Properties

Label 3969.2.a.t.1.1
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.14013.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 567)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.20800\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.20800 q^{2} +2.87525 q^{4} -3.80779 q^{5} -1.93254 q^{8} +O(q^{10})\) \(q-2.20800 q^{2} +2.87525 q^{4} -3.80779 q^{5} -1.93254 q^{8} +8.40758 q^{10} -4.32433 q^{11} +2.87525 q^{13} -1.48345 q^{16} +4.02595 q^{17} +1.60821 q^{19} -10.9483 q^{20} +9.54811 q^{22} +2.66725 q^{23} +9.49923 q^{25} -6.34853 q^{26} -0.750492 q^{29} -0.140536 q^{31} +7.14054 q^{32} -8.88928 q^{34} -8.28282 q^{37} -3.55091 q^{38} +7.35870 q^{40} -10.3745 q^{41} +0.267040 q^{43} -12.4335 q^{44} -5.88928 q^{46} -7.93254 q^{47} -20.9743 q^{50} +8.26704 q^{52} +11.2238 q^{53} +16.4661 q^{55} +1.65708 q^{58} -0.693198 q^{59} -2.10744 q^{61} +0.310302 q^{62} -12.7994 q^{64} -10.9483 q^{65} -10.7663 q^{67} +11.5756 q^{68} -3.62399 q^{71} +3.57511 q^{73} +18.2884 q^{74} +4.62399 q^{76} -15.4234 q^{79} +5.64867 q^{80} +22.9068 q^{82} +6.44067 q^{83} -15.3299 q^{85} -0.589624 q^{86} +8.35695 q^{88} +0.256874 q^{89} +7.66900 q^{92} +17.5150 q^{94} -6.12370 q^{95} -1.05856 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{8} + 7 q^{10} - 5 q^{11} + 5 q^{13} - q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} - 7 q^{22} + 12 q^{23} + 8 q^{25} + q^{26} + 10 q^{29} + 18 q^{31} + 10 q^{32} - 20 q^{38} + 18 q^{40} - 5 q^{41} - 7 q^{43} - 13 q^{44} + 12 q^{46} - 21 q^{47} - 38 q^{50} + 25 q^{52} + 12 q^{53} + 26 q^{55} - 7 q^{58} + 6 q^{59} + 20 q^{61} + 18 q^{62} - 23 q^{64} - 8 q^{65} - 5 q^{67} + 51 q^{68} + 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} - 2 q^{80} + 35 q^{82} + 9 q^{83} - 9 q^{85} + 22 q^{86} + 18 q^{88} - 22 q^{89} + 36 q^{92} + 15 q^{94} + 16 q^{95} + 9 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.20800 −1.56129 −0.780644 0.624975i \(-0.785109\pi\)
−0.780644 + 0.624975i \(0.785109\pi\)
\(3\) 0 0
\(4\) 2.87525 1.43762
\(5\) −3.80779 −1.70289 −0.851447 0.524441i \(-0.824274\pi\)
−0.851447 + 0.524441i \(0.824274\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.93254 −0.683256
\(9\) 0 0
\(10\) 8.40758 2.65871
\(11\) −4.32433 −1.30384 −0.651918 0.758290i \(-0.726035\pi\)
−0.651918 + 0.758290i \(0.726035\pi\)
\(12\) 0 0
\(13\) 2.87525 0.797450 0.398725 0.917071i \(-0.369453\pi\)
0.398725 + 0.917071i \(0.369453\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.48345 −0.370863
\(17\) 4.02595 0.976436 0.488218 0.872722i \(-0.337647\pi\)
0.488218 + 0.872722i \(0.337647\pi\)
\(18\) 0 0
\(19\) 1.60821 0.368948 0.184474 0.982837i \(-0.440942\pi\)
0.184474 + 0.982837i \(0.440942\pi\)
\(20\) −10.9483 −2.44812
\(21\) 0 0
\(22\) 9.54811 2.03566
\(23\) 2.66725 0.556160 0.278080 0.960558i \(-0.410302\pi\)
0.278080 + 0.960558i \(0.410302\pi\)
\(24\) 0 0
\(25\) 9.49923 1.89985
\(26\) −6.34853 −1.24505
\(27\) 0 0
\(28\) 0 0
\(29\) −0.750492 −0.139363 −0.0696815 0.997569i \(-0.522198\pi\)
−0.0696815 + 0.997569i \(0.522198\pi\)
\(30\) 0 0
\(31\) −0.140536 −0.0252410 −0.0126205 0.999920i \(-0.504017\pi\)
−0.0126205 + 0.999920i \(0.504017\pi\)
\(32\) 7.14054 1.26228
\(33\) 0 0
\(34\) −8.88928 −1.52450
\(35\) 0 0
\(36\) 0 0
\(37\) −8.28282 −1.36169 −0.680844 0.732429i \(-0.738387\pi\)
−0.680844 + 0.732429i \(0.738387\pi\)
\(38\) −3.55091 −0.576034
\(39\) 0 0
\(40\) 7.35870 1.16351
\(41\) −10.3745 −1.62022 −0.810111 0.586277i \(-0.800593\pi\)
−0.810111 + 0.586277i \(0.800593\pi\)
\(42\) 0 0
\(43\) 0.267040 0.0407232 0.0203616 0.999793i \(-0.493518\pi\)
0.0203616 + 0.999793i \(0.493518\pi\)
\(44\) −12.4335 −1.87442
\(45\) 0 0
\(46\) −5.88928 −0.868327
\(47\) −7.93254 −1.15708 −0.578540 0.815654i \(-0.696377\pi\)
−0.578540 + 0.815654i \(0.696377\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −20.9743 −2.96621
\(51\) 0 0
\(52\) 8.26704 1.14643
\(53\) 11.2238 1.54170 0.770852 0.637014i \(-0.219831\pi\)
0.770852 + 0.637014i \(0.219831\pi\)
\(54\) 0 0
\(55\) 16.4661 2.22029
\(56\) 0 0
\(57\) 0 0
\(58\) 1.65708 0.217586
\(59\) −0.693198 −0.0902468 −0.0451234 0.998981i \(-0.514368\pi\)
−0.0451234 + 0.998981i \(0.514368\pi\)
\(60\) 0 0
\(61\) −2.10744 −0.269830 −0.134915 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(62\) 0.310302 0.0394085
\(63\) 0 0
\(64\) −12.7994 −1.59992
\(65\) −10.9483 −1.35797
\(66\) 0 0
\(67\) −10.7663 −1.31531 −0.657655 0.753319i \(-0.728451\pi\)
−0.657655 + 0.753319i \(0.728451\pi\)
