Properties

Label 4022.2.a.c.1.12
Level $4022$
Weight $2$
Character 4022.1
Self dual yes
Analytic conductor $32.116$
Analytic rank $1$
Dimension $31$
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] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
Analytic rank: \(1\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 4022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.49586 q^{3} +1.00000 q^{4} +3.62778 q^{5} -1.49586 q^{6} -1.50310 q^{7} +1.00000 q^{8} -0.762393 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.49586 q^{3} +1.00000 q^{4} +3.62778 q^{5} -1.49586 q^{6} -1.50310 q^{7} +1.00000 q^{8} -0.762393 q^{9} +3.62778 q^{10} -3.49664 q^{11} -1.49586 q^{12} +2.70243 q^{13} -1.50310 q^{14} -5.42666 q^{15} +1.00000 q^{16} -4.55648 q^{17} -0.762393 q^{18} -4.73207 q^{19} +3.62778 q^{20} +2.24843 q^{21} -3.49664 q^{22} -6.90130 q^{23} -1.49586 q^{24} +8.16077 q^{25} +2.70243 q^{26} +5.62803 q^{27} -1.50310 q^{28} +9.73743 q^{29} -5.42666 q^{30} +9.42235 q^{31} +1.00000 q^{32} +5.23049 q^{33} -4.55648 q^{34} -5.45290 q^{35} -0.762393 q^{36} -5.74459 q^{37} -4.73207 q^{38} -4.04246 q^{39} +3.62778 q^{40} -11.9608 q^{41} +2.24843 q^{42} -10.7989 q^{43} -3.49664 q^{44} -2.76579 q^{45} -6.90130 q^{46} -5.41035 q^{47} -1.49586 q^{48} -4.74070 q^{49} +8.16077 q^{50} +6.81587 q^{51} +2.70243 q^{52} -9.69149 q^{53} +5.62803 q^{54} -12.6850 q^{55} -1.50310 q^{56} +7.07852 q^{57} +9.73743 q^{58} -5.55792 q^{59} -5.42666 q^{60} +11.2266 q^{61} +9.42235 q^{62} +1.14595 q^{63} +1.00000 q^{64} +9.80380 q^{65} +5.23049 q^{66} -12.1387 q^{67} -4.55648 q^{68} +10.3234 q^{69} -5.45290 q^{70} +10.9179 q^{71} -0.762393 q^{72} -15.2619 q^{73} -5.74459 q^{74} -12.2074 q^{75} -4.73207 q^{76} +5.25579 q^{77} -4.04246 q^{78} +3.43435 q^{79} +3.62778 q^{80} -6.13158 q^{81} -11.9608 q^{82} +6.27372 q^{83} +2.24843 q^{84} -16.5299 q^{85} -10.7989 q^{86} -14.5659 q^{87} -3.49664 q^{88} +1.43060 q^{89} -2.76579 q^{90} -4.06201 q^{91} -6.90130 q^{92} -14.0946 q^{93} -5.41035 q^{94} -17.1669 q^{95} -1.49586 q^{96} +11.2812 q^{97} -4.74070 q^{98} +2.66581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.49586 −0.863637 −0.431819 0.901960i \(-0.642128\pi\)
−0.431819 + 0.901960i \(0.642128\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.62778 1.62239 0.811196 0.584775i \(-0.198817\pi\)
0.811196 + 0.584775i \(0.198817\pi\)
\(6\) −1.49586 −0.610684
\(7\) −1.50310 −0.568117 −0.284059 0.958807i \(-0.591681\pi\)
−0.284059 + 0.958807i \(0.591681\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.762393 −0.254131
\(10\) 3.62778 1.14720
\(11\) −3.49664 −1.05428 −0.527138 0.849780i \(-0.676735\pi\)
−0.527138 + 0.849780i \(0.676735\pi\)
\(12\) −1.49586 −0.431819
\(13\) 2.70243 0.749518 0.374759 0.927122i \(-0.377725\pi\)
0.374759 + 0.927122i \(0.377725\pi\)
\(14\) −1.50310 −0.401720
\(15\) −5.42666 −1.40116
\(16\) 1.00000 0.250000
\(17\) −4.55648 −1.10511 −0.552554 0.833477i \(-0.686347\pi\)
−0.552554 + 0.833477i \(0.686347\pi\)
\(18\) −0.762393 −0.179698
\(19\) −4.73207 −1.08561 −0.542805 0.839859i \(-0.682638\pi\)
−0.542805 + 0.839859i \(0.682638\pi\)
\(20\) 3.62778 0.811196
\(21\) 2.24843 0.490647
\(22\) −3.49664 −0.745486
\(23\) −6.90130 −1.43902 −0.719511 0.694482i \(-0.755634\pi\)
−0.719511 + 0.694482i \(0.755634\pi\)
\(24\) −1.49586 −0.305342
\(25\) 8.16077 1.63215
\(26\) 2.70243 0.529989
\(27\) 5.62803 1.08311
\(28\) −1.50310 −0.284059
\(29\) 9.73743 1.80820 0.904098 0.427326i \(-0.140544\pi\)
0.904098 + 0.427326i \(0.140544\pi\)
\(30\) −5.42666 −0.990768
\(31\) 9.42235 1.69230 0.846152 0.532941i \(-0.178913\pi\)
0.846152 + 0.532941i \(0.178913\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.23049 0.910512
\(34\) −4.55648 −0.781429
\(35\) −5.45290 −0.921709
\(36\) −0.762393 −0.127065
\(37\) −5.74459 −0.944405 −0.472203 0.881490i \(-0.656541\pi\)
−0.472203 + 0.881490i \(0.656541\pi\)
\(38\) −4.73207 −0.767642
\(39\) −4.04246 −0.647312
\(40\) 3.62778 0.573602
\(41\) −11.9608 −1.86796 −0.933979 0.357328i \(-0.883688\pi\)
−0.933979 + 0.357328i \(0.883688\pi\)
\(42\) 2.24843 0.346940
\(43\) −10.7989 −1.64681 −0.823405 0.567454i \(-0.807929\pi\)
−0.823405 + 0.567454i \(0.807929\pi\)
\(44\) −3.49664 −0.527138
\(45\) −2.76579 −0.412300
\(46\) −6.90130 −1.01754
\(47\) −5.41035 −0.789180 −0.394590 0.918857i \(-0.629113\pi\)
−0.394590 + 0.918857i \(0.629113\pi\)
\(48\) −1.49586 −0.215909
\(49\) −4.74070 −0.677243
\(50\) 8.16077 1.15411
\(51\) 6.81587 0.954412
\(52\) 2.70243 0.374759
\(53\) −9.69149 −1.33123 −0.665614 0.746296i \(-0.731830\pi\)
−0.665614 + 0.746296i \(0.731830\pi\)
\(54\) 5.62803 0.765877
\(55\) −12.6850 −1.71045
\(56\) −1.50310 −0.200860
\(57\) 7.07852 0.937573
\(58\) 9.73743 1.27859
\(59\) −5.55792 −0.723580 −0.361790 0.932260i \(-0.617834\pi\)
−0.361790 + 0.932260i \(0.617834\pi\)
\(60\) −5.42666 −0.700579
\(61\) 11.2266 1.43742 0.718710 0.695310i \(-0.244733\pi\)
0.718710 + 0.695310i \(0.244733\pi\)
\(62\) 9.42235 1.19664
\(63\) 1.14595 0.144376
\(64\) 1.00000 0.125000
\(65\) 9.80380 1.21601
\(66\) 5.23049 0.643829
