Properties

Label 2646.2.d.d.2645.4
Level $2646$
Weight $2$
Character 2646.2645
Analytic conductor $21.128$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2645.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2646.2645
Dual form 2646.2.d.d.2645.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} +1.00000i q^{8} -2.44949i q^{10} +4.24264i q^{11} -0.717439i q^{13} +1.00000 q^{16} +2.44949 q^{17} -4.89898i q^{19} -2.44949 q^{20} +4.24264 q^{22} -6.00000i q^{23} +1.00000 q^{25} -0.717439 q^{26} +1.75736i q^{29} +9.08052i q^{31} -1.00000i q^{32} -2.44949i q^{34} +5.24264 q^{37} -4.89898 q^{38} +2.44949i q^{40} -2.44949 q^{41} +7.00000 q^{43} -4.24264i q^{44} -6.00000 q^{46} +12.8418 q^{47} -1.00000i q^{50} +0.717439i q^{52} -14.4853i q^{53} +10.3923i q^{55} +1.75736 q^{58} -2.44949 q^{59} +4.18154i q^{61} +9.08052 q^{62} -1.00000 q^{64} -1.75736i q^{65} +13.4853 q^{67} -2.44949 q^{68} +12.7279i q^{71} +5.49333i q^{73} -5.24264i q^{74} +4.89898i q^{76} +0.757359 q^{79} +2.44949 q^{80} +2.44949i q^{82} +15.2913 q^{83} +6.00000 q^{85} -7.00000i q^{86} -4.24264 q^{88} +3.04384 q^{89} +6.00000i q^{92} -12.8418i q^{94} -12.0000i q^{95} +3.16693i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} + 8 q^{37} + 56 q^{43} - 48 q^{46} + 48 q^{58} - 8 q^{64} + 40 q^{67} + 40 q^{79} + 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) − 2.44949i − 0.774597i
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) − 0.717439i − 0.198982i −0.995038 0.0994909i \(-0.968279\pi\)
0.995038 0.0994909i \(-0.0317214\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.44949 0.594089 0.297044 0.954864i \(-0.403999\pi\)
0.297044 + 0.954864i \(0.403999\pi\)
\(18\) 0 0
\(19\) − 4.89898i − 1.12390i −0.827170 0.561951i \(-0.810051\pi\)
0.827170 0.561951i \(-0.189949\pi\)
\(20\) −2.44949 −0.547723
\(21\) 0 0
\(22\) 4.24264 0.904534
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.717439 −0.140701
\(27\) 0 0
\(28\) 0 0
\(29\) 1.75736i 0.326333i 0.986599 + 0.163167i \(0.0521708\pi\)
−0.986599 + 0.163167i \(0.947829\pi\)
\(30\) 0 0
\(31\) 9.08052i 1.63091i 0.578821 + 0.815455i \(0.303513\pi\)
−0.578821 + 0.815455i \(0.696487\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) − 2.44949i − 0.420084i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.24264 0.861885 0.430942 0.902379i \(-0.358181\pi\)
0.430942 + 0.902379i \(0.358181\pi\)
\(38\) −4.89898 −0.794719
\(39\) 0 0
\(40\) 2.44949i 0.387298i
\(41\) −2.44949 −0.382546 −0.191273 0.981537i \(-0.561262\pi\)
−0.191273 + 0.981537i \(0.561262\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) − 4.24264i − 0.639602i
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 12.8418 1.87317 0.936584 0.350443i \(-0.113969\pi\)
0.936584 + 0.350443i \(0.113969\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 1.00000i − 0.141421i
\(51\) 0 0
\(52\) 0.717439i 0.0994909i
\(53\) − 14.4853i − 1.98971i −0.101327 0.994853i \(-0.532309\pi\)
0.101327 0.994853i \(-0.467691\pi\)
\(54\) 0 0
\(55\) 10.3923i 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.75736 0.230753
\(59\) −2.44949 −0.318896 −0.159448 0.987206i \(-0.550971\pi\)
−0.159448 + 0.987206i \(0.550971\pi\)
\(60\) 0 0
\(61\) 4.18154i 0.535391i 0.963504 + 0.267696i \(0.0862622\pi\)
−0.963504 + 0.267696i \(0.913738\pi\)
\(62\) 9.08052 1.15323
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) − 1.75736i − 0.217974i
\(66\) 0 0
\(67\) 13.4853 1.64749 0.823745 0.566961i \(-0.191881\pi\)
0.823745 + 0.566961i \(0.191881\pi\)
\(68\) −2.44949 −0.297044
\(69\) 0 0
\(70\) 0 0
\(71\) 12.7279i 1.51053i 0.655422 + 0.755263i \(0.272491\pi\)
−0.655422 + 0.755263i \(0.727509\pi\)
\(72\) 0 0
\(73\) 5.49333i 0.642945i 0.946919 + 0.321473i \(0.104178\pi\)
−0.946919 + 0.321473i \(0.895822\pi\)
\(74\) − 5.24264i − 0.609445i
\(75\) 0 0
\(76\) 4.89898i 0.561951i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.757359 0.0852096 0.0426048 0.999092i \(-0.486434\pi\)
0.0426048 + 0.999092i \(0.486434\pi\)
\(80\) 2.44949 0.273861
\(81\) 0 0
\(82\) 2.44949i 0.270501i
\(83\) 15.2913 1.67844 0.839218 0.543795i \(-0.183013\pi\)
0.839218 + 0.543795i \(0.183013\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) − 7.00000i − 0.754829i
\(87\) 0 0
\(88\) −4.24264 −0.452267
\(89\) 3.04384 0.322646 0.161323 0.986902i \(-0.448424\pi\)
0.161323 + 0.986902i \(0.448424\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000i 0.625543i
