Properties

Label 8020.2.a.f.1.4
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $37$
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: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
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)
 
\(f(q)\) \(=\) \(q-2.74673 q^{3} +1.00000 q^{5} -2.35693 q^{7} +4.54455 q^{9} +O(q^{10})\) \(q-2.74673 q^{3} +1.00000 q^{5} -2.35693 q^{7} +4.54455 q^{9} -0.675516 q^{11} +5.24160 q^{13} -2.74673 q^{15} +0.157089 q^{17} +3.38382 q^{19} +6.47385 q^{21} -7.83745 q^{23} +1.00000 q^{25} -4.24247 q^{27} -2.03674 q^{29} -2.89059 q^{31} +1.85546 q^{33} -2.35693 q^{35} +6.10459 q^{37} -14.3973 q^{39} -1.90724 q^{41} +9.82206 q^{43} +4.54455 q^{45} +12.3241 q^{47} -1.44490 q^{49} -0.431482 q^{51} +5.83339 q^{53} -0.675516 q^{55} -9.29447 q^{57} -0.257517 q^{59} +7.24169 q^{61} -10.7112 q^{63} +5.24160 q^{65} -14.7233 q^{67} +21.5274 q^{69} -3.19249 q^{71} +4.56610 q^{73} -2.74673 q^{75} +1.59214 q^{77} -16.5784 q^{79} -1.98071 q^{81} +13.8412 q^{83} +0.157089 q^{85} +5.59437 q^{87} -3.93737 q^{89} -12.3541 q^{91} +7.93969 q^{93} +3.38382 q^{95} +3.95434 q^{97} -3.06992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 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
\(3\) −2.74673 −1.58583 −0.792914 0.609334i \(-0.791437\pi\)
−0.792914 + 0.609334i \(0.791437\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.35693 −0.890835 −0.445417 0.895323i \(-0.646945\pi\)
−0.445417 + 0.895323i \(0.646945\pi\)
\(8\) 0 0
\(9\) 4.54455 1.51485
\(10\) 0 0
\(11\) −0.675516 −0.203676 −0.101838 0.994801i \(-0.532472\pi\)
−0.101838 + 0.994801i \(0.532472\pi\)
\(12\) 0 0
\(13\) 5.24160 1.45376 0.726880 0.686765i \(-0.240970\pi\)
0.726880 + 0.686765i \(0.240970\pi\)
\(14\) 0 0
\(15\) −2.74673 −0.709204
\(16\) 0 0
\(17\) 0.157089 0.0380997 0.0190499 0.999819i \(-0.493936\pi\)
0.0190499 + 0.999819i \(0.493936\pi\)
\(18\) 0 0
\(19\) 3.38382 0.776303 0.388151 0.921596i \(-0.373114\pi\)
0.388151 + 0.921596i \(0.373114\pi\)
\(20\) 0 0
\(21\) 6.47385 1.41271
\(22\) 0 0
\(23\) −7.83745 −1.63422 −0.817110 0.576481i \(-0.804425\pi\)
−0.817110 + 0.576481i \(0.804425\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.24247 −0.816464
\(28\) 0 0
\(29\) −2.03674 −0.378212 −0.189106 0.981957i \(-0.560559\pi\)
−0.189106 + 0.981957i \(0.560559\pi\)
\(30\) 0 0
\(31\) −2.89059 −0.519166 −0.259583 0.965721i \(-0.583585\pi\)
−0.259583 + 0.965721i \(0.583585\pi\)
\(32\) 0 0
\(33\) 1.85546 0.322995
\(34\) 0 0
\(35\) −2.35693 −0.398393
\(36\) 0 0
\(37\) 6.10459 1.00359 0.501794 0.864987i \(-0.332673\pi\)
0.501794 + 0.864987i \(0.332673\pi\)
\(38\) 0 0
\(39\) −14.3973 −2.30541
\(40\) 0 0
\(41\) −1.90724 −0.297860 −0.148930 0.988848i \(-0.547583\pi\)
−0.148930 + 0.988848i \(0.547583\pi\)
\(42\) 0 0
\(43\) 9.82206 1.49785 0.748925 0.662654i \(-0.230570\pi\)
0.748925 + 0.662654i \(0.230570\pi\)
\(44\) 0 0
\(45\) 4.54455 0.677462
\(46\) 0 0
\(47\) 12.3241 1.79765 0.898827 0.438304i \(-0.144421\pi\)
0.898827 + 0.438304i \(0.144421\pi\)
\(48\) 0 0
\(49\) −1.44490 −0.206414
\(50\) 0 0
\(51\) −0.431482 −0.0604196
\(52\) 0 0
\(53\) 5.83339 0.801278 0.400639 0.916236i \(-0.368788\pi\)
0.400639 + 0.916236i \(0.368788\pi\)
\(54\) 0 0
\(55\) −0.675516 −0.0910866
\(56\) 0 0
\(57\) −9.29447 −1.23108
\(58\) 0 0
\(59\) −0.257517 −0.0335258 −0.0167629 0.999859i \(-0.505336\pi\)
−0.0167629 + 0.999859i \(0.505336\pi\)
\(60\) 0 0
\(61\) 7.24169 0.927203 0.463602 0.886044i \(-0.346557\pi\)
0.463602 + 0.886044i \(0.346557\pi\)
\(62\) 0 0
\(63\) −10.7112 −1.34948
\(64\) 0 0
\(65\) 5.24160 0.650141
\(66\) 0 0
\(67\) −14.7233 −1.79873 −0.899367 0.437195i \(-0.855972\pi\)
−0.899367 + 0.437195i \(0.855972\pi\)
\(68\) 0 0
\(69\) 21.5274 2.59159
\(70\) 0 0
\(71\) −3.19249 −0.378879 −0.189439 0.981892i \(-0.560667\pi\)
−0.189439 + 0.981892i \(0.560667\pi\)
\(72\) 0 0
\(73\) 4.56610 0.534422 0.267211 0.963638i \(-0.413898\pi\)
0.267211 + 0.963638i \(0.413898\pi\)
\(74\) 0 0
\(75\) −2.74673 −0.317166
\(76\) 0 0
\(77\) 1.59214 0.181442
\(78\) 0 0
\(79\) −16.5784 −1.86521 −0.932606 0.360896i \(-0.882471\pi\)