\(67\) −12.1387 −1.48298 −0.741491 0.670963i \(-0.765881\pi\)
−0.741491 + 0.670963i \(0.765881\pi\)
\(68\) −4.55648 −0.552554
\(69\) 10.3234 1.24279
\(70\) −5.45290 −0.651746
\(71\) 10.9179 1.29572 0.647860 0.761759i \(-0.275664\pi\)
0.647860 + 0.761759i \(0.275664\pi\)
\(72\) −0.762393 −0.0898488
\(73\) −15.2619 −1.78627 −0.893135 0.449788i \(-0.851499\pi\)
−0.893135 + 0.449788i \(0.851499\pi\)
\(74\) −5.74459 −0.667795
\(75\) −12.2074 −1.40959
\(76\) −4.73207 −0.542805
\(77\) 5.25579 0.598952
\(78\) −4.04246 −0.457719
\(79\) 3.43435 0.386395 0.193197 0.981160i \(-0.438114\pi\)
0.193197 + 0.981160i \(0.438114\pi\)
\(80\) 3.62778 0.405598
\(81\) −6.13158 −0.681287
\(82\) −11.9608 −1.32085
\(83\) 6.27372 0.688631 0.344315 0.938854i \(-0.388111\pi\)
0.344315 + 0.938854i \(0.388111\pi\)
\(84\) 2.24843 0.245324
\(85\) −16.5299 −1.79292
\(86\) −10.7989 −1.16447
\(87\) −14.5659 −1.56162
\(88\) −3.49664 −0.372743
\(89\) 1.43060 0.151643 0.0758215 0.997121i \(-0.475842\pi\)
0.0758215 + 0.997121i \(0.475842\pi\)
\(90\) −2.76579 −0.291540
\(91\) −4.06201 −0.425814
\(92\) −6.90130 −0.719511
\(93\) −14.0946 −1.46154
\(94\) −5.41035 −0.558035
\(95\) −17.1669 −1.76128
\(96\) −1.49586 −0.152671
\(97\) 11.2812 1.14543 0.572717 0.819753i \(-0.305889\pi\)
0.572717 + 0.819753i \(0.305889\pi\)
\(98\) −4.74070 −0.478883
\(99\) 2.66581 0.267924
\(100\) 8.16077 0.816077
\(101\) 10.2606 1.02097 0.510485 0.859887i \(-0.329466\pi\)
0.510485 + 0.859887i \(0.329466\pi\)
\(102\) 6.81587 0.674871
\(103\) −9.36610 −0.922869 −0.461435 0.887174i \(-0.652665\pi\)
−0.461435 + 0.887174i \(0.652665\pi\)
\(104\) 2.70243 0.264995
\(105\) 8.15680 0.796022
\(106\) −9.69149 −0.941321
\(107\) 6.92796 0.669751 0.334875 0.942262i \(-0.391306\pi\)
0.334875 + 0.942262i \(0.391306\pi\)
\(108\) 5.62803 0.541557
\(109\) −7.39654 −0.708460 −0.354230 0.935158i \(-0.615257\pi\)
−0.354230 + 0.935158i \(0.615257\pi\)
\(110\) −12.6850 −1.20947
\(111\) 8.59312 0.815623
\(112\) −1.50310 −0.142029
\(113\) 19.3027 1.81584 0.907921 0.419142i \(-0.137669\pi\)
0.907921 + 0.419142i \(0.137669\pi\)
\(114\) 7.07852 0.662964
\(115\) −25.0364 −2.33466
\(116\) 9.73743 0.904098
\(117\) −2.06031 −0.190476
\(118\) −5.55792 −0.511648
\(119\) 6.84883 0.627831
\(120\) −5.42666 −0.495384
\(121\) 1.22647 0.111497
\(122\) 11.2266 1.01641
\(123\) 17.8917 1.61324
\(124\) 9.42235 0.846152
\(125\) 11.4666 1.02560
\(126\) 1.14595 0.102089
\(127\) −7.60958 −0.675241 −0.337621 0.941282i \(-0.609622\pi\)
−0.337621 + 0.941282i \(0.609622\pi\)
\(128\) 1.00000 0.0883883
\(129\) 16.1536 1.42225
\(130\) 9.80380 0.859850
\(131\) −20.8788 −1.82419 −0.912097 0.409975i \(-0.865537\pi\)
−0.912097 + 0.409975i \(0.865537\pi\)
\(132\) 5.23049 0.455256
\(133\) 7.11275 0.616754
\(134\) −12.1387 −1.04863
\(135\) 20.4172 1.75723
\(136\) −4.55648 −0.390715
\(137\) 4.99729 0.426947 0.213474 0.976949i \(-0.431522\pi\)
0.213474 + 0.976949i \(0.431522\pi\)
\(138\) 10.3234 0.878787
\(139\) 18.0601 1.53183 0.765917 0.642939i \(-0.222285\pi\)
0.765917 + 0.642939i \(0.222285\pi\)
\(140\) −5.45290 −0.460854
\(141\) 8.09314 0.681566
\(142\) 10.9179 0.916213
\(143\) −9.44940 −0.790199
\(144\) −0.762393 −0.0635327
\(145\) 35.3252 2.93360
\(146\) −15.2619 −1.26308
\(147\) 7.09144 0.584892
\(148\) −5.74459 −0.472203
\(149\) 2.22612 0.182371 0.0911855 0.995834i \(-0.470934\pi\)
0.0911855 + 0.995834i \(0.470934\pi\)
\(150\) −12.2074 −0.996729
\(151\) −14.1180 −1.14891 −0.574455 0.818536i \(-0.694786\pi\)
−0.574455 + 0.818536i \(0.694786\pi\)
\(152\) −4.73207 −0.383821
\(153\) 3.47382 0.280842
\(154\) 5.25579 0.423523
\(155\) 34.1822 2.74558
\(156\) −4.04246 −0.323656
\(157\) 4.67349 0.372985 0.186492 0.982456i \(-0.440288\pi\)
0.186492 + 0.982456i \(0.440288\pi\)
\(158\) 3.43435 0.273222
\(159\) 14.4971 1.14970
\(160\) 3.62778 0.286801
\(161\) 10.3733 0.817533
\(162\) −6.13158 −0.481742
\(163\) −20.9106 −1.63785 −0.818923 0.573903i \(-0.805429\pi\)
−0.818923 + 0.573903i \(0.805429\pi\)
\(164\) −11.9608 −0.933979
\(165\) 18.9751 1.47721
\(166\) 6.27372 0.486935
\(167\) −8.91764 −0.690067 −0.345034 0.938590i \(-0.612133\pi\)
−0.345034 + 0.938590i \(0.612133\pi\)
\(168\) 2.24843 0.173470
\(169\) −5.69689 −0.438222
\(170\) −16.5299 −1.26778
\(171\) 3.60769 0.275887
\(172\) −10.7989 −0.823405
\(173\) 12.6495 0.961723 0.480862 0.876797i \(-0.340324\pi\)
0.480862 + 0.876797i \(0.340324\pi\)
\(174\) −14.5659 −1.10424
\(175\) −12.2664 −0.927254
\(176\) −3.49664 −0.263569
\(177\) 8.31390 0.624911
\(178\) 1.43060 0.107228
\(179\) −16.5914 −1.24010 −0.620050 0.784562i \(-0.712888\pi\)
−0.620050 + 0.784562i \(0.712888\pi\)
\(180\) −2.76579 −0.206150
\(181\) −2.31522 −0.172089 −0.0860444 0.996291i \(-0.527423\pi\)
−0.0860444 + 0.996291i \(0.527423\pi\)
\(182\) −4.06201 −0.301096
\(183\) −16.7935 −1.24141
\(184\) −6.90130 −0.508771
\(185\) −20.8401 −1.53219
\(186\) −14.0946 −1.03346
\(187\) 15.9323 1.16509
\(188\) −5.41035 −0.394590
\(189\) −8.45947 −0.615336
\(190\) −17.1669 −1.24542
\(191\) −11.7056 −0.846989 −0.423495 0.905899i \(-0.639197\pi\)
−0.423495 + 0.905899i \(0.639197\pi\)
