Properties

Label 4011.2.a.j.1.6
Level $4011$
Weight $2$
Character 4011.1
Self dual yes
Analytic conductor $32.028$
Analytic rank $0$
Dimension $26$
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] = [4011,2,Mod(1,4011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, 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("4011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4011 = 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4011.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0279962507\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 4011.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.82745 q^{2} -1.00000 q^{3} +1.33956 q^{4} +3.12821 q^{5} +1.82745 q^{6} -1.00000 q^{7} +1.20691 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.82745 q^{2} -1.00000 q^{3} +1.33956 q^{4} +3.12821 q^{5} +1.82745 q^{6} -1.00000 q^{7} +1.20691 q^{8} +1.00000 q^{9} -5.71664 q^{10} +5.32117 q^{11} -1.33956 q^{12} -6.23012 q^{13} +1.82745 q^{14} -3.12821 q^{15} -4.88470 q^{16} +3.42890 q^{17} -1.82745 q^{18} +3.00445 q^{19} +4.19044 q^{20} +1.00000 q^{21} -9.72416 q^{22} -3.54138 q^{23} -1.20691 q^{24} +4.78570 q^{25} +11.3852 q^{26} -1.00000 q^{27} -1.33956 q^{28} +9.68026 q^{29} +5.71664 q^{30} -7.96130 q^{31} +6.51270 q^{32} -5.32117 q^{33} -6.26613 q^{34} -3.12821 q^{35} +1.33956 q^{36} +3.64886 q^{37} -5.49047 q^{38} +6.23012 q^{39} +3.77547 q^{40} +0.294773 q^{41} -1.82745 q^{42} -6.81775 q^{43} +7.12805 q^{44} +3.12821 q^{45} +6.47169 q^{46} -6.75725 q^{47} +4.88470 q^{48} +1.00000 q^{49} -8.74561 q^{50} -3.42890 q^{51} -8.34565 q^{52} +13.0827 q^{53} +1.82745 q^{54} +16.6457 q^{55} -1.20691 q^{56} -3.00445 q^{57} -17.6902 q^{58} +7.44207 q^{59} -4.19044 q^{60} +5.24370 q^{61} +14.5489 q^{62} -1.00000 q^{63} -2.13223 q^{64} -19.4891 q^{65} +9.72416 q^{66} -1.88125 q^{67} +4.59323 q^{68} +3.54138 q^{69} +5.71664 q^{70} +12.6404 q^{71} +1.20691 q^{72} +4.58132 q^{73} -6.66811 q^{74} -4.78570 q^{75} +4.02465 q^{76} -5.32117 q^{77} -11.3852 q^{78} +8.40838 q^{79} -15.2804 q^{80} +1.00000 q^{81} -0.538681 q^{82} +11.1938 q^{83} +1.33956 q^{84} +10.7263 q^{85} +12.4591 q^{86} -9.68026 q^{87} +6.42218 q^{88} -3.49270 q^{89} -5.71664 q^{90} +6.23012 q^{91} -4.74391 q^{92} +7.96130 q^{93} +12.3485 q^{94} +9.39855 q^{95} -6.51270 q^{96} -12.4927 q^{97} -1.82745 q^{98} +5.32117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 26 q^{3} + 34 q^{4} + 2 q^{5} - 26 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 26 q - 26 q^{3} + 34 q^{4} + 2 q^{5} - 26 q^{7} + 26 q^{9} - q^{10} + 13 q^{11} - 34 q^{12} - q^{13} - 2 q^{15} + 54 q^{16} + q^{19} - 22 q^{20} + 26 q^{21} + 17 q^{22} - 3 q^{23} + 48 q^{25} + 6 q^{26} - 26 q^{27} - 34 q^{28} + 23 q^{29} + q^{30} + 18 q^{31} + 10 q^{32} - 13 q^{33} - 19 q^{34} - 2 q^{35} + 34 q^{36} + 23 q^{37} - 15 q^{38} + q^{39} + 14 q^{40} - 4 q^{41} + 5 q^{43} + 60 q^{44} + 2 q^{45} + 8 q^{46} - 20 q^{47} - 54 q^{48} + 26 q^{49} + 26 q^{50} + 19 q^{52} + 31 q^{53} + 41 q^{55} - q^{57} + 19 q^{58} - 2 q^{59} + 22 q^{60} - 2 q^{61} - 35 q^{62} - 26 q^{63} + 132 q^{64} + 40 q^{65} - 17 q^{66} + 47 q^{67} - 60 q^{68} + 3 q^{69} + q^{70} + 16 q^{71} - 23 q^{73} + 34 q^{74} - 48 q^{75} + 72 q^{76} - 13 q^{77} - 6 q^{78} + 14 q^{79} - 21 q^{80} + 26 q^{81} + 60 q^{82} - 4 q^{83} + 34 q^{84} + 36 q^{85} + 21 q^{86} - 23 q^{87} + 67 q^{88} + 14 q^{89} - q^{90} + q^{91} + 20 q^{92} - 18 q^{93} + 58 q^{94} - 4 q^{95} - 10 q^{96} + 48 q^{97} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82745 −1.29220 −0.646100 0.763253i \(-0.723601\pi\)
−0.646100 + 0.763253i \(0.723601\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.33956 0.669782
\(5\) 3.12821 1.39898 0.699489 0.714643i \(-0.253411\pi\)
0.699489 + 0.714643i \(0.253411\pi\)
\(6\) 1.82745 0.746052
\(7\) −1.00000 −0.377964
\(8\) 1.20691 0.426708
\(9\) 1.00000 0.333333
\(10\) −5.71664 −1.80776
\(11\) 5.32117 1.60439 0.802196 0.597060i \(-0.203665\pi\)
0.802196 + 0.597060i \(0.203665\pi\)
\(12\) −1.33956 −0.386699
\(13\) −6.23012 −1.72793 −0.863963 0.503555i \(-0.832025\pi\)
−0.863963 + 0.503555i \(0.832025\pi\)
\(14\) 1.82745 0.488406
\(15\) −3.12821 −0.807700
\(16\) −4.88470 −1.22117
\(17\) 3.42890 0.831630 0.415815 0.909449i \(-0.363496\pi\)
0.415815 + 0.909449i \(0.363496\pi\)
\(18\) −1.82745 −0.430734
\(19\) 3.00445 0.689268 0.344634 0.938737i \(-0.388003\pi\)
0.344634 + 0.938737i \(0.388003\pi\)
\(20\) 4.19044 0.937010
\(21\) 1.00000 0.218218
\(22\) −9.72416 −2.07320
\(23\) −3.54138 −0.738430 −0.369215 0.929344i \(-0.620373\pi\)
−0.369215 + 0.929344i \(0.620373\pi\)
\(24\) −1.20691 −0.246360
\(25\) 4.78570 0.957139
\(26\) 11.3852 2.23283
\(27\) −1.00000 −0.192450
\(28\) −1.33956 −0.253154
\(29\) 9.68026 1.79758 0.898789 0.438381i \(-0.144448\pi\)
0.898789 + 0.438381i \(0.144448\pi\)
\(30\) 5.71664 1.04371
\(31\) −7.96130 −1.42989 −0.714946 0.699180i \(-0.753549\pi\)
−0.714946 + 0.699180i \(0.753549\pi\)
\(32\) 6.51270 1.15129
\(33\) −5.32117 −0.926296
\(34\) −6.26613 −1.07463
\(35\) −3.12821 −0.528764
\(36\) 1.33956 0.223261
\(37\) 3.64886 0.599869 0.299935 0.953960i \(-0.403035\pi\)
0.299935 + 0.953960i \(0.403035\pi\)
\(38\) −5.49047 −0.890672
\(39\) 6.23012 0.997618
\(40\) 3.77547 0.596955
