Properties

Label 8020.2.a.e.1.13
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 1.13
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)
 
\(f(q)\) \(=\) \(q-1.28894 q^{3} -1.00000 q^{5} +0.0476003 q^{7} -1.33863 q^{9} +O(q^{10})\) \(q-1.28894 q^{3} -1.00000 q^{5} +0.0476003 q^{7} -1.33863 q^{9} +1.50278 q^{11} +3.90785 q^{13} +1.28894 q^{15} -3.09866 q^{17} +8.32814 q^{19} -0.0613541 q^{21} +2.22939 q^{23} +1.00000 q^{25} +5.59224 q^{27} -5.06407 q^{29} -9.71810 q^{31} -1.93699 q^{33} -0.0476003 q^{35} +9.69898 q^{37} -5.03700 q^{39} +10.7165 q^{41} -0.420809 q^{43} +1.33863 q^{45} +0.217932 q^{47} -6.99773 q^{49} +3.99399 q^{51} -0.878938 q^{53} -1.50278 q^{55} -10.7345 q^{57} -2.11977 q^{59} -0.0739777 q^{61} -0.0637191 q^{63} -3.90785 q^{65} +0.0155387 q^{67} -2.87356 q^{69} -2.21165 q^{71} -15.0592 q^{73} -1.28894 q^{75} +0.0715327 q^{77} +2.77866 q^{79} -3.19220 q^{81} -1.20330 q^{83} +3.09866 q^{85} +6.52729 q^{87} +8.10286 q^{89} +0.186015 q^{91} +12.5261 q^{93} -8.32814 q^{95} -13.3371 q^{97} -2.01166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 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
\(3\) −1.28894 −0.744171 −0.372086 0.928198i \(-0.621357\pi\)
−0.372086 + 0.928198i \(0.621357\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.0476003 0.0179912 0.00899562 0.999960i \(-0.497137\pi\)
0.00899562 + 0.999960i \(0.497137\pi\)
\(8\) 0 0
\(9\) −1.33863 −0.446209
\(10\) 0 0
\(11\) 1.50278 0.453104 0.226552 0.973999i \(-0.427255\pi\)
0.226552 + 0.973999i \(0.427255\pi\)
\(12\) 0 0
\(13\) 3.90785 1.08384 0.541921 0.840429i \(-0.317697\pi\)
0.541921 + 0.840429i \(0.317697\pi\)
\(14\) 0 0
\(15\) 1.28894 0.332804
\(16\) 0 0
\(17\) −3.09866 −0.751535 −0.375768 0.926714i \(-0.622621\pi\)
−0.375768 + 0.926714i \(0.622621\pi\)
\(18\) 0 0
\(19\) 8.32814 1.91061 0.955304 0.295627i \(-0.0955284\pi\)
0.955304 + 0.295627i \(0.0955284\pi\)
\(20\) 0 0
\(21\) −0.0613541 −0.0133886
\(22\) 0 0
\(23\) 2.22939 0.464861 0.232430 0.972613i \(-0.425332\pi\)
0.232430 + 0.972613i \(0.425332\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.59224 1.07623
\(28\) 0 0
\(29\) −5.06407 −0.940373 −0.470187 0.882567i \(-0.655813\pi\)
−0.470187 + 0.882567i \(0.655813\pi\)
\(30\) 0 0
\(31\) −9.71810 −1.74542 −0.872711 0.488238i \(-0.837640\pi\)
−0.872711 + 0.488238i \(0.837640\pi\)
\(32\) 0 0
\(33\) −1.93699 −0.337187
\(34\) 0 0
\(35\) −0.0476003 −0.00804592
\(36\) 0 0
\(37\) 9.69898 1.59450 0.797251 0.603648i \(-0.206287\pi\)
0.797251 + 0.603648i \(0.206287\pi\)
\(38\) 0 0
\(39\) −5.03700 −0.806565
\(40\) 0 0
\(41\) 10.7165 1.67364 0.836820 0.547478i \(-0.184412\pi\)
0.836820 + 0.547478i \(0.184412\pi\)
\(42\) 0 0
\(43\) −0.420809 −0.0641728 −0.0320864 0.999485i \(-0.510215\pi\)
−0.0320864 + 0.999485i \(0.510215\pi\)
\(44\) 0 0
\(45\) 1.33863 0.199551
\(46\) 0 0
\(47\) 0.217932 0.0317886 0.0158943 0.999874i \(-0.494940\pi\)
0.0158943 + 0.999874i \(0.494940\pi\)
\(48\) 0 0
\(49\) −6.99773 −0.999676
\(50\) 0 0
\(51\) 3.99399 0.559271
\(52\) 0 0
\(53\) −0.878938 −0.120731 −0.0603657 0.998176i \(-0.519227\pi\)
−0.0603657 + 0.998176i \(0.519227\pi\)
\(54\) 0 0
\(55\) −1.50278 −0.202634
\(56\) 0 0
\(57\) −10.7345 −1.42182
\(58\) 0 0
\(59\) −2.11977 −0.275970 −0.137985 0.990434i \(-0.544063\pi\)
−0.137985 + 0.990434i \(0.544063\pi\)
\(60\) 0 0
\(61\) −0.0739777 −0.00947188 −0.00473594 0.999989i \(-0.501508\pi\)
−0.00473594 + 0.999989i \(0.501508\pi\)
\(62\) 0 0
\(63\) −0.0637191 −0.00802785
\(64\) 0 0
\(65\) −3.90785 −0.484709
\(66\) 0 0
\(67\) 0.0155387 0.00189836 0.000949178 1.00000i \(-0.499698\pi\)
0.000949178 1.00000i \(0.499698\pi\)
\(68\) 0 0
\(69\) −2.87356 −0.345936
\(70\) 0 0
\(71\) −2.21165 −0.262475 −0.131237 0.991351i \(-0.541895\pi\)
−0.131237 + 0.991351i \(0.541895\pi\)
\(72\) 0 0
\(73\) −15.0592 −1.76254 −0.881271 0.472611i \(-0.843312\pi\)
−0.881271 + 0.472611i \(0.843312\pi\)
\(74\) 0 0
\(75\) −1.28894 −0.148834
\(76\) 0 0
\(77\) 0.0715327 0.00815191
\(78\) 0 0
\(79\) 2.77866 0.312624 0.156312 0.987708i \(-0.450039\pi\)
0.156312 + 0.987708i \(0.450039\pi\)
\(80\) 0 0
