Properties

Label 8026.2.a.a.1.2
Level $8026$
Weight $2$
Character 8026.1
Self dual yes
Analytic conductor $64.088$
Analytic rank $1$
Dimension $71$
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] = [8026,2,Mod(1,8026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8026, 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("8026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8026 = 2 \cdot 4013 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8026.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: \(64.0879326623\)
Analytic rank: \(1\)
Dimension: \(71\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 8026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.18837 q^{3} +1.00000 q^{4} +1.76077 q^{5} -3.18837 q^{6} -1.56118 q^{7} +1.00000 q^{8} +7.16572 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.18837 q^{3} +1.00000 q^{4} +1.76077 q^{5} -3.18837 q^{6} -1.56118 q^{7} +1.00000 q^{8} +7.16572 q^{9} +1.76077 q^{10} -1.60222 q^{11} -3.18837 q^{12} -5.05138 q^{13} -1.56118 q^{14} -5.61398 q^{15} +1.00000 q^{16} -2.05056 q^{17} +7.16572 q^{18} +4.32464 q^{19} +1.76077 q^{20} +4.97763 q^{21} -1.60222 q^{22} +7.66530 q^{23} -3.18837 q^{24} -1.89971 q^{25} -5.05138 q^{26} -13.2819 q^{27} -1.56118 q^{28} -0.340949 q^{29} -5.61398 q^{30} -2.72687 q^{31} +1.00000 q^{32} +5.10849 q^{33} -2.05056 q^{34} -2.74887 q^{35} +7.16572 q^{36} -0.830414 q^{37} +4.32464 q^{38} +16.1057 q^{39} +1.76077 q^{40} +6.18698 q^{41} +4.97763 q^{42} +0.936856 q^{43} -1.60222 q^{44} +12.6172 q^{45} +7.66530 q^{46} +7.09986 q^{47} -3.18837 q^{48} -4.56271 q^{49} -1.89971 q^{50} +6.53796 q^{51} -5.05138 q^{52} +7.47223 q^{53} -13.2819 q^{54} -2.82114 q^{55} -1.56118 q^{56} -13.7886 q^{57} -0.340949 q^{58} -8.64683 q^{59} -5.61398 q^{60} -10.0752 q^{61} -2.72687 q^{62} -11.1870 q^{63} +1.00000 q^{64} -8.89429 q^{65} +5.10849 q^{66} +11.2757 q^{67} -2.05056 q^{68} -24.4398 q^{69} -2.74887 q^{70} -3.38843 q^{71} +7.16572 q^{72} -7.23868 q^{73} -0.830414 q^{74} +6.05697 q^{75} +4.32464 q^{76} +2.50136 q^{77} +16.1057 q^{78} -1.04892 q^{79} +1.76077 q^{80} +20.8504 q^{81} +6.18698 q^{82} +1.10682 q^{83} +4.97763 q^{84} -3.61056 q^{85} +0.936856 q^{86} +1.08707 q^{87} -1.60222 q^{88} -12.5594 q^{89} +12.6172 q^{90} +7.88612 q^{91} +7.66530 q^{92} +8.69427 q^{93} +7.09986 q^{94} +7.61468 q^{95} -3.18837 q^{96} +3.65708 q^{97} -4.56271 q^{98} -11.4811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 71 q + 71 q^{2} - 9 q^{3} + 71 q^{4} - 34 q^{5} - 9 q^{6} - 19 q^{7} + 71 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 71 q + 71 q^{2} - 9 q^{3} + 71 q^{4} - 34 q^{5} - 9 q^{6} - 19 q^{7} + 71 q^{8} + 34 q^{9} - 34 q^{10} - 37 q^{11} - 9 q^{12} - 62 q^{13} - 19 q^{14} - 29 q^{15} + 71 q^{16} - 52 q^{17} + 34 q^{18} - 30 q^{19} - 34 q^{20} - 51 q^{21} - 37 q^{22} - 45 q^{23} - 9 q^{24} + 27 q^{25} - 62 q^{26} - 27 q^{27} - 19 q^{28} - 55 q^{29} - 29 q^{30} - 61 q^{31} + 71 q^{32} - 73 q^{33} - 52 q^{34} - 33 q^{35} + 34 q^{36} - 43 q^{37} - 30 q^{38} - 40 q^{39} - 34 q^{40} - 87 q^{41} - 51 q^{42} - 4 q^{43} - 37 q^{44} - 81 q^{45} - 45 q^{46} - 89 q^{47} - 9 q^{48} - 2 q^{49} + 27 q^{50} - 19 q^{51} - 62 q^{52} - 50 q^{53} - 27 q^{54} - 66 q^{55} - 19 q^{56} - 45 q^{57} - 55 q^{58} - 118 q^{59} - 29 q^{60} - 92 q^{61} - 61 q^{62} - 54 q^{63} + 71 q^{64} - 51 q^{65} - 73 q^{66} - 17 q^{67} - 52 q^{68} - 89 q^{69} - 33 q^{70} - 95 q^{71} + 34 q^{72} - 114 q^{73} - 43 q^{74} - 38 q^{75} - 30 q^{76} - 73 q^{77} - 40 q^{78} - 47 q^{79} - 34 q^{80} - 57 q^{81} - 87 q^{82} - 68 q^{83} - 51 q^{84} - 67 q^{85} - 4 q^{86} - 55 q^{87} - 37 q^{88} - 150 q^{89} - 81 q^{90} - 23 q^{91} - 45 q^{92} - 59 q^{93} - 89 q^{94} - 47 q^{95} - 9 q^{96} - 97 q^{97} - 2 q^{98} - 57 q^{99}+O(q^{100}) \) 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
\(3\) −3.18837 −1.84081 −0.920404 0.390969i \(-0.872140\pi\)
−0.920404 + 0.390969i \(0.872140\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.76077 0.787438 0.393719 0.919231i \(-0.371188\pi\)
0.393719 + 0.919231i \(0.371188\pi\)
\(6\) −3.18837 −1.30165
\(7\) −1.56118 −0.590071 −0.295036 0.955486i \(-0.595332\pi\)
−0.295036 + 0.955486i \(0.595332\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.16572 2.38857
\(10\) 1.76077 0.556803
\(11\) −1.60222 −0.483089 −0.241544 0.970390i \(-0.577654\pi\)
−0.241544 + 0.970390i \(0.577654\pi\)
\(12\) −3.18837 −0.920404
\(13\) −5.05138 −1.40100 −0.700500 0.713652i \(-0.747040\pi\)
−0.700500 + 0.713652i \(0.747040\pi\)
\(14\) −1.56118 −0.417243
\(15\) −5.61398 −1.44952
\(16\) 1.00000 0.250000
\(17\) −2.05056 −0.497335 −0.248667 0.968589i \(-0.579993\pi\)
−0.248667 + 0.968589i \(0.579993\pi\)
\(18\) 7.16572 1.68898
\(19\) 4.32464 0.992141 0.496070 0.868282i \(-0.334776\pi\)
0.496070 + 0.868282i \(0.334776\pi\)
\(20\) 1.76077 0.393719
\(21\) 4.97763 1.08621
\(22\) −1.60222 −0.341595
\(23\) 7.66530 1.59832 0.799162 0.601115i \(-0.205277\pi\)
0.799162 + 0.601115i \(0.205277\pi\)
\(24\) −3.18837 −0.650824
\(25\) −1.89971 −0.379941
\(26\) −5.05138 −0.990657
\(27\) −13.2819 −2.55610
\(28\) −1.56118 −0.295036
\(29\) −0.340949 −0.0633127 −0.0316564 0.999499i \(-0.510078\pi\)
−0.0316564 + 0.999499i \(0.510078\pi\)
\(30\) −5.61398 −1.02497
\(31\) −2.72687 −0.489760 −0.244880 0.969553i \(-0.578749\pi\)
