Properties

Label 8550.2.a.cf.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.07838 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.07838 q^{7} -1.00000 q^{8} -3.41855 q^{11} +0.921622 q^{13} -1.07838 q^{14} +1.00000 q^{16} +0.340173 q^{17} -1.00000 q^{19} +3.41855 q^{22} -0.921622 q^{23} -0.921622 q^{26} +1.07838 q^{28} +1.41855 q^{29} +7.26180 q^{31} -1.00000 q^{32} -0.340173 q^{34} -5.60197 q^{37} +1.00000 q^{38} +1.07838 q^{41} -0.738205 q^{43} -3.41855 q^{44} +0.921622 q^{46} -7.75872 q^{47} -5.83710 q^{49} +0.921622 q^{52} +2.68035 q^{53} -1.07838 q^{56} -1.41855 q^{58} -8.34017 q^{59} +2.00000 q^{61} -7.26180 q^{62} +1.00000 q^{64} +4.68035 q^{67} +0.340173 q^{68} +10.8371 q^{71} +6.83710 q^{73} +5.60197 q^{74} -1.00000 q^{76} -3.68649 q^{77} +4.73820 q^{79} -1.07838 q^{82} +11.0205 q^{83} +0.738205 q^{86} +3.41855 q^{88} -9.75872 q^{89} +0.993857 q^{91} -0.921622 q^{92} +7.75872 q^{94} -16.2557 q^{97} +5.83710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 4 q^{11} + 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{19} - 4 q^{22} - 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} - 3 q^{32} + 10 q^{34} + 2 q^{37} + 3 q^{38} - 10 q^{43} + 4 q^{44} + 6 q^{46} + 2 q^{47} + 11 q^{49} + 6 q^{52} - 14 q^{53} + 10 q^{58} - 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 8 q^{67} - 10 q^{68} + 4 q^{71} - 8 q^{73} - 2 q^{74} - 3 q^{76} - 24 q^{77} + 22 q^{79} + 10 q^{86} - 4 q^{88} - 4 q^{89} - 32 q^{91} - 6 q^{92} - 2 q^{94} - 6 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.07838 0.407588 0.203794 0.979014i \(-0.434673\pi\)
0.203794 + 0.979014i \(0.434673\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −3.41855 −1.03073 −0.515366 0.856970i \(-0.672344\pi\)
−0.515366 + 0.856970i \(0.672344\pi\)
\(12\) 0 0
\(13\) 0.921622 0.255612 0.127806 0.991799i \(-0.459207\pi\)
0.127806 + 0.991799i \(0.459207\pi\)
\(14\) −1.07838 −0.288209
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.340173 0.0825041 0.0412520 0.999149i \(-0.486865\pi\)
0.0412520 + 0.999149i \(0.486865\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 3.41855 0.728837
\(23\) −0.921622 −0.192172 −0.0960858 0.995373i \(-0.530632\pi\)
−0.0960858 + 0.995373i \(0.530632\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.921622 −0.180745
\(27\) 0 0
\(28\) 1.07838 0.203794
\(29\) 1.41855 0.263418 0.131709 0.991288i \(-0.457954\pi\)
0.131709 + 0.991288i \(0.457954\pi\)
\(30\) 0 0
\(31\) 7.26180 1.30426 0.652128 0.758108i \(-0.273876\pi\)
0.652128 + 0.758108i \(0.273876\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.340173 −0.0583392
\(35\) 0 0
\(36\) 0 0
\(37\) −5.60197 −0.920958 −0.460479 0.887671i \(-0.652322\pi\)
−0.460479 + 0.887671i \(0.652322\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 1.07838 0.168414 0.0842072 0.996448i \(-0.473164\pi\)
0.0842072 + 0.996448i \(0.473164\pi\)
\(42\) 0 0
\(43\) −0.738205 −0.112575 −0.0562876 0.998415i \(-0.517926\pi\)
−0.0562876 + 0.998415i \(0.517926\pi\)
\(44\) −3.41855 −0.515366
\(45\) 0 0
\(46\) 0.921622 0.135886
\(47\) −7.75872 −1.13173 −0.565863 0.824499i \(-0.691457\pi\)
−0.565863 + 0.824499i \(0.691457\pi\)
\(48\) 0 0
\(49\) −5.83710 −0.833872
\(50\) 0 0
\(51\) 0 0
\(52\) 0.921622 0.127806
\(53\) 2.68035 0.368174 0.184087 0.982910i \(-0.441067\pi\)
0.184087 + 0.982910i \(0.441067\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.07838 −0.144104
\(57\) 0 0
\(58\) −1.41855 −0.186265
\(59\) −8.34017 −1.08580 −0.542899 0.839798i \(-0.682673\pi\)
−0.542899 + 0.839798i \(0.682673\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −7.26180 −0.922249
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.68035 0.571795 0.285898 0.958260i \(-0.407708\pi\)
0.285898 + 0.958260i \(0.407708\pi\)
\(68\) 0.340173 0.0412520
\(69\) 0 0
\(70\) 0 0
\(71\) 10.8371 1.28613 0.643064 0.765813i \(-0.277663\pi\)
0.643064 + 0.765813i \(0.277663\pi\)
\(72\) 0 0
\(73\) 6.83710 0.800222 0.400111 0.916467i \(-0.368972\pi\)
0.400111 + 0.916467i \(0.368972\pi\)
\(74\) 5.60197 0.651216
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −3.68649 −0.420114
\(78\) 0 0
\(79\) 4.73820 0.533090 0.266545 0.963823i \(-0.414118\pi\)
0.266545 + 0.963823i \(0.414118\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.07838 −0.119087