\(81\) −3.19220 −0.354689
\(82\) 0 0
\(83\) −1.20330 −0.132079 −0.0660395 0.997817i \(-0.521036\pi\)
−0.0660395 + 0.997817i \(0.521036\pi\)
\(84\) 0 0
\(85\) 3.09866 0.336097
\(86\) 0 0
\(87\) 6.52729 0.699799
\(88\) 0 0
\(89\) 8.10286 0.858902 0.429451 0.903090i \(-0.358707\pi\)
0.429451 + 0.903090i \(0.358707\pi\)
\(90\) 0 0
\(91\) 0.186015 0.0194997
\(92\) 0 0
\(93\) 12.5261 1.29889
\(94\) 0 0
\(95\) −8.32814 −0.854449
\(96\) 0 0
\(97\) −13.3371 −1.35418 −0.677089 0.735901i \(-0.736759\pi\)
−0.677089 + 0.735901i \(0.736759\pi\)
\(98\) 0 0
\(99\) −2.01166 −0.202179
\(100\) 0 0
\(101\) 2.34968 0.233802 0.116901 0.993144i \(-0.462704\pi\)
0.116901 + 0.993144i \(0.462704\pi\)
\(102\) 0 0
\(103\) −3.31040 −0.326184 −0.163092 0.986611i \(-0.552147\pi\)
−0.163092 + 0.986611i \(0.552147\pi\)
\(104\) 0 0
\(105\) 0.0613541 0.00598755
\(106\) 0 0
\(107\) 19.0806 1.84459 0.922297 0.386482i \(-0.126310\pi\)
0.922297 + 0.386482i \(0.126310\pi\)
\(108\) 0 0
\(109\) −1.70033 −0.162862 −0.0814308 0.996679i \(-0.525949\pi\)
−0.0814308 + 0.996679i \(0.525949\pi\)
\(110\) 0 0
\(111\) −12.5014 −1.18658
\(112\) 0 0
\(113\) 9.05493 0.851816 0.425908 0.904766i \(-0.359955\pi\)
0.425908 + 0.904766i \(0.359955\pi\)
\(114\) 0 0
\(115\) −2.22939 −0.207892
\(116\) 0 0
\(117\) −5.23115 −0.483620
\(118\) 0 0
\(119\) −0.147497 −0.0135210
\(120\) 0 0
\(121\) −8.74166 −0.794696
\(122\) 0 0
\(123\) −13.8130 −1.24548
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.979099 0.0868810 0.0434405 0.999056i \(-0.486168\pi\)
0.0434405 + 0.999056i \(0.486168\pi\)
\(128\) 0 0
\(129\) 0.542399 0.0477556
\(130\) 0 0
\(131\) 7.11612 0.621738 0.310869 0.950453i \(-0.399380\pi\)
0.310869 + 0.950453i \(0.399380\pi\)
\(132\) 0 0
\(133\) 0.396422 0.0343742
\(134\) 0 0
\(135\) −5.59224 −0.481304
\(136\) 0 0
\(137\) −20.3929 −1.74229 −0.871143 0.491029i \(-0.836621\pi\)
−0.871143 + 0.491029i \(0.836621\pi\)
\(138\) 0 0
\(139\) 5.04716 0.428095 0.214047 0.976823i \(-0.431335\pi\)
0.214047 + 0.976823i \(0.431335\pi\)
\(140\) 0 0
\(141\) −0.280902 −0.0236562
\(142\) 0 0
\(143\) 5.87263 0.491094
\(144\) 0 0
\(145\) 5.06407 0.420548
\(146\) 0 0
\(147\) 9.01968 0.743931
\(148\) 0 0
\(149\) 22.6839 1.85834 0.929168 0.369657i \(-0.120525\pi\)
0.929168 + 0.369657i \(0.120525\pi\)
\(150\) 0 0
\(151\) 4.57148 0.372022 0.186011 0.982548i \(-0.440444\pi\)
0.186011 + 0.982548i \(0.440444\pi\)
\(152\) 0 0
\(153\) 4.14795 0.335342
\(154\) 0 0
\(155\) 9.71810 0.780576
\(156\) 0 0
\(157\) 5.88672 0.469811 0.234906 0.972018i \(-0.424522\pi\)
0.234906 + 0.972018i \(0.424522\pi\)
\(158\) 0 0
\(159\) 1.13290 0.0898449
\(160\) 0 0
\(161\) 0.106120 0.00836342
\(162\) 0 0
\(163\) −10.1654 −0.796218 −0.398109 0.917338i \(-0.630333\pi\)
−0.398109 + 0.917338i \(0.630333\pi\)
\(164\) 0 0
\(165\) 1.93699 0.150795
\(166\) 0 0
\(167\) −6.54352 −0.506353 −0.253176 0.967420i \(-0.581475\pi\)
−0.253176 + 0.967420i \(0.581475\pi\)
\(168\) 0 0
\(169\) 2.27129 0.174715
\(170\) 0 0
\(171\) −11.1483 −0.852530
\(172\) 0 0
\(173\) 11.8658 0.902140 0.451070 0.892489i \(-0.351043\pi\)
0.451070 + 0.892489i \(0.351043\pi\)
\(174\) 0 0
\(175\) 0.0476003 0.00359825
\(176\) 0 0
\(177\) 2.73226 0.205369
\(178\) 0 0
\(179\) −1.47248 −0.110059 −0.0550293 0.998485i \(-0.517525\pi\)
−0.0550293 + 0.998485i \(0.517525\pi\)
\(180\) 0 0
\(181\) −10.0925 −0.750168 −0.375084 0.926991i \(-0.622386\pi\)
−0.375084 + 0.926991i \(0.622386\pi\)
\(182\) 0 0
\(183\) 0.0953530 0.00704870
\(184\) 0 0
\(185\) −9.69898 −0.713083
\(186\) 0 0
\(187\) −4.65659 −0.340524
\(188\) 0 0
\(189\) 0.266192 0.0193627
\(190\) 0 0
\(191\) 13.6202 0.985526 0.492763 0.870164i \(-0.335987\pi\)
0.492763 + 0.870164i \(0.335987\pi\)
\(192\) 0 0
\(193\) −9.10873 −0.655660 −0.327830 0.944737i \(-0.606317\pi\)
−0.327830 + 0.944737i \(0.606317\pi\)
\(194\) 0 0
\(195\) 5.03700 0.360707
\(196\) 0 0
\(197\) −17.0669 −1.21597 −0.607984 0.793949i \(-0.708022\pi\)
−0.607984 + 0.793949i \(0.708022\pi\)
\(198\) 0 0
\(199\) −24.1476 −1.71178 −0.855891 0.517157i \(-0.826990\pi\)
−0.855891 + 0.517157i \(0.826990\pi\)
\(200\) 0 0
\(201\) −0.0200285 −0.00141270