−0.244880 + 0.969553i \(0.578749\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.10849 0.889274
\(34\) −2.05056 −0.351669
\(35\) −2.74887 −0.464645
\(36\) 7.16572 1.19429
\(37\) −0.830414 −0.136519 −0.0682596 0.997668i \(-0.521745\pi\)
−0.0682596 + 0.997668i \(0.521745\pi\)
\(38\) 4.32464 0.701549
\(39\) 16.1057 2.57897
\(40\) 1.76077 0.278401
\(41\) 6.18698 0.966244 0.483122 0.875553i \(-0.339503\pi\)
0.483122 + 0.875553i \(0.339503\pi\)
\(42\) 4.97763 0.768065
\(43\) 0.936856 0.142869 0.0714346 0.997445i \(-0.477242\pi\)
0.0714346 + 0.997445i \(0.477242\pi\)
\(44\) −1.60222 −0.241544
\(45\) 12.6172 1.88085
\(46\) 7.66530 1.13019
\(47\) 7.09986 1.03562 0.517810 0.855495i \(-0.326747\pi\)
0.517810 + 0.855495i \(0.326747\pi\)
\(48\) −3.18837 −0.460202
\(49\) −4.56271 −0.651816
\(50\) −1.89971 −0.268659
\(51\) 6.53796 0.915498
\(52\) −5.05138 −0.700500
\(53\) 7.47223 1.02639 0.513195 0.858272i \(-0.328462\pi\)
0.513195 + 0.858272i \(0.328462\pi\)
\(54\) −13.2819 −1.80743
\(55\) −2.82114 −0.380402
\(56\) −1.56118 −0.208622
\(57\) −13.7886 −1.82634
\(58\) −0.340949 −0.0447688
\(59\) −8.64683 −1.12572 −0.562861 0.826552i \(-0.690299\pi\)
−0.562861 + 0.826552i \(0.690299\pi\)
\(60\) −5.61398 −0.724761
\(61\) −10.0752 −1.29000 −0.644999 0.764184i \(-0.723142\pi\)
−0.644999 + 0.764184i \(0.723142\pi\)
\(62\) −2.72687 −0.346313
\(63\) −11.1870 −1.40943
\(64\) 1.00000 0.125000
\(65\) −8.89429 −1.10320
\(66\) 5.10849 0.628811
\(67\) 11.2757 1.37755 0.688775 0.724975i \(-0.258149\pi\)
0.688775 + 0.724975i \(0.258149\pi\)
\(68\) −2.05056 −0.248667
\(69\) −24.4398 −2.94221
\(70\) −2.74887 −0.328553
\(71\) −3.38843 −0.402133 −0.201066 0.979578i \(-0.564441\pi\)
−0.201066 + 0.979578i \(0.564441\pi\)
\(72\) 7.16572 0.844488
\(73\) −7.23868 −0.847223 −0.423611 0.905844i \(-0.639238\pi\)
−0.423611 + 0.905844i \(0.639238\pi\)
\(74\) −0.830414 −0.0965337
\(75\) 6.05697 0.699399
\(76\) 4.32464 0.496070
\(77\) 2.50136 0.285057
\(78\) 16.1057 1.82361
\(79\) −1.04892 −0.118012 −0.0590062 0.998258i \(-0.518793\pi\)
−0.0590062 + 0.998258i \(0.518793\pi\)
\(80\) 1.76077 0.196860
\(81\) 20.8504 2.31671
\(82\) 6.18698 0.683238
\(83\) 1.10682 0.121489 0.0607447 0.998153i \(-0.480652\pi\)
0.0607447 + 0.998153i \(0.480652\pi\)
\(84\) 4.97763 0.543104
\(85\) −3.61056 −0.391620
\(86\) 0.936856 0.101024
\(87\) 1.08707 0.116547
\(88\) −1.60222 −0.170798
\(89\) −12.5594 −1.33129 −0.665645 0.746268i \(-0.731844\pi\)
−0.665645 + 0.746268i \(0.731844\pi\)
\(90\) 12.6172 1.32996
\(91\) 7.88612 0.826690
\(92\) 7.66530 0.799162
\(93\) 8.69427 0.901554
\(94\) 7.09986 0.732294
\(95\) 7.61468 0.781249
\(96\) −3.18837 −0.325412
\(97\) 3.65708 0.371320 0.185660 0.982614i \(-0.440558\pi\)
0.185660 + 0.982614i \(0.440558\pi\)
\(98\) −4.56271 −0.460904
\(99\) −11.4811 −1.15389
\(100\) −1.89971 −0.189971
\(101\) −9.15937 −0.911392 −0.455696 0.890136i \(-0.650610\pi\)
−0.455696 + 0.890136i \(0.650610\pi\)
\(102\) 6.53796 0.647355
\(103\) −0.368513 −0.0363106 −0.0181553 0.999835i \(-0.505779\pi\)
−0.0181553 + 0.999835i \(0.505779\pi\)
\(104\) −5.05138 −0.495329
\(105\) 8.76444 0.855321
\(106\) 7.47223 0.725767
\(107\) 13.5248 1.30749 0.653746 0.756714i \(-0.273196\pi\)
0.653746 + 0.756714i \(0.273196\pi\)
\(108\) −13.2819 −1.27805
\(109\) −2.12016 −0.203074 −0.101537 0.994832i \(-0.532376\pi\)
−0.101537 + 0.994832i \(0.532376\pi\)
\(110\) −2.82114 −0.268985
\(111\) 2.64767 0.251306
\(112\) −1.56118 −0.147518
\(113\) −13.0833 −1.23078 −0.615389 0.788224i \(-0.711001\pi\)
−0.615389 + 0.788224i \(0.711001\pi\)
\(114\) −13.7886 −1.29142
\(115\) 13.4968 1.25858
\(116\) −0.340949 −0.0316564
\(117\) −36.1968 −3.34639
\(118\) −8.64683 −0.796006
\(119\) 3.20130 0.293463
\(120\) −5.61398 −0.512483
\(121\) −8.43288 −0.766625
\(122\) −10.0752 −0.912166
\(123\) −19.7264 −1.77867
\(124\) −2.72687 −0.244880
\(125\) −12.1488 −1.08662
\(126\) −11.1870 −0.996617
\(127\) −9.61584 −0.853268 −0.426634 0.904424i \(-0.640301\pi\)
−0.426634 + 0.904424i \(0.640301\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.98705 −0.262995
\(130\) −8.89429 −0.780081
\(131\) 10.6120 0.927174 0.463587 0.886051i \(-0.346562\pi\)
0.463587 + 0.886051i \(0.346562\pi\)
\(132\) 5.10849 0.444637
\(133\) −6.75155 −0.585434
\(134\) 11.2757 0.974075
\(135\) −23.3863 −2.01277
\(136\) −2.05056 −0.175834
\(137\) 18.4358 1.57508 0.787538 0.616265i \(-0.211355\pi\)
0.787538 + 0.616265i \(0.211355\pi\)
\(138\) −24.4398 −2.08046
\(139\) 12.3160 1.04463 0.522317 0.852751i \(-0.325068\pi\)
0.522317 + 0.852751i \(0.325068\pi\)
\(140\) −2.74887 −0.232322
\(141\) −22.6370 −1.90638
\(142\) −3.38843 −0.284351
\(143\) 8.09344 0.676808
\(144\) 7.16572 0.597143
\(145\) −0.600332 −0.0498548
\(146\) −7.23868 −0.599077
\(147\) 14.5476 1.19987
\(148\) −0.830414 −0.0682596
\(149\) 8.96690 0.734597 0.367298 0.930103i \(-0.380283\pi\)
0.367298 + 0.930103i \(0.380283\pi\)
\(150\) 6.05697 0.494550
\(151\) −11.0030 −0.895412 −0.447706 0.894181i \(-0.647759\pi\)
−0.447706 + 0.894181i \(0.647759\pi\)
\(152\) 4.32464 0.350775
\(153\) −14.6938 −1.18792
\(154\) 2.50136 0.201566
\(155\) −4.80137 −0.385656
\(156\) 16.1057 1.28949
\(157\) −9.92152 −0.791824 −0.395912 0.918289i \(-0.629571\pi\)
−0.395912 + 0.918289i \(0.629571\pi\)
\(158\) −1.04892 −0.0834474
\(159\) −23.8243 −1.88939
\(160\) 1.76077 0.139201
