Properties

Label 8001.2.a.r.1.16
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} - 2747 x^{7} + 5821 x^{6} - 158 x^{5} - 3341 x^{4} + 1002 x^{3} + 416 x^{2} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-2.48981\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.48981 q^{2} +4.19916 q^{4} -1.44828 q^{5} -1.00000 q^{7} +5.47548 q^{8} +O(q^{10})\) \(q+2.48981 q^{2} +4.19916 q^{4} -1.44828 q^{5} -1.00000 q^{7} +5.47548 q^{8} -3.60594 q^{10} +3.27519 q^{11} -3.24893 q^{13} -2.48981 q^{14} +5.23459 q^{16} -2.00096 q^{17} -7.49049 q^{19} -6.08155 q^{20} +8.15461 q^{22} +5.11491 q^{23} -2.90249 q^{25} -8.08921 q^{26} -4.19916 q^{28} +2.27043 q^{29} -9.36316 q^{31} +2.08219 q^{32} -4.98200 q^{34} +1.44828 q^{35} +6.96565 q^{37} -18.6499 q^{38} -7.93002 q^{40} -11.0274 q^{41} -7.34145 q^{43} +13.7531 q^{44} +12.7352 q^{46} -3.00580 q^{47} +1.00000 q^{49} -7.22664 q^{50} -13.6427 q^{52} -10.8847 q^{53} -4.74340 q^{55} -5.47548 q^{56} +5.65295 q^{58} +8.25654 q^{59} +9.29811 q^{61} -23.3125 q^{62} -5.28494 q^{64} +4.70535 q^{65} +2.47721 q^{67} -8.40233 q^{68} +3.60594 q^{70} -6.76717 q^{71} +10.4082 q^{73} +17.3432 q^{74} -31.4537 q^{76} -3.27519 q^{77} +6.43106 q^{79} -7.58115 q^{80} -27.4562 q^{82} -15.7272 q^{83} +2.89794 q^{85} -18.2788 q^{86} +17.9333 q^{88} +9.26856 q^{89} +3.24893 q^{91} +21.4783 q^{92} -7.48388 q^{94} +10.8483 q^{95} +3.36147 q^{97} +2.48981 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48981 1.76056 0.880281 0.474453i \(-0.157354\pi\)
0.880281 + 0.474453i \(0.157354\pi\)
\(3\) 0 0
\(4\) 4.19916 2.09958
\(5\) −1.44828 −0.647690 −0.323845 0.946110i \(-0.604976\pi\)
−0.323845 + 0.946110i \(0.604976\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 5.47548 1.93587
\(9\) 0 0
\(10\) −3.60594 −1.14030
\(11\) 3.27519 0.987508 0.493754 0.869602i \(-0.335624\pi\)
0.493754 + 0.869602i \(0.335624\pi\)
\(12\) 0 0
\(13\) −3.24893 −0.901090 −0.450545 0.892754i \(-0.648770\pi\)
−0.450545 + 0.892754i \(0.648770\pi\)
\(14\) −2.48981 −0.665430
\(15\) 0 0
\(16\) 5.23459 1.30865
\(17\) −2.00096 −0.485303 −0.242652 0.970113i \(-0.578017\pi\)
−0.242652 + 0.970113i \(0.578017\pi\)
\(18\) 0 0
\(19\) −7.49049 −1.71844 −0.859218 0.511609i \(-0.829050\pi\)
−0.859218 + 0.511609i \(0.829050\pi\)
\(20\) −6.08155 −1.35988
\(21\) 0 0
\(22\) 8.15461 1.73857
\(23\) 5.11491 1.06653 0.533266 0.845947i \(-0.320964\pi\)
0.533266 + 0.845947i \(0.320964\pi\)
\(24\) 0 0
\(25\) −2.90249 −0.580498
\(26\) −8.08921 −1.58642
\(27\) 0 0
\(28\) −4.19916 −0.793566
\(29\) 2.27043 0.421609 0.210804 0.977528i \(-0.432392\pi\)
0.210804 + 0.977528i \(0.432392\pi\)
\(30\) 0 0
\(31\) −9.36316 −1.68167 −0.840837 0.541288i \(-0.817937\pi\)
−0.840837 + 0.541288i \(0.817937\pi\)
\(32\) 2.08219 0.368082
\(33\) 0 0
\(34\) −4.98200 −0.854407
\(35\) 1.44828 0.244804
\(36\) 0 0
\(37\) 6.96565 1.14515 0.572573 0.819854i \(-0.305945\pi\)
0.572573 + 0.819854i \(0.305945\pi\)
\(38\) −18.6499 −3.02541
\(39\) 0 0
\(40\) −7.93002 −1.25385
\(41\) −11.0274 −1.72219 −0.861097 0.508441i \(-0.830222\pi\)
−0.861097 + 0.508441i \(0.830222\pi\)
\(42\) 0 0
\(43\) −7.34145 −1.11956 −0.559781 0.828641i \(-0.689115\pi\)
−0.559781 + 0.828641i \(0.689115\pi\)
\(44\) 13.7531 2.07335
\(45\) 0 0
\(46\) 12.7352 1.87770
\(47\) −3.00580 −0.438442 −0.219221 0.975675i \(-0.570352\pi\)
−0.219221 + 0.975675i \(0.570352\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −7.22664 −1.02200
\(51\) 0 0
\(52\) −13.6427 −1.89191
\(53\) −10.8847 −1.49513 −0.747565 0.664189i \(-0.768777\pi\)
−0.747565 + 0.664189i \(0.768777\pi\)
\(54\) 0 0
\(55\) −4.74340 −0.639599
\(56\) −5.47548 −0.731692
\(57\) 0 0
\(58\) 5.65295 0.742268
\(59\) 8.25654 1.07491 0.537455 0.843293i \(-0.319386\pi\)
0.537455 + 0.843293i \(0.319386\pi\)
\(60\) 0 0
\(61\) 9.29811 1.19050 0.595250 0.803540i \(-0.297053\pi\)
0.595250 + 0.803540i \(0.297053\pi\)
\(62\) −23.3125 −2.96069
\(63\) 0 0
\(64\) −5.28494 −0.660617
\(65\) 4.70535 0.583627
\(66\) 0 0
\(67\) 2.47721 0.302639 0.151320 0.988485i \(-0.451648\pi\)
0.151320 + 0.988485i \(0.451648\pi\)
\(68\) −8.40233 −1.01893
\(69\) 0 0
\(70\) 3.60594 0.430992
\(71\) −6.76717 −0.803115 −0.401558 0.915834i \(-0.631531\pi\)
−0.401558 + 0.915834i \(0.631531\pi\)
\(72\) 0 0
\(73\) 10.4082 1.21818 0.609092 0.793100i \(-0.291534\pi\)
0.609092 + 0.793100i \(0.291534\pi\)
\(74\) 17.3432 2.01610
\(75\) 0 0
\(76\) −31.4537 −3.60799
