Properties

Label 8043.2.a.o.1.5
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $41$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(1\)
Dimension: \(41\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.38443 q^{2} -1.00000 q^{3} +3.68550 q^{4} +2.70331 q^{5} +2.38443 q^{6} +1.00000 q^{7} -4.01897 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.38443 q^{2} -1.00000 q^{3} +3.68550 q^{4} +2.70331 q^{5} +2.38443 q^{6} +1.00000 q^{7} -4.01897 q^{8} +1.00000 q^{9} -6.44585 q^{10} +2.12042 q^{11} -3.68550 q^{12} +0.628822 q^{13} -2.38443 q^{14} -2.70331 q^{15} +2.21194 q^{16} -0.671193 q^{17} -2.38443 q^{18} -4.69479 q^{19} +9.96306 q^{20} -1.00000 q^{21} -5.05599 q^{22} -4.25009 q^{23} +4.01897 q^{24} +2.30788 q^{25} -1.49938 q^{26} -1.00000 q^{27} +3.68550 q^{28} -8.77796 q^{29} +6.44585 q^{30} +8.26491 q^{31} +2.76373 q^{32} -2.12042 q^{33} +1.60041 q^{34} +2.70331 q^{35} +3.68550 q^{36} -2.88242 q^{37} +11.1944 q^{38} -0.628822 q^{39} -10.8645 q^{40} +0.950390 q^{41} +2.38443 q^{42} -3.42276 q^{43} +7.81482 q^{44} +2.70331 q^{45} +10.1340 q^{46} +7.41532 q^{47} -2.21194 q^{48} +1.00000 q^{49} -5.50298 q^{50} +0.671193 q^{51} +2.31753 q^{52} +10.0416 q^{53} +2.38443 q^{54} +5.73215 q^{55} -4.01897 q^{56} +4.69479 q^{57} +20.9304 q^{58} -8.47742 q^{59} -9.96306 q^{60} -8.41749 q^{61} -19.7071 q^{62} +1.00000 q^{63} -11.0138 q^{64} +1.69990 q^{65} +5.05599 q^{66} -0.248919 q^{67} -2.47368 q^{68} +4.25009 q^{69} -6.44585 q^{70} -10.5508 q^{71} -4.01897 q^{72} -6.70043 q^{73} +6.87292 q^{74} -2.30788 q^{75} -17.3027 q^{76} +2.12042 q^{77} +1.49938 q^{78} +10.9350 q^{79} +5.97955 q^{80} +1.00000 q^{81} -2.26614 q^{82} +1.52127 q^{83} -3.68550 q^{84} -1.81444 q^{85} +8.16134 q^{86} +8.77796 q^{87} -8.52190 q^{88} +9.66286 q^{89} -6.44585 q^{90} +0.628822 q^{91} -15.6637 q^{92} -8.26491 q^{93} -17.6813 q^{94} -12.6915 q^{95} -2.76373 q^{96} -0.763992 q^{97} -2.38443 q^{98} +2.12042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9} - 18 q^{10} - 17 q^{11} - 34 q^{12} - 38 q^{13} - 4 q^{14} - 5 q^{15} + 16 q^{16} + 4 q^{17} - 4 q^{18} - 15 q^{19} + 23 q^{20} - 41 q^{21} - 31 q^{22} - 8 q^{23} + 15 q^{24} + 16 q^{25} + 15 q^{26} - 41 q^{27} + 34 q^{28} - 27 q^{29} + 18 q^{30} - 23 q^{31} - 24 q^{32} + 17 q^{33} - 9 q^{34} + 5 q^{35} + 34 q^{36} - 81 q^{37} + 38 q^{39} - 52 q^{40} + 29 q^{41} + 4 q^{42} - 47 q^{43} - 20 q^{44} + 5 q^{45} - 40 q^{46} + 26 q^{47} - 16 q^{48} + 41 q^{49} - 23 q^{50} - 4 q^{51} - 64 q^{52} - 66 q^{53} + 4 q^{54} - 14 q^{55} - 15 q^{56} + 15 q^{57} - 50 q^{58} + 41 q^{59} - 23 q^{60} - 47 q^{61} + 5 q^{62} + 41 q^{63} - 37 q^{64} - 24 q^{65} + 31 q^{66} - 73 q^{67} + 10 q^{68} + 8 q^{69} - 18 q^{70} + q^{71} - 15 q^{72} - 14 q^{73} + 18 q^{74} - 16 q^{75} - 37 q^{76} - 17 q^{77} - 15 q^{78} - 80 q^{79} + 63 q^{80} + 41 q^{81} - 31 q^{82} + 20 q^{83} - 34 q^{84} - 82 q^{85} - 39 q^{86} + 27 q^{87} - 132 q^{88} + 17 q^{89} - 18 q^{90} - 38 q^{91} - 26 q^{92} + 23 q^{93} - 9 q^{94} - q^{95} + 24 q^{96} - 73 q^{97} - 4 q^{98} - 17 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38443 −1.68605 −0.843023 0.537877i \(-0.819226\pi\)
−0.843023 + 0.537877i \(0.819226\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.68550 1.84275
\(5\) 2.70331 1.20896 0.604478 0.796622i \(-0.293382\pi\)
0.604478 + 0.796622i \(0.293382\pi\)
\(6\) 2.38443 0.973439
\(7\) 1.00000 0.377964
\(8\) −4.01897 −1.42092
\(9\) 1.00000 0.333333
\(10\) −6.44585 −2.03836
\(11\) 2.12042 0.639331 0.319665 0.947531i \(-0.396430\pi\)
0.319665 + 0.947531i \(0.396430\pi\)
\(12\) −3.68550 −1.06391
\(13\) 0.628822 0.174404 0.0872019 0.996191i \(-0.472207\pi\)
0.0872019 + 0.996191i \(0.472207\pi\)
\(14\) −2.38443 −0.637266
\(15\) −2.70331 −0.697991
\(16\) 2.21194 0.552984
\(17\) −0.671193 −0.162788 −0.0813941 0.996682i \(-0.525937\pi\)
−0.0813941 + 0.996682i \(0.525937\pi\)
\(18\) −2.38443 −0.562015
\(19\) −4.69479 −1.07706 −0.538530 0.842606i \(-0.681020\pi\)
−0.538530 + 0.842606i \(0.681020\pi\)
\(20\) 9.96306 2.22781
\(21\) −1.00000 −0.218218
\(22\) −5.05599 −1.07794
\(23\) −4.25009 −0.886204 −0.443102 0.896471i \(-0.646122\pi\)
−0.443102 + 0.896471i \(0.646122\pi\)
\(24\) 4.01897 0.820368
\(25\) 2.30788 0.461576
\(26\) −1.49938 −0.294053
\(27\) −1.00000 −0.192450
\(28\) 3.68550 0.696495
\(29\) −8.77796 −1.63003 −0.815013 0.579443i \(-0.803270\pi\)
−0.815013 + 0.579443i \(0.803270\pi\)
\(30\) 6.44585 1.17685
\(31\) 8.26491 1.48442 0.742211 0.670167i \(-0.233777\pi\)
0.742211 + 0.670167i \(0.233777\pi\)
\(32\) 2.76373 0.488563
\(33\) −2.12042 −0.369118
\(34\) 1.60041 0.274468
\(35\) 2.70331 0.456943
\(36\) 3.68550 0.614251
\(37\) −2.88242 −0.473866 −0.236933 0.971526i \(-0.576142\pi\)
−0.236933 + 0.971526i \(0.576142\pi\)
\(38\) 11.1944 1.81597
\(39\) −0.628822 −0.100692
\(40\) −10.8645 −1.71783
\(41\) 0.950390 0.148426 0.0742130 0.997242i \(-0.476356\pi\)
0.0742130 + 0.997242i \(0.476356\pi\)
\(42\) 2.38443 0.367925
\(43\) −3.42276 −0.521967 −0.260983 0.965343i \(-0.584047\pi\)
−0.260983 + 0.965343i \(0.584047\pi\)
\(44\) 7.81482 1.17813
\(45\) 2.70331 0.402986
\(46\) 10.1340 1.49418
\(47\) 7.41532 1.08164 0.540818 0.841140i \(-0.318115\pi\)
