Properties

Label 4028.2.c.a.3497.10
Level $4028$
Weight $2$
Character 4028.3497
Analytic conductor $32.164$
Analytic rank $0$
Dimension $82$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4028,2,Mod(3497,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4028.3497");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3497.10
Character \(\chi\) \(=\) 4028.3497
Dual form 4028.2.c.a.3497.73

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.57074i q^{3} +2.44297i q^{5} +3.60312 q^{7} -3.60868 q^{9} +O(q^{10})\) \(q-2.57074i q^{3} +2.44297i q^{5} +3.60312 q^{7} -3.60868 q^{9} -0.948707 q^{11} -3.15492 q^{13} +6.28022 q^{15} -7.26887 q^{17} +1.00000i q^{19} -9.26266i q^{21} +6.89346i q^{23} -0.968083 q^{25} +1.56476i q^{27} -6.36961 q^{29} -2.80776i q^{31} +2.43887i q^{33} +8.80229i q^{35} +11.1740 q^{37} +8.11046i q^{39} +6.92669i q^{41} -4.16676 q^{43} -8.81589i q^{45} -5.62489 q^{47} +5.98244 q^{49} +18.6863i q^{51} +(7.24654 + 0.698304i) q^{53} -2.31766i q^{55} +2.57074 q^{57} -4.58150 q^{59} -11.5043i q^{61} -13.0025 q^{63} -7.70735i q^{65} +8.84596i q^{67} +17.7213 q^{69} +12.8529i q^{71} +4.65387i q^{73} +2.48869i q^{75} -3.41830 q^{77} -8.29138i q^{79} -6.80345 q^{81} +13.5520i q^{83} -17.7576i q^{85} +16.3746i q^{87} -10.4473 q^{89} -11.3675 q^{91} -7.21802 q^{93} -2.44297 q^{95} -15.9611 q^{97} +3.42358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times\).

\(n\) \(2015\) \(2281\) \(2757\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 2.57074i 1.48422i −0.670281 0.742108i \(-0.733826\pi\)
0.670281 0.742108i \(-0.266174\pi\)
\(4\) 0 0
\(5\) 2.44297i 1.09253i 0.837613 + 0.546264i \(0.183950\pi\)
−0.837613 + 0.546264i \(0.816050\pi\)
\(6\) 0 0
\(7\) 3.60312 1.36185 0.680925 0.732353i \(-0.261578\pi\)
0.680925 + 0.732353i \(0.261578\pi\)
\(8\) 0 0
\(9\) −3.60868 −1.20289
\(10\) 0 0
\(11\) −0.948707 −0.286046 −0.143023 0.989719i \(-0.545682\pi\)
−0.143023 + 0.989719i \(0.545682\pi\)
\(12\) 0 0
\(13\) −3.15492 −0.875016 −0.437508 0.899214i \(-0.644139\pi\)
−0.437508 + 0.899214i \(0.644139\pi\)
\(14\) 0 0
\(15\) 6.28022 1.62155
\(16\) 0 0
\(17\) −7.26887 −1.76296 −0.881480 0.472221i \(-0.843452\pi\)
−0.881480 + 0.472221i \(0.843452\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 9.26266i 2.02128i
\(22\) 0 0
\(23\) 6.89346i 1.43739i 0.695327 + 0.718693i \(0.255259\pi\)
−0.695327 + 0.718693i \(0.744741\pi\)
\(24\) 0 0
\(25\) −0.968083 −0.193617
\(26\) 0 0
