Properties

Label 441.4.c.a.440.8
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 440.8
Root \(-0.648633 + 0.374489i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.4.c.a.440.10

$q$-expansion

\(f(q)\) \(=\) \(q-0.748977i q^{2} +7.43903 q^{4} +10.8554 q^{5} -11.5635i q^{8} +O(q^{10})\) \(q-0.748977i q^{2} +7.43903 q^{4} +10.8554 q^{5} -11.5635i q^{8} -8.13042i q^{10} +51.8611i q^{11} +32.1880i q^{13} +50.8515 q^{16} +81.4649 q^{17} -0.0485737i q^{19} +80.7534 q^{20} +38.8428 q^{22} +89.2956i q^{23} -7.16119 q^{25} +24.1081 q^{26} +175.246i q^{29} -215.049i q^{31} -130.594i q^{32} -61.0153i q^{34} +64.5458 q^{37} -0.0363806 q^{38} -125.526i q^{40} -411.485 q^{41} -234.771 q^{43} +385.797i q^{44} +66.8803 q^{46} +632.152 q^{47} +5.36357i q^{50} +239.448i q^{52} +265.638i q^{53} +562.971i q^{55} +131.255 q^{58} +351.059 q^{59} -778.069i q^{61} -161.067 q^{62} +309.000 q^{64} +349.412i q^{65} -196.009 q^{67} +606.020 q^{68} -142.632i q^{71} -780.879i q^{73} -48.3433i q^{74} -0.361341i q^{76} +1289.05 q^{79} +552.011 q^{80} +308.193i q^{82} +235.123 q^{83} +884.330 q^{85} +175.838i q^{86} +599.695 q^{88} -670.780 q^{89} +664.273i q^{92} -473.468i q^{94} -0.527285i q^{95} -655.891i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 64q^{4} + O(q^{10}) \) \( 16q - 64q^{4} + 376q^{16} + 528q^{22} + 40q^{25} + 2392q^{37} + 328q^{43} + 2784q^{46} + 6744q^{58} + 5432q^{64} - 616q^{67} + 4352q^{79} - 4608q^{85} - 1416q^{88} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.748977i − 0.264803i −0.991196 0.132402i \(-0.957731\pi\)
0.991196 0.132402i \(-0.0422689\pi\)
\(3\) 0 0
\(4\) 7.43903 0.929879
\(5\) 10.8554 0.970933 0.485466 0.874255i \(-0.338650\pi\)
0.485466 + 0.874255i \(0.338650\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 11.5635i − 0.511039i
\(9\) 0 0
\(10\) − 8.13042i − 0.257106i
\(11\) 51.8611i 1.42152i 0.703435 + 0.710760i \(0.251649\pi\)
−0.703435 + 0.710760i \(0.748351\pi\)
\(12\) 0 0
\(13\) 32.1880i 0.686719i 0.939204 + 0.343360i \(0.111565\pi\)
−0.939204 + 0.343360i \(0.888435\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 50.8515 0.794554
\(17\) 81.4649 1.16224 0.581121 0.813817i \(-0.302614\pi\)
0.581121 + 0.813817i \(0.302614\pi\)
\(18\) 0 0
\(19\) − 0.0485737i 0 0.000586504i −1.00000 0.000293252i \(-0.999907\pi\)
1.00000 0.000293252i \(-9.33450e-5\pi\)
\(20\) 80.7534 0.902850
\(21\) 0 0
\(22\) 38.8428 0.376423
\(23\) 89.2956i 0.809540i 0.914419 + 0.404770i \(0.132648\pi\)
−0.914419 + 0.404770i \(0.867352\pi\)
\(24\) 0 0
\(25\) −7.16119 −0.0572896
\(26\) 24.1081 0.181846
\(27\) 0 0
\(28\) 0 0
\(29\) 175.246i 1.12215i 0.827766 + 0.561074i \(0.189612\pi\)
−0.827766 + 0.561074i \(0.810388\pi\)
\(30\) 0 0
\(31\) − 215.049i − 1.24593i −0.782249 0.622966i \(-0.785927\pi\)
0.782249 0.622966i \(-0.214073\pi\)
\(32\) − 130.594i − 0.721439i
\(33\) 0 0
\(34\) − 61.0153i − 0.307766i
\(35\) 0 0
\(36\) 0 0
\(37\) 64.5458 0.286791 0.143395 0.989665i \(-0.454198\pi\)
0.143395 + 0.989665i \(0.454198\pi\)
\(38\) −0.0363806 −0.000155308 0
\(39\) 0 0
\(40\) − 125.526i − 0.496184i
\(41\) −411.485 −1.56740 −0.783698 0.621142i \(-0.786669\pi\)
−0.783698 + 0.621142i \(0.786669\pi\)
\(42\) 0 0
\(43\) −234.771 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(44\) 385.797i 1.32184i
\(45\) 0 0
\(46\) 66.8803 0.214369
\(47\) 632.152 1.96189 0.980946 0.194283i \(-0.0622379\pi\)
0.980946 + 0.194283i \(0.0622379\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 5.36357i 0.0151705i
\(51\) 0 0
\(52\) 239.448i 0.638566i
\(53\) 265.638i 0.688456i 0.938886 + 0.344228i \(0.111859\pi\)
−0.938886 + 0.344228i \(0.888141\pi\)
\(54\) 0 0
\(55\) 562.971i 1.38020i
\(56\) 0 0
\(57\) 0 0
\(58\) 131.255 0.297149
\(59\) 351.059 0.774645 0.387322 0.921944i \(-0.373400\pi\)
0.387322 + 0.921944i \(0.373400\pi\)
\(60\) 0 0
\(61\) − 778.069i − 1.63314i −0.577247 0.816569i \(-0.695873\pi\)
0.577247 0.816569i \(-0.304127\pi\)
\(62\) −161.067 −0.329927
\(63\) 0 0
\(64\) 309.000 0.603515
\(65\) 349.412i 0.666758i
\(66\) 0 0
\(67\) −196.009 −0.357407 −0.178703 0.983903i \(-0.557190\pi\)
−0.178703 + 0.983903i \(0.557190\pi\)
\(68\) 606.020 1.08075
\(69\) 0 0
\(70\) 0 0
\(71\) − 142.632i − 0.238412i −0.992870 0.119206i \(-0.961965\pi\)
0.992870 0.119206i \(-0.0380349\pi\)
\(72\) 0 0
\(73\) − 780.879i − 1.25199i −0.779829 0.625993i \(-0.784694\pi\)
0.779829 0.625993i \(-0.215306\pi\)
\(74\) − 48.3433i − 0.0759432i
\(75\) 0 0
\(76\) − 0.361341i 0 0.000545378i
\(77\) 0 0
\(78\) 0 0
\(79\) 1289.05 1.83582 0.917908 0.396794i \(-0.129877\pi\)
