Properties

Label 784.4.f.g.783.4
Level $784$
Weight $4$
Character 784.783
Analytic conductor $46.257$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12258833328.1
Defining polynomial: \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 783.4
Root \(2.68858 - 4.65676i\) of defining polynomial
Character \(\chi\) \(=\) 784.783
Dual form 784.4.f.g.783.3

$q$-expansion

\(f(q)\) \(=\) \(q-4.76834 q^{3} +16.8950i q^{5} -4.26295 q^{9} +O(q^{10})\) \(q-4.76834 q^{3} +16.8950i q^{5} -4.26295 q^{9} +40.7424i q^{11} -56.7321i q^{13} -80.5609i q^{15} -122.834i q^{17} +75.4946 q^{19} -106.165i q^{23} -160.440 q^{25} +149.072 q^{27} -146.703 q^{29} +42.5912 q^{31} -194.274i q^{33} +80.9142 q^{37} +270.518i q^{39} +53.8052i q^{41} +341.166i q^{43} -72.0225i q^{45} +4.12785 q^{47} +585.714i q^{51} +279.697 q^{53} -688.342 q^{55} -359.984 q^{57} +174.831 q^{59} -467.087i q^{61} +958.488 q^{65} -753.251i q^{67} +506.229i q^{69} +669.583i q^{71} -835.774i q^{73} +765.033 q^{75} +1106.21i q^{79} -595.728 q^{81} -552.905 q^{83} +2075.28 q^{85} +699.530 q^{87} +122.469i q^{89} -203.089 q^{93} +1275.48i q^{95} +291.203i q^{97} -173.683i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 14q^{3} + 156q^{9} + O(q^{10}) \) \( 6q - 14q^{3} + 156q^{9} + 286q^{19} - 612q^{25} - 362q^{27} - 348q^{29} + 410q^{31} + 498q^{37} + 150q^{47} + 1290q^{53} - 918q^{55} - 6q^{57} + 642q^{59} + 1224q^{65} + 8276q^{75} - 450q^{81} - 24q^{83} + 3786q^{85} + 7284q^{87} + 5982q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 0
\(3\) −4.76834 −0.917667 −0.458834 0.888522i \(-0.651733\pi\)
−0.458834 + 0.888522i \(0.651733\pi\)
\(4\) 0 0
\(5\) 16.8950i 1.51113i 0.655072 + 0.755566i \(0.272638\pi\)
−0.655072 + 0.755566i \(0.727362\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −4.26295 −0.157887
\(10\) 0 0
\(11\) 40.7424i 1.11676i 0.829587 + 0.558378i \(0.188576\pi\)
−0.829587 + 0.558378i \(0.811424\pi\)
\(12\) 0 0
\(13\) − 56.7321i − 1.21036i −0.796089 0.605179i \(-0.793101\pi\)
0.796089 0.605179i \(-0.206899\pi\)
\(14\) 0 0
\(15\) − 80.5609i − 1.38672i
\(16\) 0 0
\(17\) − 122.834i − 1.75245i −0.481903 0.876225i \(-0.660054\pi\)
0.481903 0.876225i \(-0.339946\pi\)
\(18\) 0 0
\(19\) 75.4946 0.911561 0.455780 0.890092i \(-0.349360\pi\)
0.455780 + 0.890092i \(0.349360\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 106.165i − 0.962473i −0.876591 0.481236i \(-0.840188\pi\)
0.876591 0.481236i \(-0.159812\pi\)
\(24\) 0 0
\(25\) −160.440 −1.28352
\(26\) 0 0
\(27\) 149.072 1.06255
\(28\) 0 0
\(29\) −146.703 −0.939382 −0.469691 0.882831i \(-0.655635\pi\)
−0.469691 + 0.882831i \(0.655635\pi\)
\(30\) 0 0
\(31\) 42.5912 0.246761 0.123381 0.992359i \(-0.460626\pi\)
0.123381 + 0.992359i \(0.460626\pi\)
\(32\) 0 0
\(33\) − 194.274i − 1.02481i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 80.9142 0.359519 0.179760 0.983711i \(-0.442468\pi\)
0.179760 + 0.983711i \(0.442468\pi\)
\(38\) 0 0
\(39\) 270.518i 1.11071i
\(40\) 0 0
\(41\) 53.8052i 0.204950i 0.994736 + 0.102475i \(0.0326762\pi\)
−0.994736 + 0.102475i \(0.967324\pi\)
\(42\) 0 0
\(43\) 341.166i 1.20994i 0.796249 + 0.604970i \(0.206815\pi\)
−0.796249 + 0.604970i \(0.793185\pi\)
\(44\) 0 0
\(45\) − 72.0225i − 0.238588i
\(46\) 0 0
\(47\) 4.12785 0.0128108 0.00640542 0.999979i \(-0.497961\pi\)
0.00640542 + 0.999979i \(0.497961\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 585.714i 1.60817i
\(52\) 0 0
\(53\) 279.697 0.724894 0.362447 0.932004i \(-0.381941\pi\)
0.362447 + 0.932004i \(0.381941\pi\)
\(54\) 0 0
\(55\) −688.342 −1.68756
\(56\) 0 0
\(57\) −359.984 −0.836509
\(58\) 0 0
\(59\) 174.831 0.385780 0.192890 0.981220i \(-0.438214\pi\)
0.192890 + 0.981220i \(0.438214\pi\)
\(60\) 0 0
\(61\) − 467.087i − 0.980399i −0.871610 0.490199i \(-0.836924\pi\)
0.871610 0.490199i \(-0.163076\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 958.488 1.82901
\(66\) 0 0
\(67\) − 753.251i − 1.37350i −0.726895 0.686748i \(-0.759037\pi\)
0.726895 0.686748i \(-0.240963\pi\)
\(68\) 0 0
\(69\) 506.229i 0.883230i
\(70\) 0 0
\(71\) 669.583i 1.11922i 0.828755 + 0.559611i \(0.189050\pi\)
−0.828755 + 0.559611i \(0.810950\pi\)
\(72\) 0 0
\(73\) − 835.774i − 1.34000i −0.742361 0.670000i \(-0.766294\pi\)