\(202\) 0 0
\(203\) −0.241051 −0.0169185
\(204\) 0 0
\(205\) −10.7165 −0.748475
\(206\) 0 0
\(207\) −2.98433 −0.207425
\(208\) 0 0
\(209\) 12.5153 0.865704
\(210\) 0 0
\(211\) 16.5516 1.13946 0.569731 0.821831i \(-0.307048\pi\)
0.569731 + 0.821831i \(0.307048\pi\)
\(212\) 0 0
\(213\) 2.85069 0.195326
\(214\) 0 0
\(215\) 0.420809 0.0286990
\(216\) 0 0
\(217\) −0.462584 −0.0314023
\(218\) 0 0
\(219\) 19.4104 1.31163
\(220\) 0 0
\(221\) −12.1091 −0.814546
\(222\) 0 0
\(223\) 13.6361 0.913141 0.456570 0.889687i \(-0.349078\pi\)
0.456570 + 0.889687i \(0.349078\pi\)
\(224\) 0 0
\(225\) −1.33863 −0.0892418
\(226\) 0 0
\(227\) 11.2715 0.748115 0.374058 0.927405i \(-0.377966\pi\)
0.374058 + 0.927405i \(0.377966\pi\)
\(228\) 0 0
\(229\) 11.0109 0.727623 0.363812 0.931473i \(-0.381475\pi\)
0.363812 + 0.931473i \(0.381475\pi\)
\(230\) 0 0
\(231\) −0.0922015 −0.00606642
\(232\) 0 0
\(233\) 16.2749 1.06621 0.533103 0.846050i \(-0.321026\pi\)
0.533103 + 0.846050i \(0.321026\pi\)
\(234\) 0 0
\(235\) −0.217932 −0.0142163
\(236\) 0 0
\(237\) −3.58154 −0.232646
\(238\) 0 0
\(239\) 12.0764 0.781160 0.390580 0.920569i \(-0.372275\pi\)
0.390580 + 0.920569i \(0.372275\pi\)
\(240\) 0 0
\(241\) 14.9670 0.964107 0.482053 0.876142i \(-0.339891\pi\)
0.482053 + 0.876142i \(0.339891\pi\)
\(242\) 0 0
\(243\) −12.6622 −0.812278
\(244\) 0 0
\(245\) 6.99773 0.447069
\(246\) 0 0
\(247\) 32.5451 2.07080
\(248\) 0 0
\(249\) 1.55098 0.0982895
\(250\) 0 0
\(251\) 30.3956 1.91855 0.959277 0.282467i \(-0.0911529\pi\)
0.959277 + 0.282467i \(0.0911529\pi\)
\(252\) 0 0
\(253\) 3.35028 0.210630
\(254\) 0 0
\(255\) −3.99399 −0.250114
\(256\) 0 0
\(257\) 16.5631 1.03318 0.516590 0.856233i \(-0.327201\pi\)
0.516590 + 0.856233i \(0.327201\pi\)
\(258\) 0 0
\(259\) 0.461675 0.0286871
\(260\) 0 0
\(261\) 6.77889 0.419603
\(262\) 0 0
\(263\) −11.8184 −0.728754 −0.364377 0.931251i \(-0.618718\pi\)
−0.364377 + 0.931251i \(0.618718\pi\)
\(264\) 0 0
\(265\) 0.878938 0.0539927
\(266\) 0 0
\(267\) −10.4441 −0.639170
\(268\) 0 0
\(269\) 21.1949 1.29228 0.646138 0.763221i \(-0.276383\pi\)
0.646138 + 0.763221i \(0.276383\pi\)
\(270\) 0 0
\(271\) 12.4661 0.757260 0.378630 0.925548i \(-0.376395\pi\)
0.378630 + 0.925548i \(0.376395\pi\)
\(272\) 0 0
\(273\) −0.239763 −0.0145111
\(274\) 0 0
\(275\) 1.50278 0.0906209
\(276\) 0 0
\(277\) 6.32731 0.380171 0.190086 0.981768i \(-0.439123\pi\)
0.190086 + 0.981768i \(0.439123\pi\)
\(278\) 0 0
\(279\) 13.0089 0.778822
\(280\) 0 0
\(281\) 22.4246 1.33774 0.668869 0.743380i \(-0.266779\pi\)
0.668869 + 0.743380i \(0.266779\pi\)
\(282\) 0 0
\(283\) −27.1531 −1.61409 −0.807043 0.590493i \(-0.798934\pi\)
−0.807043 + 0.590493i \(0.798934\pi\)
\(284\) 0 0
\(285\) 10.7345 0.635857
\(286\) 0 0
\(287\) 0.510110 0.0301108
\(288\) 0 0
\(289\) −7.39832 −0.435195
\(290\) 0 0
\(291\) 17.1908 1.00774
\(292\) 0 0
\(293\) 14.4240 0.842660 0.421330 0.906907i \(-0.361563\pi\)
0.421330 + 0.906907i \(0.361563\pi\)
\(294\) 0 0
\(295\) 2.11977 0.123418
\(296\) 0 0
\(297\) 8.40389 0.487643
\(298\) 0 0
\(299\) 8.71214 0.503836
\(300\) 0 0
\(301\) −0.0200307 −0.00115455
\(302\) 0 0
\(303\) −3.02860 −0.173989
\(304\) 0 0
\(305\) 0.0739777 0.00423595
\(306\) 0 0
\(307\) −18.9377 −1.08083 −0.540415 0.841399i \(-0.681733\pi\)
−0.540415 + 0.841399i \(0.681733\pi\)
\(308\) 0 0
\(309\) 4.26692 0.242737
\(310\) 0 0
\(311\) 16.9663 0.962069 0.481034 0.876702i \(-0.340261\pi\)
0.481034 + 0.876702i \(0.340261\pi\)
\(312\) 0 0
\(313\) −9.94453 −0.562098 −0.281049 0.959693i \(-0.590682\pi\)
−0.281049 + 0.959693i \(0.590682\pi\)
\(314\) 0 0
\(315\) 0.0637191 0.00359016
\(316\) 0 0
\(317\) 35.3133 1.98339 0.991697 0.128597i \(-0.0410475\pi\)
0.991697 + 0.128597i \(0.0410475\pi\)
\(318\) 0 0
\(319\) −7.61016 −0.426087
\(320\) 0 0
\(321\) −24.5938 −1.37269
\(322\) 0 0
\(323\) −25.8061 −1.43589
\(324\) 0 0
\(325\) 3.90785 0.216769
\(326\) 0 0
\(327\) 2.19162 0.121197
\(328\) 0 0
\(329\) 0.0103736 0.000571917 0
\(330\) 0 0
\(331\) −35.5395 −1.95343 −0.976714 0.214547i \(-0.931173\pi\)
−0.976714 + 0.214547i \(0.931173\pi\)
\(332\) 0 0
\(333\) −12.9833 −0.711481