\(77\) −3.27519 −0.373243
\(78\) 0 0
\(79\) 6.43106 0.723551 0.361776 0.932265i \(-0.382171\pi\)
0.361776 + 0.932265i \(0.382171\pi\)
\(80\) −7.58115 −0.847598
\(81\) 0 0
\(82\) −27.4562 −3.03203
\(83\) −15.7272 −1.72629 −0.863144 0.504958i \(-0.831508\pi\)
−0.863144 + 0.504958i \(0.831508\pi\)
\(84\) 0 0
\(85\) 2.89794 0.314326
\(86\) −18.2788 −1.97106
\(87\) 0 0
\(88\) 17.9333 1.91169
\(89\) 9.26856 0.982466 0.491233 0.871028i \(-0.336546\pi\)
0.491233 + 0.871028i \(0.336546\pi\)
\(90\) 0 0
\(91\) 3.24893 0.340580
\(92\) 21.4783 2.23927
\(93\) 0 0
\(94\) −7.48388 −0.771903
\(95\) 10.8483 1.11301
\(96\) 0 0
\(97\) 3.36147 0.341305 0.170653 0.985331i \(-0.445412\pi\)
0.170653 + 0.985331i \(0.445412\pi\)
\(98\) 2.48981 0.251509
\(99\) 0 0
\(100\) −12.1880 −1.21880
\(101\) 3.15143 0.313579 0.156789 0.987632i \(-0.449886\pi\)
0.156789 + 0.987632i \(0.449886\pi\)
\(102\) 0 0
\(103\) −7.60402 −0.749246 −0.374623 0.927177i \(-0.622228\pi\)
−0.374623 + 0.927177i \(0.622228\pi\)
\(104\) −17.7894 −1.74440
\(105\) 0 0
\(106\) −27.1008 −2.63227
\(107\) −5.11898 −0.494871 −0.247435 0.968904i \(-0.579588\pi\)
−0.247435 + 0.968904i \(0.579588\pi\)
\(108\) 0 0
\(109\) −1.82377 −0.174686 −0.0873429 0.996178i \(-0.527838\pi\)
−0.0873429 + 0.996178i \(0.527838\pi\)
\(110\) −11.8102 −1.12605
\(111\) 0 0
\(112\) −5.23459 −0.494623
\(113\) 2.11000 0.198492 0.0992462 0.995063i \(-0.468357\pi\)
0.0992462 + 0.995063i \(0.468357\pi\)
\(114\) 0 0
\(115\) −7.40782 −0.690782
\(116\) 9.53390 0.885200
\(117\) 0 0
\(118\) 20.5572 1.89244
\(119\) 2.00096 0.183427
\(120\) 0 0
\(121\) −0.273100 −0.0248272
\(122\) 23.1505 2.09595
\(123\) 0 0
\(124\) −39.3174 −3.53081
\(125\) 11.4450 1.02367
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) −17.3229 −1.53114
\(129\) 0 0
\(130\) 11.7154 1.02751
\(131\) −5.42636 −0.474103 −0.237051 0.971497i \(-0.576181\pi\)
−0.237051 + 0.971497i \(0.576181\pi\)
\(132\) 0 0
\(133\) 7.49049 0.649508
\(134\) 6.16778 0.532815
\(135\) 0 0
\(136\) −10.9562 −0.939486
\(137\) −3.18573 −0.272175 −0.136088 0.990697i \(-0.543453\pi\)
−0.136088 + 0.990697i \(0.543453\pi\)
\(138\) 0 0
\(139\) −1.32574 −0.112448 −0.0562239 0.998418i \(-0.517906\pi\)
−0.0562239 + 0.998418i \(0.517906\pi\)
\(140\) 6.08155 0.513985
\(141\) 0 0
\(142\) −16.8490 −1.41393
\(143\) −10.6409 −0.889834
\(144\) 0 0
\(145\) −3.28822 −0.273072
\(146\) 25.9144 2.14469
\(147\) 0 0
\(148\) 29.2499 2.40432
\(149\) 0.108116 0.00885717 0.00442859 0.999990i \(-0.498590\pi\)
0.00442859 + 0.999990i \(0.498590\pi\)
\(150\) 0 0
\(151\) −14.0462 −1.14306 −0.571531 0.820581i \(-0.693650\pi\)
−0.571531 + 0.820581i \(0.693650\pi\)
\(152\) −41.0140 −3.32668
\(153\) 0 0
\(154\) −8.15461 −0.657117
\(155\) 13.5605 1.08920
\(156\) 0 0
\(157\) −16.9071 −1.34933 −0.674667 0.738122i \(-0.735713\pi\)
−0.674667 + 0.738122i \(0.735713\pi\)
\(158\) 16.0121 1.27386
\(159\) 0 0
\(160\) −3.01559 −0.238403
\(161\) −5.11491 −0.403111
\(162\) 0 0
\(163\) 18.6355 1.45965 0.729824 0.683635i \(-0.239602\pi\)
0.729824 + 0.683635i \(0.239602\pi\)
\(164\) −46.3058 −3.61588
\(165\) 0 0
\(166\) −39.1578 −3.03924
\(167\) 3.24655 0.251226 0.125613 0.992079i \(-0.459910\pi\)
0.125613 + 0.992079i \(0.459910\pi\)
\(168\) 0 0
\(169\) −2.44448 −0.188037
\(170\) 7.21533 0.553391
\(171\) 0 0
\(172\) −30.8279 −2.35061
\(173\) 2.23879 0.170212 0.0851060 0.996372i \(-0.472877\pi\)
0.0851060 + 0.996372i \(0.472877\pi\)
\(174\) 0 0
\(175\) 2.90249 0.219407
\(176\) 17.1443 1.29230
\(177\) 0 0
\(178\) 23.0770 1.72969
\(179\) −7.77245 −0.580940 −0.290470 0.956884i \(-0.593812\pi\)
−0.290470 + 0.956884i \(0.593812\pi\)
\(180\) 0 0
\(181\) 20.1554 1.49814 0.749069 0.662492i \(-0.230501\pi\)
0.749069 + 0.662492i \(0.230501\pi\)
\(182\) 8.08921 0.599612
\(183\) 0 0
\(184\) 28.0066 2.06467
\(185\) −10.0882 −0.741700
\(186\) 0 0
\(187\) −6.55352 −0.479241
\(188\) −12.6218 −0.920542
\(189\) 0 0
\(190\) 27.0103 1.95953
\(191\) −7.90565 −0.572033 −0.286016 0.958225i \(-0.592331\pi\)
−0.286016 + 0.958225i \(0.592331\pi\)
\(192\) 0 0
\(193\) −15.6092 −1.12357 −0.561786 0.827282i \(-0.689886\pi\)
−0.561786 + 0.827282i \(0.689886\pi\)
\(194\) 8.36941 0.600889
\(195\) 0 0
\(196\) 4.19916 0.299940
\(197\) −23.4362 −1.66976 −0.834881 0.550431i \(-0.814464\pi\)
−0.834881 + 0.550431i \(0.814464\pi\)
\(198\) 0 0
\(199\) 17.5614 1.24489 0.622447 0.782662i \(-0.286139\pi\)