−0.932606 + 0.360896i \(0.882471\pi\)
\(80\) 0 0
\(81\) −1.98071 −0.220079
\(82\) 0 0
\(83\) 13.8412 1.51927 0.759634 0.650351i \(-0.225378\pi\)
0.759634 + 0.650351i \(0.225378\pi\)
\(84\) 0 0
\(85\) 0.157089 0.0170387
\(86\) 0 0
\(87\) 5.59437 0.599780
\(88\) 0 0
\(89\) −3.93737 −0.417360 −0.208680 0.977984i \(-0.566917\pi\)
−0.208680 + 0.977984i \(0.566917\pi\)
\(90\) 0 0
\(91\) −12.3541 −1.29506
\(92\) 0 0
\(93\) 7.93969 0.823307
\(94\) 0 0
\(95\) 3.38382 0.347173
\(96\) 0 0
\(97\) 3.95434 0.401503 0.200751 0.979642i \(-0.435662\pi\)
0.200751 + 0.979642i \(0.435662\pi\)
\(98\) 0 0
\(99\) −3.06992 −0.308538
\(100\) 0 0
\(101\) −1.13165 −0.112603 −0.0563015 0.998414i \(-0.517931\pi\)
−0.0563015 + 0.998414i \(0.517931\pi\)
\(102\) 0 0
\(103\) −9.92119 −0.977564 −0.488782 0.872406i \(-0.662559\pi\)
−0.488782 + 0.872406i \(0.662559\pi\)
\(104\) 0 0
\(105\) 6.47385 0.631783
\(106\) 0 0
\(107\) −0.206483 −0.0199614 −0.00998072 0.999950i \(-0.503177\pi\)
−0.00998072 + 0.999950i \(0.503177\pi\)
\(108\) 0 0
\(109\) −17.7094 −1.69626 −0.848128 0.529792i \(-0.822270\pi\)
−0.848128 + 0.529792i \(0.822270\pi\)
\(110\) 0 0
\(111\) −16.7677 −1.59152
\(112\) 0 0
\(113\) −1.33331 −0.125427 −0.0627135 0.998032i \(-0.519975\pi\)
−0.0627135 + 0.998032i \(0.519975\pi\)
\(114\) 0 0
\(115\) −7.83745 −0.730846
\(116\) 0 0
\(117\) 23.8207 2.20223
\(118\) 0 0
\(119\) −0.370248 −0.0339406
\(120\) 0 0
\(121\) −10.5437 −0.958516
\(122\) 0 0
\(123\) 5.23867 0.472355
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.47071 0.485447 0.242723 0.970096i \(-0.421959\pi\)
0.242723 + 0.970096i \(0.421959\pi\)
\(128\) 0 0
\(129\) −26.9786 −2.37533
\(130\) 0 0
\(131\) −1.84144 −0.160887 −0.0804437 0.996759i \(-0.525634\pi\)
−0.0804437 + 0.996759i \(0.525634\pi\)
\(132\) 0 0
\(133\) −7.97543 −0.691557
\(134\) 0 0
\(135\) −4.24247 −0.365134
\(136\) 0 0
\(137\) 13.6340 1.16483 0.582416 0.812891i \(-0.302108\pi\)
0.582416 + 0.812891i \(0.302108\pi\)
\(138\) 0 0
\(139\) 8.18219 0.694005 0.347002 0.937864i \(-0.387200\pi\)
0.347002 + 0.937864i \(0.387200\pi\)
\(140\) 0 0
\(141\) −33.8510 −2.85077
\(142\) 0 0
\(143\) −3.54079 −0.296096
\(144\) 0 0
\(145\) −2.03674 −0.169142
\(146\) 0 0
\(147\) 3.96874 0.327336
\(148\) 0 0
\(149\) 13.3172 1.09099 0.545495 0.838114i \(-0.316342\pi\)
0.545495 + 0.838114i \(0.316342\pi\)
\(150\) 0 0
\(151\) 10.0377 0.816854 0.408427 0.912791i \(-0.366077\pi\)
0.408427 + 0.912791i \(0.366077\pi\)
\(152\) 0 0
\(153\) 0.713900 0.0577154
\(154\) 0 0
\(155\) −2.89059 −0.232178
\(156\) 0 0
\(157\) −3.44926 −0.275281 −0.137641 0.990482i \(-0.543952\pi\)
−0.137641 + 0.990482i \(0.543952\pi\)
\(158\) 0 0
\(159\) −16.0228 −1.27069
\(160\) 0 0
\(161\) 18.4723 1.45582
\(162\) 0 0
\(163\) 4.25398 0.333198 0.166599 0.986025i \(-0.446722\pi\)
0.166599 + 0.986025i \(0.446722\pi\)
\(164\) 0 0
\(165\) 1.85546 0.144448
\(166\) 0 0
\(167\) −19.2554 −1.49003 −0.745013 0.667050i \(-0.767557\pi\)
−0.745013 + 0.667050i \(0.767557\pi\)
\(168\) 0 0
\(169\) 14.4744 1.11342
\(170\) 0 0
\(171\) 15.3780 1.17598
\(172\) 0 0
\(173\) −0.949945 −0.0722230 −0.0361115 0.999348i \(-0.511497\pi\)
−0.0361115 + 0.999348i \(0.511497\pi\)
\(174\) 0 0
\(175\) −2.35693 −0.178167
\(176\) 0 0
\(177\) 0.707330 0.0531662
\(178\) 0 0
\(179\) −13.8555 −1.03561 −0.517805 0.855499i \(-0.673251\pi\)
−0.517805 + 0.855499i \(0.673251\pi\)
\(180\) 0 0
\(181\) −21.7508 −1.61672 −0.808362 0.588686i \(-0.799645\pi\)
−0.808362 + 0.588686i \(0.799645\pi\)
\(182\) 0 0
\(183\) −19.8910 −1.47038
\(184\) 0 0
\(185\) 6.10459 0.448818
\(186\) 0 0
\(187\) −0.106116 −0.00776000
\(188\) 0 0
\(189\) 9.99919 0.727334
\(190\) 0 0
\(191\) −2.89030 −0.209135 −0.104567 0.994518i \(-0.533346\pi\)
−0.104567 + 0.994518i \(0.533346\pi\)
\(192\) 0 0
\(193\) −17.0936 −1.23043 −0.615213 0.788361i \(-0.710930\pi\)
−0.615213 + 0.788361i \(0.710930\pi\)
\(194\) 0 0
\(195\) −14.3973 −1.03101
\(196\) 0 0
\(197\) 21.8684 1.55806 0.779030 0.626987i \(-0.215712\pi\)
0.779030 + 0.626987i \(0.215712\pi\)
\(198\) 0 0
\(199\) 10.1016 0.716083 0.358041 0.933706i \(-0.383445\pi\)