\(161\) −11.9669 −0.943125
\(162\) 20.8504 1.63816
\(163\) 0.00712968 0.000558439 0 0.000279220 1.00000i \(-0.499911\pi\)
0.000279220 1.00000i \(0.499911\pi\)
\(164\) 6.18698 0.483122
\(165\) 8.99485 0.700248
\(166\) 1.10682 0.0859060
\(167\) 15.8324 1.22515 0.612573 0.790414i \(-0.290135\pi\)
0.612573 + 0.790414i \(0.290135\pi\)
\(168\) 4.97763 0.384032
\(169\) 12.5164 0.962803
\(170\) −3.61056 −0.276917
\(171\) 30.9892 2.36980
\(172\) 0.936856 0.0714346
\(173\) 11.3445 0.862503 0.431251 0.902232i \(-0.358072\pi\)
0.431251 + 0.902232i \(0.358072\pi\)
\(174\) 1.08707 0.0824108
\(175\) 2.96579 0.224192
\(176\) −1.60222 −0.120772
\(177\) 27.5693 2.07224
\(178\) −12.5594 −0.941364
\(179\) 11.5861 0.865987 0.432993 0.901397i \(-0.357457\pi\)
0.432993 + 0.901397i \(0.357457\pi\)
\(180\) 12.6172 0.940427
\(181\) −3.68403 −0.273832 −0.136916 0.990583i \(-0.543719\pi\)
−0.136916 + 0.990583i \(0.543719\pi\)
\(182\) 7.88612 0.584558
\(183\) 32.1235 2.37464
\(184\) 7.66530 0.565093
\(185\) −1.46216 −0.107500
\(186\) 8.69427 0.637495
\(187\) 3.28546 0.240257
\(188\) 7.09986 0.517810
\(189\) 20.7354 1.50828
\(190\) 7.61468 0.552427
\(191\) −2.77288 −0.200639 −0.100319 0.994955i \(-0.531986\pi\)
−0.100319 + 0.994955i \(0.531986\pi\)
\(192\) −3.18837 −0.230101
\(193\) −4.13574 −0.297697 −0.148849 0.988860i \(-0.547557\pi\)
−0.148849 + 0.988860i \(0.547557\pi\)
\(194\) 3.65708 0.262563
\(195\) 28.3583 2.03078
\(196\) −4.56271 −0.325908
\(197\) −4.84442 −0.345150 −0.172575 0.984996i \(-0.555209\pi\)
−0.172575 + 0.984996i \(0.555209\pi\)
\(198\) −11.4811 −0.815926
\(199\) −12.3355 −0.874442 −0.437221 0.899354i \(-0.644037\pi\)
−0.437221 + 0.899354i \(0.644037\pi\)
\(200\) −1.89971 −0.134330
\(201\) −35.9512 −2.53581
\(202\) −9.15937 −0.644451
\(203\) 0.532284 0.0373590
\(204\) 6.53796 0.457749
\(205\) 10.8938 0.760857
\(206\) −0.368513 −0.0256755
\(207\) 54.9274 3.81772
\(208\) −5.05138 −0.350250
\(209\) −6.92904 −0.479292
\(210\) 8.76444 0.604803
\(211\) −13.4414 −0.925344 −0.462672 0.886530i \(-0.653109\pi\)
−0.462672 + 0.886530i \(0.653109\pi\)
\(212\) 7.47223 0.513195
\(213\) 10.8036 0.740250
\(214\) 13.5248 0.924537
\(215\) 1.64958 0.112501
\(216\) −13.2819 −0.903717
\(217\) 4.25714 0.288993
\(218\) −2.12016 −0.143595
\(219\) 23.0796 1.55957
\(220\) −2.82114 −0.190201
\(221\) 10.3582 0.696767
\(222\) 2.64767 0.177700
\(223\) −17.7707 −1.19001 −0.595007 0.803720i \(-0.702851\pi\)
−0.595007 + 0.803720i \(0.702851\pi\)
\(224\) −1.56118 −0.104311
\(225\) −13.6128 −0.907518
\(226\) −13.0833 −0.870291
\(227\) 13.7933 0.915491 0.457745 0.889083i \(-0.348657\pi\)
0.457745 + 0.889083i \(0.348657\pi\)
\(228\) −13.7886 −0.913170
\(229\) −13.2868 −0.878018 −0.439009 0.898483i \(-0.644670\pi\)
−0.439009 + 0.898483i \(0.644670\pi\)
\(230\) 13.4968 0.889952
\(231\) −7.97528 −0.524735
\(232\) −0.340949 −0.0223844
\(233\) −15.3937 −1.00847 −0.504236 0.863566i \(-0.668226\pi\)
−0.504236 + 0.863566i \(0.668226\pi\)
\(234\) −36.1968 −2.36626
\(235\) 12.5012 0.815487
\(236\) −8.64683 −0.562861
\(237\) 3.34434 0.217238
\(238\) 3.20130 0.207510
\(239\) −1.03722 −0.0670921 −0.0335460 0.999437i \(-0.510680\pi\)
−0.0335460 + 0.999437i \(0.510680\pi\)
\(240\) −5.61398 −0.362381
\(241\) −16.3323 −1.05206 −0.526029 0.850467i \(-0.676320\pi\)
−0.526029 + 0.850467i \(0.676320\pi\)
\(242\) −8.43288 −0.542086
\(243\) −26.6332 −1.70852
\(244\) −10.0752 −0.644999
\(245\) −8.03386 −0.513265
\(246\) −19.7264 −1.25771
\(247\) −21.8454 −1.38999
\(248\) −2.72687 −0.173156
\(249\) −3.52896 −0.223639
\(250\) −12.1488 −0.768355
\(251\) −19.2479 −1.21492 −0.607459 0.794351i \(-0.707811\pi\)
−0.607459 + 0.794351i \(0.707811\pi\)
\(252\) −11.1870 −0.704714
\(253\) −12.2815 −0.772133
\(254\) −9.61584 −0.603351
\(255\) 11.5118 0.720898
\(256\) 1.00000 0.0625000
\(257\) −11.2496 −0.701733 −0.350866 0.936426i \(-0.614113\pi\)
−0.350866 + 0.936426i \(0.614113\pi\)
\(258\) −2.98705 −0.185965
\(259\) 1.29643 0.0805561
\(260\) −8.89429 −0.551601
\(261\) −2.44315 −0.151227
\(262\) 10.6120 0.655611
\(263\) 10.3469 0.638014 0.319007 0.947752i \(-0.396651\pi\)
0.319007 + 0.947752i \(0.396651\pi\)
\(264\) 5.10849 0.314406
\(265\) 13.1568 0.808219
\(266\) −6.75155 −0.413964
\(267\) 40.0440 2.45065
\(268\) 11.2757 0.688775
\(269\) 26.3732 1.60801 0.804003 0.594626i \(-0.202700\pi\)
0.804003 + 0.594626i \(0.202700\pi\)
\(270\) −23.3863 −1.42324
\(271\) −18.2888 −1.11097 −0.555484 0.831527i \(-0.687467\pi\)
−0.555484 + 0.831527i \(0.687467\pi\)
\(272\) −2.05056 −0.124334
\(273\) −25.1439 −1.52178
\(274\) 18.4358 1.11375
\(275\) 3.04376 0.183545
\(276\) −24.4398 −1.47110
\(277\) −1.93375 −0.116188 −0.0580939 0.998311i \(-0.518502\pi\)
−0.0580939 + 0.998311i \(0.518502\pi\)
\(278\) 12.3160 0.738668
\(279\) −19.5400 −1.16983
\(280\) −2.74887 −0.164277
\(281\) −24.7920 −1.47897 −0.739484 0.673174i \(-0.764930\pi\)
−0.739484 + 0.673174i \(0.764930\pi\)
\(282\) −22.6370 −1.34801
\(283\) −26.8377 −1.59534 −0.797669 0.603096i \(-0.793934\pi\)
−0.797669 + 0.603096i \(0.793934\pi\)
\(284\) −3.38843 −0.201066
\(285\) −24.2784 −1.43813
\(286\) 8.09344 0.478575
\(287\) −9.65900 −0.570153
\(288\) 7.16572 0.422244
\(289\) −12.7952 −0.752658
\(290\) −0.600332 −0.0352527
\(291\) −11.6601 −0.683529
\(292\) −7.23868 −0.423611
\(293\) 19.2353 1.12374 0.561868 0.827227i \(-0.310083\pi\)
