Properties

Label 8006.2.a.d.1.16
Level $8006$
Weight $2$
Character 8006.1
Self dual yes
Analytic conductor $63.928$
Analytic rank $0$
Dimension $98$
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] = [8006,2,Mod(1,8006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8006 = 2 \cdot 4003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8006.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: \(63.9282318582\)
Analytic rank: \(0\)
Dimension: \(98\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 8006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.29904 q^{3} +1.00000 q^{4} -0.379667 q^{5} -2.29904 q^{6} -2.21994 q^{7} +1.00000 q^{8} +2.28559 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.29904 q^{3} +1.00000 q^{4} -0.379667 q^{5} -2.29904 q^{6} -2.21994 q^{7} +1.00000 q^{8} +2.28559 q^{9} -0.379667 q^{10} -3.25470 q^{11} -2.29904 q^{12} -0.129405 q^{13} -2.21994 q^{14} +0.872870 q^{15} +1.00000 q^{16} -3.78903 q^{17} +2.28559 q^{18} +7.10879 q^{19} -0.379667 q^{20} +5.10373 q^{21} -3.25470 q^{22} -8.35620 q^{23} -2.29904 q^{24} -4.85585 q^{25} -0.129405 q^{26} +1.64246 q^{27} -2.21994 q^{28} -0.188981 q^{29} +0.872870 q^{30} -3.93199 q^{31} +1.00000 q^{32} +7.48269 q^{33} -3.78903 q^{34} +0.842837 q^{35} +2.28559 q^{36} +2.45468 q^{37} +7.10879 q^{38} +0.297507 q^{39} -0.379667 q^{40} -2.16972 q^{41} +5.10373 q^{42} -1.66164 q^{43} -3.25470 q^{44} -0.867762 q^{45} -8.35620 q^{46} -9.30575 q^{47} -2.29904 q^{48} -2.07188 q^{49} -4.85585 q^{50} +8.71113 q^{51} -0.129405 q^{52} +6.81916 q^{53} +1.64246 q^{54} +1.23570 q^{55} -2.21994 q^{56} -16.3434 q^{57} -0.188981 q^{58} +5.28745 q^{59} +0.872870 q^{60} -0.787063 q^{61} -3.93199 q^{62} -5.07386 q^{63} +1.00000 q^{64} +0.0491307 q^{65} +7.48269 q^{66} -2.62637 q^{67} -3.78903 q^{68} +19.2112 q^{69} +0.842837 q^{70} +11.3258 q^{71} +2.28559 q^{72} -15.0058 q^{73} +2.45468 q^{74} +11.1638 q^{75} +7.10879 q^{76} +7.22523 q^{77} +0.297507 q^{78} +4.38020 q^{79} -0.379667 q^{80} -10.6329 q^{81} -2.16972 q^{82} +9.50181 q^{83} +5.10373 q^{84} +1.43857 q^{85} -1.66164 q^{86} +0.434475 q^{87} -3.25470 q^{88} -13.8358 q^{89} -0.867762 q^{90} +0.287270 q^{91} -8.35620 q^{92} +9.03979 q^{93} -9.30575 q^{94} -2.69897 q^{95} -2.29904 q^{96} -2.61324 q^{97} -2.07188 q^{98} -7.43890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9} + 4 q^{10} + 51 q^{11} + 16 q^{12} + 31 q^{13} + 29 q^{14} + 57 q^{15} + 98 q^{16} + 35 q^{17} + 130 q^{18} + 77 q^{19} + 4 q^{20} + 46 q^{21} + 51 q^{22} + 73 q^{23} + 16 q^{24} + 150 q^{25} + 31 q^{26} + 52 q^{27} + 29 q^{28} + 20 q^{29} + 57 q^{30} + 59 q^{31} + 98 q^{32} + 27 q^{33} + 35 q^{34} + 48 q^{35} + 130 q^{36} + 41 q^{37} + 77 q^{38} + 64 q^{39} + 4 q^{40} + 29 q^{41} + 46 q^{42} + 94 q^{43} + 51 q^{44} - 3 q^{45} + 73 q^{46} + 58 q^{47} + 16 q^{48} + 149 q^{49} + 150 q^{50} + 58 q^{51} + 31 q^{52} - 11 q^{53} + 52 q^{54} + 56 q^{55} + 29 q^{56} + 64 q^{57} + 20 q^{58} + 45 q^{59} + 57 q^{60} + 73 q^{61} + 59 q^{62} + 53 q^{63} + 98 q^{64} + 39 q^{65} + 27 q^{66} + 133 q^{67} + 35 q^{68} + 13 q^{69} + 48 q^{70} + 67 q^{71} + 130 q^{72} + 42 q^{73} + 41 q^{74} + 36 q^{75} + 77 q^{76} - 25 q^{77} + 64 q^{78} + 154 q^{79} + 4 q^{80} + 198 q^{81} + 29 q^{82} + 69 q^{83} + 46 q^{84} + 81 q^{85} + 94 q^{86} + 25 q^{87} + 51 q^{88} + 32 q^{89} - 3 q^{90} + 95 q^{91} + 73 q^{92} - 23 q^{93} + 58 q^{94} + 50 q^{95} + 16 q^{96} + 76 q^{97} + 149 q^{98} + 149 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.29904 −1.32735 −0.663676 0.748020i \(-0.731005\pi\)
−0.663676 + 0.748020i \(0.731005\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.379667 −0.169792 −0.0848961 0.996390i \(-0.527056\pi\)
−0.0848961 + 0.996390i \(0.527056\pi\)
\(6\) −2.29904 −0.938579
\(7\) −2.21994 −0.839058 −0.419529 0.907742i \(-0.637805\pi\)
−0.419529 + 0.907742i \(0.637805\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.28559 0.761862
\(10\) −0.379667 −0.120061
\(11\) −3.25470 −0.981329 −0.490665 0.871348i \(-0.663246\pi\)
−0.490665 + 0.871348i \(0.663246\pi\)
\(12\) −2.29904 −0.663676
\(13\) −0.129405 −0.0358904 −0.0179452 0.999839i \(-0.505712\pi\)
−0.0179452 + 0.999839i \(0.505712\pi\)
\(14\) −2.21994 −0.593303
\(15\) 0.872870 0.225374
\(16\) 1.00000 0.250000
\(17\) −3.78903 −0.918974 −0.459487 0.888184i \(-0.651967\pi\)
−0.459487 + 0.888184i \(0.651967\pi\)
\(18\) 2.28559 0.538718
\(19\) 7.10879 1.63087 0.815434 0.578850i \(-0.196498\pi\)
0.815434 + 0.578850i \(0.196498\pi\)
\(20\) −0.379667 −0.0848961
\(21\) 5.10373 1.11372
\(22\) −3.25470 −0.693905
\(23\) −8.35620 −1.74239 −0.871194 0.490938i \(-0.836654\pi\)
−0.871194 + 0.490938i \(0.836654\pi\)
\(24\) −2.29904 −0.469290
\(25\) −4.85585 −0.971171
\(26\) −0.129405 −0.0253783
\(27\) 1.64246 0.316092
\(28\) −2.21994 −0.419529
\(29\) −0.188981 −0.0350929 −0.0175464 0.999846i \(-0.505585\pi\)
−0.0175464 + 0.999846i \(0.505585\pi\)
\(30\) 0.872870 0.159363
\(31\) −3.93199 −0.706206 −0.353103 0.935585i \(-0.614873\pi\)
−0.353103 + 0.935585i \(0.614873\pi\)
\(32\) 1.00000 0.176777
\(33\) 7.48269 1.30257
\(34\) −3.78903 −0.649813
\(35\) 0.842837 0.142465
\(36\) 2.28559 0.380931
\(37\) 2.45468 0.403546 0.201773 0.979432i \(-0.435330\pi\)
0.201773 + 0.979432i \(0.435330\pi\)
\(38\) 7.10879 1.15320
\(39\) 0.297507 0.0476392
\(40\) −0.379667 −0.0600306
