Properties

Label 8020.2.a.e.1.15
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $35$
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] = [8020,2,Mod(1,8020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, 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("8020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8020.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: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 8020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.713769 q^{3} -1.00000 q^{5} -1.26919 q^{7} -2.49053 q^{9} +O(q^{10})\) \(q-0.713769 q^{3} -1.00000 q^{5} -1.26919 q^{7} -2.49053 q^{9} +1.29319 q^{11} -3.04543 q^{13} +0.713769 q^{15} -2.63853 q^{17} -4.73419 q^{19} +0.905907 q^{21} -6.57988 q^{23} +1.00000 q^{25} +3.91897 q^{27} -5.77789 q^{29} -6.97788 q^{31} -0.923041 q^{33} +1.26919 q^{35} +3.93757 q^{37} +2.17374 q^{39} -5.22630 q^{41} +2.20178 q^{43} +2.49053 q^{45} -11.7358 q^{47} -5.38916 q^{49} +1.88330 q^{51} +8.06456 q^{53} -1.29319 q^{55} +3.37912 q^{57} +11.5685 q^{59} -7.63094 q^{61} +3.16095 q^{63} +3.04543 q^{65} +10.0096 q^{67} +4.69651 q^{69} +16.0339 q^{71} +12.0878 q^{73} -0.713769 q^{75} -1.64130 q^{77} +1.86668 q^{79} +4.67436 q^{81} -16.9310 q^{83} +2.63853 q^{85} +4.12408 q^{87} -12.3893 q^{89} +3.86523 q^{91} +4.98060 q^{93} +4.73419 q^{95} -13.1561 q^{97} -3.22074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.713769 −0.412095 −0.206047 0.978542i \(-0.566060\pi\)
−0.206047 + 0.978542i \(0.566060\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.26919 −0.479708 −0.239854 0.970809i \(-0.577100\pi\)
−0.239854 + 0.970809i \(0.577100\pi\)
\(8\) 0 0
\(9\) −2.49053 −0.830178
\(10\) 0 0
\(11\) 1.29319 0.389912 0.194956 0.980812i \(-0.437544\pi\)
0.194956 + 0.980812i \(0.437544\pi\)
\(12\) 0 0
\(13\) −3.04543 −0.844651 −0.422326 0.906444i \(-0.638786\pi\)
−0.422326 + 0.906444i \(0.638786\pi\)
\(14\) 0 0
\(15\) 0.713769 0.184294
\(16\) 0 0
\(17\) −2.63853 −0.639937 −0.319969 0.947428i \(-0.603672\pi\)
−0.319969 + 0.947428i \(0.603672\pi\)
\(18\) 0 0
\(19\) −4.73419 −1.08610 −0.543049 0.839701i \(-0.682730\pi\)
−0.543049 + 0.839701i \(0.682730\pi\)
\(20\) 0 0
\(21\) 0.905907 0.197685
\(22\) 0 0
\(23\) −6.57988 −1.37200 −0.685999 0.727602i \(-0.740635\pi\)
−0.685999 + 0.727602i \(0.740635\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.91897 0.754207
\(28\) 0 0
\(29\) −5.77789 −1.07293 −0.536464 0.843923i \(-0.680240\pi\)
−0.536464 + 0.843923i \(0.680240\pi\)
\(30\) 0 0
\(31\) −6.97788 −1.25326 −0.626632 0.779315i \(-0.715567\pi\)
−0.626632 + 0.779315i \(0.715567\pi\)
\(32\) 0 0
\(33\) −0.923041 −0.160681
\(34\) 0 0
\(35\) 1.26919 0.214532
\(36\) 0 0
\(37\) 3.93757 0.647332 0.323666 0.946171i \(-0.395085\pi\)
0.323666 + 0.946171i \(0.395085\pi\)
\(38\) 0 0
\(39\) 2.17374 0.348076
\(40\) 0 0
\(41\) −5.22630 −0.816210 −0.408105 0.912935i \(-0.633810\pi\)
−0.408105 + 0.912935i \(0.633810\pi\)
\(42\) 0 0
\(43\) 2.20178 0.335769 0.167884 0.985807i \(-0.446306\pi\)
0.167884 + 0.985807i \(0.446306\pi\)
\(44\) 0 0
\(45\) 2.49053 0.371267
\(46\) 0 0
\(47\) −11.7358 −1.71184 −0.855918 0.517111i \(-0.827008\pi\)
−0.855918 + 0.517111i \(0.827008\pi\)
\(48\) 0 0
\(49\) −5.38916 −0.769880
\(50\) 0 0
\(51\) 1.88330 0.263715
\(52\) 0 0
\(53\) 8.06456 1.10775 0.553876 0.832599i \(-0.313148\pi\)
0.553876 + 0.832599i \(0.313148\pi\)
\(54\) 0 0
\(55\) −1.29319 −0.174374
\(56\) 0 0
\(57\) 3.37912 0.447576
\(58\) 0 0
\(59\) 11.5685 1.50609 0.753043 0.657971i \(-0.228585\pi\)
0.753043 + 0.657971i \(0.228585\pi\)
\(60\) 0 0
\(61\) −7.63094 −0.977042 −0.488521 0.872552i \(-0.662463\pi\)
−0.488521 + 0.872552i \(0.662463\pi\)
\(62\) 0 0
\(63\) 3.16095 0.398243
\(64\) 0 0
\(65\) 3.04543 0.377739
\(66\) 0 0
\(67\) 10.0096 1.22287 0.611436 0.791294i \(-0.290592\pi\)
0.611436 + 0.791294i \(0.290592\pi\)
\(68\) 0 0
\(69\) 4.69651 0.565394
\(70\) 0 0
\(71\) 16.0339 1.90287 0.951435 0.307851i \(-0.0996099\pi\)
0.951435 + 0.307851i \(0.0996099\pi\)
\(72\) 0 0
\(73\) 12.0878 1.41477 0.707387 0.706826i \(-0.249874\pi\)
0.707387 + 0.706826i \(0.249874\pi\)
\(74\) 0 0
