Properties

Label 882.5.c.b.685.2
Level $882$
Weight $5$
Character 882.685
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.5.c.b.685.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843 q^{2} +8.00000 q^{4} +25.3864i q^{5} -22.6274 q^{8} +O(q^{10})\) \(q-2.82843 q^{2} +8.00000 q^{4} +25.3864i q^{5} -22.6274 q^{8} -71.8036i q^{10} -43.9706 q^{11} +162.507i q^{13} +64.0000 q^{16} -108.236i q^{17} -449.568i q^{19} +203.091i q^{20} +124.368 q^{22} +435.823 q^{23} -19.4701 q^{25} -459.639i q^{26} -742.118 q^{29} -1039.43i q^{31} -181.019 q^{32} +306.139i q^{34} +986.675 q^{37} +1271.57i q^{38} -574.429i q^{40} +1143.70i q^{41} +2418.82 q^{43} -351.765 q^{44} -1232.69 q^{46} -1345.43i q^{47} +55.0698 q^{50} +1300.05i q^{52} +3566.20 q^{53} -1116.26i q^{55} +2099.03 q^{58} -4285.01i q^{59} +1396.69i q^{61} +2939.96i q^{62} +512.000 q^{64} -4125.47 q^{65} +7265.79 q^{67} -865.891i q^{68} -5987.76 q^{71} -577.493i q^{73} -2790.74 q^{74} -3596.55i q^{76} -3146.23 q^{79} +1624.73i q^{80} -3234.86i q^{82} +4729.96i q^{83} +2747.74 q^{85} -6841.46 q^{86} +994.940 q^{88} -837.909i q^{89} +3486.58 q^{92} +3805.44i q^{94} +11412.9 q^{95} +5622.23i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 108q^{11} + 256q^{16} + 192q^{22} - 972q^{23} + 1144q^{25} - 3240q^{29} + 892q^{37} + 2344q^{43} - 864q^{44} - 7680q^{46} + 3456q^{50} + 5508q^{53} - 768q^{58} + 2048q^{64} - 6048q^{65} + 10124q^{67} - 18792q^{71} - 8640q^{74} - 1588q^{79} + 10380q^{85} - 20736q^{86} + 1536q^{88} - 7776q^{92} + 17820q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 −0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 25.3864i 1.01546i 0.861517 + 0.507728i \(0.169515\pi\)
−0.861517 + 0.507728i \(0.830485\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −22.6274 −0.353553
\(9\) 0 0
\(10\) − 71.8036i − 0.718036i
\(11\) −43.9706 −0.363393 −0.181697 0.983355i \(-0.558159\pi\)
−0.181697 + 0.983355i \(0.558159\pi\)
\(12\) 0 0
\(13\) 162.507i 0.961579i 0.876836 + 0.480790i \(0.159650\pi\)
−0.876836 + 0.480790i \(0.840350\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 108.236i − 0.374521i −0.982310 0.187260i \(-0.940039\pi\)
0.982310 0.187260i \(-0.0599608\pi\)
\(18\) 0 0
\(19\) − 449.568i − 1.24534i −0.782484 0.622671i \(-0.786048\pi\)
0.782484 0.622671i \(-0.213952\pi\)
\(20\) 203.091i 0.507728i
\(21\) 0 0
\(22\) 124.368 0.256958
\(23\) 435.823 0.823861 0.411931 0.911215i \(-0.364855\pi\)
0.411931 + 0.911215i \(0.364855\pi\)
\(24\) 0 0
\(25\) −19.4701 −0.0311522
\(26\) − 459.639i − 0.679939i
\(27\) 0 0
\(28\) 0 0
\(29\) −742.118 −0.882423 −0.441212 0.897403i \(-0.645451\pi\)
−0.441212 + 0.897403i \(0.645451\pi\)
\(30\) 0 0
\(31\) − 1039.43i − 1.08162i −0.841146 0.540808i \(-0.818119\pi\)
0.841146 0.540808i \(-0.181881\pi\)
\(32\) −181.019 −0.176777
\(33\) 0 0
\(34\) 306.139i 0.264826i
\(35\) 0 0
\(36\) 0 0
\(37\) 986.675 0.720727 0.360364 0.932812i \(-0.382653\pi\)
0.360364 + 0.932812i \(0.382653\pi\)
\(38\) 1271.57i 0.880590i
\(39\) 0 0
\(40\) − 574.429i − 0.359018i
\(41\) 1143.70i 0.680366i 0.940359 + 0.340183i \(0.110489\pi\)
−0.940359 + 0.340183i \(0.889511\pi\)
\(42\) 0 0
\(43\) 2418.82 1.30818 0.654089 0.756418i \(-0.273052\pi\)
0.654089 + 0.756418i \(0.273052\pi\)
\(44\) −351.765 −0.181697
\(45\) 0 0
\(46\) −1232.69 −0.582558
\(47\) − 1345.43i − 0.609066i −0.952502 0.304533i \(-0.901500\pi\)
0.952502 0.304533i \(-0.0985004\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 55.0698 0.0220279
\(51\) 0 0
\(52\) 1300.05i 0.480790i
\(53\) 3566.20 1.26956 0.634782 0.772692i \(-0.281090\pi\)
0.634782 + 0.772692i \(0.281090\pi\)
\(54\) 0 0
\(55\) − 1116.26i − 0.369010i
\(56\) 0 0
\(57\) 0 0
\(58\) 2099.03 0.623967
\(59\) − 4285.01i − 1.23097i −0.788148 0.615485i \(-0.788960\pi\)
0.788148 0.615485i \(-0.211040\pi\)
\(60\) 0 0
\(61\) 1396.69i 0.375353i 0.982231 + 0.187677i \(0.0600957\pi\)
−0.982231 + 0.187677i \(0.939904\pi\)
\(62\) 2939.96i 0.764818i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) −4125.47 −0.976442
\(66\) 0 0
\(67\) 7265.79 1.61858 0.809288 0.587412i \(-0.199853\pi\)
0.809288 + 0.587412i \(0.199853\pi\)
\(68\) − 865.891i − 0.187260i
\(69\) 0 0
\(70\) 0 0
\(71\) −5987.76 −1.18781 −0.593906 0.804534i \(-0.702415\pi\)
−0.593906 + 0.804534i \(0.702415\pi\)
\(72\) 0 0
\(73\) − 577.493i − 0.108368i −0.998531 0.0541840i \(-0.982744\pi\)
0.998531 0.0541840i \(-0.0172557\pi\)
\(74\) −2790.74 −0.509631
\(75\) 0 0
\(76\) − 3596.55i − 0.622671i
\(77\) 0 0
\(78\) 0 0
\(79\) −3146.23 −0.504123 −0.252061 0.967711i \(-0.581109\pi\)
−0.252061 + 0.967711i \(0.581109\pi\)
