Properties

Label 4020.2.a.b.1.3
Level $4020$
Weight $2$
Character 4020.1
Self dual yes
Analytic conductor $32.100$
Analytic rank $1$
Dimension $4$
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] = [4020,2,Mod(1,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.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: \(32.0998616126\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.9301.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.68863\) of defining polynomial
Character \(\chi\) \(=\) 4020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -0.148523 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -0.148523 q^{7} +1.00000 q^{9} -0.688632 q^{11} +0.0605682 q^{13} +1.00000 q^{15} +0.628064 q^{17} -0.223414 q^{19} +0.148523 q^{21} +1.29705 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.675569 q^{29} -4.47954 q^{31} +0.688632 q^{33} +0.148523 q^{35} -0.824091 q^{37} -0.0605682 q^{39} -2.75046 q^{41} +3.23648 q^{43} -1.00000 q^{45} +6.96488 q^{47} -6.97794 q^{49} -0.628064 q^{51} -6.23407 q^{53} +0.688632 q^{55} +0.223414 q^{57} +11.7742 q^{59} +7.58521 q^{61} -0.148523 q^{63} -0.0605682 q^{65} +1.00000 q^{67} -1.29705 q^{69} +10.7669 q^{71} -12.2869 q^{73} -1.00000 q^{75} +0.102277 q^{77} -9.34214 q^{79} +1.00000 q^{81} +1.63580 q^{83} -0.628064 q^{85} -0.675569 q^{87} -11.9174 q^{89} -0.00899575 q^{91} +4.47954 q^{93} +0.223414 q^{95} -4.01432 q^{97} -0.688632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} + 5 q^{11} - q^{13} + 4 q^{15} - 4 q^{17} - 7 q^{19} + q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - q^{29} - 11 q^{31} - 5 q^{33} + q^{35} + q^{39} - 13 q^{41} + 15 q^{43} - 4 q^{45} + 7 q^{47} - 3 q^{49} + 4 q^{51} + 13 q^{53} - 5 q^{55} + 7 q^{57} + q^{59} - 15 q^{61} - q^{63} + q^{65} + 4 q^{67} - 6 q^{69} - 6 q^{71} + 8 q^{73} - 4 q^{75} + 9 q^{77} - q^{79} + 4 q^{81} + 18 q^{83} + 4 q^{85} + q^{87} - 24 q^{89} - 29 q^{91} + 11 q^{93} + 7 q^{95} - 23 q^{97} + 5 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.148523 −0.0561363 −0.0280681 0.999606i \(-0.508936\pi\)
−0.0280681 + 0.999606i \(0.508936\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.688632 −0.207630 −0.103815 0.994597i \(-0.533105\pi\)
−0.103815 + 0.994597i \(0.533105\pi\)
\(12\) 0 0
\(13\) 0.0605682 0.0167986 0.00839930 0.999965i \(-0.497326\pi\)
0.00839930 + 0.999965i \(0.497326\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.628064 0.152328 0.0761639 0.997095i \(-0.475733\pi\)
0.0761639 + 0.997095i \(0.475733\pi\)
\(18\) 0 0
\(19\) −0.223414 −0.0512546 −0.0256273 0.999672i \(-0.508158\pi\)
−0.0256273 + 0.999672i \(0.508158\pi\)
\(20\) 0 0
\(21\) 0.148523 0.0324103
\(22\) 0 0
\(23\) 1.29705 0.270453 0.135226 0.990815i \(-0.456824\pi\)
0.135226 + 0.990815i \(0.456824\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.675569 0.125450 0.0627250 0.998031i \(-0.480021\pi\)
0.0627250 + 0.998031i \(0.480021\pi\)
\(30\) 0 0
\(31\) −4.47954 −0.804549 −0.402275 0.915519i \(-0.631780\pi\)
−0.402275 + 0.915519i \(0.631780\pi\)
\(32\) 0 0
\(33\) 0.688632 0.119875
\(34\) 0 0
\(35\) 0.148523 0.0251049
\(36\) 0 0
\(37\) −0.824091 −0.135480 −0.0677399 0.997703i \(-0.521579\pi\)
−0.0677399 + 0.997703i \(0.521579\pi\)
\(38\) 0 0
\(39\) −0.0605682 −0.00969867
\(40\) 0 0
\(41\) −2.75046 −0.429550 −0.214775 0.976664i \(-0.568902\pi\)
−0.214775 + 0.976664i \(0.568902\pi\)
\(42\) 0 0
\(43\) 3.23648 0.493558 0.246779 0.969072i \(-0.420628\pi\)
0.246779 + 0.969072i \(0.420628\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.96488 1.01593 0.507966 0.861377i \(-0.330398\pi\)
0.507966 + 0.861377i \(0.330398\pi\)
\(48\) 0 0
\(49\) −6.97794 −0.996849
\(50\) 0 0
\(51\) −0.628064 −0.0879465
\(52\) 0 0
\(53\) −6.23407 −0.856315 −0.428158 0.903704i \(-0.640837\pi\)
−0.428158 + 0.903704i \(0.640837\pi\)
\(54\) 0 0
\(55\) 0.688632 0.0928551
\(56\) 0 0
\(57\) 0.223414 0.0295919
\(58\) 0 0
\(59\) 11.7742 1.53287 0.766434 0.642323i \(-0.222029\pi\)
0.766434 + 0.642323i \(0.222029\pi\)
\(60\) 0 0
\(61\) 7.58521 0.971186 0.485593 0.874185i \(-0.338604\pi\)
0.485593 + 0.874185i \(0.338604\pi\)
\(62\) 0 0
\(63\) −0.148523 −0.0187121
\(64\) 0 0
\(65\) −0.0605682 −0.00751256
\(66\) 0 0