0.358041 + 0.933706i \(0.383445\pi\)
\(200\) 0 0
\(201\) 40.4409 2.85248
\(202\) 0 0
\(203\) 4.80044 0.336925
\(204\) 0 0
\(205\) −1.90724 −0.133207
\(206\) 0 0
\(207\) −35.6177 −2.47560
\(208\) 0 0
\(209\) −2.28583 −0.158114
\(210\) 0 0
\(211\) −19.7501 −1.35965 −0.679827 0.733372i \(-0.737945\pi\)
−0.679827 + 0.733372i \(0.737945\pi\)
\(212\) 0 0
\(213\) 8.76892 0.600836
\(214\) 0 0
\(215\) 9.82206 0.669859
\(216\) 0 0
\(217\) 6.81291 0.462491
\(218\) 0 0
\(219\) −12.5419 −0.847501
\(220\) 0 0
\(221\) 0.823399 0.0553878
\(222\) 0 0
\(223\) −22.1607 −1.48399 −0.741994 0.670406i \(-0.766120\pi\)
−0.741994 + 0.670406i \(0.766120\pi\)
\(224\) 0 0
\(225\) 4.54455 0.302970
\(226\) 0 0
\(227\) −1.61799 −0.107390 −0.0536949 0.998557i \(-0.517100\pi\)
−0.0536949 + 0.998557i \(0.517100\pi\)
\(228\) 0 0
\(229\) 27.3177 1.80520 0.902602 0.430475i \(-0.141654\pi\)
0.902602 + 0.430475i \(0.141654\pi\)
\(230\) 0 0
\(231\) −4.37319 −0.287735
\(232\) 0 0
\(233\) 23.8665 1.56355 0.781773 0.623563i \(-0.214315\pi\)
0.781773 + 0.623563i \(0.214315\pi\)
\(234\) 0 0
\(235\) 12.3241 0.803935
\(236\) 0 0
\(237\) 45.5364 2.95791
\(238\) 0 0
\(239\) 6.85642 0.443505 0.221752 0.975103i \(-0.428822\pi\)
0.221752 + 0.975103i \(0.428822\pi\)
\(240\) 0 0
\(241\) −16.8260 −1.08386 −0.541930 0.840423i \(-0.682306\pi\)
−0.541930 + 0.840423i \(0.682306\pi\)
\(242\) 0 0
\(243\) 18.1679 1.16547
\(244\) 0 0
\(245\) −1.44490 −0.0923110
\(246\) 0 0
\(247\) 17.7367 1.12856
\(248\) 0 0
\(249\) −38.0181 −2.40930
\(250\) 0 0
\(251\) 26.0167 1.64216 0.821081 0.570811i \(-0.193371\pi\)
0.821081 + 0.570811i \(0.193371\pi\)
\(252\) 0 0
\(253\) 5.29432 0.332851
\(254\) 0 0
\(255\) −0.431482 −0.0270205
\(256\) 0 0
\(257\) 7.00335 0.436857 0.218429 0.975853i \(-0.429907\pi\)
0.218429 + 0.975853i \(0.429907\pi\)
\(258\) 0 0
\(259\) −14.3881 −0.894031
\(260\) 0 0
\(261\) −9.25605 −0.572935
\(262\) 0 0
\(263\) 4.64211 0.286245 0.143122 0.989705i \(-0.454286\pi\)
0.143122 + 0.989705i \(0.454286\pi\)
\(264\) 0 0
\(265\) 5.83339 0.358343
\(266\) 0 0
\(267\) 10.8149 0.661862
\(268\) 0 0
\(269\) 16.7875 1.02355 0.511774 0.859120i \(-0.328988\pi\)
0.511774 + 0.859120i \(0.328988\pi\)
\(270\) 0 0
\(271\) 15.1380 0.919565 0.459783 0.888032i \(-0.347927\pi\)
0.459783 + 0.888032i \(0.347927\pi\)
\(272\) 0 0
\(273\) 33.9334 2.05374
\(274\) 0 0
\(275\) −0.675516 −0.0407352
\(276\) 0 0
\(277\) 12.4382 0.747338 0.373669 0.927562i \(-0.378100\pi\)
0.373669 + 0.927562i \(0.378100\pi\)
\(278\) 0 0
\(279\) −13.1364 −0.786458
\(280\) 0 0
\(281\) 12.2324 0.729725 0.364863 0.931061i \(-0.381116\pi\)
0.364863 + 0.931061i \(0.381116\pi\)
\(282\) 0 0
\(283\) 9.20676 0.547285 0.273643 0.961831i \(-0.411771\pi\)
0.273643 + 0.961831i \(0.411771\pi\)
\(284\) 0 0
\(285\) −9.29447 −0.550557
\(286\) 0 0
\(287\) 4.49522 0.265344
\(288\) 0 0
\(289\) −16.9753 −0.998548
\(290\) 0 0
\(291\) −10.8615 −0.636714
\(292\) 0 0
\(293\) 23.5556 1.37613 0.688067 0.725647i \(-0.258459\pi\)
0.688067 + 0.725647i \(0.258459\pi\)
\(294\) 0 0
\(295\) −0.257517 −0.0149932
\(296\) 0 0
\(297\) 2.86586 0.166294
\(298\) 0 0
\(299\) −41.0808 −2.37576
\(300\) 0 0
\(301\) −23.1499 −1.33434
\(302\) 0 0
\(303\) 3.10833 0.178569
\(304\) 0 0
\(305\) 7.24169 0.414658
\(306\) 0 0
\(307\) 28.4008 1.62092 0.810461 0.585793i \(-0.199217\pi\)
0.810461 + 0.585793i \(0.199217\pi\)
\(308\) 0 0
\(309\) 27.2509 1.55025
\(310\) 0 0
\(311\) 14.1693 0.803467 0.401733 0.915757i \(-0.368408\pi\)
0.401733 + 0.915757i \(0.368408\pi\)
\(312\) 0 0
\(313\) −26.7246 −1.51056 −0.755282 0.655400i \(-0.772500\pi\)
−0.755282 + 0.655400i \(0.772500\pi\)
\(314\) 0 0
\(315\) −10.7112 −0.603506
\(316\) 0 0
\(317\) −20.6263 −1.15849 −0.579244 0.815154i \(-0.696652\pi\)
−0.579244 + 0.815154i \(0.696652\pi\)
\(318\) 0 0
\(319\) 1.37585 0.0770327
\(320\) 0 0
\(321\) 0.567153 0.0316554
\(322\) 0 0
\(323\) 0.531562 0.0295769
\(324\) 0 0
\(325\) 5.24160 0.290752
\(326\) 0 0
\(327\) 48.6431 2.68997
\(328\) 0 0
\(329\) −29.0470 −1.60141
\(330\) 0 0
\(331\) 20.1479 1.10743 0.553715 0.832706i \(-0.313210\pi\)