\(27\) 1.56476i 0.301139i
\(28\) 0 0
\(29\) −6.36961 −1.18281 −0.591404 0.806376i \(-0.701426\pi\)
−0.591404 + 0.806376i \(0.701426\pi\)
\(30\) 0 0
\(31\) 2.80776i 0.504289i −0.967690 0.252145i \(-0.918864\pi\)
0.967690 0.252145i \(-0.0811359\pi\)
\(32\) 0 0
\(33\) 2.43887i 0.424554i
\(34\) 0 0
\(35\) 8.80229i 1.48786i
\(36\) 0 0
\(37\) 11.1740 1.83699 0.918495 0.395433i \(-0.129405\pi\)
0.918495 + 0.395433i \(0.129405\pi\)
\(38\) 0 0
\(39\) 8.11046i 1.29871i
\(40\) 0 0
\(41\) 6.92669i 1.08177i 0.841097 + 0.540884i \(0.181910\pi\)
−0.841097 + 0.540884i \(0.818090\pi\)
\(42\) 0 0
\(43\) −4.16676 −0.635425 −0.317712 0.948187i \(-0.602915\pi\)
−0.317712 + 0.948187i \(0.602915\pi\)
\(44\) 0 0
\(45\) 8.81589i 1.31420i
\(46\) 0 0
\(47\) −5.62489 −0.820474 −0.410237 0.911979i \(-0.634554\pi\)
−0.410237 + 0.911979i \(0.634554\pi\)
\(48\) 0 0
\(49\) 5.98244 0.854635
\(50\) 0 0
\(51\) 18.6863i 2.61661i
\(52\) 0 0
\(53\) 7.24654 + 0.698304i 0.995389 + 0.0959195i
\(54\) 0 0
\(55\) 2.31766i 0.312513i
\(56\) 0 0
\(57\) 2.57074 0.340502
\(58\) 0 0
\(59\) −4.58150 −0.596460 −0.298230 0.954494i \(-0.596396\pi\)
−0.298230 + 0.954494i \(0.596396\pi\)
\(60\) 0 0
\(61\) 11.5043i 1.47297i −0.676451 0.736487i \(-0.736483\pi\)
0.676451 0.736487i \(-0.263517\pi\)
\(62\) 0 0
\(63\) −13.0025 −1.63816
\(64\) 0 0
\(65\) 7.70735i 0.955980i
\(66\) 0 0
\(67\) 8.84596i 1.08071i 0.841439 + 0.540353i \(0.181709\pi\)
−0.841439 + 0.540353i \(0.818291\pi\)
\(68\) 0 0
\(69\) 17.7213 2.13339
\(70\) 0 0
\(71\) 12.8529i 1.52536i 0.646777 + 0.762679i \(0.276116\pi\)
−0.646777 + 0.762679i \(0.723884\pi\)
\(72\) 0 0
\(73\) 4.65387i 0.544694i 0.962199 + 0.272347i \(0.0877998\pi\)
−0.962199 + 0.272347i \(0.912200\pi\)
\(74\) 0 0
\(75\) 2.48869i 0.287369i
\(76\) 0 0
\(77\) −3.41830 −0.389551
\(78\) 0 0
\(79\) 8.29138i 0.932853i −0.884560 0.466427i \(-0.845541\pi\)
0.884560 0.466427i \(-0.154459\pi\)
\(80\) 0 0
\(81\) −6.80345 −0.755939
\(82\) 0 0
\(83\) 13.5520i 1.48753i 0.668442 + 0.743764i \(0.266961\pi\)
−0.668442 + 0.743764i \(0.733039\pi\)
\(84\) 0 0
\(85\) 17.7576i 1.92608i
\(86\) 0 0
\(87\) 16.3746i 1.75554i
\(88\) 0 0
\(89\) −10.4473 −1.10741 −0.553706 0.832712i \(-0.686787\pi\)
−0.553706 + 0.832712i \(0.686787\pi\)
\(90\) 0 0
\(91\) −11.3675 −1.19164
\(92\) 0 0
\(93\) −7.21802 −0.748474
\(94\) 0 0
\(95\) −2.44297 −0.250643
\(96\) 0 0
\(97\) −15.9611 −1.62060 −0.810300 0.586016i \(-0.800696\pi\)
−0.810300 + 0.586016i \(0.800696\pi\)
\(98\) 0 0
