Properties

Label 3800.2.a.be.1.3
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
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] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.253565184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.366738\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.366738 q^{3} -3.08675 q^{7} -2.86550 q^{9} +O(q^{10})\) \(q+0.366738 q^{3} -3.08675 q^{7} -2.86550 q^{9} -4.94567 q^{11} +3.79016 q^{13} -3.97713 q^{17} +1.00000 q^{19} -1.13203 q^{21} -6.56426 q^{23} -2.15110 q^{27} +5.14084 q^{29} +4.25626 q^{31} -1.81376 q^{33} +7.75503 q^{37} +1.39000 q^{39} -2.25626 q^{41} +10.2276 q^{43} -0.935652 q^{47} +2.52804 q^{49} -1.45856 q^{51} +7.52117 q^{53} +0.366738 q^{57} -10.4356 q^{59} +10.9163 q^{61} +8.84510 q^{63} +6.80924 q^{67} -2.40736 q^{69} -2.98093 q^{71} +2.19638 q^{73} +15.2661 q^{77} -3.26652 q^{79} +7.80762 q^{81} +4.09936 q^{83} +1.88534 q^{87} -12.4134 q^{89} -11.6993 q^{91} +1.56093 q^{93} +10.6172 q^{97} +14.1718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} + 6 q^{7} + 8 q^{9} - 2 q^{11} + 14 q^{13} + 10 q^{17} + 6 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{27} - 2 q^{29} + 8 q^{31} + 8 q^{33} + 4 q^{37} - 18 q^{39} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 2 q^{49} - 34 q^{51} - 14 q^{53} + 2 q^{57} - 2 q^{59} + 10 q^{61} + 44 q^{63} + 42 q^{67} + 18 q^{69} - 8 q^{71} - 2 q^{73} + 24 q^{77} - 20 q^{79} + 6 q^{81} + 16 q^{83} + 30 q^{87} + 32 q^{89} + 10 q^{91} + 12 q^{93} + 40 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.366738 0.211736 0.105868 0.994380i \(-0.466238\pi\)
0.105868 + 0.994380i \(0.466238\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.08675 −1.16668 −0.583341 0.812227i \(-0.698255\pi\)
−0.583341 + 0.812227i \(0.698255\pi\)
\(8\) 0 0
\(9\) −2.86550 −0.955168
\(10\) 0 0
\(11\) −4.94567 −1.49118 −0.745588 0.666407i \(-0.767831\pi\)
−0.745588 + 0.666407i \(0.767831\pi\)
\(12\) 0 0
\(13\) 3.79016 1.05120 0.525601 0.850731i \(-0.323840\pi\)
0.525601 + 0.850731i \(0.323840\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.97713 −0.964596 −0.482298 0.876007i \(-0.660198\pi\)
−0.482298 + 0.876007i \(0.660198\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.13203 −0.247029
\(22\) 0 0
\(23\) −6.56426 −1.36874 −0.684372 0.729133i \(-0.739923\pi\)
−0.684372 + 0.729133i \(0.739923\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.15110 −0.413980
\(28\) 0 0
\(29\) 5.14084 0.954630 0.477315 0.878732i \(-0.341610\pi\)
0.477315 + 0.878732i \(0.341610\pi\)
\(30\) 0 0
\(31\) 4.25626 0.764447 0.382224 0.924070i \(-0.375158\pi\)
0.382224 + 0.924070i \(0.375158\pi\)
\(32\) 0 0
\(33\) −1.81376 −0.315736
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.75503 1.27492 0.637459 0.770484i \(-0.279985\pi\)
0.637459 + 0.770484i \(0.279985\pi\)
\(38\) 0 0
\(39\) 1.39000 0.222577
\(40\) 0 0
\(41\) −2.25626 −0.352369 −0.176184 0.984357i \(-0.556376\pi\)
−0.176184 + 0.984357i \(0.556376\pi\)
\(42\) 0 0
\(43\) 10.2276 1.55969 0.779846 0.625971i \(-0.215297\pi\)
0.779846 + 0.625971i \(0.215297\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.935652 −0.136479 −0.0682395 0.997669i \(-0.521738\pi\)
−0.0682395 + 0.997669i \(0.521738\pi\)
\(48\) 0 0
\(49\) 2.52804 0.361149
\(50\) 0 0
\(51\) −1.45856 −0.204240
\(52\) 0 0
\(53\) 7.52117 1.03311 0.516556 0.856253i \(-0.327214\pi\)
0.516556 + 0.856253i \(0.327214\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.366738 0.0485756
\(58\) 0 0
\(59\) −10.4356 −1.35860 −0.679298 0.733863i \(-0.737715\pi\)
−0.679298 + 0.733863i \(0.737715\pi\)
\(60\) 0 0
\(61\) 10.9163 1.39769 0.698847 0.715271i \(-0.253697\pi\)
0.698847 + 0.715271i \(0.253697\pi\)
\(62\) 0 0
\(63\) 8.84510 1.11438
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.80924 0.831881 0.415940 0.909392i \(-0.363453\pi\)
0.415940 + 0.909392i \(0.363453\pi\)
\(68\) 0 0
\(69\) −2.40736 −0.289812
\(70\) 0 0
\(71\) −2.98093 −0.353771 −0.176886 0.984231i \(-0.556602\pi\)
−0.176886 + 0.984231i \(0.556602\pi\)
\(72\) 0 0
\(73\) 2.19638 0.257066 0.128533 0.991705i \(-0.458973\pi\)