0.742361 0.670000i \(-0.233706\pi\)
\(74\) 0 0
\(75\) 765.033 1.17784
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1106.21i 1.57543i 0.616042 + 0.787713i \(0.288735\pi\)
−0.616042 + 0.787713i \(0.711265\pi\)
\(80\) 0 0
\(81\) −595.728 −0.817185
\(82\) 0 0
\(83\) −552.905 −0.731195 −0.365597 0.930773i \(-0.619135\pi\)
−0.365597 + 0.930773i \(0.619135\pi\)
\(84\) 0 0
\(85\) 2075.28 2.64818
\(86\) 0 0
\(87\) 699.530 0.862040
\(88\) 0 0
\(89\) 122.469i 0.145862i 0.997337 + 0.0729310i \(0.0232353\pi\)
−0.997337 + 0.0729310i \(0.976765\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −203.089 −0.226445
\(94\) 0 0
\(95\) 1275.48i 1.37749i
\(96\) 0 0
\(97\) 291.203i 0.304816i 0.988318 + 0.152408i \(0.0487028\pi\)
−0.988318 + 0.152408i \(0.951297\pi\)
\(98\) 0 0
\(99\) − 173.683i − 0.176321i
\(100\) 0 0
\(101\) 1349.04i 1.32905i 0.747264 + 0.664527i \(0.231367\pi\)
−0.747264 + 0.664527i \(0.768633\pi\)
\(102\) 0 0
\(103\) 1978.07 1.89228 0.946141 0.323756i \(-0.104946\pi\)
0.946141 + 0.323756i \(0.104946\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 846.316i 0.764640i 0.924030 + 0.382320i \(0.124875\pi\)
−0.924030 + 0.382320i \(0.875125\pi\)
\(108\) 0 0
\(109\) 927.077 0.814659 0.407330 0.913281i \(-0.366460\pi\)
0.407330 + 0.913281i \(0.366460\pi\)
\(110\) 0 0
\(111\) −385.826 −0.329919
\(112\) 0 0
\(113\) −599.053 −0.498709 −0.249355 0.968412i \(-0.580218\pi\)
−0.249355 + 0.968412i \(0.580218\pi\)
\(114\) 0 0
\(115\) 1793.65 1.45442
\(116\) 0 0
\(117\) 241.846i 0.191100i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −328.946 −0.247142
\(122\) 0 0
\(123\) − 256.561i − 0.188076i
\(124\) 0 0
\(125\) − 598.760i − 0.428438i
\(126\) 0 0
\(127\) − 1833.90i − 1.28136i −0.767810 0.640678i \(-0.778653\pi\)
0.767810 0.640678i \(-0.221347\pi\)
\(128\) 0 0
\(129\) − 1626.80i − 1.11032i
\(130\) 0 0
\(131\) 1379.15 0.919822 0.459911 0.887965i \(-0.347881\pi\)
0.459911 + 0.887965i \(0.347881\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2518.57i 1.60566i
\(136\) 0 0
\(137\) 1473.43 0.918857 0.459429 0.888215i \(-0.348054\pi\)
0.459429 + 0.888215i \(0.348054\pi\)
\(138\) 0 0
\(139\) 1929.87 1.17762 0.588811 0.808271i \(-0.299596\pi\)
0.588811 + 0.808271i \(0.299596\pi\)
\(140\) 0 0
\(141\) −19.6830 −0.0117561
\(142\) 0 0
\(143\) 2311.41 1.35167
\(144\) 0 0
\(145\) − 2478.54i − 1.41953i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2306.84 1.26834 0.634172 0.773192i \(-0.281341\pi\)
0.634172 + 0.773192i \(0.281341\pi\)
\(150\) 0 0
\(151\) − 1025.27i − 0.552549i −0.961079 0.276275i \(-0.910900\pi\)
0.961079 0.276275i \(-0.0891000\pi\)
\(152\) 0 0
\(153\) 523.636i 0.276689i
\(154\) 0 0
\(155\) 719.577i 0.372889i
\(156\) 0 0
\(157\) − 77.0147i − 0.0391493i −0.999808 0.0195746i \(-0.993769\pi\)
0.999808 0.0195746i \(-0.00623120\pi\)
\(158\) 0 0
\(159\) −1333.69 −0.665212
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 3887.54i − 1.86807i −0.357180 0.934036i \(-0.616262\pi\)
0.357180 0.934036i \(-0.383738\pi\)
\(164\) 0 0
\(165\) 3282.25 1.54862
\(166\) 0 0
\(167\) −117.863 −0.0546141 −0.0273070 0.999627i \(-0.508693\pi\)
−0.0273070 + 0.999627i \(0.508693\pi\)
\(168\) 0 0
\(169\) −1021.54 −0.464969
\(170\) 0 0
\(171\) −321.830 −0.143924
\(172\) 0 0
\(173\) − 2864.21i − 1.25874i −0.777105 0.629370i \(-0.783313\pi\)
0.777105 0.629370i \(-0.216687\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −833.653 −0.354018
\(178\) 0 0
\(179\) 1380.94i 0.576628i 0.957536 + 0.288314i \(0.0930947\pi\)
−0.957536 + 0.288314i \(0.906905\pi\)
\(180\) 0 0
\(181\) 204.817i 0.0841103i 0.999115 + 0.0420551i \(0.0133905\pi\)
−0.999115 + 0.0420551i \(0.986609\pi\)
\(182\) 0 0
\(183\) 2227.23i 0.899680i
\(184\) 0 0
\(185\) 1367.04i 0.543281i
\(186\) 0 0
\(187\) 5004.56 1.95706
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4167.75i 1.57889i 0.613821 + 0.789445i \(0.289631\pi\)
−0.613821 + 0.789445i \(0.710369\pi\)
\(192\) 0 0
\(193\) −4785.93 −1.78497 −0.892485 0.451078i \(-0.851040\pi\)
−0.892485 + 0.451078i \(0.851040\pi\)
\(194\) 0 0
\(195\) −4570.39 −1.67842
\(196\) 0 0
\(197\) −377.368 −0.136479 −0.0682395 0.997669i \(-0.521738\pi\)
−0.0682395 + 0.997669i \(0.521738\pi\)
\(198\) 0 0
\(199\) −1619.53 −0.576911 −0.288455 0.957493i \(-0.593142\pi\)
