Properties

Label 8001.2.a.t.1.6
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
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: \(0\)
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: \( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.532475\) 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)
 
\(f(q)\) \(=\) \(q-0.532475 q^{2} -1.71647 q^{4} +0.118172 q^{5} -1.00000 q^{7} +1.97893 q^{8} +O(q^{10})\) \(q-0.532475 q^{2} -1.71647 q^{4} +0.118172 q^{5} -1.00000 q^{7} +1.97893 q^{8} -0.0629236 q^{10} +5.22568 q^{11} -2.19166 q^{13} +0.532475 q^{14} +2.37921 q^{16} +1.28158 q^{17} +8.34525 q^{19} -0.202839 q^{20} -2.78254 q^{22} +9.05966 q^{23} -4.98604 q^{25} +1.16700 q^{26} +1.71647 q^{28} +9.26366 q^{29} +6.76707 q^{31} -5.22473 q^{32} -0.682411 q^{34} -0.118172 q^{35} +6.01829 q^{37} -4.44364 q^{38} +0.233854 q^{40} +4.32714 q^{41} +2.14731 q^{43} -8.96972 q^{44} -4.82404 q^{46} -1.79425 q^{47} +1.00000 q^{49} +2.65494 q^{50} +3.76191 q^{52} -4.27074 q^{53} +0.617528 q^{55} -1.97893 q^{56} -4.93267 q^{58} -11.1673 q^{59} -10.4330 q^{61} -3.60330 q^{62} -1.97638 q^{64} -0.258992 q^{65} -14.4350 q^{67} -2.19980 q^{68} +0.0629236 q^{70} +4.12479 q^{71} +16.5937 q^{73} -3.20459 q^{74} -14.3244 q^{76} -5.22568 q^{77} -11.3780 q^{79} +0.281156 q^{80} -2.30410 q^{82} -6.41125 q^{83} +0.151447 q^{85} -1.14339 q^{86} +10.3412 q^{88} +3.69932 q^{89} +2.19166 q^{91} -15.5506 q^{92} +0.955391 q^{94} +0.986174 q^{95} +6.10975 q^{97} -0.532475 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.532475 −0.376517 −0.188258 0.982120i \(-0.560284\pi\)
−0.188258 + 0.982120i \(0.560284\pi\)
\(3\) 0 0
\(4\) −1.71647 −0.858235
\(5\) 0.118172 0.0528481 0.0264241 0.999651i \(-0.491588\pi\)
0.0264241 + 0.999651i \(0.491588\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.97893 0.699657
\(9\) 0 0
\(10\) −0.0629236 −0.0198982
\(11\) 5.22568 1.57560 0.787800 0.615931i \(-0.211220\pi\)
0.787800 + 0.615931i \(0.211220\pi\)
\(12\) 0 0
\(13\) −2.19166 −0.607856 −0.303928 0.952695i \(-0.598298\pi\)
−0.303928 + 0.952695i \(0.598298\pi\)
\(14\) 0.532475 0.142310
\(15\) 0 0
\(16\) 2.37921 0.594803
\(17\) 1.28158 0.310829 0.155415 0.987849i \(-0.450329\pi\)
0.155415 + 0.987849i \(0.450329\pi\)
\(18\) 0 0
\(19\) 8.34525 1.91453 0.957266 0.289210i \(-0.0933925\pi\)
0.957266 + 0.289210i \(0.0933925\pi\)
\(20\) −0.202839 −0.0453561
\(21\) 0 0
\(22\) −2.78254 −0.593240
\(23\) 9.05966 1.88907 0.944535 0.328412i \(-0.106513\pi\)
0.944535 + 0.328412i \(0.106513\pi\)
\(24\) 0 0
\(25\) −4.98604 −0.997207
\(26\) 1.16700 0.228868
\(27\) 0 0
\(28\) 1.71647 0.324382
\(29\) 9.26366 1.72022 0.860109 0.510111i \(-0.170396\pi\)
0.860109 + 0.510111i \(0.170396\pi\)
\(30\) 0 0
\(31\) 6.76707 1.21540 0.607701 0.794166i \(-0.292092\pi\)
0.607701 + 0.794166i \(0.292092\pi\)
\(32\) −5.22473 −0.923610
\(33\) 0 0
\(34\) −0.682411 −0.117033
\(35\) −0.118172 −0.0199747
\(36\) 0 0
\(37\) 6.01829 0.989401 0.494700 0.869064i \(-0.335278\pi\)
0.494700 + 0.869064i \(0.335278\pi\)
\(38\) −4.44364 −0.720853
\(39\) 0 0
\(40\) 0.233854 0.0369755
\(41\) 4.32714 0.675786 0.337893 0.941184i \(-0.390286\pi\)
0.337893 + 0.941184i \(0.390286\pi\)
\(42\) 0 0
\(43\) 2.14731 0.327461 0.163730 0.986505i \(-0.447647\pi\)
0.163730 + 0.986505i \(0.447647\pi\)
\(44\) −8.96972 −1.35224
\(45\) 0 0
\(46\) −4.82404 −0.711266
\(47\) −1.79425 −0.261718 −0.130859 0.991401i \(-0.541773\pi\)
−0.130859 + 0.991401i \(0.541773\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.65494 0.375465
\(51\) 0 0
\(52\) 3.76191 0.521683
\(53\) −4.27074 −0.586631 −0.293315 0.956016i \(-0.594759\pi\)
−0.293315 + 0.956016i \(0.594759\pi\)
\(54\) 0 0
\(55\) 0.617528 0.0832675
\(56\) −1.97893 −0.264445
\(57\) 0 0
\(58\) −4.93267 −0.647691
\(59\) −11.1673 −1.45386 −0.726931 0.686710i \(-0.759054\pi\)
−0.726931 + 0.686710i \(0.759054\pi\)
\(60\) 0 0
\(61\) −10.4330 −1.33581 −0.667907 0.744245i \(-0.732810\pi\)
−0.667907 + 0.744245i \(0.732810\pi\)
\(62\) −3.60330 −0.457619
\(63\) 0 0
\(64\) −1.97638 −0.247048
\(65\) −0.258992 −0.0321240
\(66\) 0 0
\(67\) −14.4350 −1.76352 −0.881760 0.471699i \(-0.843641\pi\)
−0.881760 + 0.471699i \(0.843641\pi\)
\(68\) −2.19980 −0.266765
\(69\) 0 0
\(70\) 0.0629236 0.00752081
\(71\) 4.12479 0.489523 0.244761 0.969583i \(-0.421290\pi\)
0.244761 + 0.969583i \(0.421290\pi\)
\(72\) 0 0
\(73\) 16.5937 1.94215 0.971075 0.238774i \(-0.0767454\pi\)
0.971075 + 0.238774i \(0.0767454\pi\)