\(41\) −2.16972 −0.338853 −0.169426 0.985543i \(-0.554191\pi\)
−0.169426 + 0.985543i \(0.554191\pi\)
\(42\) 5.10373 0.787522
\(43\) −1.66164 −0.253398 −0.126699 0.991941i \(-0.540438\pi\)
−0.126699 + 0.991941i \(0.540438\pi\)
\(44\) −3.25470 −0.490665
\(45\) −0.867762 −0.129358
\(46\) −8.35620 −1.23205
\(47\) −9.30575 −1.35738 −0.678692 0.734423i \(-0.737453\pi\)
−0.678692 + 0.734423i \(0.737453\pi\)
\(48\) −2.29904 −0.331838
\(49\) −2.07188 −0.295982
\(50\) −4.85585 −0.686721
\(51\) 8.71113 1.21980
\(52\) −0.129405 −0.0179452
\(53\) 6.81916 0.936684 0.468342 0.883547i \(-0.344852\pi\)
0.468342 + 0.883547i \(0.344852\pi\)
\(54\) 1.64246 0.223511
\(55\) 1.23570 0.166622
\(56\) −2.21994 −0.296652
\(57\) −16.3434 −2.16474
\(58\) −0.188981 −0.0248144
\(59\) 5.28745 0.688367 0.344183 0.938902i \(-0.388156\pi\)
0.344183 + 0.938902i \(0.388156\pi\)
\(60\) 0.872870 0.112687
\(61\) −0.787063 −0.100773 −0.0503865 0.998730i \(-0.516045\pi\)
−0.0503865 + 0.998730i \(0.516045\pi\)
\(62\) −3.93199 −0.499363
\(63\) −5.07386 −0.639246
\(64\) 1.00000 0.125000
\(65\) 0.0491307 0.00609391
\(66\) 7.48269 0.921055
\(67\) −2.62637 −0.320863 −0.160431 0.987047i \(-0.551289\pi\)
−0.160431 + 0.987047i \(0.551289\pi\)
\(68\) −3.78903 −0.459487
\(69\) 19.2112 2.31276
\(70\) 0.842837 0.100738
\(71\) 11.3258 1.34412 0.672060 0.740497i \(-0.265410\pi\)
0.672060 + 0.740497i \(0.265410\pi\)
\(72\) 2.28559 0.269359
\(73\) −15.0058 −1.75630 −0.878148 0.478390i \(-0.841221\pi\)
−0.878148 + 0.478390i \(0.841221\pi\)
\(74\) 2.45468 0.285350
\(75\) 11.1638 1.28908
\(76\) 7.10879 0.815434
\(77\) 7.22523 0.823392
\(78\) 0.297507 0.0336860
\(79\) 4.38020 0.492812 0.246406 0.969167i \(-0.420750\pi\)
0.246406 + 0.969167i \(0.420750\pi\)
\(80\) −0.379667 −0.0424481
\(81\) −10.6329 −1.18143
\(82\) −2.16972 −0.239605
\(83\) 9.50181 1.04296 0.521480 0.853264i \(-0.325380\pi\)
0.521480 + 0.853264i \(0.325380\pi\)
\(84\) 5.10373 0.556862
\(85\) 1.43857 0.156035
\(86\) −1.66164 −0.179180
\(87\) 0.434475 0.0465806
\(88\) −3.25470 −0.346952
\(89\) −13.8358 −1.46659 −0.733293 0.679912i \(-0.762018\pi\)
−0.733293 + 0.679912i \(0.762018\pi\)
\(90\) −0.867762 −0.0914701
\(91\) 0.287270 0.0301141
\(92\) −8.35620 −0.871194
\(93\) 9.03979 0.937383
\(94\) −9.30575 −0.959815
\(95\) −2.69897 −0.276909
\(96\) −2.29904 −0.234645
\(97\) −2.61324 −0.265335 −0.132667 0.991161i \(-0.542354\pi\)
−0.132667 + 0.991161i \(0.542354\pi\)
\(98\) −2.07188 −0.209291
\(99\) −7.43890 −0.747638
\(100\) −4.85585 −0.485585
\(101\) 0.258204 0.0256922 0.0128461 0.999917i \(-0.495911\pi\)
0.0128461 + 0.999917i \(0.495911\pi\)
\(102\) 8.71113 0.862530
\(103\) 12.6133 1.24283 0.621414 0.783483i \(-0.286559\pi\)
0.621414 + 0.783483i \(0.286559\pi\)
\(104\) −0.129405 −0.0126892
\(105\) −1.93772 −0.189102
\(106\) 6.81916 0.662335
\(107\) 0.595991 0.0576166 0.0288083 0.999585i \(-0.490829\pi\)
0.0288083 + 0.999585i \(0.490829\pi\)
\(108\) 1.64246 0.158046
\(109\) −6.41836 −0.614768 −0.307384 0.951586i \(-0.599454\pi\)
−0.307384 + 0.951586i \(0.599454\pi\)
\(110\) 1.23570 0.117820
\(111\) −5.64340 −0.535648
\(112\) −2.21994 −0.209764
\(113\) −13.6994 −1.28873 −0.644364 0.764719i \(-0.722878\pi\)
−0.644364 + 0.764719i \(0.722878\pi\)
\(114\) −16.3434 −1.53070
\(115\) 3.17257 0.295844
\(116\) −0.188981 −0.0175464
\(117\) −0.295766 −0.0273435
\(118\) 5.28745 0.486749
\(119\) 8.41141 0.771072
\(120\) 0.872870 0.0796817
\(121\) −0.406920 −0.0369927
\(122\) −0.787063 −0.0712573
\(123\) 4.98827 0.449777
\(124\) −3.93199 −0.353103
\(125\) 3.74194 0.334689
\(126\) −5.07386 −0.452015
\(127\) 2.57216 0.228243 0.114121 0.993467i \(-0.463595\pi\)
0.114121 + 0.993467i \(0.463595\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.82019 0.336349
\(130\) 0.0491307 0.00430905
\(131\) −7.36658 −0.643621 −0.321811 0.946804i \(-0.604291\pi\)
−0.321811 + 0.946804i \(0.604291\pi\)
\(132\) 7.48269 0.651285
\(133\) −15.7811 −1.36839
\(134\) −2.62637 −0.226884
\(135\) −0.623589 −0.0536700
\(136\) −3.78903 −0.324906
\(137\) 3.12853 0.267288 0.133644 0.991029i \(-0.457332\pi\)
0.133644 + 0.991029i \(0.457332\pi\)
\(138\) 19.2112 1.63537
\(139\) 10.9914 0.932278 0.466139 0.884711i \(-0.345645\pi\)
0.466139 + 0.884711i \(0.345645\pi\)
\(140\) 0.842837 0.0712327
\(141\) 21.3943 1.80173
\(142\) 11.3258 0.950436
\(143\) 0.421174 0.0352203
\(144\) 2.28559 0.190466
\(145\) 0.0717498 0.00595849
\(146\) −15.0058 −1.24189
\(147\) 4.76333 0.392872
\(148\) 2.45468 0.201773
\(149\) −2.96340 −0.242771 −0.121385 0.992605i \(-0.538734\pi\)
−0.121385 + 0.992605i \(0.538734\pi\)
\(150\) 11.1638 0.911521
\(151\) 5.97398 0.486156 0.243078 0.970007i \(-0.421843\pi\)
0.243078 + 0.970007i \(0.421843\pi\)
\(152\) 7.10879 0.576599
\(153\) −8.66015 −0.700132
\(154\) 7.22523 0.582226
\(155\) 1.49284 0.119908
\(156\) 0.297507 0.0238196
\(157\) 17.5520 1.40080 0.700401 0.713750i \(-0.253005\pi\)
0.700401 + 0.713750i \(0.253005\pi\)
\(158\) 4.38020 0.348470
\(159\) −15.6775 −1.24331
\(160\) −0.379667 −0.0300153
\(161\) 18.5502 1.46196
\(162\) −10.6329 −0.835396
\(163\) 11.9347 0.934794 0.467397 0.884048i \(-0.345192\pi\)
0.467397 + 0.884048i \(0.345192\pi\)
\(164\) −2.16972 −0.169426
\(165\) −2.84093 −0.221166
\(166\) 9.50181 0.737484
\(167\) 10.6172 0.821581 0.410790 0.911730i \(-0.365253\pi\)
