Properties

Label 1521.2.a.p.1.1
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.80194 q^{2} +1.24698 q^{4} -1.44504 q^{5} +3.44504 q^{7} +1.35690 q^{8} +O(q^{10})\) \(q-1.80194 q^{2} +1.24698 q^{4} -1.44504 q^{5} +3.44504 q^{7} +1.35690 q^{8} +2.60388 q^{10} +5.18598 q^{11} -6.20775 q^{14} -4.93900 q^{16} +0.753020 q^{17} +7.96077 q^{19} -1.80194 q^{20} -9.34481 q^{22} +2.82908 q^{23} -2.91185 q^{25} +4.29590 q^{28} +3.91185 q^{29} -4.89977 q^{31} +6.18598 q^{32} -1.35690 q^{34} -4.97823 q^{35} +6.24698 q^{37} -14.3448 q^{38} -1.96077 q^{40} -1.80194 q^{41} -7.09783 q^{43} +6.46681 q^{44} -5.09783 q^{46} -10.5526 q^{47} +4.86831 q^{49} +5.24698 q^{50} +3.08815 q^{53} -7.49396 q^{55} +4.67456 q^{56} -7.04892 q^{58} -1.87800 q^{59} +3.34481 q^{61} +8.82908 q^{62} -1.26875 q^{64} -4.54288 q^{67} +0.939001 q^{68} +8.97046 q^{70} +9.11960 q^{71} +2.95108 q^{73} -11.2567 q^{74} +9.92692 q^{76} +17.8659 q^{77} -9.43296 q^{79} +7.13706 q^{80} +3.24698 q^{82} +6.46681 q^{83} -1.08815 q^{85} +12.7899 q^{86} +7.03684 q^{88} -1.15883 q^{89} +3.52781 q^{92} +19.0151 q^{94} -11.5036 q^{95} +8.65817 q^{97} -8.77240 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} - q^{4} - 4q^{5} + 10q^{7} + O(q^{10}) \) \( 3q - q^{2} - q^{4} - 4q^{5} + 10q^{7} - q^{10} + q^{11} - q^{14} - 5q^{16} + 7q^{17} + 11q^{19} - q^{20} - 5q^{22} - 2q^{23} - 5q^{25} - q^{28} + 8q^{29} + 8q^{31} + 4q^{32} - 18q^{35} + 14q^{37} - 20q^{38} + 7q^{40} - q^{41} - 3q^{43} + 16q^{44} + 3q^{46} + 9q^{47} + 17q^{49} + 11q^{50} + 13q^{53} - 13q^{55} - 7q^{56} - 12q^{58} + 14q^{59} - 13q^{61} + 16q^{62} + 4q^{64} + 5q^{67} - 7q^{68} - 8q^{70} + 6q^{71} + 18q^{73} - 7q^{74} + q^{76} + 15q^{77} - 9q^{79} + 16q^{80} + 5q^{82} + 16q^{83} - 7q^{85} + 15q^{86} - 7q^{88} + 5q^{89} + 17q^{92} + 32q^{94} - 3q^{95} + 5q^{97} + 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80194 −1.27416 −0.637081 0.770797i \(-0.719858\pi\)
−0.637081 + 0.770797i \(0.719858\pi\)
\(3\) 0 0
\(4\) 1.24698 0.623490
\(5\) −1.44504 −0.646242 −0.323121 0.946358i \(-0.604732\pi\)
−0.323121 + 0.946358i \(0.604732\pi\)
\(6\) 0 0
\(7\) 3.44504 1.30210 0.651052 0.759033i \(-0.274328\pi\)
0.651052 + 0.759033i \(0.274328\pi\)
\(8\) 1.35690 0.479735
\(9\) 0 0
\(10\) 2.60388 0.823418
\(11\) 5.18598 1.56363 0.781816 0.623509i \(-0.214294\pi\)
0.781816 + 0.623509i \(0.214294\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −6.20775 −1.65909
\(15\) 0 0
\(16\) −4.93900 −1.23475
\(17\) 0.753020 0.182634 0.0913171 0.995822i \(-0.470892\pi\)
0.0913171 + 0.995822i \(0.470892\pi\)
\(18\) 0 0
\(19\) 7.96077 1.82633 0.913163 0.407594i \(-0.133632\pi\)
0.913163 + 0.407594i \(0.133632\pi\)
\(20\) −1.80194 −0.402926
\(21\) 0 0
\(22\) −9.34481 −1.99232
\(23\) 2.82908 0.589905 0.294952 0.955512i \(-0.404696\pi\)
0.294952 + 0.955512i \(0.404696\pi\)
\(24\) 0 0
\(25\) −2.91185 −0.582371
\(26\) 0 0
\(27\) 0 0
\(28\) 4.29590 0.811848
\(29\) 3.91185 0.726413 0.363207 0.931709i \(-0.381682\pi\)
0.363207 + 0.931709i \(0.381682\pi\)
\(30\) 0 0
\(31\) −4.89977 −0.880025 −0.440013 0.897992i \(-0.645026\pi\)
−0.440013 + 0.897992i \(0.645026\pi\)
\(32\) 6.18598 1.09354
\(33\) 0 0
\(34\) −1.35690 −0.232706
\(35\) −4.97823 −0.841474
\(36\) 0 0
\(37\) 6.24698 1.02700 0.513499 0.858090i \(-0.328349\pi\)
0.513499 + 0.858090i \(0.328349\pi\)
\(38\) −14.3448 −2.32704
\(39\) 0 0
\(40\) −1.96077 −0.310025
\(41\) −1.80194 −0.281415 −0.140708 0.990051i \(-0.544938\pi\)
−0.140708 + 0.990051i \(0.544938\pi\)
\(42\) 0 0
\(43\) −7.09783 −1.08241 −0.541205 0.840891i \(-0.682032\pi\)
−0.541205 + 0.840891i \(0.682032\pi\)
\(44\) 6.46681 0.974909
\(45\) 0 0
\(46\) −5.09783 −0.751635
\(47\) −10.5526 −1.53925 −0.769625 0.638496i \(-0.779557\pi\)
−0.769625 + 0.638496i \(0.779557\pi\)
\(48\) 0 0
\(49\) 4.86831 0.695473
\(50\) 5.24698 0.742035
\(51\) 0 0
\(52\) 0 0
\(53\) 3.08815 0.424189 0.212095 0.977249i \(-0.431971\pi\)
0.212095 + 0.977249i \(0.431971\pi\)
\(54\) 0 0
\(55\) −7.49396 −1.01049
\(56\) 4.67456 0.624665
\(57\) 0 0
\(58\) −7.04892 −0.925568
\(59\) −1.87800 −0.244495 −0.122248 0.992500i \(-0.539010\pi\)
−0.122248 + 0.992500i \(0.539010\pi\)
\(60\) 0 0
\(61\) 3.34481 0.428260 0.214130 0.976805i \(-0.431308\pi\)
0.214130 + 0.976805i \(0.431308\pi\)
\(62\) 8.82908 1.12129
\(63\) 0 0
\(64\) −1.26875 −0.158594
\(65\) 0 0
\(66\) 0 0
\(67\) −4.54288 −0.555001 −0.277500 0.960726i \(-0.589506\pi\)
−0.277500 + 0.960726i \(0.589506\pi\)
\(68\) 0.939001 0.113871
\(69\) 0 0
\(70\) 8.97046 1.07218