−0.288455 + 0.957493i \(0.593142\pi\)
\(200\) 0 0
\(201\) 3591.76i 1.26041i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −909.037 −0.309707
\(206\) 0 0
\(207\) 452.575i 0.151962i
\(208\) 0 0
\(209\) 3075.83i 1.01799i
\(210\) 0 0
\(211\) 3010.54i 0.982247i 0.871090 + 0.491123i \(0.163414\pi\)
−0.871090 + 0.491123i \(0.836586\pi\)
\(212\) 0 0
\(213\) − 3192.80i − 1.02707i
\(214\) 0 0
\(215\) −5764.00 −1.82838
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3985.26i 1.22967i
\(220\) 0 0
\(221\) −6968.64 −2.12109
\(222\) 0 0
\(223\) 879.692 0.264164 0.132082 0.991239i \(-0.457834\pi\)
0.132082 + 0.991239i \(0.457834\pi\)
\(224\) 0 0
\(225\) 683.948 0.202651
\(226\) 0 0
\(227\) 1424.63 0.416547 0.208273 0.978071i \(-0.433216\pi\)
0.208273 + 0.978071i \(0.433216\pi\)
\(228\) 0 0
\(229\) 238.730i 0.0688896i 0.999407 + 0.0344448i \(0.0109663\pi\)
−0.999407 + 0.0344448i \(0.989034\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5841.72 1.64251 0.821253 0.570565i \(-0.193276\pi\)
0.821253 + 0.570565i \(0.193276\pi\)
\(234\) 0 0
\(235\) 69.7399i 0.0193589i
\(236\) 0 0
\(237\) − 5274.80i − 1.44572i
\(238\) 0 0
\(239\) − 2101.91i − 0.568876i −0.958694 0.284438i \(-0.908193\pi\)
0.958694 0.284438i \(-0.0918070\pi\)
\(240\) 0 0
\(241\) − 2175.18i − 0.581393i −0.956815 0.290697i \(-0.906113\pi\)
0.956815 0.290697i \(-0.0938871\pi\)
\(242\) 0 0
\(243\) −1184.32 −0.312652
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4282.97i − 1.10332i
\(248\) 0 0
\(249\) 2636.44 0.670993
\(250\) 0 0
\(251\) −2069.98 −0.520542 −0.260271 0.965536i \(-0.583812\pi\)
−0.260271 + 0.965536i \(0.583812\pi\)
\(252\) 0 0
\(253\) 4325.41 1.07485
\(254\) 0 0
\(255\) −9895.63 −2.43015
\(256\) 0 0
\(257\) 5194.72i 1.26085i 0.776252 + 0.630423i \(0.217119\pi\)
−0.776252 + 0.630423i \(0.782881\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 625.388 0.148316
\(262\) 0 0
\(263\) 1824.70i 0.427818i 0.976854 + 0.213909i \(0.0686196\pi\)
−0.976854 + 0.213909i \(0.931380\pi\)
\(264\) 0 0
\(265\) 4725.48i 1.09541i
\(266\) 0 0
\(267\) − 583.975i − 0.133853i
\(268\) 0 0
\(269\) − 2497.04i − 0.565974i −0.959124 0.282987i \(-0.908675\pi\)
0.959124 0.282987i \(-0.0913253\pi\)
\(270\) 0 0
\(271\) 1922.58 0.430954 0.215477 0.976509i \(-0.430869\pi\)
0.215477 + 0.976509i \(0.430869\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 6536.72i − 1.43338i
\(276\) 0 0
\(277\) −1885.43 −0.408970 −0.204485 0.978870i \(-0.565552\pi\)
−0.204485 + 0.978870i \(0.565552\pi\)
\(278\) 0 0
\(279\) −181.564 −0.0389604
\(280\) 0 0
\(281\) −2666.13 −0.566006 −0.283003 0.959119i \(-0.591331\pi\)
−0.283003 + 0.959119i \(0.591331\pi\)
\(282\) 0 0
\(283\) −3747.87 −0.787236 −0.393618 0.919274i \(-0.628777\pi\)
−0.393618 + 0.919274i \(0.628777\pi\)
\(284\) 0 0
\(285\) − 6081.92i − 1.26408i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10175.2 −2.07108
\(290\) 0 0
\(291\) − 1388.55i − 0.279720i
\(292\) 0 0
\(293\) − 6486.41i − 1.29331i −0.762782 0.646655i \(-0.776167\pi\)
0.762782 0.646655i \(-0.223833\pi\)
\(294\) 0 0
\(295\) 2953.76i 0.582965i
\(296\) 0 0
\(297\) 6073.57i 1.18661i
\(298\) 0 0
\(299\) −6022.95 −1.16494
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 6432.68i − 1.21963i
\(304\) 0 0
\(305\) 7891.42 1.48151
\(306\) 0 0
\(307\) 8884.08 1.65160 0.825800 0.563963i \(-0.190724\pi\)
0.825800 + 0.563963i \(0.190724\pi\)
\(308\) 0 0
\(309\) −9432.11 −1.73648
\(310\) 0 0
\(311\) 7001.77 1.27664 0.638318 0.769773i \(-0.279630\pi\)
0.638318 + 0.769773i \(0.279630\pi\)
\(312\) 0 0
\(313\) 4781.08i 0.863395i 0.902018 + 0.431697i \(0.142085\pi\)
−0.902018 + 0.431697i \(0.857915\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3603.43 −0.638451 −0.319226 0.947679i \(-0.603423\pi\)
−0.319226 + 0.947679i \(0.603423\pi\)
\(318\) 0 0
\(319\) − 5977.04i − 1.04906i
\(320\) 0 0
\(321\) − 4035.52i − 0.701685i
\(322\) 0 0
\(323\) − 9273.31i − 1.59746i
\(324\) 0 0
\(325\) 9102.11i 1.55352i
\(326\) 0 0
\(327\) −4420.62 −0.747586
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 3304.08i − 0.548667i −0.961635 0.274333i \(-0.911543\pi\)
0.961635 0.274333i \(-0.0884571\pi\)
\(332\) 0 0
\(333\) −344.933 −0.0567635
\(334\) 0 0
\(335\) 12726.2 2.07554
