Properties

Label 3380.2.a.r.1.9
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-2.81781\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.81781 q^{3} -1.00000 q^{5} +0.974186 q^{7} +4.94004 q^{9} +O(q^{10})\) \(q+2.81781 q^{3} -1.00000 q^{5} +0.974186 q^{7} +4.94004 q^{9} +3.42555 q^{11} -2.81781 q^{15} +0.283829 q^{17} -6.03105 q^{19} +2.74507 q^{21} +8.91511 q^{23} +1.00000 q^{25} +5.46666 q^{27} +0.461096 q^{29} -7.96959 q^{31} +9.65254 q^{33} -0.974186 q^{35} +2.95339 q^{37} +8.76466 q^{41} +8.81154 q^{43} -4.94004 q^{45} +10.3775 q^{47} -6.05096 q^{49} +0.799776 q^{51} +4.96945 q^{53} -3.42555 q^{55} -16.9943 q^{57} -7.88601 q^{59} +11.4869 q^{61} +4.81252 q^{63} +3.96340 q^{67} +25.1211 q^{69} +6.22524 q^{71} -9.94784 q^{73} +2.81781 q^{75} +3.33712 q^{77} +6.63735 q^{79} +0.583871 q^{81} -7.05049 q^{83} -0.283829 q^{85} +1.29928 q^{87} -8.66990 q^{89} -22.4568 q^{93} +6.03105 q^{95} +0.332881 q^{97} +16.9224 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + 7q^{11} + q^{15} + 13q^{17} + 4q^{19} + 3q^{21} + 12q^{23} + 9q^{25} - 4q^{27} + 16q^{29} - 13q^{31} - 34q^{33} + q^{35} - q^{37} + 6q^{41} + q^{43} - 12q^{45} + 2q^{47} + 20q^{49} + 11q^{51} + 30q^{53} - 7q^{55} - 38q^{57} - 15q^{59} + 21q^{61} + 17q^{63} + 7q^{67} + 15q^{69} + 7q^{71} + 28q^{73} - q^{75} + 46q^{77} + 31q^{79} + 41q^{81} - 45q^{83} - 13q^{85} + 28q^{87} + 41q^{89} + 11q^{93} - 4q^{95} - 8q^{97} + 81q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.81781 1.62686 0.813431 0.581661i \(-0.197597\pi\)
0.813431 + 0.581661i \(0.197597\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.974186 0.368208 0.184104 0.982907i \(-0.441062\pi\)
0.184104 + 0.982907i \(0.441062\pi\)
\(8\) 0 0
\(9\) 4.94004 1.64668
\(10\) 0 0
\(11\) 3.42555 1.03284 0.516421 0.856335i \(-0.327264\pi\)
0.516421 + 0.856335i \(0.327264\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.81781 −0.727555
\(16\) 0 0
\(17\) 0.283829 0.0688387 0.0344193 0.999407i \(-0.489042\pi\)
0.0344193 + 0.999407i \(0.489042\pi\)
\(18\) 0 0
\(19\) −6.03105 −1.38362 −0.691809 0.722081i \(-0.743186\pi\)
−0.691809 + 0.722081i \(0.743186\pi\)
\(20\) 0 0
\(21\) 2.74507 0.599023
\(22\) 0 0
\(23\) 8.91511 1.85893 0.929465 0.368911i \(-0.120269\pi\)
0.929465 + 0.368911i \(0.120269\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.46666 1.05206
\(28\) 0 0
\(29\) 0.461096 0.0856233 0.0428117 0.999083i \(-0.486368\pi\)
0.0428117 + 0.999083i \(0.486368\pi\)
\(30\) 0 0
\(31\) −7.96959 −1.43138 −0.715690 0.698418i \(-0.753888\pi\)
−0.715690 + 0.698418i \(0.753888\pi\)
\(32\) 0 0
\(33\) 9.65254 1.68029
\(34\) 0 0
\(35\) −0.974186 −0.164668
\(36\) 0 0
\(37\) 2.95339 0.485535 0.242767 0.970085i \(-0.421945\pi\)
0.242767 + 0.970085i \(0.421945\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.76466 1.36881 0.684405 0.729102i \(-0.260062\pi\)
0.684405 + 0.729102i \(0.260062\pi\)
\(42\) 0 0
\(43\) 8.81154 1.34375 0.671873 0.740666i \(-0.265490\pi\)
0.671873 + 0.740666i \(0.265490\pi\)
\(44\) 0 0
\(45\) −4.94004 −0.736418
\(46\) 0 0
\(47\) 10.3775 1.51372 0.756859 0.653578i \(-0.226733\pi\)
0.756859 + 0.653578i \(0.226733\pi\)
\(48\) 0 0
\(49\) −6.05096 −0.864423
\(50\) 0 0
\(51\) 0.799776 0.111991
\(52\) 0 0
\(53\) 4.96945 0.682606 0.341303 0.939953i \(-0.389132\pi\)
0.341303 + 0.939953i \(0.389132\pi\)
\(54\) 0 0
\(55\) −3.42555 −0.461901
\(56\) 0 0
\(57\) −16.9943 −2.25096
\(58\) 0 0
\(59\) −7.88601 −1.02667 −0.513336 0.858188i \(-0.671590\pi\)
−0.513336 + 0.858188i \(0.671590\pi\)
\(60\) 0 0
\(61\) 11.4869 1.47075 0.735375 0.677661i \(-0.237006\pi\)
0.735375 + 0.677661i \(0.237006\pi\)
\(62\) 0 0
\(63\) 4.81252 0.606320
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.96340 0.484206 0.242103 0.970251i \(-0.422163\pi\)
0.242103 + 0.970251i \(0.422163\pi\)
\(68\) 0 0
\(69\) 25.1211 3.02422
\(70\) 0 0
\(71\) 6.22524 0.738801 0.369400 0.929270i \(-0.379563\pi\)
0.369400 + 0.929270i \(0.379563\pi\)
\(72\) 0 0
\(73\) −9.94784 −1.16431 −0.582153 0.813079i \(-0.697790\pi\)
−0.582153 + 0.813079i \(0.697790\pi\)
\(74\) 0 0
\(75\) 2.81781 0.325372
\(76\) 0 0