\(74\) −3.20459 −0.372526
\(75\) 0 0
\(76\) −14.3244 −1.64312
\(77\) −5.22568 −0.595521
\(78\) 0 0
\(79\) −11.3780 −1.28012 −0.640060 0.768325i \(-0.721091\pi\)
−0.640060 + 0.768325i \(0.721091\pi\)
\(80\) 0.281156 0.0314342
\(81\) 0 0
\(82\) −2.30410 −0.254445
\(83\) −6.41125 −0.703727 −0.351863 0.936051i \(-0.614452\pi\)
−0.351863 + 0.936051i \(0.614452\pi\)
\(84\) 0 0
\(85\) 0.151447 0.0164267
\(86\) −1.14339 −0.123295
\(87\) 0 0
\(88\) 10.3412 1.10238
\(89\) 3.69932 0.392127 0.196064 0.980591i \(-0.437184\pi\)
0.196064 + 0.980591i \(0.437184\pi\)
\(90\) 0 0
\(91\) 2.19166 0.229748
\(92\) −15.5506 −1.62127
\(93\) 0 0
\(94\) 0.955391 0.0985410
\(95\) 0.986174 0.101179
\(96\) 0 0
\(97\) 6.10975 0.620351 0.310176 0.950679i \(-0.399612\pi\)
0.310176 + 0.950679i \(0.399612\pi\)
\(98\) −0.532475 −0.0537881
\(99\) 0 0
\(100\) 8.55838 0.855838
\(101\) −0.593672 −0.0590726 −0.0295363 0.999564i \(-0.509403\pi\)
−0.0295363 + 0.999564i \(0.509403\pi\)
\(102\) 0 0
\(103\) 7.82074 0.770600 0.385300 0.922791i \(-0.374098\pi\)
0.385300 + 0.922791i \(0.374098\pi\)
\(104\) −4.33713 −0.425290
\(105\) 0 0
\(106\) 2.27406 0.220876
\(107\) 11.0220 1.06553 0.532767 0.846262i \(-0.321152\pi\)
0.532767 + 0.846262i \(0.321152\pi\)
\(108\) 0 0
\(109\) −0.329550 −0.0315652 −0.0157826 0.999875i \(-0.505024\pi\)
−0.0157826 + 0.999875i \(0.505024\pi\)
\(110\) −0.328818 −0.0313516
\(111\) 0 0
\(112\) −2.37921 −0.224814
\(113\) 1.82578 0.171755 0.0858777 0.996306i \(-0.472631\pi\)
0.0858777 + 0.996306i \(0.472631\pi\)
\(114\) 0 0
\(115\) 1.07060 0.0998337
\(116\) −15.9008 −1.47635
\(117\) 0 0
\(118\) 5.94632 0.547404
\(119\) −1.28158 −0.117482
\(120\) 0 0
\(121\) 16.3077 1.48252
\(122\) 5.55533 0.502956
\(123\) 0 0
\(124\) −11.6155 −1.04310
\(125\) −1.18007 −0.105549
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 11.5018 1.01663
\(129\) 0 0
\(130\) 0.137907 0.0120952
\(131\) 8.23408 0.719415 0.359707 0.933065i \(-0.382877\pi\)
0.359707 + 0.933065i \(0.382877\pi\)
\(132\) 0 0
\(133\) −8.34525 −0.723625
\(134\) 7.68629 0.663994
\(135\) 0 0
\(136\) 2.53616 0.217474
\(137\) 3.79688 0.324390 0.162195 0.986759i \(-0.448143\pi\)
0.162195 + 0.986759i \(0.448143\pi\)
\(138\) 0 0
\(139\) −4.53910 −0.385001 −0.192501 0.981297i \(-0.561660\pi\)
−0.192501 + 0.981297i \(0.561660\pi\)
\(140\) 0.202839 0.0171430
\(141\) 0 0
\(142\) −2.19635 −0.184314
\(143\) −11.4529 −0.957738
\(144\) 0 0
\(145\) 1.09470 0.0909102
\(146\) −8.83575 −0.731252
\(147\) 0 0
\(148\) −10.3302 −0.849138
\(149\) 4.89325 0.400871 0.200435 0.979707i \(-0.435764\pi\)
0.200435 + 0.979707i \(0.435764\pi\)
\(150\) 0 0
\(151\) 11.1366 0.906286 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(152\) 16.5146 1.33951
\(153\) 0 0
\(154\) 2.78254 0.224224
\(155\) 0.799678 0.0642317
\(156\) 0 0
\(157\) −15.9627 −1.27397 −0.636983 0.770878i \(-0.719818\pi\)
−0.636983 + 0.770878i \(0.719818\pi\)
\(158\) 6.05848 0.481987
\(159\) 0 0
\(160\) −0.617416 −0.0488110
\(161\) −9.05966 −0.714001
\(162\) 0 0
\(163\) −11.4892 −0.899906 −0.449953 0.893052i \(-0.648559\pi\)
−0.449953 + 0.893052i \(0.648559\pi\)
\(164\) −7.42741 −0.579984
\(165\) 0 0
\(166\) 3.41383 0.264965
\(167\) 14.2726 1.10445 0.552223 0.833697i \(-0.313780\pi\)
0.552223 + 0.833697i \(0.313780\pi\)
\(168\) 0 0
\(169\) −8.19665 −0.630511
\(170\) −0.0806418 −0.00618495
\(171\) 0 0
\(172\) −3.68579 −0.281039
\(173\) −3.15204 −0.239645 −0.119823 0.992795i \(-0.538233\pi\)
−0.119823 + 0.992795i \(0.538233\pi\)
\(174\) 0 0
\(175\) 4.98604 0.376909
\(176\) 12.4330 0.937171
\(177\) 0 0
\(178\) −1.96980 −0.147642
\(179\) −22.0690 −1.64951 −0.824757 0.565488i \(-0.808688\pi\)
−0.824757 + 0.565488i \(0.808688\pi\)
\(180\) 0 0
\(181\) 17.7881 1.32218 0.661089 0.750307i \(-0.270094\pi\)
0.661089 + 0.750307i \(0.270094\pi\)
\(182\) −1.16700 −0.0865039
\(183\) 0 0
\(184\) 17.9284 1.32170
\(185\) 0.711193 0.0522879
\(186\) 0 0
\(187\) 6.69713 0.489743
\(188\) 3.07977 0.224615
\(189\) 0 0
\(190\) −0.525113 −0.0380957
\(191\) −9.64651 −0.697997 −0.348999 0.937123i \(-0.613478\pi\)
−0.348999 + 0.937123i \(0.613478\pi\)
\(192\) 0 0
\(193\) 12.1823 0.876901 0.438451 0.898755i \(-0.355527\pi\)
0.438451 + 0.898755i \(0.355527\pi\)
\(194\) −3.25329 −0.233573
\(195\) 0 0
\(196\) −1.71647 −0.122605
\(197\) −5.07522 −0.361595 −0.180797 0.983520i \(-0.557868\pi\)
−0.180797 + 0.983520i \(0.557868\pi\)
\(198\) 0 0
\(199\) 3.43881 0.243771 0.121885 0.992544i \(-0.461106\pi\)
0.121885 + 0.992544i \(0.461106\pi\)