\(192\) −1.49586 −0.107955
\(193\) −12.0489 −0.867300 −0.433650 0.901081i \(-0.642775\pi\)
−0.433650 + 0.901081i \(0.642775\pi\)
\(194\) 11.2812 0.809945
\(195\) −14.6651 −1.05019
\(196\) −4.74070 −0.338621
\(197\) −17.2450 −1.22865 −0.614327 0.789052i \(-0.710572\pi\)
−0.614327 + 0.789052i \(0.710572\pi\)
\(198\) 2.66581 0.189451
\(199\) 4.09423 0.290233 0.145116 0.989415i \(-0.453644\pi\)
0.145116 + 0.989415i \(0.453644\pi\)
\(200\) 8.16077 0.577053
\(201\) 18.1579 1.28076
\(202\) 10.2606 0.721934
\(203\) −14.6363 −1.02727
\(204\) 6.81587 0.477206
\(205\) −43.3910 −3.03056
\(206\) −9.36610 −0.652567
\(207\) 5.26150 0.365700
\(208\) 2.70243 0.187380
\(209\) 16.5463 1.14453
\(210\) 8.15680 0.562872
\(211\) −1.59128 −0.109548 −0.0547740 0.998499i \(-0.517444\pi\)
−0.0547740 + 0.998499i \(0.517444\pi\)
\(212\) −9.69149 −0.665614
\(213\) −16.3317 −1.11903
\(214\) 6.92796 0.473585
\(215\) −39.1759 −2.67177
\(216\) 5.62803 0.382939
\(217\) −14.1627 −0.961427
\(218\) −7.39654 −0.500957
\(219\) 22.8297 1.54269
\(220\) −12.6850 −0.855224
\(221\) −12.3135 −0.828299
\(222\) 8.59312 0.576733
\(223\) 18.5568 1.24266 0.621329 0.783550i \(-0.286593\pi\)
0.621329 + 0.783550i \(0.286593\pi\)
\(224\) −1.50310 −0.100430
\(225\) −6.22171 −0.414780
\(226\) 19.3027 1.28399
\(227\) −3.04139 −0.201864 −0.100932 0.994893i \(-0.532182\pi\)
−0.100932 + 0.994893i \(0.532182\pi\)
\(228\) 7.07852 0.468787
\(229\) 6.50956 0.430164 0.215082 0.976596i \(-0.430998\pi\)
0.215082 + 0.976596i \(0.430998\pi\)
\(230\) −25.0364 −1.65085
\(231\) −7.86194 −0.517277
\(232\) 9.73743 0.639294
\(233\) −3.98723 −0.261212 −0.130606 0.991434i \(-0.541692\pi\)
−0.130606 + 0.991434i \(0.541692\pi\)
\(234\) −2.06031 −0.134687
\(235\) −19.6275 −1.28036
\(236\) −5.55792 −0.361790
\(237\) −5.13732 −0.333705
\(238\) 6.84883 0.443944
\(239\) 7.69185 0.497544 0.248772 0.968562i \(-0.419973\pi\)
0.248772 + 0.968562i \(0.419973\pi\)
\(240\) −5.42666 −0.350289
\(241\) 26.7382 1.72236 0.861181 0.508298i \(-0.169725\pi\)
0.861181 + 0.508298i \(0.169725\pi\)
\(242\) 1.22647 0.0788405
\(243\) −7.71207 −0.494730
\(244\) 11.2266 0.718710
\(245\) −17.1982 −1.09875
\(246\) 17.8917 1.14073
\(247\) −12.7881 −0.813685
\(248\) 9.42235 0.598320
\(249\) −9.38463 −0.594727
\(250\) 11.4666 0.725208
\(251\) 6.31167 0.398389 0.199195 0.979960i \(-0.436167\pi\)
0.199195 + 0.979960i \(0.436167\pi\)
\(252\) 1.14595 0.0721881
\(253\) 24.1314 1.51713
\(254\) −7.60958 −0.477468
\(255\) 24.7264 1.54843
\(256\) 1.00000 0.0625000
\(257\) −11.7701 −0.734196 −0.367098 0.930182i \(-0.619649\pi\)
−0.367098 + 0.930182i \(0.619649\pi\)
\(258\) 16.1536 1.00568
\(259\) 8.63468 0.536533
\(260\) 9.80380 0.608006
\(261\) −7.42375 −0.459518
\(262\) −20.8788 −1.28990
\(263\) 14.7343 0.908555 0.454278 0.890860i \(-0.349897\pi\)
0.454278 + 0.890860i \(0.349897\pi\)
\(264\) 5.23049 0.321914
\(265\) −35.1586 −2.15977
\(266\) 7.11275 0.436111
\(267\) −2.13998 −0.130964
\(268\) −12.1387 −0.741491
\(269\) −17.4625 −1.06471 −0.532353 0.846522i \(-0.678692\pi\)
−0.532353 + 0.846522i \(0.678692\pi\)
\(270\) 20.4172 1.24255
\(271\) −16.2335 −0.986117 −0.493059 0.869996i \(-0.664121\pi\)
−0.493059 + 0.869996i \(0.664121\pi\)
\(272\) −4.55648 −0.276277
\(273\) 6.07621 0.367749
\(274\) 4.99729 0.301897
\(275\) −28.5352 −1.72074
\(276\) 10.3234 0.621396
\(277\) 17.6711 1.06175 0.530875 0.847450i \(-0.321863\pi\)
0.530875 + 0.847450i \(0.321863\pi\)
\(278\) 18.0601 1.08317
\(279\) −7.18353 −0.430067
\(280\) −5.45290 −0.325873
\(281\) 11.5639 0.689845 0.344922 0.938631i \(-0.387905\pi\)
0.344922 + 0.938631i \(0.387905\pi\)
\(282\) 8.09314 0.481940
\(283\) −5.54353 −0.329529 −0.164764 0.986333i \(-0.552686\pi\)
−0.164764 + 0.986333i \(0.552686\pi\)
\(284\) 10.9179 0.647860
\(285\) 25.6793 1.52111
\(286\) −9.44940 −0.558755
\(287\) 17.9782 1.06122
\(288\) −0.762393 −0.0449244
\(289\) 3.76148 0.221264
\(290\) 35.3252 2.07437
\(291\) −16.8752 −0.989240
\(292\) −15.2619 −0.893135
\(293\) −26.7201 −1.56100 −0.780502 0.625154i \(-0.785036\pi\)
−0.780502 + 0.625154i \(0.785036\pi\)
\(294\) 7.09144 0.413581
\(295\) −20.1629 −1.17393
\(296\) −5.74459 −0.333898
\(297\) −19.6792 −1.14190
\(298\) 2.22612 0.128956
\(299\) −18.6503 −1.07857
\(300\) −12.2074 −0.704794
\(301\) 16.2317 0.935582
\(302\) −14.1180 −0.812402
\(303\) −15.3485 −0.881747
\(304\) −4.73207 −0.271403
\(305\) 40.7277 2.33206
\(306\) 3.47382 0.198585
\(307\) 2.42093 0.138170 0.0690848 0.997611i \(-0.477992\pi\)
0.0690848 + 0.997611i \(0.477992\pi\)
\(308\) 5.25579 0.299476
\(309\) 14.0104 0.797024
\(310\) 34.1822 1.94142
\(311\) 26.0257 1.47578 0.737891 0.674920i \(-0.235822\pi\)
0.737891 + 0.674920i \(0.235822\pi\)
\(312\) −4.04246 −0.228859
\(313\) −2.18509 −0.123509 −0.0617544 0.998091i \(-0.519670\pi\)
−0.0617544 + 0.998091i \(0.519670\pi\)
\(314\) 4.67349 0.263740
\(315\) 4.15725 0.234235
\(316\) 3.43435 0.193197
\(317\) 28.6549 1.60942 0.804711 0.593667i \(-0.202320\pi\)
0.804711 + 0.593667i \(0.202320\pi\)
\(318\) 14.4971 0.812960
\(319\) −34.0483 −1.90634
\(320\) 3.62778 0.202799