0.917908 + 0.396794i \(0.129877\pi\)
\(80\) 552.011 0.771459
\(81\) 0 0
\(82\) 308.193i 0.415052i
\(83\) 235.123 0.310940 0.155470 0.987841i \(-0.450311\pi\)
0.155470 + 0.987841i \(0.450311\pi\)
\(84\) 0 0
\(85\) 884.330 1.12846
\(86\) 175.838i 0.220478i
\(87\) 0 0
\(88\) 599.695 0.726452
\(89\) −670.780 −0.798905 −0.399453 0.916754i \(-0.630800\pi\)
−0.399453 + 0.916754i \(0.630800\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 664.273i 0.752774i
\(93\) 0 0
\(94\) − 473.468i − 0.519515i
\(95\) − 0.527285i 0 0.000569456i
\(96\) 0 0
\(97\) − 655.891i − 0.686553i −0.939234 0.343276i \(-0.888463\pi\)
0.939234 0.343276i \(-0.111537\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −53.2724 −0.0532724
\(101\) 1163.24 1.14600 0.573002 0.819554i \(-0.305779\pi\)
0.573002 + 0.819554i \(0.305779\pi\)
\(102\) 0 0
\(103\) 26.4198i 0.0252740i 0.999920 + 0.0126370i \(0.00402258\pi\)
−0.999920 + 0.0126370i \(0.995977\pi\)
\(104\) 372.206 0.350940
\(105\) 0 0
\(106\) 198.957 0.182305
\(107\) 1467.08i 1.32549i 0.748844 + 0.662746i \(0.230609\pi\)
−0.748844 + 0.662746i \(0.769391\pi\)
\(108\) 0 0
\(109\) −135.069 −0.118690 −0.0593450 0.998238i \(-0.518901\pi\)
−0.0593450 + 0.998238i \(0.518901\pi\)
\(110\) 421.653 0.365482
\(111\) 0 0
\(112\) 0 0
\(113\) − 288.471i − 0.240151i −0.992765 0.120076i \(-0.961686\pi\)
0.992765 0.120076i \(-0.0383137\pi\)
\(114\) 0 0
\(115\) 969.335i 0.786009i
\(116\) 1303.66i 1.04346i
\(117\) 0 0
\(118\) − 262.935i − 0.205129i
\(119\) 0 0
\(120\) 0 0
\(121\) −1358.58 −1.02072
\(122\) −582.756 −0.432461
\(123\) 0 0
\(124\) − 1599.75i − 1.15857i
\(125\) −1434.66 −1.02656
\(126\) 0 0
\(127\) −2269.80 −1.58592 −0.792961 0.609273i \(-0.791461\pi\)
−0.792961 + 0.609273i \(0.791461\pi\)
\(128\) − 1276.19i − 0.881252i
\(129\) 0 0
\(130\) 261.702 0.176560
\(131\) 389.692 0.259905 0.129952 0.991520i \(-0.458518\pi\)
0.129952 + 0.991520i \(0.458518\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 146.806i 0.0946425i
\(135\) 0 0
\(136\) − 942.017i − 0.593951i
\(137\) − 1468.69i − 0.915905i −0.888977 0.457953i \(-0.848583\pi\)
0.888977 0.457953i \(-0.151417\pi\)
\(138\) 0 0
\(139\) 624.712i 0.381204i 0.981667 + 0.190602i \(0.0610440\pi\)
−0.981667 + 0.190602i \(0.938956\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −106.828 −0.0631323
\(143\) −1669.31 −0.976185
\(144\) 0 0
\(145\) 1902.35i 1.08953i
\(146\) −584.860 −0.331530
\(147\) 0 0
\(148\) 480.158 0.266681
\(149\) − 1601.86i − 0.880735i −0.897818 0.440367i \(-0.854848\pi\)
0.897818 0.440367i \(-0.145152\pi\)
\(150\) 0 0
\(151\) 404.376 0.217931 0.108966 0.994046i \(-0.465246\pi\)
0.108966 + 0.994046i \(0.465246\pi\)
\(152\) −0.561681 −0.000299726 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 2334.43i − 1.20972i
\(156\) 0 0
\(157\) 2412.07i 1.22614i 0.790028 + 0.613071i \(0.210066\pi\)
−0.790028 + 0.613071i \(0.789934\pi\)
\(158\) − 965.469i − 0.486130i
\(159\) 0 0
\(160\) − 1417.65i − 0.700469i
\(161\) 0 0
\(162\) 0 0
\(163\) −945.367 −0.454275 −0.227138 0.973863i \(-0.572937\pi\)
−0.227138 + 0.973863i \(0.572937\pi\)
\(164\) −3061.05 −1.45749
\(165\) 0 0
\(166\) − 176.101i − 0.0823381i
\(167\) 1271.18 0.589022 0.294511 0.955648i \(-0.404843\pi\)
0.294511 + 0.955648i \(0.404843\pi\)
\(168\) 0 0
\(169\) 1160.93 0.528417
\(170\) − 662.343i − 0.298820i
\(171\) 0 0
\(172\) −1746.47 −0.774228
\(173\) −4434.99 −1.94905 −0.974525 0.224278i \(-0.927998\pi\)
−0.974525 + 0.224278i \(0.927998\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2637.22i 1.12947i
\(177\) 0 0
\(178\) 502.399i 0.211553i
\(179\) − 1087.54i − 0.454114i −0.973881 0.227057i \(-0.927090\pi\)
0.973881 0.227057i \(-0.0729103\pi\)
\(180\) 0 0
\(181\) 2916.08i 1.19752i 0.800930 + 0.598758i \(0.204339\pi\)
−0.800930 + 0.598758i \(0.795661\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1032.57 0.413706
\(185\) 700.667 0.278455
\(186\) 0 0
\(187\) 4224.86i 1.65215i
\(188\) 4702.60 1.82432
\(189\) 0 0
\(190\) −0.394924 −0.000150794 0
\(191\) − 4559.87i − 1.72744i −0.503973 0.863719i \(-0.668129\pi\)
0.503973 0.863719i \(-0.331871\pi\)
\(192\) 0 0
\(193\) −3757.20 −1.40129 −0.700645 0.713510i \(-0.747104\pi\)
−0.700645 + 0.713510i \(0.747104\pi\)
\(194\) −491.247 −0.181802
\(195\) 0 0
\(196\) 0 0
\(197\) − 2014.34i − 0.728507i −0.931300 0.364253i \(-0.881324\pi\)
0.931300 0.364253i \(-0.118676\pi\)
\(198\) 0 0
\(199\) − 12.5118i − 0.00445698i −0.999998 0.00222849i \(-0.999291\pi\)
0.999998 0.00222849i \(-0.000709351\pi\)
\(200\) 82.8083i 0.0292772i
\(201\) 0 0
\(202\) − 871.238i − 0.303466i
\(203\) 0 0
\(204\) 0 0
