Properties

Label 4033.2.a.e.1.10
Level $4033$
Weight $2$
Character 4033.1
Self dual yes
Analytic conductor $32.204$
Analytic rank $0$
Dimension $82$
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] = [4033,2,Mod(1,4033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4033 = 37 \cdot 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4033.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: \(32.2036671352\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 4033.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.33134 q^{2} -0.785356 q^{3} +3.43514 q^{4} +0.539311 q^{5} +1.83093 q^{6} -3.00025 q^{7} -3.34579 q^{8} -2.38322 q^{9} +O(q^{10})\) \(q-2.33134 q^{2} -0.785356 q^{3} +3.43514 q^{4} +0.539311 q^{5} +1.83093 q^{6} -3.00025 q^{7} -3.34579 q^{8} -2.38322 q^{9} -1.25731 q^{10} -0.795769 q^{11} -2.69781 q^{12} -0.649389 q^{13} +6.99459 q^{14} -0.423551 q^{15} +0.929885 q^{16} -3.23621 q^{17} +5.55608 q^{18} -6.59290 q^{19} +1.85260 q^{20} +2.35626 q^{21} +1.85521 q^{22} -2.44607 q^{23} +2.62763 q^{24} -4.70914 q^{25} +1.51395 q^{26} +4.22774 q^{27} -10.3063 q^{28} -7.21868 q^{29} +0.987440 q^{30} +7.30019 q^{31} +4.52369 q^{32} +0.624962 q^{33} +7.54471 q^{34} -1.61806 q^{35} -8.18667 q^{36} -1.00000 q^{37} +15.3703 q^{38} +0.510002 q^{39} -1.80442 q^{40} -10.9274 q^{41} -5.49325 q^{42} -7.88053 q^{43} -2.73357 q^{44} -1.28529 q^{45} +5.70262 q^{46} -12.9550 q^{47} -0.730292 q^{48} +2.00148 q^{49} +10.9786 q^{50} +2.54158 q^{51} -2.23074 q^{52} +6.49773 q^{53} -9.85630 q^{54} -0.429166 q^{55} +10.0382 q^{56} +5.17777 q^{57} +16.8292 q^{58} -4.15850 q^{59} -1.45496 q^{60} +1.90087 q^{61} -17.0192 q^{62} +7.15023 q^{63} -12.4060 q^{64} -0.350222 q^{65} -1.45700 q^{66} +2.59315 q^{67} -11.1168 q^{68} +1.92104 q^{69} +3.77226 q^{70} +2.45964 q^{71} +7.97373 q^{72} +3.05908 q^{73} +2.33134 q^{74} +3.69836 q^{75} -22.6475 q^{76} +2.38750 q^{77} -1.18899 q^{78} -10.3354 q^{79} +0.501497 q^{80} +3.82936 q^{81} +25.4755 q^{82} +11.9288 q^{83} +8.09408 q^{84} -1.74532 q^{85} +18.3722 q^{86} +5.66923 q^{87} +2.66247 q^{88} +12.9884 q^{89} +2.99645 q^{90} +1.94833 q^{91} -8.40259 q^{92} -5.73326 q^{93} +30.2024 q^{94} -3.55562 q^{95} -3.55271 q^{96} -8.94380 q^{97} -4.66613 q^{98} +1.89649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q + 10 q^{2} + 17 q^{3} + 88 q^{4} + 22 q^{5} + 4 q^{6} + 15 q^{7} + 33 q^{8} + 91 q^{9} + 13 q^{10} + 2 q^{11} + 36 q^{12} + 23 q^{13} + 24 q^{14} + 35 q^{15} + 104 q^{16} + 43 q^{17} + 42 q^{18} + 15 q^{19} + 59 q^{20} + 9 q^{21} + 6 q^{22} + 70 q^{23} + 15 q^{24} + 98 q^{25} + 10 q^{26} + 65 q^{27} + 37 q^{28} + 27 q^{29} + 17 q^{30} + 59 q^{31} + 46 q^{32} + 16 q^{33} - 16 q^{34} + 80 q^{35} + 88 q^{36} - 82 q^{37} + 82 q^{38} + 13 q^{39} + 14 q^{40} + 3 q^{41} + 62 q^{42} + 7 q^{43} - 11 q^{44} + 42 q^{45} - 11 q^{46} + 123 q^{47} + 45 q^{48} + 105 q^{49} + 27 q^{50} + 3 q^{51} - 30 q^{52} + 82 q^{53} - 27 q^{54} + 37 q^{55} + 66 q^{56} + 29 q^{57} - 34 q^{58} + 60 q^{59} + 94 q^{60} + 9 q^{61} - 2 q^{62} + 106 q^{63} + 93 q^{64} + 9 q^{65} + 63 q^{66} + 113 q^{68} + 48 q^{69} + 47 q^{70} + 59 q^{71} + 63 q^{72} + 21 q^{73} - 10 q^{74} + 77 q^{75} + 22 q^{76} + 30 q^{77} + 29 q^{78} + 55 q^{79} + 88 q^{80} + 42 q^{81} + 4 q^{82} + 92 q^{83} + 43 q^{84} - 7 q^{85} + 17 q^{86} + 147 q^{87} - 13 q^{88} + 100 q^{89} + 91 q^{90} + 28 q^{91} + 127 q^{92} + 3 q^{93} + 30 q^{94} + 48 q^{95} - 31 q^{96} + 53 q^{97} + 101 q^{98} - 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.33134 −1.64850 −0.824252 0.566223i \(-0.808404\pi\)
−0.824252 + 0.566223i \(0.808404\pi\)
\(3\) −0.785356 −0.453426 −0.226713 0.973962i \(-0.572798\pi\)
−0.226713 + 0.973962i \(0.572798\pi\)
\(4\) 3.43514 1.71757
\(5\) 0.539311 0.241187 0.120593 0.992702i \(-0.461520\pi\)
0.120593 + 0.992702i \(0.461520\pi\)
\(6\) 1.83093 0.747475
\(7\) −3.00025 −1.13399 −0.566993 0.823722i \(-0.691894\pi\)
−0.566993 + 0.823722i \(0.691894\pi\)
\(8\) −3.34579 −1.18291
\(9\) −2.38322 −0.794405
\(10\) −1.25731 −0.397598
\(11\) −0.795769 −0.239933 −0.119967 0.992778i \(-0.538279\pi\)
−0.119967 + 0.992778i \(0.538279\pi\)
\(12\) −2.69781 −0.778789
\(13\) −0.649389 −0.180108 −0.0900541 0.995937i \(-0.528704\pi\)
−0.0900541 + 0.995937i \(0.528704\pi\)
\(14\) 6.99459 1.86938
\(15\) −0.423551 −0.109360
\(16\) 0.929885 0.232471
\(17\) −3.23621 −0.784897 −0.392448 0.919774i \(-0.628372\pi\)
−0.392448 + 0.919774i \(0.628372\pi\)
\(18\) 5.55608 1.30958
\(19\) −6.59290 −1.51251 −0.756257 0.654274i \(-0.772974\pi\)
−0.756257 + 0.654274i \(0.772974\pi\)
\(20\) 1.85260 0.414255
\(21\) 2.35626 0.514179
\(22\) 1.85521 0.395531
\(23\) −2.44607 −0.510041 −0.255021 0.966936i \(-0.582082\pi\)
−0.255021 + 0.966936i \(0.582082\pi\)
\(24\) 2.62763 0.536364
\(25\) −4.70914 −0.941829
\(26\) 1.51395 0.296909
\(27\) 4.22774 0.813629
\(28\) −10.3063 −1.94770
\(29\) −7.21868 −1.34047 −0.670237 0.742147i \(-0.733808\pi\)
−0.670237 + 0.742147i \(0.733808\pi\)
\(30\) 0.987440 0.180281
\(31\) 7.30019 1.31115 0.655577 0.755128i \(-0.272426\pi\)
0.655577 + 0.755128i \(0.272426\pi\)
\(32\) 4.52369 0.799684
\(33\) 0.624962 0.108792
\(34\) 7.54471 1.29391
\(35\) −1.61806 −0.273503
\(36\) −8.18667 −1.36444
\(37\) −1.00000 −0.164399
\(38\) 15.3703 2.49339