\(334\) 0 0
\(335\) −0.0155387 −0.000848970 0
\(336\) 0 0
\(337\) 5.66197 0.308427 0.154214 0.988038i \(-0.450716\pi\)
0.154214 + 0.988038i \(0.450716\pi\)
\(338\) 0 0
\(339\) −11.6713 −0.633897
\(340\) 0 0
\(341\) −14.6041 −0.790858
\(342\) 0 0
\(343\) −0.666297 −0.0359766
\(344\) 0 0
\(345\) 2.87356 0.154707
\(346\) 0 0
\(347\) −34.9896 −1.87834 −0.939170 0.343452i \(-0.888404\pi\)
−0.939170 + 0.343452i \(0.888404\pi\)
\(348\) 0 0
\(349\) 10.8431 0.580417 0.290209 0.956963i \(-0.406275\pi\)
0.290209 + 0.956963i \(0.406275\pi\)
\(350\) 0 0
\(351\) 21.8536 1.16646
\(352\) 0 0
\(353\) 28.3595 1.50942 0.754711 0.656058i \(-0.227777\pi\)
0.754711 + 0.656058i \(0.227777\pi\)
\(354\) 0 0
\(355\) 2.21165 0.117382
\(356\) 0 0
\(357\) 0.190115 0.0100620
\(358\) 0 0
\(359\) −25.5352 −1.34769 −0.673847 0.738871i \(-0.735359\pi\)
−0.673847 + 0.738871i \(0.735359\pi\)
\(360\) 0 0
\(361\) 50.3580 2.65042
\(362\) 0 0
\(363\) 11.2675 0.591390
\(364\) 0 0
\(365\) 15.0592 0.788233
\(366\) 0 0
\(367\) 25.9598 1.35509 0.677545 0.735481i \(-0.263044\pi\)
0.677545 + 0.735481i \(0.263044\pi\)
\(368\) 0 0
\(369\) −14.3454 −0.746793
\(370\) 0 0
\(371\) −0.0418377 −0.00217211
\(372\) 0 0
\(373\) 0.377898 0.0195668 0.00978341 0.999952i \(-0.496886\pi\)
0.00978341 + 0.999952i \(0.496886\pi\)
\(374\) 0 0
\(375\) 1.28894 0.0665607
\(376\) 0 0
\(377\) −19.7896 −1.01922
\(378\) 0 0
\(379\) 26.1000 1.34067 0.670333 0.742061i \(-0.266151\pi\)
0.670333 + 0.742061i \(0.266151\pi\)
\(380\) 0 0
\(381\) −1.26200 −0.0646544
\(382\) 0 0
\(383\) −22.9759 −1.17402 −0.587008 0.809581i \(-0.699694\pi\)
−0.587008 + 0.809581i \(0.699694\pi\)
\(384\) 0 0
\(385\) −0.0715327 −0.00364564
\(386\) 0 0
\(387\) 0.563306 0.0286345
\(388\) 0 0
\(389\) 17.3936 0.881888 0.440944 0.897535i \(-0.354644\pi\)
0.440944 + 0.897535i \(0.354644\pi\)
\(390\) 0 0
\(391\) −6.90813 −0.349359
\(392\) 0 0
\(393\) −9.17227 −0.462680
\(394\) 0 0
\(395\) −2.77866 −0.139810
\(396\) 0 0
\(397\) 24.2948 1.21932 0.609659 0.792664i \(-0.291306\pi\)
0.609659 + 0.792664i \(0.291306\pi\)
\(398\) 0 0
\(399\) −0.510966 −0.0255803
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) −37.9769 −1.89176
\(404\) 0 0
\(405\) 3.19220 0.158622
\(406\) 0 0
\(407\) 14.5754 0.722476
\(408\) 0 0
\(409\) −22.3013 −1.10273 −0.551364 0.834265i \(-0.685893\pi\)
−0.551364 + 0.834265i \(0.685893\pi\)
\(410\) 0 0
\(411\) 26.2853 1.29656
\(412\) 0 0
\(413\) −0.100902 −0.00496504
\(414\) 0 0
\(415\) 1.20330 0.0590676
\(416\) 0 0
\(417\) −6.50550 −0.318576
\(418\) 0 0
\(419\) −8.99302 −0.439338 −0.219669 0.975575i \(-0.570498\pi\)
−0.219669 + 0.975575i \(0.570498\pi\)
\(420\) 0 0
\(421\) 32.3160 1.57499 0.787494 0.616323i \(-0.211378\pi\)
0.787494 + 0.616323i \(0.211378\pi\)
\(422\) 0 0
\(423\) −0.291730 −0.0141844
\(424\) 0 0
\(425\) −3.09866 −0.150307
\(426\) 0 0
\(427\) −0.00352136 −0.000170411 0
\(428\) 0 0
\(429\) −7.56948 −0.365458
\(430\) 0 0
\(431\) −6.70614 −0.323023 −0.161512 0.986871i \(-0.551637\pi\)
−0.161512 + 0.986871i \(0.551637\pi\)
\(432\) 0 0
\(433\) −16.6776 −0.801476 −0.400738 0.916193i \(-0.631246\pi\)
−0.400738 + 0.916193i \(0.631246\pi\)
\(434\) 0 0
\(435\) −6.52729 −0.312960
\(436\) 0 0
\(437\) 18.5667 0.888166
\(438\) 0 0
\(439\) −1.42878 −0.0681917 −0.0340959 0.999419i \(-0.510855\pi\)
−0.0340959 + 0.999419i \(0.510855\pi\)
\(440\) 0 0
\(441\) 9.36735 0.446064
\(442\) 0 0
\(443\) 17.1945 0.816936 0.408468 0.912773i \(-0.366063\pi\)
0.408468 + 0.912773i \(0.366063\pi\)
\(444\) 0 0
\(445\) −8.10286 −0.384113
\(446\) 0 0
\(447\) −29.2382 −1.38292
\(448\) 0 0
\(449\) −4.03537 −0.190441 −0.0952204 0.995456i \(-0.530356\pi\)
−0.0952204 + 0.995456i \(0.530356\pi\)
\(450\) 0 0
\(451\) 16.1045 0.758334
\(452\) 0 0
\(453\) −5.89238 −0.276848
\(454\) 0 0
\(455\) −0.186015 −0.00872051
\(456\) 0 0
\(457\) 15.9332 0.745325 0.372662 0.927967i \(-0.378445\pi\)
0.372662 + 0.927967i \(0.378445\pi\)
\(458\) 0 0
\(459\) −17.3284 −0.808823
\(460\) 0 0
\(461\) 29.1828 1.35918 0.679590 0.733592i \(-0.262158\pi\)
0.679590 + 0.733592i \(0.262158\pi\)
\(462\) 0 0
\(463\) −33.4105 −1.55272 −0.776359 0.630290i \(-0.782936\pi\)
