Properties

Label 7623.2.a.cp.1.4
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.879640\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.120360 q^{2} -1.98551 q^{4} +2.80853 q^{5} +1.00000 q^{7} +0.479696 q^{8} +O(q^{10})\) \(q-0.120360 q^{2} -1.98551 q^{4} +2.80853 q^{5} +1.00000 q^{7} +0.479696 q^{8} -0.338034 q^{10} +1.07967 q^{13} -0.120360 q^{14} +3.91329 q^{16} -6.95828 q^{17} +7.54411 q^{19} -5.57636 q^{20} -4.82552 q^{23} +2.88781 q^{25} -0.129949 q^{26} -1.98551 q^{28} -1.22726 q^{29} -8.07409 q^{31} -1.43040 q^{32} +0.837498 q^{34} +2.80853 q^{35} +1.53525 q^{37} -0.908009 q^{38} +1.34724 q^{40} -9.29986 q^{41} -5.23402 q^{43} +0.580799 q^{46} +1.89387 q^{47} +1.00000 q^{49} -0.347577 q^{50} -2.14371 q^{52} +3.82552 q^{53} +0.479696 q^{56} +0.147713 q^{58} +6.66349 q^{59} -9.79952 q^{61} +0.971796 q^{62} -7.65442 q^{64} +3.03229 q^{65} -2.06100 q^{67} +13.8158 q^{68} -0.338034 q^{70} -12.5212 q^{71} +2.56708 q^{73} -0.184783 q^{74} -14.9789 q^{76} -15.3283 q^{79} +10.9906 q^{80} +1.11933 q^{82} -2.04602 q^{83} -19.5425 q^{85} +0.629967 q^{86} -4.76119 q^{89} +1.07967 q^{91} +9.58114 q^{92} -0.227945 q^{94} +21.1878 q^{95} +9.11512 q^{97} -0.120360 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} - 8q^{10} + 4q^{13} - 4q^{14} + 8q^{16} - 22q^{17} + 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} + 4q^{28} - 12q^{29} - 2q^{31} - 8q^{32} + 24q^{34} + 4q^{35} + 14q^{37} + 22q^{38} + 18q^{40} - 26q^{41} - 4q^{43} + 12q^{46} + 16q^{47} + 6q^{49} + 4q^{50} + 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} - 8q^{61} - 20q^{62} + 26q^{64} - 24q^{65} + 6q^{67} - 12q^{68} - 8q^{70} - 22q^{71} + 14q^{73} - 44q^{74} - 30q^{76} - 28q^{79} + 4q^{80} - 4q^{82} - 22q^{83} - 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} - 38q^{94} + 24q^{95} - 4q^{97} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.120360 −0.0851073 −0.0425536 0.999094i \(-0.513549\pi\)
−0.0425536 + 0.999094i \(0.513549\pi\)
\(3\) 0 0
\(4\) −1.98551 −0.992757
\(5\) 2.80853 1.25601 0.628005 0.778209i \(-0.283872\pi\)
0.628005 + 0.778209i \(0.283872\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0.479696 0.169598
\(9\) 0 0
\(10\) −0.338034 −0.106896
\(11\) 0 0
\(12\) 0 0
\(13\) 1.07967 0.299448 0.149724 0.988728i \(-0.452162\pi\)
0.149724 + 0.988728i \(0.452162\pi\)
\(14\) −0.120360 −0.0321675
\(15\) 0 0
\(16\) 3.91329 0.978323
\(17\) −6.95828 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(18\) 0 0
\(19\) 7.54411 1.73074 0.865369 0.501135i \(-0.167084\pi\)
0.865369 + 0.501135i \(0.167084\pi\)
\(20\) −5.57636 −1.24691
\(21\) 0 0
\(22\) 0 0
\(23\) −4.82552 −1.00619 −0.503095 0.864231i \(-0.667806\pi\)
−0.503095 + 0.864231i \(0.667806\pi\)
\(24\) 0 0
\(25\) 2.88781 0.577563
\(26\) −0.129949 −0.0254852
\(27\) 0 0
\(28\) −1.98551 −0.375227
\(29\) −1.22726 −0.227897 −0.113949 0.993487i \(-0.536350\pi\)
−0.113949 + 0.993487i \(0.536350\pi\)
\(30\) 0 0
\(31\) −8.07409 −1.45015 −0.725074 0.688671i \(-0.758195\pi\)
−0.725074 + 0.688671i \(0.758195\pi\)
\(32\) −1.43040 −0.252861
\(33\) 0 0
\(34\) 0.837498 0.143630
\(35\) 2.80853 0.474727
\(36\) 0 0
\(37\) 1.53525 0.252394 0.126197 0.992005i \(-0.459723\pi\)
0.126197 + 0.992005i \(0.459723\pi\)
\(38\) −0.908009 −0.147298
\(39\) 0 0
\(40\) 1.34724 0.213017
\(41\) −9.29986 −1.45239 −0.726197 0.687487i \(-0.758714\pi\)
−0.726197 + 0.687487i \(0.758714\pi\)
\(42\) 0 0
\(43\) −5.23402 −0.798181 −0.399091 0.916912i \(-0.630674\pi\)
−0.399091 + 0.916912i \(0.630674\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.580799 0.0856341
\(47\) 1.89387 0.276249 0.138124 0.990415i \(-0.455893\pi\)
0.138124 + 0.990415i \(0.455893\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −0.347577 −0.0491548
\(51\) 0 0
\(52\) −2.14371 −0.297279
\(53\) 3.82552 0.525476 0.262738 0.964867i \(-0.415375\pi\)
0.262738 + 0.964867i \(0.415375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.479696 0.0641021
\(57\) 0 0
\(58\) 0.147713 0.0193957
\(59\) 6.66349 0.867513 0.433756 0.901030i \(-0.357188\pi\)
0.433756 + 0.901030i \(0.357188\pi\)
\(60\) 0 0
\(61\) −9.79952 −1.25470 −0.627350 0.778737i \(-0.715861\pi\)
−0.627350 + 0.778737i \(0.715861\pi\)
\(62\) 0.971796 0.123418
\(63\) 0 0
\(64\) −7.65442 −0.956802
\(65\) 3.03229 0.376110
\(66\) 0 0
\(67\) −2.06100 −0.251792 −0.125896 0.992043i \(-0.540181\pi\)
−0.125896 + 0.992043i \(0.540181\pi\)
\(68\) 13.8158 1.67541
\(69\) 0 0
\(70\) −0.338034 −0.0404028
\(71\) −12.5212 −1.48599 −0.742996 0.669296i \(-0.766596\pi\)