\(321\) −10.3633 −0.578422
\(322\) 10.3733 0.578083
\(323\) 21.5615 1.19972
\(324\) −6.13158 −0.340643
\(325\) 22.0539 1.22333
\(326\) −20.9106 −1.15813
\(327\) 11.0642 0.611852
\(328\) −11.9608 −0.660423
\(329\) 8.13228 0.448347
\(330\) 18.9751 1.04454
\(331\) −19.2549 −1.05834 −0.529171 0.848515i \(-0.677497\pi\)
−0.529171 + 0.848515i \(0.677497\pi\)
\(332\) 6.27372 0.344315
\(333\) 4.37963 0.240003
\(334\) −8.91764 −0.487951
\(335\) −44.0366 −2.40598
\(336\) 2.24843 0.122662
\(337\) 13.2036 0.719246 0.359623 0.933098i \(-0.382905\pi\)
0.359623 + 0.933098i \(0.382905\pi\)
\(338\) −5.69689 −0.309870
\(339\) −28.8741 −1.56823
\(340\) −16.5299 −0.896459
\(341\) −32.9465 −1.78416
\(342\) 3.60769 0.195082
\(343\) 17.6474 0.952871
\(344\) −10.7989 −0.582236
\(345\) 37.4510 2.01629
\(346\) 12.6495 0.680041
\(347\) −11.0019 −0.590613 −0.295306 0.955403i \(-0.595422\pi\)
−0.295306 + 0.955403i \(0.595422\pi\)
\(348\) −14.5659 −0.780812
\(349\) −12.1730 −0.651607 −0.325803 0.945438i \(-0.605635\pi\)
−0.325803 + 0.945438i \(0.605635\pi\)
\(350\) −12.2664 −0.655668
\(351\) 15.2093 0.811814
\(352\) −3.49664 −0.186371
\(353\) 22.3049 1.18717 0.593586 0.804770i \(-0.297712\pi\)
0.593586 + 0.804770i \(0.297712\pi\)
\(354\) 8.31390 0.441879
\(355\) 39.6078 2.10217
\(356\) 1.43060 0.0758215
\(357\) −10.2449 −0.542218
\(358\) −16.5914 −0.876883
\(359\) 9.94994 0.525138 0.262569 0.964913i \(-0.415430\pi\)
0.262569 + 0.964913i \(0.415430\pi\)
\(360\) −2.76579 −0.145770
\(361\) 3.39244 0.178550
\(362\) −2.31522 −0.121685
\(363\) −1.83463 −0.0962933
\(364\) −4.06201 −0.212907
\(365\) −55.3668 −2.89803
\(366\) −16.7935 −0.877809
\(367\) 21.2306 1.10823 0.554115 0.832440i \(-0.313057\pi\)
0.554115 + 0.832440i \(0.313057\pi\)
\(368\) −6.90130 −0.359755
\(369\) 9.11880 0.474706
\(370\) −20.8401 −1.08343
\(371\) 14.5673 0.756294
\(372\) −14.0946 −0.730768
\(373\) 17.8721 0.925380 0.462690 0.886520i \(-0.346884\pi\)
0.462690 + 0.886520i \(0.346884\pi\)
\(374\) 15.9323 0.823842
\(375\) −17.1524 −0.885746
\(376\) −5.41035 −0.279017
\(377\) 26.3147 1.35528
\(378\) −8.45947 −0.435108
\(379\) 20.5879 1.05753 0.528764 0.848769i \(-0.322656\pi\)
0.528764 + 0.848769i \(0.322656\pi\)
\(380\) −17.1669 −0.880642
\(381\) 11.3829 0.583163
\(382\) −11.7056 −0.598912
\(383\) −28.5254 −1.45758 −0.728789 0.684739i \(-0.759916\pi\)
−0.728789 + 0.684739i \(0.759916\pi\)
\(384\) −1.49586 −0.0763355
\(385\) 19.0668 0.971735
\(386\) −12.0489 −0.613274
\(387\) 8.23297 0.418506
\(388\) 11.2812 0.572717
\(389\) −17.2629 −0.875261 −0.437631 0.899155i \(-0.644182\pi\)
−0.437631 + 0.899155i \(0.644182\pi\)
\(390\) −14.6651 −0.742598
\(391\) 31.4456 1.59027
\(392\) −4.74070 −0.239441
\(393\) 31.2319 1.57544
\(394\) −17.2450 −0.868789
\(395\) 12.4591 0.626884
\(396\) 2.66581 0.133962
\(397\) −33.6567 −1.68918 −0.844591 0.535411i \(-0.820157\pi\)
−0.844591 + 0.535411i \(0.820157\pi\)
\(398\) 4.09423 0.205225
\(399\) −10.6397 −0.532652
\(400\) 8.16077 0.408038
\(401\) −21.0584 −1.05161 −0.525803 0.850607i \(-0.676235\pi\)
−0.525803 + 0.850607i \(0.676235\pi\)
\(402\) 18.1579 0.905633
\(403\) 25.4632 1.26841
\(404\) 10.2606 0.510485
\(405\) −22.2440 −1.10531
\(406\) −14.6363 −0.726388
\(407\) 20.0868 0.995663
\(408\) 6.81587 0.337436
\(409\) −14.8304 −0.733318 −0.366659 0.930355i \(-0.619498\pi\)
−0.366659 + 0.930355i \(0.619498\pi\)
\(410\) −43.3910 −2.14293
\(411\) −7.47526 −0.368728
\(412\) −9.36610 −0.461435
\(413\) 8.35410 0.411078
\(414\) 5.26150 0.258589
\(415\) 22.7597 1.11723
\(416\) 2.70243 0.132497
\(417\) −27.0154 −1.32295
\(418\) 16.5463 0.809307
\(419\) 11.9369 0.583157 0.291579 0.956547i \(-0.405819\pi\)
0.291579 + 0.956547i \(0.405819\pi\)
\(420\) 8.15680 0.398011
\(421\) −26.9131 −1.31167 −0.655833 0.754906i \(-0.727683\pi\)
−0.655833 + 0.754906i \(0.727683\pi\)
\(422\) −1.59128 −0.0774622
\(423\) 4.12481 0.200555
\(424\) −9.69149 −0.470661
\(425\) −37.1843 −1.80371
\(426\) −16.3317 −0.791276
\(427\) −16.8747 −0.816624
\(428\) 6.92796 0.334875
\(429\) 14.1350 0.682445
\(430\) −39.1759 −1.88923
\(431\) 8.13486 0.391842 0.195921 0.980620i \(-0.437230\pi\)
0.195921 + 0.980620i \(0.437230\pi\)
\(432\) 5.62803 0.270779
\(433\) 5.64742 0.271398 0.135699 0.990750i \(-0.456672\pi\)
0.135699 + 0.990750i \(0.456672\pi\)
\(434\) −14.1627 −0.679832
\(435\) −52.8417 −2.53357
\(436\) −7.39654 −0.354230
\(437\) 32.6574 1.56222
\(438\) 22.8297 1.09085
\(439\) 12.9933 0.620136 0.310068 0.950714i \(-0.399648\pi\)
0.310068 + 0.950714i \(0.399648\pi\)
\(440\) −12.6850 −0.604735
\(441\) 3.61427 0.172108
\(442\) −12.3135 −0.585696
\(443\) 5.55165 0.263767 0.131883 0.991265i \(-0.457898\pi\)
0.131883 + 0.991265i \(0.457898\pi\)
\(444\) 8.59312 0.407812
\(445\) 5.18988 0.246024
\(446\) 18.5568 0.878692
\(447\) −3.32998 −0.157502
\(448\) −1.50310 −0.0710147
\(449\) −12.9881 −0.612945 −0.306472 0.951880i \(-0.599149\pi\)
−0.306472 + 0.951880i \(0.599149\pi\)
\(450\) −6.22171 −0.293294
\(451\) 41.8225 1.96934
\(452\) 19.3027 0.907921
\(453\) 21.1187 0.992241
\(454\) −3.04139 −0.142739
\(455\) −14.7361 −0.690837