0.128533 + 0.991705i \(0.458973\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.2661 1.73973
\(78\) 0 0
\(79\) −3.26652 −0.367513 −0.183756 0.982972i \(-0.558826\pi\)
−0.183756 + 0.982972i \(0.558826\pi\)
\(80\) 0 0
\(81\) 7.80762 0.867513
\(82\) 0 0
\(83\) 4.09936 0.449963 0.224982 0.974363i \(-0.427768\pi\)
0.224982 + 0.974363i \(0.427768\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.88534 0.202129
\(88\) 0 0
\(89\) −12.4134 −1.31581 −0.657907 0.753099i \(-0.728558\pi\)
−0.657907 + 0.753099i \(0.728558\pi\)
\(90\) 0 0
\(91\) −11.6993 −1.22642
\(92\) 0 0
\(93\) 1.56093 0.161861
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.6172 1.07801 0.539007 0.842302i \(-0.318800\pi\)
0.539007 + 0.842302i \(0.318800\pi\)
\(98\) 0 0
\(99\) 14.1718 1.42432
\(100\) 0 0
\(101\) −13.1119 −1.30469 −0.652343 0.757924i \(-0.726214\pi\)
−0.652343 + 0.757924i \(0.726214\pi\)
\(102\) 0 0
\(103\) 4.95691 0.488419 0.244209 0.969723i \(-0.421472\pi\)
0.244209 + 0.969723i \(0.421472\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.9142 1.82850 0.914251 0.405148i \(-0.132780\pi\)
0.914251 + 0.405148i \(0.132780\pi\)
\(108\) 0 0
\(109\) 8.90118 0.852578 0.426289 0.904587i \(-0.359821\pi\)
0.426289 + 0.904587i \(0.359821\pi\)
\(110\) 0 0
\(111\) 2.84406 0.269946
\(112\) 0 0
\(113\) −9.85071 −0.926677 −0.463339 0.886181i \(-0.653349\pi\)
−0.463339 + 0.886181i \(0.653349\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.8607 −1.00407
\(118\) 0 0
\(119\) 12.2764 1.12538
\(120\) 0 0
\(121\) 13.4596 1.22360
\(122\) 0 0
\(123\) −0.827456 −0.0746092
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.76384 −0.333987 −0.166993 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(128\) 0 0
\(129\) 3.75084 0.330243
\(130\) 0 0
\(131\) 10.0488 0.877966 0.438983 0.898495i \(-0.355339\pi\)
0.438983 + 0.898495i \(0.355339\pi\)
\(132\) 0 0
\(133\) −3.08675 −0.267655
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.6244 1.24945 0.624724 0.780845i \(-0.285211\pi\)
0.624724 + 0.780845i \(0.285211\pi\)
\(138\) 0 0
\(139\) −12.0773 −1.02438 −0.512191 0.858872i \(-0.671166\pi\)
−0.512191 + 0.858872i \(0.671166\pi\)
\(140\) 0 0
\(141\) −0.343139 −0.0288975
\(142\) 0 0
\(143\) −18.7449 −1.56753
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.927129 0.0764683
\(148\) 0 0
\(149\) 12.5739 1.03009 0.515046 0.857162i \(-0.327775\pi\)
0.515046 + 0.857162i \(0.327775\pi\)
\(150\) 0 0
\(151\) −0.390336 −0.0317651 −0.0158826 0.999874i \(-0.505056\pi\)
−0.0158826 + 0.999874i \(0.505056\pi\)
\(152\) 0 0
\(153\) 11.3965 0.921351
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.6517 1.40876 0.704379 0.709824i \(-0.251226\pi\)
0.704379 + 0.709824i \(0.251226\pi\)
\(158\) 0 0
\(159\) 2.75830 0.218747
\(160\) 0 0
\(161\) 20.2623 1.59689
\(162\) 0 0
\(163\) 5.72733 0.448599 0.224300 0.974520i \(-0.427991\pi\)
0.224300 + 0.974520i \(0.427991\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.22343 0.636348 0.318174 0.948032i \(-0.396930\pi\)
0.318174 + 0.948032i \(0.396930\pi\)
\(168\) 0 0
\(169\) 1.36534 0.105027
\(170\) 0 0
\(171\) −2.86550 −0.219131
\(172\) 0 0
\(173\) −15.1427 −1.15127 −0.575637 0.817705i \(-0.695246\pi\)
−0.575637 + 0.817705i \(0.695246\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.82712 −0.287664
\(178\) 0 0
\(179\) 4.07415 0.304516 0.152258 0.988341i \(-0.451346\pi\)
0.152258 + 0.988341i \(0.451346\pi\)
\(180\) 0 0
\(181\) 2.92981 0.217771 0.108885 0.994054i \(-0.465272\pi\)
0.108885 + 0.994054i \(0.465272\pi\)
\(182\) 0 0
\(183\) 4.00343 0.295942
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 19.6696 1.43838
\(188\) 0 0
\(189\) 6.63992 0.482983
\(190\) 0 0
\(191\) −5.41644 −0.391920 −0.195960 0.980612i \(-0.562782\pi\)
−0.195960 + 0.980612i \(0.562782\pi\)
\(192\) 0 0
\(193\) −24.6275 −1.77273 −0.886363 0.462991i \(-0.846776\pi\)
−0.886363 + 0.462991i \(0.846776\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.93923 −0.280658 −0.140329 0.990105i \(-0.544816\pi\)
−0.140329 + 0.990105i \(0.544816\pi\)
\(198\) 0 0
