Properties

Label 855.2.a.d.1.1
Level $855$
Weight $2$
Character 855.1
Self dual yes
Analytic conductor $6.827$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.82720937282\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 855.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -1.41421 q^{7} -4.41421 q^{8} +O(q^{10})\) \(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -1.41421 q^{7} -4.41421 q^{8} -2.41421 q^{10} +2.24264 q^{11} -3.41421 q^{13} +3.41421 q^{14} +3.00000 q^{16} -1.17157 q^{17} -1.00000 q^{19} +3.82843 q^{20} -5.41421 q^{22} -7.65685 q^{23} +1.00000 q^{25} +8.24264 q^{26} -5.41421 q^{28} -1.41421 q^{29} -3.17157 q^{31} +1.58579 q^{32} +2.82843 q^{34} -1.41421 q^{35} -3.41421 q^{37} +2.41421 q^{38} -4.41421 q^{40} +0.242641 q^{41} +12.2426 q^{43} +8.58579 q^{44} +18.4853 q^{46} +7.65685 q^{47} -5.00000 q^{49} -2.41421 q^{50} -13.0711 q^{52} -8.00000 q^{53} +2.24264 q^{55} +6.24264 q^{56} +3.41421 q^{58} -12.4853 q^{59} +7.31371 q^{61} +7.65685 q^{62} -9.82843 q^{64} -3.41421 q^{65} -9.65685 q^{67} -4.48528 q^{68} +3.41421 q^{70} +10.8284 q^{71} -7.65685 q^{73} +8.24264 q^{74} -3.82843 q^{76} -3.17157 q^{77} +3.00000 q^{80} -0.585786 q^{82} -12.8284 q^{83} -1.17157 q^{85} -29.5563 q^{86} -9.89949 q^{88} -5.89949 q^{89} +4.82843 q^{91} -29.3137 q^{92} -18.4853 q^{94} -1.00000 q^{95} -9.75736 q^{97} +12.0711 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 6q^{8} - 2q^{10} - 4q^{11} - 4q^{13} + 4q^{14} + 6q^{16} - 8q^{17} - 2q^{19} + 2q^{20} - 8q^{22} - 4q^{23} + 2q^{25} + 8q^{26} - 8q^{28} - 12q^{31} + 6q^{32} - 4q^{37} + 2q^{38} - 6q^{40} - 8q^{41} + 16q^{43} + 20q^{44} + 20q^{46} + 4q^{47} - 10q^{49} - 2q^{50} - 12q^{52} - 16q^{53} - 4q^{55} + 4q^{56} + 4q^{58} - 8q^{59} - 8q^{61} + 4q^{62} - 14q^{64} - 4q^{65} - 8q^{67} + 8q^{68} + 4q^{70} + 16q^{71} - 4q^{73} + 8q^{74} - 2q^{76} - 12q^{77} + 6q^{80} - 4q^{82} - 20q^{83} - 8q^{85} - 28q^{86} + 8q^{89} + 4q^{91} - 36q^{92} - 20q^{94} - 2q^{95} - 28q^{97} + 10q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) −4.41421 −1.56066
\(9\) 0 0
\(10\) −2.41421 −0.763441
\(11\) 2.24264 0.676182 0.338091 0.941113i \(-0.390219\pi\)
0.338091 + 0.941113i \(0.390219\pi\)
\(12\) 0 0
\(13\) −3.41421 −0.946932 −0.473466 0.880812i \(-0.656997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(14\) 3.41421 0.912487
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −1.17157 −0.284148 −0.142074 0.989856i \(-0.545377\pi\)
−0.142074 + 0.989856i \(0.545377\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) −5.41421 −1.15431
\(23\) −7.65685 −1.59656 −0.798282 0.602284i \(-0.794258\pi\)
−0.798282 + 0.602284i \(0.794258\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 8.24264 1.61651
\(27\) 0 0
\(28\) −5.41421 −1.02319
\(29\) −1.41421 −0.262613 −0.131306 0.991342i \(-0.541917\pi\)
−0.131306 + 0.991342i \(0.541917\pi\)
\(30\) 0 0
\(31\) −3.17157 −0.569631 −0.284816 0.958582i \(-0.591932\pi\)
−0.284816 + 0.958582i \(0.591932\pi\)
\(32\) 1.58579 0.280330
\(33\) 0 0
\(34\) 2.82843 0.485071
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) −3.41421 −0.561293 −0.280647 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) 2.41421 0.391637
\(39\) 0 0
\(40\) −4.41421 −0.697948
\(41\) 0.242641 0.0378941 0.0189471 0.999820i \(-0.493969\pi\)
0.0189471 + 0.999820i \(0.493969\pi\)
\(42\) 0 0
\(43\) 12.2426 1.86699 0.933493 0.358597i \(-0.116745\pi\)
0.933493 + 0.358597i \(0.116745\pi\)
\(44\) 8.58579 1.29436
\(45\) 0 0
\(46\) 18.4853 2.72551
\(47\) 7.65685 1.11687 0.558433 0.829549i \(-0.311403\pi\)
0.558433 + 0.829549i \(0.311403\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) −2.41421 −0.341421
\(51\) 0 0
\(52\) −13.0711 −1.81263
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 2.24264 0.302398
\(56\) 6.24264 0.834208
\(57\) 0 0
\(58\) 3.41421 0.448308
\(59\) −12.4853 −1.62545 −0.812723 0.582651i \(-0.802016\pi\)
−0.812723 + 0.582651i \(0.802016\pi\)
\(60\) 0 0
\(61\) 7.31371 0.936424 0.468212 0.883616i \(-0.344898\pi\)
0.468212 + 0.883616i \(0.344898\pi\)
\(62\) 7.65685 0.972421
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) −3.41421 −0.423481
\(66\) 0 0
\(67\) −9.65685 −1.17977 −0.589886 0.807486i \(-0.700827\pi\)
−0.589886 + 0.807486i \(0.700827\pi\)
\(68\) −4.48528 −0.543920
\(69\) 0 0
\(70\) 3.41421 0.408077