\(336\) 0 0
\(337\) −6650.35 −1.07498 −0.537489 0.843271i \(-0.680627\pi\)
−0.537489 + 0.843271i \(0.680627\pi\)
\(338\) 0 0
\(339\) 2856.49 0.457649
\(340\) 0 0
\(341\) 1735.27i 0.275572i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −8552.73 −1.33468
\(346\) 0 0
\(347\) − 235.365i − 0.0364123i −0.999834 0.0182061i \(-0.994204\pi\)
0.999834 0.0182061i \(-0.00579551\pi\)
\(348\) 0 0
\(349\) 2984.02i 0.457681i 0.973464 + 0.228841i \(0.0734935\pi\)
−0.973464 + 0.228841i \(0.926507\pi\)
\(350\) 0 0
\(351\) − 8457.19i − 1.28607i
\(352\) 0 0
\(353\) 2446.28i 0.368845i 0.982847 + 0.184422i \(0.0590414\pi\)
−0.982847 + 0.184422i \(0.940959\pi\)
\(354\) 0 0
\(355\) −11312.6 −1.69129
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 5694.08i − 0.837108i −0.908192 0.418554i \(-0.862537\pi\)
0.908192 0.418554i \(-0.137463\pi\)
\(360\) 0 0
\(361\) −1159.56 −0.169057
\(362\) 0 0
\(363\) 1568.53 0.226794
\(364\) 0 0
\(365\) 14120.4 2.02492
\(366\) 0 0
\(367\) −336.321 −0.0478361 −0.0239180 0.999714i \(-0.507614\pi\)
−0.0239180 + 0.999714i \(0.507614\pi\)
\(368\) 0 0
\(369\) − 229.369i − 0.0323590i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9816.84 1.36273 0.681363 0.731945i \(-0.261387\pi\)
0.681363 + 0.731945i \(0.261387\pi\)
\(374\) 0 0
\(375\) 2855.09i 0.393163i
\(376\) 0 0
\(377\) 8322.78i 1.13699i
\(378\) 0 0
\(379\) − 3978.71i − 0.539242i −0.962967 0.269621i \(-0.913102\pi\)
0.962967 0.269621i \(-0.0868984\pi\)
\(380\) 0 0
\(381\) 8744.65i 1.17586i
\(382\) 0 0
\(383\) 3725.99 0.497100 0.248550 0.968619i \(-0.420046\pi\)
0.248550 + 0.968619i \(0.420046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1454.38i − 0.191034i
\(388\) 0 0
\(389\) 12693.8 1.65450 0.827250 0.561834i \(-0.189904\pi\)
0.827250 + 0.561834i \(0.189904\pi\)
\(390\) 0 0
\(391\) −13040.6 −1.68669
\(392\) 0 0
\(393\) −6576.24 −0.844091
\(394\) 0 0
\(395\) −18689.4 −2.38068
\(396\) 0 0
\(397\) 2072.26i 0.261974i 0.991384 + 0.130987i \(0.0418147\pi\)
−0.991384 + 0.130987i \(0.958185\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9857.39 −1.22757 −0.613784 0.789474i \(-0.710353\pi\)
−0.613784 + 0.789474i \(0.710353\pi\)
\(402\) 0 0
\(403\) − 2416.29i − 0.298670i
\(404\) 0 0
\(405\) − 10064.8i − 1.23487i
\(406\) 0 0
\(407\) 3296.64i 0.401495i
\(408\) 0 0
\(409\) − 1374.47i − 0.166169i −0.996542 0.0830847i \(-0.973523\pi\)
0.996542 0.0830847i \(-0.0264772\pi\)
\(410\) 0 0
\(411\) −7025.80 −0.843205
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 9341.31i − 1.10493i
\(416\) 0 0
\(417\) −9202.27 −1.08066
\(418\) 0 0
\(419\) 11066.7 1.29032 0.645158 0.764049i \(-0.276791\pi\)
0.645158 + 0.764049i \(0.276791\pi\)
\(420\) 0 0
\(421\) 15288.1 1.76982 0.884910 0.465761i \(-0.154219\pi\)
0.884910 + 0.465761i \(0.154219\pi\)
\(422\) 0 0
\(423\) −17.5968 −0.00202266
\(424\) 0 0
\(425\) 19707.5i 2.24931i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −11021.6 −1.24039
\(430\) 0 0
\(431\) 2615.48i 0.292305i 0.989262 + 0.146152i \(0.0466890\pi\)
−0.989262 + 0.146152i \(0.953311\pi\)
\(432\) 0 0
\(433\) − 3152.43i − 0.349875i −0.984580 0.174938i \(-0.944028\pi\)
0.984580 0.174938i \(-0.0559723\pi\)
\(434\) 0 0
\(435\) 11818.5i 1.30266i
\(436\) 0 0
\(437\) − 8014.87i − 0.877352i
\(438\) 0 0
\(439\) 6940.47 0.754557 0.377278 0.926100i \(-0.376860\pi\)
0.377278 + 0.926100i \(0.376860\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12533.9i 1.34425i 0.740439 + 0.672123i \(0.234618\pi\)
−0.740439 + 0.672123i \(0.765382\pi\)
\(444\) 0 0
\(445\) −2069.12 −0.220417
\(446\) 0 0
\(447\) −10999.8 −1.16392
\(448\) 0 0
\(449\) 8637.25 0.907833 0.453917 0.891044i \(-0.350026\pi\)
0.453917 + 0.891044i \(0.350026\pi\)
\(450\) 0 0
\(451\) −2192.15 −0.228879
\(452\) 0 0
\(453\) 4888.82i 0.507056i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16496.7 1.68858 0.844291 0.535885i \(-0.180022\pi\)
0.844291 + 0.535885i \(0.180022\pi\)
\(458\) 0 0
\(459\) − 18311.2i − 1.86207i
\(460\) 0 0
\(461\) − 6423.89i − 0.649003i −0.945885 0.324501i \(-0.894803\pi\)
0.945885 0.324501i \(-0.105197\pi\)
\(462\) 0 0
\(463\) − 3390.22i − 0.340295i −0.985419 0.170148i \(-0.945576\pi\)
0.985419 0.170148i \(-0.0544245\pi\)
\(464\) 0 0
\(465\) − 3431.19i − 0.342188i
