Properties

Label 4001.2.a.a.1.8
Level $4001$
Weight $2$
Character 4001.1
Self dual yes
Analytic conductor $31.948$
Analytic rank $1$
Dimension $149$
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] = [4001,2,Mod(1,4001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4001, 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("4001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4001.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: \(31.9481458487\)
Analytic rank: \(1\)
Dimension: \(149\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 4001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.58646 q^{2} -1.10071 q^{3} +4.68977 q^{4} -0.828720 q^{5} +2.84694 q^{6} -1.43853 q^{7} -6.95698 q^{8} -1.78844 q^{9} +O(q^{10})\) \(q-2.58646 q^{2} -1.10071 q^{3} +4.68977 q^{4} -0.828720 q^{5} +2.84694 q^{6} -1.43853 q^{7} -6.95698 q^{8} -1.78844 q^{9} +2.14345 q^{10} +4.30556 q^{11} -5.16208 q^{12} +5.68051 q^{13} +3.72070 q^{14} +0.912180 q^{15} +8.61440 q^{16} +0.204728 q^{17} +4.62572 q^{18} -1.11011 q^{19} -3.88651 q^{20} +1.58341 q^{21} -11.1362 q^{22} -1.09978 q^{23} +7.65761 q^{24} -4.31322 q^{25} -14.6924 q^{26} +5.27068 q^{27} -6.74638 q^{28} +0.550487 q^{29} -2.35932 q^{30} -5.70327 q^{31} -8.36683 q^{32} -4.73917 q^{33} -0.529520 q^{34} +1.19214 q^{35} -8.38736 q^{36} -10.8782 q^{37} +2.87126 q^{38} -6.25259 q^{39} +5.76539 q^{40} -9.68391 q^{41} -4.09541 q^{42} +7.33515 q^{43} +20.1921 q^{44} +1.48211 q^{45} +2.84454 q^{46} +12.3431 q^{47} -9.48195 q^{48} -4.93063 q^{49} +11.1560 q^{50} -0.225346 q^{51} +26.6403 q^{52} +3.70866 q^{53} -13.6324 q^{54} -3.56810 q^{55} +10.0078 q^{56} +1.22191 q^{57} -1.42381 q^{58} +13.3936 q^{59} +4.27791 q^{60} +2.02183 q^{61} +14.7513 q^{62} +2.57272 q^{63} +4.41167 q^{64} -4.70755 q^{65} +12.2577 q^{66} -16.1756 q^{67} +0.960127 q^{68} +1.21054 q^{69} -3.08342 q^{70} -2.30638 q^{71} +12.4421 q^{72} -0.0880236 q^{73} +28.1360 q^{74} +4.74761 q^{75} -5.20616 q^{76} -6.19369 q^{77} +16.1721 q^{78} -2.05059 q^{79} -7.13892 q^{80} -0.436175 q^{81} +25.0470 q^{82} +9.20626 q^{83} +7.42581 q^{84} -0.169662 q^{85} -18.9721 q^{86} -0.605927 q^{87} -29.9537 q^{88} -3.27083 q^{89} -3.83343 q^{90} -8.17159 q^{91} -5.15773 q^{92} +6.27764 q^{93} -31.9249 q^{94} +0.919971 q^{95} +9.20945 q^{96} +9.21504 q^{97} +12.7529 q^{98} -7.70023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58646 −1.82890 −0.914451 0.404696i \(-0.867377\pi\)
−0.914451 + 0.404696i \(0.867377\pi\)
\(3\) −1.10071 −0.635495 −0.317748 0.948175i \(-0.602926\pi\)
−0.317748 + 0.948175i \(0.602926\pi\)
\(4\) 4.68977 2.34488
\(5\) −0.828720 −0.370615 −0.185307 0.982681i \(-0.559328\pi\)
−0.185307 + 0.982681i \(0.559328\pi\)
\(6\) 2.84694 1.16226
\(7\) −1.43853 −0.543714 −0.271857 0.962338i \(-0.587638\pi\)
−0.271857 + 0.962338i \(0.587638\pi\)
\(8\) −6.95698 −2.45966
\(9\) −1.78844 −0.596146
\(10\) 2.14345 0.677818
\(11\) 4.30556 1.29818 0.649088 0.760713i \(-0.275151\pi\)
0.649088 + 0.760713i \(0.275151\pi\)
\(12\) −5.16208 −1.49016
\(13\) 5.68051 1.57549 0.787744 0.616002i \(-0.211249\pi\)
0.787744 + 0.616002i \(0.211249\pi\)
\(14\) 3.72070 0.994400
\(15\) 0.912180 0.235524
\(16\) 8.61440 2.15360
\(17\) 0.204728 0.0496538 0.0248269 0.999692i \(-0.492097\pi\)
0.0248269 + 0.999692i \(0.492097\pi\)
\(18\) 4.62572 1.09029
\(19\) −1.11011 −0.254677 −0.127338 0.991859i \(-0.540643\pi\)
−0.127338 + 0.991859i \(0.540643\pi\)
\(20\) −3.88651 −0.869049
\(21\) 1.58341 0.345527
\(22\) −11.1362 −2.37424
\(23\) −1.09978 −0.229320 −0.114660 0.993405i \(-0.536578\pi\)
−0.114660 + 0.993405i \(0.536578\pi\)
\(24\) 7.65761 1.56310
\(25\) −4.31322 −0.862645
\(26\) −14.6924 −2.88142
\(27\) 5.27068 1.01434
\(28\) −6.74638 −1.27495
\(29\) 0.550487 0.102223 0.0511114 0.998693i \(-0.483724\pi\)
0.0511114 + 0.998693i \(0.483724\pi\)
\(30\) −2.35932 −0.430750
\(31\) −5.70327 −1.02434 −0.512169 0.858885i \(-0.671158\pi\)
−0.512169 + 0.858885i \(0.671158\pi\)
\(32\) −8.36683 −1.47906
\(33\) −4.73917 −0.824984
\(34\) −0.529520 −0.0908120
\(35\) 1.19214 0.201508
\(36\) −8.38736 −1.39789
\(37\) −10.8782 −1.78837 −0.894183 0.447701i \(-0.852243\pi\)
−0.894183 + 0.447701i \(0.852243\pi\)
\(38\) 2.87126 0.465779
\(39\) −6.25259 −1.00122
\(40\) 5.76539 0.911588
\(41\) −9.68391 −1.51237 −0.756186 0.654357i \(-0.772939\pi\)
−0.756186 + 0.654357i \(0.772939\pi\)
\(42\) −4.09541 −0.631936
\(43\) 7.33515 1.11860 0.559300 0.828966i \(-0.311070\pi\)
0.559300 + 0.828966i \(0.311070\pi\)
\(44\) 20.1921 3.04407
\(45\) 1.48211 0.220941
\(46\) 2.84454 0.419405
\(47\) 12.3431 1.80042 0.900212 0.435452i \(-0.143411\pi\)
0.900212 + 0.435452i \(0.143411\pi\)
\(48\) −9.48195 −1.36860
\(49\) −4.93063 −0.704375
\(50\) 11.1560 1.57769
\(51\) −0.225346 −0.0315547
\(52\) 26.6403 3.69434
\(53\) 3.70866 0.509424 0.254712 0.967017i \(-0.418019\pi\)
0.254712 + 0.967017i \(0.418019\pi\)
\(54\) −13.6324 −1.85513
\(55\) −3.56810 −0.481123
\(56\) 10.0078 1.33735
\(57\) 1.22191 0.161846
\(58\) −1.42381 −0.186956
\(59\) 13.3936 1.74370 0.871851 0.489771i \(-0.162919\pi\)
0.871851 + 0.489771i \(0.162919\pi\)
\(60\) 4.27791 0.552276