\(68\) 11.5756 1.40375
\(69\) 0 0
\(70\) 0 0
\(71\) −3.62399 −0.430088 −0.215044 0.976604i \(-0.568990\pi\)
−0.215044 + 0.976604i \(0.568990\pi\)
\(72\) 0 0
\(73\) 3.57511 0.418435 0.209217 0.977869i \(-0.432908\pi\)
0.209217 + 0.977869i \(0.432908\pi\)
\(74\) 18.2884 2.12599
\(75\) 0 0
\(76\) 4.62399 0.530408
\(77\) 0 0
\(78\) 0 0
\(79\) −15.4234 −1.73526 −0.867632 0.497207i \(-0.834359\pi\)
−0.867632 + 0.497207i \(0.834359\pi\)
\(80\) 5.64867 0.631540
\(81\) 0 0
\(82\) 22.9068 2.52963
\(83\) 6.44067 0.706956 0.353478 0.935443i \(-0.384999\pi\)
0.353478 + 0.935443i \(0.384999\pi\)
\(84\) 0 0
\(85\) −15.3299 −1.66277
\(86\) −0.589624 −0.0635808
\(87\) 0 0
\(88\) 8.35695 0.890854
\(89\) 0.256874 0.0272286 0.0136143 0.999907i \(-0.495666\pi\)
0.0136143 + 0.999907i \(0.495666\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.66900 0.799549
\(93\) 0 0
\(94\) 17.5150 1.80654
\(95\) −6.12370 −0.628279
\(96\) 0 0
\(97\) −1.05856 −0.107481 −0.0537403 0.998555i \(-0.517114\pi\)
−0.0537403 + 0.998555i \(0.517114\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 27.3126 2.73126
\(101\) −1.36585 −0.135907 −0.0679534 0.997688i \(-0.521647\pi\)
−0.0679534 + 0.997688i \(0.521647\pi\)
\(102\) 0 0
\(103\) 14.9713 1.47516 0.737581 0.675259i \(-0.235968\pi\)
0.737581 + 0.675259i \(0.235968\pi\)
\(104\) −5.55653 −0.544862
\(105\) 0 0
\(106\) −24.7821 −2.40705
\(107\) −8.08149 −0.781267 −0.390634 0.920546i \(-0.627744\pi\)
−0.390634 + 0.920546i \(0.627744\pi\)
\(108\) 0 0
\(109\) −9.10919 −0.872502 −0.436251 0.899825i \(-0.643694\pi\)
−0.436251 + 0.899825i \(0.643694\pi\)
\(110\) −36.3572 −3.46652
\(111\) 0 0
\(112\) 0 0
\(113\) −2.76606 −0.260209 −0.130104 0.991500i \(-0.541531\pi\)
−0.130104 + 0.991500i \(0.541531\pi\)
\(114\) 0 0
\(115\) −10.1563 −0.947082
\(116\) −2.15785 −0.200351
\(117\) 0 0
\(118\) 1.53058 0.140901
\(119\) 0 0
\(120\) 0 0
\(121\) 7.69986 0.699988
\(122\) 4.65322 0.421283
\(123\) 0 0
\(124\) −0.404075 −0.0362870
\(125\) −17.1321 −1.53234
\(126\) 0 0
\(127\) 13.6573 1.21189 0.605945 0.795507i \(-0.292795\pi\)
0.605945 + 0.795507i \(0.292795\pi\)
\(128\) 13.9799 1.23566
\(129\) 0 0
\(130\) 24.1739 2.12019
\(131\) 17.6138 1.53893 0.769463 0.638691i \(-0.220524\pi\)
0.769463 + 0.638691i \(0.220524\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 23.7719 2.05358
\(135\) 0 0
\(136\) −7.78031 −0.667156
\(137\) −8.66725 −0.740493 −0.370247 0.928934i \(-0.620727\pi\)
−0.370247 + 0.928934i \(0.620727\pi\)
\(138\) 0 0
\(139\) 3.76606 0.319433 0.159716 0.987163i \(-0.448942\pi\)
0.159716 + 0.987163i \(0.448942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00175 0.671492
\(143\) −12.4335 −1.03974
\(144\) 0 0
\(145\) 2.85771 0.237320
\(146\) −7.89383 −0.653298
\(147\) 0 0
\(148\) −23.8152 −1.95759
\(149\) −4.61122 −0.377766 −0.188883 0.982000i \(-0.560487\pi\)
−0.188883 + 0.982000i \(0.560487\pi\)
\(150\) 0 0
\(151\) 7.79762 0.634561 0.317281 0.948332i \(-0.397230\pi\)
0.317281 + 0.948332i \(0.397230\pi\)
\(152\) −3.10792 −0.252086
\(153\) 0 0
\(154\) 0 0
\(155\) 0.535130 0.0429827
\(156\) 0 0
\(157\) −23.2655 −1.85679 −0.928395 0.371595i \(-0.878811\pi\)
−0.928395 + 0.371595i \(0.878811\pi\)
\(158\) 34.0547 2.70925
\(159\) 0 0
\(160\) −27.1896 −2.14953
\(161\) 0 0
\(162\) 0 0
\(163\) 6.11094 0.478646 0.239323 0.970940i \(-0.423075\pi\)
0.239323 + 0.970940i \(0.423075\pi\)
\(164\) −29.8292 −2.32927
\(165\) 0 0
\(166\) −14.2210 −1.10376
\(167\) 11.6415 0.900848 0.450424 0.892815i \(-0.351273\pi\)
0.450424 + 0.892815i \(0.351273\pi\)
\(168\) 0 0
\(169\) −4.73296 −0.364074
\(170\) 33.8485 2.59606
\(171\) 0 0
\(172\) 0.767806 0.0585447
\(173\) 19.2571 1.46409 0.732045 0.681256i \(-0.238566\pi\)
0.732045 + 0.681256i \(0.238566\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.41494 0.483544
\(177\) 0 0
\(178\) −0.567176 −0.0425117
\(179\) −3.34187 −0.249783 −0.124891 0.992170i \(-0.539858\pi\)
−0.124891 + 0.992170i \(0.539858\pi\)
\(180\) 0 0
\(181\) 19.7358 1.46695 0.733474 0.679717i \(-0.237897\pi\)
0.733474 + 0.679717i \(0.237897\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5.15457 −0.380000
\(185\) 31.5392 2.31881
\(186\) 0 0
\(187\) −17.4095 −1.27311
\(188\) −22.8080 −1.66344
\(189\) 0 0
\(190\) 13.5211 0.980925
\(191\) −4.17490 −0.302085 −0.151043 0.988527i \(-0.548263\pi\)
−0.151043 + 0.988527i \(0.548263\pi\)
\(192\) 0 0
\(193\) −13.8737 −0.998652 −0.499326 0.866414i \(-0.666419\pi\)