0.410790 + 0.911730i \(0.365253\pi\)
\(168\) 5.10373 0.393761
\(169\) −12.9833 −0.998712
\(170\) 1.43857 0.110333
\(171\) 16.2478 1.24250
\(172\) −1.66164 −0.126699
\(173\) 12.1446 0.923338 0.461669 0.887052i \(-0.347251\pi\)
0.461669 + 0.887052i \(0.347251\pi\)
\(174\) 0.434475 0.0329374
\(175\) 10.7797 0.814868
\(176\) −3.25470 −0.245332
\(177\) −12.1561 −0.913705
\(178\) −13.8358 −1.03703
\(179\) −5.31074 −0.396944 −0.198472 0.980107i \(-0.563598\pi\)
−0.198472 + 0.980107i \(0.563598\pi\)
\(180\) −0.867762 −0.0646791
\(181\) 18.4867 1.37410 0.687052 0.726608i \(-0.258905\pi\)
0.687052 + 0.726608i \(0.258905\pi\)
\(182\) 0.287270 0.0212939
\(183\) 1.80949 0.133761
\(184\) −8.35620 −0.616027
\(185\) −0.931959 −0.0685190
\(186\) 9.03979 0.662830
\(187\) 12.3322 0.901816
\(188\) −9.30575 −0.678692
\(189\) −3.64617 −0.265220
\(190\) −2.69897 −0.195804
\(191\) −3.66049 −0.264864 −0.132432 0.991192i \(-0.542279\pi\)
−0.132432 + 0.991192i \(0.542279\pi\)
\(192\) −2.29904 −0.165919
\(193\) −6.83330 −0.491872 −0.245936 0.969286i \(-0.579095\pi\)
−0.245936 + 0.969286i \(0.579095\pi\)
\(194\) −2.61324 −0.187620
\(195\) −0.112953 −0.00808876
\(196\) −2.07188 −0.147991
\(197\) −0.268417 −0.0191239 −0.00956196 0.999954i \(-0.503044\pi\)
−0.00956196 + 0.999954i \(0.503044\pi\)
\(198\) −7.43890 −0.528660
\(199\) 16.9192 1.19937 0.599687 0.800235i \(-0.295292\pi\)
0.599687 + 0.800235i \(0.295292\pi\)
\(200\) −4.85585 −0.343361
\(201\) 6.03814 0.425898
\(202\) 0.258204 0.0181672
\(203\) 0.419526 0.0294449
\(204\) 8.71113 0.609901
\(205\) 0.823770 0.0575346
\(206\) 12.6133 0.878812
\(207\) −19.0988 −1.32746
\(208\) −0.129405 −0.00897260
\(209\) −23.1370 −1.60042
\(210\) −1.93772 −0.133715
\(211\) 5.39436 0.371363 0.185682 0.982610i \(-0.440551\pi\)
0.185682 + 0.982610i \(0.440551\pi\)
\(212\) 6.81916 0.468342
\(213\) −26.0384 −1.78412
\(214\) 0.595991 0.0407411
\(215\) 0.630871 0.0430251
\(216\) 1.64246 0.111756
\(217\) 8.72876 0.592547
\(218\) −6.41836 −0.434706
\(219\) 34.4989 2.33122
\(220\) 1.23570 0.0833110
\(221\) 0.490318 0.0329824
\(222\) −5.64340 −0.378760
\(223\) 13.4992 0.903977 0.451988 0.892024i \(-0.350715\pi\)
0.451988 + 0.892024i \(0.350715\pi\)
\(224\) −2.21994 −0.148326
\(225\) −11.0985 −0.739898
\(226\) −13.6994 −0.911268
\(227\) −21.3499 −1.41704 −0.708520 0.705690i \(-0.750637\pi\)
−0.708520 + 0.705690i \(0.750637\pi\)
\(228\) −16.3434 −1.08237
\(229\) 6.46004 0.426891 0.213446 0.976955i \(-0.431531\pi\)
0.213446 + 0.976955i \(0.431531\pi\)
\(230\) 3.17257 0.209193
\(231\) −16.6111 −1.09293
\(232\) −0.188981 −0.0124072
\(233\) 3.80737 0.249429 0.124715 0.992193i \(-0.460198\pi\)
0.124715 + 0.992193i \(0.460198\pi\)
\(234\) −0.295766 −0.0193348
\(235\) 3.53309 0.230473
\(236\) 5.28745 0.344183
\(237\) −10.0703 −0.654134
\(238\) 8.41141 0.545231
\(239\) 24.4880 1.58400 0.791999 0.610523i \(-0.209041\pi\)
0.791999 + 0.610523i \(0.209041\pi\)
\(240\) 0.872870 0.0563435
\(241\) 17.3310 1.11639 0.558195 0.829710i \(-0.311494\pi\)
0.558195 + 0.829710i \(0.311494\pi\)
\(242\) −0.406920 −0.0261578
\(243\) 19.5180 1.25208
\(244\) −0.787063 −0.0503865
\(245\) 0.786623 0.0502555
\(246\) 4.98827 0.318040
\(247\) −0.919911 −0.0585325
\(248\) −3.93199 −0.249681
\(249\) −21.8451 −1.38437
\(250\) 3.74194 0.236661
\(251\) −7.93796 −0.501040 −0.250520 0.968111i \(-0.580602\pi\)
−0.250520 + 0.968111i \(0.580602\pi\)
\(252\) −5.07386 −0.319623
\(253\) 27.1969 1.70986
\(254\) 2.57216 0.161392
\(255\) −3.30733 −0.207113
\(256\) 1.00000 0.0625000
\(257\) −20.2607 −1.26383 −0.631913 0.775039i \(-0.717730\pi\)
−0.631913 + 0.775039i \(0.717730\pi\)
\(258\) 3.82019 0.237834
\(259\) −5.44923 −0.338599
\(260\) 0.0491307 0.00304696
\(261\) −0.431932 −0.0267359
\(262\) −7.36658 −0.455109
\(263\) −20.8160 −1.28357 −0.641783 0.766886i \(-0.721805\pi\)
−0.641783 + 0.766886i \(0.721805\pi\)
\(264\) 7.48269 0.460528
\(265\) −2.58901 −0.159042
\(266\) −15.7811 −0.967599
\(267\) 31.8090 1.94668
\(268\) −2.62637 −0.160431
\(269\) 12.9346 0.788635 0.394317 0.918974i \(-0.370981\pi\)
0.394317 + 0.918974i \(0.370981\pi\)
\(270\) −0.623589 −0.0379504
\(271\) 17.2294 1.04661 0.523306 0.852145i \(-0.324699\pi\)
0.523306 + 0.852145i \(0.324699\pi\)
\(272\) −3.78903 −0.229744
\(273\) −0.660446 −0.0399720
\(274\) 3.12853 0.189001
\(275\) 15.8044 0.953038
\(276\) 19.2112 1.15638
\(277\) −2.29192 −0.137708 −0.0688541 0.997627i \(-0.521934\pi\)
−0.0688541 + 0.997627i \(0.521934\pi\)
\(278\) 10.9914 0.659220
\(279\) −8.98690 −0.538031
\(280\) 0.842837 0.0503691
\(281\) 21.7249 1.29600 0.647998 0.761642i \(-0.275606\pi\)
0.647998 + 0.761642i \(0.275606\pi\)
\(282\) 21.3943 1.27401
\(283\) −4.54744 −0.270317 −0.135159 0.990824i \(-0.543154\pi\)
−0.135159 + 0.990824i \(0.543154\pi\)
\(284\) 11.3258 0.672060
\(285\) 6.20505 0.367555
\(286\) 0.421174 0.0249045
\(287\) 4.81664 0.284317
\(288\) 2.28559 0.134679
\(289\) −2.64327 −0.155486
\(290\) 0.0717498 0.00421329
\(291\) 6.00795 0.352192
\(292\) −15.0058 −0.878148
\(293\) −14.7163 −0.859737 −0.429869 0.902891i \(-0.641440\pi\)
−0.429869 + 0.902891i \(0.641440\pi\)
\(294\) 4.76333 0.277803
\(295\) −2.00747 −0.116879
\(296\) 2.45468 0.142675
\(297\) −5.34573 −0.310191
\(298\) −2.96340 −0.171665
\(299\) 1.08133 0.0625350
\(300\) 11.1638 0.644542
\(301\) 3.68875 0.212616
\(302\) 5.97398 0.343764