\(61\) 2.02183 0.258868 0.129434 0.991588i \(-0.458684\pi\)
0.129434 + 0.991588i \(0.458684\pi\)
\(62\) 14.7513 1.87341
\(63\) 2.57272 0.324133
\(64\) 4.41167 0.551459
\(65\) −4.70755 −0.583899
\(66\) 12.2577 1.50882
\(67\) −16.1756 −1.97616 −0.988080 0.153942i \(-0.950803\pi\)
−0.988080 + 0.153942i \(0.950803\pi\)
\(68\) 0.960127 0.116432
\(69\) 1.21054 0.145732
\(70\) −3.08342 −0.368539
\(71\) −2.30638 −0.273717 −0.136859 0.990591i \(-0.543701\pi\)
−0.136859 + 0.990591i \(0.543701\pi\)
\(72\) 12.4421 1.46632
\(73\) −0.0880236 −0.0103024 −0.00515119 0.999987i \(-0.501640\pi\)
−0.00515119 + 0.999987i \(0.501640\pi\)
\(74\) 28.1360 3.27075
\(75\) 4.74761 0.548206
\(76\) −5.20616 −0.597188
\(77\) −6.19369 −0.705836
\(78\) 16.1721 1.83113
\(79\) −2.05059 −0.230709 −0.115355 0.993324i \(-0.536800\pi\)
−0.115355 + 0.993324i \(0.536800\pi\)
\(80\) −7.13892 −0.798156
\(81\) −0.436175 −0.0484639
\(82\) 25.0470 2.76598
\(83\) 9.20626 1.01052 0.505259 0.862968i \(-0.331397\pi\)
0.505259 + 0.862968i \(0.331397\pi\)
\(84\) 7.42581 0.810222
\(85\) −0.169662 −0.0184024
\(86\) −18.9721 −2.04581
\(87\) −0.605927 −0.0649621
\(88\) −29.9537 −3.19307
\(89\) −3.27083 −0.346708 −0.173354 0.984860i \(-0.555460\pi\)
−0.173354 + 0.984860i \(0.555460\pi\)
\(90\) −3.83343 −0.404079
\(91\) −8.17159 −0.856615
\(92\) −5.15773 −0.537730
\(93\) 6.27764 0.650961
\(94\) −31.9249 −3.29280
\(95\) 0.919971 0.0943870
\(96\) 9.20945 0.939936
\(97\) 9.21504 0.935646 0.467823 0.883822i \(-0.345039\pi\)
0.467823 + 0.883822i \(0.345039\pi\)
\(98\) 12.7529 1.28823
\(99\) −7.70023 −0.773902
\(100\) −20.2280 −2.02280
\(101\) −8.92869 −0.888438 −0.444219 0.895918i \(-0.646519\pi\)
−0.444219 + 0.895918i \(0.646519\pi\)
\(102\) 0.582848 0.0577106
\(103\) 13.3121 1.31168 0.655842 0.754898i \(-0.272314\pi\)
0.655842 + 0.754898i \(0.272314\pi\)
\(104\) −39.5192 −3.87517
\(105\) −1.31220 −0.128058
\(106\) −9.59231 −0.931687
\(107\) −11.3041 −1.09281 −0.546407 0.837520i \(-0.684005\pi\)
−0.546407 + 0.837520i \(0.684005\pi\)
\(108\) 24.7183 2.37852
\(109\) −3.74546 −0.358750 −0.179375 0.983781i \(-0.557408\pi\)
−0.179375 + 0.983781i \(0.557408\pi\)
\(110\) 9.22876 0.879927
\(111\) 11.9738 1.13650
\(112\) −12.3921 −1.17094
\(113\) 16.0777 1.51246 0.756231 0.654305i \(-0.227039\pi\)
0.756231 + 0.654305i \(0.227039\pi\)
\(114\) −3.16042 −0.296000
\(115\) 0.911412 0.0849896
\(116\) 2.58166 0.239701
\(117\) −10.1592 −0.939221
\(118\) −34.6421 −3.18906
\(119\) −0.294507 −0.0269975
\(120\) −6.34602 −0.579309
\(121\) 7.53786 0.685260
\(122\) −5.22937 −0.473445
\(123\) 10.6592 0.961105
\(124\) −26.7470 −2.40195
\(125\) 7.71805 0.690324
\(126\) −6.65425 −0.592807
\(127\) −7.00536 −0.621625 −0.310812 0.950471i \(-0.600601\pi\)
−0.310812 + 0.950471i \(0.600601\pi\)
\(128\) 5.32306 0.470497
\(129\) −8.07387 −0.710864
\(130\) 12.1759 1.06790
\(131\) −5.71902 −0.499673 −0.249836 0.968288i \(-0.580377\pi\)
−0.249836 + 0.968288i \(0.580377\pi\)
\(132\) −22.2256 −1.93449
\(133\) 1.59693 0.138471
\(134\) 41.8374 3.61420
\(135\) −4.36792 −0.375930
\(136\) −1.42429 −0.122132
\(137\) 10.4831 0.895629 0.447814 0.894127i \(-0.352203\pi\)
0.447814 + 0.894127i \(0.352203\pi\)
\(138\) −3.13102 −0.266530
\(139\) −15.8050 −1.34056 −0.670282 0.742107i \(-0.733827\pi\)
−0.670282 + 0.742107i \(0.733827\pi\)
\(140\) 5.59086 0.472514
\(141\) −13.5862 −1.14416
\(142\) 5.96536 0.500602
\(143\) 24.4578 2.04526
\(144\) −15.4063 −1.28386
\(145\) −0.456200 −0.0378853
\(146\) 0.227669 0.0188421
\(147\) 5.42719 0.447627
\(148\) −51.0163 −4.19351
\(149\) 4.04694 0.331538 0.165769 0.986165i \(-0.446989\pi\)
0.165769 + 0.986165i \(0.446989\pi\)
\(150\) −12.2795 −1.00262
\(151\) 14.4336 1.17459 0.587293 0.809374i \(-0.300194\pi\)
0.587293 + 0.809374i \(0.300194\pi\)
\(152\) 7.72301 0.626419
\(153\) −0.366143 −0.0296009
\(154\) 16.0197 1.29091
\(155\) 4.72641 0.379635
\(156\) −29.3232 −2.34773
\(157\) −9.68420 −0.772883 −0.386442 0.922314i \(-0.626296\pi\)
−0.386442 + 0.922314i \(0.626296\pi\)
\(158\) 5.30377 0.421945
\(159\) −4.08216 −0.323737
\(160\) 6.93376 0.548162
\(161\) 1.58207 0.124685
\(162\) 1.12815 0.0886357
\(163\) 7.28419 0.570542 0.285271 0.958447i \(-0.407916\pi\)
0.285271 + 0.958447i \(0.407916\pi\)
\(164\) −45.4153 −3.54634
\(165\) 3.92745 0.305751
\(166\) −23.8116 −1.84814
\(167\) −12.1871 −0.943069 −0.471534 0.881848i \(-0.656300\pi\)
−0.471534 + 0.881848i \(0.656300\pi\)
\(168\) −11.0157 −0.849881
\(169\) 19.2681 1.48217
\(170\) 0.438824 0.0336563
\(171\) 1.98536 0.151825
\(172\) 34.4001 2.62299
\(173\) −0.170098 −0.0129323 −0.00646616 0.999979i \(-0.502058\pi\)
−0.00646616 + 0.999979i \(0.502058\pi\)
\(174\) 1.56720 0.118809
\(175\) 6.20471 0.469032
\(176\) 37.0898 2.79575
\(177\) −14.7425 −1.10811
\(178\) 8.45987 0.634094
\(179\) 3.47482 0.259720 0.129860 0.991532i \(-0.458547\pi\)
0.129860 + 0.991532i \(0.458547\pi\)
\(180\) 6.95077 0.518080
\(181\) 0.324027 0.0240847 0.0120424 0.999927i \(-0.496167\pi\)
0.0120424 + 0.999927i \(0.496167\pi\)
\(182\) 21.1355 1.56667
\(183\) −2.22544 −0.164510
\(184\) 7.65116 0.564051
\(185\) 9.01499 0.662795
\(186\) −16.2369 −1.19054