\(200\) −9.86700 −0.697703
\(201\) 0 0
\(202\) 0.316116 0.0222418
\(203\) −9.26366 −0.650181
\(204\) 0 0
\(205\) 0.511347 0.0357140
\(206\) −4.16435 −0.290144
\(207\) 0 0
\(208\) −5.21441 −0.361554
\(209\) 43.6096 3.01654
\(210\) 0 0
\(211\) −13.3818 −0.921243 −0.460621 0.887597i \(-0.652373\pi\)
−0.460621 + 0.887597i \(0.652373\pi\)
\(212\) 7.33059 0.503467
\(213\) 0 0
\(214\) −5.86892 −0.401191
\(215\) 0.253751 0.0173057
\(216\) 0 0
\(217\) −6.76707 −0.459379
\(218\) 0.175477 0.0118848
\(219\) 0 0
\(220\) −1.05997 −0.0714631
\(221\) −2.80879 −0.188940
\(222\) 0 0
\(223\) −3.72789 −0.249638 −0.124819 0.992180i \(-0.539835\pi\)
−0.124819 + 0.992180i \(0.539835\pi\)
\(224\) 5.22473 0.349092
\(225\) 0 0
\(226\) −0.972184 −0.0646687
\(227\) −7.52598 −0.499517 −0.249759 0.968308i \(-0.580351\pi\)
−0.249759 + 0.968308i \(0.580351\pi\)
\(228\) 0 0
\(229\) −0.811157 −0.0536027 −0.0268014 0.999641i \(-0.508532\pi\)
−0.0268014 + 0.999641i \(0.508532\pi\)
\(230\) −0.570066 −0.0375891
\(231\) 0 0
\(232\) 18.3321 1.20356
\(233\) −21.7672 −1.42602 −0.713008 0.701156i \(-0.752668\pi\)
−0.713008 + 0.701156i \(0.752668\pi\)
\(234\) 0 0
\(235\) −0.212029 −0.0138313
\(236\) 19.1684 1.24776
\(237\) 0 0
\(238\) 0.682411 0.0442341
\(239\) −10.2516 −0.663119 −0.331559 0.943434i \(-0.607575\pi\)
−0.331559 + 0.943434i \(0.607575\pi\)
\(240\) 0 0
\(241\) 12.4960 0.804938 0.402469 0.915434i \(-0.368152\pi\)
0.402469 + 0.915434i \(0.368152\pi\)
\(242\) −8.68343 −0.558192
\(243\) 0 0
\(244\) 17.9080 1.14644
\(245\) 0.118172 0.00754973
\(246\) 0 0
\(247\) −18.2899 −1.16376
\(248\) 13.3915 0.850364
\(249\) 0 0
\(250\) 0.628358 0.0397408
\(251\) 21.5708 1.36154 0.680769 0.732498i \(-0.261646\pi\)
0.680769 + 0.732498i \(0.261646\pi\)
\(252\) 0 0
\(253\) 47.3428 2.97642
\(254\) 0.532475 0.0334105
\(255\) 0 0
\(256\) −2.17167 −0.135729
\(257\) 19.5843 1.22163 0.610817 0.791772i \(-0.290841\pi\)
0.610817 + 0.791772i \(0.290841\pi\)
\(258\) 0 0
\(259\) −6.01829 −0.373958
\(260\) 0.444552 0.0275700
\(261\) 0 0
\(262\) −4.38444 −0.270872
\(263\) 7.21804 0.445083 0.222542 0.974923i \(-0.428565\pi\)
0.222542 + 0.974923i \(0.428565\pi\)
\(264\) 0 0
\(265\) −0.504681 −0.0310023
\(266\) 4.44364 0.272457
\(267\) 0 0
\(268\) 24.7773 1.51351
\(269\) 14.4911 0.883538 0.441769 0.897129i \(-0.354351\pi\)
0.441769 + 0.897129i \(0.354351\pi\)
\(270\) 0 0
\(271\) −17.3584 −1.05445 −0.527224 0.849726i \(-0.676767\pi\)
−0.527224 + 0.849726i \(0.676767\pi\)
\(272\) 3.04916 0.184882
\(273\) 0 0
\(274\) −2.02175 −0.122138
\(275\) −26.0554 −1.57120
\(276\) 0 0
\(277\) −25.4371 −1.52836 −0.764182 0.645000i \(-0.776857\pi\)
−0.764182 + 0.645000i \(0.776857\pi\)
\(278\) 2.41696 0.144959
\(279\) 0 0
\(280\) −0.233854 −0.0139754
\(281\) −10.7927 −0.643837 −0.321918 0.946767i \(-0.604328\pi\)
−0.321918 + 0.946767i \(0.604328\pi\)
\(282\) 0 0
\(283\) 1.47440 0.0876441 0.0438220 0.999039i \(-0.486047\pi\)
0.0438220 + 0.999039i \(0.486047\pi\)
\(284\) −7.08009 −0.420126
\(285\) 0 0
\(286\) 6.09837 0.360604
\(287\) −4.32714 −0.255423
\(288\) 0 0
\(289\) −15.3575 −0.903385
\(290\) −0.582903 −0.0342292
\(291\) 0 0
\(292\) −28.4827 −1.66682
\(293\) −12.6888 −0.741290 −0.370645 0.928775i \(-0.620863\pi\)
−0.370645 + 0.928775i \(0.620863\pi\)
\(294\) 0 0
\(295\) −1.31966 −0.0768339
\(296\) 11.9098 0.692241
\(297\) 0 0
\(298\) −2.60553 −0.150934
\(299\) −19.8557 −1.14828
\(300\) 0 0
\(301\) −2.14731 −0.123769
\(302\) −5.92998 −0.341232
\(303\) 0 0
\(304\) 19.8551 1.13877
\(305\) −1.23289 −0.0705952
\(306\) 0 0
\(307\) −7.60261 −0.433904 −0.216952 0.976182i \(-0.569612\pi\)
−0.216952 + 0.976182i \(0.569612\pi\)
\(308\) 8.96972 0.511097
\(309\) 0 0
\(310\) −0.425808 −0.0241843
\(311\) 34.6469 1.96465 0.982324 0.187190i \(-0.0599381\pi\)
0.982324 + 0.187190i \(0.0599381\pi\)
\(312\) 0 0
\(313\) −13.4142 −0.758216 −0.379108 0.925352i \(-0.623769\pi\)
−0.379108 + 0.925352i \(0.623769\pi\)
\(314\) 8.49977 0.479670
\(315\) 0 0
\(316\) 19.5299 1.09864
\(317\) 25.3786 1.42540 0.712702 0.701467i \(-0.247471\pi\)
0.712702 + 0.701467i \(0.247471\pi\)
\(318\) 0 0
\(319\) 48.4089 2.71038
\(320\) −0.233553 −0.0130560
\(321\) 0 0
\(322\) 4.82404 0.268833
\(323\) 10.6951 0.595093
\(324\) 0 0
\(325\) 10.9277 0.606158
\(326\) 6.11773 0.338830
\(327\) 0 0
\(328\) 8.56311 0.472818
\(329\) 1.79425 0.0989199
\(330\) 0 0
\(331\) 4.32910 0.237949 0.118974 0.992897i \(-0.462039\pi\)
0.118974 + 0.992897i \(0.462039\pi\)