−0.776359 + 0.630290i \(0.782936\pi\)
\(464\) 0 0
\(465\) −12.5261 −0.580883
\(466\) 0 0
\(467\) 2.37186 0.109757 0.0548784 0.998493i \(-0.482523\pi\)
0.0548784 + 0.998493i \(0.482523\pi\)
\(468\) 0 0
\(469\) 0.000739648 0 3.41537e−5 0
\(470\) 0 0
\(471\) −7.58764 −0.349620
\(472\) 0 0
\(473\) −0.632383 −0.0290770
\(474\) 0 0
\(475\) 8.32814 0.382121
\(476\) 0 0
\(477\) 1.17657 0.0538714
\(478\) 0 0
\(479\) −24.0198 −1.09749 −0.548745 0.835990i \(-0.684894\pi\)
−0.548745 + 0.835990i \(0.684894\pi\)
\(480\) 0 0
\(481\) 37.9022 1.72819
\(482\) 0 0
\(483\) −0.136782 −0.00622382
\(484\) 0 0
\(485\) 13.3371 0.605607
\(486\) 0 0
\(487\) 10.4570 0.473854 0.236927 0.971527i \(-0.423860\pi\)
0.236927 + 0.971527i \(0.423860\pi\)
\(488\) 0 0
\(489\) 13.1027 0.592522
\(490\) 0 0
\(491\) 14.8963 0.672262 0.336131 0.941815i \(-0.390881\pi\)
0.336131 + 0.941815i \(0.390881\pi\)
\(492\) 0 0
\(493\) 15.6918 0.706723
\(494\) 0 0
\(495\) 2.01166 0.0904173
\(496\) 0 0
\(497\) −0.105275 −0.00472225
\(498\) 0 0
\(499\) −3.46705 −0.155207 −0.0776033 0.996984i \(-0.524727\pi\)
−0.0776033 + 0.996984i \(0.524727\pi\)
\(500\) 0 0
\(501\) 8.43422 0.376813
\(502\) 0 0
\(503\) −19.8262 −0.884009 −0.442004 0.897013i \(-0.645732\pi\)
−0.442004 + 0.897013i \(0.645732\pi\)
\(504\) 0 0
\(505\) −2.34968 −0.104559
\(506\) 0 0
\(507\) −2.92757 −0.130018
\(508\) 0 0
\(509\) 14.3931 0.637964 0.318982 0.947761i \(-0.396659\pi\)
0.318982 + 0.947761i \(0.396659\pi\)
\(510\) 0 0
\(511\) −0.716821 −0.0317103
\(512\) 0 0
\(513\) 46.5730 2.05625
\(514\) 0 0
\(515\) 3.31040 0.145874
\(516\) 0 0
\(517\) 0.327503 0.0144036
\(518\) 0 0
\(519\) −15.2943 −0.671347
\(520\) 0 0
\(521\) 39.1517 1.71527 0.857633 0.514262i \(-0.171934\pi\)
0.857633 + 0.514262i \(0.171934\pi\)
\(522\) 0 0
\(523\) −19.2954 −0.843728 −0.421864 0.906659i \(-0.638624\pi\)
−0.421864 + 0.906659i \(0.638624\pi\)
\(524\) 0 0
\(525\) −0.0613541 −0.00267771
\(526\) 0 0
\(527\) 30.1131 1.31175
\(528\) 0 0
\(529\) −18.0298 −0.783904
\(530\) 0 0
\(531\) 2.83757 0.123140
\(532\) 0 0
\(533\) 41.8786 1.81396
\(534\) 0 0
\(535\) −19.0806 −0.824927
\(536\) 0 0
\(537\) 1.89795 0.0819025
\(538\) 0 0
\(539\) −10.5160 −0.452958
\(540\) 0 0
\(541\) −38.3387 −1.64831 −0.824154 0.566365i \(-0.808349\pi\)
−0.824154 + 0.566365i \(0.808349\pi\)
\(542\) 0 0
\(543\) 13.0086 0.558253
\(544\) 0 0
\(545\) 1.70033 0.0728340
\(546\) 0 0
\(547\) 32.2043 1.37695 0.688477 0.725258i \(-0.258279\pi\)
0.688477 + 0.725258i \(0.258279\pi\)
\(548\) 0 0
\(549\) 0.0990285 0.00422643
\(550\) 0 0
\(551\) −42.1743 −1.79668
\(552\) 0 0
\(553\) 0.132265 0.00562449
\(554\) 0 0
\(555\) 12.5014 0.530656
\(556\) 0 0
\(557\) −11.7424 −0.497542 −0.248771 0.968562i \(-0.580027\pi\)
−0.248771 + 0.968562i \(0.580027\pi\)
\(558\) 0 0
\(559\) −1.64446 −0.0695532
\(560\) 0 0
\(561\) 6.00208 0.253408
\(562\) 0 0
\(563\) 33.9362 1.43024 0.715120 0.699002i \(-0.246372\pi\)
0.715120 + 0.699002i \(0.246372\pi\)
\(564\) 0 0
\(565\) −9.05493 −0.380944
\(566\) 0 0
\(567\) −0.151950 −0.00638129
\(568\) 0 0
\(569\) 24.2860 1.01812 0.509061 0.860730i \(-0.329993\pi\)
0.509061 + 0.860730i \(0.329993\pi\)
\(570\) 0 0
\(571\) −15.9144 −0.665999 −0.332999 0.942927i \(-0.608061\pi\)
−0.332999 + 0.942927i \(0.608061\pi\)
\(572\) 0 0
\(573\) −17.5557 −0.733400
\(574\) 0 0
\(575\) 2.22939 0.0929721
\(576\) 0 0
\(577\) 44.8834 1.86852 0.934260 0.356592i \(-0.116061\pi\)
0.934260 + 0.356592i \(0.116061\pi\)
\(578\) 0 0
\(579\) 11.7406 0.487924
\(580\) 0 0
\(581\) −0.0572773 −0.00237626
\(582\) 0 0
\(583\) −1.32085 −0.0547039
\(584\) 0 0
\(585\) 5.23115 0.216281
\(586\) 0 0
\(587\) −15.4990 −0.639711 −0.319856 0.947466i \(-0.603634\pi\)
−0.319856 + 0.947466i \(0.603634\pi\)
\(588\) 0 0
\(589\) −80.9337 −3.33481
\(590\) 0 0
\(591\) 21.9983 0.904889
\(592\) 0 0
\(593\) 26.9706 1.10755 0.553774 0.832667i \(-0.313187\pi\)
0.553774 + 0.832667i \(0.313187\pi\)
\(594\) 0 0
\(595\) 0.147497 0.00604679
\(596\) 0 0
\(597\) 31.1249 1.27386
\(598\) 0 0
\(599\) 19.4884 0.796274 0.398137 0.917326i \(-0.369657\pi\)
0.398137 + 0.917326i \(0.369657\pi\)