0.553715 + 0.832706i \(0.313210\pi\)
\(332\) 0 0
\(333\) 27.7426 1.52029
\(334\) 0 0
\(335\) −14.7233 −0.804418
\(336\) 0 0
\(337\) 4.77718 0.260229 0.130115 0.991499i \(-0.458465\pi\)
0.130115 + 0.991499i \(0.458465\pi\)
\(338\) 0 0
\(339\) 3.66224 0.198906
\(340\) 0 0
\(341\) 1.95264 0.105742
\(342\) 0 0
\(343\) 19.9040 1.07472
\(344\) 0 0
\(345\) 21.5274 1.15900
\(346\) 0 0
\(347\) −19.1422 −1.02761 −0.513803 0.857908i \(-0.671764\pi\)
−0.513803 + 0.857908i \(0.671764\pi\)
\(348\) 0 0
\(349\) 9.80676 0.524944 0.262472 0.964940i \(-0.415462\pi\)
0.262472 + 0.964940i \(0.415462\pi\)
\(350\) 0 0
\(351\) −22.2373 −1.18694
\(352\) 0 0
\(353\) −10.8820 −0.579191 −0.289596 0.957149i \(-0.593521\pi\)
−0.289596 + 0.957149i \(0.593521\pi\)
\(354\) 0 0
\(355\) −3.19249 −0.169440
\(356\) 0 0
\(357\) 1.01697 0.0538239
\(358\) 0 0
\(359\) −6.58874 −0.347740 −0.173870 0.984769i \(-0.555627\pi\)
−0.173870 + 0.984769i \(0.555627\pi\)
\(360\) 0 0
\(361\) −7.54973 −0.397354
\(362\) 0 0
\(363\) 28.9607 1.52004
\(364\) 0 0
\(365\) 4.56610 0.239001
\(366\) 0 0
\(367\) 11.4470 0.597529 0.298764 0.954327i \(-0.403426\pi\)
0.298764 + 0.954327i \(0.403426\pi\)
\(368\) 0 0
\(369\) −8.66753 −0.451214
\(370\) 0 0
\(371\) −13.7489 −0.713807
\(372\) 0 0
\(373\) −32.8529 −1.70106 −0.850529 0.525928i \(-0.823718\pi\)
−0.850529 + 0.525928i \(0.823718\pi\)
\(374\) 0 0
\(375\) −2.74673 −0.141841
\(376\) 0 0
\(377\) −10.6758 −0.549830
\(378\) 0 0
\(379\) −16.6223 −0.853831 −0.426916 0.904291i \(-0.640400\pi\)
−0.426916 + 0.904291i \(0.640400\pi\)
\(380\) 0 0
\(381\) −15.0266 −0.769835
\(382\) 0 0
\(383\) 27.3113 1.39554 0.697771 0.716320i \(-0.254175\pi\)
0.697771 + 0.716320i \(0.254175\pi\)
\(384\) 0 0
\(385\) 1.59214 0.0811431
\(386\) 0 0
\(387\) 44.6369 2.26902
\(388\) 0 0
\(389\) −0.869264 −0.0440734 −0.0220367 0.999757i \(-0.507015\pi\)
−0.0220367 + 0.999757i \(0.507015\pi\)
\(390\) 0 0
\(391\) −1.23118 −0.0622634
\(392\) 0 0
\(393\) 5.05795 0.255140
\(394\) 0 0
\(395\) −16.5784 −0.834148
\(396\) 0 0
\(397\) −1.64722 −0.0826717 −0.0413359 0.999145i \(-0.513161\pi\)
−0.0413359 + 0.999145i \(0.513161\pi\)
\(398\) 0 0
\(399\) 21.9064 1.09669
\(400\) 0 0
\(401\) 1.00000 0.0499376
\(402\) 0 0
\(403\) −15.1513 −0.754742
\(404\) 0 0
\(405\) −1.98071 −0.0984224
\(406\) 0 0
\(407\) −4.12375 −0.204407
\(408\) 0 0
\(409\) 31.9538 1.58001 0.790007 0.613098i \(-0.210077\pi\)
0.790007 + 0.613098i \(0.210077\pi\)
\(410\) 0 0
\(411\) −37.4490 −1.84722
\(412\) 0 0
\(413\) 0.606948 0.0298660
\(414\) 0 0
\(415\) 13.8412 0.679437
\(416\) 0 0
\(417\) −22.4743 −1.10057
\(418\) 0 0
\(419\) 20.5057 1.00177 0.500885 0.865514i \(-0.333008\pi\)
0.500885 + 0.865514i \(0.333008\pi\)
\(420\) 0 0
\(421\) 26.5213 1.29257 0.646285 0.763096i \(-0.276322\pi\)
0.646285 + 0.763096i \(0.276322\pi\)
\(422\) 0 0
\(423\) 56.0075 2.72318
\(424\) 0 0
\(425\) 0.157089 0.00761995
\(426\) 0 0
\(427\) −17.0681 −0.825985
\(428\) 0 0
\(429\) 9.72561 0.469557
\(430\) 0 0
\(431\) 8.89089 0.428259 0.214130 0.976805i \(-0.431309\pi\)
0.214130 + 0.976805i \(0.431309\pi\)
\(432\) 0 0
\(433\) −3.33574 −0.160305 −0.0801527 0.996783i \(-0.525541\pi\)
−0.0801527 + 0.996783i \(0.525541\pi\)
\(434\) 0 0
\(435\) 5.59437 0.268230
\(436\) 0 0
\(437\) −26.5205 −1.26865
\(438\) 0 0
\(439\) 4.57491 0.218349 0.109174 0.994023i \(-0.465179\pi\)
0.109174 + 0.994023i \(0.465179\pi\)
\(440\) 0 0
\(441\) −6.56640 −0.312686
\(442\) 0 0
\(443\) −4.66705 −0.221738 −0.110869 0.993835i \(-0.535363\pi\)
−0.110869 + 0.993835i \(0.535363\pi\)
\(444\) 0 0
\(445\) −3.93737 −0.186649
\(446\) 0 0
\(447\) −36.5789 −1.73012
\(448\) 0 0
\(449\) 34.0697 1.60785 0.803925 0.594731i \(-0.202741\pi\)
0.803925 + 0.594731i \(0.202741\pi\)
\(450\) 0 0
\(451\) 1.28837 0.0606669
\(452\) 0 0
\(453\) −27.5708 −1.29539
\(454\) 0 0
\(455\) −12.3541 −0.579168
\(456\) 0 0
\(457\) 24.8840 1.16402 0.582011 0.813181i \(-0.302266\pi\)
0.582011 + 0.813181i \(0.302266\pi\)
\(458\) 0 0
\(459\) −0.666446 −0.0311070
\(460\) 0 0
\(461\) 1.37550 0.0640635 0.0320318 0.999487i \(-0.489802\pi\)
0.0320318 + 0.999487i \(0.489802\pi\)