\(332\) 11.0047 0.603963
\(333\) 0 0
\(334\) −7.59979 −0.415842
\(335\) −1.70582 −0.0931986
\(336\) 0 0
\(337\) −30.9193 −1.68428 −0.842141 0.539257i \(-0.818705\pi\)
−0.842141 + 0.539257i \(0.818705\pi\)
\(338\) 4.36451 0.237398
\(339\) 0 0
\(340\) −0.259954 −0.0140980
\(341\) 35.3625 1.91499
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 4.24936 0.229110
\(345\) 0 0
\(346\) 1.67838 0.0902304
\(347\) −11.4166 −0.612873 −0.306436 0.951891i \(-0.599137\pi\)
−0.306436 + 0.951891i \(0.599137\pi\)
\(348\) 0 0
\(349\) −25.1259 −1.34496 −0.672479 0.740116i \(-0.734771\pi\)
−0.672479 + 0.740116i \(0.734771\pi\)
\(350\) −2.65494 −0.141912
\(351\) 0 0
\(352\) −27.3027 −1.45524
\(353\) −16.4066 −0.873234 −0.436617 0.899647i \(-0.643824\pi\)
−0.436617 + 0.899647i \(0.643824\pi\)
\(354\) 0 0
\(355\) 0.487435 0.0258704
\(356\) −6.34977 −0.336537
\(357\) 0 0
\(358\) 11.7512 0.621069
\(359\) 17.7879 0.938809 0.469404 0.882983i \(-0.344469\pi\)
0.469404 + 0.882983i \(0.344469\pi\)
\(360\) 0 0
\(361\) 50.6432 2.66543
\(362\) −9.47172 −0.497822
\(363\) 0 0
\(364\) −3.76191 −0.197178
\(365\) 1.96091 0.102639
\(366\) 0 0
\(367\) −23.9393 −1.24962 −0.624812 0.780776i \(-0.714824\pi\)
−0.624812 + 0.780776i \(0.714824\pi\)
\(368\) 21.5548 1.12362
\(369\) 0 0
\(370\) −0.378693 −0.0196873
\(371\) 4.27074 0.221726
\(372\) 0 0
\(373\) 24.1062 1.24817 0.624085 0.781356i \(-0.285472\pi\)
0.624085 + 0.781356i \(0.285472\pi\)
\(374\) −3.56606 −0.184396
\(375\) 0 0
\(376\) −3.55068 −0.183112
\(377\) −20.3027 −1.04564
\(378\) 0 0
\(379\) −9.74899 −0.500772 −0.250386 0.968146i \(-0.580558\pi\)
−0.250386 + 0.968146i \(0.580558\pi\)
\(380\) −1.69274 −0.0868357
\(381\) 0 0
\(382\) 5.13653 0.262808
\(383\) −21.0940 −1.07785 −0.538925 0.842354i \(-0.681169\pi\)
−0.538925 + 0.842354i \(0.681169\pi\)
\(384\) 0 0
\(385\) −0.617528 −0.0314722
\(386\) −6.48677 −0.330168
\(387\) 0 0
\(388\) −10.4872 −0.532407
\(389\) −11.4795 −0.582032 −0.291016 0.956718i \(-0.593993\pi\)
−0.291016 + 0.956718i \(0.593993\pi\)
\(390\) 0 0
\(391\) 11.6107 0.587178
\(392\) 1.97893 0.0999510
\(393\) 0 0
\(394\) 2.70243 0.136146
\(395\) −1.34456 −0.0676520
\(396\) 0 0
\(397\) 33.4207 1.67734 0.838668 0.544643i \(-0.183335\pi\)
0.838668 + 0.544643i \(0.183335\pi\)
\(398\) −1.83108 −0.0917838
\(399\) 0 0
\(400\) −11.8628 −0.593141
\(401\) −0.506322 −0.0252845 −0.0126423 0.999920i \(-0.504024\pi\)
−0.0126423 + 0.999920i \(0.504024\pi\)
\(402\) 0 0
\(403\) −14.8311 −0.738789
\(404\) 1.01902 0.0506982
\(405\) 0 0
\(406\) 4.93267 0.244804
\(407\) 31.4496 1.55890
\(408\) 0 0
\(409\) 27.6889 1.36913 0.684563 0.728953i \(-0.259993\pi\)
0.684563 + 0.728953i \(0.259993\pi\)
\(410\) −0.272280 −0.0134469
\(411\) 0 0
\(412\) −13.4241 −0.661356
\(413\) 11.1673 0.549508
\(414\) 0 0
\(415\) −0.757631 −0.0371906
\(416\) 11.4508 0.561422
\(417\) 0 0
\(418\) −23.2210 −1.13578
\(419\) −5.22108 −0.255067 −0.127533 0.991834i \(-0.540706\pi\)
−0.127533 + 0.991834i \(0.540706\pi\)
\(420\) 0 0
\(421\) −30.2687 −1.47521 −0.737603 0.675235i \(-0.764042\pi\)
−0.737603 + 0.675235i \(0.764042\pi\)
\(422\) 7.12549 0.346863
\(423\) 0 0
\(424\) −8.45148 −0.410440
\(425\) −6.39002 −0.309961
\(426\) 0 0
\(427\) 10.4330 0.504890
\(428\) −18.9189 −0.914479
\(429\) 0 0
\(430\) −0.135116 −0.00651588
\(431\) 12.4316 0.598807 0.299403 0.954127i \(-0.403212\pi\)
0.299403 + 0.954127i \(0.403212\pi\)
\(432\) 0 0
\(433\) 27.6679 1.32964 0.664818 0.747006i \(-0.268509\pi\)
0.664818 + 0.747006i \(0.268509\pi\)
\(434\) 3.60330 0.172964
\(435\) 0 0
\(436\) 0.565664 0.0270904
\(437\) 75.6051 3.61668
\(438\) 0 0
\(439\) 5.41016 0.258213 0.129106 0.991631i \(-0.458789\pi\)
0.129106 + 0.991631i \(0.458789\pi\)
\(440\) 1.22204 0.0582587
\(441\) 0 0
\(442\) 1.49561 0.0711389
\(443\) 11.5485 0.548686 0.274343 0.961632i \(-0.411540\pi\)
0.274343 + 0.961632i \(0.411540\pi\)
\(444\) 0 0
\(445\) 0.437156 0.0207232
\(446\) 1.98501 0.0939930
\(447\) 0 0
\(448\) 1.97638 0.0933754
\(449\) 37.7469 1.78138 0.890692 0.454607i \(-0.150220\pi\)
0.890692 + 0.454607i \(0.150220\pi\)
\(450\) 0 0
\(451\) 22.6123 1.06477
\(452\) −3.13390 −0.147406
\(453\) 0 0
\(454\) 4.00740 0.188077
\(455\) 0.258992 0.0121417
\(456\) 0 0
\(457\) −4.86171 −0.227421 −0.113711 0.993514i \(-0.536274\pi\)
−0.113711 + 0.993514i \(0.536274\pi\)
\(458\) 0.431921 0.0201823
\(459\) 0 0
\(460\) −1.83765 −0.0856808
\(461\) 22.2371 1.03569 0.517843 0.855475i \(-0.326735\pi\)
0.517843 + 0.855475i \(0.326735\pi\)