\(75\) −0.713769 −0.0824190
\(76\) 0 0
\(77\) −1.64130 −0.187044
\(78\) 0 0
\(79\) 1.86668 0.210018 0.105009 0.994471i \(-0.466513\pi\)
0.105009 + 0.994471i \(0.466513\pi\)
\(80\) 0 0
\(81\) 4.67436 0.519373
\(82\) 0 0
\(83\) −16.9310 −1.85842 −0.929209 0.369556i \(-0.879510\pi\)
−0.929209 + 0.369556i \(0.879510\pi\)
\(84\) 0 0
\(85\) 2.63853 0.286189
\(86\) 0 0
\(87\) 4.12408 0.442148
\(88\) 0 0
\(89\) −12.3893 −1.31326 −0.656630 0.754212i \(-0.728019\pi\)
−0.656630 + 0.754212i \(0.728019\pi\)
\(90\) 0 0
\(91\) 3.86523 0.405186
\(92\) 0 0
\(93\) 4.98060 0.516464
\(94\) 0 0
\(95\) 4.73419 0.485718
\(96\) 0 0
\(97\) −13.1561 −1.33580 −0.667898 0.744253i \(-0.732806\pi\)
−0.667898 + 0.744253i \(0.732806\pi\)
\(98\) 0 0
\(99\) −3.22074 −0.323697
\(100\) 0 0
\(101\) 14.6448 1.45721 0.728605 0.684934i \(-0.240169\pi\)
0.728605 + 0.684934i \(0.240169\pi\)
\(102\) 0 0
\(103\) −19.0490 −1.87696 −0.938478 0.345340i \(-0.887763\pi\)
−0.938478 + 0.345340i \(0.887763\pi\)
\(104\) 0 0
\(105\) −0.905907 −0.0884075
\(106\) 0 0
\(107\) 0.104128 0.0100664 0.00503320 0.999987i \(-0.498398\pi\)
0.00503320 + 0.999987i \(0.498398\pi\)
\(108\) 0 0
\(109\) −9.07641 −0.869362 −0.434681 0.900584i \(-0.643139\pi\)
−0.434681 + 0.900584i \(0.643139\pi\)
\(110\) 0 0
\(111\) −2.81052 −0.266762
\(112\) 0 0
\(113\) −16.2232 −1.52615 −0.763073 0.646312i \(-0.776310\pi\)
−0.763073 + 0.646312i \(0.776310\pi\)
\(114\) 0 0
\(115\) 6.57988 0.613577
\(116\) 0 0
\(117\) 7.58475 0.701210
\(118\) 0 0
\(119\) 3.34879 0.306983
\(120\) 0 0
\(121\) −9.32765 −0.847968
\(122\) 0 0
\(123\) 3.73037 0.336356
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.425408 0.0377489 0.0188744 0.999822i \(-0.493992\pi\)
0.0188744 + 0.999822i \(0.493992\pi\)
\(128\) 0 0
\(129\) −1.57157 −0.138369
\(130\) 0 0
\(131\) −15.5119 −1.35528 −0.677639 0.735395i \(-0.736997\pi\)
−0.677639 + 0.735395i \(0.736997\pi\)
\(132\) 0 0
\(133\) 6.00858 0.521010
\(134\) 0 0
\(135\) −3.91897 −0.337292
\(136\) 0 0
\(137\) 21.6285 1.84784 0.923922 0.382581i \(-0.124965\pi\)
0.923922 + 0.382581i \(0.124965\pi\)
\(138\) 0 0
\(139\) −6.58707 −0.558708 −0.279354 0.960188i \(-0.590120\pi\)
−0.279354 + 0.960188i \(0.590120\pi\)
\(140\) 0 0
\(141\) 8.37663 0.705439
\(142\) 0 0
\(143\) −3.93833 −0.329340
\(144\) 0 0
\(145\) 5.77789 0.479828
\(146\) 0 0
\(147\) 3.84662 0.317264
\(148\) 0 0
\(149\) 11.5561 0.946712 0.473356 0.880871i \(-0.343042\pi\)
0.473356 + 0.880871i \(0.343042\pi\)
\(150\) 0 0
\(151\) 18.1816 1.47960 0.739800 0.672827i \(-0.234920\pi\)
0.739800 + 0.672827i \(0.234920\pi\)
\(152\) 0 0
\(153\) 6.57134 0.531262
\(154\) 0 0
\(155\) 6.97788 0.560477
\(156\) 0 0
\(157\) −4.37146 −0.348880 −0.174440 0.984668i \(-0.555812\pi\)
−0.174440 + 0.984668i \(0.555812\pi\)
\(158\) 0 0
\(159\) −5.75623 −0.456499
\(160\) 0 0
\(161\) 8.35110 0.658159
\(162\) 0 0
\(163\) −2.46160 −0.192807 −0.0964035 0.995342i \(-0.530734\pi\)
−0.0964035 + 0.995342i \(0.530734\pi\)
\(164\) 0 0
\(165\) 0.923041 0.0718587
\(166\) 0 0
\(167\) 0.932627 0.0721688 0.0360844 0.999349i \(-0.488511\pi\)
0.0360844 + 0.999349i \(0.488511\pi\)
\(168\) 0 0
\(169\) −3.72534 −0.286565
\(170\) 0 0
\(171\) 11.7907 0.901654
\(172\) 0 0
\(173\) −1.70436 −0.129580 −0.0647899 0.997899i \(-0.520638\pi\)
−0.0647899 + 0.997899i \(0.520638\pi\)
\(174\) 0 0
\(175\) −1.26919 −0.0959416
\(176\) 0 0
\(177\) −8.25722 −0.620651
\(178\) 0 0
\(179\) 7.97102 0.595782 0.297891 0.954600i \(-0.403717\pi\)
0.297891 + 0.954600i \(0.403717\pi\)
\(180\) 0 0
\(181\) 14.0841 1.04686 0.523432 0.852067i \(-0.324651\pi\)
0.523432 + 0.852067i \(0.324651\pi\)
\(182\) 0 0
\(183\) 5.44673 0.402634
\(184\) 0 0
\(185\) −3.93757 −0.289496
\(186\) 0 0
\(187\) −3.41213 −0.249519
\(188\) 0 0
\(189\) −4.97391 −0.361799
\(190\) 0 0
\(191\) 19.9750 1.44534 0.722669 0.691195i \(-0.242915\pi\)
0.722669 + 0.691195i \(0.242915\pi\)
\(192\) 0 0
\(193\) −0.00907017 −0.000652885 0 −0.000326443 1.00000i \(-0.500104\pi\)
−0.000326443 1.00000i \(0.500104\pi\)
\(194\) 0 0
\(195\) −2.17374 −0.155665
\(196\) 0 0
\(197\) 0.828902 0.0590568 0.0295284 0.999564i \(-0.490599\pi\)