\(71\) 9.11960 1.08230 0.541149 0.840927i \(-0.317989\pi\)
0.541149 + 0.840927i \(0.317989\pi\)
\(72\) 0 0
\(73\) 2.95108 0.345398 0.172699 0.984975i \(-0.444751\pi\)
0.172699 + 0.984975i \(0.444751\pi\)
\(74\) −11.2567 −1.30856
\(75\) 0 0
\(76\) 9.92692 1.13870
\(77\) 17.8659 2.03601
\(78\) 0 0
\(79\) −9.43296 −1.06129 −0.530645 0.847594i \(-0.678050\pi\)
−0.530645 + 0.847594i \(0.678050\pi\)
\(80\) 7.13706 0.797948
\(81\) 0 0
\(82\) 3.24698 0.358569
\(83\) 6.46681 0.709825 0.354912 0.934900i \(-0.384511\pi\)
0.354912 + 0.934900i \(0.384511\pi\)
\(84\) 0 0
\(85\) −1.08815 −0.118026
\(86\) 12.7899 1.37917
\(87\) 0 0
\(88\) 7.03684 0.750129
\(89\) −1.15883 −0.122836 −0.0614181 0.998112i \(-0.519562\pi\)
−0.0614181 + 0.998112i \(0.519562\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.52781 0.367800
\(93\) 0 0
\(94\) 19.0151 1.96125
\(95\) −11.5036 −1.18025
\(96\) 0 0
\(97\) 8.65817 0.879104 0.439552 0.898217i \(-0.355137\pi\)
0.439552 + 0.898217i \(0.355137\pi\)
\(98\) −8.77240 −0.886146
\(99\) 0 0
\(100\) −3.63102 −0.363102
\(101\) 8.47650 0.843443 0.421722 0.906725i \(-0.361426\pi\)
0.421722 + 0.906725i \(0.361426\pi\)
\(102\) 0 0
\(103\) 5.64742 0.556456 0.278228 0.960515i \(-0.410253\pi\)
0.278228 + 0.960515i \(0.410253\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −5.56465 −0.540486
\(107\) 6.73556 0.651151 0.325576 0.945516i \(-0.394442\pi\)
0.325576 + 0.945516i \(0.394442\pi\)
\(108\) 0 0
\(109\) −2.07606 −0.198851 −0.0994255 0.995045i \(-0.531700\pi\)
−0.0994255 + 0.995045i \(0.531700\pi\)
\(110\) 13.5036 1.28752
\(111\) 0 0
\(112\) −17.0151 −1.60777
\(113\) −6.16852 −0.580286 −0.290143 0.956983i \(-0.593703\pi\)
−0.290143 + 0.956983i \(0.593703\pi\)
\(114\) 0 0
\(115\) −4.08815 −0.381222
\(116\) 4.87800 0.452911
\(117\) 0 0
\(118\) 3.38404 0.311526
\(119\) 2.59419 0.237809
\(120\) 0 0
\(121\) 15.8944 1.44495
\(122\) −6.02715 −0.545672
\(123\) 0 0
\(124\) −6.10992 −0.548687
\(125\) 11.4330 1.02260
\(126\) 0 0
\(127\) −14.2620 −1.26555 −0.632776 0.774335i \(-0.718085\pi\)
−0.632776 + 0.774335i \(0.718085\pi\)
\(128\) −10.0858 −0.891463
\(129\) 0 0
\(130\) 0 0
\(131\) −22.6015 −1.97470 −0.987350 0.158554i \(-0.949317\pi\)
−0.987350 + 0.158554i \(0.949317\pi\)
\(132\) 0 0
\(133\) 27.4252 2.37807
\(134\) 8.18598 0.707161
\(135\) 0 0
\(136\) 1.02177 0.0876161
\(137\) 13.6353 1.16495 0.582473 0.812850i \(-0.302085\pi\)
0.582473 + 0.812850i \(0.302085\pi\)
\(138\) 0 0
\(139\) 17.6015 1.49294 0.746469 0.665420i \(-0.231748\pi\)
0.746469 + 0.665420i \(0.231748\pi\)
\(140\) −6.20775 −0.524651
\(141\) 0 0
\(142\) −16.4330 −1.37902
\(143\) 0 0
\(144\) 0 0
\(145\) −5.65279 −0.469439
\(146\) −5.31767 −0.440093
\(147\) 0 0
\(148\) 7.78986 0.640322
\(149\) −12.7385 −1.04358 −0.521791 0.853073i \(-0.674736\pi\)
−0.521791 + 0.853073i \(0.674736\pi\)
\(150\) 0 0
\(151\) 15.6407 1.27282 0.636412 0.771350i \(-0.280418\pi\)
0.636412 + 0.771350i \(0.280418\pi\)
\(152\) 10.8019 0.876153
\(153\) 0 0
\(154\) −32.1933 −2.59421
\(155\) 7.08038 0.568710
\(156\) 0 0
\(157\) −0.823708 −0.0657391 −0.0328695 0.999460i \(-0.510465\pi\)
−0.0328695 + 0.999460i \(0.510465\pi\)
\(158\) 16.9976 1.35226
\(159\) 0 0
\(160\) −8.93900 −0.706690
\(161\) 9.74632 0.768117
\(162\) 0 0
\(163\) 6.26875 0.491006 0.245503 0.969396i \(-0.421047\pi\)
0.245503 + 0.969396i \(0.421047\pi\)
\(164\) −2.24698 −0.175460
\(165\) 0 0
\(166\) −11.6528 −0.904432
\(167\) 7.45042 0.576531 0.288265 0.957551i \(-0.406921\pi\)
0.288265 + 0.957551i \(0.406921\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.96077 0.150384
\(171\) 0 0
\(172\) −8.85086 −0.674871
\(173\) 2.00969 0.152794 0.0763969 0.997077i \(-0.475658\pi\)
0.0763969 + 0.997077i \(0.475658\pi\)
\(174\) 0 0
\(175\) −10.0315 −0.758307
\(176\) −25.6136 −1.93070
\(177\) 0 0
\(178\) 2.08815 0.156513
\(179\) −20.0368 −1.49762 −0.748812 0.662783i \(-0.769375\pi\)
−0.748812 + 0.662783i \(0.769375\pi\)
\(180\) 0 0
\(181\) 24.1226 1.79302 0.896509 0.443026i \(-0.146095\pi\)
0.896509 + 0.443026i \(0.146095\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.83877 0.282998
\(185\) −9.02715 −0.663689
\(186\) 0 0
\(187\) 3.90515 0.285573
\(188\) −13.1588 −0.959707
\(189\) 0 0
\(190\) 20.7289 1.50383
\(191\) 7.08038 0.512318 0.256159 0.966635i \(-0.417543\pi\)
0.256159 + 0.966635i \(0.417543\pi\)
\(192\) 0 0
\(193\) 9.76809 0.703122 0.351561 0.936165i \(-0.385651\pi\)
0.351561 + 0.936165i \(0.385651\pi\)
\(194\) −15.6015 −1.12012
\(195\) 0 0
\(196\) 6.07069 0.433621
\(197\) −23.4112 −1.66798 −0.833989 0.551781i \(-0.813948\pi\)
