Properties

Label 8043.2.a.o.1.14
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $41$
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] = [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 1.14
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)
 
\(f(q)\) \(=\) \(q-1.32430 q^{2} -1.00000 q^{3} -0.246218 q^{4} -1.21552 q^{5} +1.32430 q^{6} +1.00000 q^{7} +2.97468 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.32430 q^{2} -1.00000 q^{3} -0.246218 q^{4} -1.21552 q^{5} +1.32430 q^{6} +1.00000 q^{7} +2.97468 q^{8} +1.00000 q^{9} +1.60972 q^{10} -3.63830 q^{11} +0.246218 q^{12} +0.340762 q^{13} -1.32430 q^{14} +1.21552 q^{15} -3.44694 q^{16} -5.62220 q^{17} -1.32430 q^{18} +3.82899 q^{19} +0.299284 q^{20} -1.00000 q^{21} +4.81821 q^{22} +3.44689 q^{23} -2.97468 q^{24} -3.52251 q^{25} -0.451273 q^{26} -1.00000 q^{27} -0.246218 q^{28} +10.3910 q^{29} -1.60972 q^{30} -2.06533 q^{31} -1.38456 q^{32} +3.63830 q^{33} +7.44551 q^{34} -1.21552 q^{35} -0.246218 q^{36} -1.93053 q^{37} -5.07074 q^{38} -0.340762 q^{39} -3.61578 q^{40} -3.22491 q^{41} +1.32430 q^{42} -4.17446 q^{43} +0.895816 q^{44} -1.21552 q^{45} -4.56473 q^{46} +11.1954 q^{47} +3.44694 q^{48} +1.00000 q^{49} +4.66488 q^{50} +5.62220 q^{51} -0.0839020 q^{52} -1.83376 q^{53} +1.32430 q^{54} +4.42242 q^{55} +2.97468 q^{56} -3.82899 q^{57} -13.7608 q^{58} +3.71579 q^{59} -0.299284 q^{60} -15.1855 q^{61} +2.73513 q^{62} +1.00000 q^{63} +8.72745 q^{64} -0.414203 q^{65} -4.81821 q^{66} -3.40054 q^{67} +1.38429 q^{68} -3.44689 q^{69} +1.60972 q^{70} +8.89396 q^{71} +2.97468 q^{72} +3.73549 q^{73} +2.55660 q^{74} +3.52251 q^{75} -0.942767 q^{76} -3.63830 q^{77} +0.451273 q^{78} -11.2227 q^{79} +4.18982 q^{80} +1.00000 q^{81} +4.27077 q^{82} -5.29508 q^{83} +0.246218 q^{84} +6.83390 q^{85} +5.52826 q^{86} -10.3910 q^{87} -10.8228 q^{88} +5.11647 q^{89} +1.60972 q^{90} +0.340762 q^{91} -0.848688 q^{92} +2.06533 q^{93} -14.8261 q^{94} -4.65421 q^{95} +1.38456 q^{96} +15.1531 q^{97} -1.32430 q^{98} -3.63830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32430 −0.936424 −0.468212 0.883616i \(-0.655102\pi\)
−0.468212 + 0.883616i \(0.655102\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.246218 −0.123109
\(5\) −1.21552 −0.543597 −0.271799 0.962354i \(-0.587618\pi\)
−0.271799 + 0.962354i \(0.587618\pi\)
\(6\) 1.32430 0.540645
\(7\) 1.00000 0.377964
\(8\) 2.97468 1.05171
\(9\) 1.00000 0.333333
\(10\) 1.60972 0.509038
\(11\) −3.63830 −1.09699 −0.548494 0.836155i \(-0.684799\pi\)
−0.548494 + 0.836155i \(0.684799\pi\)
\(12\) 0.246218 0.0710771
\(13\) 0.340762 0.0945105 0.0472552 0.998883i \(-0.484953\pi\)
0.0472552 + 0.998883i \(0.484953\pi\)
\(14\) −1.32430 −0.353935
\(15\) 1.21552 0.313846
\(16\) −3.44694 −0.861735
\(17\) −5.62220 −1.36358 −0.681792 0.731546i \(-0.738799\pi\)
−0.681792 + 0.731546i \(0.738799\pi\)
\(18\) −1.32430 −0.312141
\(19\) 3.82899 0.878430 0.439215 0.898382i \(-0.355257\pi\)
0.439215 + 0.898382i \(0.355257\pi\)
\(20\) 0.299284 0.0669218
\(21\) −1.00000 −0.218218
\(22\) 4.81821 1.02725
\(23\) 3.44689 0.718727 0.359363 0.933198i \(-0.382994\pi\)
0.359363 + 0.933198i \(0.382994\pi\)
\(24\) −2.97468 −0.607203
\(25\) −3.52251 −0.704502
\(26\) −0.451273 −0.0885019
\(27\) −1.00000 −0.192450
\(28\) −0.246218 −0.0465309
\(29\) 10.3910 1.92955 0.964777 0.263068i \(-0.0847345\pi\)
0.964777 + 0.263068i \(0.0847345\pi\)
\(30\) −1.60972 −0.293893
\(31\) −2.06533 −0.370945 −0.185473 0.982649i \(-0.559382\pi\)
−0.185473 + 0.982649i \(0.559382\pi\)
\(32\) −1.38456 −0.244757
\(33\) 3.63830 0.633346
\(34\) 7.44551 1.27689
\(35\) −1.21552 −0.205460
\(36\) −0.246218 −0.0410364
\(37\) −1.93053 −0.317377 −0.158688 0.987329i \(-0.550726\pi\)
−0.158688 + 0.987329i \(0.550726\pi\)
\(38\) −5.07074 −0.822583
\(39\) −0.340762 −0.0545656
\(40\) −3.61578 −0.571705
\(41\) −3.22491 −0.503647 −0.251824 0.967773i \(-0.581030\pi\)
−0.251824 + 0.967773i \(0.581030\pi\)
\(42\) 1.32430 0.204345
\(43\) −4.17446 −0.636600 −0.318300 0.947990i \(-0.603112\pi\)
−0.318300 + 0.947990i \(0.603112\pi\)
\(44\) 0.895816 0.135049
\(45\) −1.21552 −0.181199
\(46\) −4.56473 −0.673033
\(47\) 11.1954 1.63301 0.816506 0.577337i \(-0.195908\pi\)
0.816506 + 0.577337i \(0.195908\pi\)
\(48\) 3.44694 0.497523
\(49\) 1.00000 0.142857
\(50\) 4.66488 0.659713
\(51\) 5.62220 0.787266
\(52\) −0.0839020 −0.0116351
\(53\) −1.83376 −0.251886 −0.125943 0.992037i \(-0.540196\pi\)
−0.125943 + 0.992037i \(0.540196\pi\)
\(54\) 1.32430 0.180215
\(55\) 4.42242 0.596319
\(56\) 2.97468 0.397508
\(57\) −3.82899 −0.507162
\(58\) −13.7608 −1.80688
\(59\) 3.71579 0.483754 0.241877 0.970307i \(-0.422237\pi\)
0.241877 + 0.970307i \(0.422237\pi\)
\(60\) −0.299284 −0.0386373
\(61\) −15.1855 −1.94431 −0.972155 0.234338i \(-0.924708\pi\)
−0.972155 + 0.234338i \(0.924708\pi\)
\(62\) 2.73513 0.347362
\(63\) 1.00000 0.125988
\(64\) 8.72745 1.09093
\(65\) −0.414203 −0.0513756
\(66\) −4.81821 −0.593081
\(67\) −3.40054 −0.415443 −0.207721 0.978188i \(-0.566605\pi\)
−0.207721 + 0.978188i \(0.566605\pi\)
\(68\) 1.38429 0.167870
\(69\) −3.44689 −0.414957
\(70\) 1.60972 0.192398