\(187\) 0.881468 0.0644594
\(188\) 57.8862 4.22179
\(189\) −7.58204 −0.551512
\(190\) −2.37947 −0.172625
\(191\) −15.4486 −1.11782 −0.558910 0.829228i \(-0.688780\pi\)
−0.558910 + 0.829228i \(0.688780\pi\)
\(192\) −4.85597 −0.350449
\(193\) 12.2829 0.884141 0.442070 0.896980i \(-0.354244\pi\)
0.442070 + 0.896980i \(0.354244\pi\)
\(194\) −23.8343 −1.71120
\(195\) 5.18164 0.371065
\(196\) −23.1235 −1.65168
\(197\) 7.37020 0.525105 0.262552 0.964918i \(-0.415436\pi\)
0.262552 + 0.964918i \(0.415436\pi\)
\(198\) 19.9163 1.41539
\(199\) 17.4053 1.23383 0.616916 0.787029i \(-0.288382\pi\)
0.616916 + 0.787029i \(0.288382\pi\)
\(200\) 30.0070 2.12182
\(201\) 17.8046 1.25584
\(202\) 23.0937 1.62487
\(203\) −0.791893 −0.0555800
\(204\) −1.05682 −0.0739922
\(205\) 8.02525 0.560507
\(206\) −34.4313 −2.39894
\(207\) 1.96689 0.136709
\(208\) 48.9341 3.39297
\(209\) −4.77965 −0.330615
\(210\) 3.39395 0.234205
\(211\) −19.8550 −1.36687 −0.683437 0.730010i \(-0.739515\pi\)
−0.683437 + 0.730010i \(0.739515\pi\)
\(212\) 17.3928 1.19454
\(213\) 2.53866 0.173946
\(214\) 29.2377 1.99865
\(215\) −6.07878 −0.414569
\(216\) −36.6680 −2.49494
\(217\) 8.20433 0.556946
\(218\) 9.68748 0.656119
\(219\) 0.0968884 0.00654711
\(220\) −16.7336 −1.12818
\(221\) 1.16296 0.0782290
\(222\) −30.9696 −2.07854
\(223\) −19.5533 −1.30939 −0.654693 0.755894i \(-0.727202\pi\)
−0.654693 + 0.755894i \(0.727202\pi\)
\(224\) 12.0360 0.804186
\(225\) 7.71393 0.514262
\(226\) −41.5843 −2.76614
\(227\) −10.2233 −0.678547 −0.339273 0.940688i \(-0.610181\pi\)
−0.339273 + 0.940688i \(0.610181\pi\)
\(228\) 5.73047 0.379510
\(229\) 12.8232 0.847384 0.423692 0.905806i \(-0.360734\pi\)
0.423692 + 0.905806i \(0.360734\pi\)
\(230\) −2.35733 −0.155438
\(231\) 6.81745 0.448555
\(232\) −3.82973 −0.251434
\(233\) −14.9323 −0.978248 −0.489124 0.872214i \(-0.662683\pi\)
−0.489124 + 0.872214i \(0.662683\pi\)
\(234\) 26.2764 1.71774
\(235\) −10.2290 −0.667264
\(236\) 62.8131 4.08878
\(237\) 2.25710 0.146615
\(238\) 0.761731 0.0493757
\(239\) −4.65723 −0.301251 −0.150626 0.988591i \(-0.548129\pi\)
−0.150626 + 0.988591i \(0.548129\pi\)
\(240\) 7.85788 0.507224
\(241\) 9.73791 0.627274 0.313637 0.949543i \(-0.398452\pi\)
0.313637 + 0.949543i \(0.398452\pi\)
\(242\) −19.4964 −1.25327
\(243\) −15.3319 −0.983544
\(244\) 9.48190 0.607017
\(245\) 4.08611 0.261052
\(246\) −27.5695 −1.75777
\(247\) −6.30599 −0.401241
\(248\) 39.6775 2.51952
\(249\) −10.1334 −0.642179
\(250\) −19.9624 −1.26253
\(251\) 6.50042 0.410303 0.205152 0.978730i \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(252\) 12.0655 0.760054
\(253\) −4.73518 −0.297698
\(254\) 18.1191 1.13689
\(255\) 0.186749 0.0116947
\(256\) −22.5912 −1.41195
\(257\) 29.5625 1.84406 0.922030 0.387118i \(-0.126529\pi\)
0.922030 + 0.387118i \(0.126529\pi\)
\(258\) 20.8827 1.30010
\(259\) 15.6487 0.972360
\(260\) −22.0773 −1.36918
\(261\) −0.984512 −0.0609398
\(262\) 14.7920 0.913853
\(263\) 16.2205 1.00020 0.500100 0.865968i \(-0.333297\pi\)
0.500100 + 0.865968i \(0.333297\pi\)
\(264\) 32.9703 2.02918
\(265\) −3.07344 −0.188800
\(266\) −4.13039 −0.253251
\(267\) 3.60024 0.220331
\(268\) −75.8597 −4.63387
\(269\) −23.4511 −1.42984 −0.714920 0.699206i \(-0.753537\pi\)
−0.714920 + 0.699206i \(0.753537\pi\)
\(270\) 11.2974 0.687540
\(271\) −14.5396 −0.883218 −0.441609 0.897208i \(-0.645592\pi\)
−0.441609 + 0.897208i \(0.645592\pi\)
\(272\) 1.76361 0.106934
\(273\) 8.99455 0.544375
\(274\) −27.1140 −1.63802
\(275\) −18.5708 −1.11986
\(276\) 5.67716 0.341725
\(277\) 17.6860 1.06265 0.531324 0.847169i \(-0.321695\pi\)
0.531324 + 0.847169i \(0.321695\pi\)
\(278\) 40.8790 2.45176
\(279\) 10.1999 0.610655
\(280\) −8.29369 −0.495643
\(281\) −15.6611 −0.934260 −0.467130 0.884189i \(-0.654712\pi\)
−0.467130 + 0.884189i \(0.654712\pi\)
\(282\) 35.1400 2.09256
\(283\) −25.9833 −1.54455 −0.772275 0.635289i \(-0.780881\pi\)
−0.772275 + 0.635289i \(0.780881\pi\)
\(284\) −10.8164 −0.641835
\(285\) −1.01262 −0.0599825
\(286\) −63.2590 −3.74058
\(287\) 13.9306 0.822298
\(288\) 14.9636 0.881736
\(289\) −16.9581 −0.997535
\(290\) 1.17994 0.0692886
\(291\) −10.1431 −0.594598
\(292\) −0.412810 −0.0241579
\(293\) −11.8971 −0.695035 −0.347517 0.937674i \(-0.612975\pi\)
−0.347517 + 0.937674i \(0.612975\pi\)
\(294\) −14.0372 −0.818666
\(295\) −11.0996 −0.646242
\(296\) 75.6795 4.39878
\(297\) 22.6932 1.31680
\(298\) −10.4672 −0.606351
\(299\) −6.24732 −0.361292
\(300\) 22.2652 1.28548
\(301\) −10.5518 −0.608198
\(302\) −37.3318 −2.14820
\(303\) 9.82789 0.564598
\(304\) −9.56294 −0.548472
\(305\) −1.67553 −0.0959405
\(306\) 0.947014 0.0541372
\(307\) 6.79767 0.387964 0.193982 0.981005i \(-0.437860\pi\)
0.193982 + 0.981005i \(0.437860\pi\)
\(308\) −29.0470 −1.65510
\(309\) −14.6528 −0.833569
\(310\) −12.2247 −0.694315
\(311\) 30.3767 1.72251 0.861254 0.508175i \(-0.169680\pi\)
0.861254 + 0.508175i \(0.169680\pi\)
\(312\) 43.4991 2.46265
\(313\) 16.5336 0.934532 0.467266 0.884117i \(-0.345239\pi\)
0.467266 + 0.884117i \(0.345239\pi\)
\(314\) 25.0478 1.41353
\(315\) −2.13207 −0.120128
\(316\) −9.61679 −0.540987
\(317\) −7.18606 −0.403609 −0.201805 0.979426i \(-0.564681\pi\)