−0.742996 + 0.669296i \(0.766596\pi\)
\(72\) 0 0
\(73\) 2.56708 0.300454 0.150227 0.988652i \(-0.452000\pi\)
0.150227 + 0.988652i \(0.452000\pi\)
\(74\) −0.184783 −0.0214806
\(75\) 0 0
\(76\) −14.9789 −1.71820
\(77\) 0 0
\(78\) 0 0
\(79\) −15.3283 −1.72457 −0.862287 0.506420i \(-0.830969\pi\)
−0.862287 + 0.506420i \(0.830969\pi\)
\(80\) 10.9906 1.22878
\(81\) 0 0
\(82\) 1.11933 0.123609
\(83\) −2.04602 −0.224580 −0.112290 0.993675i \(-0.535819\pi\)
−0.112290 + 0.993675i \(0.535819\pi\)
\(84\) 0 0
\(85\) −19.5425 −2.11968
\(86\) 0.629967 0.0679310
\(87\) 0 0
\(88\) 0 0
\(89\) −4.76119 −0.504685 −0.252342 0.967638i \(-0.581201\pi\)
−0.252342 + 0.967638i \(0.581201\pi\)
\(90\) 0 0
\(91\) 1.07967 0.113181
\(92\) 9.58114 0.998902
\(93\) 0 0
\(94\) −0.227945 −0.0235108
\(95\) 21.1878 2.17383
\(96\) 0 0
\(97\) 9.11512 0.925500 0.462750 0.886489i \(-0.346863\pi\)
0.462750 + 0.886489i \(0.346863\pi\)
\(98\) −0.120360 −0.0121582
\(99\) 0 0
\(100\) −5.73379 −0.573379
\(101\) 4.74385 0.472031 0.236015 0.971749i \(-0.424158\pi\)
0.236015 + 0.971749i \(0.424158\pi\)
\(102\) 0 0
\(103\) 0.350901 0.0345753 0.0172876 0.999851i \(-0.494497\pi\)
0.0172876 + 0.999851i \(0.494497\pi\)
\(104\) 0.517915 0.0507858
\(105\) 0 0
\(106\) −0.460439 −0.0447218
\(107\) −2.54774 −0.246299 −0.123149 0.992388i \(-0.539299\pi\)
−0.123149 + 0.992388i \(0.539299\pi\)
\(108\) 0 0
\(109\) −3.81522 −0.365432 −0.182716 0.983166i \(-0.558489\pi\)
−0.182716 + 0.983166i \(0.558489\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.91329 0.369771
\(113\) −2.31468 −0.217747 −0.108873 0.994056i \(-0.534724\pi\)
−0.108873 + 0.994056i \(0.534724\pi\)
\(114\) 0 0
\(115\) −13.5526 −1.26379
\(116\) 2.43675 0.226246
\(117\) 0 0
\(118\) −0.802017 −0.0738316
\(119\) −6.95828 −0.637865
\(120\) 0 0
\(121\) 0 0
\(122\) 1.17947 0.106784
\(123\) 0 0
\(124\) 16.0312 1.43964
\(125\) −5.93213 −0.530586
\(126\) 0 0
\(127\) 9.84644 0.873730 0.436865 0.899527i \(-0.356089\pi\)
0.436865 + 0.899527i \(0.356089\pi\)
\(128\) 3.78208 0.334291
\(129\) 0 0
\(130\) −0.364966 −0.0320097
\(131\) −16.3782 −1.43097 −0.715484 0.698629i \(-0.753794\pi\)
−0.715484 + 0.698629i \(0.753794\pi\)
\(132\) 0 0
\(133\) 7.54411 0.654158
\(134\) 0.248062 0.0214293
\(135\) 0 0
\(136\) −3.33786 −0.286219
\(137\) 14.2142 1.21440 0.607199 0.794550i \(-0.292293\pi\)
0.607199 + 0.794550i \(0.292293\pi\)
\(138\) 0 0
\(139\) −9.17942 −0.778588 −0.389294 0.921114i \(-0.627281\pi\)
−0.389294 + 0.921114i \(0.627281\pi\)
\(140\) −5.57636 −0.471289
\(141\) 0 0
\(142\) 1.50705 0.126469
\(143\) 0 0
\(144\) 0 0
\(145\) −3.44680 −0.286241
\(146\) −0.308974 −0.0255708
\(147\) 0 0
\(148\) −3.04827 −0.250566
\(149\) −11.6200 −0.951947 −0.475973 0.879460i \(-0.657904\pi\)
−0.475973 + 0.879460i \(0.657904\pi\)
\(150\) 0 0
\(151\) 17.9150 1.45790 0.728952 0.684565i \(-0.240008\pi\)
0.728952 + 0.684565i \(0.240008\pi\)
\(152\) 3.61888 0.293530
\(153\) 0 0
\(154\) 0 0
\(155\) −22.6763 −1.82140
\(156\) 0 0
\(157\) 7.42805 0.592823 0.296412 0.955060i \(-0.404210\pi\)
0.296412 + 0.955060i \(0.404210\pi\)
\(158\) 1.84492 0.146774
\(159\) 0 0
\(160\) −4.01730 −0.317596
\(161\) −4.82552 −0.380304
\(162\) 0 0
\(163\) 7.85296 0.615091 0.307546 0.951533i \(-0.400492\pi\)
0.307546 + 0.951533i \(0.400492\pi\)
\(164\) 18.4650 1.44187
\(165\) 0 0
\(166\) 0.246259 0.0191134
\(167\) 5.40259 0.418065 0.209032 0.977909i \(-0.432969\pi\)
0.209032 + 0.977909i \(0.432969\pi\)
\(168\) 0 0
\(169\) −11.8343 −0.910331
\(170\) 2.35214 0.180401
\(171\) 0 0
\(172\) 10.3922 0.792400
\(173\) 19.7211 1.49937 0.749685 0.661795i \(-0.230205\pi\)
0.749685 + 0.661795i \(0.230205\pi\)
\(174\) 0 0
\(175\) 2.88781 0.218298
\(176\) 0 0
\(177\) 0 0
\(178\) 0.573056 0.0429524
\(179\) 23.3292 1.74370 0.871851 0.489770i \(-0.162919\pi\)
0.871851 + 0.489770i \(0.162919\pi\)
\(180\) 0 0
\(181\) 3.08500 0.229306 0.114653 0.993406i \(-0.463424\pi\)
0.114653 + 0.993406i \(0.463424\pi\)
\(182\) −0.129949 −0.00963250
\(183\) 0 0
\(184\) −2.31478 −0.170648
\(185\) 4.31180 0.317010
\(186\) 0 0
\(187\) 0 0
\(188\) −3.76030 −0.274248
\(189\) 0 0
\(190\) −2.55017 −0.185008
\(191\) −12.5715 −0.909640 −0.454820 0.890583i \(-0.650296\pi\)
−0.454820 + 0.890583i \(0.650296\pi\)
\(192\) 0 0
\(193\) −20.6685 −1.48775 −0.743877 0.668317i \(-0.767015\pi\)
−0.743877 + 0.668317i \(0.767015\pi\)
\(194\) −1.09709 −0.0787668
\(195\) 0 0
\(196\) −1.98551 −0.141822
\(197\) −6.27954 −0.447399 −0.223699 0.974658i \(-0.571813\pi\)
−0.223699 + 0.974658i \(0.571813\pi\)