\(456\) 7.07852 0.331482
\(457\) −2.34198 −0.109553 −0.0547765 0.998499i \(-0.517445\pi\)
−0.0547765 + 0.998499i \(0.517445\pi\)
\(458\) 6.50956 0.304172
\(459\) −25.6440 −1.19696
\(460\) −25.0364 −1.16733
\(461\) 28.0100 1.30456 0.652278 0.757979i \(-0.273813\pi\)
0.652278 + 0.757979i \(0.273813\pi\)
\(462\) −7.86194 −0.365770
\(463\) 24.0740 1.11881 0.559407 0.828893i \(-0.311029\pi\)
0.559407 + 0.828893i \(0.311029\pi\)
\(464\) 9.73743 0.452049
\(465\) −51.1319 −2.37118
\(466\) −3.98723 −0.184705
\(467\) 8.58512 0.397272 0.198636 0.980073i \(-0.436349\pi\)
0.198636 + 0.980073i \(0.436349\pi\)
\(468\) −2.06031 −0.0952379
\(469\) 18.2457 0.842508
\(470\) −19.6275 −0.905351
\(471\) −6.99090 −0.322124
\(472\) −5.55792 −0.255824
\(473\) 37.7597 1.73619
\(474\) −5.13732 −0.235965
\(475\) −38.6173 −1.77188
\(476\) 6.84883 0.313916
\(477\) 7.38872 0.338306
\(478\) 7.69185 0.351817
\(479\) 31.3404 1.43198 0.715991 0.698110i \(-0.245975\pi\)
0.715991 + 0.698110i \(0.245975\pi\)
\(480\) −5.42666 −0.247692
\(481\) −15.5243 −0.707849
\(482\) 26.7382 1.21789
\(483\) −15.5171 −0.706052
\(484\) 1.22647 0.0557487
\(485\) 40.9258 1.85834
\(486\) −7.71207 −0.349827
\(487\) −31.7625 −1.43929 −0.719647 0.694340i \(-0.755697\pi\)
−0.719647 + 0.694340i \(0.755697\pi\)
\(488\) 11.2266 0.508205
\(489\) 31.2794 1.41451
\(490\) −17.1982 −0.776935
\(491\) −11.7168 −0.528771 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(492\) 17.8917 0.806619
\(493\) −44.3684 −1.99825
\(494\) −12.7881 −0.575362
\(495\) 9.67097 0.434678
\(496\) 9.42235 0.423076
\(497\) −16.4107 −0.736122
\(498\) −9.38463 −0.420536
\(499\) 3.23553 0.144842 0.0724210 0.997374i \(-0.476927\pi\)
0.0724210 + 0.997374i \(0.476927\pi\)
\(500\) 11.4666 0.512800
\(501\) 13.3396 0.595968
\(502\) 6.31167 0.281704
\(503\) 34.5273 1.53949 0.769747 0.638349i \(-0.220382\pi\)
0.769747 + 0.638349i \(0.220382\pi\)
\(504\) 1.14595 0.0510447
\(505\) 37.2232 1.65641
\(506\) 24.1314 1.07277
\(507\) 8.52177 0.378465
\(508\) −7.60958 −0.337621
\(509\) 8.43618 0.373927 0.186964 0.982367i \(-0.440135\pi\)
0.186964 + 0.982367i \(0.440135\pi\)
\(510\) 24.7264 1.09491
\(511\) 22.9401 1.01481
\(512\) 1.00000 0.0441942
\(513\) −26.6322 −1.17584
\(514\) −11.7701 −0.519155
\(515\) −33.9781 −1.49725
\(516\) 16.1536 0.711124
\(517\) 18.9180 0.832014
\(518\) 8.63468 0.379386
\(519\) −18.9219 −0.830580
\(520\) 9.80380 0.429925
\(521\) 13.1006 0.573948 0.286974 0.957938i \(-0.407351\pi\)
0.286974 + 0.957938i \(0.407351\pi\)
\(522\) −7.42375 −0.324929
\(523\) −35.7014 −1.56112 −0.780558 0.625084i \(-0.785065\pi\)
−0.780558 + 0.625084i \(0.785065\pi\)
\(524\) −20.8788 −0.912097
\(525\) 18.3489 0.800811
\(526\) 14.7343 0.642446
\(527\) −42.9327 −1.87018
\(528\) 5.23049 0.227628
\(529\) 24.6280 1.07078
\(530\) −35.1586 −1.52719
\(531\) 4.23732 0.183884
\(532\) 7.11275 0.308377
\(533\) −32.3231 −1.40007
\(534\) −2.13998 −0.0926059
\(535\) 25.1331 1.08660
\(536\) −12.1387 −0.524313
\(537\) 24.8185 1.07100
\(538\) −17.4625 −0.752861
\(539\) 16.5765 0.714001
\(540\) 20.4172 0.878617
\(541\) −6.97435 −0.299851 −0.149925 0.988697i \(-0.547903\pi\)
−0.149925 + 0.988697i \(0.547903\pi\)
\(542\) −16.2335 −0.697290
\(543\) 3.46325 0.148622
\(544\) −4.55648 −0.195357
\(545\) −26.8330 −1.14940
\(546\) 6.07621 0.260038
\(547\) 2.06812 0.0884263 0.0442132 0.999022i \(-0.485922\pi\)
0.0442132 + 0.999022i \(0.485922\pi\)
\(548\) 4.99729 0.213474
\(549\) −8.55909 −0.365293
\(550\) −28.5352 −1.21675
\(551\) −46.0782 −1.96300
\(552\) 10.3234 0.439393
\(553\) −5.16217 −0.219518
\(554\) 17.6711 0.750771
\(555\) 31.1739 1.32326
\(556\) 18.0601 0.765917
\(557\) −17.7347 −0.751443 −0.375721 0.926733i \(-0.622605\pi\)
−0.375721 + 0.926733i \(0.622605\pi\)
\(558\) −7.18353 −0.304103
\(559\) −29.1831 −1.23431
\(560\) −5.45290 −0.230427
\(561\) −23.8326 −1.00621
\(562\) 11.5639 0.487794
\(563\) 30.3753 1.28017 0.640083 0.768306i \(-0.278900\pi\)
0.640083 + 0.768306i \(0.278900\pi\)
\(564\) 8.09314 0.340783
\(565\) 70.0257 2.94601
\(566\) −5.54353 −0.233012
\(567\) 9.21636 0.387051
\(568\) 10.9179 0.458107
\(569\) −20.3967 −0.855073 −0.427537 0.903998i \(-0.640619\pi\)
−0.427537 + 0.903998i \(0.640619\pi\)
\(570\) 25.6793 1.07559
\(571\) 18.2569 0.764027 0.382014 0.924157i \(-0.375231\pi\)
0.382014 + 0.924157i \(0.375231\pi\)
\(572\) −9.44940 −0.395099
\(573\) 17.5100 0.731492
\(574\) 17.9782 0.750395
\(575\) −56.3199 −2.34870
\(576\) −0.762393 −0.0317664
\(577\) 38.2990 1.59441 0.797203 0.603711i \(-0.206312\pi\)
0.797203 + 0.603711i \(0.206312\pi\)
\(578\) 3.76148 0.156457
\(579\) 18.0235 0.749032
\(580\) 35.3252 1.46680
\(581\) −9.43002 −0.391223
\(582\) −16.8752 −0.699498
\(583\) 33.8876 1.40348
\(584\) −15.2619 −0.631542
\(585\) −7.47434 −0.309026
\(586\) −26.7201 −1.10380
\(587\) −3.26606 −0.134805 −0.0674023 0.997726i \(-0.521471\pi\)
−0.0674023 + 0.997726i \(0.521471\pi\)
\(588\) 7.09144 0.292446
\(589\) −44.5872 −1.83718
\(590\) −20.1629 −0.830094
\(591\) 25.7961 1.06111
\(592\) −5.74459 −0.236101
\(593\) −28.4117 −1.16673 −0.583365 0.812210i \(-0.698264\pi\)