−0.201805 + 0.979426i \(0.564681\pi\)
\(318\) 10.5583 0.592083
\(319\) 2.37016 0.132703
\(320\) −3.65604 −0.204379
\(321\) 12.4426 0.694477
\(322\) −4.09196 −0.228036
\(323\) −0.227271 −0.0126457
\(324\) −2.04556 −0.113642
\(325\) −24.5013 −1.35909
\(326\) −18.8403 −1.04346
\(327\) 4.12266 0.227984
\(328\) 67.3707 3.71993
\(329\) −17.7559 −0.978915
\(330\) −10.1582 −0.559189
\(331\) −20.5435 −1.12917 −0.564586 0.825374i \(-0.690964\pi\)
−0.564586 + 0.825374i \(0.690964\pi\)
\(332\) 43.1752 2.36955
\(333\) 19.4550 1.06613
\(334\) 31.5215 1.72478
\(335\) 13.4050 0.732394
\(336\) 13.6401 0.744128
\(337\) −20.1185 −1.09592 −0.547962 0.836503i \(-0.684596\pi\)
−0.547962 + 0.836503i \(0.684596\pi\)
\(338\) −49.8363 −2.71074
\(339\) −17.6969 −0.961162
\(340\) −0.795676 −0.0431516
\(341\) −24.5558 −1.32977
\(342\) −5.13506 −0.277672
\(343\) 17.1626 0.926692
\(344\) −51.0305 −2.75138
\(345\) −1.00320 −0.0540105
\(346\) 0.439952 0.0236520
\(347\) 11.9334 0.640618 0.320309 0.947313i \(-0.396213\pi\)
0.320309 + 0.947313i \(0.396213\pi\)
\(348\) −2.84166 −0.152329
\(349\) 18.3110 0.980167 0.490083 0.871676i \(-0.336966\pi\)
0.490083 + 0.871676i \(0.336966\pi\)
\(350\) −16.0482 −0.857814
\(351\) 29.9401 1.59809
\(352\) −36.0239 −1.92008
\(353\) −14.6976 −0.782273 −0.391137 0.920333i \(-0.627918\pi\)
−0.391137 + 0.920333i \(0.627918\pi\)
\(354\) 38.1309 2.02663
\(355\) 1.91134 0.101444
\(356\) −15.3394 −0.812989
\(357\) 0.324167 0.0171568
\(358\) −8.98748 −0.475003
\(359\) −32.3422 −1.70695 −0.853477 0.521131i \(-0.825510\pi\)
−0.853477 + 0.521131i \(0.825510\pi\)
\(360\) −10.3110 −0.543439
\(361\) −17.7677 −0.935140
\(362\) −0.838081 −0.0440486
\(363\) −8.29700 −0.435479
\(364\) −38.3229 −2.00866
\(365\) 0.0729469 0.00381822
\(366\) 5.75602 0.300872
\(367\) −17.0680 −0.890944 −0.445472 0.895296i \(-0.646964\pi\)
−0.445472 + 0.895296i \(0.646964\pi\)
\(368\) −9.47397 −0.493865
\(369\) 17.3191 0.901595
\(370\) −23.3169 −1.21219
\(371\) −5.33503 −0.276981
\(372\) 29.4407 1.52643
\(373\) −9.49606 −0.491687 −0.245844 0.969309i \(-0.579065\pi\)
−0.245844 + 0.969309i \(0.579065\pi\)
\(374\) −2.27988 −0.117890
\(375\) −8.49534 −0.438697
\(376\) −85.8706 −4.42844
\(377\) 3.12705 0.161051
\(378\) 19.6106 1.00866
\(379\) −24.4920 −1.25807 −0.629034 0.777378i \(-0.716550\pi\)
−0.629034 + 0.777378i \(0.716550\pi\)
\(380\) 4.31445 0.221327
\(381\) 7.71087 0.395040
\(382\) 39.9571 2.04438
\(383\) −32.8566 −1.67890 −0.839448 0.543440i \(-0.817121\pi\)
−0.839448 + 0.543440i \(0.817121\pi\)
\(384\) −5.85915 −0.298998
\(385\) 5.13283 0.261593
\(386\) −31.7691 −1.61701
\(387\) −13.1185 −0.666849
\(388\) 43.2164 2.19398
\(389\) 2.75465 0.139666 0.0698330 0.997559i \(-0.477753\pi\)
0.0698330 + 0.997559i \(0.477753\pi\)
\(390\) −13.4021 −0.678642
\(391\) −0.225156 −0.0113866
\(392\) 34.3023 1.73253
\(393\) 6.29498 0.317540
\(394\) −19.0627 −0.960366
\(395\) 1.69936 0.0855043
\(396\) −36.1123 −1.81471
\(397\) 9.21044 0.462259 0.231129 0.972923i \(-0.425758\pi\)
0.231129 + 0.972923i \(0.425758\pi\)
\(398\) −45.0182 −2.25656
\(399\) −1.75776 −0.0879978
\(400\) −37.1558 −1.85779
\(401\) 17.7197 0.884879 0.442440 0.896798i \(-0.354113\pi\)
0.442440 + 0.896798i \(0.354113\pi\)
\(402\) −46.0509 −2.29681
\(403\) −32.3974 −1.61383
\(404\) −41.8735 −2.08328
\(405\) 0.361467 0.0179614
\(406\) 2.04820 0.101650
\(407\) −46.8368 −2.32161
\(408\) 1.56773 0.0776140
\(409\) 15.3811 0.760549 0.380274 0.924874i \(-0.375830\pi\)
0.380274 + 0.924874i \(0.375830\pi\)
\(410\) −20.7570 −1.02511
\(411\) −11.5388 −0.569168
\(412\) 62.4309 3.07575
\(413\) −19.2672 −0.948075
\(414\) −5.08729 −0.250027
\(415\) −7.62941 −0.374513
\(416\) −47.5278 −2.33024
\(417\) 17.3967 0.851922
\(418\) 12.3624 0.604663
\(419\) −30.6074 −1.49527 −0.747634 0.664111i \(-0.768810\pi\)
−0.747634 + 0.664111i \(0.768810\pi\)
\(420\) −6.15391 −0.300280
\(421\) −7.35995 −0.358702 −0.179351 0.983785i \(-0.557400\pi\)
−0.179351 + 0.983785i \(0.557400\pi\)
\(422\) 51.3541 2.49988
\(423\) −22.0748 −1.07332
\(424\) −25.8011 −1.25301
\(425\) −0.883037 −0.0428336
\(426\) −6.56613 −0.318130
\(427\) −2.90846 −0.140750
\(428\) −53.0138 −2.56252
\(429\) −26.9209 −1.29975
\(430\) 15.7225 0.758207
\(431\) 13.3903 0.644987 0.322494 0.946572i \(-0.395479\pi\)
0.322494 + 0.946572i \(0.395479\pi\)
\(432\) 45.4037 2.18449
\(433\) 2.29984 0.110523 0.0552616 0.998472i \(-0.482401\pi\)
0.0552616 + 0.998472i \(0.482401\pi\)
\(434\) −21.2202 −1.01860
\(435\) 0.502143 0.0240759
\(436\) −17.5653 −0.841227
\(437\) 1.22088 0.0584026
\(438\) −0.250598 −0.0119740
\(439\) −34.6791 −1.65514 −0.827571 0.561361i \(-0.810278\pi\)
−0.827571 + 0.561361i \(0.810278\pi\)
\(440\) 24.8232 1.18340
\(441\) 8.81812 0.419911
\(442\) −3.00794 −0.143073
\(443\) −23.9884 −1.13972 −0.569861 0.821741i \(-0.693003\pi\)
−0.569861 + 0.821741i \(0.693003\pi\)
\(444\) 56.1541 2.66496
\(445\) 2.71060 0.128495
\(446\) 50.5738 2.39474
\(447\) −4.45451 −0.210691
\(448\) −6.34633 −0.299836
\(449\) −3.33285 −0.157287 −0.0786435 0.996903i \(-0.525059\pi\)
−0.0786435 + 0.996903i \(0.525059\pi\)
\(450\) −19.9518 −0.940535