\(205\) −4466.82 −1.52184
\(206\) 19.7878 0.00669263
\(207\) 0 0
\(208\) 1636.81i 0.545636i
\(209\) 2.51909 0.000833727 0
\(210\) 0 0
\(211\) −2915.84 −0.951349 −0.475675 0.879621i \(-0.657796\pi\)
−0.475675 + 0.879621i \(0.657796\pi\)
\(212\) 1976.09i 0.640181i
\(213\) 0 0
\(214\) 1098.81 0.350995
\(215\) −2548.53 −0.808409
\(216\) 0 0
\(217\) 0 0
\(218\) 101.163i 0.0314295i
\(219\) 0 0
\(220\) 4187.96i 1.28342i
\(221\) 2622.19i 0.798135i
\(222\) 0 0
\(223\) − 1097.87i − 0.329681i −0.986320 0.164841i \(-0.947289\pi\)
0.986320 0.164841i \(-0.0527110\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −216.058 −0.0635929
\(227\) −500.594 −0.146368 −0.0731841 0.997318i \(-0.523316\pi\)
−0.0731841 + 0.997318i \(0.523316\pi\)
\(228\) 0 0
\(229\) − 1133.53i − 0.327099i −0.986535 0.163549i \(-0.947706\pi\)
0.986535 0.163549i \(-0.0522943\pi\)
\(230\) 726.010 0.208138
\(231\) 0 0
\(232\) 2026.45 0.573461
\(233\) − 3435.37i − 0.965916i −0.875643 0.482958i \(-0.839562\pi\)
0.875643 0.482958i \(-0.160438\pi\)
\(234\) 0 0
\(235\) 6862.24 1.90486
\(236\) 2611.54 0.720326
\(237\) 0 0
\(238\) 0 0
\(239\) − 2213.97i − 0.599203i −0.954064 0.299602i \(-0.903146\pi\)
0.954064 0.299602i \(-0.0968537\pi\)
\(240\) 0 0
\(241\) − 5951.96i − 1.59087i −0.606040 0.795435i \(-0.707243\pi\)
0.606040 0.795435i \(-0.292757\pi\)
\(242\) 1017.54i 0.270290i
\(243\) 0 0
\(244\) − 5788.08i − 1.51862i
\(245\) 0 0
\(246\) 0 0
\(247\) 1.56349 0.000402763 0
\(248\) −2486.71 −0.636719
\(249\) 0 0
\(250\) 1074.53i 0.271836i
\(251\) −4889.86 −1.22966 −0.614831 0.788659i \(-0.710776\pi\)
−0.614831 + 0.788659i \(0.710776\pi\)
\(252\) 0 0
\(253\) −4630.97 −1.15078
\(254\) 1700.03i 0.419957i
\(255\) 0 0
\(256\) 1516.16 0.370156
\(257\) −3196.42 −0.775825 −0.387913 0.921696i \(-0.626804\pi\)
−0.387913 + 0.921696i \(0.626804\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2599.29i 0.620005i
\(261\) 0 0
\(262\) − 291.870i − 0.0688237i
\(263\) 748.464i 0.175484i 0.996143 + 0.0877419i \(0.0279651\pi\)
−0.996143 + 0.0877419i \(0.972035\pi\)
\(264\) 0 0
\(265\) 2883.59i 0.668444i
\(266\) 0 0
\(267\) 0 0
\(268\) −1458.11 −0.332345
\(269\) −1299.26 −0.294487 −0.147244 0.989100i \(-0.547040\pi\)
−0.147244 + 0.989100i \(0.547040\pi\)
\(270\) 0 0
\(271\) 83.5611i 0.0187305i 0.999956 + 0.00936527i \(0.00298110\pi\)
−0.999956 + 0.00936527i \(0.997019\pi\)
\(272\) 4142.61 0.923465
\(273\) 0 0
\(274\) −1100.02 −0.242535
\(275\) − 371.388i − 0.0814382i
\(276\) 0 0
\(277\) −4641.86 −1.00687 −0.503434 0.864034i \(-0.667930\pi\)
−0.503434 + 0.864034i \(0.667930\pi\)
\(278\) 467.895 0.100944
\(279\) 0 0
\(280\) 0 0
\(281\) 179.289i 0.0380622i 0.999819 + 0.0190311i \(0.00605816\pi\)
−0.999819 + 0.0190311i \(0.993942\pi\)
\(282\) 0 0
\(283\) 4048.54i 0.850391i 0.905101 + 0.425196i \(0.139795\pi\)
−0.905101 + 0.425196i \(0.860205\pi\)
\(284\) − 1061.04i − 0.221694i
\(285\) 0 0
\(286\) 1250.27i 0.258497i
\(287\) 0 0
\(288\) 0 0
\(289\) 1723.52 0.350808
\(290\) 1424.82 0.288511
\(291\) 0 0
\(292\) − 5808.98i − 1.16420i
\(293\) −3389.52 −0.675828 −0.337914 0.941177i \(-0.609721\pi\)
−0.337914 + 0.941177i \(0.609721\pi\)
\(294\) 0 0
\(295\) 3810.88 0.752128
\(296\) − 746.374i − 0.146561i
\(297\) 0 0
\(298\) −1199.76 −0.233222
\(299\) −2874.25 −0.555927
\(300\) 0 0
\(301\) 0 0
\(302\) − 302.868i − 0.0577089i
\(303\) 0 0
\(304\) − 2.47004i 0 0.000466009i
\(305\) − 8446.21i − 1.58567i
\(306\) 0 0
\(307\) − 2014.64i − 0.374534i −0.982309 0.187267i \(-0.940037\pi\)
0.982309 0.187267i \(-0.0599629\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1748.44 −0.320337
\(311\) 5844.02 1.06554 0.532772 0.846259i \(-0.321150\pi\)
0.532772 + 0.846259i \(0.321150\pi\)
\(312\) 0 0
\(313\) 2449.45i 0.442336i 0.975236 + 0.221168i \(0.0709870\pi\)
−0.975236 + 0.221168i \(0.929013\pi\)
\(314\) 1806.59 0.324686
\(315\) 0 0
\(316\) 9589.28 1.70709
\(317\) 6123.66i 1.08498i 0.840062 + 0.542490i \(0.182519\pi\)
−0.840062 + 0.542490i \(0.817481\pi\)
\(318\) 0 0
\(319\) −9088.44 −1.59516
\(320\) 3354.30 0.585972
\(321\) 0 0
\(322\) 0 0
\(323\) − 3.95705i 0 0.000681660i
\(324\) 0 0
\(325\) − 230.505i − 0.0393418i
\(326\) 708.059i 0.120294i
\(327\) 0 0
\(328\) 4758.20i 0.801000i
\(329\) 0 0
\(330\) 0 0
\(331\) 7597.04 1.26154 0.630772 0.775968i \(-0.282738\pi\)
0.630772 + 0.775968i \(0.282738\pi\)
\(332\) 1749.08 0.289137
\(333\) 0 0
\(334\) − 952.083i − 0.155975i
\(335\) −2127.74 −0.347018
\(336\) 0 0
\(337\) −3863.22 −0.624460 −0.312230 0.950007i \(-0.601076\pi\)
−0.312230 + 0.950007i \(0.601076\pi\)
\(338\) − 869.511i − 0.139927i