\(99\) 3.42358 0.344083
\(100\) 0 0
\(101\) 5.15664i 0.513104i 0.966530 + 0.256552i \(0.0825866\pi\)
−0.966530 + 0.256552i \(0.917413\pi\)
\(102\) 0 0
\(103\) 0.888961i 0.0875919i −0.999040 0.0437960i \(-0.986055\pi\)
0.999040 0.0437960i \(-0.0139451\pi\)
\(104\) 0 0
\(105\) 22.6284 2.20830
\(106\) 0 0
\(107\) 4.55582 0.440428 0.220214 0.975452i \(-0.429324\pi\)
0.220214 + 0.975452i \(0.429324\pi\)
\(108\) 0 0
\(109\) 16.1576i 1.54762i 0.633419 + 0.773809i \(0.281651\pi\)
−0.633419 + 0.773809i \(0.718349\pi\)
\(110\) 0 0
\(111\) 28.7253i 2.72649i
\(112\) 0 0
\(113\) −1.11319 −0.104720 −0.0523602 0.998628i \(-0.516674\pi\)
−0.0523602 + 0.998628i \(0.516674\pi\)
\(114\) 0 0
\(115\) −16.8405 −1.57038
\(116\) 0 0
\(117\) 11.3851 1.05255
\(118\) 0 0
\(119\) −26.1906 −2.40089
\(120\) 0 0
\(121\) −10.1000 −0.918178
\(122\) 0 0
\(123\) 17.8067 1.60558
\(124\) 0 0
\(125\) 9.84984i 0.880996i
\(126\) 0 0
\(127\) 12.2788i 1.08957i 0.838576 + 0.544784i \(0.183389\pi\)
−0.838576 + 0.544784i \(0.816611\pi\)
\(128\) 0 0
\(129\) 10.7116i 0.943107i
\(130\) 0 0
\(131\) −21.7383 −1.89928 −0.949642 0.313337i \(-0.898553\pi\)
−0.949642 + 0.313337i \(0.898553\pi\)
\(132\) 0 0
\(133\) 3.60312i 0.312430i
\(134\) 0 0
\(135\) −3.82267 −0.329003
\(136\) 0 0
\(137\) 12.6929i 1.08443i −0.840240 0.542214i \(-0.817586\pi\)
0.840240 0.542214i \(-0.182414\pi\)
\(138\) 0 0
\(139\) 9.58144i 0.812687i 0.913720 + 0.406343i \(0.133196\pi\)
−0.913720 + 0.406343i \(0.866804\pi\)
\(140\) 0 0
\(141\) 14.4601i 1.21776i
\(142\) 0 0
\(143\) 2.99309 0.250295
\(144\) 0 0
\(145\) 15.5608i 1.29225i
\(146\) 0 0
\(147\) 15.3793i 1.26846i
\(148\) 0 0
\(149\) 17.0973 1.40066 0.700332 0.713817i \(-0.253035\pi\)
0.700332 + 0.713817i \(0.253035\pi\)
\(150\) 0 0
\(151\) 13.8359i 1.12595i 0.826475 + 0.562974i \(0.190343\pi\)
−0.826475 + 0.562974i \(0.809657\pi\)
\(152\) 0 0
\(153\) 26.2311 2.12066
\(154\) 0 0
\(155\) 6.85927 0.550950
\(156\) 0 0
\(157\) 19.0964i 1.52406i −0.647541 0.762031i \(-0.724202\pi\)
0.647541 0.762031i \(-0.275798\pi\)
\(158\) 0 0
\(159\) 1.79516 18.6289i 0.142365 1.47737i
\(160\) 0 0
\(161\) 24.8379i 1.95750i
\(162\) 0 0
\(163\) 14.8464 1.16286 0.581428 0.813598i \(-0.302494\pi\)
0.581428 + 0.813598i \(0.302494\pi\)
\(164\) 0 0
\(165\) −5.95809 −0.463837
\(166\) 0 0
\(167\) 19.0043i 1.47060i −0.677742 0.735299i \(-0.737042\pi\)
0.677742 0.735299i \(-0.262958\pi\)
\(168\) 0 0
\(169\) −3.04650 −0.234346
\(170\) 0 0