0.540818 + 0.841140i \(0.318115\pi\)
\(48\) −2.21194 −0.319265
\(49\) 1.00000 0.142857
\(50\) −5.50298 −0.778239
\(51\) 0.671193 0.0939858
\(52\) 2.31753 0.321383
\(53\) 10.0416 1.37932 0.689659 0.724134i \(-0.257760\pi\)
0.689659 + 0.724134i \(0.257760\pi\)
\(54\) 2.38443 0.324480
\(55\) 5.73215 0.772923
\(56\) −4.01897 −0.537057
\(57\) 4.69479 0.621841
\(58\) 20.9304 2.74830
\(59\) −8.47742 −1.10367 −0.551833 0.833955i \(-0.686071\pi\)
−0.551833 + 0.833955i \(0.686071\pi\)
\(60\) −9.96306 −1.28623
\(61\) −8.41749 −1.07775 −0.538875 0.842386i \(-0.681150\pi\)
−0.538875 + 0.842386i \(0.681150\pi\)
\(62\) −19.7071 −2.50280
\(63\) 1.00000 0.125988
\(64\) −11.0138 −1.37672
\(65\) 1.69990 0.210847
\(66\) 5.05599 0.622350
\(67\) −0.248919 −0.0304103 −0.0152051 0.999884i \(-0.504840\pi\)
−0.0152051 + 0.999884i \(0.504840\pi\)
\(68\) −2.47368 −0.299978
\(69\) 4.25009 0.511650
\(70\) −6.44585 −0.770427
\(71\) −10.5508 −1.25215 −0.626073 0.779764i \(-0.715339\pi\)
−0.626073 + 0.779764i \(0.715339\pi\)
\(72\) −4.01897 −0.473640
\(73\) −6.70043 −0.784226 −0.392113 0.919917i \(-0.628256\pi\)
−0.392113 + 0.919917i \(0.628256\pi\)
\(74\) 6.87292 0.798960
\(75\) −2.30788 −0.266491
\(76\) −17.3027 −1.98475
\(77\) 2.12042 0.241644
\(78\) 1.49938 0.169771
\(79\) 10.9350 1.23029 0.615143 0.788415i \(-0.289098\pi\)
0.615143 + 0.788415i \(0.289098\pi\)
\(80\) 5.97955 0.668534
\(81\) 1.00000 0.111111
\(82\) −2.26614 −0.250253
\(83\) 1.52127 0.166981 0.0834907 0.996509i \(-0.473393\pi\)
0.0834907 + 0.996509i \(0.473393\pi\)
\(84\) −3.68550 −0.402122
\(85\) −1.81444 −0.196804
\(86\) 8.16134 0.880060
\(87\) 8.77796 0.941096
\(88\) −8.52190 −0.908437
\(89\) 9.66286 1.02426 0.512130 0.858908i \(-0.328856\pi\)
0.512130 + 0.858908i \(0.328856\pi\)
\(90\) −6.44585 −0.679452
\(91\) 0.628822 0.0659184
\(92\) −15.6637 −1.63305
\(93\) −8.26491 −0.857031
\(94\) −17.6813 −1.82369
\(95\) −12.6915 −1.30212
\(96\) −2.76373 −0.282072
\(97\) −0.763992 −0.0775716 −0.0387858 0.999248i \(-0.512349\pi\)
−0.0387858 + 0.999248i \(0.512349\pi\)
\(98\) −2.38443 −0.240864
\(99\) 2.12042 0.213110
\(100\) 8.50571 0.850571
\(101\) −1.38949 −0.138259 −0.0691295 0.997608i \(-0.522022\pi\)
−0.0691295 + 0.997608i \(0.522022\pi\)
\(102\) −1.60041 −0.158464
\(103\) −17.9914 −1.77274 −0.886371 0.462975i \(-0.846782\pi\)
−0.886371 + 0.462975i \(0.846782\pi\)
\(104\) −2.52721 −0.247814
\(105\) −2.70331 −0.263816
\(106\) −23.9435 −2.32559
\(107\) −5.00643 −0.483990 −0.241995 0.970277i \(-0.577802\pi\)
−0.241995 + 0.970277i \(0.577802\pi\)
\(108\) −3.68550 −0.354638
\(109\) 3.95320 0.378648 0.189324 0.981915i \(-0.439370\pi\)
0.189324 + 0.981915i \(0.439370\pi\)
\(110\) −13.6679 −1.30318
\(111\) 2.88242 0.273587
\(112\) 2.21194 0.209008
\(113\) 5.42209 0.510067 0.255034 0.966932i \(-0.417913\pi\)
0.255034 + 0.966932i \(0.417913\pi\)
\(114\) −11.1944 −1.04845
\(115\) −11.4893 −1.07138
\(116\) −32.3512 −3.00373
\(117\) 0.628822 0.0581346
\(118\) 20.2138 1.86083
\(119\) −0.671193 −0.0615282
\(120\) 10.8645 0.991790
\(121\) −6.50382 −0.591256
\(122\) 20.0709 1.81714
\(123\) −0.950390 −0.0856938
\(124\) 30.4604 2.73542
\(125\) −7.27763 −0.650931
\(126\) −2.38443 −0.212422
\(127\) −4.21917 −0.374391 −0.187196 0.982323i \(-0.559940\pi\)
−0.187196 + 0.982323i \(0.559940\pi\)
\(128\) 20.7341 1.83266
\(129\) 3.42276 0.301358
\(130\) −4.05329 −0.355497
\(131\) 8.52713 0.745018 0.372509 0.928028i \(-0.378497\pi\)
0.372509 + 0.928028i \(0.378497\pi\)
\(132\) −7.81482 −0.680193
\(133\) −4.69479 −0.407090
\(134\) 0.593529 0.0512731
\(135\) −2.70331 −0.232664
\(136\) 2.69750 0.231309
\(137\) −7.13079 −0.609225 −0.304612 0.952476i \(-0.598527\pi\)
−0.304612 + 0.952476i \(0.598527\pi\)
\(138\) −10.1340 −0.862666
\(139\) 4.66223 0.395445 0.197722 0.980258i \(-0.436646\pi\)
0.197722 + 0.980258i \(0.436646\pi\)
\(140\) 9.96306 0.842032
\(141\) −7.41532 −0.624483
\(142\) 25.1576 2.11118
\(143\) 1.33337 0.111502
\(144\) 2.21194 0.184328
\(145\) −23.7295 −1.97063
\(146\) 15.9767 1.32224
\(147\) −1.00000 −0.0824786
\(148\) −10.6232 −0.873218
\(149\) 15.9461 1.30635 0.653177 0.757206i \(-0.273436\pi\)
0.653177 + 0.757206i \(0.273436\pi\)
\(150\) 5.50298 0.449317
\(151\) 19.9621 1.62449 0.812247 0.583314i \(-0.198244\pi\)
0.812247 + 0.583314i \(0.198244\pi\)
\(152\) 18.8682 1.53042
\(153\) −0.671193 −0.0542627
\(154\) −5.05599 −0.407423
\(155\) 22.3426 1.79460
\(156\) −2.31753 −0.185551
\(157\) −15.4079 −1.22968 −0.614841 0.788651i \(-0.710780\pi\)
−0.614841 + 0.788651i \(0.710780\pi\)
\(158\) −26.0738 −2.07432
\(159\) −10.0416 −0.796350
\(160\) 7.47122 0.590652
\(161\) −4.25009 −0.334954
\(162\) −2.38443 −0.187338
\(163\) −23.8685 −1.86952 −0.934761 0.355276i \(-0.884387\pi\)
−0.934761 + 0.355276i \(0.884387\pi\)
\(164\) 3.50267 0.273512
\(165\) −5.73215 −0.446247
\(166\) −3.62737 −0.281538
\(167\) 11.1535 0.863080 0.431540 0.902094i \(-0.357970\pi\)
0.431540 + 0.902094i \(0.357970\pi\)
\(168\) 4.01897 0.310070
\(169\) −12.6046 −0.969583
\(170\) 4.32641 0.331820
\(171\) −4.69479 −0.359020
\(172\) −12.6146 −0.961855
\(173\) −22.9710 −1.74645 −0.873227 0.487314i \(-0.837977\pi\)
−0.873227 + 0.487314i \(0.837977\pi\)
\(174\) −20.9304 −1.58673
\(175\) 2.30788 0.174459