\(303\) −0.593621 −0.0341026
\(304\) 7.10879 0.407717
\(305\) 0.298822 0.0171105
\(306\) −8.66015 −0.495068
\(307\) 8.51726 0.486106 0.243053 0.970013i \(-0.421851\pi\)
0.243053 + 0.970013i \(0.421851\pi\)
\(308\) 7.22523 0.411696
\(309\) −28.9985 −1.64967
\(310\) 1.49284 0.0847879
\(311\) 6.96597 0.395004 0.197502 0.980302i \(-0.436717\pi\)
0.197502 + 0.980302i \(0.436717\pi\)
\(312\) 0.297507 0.0168430
\(313\) −12.1299 −0.685623 −0.342811 0.939404i \(-0.611379\pi\)
−0.342811 + 0.939404i \(0.611379\pi\)
\(314\) 17.5520 0.990516
\(315\) 1.92638 0.108539
\(316\) 4.38020 0.246406
\(317\) 24.8250 1.39431 0.697154 0.716921i \(-0.254449\pi\)
0.697154 + 0.716921i \(0.254449\pi\)
\(318\) −15.6775 −0.879152
\(319\) 0.615076 0.0344376
\(320\) −0.379667 −0.0212240
\(321\) −1.37021 −0.0764775
\(322\) 18.5502 1.03377
\(323\) −26.9354 −1.49873
\(324\) −10.6329 −0.590714
\(325\) 0.628370 0.0348557
\(326\) 11.9347 0.660999
\(327\) 14.7561 0.816013
\(328\) −2.16972 −0.119803
\(329\) 20.6582 1.13892
\(330\) −2.84093 −0.156388
\(331\) 21.9163 1.20463 0.602314 0.798259i \(-0.294246\pi\)
0.602314 + 0.798259i \(0.294246\pi\)
\(332\) 9.50181 0.521480
\(333\) 5.61038 0.307447
\(334\) 10.6172 0.580945
\(335\) 0.997148 0.0544800
\(336\) 5.10373 0.278431
\(337\) 32.4362 1.76691 0.883456 0.468515i \(-0.155211\pi\)
0.883456 + 0.468515i \(0.155211\pi\)
\(338\) −12.9833 −0.706196
\(339\) 31.4954 1.71059
\(340\) 1.43857 0.0780173
\(341\) 12.7974 0.693020
\(342\) 16.2478 0.878578
\(343\) 20.1390 1.08740
\(344\) −1.66164 −0.0895898
\(345\) −7.29387 −0.392689
\(346\) 12.1446 0.652899
\(347\) −18.7842 −1.00839 −0.504193 0.863591i \(-0.668210\pi\)
−0.504193 + 0.863591i \(0.668210\pi\)
\(348\) 0.434475 0.0232903
\(349\) −5.81954 −0.311513 −0.155756 0.987795i \(-0.549782\pi\)
−0.155756 + 0.987795i \(0.549782\pi\)
\(350\) 10.7797 0.576199
\(351\) −0.212543 −0.0113447
\(352\) −3.25470 −0.173476
\(353\) 13.3356 0.709781 0.354891 0.934908i \(-0.384518\pi\)
0.354891 + 0.934908i \(0.384518\pi\)
\(354\) −12.1561 −0.646087
\(355\) −4.30001 −0.228221
\(356\) −13.8358 −0.733293
\(357\) −19.3382 −1.02348
\(358\) −5.31074 −0.280682
\(359\) −22.4544 −1.18510 −0.592549 0.805534i \(-0.701878\pi\)
−0.592549 + 0.805534i \(0.701878\pi\)
\(360\) −0.867762 −0.0457351
\(361\) 31.5349 1.65973
\(362\) 18.4867 0.971638
\(363\) 0.935525 0.0491023
\(364\) 0.287270 0.0150571
\(365\) 5.69720 0.298205
\(366\) 1.80949 0.0945835
\(367\) −14.2920 −0.746038 −0.373019 0.927824i \(-0.621677\pi\)
−0.373019 + 0.927824i \(0.621677\pi\)
\(368\) −8.35620 −0.435597
\(369\) −4.95908 −0.258159
\(370\) −0.931959 −0.0484503
\(371\) −15.1381 −0.785932
\(372\) 9.03979 0.468692
\(373\) −0.0528215 −0.00273499 −0.00136750 0.999999i \(-0.500435\pi\)
−0.00136750 + 0.999999i \(0.500435\pi\)
\(374\) 12.3322 0.637680
\(375\) −8.60287 −0.444251
\(376\) −9.30575 −0.479908
\(377\) 0.0244550 0.00125950
\(378\) −3.64617 −0.187539
\(379\) 11.7126 0.601636 0.300818 0.953682i \(-0.402740\pi\)
0.300818 + 0.953682i \(0.402740\pi\)
\(380\) −2.69897 −0.138454
\(381\) −5.91351 −0.302958
\(382\) −3.66049 −0.187287
\(383\) −28.1142 −1.43657 −0.718284 0.695750i \(-0.755072\pi\)
−0.718284 + 0.695750i \(0.755072\pi\)
\(384\) −2.29904 −0.117322
\(385\) −2.74318 −0.139806
\(386\) −6.83330 −0.347806
\(387\) −3.79783 −0.193055
\(388\) −2.61324 −0.132667
\(389\) 8.14156 0.412794 0.206397 0.978468i \(-0.433826\pi\)
0.206397 + 0.978468i \(0.433826\pi\)
\(390\) −0.112953 −0.00571962
\(391\) 31.6619 1.60121
\(392\) −2.07188 −0.104646
\(393\) 16.9361 0.854311
\(394\) −0.268417 −0.0135226
\(395\) −1.66302 −0.0836756
\(396\) −7.43890 −0.373819
\(397\) −35.2797 −1.77064 −0.885319 0.464984i \(-0.846060\pi\)
−0.885319 + 0.464984i \(0.846060\pi\)
\(398\) 16.9192 0.848085
\(399\) 36.2813 1.81634
\(400\) −4.85585 −0.242793
\(401\) −4.54194 −0.226813 −0.113407 0.993549i \(-0.536176\pi\)
−0.113407 + 0.993549i \(0.536176\pi\)
\(402\) 6.03814 0.301155
\(403\) 0.508817 0.0253460
\(404\) 0.258204 0.0128461
\(405\) 4.03694 0.200597
\(406\) 0.419526 0.0208207
\(407\) −7.98924 −0.396012
\(408\) 8.71113 0.431265
\(409\) 26.4967 1.31018 0.655088 0.755553i \(-0.272632\pi\)
0.655088 + 0.755553i \(0.272632\pi\)
\(410\) 0.823770 0.0406831
\(411\) −7.19262 −0.354786
\(412\) 12.6133 0.621414
\(413\) −11.7378 −0.577580
\(414\) −19.0988 −0.938656
\(415\) −3.60752 −0.177086
\(416\) −0.129405 −0.00634459
\(417\) −25.2697 −1.23746
\(418\) −23.1370 −1.13167
\(419\) 18.0410 0.881362 0.440681 0.897664i \(-0.354737\pi\)
0.440681 + 0.897664i \(0.354737\pi\)
\(420\) −1.93772 −0.0945509
\(421\) 23.7255 1.15631 0.578154 0.815927i \(-0.303773\pi\)
0.578154 + 0.815927i \(0.303773\pi\)
\(422\) 5.39436 0.262593
\(423\) −21.2691 −1.03414
\(424\) 6.81916 0.331168
\(425\) 18.3990 0.892481
\(426\) −26.0384 −1.26156
\(427\) 1.74723 0.0845544
\(428\) 0.595991 0.0288083
\(429\) −0.968295 −0.0467497
\(430\) 0.630871 0.0304233
\(431\) 8.25690 0.397721 0.198860 0.980028i \(-0.436276\pi\)
0.198860 + 0.980028i \(0.436276\pi\)
\(432\) 1.64246 0.0790231
\(433\) −16.6226 −0.798833 −0.399416 0.916770i \(-0.630787\pi\)
−0.399416 + 0.916770i \(0.630787\pi\)
\(434\) 8.72876 0.418994
\(435\) −0.164956 −0.00790902
\(436\) −6.41836 −0.307384
\(437\) −59.4025 −2.84161
\(438\) 34.4989 1.64842
\(439\) −39.1742 −1.86968 −0.934842 0.355064i \(-0.884459\pi\)