0.561868 + 0.827227i \(0.310083\pi\)
\(294\) 14.5476 0.848435
\(295\) −15.2250 −0.886436
\(296\) −0.830414 −0.0482668
\(297\) 21.2805 1.23482
\(298\) 8.96690 0.519438
\(299\) −38.7203 −2.23925
\(300\) 6.05697 0.349699
\(301\) −1.46260 −0.0843030
\(302\) −11.0030 −0.633152
\(303\) 29.2035 1.67770
\(304\) 4.32464 0.248035
\(305\) −17.7401 −1.01579
\(306\) −14.6938 −0.839987
\(307\) −24.6003 −1.40402 −0.702008 0.712169i \(-0.747713\pi\)
−0.702008 + 0.712169i \(0.747713\pi\)
\(308\) 2.50136 0.142528
\(309\) 1.17496 0.0668409
\(310\) −4.80137 −0.272700
\(311\) −32.0503 −1.81741 −0.908704 0.417441i \(-0.862927\pi\)
−0.908704 + 0.417441i \(0.862927\pi\)
\(312\) 16.1057 0.911805
\(313\) −3.16599 −0.178952 −0.0894762 0.995989i \(-0.528519\pi\)
−0.0894762 + 0.995989i \(0.528519\pi\)
\(314\) −9.92152 −0.559904
\(315\) −19.6977 −1.10984
\(316\) −1.04892 −0.0590062
\(317\) −8.51115 −0.478034 −0.239017 0.971015i \(-0.576825\pi\)
−0.239017 + 0.971015i \(0.576825\pi\)
\(318\) −23.8243 −1.33600
\(319\) 0.546277 0.0305857
\(320\) 1.76077 0.0984298
\(321\) −43.1221 −2.40684
\(322\) −11.9669 −0.666890
\(323\) −8.86796 −0.493426
\(324\) 20.8504 1.15836
\(325\) 9.59614 0.532298
\(326\) 0.00712968 0.000394876 0
\(327\) 6.75985 0.373821
\(328\) 6.18698 0.341619
\(329\) −11.0842 −0.611090
\(330\) 8.99485 0.495150
\(331\) 27.2826 1.49959 0.749794 0.661671i \(-0.230152\pi\)
0.749794 + 0.661671i \(0.230152\pi\)
\(332\) 1.10682 0.0607447
\(333\) −5.95052 −0.326086
\(334\) 15.8324 0.866309
\(335\) 19.8539 1.08474
\(336\) 4.97763 0.271552
\(337\) 3.19555 0.174073 0.0870363 0.996205i \(-0.472260\pi\)
0.0870363 + 0.996205i \(0.472260\pi\)
\(338\) 12.5164 0.680805
\(339\) 41.7146 2.26563
\(340\) −3.61056 −0.195810
\(341\) 4.36905 0.236598
\(342\) 30.9892 1.67570
\(343\) 18.0515 0.974689
\(344\) 0.936856 0.0505119
\(345\) −43.0328 −2.31681
\(346\) 11.3445 0.609881
\(347\) −2.86923 −0.154028 −0.0770142 0.997030i \(-0.524539\pi\)
−0.0770142 + 0.997030i \(0.524539\pi\)
\(348\) 1.08707 0.0582733
\(349\) −11.0513 −0.591565 −0.295782 0.955255i \(-0.595580\pi\)
−0.295782 + 0.955255i \(0.595580\pi\)
\(350\) 2.96579 0.158528
\(351\) 67.0918 3.58110
\(352\) −1.60222 −0.0853988
\(353\) −6.36400 −0.338722 −0.169361 0.985554i \(-0.554170\pi\)
−0.169361 + 0.985554i \(0.554170\pi\)
\(354\) 27.5693 1.46529
\(355\) −5.96623 −0.316655
\(356\) −12.5594 −0.665645
\(357\) −10.2069 −0.540209
\(358\) 11.5861 0.612345
\(359\) −10.9510 −0.577974 −0.288987 0.957333i \(-0.593318\pi\)
−0.288987 + 0.957333i \(0.593318\pi\)
\(360\) 12.6172 0.664982
\(361\) −0.297477 −0.0156567
\(362\) −3.68403 −0.193629
\(363\) 26.8872 1.41121
\(364\) 7.88612 0.413345
\(365\) −12.7456 −0.667136
\(366\) 32.1235 1.67912
\(367\) −2.58644 −0.135011 −0.0675055 0.997719i \(-0.521504\pi\)
−0.0675055 + 0.997719i \(0.521504\pi\)
\(368\) 7.66530 0.399581
\(369\) 44.3342 2.30794
\(370\) −1.46216 −0.0760143
\(371\) −11.6655 −0.605643
\(372\) 8.69427 0.450777
\(373\) −35.1217 −1.81853 −0.909266 0.416216i \(-0.863356\pi\)
−0.909266 + 0.416216i \(0.863356\pi\)
\(374\) 3.28546 0.169887
\(375\) 38.7348 2.00026
\(376\) 7.09986 0.366147
\(377\) 1.72226 0.0887012
\(378\) 20.7354 1.06651
\(379\) 6.14724 0.315762 0.157881 0.987458i \(-0.449534\pi\)
0.157881 + 0.987458i \(0.449534\pi\)
\(380\) 7.61468 0.390625
\(381\) 30.6589 1.57070
\(382\) −2.77288 −0.141873
\(383\) −22.1836 −1.13353 −0.566764 0.823880i \(-0.691805\pi\)
−0.566764 + 0.823880i \(0.691805\pi\)
\(384\) −3.18837 −0.162706
\(385\) 4.40431 0.224465
\(386\) −4.13574 −0.210504
\(387\) 6.71325 0.341254
\(388\) 3.65708 0.185660
\(389\) 8.36423 0.424083 0.212042 0.977261i \(-0.431989\pi\)
0.212042 + 0.977261i \(0.431989\pi\)
\(390\) 28.3583 1.43598
\(391\) −15.7182 −0.794903
\(392\) −4.56271 −0.230452
\(393\) −33.8350 −1.70675
\(394\) −4.84442 −0.244058
\(395\) −1.84690 −0.0929275
\(396\) −11.4811 −0.576947
\(397\) 10.4305 0.523494 0.261747 0.965137i \(-0.415701\pi\)
0.261747 + 0.965137i \(0.415701\pi\)
\(398\) −12.3355 −0.618324
\(399\) 21.5265 1.07767
\(400\) −1.89971 −0.0949853
\(401\) −10.7377 −0.536213 −0.268107 0.963389i \(-0.586398\pi\)
−0.268107 + 0.963389i \(0.586398\pi\)
\(402\) −35.9512 −1.79309
\(403\) 13.7745 0.686154
\(404\) −9.15937 −0.455696
\(405\) 36.7127 1.82427
\(406\) 0.532284 0.0264168
\(407\) 1.33051 0.0659509
\(408\) 6.53796 0.323677
\(409\) −16.4325 −0.812534 −0.406267 0.913754i \(-0.633170\pi\)
−0.406267 + 0.913754i \(0.633170\pi\)
\(410\) 10.8938 0.538007
\(411\) −58.7802 −2.89941
\(412\) −0.368513 −0.0181553
\(413\) 13.4993 0.664256
\(414\) 54.9274 2.69953
\(415\) 1.94885 0.0956653
\(416\) −5.05138 −0.247664
\(417\) −39.2682 −1.92297
\(418\) −6.92904 −0.338911
\(419\) 0.515672 0.0251922 0.0125961 0.999921i \(-0.495990\pi\)
0.0125961 + 0.999921i \(0.495990\pi\)
\(420\) 8.76444 0.427661
\(421\) 18.2572 0.889802 0.444901 0.895580i \(-0.353239\pi\)
0.444901 + 0.895580i \(0.353239\pi\)
\(422\) −13.4414 −0.654317
\(423\) 50.8756 2.47366
\(424\) 7.47223 0.362884
\(425\) 3.89547 0.188958
\(426\) 10.8036 0.523435
\(427\) 15.7292 0.761190
\(428\) 13.5248 0.653746
\(429\) −25.8049 −1.24587
\(430\) 1.64958 0.0795500
\(431\) −17.1360 −0.825412 −0.412706 0.910864i \(-0.635416\pi\)
−0.412706 + 0.910864i \(0.635416\pi\)
\(432\) −13.2819 −0.639024
\(433\) 30.8145 1.48085 0.740426 0.672138i \(-0.234624\pi\)