\(77\) 3.33712 0.380301
\(78\) 0 0
\(79\) 6.63735 0.746760 0.373380 0.927679i \(-0.378199\pi\)
0.373380 + 0.927679i \(0.378199\pi\)
\(80\) 0 0
\(81\) 0.583871 0.0648745
\(82\) 0 0
\(83\) −7.05049 −0.773891 −0.386946 0.922103i \(-0.626470\pi\)
−0.386946 + 0.922103i \(0.626470\pi\)
\(84\) 0 0
\(85\) −0.283829 −0.0307856
\(86\) 0 0
\(87\) 1.29928 0.139297
\(88\) 0 0
\(89\) −8.66990 −0.919007 −0.459504 0.888176i \(-0.651973\pi\)
−0.459504 + 0.888176i \(0.651973\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −22.4568 −2.32866
\(94\) 0 0
\(95\) 6.03105 0.618773
\(96\) 0 0
\(97\) 0.332881 0.0337990 0.0168995 0.999857i \(-0.494620\pi\)
0.0168995 + 0.999857i \(0.494620\pi\)
\(98\) 0 0
\(99\) 16.9224 1.70076
\(100\) 0 0
\(101\) −14.0197 −1.39501 −0.697506 0.716579i \(-0.745707\pi\)
−0.697506 + 0.716579i \(0.745707\pi\)
\(102\) 0 0
\(103\) −18.6039 −1.83310 −0.916549 0.399923i \(-0.869037\pi\)
−0.916549 + 0.399923i \(0.869037\pi\)
\(104\) 0 0
\(105\) −2.74507 −0.267891
\(106\) 0 0
\(107\) −10.8007 −1.04414 −0.522071 0.852902i \(-0.674840\pi\)
−0.522071 + 0.852902i \(0.674840\pi\)
\(108\) 0 0
\(109\) −2.87817 −0.275679 −0.137839 0.990455i \(-0.544016\pi\)
−0.137839 + 0.990455i \(0.544016\pi\)
\(110\) 0 0
\(111\) 8.32209 0.789898
\(112\) 0 0
\(113\) 9.77628 0.919675 0.459838 0.888003i \(-0.347908\pi\)
0.459838 + 0.888003i \(0.347908\pi\)
\(114\) 0 0
\(115\) −8.91511 −0.831338
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.276502 0.0253469
\(120\) 0 0
\(121\) 0.734394 0.0667631
\(122\) 0 0
\(123\) 24.6971 2.22686
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.3824 1.27623 0.638114 0.769942i \(-0.279715\pi\)
0.638114 + 0.769942i \(0.279715\pi\)
\(128\) 0 0
\(129\) 24.8292 2.18609
\(130\) 0 0
\(131\) 16.9106 1.47749 0.738745 0.673985i \(-0.235419\pi\)
0.738745 + 0.673985i \(0.235419\pi\)
\(132\) 0 0
\(133\) −5.87537 −0.509459
\(134\) 0 0
\(135\) −5.46666 −0.470495
\(136\) 0 0
\(137\) −7.71889 −0.659469 −0.329735 0.944074i \(-0.606959\pi\)
−0.329735 + 0.944074i \(0.606959\pi\)
\(138\) 0 0
\(139\) −18.7958 −1.59424 −0.797120 0.603821i \(-0.793644\pi\)
−0.797120 + 0.603821i \(0.793644\pi\)
\(140\) 0 0
\(141\) 29.2419 2.46261
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.461096 −0.0382919
\(146\) 0 0
\(147\) −17.0504 −1.40630
\(148\) 0 0
\(149\) −3.52113 −0.288462 −0.144231 0.989544i \(-0.546071\pi\)
−0.144231 + 0.989544i \(0.546071\pi\)
\(150\) 0 0
\(151\) 11.0498 0.899217 0.449608 0.893226i \(-0.351564\pi\)
0.449608 + 0.893226i \(0.351564\pi\)
\(152\) 0 0
\(153\) 1.40213 0.113355
\(154\) 0 0
\(155\) 7.96959 0.640133
\(156\) 0 0
\(157\) 4.47826 0.357404 0.178702 0.983903i \(-0.442810\pi\)
0.178702 + 0.983903i \(0.442810\pi\)
\(158\) 0 0
\(159\) 14.0029 1.11051
\(160\) 0 0
\(161\) 8.68498 0.684472
\(162\) 0 0
\(163\) 11.3490 0.888923 0.444462 0.895798i \(-0.353395\pi\)
0.444462 + 0.895798i \(0.353395\pi\)
\(164\) 0 0
\(165\) −9.65254 −0.751449
\(166\) 0 0
\(167\) 8.82022 0.682529 0.341265 0.939967i \(-0.389145\pi\)
0.341265 + 0.939967i \(0.389145\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −29.7936 −2.27838
\(172\) 0 0
\(173\) −4.64718 −0.353319 −0.176659 0.984272i \(-0.556529\pi\)
−0.176659 + 0.984272i \(0.556529\pi\)
\(174\) 0 0
\(175\) 0.974186 0.0736416
\(176\) 0 0
\(177\) −22.2213 −1.67025
\(178\) 0 0
\(179\) −25.3186 −1.89240 −0.946200 0.323581i \(-0.895113\pi\)
−0.946200 + 0.323581i \(0.895113\pi\)
\(180\) 0 0
\(181\) 4.32715 0.321635 0.160817 0.986984i \(-0.448587\pi\)
0.160817 + 0.986984i \(0.448587\pi\)
\(182\) 0 0
\(183\) 32.3679 2.39271
\(184\) 0 0
\(185\) −2.95339 −0.217138
\(186\) 0 0
\(187\) 0.972271 0.0710995
\(188\) 0 0
\(189\) 5.32554 0.387376
\(190\) 0 0
\(191\) −2.62340 −0.189823 −0.0949114 0.995486i \(-0.530257\pi\)
−0.0949114 + 0.995486i \(0.530257\pi\)
\(192\) 0 0
\(193\) 21.6639 1.55940 0.779700 0.626153i \(-0.215371\pi\)
0.779700 + 0.626153i \(0.215371\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.8601 −1.48622 −0.743110 0.669169i \(-0.766650\pi\)
−0.743110 + 0.669169i \(0.766650\pi\)
\(198\) 0 0
\(199\) 22.7819 1.61497 0.807485 0.589888i \(-0.200828\pi\)