\(451\) −41.6947 −1.96332
\(452\) 75.4006 3.54655
\(453\) −15.8872 −0.746444
\(454\) 26.4422 1.24100
\(455\) 6.77196 0.317474
\(456\) −8.50080 −0.398086
\(457\) −17.6869 −0.827360 −0.413680 0.910422i \(-0.635757\pi\)
−0.413680 + 0.910422i \(0.635757\pi\)
\(458\) −33.1668 −1.54978
\(459\) 1.07906 0.0503660
\(460\) 4.27431 0.199291
\(461\) −16.0136 −0.745826 −0.372913 0.927866i \(-0.621641\pi\)
−0.372913 + 0.927866i \(0.621641\pi\)
\(462\) −17.6331 −0.820364
\(463\) −24.9357 −1.15886 −0.579431 0.815021i \(-0.696725\pi\)
−0.579431 + 0.815021i \(0.696725\pi\)
\(464\) 4.74212 0.220147
\(465\) −5.20241 −0.241256
\(466\) 38.6218 1.78912
\(467\) −0.946708 −0.0438084 −0.0219042 0.999760i \(-0.506973\pi\)
−0.0219042 + 0.999760i \(0.506973\pi\)
\(468\) −47.6445 −2.20237
\(469\) 23.2691 1.07447
\(470\) 26.4568 1.22036
\(471\) 10.6595 0.491163
\(472\) −93.1792 −4.28892
\(473\) 31.5819 1.45214
\(474\) −5.83791 −0.268144
\(475\) 4.78815 0.219696
\(476\) −1.38117 −0.0633059
\(477\) −6.63272 −0.303691
\(478\) 12.0457 0.550960
\(479\) 19.5675 0.894063 0.447032 0.894518i \(-0.352481\pi\)
0.447032 + 0.894518i \(0.352481\pi\)
\(480\) −7.63206 −0.348354
\(481\) −61.7937 −2.81755
\(482\) −25.1867 −1.14722
\(483\) −1.74140 −0.0792365
\(484\) 35.3508 1.60686
\(485\) −7.63669 −0.346764
\(486\) 39.6554 1.79881
\(487\) −5.92462 −0.268470 −0.134235 0.990950i \(-0.542858\pi\)
−0.134235 + 0.990950i \(0.542858\pi\)
\(488\) −14.0658 −0.636729
\(489\) −8.01778 −0.362576
\(490\) −10.5686 −0.477439
\(491\) −14.8770 −0.671391 −0.335695 0.941971i \(-0.608971\pi\)
−0.335695 + 0.941971i \(0.608971\pi\)
\(492\) 49.9890 2.25368
\(493\) 0.112700 0.00507576
\(494\) 16.3102 0.733830
\(495\) 6.38133 0.286820
\(496\) −49.1302 −2.20601
\(497\) 3.31780 0.148824
\(498\) 26.2097 1.17448
\(499\) −21.0463 −0.942162 −0.471081 0.882090i \(-0.656136\pi\)
−0.471081 + 0.882090i \(0.656136\pi\)
\(500\) 36.1959 1.61873
\(501\) 13.4145 0.599315
\(502\) −16.8131 −0.750404
\(503\) −12.9687 −0.578244 −0.289122 0.957292i \(-0.593363\pi\)
−0.289122 + 0.957292i \(0.593363\pi\)
\(504\) −17.8984 −0.797258
\(505\) 7.39938 0.329268
\(506\) 12.2474 0.544461
\(507\) −21.2086 −0.941909
\(508\) −32.8535 −1.45764
\(509\) 9.53843 0.422784 0.211392 0.977401i \(-0.432200\pi\)
0.211392 + 0.977401i \(0.432200\pi\)
\(510\) −0.483018 −0.0213884
\(511\) 0.126625 0.00560155
\(512\) 47.7851 2.11182
\(513\) −5.85104 −0.258330
\(514\) −76.4623 −3.37261
\(515\) −11.0320 −0.486130
\(516\) −37.8646 −1.66690
\(517\) 53.1439 2.33727
\(518\) −40.4746 −1.77835
\(519\) 0.187229 0.00821843
\(520\) 32.7503 1.43620
\(521\) 9.19325 0.402764 0.201382 0.979513i \(-0.435457\pi\)
0.201382 + 0.979513i \(0.435457\pi\)
\(522\) 2.54640 0.111453
\(523\) 32.7411 1.43167 0.715834 0.698270i \(-0.246047\pi\)
0.715834 + 0.698270i \(0.246047\pi\)
\(524\) −26.8209 −1.17168
\(525\) −6.82958 −0.298067
\(526\) −41.9537 −1.82927
\(527\) −1.16762 −0.0508622
\(528\) −40.8251 −1.77669
\(529\) −21.7905 −0.947412
\(530\) 7.94934 0.345297
\(531\) −23.9537 −1.03950
\(532\) 7.48923 0.324699
\(533\) −55.0095 −2.38273
\(534\) −9.31186 −0.402964
\(535\) 9.36797 0.405013
\(536\) 112.533 4.86069
\(537\) −3.82477 −0.165051
\(538\) 60.6553 2.61504
\(539\) −21.2291 −0.914403
\(540\) −20.4845 −0.881514
\(541\) −38.7962 −1.66798 −0.833989 0.551781i \(-0.813948\pi\)
−0.833989 + 0.551781i \(0.813948\pi\)
\(542\) 37.6061 1.61532
\(543\) −0.356659 −0.0153057
\(544\) −1.71292 −0.0734410
\(545\) 3.10394 0.132958
\(546\) −23.2640 −0.995608
\(547\) 7.41686 0.317122 0.158561 0.987349i \(-0.449315\pi\)
0.158561 + 0.987349i \(0.449315\pi\)
\(548\) 49.1632 2.10015
\(549\) −3.61591 −0.154323
\(550\) 48.0327 2.04812
\(551\) −0.611102 −0.0260338
\(552\) −8.42171 −0.358452
\(553\) 2.94984 0.125440
\(554\) −45.7441 −1.94348
\(555\) −9.92289 −0.421203
\(556\) −74.1219 −3.14347
\(557\) −21.8107 −0.924150 −0.462075 0.886841i \(-0.652895\pi\)
−0.462075 + 0.886841i \(0.652895\pi\)
\(558\) −26.3817 −1.11683
\(559\) 41.6673 1.76234
\(560\) 10.2696 0.433968
\(561\) −0.970241 −0.0409636
\(562\) 40.5067 1.70867
\(563\) −10.1224 −0.426608 −0.213304 0.976986i \(-0.568423\pi\)
−0.213304 + 0.976986i \(0.568423\pi\)
\(564\) −63.7159 −2.68292
\(565\) −13.3239 −0.560541
\(566\) 67.2048 2.82483
\(567\) 0.627452 0.0263505
\(568\) 16.0454 0.673252
\(569\) 30.7376 1.28859 0.644293 0.764778i \(-0.277152\pi\)
0.644293 + 0.764778i \(0.277152\pi\)
\(570\) 2.61910 0.109702
\(571\) 25.5805 1.07051 0.535256 0.844690i \(-0.320215\pi\)
0.535256 + 0.844690i \(0.320215\pi\)
\(572\) 114.701 4.79590
\(573\) 17.0044 0.710369
\(574\) −36.0309 −1.50390
\(575\) 4.74361 0.197822
\(576\) −7.89000 −0.328750
\(577\) 32.1704 1.33927 0.669636 0.742690i \(-0.266450\pi\)
0.669636 + 0.742690i \(0.266450\pi\)
\(578\) 43.8614 1.82439
\(579\) −13.5199 −0.561867
\(580\) −2.13947 −0.0888367
\(581\) −13.2435 −0.549433
\(582\) 26.2347 1.08746
\(583\) 15.9679 0.661322
\(584\) 0.612378 0.0253404
\(585\) 8.41916 0.348089
\(586\) 30.7713 1.27115
\(587\) 40.9828 1.69154 0.845771 0.533546i \(-0.179141\pi\)
0.845771 + 0.533546i \(0.179141\pi\)
\(588\) 25.4523 1.04963
\(589\) 6.33126 0.260875