\(171\) 3.60868i 0.275963i
\(172\) 0 0
\(173\) 21.4866i 1.63360i 0.576923 + 0.816798i \(0.304253\pi\)
−0.576923 + 0.816798i \(0.695747\pi\)
\(174\) 0 0
\(175\) −3.48812 −0.263677
\(176\) 0 0
\(177\) 11.7778i 0.885275i
\(178\) 0 0
\(179\) 3.72190i 0.278188i 0.990279 + 0.139094i \(0.0444190\pi\)
−0.990279 + 0.139094i \(0.955581\pi\)
\(180\) 0 0
\(181\) 7.11585i 0.528917i 0.964397 + 0.264459i \(0.0851933\pi\)
−0.964397 + 0.264459i \(0.914807\pi\)
\(182\) 0 0
\(183\) −29.5745 −2.18621
\(184\) 0 0
\(185\) 27.2976i 2.00696i
\(186\) 0 0
\(187\) 6.89603 0.504287
\(188\) 0 0
\(189\) 5.63803i 0.410106i
\(190\) 0 0
\(191\) 9.99012i 0.722859i −0.932399 0.361430i \(-0.882289\pi\)
0.932399 0.361430i \(-0.117711\pi\)
\(192\) 0 0
\(193\) 14.5860i 1.04992i 0.851126 + 0.524962i \(0.175920\pi\)
−0.851126 + 0.524962i \(0.824080\pi\)
\(194\) 0 0
\(195\) −19.8136 −1.41888
\(196\) 0 0
\(197\) 12.0710 0.860023 0.430011 0.902824i \(-0.358510\pi\)
0.430011 + 0.902824i \(0.358510\pi\)
\(198\) 0 0
\(199\) 2.40006 0.170136 0.0850679 0.996375i \(-0.472889\pi\)
0.0850679 + 0.996375i \(0.472889\pi\)
\(200\) 0 0
\(201\) 22.7406 1.60400
\(202\) 0 0
\(203\) −22.9505 −1.61081
\(204\) 0 0
\(205\) −16.9217 −1.18186
\(206\) 0 0
\(207\) 24.8763i 1.72902i
\(208\) 0 0
\(209\) 0.948707i 0.0656234i
\(210\) 0 0
\(211\) −1.93651 −0.133315 −0.0666573 0.997776i \(-0.521233\pi\)
−0.0666573 + 0.997776i \(0.521233\pi\)
\(212\) 0 0
\(213\) 33.0414 2.26396
\(214\) 0 0
\(215\) 10.1793i 0.694219i
\(216\) 0 0
\(217\) 10.1167i 0.686766i
\(218\) 0 0
\(219\) 11.9639 0.808443
\(220\) 0 0
\(221\) 22.9327 1.54262
\(222\) 0 0
\(223\) −5.85826 −0.392298 −0.196149 0.980574i \(-0.562844\pi\)
−0.196149 + 0.980574i \(0.562844\pi\)
\(224\) 0 0
\(225\) 3.49351 0.232900
\(226\) 0 0
\(227\) 2.75394 0.182785 0.0913927 0.995815i \(-0.470868\pi\)
0.0913927 + 0.995815i \(0.470868\pi\)
\(228\) 0 0
\(229\) −24.5488 −1.62223 −0.811116 0.584886i \(-0.801139\pi\)
−0.811116 + 0.584886i \(0.801139\pi\)
\(230\) 0 0
\(231\) 8.78755i 0.578178i
\(232\) 0 0
\(233\) 21.7655i 1.42590i −0.701214 0.712951i \(-0.747358\pi\)
0.701214 0.712951i \(-0.252642\pi\)
\(234\) 0 0
\(235\) 13.7414i 0.896390i
\(236\) 0 0
\(237\) −21.3150 −1.38456
\(238\) 0 0
\(239\) 18.1693i 1.17527i −0.809125 0.587636i \(-0.800059\pi\)
0.809125 0.587636i \(-0.199941\pi\)
\(240\) 0 0
\(241\) 24.0911 1.55184 0.775921 0.630831i \(-0.217286\pi\)
0.775921 + 0.630831i \(0.217286\pi\)
\(242\) 0 0
\(243\) 22.1842i 1.42312i