\(466\) 0 0
\(467\) 1160.52 0.114995 0.0574975 0.998346i \(-0.481688\pi\)
0.0574975 + 0.998346i \(0.481688\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 367.232i 0.0359260i
\(472\) 0 0
\(473\) −13899.9 −1.35121
\(474\) 0 0
\(475\) −12112.4 −1.17001
\(476\) 0 0
\(477\) −1192.34 −0.114451
\(478\) 0 0
\(479\) 4084.93 0.389656 0.194828 0.980837i \(-0.437585\pi\)
0.194828 + 0.980837i \(0.437585\pi\)
\(480\) 0 0
\(481\) − 4590.44i − 0.435147i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4919.86 −0.460617
\(486\) 0 0
\(487\) − 7320.89i − 0.681193i −0.940210 0.340597i \(-0.889371\pi\)
0.940210 0.340597i \(-0.110629\pi\)
\(488\) 0 0
\(489\) 18537.1i 1.71427i
\(490\) 0 0
\(491\) 32.0507i 0.00294588i 0.999999 + 0.00147294i \(0.000468851\pi\)
−0.999999 + 0.00147294i \(0.999531\pi\)
\(492\) 0 0
\(493\) 18020.1i 1.64622i
\(494\) 0 0
\(495\) 2934.37 0.266445
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7521.66i 0.674781i 0.941365 + 0.337391i \(0.109544\pi\)
−0.941365 + 0.337391i \(0.890456\pi\)
\(500\) 0 0
\(501\) 562.013 0.0501175
\(502\) 0 0
\(503\) 4599.64 0.407729 0.203865 0.978999i \(-0.434650\pi\)
0.203865 + 0.978999i \(0.434650\pi\)
\(504\) 0 0
\(505\) −22792.0 −2.00838
\(506\) 0 0
\(507\) 4871.03 0.426686
\(508\) 0 0
\(509\) − 20791.8i − 1.81057i −0.424803 0.905286i \(-0.639657\pi\)
0.424803 0.905286i \(-0.360343\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 11254.2 0.968583
\(514\) 0 0
\(515\) 33419.4i 2.85949i
\(516\) 0 0
\(517\) 168.179i 0.0143066i
\(518\) 0 0
\(519\) 13657.5i 1.15511i
\(520\) 0 0
\(521\) − 19292.6i − 1.62231i −0.584829 0.811156i \(-0.698839\pi\)
0.584829 0.811156i \(-0.301161\pi\)
\(522\) 0 0
\(523\) −16688.0 −1.39525 −0.697626 0.716462i \(-0.745760\pi\)
−0.697626 + 0.716462i \(0.745760\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5231.65i − 0.432437i
\(528\) 0 0
\(529\) 896.048 0.0736458
\(530\) 0 0
\(531\) −745.296 −0.0609098
\(532\) 0 0
\(533\) 3052.48 0.248063
\(534\) 0 0
\(535\) −14298.5 −1.15547
\(536\) 0 0
\(537\) − 6584.80i − 0.529153i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3051.34 0.242491 0.121245 0.992623i \(-0.461311\pi\)
0.121245 + 0.992623i \(0.461311\pi\)
\(542\) 0 0
\(543\) − 976.639i − 0.0771853i
\(544\) 0 0
\(545\) 15662.9i 1.23106i
\(546\) 0 0
\(547\) − 12022.3i − 0.939740i −0.882736 0.469870i \(-0.844301\pi\)
0.882736 0.469870i \(-0.155699\pi\)
\(548\) 0 0
\(549\) 1991.17i 0.154792i
\(550\) 0 0
\(551\) −11075.3 −0.856304
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 6518.53i − 0.498551i
\(556\) 0 0
\(557\) 11827.8 0.899747 0.449873 0.893092i \(-0.351469\pi\)
0.449873 + 0.893092i \(0.351469\pi\)
\(558\) 0 0
\(559\) 19355.1 1.46446
\(560\) 0 0
\(561\) −23863.4 −1.79593
\(562\) 0 0
\(563\) 13682.6 1.02425 0.512127 0.858910i \(-0.328858\pi\)
0.512127 + 0.858910i \(0.328858\pi\)
\(564\) 0 0
\(565\) − 10121.0i − 0.753616i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1633.47 0.120349 0.0601747 0.998188i \(-0.480834\pi\)
0.0601747 + 0.998188i \(0.480834\pi\)
\(570\) 0 0
\(571\) 14967.3i 1.09696i 0.836164 + 0.548480i \(0.184793\pi\)
−0.836164 + 0.548480i \(0.815207\pi\)
\(572\) 0 0
\(573\) − 19873.3i − 1.44890i
\(574\) 0 0
\(575\) 17033.1i 1.23535i
\(576\) 0 0
\(577\) − 12929.6i − 0.932873i −0.884555 0.466436i \(-0.845538\pi\)
0.884555 0.466436i \(-0.154462\pi\)
\(578\) 0 0
\(579\) 22821.0 1.63801
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11395.6i 0.809530i
\(584\) 0 0
\(585\) −4085.99 −0.288777
\(586\) 0 0
\(587\) −7486.36 −0.526397 −0.263199 0.964742i \(-0.584777\pi\)
−0.263199 + 0.964742i \(0.584777\pi\)
\(588\) 0 0
\(589\) 3215.40 0.224938
\(590\) 0 0
\(591\) 1799.42 0.125242
\(592\) 0 0
\(593\) 2649.26i 0.183461i 0.995784 + 0.0917304i \(0.0292398\pi\)
−0.995784 + 0.0917304i \(0.970760\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7722.46 0.529412
\(598\) 0 0
\(599\) − 17510.6i − 1.19443i −0.802082 0.597213i \(-0.796275\pi\)
0.802082 0.597213i \(-0.203725\pi\)
\(600\) 0 0
\(601\) − 12124.7i − 0.822922i −0.911427 0.411461i \(-0.865019\pi\)
0.911427 0.411461i \(-0.134981\pi\)
\(602\) 0 0
\(603\) 3211.07i 0.216857i
\(604\) 0 0
\(605\) − 5557.53i − 0.373464i
\(606\) 0 0
\(607\) 11412.5 0.763131 0.381566 0.924342i \(-0.375385\pi\)