\(71\) 10.8284 1.28510 0.642549 0.766245i \(-0.277877\pi\)
0.642549 + 0.766245i \(0.277877\pi\)
\(72\) 0 0
\(73\) −7.65685 −0.896167 −0.448084 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) 8.24264 0.958188
\(75\) 0 0
\(76\) −3.82843 −0.439151
\(77\) −3.17157 −0.361434
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) −0.585786 −0.0646893
\(83\) −12.8284 −1.40810 −0.704051 0.710149i \(-0.748628\pi\)
−0.704051 + 0.710149i \(0.748628\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) −29.5563 −3.18714
\(87\) 0 0
\(88\) −9.89949 −1.05529
\(89\) −5.89949 −0.625345 −0.312673 0.949861i \(-0.601224\pi\)
−0.312673 + 0.949861i \(0.601224\pi\)
\(90\) 0 0
\(91\) 4.82843 0.506157
\(92\) −29.3137 −3.05617
\(93\) 0 0
\(94\) −18.4853 −1.90661
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −9.75736 −0.990710 −0.495355 0.868691i \(-0.664962\pi\)
−0.495355 + 0.868691i \(0.664962\pi\)
\(98\) 12.0711 1.21936
\(99\) 0 0
\(100\) 3.82843 0.382843
\(101\) 3.17157 0.315583 0.157792 0.987472i \(-0.449563\pi\)
0.157792 + 0.987472i \(0.449563\pi\)
\(102\) 0 0
\(103\) −7.31371 −0.720641 −0.360321 0.932829i \(-0.617333\pi\)
−0.360321 + 0.932829i \(0.617333\pi\)
\(104\) 15.0711 1.47784
\(105\) 0 0
\(106\) 19.3137 1.87591
\(107\) −19.3137 −1.86713 −0.933563 0.358412i \(-0.883318\pi\)
−0.933563 + 0.358412i \(0.883318\pi\)
\(108\) 0 0
\(109\) −6.48528 −0.621177 −0.310589 0.950544i \(-0.600526\pi\)
−0.310589 + 0.950544i \(0.600526\pi\)
\(110\) −5.41421 −0.516225
\(111\) 0 0
\(112\) −4.24264 −0.400892
\(113\) −10.1421 −0.954092 −0.477046 0.878878i \(-0.658292\pi\)
−0.477046 + 0.878878i \(0.658292\pi\)
\(114\) 0 0
\(115\) −7.65685 −0.714005
\(116\) −5.41421 −0.502697
\(117\) 0 0
\(118\) 30.1421 2.77481
\(119\) 1.65685 0.151884
\(120\) 0 0
\(121\) −5.97056 −0.542778
\(122\) −17.6569 −1.59858
\(123\) 0 0
\(124\) −12.1421 −1.09040
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 19.3137 1.71381 0.856907 0.515471i \(-0.172383\pi\)
0.856907 + 0.515471i \(0.172383\pi\)
\(128\) 20.5563 1.81694
\(129\) 0 0
\(130\) 8.24264 0.722927
\(131\) 8.58579 0.750144 0.375072 0.926996i \(-0.377618\pi\)
0.375072 + 0.926996i \(0.377618\pi\)
\(132\) 0 0
\(133\) 1.41421 0.122628
\(134\) 23.3137 2.01400
\(135\) 0 0
\(136\) 5.17157 0.443459
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −14.1421 −1.19952 −0.599760 0.800180i \(-0.704737\pi\)
−0.599760 + 0.800180i \(0.704737\pi\)
\(140\) −5.41421 −0.457585
\(141\) 0 0
\(142\) −26.1421 −2.19380
\(143\) −7.65685 −0.640298
\(144\) 0 0
\(145\) −1.41421 −0.117444
\(146\) 18.4853 1.52985
\(147\) 0 0
\(148\) −13.0711 −1.07444
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −6.48528 −0.527765 −0.263882 0.964555i \(-0.585003\pi\)
−0.263882 + 0.964555i \(0.585003\pi\)
\(152\) 4.41421 0.358040
\(153\) 0 0
\(154\) 7.65685 0.617007
\(155\) −3.17157 −0.254747
\(156\) 0 0
\(157\) 10.4853 0.836817 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.58579 0.125367
\(161\) 10.8284 0.853400
\(162\) 0 0
\(163\) 21.8995 1.71530 0.857650 0.514233i \(-0.171923\pi\)
0.857650 + 0.514233i \(0.171923\pi\)
\(164\) 0.928932 0.0725374
\(165\) 0 0
\(166\) 30.9706 2.40378
\(167\) 17.3137 1.33977 0.669887 0.742463i \(-0.266342\pi\)
0.669887 + 0.742463i \(0.266342\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 2.82843 0.216930
\(171\) 0 0
\(172\) 46.8701 3.57381
\(173\) 19.7990 1.50529 0.752645 0.658427i \(-0.228778\pi\)
0.752645 + 0.658427i \(0.228778\pi\)
\(174\) 0 0
\(175\) −1.41421 −0.106904
\(176\) 6.72792 0.507136
\(177\) 0 0
\(178\) 14.2426 1.06753
\(179\) −16.4853 −1.23217 −0.616084 0.787681i \(-0.711282\pi\)
−0.616084 + 0.787681i \(0.711282\pi\)
\(180\) 0 0
\(181\) −20.8284 −1.54816 −0.774082 0.633085i \(-0.781788\pi\)
−0.774082 + 0.633085i \(0.781788\pi\)
\(182\) −11.6569 −0.864064
\(183\) 0 0
\(184\) 33.7990 2.49169
\(185\) −3.41421 −0.251018
\(186\) 0 0
\(187\) −2.62742 −0.192136
\(188\) 29.3137 2.13792
\(189\) 0 0
\(190\) 2.41421 0.175145
\(191\) −10.2426 −0.741131 −0.370566 0.928806i \(-0.620836\pi\)
−0.370566 + 0.928806i \(0.620836\pi\)
\(192\) 0 0
\(193\) 5.07107 0.365023 0.182512 0.983204i \(-0.441577\pi\)
0.182512 + 0.983204i \(0.441577\pi\)
\(194\) 23.5563 1.69125
\(195\) 0 0
\(196\) −19.1421 −1.36730
\(197\) −6.82843 −0.486505 −0.243253 0.969963i \(-0.578214\pi\)
−0.243253 + 0.969963i \(0.578214\pi\)
\(198\) 0 0