\(80\) 1624.73i 0.253864i
\(81\) 0 0
\(82\) − 3234.86i − 0.481091i
\(83\) 4729.96i 0.686596i 0.939227 + 0.343298i \(0.111544\pi\)
−0.939227 + 0.343298i \(0.888456\pi\)
\(84\) 0 0
\(85\) 2747.74 0.380309
\(86\) −6841.46 −0.925021
\(87\) 0 0
\(88\) 994.940 0.128479
\(89\) − 837.909i − 0.105783i −0.998600 0.0528916i \(-0.983156\pi\)
0.998600 0.0528916i \(-0.0168438\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3486.58 0.411931
\(93\) 0 0
\(94\) 3805.44i 0.430675i
\(95\) 11412.9 1.26459
\(96\) 0 0
\(97\) 5622.23i 0.597537i 0.954326 + 0.298769i \(0.0965759\pi\)
−0.954326 + 0.298769i \(0.903424\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −155.761 −0.0155761
\(101\) 6585.50i 0.645574i 0.946472 + 0.322787i \(0.104620\pi\)
−0.946472 + 0.322787i \(0.895380\pi\)
\(102\) 0 0
\(103\) − 1524.50i − 0.143698i −0.997416 0.0718492i \(-0.977110\pi\)
0.997416 0.0718492i \(-0.0228900\pi\)
\(104\) − 3677.11i − 0.339970i
\(105\) 0 0
\(106\) −10086.7 −0.897717
\(107\) −10606.6 −0.926424 −0.463212 0.886248i \(-0.653303\pi\)
−0.463212 + 0.886248i \(0.653303\pi\)
\(108\) 0 0
\(109\) −13952.3 −1.17433 −0.587167 0.809466i \(-0.699757\pi\)
−0.587167 + 0.809466i \(0.699757\pi\)
\(110\) 3157.25i 0.260929i
\(111\) 0 0
\(112\) 0 0
\(113\) 8811.30 0.690054 0.345027 0.938593i \(-0.387870\pi\)
0.345027 + 0.938593i \(0.387870\pi\)
\(114\) 0 0
\(115\) 11064.0i 0.836595i
\(116\) −5936.94 −0.441212
\(117\) 0 0
\(118\) 12119.8i 0.870427i
\(119\) 0 0
\(120\) 0 0
\(121\) −12707.6 −0.867945
\(122\) − 3950.44i − 0.265415i
\(123\) 0 0
\(124\) − 8315.46i − 0.540808i
\(125\) 15372.2i 0.983823i
\(126\) 0 0
\(127\) 3992.70 0.247548 0.123774 0.992310i \(-0.460500\pi\)
0.123774 + 0.992310i \(0.460500\pi\)
\(128\) −1448.15 −0.0883883
\(129\) 0 0
\(130\) 11668.6 0.690449
\(131\) − 20225.0i − 1.17854i −0.807935 0.589272i \(-0.799415\pi\)
0.807935 0.589272i \(-0.200585\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −20550.7 −1.14451
\(135\) 0 0
\(136\) 2449.11i 0.132413i
\(137\) −25057.7 −1.33506 −0.667530 0.744583i \(-0.732648\pi\)
−0.667530 + 0.744583i \(0.732648\pi\)
\(138\) 0 0
\(139\) 20070.1i 1.03877i 0.854541 + 0.519385i \(0.173839\pi\)
−0.854541 + 0.519385i \(0.826161\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16936.0 0.839910
\(143\) − 7145.52i − 0.349431i
\(144\) 0 0
\(145\) − 18839.7i − 0.896062i
\(146\) 1633.40i 0.0766277i
\(147\) 0 0
\(148\) 7893.40 0.360364
\(149\) 42707.9 1.92369 0.961846 0.273591i \(-0.0882115\pi\)
0.961846 + 0.273591i \(0.0882115\pi\)
\(150\) 0 0
\(151\) 29607.4 1.29852 0.649258 0.760569i \(-0.275080\pi\)
0.649258 + 0.760569i \(0.275080\pi\)
\(152\) 10172.6i 0.440295i
\(153\) 0 0
\(154\) 0 0
\(155\) 26387.5 1.09833
\(156\) 0 0
\(157\) 12237.1i 0.496452i 0.968702 + 0.248226i \(0.0798476\pi\)
−0.968702 + 0.248226i \(0.920152\pi\)
\(158\) 8898.89 0.356469
\(159\) 0 0
\(160\) − 4595.43i − 0.179509i
\(161\) 0 0
\(162\) 0 0
\(163\) 21531.1 0.810385 0.405193 0.914231i \(-0.367204\pi\)
0.405193 + 0.914231i \(0.367204\pi\)
\(164\) 9149.56i 0.340183i
\(165\) 0 0
\(166\) − 13378.3i − 0.485497i
\(167\) − 16578.9i − 0.594461i −0.954806 0.297230i \(-0.903937\pi\)
0.954806 0.297230i \(-0.0960629\pi\)
\(168\) 0 0
\(169\) 2152.52 0.0753658
\(170\) −7771.77 −0.268919
\(171\) 0 0
\(172\) 19350.6 0.654089
\(173\) − 40383.1i − 1.34930i −0.738140 0.674648i \(-0.764296\pi\)
0.738140 0.674648i \(-0.235704\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2814.12 −0.0908483
\(177\) 0 0
\(178\) 2369.97i 0.0748001i
\(179\) 53494.4 1.66956 0.834780 0.550583i \(-0.185595\pi\)
0.834780 + 0.550583i \(0.185595\pi\)
\(180\) 0 0
\(181\) 54202.1i 1.65447i 0.561857 + 0.827235i \(0.310087\pi\)
−0.561857 + 0.827235i \(0.689913\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −9861.54 −0.291279
\(185\) 25048.2i 0.731867i
\(186\) 0 0
\(187\) 4759.22i 0.136098i
\(188\) − 10763.4i − 0.304533i
\(189\) 0 0
\(190\) −32280.6 −0.894201
\(191\) 1203.87 0.0330000 0.0165000 0.999864i \(-0.494748\pi\)
0.0165000 + 0.999864i \(0.494748\pi\)
\(192\) 0 0
\(193\) 51312.9 1.37757 0.688783 0.724968i \(-0.258145\pi\)
0.688783 + 0.724968i \(0.258145\pi\)
\(194\) − 15902.1i − 0.422523i
\(195\) 0 0
\(196\) 0 0
\(197\) 1456.28 0.0375242 0.0187621 0.999824i \(-0.494027\pi\)
0.0187621 + 0.999824i \(0.494027\pi\)
\(198\) 0 0
\(199\) 68758.2i 1.73627i 0.496325 + 0.868137i \(0.334683\pi\)
−0.496325 + 0.868137i \(0.665317\pi\)
\(200\) 440.559 0.0110140
\(201\) 0 0
\(202\) − 18626.6i − 0.456490i
\(203\) 0 0
\(204\) 0 0
\(205\) −29034.3 −0.690882
\(206\) 4311.93i 0.101610i
\(207\) 0 0
\(208\) 10400.4i 0.240395i