0.0295284 + 0.999564i \(0.490599\pi\)
\(198\) 0 0
\(199\) 5.85467 0.415027 0.207513 0.978232i \(-0.433463\pi\)
0.207513 + 0.978232i \(0.433463\pi\)
\(200\) 0 0
\(201\) −7.14458 −0.503940
\(202\) 0 0
\(203\) 7.33323 0.514692
\(204\) 0 0
\(205\) 5.22630 0.365020
\(206\) 0 0
\(207\) 16.3874 1.13900
\(208\) 0 0
\(209\) −6.12222 −0.423483
\(210\) 0 0
\(211\) 9.55210 0.657594 0.328797 0.944401i \(-0.393357\pi\)
0.328797 + 0.944401i \(0.393357\pi\)
\(212\) 0 0
\(213\) −11.4445 −0.784163
\(214\) 0 0
\(215\) −2.20178 −0.150160
\(216\) 0 0
\(217\) 8.85624 0.601201
\(218\) 0 0
\(219\) −8.62793 −0.583021
\(220\) 0 0
\(221\) 8.03546 0.540524
\(222\) 0 0
\(223\) −3.31204 −0.221791 −0.110895 0.993832i \(-0.535372\pi\)
−0.110895 + 0.993832i \(0.535372\pi\)
\(224\) 0 0
\(225\) −2.49053 −0.166036
\(226\) 0 0
\(227\) −28.0737 −1.86332 −0.931659 0.363334i \(-0.881638\pi\)
−0.931659 + 0.363334i \(0.881638\pi\)
\(228\) 0 0
\(229\) 2.37671 0.157058 0.0785288 0.996912i \(-0.474978\pi\)
0.0785288 + 0.996912i \(0.474978\pi\)
\(230\) 0 0
\(231\) 1.17151 0.0770799
\(232\) 0 0
\(233\) −23.3565 −1.53014 −0.765068 0.643949i \(-0.777295\pi\)
−0.765068 + 0.643949i \(0.777295\pi\)
\(234\) 0 0
\(235\) 11.7358 0.765557
\(236\) 0 0
\(237\) −1.33238 −0.0865474
\(238\) 0 0
\(239\) 9.80252 0.634072 0.317036 0.948413i \(-0.397312\pi\)
0.317036 + 0.948413i \(0.397312\pi\)
\(240\) 0 0
\(241\) −9.60175 −0.618503 −0.309251 0.950980i \(-0.600078\pi\)
−0.309251 + 0.950980i \(0.600078\pi\)
\(242\) 0 0
\(243\) −15.0933 −0.968238
\(244\) 0 0
\(245\) 5.38916 0.344301
\(246\) 0 0
\(247\) 14.4177 0.917374
\(248\) 0 0
\(249\) 12.0848 0.765844
\(250\) 0 0
\(251\) 9.87191 0.623109 0.311555 0.950228i \(-0.399150\pi\)
0.311555 + 0.950228i \(0.399150\pi\)
\(252\) 0 0
\(253\) −8.50905 −0.534959
\(254\) 0 0
\(255\) −1.88330 −0.117937
\(256\) 0 0
\(257\) −21.1640 −1.32017 −0.660086 0.751190i \(-0.729480\pi\)
−0.660086 + 0.751190i \(0.729480\pi\)
\(258\) 0 0
\(259\) −4.99751 −0.310530
\(260\) 0 0
\(261\) 14.3900 0.890721
\(262\) 0 0
\(263\) −6.10690 −0.376567 −0.188284 0.982115i \(-0.560292\pi\)
−0.188284 + 0.982115i \(0.560292\pi\)
\(264\) 0 0
\(265\) −8.06456 −0.495402
\(266\) 0 0
\(267\) 8.84309 0.541188
\(268\) 0 0
\(269\) 8.13634 0.496081 0.248041 0.968750i \(-0.420213\pi\)
0.248041 + 0.968750i \(0.420213\pi\)
\(270\) 0 0
\(271\) 1.98229 0.120415 0.0602077 0.998186i \(-0.480824\pi\)
0.0602077 + 0.998186i \(0.480824\pi\)
\(272\) 0 0
\(273\) −2.75888 −0.166975
\(274\) 0 0
\(275\) 1.29319 0.0779825
\(276\) 0 0
\(277\) 19.8515 1.19276 0.596379 0.802703i \(-0.296606\pi\)
0.596379 + 0.802703i \(0.296606\pi\)
\(278\) 0 0
\(279\) 17.3786 1.04043
\(280\) 0 0
\(281\) 17.2799 1.03083 0.515417 0.856939i \(-0.327637\pi\)
0.515417 + 0.856939i \(0.327637\pi\)
\(282\) 0 0
\(283\) −23.7464 −1.41158 −0.705789 0.708422i \(-0.749407\pi\)
−0.705789 + 0.708422i \(0.749407\pi\)
\(284\) 0 0
\(285\) −3.37912 −0.200162
\(286\) 0 0
\(287\) 6.63315 0.391543
\(288\) 0 0
\(289\) −10.0382 −0.590481
\(290\) 0 0
\(291\) 9.39039 0.550474
\(292\) 0 0
\(293\) 17.0950 0.998698 0.499349 0.866401i \(-0.333572\pi\)
0.499349 + 0.866401i \(0.333572\pi\)
\(294\) 0 0
\(295\) −11.5685 −0.673542
\(296\) 0 0
\(297\) 5.06799 0.294075
\(298\) 0 0
\(299\) 20.0386 1.15886
\(300\) 0 0
\(301\) −2.79448 −0.161071
\(302\) 0 0
\(303\) −10.4530 −0.600509
\(304\) 0 0
\(305\) 7.63094 0.436947
\(306\) 0 0
\(307\) 22.2256 1.26848 0.634240 0.773136i \(-0.281313\pi\)
0.634240 + 0.773136i \(0.281313\pi\)
\(308\) 0 0
\(309\) 13.5966 0.773484
\(310\) 0 0
\(311\) −19.0600 −1.08079 −0.540396 0.841411i \(-0.681725\pi\)
−0.540396 + 0.841411i \(0.681725\pi\)
\(312\) 0 0
\(313\) 13.6001 0.768726 0.384363 0.923182i \(-0.374421\pi\)
0.384363 + 0.923182i \(0.374421\pi\)
\(314\) 0 0
\(315\) −3.16095 −0.178100
\(316\) 0 0
\(317\) −5.74631 −0.322745 −0.161372 0.986894i \(-0.551592\pi\)
−0.161372 + 0.986894i \(0.551592\pi\)
\(318\) 0 0
\(319\) −7.47193 −0.418348
\(320\) 0 0
\(321\) −0.0743231 −0.00414831
\(322\) 0 0
\(323\) 12.4913 0.695034
\(324\) 0 0
\(325\) −3.04543 −0.168930
\(326\) 0 0
\(327\) 6.47846 0.358260
\(328\) 0 0
