Properties

Label 4022.2.a.b.1.3
Level $4022$
Weight $2$
Character 4022.1
Self dual yes
Analytic conductor $32.116$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, 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("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.65109\) of defining polynomial
Character \(\chi\) \(=\) 4022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.65109 q^{5} -1.00000 q^{6} +0.273891 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.65109 q^{5} -1.00000 q^{6} +0.273891 q^{7} +1.00000 q^{8} -2.00000 q^{9} +2.65109 q^{10} +4.10331 q^{11} -1.00000 q^{12} -4.75441 q^{13} +0.273891 q^{14} -2.65109 q^{15} +1.00000 q^{16} -7.57608 q^{17} -2.00000 q^{18} -1.65109 q^{19} +2.65109 q^{20} -0.273891 q^{21} +4.10331 q^{22} -3.27389 q^{23} -1.00000 q^{24} +2.02830 q^{25} -4.75441 q^{26} +5.00000 q^{27} +0.273891 q^{28} -7.27389 q^{29} -2.65109 q^{30} -8.12386 q^{31} +1.00000 q^{32} -4.10331 q^{33} -7.57608 q^{34} +0.726109 q^{35} -2.00000 q^{36} -4.10331 q^{37} -1.65109 q^{38} +4.75441 q^{39} +2.65109 q^{40} +3.36945 q^{41} -0.273891 q^{42} -5.37720 q^{43} +4.10331 q^{44} -5.30219 q^{45} -3.27389 q^{46} -7.35878 q^{47} -1.00000 q^{48} -6.92498 q^{49} +2.02830 q^{50} +7.57608 q^{51} -4.75441 q^{52} -3.10331 q^{53} +5.00000 q^{54} +10.8783 q^{55} +0.273891 q^{56} +1.65109 q^{57} -7.27389 q^{58} +12.4621 q^{59} -2.65109 q^{60} +5.13161 q^{61} -8.12386 q^{62} -0.547781 q^{63} +1.00000 q^{64} -12.6044 q^{65} -4.10331 q^{66} -1.26614 q^{67} -7.57608 q^{68} +3.27389 q^{69} +0.726109 q^{70} +0.754406 q^{71} -2.00000 q^{72} +7.70769 q^{73} -4.10331 q^{74} -2.02830 q^{75} -1.65109 q^{76} +1.12386 q^{77} +4.75441 q^{78} +11.3695 q^{79} +2.65109 q^{80} +1.00000 q^{81} +3.36945 q^{82} -5.03605 q^{83} -0.273891 q^{84} -20.0849 q^{85} -5.37720 q^{86} +7.27389 q^{87} +4.10331 q^{88} +3.25547 q^{89} -5.30219 q^{90} -1.30219 q^{91} -3.27389 q^{92} +8.12386 q^{93} -7.35878 q^{94} -4.37720 q^{95} -1.00000 q^{96} +13.1239 q^{97} -6.92498 q^{98} -8.20662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} - q^{7} + 3 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + q^{5} - 3 q^{6} - q^{7} + 3 q^{8} - 6 q^{9} + q^{10} + 9 q^{11} - 3 q^{12} - 4 q^{13} - q^{14} - q^{15} + 3 q^{16} - 7 q^{17} - 6 q^{18} + 2 q^{19} + q^{20} + q^{21} + 9 q^{22} - 8 q^{23} - 3 q^{24} - 6 q^{25} - 4 q^{26} + 15 q^{27} - q^{28} - 20 q^{29} - q^{30} - 5 q^{31} + 3 q^{32} - 9 q^{33} - 7 q^{34} + 4 q^{35} - 6 q^{36} - 9 q^{37} + 2 q^{38} + 4 q^{39} + q^{40} + q^{41} + q^{42} - 11 q^{43} + 9 q^{44} - 2 q^{45} - 8 q^{46} + 16 q^{47} - 3 q^{48} - 12 q^{49} - 6 q^{50} + 7 q^{51} - 4 q^{52} - 6 q^{53} + 15 q^{54} + 3 q^{55} - q^{56} - 2 q^{57} - 20 q^{58} - 4 q^{59} - q^{60} - 5 q^{62} + 2 q^{63} + 3 q^{64} - 10 q^{65} - 9 q^{66} + 2 q^{67} - 7 q^{68} + 8 q^{69} + 4 q^{70} - 8 q^{71} - 6 q^{72} - 8 q^{73} - 9 q^{74} + 6 q^{75} + 2 q^{76} - 16 q^{77} + 4 q^{78} + 25 q^{79} + q^{80} + 3 q^{81} + q^{82} - 7 q^{83} + q^{84} - 24 q^{85} - 11 q^{86} + 20 q^{87} + 9 q^{88} - 25 q^{89} - 2 q^{90} + 10 q^{91} - 8 q^{92} + 5 q^{93} + 16 q^{94} - 8 q^{95} - 3 q^{96} + 20 q^{97} - 12 q^{98} - 18 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.65109 1.18561 0.592803 0.805348i \(-0.298021\pi\)
0.592803 + 0.805348i \(0.298021\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.273891 0.103521 0.0517604 0.998660i \(-0.483517\pi\)
0.0517604 + 0.998660i \(0.483517\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 2.65109 0.838349
\(11\) 4.10331 1.23720 0.618598 0.785708i \(-0.287701\pi\)
0.618598 + 0.785708i \(0.287701\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.75441 −1.31863 −0.659317 0.751865i \(-0.729155\pi\)
−0.659317 + 0.751865i \(0.729155\pi\)
\(14\) 0.273891 0.0732003
\(15\) −2.65109 −0.684509
\(16\) 1.00000 0.250000
\(17\) −7.57608 −1.83747 −0.918734 0.394876i \(-0.870787\pi\)
−0.918734 + 0.394876i \(0.870787\pi\)
\(18\) −2.00000 −0.471405
\(19\) −1.65109 −0.378787 −0.189393 0.981901i \(-0.560652\pi\)
−0.189393 + 0.981901i \(0.560652\pi\)
\(20\) 2.65109 0.592803
\(21\) −0.273891 −0.0597678
\(22\) 4.10331 0.874829
\(23\) −3.27389 −0.682653 −0.341327 0.939945i \(-0.610876\pi\)
−0.341327 + 0.939945i \(0.610876\pi\)
\(24\) −1.00000 −0.204124
\(25\) 2.02830 0.405659
\(26\) −4.75441 −0.932416
\(27\) 5.00000 0.962250
\(28\) 0.273891 0.0517604
\(29\) −7.27389 −1.35073 −0.675364 0.737485i \(-0.736013\pi\)
−0.675364 + 0.737485i \(0.736013\pi\)
\(30\) −2.65109 −0.484021
\(31\) −8.12386 −1.45909 −0.729544 0.683934i \(-0.760268\pi\)
−0.729544 + 0.683934i \(0.760268\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.10331 −0.714295
\(34\) −7.57608 −1.29929
\(35\) 0.726109 0.122735
\(36\) −2.00000 −0.333333
\(37\) −4.10331 −0.674580 −0.337290 0.941401i \(-0.609510\pi\)
−0.337290 + 0.941401i \(0.609510\pi\)
\(38\) −1.65109 −0.267843
\(39\) 4.75441 0.761314
\(40\) 2.65109 0.419175
\(41\) 3.36945 0.526220 0.263110 0.964766i \(-0.415252\pi\)
0.263110 + 0.964766i \(0.415252\pi\)
\(42\) −0.273891 −0.0422622
\(43\) −5.37720 −0.820016 −0.410008 0.912082i \(-0.634474\pi\)
−0.410008 + 0.912082i \(0.634474\pi\)
\(44\) 4.10331 0.618598
\(45\) −5.30219 −0.790403
\(46\) −3.27389 −0.482709
\(47\) −7.35878 −1.07339 −0.536694 0.843777i \(-0.680327\pi\)