\(199\) 18.1421 1.28606 0.643031 0.765840i \(-0.277677\pi\)
0.643031 + 0.765840i \(0.277677\pi\)
\(200\) −4.41421 −0.312132
\(201\) 0 0
\(202\) −7.65685 −0.538734
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0.242641 0.0169468
\(206\) 17.6569 1.23021
\(207\) 0 0
\(208\) −10.2426 −0.710199
\(209\) −2.24264 −0.155127
\(210\) 0 0
\(211\) 7.31371 0.503496 0.251748 0.967793i \(-0.418995\pi\)
0.251748 + 0.967793i \(0.418995\pi\)
\(212\) −30.6274 −2.10350
\(213\) 0 0
\(214\) 46.6274 3.18738
\(215\) 12.2426 0.834941
\(216\) 0 0
\(217\) 4.48528 0.304481
\(218\) 15.6569 1.06042
\(219\) 0 0
\(220\) 8.58579 0.578854
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 18.6274 1.24738 0.623692 0.781670i \(-0.285632\pi\)
0.623692 + 0.781670i \(0.285632\pi\)
\(224\) −2.24264 −0.149843
\(225\) 0 0
\(226\) 24.4853 1.62874
\(227\) −14.9706 −0.993631 −0.496816 0.867856i \(-0.665497\pi\)
−0.496816 + 0.867856i \(0.665497\pi\)
\(228\) 0 0
\(229\) −22.6274 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(230\) 18.4853 1.21888
\(231\) 0 0
\(232\) 6.24264 0.409849
\(233\) 0.343146 0.0224802 0.0112401 0.999937i \(-0.496422\pi\)
0.0112401 + 0.999937i \(0.496422\pi\)
\(234\) 0 0
\(235\) 7.65685 0.499478
\(236\) −47.7990 −3.11145
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 26.7279 1.72889 0.864443 0.502731i \(-0.167671\pi\)
0.864443 + 0.502731i \(0.167671\pi\)
\(240\) 0 0
\(241\) −19.6569 −1.26621 −0.633105 0.774066i \(-0.718220\pi\)
−0.633105 + 0.774066i \(0.718220\pi\)
\(242\) 14.4142 0.926581
\(243\) 0 0
\(244\) 28.0000 1.79252
\(245\) −5.00000 −0.319438
\(246\) 0 0
\(247\) 3.41421 0.217241
\(248\) 14.0000 0.889001
\(249\) 0 0
\(250\) −2.41421 −0.152688
\(251\) 1.75736 0.110924 0.0554618 0.998461i \(-0.482337\pi\)
0.0554618 + 0.998461i \(0.482337\pi\)
\(252\) 0 0
\(253\) −17.1716 −1.07957
\(254\) −46.6274 −2.92566
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 4.48528 0.279784 0.139892 0.990167i \(-0.455324\pi\)
0.139892 + 0.990167i \(0.455324\pi\)
\(258\) 0 0
\(259\) 4.82843 0.300024
\(260\) −13.0711 −0.810633
\(261\) 0 0
\(262\) −20.7279 −1.28058
\(263\) 23.4558 1.44635 0.723175 0.690665i \(-0.242682\pi\)
0.723175 + 0.690665i \(0.242682\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) −3.41421 −0.209339
\(267\) 0 0
\(268\) −36.9706 −2.25834
\(269\) 3.07107 0.187246 0.0936232 0.995608i \(-0.470155\pi\)
0.0936232 + 0.995608i \(0.470155\pi\)
\(270\) 0 0
\(271\) −10.8284 −0.657780 −0.328890 0.944368i \(-0.606675\pi\)
−0.328890 + 0.944368i \(0.606675\pi\)
\(272\) −3.51472 −0.213111
\(273\) 0 0
\(274\) 24.1421 1.45848
\(275\) 2.24264 0.135236
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 34.1421 2.04771
\(279\) 0 0
\(280\) 6.24264 0.373069
\(281\) 14.5858 0.870115 0.435058 0.900403i \(-0.356728\pi\)
0.435058 + 0.900403i \(0.356728\pi\)
\(282\) 0 0
\(283\) −10.3848 −0.617311 −0.308655 0.951174i \(-0.599879\pi\)
−0.308655 + 0.951174i \(0.599879\pi\)
\(284\) 41.4558 2.45995
\(285\) 0 0
\(286\) 18.4853 1.09306
\(287\) −0.343146 −0.0202553
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 3.41421 0.200490
\(291\) 0 0
\(292\) −29.3137 −1.71546
\(293\) 11.5147 0.672697 0.336349 0.941738i \(-0.390808\pi\)
0.336349 + 0.941738i \(0.390808\pi\)
\(294\) 0 0
\(295\) −12.4853 −0.726921
\(296\) 15.0711 0.875988
\(297\) 0 0
\(298\) −14.4853 −0.839110
\(299\) 26.1421 1.51184
\(300\) 0 0
\(301\) −17.3137 −0.997946
\(302\) 15.6569 0.900951
\(303\) 0 0
\(304\) −3.00000 −0.172062
\(305\) 7.31371 0.418782
\(306\) 0 0
\(307\) 21.1716 1.20833 0.604163 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(308\) −12.1421 −0.691862
\(309\) 0 0
\(310\) 7.65685 0.434880
\(311\) 6.24264 0.353988 0.176994 0.984212i \(-0.443363\pi\)
0.176994 + 0.984212i \(0.443363\pi\)
\(312\) 0 0
\(313\) 5.79899 0.327778 0.163889 0.986479i \(-0.447596\pi\)
0.163889 + 0.986479i \(0.447596\pi\)
\(314\) −25.3137 −1.42854
\(315\) 0 0
\(316\) 0 0
\(317\) 26.6274 1.49554 0.747772 0.663955i \(-0.231123\pi\)
0.747772 + 0.663955i \(0.231123\pi\)
\(318\) 0 0
\(319\) −3.17157 −0.177574
\(320\) −9.82843 −0.549426
\(321\) 0 0
\(322\) −26.1421 −1.45684
\(323\) 1.17157 0.0651881
\(324\) 0 0
\(325\) −3.41421 −0.189386
\(326\) −52.8701 −2.92820
\(327\) 0 0
\(328\) −1.07107 −0.0591398
\(329\) −10.8284 −0.596991
\(330\) 0 0
\(331\) 28.1421 1.54683 0.773416 0.633899i \(-0.218547\pi\)