\(199\) −20.8079 −1.47504 −0.737518 0.675328i \(-0.764002\pi\)
−0.737518 + 0.675328i \(0.764002\pi\)
\(200\) 0 0
\(201\) 2.49720 0.176139
\(202\) 0 0
\(203\) −15.8685 −1.11375
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 18.8099 1.30738
\(208\) 0 0
\(209\) −4.94567 −0.342099
\(210\) 0 0
\(211\) −22.9069 −1.57698 −0.788488 0.615050i \(-0.789136\pi\)
−0.788488 + 0.615050i \(0.789136\pi\)
\(212\) 0 0
\(213\) −1.09322 −0.0749061
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −13.1380 −0.891868
\(218\) 0 0
\(219\) 0.805494 0.0544302
\(220\) 0 0
\(221\) −15.0740 −1.01399
\(222\) 0 0
\(223\) 16.6488 1.11489 0.557444 0.830214i \(-0.311782\pi\)
0.557444 + 0.830214i \(0.311782\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.8343 −0.785469 −0.392735 0.919652i \(-0.628471\pi\)
−0.392735 + 0.919652i \(0.628471\pi\)
\(228\) 0 0
\(229\) 4.84288 0.320026 0.160013 0.987115i \(-0.448846\pi\)
0.160013 + 0.987115i \(0.448846\pi\)
\(230\) 0 0
\(231\) 5.59864 0.368363
\(232\) 0 0
\(233\) 9.60851 0.629474 0.314737 0.949179i \(-0.398084\pi\)
0.314737 + 0.949179i \(0.398084\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.19796 −0.0778157
\(238\) 0 0
\(239\) −9.49984 −0.614493 −0.307247 0.951630i \(-0.599408\pi\)
−0.307247 + 0.951630i \(0.599408\pi\)
\(240\) 0 0
\(241\) −21.8687 −1.40868 −0.704342 0.709860i \(-0.748758\pi\)
−0.704342 + 0.709860i \(0.748758\pi\)
\(242\) 0 0
\(243\) 9.31665 0.597663
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.79016 0.241162
\(248\) 0 0
\(249\) 1.50339 0.0952734
\(250\) 0 0
\(251\) 2.03109 0.128201 0.0641007 0.997943i \(-0.479582\pi\)
0.0641007 + 0.997943i \(0.479582\pi\)
\(252\) 0 0
\(253\) 32.4647 2.04104
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.62917 0.475894 0.237947 0.971278i \(-0.423525\pi\)
0.237947 + 0.971278i \(0.423525\pi\)
\(258\) 0 0
\(259\) −23.9379 −1.48743
\(260\) 0 0
\(261\) −14.7311 −0.911831
\(262\) 0 0
\(263\) −30.3299 −1.87022 −0.935112 0.354352i \(-0.884702\pi\)
−0.935112 + 0.354352i \(0.884702\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4.55245 −0.278605
\(268\) 0 0
\(269\) 24.3382 1.48393 0.741963 0.670441i \(-0.233895\pi\)
0.741963 + 0.670441i \(0.233895\pi\)
\(270\) 0 0
\(271\) 3.62084 0.219951 0.109975 0.993934i \(-0.464923\pi\)
0.109975 + 0.993934i \(0.464923\pi\)
\(272\) 0 0
\(273\) −4.29057 −0.259677
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −29.6842 −1.78355 −0.891776 0.452477i \(-0.850541\pi\)
−0.891776 + 0.452477i \(0.850541\pi\)
\(278\) 0 0
\(279\) −12.1963 −0.730175
\(280\) 0 0
\(281\) 29.1363 1.73813 0.869064 0.494700i \(-0.164722\pi\)
0.869064 + 0.494700i \(0.164722\pi\)
\(282\) 0 0
\(283\) 24.2081 1.43902 0.719511 0.694481i \(-0.244366\pi\)
0.719511 + 0.694481i \(0.244366\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.96452 0.411103
\(288\) 0 0
\(289\) −1.18244 −0.0695551
\(290\) 0 0
\(291\) 3.89373 0.228254
\(292\) 0 0
\(293\) −15.9338 −0.930865 −0.465433 0.885083i \(-0.654101\pi\)
−0.465433 + 0.885083i \(0.654101\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.6386 0.617316
\(298\) 0 0
\(299\) −24.8796 −1.43883
\(300\) 0 0
\(301\) −31.5700 −1.81967
\(302\) 0 0
\(303\) −4.80864 −0.276249
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 25.9373 1.48032 0.740161 0.672429i \(-0.234749\pi\)
0.740161 + 0.672429i \(0.234749\pi\)
\(308\) 0 0
\(309\) 1.81788 0.103416
\(310\) 0 0
\(311\) −2.58526 −0.146597 −0.0732983 0.997310i \(-0.523353\pi\)
−0.0732983 + 0.997310i \(0.523353\pi\)
\(312\) 0 0
\(313\) 17.1027 0.966701 0.483351 0.875427i \(-0.339420\pi\)
0.483351 + 0.875427i \(0.339420\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.5591 1.37938 0.689690 0.724105i \(-0.257747\pi\)
0.689690 + 0.724105i \(0.257747\pi\)
\(318\) 0 0
\(319\) −25.4249 −1.42352
\(320\) 0 0
\(321\) 6.93654 0.387160
\(322\) 0 0
\(323\) −3.97713 −0.221293
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.26440 0.180522
\(328\) 0 0
\(329\) 2.88813 0.159228
\(330\) 0 0
\(331\) 16.4328 0.903227 0.451614 0.892214i \(-0.350849\pi\)
0.451614 + 0.892214i \(0.350849\pi\)