−0.536694 + 0.843777i \(0.680327\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.92498 −0.989283
\(50\) 2.02830 0.286844
\(51\) 7.57608 1.06086
\(52\) −4.75441 −0.659317
\(53\) −3.10331 −0.426273 −0.213136 0.977022i \(-0.568368\pi\)
−0.213136 + 0.977022i \(0.568368\pi\)
\(54\) 5.00000 0.680414
\(55\) 10.8783 1.46682
\(56\) 0.273891 0.0366002
\(57\) 1.65109 0.218693
\(58\) −7.27389 −0.955109
\(59\) 12.4621 1.62243 0.811213 0.584751i \(-0.198808\pi\)
0.811213 + 0.584751i \(0.198808\pi\)
\(60\) −2.65109 −0.342255
\(61\) 5.13161 0.657035 0.328518 0.944498i \(-0.393451\pi\)
0.328518 + 0.944498i \(0.393451\pi\)
\(62\) −8.12386 −1.03173
\(63\) −0.547781 −0.0690139
\(64\) 1.00000 0.125000
\(65\) −12.6044 −1.56338
\(66\) −4.10331 −0.505083
\(67\) −1.26614 −0.154684 −0.0773418 0.997005i \(-0.524643\pi\)
−0.0773418 + 0.997005i \(0.524643\pi\)
\(68\) −7.57608 −0.918734
\(69\) 3.27389 0.394130
\(70\) 0.726109 0.0867867
\(71\) 0.754406 0.0895315 0.0447657 0.998998i \(-0.485746\pi\)
0.0447657 + 0.998998i \(0.485746\pi\)
\(72\) −2.00000 −0.235702
\(73\) 7.70769 0.902116 0.451058 0.892495i \(-0.351047\pi\)
0.451058 + 0.892495i \(0.351047\pi\)
\(74\) −4.10331 −0.477000
\(75\) −2.02830 −0.234207
\(76\) −1.65109 −0.189393
\(77\) 1.12386 0.128076
\(78\) 4.75441 0.538330
\(79\) 11.3695 1.27916 0.639582 0.768723i \(-0.279108\pi\)
0.639582 + 0.768723i \(0.279108\pi\)
\(80\) 2.65109 0.296401
\(81\) 1.00000 0.111111
\(82\) 3.36945 0.372094
\(83\) −5.03605 −0.552778 −0.276389 0.961046i \(-0.589138\pi\)
−0.276389 + 0.961046i \(0.589138\pi\)
\(84\) −0.273891 −0.0298839
\(85\) −20.0849 −2.17851
\(86\) −5.37720 −0.579839
\(87\) 7.27389 0.779843
\(88\) 4.10331 0.437415
\(89\) 3.25547 0.345079 0.172539 0.985003i \(-0.444803\pi\)
0.172539 + 0.985003i \(0.444803\pi\)
\(90\) −5.30219 −0.558900
\(91\) −1.30219 −0.136506
\(92\) −3.27389 −0.341327
\(93\) 8.12386 0.842405
\(94\) −7.35878 −0.759000
\(95\) −4.37720 −0.449092
\(96\) −1.00000 −0.102062
\(97\) 13.1239 1.33253 0.666263 0.745717i \(-0.267893\pi\)
0.666263 + 0.745717i \(0.267893\pi\)
\(98\) −6.92498 −0.699529
\(99\) −8.20662 −0.824797
\(100\) 2.02830 0.202830
\(101\) 4.72611 0.470265 0.235133 0.971963i \(-0.424448\pi\)
0.235133 + 0.971963i \(0.424448\pi\)
\(102\) 7.57608 0.750143
\(103\) −12.0750 −1.18979 −0.594893 0.803805i \(-0.702806\pi\)
−0.594893 + 0.803805i \(0.702806\pi\)
\(104\) −4.75441 −0.466208
\(105\) −0.726109 −0.0708610
\(106\) −3.10331 −0.301420
\(107\) −10.1783 −0.983976 −0.491988 0.870602i \(-0.663730\pi\)
−0.491988 + 0.870602i \(0.663730\pi\)
\(108\) 5.00000 0.481125
\(109\) −2.27389 −0.217799 −0.108900 0.994053i \(-0.534733\pi\)
−0.108900 + 0.994053i \(0.534733\pi\)
\(110\) 10.8783 1.03720
\(111\) 4.10331 0.389469
\(112\) 0.273891 0.0258802
\(113\) −4.58383 −0.431210 −0.215605 0.976481i \(-0.569172\pi\)
−0.215605 + 0.976481i \(0.569172\pi\)
\(114\) 1.65109 0.154639
\(115\) −8.67939 −0.809357
\(116\) −7.27389 −0.675364
\(117\) 9.50881 0.879090
\(118\) 12.4621 1.14723
\(119\) −2.07502 −0.190216
\(120\) −2.65109 −0.242011
\(121\) 5.83717 0.530652
\(122\) 5.13161 0.464594
\(123\) −3.36945 −0.303813
\(124\) −8.12386 −0.729544
\(125\) −7.87826 −0.704653
\(126\) −0.547781 −0.0488002
\(127\) 12.1132 1.07487 0.537436 0.843305i \(-0.319393\pi\)
0.537436 + 0.843305i \(0.319393\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.37720 0.473436
\(130\) −12.6044 −1.10548
\(131\) 2.57608 0.225073 0.112536 0.993648i \(-0.464102\pi\)
0.112536 + 0.993648i \(0.464102\pi\)
\(132\) −4.10331 −0.357147
\(133\) −0.452219 −0.0392124
\(134\) −1.26614 −0.109378
\(135\) 13.2555 1.14085
\(136\) −7.57608 −0.649643
\(137\) −1.95328 −0.166880 −0.0834400 0.996513i \(-0.526591\pi\)
−0.0834400 + 0.996513i \(0.526591\pi\)
\(138\) 3.27389 0.278692
\(139\) −16.4805 −1.39786 −0.698930 0.715191i \(-0.746340\pi\)
−0.698930 + 0.715191i \(0.746340\pi\)
\(140\) 0.726109 0.0613674
\(141\) 7.35878 0.619721
\(142\) 0.754406 0.0633083
\(143\) −19.5088 −1.63141
\(144\) −2.00000 −0.166667
\(145\) −19.2838 −1.60143
\(146\) 7.70769 0.637893
\(147\) 6.92498 0.571163
\(148\) −4.10331 −0.337290
\(149\) 15.8110 1.29529 0.647644 0.761943i \(-0.275755\pi\)
0.647644 + 0.761943i \(0.275755\pi\)
\(150\) −2.02830 −0.165610
\(151\) 17.7643 1.44564 0.722818 0.691038i \(-0.242846\pi\)
0.722818 + 0.691038i \(0.242846\pi\)
\(152\) −1.65109 −0.133921
\(153\) 15.1522 1.22498
\(154\) 1.12386 0.0905631
\(155\) −21.5371 −1.72990
\(156\) 4.75441 0.380657
\(157\) 11.5371 0.920761 0.460381 0.887722i \(-0.347713\pi\)
0.460381 + 0.887722i \(0.347713\pi\)
\(158\) 11.3695 0.904505
\(159\) 3.10331 0.246109
\(160\) 2.65109 0.209587
\(161\) −0.896688 −0.0706689
\(162\) 1.00000 0.0785674
\(163\) 9.18820 0.719675 0.359838 0.933015i \(-0.382832\pi\)
0.359838 + 0.933015i \(0.382832\pi\)
\(164\) 3.36945 0.263110
\(165\) −10.8783 −0.846872
\(166\) −5.03605 −0.390873
\(167\) 5.44447 0.421306 0.210653 0.977561i \(-0.432441\pi\)
0.210653 + 0.977561i \(0.432441\pi\)
\(168\) −0.273891 −0.0211311
\(169\) 9.60437 0.738798
\(170\) −20.0849 −1.54044
\(171\) 3.30219 0.252525
\(172\) −5.37720 −0.410008
\(173\) 18.6893 1.42092 0.710459 0.703738i \(-0.248487\pi\)
0.710459 + 0.703738i \(0.248487\pi\)
\(174\) 7.27389 0.551432