\(329\) 14.8949 0.821182
\(330\) 0 0
\(331\) 15.2249 0.836838 0.418419 0.908254i \(-0.362584\pi\)
0.418419 + 0.908254i \(0.362584\pi\)
\(332\) 0 0
\(333\) −9.80665 −0.537401
\(334\) 0 0
\(335\) −10.0096 −0.546885
\(336\) 0 0
\(337\) 19.0754 1.03910 0.519551 0.854440i \(-0.326099\pi\)
0.519551 + 0.854440i \(0.326099\pi\)
\(338\) 0 0
\(339\) 11.5796 0.628917
\(340\) 0 0
\(341\) −9.02374 −0.488663
\(342\) 0 0
\(343\) 15.7242 0.849025
\(344\) 0 0
\(345\) −4.69651 −0.252852
\(346\) 0 0
\(347\) −11.6976 −0.627962 −0.313981 0.949429i \(-0.601663\pi\)
−0.313981 + 0.949429i \(0.601663\pi\)
\(348\) 0 0
\(349\) 31.9057 1.70787 0.853937 0.520377i \(-0.174209\pi\)
0.853937 + 0.520377i \(0.174209\pi\)
\(350\) 0 0
\(351\) −11.9350 −0.637042
\(352\) 0 0
\(353\) 27.0249 1.43839 0.719195 0.694808i \(-0.244511\pi\)
0.719195 + 0.694808i \(0.244511\pi\)
\(354\) 0 0
\(355\) −16.0339 −0.850989
\(356\) 0 0
\(357\) −2.39026 −0.126506
\(358\) 0 0
\(359\) 8.56540 0.452064 0.226032 0.974120i \(-0.427425\pi\)
0.226032 + 0.974120i \(0.427425\pi\)
\(360\) 0 0
\(361\) 3.41257 0.179609
\(362\) 0 0
\(363\) 6.65779 0.349444
\(364\) 0 0
\(365\) −12.0878 −0.632706
\(366\) 0 0
\(367\) 0.0909754 0.00474888 0.00237444 0.999997i \(-0.499244\pi\)
0.00237444 + 0.999997i \(0.499244\pi\)
\(368\) 0 0
\(369\) 13.0163 0.677600
\(370\) 0 0
\(371\) −10.2354 −0.531397
\(372\) 0 0
\(373\) −16.4955 −0.854104 −0.427052 0.904227i \(-0.640448\pi\)
−0.427052 + 0.904227i \(0.640448\pi\)
\(374\) 0 0
\(375\) 0.713769 0.0368589
\(376\) 0 0
\(377\) 17.5962 0.906250
\(378\) 0 0
\(379\) 15.1710 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(380\) 0 0
\(381\) −0.303643 −0.0155561
\(382\) 0 0
\(383\) −16.9960 −0.868454 −0.434227 0.900804i \(-0.642978\pi\)
−0.434227 + 0.900804i \(0.642978\pi\)
\(384\) 0 0
\(385\) 1.64130 0.0836486
\(386\) 0 0
\(387\) −5.48362 −0.278748
\(388\) 0 0
\(389\) 31.6580 1.60513 0.802563 0.596568i \(-0.203469\pi\)
0.802563 + 0.596568i \(0.203469\pi\)
\(390\) 0 0
\(391\) 17.3612 0.877993
\(392\) 0 0
\(393\) 11.0719 0.558503
\(394\) 0 0
\(395\) −1.86668 −0.0939229
\(396\) 0 0
\(397\) −3.98075 −0.199788 −0.0998941 0.994998i \(-0.531850\pi\)
−0.0998941 + 0.994998i \(0.531850\pi\)
\(398\) 0 0
\(399\) −4.28874 −0.214705
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) 21.2507 1.05857
\(404\) 0 0
\(405\) −4.67436 −0.232271
\(406\) 0 0
\(407\) 5.09204 0.252403
\(408\) 0 0
\(409\) 21.9846 1.08707 0.543534 0.839387i \(-0.317086\pi\)
0.543534 + 0.839387i \(0.317086\pi\)
\(410\) 0 0
\(411\) −15.4377 −0.761487
\(412\) 0 0
\(413\) −14.6826 −0.722481
\(414\) 0 0
\(415\) 16.9310 0.831109
\(416\) 0 0
\(417\) 4.70165 0.230241
\(418\) 0 0
\(419\) −23.6768 −1.15669 −0.578344 0.815793i \(-0.696301\pi\)
−0.578344 + 0.815793i \(0.696301\pi\)
\(420\) 0 0
\(421\) −22.3551 −1.08952 −0.544761 0.838591i \(-0.683380\pi\)
−0.544761 + 0.838591i \(0.683380\pi\)
\(422\) 0 0
\(423\) 29.2283 1.42113
\(424\) 0 0
\(425\) −2.63853 −0.127987
\(426\) 0 0
\(427\) 9.68510 0.468695
\(428\) 0 0
\(429\) 2.81106 0.135719
\(430\) 0 0
\(431\) 1.77421 0.0854606 0.0427303 0.999087i \(-0.486394\pi\)
0.0427303 + 0.999087i \(0.486394\pi\)
\(432\) 0 0
\(433\) 16.0472 0.771178 0.385589 0.922671i \(-0.373998\pi\)
0.385589 + 0.922671i \(0.373998\pi\)
\(434\) 0 0
\(435\) −4.12408 −0.197735
\(436\) 0 0
\(437\) 31.1504 1.49013
\(438\) 0 0
\(439\) −22.8644 −1.09126 −0.545630 0.838026i \(-0.683709\pi\)
−0.545630 + 0.838026i \(0.683709\pi\)
\(440\) 0 0
\(441\) 13.4219 0.639138
\(442\) 0 0
\(443\) −10.3451 −0.491510 −0.245755 0.969332i \(-0.579036\pi\)
−0.245755 + 0.969332i \(0.579036\pi\)
\(444\) 0 0
\(445\) 12.3893 0.587308
\(446\) 0 0
\(447\) −8.24838 −0.390135
\(448\) 0 0
\(449\) −30.3356 −1.43163 −0.715813 0.698292i \(-0.753944\pi\)
−0.715813 + 0.698292i \(0.753944\pi\)
\(450\) 0 0
\(451\) −6.75861 −0.318250
\(452\) 0 0
\(453\) −12.9775 −0.609735
\(454\) 0 0
\(455\) −3.86523 −0.181205
\(456\) 0 0
\(457\) −4.34804 −0.203393 −0.101696 0.994815i \(-0.532427\pi\)
−0.101696 + 0.994815i \(0.532427\pi\)
\(458\) 0 0
\(459\) −10.3403 −0.482645
\(460\) 0 0
\(461\) 9.81743 0.457243 0.228622 0.973515i \(-0.426578\pi\)