\(244\) 0 0
\(245\) 14.6149i 0.933712i
\(246\) 0 0
\(247\) 3.15492i 0.200743i
\(248\) 0 0
\(249\) 34.8387 2.20781
\(250\) 0 0
\(251\) 12.2126i 0.770856i 0.922738 + 0.385428i \(0.125946\pi\)
−0.922738 + 0.385428i \(0.874054\pi\)
\(252\) 0 0
\(253\) 6.53987i 0.411158i
\(254\) 0 0
\(255\) −45.6501 −2.85872
\(256\) 0 0
\(257\) 9.45904i 0.590039i −0.955491 0.295019i \(-0.904674\pi\)
0.955491 0.295019i \(-0.0953261\pi\)
\(258\) 0 0
\(259\) 40.2611 2.50170
\(260\) 0 0
\(261\) 22.9859 1.42279
\(262\) 0 0
\(263\) 7.16668i 0.441916i −0.975283 0.220958i \(-0.929082\pi\)
0.975283 0.220958i \(-0.0709184\pi\)
\(264\) 0 0
\(265\) −1.70593 + 17.7031i −0.104795 + 1.08749i
\(266\) 0 0
\(267\) 26.8573i 1.64364i
\(268\) 0 0
\(269\) −8.56790 −0.522394 −0.261197 0.965285i \(-0.584117\pi\)
−0.261197 + 0.965285i \(0.584117\pi\)
\(270\) 0 0
\(271\) −17.5952 −1.06883 −0.534415 0.845222i \(-0.679468\pi\)
−0.534415 + 0.845222i \(0.679468\pi\)
\(272\) 0 0
\(273\) 29.2229i 1.76865i
\(274\) 0 0
\(275\) 0.918427 0.0553832
\(276\) 0 0
\(277\) 15.5039i 0.931537i 0.884907 + 0.465768i \(0.154222\pi\)
−0.884907 + 0.465768i \(0.845778\pi\)
\(278\) 0 0
\(279\) 10.1323i 0.606607i
\(280\) 0 0
\(281\) −26.7959 −1.59851 −0.799255 0.600992i \(-0.794772\pi\)
−0.799255 + 0.600992i \(0.794772\pi\)
\(282\) 0 0
\(283\) 12.6246i 0.750454i −0.926933 0.375227i \(-0.877565\pi\)
0.926933 0.375227i \(-0.122435\pi\)
\(284\) 0 0
\(285\) 6.28022i 0.372008i
\(286\) 0 0
\(287\) 24.9577i 1.47320i
\(288\) 0 0
\(289\) 35.8365 2.10803
\(290\) 0 0
\(291\) 41.0317i 2.40532i
\(292\) 0 0
\(293\) −5.83415 −0.340835 −0.170417 0.985372i \(-0.554512\pi\)
−0.170417 + 0.985372i \(0.554512\pi\)
\(294\) 0 0
\(295\) 11.1924i 0.651649i
\(296\) 0 0
\(297\) 1.48450i 0.0861396i
\(298\) 0 0
\(299\) 21.7483i 1.25774i
\(300\) 0 0
\(301\) −15.0133 −0.865353
\(302\) 0 0
\(303\) 13.2563 0.761557
\(304\) 0 0
\(305\) 28.1046 1.60927
\(306\) 0 0
\(307\) 31.0926 1.77455 0.887276 0.461240i \(-0.152595\pi\)
0.887276 + 0.461240i \(0.152595\pi\)
\(308\) 0 0
\(309\) −2.28528 −0.130005
\(310\) 0 0
\(311\) 8.31053 0.471247 0.235623 0.971844i \(-0.424287\pi\)
0.235623 + 0.971844i \(0.424287\pi\)
\(312\) 0 0
\(313\) 1.49511i 0.0845085i 0.999107 + 0.0422543i \(0.0134540\pi\)
−0.999107 + 0.0422543i \(0.986546\pi\)
\(314\) 0 0
\(315\) 31.7647i 1.78974i
\(316\) 0 0
\(317\) 10.0874 0.566566 0.283283 0.959036i \(-0.408576\pi\)
0.283283 + 0.959036i \(0.408576\pi\)
\(318\) 0 0
\(319\) 6.04290 0.338337