0.807485 + 0.589888i \(0.200828\pi\)
\(200\) 0 0
\(201\) 11.1681 0.787736
\(202\) 0 0
\(203\) 0.449193 0.0315272
\(204\) 0 0
\(205\) −8.76466 −0.612150
\(206\) 0 0
\(207\) 44.0410 3.06106
\(208\) 0 0
\(209\) −20.6597 −1.42906
\(210\) 0 0
\(211\) 15.9891 1.10073 0.550366 0.834923i \(-0.314488\pi\)
0.550366 + 0.834923i \(0.314488\pi\)
\(212\) 0 0
\(213\) 17.5415 1.20193
\(214\) 0 0
\(215\) −8.81154 −0.600942
\(216\) 0 0
\(217\) −7.76387 −0.527046
\(218\) 0 0
\(219\) −28.0311 −1.89417
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.54613 0.505326 0.252663 0.967554i \(-0.418694\pi\)
0.252663 + 0.967554i \(0.418694\pi\)
\(224\) 0 0
\(225\) 4.94004 0.329336
\(226\) 0 0
\(227\) 28.4988 1.89153 0.945765 0.324853i \(-0.105315\pi\)
0.945765 + 0.324853i \(0.105315\pi\)
\(228\) 0 0
\(229\) 19.3902 1.28134 0.640671 0.767815i \(-0.278656\pi\)
0.640671 + 0.767815i \(0.278656\pi\)
\(230\) 0 0
\(231\) 9.40337 0.618696
\(232\) 0 0
\(233\) −12.7622 −0.836080 −0.418040 0.908429i \(-0.637283\pi\)
−0.418040 + 0.908429i \(0.637283\pi\)
\(234\) 0 0
\(235\) −10.3775 −0.676955
\(236\) 0 0
\(237\) 18.7028 1.21488
\(238\) 0 0
\(239\) 11.1424 0.720741 0.360371 0.932809i \(-0.382650\pi\)
0.360371 + 0.932809i \(0.382650\pi\)
\(240\) 0 0
\(241\) 5.48277 0.353176 0.176588 0.984285i \(-0.443494\pi\)
0.176588 + 0.984285i \(0.443494\pi\)
\(242\) 0 0
\(243\) −14.7547 −0.946517
\(244\) 0 0
\(245\) 6.05096 0.386582
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −19.8669 −1.25901
\(250\) 0 0
\(251\) −24.4904 −1.54582 −0.772910 0.634516i \(-0.781199\pi\)
−0.772910 + 0.634516i \(0.781199\pi\)
\(252\) 0 0
\(253\) 30.5392 1.91998
\(254\) 0 0
\(255\) −0.799776 −0.0500839
\(256\) 0 0
\(257\) −14.7828 −0.922128 −0.461064 0.887367i \(-0.652532\pi\)
−0.461064 + 0.887367i \(0.652532\pi\)
\(258\) 0 0
\(259\) 2.87715 0.178778
\(260\) 0 0
\(261\) 2.27783 0.140994
\(262\) 0 0
\(263\) 3.01732 0.186056 0.0930279 0.995664i \(-0.470345\pi\)
0.0930279 + 0.995664i \(0.470345\pi\)
\(264\) 0 0
\(265\) −4.96945 −0.305271
\(266\) 0 0
\(267\) −24.4301 −1.49510
\(268\) 0 0
\(269\) −8.46709 −0.516248 −0.258124 0.966112i \(-0.583104\pi\)
−0.258124 + 0.966112i \(0.583104\pi\)
\(270\) 0 0
\(271\) −25.7542 −1.56445 −0.782227 0.622994i \(-0.785916\pi\)
−0.782227 + 0.622994i \(0.785916\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.42555 0.206568
\(276\) 0 0
\(277\) 14.6848 0.882322 0.441161 0.897428i \(-0.354567\pi\)
0.441161 + 0.897428i \(0.354567\pi\)
\(278\) 0 0
\(279\) −39.3701 −2.35703
\(280\) 0 0
\(281\) −25.1933 −1.50290 −0.751452 0.659788i \(-0.770646\pi\)
−0.751452 + 0.659788i \(0.770646\pi\)
\(282\) 0 0
\(283\) −19.3058 −1.14761 −0.573807 0.818991i \(-0.694534\pi\)
−0.573807 + 0.818991i \(0.694534\pi\)
\(284\) 0 0
\(285\) 16.9943 1.00666
\(286\) 0 0
\(287\) 8.53841 0.504006
\(288\) 0 0
\(289\) −16.9194 −0.995261
\(290\) 0 0
\(291\) 0.937995 0.0549862
\(292\) 0 0
\(293\) −1.23340 −0.0720561 −0.0360281 0.999351i \(-0.511471\pi\)
−0.0360281 + 0.999351i \(0.511471\pi\)
\(294\) 0 0
\(295\) 7.88601 0.459141
\(296\) 0 0
\(297\) 18.7263 1.08661
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.58408 0.494778
\(302\) 0 0
\(303\) −39.5048 −2.26949
\(304\) 0 0
\(305\) −11.4869 −0.657739
\(306\) 0 0
\(307\) −7.36503 −0.420344 −0.210172 0.977664i \(-0.567402\pi\)
−0.210172 + 0.977664i \(0.567402\pi\)
\(308\) 0 0
\(309\) −52.4222 −2.98220
\(310\) 0 0
\(311\) −7.77672 −0.440977 −0.220489 0.975390i \(-0.570765\pi\)
−0.220489 + 0.975390i \(0.570765\pi\)
\(312\) 0 0
\(313\) 0.104459 0.00590438 0.00295219 0.999996i \(-0.499060\pi\)
0.00295219 + 0.999996i \(0.499060\pi\)
\(314\) 0 0
\(315\) −4.81252 −0.271155
\(316\) 0 0
\(317\) −32.3972 −1.81961 −0.909803 0.415040i \(-0.863768\pi\)
−0.909803 + 0.415040i \(0.863768\pi\)
\(318\) 0 0
\(319\) 1.57951 0.0884354
\(320\) 0 0
\(321\) −30.4342 −1.69867
\(322\) 0 0
\(323\) −1.71179 −0.0952464
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −8.11013 −0.448491
\(328\) 0 0
\(329\) 10.1096 0.557363
\(330\) 0 0
\(331\) 19.1973 1.05518 0.527589 0.849500i \(-0.323096\pi\)
0.527589 + 0.849500i \(0.323096\pi\)