−0.833989 + 0.551781i \(0.813948\pi\)
\(198\) 0 0
\(199\) 4.02475 0.285307 0.142654 0.989773i \(-0.454437\pi\)
0.142654 + 0.989773i \(0.454437\pi\)
\(200\) −3.95108 −0.279384
\(201\) 0 0
\(202\) −15.2741 −1.07468
\(203\) 13.4765 0.945865
\(204\) 0 0
\(205\) 2.60388 0.181863
\(206\) −10.1763 −0.709016
\(207\) 0 0
\(208\) 0 0
\(209\) 41.2844 2.85570
\(210\) 0 0
\(211\) −3.91185 −0.269303 −0.134652 0.990893i \(-0.542992\pi\)
−0.134652 + 0.990893i \(0.542992\pi\)
\(212\) 3.85086 0.264478
\(213\) 0 0
\(214\) −12.1371 −0.829673
\(215\) 10.2567 0.699499
\(216\) 0 0
\(217\) −16.8799 −1.14588
\(218\) 3.74094 0.253368
\(219\) 0 0
\(220\) −9.34481 −0.630027
\(221\) 0 0
\(222\) 0 0
\(223\) 7.44935 0.498846 0.249423 0.968395i \(-0.419759\pi\)
0.249423 + 0.968395i \(0.419759\pi\)
\(224\) 21.3110 1.42390
\(225\) 0 0
\(226\) 11.1153 0.739378
\(227\) 21.2500 1.41041 0.705205 0.709004i \(-0.250855\pi\)
0.705205 + 0.709004i \(0.250855\pi\)
\(228\) 0 0
\(229\) 9.29590 0.614290 0.307145 0.951663i \(-0.400626\pi\)
0.307145 + 0.951663i \(0.400626\pi\)
\(230\) 7.36658 0.485738
\(231\) 0 0
\(232\) 5.30798 0.348486
\(233\) −16.2107 −1.06200 −0.531000 0.847372i \(-0.678184\pi\)
−0.531000 + 0.847372i \(0.678184\pi\)
\(234\) 0 0
\(235\) 15.2489 0.994728
\(236\) −2.34183 −0.152440
\(237\) 0 0
\(238\) −4.67456 −0.303007
\(239\) 13.5090 0.873826 0.436913 0.899504i \(-0.356072\pi\)
0.436913 + 0.899504i \(0.356072\pi\)
\(240\) 0 0
\(241\) 6.26875 0.403806 0.201903 0.979406i \(-0.435287\pi\)
0.201903 + 0.979406i \(0.435287\pi\)
\(242\) −28.6407 −1.84109
\(243\) 0 0
\(244\) 4.17092 0.267015
\(245\) −7.03492 −0.449444
\(246\) 0 0
\(247\) 0 0
\(248\) −6.64848 −0.422179
\(249\) 0 0
\(250\) −20.6015 −1.30295
\(251\) 0.753020 0.0475302 0.0237651 0.999718i \(-0.492435\pi\)
0.0237651 + 0.999718i \(0.492435\pi\)
\(252\) 0 0
\(253\) 14.6716 0.922394
\(254\) 25.6993 1.61252
\(255\) 0 0
\(256\) 20.7114 1.29446
\(257\) −19.7265 −1.23050 −0.615252 0.788331i \(-0.710946\pi\)
−0.615252 + 0.788331i \(0.710946\pi\)
\(258\) 0 0
\(259\) 21.5211 1.33726
\(260\) 0 0
\(261\) 0 0
\(262\) 40.7265 2.51609
\(263\) 17.6093 1.08583 0.542917 0.839787i \(-0.317320\pi\)
0.542917 + 0.839787i \(0.317320\pi\)
\(264\) 0 0
\(265\) −4.46250 −0.274129
\(266\) −49.4185 −3.03004
\(267\) 0 0
\(268\) −5.66487 −0.346037
\(269\) 16.3870 0.999135 0.499567 0.866275i \(-0.333492\pi\)
0.499567 + 0.866275i \(0.333492\pi\)
\(270\) 0 0
\(271\) −0.795233 −0.0483070 −0.0241535 0.999708i \(-0.507689\pi\)
−0.0241535 + 0.999708i \(0.507689\pi\)
\(272\) −3.71917 −0.225508
\(273\) 0 0
\(274\) −24.5700 −1.48433
\(275\) −15.1008 −0.910614
\(276\) 0 0
\(277\) −4.83340 −0.290411 −0.145205 0.989402i \(-0.546384\pi\)
−0.145205 + 0.989402i \(0.546384\pi\)
\(278\) −31.7168 −1.90225
\(279\) 0 0
\(280\) −6.75494 −0.403685
\(281\) −18.7748 −1.12001 −0.560005 0.828489i \(-0.689201\pi\)
−0.560005 + 0.828489i \(0.689201\pi\)
\(282\) 0 0
\(283\) −7.91723 −0.470631 −0.235315 0.971919i \(-0.575612\pi\)
−0.235315 + 0.971919i \(0.575612\pi\)
\(284\) 11.3720 0.674802
\(285\) 0 0
\(286\) 0 0
\(287\) −6.20775 −0.366432
\(288\) 0 0
\(289\) −16.4330 −0.966645
\(290\) 10.1860 0.598141
\(291\) 0 0
\(292\) 3.67994 0.215352
\(293\) −6.57912 −0.384356 −0.192178 0.981360i \(-0.561555\pi\)
−0.192178 + 0.981360i \(0.561555\pi\)
\(294\) 0 0
\(295\) 2.71379 0.158003
\(296\) 8.47650 0.492687
\(297\) 0 0
\(298\) 22.9541 1.32969
\(299\) 0 0
\(300\) 0 0
\(301\) −24.4523 −1.40941
\(302\) −28.1836 −1.62178
\(303\) 0 0
\(304\) −39.3183 −2.25506
\(305\) −4.83340 −0.276759
\(306\) 0 0
\(307\) −24.8649 −1.41911 −0.709556 0.704649i \(-0.751105\pi\)
−0.709556 + 0.704649i \(0.751105\pi\)
\(308\) 22.2784 1.26943
\(309\) 0 0
\(310\) −12.7584 −0.724628
\(311\) 17.0804 0.968539 0.484270 0.874919i \(-0.339085\pi\)
0.484270 + 0.874919i \(0.339085\pi\)
\(312\) 0 0
\(313\) 15.6974 0.887269 0.443635 0.896208i \(-0.353689\pi\)
0.443635 + 0.896208i \(0.353689\pi\)
\(314\) 1.48427 0.0837622
\(315\) 0 0
\(316\) −11.7627 −0.661704
\(317\) −32.7821 −1.84123 −0.920613 0.390477i \(-0.872310\pi\)
−0.920613 + 0.390477i \(0.872310\pi\)
\(318\) 0 0
\(319\) 20.2868 1.13584
\(320\) 1.83340 0.102490
\(321\) 0 0
\(322\) −17.5623 −0.978706
\(323\) 5.99462 0.333550
\(324\) 0 0
\(325\) 0 0
\(326\) −11.2959 −0.625622
\(327\) 0 0
\(328\) −2.44504 −0.135005
\(329\) −36.3540 −2.00426
\(330\) 0 0
\(331\) −29.1618 −1.60288 −0.801439 0.598076i \(-0.795932\pi\)
−0.801439 + 0.598076i \(0.795932\pi\)
\(332\) 8.06398 0.442569
\(333\) 0 0