\(71\) 8.89396 1.05552 0.527759 0.849394i \(-0.323032\pi\)
0.527759 + 0.849394i \(0.323032\pi\)
\(72\) 2.97468 0.350569
\(73\) 3.73549 0.437206 0.218603 0.975814i \(-0.429850\pi\)
0.218603 + 0.975814i \(0.429850\pi\)
\(74\) 2.55660 0.297199
\(75\) 3.52251 0.406744
\(76\) −0.942767 −0.108143
\(77\) −3.63830 −0.414622
\(78\) 0.451273 0.0510966
\(79\) −11.2227 −1.26265 −0.631327 0.775517i \(-0.717489\pi\)
−0.631327 + 0.775517i \(0.717489\pi\)
\(80\) 4.18982 0.468437
\(81\) 1.00000 0.111111
\(82\) 4.27077 0.471627
\(83\) −5.29508 −0.581211 −0.290605 0.956843i \(-0.593857\pi\)
−0.290605 + 0.956843i \(0.593857\pi\)
\(84\) 0.246218 0.0268646
\(85\) 6.83390 0.741241
\(86\) 5.52826 0.596127
\(87\) −10.3910 −1.11403
\(88\) −10.8228 −1.15371
\(89\) 5.11647 0.542345 0.271172 0.962531i \(-0.412589\pi\)
0.271172 + 0.962531i \(0.412589\pi\)
\(90\) 1.60972 0.169679
\(91\) 0.340762 0.0357216
\(92\) −0.848688 −0.0884819
\(93\) 2.06533 0.214165
\(94\) −14.8261 −1.52919
\(95\) −4.65421 −0.477512
\(96\) 1.38456 0.141311
\(97\) 15.1531 1.53856 0.769280 0.638912i \(-0.220615\pi\)
0.769280 + 0.638912i \(0.220615\pi\)
\(98\) −1.32430 −0.133775
\(99\) −3.63830 −0.365663
\(100\) 0.867307 0.0867307
\(101\) 3.73324 0.371471 0.185736 0.982600i \(-0.440533\pi\)
0.185736 + 0.982600i \(0.440533\pi\)
\(102\) −7.44551 −0.737215
\(103\) −8.36010 −0.823745 −0.411873 0.911241i \(-0.635125\pi\)
−0.411873 + 0.911241i \(0.635125\pi\)
\(104\) 1.01366 0.0993973
\(105\) 1.21552 0.118623
\(106\) 2.42846 0.235873
\(107\) −12.4964 −1.20807 −0.604035 0.796958i \(-0.706441\pi\)
−0.604035 + 0.796958i \(0.706441\pi\)
\(108\) 0.246218 0.0236924
\(109\) 6.07106 0.581502 0.290751 0.956799i \(-0.406095\pi\)
0.290751 + 0.956799i \(0.406095\pi\)
\(110\) −5.85663 −0.558408
\(111\) 1.93053 0.183237
\(112\) −3.44694 −0.325705
\(113\) 12.7669 1.20101 0.600506 0.799620i \(-0.294966\pi\)
0.600506 + 0.799620i \(0.294966\pi\)
\(114\) 5.07074 0.474919
\(115\) −4.18977 −0.390698
\(116\) −2.55845 −0.237546
\(117\) 0.340762 0.0315035
\(118\) −4.92083 −0.452999
\(119\) −5.62220 −0.515386
\(120\) 3.61578 0.330074
\(121\) 2.23720 0.203382
\(122\) 20.1103 1.82070
\(123\) 3.22491 0.290781
\(124\) 0.508523 0.0456668
\(125\) 10.3593 0.926563
\(126\) −1.32430 −0.117978
\(127\) 1.83416 0.162756 0.0813778 0.996683i \(-0.474068\pi\)
0.0813778 + 0.996683i \(0.474068\pi\)
\(128\) −8.78869 −0.776818
\(129\) 4.17446 0.367541
\(130\) 0.548531 0.0481094
\(131\) 4.33816 0.379026 0.189513 0.981878i \(-0.439309\pi\)
0.189513 + 0.981878i \(0.439309\pi\)
\(132\) −0.895816 −0.0779708
\(133\) 3.82899 0.332015
\(134\) 4.50336 0.389031
\(135\) 1.21552 0.104615
\(136\) −16.7242 −1.43409
\(137\) −5.05319 −0.431723 −0.215862 0.976424i \(-0.569256\pi\)
−0.215862 + 0.976424i \(0.569256\pi\)
\(138\) 4.56473 0.388576
\(139\) 13.8980 1.17881 0.589405 0.807838i \(-0.299362\pi\)
0.589405 + 0.807838i \(0.299362\pi\)
\(140\) 0.299284 0.0252941
\(141\) −11.1954 −0.942820
\(142\) −11.7783 −0.988413
\(143\) −1.23979 −0.103677
\(144\) −3.44694 −0.287245
\(145\) −12.6304 −1.04890
\(146\) −4.94692 −0.409410
\(147\) −1.00000 −0.0824786
\(148\) 0.475331 0.0390720
\(149\) 22.0493 1.80635 0.903176 0.429271i \(-0.141230\pi\)
0.903176 + 0.429271i \(0.141230\pi\)
\(150\) −4.66488 −0.380885
\(151\) −9.98536 −0.812597 −0.406298 0.913740i \(-0.633181\pi\)
−0.406298 + 0.913740i \(0.633181\pi\)
\(152\) 11.3900 0.923851
\(153\) −5.62220 −0.454528
\(154\) 4.81821 0.388263
\(155\) 2.51046 0.201645
\(156\) 0.0839020 0.00671753
\(157\) 5.55643 0.443451 0.221726 0.975109i \(-0.428831\pi\)
0.221726 + 0.975109i \(0.428831\pi\)
\(158\) 14.8623 1.18238
\(159\) 1.83376 0.145427
\(160\) 1.68296 0.133049
\(161\) 3.44689 0.271653
\(162\) −1.32430 −0.104047
\(163\) −13.9127 −1.08972 −0.544862 0.838525i \(-0.683418\pi\)
−0.544862 + 0.838525i \(0.683418\pi\)
\(164\) 0.794034 0.0620036
\(165\) −4.42242 −0.344285
\(166\) 7.01230 0.544260
\(167\) −12.2355 −0.946809 −0.473404 0.880845i \(-0.656975\pi\)
−0.473404 + 0.880845i \(0.656975\pi\)
\(168\) −2.97468 −0.229501
\(169\) −12.8839 −0.991068
\(170\) −9.05016 −0.694116
\(171\) 3.82899 0.292810
\(172\) 1.02783 0.0783713
\(173\) 7.46943 0.567891 0.283945 0.958840i \(-0.408357\pi\)
0.283945 + 0.958840i \(0.408357\pi\)
\(174\) 13.7608 1.04320
\(175\) −3.52251 −0.266277
\(176\) 12.5410 0.945313
\(177\) −3.71579 −0.279296
\(178\) −6.77576 −0.507865
\(179\) 12.9833 0.970417 0.485208 0.874399i \(-0.338744\pi\)
0.485208 + 0.874399i \(0.338744\pi\)
\(180\) 0.299284 0.0223073
\(181\) −7.85925 −0.584174 −0.292087 0.956392i \(-0.594350\pi\)
−0.292087 + 0.956392i \(0.594350\pi\)
\(182\) −0.451273 −0.0334506
\(183\) 15.1855 1.12255
\(184\) 10.2534 0.755890
\(185\) 2.34659 0.172525
\(186\) −2.73513 −0.200550
\(187\) 20.4552 1.49584
\(188\) −2.75651 −0.201039
\(189\) −1.00000 −0.0727393
\(190\) 6.16359 0.447154
\(191\) −11.4337 −0.827316 −0.413658 0.910432i \(-0.635749\pi\)
−0.413658 + 0.910432i \(0.635749\pi\)
\(192\) −8.72745 −0.629850
\(193\) −20.5183 −1.47694 −0.738469 0.674287i \(-0.764451\pi\)
−0.738469 + 0.674287i \(0.764451\pi\)
\(194\) −20.0673 −1.44075
\(195\) 0.414203 0.0296617