\(67\) 1.00000 0.122169
\(68\) 0 0
\(69\) −1.29705 −0.156146
\(70\) 0 0
\(71\) 10.7669 1.27780 0.638899 0.769291i \(-0.279390\pi\)
0.638899 + 0.769291i \(0.279390\pi\)
\(72\) 0 0
\(73\) −12.2869 −1.43807 −0.719036 0.694972i \(-0.755417\pi\)
−0.719036 + 0.694972i \(0.755417\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0.102277 0.0116556
\(78\) 0 0
\(79\) −9.34214 −1.05107 −0.525537 0.850771i \(-0.676135\pi\)
−0.525537 + 0.850771i \(0.676135\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.63580 0.179552 0.0897762 0.995962i \(-0.471385\pi\)
0.0897762 + 0.995962i \(0.471385\pi\)
\(84\) 0 0
\(85\) −0.628064 −0.0681231
\(86\) 0 0
\(87\) −0.675569 −0.0724286
\(88\) 0 0
\(89\) −11.9174 −1.26324 −0.631619 0.775279i \(-0.717609\pi\)
−0.631619 + 0.775279i \(0.717609\pi\)
\(90\) 0 0
\(91\) −0.00899575 −0.000943010 0
\(92\) 0 0
\(93\) 4.47954 0.464507
\(94\) 0 0
\(95\) 0.223414 0.0229218
\(96\) 0 0
\(97\) −4.01432 −0.407593 −0.203796 0.979013i \(-0.565328\pi\)
−0.203796 + 0.979013i \(0.565328\pi\)
\(98\) 0 0
\(99\) −0.688632 −0.0692101
\(100\) 0 0
\(101\) −17.1057 −1.70208 −0.851039 0.525103i \(-0.824027\pi\)
−0.851039 + 0.525103i \(0.824027\pi\)
\(102\) 0 0
\(103\) −3.23648 −0.318900 −0.159450 0.987206i \(-0.550972\pi\)
−0.159450 + 0.987206i \(0.550972\pi\)
\(104\) 0 0
\(105\) −0.148523 −0.0144943
\(106\) 0 0
\(107\) −17.7313 −1.71415 −0.857076 0.515191i \(-0.827721\pi\)
−0.857076 + 0.515191i \(0.827721\pi\)
\(108\) 0 0
\(109\) 8.37485 0.802166 0.401083 0.916042i \(-0.368634\pi\)
0.401083 + 0.916042i \(0.368634\pi\)
\(110\) 0 0
\(111\) 0.824091 0.0782193
\(112\) 0 0
\(113\) 18.2935 1.72091 0.860453 0.509529i \(-0.170180\pi\)
0.860453 + 0.509529i \(0.170180\pi\)
\(114\) 0 0
\(115\) −1.29705 −0.120950
\(116\) 0 0
\(117\) 0.0605682 0.00559953
\(118\) 0 0
\(119\) −0.0932816 −0.00855111
\(120\) 0 0
\(121\) −10.5258 −0.956890
\(122\) 0 0
\(123\) 2.75046 0.248001
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.88660 −0.167409 −0.0837043 0.996491i \(-0.526675\pi\)
−0.0837043 + 0.996491i \(0.526675\pi\)
\(128\) 0 0
\(129\) −3.23648 −0.284956
\(130\) 0 0
\(131\) −16.7051 −1.45953 −0.729765 0.683698i \(-0.760370\pi\)
−0.729765 + 0.683698i \(0.760370\pi\)
\(132\) 0 0
\(133\) 0.0331820 0.00287724
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 22.5881 1.92983 0.964917 0.262556i \(-0.0845654\pi\)
0.964917 + 0.262556i \(0.0845654\pi\)
\(138\) 0 0
\(139\) −5.70170 −0.483611 −0.241806 0.970325i \(-0.577740\pi\)
−0.241806 + 0.970325i \(0.577740\pi\)
\(140\) 0 0
\(141\) −6.96488 −0.586549
\(142\) 0 0
\(143\) −0.0417092 −0.00348790
\(144\) 0 0
\(145\) −0.675569 −0.0561029
\(146\) 0 0
\(147\) 6.97794 0.575531
\(148\) 0 0
\(149\) −3.62680 −0.297119 −0.148560 0.988903i \(-0.547464\pi\)
−0.148560 + 0.988903i \(0.547464\pi\)
\(150\) 0 0
\(151\) −21.7267 −1.76809 −0.884046 0.467400i \(-0.845191\pi\)
−0.884046 + 0.467400i \(0.845191\pi\)
\(152\) 0 0
\(153\) 0.628064 0.0507759
\(154\) 0 0
\(155\) 4.47954 0.359805
\(156\) 0 0
\(157\) −5.12240 −0.408812 −0.204406 0.978886i \(-0.565526\pi\)
−0.204406 + 0.978886i \(0.565526\pi\)
\(158\) 0 0
\(159\) 6.23407 0.494394
\(160\) 0 0
\(161\) −0.192641 −0.0151822
\(162\) 0 0
\(163\) 8.77292 0.687148 0.343574 0.939126i \(-0.388362\pi\)
0.343574 + 0.939126i \(0.388362\pi\)
\(164\) 0 0
\(165\) −0.688632 −0.0536099
\(166\) 0 0
\(167\) −8.01771 −0.620429 −0.310215 0.950667i \(-0.600401\pi\)
−0.310215 + 0.950667i \(0.600401\pi\)
\(168\) 0 0
\(169\) −12.9963 −0.999718
\(170\) 0 0
\(171\) −0.223414 −0.0170849
\(172\) 0 0
\(173\) 7.40930 0.563318 0.281659 0.959515i \(-0.409115\pi\)
0.281659 + 0.959515i \(0.409115\pi\)
\(174\) 0 0
\(175\) −0.148523 −0.0112273
\(176\) 0 0
\(177\) −11.7742 −0.885001
\(178\) 0 0
\(179\) 10.7676 0.804807 0.402404 0.915462i \(-0.368175\pi\)
0.402404 + 0.915462i \(0.368175\pi\)
\(180\) 0 0
\(181\) −23.7403 −1.76460 −0.882302 0.470684i \(-0.844007\pi\)
−0.882302 + 0.470684i \(0.844007\pi\)
\(182\) 0 0
\(183\) −7.58521 −0.560715
\(184\) 0 0
\(185\) 0.824091 0.0605884
\(186\) 0 0
\(187\) −0.432505 −0.0316279
\(188\) 0 0
\(189\) 0.148523 0.0108034
\(190\) 0 0
\(191\) −9.00465 −0.651553 −0.325777 0.945447i \(-0.605626\pi\)
−0.325777 + 0.945447i \(0.605626\pi\)