\(93\) 0 0
\(94\) − 12.8418i − 1.32453i
\(95\) − 12.0000i − 1.23117i
\(96\) 0 0
\(97\) 3.16693i 0.321553i 0.986991 + 0.160776i \(0.0513998\pi\)
−0.986991 + 0.160776i \(0.948600\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 7.34847 0.731200 0.365600 0.930772i \(-0.380864\pi\)
0.365600 + 0.930772i \(0.380864\pi\)
\(102\) 0 0
\(103\) 11.1097i 1.09468i 0.836912 + 0.547338i \(0.184359\pi\)
−0.836912 + 0.547338i \(0.815641\pi\)
\(104\) 0.717439 0.0703507
\(105\) 0 0
\(106\) −14.4853 −1.40693
\(107\) 2.48528i 0.240261i 0.992758 + 0.120131i \(0.0383313\pi\)
−0.992758 + 0.120131i \(0.961669\pi\)
\(108\) 0 0
\(109\) −17.7279 −1.69803 −0.849013 0.528371i \(-0.822803\pi\)
−0.849013 + 0.528371i \(0.822803\pi\)
\(110\) 10.3923 0.990867
\(111\) 0 0
\(112\) 0 0
\(113\) − 10.2426i − 0.963547i −0.876296 0.481773i \(-0.839993\pi\)
0.876296 0.481773i \(-0.160007\pi\)
\(114\) 0 0
\(115\) − 14.6969i − 1.37050i
\(116\) − 1.75736i − 0.163167i
\(117\) 0 0
\(118\) 2.44949i 0.225494i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 4.18154 0.378579
\(123\) 0 0
\(124\) − 9.08052i − 0.815455i
\(125\) −9.79796 −0.876356
\(126\) 0 0
\(127\) −7.72792 −0.685742 −0.342871 0.939382i \(-0.611399\pi\)
−0.342871 + 0.939382i \(0.611399\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −1.75736 −0.154131
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 13.4853i − 1.16495i
\(135\) 0 0
\(136\) 2.44949i 0.210042i
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 8.06591i 0.684141i 0.939674 + 0.342071i \(0.111128\pi\)
−0.939674 + 0.342071i \(0.888872\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.7279 1.06810
\(143\) 3.04384 0.254538
\(144\) 0 0
\(145\) 4.30463i 0.357480i
\(146\) 5.49333 0.454631
\(147\) 0 0
\(148\) −5.24264 −0.430942
\(149\) − 16.2426i − 1.33065i −0.746554 0.665324i \(-0.768293\pi\)
0.746554 0.665324i \(-0.231707\pi\)
\(150\) 0 0
\(151\) 8.75736 0.712664 0.356332 0.934359i \(-0.384027\pi\)
0.356332 + 0.934359i \(0.384027\pi\)
\(152\) 4.89898 0.397360
\(153\) 0 0
\(154\) 0 0
\(155\) 22.2426i 1.78657i
\(156\) 0 0
\(157\) 10.3923i 0.829396i 0.909959 + 0.414698i \(0.136113\pi\)
−0.909959 + 0.414698i \(0.863887\pi\)
\(158\) − 0.757359i − 0.0602523i
\(159\) 0 0
\(160\) − 2.44949i − 0.193649i
\(161\) 0 0
\(162\) 0 0
\(163\) −9.48528 −0.742945 −0.371472 0.928444i \(-0.621147\pi\)
−0.371472 + 0.928444i \(0.621147\pi\)
\(164\) 2.44949 0.191273
\(165\) 0 0
\(166\) − 15.2913i − 1.18683i
\(167\) −0.594346 −0.0459919 −0.0229959 0.999736i \(-0.507320\pi\)
−0.0229959 + 0.999736i \(0.507320\pi\)
\(168\) 0 0
\(169\) 12.4853 0.960406
\(170\) − 6.00000i − 0.460179i
\(171\) 0 0
\(172\) −7.00000 −0.533745
\(173\) 20.7846 1.58022 0.790112 0.612962i \(-0.210022\pi\)
0.790112 + 0.612962i \(0.210022\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.24264i 0.319801i
\(177\) 0 0
\(178\) − 3.04384i − 0.228145i
\(179\) 6.72792i 0.502869i 0.967874 + 0.251434i \(0.0809022\pi\)
−0.967874 + 0.251434i \(0.919098\pi\)
\(180\) 0 0
\(181\) − 9.79796i − 0.728277i −0.931345 0.364138i \(-0.881364\pi\)
0.931345 0.364138i \(-0.118636\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) 12.8418 0.944148
\(186\) 0 0
\(187\) 10.3923i 0.759961i
\(188\) −12.8418 −0.936584
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) − 21.2132i − 1.53493i −0.641089 0.767467i \(-0.721517\pi\)
0.641089 0.767467i \(-0.278483\pi\)
\(192\) 0 0
\(193\) −1.48528 −0.106913 −0.0534564 0.998570i \(-0.517024\pi\)
−0.0534564 + 0.998570i \(0.517024\pi\)
\(194\) 3.16693 0.227372
\(195\) 0 0
\(196\) 0 0
\(197\) − 16.9706i − 1.20910i −0.796566 0.604551i \(-0.793352\pi\)
0.796566 0.604551i \(-0.206648\pi\)
\(198\) 0 0
\(199\) 20.9077i 1.48211i 0.671446 + 0.741054i \(0.265674\pi\)
−0.671446 + 0.741054i \(0.734326\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 0 0
\(202\) − 7.34847i − 0.517036i
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 11.1097 0.774053
\(207\) 0 0
\(208\) − 0.717439i − 0.0497454i
\(209\) 20.7846 1.43770
\(210\) 0 0
\(211\) 3.48528 0.239937 0.119968 0.992778i \(-0.461721\pi\)
0.119968 + 0.992778i \(0.461721\pi\)
\(212\) 14.4853i 0.994853i
\(213\) 0 0
\(214\) 2.48528 0.169890
\(215\) 17.1464 1.16938
\(216\) 0 0
\(217\) 0 0
\(218\) 17.7279i 1.20069i
\(219\) 0 0
\(220\) − 10.3923i − 0.700649i
\(221\) − 1.75736i − 0.118213i
\(222\) 0 0
\(223\) − 10.3923i − 0.695920i −0.937509 0.347960i \(-0.886874\pi\)