\(320\) 0 0
\(321\) 11.7118i 0.653690i
\(322\) 0 0
\(323\) 7.26887i 0.404451i
\(324\) 0 0
\(325\) 3.05422 0.169418
\(326\) 0 0
\(327\) 41.5370 2.29700
\(328\) 0 0
\(329\) −20.2671 −1.11736
\(330\) 0 0
\(331\) 20.9475 1.15138 0.575688 0.817669i \(-0.304734\pi\)
0.575688 + 0.817669i \(0.304734\pi\)
\(332\) 0 0
\(333\) −40.3233 −2.20971
\(334\) 0 0
\(335\) −21.6104 −1.18070
\(336\) 0 0
\(337\) 5.92030i 0.322500i −0.986914 0.161250i \(-0.948448\pi\)
0.986914 0.161250i \(-0.0515525\pi\)
\(338\) 0 0
\(339\) 2.86172i 0.155427i
\(340\) 0 0
\(341\) 2.66374i 0.144250i
\(342\) 0 0
\(343\) −3.66637 −0.197965
\(344\) 0 0
\(345\) 43.2925i 2.33079i
\(346\) 0 0
\(347\) 33.8245 1.81579 0.907897 0.419193i \(-0.137687\pi\)
0.907897 + 0.419193i \(0.137687\pi\)
\(348\) 0 0
\(349\) 35.3581i 1.89268i −0.323178 0.946338i \(-0.604751\pi\)
0.323178 0.946338i \(-0.395249\pi\)
\(350\) 0 0
\(351\) 4.93670i 0.263502i
\(352\) 0 0
\(353\) 7.04080i 0.374744i 0.982289 + 0.187372i \(0.0599969\pi\)
−0.982289 + 0.187372i \(0.940003\pi\)
\(354\) 0 0
\(355\) −31.3992 −1.66650
\(356\) 0 0
\(357\) 67.3291i 3.56343i
\(358\) 0 0
\(359\) 20.4866i 1.08124i 0.841267 + 0.540620i \(0.181810\pi\)
−0.841267 + 0.540620i \(0.818190\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 25.9643i 1.36277i
\(364\) 0 0
\(365\) −11.3692 −0.595093
\(366\) 0 0
\(367\) −28.6024 −1.49303 −0.746516 0.665368i \(-0.768275\pi\)
−0.746516 + 0.665368i \(0.768275\pi\)
\(368\) 0 0
\(369\) 24.9962i 1.30125i
\(370\) 0 0
\(371\) 26.1101 + 2.51607i 1.35557 + 0.130628i
\(372\) 0 0
\(373\) 13.8995i 0.719690i 0.933012 + 0.359845i \(0.117170\pi\)
−0.933012 + 0.359845i \(0.882830\pi\)
\(374\) 0 0
\(375\) 25.3213 1.30759
\(376\) 0 0
\(377\) 20.0956 1.03498
\(378\) 0 0
\(379\) 24.8767i 1.27783i −0.769278 0.638915i \(-0.779384\pi\)
0.769278 0.638915i \(-0.220616\pi\)
\(380\) 0 0
\(381\) 31.5656 1.61715
\(382\) 0 0
\(383\) 11.9379i 0.609998i −0.952353 0.304999i \(-0.901344\pi\)
0.952353 0.304999i \(-0.0986562\pi\)
\(384\) 0 0
\(385\) 8.35079i 0.425596i
\(386\) 0 0
\(387\) 15.0365 0.764349
\(388\) 0 0
\(389\) 7.34772i 0.372544i −0.982498 0.186272i \(-0.940359\pi\)
0.982498 0.186272i \(-0.0596406\pi\)
\(390\) 0 0
\(391\) 50.1077i 2.53406i
\(392\) 0 0
\(393\) 55.8834i 2.81895i
\(394\) 0 0
\(395\) 20.2556 1.01917
\(396\) 0 0
\(397\) 21.8014i 1.09418i 0.837073 + 0.547091i \(0.184265\pi\)
−0.837073 + 0.547091i \(0.815735\pi\)
\(398\) 0 0
\(399\) 9.26266 0.463713
\(400\) 0 0