\(600\) 0 0
\(601\) 17.6204 0.718751 0.359375 0.933193i \(-0.382990\pi\)
0.359375 + 0.933193i \(0.382990\pi\)
\(602\) 0 0
\(603\) −0.0208005 −0.000847063 0
\(604\) 0 0
\(605\) 8.74166 0.355399
\(606\) 0 0
\(607\) 30.5996 1.24200 0.621000 0.783811i \(-0.286727\pi\)
0.621000 + 0.783811i \(0.286727\pi\)
\(608\) 0 0
\(609\) 0.310701 0.0125902
\(610\) 0 0
\(611\) 0.851646 0.0344539
\(612\) 0 0
\(613\) 41.2528 1.66618 0.833092 0.553134i \(-0.186568\pi\)
0.833092 + 0.553134i \(0.186568\pi\)
\(614\) 0 0
\(615\) 13.8130 0.556993
\(616\) 0 0
\(617\) −0.809268 −0.0325799 −0.0162899 0.999867i \(-0.505185\pi\)
−0.0162899 + 0.999867i \(0.505185\pi\)
\(618\) 0 0
\(619\) 31.2725 1.25695 0.628474 0.777831i \(-0.283680\pi\)
0.628474 + 0.777831i \(0.283680\pi\)
\(620\) 0 0
\(621\) 12.4673 0.500296
\(622\) 0 0
\(623\) 0.385699 0.0154527
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −16.1316 −0.644233
\(628\) 0 0
\(629\) −30.0538 −1.19832
\(630\) 0 0
\(631\) 1.03218 0.0410906 0.0205453 0.999789i \(-0.493460\pi\)
0.0205453 + 0.999789i \(0.493460\pi\)
\(632\) 0 0
\(633\) −21.3341 −0.847955
\(634\) 0 0
\(635\) −0.979099 −0.0388544
\(636\) 0 0
\(637\) −27.3461 −1.08349
\(638\) 0 0
\(639\) 2.96058 0.117119
\(640\) 0 0
\(641\) 25.9666 1.02562 0.512809 0.858503i \(-0.328605\pi\)
0.512809 + 0.858503i \(0.328605\pi\)
\(642\) 0 0
\(643\) −47.5143 −1.87378 −0.936890 0.349624i \(-0.886309\pi\)
−0.936890 + 0.349624i \(0.886309\pi\)
\(644\) 0 0
\(645\) −0.542399 −0.0213569
\(646\) 0 0
\(647\) −29.0205 −1.14092 −0.570458 0.821327i \(-0.693234\pi\)
−0.570458 + 0.821327i \(0.693234\pi\)
\(648\) 0 0
\(649\) −3.18554 −0.125043
\(650\) 0 0
\(651\) 0.596245 0.0233687
\(652\) 0 0
\(653\) −43.1424 −1.68829 −0.844146 0.536113i \(-0.819892\pi\)
−0.844146 + 0.536113i \(0.819892\pi\)
\(654\) 0 0
\(655\) −7.11612 −0.278050
\(656\) 0 0
\(657\) 20.1586 0.786462
\(658\) 0 0
\(659\) −12.3354 −0.480521 −0.240260 0.970708i \(-0.577233\pi\)
−0.240260 + 0.970708i \(0.577233\pi\)
\(660\) 0 0
\(661\) 15.0098 0.583813 0.291907 0.956447i \(-0.405710\pi\)
0.291907 + 0.956447i \(0.405710\pi\)
\(662\) 0 0
\(663\) 15.6079 0.606162
\(664\) 0 0
\(665\) −0.396422 −0.0153726
\(666\) 0 0
\(667\) −11.2898 −0.437143
\(668\) 0 0
\(669\) −17.5762 −0.679533
\(670\) 0 0
\(671\) −0.111172 −0.00429175
\(672\) 0 0
\(673\) 20.2319 0.779883 0.389942 0.920840i \(-0.372495\pi\)
0.389942 + 0.920840i \(0.372495\pi\)
\(674\) 0 0
\(675\) 5.59224 0.215245
\(676\) 0 0
\(677\) −25.2715 −0.971263 −0.485631 0.874164i \(-0.661410\pi\)
−0.485631 + 0.874164i \(0.661410\pi\)
\(678\) 0 0
\(679\) −0.634851 −0.0243633
\(680\) 0 0
\(681\) −14.5283 −0.556726
\(682\) 0 0
\(683\) −3.90956 −0.149595 −0.0747976 0.997199i \(-0.523831\pi\)
−0.0747976 + 0.997199i \(0.523831\pi\)
\(684\) 0 0
\(685\) 20.3929 0.779174
\(686\) 0 0
\(687\) −14.1925 −0.541476
\(688\) 0 0
\(689\) −3.43476 −0.130854
\(690\) 0 0
\(691\) 43.7344 1.66373 0.831867 0.554974i \(-0.187272\pi\)
0.831867 + 0.554974i \(0.187272\pi\)
\(692\) 0 0
\(693\) −0.0957555 −0.00363745
\(694\) 0 0
\(695\) −5.04716 −0.191450
\(696\) 0 0
\(697\) −33.2068 −1.25780
\(698\) 0 0
\(699\) −20.9775 −0.793440
\(700\) 0 0
\(701\) 37.3707 1.41147 0.705735 0.708476i \(-0.250617\pi\)
0.705735 + 0.708476i \(0.250617\pi\)
\(702\) 0 0
\(703\) 80.7745 3.04647
\(704\) 0 0
\(705\) 0.280902 0.0105794
\(706\) 0 0
\(707\) 0.111846 0.00420638
\(708\) 0 0
\(709\) −27.6858 −1.03976 −0.519882 0.854238i \(-0.674024\pi\)
−0.519882 + 0.854238i \(0.674024\pi\)
\(710\) 0 0
\(711\) −3.71959 −0.139496
\(712\) 0 0
\(713\) −21.6655 −0.811378
\(714\) 0 0
\(715\) −5.87263 −0.219624
\(716\) 0 0
\(717\) −15.5658 −0.581317
\(718\) 0 0
\(719\) −22.6700 −0.845450 −0.422725 0.906258i \(-0.638926\pi\)
−0.422725 + 0.906258i \(0.638926\pi\)
\(720\) 0 0
\(721\) −0.157576 −0.00586844
\(722\) 0 0
\(723\) −19.2916 −0.717461
\(724\) 0 0
\(725\) −5.06407 −0.188075
\(726\) 0 0
\(727\) 39.3281 1.45860 0.729299 0.684196i \(-0.239847\pi\)
0.729299 + 0.684196i \(0.239847\pi\)
\(728\) 0 0
\(729\) 25.8974 0.959163
\(730\) 0 0
\(731\) 1.30394 0.0482281
\(732\) 0 0