\(332\) 0 0
\(333\) −22.2221 −1.21776
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.0174 −0.654627 −0.327314 0.944916i \(-0.606143\pi\)
−0.327314 + 0.944916i \(0.606143\pi\)
\(338\) 0 0
\(339\) −3.61263 −0.196211
\(340\) 0 0
\(341\) −21.0501 −1.13992
\(342\) 0 0
\(343\) 13.8038 0.745336
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.3681 1.52288 0.761439 0.648236i \(-0.224493\pi\)
0.761439 + 0.648236i \(0.224493\pi\)
\(348\) 0 0
\(349\) 33.9733 1.81855 0.909275 0.416195i \(-0.136637\pi\)
0.909275 + 0.416195i \(0.136637\pi\)
\(350\) 0 0
\(351\) −8.15302 −0.435176
\(352\) 0 0
\(353\) 26.6181 1.41674 0.708369 0.705842i \(-0.249431\pi\)
0.708369 + 0.705842i \(0.249431\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.50222 0.238283
\(358\) 0 0
\(359\) 5.84964 0.308732 0.154366 0.988014i \(-0.450666\pi\)
0.154366 + 0.988014i \(0.450666\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 4.93616 0.259081
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.41146 0.125877 0.0629385 0.998017i \(-0.479953\pi\)
0.0629385 + 0.998017i \(0.479953\pi\)
\(368\) 0 0
\(369\) 6.46533 0.336571
\(370\) 0 0
\(371\) −23.2160 −1.20531
\(372\) 0 0
\(373\) 27.2074 1.40874 0.704372 0.709831i \(-0.251229\pi\)
0.704372 + 0.709831i \(0.251229\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 19.4846 1.00351
\(378\) 0 0
\(379\) −37.0749 −1.90441 −0.952204 0.305461i \(-0.901189\pi\)
−0.952204 + 0.305461i \(0.901189\pi\)
\(380\) 0 0
\(381\) −1.38034 −0.0707170
\(382\) 0 0
\(383\) −0.238749 −0.0121995 −0.00609975 0.999981i \(-0.501942\pi\)
−0.00609975 + 0.999981i \(0.501942\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −29.3072 −1.48977
\(388\) 0 0
\(389\) 21.3399 1.08197 0.540987 0.841031i \(-0.318051\pi\)
0.540987 + 0.841031i \(0.318051\pi\)
\(390\) 0 0
\(391\) 26.1069 1.32028
\(392\) 0 0
\(393\) 3.68527 0.185897
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −17.7416 −0.890424 −0.445212 0.895425i \(-0.646872\pi\)
−0.445212 + 0.895425i \(0.646872\pi\)
\(398\) 0 0
\(399\) −1.13203 −0.0566723
\(400\) 0 0
\(401\) 36.4310 1.81928 0.909638 0.415401i \(-0.136359\pi\)
0.909638 + 0.415401i \(0.136359\pi\)
\(402\) 0 0
\(403\) 16.1319 0.803589
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −38.3538 −1.90113
\(408\) 0 0
\(409\) −0.530833 −0.0262480 −0.0131240 0.999914i \(-0.504178\pi\)
−0.0131240 + 0.999914i \(0.504178\pi\)
\(410\) 0 0
\(411\) 5.36332 0.264553
\(412\) 0 0
\(413\) 32.2120 1.58505
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.42919 −0.216898
\(418\) 0 0
\(419\) 29.3816 1.43538 0.717692 0.696361i \(-0.245199\pi\)
0.717692 + 0.696361i \(0.245199\pi\)
\(420\) 0 0
\(421\) 2.95912 0.144219 0.0721093 0.997397i \(-0.477027\pi\)
0.0721093 + 0.997397i \(0.477027\pi\)
\(422\) 0 0
\(423\) 2.68112 0.130360
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −33.6960 −1.63067
\(428\) 0 0
\(429\) −6.87446 −0.331902
\(430\) 0 0
\(431\) −32.4518 −1.56315 −0.781573 0.623814i \(-0.785582\pi\)
−0.781573 + 0.623814i \(0.785582\pi\)
\(432\) 0 0
\(433\) 27.7821 1.33512 0.667562 0.744554i \(-0.267338\pi\)
0.667562 + 0.744554i \(0.267338\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.56426 −0.314011
\(438\) 0 0
\(439\) −0.0388357 −0.00185352 −0.000926762 1.00000i \(-0.500295\pi\)
−0.000926762 1.00000i \(0.500295\pi\)
\(440\) 0 0
\(441\) −7.24412 −0.344958
\(442\) 0 0
\(443\) 3.98842 0.189496 0.0947479 0.995501i \(-0.469796\pi\)
0.0947479 + 0.995501i \(0.469796\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.61132 0.218108
\(448\) 0 0
\(449\) 23.1695 1.09343 0.546717 0.837317i \(-0.315877\pi\)
0.546717 + 0.837317i \(0.315877\pi\)
\(450\) 0 0
\(451\) 11.1587 0.525444
\(452\) 0 0
\(453\) −0.143151 −0.00672582
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.5398 0.539811 0.269905 0.962887i \(-0.413008\pi\)
0.269905 + 0.962887i \(0.413008\pi\)
\(458\) 0 0
\(459\) 8.55521 0.399323
\(460\) 0 0
\(461\) 8.96090 0.417351 0.208675 0.977985i \(-0.433085\pi\)
0.208675 + 0.977985i \(0.433085\pi\)
\(462\) 0 0
\(463\) −4.43012 −0.205885 −0.102943 0.994687i \(-0.532826\pi\)