\(83\) 11.0205 1.20966 0.604830 0.796355i \(-0.293241\pi\)
0.604830 + 0.796355i \(0.293241\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.738205 0.0796027
\(87\) 0 0
\(88\) 3.41855 0.364419
\(89\) −9.75872 −1.03442 −0.517211 0.855858i \(-0.673030\pi\)
−0.517211 + 0.855858i \(0.673030\pi\)
\(90\) 0 0
\(91\) 0.993857 0.104185
\(92\) −0.921622 −0.0960858
\(93\) 0 0
\(94\) 7.75872 0.800251
\(95\) 0 0
\(96\) 0 0
\(97\) −16.2557 −1.65051 −0.825256 0.564759i \(-0.808969\pi\)
−0.825256 + 0.564759i \(0.808969\pi\)
\(98\) 5.83710 0.589636
\(99\) 0 0
\(100\) 0 0
\(101\) 13.0205 1.29559 0.647795 0.761815i \(-0.275691\pi\)
0.647795 + 0.761815i \(0.275691\pi\)
\(102\) 0 0
\(103\) −14.3402 −1.41298 −0.706490 0.707723i \(-0.749722\pi\)
−0.706490 + 0.707723i \(0.749722\pi\)
\(104\) −0.921622 −0.0903725
\(105\) 0 0
\(106\) −2.68035 −0.260338
\(107\) −10.8371 −1.04766 −0.523831 0.851822i \(-0.675498\pi\)
−0.523831 + 0.851822i \(0.675498\pi\)
\(108\) 0 0
\(109\) 18.6225 1.78371 0.891855 0.452321i \(-0.149404\pi\)
0.891855 + 0.452321i \(0.149404\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.07838 0.101897
\(113\) −13.5174 −1.27161 −0.635807 0.771848i \(-0.719333\pi\)
−0.635807 + 0.771848i \(0.719333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.41855 0.131709
\(117\) 0 0
\(118\) 8.34017 0.767775
\(119\) 0.366835 0.0336277
\(120\) 0 0
\(121\) 0.686489 0.0624081
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 7.26180 0.652128
\(125\) 0 0
\(126\) 0 0
\(127\) −16.4969 −1.46387 −0.731933 0.681377i \(-0.761382\pi\)
−0.731933 + 0.681377i \(0.761382\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −0.581449 −0.0508015 −0.0254007 0.999677i \(-0.508086\pi\)
−0.0254007 + 0.999677i \(0.508086\pi\)
\(132\) 0 0
\(133\) −1.07838 −0.0935072
\(134\) −4.68035 −0.404320
\(135\) 0 0
\(136\) −0.340173 −0.0291696
\(137\) −5.81658 −0.496944 −0.248472 0.968639i \(-0.579928\pi\)
−0.248472 + 0.968639i \(0.579928\pi\)
\(138\) 0 0
\(139\) 6.15676 0.522209 0.261105 0.965311i \(-0.415913\pi\)
0.261105 + 0.965311i \(0.415913\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.8371 −0.909429
\(143\) −3.15061 −0.263467
\(144\) 0 0
\(145\) 0 0
\(146\) −6.83710 −0.565843
\(147\) 0 0
\(148\) −5.60197 −0.460479
\(149\) −8.34017 −0.683254 −0.341627 0.939836i \(-0.610978\pi\)
−0.341627 + 0.939836i \(0.610978\pi\)
\(150\) 0 0
\(151\) 16.2557 1.32287 0.661433 0.750004i \(-0.269949\pi\)
0.661433 + 0.750004i \(0.269949\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 3.68649 0.297066
\(155\) 0 0
\(156\) 0 0
\(157\) −3.65983 −0.292086 −0.146043 0.989278i \(-0.546654\pi\)
−0.146043 + 0.989278i \(0.546654\pi\)
\(158\) −4.73820 −0.376951
\(159\) 0 0
\(160\) 0 0
\(161\) −0.993857 −0.0783269
\(162\) 0 0
\(163\) −6.89496 −0.540055 −0.270027 0.962853i \(-0.587033\pi\)
−0.270027 + 0.962853i \(0.587033\pi\)
\(164\) 1.07838 0.0842072
\(165\) 0 0
\(166\) −11.0205 −0.855358
\(167\) −4.68035 −0.362176 −0.181088 0.983467i \(-0.557962\pi\)
−0.181088 + 0.983467i \(0.557962\pi\)
\(168\) 0 0
\(169\) −12.1506 −0.934662
\(170\) 0 0
\(171\) 0 0
\(172\) −0.738205 −0.0562876
\(173\) −4.52359 −0.343922 −0.171961 0.985104i \(-0.555010\pi\)
−0.171961 + 0.985104i \(0.555010\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.41855 −0.257683
\(177\) 0 0
\(178\) 9.75872 0.731447
\(179\) −14.4969 −1.08355 −0.541776 0.840523i \(-0.682248\pi\)
−0.541776 + 0.840523i \(0.682248\pi\)
\(180\) 0 0
\(181\) −12.7792 −0.949874 −0.474937 0.880020i \(-0.657529\pi\)
−0.474937 + 0.880020i \(0.657529\pi\)
\(182\) −0.993857 −0.0736696
\(183\) 0 0
\(184\) 0.921622 0.0679429
\(185\) 0 0
\(186\) 0 0
\(187\) −1.16290 −0.0850396
\(188\) −7.75872 −0.565863
\(189\) 0 0
\(190\) 0 0
\(191\) 4.39803 0.318230 0.159115 0.987260i \(-0.449136\pi\)
0.159115 + 0.987260i \(0.449136\pi\)
\(192\) 0 0
\(193\) −24.6225 −1.77237 −0.886183 0.463336i \(-0.846652\pi\)
−0.886183 + 0.463336i \(0.846652\pi\)
\(194\) 16.2557 1.16709
\(195\) 0 0
\(196\) −5.83710 −0.416936
\(197\) −3.91548 −0.278966 −0.139483 0.990224i \(-0.544544\pi\)
−0.139483 + 0.990224i \(0.544544\pi\)
\(198\) 0 0
\(199\) 8.99386 0.637558 0.318779 0.947829i \(-0.396727\pi\)
0.318779 + 0.947829i \(0.396727\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −13.0205 −0.916121
\(203\) 1.52973 0.107366
\(204\) 0 0