\(41\) 0.294773 0.0460357 0.0230179 0.999735i \(-0.492673\pi\)
0.0230179 + 0.999735i \(0.492673\pi\)
\(42\) −1.82745 −0.281981
\(43\) −6.81775 −1.03970 −0.519849 0.854258i \(-0.674012\pi\)
−0.519849 + 0.854258i \(0.674012\pi\)
\(44\) 7.12805 1.07459
\(45\) 3.12821 0.466326
\(46\) 6.47169 0.954199
\(47\) −6.75725 −0.985647 −0.492823 0.870129i \(-0.664035\pi\)
−0.492823 + 0.870129i \(0.664035\pi\)
\(48\) 4.88470 0.705045
\(49\) 1.00000 0.142857
\(50\) −8.74561 −1.23682
\(51\) −3.42890 −0.480142
\(52\) −8.34565 −1.15733
\(53\) 13.0827 1.79705 0.898526 0.438921i \(-0.144639\pi\)
0.898526 + 0.438921i \(0.144639\pi\)
\(54\) 1.82745 0.248684
\(55\) 16.6457 2.24451
\(56\) −1.20691 −0.161280
\(57\) −3.00445 −0.397949
\(58\) −17.6902 −2.32283
\(59\) 7.44207 0.968875 0.484438 0.874826i \(-0.339024\pi\)
0.484438 + 0.874826i \(0.339024\pi\)
\(60\) −4.19044 −0.540983
\(61\) 5.24370 0.671387 0.335694 0.941971i \(-0.391029\pi\)
0.335694 + 0.941971i \(0.391029\pi\)
\(62\) 14.5489 1.84771
\(63\) −1.00000 −0.125988
\(64\) −2.13223 −0.266529
\(65\) −19.4891 −2.41733
\(66\) 9.72416 1.19696
\(67\) −1.88125 −0.229831 −0.114916 0.993375i \(-0.536660\pi\)
−0.114916 + 0.993375i \(0.536660\pi\)
\(68\) 4.59323 0.557011
\(69\) 3.54138 0.426333
\(70\) 5.71664 0.683269
\(71\) 12.6404 1.50014 0.750070 0.661359i \(-0.230020\pi\)
0.750070 + 0.661359i \(0.230020\pi\)
\(72\) 1.20691 0.142236
\(73\) 4.58132 0.536203 0.268102 0.963391i \(-0.413604\pi\)
0.268102 + 0.963391i \(0.413604\pi\)
\(74\) −6.66811 −0.775152
\(75\) −4.78570 −0.552604
\(76\) 4.02465 0.461659
\(77\) −5.32117 −0.606403
\(78\) −11.3852 −1.28912
\(79\) 8.40838 0.946017 0.473009 0.881058i \(-0.343168\pi\)
0.473009 + 0.881058i \(0.343168\pi\)
\(80\) −15.2804 −1.70840
\(81\) 1.00000 0.111111
\(82\) −0.538681 −0.0594874
\(83\) 11.1938 1.22868 0.614338 0.789043i \(-0.289423\pi\)
0.614338 + 0.789043i \(0.289423\pi\)
\(84\) 1.33956 0.146158
\(85\) 10.7263 1.16343
\(86\) 12.4591 1.34350
\(87\) −9.68026 −1.03783
\(88\) 6.42218 0.684607
\(89\) −3.49270 −0.370226 −0.185113 0.982717i \(-0.559265\pi\)
−0.185113 + 0.982717i \(0.559265\pi\)
\(90\) −5.71664 −0.602587
\(91\) 6.23012 0.653095
\(92\) −4.74391 −0.494587
\(93\) 7.96130 0.825548
\(94\) 12.3485 1.27365
\(95\) 9.39855 0.964271
\(96\) −6.51270 −0.664700
\(97\) −12.4927 −1.26844 −0.634222 0.773151i \(-0.718680\pi\)
−0.634222 + 0.773151i \(0.718680\pi\)
\(98\) −1.82745 −0.184600
\(99\) 5.32117 0.534798
\(100\) 6.41075 0.641075
\(101\) −6.14241 −0.611193 −0.305597 0.952161i \(-0.598856\pi\)
−0.305597 + 0.952161i \(0.598856\pi\)
\(102\) 6.26613 0.620439
\(103\) 6.42298 0.632875 0.316438 0.948613i \(-0.397513\pi\)
0.316438 + 0.948613i \(0.397513\pi\)
\(104\) −7.51921 −0.737319
\(105\) 3.12821 0.305282
\(106\) −23.9080 −2.32215
\(107\) −9.48484 −0.916934 −0.458467 0.888711i \(-0.651601\pi\)
−0.458467 + 0.888711i \(0.651601\pi\)
\(108\) −1.33956 −0.128900
\(109\) −10.5245 −1.00806 −0.504032 0.863685i \(-0.668151\pi\)
−0.504032 + 0.863685i \(0.668151\pi\)
\(110\) −30.4192 −2.90036
\(111\) −3.64886 −0.346335
\(112\) 4.88470 0.461560
\(113\) −8.33553 −0.784140 −0.392070 0.919935i \(-0.628241\pi\)
−0.392070 + 0.919935i \(0.628241\pi\)
\(114\) 5.49047 0.514230
\(115\) −11.0782 −1.03305
\(116\) 12.9673 1.20399
\(117\) −6.23012 −0.575975
\(118\) −13.6000 −1.25198
\(119\) −3.42890 −0.314326
\(120\) −3.77547 −0.344652
\(121\) 17.3148 1.57408
\(122\) −9.58259 −0.867567
\(123\) −0.294773 −0.0265787
\(124\) −10.6647 −0.957716
\(125\) −0.670390 −0.0599615
\(126\) 1.82745 0.162802
\(127\) 12.3902 1.09945 0.549726 0.835345i \(-0.314732\pi\)
0.549726 + 0.835345i \(0.314732\pi\)
\(128\) −9.12887 −0.806886
\(129\) 6.81775 0.600270
\(130\) 35.6154 3.12367
\(131\) −2.82551 −0.246866 −0.123433 0.992353i \(-0.539390\pi\)
−0.123433 + 0.992353i \(0.539390\pi\)
\(132\) −7.12805 −0.620417
\(133\) −3.00445 −0.260519
\(134\) 3.43789 0.296988
\(135\) −3.12821 −0.269233
\(136\) 4.13838 0.354863
\(137\) 13.7223 1.17238 0.586188 0.810175i \(-0.300628\pi\)
0.586188 + 0.810175i \(0.300628\pi\)
\(138\) −6.47169 −0.550907
\(139\) −8.46681 −0.718145 −0.359073 0.933310i \(-0.616907\pi\)
−0.359073 + 0.933310i \(0.616907\pi\)
\(140\) −4.19044 −0.354157
\(141\) 6.75725 0.569063
\(142\) −23.0997 −1.93848
\(143\) −33.1515 −2.77227
\(144\) −4.88470 −0.407058
\(145\) 30.2819 2.51477
\(146\) −8.37213 −0.692882
\(147\) −1.00000 −0.0824786
\(148\) 4.88789 0.401782
\(149\) 19.5588 1.60232 0.801161 0.598449i \(-0.204216\pi\)
0.801161 + 0.598449i \(0.204216\pi\)
\(150\) 8.74561 0.714076
\(151\) −15.4440 −1.25681 −0.628407 0.777885i \(-0.716293\pi\)
−0.628407 + 0.777885i \(0.716293\pi\)
\(152\) 3.62610 0.294116
\(153\) 3.42890 0.277210
\(154\) 9.72416 0.783595
\(155\) −24.9046 −2.00039
\(156\) 8.34565 0.668187
\(157\) −19.4360 −1.55116 −0.775582 0.631246i \(-0.782544\pi\)
−0.775582 + 0.631246i \(0.782544\pi\)
\(158\) −15.3659 −1.22244
\(159\) −13.0827 −1.03753
\(160\) 20.3731 1.61063
\(161\) 3.54138 0.279100
\(162\) −1.82745 −0.143578
\(163\) 5.76187 0.451304 0.225652 0.974208i \(-0.427549\pi\)
0.225652 + 0.974208i \(0.427549\pi\)
\(164\) 0.394867 0.0308339
\(165\) −16.6457 −1.29587
\(166\) −20.4560 −1.58769
\(167\) 7.03899 0.544693 0.272347 0.962199i \(-0.412200\pi\)
0.272347 + 0.962199i \(0.412200\pi\)