\(339\) 0 0
\(340\) 6578.56 1.04933
\(341\) 11152.7 1.77112
\(342\) 0 0
\(343\) 0 0
\(344\) 2714.77i 0.425496i
\(345\) 0 0
\(346\) 3321.70i 0.516115i
\(347\) 1824.41i 0.282246i 0.989992 + 0.141123i \(0.0450713\pi\)
−0.989992 + 0.141123i \(0.954929\pi\)
\(348\) 0 0
\(349\) 1537.52i 0.235822i 0.993024 + 0.117911i \(0.0376197\pi\)
−0.993024 + 0.117911i \(0.962380\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6772.78 1.02554
\(353\) −5927.33 −0.893711 −0.446855 0.894606i \(-0.647456\pi\)
−0.446855 + 0.894606i \(0.647456\pi\)
\(354\) 0 0
\(355\) − 1548.32i − 0.231482i
\(356\) −4989.96 −0.742885
\(357\) 0 0
\(358\) −814.541 −0.120251
\(359\) 4839.49i 0.711472i 0.934587 + 0.355736i \(0.115770\pi\)
−0.934587 + 0.355736i \(0.884230\pi\)
\(360\) 0 0
\(361\) 6859.00 1.00000
\(362\) 2184.08 0.317106
\(363\) 0 0
\(364\) 0 0
\(365\) − 8476.72i − 1.21559i
\(366\) 0 0
\(367\) − 11509.1i − 1.63698i −0.574519 0.818491i \(-0.694811\pi\)
0.574519 0.818491i \(-0.305189\pi\)
\(368\) 4540.81i 0.643223i
\(369\) 0 0
\(370\) − 524.784i − 0.0737357i
\(371\) 0 0
\(372\) 0 0
\(373\) 187.497 0.0260275 0.0130137 0.999915i \(-0.495857\pi\)
0.0130137 + 0.999915i \(0.495857\pi\)
\(374\) 3164.32 0.437495
\(375\) 0 0
\(376\) − 7309.88i − 1.00260i
\(377\) −5640.81 −0.770601
\(378\) 0 0
\(379\) 3515.82 0.476506 0.238253 0.971203i \(-0.423425\pi\)
0.238253 + 0.971203i \(0.423425\pi\)
\(380\) − 3.92249i 0 0.000529525i
\(381\) 0 0
\(382\) −3415.24 −0.457432
\(383\) 2029.38 0.270748 0.135374 0.990795i \(-0.456776\pi\)
0.135374 + 0.990795i \(0.456776\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2814.06i 0.371067i
\(387\) 0 0
\(388\) − 4879.19i − 0.638411i
\(389\) 6439.94i 0.839378i 0.907668 + 0.419689i \(0.137861\pi\)
−0.907668 + 0.419689i \(0.862139\pi\)
\(390\) 0 0
\(391\) 7274.45i 0.940882i
\(392\) 0 0
\(393\) 0 0
\(394\) −1508.70 −0.192911
\(395\) 13993.1 1.78245
\(396\) 0 0
\(397\) − 9664.61i − 1.22180i −0.791709 0.610898i \(-0.790809\pi\)
0.791709 0.610898i \(-0.209191\pi\)
\(398\) −9.37106 −0.00118022
\(399\) 0 0
\(400\) −364.157 −0.0455197
\(401\) 11762.8i 1.46485i 0.680846 + 0.732426i \(0.261612\pi\)
−0.680846 + 0.732426i \(0.738388\pi\)
\(402\) 0 0
\(403\) 6921.99 0.855606
\(404\) 8653.36 1.06565
\(405\) 0 0
\(406\) 0 0
\(407\) 3347.42i 0.407679i
\(408\) 0 0
\(409\) 5272.03i 0.637371i 0.947860 + 0.318686i \(0.103241\pi\)
−0.947860 + 0.318686i \(0.896759\pi\)
\(410\) 3345.55i 0.402987i
\(411\) 0 0
\(412\) 196.538i 0.0235017i
\(413\) 0 0
\(414\) 0 0
\(415\) 2552.34 0.301902
\(416\) 4203.58 0.495426
\(417\) 0 0
\(418\) − 1.88674i 0 0.000220774i
\(419\) 5103.18 0.595003 0.297502 0.954721i \(-0.403847\pi\)
0.297502 + 0.954721i \(0.403847\pi\)
\(420\) 0 0
\(421\) −8395.31 −0.971882 −0.485941 0.873992i \(-0.661523\pi\)
−0.485941 + 0.873992i \(0.661523\pi\)
\(422\) 2183.90i 0.251921i
\(423\) 0 0
\(424\) 3071.70 0.351827
\(425\) −583.386 −0.0665844
\(426\) 0 0
\(427\) 0 0
\(428\) 10913.6i 1.23255i
\(429\) 0 0
\(430\) 1908.79i 0.214070i
\(431\) 2088.48i 0.233408i 0.993167 + 0.116704i \(0.0372329\pi\)
−0.993167 + 0.116704i \(0.962767\pi\)
\(432\) 0 0
\(433\) − 11495.3i − 1.27582i −0.770111 0.637910i \(-0.779799\pi\)
0.770111 0.637910i \(-0.220201\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1004.78 −0.110367
\(437\) 4.33742 0.000474798 0
\(438\) 0 0
\(439\) 4252.70i 0.462347i 0.972913 + 0.231173i \(0.0742565\pi\)
−0.972913 + 0.231173i \(0.925744\pi\)
\(440\) 6509.91 0.705336
\(441\) 0 0
\(442\) 1963.96 0.211349
\(443\) 3304.79i 0.354437i 0.984172 + 0.177218i \(0.0567098\pi\)
−0.984172 + 0.177218i \(0.943290\pi\)
\(444\) 0 0
\(445\) −7281.56 −0.775683
\(446\) −822.280 −0.0873007
\(447\) 0 0
\(448\) 0 0
\(449\) 6952.63i 0.730768i 0.930857 + 0.365384i \(0.119062\pi\)
−0.930857 + 0.365384i \(0.880938\pi\)
\(450\) 0 0
\(451\) − 21340.1i − 2.22808i
\(452\) − 2145.95i − 0.223312i
\(453\) 0 0
\(454\) 374.933i 0.0387588i
\(455\) 0 0
\(456\) 0 0
\(457\) 9741.14 0.997093 0.498546 0.866863i \(-0.333867\pi\)
0.498546 + 0.866863i \(0.333867\pi\)
\(458\) −848.987 −0.0866169
\(459\) 0 0
\(460\) 7210.92i 0.730893i
\(461\) −5563.15 −0.562043 −0.281021 0.959702i \(-0.590673\pi\)
−0.281021 + 0.959702i \(0.590673\pi\)
\(462\) 0 0
\(463\) 4114.02 0.412948 0.206474 0.978452i \(-0.433801\pi\)
0.206474 + 0.978452i \(0.433801\pi\)
\(464\) 8911.50i 0.891608i
\(465\) 0 0
\(466\) −2573.01 −0.255778
\(467\) −6061.59 −0.600636 −0.300318 0.953839i \(-0.597093\pi\)
−0.300318 + 0.953839i \(0.597093\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 5139.66i − 0.504415i
\(471\) 0 0