\(205\) 0 0
\(206\) 14.3402 0.999127
\(207\) 0 0
\(208\) 0.921622 0.0639030
\(209\) 3.41855 0.236466
\(210\) 0 0
\(211\) −4.68035 −0.322208 −0.161104 0.986937i \(-0.551506\pi\)
−0.161104 + 0.986937i \(0.551506\pi\)
\(212\) 2.68035 0.184087
\(213\) 0 0
\(214\) 10.8371 0.740809
\(215\) 0 0
\(216\) 0 0
\(217\) 7.83096 0.531600
\(218\) −18.6225 −1.26127
\(219\) 0 0
\(220\) 0 0
\(221\) 0.313511 0.0210890
\(222\) 0 0
\(223\) −14.3402 −0.960289 −0.480145 0.877189i \(-0.659416\pi\)
−0.480145 + 0.877189i \(0.659416\pi\)
\(224\) −1.07838 −0.0720521
\(225\) 0 0
\(226\) 13.5174 0.899167
\(227\) 19.2039 1.27461 0.637305 0.770612i \(-0.280049\pi\)
0.637305 + 0.770612i \(0.280049\pi\)
\(228\) 0 0
\(229\) 2.31351 0.152881 0.0764406 0.997074i \(-0.475644\pi\)
0.0764406 + 0.997074i \(0.475644\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.41855 −0.0931324
\(233\) 25.6475 1.68023 0.840113 0.542411i \(-0.182489\pi\)
0.840113 + 0.542411i \(0.182489\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.34017 −0.542899
\(237\) 0 0
\(238\) −0.366835 −0.0237784
\(239\) −3.60197 −0.232992 −0.116496 0.993191i \(-0.537166\pi\)
−0.116496 + 0.993191i \(0.537166\pi\)
\(240\) 0 0
\(241\) −2.36683 −0.152461 −0.0762306 0.997090i \(-0.524289\pi\)
−0.0762306 + 0.997090i \(0.524289\pi\)
\(242\) −0.686489 −0.0441292
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −0.921622 −0.0586414
\(248\) −7.26180 −0.461124
\(249\) 0 0
\(250\) 0 0
\(251\) −6.73820 −0.425312 −0.212656 0.977127i \(-0.568211\pi\)
−0.212656 + 0.977127i \(0.568211\pi\)
\(252\) 0 0
\(253\) 3.15061 0.198077
\(254\) 16.4969 1.03511
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.20394 0.574126 0.287063 0.957912i \(-0.407321\pi\)
0.287063 + 0.957912i \(0.407321\pi\)
\(258\) 0 0
\(259\) −6.04104 −0.375372
\(260\) 0 0
\(261\) 0 0
\(262\) 0.581449 0.0359221
\(263\) 25.7998 1.59088 0.795441 0.606031i \(-0.207239\pi\)
0.795441 + 0.606031i \(0.207239\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.07838 0.0661196
\(267\) 0 0
\(268\) 4.68035 0.285898
\(269\) −4.73820 −0.288893 −0.144447 0.989513i \(-0.546140\pi\)
−0.144447 + 0.989513i \(0.546140\pi\)
\(270\) 0 0
\(271\) 16.3668 0.994214 0.497107 0.867689i \(-0.334396\pi\)
0.497107 + 0.867689i \(0.334396\pi\)
\(272\) 0.340173 0.0206260
\(273\) 0 0
\(274\) 5.81658 0.351393
\(275\) 0 0
\(276\) 0 0
\(277\) −12.6537 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(278\) −6.15676 −0.369258
\(279\) 0 0
\(280\) 0 0
\(281\) −9.44521 −0.563454 −0.281727 0.959495i \(-0.590907\pi\)
−0.281727 + 0.959495i \(0.590907\pi\)
\(282\) 0 0
\(283\) −15.8888 −0.944492 −0.472246 0.881467i \(-0.656557\pi\)
−0.472246 + 0.881467i \(0.656557\pi\)
\(284\) 10.8371 0.643064
\(285\) 0 0
\(286\) 3.15061 0.186300
\(287\) 1.16290 0.0686437
\(288\) 0 0
\(289\) −16.8843 −0.993193
\(290\) 0 0
\(291\) 0 0
\(292\) 6.83710 0.400111
\(293\) 4.47027 0.261156 0.130578 0.991438i \(-0.458317\pi\)
0.130578 + 0.991438i \(0.458317\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.60197 0.325608
\(297\) 0 0
\(298\) 8.34017 0.483133
\(299\) −0.849388 −0.0491214
\(300\) 0 0
\(301\) −0.796064 −0.0458843
\(302\) −16.2557 −0.935408
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −15.3197 −0.874339 −0.437169 0.899379i \(-0.644019\pi\)
−0.437169 + 0.899379i \(0.644019\pi\)
\(308\) −3.68649 −0.210057
\(309\) 0 0
\(310\) 0 0
\(311\) 4.39803 0.249390 0.124695 0.992195i \(-0.460205\pi\)
0.124695 + 0.992195i \(0.460205\pi\)
\(312\) 0 0
\(313\) −8.31351 −0.469907 −0.234954 0.972007i \(-0.575494\pi\)
−0.234954 + 0.972007i \(0.575494\pi\)
\(314\) 3.65983 0.206536
\(315\) 0 0
\(316\) 4.73820 0.266545
\(317\) 0.470266 0.0264128 0.0132064 0.999913i \(-0.495796\pi\)
0.0132064 + 0.999913i \(0.495796\pi\)
\(318\) 0 0
\(319\) −4.84939 −0.271514
\(320\) 0 0
\(321\) 0 0
\(322\) 0.993857 0.0553855
\(323\) −0.340173 −0.0189277
\(324\) 0 0
\(325\) 0 0
\(326\) 6.89496 0.381877
\(327\) 0 0
\(328\) −1.07838 −0.0595435
\(329\) −8.36683 −0.461279
\(330\) 0 0
\(331\) 31.5174 1.73236 0.866178 0.499736i \(-0.166570\pi\)
0.866178 + 0.499736i \(0.166570\pi\)
\(332\) 11.0205 0.604830
\(333\) 0 0
\(334\) 4.68035 0.256097
\(335\) 0 0
\(336\) 0 0
\(337\) −0.0578588 −0.00315177 −0.00157589 0.999999i \(-0.500502\pi\)
−0.00157589 + 0.999999i \(0.500502\pi\)