\(39\) 0.510002 0.0816657
\(40\) −1.80442 −0.285303
\(41\) −10.9274 −1.70657 −0.853287 0.521442i \(-0.825394\pi\)
−0.853287 + 0.521442i \(0.825394\pi\)
\(42\) −5.49325 −0.847626
\(43\) −7.88053 −1.20177 −0.600885 0.799335i \(-0.705185\pi\)
−0.600885 + 0.799335i \(0.705185\pi\)
\(44\) −2.73357 −0.412102
\(45\) −1.28529 −0.191600
\(46\) 5.70262 0.840806
\(47\) −12.9550 −1.88967 −0.944837 0.327539i \(-0.893781\pi\)
−0.944837 + 0.327539i \(0.893781\pi\)
\(48\) −0.730292 −0.105408
\(49\) 2.00148 0.285926
\(50\) 10.9786 1.55261
\(51\) 2.54158 0.355893
\(52\) −2.23074 −0.309348
\(53\) 6.49773 0.892532 0.446266 0.894900i \(-0.352754\pi\)
0.446266 + 0.894900i \(0.352754\pi\)
\(54\) −9.85630 −1.34127
\(55\) −0.429166 −0.0578688
\(56\) 10.0382 1.34141
\(57\) 5.17777 0.685813
\(58\) 16.8292 2.20978
\(59\) −4.15850 −0.541391 −0.270695 0.962665i \(-0.587254\pi\)
−0.270695 + 0.962665i \(0.587254\pi\)
\(60\) −1.45496 −0.187834
\(61\) 1.90087 0.243382 0.121691 0.992568i \(-0.461168\pi\)
0.121691 + 0.992568i \(0.461168\pi\)
\(62\) −17.0192 −2.16144
\(63\) 7.15023 0.900845
\(64\) −12.4060 −1.55075
\(65\) −0.350222 −0.0434397
\(66\) −1.45700 −0.179344
\(67\) 2.59315 0.316804 0.158402 0.987375i \(-0.449366\pi\)
0.158402 + 0.987375i \(0.449366\pi\)
\(68\) −11.1168 −1.34811
\(69\) 1.92104 0.231266
\(70\) 3.77226 0.450871
\(71\) 2.45964 0.291905 0.145953 0.989292i \(-0.453375\pi\)
0.145953 + 0.989292i \(0.453375\pi\)
\(72\) 7.97373 0.939713
\(73\) 3.05908 0.358039 0.179019 0.983846i \(-0.442708\pi\)
0.179019 + 0.983846i \(0.442708\pi\)
\(74\) 2.33134 0.271013
\(75\) 3.69836 0.427049
\(76\) −22.6475 −2.59785
\(77\) 2.38750 0.272081
\(78\) −1.18899 −0.134626
\(79\) −10.3354 −1.16282 −0.581412 0.813609i \(-0.697500\pi\)
−0.581412 + 0.813609i \(0.697500\pi\)
\(80\) 0.501497 0.0560691
\(81\) 3.82936 0.425485
\(82\) 25.4755 2.81329
\(83\) 11.9288 1.30936 0.654681 0.755906i \(-0.272803\pi\)
0.654681 + 0.755906i \(0.272803\pi\)
\(84\) 8.09408 0.883137
\(85\) −1.74532 −0.189307
\(86\) 18.3722 1.98112
\(87\) 5.66923 0.607806
\(88\) 2.66247 0.283820
\(89\) 12.9884 1.37677 0.688384 0.725346i \(-0.258320\pi\)
0.688384 + 0.725346i \(0.258320\pi\)
\(90\) 2.99645 0.315854
\(91\) 1.94833 0.204240
\(92\) −8.40259 −0.876031
\(93\) −5.73326 −0.594511
\(94\) 30.2024 3.11514
\(95\) −3.55562 −0.364799
\(96\) −3.55271 −0.362597
\(97\) −8.94380 −0.908105 −0.454052 0.890975i \(-0.650022\pi\)
−0.454052 + 0.890975i \(0.650022\pi\)
\(98\) −4.66613 −0.471350
\(99\) 1.89649 0.190604
\(100\) −16.1765 −1.61765
\(101\) −17.1608 −1.70757 −0.853783 0.520629i \(-0.825698\pi\)
−0.853783 + 0.520629i \(0.825698\pi\)
\(102\) −5.92528 −0.586690
\(103\) 1.22033 0.120243 0.0601215 0.998191i \(-0.480851\pi\)
0.0601215 + 0.998191i \(0.480851\pi\)
\(104\) 2.17272 0.213052
\(105\) 1.27076 0.124013
\(106\) −15.1484 −1.47134
\(107\) −5.50039 −0.531743 −0.265871 0.964008i \(-0.585660\pi\)
−0.265871 + 0.964008i \(0.585660\pi\)
\(108\) 14.5229 1.39746
\(109\) 1.00000 0.0957826
\(110\) 1.00053 0.0953970
\(111\) 0.785356 0.0745427
\(112\) −2.78989 −0.263619
\(113\) −7.67104 −0.721631 −0.360816 0.932637i \(-0.617502\pi\)
−0.360816 + 0.932637i \(0.617502\pi\)
\(114\) −12.0711 −1.13057
\(115\) −1.31919 −0.123015
\(116\) −24.7971 −2.30236
\(117\) 1.54763 0.143079
\(118\) 9.69487 0.892485
\(119\) 9.70944 0.890063
\(120\) 1.41711 0.129364
\(121\) −10.3668 −0.942432
\(122\) −4.43157 −0.401216
\(123\) 8.58190 0.773804
\(124\) 25.0772 2.25200
\(125\) −5.23624 −0.468344
\(126\) −16.6696 −1.48505
\(127\) −4.06695 −0.360884 −0.180442 0.983586i \(-0.557753\pi\)
−0.180442 + 0.983586i \(0.557753\pi\)
\(128\) 19.8753 1.75674
\(129\) 6.18903 0.544913
\(130\) 0.816487 0.0716106
\(131\) 12.8962 1.12675 0.563373 0.826203i \(-0.309503\pi\)
0.563373 + 0.826203i \(0.309503\pi\)
\(132\) 2.14683 0.186858
\(133\) 19.7803 1.71517
\(134\) −6.04551 −0.522253
\(135\) 2.28007 0.196237
\(136\) 10.8277 0.928466
\(137\) 11.6034 0.991348 0.495674 0.868509i \(-0.334921\pi\)
0.495674 + 0.868509i \(0.334921\pi\)
\(138\) −4.47859 −0.381243
\(139\) −12.4435 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(140\) −5.55827 −0.469760
\(141\) 10.1743 0.856827
\(142\) −5.73425 −0.481207
\(143\) 0.516763 0.0432139
\(144\) −2.21612 −0.184676
\(145\) −3.89311 −0.323305
\(146\) −7.13176 −0.590229
\(147\) −1.57188 −0.129646
\(148\) −3.43514 −0.282366
\(149\) 22.3239 1.82884 0.914421 0.404765i \(-0.132647\pi\)
0.914421 + 0.404765i \(0.132647\pi\)
\(150\) −8.62212 −0.703993
\(151\) −0.434227 −0.0353369 −0.0176685 0.999844i \(-0.505624\pi\)
−0.0176685 + 0.999844i \(0.505624\pi\)
\(152\) 22.0584 1.78917
\(153\) 7.71259 0.623526
\(154\) −5.56607 −0.448527
\(155\) 3.93707 0.316233
\(156\) 1.75193 0.140266
\(157\) 16.6221 1.32659 0.663296 0.748357i \(-0.269157\pi\)
0.663296 + 0.748357i \(0.269157\pi\)
\(158\) 24.0953 1.91692
\(159\) −5.10303 −0.404697
\(160\) 2.43968 0.192873
\(161\) 7.33882 0.578380
\(162\) −8.92753 −0.701413
\(163\) −0.309575 −0.0242478 −0.0121239 0.999927i \(-0.503859\pi\)
−0.0121239 + 0.999927i \(0.503859\pi\)
\(164\) −37.5371 −2.93116
\(165\) 0.337049 0.0262392
\(166\) −27.8102 −2.15849
\(167\) −1.37567 −0.106453 −0.0532263 0.998582i \(-0.516950\pi\)
−0.0532263 + 0.998582i \(0.516950\pi\)
\(168\) −7.88355 −0.608229