\(590\) 28.7086 1.18191
\(591\) −8.11245 −0.333702
\(592\) −93.7093 −3.85143
\(593\) −11.7325 −0.481797 −0.240898 0.970550i \(-0.577442\pi\)
−0.240898 + 0.970550i \(0.577442\pi\)
\(594\) −58.6951 −2.40829
\(595\) 0.244064 0.0100057
\(596\) 18.9792 0.777419
\(597\) −19.1582 −0.784094
\(598\) 16.1584 0.660768
\(599\) 36.1695 1.47785 0.738923 0.673790i \(-0.235335\pi\)
0.738923 + 0.673790i \(0.235335\pi\)
\(600\) −33.0290 −1.34840
\(601\) −31.9739 −1.30424 −0.652120 0.758115i \(-0.726120\pi\)
−0.652120 + 0.758115i \(0.726120\pi\)
\(602\) 27.2919 1.11233
\(603\) 28.9290 1.17808
\(604\) 67.6901 2.75427
\(605\) −6.24677 −0.253967
\(606\) −25.4194 −1.03259
\(607\) 16.2795 0.660764 0.330382 0.943847i \(-0.392822\pi\)
0.330382 + 0.943847i \(0.392822\pi\)
\(608\) 9.28811 0.376683
\(609\) 0.871644 0.0353208
\(610\) 4.33368 0.175466
\(611\) 70.1150 2.83655
\(612\) −1.71713 −0.0694107
\(613\) −33.9852 −1.37265 −0.686325 0.727295i \(-0.740777\pi\)
−0.686325 + 0.727295i \(0.740777\pi\)
\(614\) −17.5819 −0.709547
\(615\) −8.83347 −0.356200
\(616\) 43.0893 1.73612
\(617\) −36.4428 −1.46713 −0.733566 0.679618i \(-0.762146\pi\)
−0.733566 + 0.679618i \(0.762146\pi\)
\(618\) 37.8989 1.52452
\(619\) 19.6736 0.790750 0.395375 0.918520i \(-0.370615\pi\)
0.395375 + 0.918520i \(0.370615\pi\)
\(620\) 22.1658 0.890199
\(621\) −5.79660 −0.232610
\(622\) −78.5682 −3.15030
\(623\) 4.70519 0.188510
\(624\) −53.8623 −2.15622
\(625\) 15.1700 0.606801
\(626\) −42.7634 −1.70917
\(627\) 5.26101 0.210104
\(628\) −45.4167 −1.81232
\(629\) −2.22707 −0.0887992
\(630\) 5.51451 0.219703
\(631\) 1.06728 0.0424877 0.0212439 0.999774i \(-0.493237\pi\)
0.0212439 + 0.999774i \(0.493237\pi\)
\(632\) 14.2659 0.567467
\(633\) 21.8546 0.868641
\(634\) 18.5864 0.738162
\(635\) 5.80548 0.230383
\(636\) −19.1444 −0.759125
\(637\) −28.0085 −1.10974
\(638\) −6.13031 −0.242701
\(639\) 4.12482 0.163175
\(640\) −4.41133 −0.174373
\(641\) −3.67957 −0.145334 −0.0726672 0.997356i \(-0.523151\pi\)
−0.0726672 + 0.997356i \(0.523151\pi\)
\(642\) −32.1822 −1.27013
\(643\) 12.7689 0.503557 0.251778 0.967785i \(-0.418985\pi\)
0.251778 + 0.967785i \(0.418985\pi\)
\(644\) 7.41955 0.292371
\(645\) 6.69097 0.263457
\(646\) 0.587826 0.0231277
\(647\) 2.14580 0.0843601 0.0421801 0.999110i \(-0.486570\pi\)
0.0421801 + 0.999110i \(0.486570\pi\)
\(648\) 3.03446 0.119205
\(649\) 57.6671 2.26363
\(650\) 63.3716 2.48564
\(651\) −9.03059 −0.353937
\(652\) 34.1612 1.33785
\(653\) 9.45981 0.370191 0.185096 0.982721i \(-0.440741\pi\)
0.185096 + 0.982721i \(0.440741\pi\)
\(654\) −10.6631 −0.416960
\(655\) 4.73946 0.185186
\(656\) −83.4210 −3.25704
\(657\) 0.157425 0.00614173
\(658\) 45.9250 1.79034
\(659\) −36.6516 −1.42774 −0.713872 0.700276i \(-0.753060\pi\)
−0.713872 + 0.700276i \(0.753060\pi\)
\(660\) 18.4188 0.716952
\(661\) 19.9342 0.775350 0.387675 0.921796i \(-0.373278\pi\)
0.387675 + 0.921796i \(0.373278\pi\)
\(662\) 53.1349 2.06515
\(663\) −1.28008 −0.0497142
\(664\) −64.0477 −2.48553
\(665\) −1.32341 −0.0513195
\(666\) −50.3196 −1.94984
\(667\) −0.605416 −0.0234418
\(668\) −57.1548 −2.21139
\(669\) 21.5225 0.832109
\(670\) −34.6715 −1.33948
\(671\) 8.70510 0.336057
\(672\) −13.2481 −0.511056
\(673\) −25.7282 −0.991751 −0.495875 0.868394i \(-0.665153\pi\)
−0.495875 + 0.868394i \(0.665153\pi\)
\(674\) 52.0357 2.00434
\(675\) −22.7336 −0.875018
\(676\) 90.3632 3.47551
\(677\) −31.4474 −1.20862 −0.604310 0.796749i \(-0.706551\pi\)
−0.604310 + 0.796749i \(0.706551\pi\)
\(678\) 45.7722 1.75787
\(679\) −13.2561 −0.508723
\(680\) 1.18034 0.0452638
\(681\) 11.2529 0.431213
\(682\) 63.5125 2.43202
\(683\) 8.39234 0.321124 0.160562 0.987026i \(-0.448669\pi\)
0.160562 + 0.987026i \(0.448669\pi\)
\(684\) 9.31090 0.356011
\(685\) −8.68753 −0.331933
\(686\) −44.3903 −1.69483
\(687\) −14.1147 −0.538508
\(688\) 63.1879 2.40902
\(689\) 21.0671 0.802592
\(690\) 2.59473 0.0987799
\(691\) −50.5503 −1.92302 −0.961512 0.274764i \(-0.911400\pi\)
−0.961512 + 0.274764i \(0.911400\pi\)
\(692\) −0.797721 −0.0303248
\(693\) 11.0770 0.420781
\(694\) −30.8652 −1.17163
\(695\) 13.0979 0.496833
\(696\) 4.21542 0.159785
\(697\) −1.98257 −0.0750950
\(698\) −47.3607 −1.79263
\(699\) 16.4361 0.621672
\(700\) 29.0986 1.09983
\(701\) 22.1506 0.836618 0.418309 0.908305i \(-0.362623\pi\)
0.418309 + 0.908305i \(0.362623\pi\)
\(702\) −77.4389 −2.92274
\(703\) 12.0760 0.455456
\(704\) 18.9947 0.715890
\(705\) 11.2591 0.424043
\(706\) 38.0147 1.43070
\(707\) 12.8442 0.483056
\(708\) −69.1389 −2.59840
\(709\) −42.2098 −1.58522 −0.792610 0.609728i \(-0.791279\pi\)
−0.792610 + 0.609728i \(0.791279\pi\)
\(710\) −4.94361 −0.185530
\(711\) 3.66735 0.137536
\(712\) 22.7551 0.852784
\(713\) 6.27235 0.234902
\(714\) −0.838445 −0.0313780
\(715\) −20.2686 −0.758004
\(716\) 16.2961 0.609014
\(717\) 5.12626 0.191444
\(718\) 83.6517 3.12185
\(719\) −26.0383 −0.971063 −0.485532 0.874219i \(-0.661374\pi\)
−0.485532 + 0.874219i \(0.661374\pi\)
\(720\) 12.7675 0.475817
\(721\) −19.1499 −0.713181
\(722\) 45.9553 1.71028
\(723\) −10.7186 −0.398630
\(724\) 1.51961 0.0564759
\(725\) −2.37437 −0.0881820
\(726\) 21.4598 0.796449