0.937509 0.347960i \(-0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −10.2426 −0.681330
\(227\) −25.0892 −1.66523 −0.832616 0.553851i \(-0.813158\pi\)
−0.832616 + 0.553851i \(0.813158\pi\)
\(228\) 0 0
\(229\) − 5.61642i − 0.371143i −0.982631 0.185572i \(-0.940586\pi\)
0.982631 0.185572i \(-0.0594137\pi\)
\(230\) −14.6969 −0.969087
\(231\) 0 0
\(232\) −1.75736 −0.115376
\(233\) − 20.4853i − 1.34204i −0.741441 0.671018i \(-0.765857\pi\)
0.741441 0.671018i \(-0.234143\pi\)
\(234\) 0 0
\(235\) 31.4558 2.05195
\(236\) 2.44949 0.159448
\(237\) 0 0
\(238\) 0 0
\(239\) − 16.2426i − 1.05065i −0.850902 0.525325i \(-0.823944\pi\)
0.850902 0.525325i \(-0.176056\pi\)
\(240\) 0 0
\(241\) 1.13770i 0.0732860i 0.999328 + 0.0366430i \(0.0116664\pi\)
−0.999328 + 0.0366430i \(0.988334\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 0 0
\(244\) − 4.18154i − 0.267696i
\(245\) 0 0
\(246\) 0 0
\(247\) −3.51472 −0.223636
\(248\) −9.08052 −0.576614
\(249\) 0 0
\(250\) 9.79796i 0.619677i
\(251\) −15.2913 −0.965177 −0.482589 0.875847i \(-0.660303\pi\)
−0.482589 + 0.875847i \(0.660303\pi\)
\(252\) 0 0
\(253\) 25.4558 1.60040
\(254\) 7.72792i 0.484893i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −25.0892 −1.56502 −0.782512 0.622636i \(-0.786062\pi\)
−0.782512 + 0.622636i \(0.786062\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.75736i 0.108987i
\(261\) 0 0
\(262\) 0 0
\(263\) − 6.72792i − 0.414861i −0.978250 0.207431i \(-0.933490\pi\)
0.978250 0.207431i \(-0.0665102\pi\)
\(264\) 0 0
\(265\) − 35.4815i − 2.17961i
\(266\) 0 0
\(267\) 0 0
\(268\) −13.4853 −0.823745
\(269\) −9.79796 −0.597392 −0.298696 0.954348i \(-0.596552\pi\)
−0.298696 + 0.954348i \(0.596552\pi\)
\(270\) 0 0
\(271\) − 18.8785i − 1.14679i −0.819280 0.573393i \(-0.805627\pi\)
0.819280 0.573393i \(-0.194373\pi\)
\(272\) 2.44949 0.148522
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 4.24264i 0.255841i
\(276\) 0 0
\(277\) 23.7279 1.42567 0.712836 0.701330i \(-0.247410\pi\)
0.712836 + 0.701330i \(0.247410\pi\)
\(278\) 8.06591 0.483761
\(279\) 0 0
\(280\) 0 0
\(281\) 13.7574i 0.820695i 0.911929 + 0.410348i \(0.134593\pi\)
−0.911929 + 0.410348i \(0.865407\pi\)
\(282\) 0 0
\(283\) − 6.03668i − 0.358844i −0.983772 0.179422i \(-0.942577\pi\)
0.983772 0.179422i \(-0.0574227\pi\)
\(284\) − 12.7279i − 0.755263i
\(285\) 0 0
\(286\) − 3.04384i − 0.179986i
\(287\) 0 0
\(288\) 0 0
\(289\) −11.0000 −0.647059
\(290\) 4.30463 0.252777
\(291\) 0 0
\(292\) − 5.49333i − 0.321473i
\(293\) −12.8418 −0.750226 −0.375113 0.926979i \(-0.622396\pi\)
−0.375113 + 0.926979i \(0.622396\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 5.24264i 0.304722i
\(297\) 0 0
\(298\) −16.2426 −0.940911
\(299\) −4.30463 −0.248943
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.75736i − 0.503929i
\(303\) 0 0
\(304\) − 4.89898i − 0.280976i
\(305\) 10.2426i 0.586492i
\(306\) 0 0
\(307\) 26.8213i 1.53077i 0.643571 + 0.765386i \(0.277452\pi\)
−0.643571 + 0.765386i \(0.722548\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 22.2426 1.26330
\(311\) 17.1464 0.972285 0.486142 0.873880i \(-0.338404\pi\)
0.486142 + 0.873880i \(0.338404\pi\)
\(312\) 0 0
\(313\) − 20.1903i − 1.14122i −0.821221 0.570611i \(-0.806707\pi\)
0.821221 0.570611i \(-0.193293\pi\)
\(314\) 10.3923 0.586472
\(315\) 0 0
\(316\) −0.757359 −0.0426048
\(317\) 22.2426i 1.24927i 0.780916 + 0.624636i \(0.214752\pi\)
−0.780916 + 0.624636i \(0.785248\pi\)
\(318\) 0 0
\(319\) −7.45584 −0.417447
\(320\) −2.44949 −0.136931
\(321\) 0 0
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 0 0
\(325\) − 0.717439i − 0.0397964i
\(326\) 9.48528i 0.525341i
\(327\) 0 0
\(328\) − 2.44949i − 0.135250i
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −15.2913 −0.839218
\(333\) 0 0
\(334\) 0.594346i 0.0325212i
\(335\) 33.0321 1.80473
\(336\) 0 0
\(337\) −21.4558 −1.16877 −0.584387 0.811475i \(-0.698665\pi\)
−0.584387 + 0.811475i \(0.698665\pi\)
\(338\) − 12.4853i − 0.679110i
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −38.5254 −2.08627
\(342\) 0 0
\(343\) 0 0
\(344\) 7.00000i 0.377415i
\(345\) 0 0
\(346\) − 20.7846i − 1.11739i
\(347\) − 4.97056i − 0.266834i −0.991060 0.133417i \(-0.957405\pi\)
0.991060 0.133417i \(-0.0425949\pi\)
\(348\) 0 0
\(349\) − 0.123093i − 0.00658902i −0.999995 0.00329451i \(-0.998951\pi\)