0.622447 + 0.782662i \(0.286139\pi\)
\(200\) −15.8925 −1.12377
\(201\) 0 0
\(202\) 7.84646 0.552075
\(203\) −2.27043 −0.159353
\(204\) 0 0
\(205\) 15.9708 1.11545
\(206\) −18.9326 −1.31909
\(207\) 0 0
\(208\) −17.0068 −1.17921
\(209\) −24.5328 −1.69697
\(210\) 0 0
\(211\) 15.1757 1.04474 0.522371 0.852719i \(-0.325048\pi\)
0.522371 + 0.852719i \(0.325048\pi\)
\(212\) −45.7066 −3.13914
\(213\) 0 0
\(214\) −12.7453 −0.871250
\(215\) 10.6325 0.725129
\(216\) 0 0
\(217\) 9.36316 0.635613
\(218\) −4.54085 −0.307545
\(219\) 0 0
\(220\) −19.9183 −1.34289
\(221\) 6.50096 0.437302
\(222\) 0 0
\(223\) 15.5556 1.04168 0.520841 0.853654i \(-0.325619\pi\)
0.520841 + 0.853654i \(0.325619\pi\)
\(224\) −2.08219 −0.139122
\(225\) 0 0
\(226\) 5.25351 0.349458
\(227\) −24.0255 −1.59463 −0.797314 0.603564i \(-0.793747\pi\)
−0.797314 + 0.603564i \(0.793747\pi\)
\(228\) 0 0
\(229\) −25.7078 −1.69882 −0.849409 0.527735i \(-0.823041\pi\)
−0.849409 + 0.527735i \(0.823041\pi\)
\(230\) −18.4441 −1.21617
\(231\) 0 0
\(232\) 12.4317 0.816181
\(233\) −4.47472 −0.293149 −0.146574 0.989200i \(-0.546825\pi\)
−0.146574 + 0.989200i \(0.546825\pi\)
\(234\) 0 0
\(235\) 4.35324 0.283974
\(236\) 34.6705 2.25686
\(237\) 0 0
\(238\) 4.98200 0.322935
\(239\) −3.71050 −0.240012 −0.120006 0.992773i \(-0.538291\pi\)
−0.120006 + 0.992773i \(0.538291\pi\)
\(240\) 0 0
\(241\) −23.5929 −1.51975 −0.759875 0.650069i \(-0.774740\pi\)
−0.759875 + 0.650069i \(0.774740\pi\)
\(242\) −0.679966 −0.0437099
\(243\) 0 0
\(244\) 39.0442 2.49955
\(245\) −1.44828 −0.0925271
\(246\) 0 0
\(247\) 24.3361 1.54847
\(248\) −51.2678 −3.25551
\(249\) 0 0
\(250\) 28.4959 1.80224
\(251\) −16.0972 −1.01604 −0.508022 0.861344i \(-0.669623\pi\)
−0.508022 + 0.861344i \(0.669623\pi\)
\(252\) 0 0
\(253\) 16.7523 1.05321
\(254\) 2.48981 0.156225
\(255\) 0 0
\(256\) −32.5608 −2.03505
\(257\) 15.7376 0.981682 0.490841 0.871249i \(-0.336690\pi\)
0.490841 + 0.871249i \(0.336690\pi\)
\(258\) 0 0
\(259\) −6.96565 −0.432825
\(260\) 19.7585 1.22537
\(261\) 0 0
\(262\) −13.5106 −0.834687
\(263\) −4.31962 −0.266359 −0.133180 0.991092i \(-0.542519\pi\)
−0.133180 + 0.991092i \(0.542519\pi\)
\(264\) 0 0
\(265\) 15.7641 0.968380
\(266\) 18.6499 1.14350
\(267\) 0 0
\(268\) 10.4022 0.635415
\(269\) 30.7930 1.87748 0.938741 0.344623i \(-0.111993\pi\)
0.938741 + 0.344623i \(0.111993\pi\)
\(270\) 0 0
\(271\) 2.75638 0.167438 0.0837190 0.996489i \(-0.473320\pi\)
0.0837190 + 0.996489i \(0.473320\pi\)
\(272\) −10.4742 −0.635091
\(273\) 0 0
\(274\) −7.93186 −0.479181
\(275\) −9.50621 −0.573246
\(276\) 0 0
\(277\) −15.0763 −0.905845 −0.452922 0.891550i \(-0.649619\pi\)
−0.452922 + 0.891550i \(0.649619\pi\)
\(278\) −3.30084 −0.197971
\(279\) 0 0
\(280\) 7.93002 0.473909
\(281\) 4.56741 0.272469 0.136234 0.990677i \(-0.456500\pi\)
0.136234 + 0.990677i \(0.456500\pi\)
\(282\) 0 0
\(283\) 22.2507 1.32267 0.661334 0.750091i \(-0.269991\pi\)
0.661334 + 0.750091i \(0.269991\pi\)
\(284\) −28.4164 −1.68620
\(285\) 0 0
\(286\) −26.4937 −1.56661
\(287\) 11.0274 0.650928
\(288\) 0 0
\(289\) −12.9962 −0.764481
\(290\) −8.18704 −0.480760
\(291\) 0 0
\(292\) 43.7055 2.55767
\(293\) −22.4715 −1.31280 −0.656399 0.754414i \(-0.727921\pi\)
−0.656399 + 0.754414i \(0.727921\pi\)
\(294\) 0 0
\(295\) −11.9578 −0.696208
\(296\) 38.1403 2.21686
\(297\) 0 0
\(298\) 0.269187 0.0155936
\(299\) −16.6180 −0.961042
\(300\) 0 0
\(301\) 7.34145 0.423154
\(302\) −34.9723 −2.01243
\(303\) 0 0
\(304\) −39.2097 −2.24883
\(305\) −13.4663 −0.771075
\(306\) 0 0
\(307\) 15.5752 0.888922 0.444461 0.895798i \(-0.353395\pi\)
0.444461 + 0.895798i \(0.353395\pi\)
\(308\) −13.7531 −0.783653
\(309\) 0 0
\(310\) 33.7630 1.91761
\(311\) 20.5657 1.16617 0.583086 0.812410i \(-0.301845\pi\)
0.583086 + 0.812410i \(0.301845\pi\)
\(312\) 0 0
\(313\) 21.9001 1.23787 0.618935 0.785442i \(-0.287565\pi\)
0.618935 + 0.785442i \(0.287565\pi\)
\(314\) −42.0955 −2.37559
\(315\) 0 0
\(316\) 27.0050 1.51915
\(317\) −12.3293 −0.692482 −0.346241 0.938146i \(-0.612542\pi\)
−0.346241 + 0.938146i \(0.612542\pi\)
\(318\) 0 0
\(319\) 7.43611 0.416342
\(320\) 7.65407 0.427875
\(321\) 0 0
\(322\) −12.7352 −0.709703
\(323\) 14.9882 0.833963
\(324\) 0 0
\(325\) 9.42997 0.523081
\(326\) 46.3990 2.56980
\(327\) 0 0
\(328\) −60.3804 −3.33395
\(329\) 3.00580 0.165715
\(330\) 0 0
\(331\) −25.5146 −1.40241 −0.701206 0.712959i \(-0.747355\pi\)
−0.701206 + 0.712959i \(0.747355\pi\)