\(168\) 1.20691 0.0931152
\(169\) 25.8145 1.98573
\(170\) −19.6018 −1.50339
\(171\) 3.00445 0.229756
\(172\) −9.13282 −0.696371
\(173\) −10.5039 −0.798600 −0.399300 0.916820i \(-0.630747\pi\)
−0.399300 + 0.916820i \(0.630747\pi\)
\(174\) 17.6902 1.34109
\(175\) −4.78570 −0.361765
\(176\) −25.9923 −1.95924
\(177\) −7.44207 −0.559380
\(178\) 6.38273 0.478406
\(179\) −4.17244 −0.311863 −0.155931 0.987768i \(-0.549838\pi\)
−0.155931 + 0.987768i \(0.549838\pi\)
\(180\) 4.19044 0.312337
\(181\) 4.66313 0.346607 0.173304 0.984868i \(-0.444556\pi\)
0.173304 + 0.984868i \(0.444556\pi\)
\(182\) −11.3852 −0.843929
\(183\) −5.24370 −0.387626
\(184\) −4.27414 −0.315094
\(185\) 11.4144 0.839204
\(186\) −14.5489 −1.06677
\(187\) 18.2457 1.33426
\(188\) −9.05178 −0.660169
\(189\) 1.00000 0.0727393
\(190\) −17.1754 −1.24603
\(191\) 1.00000 0.0723575
\(192\) 2.13223 0.153880
\(193\) −3.70861 −0.266952 −0.133476 0.991052i \(-0.542614\pi\)
−0.133476 + 0.991052i \(0.542614\pi\)
\(194\) 22.8298 1.63908
\(195\) 19.4891 1.39565
\(196\) 1.33956 0.0956832
\(197\) 3.25792 0.232117 0.116059 0.993242i \(-0.462974\pi\)
0.116059 + 0.993242i \(0.462974\pi\)
\(198\) −9.72416 −0.691066
\(199\) −9.02179 −0.639538 −0.319769 0.947496i \(-0.603605\pi\)
−0.319769 + 0.947496i \(0.603605\pi\)
\(200\) 5.77591 0.408419
\(201\) 1.88125 0.132693
\(202\) 11.2249 0.789784
\(203\) −9.68026 −0.679421
\(204\) −4.59323 −0.321590
\(205\) 0.922110 0.0644030
\(206\) −11.7377 −0.817802
\(207\) −3.54138 −0.246143
\(208\) 30.4323 2.11010
\(209\) 15.9872 1.10586
\(210\) −5.71664 −0.394486
\(211\) 2.27137 0.156368 0.0781839 0.996939i \(-0.475088\pi\)
0.0781839 + 0.996939i \(0.475088\pi\)
\(212\) 17.5252 1.20363
\(213\) −12.6404 −0.866106
\(214\) 17.3330 1.18486
\(215\) −21.3274 −1.45451
\(216\) −1.20691 −0.0821199
\(217\) 7.96130 0.540448
\(218\) 19.2330 1.30262
\(219\) −4.58132 −0.309577
\(220\) 22.2980 1.50333
\(221\) −21.3625 −1.43699
\(222\) 6.66811 0.447534
\(223\) 0.511803 0.0342729 0.0171364 0.999853i \(-0.494545\pi\)
0.0171364 + 0.999853i \(0.494545\pi\)
\(224\) −6.51270 −0.435148
\(225\) 4.78570 0.319046
\(226\) 15.2327 1.01327
\(227\) −17.3747 −1.15320 −0.576600 0.817026i \(-0.695621\pi\)
−0.576600 + 0.817026i \(0.695621\pi\)
\(228\) −4.02465 −0.266539
\(229\) 21.6297 1.42933 0.714664 0.699468i \(-0.246579\pi\)
0.714664 + 0.699468i \(0.246579\pi\)
\(230\) 20.2448 1.33490
\(231\) 5.32117 0.350107
\(232\) 11.6832 0.767040
\(233\) 13.5171 0.885535 0.442768 0.896636i \(-0.353997\pi\)
0.442768 + 0.896636i \(0.353997\pi\)
\(234\) 11.3852 0.744276
\(235\) −21.1381 −1.37890
\(236\) 9.96913 0.648935
\(237\) −8.40838 −0.546183
\(238\) 6.26613 0.406173
\(239\) 17.1086 1.10666 0.553331 0.832962i \(-0.313357\pi\)
0.553331 + 0.832962i \(0.313357\pi\)
\(240\) 15.2804 0.986343
\(241\) 21.6264 1.39308 0.696541 0.717517i \(-0.254721\pi\)
0.696541 + 0.717517i \(0.254721\pi\)
\(242\) −31.6419 −2.03402
\(243\) −1.00000 −0.0641500
\(244\) 7.02428 0.449683
\(245\) 3.12821 0.199854
\(246\) 0.538681 0.0343451
\(247\) −18.7181 −1.19100
\(248\) −9.60858 −0.610146
\(249\) −11.1938 −0.709376
\(250\) 1.22510 0.0774823
\(251\) 7.73773 0.488401 0.244201 0.969725i \(-0.421474\pi\)
0.244201 + 0.969725i \(0.421474\pi\)
\(252\) −1.33956 −0.0843846
\(253\) −18.8443 −1.18473
\(254\) −22.6424 −1.42071
\(255\) −10.7263 −0.671707
\(256\) 20.9470 1.30919
\(257\) 25.9076 1.61607 0.808037 0.589132i \(-0.200530\pi\)
0.808037 + 0.589132i \(0.200530\pi\)
\(258\) −12.4591 −0.775669
\(259\) −3.64886 −0.226729
\(260\) −26.1070 −1.61908
\(261\) 9.68026 0.599193
\(262\) 5.16346 0.319000
\(263\) 20.7483 1.27940 0.639699 0.768626i \(-0.279059\pi\)
0.639699 + 0.768626i \(0.279059\pi\)
\(264\) −6.42218 −0.395258
\(265\) 40.9255 2.51404
\(266\) 5.49047 0.336643
\(267\) 3.49270 0.213750
\(268\) −2.52006 −0.153937
\(269\) −7.05197 −0.429966 −0.214983 0.976618i \(-0.568970\pi\)
−0.214983 + 0.976618i \(0.568970\pi\)
\(270\) 5.71664 0.347904
\(271\) 23.2104 1.40993 0.704964 0.709243i \(-0.250963\pi\)
0.704964 + 0.709243i \(0.250963\pi\)
\(272\) −16.7491 −1.01556
\(273\) −6.23012 −0.377064
\(274\) −25.0768 −1.51495
\(275\) 25.4655 1.53563
\(276\) 4.74391 0.285550
\(277\) −2.12246 −0.127526 −0.0637632 0.997965i \(-0.520310\pi\)
−0.0637632 + 0.997965i \(0.520310\pi\)
\(278\) 15.4727 0.927988
\(279\) −7.96130 −0.476630
\(280\) −3.77547 −0.225628
\(281\) −10.5544 −0.629625 −0.314812 0.949154i \(-0.601942\pi\)
−0.314812 + 0.949154i \(0.601942\pi\)
\(282\) −12.3485 −0.735344
\(283\) −31.5098 −1.87306 −0.936532 0.350583i \(-0.885984\pi\)
−0.936532 + 0.350583i \(0.885984\pi\)
\(284\) 16.9326 1.00477
\(285\) −9.39855 −0.556722
\(286\) 60.5827 3.58233
\(287\) −0.294773 −0.0173999
\(288\) 6.51270 0.383765
\(289\) −5.24267 −0.308392
\(290\) −55.3385 −3.24959
\(291\) 12.4927 0.732337
\(292\) 6.13698 0.359139
\(293\) −21.2010 −1.23858 −0.619289 0.785163i \(-0.712579\pi\)
−0.619289 + 0.785163i \(0.712579\pi\)
\(294\) 1.82745 0.106579
\(295\) 23.2804 1.35543
\(296\) 4.40386 0.255969
\(297\) −5.32117 −0.308765
\(298\) −35.7427 −2.07052
\(299\) 22.0633 1.27595
\(300\) −6.41075 −0.370125
\(301\) 6.81775 0.392969
\(302\) 28.2231 1.62406
\(303\) 6.14241 0.352873
\(304\) −14.6758 −0.841716
\(305\) 16.4034 0.939256