\(462\) 0 0
\(463\) −34.0919 −1.58439 −0.792193 0.610271i \(-0.791061\pi\)
−0.792193 + 0.610271i \(0.791061\pi\)
\(464\) 22.0402 1.02319
\(465\) 0 0
\(466\) 11.5905 0.536919
\(467\) 18.0906 0.837131 0.418566 0.908187i \(-0.362533\pi\)
0.418566 + 0.908187i \(0.362533\pi\)
\(468\) 0 0
\(469\) 14.4350 0.666548
\(470\) 0.112900 0.00520771
\(471\) 0 0
\(472\) −22.0993 −1.01720
\(473\) 11.2211 0.515948
\(474\) 0 0
\(475\) −41.6097 −1.90918
\(476\) 2.19980 0.100828
\(477\) 0 0
\(478\) 5.45871 0.249675
\(479\) −13.8985 −0.635037 −0.317519 0.948252i \(-0.602850\pi\)
−0.317519 + 0.948252i \(0.602850\pi\)
\(480\) 0 0
\(481\) −13.1900 −0.601413
\(482\) −6.65380 −0.303072
\(483\) 0 0
\(484\) −27.9916 −1.27235
\(485\) 0.722001 0.0327844
\(486\) 0 0
\(487\) −1.46671 −0.0664628 −0.0332314 0.999448i \(-0.510580\pi\)
−0.0332314 + 0.999448i \(0.510580\pi\)
\(488\) −20.6462 −0.934611
\(489\) 0 0
\(490\) −0.0629236 −0.00284260
\(491\) −13.3834 −0.603983 −0.301992 0.953311i \(-0.597651\pi\)
−0.301992 + 0.953311i \(0.597651\pi\)
\(492\) 0 0
\(493\) 11.8721 0.534694
\(494\) 9.73892 0.438175
\(495\) 0 0
\(496\) 16.1003 0.722924
\(497\) −4.12479 −0.185022
\(498\) 0 0
\(499\) 21.8737 0.979201 0.489601 0.871947i \(-0.337143\pi\)
0.489601 + 0.871947i \(0.337143\pi\)
\(500\) 2.02555 0.0905855
\(501\) 0 0
\(502\) −11.4859 −0.512642
\(503\) 41.4018 1.84601 0.923007 0.384784i \(-0.125724\pi\)
0.923007 + 0.384784i \(0.125724\pi\)
\(504\) 0 0
\(505\) −0.0701554 −0.00312187
\(506\) −25.2089 −1.12067
\(507\) 0 0
\(508\) 1.71647 0.0761561
\(509\) −12.5384 −0.555757 −0.277878 0.960616i \(-0.589631\pi\)
−0.277878 + 0.960616i \(0.589631\pi\)
\(510\) 0 0
\(511\) −16.5937 −0.734064
\(512\) −21.8473 −0.965523
\(513\) 0 0
\(514\) −10.4281 −0.459965
\(515\) 0.924192 0.0407248
\(516\) 0 0
\(517\) −9.37614 −0.412362
\(518\) 3.20459 0.140802
\(519\) 0 0
\(520\) −0.512527 −0.0224758
\(521\) 25.5657 1.12006 0.560028 0.828474i \(-0.310790\pi\)
0.560028 + 0.828474i \(0.310790\pi\)
\(522\) 0 0
\(523\) −0.965570 −0.0422214 −0.0211107 0.999777i \(-0.506720\pi\)
−0.0211107 + 0.999777i \(0.506720\pi\)
\(524\) −14.1335 −0.617427
\(525\) 0 0
\(526\) −3.84343 −0.167581
\(527\) 8.67256 0.377783
\(528\) 0 0
\(529\) 59.0774 2.56858
\(530\) 0.268730 0.0116729
\(531\) 0 0
\(532\) 14.3244 0.621040
\(533\) −9.48361 −0.410781
\(534\) 0 0
\(535\) 1.30249 0.0563115
\(536\) −28.5659 −1.23386
\(537\) 0 0
\(538\) −7.71615 −0.332667
\(539\) 5.22568 0.225086
\(540\) 0 0
\(541\) −1.43271 −0.0615970 −0.0307985 0.999526i \(-0.509805\pi\)
−0.0307985 + 0.999526i \(0.509805\pi\)
\(542\) 9.24292 0.397017
\(543\) 0 0
\(544\) −6.69592 −0.287085
\(545\) −0.0389436 −0.00166816
\(546\) 0 0
\(547\) −13.4988 −0.577167 −0.288584 0.957455i \(-0.593184\pi\)
−0.288584 + 0.957455i \(0.593184\pi\)
\(548\) −6.51724 −0.278403
\(549\) 0 0
\(550\) 13.8739 0.591583
\(551\) 77.3075 3.29341
\(552\) 0 0
\(553\) 11.3780 0.483840
\(554\) 13.5446 0.575455
\(555\) 0 0
\(556\) 7.79122 0.330421
\(557\) −29.6076 −1.25451 −0.627257 0.778812i \(-0.715822\pi\)
−0.627257 + 0.778812i \(0.715822\pi\)
\(558\) 0 0
\(559\) −4.70615 −0.199049
\(560\) −0.281156 −0.0118810
\(561\) 0 0
\(562\) 5.74683 0.242415
\(563\) −18.3169 −0.771963 −0.385982 0.922506i \(-0.626137\pi\)
−0.385982 + 0.922506i \(0.626137\pi\)
\(564\) 0 0
\(565\) 0.215756 0.00907694
\(566\) −0.785082 −0.0329995
\(567\) 0 0
\(568\) 8.16267 0.342498
\(569\) 35.3075 1.48017 0.740083 0.672516i \(-0.234786\pi\)
0.740083 + 0.672516i \(0.234786\pi\)
\(570\) 0 0
\(571\) 3.93893 0.164839 0.0824196 0.996598i \(-0.473735\pi\)
0.0824196 + 0.996598i \(0.473735\pi\)
\(572\) 19.6585 0.821964
\(573\) 0 0
\(574\) 2.30410 0.0961711
\(575\) −45.1718 −1.88379
\(576\) 0 0
\(577\) 33.2150 1.38276 0.691380 0.722492i \(-0.257003\pi\)
0.691380 + 0.722492i \(0.257003\pi\)
\(578\) 8.17751 0.340140
\(579\) 0 0
\(580\) −1.87903 −0.0780224
\(581\) 6.41125 0.265984
\(582\) 0 0
\(583\) −22.3175 −0.924296
\(584\) 32.8378 1.35884
\(585\) 0 0
\(586\) 6.75649 0.279108
\(587\) −24.6822 −1.01874 −0.509372 0.860546i \(-0.670122\pi\)
−0.509372 + 0.860546i \(0.670122\pi\)
\(588\) 0 0
\(589\) 56.4729 2.32692
\(590\) 0.702689 0.0289292
\(591\) 0 0
\(592\) 14.3188 0.588498
\(593\) −6.61375 −0.271594 −0.135797 0.990737i \(-0.543360\pi\)
−0.135797 + 0.990737i \(0.543360\pi\)
\(594\) 0 0
\(595\) −0.151447 −0.00620873
\(596\) −8.39912 −0.344041
\(597\) 0 0
\(598\) 10.5726 0.432347
\(599\) 25.3672 1.03648 0.518239 0.855236i \(-0.326588\pi\)