\(334\) −13.4252 −0.734594
\(335\) 6.56465 0.358665
\(336\) 0 0
\(337\) −33.2911 −1.81348 −0.906741 0.421688i \(-0.861438\pi\)
−0.906741 + 0.421688i \(0.861438\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1.35690 −0.0735880
\(341\) −25.4101 −1.37604
\(342\) 0 0
\(343\) −7.34375 −0.396525
\(344\) −9.63102 −0.519270
\(345\) 0 0
\(346\) −3.62133 −0.194684
\(347\) 0.873690 0.0469022 0.0234511 0.999725i \(-0.492535\pi\)
0.0234511 + 0.999725i \(0.492535\pi\)
\(348\) 0 0
\(349\) −3.23191 −0.173000 −0.0865002 0.996252i \(-0.527568\pi\)
−0.0865002 + 0.996252i \(0.527568\pi\)
\(350\) 18.0761 0.966206
\(351\) 0 0
\(352\) 32.0804 1.70989
\(353\) −8.14675 −0.433608 −0.216804 0.976215i \(-0.569563\pi\)
−0.216804 + 0.976215i \(0.569563\pi\)
\(354\) 0 0
\(355\) −13.1782 −0.699427
\(356\) −1.44504 −0.0765871
\(357\) 0 0
\(358\) 36.1051 1.90822
\(359\) −2.64071 −0.139371 −0.0696857 0.997569i \(-0.522200\pi\)
−0.0696857 + 0.997569i \(0.522200\pi\)
\(360\) 0 0
\(361\) 44.3739 2.33547
\(362\) −43.4674 −2.28460
\(363\) 0 0
\(364\) 0 0
\(365\) −4.26444 −0.223211
\(366\) 0 0
\(367\) −2.90408 −0.151592 −0.0757960 0.997123i \(-0.524150\pi\)
−0.0757960 + 0.997123i \(0.524150\pi\)
\(368\) −13.9729 −0.728385
\(369\) 0 0
\(370\) 16.2664 0.845648
\(371\) 10.6388 0.552339
\(372\) 0 0
\(373\) −8.39852 −0.434859 −0.217429 0.976076i \(-0.569767\pi\)
−0.217429 + 0.976076i \(0.569767\pi\)
\(374\) −7.03684 −0.363866
\(375\) 0 0
\(376\) −14.3187 −0.738432
\(377\) 0 0
\(378\) 0 0
\(379\) 15.7482 0.808932 0.404466 0.914553i \(-0.367457\pi\)
0.404466 + 0.914553i \(0.367457\pi\)
\(380\) −14.3448 −0.735873
\(381\) 0 0
\(382\) −12.7584 −0.652776
\(383\) −12.7385 −0.650909 −0.325455 0.945558i \(-0.605517\pi\)
−0.325455 + 0.945558i \(0.605517\pi\)
\(384\) 0 0
\(385\) −25.8170 −1.31576
\(386\) −17.6015 −0.895892
\(387\) 0 0
\(388\) 10.7966 0.548112
\(389\) 0.310371 0.0157365 0.00786823 0.999969i \(-0.497495\pi\)
0.00786823 + 0.999969i \(0.497495\pi\)
\(390\) 0 0
\(391\) 2.13036 0.107737
\(392\) 6.60579 0.333643
\(393\) 0 0
\(394\) 42.1855 2.12528
\(395\) 13.6310 0.685851
\(396\) 0 0
\(397\) 1.49098 0.0748299 0.0374150 0.999300i \(-0.488088\pi\)
0.0374150 + 0.999300i \(0.488088\pi\)
\(398\) −7.25236 −0.363528
\(399\) 0 0
\(400\) 14.3817 0.719083
\(401\) 23.8334 1.19018 0.595092 0.803658i \(-0.297116\pi\)
0.595092 + 0.803658i \(0.297116\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.5700 0.525878
\(405\) 0 0
\(406\) −24.2838 −1.20519
\(407\) 32.3967 1.60585
\(408\) 0 0
\(409\) 4.26742 0.211010 0.105505 0.994419i \(-0.466354\pi\)
0.105505 + 0.994419i \(0.466354\pi\)
\(410\) −4.69202 −0.231722
\(411\) 0 0
\(412\) 7.04221 0.346945
\(413\) −6.46980 −0.318358
\(414\) 0 0
\(415\) −9.34481 −0.458719
\(416\) 0 0
\(417\) 0 0
\(418\) −74.3919 −3.63863
\(419\) −29.6896 −1.45043 −0.725217 0.688521i \(-0.758260\pi\)
−0.725217 + 0.688521i \(0.758260\pi\)
\(420\) 0 0
\(421\) 29.3991 1.43282 0.716412 0.697677i \(-0.245783\pi\)
0.716412 + 0.697677i \(0.245783\pi\)
\(422\) 7.04892 0.343136
\(423\) 0 0
\(424\) 4.19029 0.203499
\(425\) −2.19269 −0.106361
\(426\) 0 0
\(427\) 11.5230 0.557638
\(428\) 8.39911 0.405986
\(429\) 0 0
\(430\) −18.4819 −0.891275
\(431\) 33.0562 1.59226 0.796131 0.605124i \(-0.206877\pi\)
0.796131 + 0.605124i \(0.206877\pi\)
\(432\) 0 0
\(433\) −29.2664 −1.40645 −0.703226 0.710967i \(-0.748258\pi\)
−0.703226 + 0.710967i \(0.748258\pi\)
\(434\) 30.4166 1.46004
\(435\) 0 0
\(436\) −2.58881 −0.123982
\(437\) 22.5217 1.07736
\(438\) 0 0
\(439\) −2.13169 −0.101740 −0.0508699 0.998705i \(-0.516199\pi\)
−0.0508699 + 0.998705i \(0.516199\pi\)
\(440\) −10.1685 −0.484765
\(441\) 0 0
\(442\) 0 0
\(443\) 22.9922 1.09239 0.546197 0.837657i \(-0.316075\pi\)
0.546197 + 0.837657i \(0.316075\pi\)
\(444\) 0 0
\(445\) 1.67456 0.0793819
\(446\) −13.4233 −0.635610
\(447\) 0 0
\(448\) −4.37090 −0.206505
\(449\) 12.9379 0.610579 0.305289 0.952260i \(-0.401247\pi\)
0.305289 + 0.952260i \(0.401247\pi\)
\(450\) 0 0
\(451\) −9.34481 −0.440030
\(452\) −7.69202 −0.361802
\(453\) 0 0
\(454\) −38.2911 −1.79709
\(455\) 0 0
\(456\) 0 0
\(457\) −4.85325 −0.227025 −0.113513 0.993537i \(-0.536210\pi\)
−0.113513 + 0.993537i \(0.536210\pi\)
\(458\) −16.7506 −0.782705
\(459\) 0 0
\(460\) −5.09783 −0.237688
\(461\) −18.8345 −0.877208 −0.438604 0.898680i \(-0.644527\pi\)
−0.438604 + 0.898680i \(0.644527\pi\)
\(462\) 0 0
\(463\) −22.8767 −1.06317 −0.531585 0.847005i \(-0.678403\pi\)
−0.531585 + 0.847005i \(0.678403\pi\)
\(464\) −19.3207 −0.896939
\(465\) 0 0