\(306\) −6.26613 −0.358211
\(307\) −3.93876 −0.224797 −0.112399 0.993663i \(-0.535853\pi\)
−0.112399 + 0.993663i \(0.535853\pi\)
\(308\) −7.12805 −0.406158
\(309\) −6.42298 −0.365391
\(310\) 45.5119 2.58490
\(311\) −7.83859 −0.444486 −0.222243 0.974991i \(-0.571338\pi\)
−0.222243 + 0.974991i \(0.571338\pi\)
\(312\) 7.51921 0.425691
\(313\) 5.71415 0.322983 0.161491 0.986874i \(-0.448370\pi\)
0.161491 + 0.986874i \(0.448370\pi\)
\(314\) 35.5183 2.00442
\(315\) −3.12821 −0.176255
\(316\) 11.2636 0.633625
\(317\) 15.6031 0.876356 0.438178 0.898888i \(-0.355624\pi\)
0.438178 + 0.898888i \(0.355624\pi\)
\(318\) 23.9080 1.34069
\(319\) 51.5103 2.88402
\(320\) −6.67006 −0.372868
\(321\) 9.48484 0.529392
\(322\) −6.47169 −0.360653
\(323\) 10.3019 0.573216
\(324\) 1.33956 0.0744202
\(325\) −29.8155 −1.65387
\(326\) −10.5295 −0.583176
\(327\) 10.5245 0.582006
\(328\) 0.355764 0.0196438
\(329\) 6.75725 0.372539
\(330\) 30.4192 1.67452
\(331\) −22.3680 −1.22946 −0.614729 0.788738i \(-0.710735\pi\)
−0.614729 + 0.788738i \(0.710735\pi\)
\(332\) 14.9948 0.822945
\(333\) 3.64886 0.199956
\(334\) −12.8634 −0.703853
\(335\) −5.88495 −0.321529
\(336\) −4.88470 −0.266482
\(337\) −30.9508 −1.68600 −0.843000 0.537914i \(-0.819213\pi\)
−0.843000 + 0.537914i \(0.819213\pi\)
\(338\) −47.1746 −2.56596
\(339\) 8.33553 0.452724
\(340\) 14.3686 0.779246
\(341\) −42.3634 −2.29411
\(342\) −5.49047 −0.296891
\(343\) −1.00000 −0.0539949
\(344\) −8.22842 −0.443647
\(345\) 11.0782 0.596430
\(346\) 19.1954 1.03195
\(347\) 20.6092 1.10636 0.553179 0.833062i \(-0.313415\pi\)
0.553179 + 0.833062i \(0.313415\pi\)
\(348\) −12.9673 −0.695122
\(349\) −13.3692 −0.715636 −0.357818 0.933791i \(-0.616479\pi\)
−0.357818 + 0.933791i \(0.616479\pi\)
\(350\) 8.74561 0.467472
\(351\) 6.23012 0.332539
\(352\) 34.6552 1.84713
\(353\) 14.2895 0.760554 0.380277 0.924873i \(-0.375829\pi\)
0.380277 + 0.924873i \(0.375829\pi\)
\(354\) 13.6000 0.722831
\(355\) 39.5418 2.09866
\(356\) −4.67870 −0.247971
\(357\) 3.42890 0.181476
\(358\) 7.62491 0.402989
\(359\) 0.666146 0.0351579 0.0175789 0.999845i \(-0.494404\pi\)
0.0175789 + 0.999845i \(0.494404\pi\)
\(360\) 3.77547 0.198985
\(361\) −9.97328 −0.524910
\(362\) −8.52162 −0.447886
\(363\) −17.3148 −0.908793
\(364\) 8.34565 0.437431
\(365\) 14.3313 0.750136
\(366\) 9.58259 0.500890
\(367\) 29.4696 1.53830 0.769150 0.639068i \(-0.220680\pi\)
0.769150 + 0.639068i \(0.220680\pi\)
\(368\) 17.2986 0.901751
\(369\) 0.294773 0.0153452
\(370\) −20.8592 −1.08442
\(371\) −13.0827 −0.679222
\(372\) 10.6647 0.552937
\(373\) 31.2192 1.61647 0.808233 0.588862i \(-0.200424\pi\)
0.808233 + 0.588862i \(0.200424\pi\)
\(374\) −33.3431 −1.72413
\(375\) 0.670390 0.0346188
\(376\) −8.15541 −0.420583
\(377\) −60.3092 −3.10608
\(378\) −1.82745 −0.0939938
\(379\) −7.11163 −0.365300 −0.182650 0.983178i \(-0.558467\pi\)
−0.182650 + 0.983178i \(0.558467\pi\)
\(380\) 12.5900 0.645851
\(381\) −12.3902 −0.634769
\(382\) −1.82745 −0.0935003
\(383\) −23.0648 −1.17855 −0.589277 0.807931i \(-0.700587\pi\)
−0.589277 + 0.807931i \(0.700587\pi\)
\(384\) 9.12887 0.465856
\(385\) −16.6457 −0.848345
\(386\) 6.77729 0.344955
\(387\) −6.81775 −0.346566
\(388\) −16.7348 −0.849582
\(389\) −1.55520 −0.0788520 −0.0394260 0.999222i \(-0.512553\pi\)
−0.0394260 + 0.999222i \(0.512553\pi\)
\(390\) −35.6154 −1.80345
\(391\) −12.1430 −0.614100
\(392\) 1.20691 0.0609582
\(393\) 2.82551 0.142528
\(394\) −5.95368 −0.299942
\(395\) 26.3032 1.32346
\(396\) 7.12805 0.358198
\(397\) 33.9256 1.70268 0.851339 0.524615i \(-0.175791\pi\)
0.851339 + 0.524615i \(0.175791\pi\)
\(398\) 16.4868 0.826411
\(399\) 3.00445 0.150411
\(400\) −23.3767 −1.16883
\(401\) 23.1948 1.15830 0.579148 0.815223i \(-0.303385\pi\)
0.579148 + 0.815223i \(0.303385\pi\)
\(402\) −3.43789 −0.171466
\(403\) 49.5999 2.47075
\(404\) −8.22816 −0.409366
\(405\) 3.12821 0.155442
\(406\) 17.6902 0.877948
\(407\) 19.4162 0.962426
\(408\) −4.13838 −0.204880
\(409\) 17.2077 0.850864 0.425432 0.904990i \(-0.360122\pi\)
0.425432 + 0.904990i \(0.360122\pi\)
\(410\) −1.68511 −0.0832216
\(411\) −13.7223 −0.676872
\(412\) 8.60400 0.423889
\(413\) −7.44207 −0.366200
\(414\) 6.47169 0.318066
\(415\) 35.0164 1.71889
\(416\) −40.5749 −1.98935
\(417\) 8.46681 0.414621
\(418\) −29.2157 −1.42899
\(419\) 24.1906 1.18179 0.590895 0.806749i \(-0.298775\pi\)
0.590895 + 0.806749i \(0.298775\pi\)
\(420\) 4.19044 0.204472
\(421\) 18.9170 0.921959 0.460979 0.887411i \(-0.347498\pi\)
0.460979 + 0.887411i \(0.347498\pi\)
\(422\) −4.15082 −0.202059
\(423\) −6.75725 −0.328549
\(424\) 15.7897 0.766816
\(425\) 16.4097 0.795985
\(426\) 23.0997 1.11918
\(427\) −5.24370 −0.253761
\(428\) −12.7055 −0.614146
\(429\) 33.1515 1.60057
\(430\) 38.9746 1.87952
\(431\) 13.7526 0.662440 0.331220 0.943553i \(-0.392540\pi\)
0.331220 + 0.943553i \(0.392540\pi\)
\(432\) 4.88470 0.235015
\(433\) 17.7825 0.854574 0.427287 0.904116i \(-0.359469\pi\)
0.427287 + 0.904116i \(0.359469\pi\)
\(434\) −14.5489 −0.698367
\(435\) −30.2819 −1.45190
\(436\) −14.0983 −0.675184
\(437\) −10.6399 −0.508976
\(438\) 8.37213 0.400036
\(439\) −12.7244 −0.607303 −0.303651 0.952783i \(-0.598206\pi\)
−0.303651 + 0.952783i \(0.598206\pi\)
\(440\) 20.0899 0.957750
\(441\) 1.00000 0.0476190