0.381566 + 0.924342i \(0.375385\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 234.182i − 0.0155057i
\(612\) 0 0
\(613\) 6758.76 0.445324 0.222662 0.974896i \(-0.428525\pi\)
0.222662 + 0.974896i \(0.428525\pi\)
\(614\) 0 0
\(615\) 4334.60 0.284208
\(616\) 0 0
\(617\) −9991.14 −0.651909 −0.325955 0.945385i \(-0.605686\pi\)
−0.325955 + 0.945385i \(0.605686\pi\)
\(618\) 0 0
\(619\) 20664.1 1.34178 0.670889 0.741558i \(-0.265913\pi\)
0.670889 + 0.741558i \(0.265913\pi\)
\(620\) 0 0
\(621\) − 15826.2i − 1.02268i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9938.98 −0.636095
\(626\) 0 0
\(627\) − 14666.6i − 0.934176i
\(628\) 0 0
\(629\) − 9939.02i − 0.630039i
\(630\) 0 0
\(631\) − 28295.8i − 1.78516i −0.450887 0.892581i \(-0.648892\pi\)
0.450887 0.892581i \(-0.351108\pi\)
\(632\) 0 0
\(633\) − 14355.3i − 0.901376i
\(634\) 0 0
\(635\) 30983.7 1.93630
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 2854.40i − 0.176711i
\(640\) 0 0
\(641\) −16415.7 −1.01152 −0.505758 0.862676i \(-0.668787\pi\)
−0.505758 + 0.862676i \(0.668787\pi\)
\(642\) 0 0
\(643\) −14139.9 −0.867220 −0.433610 0.901101i \(-0.642760\pi\)
−0.433610 + 0.901101i \(0.642760\pi\)
\(644\) 0 0
\(645\) 27484.7 1.67784
\(646\) 0 0
\(647\) 21949.1 1.33371 0.666853 0.745189i \(-0.267641\pi\)
0.666853 + 0.745189i \(0.267641\pi\)
\(648\) 0 0
\(649\) 7123.04i 0.430822i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5877.76 0.352243 0.176121 0.984368i \(-0.443645\pi\)
0.176121 + 0.984368i \(0.443645\pi\)
\(654\) 0 0
\(655\) 23300.7i 1.38997i
\(656\) 0 0
\(657\) 3562.87i 0.211569i
\(658\) 0 0
\(659\) − 11002.0i − 0.650344i −0.945655 0.325172i \(-0.894578\pi\)
0.945655 0.325172i \(-0.105422\pi\)
\(660\) 0 0
\(661\) − 3691.25i − 0.217206i −0.994085 0.108603i \(-0.965362\pi\)
0.994085 0.108603i \(-0.0346377\pi\)
\(662\) 0 0
\(663\) 33228.8 1.94646
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15574.7i 0.904130i
\(668\) 0 0
\(669\) −4194.67 −0.242414
\(670\) 0 0
\(671\) 19030.2 1.09487
\(672\) 0 0
\(673\) 6819.19 0.390580 0.195290 0.980746i \(-0.437435\pi\)
0.195290 + 0.980746i \(0.437435\pi\)
\(674\) 0 0
\(675\) −23917.2 −1.36381
\(676\) 0 0
\(677\) 15800.2i 0.896973i 0.893790 + 0.448487i \(0.148037\pi\)
−0.893790 + 0.448487i \(0.851963\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6793.12 −0.382251
\(682\) 0 0
\(683\) 21534.7i 1.20644i 0.797573 + 0.603222i \(0.206117\pi\)
−0.797573 + 0.603222i \(0.793883\pi\)
\(684\) 0 0
\(685\) 24893.5i 1.38851i
\(686\) 0 0
\(687\) − 1138.35i − 0.0632177i
\(688\) 0 0
\(689\) − 15867.8i − 0.877382i
\(690\) 0 0
\(691\) −4038.54 −0.222335 −0.111167 0.993802i \(-0.535459\pi\)
−0.111167 + 0.993802i \(0.535459\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 32605.1i 1.77954i
\(696\) 0 0
\(697\) 6609.11 0.359165
\(698\) 0 0
\(699\) −27855.3 −1.50727
\(700\) 0 0
\(701\) 11823.9 0.637066 0.318533 0.947912i \(-0.396810\pi\)
0.318533 + 0.947912i \(0.396810\pi\)
\(702\) 0 0
\(703\) 6108.59 0.327724
\(704\) 0 0
\(705\) − 332.544i − 0.0177650i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26286.5 1.39240 0.696200 0.717848i \(-0.254873\pi\)
0.696200 + 0.717848i \(0.254873\pi\)
\(710\) 0 0
\(711\) − 4715.73i − 0.248739i
\(712\) 0 0
\(713\) − 4521.68i − 0.237501i
\(714\) 0 0
\(715\) 39051.1i 2.04256i
\(716\) 0 0
\(717\) 10022.6i 0.522039i
\(718\) 0 0
\(719\) −19229.9 −0.997435 −0.498717 0.866765i \(-0.666195\pi\)
−0.498717 + 0.866765i \(0.666195\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 10372.0i 0.533526i
\(724\) 0 0
\(725\) 23537.1 1.20572
\(726\) 0 0
\(727\) −33771.2 −1.72284 −0.861419 0.507895i \(-0.830424\pi\)
−0.861419 + 0.507895i \(0.830424\pi\)
\(728\) 0 0
\(729\) 21731.9 1.10409
\(730\) 0 0
\(731\) 41906.9 2.12036
\(732\) 0 0
\(733\) 2119.95i 0.106824i 0.998573 + 0.0534121i \(0.0170097\pi\)
−0.998573 + 0.0534121i \(0.982990\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30689.3 1.53386
\(738\) 0 0
\(739\) − 10667.5i − 0.531004i −0.964110 0.265502i \(-0.914462\pi\)
0.964110 0.265502i \(-0.0855377\pi\)
\(740\) 0 0
\(741\) 20422.7i 1.01248i
\(742\) 0 0
\(743\) 26684.2i 1.31756i 0.752335 + 0.658781i \(0.228928\pi\)
−0.752335 + 0.658781i \(0.771072\pi\)
\(744\) 0 0