\(466\) 29.2107 1.35316
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) −15.6504 −0.722668
\(470\) −27.4776 −1.26745
\(471\) 0 0
\(472\) −2.54825 −0.117293
\(473\) −36.8092 −1.69249
\(474\) 0 0
\(475\) −23.1806 −1.06360
\(476\) 3.23490 0.148271
\(477\) 0 0
\(478\) −24.3424 −1.11340
\(479\) 38.0901 1.74038 0.870190 0.492717i \(-0.163996\pi\)
0.870190 + 0.492717i \(0.163996\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −11.2959 −0.514514
\(483\) 0 0
\(484\) 19.8200 0.900909
\(485\) −12.5114 −0.568114
\(486\) 0 0
\(487\) −21.2500 −0.962928 −0.481464 0.876466i \(-0.659895\pi\)
−0.481464 + 0.876466i \(0.659895\pi\)
\(488\) 4.53856 0.205451
\(489\) 0 0
\(490\) 12.6765 0.572665
\(491\) 6.35019 0.286580 0.143290 0.989681i \(-0.454232\pi\)
0.143290 + 0.989681i \(0.454232\pi\)
\(492\) 0 0
\(493\) 2.94571 0.132668
\(494\) 0 0
\(495\) 0 0
\(496\) 24.2000 1.08661
\(497\) 31.4174 1.40926
\(498\) 0 0
\(499\) −4.65087 −0.208202 −0.104101 0.994567i \(-0.533196\pi\)
−0.104101 + 0.994567i \(0.533196\pi\)
\(500\) 14.2567 0.637578
\(501\) 0 0
\(502\) −1.35690 −0.0605612
\(503\) −15.4752 −0.690004 −0.345002 0.938602i \(-0.612122\pi\)
−0.345002 + 0.938602i \(0.612122\pi\)
\(504\) 0 0
\(505\) −12.2489 −0.545069
\(506\) −26.4373 −1.17528
\(507\) 0 0
\(508\) −17.7845 −0.789059
\(509\) 20.5047 0.908855 0.454428 0.890784i \(-0.349844\pi\)
0.454428 + 0.890784i \(0.349844\pi\)
\(510\) 0 0
\(511\) 10.1666 0.449744
\(512\) −17.1491 −0.757892
\(513\) 0 0
\(514\) 35.5459 1.56786
\(515\) −8.16075 −0.359606
\(516\) 0 0
\(517\) −54.7254 −2.40682
\(518\) −38.7797 −1.70388
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0267 1.84122 0.920611 0.390481i \(-0.127691\pi\)
0.920611 + 0.390481i \(0.127691\pi\)
\(522\) 0 0
\(523\) −29.9885 −1.31131 −0.655653 0.755062i \(-0.727607\pi\)
−0.655653 + 0.755062i \(0.727607\pi\)
\(524\) −28.1836 −1.23121
\(525\) 0 0
\(526\) −31.7308 −1.38353
\(527\) −3.68963 −0.160723
\(528\) 0 0
\(529\) −14.9963 −0.652012
\(530\) 8.04115 0.349285
\(531\) 0 0
\(532\) 34.1987 1.48270
\(533\) 0 0
\(534\) 0 0
\(535\) −9.73317 −0.420802
\(536\) −6.16421 −0.266253
\(537\) 0 0
\(538\) −29.5284 −1.27306
\(539\) 25.2470 1.08746
\(540\) 0 0
\(541\) 36.3803 1.56411 0.782056 0.623208i \(-0.214171\pi\)
0.782056 + 0.623208i \(0.214171\pi\)
\(542\) 1.43296 0.0615509
\(543\) 0 0
\(544\) 4.65817 0.199717
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −25.8159 −1.10381 −0.551905 0.833907i \(-0.686099\pi\)
−0.551905 + 0.833907i \(0.686099\pi\)
\(548\) 17.0030 0.726331
\(549\) 0 0
\(550\) 27.2107 1.16027
\(551\) 31.1414 1.32667
\(552\) 0 0
\(553\) −32.4969 −1.38191
\(554\) 8.70948 0.370030
\(555\) 0 0
\(556\) 21.9487 0.930832
\(557\) −17.9903 −0.762274 −0.381137 0.924519i \(-0.624467\pi\)
−0.381137 + 0.924519i \(0.624467\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 24.5875 1.03901
\(561\) 0 0
\(562\) 33.8310 1.42707
\(563\) 39.1323 1.64923 0.824614 0.565695i \(-0.191392\pi\)
0.824614 + 0.565695i \(0.191392\pi\)
\(564\) 0 0
\(565\) 8.91377 0.375005
\(566\) 14.2664 0.599660
\(567\) 0 0
\(568\) 12.3744 0.519216
\(569\) −30.6002 −1.28283 −0.641413 0.767196i \(-0.721651\pi\)
−0.641413 + 0.767196i \(0.721651\pi\)
\(570\) 0 0
\(571\) −2.96184 −0.123949 −0.0619745 0.998078i \(-0.519740\pi\)
−0.0619745 + 0.998078i \(0.519740\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 11.1860 0.466894
\(575\) −8.23788 −0.343543
\(576\) 0 0
\(577\) −0.819396 −0.0341119 −0.0170560 0.999855i \(-0.505429\pi\)
−0.0170560 + 0.999855i \(0.505429\pi\)
\(578\) 29.6112 1.23166
\(579\) 0 0
\(580\) −7.04892 −0.292690
\(581\) 22.2784 0.924265
\(582\) 0 0
\(583\) 16.0151 0.663276
\(584\) 4.00431 0.165700
\(585\) 0 0
\(586\) 11.8552 0.489732
\(587\) 31.7995 1.31251 0.656254 0.754540i \(-0.272140\pi\)
0.656254 + 0.754540i \(0.272140\pi\)
\(588\) 0 0
\(589\) −39.0060 −1.60721
\(590\) −4.89008 −0.201322
\(591\) 0 0
\(592\) −30.8538 −1.26808
\(593\) 4.26337 0.175076 0.0875379 0.996161i \(-0.472100\pi\)
0.0875379 + 0.996161i \(0.472100\pi\)
\(594\) 0 0
\(595\) −3.74871 −0.153682
\(596\) −15.8847 −0.650663
\(597\) 0 0
\(598\) 0 0
\(599\) −24.7278 −1.01035 −0.505175 0.863017i \(-0.668572\pi\)
−0.505175 + 0.863017i \(0.668572\pi\)
\(600\) 0 0
\(601\) 6.82371 0.278345 0.139172 0.990268i \(-0.455556\pi\)
0.139172 + 0.990268i \(0.455556\pi\)
\(602\) 44.0616 1.79582
\(603\) 0 0
\(604\) 19.5036 0.793592
\(605\) −22.9681 −0.933785
\(606\) 0 0
\(607\) 31.9963 1.29869 0.649344 0.760494i \(-0.275043\pi\)
0.649344 + 0.760494i \(0.275043\pi\)