0.228622 + 0.973515i \(0.426578\pi\)
\(462\) 0 0
\(463\) 9.39883 0.436801 0.218400 0.975859i \(-0.429916\pi\)
0.218400 + 0.975859i \(0.429916\pi\)
\(464\) 0 0
\(465\) −4.98060 −0.230970
\(466\) 0 0
\(467\) −4.11104 −0.190236 −0.0951182 0.995466i \(-0.530323\pi\)
−0.0951182 + 0.995466i \(0.530323\pi\)
\(468\) 0 0
\(469\) −12.7041 −0.586622
\(470\) 0 0
\(471\) 3.12021 0.143772
\(472\) 0 0
\(473\) 2.84733 0.130920
\(474\) 0 0
\(475\) −4.73419 −0.217220
\(476\) 0 0
\(477\) −20.0850 −0.919631
\(478\) 0 0
\(479\) 10.7238 0.489981 0.244990 0.969525i \(-0.421215\pi\)
0.244990 + 0.969525i \(0.421215\pi\)
\(480\) 0 0
\(481\) −11.9916 −0.546770
\(482\) 0 0
\(483\) −5.96076 −0.271224
\(484\) 0 0
\(485\) 13.1561 0.597386
\(486\) 0 0
\(487\) −25.4659 −1.15397 −0.576985 0.816755i \(-0.695771\pi\)
−0.576985 + 0.816755i \(0.695771\pi\)
\(488\) 0 0
\(489\) 1.75701 0.0794548
\(490\) 0 0
\(491\) −11.5251 −0.520121 −0.260061 0.965592i \(-0.583743\pi\)
−0.260061 + 0.965592i \(0.583743\pi\)
\(492\) 0 0
\(493\) 15.2451 0.686606
\(494\) 0 0
\(495\) 3.22074 0.144761
\(496\) 0 0
\(497\) −20.3500 −0.912821
\(498\) 0 0
\(499\) −22.7293 −1.01750 −0.508752 0.860913i \(-0.669893\pi\)
−0.508752 + 0.860913i \(0.669893\pi\)
\(500\) 0 0
\(501\) −0.665680 −0.0297404
\(502\) 0 0
\(503\) 20.1058 0.896472 0.448236 0.893915i \(-0.352052\pi\)
0.448236 + 0.893915i \(0.352052\pi\)
\(504\) 0 0
\(505\) −14.6448 −0.651684
\(506\) 0 0
\(507\) 2.65903 0.118092
\(508\) 0 0
\(509\) 28.2914 1.25399 0.626997 0.779022i \(-0.284284\pi\)
0.626997 + 0.779022i \(0.284284\pi\)
\(510\) 0 0
\(511\) −15.3417 −0.678678
\(512\) 0 0
\(513\) −18.5532 −0.819143
\(514\) 0 0
\(515\) 19.0490 0.839400
\(516\) 0 0
\(517\) −15.1766 −0.667466
\(518\) 0 0
\(519\) 1.21652 0.0533992
\(520\) 0 0
\(521\) 27.3444 1.19798 0.598990 0.800756i \(-0.295569\pi\)
0.598990 + 0.800756i \(0.295569\pi\)
\(522\) 0 0
\(523\) 29.1402 1.27421 0.637106 0.770776i \(-0.280131\pi\)
0.637106 + 0.770776i \(0.280131\pi\)
\(524\) 0 0
\(525\) 0.905907 0.0395370
\(526\) 0 0
\(527\) 18.4113 0.802010
\(528\) 0 0
\(529\) 20.2948 0.882381
\(530\) 0 0
\(531\) −28.8117 −1.25032
\(532\) 0 0
\(533\) 15.9163 0.689413
\(534\) 0 0
\(535\) −0.104128 −0.00450183
\(536\) 0 0
\(537\) −5.68947 −0.245519
\(538\) 0 0
\(539\) −6.96923 −0.300186
\(540\) 0 0
\(541\) 13.8977 0.597508 0.298754 0.954330i \(-0.403429\pi\)
0.298754 + 0.954330i \(0.403429\pi\)
\(542\) 0 0
\(543\) −10.0528 −0.431408
\(544\) 0 0
\(545\) 9.07641 0.388791
\(546\) 0 0
\(547\) −3.92237 −0.167708 −0.0838541 0.996478i \(-0.526723\pi\)
−0.0838541 + 0.996478i \(0.526723\pi\)
\(548\) 0 0
\(549\) 19.0051 0.811119
\(550\) 0 0
\(551\) 27.3537 1.16530
\(552\) 0 0
\(553\) −2.36917 −0.100747
\(554\) 0 0
\(555\) 2.81052 0.119300
\(556\) 0 0
\(557\) −19.8072 −0.839258 −0.419629 0.907696i \(-0.637840\pi\)
−0.419629 + 0.907696i \(0.637840\pi\)
\(558\) 0 0
\(559\) −6.70539 −0.283608
\(560\) 0 0
\(561\) 2.43547 0.102826
\(562\) 0 0
\(563\) −26.3190 −1.10921 −0.554606 0.832113i \(-0.687131\pi\)
−0.554606 + 0.832113i \(0.687131\pi\)
\(564\) 0 0
\(565\) 16.2232 0.682513
\(566\) 0 0
\(567\) −5.93263 −0.249147
\(568\) 0 0
\(569\) −2.41511 −0.101247 −0.0506234 0.998718i \(-0.516121\pi\)
−0.0506234 + 0.998718i \(0.516121\pi\)
\(570\) 0 0
\(571\) 6.47436 0.270943 0.135472 0.990781i \(-0.456745\pi\)
0.135472 + 0.990781i \(0.456745\pi\)
\(572\) 0 0
\(573\) −14.2575 −0.595616
\(574\) 0 0
\(575\) −6.57988 −0.274400
\(576\) 0 0
\(577\) 35.4985 1.47782 0.738911 0.673803i \(-0.235340\pi\)
0.738911 + 0.673803i \(0.235340\pi\)
\(578\) 0 0
\(579\) 0.00647401 0.000269051 0
\(580\) 0 0
\(581\) 21.4886 0.891497
\(582\) 0 0
\(583\) 10.4290 0.431926
\(584\) 0 0
\(585\) −7.58475 −0.313591
\(586\) 0 0
\(587\) −11.2013 −0.462327 −0.231164 0.972915i \(-0.574253\pi\)
−0.231164 + 0.972915i \(0.574253\pi\)
\(588\) 0 0
\(589\) 33.0346 1.36117
\(590\) 0 0
\(591\) −0.591645 −0.0243370
\(592\) 0 0
\(593\) 24.8629 1.02100 0.510498 0.859879i \(-0.329461\pi\)
0.510498 + 0.859879i \(0.329461\pi\)
\(594\) 0 0
\(595\) −3.34879 −0.137287
\(596\) 0 0
\(597\) −4.17889 −0.171030
\(598\) 0 0