−0.499326 + 0.866414i \(0.666419\pi\)
\(194\) 2.33730 0.167808
\(195\) 0 0
\(196\) 0 0
\(197\) −7.48520 −0.533299 −0.266649 0.963794i \(-0.585917\pi\)
−0.266649 + 0.963794i \(0.585917\pi\)
\(198\) 0 0
\(199\) 12.6553 0.897113 0.448556 0.893755i \(-0.351938\pi\)
0.448556 + 0.893755i \(0.351938\pi\)
\(200\) −18.3576 −1.29808
\(201\) 0 0
\(202\) 3.01578 0.212190
\(203\) 0 0
\(204\) 0 0
\(205\) 39.5038 2.75907
\(206\) −33.0565 −2.30315
\(207\) 0 0
\(208\) −4.26529 −0.295745
\(209\) −6.95442 −0.481047
\(210\) 0 0
\(211\) −5.15197 −0.354676 −0.177338 0.984150i \(-0.556749\pi\)
−0.177338 + 0.984150i \(0.556749\pi\)
\(212\) 32.2711 2.21639
\(213\) 0 0
\(214\) 17.8439 1.21978
\(215\) −1.01683 −0.0693474
\(216\) 0 0
\(217\) 0 0
\(218\) 20.1131 1.36223
\(219\) 0 0
\(220\) 47.3442 3.19195
\(221\) 11.5756 0.778659
\(222\) 0 0
\(223\) 15.2166 1.01898 0.509490 0.860476i \(-0.329834\pi\)
0.509490 + 0.860476i \(0.329834\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.10744 0.406261
\(227\) 4.28949 0.284703 0.142352 0.989816i \(-0.454534\pi\)
0.142352 + 0.989816i \(0.454534\pi\)
\(228\) 0 0
\(229\) −0.971251 −0.0641821 −0.0320910 0.999485i \(-0.510217\pi\)
−0.0320910 + 0.999485i \(0.510217\pi\)
\(230\) 22.4251 1.47867
\(231\) 0 0
\(232\) 1.45036 0.0952205
\(233\) −3.77342 −0.247205 −0.123603 0.992332i \(-0.539445\pi\)
−0.123603 + 0.992332i \(0.539445\pi\)
\(234\) 0 0
\(235\) 30.2054 1.97038
\(236\) −1.99312 −0.129741
\(237\) 0 0
\(238\) 0 0
\(239\) 6.69040 0.432766 0.216383 0.976309i \(-0.430574\pi\)
0.216383 + 0.976309i \(0.430574\pi\)
\(240\) 0 0
\(241\) 14.9713 0.964383 0.482192 0.876066i \(-0.339841\pi\)
0.482192 + 0.876066i \(0.339841\pi\)
\(242\) −17.0013 −1.09288
\(243\) 0 0
\(244\) −6.05941 −0.387914
\(245\) 0 0
\(246\) 0 0
\(247\) 4.62399 0.294217
\(248\) 0.271591 0.0172460
\(249\) 0 0
\(250\) 37.8277 2.39243
\(251\) 17.0787 1.07800 0.538999 0.842307i \(-0.318803\pi\)
0.538999 + 0.842307i \(0.318803\pi\)
\(252\) 0 0
\(253\) −11.5341 −0.725141
\(254\) −30.1553 −1.89211
\(255\) 0 0
\(256\) −5.26879 −0.329299
\(257\) −8.57231 −0.534726 −0.267363 0.963596i \(-0.586152\pi\)
−0.267363 + 0.963596i \(0.586152\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −31.4791 −1.95225
\(261\) 0 0
\(262\) −38.8912 −2.40271
\(263\) −5.69685 −0.351283 −0.175641 0.984454i \(-0.556200\pi\)
−0.175641 + 0.984454i \(0.556200\pi\)
\(264\) 0 0
\(265\) −42.7377 −2.62536
\(266\) 0 0
\(267\) 0 0
\(268\) −30.9557 −1.89092
\(269\) −15.6015 −0.951243 −0.475621 0.879650i \(-0.657777\pi\)
−0.475621 + 0.879650i \(0.657777\pi\)
\(270\) 0 0
\(271\) 30.7375 1.86717 0.933586 0.358354i \(-0.116662\pi\)
0.933586 + 0.358354i \(0.116662\pi\)
\(272\) −5.97230 −0.362124
\(273\) 0 0
\(274\) 19.1373 1.15612
\(275\) −41.0779 −2.47709
\(276\) 0 0
\(277\) −26.5926 −1.59780 −0.798899 0.601466i \(-0.794584\pi\)
−0.798899 + 0.601466i \(0.794584\pi\)
\(278\) −8.31544 −0.498727
\(279\) 0 0
\(280\) 0 0
\(281\) 17.3613 1.03569 0.517844 0.855475i \(-0.326735\pi\)
0.517844 + 0.855475i \(0.326735\pi\)
\(282\) 0 0
\(283\) −18.3554 −1.09112 −0.545558 0.838073i \(-0.683682\pi\)
−0.545558 + 0.838073i \(0.683682\pi\)
\(284\) −10.4199 −0.618305
\(285\) 0 0
\(286\) 27.4532 1.62334
\(287\) 0 0
\(288\) 0 0
\(289\) −0.791740 −0.0465730
\(290\) −6.30982 −0.370525
\(291\) 0 0
\(292\) 10.2793 0.601552
\(293\) 10.6981 0.624990 0.312495 0.949919i \(-0.398835\pi\)
0.312495 + 0.949919i \(0.398835\pi\)
\(294\) 0 0
\(295\) 2.63955 0.153681
\(296\) 16.0069 0.930381
\(297\) 0 0
\(298\) 10.1816 0.589802
\(299\) 7.66900 0.443510
\(300\) 0 0
\(301\) 0 0
\(302\) −17.2171 −0.990733
\(303\) 0 0
\(304\) −2.38570 −0.136829
\(305\) 8.02468 0.459492
\(306\) 0 0
\(307\) −31.3948 −1.79180 −0.895899 0.444258i \(-0.853467\pi\)
−0.895899 + 0.444258i \(0.853467\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.18157 −0.0671084
\(311\) −13.4891 −0.764895 −0.382447 0.923977i \(-0.624919\pi\)
−0.382447 + 0.923977i \(0.624919\pi\)
\(312\) 0 0
\(313\) 19.3414 1.09324 0.546620 0.837381i \(-0.315914\pi\)
0.546620 + 0.837381i \(0.315914\pi\)
\(314\) 51.3701 2.89899
\(315\) 0 0
\(316\) −44.3459 −2.49465
\(317\) 15.3576 0.862571 0.431286 0.902215i \(-0.358060\pi\)
0.431286 + 0.902215i \(0.358060\pi\)
\(318\) 0 0
\(319\) 3.24538 0.181706
\(320\) 48.7373 2.72450
\(321\) 0 0
\(322\) 0 0
\(323\) 6.47455 0.360254
\(324\) 0 0
\(325\) 27.3126 1.51503
\(326\) −13.4929 −0.747304
\(327\) 0 0
\(328\) 20.0491 1.10703
\(329\) 0 0
\(330\) 0 0