0.773416 + 0.633899i \(0.218547\pi\)
\(332\) −49.1127 −2.69541
\(333\) 0 0
\(334\) −41.7990 −2.28714
\(335\) −9.65685 −0.527610
\(336\) 0 0
\(337\) −24.5858 −1.33927 −0.669637 0.742689i \(-0.733550\pi\)
−0.669637 + 0.742689i \(0.733550\pi\)
\(338\) 3.24264 0.176376
\(339\) 0 0
\(340\) −4.48528 −0.243249
\(341\) −7.11270 −0.385174
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) −54.0416 −2.91373
\(345\) 0 0
\(346\) −47.7990 −2.56969
\(347\) 27.4558 1.47391 0.736953 0.675943i \(-0.236264\pi\)
0.736953 + 0.675943i \(0.236264\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 3.41421 0.182497
\(351\) 0 0
\(352\) 3.55635 0.189554
\(353\) −3.65685 −0.194635 −0.0973174 0.995253i \(-0.531026\pi\)
−0.0973174 + 0.995253i \(0.531026\pi\)
\(354\) 0 0
\(355\) 10.8284 0.574713
\(356\) −22.5858 −1.19704
\(357\) 0 0
\(358\) 39.7990 2.10344
\(359\) −24.8701 −1.31259 −0.656296 0.754504i \(-0.727878\pi\)
−0.656296 + 0.754504i \(0.727878\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 50.2843 2.64288
\(363\) 0 0
\(364\) 18.4853 0.968892
\(365\) −7.65685 −0.400778
\(366\) 0 0
\(367\) −14.3848 −0.750879 −0.375440 0.926847i \(-0.622508\pi\)
−0.375440 + 0.926847i \(0.622508\pi\)
\(368\) −22.9706 −1.19742
\(369\) 0 0
\(370\) 8.24264 0.428514
\(371\) 11.3137 0.587378
\(372\) 0 0
\(373\) 0.585786 0.0303309 0.0151654 0.999885i \(-0.495173\pi\)
0.0151654 + 0.999885i \(0.495173\pi\)
\(374\) 6.34315 0.327996
\(375\) 0 0
\(376\) −33.7990 −1.74305
\(377\) 4.82843 0.248677
\(378\) 0 0
\(379\) 3.17157 0.162913 0.0814564 0.996677i \(-0.474043\pi\)
0.0814564 + 0.996677i \(0.474043\pi\)
\(380\) −3.82843 −0.196394
\(381\) 0 0
\(382\) 24.7279 1.26519
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 0 0
\(385\) −3.17157 −0.161638
\(386\) −12.2426 −0.623134
\(387\) 0 0
\(388\) −37.3553 −1.89643
\(389\) −30.9706 −1.57027 −0.785135 0.619325i \(-0.787406\pi\)
−0.785135 + 0.619325i \(0.787406\pi\)
\(390\) 0 0
\(391\) 8.97056 0.453661
\(392\) 22.0711 1.11476
\(393\) 0 0
\(394\) 16.4853 0.830516
\(395\) 0 0
\(396\) 0 0
\(397\) 28.6274 1.43677 0.718384 0.695646i \(-0.244882\pi\)
0.718384 + 0.695646i \(0.244882\pi\)
\(398\) −43.7990 −2.19544
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 7.75736 0.387384 0.193692 0.981062i \(-0.437954\pi\)
0.193692 + 0.981062i \(0.437954\pi\)
\(402\) 0 0
\(403\) 10.8284 0.539402
\(404\) 12.1421 0.604094
\(405\) 0 0
\(406\) −4.82843 −0.239631
\(407\) −7.65685 −0.379536
\(408\) 0 0
\(409\) −12.8284 −0.634325 −0.317162 0.948371i \(-0.602730\pi\)
−0.317162 + 0.948371i \(0.602730\pi\)
\(410\) −0.585786 −0.0289299
\(411\) 0 0
\(412\) −28.0000 −1.37946
\(413\) 17.6569 0.868837
\(414\) 0 0
\(415\) −12.8284 −0.629723
\(416\) −5.41421 −0.265454
\(417\) 0 0
\(418\) 5.41421 0.264818
\(419\) −3.41421 −0.166795 −0.0833976 0.996516i \(-0.526577\pi\)
−0.0833976 + 0.996516i \(0.526577\pi\)
\(420\) 0 0
\(421\) 9.31371 0.453922 0.226961 0.973904i \(-0.427121\pi\)
0.226961 + 0.973904i \(0.427121\pi\)
\(422\) −17.6569 −0.859522
\(423\) 0 0
\(424\) 35.3137 1.71499
\(425\) −1.17157 −0.0568296
\(426\) 0 0
\(427\) −10.3431 −0.500540
\(428\) −73.9411 −3.57408
\(429\) 0 0
\(430\) −29.5563 −1.42533
\(431\) 31.1127 1.49865 0.749323 0.662205i \(-0.230379\pi\)
0.749323 + 0.662205i \(0.230379\pi\)
\(432\) 0 0
\(433\) −10.9289 −0.525211 −0.262605 0.964903i \(-0.584582\pi\)
−0.262605 + 0.964903i \(0.584582\pi\)
\(434\) −10.8284 −0.519781
\(435\) 0 0
\(436\) −24.8284 −1.18907
\(437\) 7.65685 0.366277
\(438\) 0 0
\(439\) −21.6569 −1.03363 −0.516813 0.856099i \(-0.672882\pi\)
−0.516813 + 0.856099i \(0.672882\pi\)
\(440\) −9.89949 −0.471940
\(441\) 0 0
\(442\) −9.65685 −0.459330
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) −5.89949 −0.279663
\(446\) −44.9706 −2.12942
\(447\) 0 0
\(448\) 13.8995 0.656689
\(449\) 26.8701 1.26808 0.634038 0.773302i \(-0.281396\pi\)
0.634038 + 0.773302i \(0.281396\pi\)
\(450\) 0 0
\(451\) 0.544156 0.0256233
\(452\) −38.8284 −1.82634
\(453\) 0 0
\(454\) 36.1421 1.69623
\(455\) 4.82843 0.226360
\(456\) 0 0
\(457\) 32.8284 1.53565 0.767825 0.640660i \(-0.221339\pi\)
0.767825 + 0.640660i \(0.221339\pi\)
\(458\) 54.6274 2.55257
\(459\) 0 0
\(460\) −29.3137 −1.36676
\(461\) −19.6569 −0.915511 −0.457755 0.889078i \(-0.651346\pi\)
−0.457755 + 0.889078i \(0.651346\pi\)
\(462\) 0 0