\(176\) 4.69023 0.353540
\(177\) 8.47742 0.637202
\(178\) −23.0404 −1.72695
\(179\) −26.0779 −1.94916 −0.974578 0.224049i \(-0.928072\pi\)
−0.974578 + 0.224049i \(0.928072\pi\)
\(180\) 9.96306 0.742603
\(181\) −22.8074 −1.69526 −0.847631 0.530586i \(-0.821972\pi\)
−0.847631 + 0.530586i \(0.821972\pi\)
\(182\) −1.49938 −0.111142
\(183\) 8.41749 0.622239
\(184\) 17.0810 1.25922
\(185\) −7.79206 −0.572884
\(186\) 19.7071 1.44499
\(187\) −1.42321 −0.104075
\(188\) 27.3292 1.99319
\(189\) −1.00000 −0.0727393
\(190\) 30.2619 2.19543
\(191\) 6.76855 0.489755 0.244877 0.969554i \(-0.421252\pi\)
0.244877 + 0.969554i \(0.421252\pi\)
\(192\) 11.0138 0.794852
\(193\) −21.9063 −1.57685 −0.788424 0.615133i \(-0.789102\pi\)
−0.788424 + 0.615133i \(0.789102\pi\)
\(194\) 1.82169 0.130789
\(195\) −1.69990 −0.121732
\(196\) 3.68550 0.263250
\(197\) −4.14688 −0.295453 −0.147726 0.989028i \(-0.547195\pi\)
−0.147726 + 0.989028i \(0.547195\pi\)
\(198\) −5.05599 −0.359314
\(199\) −14.8300 −1.05127 −0.525635 0.850710i \(-0.676172\pi\)
−0.525635 + 0.850710i \(0.676172\pi\)
\(200\) −9.27530 −0.655863
\(201\) 0.248919 0.0175574
\(202\) 3.31313 0.233111
\(203\) −8.77796 −0.616092
\(204\) 2.47368 0.173193
\(205\) 2.56920 0.179441
\(206\) 42.8992 2.98893
\(207\) −4.25009 −0.295401
\(208\) 1.39091 0.0964425
\(209\) −9.95494 −0.688597
\(210\) 6.44585 0.444806
\(211\) 19.0732 1.31306 0.656528 0.754302i \(-0.272025\pi\)
0.656528 + 0.754302i \(0.272025\pi\)
\(212\) 37.0083 2.54174
\(213\) 10.5508 0.722927
\(214\) 11.9375 0.816030
\(215\) −9.25279 −0.631035
\(216\) 4.01897 0.273456
\(217\) 8.26491 0.561058
\(218\) −9.42613 −0.638418
\(219\) 6.70043 0.452773
\(220\) 21.1259 1.42431
\(221\) −0.422061 −0.0283909
\(222\) −6.87292 −0.461280
\(223\) −13.9926 −0.937014 −0.468507 0.883460i \(-0.655208\pi\)
−0.468507 + 0.883460i \(0.655208\pi\)
\(224\) 2.76373 0.184659
\(225\) 2.30788 0.153859
\(226\) −12.9286 −0.859997
\(227\) −26.7515 −1.77556 −0.887780 0.460268i \(-0.847753\pi\)
−0.887780 + 0.460268i \(0.847753\pi\)
\(228\) 17.3027 1.14590
\(229\) −20.5408 −1.35737 −0.678686 0.734428i \(-0.737450\pi\)
−0.678686 + 0.734428i \(0.737450\pi\)
\(230\) 27.3954 1.80640
\(231\) −2.12042 −0.139513
\(232\) 35.2783 2.31614
\(233\) 28.9276 1.89511 0.947556 0.319589i \(-0.103545\pi\)
0.947556 + 0.319589i \(0.103545\pi\)
\(234\) −1.49938 −0.0980176
\(235\) 20.0459 1.30765
\(236\) −31.2436 −2.03378
\(237\) −10.9350 −0.710306
\(238\) 1.60041 0.103739
\(239\) 11.0244 0.713108 0.356554 0.934275i \(-0.383952\pi\)
0.356554 + 0.934275i \(0.383952\pi\)
\(240\) −5.97955 −0.385978
\(241\) −0.651087 −0.0419402 −0.0209701 0.999780i \(-0.506675\pi\)
−0.0209701 + 0.999780i \(0.506675\pi\)
\(242\) 15.5079 0.996885
\(243\) −1.00000 −0.0641500
\(244\) −31.0227 −1.98602
\(245\) 2.70331 0.172708
\(246\) 2.26614 0.144484
\(247\) −2.95219 −0.187843
\(248\) −33.2164 −2.10924
\(249\) −1.52127 −0.0964067
\(250\) 17.3530 1.09750
\(251\) 25.1078 1.58479 0.792396 0.610007i \(-0.208833\pi\)
0.792396 + 0.610007i \(0.208833\pi\)
\(252\) 3.68550 0.232165
\(253\) −9.01197 −0.566577
\(254\) 10.0603 0.631241
\(255\) 1.81444 0.113625
\(256\) −27.4115 −1.71322
\(257\) 21.4888 1.34044 0.670218 0.742164i \(-0.266201\pi\)
0.670218 + 0.742164i \(0.266201\pi\)
\(258\) −8.16134 −0.508103
\(259\) −2.88242 −0.179105
\(260\) 6.26499 0.388538
\(261\) −8.77796 −0.543342
\(262\) −20.3323 −1.25614
\(263\) −8.15338 −0.502759 −0.251380 0.967889i \(-0.580884\pi\)
−0.251380 + 0.967889i \(0.580884\pi\)
\(264\) 8.52190 0.524487
\(265\) 27.1455 1.66754
\(266\) 11.1944 0.686373
\(267\) −9.66286 −0.591357
\(268\) −0.917391 −0.0560386
\(269\) 4.11605 0.250960 0.125480 0.992096i \(-0.459953\pi\)
0.125480 + 0.992096i \(0.459953\pi\)
\(270\) 6.44585 0.392282
\(271\) 6.69170 0.406492 0.203246 0.979128i \(-0.434851\pi\)
0.203246 + 0.979128i \(0.434851\pi\)
\(272\) −1.48464 −0.0900192
\(273\) −0.628822 −0.0380580
\(274\) 17.0029 1.02718
\(275\) 4.89368 0.295100
\(276\) 15.6637 0.942844
\(277\) −3.57687 −0.214913 −0.107457 0.994210i \(-0.534271\pi\)
−0.107457 + 0.994210i \(0.534271\pi\)
\(278\) −11.1168 −0.666739
\(279\) 8.26491 0.494807
\(280\) −10.8645 −0.649279
\(281\) 19.9376 1.18938 0.594688 0.803957i \(-0.297276\pi\)
0.594688 + 0.803957i \(0.297276\pi\)
\(282\) 17.6813 1.05291
\(283\) 4.01533 0.238687 0.119343 0.992853i \(-0.461921\pi\)
0.119343 + 0.992853i \(0.461921\pi\)
\(284\) −38.8849 −2.30740
\(285\) 12.6915 0.751779
\(286\) −3.17932 −0.187997
\(287\) 0.950390 0.0560998
\(288\) 2.76373 0.162854
\(289\) −16.5495 −0.973500
\(290\) 56.5814 3.32258
\(291\) 0.763992 0.0447860
\(292\) −24.6945 −1.44513
\(293\) −26.9013 −1.57159 −0.785795 0.618487i \(-0.787746\pi\)
−0.785795 + 0.618487i \(0.787746\pi\)
\(294\) 2.38443 0.139063
\(295\) −22.9171 −1.33428
\(296\) 11.5843 0.673326
\(297\) −2.12042 −0.123039
\(298\) −38.0223 −2.20257
\(299\) −2.67255 −0.154557
\(300\) −8.50571 −0.491077
\(301\) −3.42276 −0.197285
\(302\) −47.5983 −2.73897
\(303\) 1.38949 0.0798239
\(304\) −10.3846 −0.595597
\(305\) −22.7551 −1.30295
\(306\) 1.60041 0.0914895
\(307\) −24.9984 −1.42673 −0.713367 0.700791i \(-0.752831\pi\)
−0.713367 + 0.700791i \(0.752831\pi\)
\(308\) 7.81482 0.445291