\(462\) 0 0
\(463\) 23.4902 1.09168 0.545840 0.837889i \(-0.316210\pi\)
0.545840 + 0.837889i \(0.316210\pi\)
\(464\) 0 0
\(465\) 7.93969 0.368194
\(466\) 0 0
\(467\) 13.7970 0.638450 0.319225 0.947679i \(-0.396577\pi\)
0.319225 + 0.947679i \(0.396577\pi\)
\(468\) 0 0
\(469\) 34.7017 1.60237
\(470\) 0 0
\(471\) 9.47421 0.436549
\(472\) 0 0
\(473\) −6.63497 −0.305076
\(474\) 0 0
\(475\) 3.38382 0.155261
\(476\) 0 0
\(477\) 26.5102 1.21382
\(478\) 0 0
\(479\) 1.65571 0.0756515 0.0378258 0.999284i \(-0.487957\pi\)
0.0378258 + 0.999284i \(0.487957\pi\)
\(480\) 0 0
\(481\) 31.9978 1.45898
\(482\) 0 0
\(483\) −50.7385 −2.30868
\(484\) 0 0
\(485\) 3.95434 0.179558
\(486\) 0 0
\(487\) 20.0482 0.908470 0.454235 0.890882i \(-0.349913\pi\)
0.454235 + 0.890882i \(0.349913\pi\)
\(488\) 0 0
\(489\) −11.6846 −0.528394
\(490\) 0 0
\(491\) −27.5603 −1.24378 −0.621888 0.783106i \(-0.713634\pi\)
−0.621888 + 0.783106i \(0.713634\pi\)
\(492\) 0 0
\(493\) −0.319949 −0.0144098
\(494\) 0 0
\(495\) −3.06992 −0.137983
\(496\) 0 0
\(497\) 7.52446 0.337518
\(498\) 0 0
\(499\) 42.5205 1.90348 0.951739 0.306909i \(-0.0992946\pi\)
0.951739 + 0.306909i \(0.0992946\pi\)
\(500\) 0 0
\(501\) 52.8894 2.36292
\(502\) 0 0
\(503\) −23.8194 −1.06206 −0.531028 0.847354i \(-0.678194\pi\)
−0.531028 + 0.847354i \(0.678194\pi\)
\(504\) 0 0
\(505\) −1.13165 −0.0503576
\(506\) 0 0
\(507\) −39.7573 −1.76569
\(508\) 0 0
\(509\) 42.1383 1.86775 0.933873 0.357605i \(-0.116406\pi\)
0.933873 + 0.357605i \(0.116406\pi\)
\(510\) 0 0
\(511\) −10.7620 −0.476081
\(512\) 0 0
\(513\) −14.3558 −0.633823
\(514\) 0 0
\(515\) −9.92119 −0.437180
\(516\) 0 0
\(517\) −8.32513 −0.366139
\(518\) 0 0
\(519\) 2.60925 0.114533
\(520\) 0 0
\(521\) 24.8252 1.08761 0.543806 0.839211i \(-0.316983\pi\)
0.543806 + 0.839211i \(0.316983\pi\)
\(522\) 0 0
\(523\) −2.29873 −0.100516 −0.0502582 0.998736i \(-0.516004\pi\)
−0.0502582 + 0.998736i \(0.516004\pi\)
\(524\) 0 0
\(525\) 6.47385 0.282542
\(526\) 0 0
\(527\) −0.454081 −0.0197801
\(528\) 0 0
\(529\) 38.4256 1.67068
\(530\) 0 0
\(531\) −1.17030 −0.0507866
\(532\) 0 0
\(533\) −9.99697 −0.433017
\(534\) 0 0
\(535\) −0.206483 −0.00892703
\(536\) 0 0
\(537\) 38.0574 1.64230
\(538\) 0 0
\(539\) 0.976051 0.0420415
\(540\) 0 0
\(541\) −7.97452 −0.342851 −0.171426 0.985197i \(-0.554837\pi\)
−0.171426 + 0.985197i \(0.554837\pi\)
\(542\) 0 0
\(543\) 59.7436 2.56385
\(544\) 0 0
\(545\) −17.7094 −0.758589
\(546\) 0 0
\(547\) −6.69184 −0.286122 −0.143061 0.989714i \(-0.545695\pi\)
−0.143061 + 0.989714i \(0.545695\pi\)
\(548\) 0 0
\(549\) 32.9102 1.40457
\(550\) 0 0
\(551\) −6.89196 −0.293607
\(552\) 0 0
\(553\) 39.0740 1.66160
\(554\) 0 0
\(555\) −16.7677 −0.711749
\(556\) 0 0
\(557\) −22.1841 −0.939972 −0.469986 0.882674i \(-0.655741\pi\)
−0.469986 + 0.882674i \(0.655741\pi\)
\(558\) 0 0
\(559\) 51.4834 2.17751
\(560\) 0 0
\(561\) 0.291473 0.0123060
\(562\) 0 0
\(563\) 20.4256 0.860838 0.430419 0.902629i \(-0.358366\pi\)
0.430419 + 0.902629i \(0.358366\pi\)
\(564\) 0 0
\(565\) −1.33331 −0.0560927
\(566\) 0 0
\(567\) 4.66840 0.196054
\(568\) 0 0
\(569\) −35.9291 −1.50622 −0.753112 0.657892i \(-0.771448\pi\)
−0.753112 + 0.657892i \(0.771448\pi\)
\(570\) 0 0
\(571\) −23.0430 −0.964319 −0.482160 0.876083i \(-0.660147\pi\)
−0.482160 + 0.876083i \(0.660147\pi\)
\(572\) 0 0
\(573\) 7.93889 0.331652
\(574\) 0 0
\(575\) −7.83745 −0.326844
\(576\) 0 0
\(577\) −3.07596 −0.128054 −0.0640269 0.997948i \(-0.520394\pi\)
−0.0640269 + 0.997948i \(0.520394\pi\)
\(578\) 0 0
\(579\) 46.9516 1.95124
\(580\) 0 0
\(581\) −32.6227 −1.35342
\(582\) 0 0
\(583\) −3.94055 −0.163201
\(584\) 0 0
\(585\) 23.8207 0.984866
\(586\) 0 0
\(587\) 27.6217 1.14007 0.570034 0.821621i \(-0.306930\pi\)
0.570034 + 0.821621i \(0.306930\pi\)
\(588\) 0 0
\(589\) −9.78126 −0.403030
\(590\) 0 0
\(591\) −60.0667 −2.47081
\(592\) 0 0
\(593\) −8.67058 −0.356058 −0.178029 0.984025i \(-0.556972\pi\)
−0.178029 + 0.984025i \(0.556972\pi\)
\(594\) 0 0
\(595\) −0.370248 −0.0151787
\(596\) 0 0
\(597\) −27.7464 −1.13558
\(598\) 0 0