\(727\) −47.7304 −1.77022 −0.885111 0.465381i \(-0.845917\pi\)
−0.885111 + 0.465381i \(0.845917\pi\)
\(728\) 56.8496 2.10698
\(729\) 18.1845 0.673501
\(730\) −0.188674 −0.00698315
\(731\) 1.50171 0.0555427
\(732\) −10.4368 −0.385756
\(733\) −1.42247 −0.0525400 −0.0262700 0.999655i \(-0.508363\pi\)
−0.0262700 + 0.999655i \(0.508363\pi\)
\(734\) 44.1458 1.62945
\(735\) −4.49762 −0.165897
\(736\) 9.20170 0.339179
\(737\) −69.6449 −2.56540
\(738\) −44.7951 −1.64893
\(739\) 27.9718 1.02896 0.514479 0.857503i \(-0.327985\pi\)
0.514479 + 0.857503i \(0.327985\pi\)
\(740\) 42.2782 1.55418
\(741\) 6.94106 0.254986
\(742\) 13.7988 0.506571
\(743\) −8.19505 −0.300647 −0.150324 0.988637i \(-0.548032\pi\)
−0.150324 + 0.988637i \(0.548032\pi\)
\(744\) −43.6734 −1.60115
\(745\) −3.35378 −0.122873
\(746\) 24.5612 0.899249
\(747\) −16.4648 −0.602416
\(748\) 4.13388 0.151150
\(749\) 16.2614 0.594178
\(750\) 21.9728 0.802335
\(751\) −49.2692 −1.79786 −0.898929 0.438093i \(-0.855654\pi\)
−0.898929 + 0.438093i \(0.855654\pi\)
\(752\) 106.328 3.87739
\(753\) −7.15508 −0.260746
\(754\) −8.08797 −0.294547
\(755\) −11.9614 −0.435319
\(756\) −35.5580 −1.29323
\(757\) −10.2093 −0.371064 −0.185532 0.982638i \(-0.559401\pi\)
−0.185532 + 0.982638i \(0.559401\pi\)
\(758\) 63.3475 2.30088
\(759\) 5.21206 0.189186
\(760\) −6.40022 −0.232160
\(761\) 15.8815 0.575702 0.287851 0.957675i \(-0.407059\pi\)
0.287851 + 0.957675i \(0.407059\pi\)
\(762\) −19.9438 −0.722489
\(763\) 5.38796 0.195057
\(764\) −72.4503 −2.62116
\(765\) 0.303430 0.0109705
\(766\) 84.9823 3.07054
\(767\) 76.0826 2.74718
\(768\) 24.8664 0.897288
\(769\) 6.70360 0.241738 0.120869 0.992668i \(-0.461432\pi\)
0.120869 + 0.992668i \(0.461432\pi\)
\(770\) −13.2759 −0.478429
\(771\) −32.5398 −1.17189
\(772\) 57.6038 2.07321
\(773\) 14.7305 0.529818 0.264909 0.964273i \(-0.414658\pi\)
0.264909 + 0.964273i \(0.414658\pi\)
\(774\) 33.9303 1.21960
\(775\) 24.5995 0.883639
\(776\) −64.1088 −2.30137
\(777\) −17.2246 −0.617930
\(778\) −7.12478 −0.255436
\(779\) 10.7502 0.385166
\(780\) 24.3007 0.870105
\(781\) −9.93026 −0.355333
\(782\) 0.582357 0.0208250
\(783\) 2.90144 0.103689
\(784\) −42.4744 −1.51694
\(785\) 8.02549 0.286442
\(786\) −16.2817 −0.580749
\(787\) −40.9702 −1.46043 −0.730214 0.683218i \(-0.760580\pi\)
−0.730214 + 0.683218i \(0.760580\pi\)
\(788\) 34.5645 1.23131
\(789\) −17.8541 −0.635622
\(790\) −4.39534 −0.156379
\(791\) −23.1283 −0.822346
\(792\) 53.5703 1.90354
\(793\) 11.4850 0.407844
\(794\) −23.8224 −0.845426
\(795\) 3.38297 0.119982
\(796\) 81.6270 2.89319
\(797\) −3.13785 −0.111148 −0.0555742 0.998455i \(-0.517699\pi\)
−0.0555742 + 0.998455i \(0.517699\pi\)
\(798\) 4.54636 0.160939
\(799\) 2.52697 0.0893979
\(800\) 36.0880 1.27590
\(801\) 5.84968 0.206688
\(802\) −45.8312 −1.61836
\(803\) −0.378991 −0.0133743
\(804\) 83.4995 2.94480
\(805\) −1.31109 −0.0462100
\(806\) 83.7947 2.95154
\(807\) 25.8129 0.908656
\(808\) 62.1167 2.18526
\(809\) −6.64458 −0.233611 −0.116805 0.993155i \(-0.537265\pi\)
−0.116805 + 0.993155i \(0.537265\pi\)
\(810\) −0.934919 −0.0328497
\(811\) −17.1886 −0.603572 −0.301786 0.953376i \(-0.597583\pi\)
−0.301786 + 0.953376i \(0.597583\pi\)
\(812\) −3.71380 −0.130329
\(813\) 16.0039 0.561281
\(814\) 121.141 4.24601
\(815\) −6.03655 −0.211451
\(816\) −1.94122 −0.0679563
\(817\) −8.14282 −0.284881
\(818\) −39.7827 −1.39097
\(819\) 14.6144 0.510668
\(820\) 37.6366 1.31433
\(821\) 23.1309 0.807273 0.403637 0.914919i \(-0.367746\pi\)
0.403637 + 0.914919i \(0.367746\pi\)
\(822\) 29.8447 1.04095
\(823\) −16.1841 −0.564143 −0.282072 0.959393i \(-0.591022\pi\)
−0.282072 + 0.959393i \(0.591022\pi\)
\(824\) −92.6123 −3.22630
\(825\) 20.4411 0.711668
\(826\) 49.8337 1.73394
\(827\) −7.31883 −0.254501 −0.127250 0.991871i \(-0.540615\pi\)
−0.127250 + 0.991871i \(0.540615\pi\)
\(828\) 9.22427 0.320566
\(829\) −13.5660 −0.471168 −0.235584 0.971854i \(-0.575700\pi\)
−0.235584 + 0.971854i \(0.575700\pi\)
\(830\) 19.7332 0.684948
\(831\) −19.4671 −0.675307
\(832\) 25.0605 0.868817
\(833\) −1.00944 −0.0349749
\(834\) −44.9959 −1.55808
\(835\) 10.0997 0.349515
\(836\) −22.4155 −0.775255
\(837\) −30.0601 −1.03903
\(838\) 79.1647 2.73470
\(839\) 33.2951 1.14947 0.574737 0.818338i \(-0.305104\pi\)
0.574737 + 0.818338i \(0.305104\pi\)
\(840\) 9.12894 0.314979
\(841\) −28.6970 −0.989550
\(842\) 19.0362 0.656031
\(843\) 17.2383 0.593718
\(844\) −93.1153 −3.20516
\(845\) −15.9679 −0.549312
\(846\) 57.0957 1.96299
\(847\) −10.8434 −0.372585
\(848\) 31.9479 1.09710
\(849\) 28.6001 0.981553
\(850\) 2.28394 0.0783385
\(851\) 11.9637 0.410109
\(852\) 11.9057 0.407883
\(853\) −21.7475 −0.744619 −0.372310 0.928109i \(-0.621434\pi\)
−0.372310 + 0.928109i \(0.621434\pi\)
\(854\) 7.52262 0.257419
\(855\) −1.64531 −0.0562684
\(856\) 78.6427 2.68795
\(857\) −10.2882 −0.351439 −0.175719 0.984440i \(-0.556225\pi\)
−0.175719 + 0.984440i \(0.556225\pi\)
\(858\) 69.6298 2.37712
\(859\) 37.1281 1.26679 0.633397 0.773827i \(-0.281660\pi\)
0.633397 + 0.773827i \(0.281660\pi\)
\(860\) −28.5081 −0.972118
\(861\) −15.3336 −0.522566
\(862\) −34.6334 −1.17962