\(169\) −12.5783 −0.967561
\(170\) 4.06894 0.312073
\(171\) 15.7123 1.20155
\(172\) −27.0707 −2.06412
\(173\) −17.6238 −1.33991 −0.669955 0.742402i \(-0.733686\pi\)
−0.669955 + 0.742402i \(0.733686\pi\)
\(174\) −13.2169 −1.00197
\(175\) 14.1286 1.06802
\(176\) −0.739974 −0.0557776
\(177\) 3.26591 0.245480
\(178\) −30.2804 −2.26961
\(179\) −0.167386 −0.0125110 −0.00625552 0.999980i \(-0.501991\pi\)
−0.00625552 + 0.999980i \(0.501991\pi\)
\(180\) −4.41516 −0.329086
\(181\) −17.0999 −1.27103 −0.635513 0.772090i \(-0.719211\pi\)
−0.635513 + 0.772090i \(0.719211\pi\)
\(182\) −4.54221 −0.336691
\(183\) −1.49286 −0.110356
\(184\) 8.18403 0.603335
\(185\) −0.539311 −0.0396509
\(186\) 13.3662 0.980054
\(187\) 2.57528 0.188323
\(188\) −44.5020 −3.24564
\(189\) −12.6843 −0.922645
\(190\) 8.28935 0.601373
\(191\) −14.4227 −1.04359 −0.521796 0.853070i \(-0.674738\pi\)
−0.521796 + 0.853070i \(0.674738\pi\)
\(192\) 9.74316 0.703152
\(193\) −5.74160 −0.413289 −0.206645 0.978416i \(-0.566254\pi\)
−0.206645 + 0.978416i \(0.566254\pi\)
\(194\) 20.8510 1.49702
\(195\) 0.275049 0.0196967
\(196\) 6.87536 0.491097
\(197\) −0.439451 −0.0313096 −0.0156548 0.999877i \(-0.504983\pi\)
−0.0156548 + 0.999877i \(0.504983\pi\)
\(198\) −4.42135 −0.314212
\(199\) −14.5105 −1.02862 −0.514311 0.857604i \(-0.671952\pi\)
−0.514311 + 0.857604i \(0.671952\pi\)
\(200\) 15.7558 1.11410
\(201\) −2.03655 −0.143647
\(202\) 40.0077 2.81493
\(203\) 21.6578 1.52008
\(204\) 8.73067 0.611270
\(205\) −5.89326 −0.411603
\(206\) −2.84501 −0.198221
\(207\) 5.82952 0.405179
\(208\) −0.603857 −0.0418700
\(209\) 5.24642 0.362902
\(210\) −2.96257 −0.204436
\(211\) −12.2480 −0.843188 −0.421594 0.906785i \(-0.638529\pi\)
−0.421594 + 0.906785i \(0.638529\pi\)
\(212\) 22.3206 1.53298
\(213\) −1.93169 −0.132357
\(214\) 12.8233 0.876581
\(215\) −4.25005 −0.289851
\(216\) −14.1451 −0.962454
\(217\) −21.9024 −1.48683
\(218\) −2.33134 −0.157898
\(219\) −2.40247 −0.162344
\(220\) −1.47424 −0.0993936
\(221\) 2.10156 0.141366
\(222\) −1.83093 −0.122884
\(223\) 6.60184 0.442092 0.221046 0.975263i \(-0.429053\pi\)
0.221046 + 0.975263i \(0.429053\pi\)
\(224\) −13.5722 −0.906831
\(225\) 11.2229 0.748194
\(226\) 17.8838 1.18961
\(227\) −3.89368 −0.258432 −0.129216 0.991616i \(-0.541246\pi\)
−0.129216 + 0.991616i \(0.541246\pi\)
\(228\) 17.7864 1.17793
\(229\) −23.3583 −1.54356 −0.771781 0.635889i \(-0.780634\pi\)
−0.771781 + 0.635889i \(0.780634\pi\)
\(230\) 3.07548 0.202791
\(231\) −1.87504 −0.123369
\(232\) 24.1521 1.58567
\(233\) 24.6770 1.61664 0.808321 0.588742i \(-0.200376\pi\)
0.808321 + 0.588742i \(0.200376\pi\)
\(234\) −3.60806 −0.235866
\(235\) −6.98675 −0.455765
\(236\) −14.2850 −0.929875
\(237\) 8.11698 0.527254
\(238\) −22.6360 −1.46727
\(239\) −17.5349 −1.13424 −0.567121 0.823635i \(-0.691943\pi\)
−0.567121 + 0.823635i \(0.691943\pi\)
\(240\) −0.393854 −0.0254232
\(241\) −22.1761 −1.42849 −0.714245 0.699896i \(-0.753230\pi\)
−0.714245 + 0.699896i \(0.753230\pi\)
\(242\) 24.1684 1.55360
\(243\) −15.6906 −1.00656
\(244\) 6.52975 0.418025
\(245\) 1.07942 0.0689616
\(246\) −20.0073 −1.27562
\(247\) 4.28136 0.272416
\(248\) −24.4249 −1.55098
\(249\) −9.36840 −0.593698
\(250\) 12.2075 0.772067
\(251\) 20.2005 1.27504 0.637522 0.770432i \(-0.279959\pi\)
0.637522 + 0.770432i \(0.279959\pi\)
\(252\) 24.5620 1.54726
\(253\) 1.94651 0.122376
\(254\) 9.48144 0.594919
\(255\) 1.37070 0.0858366
\(256\) −21.5239 −1.34524
\(257\) −1.24949 −0.0779411 −0.0389705 0.999240i \(-0.512408\pi\)
−0.0389705 + 0.999240i \(0.512408\pi\)
\(258\) −14.4287 −0.898292
\(259\) 3.00025 0.186426
\(260\) −1.20306 −0.0746107
\(261\) 17.2037 1.06488
\(262\) −30.0654 −1.85745
\(263\) 5.30809 0.327311 0.163656 0.986518i \(-0.447671\pi\)
0.163656 + 0.986518i \(0.447671\pi\)
\(264\) −2.09099 −0.128691
\(265\) 3.50429 0.215267
\(266\) −46.1146 −2.82747
\(267\) −10.2005 −0.624262
\(268\) 8.90783 0.544132
\(269\) 12.7805 0.779240 0.389620 0.920976i \(-0.372606\pi\)
0.389620 + 0.920976i \(0.372606\pi\)
\(270\) −5.31560 −0.323497
\(271\) 7.20552 0.437704 0.218852 0.975758i \(-0.429769\pi\)
0.218852 + 0.975758i \(0.429769\pi\)
\(272\) −3.00931 −0.182466
\(273\) −1.53013 −0.0926078
\(274\) −27.0515 −1.63424
\(275\) 3.74739 0.225976
\(276\) 6.59903 0.397215
\(277\) −10.7882 −0.648202 −0.324101 0.946022i \(-0.605062\pi\)
−0.324101 + 0.946022i \(0.605062\pi\)
\(278\) 29.0099 1.73990
\(279\) −17.3979 −1.04159
\(280\) 5.41370 0.323530
\(281\) −30.0396 −1.79201 −0.896007 0.444041i \(-0.853545\pi\)
−0.896007 + 0.444041i \(0.853545\pi\)
\(282\) −23.7196 −1.41248
\(283\) −17.3914 −1.03381 −0.516905 0.856042i \(-0.672916\pi\)
−0.516905 + 0.856042i \(0.672916\pi\)
\(284\) 8.44919 0.501367
\(285\) 2.79243 0.165409
\(286\) −1.20475 −0.0712384
\(287\) 32.7849 1.93523
\(288\) −10.7809 −0.635273
\(289\) −6.52693 −0.383937
\(290\) 9.07615 0.532970
\(291\) 7.02407 0.411758
\(292\) 10.5084 0.614956
\(293\) −15.9269 −0.930458 −0.465229 0.885190i \(-0.654028\pi\)
−0.465229 + 0.885190i \(0.654028\pi\)
\(294\) 3.66458 0.213722
\(295\) −2.24272 −0.130576
\(296\) 3.34579 0.194470
\(297\) −3.36431 −0.195217
\(298\) −52.0445 −3.01485
\(299\) 1.58845 0.0918626
\(300\) 12.7044 0.733486
\(301\) 23.6435 1.36279
\(302\) 1.01233 0.0582531
\(303\) 13.4774 0.774255