\(209\) 19767.8i 0.452549i
\(210\) 0 0
\(211\) −622.821 −0.0139894 −0.00699469 0.999976i \(-0.502226\pi\)
−0.00699469 + 0.999976i \(0.502226\pi\)
\(212\) 28529.6 0.634782
\(213\) 0 0
\(214\) 30000.1 0.655080
\(215\) 61405.2i 1.32840i
\(216\) 0 0
\(217\) 0 0
\(218\) 39463.0 0.830380
\(219\) 0 0
\(220\) − 8930.04i − 0.184505i
\(221\) 17589.2 0.360131
\(222\) 0 0
\(223\) 11509.7i 0.231449i 0.993281 + 0.115725i \(0.0369190\pi\)
−0.993281 + 0.115725i \(0.963081\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −24922.1 −0.487942
\(227\) 56367.2i 1.09389i 0.837168 + 0.546946i \(0.184210\pi\)
−0.837168 + 0.546946i \(0.815790\pi\)
\(228\) 0 0
\(229\) 42128.5i 0.803351i 0.915782 + 0.401675i \(0.131572\pi\)
−0.915782 + 0.401675i \(0.868428\pi\)
\(230\) − 31293.6i − 0.591562i
\(231\) 0 0
\(232\) 16792.2 0.311984
\(233\) 55101.9 1.01497 0.507486 0.861660i \(-0.330575\pi\)
0.507486 + 0.861660i \(0.330575\pi\)
\(234\) 0 0
\(235\) 34155.6 0.618480
\(236\) − 34280.1i − 0.615485i
\(237\) 0 0
\(238\) 0 0
\(239\) −23082.5 −0.404098 −0.202049 0.979375i \(-0.564760\pi\)
−0.202049 + 0.979375i \(0.564760\pi\)
\(240\) 0 0
\(241\) 81572.5i 1.40446i 0.711950 + 0.702231i \(0.247812\pi\)
−0.711950 + 0.702231i \(0.752188\pi\)
\(242\) 35942.5 0.613730
\(243\) 0 0
\(244\) 11173.5i 0.187677i
\(245\) 0 0
\(246\) 0 0
\(247\) 73057.9 1.19749
\(248\) 23519.7i 0.382409i
\(249\) 0 0
\(250\) − 43479.2i − 0.695668i
\(251\) 27207.6i 0.431859i 0.976409 + 0.215930i \(0.0692782\pi\)
−0.976409 + 0.215930i \(0.930722\pi\)
\(252\) 0 0
\(253\) −19163.4 −0.299385
\(254\) −11293.1 −0.175043
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 107929.i 1.63408i 0.576584 + 0.817038i \(0.304385\pi\)
−0.576584 + 0.817038i \(0.695615\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −33003.7 −0.488221
\(261\) 0 0
\(262\) 57204.9i 0.833356i
\(263\) 40144.1 0.580377 0.290188 0.956970i \(-0.406282\pi\)
0.290188 + 0.956970i \(0.406282\pi\)
\(264\) 0 0
\(265\) 90533.1i 1.28919i
\(266\) 0 0
\(267\) 0 0
\(268\) 58126.3 0.809288
\(269\) − 105354.i − 1.45595i −0.685606 0.727973i \(-0.740462\pi\)
0.685606 0.727973i \(-0.259538\pi\)
\(270\) 0 0
\(271\) 15229.7i 0.207373i 0.994610 + 0.103687i \(0.0330639\pi\)
−0.994610 + 0.103687i \(0.966936\pi\)
\(272\) − 6927.13i − 0.0936301i
\(273\) 0 0
\(274\) 70874.0 0.944030
\(275\) 856.113 0.0113205
\(276\) 0 0
\(277\) −9871.62 −0.128656 −0.0643278 0.997929i \(-0.520490\pi\)
−0.0643278 + 0.997929i \(0.520490\pi\)
\(278\) − 56766.7i − 0.734521i
\(279\) 0 0
\(280\) 0 0
\(281\) 155117. 1.96448 0.982240 0.187631i \(-0.0600810\pi\)
0.982240 + 0.187631i \(0.0600810\pi\)
\(282\) 0 0
\(283\) − 109662.i − 1.36925i −0.728895 0.684626i \(-0.759966\pi\)
0.728895 0.684626i \(-0.240034\pi\)
\(284\) −47902.1 −0.593906
\(285\) 0 0
\(286\) 20210.6i 0.247085i
\(287\) 0 0
\(288\) 0 0
\(289\) 71805.9 0.859734
\(290\) 53286.7i 0.633612i
\(291\) 0 0
\(292\) − 4619.94i − 0.0541840i
\(293\) 89912.4i 1.04733i 0.851924 + 0.523666i \(0.175436\pi\)
−0.851924 + 0.523666i \(0.824564\pi\)
\(294\) 0 0
\(295\) 108781. 1.25000
\(296\) −22325.9 −0.254815
\(297\) 0 0
\(298\) −120796. −1.36026
\(299\) 70824.1i 0.792208i
\(300\) 0 0
\(301\) 0 0
\(302\) −83742.5 −0.918189
\(303\) 0 0
\(304\) − 28772.4i − 0.311335i
\(305\) −35456.9 −0.381155
\(306\) 0 0
\(307\) − 141681.i − 1.50327i −0.659582 0.751633i \(-0.729267\pi\)
0.659582 0.751633i \(-0.270733\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −74635.1 −0.776639
\(311\) 85645.6i 0.885491i 0.896647 + 0.442746i \(0.145995\pi\)
−0.896647 + 0.442746i \(0.854005\pi\)
\(312\) 0 0
\(313\) − 17831.6i − 0.182012i −0.995850 0.0910061i \(-0.970992\pi\)
0.995850 0.0910061i \(-0.0290083\pi\)
\(314\) − 34611.6i − 0.351045i
\(315\) 0 0
\(316\) −25169.8 −0.252061
\(317\) −80805.2 −0.804120 −0.402060 0.915613i \(-0.631706\pi\)
−0.402060 + 0.915613i \(0.631706\pi\)
\(318\) 0 0
\(319\) 32631.3 0.320666
\(320\) 12997.8i 0.126932i
\(321\) 0 0
\(322\) 0 0
\(323\) −48659.7 −0.466406
\(324\) 0 0
\(325\) − 3164.03i − 0.0299553i
\(326\) −60899.2 −0.573029
\(327\) 0 0
\(328\) − 25878.9i − 0.240546i
\(329\) 0 0
\(330\) 0 0
\(331\) 72804.3 0.664509 0.332254 0.943190i \(-0.392191\pi\)
0.332254 + 0.943190i \(0.392191\pi\)
\(332\) 37839.7i 0.343298i
\(333\) 0 0
\(334\) 46892.3i 0.420347i
\(335\) 184452.i 1.64359i
\(336\) 0 0
\(337\) 126538. 1.11419 0.557095 0.830449i \(-0.311916\pi\)
0.557095 + 0.830449i \(0.311916\pi\)
\(338\) −6088.25 −0.0532917
\(339\) 0 0
\(340\) 21981.9 0.190155
\(341\) 45704.5i 0.393052i
\(342\) 0 0
\(343\) 0 0
\(344\) −54731.7 −0.462511
\(345\) 0 0