0.518239 + 0.855236i \(0.326588\pi\)
\(600\) 0 0
\(601\) −10.2704 −0.418940 −0.209470 0.977815i \(-0.567174\pi\)
−0.209470 + 0.977815i \(0.567174\pi\)
\(602\) 1.14339 0.0466010
\(603\) 0 0
\(604\) −19.1157 −0.777807
\(605\) 1.92711 0.0783482
\(606\) 0 0
\(607\) −0.280009 −0.0113652 −0.00568261 0.999984i \(-0.501809\pi\)
−0.00568261 + 0.999984i \(0.501809\pi\)
\(608\) −43.6016 −1.76828
\(609\) 0 0
\(610\) 0.656485 0.0265803
\(611\) 3.93237 0.159087
\(612\) 0 0
\(613\) −34.3249 −1.38637 −0.693185 0.720760i \(-0.743793\pi\)
−0.693185 + 0.720760i \(0.743793\pi\)
\(614\) 4.04820 0.163372
\(615\) 0 0
\(616\) −10.3412 −0.416660
\(617\) 7.14920 0.287816 0.143908 0.989591i \(-0.454033\pi\)
0.143908 + 0.989591i \(0.454033\pi\)
\(618\) 0 0
\(619\) 10.8686 0.436846 0.218423 0.975854i \(-0.429909\pi\)
0.218423 + 0.975854i \(0.429909\pi\)
\(620\) −1.37262 −0.0551259
\(621\) 0 0
\(622\) −18.4486 −0.739723
\(623\) −3.69932 −0.148210
\(624\) 0 0
\(625\) 24.7907 0.991629
\(626\) 7.14273 0.285481
\(627\) 0 0
\(628\) 27.3996 1.09336
\(629\) 7.71293 0.307535
\(630\) 0 0
\(631\) −46.4839 −1.85050 −0.925248 0.379362i \(-0.876143\pi\)
−0.925248 + 0.379362i \(0.876143\pi\)
\(632\) −22.5162 −0.895645
\(633\) 0 0
\(634\) −13.5135 −0.536689
\(635\) −0.118172 −0.00468951
\(636\) 0 0
\(637\) −2.19166 −0.0868366
\(638\) −25.7765 −1.02050
\(639\) 0 0
\(640\) 1.35919 0.0537268
\(641\) −13.4471 −0.531127 −0.265563 0.964093i \(-0.585558\pi\)
−0.265563 + 0.964093i \(0.585558\pi\)
\(642\) 0 0
\(643\) −40.9464 −1.61477 −0.807383 0.590027i \(-0.799117\pi\)
−0.807383 + 0.590027i \(0.799117\pi\)
\(644\) 15.5506 0.612781
\(645\) 0 0
\(646\) −5.69489 −0.224062
\(647\) 9.04141 0.355454 0.177727 0.984080i \(-0.443126\pi\)
0.177727 + 0.984080i \(0.443126\pi\)
\(648\) 0 0
\(649\) −58.3568 −2.29071
\(650\) −5.81871 −0.228229
\(651\) 0 0
\(652\) 19.7209 0.772331
\(653\) 9.65099 0.377672 0.188836 0.982009i \(-0.439528\pi\)
0.188836 + 0.982009i \(0.439528\pi\)
\(654\) 0 0
\(655\) 0.973037 0.0380197
\(656\) 10.2952 0.401960
\(657\) 0 0
\(658\) −0.955391 −0.0372450
\(659\) 25.6398 0.998786 0.499393 0.866376i \(-0.333556\pi\)
0.499393 + 0.866376i \(0.333556\pi\)
\(660\) 0 0
\(661\) 5.36192 0.208555 0.104277 0.994548i \(-0.466747\pi\)
0.104277 + 0.994548i \(0.466747\pi\)
\(662\) −2.30514 −0.0895918
\(663\) 0 0
\(664\) −12.6874 −0.492367
\(665\) −0.986174 −0.0382422
\(666\) 0 0
\(667\) 83.9256 3.24961
\(668\) −24.4985 −0.947874
\(669\) 0 0
\(670\) 0.908304 0.0350908
\(671\) −54.5197 −2.10471
\(672\) 0 0
\(673\) 10.4649 0.403392 0.201696 0.979448i \(-0.435355\pi\)
0.201696 + 0.979448i \(0.435355\pi\)
\(674\) 16.4638 0.634161
\(675\) 0 0
\(676\) 14.0693 0.541127
\(677\) −9.28929 −0.357017 −0.178508 0.983938i \(-0.557127\pi\)
−0.178508 + 0.983938i \(0.557127\pi\)
\(678\) 0 0
\(679\) −6.10975 −0.234471
\(680\) 0.299703 0.0114931
\(681\) 0 0
\(682\) −18.8297 −0.721025
\(683\) 16.7140 0.639544 0.319772 0.947494i \(-0.396394\pi\)
0.319772 + 0.947494i \(0.396394\pi\)
\(684\) 0 0
\(685\) 0.448685 0.0171434
\(686\) 0.532475 0.0203300
\(687\) 0 0
\(688\) 5.10889 0.194775
\(689\) 9.35998 0.356587
\(690\) 0 0
\(691\) 32.0986 1.22109 0.610544 0.791982i \(-0.290951\pi\)
0.610544 + 0.791982i \(0.290951\pi\)
\(692\) 5.41038 0.205672
\(693\) 0 0
\(694\) 6.07903 0.230757
\(695\) −0.536394 −0.0203466
\(696\) 0 0
\(697\) 5.54559 0.210054
\(698\) 13.3789 0.506399
\(699\) 0 0
\(700\) −8.55838 −0.323476
\(701\) −8.24121 −0.311266 −0.155633 0.987815i \(-0.549742\pi\)
−0.155633 + 0.987815i \(0.549742\pi\)
\(702\) 0 0
\(703\) 50.2241 1.89424
\(704\) −10.3279 −0.389249
\(705\) 0 0
\(706\) 8.73610 0.328787
\(707\) 0.593672 0.0223273
\(708\) 0 0
\(709\) 18.5251 0.695723 0.347862 0.937546i \(-0.386908\pi\)
0.347862 + 0.937546i \(0.386908\pi\)
\(710\) −0.259547 −0.00974062
\(711\) 0 0
\(712\) 7.32069 0.274354
\(713\) 61.3073 2.29598
\(714\) 0 0
\(715\) −1.35341 −0.0506146
\(716\) 37.8807 1.41567
\(717\) 0 0
\(718\) −9.47161 −0.353477
\(719\) 29.8462 1.11308 0.556538 0.830822i \(-0.312129\pi\)
0.556538 + 0.830822i \(0.312129\pi\)
\(720\) 0 0
\(721\) −7.82074 −0.291260
\(722\) −26.9662 −1.00358
\(723\) 0 0
\(724\) −30.5327 −1.13474
\(725\) −46.1889 −1.71541
\(726\) 0 0
\(727\) −42.7041 −1.58381 −0.791904 0.610645i \(-0.790910\pi\)
−0.791904 + 0.610645i \(0.790910\pi\)
\(728\) 4.33713 0.160745
\(729\) 0 0
\(730\) −1.04414 −0.0386453
\(731\) 2.75195 0.101785
\(732\) 0 0
\(733\) 13.0748 0.482930 0.241465 0.970409i \(-0.422372\pi\)