\(196\) −0.246218 −0.0175870
\(197\) 1.38685 0.0988092 0.0494046 0.998779i \(-0.484268\pi\)
0.0494046 + 0.998779i \(0.484268\pi\)
\(198\) 4.81821 0.342415
\(199\) 6.38280 0.452464 0.226232 0.974073i \(-0.427359\pi\)
0.226232 + 0.974073i \(0.427359\pi\)
\(200\) −10.4783 −0.740930
\(201\) 3.40054 0.239856
\(202\) −4.94394 −0.347855
\(203\) 10.3910 0.729303
\(204\) −1.38429 −0.0969197
\(205\) 3.91995 0.273781
\(206\) 11.0713 0.771375
\(207\) 3.44689 0.239576
\(208\) −1.17459 −0.0814430
\(209\) −13.9310 −0.963627
\(210\) −1.60972 −0.111081
\(211\) 6.95383 0.478722 0.239361 0.970931i \(-0.423062\pi\)
0.239361 + 0.970931i \(0.423062\pi\)
\(212\) 0.451506 0.0310095
\(213\) −8.89396 −0.609404
\(214\) 16.5490 1.13127
\(215\) 5.07414 0.346054
\(216\) −2.97468 −0.202401
\(217\) −2.06533 −0.140204
\(218\) −8.03994 −0.544533
\(219\) −3.73549 −0.252421
\(220\) −1.08888 −0.0734124
\(221\) −1.91583 −0.128873
\(222\) −2.55660 −0.171588
\(223\) 27.2264 1.82322 0.911608 0.411060i \(-0.134841\pi\)
0.911608 + 0.411060i \(0.134841\pi\)
\(224\) −1.38456 −0.0925096
\(225\) −3.52251 −0.234834
\(226\) −16.9073 −1.12466
\(227\) −22.8224 −1.51478 −0.757388 0.652965i \(-0.773525\pi\)
−0.757388 + 0.652965i \(0.773525\pi\)
\(228\) 0.942767 0.0624363
\(229\) 20.0672 1.32608 0.663039 0.748585i \(-0.269267\pi\)
0.663039 + 0.748585i \(0.269267\pi\)
\(230\) 5.54853 0.365859
\(231\) 3.63830 0.239382
\(232\) 30.9098 2.02933
\(233\) 24.9561 1.63493 0.817463 0.575981i \(-0.195380\pi\)
0.817463 + 0.575981i \(0.195380\pi\)
\(234\) −0.451273 −0.0295006
\(235\) −13.6082 −0.887701
\(236\) −0.914895 −0.0595546
\(237\) 11.2227 0.728993
\(238\) 7.44551 0.482621
\(239\) 13.1443 0.850237 0.425118 0.905138i \(-0.360232\pi\)
0.425118 + 0.905138i \(0.360232\pi\)
\(240\) −4.18982 −0.270452
\(241\) 22.6920 1.46172 0.730860 0.682528i \(-0.239119\pi\)
0.730860 + 0.682528i \(0.239119\pi\)
\(242\) −2.96274 −0.190452
\(243\) −1.00000 −0.0641500
\(244\) 3.73896 0.239363
\(245\) −1.21552 −0.0776567
\(246\) −4.27077 −0.272294
\(247\) 1.30477 0.0830208
\(248\) −6.14370 −0.390125
\(249\) 5.29508 0.335562
\(250\) −13.7188 −0.867656
\(251\) −13.4776 −0.850700 −0.425350 0.905029i \(-0.639849\pi\)
−0.425350 + 0.905029i \(0.639849\pi\)
\(252\) −0.246218 −0.0155103
\(253\) −12.5408 −0.788434
\(254\) −2.42899 −0.152408
\(255\) −6.83390 −0.427955
\(256\) −5.81601 −0.363500
\(257\) −1.01377 −0.0632373 −0.0316186 0.999500i \(-0.510066\pi\)
−0.0316186 + 0.999500i \(0.510066\pi\)
\(258\) −5.52826 −0.344174
\(259\) −1.93053 −0.119957
\(260\) 0.101985 0.00632481
\(261\) 10.3910 0.643185
\(262\) −5.74504 −0.354930
\(263\) 18.1756 1.12076 0.560378 0.828237i \(-0.310656\pi\)
0.560378 + 0.828237i \(0.310656\pi\)
\(264\) 10.8228 0.666095
\(265\) 2.22897 0.136925
\(266\) −5.07074 −0.310907
\(267\) −5.11647 −0.313123
\(268\) 0.837277 0.0511448
\(269\) −8.78753 −0.535785 −0.267893 0.963449i \(-0.586327\pi\)
−0.267893 + 0.963449i \(0.586327\pi\)
\(270\) −1.60972 −0.0979643
\(271\) 10.0417 0.609992 0.304996 0.952354i \(-0.401345\pi\)
0.304996 + 0.952354i \(0.401345\pi\)
\(272\) 19.3794 1.17505
\(273\) −0.340762 −0.0206239
\(274\) 6.69196 0.404276
\(275\) 12.8159 0.772830
\(276\) 0.848688 0.0510850
\(277\) 17.0788 1.02617 0.513084 0.858338i \(-0.328503\pi\)
0.513084 + 0.858338i \(0.328503\pi\)
\(278\) −18.4051 −1.10387
\(279\) −2.06533 −0.123648
\(280\) −3.61578 −0.216084
\(281\) −17.5354 −1.04607 −0.523037 0.852310i \(-0.675201\pi\)
−0.523037 + 0.852310i \(0.675201\pi\)
\(282\) 14.8261 0.882880
\(283\) 1.96482 0.116797 0.0583983 0.998293i \(-0.481401\pi\)
0.0583983 + 0.998293i \(0.481401\pi\)
\(284\) −2.18986 −0.129944
\(285\) 4.65421 0.275692
\(286\) 1.64186 0.0970855
\(287\) −3.22491 −0.190361
\(288\) −1.38456 −0.0815858
\(289\) 14.6092 0.859363
\(290\) 16.7265 0.982216
\(291\) −15.1531 −0.888288
\(292\) −0.919746 −0.0538241
\(293\) 24.6510 1.44012 0.720062 0.693910i \(-0.244113\pi\)
0.720062 + 0.693910i \(0.244113\pi\)
\(294\) 1.32430 0.0772350
\(295\) −4.51661 −0.262967
\(296\) −5.74269 −0.333787
\(297\) 3.63830 0.211115
\(298\) −29.2000 −1.69151
\(299\) 1.17457 0.0679272
\(300\) −0.867307 −0.0500740
\(301\) −4.17446 −0.240612
\(302\) 13.2236 0.760935
\(303\) −3.73324 −0.214469
\(304\) −13.1983 −0.756974
\(305\) 18.4583 1.05692
\(306\) 7.44551 0.425631
\(307\) −10.7166 −0.611631 −0.305816 0.952091i \(-0.598929\pi\)
−0.305816 + 0.952091i \(0.598929\pi\)
\(308\) 0.895816 0.0510438
\(309\) 8.36010 0.475590
\(310\) −3.32461 −0.188825
\(311\) 9.36871 0.531251 0.265625 0.964076i \(-0.414422\pi\)
0.265625 + 0.964076i \(0.414422\pi\)
\(312\) −1.01366 −0.0573871
\(313\) 19.8408 1.12147 0.560734 0.827996i \(-0.310519\pi\)
0.560734 + 0.827996i \(0.310519\pi\)
\(314\) −7.35840 −0.415259
\(315\) −1.21552 −0.0684868
\(316\) 2.76324 0.155444
\(317\) −11.2125 −0.629755 −0.314877 0.949132i \(-0.601963\pi\)
−0.314877 + 0.949132i \(0.601963\pi\)
\(318\) −2.42846 −0.136181
\(319\) −37.8054 −2.11670
\(320\) −10.6084 −0.593027
\(321\) 12.4964 0.697480
\(322\) −4.56473 −0.254383
\(323\) −21.5273 −1.19781
\(324\) −0.246218 −0.0136788
\(325\) −1.20034 −0.0665828