\(304\) −6.13064 −0.351616
\(305\) 1.02516 0.0587005
\(306\) −17.9807 −1.02789
\(307\) 3.28079 0.187245 0.0936224 0.995608i \(-0.470155\pi\)
0.0936224 + 0.995608i \(0.470155\pi\)
\(308\) 8.20139 0.467318
\(309\) −0.958397 −0.0545213
\(310\) −9.17864 −0.521312
\(311\) 14.3857 0.815738 0.407869 0.913040i \(-0.366272\pi\)
0.407869 + 0.913040i \(0.366272\pi\)
\(312\) −1.70636 −0.0966034
\(313\) 17.8657 1.00983 0.504914 0.863170i \(-0.331524\pi\)
0.504914 + 0.863170i \(0.331524\pi\)
\(314\) −38.7518 −2.18689
\(315\) 3.85620 0.217272
\(316\) −35.5035 −1.99723
\(317\) 4.28569 0.240708 0.120354 0.992731i \(-0.461597\pi\)
0.120354 + 0.992731i \(0.461597\pi\)
\(318\) 11.8969 0.667145
\(319\) 5.74440 0.321624
\(320\) −6.69070 −0.374022
\(321\) 4.31977 0.241106
\(322\) −17.1093 −0.953462
\(323\) 21.3360 1.18717
\(324\) 13.1544 0.730798
\(325\) 3.05807 0.169631
\(326\) 0.721725 0.0399726
\(327\) −0.785356 −0.0434303
\(328\) 36.5607 2.01873
\(329\) 38.8681 2.14287
\(330\) −0.785774 −0.0432554
\(331\) 14.8114 0.814107 0.407053 0.913404i \(-0.366556\pi\)
0.407053 + 0.913404i \(0.366556\pi\)
\(332\) 40.9772 2.24892
\(333\) 2.38322 0.130599
\(334\) 3.20715 0.175487
\(335\) 1.39851 0.0764090
\(336\) 2.19105 0.119532
\(337\) 10.6619 0.580793 0.290397 0.956906i \(-0.406213\pi\)
0.290397 + 0.956906i \(0.406213\pi\)
\(338\) 29.3243 1.59503
\(339\) 6.02450 0.327206
\(340\) −5.99542 −0.325148
\(341\) −5.80927 −0.314589
\(342\) −36.6307 −1.98076
\(343\) 14.9968 0.809750
\(344\) 26.3666 1.42159
\(345\) 1.03604 0.0557783
\(346\) 41.0869 2.20885
\(347\) 21.3224 1.14465 0.572323 0.820028i \(-0.306042\pi\)
0.572323 + 0.820028i \(0.306042\pi\)
\(348\) 19.4746 1.04395
\(349\) 19.0377 1.01907 0.509533 0.860451i \(-0.329818\pi\)
0.509533 + 0.860451i \(0.329818\pi\)
\(350\) −32.9385 −1.76064
\(351\) −2.74545 −0.146541
\(352\) −3.59981 −0.191871
\(353\) 13.5897 0.723308 0.361654 0.932312i \(-0.382212\pi\)
0.361654 + 0.932312i \(0.382212\pi\)
\(354\) −7.61393 −0.404676
\(355\) 1.32651 0.0704038
\(356\) 44.6169 2.36469
\(357\) −7.62537 −0.403577
\(358\) 0.390234 0.0206245
\(359\) 18.8696 0.995901 0.497951 0.867205i \(-0.334086\pi\)
0.497951 + 0.867205i \(0.334086\pi\)
\(360\) 4.30032 0.226647
\(361\) 24.4663 1.28770
\(362\) 39.8656 2.09529
\(363\) 8.14160 0.427323
\(364\) 6.69277 0.350796
\(365\) 1.64980 0.0863543
\(366\) 3.48037 0.181922
\(367\) 7.02187 0.366539 0.183269 0.983063i \(-0.441332\pi\)
0.183269 + 0.983063i \(0.441332\pi\)
\(368\) −2.27457 −0.118570
\(369\) 26.0423 1.35571
\(370\) 1.25731 0.0653647
\(371\) −19.4948 −1.01212
\(372\) −19.6945 −1.02111
\(373\) 9.72342 0.503460 0.251730 0.967798i \(-0.419001\pi\)
0.251730 + 0.967798i \(0.419001\pi\)
\(374\) −6.00384 −0.310451
\(375\) 4.11232 0.212359
\(376\) 43.3445 2.23532
\(377\) 4.68773 0.241430
\(378\) 29.5713 1.52098
\(379\) 8.41766 0.432386 0.216193 0.976351i \(-0.430636\pi\)
0.216193 + 0.976351i \(0.430636\pi\)
\(380\) −12.2140 −0.626567
\(381\) 3.19401 0.163634
\(382\) 33.6242 1.72037
\(383\) 26.3782 1.34787 0.673933 0.738793i \(-0.264604\pi\)
0.673933 + 0.738793i \(0.264604\pi\)
\(384\) −15.6092 −0.796552
\(385\) 1.28761 0.0656224
\(386\) 13.3856 0.681309
\(387\) 18.7810 0.954692
\(388\) −30.7232 −1.55973
\(389\) −23.5909 −1.19610 −0.598052 0.801457i \(-0.704058\pi\)
−0.598052 + 0.801457i \(0.704058\pi\)
\(390\) −0.641233 −0.0324701
\(391\) 7.91601 0.400330
\(392\) −6.69653 −0.338226
\(393\) −10.1281 −0.510896
\(394\) 1.02451 0.0516139
\(395\) −5.57399 −0.280458
\(396\) 6.51469 0.327376
\(397\) −24.3525 −1.22222 −0.611109 0.791546i \(-0.709276\pi\)
−0.611109 + 0.791546i \(0.709276\pi\)
\(398\) 33.8289 1.69569
\(399\) −15.5346 −0.777703
\(400\) −4.37896 −0.218948
\(401\) 21.2851 1.06293 0.531464 0.847081i \(-0.321642\pi\)
0.531464 + 0.847081i \(0.321642\pi\)
\(402\) 4.74788 0.236803
\(403\) −4.74067 −0.236149
\(404\) −58.9498 −2.93286
\(405\) 2.06521 0.102621
\(406\) −50.4917 −2.50586
\(407\) 0.795769 0.0394448
\(408\) −8.50358 −0.420990
\(409\) −15.6659 −0.774630 −0.387315 0.921948i \(-0.626597\pi\)
−0.387315 + 0.921948i \(0.626597\pi\)
\(410\) 13.7392 0.678530
\(411\) −9.11283 −0.449503
\(412\) 4.19201 0.206526
\(413\) 12.4765 0.613930
\(414\) −13.5906 −0.667940
\(415\) 6.43335 0.315801
\(416\) −2.93764 −0.144030
\(417\) 9.77255 0.478564
\(418\) −12.2312 −0.598246
\(419\) 22.1978 1.08443 0.542216 0.840239i \(-0.317585\pi\)
0.542216 + 0.840239i \(0.317585\pi\)
\(420\) 4.36522 0.213001
\(421\) −10.1692 −0.495617 −0.247809 0.968809i \(-0.579710\pi\)
−0.247809 + 0.968809i \(0.579710\pi\)
\(422\) 28.5543 1.39000
\(423\) 30.8745 1.50117
\(424\) −21.7400 −1.05579
\(425\) 15.2398 0.739239
\(426\) 4.50343 0.218192
\(427\) −5.70308 −0.275992
\(428\) −18.8946 −0.913304
\(429\) −0.405844 −0.0195943
\(430\) 9.90831 0.477821
\(431\) −23.0489 −1.11023 −0.555114 0.831774i \(-0.687325\pi\)
−0.555114 + 0.831774i \(0.687325\pi\)
\(432\) 3.93132 0.189146
\(433\) −11.3144 −0.543736 −0.271868 0.962335i \(-0.587641\pi\)
−0.271868 + 0.962335i \(0.587641\pi\)
\(434\) 51.0619 2.45105
\(435\) 3.05748 0.146595
\(436\) 3.43514 0.164513
\(437\) 16.1267 0.771445
\(438\) 5.60097 0.267625
\(439\) −12.2506 −0.584688 −0.292344 0.956313i \(-0.594435\pi\)
−0.292344 + 0.956313i \(0.594435\pi\)
\(440\) 1.43590 0.0684538
\(441\) −4.76996 −0.227141