\(198\) 0 0
\(199\) −6.86896 −0.486927 −0.243464 0.969910i \(-0.578284\pi\)
−0.243464 + 0.969910i \(0.578284\pi\)
\(200\) 1.38527 0.0979536
\(201\) 0 0
\(202\) −0.570969 −0.0401733
\(203\) −1.22726 −0.0861370
\(204\) 0 0
\(205\) −26.1189 −1.82422
\(206\) −0.0422344 −0.00294261
\(207\) 0 0
\(208\) 4.22508 0.292957
\(209\) 0 0
\(210\) 0 0
\(211\) 10.2586 0.706232 0.353116 0.935580i \(-0.385122\pi\)
0.353116 + 0.935580i \(0.385122\pi\)
\(212\) −7.59562 −0.521669
\(213\) 0 0
\(214\) 0.306645 0.0209618
\(215\) −14.6999 −1.00252
\(216\) 0 0
\(217\) −8.07409 −0.548105
\(218\) 0.459199 0.0311009
\(219\) 0 0
\(220\) 0 0
\(221\) −7.51268 −0.505358
\(222\) 0 0
\(223\) −13.5221 −0.905506 −0.452753 0.891636i \(-0.649558\pi\)
−0.452753 + 0.891636i \(0.649558\pi\)
\(224\) −1.43040 −0.0955723
\(225\) 0 0
\(226\) 0.278595 0.0185319
\(227\) −13.5764 −0.901100 −0.450550 0.892751i \(-0.648772\pi\)
−0.450550 + 0.892751i \(0.648772\pi\)
\(228\) 0 0
\(229\) −1.45296 −0.0960141 −0.0480070 0.998847i \(-0.515287\pi\)
−0.0480070 + 0.998847i \(0.515287\pi\)
\(230\) 1.63119 0.107557
\(231\) 0 0
\(232\) −0.588713 −0.0386509
\(233\) −8.20387 −0.537453 −0.268727 0.963216i \(-0.586603\pi\)
−0.268727 + 0.963216i \(0.586603\pi\)
\(234\) 0 0
\(235\) 5.31897 0.346971
\(236\) −13.2305 −0.861229
\(237\) 0 0
\(238\) 0.837498 0.0542870
\(239\) −10.3835 −0.671655 −0.335827 0.941924i \(-0.609016\pi\)
−0.335827 + 0.941924i \(0.609016\pi\)
\(240\) 0 0
\(241\) −8.20445 −0.528495 −0.264248 0.964455i \(-0.585124\pi\)
−0.264248 + 0.964455i \(0.585124\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 19.4571 1.24561
\(245\) 2.80853 0.179430
\(246\) 0 0
\(247\) 8.14519 0.518266
\(248\) −3.87311 −0.245943
\(249\) 0 0
\(250\) 0.713990 0.0451567
\(251\) 16.4452 1.03801 0.519005 0.854771i \(-0.326302\pi\)
0.519005 + 0.854771i \(0.326302\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.18512 −0.0743608
\(255\) 0 0
\(256\) 14.8536 0.928352
\(257\) 12.0440 0.751282 0.375641 0.926765i \(-0.377423\pi\)
0.375641 + 0.926765i \(0.377423\pi\)
\(258\) 0 0
\(259\) 1.53525 0.0953960
\(260\) −6.02066 −0.373385
\(261\) 0 0
\(262\) 1.97127 0.121786
\(263\) −1.87602 −0.115681 −0.0578403 0.998326i \(-0.518421\pi\)
−0.0578403 + 0.998326i \(0.518421\pi\)
\(264\) 0 0
\(265\) 10.7441 0.660003
\(266\) −0.908009 −0.0556736
\(267\) 0 0
\(268\) 4.09215 0.249968
\(269\) 14.0290 0.855362 0.427681 0.903930i \(-0.359331\pi\)
0.427681 + 0.903930i \(0.359331\pi\)
\(270\) 0 0
\(271\) 1.35328 0.0822056 0.0411028 0.999155i \(-0.486913\pi\)
0.0411028 + 0.999155i \(0.486913\pi\)
\(272\) −27.2298 −1.65105
\(273\) 0 0
\(274\) −1.71081 −0.103354
\(275\) 0 0
\(276\) 0 0
\(277\) −13.3835 −0.804138 −0.402069 0.915609i \(-0.631709\pi\)
−0.402069 + 0.915609i \(0.631709\pi\)
\(278\) 1.10483 0.0662635
\(279\) 0 0
\(280\) 1.34724 0.0805129
\(281\) −2.31887 −0.138332 −0.0691661 0.997605i \(-0.522034\pi\)
−0.0691661 + 0.997605i \(0.522034\pi\)
\(282\) 0 0
\(283\) −22.2679 −1.32369 −0.661845 0.749641i \(-0.730226\pi\)
−0.661845 + 0.749641i \(0.730226\pi\)
\(284\) 24.8610 1.47523
\(285\) 0 0
\(286\) 0 0
\(287\) −9.29986 −0.548953
\(288\) 0 0
\(289\) 31.4177 1.84810
\(290\) 0.414857 0.0243612
\(291\) 0 0
\(292\) −5.09698 −0.298278
\(293\) 2.43981 0.142535 0.0712675 0.997457i \(-0.477296\pi\)
0.0712675 + 0.997457i \(0.477296\pi\)
\(294\) 0 0
\(295\) 18.7146 1.08961
\(296\) 0.736455 0.0428056
\(297\) 0 0
\(298\) 1.39858 0.0810176
\(299\) −5.20999 −0.301301
\(300\) 0 0
\(301\) −5.23402 −0.301684
\(302\) −2.15625 −0.124078
\(303\) 0 0
\(304\) 29.5223 1.69322
\(305\) −27.5222 −1.57592
\(306\) 0 0
\(307\) −8.89055 −0.507410 −0.253705 0.967282i \(-0.581649\pi\)
−0.253705 + 0.967282i \(0.581649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.72931 0.155015
\(311\) 11.8347 0.671085 0.335543 0.942025i \(-0.391080\pi\)
0.335543 + 0.942025i \(0.391080\pi\)
\(312\) 0 0
\(313\) −29.5406 −1.66973 −0.834865 0.550454i \(-0.814455\pi\)
−0.834865 + 0.550454i \(0.814455\pi\)
\(314\) −0.894039 −0.0504536
\(315\) 0 0
\(316\) 30.4346 1.71208
\(317\) 25.1545 1.41282 0.706409 0.707804i \(-0.250314\pi\)
0.706409 + 0.707804i \(0.250314\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −21.4976 −1.20175
\(321\) 0 0
\(322\) 0.580799 0.0323667
\(323\) −52.4941 −2.92085
\(324\) 0 0
\(325\) 3.11790 0.172950
\(326\) −0.945181 −0.0523488
\(327\) 0 0
\(328\) −4.46110 −0.246323
\(329\) 1.89387 0.104412
\(330\) 0 0
\(331\) −9.46333 −0.520152 −0.260076 0.965588i \(-0.583748\pi\)
−0.260076 + 0.965588i \(0.583748\pi\)