\(338\) 12.1506 0.660906
\(339\) 0 0
\(340\) 0 0
\(341\) −24.8248 −1.34434
\(342\) 0 0
\(343\) −13.8432 −0.747465
\(344\) 0.738205 0.0398013
\(345\) 0 0
\(346\) 4.52359 0.243190
\(347\) −6.34017 −0.340358 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(348\) 0 0
\(349\) −20.5236 −1.09860 −0.549301 0.835624i \(-0.685106\pi\)
−0.549301 + 0.835624i \(0.685106\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.41855 0.182209
\(353\) −22.3812 −1.19123 −0.595616 0.803269i \(-0.703092\pi\)
−0.595616 + 0.803269i \(0.703092\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.75872 −0.517211
\(357\) 0 0
\(358\) 14.4969 0.766186
\(359\) 0.282314 0.0149000 0.00744999 0.999972i \(-0.497629\pi\)
0.00744999 + 0.999972i \(0.497629\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 12.7792 0.671662
\(363\) 0 0
\(364\) 0.993857 0.0520923
\(365\) 0 0
\(366\) 0 0
\(367\) −27.9155 −1.45718 −0.728588 0.684952i \(-0.759823\pi\)
−0.728588 + 0.684952i \(0.759823\pi\)
\(368\) −0.921622 −0.0480429
\(369\) 0 0
\(370\) 0 0
\(371\) 2.89043 0.150063
\(372\) 0 0
\(373\) −3.44521 −0.178386 −0.0891932 0.996014i \(-0.528429\pi\)
−0.0891932 + 0.996014i \(0.528429\pi\)
\(374\) 1.16290 0.0601321
\(375\) 0 0
\(376\) 7.75872 0.400126
\(377\) 1.30737 0.0673329
\(378\) 0 0
\(379\) −28.5113 −1.46453 −0.732264 0.681021i \(-0.761536\pi\)
−0.732264 + 0.681021i \(0.761536\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.39803 −0.225023
\(383\) 2.63931 0.134862 0.0674312 0.997724i \(-0.478520\pi\)
0.0674312 + 0.997724i \(0.478520\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.6225 1.25325
\(387\) 0 0
\(388\) −16.2557 −0.825256
\(389\) 22.1834 1.12474 0.562372 0.826884i \(-0.309889\pi\)
0.562372 + 0.826884i \(0.309889\pi\)
\(390\) 0 0
\(391\) −0.313511 −0.0158549
\(392\) 5.83710 0.294818
\(393\) 0 0
\(394\) 3.91548 0.197259
\(395\) 0 0
\(396\) 0 0
\(397\) −24.2245 −1.21579 −0.607895 0.794017i \(-0.707986\pi\)
−0.607895 + 0.794017i \(0.707986\pi\)
\(398\) −8.99386 −0.450821
\(399\) 0 0
\(400\) 0 0
\(401\) −12.7649 −0.637447 −0.318724 0.947848i \(-0.603254\pi\)
−0.318724 + 0.947848i \(0.603254\pi\)
\(402\) 0 0
\(403\) 6.69263 0.333384
\(404\) 13.0205 0.647795
\(405\) 0 0
\(406\) −1.52973 −0.0759194
\(407\) 19.1506 0.949261
\(408\) 0 0
\(409\) 22.8781 1.13125 0.565626 0.824662i \(-0.308635\pi\)
0.565626 + 0.824662i \(0.308635\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −14.3402 −0.706490
\(413\) −8.99386 −0.442559
\(414\) 0 0
\(415\) 0 0
\(416\) −0.921622 −0.0451862
\(417\) 0 0
\(418\) −3.41855 −0.167207
\(419\) −3.47187 −0.169612 −0.0848061 0.996397i \(-0.527027\pi\)
−0.0848061 + 0.996397i \(0.527027\pi\)
\(420\) 0 0
\(421\) −20.4657 −0.997439 −0.498719 0.866764i \(-0.666196\pi\)
−0.498719 + 0.866764i \(0.666196\pi\)
\(422\) 4.68035 0.227836
\(423\) 0 0
\(424\) −2.68035 −0.130169
\(425\) 0 0
\(426\) 0 0
\(427\) 2.15676 0.104373
\(428\) −10.8371 −0.523831
\(429\) 0 0
\(430\) 0 0
\(431\) 33.1917 1.59879 0.799393 0.600809i \(-0.205155\pi\)
0.799393 + 0.600809i \(0.205155\pi\)
\(432\) 0 0
\(433\) 41.1338 1.97676 0.988382 0.151991i \(-0.0485684\pi\)
0.988382 + 0.151991i \(0.0485684\pi\)
\(434\) −7.83096 −0.375898
\(435\) 0 0
\(436\) 18.6225 0.891855
\(437\) 0.921622 0.0440872
\(438\) 0 0
\(439\) 19.1461 0.913792 0.456896 0.889520i \(-0.348961\pi\)
0.456896 + 0.889520i \(0.348961\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.313511 −0.0149122
\(443\) −37.4908 −1.78124 −0.890620 0.454747i \(-0.849730\pi\)
−0.890620 + 0.454747i \(0.849730\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.3402 0.679027
\(447\) 0 0
\(448\) 1.07838 0.0509486
\(449\) −6.43907 −0.303878 −0.151939 0.988390i \(-0.548552\pi\)
−0.151939 + 0.988390i \(0.548552\pi\)
\(450\) 0 0
\(451\) −3.68649 −0.173590
\(452\) −13.5174 −0.635807
\(453\) 0 0
\(454\) −19.2039 −0.901285
\(455\) 0 0
\(456\) 0 0
\(457\) 13.6742 0.639652 0.319826 0.947476i \(-0.396376\pi\)
0.319826 + 0.947476i \(0.396376\pi\)
\(458\) −2.31351 −0.108103
\(459\) 0 0
\(460\) 0 0
\(461\) −27.5441 −1.28286 −0.641429 0.767183i \(-0.721658\pi\)
−0.641429 + 0.767183i \(0.721658\pi\)
\(462\) 0 0
\(463\) −28.5958 −1.32896 −0.664480 0.747306i \(-0.731347\pi\)
−0.664480 + 0.747306i \(0.731347\pi\)
\(464\) 1.41855 0.0658546
\(465\) 0 0
\(466\) −25.6475 −1.18810