\(192\) 0 0
\(193\) −5.24180 −0.377313 −0.188657 0.982043i \(-0.560413\pi\)
−0.188657 + 0.982043i \(0.560413\pi\)
\(194\) 0 0
\(195\) 0.0605682 0.00433738
\(196\) 0 0
\(197\) −5.74020 −0.408973 −0.204486 0.978869i \(-0.565552\pi\)
−0.204486 + 0.978869i \(0.565552\pi\)
\(198\) 0 0
\(199\) −5.09695 −0.361313 −0.180657 0.983546i \(-0.557822\pi\)
−0.180657 + 0.983546i \(0.557822\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) 0 0
\(203\) −0.100337 −0.00704229
\(204\) 0 0
\(205\) 2.75046 0.192100
\(206\) 0 0
\(207\) 1.29705 0.0901509
\(208\) 0 0
\(209\) 0.153850 0.0106420
\(210\) 0 0
\(211\) 9.60486 0.661226 0.330613 0.943766i \(-0.392745\pi\)
0.330613 + 0.943766i \(0.392745\pi\)
\(212\) 0 0
\(213\) −10.7669 −0.737737
\(214\) 0 0
\(215\) −3.23648 −0.220726
\(216\) 0 0
\(217\) 0.665313 0.0451644
\(218\) 0 0
\(219\) 12.2869 0.830272
\(220\) 0 0
\(221\) 0.0380407 0.00255889
\(222\) 0 0
\(223\) 0.314287 0.0210462 0.0105231 0.999945i \(-0.496650\pi\)
0.0105231 + 0.999945i \(0.496650\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0.298305 0.0197992 0.00989960 0.999951i \(-0.496849\pi\)
0.00989960 + 0.999951i \(0.496849\pi\)
\(228\) 0 0
\(229\) −0.150931 −0.00997382 −0.00498691 0.999988i \(-0.501587\pi\)
−0.00498691 + 0.999988i \(0.501587\pi\)
\(230\) 0 0
\(231\) −0.102277 −0.00672936
\(232\) 0 0
\(233\) −26.1762 −1.71486 −0.857430 0.514600i \(-0.827940\pi\)
−0.857430 + 0.514600i \(0.827940\pi\)
\(234\) 0 0
\(235\) −6.96488 −0.454339
\(236\) 0 0
\(237\) 9.34214 0.606837
\(238\) 0 0
\(239\) −15.2727 −0.987908 −0.493954 0.869488i \(-0.664449\pi\)
−0.493954 + 0.869488i \(0.664449\pi\)
\(240\) 0 0
\(241\) −6.37388 −0.410577 −0.205289 0.978701i \(-0.565813\pi\)
−0.205289 + 0.978701i \(0.565813\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.97794 0.445804
\(246\) 0 0
\(247\) −0.0135318 −0.000861006 0
\(248\) 0 0
\(249\) −1.63580 −0.103665
\(250\) 0 0
\(251\) 22.9065 1.44585 0.722924 0.690927i \(-0.242798\pi\)
0.722924 + 0.690927i \(0.242798\pi\)
\(252\) 0 0
\(253\) −0.893187 −0.0561542
\(254\) 0 0
\(255\) 0.628064 0.0393309
\(256\) 0 0
\(257\) −17.6334 −1.09994 −0.549970 0.835184i \(-0.685361\pi\)
−0.549970 + 0.835184i \(0.685361\pi\)
\(258\) 0 0
\(259\) 0.122396 0.00760533
\(260\) 0 0
\(261\) 0.675569 0.0418166
\(262\) 0 0
\(263\) −1.12840 −0.0695804 −0.0347902 0.999395i \(-0.511076\pi\)
−0.0347902 + 0.999395i \(0.511076\pi\)
\(264\) 0 0
\(265\) 6.23407 0.382956
\(266\) 0 0
\(267\) 11.9174 0.729331
\(268\) 0 0
\(269\) 0.450895 0.0274916 0.0137458 0.999906i \(-0.495624\pi\)
0.0137458 + 0.999906i \(0.495624\pi\)
\(270\) 0 0
\(271\) −27.0615 −1.64387 −0.821936 0.569580i \(-0.807106\pi\)
−0.821936 + 0.569580i \(0.807106\pi\)
\(272\) 0 0
\(273\) 0.00899575 0.000544447 0
\(274\) 0 0
\(275\) −0.688632 −0.0415261
\(276\) 0 0
\(277\) −2.45295 −0.147383 −0.0736916 0.997281i \(-0.523478\pi\)
−0.0736916 + 0.997281i \(0.523478\pi\)
\(278\) 0 0
\(279\) −4.47954 −0.268183
\(280\) 0 0
\(281\) −31.1442 −1.85791 −0.928953 0.370198i \(-0.879290\pi\)
−0.928953 + 0.370198i \(0.879290\pi\)
\(282\) 0 0
\(283\) 11.9775 0.711987 0.355993 0.934488i \(-0.384143\pi\)
0.355993 + 0.934488i \(0.384143\pi\)
\(284\) 0 0
\(285\) −0.223414 −0.0132339
\(286\) 0 0
\(287\) 0.408505 0.0241133
\(288\) 0 0
\(289\) −16.6055 −0.976796
\(290\) 0 0
\(291\) 4.01432 0.235324
\(292\) 0 0
\(293\) 7.02951 0.410669 0.205334 0.978692i \(-0.434172\pi\)
0.205334 + 0.978692i \(0.434172\pi\)
\(294\) 0 0
\(295\) −11.7742 −0.685519
\(296\) 0 0
\(297\) 0.688632 0.0399585
\(298\) 0 0
\(299\) 0.0785597 0.00454322
\(300\) 0 0
\(301\) −0.480690 −0.0277065
\(302\) 0 0
\(303\) 17.1057 0.982695
\(304\) 0 0
\(305\) −7.58521 −0.434328
\(306\) 0 0
\(307\) −23.5548 −1.34435 −0.672173 0.740394i \(-0.734639\pi\)
−0.672173 + 0.740394i \(0.734639\pi\)
\(308\) 0 0
\(309\) 3.23648 0.184117
\(310\) 0 0
\(311\) 22.5721 1.27995 0.639974 0.768396i \(-0.278945\pi\)
0.639974 + 0.768396i \(0.278945\pi\)
\(312\) 0 0
\(313\) 8.70702 0.492150 0.246075 0.969251i \(-0.420859\pi\)
0.246075 + 0.969251i \(0.420859\pi\)
\(314\) 0 0
\(315\) 0.148523 0.00836830
\(316\) 0 0
\(317\) −12.0278 −0.675548 −0.337774 0.941227i \(-0.609674\pi\)
−0.337774 + 0.941227i \(0.609674\pi\)