0.241465 + 0.970409i \(0.422372\pi\)
\(734\) 12.7471 0.470504
\(735\) 0 0
\(736\) −47.3342 −1.74476
\(737\) −75.4328 −2.77860
\(738\) 0 0
\(739\) −11.8892 −0.437351 −0.218676 0.975798i \(-0.570174\pi\)
−0.218676 + 0.975798i \(0.570174\pi\)
\(740\) −1.22074 −0.0448753
\(741\) 0 0
\(742\) −2.27406 −0.0834834
\(743\) −24.3686 −0.893998 −0.446999 0.894534i \(-0.647507\pi\)
−0.446999 + 0.894534i \(0.647507\pi\)
\(744\) 0 0
\(745\) 0.578245 0.0211853
\(746\) −12.8359 −0.469957
\(747\) 0 0
\(748\) −11.4954 −0.420315
\(749\) −11.0220 −0.402734
\(750\) 0 0
\(751\) 14.6876 0.535957 0.267978 0.963425i \(-0.413644\pi\)
0.267978 + 0.963425i \(0.413644\pi\)
\(752\) −4.26889 −0.155670
\(753\) 0 0
\(754\) 10.8107 0.393703
\(755\) 1.31604 0.0478955
\(756\) 0 0
\(757\) −2.79722 −0.101667 −0.0508334 0.998707i \(-0.516188\pi\)
−0.0508334 + 0.998707i \(0.516188\pi\)
\(758\) 5.19110 0.188549
\(759\) 0 0
\(760\) 1.95157 0.0707908
\(761\) −36.1872 −1.31179 −0.655893 0.754854i \(-0.727708\pi\)
−0.655893 + 0.754854i \(0.727708\pi\)
\(762\) 0 0
\(763\) 0.329550 0.0119305
\(764\) 16.5580 0.599046
\(765\) 0 0
\(766\) 11.2320 0.405829
\(767\) 24.4749 0.883739
\(768\) 0 0
\(769\) −9.22886 −0.332801 −0.166401 0.986058i \(-0.553214\pi\)
−0.166401 + 0.986058i \(0.553214\pi\)
\(770\) 0.328818 0.0118498
\(771\) 0 0
\(772\) −20.9106 −0.752588
\(773\) 4.09236 0.147192 0.0735959 0.997288i \(-0.476552\pi\)
0.0735959 + 0.997288i \(0.476552\pi\)
\(774\) 0 0
\(775\) −33.7408 −1.21201
\(776\) 12.0908 0.434033
\(777\) 0 0
\(778\) 6.11253 0.219145
\(779\) 36.1111 1.29381
\(780\) 0 0
\(781\) 21.5548 0.771293
\(782\) −6.18241 −0.221083
\(783\) 0 0
\(784\) 2.37921 0.0849718
\(785\) −1.88635 −0.0673267
\(786\) 0 0
\(787\) −10.3020 −0.367225 −0.183613 0.982999i \(-0.558779\pi\)
−0.183613 + 0.982999i \(0.558779\pi\)
\(788\) 8.71147 0.310333
\(789\) 0 0
\(790\) 0.715943 0.0254721
\(791\) −1.82578 −0.0649174
\(792\) 0 0
\(793\) 22.8656 0.811982
\(794\) −17.7957 −0.631545
\(795\) 0 0
\(796\) −5.90262 −0.209213
\(797\) −40.9311 −1.44986 −0.724928 0.688825i \(-0.758127\pi\)
−0.724928 + 0.688825i \(0.758127\pi\)
\(798\) 0 0
\(799\) −2.29947 −0.0813495
\(800\) 26.0507 0.921030
\(801\) 0 0
\(802\) 0.269604 0.00952004
\(803\) 86.7135 3.06005
\(804\) 0 0
\(805\) −1.07060 −0.0377336
\(806\) 7.89718 0.278166
\(807\) 0 0
\(808\) −1.17483 −0.0413305
\(809\) −9.34786 −0.328653 −0.164327 0.986406i \(-0.552545\pi\)
−0.164327 + 0.986406i \(0.552545\pi\)
\(810\) 0 0
\(811\) −10.7341 −0.376925 −0.188463 0.982080i \(-0.560350\pi\)
−0.188463 + 0.982080i \(0.560350\pi\)
\(812\) 15.9008 0.558008
\(813\) 0 0
\(814\) −16.7461 −0.586952
\(815\) −1.35770 −0.0475583
\(816\) 0 0
\(817\) 17.9198 0.626934
\(818\) −14.7436 −0.515499
\(819\) 0 0
\(820\) −0.877712 −0.0306510
\(821\) −13.9500 −0.486859 −0.243430 0.969919i \(-0.578272\pi\)
−0.243430 + 0.969919i \(0.578272\pi\)
\(822\) 0 0
\(823\) −7.62995 −0.265963 −0.132982 0.991119i \(-0.542455\pi\)
−0.132982 + 0.991119i \(0.542455\pi\)
\(824\) 15.4767 0.539156
\(825\) 0 0
\(826\) −5.94632 −0.206899
\(827\) 50.6423 1.76101 0.880503 0.474040i \(-0.157205\pi\)
0.880503 + 0.474040i \(0.157205\pi\)
\(828\) 0 0
\(829\) 27.1635 0.943428 0.471714 0.881752i \(-0.343635\pi\)
0.471714 + 0.881752i \(0.343635\pi\)
\(830\) 0.403419 0.0140029
\(831\) 0 0
\(832\) 4.33155 0.150170
\(833\) 1.28158 0.0444042
\(834\) 0 0
\(835\) 1.68662 0.0583678
\(836\) −74.8545 −2.58890
\(837\) 0 0
\(838\) 2.78010 0.0960368
\(839\) −18.7378 −0.646900 −0.323450 0.946245i \(-0.604843\pi\)
−0.323450 + 0.946245i \(0.604843\pi\)
\(840\) 0 0
\(841\) 56.8153 1.95915
\(842\) 16.1173 0.555440
\(843\) 0 0
\(844\) 22.9695 0.790643
\(845\) −0.968614 −0.0333213
\(846\) 0 0
\(847\) −16.3077 −0.560339
\(848\) −10.1610 −0.348930
\(849\) 0 0
\(850\) 3.40252 0.116706
\(851\) 54.5236 1.86905
\(852\) 0 0
\(853\) 49.8498 1.70682 0.853412 0.521238i \(-0.174530\pi\)
0.853412 + 0.521238i \(0.174530\pi\)
\(854\) −5.55533 −0.190100
\(855\) 0 0
\(856\) 21.8117 0.745508
\(857\) −39.2898 −1.34211 −0.671057 0.741406i \(-0.734159\pi\)
−0.671057 + 0.741406i \(0.734159\pi\)
\(858\) 0 0
\(859\) 37.3922 1.27581 0.637903 0.770117i \(-0.279802\pi\)
0.637903 + 0.770117i \(0.279802\pi\)
\(860\) −0.435556 −0.0148524
\(861\) 0 0
\(862\) −6.61949 −0.225461
\(863\) −41.1465 −1.40064 −0.700322 0.713827i \(-0.746960\pi\)
−0.700322 + 0.713827i \(0.746960\pi\)
\(864\) 0 0
\(865\) −0.372483 −0.0126648
\(866\) −14.7325 −0.500630
\(867\) 0 0