\(467\) 35.5318 1.64422 0.822108 0.569331i \(-0.192798\pi\)
0.822108 + 0.569331i \(0.192798\pi\)
\(468\) 0 0
\(469\) 5.04718 0.233057
\(470\) 0 0
\(471\) 0 0
\(472\) 8.34017 0.383888
\(473\) 2.52359 0.116035
\(474\) 0 0
\(475\) 0 0
\(476\) 0.366835 0.0168139
\(477\) 0 0
\(478\) 3.60197 0.164750
\(479\) −24.6491 −1.12625 −0.563124 0.826372i \(-0.690401\pi\)
−0.563124 + 0.826372i \(0.690401\pi\)
\(480\) 0 0
\(481\) −5.16290 −0.235408
\(482\) 2.36683 0.107806
\(483\) 0 0
\(484\) 0.686489 0.0312040
\(485\) 0 0
\(486\) 0 0
\(487\) 15.3340 0.694851 0.347426 0.937708i \(-0.387056\pi\)
0.347426 + 0.937708i \(0.387056\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) 32.7792 1.47931 0.739653 0.672988i \(-0.234990\pi\)
0.739653 + 0.672988i \(0.234990\pi\)
\(492\) 0 0
\(493\) 0.482553 0.0217331
\(494\) 0.921622 0.0414657
\(495\) 0 0
\(496\) 7.26180 0.326064
\(497\) 11.6865 0.524211
\(498\) 0 0
\(499\) −25.6742 −1.14934 −0.574668 0.818387i \(-0.694869\pi\)
−0.574668 + 0.818387i \(0.694869\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 6.73820 0.300741
\(503\) 24.7526 1.10366 0.551832 0.833956i \(-0.313929\pi\)
0.551832 + 0.833956i \(0.313929\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.15061 −0.140062
\(507\) 0 0
\(508\) −16.4969 −0.731933
\(509\) 20.9360 0.927972 0.463986 0.885843i \(-0.346419\pi\)
0.463986 + 0.885843i \(0.346419\pi\)
\(510\) 0 0
\(511\) 7.37298 0.326161
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −9.20394 −0.405968
\(515\) 0 0
\(516\) 0 0
\(517\) 26.5236 1.16651
\(518\) 6.04104 0.265428
\(519\) 0 0
\(520\) 0 0
\(521\) −23.5486 −1.03168 −0.515842 0.856683i \(-0.672521\pi\)
−0.515842 + 0.856683i \(0.672521\pi\)
\(522\) 0 0
\(523\) 14.7838 0.646449 0.323225 0.946322i \(-0.395233\pi\)
0.323225 + 0.946322i \(0.395233\pi\)
\(524\) −0.581449 −0.0254007
\(525\) 0 0
\(526\) −25.7998 −1.12492
\(527\) 2.47027 0.107606
\(528\) 0 0
\(529\) −22.1506 −0.963070
\(530\) 0 0
\(531\) 0 0
\(532\) −1.07838 −0.0467536
\(533\) 0.993857 0.0430487
\(534\) 0 0
\(535\) 0 0
\(536\) −4.68035 −0.202160
\(537\) 0 0
\(538\) 4.73820 0.204279
\(539\) 19.9544 0.859498
\(540\) 0 0
\(541\) −40.8371 −1.75572 −0.877862 0.478914i \(-0.841031\pi\)
−0.877862 + 0.478914i \(0.841031\pi\)
\(542\) −16.3668 −0.703016
\(543\) 0 0
\(544\) −0.340173 −0.0145848
\(545\) 0 0
\(546\) 0 0
\(547\) −42.0410 −1.79754 −0.898772 0.438415i \(-0.855540\pi\)
−0.898772 + 0.438415i \(0.855540\pi\)
\(548\) −5.81658 −0.248472
\(549\) 0 0
\(550\) 0 0
\(551\) −1.41855 −0.0604323
\(552\) 0 0
\(553\) 5.10957 0.217281
\(554\) 12.6537 0.537604
\(555\) 0 0
\(556\) 6.15676 0.261105
\(557\) −26.7526 −1.13354 −0.566772 0.823875i \(-0.691808\pi\)
−0.566772 + 0.823875i \(0.691808\pi\)
\(558\) 0 0
\(559\) −0.680346 −0.0287756
\(560\) 0 0
\(561\) 0 0
\(562\) 9.44521 0.398422
\(563\) −21.9877 −0.926672 −0.463336 0.886183i \(-0.653348\pi\)
−0.463336 + 0.886183i \(0.653348\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 15.8888 0.667857
\(567\) 0 0
\(568\) −10.8371 −0.454715
\(569\) 32.5958 1.36649 0.683244 0.730190i \(-0.260569\pi\)
0.683244 + 0.730190i \(0.260569\pi\)
\(570\) 0 0
\(571\) −32.1445 −1.34520 −0.672602 0.740004i \(-0.734823\pi\)
−0.672602 + 0.740004i \(0.734823\pi\)
\(572\) −3.15061 −0.131734
\(573\) 0 0
\(574\) −1.16290 −0.0485384
\(575\) 0 0
\(576\) 0 0
\(577\) 13.6742 0.569265 0.284632 0.958637i \(-0.408129\pi\)
0.284632 + 0.958637i \(0.408129\pi\)
\(578\) 16.8843 0.702294
\(579\) 0 0
\(580\) 0 0
\(581\) 11.8843 0.493043
\(582\) 0 0
\(583\) −9.16290 −0.379488
\(584\) −6.83710 −0.282921
\(585\) 0 0
\(586\) −4.47027 −0.184665
\(587\) −30.9672 −1.27815 −0.639076 0.769143i \(-0.720683\pi\)
−0.639076 + 0.769143i \(0.720683\pi\)
\(588\) 0 0
\(589\) −7.26180 −0.299217
\(590\) 0 0
\(591\) 0 0
\(592\) −5.60197 −0.230239
\(593\) −42.3812 −1.74039 −0.870194 0.492709i \(-0.836007\pi\)
−0.870194 + 0.492709i \(0.836007\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −8.34017 −0.341627
\(597\) 0 0
\(598\) 0.849388 0.0347340
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 11.4764 0.468133 0.234066 0.972221i \(-0.424797\pi\)
0.234066 + 0.972221i \(0.424797\pi\)
\(602\) 0.796064 0.0324451
\(603\) 0 0
\(604\) 16.2557 0.661433
\(605\) 0 0
\(606\) 0 0
\(607\) 15.3340 0.622389 0.311195 0.950346i \(-0.399271\pi\)