\(442\) −4.89945 −0.233043
\(443\) 35.3529 1.67966 0.839832 0.542846i \(-0.182653\pi\)
0.839832 + 0.542846i \(0.182653\pi\)
\(444\) 2.69781 0.128032
\(445\) 7.00479 0.332059
\(446\) −15.3911 −0.728791
\(447\) −17.5322 −0.829244
\(448\) 37.2212 1.75853
\(449\) 16.7119 0.788682 0.394341 0.918964i \(-0.370973\pi\)
0.394341 + 0.918964i \(0.370973\pi\)
\(450\) −26.1644 −1.23340
\(451\) 8.69568 0.409464
\(452\) −26.3511 −1.23945
\(453\) 0.341023 0.0160227
\(454\) 9.07748 0.426027
\(455\) 1.05075 0.0492601
\(456\) −17.3237 −0.811258
\(457\) −7.40351 −0.346322 −0.173161 0.984894i \(-0.555398\pi\)
−0.173161 + 0.984894i \(0.555398\pi\)
\(458\) 54.4562 2.54457
\(459\) −13.6819 −0.638615
\(460\) −4.53161 −0.211287
\(461\) −16.9742 −0.790569 −0.395284 0.918559i \(-0.629354\pi\)
−0.395284 + 0.918559i \(0.629354\pi\)
\(462\) 4.37135 0.203374
\(463\) 16.6781 0.775097 0.387548 0.921849i \(-0.373322\pi\)
0.387548 + 0.921849i \(0.373322\pi\)
\(464\) −6.71254 −0.311622
\(465\) −3.09200 −0.143388
\(466\) −57.5304 −2.66504
\(467\) 33.8684 1.56724 0.783621 0.621239i \(-0.213370\pi\)
0.783621 + 0.621239i \(0.213370\pi\)
\(468\) 5.31633 0.245748
\(469\) −7.78010 −0.359251
\(470\) 16.2885 0.751331
\(471\) −13.0543 −0.601511
\(472\) 13.9135 0.640419
\(473\) 6.27108 0.288345
\(474\) −18.9234 −0.869181
\(475\) 31.0469 1.42453
\(476\) 33.3532 1.52874
\(477\) −15.4855 −0.709032
\(478\) 40.8799 1.86980
\(479\) 4.14348 0.189321 0.0946603 0.995510i \(-0.469824\pi\)
0.0946603 + 0.995510i \(0.469824\pi\)
\(480\) −1.91602 −0.0874537
\(481\) 0.649389 0.0296096
\(482\) 51.7000 2.35487
\(483\) −5.76359 −0.262252
\(484\) −35.6112 −1.61869
\(485\) −4.82348 −0.219023
\(486\) 36.5802 1.65931
\(487\) −36.4013 −1.64950 −0.824751 0.565496i \(-0.808685\pi\)
−0.824751 + 0.565496i \(0.808685\pi\)
\(488\) −6.35991 −0.287900
\(489\) 0.243127 0.0109946
\(490\) −2.51649 −0.113684
\(491\) 33.5646 1.51475 0.757375 0.652980i \(-0.226481\pi\)
0.757375 + 0.652980i \(0.226481\pi\)
\(492\) 29.4800 1.32906
\(493\) 23.3612 1.05213
\(494\) −9.98129 −0.449079
\(495\) 1.02280 0.0459713
\(496\) 6.78834 0.304806
\(497\) −7.37952 −0.331017
\(498\) 21.8409 0.978714
\(499\) −17.1110 −0.765994 −0.382997 0.923750i \(-0.625108\pi\)
−0.382997 + 0.923750i \(0.625108\pi\)
\(500\) −17.9872 −0.804412
\(501\) 1.08039 0.0482683
\(502\) −47.0942 −2.10192
\(503\) 29.1668 1.30048 0.650241 0.759728i \(-0.274668\pi\)
0.650241 + 0.759728i \(0.274668\pi\)
\(504\) −23.9232 −1.06562
\(505\) −9.25502 −0.411843
\(506\) −4.53797 −0.201737
\(507\) 9.87844 0.438717
\(508\) −13.9705 −0.619842
\(509\) 14.7919 0.655640 0.327820 0.944740i \(-0.393686\pi\)
0.327820 + 0.944740i \(0.393686\pi\)
\(510\) −3.19557 −0.141502
\(511\) −9.17801 −0.406011
\(512\) 10.4289 0.460897
\(513\) −27.8731 −1.23063
\(514\) 2.91298 0.128486
\(515\) 0.658139 0.0290011
\(516\) 21.2601 0.935926
\(517\) 10.3091 0.453396
\(518\) −6.99459 −0.307325
\(519\) 13.8409 0.607549
\(520\) 1.17177 0.0513855
\(521\) −12.6683 −0.555008 −0.277504 0.960724i \(-0.589507\pi\)
−0.277504 + 0.960724i \(0.589507\pi\)
\(522\) −40.1075 −1.75546
\(523\) −4.80546 −0.210128 −0.105064 0.994465i \(-0.533505\pi\)
−0.105064 + 0.994465i \(0.533505\pi\)
\(524\) 44.3002 1.93526
\(525\) −11.0960 −0.484268
\(526\) −12.3750 −0.539574
\(527\) −23.6250 −1.02912
\(528\) 0.581143 0.0252910
\(529\) −17.0167 −0.739858
\(530\) −8.16969 −0.354869
\(531\) 9.91060 0.430083
\(532\) 67.9481 2.94592
\(533\) 7.09613 0.307368
\(534\) 23.7809 1.02910
\(535\) −2.96642 −0.128249
\(536\) −8.67613 −0.374752
\(537\) 0.131458 0.00567283
\(538\) −29.7956 −1.28458
\(539\) −1.59272 −0.0686032
\(540\) 7.83234 0.337050
\(541\) 6.77689 0.291361 0.145681 0.989332i \(-0.453463\pi\)
0.145681 + 0.989332i \(0.453463\pi\)
\(542\) −16.7985 −0.721557
\(543\) 13.4295 0.576316
\(544\) −14.6396 −0.627669
\(545\) 0.539311 0.0231015
\(546\) 3.56725 0.152664
\(547\) 24.4096 1.04368 0.521840 0.853043i \(-0.325246\pi\)
0.521840 + 0.853043i \(0.325246\pi\)
\(548\) 39.8594 1.70271
\(549\) −4.53019 −0.193344
\(550\) −8.73643 −0.372523
\(551\) 47.5920 2.02749
\(552\) −6.42738 −0.273568
\(553\) 31.0088 1.31863
\(554\) 25.1510 1.06856
\(555\) 0.423551 0.0179787
\(556\) −42.7449 −1.81279
\(557\) −33.8449 −1.43405 −0.717027 0.697046i \(-0.754497\pi\)
−0.717027 + 0.697046i \(0.754497\pi\)
\(558\) 40.5605 1.71706
\(559\) 5.11753 0.216449
\(560\) −1.50461 −0.0635816
\(561\) −2.02251 −0.0853904
\(562\) 70.0325 2.95414
\(563\) 10.8104 0.455605 0.227803 0.973707i \(-0.426846\pi\)
0.227803 + 0.973707i \(0.426846\pi\)
\(564\) 34.9500 1.47166
\(565\) −4.13707 −0.174048
\(566\) 40.5452 1.70424
\(567\) −11.4890 −0.482494
\(568\) −8.22942 −0.345299
\(569\) −19.2197 −0.805731 −0.402866 0.915259i \(-0.631986\pi\)
−0.402866 + 0.915259i \(0.631986\pi\)
\(570\) −6.51009 −0.272678
\(571\) 9.19600 0.384841 0.192420 0.981313i \(-0.438366\pi\)
0.192420 + 0.981313i \(0.438366\pi\)
\(572\) 1.77515 0.0742229
\(573\) 11.3270 0.473191
\(574\) −76.4327 −3.19024
\(575\) 11.5189 0.480372
\(576\) 29.5662 1.23193
\(577\) −6.90358 −0.287400 −0.143700 0.989621i \(-0.545900\pi\)
−0.143700 + 0.989621i \(0.545900\pi\)
\(578\) 15.2165 0.632922
\(579\) 4.50920 0.187396
\(580\) −13.3734 −0.555298
\(581\) −35.7895 −1.48480
\(582\) −16.3755 −0.678785