−0.102943 + 0.994687i \(0.532826\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.6874 −1.42004 −0.710021 0.704180i \(-0.751315\pi\)
−0.710021 + 0.704180i \(0.751315\pi\)
\(468\) 0 0
\(469\) −21.0184 −0.970541
\(470\) 0 0
\(471\) 6.47354 0.298285
\(472\) 0 0
\(473\) −50.5823 −2.32578
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −21.5519 −0.986796
\(478\) 0 0
\(479\) −18.0047 −0.822657 −0.411328 0.911487i \(-0.634935\pi\)
−0.411328 + 0.911487i \(0.634935\pi\)
\(480\) 0 0
\(481\) 29.3928 1.34020
\(482\) 0 0
\(483\) 7.43093 0.338119
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.7434 −0.668087 −0.334043 0.942558i \(-0.608413\pi\)
−0.334043 + 0.942558i \(0.608413\pi\)
\(488\) 0 0
\(489\) 2.10043 0.0949847
\(490\) 0 0
\(491\) 40.0805 1.80881 0.904404 0.426677i \(-0.140316\pi\)
0.904404 + 0.426677i \(0.140316\pi\)
\(492\) 0 0
\(493\) −20.4458 −0.920832
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.20139 0.412739
\(498\) 0 0
\(499\) 12.7838 0.572282 0.286141 0.958188i \(-0.407627\pi\)
0.286141 + 0.958188i \(0.407627\pi\)
\(500\) 0 0
\(501\) 3.01584 0.134738
\(502\) 0 0
\(503\) −11.9101 −0.531046 −0.265523 0.964105i \(-0.585545\pi\)
−0.265523 + 0.964105i \(0.585545\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.500723 0.0222379
\(508\) 0 0
\(509\) 15.7374 0.697549 0.348775 0.937207i \(-0.386598\pi\)
0.348775 + 0.937207i \(0.386598\pi\)
\(510\) 0 0
\(511\) −6.77967 −0.299915
\(512\) 0 0
\(513\) −2.15110 −0.0949734
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.62743 0.203514
\(518\) 0 0
\(519\) −5.55338 −0.243766
\(520\) 0 0
\(521\) 5.67909 0.248806 0.124403 0.992232i \(-0.460299\pi\)
0.124403 + 0.992232i \(0.460299\pi\)
\(522\) 0 0
\(523\) −36.8094 −1.60956 −0.804781 0.593572i \(-0.797717\pi\)
−0.804781 + 0.593572i \(0.797717\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.9277 −0.737382
\(528\) 0 0
\(529\) 20.0896 0.873459
\(530\) 0 0
\(531\) 29.9032 1.29769
\(532\) 0 0
\(533\) −8.55160 −0.370411
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.49414 0.0644770
\(538\) 0 0
\(539\) −12.5029 −0.538537
\(540\) 0 0
\(541\) −24.9051 −1.07076 −0.535378 0.844613i \(-0.679831\pi\)
−0.535378 + 0.844613i \(0.679831\pi\)
\(542\) 0 0
\(543\) 1.07447 0.0461100
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.17066 0.392109 0.196055 0.980593i \(-0.437187\pi\)
0.196055 + 0.980593i \(0.437187\pi\)
\(548\) 0 0
\(549\) −31.2808 −1.33503
\(550\) 0 0
\(551\) 5.14084 0.219007
\(552\) 0 0
\(553\) 10.0830 0.428771
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.3168 −0.691364 −0.345682 0.938352i \(-0.612352\pi\)
−0.345682 + 0.938352i \(0.612352\pi\)
\(558\) 0 0
\(559\) 38.7642 1.63955
\(560\) 0 0
\(561\) 7.21357 0.304557
\(562\) 0 0
\(563\) 23.4254 0.987264 0.493632 0.869671i \(-0.335669\pi\)
0.493632 + 0.869671i \(0.335669\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −24.1002 −1.01211
\(568\) 0 0
\(569\) −21.5994 −0.905495 −0.452748 0.891639i \(-0.649556\pi\)
−0.452748 + 0.891639i \(0.649556\pi\)
\(570\) 0 0
\(571\) −13.9099 −0.582111 −0.291055 0.956706i \(-0.594006\pi\)
−0.291055 + 0.956706i \(0.594006\pi\)
\(572\) 0 0
\(573\) −1.98641 −0.0829836
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −27.0208 −1.12489 −0.562445 0.826834i \(-0.690139\pi\)
−0.562445 + 0.826834i \(0.690139\pi\)
\(578\) 0 0
\(579\) −9.03183 −0.375350
\(580\) 0 0
\(581\) −12.6537 −0.524964
\(582\) 0 0
\(583\) −37.1972 −1.54055
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.29463 0.0947096 0.0473548 0.998878i \(-0.484921\pi\)
0.0473548 + 0.998878i \(0.484921\pi\)
\(588\) 0 0
\(589\) 4.25626 0.175376
\(590\) 0 0
\(591\) −1.44466 −0.0594255
\(592\) 0 0
\(593\) −22.6929 −0.931888 −0.465944 0.884814i \(-0.654285\pi\)
−0.465944 + 0.884814i \(0.654285\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.63105 −0.312318
\(598\) 0 0
\(599\) −4.95675 −0.202527 −0.101264 0.994860i \(-0.532289\pi\)
−0.101264 + 0.994860i \(0.532289\pi\)
\(600\) 0 0
\(601\) −33.5372 −1.36801 −0.684005 0.729477i \(-0.739763\pi\)
−0.684005 + 0.729477i \(0.739763\pi\)