0.311195 + 0.950346i \(0.399271\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −7.15061 −0.289283
\(612\) 0 0
\(613\) −4.34017 −0.175298 −0.0876490 0.996151i \(-0.527935\pi\)
−0.0876490 + 0.996151i \(0.527935\pi\)
\(614\) 15.3197 0.618251
\(615\) 0 0
\(616\) 3.68649 0.148533
\(617\) −30.1834 −1.21514 −0.607569 0.794267i \(-0.707855\pi\)
−0.607569 + 0.794267i \(0.707855\pi\)
\(618\) 0 0
\(619\) −7.94668 −0.319404 −0.159702 0.987165i \(-0.551053\pi\)
−0.159702 + 0.987165i \(0.551053\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.39803 −0.176345
\(623\) −10.5236 −0.421619
\(624\) 0 0
\(625\) 0 0
\(626\) 8.31351 0.332275
\(627\) 0 0
\(628\) −3.65983 −0.146043
\(629\) −1.90564 −0.0759828
\(630\) 0 0
\(631\) 8.68035 0.345559 0.172780 0.984961i \(-0.444725\pi\)
0.172780 + 0.984961i \(0.444725\pi\)
\(632\) −4.73820 −0.188476
\(633\) 0 0
\(634\) −0.470266 −0.0186767
\(635\) 0 0
\(636\) 0 0
\(637\) −5.37960 −0.213148
\(638\) 4.84939 0.191989
\(639\) 0 0
\(640\) 0 0
\(641\) 41.6430 1.64480 0.822400 0.568910i \(-0.192635\pi\)
0.822400 + 0.568910i \(0.192635\pi\)
\(642\) 0 0
\(643\) 29.7854 1.17462 0.587310 0.809362i \(-0.300187\pi\)
0.587310 + 0.809362i \(0.300187\pi\)
\(644\) −0.993857 −0.0391634
\(645\) 0 0
\(646\) 0.340173 0.0133839
\(647\) 28.9216 1.13703 0.568513 0.822674i \(-0.307519\pi\)
0.568513 + 0.822674i \(0.307519\pi\)
\(648\) 0 0
\(649\) 28.5113 1.11917
\(650\) 0 0
\(651\) 0 0
\(652\) −6.89496 −0.270027
\(653\) 41.9565 1.64189 0.820943 0.571011i \(-0.193448\pi\)
0.820943 + 0.571011i \(0.193448\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.07838 0.0421036
\(657\) 0 0
\(658\) 8.36683 0.326173
\(659\) −43.0082 −1.67536 −0.837681 0.546159i \(-0.816089\pi\)
−0.837681 + 0.546159i \(0.816089\pi\)
\(660\) 0 0
\(661\) −34.3090 −1.33446 −0.667232 0.744850i \(-0.732521\pi\)
−0.667232 + 0.744850i \(0.732521\pi\)
\(662\) −31.5174 −1.22496
\(663\) 0 0
\(664\) −11.0205 −0.427679
\(665\) 0 0
\(666\) 0 0
\(667\) −1.30737 −0.0506215
\(668\) −4.68035 −0.181088
\(669\) 0 0
\(670\) 0 0
\(671\) −6.83710 −0.263943
\(672\) 0 0
\(673\) 31.0928 1.19854 0.599269 0.800548i \(-0.295458\pi\)
0.599269 + 0.800548i \(0.295458\pi\)
\(674\) 0.0578588 0.00222864
\(675\) 0 0
\(676\) −12.1506 −0.467331
\(677\) −35.9877 −1.38312 −0.691560 0.722319i \(-0.743076\pi\)
−0.691560 + 0.722319i \(0.743076\pi\)
\(678\) 0 0
\(679\) −17.5297 −0.672729
\(680\) 0 0
\(681\) 0 0
\(682\) 24.8248 0.950591
\(683\) −27.4017 −1.04850 −0.524249 0.851565i \(-0.675654\pi\)
−0.524249 + 0.851565i \(0.675654\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.8432 0.528538
\(687\) 0 0
\(688\) −0.738205 −0.0281438
\(689\) 2.47027 0.0941097
\(690\) 0 0
\(691\) 3.83096 0.145737 0.0728683 0.997342i \(-0.476785\pi\)
0.0728683 + 0.997342i \(0.476785\pi\)
\(692\) −4.52359 −0.171961
\(693\) 0 0
\(694\) 6.34017 0.240670
\(695\) 0 0
\(696\) 0 0
\(697\) 0.366835 0.0138949
\(698\) 20.5236 0.776829
\(699\) 0 0
\(700\) 0 0
\(701\) −46.5790 −1.75926 −0.879632 0.475654i \(-0.842211\pi\)
−0.879632 + 0.475654i \(0.842211\pi\)
\(702\) 0 0
\(703\) 5.60197 0.211282
\(704\) −3.41855 −0.128841
\(705\) 0 0
\(706\) 22.3812 0.842328
\(707\) 14.0410 0.528068
\(708\) 0 0
\(709\) 15.1917 0.570534 0.285267 0.958448i \(-0.407918\pi\)
0.285267 + 0.958448i \(0.407918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.75872 0.365724
\(713\) −6.69263 −0.250641
\(714\) 0 0
\(715\) 0 0
\(716\) −14.4969 −0.541776
\(717\) 0 0
\(718\) −0.282314 −0.0105359
\(719\) 8.45136 0.315182 0.157591 0.987504i \(-0.449627\pi\)
0.157591 + 0.987504i \(0.449627\pi\)
\(720\) 0 0
\(721\) −15.4641 −0.575914
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −12.7792 −0.474937
\(725\) 0 0
\(726\) 0 0
\(727\) −39.9688 −1.48236 −0.741180 0.671306i \(-0.765734\pi\)
−0.741180 + 0.671306i \(0.765734\pi\)
\(728\) −0.993857 −0.0368348
\(729\) 0 0
\(730\) 0 0
\(731\) −0.251117 −0.00928791
\(732\) 0 0
\(733\) −38.7480 −1.43119 −0.715596 0.698515i \(-0.753845\pi\)
−0.715596 + 0.698515i \(0.753845\pi\)
\(734\) 27.9155 1.03038
\(735\) 0 0
\(736\) 0.921622 0.0339714
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −15.1506 −0.557324 −0.278662 0.960389i \(-0.589891\pi\)
−0.278662 + 0.960389i \(0.589891\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.89043 −0.106111