\(346\) 114221.i 0.954096i
\(347\) 66949.3 0.556016 0.278008 0.960579i \(-0.410326\pi\)
0.278008 + 0.960579i \(0.410326\pi\)
\(348\) 0 0
\(349\) 25527.5i 0.209583i 0.994494 + 0.104792i \(0.0334176\pi\)
−0.994494 + 0.104792i \(0.966582\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7959.52 0.0642394
\(353\) 56575.6i 0.454025i 0.973892 + 0.227013i \(0.0728959\pi\)
−0.973892 + 0.227013i \(0.927104\pi\)
\(354\) 0 0
\(355\) − 152008.i − 1.20617i
\(356\) − 6703.27i − 0.0528916i
\(357\) 0 0
\(358\) −151305. −1.18056
\(359\) −37804.8 −0.293331 −0.146665 0.989186i \(-0.546854\pi\)
−0.146665 + 0.989186i \(0.546854\pi\)
\(360\) 0 0
\(361\) −71790.8 −0.550876
\(362\) − 153307.i − 1.16989i
\(363\) 0 0
\(364\) 0 0
\(365\) 14660.5 0.110043
\(366\) 0 0
\(367\) 10816.4i 0.0803065i 0.999194 + 0.0401532i \(0.0127846\pi\)
−0.999194 + 0.0401532i \(0.987215\pi\)
\(368\) 27892.6 0.205965
\(369\) 0 0
\(370\) − 70846.9i − 0.517508i
\(371\) 0 0
\(372\) 0 0
\(373\) −31023.4 −0.222983 −0.111491 0.993765i \(-0.535563\pi\)
−0.111491 + 0.993765i \(0.535563\pi\)
\(374\) − 13461.1i − 0.0962359i
\(375\) 0 0
\(376\) 30443.6i 0.215337i
\(377\) − 120599.i − 0.848519i
\(378\) 0 0
\(379\) −98527.5 −0.685929 −0.342964 0.939348i \(-0.611431\pi\)
−0.342964 + 0.939348i \(0.611431\pi\)
\(380\) 91303.4 0.632295
\(381\) 0 0
\(382\) −3405.07 −0.0233345
\(383\) − 5143.33i − 0.0350628i −0.999846 0.0175314i \(-0.994419\pi\)
0.999846 0.0175314i \(-0.00558071\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −145135. −0.974086
\(387\) 0 0
\(388\) 44977.8i 0.298769i
\(389\) −86282.0 −0.570192 −0.285096 0.958499i \(-0.592025\pi\)
−0.285096 + 0.958499i \(0.592025\pi\)
\(390\) 0 0
\(391\) − 47171.9i − 0.308553i
\(392\) 0 0
\(393\) 0 0
\(394\) −4118.97 −0.0265336
\(395\) − 79871.5i − 0.511915i
\(396\) 0 0
\(397\) 155926.i 0.989323i 0.869086 + 0.494661i \(0.164708\pi\)
−0.869086 + 0.494661i \(0.835292\pi\)
\(398\) − 194478.i − 1.22773i
\(399\) 0 0
\(400\) −1246.09 −0.00778805
\(401\) −34614.5 −0.215263 −0.107631 0.994191i \(-0.534327\pi\)
−0.107631 + 0.994191i \(0.534327\pi\)
\(402\) 0 0
\(403\) 168915. 1.04006
\(404\) 52684.0i 0.322787i
\(405\) 0 0
\(406\) 0 0
\(407\) −43384.7 −0.261907
\(408\) 0 0
\(409\) − 313788.i − 1.87581i −0.346886 0.937907i \(-0.612761\pi\)
0.346886 0.937907i \(-0.387239\pi\)
\(410\) 82121.5 0.488528
\(411\) 0 0
\(412\) − 12196.0i − 0.0718492i
\(413\) 0 0
\(414\) 0 0
\(415\) −120077. −0.697208
\(416\) − 29416.9i − 0.169985i
\(417\) 0 0
\(418\) − 55911.7i − 0.320000i
\(419\) 7324.74i 0.0417219i 0.999782 + 0.0208609i \(0.00664073\pi\)
−0.999782 + 0.0208609i \(0.993359\pi\)
\(420\) 0 0
\(421\) −25178.3 −0.142057 −0.0710285 0.997474i \(-0.522628\pi\)
−0.0710285 + 0.997474i \(0.522628\pi\)
\(422\) 1761.60 0.00989198
\(423\) 0 0
\(424\) −80694.0 −0.448858
\(425\) 2107.38i 0.0116671i
\(426\) 0 0
\(427\) 0 0
\(428\) −84853.0 −0.463212
\(429\) 0 0
\(430\) − 173680.i − 0.939319i
\(431\) −13827.2 −0.0744353 −0.0372176 0.999307i \(-0.511849\pi\)
−0.0372176 + 0.999307i \(0.511849\pi\)
\(432\) 0 0
\(433\) − 47438.8i − 0.253022i −0.991965 0.126511i \(-0.959622\pi\)
0.991965 0.126511i \(-0.0403779\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −111618. −0.587167
\(437\) − 195932.i − 1.02599i
\(438\) 0 0
\(439\) − 109814.i − 0.569808i −0.958556 0.284904i \(-0.908038\pi\)
0.958556 0.284904i \(-0.0919618\pi\)
\(440\) 25258.0i 0.130465i
\(441\) 0 0
\(442\) −49749.7 −0.254651
\(443\) 82560.7 0.420694 0.210347 0.977627i \(-0.432541\pi\)
0.210347 + 0.977627i \(0.432541\pi\)
\(444\) 0 0
\(445\) 21271.5 0.107418
\(446\) − 32554.5i − 0.163659i
\(447\) 0 0
\(448\) 0 0
\(449\) 330438. 1.63907 0.819534 0.573030i \(-0.194232\pi\)
0.819534 + 0.573030i \(0.194232\pi\)
\(450\) 0 0
\(451\) − 50288.9i − 0.247240i
\(452\) 70490.4 0.345027
\(453\) 0 0
\(454\) − 159430.i − 0.773499i
\(455\) 0 0
\(456\) 0 0
\(457\) −373287. −1.78735 −0.893677 0.448711i \(-0.851883\pi\)
−0.893677 + 0.448711i \(0.851883\pi\)
\(458\) − 119157.i − 0.568055i
\(459\) 0 0
\(460\) 88511.8i 0.418298i
\(461\) 20355.2i 0.0957798i 0.998853 + 0.0478899i \(0.0152497\pi\)
−0.998853 + 0.0478899i \(0.984750\pi\)
\(462\) 0 0
\(463\) −31552.7 −0.147189 −0.0735944 0.997288i \(-0.523447\pi\)
−0.0735944 + 0.997288i \(0.523447\pi\)
\(464\) −47495.5 −0.220606
\(465\) 0 0
\(466\) −155852. −0.717694
\(467\) − 241854.i − 1.10897i −0.832193 0.554486i \(-0.812915\pi\)
0.832193 0.554486i \(-0.187085\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −96606.6 −0.437332
\(471\) 0 0
\(472\) 96958.7i 0.435214i
\(473\) −106357. −0.475383
\(474\) 0 0
\(475\) 8753.16i 0.0387951i