\(318\) 0 0
\(319\) −0.465218 −0.0260472
\(320\) 0 0
\(321\) 17.7313 0.989666
\(322\) 0 0
\(323\) −0.140318 −0.00780751
\(324\) 0 0
\(325\) 0.0605682 0.00335972
\(326\) 0 0
\(327\) −8.37485 −0.463131
\(328\) 0 0
\(329\) −1.03444 −0.0570306
\(330\) 0 0
\(331\) −6.08022 −0.334199 −0.167100 0.985940i \(-0.553440\pi\)
−0.167100 + 0.985940i \(0.553440\pi\)
\(332\) 0 0
\(333\) −0.824091 −0.0451599
\(334\) 0 0
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) −14.5651 −0.793411 −0.396705 0.917946i \(-0.629847\pi\)
−0.396705 + 0.917946i \(0.629847\pi\)
\(338\) 0 0
\(339\) −18.2935 −0.993566
\(340\) 0 0
\(341\) 3.08475 0.167049
\(342\) 0 0
\(343\) 2.07604 0.112096
\(344\) 0 0
\(345\) 1.29705 0.0698306
\(346\) 0 0
\(347\) 18.5925 0.998096 0.499048 0.866574i \(-0.333683\pi\)
0.499048 + 0.866574i \(0.333683\pi\)
\(348\) 0 0
\(349\) −23.1294 −1.23809 −0.619044 0.785357i \(-0.712480\pi\)
−0.619044 + 0.785357i \(0.712480\pi\)
\(350\) 0 0
\(351\) −0.0605682 −0.00323289
\(352\) 0 0
\(353\) −29.5152 −1.57093 −0.785467 0.618903i \(-0.787577\pi\)
−0.785467 + 0.618903i \(0.787577\pi\)
\(354\) 0 0
\(355\) −10.7669 −0.571448
\(356\) 0 0
\(357\) 0.0932816 0.00493699
\(358\) 0 0
\(359\) 6.03919 0.318736 0.159368 0.987219i \(-0.449054\pi\)
0.159368 + 0.987219i \(0.449054\pi\)
\(360\) 0 0
\(361\) −18.9501 −0.997373
\(362\) 0 0
\(363\) 10.5258 0.552461
\(364\) 0 0
\(365\) 12.2869 0.643126
\(366\) 0 0
\(367\) 22.0076 1.14879 0.574393 0.818580i \(-0.305238\pi\)
0.574393 + 0.818580i \(0.305238\pi\)
\(368\) 0 0
\(369\) −2.75046 −0.143183
\(370\) 0 0
\(371\) 0.925900 0.0480703
\(372\) 0 0
\(373\) 10.5561 0.546574 0.273287 0.961933i \(-0.411889\pi\)
0.273287 + 0.961933i \(0.411889\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0.0409180 0.00210738
\(378\) 0 0
\(379\) 21.2199 1.08999 0.544995 0.838439i \(-0.316531\pi\)
0.544995 + 0.838439i \(0.316531\pi\)
\(380\) 0 0
\(381\) 1.88660 0.0966534
\(382\) 0 0
\(383\) 27.0427 1.38182 0.690908 0.722942i \(-0.257211\pi\)
0.690908 + 0.722942i \(0.257211\pi\)
\(384\) 0 0
\(385\) −0.102277 −0.00521254
\(386\) 0 0
\(387\) 3.23648 0.164519
\(388\) 0 0
\(389\) 33.9113 1.71937 0.859685 0.510824i \(-0.170660\pi\)
0.859685 + 0.510824i \(0.170660\pi\)
\(390\) 0 0
\(391\) 0.814627 0.0411975
\(392\) 0 0
\(393\) 16.7051 0.842660
\(394\) 0 0
\(395\) 9.34214 0.470054
\(396\) 0 0
\(397\) −31.8921 −1.60062 −0.800310 0.599587i \(-0.795332\pi\)
−0.800310 + 0.599587i \(0.795332\pi\)
\(398\) 0 0
\(399\) −0.0331820 −0.00166118
\(400\) 0 0
\(401\) −3.54670 −0.177114 −0.0885568 0.996071i \(-0.528225\pi\)
−0.0885568 + 0.996071i \(0.528225\pi\)
\(402\) 0 0
\(403\) −0.271318 −0.0135153
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0.567495 0.0281297
\(408\) 0 0
\(409\) 8.71109 0.430736 0.215368 0.976533i \(-0.430905\pi\)
0.215368 + 0.976533i \(0.430905\pi\)
\(410\) 0 0
\(411\) −22.5881 −1.11419
\(412\) 0 0
\(413\) −1.74873 −0.0860494
\(414\) 0 0
\(415\) −1.63580 −0.0802982
\(416\) 0 0
\(417\) 5.70170 0.279213
\(418\) 0 0
\(419\) 12.6612 0.618542 0.309271 0.950974i \(-0.399915\pi\)
0.309271 + 0.950974i \(0.399915\pi\)
\(420\) 0 0
\(421\) −23.3943 −1.14017 −0.570084 0.821586i \(-0.693089\pi\)
−0.570084 + 0.821586i \(0.693089\pi\)
\(422\) 0 0
\(423\) 6.96488 0.338644
\(424\) 0 0
\(425\) 0.628064 0.0304656
\(426\) 0 0
\(427\) −1.12657 −0.0545188
\(428\) 0 0
\(429\) 0.0417092 0.00201374
\(430\) 0 0
\(431\) −27.4009 −1.31985 −0.659927 0.751330i \(-0.729413\pi\)
−0.659927 + 0.751330i \(0.729413\pi\)
\(432\) 0 0
\(433\) 33.6696 1.61806 0.809029 0.587769i \(-0.199994\pi\)
0.809029 + 0.587769i \(0.199994\pi\)
\(434\) 0 0
\(435\) 0.675569 0.0323910
\(436\) 0 0
\(437\) −0.289778 −0.0138620
\(438\) 0 0
\(439\) 9.64462 0.460313 0.230156 0.973154i \(-0.426076\pi\)
0.230156 + 0.973154i \(0.426076\pi\)
\(440\) 0 0
\(441\) −6.97794 −0.332283
\(442\) 0 0
\(443\) 3.04333 0.144593 0.0722964 0.997383i \(-0.476967\pi\)
0.0722964 + 0.997383i \(0.476967\pi\)
\(444\) 0 0
\(445\) 11.9174 0.564938
\(446\) 0 0
\(447\) 3.62680 0.171542
\(448\) 0 0
\(449\) 9.10343 0.429617 0.214809 0.976656i \(-0.431087\pi\)
0.214809 + 0.976656i \(0.431087\pi\)
\(450\) 0 0
\(451\) 1.89405 0.0891875
\(452\) 0 0
\(453\) 21.7267 1.02081