\(599\) −13.5793 −0.554836 −0.277418 0.960749i \(-0.589479\pi\)
−0.277418 + 0.960749i \(0.589479\pi\)
\(600\) 0 0
\(601\) −16.1672 −0.659475 −0.329737 0.944073i \(-0.606960\pi\)
−0.329737 + 0.944073i \(0.606960\pi\)
\(602\) 0 0
\(603\) −24.9294 −1.01520
\(604\) 0 0
\(605\) 9.32765 0.379223
\(606\) 0 0
\(607\) 35.5503 1.44294 0.721470 0.692445i \(-0.243467\pi\)
0.721470 + 0.692445i \(0.243467\pi\)
\(608\) 0 0
\(609\) −5.23424 −0.212102
\(610\) 0 0
\(611\) 35.7405 1.44590
\(612\) 0 0
\(613\) −7.80038 −0.315054 −0.157527 0.987515i \(-0.550352\pi\)
−0.157527 + 0.987515i \(0.550352\pi\)
\(614\) 0 0
\(615\) −3.73037 −0.150423
\(616\) 0 0
\(617\) 37.1170 1.49427 0.747137 0.664670i \(-0.231428\pi\)
0.747137 + 0.664670i \(0.231428\pi\)
\(618\) 0 0
\(619\) 2.52549 0.101508 0.0507540 0.998711i \(-0.483838\pi\)
0.0507540 + 0.998711i \(0.483838\pi\)
\(620\) 0 0
\(621\) −25.7864 −1.03477
\(622\) 0 0
\(623\) 15.7243 0.629982
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.36986 0.174515
\(628\) 0 0
\(629\) −10.3894 −0.414252
\(630\) 0 0
\(631\) −24.5770 −0.978394 −0.489197 0.872173i \(-0.662710\pi\)
−0.489197 + 0.872173i \(0.662710\pi\)
\(632\) 0 0
\(633\) −6.81800 −0.270991
\(634\) 0 0
\(635\) −0.425408 −0.0168818
\(636\) 0 0
\(637\) 16.4123 0.650280
\(638\) 0 0
\(639\) −39.9329 −1.57972
\(640\) 0 0
\(641\) 39.9876 1.57942 0.789708 0.613483i \(-0.210232\pi\)
0.789708 + 0.613483i \(0.210232\pi\)
\(642\) 0 0
\(643\) −32.4864 −1.28114 −0.640568 0.767901i \(-0.721301\pi\)
−0.640568 + 0.767901i \(0.721301\pi\)
\(644\) 0 0
\(645\) 1.57157 0.0618804
\(646\) 0 0
\(647\) −28.6263 −1.12542 −0.562708 0.826655i \(-0.690241\pi\)
−0.562708 + 0.826655i \(0.690241\pi\)
\(648\) 0 0
\(649\) 14.9603 0.587242
\(650\) 0 0
\(651\) −6.32131 −0.247752
\(652\) 0 0
\(653\) −6.94845 −0.271914 −0.135957 0.990715i \(-0.543411\pi\)
−0.135957 + 0.990715i \(0.543411\pi\)
\(654\) 0 0
\(655\) 15.5119 0.606099
\(656\) 0 0
\(657\) −30.1052 −1.17451
\(658\) 0 0
\(659\) −41.3012 −1.60887 −0.804433 0.594043i \(-0.797531\pi\)
−0.804433 + 0.594043i \(0.797531\pi\)
\(660\) 0 0
\(661\) 9.83665 0.382601 0.191301 0.981531i \(-0.438729\pi\)
0.191301 + 0.981531i \(0.438729\pi\)
\(662\) 0 0
\(663\) −5.73547 −0.222747
\(664\) 0 0
\(665\) −6.00858 −0.233003
\(666\) 0 0
\(667\) 38.0178 1.47206
\(668\) 0 0
\(669\) 2.36403 0.0913988
\(670\) 0 0
\(671\) −9.86828 −0.380961
\(672\) 0 0
\(673\) 6.51047 0.250960 0.125480 0.992096i \(-0.459953\pi\)
0.125480 + 0.992096i \(0.459953\pi\)
\(674\) 0 0
\(675\) 3.91897 0.150841
\(676\) 0 0
\(677\) 13.7856 0.529823 0.264911 0.964273i \(-0.414657\pi\)
0.264911 + 0.964273i \(0.414657\pi\)
\(678\) 0 0
\(679\) 16.6975 0.640791
\(680\) 0 0
\(681\) 20.0382 0.767864
\(682\) 0 0
\(683\) −19.1093 −0.731197 −0.365599 0.930773i \(-0.619136\pi\)
−0.365599 + 0.930773i \(0.619136\pi\)
\(684\) 0 0
\(685\) −21.6285 −0.826381
\(686\) 0 0
\(687\) −1.69642 −0.0647226
\(688\) 0 0
\(689\) −24.5601 −0.935664
\(690\) 0 0
\(691\) 16.2019 0.616348 0.308174 0.951330i \(-0.400282\pi\)
0.308174 + 0.951330i \(0.400282\pi\)
\(692\) 0 0
\(693\) 4.08772 0.155280
\(694\) 0 0
\(695\) 6.58707 0.249862
\(696\) 0 0
\(697\) 13.7897 0.522323
\(698\) 0 0
\(699\) 16.6712 0.630561
\(700\) 0 0
\(701\) 33.1986 1.25389 0.626947 0.779062i \(-0.284304\pi\)
0.626947 + 0.779062i \(0.284304\pi\)
\(702\) 0 0
\(703\) −18.6412 −0.703066
\(704\) 0 0
\(705\) −8.37663 −0.315482
\(706\) 0 0
\(707\) −18.5870 −0.699035
\(708\) 0 0
\(709\) −7.03245 −0.264109 −0.132055 0.991242i \(-0.542157\pi\)
−0.132055 + 0.991242i \(0.542157\pi\)
\(710\) 0 0
\(711\) −4.64903 −0.174352
\(712\) 0 0
\(713\) 45.9136 1.71948
\(714\) 0 0
\(715\) 3.93833 0.147285
\(716\) 0 0
\(717\) −6.99674 −0.261298
\(718\) 0 0
\(719\) −27.8501 −1.03863 −0.519316 0.854582i \(-0.673813\pi\)
−0.519316 + 0.854582i \(0.673813\pi\)
\(720\) 0 0
\(721\) 24.1768 0.900390
\(722\) 0 0
\(723\) 6.85343 0.254882
\(724\) 0 0
\(725\) −5.77789 −0.214586
\(726\) 0 0
\(727\) −40.5901 −1.50540 −0.752702 0.658362i \(-0.771250\pi\)
−0.752702 + 0.658362i \(0.771250\pi\)
\(728\) 0 0
\(729\) −3.24991 −0.120367
\(730\) 0 0
\(731\) −5.80947 −0.214871
\(732\) 0 0