\(602\) 0 0
\(603\) −19.5119 −0.794586
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 32.9103 1.33579 0.667893 0.744257i \(-0.267196\pi\)
0.667893 + 0.744257i \(0.267196\pi\)
\(608\) 0 0
\(609\) −5.81957 −0.235821
\(610\) 0 0
\(611\) −3.54628 −0.143467
\(612\) 0 0
\(613\) −44.4090 −1.79366 −0.896831 0.442374i \(-0.854137\pi\)
−0.896831 + 0.442374i \(0.854137\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.2962 0.736579 0.368290 0.929711i \(-0.379944\pi\)
0.368290 + 0.929711i \(0.379944\pi\)
\(618\) 0 0
\(619\) 2.76770 0.111243 0.0556216 0.998452i \(-0.482286\pi\)
0.0556216 + 0.998452i \(0.482286\pi\)
\(620\) 0 0
\(621\) 14.1204 0.566632
\(622\) 0 0
\(623\) 38.3170 1.53514
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.81376 −0.0724347
\(628\) 0 0
\(629\) −30.8428 −1.22978
\(630\) 0 0
\(631\) −5.74488 −0.228700 −0.114350 0.993441i \(-0.536479\pi\)
−0.114350 + 0.993441i \(0.536479\pi\)
\(632\) 0 0
\(633\) −8.40083 −0.333903
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.58170 0.379641
\(638\) 0 0
\(639\) 8.54186 0.337911
\(640\) 0 0
\(641\) 19.3436 0.764028 0.382014 0.924157i \(-0.375231\pi\)
0.382014 + 0.924157i \(0.375231\pi\)
\(642\) 0 0
\(643\) −11.0400 −0.435375 −0.217687 0.976019i \(-0.569851\pi\)
−0.217687 + 0.976019i \(0.569851\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.35938 0.132071 0.0660353 0.997817i \(-0.478965\pi\)
0.0660353 + 0.997817i \(0.478965\pi\)
\(648\) 0 0
\(649\) 51.6109 2.02590
\(650\) 0 0
\(651\) −4.81821 −0.188841
\(652\) 0 0
\(653\) 44.1434 1.72747 0.863733 0.503950i \(-0.168121\pi\)
0.863733 + 0.503950i \(0.168121\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.29372 −0.245542
\(658\) 0 0
\(659\) 35.1874 1.37071 0.685354 0.728210i \(-0.259648\pi\)
0.685354 + 0.728210i \(0.259648\pi\)
\(660\) 0 0
\(661\) 10.3574 0.402855 0.201428 0.979503i \(-0.435442\pi\)
0.201428 + 0.979503i \(0.435442\pi\)
\(662\) 0 0
\(663\) −5.52819 −0.214697
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −33.7458 −1.30664
\(668\) 0 0
\(669\) 6.10575 0.236062
\(670\) 0 0
\(671\) −53.9886 −2.08421
\(672\) 0 0
\(673\) 36.0384 1.38918 0.694589 0.719407i \(-0.255586\pi\)
0.694589 + 0.719407i \(0.255586\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.4531 0.440177 0.220089 0.975480i \(-0.429365\pi\)
0.220089 + 0.975480i \(0.429365\pi\)
\(678\) 0 0
\(679\) −32.7727 −1.25770
\(680\) 0 0
\(681\) −4.34008 −0.166312
\(682\) 0 0
\(683\) −42.9554 −1.64364 −0.821821 0.569745i \(-0.807042\pi\)
−0.821821 + 0.569745i \(0.807042\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.77607 0.0677611
\(688\) 0 0
\(689\) 28.5065 1.08601
\(690\) 0 0
\(691\) 5.23235 0.199048 0.0995241 0.995035i \(-0.468268\pi\)
0.0995241 + 0.995035i \(0.468268\pi\)
\(692\) 0 0
\(693\) −43.7449 −1.66173
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.97345 0.339894
\(698\) 0 0
\(699\) 3.52380 0.133282
\(700\) 0 0
\(701\) −17.3930 −0.656925 −0.328463 0.944517i \(-0.606531\pi\)
−0.328463 + 0.944517i \(0.606531\pi\)
\(702\) 0 0
\(703\) 7.75503 0.292486
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 40.4733 1.52215
\(708\) 0 0
\(709\) −21.5383 −0.808888 −0.404444 0.914563i \(-0.632535\pi\)
−0.404444 + 0.914563i \(0.632535\pi\)
\(710\) 0 0
\(711\) 9.36024 0.351036
\(712\) 0 0
\(713\) −27.9392 −1.04633
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.48395 −0.130110
\(718\) 0 0
\(719\) 30.8923 1.15209 0.576044 0.817418i \(-0.304596\pi\)
0.576044 + 0.817418i \(0.304596\pi\)
\(720\) 0 0
\(721\) −15.3007 −0.569830
\(722\) 0 0
\(723\) −8.02006 −0.298269
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20.9947 0.778649 0.389325 0.921101i \(-0.372708\pi\)
0.389325 + 0.921101i \(0.372708\pi\)
\(728\) 0 0
\(729\) −20.0061 −0.740967
\(730\) 0 0
\(731\) −40.6765 −1.50447
\(732\) 0 0
\(733\) −22.0911 −0.815955 −0.407978 0.912992i \(-0.633766\pi\)
−0.407978 + 0.912992i \(0.633766\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −33.6762 −1.24048
\(738\) 0 0
\(739\) −7.81209 −0.287372 −0.143686 0.989623i \(-0.545896\pi\)
−0.143686 + 0.989623i \(0.545896\pi\)