\(745\) 38973.9i 1.91664i
\(746\) 0 0
\(747\) 2357.01 0.115446
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 35120.5i 1.70648i 0.521519 + 0.853240i \(0.325366\pi\)
−0.521519 + 0.853240i \(0.674634\pi\)
\(752\) 0 0
\(753\) 9870.37 0.477684
\(754\) 0 0
\(755\) 17321.8 0.834975
\(756\) 0 0
\(757\) 10060.1 0.483011 0.241506 0.970399i \(-0.422359\pi\)
0.241506 + 0.970399i \(0.422359\pi\)
\(758\) 0 0
\(759\) −20625.0 −0.986351
\(760\) 0 0
\(761\) − 36350.6i − 1.73155i −0.500435 0.865774i \(-0.666826\pi\)
0.500435 0.865774i \(-0.333174\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −8846.81 −0.418114
\(766\) 0 0
\(767\) − 9918.53i − 0.466933i
\(768\) 0 0
\(769\) 17664.9i 0.828366i 0.910194 + 0.414183i \(0.135933\pi\)
−0.910194 + 0.414183i \(0.864067\pi\)
\(770\) 0 0
\(771\) − 24770.2i − 1.15704i
\(772\) 0 0
\(773\) 7908.42i 0.367977i 0.982928 + 0.183988i \(0.0589009\pi\)
−0.982928 + 0.183988i \(0.941099\pi\)
\(774\) 0 0
\(775\) −6833.33 −0.316723
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4062.00i 0.186825i
\(780\) 0 0
\(781\) −27280.4 −1.24990
\(782\) 0 0
\(783\) −21869.4 −0.998145
\(784\) 0 0
\(785\) 1301.16 0.0591598
\(786\) 0 0
\(787\) −11881.1 −0.538140 −0.269070 0.963121i \(-0.586716\pi\)
−0.269070 + 0.963121i \(0.586716\pi\)
\(788\) 0 0
\(789\) − 8700.81i − 0.392594i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −26498.8 −1.18663
\(794\) 0 0
\(795\) − 22532.7i − 1.00522i
\(796\) 0 0
\(797\) − 10203.4i − 0.453480i −0.973955 0.226740i \(-0.927193\pi\)
0.973955 0.226740i \(-0.0728068\pi\)
\(798\) 0 0
\(799\) − 507.041i − 0.0224503i
\(800\) 0 0
\(801\) − 522.081i − 0.0230297i
\(802\) 0 0
\(803\) 34051.5 1.49645
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11906.7i 0.519376i
\(808\) 0 0
\(809\) 20942.1 0.910117 0.455059 0.890461i \(-0.349618\pi\)
0.455059 + 0.890461i \(0.349618\pi\)
\(810\) 0 0
\(811\) −5434.74 −0.235314 −0.117657 0.993054i \(-0.537538\pi\)
−0.117657 + 0.993054i \(0.537538\pi\)
\(812\) 0 0
\(813\) −9167.53 −0.395473
\(814\) 0 0
\(815\) 65679.9 2.82290
\(816\) 0 0
\(817\) 25756.2i 1.10293i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2132.86 −0.0906667 −0.0453334 0.998972i \(-0.514435\pi\)
−0.0453334 + 0.998972i \(0.514435\pi\)
\(822\) 0 0
\(823\) − 19157.9i − 0.811424i −0.914001 0.405712i \(-0.867024\pi\)
0.914001 0.405712i \(-0.132976\pi\)
\(824\) 0 0
\(825\) 31169.3i 1.31536i
\(826\) 0 0
\(827\) 27208.2i 1.14404i 0.820239 + 0.572021i \(0.193840\pi\)
−0.820239 + 0.572021i \(0.806160\pi\)
\(828\) 0 0
\(829\) − 17206.0i − 0.720857i −0.932787 0.360429i \(-0.882630\pi\)
0.932787 0.360429i \(-0.117370\pi\)
\(830\) 0 0
\(831\) 8990.39 0.375299
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 1991.30i − 0.0825291i
\(836\) 0 0
\(837\) 6349.17 0.262197
\(838\) 0 0
\(839\) −9526.10 −0.391988 −0.195994 0.980605i \(-0.562793\pi\)
−0.195994 + 0.980605i \(0.562793\pi\)
\(840\) 0 0
\(841\) −2867.21 −0.117562
\(842\) 0 0
\(843\) 12713.0 0.519405
\(844\) 0 0
\(845\) − 17258.8i − 0.702629i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 17871.1 0.722421
\(850\) 0 0
\(851\) − 8590.24i − 0.346028i
\(852\) 0 0
\(853\) − 8447.62i − 0.339087i −0.985523 0.169543i \(-0.945771\pi\)
0.985523 0.169543i \(-0.0542293\pi\)
\(854\) 0 0
\(855\) − 5437.31i − 0.217488i
\(856\) 0 0
\(857\) 25647.7i 1.02230i 0.859492 + 0.511150i \(0.170780\pi\)
−0.859492 + 0.511150i \(0.829220\pi\)
\(858\) 0 0
\(859\) 7477.32 0.297000 0.148500 0.988912i \(-0.452556\pi\)
0.148500 + 0.988912i \(0.452556\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 23203.2i − 0.915234i −0.889150 0.457617i \(-0.848703\pi\)
0.889150 0.457617i \(-0.151297\pi\)
\(864\) 0 0
\(865\) 48390.8 1.90212
\(866\) 0 0
\(867\) 48518.9 1.90056
\(868\) 0 0
\(869\) −45069.8 −1.75937
\(870\) 0 0
\(871\) −42733.6 −1.66242
\(872\) 0 0
\(873\) − 1241.38i − 0.0481265i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12283.2 0.472947 0.236474 0.971638i \(-0.424008\pi\)
0.236474 + 0.971638i \(0.424008\pi\)
\(878\) 0 0
\(879\) 30929.4i 1.18683i
\(880\) 0 0
\(881\) − 44697.8i − 1.70931i −0.519193 0.854657i \(-0.673767\pi\)
0.519193 0.854657i \(-0.326233\pi\)
\(882\) 0 0
\(883\) 9474.88i 0.361104i 0.983565 + 0.180552i \(0.0577884\pi\)