\(608\) 49.2452 1.99716
\(609\) 0 0
\(610\) 8.70948 0.352637
\(611\) 0 0
\(612\) 0 0
\(613\) 33.5875 1.35659 0.678293 0.734792i \(-0.262720\pi\)
0.678293 + 0.734792i \(0.262720\pi\)
\(614\) 44.8049 1.80818
\(615\) 0 0
\(616\) 24.2422 0.976746
\(617\) 26.5870 1.07035 0.535176 0.844740i \(-0.320245\pi\)
0.535176 + 0.844740i \(0.320245\pi\)
\(618\) 0 0
\(619\) −9.17928 −0.368946 −0.184473 0.982838i \(-0.559058\pi\)
−0.184473 + 0.982838i \(0.559058\pi\)
\(620\) 8.82908 0.354585
\(621\) 0 0
\(622\) −30.7778 −1.23408
\(623\) −3.99223 −0.159945
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) −28.2857 −1.13053
\(627\) 0 0
\(628\) −1.02715 −0.0409876
\(629\) 4.70410 0.187565
\(630\) 0 0
\(631\) 17.0043 0.676931 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(632\) −12.7995 −0.509139
\(633\) 0 0
\(634\) 59.0713 2.34602
\(635\) 20.6093 0.817853
\(636\) 0 0
\(637\) 0 0
\(638\) −36.5555 −1.44725
\(639\) 0 0
\(640\) 14.5743 0.576101
\(641\) 21.6649 0.855711 0.427856 0.903847i \(-0.359269\pi\)
0.427856 + 0.903847i \(0.359269\pi\)
\(642\) 0 0
\(643\) 9.35557 0.368948 0.184474 0.982837i \(-0.440942\pi\)
0.184474 + 0.982837i \(0.440942\pi\)
\(644\) 12.1535 0.478913
\(645\) 0 0
\(646\) −10.8019 −0.424997
\(647\) 0.702775 0.0276289 0.0138145 0.999905i \(-0.495603\pi\)
0.0138145 + 0.999905i \(0.495603\pi\)
\(648\) 0 0
\(649\) −9.73928 −0.382300
\(650\) 0 0
\(651\) 0 0
\(652\) 7.81700 0.306137
\(653\) 37.3411 1.46127 0.730635 0.682768i \(-0.239224\pi\)
0.730635 + 0.682768i \(0.239224\pi\)
\(654\) 0 0
\(655\) 32.6601 1.27614
\(656\) 8.89977 0.347478
\(657\) 0 0
\(658\) 65.5077 2.55376
\(659\) 0.735562 0.0286534 0.0143267 0.999897i \(-0.495440\pi\)
0.0143267 + 0.999897i \(0.495440\pi\)
\(660\) 0 0
\(661\) 13.8485 0.538643 0.269321 0.963050i \(-0.413201\pi\)
0.269321 + 0.963050i \(0.413201\pi\)
\(662\) 52.5478 2.04233
\(663\) 0 0
\(664\) 8.77479 0.340528
\(665\) −39.6305 −1.53681
\(666\) 0 0
\(667\) 11.0670 0.428515
\(668\) 9.29052 0.359461
\(669\) 0 0
\(670\) −11.8291 −0.456997
\(671\) 17.3461 0.669640
\(672\) 0 0
\(673\) −6.35019 −0.244782 −0.122391 0.992482i \(-0.539056\pi\)
−0.122391 + 0.992482i \(0.539056\pi\)
\(674\) 59.9885 2.31067
\(675\) 0 0
\(676\) 0 0
\(677\) −33.7241 −1.29612 −0.648061 0.761589i \(-0.724420\pi\)
−0.648061 + 0.761589i \(0.724420\pi\)
\(678\) 0 0
\(679\) 29.8278 1.14468
\(680\) −1.47650 −0.0566212
\(681\) 0 0
\(682\) 45.7875 1.75329
\(683\) −19.2687 −0.737298 −0.368649 0.929569i \(-0.620180\pi\)
−0.368649 + 0.929569i \(0.620180\pi\)
\(684\) 0 0
\(685\) −19.7036 −0.752837
\(686\) 13.2330 0.505237
\(687\) 0 0
\(688\) 35.0562 1.33651
\(689\) 0 0
\(690\) 0 0
\(691\) −39.4010 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(692\) 2.50604 0.0952654
\(693\) 0 0
\(694\) −1.57434 −0.0597610
\(695\) −25.4349 −0.964800
\(696\) 0 0
\(697\) −1.35690 −0.0513961
\(698\) 5.82371 0.220431
\(699\) 0 0
\(700\) −12.5090 −0.472797
\(701\) −18.3985 −0.694902 −0.347451 0.937698i \(-0.612953\pi\)
−0.347451 + 0.937698i \(0.612953\pi\)
\(702\) 0 0
\(703\) 49.7308 1.87563
\(704\) −6.57971 −0.247982
\(705\) 0 0
\(706\) 14.6799 0.552487
\(707\) 29.2019 1.09825
\(708\) 0 0
\(709\) −38.4553 −1.44422 −0.722110 0.691778i \(-0.756828\pi\)
−0.722110 + 0.691778i \(0.756828\pi\)
\(710\) 23.7463 0.891183
\(711\) 0 0
\(712\) −1.57242 −0.0589288
\(713\) −13.8619 −0.519131
\(714\) 0 0
\(715\) 0 0
\(716\) −24.9855 −0.933753
\(717\) 0 0
\(718\) 4.75840 0.177582
\(719\) −48.4999 −1.80874 −0.904371 0.426747i \(-0.859659\pi\)
−0.904371 + 0.426747i \(0.859659\pi\)
\(720\) 0 0
\(721\) 19.4556 0.724564
\(722\) −79.9590 −2.97576
\(723\) 0 0
\(724\) 30.0804 1.11793
\(725\) −11.3907 −0.423042
\(726\) 0 0
\(727\) 19.0344 0.705948 0.352974 0.935633i \(-0.385170\pi\)
0.352974 + 0.935633i \(0.385170\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7.68425 0.284407
\(731\) −5.34481 −0.197685
\(732\) 0 0
\(733\) 24.6213 0.909410 0.454705 0.890642i \(-0.349745\pi\)
0.454705 + 0.890642i \(0.349745\pi\)
\(734\) 5.23298 0.193153
\(735\) 0 0
\(736\) 17.5007 0.645083
\(737\) −23.5593 −0.867817
\(738\) 0 0
\(739\) −44.5115 −1.63738 −0.818692 0.574233i \(-0.805300\pi\)
−0.818692 + 0.574233i \(0.805300\pi\)
\(740\) −11.2567 −0.413803
\(741\) 0 0
\(742\) −19.1704 −0.703769
\(743\) −10.4112 −0.381950 −0.190975 0.981595i \(-0.561165\pi\)
−0.190975 + 0.981595i \(0.561165\pi\)
\(744\) 0 0
\(745\) 18.4077 0.674407
\(746\) 15.1336 0.554081
\(747\) 0 0
\(748\) 4.86964 0.178052
\(749\) 23.2043 0.847866
\(750\) 0 0
\(751\) 1.69979 0.0620263 0.0310131 0.999519i \(-0.490127\pi\)