\(332\) −66.0411 −3.62447
\(333\) 0 0
\(334\) 8.08331 0.442299
\(335\) −3.58769 −0.196016
\(336\) 0 0
\(337\) 16.8065 0.915507 0.457753 0.889079i \(-0.348654\pi\)
0.457753 + 0.889079i \(0.348654\pi\)
\(338\) −6.08629 −0.331051
\(339\) 0 0
\(340\) 12.1689 0.659952
\(341\) −30.6662 −1.66067
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −40.1980 −2.16733
\(345\) 0 0
\(346\) 5.57416 0.299669
\(347\) 27.1445 1.45719 0.728596 0.684944i \(-0.240173\pi\)
0.728596 + 0.684944i \(0.240173\pi\)
\(348\) 0 0
\(349\) 21.2339 1.13662 0.568311 0.822814i \(-0.307597\pi\)
0.568311 + 0.822814i \(0.307597\pi\)
\(350\) 7.22664 0.386280
\(351\) 0 0
\(352\) 6.81956 0.363484
\(353\) 27.5779 1.46782 0.733911 0.679246i \(-0.237693\pi\)
0.733911 + 0.679246i \(0.237693\pi\)
\(354\) 0 0
\(355\) 9.80075 0.520170
\(356\) 38.9201 2.06276
\(357\) 0 0
\(358\) −19.3519 −1.02278
\(359\) −11.4069 −0.602035 −0.301017 0.953619i \(-0.597326\pi\)
−0.301017 + 0.953619i \(0.597326\pi\)
\(360\) 0 0
\(361\) 37.1075 1.95302
\(362\) 50.1831 2.63757
\(363\) 0 0
\(364\) 13.6427 0.715074
\(365\) −15.0739 −0.789005
\(366\) 0 0
\(367\) 24.6520 1.28682 0.643412 0.765520i \(-0.277518\pi\)
0.643412 + 0.765520i \(0.277518\pi\)
\(368\) 26.7745 1.39572
\(369\) 0 0
\(370\) −25.1177 −1.30581
\(371\) 10.8847 0.565106
\(372\) 0 0
\(373\) −31.0058 −1.60542 −0.802711 0.596368i \(-0.796610\pi\)
−0.802711 + 0.596368i \(0.796610\pi\)
\(374\) −16.3170 −0.843734
\(375\) 0 0
\(376\) −16.4582 −0.848768
\(377\) −7.37647 −0.379907
\(378\) 0 0
\(379\) 22.9048 1.17654 0.588270 0.808665i \(-0.299809\pi\)
0.588270 + 0.808665i \(0.299809\pi\)
\(380\) 45.5538 2.33686
\(381\) 0 0
\(382\) −19.6836 −1.00710
\(383\) 29.0013 1.48190 0.740948 0.671562i \(-0.234376\pi\)
0.740948 + 0.671562i \(0.234376\pi\)
\(384\) 0 0
\(385\) 4.74340 0.241746
\(386\) −38.8639 −1.97812
\(387\) 0 0
\(388\) 14.1153 0.716597
\(389\) −16.5467 −0.838948 −0.419474 0.907767i \(-0.637786\pi\)
−0.419474 + 0.907767i \(0.637786\pi\)
\(390\) 0 0
\(391\) −10.2347 −0.517592
\(392\) 5.47548 0.276553
\(393\) 0 0
\(394\) −58.3517 −2.93972
\(395\) −9.31398 −0.468637
\(396\) 0 0
\(397\) 8.20972 0.412034 0.206017 0.978548i \(-0.433950\pi\)
0.206017 + 0.978548i \(0.433950\pi\)
\(398\) 43.7245 2.19171
\(399\) 0 0
\(400\) −15.1933 −0.759667
\(401\) −30.2608 −1.51115 −0.755577 0.655060i \(-0.772643\pi\)
−0.755577 + 0.655060i \(0.772643\pi\)
\(402\) 0 0
\(403\) 30.4202 1.51534
\(404\) 13.2333 0.658383
\(405\) 0 0
\(406\) −5.65295 −0.280551
\(407\) 22.8139 1.13084
\(408\) 0 0
\(409\) −8.23789 −0.407338 −0.203669 0.979040i \(-0.565287\pi\)
−0.203669 + 0.979040i \(0.565287\pi\)
\(410\) 39.7642 1.96381
\(411\) 0 0
\(412\) −31.9305 −1.57310
\(413\) −8.25654 −0.406278
\(414\) 0 0
\(415\) 22.7774 1.11810
\(416\) −6.76487 −0.331675
\(417\) 0 0
\(418\) −61.0821 −2.98762
\(419\) −26.3050 −1.28508 −0.642540 0.766252i \(-0.722120\pi\)
−0.642540 + 0.766252i \(0.722120\pi\)
\(420\) 0 0
\(421\) 17.1178 0.834269 0.417135 0.908845i \(-0.363034\pi\)
0.417135 + 0.908845i \(0.363034\pi\)
\(422\) 37.7847 1.83933
\(423\) 0 0
\(424\) −59.5990 −2.89438
\(425\) 5.80775 0.281717
\(426\) 0 0
\(427\) −9.29811 −0.449967
\(428\) −21.4954 −1.03902
\(429\) 0 0
\(430\) 26.4728 1.27663
\(431\) −25.7041 −1.23812 −0.619061 0.785343i \(-0.712486\pi\)
−0.619061 + 0.785343i \(0.712486\pi\)
\(432\) 0 0
\(433\) 19.9912 0.960715 0.480358 0.877073i \(-0.340507\pi\)
0.480358 + 0.877073i \(0.340507\pi\)
\(434\) 23.3125 1.11904
\(435\) 0 0
\(436\) −7.65831 −0.366766
\(437\) −38.3132 −1.83277
\(438\) 0 0
\(439\) −23.7707 −1.13451 −0.567257 0.823541i \(-0.691996\pi\)
−0.567257 + 0.823541i \(0.691996\pi\)
\(440\) −25.9724 −1.23818
\(441\) 0 0
\(442\) 16.1862 0.769897
\(443\) 40.7424 1.93573 0.967865 0.251469i \(-0.0809137\pi\)
0.967865 + 0.251469i \(0.0809137\pi\)
\(444\) 0 0
\(445\) −13.4235 −0.636333
\(446\) 38.7305 1.83394
\(447\) 0 0
\(448\) 5.28494 0.249690
\(449\) 24.5392 1.15808 0.579038 0.815301i \(-0.303428\pi\)
0.579038 + 0.815301i \(0.303428\pi\)
\(450\) 0 0
\(451\) −36.1169 −1.70068
\(452\) 8.86023 0.416750
\(453\) 0 0
\(454\) −59.8189 −2.80744
\(455\) −4.70535 −0.220590
\(456\) 0 0
\(457\) 13.4338 0.628407 0.314203 0.949356i \(-0.398263\pi\)
0.314203 + 0.949356i \(0.398263\pi\)
\(458\) −64.0075 −2.99087
\(459\) 0 0
\(460\) −31.1066 −1.45035
\(461\) 26.8553 1.25078 0.625389 0.780313i \(-0.284940\pi\)