−0.583365 + 0.812210i \(0.698264\pi\)
\(594\) −19.6792 −0.807446
\(595\) 24.8460 1.01859
\(596\) 2.22612 0.0911855
\(597\) −6.12442 −0.250656
\(598\) −18.6503 −0.762666
\(599\) 37.2628 1.52252 0.761259 0.648448i \(-0.224581\pi\)
0.761259 + 0.648448i \(0.224581\pi\)
\(600\) −12.2074 −0.498365
\(601\) 20.3225 0.828970 0.414485 0.910056i \(-0.363962\pi\)
0.414485 + 0.910056i \(0.363962\pi\)
\(602\) 16.2317 0.661556
\(603\) 9.25448 0.376871
\(604\) −14.1180 −0.574455
\(605\) 4.44936 0.180892
\(606\) −15.3485 −0.623489
\(607\) −13.7892 −0.559685 −0.279842 0.960046i \(-0.590282\pi\)
−0.279842 + 0.960046i \(0.590282\pi\)
\(608\) −4.73207 −0.191911
\(609\) 21.8939 0.887186
\(610\) 40.7277 1.64901
\(611\) −14.6211 −0.591505
\(612\) 3.47382 0.140421
\(613\) −15.3938 −0.621751 −0.310876 0.950451i \(-0.600622\pi\)
−0.310876 + 0.950451i \(0.600622\pi\)
\(614\) 2.42093 0.0977006
\(615\) 64.9070 2.61730
\(616\) 5.25579 0.211762
\(617\) −39.0794 −1.57328 −0.786639 0.617414i \(-0.788181\pi\)
−0.786639 + 0.617414i \(0.788181\pi\)
\(618\) 14.0104 0.563581
\(619\) 35.4401 1.42446 0.712229 0.701947i \(-0.247686\pi\)
0.712229 + 0.701947i \(0.247686\pi\)
\(620\) 34.1822 1.37279
\(621\) −38.8407 −1.55862
\(622\) 26.0257 1.04354
\(623\) −2.15033 −0.0861510
\(624\) −4.04246 −0.161828
\(625\) 0.794261 0.0317704
\(626\) −2.18509 −0.0873339
\(627\) −24.7510 −0.988461
\(628\) 4.67349 0.186492
\(629\) 26.1751 1.04367
\(630\) 4.15725 0.165629
\(631\) −14.8991 −0.593125 −0.296563 0.955013i \(-0.595840\pi\)
−0.296563 + 0.955013i \(0.595840\pi\)
\(632\) 3.43435 0.136611
\(633\) 2.38033 0.0946098
\(634\) 28.6549 1.13803
\(635\) −27.6059 −1.09551
\(636\) 14.4971 0.574849
\(637\) −12.8114 −0.507606
\(638\) −34.0483 −1.34798
\(639\) −8.32376 −0.329283
\(640\) 3.62778 0.143400
\(641\) 36.3219 1.43463 0.717314 0.696750i \(-0.245371\pi\)
0.717314 + 0.696750i \(0.245371\pi\)
\(642\) −10.3633 −0.409006
\(643\) −20.6587 −0.814701 −0.407350 0.913272i \(-0.633547\pi\)
−0.407350 + 0.913272i \(0.633547\pi\)
\(644\) 10.3733 0.408766
\(645\) 58.6017 2.30744
\(646\) 21.5615 0.848328
\(647\) −35.1471 −1.38177 −0.690887 0.722963i \(-0.742780\pi\)
−0.690887 + 0.722963i \(0.742780\pi\)
\(648\) −6.13158 −0.240871
\(649\) 19.4340 0.762853
\(650\) 22.0539 0.865024
\(651\) 21.1855 0.830324
\(652\) −20.9106 −0.818923
\(653\) 26.2863 1.02866 0.514331 0.857592i \(-0.328040\pi\)
0.514331 + 0.857592i \(0.328040\pi\)
\(654\) 11.0642 0.432645
\(655\) −75.7438 −2.95955
\(656\) −11.9608 −0.466990
\(657\) 11.6356 0.453947
\(658\) 8.13228 0.317029
\(659\) 5.99504 0.233534 0.116767 0.993159i \(-0.462747\pi\)
0.116767 + 0.993159i \(0.462747\pi\)
\(660\) 18.9751 0.738603
\(661\) −36.9297 −1.43640 −0.718199 0.695838i \(-0.755033\pi\)
−0.718199 + 0.695838i \(0.755033\pi\)
\(662\) −19.2549 −0.748361
\(663\) 18.4194 0.715349
\(664\) 6.27372 0.243468
\(665\) 25.8035 1.00062
\(666\) 4.37963 0.169707
\(667\) −67.2010 −2.60203
\(668\) −8.91764 −0.345034
\(669\) −27.7585 −1.07321
\(670\) −44.0366 −1.70128
\(671\) −39.2554 −1.51544
\(672\) 2.24843 0.0867350
\(673\) 16.4358 0.633555 0.316778 0.948500i \(-0.397399\pi\)
0.316778 + 0.948500i \(0.397399\pi\)
\(674\) 13.2036 0.508584
\(675\) 45.9290 1.76781
\(676\) −5.69689 −0.219111
\(677\) −18.5996 −0.714841 −0.357421 0.933944i \(-0.616344\pi\)
−0.357421 + 0.933944i \(0.616344\pi\)
\(678\) −28.8741 −1.10890
\(679\) −16.9568 −0.650741
\(680\) −16.5299 −0.633892
\(681\) 4.54950 0.174337
\(682\) −32.9465 −1.26159
\(683\) −15.8606 −0.606889 −0.303444 0.952849i \(-0.598137\pi\)
−0.303444 + 0.952849i \(0.598137\pi\)
\(684\) 3.60769 0.137944
\(685\) 18.1291 0.692676
\(686\) 17.6474 0.673781
\(687\) −9.73741 −0.371505
\(688\) −10.7989 −0.411703
\(689\) −26.1905 −0.997780
\(690\) 37.4510 1.42574
\(691\) 21.3595 0.812552 0.406276 0.913750i \(-0.366827\pi\)
0.406276 + 0.913750i \(0.366827\pi\)
\(692\) 12.6495 0.480862
\(693\) −4.00697 −0.152212
\(694\) −11.0019 −0.417626
\(695\) 65.5179 2.48524
\(696\) −14.5659 −0.552118
\(697\) 54.4990 2.06430
\(698\) −12.1730 −0.460756
\(699\) 5.96436 0.225593
\(700\) −12.2664 −0.463627
\(701\) 7.81487 0.295163 0.147582 0.989050i \(-0.452851\pi\)
0.147582 + 0.989050i \(0.452851\pi\)
\(702\) 15.2093 0.574039
\(703\) 27.1838 1.02526
\(704\) −3.49664 −0.131784
\(705\) 29.3601 1.10577
\(706\) 22.3049 0.839458
\(707\) −15.4227 −0.580030
\(708\) 8.31390 0.312455
\(709\) 18.6148 0.699092 0.349546 0.936919i \(-0.386336\pi\)
0.349546 + 0.936919i \(0.386336\pi\)
\(710\) 39.6078 1.48646
\(711\) −2.61833 −0.0981949
\(712\) 1.43060 0.0536139
\(713\) −65.0265 −2.43526
\(714\) −10.2449 −0.383406
\(715\) −34.2803 −1.28201
\(716\) −16.5914 −0.620050
\(717\) −11.5060 −0.429698
\(718\) 9.94994 0.371328
\(719\) −28.8184 −1.07474 −0.537372 0.843345i \(-0.680583\pi\)
−0.537372 + 0.843345i \(0.680583\pi\)
\(720\) −2.76579 −0.103075
\(721\) 14.0782 0.524298
\(722\) 3.39244 0.126254
\(723\) −39.9968 −1.48750
\(724\) −2.31522 −0.0860444
\(725\) 79.4649 2.95125
\(726\) −1.83463 −0.0680896
\(727\) 42.9152 1.59164 0.795818 0.605536i \(-0.207041\pi\)
0.795818 + 0.605536i \(0.207041\pi\)