\(326\) 18.4246 1.02044
\(327\) −6.07106 −0.335731
\(328\) −9.59308 −0.529689
\(329\) 11.1954 0.617221
\(330\) 5.85663 0.322397
\(331\) −26.9205 −1.47969 −0.739843 0.672780i \(-0.765100\pi\)
−0.739843 + 0.672780i \(0.765100\pi\)
\(332\) 1.30375 0.0715524
\(333\) −1.93053 −0.105792
\(334\) 16.2035 0.886615
\(335\) 4.13343 0.225833
\(336\) 3.44694 0.188046
\(337\) −7.28486 −0.396832 −0.198416 0.980118i \(-0.563580\pi\)
−0.198416 + 0.980118i \(0.563580\pi\)
\(338\) 17.0622 0.928060
\(339\) −12.7669 −0.693405
\(340\) −1.68263 −0.0912536
\(341\) 7.51430 0.406922
\(342\) −5.07074 −0.274194
\(343\) 1.00000 0.0539949
\(344\) −12.4177 −0.669516
\(345\) 4.18977 0.225569
\(346\) −9.89180 −0.531787
\(347\) 28.6543 1.53824 0.769121 0.639103i \(-0.220694\pi\)
0.769121 + 0.639103i \(0.220694\pi\)
\(348\) 2.55845 0.137147
\(349\) 2.15367 0.115283 0.0576416 0.998337i \(-0.481642\pi\)
0.0576416 + 0.998337i \(0.481642\pi\)
\(350\) 4.66488 0.249348
\(351\) −0.340762 −0.0181885
\(352\) 5.03743 0.268496
\(353\) −26.1391 −1.39124 −0.695621 0.718409i \(-0.744871\pi\)
−0.695621 + 0.718409i \(0.744871\pi\)
\(354\) 4.92083 0.261539
\(355\) −10.8108 −0.573777
\(356\) −1.25977 −0.0667677
\(357\) 5.62220 0.297559
\(358\) −17.1938 −0.908722
\(359\) −1.05157 −0.0554998 −0.0277499 0.999615i \(-0.508834\pi\)
−0.0277499 + 0.999615i \(0.508834\pi\)
\(360\) −3.61578 −0.190568
\(361\) −4.33886 −0.228361
\(362\) 10.4080 0.547035
\(363\) −2.23720 −0.117423
\(364\) −0.0839020 −0.00439766
\(365\) −4.54056 −0.237664
\(366\) −20.1103 −1.05118
\(367\) −10.5881 −0.552695 −0.276347 0.961058i \(-0.589124\pi\)
−0.276347 + 0.961058i \(0.589124\pi\)
\(368\) −11.8812 −0.619352
\(369\) −3.22491 −0.167882
\(370\) −3.10760 −0.161557
\(371\) −1.83376 −0.0952041
\(372\) −0.508523 −0.0263657
\(373\) 7.15747 0.370600 0.185300 0.982682i \(-0.440674\pi\)
0.185300 + 0.982682i \(0.440674\pi\)
\(374\) −27.0890 −1.40074
\(375\) −10.3593 −0.534951
\(376\) 33.3026 1.71745
\(377\) 3.54085 0.182363
\(378\) 1.32430 0.0681149
\(379\) 9.38840 0.482250 0.241125 0.970494i \(-0.422484\pi\)
0.241125 + 0.970494i \(0.422484\pi\)
\(380\) 1.14595 0.0587861
\(381\) −1.83416 −0.0939670
\(382\) 15.1417 0.774719
\(383\) 1.00000 0.0510976
\(384\) 8.78869 0.448496
\(385\) 4.42242 0.225388
\(386\) 27.1724 1.38304
\(387\) −4.17446 −0.212200
\(388\) −3.73096 −0.189411
\(389\) −19.1768 −0.972301 −0.486150 0.873875i \(-0.661599\pi\)
−0.486150 + 0.873875i \(0.661599\pi\)
\(390\) −0.548531 −0.0277760
\(391\) −19.3791 −0.980045
\(392\) 2.97468 0.150244
\(393\) −4.33816 −0.218831
\(394\) −1.83661 −0.0925273
\(395\) 13.6414 0.686375
\(396\) 0.895816 0.0450164
\(397\) −27.8058 −1.39553 −0.697766 0.716326i \(-0.745822\pi\)
−0.697766 + 0.716326i \(0.745822\pi\)
\(398\) −8.45276 −0.423699
\(399\) −3.82899 −0.191689
\(400\) 12.1419 0.607094
\(401\) −13.9664 −0.697450 −0.348725 0.937225i \(-0.613385\pi\)
−0.348725 + 0.937225i \(0.613385\pi\)
\(402\) −4.50336 −0.224607
\(403\) −0.703788 −0.0350582
\(404\) −0.919192 −0.0457315
\(405\) −1.21552 −0.0603997
\(406\) −13.7608 −0.682937
\(407\) 7.02383 0.348158
\(408\) 16.7242 0.827973
\(409\) 0.818225 0.0404586 0.0202293 0.999795i \(-0.493560\pi\)
0.0202293 + 0.999795i \(0.493560\pi\)
\(410\) −5.19121 −0.256375
\(411\) 5.05319 0.249256
\(412\) 2.05841 0.101411
\(413\) 3.71579 0.182842
\(414\) −4.56473 −0.224344
\(415\) 6.43628 0.315945
\(416\) −0.471805 −0.0231321
\(417\) −13.8980 −0.680586
\(418\) 18.4489 0.902364
\(419\) −19.1508 −0.935576 −0.467788 0.883841i \(-0.654949\pi\)
−0.467788 + 0.883841i \(0.654949\pi\)
\(420\) −0.299284 −0.0146035
\(421\) −28.0335 −1.36627 −0.683135 0.730292i \(-0.739384\pi\)
−0.683135 + 0.730292i \(0.739384\pi\)
\(422\) −9.20899 −0.448287
\(423\) 11.1954 0.544337
\(424\) −5.45485 −0.264911
\(425\) 19.8043 0.960648
\(426\) 11.7783 0.570661
\(427\) −15.1855 −0.734880
\(428\) 3.07684 0.148725
\(429\) 1.23979 0.0598578
\(430\) −6.71971 −0.324053
\(431\) −22.4666 −1.08218 −0.541090 0.840965i \(-0.681988\pi\)
−0.541090 + 0.840965i \(0.681988\pi\)
\(432\) 3.44694 0.165841
\(433\) −37.5483 −1.80446 −0.902229 0.431257i \(-0.858070\pi\)
−0.902229 + 0.431257i \(0.858070\pi\)
\(434\) 2.73513 0.131290
\(435\) 12.6304 0.605583
\(436\) −1.49481 −0.0715883
\(437\) 13.1981 0.631351
\(438\) 4.94692 0.236373
\(439\) 4.23051 0.201911 0.100956 0.994891i \(-0.467810\pi\)
0.100956 + 0.994891i \(0.467810\pi\)
\(440\) 13.1553 0.627153
\(441\) 1.00000 0.0476190
\(442\) 2.53715 0.120680
\(443\) 24.1931 1.14945 0.574724 0.818347i \(-0.305109\pi\)
0.574724 + 0.818347i \(0.305109\pi\)
\(444\) −0.475331 −0.0225582
\(445\) −6.21917 −0.294817
\(446\) −36.0561 −1.70730
\(447\) −22.0493 −1.04290
\(448\) 8.72745 0.412333
\(449\) −1.90456 −0.0898819 −0.0449410 0.998990i \(-0.514310\pi\)
−0.0449410 + 0.998990i \(0.514310\pi\)
\(450\) 4.66488 0.219904
\(451\) 11.7332 0.552495
\(452\) −3.14346 −0.147856
\(453\) 9.98536 0.469153
\(454\) 30.2238 1.41847
\(455\) −0.414203 −0.0194182
\(456\) −11.3900 −0.533385
\(457\) −16.9744 −0.794029 −0.397015 0.917812i \(-0.629954\pi\)
−0.397015 + 0.917812i \(0.629954\pi\)
\(458\) −26.5751 −1.24177