\(476\) 0 0
\(477\) 0 0
\(478\) 65287.1 0.285740
\(479\) − 272083.i − 1.18585i −0.805257 0.592926i \(-0.797973\pi\)
0.805257 0.592926i \(-0.202027\pi\)
\(480\) 0 0
\(481\) 160342.i 0.693036i
\(482\) − 230722.i − 0.993104i
\(483\) 0 0
\(484\) −101661. −0.433973
\(485\) −142728. −0.606773
\(486\) 0 0
\(487\) −407746. −1.71922 −0.859611 0.510949i \(-0.829294\pi\)
−0.859611 + 0.510949i \(0.829294\pi\)
\(488\) − 31603.5i − 0.132707i
\(489\) 0 0
\(490\) 0 0
\(491\) 286772. 1.18953 0.594763 0.803901i \(-0.297246\pi\)
0.594763 + 0.803901i \(0.297246\pi\)
\(492\) 0 0
\(493\) 80324.2i 0.330486i
\(494\) −206639. −0.846757
\(495\) 0 0
\(496\) − 66523.7i − 0.270404i
\(497\) 0 0
\(498\) 0 0
\(499\) 351435. 1.41138 0.705690 0.708520i \(-0.250637\pi\)
0.705690 + 0.708520i \(0.250637\pi\)
\(500\) 122978.i 0.491911i
\(501\) 0 0
\(502\) − 76954.6i − 0.305371i
\(503\) 116045.i 0.458660i 0.973349 + 0.229330i \(0.0736535\pi\)
−0.973349 + 0.229330i \(0.926347\pi\)
\(504\) 0 0
\(505\) −167182. −0.655552
\(506\) 54202.2 0.211697
\(507\) 0 0
\(508\) 31941.6 0.123774
\(509\) 83173.8i 0.321034i 0.987033 + 0.160517i \(0.0513162\pi\)
−0.987033 + 0.160517i \(0.948684\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11585.2 −0.0441942
\(513\) 0 0
\(514\) − 305269.i − 1.15547i
\(515\) 38701.5 0.145920
\(516\) 0 0
\(517\) 59159.2i 0.221330i
\(518\) 0 0
\(519\) 0 0
\(520\) 93348.7 0.345224
\(521\) − 202013.i − 0.744224i −0.928188 0.372112i \(-0.878634\pi\)
0.928188 0.372112i \(-0.121366\pi\)
\(522\) 0 0
\(523\) 27867.6i 0.101882i 0.998702 + 0.0509408i \(0.0162220\pi\)
−0.998702 + 0.0509408i \(0.983778\pi\)
\(524\) − 161800.i − 0.589272i
\(525\) 0 0
\(526\) −113545. −0.410389
\(527\) −112505. −0.405087
\(528\) 0 0
\(529\) −89899.7 −0.321253
\(530\) − 256066.i − 0.911592i
\(531\) 0 0
\(532\) 0 0
\(533\) −185858. −0.654226
\(534\) 0 0
\(535\) − 269264.i − 0.940743i
\(536\) −164406. −0.572253
\(537\) 0 0
\(538\) 297985.i 1.02951i
\(539\) 0 0
\(540\) 0 0
\(541\) −37993.6 −0.129812 −0.0649062 0.997891i \(-0.520675\pi\)
−0.0649062 + 0.997891i \(0.520675\pi\)
\(542\) − 43076.1i − 0.146635i
\(543\) 0 0
\(544\) 19592.9i 0.0662065i
\(545\) − 354198.i − 1.19249i
\(546\) 0 0
\(547\) 360160. 1.20371 0.601854 0.798606i \(-0.294429\pi\)
0.601854 + 0.798606i \(0.294429\pi\)
\(548\) −200462. −0.667530
\(549\) 0 0
\(550\) −2421.45 −0.00800480
\(551\) 333633.i 1.09892i
\(552\) 0 0
\(553\) 0 0
\(554\) 27921.1 0.0909732
\(555\) 0 0
\(556\) 160560.i 0.519385i
\(557\) −310543. −1.00095 −0.500474 0.865751i \(-0.666841\pi\)
−0.500474 + 0.865751i \(0.666841\pi\)
\(558\) 0 0
\(559\) 393075.i 1.25792i
\(560\) 0 0
\(561\) 0 0
\(562\) −438738. −1.38910
\(563\) − 16689.4i − 0.0526530i −0.999653 0.0263265i \(-0.991619\pi\)
0.999653 0.0263265i \(-0.00838095\pi\)
\(564\) 0 0
\(565\) 223687.i 0.700720i
\(566\) 310171.i 0.968207i
\(567\) 0 0
\(568\) 135488. 0.419955
\(569\) 130262. 0.402341 0.201170 0.979556i \(-0.435526\pi\)
0.201170 + 0.979556i \(0.435526\pi\)
\(570\) 0 0
\(571\) −498239. −1.52815 −0.764073 0.645129i \(-0.776804\pi\)
−0.764073 + 0.645129i \(0.776804\pi\)
\(572\) − 57164.1i − 0.174716i
\(573\) 0 0
\(574\) 0 0
\(575\) −8485.52 −0.0256651
\(576\) 0 0
\(577\) − 33716.2i − 0.101271i −0.998717 0.0506357i \(-0.983875\pi\)
0.998717 0.0506357i \(-0.0161247\pi\)
\(578\) −203098. −0.607924
\(579\) 0 0
\(580\) − 150718.i − 0.448031i
\(581\) 0 0
\(582\) 0 0
\(583\) −156808. −0.461350
\(584\) 13067.2i 0.0383139i
\(585\) 0 0
\(586\) − 254311.i − 0.740576i
\(587\) 356809.i 1.03552i 0.855525 + 0.517762i \(0.173235\pi\)
−0.855525 + 0.517762i \(0.826765\pi\)
\(588\) 0 0
\(589\) −467296. −1.34698
\(590\) −307679. −0.883881
\(591\) 0 0
\(592\) 63147.2 0.180182
\(593\) − 33339.2i − 0.0948082i −0.998876 0.0474041i \(-0.984905\pi\)
0.998876 0.0474041i \(-0.0150949\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 341663. 0.961846
\(597\) 0 0
\(598\) − 200321.i − 0.560175i
\(599\) −324330. −0.903927 −0.451963 0.892036i \(-0.649276\pi\)
−0.451963 + 0.892036i \(0.649276\pi\)
\(600\) 0 0
\(601\) − 66322.0i − 0.183615i −0.995777 0.0918076i \(-0.970736\pi\)
0.995777 0.0918076i \(-0.0292645\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 236860. 0.649258
\(605\) − 322600.i − 0.881361i
\(606\) 0 0
\(607\) 487655.i 1.32354i 0.749709 + 0.661768i \(0.230194\pi\)
−0.749709 + 0.661768i \(0.769806\pi\)
\(608\) 81380.6i 0.220147i
\(609\) 0 0
\(610\) 100287. 0.269517
\(611\) 218641. 0.585665
\(612\) 0 0
\(613\) −341510. −0.908831 −0.454415 0.890790i \(-0.650152\pi\)
−0.454415 + 0.890790i \(0.650152\pi\)
\(614\) 400735.i 1.06297i