\(728\) −4.06201 −0.150548
\(729\) 29.9309 1.10855
\(730\) −55.3668 −2.04922
\(731\) 49.2048 1.81990
\(732\) −16.7935 −0.620705
\(733\) −1.99756 −0.0737816 −0.0368908 0.999319i \(-0.511745\pi\)
−0.0368908 + 0.999319i \(0.511745\pi\)
\(734\) 21.2306 0.783637
\(735\) 25.7262 0.948924
\(736\) −6.90130 −0.254385
\(737\) 42.4447 1.56347
\(738\) 9.11880 0.335668
\(739\) −11.4165 −0.419964 −0.209982 0.977705i \(-0.567341\pi\)
−0.209982 + 0.977705i \(0.567341\pi\)
\(740\) −20.8401 −0.766097
\(741\) 19.1292 0.702728
\(742\) 14.5673 0.534781
\(743\) 5.85467 0.214787 0.107393 0.994217i \(-0.465750\pi\)
0.107393 + 0.994217i \(0.465750\pi\)
\(744\) −14.0946 −0.516731
\(745\) 8.07588 0.295877
\(746\) 17.8721 0.654342
\(747\) −4.78304 −0.175002
\(748\) 15.9323 0.582544
\(749\) −10.4134 −0.380497
\(750\) −17.1524 −0.626317
\(751\) −8.16013 −0.297767 −0.148884 0.988855i \(-0.547568\pi\)
−0.148884 + 0.988855i \(0.547568\pi\)
\(752\) −5.41035 −0.197295
\(753\) −9.44140 −0.344064
\(754\) 26.3147 0.958324
\(755\) −51.2171 −1.86398
\(756\) −8.45947 −0.307668
\(757\) 9.09891 0.330706 0.165353 0.986234i \(-0.447124\pi\)
0.165353 + 0.986234i \(0.447124\pi\)
\(758\) 20.5879 0.747785
\(759\) −36.0972 −1.31025
\(760\) −17.1669 −0.622708
\(761\) 30.2631 1.09703 0.548517 0.836139i \(-0.315192\pi\)
0.548517 + 0.836139i \(0.315192\pi\)
\(762\) 11.3829 0.412359
\(763\) 11.1177 0.402488
\(764\) −11.7056 −0.423495
\(765\) 12.6023 0.455636
\(766\) −28.5254 −1.03066
\(767\) −15.0199 −0.542336
\(768\) −1.49586 −0.0539773
\(769\) −24.3360 −0.877579 −0.438790 0.898590i \(-0.644593\pi\)
−0.438790 + 0.898590i \(0.644593\pi\)
\(770\) 19.0668 0.687120
\(771\) 17.6064 0.634079
\(772\) −12.0489 −0.433650
\(773\) 27.4969 0.988994 0.494497 0.869179i \(-0.335352\pi\)
0.494497 + 0.869179i \(0.335352\pi\)
\(774\) 8.23297 0.295928
\(775\) 76.8936 2.76210
\(776\) 11.2812 0.404972
\(777\) −12.9163 −0.463370
\(778\) −17.2629 −0.618903
\(779\) 56.5991 2.02787
\(780\) −14.6651 −0.525096
\(781\) −38.1761 −1.36605
\(782\) 31.4456 1.12449
\(783\) 54.8025 1.95848
\(784\) −4.74070 −0.169311
\(785\) 16.9544 0.605127
\(786\) 31.2319 1.11400
\(787\) 29.9692 1.06829 0.534143 0.845394i \(-0.320634\pi\)
0.534143 + 0.845394i \(0.320634\pi\)
\(788\) −17.2450 −0.614327
\(789\) −22.0405 −0.784662
\(790\) 12.4591 0.443274
\(791\) −29.0138 −1.03161
\(792\) 2.66581 0.0947254
\(793\) 30.3391 1.07737
\(794\) −33.6567 −1.19443
\(795\) 52.5924 1.86526
\(796\) 4.09423 0.145116
\(797\) 31.7567 1.12488 0.562440 0.826838i \(-0.309863\pi\)
0.562440 + 0.826838i \(0.309863\pi\)
\(798\) −10.6397 −0.376642
\(799\) 24.6521 0.872130
\(800\) 8.16077 0.288527
\(801\) −1.09068 −0.0385372
\(802\) −21.0584 −0.743597
\(803\) 53.3653 1.88322
\(804\) 18.1579 0.640379
\(805\) 37.6321 1.32636
\(806\) 25.4632 0.896903
\(807\) 26.1215 0.919520
\(808\) 10.2606 0.360967
\(809\) −2.23254 −0.0784921 −0.0392460 0.999230i \(-0.512496\pi\)
−0.0392460 + 0.999230i \(0.512496\pi\)
\(810\) −22.2440 −0.781575
\(811\) 7.67771 0.269601 0.134800 0.990873i \(-0.456961\pi\)
0.134800 + 0.990873i \(0.456961\pi\)
\(812\) −14.6363 −0.513634
\(813\) 24.2832 0.851647
\(814\) 20.0868 0.704040
\(815\) −75.8591 −2.65723
\(816\) 6.81587 0.238603
\(817\) 51.1009 1.78779
\(818\) −14.8304 −0.518534
\(819\) 3.09685 0.108213
\(820\) −43.3910 −1.51528
\(821\) −7.52119 −0.262491 −0.131246 0.991350i \(-0.541898\pi\)
−0.131246 + 0.991350i \(0.541898\pi\)
\(822\) −7.47526 −0.260730
\(823\) −38.2704 −1.33402 −0.667011 0.745048i \(-0.732427\pi\)
−0.667011 + 0.745048i \(0.732427\pi\)
\(824\) −9.36610 −0.326283
\(825\) 42.6848 1.48609
\(826\) 8.35410 0.290676
\(827\) −2.52723 −0.0878804 −0.0439402 0.999034i \(-0.513991\pi\)
−0.0439402 + 0.999034i \(0.513991\pi\)
\(828\) 5.26150 0.182850
\(829\) −5.76684 −0.200291 −0.100145 0.994973i \(-0.531931\pi\)
−0.100145 + 0.994973i \(0.531931\pi\)
\(830\) 22.7597 0.790000
\(831\) −26.4335 −0.916967
\(832\) 2.70243 0.0936898
\(833\) 21.6009 0.748426
\(834\) −27.0154 −0.935467
\(835\) −32.3512 −1.11956
\(836\) 16.5463 0.572266
\(837\) 53.0292 1.83296
\(838\) 11.9369 0.412355
\(839\) −9.20909 −0.317933 −0.158967 0.987284i \(-0.550816\pi\)
−0.158967 + 0.987284i \(0.550816\pi\)
\(840\) 8.15680 0.281436
\(841\) 65.8175 2.26957
\(842\) −26.9131 −0.927488
\(843\) −17.2980 −0.595775
\(844\) −1.59128 −0.0547740
\(845\) −20.6671 −0.710968
\(846\) 4.12481 0.141814
\(847\) −1.84350 −0.0633436
\(848\) −9.69149 −0.332807
\(849\) 8.29237 0.284593
\(850\) −37.1843 −1.27541
\(851\) 39.6452 1.35902
\(852\) −16.3317 −0.559516
\(853\) −25.1629 −0.861562 −0.430781 0.902457i \(-0.641762\pi\)
−0.430781 + 0.902457i \(0.641762\pi\)
\(854\) −16.8747 −0.577440
\(855\) 13.0879 0.447597
\(856\) 6.92796 0.236793
\(857\) 40.8657 1.39594 0.697972 0.716125i \(-0.254086\pi\)
0.697972 + 0.716125i \(0.254086\pi\)
\(858\) 14.1350 0.482562
\(859\) 13.3540 0.455632 0.227816 0.973704i \(-0.426841\pi\)
0.227816 + 0.973704i \(0.426841\pi\)
\(860\) −39.1759 −1.33589
\(861\) −26.8929 −0.916509
\(862\) 8.13486 0.277074
\(863\) −34.3408 −1.16898 −0.584488 0.811403i \(-0.698704\pi\)