\(599\) 27.5487 1.12561 0.562805 0.826590i \(-0.309722\pi\)
0.562805 + 0.826590i \(0.309722\pi\)
\(600\) 0 0
\(601\) −27.1909 −1.10914 −0.554569 0.832138i \(-0.687117\pi\)
−0.554569 + 0.832138i \(0.687117\pi\)
\(602\) 0 0
\(603\) −66.9106 −2.72481
\(604\) 0 0
\(605\) −10.5437 −0.428661
\(606\) 0 0
\(607\) −7.75401 −0.314726 −0.157363 0.987541i \(-0.550299\pi\)
−0.157363 + 0.987541i \(0.550299\pi\)
\(608\) 0 0
\(609\) −13.1855 −0.534305
\(610\) 0 0
\(611\) 64.5980 2.61336
\(612\) 0 0
\(613\) 13.1104 0.529525 0.264762 0.964314i \(-0.414706\pi\)
0.264762 + 0.964314i \(0.414706\pi\)
\(614\) 0 0
\(615\) 5.23867 0.211244
\(616\) 0 0
\(617\) 14.4737 0.582689 0.291344 0.956618i \(-0.405897\pi\)
0.291344 + 0.956618i \(0.405897\pi\)
\(618\) 0 0
\(619\) 4.72697 0.189993 0.0949966 0.995478i \(-0.469716\pi\)
0.0949966 + 0.995478i \(0.469716\pi\)
\(620\) 0 0
\(621\) 33.2501 1.33428
\(622\) 0 0
\(623\) 9.28009 0.371799
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.27857 0.250742
\(628\) 0 0
\(629\) 0.958965 0.0382365
\(630\) 0 0
\(631\) −12.4211 −0.494475 −0.247237 0.968955i \(-0.579523\pi\)
−0.247237 + 0.968955i \(0.579523\pi\)
\(632\) 0 0
\(633\) 54.2484 2.15618
\(634\) 0 0
\(635\) 5.47071 0.217098
\(636\) 0 0
\(637\) −7.57357 −0.300076
\(638\) 0 0
\(639\) −14.5084 −0.573944
\(640\) 0 0
\(641\) −35.7557 −1.41227 −0.706133 0.708079i \(-0.749562\pi\)
−0.706133 + 0.708079i \(0.749562\pi\)
\(642\) 0 0
\(643\) 3.62857 0.143097 0.0715483 0.997437i \(-0.477206\pi\)
0.0715483 + 0.997437i \(0.477206\pi\)
\(644\) 0 0
\(645\) −26.9786 −1.06228
\(646\) 0 0
\(647\) −7.23846 −0.284573 −0.142287 0.989825i \(-0.545445\pi\)
−0.142287 + 0.989825i \(0.545445\pi\)
\(648\) 0 0
\(649\) 0.173957 0.00682840
\(650\) 0 0
\(651\) −18.7133 −0.733431
\(652\) 0 0
\(653\) 26.7184 1.04557 0.522787 0.852463i \(-0.324892\pi\)
0.522787 + 0.852463i \(0.324892\pi\)
\(654\) 0 0
\(655\) −1.84144 −0.0719510
\(656\) 0 0
\(657\) 20.7509 0.809569
\(658\) 0 0
\(659\) −10.2410 −0.398932 −0.199466 0.979905i \(-0.563921\pi\)
−0.199466 + 0.979905i \(0.563921\pi\)
\(660\) 0 0
\(661\) −33.3705 −1.29796 −0.648981 0.760804i \(-0.724805\pi\)
−0.648981 + 0.760804i \(0.724805\pi\)
\(662\) 0 0
\(663\) −2.26166 −0.0878356
\(664\) 0 0
\(665\) −7.97543 −0.309274
\(666\) 0 0
\(667\) 15.9628 0.618082
\(668\) 0 0
\(669\) 60.8695 2.35335
\(670\) 0 0
\(671\) −4.89188 −0.188849
\(672\) 0 0
\(673\) 31.1827 1.20201 0.601003 0.799247i \(-0.294768\pi\)
0.601003 + 0.799247i \(0.294768\pi\)
\(674\) 0 0
\(675\) −4.24247 −0.163293
\(676\) 0 0
\(677\) 34.9394 1.34283 0.671415 0.741082i \(-0.265687\pi\)
0.671415 + 0.741082i \(0.265687\pi\)
\(678\) 0 0
\(679\) −9.32010 −0.357673
\(680\) 0 0
\(681\) 4.44419 0.170302
\(682\) 0 0
\(683\) 9.69137 0.370830 0.185415 0.982660i \(-0.440637\pi\)
0.185415 + 0.982660i \(0.440637\pi\)
\(684\) 0 0
\(685\) 13.6340 0.520929
\(686\) 0 0
\(687\) −75.0345 −2.86274
\(688\) 0 0
\(689\) 30.5763 1.16487
\(690\) 0 0
\(691\) 2.59884 0.0988647 0.0494324 0.998777i \(-0.484259\pi\)
0.0494324 + 0.998777i \(0.484259\pi\)
\(692\) 0 0
\(693\) 7.23557 0.274857
\(694\) 0 0
\(695\) 8.18219 0.310368
\(696\) 0 0
\(697\) −0.299606 −0.0113484
\(698\) 0 0
\(699\) −65.5550 −2.47952
\(700\) 0 0
\(701\) 5.36879 0.202776 0.101388 0.994847i \(-0.467672\pi\)
0.101388 + 0.994847i \(0.467672\pi\)
\(702\) 0 0
\(703\) 20.6569 0.779088
\(704\) 0 0
\(705\) −33.8510 −1.27490
\(706\) 0 0
\(707\) 2.66721 0.100311
\(708\) 0 0
\(709\) 19.3711 0.727497 0.363748 0.931497i \(-0.381497\pi\)
0.363748 + 0.931497i \(0.381497\pi\)
\(710\) 0 0
\(711\) −75.3412 −2.82552
\(712\) 0 0
\(713\) 22.6549 0.848431
\(714\) 0 0
\(715\) −3.54079 −0.132418
\(716\) 0 0
\(717\) −18.8328 −0.703322
\(718\) 0 0
\(719\) 14.2067 0.529821 0.264911 0.964273i \(-0.414658\pi\)
0.264911 + 0.964273i \(0.414658\pi\)
\(720\) 0 0
\(721\) 23.3835 0.870848
\(722\) 0 0
\(723\) 46.2167 1.71882
\(724\) 0 0
\(725\) −2.03674 −0.0756425
\(726\) 0 0
\(727\) 6.92275 0.256751 0.128375 0.991726i \(-0.459024\pi\)
0.128375 + 0.991726i \(0.459024\pi\)
\(728\) 0 0
\(729\) −43.9603 −1.62816
\(730\) 0 0
\(731\) 1.54294 0.0570677
\(732\) 0 0