\(331\) −1.23829 −0.0680627 −0.0340313 0.999421i \(-0.510835\pi\)
−0.0340313 + 0.999421i \(0.510835\pi\)
\(332\) 18.5185 1.01634
\(333\) 0 0
\(334\) −25.7044 −1.40648
\(335\) 40.9957 2.23983
\(336\) 0 0
\(337\) 11.9086 0.648701 0.324350 0.945937i \(-0.394854\pi\)
0.324350 + 0.945937i \(0.394854\pi\)
\(338\) 10.4504 0.568424
\(339\) 0 0
\(340\) −44.0774 −2.39043
\(341\) 0.607724 0.0329101
\(342\) 0 0
\(343\) 0 0
\(344\) −0.516066 −0.0278244
\(345\) 0 0
\(346\) −42.5196 −2.28587
\(347\) 4.34678 0.233347 0.116674 0.993170i \(-0.462777\pi\)
0.116674 + 0.993170i \(0.462777\pi\)
\(348\) 0 0
\(349\) −7.17560 −0.384101 −0.192051 0.981385i \(-0.561514\pi\)
−0.192051 + 0.981385i \(0.561514\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −30.8781 −1.64581
\(353\) 25.0967 1.33576 0.667881 0.744268i \(-0.267201\pi\)
0.667881 + 0.744268i \(0.267201\pi\)
\(354\) 0 0
\(355\) 13.7994 0.732395
\(356\) 0.738575 0.0391444
\(357\) 0 0
\(358\) 7.37883 0.389983
\(359\) −31.9918 −1.68846 −0.844231 0.535979i \(-0.819943\pi\)
−0.844231 + 0.535979i \(0.819943\pi\)
\(360\) 0 0
\(361\) −16.4137 −0.863878
\(362\) −43.5765 −2.29033
\(363\) 0 0
\(364\) 0 0
\(365\) −13.6133 −0.712550
\(366\) 0 0
\(367\) 11.1248 0.580707 0.290354 0.956919i \(-0.406227\pi\)
0.290354 + 0.956919i \(0.406227\pi\)
\(368\) −3.95674 −0.206259
\(369\) 0 0
\(370\) −69.6385 −3.62033
\(371\) 0 0
\(372\) 0 0
\(373\) 16.1546 0.836452 0.418226 0.908343i \(-0.362652\pi\)
0.418226 + 0.908343i \(0.362652\pi\)
\(374\) 38.4402 1.98770
\(375\) 0 0
\(376\) 15.3299 0.790582
\(377\) −2.15785 −0.111135
\(378\) 0 0
\(379\) 3.18485 0.163595 0.0817973 0.996649i \(-0.473934\pi\)
0.0817973 + 0.996649i \(0.473934\pi\)
\(380\) −17.6072 −0.903228
\(381\) 0 0
\(382\) 9.21816 0.471642
\(383\) 16.4776 0.841968 0.420984 0.907068i \(-0.361685\pi\)
0.420984 + 0.907068i \(0.361685\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 30.6331 1.55918
\(387\) 0 0
\(388\) −3.04363 −0.154517
\(389\) 31.9771 1.62130 0.810651 0.585529i \(-0.199113\pi\)
0.810651 + 0.585529i \(0.199113\pi\)
\(390\) 0 0
\(391\) 10.7382 0.543055
\(392\) 0 0
\(393\) 0 0
\(394\) 16.5273 0.832633
\(395\) 58.7288 2.95497
\(396\) 0 0
\(397\) −27.8119 −1.39584 −0.697919 0.716177i \(-0.745890\pi\)
−0.697919 + 0.716177i \(0.745890\pi\)
\(398\) −27.9429 −1.40065
\(399\) 0 0
\(400\) −14.0917 −0.704583
\(401\) 27.5883 1.37769 0.688847 0.724907i \(-0.258117\pi\)
0.688847 + 0.724907i \(0.258117\pi\)
\(402\) 0 0
\(403\) −0.404075 −0.0201284
\(404\) −3.92714 −0.195383
\(405\) 0 0
\(406\) 0 0
\(407\) 35.8177 1.77542
\(408\) 0 0
\(409\) 11.8246 0.584690 0.292345 0.956313i \(-0.405564\pi\)
0.292345 + 0.956313i \(0.405564\pi\)
\(410\) −87.2242 −4.30770
\(411\) 0 0
\(412\) 43.0460 2.12073
\(413\) 0 0
\(414\) 0 0
\(415\) −24.5247 −1.20387
\(416\) 20.5308 1.00661
\(417\) 0 0
\(418\) 15.3553 0.751054
\(419\) −8.81668 −0.430723 −0.215362 0.976534i \(-0.569093\pi\)
−0.215362 + 0.976534i \(0.569093\pi\)
\(420\) 0 0
\(421\) 18.7001 0.911386 0.455693 0.890137i \(-0.349391\pi\)
0.455693 + 0.890137i \(0.349391\pi\)
\(422\) 11.3755 0.553752
\(423\) 0 0
\(424\) −21.6904 −1.05338
\(425\) 38.2434 1.85508
\(426\) 0 0
\(427\) 0 0
\(428\) −23.2363 −1.12317
\(429\) 0 0
\(430\) 2.24516 0.108271
\(431\) −16.6194 −0.800530 −0.400265 0.916399i \(-0.631082\pi\)
−0.400265 + 0.916399i \(0.631082\pi\)
\(432\) 0 0
\(433\) 25.3004 1.21586 0.607929 0.793992i \(-0.292001\pi\)
0.607929 + 0.793992i \(0.292001\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −26.1912 −1.25433
\(437\) 4.28949 0.205194
\(438\) 0 0
\(439\) 29.4523 1.40568 0.702841 0.711347i \(-0.251914\pi\)
0.702841 + 0.711347i \(0.251914\pi\)
\(440\) −31.8215 −1.51703
\(441\) 0 0
\(442\) −25.5589 −1.21571
\(443\) 32.5565 1.54681 0.773404 0.633914i \(-0.218553\pi\)
0.773404 + 0.633914i \(0.218553\pi\)
\(444\) 0 0
\(445\) −0.978120 −0.0463674
\(446\) −33.5983 −1.59092
\(447\) 0 0
\(448\) 0 0
\(449\) −0.171881 −0.00811158 −0.00405579 0.999992i \(-0.501291\pi\)
−0.00405579 + 0.999992i \(0.501291\pi\)
\(450\) 0 0
\(451\) 44.8627 2.11250
\(452\) −7.95309 −0.374082
\(453\) 0 0
\(454\) −9.47117 −0.444504
\(455\) 0 0
\(456\) 0 0
\(457\) 24.5020 1.14616 0.573079 0.819500i \(-0.305749\pi\)
0.573079 + 0.819500i \(0.305749\pi\)
\(458\) 2.14452 0.100207
\(459\) 0 0
\(460\) −29.2019 −1.36155
\(461\) 9.43107 0.439249 0.219624 0.975584i \(-0.429517\pi\)
0.219624 + 0.975584i \(0.429517\pi\)
\(462\) 0 0
\(463\) 2.86121 0.132972 0.0664860 0.997787i \(-0.478821\pi\)
0.0664860 + 0.997787i \(0.478821\pi\)