\(733\) −41.6563 −1.53861 −0.769305 0.638882i \(-0.779397\pi\)
−0.769305 + 0.638882i \(0.779397\pi\)
\(734\) 0 0
\(735\) −3.84662 −0.141885
\(736\) 0 0
\(737\) 12.9444 0.476813
\(738\) 0 0
\(739\) −5.66063 −0.208230 −0.104115 0.994565i \(-0.533201\pi\)
−0.104115 + 0.994565i \(0.533201\pi\)
\(740\) 0 0
\(741\) −10.2909 −0.378045
\(742\) 0 0
\(743\) −42.5400 −1.56064 −0.780320 0.625380i \(-0.784944\pi\)
−0.780320 + 0.625380i \(0.784944\pi\)
\(744\) 0 0
\(745\) −11.5561 −0.423382
\(746\) 0 0
\(747\) 42.1672 1.54282
\(748\) 0 0
\(749\) −0.132158 −0.00482893
\(750\) 0 0
\(751\) 33.2386 1.21289 0.606446 0.795125i \(-0.292595\pi\)
0.606446 + 0.795125i \(0.292595\pi\)
\(752\) 0 0
\(753\) −7.04627 −0.256780
\(754\) 0 0
\(755\) −18.1816 −0.661697
\(756\) 0 0
\(757\) 7.15003 0.259872 0.129936 0.991522i \(-0.458523\pi\)
0.129936 + 0.991522i \(0.458523\pi\)
\(758\) 0 0
\(759\) 6.07350 0.220454
\(760\) 0 0
\(761\) 30.4653 1.10437 0.552183 0.833723i \(-0.313795\pi\)
0.552183 + 0.833723i \(0.313795\pi\)
\(762\) 0 0
\(763\) 11.5197 0.417040
\(764\) 0 0
\(765\) −6.57134 −0.237587
\(766\) 0 0
\(767\) −35.2310 −1.27212
\(768\) 0 0
\(769\) −15.6977 −0.566072 −0.283036 0.959109i \(-0.591342\pi\)
−0.283036 + 0.959109i \(0.591342\pi\)
\(770\) 0 0
\(771\) 15.1062 0.544036
\(772\) 0 0
\(773\) −19.2779 −0.693378 −0.346689 0.937980i \(-0.612694\pi\)
−0.346689 + 0.937980i \(0.612694\pi\)
\(774\) 0 0
\(775\) −6.97788 −0.250653
\(776\) 0 0
\(777\) 3.56707 0.127968
\(778\) 0 0
\(779\) 24.7423 0.886485
\(780\) 0 0
\(781\) 20.7349 0.741952
\(782\) 0 0
\(783\) −22.6434 −0.809210
\(784\) 0 0
\(785\) 4.37146 0.156024
\(786\) 0 0
\(787\) −0.271020 −0.00966082 −0.00483041 0.999988i \(-0.501538\pi\)
−0.00483041 + 0.999988i \(0.501538\pi\)
\(788\) 0 0
\(789\) 4.35892 0.155181
\(790\) 0 0
\(791\) 20.5902 0.732104
\(792\) 0 0
\(793\) 23.2395 0.825260
\(794\) 0 0
\(795\) 5.75623 0.204153
\(796\) 0 0
\(797\) −42.4376 −1.50322 −0.751608 0.659610i \(-0.770722\pi\)
−0.751608 + 0.659610i \(0.770722\pi\)
\(798\) 0 0
\(799\) 30.9651 1.09547
\(800\) 0 0
\(801\) 30.8559 1.09024
\(802\) 0 0
\(803\) 15.6319 0.551638
\(804\) 0 0
\(805\) −8.35110 −0.294337
\(806\) 0 0
\(807\) −5.80747 −0.204433
\(808\) 0 0
\(809\) −22.0743 −0.776091 −0.388046 0.921640i \(-0.626850\pi\)
−0.388046 + 0.921640i \(0.626850\pi\)
\(810\) 0 0
\(811\) −49.9990 −1.75570 −0.877851 0.478933i \(-0.841024\pi\)
−0.877851 + 0.478933i \(0.841024\pi\)
\(812\) 0 0
\(813\) −1.41490 −0.0496226
\(814\) 0 0
\(815\) 2.46160 0.0862259
\(816\) 0 0
\(817\) −10.4237 −0.364678
\(818\) 0 0
\(819\) −9.62647 −0.336376
\(820\) 0 0
\(821\) −29.3871 −1.02562 −0.512809 0.858503i \(-0.671395\pi\)
−0.512809 + 0.858503i \(0.671395\pi\)
\(822\) 0 0
\(823\) 10.3806 0.361845 0.180922 0.983497i \(-0.442092\pi\)
0.180922 + 0.983497i \(0.442092\pi\)
\(824\) 0 0
\(825\) −0.923041 −0.0321362
\(826\) 0 0
\(827\) 16.0873 0.559411 0.279706 0.960086i \(-0.409763\pi\)
0.279706 + 0.960086i \(0.409763\pi\)
\(828\) 0 0
\(829\) 9.89833 0.343783 0.171892 0.985116i \(-0.445012\pi\)
0.171892 + 0.985116i \(0.445012\pi\)
\(830\) 0 0
\(831\) −14.1694 −0.491530
\(832\) 0 0
\(833\) 14.2195 0.492675
\(834\) 0 0
\(835\) −0.932627 −0.0322749
\(836\) 0 0
\(837\) −27.3461 −0.945220
\(838\) 0 0
\(839\) −13.0686 −0.451179 −0.225589 0.974222i \(-0.572431\pi\)
−0.225589 + 0.974222i \(0.572431\pi\)
\(840\) 0 0
\(841\) 4.38405 0.151174
\(842\) 0 0
\(843\) −12.3339 −0.424802
\(844\) 0 0
\(845\) 3.72534 0.128156
\(846\) 0 0
\(847\) 11.8385 0.406777
\(848\) 0 0
\(849\) 16.9495 0.581704
\(850\) 0 0
\(851\) −25.9087 −0.888139
\(852\) 0 0
\(853\) 7.49114 0.256492 0.128246 0.991742i \(-0.459065\pi\)
0.128246 + 0.991742i \(0.459065\pi\)
\(854\) 0 0
\(855\) −11.7907 −0.403232
\(856\) 0 0
\(857\) 11.2677 0.384897 0.192448 0.981307i \(-0.438357\pi\)
0.192448 + 0.981307i \(0.438357\pi\)
\(858\) 0 0
\(859\) −17.2435 −0.588341 −0.294171 0.955753i \(-0.595043\pi\)
−0.294171 + 0.955753i \(0.595043\pi\)
\(860\) 0 0
\(861\) −4.73454 −0.161353
\(862\) 0 0
\(863\) 22.2898 0.758754 0.379377 0.925242i \(-0.376138\pi\)
0.379377 + 0.925242i \(0.376138\pi\)
\(864\) 0 0