\(743\) 28.8781 1.05944 0.529718 0.848174i \(-0.322298\pi\)
0.529718 + 0.848174i \(0.322298\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3.44521 0.126138
\(747\) 0 0
\(748\) −1.16290 −0.0425198
\(749\) −11.6865 −0.427015
\(750\) 0 0
\(751\) −14.2679 −0.520644 −0.260322 0.965522i \(-0.583829\pi\)
−0.260322 + 0.965522i \(0.583829\pi\)
\(752\) −7.75872 −0.282932
\(753\) 0 0
\(754\) −1.30737 −0.0476115
\(755\) 0 0
\(756\) 0 0
\(757\) −22.2122 −0.807315 −0.403658 0.914910i \(-0.632261\pi\)
−0.403658 + 0.914910i \(0.632261\pi\)
\(758\) 28.5113 1.03558
\(759\) 0 0
\(760\) 0 0
\(761\) −2.48255 −0.0899925 −0.0449962 0.998987i \(-0.514328\pi\)
−0.0449962 + 0.998987i \(0.514328\pi\)
\(762\) 0 0
\(763\) 20.0821 0.727020
\(764\) 4.39803 0.159115
\(765\) 0 0
\(766\) −2.63931 −0.0953621
\(767\) −7.68649 −0.277543
\(768\) 0 0
\(769\) −22.3135 −0.804646 −0.402323 0.915498i \(-0.631797\pi\)
−0.402323 + 0.915498i \(0.631797\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.6225 −0.886183
\(773\) 24.2101 0.870776 0.435388 0.900243i \(-0.356611\pi\)
0.435388 + 0.900243i \(0.356611\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.2557 0.583544
\(777\) 0 0
\(778\) −22.1834 −0.795314
\(779\) −1.07838 −0.0386369
\(780\) 0 0
\(781\) −37.0472 −1.32565
\(782\) 0.313511 0.0112111
\(783\) 0 0
\(784\) −5.83710 −0.208468
\(785\) 0 0
\(786\) 0 0
\(787\) −31.0349 −1.10627 −0.553137 0.833090i \(-0.686569\pi\)
−0.553137 + 0.833090i \(0.686569\pi\)
\(788\) −3.91548 −0.139483
\(789\) 0 0
\(790\) 0 0
\(791\) −14.5769 −0.518295
\(792\) 0 0
\(793\) 1.84324 0.0654555
\(794\) 24.2245 0.859694
\(795\) 0 0
\(796\) 8.99386 0.318779
\(797\) −25.2039 −0.892769 −0.446385 0.894841i \(-0.647289\pi\)
−0.446385 + 0.894841i \(0.647289\pi\)
\(798\) 0 0
\(799\) −2.63931 −0.0933720
\(800\) 0 0
\(801\) 0 0
\(802\) 12.7649 0.450743
\(803\) −23.3730 −0.824814
\(804\) 0 0
\(805\) 0 0
\(806\) −6.69263 −0.235738
\(807\) 0 0
\(808\) −13.0205 −0.458060
\(809\) −17.9467 −0.630972 −0.315486 0.948930i \(-0.602167\pi\)
−0.315486 + 0.948930i \(0.602167\pi\)
\(810\) 0 0
\(811\) 40.5113 1.42254 0.711272 0.702917i \(-0.248119\pi\)
0.711272 + 0.702917i \(0.248119\pi\)
\(812\) 1.52973 0.0536831
\(813\) 0 0
\(814\) −19.1506 −0.671229
\(815\) 0 0
\(816\) 0 0
\(817\) 0.738205 0.0258265
\(818\) −22.8781 −0.799915
\(819\) 0 0
\(820\) 0 0
\(821\) −49.3340 −1.72177 −0.860885 0.508800i \(-0.830089\pi\)
−0.860885 + 0.508800i \(0.830089\pi\)
\(822\) 0 0
\(823\) 5.24742 0.182914 0.0914568 0.995809i \(-0.470848\pi\)
0.0914568 + 0.995809i \(0.470848\pi\)
\(824\) 14.3402 0.499564
\(825\) 0 0
\(826\) 8.99386 0.312936
\(827\) 22.5236 0.783222 0.391611 0.920131i \(-0.371918\pi\)
0.391611 + 0.920131i \(0.371918\pi\)
\(828\) 0 0
\(829\) 43.2495 1.50212 0.751059 0.660235i \(-0.229543\pi\)
0.751059 + 0.660235i \(0.229543\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.921622 0.0319515
\(833\) −1.98562 −0.0687978
\(834\) 0 0
\(835\) 0 0
\(836\) 3.41855 0.118233
\(837\) 0 0
\(838\) 3.47187 0.119934
\(839\) 34.6681 1.19687 0.598437 0.801170i \(-0.295789\pi\)
0.598437 + 0.801170i \(0.295789\pi\)
\(840\) 0 0
\(841\) −26.9877 −0.930611
\(842\) 20.4657 0.705296
\(843\) 0 0
\(844\) −4.68035 −0.161104
\(845\) 0 0
\(846\) 0 0
\(847\) 0.740294 0.0254368
\(848\) 2.68035 0.0920435
\(849\) 0 0
\(850\) 0 0
\(851\) 5.16290 0.176982
\(852\) 0 0
\(853\) 28.0554 0.960599 0.480300 0.877105i \(-0.340528\pi\)
0.480300 + 0.877105i \(0.340528\pi\)
\(854\) −2.15676 −0.0738027
\(855\) 0 0
\(856\) 10.8371 0.370405
\(857\) −49.0882 −1.67682 −0.838411 0.545039i \(-0.816515\pi\)
−0.838411 + 0.545039i \(0.816515\pi\)
\(858\) 0 0
\(859\) −37.7275 −1.28725 −0.643623 0.765342i \(-0.722570\pi\)
−0.643623 + 0.765342i \(0.722570\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −33.1917 −1.13051
\(863\) 44.9939 1.53161 0.765804 0.643074i \(-0.222341\pi\)
0.765804 + 0.643074i \(0.222341\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −41.1338 −1.39778
\(867\) 0 0
\(868\) 7.83096 0.265800
\(869\) −16.1978 −0.549473
\(870\) 0 0
\(871\) 4.31351 0.146158
\(872\) −18.6225 −0.630637
\(873\) 0 0
\(874\) −0.921622 −0.0311743
\(875\) 0 0
\(876\) 0 0
\(877\) 2.64915 0.0894554 0.0447277 0.998999i \(-0.485758\pi\)
0.0447277 + 0.998999i \(0.485758\pi\)
\(878\) −19.1461 −0.646149
\(879\) 0 0