\(454\) 0 0
\(455\) 0.00899575 0.000421727 0
\(456\) 0 0
\(457\) 14.1659 0.662652 0.331326 0.943516i \(-0.392504\pi\)
0.331326 + 0.943516i \(0.392504\pi\)
\(458\) 0 0
\(459\) −0.628064 −0.0293155
\(460\) 0 0
\(461\) −40.6945 −1.89533 −0.947665 0.319266i \(-0.896564\pi\)
−0.947665 + 0.319266i \(0.896564\pi\)
\(462\) 0 0
\(463\) −32.9445 −1.53106 −0.765530 0.643400i \(-0.777523\pi\)
−0.765530 + 0.643400i \(0.777523\pi\)
\(464\) 0 0
\(465\) −4.47954 −0.207734
\(466\) 0 0
\(467\) 20.5203 0.949566 0.474783 0.880103i \(-0.342527\pi\)
0.474783 + 0.880103i \(0.342527\pi\)
\(468\) 0 0
\(469\) −0.148523 −0.00685814
\(470\) 0 0
\(471\) 5.12240 0.236028
\(472\) 0 0
\(473\) −2.22874 −0.102478
\(474\) 0 0
\(475\) −0.223414 −0.0102509
\(476\) 0 0
\(477\) −6.23407 −0.285438
\(478\) 0 0
\(479\) −13.5904 −0.620962 −0.310481 0.950580i \(-0.600490\pi\)
−0.310481 + 0.950580i \(0.600490\pi\)
\(480\) 0 0
\(481\) −0.0499137 −0.00227587
\(482\) 0 0
\(483\) 0.192641 0.00876545
\(484\) 0 0
\(485\) 4.01432 0.182281
\(486\) 0 0
\(487\) −8.93284 −0.404786 −0.202393 0.979304i \(-0.564872\pi\)
−0.202393 + 0.979304i \(0.564872\pi\)
\(488\) 0 0
\(489\) −8.77292 −0.396725
\(490\) 0 0
\(491\) −10.0961 −0.455630 −0.227815 0.973704i \(-0.573158\pi\)
−0.227815 + 0.973704i \(0.573158\pi\)
\(492\) 0 0
\(493\) 0.424300 0.0191095
\(494\) 0 0
\(495\) 0.688632 0.0309517
\(496\) 0 0
\(497\) −1.59913 −0.0717308
\(498\) 0 0
\(499\) −29.5242 −1.32168 −0.660842 0.750525i \(-0.729801\pi\)
−0.660842 + 0.750525i \(0.729801\pi\)
\(500\) 0 0
\(501\) 8.01771 0.358205
\(502\) 0 0
\(503\) 26.2721 1.17142 0.585708 0.810522i \(-0.300817\pi\)
0.585708 + 0.810522i \(0.300817\pi\)
\(504\) 0 0
\(505\) 17.1057 0.761192
\(506\) 0 0
\(507\) 12.9963 0.577187
\(508\) 0 0
\(509\) −6.10714 −0.270694 −0.135347 0.990798i \(-0.543215\pi\)
−0.135347 + 0.990798i \(0.543215\pi\)
\(510\) 0 0
\(511\) 1.82488 0.0807280
\(512\) 0 0
\(513\) 0.223414 0.00986396
\(514\) 0 0
\(515\) 3.23648 0.142616
\(516\) 0 0
\(517\) −4.79624 −0.210938
\(518\) 0 0
\(519\) −7.40930 −0.325232
\(520\) 0 0
\(521\) −3.30625 −0.144850 −0.0724248 0.997374i \(-0.523074\pi\)
−0.0724248 + 0.997374i \(0.523074\pi\)
\(522\) 0 0
\(523\) 2.90218 0.126904 0.0634518 0.997985i \(-0.479789\pi\)
0.0634518 + 0.997985i \(0.479789\pi\)
\(524\) 0 0
\(525\) 0.148523 0.00648206
\(526\) 0 0
\(527\) −2.81344 −0.122555
\(528\) 0 0
\(529\) −21.3177 −0.926855
\(530\) 0 0
\(531\) 11.7742 0.510956
\(532\) 0 0
\(533\) −0.166590 −0.00721583
\(534\) 0 0
\(535\) 17.7313 0.766592
\(536\) 0 0
\(537\) −10.7676 −0.464656
\(538\) 0 0
\(539\) 4.80523 0.206976
\(540\) 0 0
\(541\) 41.2659 1.77416 0.887080 0.461615i \(-0.152730\pi\)
0.887080 + 0.461615i \(0.152730\pi\)
\(542\) 0 0
\(543\) 23.7403 1.01879
\(544\) 0 0
\(545\) −8.37485 −0.358739
\(546\) 0 0
\(547\) 2.06819 0.0884296 0.0442148 0.999022i \(-0.485921\pi\)
0.0442148 + 0.999022i \(0.485921\pi\)
\(548\) 0 0
\(549\) 7.58521 0.323729
\(550\) 0 0
\(551\) −0.150931 −0.00642989
\(552\) 0 0
\(553\) 1.38752 0.0590033
\(554\) 0 0
\(555\) −0.824091 −0.0349807
\(556\) 0 0
\(557\) 23.9281 1.01387 0.506934 0.861985i \(-0.330779\pi\)
0.506934 + 0.861985i \(0.330779\pi\)
\(558\) 0 0
\(559\) 0.196028 0.00829108
\(560\) 0 0
\(561\) 0.432505 0.0182604
\(562\) 0 0
\(563\) 18.2267 0.768163 0.384082 0.923299i \(-0.374518\pi\)
0.384082 + 0.923299i \(0.374518\pi\)
\(564\) 0 0
\(565\) −18.2935 −0.769613
\(566\) 0 0
\(567\) −0.148523 −0.00623736
\(568\) 0 0
\(569\) 27.8499 1.16753 0.583765 0.811923i \(-0.301579\pi\)
0.583765 + 0.811923i \(0.301579\pi\)
\(570\) 0 0
\(571\) 15.7090 0.657400 0.328700 0.944434i \(-0.393390\pi\)
0.328700 + 0.944434i \(0.393390\pi\)
\(572\) 0 0
\(573\) 9.00465 0.376175
\(574\) 0 0
\(575\) 1.29705 0.0540905
\(576\) 0 0
\(577\) 19.3066 0.803742 0.401871 0.915696i \(-0.368360\pi\)
0.401871 + 0.915696i \(0.368360\pi\)
\(578\) 0 0
\(579\) 5.24180 0.217842
\(580\) 0 0
\(581\) −0.242953 −0.0100794
\(582\) 0 0
\(583\) 4.29298 0.177797
\(584\) 0 0
\(585\) −0.0605682 −0.00250419
\(586\) 0 0
\(587\) 2.56750 0.105972 0.0529859 0.998595i \(-0.483126\pi\)
0.0529859 + 0.998595i \(0.483126\pi\)
\(588\) 0 0
\(589\) 1.00079 0.0412369
\(590\) 0 0
\(591\) 5.74020 0.236120
\(592\) 0 0