\(865\) 1.70436 0.0579498
\(866\) 0 0
\(867\) 7.16494 0.243334
\(868\) 0 0
\(869\) 2.41398 0.0818886
\(870\) 0 0
\(871\) −30.4837 −1.03290
\(872\) 0 0
\(873\) 32.7656 1.10895
\(874\) 0 0
\(875\) 1.26919 0.0429064
\(876\) 0 0
\(877\) −33.7198 −1.13864 −0.569318 0.822117i \(-0.692793\pi\)
−0.569318 + 0.822117i \(0.692793\pi\)
\(878\) 0 0
\(879\) −12.2019 −0.411559
\(880\) 0 0
\(881\) 39.9805 1.34698 0.673489 0.739197i \(-0.264795\pi\)
0.673489 + 0.739197i \(0.264795\pi\)
\(882\) 0 0
\(883\) −1.78565 −0.0600920 −0.0300460 0.999549i \(-0.509565\pi\)
−0.0300460 + 0.999549i \(0.509565\pi\)
\(884\) 0 0
\(885\) 8.25722 0.277563
\(886\) 0 0
\(887\) 42.5734 1.42948 0.714738 0.699392i \(-0.246546\pi\)
0.714738 + 0.699392i \(0.246546\pi\)
\(888\) 0 0
\(889\) −0.539923 −0.0181084
\(890\) 0 0
\(891\) 6.04484 0.202510
\(892\) 0 0
\(893\) 55.5594 1.85922
\(894\) 0 0
\(895\) −7.97102 −0.266442
\(896\) 0 0
\(897\) −14.3029 −0.477560
\(898\) 0 0
\(899\) 40.3174 1.34466
\(900\) 0 0
\(901\) −21.2786 −0.708892
\(902\) 0 0
\(903\) 1.99461 0.0663765
\(904\) 0 0
\(905\) −14.0841 −0.468172
\(906\) 0 0
\(907\) −21.4336 −0.711691 −0.355845 0.934545i \(-0.615807\pi\)
−0.355845 + 0.934545i \(0.615807\pi\)
\(908\) 0 0
\(909\) −36.4733 −1.20974
\(910\) 0 0
\(911\) 52.3406 1.73412 0.867060 0.498203i \(-0.166007\pi\)
0.867060 + 0.498203i \(0.166007\pi\)
\(912\) 0 0
\(913\) −21.8950 −0.724620
\(914\) 0 0
\(915\) −5.44673 −0.180063
\(916\) 0 0
\(917\) 19.6875 0.650137
\(918\) 0 0
\(919\) −6.10491 −0.201383 −0.100691 0.994918i \(-0.532105\pi\)
−0.100691 + 0.994918i \(0.532105\pi\)
\(920\) 0 0
\(921\) −15.8639 −0.522735
\(922\) 0 0
\(923\) −48.8300 −1.60726
\(924\) 0 0
\(925\) 3.93757 0.129466
\(926\) 0 0
\(927\) 47.4422 1.55821
\(928\) 0 0
\(929\) −1.75417 −0.0575526 −0.0287763 0.999586i \(-0.509161\pi\)
−0.0287763 + 0.999586i \(0.509161\pi\)
\(930\) 0 0
\(931\) 25.5133 0.836166
\(932\) 0 0
\(933\) 13.6044 0.445389
\(934\) 0 0
\(935\) 3.41213 0.111588
\(936\) 0 0
\(937\) −6.16858 −0.201519 −0.100759 0.994911i \(-0.532127\pi\)
−0.100759 + 0.994911i \(0.532127\pi\)
\(938\) 0 0
\(939\) −9.70737 −0.316788
\(940\) 0 0
\(941\) 29.0330 0.946450 0.473225 0.880942i \(-0.343090\pi\)
0.473225 + 0.880942i \(0.343090\pi\)
\(942\) 0 0
\(943\) 34.3884 1.11984
\(944\) 0 0
\(945\) 4.97391 0.161801
\(946\) 0 0
\(947\) −24.8720 −0.808232 −0.404116 0.914708i \(-0.632421\pi\)
−0.404116 + 0.914708i \(0.632421\pi\)
\(948\) 0 0
\(949\) −36.8127 −1.19499
\(950\) 0 0
\(951\) 4.10154 0.133001
\(952\) 0 0
\(953\) −5.96771 −0.193313 −0.0966566 0.995318i \(-0.530815\pi\)
−0.0966566 + 0.995318i \(0.530815\pi\)
\(954\) 0 0
\(955\) −19.9750 −0.646374
\(956\) 0 0
\(957\) 5.33323 0.172399
\(958\) 0 0
\(959\) −27.4506 −0.886425
\(960\) 0 0
\(961\) 17.6908 0.570671
\(962\) 0 0
\(963\) −0.259333 −0.00835690
\(964\) 0 0
\(965\) 0.00907017 0.000291979 0
\(966\) 0 0
\(967\) 35.6316 1.14583 0.572917 0.819613i \(-0.305812\pi\)
0.572917 + 0.819613i \(0.305812\pi\)
\(968\) 0 0
\(969\) −8.91591 −0.286420
\(970\) 0 0
\(971\) −25.9878 −0.833989 −0.416994 0.908909i \(-0.636917\pi\)
−0.416994 + 0.908909i \(0.636917\pi\)
\(972\) 0 0
\(973\) 8.36022 0.268016
\(974\) 0 0
\(975\) 2.17374 0.0696153
\(976\) 0 0
\(977\) 28.8423 0.922746 0.461373 0.887206i \(-0.347357\pi\)
0.461373 + 0.887206i \(0.347357\pi\)
\(978\) 0 0
\(979\) −16.0217 −0.512057
\(980\) 0 0
\(981\) 22.6051 0.721725
\(982\) 0 0
\(983\) 18.2422 0.581835 0.290918 0.956748i \(-0.406039\pi\)
0.290918 + 0.956748i \(0.406039\pi\)
\(984\) 0 0
\(985\) −0.828902 −0.0264110
\(986\) 0 0
\(987\) −10.6315 −0.338405
\(988\) 0 0
\(989\) −14.4875 −0.460675
\(990\) 0 0
\(991\) −61.8319 −1.96416 −0.982078 0.188477i \(-0.939645\pi\)
−0.982078 + 0.188477i \(0.939645\pi\)
\(992\) 0 0
\(993\) −10.8671 −0.344857
\(994\) 0 0
\(995\) −5.85467 −0.185606
\(996\) 0 0
\(997\) −34.7987 −1.10209 −0.551043 0.834477i \(-0.685770\pi\)
−0.551043 + 0.834477i \(0.685770\pi\)
\(998\) 0 0
\(999\) 15.4312 0.488223
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 8020.2.a.e.1.15 35
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.e.1.15 35 1.1 even 1 trivial