\(733\) 42.9086 1.58487 0.792433 0.609959i \(-0.208814\pi\)
0.792433 + 0.609959i \(0.208814\pi\)
\(734\) 0 0
\(735\) −9.01968 −0.332696
\(736\) 0 0
\(737\) 0.0233512 0.000860153 0
\(738\) 0 0
\(739\) 9.75612 0.358885 0.179442 0.983768i \(-0.442571\pi\)
0.179442 + 0.983768i \(0.442571\pi\)
\(740\) 0 0
\(741\) −41.9488 −1.54103
\(742\) 0 0
\(743\) 36.7509 1.34826 0.674130 0.738613i \(-0.264519\pi\)
0.674130 + 0.738613i \(0.264519\pi\)
\(744\) 0 0
\(745\) −22.6839 −0.831073
\(746\) 0 0
\(747\) 1.61077 0.0589348
\(748\) 0 0
\(749\) 0.908244 0.0331865
\(750\) 0 0
\(751\) 44.7414 1.63264 0.816318 0.577602i \(-0.196011\pi\)
0.816318 + 0.577602i \(0.196011\pi\)
\(752\) 0 0
\(753\) −39.1782 −1.42773
\(754\) 0 0
\(755\) −4.57148 −0.166373
\(756\) 0 0
\(757\) −23.4818 −0.853461 −0.426730 0.904379i \(-0.640335\pi\)
−0.426730 + 0.904379i \(0.640335\pi\)
\(758\) 0 0
\(759\) −4.31832 −0.156745
\(760\) 0 0
\(761\) −26.3747 −0.956082 −0.478041 0.878337i \(-0.658653\pi\)
−0.478041 + 0.878337i \(0.658653\pi\)
\(762\) 0 0
\(763\) −0.0809361 −0.00293008
\(764\) 0 0
\(765\) −4.14795 −0.149969
\(766\) 0 0
\(767\) −8.28373 −0.299108
\(768\) 0 0
\(769\) −43.4834 −1.56805 −0.784025 0.620729i \(-0.786837\pi\)
−0.784025 + 0.620729i \(0.786837\pi\)
\(770\) 0 0
\(771\) −21.3489 −0.768863
\(772\) 0 0
\(773\) 35.2137 1.26655 0.633275 0.773927i \(-0.281710\pi\)
0.633275 + 0.773927i \(0.281710\pi\)
\(774\) 0 0
\(775\) −9.71810 −0.349084
\(776\) 0 0
\(777\) −0.595072 −0.0213481
\(778\) 0 0
\(779\) 89.2488 3.19767
\(780\) 0 0
\(781\) −3.32362 −0.118929
\(782\) 0 0
\(783\) −28.3195 −1.01206
\(784\) 0 0
\(785\) −5.88672 −0.210106
\(786\) 0 0
\(787\) −37.3793 −1.33243 −0.666215 0.745760i \(-0.732087\pi\)
−0.666215 + 0.745760i \(0.732087\pi\)
\(788\) 0 0
\(789\) 15.2333 0.542318
\(790\) 0 0
\(791\) 0.431018 0.0153252
\(792\) 0 0
\(793\) −0.289094 −0.0102660
\(794\) 0 0
\(795\) −1.13290 −0.0401799
\(796\) 0 0
\(797\) −29.6706 −1.05099 −0.525493 0.850798i \(-0.676119\pi\)
−0.525493 + 0.850798i \(0.676119\pi\)
\(798\) 0 0
\(799\) −0.675297 −0.0238903
\(800\) 0 0
\(801\) −10.8467 −0.383250
\(802\) 0 0
\(803\) −22.6306 −0.798616
\(804\) 0 0
\(805\) −0.106120 −0.00374023
\(806\) 0 0
\(807\) −27.3190 −0.961674
\(808\) 0 0
\(809\) −54.8870 −1.92972 −0.964861 0.262760i \(-0.915367\pi\)
−0.964861 + 0.262760i \(0.915367\pi\)
\(810\) 0 0
\(811\) 23.8967 0.839125 0.419563 0.907726i \(-0.362183\pi\)
0.419563 + 0.907726i \(0.362183\pi\)
\(812\) 0 0
\(813\) −16.0681 −0.563532
\(814\) 0 0
\(815\) 10.1654 0.356079
\(816\) 0 0
\(817\) −3.50456 −0.122609
\(818\) 0 0
\(819\) −0.249004 −0.00870092
\(820\) 0 0
\(821\) 19.5078 0.680826 0.340413 0.940276i \(-0.389433\pi\)
0.340413 + 0.940276i \(0.389433\pi\)
\(822\) 0 0
\(823\) −3.42346 −0.119334 −0.0596672 0.998218i \(-0.519004\pi\)
−0.0596672 + 0.998218i \(0.519004\pi\)
\(824\) 0 0
\(825\) −1.93699 −0.0674375
\(826\) 0 0
\(827\) 3.17349 0.110353 0.0551766 0.998477i \(-0.482428\pi\)
0.0551766 + 0.998477i \(0.482428\pi\)
\(828\) 0 0
\(829\) −6.92777 −0.240612 −0.120306 0.992737i \(-0.538388\pi\)
−0.120306 + 0.992737i \(0.538388\pi\)
\(830\) 0 0
\(831\) −8.15554 −0.282913
\(832\) 0 0
\(833\) 21.6836 0.751292
\(834\) 0 0
\(835\) 6.54352 0.226448
\(836\) 0 0
\(837\) −54.3459 −1.87847
\(838\) 0 0
\(839\) −21.4946 −0.742077 −0.371039 0.928617i \(-0.620998\pi\)
−0.371039 + 0.928617i \(0.620998\pi\)
\(840\) 0 0
\(841\) −3.35524 −0.115698
\(842\) 0 0
\(843\) −28.9040 −0.995507
\(844\) 0 0
\(845\) −2.27129 −0.0781348
\(846\) 0 0
\(847\) −0.416106 −0.0142976
\(848\) 0 0
\(849\) 34.9988 1.20116
\(850\) 0 0
\(851\) 21.6228 0.741222
\(852\) 0 0
\(853\) 9.72028 0.332816 0.166408 0.986057i \(-0.446783\pi\)
0.166408 + 0.986057i \(0.446783\pi\)
\(854\) 0 0
\(855\) 11.1483 0.381263
\(856\) 0 0
\(857\) 13.1493 0.449172 0.224586 0.974454i \(-0.427897\pi\)
0.224586 + 0.974454i \(0.427897\pi\)
\(858\) 0 0
\(859\) 32.6717 1.11474 0.557372 0.830263i \(-0.311810\pi\)
0.557372 + 0.830263i \(0.311810\pi\)
\(860\) 0 0
\(861\) −0.657503 −0.0224076
\(862\) 0 0
\(863\) −41.0210 −1.39637 −0.698186 0.715917i \(-0.746009\pi\)
−0.698186 + 0.715917i \(0.746009\pi\)