\(463\) 24.2426 1.12665 0.563326 0.826235i \(-0.309522\pi\)
0.563326 + 0.826235i \(0.309522\pi\)
\(464\) −4.24264 −0.196960
\(465\) 0 0
\(466\) −0.828427 −0.0383761
\(467\) −35.6569 −1.65000 −0.825001 0.565131i \(-0.808826\pi\)
−0.825001 + 0.565131i \(0.808826\pi\)
\(468\) 0 0
\(469\) 13.6569 0.630615
\(470\) −18.4853 −0.852662
\(471\) 0 0
\(472\) 55.1127 2.53677
\(473\) 27.4558 1.26242
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 6.34315 0.290738
\(477\) 0 0
\(478\) −64.5269 −2.95139
\(479\) −17.0711 −0.779997 −0.389998 0.920815i \(-0.627524\pi\)
−0.389998 + 0.920815i \(0.627524\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) 47.4558 2.16155
\(483\) 0 0
\(484\) −22.8579 −1.03899
\(485\) −9.75736 −0.443059
\(486\) 0 0
\(487\) 20.4853 0.928277 0.464138 0.885763i \(-0.346364\pi\)
0.464138 + 0.885763i \(0.346364\pi\)
\(488\) −32.2843 −1.46144
\(489\) 0 0
\(490\) 12.0711 0.545315
\(491\) 1.75736 0.0793085 0.0396543 0.999213i \(-0.487374\pi\)
0.0396543 + 0.999213i \(0.487374\pi\)
\(492\) 0 0
\(493\) 1.65685 0.0746210
\(494\) −8.24264 −0.370854
\(495\) 0 0
\(496\) −9.51472 −0.427223
\(497\) −15.3137 −0.686914
\(498\) 0 0
\(499\) −5.17157 −0.231511 −0.115756 0.993278i \(-0.536929\pi\)
−0.115756 + 0.993278i \(0.536929\pi\)
\(500\) 3.82843 0.171212
\(501\) 0 0
\(502\) −4.24264 −0.189358
\(503\) −4.82843 −0.215289 −0.107644 0.994189i \(-0.534331\pi\)
−0.107644 + 0.994189i \(0.534331\pi\)
\(504\) 0 0
\(505\) 3.17157 0.141133
\(506\) 41.4558 1.84294
\(507\) 0 0
\(508\) 73.9411 3.28061
\(509\) 0.727922 0.0322646 0.0161323 0.999870i \(-0.494865\pi\)
0.0161323 + 0.999870i \(0.494865\pi\)
\(510\) 0 0
\(511\) 10.8284 0.479021
\(512\) 31.2426 1.38074
\(513\) 0 0
\(514\) −10.8284 −0.477621
\(515\) −7.31371 −0.322281
\(516\) 0 0
\(517\) 17.1716 0.755205
\(518\) −11.6569 −0.512173
\(519\) 0 0
\(520\) 15.0711 0.660910
\(521\) −7.75736 −0.339856 −0.169928 0.985456i \(-0.554354\pi\)
−0.169928 + 0.985456i \(0.554354\pi\)
\(522\) 0 0
\(523\) −15.5147 −0.678411 −0.339206 0.940712i \(-0.610158\pi\)
−0.339206 + 0.940712i \(0.610158\pi\)
\(524\) 32.8701 1.43594
\(525\) 0 0
\(526\) −56.6274 −2.46907
\(527\) 3.71573 0.161860
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 19.3137 0.838934
\(531\) 0 0
\(532\) 5.41421 0.234736
\(533\) −0.828427 −0.0358832
\(534\) 0 0
\(535\) −19.3137 −0.835004
\(536\) 42.6274 1.84122
\(537\) 0 0
\(538\) −7.41421 −0.319649
\(539\) −11.2132 −0.482987
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 26.1421 1.12290
\(543\) 0 0
\(544\) −1.85786 −0.0796553
\(545\) −6.48528 −0.277799
\(546\) 0 0
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) −38.2843 −1.63542
\(549\) 0 0
\(550\) −5.41421 −0.230863
\(551\) 1.41421 0.0602475
\(552\) 0 0
\(553\) 0 0
\(554\) 53.1127 2.25654
\(555\) 0 0
\(556\) −54.1421 −2.29614
\(557\) 25.3137 1.07258 0.536288 0.844035i \(-0.319826\pi\)
0.536288 + 0.844035i \(0.319826\pi\)
\(558\) 0 0
\(559\) −41.7990 −1.76791
\(560\) −4.24264 −0.179284
\(561\) 0 0
\(562\) −35.2132 −1.48538
\(563\) 30.2843 1.27633 0.638165 0.769900i \(-0.279694\pi\)
0.638165 + 0.769900i \(0.279694\pi\)
\(564\) 0 0
\(565\) −10.1421 −0.426683
\(566\) 25.0711 1.05382
\(567\) 0 0
\(568\) −47.7990 −2.00560
\(569\) −31.0711 −1.30257 −0.651283 0.758835i \(-0.725769\pi\)
−0.651283 + 0.758835i \(0.725769\pi\)
\(570\) 0 0
\(571\) 9.17157 0.383818 0.191909 0.981413i \(-0.438532\pi\)
0.191909 + 0.981413i \(0.438532\pi\)
\(572\) −29.3137 −1.22567
\(573\) 0 0
\(574\) 0.828427 0.0345779
\(575\) −7.65685 −0.319313
\(576\) 0 0
\(577\) 19.4558 0.809957 0.404979 0.914326i \(-0.367279\pi\)
0.404979 + 0.914326i \(0.367279\pi\)
\(578\) 37.7279 1.56927
\(579\) 0 0
\(580\) −5.41421 −0.224813
\(581\) 18.1421 0.752663
\(582\) 0 0
\(583\) −17.9411 −0.743045
\(584\) 33.7990 1.39861
\(585\) 0 0
\(586\) −27.7990 −1.14837
\(587\) 12.3431 0.509456 0.254728 0.967013i \(-0.418014\pi\)
0.254728 + 0.967013i \(0.418014\pi\)
\(588\) 0 0
\(589\) 3.17157 0.130682
\(590\) 30.1421 1.24093
\(591\) 0 0
\(592\) −10.2426 −0.420970
\(593\) −36.6274 −1.50411 −0.752054 0.659102i \(-0.770937\pi\)
−0.752054 + 0.659102i \(0.770937\pi\)
\(594\) 0 0
\(595\) 1.65685 0.0679244
\(596\) 22.9706 0.940911
\(597\) 0 0
\(598\) −63.1127 −2.58087
\(599\) 3.02944 0.123779 0.0618897 0.998083i \(-0.480287\pi\)
0.0618897 + 0.998083i \(0.480287\pi\)
\(600\) 0 0