\(459\) 5.62220 0.262422
\(460\) 1.03160 0.0480985
\(461\) −36.6093 −1.70507 −0.852533 0.522674i \(-0.824935\pi\)
−0.852533 + 0.522674i \(0.824935\pi\)
\(462\) −4.81821 −0.224163
\(463\) −37.7860 −1.75607 −0.878033 0.478601i \(-0.841144\pi\)
−0.878033 + 0.478601i \(0.841144\pi\)
\(464\) −35.8170 −1.66276
\(465\) −2.51046 −0.116420
\(466\) −33.0494 −1.53098
\(467\) 5.16106 0.238825 0.119413 0.992845i \(-0.461899\pi\)
0.119413 + 0.992845i \(0.461899\pi\)
\(468\) −0.0839020 −0.00387837
\(469\) −3.40054 −0.157023
\(470\) 18.0214 0.831265
\(471\) −5.55643 −0.256027
\(472\) 11.0533 0.508768
\(473\) 15.1879 0.698342
\(474\) −14.8623 −0.682647
\(475\) −13.4876 −0.618856
\(476\) 1.38429 0.0634488
\(477\) −1.83376 −0.0839622
\(478\) −17.4071 −0.796182
\(479\) 16.7221 0.764054 0.382027 0.924151i \(-0.375226\pi\)
0.382027 + 0.924151i \(0.375226\pi\)
\(480\) −1.68296 −0.0768161
\(481\) −0.657851 −0.0299954
\(482\) −30.0511 −1.36879
\(483\) −3.44689 −0.156839
\(484\) −0.550841 −0.0250382
\(485\) −18.4189 −0.836357
\(486\) 1.32430 0.0600717
\(487\) −2.75869 −0.125008 −0.0625040 0.998045i \(-0.519909\pi\)
−0.0625040 + 0.998045i \(0.519909\pi\)
\(488\) −45.1721 −2.04484
\(489\) 13.9127 0.629153
\(490\) 1.60972 0.0727197
\(491\) −32.4571 −1.46477 −0.732385 0.680890i \(-0.761593\pi\)
−0.732385 + 0.680890i \(0.761593\pi\)
\(492\) −0.794034 −0.0357978
\(493\) −58.4201 −2.63111
\(494\) −1.72792 −0.0777427
\(495\) 4.42242 0.198773
\(496\) 7.11908 0.319656
\(497\) 8.89396 0.398949
\(498\) −7.01230 −0.314229
\(499\) −25.0944 −1.12338 −0.561689 0.827348i \(-0.689848\pi\)
−0.561689 + 0.827348i \(0.689848\pi\)
\(500\) −2.55065 −0.114068
\(501\) 12.2355 0.546640
\(502\) 17.8485 0.796616
\(503\) −26.4315 −1.17852 −0.589260 0.807943i \(-0.700581\pi\)
−0.589260 + 0.807943i \(0.700581\pi\)
\(504\) 2.97468 0.132503
\(505\) −4.53783 −0.201931
\(506\) 16.6079 0.738309
\(507\) 12.8839 0.572193
\(508\) −0.451605 −0.0200367
\(509\) −5.45591 −0.241829 −0.120915 0.992663i \(-0.538583\pi\)
−0.120915 + 0.992663i \(0.538583\pi\)
\(510\) 9.05016 0.400748
\(511\) 3.73549 0.165248
\(512\) 25.2795 1.11721
\(513\) −3.82899 −0.169054
\(514\) 1.34254 0.0592169
\(515\) 10.1619 0.447786
\(516\) −1.02783 −0.0452477
\(517\) −40.7321 −1.79139
\(518\) 2.55660 0.112331
\(519\) −7.46943 −0.327872
\(520\) −1.23212 −0.0540321
\(521\) −37.7111 −1.65215 −0.826076 0.563559i \(-0.809432\pi\)
−0.826076 + 0.563559i \(0.809432\pi\)
\(522\) −13.7608 −0.602294
\(523\) −28.6262 −1.25174 −0.625869 0.779928i \(-0.715256\pi\)
−0.625869 + 0.779928i \(0.715256\pi\)
\(524\) −1.06813 −0.0466617
\(525\) 3.52251 0.153735
\(526\) −24.0700 −1.04950
\(527\) 11.6117 0.505815
\(528\) −12.5410 −0.545777
\(529\) −11.1189 −0.483432
\(530\) −2.95184 −0.128220
\(531\) 3.71579 0.161251
\(532\) −0.942767 −0.0408741
\(533\) −1.09893 −0.0475999
\(534\) 6.77576 0.293216
\(535\) 15.1896 0.656704
\(536\) −10.1155 −0.436924
\(537\) −12.9833 −0.560270
\(538\) 11.6374 0.501723
\(539\) −3.63830 −0.156713
\(540\) −0.299284 −0.0128791
\(541\) 32.1296 1.38136 0.690680 0.723161i \(-0.257311\pi\)
0.690680 + 0.723161i \(0.257311\pi\)
\(542\) −13.2983 −0.571211
\(543\) 7.85925 0.337273
\(544\) 7.78426 0.333747
\(545\) −7.37950 −0.316103
\(546\) 0.451273 0.0193127
\(547\) −3.57051 −0.152664 −0.0763319 0.997082i \(-0.524321\pi\)
−0.0763319 + 0.997082i \(0.524321\pi\)
\(548\) 1.24419 0.0531491
\(549\) −15.1855 −0.648103
\(550\) −16.9722 −0.723697
\(551\) 39.7869 1.69498
\(552\) −10.2534 −0.436413
\(553\) −11.2227 −0.477238
\(554\) −22.6176 −0.960929
\(555\) −2.34659 −0.0996074
\(556\) −3.42194 −0.145122
\(557\) 11.0377 0.467682 0.233841 0.972275i \(-0.424871\pi\)
0.233841 + 0.972275i \(0.424871\pi\)
\(558\) 2.73513 0.115787
\(559\) −1.42250 −0.0601653
\(560\) 4.18982 0.177052
\(561\) −20.4552 −0.863621
\(562\) 23.2222 0.979570
\(563\) 16.1678 0.681390 0.340695 0.940174i \(-0.389338\pi\)
0.340695 + 0.940174i \(0.389338\pi\)
\(564\) 2.75651 0.116070
\(565\) −15.5185 −0.652867
\(566\) −2.60202 −0.109371
\(567\) 1.00000 0.0419961
\(568\) 26.4566 1.11010
\(569\) 27.4893 1.15241 0.576206 0.817305i \(-0.304533\pi\)
0.576206 + 0.817305i \(0.304533\pi\)
\(570\) −6.16359 −0.258164
\(571\) −11.9699 −0.500925 −0.250463 0.968126i \(-0.580583\pi\)
−0.250463 + 0.968126i \(0.580583\pi\)
\(572\) 0.305260 0.0127636
\(573\) 11.4337 0.477651
\(574\) 4.27077 0.178258
\(575\) −12.1417 −0.506344
\(576\) 8.72745 0.363644
\(577\) 12.1567 0.506091 0.253045 0.967454i \(-0.418568\pi\)
0.253045 + 0.967454i \(0.418568\pi\)
\(578\) −19.3470 −0.804728
\(579\) 20.5183 0.852711
\(580\) 3.10985 0.129129
\(581\) −5.29508 −0.219677
\(582\) 20.0673 0.831815
\(583\) 6.67177 0.276316
\(584\) 11.1119 0.459812
\(585\) −0.414203 −0.0171252
\(586\) −32.6454 −1.34857
\(587\) 10.6062 0.437763 0.218882 0.975751i \(-0.429759\pi\)
0.218882 + 0.975751i \(0.429759\pi\)
\(588\) 0.246218 0.0101539
\(589\) −7.90814 −0.325849
\(590\) 5.98137 0.246249
\(591\) −1.38685 −0.0570475
\(592\) 6.65441 0.273495
\(593\) −28.5874 −1.17394 −0.586972 0.809607i \(-0.699680\pi\)
−0.586972 + 0.809607i \(0.699680\pi\)
\(594\) −4.81821 −0.197694
\(595\) 6.83390 0.280163