\(332\) 0 0
\(333\) 14.5899 0.799520
\(334\) 0 0
\(335\) −3.96340 −0.216543
\(336\) 0 0
\(337\) −21.6042 −1.17686 −0.588429 0.808549i \(-0.700253\pi\)
−0.588429 + 0.808549i \(0.700253\pi\)
\(338\) 0 0
\(339\) 27.5477 1.49618
\(340\) 0 0
\(341\) −27.3002 −1.47839
\(342\) 0 0
\(343\) −12.7141 −0.686495
\(344\) 0 0
\(345\) −25.1211 −1.35247
\(346\) 0 0
\(347\) −12.4830 −0.670120 −0.335060 0.942197i \(-0.608757\pi\)
−0.335060 + 0.942197i \(0.608757\pi\)
\(348\) 0 0
\(349\) −26.1232 −1.39834 −0.699171 0.714954i \(-0.746448\pi\)
−0.699171 + 0.714954i \(0.746448\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.7057 1.04883 0.524413 0.851464i \(-0.324285\pi\)
0.524413 + 0.851464i \(0.324285\pi\)
\(354\) 0 0
\(355\) −6.22524 −0.330402
\(356\) 0 0
\(357\) 0.779130 0.0412360
\(358\) 0 0
\(359\) 6.56325 0.346395 0.173198 0.984887i \(-0.444590\pi\)
0.173198 + 0.984887i \(0.444590\pi\)
\(360\) 0 0
\(361\) 17.3736 0.914399
\(362\) 0 0
\(363\) 2.06938 0.108614
\(364\) 0 0
\(365\) 9.94784 0.520694
\(366\) 0 0
\(367\) −33.1532 −1.73058 −0.865291 0.501270i \(-0.832866\pi\)
−0.865291 + 0.501270i \(0.832866\pi\)
\(368\) 0 0
\(369\) 43.2978 2.25399
\(370\) 0 0
\(371\) 4.84117 0.251341
\(372\) 0 0
\(373\) 30.0243 1.55460 0.777300 0.629130i \(-0.216589\pi\)
0.777300 + 0.629130i \(0.216589\pi\)
\(374\) 0 0
\(375\) −2.81781 −0.145511
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11.1570 0.573097 0.286549 0.958066i \(-0.407492\pi\)
0.286549 + 0.958066i \(0.407492\pi\)
\(380\) 0 0
\(381\) 40.5267 2.07625
\(382\) 0 0
\(383\) 11.5707 0.591233 0.295616 0.955307i \(-0.404475\pi\)
0.295616 + 0.955307i \(0.404475\pi\)
\(384\) 0 0
\(385\) −3.33712 −0.170076
\(386\) 0 0
\(387\) 43.5293 2.21272
\(388\) 0 0
\(389\) −18.6724 −0.946727 −0.473363 0.880867i \(-0.656960\pi\)
−0.473363 + 0.880867i \(0.656960\pi\)
\(390\) 0 0
\(391\) 2.53037 0.127966
\(392\) 0 0
\(393\) 47.6509 2.40367
\(394\) 0 0
\(395\) −6.63735 −0.333961
\(396\) 0 0
\(397\) −14.7838 −0.741977 −0.370988 0.928638i \(-0.620981\pi\)
−0.370988 + 0.928638i \(0.620981\pi\)
\(398\) 0 0
\(399\) −16.5557 −0.828819
\(400\) 0 0
\(401\) 13.5374 0.676025 0.338012 0.941142i \(-0.390245\pi\)
0.338012 + 0.941142i \(0.390245\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.583871 −0.0290128
\(406\) 0 0
\(407\) 10.1170 0.501481
\(408\) 0 0
\(409\) −9.79211 −0.484189 −0.242094 0.970253i \(-0.577834\pi\)
−0.242094 + 0.970253i \(0.577834\pi\)
\(410\) 0 0
\(411\) −21.7504 −1.07287
\(412\) 0 0
\(413\) −7.68244 −0.378028
\(414\) 0 0
\(415\) 7.05049 0.346095
\(416\) 0 0
\(417\) −52.9630 −2.59361
\(418\) 0 0
\(419\) −19.3450 −0.945065 −0.472533 0.881313i \(-0.656660\pi\)
−0.472533 + 0.881313i \(0.656660\pi\)
\(420\) 0 0
\(421\) −10.8582 −0.529199 −0.264599 0.964358i \(-0.585240\pi\)
−0.264599 + 0.964358i \(0.585240\pi\)
\(422\) 0 0
\(423\) 51.2654 2.49261
\(424\) 0 0
\(425\) 0.283829 0.0137677
\(426\) 0 0
\(427\) 11.1904 0.541541
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.47538 −0.408245 −0.204122 0.978945i \(-0.565434\pi\)
−0.204122 + 0.978945i \(0.565434\pi\)
\(432\) 0 0
\(433\) −14.6325 −0.703193 −0.351596 0.936152i \(-0.614361\pi\)
−0.351596 + 0.936152i \(0.614361\pi\)
\(434\) 0 0
\(435\) −1.29928 −0.0622957
\(436\) 0 0
\(437\) −53.7675 −2.57205
\(438\) 0 0
\(439\) 10.2299 0.488245 0.244123 0.969744i \(-0.421500\pi\)
0.244123 + 0.969744i \(0.421500\pi\)
\(440\) 0 0
\(441\) −29.8920 −1.42343
\(442\) 0 0
\(443\) 1.89017 0.0898045 0.0449023 0.998991i \(-0.485702\pi\)
0.0449023 + 0.998991i \(0.485702\pi\)
\(444\) 0 0
\(445\) 8.66990 0.410993
\(446\) 0 0
\(447\) −9.92185 −0.469288
\(448\) 0 0
\(449\) −10.6031 −0.500393 −0.250196 0.968195i \(-0.580495\pi\)
−0.250196 + 0.968195i \(0.580495\pi\)
\(450\) 0 0
\(451\) 30.0238 1.41376
\(452\) 0 0
\(453\) 31.1361 1.46290
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.2158 1.31988 0.659939 0.751319i \(-0.270582\pi\)
0.659939 + 0.751319i \(0.270582\pi\)
\(458\) 0 0
\(459\) 1.55160 0.0724223
\(460\) 0 0
\(461\) 14.1918 0.660976 0.330488 0.943810i \(-0.392787\pi\)
0.330488 + 0.943810i \(0.392787\pi\)
\(462\) 0 0