\(601\) 8.14214 0.332125 0.166062 0.986115i \(-0.446895\pi\)
0.166062 + 0.986115i \(0.446895\pi\)
\(602\) 41.7990 1.70360
\(603\) 0 0
\(604\) −24.8284 −1.01025
\(605\) −5.97056 −0.242738
\(606\) 0 0
\(607\) 16.4853 0.669117 0.334558 0.942375i \(-0.391413\pi\)
0.334558 + 0.942375i \(0.391413\pi\)
\(608\) −1.58579 −0.0643121
\(609\) 0 0
\(610\) −17.6569 −0.714905
\(611\) −26.1421 −1.05760
\(612\) 0 0
\(613\) −10.4853 −0.423497 −0.211748 0.977324i \(-0.567916\pi\)
−0.211748 + 0.977324i \(0.567916\pi\)
\(614\) −51.1127 −2.06274
\(615\) 0 0
\(616\) 14.0000 0.564076
\(617\) −28.4853 −1.14677 −0.573387 0.819285i \(-0.694371\pi\)
−0.573387 + 0.819285i \(0.694371\pi\)
\(618\) 0 0
\(619\) −20.4853 −0.823373 −0.411686 0.911326i \(-0.635060\pi\)
−0.411686 + 0.911326i \(0.635060\pi\)
\(620\) −12.1421 −0.487640
\(621\) 0 0
\(622\) −15.0711 −0.604295
\(623\) 8.34315 0.334261
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 40.1421 1.60185
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −64.2843 −2.55305
\(635\) 19.3137 0.766441
\(636\) 0 0
\(637\) 17.0711 0.676380
\(638\) 7.65685 0.303138
\(639\) 0 0
\(640\) 20.5563 0.812561
\(641\) 31.3553 1.23846 0.619231 0.785209i \(-0.287445\pi\)
0.619231 + 0.785209i \(0.287445\pi\)
\(642\) 0 0
\(643\) 42.3848 1.67149 0.835746 0.549116i \(-0.185035\pi\)
0.835746 + 0.549116i \(0.185035\pi\)
\(644\) 41.4558 1.63359
\(645\) 0 0
\(646\) −2.82843 −0.111283
\(647\) 36.1421 1.42089 0.710447 0.703751i \(-0.248493\pi\)
0.710447 + 0.703751i \(0.248493\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 8.24264 0.323303
\(651\) 0 0
\(652\) 83.8406 3.28345
\(653\) −42.8284 −1.67601 −0.838003 0.545666i \(-0.816277\pi\)
−0.838003 + 0.545666i \(0.816277\pi\)
\(654\) 0 0
\(655\) 8.58579 0.335474
\(656\) 0.727922 0.0284206
\(657\) 0 0
\(658\) 26.1421 1.01913
\(659\) 18.6274 0.725621 0.362811 0.931863i \(-0.381817\pi\)
0.362811 + 0.931863i \(0.381817\pi\)
\(660\) 0 0
\(661\) 7.45584 0.289999 0.144999 0.989432i \(-0.453682\pi\)
0.144999 + 0.989432i \(0.453682\pi\)
\(662\) −67.9411 −2.64061
\(663\) 0 0
\(664\) 56.6274 2.19757
\(665\) 1.41421 0.0548408
\(666\) 0 0
\(667\) 10.8284 0.419278
\(668\) 66.2843 2.56462
\(669\) 0 0
\(670\) 23.3137 0.900687
\(671\) 16.4020 0.633193
\(672\) 0 0
\(673\) −44.1838 −1.70316 −0.851580 0.524225i \(-0.824355\pi\)
−0.851580 + 0.524225i \(0.824355\pi\)
\(674\) 59.3553 2.28628
\(675\) 0 0
\(676\) −5.14214 −0.197774
\(677\) −16.9706 −0.652232 −0.326116 0.945330i \(-0.605740\pi\)
−0.326116 + 0.945330i \(0.605740\pi\)
\(678\) 0 0
\(679\) 13.7990 0.529557
\(680\) 5.17157 0.198321
\(681\) 0 0
\(682\) 17.1716 0.657534
\(683\) −18.3431 −0.701881 −0.350940 0.936398i \(-0.614138\pi\)
−0.350940 + 0.936398i \(0.614138\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −40.9706 −1.56426
\(687\) 0 0
\(688\) 36.7279 1.40024
\(689\) 27.3137 1.04057
\(690\) 0 0
\(691\) 46.8284 1.78144 0.890719 0.454555i \(-0.150202\pi\)
0.890719 + 0.454555i \(0.150202\pi\)
\(692\) 75.7990 2.88145
\(693\) 0 0
\(694\) −66.2843 −2.51612
\(695\) −14.1421 −0.536442
\(696\) 0 0
\(697\) −0.284271 −0.0107675
\(698\) −43.4558 −1.64483
\(699\) 0 0
\(700\) −5.41421 −0.204638
\(701\) −6.68629 −0.252538 −0.126269 0.991996i \(-0.540300\pi\)
−0.126269 + 0.991996i \(0.540300\pi\)
\(702\) 0 0
\(703\) 3.41421 0.128770
\(704\) −22.0416 −0.830725
\(705\) 0 0
\(706\) 8.82843 0.332262
\(707\) −4.48528 −0.168686
\(708\) 0 0
\(709\) 28.9706 1.08801 0.544006 0.839081i \(-0.316907\pi\)
0.544006 + 0.839081i \(0.316907\pi\)
\(710\) −26.1421 −0.981097
\(711\) 0 0
\(712\) 26.0416 0.975951
\(713\) 24.2843 0.909453
\(714\) 0 0
\(715\) −7.65685 −0.286350
\(716\) −63.1127 −2.35863
\(717\) 0 0
\(718\) 60.0416 2.24073
\(719\) 1.07107 0.0399441 0.0199720 0.999801i \(-0.493642\pi\)
0.0199720 + 0.999801i \(0.493642\pi\)
\(720\) 0 0
\(721\) 10.3431 0.385199
\(722\) −2.41421 −0.0898477
\(723\) 0 0
\(724\) −79.7401 −2.96352
\(725\) −1.41421 −0.0525226
\(726\) 0 0
\(727\) 47.3553 1.75631 0.878156 0.478374i \(-0.158774\pi\)
0.878156 + 0.478374i \(0.158774\pi\)
\(728\) −21.3137 −0.789939
\(729\) 0 0
\(730\) 18.4853 0.684171
\(731\) −14.3431 −0.530500
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 34.7279 1.28183
\(735\) 0 0
\(736\) −12.1421 −0.447565
\(737\) −21.6569 −0.797740
\(738\) 0 0
\(739\) 14.3431 0.527621 0.263811 0.964575i \(-0.415021\pi\)