−0.934842 + 0.355064i \(0.884459\pi\)
\(440\) 1.23570 0.0589098
\(441\) −4.73545 −0.225498
\(442\) 0.490318 0.0233220
\(443\) 39.7801 1.89001 0.945005 0.327056i \(-0.106056\pi\)
0.945005 + 0.327056i \(0.106056\pi\)
\(444\) −5.64340 −0.267824
\(445\) 5.25298 0.249015
\(446\) 13.4992 0.639208
\(447\) 6.81297 0.322242
\(448\) −2.21994 −0.104882
\(449\) −21.3558 −1.00784 −0.503920 0.863750i \(-0.668109\pi\)
−0.503920 + 0.863750i \(0.668109\pi\)
\(450\) −11.0985 −0.523187
\(451\) 7.06178 0.332526
\(452\) −13.6994 −0.644364
\(453\) −13.7344 −0.645300
\(454\) −21.3499 −1.00200
\(455\) −0.109067 −0.00511314
\(456\) −16.3434 −0.765349
\(457\) −2.29166 −0.107199 −0.0535997 0.998563i \(-0.517069\pi\)
−0.0535997 + 0.998563i \(0.517069\pi\)
\(458\) 6.46004 0.301858
\(459\) −6.22334 −0.290481
\(460\) 3.17257 0.147922
\(461\) 4.84273 0.225548 0.112774 0.993621i \(-0.464026\pi\)
0.112774 + 0.993621i \(0.464026\pi\)
\(462\) −16.6111 −0.772819
\(463\) 19.7826 0.919376 0.459688 0.888081i \(-0.347961\pi\)
0.459688 + 0.888081i \(0.347961\pi\)
\(464\) −0.188981 −0.00877321
\(465\) −3.43211 −0.159160
\(466\) 3.80737 0.176373
\(467\) 10.9655 0.507421 0.253710 0.967280i \(-0.418349\pi\)
0.253710 + 0.967280i \(0.418349\pi\)
\(468\) −0.295766 −0.0136718
\(469\) 5.83039 0.269222
\(470\) 3.53309 0.162969
\(471\) −40.3527 −1.85936
\(472\) 5.28745 0.243374
\(473\) 5.40815 0.248667
\(474\) −10.0703 −0.462543
\(475\) −34.5192 −1.58385
\(476\) 8.41141 0.385536
\(477\) 15.5858 0.713624
\(478\) 24.4880 1.12006
\(479\) −17.7629 −0.811608 −0.405804 0.913960i \(-0.633009\pi\)
−0.405804 + 0.913960i \(0.633009\pi\)
\(480\) 0.872870 0.0398409
\(481\) −0.317647 −0.0144834
\(482\) 17.3310 0.789407
\(483\) −42.6478 −1.94054
\(484\) −0.406920 −0.0184963
\(485\) 0.992162 0.0450518
\(486\) 19.5180 0.885353
\(487\) −11.2771 −0.511015 −0.255508 0.966807i \(-0.582243\pi\)
−0.255508 + 0.966807i \(0.582243\pi\)
\(488\) −0.787063 −0.0356287
\(489\) −27.4382 −1.24080
\(490\) 0.786623 0.0355360
\(491\) 26.9438 1.21596 0.607978 0.793954i \(-0.291981\pi\)
0.607978 + 0.793954i \(0.291981\pi\)
\(492\) 4.98827 0.224888
\(493\) 0.716054 0.0322494
\(494\) −0.919911 −0.0413887
\(495\) 2.82431 0.126943
\(496\) −3.93199 −0.176551
\(497\) −25.1425 −1.12779
\(498\) −21.8451 −0.978900
\(499\) −4.45445 −0.199409 −0.0997043 0.995017i \(-0.531790\pi\)
−0.0997043 + 0.995017i \(0.531790\pi\)
\(500\) 3.74194 0.167345
\(501\) −24.4093 −1.09053
\(502\) −7.93796 −0.354288
\(503\) −6.89740 −0.307540 −0.153770 0.988107i \(-0.549141\pi\)
−0.153770 + 0.988107i \(0.549141\pi\)
\(504\) −5.07386 −0.226008
\(505\) −0.0980315 −0.00436234
\(506\) 27.1969 1.20905
\(507\) 29.8490 1.32564
\(508\) 2.57216 0.114121
\(509\) 41.1512 1.82399 0.911997 0.410196i \(-0.134540\pi\)
0.911997 + 0.410196i \(0.134540\pi\)
\(510\) −3.30733 −0.146451
\(511\) 33.3119 1.47363
\(512\) 1.00000 0.0441942
\(513\) 11.6759 0.515505
\(514\) −20.2607 −0.893660
\(515\) −4.78886 −0.211022
\(516\) 3.82019 0.168174
\(517\) 30.2874 1.33204
\(518\) −5.44923 −0.239425
\(519\) −27.9210 −1.22559
\(520\) 0.0491307 0.00215452
\(521\) 25.4981 1.11709 0.558546 0.829473i \(-0.311359\pi\)
0.558546 + 0.829473i \(0.311359\pi\)
\(522\) −0.431932 −0.0189052
\(523\) −7.20845 −0.315204 −0.157602 0.987503i \(-0.550376\pi\)
−0.157602 + 0.987503i \(0.550376\pi\)
\(524\) −7.36658 −0.321811
\(525\) −24.7829 −1.08162
\(526\) −20.8160 −0.907619
\(527\) 14.8984 0.648985
\(528\) 7.48269 0.325642
\(529\) 46.8261 2.03592
\(530\) −2.58901 −0.112459
\(531\) 12.0849 0.524441
\(532\) −15.7811 −0.684196
\(533\) 0.280771 0.0121616
\(534\) 31.8090 1.37651
\(535\) −0.226278 −0.00978285
\(536\) −2.62637 −0.113442
\(537\) 12.2096 0.526884
\(538\) 12.9346 0.557649
\(539\) 6.74334 0.290456
\(540\) −0.623589 −0.0268350
\(541\) 27.7047 1.19112 0.595558 0.803312i \(-0.296931\pi\)
0.595558 + 0.803312i \(0.296931\pi\)
\(542\) 17.2294 0.740066
\(543\) −42.5016 −1.82392
\(544\) −3.78903 −0.162453
\(545\) 2.43684 0.104383
\(546\) −0.660446 −0.0282645
\(547\) 29.1504 1.24638 0.623190 0.782071i \(-0.285837\pi\)
0.623190 + 0.782071i \(0.285837\pi\)
\(548\) 3.12853 0.133644
\(549\) −1.79890 −0.0767752
\(550\) 15.8044 0.673900
\(551\) −1.34342 −0.0572318
\(552\) 19.2112 0.817685
\(553\) −9.72378 −0.413497
\(554\) −2.29192 −0.0973744
\(555\) 2.14261 0.0909488
\(556\) 10.9914 0.466139
\(557\) −42.3723 −1.79537 −0.897687 0.440634i \(-0.854754\pi\)
−0.897687 + 0.440634i \(0.854754\pi\)
\(558\) −8.98690 −0.380446
\(559\) 0.215025 0.00909457
\(560\) 0.842837 0.0356164
\(561\) −28.3521 −1.19703
\(562\) 21.7249 0.916408
\(563\) −13.7209 −0.578268 −0.289134 0.957289i \(-0.593367\pi\)
−0.289134 + 0.957289i \(0.593367\pi\)
\(564\) 21.3943 0.900863
\(565\) 5.20119 0.218816
\(566\) −4.54744 −0.191143
\(567\) 23.6043 0.991286
\(568\) 11.3258 0.475218
\(569\) −13.9442 −0.584570 −0.292285 0.956331i \(-0.594416\pi\)
−0.292285 + 0.956331i \(0.594416\pi\)
\(570\) 6.20505 0.259901
\(571\) 22.0617 0.923251 0.461626 0.887075i \(-0.347266\pi\)
0.461626 + 0.887075i \(0.347266\pi\)
\(572\) 0.421174 0.0176102
\(573\) 8.41562 0.351568
\(574\) 4.81664 0.201042
\(575\) 40.5765 1.69216
\(576\) 2.28559 0.0952328
\(577\) −36.9418 −1.53791 −0.768953 0.639306i \(-0.779222\pi\)
−0.768953 + 0.639306i \(0.779222\pi\)
\(578\) −2.64327 −0.109945
\(579\) 15.7100 0.652887