\(472\) − 4059.47i − 0.395873i
\(473\) − 12175.5i − 1.18357i
\(474\) 0 0
\(475\) 0.347846i 0 3.36005e-5i
\(476\) 0 0
\(477\) 0 0
\(478\) −1658.21 −0.158671
\(479\) 8246.66 0.786638 0.393319 0.919402i \(-0.371327\pi\)
0.393319 + 0.919402i \(0.371327\pi\)
\(480\) 0 0
\(481\) 2077.60i 0.196945i
\(482\) −4457.88 −0.421268
\(483\) 0 0
\(484\) −10106.5 −0.949145
\(485\) − 7119.93i − 0.666597i
\(486\) 0 0
\(487\) 11744.2 1.09277 0.546385 0.837534i \(-0.316004\pi\)
0.546385 + 0.837534i \(0.316004\pi\)
\(488\) −8997.18 −0.834597
\(489\) 0 0
\(490\) 0 0
\(491\) 6008.34i 0.552246i 0.961122 + 0.276123i \(0.0890497\pi\)
−0.961122 + 0.276123i \(0.910950\pi\)
\(492\) 0 0
\(493\) 14276.4i 1.30421i
\(494\) − 1.17102i 0 0.000106653i
\(495\) 0 0
\(496\) − 10935.5i − 0.989961i
\(497\) 0 0
\(498\) 0 0
\(499\) −13649.9 −1.22455 −0.612276 0.790644i \(-0.709746\pi\)
−0.612276 + 0.790644i \(0.709746\pi\)
\(500\) −10672.5 −0.954574
\(501\) 0 0
\(502\) 3662.39i 0.325619i
\(503\) 4862.69 0.431047 0.215524 0.976499i \(-0.430854\pi\)
0.215524 + 0.976499i \(0.430854\pi\)
\(504\) 0 0
\(505\) 12627.4 1.11269
\(506\) 3468.49i 0.304730i
\(507\) 0 0
\(508\) −16885.1 −1.47472
\(509\) 17722.7 1.54331 0.771653 0.636043i \(-0.219430\pi\)
0.771653 + 0.636043i \(0.219430\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 11345.1i − 0.979271i
\(513\) 0 0
\(514\) 2394.04i 0.205441i
\(515\) 286.796i 0.0245393i
\(516\) 0 0
\(517\) 32784.1i 2.78887i
\(518\) 0 0
\(519\) 0 0
\(520\) 4040.43 0.340739
\(521\) −13755.9 −1.15673 −0.578366 0.815777i \(-0.696309\pi\)
−0.578366 + 0.815777i \(0.696309\pi\)
\(522\) 0 0
\(523\) 1311.94i 0.109689i 0.998495 + 0.0548443i \(0.0174663\pi\)
−0.998495 + 0.0548443i \(0.982534\pi\)
\(524\) 2898.93 0.241680
\(525\) 0 0
\(526\) 560.582 0.0464687
\(527\) − 17518.9i − 1.44808i
\(528\) 0 0
\(529\) 4193.30 0.344645
\(530\) 2159.75 0.177006
\(531\) 0 0
\(532\) 0 0
\(533\) − 13244.9i − 1.07636i
\(534\) 0 0
\(535\) 15925.6i 1.28696i
\(536\) 2266.54i 0.182649i
\(537\) 0 0
\(538\) 973.114i 0.0779812i
\(539\) 0 0
\(540\) 0 0
\(541\) 1195.91 0.0950389 0.0475195 0.998870i \(-0.484868\pi\)
0.0475195 + 0.998870i \(0.484868\pi\)
\(542\) 62.5854 0.00495991
\(543\) 0 0
\(544\) − 10638.9i − 0.838488i
\(545\) −1466.22 −0.115240
\(546\) 0 0
\(547\) −6178.59 −0.482957 −0.241478 0.970406i \(-0.577632\pi\)
−0.241478 + 0.970406i \(0.577632\pi\)
\(548\) − 10925.7i − 0.851681i
\(549\) 0 0
\(550\) −278.161 −0.0215651
\(551\) 8.51233 0.000658144 0
\(552\) 0 0
\(553\) 0 0
\(554\) 3476.65i 0.266622i
\(555\) 0 0
\(556\) 4647.26i 0.354474i
\(557\) 3356.20i 0.255309i 0.991819 + 0.127654i \(0.0407448\pi\)
−0.991819 + 0.127654i \(0.959255\pi\)
\(558\) 0 0
\(559\) − 7556.82i − 0.571770i
\(560\) 0 0
\(561\) 0 0
\(562\) 134.283 0.0100790
\(563\) −3585.28 −0.268386 −0.134193 0.990955i \(-0.542844\pi\)
−0.134193 + 0.990955i \(0.542844\pi\)
\(564\) 0 0
\(565\) − 3131.46i − 0.233171i
\(566\) 3032.26 0.225187
\(567\) 0 0
\(568\) −1649.32 −0.121838
\(569\) − 18285.7i − 1.34723i −0.739081 0.673617i \(-0.764740\pi\)
0.739081 0.673617i \(-0.235260\pi\)
\(570\) 0 0
\(571\) −16363.0 −1.19925 −0.599624 0.800282i \(-0.704683\pi\)
−0.599624 + 0.800282i \(0.704683\pi\)
\(572\) −12418.0 −0.907734
\(573\) 0 0
\(574\) 0 0
\(575\) − 639.463i − 0.0463782i
\(576\) 0 0
\(577\) 7711.39i 0.556376i 0.960527 + 0.278188i \(0.0897339\pi\)
−0.960527 + 0.278188i \(0.910266\pi\)
\(578\) − 1290.88i − 0.0928953i
\(579\) 0 0
\(580\) 14151.7i 1.01313i
\(581\) 0 0
\(582\) 0 0
\(583\) −13776.3 −0.978653
\(584\) −9029.68 −0.639813
\(585\) 0 0
\(586\) 2538.67i 0.178962i
\(587\) −4182.21 −0.294069 −0.147034 0.989131i \(-0.546973\pi\)
−0.147034 + 0.989131i \(0.546973\pi\)
\(588\) 0 0
\(589\) −10.4457 −0.000730744 0
\(590\) − 2854.26i − 0.199166i
\(591\) 0 0
\(592\) 3282.25 0.227871
\(593\) −16189.3 −1.12110 −0.560552 0.828119i \(-0.689411\pi\)
−0.560552 + 0.828119i \(0.689411\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 11916.3i − 0.818977i
\(597\) 0 0
\(598\) 2152.75i 0.147211i
\(599\) 12777.3i 0.871566i 0.900052 + 0.435783i \(0.143529\pi\)
−0.900052 + 0.435783i \(0.856471\pi\)
\(600\) 0 0
\(601\) − 24022.7i − 1.63046i −0.579137 0.815231i \(-0.696610\pi\)
0.579137 0.815231i \(-0.303390\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3008.16 0.202650
\(605\) −14747.8 −0.991050
\(606\) 0 0
\(607\) 9666.83i 0.646400i 0.946331 + 0.323200i \(0.104759\pi\)
−0.946331 + 0.323200i \(0.895241\pi\)
\(608\) −6.34346 −0.000423127 0
\(609\) 0 0
\(610\) −6326.02 −0.419890
\(611\) 20347.7i 1.34727i
\(612\) 0 0
\(613\) 14385.5 0.947836 0.473918 0.880569i \(-0.342839\pi\)