\(442\) 39.0388 1.85688
\(443\) 34.8267 1.65466 0.827332 0.561713i \(-0.189857\pi\)
0.827332 + 0.561713i \(0.189857\pi\)
\(444\) −4.88789 −0.231969
\(445\) −10.9259 −0.517938
\(446\) −0.935293 −0.0442874
\(447\) −19.5588 −0.925101
\(448\) 2.13223 0.100738
\(449\) 20.1055 0.948838 0.474419 0.880299i \(-0.342658\pi\)
0.474419 + 0.880299i \(0.342658\pi\)
\(450\) −8.74561 −0.412272
\(451\) 1.56853 0.0738594
\(452\) −11.1660 −0.525203
\(453\) 15.4440 0.725622
\(454\) 31.7514 1.49017
\(455\) 19.4891 0.913665
\(456\) −3.62610 −0.169808
\(457\) 3.08405 0.144266 0.0721328 0.997395i \(-0.477019\pi\)
0.0721328 + 0.997395i \(0.477019\pi\)
\(458\) −39.5271 −1.84698
\(459\) −3.42890 −0.160047
\(460\) −14.8400 −0.691916
\(461\) 39.0986 1.82101 0.910503 0.413503i \(-0.135695\pi\)
0.910503 + 0.413503i \(0.135695\pi\)
\(462\) −9.72416 −0.452409
\(463\) 30.5919 1.42173 0.710863 0.703330i \(-0.248304\pi\)
0.710863 + 0.703330i \(0.248304\pi\)
\(464\) −47.2851 −2.19516
\(465\) 24.9046 1.15492
\(466\) −24.7018 −1.14429
\(467\) 28.0040 1.29587 0.647935 0.761695i \(-0.275633\pi\)
0.647935 + 0.761695i \(0.275633\pi\)
\(468\) −8.34565 −0.385778
\(469\) 1.88125 0.0868681
\(470\) 38.6288 1.78181
\(471\) 19.4360 0.895565
\(472\) 8.98192 0.413426
\(473\) −36.2784 −1.66808
\(474\) 15.3659 0.705778
\(475\) 14.3784 0.659725
\(476\) −4.59323 −0.210530
\(477\) 13.0827 0.599017
\(478\) −31.2650 −1.43003
\(479\) −1.49664 −0.0683833 −0.0341917 0.999415i \(-0.510886\pi\)
−0.0341917 + 0.999415i \(0.510886\pi\)
\(480\) −20.3731 −0.929900
\(481\) −22.7329 −1.03653
\(482\) −39.5212 −1.80014
\(483\) −3.54138 −0.161139
\(484\) 23.1943 1.05429
\(485\) −39.0799 −1.77453
\(486\) 1.82745 0.0828947
\(487\) −16.1889 −0.733590 −0.366795 0.930302i \(-0.619545\pi\)
−0.366795 + 0.930302i \(0.619545\pi\)
\(488\) 6.32868 0.286486
\(489\) −5.76187 −0.260561
\(490\) −5.71664 −0.258251
\(491\) −19.9786 −0.901621 −0.450810 0.892620i \(-0.648865\pi\)
−0.450810 + 0.892620i \(0.648865\pi\)
\(492\) −0.394867 −0.0178020
\(493\) 33.1926 1.49492
\(494\) 34.2063 1.53902
\(495\) 16.6457 0.748170
\(496\) 38.8885 1.74615
\(497\) −12.6404 −0.566999
\(498\) 20.4560 0.916656
\(499\) −6.36633 −0.284996 −0.142498 0.989795i \(-0.545513\pi\)
−0.142498 + 0.989795i \(0.545513\pi\)
\(500\) −0.898030 −0.0401611
\(501\) −7.03899 −0.314479
\(502\) −14.1403 −0.631112
\(503\) 26.1278 1.16498 0.582490 0.812838i \(-0.302079\pi\)
0.582490 + 0.812838i \(0.302079\pi\)
\(504\) −1.20691 −0.0537601
\(505\) −19.2148 −0.855046
\(506\) 34.4370 1.53091
\(507\) −25.8145 −1.14646
\(508\) 16.5975 0.736394
\(509\) −10.3430 −0.458447 −0.229224 0.973374i \(-0.573619\pi\)
−0.229224 + 0.973374i \(0.573619\pi\)
\(510\) 19.6018 0.867981
\(511\) −4.58132 −0.202666
\(512\) −20.0218 −0.884846
\(513\) −3.00445 −0.132650
\(514\) −47.3448 −2.08829
\(515\) 20.0924 0.885379
\(516\) 9.13282 0.402050
\(517\) −35.9565 −1.58136
\(518\) 6.66811 0.292980
\(519\) 10.5039 0.461072
\(520\) −23.5217 −1.03149
\(521\) 29.5916 1.29643 0.648215 0.761457i \(-0.275516\pi\)
0.648215 + 0.761457i \(0.275516\pi\)
\(522\) −17.6902 −0.774277
\(523\) −29.9079 −1.30778 −0.653890 0.756590i \(-0.726864\pi\)
−0.653890 + 0.756590i \(0.726864\pi\)
\(524\) −3.78495 −0.165346
\(525\) 4.78570 0.208865
\(526\) −37.9165 −1.65324
\(527\) −27.2985 −1.18914
\(528\) 25.9923 1.13117
\(529\) −10.4586 −0.454722
\(530\) −74.7893 −3.24864
\(531\) 7.44207 0.322958
\(532\) −4.02465 −0.174491
\(533\) −1.83647 −0.0795463
\(534\) −6.38273 −0.276208
\(535\) −29.6706 −1.28277
\(536\) −2.27050 −0.0980708
\(537\) 4.17244 0.180054
\(538\) 12.8871 0.555603
\(539\) 5.32117 0.229199
\(540\) −4.19044 −0.180328
\(541\) 21.5157 0.925030 0.462515 0.886611i \(-0.346947\pi\)
0.462515 + 0.886611i \(0.346947\pi\)
\(542\) −42.4157 −1.82191
\(543\) −4.66313 −0.200114
\(544\) 22.3314 0.957450
\(545\) −32.9229 −1.41026
\(546\) 11.3852 0.487243
\(547\) −36.7028 −1.56930 −0.784650 0.619939i \(-0.787157\pi\)
−0.784650 + 0.619939i \(0.787157\pi\)
\(548\) 18.3819 0.785237
\(549\) 5.24370 0.223796
\(550\) −46.5368 −1.98434
\(551\) 29.0838 1.23901
\(552\) 4.27414 0.181919
\(553\) −8.40838 −0.357561
\(554\) 3.87869 0.164790
\(555\) −11.4144 −0.484515
\(556\) −11.3418 −0.481001
\(557\) 8.09376 0.342944 0.171472 0.985189i \(-0.445148\pi\)
0.171472 + 0.985189i \(0.445148\pi\)
\(558\) 14.5489 0.615902
\(559\) 42.4754 1.79652
\(560\) 15.2804 0.645713
\(561\) −18.2457 −0.770336
\(562\) 19.2877 0.813601
\(563\) 37.0310 1.56067 0.780334 0.625362i \(-0.215049\pi\)
0.780334 + 0.625362i \(0.215049\pi\)
\(564\) 9.05178 0.381149
\(565\) −26.0753 −1.09700
\(566\) 57.5825 2.42037
\(567\) −1.00000 −0.0419961
\(568\) 15.2558 0.640121
\(569\) 24.5532 1.02932 0.514661 0.857394i \(-0.327918\pi\)
0.514661 + 0.857394i \(0.327918\pi\)
\(570\) 17.1754 0.719396
\(571\) −32.3888 −1.35543 −0.677715 0.735325i \(-0.737030\pi\)
−0.677715 + 0.735325i \(0.737030\pi\)
\(572\) −44.4086 −1.85682
\(573\) −1.00000 −0.0417756
\(574\) 0.538681 0.0224841
\(575\) −16.9480 −0.706780
\(576\) −2.13223 −0.0888429
\(577\) 2.67247 0.111256 0.0556281 0.998452i \(-0.482284\pi\)
0.0556281 + 0.998452i \(0.482284\pi\)
\(578\) 9.58070 0.398504
\(579\) 3.70861 0.154125
\(580\) 40.5645 1.68435
\(581\) −11.1938 −0.464396
\(582\) −22.8298 −0.946326