\(464\) 1.11332 0.0516845
\(465\) 0 0
\(466\) 8.33170 0.385959
\(467\) 7.70288 0.356447 0.178223 0.983990i \(-0.442965\pi\)
0.178223 + 0.983990i \(0.442965\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −66.6934 −3.07634
\(471\) 0 0
\(472\) 1.33963 0.0616616
\(473\) −1.15477 −0.0530964
\(474\) 0 0
\(475\) 15.2767 0.700944
\(476\) 0 0
\(477\) 0 0
\(478\) −14.7724 −0.675673
\(479\) 34.5822 1.58010 0.790051 0.613041i \(-0.210054\pi\)
0.790051 + 0.613041i \(0.210054\pi\)
\(480\) 0 0
\(481\) −23.8152 −1.08588
\(482\) −33.0565 −1.50568
\(483\) 0 0
\(484\) 22.1390 1.00632
\(485\) 4.03078 0.183028
\(486\) 0 0
\(487\) 18.4356 0.835399 0.417699 0.908585i \(-0.362837\pi\)
0.417699 + 0.908585i \(0.362837\pi\)
\(488\) 4.07271 0.184363
\(489\) 0 0
\(490\) 0 0
\(491\) −3.15372 −0.142325 −0.0711627 0.997465i \(-0.522671\pi\)
−0.0711627 + 0.997465i \(0.522671\pi\)
\(492\) 0 0
\(493\) −3.02144 −0.136079
\(494\) −10.2097 −0.459358
\(495\) 0 0
\(496\) 0.208478 0.00936094
\(497\) 0 0
\(498\) 0 0
\(499\) 28.8798 1.29284 0.646419 0.762983i \(-0.276266\pi\)
0.646419 + 0.762983i \(0.276266\pi\)
\(500\) −49.2591 −2.20293
\(501\) 0 0
\(502\) −37.7097 −1.68307
\(503\) 21.4742 0.957487 0.478744 0.877955i \(-0.341092\pi\)
0.478744 + 0.877955i \(0.341092\pi\)
\(504\) 0 0
\(505\) 5.20085 0.231435
\(506\) 25.4672 1.13216
\(507\) 0 0
\(508\) 39.2681 1.74224
\(509\) 8.82435 0.391133 0.195566 0.980690i \(-0.437346\pi\)
0.195566 + 0.980690i \(0.437346\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −16.3263 −0.721527
\(513\) 0 0
\(514\) 18.9276 0.834862
\(515\) −57.0073 −2.51204
\(516\) 0 0
\(517\) 34.3030 1.50864
\(518\) 0 0
\(519\) 0 0
\(520\) 21.1581 0.927843
\(521\) 29.6703 1.29988 0.649939 0.759986i \(-0.274794\pi\)
0.649939 + 0.759986i \(0.274794\pi\)
\(522\) 0 0
\(523\) 6.93991 0.303461 0.151730 0.988422i \(-0.451515\pi\)
0.151730 + 0.988422i \(0.451515\pi\)
\(524\) 50.6441 2.21240
\(525\) 0 0
\(526\) 12.5786 0.548454
\(527\) −0.565790 −0.0246462
\(528\) 0 0
\(529\) −15.8858 −0.690686
\(530\) 94.3648 4.09894
\(531\) 0 0
\(532\) 0 0
\(533\) −29.8292 −1.29205
\(534\) 0 0
\(535\) 30.7726 1.33042
\(536\) 20.8063 0.898693
\(537\) 0 0
\(538\) 34.4481 1.48516
\(539\) 0 0
\(540\) 0 0
\(541\) −16.6904 −0.717576 −0.358788 0.933419i \(-0.616810\pi\)
−0.358788 + 0.933419i \(0.616810\pi\)
\(542\) −67.8683 −2.91519
\(543\) 0 0
\(544\) 28.7474 1.23254
\(545\) 34.6858 1.48578
\(546\) 0 0
\(547\) 3.77924 0.161589 0.0807943 0.996731i \(-0.474254\pi\)
0.0807943 + 0.996731i \(0.474254\pi\)
\(548\) −24.9205 −1.06455
\(549\) 0 0
\(550\) 90.6997 3.86745
\(551\) −1.20695 −0.0514176
\(552\) 0 0
\(553\) 0 0
\(554\) 58.7164 2.49462
\(555\) 0 0
\(556\) 10.8283 0.459224
\(557\) 11.4692 0.485964 0.242982 0.970031i \(-0.421874\pi\)
0.242982 + 0.970031i \(0.421874\pi\)
\(558\) 0 0
\(559\) 0.767806 0.0324747
\(560\) 0 0
\(561\) 0 0
\(562\) −38.3337 −1.61701
\(563\) −25.3816 −1.06971 −0.534854 0.844944i \(-0.679633\pi\)
−0.534854 + 0.844944i \(0.679633\pi\)
\(564\) 0 0
\(565\) 10.5325 0.443108
\(566\) 40.5287 1.70355
\(567\) 0 0
\(568\) 7.00350 0.293860
\(569\) −8.69404 −0.364473 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(570\) 0 0
\(571\) 8.12757 0.340128 0.170064 0.985433i \(-0.445603\pi\)
0.170064 + 0.985433i \(0.445603\pi\)
\(572\) −35.7494 −1.49476
\(573\) 0 0
\(574\) 0 0
\(575\) 25.3368 1.05662
\(576\) 0 0
\(577\) −22.0839 −0.919367 −0.459683 0.888083i \(-0.652037\pi\)
−0.459683 + 0.888083i \(0.652037\pi\)
\(578\) 1.74816 0.0727138
\(579\) 0 0
\(580\) 8.21663 0.341177
\(581\) 0 0
\(582\) 0 0
\(583\) −48.5354 −2.01013
\(584\) −6.90904 −0.285898
\(585\) 0 0
\(586\) −23.6214 −0.975791
\(587\) 30.7200 1.26795 0.633975 0.773354i \(-0.281422\pi\)
0.633975 + 0.773354i \(0.281422\pi\)
\(588\) 0 0
\(589\) −0.226011 −0.00931260
\(590\) −5.82812 −0.239940
\(591\) 0 0
\(592\) 12.2872 0.505000
\(593\) −44.0715 −1.80980 −0.904901 0.425623i \(-0.860055\pi\)
−0.904901 + 0.425623i \(0.860055\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13.2584 −0.543085
\(597\) 0 0
\(598\) −16.9331 −0.692447
\(599\) 27.3663 1.11816 0.559078 0.829115i \(-0.311155\pi\)
0.559078 + 0.829115i \(0.311155\pi\)
\(600\) 0 0
\(601\) 7.21816 0.294435 0.147217 0.989104i \(-0.452968\pi\)
0.147217 + 0.989104i \(0.452968\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 22.4201 0.912260
\(605\) −29.3194 −1.19200
\(606\) 0 0
\(607\) −37.0048 −1.50198 −0.750989 0.660315i \(-0.770423\pi\)