\(175\) 0.555531 0.0419942
\(176\) 4.10331 0.309299
\(177\) −12.4621 −0.936708
\(178\) 3.25547 0.244008
\(179\) −2.96103 −0.221318 −0.110659 0.993858i \(-0.535296\pi\)
−0.110659 + 0.993858i \(0.535296\pi\)
\(180\) −5.30219 −0.395202
\(181\) 1.54778 0.115046 0.0575228 0.998344i \(-0.481680\pi\)
0.0575228 + 0.998344i \(0.481680\pi\)
\(182\) −1.30219 −0.0965245
\(183\) −5.13161 −0.379339
\(184\) −3.27389 −0.241354
\(185\) −10.8783 −0.799786
\(186\) 8.12386 0.595670
\(187\) −31.0870 −2.27331
\(188\) −7.35878 −0.536694
\(189\) 1.36945 0.0996130
\(190\) −4.37720 −0.317556
\(191\) 2.90444 0.210158 0.105079 0.994464i \(-0.466490\pi\)
0.105079 + 0.994464i \(0.466490\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −11.9172 −0.857821 −0.428911 0.903347i \(-0.641102\pi\)
−0.428911 + 0.903347i \(0.641102\pi\)
\(194\) 13.1239 0.942238
\(195\) 12.6044 0.902618
\(196\) −6.92498 −0.494642
\(197\) 6.63055 0.472407 0.236203 0.971704i \(-0.424097\pi\)
0.236203 + 0.971704i \(0.424097\pi\)
\(198\) −8.20662 −0.583219
\(199\) 14.4415 1.02373 0.511867 0.859065i \(-0.328954\pi\)
0.511867 + 0.859065i \(0.328954\pi\)
\(200\) 2.02830 0.143422
\(201\) 1.26614 0.0893067
\(202\) 4.72611 0.332528
\(203\) −1.99225 −0.139829
\(204\) 7.57608 0.530432
\(205\) 8.93273 0.623889
\(206\) −12.0750 −0.841306
\(207\) 6.54778 0.455102
\(208\) −4.75441 −0.329659
\(209\) −6.77495 −0.468633
\(210\) −0.726109 −0.0501063
\(211\) 3.43380 0.236392 0.118196 0.992990i \(-0.462289\pi\)
0.118196 + 0.992990i \(0.462289\pi\)
\(212\) −3.10331 −0.213136
\(213\) −0.754406 −0.0516910
\(214\) −10.1783 −0.695776
\(215\) −14.2555 −0.972215
\(216\) 5.00000 0.340207
\(217\) −2.22505 −0.151046
\(218\) −2.27389 −0.154007
\(219\) −7.70769 −0.520837
\(220\) 10.8783 0.733412
\(221\) 36.0197 2.42295
\(222\) 4.10331 0.275396
\(223\) −10.1599 −0.680358 −0.340179 0.940361i \(-0.610488\pi\)
−0.340179 + 0.940361i \(0.610488\pi\)
\(224\) 0.273891 0.0183001
\(225\) −4.05659 −0.270440
\(226\) −4.58383 −0.304912
\(227\) −4.96103 −0.329275 −0.164638 0.986354i \(-0.552645\pi\)
−0.164638 + 0.986354i \(0.552645\pi\)
\(228\) 1.65109 0.109346
\(229\) −14.7184 −0.972617 −0.486308 0.873787i \(-0.661657\pi\)
−0.486308 + 0.873787i \(0.661657\pi\)
\(230\) −8.67939 −0.572302
\(231\) −1.12386 −0.0739445
\(232\) −7.27389 −0.477554
\(233\) −14.9426 −0.978923 −0.489461 0.872025i \(-0.662807\pi\)
−0.489461 + 0.872025i \(0.662807\pi\)
\(234\) 9.50881 0.621610
\(235\) −19.5088 −1.27261
\(236\) 12.4621 0.811213
\(237\) −11.3695 −0.738525
\(238\) −2.07502 −0.134503
\(239\) 8.02055 0.518806 0.259403 0.965769i \(-0.416474\pi\)
0.259403 + 0.965769i \(0.416474\pi\)
\(240\) −2.65109 −0.171127
\(241\) −11.5499 −0.743995 −0.371997 0.928234i \(-0.621327\pi\)
−0.371997 + 0.928234i \(0.621327\pi\)
\(242\) 5.83717 0.375228
\(243\) −16.0000 −1.02640
\(244\) 5.13161 0.328518
\(245\) −18.3588 −1.17290
\(246\) −3.36945 −0.214828
\(247\) 7.84997 0.499481
\(248\) −8.12386 −0.515866
\(249\) 5.03605 0.319146
\(250\) −7.87826 −0.498265
\(251\) −9.18608 −0.579820 −0.289910 0.957054i \(-0.593625\pi\)
−0.289910 + 0.957054i \(0.593625\pi\)
\(252\) −0.547781 −0.0345070
\(253\) −13.4338 −0.844575
\(254\) 12.1132 0.760049
\(255\) 20.0849 1.25776
\(256\) 1.00000 0.0625000
\(257\) −3.42100 −0.213396 −0.106698 0.994291i \(-0.534028\pi\)
−0.106698 + 0.994291i \(0.534028\pi\)
\(258\) 5.37720 0.334770
\(259\) −1.12386 −0.0698332
\(260\) −12.6044 −0.781690
\(261\) 14.5478 0.900485
\(262\) 2.57608 0.159151
\(263\) −19.1444 −1.18049 −0.590247 0.807222i \(-0.700970\pi\)
−0.590247 + 0.807222i \(0.700970\pi\)
\(264\) −4.10331 −0.252541
\(265\) −8.22717 −0.505391
\(266\) −0.452219 −0.0277273
\(267\) −3.25547 −0.199231
\(268\) −1.26614 −0.0773418
\(269\) −27.7565 −1.69235 −0.846173 0.532909i \(-0.821099\pi\)
−0.846173 + 0.532909i \(0.821099\pi\)
\(270\) 13.2555 0.806702
\(271\) −13.6999 −0.832212 −0.416106 0.909316i \(-0.636605\pi\)
−0.416106 + 0.909316i \(0.636605\pi\)
\(272\) −7.57608 −0.459367
\(273\) 1.30219 0.0788119
\(274\) −1.95328 −0.118002
\(275\) 8.32273 0.501880
\(276\) 3.27389 0.197065
\(277\) −6.48827 −0.389842 −0.194921 0.980819i \(-0.562445\pi\)
−0.194921 + 0.980819i \(0.562445\pi\)
\(278\) −16.4805 −0.988436
\(279\) 16.2477 0.972725
\(280\) 0.726109 0.0433933
\(281\) −28.5937 −1.70576 −0.852879 0.522109i \(-0.825145\pi\)
−0.852879 + 0.522109i \(0.825145\pi\)
\(282\) 7.35878 0.438209
\(283\) −14.8959 −0.885469 −0.442734 0.896653i \(-0.645991\pi\)
−0.442734 + 0.896653i \(0.645991\pi\)
\(284\) 0.754406 0.0447657
\(285\) 4.37720 0.259283
\(286\) −19.5088 −1.15358
\(287\) 0.922861 0.0544748
\(288\) −2.00000 −0.117851
\(289\) 40.3969 2.37629
\(290\) −19.2838 −1.13238
\(291\) −13.1239 −0.769334
\(292\) 7.70769 0.451058
\(293\) −14.4776 −0.845790 −0.422895 0.906179i \(-0.638986\pi\)
−0.422895 + 0.906179i \(0.638986\pi\)
\(294\) 6.92498 0.403873
\(295\) 33.0382 1.92356
\(296\) −4.10331 −0.238500
\(297\) 20.5166 1.19049
\(298\) 15.8110 0.915907
\(299\) 15.5654 0.900171
\(300\) −2.02830 −0.117104
\(301\) −1.47277 −0.0848888
\(302\) 17.7643 1.02222
\(303\) −4.72611 −0.271508
\(304\) −1.65109 −0.0946967
\(305\) 13.6044 0.778984
\(306\) 15.1522 0.866191
\(307\) −20.8315 −1.18892 −0.594460 0.804126i \(-0.702634\pi\)