\(583\) 69.6154 2.88318
\(584\) 5.52925 0.228802
\(585\) −19.4891 −0.805777
\(586\) 38.7438 1.60049
\(587\) −25.6696 −1.05950 −0.529750 0.848154i \(-0.677714\pi\)
−0.529750 + 0.848154i \(0.677714\pi\)
\(588\) −1.33956 −0.0552427
\(589\) −23.9193 −0.985578
\(590\) −42.5436 −1.75149
\(591\) −3.25792 −0.134013
\(592\) −17.8236 −0.732545
\(593\) 29.3254 1.20425 0.602125 0.798402i \(-0.294321\pi\)
0.602125 + 0.798402i \(0.294321\pi\)
\(594\) 9.72416 0.398987
\(595\) −10.7263 −0.439736
\(596\) 26.2003 1.07321
\(597\) 9.02179 0.369237
\(598\) −40.3195 −1.64879
\(599\) −16.7040 −0.682508 −0.341254 0.939971i \(-0.610852\pi\)
−0.341254 + 0.939971i \(0.610852\pi\)
\(600\) −5.77591 −0.235801
\(601\) 31.4715 1.28375 0.641875 0.766809i \(-0.278157\pi\)
0.641875 + 0.766809i \(0.278157\pi\)
\(602\) −12.4591 −0.507794
\(603\) −1.88125 −0.0766105
\(604\) −20.6882 −0.841792
\(605\) 54.1644 2.20210
\(606\) −11.2249 −0.455982
\(607\) −39.7592 −1.61377 −0.806887 0.590705i \(-0.798850\pi\)
−0.806887 + 0.590705i \(0.798850\pi\)
\(608\) 19.5671 0.793550
\(609\) 9.68026 0.392264
\(610\) −29.9764 −1.21371
\(611\) 42.0985 1.70312
\(612\) 4.59323 0.185670
\(613\) 0.219957 0.00888397 0.00444199 0.999990i \(-0.498586\pi\)
0.00444199 + 0.999990i \(0.498586\pi\)
\(614\) 7.19788 0.290483
\(615\) −0.922110 −0.0371831
\(616\) −6.42218 −0.258757
\(617\) −13.7439 −0.553308 −0.276654 0.960970i \(-0.589226\pi\)
−0.276654 + 0.960970i \(0.589226\pi\)
\(618\) 11.7377 0.472158
\(619\) −29.9646 −1.20438 −0.602190 0.798353i \(-0.705705\pi\)
−0.602190 + 0.798353i \(0.705705\pi\)
\(620\) −33.3613 −1.33982
\(621\) 3.54138 0.142111
\(622\) 14.3246 0.574365
\(623\) 3.49270 0.139932
\(624\) −30.4323 −1.21827
\(625\) −26.0256 −1.04102
\(626\) −10.4423 −0.417359
\(627\) −15.9872 −0.638466
\(628\) −26.0358 −1.03894
\(629\) 12.5116 0.498869
\(630\) 5.71664 0.227756
\(631\) −10.3776 −0.413128 −0.206564 0.978433i \(-0.566228\pi\)
−0.206564 + 0.978433i \(0.566228\pi\)
\(632\) 10.1482 0.403673
\(633\) −2.27137 −0.0902790
\(634\) −28.5138 −1.13243
\(635\) 38.7592 1.53811
\(636\) −17.5252 −0.694918
\(637\) −6.23012 −0.246847
\(638\) −94.1323 −3.72673
\(639\) 12.6404 0.500046
\(640\) −28.5570 −1.12881
\(641\) −44.3334 −1.75107 −0.875533 0.483159i \(-0.839489\pi\)
−0.875533 + 0.483159i \(0.839489\pi\)
\(642\) −17.3330 −0.684081
\(643\) −29.4003 −1.15943 −0.579717 0.814818i \(-0.696837\pi\)
−0.579717 + 0.814818i \(0.696837\pi\)
\(644\) 4.74391 0.186936
\(645\) 21.3274 0.839764
\(646\) −18.8263 −0.740710
\(647\) −6.96071 −0.273654 −0.136827 0.990595i \(-0.543690\pi\)
−0.136827 + 0.990595i \(0.543690\pi\)
\(648\) 1.20691 0.0474120
\(649\) 39.6005 1.55446
\(650\) 54.4862 2.13713
\(651\) −7.96130 −0.312028
\(652\) 7.71839 0.302276
\(653\) 30.7488 1.20330 0.601648 0.798762i \(-0.294511\pi\)
0.601648 + 0.798762i \(0.294511\pi\)
\(654\) −19.2330 −0.752069
\(655\) −8.83877 −0.345359
\(656\) −1.43987 −0.0562177
\(657\) 4.58132 0.178734
\(658\) −12.3485 −0.481396
\(659\) 0.138118 0.00538029 0.00269015 0.999996i \(-0.499144\pi\)
0.00269015 + 0.999996i \(0.499144\pi\)
\(660\) −22.2980 −0.867949
\(661\) 25.7912 1.00316 0.501581 0.865111i \(-0.332752\pi\)
0.501581 + 0.865111i \(0.332752\pi\)
\(662\) 40.8764 1.58871
\(663\) 21.3625 0.829649
\(664\) 13.5099 0.524285
\(665\) −9.39855 −0.364460
\(666\) −6.66811 −0.258384
\(667\) −34.2815 −1.32739
\(668\) 9.42918 0.364826
\(669\) −0.511803 −0.0197874
\(670\) 10.7544 0.415480
\(671\) 27.9026 1.07717
\(672\) 6.51270 0.251233
\(673\) 7.16235 0.276088 0.138044 0.990426i \(-0.455918\pi\)
0.138044 + 0.990426i \(0.455918\pi\)
\(674\) 56.5610 2.17865
\(675\) −4.78570 −0.184201
\(676\) 34.5801 1.33000
\(677\) 33.2004 1.27600 0.637998 0.770038i \(-0.279763\pi\)
0.637998 + 0.770038i \(0.279763\pi\)
\(678\) −15.2327 −0.585010
\(679\) 12.4927 0.479427
\(680\) 12.9457 0.496445
\(681\) 17.3747 0.665801
\(682\) 77.4169 2.96445
\(683\) −32.3402 −1.23746 −0.618732 0.785602i \(-0.712353\pi\)
−0.618732 + 0.785602i \(0.712353\pi\)
\(684\) 4.02465 0.153886
\(685\) 42.9263 1.64013
\(686\) 1.82745 0.0697723
\(687\) −21.6297 −0.825223
\(688\) 33.3026 1.26965
\(689\) −81.5071 −3.10517
\(690\) −20.2448 −0.770707
\(691\) −0.421706 −0.0160424 −0.00802122 0.999968i \(-0.502553\pi\)
−0.00802122 + 0.999968i \(0.502553\pi\)
\(692\) −14.0707 −0.534888
\(693\) −5.32117 −0.202134
\(694\) −37.6622 −1.42964
\(695\) −26.4860 −1.00467
\(696\) −11.6832 −0.442851
\(697\) 1.01074 0.0382847
\(698\) 24.4315 0.924745
\(699\) −13.5171 −0.511264
\(700\) −6.41075 −0.242303
\(701\) 23.6163 0.891974 0.445987 0.895039i \(-0.352853\pi\)
0.445987 + 0.895039i \(0.352853\pi\)
\(702\) −11.3852 −0.429708
\(703\) 10.9628 0.413471
\(704\) −11.3460 −0.427617
\(705\) 21.1381 0.796107
\(706\) −26.1133 −0.982789
\(707\) 6.14241 0.231009
\(708\) −9.96913 −0.374663
\(709\) −39.0828 −1.46779 −0.733893 0.679265i \(-0.762299\pi\)
−0.733893 + 0.679265i \(0.762299\pi\)
\(710\) −72.2606 −2.71189
\(711\) 8.40838 0.315339
\(712\) −4.21539 −0.157978
\(713\) 28.1940 1.05587
\(714\) −6.26613 −0.234504
\(715\) −103.705 −3.87835
\(716\) −5.58925 −0.208880
\(717\) −17.1086 −0.638931
\(718\) −1.21735 −0.0454310
\(719\) −18.2086 −0.679066 −0.339533 0.940594i \(-0.610269\pi\)
−0.339533 + 0.940594i \(0.610269\pi\)