0.473918 + 0.880569i \(0.342839\pi\)
\(614\) −1508.92 −0.0991778
\(615\) 0 0
\(616\) 0 0
\(617\) − 7712.69i − 0.503244i −0.967826 0.251622i \(-0.919036\pi\)
0.967826 0.251622i \(-0.0809639\pi\)
\(618\) 0 0
\(619\) − 14317.0i − 0.929645i −0.885404 0.464822i \(-0.846118\pi\)
0.885404 0.464822i \(-0.153882\pi\)
\(620\) − 17365.9i − 1.12489i
\(621\) 0 0
\(622\) − 4377.04i − 0.282160i
\(623\) 0 0
\(624\) 0 0
\(625\) −14678.6 −0.939428
\(626\) 1834.58 0.117132
\(627\) 0 0
\(628\) 17943.5i 1.14016i
\(629\) 5258.21 0.333320
\(630\) 0 0
\(631\) 4971.96 0.313678 0.156839 0.987624i \(-0.449870\pi\)
0.156839 + 0.987624i \(0.449870\pi\)
\(632\) − 14905.9i − 0.938172i
\(633\) 0 0
\(634\) 4586.48 0.287307
\(635\) −24639.5 −1.53982
\(636\) 0 0
\(637\) 0 0
\(638\) 6807.03i 0.422403i
\(639\) 0 0
\(640\) − 13853.5i − 0.855637i
\(641\) − 29423.8i − 1.81306i −0.422145 0.906529i \(-0.638723\pi\)
0.422145 0.906529i \(-0.361277\pi\)
\(642\) 0 0
\(643\) − 31273.9i − 1.91807i −0.283283 0.959036i \(-0.591424\pi\)
0.283283 0.959036i \(-0.408576\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.96374 −0.000180506 0
\(647\) 8011.64 0.486816 0.243408 0.969924i \(-0.421735\pi\)
0.243408 + 0.969924i \(0.421735\pi\)
\(648\) 0 0
\(649\) 18206.3i 1.10117i
\(650\) −172.643 −0.0104179
\(651\) 0 0
\(652\) −7032.62 −0.422421
\(653\) − 15440.6i − 0.925327i −0.886534 0.462664i \(-0.846894\pi\)
0.886534 0.462664i \(-0.153106\pi\)
\(654\) 0 0
\(655\) 4230.24 0.252350
\(656\) −20924.6 −1.24538
\(657\) 0 0
\(658\) 0 0
\(659\) 31288.9i 1.84953i 0.380537 + 0.924766i \(0.375739\pi\)
−0.380537 + 0.924766i \(0.624261\pi\)
\(660\) 0 0
\(661\) 30326.1i 1.78449i 0.451550 + 0.892246i \(0.350871\pi\)
−0.451550 + 0.892246i \(0.649129\pi\)
\(662\) − 5690.01i − 0.334061i
\(663\) 0 0
\(664\) − 2718.84i − 0.158903i
\(665\) 0 0
\(666\) 0 0
\(667\) −15648.7 −0.908424
\(668\) 9456.33 0.547719
\(669\) 0 0
\(670\) 1593.63i 0.0918915i
\(671\) 40351.5 2.32154
\(672\) 0 0
\(673\) 12067.9 0.691207 0.345604 0.938381i \(-0.387674\pi\)
0.345604 + 0.938381i \(0.387674\pi\)
\(674\) 2893.46i 0.165359i
\(675\) 0 0
\(676\) 8636.21 0.491364
\(677\) 6544.82 0.371548 0.185774 0.982593i \(-0.440521\pi\)
0.185774 + 0.982593i \(0.440521\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 10225.9i − 0.576686i
\(681\) 0 0
\(682\) − 8353.10i − 0.468998i
\(683\) 31485.3i 1.76391i 0.471332 + 0.881956i \(0.343773\pi\)
−0.471332 + 0.881956i \(0.656227\pi\)
\(684\) 0 0
\(685\) − 15943.2i − 0.889282i
\(686\) 0 0
\(687\) 0 0
\(688\) −11938.5 −0.661555
\(689\) −8550.35 −0.472776
\(690\) 0 0
\(691\) − 10035.3i − 0.552476i −0.961089 0.276238i \(-0.910912\pi\)
0.961089 0.276238i \(-0.0890879\pi\)
\(692\) −32992.0 −1.81238
\(693\) 0 0
\(694\) 1366.44 0.0747397
\(695\) 6781.48i 0.370124i
\(696\) 0 0
\(697\) −33521.6 −1.82169
\(698\) 1151.57 0.0624464
\(699\) 0 0
\(700\) 0 0
\(701\) 768.196i 0.0413900i 0.999786 + 0.0206950i \(0.00658789\pi\)
−0.999786 + 0.0206950i \(0.993412\pi\)
\(702\) 0 0
\(703\) − 3.13523i 0 0.000168204i
\(704\) 16025.1i 0.857908i
\(705\) 0 0
\(706\) 4439.43i 0.236658i
\(707\) 0 0
\(708\) 0 0
\(709\) 13968.6 0.739918 0.369959 0.929048i \(-0.379372\pi\)
0.369959 + 0.929048i \(0.379372\pi\)
\(710\) −1159.65 −0.0612973
\(711\) 0 0
\(712\) 7756.56i 0.408271i
\(713\) 19202.9 1.00863
\(714\) 0 0
\(715\) −18120.9 −0.947810
\(716\) − 8090.23i − 0.422271i
\(717\) 0 0
\(718\) 3624.67 0.188400
\(719\) −11969.6 −0.620847 −0.310424 0.950598i \(-0.600471\pi\)
−0.310424 + 0.950598i \(0.600471\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 5137.23i − 0.264803i
\(723\) 0 0
\(724\) 21692.8i 1.11354i
\(725\) − 1254.97i − 0.0642874i
\(726\) 0 0
\(727\) 12223.3i 0.623575i 0.950152 + 0.311787i \(0.100928\pi\)
−0.950152 + 0.311787i \(0.899072\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −6348.87 −0.321893
\(731\) −19125.6 −0.967696
\(732\) 0 0
\(733\) − 23785.0i − 1.19853i −0.800552 0.599264i \(-0.795460\pi\)
0.800552 0.599264i \(-0.204540\pi\)
\(734\) −8620.09 −0.433478
\(735\) 0 0
\(736\) 11661.5 0.584034
\(737\) − 10165.2i − 0.508061i
\(738\) 0 0
\(739\) 11478.1 0.571351 0.285675 0.958326i \(-0.407782\pi\)
0.285675 + 0.958326i \(0.407782\pi\)
\(740\) 5212.29 0.258929
\(741\) 0 0
\(742\) 0 0
\(743\) − 18604.0i − 0.918593i −0.888283 0.459297i \(-0.848101\pi\)
0.888283 0.459297i \(-0.151899\pi\)
\(744\) 0 0
\(745\) − 17388.8i − 0.855134i
\(746\) − 140.431i − 0.00689217i
\(747\) 0 0
\(748\) 31428.9i 1.53630i
\(749\) 0 0
\(750\) 0 0
\(751\) 31012.5 1.50687 0.753436 0.657522i \(-0.228395\pi\)