\(593\) 32.5394 1.33623 0.668116 0.744057i \(-0.267101\pi\)
0.668116 + 0.744057i \(0.267101\pi\)
\(594\) 0 0
\(595\) 0.0932816 0.00382417
\(596\) 0 0
\(597\) 5.09695 0.208604
\(598\) 0 0
\(599\) 35.3119 1.44280 0.721402 0.692516i \(-0.243498\pi\)
0.721402 + 0.692516i \(0.243498\pi\)
\(600\) 0 0
\(601\) −43.4173 −1.77103 −0.885515 0.464611i \(-0.846194\pi\)
−0.885515 + 0.464611i \(0.846194\pi\)
\(602\) 0 0
\(603\) 1.00000 0.0407231
\(604\) 0 0
\(605\) 10.5258 0.427934
\(606\) 0 0
\(607\) −20.5844 −0.835496 −0.417748 0.908563i \(-0.637180\pi\)
−0.417748 + 0.908563i \(0.637180\pi\)
\(608\) 0 0
\(609\) 0.100337 0.00406587
\(610\) 0 0
\(611\) 0.421850 0.0170662
\(612\) 0 0
\(613\) −31.9712 −1.29130 −0.645652 0.763632i \(-0.723414\pi\)
−0.645652 + 0.763632i \(0.723414\pi\)
\(614\) 0 0
\(615\) −2.75046 −0.110909
\(616\) 0 0
\(617\) 2.16479 0.0871510 0.0435755 0.999050i \(-0.486125\pi\)
0.0435755 + 0.999050i \(0.486125\pi\)
\(618\) 0 0
\(619\) −8.12097 −0.326409 −0.163205 0.986592i \(-0.552183\pi\)
−0.163205 + 0.986592i \(0.552183\pi\)
\(620\) 0 0
\(621\) −1.29705 −0.0520486
\(622\) 0 0
\(623\) 1.77000 0.0709135
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.153850 −0.00614417
\(628\) 0 0
\(629\) −0.517582 −0.0206373
\(630\) 0 0
\(631\) −5.80224 −0.230984 −0.115492 0.993308i \(-0.536844\pi\)
−0.115492 + 0.993308i \(0.536844\pi\)
\(632\) 0 0
\(633\) −9.60486 −0.381759
\(634\) 0 0
\(635\) 1.88660 0.0748674
\(636\) 0 0
\(637\) −0.422641 −0.0167457
\(638\) 0 0
\(639\) 10.7669 0.425932
\(640\) 0 0
\(641\) −11.2254 −0.443375 −0.221687 0.975118i \(-0.571156\pi\)
−0.221687 + 0.975118i \(0.571156\pi\)
\(642\) 0 0
\(643\) 32.0337 1.26329 0.631643 0.775260i \(-0.282381\pi\)
0.631643 + 0.775260i \(0.282381\pi\)
\(644\) 0 0
\(645\) 3.23648 0.127436
\(646\) 0 0
\(647\) −35.6385 −1.40110 −0.700548 0.713605i \(-0.747061\pi\)
−0.700548 + 0.713605i \(0.747061\pi\)
\(648\) 0 0
\(649\) −8.10807 −0.318270
\(650\) 0 0
\(651\) −0.665313 −0.0260757
\(652\) 0 0
\(653\) −1.38637 −0.0542529 −0.0271264 0.999632i \(-0.508636\pi\)
−0.0271264 + 0.999632i \(0.508636\pi\)
\(654\) 0 0
\(655\) 16.7051 0.652722
\(656\) 0 0
\(657\) −12.2869 −0.479358
\(658\) 0 0
\(659\) 20.4088 0.795015 0.397508 0.917599i \(-0.369875\pi\)
0.397508 + 0.917599i \(0.369875\pi\)
\(660\) 0 0
\(661\) 15.6649 0.609295 0.304647 0.952465i \(-0.401461\pi\)
0.304647 + 0.952465i \(0.401461\pi\)
\(662\) 0 0
\(663\) −0.0380407 −0.00147738
\(664\) 0 0
\(665\) −0.0331820 −0.00128674
\(666\) 0 0
\(667\) 0.876243 0.0339283
\(668\) 0 0
\(669\) −0.314287 −0.0121510
\(670\) 0 0
\(671\) −5.22341 −0.201648
\(672\) 0 0
\(673\) 28.4735 1.09757 0.548787 0.835962i \(-0.315090\pi\)
0.548787 + 0.835962i \(0.315090\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −27.2279 −1.04646 −0.523228 0.852193i \(-0.675272\pi\)
−0.523228 + 0.852193i \(0.675272\pi\)
\(678\) 0 0
\(679\) 0.596218 0.0228807
\(680\) 0 0
\(681\) −0.298305 −0.0114311
\(682\) 0 0
\(683\) 20.2680 0.775535 0.387768 0.921757i \(-0.373246\pi\)
0.387768 + 0.921757i \(0.373246\pi\)
\(684\) 0 0
\(685\) −22.5881 −0.863048
\(686\) 0 0
\(687\) 0.150931 0.00575839
\(688\) 0 0
\(689\) −0.377586 −0.0143849
\(690\) 0 0
\(691\) 10.5356 0.400795 0.200397 0.979715i \(-0.435777\pi\)
0.200397 + 0.979715i \(0.435777\pi\)
\(692\) 0 0
\(693\) 0.102277 0.00388520
\(694\) 0 0
\(695\) 5.70170 0.216278
\(696\) 0 0
\(697\) −1.72746 −0.0654324
\(698\) 0 0
\(699\) 26.1762 0.990075
\(700\) 0 0
\(701\) −5.91942 −0.223574 −0.111787 0.993732i \(-0.535657\pi\)
−0.111787 + 0.993732i \(0.535657\pi\)
\(702\) 0 0
\(703\) 0.184113 0.00694397
\(704\) 0 0
\(705\) 6.96488 0.262312
\(706\) 0 0
\(707\) 2.54058 0.0955483
\(708\) 0 0
\(709\) −7.12819 −0.267705 −0.133852 0.991001i \(-0.542735\pi\)
−0.133852 + 0.991001i \(0.542735\pi\)
\(710\) 0 0
\(711\) −9.34214 −0.350358
\(712\) 0 0
\(713\) −5.81017 −0.217592
\(714\) 0 0
\(715\) 0.0417092 0.00155984
\(716\) 0 0
\(717\) 15.2727 0.570369
\(718\) 0 0
\(719\) 38.3704 1.43097 0.715487 0.698626i \(-0.246205\pi\)
0.715487 + 0.698626i \(0.246205\pi\)
\(720\) 0 0
\(721\) 0.480690 0.0179018
\(722\) 0 0
\(723\) 6.37388 0.237047
\(724\) 0 0
\(725\) 0.675569 0.0250900
\(726\) 0 0