0.740426 + 0.672138i \(0.234624\pi\)
\(434\) 4.25714 0.204349
\(435\) 1.91408 0.0917732
\(436\) −2.12016 −0.101537
\(437\) 33.1497 1.58576
\(438\) 23.0796 1.10279
\(439\) −22.0206 −1.05099 −0.525494 0.850797i \(-0.676119\pi\)
−0.525494 + 0.850797i \(0.676119\pi\)
\(440\) −2.82114 −0.134493
\(441\) −32.6951 −1.55691
\(442\) 10.3582 0.492688
\(443\) −0.701050 −0.0333079 −0.0166540 0.999861i \(-0.505301\pi\)
−0.0166540 + 0.999861i \(0.505301\pi\)
\(444\) 2.64767 0.125653
\(445\) −22.1141 −1.04831
\(446\) −17.7707 −0.841467
\(447\) −28.5898 −1.35225
\(448\) −1.56118 −0.0737589
\(449\) −22.0338 −1.03984 −0.519919 0.854216i \(-0.674038\pi\)
−0.519919 + 0.854216i \(0.674038\pi\)
\(450\) −13.6128 −0.641712
\(451\) −9.91293 −0.466781
\(452\) −13.0833 −0.615389
\(453\) 35.0817 1.64828
\(454\) 13.7933 0.647350
\(455\) 13.8856 0.650967
\(456\) −13.7886 −0.645709
\(457\) −4.53353 −0.212070 −0.106035 0.994362i \(-0.533815\pi\)
−0.106035 + 0.994362i \(0.533815\pi\)
\(458\) −13.2868 −0.620853
\(459\) 27.2353 1.27124
\(460\) 13.4968 0.629291
\(461\) 25.5224 1.18870 0.594349 0.804207i \(-0.297410\pi\)
0.594349 + 0.804207i \(0.297410\pi\)
\(462\) −7.97528 −0.371043
\(463\) 16.1435 0.750250 0.375125 0.926974i \(-0.377600\pi\)
0.375125 + 0.926974i \(0.377600\pi\)
\(464\) −0.340949 −0.0158282
\(465\) 15.3086 0.709918
\(466\) −15.3937 −0.713098
\(467\) −33.1085 −1.53208 −0.766040 0.642793i \(-0.777776\pi\)
−0.766040 + 0.642793i \(0.777776\pi\)
\(468\) −36.1968 −1.67320
\(469\) −17.6035 −0.812853
\(470\) 12.5012 0.576636
\(471\) 31.6335 1.45760
\(472\) −8.64683 −0.398003
\(473\) −1.50105 −0.0690185
\(474\) 3.34434 0.153611
\(475\) −8.21555 −0.376955
\(476\) 3.20130 0.146731
\(477\) 53.5439 2.45161
\(478\) −1.03722 −0.0474413
\(479\) 36.8589 1.68412 0.842062 0.539380i \(-0.181341\pi\)
0.842062 + 0.539380i \(0.181341\pi\)
\(480\) −5.61398 −0.256242
\(481\) 4.19474 0.191264
\(482\) −16.3323 −0.743917
\(483\) 38.1550 1.73611
\(484\) −8.43288 −0.383313
\(485\) 6.43926 0.292392
\(486\) −26.6332 −1.20811
\(487\) −42.8576 −1.94206 −0.971032 0.238948i \(-0.923197\pi\)
−0.971032 + 0.238948i \(0.923197\pi\)
\(488\) −10.0752 −0.456083
\(489\) −0.0227321 −0.00102798
\(490\) −8.03386 −0.362933
\(491\) −19.4186 −0.876348 −0.438174 0.898890i \(-0.644375\pi\)
−0.438174 + 0.898890i \(0.644375\pi\)
\(492\) −19.7264 −0.889335
\(493\) 0.699139 0.0314876
\(494\) −21.8454 −0.982871
\(495\) −20.2155 −0.908619
\(496\) −2.72687 −0.122440
\(497\) 5.28996 0.237287
\(498\) −3.52896 −0.158136
\(499\) −12.3810 −0.554252 −0.277126 0.960834i \(-0.589382\pi\)
−0.277126 + 0.960834i \(0.589382\pi\)
\(500\) −12.1488 −0.543309
\(501\) −50.4795 −2.25526
\(502\) −19.2479 −0.859077
\(503\) −9.87872 −0.440470 −0.220235 0.975447i \(-0.570682\pi\)
−0.220235 + 0.975447i \(0.570682\pi\)
\(504\) −11.1870 −0.498308
\(505\) −16.1275 −0.717665
\(506\) −12.2815 −0.545980
\(507\) −39.9071 −1.77234
\(508\) −9.61584 −0.426634
\(509\) −30.3055 −1.34327 −0.671634 0.740883i \(-0.734407\pi\)
−0.671634 + 0.740883i \(0.734407\pi\)
\(510\) 11.5118 0.509752
\(511\) 11.3009 0.499922
\(512\) 1.00000 0.0441942
\(513\) −57.4393 −2.53601
\(514\) −11.2496 −0.496200
\(515\) −0.648864 −0.0285924
\(516\) −2.98705 −0.131497
\(517\) −11.3756 −0.500297
\(518\) 1.29643 0.0569617
\(519\) −36.1703 −1.58770
\(520\) −8.89429 −0.390041
\(521\) −0.828053 −0.0362777 −0.0181388 0.999835i \(-0.505774\pi\)
−0.0181388 + 0.999835i \(0.505774\pi\)
\(522\) −2.44315 −0.106934
\(523\) 21.1567 0.925116 0.462558 0.886589i \(-0.346932\pi\)
0.462558 + 0.886589i \(0.346932\pi\)
\(524\) 10.6120 0.463587
\(525\) −9.45603 −0.412695
\(526\) 10.3469 0.451144
\(527\) 5.59162 0.243575
\(528\) 5.10849 0.222318
\(529\) 35.7568 1.55464
\(530\) 13.1568 0.571497
\(531\) −61.9608 −2.68887
\(532\) −6.75155 −0.292717
\(533\) −31.2528 −1.35371
\(534\) 40.0440 1.73287
\(535\) 23.8140 1.02957
\(536\) 11.2757 0.487038
\(537\) −36.9408 −1.59412
\(538\) 26.3732 1.13703
\(539\) 7.31049 0.314885
\(540\) −23.3863 −1.00638
\(541\) 8.10695 0.348545 0.174272 0.984697i \(-0.444243\pi\)
0.174272 + 0.984697i \(0.444243\pi\)
\(542\) −18.2888 −0.785573
\(543\) 11.7461 0.504072
\(544\) −2.05056 −0.0879172
\(545\) −3.73310 −0.159908
\(546\) −25.1439 −1.07606
\(547\) −2.53565 −0.108417 −0.0542084 0.998530i \(-0.517264\pi\)
−0.0542084 + 0.998530i \(0.517264\pi\)
\(548\) 18.4358 0.787538
\(549\) −72.1961 −3.08125
\(550\) 3.04376 0.129786
\(551\) −1.47448 −0.0628151
\(552\) −24.4398 −1.04023
\(553\) 1.63755 0.0696357
\(554\) −1.93375 −0.0821572
\(555\) 4.66192 0.197888
\(556\) 12.3160 0.522317
\(557\) 9.34584 0.395996 0.197998 0.980202i \(-0.436556\pi\)
0.197998 + 0.980202i \(0.436556\pi\)
\(558\) −19.5400 −0.827193
\(559\) −4.73242 −0.200160
\(560\) −2.74887 −0.116161
\(561\) −10.4753 −0.442267
\(562\) −24.7920 −1.04579
\(563\) 16.8886 0.711771 0.355886 0.934529i \(-0.384179\pi\)
0.355886 + 0.934529i \(0.384179\pi\)
\(564\) −22.6370 −0.953189
\(565\) −23.0367 −0.969161
\(566\) −26.8377 −1.12807
\(567\) −32.5513 −1.36702
\(568\) −3.38843 −0.142175
\(569\) −27.0105 −1.13234 −0.566169 0.824289i \(-0.691575\pi\)
−0.566169 + 0.824289i \(0.691575\pi\)
\(570\) −24.2784 −1.01691
\(571\) −30.3919 −1.27186 −0.635931 0.771746i \(-0.719384\pi\)
−0.635931 + 0.771746i \(0.719384\pi\)
\(572\) 8.09344 0.338404
\(573\) 8.84098 0.369337
\(574\) −9.65900 −0.403159