\(863\) −51.1954 −1.74271 −0.871355 0.490653i \(-0.836758\pi\)
−0.871355 + 0.490653i \(0.836758\pi\)
\(864\) −44.0989 −1.50028
\(865\) 0.140964 0.00479291
\(866\) −5.94844 −0.202136
\(867\) 18.6659 0.633928
\(868\) 38.4764 1.30597
\(869\) −8.82894 −0.299501
\(870\) −1.29877 −0.0440325
\(871\) −91.8854 −3.11342
\(872\) 26.0571 0.882404
\(873\) −16.4805 −0.557781
\(874\) −3.15776 −0.106813
\(875\) −11.1027 −0.375339
\(876\) 0.454384 0.0153522
\(877\) 33.9883 1.14770 0.573852 0.818959i \(-0.305448\pi\)
0.573852 + 0.818959i \(0.305448\pi\)
\(878\) 89.6960 3.02709
\(879\) 13.0952 0.441691
\(880\) −30.7371 −1.03615
\(881\) −10.0506 −0.338614 −0.169307 0.985563i \(-0.554153\pi\)
−0.169307 + 0.985563i \(0.554153\pi\)
\(882\) −22.8077 −0.767975
\(883\) −34.7134 −1.16820 −0.584100 0.811682i \(-0.698552\pi\)
−0.584100 + 0.811682i \(0.698552\pi\)
\(884\) 5.45400 0.183438
\(885\) 12.2174 0.410684
\(886\) 62.0449 2.08444
\(887\) 25.7238 0.863722 0.431861 0.901940i \(-0.357857\pi\)
0.431861 + 0.901940i \(0.357857\pi\)
\(888\) −83.3011 −2.79540
\(889\) 10.0774 0.337986
\(890\) −7.01086 −0.235005
\(891\) −1.87798 −0.0629146
\(892\) −91.7005 −3.07036
\(893\) −13.7022 −0.458526
\(894\) 11.5214 0.385333
\(895\) −2.87965 −0.0962562
\(896\) −7.65739 −0.255815
\(897\) 6.87649 0.229599
\(898\) 8.62029 0.287663
\(899\) −3.13958 −0.104711
\(900\) 36.1766 1.20589
\(901\) 0.759267 0.0252948
\(902\) 107.842 3.59073
\(903\) 11.6145 0.386507
\(904\) −111.852 −3.72015
\(905\) −0.268527 −0.00892615
\(906\) 41.0915 1.36517
\(907\) 10.4991 0.348617 0.174309 0.984691i \(-0.444231\pi\)
0.174309 + 0.984691i \(0.444231\pi\)
\(908\) −47.9451 −1.59111
\(909\) 15.9684 0.529639
\(910\) −17.5154 −0.580629
\(911\) 6.79969 0.225284 0.112642 0.993636i \(-0.464069\pi\)
0.112642 + 0.993636i \(0.464069\pi\)
\(912\) 10.5260 0.348551
\(913\) 39.6381 1.31183
\(914\) 45.7465 1.51316
\(915\) 1.84427 0.0609697
\(916\) 60.1380 1.98702
\(917\) 8.22699 0.271679
\(918\) −2.79093 −0.0921145
\(919\) −22.1483 −0.730604 −0.365302 0.930889i \(-0.619034\pi\)
−0.365302 + 0.930889i \(0.619034\pi\)
\(920\) −6.34067 −0.209046
\(921\) −7.48226 −0.246549
\(922\) 41.4184 1.36404
\(923\) −13.1014 −0.431238
\(924\) 31.9723 1.05181
\(925\) 46.9202 1.54273
\(926\) 64.4953 2.11945
\(927\) −23.8079 −0.781955
\(928\) −4.60583 −0.151194
\(929\) −55.9373 −1.83524 −0.917621 0.397456i \(-0.869893\pi\)
−0.917621 + 0.397456i \(0.869893\pi\)
\(930\) 13.4558 0.441233
\(931\) 5.47354 0.179388
\(932\) −70.0291 −2.29388
\(933\) −33.4360 −1.09464
\(934\) 2.44862 0.0801213
\(935\) −0.730490 −0.0238896
\(936\) 70.6776 2.31017
\(937\) 52.0302 1.69975 0.849876 0.526983i \(-0.176677\pi\)
0.849876 + 0.526983i \(0.176677\pi\)
\(938\) −60.1845 −1.96509
\(939\) −18.1986 −0.593890
\(940\) −47.9715 −1.56466
\(941\) −37.2946 −1.21577 −0.607885 0.794025i \(-0.707982\pi\)
−0.607885 + 0.794025i \(0.707982\pi\)
\(942\) −27.5703 −0.898290
\(943\) 10.6502 0.346818
\(944\) 115.378 3.75524
\(945\) 6.28339 0.204399
\(946\) −81.6853 −2.65582
\(947\) −50.8260 −1.65162 −0.825812 0.563946i \(-0.809283\pi\)
−0.825812 + 0.563946i \(0.809283\pi\)
\(948\) 10.5853 0.343794
\(949\) −0.500019 −0.0162313
\(950\) −12.3844 −0.401802
\(951\) 7.90976 0.256492
\(952\) 2.04888 0.0664047
\(953\) 40.7835 1.32111 0.660553 0.750779i \(-0.270322\pi\)
0.660553 + 0.750779i \(0.270322\pi\)
\(954\) 17.1552 0.555422
\(955\) 12.8025 0.414281
\(956\) −21.8414 −0.706400
\(957\) −2.60885 −0.0843323
\(958\) −50.6106 −1.63515
\(959\) −15.0802 −0.486966
\(960\) 4.02424 0.129882
\(961\) 1.52727 0.0492667
\(962\) 159.827 5.15303
\(963\) 20.2168 0.651476
\(964\) 45.6686 1.47089
\(965\) −10.1791 −0.327676
\(966\) 4.50406 0.144916
\(967\) 49.4076 1.58884 0.794420 0.607369i \(-0.207775\pi\)
0.794420 + 0.607369i \(0.207775\pi\)
\(968\) −52.4407 −1.68551
\(969\) 0.250159 0.00803626
\(970\) 19.7520 0.634198
\(971\) 21.4751 0.689167 0.344584 0.938756i \(-0.388020\pi\)
0.344584 + 0.938756i \(0.388020\pi\)
\(972\) −71.9033 −2.30630
\(973\) 22.7360 0.728883
\(974\) 15.3238 0.491006
\(975\) 26.9688 0.863693
\(976\) 17.4168 0.557499
\(977\) −15.2844 −0.488992 −0.244496 0.969650i \(-0.578622\pi\)
−0.244496 + 0.969650i \(0.578622\pi\)
\(978\) 20.7376 0.663117
\(979\) −14.0828 −0.450087
\(980\) 19.1629 0.612137
\(981\) 6.69852 0.213867
\(982\) 38.4788 1.22791
\(983\) 44.4677 1.41830 0.709150 0.705058i \(-0.249079\pi\)
0.709150 + 0.705058i \(0.249079\pi\)
\(984\) −74.1556 −2.36399
\(985\) −6.10783 −0.194612
\(986\) −0.291494 −0.00928306
\(987\) 19.5441 0.622096
\(988\) −29.5736 −0.940863
\(989\) −8.06707 −0.256518
\(990\) −16.5051 −0.524565
\(991\) −40.4447 −1.28477 −0.642384 0.766383i \(-0.722054\pi\)
−0.642384 + 0.766383i \(0.722054\pi\)
\(992\) 47.7183 1.51506
\(993\) 22.6124 0.717584
\(994\) −8.58136 −0.272184
\(995\) −14.4242 −0.457276
\(996\) −47.5234 −1.50584
\(997\) 31.6645 1.00282 0.501412 0.865209i \(-0.332814\pi\)
0.501412 + 0.865209i \(0.332814\pi\)
\(998\) 54.4354 1.72312
\(999\) −57.3356 −1.81402
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 4001.2.a.a.1.8 149
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.a.1.8 149 1.1 even 1 trivial