\(463\) 18.9826 0.882195 0.441098 0.897459i \(-0.354589\pi\)
0.441098 + 0.897459i \(0.354589\pi\)
\(464\) 0 0
\(465\) 22.4568 1.04141
\(466\) 0 0
\(467\) −9.88823 −0.457573 −0.228786 0.973477i \(-0.573476\pi\)
−0.228786 + 0.973477i \(0.573476\pi\)
\(468\) 0 0
\(469\) 3.86109 0.178288
\(470\) 0 0
\(471\) 12.6189 0.581447
\(472\) 0 0
\(473\) 30.1844 1.38788
\(474\) 0 0
\(475\) −6.03105 −0.276724
\(476\) 0 0
\(477\) 24.5493 1.12403
\(478\) 0 0
\(479\) −26.4914 −1.21042 −0.605211 0.796065i \(-0.706911\pi\)
−0.605211 + 0.796065i \(0.706911\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 24.4726 1.11354
\(484\) 0 0
\(485\) −0.332881 −0.0151154
\(486\) 0 0
\(487\) −34.8349 −1.57852 −0.789261 0.614058i \(-0.789536\pi\)
−0.789261 + 0.614058i \(0.789536\pi\)
\(488\) 0 0
\(489\) 31.9793 1.44616
\(490\) 0 0
\(491\) −0.113245 −0.00511068 −0.00255534 0.999997i \(-0.500813\pi\)
−0.00255534 + 0.999997i \(0.500813\pi\)
\(492\) 0 0
\(493\) 0.130872 0.00589420
\(494\) 0 0
\(495\) −16.9224 −0.760603
\(496\) 0 0
\(497\) 6.06455 0.272032
\(498\) 0 0
\(499\) −7.69924 −0.344665 −0.172333 0.985039i \(-0.555130\pi\)
−0.172333 + 0.985039i \(0.555130\pi\)
\(500\) 0 0
\(501\) 24.8537 1.11038
\(502\) 0 0
\(503\) −1.09165 −0.0486742 −0.0243371 0.999704i \(-0.507748\pi\)
−0.0243371 + 0.999704i \(0.507748\pi\)
\(504\) 0 0
\(505\) 14.0197 0.623869
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.78054 −0.167569 −0.0837847 0.996484i \(-0.526701\pi\)
−0.0837847 + 0.996484i \(0.526701\pi\)
\(510\) 0 0
\(511\) −9.69105 −0.428707
\(512\) 0 0
\(513\) −32.9697 −1.45565
\(514\) 0 0
\(515\) 18.6039 0.819786
\(516\) 0 0
\(517\) 35.5487 1.56343
\(518\) 0 0
\(519\) −13.0949 −0.574800
\(520\) 0 0
\(521\) −8.70631 −0.381430 −0.190715 0.981645i \(-0.561081\pi\)
−0.190715 + 0.981645i \(0.561081\pi\)
\(522\) 0 0
\(523\) −11.8023 −0.516077 −0.258038 0.966135i \(-0.583076\pi\)
−0.258038 + 0.966135i \(0.583076\pi\)
\(524\) 0 0
\(525\) 2.74507 0.119805
\(526\) 0 0
\(527\) −2.26200 −0.0985343
\(528\) 0 0
\(529\) 56.4792 2.45562
\(530\) 0 0
\(531\) −38.9572 −1.69060
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.8007 0.466954
\(536\) 0 0
\(537\) −71.3429 −3.07867
\(538\) 0 0
\(539\) −20.7279 −0.892813
\(540\) 0 0
\(541\) 21.0010 0.902902 0.451451 0.892296i \(-0.350907\pi\)
0.451451 + 0.892296i \(0.350907\pi\)
\(542\) 0 0
\(543\) 12.1931 0.523255
\(544\) 0 0
\(545\) 2.87817 0.123287
\(546\) 0 0
\(547\) −19.6759 −0.841282 −0.420641 0.907227i \(-0.638195\pi\)
−0.420641 + 0.907227i \(0.638195\pi\)
\(548\) 0 0
\(549\) 56.7458 2.42185
\(550\) 0 0
\(551\) −2.78089 −0.118470
\(552\) 0 0
\(553\) 6.46601 0.274963
\(554\) 0 0
\(555\) −8.32209 −0.353253
\(556\) 0 0
\(557\) −21.5516 −0.913172 −0.456586 0.889679i \(-0.650928\pi\)
−0.456586 + 0.889679i \(0.650928\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2.73967 0.115669
\(562\) 0 0
\(563\) 10.6586 0.449205 0.224602 0.974450i \(-0.427892\pi\)
0.224602 + 0.974450i \(0.427892\pi\)
\(564\) 0 0
\(565\) −9.77628 −0.411291
\(566\) 0 0
\(567\) 0.568799 0.0238873
\(568\) 0 0
\(569\) 27.5445 1.15473 0.577363 0.816488i \(-0.304082\pi\)
0.577363 + 0.816488i \(0.304082\pi\)
\(570\) 0 0
\(571\) −42.1038 −1.76199 −0.880994 0.473127i \(-0.843125\pi\)
−0.880994 + 0.473127i \(0.843125\pi\)
\(572\) 0 0
\(573\) −7.39224 −0.308815
\(574\) 0 0
\(575\) 8.91511 0.371786
\(576\) 0 0
\(577\) −33.7205 −1.40380 −0.701901 0.712274i \(-0.747665\pi\)
−0.701901 + 0.712274i \(0.747665\pi\)
\(578\) 0 0
\(579\) 61.0447 2.53693
\(580\) 0 0
\(581\) −6.86849 −0.284953
\(582\) 0 0
\(583\) 17.0231 0.705024
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.3282 −0.550114 −0.275057 0.961428i \(-0.588697\pi\)
−0.275057 + 0.961428i \(0.588697\pi\)
\(588\) 0 0
\(589\) 48.0650 1.98048
\(590\) 0 0
\(591\) −58.7797 −2.41788
\(592\) 0 0
\(593\) 1.07378 0.0440949 0.0220474 0.999757i \(-0.492982\pi\)
0.0220474 + 0.999757i \(0.492982\pi\)
\(594\) 0 0
\(595\) −0.276502 −0.0113355
\(596\) 0 0
\(597\) 64.1951 2.62733
\(598\) 0 0
\(599\) −18.3111 −0.748169 −0.374085 0.927395i \(-0.622043\pi\)
−0.374085 + 0.927395i \(0.622043\pi\)