0.999995 0.00329451i \(-0.00104868\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.24264 0.226134
\(353\) −10.9867 −0.584760 −0.292380 0.956302i \(-0.594447\pi\)
−0.292380 + 0.956302i \(0.594447\pi\)
\(354\) 0 0
\(355\) 31.1769i 1.65470i
\(356\) −3.04384 −0.161323
\(357\) 0 0
\(358\) 6.72792 0.355582
\(359\) 14.4853i 0.764504i 0.924058 + 0.382252i \(0.124851\pi\)
−0.924058 + 0.382252i \(0.875149\pi\)
\(360\) 0 0
\(361\) −5.00000 −0.263158
\(362\) −9.79796 −0.514969
\(363\) 0 0
\(364\) 0 0
\(365\) 13.4558i 0.704311i
\(366\) 0 0
\(367\) 29.9882i 1.56537i 0.622416 + 0.782686i \(0.286151\pi\)
−0.622416 + 0.782686i \(0.713849\pi\)
\(368\) − 6.00000i − 0.312772i
\(369\) 0 0
\(370\) − 12.8418i − 0.667613i
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 10.3923 0.537373
\(375\) 0 0
\(376\) 12.8418i 0.662265i
\(377\) 1.26080 0.0649344
\(378\) 0 0
\(379\) −7.48528 −0.384493 −0.192247 0.981347i \(-0.561577\pi\)
−0.192247 + 0.981347i \(0.561577\pi\)
\(380\) 12.0000i 0.615587i
\(381\) 0 0
\(382\) −21.2132 −1.08536
\(383\) −5.49333 −0.280696 −0.140348 0.990102i \(-0.544822\pi\)
−0.140348 + 0.990102i \(0.544822\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.48528i 0.0755988i
\(387\) 0 0
\(388\) − 3.16693i − 0.160776i
\(389\) 15.5147i 0.786627i 0.919404 + 0.393314i \(0.128671\pi\)
−0.919404 + 0.393314i \(0.871329\pi\)
\(390\) 0 0
\(391\) − 14.6969i − 0.743256i
\(392\) 0 0
\(393\) 0 0
\(394\) −16.9706 −0.854965
\(395\) 1.85514 0.0933424
\(396\) 0 0
\(397\) − 15.1682i − 0.761270i −0.924725 0.380635i \(-0.875705\pi\)
0.924725 0.380635i \(-0.124295\pi\)
\(398\) 20.9077 1.04801
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 19.7574i 0.986635i 0.869849 + 0.493318i \(0.164216\pi\)
−0.869849 + 0.493318i \(0.835784\pi\)
\(402\) 0 0
\(403\) 6.51472 0.324521
\(404\) −7.34847 −0.365600
\(405\) 0 0
\(406\) 0 0
\(407\) 22.2426i 1.10253i
\(408\) 0 0
\(409\) 3.76127i 0.185983i 0.995667 + 0.0929915i \(0.0296430\pi\)
−0.995667 + 0.0929915i \(0.970357\pi\)
\(410\) 6.00000i 0.296319i
\(411\) 0 0
\(412\) − 11.1097i − 0.547338i
\(413\) 0 0
\(414\) 0 0
\(415\) 37.4558 1.83864
\(416\) −0.717439 −0.0351753
\(417\) 0 0
\(418\) − 20.7846i − 1.01661i
\(419\) −7.94282 −0.388032 −0.194016 0.980998i \(-0.562151\pi\)
−0.194016 + 0.980998i \(0.562151\pi\)
\(420\) 0 0
\(421\) −23.4558 −1.14317 −0.571584 0.820544i \(-0.693671\pi\)
−0.571584 + 0.820544i \(0.693671\pi\)
\(422\) − 3.48528i − 0.169661i
\(423\) 0 0
\(424\) 14.4853 0.703467
\(425\) 2.44949 0.118818
\(426\) 0 0
\(427\) 0 0
\(428\) − 2.48528i − 0.120131i
\(429\) 0 0
\(430\) − 17.1464i − 0.826874i
\(431\) 1.75736i 0.0846490i 0.999104 + 0.0423245i \(0.0134763\pi\)
−0.999104 + 0.0423245i \(0.986524\pi\)
\(432\) 0 0
\(433\) 2.57258i 0.123630i 0.998088 + 0.0618152i \(0.0196889\pi\)
−0.998088 + 0.0618152i \(0.980311\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 17.7279 0.849013
\(437\) −29.3939 −1.40610
\(438\) 0 0
\(439\) 4.30463i 0.205449i 0.994710 + 0.102724i \(0.0327560\pi\)
−0.994710 + 0.102724i \(0.967244\pi\)
\(440\) −10.3923 −0.495434
\(441\) 0 0
\(442\) −1.75736 −0.0835891
\(443\) 25.4558i 1.20944i 0.796437 + 0.604722i \(0.206716\pi\)
−0.796437 + 0.604722i \(0.793284\pi\)
\(444\) 0 0
\(445\) 7.45584 0.353441
\(446\) −10.3923 −0.492090
\(447\) 0 0
\(448\) 0 0
\(449\) − 5.27208i − 0.248805i −0.992232 0.124402i \(-0.960299\pi\)
0.992232 0.124402i \(-0.0397014\pi\)
\(450\) 0 0
\(451\) − 10.3923i − 0.489355i
\(452\) 10.2426i 0.481773i
\(453\) 0 0
\(454\) 25.0892i 1.17750i
\(455\) 0 0
\(456\) 0 0
\(457\) −23.0000 −1.07589 −0.537947 0.842978i \(-0.680800\pi\)
−0.537947 + 0.842978i \(0.680800\pi\)
\(458\) −5.61642 −0.262438
\(459\) 0 0
\(460\) 14.6969i 0.685248i
\(461\) −21.4511 −0.999076 −0.499538 0.866292i \(-0.666497\pi\)
−0.499538 + 0.866292i \(0.666497\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 1.75736i 0.0815834i
\(465\) 0 0
\(466\) −20.4853 −0.948962
\(467\) −17.7408 −0.820945 −0.410473 0.911873i \(-0.634636\pi\)
−0.410473 + 0.911873i \(0.634636\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 31.4558i − 1.45095i
\(471\) 0 0
\(472\) − 2.44949i − 0.112747i
\(473\) 29.6985i 1.36554i
\(474\) 0 0
\(475\) − 4.89898i − 0.224781i
\(476\) 0 0
\(477\) 0 0
\(478\) −16.2426 −0.742921
\(479\) −2.44949 −0.111920 −0.0559600 0.998433i \(-0.517822\pi\)
−0.0559600 + 0.998433i \(0.517822\pi\)
\(480\) 0 0
\(481\) − 3.76127i − 0.171499i
\(482\) 1.13770 0.0518210