\(583\) −5.17069 −0.214148
\(584\) −10.2350 −0.423529
\(585\) 0.834655 0.0345087
\(586\) 37.1309 1.53386
\(587\) −29.5311 −1.21888 −0.609440 0.792832i \(-0.708606\pi\)
−0.609440 + 0.792832i \(0.708606\pi\)
\(588\) −5.39961 −0.222676
\(589\) −48.1294 −1.98314
\(590\) 5.22854 0.215256
\(591\) 0.345125 0.0141966
\(592\) −0.929885 −0.0382181
\(593\) −11.0037 −0.451869 −0.225934 0.974143i \(-0.572544\pi\)
−0.225934 + 0.974143i \(0.572544\pi\)
\(594\) 7.84333 0.321816
\(595\) 5.23640 0.214672
\(596\) 76.6855 3.14116
\(597\) 11.3959 0.466403
\(598\) −3.70322 −0.151436
\(599\) −8.35889 −0.341535 −0.170768 0.985311i \(-0.554625\pi\)
−0.170768 + 0.985311i \(0.554625\pi\)
\(600\) −12.3739 −0.505163
\(601\) −2.16462 −0.0882966 −0.0441483 0.999025i \(-0.514057\pi\)
−0.0441483 + 0.999025i \(0.514057\pi\)
\(602\) −55.1211 −2.24657
\(603\) −6.18004 −0.251671
\(604\) −1.49163 −0.0606936
\(605\) −5.59090 −0.227302
\(606\) −31.4203 −1.27636
\(607\) 39.6869 1.61084 0.805421 0.592703i \(-0.201939\pi\)
0.805421 + 0.592703i \(0.201939\pi\)
\(608\) −29.8243 −1.20953
\(609\) −17.0091 −0.689244
\(610\) −2.38999 −0.0967680
\(611\) 8.41281 0.340346
\(612\) 26.4938 1.07095
\(613\) 14.7180 0.594455 0.297228 0.954807i \(-0.403938\pi\)
0.297228 + 0.954807i \(0.403938\pi\)
\(614\) −7.64864 −0.308674
\(615\) 4.62831 0.186632
\(616\) −7.98807 −0.321849
\(617\) 14.7349 0.593207 0.296603 0.955001i \(-0.404146\pi\)
0.296603 + 0.955001i \(0.404146\pi\)
\(618\) 2.23435 0.0898786
\(619\) −30.8137 −1.23851 −0.619254 0.785191i \(-0.712565\pi\)
−0.619254 + 0.785191i \(0.712565\pi\)
\(620\) 13.5244 0.543152
\(621\) −10.3414 −0.414985
\(622\) −33.5379 −1.34475
\(623\) −38.9684 −1.56124
\(624\) 0.474243 0.0189849
\(625\) 20.7218 0.828870
\(626\) −41.6509 −1.66471
\(627\) −4.12031 −0.164549
\(628\) 57.0993 2.27851
\(629\) 3.23621 0.129036
\(630\) −8.99010 −0.358174
\(631\) 6.71897 0.267478 0.133739 0.991017i \(-0.457302\pi\)
0.133739 + 0.991017i \(0.457302\pi\)
\(632\) 34.5801 1.37552
\(633\) 9.61905 0.382323
\(634\) −9.99139 −0.396809
\(635\) −2.19335 −0.0870405
\(636\) −17.5296 −0.695094
\(637\) −1.29974 −0.0514976
\(638\) −13.3921 −0.530199
\(639\) −5.86185 −0.231891
\(640\) 10.7189 0.423703
\(641\) 3.79314 0.149820 0.0749100 0.997190i \(-0.476133\pi\)
0.0749100 + 0.997190i \(0.476133\pi\)
\(642\) −10.0708 −0.397464
\(643\) 36.1832 1.42693 0.713463 0.700693i \(-0.247126\pi\)
0.713463 + 0.700693i \(0.247126\pi\)
\(644\) 25.2098 0.993407
\(645\) 3.33781 0.131426
\(646\) −49.7415 −1.95705
\(647\) −9.03471 −0.355191 −0.177596 0.984104i \(-0.556832\pi\)
−0.177596 + 0.984104i \(0.556832\pi\)
\(648\) −12.8122 −0.503312
\(649\) 3.30920 0.129898
\(650\) −7.12939 −0.279638
\(651\) 17.2012 0.674167
\(652\) −1.06343 −0.0416473
\(653\) 3.65802 0.143149 0.0715746 0.997435i \(-0.477198\pi\)
0.0715746 + 0.997435i \(0.477198\pi\)
\(654\) 1.83093 0.0715951
\(655\) 6.95505 0.271756
\(656\) −10.1612 −0.396729
\(657\) −7.29046 −0.284428
\(658\) −90.6146 −3.53253
\(659\) −2.70982 −0.105559 −0.0527797 0.998606i \(-0.516808\pi\)
−0.0527797 + 0.998606i \(0.516808\pi\)
\(660\) 1.15781 0.0450676
\(661\) −15.5150 −0.603463 −0.301731 0.953393i \(-0.597565\pi\)
−0.301731 + 0.953393i \(0.597565\pi\)
\(662\) −34.5303 −1.34206
\(663\) −1.65047 −0.0640991
\(664\) −39.9114 −1.54886
\(665\) 10.6677 0.413677
\(666\) −5.55608 −0.215294
\(667\) 17.6574 0.683698
\(668\) −4.72561 −0.182839
\(669\) −5.18480 −0.200456
\(670\) −3.26041 −0.125961
\(671\) −1.51265 −0.0583954
\(672\) 10.6590 0.411180
\(673\) −38.7570 −1.49397 −0.746986 0.664839i \(-0.768500\pi\)
−0.746986 + 0.664839i \(0.768500\pi\)
\(674\) −24.8566 −0.957441
\(675\) −19.9091 −0.766300
\(676\) −43.2081 −1.66185
\(677\) −38.0659 −1.46299 −0.731495 0.681847i \(-0.761177\pi\)
−0.731495 + 0.681847i \(0.761177\pi\)
\(678\) −14.0452 −0.539401
\(679\) 26.8336 1.02978
\(680\) 5.83948 0.223934
\(681\) 3.05792 0.117180
\(682\) 13.5434 0.518602
\(683\) 43.9239 1.68070 0.840350 0.542044i \(-0.182350\pi\)
0.840350 + 0.542044i \(0.182350\pi\)
\(684\) 53.9739 2.06374
\(685\) 6.25785 0.239100
\(686\) −34.9626 −1.33488
\(687\) 18.3446 0.699891
\(688\) −7.32799 −0.279377
\(689\) −4.21955 −0.160752
\(690\) −2.41535 −0.0919508
\(691\) 1.59591 0.0607113 0.0303556 0.999539i \(-0.490336\pi\)
0.0303556 + 0.999539i \(0.490336\pi\)
\(692\) −60.5400 −2.30139
\(693\) −5.68993 −0.216143
\(694\) −49.7097 −1.88695
\(695\) −6.71088 −0.254558
\(696\) −18.9680 −0.718982
\(697\) 35.3634 1.33948
\(698\) −44.3833 −1.67993
\(699\) −19.3802 −0.733027
\(700\) 48.5336 1.83440
\(701\) 2.87514 0.108593 0.0542963 0.998525i \(-0.482708\pi\)
0.0542963 + 0.998525i \(0.482708\pi\)
\(702\) 6.40057 0.241574
\(703\) 6.59290 0.248656
\(704\) 9.87233 0.372077
\(705\) 5.48709 0.206656
\(706\) −31.6822 −1.19238
\(707\) 51.4867 1.93636
\(708\) 11.2188 0.421629
\(709\) −32.3808 −1.21609 −0.608043 0.793904i \(-0.708045\pi\)
−0.608043 + 0.793904i \(0.708045\pi\)
\(710\) −3.09254 −0.116061
\(711\) 24.6315 0.923753
\(712\) −43.4564 −1.62860
\(713\) −17.8568 −0.668743
\(714\) 17.7773 0.665299
\(715\) 0.278696 0.0104226
\(716\) −0.574995 −0.0214886
\(717\) 13.7712 0.514294
\(718\) −43.9915 −1.64175
\(719\) −17.3136 −0.645686 −0.322843 0.946452i \(-0.604639\pi\)
−0.322843 + 0.946452i \(0.604639\pi\)