\(600\) 0 0
\(601\) −1.14497 −0.0467043 −0.0233521 0.999727i \(-0.507434\pi\)
−0.0233521 + 0.999727i \(0.507434\pi\)
\(602\) 0 0
\(603\) 19.5793 0.797332
\(604\) 0 0
\(605\) −0.734394 −0.0298573
\(606\) 0 0
\(607\) −11.3920 −0.462385 −0.231193 0.972908i \(-0.574263\pi\)
−0.231193 + 0.972908i \(0.574263\pi\)
\(608\) 0 0
\(609\) 1.26574 0.0512904
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.51518 −0.263146 −0.131573 0.991306i \(-0.542003\pi\)
−0.131573 + 0.991306i \(0.542003\pi\)
\(614\) 0 0
\(615\) −24.6971 −0.995884
\(616\) 0 0
\(617\) 4.43338 0.178481 0.0892405 0.996010i \(-0.471556\pi\)
0.0892405 + 0.996010i \(0.471556\pi\)
\(618\) 0 0
\(619\) −2.32341 −0.0933857 −0.0466929 0.998909i \(-0.514868\pi\)
−0.0466929 + 0.998909i \(0.514868\pi\)
\(620\) 0 0
\(621\) 48.7359 1.95570
\(622\) 0 0
\(623\) −8.44610 −0.338386
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −58.2150 −2.32488
\(628\) 0 0
\(629\) 0.838259 0.0334236
\(630\) 0 0
\(631\) −25.4463 −1.01300 −0.506501 0.862239i \(-0.669061\pi\)
−0.506501 + 0.862239i \(0.669061\pi\)
\(632\) 0 0
\(633\) 45.0541 1.79074
\(634\) 0 0
\(635\) −14.3824 −0.570746
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 30.7530 1.21657
\(640\) 0 0
\(641\) 9.30285 0.367440 0.183720 0.982979i \(-0.441186\pi\)
0.183720 + 0.982979i \(0.441186\pi\)
\(642\) 0 0
\(643\) −49.3771 −1.94724 −0.973621 0.228172i \(-0.926725\pi\)
−0.973621 + 0.228172i \(0.926725\pi\)
\(644\) 0 0
\(645\) −24.8292 −0.977649
\(646\) 0 0
\(647\) 26.1654 1.02867 0.514334 0.857590i \(-0.328039\pi\)
0.514334 + 0.857590i \(0.328039\pi\)
\(648\) 0 0
\(649\) −27.0139 −1.06039
\(650\) 0 0
\(651\) −21.8771 −0.857430
\(652\) 0 0
\(653\) −12.5177 −0.489857 −0.244929 0.969541i \(-0.578765\pi\)
−0.244929 + 0.969541i \(0.578765\pi\)
\(654\) 0 0
\(655\) −16.9106 −0.660753
\(656\) 0 0
\(657\) −49.1427 −1.91724
\(658\) 0 0
\(659\) −9.30544 −0.362488 −0.181244 0.983438i \(-0.558012\pi\)
−0.181244 + 0.983438i \(0.558012\pi\)
\(660\) 0 0
\(661\) 0.778487 0.0302796 0.0151398 0.999885i \(-0.495181\pi\)
0.0151398 + 0.999885i \(0.495181\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.87537 0.227837
\(666\) 0 0
\(667\) 4.11072 0.159168
\(668\) 0 0
\(669\) 21.2635 0.822096
\(670\) 0 0
\(671\) 39.3490 1.51905
\(672\) 0 0
\(673\) 18.6069 0.717243 0.358621 0.933483i \(-0.383247\pi\)
0.358621 + 0.933483i \(0.383247\pi\)
\(674\) 0 0
\(675\) 5.46666 0.210412
\(676\) 0 0
\(677\) 0.451238 0.0173425 0.00867125 0.999962i \(-0.497240\pi\)
0.00867125 + 0.999962i \(0.497240\pi\)
\(678\) 0 0
\(679\) 0.324288 0.0124450
\(680\) 0 0
\(681\) 80.3040 3.07726
\(682\) 0 0
\(683\) −35.3929 −1.35427 −0.677136 0.735858i \(-0.736779\pi\)
−0.677136 + 0.735858i \(0.736779\pi\)
\(684\) 0 0
\(685\) 7.71889 0.294924
\(686\) 0 0
\(687\) 54.6379 2.08457
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 39.0369 1.48503 0.742516 0.669828i \(-0.233632\pi\)
0.742516 + 0.669828i \(0.233632\pi\)
\(692\) 0 0
\(693\) 16.4855 0.626233
\(694\) 0 0
\(695\) 18.7958 0.712966
\(696\) 0 0
\(697\) 2.48767 0.0942270
\(698\) 0 0
\(699\) −35.9614 −1.36019
\(700\) 0 0
\(701\) 18.0395 0.681344 0.340672 0.940182i \(-0.389345\pi\)
0.340672 + 0.940182i \(0.389345\pi\)
\(702\) 0 0
\(703\) −17.8121 −0.671795
\(704\) 0 0
\(705\) −29.2419 −1.10131
\(706\) 0 0
\(707\) −13.6578 −0.513654
\(708\) 0 0
\(709\) 19.7805 0.742871 0.371436 0.928459i \(-0.378866\pi\)
0.371436 + 0.928459i \(0.378866\pi\)
\(710\) 0 0
\(711\) 32.7888 1.22967
\(712\) 0 0
\(713\) −71.0498 −2.66084
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 31.3971 1.17255
\(718\) 0 0
\(719\) −10.2849 −0.383562 −0.191781 0.981438i \(-0.561426\pi\)
−0.191781 + 0.981438i \(0.561426\pi\)
\(720\) 0 0
\(721\) −18.1237 −0.674961
\(722\) 0 0
\(723\) 15.4494 0.574569
\(724\) 0 0
\(725\) 0.461096 0.0171247
\(726\) 0 0
\(727\) −7.08016 −0.262589 −0.131294 0.991343i \(-0.541913\pi\)
−0.131294 + 0.991343i \(0.541913\pi\)
\(728\) 0 0
\(729\) −43.3276 −1.60473
\(730\) 0 0
\(731\) 2.50097 0.0925017
\(732\) 0 0
\(733\) −6.29532 −0.232523 −0.116261 0.993219i \(-0.537091\pi\)
−0.116261 + 0.993219i \(0.537091\pi\)