\(880\) 0 0
\(881\) −30.2511 −1.01919 −0.509593 0.860416i \(-0.670204\pi\)
−0.509593 + 0.860416i \(0.670204\pi\)
\(882\) 0 0
\(883\) −19.9955 −0.672901 −0.336450 0.941701i \(-0.609226\pi\)
−0.336450 + 0.941701i \(0.609226\pi\)
\(884\) 0.313511 0.0105445
\(885\) 0 0
\(886\) 37.4908 1.25953
\(887\) −57.0759 −1.91642 −0.958211 0.286062i \(-0.907654\pi\)
−0.958211 + 0.286062i \(0.907654\pi\)
\(888\) 0 0
\(889\) −17.7899 −0.596655
\(890\) 0 0
\(891\) 0 0
\(892\) −14.3402 −0.480145
\(893\) 7.75872 0.259636
\(894\) 0 0
\(895\) 0 0
\(896\) −1.07838 −0.0360261
\(897\) 0 0
\(898\) 6.43907 0.214875
\(899\) 10.3012 0.343565
\(900\) 0 0
\(901\) 0.911781 0.0303758
\(902\) 3.68649 0.122747
\(903\) 0 0
\(904\) 13.5174 0.449584
\(905\) 0 0
\(906\) 0 0
\(907\) −15.4017 −0.511406 −0.255703 0.966755i \(-0.582307\pi\)
−0.255703 + 0.966755i \(0.582307\pi\)
\(908\) 19.2039 0.637305
\(909\) 0 0
\(910\) 0 0
\(911\) 37.5585 1.24437 0.622184 0.782871i \(-0.286246\pi\)
0.622184 + 0.782871i \(0.286246\pi\)
\(912\) 0 0
\(913\) −37.6742 −1.24683
\(914\) −13.6742 −0.452302
\(915\) 0 0
\(916\) 2.31351 0.0764406
\(917\) −0.627022 −0.0207061
\(918\) 0 0
\(919\) −10.0410 −0.331223 −0.165612 0.986191i \(-0.552960\pi\)
−0.165612 + 0.986191i \(0.552960\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 27.5441 0.907117
\(923\) 9.98771 0.328750
\(924\) 0 0
\(925\) 0 0
\(926\) 28.5958 0.939717
\(927\) 0 0
\(928\) −1.41855 −0.0465662
\(929\) −1.26633 −0.0415469 −0.0207735 0.999784i \(-0.506613\pi\)
−0.0207735 + 0.999784i \(0.506613\pi\)
\(930\) 0 0
\(931\) 5.83710 0.191303
\(932\) 25.6475 0.840113
\(933\) 0 0
\(934\) −35.5318 −1.16264
\(935\) 0 0
\(936\) 0 0
\(937\) 14.1568 0.462481 0.231241 0.972897i \(-0.425722\pi\)
0.231241 + 0.972897i \(0.425722\pi\)
\(938\) −5.04718 −0.164796
\(939\) 0 0
\(940\) 0 0
\(941\) −15.2085 −0.495782 −0.247891 0.968788i \(-0.579737\pi\)
−0.247891 + 0.968788i \(0.579737\pi\)
\(942\) 0 0
\(943\) −0.993857 −0.0323644
\(944\) −8.34017 −0.271450
\(945\) 0 0
\(946\) −2.52359 −0.0820490
\(947\) −2.53797 −0.0824728 −0.0412364 0.999149i \(-0.513130\pi\)
−0.0412364 + 0.999149i \(0.513130\pi\)
\(948\) 0 0
\(949\) 6.30122 0.204546
\(950\) 0 0
\(951\) 0 0
\(952\) −0.366835 −0.0118892
\(953\) 55.0759 1.78408 0.892042 0.451952i \(-0.149272\pi\)
0.892042 + 0.451952i \(0.149272\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −3.60197 −0.116496
\(957\) 0 0
\(958\) 24.6491 0.796378
\(959\) −6.27247 −0.202549
\(960\) 0 0
\(961\) 21.7337 0.701086
\(962\) 5.16290 0.166459
\(963\) 0 0
\(964\) −2.36683 −0.0762306
\(965\) 0 0
\(966\) 0 0
\(967\) 12.6491 0.406769 0.203385 0.979099i \(-0.434806\pi\)
0.203385 + 0.979099i \(0.434806\pi\)
\(968\) −0.686489 −0.0220646
\(969\) 0 0
\(970\) 0 0
\(971\) −38.0677 −1.22165 −0.610825 0.791765i \(-0.709162\pi\)
−0.610825 + 0.791765i \(0.709162\pi\)
\(972\) 0 0
\(973\) 6.63931 0.212846
\(974\) −15.3340 −0.491334
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −6.16904 −0.197365 −0.0986826 0.995119i \(-0.531463\pi\)
−0.0986826 + 0.995119i \(0.531463\pi\)
\(978\) 0 0
\(979\) 33.3607 1.06621
\(980\) 0 0
\(981\) 0 0
\(982\) −32.7792 −1.04603
\(983\) −1.98771 −0.0633982 −0.0316991 0.999497i \(-0.510092\pi\)
−0.0316991 + 0.999497i \(0.510092\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.482553 −0.0153676
\(987\) 0 0
\(988\) −0.921622 −0.0293207
\(989\) 0.680346 0.0216337
\(990\) 0 0
\(991\) −0.907246 −0.0288196 −0.0144098 0.999896i \(-0.504587\pi\)
−0.0144098 + 0.999896i \(0.504587\pi\)
\(992\) −7.26180 −0.230562
\(993\) 0 0
\(994\) −11.6865 −0.370673
\(995\) 0 0
\(996\) 0 0
\(997\) 17.3874 0.550663 0.275332 0.961349i \(-0.411212\pi\)
0.275332 + 0.961349i \(0.411212\pi\)
\(998\) 25.6742 0.812703
\(999\) 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 8550.2.a.cf.1.2 3
3.2 odd 2 2850.2.a.bn.1.2 3
5.2 odd 4 1710.2.d.e.1369.1 6
5.3 odd 4 1710.2.d.e.1369.4 6
5.4 even 2 8550.2.a.cr.1.2 3
15.2 even 4 570.2.d.d.229.6 yes 6
15.8 even 4 570.2.d.d.229.3 6
15.14 odd 2 2850.2.a.bk.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.d.229.3 6 15.8 even 4
570.2.d.d.229.6 yes 6 15.2 even 4
1710.2.d.e.1369.1 6 5.2 odd 4
1710.2.d.e.1369.4 6 5.3 odd 4
2850.2.a.bk.1.2 3 15.14 odd 2
2850.2.a.bn.1.2 3 3.2 odd 2
8550.2.a.cf.1.2 3 1.1 even 1 trivial
8550.2.a.cr.1.2 3 5.4 even 2