\(575\) −14.5618 −0.607270
\(576\) 7.16572 0.298572
\(577\) 27.6921 1.15284 0.576419 0.817155i \(-0.304450\pi\)
0.576419 + 0.817155i \(0.304450\pi\)
\(578\) −12.7952 −0.532210
\(579\) 13.1863 0.548003
\(580\) −0.600332 −0.0249274
\(581\) −1.72795 −0.0716874
\(582\) −11.6601 −0.483328
\(583\) −11.9722 −0.495838
\(584\) −7.23868 −0.299539
\(585\) −63.7340 −2.63508
\(586\) 19.2353 0.794601
\(587\) −3.17361 −0.130989 −0.0654945 0.997853i \(-0.520862\pi\)
−0.0654945 + 0.997853i \(0.520862\pi\)
\(588\) 14.5476 0.599934
\(589\) −11.7927 −0.485911
\(590\) −15.2250 −0.626805
\(591\) 15.4458 0.635356
\(592\) −0.830414 −0.0341298
\(593\) 19.3394 0.794175 0.397087 0.917781i \(-0.370021\pi\)
0.397087 + 0.917781i \(0.370021\pi\)
\(594\) 21.2805 0.873151
\(595\) 5.63674 0.231084
\(596\) 8.96690 0.367298
\(597\) 39.3302 1.60968
\(598\) −38.7203 −1.58339
\(599\) 15.7256 0.642530 0.321265 0.946989i \(-0.395892\pi\)
0.321265 + 0.946989i \(0.395892\pi\)
\(600\) 6.05697 0.247275
\(601\) 19.8293 0.808853 0.404427 0.914570i \(-0.367471\pi\)
0.404427 + 0.914570i \(0.367471\pi\)
\(602\) −1.46260 −0.0596112
\(603\) 80.7988 3.29038
\(604\) −11.0030 −0.447706
\(605\) −14.8483 −0.603670
\(606\) 29.2035 1.18631
\(607\) 5.94844 0.241440 0.120720 0.992687i \(-0.461480\pi\)
0.120720 + 0.992687i \(0.461480\pi\)
\(608\) 4.32464 0.175387
\(609\) −1.69712 −0.0687708
\(610\) −17.7401 −0.718274
\(611\) −35.8641 −1.45091
\(612\) −14.6938 −0.593961
\(613\) −23.7062 −0.957483 −0.478742 0.877956i \(-0.658907\pi\)
−0.478742 + 0.877956i \(0.658907\pi\)
\(614\) −24.6003 −0.992789
\(615\) −34.7335 −1.40059
\(616\) 2.50136 0.100783
\(617\) 30.2774 1.21892 0.609461 0.792816i \(-0.291386\pi\)
0.609461 + 0.792816i \(0.291386\pi\)
\(618\) 1.17496 0.0472636
\(619\) −22.9972 −0.924334 −0.462167 0.886793i \(-0.652928\pi\)
−0.462167 + 0.886793i \(0.652928\pi\)
\(620\) −4.80137 −0.192828
\(621\) −101.810 −4.08547
\(622\) −32.0503 −1.28510
\(623\) 19.6075 0.785556
\(624\) 16.1057 0.644743
\(625\) −11.8926 −0.475703
\(626\) −3.16599 −0.126538
\(627\) 22.0924 0.882285
\(628\) −9.92152 −0.395912
\(629\) 1.70282 0.0678958
\(630\) −19.6977 −0.784774
\(631\) −21.1793 −0.843133 −0.421566 0.906797i \(-0.638520\pi\)
−0.421566 + 0.906797i \(0.638520\pi\)
\(632\) −1.04892 −0.0417237
\(633\) 42.8562 1.70338
\(634\) −8.51115 −0.338021
\(635\) −16.9312 −0.671895
\(636\) −23.8243 −0.944694
\(637\) 23.0480 0.913195
\(638\) 0.546277 0.0216273
\(639\) −24.2806 −0.960524
\(640\) 1.76077 0.0696003
\(641\) −9.74469 −0.384892 −0.192446 0.981308i \(-0.561642\pi\)
−0.192446 + 0.981308i \(0.561642\pi\)
\(642\) −43.1221 −1.70189
\(643\) −26.8569 −1.05913 −0.529566 0.848269i \(-0.677645\pi\)
−0.529566 + 0.848269i \(0.677645\pi\)
\(644\) −11.9669 −0.471563
\(645\) −5.25949 −0.207092
\(646\) −8.86796 −0.348905
\(647\) −21.9696 −0.863713 −0.431857 0.901942i \(-0.642141\pi\)
−0.431857 + 0.901942i \(0.642141\pi\)
\(648\) 20.8504 0.819081
\(649\) 13.8542 0.543824
\(650\) 9.59614 0.376392
\(651\) −13.5733 −0.531981
\(652\) 0.00712968 0.000279220 0
\(653\) −7.03756 −0.275401 −0.137700 0.990474i \(-0.543971\pi\)
−0.137700 + 0.990474i \(0.543971\pi\)
\(654\) 6.75985 0.264331
\(655\) 18.6852 0.730092
\(656\) 6.18698 0.241561
\(657\) −51.8703 −2.02365
\(658\) −11.0842 −0.432106
\(659\) −11.4398 −0.445631 −0.222816 0.974861i \(-0.571525\pi\)
−0.222816 + 0.974861i \(0.571525\pi\)
\(660\) 8.99485 0.350124
\(661\) 47.0485 1.82998 0.914988 0.403481i \(-0.132200\pi\)
0.914988 + 0.403481i \(0.132200\pi\)
\(662\) 27.2826 1.06037
\(663\) −33.0257 −1.28261
\(664\) 1.10682 0.0429530
\(665\) −11.8879 −0.460993
\(666\) −5.95052 −0.230578
\(667\) −2.61348 −0.101194
\(668\) 15.8324 0.612573
\(669\) 56.6596 2.19059
\(670\) 19.8539 0.767024
\(671\) 16.1427 0.623183
\(672\) 4.97763 0.192016
\(673\) 2.47436 0.0953794 0.0476897 0.998862i \(-0.484814\pi\)
0.0476897 + 0.998862i \(0.484814\pi\)
\(674\) 3.19555 0.123088
\(675\) 25.2317 0.971167
\(676\) 12.5164 0.481402
\(677\) −40.4942 −1.55632 −0.778160 0.628066i \(-0.783847\pi\)
−0.778160 + 0.628066i \(0.783847\pi\)
\(678\) 41.7146 1.60204
\(679\) −5.70937 −0.219105
\(680\) −3.61056 −0.138459
\(681\) −43.9780 −1.68524
\(682\) 4.36905 0.167300
\(683\) 38.5565 1.47532 0.737662 0.675170i \(-0.235930\pi\)
0.737662 + 0.675170i \(0.235930\pi\)
\(684\) 30.9892 1.18490
\(685\) 32.4611 1.24028
\(686\) 18.0515 0.689209
\(687\) 42.3633 1.61626
\(688\) 0.936856 0.0357173
\(689\) −37.7451 −1.43797
\(690\) −43.0328 −1.63823
\(691\) −39.1794 −1.49045 −0.745227 0.666811i \(-0.767659\pi\)
−0.745227 + 0.666811i \(0.767659\pi\)
\(692\) 11.3445 0.431251
\(693\) 17.9241 0.680879
\(694\) −2.86923 −0.108915
\(695\) 21.6857 0.822584
\(696\) 1.08707 0.0412054
\(697\) −12.6868 −0.480547
\(698\) −11.0513 −0.418300
\(699\) 49.0807 1.85640
\(700\) 2.96579 0.112096
\(701\) −22.3199 −0.843012 −0.421506 0.906826i \(-0.638498\pi\)
−0.421506 + 0.906826i \(0.638498\pi\)
\(702\) 67.0918 2.53222
\(703\) −3.59124 −0.135446
\(704\) −1.60222 −0.0603861
\(705\) −39.8584 −1.50115
\(706\) −6.36400 −0.239512
\(707\) 14.2994 0.537786
\(708\) 27.5693 1.03612
\(709\) −39.3745 −1.47874 −0.739370 0.673300i \(-0.764876\pi\)
−0.739370 + 0.673300i \(0.764876\pi\)
\(710\) −5.96623 −0.223909
\(711\) −7.51625 −0.281881
\(712\) −12.5594 −0.470682
\(713\) −20.9023 −0.782796
\(714\) −10.2069 −0.381985