−0.750989 + 0.660315i \(0.770423\pi\)
\(608\) 11.4835 0.465715
\(609\) 0 0
\(610\) −17.7185 −0.717400
\(611\) −22.8080 −0.922713
\(612\) 0 0
\(613\) 1.66058 0.0670704 0.0335352 0.999438i \(-0.489323\pi\)
0.0335352 + 0.999438i \(0.489323\pi\)
\(614\) 69.3197 2.79751
\(615\) 0 0
\(616\) 0 0
\(617\) 22.2418 0.895421 0.447710 0.894179i \(-0.352240\pi\)
0.447710 + 0.894179i \(0.352240\pi\)
\(618\) 0 0
\(619\) −19.7805 −0.795046 −0.397523 0.917592i \(-0.630130\pi\)
−0.397523 + 0.917592i \(0.630130\pi\)
\(620\) 1.53863 0.0617929
\(621\) 0 0
\(622\) 29.7838 1.19422
\(623\) 0 0
\(624\) 0 0
\(625\) 17.7393 0.709571
\(626\) −42.7057 −1.70686
\(627\) 0 0
\(628\) −66.8941 −2.66936
\(629\) −33.3462 −1.32960
\(630\) 0 0
\(631\) 49.2569 1.96089 0.980443 0.196804i \(-0.0630562\pi\)
0.980443 + 0.196804i \(0.0630562\pi\)
\(632\) 29.8063 1.18563
\(633\) 0 0
\(634\) −33.9096 −1.34672
\(635\) −52.0041 −2.06372
\(636\) 0 0
\(637\) 0 0
\(638\) −7.16578 −0.283696
\(639\) 0 0
\(640\) −53.2324 −2.10420
\(641\) 8.65682 0.341924 0.170962 0.985278i \(-0.445312\pi\)
0.170962 + 0.985278i \(0.445312\pi\)
\(642\) 0 0
\(643\) −17.7156 −0.698637 −0.349318 0.937004i \(-0.613587\pi\)
−0.349318 + 0.937004i \(0.613587\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −14.2958 −0.562460
\(647\) −12.1018 −0.475772 −0.237886 0.971293i \(-0.576455\pi\)
−0.237886 + 0.971293i \(0.576455\pi\)
\(648\) 0 0
\(649\) 2.99762 0.117667
\(650\) −60.3062 −2.36540
\(651\) 0 0
\(652\) 17.5705 0.688112
\(653\) 36.2140 1.41717 0.708583 0.705628i \(-0.249335\pi\)
0.708583 + 0.705628i \(0.249335\pi\)
\(654\) 0 0
\(655\) −67.0697 −2.62063
\(656\) 15.3900 0.600880
\(657\) 0 0
\(658\) 0 0
\(659\) −39.3414 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(660\) 0 0
\(661\) 1.66080 0.0645978 0.0322989 0.999478i \(-0.489717\pi\)
0.0322989 + 0.999478i \(0.489717\pi\)
\(662\) 2.73414 0.106265
\(663\) 0 0
\(664\) −12.4469 −0.483032
\(665\) 0 0
\(666\) 0 0
\(667\) −2.00175 −0.0775081
\(668\) 33.4722 1.29508
\(669\) 0 0
\(670\) −90.5183 −3.49703
\(671\) 9.11327 0.351814
\(672\) 0 0
\(673\) 20.8307 0.802965 0.401483 0.915867i \(-0.368495\pi\)
0.401483 + 0.915867i \(0.368495\pi\)
\(674\) −26.2941 −1.01281
\(675\) 0 0
\(676\) −13.6084 −0.523401
\(677\) −46.7308 −1.79601 −0.898006 0.439984i \(-0.854984\pi\)
−0.898006 + 0.439984i \(0.854984\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 29.6257 1.13610
\(681\) 0 0
\(682\) −1.34185 −0.0513822
\(683\) 21.1494 0.809261 0.404630 0.914480i \(-0.367400\pi\)
0.404630 + 0.914480i \(0.367400\pi\)
\(684\) 0 0
\(685\) 33.0030 1.26098
\(686\) 0 0
\(687\) 0 0
\(688\) −0.396141 −0.0151027
\(689\) 32.2711 1.22943
\(690\) 0 0
\(691\) 23.5970 0.897672 0.448836 0.893614i \(-0.351839\pi\)
0.448836 + 0.893614i \(0.351839\pi\)
\(692\) 55.3689 2.10481
\(693\) 0 0
\(694\) −9.59768 −0.364323
\(695\) −14.3403 −0.543960
\(696\) 0 0
\(697\) −41.7671 −1.58204
\(698\) 15.8437 0.599693
\(699\) 0 0
\(700\) 0 0
\(701\) 42.1420 1.59168 0.795841 0.605505i \(-0.207029\pi\)
0.795841 + 0.605505i \(0.207029\pi\)
\(702\) 0 0
\(703\) −13.3205 −0.502392
\(704\) 55.3488 2.08603
\(705\) 0 0
\(706\) −55.4134 −2.08551
\(707\) 0 0
\(708\) 0 0
\(709\) 45.9838 1.72696 0.863478 0.504386i \(-0.168281\pi\)
0.863478 + 0.504386i \(0.168281\pi\)
\(710\) −30.4690 −1.14348
\(711\) 0 0
\(712\) −0.496419 −0.0186041
\(713\) −0.374844 −0.0140380
\(714\) 0 0
\(715\) 47.3442 1.77057
\(716\) −9.60869 −0.359094
\(717\) 0 0
\(718\) 70.6378 2.63618
\(719\) −33.0498 −1.23255 −0.616275 0.787531i \(-0.711359\pi\)
−0.616275 + 0.787531i \(0.711359\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 36.2413 1.34876
\(723\) 0 0
\(724\) 56.7452 2.10892
\(725\) −7.12910 −0.264768
\(726\) 0 0
\(727\) 25.0550 0.929238 0.464619 0.885511i \(-0.346191\pi\)
0.464619 + 0.885511i \(0.346191\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 30.0580 1.11250
\(731\) 1.07509 0.0397636
\(732\) 0 0
\(733\) −14.0125 −0.517563 −0.258782 0.965936i \(-0.583321\pi\)
−0.258782 + 0.965936i \(0.583321\pi\)
\(734\) −24.5634 −0.906652
\(735\) 0 0
\(736\) 19.0456 0.702030
\(737\) 46.5570 1.71495
\(738\) 0 0
\(739\) −38.6013 −1.41997 −0.709987 0.704215i \(-0.751299\pi\)
−0.709987 + 0.704215i \(0.751299\pi\)
\(740\) 90.6830 3.33357
\(741\) 0 0
\(742\) 0 0
\(743\) 1.81318 0.0665192 0.0332596 0.999447i \(-0.489411\pi\)
0.0332596 + 0.999447i \(0.489411\pi\)
\(744\) 0 0
\(745\) 17.5586 0.643296
\(746\) −35.6692 −1.30594
\(747\) 0 0
\(748\) −50.0567 −1.83026