−0.594460 + 0.804126i \(0.702634\pi\)
\(308\) 1.12386 0.0640378
\(309\) 12.0750 0.686924
\(310\) −21.5371 −1.22323
\(311\) −14.9795 −0.849407 −0.424703 0.905333i \(-0.639622\pi\)
−0.424703 + 0.905333i \(0.639622\pi\)
\(312\) 4.75441 0.269165
\(313\) 21.5216 1.21647 0.608237 0.793756i \(-0.291877\pi\)
0.608237 + 0.793756i \(0.291877\pi\)
\(314\) 11.5371 0.651077
\(315\) −1.45222 −0.0818233
\(316\) 11.3695 0.639582
\(317\) −17.5470 −0.985537 −0.492768 0.870161i \(-0.664015\pi\)
−0.492768 + 0.870161i \(0.664015\pi\)
\(318\) 3.10331 0.174025
\(319\) −29.8470 −1.67111
\(320\) 2.65109 0.148201
\(321\) 10.1783 0.568099
\(322\) −0.896688 −0.0499705
\(323\) 12.5088 0.696009
\(324\) 1.00000 0.0555556
\(325\) −9.64334 −0.534916
\(326\) 9.18820 0.508887
\(327\) 2.27389 0.125746
\(328\) 3.36945 0.186047
\(329\) −2.01550 −0.111118
\(330\) −10.8783 −0.598829
\(331\) 3.80888 0.209355 0.104677 0.994506i \(-0.466619\pi\)
0.104677 + 0.994506i \(0.466619\pi\)
\(332\) −5.03605 −0.276389
\(333\) 8.20662 0.449720
\(334\) 5.44447 0.297908
\(335\) −3.35666 −0.183394
\(336\) −0.273891 −0.0149420
\(337\) 34.9036 1.90132 0.950661 0.310231i \(-0.100406\pi\)
0.950661 + 0.310231i \(0.100406\pi\)
\(338\) 9.60437 0.522409
\(339\) 4.58383 0.248959
\(340\) −20.0849 −1.08926
\(341\) −33.3347 −1.80518
\(342\) 3.30219 0.178562
\(343\) −3.81392 −0.205932
\(344\) −5.37720 −0.289919
\(345\) 8.67939 0.467283
\(346\) 18.6893 1.00474
\(347\) −16.5478 −0.888331 −0.444166 0.895945i \(-0.646500\pi\)
−0.444166 + 0.895945i \(0.646500\pi\)
\(348\) 7.27389 0.389921
\(349\) −31.2341 −1.67192 −0.835962 0.548787i \(-0.815090\pi\)
−0.835962 + 0.548787i \(0.815090\pi\)
\(350\) 0.555531 0.0296944
\(351\) −23.7720 −1.26886
\(352\) 4.10331 0.218707
\(353\) 11.6327 0.619145 0.309572 0.950876i \(-0.399814\pi\)
0.309572 + 0.950876i \(0.399814\pi\)
\(354\) −12.4621 −0.662353
\(355\) 2.00000 0.106149
\(356\) 3.25547 0.172539
\(357\) 2.07502 0.109821
\(358\) −2.96103 −0.156495
\(359\) 15.8753 0.837868 0.418934 0.908017i \(-0.362404\pi\)
0.418934 + 0.908017i \(0.362404\pi\)
\(360\) −5.30219 −0.279450
\(361\) −16.2739 −0.856521
\(362\) 1.54778 0.0813496
\(363\) −5.83717 −0.306372
\(364\) −1.30219 −0.0682531
\(365\) 20.4338 1.06955
\(366\) −5.13161 −0.268233
\(367\) 31.6142 1.65025 0.825125 0.564950i \(-0.191105\pi\)
0.825125 + 0.564950i \(0.191105\pi\)
\(368\) −3.27389 −0.170663
\(369\) −6.73891 −0.350813
\(370\) −10.8783 −0.565534
\(371\) −0.849968 −0.0441281
\(372\) 8.12386 0.421202
\(373\) 33.6249 1.74103 0.870516 0.492140i \(-0.163785\pi\)
0.870516 + 0.492140i \(0.163785\pi\)
\(374\) −31.0870 −1.60747
\(375\) 7.87826 0.406832
\(376\) −7.35878 −0.379500
\(377\) 34.5830 1.78112
\(378\) 1.36945 0.0704371
\(379\) −1.42897 −0.0734012 −0.0367006 0.999326i \(-0.511685\pi\)
−0.0367006 + 0.999326i \(0.511685\pi\)
\(380\) −4.37720 −0.224546
\(381\) −12.1132 −0.620577
\(382\) 2.90444 0.148604
\(383\) −36.1076 −1.84501 −0.922505 0.385985i \(-0.873862\pi\)
−0.922505 + 0.385985i \(0.873862\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.97945 0.151847
\(386\) −11.9172 −0.606571
\(387\) 10.7544 0.546677
\(388\) 13.1239 0.666263
\(389\) 32.8081 1.66344 0.831718 0.555199i \(-0.187358\pi\)
0.831718 + 0.555199i \(0.187358\pi\)
\(390\) 12.6044 0.638247
\(391\) 24.8032 1.25435
\(392\) −6.92498 −0.349765
\(393\) −2.57608 −0.129946
\(394\) 6.63055 0.334042
\(395\) 30.1415 1.51658
\(396\) −8.20662 −0.412398
\(397\) −18.7622 −0.941646 −0.470823 0.882228i \(-0.656043\pi\)
−0.470823 + 0.882228i \(0.656043\pi\)
\(398\) 14.4415 0.723889
\(399\) 0.452219 0.0226393
\(400\) 2.02830 0.101415
\(401\) 35.9730 1.79641 0.898204 0.439580i \(-0.144873\pi\)
0.898204 + 0.439580i \(0.144873\pi\)
\(402\) 1.26614 0.0631493
\(403\) 38.6241 1.92400
\(404\) 4.72611 0.235133
\(405\) 2.65109 0.131734
\(406\) −1.99225 −0.0988737
\(407\) −16.8372 −0.834588
\(408\) 7.57608 0.375072
\(409\) 20.4989 1.01361 0.506804 0.862062i \(-0.330827\pi\)
0.506804 + 0.862062i \(0.330827\pi\)
\(410\) 8.93273 0.441156
\(411\) 1.95328 0.0963482
\(412\) −12.0750 −0.594893
\(413\) 3.41325 0.167955
\(414\) 6.54778 0.321806
\(415\) −13.3510 −0.655376
\(416\) −4.75441 −0.233104
\(417\) 16.4805 0.807054
\(418\) −6.77495 −0.331374
\(419\) −12.4445 −0.607952 −0.303976 0.952680i \(-0.598314\pi\)
−0.303976 + 0.952680i \(0.598314\pi\)
\(420\) −0.726109 −0.0354305
\(421\) −33.8881 −1.65161 −0.825803 0.563958i \(-0.809278\pi\)
−0.825803 + 0.563958i \(0.809278\pi\)
\(422\) 3.43380 0.167155
\(423\) 14.7176 0.715592
\(424\) −3.10331 −0.150710
\(425\) −15.3665 −0.745386
\(426\) −0.754406 −0.0365511
\(427\) 1.40550 0.0680169
\(428\) −10.1783 −0.491988
\(429\) 19.5088 0.941894
\(430\) −14.2555 −0.687460
\(431\) −13.5470 −0.652535 −0.326268 0.945277i \(-0.605791\pi\)
−0.326268 + 0.945277i \(0.605791\pi\)
\(432\) 5.00000 0.240563
\(433\) −3.49602 −0.168008 −0.0840039 0.996465i \(-0.526771\pi\)
−0.0840039 + 0.996465i \(0.526771\pi\)
\(434\) −2.22505 −0.106806
\(435\) 19.2838 0.924586
\(436\) −2.27389 −0.108900
\(437\) 5.40550 0.258580
\(438\) −7.70769 −0.368287
\(439\) 21.1394 1.00893 0.504463 0.863433i \(-0.331690\pi\)
0.504463 + 0.863433i \(0.331690\pi\)
\(440\) 10.8783 0.518601
\(441\) 13.8500 0.659522
\(442\) 36.0197 1.71328