\(715\) 14.2507 0.532944
\(716\) 11.5861 0.432993
\(717\) 3.30704 0.123504
\(718\) −10.9510 −0.408689
\(719\) 36.1666 1.34879 0.674393 0.738372i \(-0.264405\pi\)
0.674393 + 0.738372i \(0.264405\pi\)
\(720\) 12.6172 0.470213
\(721\) 0.575315 0.0214259
\(722\) −0.297477 −0.0110709
\(723\) 52.0736 1.93664
\(724\) −3.68403 −0.136916
\(725\) 0.647704 0.0240551
\(726\) 26.8872 0.997876
\(727\) 1.01733 0.0377308 0.0188654 0.999822i \(-0.493995\pi\)
0.0188654 + 0.999822i \(0.493995\pi\)
\(728\) 7.88612 0.292279
\(729\) 22.3655 0.828351
\(730\) −12.7456 −0.471736
\(731\) −1.92108 −0.0710539
\(732\) 32.1235 1.18732
\(733\) −33.6959 −1.24459 −0.622294 0.782784i \(-0.713799\pi\)
−0.622294 + 0.782784i \(0.713799\pi\)
\(734\) −2.58644 −0.0954673
\(735\) 25.6150 0.944822
\(736\) 7.66530 0.282547
\(737\) −18.0663 −0.665479
\(738\) 44.3342 1.63196
\(739\) 22.0748 0.812036 0.406018 0.913865i \(-0.366917\pi\)
0.406018 + 0.913865i \(0.366917\pi\)
\(740\) −1.46216 −0.0537502
\(741\) 69.6513 2.55870
\(742\) −11.6655 −0.428254
\(743\) 40.0012 1.46750 0.733751 0.679418i \(-0.237768\pi\)
0.733751 + 0.679418i \(0.237768\pi\)
\(744\) 8.69427 0.318748
\(745\) 15.7886 0.578449
\(746\) −35.1217 −1.28590
\(747\) 7.93117 0.290186
\(748\) 3.28546 0.120128
\(749\) −21.1147 −0.771514
\(750\) 38.7348 1.41439
\(751\) −8.78921 −0.320723 −0.160361 0.987058i \(-0.551266\pi\)
−0.160361 + 0.987058i \(0.551266\pi\)
\(752\) 7.09986 0.258905
\(753\) 61.3696 2.23643
\(754\) 1.72226 0.0627212
\(755\) −19.3737 −0.705081
\(756\) 20.7354 0.754140
\(757\) −25.1400 −0.913729 −0.456864 0.889536i \(-0.651028\pi\)
−0.456864 + 0.889536i \(0.651028\pi\)
\(758\) 6.14724 0.223278
\(759\) 39.1581 1.42135
\(760\) 7.61468 0.276213
\(761\) −35.6544 −1.29247 −0.646235 0.763138i \(-0.723658\pi\)
−0.646235 + 0.763138i \(0.723658\pi\)
\(762\) 30.6589 1.11065
\(763\) 3.30995 0.119828
\(764\) −2.77288 −0.100319
\(765\) −25.8723 −0.935414
\(766\) −22.1836 −0.801526
\(767\) 43.6784 1.57714
\(768\) −3.18837 −0.115050
\(769\) 4.25159 0.153316 0.0766580 0.997057i \(-0.475575\pi\)
0.0766580 + 0.997057i \(0.475575\pi\)
\(770\) 4.40431 0.158720
\(771\) 35.8680 1.29175
\(772\) −4.13574 −0.148849
\(773\) −39.0701 −1.40525 −0.702626 0.711559i \(-0.747989\pi\)
−0.702626 + 0.711559i \(0.747989\pi\)
\(774\) 6.71325 0.241303
\(775\) 5.18025 0.186080
\(776\) 3.65708 0.131282
\(777\) −4.13349 −0.148288
\(778\) 8.36423 0.299872
\(779\) 26.7565 0.958650
\(780\) 28.3583 1.01539
\(781\) 5.42903 0.194266
\(782\) −15.7182 −0.562081
\(783\) 4.52845 0.161833
\(784\) −4.56271 −0.162954
\(785\) −17.4695 −0.623512
\(786\) −33.8350 −1.20685
\(787\) −16.4952 −0.587991 −0.293995 0.955807i \(-0.594985\pi\)
−0.293995 + 0.955807i \(0.594985\pi\)
\(788\) −4.84442 −0.172575
\(789\) −32.9896 −1.17446
\(790\) −1.84690 −0.0657096
\(791\) 20.4255 0.726247
\(792\) −11.4811 −0.407963
\(793\) 50.8937 1.80729
\(794\) 10.4305 0.370166
\(795\) −41.9489 −1.48778
\(796\) −12.3355 −0.437221
\(797\) 22.5708 0.799499 0.399749 0.916624i \(-0.369097\pi\)
0.399749 + 0.916624i \(0.369097\pi\)
\(798\) 21.5265 0.762028
\(799\) −14.5587 −0.515050
\(800\) −1.89971 −0.0671648
\(801\) −89.9969 −3.17989
\(802\) −10.7377 −0.379160
\(803\) 11.5980 0.409284
\(804\) −35.9512 −1.26790
\(805\) −21.0709 −0.742653
\(806\) 13.7745 0.485184
\(807\) −84.0877 −2.96003
\(808\) −9.15937 −0.322226
\(809\) 6.13926 0.215845 0.107923 0.994159i \(-0.465580\pi\)
0.107923 + 0.994159i \(0.465580\pi\)
\(810\) 36.7127 1.28995
\(811\) −12.6099 −0.442793 −0.221397 0.975184i \(-0.571062\pi\)
−0.221397 + 0.975184i \(0.571062\pi\)
\(812\) 0.532284 0.0186795
\(813\) 58.3116 2.04508
\(814\) 1.33051 0.0466343
\(815\) 0.0125537 0.000439736 0
\(816\) 6.53796 0.228875
\(817\) 4.05157 0.141746
\(818\) −16.4325 −0.574549
\(819\) 56.5098 1.97461
\(820\) 10.8938 0.380429
\(821\) −2.43571 −0.0850067 −0.0425034 0.999096i \(-0.513533\pi\)
−0.0425034 + 0.999096i \(0.513533\pi\)
\(822\) −58.7802 −2.05020
\(823\) −38.3762 −1.33771 −0.668855 0.743393i \(-0.733215\pi\)
−0.668855 + 0.743393i \(0.733215\pi\)
\(824\) −0.368513 −0.0128377
\(825\) −9.70463 −0.337872
\(826\) 13.4993 0.469700
\(827\) 42.0778 1.46319 0.731594 0.681740i \(-0.238777\pi\)
0.731594 + 0.681740i \(0.238777\pi\)
\(828\) 54.9274 1.90886
\(829\) 31.4243 1.09141 0.545705 0.837977i \(-0.316262\pi\)
0.545705 + 0.837977i \(0.316262\pi\)
\(830\) 1.94885 0.0676456
\(831\) 6.16552 0.213880
\(832\) −5.05138 −0.175125
\(833\) 9.35613 0.324171
\(834\) −39.2682 −1.35975
\(835\) 27.8771 0.964727
\(836\) −6.92904 −0.239646
\(837\) 36.2179 1.25187
\(838\) 0.515672 0.0178136
\(839\) −49.1624 −1.69728 −0.848638 0.528974i \(-0.822577\pi\)
−0.848638 + 0.528974i \(0.822577\pi\)
\(840\) 8.76444 0.302402
\(841\) −28.8838 −0.995992
\(842\) 18.2572 0.629185
\(843\) 79.0462 2.72250
\(844\) −13.4414 −0.462672
\(845\) 22.0385 0.758148
\(846\) 50.8756 1.74914
\(847\) 13.1653 0.452363
\(848\) 7.47223 0.256598
\(849\) 85.5687 2.93671
\(850\) 3.89547 0.133614
\(851\) −6.36537 −0.218202
\(852\) 10.8036 0.370125
\(853\) −7.35791 −0.251930 −0.125965 0.992035i \(-0.540203\pi\)
−0.125965 + 0.992035i \(0.540203\pi\)
\(854\) 15.7292 0.538243
\(855\) 54.5647 1.86607
\(856\) 13.5248 0.462268
\(857\) 36.8397 1.25842 0.629210 0.777236i \(-0.283379\pi\)
0.629210 + 0.777236i \(0.283379\pi\)
\(858\) −25.8049 −0.880965