0.625389 + 0.780313i \(0.284940\pi\)
\(462\) 0 0
\(463\) −9.24734 −0.429760 −0.214880 0.976640i \(-0.568936\pi\)
−0.214880 + 0.976640i \(0.568936\pi\)
\(464\) 11.8848 0.551738
\(465\) 0 0
\(466\) −11.1412 −0.516106
\(467\) 21.5833 0.998754 0.499377 0.866385i \(-0.333562\pi\)
0.499377 + 0.866385i \(0.333562\pi\)
\(468\) 0 0
\(469\) −2.47721 −0.114387
\(470\) 10.8387 0.499954
\(471\) 0 0
\(472\) 45.2085 2.08089
\(473\) −24.0447 −1.10558
\(474\) 0 0
\(475\) 21.7411 0.997548
\(476\) 8.40233 0.385120
\(477\) 0 0
\(478\) −9.23844 −0.422556
\(479\) 9.92648 0.453553 0.226776 0.973947i \(-0.427181\pi\)
0.226776 + 0.973947i \(0.427181\pi\)
\(480\) 0 0
\(481\) −22.6309 −1.03188
\(482\) −58.7418 −2.67561
\(483\) 0 0
\(484\) −1.14679 −0.0521267
\(485\) −4.86834 −0.221060
\(486\) 0 0
\(487\) 17.5491 0.795227 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(488\) 50.9116 2.30466
\(489\) 0 0
\(490\) −3.60594 −0.162900
\(491\) −11.9827 −0.540770 −0.270385 0.962752i \(-0.587151\pi\)
−0.270385 + 0.962752i \(0.587151\pi\)
\(492\) 0 0
\(493\) −4.54304 −0.204608
\(494\) 60.5922 2.72617
\(495\) 0 0
\(496\) −49.0124 −2.20072
\(497\) 6.76717 0.303549
\(498\) 0 0
\(499\) −17.7117 −0.792885 −0.396442 0.918060i \(-0.629755\pi\)
−0.396442 + 0.918060i \(0.629755\pi\)
\(500\) 48.0594 2.14928
\(501\) 0 0
\(502\) −40.0789 −1.78881
\(503\) −8.29004 −0.369635 −0.184817 0.982773i \(-0.559169\pi\)
−0.184817 + 0.982773i \(0.559169\pi\)
\(504\) 0 0
\(505\) −4.56415 −0.203102
\(506\) 41.7101 1.85424
\(507\) 0 0
\(508\) 4.19916 0.186307
\(509\) −18.1248 −0.803365 −0.401683 0.915779i \(-0.631575\pi\)
−0.401683 + 0.915779i \(0.631575\pi\)
\(510\) 0 0
\(511\) −10.4082 −0.460430
\(512\) −46.4244 −2.05169
\(513\) 0 0
\(514\) 39.1835 1.72831
\(515\) 11.0127 0.485279
\(516\) 0 0
\(517\) −9.84459 −0.432965
\(518\) −17.3432 −0.762015
\(519\) 0 0
\(520\) 25.7640 1.12983
\(521\) 26.6839 1.16904 0.584522 0.811378i \(-0.301282\pi\)
0.584522 + 0.811378i \(0.301282\pi\)
\(522\) 0 0
\(523\) −3.04549 −0.133170 −0.0665849 0.997781i \(-0.521210\pi\)
−0.0665849 + 0.997781i \(0.521210\pi\)
\(524\) −22.7861 −0.995416
\(525\) 0 0
\(526\) −10.7550 −0.468942
\(527\) 18.7353 0.816122
\(528\) 0 0
\(529\) 3.16231 0.137492
\(530\) 39.2496 1.70489
\(531\) 0 0
\(532\) 31.4537 1.36369
\(533\) 35.8273 1.55185
\(534\) 0 0
\(535\) 7.41371 0.320523
\(536\) 13.5639 0.585872
\(537\) 0 0
\(538\) 76.6687 3.30542
\(539\) 3.27519 0.141073
\(540\) 0 0
\(541\) 26.4448 1.13695 0.568474 0.822701i \(-0.307534\pi\)
0.568474 + 0.822701i \(0.307534\pi\)
\(542\) 6.86286 0.294785
\(543\) 0 0
\(544\) −4.16636 −0.178631
\(545\) 2.64133 0.113142
\(546\) 0 0
\(547\) 25.9693 1.11037 0.555183 0.831728i \(-0.312648\pi\)
0.555183 + 0.831728i \(0.312648\pi\)
\(548\) −13.3774 −0.571453
\(549\) 0 0
\(550\) −23.6687 −1.00924
\(551\) −17.0067 −0.724508
\(552\) 0 0
\(553\) −6.43106 −0.273477
\(554\) −37.5370 −1.59480
\(555\) 0 0
\(556\) −5.56698 −0.236093
\(557\) −31.5443 −1.33657 −0.668287 0.743904i \(-0.732972\pi\)
−0.668287 + 0.743904i \(0.732972\pi\)
\(558\) 0 0
\(559\) 23.8518 1.00883
\(560\) 7.58115 0.320362
\(561\) 0 0
\(562\) 11.3720 0.479698
\(563\) −33.1892 −1.39876 −0.699378 0.714752i \(-0.746540\pi\)
−0.699378 + 0.714752i \(0.746540\pi\)
\(564\) 0 0
\(565\) −3.05587 −0.128562
\(566\) 55.4001 2.32864
\(567\) 0 0
\(568\) −37.0535 −1.55473
\(569\) −10.6871 −0.448026 −0.224013 0.974586i \(-0.571916\pi\)
−0.224013 + 0.974586i \(0.571916\pi\)
\(570\) 0 0
\(571\) 5.13536 0.214908 0.107454 0.994210i \(-0.465730\pi\)
0.107454 + 0.994210i \(0.465730\pi\)
\(572\) −44.6826 −1.86828
\(573\) 0 0
\(574\) 27.4562 1.14600
\(575\) −14.8460 −0.619120
\(576\) 0 0
\(577\) −34.0207 −1.41630 −0.708150 0.706062i \(-0.750470\pi\)
−0.708150 + 0.706062i \(0.750470\pi\)
\(578\) −32.3580 −1.34592
\(579\) 0 0
\(580\) −13.8077 −0.573335
\(581\) 15.7272 0.652475
\(582\) 0 0
\(583\) −35.6495 −1.47645
\(584\) 56.9897 2.35825
\(585\) 0 0
\(586\) −55.9498 −2.31126
\(587\) −24.3928 −1.00680 −0.503400 0.864053i \(-0.667918\pi\)
−0.503400 + 0.864053i \(0.667918\pi\)
\(588\) 0 0
\(589\) 70.1347 2.88985
\(590\) −29.7726 −1.22572
\(591\) 0 0
\(592\) 36.4624 1.49859
\(593\) −9.66401 −0.396853 −0.198427 0.980116i \(-0.563583\pi\)
−0.198427 + 0.980116i \(0.563583\pi\)
\(594\) 0 0
\(595\) −2.89794 −0.118804
\(596\) 0.453994 0.0185963
\(597\) 0 0
\(598\) −41.3756 −1.69197
\(599\) 40.5869 1.65834 0.829168 0.558999i \(-0.188815\pi\)