\(332\) 4.06241 0.222954
\(333\) 0 0
\(334\) −0.650255 −0.0355804
\(335\) −5.78838 −0.316253
\(336\) 0 0
\(337\) −17.2248 −0.938297 −0.469148 0.883119i \(-0.655439\pi\)
−0.469148 + 0.883119i \(0.655439\pi\)
\(338\) 1.42438 0.0774758
\(339\) 0 0
\(340\) 38.8019 2.10433
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.51074 −0.135370
\(345\) 0 0
\(346\) −2.37363 −0.127607
\(347\) 8.73485 0.468911 0.234456 0.972127i \(-0.424669\pi\)
0.234456 + 0.972127i \(0.424669\pi\)
\(348\) 0 0
\(349\) −29.9851 −1.60506 −0.802532 0.596609i \(-0.796514\pi\)
−0.802532 + 0.596609i \(0.796514\pi\)
\(350\) −0.347577 −0.0185788
\(351\) 0 0
\(352\) 0 0
\(353\) −7.31999 −0.389604 −0.194802 0.980843i \(-0.562406\pi\)
−0.194802 + 0.980843i \(0.562406\pi\)
\(354\) 0 0
\(355\) −35.1661 −1.86642
\(356\) 9.45340 0.501029
\(357\) 0 0
\(358\) −2.80789 −0.148402
\(359\) −4.96996 −0.262304 −0.131152 0.991362i \(-0.541868\pi\)
−0.131152 + 0.991362i \(0.541868\pi\)
\(360\) 0 0
\(361\) 37.9137 1.99546
\(362\) −0.371311 −0.0195156
\(363\) 0 0
\(364\) −2.14371 −0.112361
\(365\) 7.20971 0.377374
\(366\) 0 0
\(367\) −14.2042 −0.741455 −0.370727 0.928742i \(-0.620892\pi\)
−0.370727 + 0.928742i \(0.620892\pi\)
\(368\) −18.8837 −0.984379
\(369\) 0 0
\(370\) −0.518967 −0.0269798
\(371\) 3.82552 0.198611
\(372\) 0 0
\(373\) 8.97781 0.464853 0.232427 0.972614i \(-0.425333\pi\)
0.232427 + 0.972614i \(0.425333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.908480 0.0468513
\(377\) −1.32504 −0.0682433
\(378\) 0 0
\(379\) 10.7896 0.554223 0.277112 0.960838i \(-0.410623\pi\)
0.277112 + 0.960838i \(0.410623\pi\)
\(380\) −42.0687 −2.15808
\(381\) 0 0
\(382\) 1.51310 0.0774170
\(383\) 23.5100 1.20130 0.600652 0.799511i \(-0.294908\pi\)
0.600652 + 0.799511i \(0.294908\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.48766 0.126619
\(387\) 0 0
\(388\) −18.0982 −0.918797
\(389\) 11.8745 0.602063 0.301032 0.953614i \(-0.402669\pi\)
0.301032 + 0.953614i \(0.402669\pi\)
\(390\) 0 0
\(391\) 33.5773 1.69808
\(392\) 0.479696 0.0242283
\(393\) 0 0
\(394\) 0.755804 0.0380769
\(395\) −43.0501 −2.16608
\(396\) 0 0
\(397\) −23.7264 −1.19079 −0.595397 0.803431i \(-0.703005\pi\)
−0.595397 + 0.803431i \(0.703005\pi\)
\(398\) 0.826747 0.0414411
\(399\) 0 0
\(400\) 11.3009 0.565043
\(401\) 3.80121 0.189823 0.0949117 0.995486i \(-0.469743\pi\)
0.0949117 + 0.995486i \(0.469743\pi\)
\(402\) 0 0
\(403\) −8.71738 −0.434244
\(404\) −9.41898 −0.468612
\(405\) 0 0
\(406\) 0.147713 0.00733089
\(407\) 0 0
\(408\) 0 0
\(409\) −5.98291 −0.295836 −0.147918 0.989000i \(-0.547257\pi\)
−0.147918 + 0.989000i \(0.547257\pi\)
\(410\) 3.14367 0.155255
\(411\) 0 0
\(412\) −0.696718 −0.0343248
\(413\) 6.66349 0.327889
\(414\) 0 0
\(415\) −5.74631 −0.282075
\(416\) −1.54436 −0.0757185
\(417\) 0 0
\(418\) 0 0
\(419\) 4.16889 0.203664 0.101832 0.994802i \(-0.467530\pi\)
0.101832 + 0.994802i \(0.467530\pi\)
\(420\) 0 0
\(421\) 23.4555 1.14315 0.571575 0.820550i \(-0.306333\pi\)
0.571575 + 0.820550i \(0.306333\pi\)
\(422\) −1.23473 −0.0601055
\(423\) 0 0
\(424\) 1.83509 0.0891197
\(425\) −20.0942 −0.974713
\(426\) 0 0
\(427\) −9.79952 −0.474232
\(428\) 5.05856 0.244515
\(429\) 0 0
\(430\) 1.76928 0.0853221
\(431\) −37.2730 −1.79538 −0.897689 0.440630i \(-0.854755\pi\)
−0.897689 + 0.440630i \(0.854755\pi\)
\(432\) 0 0
\(433\) −22.7863 −1.09504 −0.547520 0.836793i \(-0.684428\pi\)
−0.547520 + 0.836793i \(0.684428\pi\)
\(434\) 0.971796 0.0466477
\(435\) 0 0
\(436\) 7.57517 0.362785
\(437\) −36.4043 −1.74145
\(438\) 0 0
\(439\) −1.42974 −0.0682379 −0.0341189 0.999418i \(-0.510863\pi\)
−0.0341189 + 0.999418i \(0.510863\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.904225 0.0430096
\(443\) 26.4301 1.25573 0.627866 0.778321i \(-0.283928\pi\)
0.627866 + 0.778321i \(0.283928\pi\)
\(444\) 0 0
\(445\) −13.3719 −0.633890
\(446\) 1.62752 0.0770651
\(447\) 0 0
\(448\) −7.65442 −0.361637
\(449\) −14.0870 −0.664806 −0.332403 0.943137i \(-0.607859\pi\)
−0.332403 + 0.943137i \(0.607859\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.59583 0.216170
\(453\) 0 0
\(454\) 1.63406 0.0766902
\(455\) 3.03229 0.142156
\(456\) 0 0
\(457\) 29.4829 1.37915 0.689575 0.724214i \(-0.257797\pi\)
0.689575 + 0.724214i \(0.257797\pi\)
\(458\) 0.174878 0.00817150
\(459\) 0 0
\(460\) 26.9089 1.25463
\(461\) −31.7282 −1.47773 −0.738866 0.673853i \(-0.764638\pi\)
−0.738866 + 0.673853i \(0.764638\pi\)
\(462\) 0 0
\(463\) −12.7839 −0.594117 −0.297059 0.954859i \(-0.596006\pi\)
−0.297059 + 0.954859i \(0.596006\pi\)