\(720\) −15.2804 −0.569465
\(721\) −6.42298 −0.239204
\(722\) 18.2257 0.678289
\(723\) −21.6264 −0.804296
\(724\) 6.24656 0.232151
\(725\) 46.3268 1.72053
\(726\) 31.6419 1.17434
\(727\) 22.8951 0.849134 0.424567 0.905397i \(-0.360426\pi\)
0.424567 + 0.905397i \(0.360426\pi\)
\(728\) 7.51921 0.278680
\(729\) 1.00000 0.0370370
\(730\) −26.1898 −0.969327
\(731\) −23.3774 −0.864643
\(732\) −7.02428 −0.259625
\(733\) 52.7136 1.94702 0.973511 0.228639i \(-0.0734275\pi\)
0.973511 + 0.228639i \(0.0734275\pi\)
\(734\) −53.8542 −1.98779
\(735\) −3.12821 −0.115386
\(736\) −23.0640 −0.850150
\(737\) −10.0105 −0.368740
\(738\) −0.538681 −0.0198291
\(739\) −6.13486 −0.225674 −0.112837 0.993613i \(-0.535994\pi\)
−0.112837 + 0.993613i \(0.535994\pi\)
\(740\) 15.2903 0.562084
\(741\) 18.7181 0.687626
\(742\) 23.9080 0.877691
\(743\) 2.51718 0.0923465 0.0461733 0.998933i \(-0.485297\pi\)
0.0461733 + 0.998933i \(0.485297\pi\)
\(744\) 9.60858 0.352268
\(745\) 61.1841 2.24161
\(746\) −57.0514 −2.08880
\(747\) 11.1938 0.409558
\(748\) 24.4413 0.893664
\(749\) 9.48484 0.346568
\(750\) −1.22510 −0.0447344
\(751\) −39.5923 −1.44475 −0.722373 0.691504i \(-0.756948\pi\)
−0.722373 + 0.691504i \(0.756948\pi\)
\(752\) 33.0071 1.20365
\(753\) −7.73773 −0.281978
\(754\) 110.212 4.01368
\(755\) −48.3120 −1.75826
\(756\) 1.33956 0.0487195
\(757\) 0.850165 0.0308998 0.0154499 0.999881i \(-0.495082\pi\)
0.0154499 + 0.999881i \(0.495082\pi\)
\(758\) 12.9961 0.472041
\(759\) 18.8443 0.684005
\(760\) 11.3432 0.411462
\(761\) −26.2249 −0.950652 −0.475326 0.879810i \(-0.657670\pi\)
−0.475326 + 0.879810i \(0.657670\pi\)
\(762\) 22.6424 0.820249
\(763\) 10.5245 0.381013
\(764\) 1.33956 0.0484637
\(765\) 10.7263 0.387811
\(766\) 42.1496 1.52293
\(767\) −46.3650 −1.67414
\(768\) −20.9470 −0.755859
\(769\) −50.5509 −1.82291 −0.911455 0.411399i \(-0.865040\pi\)
−0.911455 + 0.411399i \(0.865040\pi\)
\(770\) 30.4192 1.09623
\(771\) −25.9076 −0.933040
\(772\) −4.96792 −0.178799
\(773\) 12.6438 0.454767 0.227384 0.973805i \(-0.426983\pi\)
0.227384 + 0.973805i \(0.426983\pi\)
\(774\) 12.4591 0.447833
\(775\) −38.1003 −1.36860
\(776\) −15.0776 −0.541255
\(777\) 3.64886 0.130902
\(778\) 2.84205 0.101893
\(779\) 0.885629 0.0317310
\(780\) 26.1070 0.934779
\(781\) 67.2617 2.40681
\(782\) 22.1908 0.793540
\(783\) −9.68026 −0.345944
\(784\) −4.88470 −0.174453
\(785\) −60.8000 −2.17005
\(786\) −5.16346 −0.184175
\(787\) −23.3402 −0.831989 −0.415995 0.909367i \(-0.636567\pi\)
−0.415995 + 0.909367i \(0.636567\pi\)
\(788\) 4.36419 0.155468
\(789\) −20.7483 −0.738660
\(790\) −48.0677 −1.71017
\(791\) 8.33553 0.296377
\(792\) 6.42218 0.228202
\(793\) −32.6689 −1.16011
\(794\) −61.9973 −2.20020
\(795\) −40.9255 −1.45148
\(796\) −12.0853 −0.428351
\(797\) 9.55645 0.338507 0.169253 0.985573i \(-0.445864\pi\)
0.169253 + 0.985573i \(0.445864\pi\)
\(798\) −5.49047 −0.194361
\(799\) −23.1699 −0.819693
\(800\) 31.1678 1.10195
\(801\) −3.49270 −0.123409
\(802\) −42.3874 −1.49675
\(803\) 24.3780 0.860281
\(804\) 2.52006 0.0888755
\(805\) 11.0782 0.390455
\(806\) −90.6412 −3.19270
\(807\) 7.05197 0.248241
\(808\) −7.41335 −0.260801
\(809\) −40.8987 −1.43792 −0.718961 0.695050i \(-0.755382\pi\)
−0.718961 + 0.695050i \(0.755382\pi\)
\(810\) −5.71664 −0.200862
\(811\) 9.69394 0.340400 0.170200 0.985410i \(-0.445559\pi\)
0.170200 + 0.985410i \(0.445559\pi\)
\(812\) −12.9673 −0.455064
\(813\) −23.2104 −0.814023
\(814\) −35.4821 −1.24365
\(815\) 18.0243 0.631365
\(816\) 16.7491 0.586336
\(817\) −20.4836 −0.716630
\(818\) −31.4461 −1.09949
\(819\) 6.23012 0.217698
\(820\) 1.23523 0.0431360
\(821\) 31.3298 1.09342 0.546708 0.837323i \(-0.315881\pi\)
0.546708 + 0.837323i \(0.315881\pi\)
\(822\) 25.0768 0.874654
\(823\) −28.0131 −0.976474 −0.488237 0.872711i \(-0.662360\pi\)
−0.488237 + 0.872711i \(0.662360\pi\)
\(824\) 7.75197 0.270053
\(825\) −25.4655 −0.886595
\(826\) 13.6000 0.473204
\(827\) −37.3982 −1.30046 −0.650232 0.759736i \(-0.725328\pi\)
−0.650232 + 0.759736i \(0.725328\pi\)
\(828\) −4.74391 −0.164862
\(829\) 35.5639 1.23519 0.617594 0.786497i \(-0.288108\pi\)
0.617594 + 0.786497i \(0.288108\pi\)
\(830\) −63.9907 −2.22115
\(831\) 2.12246 0.0736274
\(832\) 13.2841 0.460542
\(833\) 3.42890 0.118804
\(834\) −15.4727 −0.535774
\(835\) 22.0194 0.762014
\(836\) 21.4159 0.740683
\(837\) 7.96130 0.275183
\(838\) −44.2071 −1.52711
\(839\) −7.11969 −0.245799 −0.122899 0.992419i \(-0.539219\pi\)
−0.122899 + 0.992419i \(0.539219\pi\)
\(840\) 3.77547 0.130266
\(841\) 64.7073 2.23129
\(842\) −34.5699 −1.19136
\(843\) 10.5544 0.363514
\(844\) 3.04265 0.104732
\(845\) 80.7530 2.77799
\(846\) 12.3485 0.424551
\(847\) −17.3148 −0.594945
\(848\) −63.9052 −2.19451
\(849\) 31.5098 1.08141
\(850\) −29.9878 −1.02857
\(851\) −12.9220 −0.442961
\(852\) −16.9326 −0.580102
\(853\) −30.3904 −1.04055 −0.520273 0.854000i \(-0.674170\pi\)
−0.520273 + 0.854000i \(0.674170\pi\)
\(854\) 9.58259 0.327909
\(855\) 9.39855 0.321424
\(856\) −11.4474 −0.391263
\(857\) −0.620340 −0.0211904 −0.0105952 0.999944i \(-0.503373\pi\)
−0.0105952 + 0.999944i \(0.503373\pi\)
\(858\) −60.5827 −2.06826
\(859\) −12.7467 −0.434910 −0.217455 0.976070i \(-0.569776\pi\)
−0.217455 + 0.976070i \(0.569776\pi\)