0.829168 + 0.558999i \(0.188815\pi\)
\(600\) 0 0
\(601\) −39.6171 −1.61602 −0.808008 0.589172i \(-0.799454\pi\)
−0.808008 + 0.589172i \(0.799454\pi\)
\(602\) 18.2788 0.744989
\(603\) 0 0
\(604\) −58.9821 −2.39995
\(605\) 0.395524 0.0160804
\(606\) 0 0
\(607\) −34.1486 −1.38605 −0.693023 0.720915i \(-0.743722\pi\)
−0.693023 + 0.720915i \(0.743722\pi\)
\(608\) −15.5966 −0.632525
\(609\) 0 0
\(610\) −33.5284 −1.35753
\(611\) 9.76563 0.395075
\(612\) 0 0
\(613\) −10.6189 −0.428894 −0.214447 0.976736i \(-0.568795\pi\)
−0.214447 + 0.976736i \(0.568795\pi\)
\(614\) 38.7792 1.56500
\(615\) 0 0
\(616\) −17.9333 −0.722552
\(617\) 12.6928 0.510994 0.255497 0.966810i \(-0.417761\pi\)
0.255497 + 0.966810i \(0.417761\pi\)
\(618\) 0 0
\(619\) −16.8524 −0.677357 −0.338678 0.940902i \(-0.609980\pi\)
−0.338678 + 0.940902i \(0.609980\pi\)
\(620\) 56.9425 2.28687
\(621\) 0 0
\(622\) 51.2046 2.05312
\(623\) −9.26856 −0.371337
\(624\) 0 0
\(625\) −2.06312 −0.0825249
\(626\) 54.5272 2.17935
\(627\) 0 0
\(628\) −70.9955 −2.83303
\(629\) −13.9380 −0.555744
\(630\) 0 0
\(631\) −25.6820 −1.02238 −0.511191 0.859467i \(-0.670796\pi\)
−0.511191 + 0.859467i \(0.670796\pi\)
\(632\) 35.2132 1.40070
\(633\) 0 0
\(634\) −30.6976 −1.21916
\(635\) −1.44828 −0.0574732
\(636\) 0 0
\(637\) −3.24893 −0.128727
\(638\) 18.5145 0.732996
\(639\) 0 0
\(640\) 25.0883 0.991704
\(641\) −3.88658 −0.153511 −0.0767554 0.997050i \(-0.524456\pi\)
−0.0767554 + 0.997050i \(0.524456\pi\)
\(642\) 0 0
\(643\) −21.5878 −0.851339 −0.425669 0.904879i \(-0.639961\pi\)
−0.425669 + 0.904879i \(0.639961\pi\)
\(644\) −21.4783 −0.846364
\(645\) 0 0
\(646\) 37.3177 1.46824
\(647\) 43.9149 1.72647 0.863236 0.504801i \(-0.168434\pi\)
0.863236 + 0.504801i \(0.168434\pi\)
\(648\) 0 0
\(649\) 27.0418 1.06148
\(650\) 23.4788 0.920916
\(651\) 0 0
\(652\) 78.2535 3.06464
\(653\) 6.36272 0.248992 0.124496 0.992220i \(-0.460269\pi\)
0.124496 + 0.992220i \(0.460269\pi\)
\(654\) 0 0
\(655\) 7.85888 0.307072
\(656\) −57.7240 −2.25375
\(657\) 0 0
\(658\) 7.48388 0.291752
\(659\) −13.5416 −0.527505 −0.263752 0.964590i \(-0.584960\pi\)
−0.263752 + 0.964590i \(0.584960\pi\)
\(660\) 0 0
\(661\) 45.4713 1.76863 0.884315 0.466892i \(-0.154626\pi\)
0.884315 + 0.466892i \(0.154626\pi\)
\(662\) −63.5266 −2.46903
\(663\) 0 0
\(664\) −86.1141 −3.34188
\(665\) −10.8483 −0.420680
\(666\) 0 0
\(667\) 11.6131 0.449659
\(668\) 13.6328 0.527468
\(669\) 0 0
\(670\) −8.93267 −0.345099
\(671\) 30.4531 1.17563
\(672\) 0 0
\(673\) 33.3052 1.28382 0.641911 0.766779i \(-0.278142\pi\)
0.641911 + 0.766779i \(0.278142\pi\)
\(674\) 41.8449 1.61181
\(675\) 0 0
\(676\) −10.2648 −0.394798
\(677\) 13.7600 0.528841 0.264421 0.964407i \(-0.414819\pi\)
0.264421 + 0.964407i \(0.414819\pi\)
\(678\) 0 0
\(679\) −3.36147 −0.129001
\(680\) 15.8676 0.608496
\(681\) 0 0
\(682\) −76.3530 −2.92371
\(683\) 3.68291 0.140922 0.0704612 0.997515i \(-0.477553\pi\)
0.0704612 + 0.997515i \(0.477553\pi\)
\(684\) 0 0
\(685\) 4.61382 0.176285
\(686\) −2.48981 −0.0950614
\(687\) 0 0
\(688\) −38.4295 −1.46511
\(689\) 35.3636 1.34725
\(690\) 0 0
\(691\) 17.0240 0.647622 0.323811 0.946122i \(-0.395036\pi\)
0.323811 + 0.946122i \(0.395036\pi\)
\(692\) 9.40102 0.357373
\(693\) 0 0
\(694\) 67.5846 2.56548
\(695\) 1.92004 0.0728313
\(696\) 0 0
\(697\) 22.0654 0.835786
\(698\) 52.8683 2.00109
\(699\) 0 0
\(700\) 12.1880 0.460663
\(701\) −21.8960 −0.827001 −0.413501 0.910504i \(-0.635694\pi\)
−0.413501 + 0.910504i \(0.635694\pi\)
\(702\) 0 0
\(703\) −52.1762 −1.96786
\(704\) −17.3092 −0.652365
\(705\) 0 0
\(706\) 68.6637 2.58419
\(707\) −3.15143 −0.118522
\(708\) 0 0
\(709\) 4.17525 0.156805 0.0784024 0.996922i \(-0.475018\pi\)
0.0784024 + 0.996922i \(0.475018\pi\)
\(710\) 24.4020 0.915791
\(711\) 0 0
\(712\) 50.7498 1.90193
\(713\) −47.8917 −1.79356
\(714\) 0 0
\(715\) 15.4109 0.576336
\(716\) −32.6377 −1.21973
\(717\) 0 0
\(718\) −28.4011 −1.05992
\(719\) −18.8713 −0.703781 −0.351890 0.936041i \(-0.614461\pi\)
−0.351890 + 0.936041i \(0.614461\pi\)
\(720\) 0 0
\(721\) 7.60402 0.283188
\(722\) 92.3906 3.43842
\(723\) 0 0
\(724\) 84.6356 3.14546
\(725\) −6.58990 −0.244743
\(726\) 0 0
\(727\) −28.0662 −1.04092 −0.520458 0.853887i \(-0.674239\pi\)
−0.520458 + 0.853887i \(0.674239\pi\)
\(728\) 17.7894 0.659320
\(729\) 0 0
\(730\) −37.5312 −1.38909
\(731\) 14.6899 0.543327
\(732\) 0 0