\(720\) −1.19518 −0.0445416
\(721\) −3.66130 −0.136354
\(722\) −57.0392 −2.12278
\(723\) 17.4162 0.647714
\(724\) −58.7405 −2.18307
\(725\) 33.9938 1.26250
\(726\) −18.9808 −0.704444
\(727\) −0.538076 −0.0199562 −0.00997808 0.999950i \(-0.503176\pi\)
−0.00997808 + 0.999950i \(0.503176\pi\)
\(728\) −6.51869 −0.241599
\(729\) 0.834665 0.0309135
\(730\) −3.84623 −0.142355
\(731\) 25.5031 0.943266
\(732\) −5.12818 −0.189543
\(733\) 5.43035 0.200575 0.100287 0.994959i \(-0.468024\pi\)
0.100287 + 0.994959i \(0.468024\pi\)
\(734\) −16.3704 −0.604241
\(735\) −0.847730 −0.0312690
\(736\) −11.0653 −0.407872
\(737\) −2.06355 −0.0760118
\(738\) −60.7135 −2.23489
\(739\) −27.5295 −1.01269 −0.506344 0.862332i \(-0.669003\pi\)
−0.506344 + 0.862332i \(0.669003\pi\)
\(740\) −1.85260 −0.0681031
\(741\) −3.36239 −0.123520
\(742\) 45.4490 1.66848
\(743\) 23.6459 0.867483 0.433742 0.901037i \(-0.357193\pi\)
0.433742 + 0.901037i \(0.357193\pi\)
\(744\) 19.1822 0.703255
\(745\) 12.0395 0.441093
\(746\) −22.6686 −0.829956
\(747\) −28.4290 −1.04016
\(748\) 8.84642 0.323457
\(749\) 16.5025 0.602989
\(750\) −9.58720 −0.350075
\(751\) −3.80026 −0.138674 −0.0693368 0.997593i \(-0.522088\pi\)
−0.0693368 + 0.997593i \(0.522088\pi\)
\(752\) −12.0466 −0.439295
\(753\) −15.8646 −0.578138
\(754\) −10.9287 −0.397999
\(755\) −0.234183 −0.00852280
\(756\) −43.5722 −1.58471
\(757\) 40.1975 1.46100 0.730502 0.682910i \(-0.239286\pi\)
0.730502 + 0.682910i \(0.239286\pi\)
\(758\) −19.6244 −0.712791
\(759\) −1.52870 −0.0554884
\(760\) 11.8963 0.431526
\(761\) −50.5236 −1.83148 −0.915739 0.401774i \(-0.868394\pi\)
−0.915739 + 0.401774i \(0.868394\pi\)
\(762\) −7.44631 −0.269751
\(763\) −3.00025 −0.108616
\(764\) −49.5440 −1.79244
\(765\) 4.15948 0.150386
\(766\) −61.4966 −2.22196
\(767\) 2.70048 0.0975089
\(768\) 16.9039 0.609968
\(769\) −37.9780 −1.36952 −0.684760 0.728769i \(-0.740093\pi\)
−0.684760 + 0.728769i \(0.740093\pi\)
\(770\) −3.00184 −0.108179
\(771\) 0.981295 0.0353405
\(772\) −19.7232 −0.709852
\(773\) −42.7526 −1.53770 −0.768851 0.639427i \(-0.779171\pi\)
−0.768851 + 0.639427i \(0.779171\pi\)
\(774\) −43.7849 −1.57381
\(775\) −34.3777 −1.23488
\(776\) 29.9240 1.07421
\(777\) −2.35626 −0.0845305
\(778\) 54.9983 1.97178
\(779\) 72.0432 2.58122
\(780\) 0.944832 0.0338304
\(781\) −1.95730 −0.0700378
\(782\) −18.4549 −0.659946
\(783\) −30.5187 −1.09065
\(784\) 1.86115 0.0664696
\(785\) 8.96450 0.319957
\(786\) 23.6120 0.842214
\(787\) −17.4328 −0.621413 −0.310706 0.950506i \(-0.600566\pi\)
−0.310706 + 0.950506i \(0.600566\pi\)
\(788\) −1.50957 −0.0537763
\(789\) −4.16875 −0.148411
\(790\) 12.9949 0.462336
\(791\) 23.0150 0.818320
\(792\) −6.34524 −0.225468
\(793\) −1.23441 −0.0438350
\(794\) 56.7740 2.01483
\(795\) −2.75212 −0.0976076
\(796\) −49.8455 −1.76673
\(797\) −22.6239 −0.801380 −0.400690 0.916214i \(-0.631230\pi\)
−0.400690 + 0.916214i \(0.631230\pi\)
\(798\) 36.2164 1.28205
\(799\) 41.9250 1.48320
\(800\) −21.3027 −0.753165
\(801\) −30.9542 −1.09371
\(802\) −49.6228 −1.75224
\(803\) −2.43432 −0.0859054
\(804\) −6.99582 −0.246724
\(805\) 3.95790 0.139498
\(806\) 11.0521 0.389293
\(807\) −10.0372 −0.353327
\(808\) 57.4165 2.01990
\(809\) 36.3020 1.27631 0.638155 0.769908i \(-0.279698\pi\)
0.638155 + 0.769908i \(0.279698\pi\)
\(810\) −4.81471 −0.169172
\(811\) 47.2757 1.66007 0.830036 0.557709i \(-0.188320\pi\)
0.830036 + 0.557709i \(0.188320\pi\)
\(812\) 74.3975 2.61084
\(813\) −5.65890 −0.198466
\(814\) −1.85521 −0.0650249
\(815\) −0.166957 −0.00584826
\(816\) 2.36338 0.0827348
\(817\) 51.9555 1.81769
\(818\) 36.5225 1.27698
\(819\) −4.64328 −0.162249
\(820\) −20.2442 −0.706957
\(821\) −13.7154 −0.478670 −0.239335 0.970937i \(-0.576929\pi\)
−0.239335 + 0.970937i \(0.576929\pi\)
\(822\) 21.2451 0.741007
\(823\) −51.4044 −1.79185 −0.895923 0.444210i \(-0.853485\pi\)
−0.895923 + 0.444210i \(0.853485\pi\)
\(824\) −4.08297 −0.142237
\(825\) −2.94304 −0.102463
\(826\) −29.0870 −1.01207
\(827\) −12.0718 −0.419777 −0.209888 0.977725i \(-0.567310\pi\)
−0.209888 + 0.977725i \(0.567310\pi\)
\(828\) 20.0252 0.695923
\(829\) 34.6406 1.20312 0.601558 0.798829i \(-0.294547\pi\)
0.601558 + 0.798829i \(0.294547\pi\)
\(830\) −14.9983 −0.520599
\(831\) 8.47261 0.293912
\(832\) 8.05634 0.279303
\(833\) −6.47722 −0.224422
\(834\) −22.7831 −0.788914
\(835\) −0.741913 −0.0256750
\(836\) 18.0222 0.623310
\(837\) 30.8633 1.06679
\(838\) −51.7505 −1.78769
\(839\) −23.2399 −0.802329 −0.401164 0.916006i \(-0.631394\pi\)
−0.401164 + 0.916006i \(0.631394\pi\)
\(840\) −4.25168 −0.146697
\(841\) 23.1093 0.796873
\(842\) 23.7079 0.817027
\(843\) 23.5918 0.812545
\(844\) −42.0736 −1.44823
\(845\) −6.78361 −0.233363
\(846\) −71.9788 −2.47468
\(847\) 31.1028 1.06871
\(848\) 6.04214 0.207488
\(849\) 13.6584 0.468757
\(850\) −35.5291 −1.21864
\(851\) 2.44607 0.0838503
\(852\) −6.63563 −0.227333
\(853\) 13.4371 0.460079 0.230039 0.973181i \(-0.426114\pi\)
0.230039 + 0.973181i \(0.426114\pi\)
\(854\) 13.2958 0.454973
\(855\) 8.47380 0.289798
\(856\) 18.4031 0.629006
\(857\) 12.0581 0.411897 0.205949 0.978563i \(-0.433972\pi\)
0.205949 + 0.978563i \(0.433972\pi\)
\(858\) 0.946158 0.0323013
\(859\) 26.2382 0.895236 0.447618 0.894225i \(-0.352272\pi\)
0.447618 + 0.894225i \(0.352272\pi\)