\(615\) 0 0
\(616\) 0 0
\(617\) −240873. −0.632728 −0.316364 0.948638i \(-0.602462\pi\)
−0.316364 + 0.948638i \(0.602462\pi\)
\(618\) 0 0
\(619\) − 154335.i − 0.402794i −0.979510 0.201397i \(-0.935452\pi\)
0.979510 0.201397i \(-0.0645481\pi\)
\(620\) 211100. 0.549167
\(621\) 0 0
\(622\) − 242242.i − 0.626137i
\(623\) 0 0
\(624\) 0 0
\(625\) −402415. −1.03018
\(626\) 50435.2i 0.128702i
\(627\) 0 0
\(628\) 97896.4i 0.248226i
\(629\) − 106794.i − 0.269927i
\(630\) 0 0
\(631\) 220248. 0.553164 0.276582 0.960990i \(-0.410798\pi\)
0.276582 + 0.960990i \(0.410798\pi\)
\(632\) 71191.1 0.178234
\(633\) 0 0
\(634\) 228552. 0.568599
\(635\) 101360.i 0.251374i
\(636\) 0 0
\(637\) 0 0
\(638\) −92295.4 −0.226745
\(639\) 0 0
\(640\) − 36763.5i − 0.0897545i
\(641\) 681208. 1.65792 0.828960 0.559308i \(-0.188933\pi\)
0.828960 + 0.559308i \(0.188933\pi\)
\(642\) 0 0
\(643\) − 572102.i − 1.38373i −0.722027 0.691865i \(-0.756789\pi\)
0.722027 0.691865i \(-0.243211\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 137630. 0.329799
\(647\) 607275.i 1.45070i 0.688382 + 0.725348i \(0.258321\pi\)
−0.688382 + 0.725348i \(0.741679\pi\)
\(648\) 0 0
\(649\) 188414.i 0.447326i
\(650\) 8949.23i 0.0211816i
\(651\) 0 0
\(652\) 172249. 0.405193
\(653\) 484670. 1.13663 0.568316 0.822810i \(-0.307595\pi\)
0.568316 + 0.822810i \(0.307595\pi\)
\(654\) 0 0
\(655\) 513440. 1.19676
\(656\) 73196.5i 0.170092i
\(657\) 0 0
\(658\) 0 0
\(659\) −329627. −0.759017 −0.379509 0.925188i \(-0.623907\pi\)
−0.379509 + 0.925188i \(0.623907\pi\)
\(660\) 0 0
\(661\) − 210608.i − 0.482028i −0.970522 0.241014i \(-0.922520\pi\)
0.970522 0.241014i \(-0.0774800\pi\)
\(662\) −205922. −0.469879
\(663\) 0 0
\(664\) − 107027.i − 0.242748i
\(665\) 0 0
\(666\) 0 0
\(667\) −323432. −0.726994
\(668\) − 132631.i − 0.297230i
\(669\) 0 0
\(670\) − 521710.i − 1.16220i
\(671\) − 61413.2i − 0.136401i
\(672\) 0 0
\(673\) 94709.8 0.209105 0.104553 0.994519i \(-0.466659\pi\)
0.104553 + 0.994519i \(0.466659\pi\)
\(674\) −357902. −0.787852
\(675\) 0 0
\(676\) 17220.2 0.0376829
\(677\) 538552.i 1.17503i 0.809212 + 0.587517i \(0.199894\pi\)
−0.809212 + 0.587517i \(0.800106\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −62174.1 −0.134460
\(681\) 0 0
\(682\) − 129272.i − 0.277930i
\(683\) 129072. 0.276688 0.138344 0.990384i \(-0.455822\pi\)
0.138344 + 0.990384i \(0.455822\pi\)
\(684\) 0 0
\(685\) − 636126.i − 1.35570i
\(686\) 0 0
\(687\) 0 0
\(688\) 154805. 0.327044
\(689\) 579532.i 1.22079i
\(690\) 0 0
\(691\) − 130713.i − 0.273756i −0.990588 0.136878i \(-0.956293\pi\)
0.990588 0.136878i \(-0.0437068\pi\)
\(692\) − 323064.i − 0.674648i
\(693\) 0 0
\(694\) −189361. −0.393162
\(695\) −509507. −1.05483
\(696\) 0 0
\(697\) 123790. 0.254811
\(698\) − 72202.6i − 0.148198i
\(699\) 0 0
\(700\) 0 0
\(701\) 182501. 0.371389 0.185694 0.982608i \(-0.440547\pi\)
0.185694 + 0.982608i \(0.440547\pi\)
\(702\) 0 0
\(703\) − 443578.i − 0.897552i
\(704\) −22512.9 −0.0454241
\(705\) 0 0
\(706\) − 160020.i − 0.321044i
\(707\) 0 0
\(708\) 0 0
\(709\) 813798. 1.61892 0.809458 0.587178i \(-0.199761\pi\)
0.809458 + 0.587178i \(0.199761\pi\)
\(710\) 429943.i 0.852892i
\(711\) 0 0
\(712\) 18959.7i 0.0374000i
\(713\) − 453008.i − 0.891101i
\(714\) 0 0
\(715\) 181399. 0.354832
\(716\) 427955. 0.834780
\(717\) 0 0
\(718\) 106928. 0.207416
\(719\) − 420567.i − 0.813537i −0.913531 0.406769i \(-0.866656\pi\)
0.913531 0.406769i \(-0.133344\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 203055. 0.389528
\(723\) 0 0
\(724\) 433616.i 0.827235i
\(725\) 14449.1 0.0274894
\(726\) 0 0
\(727\) − 172948.i − 0.327225i −0.986525 0.163613i \(-0.947685\pi\)
0.986525 0.163613i \(-0.0523147\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −41466.1 −0.0778121
\(731\) − 261805.i − 0.489939i
\(732\) 0 0
\(733\) − 131562.i − 0.244862i −0.992477 0.122431i \(-0.960931\pi\)
0.992477 0.122431i \(-0.0390690\pi\)
\(734\) − 30593.4i − 0.0567852i
\(735\) 0 0
\(736\) −78892.3 −0.145639
\(737\) −319481. −0.588179
\(738\) 0 0
\(739\) −489768. −0.896812 −0.448406 0.893830i \(-0.648008\pi\)
−0.448406 + 0.893830i \(0.648008\pi\)
\(740\) 200385.i 0.365934i
\(741\) 0 0
\(742\) 0 0
\(743\) −580258. −1.05110 −0.525549 0.850763i \(-0.676140\pi\)
−0.525549 + 0.850763i \(0.676140\pi\)
\(744\) 0 0
\(745\) 1.08420e6i 1.95343i
\(746\) 87747.4 0.157673
\(747\) 0 0
\(748\) 38073.7i 0.0680491i
\(749\) 0 0
\(750\) 0 0
\(751\) 437397. 0.775525 0.387763 0.921759i \(-0.373248\pi\)
0.387763 + 0.921759i \(0.373248\pi\)
\(752\) − 86107.4i − 0.152267i
\(753\) 0 0
\(754\) 341106.i 0.599994i
\(755\) 751627.i 1.31859i