\(864\) 0 0
\(865\) −11.8658 −0.403449
\(866\) 0 0
\(867\) 9.53601 0.323860
\(868\) 0 0
\(869\) 4.17571 0.141651
\(870\) 0 0
\(871\) 0.0607229 0.00205752
\(872\) 0 0
\(873\) 17.8534 0.604246
\(874\) 0 0
\(875\) −0.0476003 −0.00160918
\(876\) 0 0
\(877\) 2.21474 0.0747864 0.0373932 0.999301i \(-0.488095\pi\)
0.0373932 + 0.999301i \(0.488095\pi\)
\(878\) 0 0
\(879\) −18.5917 −0.627084
\(880\) 0 0
\(881\) 26.5186 0.893435 0.446717 0.894675i \(-0.352593\pi\)
0.446717 + 0.894675i \(0.352593\pi\)
\(882\) 0 0
\(883\) −16.5822 −0.558035 −0.279017 0.960286i \(-0.590009\pi\)
−0.279017 + 0.960286i \(0.590009\pi\)
\(884\) 0 0
\(885\) −2.73226 −0.0918438
\(886\) 0 0
\(887\) −51.6449 −1.73407 −0.867033 0.498250i \(-0.833976\pi\)
−0.867033 + 0.498250i \(0.833976\pi\)
\(888\) 0 0
\(889\) 0.0466055 0.00156310
\(890\) 0 0
\(891\) −4.79717 −0.160711
\(892\) 0 0
\(893\) 1.81497 0.0607356
\(894\) 0 0
\(895\) 1.47248 0.0492197
\(896\) 0 0
\(897\) −11.2294 −0.374940
\(898\) 0 0
\(899\) 49.2131 1.64135
\(900\) 0 0
\(901\) 2.72353 0.0907339
\(902\) 0 0
\(903\) 0.0258184 0.000859182 0
\(904\) 0 0
\(905\) 10.0925 0.335485
\(906\) 0 0
\(907\) 48.3896 1.60675 0.803376 0.595472i \(-0.203035\pi\)
0.803376 + 0.595472i \(0.203035\pi\)
\(908\) 0 0
\(909\) −3.14534 −0.104324
\(910\) 0 0
\(911\) −31.9760 −1.05941 −0.529705 0.848182i \(-0.677698\pi\)
−0.529705 + 0.848182i \(0.677698\pi\)
\(912\) 0 0
\(913\) −1.80829 −0.0598456
\(914\) 0 0
\(915\) −0.0953530 −0.00315227
\(916\) 0 0
\(917\) 0.338730 0.0111858
\(918\) 0 0
\(919\) 40.9970 1.35237 0.676183 0.736734i \(-0.263633\pi\)
0.676183 + 0.736734i \(0.263633\pi\)
\(920\) 0 0
\(921\) 24.4096 0.804322
\(922\) 0 0
\(923\) −8.64281 −0.284482
\(924\) 0 0
\(925\) 9.69898 0.318900
\(926\) 0 0
\(927\) 4.43139 0.145546
\(928\) 0 0
\(929\) 35.0422 1.14970 0.574849 0.818260i \(-0.305061\pi\)
0.574849 + 0.818260i \(0.305061\pi\)
\(930\) 0 0
\(931\) −58.2781 −1.90999
\(932\) 0 0
\(933\) −21.8686 −0.715944
\(934\) 0 0
\(935\) 4.65659 0.152287
\(936\) 0 0
\(937\) 1.40509 0.0459024 0.0229512 0.999737i \(-0.492694\pi\)
0.0229512 + 0.999737i \(0.492694\pi\)
\(938\) 0 0
\(939\) 12.8179 0.418297
\(940\) 0 0
\(941\) 37.5816 1.22512 0.612562 0.790422i \(-0.290139\pi\)
0.612562 + 0.790422i \(0.290139\pi\)
\(942\) 0 0
\(943\) 23.8914 0.778010
\(944\) 0 0
\(945\) −0.266192 −0.00865924
\(946\) 0 0
\(947\) −37.3815 −1.21473 −0.607367 0.794421i \(-0.707774\pi\)
−0.607367 + 0.794421i \(0.707774\pi\)
\(948\) 0 0
\(949\) −58.8490 −1.91032
\(950\) 0 0
\(951\) −45.5169 −1.47599
\(952\) 0 0
\(953\) 36.2683 1.17485 0.587423 0.809280i \(-0.300142\pi\)
0.587423 + 0.809280i \(0.300142\pi\)
\(954\) 0 0
\(955\) −13.6202 −0.440741
\(956\) 0 0
\(957\) 9.80906 0.317082
\(958\) 0 0
\(959\) −0.970711 −0.0313459
\(960\) 0 0
\(961\) 63.4414 2.04650
\(962\) 0 0
\(963\) −25.5418 −0.823074
\(964\) 0 0
\(965\) 9.10873 0.293220
\(966\) 0 0
\(967\) −8.93926 −0.287467 −0.143734 0.989616i \(-0.545911\pi\)
−0.143734 + 0.989616i \(0.545911\pi\)
\(968\) 0 0
\(969\) 33.2625 1.06855
\(970\) 0 0
\(971\) 3.01371 0.0967147 0.0483574 0.998830i \(-0.484601\pi\)
0.0483574 + 0.998830i \(0.484601\pi\)
\(972\) 0 0
\(973\) 0.240247 0.00770195
\(974\) 0 0
\(975\) −5.03700 −0.161313
\(976\) 0 0
\(977\) 10.0313 0.320930 0.160465 0.987042i \(-0.448701\pi\)
0.160465 + 0.987042i \(0.448701\pi\)
\(978\) 0 0
\(979\) 12.1768 0.389172
\(980\) 0 0
\(981\) 2.27610 0.0726703
\(982\) 0 0
\(983\) 44.0557 1.40516 0.702579 0.711606i \(-0.252032\pi\)
0.702579 + 0.711606i \(0.252032\pi\)
\(984\) 0 0
\(985\) 17.0669 0.543797
\(986\) 0 0
\(987\) −0.0133710 −0.000425604 0
\(988\) 0 0
\(989\) −0.938149 −0.0298314
\(990\) 0 0
\(991\) 41.7710 1.32690 0.663450 0.748220i \(-0.269091\pi\)
0.663450 + 0.748220i \(0.269091\pi\)
\(992\) 0 0
\(993\) 45.8084 1.45368
\(994\) 0 0
\(995\) 24.1476 0.765532
\(996\) 0 0
\(997\) 46.6004 1.47585 0.737925 0.674882i \(-0.235806\pi\)
0.737925 + 0.674882i \(0.235806\pi\)
\(998\) 0 0
\(999\) 54.2390 1.71605
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.1.13 35
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.e.1.13 35 1.1 even 1 trivial