0.0310131 + 0.999519i \(0.490127\pi\)
\(752\) 52.1191 1.90059
\(753\) 0 0
\(754\) 0 0
\(755\) −22.6015 −0.822552
\(756\) 0 0
\(757\) 27.4252 0.996786 0.498393 0.866951i \(-0.333924\pi\)
0.498393 + 0.866951i \(0.333924\pi\)
\(758\) −28.3773 −1.03071
\(759\) 0 0
\(760\) −15.6093 −0.566207
\(761\) 5.02608 0.182195 0.0910977 0.995842i \(-0.470962\pi\)
0.0910977 + 0.995842i \(0.470962\pi\)
\(762\) 0 0
\(763\) −7.15213 −0.258924
\(764\) 8.82908 0.319425
\(765\) 0 0
\(766\) 22.9541 0.829364
\(767\) 0 0
\(768\) 0 0
\(769\) 42.4456 1.53063 0.765314 0.643657i \(-0.222584\pi\)
0.765314 + 0.643657i \(0.222584\pi\)
\(770\) 46.5206 1.67649
\(771\) 0 0
\(772\) 12.1806 0.438390
\(773\) −26.3593 −0.948078 −0.474039 0.880504i \(-0.657204\pi\)
−0.474039 + 0.880504i \(0.657204\pi\)
\(774\) 0 0
\(775\) 14.2674 0.512501
\(776\) 11.7482 0.421737
\(777\) 0 0
\(778\) −0.559270 −0.0200508
\(779\) −14.3448 −0.513956
\(780\) 0 0
\(781\) 47.2941 1.69232
\(782\) −3.83877 −0.137274
\(783\) 0 0
\(784\) −24.0446 −0.858736
\(785\) 1.19029 0.0424834
\(786\) 0 0
\(787\) 17.1424 0.611062 0.305531 0.952182i \(-0.401166\pi\)
0.305531 + 0.952182i \(0.401166\pi\)
\(788\) −29.1933 −1.03997
\(789\) 0 0
\(790\) −24.5623 −0.873886
\(791\) −21.2508 −0.755592
\(792\) 0 0
\(793\) 0 0
\(794\) −2.68664 −0.0953455
\(795\) 0 0
\(796\) 5.01879 0.177886
\(797\) −30.1629 −1.06842 −0.534212 0.845351i \(-0.679392\pi\)
−0.534212 + 0.845351i \(0.679392\pi\)
\(798\) 0 0
\(799\) −7.94630 −0.281120
\(800\) −18.0127 −0.636844
\(801\) 0 0
\(802\) −42.9463 −1.51649
\(803\) 15.3043 0.540076
\(804\) 0 0
\(805\) −14.0838 −0.496390
\(806\) 0 0
\(807\) 0 0
\(808\) 11.5017 0.404629
\(809\) 29.8504 1.04948 0.524742 0.851261i \(-0.324162\pi\)
0.524742 + 0.851261i \(0.324162\pi\)
\(810\) 0 0
\(811\) 47.7362 1.67624 0.838122 0.545484i \(-0.183654\pi\)
0.838122 + 0.545484i \(0.183654\pi\)
\(812\) 16.8049 0.589737
\(813\) 0 0
\(814\) −58.3769 −2.04611
\(815\) −9.05861 −0.317309
\(816\) 0 0
\(817\) −56.5042 −1.97683
\(818\) −7.68963 −0.268862
\(819\) 0 0
\(820\) 3.24698 0.113389
\(821\) 17.9299 0.625758 0.312879 0.949793i \(-0.398707\pi\)
0.312879 + 0.949793i \(0.398707\pi\)
\(822\) 0 0
\(823\) 54.3196 1.89346 0.946731 0.322026i \(-0.104364\pi\)
0.946731 + 0.322026i \(0.104364\pi\)
\(824\) 7.66296 0.266952
\(825\) 0 0
\(826\) 11.6582 0.405640
\(827\) −49.1041 −1.70752 −0.853758 0.520670i \(-0.825682\pi\)
−0.853758 + 0.520670i \(0.825682\pi\)
\(828\) 0 0
\(829\) −7.35796 −0.255553 −0.127776 0.991803i \(-0.540784\pi\)
−0.127776 + 0.991803i \(0.540784\pi\)
\(830\) 16.8388 0.584482
\(831\) 0 0
\(832\) 0 0
\(833\) 3.66594 0.127017
\(834\) 0 0
\(835\) −10.7662 −0.372579
\(836\) 51.4808 1.78050
\(837\) 0 0
\(838\) 53.4989 1.84809
\(839\) 36.5013 1.26016 0.630082 0.776529i \(-0.283021\pi\)
0.630082 + 0.776529i \(0.283021\pi\)
\(840\) 0 0
\(841\) −13.6974 −0.472324
\(842\) −52.9754 −1.82565
\(843\) 0 0
\(844\) −4.87800 −0.167908
\(845\) 0 0
\(846\) 0 0
\(847\) 54.7569 1.88147
\(848\) −15.2524 −0.523768
\(849\) 0 0
\(850\) 3.95108 0.135521
\(851\) 17.6732 0.605831
\(852\) 0 0
\(853\) −9.73855 −0.333441 −0.166721 0.986004i \(-0.553318\pi\)
−0.166721 + 0.986004i \(0.553318\pi\)
\(854\) −20.7638 −0.710522
\(855\) 0 0
\(856\) 9.13946 0.312380
\(857\) 15.2030 0.519323 0.259662 0.965700i \(-0.416389\pi\)
0.259662 + 0.965700i \(0.416389\pi\)
\(858\) 0 0
\(859\) −31.9885 −1.09143 −0.545717 0.837970i \(-0.683743\pi\)
−0.545717 + 0.837970i \(0.683743\pi\)
\(860\) 12.7899 0.436130
\(861\) 0 0
\(862\) −59.5652 −2.02880
\(863\) 35.2905 1.20130 0.600652 0.799511i \(-0.294908\pi\)
0.600652 + 0.799511i \(0.294908\pi\)
\(864\) 0 0
\(865\) −2.90408 −0.0987418
\(866\) 52.7362 1.79205
\(867\) 0 0
\(868\) −21.0489 −0.714447
\(869\) −48.9191 −1.65947
\(870\) 0 0
\(871\) 0 0
\(872\) −2.81700 −0.0953958
\(873\) 0 0
\(874\) −40.5827 −1.37273
\(875\) 39.3870 1.33152
\(876\) 0 0
\(877\) −15.3263 −0.517532 −0.258766 0.965940i \(-0.583316\pi\)
−0.258766 + 0.965940i \(0.583316\pi\)
\(878\) 3.84117 0.129633
\(879\) 0 0
\(880\) 37.0127 1.24770
\(881\) 36.4306 1.22738 0.613689 0.789548i \(-0.289685\pi\)
0.613689 + 0.789548i \(0.289685\pi\)
\(882\) 0 0
\(883\) −37.6819 −1.26810 −0.634048 0.773294i \(-0.718608\pi\)
−0.634048 + 0.773294i \(0.718608\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −41.4306 −1.39189
\(887\) −4.24890 −0.142664 −0.0713320 0.997453i \(-0.522725\pi\)
−0.0713320 + 0.997453i \(0.522725\pi\)
\(888\) 0 0
\(889\) −49.1333 −1.64788
\(890\) −3.01746 −0.101145