0.263811 + 0.964575i \(0.415021\pi\)
\(740\) −13.0711 −0.480502
\(741\) 0 0
\(742\) −27.3137 −1.00272
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) −1.41421 −0.0517780
\(747\) 0 0
\(748\) −10.0589 −0.367789
\(749\) 27.3137 0.998021
\(750\) 0 0
\(751\) −24.1421 −0.880959 −0.440480 0.897763i \(-0.645192\pi\)
−0.440480 + 0.897763i \(0.645192\pi\)
\(752\) 22.9706 0.837650
\(753\) 0 0
\(754\) −11.6569 −0.424518
\(755\) −6.48528 −0.236024
\(756\) 0 0
\(757\) −32.4264 −1.17856 −0.589279 0.807930i \(-0.700588\pi\)
−0.589279 + 0.807930i \(0.700588\pi\)
\(758\) −7.65685 −0.278109
\(759\) 0 0
\(760\) 4.41421 0.160120
\(761\) −43.9411 −1.59286 −0.796432 0.604728i \(-0.793282\pi\)
−0.796432 + 0.604728i \(0.793282\pi\)
\(762\) 0 0
\(763\) 9.17157 0.332033
\(764\) −39.2132 −1.41868
\(765\) 0 0
\(766\) 67.5980 2.44241
\(767\) 42.6274 1.53919
\(768\) 0 0
\(769\) 42.9706 1.54956 0.774779 0.632232i \(-0.217861\pi\)
0.774779 + 0.632232i \(0.217861\pi\)
\(770\) 7.65685 0.275934
\(771\) 0 0
\(772\) 19.4142 0.698733
\(773\) 25.6569 0.922813 0.461406 0.887189i \(-0.347345\pi\)
0.461406 + 0.887189i \(0.347345\pi\)
\(774\) 0 0
\(775\) −3.17157 −0.113926
\(776\) 43.0711 1.54616
\(777\) 0 0
\(778\) 74.7696 2.68062
\(779\) −0.242641 −0.00869350
\(780\) 0 0
\(781\) 24.2843 0.868960
\(782\) −21.6569 −0.774448
\(783\) 0 0
\(784\) −15.0000 −0.535714
\(785\) 10.4853 0.374236
\(786\) 0 0
\(787\) −42.8284 −1.52667 −0.763334 0.646004i \(-0.776439\pi\)
−0.763334 + 0.646004i \(0.776439\pi\)
\(788\) −26.1421 −0.931275
\(789\) 0 0
\(790\) 0 0
\(791\) 14.3431 0.509984
\(792\) 0 0
\(793\) −24.9706 −0.886731
\(794\) −69.1127 −2.45272
\(795\) 0 0
\(796\) 69.4558 2.46180
\(797\) 14.1421 0.500940 0.250470 0.968124i \(-0.419415\pi\)
0.250470 + 0.968124i \(0.419415\pi\)
\(798\) 0 0
\(799\) −8.97056 −0.317356
\(800\) 1.58579 0.0560660
\(801\) 0 0
\(802\) −18.7279 −0.661306
\(803\) −17.1716 −0.605972
\(804\) 0 0
\(805\) 10.8284 0.381652
\(806\) −26.1421 −0.920817
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 8.68629 0.305017 0.152508 0.988302i \(-0.451265\pi\)
0.152508 + 0.988302i \(0.451265\pi\)
\(812\) 7.65685 0.268703
\(813\) 0 0
\(814\) 18.4853 0.647909
\(815\) 21.8995 0.767106
\(816\) 0 0
\(817\) −12.2426 −0.428316
\(818\) 30.9706 1.08286
\(819\) 0 0
\(820\) 0.928932 0.0324397
\(821\) −14.4853 −0.505540 −0.252770 0.967526i \(-0.581342\pi\)
−0.252770 + 0.967526i \(0.581342\pi\)
\(822\) 0 0
\(823\) −25.0122 −0.871870 −0.435935 0.899978i \(-0.643582\pi\)
−0.435935 + 0.899978i \(0.643582\pi\)
\(824\) 32.2843 1.12468
\(825\) 0 0
\(826\) −42.6274 −1.48320
\(827\) 2.68629 0.0934115 0.0467058 0.998909i \(-0.485128\pi\)
0.0467058 + 0.998909i \(0.485128\pi\)
\(828\) 0 0
\(829\) 46.4853 1.61450 0.807250 0.590209i \(-0.200955\pi\)
0.807250 + 0.590209i \(0.200955\pi\)
\(830\) 30.9706 1.07500
\(831\) 0 0
\(832\) 33.5563 1.16336
\(833\) 5.85786 0.202963
\(834\) 0 0
\(835\) 17.3137 0.599166
\(836\) −8.58579 −0.296946
\(837\) 0 0
\(838\) 8.24264 0.284737
\(839\) 9.85786 0.340331 0.170166 0.985415i \(-0.445570\pi\)
0.170166 + 0.985415i \(0.445570\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) −22.4853 −0.774894
\(843\) 0 0
\(844\) 28.0000 0.963800
\(845\) −1.34315 −0.0462056
\(846\) 0 0
\(847\) 8.44365 0.290127
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) 2.82843 0.0970143
\(851\) 26.1421 0.896141
\(852\) 0 0
\(853\) −16.8284 −0.576194 −0.288097 0.957601i \(-0.593023\pi\)
−0.288097 + 0.957601i \(0.593023\pi\)
\(854\) 24.9706 0.854475
\(855\) 0 0
\(856\) 85.2548 2.91395
\(857\) −12.6863 −0.433355 −0.216678 0.976243i \(-0.569522\pi\)
−0.216678 + 0.976243i \(0.569522\pi\)
\(858\) 0 0
\(859\) −49.9411 −1.70397 −0.851985 0.523567i \(-0.824601\pi\)
−0.851985 + 0.523567i \(0.824601\pi\)
\(860\) 46.8701 1.59826
\(861\) 0 0
\(862\) −75.1127 −2.55835
\(863\) −8.68629 −0.295685 −0.147842 0.989011i \(-0.547233\pi\)
−0.147842 + 0.989011i \(0.547233\pi\)
\(864\) 0 0
\(865\) 19.7990 0.673186
\(866\) 26.3848 0.896591
\(867\) 0 0
\(868\) 17.1716 0.582841
\(869\) 0 0
\(870\) 0 0
\(871\) 32.9706 1.11716
\(872\) 28.6274 0.969447
\(873\) 0 0
\(874\) −18.4853 −0.625274
\(875\) −1.41421 −0.0478091
\(876\) 0 0
\(877\) −34.9289 −1.17947 −0.589733 0.807598i \(-0.700767\pi\)
−0.589733 + 0.807598i \(0.700767\pi\)
\(878\) 52.2843 1.76451
\(879\) 0 0
\(880\) 6.72792 0.226798