\(309\) 17.9914 1.02349
\(310\) −53.2744 −3.02578
\(311\) 7.01552 0.397814 0.198907 0.980018i \(-0.436261\pi\)
0.198907 + 0.980018i \(0.436261\pi\)
\(312\) 2.52721 0.143075
\(313\) −8.08016 −0.456717 −0.228359 0.973577i \(-0.573336\pi\)
−0.228359 + 0.973577i \(0.573336\pi\)
\(314\) 36.7390 2.07330
\(315\) 2.70331 0.152314
\(316\) 40.3011 2.26711
\(317\) −8.21015 −0.461128 −0.230564 0.973057i \(-0.574057\pi\)
−0.230564 + 0.973057i \(0.574057\pi\)
\(318\) 23.9435 1.34268
\(319\) −18.6130 −1.04213
\(320\) −29.7737 −1.66440
\(321\) 5.00643 0.279432
\(322\) 10.1340 0.564747
\(323\) 3.15111 0.175333
\(324\) 3.68550 0.204750
\(325\) 1.45125 0.0805006
\(326\) 56.9127 3.15210
\(327\) −3.95320 −0.218613
\(328\) −3.81959 −0.210901
\(329\) 7.41532 0.408820
\(330\) 13.6679 0.752394
\(331\) 33.0162 1.81474 0.907369 0.420336i \(-0.138088\pi\)
0.907369 + 0.420336i \(0.138088\pi\)
\(332\) 5.60666 0.307705
\(333\) −2.88242 −0.157955
\(334\) −26.5946 −1.45519
\(335\) −0.672904 −0.0367647
\(336\) −2.21194 −0.120671
\(337\) 6.59010 0.358986 0.179493 0.983759i \(-0.442554\pi\)
0.179493 + 0.983759i \(0.442554\pi\)
\(338\) 30.0547 1.63476
\(339\) −5.42209 −0.294488
\(340\) −6.68714 −0.362661
\(341\) 17.5251 0.949036
\(342\) 11.1944 0.605324
\(343\) 1.00000 0.0539949
\(344\) 13.7560 0.741673
\(345\) 11.4893 0.618563
\(346\) 54.7727 2.94460
\(347\) −10.2143 −0.548330 −0.274165 0.961683i \(-0.588401\pi\)
−0.274165 + 0.961683i \(0.588401\pi\)
\(348\) 32.3512 1.73421
\(349\) 0.804067 0.0430407 0.0215204 0.999768i \(-0.493149\pi\)
0.0215204 + 0.999768i \(0.493149\pi\)
\(350\) −5.50298 −0.294147
\(351\) −0.628822 −0.0335640
\(352\) 5.86027 0.312353
\(353\) 8.50057 0.452440 0.226220 0.974076i \(-0.427363\pi\)
0.226220 + 0.974076i \(0.427363\pi\)
\(354\) −20.2138 −1.07435
\(355\) −28.5220 −1.51379
\(356\) 35.6125 1.88746
\(357\) 0.671193 0.0355233
\(358\) 62.1810 3.28637
\(359\) 19.1932 1.01298 0.506489 0.862247i \(-0.330943\pi\)
0.506489 + 0.862247i \(0.330943\pi\)
\(360\) −10.8645 −0.572610
\(361\) 3.04109 0.160058
\(362\) 54.3827 2.85829
\(363\) 6.50382 0.341362
\(364\) 2.31753 0.121471
\(365\) −18.1133 −0.948095
\(366\) −20.0709 −1.04912
\(367\) 28.5258 1.48903 0.744516 0.667604i \(-0.232680\pi\)
0.744516 + 0.667604i \(0.232680\pi\)
\(368\) −9.40091 −0.490057
\(369\) 0.950390 0.0494753
\(370\) 18.5796 0.965909
\(371\) 10.0416 0.521333
\(372\) −30.4604 −1.57930
\(373\) 29.2428 1.51414 0.757068 0.653336i \(-0.226631\pi\)
0.757068 + 0.653336i \(0.226631\pi\)
\(374\) 3.39355 0.175476
\(375\) 7.27763 0.375815
\(376\) −29.8019 −1.53692
\(377\) −5.51977 −0.284283
\(378\) 2.38443 0.122642
\(379\) −12.7962 −0.657297 −0.328649 0.944452i \(-0.606593\pi\)
−0.328649 + 0.944452i \(0.606593\pi\)
\(380\) −46.7745 −2.39948
\(381\) 4.21917 0.216155
\(382\) −16.1391 −0.825749
\(383\) 1.00000 0.0510976
\(384\) −20.7341 −1.05809
\(385\) 5.73215 0.292137
\(386\) 52.2339 2.65864
\(387\) −3.42276 −0.173989
\(388\) −2.81570 −0.142945
\(389\) 33.3570 1.69127 0.845634 0.533763i \(-0.179222\pi\)
0.845634 + 0.533763i \(0.179222\pi\)
\(390\) 4.05329 0.205246
\(391\) 2.85263 0.144264
\(392\) −4.01897 −0.202989
\(393\) −8.52713 −0.430137
\(394\) 9.88794 0.498147
\(395\) 29.5608 1.48736
\(396\) 7.81482 0.392709
\(397\) 27.3629 1.37330 0.686651 0.726987i \(-0.259080\pi\)
0.686651 + 0.726987i \(0.259080\pi\)
\(398\) 35.3611 1.77249
\(399\) 4.69479 0.235034
\(400\) 5.10489 0.255244
\(401\) −8.80353 −0.439627 −0.219814 0.975542i \(-0.570545\pi\)
−0.219814 + 0.975542i \(0.570545\pi\)
\(402\) −0.593529 −0.0296025
\(403\) 5.19715 0.258889
\(404\) −5.12096 −0.254777
\(405\) 2.70331 0.134329
\(406\) 20.9304 1.03876
\(407\) −6.11193 −0.302957
\(408\) −2.69750 −0.133546
\(409\) −1.53212 −0.0757586 −0.0378793 0.999282i \(-0.512060\pi\)
−0.0378793 + 0.999282i \(0.512060\pi\)
\(410\) −6.12607 −0.302545
\(411\) 7.13079 0.351736
\(412\) −66.3073 −3.26672
\(413\) −8.47742 −0.417147
\(414\) 10.1340 0.498060
\(415\) 4.11247 0.201873
\(416\) 1.73789 0.0852072
\(417\) −4.66223 −0.228310
\(418\) 23.7368 1.16101
\(419\) −13.4647 −0.657796 −0.328898 0.944365i \(-0.606677\pi\)
−0.328898 + 0.944365i \(0.606677\pi\)
\(420\) −9.96306 −0.486148
\(421\) −29.2953 −1.42777 −0.713883 0.700265i \(-0.753065\pi\)
−0.713883 + 0.700265i \(0.753065\pi\)
\(422\) −45.4788 −2.21387
\(423\) 7.41532 0.360545
\(424\) −40.3568 −1.95990
\(425\) −1.54903 −0.0751392
\(426\) −25.1576 −1.21889
\(427\) −8.41749 −0.407351
\(428\) −18.4512 −0.891874
\(429\) −1.33337 −0.0643755
\(430\) 22.0626 1.06395
\(431\) −23.9157 −1.15198 −0.575990 0.817457i \(-0.695383\pi\)
−0.575990 + 0.817457i \(0.695383\pi\)
\(432\) −2.21194 −0.106422
\(433\) −32.7290 −1.57286 −0.786429 0.617681i \(-0.788072\pi\)
−0.786429 + 0.617681i \(0.788072\pi\)
\(434\) −19.7071 −0.945971
\(435\) 23.7295 1.13774
\(436\) 14.5695 0.697755
\(437\) 19.9533 0.954495
\(438\) −15.9767 −0.763396
\(439\) 33.1445 1.58190 0.790951 0.611879i \(-0.209586\pi\)
0.790951 + 0.611879i \(0.209586\pi\)
\(440\) −23.0373 −1.09826
\(441\) 1.00000 0.0476190
\(442\) 1.00637 0.0478683
\(443\) −29.5375 −1.40337 −0.701685 0.712487i \(-0.747569\pi\)
−0.701685 + 0.712487i \(0.747569\pi\)
\(444\) 10.6232 0.504153
\(445\) 26.1217 1.23829