\(464\) −4.80264 −0.222957
\(465\) 0 0
\(466\) 0.987417 0.0457412
\(467\) −29.5768 −1.36865 −0.684325 0.729177i \(-0.739903\pi\)
−0.684325 + 0.729177i \(0.739903\pi\)
\(468\) 0 0
\(469\) −2.06100 −0.0951683
\(470\) −0.640191 −0.0295298
\(471\) 0 0
\(472\) 3.19645 0.147129
\(473\) 0 0
\(474\) 0 0
\(475\) 21.7860 0.999610
\(476\) 13.8158 0.633245
\(477\) 0 0
\(478\) 1.24976 0.0571627
\(479\) 28.5574 1.30482 0.652411 0.757866i \(-0.273758\pi\)
0.652411 + 0.757866i \(0.273758\pi\)
\(480\) 0 0
\(481\) 1.65757 0.0755788
\(482\) 0.987487 0.0449788
\(483\) 0 0
\(484\) 0 0
\(485\) 25.6000 1.16244
\(486\) 0 0
\(487\) 1.05828 0.0479552 0.0239776 0.999712i \(-0.492367\pi\)
0.0239776 + 0.999712i \(0.492367\pi\)
\(488\) −4.70079 −0.212795
\(489\) 0 0
\(490\) −0.338034 −0.0152708
\(491\) −18.4489 −0.832586 −0.416293 0.909230i \(-0.636671\pi\)
−0.416293 + 0.909230i \(0.636671\pi\)
\(492\) 0 0
\(493\) 8.53965 0.384606
\(494\) −0.980354 −0.0441082
\(495\) 0 0
\(496\) −31.5962 −1.41871
\(497\) −12.5212 −0.561652
\(498\) 0 0
\(499\) 31.7293 1.42040 0.710199 0.704001i \(-0.248605\pi\)
0.710199 + 0.704001i \(0.248605\pi\)
\(500\) 11.7783 0.526742
\(501\) 0 0
\(502\) −1.97934 −0.0883423
\(503\) 29.0283 1.29431 0.647154 0.762359i \(-0.275959\pi\)
0.647154 + 0.762359i \(0.275959\pi\)
\(504\) 0 0
\(505\) 13.3232 0.592876
\(506\) 0 0
\(507\) 0 0
\(508\) −19.5502 −0.867401
\(509\) −2.88831 −0.128022 −0.0640110 0.997949i \(-0.520389\pi\)
−0.0640110 + 0.997949i \(0.520389\pi\)
\(510\) 0 0
\(511\) 2.56708 0.113561
\(512\) −9.35193 −0.413301
\(513\) 0 0
\(514\) −1.44961 −0.0639396
\(515\) 0.985513 0.0434269
\(516\) 0 0
\(517\) 0 0
\(518\) −0.184783 −0.00811890
\(519\) 0 0
\(520\) 1.45458 0.0637875
\(521\) −31.6708 −1.38752 −0.693762 0.720204i \(-0.744048\pi\)
−0.693762 + 0.720204i \(0.744048\pi\)
\(522\) 0 0
\(523\) −16.0380 −0.701295 −0.350647 0.936508i \(-0.614038\pi\)
−0.350647 + 0.936508i \(0.614038\pi\)
\(524\) 32.5191 1.42060
\(525\) 0 0
\(526\) 0.225798 0.00984526
\(527\) 56.1818 2.44732
\(528\) 0 0
\(529\) 0.285644 0.0124193
\(530\) −1.29316 −0.0561711
\(531\) 0 0
\(532\) −14.9789 −0.649419
\(533\) −10.0408 −0.434916
\(534\) 0 0
\(535\) −7.15538 −0.309354
\(536\) −0.988655 −0.0427034
\(537\) 0 0
\(538\) −1.68853 −0.0727976
\(539\) 0 0
\(540\) 0 0
\(541\) 13.5946 0.584480 0.292240 0.956345i \(-0.405599\pi\)
0.292240 + 0.956345i \(0.405599\pi\)
\(542\) −0.162880 −0.00699630
\(543\) 0 0
\(544\) 9.95310 0.426735
\(545\) −10.7151 −0.458986
\(546\) 0 0
\(547\) −8.82486 −0.377324 −0.188662 0.982042i \(-0.560415\pi\)
−0.188662 + 0.982042i \(0.560415\pi\)
\(548\) −28.2224 −1.20560
\(549\) 0 0
\(550\) 0 0
\(551\) −9.25862 −0.394430
\(552\) 0 0
\(553\) −15.3283 −0.651828
\(554\) 1.61084 0.0684380
\(555\) 0 0
\(556\) 18.2259 0.772949
\(557\) −33.9920 −1.44029 −0.720145 0.693824i \(-0.755925\pi\)
−0.720145 + 0.693824i \(0.755925\pi\)
\(558\) 0 0
\(559\) −5.65104 −0.239014
\(560\) 10.9906 0.464437
\(561\) 0 0
\(562\) 0.279099 0.0117731
\(563\) 20.2256 0.852406 0.426203 0.904628i \(-0.359851\pi\)
0.426203 + 0.904628i \(0.359851\pi\)
\(564\) 0 0
\(565\) −6.50084 −0.273492
\(566\) 2.68016 0.112656
\(567\) 0 0
\(568\) −6.00637 −0.252022
\(569\) −28.9330 −1.21293 −0.606467 0.795109i \(-0.707414\pi\)
−0.606467 + 0.795109i \(0.707414\pi\)
\(570\) 0 0
\(571\) −7.51312 −0.314414 −0.157207 0.987566i \(-0.550249\pi\)
−0.157207 + 0.987566i \(0.550249\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.11933 0.0467199
\(575\) −13.9352 −0.581138
\(576\) 0 0
\(577\) 34.3748 1.43104 0.715521 0.698592i \(-0.246190\pi\)
0.715521 + 0.698592i \(0.246190\pi\)
\(578\) −3.78143 −0.157287
\(579\) 0 0
\(580\) 6.84367 0.284168
\(581\) −2.04602 −0.0848834
\(582\) 0 0
\(583\) 0 0
\(584\) 1.23142 0.0509565
\(585\) 0 0
\(586\) −0.293655 −0.0121308
\(587\) −42.0392 −1.73514 −0.867571 0.497313i \(-0.834320\pi\)
−0.867571 + 0.497313i \(0.834320\pi\)
\(588\) 0 0
\(589\) −60.9118 −2.50983
\(590\) −2.25249 −0.0927333
\(591\) 0 0
\(592\) 6.00789 0.246923
\(593\) −29.2867 −1.20266 −0.601331 0.799000i \(-0.705363\pi\)
−0.601331 + 0.799000i \(0.705363\pi\)
\(594\) 0 0
\(595\) −19.5425 −0.801165
\(596\) 23.0716 0.945051
\(597\) 0 0
\(598\) 0.627074 0.0256430
\(599\) 15.1577 0.619327 0.309663 0.950846i \(-0.399784\pi\)
0.309663 + 0.950846i \(0.399784\pi\)
\(600\) 0 0
\(601\) 31.8735 1.30015 0.650073 0.759872i \(-0.274738\pi\)
0.650073 + 0.759872i \(0.274738\pi\)
\(602\) 0.629967 0.0256755
\(603\) 0 0
\(604\) −35.5705 −1.44734
\(605\) 0 0
\(606\) 0 0