\(756\) 0 0
\(757\) −24171.7 −0.0421809 −0.0210905 0.999778i \(-0.506714\pi\)
−0.0210905 + 0.999778i \(0.506714\pi\)
\(758\) 278678. 0.485025
\(759\) 0 0
\(760\) −258245. −0.447100
\(761\) − 648221.i − 1.11932i −0.828723 0.559659i \(-0.810932\pi\)
0.828723 0.559659i \(-0.189068\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 9630.98 0.0165000
\(765\) 0 0
\(766\) 14547.5i 0.0247932i
\(767\) 696343. 1.18368
\(768\) 0 0
\(769\) 179564.i 0.303646i 0.988408 + 0.151823i \(0.0485143\pi\)
−0.988408 + 0.151823i \(0.951486\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 410503. 0.688783
\(773\) − 610915.i − 1.02240i −0.859461 0.511201i \(-0.829201\pi\)
0.859461 0.511201i \(-0.170799\pi\)
\(774\) 0 0
\(775\) 20237.9i 0.0336947i
\(776\) − 127216.i − 0.211261i
\(777\) 0 0
\(778\) 244042. 0.403186
\(779\) 514169. 0.847288
\(780\) 0 0
\(781\) 263285. 0.431643
\(782\) 133422.i 0.218180i
\(783\) 0 0
\(784\) 0 0
\(785\) −310655. −0.504126
\(786\) 0 0
\(787\) 1.19155e6i 1.92381i 0.273384 + 0.961905i \(0.411857\pi\)
−0.273384 + 0.961905i \(0.588143\pi\)
\(788\) 11650.2 0.0187621
\(789\) 0 0
\(790\) 225911.i 0.361979i
\(791\) 0 0
\(792\) 0 0
\(793\) −226972. −0.360932
\(794\) − 441026.i − 0.699557i
\(795\) 0 0
\(796\) 550066.i 0.868137i
\(797\) − 900495.i − 1.41764i −0.705392 0.708818i \(-0.749229\pi\)
0.705392 0.708818i \(-0.250771\pi\)
\(798\) 0 0
\(799\) −145624. −0.228108
\(800\) 3524.47 0.00550698
\(801\) 0 0
\(802\) 97904.5 0.152214
\(803\) 25392.7i 0.0393802i
\(804\) 0 0
\(805\) 0 0
\(806\) −477764. −0.735433
\(807\) 0 0
\(808\) − 149013.i − 0.228245i
\(809\) −555764. −0.849168 −0.424584 0.905389i \(-0.639580\pi\)
−0.424584 + 0.905389i \(0.639580\pi\)
\(810\) 0 0
\(811\) − 70954.2i − 0.107879i −0.998544 0.0539395i \(-0.982822\pi\)
0.998544 0.0539395i \(-0.0171778\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 122710. 0.185196
\(815\) 546598.i 0.822911i
\(816\) 0 0
\(817\) − 1.08743e6i − 1.62913i
\(818\) 887527.i 1.32640i
\(819\) 0 0
\(820\) −232275. −0.345441
\(821\) 624712. 0.926816 0.463408 0.886145i \(-0.346626\pi\)
0.463408 + 0.886145i \(0.346626\pi\)
\(822\) 0 0
\(823\) −210059. −0.310128 −0.155064 0.987904i \(-0.549558\pi\)
−0.155064 + 0.987904i \(0.549558\pi\)
\(824\) 34495.4i 0.0508051i
\(825\) 0 0
\(826\) 0 0
\(827\) 880910. 1.28801 0.644007 0.765020i \(-0.277271\pi\)
0.644007 + 0.765020i \(0.277271\pi\)
\(828\) 0 0
\(829\) − 1.29270e6i − 1.88100i −0.339799 0.940498i \(-0.610359\pi\)
0.339799 0.940498i \(-0.389641\pi\)
\(830\) 339628. 0.493001
\(831\) 0 0
\(832\) 83203.5i 0.120197i
\(833\) 0 0
\(834\) 0 0
\(835\) 420879. 0.603649
\(836\) 158142.i 0.226274i
\(837\) 0 0
\(838\) − 20717.5i − 0.0295018i
\(839\) 1.11586e6i 1.58520i 0.609741 + 0.792601i \(0.291274\pi\)
−0.609741 + 0.792601i \(0.708726\pi\)
\(840\) 0 0
\(841\) −156542. −0.221330
\(842\) 71215.1 0.100449
\(843\) 0 0
\(844\) −4982.57 −0.00699469
\(845\) 54644.8i 0.0765307i
\(846\) 0 0
\(847\) 0 0
\(848\) 228237. 0.317391
\(849\) 0 0
\(850\) − 5960.56i − 0.00824991i
\(851\) 430015. 0.593779
\(852\) 0 0
\(853\) 1.28959e6i 1.77236i 0.463341 + 0.886180i \(0.346650\pi\)
−0.463341 + 0.886180i \(0.653350\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 240001. 0.327540
\(857\) 654547.i 0.891208i 0.895230 + 0.445604i \(0.147011\pi\)
−0.895230 + 0.445604i \(0.852989\pi\)
\(858\) 0 0
\(859\) − 244561.i − 0.331437i −0.986173 0.165718i \(-0.947006\pi\)
0.986173 0.165718i \(-0.0529943\pi\)
\(860\) 491242.i 0.664199i
\(861\) 0 0
\(862\) 39109.2 0.0526337
\(863\) 465277. 0.624726 0.312363 0.949963i \(-0.398879\pi\)
0.312363 + 0.949963i \(0.398879\pi\)
\(864\) 0 0
\(865\) 1.02518e6 1.37015
\(866\) 134177.i 0.178913i
\(867\) 0 0
\(868\) 0 0
\(869\) 138342. 0.183195
\(870\) 0 0
\(871\) 1.18074e6i 1.55639i
\(872\) 315704. 0.415190
\(873\) 0 0
\(874\) 554179.i 0.725484i
\(875\) 0 0
\(876\) 0 0
\(877\) 304172. 0.395475 0.197738 0.980255i \(-0.436641\pi\)
0.197738 + 0.980255i \(0.436641\pi\)
\(878\) 310601.i 0.402915i
\(879\) 0 0
\(880\) − 71440.3i − 0.0922525i
\(881\) − 697876.i − 0.899138i −0.893246 0.449569i \(-0.851578\pi\)
0.893246 0.449569i \(-0.148422\pi\)
\(882\) 0 0
\(883\) 891773. 1.14375 0.571877 0.820339i \(-0.306215\pi\)
0.571877 + 0.820339i \(0.306215\pi\)
\(884\) 140713. 0.180066
\(885\) 0 0
\(886\) −233517. −0.297475
\(887\) − 1.26256e6i − 1.60473i −0.596831 0.802367i \(-0.703574\pi\)
0.596831 0.802367i \(-0.296426\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −60164.9 −0.0759562
\(891\) 0 0
\(892\) 92077.9i 0.115725i
\(893\) −604862. −0.758496
\(894\) 0 0
\(895\) 1.35803e6i 1.69537i