\(443\) −37.5158 −1.78243 −0.891214 0.453583i \(-0.850145\pi\)
−0.891214 + 0.453583i \(0.850145\pi\)
\(444\) 4.10331 0.194735
\(445\) 8.63055 0.409127
\(446\) −10.1599 −0.481085
\(447\) −15.8110 −0.747835
\(448\) 0.273891 0.0129401
\(449\) 9.81180 0.463047 0.231524 0.972829i \(-0.425629\pi\)
0.231524 + 0.972829i \(0.425629\pi\)
\(450\) −4.05659 −0.191230
\(451\) 13.8259 0.651037
\(452\) −4.58383 −0.215605
\(453\) −17.7643 −0.834639
\(454\) −4.96103 −0.232833
\(455\) −3.45222 −0.161843
\(456\) 1.65109 0.0773195
\(457\) 28.5598 1.33597 0.667985 0.744175i \(-0.267157\pi\)
0.667985 + 0.744175i \(0.267157\pi\)
\(458\) −14.7184 −0.687744
\(459\) −37.8804 −1.76811
\(460\) −8.67939 −0.404679
\(461\) 38.1826 1.77834 0.889170 0.457577i \(-0.151283\pi\)
0.889170 + 0.457577i \(0.151283\pi\)
\(462\) −1.12386 −0.0522866
\(463\) −1.85772 −0.0863355 −0.0431677 0.999068i \(-0.513745\pi\)
−0.0431677 + 0.999068i \(0.513745\pi\)
\(464\) −7.27389 −0.337682
\(465\) 21.5371 0.998759
\(466\) −14.9426 −0.692203
\(467\) −11.0360 −0.510687 −0.255344 0.966850i \(-0.582189\pi\)
−0.255344 + 0.966850i \(0.582189\pi\)
\(468\) 9.50881 0.439545
\(469\) −0.346784 −0.0160130
\(470\) −19.5088 −0.899874
\(471\) −11.5371 −0.531602
\(472\) 12.4621 0.573614
\(473\) −22.0643 −1.01452
\(474\) −11.3695 −0.522216
\(475\) −3.34891 −0.153658
\(476\) −2.07502 −0.0951082
\(477\) 6.20662 0.284182
\(478\) 8.02055 0.366851
\(479\) −24.9194 −1.13859 −0.569297 0.822132i \(-0.692785\pi\)
−0.569297 + 0.822132i \(0.692785\pi\)
\(480\) −2.65109 −0.121005
\(481\) 19.5088 0.889525
\(482\) −11.5499 −0.526084
\(483\) 0.896688 0.0408007
\(484\) 5.83717 0.265326
\(485\) 34.7926 1.57985
\(486\) −16.0000 −0.725775
\(487\) 17.6588 0.800199 0.400099 0.916472i \(-0.368976\pi\)
0.400099 + 0.916472i \(0.368976\pi\)
\(488\) 5.13161 0.232297
\(489\) −9.18820 −0.415505
\(490\) −18.3588 −0.829365
\(491\) −24.1231 −1.08866 −0.544329 0.838872i \(-0.683216\pi\)
−0.544329 + 0.838872i \(0.683216\pi\)
\(492\) −3.36945 −0.151907
\(493\) 55.1076 2.48192
\(494\) 7.84997 0.353187
\(495\) −21.7565 −0.977883
\(496\) −8.12386 −0.364772
\(497\) 0.206625 0.00926838
\(498\) 5.03605 0.225671
\(499\) 17.7728 0.795621 0.397811 0.917468i \(-0.369770\pi\)
0.397811 + 0.917468i \(0.369770\pi\)
\(500\) −7.87826 −0.352327
\(501\) −5.44447 −0.243241
\(502\) −9.18608 −0.409995
\(503\) −26.8881 −1.19888 −0.599441 0.800419i \(-0.704611\pi\)
−0.599441 + 0.800419i \(0.704611\pi\)
\(504\) −0.547781 −0.0244001
\(505\) 12.5294 0.557549
\(506\) −13.4338 −0.597205
\(507\) −9.60437 −0.426545
\(508\) 12.1132 0.537436
\(509\) 3.36653 0.149219 0.0746094 0.997213i \(-0.476229\pi\)
0.0746094 + 0.997213i \(0.476229\pi\)
\(510\) 20.0849 0.889374
\(511\) 2.11106 0.0933879
\(512\) 1.00000 0.0441942
\(513\) −8.25547 −0.364488
\(514\) −3.42100 −0.150894
\(515\) −32.0120 −1.41062
\(516\) 5.37720 0.236718
\(517\) −30.1954 −1.32799
\(518\) −1.12386 −0.0493795
\(519\) −18.6893 −0.820368
\(520\) −12.6044 −0.552738
\(521\) −17.6588 −0.773648 −0.386824 0.922154i \(-0.626428\pi\)
−0.386824 + 0.922154i \(0.626428\pi\)
\(522\) 14.5478 0.636739
\(523\) −40.9066 −1.78872 −0.894359 0.447349i \(-0.852368\pi\)
−0.894359 + 0.447349i \(0.852368\pi\)
\(524\) 2.57608 0.112536
\(525\) −0.555531 −0.0242454
\(526\) −19.1444 −0.834736
\(527\) 61.5470 2.68103
\(528\) −4.10331 −0.178574
\(529\) −12.2816 −0.533984
\(530\) −8.22717 −0.357365
\(531\) −24.9242 −1.08162
\(532\) −0.452219 −0.0196062
\(533\) −16.0197 −0.693892
\(534\) −3.25547 −0.140878
\(535\) −26.9837 −1.16661
\(536\) −1.26614 −0.0546889
\(537\) 2.96103 0.127778
\(538\) −27.7565 −1.19667
\(539\) −28.4154 −1.22394
\(540\) 13.2555 0.570424
\(541\) 34.6455 1.48952 0.744762 0.667330i \(-0.232563\pi\)
0.744762 + 0.667330i \(0.232563\pi\)
\(542\) −13.6999 −0.588463
\(543\) −1.54778 −0.0664216
\(544\) −7.57608 −0.324822
\(545\) −6.02830 −0.258224
\(546\) 1.30219 0.0557285
\(547\) −22.6404 −0.968034 −0.484017 0.875058i \(-0.660823\pi\)
−0.484017 + 0.875058i \(0.660823\pi\)
\(548\) −1.95328 −0.0834400
\(549\) −10.2632 −0.438023
\(550\) 8.32273 0.354883
\(551\) 12.0099 0.511638
\(552\) 3.27389 0.139346
\(553\) 3.11399 0.132420
\(554\) −6.48827 −0.275660
\(555\) 10.8783 0.461757
\(556\) −16.4805 −0.698930
\(557\) 41.0219 1.73815 0.869076 0.494678i \(-0.164714\pi\)
0.869076 + 0.494678i \(0.164714\pi\)
\(558\) 16.2477 0.687821
\(559\) 25.5654 1.08130
\(560\) 0.726109 0.0306837
\(561\) 31.0870 1.31249
\(562\) −28.5937 −1.20615
\(563\) −24.9378 −1.05100 −0.525501 0.850793i \(-0.676122\pi\)
−0.525501 + 0.850793i \(0.676122\pi\)
\(564\) 7.35878 0.309861
\(565\) −12.1522 −0.511245
\(566\) −14.8959 −0.626121
\(567\) 0.273891 0.0115023
\(568\) 0.754406 0.0316542
\(569\) −14.0616 −0.589495 −0.294747 0.955575i \(-0.595235\pi\)
−0.294747 + 0.955575i \(0.595235\pi\)
\(570\) 4.37720 0.183341
\(571\) 32.0304 1.34043 0.670216 0.742166i \(-0.266202\pi\)
0.670216 + 0.742166i \(0.266202\pi\)
\(572\) −19.5088 −0.815704
\(573\) −2.90444 −0.121335
\(574\) 0.922861 0.0385195
\(575\) −6.64042 −0.276925
\(576\) −2.00000 −0.0833333
\(577\) −41.6941 −1.73575 −0.867874 0.496785i \(-0.834514\pi\)
−0.867874 + 0.496785i \(0.834514\pi\)
\(578\) 40.3969 1.68029
\(579\) 11.9172 0.495263
\(580\) −19.2838 −0.800715
\(581\) −1.37933 −0.0572241
