064.9i − 1.74570i
\(975\) 0 0
\(976\) − 112.369i − 0.00368528i
\(977\) 38403.8i 1.25757i 0.777579 + 0.628785i \(0.216447\pi\)
−0.777579 + 0.628785i \(0.783553\pi\)
\(978\) 0 0
\(979\) − 1235.38i − 0.0403297i
\(980\) 0 0
\(981\) 0 0
\(982\) −15896.4 −0.516572
\(983\) −27418.2 −0.889627 −0.444814 0.895623i \(-0.646730\pi\)
−0.444814 + 0.895623i \(0.646730\pi\)
\(984\) 0 0
\(985\) − 2058.49i − 0.0665878i
\(986\) 53189.1 1.71794
\(987\) 0 0
\(988\) 51982.4 1.67387
\(989\) 20726.7i 0.666401i
\(990\) 0 0
\(991\) 43715.9 1.40129 0.700646 0.713509i \(-0.252895\pi\)
0.700646 + 0.713509i \(0.252895\pi\)
\(992\) −17613.9 −0.563751
\(993\) 0 0
\(994\) 0 0
\(995\) 220.484i 0.00702495i
\(996\) 0 0
\(997\) 25323.5i 0.804415i 0.915548 + 0.402208i \(0.131757\pi\)
−0.915548 + 0.402208i \(0.868243\pi\)
\(998\) 44723.3i 1.41853i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.4.c.a.440.13 16
3.2 odd 2 inner 441.4.c.a.440.4 16
7.2 even 3 63.4.p.a.17.7 yes 16
7.3 odd 6 63.4.p.a.26.2 yes 16
7.4 even 3 441.4.p.c.215.2 16
7.5 odd 6 441.4.p.c.80.7 16
7.6 odd 2 inner 441.4.c.a.440.14 16
21.2 odd 6 63.4.p.a.17.2 16
21.5 even 6 441.4.p.c.80.2 16
21.11 odd 6 441.4.p.c.215.7 16
21.17 even 6 63.4.p.a.26.7 yes 16
21.20 even 2 inner 441.4.c.a.440.3 16
28.3 even 6 1008.4.bt.a.593.4 16
28.23 odd 6 1008.4.bt.a.17.5 16
84.23 even 6 1008.4.bt.a.17.4 16
84.59 odd 6 1008.4.bt.a.593.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.p.a.17.2 16 21.2 odd 6
63.4.p.a.17.7 yes 16 7.2 even 3
63.4.p.a.26.2 yes 16 7.3 odd 6
63.4.p.a.26.7 yes 16 21.17 even 6
441.4.c.a.440.3 16 21.20 even 2 inner
441.4.c.a.440.4 16 3.2 odd 2 inner
441.4.c.a.440.13 16 1.1 even 1 trivial
441.4.c.a.440.14 16 7.6 odd 2 inner
441.4.p.c.80.2 16 21.5 even 6
441.4.p.c.80.7 16 7.5 odd 6
441.4.p.c.215.2 16 7.4 even 3
441.4.p.c.215.7 16 21.11 odd 6
1008.4.bt.a.17.4 16 84.23 even 6
1008.4.bt.a.17.5 16 28.23 odd 6
1008.4.bt.a.593.4 16 28.3 even 6
1008.4.bt.a.593.5 16 84.59 odd 6