\(580\) 0.0717498 0.00297925
\(581\) −21.0934 −0.875103
\(582\) 6.00795 0.249038
\(583\) −22.1943 −0.919195
\(584\) −15.0058 −0.620944
\(585\) 0.112292 0.00464272
\(586\) −14.7163 −0.607926
\(587\) −11.7370 −0.484436 −0.242218 0.970222i \(-0.577875\pi\)
−0.242218 + 0.970222i \(0.577875\pi\)
\(588\) 4.76333 0.196436
\(589\) −27.9517 −1.15173
\(590\) −2.00747 −0.0826462
\(591\) 0.617101 0.0253842
\(592\) 2.45468 0.100887
\(593\) 31.9252 1.31101 0.655505 0.755191i \(-0.272456\pi\)
0.655505 + 0.755191i \(0.272456\pi\)
\(594\) −5.34573 −0.219338
\(595\) −3.19353 −0.130922
\(596\) −2.96340 −0.121385
\(597\) −38.8980 −1.59199
\(598\) 1.08133 0.0442189
\(599\) 46.3703 1.89464 0.947320 0.320289i \(-0.103780\pi\)
0.947320 + 0.320289i \(0.103780\pi\)
\(600\) 11.1638 0.455760
\(601\) 6.53899 0.266731 0.133366 0.991067i \(-0.457422\pi\)
0.133366 + 0.991067i \(0.457422\pi\)
\(602\) 3.68875 0.150342
\(603\) −6.00281 −0.244453
\(604\) 5.97398 0.243078
\(605\) 0.154494 0.00628107
\(606\) −0.593621 −0.0241142
\(607\) 21.3440 0.866327 0.433164 0.901315i \(-0.357397\pi\)
0.433164 + 0.901315i \(0.357397\pi\)
\(608\) 7.10879 0.288299
\(609\) −0.964506 −0.0390838
\(610\) 0.298822 0.0120989
\(611\) 1.20421 0.0487170
\(612\) −8.66015 −0.350066
\(613\) 24.8282 1.00280 0.501401 0.865215i \(-0.332818\pi\)
0.501401 + 0.865215i \(0.332818\pi\)
\(614\) 8.51726 0.343729
\(615\) −1.89388 −0.0763686
\(616\) 7.22523 0.291113
\(617\) 8.51019 0.342607 0.171304 0.985218i \(-0.445202\pi\)
0.171304 + 0.985218i \(0.445202\pi\)
\(618\) −28.9985 −1.16649
\(619\) 30.7570 1.23623 0.618114 0.786088i \(-0.287897\pi\)
0.618114 + 0.786088i \(0.287897\pi\)
\(620\) 1.49284 0.0599541
\(621\) −13.7248 −0.550756
\(622\) 6.96597 0.279310
\(623\) 30.7145 1.23055
\(624\) 0.297507 0.0119098
\(625\) 22.8586 0.914343
\(626\) −12.1299 −0.484808
\(627\) 53.1929 2.12432
\(628\) 17.5520 0.700401
\(629\) −9.30084 −0.370849
\(630\) 1.92638 0.0767487
\(631\) −15.7094 −0.625380 −0.312690 0.949855i \(-0.601230\pi\)
−0.312690 + 0.949855i \(0.601230\pi\)
\(632\) 4.38020 0.174235
\(633\) −12.4019 −0.492929
\(634\) 24.8250 0.985925
\(635\) −0.976565 −0.0387538
\(636\) −15.6775 −0.621654
\(637\) 0.268110 0.0106229
\(638\) 0.615076 0.0243511
\(639\) 25.8860 1.02403
\(640\) −0.379667 −0.0150077
\(641\) −9.75381 −0.385252 −0.192626 0.981272i \(-0.561700\pi\)
−0.192626 + 0.981272i \(0.561700\pi\)
\(642\) −1.37021 −0.0540777
\(643\) −18.7144 −0.738022 −0.369011 0.929425i \(-0.620304\pi\)
−0.369011 + 0.929425i \(0.620304\pi\)
\(644\) 18.5502 0.730982
\(645\) −1.45040 −0.0571094
\(646\) −26.9354 −1.05976
\(647\) −1.48690 −0.0584560 −0.0292280 0.999573i \(-0.509305\pi\)
−0.0292280 + 0.999573i \(0.509305\pi\)
\(648\) −10.6329 −0.417698
\(649\) −17.2091 −0.675515
\(650\) 0.628370 0.0246467
\(651\) −20.0678 −0.786518
\(652\) 11.9347 0.467397
\(653\) −9.77388 −0.382481 −0.191241 0.981543i \(-0.561251\pi\)
−0.191241 + 0.981543i \(0.561251\pi\)
\(654\) 14.7561 0.577008
\(655\) 2.79685 0.109282
\(656\) −2.16972 −0.0847132
\(657\) −34.2970 −1.33806
\(658\) 20.6582 0.805340
\(659\) 26.0386 1.01432 0.507160 0.861852i \(-0.330695\pi\)
0.507160 + 0.861852i \(0.330695\pi\)
\(660\) −2.84093 −0.110583
\(661\) −32.5010 −1.26414 −0.632071 0.774910i \(-0.717795\pi\)
−0.632071 + 0.774910i \(0.717795\pi\)
\(662\) 21.9163 0.851801
\(663\) −1.12726 −0.0437792
\(664\) 9.50181 0.368742
\(665\) 5.99155 0.232342
\(666\) 5.61038 0.217398
\(667\) 1.57916 0.0611454
\(668\) 10.6172 0.410790
\(669\) −31.0353 −1.19989
\(670\) 0.997148 0.0385232
\(671\) 2.56165 0.0988916
\(672\) 5.10373 0.196881
\(673\) −23.0520 −0.888590 −0.444295 0.895881i \(-0.646546\pi\)
−0.444295 + 0.895881i \(0.646546\pi\)
\(674\) 32.4362 1.24940
\(675\) −7.97557 −0.306980
\(676\) −12.9833 −0.499356
\(677\) −7.79858 −0.299724 −0.149862 0.988707i \(-0.547883\pi\)
−0.149862 + 0.988707i \(0.547883\pi\)
\(678\) 31.4954 1.20957
\(679\) 5.80124 0.222631
\(680\) 1.43857 0.0551666
\(681\) 49.0842 1.88091
\(682\) 12.7974 0.490039
\(683\) 1.74670 0.0668357 0.0334178 0.999441i \(-0.489361\pi\)
0.0334178 + 0.999441i \(0.489361\pi\)
\(684\) 16.2478 0.621248
\(685\) −1.18780 −0.0453835
\(686\) 20.1390 0.768911
\(687\) −14.8519 −0.566635
\(688\) −1.66164 −0.0633496
\(689\) −0.882431 −0.0336180
\(690\) −7.29387 −0.277673
\(691\) 6.87317 0.261468 0.130734 0.991417i \(-0.458267\pi\)
0.130734 + 0.991417i \(0.458267\pi\)
\(692\) 12.1446 0.461669
\(693\) 16.5139 0.627311
\(694\) −18.7842 −0.713037
\(695\) −4.17307 −0.158294
\(696\) 0.434475 0.0164687
\(697\) 8.22112 0.311397
\(698\) −5.81954 −0.220273
\(699\) −8.75330 −0.331080
\(700\) 10.7797 0.407434
\(701\) 5.00366 0.188986 0.0944928 0.995526i \(-0.469877\pi\)
0.0944928 + 0.995526i \(0.469877\pi\)
\(702\) −0.212543 −0.00802190
\(703\) 17.4498 0.658131
\(704\) −3.25470 −0.122666
\(705\) −8.12271 −0.305919
\(706\) 13.3356 0.501891
\(707\) −0.573197 −0.0215573
\(708\) −12.1561 −0.456852
\(709\) −36.7030 −1.37841 −0.689204 0.724567i \(-0.742040\pi\)
−0.689204 + 0.724567i \(0.742040\pi\)
\(710\) −4.30001 −0.161377
\(711\) 10.0113 0.375455
\(712\) −13.8358 −0.518517
\(713\) 32.8565 1.23048
\(714\) −19.3382 −0.723713
\(715\) −0.159906 −0.00598013
\(716\) −5.31074 −0.198472
\(717\) −56.2989 −2.10252
\(718\) −22.4544 −0.837991
\(719\) −23.4988 −0.876359 −0.438180 0.898887i \(-0.644377\pi\)