0.753436 + 0.657522i \(0.228395\pi\)
\(752\) 32145.9 1.55883
\(753\) 0 0
\(754\) 4224.84i 0.204058i
\(755\) 4389.64 0.211597
\(756\) 0 0
\(757\) −19065.9 −0.915407 −0.457703 0.889105i \(-0.651328\pi\)
−0.457703 + 0.889105i \(0.651328\pi\)
\(758\) − 2633.27i − 0.126180i
\(759\) 0 0
\(760\) −6.09725 −0.000291014 0
\(761\) 14679.2 0.699236 0.349618 0.936892i \(-0.386311\pi\)
0.349618 + 0.936892i \(0.386311\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 33921.0i − 1.60631i
\(765\) 0 0
\(766\) − 1519.96i − 0.0716949i
\(767\) 11299.9i 0.531963i
\(768\) 0 0
\(769\) 29972.5i 1.40551i 0.711434 + 0.702753i \(0.248046\pi\)
−0.711434 + 0.702753i \(0.751954\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −27949.9 −1.30303
\(773\) −22687.8 −1.05566 −0.527828 0.849351i \(-0.676993\pi\)
−0.527828 + 0.849351i \(0.676993\pi\)
\(774\) 0 0
\(775\) 1540.01i 0.0713789i
\(776\) −7584.38 −0.350855
\(777\) 0 0
\(778\) 4823.37 0.222270
\(779\) 19.9874i 0 0.000919284i
\(780\) 0 0
\(781\) 7397.04 0.338908
\(782\) 5448.40 0.249149
\(783\) 0 0
\(784\) 0 0
\(785\) 26183.9i 1.19050i
\(786\) 0 0
\(787\) 20937.9i 0.948357i 0.880429 + 0.474178i \(0.157255\pi\)
−0.880429 + 0.474178i \(0.842745\pi\)
\(788\) − 14984.7i − 0.677423i
\(789\) 0 0
\(790\) − 10480.5i − 0.472000i
\(791\) 0 0
\(792\) 0 0
\(793\) 25044.5 1.12151
\(794\) −7238.57 −0.323536
\(795\) 0 0
\(796\) − 93.0758i − 0.00414445i
\(797\) −27393.4 −1.21747 −0.608735 0.793373i \(-0.708323\pi\)
−0.608735 + 0.793373i \(0.708323\pi\)
\(798\) 0 0
\(799\) 51498.2 2.28019
\(800\) 935.212i 0.0413309i
\(801\) 0 0
\(802\) 8810.06 0.387898
\(803\) 40497.3 1.77972
\(804\) 0 0
\(805\) 0 0
\(806\) − 5184.41i − 0.226567i
\(807\) 0 0
\(808\) − 13451.1i − 0.585652i
\(809\) − 26805.4i − 1.16493i −0.812856 0.582465i \(-0.802088\pi\)
0.812856 0.582465i \(-0.197912\pi\)
\(810\) 0 0
\(811\) − 17139.6i − 0.742113i −0.928610 0.371056i \(-0.878996\pi\)
0.928610 0.371056i \(-0.121004\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2507.14 0.107955
\(815\) −10262.3 −0.441071
\(816\) 0 0
\(817\) 11.4037i 0 0.000488329i
\(818\) 3948.63 0.168778
\(819\) 0 0
\(820\) −33228.8 −1.41512
\(821\) 8629.28i 0.366826i 0.983036 + 0.183413i \(0.0587145\pi\)
−0.983036 + 0.183413i \(0.941285\pi\)
\(822\) 0 0
\(823\) −5284.96 −0.223842 −0.111921 0.993717i \(-0.535700\pi\)
−0.111921 + 0.993717i \(0.535700\pi\)
\(824\) 305.505 0.0129160
\(825\) 0 0
\(826\) 0 0
\(827\) − 1845.35i − 0.0775926i −0.999247 0.0387963i \(-0.987648\pi\)
0.999247 0.0387963i \(-0.0123523\pi\)
\(828\) 0 0
\(829\) − 19493.2i − 0.816677i −0.912831 0.408339i \(-0.866108\pi\)
0.912831 0.408339i \(-0.133892\pi\)
\(830\) − 1911.64i − 0.0799447i
\(831\) 0 0
\(832\) 9946.08i 0.414445i
\(833\) 0 0
\(834\) 0 0
\(835\) 13799.1 0.571901
\(836\) 18.7396 0.000775265 0
\(837\) 0 0
\(838\) − 3822.16i − 0.157559i
\(839\) 27816.7 1.14462 0.572312 0.820036i \(-0.306047\pi\)
0.572312 + 0.820036i \(0.306047\pi\)
\(840\) 0 0
\(841\) −6322.03 −0.259217
\(842\) 6287.90i 0.257358i
\(843\) 0 0
\(844\) −21691.0 −0.884640
\(845\) 12602.3 0.513057
\(846\) 0 0
\(847\) 0 0
\(848\) 13508.1i 0.547015i
\(849\) 0 0
\(850\) 436.942i 0.0176318i
\(851\) 5763.65i 0.232168i
\(852\) 0 0
\(853\) 9210.41i 0.369705i 0.982766 + 0.184852i \(0.0591807\pi\)
−0.982766 + 0.184852i \(0.940819\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16964.5 0.677377
\(857\) 34500.9 1.37518 0.687589 0.726100i \(-0.258669\pi\)
0.687589 + 0.726100i \(0.258669\pi\)
\(858\) 0 0
\(859\) − 43698.0i − 1.73569i −0.496836 0.867845i \(-0.665505\pi\)
0.496836 0.867845i \(-0.334495\pi\)
\(860\) −18958.6 −0.751723
\(861\) 0 0
\(862\) 1564.23 0.0618072
\(863\) 28333.6i 1.11760i 0.829303 + 0.558800i \(0.188738\pi\)
−0.829303 + 0.558800i \(0.811262\pi\)
\(864\) 0 0
\(865\) −48143.4 −1.89240
\(866\) −8609.73 −0.337841
\(867\) 0 0
\(868\) 0 0
\(869\) 66851.6i 2.60965i
\(870\) 0 0
\(871\) − 6309.13i − 0.245438i
\(872\) 1561.86i 0.0606552i
\(873\) 0 0
\(874\) − 3.24863i 0 0.000125728i
\(875\) 0 0
\(876\) 0 0
\(877\) −45125.8 −1.73750 −0.868751 0.495249i \(-0.835077\pi\)
−0.868751 + 0.495249i \(0.835077\pi\)
\(878\) 3185.17 0.122431
\(879\) 0 0
\(880\) 28627.9i 1.09664i
\(881\) −4244.09 −0.162301 −0.0811504 0.996702i \(-0.525859\pi\)
−0.0811504 + 0.996702i \(0.525859\pi\)
\(882\) 0 0
\(883\) −21.9077 −0.000834939 0 −0.000417470 1.00000i \(-0.500133\pi\)
−0.000417470 1.00000i \(0.500133\pi\)
\(884\) 19506.6i 0.742169i
\(885\) 0 0
\(886\) 2475.21 0.0938560
\(887\) −7501.63 −0.283969 −0.141984 0.989869i \(-0.545348\pi\)