\(596\) −5.42895 −0.222379
\(597\) −6.38280 −0.261230
\(598\) −1.55549 −0.0636087
\(599\) 36.3073 1.48348 0.741738 0.670690i \(-0.234002\pi\)
0.741738 + 0.670690i \(0.234002\pi\)
\(600\) 10.4783 0.427776
\(601\) −11.4172 −0.465718 −0.232859 0.972510i \(-0.574808\pi\)
−0.232859 + 0.972510i \(0.574808\pi\)
\(602\) 5.52826 0.225315
\(603\) −3.40054 −0.138481
\(604\) 2.45858 0.100038
\(605\) −2.71937 −0.110558
\(606\) 4.94394 0.200834
\(607\) 24.5839 0.997830 0.498915 0.866651i \(-0.333732\pi\)
0.498915 + 0.866651i \(0.333732\pi\)
\(608\) −5.30145 −0.215002
\(609\) −10.3910 −0.421063
\(610\) −24.4445 −0.989727
\(611\) 3.81496 0.154337
\(612\) 1.38429 0.0559566
\(613\) −25.1341 −1.01516 −0.507578 0.861606i \(-0.669459\pi\)
−0.507578 + 0.861606i \(0.669459\pi\)
\(614\) 14.1921 0.572747
\(615\) −3.91995 −0.158068
\(616\) −10.8228 −0.436061
\(617\) −28.9376 −1.16499 −0.582493 0.812836i \(-0.697923\pi\)
−0.582493 + 0.812836i \(0.697923\pi\)
\(618\) −11.0713 −0.445354
\(619\) 0.0991748 0.00398617 0.00199309 0.999998i \(-0.499366\pi\)
0.00199309 + 0.999998i \(0.499366\pi\)
\(620\) −0.618121 −0.0248243
\(621\) −3.44689 −0.138319
\(622\) −12.4070 −0.497476
\(623\) 5.11647 0.204987
\(624\) 1.17459 0.0470211
\(625\) 5.02063 0.200825
\(626\) −26.2752 −1.05017
\(627\) 13.9310 0.556350
\(628\) −1.36810 −0.0545930
\(629\) 10.8538 0.432770
\(630\) 1.60972 0.0641327
\(631\) −1.39530 −0.0555459 −0.0277730 0.999614i \(-0.508842\pi\)
−0.0277730 + 0.999614i \(0.508842\pi\)
\(632\) −33.3839 −1.32794
\(633\) −6.95383 −0.276390
\(634\) 14.8487 0.589718
\(635\) −2.22946 −0.0884735
\(636\) −0.451506 −0.0179034
\(637\) 0.340762 0.0135015
\(638\) 50.0659 1.98213
\(639\) 8.89396 0.351840
\(640\) 10.6828 0.422276
\(641\) −12.8855 −0.508945 −0.254473 0.967080i \(-0.581902\pi\)
−0.254473 + 0.967080i \(0.581902\pi\)
\(642\) −16.5490 −0.653137
\(643\) −30.4827 −1.20212 −0.601061 0.799203i \(-0.705255\pi\)
−0.601061 + 0.799203i \(0.705255\pi\)
\(644\) −0.848688 −0.0334430
\(645\) −5.07414 −0.199794
\(646\) 28.5087 1.12166
\(647\) 28.6424 1.12605 0.563025 0.826440i \(-0.309637\pi\)
0.563025 + 0.826440i \(0.309637\pi\)
\(648\) 2.97468 0.116856
\(649\) −13.5191 −0.530672
\(650\) 1.58961 0.0623498
\(651\) 2.06533 0.0809468
\(652\) 3.42556 0.134155
\(653\) −43.3526 −1.69652 −0.848260 0.529580i \(-0.822349\pi\)
−0.848260 + 0.529580i \(0.822349\pi\)
\(654\) 8.03994 0.314386
\(655\) −5.27312 −0.206038
\(656\) 11.1161 0.434010
\(657\) 3.73549 0.145735
\(658\) −14.8261 −0.577981
\(659\) −9.04320 −0.352273 −0.176136 0.984366i \(-0.556360\pi\)
−0.176136 + 0.984366i \(0.556360\pi\)
\(660\) 1.08888 0.0423847
\(661\) −38.6904 −1.50488 −0.752440 0.658660i \(-0.771123\pi\)
−0.752440 + 0.658660i \(0.771123\pi\)
\(662\) 35.6510 1.38561
\(663\) 1.91583 0.0744049
\(664\) −15.7512 −0.611264
\(665\) −4.65421 −0.180483
\(666\) 2.55660 0.0990664
\(667\) 35.8165 1.38682
\(668\) 3.01260 0.116561
\(669\) −27.2264 −1.05263
\(670\) −5.47392 −0.211476
\(671\) 55.2495 2.13288
\(672\) 1.38456 0.0534104
\(673\) −23.2958 −0.897988 −0.448994 0.893535i \(-0.648218\pi\)
−0.448994 + 0.893535i \(0.648218\pi\)
\(674\) 9.64737 0.371603
\(675\) 3.52251 0.135581
\(676\) 3.17225 0.122010
\(677\) 7.34575 0.282320 0.141160 0.989987i \(-0.454917\pi\)
0.141160 + 0.989987i \(0.454917\pi\)
\(678\) 16.9073 0.649321
\(679\) 15.1531 0.581521
\(680\) 20.3286 0.779568
\(681\) 22.8224 0.874556
\(682\) −9.95122 −0.381052
\(683\) −11.0789 −0.423921 −0.211961 0.977278i \(-0.567985\pi\)
−0.211961 + 0.977278i \(0.567985\pi\)
\(684\) −0.942767 −0.0360476
\(685\) 6.14226 0.234684
\(686\) −1.32430 −0.0505622
\(687\) −20.0672 −0.765611
\(688\) 14.3891 0.548580
\(689\) −0.624877 −0.0238059
\(690\) −5.54853 −0.211229
\(691\) 5.25568 0.199935 0.0999677 0.994991i \(-0.468126\pi\)
0.0999677 + 0.994991i \(0.468126\pi\)
\(692\) −1.83911 −0.0699126
\(693\) −3.63830 −0.138207
\(694\) −37.9470 −1.44045
\(695\) −16.8933 −0.640798
\(696\) −30.9098 −1.17163
\(697\) 18.1311 0.686765
\(698\) −2.85211 −0.107954
\(699\) −24.9561 −0.943925
\(700\) 0.867307 0.0327811
\(701\) 34.7457 1.31233 0.656164 0.754619i \(-0.272178\pi\)
0.656164 + 0.754619i \(0.272178\pi\)
\(702\) 0.451273 0.0170322
\(703\) −7.39196 −0.278793
\(704\) −31.7531 −1.19674
\(705\) 13.6082 0.512514
\(706\) 34.6161 1.30279
\(707\) 3.73324 0.140403
\(708\) 0.914895 0.0343839
\(709\) 0.527890 0.0198253 0.00991267 0.999951i \(-0.496845\pi\)
0.00991267 + 0.999951i \(0.496845\pi\)
\(710\) 14.3168 0.537299
\(711\) −11.2227 −0.420884
\(712\) 15.2198 0.570388
\(713\) −7.11898 −0.266608
\(714\) −7.44551 −0.278641
\(715\) 1.50700 0.0563584
\(716\) −3.19673 −0.119467
\(717\) −13.1443 −0.490884
\(718\) 1.39260 0.0519713
\(719\) 24.5972 0.917322 0.458661 0.888611i \(-0.348329\pi\)
0.458661 + 0.888611i \(0.348329\pi\)
\(720\) 4.18982 0.156146
\(721\) −8.36010 −0.311346
\(722\) 5.74598 0.213843
\(723\) −22.6920 −0.843924
\(724\) 1.93509 0.0719172
\(725\) −36.6023 −1.35938
\(726\) 2.96274 0.109958
\(727\) −8.37422 −0.310582 −0.155291 0.987869i \(-0.549632\pi\)
−0.155291 + 0.987869i \(0.549632\pi\)
\(728\) 1.01366 0.0375687