−0.983565 + 0.180552i \(0.942212\pi\)
\(884\) 0 0
\(885\) − 14084.5i − 0.534968i
\(886\) 0 0
\(887\) −15004.6 −0.567987 −0.283993 0.958826i \(-0.591659\pi\)
−0.283993 + 0.958826i \(0.591659\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 24271.4i − 0.912595i
\(892\) 0 0
\(893\) 311.631 0.0116778
\(894\) 0 0
\(895\) −23331.0 −0.871361
\(896\) 0 0
\(897\) 28719.5 1.06902
\(898\) 0 0
\(899\) −6248.26 −0.231803
\(900\) 0 0
\(901\) − 34356.4i − 1.27034i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3460.39 −0.127102
\(906\) 0 0
\(907\) − 27994.5i − 1.02485i −0.858731 0.512427i \(-0.828747\pi\)
0.858731 0.512427i \(-0.171253\pi\)
\(908\) 0 0
\(909\) − 5750.89i − 0.209840i
\(910\) 0 0
\(911\) − 27683.1i − 1.00679i −0.864057 0.503393i \(-0.832085\pi\)
0.864057 0.503393i \(-0.167915\pi\)
\(912\) 0 0
\(913\) − 22526.7i − 0.816566i
\(914\) 0 0
\(915\) −37628.9 −1.35953
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 17.4351i 0 0.000625822i 1.00000 0.000312911i \(9.96027e-5\pi\)
−1.00000 0.000312911i \(0.999900\pi\)
\(920\) 0 0
\(921\) −42362.3 −1.51562
\(922\) 0 0
\(923\) 37986.9 1.35466
\(924\) 0 0
\(925\) −12981.9 −0.461451
\(926\) 0 0
\(927\) −8432.42 −0.298767
\(928\) 0 0
\(929\) 42595.2i 1.50431i 0.658986 + 0.752155i \(0.270985\pi\)
−0.658986 + 0.752155i \(0.729015\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −33386.8 −1.17153
\(934\) 0 0
\(935\) 84551.9i 2.95737i
\(936\) 0 0
\(937\) 26316.4i 0.917522i 0.888560 + 0.458761i \(0.151707\pi\)
−0.888560 + 0.458761i \(0.848293\pi\)
\(938\) 0 0
\(939\) − 22797.8i − 0.792309i
\(940\) 0 0
\(941\) − 29453.0i − 1.02034i −0.860074 0.510170i \(-0.829583\pi\)
0.860074 0.510170i \(-0.170417\pi\)
\(942\) 0 0
\(943\) 5712.21 0.197259
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 21005.0i − 0.720770i −0.932804 0.360385i \(-0.882645\pi\)
0.932804 0.360385i \(-0.117355\pi\)
\(948\) 0 0
\(949\) −47415.3 −1.62188
\(950\) 0 0
\(951\) 17182.4 0.585885
\(952\) 0 0
\(953\) 10707.4 0.363952 0.181976 0.983303i \(-0.441751\pi\)
0.181976 + 0.983303i \(0.441751\pi\)
\(954\) 0 0
\(955\) −70414.1 −2.38591
\(956\) 0 0
\(957\) 28500.5i 0.962687i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27977.0 −0.939109
\(962\) 0 0
\(963\) − 3607.81i − 0.120727i
\(964\) 0 0
\(965\) − 80858.2i − 2.69733i
\(966\) 0 0
\(967\) − 20353.4i − 0.676859i −0.940992 0.338429i \(-0.890104\pi\)
0.940992 0.338429i \(-0.109896\pi\)
\(968\) 0 0
\(969\) 44218.3i 1.46594i
\(970\) 0 0
\(971\) −55329.3 −1.82863 −0.914316 0.405002i \(-0.867271\pi\)
−0.914316 + 0.405002i \(0.867271\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 43401.9i − 1.42562i
\(976\) 0 0
\(977\) 29925.0 0.979925 0.489963 0.871743i \(-0.337010\pi\)
0.489963 + 0.871743i \(0.337010\pi\)
\(978\) 0 0
\(979\) −4989.70 −0.162892
\(980\) 0 0
\(981\) −3952.08 −0.128624
\(982\) 0 0
\(983\) 4562.19 0.148028 0.0740139 0.997257i \(-0.476419\pi\)
0.0740139 + 0.997257i \(0.476419\pi\)
\(984\) 0 0
\(985\) − 6375.63i − 0.206238i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 36219.8 1.16453
\(990\) 0 0
\(991\) − 1224.37i − 0.0392467i −0.999807 0.0196233i \(-0.993753\pi\)
0.999807 0.0196233i \(-0.00624671\pi\)
\(992\) 0 0
\(993\) 15755.0i 0.503493i
\(994\) 0 0
\(995\) − 27361.9i − 0.871789i
\(996\) 0 0
\(997\) − 46098.0i − 1.46433i −0.681127 0.732165i \(-0.738510\pi\)
0.681127 0.732165i \(-0.261490\pi\)
\(998\) 0 0
\(999\) 12062.1 0.382009
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 784.4.f.g.783.4 6
4.3 odd 2 784.4.f.h.783.4 6
7.2 even 3 112.4.p.g.31.2 yes 6
7.3 odd 6 112.4.p.f.47.2 yes 6
7.6 odd 2 784.4.f.h.783.3 6
28.3 even 6 112.4.p.g.47.2 yes 6
28.23 odd 6 112.4.p.f.31.2 6
28.27 even 2 inner 784.4.f.g.783.3 6
56.3 even 6 448.4.p.f.383.2 6
56.37 even 6 448.4.p.f.255.2 6
56.45 odd 6 448.4.p.g.383.2 6
56.51 odd 6 448.4.p.g.255.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.4.p.f.31.2 6 28.23 odd 6
112.4.p.f.47.2 yes 6 7.3 odd 6
112.4.p.g.31.2 yes 6 7.2 even 3
112.4.p.g.47.2 yes 6 28.3 even 6
448.4.p.f.255.2 6 56.37 even 6
448.4.p.f.383.2 6 56.3 even 6
448.4.p.g.255.2 6 56.51 odd 6
448.4.p.g.383.2 6 56.45 odd 6
784.4.f.g.783.3 6 28.27 even 2 inner
784.4.f.g.783.4 6 1.1 even 1 trivial
784.4.f.h.783.3 6 7.6 odd 2
784.4.f.h.783.4 6 4.3 odd 2