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 7.75736i 0.352244i
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) −4.18154 −0.189289
\(489\) 0 0
\(490\) 0 0
\(491\) − 34.9706i − 1.57820i −0.614265 0.789100i \(-0.710547\pi\)
0.614265 0.789100i \(-0.289453\pi\)
\(492\) 0 0
\(493\) 4.30463i 0.193871i
\(494\) 3.51472i 0.158135i
\(495\) 0 0
\(496\) 9.08052i 0.407727i
\(497\) 0 0
\(498\) 0 0
\(499\) −24.4558 −1.09479 −0.547397 0.836873i \(-0.684381\pi\)
−0.547397 + 0.836873i \(0.684381\pi\)
\(500\) 9.79796 0.438178
\(501\) 0 0
\(502\) 15.2913i 0.682483i
\(503\) 0.594346 0.0265006 0.0132503 0.999912i \(-0.495782\pi\)
0.0132503 + 0.999912i \(0.495782\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) − 25.4558i − 1.13165i
\(507\) 0 0
\(508\) 7.72792 0.342871
\(509\) 9.20361 0.407943 0.203971 0.978977i \(-0.434615\pi\)
0.203971 + 0.978977i \(0.434615\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 25.0892i 1.10664i
\(515\) 27.2132i 1.19916i
\(516\) 0 0
\(517\) 54.4831i 2.39616i
\(518\) 0 0
\(519\) 0 0
\(520\) 1.75736 0.0770653
\(521\) 29.9882 1.31381 0.656904 0.753974i \(-0.271866\pi\)
0.656904 + 0.753974i \(0.271866\pi\)
\(522\) 0 0
\(523\) 27.4156i 1.19880i 0.800449 + 0.599401i \(0.204595\pi\)
−0.800449 + 0.599401i \(0.795405\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −6.72792 −0.293351
\(527\) 22.2426i 0.968905i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) −35.4815 −1.54122
\(531\) 0 0
\(532\) 0 0
\(533\) 1.75736i 0.0761197i
\(534\) 0 0
\(535\) 6.08767i 0.263193i
\(536\) 13.4853i 0.582475i
\(537\) 0 0
\(538\) 9.79796i 0.422420i
\(539\) 0 0
\(540\) 0 0
\(541\) −5.45584 −0.234565 −0.117283 0.993099i \(-0.537418\pi\)
−0.117283 + 0.993099i \(0.537418\pi\)
\(542\) −18.8785 −0.810900
\(543\) 0 0
\(544\) − 2.44949i − 0.105021i
\(545\) −43.4244 −1.86010
\(546\) 0 0
\(547\) 39.9706 1.70902 0.854509 0.519437i \(-0.173858\pi\)
0.854509 + 0.519437i \(0.173858\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 0 0
\(550\) 4.24264 0.180907
\(551\) 8.60927 0.366767
\(552\) 0 0
\(553\) 0 0
\(554\) − 23.7279i − 1.00810i
\(555\) 0 0
\(556\) − 8.06591i − 0.342071i
\(557\) − 21.2132i − 0.898832i −0.893323 0.449416i \(-0.851632\pi\)
0.893323 0.449416i \(-0.148368\pi\)
\(558\) 0 0
\(559\) − 5.02207i − 0.212411i
\(560\) 0 0
\(561\) 0 0
\(562\) 13.7574 0.580319
\(563\) 45.8739 1.93335 0.966676 0.256002i \(-0.0824054\pi\)
0.966676 + 0.256002i \(0.0824054\pi\)
\(564\) 0 0
\(565\) − 25.0892i − 1.05551i
\(566\) −6.03668 −0.253741
\(567\) 0 0
\(568\) −12.7279 −0.534052
\(569\) 10.2426i 0.429394i 0.976681 + 0.214697i \(0.0688764\pi\)
−0.976681 + 0.214697i \(0.931124\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) −3.04384 −0.127269
\(573\) 0 0
\(574\) 0 0
\(575\) − 6.00000i − 0.250217i
\(576\) 0 0
\(577\) − 27.4156i − 1.14133i −0.821184 0.570664i \(-0.806686\pi\)
0.821184 0.570664i \(-0.193314\pi\)
\(578\) 11.0000i 0.457540i
\(579\) 0 0
\(580\) − 4.30463i − 0.178740i
\(581\) 0 0
\(582\) 0 0
\(583\) 61.4558 2.54524
\(584\) −5.49333 −0.227315
\(585\) 0 0
\(586\) 12.8418i 0.530490i
\(587\) −32.4377 −1.33885 −0.669424 0.742881i \(-0.733459\pi\)
−0.669424 + 0.742881i \(0.733459\pi\)
\(588\) 0 0
\(589\) 44.4853 1.83298
\(590\) 6.00000i 0.247016i
\(591\) 0 0
\(592\) 5.24264 0.215471
\(593\) 1.85514 0.0761816 0.0380908 0.999274i \(-0.487872\pi\)
0.0380908 + 0.999274i \(0.487872\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.2426i 0.665324i
\(597\) 0 0
\(598\) 4.30463i 0.176030i
\(599\) 3.51472i 0.143608i 0.997419 + 0.0718038i \(0.0228755\pi\)
−0.997419 + 0.0718038i \(0.977124\pi\)
\(600\) 0 0
\(601\) 41.5182i 1.69356i 0.531940 + 0.846782i \(0.321463\pi\)
−0.531940 + 0.846782i \(0.678537\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.75736 −0.356332
\(605\) −17.1464 −0.697101
\(606\) 0 0
\(607\) 36.0759i 1.46428i 0.681157 + 0.732138i \(0.261477\pi\)
−0.681157 + 0.732138i \(0.738523\pi\)
\(608\) −4.89898 −0.198680
\(609\) 0 0
\(610\) 10.2426 0.414712
\(611\) − 9.21320i − 0.372726i
\(612\) 0 0
\(613\) −28.2132 −1.13952 −0.569760 0.821811i \(-0.692964\pi\)
−0.569760 + 0.821811i \(0.692964\pi\)
\(614\) 26.8213 1.08242
\(615\) 0 0
\(616\) 0 0
\(617\) − 4.24264i − 0.170802i −0.996347 0.0854011i \(-0.972783\pi\)
0.996347 0.0854011i \(-0.0272172\pi\)
\(618\) 0 0
\(619\) 12.7187i 0.511208i 0.966782 + 0.255604i \(0.0822743\pi\)
−0.966782 + 0.255604i \(0.917726\pi\)