\(727\) −26.7895 −0.993567 −0.496783 0.867875i \(-0.665486\pi\)
−0.496783 + 0.867875i \(0.665486\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.03271 0.0751826
\(732\) 0 0
\(733\) −16.6819 −0.616159 −0.308080 0.951361i \(-0.599686\pi\)
−0.308080 + 0.951361i \(0.599686\pi\)
\(734\) 0 0
\(735\) −6.97794 −0.257385
\(736\) 0 0
\(737\) −0.688632 −0.0253661
\(738\) 0 0
\(739\) 37.0022 1.36115 0.680573 0.732680i \(-0.261731\pi\)
0.680573 + 0.732680i \(0.261731\pi\)
\(740\) 0 0
\(741\) 0.0135318 0.000497102 0
\(742\) 0 0
\(743\) 11.7180 0.429890 0.214945 0.976626i \(-0.431043\pi\)
0.214945 + 0.976626i \(0.431043\pi\)
\(744\) 0 0
\(745\) 3.62680 0.132876
\(746\) 0 0
\(747\) 1.63580 0.0598508
\(748\) 0 0
\(749\) 2.63350 0.0962261
\(750\) 0 0
\(751\) −4.95638 −0.180861 −0.0904304 0.995903i \(-0.528824\pi\)
−0.0904304 + 0.995903i \(0.528824\pi\)
\(752\) 0 0
\(753\) −22.9065 −0.834761
\(754\) 0 0
\(755\) 21.7267 0.790715
\(756\) 0 0
\(757\) 42.0672 1.52896 0.764479 0.644649i \(-0.222996\pi\)
0.764479 + 0.644649i \(0.222996\pi\)
\(758\) 0 0
\(759\) 0.893187 0.0324206
\(760\) 0 0
\(761\) −8.54000 −0.309575 −0.154787 0.987948i \(-0.549469\pi\)
−0.154787 + 0.987948i \(0.549469\pi\)
\(762\) 0 0
\(763\) −1.24386 −0.0450306
\(764\) 0 0
\(765\) −0.628064 −0.0227077
\(766\) 0 0
\(767\) 0.713141 0.0257500
\(768\) 0 0
\(769\) 25.5867 0.922678 0.461339 0.887224i \(-0.347369\pi\)
0.461339 + 0.887224i \(0.347369\pi\)
\(770\) 0 0
\(771\) 17.6334 0.635051
\(772\) 0 0
\(773\) −26.4563 −0.951568 −0.475784 0.879562i \(-0.657836\pi\)
−0.475784 + 0.879562i \(0.657836\pi\)
\(774\) 0 0
\(775\) −4.47954 −0.160910
\(776\) 0 0
\(777\) −0.122396 −0.00439094
\(778\) 0 0
\(779\) 0.614491 0.0220164
\(780\) 0 0
\(781\) −7.41444 −0.265309
\(782\) 0 0
\(783\) −0.675569 −0.0241429
\(784\) 0 0
\(785\) 5.12240 0.182826
\(786\) 0 0
\(787\) −25.6851 −0.915574 −0.457787 0.889062i \(-0.651358\pi\)
−0.457787 + 0.889062i \(0.651358\pi\)
\(788\) 0 0
\(789\) 1.12840 0.0401722
\(790\) 0 0
\(791\) −2.71700 −0.0966053
\(792\) 0 0
\(793\) 0.459422 0.0163146
\(794\) 0 0
\(795\) −6.23407 −0.221100
\(796\) 0 0
\(797\) −21.6939 −0.768436 −0.384218 0.923242i \(-0.625529\pi\)
−0.384218 + 0.923242i \(0.625529\pi\)
\(798\) 0 0
\(799\) 4.37439 0.154755
\(800\) 0 0
\(801\) −11.9174 −0.421080
\(802\) 0 0
\(803\) 8.46115 0.298588
\(804\) 0 0
\(805\) 0.192641 0.00678969
\(806\) 0 0
\(807\) −0.450895 −0.0158723
\(808\) 0 0
\(809\) −7.59100 −0.266885 −0.133443 0.991057i \(-0.542603\pi\)
−0.133443 + 0.991057i \(0.542603\pi\)
\(810\) 0 0
\(811\) 4.11978 0.144665 0.0723324 0.997381i \(-0.476956\pi\)
0.0723324 + 0.997381i \(0.476956\pi\)
\(812\) 0 0
\(813\) 27.0615 0.949090
\(814\) 0 0
\(815\) −8.77292 −0.307302
\(816\) 0 0
\(817\) −0.723074 −0.0252971
\(818\) 0 0
\(819\) −0.00899575 −0.000314337 0
\(820\) 0 0
\(821\) 30.4111 1.06135 0.530677 0.847574i \(-0.321938\pi\)
0.530677 + 0.847574i \(0.321938\pi\)
\(822\) 0 0
\(823\) 36.9671 1.28859 0.644296 0.764776i \(-0.277150\pi\)
0.644296 + 0.764776i \(0.277150\pi\)
\(824\) 0 0
\(825\) 0.688632 0.0239751
\(826\) 0 0
\(827\) 13.6727 0.475445 0.237722 0.971333i \(-0.423599\pi\)
0.237722 + 0.971333i \(0.423599\pi\)
\(828\) 0 0
\(829\) −42.2070 −1.46591 −0.732955 0.680278i \(-0.761859\pi\)
−0.732955 + 0.680278i \(0.761859\pi\)
\(830\) 0 0
\(831\) 2.45295 0.0850918
\(832\) 0 0
\(833\) −4.38259 −0.151848
\(834\) 0 0
\(835\) 8.01771 0.277464
\(836\) 0 0
\(837\) 4.47954 0.154836
\(838\) 0 0
\(839\) 5.28185 0.182350 0.0911749 0.995835i \(-0.470938\pi\)
0.0911749 + 0.995835i \(0.470938\pi\)
\(840\) 0 0
\(841\) −28.5436 −0.984262
\(842\) 0 0
\(843\) 31.1442 1.07266
\(844\) 0 0
\(845\) 12.9963 0.447087
\(846\) 0 0
\(847\) 1.56332 0.0537162
\(848\) 0 0
\(849\) −11.9775 −0.411066
\(850\) 0 0
\(851\) −1.06888 −0.0366409
\(852\) 0 0
\(853\) 30.6140 1.04820 0.524102 0.851655i \(-0.324401\pi\)
0.524102 + 0.851655i \(0.324401\pi\)
\(854\) 0 0
\(855\) 0.223414 0.00764059
\(856\) 0 0
\(857\) −12.9493 −0.442339 −0.221170 0.975235i \(-0.570987\pi\)
−0.221170 + 0.975235i \(0.570987\pi\)
\(858\) 0 0
\(859\) −39.7777 −1.35720 −0.678598 0.734510i \(-0.737412\pi\)
−0.678598 + 0.734510i \(0.737412\pi\)
\(860\) 0 0
\(861\) −0.408505 −0.0139218
\(862\) 0 0