\(881\) 5.79899 0.195373 0.0976865 0.995217i \(-0.468856\pi\)
0.0976865 + 0.995217i \(0.468856\pi\)
\(882\) 0 0
\(883\) −12.0416 −0.405233 −0.202617 0.979258i \(-0.564945\pi\)
−0.202617 + 0.979258i \(0.564945\pi\)
\(884\) 15.3137 0.515056
\(885\) 0 0
\(886\) 43.4558 1.45993
\(887\) −25.9411 −0.871018 −0.435509 0.900184i \(-0.643432\pi\)
−0.435509 + 0.900184i \(0.643432\pi\)
\(888\) 0 0
\(889\) −27.3137 −0.916072
\(890\) 14.2426 0.477414
\(891\) 0 0
\(892\) 71.3137 2.38776
\(893\) −7.65685 −0.256227
\(894\) 0 0
\(895\) −16.4853 −0.551042
\(896\) −29.0711 −0.971196
\(897\) 0 0
\(898\) −64.8701 −2.16474
\(899\) 4.48528 0.149593
\(900\) 0 0
\(901\) 9.37258 0.312246
\(902\) −1.31371 −0.0437417
\(903\) 0 0
\(904\) 44.7696 1.48901
\(905\) −20.8284 −0.692360
\(906\) 0 0
\(907\) −18.1421 −0.602400 −0.301200 0.953561i \(-0.597387\pi\)
−0.301200 + 0.953561i \(0.597387\pi\)
\(908\) −57.3137 −1.90202
\(909\) 0 0
\(910\) −11.6569 −0.386421
\(911\) −8.68629 −0.287790 −0.143895 0.989593i \(-0.545963\pi\)
−0.143895 + 0.989593i \(0.545963\pi\)
\(912\) 0 0
\(913\) −28.7696 −0.952133
\(914\) −79.2548 −2.62152
\(915\) 0 0
\(916\) −86.6274 −2.86225
\(917\) −12.1421 −0.400969
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 33.7990 1.11432
\(921\) 0 0
\(922\) 47.4558 1.56287
\(923\) −36.9706 −1.21690
\(924\) 0 0
\(925\) −3.41421 −0.112259
\(926\) −58.5269 −1.92331
\(927\) 0 0
\(928\) −2.24264 −0.0736183
\(929\) −33.1127 −1.08639 −0.543196 0.839606i \(-0.682786\pi\)
−0.543196 + 0.839606i \(0.682786\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 1.31371 0.0430320
\(933\) 0 0
\(934\) 86.0833 2.81673
\(935\) −2.62742 −0.0859257
\(936\) 0 0
\(937\) −29.7990 −0.973491 −0.486745 0.873544i \(-0.661816\pi\)
−0.486745 + 0.873544i \(0.661816\pi\)
\(938\) −32.9706 −1.07653
\(939\) 0 0
\(940\) 29.3137 0.956108
\(941\) −22.8701 −0.745543 −0.372771 0.927923i \(-0.621592\pi\)
−0.372771 + 0.927923i \(0.621592\pi\)
\(942\) 0 0
\(943\) −1.85786 −0.0605004
\(944\) −37.4558 −1.21908
\(945\) 0 0
\(946\) −66.2843 −2.15509
\(947\) 47.1716 1.53287 0.766435 0.642322i \(-0.222029\pi\)
0.766435 + 0.642322i \(0.222029\pi\)
\(948\) 0 0
\(949\) 26.1421 0.848610
\(950\) 2.41421 0.0783274
\(951\) 0 0
\(952\) −7.31371 −0.237039
\(953\) 53.4558 1.73160 0.865802 0.500386i \(-0.166809\pi\)
0.865802 + 0.500386i \(0.166809\pi\)
\(954\) 0 0
\(955\) −10.2426 −0.331444
\(956\) 102.326 3.30946
\(957\) 0 0
\(958\) 41.2132 1.33154
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) −20.9411 −0.675520
\(962\) −28.1421 −0.907339
\(963\) 0 0
\(964\) −75.2548 −2.42379
\(965\) 5.07107 0.163243
\(966\) 0 0
\(967\) 8.04163 0.258601 0.129301 0.991605i \(-0.458727\pi\)
0.129301 + 0.991605i \(0.458727\pi\)
\(968\) 26.3553 0.847093
\(969\) 0 0
\(970\) 23.5563 0.756349
\(971\) −22.3431 −0.717026 −0.358513 0.933525i \(-0.616716\pi\)
−0.358513 + 0.933525i \(0.616716\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) −49.4558 −1.58467
\(975\) 0 0
\(976\) 21.9411 0.702318
\(977\) −32.2843 −1.03287 −0.516433 0.856328i \(-0.672740\pi\)
−0.516433 + 0.856328i \(0.672740\pi\)
\(978\) 0 0
\(979\) −13.2304 −0.422847
\(980\) −19.1421 −0.611473
\(981\) 0 0
\(982\) −4.24264 −0.135388
\(983\) 24.6274 0.785493 0.392746 0.919647i \(-0.371525\pi\)
0.392746 + 0.919647i \(0.371525\pi\)
\(984\) 0 0
\(985\) −6.82843 −0.217572
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 13.0711 0.415846
\(989\) −93.7401 −2.98076
\(990\) 0 0
\(991\) 2.34315 0.0744325 0.0372162 0.999307i \(-0.488151\pi\)
0.0372162 + 0.999307i \(0.488151\pi\)
\(992\) −5.02944 −0.159685
\(993\) 0 0
\(994\) 36.9706 1.17264
\(995\) 18.1421 0.575144
\(996\) 0 0
\(997\) −38.0833 −1.20611 −0.603054 0.797700i \(-0.706050\pi\)
−0.603054 + 0.797700i \(0.706050\pi\)
\(998\) 12.4853 0.395215
\(999\) 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 855.2.a.d.1.1 2
3.2 odd 2 285.2.a.g.1.2 2
5.4 even 2 4275.2.a.y.1.2 2
12.11 even 2 4560.2.a.bf.1.2 2
15.2 even 4 1425.2.c.l.799.4 4
15.8 even 4 1425.2.c.l.799.1 4
15.14 odd 2 1425.2.a.k.1.1 2
57.56 even 2 5415.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.2 2 3.2 odd 2
855.2.a.d.1.1 2 1.1 even 1 trivial
1425.2.a.k.1.1 2 15.14 odd 2
1425.2.c.l.799.1 4 15.8 even 4
1425.2.c.l.799.4 4 15.2 even 4
4275.2.a.y.1.2 2 5.4 even 2
4560.2.a.bf.1.2 2 12.11 even 2
5415.2.a.n.1.1 2 57.56 even 2