\(446\) 33.3644 1.57985
\(447\) −15.9461 −0.754223
\(448\) −11.0138 −0.520353
\(449\) 8.53233 0.402666 0.201333 0.979523i \(-0.435473\pi\)
0.201333 + 0.979523i \(0.435473\pi\)
\(450\) −5.50298 −0.259413
\(451\) 2.01523 0.0948933
\(452\) 19.9831 0.939928
\(453\) −19.9621 −0.937902
\(454\) 63.7871 2.99368
\(455\) 1.69990 0.0796925
\(456\) −18.8682 −0.883586
\(457\) −2.88815 −0.135102 −0.0675510 0.997716i \(-0.521519\pi\)
−0.0675510 + 0.997716i \(0.521519\pi\)
\(458\) 48.9780 2.28859
\(459\) 0.671193 0.0313286
\(460\) −42.3439 −1.97429
\(461\) 16.8996 0.787095 0.393547 0.919304i \(-0.371248\pi\)
0.393547 + 0.919304i \(0.371248\pi\)
\(462\) 5.05599 0.235226
\(463\) 13.0716 0.607490 0.303745 0.952753i \(-0.401763\pi\)
0.303745 + 0.952753i \(0.401763\pi\)
\(464\) −19.4163 −0.901378
\(465\) −22.3426 −1.03611
\(466\) −68.9759 −3.19525
\(467\) −3.86656 −0.178923 −0.0894616 0.995990i \(-0.528515\pi\)
−0.0894616 + 0.995990i \(0.528515\pi\)
\(468\) 2.31753 0.107128
\(469\) −0.248919 −0.0114940
\(470\) −47.7981 −2.20476
\(471\) 15.4079 0.709957
\(472\) 34.0705 1.56822
\(473\) −7.25770 −0.333709
\(474\) 26.0738 1.19761
\(475\) −10.8350 −0.497145
\(476\) −2.47368 −0.113381
\(477\) 10.0416 0.459773
\(478\) −26.2869 −1.20233
\(479\) 34.4647 1.57473 0.787367 0.616485i \(-0.211444\pi\)
0.787367 + 0.616485i \(0.211444\pi\)
\(480\) −7.47122 −0.341013
\(481\) −1.81253 −0.0826440
\(482\) 1.55247 0.0707132
\(483\) 4.25009 0.193386
\(484\) −23.9699 −1.08954
\(485\) −2.06531 −0.0937808
\(486\) 2.38443 0.108160
\(487\) 10.0622 0.455964 0.227982 0.973665i \(-0.426787\pi\)
0.227982 + 0.973665i \(0.426787\pi\)
\(488\) 33.8296 1.53139
\(489\) 23.8685 1.07937
\(490\) −6.44585 −0.291194
\(491\) 28.8117 1.30026 0.650128 0.759825i \(-0.274715\pi\)
0.650128 + 0.759825i \(0.274715\pi\)
\(492\) −3.50267 −0.157912
\(493\) 5.89170 0.265349
\(494\) 7.03929 0.316712
\(495\) 5.73215 0.257641
\(496\) 18.2814 0.820861
\(497\) −10.5508 −0.473267
\(498\) 3.62737 0.162546
\(499\) −34.4603 −1.54265 −0.771327 0.636439i \(-0.780407\pi\)
−0.771327 + 0.636439i \(0.780407\pi\)
\(500\) −26.8217 −1.19950
\(501\) −11.1535 −0.498300
\(502\) −59.8678 −2.67203
\(503\) −11.2997 −0.503831 −0.251915 0.967749i \(-0.581060\pi\)
−0.251915 + 0.967749i \(0.581060\pi\)
\(504\) −4.01897 −0.179019
\(505\) −3.75621 −0.167149
\(506\) 21.4884 0.955276
\(507\) 12.6046 0.559789
\(508\) −15.5498 −0.689910
\(509\) −23.9688 −1.06240 −0.531200 0.847247i \(-0.678259\pi\)
−0.531200 + 0.847247i \(0.678259\pi\)
\(510\) −4.32641 −0.191577
\(511\) −6.70043 −0.296410
\(512\) 23.8926 1.05591
\(513\) 4.69479 0.207280
\(514\) −51.2386 −2.26004
\(515\) −48.6362 −2.14317
\(516\) 12.6146 0.555327
\(517\) 15.7236 0.691523
\(518\) 6.87292 0.301979
\(519\) 22.9710 1.00832
\(520\) −6.83184 −0.299596
\(521\) 13.4663 0.589968 0.294984 0.955502i \(-0.404686\pi\)
0.294984 + 0.955502i \(0.404686\pi\)
\(522\) 20.9304 0.916100
\(523\) −29.1859 −1.27621 −0.638106 0.769949i \(-0.720282\pi\)
−0.638106 + 0.769949i \(0.720282\pi\)
\(524\) 31.4268 1.37288
\(525\) −2.30788 −0.100724
\(526\) 19.4412 0.847675
\(527\) −5.54735 −0.241646
\(528\) −4.69023 −0.204116
\(529\) −4.93678 −0.214642
\(530\) −64.7266 −2.81154
\(531\) −8.47742 −0.367889
\(532\) −17.3027 −0.750167
\(533\) 0.597626 0.0258861
\(534\) 23.0404 0.997056
\(535\) −13.5339 −0.585123
\(536\) 1.00040 0.0432105
\(537\) 26.0779 1.12535
\(538\) −9.81443 −0.423130
\(539\) 2.12042 0.0913330
\(540\) −9.96306 −0.428742
\(541\) 41.4908 1.78383 0.891914 0.452205i \(-0.149362\pi\)
0.891914 + 0.452205i \(0.149362\pi\)
\(542\) −15.9559 −0.685364
\(543\) 22.8074 0.978760
\(544\) −1.85500 −0.0795323
\(545\) 10.6867 0.457769
\(546\) 1.49938 0.0641676
\(547\) −41.8908 −1.79112 −0.895561 0.444939i \(-0.853225\pi\)
−0.895561 + 0.444939i \(0.853225\pi\)
\(548\) −26.2806 −1.12265
\(549\) −8.41749 −0.359250
\(550\) −11.6686 −0.497552
\(551\) 41.2107 1.75564
\(552\) −17.0810 −0.727014
\(553\) 10.9350 0.465005
\(554\) 8.52880 0.362354
\(555\) 7.79206 0.330755
\(556\) 17.1827 0.728707
\(557\) −19.9302 −0.844470 −0.422235 0.906486i \(-0.638754\pi\)
−0.422235 + 0.906486i \(0.638754\pi\)
\(558\) −19.7071 −0.834268
\(559\) −2.15231 −0.0910329
\(560\) 5.97955 0.252682
\(561\) 1.42321 0.0600880
\(562\) −47.5397 −2.00534
\(563\) 22.5664 0.951062 0.475531 0.879699i \(-0.342256\pi\)
0.475531 + 0.879699i \(0.342256\pi\)
\(564\) −27.3292 −1.15077
\(565\) 14.6576 0.616649
\(566\) −9.57427 −0.402437
\(567\) 1.00000 0.0419961
\(568\) 42.4032 1.77920
\(569\) 4.21106 0.176537 0.0882685 0.996097i \(-0.471867\pi\)
0.0882685 + 0.996097i \(0.471867\pi\)
\(570\) −30.2619 −1.26753
\(571\) 1.55138 0.0649231 0.0324615 0.999473i \(-0.489665\pi\)
0.0324615 + 0.999473i \(0.489665\pi\)
\(572\) 4.91413 0.205470
\(573\) −6.76855 −0.282760
\(574\) −2.26614 −0.0945868
\(575\) −9.80869 −0.409051
\(576\) −11.0138 −0.458908
\(577\) −32.8411 −1.36719 −0.683597 0.729859i \(-0.739586\pi\)
−0.683597 + 0.729859i \(0.739586\pi\)
\(578\) 39.4611 1.64137
\(579\) 21.9063 0.910393
\(580\) −87.4553 −3.63139
\(581\) 1.52127 0.0631130
\(582\) −1.82169 −0.0755113
\(583\) 21.2924 0.881840
\(584\) 26.9288 1.11432
\(585\) 1.69990 0.0702822
\(586\) 64.1443 2.64977
\(587\) 38.4391 1.58655 0.793277 0.608861i \(-0.208373\pi\)