−0.438180 + 0.898887i \(0.644377\pi\)
\(720\) −0.867762 −0.0323396
\(721\) −28.0008 −1.04280
\(722\) 31.5349 1.17361
\(723\) −39.8448 −1.48184
\(724\) 18.4867 0.687052
\(725\) 0.917663 0.0340811
\(726\) 0.935525 0.0347206
\(727\) −43.8647 −1.62685 −0.813426 0.581668i \(-0.802400\pi\)
−0.813426 + 0.581668i \(0.802400\pi\)
\(728\) 0.287270 0.0106469
\(729\) −12.9740 −0.480520
\(730\) 5.69720 0.210863
\(731\) 6.29602 0.232867
\(732\) 1.80949 0.0668807
\(733\) 37.8623 1.39847 0.699237 0.714890i \(-0.253523\pi\)
0.699237 + 0.714890i \(0.253523\pi\)
\(734\) −14.2920 −0.527529
\(735\) −1.80848 −0.0667067
\(736\) −8.35620 −0.308014
\(737\) 8.54807 0.314872
\(738\) −4.95908 −0.182546
\(739\) −10.4108 −0.382969 −0.191484 0.981496i \(-0.561330\pi\)
−0.191484 + 0.981496i \(0.561330\pi\)
\(740\) −0.931959 −0.0342595
\(741\) 2.11491 0.0776932
\(742\) −15.1381 −0.555738
\(743\) 49.9992 1.83429 0.917147 0.398549i \(-0.130486\pi\)
0.917147 + 0.398549i \(0.130486\pi\)
\(744\) 9.03979 0.331415
\(745\) 1.12510 0.0412206
\(746\) −0.0528215 −0.00193393
\(747\) 21.7172 0.794592
\(748\) 12.3322 0.450908
\(749\) −1.32306 −0.0483436
\(750\) −8.60287 −0.314133
\(751\) 46.8122 1.70820 0.854101 0.520107i \(-0.174108\pi\)
0.854101 + 0.520107i \(0.174108\pi\)
\(752\) −9.30575 −0.339346
\(753\) 18.2497 0.665056
\(754\) 0.0244550 0.000890599 0
\(755\) −2.26812 −0.0825455
\(756\) −3.64617 −0.132610
\(757\) 27.8507 1.01225 0.506126 0.862460i \(-0.331077\pi\)
0.506126 + 0.862460i \(0.331077\pi\)
\(758\) 11.7126 0.425421
\(759\) −62.5269 −2.26958
\(760\) −2.69897 −0.0979020
\(761\) 3.95804 0.143479 0.0717395 0.997423i \(-0.477145\pi\)
0.0717395 + 0.997423i \(0.477145\pi\)
\(762\) −5.91351 −0.214224
\(763\) 14.2484 0.515826
\(764\) −3.66049 −0.132432
\(765\) 3.28797 0.118877
\(766\) −28.1142 −1.01581
\(767\) −0.684220 −0.0247058
\(768\) −2.29904 −0.0829595
\(769\) 48.9333 1.76458 0.882289 0.470708i \(-0.156001\pi\)
0.882289 + 0.470708i \(0.156001\pi\)
\(770\) −2.74318 −0.0988574
\(771\) 46.5801 1.67754
\(772\) −6.83330 −0.245936
\(773\) 2.17902 0.0783740 0.0391870 0.999232i \(-0.487523\pi\)
0.0391870 + 0.999232i \(0.487523\pi\)
\(774\) −3.79783 −0.136510
\(775\) 19.0931 0.685846
\(776\) −2.61324 −0.0938100
\(777\) 12.5280 0.449439
\(778\) 8.14156 0.291889
\(779\) −15.4241 −0.552624
\(780\) −0.112953 −0.00404438
\(781\) −36.8619 −1.31902
\(782\) 31.6619 1.13223
\(783\) −0.310394 −0.0110926
\(784\) −2.07188 −0.0739956
\(785\) −6.66391 −0.237845
\(786\) 16.9361 0.604089
\(787\) −22.6975 −0.809077 −0.404539 0.914521i \(-0.632568\pi\)
−0.404539 + 0.914521i \(0.632568\pi\)
\(788\) −0.268417 −0.00956196
\(789\) 47.8567 1.70374
\(790\) −1.66302 −0.0591675
\(791\) 30.4117 1.08132
\(792\) −7.43890 −0.264330
\(793\) 0.101850 0.00361679
\(794\) −35.2797 −1.25203
\(795\) 5.95224 0.211104
\(796\) 16.9192 0.599687
\(797\) 42.6147 1.50949 0.754745 0.656019i \(-0.227761\pi\)
0.754745 + 0.656019i \(0.227761\pi\)
\(798\) 36.2813 1.28434
\(799\) 35.2598 1.24740
\(800\) −4.85585 −0.171680
\(801\) −31.6228 −1.11734
\(802\) −4.54194 −0.160381
\(803\) 48.8394 1.72350
\(804\) 6.03814 0.212949
\(805\) −7.04292 −0.248230
\(806\) 0.508817 0.0179223
\(807\) −29.7371 −1.04680
\(808\) 0.258204 0.00908358
\(809\) −24.6488 −0.866606 −0.433303 0.901248i \(-0.642652\pi\)
−0.433303 + 0.901248i \(0.642652\pi\)
\(810\) 4.03694 0.141844
\(811\) −26.2859 −0.923021 −0.461511 0.887135i \(-0.652692\pi\)
−0.461511 + 0.887135i \(0.652692\pi\)
\(812\) 0.419526 0.0147225
\(813\) −39.6111 −1.38922
\(814\) −7.98924 −0.280023
\(815\) −4.53119 −0.158721
\(816\) 8.71113 0.304950
\(817\) −11.8123 −0.413259
\(818\) 26.4967 0.926434
\(819\) 0.656581 0.0229428
\(820\) 0.823770 0.0287673
\(821\) −10.1238 −0.353322 −0.176661 0.984272i \(-0.556530\pi\)
−0.176661 + 0.984272i \(0.556530\pi\)
\(822\) −7.19262 −0.250871
\(823\) −43.7943 −1.52657 −0.763286 0.646061i \(-0.776415\pi\)
−0.763286 + 0.646061i \(0.776415\pi\)
\(824\) 12.6133 0.439406
\(825\) −36.3348 −1.26502
\(826\) −11.7378 −0.408410
\(827\) 6.47610 0.225196 0.112598 0.993641i \(-0.464083\pi\)
0.112598 + 0.993641i \(0.464083\pi\)
\(828\) −19.0988 −0.663730
\(829\) −52.2306 −1.81404 −0.907022 0.421083i \(-0.861650\pi\)
−0.907022 + 0.421083i \(0.861650\pi\)
\(830\) −3.60752 −0.125219
\(831\) 5.26922 0.182787
\(832\) −0.129405 −0.00448630
\(833\) 7.85039 0.272000
\(834\) −25.2697 −0.875017
\(835\) −4.03099 −0.139498
\(836\) −23.1370 −0.800209
\(837\) −6.45815 −0.223226
\(838\) 18.0410 0.623217
\(839\) −49.4850 −1.70841 −0.854206 0.519935i \(-0.825956\pi\)
−0.854206 + 0.519935i \(0.825956\pi\)
\(840\) −1.93772 −0.0668576
\(841\) −28.9643 −0.998768
\(842\) 23.7255 0.817634
\(843\) −49.9463 −1.72024
\(844\) 5.39436 0.185682
\(845\) 4.92931 0.169573
\(846\) −21.2691 −0.731247
\(847\) 0.903336 0.0310390
\(848\) 6.81916 0.234171
\(849\) 10.4547 0.358806
\(850\) 18.3990 0.631079
\(851\) −20.5118 −0.703134
\(852\) −26.0384 −0.892060
\(853\) −24.8949 −0.852386 −0.426193 0.904632i \(-0.640146\pi\)
−0.426193 + 0.904632i \(0.640146\pi\)
\(854\) 1.74723 0.0597890
\(855\) −6.16873 −0.210966
\(856\) 0.595991 0.0203705
\(857\) −6.78744 −0.231854 −0.115927 0.993258i \(-0.536984\pi\)
−0.115927 + 0.993258i \(0.536984\pi\)
\(858\) −0.968295 −0.0330570
\(859\) 9.04897 0.308747 0.154373 0.988013i \(-0.450664\pi\)
0.154373 + 0.988013i \(0.450664\pi\)