\(582\) −13.1239 −0.544001
\(583\) −12.7339 −0.527383
\(584\) 7.70769 0.318946
\(585\) 25.2087 1.04225
\(586\) −14.4776 −0.598064
\(587\) 35.1642 1.45138 0.725690 0.688022i \(-0.241521\pi\)
0.725690 + 0.688022i \(0.241521\pi\)
\(588\) 6.92498 0.285582
\(589\) 13.4132 0.552683
\(590\) 33.0382 1.36016
\(591\) −6.63055 −0.272744
\(592\) −4.10331 −0.168645
\(593\) −3.18045 −0.130605 −0.0653027 0.997865i \(-0.520801\pi\)
−0.0653027 + 0.997865i \(0.520801\pi\)
\(594\) 20.5166 0.841805
\(595\) −5.50106 −0.225522
\(596\) 15.8110 0.647644
\(597\) −14.4415 −0.591053
\(598\) 15.5654 0.636517
\(599\) −20.0013 −0.817232 −0.408616 0.912706i \(-0.633988\pi\)
−0.408616 + 0.912706i \(0.633988\pi\)
\(600\) −2.02830 −0.0828048
\(601\) −7.59450 −0.309786 −0.154893 0.987931i \(-0.549503\pi\)
−0.154893 + 0.987931i \(0.549503\pi\)
\(602\) −1.47277 −0.0600254
\(603\) 2.53228 0.103122
\(604\) 17.7643 0.722818
\(605\) 15.4749 0.629144
\(606\) −4.72611 −0.191985
\(607\) 4.34036 0.176170 0.0880848 0.996113i \(-0.471925\pi\)
0.0880848 + 0.996113i \(0.471925\pi\)
\(608\) −1.65109 −0.0669607
\(609\) 1.99225 0.0807300
\(610\) 13.6044 0.550825
\(611\) 34.9866 1.41541
\(612\) 15.1522 0.612490
\(613\) 9.64042 0.389373 0.194686 0.980866i \(-0.437631\pi\)
0.194686 + 0.980866i \(0.437631\pi\)
\(614\) −20.8315 −0.840693
\(615\) −8.93273 −0.360203
\(616\) 1.12386 0.0452815
\(617\) 15.8500 0.638096 0.319048 0.947739i \(-0.396637\pi\)
0.319048 + 0.947739i \(0.396637\pi\)
\(618\) 12.0750 0.485728
\(619\) −7.66659 −0.308146 −0.154073 0.988059i \(-0.549239\pi\)
−0.154073 + 0.988059i \(0.549239\pi\)
\(620\) −21.5371 −0.864951
\(621\) −16.3695 −0.656884
\(622\) −14.9795 −0.600621
\(623\) 0.891642 0.0357229
\(624\) 4.75441 0.190329
\(625\) −31.0275 −1.24110
\(626\) 21.5216 0.860176
\(627\) 6.77495 0.270566
\(628\) 11.5371 0.460381
\(629\) 31.0870 1.23952
\(630\) −1.45222 −0.0578578
\(631\) 18.4466 0.734347 0.367174 0.930152i \(-0.380325\pi\)
0.367174 + 0.930152i \(0.380325\pi\)
\(632\) 11.3695 0.452253
\(633\) −3.43380 −0.136481
\(634\) −17.5470 −0.696880
\(635\) 32.1132 1.27437
\(636\) 3.10331 0.123054
\(637\) 32.9242 1.30450
\(638\) −29.8470 −1.18166
\(639\) −1.50881 −0.0596877
\(640\) 2.65109 0.104794
\(641\) 47.9944 1.89566 0.947832 0.318771i \(-0.103270\pi\)
0.947832 + 0.318771i \(0.103270\pi\)
\(642\) 10.1783 0.401707
\(643\) 12.9554 0.510911 0.255456 0.966821i \(-0.417775\pi\)
0.255456 + 0.966821i \(0.417775\pi\)
\(644\) −0.896688 −0.0353344
\(645\) 14.2555 0.561308
\(646\) 12.5088 0.492153
\(647\) 9.29231 0.365319 0.182659 0.983176i \(-0.441529\pi\)
0.182659 + 0.983176i \(0.441529\pi\)
\(648\) 1.00000 0.0392837
\(649\) 51.1359 2.00726
\(650\) −9.64334 −0.378243
\(651\) 2.22505 0.0872065
\(652\) 9.18820 0.359838
\(653\) 43.4768 1.70138 0.850689 0.525669i \(-0.176185\pi\)
0.850689 + 0.525669i \(0.176185\pi\)
\(654\) 2.27389 0.0889162
\(655\) 6.82942 0.266848
\(656\) 3.36945 0.131555
\(657\) −15.4154 −0.601411
\(658\) −2.01550 −0.0785724
\(659\) 25.4875 0.992851 0.496425 0.868079i \(-0.334646\pi\)
0.496425 + 0.868079i \(0.334646\pi\)
\(660\) −10.8783 −0.423436
\(661\) 29.1805 1.13499 0.567494 0.823378i \(-0.307913\pi\)
0.567494 + 0.823378i \(0.307913\pi\)
\(662\) 3.80888 0.148036
\(663\) −36.0197 −1.39889
\(664\) −5.03605 −0.195437
\(665\) −1.19887 −0.0464904
\(666\) 8.20662 0.318000
\(667\) 23.8139 0.922079
\(668\) 5.44447 0.210653
\(669\) 10.1599 0.392805
\(670\) −3.35666 −0.129679
\(671\) 21.0566 0.812881
\(672\) −0.273891 −0.0105656
\(673\) −10.0184 −0.386182 −0.193091 0.981181i \(-0.561851\pi\)
−0.193091 + 0.981181i \(0.561851\pi\)
\(674\) 34.9036 1.34444
\(675\) 10.1415 0.390346
\(676\) 9.60437 0.369399
\(677\) −7.40550 −0.284616 −0.142308 0.989822i \(-0.545452\pi\)
−0.142308 + 0.989822i \(0.545452\pi\)
\(678\) 4.58383 0.176041
\(679\) 3.59450 0.137944
\(680\) −20.0849 −0.770220
\(681\) 4.96103 0.190107
\(682\) −33.3347 −1.27645
\(683\) −32.4514 −1.24172 −0.620860 0.783922i \(-0.713216\pi\)
−0.620860 + 0.783922i \(0.713216\pi\)
\(684\) 3.30219 0.126262
\(685\) −5.17833 −0.197854
\(686\) −3.81392 −0.145616
\(687\) 14.7184 0.561540
\(688\) −5.37720 −0.205004
\(689\) 14.7544 0.562098
\(690\) 8.67939 0.330419
\(691\) 8.71544 0.331551 0.165775 0.986164i \(-0.446987\pi\)
0.165775 + 0.986164i \(0.446987\pi\)
\(692\) 18.6893 0.710459
\(693\) −2.24772 −0.0853837
\(694\) −16.5478 −0.628145
\(695\) −43.6914 −1.65731
\(696\) 7.27389 0.275716
\(697\) −25.5272 −0.966913
\(698\) −31.2341 −1.18223
\(699\) 14.9426 0.565181
\(700\) 0.555531 0.0209971
\(701\) 39.6794 1.49867 0.749335 0.662191i \(-0.230373\pi\)
0.749335 + 0.662191i \(0.230373\pi\)
\(702\) −23.7720 −0.897217
\(703\) 6.77495 0.255522
\(704\) 4.10331 0.154649
\(705\) 19.5088 0.734744
\(706\) 11.6327 0.437801
\(707\) 1.29444 0.0486823
\(708\) −12.4621 −0.468354
\(709\) −30.2653 −1.13664 −0.568319 0.822808i \(-0.692406\pi\)
−0.568319 + 0.822808i \(0.692406\pi\)
\(710\) 2.00000 0.0750587
\(711\) −22.7389 −0.852776
\(712\) 3.25547 0.122004
\(713\) 26.5966 0.996051
\(714\) 2.07502 0.0776555
\(715\) −51.7197 −1.93421
\(716\) −2.96103 −0.110659
\(717\) −8.02055 −0.299533
\(718\) 15.8753 0.592462
\(719\) 22.2109 0.828326 0.414163 0.910203i \(-0.364074\pi\)
0.414163 + 0.910203i \(0.364074\pi\)