\(860\) −14.5995 −0.497839
\(861\) −25.7478 −0.877484
\(862\) 53.7348 1.83022
\(863\) −14.0937 −0.479756 −0.239878 0.970803i \(-0.577108\pi\)
−0.239878 + 0.970803i \(0.577108\pi\)
\(864\) 19.1250 0.650646
\(865\) −9.50468 −0.323169
\(866\) 26.3777 0.896352
\(867\) 5.12596 0.174087
\(868\) −75.2377 −2.55373
\(869\) 8.22459 0.279000
\(870\) −7.12801 −0.241662
\(871\) −1.68396 −0.0570590
\(872\) −3.34579 −0.113303
\(873\) 21.3150 0.721403
\(874\) −37.5968 −1.27173
\(875\) 15.7100 0.531096
\(876\) −8.25282 −0.278837
\(877\) −6.93409 −0.234148 −0.117074 0.993123i \(-0.537351\pi\)
−0.117074 + 0.993123i \(0.537351\pi\)
\(878\) 28.5602 0.963862
\(879\) 12.5083 0.421894
\(880\) −0.399076 −0.0134528
\(881\) −43.8606 −1.47770 −0.738851 0.673868i \(-0.764631\pi\)
−0.738851 + 0.673868i \(0.764631\pi\)
\(882\) 11.1204 0.374443
\(883\) 12.5169 0.421227 0.210614 0.977569i \(-0.432454\pi\)
0.210614 + 0.977569i \(0.432454\pi\)
\(884\) 7.21915 0.242806
\(885\) 1.76134 0.0592067
\(886\) −82.4195 −2.76894
\(887\) −41.3722 −1.38914 −0.694571 0.719424i \(-0.744406\pi\)
−0.694571 + 0.719424i \(0.744406\pi\)
\(888\) −2.62763 −0.0881776
\(889\) 12.2019 0.409237
\(890\) −16.3305 −0.547400
\(891\) −3.04728 −0.102088
\(892\) 22.6782 0.759323
\(893\) 85.4107 2.85816
\(894\) 40.8734 1.36701
\(895\) −0.0902732 −0.00301750
\(896\) −59.6307 −1.99212
\(897\) −1.24750 −0.0416529
\(898\) −38.9610 −1.30015
\(899\) −52.6978 −1.75757
\(900\) 38.5522 1.28507
\(901\) −21.0280 −0.700546
\(902\) −20.2726 −0.675003
\(903\) −18.5686 −0.617925
\(904\) 25.6657 0.853627
\(905\) −9.22216 −0.306555
\(906\) −0.795040 −0.0264134
\(907\) −40.6815 −1.35081 −0.675404 0.737448i \(-0.736031\pi\)
−0.675404 + 0.737448i \(0.736031\pi\)
\(908\) −13.3753 −0.443875
\(909\) 40.8980 1.35650
\(910\) −2.44966 −0.0812055
\(911\) −50.7882 −1.68269 −0.841344 0.540500i \(-0.818235\pi\)
−0.841344 + 0.540500i \(0.818235\pi\)
\(912\) 4.81474 0.159432
\(913\) −9.49260 −0.314159
\(914\) 17.2601 0.570913
\(915\) −0.805116 −0.0266163
\(916\) −80.2390 −2.65117
\(917\) −38.6918 −1.27771
\(918\) 31.8971 1.05276
\(919\) 10.6720 0.352036 0.176018 0.984387i \(-0.443678\pi\)
0.176018 + 0.984387i \(0.443678\pi\)
\(920\) 4.41374 0.145517
\(921\) −2.57659 −0.0849016
\(922\) 39.5727 1.30326
\(923\) −1.59726 −0.0525745
\(924\) −6.44102 −0.211894
\(925\) 4.70914 0.154836
\(926\) −38.8823 −1.27775
\(927\) −2.90832 −0.0955217
\(928\) −32.6551 −1.07196
\(929\) −23.5478 −0.772580 −0.386290 0.922377i \(-0.626244\pi\)
−0.386290 + 0.922377i \(0.626244\pi\)
\(930\) 7.20851 0.236376
\(931\) −13.1956 −0.432467
\(932\) 84.7688 2.77669
\(933\) −11.2979 −0.369877
\(934\) −78.9587 −2.58361
\(935\) 1.38887 0.0454210
\(936\) −5.17805 −0.169250
\(937\) −36.1802 −1.18196 −0.590979 0.806687i \(-0.701258\pi\)
−0.590979 + 0.806687i \(0.701258\pi\)
\(938\) 18.1380 0.592228
\(939\) −14.0309 −0.457882
\(940\) −24.0004 −0.782807
\(941\) 31.0330 1.01165 0.505824 0.862637i \(-0.331189\pi\)
0.505824 + 0.862637i \(0.331189\pi\)
\(942\) 30.4340 0.991593
\(943\) 26.7292 0.870423
\(944\) −3.86693 −0.125858
\(945\) −6.84076 −0.222530
\(946\) −14.6200 −0.475337
\(947\) −36.6051 −1.18950 −0.594752 0.803909i \(-0.702750\pi\)
−0.594752 + 0.803909i \(0.702750\pi\)
\(948\) 27.8829 0.905595
\(949\) −1.98654 −0.0644857
\(950\) −72.3808 −2.34834
\(951\) −3.36579 −0.109143
\(952\) −32.4857 −1.05287
\(953\) 15.1993 0.492354 0.246177 0.969225i \(-0.420826\pi\)
0.246177 + 0.969225i \(0.420826\pi\)
\(954\) 36.1019 1.16884
\(955\) −7.77832 −0.251701
\(956\) −60.2349 −1.94814
\(957\) −4.51140 −0.145833
\(958\) −9.65986 −0.312096
\(959\) −34.8132 −1.12418
\(960\) 5.25459 0.169591
\(961\) 22.2928 0.719124
\(962\) −1.51395 −0.0488116
\(963\) 13.1086 0.422419
\(964\) −76.1780 −2.45353
\(965\) −3.09650 −0.0996800
\(966\) 13.4369 0.432324
\(967\) −20.9316 −0.673115 −0.336558 0.941663i \(-0.609263\pi\)
−0.336558 + 0.941663i \(0.609263\pi\)
\(968\) 34.6849 1.11482
\(969\) −16.7564 −0.538293
\(970\) 11.2452 0.361061
\(971\) 17.3852 0.557918 0.278959 0.960303i \(-0.410011\pi\)
0.278959 + 0.960303i \(0.410011\pi\)
\(972\) −53.8995 −1.72883
\(973\) 37.3334 1.19685
\(974\) 84.8638 2.71921
\(975\) −2.40167 −0.0769151
\(976\) 1.76759 0.0565793
\(977\) −12.7447 −0.407739 −0.203870 0.978998i \(-0.565352\pi\)
−0.203870 + 0.978998i \(0.565352\pi\)
\(978\) −0.566811 −0.0181246
\(979\) −10.3358 −0.330333
\(980\) 3.70795 0.118446
\(981\) −2.38322 −0.0760902
\(982\) −78.2505 −2.49707
\(983\) 19.6734 0.627482 0.313741 0.949509i \(-0.398418\pi\)
0.313741 + 0.949509i \(0.398418\pi\)
\(984\) −28.7132 −0.915344
\(985\) −0.237000 −0.00755146
\(986\) −54.4628 −1.73445
\(987\) −30.5253 −0.971631
\(988\) 14.7070 0.467893
\(989\) 19.2764 0.612952
\(990\) −2.38448 −0.0757838
\(991\) 22.5825 0.717356 0.358678 0.933461i \(-0.383228\pi\)
0.358678 + 0.933461i \(0.383228\pi\)
\(992\) 33.0239 1.04851
\(993\) −11.6322 −0.369137
\(994\) 17.2042 0.545683
\(995\) −7.82566 −0.248090
\(996\) −32.1817 −1.01972
\(997\) 4.04191 0.128008 0.0640042 0.997950i \(-0.479613\pi\)
0.0640042 + 0.997950i \(0.479613\pi\)
\(998\) 39.8915 1.26274
\(999\) −4.22774 −0.133760
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 4033.2.a.e.1.10 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.e.1.10 82 1.1 even 1 trivial