\(733\) 25.7440 0.950875 0.475437 0.879750i \(-0.342290\pi\)
0.475437 + 0.879750i \(0.342290\pi\)
\(734\) 61.3788 2.26553
\(735\) 0 0
\(736\) 10.6502 0.392571
\(737\) 8.11334 0.298859
\(738\) 0 0
\(739\) −13.7151 −0.504517 −0.252259 0.967660i \(-0.581173\pi\)
−0.252259 + 0.967660i \(0.581173\pi\)
\(740\) −42.3620 −1.55726
\(741\) 0 0
\(742\) 27.1008 0.994903
\(743\) −44.3812 −1.62819 −0.814094 0.580733i \(-0.802766\pi\)
−0.814094 + 0.580733i \(0.802766\pi\)
\(744\) 0 0
\(745\) −0.156582 −0.00573670
\(746\) −77.1987 −2.82644
\(747\) 0 0
\(748\) −27.5193 −1.00620
\(749\) 5.11898 0.187043
\(750\) 0 0
\(751\) 38.9092 1.41982 0.709909 0.704294i \(-0.248736\pi\)
0.709909 + 0.704294i \(0.248736\pi\)
\(752\) −15.7342 −0.573766
\(753\) 0 0
\(754\) −18.3660 −0.668850
\(755\) 20.3428 0.740350
\(756\) 0 0
\(757\) −7.16594 −0.260451 −0.130225 0.991484i \(-0.541570\pi\)
−0.130225 + 0.991484i \(0.541570\pi\)
\(758\) 57.0286 2.07137
\(759\) 0 0
\(760\) 59.3998 2.15466
\(761\) 3.29004 0.119264 0.0596318 0.998220i \(-0.481007\pi\)
0.0596318 + 0.998220i \(0.481007\pi\)
\(762\) 0 0
\(763\) 1.82377 0.0660250
\(764\) −33.1970 −1.20103
\(765\) 0 0
\(766\) 72.2077 2.60897
\(767\) −26.8249 −0.968590
\(768\) 0 0
\(769\) −8.91144 −0.321355 −0.160677 0.987007i \(-0.551368\pi\)
−0.160677 + 0.987007i \(0.551368\pi\)
\(770\) 11.8102 0.425608
\(771\) 0 0
\(772\) −65.5453 −2.35903
\(773\) 33.6491 1.21027 0.605137 0.796121i \(-0.293118\pi\)
0.605137 + 0.796121i \(0.293118\pi\)
\(774\) 0 0
\(775\) 27.1765 0.976208
\(776\) 18.4056 0.660724
\(777\) 0 0
\(778\) −41.1980 −1.47702
\(779\) 82.6008 2.95948
\(780\) 0 0
\(781\) −22.1638 −0.793083
\(782\) −25.4825 −0.911252
\(783\) 0 0
\(784\) 5.23459 0.186950
\(785\) 24.4862 0.873950
\(786\) 0 0
\(787\) 24.5016 0.873387 0.436694 0.899610i \(-0.356149\pi\)
0.436694 + 0.899610i \(0.356149\pi\)
\(788\) −98.4123 −3.50579
\(789\) 0 0
\(790\) −23.1900 −0.825064
\(791\) −2.11000 −0.0750231
\(792\) 0 0
\(793\) −30.2089 −1.07275
\(794\) 20.4406 0.725411
\(795\) 0 0
\(796\) 73.7429 2.61375
\(797\) −11.5574 −0.409384 −0.204692 0.978826i \(-0.565619\pi\)
−0.204692 + 0.978826i \(0.565619\pi\)
\(798\) 0 0
\(799\) 6.01449 0.212777
\(800\) −6.04352 −0.213671
\(801\) 0 0
\(802\) −75.3438 −2.66048
\(803\) 34.0888 1.20297
\(804\) 0 0
\(805\) 7.40782 0.261091
\(806\) 75.7406 2.66785
\(807\) 0 0
\(808\) 17.2556 0.607049
\(809\) 28.4257 0.999395 0.499697 0.866200i \(-0.333445\pi\)
0.499697 + 0.866200i \(0.333445\pi\)
\(810\) 0 0
\(811\) 43.5857 1.53050 0.765251 0.643732i \(-0.222615\pi\)
0.765251 + 0.643732i \(0.222615\pi\)
\(812\) −9.53390 −0.334574
\(813\) 0 0
\(814\) 56.8022 1.99092
\(815\) −26.9895 −0.945400
\(816\) 0 0
\(817\) 54.9911 1.92389
\(818\) −20.5108 −0.717143
\(819\) 0 0
\(820\) 67.0638 2.34197
\(821\) −16.6400 −0.580738 −0.290369 0.956915i \(-0.593778\pi\)
−0.290369 + 0.956915i \(0.593778\pi\)
\(822\) 0 0
\(823\) 41.4873 1.44616 0.723078 0.690767i \(-0.242727\pi\)
0.723078 + 0.690767i \(0.242727\pi\)
\(824\) −41.6356 −1.45045
\(825\) 0 0
\(826\) −20.5572 −0.715277
\(827\) 10.8714 0.378035 0.189018 0.981974i \(-0.439470\pi\)
0.189018 + 0.981974i \(0.439470\pi\)
\(828\) 0 0
\(829\) −29.5467 −1.02620 −0.513099 0.858330i \(-0.671503\pi\)
−0.513099 + 0.858330i \(0.671503\pi\)
\(830\) 56.7114 1.96848
\(831\) 0 0
\(832\) 17.1704 0.595276
\(833\) −2.00096 −0.0693291
\(834\) 0 0
\(835\) −4.70192 −0.162717
\(836\) −103.017 −3.56292
\(837\) 0 0
\(838\) −65.4943 −2.26246
\(839\) −32.2794 −1.11441 −0.557204 0.830376i \(-0.688126\pi\)
−0.557204 + 0.830376i \(0.688126\pi\)
\(840\) 0 0
\(841\) −23.8451 −0.822246
\(842\) 42.6200 1.46878
\(843\) 0 0
\(844\) 63.7253 2.19352
\(845\) 3.54029 0.121790
\(846\) 0 0
\(847\) 0.273100 0.00938382
\(848\) −56.9770 −1.95660
\(849\) 0 0
\(850\) 14.4602 0.495981
\(851\) 35.6287 1.22134
\(852\) 0 0
\(853\) −33.2528 −1.13855 −0.569277 0.822145i \(-0.692777\pi\)
−0.569277 + 0.822145i \(0.692777\pi\)
\(854\) −23.1505 −0.792194
\(855\) 0 0
\(856\) −28.0289 −0.958007
\(857\) −37.1866 −1.27027 −0.635135 0.772401i \(-0.719055\pi\)
−0.635135 + 0.772401i \(0.719055\pi\)
\(858\) 0 0
\(859\) −57.8946 −1.97534 −0.987668 0.156561i \(-0.949959\pi\)
−0.987668 + 0.156561i \(0.949959\pi\)
\(860\) 44.6474 1.52246
\(861\) 0 0
\(862\) −63.9982 −2.17979
\(863\) −46.7651 −1.59190 −0.795951 0.605362i \(-0.793028\pi\)
−0.795951 + 0.605362i \(0.793028\pi\)
\(864\) 0 0
\(865\) −3.24239 −0.110245
\(866\) 49.7743 1.69140