\(620\) − 22.2426i − 0.893286i
\(621\) 0 0
\(622\) − 17.1464i − 0.687509i
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −20.1903 −0.806965
\(627\) 0 0
\(628\) − 10.3923i − 0.414698i
\(629\) 12.8418 0.512036
\(630\) 0 0
\(631\) −14.7574 −0.587481 −0.293741 0.955885i \(-0.594900\pi\)
−0.293741 + 0.955885i \(0.594900\pi\)
\(632\) 0.757359i 0.0301261i
\(633\) 0 0
\(634\) 22.2426 0.883368
\(635\) −18.9295 −0.751193
\(636\) 0 0
\(637\) 0 0
\(638\) 7.45584i 0.295180i
\(639\) 0 0
\(640\) 2.44949i 0.0968246i
\(641\) 33.2132i 1.31184i 0.754829 + 0.655921i \(0.227720\pi\)
−0.754829 + 0.655921i \(0.772280\pi\)
\(642\) 0 0
\(643\) − 1.73205i − 0.0683054i −0.999417 0.0341527i \(-0.989127\pi\)
0.999417 0.0341527i \(-0.0108733\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) −20.7846 −0.817127 −0.408564 0.912730i \(-0.633970\pi\)
−0.408564 + 0.912730i \(0.633970\pi\)
\(648\) 0 0
\(649\) − 10.3923i − 0.407934i
\(650\) −0.717439 −0.0281403
\(651\) 0 0
\(652\) 9.48528 0.371472
\(653\) 2.48528i 0.0972566i 0.998817 + 0.0486283i \(0.0154850\pi\)
−0.998817 + 0.0486283i \(0.984515\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.44949 −0.0956365
\(657\) 0 0
\(658\) 0 0
\(659\) − 22.2426i − 0.866450i −0.901286 0.433225i \(-0.857376\pi\)
0.901286 0.433225i \(-0.142624\pi\)
\(660\) 0 0
\(661\) 4.89898i 0.190548i 0.995451 + 0.0952741i \(0.0303728\pi\)
−0.995451 + 0.0952741i \(0.969627\pi\)
\(662\) − 10.0000i − 0.388661i
\(663\) 0 0
\(664\) 15.2913i 0.593417i
\(665\) 0 0
\(666\) 0 0
\(667\) 10.5442 0.408271
\(668\) 0.594346 0.0229959
\(669\) 0 0
\(670\) − 33.0321i − 1.27614i
\(671\) −17.7408 −0.684875
\(672\) 0 0
\(673\) −45.4558 −1.75219 −0.876097 0.482135i \(-0.839862\pi\)
−0.876097 + 0.482135i \(0.839862\pi\)
\(674\) 21.4558i 0.826448i
\(675\) 0 0
\(676\) −12.4853 −0.480203
\(677\) 14.6969 0.564849 0.282425 0.959289i \(-0.408861\pi\)
0.282425 + 0.959289i \(0.408861\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.00000i 0.230089i
\(681\) 0 0
\(682\) 38.5254i 1.47521i
\(683\) 10.2426i 0.391924i 0.980612 + 0.195962i \(0.0627829\pi\)
−0.980612 + 0.195962i \(0.937217\pi\)
\(684\) 0 0
\(685\) − 14.6969i − 0.561541i
\(686\) 0 0
\(687\) 0 0
\(688\) 7.00000 0.266872
\(689\) −10.3923 −0.395915
\(690\) 0 0
\(691\) 2.57258i 0.0978657i 0.998802 + 0.0489328i \(0.0155820\pi\)
−0.998802 + 0.0489328i \(0.984418\pi\)
\(692\) −20.7846 −0.790112
\(693\) 0 0
\(694\) −4.97056 −0.188680
\(695\) 19.7574i 0.749439i
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) −0.123093 −0.00465914
\(699\) 0 0
\(700\) 0 0
\(701\) − 20.4853i − 0.773718i −0.922139 0.386859i \(-0.873560\pi\)
0.922139 0.386859i \(-0.126440\pi\)
\(702\) 0 0
\(703\) − 25.6836i − 0.968675i
\(704\) − 4.24264i − 0.159901i
\(705\) 0 0
\(706\) 10.9867i 0.413488i
\(707\) 0 0
\(708\) 0 0
\(709\) 16.2132 0.608900 0.304450 0.952528i \(-0.401527\pi\)
0.304450 + 0.952528i \(0.401527\pi\)
\(710\) 31.1769 1.17005
\(711\) 0 0
\(712\) 3.04384i 0.114073i
\(713\) 54.4831 2.04041
\(714\) 0 0
\(715\) 7.45584 0.278833
\(716\) − 6.72792i − 0.251434i
\(717\) 0 0
\(718\) 14.4853 0.540586
\(719\) −52.6280 −1.96269 −0.981346 0.192249i \(-0.938422\pi\)
−0.981346 + 0.192249i \(0.938422\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5.00000i 0.186081i
\(723\) 0 0
\(724\) 9.79796i 0.364138i
\(725\) 1.75736i 0.0652667i
\(726\) 0 0
\(727\) − 28.0821i − 1.04151i −0.853707 0.520754i \(-0.825651\pi\)
0.853707 0.520754i \(-0.174349\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 13.4558 0.498023
\(731\) 17.1464 0.634184
\(732\) 0 0
\(733\) 1.31178i 0.0484519i 0.999707 + 0.0242259i \(0.00771211\pi\)
−0.999707 + 0.0242259i \(0.992288\pi\)
\(734\) 29.9882 1.10689
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 57.2132i 2.10748i
\(738\) 0 0
\(739\) −8.45584 −0.311053 −0.155527 0.987832i \(-0.549707\pi\)
−0.155527 + 0.987832i \(0.549707\pi\)
\(740\) −12.8418 −0.472074
\(741\) 0 0
\(742\) 0 0
\(743\) − 18.7279i − 0.687061i −0.939142 0.343530i \(-0.888377\pi\)
0.939142 0.343530i \(-0.111623\pi\)
\(744\) 0 0
\(745\) − 39.7862i − 1.45765i
\(746\) 22.0000i 0.805477i
\(747\) 0 0
\(748\) − 10.3923i − 0.379980i
\(749\) 0 0
\(750\) 0 0
\(751\) 53.4558 1.95063 0.975316 0.220815i \(-0.0708717\pi\)
0.975316 + 0.220815i \(0.0708717\pi\)
\(752\) 12.8418 0.468292
\(753\) 0 0
\(754\) − 1.26080i − 0.0459156i
\(755\) 21.4511 0.780684
\(756\) 0 0