−0.141984 + 0.989869i \(0.545348\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 5453.72i 0.205404i
\(891\) 0 0
\(892\) − 8167.10i − 0.306564i
\(893\) − 30.7060i − 0.00115066i
\(894\) 0 0
\(895\) − 11805.6i − 0.440914i
\(896\) 0 0
\(897\) 0 0
\(898\) 5207.36 0.193510
\(899\) 37686.4 1.39812
\(900\) 0 0
\(901\) 21640.1i 0.800153i
\(902\) −15983.2 −0.590004
\(903\) 0 0
\(904\) −3335.73 −0.122727
\(905\) 31655.1i 1.16271i
\(906\) 0 0
\(907\) 41137.3 1.50600 0.753001 0.658020i \(-0.228606\pi\)
0.753001 + 0.658020i \(0.228606\pi\)
\(908\) −3723.94 −0.136105
\(909\) 0 0
\(910\) 0 0
\(911\) − 14080.6i − 0.512086i −0.966665 0.256043i \(-0.917581\pi\)
0.966665 0.256043i \(-0.0824189\pi\)
\(912\) 0 0
\(913\) 12193.7i 0.442008i
\(914\) − 7295.89i − 0.264034i
\(915\) 0 0
\(916\) − 8432.35i − 0.304162i
\(917\) 0 0
\(918\) 0 0
\(919\) 12623.2 0.453101 0.226550 0.973999i \(-0.427255\pi\)
0.226550 + 0.973999i \(0.427255\pi\)
\(920\) 11208.9 0.401681
\(921\) 0 0
\(922\) 4166.67i 0.148831i
\(923\) 4591.03 0.163722
\(924\) 0 0
\(925\) −462.225 −0.0164301
\(926\) − 3081.31i − 0.109350i
\(927\) 0 0
\(928\) 22886.1 0.809562
\(929\) 33473.7 1.18217 0.591085 0.806609i \(-0.298700\pi\)
0.591085 + 0.806609i \(0.298700\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 25555.8i − 0.898186i
\(933\) 0 0
\(934\) 4539.99i 0.159050i
\(935\) 45862.4i 1.60413i
\(936\) 0 0
\(937\) 25652.3i 0.894371i 0.894441 + 0.447185i \(0.147574\pi\)
−0.894441 + 0.447185i \(0.852426\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 51048.4 1.77129
\(941\) −2602.96 −0.0901744 −0.0450872 0.998983i \(-0.514357\pi\)
−0.0450872 + 0.998983i \(0.514357\pi\)
\(942\) 0 0
\(943\) − 36743.8i − 1.26887i
\(944\) 17851.9 0.615497
\(945\) 0 0
\(946\) −9119.17 −0.313414
\(947\) 37616.8i 1.29079i 0.763847 + 0.645397i \(0.223308\pi\)
−0.763847 + 0.645397i \(0.776692\pi\)
\(948\) 0 0
\(949\) 25134.9 0.859763
\(950\) 0.260528 8.89754e−6 0
\(951\) 0 0
\(952\) 0 0
\(953\) 23444.5i 0.796896i 0.917191 + 0.398448i \(0.130451\pi\)
−0.917191 + 0.398448i \(0.869549\pi\)
\(954\) 0 0
\(955\) − 49499.1i − 1.67723i
\(956\) − 16469.8i − 0.557187i
\(957\) 0 0
\(958\) − 6176.56i − 0.208304i
\(959\) 0 0
\(960\) 0 0
\(961\) −16455.0 −0.552347
\(962\) 1556.07 0.0521516
\(963\) 0 0
\(964\) − 44276.8i − 1.47932i
\(965\) −40785.7 −1.36056
\(966\) 0 0
\(967\) 4312.24 0.143405 0.0717023 0.997426i \(-0.477157\pi\)
0.0717023 + 0.997426i \(0.477157\pi\)
\(968\) 15709.9i 0.521627i
\(969\) 0 0
\(970\) −5332.66 −0.176517
\(971\) 47052.7 1.55509 0.777545 0.628827i \(-0.216465\pi\)
0.777545 + 0.628827i \(0.216465\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 8796.11i − 0.289369i
\(975\) 0 0
\(976\) − 39565.9i − 1.29762i
\(977\) 20463.0i 0.670082i 0.942204 + 0.335041i \(0.108750\pi\)
−0.942204 + 0.335041i \(0.891250\pi\)
\(978\) 0 0
\(979\) − 34787.4i − 1.13566i
\(980\) 0 0
\(981\) 0 0
\(982\) 4500.11 0.146237
\(983\) 44629.0 1.44806 0.724031 0.689767i \(-0.242287\pi\)
0.724031 + 0.689767i \(0.242287\pi\)
\(984\) 0 0
\(985\) − 21866.4i − 0.707331i
\(986\) 10692.7 0.345359
\(987\) 0 0
\(988\) 11.6309 0.000374521 0
\(989\) − 20964.0i − 0.674032i
\(990\) 0 0
\(991\) 43131.0 1.38254 0.691272 0.722594i \(-0.257051\pi\)
0.691272 + 0.722594i \(0.257051\pi\)
\(992\) −28084.2 −0.898864
\(993\) 0 0
\(994\) 0 0
\(995\) − 135.820i − 0.00432743i
\(996\) 0 0
\(997\) − 3622.25i − 0.115063i −0.998344 0.0575315i \(-0.981677\pi\)
0.998344 0.0575315i \(-0.0183230\pi\)
\(998\) 10223.4i 0.324266i
\(999\) 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.440.8 16
3.2 odd 2 inner 441.4.c.a.440.9 16
7.2 even 3 63.4.p.a.17.4 16
7.3 odd 6 63.4.p.a.26.5 yes 16
7.4 even 3 441.4.p.c.215.5 16
7.5 odd 6 441.4.p.c.80.4 16
7.6 odd 2 inner 441.4.c.a.440.7 16
21.2 odd 6 63.4.p.a.17.5 yes 16
21.5 even 6 441.4.p.c.80.5 16
21.11 odd 6 441.4.p.c.215.4 16
21.17 even 6 63.4.p.a.26.4 yes 16
21.20 even 2 inner 441.4.c.a.440.10 16
28.3 even 6 1008.4.bt.a.593.6 16
28.23 odd 6 1008.4.bt.a.17.3 16
84.23 even 6 1008.4.bt.a.17.6 16
84.59 odd 6 1008.4.bt.a.593.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.p.a.17.4 16 7.2 even 3
63.4.p.a.17.5 yes 16 21.2 odd 6
63.4.p.a.26.4 yes 16 21.17 even 6
63.4.p.a.26.5 yes 16 7.3 odd 6
441.4.c.a.440.7 16 7.6 odd 2 inner
441.4.c.a.440.8 16 1.1 even 1 trivial
441.4.c.a.440.9 16 3.2 odd 2 inner
441.4.c.a.440.10 16 21.20 even 2 inner
441.4.p.c.80.4 16 7.5 odd 6
441.4.p.c.80.5 16 21.5 even 6
441.4.p.c.215.4 16 21.11 odd 6
441.4.p.c.215.5 16 7.4 even 3
1008.4.bt.a.17.3 16 28.23 odd 6
1008.4.bt.a.17.6 16 84.23 even 6
1008.4.bt.a.593.3 16 84.59 odd 6
1008.4.bt.a.593.6 16 28.3 even 6