\(863\) 31.7486 1.08073 0.540367 0.841430i \(-0.318285\pi\)
0.540367 + 0.841430i \(0.318285\pi\)
\(864\) 0 0
\(865\) −7.40930 −0.251924
\(866\) 0 0
\(867\) 16.6055 0.563954
\(868\) 0 0
\(869\) 6.43330 0.218235
\(870\) 0 0
\(871\) 0.0605682 0.00205228
\(872\) 0 0
\(873\) −4.01432 −0.135864
\(874\) 0 0
\(875\) 0.148523 0.00502098
\(876\) 0 0
\(877\) 51.9908 1.75561 0.877803 0.479022i \(-0.159009\pi\)
0.877803 + 0.479022i \(0.159009\pi\)
\(878\) 0 0
\(879\) −7.02951 −0.237100
\(880\) 0 0
\(881\) −5.79447 −0.195221 −0.0976103 0.995225i \(-0.531120\pi\)
−0.0976103 + 0.995225i \(0.531120\pi\)
\(882\) 0 0
\(883\) 53.2763 1.79289 0.896444 0.443156i \(-0.146141\pi\)
0.896444 + 0.443156i \(0.146141\pi\)
\(884\) 0 0
\(885\) 11.7742 0.395785
\(886\) 0 0
\(887\) 20.6087 0.691973 0.345986 0.938240i \(-0.387544\pi\)
0.345986 + 0.938240i \(0.387544\pi\)
\(888\) 0 0
\(889\) 0.280203 0.00939770
\(890\) 0 0
\(891\) −0.688632 −0.0230700
\(892\) 0 0
\(893\) −1.55605 −0.0520712
\(894\) 0 0
\(895\) −10.7676 −0.359921
\(896\) 0 0
\(897\) −0.0785597 −0.00262303
\(898\) 0 0
\(899\) −3.02624 −0.100931
\(900\) 0 0
\(901\) −3.91539 −0.130441
\(902\) 0 0
\(903\) 0.480690 0.0159964
\(904\) 0 0
\(905\) 23.7403 0.789155
\(906\) 0 0
\(907\) −17.8440 −0.592502 −0.296251 0.955110i \(-0.595736\pi\)
−0.296251 + 0.955110i \(0.595736\pi\)
\(908\) 0 0
\(909\) −17.1057 −0.567359
\(910\) 0 0
\(911\) 25.0058 0.828480 0.414240 0.910168i \(-0.364047\pi\)
0.414240 + 0.910168i \(0.364047\pi\)
\(912\) 0 0
\(913\) −1.12646 −0.0372805
\(914\) 0 0
\(915\) 7.58521 0.250759
\(916\) 0 0
\(917\) 2.48108 0.0819326
\(918\) 0 0
\(919\) −52.2586 −1.72385 −0.861925 0.507035i \(-0.830741\pi\)
−0.861925 + 0.507035i \(0.830741\pi\)
\(920\) 0 0
\(921\) 23.5548 0.776158
\(922\) 0 0
\(923\) 0.652132 0.0214652
\(924\) 0 0
\(925\) −0.824091 −0.0270960
\(926\) 0 0
\(927\) −3.23648 −0.106300
\(928\) 0 0
\(929\) −38.2212 −1.25400 −0.626999 0.779020i \(-0.715717\pi\)
−0.626999 + 0.779020i \(0.715717\pi\)
\(930\) 0 0
\(931\) 1.55897 0.0510931
\(932\) 0 0
\(933\) −22.5721 −0.738979
\(934\) 0 0
\(935\) 0.432505 0.0141444
\(936\) 0 0
\(937\) −24.9315 −0.814477 −0.407239 0.913322i \(-0.633508\pi\)
−0.407239 + 0.913322i \(0.633508\pi\)
\(938\) 0 0
\(939\) −8.70702 −0.284143
\(940\) 0 0
\(941\) −19.2412 −0.627244 −0.313622 0.949548i \(-0.601542\pi\)
−0.313622 + 0.949548i \(0.601542\pi\)
\(942\) 0 0
\(943\) −3.56747 −0.116173
\(944\) 0 0
\(945\) −0.148523 −0.00483144
\(946\) 0 0
\(947\) 36.3169 1.18014 0.590070 0.807352i \(-0.299100\pi\)
0.590070 + 0.807352i \(0.299100\pi\)
\(948\) 0 0
\(949\) −0.744195 −0.0241576
\(950\) 0 0
\(951\) 12.0278 0.390028
\(952\) 0 0
\(953\) 29.8461 0.966810 0.483405 0.875397i \(-0.339400\pi\)
0.483405 + 0.875397i \(0.339400\pi\)
\(954\) 0 0
\(955\) 9.00465 0.291384
\(956\) 0 0
\(957\) 0.465218 0.0150384
\(958\) 0 0
\(959\) −3.35485 −0.108334
\(960\) 0 0
\(961\) −10.9337 −0.352700
\(962\) 0 0
\(963\) −17.7313 −0.571384
\(964\) 0 0
\(965\) 5.24180 0.168740
\(966\) 0 0
\(967\) −19.7932 −0.636508 −0.318254 0.948005i \(-0.603096\pi\)
−0.318254 + 0.948005i \(0.603096\pi\)
\(968\) 0 0
\(969\) 0.140318 0.00450767
\(970\) 0 0
\(971\) −13.2682 −0.425798 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(972\) 0 0
\(973\) 0.846831 0.0271481
\(974\) 0 0
\(975\) −0.0605682 −0.00193973
\(976\) 0 0
\(977\) 24.5787 0.786343 0.393172 0.919465i \(-0.371378\pi\)
0.393172 + 0.919465i \(0.371378\pi\)
\(978\) 0 0
\(979\) 8.20668 0.262287
\(980\) 0 0
\(981\) 8.37485 0.267389
\(982\) 0 0
\(983\) −27.8773 −0.889148 −0.444574 0.895742i \(-0.646645\pi\)
−0.444574 + 0.895742i \(0.646645\pi\)
\(984\) 0 0
\(985\) 5.74020 0.182898
\(986\) 0 0
\(987\) 1.03444 0.0329266
\(988\) 0 0
\(989\) 4.19786 0.133484
\(990\) 0 0
\(991\) −47.2253 −1.50016 −0.750080 0.661347i \(-0.769985\pi\)
−0.750080 + 0.661347i \(0.769985\pi\)
\(992\) 0 0
\(993\) 6.08022 0.192950
\(994\) 0 0
\(995\) 5.09695 0.161584
\(996\) 0 0
\(997\) 25.5349 0.808698 0.404349 0.914605i \(-0.367498\pi\)
0.404349 + 0.914605i \(0.367498\pi\)
\(998\) 0 0
\(999\) 0.824091 0.0260731
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 4020.2.a.b.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.a.b.1.3 4 1.1 even 1 trivial