\(868\) 11.6155 0.394255
\(869\) −59.4575 −2.01696
\(870\) 0 0
\(871\) 31.6366 1.07197
\(872\) −0.652157 −0.0220848
\(873\) 0 0
\(874\) −40.2578 −1.36174
\(875\) 1.18007 0.0398936
\(876\) 0 0
\(877\) −40.2479 −1.35907 −0.679537 0.733641i \(-0.737819\pi\)
−0.679537 + 0.733641i \(0.737819\pi\)
\(878\) −2.88077 −0.0972214
\(879\) 0 0
\(880\) 1.46923 0.0495277
\(881\) −45.5773 −1.53554 −0.767769 0.640727i \(-0.778633\pi\)
−0.767769 + 0.640727i \(0.778633\pi\)
\(882\) 0 0
\(883\) 11.2401 0.378259 0.189130 0.981952i \(-0.439433\pi\)
0.189130 + 0.981952i \(0.439433\pi\)
\(884\) 4.82120 0.162155
\(885\) 0 0
\(886\) −6.14929 −0.206589
\(887\) −18.7760 −0.630437 −0.315219 0.949019i \(-0.602078\pi\)
−0.315219 + 0.949019i \(0.602078\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −0.232775 −0.00780262
\(891\) 0 0
\(892\) 6.39882 0.214248
\(893\) −14.9734 −0.501066
\(894\) 0 0
\(895\) −2.60793 −0.0871736
\(896\) −11.5018 −0.384249
\(897\) 0 0
\(898\) −20.0993 −0.670721
\(899\) 62.6878 2.09076
\(900\) 0 0
\(901\) −5.47330 −0.182342
\(902\) −12.0405 −0.400903
\(903\) 0 0
\(904\) 3.61309 0.120170
\(905\) 2.10205 0.0698746
\(906\) 0 0
\(907\) −28.0916 −0.932767 −0.466383 0.884583i \(-0.654443\pi\)
−0.466383 + 0.884583i \(0.654443\pi\)
\(908\) 12.9181 0.428703
\(909\) 0 0
\(910\) −0.137907 −0.00457157
\(911\) 11.5012 0.381053 0.190526 0.981682i \(-0.438981\pi\)
0.190526 + 0.981682i \(0.438981\pi\)
\(912\) 0 0
\(913\) −33.5031 −1.10879
\(914\) 2.58874 0.0856279
\(915\) 0 0
\(916\) 1.39233 0.0460038
\(917\) −8.23408 −0.271913
\(918\) 0 0
\(919\) −21.4575 −0.707818 −0.353909 0.935280i \(-0.615148\pi\)
−0.353909 + 0.935280i \(0.615148\pi\)
\(920\) 2.11864 0.0698493
\(921\) 0 0
\(922\) −11.8407 −0.389953
\(923\) −9.04013 −0.297559
\(924\) 0 0
\(925\) −30.0074 −0.986637
\(926\) 18.1531 0.596548
\(927\) 0 0
\(928\) −48.4001 −1.58881
\(929\) −58.5953 −1.92245 −0.961225 0.275766i \(-0.911068\pi\)
−0.961225 + 0.275766i \(0.911068\pi\)
\(930\) 0 0
\(931\) 8.34525 0.273504
\(932\) 37.3628 1.22386
\(933\) 0 0
\(934\) −9.63277 −0.315194
\(935\) 0.791413 0.0258820
\(936\) 0 0
\(937\) −37.4295 −1.22277 −0.611384 0.791334i \(-0.709387\pi\)
−0.611384 + 0.791334i \(0.709387\pi\)
\(938\) −7.68629 −0.250966
\(939\) 0 0
\(940\) 0.363942 0.0118705
\(941\) 44.8032 1.46054 0.730272 0.683157i \(-0.239393\pi\)
0.730272 + 0.683157i \(0.239393\pi\)
\(942\) 0 0
\(943\) 39.2024 1.27661
\(944\) −26.5694 −0.864761
\(945\) 0 0
\(946\) −5.97497 −0.194263
\(947\) −30.9990 −1.00733 −0.503666 0.863898i \(-0.668016\pi\)
−0.503666 + 0.863898i \(0.668016\pi\)
\(948\) 0 0
\(949\) −36.3678 −1.18055
\(950\) 22.1561 0.718840
\(951\) 0 0
\(952\) −2.53616 −0.0821974
\(953\) −45.1029 −1.46103 −0.730513 0.682899i \(-0.760719\pi\)
−0.730513 + 0.682899i \(0.760719\pi\)
\(954\) 0 0
\(955\) −1.13995 −0.0368878
\(956\) 17.5965 0.569112
\(957\) 0 0
\(958\) 7.40059 0.239102
\(959\) −3.79688 −0.122608
\(960\) 0 0
\(961\) 14.7932 0.477201
\(962\) 7.02335 0.226442
\(963\) 0 0
\(964\) −21.4490 −0.690826
\(965\) 1.43961 0.0463426
\(966\) 0 0
\(967\) 57.9650 1.86403 0.932015 0.362420i \(-0.118049\pi\)
0.932015 + 0.362420i \(0.118049\pi\)
\(968\) 32.2717 1.03725
\(969\) 0 0
\(970\) −0.384448 −0.0123439
\(971\) −48.9510 −1.57091 −0.785457 0.618916i \(-0.787572\pi\)
−0.785457 + 0.618916i \(0.787572\pi\)
\(972\) 0 0
\(973\) 4.53910 0.145517
\(974\) 0.780984 0.0250243
\(975\) 0 0
\(976\) −24.8224 −0.794546
\(977\) −38.2915 −1.22505 −0.612526 0.790450i \(-0.709847\pi\)
−0.612526 + 0.790450i \(0.709847\pi\)
\(978\) 0 0
\(979\) 19.3314 0.617836
\(980\) −0.202839 −0.00647944
\(981\) 0 0
\(982\) 7.12632 0.227410
\(983\) 4.26394 0.135999 0.0679993 0.997685i \(-0.478338\pi\)
0.0679993 + 0.997685i \(0.478338\pi\)
\(984\) 0 0
\(985\) −0.599749 −0.0191096
\(986\) −6.32162 −0.201321
\(987\) 0 0
\(988\) 31.3941 0.998779
\(989\) 19.4539 0.618596
\(990\) 0 0
\(991\) 8.67251 0.275491 0.137746 0.990468i \(-0.456014\pi\)
0.137746 + 0.990468i \(0.456014\pi\)
\(992\) −35.3561 −1.12256
\(993\) 0 0
\(994\) 2.19635 0.0696640
\(995\) 0.406371 0.0128828
\(996\) 0 0
\(997\) 31.2119 0.988492 0.494246 0.869322i \(-0.335444\pi\)
0.494246 + 0.869322i \(0.335444\pi\)
\(998\) −11.6472 −0.368686
\(999\) 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.t.1.6 16
3.2 odd 2 889.2.a.c.1.11 16
21.20 even 2 6223.2.a.k.1.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.c.1.11 16 3.2 odd 2
6223.2.a.k.1.11 16 21.20 even 2
8001.2.a.t.1.6 16 1.1 even 1 trivial