\(860\) 0.630871 0.0215125
\(861\) −11.0736 −0.377389
\(862\) 8.25690 0.281231
\(863\) −8.11789 −0.276336 −0.138168 0.990409i \(-0.544121\pi\)
−0.138168 + 0.990409i \(0.544121\pi\)
\(864\) 1.64246 0.0558778
\(865\) −4.61091 −0.156776
\(866\) −16.6226 −0.564860
\(867\) 6.07698 0.206385
\(868\) 8.72876 0.296274
\(869\) −14.2563 −0.483610
\(870\) −0.164956 −0.00559252
\(871\) 0.339865 0.0115159
\(872\) −6.41836 −0.217353
\(873\) −5.97280 −0.202148
\(874\) −59.4025 −2.00932
\(875\) −8.30688 −0.280824
\(876\) 34.4989 1.16561
\(877\) −17.7430 −0.599137 −0.299569 0.954075i \(-0.596843\pi\)
−0.299569 + 0.954075i \(0.596843\pi\)
\(878\) −39.1742 −1.32207
\(879\) 33.8334 1.14117
\(880\) 1.23570 0.0416555
\(881\) −0.380184 −0.0128087 −0.00640436 0.999979i \(-0.502039\pi\)
−0.00640436 + 0.999979i \(0.502039\pi\)
\(882\) −4.73545 −0.159451
\(883\) 27.7162 0.932726 0.466363 0.884593i \(-0.345564\pi\)
0.466363 + 0.884593i \(0.345564\pi\)
\(884\) 0.490318 0.0164912
\(885\) 4.61525 0.155140
\(886\) 39.7801 1.33644
\(887\) 35.2256 1.18276 0.591381 0.806393i \(-0.298583\pi\)
0.591381 + 0.806393i \(0.298583\pi\)
\(888\) −5.64340 −0.189380
\(889\) −5.71004 −0.191509
\(890\) 5.25298 0.176080
\(891\) 34.6068 1.15937
\(892\) 13.4992 0.451988
\(893\) −66.1526 −2.21371
\(894\) 6.81297 0.227860
\(895\) 2.01631 0.0673979
\(896\) −2.21994 −0.0741629
\(897\) −2.48603 −0.0830060
\(898\) −21.3558 −0.712651
\(899\) 0.743070 0.0247828
\(900\) −11.0985 −0.369949
\(901\) −25.8380 −0.860788
\(902\) 7.06178 0.235132
\(903\) −8.48058 −0.282216
\(904\) −13.6994 −0.455634
\(905\) −7.01878 −0.233312
\(906\) −13.7344 −0.456296
\(907\) 18.1720 0.603390 0.301695 0.953404i \(-0.402448\pi\)
0.301695 + 0.953404i \(0.402448\pi\)
\(908\) −21.3499 −0.708520
\(909\) 0.590147 0.0195740
\(910\) −0.109067 −0.00361554
\(911\) −34.6244 −1.14716 −0.573579 0.819151i \(-0.694445\pi\)
−0.573579 + 0.819151i \(0.694445\pi\)
\(912\) −16.3434 −0.541184
\(913\) −30.9256 −1.02349
\(914\) −2.29166 −0.0758014
\(915\) −0.687003 −0.0227116
\(916\) 6.46004 0.213446
\(917\) 16.3533 0.540035
\(918\) −6.22334 −0.205401
\(919\) −12.1005 −0.399159 −0.199579 0.979882i \(-0.563958\pi\)
−0.199579 + 0.979882i \(0.563958\pi\)
\(920\) 3.17257 0.104597
\(921\) −19.5815 −0.645233
\(922\) 4.84273 0.159487
\(923\) −1.46561 −0.0482410
\(924\) −16.6111 −0.546465
\(925\) −11.9195 −0.391912
\(926\) 19.7826 0.650097
\(927\) 28.8288 0.946863
\(928\) −0.188981 −0.00620360
\(929\) −20.5697 −0.674869 −0.337434 0.941349i \(-0.609559\pi\)
−0.337434 + 0.941349i \(0.609559\pi\)
\(930\) −3.43211 −0.112543
\(931\) −14.7285 −0.482708
\(932\) 3.80737 0.124715
\(933\) −16.0150 −0.524309
\(934\) 10.9655 0.358801
\(935\) −4.68211 −0.153121
\(936\) −0.295766 −0.00966740
\(937\) −28.8771 −0.943374 −0.471687 0.881766i \(-0.656355\pi\)
−0.471687 + 0.881766i \(0.656355\pi\)
\(938\) 5.83039 0.190369
\(939\) 27.8871 0.910062
\(940\) 3.53309 0.115237
\(941\) −46.9007 −1.52892 −0.764460 0.644671i \(-0.776994\pi\)
−0.764460 + 0.644671i \(0.776994\pi\)
\(942\) −40.3527 −1.31476
\(943\) 18.1306 0.590413
\(944\) 5.28745 0.172092
\(945\) 1.38433 0.0450323
\(946\) 5.40815 0.175834
\(947\) −45.5153 −1.47905 −0.739523 0.673131i \(-0.764949\pi\)
−0.739523 + 0.673131i \(0.764949\pi\)
\(948\) −10.0703 −0.327067
\(949\) 1.94182 0.0630341
\(950\) −34.5192 −1.11995
\(951\) −57.0736 −1.85074
\(952\) 8.41141 0.272615
\(953\) 15.1338 0.490233 0.245116 0.969494i \(-0.421174\pi\)
0.245116 + 0.969494i \(0.421174\pi\)
\(954\) 15.5858 0.504608
\(955\) 1.38977 0.0449718
\(956\) 24.4880 0.791999
\(957\) −1.41408 −0.0457109
\(958\) −17.7629 −0.573894
\(959\) −6.94514 −0.224270
\(960\) 0.872870 0.0281717
\(961\) −15.5395 −0.501274
\(962\) −0.317647 −0.0102413
\(963\) 1.36219 0.0438959
\(964\) 17.3310 0.558195
\(965\) 2.59438 0.0835160
\(966\) −42.6478 −1.37217
\(967\) 14.4144 0.463536 0.231768 0.972771i \(-0.425549\pi\)
0.231768 + 0.972771i \(0.425549\pi\)
\(968\) −0.406920 −0.0130789
\(969\) 61.9256 1.98934
\(970\) 0.992162 0.0318564
\(971\) −31.1751 −1.00046 −0.500228 0.865894i \(-0.666751\pi\)
−0.500228 + 0.865894i \(0.666751\pi\)
\(972\) 19.5180 0.626039
\(973\) −24.4002 −0.782235
\(974\) −11.2771 −0.361342
\(975\) −1.44465 −0.0462658
\(976\) −0.787063 −0.0251933
\(977\) 39.5180 1.26429 0.632146 0.774849i \(-0.282174\pi\)
0.632146 + 0.774849i \(0.282174\pi\)
\(978\) −27.4382 −0.877378
\(979\) 45.0312 1.43920
\(980\) 0.786623 0.0251277
\(981\) −14.6697 −0.468368
\(982\) 26.9438 0.859811
\(983\) −52.2456 −1.66637 −0.833187 0.552991i \(-0.813486\pi\)
−0.833187 + 0.552991i \(0.813486\pi\)
\(984\) 4.98827 0.159020
\(985\) 0.101909 0.00324709
\(986\) 0.716054 0.0228038
\(987\) −47.4940 −1.51175
\(988\) −0.919911 −0.0292663
\(989\) 13.8850 0.441518
\(990\) 2.82431 0.0897623
\(991\) 12.2889 0.390371 0.195185 0.980766i \(-0.437469\pi\)
0.195185 + 0.980766i \(0.437469\pi\)
\(992\) −3.93199 −0.124841
\(993\) −50.3864 −1.59896
\(994\) −25.1425 −0.797471
\(995\) −6.42368 −0.203644
\(996\) −21.8451 −0.692187
\(997\) −38.7950 −1.22865 −0.614324 0.789054i \(-0.710571\pi\)
−0.614324 + 0.789054i \(0.710571\pi\)
\(998\) −4.45445 −0.141003
\(999\) 4.03172 0.127558
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 8006.2.a.d.1.16 98
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8006.2.a.d.1.16 98 1.1 even 1 trivial