\(733\) −41.2470 −1.52349 −0.761747 0.647875i \(-0.775658\pi\)
−0.761747 + 0.647875i \(0.775658\pi\)
\(734\) 0 0
\(735\) 3.96874 0.146389
\(736\) 0 0
\(737\) 9.94581 0.366359
\(738\) 0 0
\(739\) −47.2304 −1.73740 −0.868699 0.495341i \(-0.835043\pi\)
−0.868699 + 0.495341i \(0.835043\pi\)
\(740\) 0 0
\(741\) −48.7179 −1.78970
\(742\) 0 0
\(743\) 49.5974 1.81955 0.909777 0.415098i \(-0.136253\pi\)
0.909777 + 0.415098i \(0.136253\pi\)
\(744\) 0 0
\(745\) 13.3172 0.487905
\(746\) 0 0
\(747\) 62.9020 2.30146
\(748\) 0 0
\(749\) 0.486665 0.0177823
\(750\) 0 0
\(751\) 41.3945 1.51051 0.755254 0.655432i \(-0.227514\pi\)
0.755254 + 0.655432i \(0.227514\pi\)
\(752\) 0 0
\(753\) −71.4611 −2.60419
\(754\) 0 0
\(755\) 10.0377 0.365308
\(756\) 0 0
\(757\) −21.3215 −0.774943 −0.387472 0.921882i \(-0.626652\pi\)
−0.387472 + 0.921882i \(0.626652\pi\)
\(758\) 0 0
\(759\) −14.5421 −0.527845
\(760\) 0 0
\(761\) 39.4007 1.42827 0.714136 0.700007i \(-0.246820\pi\)
0.714136 + 0.700007i \(0.246820\pi\)
\(762\) 0 0
\(763\) 41.7398 1.51108
\(764\) 0 0
\(765\) 0.713900 0.0258111
\(766\) 0 0
\(767\) −1.34980 −0.0487385
\(768\) 0 0
\(769\) 50.9959 1.83896 0.919479 0.393138i \(-0.128611\pi\)
0.919479 + 0.393138i \(0.128611\pi\)
\(770\) 0 0
\(771\) −19.2363 −0.692780
\(772\) 0 0
\(773\) −11.0573 −0.397705 −0.198852 0.980029i \(-0.563721\pi\)
−0.198852 + 0.980029i \(0.563721\pi\)
\(774\) 0 0
\(775\) −2.89059 −0.103833
\(776\) 0 0
\(777\) 39.5202 1.41778
\(778\) 0 0
\(779\) −6.45375 −0.231230
\(780\) 0 0
\(781\) 2.15658 0.0771684
\(782\) 0 0
\(783\) 8.64079 0.308797
\(784\) 0 0
\(785\) −3.44926 −0.123110
\(786\) 0 0
\(787\) −23.1182 −0.824073 −0.412037 0.911167i \(-0.635182\pi\)
−0.412037 + 0.911167i \(0.635182\pi\)
\(788\) 0 0
\(789\) −12.7506 −0.453935
\(790\) 0 0
\(791\) 3.14251 0.111735
\(792\) 0 0
\(793\) 37.9581 1.34793
\(794\) 0 0
\(795\) −16.0228 −0.568270
\(796\) 0 0
\(797\) −24.7864 −0.877978 −0.438989 0.898492i \(-0.644663\pi\)
−0.438989 + 0.898492i \(0.644663\pi\)
\(798\) 0 0
\(799\) 1.93598 0.0684901
\(800\) 0 0
\(801\) −17.8936 −0.632238
\(802\) 0 0
\(803\) −3.08448 −0.108849
\(804\) 0 0
\(805\) 18.4723 0.651063
\(806\) 0 0
\(807\) −46.1107 −1.62317
\(808\) 0 0
\(809\) 21.5187 0.756557 0.378279 0.925692i \(-0.376516\pi\)
0.378279 + 0.925692i \(0.376516\pi\)
\(810\) 0 0
\(811\) 11.0224 0.387050 0.193525 0.981095i \(-0.438008\pi\)
0.193525 + 0.981095i \(0.438008\pi\)
\(812\) 0 0
\(813\) −41.5799 −1.45827
\(814\) 0 0
\(815\) 4.25398 0.149011
\(816\) 0 0
\(817\) 33.2361 1.16279
\(818\) 0 0
\(819\) −56.1437 −1.96182
\(820\) 0 0
\(821\) −53.0765 −1.85238 −0.926191 0.377055i \(-0.876937\pi\)
−0.926191 + 0.377055i \(0.876937\pi\)
\(822\) 0 0
\(823\) 15.7220 0.548035 0.274018 0.961725i \(-0.411647\pi\)
0.274018 + 0.961725i \(0.411647\pi\)
\(824\) 0 0
\(825\) 1.85546 0.0645990
\(826\) 0 0
\(827\) 2.15905 0.0750774 0.0375387 0.999295i \(-0.488048\pi\)
0.0375387 + 0.999295i \(0.488048\pi\)
\(828\) 0 0
\(829\) 52.3343 1.81764 0.908822 0.417184i \(-0.136983\pi\)
0.908822 + 0.417184i \(0.136983\pi\)
\(830\) 0 0
\(831\) −34.1644 −1.18515
\(832\) 0 0
\(833\) −0.226978 −0.00786430
\(834\) 0 0
\(835\) −19.2554 −0.666360
\(836\) 0 0
\(837\) 12.2632 0.423880
\(838\) 0 0
\(839\) 13.2239 0.456539 0.228269 0.973598i \(-0.426693\pi\)
0.228269 + 0.973598i \(0.426693\pi\)
\(840\) 0 0
\(841\) −24.8517 −0.856955
\(842\) 0 0
\(843\) −33.5992 −1.15722
\(844\) 0 0
\(845\) 14.4744 0.497935
\(846\) 0 0
\(847\) 24.8507 0.853879
\(848\) 0 0
\(849\) −25.2885 −0.867900
\(850\) 0 0
\(851\) −47.8444 −1.64008
\(852\) 0 0
\(853\) 37.4094 1.28087 0.640436 0.768011i \(-0.278754\pi\)
0.640436 + 0.768011i \(0.278754\pi\)
\(854\) 0 0
\(855\) 15.3780 0.525915
\(856\) 0 0
\(857\) −0.587197 −0.0200583 −0.0100291 0.999950i \(-0.503192\pi\)
−0.0100291 + 0.999950i \(0.503192\pi\)
\(858\) 0 0
\(859\) 34.6017 1.18060 0.590298 0.807186i \(-0.299010\pi\)
0.590298 + 0.807186i \(0.299010\pi\)
\(860\) 0 0
\(861\) −12.3472 −0.420790
\(862\) 0 0
\(863\) 9.56387 0.325558 0.162779 0.986663i \(-0.447954\pi\)
0.162779 + 0.986663i \(0.447954\pi\)
\(864\) 0 0