\(749\) 0 0
\(750\) 0 0
\(751\) −29.4262 −1.07378 −0.536888 0.843653i \(-0.680400\pi\)
−0.536888 + 0.843653i \(0.680400\pi\)
\(752\) 11.7675 0.429118
\(753\) 0 0
\(754\) 4.76452 0.173514
\(755\) −29.6917 −1.08059
\(756\) 0 0
\(757\) 22.5455 0.819431 0.409715 0.912213i \(-0.365628\pi\)
0.409715 + 0.912213i \(0.365628\pi\)
\(758\) −7.03213 −0.255419
\(759\) 0 0
\(760\) 11.8343 0.429275
\(761\) −12.8626 −0.466270 −0.233135 0.972444i \(-0.574898\pi\)
−0.233135 + 0.972444i \(0.574898\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0039 −0.434285
\(765\) 0 0
\(766\) −36.3825 −1.31455
\(767\) −1.99312 −0.0719673
\(768\) 0 0
\(769\) 44.7222 1.61272 0.806362 0.591423i \(-0.201434\pi\)
0.806362 + 0.591423i \(0.201434\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −39.8903 −1.43568
\(773\) −10.4880 −0.377227 −0.188614 0.982051i \(-0.560399\pi\)
−0.188614 + 0.982051i \(0.560399\pi\)
\(774\) 0 0
\(775\) −1.33498 −0.0479540
\(776\) 2.04571 0.0734368
\(777\) 0 0
\(778\) −70.6053 −2.53132
\(779\) −16.6843 −0.597777
\(780\) 0 0
\(781\) 15.6713 0.560764
\(782\) −23.7099 −0.847865
\(783\) 0 0
\(784\) 0 0
\(785\) 88.5901 3.16192
\(786\) 0 0
\(787\) −32.2074 −1.14807 −0.574035 0.818831i \(-0.694623\pi\)
−0.574035 + 0.818831i \(0.694623\pi\)
\(788\) −21.5218 −0.766682
\(789\) 0 0
\(790\) −129.673 −4.61356
\(791\) 0 0
\(792\) 0 0
\(793\) −6.05941 −0.215176
\(794\) 61.4085 2.17931
\(795\) 0 0
\(796\) 36.3872 1.28971
\(797\) −32.2778 −1.14334 −0.571669 0.820484i \(-0.693704\pi\)
−0.571669 + 0.820484i \(0.693704\pi\)
\(798\) 0 0
\(799\) −31.9360 −1.12981
\(800\) 67.8296 2.39814
\(801\) 0 0
\(802\) −60.9149 −2.15098
\(803\) −15.4600 −0.545570
\(804\) 0 0
\(805\) 0 0
\(806\) 0.892196 0.0314263
\(807\) 0 0
\(808\) 2.63955 0.0928591
\(809\) 29.9639 1.05347 0.526737 0.850028i \(-0.323415\pi\)
0.526737 + 0.850028i \(0.323415\pi\)
\(810\) 0 0
\(811\) 13.1971 0.463414 0.231707 0.972786i \(-0.425569\pi\)
0.231707 + 0.972786i \(0.425569\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −79.0853 −2.77194
\(815\) −23.2692 −0.815083
\(816\) 0 0
\(817\) 0.429456 0.0150247
\(818\) −26.1087 −0.912870
\(819\) 0 0
\(820\) 113.583 3.96650
\(821\) −8.03611 −0.280462 −0.140231 0.990119i \(-0.544785\pi\)
−0.140231 + 0.990119i \(0.544785\pi\)
\(822\) 0 0
\(823\) 19.5656 0.682015 0.341008 0.940060i \(-0.389232\pi\)
0.341008 + 0.940060i \(0.389232\pi\)
\(824\) −28.9325 −1.00791
\(825\) 0 0
\(826\) 0 0
\(827\) −45.8218 −1.59338 −0.796690 0.604389i \(-0.793417\pi\)
−0.796690 + 0.604389i \(0.793417\pi\)
\(828\) 0 0
\(829\) −34.1546 −1.18624 −0.593119 0.805115i \(-0.702103\pi\)
−0.593119 + 0.805115i \(0.702103\pi\)
\(830\) 54.1504 1.87959
\(831\) 0 0
\(832\) −36.8013 −1.27586
\(833\) 0 0
\(834\) 0 0
\(835\) −44.3284 −1.53405
\(836\) −19.9957 −0.691565
\(837\) 0 0
\(838\) 19.4672 0.672483
\(839\) 35.5972 1.22895 0.614476 0.788935i \(-0.289367\pi\)
0.614476 + 0.788935i \(0.289367\pi\)
\(840\) 0 0
\(841\) −28.4368 −0.980578
\(842\) −41.2897 −1.42294
\(843\) 0 0
\(844\) −14.8132 −0.509891
\(845\) 18.0221 0.619979
\(846\) 0 0
\(847\) 0 0
\(848\) −16.6499 −0.571761
\(849\) 0 0
\(850\) −84.4413 −2.89631
\(851\) −22.0924 −0.757316
\(852\) 0 0
\(853\) −33.3946 −1.14341 −0.571705 0.820459i \(-0.693718\pi\)
−0.571705 + 0.820459i \(0.693718\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 15.6178 0.533806
\(857\) 25.8147 0.881814 0.440907 0.897553i \(-0.354657\pi\)
0.440907 + 0.897553i \(0.354657\pi\)
\(858\) 0 0
\(859\) 43.0007 1.46716 0.733582 0.679601i \(-0.237847\pi\)
0.733582 + 0.679601i \(0.237847\pi\)
\(860\) −2.92364 −0.0996954
\(861\) 0 0
\(862\) 36.6956 1.24986
\(863\) 21.4967 0.731755 0.365877 0.930663i \(-0.380769\pi\)
0.365877 + 0.930663i \(0.380769\pi\)
\(864\) 0 0
\(865\) −73.3269 −2.49319
\(866\) −55.8631 −1.89830
\(867\) 0 0
\(868\) 0 0
\(869\) 66.6957 2.26250
\(870\) 0 0
\(871\) −30.9557 −1.04889
\(872\) 17.6039 0.596142
\(873\) 0 0
\(874\) −9.47117 −0.320367
\(875\) 0 0
\(876\) 0 0
\(877\) 14.6841 0.495846 0.247923 0.968780i \(-0.420252\pi\)
0.247923 + 0.968780i \(0.420252\pi\)
\(878\) −65.0306 −2.19468
\(879\) 0 0
\(880\) −24.4267 −0.823425
\(881\) −37.6060 −1.26698 −0.633490 0.773751i \(-0.718378\pi\)
−0.633490 + 0.773751i \(0.718378\pi\)
\(882\) 0 0
\(883\) −46.9989 −1.58164 −0.790820 0.612049i \(-0.790345\pi\)
−0.790820 + 0.612049i \(0.790345\pi\)
\(884\) 33.2827 1.11942
\(885\) 0 0
\(886\) −71.8847 −2.41501
\(887\) 26.4248 0.887259 0.443630 0.896210i \(-0.353691\pi\)