\(740\) 0 0
\(741\) 1.39000 0.0510628
\(742\) 0 0
\(743\) −17.9012 −0.656731 −0.328365 0.944551i \(-0.606498\pi\)
−0.328365 + 0.944551i \(0.606498\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.7467 −0.429790
\(748\) 0 0
\(749\) −58.3834 −2.13328
\(750\) 0 0
\(751\) −48.8802 −1.78366 −0.891832 0.452367i \(-0.850580\pi\)
−0.891832 + 0.452367i \(0.850580\pi\)
\(752\) 0 0
\(753\) 0.744878 0.0271449
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.75011 −0.245337 −0.122668 0.992448i \(-0.539145\pi\)
−0.122668 + 0.992448i \(0.539145\pi\)
\(758\) 0 0
\(759\) 11.9060 0.432161
\(760\) 0 0
\(761\) 47.5563 1.72392 0.861958 0.506980i \(-0.169238\pi\)
0.861958 + 0.506980i \(0.169238\pi\)
\(762\) 0 0
\(763\) −27.4757 −0.994689
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −39.5525 −1.42816
\(768\) 0 0
\(769\) −0.849714 −0.0306415 −0.0153207 0.999883i \(-0.504877\pi\)
−0.0153207 + 0.999883i \(0.504877\pi\)
\(770\) 0 0
\(771\) 2.79790 0.100764
\(772\) 0 0
\(773\) 4.74109 0.170525 0.0852627 0.996359i \(-0.472827\pi\)
0.0852627 + 0.996359i \(0.472827\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −8.77891 −0.314942
\(778\) 0 0
\(779\) −2.25626 −0.0808390
\(780\) 0 0
\(781\) 14.7427 0.527535
\(782\) 0 0
\(783\) −11.0585 −0.395197
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 19.0208 0.678018 0.339009 0.940783i \(-0.389908\pi\)
0.339009 + 0.940783i \(0.389908\pi\)
\(788\) 0 0
\(789\) −11.1231 −0.395994
\(790\) 0 0
\(791\) 30.4067 1.08114
\(792\) 0 0
\(793\) 41.3747 1.46926
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.8784 0.668706 0.334353 0.942448i \(-0.391482\pi\)
0.334353 + 0.942448i \(0.391482\pi\)
\(798\) 0 0
\(799\) 3.72121 0.131647
\(800\) 0 0
\(801\) 35.5705 1.25682
\(802\) 0 0
\(803\) −10.8625 −0.383331
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.92573 0.314201
\(808\) 0 0
\(809\) −1.68753 −0.0593305 −0.0296652 0.999560i \(-0.509444\pi\)
−0.0296652 + 0.999560i \(0.509444\pi\)
\(810\) 0 0
\(811\) 38.3960 1.34827 0.674133 0.738610i \(-0.264517\pi\)
0.674133 + 0.738610i \(0.264517\pi\)
\(812\) 0 0
\(813\) 1.32790 0.0465715
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 10.2276 0.357818
\(818\) 0 0
\(819\) 33.5244 1.17144
\(820\) 0 0
\(821\) −11.4671 −0.400204 −0.200102 0.979775i \(-0.564127\pi\)
−0.200102 + 0.979775i \(0.564127\pi\)
\(822\) 0 0
\(823\) 46.0927 1.60669 0.803345 0.595515i \(-0.203052\pi\)
0.803345 + 0.595515i \(0.203052\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.44674 0.0503081 0.0251540 0.999684i \(-0.491992\pi\)
0.0251540 + 0.999684i \(0.491992\pi\)
\(828\) 0 0
\(829\) −43.8553 −1.52316 −0.761578 0.648073i \(-0.775575\pi\)
−0.761578 + 0.648073i \(0.775575\pi\)
\(830\) 0 0
\(831\) −10.8863 −0.377642
\(832\) 0 0
\(833\) −10.0544 −0.348363
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −9.15565 −0.316465
\(838\) 0 0
\(839\) −45.7264 −1.57865 −0.789325 0.613976i \(-0.789569\pi\)
−0.789325 + 0.613976i \(0.789569\pi\)
\(840\) 0 0
\(841\) −2.57179 −0.0886824
\(842\) 0 0
\(843\) 10.6854 0.368024
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −41.5466 −1.42756
\(848\) 0 0
\(849\) 8.87802 0.304693
\(850\) 0 0
\(851\) −50.9061 −1.74504
\(852\) 0 0
\(853\) −28.3388 −0.970303 −0.485151 0.874430i \(-0.661236\pi\)
−0.485151 + 0.874430i \(0.661236\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.7526 −0.640575 −0.320288 0.947320i \(-0.603780\pi\)
−0.320288 + 0.947320i \(0.603780\pi\)
\(858\) 0 0
\(859\) −14.2675 −0.486801 −0.243401 0.969926i \(-0.578263\pi\)
−0.243401 + 0.969926i \(0.578263\pi\)
\(860\) 0 0
\(861\) 2.55415 0.0870453
\(862\) 0 0
\(863\) 32.9573 1.12188 0.560940 0.827857i \(-0.310440\pi\)
0.560940 + 0.827857i \(0.310440\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.433644 −0.0147273
\(868\) 0 0
\(869\) 16.1552 0.548026
\(870\) 0 0
\(871\) 25.8081 0.874475
\(872\) 0 0
\(873\) −30.4236 −1.02968
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.9346 1.41603 0.708016 0.706197i \(-0.249591\pi\)
0.708016 + 0.706197i \(0.249591\pi\)
\(878\) 0 0