0.793277 + 0.608861i \(0.208373\pi\)
\(588\) −3.68550 −0.151988
\(589\) −38.8020 −1.59881
\(590\) 54.6442 2.24967
\(591\) 4.14688 0.170580
\(592\) −6.37572 −0.262040
\(593\) 15.7194 0.645520 0.322760 0.946481i \(-0.395389\pi\)
0.322760 + 0.946481i \(0.395389\pi\)
\(594\) 5.05599 0.207450
\(595\) −1.81444 −0.0743849
\(596\) 58.7693 2.40729
\(597\) 14.8300 0.606952
\(598\) 6.37250 0.260591
\(599\) −25.2442 −1.03145 −0.515724 0.856755i \(-0.672477\pi\)
−0.515724 + 0.856755i \(0.672477\pi\)
\(600\) 9.27530 0.378663
\(601\) −40.7117 −1.66066 −0.830332 0.557269i \(-0.811849\pi\)
−0.830332 + 0.557269i \(0.811849\pi\)
\(602\) 8.16134 0.332631
\(603\) −0.248919 −0.0101368
\(604\) 73.5705 2.99354
\(605\) −17.5818 −0.714803
\(606\) −3.31313 −0.134587
\(607\) −39.9451 −1.62132 −0.810661 0.585516i \(-0.800892\pi\)
−0.810661 + 0.585516i \(0.800892\pi\)
\(608\) −12.9751 −0.526212
\(609\) 8.77796 0.355701
\(610\) 54.2579 2.19684
\(611\) 4.66292 0.188641
\(612\) −2.47368 −0.0999928
\(613\) −5.38078 −0.217328 −0.108664 0.994079i \(-0.534657\pi\)
−0.108664 + 0.994079i \(0.534657\pi\)
\(614\) 59.6069 2.40554
\(615\) −2.56920 −0.103600
\(616\) −8.52190 −0.343357
\(617\) −37.4738 −1.50864 −0.754320 0.656507i \(-0.772033\pi\)
−0.754320 + 0.656507i \(0.772033\pi\)
\(618\) −42.8992 −1.72566
\(619\) −6.64696 −0.267164 −0.133582 0.991038i \(-0.542648\pi\)
−0.133582 + 0.991038i \(0.542648\pi\)
\(620\) 82.3438 3.30701
\(621\) 4.25009 0.170550
\(622\) −16.7280 −0.670732
\(623\) 9.66286 0.387134
\(624\) −1.39091 −0.0556811
\(625\) −31.2131 −1.24852
\(626\) 19.2666 0.770047
\(627\) 9.95494 0.397562
\(628\) −56.7858 −2.26600
\(629\) 1.93466 0.0771398
\(630\) −6.44585 −0.256809
\(631\) −35.4223 −1.41014 −0.705070 0.709137i \(-0.749085\pi\)
−0.705070 + 0.709137i \(0.749085\pi\)
\(632\) −43.9475 −1.74814
\(633\) −19.0732 −0.758093
\(634\) 19.5765 0.777483
\(635\) −11.4057 −0.452623
\(636\) −37.0083 −1.46748
\(637\) 0.628822 0.0249148
\(638\) 44.3813 1.75707
\(639\) −10.5508 −0.417382
\(640\) 56.0508 2.21560
\(641\) −35.4173 −1.39890 −0.699450 0.714682i \(-0.746572\pi\)
−0.699450 + 0.714682i \(0.746572\pi\)
\(642\) −11.9375 −0.471135
\(643\) 0.0772237 0.00304541 0.00152270 0.999999i \(-0.499515\pi\)
0.00152270 + 0.999999i \(0.499515\pi\)
\(644\) −15.6637 −0.617237
\(645\) 9.25279 0.364328
\(646\) −7.51361 −0.295619
\(647\) 36.9595 1.45303 0.726513 0.687153i \(-0.241140\pi\)
0.726513 + 0.687153i \(0.241140\pi\)
\(648\) −4.01897 −0.157880
\(649\) −17.9757 −0.705608
\(650\) −3.46039 −0.135728
\(651\) −8.26491 −0.323927
\(652\) −87.9673 −3.44507
\(653\) −21.2583 −0.831900 −0.415950 0.909388i \(-0.636551\pi\)
−0.415950 + 0.909388i \(0.636551\pi\)
\(654\) 9.42613 0.368591
\(655\) 23.0515 0.900695
\(656\) 2.10220 0.0820772
\(657\) −6.70043 −0.261409
\(658\) −17.6813 −0.689289
\(659\) −15.4509 −0.601882 −0.300941 0.953643i \(-0.597301\pi\)
−0.300941 + 0.953643i \(0.597301\pi\)
\(660\) −21.1259 −0.822323
\(661\) −36.3074 −1.41220 −0.706098 0.708114i \(-0.749546\pi\)
−0.706098 + 0.708114i \(0.749546\pi\)
\(662\) −78.7249 −3.05973
\(663\) 0.422061 0.0163915
\(664\) −6.11394 −0.237267
\(665\) −12.6915 −0.492155
\(666\) 6.87292 0.266320
\(667\) 37.3071 1.44454
\(668\) 41.1061 1.59044
\(669\) 13.9926 0.540985
\(670\) 1.60449 0.0619870
\(671\) −17.8486 −0.689038
\(672\) −2.76373 −0.106613
\(673\) 31.8991 1.22962 0.614810 0.788675i \(-0.289233\pi\)
0.614810 + 0.788675i \(0.289233\pi\)
\(674\) −15.7136 −0.605267
\(675\) −2.30788 −0.0888304
\(676\) −46.4543 −1.78670
\(677\) −3.10925 −0.119498 −0.0597491 0.998213i \(-0.519030\pi\)
−0.0597491 + 0.998213i \(0.519030\pi\)
\(678\) 12.9286 0.496520
\(679\) −0.763992 −0.0293193
\(680\) 7.29218 0.279642
\(681\) 26.7515 1.02512
\(682\) −41.7873 −1.60012
\(683\) −20.2933 −0.776502 −0.388251 0.921554i \(-0.626921\pi\)
−0.388251 + 0.921554i \(0.626921\pi\)
\(684\) −17.3027 −0.661585
\(685\) −19.2767 −0.736526
\(686\) −2.38443 −0.0910379
\(687\) 20.5408 0.783679
\(688\) −7.57093 −0.288639
\(689\) 6.31437 0.240558
\(690\) −27.3954 −1.04293
\(691\) 44.0194 1.67458 0.837288 0.546763i \(-0.184140\pi\)
0.837288 + 0.546763i \(0.184140\pi\)
\(692\) −84.6597 −3.21828
\(693\) 2.12042 0.0805481
\(694\) 24.3552 0.924510
\(695\) 12.6034 0.478076
\(696\) −35.2783 −1.33722
\(697\) −0.637895 −0.0241620
\(698\) −1.91724 −0.0725686
\(699\) −28.9276 −1.09414
\(700\) 8.50571 0.321486
\(701\) 21.2660 0.803206 0.401603 0.915814i \(-0.368453\pi\)
0.401603 + 0.915814i \(0.368453\pi\)
\(702\) 1.49938 0.0565905
\(703\) 13.5323 0.510382
\(704\) −23.3539 −0.880182
\(705\) −20.0459 −0.754973
\(706\) −20.2690 −0.762835
\(707\) −1.38949 −0.0522570
\(708\) 31.2436 1.17421
\(709\) −20.4536 −0.768150 −0.384075 0.923302i \(-0.625480\pi\)
−0.384075 + 0.923302i \(0.625480\pi\)
\(710\) 68.0087 2.55232
\(711\) 10.9350 0.410095
\(712\) −38.8347 −1.45539
\(713\) −35.1266 −1.31550
\(714\) −1.60041 −0.0598939
\(715\) 3.60450 0.134801
\(716\) −96.1103 −3.59181
\(717\) −11.0244 −0.411713
\(718\) −45.7648 −1.70793
\(719\) 47.4720 1.77041 0.885203 0.465205i \(-0.154020\pi\)
0.885203 + 0.465205i \(0.154020\pi\)
\(720\) 5.97955 0.222845
\(721\) −17.9914 −0.670034
\(722\) −7.25128 −0.269865