\(865\) −0.949945 −0.0322991
\(866\) 0 0
\(867\) 46.6267 1.58353
\(868\) 0 0
\(869\) 11.1990 0.379899
\(870\) 0 0
\(871\) −77.1735 −2.61493
\(872\) 0 0
\(873\) 17.9707 0.608217
\(874\) 0 0
\(875\) −2.35693 −0.0796787
\(876\) 0 0
\(877\) −5.59274 −0.188853 −0.0944266 0.995532i \(-0.530102\pi\)
−0.0944266 + 0.995532i \(0.530102\pi\)
\(878\) 0 0
\(879\) −64.7011 −2.18231
\(880\) 0 0
\(881\) 29.9843 1.01020 0.505098 0.863062i \(-0.331456\pi\)
0.505098 + 0.863062i \(0.331456\pi\)
\(882\) 0 0
\(883\) 28.2756 0.951551 0.475775 0.879567i \(-0.342168\pi\)
0.475775 + 0.879567i \(0.342168\pi\)
\(884\) 0 0
\(885\) 0.707330 0.0237766
\(886\) 0 0
\(887\) 26.5686 0.892087 0.446044 0.895011i \(-0.352833\pi\)
0.446044 + 0.895011i \(0.352833\pi\)
\(888\) 0 0
\(889\) −12.8941 −0.432453
\(890\) 0 0
\(891\) 1.33800 0.0448248
\(892\) 0 0
\(893\) 41.7026 1.39552
\(894\) 0 0
\(895\) −13.8555 −0.463139
\(896\) 0 0
\(897\) 112.838 3.76755
\(898\) 0 0
\(899\) 5.88737 0.196355
\(900\) 0 0
\(901\) 0.916364 0.0305285
\(902\) 0 0
\(903\) 63.5866 2.11603
\(904\) 0 0
\(905\) −21.7508 −0.723021
\(906\) 0 0
\(907\) 21.9310 0.728205 0.364103 0.931359i \(-0.381376\pi\)
0.364103 + 0.931359i \(0.381376\pi\)
\(908\) 0 0
\(909\) −5.14283 −0.170577
\(910\) 0 0
\(911\) 29.1605 0.966130 0.483065 0.875585i \(-0.339524\pi\)
0.483065 + 0.875585i \(0.339524\pi\)
\(912\) 0 0
\(913\) −9.34995 −0.309438
\(914\) 0 0
\(915\) −19.8910 −0.657576
\(916\) 0 0
\(917\) 4.34014 0.143324
\(918\) 0 0
\(919\) 30.8207 1.01668 0.508341 0.861156i \(-0.330259\pi\)
0.508341 + 0.861156i \(0.330259\pi\)
\(920\) 0 0
\(921\) −78.0096 −2.57050
\(922\) 0 0
\(923\) −16.7338 −0.550798
\(924\) 0 0
\(925\) 6.10459 0.200718
\(926\) 0 0
\(927\) −45.0873 −1.48086
\(928\) 0 0
\(929\) −46.0881 −1.51210 −0.756050 0.654514i \(-0.772873\pi\)
−0.756050 + 0.654514i \(0.772873\pi\)
\(930\) 0 0
\(931\) −4.88927 −0.160239
\(932\) 0 0
\(933\) −38.9193 −1.27416
\(934\) 0 0
\(935\) −0.106116 −0.00347038
\(936\) 0 0
\(937\) −41.5416 −1.35711 −0.678553 0.734551i \(-0.737393\pi\)
−0.678553 + 0.734551i \(0.737393\pi\)
\(938\) 0 0
\(939\) 73.4054 2.39549
\(940\) 0 0
\(941\) −9.82853 −0.320401 −0.160201 0.987084i \(-0.551214\pi\)
−0.160201 + 0.987084i \(0.551214\pi\)
\(942\) 0 0
\(943\) 14.9479 0.486769
\(944\) 0 0
\(945\) 9.99919 0.325274
\(946\) 0 0
\(947\) 30.1718 0.980453 0.490226 0.871595i \(-0.336914\pi\)
0.490226 + 0.871595i \(0.336914\pi\)
\(948\) 0 0
\(949\) 23.9337 0.776920
\(950\) 0 0
\(951\) 56.6550 1.83716
\(952\) 0 0
\(953\) 33.4222 1.08265 0.541326 0.840813i \(-0.317922\pi\)
0.541326 + 0.840813i \(0.317922\pi\)
\(954\) 0 0
\(955\) −2.89030 −0.0935280
\(956\) 0 0
\(957\) −3.77909 −0.122161
\(958\) 0 0
\(959\) −32.1344 −1.03767
\(960\) 0 0
\(961\) −22.6445 −0.730467
\(962\) 0 0
\(963\) −0.938372 −0.0302386
\(964\) 0 0
\(965\) −17.0936 −0.550263
\(966\) 0 0
\(967\) 29.1891 0.938659 0.469329 0.883023i \(-0.344496\pi\)
0.469329 + 0.883023i \(0.344496\pi\)
\(968\) 0 0
\(969\) −1.46006 −0.0469039
\(970\) 0 0
\(971\) −53.2958 −1.71034 −0.855172 0.518344i \(-0.826549\pi\)
−0.855172 + 0.518344i \(0.826549\pi\)
\(972\) 0 0
\(973\) −19.2848 −0.618243
\(974\) 0 0
\(975\) −14.3973 −0.461082
\(976\) 0 0
\(977\) −38.7273 −1.23900 −0.619499 0.784998i \(-0.712664\pi\)
−0.619499 + 0.784998i \(0.712664\pi\)
\(978\) 0 0
\(979\) 2.65976 0.0850063
\(980\) 0 0
\(981\) −80.4814 −2.56957
\(982\) 0 0
\(983\) 59.0720 1.88410 0.942052 0.335468i \(-0.108894\pi\)
0.942052 + 0.335468i \(0.108894\pi\)
\(984\) 0 0
\(985\) 21.8684 0.696785
\(986\) 0 0
\(987\) 79.7843 2.53956
\(988\) 0 0
\(989\) −76.9799 −2.44782
\(990\) 0 0
\(991\) −38.7595 −1.23124 −0.615618 0.788045i \(-0.711093\pi\)
−0.615618 + 0.788045i \(0.711093\pi\)
\(992\) 0 0
\(993\) −55.3410 −1.75619
\(994\) 0 0
\(995\) 10.1016 0.320242
\(996\) 0 0
\(997\) −16.4228 −0.520115 −0.260058 0.965593i \(-0.583742\pi\)
−0.260058 + 0.965593i \(0.583742\pi\)
\(998\) 0 0
\(999\) −25.8985 −0.819394
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.f.1.4 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.f.1.4 37 1.1 even 1 trivial