\(734\) 0 0
\(735\) 17.0504 0.628915
\(736\) 0 0
\(737\) 13.5768 0.500108
\(738\) 0 0
\(739\) 33.4301 1.22975 0.614873 0.788626i \(-0.289207\pi\)
0.614873 + 0.788626i \(0.289207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.84274 0.140976 0.0704882 0.997513i \(-0.477544\pi\)
0.0704882 + 0.997513i \(0.477544\pi\)
\(744\) 0 0
\(745\) 3.52113 0.129004
\(746\) 0 0
\(747\) −34.8297 −1.27435
\(748\) 0 0
\(749\) −10.5219 −0.384461
\(750\) 0 0
\(751\) −34.7285 −1.26726 −0.633630 0.773636i \(-0.718436\pi\)
−0.633630 + 0.773636i \(0.718436\pi\)
\(752\) 0 0
\(753\) −69.0092 −2.51483
\(754\) 0 0
\(755\) −11.0498 −0.402142
\(756\) 0 0
\(757\) 8.10935 0.294739 0.147370 0.989081i \(-0.452919\pi\)
0.147370 + 0.989081i \(0.452919\pi\)
\(758\) 0 0
\(759\) 86.0535 3.12354
\(760\) 0 0
\(761\) −10.4717 −0.379598 −0.189799 0.981823i \(-0.560784\pi\)
−0.189799 + 0.981823i \(0.560784\pi\)
\(762\) 0 0
\(763\) −2.80388 −0.101507
\(764\) 0 0
\(765\) −1.40213 −0.0506940
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 44.2939 1.59728 0.798639 0.601810i \(-0.205554\pi\)
0.798639 + 0.601810i \(0.205554\pi\)
\(770\) 0 0
\(771\) −41.6552 −1.50017
\(772\) 0 0
\(773\) 12.2760 0.441537 0.220769 0.975326i \(-0.429143\pi\)
0.220769 + 0.975326i \(0.429143\pi\)
\(774\) 0 0
\(775\) −7.96959 −0.286276
\(776\) 0 0
\(777\) 8.10727 0.290847
\(778\) 0 0
\(779\) −52.8601 −1.89391
\(780\) 0 0
\(781\) 21.3249 0.763064
\(782\) 0 0
\(783\) 2.52065 0.0900808
\(784\) 0 0
\(785\) −4.47826 −0.159836
\(786\) 0 0
\(787\) 7.08241 0.252461 0.126230 0.992001i \(-0.459712\pi\)
0.126230 + 0.992001i \(0.459712\pi\)
\(788\) 0 0
\(789\) 8.50222 0.302687
\(790\) 0 0
\(791\) 9.52392 0.338632
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −14.0029 −0.496633
\(796\) 0 0
\(797\) −19.6799 −0.697099 −0.348550 0.937290i \(-0.613326\pi\)
−0.348550 + 0.937290i \(0.613326\pi\)
\(798\) 0 0
\(799\) 2.94544 0.104202
\(800\) 0 0
\(801\) −42.8296 −1.51331
\(802\) 0 0
\(803\) −34.0768 −1.20254
\(804\) 0 0
\(805\) −8.68498 −0.306105
\(806\) 0 0
\(807\) −23.8586 −0.839864
\(808\) 0 0
\(809\) 26.3066 0.924890 0.462445 0.886648i \(-0.346972\pi\)
0.462445 + 0.886648i \(0.346972\pi\)
\(810\) 0 0
\(811\) −52.2096 −1.83333 −0.916663 0.399662i \(-0.869128\pi\)
−0.916663 + 0.399662i \(0.869128\pi\)
\(812\) 0 0
\(813\) −72.5702 −2.54515
\(814\) 0 0
\(815\) −11.3490 −0.397539
\(816\) 0 0
\(817\) −53.1428 −1.85923
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 48.9551 1.70855 0.854273 0.519825i \(-0.174003\pi\)
0.854273 + 0.519825i \(0.174003\pi\)
\(822\) 0 0
\(823\) 28.3073 0.986731 0.493366 0.869822i \(-0.335767\pi\)
0.493366 + 0.869822i \(0.335767\pi\)
\(824\) 0 0
\(825\) 9.65254 0.336058
\(826\) 0 0
\(827\) −11.1223 −0.386759 −0.193380 0.981124i \(-0.561945\pi\)
−0.193380 + 0.981124i \(0.561945\pi\)
\(828\) 0 0
\(829\) −1.14566 −0.0397905 −0.0198952 0.999802i \(-0.506333\pi\)
−0.0198952 + 0.999802i \(0.506333\pi\)
\(830\) 0 0
\(831\) 41.3788 1.43542
\(832\) 0 0
\(833\) −1.71744 −0.0595057
\(834\) 0 0
\(835\) −8.82022 −0.305236
\(836\) 0 0
\(837\) −43.5670 −1.50590
\(838\) 0 0
\(839\) 28.3167 0.977602 0.488801 0.872395i \(-0.337434\pi\)
0.488801 + 0.872395i \(0.337434\pi\)
\(840\) 0 0
\(841\) −28.7874 −0.992669
\(842\) 0 0
\(843\) −70.9898 −2.44502
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.715436 0.0245827
\(848\) 0 0
\(849\) −54.4002 −1.86701
\(850\) 0 0
\(851\) 26.3298 0.902574
\(852\) 0 0
\(853\) −29.6180 −1.01410 −0.507051 0.861916i \(-0.669264\pi\)
−0.507051 + 0.861916i \(0.669264\pi\)
\(854\) 0 0
\(855\) 29.7936 1.01892
\(856\) 0 0
\(857\) 14.3690 0.490835 0.245417 0.969418i \(-0.421075\pi\)
0.245417 + 0.969418i \(0.421075\pi\)
\(858\) 0 0
\(859\) −9.17850 −0.313166 −0.156583 0.987665i \(-0.550048\pi\)
−0.156583 + 0.987665i \(0.550048\pi\)
\(860\) 0 0
\(861\) 24.0596 0.819949
\(862\) 0 0
\(863\) 32.1458 1.09426 0.547128 0.837049i \(-0.315721\pi\)
0.547128 + 0.837049i \(0.315721\pi\)
\(864\) 0 0
\(865\) 4.64718 0.158009
\(866\) 0 0
\(867\) −47.6757 −1.61915
\(868\) 0 0
\(869\) 22.7366 0.771285