\(891\) 0 0
\(892\) 9.28919 0.311025
\(893\) −84.0066 −2.81117
\(894\) 0 0
\(895\) 28.9541 0.967828
\(896\) −34.7458 −1.16078
\(897\) 0 0
\(898\) −23.3134 −0.777977
\(899\) −19.1672 −0.639262
\(900\) 0 0
\(901\) 2.32544 0.0774715
\(902\) 16.8388 0.560670
\(903\) 0 0
\(904\) −8.37004 −0.278383
\(905\) −34.8582 −1.15872
\(906\) 0 0
\(907\) −12.6183 −0.418985 −0.209493 0.977810i \(-0.567181\pi\)
−0.209493 + 0.977810i \(0.567181\pi\)
\(908\) 26.4983 0.879376
\(909\) 0 0
\(910\) 0 0
\(911\) −6.77777 −0.224558 −0.112279 0.993677i \(-0.535815\pi\)
−0.112279 + 0.993677i \(0.535815\pi\)
\(912\) 0 0
\(913\) 33.5368 1.10990
\(914\) 8.74525 0.289267
\(915\) 0 0
\(916\) 11.5918 0.383004
\(917\) −77.8631 −2.57126
\(918\) 0 0
\(919\) −20.4674 −0.675157 −0.337579 0.941297i \(-0.609608\pi\)
−0.337579 + 0.941297i \(0.609608\pi\)
\(920\) −5.54719 −0.182885
\(921\) 0 0
\(922\) 33.9385 1.11771
\(923\) 0 0
\(924\) 0 0
\(925\) −18.1903 −0.598093
\(926\) 41.2223 1.35465
\(927\) 0 0
\(928\) 24.1987 0.794360
\(929\) 4.65220 0.152634 0.0763169 0.997084i \(-0.475684\pi\)
0.0763169 + 0.997084i \(0.475684\pi\)
\(930\) 0 0
\(931\) 38.7555 1.27016
\(932\) −20.2145 −0.662147
\(933\) 0 0
\(934\) 23.4252 0.766496
\(935\) −5.64310 −0.184549
\(936\) 0 0
\(937\) −41.8544 −1.36732 −0.683662 0.729798i \(-0.739614\pi\)
−0.683662 + 0.729798i \(0.739614\pi\)
\(938\) 28.2010 0.920797
\(939\) 0 0
\(940\) 19.0151 0.620203
\(941\) −30.3454 −0.989232 −0.494616 0.869112i \(-0.664691\pi\)
−0.494616 + 0.869112i \(0.664691\pi\)
\(942\) 0 0
\(943\) −5.09783 −0.166008
\(944\) 9.27545 0.301890
\(945\) 0 0
\(946\) 66.3279 2.15651
\(947\) −12.0325 −0.391004 −0.195502 0.980703i \(-0.562634\pi\)
−0.195502 + 0.980703i \(0.562634\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 41.7700 1.35520
\(951\) 0 0
\(952\) 3.52004 0.114085
\(953\) −22.9825 −0.744478 −0.372239 0.928137i \(-0.621410\pi\)
−0.372239 + 0.928137i \(0.621410\pi\)
\(954\) 0 0
\(955\) −10.2314 −0.331082
\(956\) 16.8455 0.544822
\(957\) 0 0
\(958\) −68.6359 −2.21753
\(959\) 46.9743 1.51688
\(960\) 0 0
\(961\) −6.99223 −0.225556
\(962\) 0 0
\(963\) 0 0
\(964\) 7.81700 0.251769
\(965\) −14.1153 −0.454387
\(966\) 0 0
\(967\) −38.8883 −1.25056 −0.625281 0.780399i \(-0.715016\pi\)
−0.625281 + 0.780399i \(0.715016\pi\)
\(968\) 21.5670 0.693191
\(969\) 0 0
\(970\) 22.5448 0.723870
\(971\) 57.5133 1.84569 0.922845 0.385171i \(-0.125857\pi\)
0.922845 + 0.385171i \(0.125857\pi\)
\(972\) 0 0
\(973\) 60.6378 1.94396
\(974\) 38.2911 1.22693
\(975\) 0 0
\(976\) −16.5200 −0.528794
\(977\) −16.3690 −0.523690 −0.261845 0.965110i \(-0.584331\pi\)
−0.261845 + 0.965110i \(0.584331\pi\)
\(978\) 0 0
\(979\) −6.00969 −0.192070
\(980\) −8.77240 −0.280224
\(981\) 0 0
\(982\) −11.4426 −0.365150
\(983\) 15.6963 0.500635 0.250318 0.968164i \(-0.419465\pi\)
0.250318 + 0.968164i \(0.419465\pi\)
\(984\) 0 0
\(985\) 33.8301 1.07792
\(986\) −5.30798 −0.169040
\(987\) 0 0
\(988\) 0 0
\(989\) −20.0804 −0.638519
\(990\) 0 0
\(991\) −11.2644 −0.357827 −0.178913 0.983865i \(-0.557258\pi\)
−0.178913 + 0.983865i \(0.557258\pi\)
\(992\) −30.3099 −0.962340
\(993\) 0 0
\(994\) −56.6122 −1.79563
\(995\) −5.81594 −0.184378
\(996\) 0 0
\(997\) −7.70112 −0.243897 −0.121948 0.992536i \(-0.538914\pi\)
−0.121948 + 0.992536i \(0.538914\pi\)
\(998\) 8.38059 0.265283
\(999\) 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 1521.2.a.p.1.1 3
3.2 odd 2 507.2.a.k.1.3 yes 3
12.11 even 2 8112.2.a.cf.1.2 3
13.5 odd 4 1521.2.b.m.1351.6 6
13.8 odd 4 1521.2.b.m.1351.1 6
13.12 even 2 1521.2.a.q.1.3 3
39.2 even 12 507.2.j.h.316.6 12
39.5 even 4 507.2.b.g.337.1 6
39.8 even 4 507.2.b.g.337.6 6
39.11 even 12 507.2.j.h.316.1 12
39.17 odd 6 507.2.e.k.484.3 6
39.20 even 12 507.2.j.h.361.6 12
39.23 odd 6 507.2.e.k.22.3 6
39.29 odd 6 507.2.e.j.22.1 6
39.32 even 12 507.2.j.h.361.1 12
39.35 odd 6 507.2.e.j.484.1 6
39.38 odd 2 507.2.a.j.1.1 3
156.155 even 2 8112.2.a.by.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.1 3 39.38 odd 2
507.2.a.k.1.3 yes 3 3.2 odd 2
507.2.b.g.337.1 6 39.5 even 4
507.2.b.g.337.6 6 39.8 even 4
507.2.e.j.22.1 6 39.29 odd 6
507.2.e.j.484.1 6 39.35 odd 6
507.2.e.k.22.3 6 39.23 odd 6
507.2.e.k.484.3 6 39.17 odd 6
507.2.j.h.316.1 12 39.11 even 12
507.2.j.h.316.6 12 39.2 even 12
507.2.j.h.361.1 12 39.32 even 12
507.2.j.h.361.6 12 39.20 even 12
1521.2.a.p.1.1 3 1.1 even 1 trivial
1521.2.a.q.1.3 3 13.12 even 2
1521.2.b.m.1351.1 6 13.8 odd 4
1521.2.b.m.1351.6 6 13.5 odd 4
8112.2.a.by.1.2 3 156.155 even 2
8112.2.a.cf.1.2 3 12.11 even 2