0.443630 + 0.896210i \(0.353691\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.15969 0.0723928
\(891\) 0 0
\(892\) 43.7516 1.46491
\(893\) −12.7572 −0.426902
\(894\) 0 0
\(895\) 12.7251 0.425354
\(896\) 0 0
\(897\) 0 0
\(898\) 0.379513 0.0126645
\(899\) 0.105471 0.00351766
\(900\) 0 0
\(901\) 45.1863 1.50538
\(902\) −99.0567 −3.29823
\(903\) 0 0
\(904\) 5.34551 0.177789
\(905\) −75.1496 −2.49806
\(906\) 0 0
\(907\) −0.450052 −0.0149437 −0.00747187 0.999972i \(-0.502378\pi\)
−0.00747187 + 0.999972i \(0.502378\pi\)
\(908\) 12.3333 0.409296
\(909\) 0 0
\(910\) 0 0
\(911\) −4.74439 −0.157189 −0.0785944 0.996907i \(-0.525043\pi\)
−0.0785944 + 0.996907i \(0.525043\pi\)
\(912\) 0 0
\(913\) −27.8516 −0.921754
\(914\) −54.1004 −1.78948
\(915\) 0 0
\(916\) −2.79259 −0.0922697
\(917\) 0 0
\(918\) 0 0
\(919\) 42.4205 1.39932 0.699662 0.714474i \(-0.253334\pi\)
0.699662 + 0.714474i \(0.253334\pi\)
\(920\) 19.6275 0.647099
\(921\) 0 0
\(922\) −20.8238 −0.685794
\(923\) −10.4199 −0.342974
\(924\) 0 0
\(925\) −78.6805 −2.58700
\(926\) −6.31755 −0.207608
\(927\) 0 0
\(928\) −5.35892 −0.175915
\(929\) 45.0817 1.47908 0.739541 0.673111i \(-0.235042\pi\)
0.739541 + 0.673111i \(0.235042\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.8495 −0.355388
\(933\) 0 0
\(934\) −17.0079 −0.556517
\(935\) 66.2918 2.16797
\(936\) 0 0
\(937\) −19.3045 −0.630650 −0.315325 0.948984i \(-0.602114\pi\)
−0.315325 + 0.948984i \(0.602114\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 86.8480 2.83267
\(941\) 10.5539 0.344048 0.172024 0.985093i \(-0.444969\pi\)
0.172024 + 0.985093i \(0.444969\pi\)
\(942\) 0 0
\(943\) −27.6713 −0.901103
\(944\) 1.02833 0.0334692
\(945\) 0 0
\(946\) 2.54973 0.0828989
\(947\) −47.7600 −1.55199 −0.775996 0.630737i \(-0.782753\pi\)
−0.775996 + 0.630737i \(0.782753\pi\)
\(948\) 0 0
\(949\) 10.2793 0.333681
\(950\) −33.7309 −1.09438
\(951\) 0 0
\(952\) 0 0
\(953\) −53.8101 −1.74308 −0.871540 0.490324i \(-0.836879\pi\)
−0.871540 + 0.490324i \(0.836879\pi\)
\(954\) 0 0
\(955\) 15.8971 0.514419
\(956\) 19.2365 0.622154
\(957\) 0 0
\(958\) −76.3574 −2.46700
\(959\) 0 0
\(960\) 0 0
\(961\) −30.9802 −0.999363
\(962\) 52.5838 1.69537
\(963\) 0 0
\(964\) 43.0460 1.38642
\(965\) 52.8281 1.70060
\(966\) 0 0
\(967\) −9.81775 −0.315718 −0.157859 0.987462i \(-0.550459\pi\)
−0.157859 + 0.987462i \(0.550459\pi\)
\(968\) −14.8803 −0.478271
\(969\) 0 0
\(970\) −8.89994 −0.285760
\(971\) −23.7986 −0.763734 −0.381867 0.924217i \(-0.624719\pi\)
−0.381867 + 0.924217i \(0.624719\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −40.7058 −1.30430
\(975\) 0 0
\(976\) 3.12629 0.100070
\(977\) 39.9903 1.27940 0.639701 0.768624i \(-0.279058\pi\)
0.639701 + 0.768624i \(0.279058\pi\)
\(978\) 0 0
\(979\) −1.11081 −0.0355016
\(980\) 0 0
\(981\) 0 0
\(982\) 6.96340 0.222211
\(983\) −46.2286 −1.47446 −0.737231 0.675641i \(-0.763867\pi\)
−0.737231 + 0.675641i \(0.763867\pi\)
\(984\) 0 0
\(985\) 28.5020 0.908151
\(986\) 6.67133 0.212459
\(987\) 0 0
\(988\) 13.2951 0.422974
\(989\) 0.712263 0.0226486
\(990\) 0 0
\(991\) 14.3964 0.457315 0.228658 0.973507i \(-0.426566\pi\)
0.228658 + 0.973507i \(0.426566\pi\)
\(992\) −1.00350 −0.0318612
\(993\) 0 0
\(994\) 0 0
\(995\) −48.1888 −1.52769
\(996\) 0 0
\(997\) −43.8546 −1.38889 −0.694444 0.719547i \(-0.744350\pi\)
−0.694444 + 0.719547i \(0.744350\pi\)
\(998\) −63.7665 −2.01849
\(999\) 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 3969.2.a.t.1.1 4
3.2 odd 2 3969.2.a.w.1.4 4
7.3 odd 6 567.2.e.d.163.4 yes 8
7.5 odd 6 567.2.e.d.487.4 yes 8
7.6 odd 2 3969.2.a.s.1.1 4
21.5 even 6 567.2.e.c.487.1 yes 8
21.17 even 6 567.2.e.c.163.1 8
21.20 even 2 3969.2.a.x.1.4 4
63.5 even 6 567.2.h.k.298.4 8
63.31 odd 6 567.2.g.k.541.4 8
63.38 even 6 567.2.h.k.352.4 8
63.40 odd 6 567.2.h.j.298.1 8
63.47 even 6 567.2.g.j.109.1 8
63.52 odd 6 567.2.h.j.352.1 8
63.59 even 6 567.2.g.j.541.1 8
63.61 odd 6 567.2.g.k.109.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.e.c.163.1 8 21.17 even 6
567.2.e.c.487.1 yes 8 21.5 even 6
567.2.e.d.163.4 yes 8 7.3 odd 6
567.2.e.d.487.4 yes 8 7.5 odd 6
567.2.g.j.109.1 8 63.47 even 6
567.2.g.j.541.1 8 63.59 even 6
567.2.g.k.109.4 8 63.61 odd 6
567.2.g.k.541.4 8 63.31 odd 6
567.2.h.j.298.1 8 63.40 odd 6
567.2.h.j.352.1 8 63.52 odd 6
567.2.h.k.298.4 8 63.5 even 6
567.2.h.k.352.4 8 63.38 even 6
3969.2.a.s.1.1 4 7.6 odd 2
3969.2.a.t.1.1 4 1.1 even 1 trivial
3969.2.a.w.1.4 4 3.2 odd 2
3969.2.a.x.1.4 4 21.20 even 2