−0.584488 + 0.811403i \(0.698704\pi\)
\(864\) 5.62803 0.191469
\(865\) 45.8895 1.56029
\(866\) 5.64742 0.191907
\(867\) −5.62666 −0.191092
\(868\) −14.1627 −0.480714
\(869\) −12.0087 −0.407367
\(870\) −52.8417 −1.79150
\(871\) −32.8040 −1.11152
\(872\) −7.39654 −0.250478
\(873\) −8.60072 −0.291090
\(874\) 32.6574 1.10465
\(875\) −17.2353 −0.582661
\(876\) 22.8297 0.771345
\(877\) −44.9398 −1.51751 −0.758755 0.651376i \(-0.774192\pi\)
−0.758755 + 0.651376i \(0.774192\pi\)
\(878\) 12.9933 0.438503
\(879\) 39.9696 1.34814
\(880\) −12.6850 −0.427612
\(881\) −35.1303 −1.18357 −0.591785 0.806096i \(-0.701577\pi\)
−0.591785 + 0.806096i \(0.701577\pi\)
\(882\) 3.61427 0.121699
\(883\) −13.8957 −0.467628 −0.233814 0.972281i \(-0.575121\pi\)
−0.233814 + 0.972281i \(0.575121\pi\)
\(884\) −12.3135 −0.414149
\(885\) 30.1610 1.01385
\(886\) 5.55165 0.186511
\(887\) −29.6682 −0.996162 −0.498081 0.867130i \(-0.665962\pi\)
−0.498081 + 0.867130i \(0.665962\pi\)
\(888\) 8.59312 0.288366
\(889\) 11.4379 0.383616
\(890\) 5.18988 0.173965
\(891\) 21.4399 0.718264
\(892\) 18.5568 0.621329
\(893\) 25.6021 0.856742
\(894\) −3.32998 −0.111371
\(895\) −60.1899 −2.01193
\(896\) −1.50310 −0.0502150
\(897\) 27.8982 0.931495
\(898\) −12.9881 −0.433417
\(899\) 91.7495 3.06002
\(900\) −6.22171 −0.207390
\(901\) 44.1591 1.47115
\(902\) 41.8225 1.39254
\(903\) −24.2805 −0.808003
\(904\) 19.3027 0.641997
\(905\) −8.39909 −0.279195
\(906\) 21.1187 0.701620
\(907\) 28.1012 0.933086 0.466543 0.884499i \(-0.345499\pi\)
0.466543 + 0.884499i \(0.345499\pi\)
\(908\) −3.04139 −0.100932
\(909\) −7.82262 −0.259460
\(910\) −14.7361 −0.488496
\(911\) 0.978883 0.0324318 0.0162159 0.999869i \(-0.494838\pi\)
0.0162159 + 0.999869i \(0.494838\pi\)
\(912\) 7.07852 0.234393
\(913\) −21.9369 −0.726007
\(914\) −2.34198 −0.0774657
\(915\) −60.9230 −2.01405
\(916\) 6.50956 0.215082
\(917\) 31.3829 1.03636
\(918\) −25.6440 −0.846377
\(919\) 11.0676 0.365088 0.182544 0.983198i \(-0.441567\pi\)
0.182544 + 0.983198i \(0.441567\pi\)
\(920\) −25.0364 −0.825425
\(921\) −3.62137 −0.119328
\(922\) 28.0100 0.922461
\(923\) 29.5049 0.971166
\(924\) −7.86194 −0.258639
\(925\) −46.8803 −1.54141
\(926\) 24.0740 0.791121
\(927\) 7.14064 0.234530
\(928\) 9.73743 0.319647
\(929\) −31.6228 −1.03751 −0.518755 0.854923i \(-0.673604\pi\)
−0.518755 + 0.854923i \(0.673604\pi\)
\(930\) −51.1319 −1.67668
\(931\) 22.4333 0.735222
\(932\) −3.98723 −0.130606
\(933\) −38.9309 −1.27454
\(934\) 8.58512 0.280914
\(935\) 57.7990 1.89023
\(936\) −2.06031 −0.0673433
\(937\) 32.1481 1.05023 0.525116 0.851031i \(-0.324022\pi\)
0.525116 + 0.851031i \(0.324022\pi\)
\(938\) 18.2457 0.595743
\(939\) 3.26860 0.106667
\(940\) −19.6275 −0.640180
\(941\) 15.9760 0.520803 0.260402 0.965500i \(-0.416145\pi\)
0.260402 + 0.965500i \(0.416145\pi\)
\(942\) −6.99090 −0.227776
\(943\) 82.5449 2.68803
\(944\) −5.55792 −0.180895
\(945\) −30.6891 −0.998315
\(946\) 37.7597 1.22767
\(947\) 15.9632 0.518734 0.259367 0.965779i \(-0.416486\pi\)
0.259367 + 0.965779i \(0.416486\pi\)
\(948\) −5.13732 −0.166852
\(949\) −41.2442 −1.33884
\(950\) −38.6173 −1.25291
\(951\) −42.8639 −1.38996
\(952\) 6.84883 0.221972
\(953\) −33.4673 −1.08411 −0.542056 0.840342i \(-0.682354\pi\)
−0.542056 + 0.840342i \(0.682354\pi\)
\(954\) 7.38872 0.239219
\(955\) −42.4654 −1.37415
\(956\) 7.69185 0.248772
\(957\) 50.9315 1.64638
\(958\) 31.3404 1.01256
\(959\) −7.51141 −0.242556
\(960\) −5.42666 −0.175145
\(961\) 57.7807 1.86389
\(962\) −15.5243 −0.500525
\(963\) −5.28182 −0.170204
\(964\) 26.7382 0.861181
\(965\) −43.7108 −1.40710
\(966\) −15.5171 −0.499254
\(967\) 33.4059 1.07426 0.537130 0.843499i \(-0.319508\pi\)
0.537130 + 0.843499i \(0.319508\pi\)
\(968\) 1.22647 0.0394203
\(969\) −32.2531 −1.03612
\(970\) 40.9258 1.31405
\(971\) −39.6081 −1.27109 −0.635543 0.772066i \(-0.719224\pi\)
−0.635543 + 0.772066i \(0.719224\pi\)
\(972\) −7.71207 −0.247365
\(973\) −27.1460 −0.870262
\(974\) −31.7625 −1.01774
\(975\) −32.9896 −1.05651
\(976\) 11.2266 0.359355
\(977\) 58.5224 1.87230 0.936148 0.351605i \(-0.114364\pi\)
0.936148 + 0.351605i \(0.114364\pi\)
\(978\) 31.2794 1.00021
\(979\) −5.00228 −0.159873
\(980\) −17.1982 −0.549376
\(981\) 5.63907 0.180042
\(982\) −11.7168 −0.373898
\(983\) 4.85826 0.154955 0.0774773 0.996994i \(-0.475313\pi\)
0.0774773 + 0.996994i \(0.475313\pi\)
\(984\) 17.8917 0.570366
\(985\) −62.5609 −1.99336
\(986\) −44.3684 −1.41298
\(987\) −12.1648 −0.387209
\(988\) −12.7881 −0.406842
\(989\) 74.5262 2.36980
\(990\) 9.67097 0.307363
\(991\) 10.2855 0.326729 0.163365 0.986566i \(-0.447765\pi\)
0.163365 + 0.986566i \(0.447765\pi\)
\(992\) 9.42235 0.299160
\(993\) 28.8026 0.914024
\(994\) −16.4107 −0.520517
\(995\) 14.8530 0.470871
\(996\) −9.38463 −0.297364
\(997\) 21.4121 0.678130 0.339065 0.940763i \(-0.389889\pi\)
0.339065 + 0.940763i \(0.389889\pi\)
\(998\) 3.23553 0.102419
\(999\) −32.3307 −1.02290
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 4022.2.a.c.1.12 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4022.2.a.c.1.12 31 1.1 even 1 trivial