\(723\) 0.651087 0.0242142
\(724\) −84.0568 −3.12395
\(725\) −20.2585 −0.752382
\(726\) −15.5079 −0.575552
\(727\) 36.1902 1.34222 0.671110 0.741358i \(-0.265818\pi\)
0.671110 + 0.741358i \(0.265818\pi\)
\(728\) −2.52721 −0.0936648
\(729\) 1.00000 0.0370370
\(730\) 43.1900 1.59853
\(731\) 2.29734 0.0849700
\(732\) 31.0227 1.14663
\(733\) −22.8890 −0.845424 −0.422712 0.906264i \(-0.638922\pi\)
−0.422712 + 0.906264i \(0.638922\pi\)
\(734\) −68.0177 −2.51058
\(735\) −2.70331 −0.0997131
\(736\) −11.7461 −0.432967
\(737\) −0.527812 −0.0194422
\(738\) −2.26614 −0.0834177
\(739\) 3.38323 0.124454 0.0622271 0.998062i \(-0.480180\pi\)
0.0622271 + 0.998062i \(0.480180\pi\)
\(740\) −28.7177 −1.05568
\(741\) 2.95219 0.108451
\(742\) −23.9435 −0.878992
\(743\) −47.2702 −1.73418 −0.867088 0.498156i \(-0.834011\pi\)
−0.867088 + 0.498156i \(0.834011\pi\)
\(744\) 33.2164 1.21777
\(745\) 43.1072 1.57932
\(746\) −69.7275 −2.55290
\(747\) 1.52127 0.0556604
\(748\) −5.24525 −0.191785
\(749\) −5.00643 −0.182931
\(750\) −17.3530 −0.633642
\(751\) 23.3447 0.851860 0.425930 0.904756i \(-0.359947\pi\)
0.425930 + 0.904756i \(0.359947\pi\)
\(752\) 16.4022 0.598127
\(753\) −25.1078 −0.914980
\(754\) 13.1615 0.479314
\(755\) 53.9638 1.96394
\(756\) −3.68550 −0.134041
\(757\) −3.83114 −0.139245 −0.0696226 0.997573i \(-0.522180\pi\)
−0.0696226 + 0.997573i \(0.522180\pi\)
\(758\) 30.5117 1.10823
\(759\) 9.01197 0.327114
\(760\) 51.0067 1.85021
\(761\) 12.6167 0.457357 0.228678 0.973502i \(-0.426560\pi\)
0.228678 + 0.973502i \(0.426560\pi\)
\(762\) −10.0603 −0.364447
\(763\) 3.95320 0.143116
\(764\) 24.9455 0.902497
\(765\) −1.81444 −0.0656013
\(766\) −2.38443 −0.0861529
\(767\) −5.33079 −0.192484
\(768\) 27.4115 0.989129
\(769\) −14.4670 −0.521695 −0.260848 0.965380i \(-0.584002\pi\)
−0.260848 + 0.965380i \(0.584002\pi\)
\(770\) −13.6679 −0.492557
\(771\) −21.4888 −0.773901
\(772\) −80.7356 −2.90574
\(773\) −49.7595 −1.78972 −0.894862 0.446344i \(-0.852726\pi\)
−0.894862 + 0.446344i \(0.852726\pi\)
\(774\) 8.16134 0.293353
\(775\) 19.0744 0.685174
\(776\) 3.07046 0.110223
\(777\) 2.88242 0.103406
\(778\) −79.5375 −2.85156
\(779\) −4.46189 −0.159864
\(780\) −6.26499 −0.224323
\(781\) −22.3721 −0.800536
\(782\) −6.80189 −0.243235
\(783\) 8.77796 0.313699
\(784\) 2.21194 0.0789977
\(785\) −41.6522 −1.48663
\(786\) 20.3323 0.725230
\(787\) 2.56648 0.0914851 0.0457425 0.998953i \(-0.485435\pi\)
0.0457425 + 0.998953i \(0.485435\pi\)
\(788\) −15.2833 −0.544446
\(789\) 8.15338 0.290268
\(790\) −70.4855 −2.50776
\(791\) 5.42209 0.192787
\(792\) −8.52190 −0.302812
\(793\) −5.29310 −0.187963
\(794\) −65.2448 −2.31545
\(795\) −27.1455 −0.962752
\(796\) −54.6560 −1.93723
\(797\) −27.0618 −0.958580 −0.479290 0.877657i \(-0.659106\pi\)
−0.479290 + 0.877657i \(0.659106\pi\)
\(798\) −11.1944 −0.396278
\(799\) −4.97711 −0.176078
\(800\) 6.37836 0.225509
\(801\) 9.66286 0.341420
\(802\) 20.9914 0.741232
\(803\) −14.2077 −0.501380
\(804\) 0.917391 0.0323539
\(805\) −11.4893 −0.404944
\(806\) −12.3922 −0.436498
\(807\) −4.11605 −0.144892
\(808\) 5.58430 0.196455
\(809\) −26.4470 −0.929826 −0.464913 0.885356i \(-0.653914\pi\)
−0.464913 + 0.885356i \(0.653914\pi\)
\(810\) −6.44585 −0.226484
\(811\) 32.7868 1.15130 0.575650 0.817696i \(-0.304749\pi\)
0.575650 + 0.817696i \(0.304749\pi\)
\(812\) −32.3512 −1.13530
\(813\) −6.69170 −0.234688
\(814\) 14.5735 0.510800
\(815\) −64.5238 −2.26017
\(816\) 1.48464 0.0519726
\(817\) 16.0692 0.562189
\(818\) 3.65324 0.127733
\(819\) 0.628822 0.0219728
\(820\) 9.46880 0.330665
\(821\) −44.3537 −1.54795 −0.773977 0.633214i \(-0.781735\pi\)
−0.773977 + 0.633214i \(0.781735\pi\)
\(822\) −17.0029 −0.593043
\(823\) −23.3414 −0.813629 −0.406815 0.913511i \(-0.633360\pi\)
−0.406815 + 0.913511i \(0.633360\pi\)
\(824\) 72.3067 2.51892
\(825\) −4.89368 −0.170376
\(826\) 20.2138 0.703328
\(827\) −8.07449 −0.280777 −0.140389 0.990096i \(-0.544835\pi\)
−0.140389 + 0.990096i \(0.544835\pi\)
\(828\) −15.6637 −0.544352
\(829\) −6.30212 −0.218882 −0.109441 0.993993i \(-0.534906\pi\)
−0.109441 + 0.993993i \(0.534906\pi\)
\(830\) −9.80589 −0.340368
\(831\) 3.57687 0.124080
\(832\) −6.92571 −0.240106
\(833\) −0.671193 −0.0232555
\(834\) 11.1168 0.384942
\(835\) 30.1512 1.04343
\(836\) −36.6890 −1.26891
\(837\) −8.26491 −0.285677
\(838\) 32.1057 1.10907
\(839\) 29.0067 1.00142 0.500711 0.865615i \(-0.333072\pi\)
0.500711 + 0.865615i \(0.333072\pi\)
\(840\) 10.8645 0.374861
\(841\) 48.0526 1.65699
\(842\) 69.8526 2.40728
\(843\) −19.9376 −0.686686
\(844\) 70.2945 2.41964
\(845\) −34.0741 −1.17218
\(846\) −17.6813 −0.607896
\(847\) −6.50382 −0.223474
\(848\) 22.2113 0.762741
\(849\) −4.01533 −0.137806
\(850\) 3.69356 0.126688
\(851\) 12.2505 0.419942
\(852\) 38.8849 1.33218
\(853\) 45.1348 1.54539 0.772693 0.634780i \(-0.218909\pi\)
0.772693 + 0.634780i \(0.218909\pi\)
\(854\) 20.0709 0.686813
\(855\) −12.6915 −0.434040
\(856\) 20.1207 0.687711
\(857\) 40.1085 1.37008 0.685041 0.728505i \(-0.259784\pi\)
0.685041 + 0.728505i \(0.259784\pi\)
\(858\) 3.17932 0.108540
\(859\) 21.1098 0.720257 0.360129 0.932903i \(-0.382733\pi\)
0.360129 + 0.932903i \(0.382733\pi\)
\(860\) −34.1012 −1.16284
\(861\) −0.950390 −0.0323892