\(607\) −20.1463 −0.817714 −0.408857 0.912598i \(-0.634073\pi\)
−0.408857 + 0.912598i \(0.634073\pi\)
\(608\) −10.7911 −0.437635
\(609\) 0 0
\(610\) 3.31257 0.134122
\(611\) 2.04476 0.0827220
\(612\) 0 0
\(613\) −3.89132 −0.157169 −0.0785844 0.996907i \(-0.525040\pi\)
−0.0785844 + 0.996907i \(0.525040\pi\)
\(614\) 1.07007 0.0431843
\(615\) 0 0
\(616\) 0 0
\(617\) −28.6122 −1.15189 −0.575943 0.817490i \(-0.695365\pi\)
−0.575943 + 0.817490i \(0.695365\pi\)
\(618\) 0 0
\(619\) −26.1546 −1.05124 −0.525621 0.850719i \(-0.676167\pi\)
−0.525621 + 0.850719i \(0.676167\pi\)
\(620\) 45.0240 1.80821
\(621\) 0 0
\(622\) −1.42443 −0.0571143
\(623\) −4.76119 −0.190753
\(624\) 0 0
\(625\) −31.0996 −1.24398
\(626\) 3.55550 0.142106
\(627\) 0 0
\(628\) −14.7485 −0.588529
\(629\) −10.6827 −0.425948
\(630\) 0 0
\(631\) −5.68272 −0.226225 −0.113113 0.993582i \(-0.536082\pi\)
−0.113113 + 0.993582i \(0.536082\pi\)
\(632\) −7.35295 −0.292485
\(633\) 0 0
\(634\) −3.02759 −0.120241
\(635\) 27.6540 1.09741
\(636\) 0 0
\(637\) 1.07967 0.0427782
\(638\) 0 0
\(639\) 0 0
\(640\) 10.6221 0.419874
\(641\) 20.7292 0.818756 0.409378 0.912365i \(-0.365746\pi\)
0.409378 + 0.912365i \(0.365746\pi\)
\(642\) 0 0
\(643\) 21.2756 0.839029 0.419515 0.907749i \(-0.362200\pi\)
0.419515 + 0.907749i \(0.362200\pi\)
\(644\) 9.58114 0.377550
\(645\) 0 0
\(646\) 6.31818 0.248586
\(647\) 17.2718 0.679023 0.339512 0.940602i \(-0.389738\pi\)
0.339512 + 0.940602i \(0.389738\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.375270 −0.0147193
\(651\) 0 0
\(652\) −15.5922 −0.610636
\(653\) −41.0788 −1.60754 −0.803768 0.594942i \(-0.797175\pi\)
−0.803768 + 0.594942i \(0.797175\pi\)
\(654\) 0 0
\(655\) −45.9985 −1.79731
\(656\) −36.3930 −1.42091
\(657\) 0 0
\(658\) −0.227945 −0.00888624
\(659\) −6.00410 −0.233887 −0.116943 0.993139i \(-0.537310\pi\)
−0.116943 + 0.993139i \(0.537310\pi\)
\(660\) 0 0
\(661\) −1.83502 −0.0713739 −0.0356870 0.999363i \(-0.511362\pi\)
−0.0356870 + 0.999363i \(0.511362\pi\)
\(662\) 1.13901 0.0442687
\(663\) 0 0
\(664\) −0.981469 −0.0380884
\(665\) 21.1878 0.821629
\(666\) 0 0
\(667\) 5.92218 0.229308
\(668\) −10.7269 −0.415037
\(669\) 0 0
\(670\) 0.696689 0.0269154
\(671\) 0 0
\(672\) 0 0
\(673\) 44.4403 1.71305 0.856524 0.516107i \(-0.172619\pi\)
0.856524 + 0.516107i \(0.172619\pi\)
\(674\) 2.07318 0.0798559
\(675\) 0 0
\(676\) 23.4972 0.903737
\(677\) −15.5471 −0.597525 −0.298762 0.954327i \(-0.596574\pi\)
−0.298762 + 0.954327i \(0.596574\pi\)
\(678\) 0 0
\(679\) 9.11512 0.349806
\(680\) −9.37447 −0.359494
\(681\) 0 0
\(682\) 0 0
\(683\) 36.8979 1.41186 0.705930 0.708282i \(-0.250529\pi\)
0.705930 + 0.708282i \(0.250529\pi\)
\(684\) 0 0
\(685\) 39.9208 1.52530
\(686\) −0.120360 −0.00459536
\(687\) 0 0
\(688\) −20.4823 −0.780879
\(689\) 4.13032 0.157352
\(690\) 0 0
\(691\) −18.6726 −0.710339 −0.355169 0.934802i \(-0.615577\pi\)
−0.355169 + 0.934802i \(0.615577\pi\)
\(692\) −39.1566 −1.48851
\(693\) 0 0
\(694\) −1.05133 −0.0399078
\(695\) −25.7806 −0.977915
\(696\) 0 0
\(697\) 64.7110 2.45111
\(698\) 3.60900 0.136603
\(699\) 0 0
\(700\) −5.73379 −0.216717
\(701\) 26.1328 0.987022 0.493511 0.869739i \(-0.335713\pi\)
0.493511 + 0.869739i \(0.335713\pi\)
\(702\) 0 0
\(703\) 11.5821 0.436828
\(704\) 0 0
\(705\) 0 0
\(706\) 0.881033 0.0331581
\(707\) 4.74385 0.178411
\(708\) 0 0
\(709\) −16.4449 −0.617602 −0.308801 0.951127i \(-0.599928\pi\)
−0.308801 + 0.951127i \(0.599928\pi\)
\(710\) 4.23259 0.158846
\(711\) 0 0
\(712\) −2.28392 −0.0855936
\(713\) 38.9617 1.45913
\(714\) 0 0
\(715\) 0 0
\(716\) −46.3204 −1.73107
\(717\) 0 0
\(718\) 0.598184 0.0223240
\(719\) 9.34913 0.348664 0.174332 0.984687i \(-0.444223\pi\)
0.174332 + 0.984687i \(0.444223\pi\)
\(720\) 0 0
\(721\) 0.350901 0.0130682
\(722\) −4.56328 −0.169828
\(723\) 0 0
\(724\) −6.12531 −0.227646
\(725\) −3.54411 −0.131625
\(726\) 0 0
\(727\) 27.7523 1.02928 0.514638 0.857408i \(-0.327927\pi\)
0.514638 + 0.857408i \(0.327927\pi\)
\(728\) 0.517915 0.0191952
\(729\) 0 0
\(730\) −0.867760 −0.0321173
\(731\) 36.4198 1.34704
\(732\) 0 0
\(733\) 1.45630 0.0537896 0.0268948 0.999638i \(-0.491438\pi\)
0.0268948 + 0.999638i \(0.491438\pi\)
\(734\) 1.70962 0.0631032
\(735\) 0 0
\(736\) 6.90240 0.254426
\(737\) 0 0
\(738\) 0 0
\(739\) −27.5966 −1.01516 −0.507579 0.861605i \(-0.669459\pi\)
−0.507579 + 0.861605i \(0.669459\pi\)
\(740\) −8.56113 −0.314713
\(741\) 0 0
\(742\) −0.460439 −0.0169033
\(743\) −27.9773 −1.02639 −0.513194 0.858273i \(-0.671538\pi\)
−0.513194 + 0.858273i \(0.671538\pi\)