\(879\) −5.84354 −0.197098
\(880\) 0 0
\(881\) 7.69223 0.259158 0.129579 0.991569i \(-0.458637\pi\)
0.129579 + 0.991569i \(0.458637\pi\)
\(882\) 0 0
\(883\) 32.8546 1.10565 0.552823 0.833299i \(-0.313551\pi\)
0.552823 + 0.833299i \(0.313551\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.97358 0.0662664 0.0331332 0.999451i \(-0.489451\pi\)
0.0331332 + 0.999451i \(0.489451\pi\)
\(888\) 0 0
\(889\) 11.6180 0.389656
\(890\) 0 0
\(891\) −38.6139 −1.29361
\(892\) 0 0
\(893\) −0.935652 −0.0313104
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −9.12430 −0.304651
\(898\) 0 0
\(899\) 21.8808 0.729764
\(900\) 0 0
\(901\) −29.9127 −0.996536
\(902\) 0 0
\(903\) −11.5779 −0.385289
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −6.21342 −0.206313 −0.103157 0.994665i \(-0.532894\pi\)
−0.103157 + 0.994665i \(0.532894\pi\)
\(908\) 0 0
\(909\) 37.5723 1.24619
\(910\) 0 0
\(911\) −40.4200 −1.33917 −0.669586 0.742734i \(-0.733529\pi\)
−0.669586 + 0.742734i \(0.733529\pi\)
\(912\) 0 0
\(913\) −20.2741 −0.670974
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −31.0181 −1.02431
\(918\) 0 0
\(919\) 53.0177 1.74889 0.874446 0.485122i \(-0.161225\pi\)
0.874446 + 0.485122i \(0.161225\pi\)
\(920\) 0 0
\(921\) 9.51220 0.313438
\(922\) 0 0
\(923\) −11.2982 −0.371885
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −14.2040 −0.466522
\(928\) 0 0
\(929\) 38.0520 1.24845 0.624224 0.781246i \(-0.285415\pi\)
0.624224 + 0.781246i \(0.285415\pi\)
\(930\) 0 0
\(931\) 2.52804 0.0828533
\(932\) 0 0
\(933\) −0.948112 −0.0310398
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.8035 0.418271 0.209135 0.977887i \(-0.432935\pi\)
0.209135 + 0.977887i \(0.432935\pi\)
\(938\) 0 0
\(939\) 6.27220 0.204686
\(940\) 0 0
\(941\) 40.1622 1.30925 0.654625 0.755954i \(-0.272827\pi\)
0.654625 + 0.755954i \(0.272827\pi\)
\(942\) 0 0
\(943\) 14.8107 0.482303
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.3713 −0.824458 −0.412229 0.911080i \(-0.635250\pi\)
−0.412229 + 0.911080i \(0.635250\pi\)
\(948\) 0 0
\(949\) 8.32463 0.270229
\(950\) 0 0
\(951\) 9.00676 0.292064
\(952\) 0 0
\(953\) 6.25884 0.202744 0.101372 0.994849i \(-0.467677\pi\)
0.101372 + 0.994849i \(0.467677\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −9.32426 −0.301411
\(958\) 0 0
\(959\) −45.1420 −1.45771
\(960\) 0 0
\(961\) −12.8842 −0.415620
\(962\) 0 0
\(963\) −54.1986 −1.74653
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −16.6425 −0.535186 −0.267593 0.963532i \(-0.586228\pi\)
−0.267593 + 0.963532i \(0.586228\pi\)
\(968\) 0 0
\(969\) −1.45856 −0.0468558
\(970\) 0 0
\(971\) −20.5468 −0.659377 −0.329688 0.944090i \(-0.606944\pi\)
−0.329688 + 0.944090i \(0.606944\pi\)
\(972\) 0 0
\(973\) 37.2796 1.19513
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.7487 −1.55961 −0.779804 0.626024i \(-0.784681\pi\)
−0.779804 + 0.626024i \(0.784681\pi\)
\(978\) 0 0
\(979\) 61.3924 1.96211
\(980\) 0 0
\(981\) −25.5064 −0.814355
\(982\) 0 0
\(983\) −34.9881 −1.11595 −0.557973 0.829859i \(-0.688421\pi\)
−0.557973 + 0.829859i \(0.688421\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.05919 0.0337142
\(988\) 0 0
\(989\) −67.1366 −2.13482
\(990\) 0 0
\(991\) 12.4515 0.395535 0.197768 0.980249i \(-0.436631\pi\)
0.197768 + 0.980249i \(0.436631\pi\)
\(992\) 0 0
\(993\) 6.02652 0.191246
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −13.4889 −0.427199 −0.213599 0.976921i \(-0.568519\pi\)
−0.213599 + 0.976921i \(0.568519\pi\)
\(998\) 0 0
\(999\) −16.6818 −0.527790
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 3800.2.a.be.1.3 6
4.3 odd 2 7600.2.a.cg.1.4 6
5.2 odd 4 760.2.d.e.609.6 12
5.3 odd 4 760.2.d.e.609.7 yes 12
5.4 even 2 3800.2.a.z.1.4 6
20.3 even 4 1520.2.d.k.609.6 12
20.7 even 4 1520.2.d.k.609.7 12
20.19 odd 2 7600.2.a.cn.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.e.609.6 12 5.2 odd 4
760.2.d.e.609.7 yes 12 5.3 odd 4
1520.2.d.k.609.6 12 20.3 even 4
1520.2.d.k.609.7 12 20.7 even 4
3800.2.a.z.1.4 6 5.4 even 2
3800.2.a.be.1.3 6 1.1 even 1 trivial
7600.2.a.cg.1.4 6 4.3 odd 2
7600.2.a.cn.1.3 6 20.19 odd 2