\(757\) 32.7574 1.19059 0.595293 0.803509i \(-0.297036\pi\)
0.595293 + 0.803509i \(0.297036\pi\)
\(758\) 7.48528i 0.271878i
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −24.4949 −0.887939 −0.443970 0.896042i \(-0.646430\pi\)
−0.443970 + 0.896042i \(0.646430\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 21.2132i 0.767467i
\(765\) 0 0
\(766\) 5.49333i 0.198482i
\(767\) 1.75736i 0.0634546i
\(768\) 0 0
\(769\) − 40.3805i − 1.45616i −0.685493 0.728080i \(-0.740413\pi\)
0.685493 0.728080i \(-0.259587\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.48528 0.0534564
\(773\) 9.79796 0.352408 0.176204 0.984354i \(-0.443618\pi\)
0.176204 + 0.984354i \(0.443618\pi\)
\(774\) 0 0
\(775\) 9.08052i 0.326182i
\(776\) −3.16693 −0.113686
\(777\) 0 0
\(778\) 15.5147 0.556230
\(779\) 12.0000i 0.429945i
\(780\) 0 0
\(781\) −54.0000 −1.93227
\(782\) −14.6969 −0.525561
\(783\) 0 0
\(784\) 0 0
\(785\) 25.4558i 0.908558i
\(786\) 0 0
\(787\) 28.2562i 1.00722i 0.863930 + 0.503612i \(0.167996\pi\)
−0.863930 + 0.503612i \(0.832004\pi\)
\(788\) 16.9706i 0.604551i
\(789\) 0 0
\(790\) − 1.85514i − 0.0660031i
\(791\) 0 0
\(792\) 0 0
\(793\) 3.00000 0.106533
\(794\) −15.1682 −0.538299
\(795\) 0 0
\(796\) − 20.9077i − 0.741054i
\(797\) 17.7408 0.628410 0.314205 0.949355i \(-0.398262\pi\)
0.314205 + 0.949355i \(0.398262\pi\)
\(798\) 0 0
\(799\) 31.4558 1.11283
\(800\) − 1.00000i − 0.0353553i
\(801\) 0 0
\(802\) 19.7574 0.697657
\(803\) −23.3062 −0.822458
\(804\) 0 0
\(805\) 0 0
\(806\) − 6.51472i − 0.229471i
\(807\) 0 0
\(808\) 7.34847i 0.258518i
\(809\) 14.4853i 0.509275i 0.967037 + 0.254638i \(0.0819562\pi\)
−0.967037 + 0.254638i \(0.918044\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 22.2426 0.779604
\(815\) −23.2341 −0.813855
\(816\) 0 0
\(817\) − 34.2929i − 1.19976i
\(818\) 3.76127 0.131510
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) − 22.9706i − 0.801678i −0.916148 0.400839i \(-0.868719\pi\)
0.916148 0.400839i \(-0.131281\pi\)
\(822\) 0 0
\(823\) −2.75736 −0.0961155 −0.0480578 0.998845i \(-0.515303\pi\)
−0.0480578 + 0.998845i \(0.515303\pi\)
\(824\) −11.1097 −0.387026
\(825\) 0 0
\(826\) 0 0
\(827\) 3.51472i 0.122219i 0.998131 + 0.0611094i \(0.0194638\pi\)
−0.998131 + 0.0611094i \(0.980536\pi\)
\(828\) 0 0
\(829\) − 9.79796i − 0.340297i −0.985418 0.170149i \(-0.945575\pi\)
0.985418 0.170149i \(-0.0544248\pi\)
\(830\) − 37.4558i − 1.30011i
\(831\) 0 0
\(832\) 0.717439i 0.0248727i
\(833\) 0 0
\(834\) 0 0
\(835\) −1.45584 −0.0503816
\(836\) −20.7846 −0.718851
\(837\) 0 0
\(838\) 7.94282i 0.274380i
\(839\) −15.2913 −0.527914 −0.263957 0.964534i \(-0.585028\pi\)
−0.263957 + 0.964534i \(0.585028\pi\)
\(840\) 0 0
\(841\) 25.9117 0.893506
\(842\) 23.4558i 0.808342i
\(843\) 0 0
\(844\) −3.48528 −0.119968
\(845\) 30.5826 1.05207
\(846\) 0 0
\(847\) 0 0
\(848\) − 14.4853i − 0.497427i
\(849\) 0 0
\(850\) − 2.44949i − 0.0840168i
\(851\) − 31.4558i − 1.07829i
\(852\) 0 0
\(853\) 31.7713i 1.08783i 0.839141 + 0.543914i \(0.183058\pi\)
−0.839141 + 0.543914i \(0.816942\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.48528 −0.0849452
\(857\) 27.5387 0.940705 0.470353 0.882479i \(-0.344127\pi\)
0.470353 + 0.882479i \(0.344127\pi\)
\(858\) 0 0
\(859\) 28.2562i 0.964088i 0.876147 + 0.482044i \(0.160105\pi\)
−0.876147 + 0.482044i \(0.839895\pi\)
\(860\) −17.1464 −0.584688
\(861\) 0 0
\(862\) 1.75736 0.0598559
\(863\) − 37.4558i − 1.27501i −0.770445 0.637506i \(-0.779966\pi\)
0.770445 0.637506i \(-0.220034\pi\)
\(864\) 0 0
\(865\) 50.9117 1.73105
\(866\) 2.57258 0.0874199
\(867\) 0 0
\(868\) 0 0
\(869\) 3.21320i 0.109000i
\(870\) 0 0
\(871\) − 9.67487i − 0.327820i
\(872\) − 17.7279i − 0.600343i
\(873\) 0 0
\(874\) 29.3939i 0.994263i
\(875\) 0 0
\(876\) 0 0
\(877\) −46.2132 −1.56051 −0.780254 0.625462i \(-0.784910\pi\)
−0.780254 + 0.625462i \(0.784910\pi\)
\(878\) 4.30463 0.145274
\(879\) 0 0
\(880\) 10.3923i 0.350325i
\(881\) 25.0892 0.845278 0.422639 0.906298i \(-0.361104\pi\)
0.422639 + 0.906298i \(0.361104\pi\)
\(882\) 0 0
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 1.75736i 0.0591064i
\(885\) 0 0
\(886\) 25.4558 0.855206
\(887\) −9.20361 −0.309027 −0.154514 0.987991i \(-0.549381\pi\)
−0.154514 + 0.987991i \(0.549381\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 7.45584i − 0.249920i
\(891\) 0 0
\(892\) 10.3923i 0.347960i
\(893\) − 62.9117i − 2.10526i
\(894\) 0 0