\(744\) 0 0
\(745\) −32.6350 −1.19566
\(746\) −1.08057 −0.0395624
\(747\) 0 0
\(748\) 0 0
\(749\) −2.54774 −0.0930922
\(750\) 0 0
\(751\) 34.0957 1.24417 0.622085 0.782950i \(-0.286286\pi\)
0.622085 + 0.782950i \(0.286286\pi\)
\(752\) 7.41125 0.270260
\(753\) 0 0
\(754\) 0.159482 0.00580800
\(755\) 50.3148 1.83114
\(756\) 0 0
\(757\) −39.7629 −1.44521 −0.722604 0.691262i \(-0.757055\pi\)
−0.722604 + 0.691262i \(0.757055\pi\)
\(758\) −1.29863 −0.0471684
\(759\) 0 0
\(760\) 10.1637 0.368677
\(761\) 3.82415 0.138625 0.0693127 0.997595i \(-0.477919\pi\)
0.0693127 + 0.997595i \(0.477919\pi\)
\(762\) 0 0
\(763\) −3.81522 −0.138120
\(764\) 24.9608 0.903051
\(765\) 0 0
\(766\) −2.82966 −0.102240
\(767\) 7.19440 0.259775
\(768\) 0 0
\(769\) −19.6583 −0.708897 −0.354449 0.935075i \(-0.615331\pi\)
−0.354449 + 0.935075i \(0.615331\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 41.0376 1.47698
\(773\) 17.5966 0.632905 0.316452 0.948608i \(-0.397508\pi\)
0.316452 + 0.948608i \(0.397508\pi\)
\(774\) 0 0
\(775\) −23.3165 −0.837552
\(776\) 4.37249 0.156963
\(777\) 0 0
\(778\) −1.42922 −0.0512400
\(779\) −70.1592 −2.51371
\(780\) 0 0
\(781\) 0 0
\(782\) −4.04136 −0.144519
\(783\) 0 0
\(784\) 3.91329 0.139760
\(785\) 20.8619 0.744592
\(786\) 0 0
\(787\) −54.0967 −1.92834 −0.964169 0.265287i \(-0.914533\pi\)
−0.964169 + 0.265287i \(0.914533\pi\)
\(788\) 12.4681 0.444158
\(789\) 0 0
\(790\) 5.18150 0.184349
\(791\) −2.31468 −0.0823006
\(792\) 0 0
\(793\) −10.5803 −0.375717
\(794\) 2.85571 0.101345
\(795\) 0 0
\(796\) 13.6384 0.483401
\(797\) −36.5244 −1.29376 −0.646881 0.762591i \(-0.723927\pi\)
−0.646881 + 0.762591i \(0.723927\pi\)
\(798\) 0 0
\(799\) −13.1781 −0.466206
\(800\) −4.13072 −0.146043
\(801\) 0 0
\(802\) −0.457513 −0.0161554
\(803\) 0 0
\(804\) 0 0
\(805\) −13.5526 −0.477666
\(806\) 1.04922 0.0369573
\(807\) 0 0
\(808\) 2.27561 0.0800555
\(809\) −46.9354 −1.65016 −0.825081 0.565015i \(-0.808870\pi\)
−0.825081 + 0.565015i \(0.808870\pi\)
\(810\) 0 0
\(811\) −12.6615 −0.444605 −0.222303 0.974978i \(-0.571357\pi\)
−0.222303 + 0.974978i \(0.571357\pi\)
\(812\) 2.43675 0.0855131
\(813\) 0 0
\(814\) 0 0
\(815\) 22.0552 0.772561
\(816\) 0 0
\(817\) −39.4861 −1.38144
\(818\) 0.720103 0.0251778
\(819\) 0 0
\(820\) 51.8594 1.81101
\(821\) 49.2938 1.72037 0.860183 0.509986i \(-0.170349\pi\)
0.860183 + 0.509986i \(0.170349\pi\)
\(822\) 0 0
\(823\) 19.2259 0.670173 0.335087 0.942187i \(-0.391234\pi\)
0.335087 + 0.942187i \(0.391234\pi\)
\(824\) 0.168326 0.00586390
\(825\) 0 0
\(826\) −0.802017 −0.0279057
\(827\) −43.6419 −1.51758 −0.758789 0.651337i \(-0.774208\pi\)
−0.758789 + 0.651337i \(0.774208\pi\)
\(828\) 0 0
\(829\) −36.7072 −1.27489 −0.637447 0.770494i \(-0.720010\pi\)
−0.637447 + 0.770494i \(0.720010\pi\)
\(830\) 0.691625 0.0240067
\(831\) 0 0
\(832\) −8.26428 −0.286512
\(833\) −6.95828 −0.241090
\(834\) 0 0
\(835\) 15.1733 0.525094
\(836\) 0 0
\(837\) 0 0
\(838\) −0.501768 −0.0173333
\(839\) −40.4545 −1.39665 −0.698323 0.715783i \(-0.746070\pi\)
−0.698323 + 0.715783i \(0.746070\pi\)
\(840\) 0 0
\(841\) −27.4938 −0.948063
\(842\) −2.82310 −0.0972904
\(843\) 0 0
\(844\) −20.3686 −0.701117
\(845\) −33.2369 −1.14339
\(846\) 0 0
\(847\) 0 0
\(848\) 14.9704 0.514085
\(849\) 0 0
\(850\) 2.41854 0.0829552
\(851\) −7.40840 −0.253957
\(852\) 0 0
\(853\) 25.4948 0.872927 0.436463 0.899722i \(-0.356231\pi\)
0.436463 + 0.899722i \(0.356231\pi\)
\(854\) 1.17947 0.0403606
\(855\) 0 0
\(856\) −1.22214 −0.0417718
\(857\) −49.4756 −1.69005 −0.845027 0.534723i \(-0.820416\pi\)
−0.845027 + 0.534723i \(0.820416\pi\)
\(858\) 0 0
\(859\) 21.6779 0.739641 0.369821 0.929103i \(-0.379419\pi\)
0.369821 + 0.929103i \(0.379419\pi\)
\(860\) 29.1868 0.995262
\(861\) 0 0
\(862\) 4.48618 0.152800
\(863\) −13.6398 −0.464306 −0.232153 0.972679i \(-0.574577\pi\)
−0.232153 + 0.972679i \(0.574577\pi\)
\(864\) 0 0
\(865\) 55.3873 1.88322
\(866\) 2.74256 0.0931959
\(867\) 0 0
\(868\) 16.0312 0.544135
\(869\) 0 0
\(870\) 0 0
\(871\) −2.22521 −0.0753985
\(872\) −1.83014 −0.0619765
\(873\) 0 0
\(874\) 4.38161 0.148210
\(875\) −5.93213 −0.200542
\(876\) 0 0
\(877\) 29.1290 0.983618 0.491809 0.870703i \(-0.336336\pi\)
0.491809 + 0.870703i \(0.336336\pi\)
\(878\) 0.172084 0.00580754
\(879\) 0 0
\(880\) 0 0
\(881\) 48.8256 1.64498 0.822488 0.568783i \(-0.192586\pi\)
0.822488 + 0.568783i \(0.192586\pi\)
\(882\) 0 0
\(883\) −24.2131 −0.814837 −0.407419 0.913242i \(-0.633571\pi\)
−0.407419 + 0.913242i \(0.633571\pi\)
\(884\) 14.9165 0.501697