\(896\) 0 0
\(897\) 0 0
\(898\) −934619. −1.15900
\(899\) 771382.i 0.954443i
\(900\) 0 0
\(901\) − 385993.i − 0.475477i
\(902\) 142239.i 0.174825i
\(903\) 0 0
\(904\) −199377. −0.243971
\(905\) −1.37600e6 −1.68004
\(906\) 0 0
\(907\) 951981. 1.15721 0.578607 0.815607i \(-0.303597\pi\)
0.578607 + 0.815607i \(0.303597\pi\)
\(908\) 450938.i 0.546946i
\(909\) 0 0
\(910\) 0 0
\(911\) 743243. 0.895559 0.447779 0.894144i \(-0.352215\pi\)
0.447779 + 0.894144i \(0.352215\pi\)
\(912\) 0 0
\(913\) − 207979.i − 0.249504i
\(914\) 1.05582e6 1.26385
\(915\) 0 0
\(916\) 337028.i 0.401675i
\(917\) 0 0
\(918\) 0 0
\(919\) −524300. −0.620796 −0.310398 0.950607i \(-0.600462\pi\)
−0.310398 + 0.950607i \(0.600462\pi\)
\(920\) − 250349.i − 0.295781i
\(921\) 0 0
\(922\) − 57573.3i − 0.0677266i
\(923\) − 973052.i − 1.14218i
\(924\) 0 0
\(925\) −19210.7 −0.0224522
\(926\) 89244.6 0.104078
\(927\) 0 0
\(928\) 134338. 0.155992
\(929\) 979392.i 1.13482i 0.823437 + 0.567408i \(0.192054\pi\)
−0.823437 + 0.567408i \(0.807946\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 440815. 0.507486
\(933\) 0 0
\(934\) 684068.i 0.784161i
\(935\) −120819. −0.138202
\(936\) 0 0
\(937\) − 1.13988e6i − 1.29831i −0.760656 0.649155i \(-0.775123\pi\)
0.760656 0.649155i \(-0.224877\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 273245. 0.309240
\(941\) − 801121.i − 0.904730i −0.891833 0.452365i \(-0.850580\pi\)
0.891833 0.452365i \(-0.149420\pi\)
\(942\) 0 0
\(943\) 498448.i 0.560527i
\(944\) − 274240.i − 0.307743i
\(945\) 0 0
\(946\) 300823. 0.336146
\(947\) −680286. −0.758563 −0.379282 0.925281i \(-0.623829\pi\)
−0.379282 + 0.925281i \(0.623829\pi\)
\(948\) 0 0
\(949\) 93846.5 0.104204
\(950\) − 24757.7i − 0.0274323i
\(951\) 0 0
\(952\) 0 0
\(953\) 394774. 0.434673 0.217337 0.976097i \(-0.430263\pi\)
0.217337 + 0.976097i \(0.430263\pi\)
\(954\) 0 0
\(955\) 30562.0i 0.0335101i
\(956\) −184660. −0.202049
\(957\) 0 0
\(958\) 769567.i 0.838524i
\(959\) 0 0
\(960\) 0 0
\(961\) −156900. −0.169893
\(962\) − 453514.i − 0.490050i
\(963\) 0 0
\(964\) 652580.i 0.702231i
\(965\) 1.30265e6i 1.39886i
\(966\) 0 0
\(967\) −604328. −0.646279 −0.323140 0.946351i \(-0.604738\pi\)
−0.323140 + 0.946351i \(0.604738\pi\)
\(968\) 287540. 0.306865
\(969\) 0 0
\(970\) 403696. 0.429053
\(971\) 758986.i 0.804999i 0.915420 + 0.402499i \(0.131858\pi\)
−0.915420 + 0.402499i \(0.868142\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.15328e6 1.21567
\(975\) 0 0
\(976\) 89388.1i 0.0938383i
\(977\) −1.22518e6 −1.28355 −0.641773 0.766894i \(-0.721801\pi\)
−0.641773 + 0.766894i \(0.721801\pi\)
\(978\) 0 0
\(979\) 36843.3i 0.0384409i
\(980\) 0 0
\(981\) 0 0
\(982\) −811115. −0.841123
\(983\) 1.20047e6i 1.24235i 0.783670 + 0.621177i \(0.213345\pi\)
−0.783670 + 0.621177i \(0.786655\pi\)
\(984\) 0 0
\(985\) 36969.6i 0.0381042i
\(986\) − 227191.i − 0.233689i
\(987\) 0 0
\(988\) 584464. 0.598747
\(989\) 1.05418e6 1.07776
\(990\) 0 0
\(991\) 1.80993e6 1.84295 0.921475 0.388438i \(-0.126985\pi\)
0.921475 + 0.388438i \(0.126985\pi\)
\(992\) 188157.i 0.191204i
\(993\) 0 0
\(994\) 0 0
\(995\) −1.74552e6 −1.76311
\(996\) 0 0
\(997\) 749289.i 0.753805i 0.926253 + 0.376902i \(0.123011\pi\)
−0.926253 + 0.376902i \(0.876989\pi\)
\(998\) −994009. −0.997997
\(999\) 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 882.5.c.b.685.2 4
3.2 odd 2 98.5.b.b.97.3 4
7.4 even 3 126.5.n.a.19.2 4
7.5 odd 6 126.5.n.a.73.2 4
7.6 odd 2 inner 882.5.c.b.685.1 4
12.11 even 2 784.5.c.b.97.4 4
21.2 odd 6 98.5.d.a.31.1 4
21.5 even 6 14.5.d.a.3.1 4
21.11 odd 6 14.5.d.a.5.1 yes 4
21.17 even 6 98.5.d.a.19.1 4
21.20 even 2 98.5.b.b.97.4 4
84.11 even 6 112.5.s.b.33.2 4
84.47 odd 6 112.5.s.b.17.2 4
84.83 odd 2 784.5.c.b.97.1 4
105.32 even 12 350.5.i.a.299.2 8
105.47 odd 12 350.5.i.a.199.3 8
105.53 even 12 350.5.i.a.299.3 8
105.68 odd 12 350.5.i.a.199.2 8
105.74 odd 6 350.5.k.a.201.2 4
105.89 even 6 350.5.k.a.101.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.5.d.a.3.1 4 21.5 even 6
14.5.d.a.5.1 yes 4 21.11 odd 6
98.5.b.b.97.3 4 3.2 odd 2
98.5.b.b.97.4 4 21.20 even 2
98.5.d.a.19.1 4 21.17 even 6
98.5.d.a.31.1 4 21.2 odd 6
112.5.s.b.17.2 4 84.47 odd 6
112.5.s.b.33.2 4 84.11 even 6
126.5.n.a.19.2 4 7.4 even 3
126.5.n.a.73.2 4 7.5 odd 6
350.5.i.a.199.2 8 105.68 odd 12
350.5.i.a.199.3 8 105.47 odd 12
350.5.i.a.299.2 8 105.32 even 12
350.5.i.a.299.3 8 105.53 even 12
350.5.k.a.101.2 4 105.89 even 6
350.5.k.a.201.2 4 105.74 odd 6
784.5.c.b.97.1 4 84.83 odd 2
784.5.c.b.97.4 4 12.11 even 2
882.5.c.b.685.1 4 7.6 odd 2 inner
882.5.c.b.685.2 4 1.1 even 1 trivial