\(895\) 16.4800i 0.550865i
\(896\) 0 0
\(897\) 0 0
\(898\) −5.27208 −0.175932
\(899\) −15.9577 −0.532220
\(900\) 0 0
\(901\) − 35.4815i − 1.18206i
\(902\) −10.3923 −0.346026
\(903\) 0 0
\(904\) 10.2426 0.340665
\(905\) − 24.0000i − 0.797787i
\(906\) 0 0
\(907\) −13.4853 −0.447771 −0.223886 0.974615i \(-0.571874\pi\)
−0.223886 + 0.974615i \(0.571874\pi\)
\(908\) 25.0892 0.832616
\(909\) 0 0
\(910\) 0 0
\(911\) 10.2426i 0.339354i 0.985500 + 0.169677i \(0.0542724\pi\)
−0.985500 + 0.169677i \(0.945728\pi\)
\(912\) 0 0
\(913\) 64.8754i 2.14706i
\(914\) 23.0000i 0.760772i
\(915\) 0 0
\(916\) 5.61642i 0.185572i
\(917\) 0 0
\(918\) 0 0
\(919\) 25.7279 0.848686 0.424343 0.905502i \(-0.360505\pi\)
0.424343 + 0.905502i \(0.360505\pi\)
\(920\) 14.6969 0.484544
\(921\) 0 0
\(922\) 21.4511i 0.706453i
\(923\) 9.13151 0.300567
\(924\) 0 0
\(925\) 5.24264 0.172377
\(926\) 22.0000i 0.722965i
\(927\) 0 0
\(928\) 1.75736 0.0576881
\(929\) −28.7274 −0.942516 −0.471258 0.881995i \(-0.656200\pi\)
−0.471258 + 0.881995i \(0.656200\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 20.4853i 0.671018i
\(933\) 0 0
\(934\) 17.7408i 0.580496i
\(935\) 25.4558i 0.832495i
\(936\) 0 0
\(937\) − 33.5033i − 1.09451i −0.836967 0.547253i \(-0.815674\pi\)
0.836967 0.547253i \(-0.184326\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −31.4558 −1.02598
\(941\) −42.2357 −1.37684 −0.688422 0.725311i \(-0.741696\pi\)
−0.688422 + 0.725311i \(0.741696\pi\)
\(942\) 0 0
\(943\) 14.6969i 0.478598i
\(944\) −2.44949 −0.0797241
\(945\) 0 0
\(946\) 29.6985 0.965581
\(947\) 16.2426i 0.527815i 0.964548 + 0.263907i \(0.0850114\pi\)
−0.964548 + 0.263907i \(0.914989\pi\)
\(948\) 0 0
\(949\) 3.94113 0.127934
\(950\) −4.89898 −0.158944
\(951\) 0 0
\(952\) 0 0
\(953\) 1.02944i 0.0333467i 0.999861 + 0.0166734i \(0.00530755\pi\)
−0.999861 + 0.0166734i \(0.994692\pi\)
\(954\) 0 0
\(955\) − 51.9615i − 1.68144i
\(956\) 16.2426i 0.525325i
\(957\) 0 0
\(958\) 2.44949i 0.0791394i
\(959\) 0 0
\(960\) 0 0
\(961\) −51.4558 −1.65987
\(962\) −3.76127 −0.121268
\(963\) 0 0
\(964\) − 1.13770i − 0.0366430i
\(965\) −3.63818 −0.117117
\(966\) 0 0
\(967\) −42.6985 −1.37309 −0.686545 0.727087i \(-0.740874\pi\)
−0.686545 + 0.727087i \(0.740874\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) 0 0
\(970\) 7.75736 0.249074
\(971\) 40.3805 1.29587 0.647936 0.761694i \(-0.275632\pi\)
0.647936 + 0.761694i \(0.275632\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 22.0000i − 0.704925i
\(975\) 0 0
\(976\) 4.18154i 0.133848i
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) 0 0
\(979\) 12.9139i 0.412730i
\(980\) 0 0
\(981\) 0 0
\(982\) −34.9706 −1.11596
\(983\) −23.2341 −0.741053 −0.370526 0.928822i \(-0.620823\pi\)
−0.370526 + 0.928822i \(0.620823\pi\)
\(984\) 0 0
\(985\) − 41.5692i − 1.32451i
\(986\) 4.30463 0.137087
\(987\) 0 0
\(988\) 3.51472 0.111818
\(989\) − 42.0000i − 1.33552i
\(990\) 0 0
\(991\) −20.2132 −0.642094 −0.321047 0.947063i \(-0.604035\pi\)
−0.321047 + 0.947063i \(0.604035\pi\)
\(992\) 9.08052 0.288307
\(993\) 0 0
\(994\) 0 0
\(995\) 51.2132i 1.62357i
\(996\) 0 0
\(997\) − 19.4728i − 0.616711i −0.951271 0.308355i \(-0.900221\pi\)
0.951271 0.308355i \(-0.0997786\pi\)
\(998\) 24.4558i 0.774136i
\(999\) 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 2646.2.d.d.2645.4 8
3.2 odd 2 inner 2646.2.d.d.2645.5 8
7.2 even 3 378.2.k.d.269.1 yes 8
7.3 odd 6 378.2.k.d.215.4 yes 8
7.6 odd 2 inner 2646.2.d.d.2645.2 8
21.2 odd 6 378.2.k.d.269.4 yes 8
21.17 even 6 378.2.k.d.215.1 8
21.20 even 2 inner 2646.2.d.d.2645.7 8
63.2 odd 6 1134.2.t.f.1025.1 8
63.16 even 3 1134.2.t.f.1025.4 8
63.23 odd 6 1134.2.l.e.269.4 8
63.31 odd 6 1134.2.t.f.593.1 8
63.38 even 6 1134.2.l.e.215.3 8
63.52 odd 6 1134.2.l.e.215.2 8
63.58 even 3 1134.2.l.e.269.1 8
63.59 even 6 1134.2.t.f.593.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.d.215.1 8 21.17 even 6
378.2.k.d.215.4 yes 8 7.3 odd 6
378.2.k.d.269.1 yes 8 7.2 even 3
378.2.k.d.269.4 yes 8 21.2 odd 6
1134.2.l.e.215.2 8 63.52 odd 6
1134.2.l.e.215.3 8 63.38 even 6
1134.2.l.e.269.1 8 63.58 even 3
1134.2.l.e.269.4 8 63.23 odd 6
1134.2.t.f.593.1 8 63.31 odd 6
1134.2.t.f.593.4 8 63.59 even 6
1134.2.t.f.1025.1 8 63.2 odd 6
1134.2.t.f.1025.4 8 63.16 even 3
2646.2.d.d.2645.2 8 7.6 odd 2 inner
2646.2.d.d.2645.4 8 1.1 even 1 trivial
2646.2.d.d.2645.5 8 3.2 odd 2 inner
2646.2.d.d.2645.7 8 21.20 even 2 inner