\(867\) 0 0
\(868\) 39.3174 1.33452
\(869\) 21.0630 0.714513
\(870\) 0 0
\(871\) −8.04827 −0.272705
\(872\) −9.98603 −0.338170
\(873\) 0 0
\(874\) −95.3926 −3.22670
\(875\) −11.4450 −0.386912
\(876\) 0 0
\(877\) 43.8243 1.47984 0.739921 0.672694i \(-0.234863\pi\)
0.739921 + 0.672694i \(0.234863\pi\)
\(878\) −59.1846 −1.99738
\(879\) 0 0
\(880\) −24.8297 −0.837011
\(881\) 17.1037 0.576239 0.288119 0.957594i \(-0.406970\pi\)
0.288119 + 0.957594i \(0.406970\pi\)
\(882\) 0 0
\(883\) −1.94657 −0.0655073 −0.0327537 0.999463i \(-0.510428\pi\)
−0.0327537 + 0.999463i \(0.510428\pi\)
\(884\) 27.2985 0.918149
\(885\) 0 0
\(886\) 101.441 3.40797
\(887\) −3.52396 −0.118323 −0.0591615 0.998248i \(-0.518843\pi\)
−0.0591615 + 0.998248i \(0.518843\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) −33.4219 −1.12030
\(891\) 0 0
\(892\) 65.3205 2.18709
\(893\) 22.5150 0.753434
\(894\) 0 0
\(895\) 11.2567 0.376269
\(896\) 17.3229 0.578716
\(897\) 0 0
\(898\) 61.0979 2.03886
\(899\) −21.2584 −0.709008
\(900\) 0 0
\(901\) 21.7798 0.725591
\(902\) −89.9243 −2.99415
\(903\) 0 0
\(904\) 11.5533 0.384256
\(905\) −29.1906 −0.970330
\(906\) 0 0
\(907\) 44.0478 1.46258 0.731292 0.682065i \(-0.238918\pi\)
0.731292 + 0.682065i \(0.238918\pi\)
\(908\) −100.887 −3.34805
\(909\) 0 0
\(910\) −11.7154 −0.388363
\(911\) 35.9406 1.19077 0.595383 0.803442i \(-0.297000\pi\)
0.595383 + 0.803442i \(0.297000\pi\)
\(912\) 0 0
\(913\) −51.5097 −1.70472
\(914\) 33.4476 1.10635
\(915\) 0 0
\(916\) −107.951 −3.56680
\(917\) 5.42636 0.179194
\(918\) 0 0
\(919\) 26.8856 0.886874 0.443437 0.896305i \(-0.353759\pi\)
0.443437 + 0.896305i \(0.353759\pi\)
\(920\) −40.5613 −1.33727
\(921\) 0 0
\(922\) 66.8647 2.20207
\(923\) 21.9860 0.723679
\(924\) 0 0
\(925\) −20.2177 −0.664755
\(926\) −23.0241 −0.756619
\(927\) 0 0
\(928\) 4.72746 0.155187
\(929\) −25.1460 −0.825013 −0.412506 0.910955i \(-0.635347\pi\)
−0.412506 + 0.910955i \(0.635347\pi\)
\(930\) 0 0
\(931\) −7.49049 −0.245491
\(932\) −18.7900 −0.615488
\(933\) 0 0
\(934\) 53.7382 1.75837
\(935\) 9.49133 0.310400
\(936\) 0 0
\(937\) −3.58973 −0.117271 −0.0586357 0.998279i \(-0.518675\pi\)
−0.0586357 + 0.998279i \(0.518675\pi\)
\(938\) −6.16778 −0.201385
\(939\) 0 0
\(940\) 18.2799 0.596226
\(941\) −19.4890 −0.635323 −0.317662 0.948204i \(-0.602898\pi\)
−0.317662 + 0.948204i \(0.602898\pi\)
\(942\) 0 0
\(943\) −56.4042 −1.83678
\(944\) 43.2196 1.40668
\(945\) 0 0
\(946\) −59.8667 −1.94643
\(947\) 11.2713 0.366268 0.183134 0.983088i \(-0.441376\pi\)
0.183134 + 0.983088i \(0.441376\pi\)
\(948\) 0 0
\(949\) −33.8154 −1.09769
\(950\) 54.1311 1.75625
\(951\) 0 0
\(952\) 10.9562 0.355092
\(953\) 34.6856 1.12358 0.561788 0.827281i \(-0.310114\pi\)
0.561788 + 0.827281i \(0.310114\pi\)
\(954\) 0 0
\(955\) 11.4496 0.370500
\(956\) −15.5810 −0.503924
\(957\) 0 0
\(958\) 24.7151 0.798507
\(959\) 3.18573 0.102873
\(960\) 0 0
\(961\) 56.6689 1.82803
\(962\) −56.3466 −1.81669
\(963\) 0 0
\(964\) −99.0701 −3.19083
\(965\) 22.6064 0.727727
\(966\) 0 0
\(967\) −34.2544 −1.10155 −0.550774 0.834655i \(-0.685667\pi\)
−0.550774 + 0.834655i \(0.685667\pi\)
\(968\) −1.49535 −0.0480624
\(969\) 0 0
\(970\) −12.1212 −0.389190
\(971\) −25.1782 −0.808007 −0.404004 0.914757i \(-0.632382\pi\)
−0.404004 + 0.914757i \(0.632382\pi\)
\(972\) 0 0
\(973\) 1.32574 0.0425012
\(974\) 43.6940 1.40005
\(975\) 0 0
\(976\) 48.6718 1.55795
\(977\) −52.0600 −1.66555 −0.832773 0.553614i \(-0.813248\pi\)
−0.832773 + 0.553614i \(0.813248\pi\)
\(978\) 0 0
\(979\) 30.3563 0.970193
\(980\) −6.08155 −0.194268
\(981\) 0 0
\(982\) −29.8346 −0.952059
\(983\) 2.90281 0.0925852 0.0462926 0.998928i \(-0.485259\pi\)
0.0462926 + 0.998928i \(0.485259\pi\)
\(984\) 0 0
\(985\) 33.9422 1.08149
\(986\) −11.3113 −0.360225
\(987\) 0 0
\(988\) 102.191 3.25112
\(989\) −37.5509 −1.19405
\(990\) 0 0
\(991\) 49.2426 1.56424 0.782121 0.623127i \(-0.214138\pi\)
0.782121 + 0.623127i \(0.214138\pi\)
\(992\) −19.4958 −0.618994
\(993\) 0 0
\(994\) 16.8490 0.534417
\(995\) −25.4338 −0.806305
\(996\) 0 0
\(997\) −24.3407 −0.770878 −0.385439 0.922733i \(-0.625950\pi\)
−0.385439 + 0.922733i \(0.625950\pi\)
\(998\) −44.0988 −1.39592
\(999\) 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 8001.2.a.r.1.16 16
3.2 odd 2 2667.2.a.o.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.o.1.1 16 3.2 odd 2
8001.2.a.r.1.16 16 1.1 even 1 trivial