\(729\) 1.00000 0.0370370
\(730\) 6.01308 0.222554
\(731\) 23.4697 0.868057
\(732\) −3.73896 −0.138196
\(733\) −12.9112 −0.476886 −0.238443 0.971156i \(-0.576637\pi\)
−0.238443 + 0.971156i \(0.576637\pi\)
\(734\) 14.0219 0.517557
\(735\) 1.21552 0.0448351
\(736\) −4.77242 −0.175914
\(737\) 12.3722 0.455735
\(738\) 4.27077 0.157209
\(739\) −3.62813 −0.133463 −0.0667314 0.997771i \(-0.521257\pi\)
−0.0667314 + 0.997771i \(0.521257\pi\)
\(740\) −0.577775 −0.0212394
\(741\) −1.30477 −0.0479321
\(742\) 2.42846 0.0891515
\(743\) 6.94403 0.254752 0.127376 0.991855i \(-0.459345\pi\)
0.127376 + 0.991855i \(0.459345\pi\)
\(744\) 6.14370 0.225239
\(745\) −26.8014 −0.981927
\(746\) −9.47866 −0.347039
\(747\) −5.29508 −0.193737
\(748\) −5.03646 −0.184151
\(749\) −12.4964 −0.456608
\(750\) 13.7188 0.500941
\(751\) −3.65813 −0.133487 −0.0667435 0.997770i \(-0.521261\pi\)
−0.0667435 + 0.997770i \(0.521261\pi\)
\(752\) −38.5898 −1.40722
\(753\) 13.4776 0.491152
\(754\) −4.68916 −0.170769
\(755\) 12.1374 0.441725
\(756\) 0.246218 0.00895488
\(757\) 3.64600 0.132516 0.0662580 0.997803i \(-0.478894\pi\)
0.0662580 + 0.997803i \(0.478894\pi\)
\(758\) −12.4331 −0.451591
\(759\) 12.5408 0.455203
\(760\) −13.8448 −0.502203
\(761\) −37.8893 −1.37348 −0.686742 0.726901i \(-0.740960\pi\)
−0.686742 + 0.726901i \(0.740960\pi\)
\(762\) 2.42899 0.0879930
\(763\) 6.07106 0.219787
\(764\) 2.81520 0.101850
\(765\) 6.83390 0.247080
\(766\) −1.32430 −0.0478491
\(767\) 1.26620 0.0457198
\(768\) 5.81601 0.209867
\(769\) −41.2348 −1.48696 −0.743482 0.668756i \(-0.766827\pi\)
−0.743482 + 0.668756i \(0.766827\pi\)
\(770\) −5.85663 −0.211058
\(771\) 1.01377 0.0365101
\(772\) 5.05198 0.181825
\(773\) −12.5336 −0.450802 −0.225401 0.974266i \(-0.572369\pi\)
−0.225401 + 0.974266i \(0.572369\pi\)
\(774\) 5.52826 0.198709
\(775\) 7.27516 0.261332
\(776\) 45.0755 1.61811
\(777\) 1.93053 0.0692573
\(778\) 25.3959 0.910486
\(779\) −12.3482 −0.442419
\(780\) −0.101985 −0.00365163
\(781\) −32.3589 −1.15789
\(782\) 25.6639 0.917738
\(783\) −10.3910 −0.371343
\(784\) −3.44694 −0.123105
\(785\) −6.75395 −0.241059
\(786\) 5.74504 0.204919
\(787\) −5.99542 −0.213714 −0.106857 0.994274i \(-0.534079\pi\)
−0.106857 + 0.994274i \(0.534079\pi\)
\(788\) −0.341469 −0.0121643
\(789\) −18.1756 −0.647068
\(790\) −18.0654 −0.642738
\(791\) 12.7669 0.453940
\(792\) −10.8228 −0.384570
\(793\) −5.17466 −0.183758
\(794\) 36.8233 1.30681
\(795\) −2.22897 −0.0790536
\(796\) −1.57156 −0.0557025
\(797\) 25.4076 0.899983 0.449992 0.893033i \(-0.351427\pi\)
0.449992 + 0.893033i \(0.351427\pi\)
\(798\) 5.07074 0.179502
\(799\) −62.9426 −2.22675
\(800\) 4.87711 0.172432
\(801\) 5.11647 0.180782
\(802\) 18.4958 0.653109
\(803\) −13.5908 −0.479609
\(804\) −0.837277 −0.0295285
\(805\) −4.18977 −0.147670
\(806\) 0.932030 0.0328293
\(807\) 8.78753 0.309336
\(808\) 11.1052 0.390679
\(809\) −44.8406 −1.57651 −0.788255 0.615348i \(-0.789015\pi\)
−0.788255 + 0.615348i \(0.789015\pi\)
\(810\) 1.60972 0.0565597
\(811\) 15.6957 0.551151 0.275575 0.961279i \(-0.411132\pi\)
0.275575 + 0.961279i \(0.411132\pi\)
\(812\) −2.55845 −0.0897839
\(813\) −10.0417 −0.352179
\(814\) −9.30169 −0.326024
\(815\) 16.9111 0.592371
\(816\) −19.3794 −0.678414
\(817\) −15.9840 −0.559208
\(818\) −1.08358 −0.0378864
\(819\) 0.340762 0.0119072
\(820\) −0.965164 −0.0337050
\(821\) −26.9222 −0.939590 −0.469795 0.882776i \(-0.655672\pi\)
−0.469795 + 0.882776i \(0.655672\pi\)
\(822\) −6.69196 −0.233409
\(823\) −8.84287 −0.308243 −0.154122 0.988052i \(-0.549255\pi\)
−0.154122 + 0.988052i \(0.549255\pi\)
\(824\) −24.8686 −0.866339
\(825\) −12.8159 −0.446194
\(826\) −4.92083 −0.171218
\(827\) −26.5690 −0.923894 −0.461947 0.886908i \(-0.652849\pi\)
−0.461947 + 0.886908i \(0.652849\pi\)
\(828\) −0.848688 −0.0294940
\(829\) 42.3942 1.47241 0.736206 0.676757i \(-0.236615\pi\)
0.736206 + 0.676757i \(0.236615\pi\)
\(830\) −8.52359 −0.295858
\(831\) −17.0788 −0.592458
\(832\) 2.97399 0.103104
\(833\) −5.62220 −0.194798
\(834\) 18.4051 0.637318
\(835\) 14.8725 0.514682
\(836\) 3.43007 0.118631
\(837\) 2.06533 0.0713884
\(838\) 25.3614 0.876096
\(839\) 0.577628 0.0199419 0.00997097 0.999950i \(-0.496826\pi\)
0.00997097 + 0.999950i \(0.496826\pi\)
\(840\) 3.61578 0.124756
\(841\) 78.9722 2.72318
\(842\) 37.1249 1.27941
\(843\) 17.5354 0.603951
\(844\) −1.71216 −0.0589350
\(845\) 15.6606 0.538742
\(846\) −14.8261 −0.509731
\(847\) 2.23720 0.0768712
\(848\) 6.32086 0.217059
\(849\) −1.96482 −0.0674326
\(850\) −26.2269 −0.899574
\(851\) −6.65432 −0.228107
\(852\) 2.18986 0.0750232
\(853\) −2.86371 −0.0980517 −0.0490258 0.998798i \(-0.515612\pi\)
−0.0490258 + 0.998798i \(0.515612\pi\)
\(854\) 20.1103 0.688160
\(855\) −4.65421 −0.159171
\(856\) −37.1727 −1.27054
\(857\) −20.8218 −0.711259 −0.355630 0.934627i \(-0.615734\pi\)
−0.355630 + 0.934627i \(0.615734\pi\)
\(858\) −1.64186 −0.0560523
\(859\) −47.2524 −1.61223 −0.806115 0.591758i \(-0.798434\pi\)
−0.806115 + 0.591758i \(0.798434\pi\)
\(860\) −1.24935 −0.0426024
\(861\) 3.22491 0.109905
\(862\) 29.7526 1.01338
\(863\) 23.2301 0.790761 0.395380 0.918517i \(-0.370613\pi\)