\(859\) −25.3202 −0.863915 −0.431957 0.901894i \(-0.642177\pi\)
−0.431957 + 0.901894i \(0.642177\pi\)
\(860\) 1.64958 0.0562503
\(861\) 30.7965 1.04954
\(862\) −17.1360 −0.583654
\(863\) −8.16205 −0.277839 −0.138920 0.990304i \(-0.544363\pi\)
−0.138920 + 0.990304i \(0.544363\pi\)
\(864\) −13.2819 −0.451859
\(865\) 19.9749 0.679167
\(866\) 30.8145 1.04712
\(867\) 40.7958 1.38550
\(868\) 4.25714 0.144497
\(869\) 1.68060 0.0570105
\(870\) 1.91408 0.0648934
\(871\) −56.9580 −1.92995
\(872\) −2.12016 −0.0717976
\(873\) 26.2056 0.886926
\(874\) 33.1497 1.12130
\(875\) 18.9664 0.641182
\(876\) 23.0796 0.779787
\(877\) 7.84546 0.264922 0.132461 0.991188i \(-0.457712\pi\)
0.132461 + 0.991188i \(0.457712\pi\)
\(878\) −22.0206 −0.743160
\(879\) −61.3292 −2.06858
\(880\) −2.82114 −0.0951006
\(881\) 6.90796 0.232735 0.116368 0.993206i \(-0.462875\pi\)
0.116368 + 0.993206i \(0.462875\pi\)
\(882\) −32.6951 −1.10090
\(883\) 33.7164 1.13465 0.567324 0.823495i \(-0.307979\pi\)
0.567324 + 0.823495i \(0.307979\pi\)
\(884\) 10.3582 0.348383
\(885\) 48.5431 1.63176
\(886\) −0.701050 −0.0235523
\(887\) 34.3899 1.15470 0.577349 0.816497i \(-0.304087\pi\)
0.577349 + 0.816497i \(0.304087\pi\)
\(888\) 2.64767 0.0888500
\(889\) 15.0121 0.503489
\(890\) −22.1141 −0.741266
\(891\) −33.4070 −1.11918
\(892\) −17.7707 −0.595007
\(893\) 30.7043 1.02748
\(894\) −28.5898 −0.956186
\(895\) 20.4004 0.681911
\(896\) −1.56118 −0.0521554
\(897\) 123.455 4.12204
\(898\) −22.0338 −0.735276
\(899\) 0.929724 0.0310080
\(900\) −13.6128 −0.453759
\(901\) −15.3223 −0.510460
\(902\) −9.91293 −0.330064
\(903\) 4.66332 0.155186
\(904\) −13.0833 −0.435146
\(905\) −6.48672 −0.215626
\(906\) 35.0817 1.16551
\(907\) 5.39802 0.179238 0.0896192 0.995976i \(-0.471435\pi\)
0.0896192 + 0.995976i \(0.471435\pi\)
\(908\) 13.7933 0.457745
\(909\) −65.6335 −2.17693
\(910\) 13.8856 0.460303
\(911\) 30.2896 1.00354 0.501770 0.865001i \(-0.332682\pi\)
0.501770 + 0.865001i \(0.332682\pi\)
\(912\) −13.7886 −0.456585
\(913\) −1.77337 −0.0586901
\(914\) −4.53353 −0.149956
\(915\) 56.5619 1.86988
\(916\) −13.2868 −0.439009
\(917\) −16.5672 −0.547099
\(918\) 27.2353 0.898900
\(919\) 13.3834 0.441477 0.220739 0.975333i \(-0.429153\pi\)
0.220739 + 0.975333i \(0.429153\pi\)
\(920\) 13.4968 0.444976
\(921\) 78.4350 2.58452
\(922\) 25.5224 0.840537
\(923\) 17.1163 0.563389
\(924\) −7.97528 −0.262367
\(925\) 1.57754 0.0518693
\(926\) 16.1435 0.530507
\(927\) −2.64066 −0.0867306
\(928\) −0.340949 −0.0111922
\(929\) −5.85105 −0.191967 −0.0959834 0.995383i \(-0.530600\pi\)
−0.0959834 + 0.995383i \(0.530600\pi\)
\(930\) 15.3086 0.501988
\(931\) −19.7321 −0.646693
\(932\) −15.3937 −0.504236
\(933\) 102.188 3.34550
\(934\) −33.1085 −1.08334
\(935\) 5.78493 0.189187
\(936\) −36.1968 −1.18313
\(937\) 7.55557 0.246830 0.123415 0.992355i \(-0.460615\pi\)
0.123415 + 0.992355i \(0.460615\pi\)
\(938\) −17.6035 −0.574774
\(939\) 10.0944 0.329417
\(940\) 12.5012 0.407743
\(941\) 19.0297 0.620352 0.310176 0.950679i \(-0.399612\pi\)
0.310176 + 0.950679i \(0.399612\pi\)
\(942\) 31.6335 1.03068
\(943\) 47.4250 1.54437
\(944\) −8.64683 −0.281430
\(945\) 36.5102 1.18768
\(946\) −1.50105 −0.0488035
\(947\) −1.35220 −0.0439407 −0.0219703 0.999759i \(-0.506994\pi\)
−0.0219703 + 0.999759i \(0.506994\pi\)
\(948\) 3.34434 0.108619
\(949\) 36.5653 1.18696
\(950\) −8.21555 −0.266548
\(951\) 27.1367 0.879969
\(952\) 3.20130 0.103755
\(953\) 39.3768 1.27554 0.637770 0.770227i \(-0.279857\pi\)
0.637770 + 0.770227i \(0.279857\pi\)
\(954\) 53.5439 1.73355
\(955\) −4.88239 −0.157991
\(956\) −1.03722 −0.0335460
\(957\) −1.74174 −0.0563023
\(958\) 36.8589 1.19086
\(959\) −28.7816 −0.929408
\(960\) −5.61398 −0.181190
\(961\) −23.5642 −0.760135
\(962\) 4.19474 0.135244
\(963\) 96.9150 3.12304
\(964\) −16.3323 −0.526029
\(965\) −7.28207 −0.234418
\(966\) 38.1550 1.22762
\(967\) −21.2109 −0.682097 −0.341049 0.940046i \(-0.610782\pi\)
−0.341049 + 0.940046i \(0.610782\pi\)
\(968\) −8.43288 −0.271043
\(969\) 28.2743 0.908303
\(970\) 6.43926 0.206752
\(971\) −18.4749 −0.592889 −0.296444 0.955050i \(-0.595801\pi\)
−0.296444 + 0.955050i \(0.595801\pi\)
\(972\) −26.6332 −0.854261
\(973\) −19.2276 −0.616408
\(974\) −42.8576 −1.37325
\(975\) −30.5961 −0.979859
\(976\) −10.0752 −0.322499
\(977\) 14.4216 0.461389 0.230694 0.973026i \(-0.425900\pi\)
0.230694 + 0.973026i \(0.425900\pi\)
\(978\) −0.0227321 −0.000726891 0
\(979\) 20.1229 0.643131
\(980\) −8.03386 −0.256632
\(981\) −15.1925 −0.485058
\(982\) −19.4186 −0.619672
\(983\) −24.7203 −0.788455 −0.394227 0.919013i \(-0.628988\pi\)
−0.394227 + 0.919013i \(0.628988\pi\)
\(984\) −19.7264 −0.628855
\(985\) −8.52988 −0.271785
\(986\) 0.699139 0.0222651
\(987\) 35.3405 1.12490
\(988\) −21.8454 −0.694995
\(989\) 7.18128 0.228351
\(990\) −20.2155 −0.642491
\(991\) 37.9665 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(992\) −2.72687 −0.0865782
\(993\) −86.9872 −2.76046
\(994\) 5.28996 0.167787
\(995\) −21.7200 −0.688569
\(996\) −3.52896 −0.111819
\(997\) 1.96182 0.0621313 0.0310657 0.999517i \(-0.490110\pi\)
0.0310657 + 0.999517i \(0.490110\pi\)
\(998\) −12.3810 −0.391915
\(999\) 11.0295 0.348957
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 8026.2.a.a.1.2 71
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8026.2.a.a.1.2 71 1.1 even 1 trivial