\(862\) 57.0254 1.94229
\(863\) 29.3890 1.00041 0.500207 0.865906i \(-0.333257\pi\)
0.500207 + 0.865906i \(0.333257\pi\)
\(864\) −2.76373 −0.0940240
\(865\) −62.0977 −2.11139
\(866\) 78.0401 2.65191
\(867\) 16.5495 0.562050
\(868\) 30.4604 1.03389
\(869\) 23.1868 0.786560
\(870\) −56.5814 −1.91829
\(871\) −0.156525 −0.00530366
\(872\) −15.8878 −0.538028
\(873\) −0.763992 −0.0258572
\(874\) −47.5772 −1.60932
\(875\) −7.27763 −0.246029
\(876\) 24.6945 0.834349
\(877\) 47.0360 1.58829 0.794146 0.607727i \(-0.207918\pi\)
0.794146 + 0.607727i \(0.207918\pi\)
\(878\) −79.0308 −2.66716
\(879\) 26.9013 0.907358
\(880\) 12.6791 0.427414
\(881\) −48.3447 −1.62877 −0.814387 0.580321i \(-0.802927\pi\)
−0.814387 + 0.580321i \(0.802927\pi\)
\(882\) −2.38443 −0.0802879
\(883\) 4.66484 0.156984 0.0784922 0.996915i \(-0.474989\pi\)
0.0784922 + 0.996915i \(0.474989\pi\)
\(884\) −1.55551 −0.0523173
\(885\) 22.9171 0.770350
\(886\) 70.4302 2.36615
\(887\) −14.4277 −0.484436 −0.242218 0.970222i \(-0.577875\pi\)
−0.242218 + 0.970222i \(0.577875\pi\)
\(888\) −11.5843 −0.388745
\(889\) −4.21917 −0.141507
\(890\) −62.2853 −2.08781
\(891\) 2.12042 0.0710367
\(892\) −51.5698 −1.72668
\(893\) −34.8134 −1.16499
\(894\) 38.0223 1.27166
\(895\) −70.4967 −2.35644
\(896\) 20.7341 0.692679
\(897\) 2.67255 0.0892337
\(898\) −20.3448 −0.678913
\(899\) −72.5490 −2.41965
\(900\) 8.50571 0.283524
\(901\) −6.73984 −0.224537
\(902\) −4.80517 −0.159995
\(903\) 3.42276 0.113902
\(904\) −21.7912 −0.724765
\(905\) −61.6555 −2.04950
\(906\) 47.5983 1.58135
\(907\) −24.0267 −0.797792 −0.398896 0.916996i \(-0.630607\pi\)
−0.398896 + 0.916996i \(0.630607\pi\)
\(908\) −98.5928 −3.27192
\(909\) −1.38949 −0.0460863
\(910\) −4.05329 −0.134365
\(911\) −24.2309 −0.802804 −0.401402 0.915902i \(-0.631477\pi\)
−0.401402 + 0.915902i \(0.631477\pi\)
\(912\) 10.3846 0.343868
\(913\) 3.22574 0.106756
\(914\) 6.88659 0.227788
\(915\) 22.7551 0.752260
\(916\) −75.7031 −2.50130
\(917\) 8.52713 0.281591
\(918\) −1.60041 −0.0528215
\(919\) −7.27817 −0.240085 −0.120042 0.992769i \(-0.538303\pi\)
−0.120042 + 0.992769i \(0.538303\pi\)
\(920\) 46.1751 1.52235
\(921\) 24.9984 0.823725
\(922\) −40.2960 −1.32708
\(923\) −6.63455 −0.218379
\(924\) −7.81482 −0.257089
\(925\) −6.65227 −0.218725
\(926\) −31.1684 −1.02426
\(927\) −17.9914 −0.590914
\(928\) −24.2599 −0.796371
\(929\) 37.6980 1.23683 0.618415 0.785852i \(-0.287775\pi\)
0.618415 + 0.785852i \(0.287775\pi\)
\(930\) 53.2744 1.74694
\(931\) −4.69479 −0.153866
\(932\) 106.613 3.49222
\(933\) −7.01552 −0.229678
\(934\) 9.21954 0.301673
\(935\) −3.84738 −0.125823
\(936\) −2.52721 −0.0826046
\(937\) 21.5519 0.704070 0.352035 0.935987i \(-0.385490\pi\)
0.352035 + 0.935987i \(0.385490\pi\)
\(938\) 0.593529 0.0193794
\(939\) 8.08016 0.263686
\(940\) 73.8793 2.40968
\(941\) −31.4780 −1.02615 −0.513077 0.858343i \(-0.671494\pi\)
−0.513077 + 0.858343i \(0.671494\pi\)
\(942\) −36.7390 −1.19702
\(943\) −4.03924 −0.131536
\(944\) −18.7515 −0.610310
\(945\) −2.70331 −0.0879387
\(946\) 17.3055 0.562649
\(947\) 5.02412 0.163262 0.0816309 0.996663i \(-0.473987\pi\)
0.0816309 + 0.996663i \(0.473987\pi\)
\(948\) −40.3011 −1.30892
\(949\) −4.21337 −0.136772
\(950\) 25.8354 0.838210
\(951\) 8.21015 0.266232
\(952\) 2.69750 0.0874266
\(953\) −35.5182 −1.15055 −0.575273 0.817961i \(-0.695104\pi\)
−0.575273 + 0.817961i \(0.695104\pi\)
\(954\) −23.9435 −0.775198
\(955\) 18.2975 0.592092
\(956\) 40.6304 1.31408
\(957\) 18.6130 0.601672
\(958\) −82.1787 −2.65507
\(959\) −7.13079 −0.230265
\(960\) 29.7737 0.960942
\(961\) 37.3087 1.20351
\(962\) 4.32184 0.139342
\(963\) −5.00643 −0.161330
\(964\) −2.39959 −0.0772855
\(965\) −59.2194 −1.90634
\(966\) −10.1340 −0.326057
\(967\) −36.8541 −1.18515 −0.592574 0.805516i \(-0.701888\pi\)
−0.592574 + 0.805516i \(0.701888\pi\)
\(968\) 26.1386 0.840128
\(969\) −3.15111 −0.101228
\(970\) 4.92458 0.158119
\(971\) 33.7414 1.08281 0.541406 0.840761i \(-0.317892\pi\)
0.541406 + 0.840761i \(0.317892\pi\)
\(972\) −3.68550 −0.118213
\(973\) 4.66223 0.149464
\(974\) −23.9927 −0.768776
\(975\) −1.45125 −0.0464771
\(976\) −18.6189 −0.595978
\(977\) 29.8470 0.954891 0.477445 0.878661i \(-0.341563\pi\)
0.477445 + 0.878661i \(0.341563\pi\)
\(978\) −56.9127 −1.81987
\(979\) 20.4893 0.654841
\(980\) 9.96306 0.318258
\(981\) 3.95320 0.126216
\(982\) −68.6995 −2.19229
\(983\) −34.6141 −1.10402 −0.552010 0.833838i \(-0.686139\pi\)
−0.552010 + 0.833838i \(0.686139\pi\)
\(984\) 3.81959 0.121764
\(985\) −11.2103 −0.357190
\(986\) −14.0484 −0.447391
\(987\) −7.41532 −0.236032
\(988\) −10.8803 −0.346149
\(989\) 14.5470 0.462569
\(990\) −13.6679 −0.434395
\(991\) 36.3178 1.15367 0.576836 0.816860i \(-0.304287\pi\)
0.576836 + 0.816860i \(0.304287\pi\)
\(992\) 22.8420 0.725233
\(993\) −33.0162 −1.04774
\(994\) 25.1576 0.797950
\(995\) −40.0901 −1.27094
\(996\) −5.60666 −0.177654
\(997\) −32.5037 −1.02940 −0.514702 0.857369i \(-0.672097\pi\)
−0.514702 + 0.857369i \(0.672097\pi\)
\(998\) 82.1681 2.60099
\(999\) 2.88242 0.0911956
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 8043.2.a.o.1.5 41
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.o.1.5 41 1.1 even 1 trivial