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.64445 0.0556561
\(874\) 0 0
\(875\) −0.974186 −0.0329335
\(876\) 0 0
\(877\) 25.9633 0.876720 0.438360 0.898800i \(-0.355560\pi\)
0.438360 + 0.898800i \(0.355560\pi\)
\(878\) 0 0
\(879\) −3.47549 −0.117225
\(880\) 0 0
\(881\) −17.1165 −0.576671 −0.288335 0.957529i \(-0.593102\pi\)
−0.288335 + 0.957529i \(0.593102\pi\)
\(882\) 0 0
\(883\) −18.3273 −0.616762 −0.308381 0.951263i \(-0.599787\pi\)
−0.308381 + 0.951263i \(0.599787\pi\)
\(884\) 0 0
\(885\) 22.2213 0.746960
\(886\) 0 0
\(887\) −36.2866 −1.21838 −0.609192 0.793023i \(-0.708506\pi\)
−0.609192 + 0.793023i \(0.708506\pi\)
\(888\) 0 0
\(889\) 14.0111 0.469917
\(890\) 0 0
\(891\) 2.00008 0.0670051
\(892\) 0 0
\(893\) −62.5874 −2.09441
\(894\) 0 0
\(895\) 25.3186 0.846307
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.67475 −0.122560
\(900\) 0 0
\(901\) 1.41047 0.0469897
\(902\) 0 0
\(903\) 24.1883 0.804935
\(904\) 0 0
\(905\) −4.32715 −0.143839
\(906\) 0 0
\(907\) 10.7409 0.356645 0.178323 0.983972i \(-0.442933\pi\)
0.178323 + 0.983972i \(0.442933\pi\)
\(908\) 0 0
\(909\) −69.2579 −2.29714
\(910\) 0 0
\(911\) −31.8649 −1.05573 −0.527866 0.849327i \(-0.677008\pi\)
−0.527866 + 0.849327i \(0.677008\pi\)
\(912\) 0 0
\(913\) −24.1518 −0.799308
\(914\) 0 0
\(915\) −32.3679 −1.07005
\(916\) 0 0
\(917\) 16.4741 0.544023
\(918\) 0 0
\(919\) 14.6467 0.483149 0.241574 0.970382i \(-0.422336\pi\)
0.241574 + 0.970382i \(0.422336\pi\)
\(920\) 0 0
\(921\) −20.7532 −0.683842
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.95339 0.0971069
\(926\) 0 0
\(927\) −91.9040 −3.01852
\(928\) 0 0
\(929\) 56.9334 1.86792 0.933962 0.357373i \(-0.116328\pi\)
0.933962 + 0.357373i \(0.116328\pi\)
\(930\) 0 0
\(931\) 36.4937 1.19603
\(932\) 0 0
\(933\) −21.9133 −0.717409
\(934\) 0 0
\(935\) −0.972271 −0.0317967
\(936\) 0 0
\(937\) 3.76516 0.123002 0.0615012 0.998107i \(-0.480411\pi\)
0.0615012 + 0.998107i \(0.480411\pi\)
\(938\) 0 0
\(939\) 0.294346 0.00960562
\(940\) 0 0
\(941\) 28.9802 0.944728 0.472364 0.881404i \(-0.343401\pi\)
0.472364 + 0.881404i \(0.343401\pi\)
\(942\) 0 0
\(943\) 78.1379 2.54452
\(944\) 0 0
\(945\) −5.32554 −0.173240
\(946\) 0 0
\(947\) 9.77047 0.317498 0.158749 0.987319i \(-0.449254\pi\)
0.158749 + 0.987319i \(0.449254\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −91.2890 −2.96025
\(952\) 0 0
\(953\) 42.3749 1.37266 0.686330 0.727291i \(-0.259221\pi\)
0.686330 + 0.727291i \(0.259221\pi\)
\(954\) 0 0
\(955\) 2.62340 0.0848913
\(956\) 0 0
\(957\) 4.45075 0.143872
\(958\) 0 0
\(959\) −7.51964 −0.242822
\(960\) 0 0
\(961\) 32.5144 1.04885
\(962\) 0 0
\(963\) −53.3558 −1.71937
\(964\) 0 0
\(965\) −21.6639 −0.697385
\(966\) 0 0
\(967\) 5.65301 0.181788 0.0908942 0.995861i \(-0.471027\pi\)
0.0908942 + 0.995861i \(0.471027\pi\)
\(968\) 0 0
\(969\) −4.82349 −0.154953
\(970\) 0 0
\(971\) 40.0547 1.28542 0.642709 0.766111i \(-0.277811\pi\)
0.642709 + 0.766111i \(0.277811\pi\)
\(972\) 0 0
\(973\) −18.3106 −0.587012
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −15.7119 −0.502668 −0.251334 0.967900i \(-0.580869\pi\)
−0.251334 + 0.967900i \(0.580869\pi\)
\(978\) 0 0
\(979\) −29.6992 −0.949190
\(980\) 0 0
\(981\) −14.2183 −0.453955
\(982\) 0 0
\(983\) −46.6671 −1.48845 −0.744225 0.667929i \(-0.767181\pi\)
−0.744225 + 0.667929i \(0.767181\pi\)
\(984\) 0 0
\(985\) 20.8601 0.664658
\(986\) 0 0
\(987\) 28.4870 0.906752
\(988\) 0 0
\(989\) 78.5558 2.49793
\(990\) 0 0
\(991\) 1.23358 0.0391859 0.0195930 0.999808i \(-0.493763\pi\)
0.0195930 + 0.999808i \(0.493763\pi\)
\(992\) 0 0
\(993\) 54.0942 1.71663
\(994\) 0 0
\(995\) −22.7819 −0.722236
\(996\) 0 0
\(997\) 32.9285 1.04286 0.521428 0.853295i \(-0.325400\pi\)
0.521428 + 0.853295i \(0.325400\pi\)
\(998\) 0 0
\(999\) 16.1452 0.510811
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 3380.2.a.r.1.9 9
13.5 odd 4 3380.2.f.j.3041.18 18
13.8 odd 4 3380.2.f.j.3041.17 18
13.12 even 2 3380.2.a.s.1.9 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.9 9 1.1 even 1 trivial
3380.2.a.s.1.9 yes 9 13.12 even 2
3380.2.f.j.3041.17 18 13.8 odd 4
3380.2.f.j.3041.18 18 13.5 odd 4