\(885\) 0 0
\(886\) −3.18113 −0.106872
\(887\) −6.64749 −0.223201 −0.111600 0.993753i \(-0.535598\pi\)
−0.111600 + 0.993753i \(0.535598\pi\)
\(888\) 0 0
\(889\) 9.84644 0.330239
\(890\) 1.60944 0.0539486
\(891\) 0 0
\(892\) 26.8483 0.898947
\(893\) 14.2875 0.478114
\(894\) 0 0
\(895\) 65.5205 2.19011
\(896\) 3.78208 0.126350
\(897\) 0 0
\(898\) 1.69551 0.0565799
\(899\) 9.90903 0.330485
\(900\) 0 0
\(901\) −26.6191 −0.886809
\(902\) 0 0
\(903\) 0 0
\(904\) −1.11034 −0.0369295
\(905\) 8.66431 0.288011
\(906\) 0 0
\(907\) −8.37806 −0.278189 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(908\) 26.9562 0.894573
\(909\) 0 0
\(910\) −0.364966 −0.0120985
\(911\) 18.7868 0.622434 0.311217 0.950339i \(-0.399263\pi\)
0.311217 + 0.950339i \(0.399263\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.54856 −0.117376
\(915\) 0 0
\(916\) 2.88486 0.0953186
\(917\) −16.3782 −0.540855
\(918\) 0 0
\(919\) 18.0039 0.593894 0.296947 0.954894i \(-0.404032\pi\)
0.296947 + 0.954894i \(0.404032\pi\)
\(920\) −6.50113 −0.214336
\(921\) 0 0
\(922\) 3.81881 0.125766
\(923\) −13.5188 −0.444977
\(924\) 0 0
\(925\) 4.43353 0.145773
\(926\) 1.53867 0.0505637
\(927\) 0 0
\(928\) 1.75547 0.0576262
\(929\) 10.1225 0.332109 0.166054 0.986117i \(-0.446897\pi\)
0.166054 + 0.986117i \(0.446897\pi\)
\(930\) 0 0
\(931\) 7.54411 0.247248
\(932\) 16.2889 0.533560
\(933\) 0 0
\(934\) 3.55986 0.116482
\(935\) 0 0
\(936\) 0 0
\(937\) −28.2113 −0.921622 −0.460811 0.887498i \(-0.652441\pi\)
−0.460811 + 0.887498i \(0.652441\pi\)
\(938\) 0.248062 0.00809952
\(939\) 0 0
\(940\) −10.5609 −0.344458
\(941\) 20.1501 0.656875 0.328437 0.944526i \(-0.393478\pi\)
0.328437 + 0.944526i \(0.393478\pi\)
\(942\) 0 0
\(943\) 44.8766 1.46138
\(944\) 26.0762 0.848707
\(945\) 0 0
\(946\) 0 0
\(947\) −0.125141 −0.00406653 −0.00203326 0.999998i \(-0.500647\pi\)
−0.00203326 + 0.999998i \(0.500647\pi\)
\(948\) 0 0
\(949\) 2.77161 0.0899703
\(950\) −2.62216 −0.0850741
\(951\) 0 0
\(952\) −3.33786 −0.108181
\(953\) 29.3495 0.950723 0.475362 0.879790i \(-0.342317\pi\)
0.475362 + 0.879790i \(0.342317\pi\)
\(954\) 0 0
\(955\) −35.3073 −1.14252
\(956\) 20.6166 0.666790
\(957\) 0 0
\(958\) −3.43717 −0.111050
\(959\) 14.2142 0.458999
\(960\) 0 0
\(961\) 34.1909 1.10293
\(962\) −0.199505 −0.00643231
\(963\) 0 0
\(964\) 16.2900 0.524667
\(965\) −58.0481 −1.86863
\(966\) 0 0
\(967\) −51.9463 −1.67048 −0.835240 0.549885i \(-0.814672\pi\)
−0.835240 + 0.549885i \(0.814672\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −3.08122 −0.0989320
\(971\) 25.0066 0.802500 0.401250 0.915969i \(-0.368576\pi\)
0.401250 + 0.915969i \(0.368576\pi\)
\(972\) 0 0
\(973\) −9.17942 −0.294279
\(974\) −0.127374 −0.00408134
\(975\) 0 0
\(976\) −38.3484 −1.22750
\(977\) 17.4156 0.557176 0.278588 0.960411i \(-0.410134\pi\)
0.278588 + 0.960411i \(0.410134\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.57636 −0.178130
\(981\) 0 0
\(982\) 2.22051 0.0708592
\(983\) 42.3349 1.35028 0.675138 0.737692i \(-0.264084\pi\)
0.675138 + 0.737692i \(0.264084\pi\)
\(984\) 0 0
\(985\) −17.6362 −0.561937
\(986\) −1.02783 −0.0327328
\(987\) 0 0
\(988\) −16.1724 −0.514512
\(989\) 25.2569 0.803122
\(990\) 0 0
\(991\) 55.6263 1.76703 0.883513 0.468406i \(-0.155172\pi\)
0.883513 + 0.468406i \(0.155172\pi\)
\(992\) 11.5491 0.366685
\(993\) 0 0
\(994\) 1.50705 0.0478007
\(995\) −19.2916 −0.611586
\(996\) 0 0
\(997\) −16.4267 −0.520240 −0.260120 0.965576i \(-0.583762\pi\)
−0.260120 + 0.965576i \(0.583762\pi\)
\(998\) −3.81894 −0.120886
\(999\) 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 7623.2.a.cp.1.4 6
3.2 odd 2 847.2.a.n.1.3 yes 6
11.10 odd 2 7623.2.a.cs.1.3 6
21.20 even 2 5929.2.a.bm.1.3 6
33.2 even 10 847.2.f.z.323.3 24
33.5 odd 10 847.2.f.y.729.4 24
33.8 even 10 847.2.f.z.372.4 24
33.14 odd 10 847.2.f.y.372.3 24
33.17 even 10 847.2.f.z.729.3 24
33.20 odd 10 847.2.f.y.323.4 24
33.26 odd 10 847.2.f.y.148.3 24
33.29 even 10 847.2.f.z.148.4 24
33.32 even 2 847.2.a.m.1.4 6
231.230 odd 2 5929.2.a.bj.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.m.1.4 6 33.32 even 2
847.2.a.n.1.3 yes 6 3.2 odd 2
847.2.f.y.148.3 24 33.26 odd 10
847.2.f.y.323.4 24 33.20 odd 10
847.2.f.y.372.3 24 33.14 odd 10
847.2.f.y.729.4 24 33.5 odd 10
847.2.f.z.148.4 24 33.29 even 10
847.2.f.z.323.3 24 33.2 even 10
847.2.f.z.372.4 24 33.8 even 10
847.2.f.z.729.3 24 33.17 even 10
5929.2.a.bj.1.4 6 231.230 odd 2
5929.2.a.bm.1.3 6 21.20 even 2
7623.2.a.cp.1.4 6 1.1 even 1 trivial
7623.2.a.cs.1.3 6 11.10 odd 2