\(860\) −28.5694 −0.974207
\(861\) 0.294773 0.0100458
\(862\) −25.1322 −0.856006
\(863\) −41.9397 −1.42765 −0.713823 0.700327i \(-0.753038\pi\)
−0.713823 + 0.700327i \(0.753038\pi\)
\(864\) −6.51270 −0.221567
\(865\) −32.8585 −1.11722
\(866\) −32.4966 −1.10428
\(867\) 5.24267 0.178050
\(868\) 10.6647 0.361982
\(869\) 44.7424 1.51778
\(870\) 55.3385 1.87615
\(871\) 11.7204 0.397132
\(872\) −12.7021 −0.430149
\(873\) −12.4927 −0.422815
\(874\) 19.4439 0.657699
\(875\) 0.670390 0.0226633
\(876\) −6.13698 −0.207349
\(877\) −37.0536 −1.25121 −0.625606 0.780139i \(-0.715148\pi\)
−0.625606 + 0.780139i \(0.715148\pi\)
\(878\) 23.2532 0.784757
\(879\) 21.2010 0.715093
\(880\) −81.3093 −2.74094
\(881\) −12.8619 −0.433328 −0.216664 0.976246i \(-0.569518\pi\)
−0.216664 + 0.976246i \(0.569518\pi\)
\(882\) −1.82745 −0.0615334
\(883\) 17.5336 0.590052 0.295026 0.955489i \(-0.404672\pi\)
0.295026 + 0.955489i \(0.404672\pi\)
\(884\) −28.6164 −0.962473
\(885\) −23.2804 −0.782561
\(886\) −63.6439 −2.13816
\(887\) 48.9193 1.64255 0.821275 0.570533i \(-0.193263\pi\)
0.821275 + 0.570533i \(0.193263\pi\)
\(888\) −4.40386 −0.147784
\(889\) −12.3902 −0.415554
\(890\) 19.9665 0.669280
\(891\) 5.32117 0.178266
\(892\) 0.685593 0.0229553
\(893\) −20.3018 −0.679375
\(894\) 35.7427 1.19542
\(895\) −13.0523 −0.436289
\(896\) 9.12887 0.304974
\(897\) −22.0633 −0.736671
\(898\) −36.7418 −1.22609
\(899\) −77.0674 −2.57034
\(900\) 6.41075 0.213692
\(901\) 44.8593 1.49448
\(902\) −2.86641 −0.0954412
\(903\) −6.81775 −0.226881
\(904\) −10.0602 −0.334599
\(905\) 14.5872 0.484896
\(906\) −28.2231 −0.937649
\(907\) −15.8634 −0.526736 −0.263368 0.964696i \(-0.584833\pi\)
−0.263368 + 0.964696i \(0.584833\pi\)
\(908\) −23.2746 −0.772393
\(909\) −6.14241 −0.203731
\(910\) −35.6154 −1.18064
\(911\) 22.4983 0.745402 0.372701 0.927951i \(-0.378432\pi\)
0.372701 + 0.927951i \(0.378432\pi\)
\(912\) 14.6758 0.485965
\(913\) 59.5639 1.97128
\(914\) −5.63594 −0.186420
\(915\) −16.4034 −0.542280
\(916\) 28.9743 0.957339
\(917\) 2.82551 0.0933064
\(918\) 6.26613 0.206813
\(919\) 34.5130 1.13848 0.569240 0.822172i \(-0.307238\pi\)
0.569240 + 0.822172i \(0.307238\pi\)
\(920\) −13.3704 −0.440809
\(921\) 3.93876 0.129787
\(922\) −71.4507 −2.35310
\(923\) −78.7513 −2.59213
\(924\) 7.12805 0.234496
\(925\) 17.4623 0.574158
\(926\) −55.9051 −1.83716
\(927\) 6.42298 0.210958
\(928\) 63.0446 2.06954
\(929\) −25.1880 −0.826393 −0.413196 0.910642i \(-0.635588\pi\)
−0.413196 + 0.910642i \(0.635588\pi\)
\(930\) −45.5119 −1.49239
\(931\) 3.00445 0.0984668
\(932\) 18.1070 0.593116
\(933\) 7.83859 0.256624
\(934\) −51.1759 −1.67453
\(935\) 57.0765 1.86660
\(936\) −7.51921 −0.245773
\(937\) −1.46214 −0.0477661 −0.0238830 0.999715i \(-0.507603\pi\)
−0.0238830 + 0.999715i \(0.507603\pi\)
\(938\) −3.43789 −0.112251
\(939\) −5.71415 −0.186474
\(940\) −28.3159 −0.923561
\(941\) −30.4128 −0.991428 −0.495714 0.868486i \(-0.665094\pi\)
−0.495714 + 0.868486i \(0.665094\pi\)
\(942\) −35.5183 −1.15725
\(943\) −1.04390 −0.0339942
\(944\) −36.3523 −1.18317
\(945\) 3.12821 0.101761
\(946\) 66.2969 2.15550
\(947\) 19.6335 0.638003 0.319001 0.947754i \(-0.396653\pi\)
0.319001 + 0.947754i \(0.396653\pi\)
\(948\) −11.2636 −0.365824
\(949\) −28.5422 −0.926519
\(950\) −26.2757 −0.852497
\(951\) −15.6031 −0.505965
\(952\) −4.13838 −0.134126
\(953\) −11.9892 −0.388367 −0.194183 0.980965i \(-0.562206\pi\)
−0.194183 + 0.980965i \(0.562206\pi\)
\(954\) −23.9080 −0.774050
\(955\) 3.12821 0.101226
\(956\) 22.9180 0.741222
\(957\) −51.5103 −1.66509
\(958\) 2.73503 0.0883650
\(959\) −13.7223 −0.443117
\(960\) 6.67006 0.215275
\(961\) 32.3822 1.04459
\(962\) 41.5431 1.33940
\(963\) −9.48484 −0.305645
\(964\) 28.9700 0.933061
\(965\) −11.6013 −0.373459
\(966\) 6.47169 0.208223
\(967\) 41.4489 1.33291 0.666454 0.745546i \(-0.267811\pi\)
0.666454 + 0.745546i \(0.267811\pi\)
\(968\) 20.8975 0.671670
\(969\) −10.3019 −0.330946
\(970\) 71.4164 2.29304
\(971\) −1.60436 −0.0514863 −0.0257431 0.999669i \(-0.508195\pi\)
−0.0257431 + 0.999669i \(0.508195\pi\)
\(972\) −1.33956 −0.0429665
\(973\) 8.46681 0.271433
\(974\) 29.5844 0.947946
\(975\) 29.8155 0.954860
\(976\) −25.6139 −0.819881
\(977\) 47.0685 1.50585 0.752927 0.658104i \(-0.228641\pi\)
0.752927 + 0.658104i \(0.228641\pi\)
\(978\) 10.5295 0.336697
\(979\) −18.5853 −0.593988
\(980\) 4.19044 0.133859
\(981\) −10.5245 −0.336022
\(982\) 36.5098 1.16507
\(983\) −45.3101 −1.44517 −0.722584 0.691283i \(-0.757046\pi\)
−0.722584 + 0.691283i \(0.757046\pi\)
\(984\) −0.355764 −0.0113414
\(985\) 10.1915 0.324727
\(986\) −60.6577 −1.93174
\(987\) −6.75725 −0.215086
\(988\) −25.0741 −0.797713
\(989\) 24.1443 0.767743
\(990\) −30.4192 −0.966786
\(991\) 34.9225 1.10935 0.554674 0.832068i \(-0.312843\pi\)
0.554674 + 0.832068i \(0.312843\pi\)
\(992\) −51.8496 −1.64623
\(993\) 22.3680 0.709828
\(994\) 23.0997 0.732677
\(995\) −28.2220 −0.894699
\(996\) −14.9948 −0.475127
\(997\) −59.9137 −1.89749 −0.948743 0.316050i \(-0.897643\pi\)
−0.948743 + 0.316050i \(0.897643\pi\)
\(998\) 11.6341 0.368272
\(999\) −3.64886 −0.115445
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 4011.2.a.j.1.6 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4011.2.a.j.1.6 26 1.1 even 1 trivial