0.395380 + 0.918517i \(0.370613\pi\)
\(864\) 1.38456 0.0471036
\(865\) −9.07925 −0.308704
\(866\) 49.7254 1.68974
\(867\) −14.6092 −0.496153
\(868\) 0.508523 0.0172604
\(869\) 40.8315 1.38512
\(870\) −16.7265 −0.567083
\(871\) −1.15878 −0.0392637
\(872\) 18.0595 0.611570
\(873\) 15.1531 0.512853
\(874\) −17.4783 −0.591212
\(875\) 10.3593 0.350208
\(876\) 0.919746 0.0310753
\(877\) −40.0216 −1.35143 −0.675716 0.737162i \(-0.736166\pi\)
−0.675716 + 0.737162i \(0.736166\pi\)
\(878\) −5.60248 −0.189074
\(879\) −24.6510 −0.831456
\(880\) −15.2438 −0.513869
\(881\) 50.0895 1.68756 0.843779 0.536690i \(-0.180326\pi\)
0.843779 + 0.536690i \(0.180326\pi\)
\(882\) −1.32430 −0.0445916
\(883\) 13.5501 0.455999 0.227999 0.973661i \(-0.426782\pi\)
0.227999 + 0.973661i \(0.426782\pi\)
\(884\) 0.471714 0.0158655
\(885\) 4.51661 0.151824
\(886\) −32.0390 −1.07637
\(887\) 38.2009 1.28266 0.641330 0.767265i \(-0.278383\pi\)
0.641330 + 0.767265i \(0.278383\pi\)
\(888\) 5.74269 0.192712
\(889\) 1.83416 0.0615159
\(890\) 8.23608 0.276074
\(891\) −3.63830 −0.121888
\(892\) −6.70365 −0.224455
\(893\) 42.8669 1.43449
\(894\) 29.2000 0.976595
\(895\) −15.7815 −0.527516
\(896\) −8.78869 −0.293610
\(897\) −1.17457 −0.0392178
\(898\) 2.52222 0.0841676
\(899\) −21.4608 −0.715759
\(900\) 0.867307 0.0289102
\(901\) 10.3098 0.343468
\(902\) −15.5383 −0.517370
\(903\) 4.17446 0.138917
\(904\) 37.9775 1.26311
\(905\) 9.55308 0.317555
\(906\) −13.2236 −0.439326
\(907\) −41.4321 −1.37573 −0.687865 0.725838i \(-0.741452\pi\)
−0.687865 + 0.725838i \(0.741452\pi\)
\(908\) 5.61930 0.186483
\(909\) 3.73324 0.123824
\(910\) 0.548531 0.0181836
\(911\) −7.02975 −0.232906 −0.116453 0.993196i \(-0.537152\pi\)
−0.116453 + 0.993196i \(0.537152\pi\)
\(912\) 13.1983 0.437039
\(913\) 19.2651 0.637581
\(914\) 22.4793 0.743548
\(915\) −18.4583 −0.610214
\(916\) −4.94091 −0.163252
\(917\) 4.33816 0.143259
\(918\) −7.44551 −0.245738
\(919\) 13.5923 0.448368 0.224184 0.974547i \(-0.428028\pi\)
0.224184 + 0.974547i \(0.428028\pi\)
\(920\) −12.4632 −0.410900
\(921\) 10.7166 0.353126
\(922\) 48.4819 1.59666
\(923\) 3.03073 0.0997575
\(924\) −0.895816 −0.0294702
\(925\) 6.80030 0.223593
\(926\) 50.0402 1.64442
\(927\) −8.36010 −0.274582
\(928\) −14.3869 −0.472273
\(929\) −12.3717 −0.405903 −0.202951 0.979189i \(-0.565053\pi\)
−0.202951 + 0.979189i \(0.565053\pi\)
\(930\) 3.32461 0.109018
\(931\) 3.82899 0.125490
\(932\) −6.14464 −0.201274
\(933\) −9.36871 −0.306718
\(934\) −6.83481 −0.223642
\(935\) −24.8638 −0.813132
\(936\) 1.01366 0.0331324
\(937\) −41.3937 −1.35227 −0.676137 0.736776i \(-0.736347\pi\)
−0.676137 + 0.736776i \(0.736347\pi\)
\(938\) 4.50336 0.147040
\(939\) −19.8408 −0.647479
\(940\) 3.35059 0.109284
\(941\) 14.4420 0.470795 0.235398 0.971899i \(-0.424361\pi\)
0.235398 + 0.971899i \(0.424361\pi\)
\(942\) 7.35840 0.239750
\(943\) −11.1159 −0.361985
\(944\) −12.8081 −0.416868
\(945\) 1.21552 0.0395409
\(946\) −20.1134 −0.653944
\(947\) −50.7602 −1.64949 −0.824743 0.565508i \(-0.808680\pi\)
−0.824743 + 0.565508i \(0.808680\pi\)
\(948\) −2.76324 −0.0897458
\(949\) 1.27291 0.0413205
\(950\) 17.8617 0.579512
\(951\) 11.2125 0.363589
\(952\) −16.7242 −0.542036
\(953\) −2.46114 −0.0797240 −0.0398620 0.999205i \(-0.512692\pi\)
−0.0398620 + 0.999205i \(0.512692\pi\)
\(954\) 2.42846 0.0786242
\(955\) 13.8979 0.449727
\(956\) −3.23638 −0.104672
\(957\) 37.8054 1.22208
\(958\) −22.1452 −0.715479
\(959\) −5.05319 −0.163176
\(960\) 10.6084 0.342385
\(961\) −26.7344 −0.862400
\(962\) 0.871194 0.0280884
\(963\) −12.4964 −0.402690
\(964\) −5.58719 −0.179951
\(965\) 24.9404 0.802859
\(966\) 4.56473 0.146868
\(967\) −4.88071 −0.156953 −0.0784765 0.996916i \(-0.525006\pi\)
−0.0784765 + 0.996916i \(0.525006\pi\)
\(968\) 6.65496 0.213898
\(969\) 21.5273 0.691558
\(970\) 24.3922 0.783185
\(971\) −9.65980 −0.309998 −0.154999 0.987915i \(-0.549537\pi\)
−0.154999 + 0.987915i \(0.549537\pi\)
\(972\) 0.246218 0.00789746
\(973\) 13.8980 0.445548
\(974\) 3.65334 0.117061
\(975\) 1.20034 0.0384416
\(976\) 52.3437 1.67548
\(977\) 47.8634 1.53129 0.765643 0.643266i \(-0.222421\pi\)
0.765643 + 0.643266i \(0.222421\pi\)
\(978\) −18.4246 −0.589154
\(979\) −18.6152 −0.594946
\(980\) 0.299284 0.00956026
\(981\) 6.07106 0.193834
\(982\) 42.9831 1.37165
\(983\) 27.5373 0.878302 0.439151 0.898413i \(-0.355279\pi\)
0.439151 + 0.898413i \(0.355279\pi\)
\(984\) 9.59308 0.305816
\(985\) −1.68575 −0.0537124
\(986\) 77.3660 2.46384
\(987\) −11.1954 −0.356353
\(988\) −0.321259 −0.0102206
\(989\) −14.3889 −0.457541
\(990\) −5.85663 −0.186136
\(991\) 58.4210 1.85580 0.927901 0.372825i \(-0.121611\pi\)
0.927901 + 0.372825i \(0.121611\pi\)
\(992\) 2.85957 0.0907915
\(993\) 26.9205 0.854297
\(994\) −11.7783 −0.373585
\(995\) −7.75842 −0.245958
\(996\) −1.30375 −0.0413108
\(997\) −52.1960 −1.65307 −0.826533 0.562889i \(-0.809690\pi\)
−0.826533 + 0.562889i \(0.809690\pi\)
\(998\) 33.2326 1.05196
\(999\) 1.93053 0.0610792
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.1.14 41
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.o.1.14 41 1.1 even 1 trivial