\(720\) −5.30219 −0.197601
\(721\) −3.30723 −0.123168
\(722\) −16.2739 −0.605651
\(723\) 11.5499 0.429546
\(724\) 1.54778 0.0575228
\(725\) −14.7536 −0.547935
\(726\) −5.83717 −0.216638
\(727\) 7.19032 0.266674 0.133337 0.991071i \(-0.457431\pi\)
0.133337 + 0.991071i \(0.457431\pi\)
\(728\) −1.30219 −0.0482623
\(729\) 13.0000 0.481481
\(730\) 20.4338 0.756289
\(731\) 40.7381 1.50675
\(732\) −5.13161 −0.189670
\(733\) −27.3532 −1.01031 −0.505156 0.863028i \(-0.668565\pi\)
−0.505156 + 0.863028i \(0.668565\pi\)
\(734\) 31.6142 1.16690
\(735\) 18.3588 0.677174
\(736\) −3.27389 −0.120677
\(737\) −5.19537 −0.191374
\(738\) −6.73891 −0.248063
\(739\) −19.0841 −0.702020 −0.351010 0.936372i \(-0.614162\pi\)
−0.351010 + 0.936372i \(0.614162\pi\)
\(740\) −10.8783 −0.399893
\(741\) −7.84997 −0.288376
\(742\) −0.849968 −0.0312033
\(743\) −40.1132 −1.47161 −0.735805 0.677193i \(-0.763196\pi\)
−0.735805 + 0.677193i \(0.763196\pi\)
\(744\) 8.12386 0.297835
\(745\) 41.9164 1.53570
\(746\) 33.6249 1.23110
\(747\) 10.0721 0.368519
\(748\) −31.0870 −1.13665
\(749\) −2.78775 −0.101862
\(750\) 7.87826 0.287674
\(751\) −53.7819 −1.96253 −0.981265 0.192664i \(-0.938287\pi\)
−0.981265 + 0.192664i \(0.938287\pi\)
\(752\) −7.35878 −0.268347
\(753\) 9.18608 0.334759
\(754\) 34.5830 1.25944
\(755\) 47.0948 1.71395
\(756\) 1.36945 0.0498065
\(757\) −20.6786 −0.751576 −0.375788 0.926706i \(-0.622628\pi\)
−0.375788 + 0.926706i \(0.622628\pi\)
\(758\) −1.42897 −0.0519025
\(759\) 13.4338 0.487616
\(760\) −4.37720 −0.158778
\(761\) 10.3313 0.374509 0.187254 0.982311i \(-0.440041\pi\)
0.187254 + 0.982311i \(0.440041\pi\)
\(762\) −12.1132 −0.438814
\(763\) −0.622797 −0.0225468
\(764\) 2.90444 0.105079
\(765\) 40.1698 1.45234
\(766\) −36.1076 −1.30462
\(767\) −59.2498 −2.13939
\(768\) −1.00000 −0.0360844
\(769\) −49.5235 −1.78586 −0.892932 0.450192i \(-0.851356\pi\)
−0.892932 + 0.450192i \(0.851356\pi\)
\(770\) 2.97945 0.107372
\(771\) 3.42100 0.123204
\(772\) −11.9172 −0.428911
\(773\) −5.56058 −0.200000 −0.100000 0.994987i \(-0.531884\pi\)
−0.100000 + 0.994987i \(0.531884\pi\)
\(774\) 10.7544 0.386559
\(775\) −16.4776 −0.591893
\(776\) 13.1239 0.471119
\(777\) 1.12386 0.0403182
\(778\) 32.8081 1.17623
\(779\) −5.56328 −0.199325
\(780\) 12.6044 0.451309
\(781\) 3.09556 0.110768
\(782\) 24.8032 0.886962
\(783\) −36.3695 −1.29974
\(784\) −6.92498 −0.247321
\(785\) 30.5860 1.09166
\(786\) −2.57608 −0.0918856
\(787\) −21.1471 −0.753813 −0.376907 0.926251i \(-0.623012\pi\)
−0.376907 + 0.926251i \(0.623012\pi\)
\(788\) 6.63055 0.236203
\(789\) 19.1444 0.681559
\(790\) 30.1415 1.07239
\(791\) −1.25547 −0.0446393
\(792\) −8.20662 −0.291610
\(793\) −24.3977 −0.866389
\(794\) −18.7622 −0.665844
\(795\) 8.22717 0.291788
\(796\) 14.4415 0.511867
\(797\) −14.6327 −0.518316 −0.259158 0.965835i \(-0.583445\pi\)
−0.259158 + 0.965835i \(0.583445\pi\)
\(798\) 0.452219 0.0160084
\(799\) 55.7507 1.97232
\(800\) 2.02830 0.0717111
\(801\) −6.51093 −0.230053
\(802\) 35.9730 1.27025
\(803\) 31.6270 1.11609
\(804\) 1.26614 0.0446533
\(805\) −2.37720 −0.0837854
\(806\) 38.6241 1.36048
\(807\) 27.7565 0.977076
\(808\) 4.72611 0.166264
\(809\) −4.35958 −0.153275 −0.0766373 0.997059i \(-0.524418\pi\)
−0.0766373 + 0.997059i \(0.524418\pi\)
\(810\) 2.65109 0.0931499
\(811\) −30.3347 −1.06520 −0.532598 0.846368i \(-0.678784\pi\)
−0.532598 + 0.846368i \(0.678784\pi\)
\(812\) −1.99225 −0.0699143
\(813\) 13.6999 0.480478
\(814\) −16.8372 −0.590143
\(815\) 24.3588 0.853251
\(816\) 7.57608 0.265216
\(817\) 8.87826 0.310611
\(818\) 20.4989 0.716729
\(819\) 2.60437 0.0910042
\(820\) 8.93273 0.311945
\(821\) −51.2440 −1.78843 −0.894214 0.447640i \(-0.852265\pi\)
−0.894214 + 0.447640i \(0.852265\pi\)
\(822\) 1.95328 0.0681285
\(823\) −23.6036 −0.822769 −0.411384 0.911462i \(-0.634955\pi\)
−0.411384 + 0.911462i \(0.634955\pi\)
\(824\) −12.0750 −0.420653
\(825\) −8.32273 −0.289760
\(826\) 3.41325 0.118762
\(827\) 10.9659 0.381320 0.190660 0.981656i \(-0.438937\pi\)
0.190660 + 0.981656i \(0.438937\pi\)
\(828\) 6.54778 0.227551
\(829\) 1.04460 0.0362804 0.0181402 0.999835i \(-0.494225\pi\)
0.0181402 + 0.999835i \(0.494225\pi\)
\(830\) −13.3510 −0.463421
\(831\) 6.48827 0.225075
\(832\) −4.75441 −0.164829
\(833\) 52.4642 1.81778
\(834\) 16.4805 0.570674
\(835\) 14.4338 0.499502
\(836\) −6.77495 −0.234317
\(837\) −40.6193 −1.40401
\(838\) −12.4445 −0.429887
\(839\) 25.9992 0.897592 0.448796 0.893634i \(-0.351853\pi\)
0.448796 + 0.893634i \(0.351853\pi\)
\(840\) −0.726109 −0.0250532
\(841\) 23.9095 0.824465
\(842\) −33.8881 −1.16786
\(843\) 28.5937 0.984819
\(844\) 3.43380 0.118196
\(845\) 25.4621 0.875923
\(846\) 14.7176 0.506000
\(847\) 1.59875 0.0549336
\(848\) −3.10331 −0.106568
\(849\) 14.8959 0.511226
\(850\) −15.3665 −0.527068
\(851\) 13.4338 0.460505
\(852\) −0.754406 −0.0258455
\(853\) 37.6786 1.29009 0.645045 0.764144i \(-0.276839\pi\)
0.645045 + 0.764144i \(0.276839\pi\)
\(854\) 1.40550 0.0480952
\(855\) 8.75441 0.299394
\(856\) −10.1783 −0.347888
\(857\) 12.4239 0.424393 0.212197 0.977227i \(-0.431938\pi\)
0.212197 + 0.977227i \(0.431938\pi\)
\(858\) 19.5088 0.666020
\(859\) −29.0969 −0.992773 −0.496386 0.868102i \(-0.665340\pi\)
−0.496386 + 0.868102i \(0.665340\pi\)
\(860\) −14.2555 −0.486107
\(861\) −0.922861 −0.0314510
\(862\) −13.5470 −0.461412
\(863\) 55.8873 1.90243 0.951214 0.308533i \(-0.0998381\pi\)
0.951214 + 0.308533i \(0.0998381\pi\)
\(864\) 5.00000 0.170103
\(865\) 49.5470 1.68465
\(866\) −3.49602 −0.118799
\(867\) −40.3969 −1.37195
\(868\) −2.22505 −0.0755231
\(869\) 46.6524 1.58257
\(870\) 19.2838 0.653781
\(871\) 6.01975 0.203971
\(872\) −2.27389 −0.0770037
\(873\) −26.2477 −0.888351
\(874\) 5.40550 0.182844
\(875\) −2.15778 −0.0729464
\(876\) −7.70769 −0.260419
\(877\) 14.5995 0.492992 0.246496 0.969144i \(-0.420721\pi\)
0.246496 + 0.969144i \(0.420721\pi\)
\(878\) 21.1394 0.713419
\(879\) 14.4776 0.488317
\(880\) 10.8783 0.366706
\(881\) 22.6871 0.764349 0.382175 0.924090i \(-0.375175\pi\)
0.382175 + 0.924090i \(0.375175\pi\)
\(882\) 13.8500 0.466353
\(883\) 45.0072 1.51461 0.757306 0.653060i \(-0.226515\pi\)
0.757306 + 0.653060i \(0.226515\pi\)
\(884\) 36.0197 1.21148
\(885\) −33.0382 −1.11057
\(886\) −37.5158 −1.26037
\(887\) −45.9730 −1.54362 −0.771812 0.635851i \(-0.780649\pi\)
−0.771812 + 0.635851i \(0.780649\pi\)
\(888\) 4.10331 0.137698
\(889\) 3.31769 0.111272
\(890\) 8.63055 0.289297
\(891\) 4.10331 0.137466
\(892\) −10.1599 −0.340179
\(893\) 12.1500 0.406585
\(894\) −15.8110 −0.528799
\(895\) −7.84997 −0.262395
\(896\) 0.273891 0.00915004
\(897\) −15.5654 −0.519714
\(898\) 9.81180 0.327424
\(899\) 59.0921 1.97083
\(900\) −4.05659 −0.135220
\(901\) 23.5109 0.783263
\(902\) 13.8259 0.460353
\(903\) 1.47277 0.0490106
\(904\) −4.58383 −0.152456
\(905\) 4.10331 0.136399
\(906\) −17.7643 −0.590179
\(907\) 3.69861 0.122810 0.0614052 0.998113i \(-0.480442\pi\)
0.0614052 + 0.998113i \(0.480442\pi\)
\(908\) −4.96103 −0.164638
\(909\) −9.45222 −0.313510
\(910\) −3.45222 −0.114440
\(911\) 37.5342 1.24356 0.621782 0.783191i \(-0.286409\pi\)
0.621782 + 0.783191i \(0.286409\pi\)
\(912\) 1.65109 0.0546732
\(913\) −20.6645 −0.683894
\(914\) 28.5598 0.944673
\(915\) −13.6044 −0.449747
\(916\) −14.7184 −0.486308
\(917\) 0.705563 0.0232998
\(918\) −37.8804 −1.25024
\(919\) −22.1677 −0.731243 −0.365622 0.930764i \(-0.619144\pi\)
−0.365622 + 0.930764i \(0.619144\pi\)
\(920\) −8.67939 −0.286151
\(921\) 20.8315 0.686423
\(922\) 38.1826 1.25748
\(923\) −3.58675 −0.118059
\(924\) −1.12386 −0.0369722
\(925\) −8.32273 −0.273650
\(926\) −1.85772 −0.0610484
\(927\) 24.1500 0.793191
\(928\) −7.27389 −0.238777
\(929\) −37.6503 −1.23527 −0.617633 0.786466i \(-0.711908\pi\)
−0.617633 + 0.786466i \(0.711908\pi\)
\(930\) 21.5371 0.706230
\(931\) 11.4338 0.374728
\(932\) −14.9426 −0.489461
\(933\) 14.9795 0.490405
\(934\) −11.0360 −0.361110
\(935\) −82.4146 −2.69524
\(936\) 9.50881 0.310805
\(937\) 24.4047 0.797267 0.398633 0.917110i \(-0.369485\pi\)
0.398633 + 0.917110i \(0.369485\pi\)
\(938\) −0.346784 −0.0113229
\(939\) −21.5216 −0.702331
\(940\) −19.5088 −0.636307
\(941\) −12.5491 −0.409089 −0.204545 0.978857i \(-0.565571\pi\)
−0.204545 + 0.978857i \(0.565571\pi\)
\(942\) −11.5371 −0.375899
\(943\) −11.0312 −0.359226
\(944\) 12.4621 0.405607
\(945\) 3.63055 0.118102
\(946\) −22.0643 −0.717374
\(947\) −39.4749 −1.28276 −0.641381 0.767223i \(-0.721638\pi\)
−0.641381 + 0.767223i \(0.721638\pi\)
\(948\) −11.3695 −0.369263
\(949\) −36.6455 −1.18956
\(950\) −3.34891 −0.108653
\(951\) 17.5470 0.569000
\(952\) −2.07502 −0.0672517
\(953\) 10.3646 0.335743 0.167872 0.985809i \(-0.446311\pi\)
0.167872 + 0.985809i \(0.446311\pi\)
\(954\) 6.20662 0.200947
\(955\) 7.69994 0.249164
\(956\) 8.02055 0.259403
\(957\) 29.8470 0.964818
\(958\) −24.9194 −0.805108
\(959\) −0.534985 −0.0172756
\(960\) −2.65109 −0.0855637
\(961\) 34.9971 1.12894
\(962\) 19.5088 0.628989
\(963\) 20.3567 0.655984
\(964\) −11.5499 −0.371997
\(965\) −31.5937 −1.01704
\(966\) 0.896688 0.0288505
\(967\) −10.7029 −0.344181 −0.172090 0.985081i \(-0.555052\pi\)
−0.172090 + 0.985081i \(0.555052\pi\)
\(968\) 5.83717 0.187614
\(969\) −12.5088 −0.401841
\(970\) 34.7926 1.11712
\(971\) −16.6015 −0.532766 −0.266383 0.963867i \(-0.585829\pi\)
−0.266383 + 0.963867i \(0.585829\pi\)
\(972\) −16.0000 −0.513200
\(973\) −4.51386 −0.144708
\(974\) 17.6588 0.565826
\(975\) 9.64334 0.308834
\(976\) 5.13161 0.164259
\(977\) 9.46289 0.302745 0.151372 0.988477i \(-0.451631\pi\)
0.151372 + 0.988477i \(0.451631\pi\)
\(978\) −9.18820 −0.293806
\(979\) 13.3582 0.426930
\(980\) −18.3588 −0.586450
\(981\) 4.54778 0.145199
\(982\) −24.1231 −0.769798
\(983\) 47.7429 1.52276 0.761382 0.648304i \(-0.224521\pi\)
0.761382 + 0.648304i \(0.224521\pi\)
\(984\) −3.36945 −0.107414
\(985\) 17.5782 0.560088
\(986\) 55.1076 1.75498
\(987\) 2.01550 0.0641541
\(988\) 7.84997 0.249741
\(989\) 17.6044 0.559787
\(990\) −21.7565 −0.691468
\(991\) 2.88331 0.0915914 0.0457957 0.998951i \(-0.485418\pi\)
0.0457957 + 0.998951i \(0.485418\pi\)
\(992\) −8.12386 −0.257933
\(993\) −3.80888 −0.120871
\(994\) 0.206625 0.00655373
\(995\) 38.2859 1.21374
\(996\) 5.03605 0.159573
\(997\) −5.20392 −0.164810 −0.0824049 0.996599i \(-0.526260\pi\)
−0.0824049 + 0.996599i \(0.526260\pi\)
\(998\) 17.7728 0.562589
\(999\) −20.5166 −0.649115
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 4022.2.a.b.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4022.2.a.b.1.3 3 1.1 even 1 trivial