\(401\) 18.9816i 0.947897i 0.880553 + 0.473948i \(0.157172\pi\)
−0.880553 + 0.473948i \(0.842828\pi\)
\(402\) 0 0
\(403\) 8.85826i 0.441261i
\(404\) 0 0
\(405\) 16.6206i 0.825885i
\(406\) 0 0
\(407\) −10.6008 −0.525463
\(408\) 0 0
\(409\) 32.1676 1.59058 0.795292 0.606226i \(-0.207317\pi\)
0.795292 + 0.606226i \(0.207317\pi\)
\(410\) 0 0
\(411\) −32.6301 −1.60952
\(412\) 0 0
\(413\) −16.5077 −0.812289
\(414\) 0 0
\(415\) −33.1071 −1.62517
\(416\) 0 0
\(417\) 24.6313 1.20620
\(418\) 0 0
\(419\) 29.6421i 1.44811i 0.689741 + 0.724056i \(0.257725\pi\)
−0.689741 + 0.724056i \(0.742275\pi\)
\(420\) 0 0
\(421\) 22.8937i 1.11577i 0.829917 + 0.557886i \(0.188388\pi\)
−0.829917 + 0.557886i \(0.811612\pi\)
\(422\) 0 0
\(423\) 20.2984 0.986943
\(424\) 0 0
\(425\) 7.03687 0.341338
\(426\) 0 0
\(427\) 41.4513i 2.00597i
\(428\) 0 0
\(429\) 7.69445i 0.371491i
\(430\) 0 0
\(431\) 34.7222 1.67251 0.836255 0.548341i \(-0.184740\pi\)
0.836255 + 0.548341i \(0.184740\pi\)
\(432\) 0 0
\(433\) 30.1939 1.45102 0.725512 0.688209i \(-0.241603\pi\)
0.725512 + 0.688209i \(0.241603\pi\)
\(434\) 0 0
\(435\) −40.0026 −1.91798
\(436\) 0 0
\(437\) −6.89346 −0.329759
\(438\) 0 0
\(439\) 24.7509 1.18130 0.590648 0.806930i \(-0.298872\pi\)
0.590648 + 0.806930i \(0.298872\pi\)
\(440\) 0 0
\(441\) −21.5887 −1.02804
\(442\) 0 0
\(443\) 11.6006i 0.551159i 0.961278 + 0.275580i \(0.0888698\pi\)
−0.961278 + 0.275580i \(0.911130\pi\)
\(444\) 0 0
\(445\) 25.5224i 1.20988i
\(446\) 0 0
\(447\) 43.9526i 2.07889i
\(448\) 0 0
\(449\) −10.4105 −0.491303 −0.245652 0.969358i \(-0.579002\pi\)
−0.245652 + 0.969358i \(0.579002\pi\)
\(450\) 0 0
\(451\) 6.57140i 0.309435i
\(452\) 0 0
\(453\) 35.5684 1.67115
\(454\) 0 0
\(455\) 27.7705i 1.30190i
\(456\) 0 0
\(457\) 25.7431i 1.20421i −0.798417 0.602105i \(-0.794329\pi\)
0.798417 0.602105i \(-0.205671\pi\)
\(458\) 0 0
\(459\) 11.3741i 0.530896i
\(460\) 0 0
\(461\) −6.85631 −0.319330 −0.159665 0.987171i \(-0.551041\pi\)
−0.159665 + 0.987171i \(0.551041\pi\)
\(462\) 0 0
\(463\) 32.8067i 1.52466i 0.647190 + 0.762329i \(0.275944\pi\)
−0.647190 + 0.762329i \(0.724056\pi\)
\(464\) 0 0
\(465\) 17.6334i 0.817728i
\(466\) 0 0
\(467\) −29.8687 −1.38216 −0.691078 0.722780i \(-0.742864\pi\)
−0.691078 + 0.722780i \(0.742864\pi\)
\(468\) 0 0
\(469\) 31.8730i 1.47176i
\(470\) 0 0
\(471\) −49.0919 −2.26204
\(472\) 0 0
\(473\) 3.95303 0.181761
\(474\) 0 0
\(475\) 0.968083i 0.0444187i
\(476\) 0 0
\(477\) −26.1505 2.51996i −1.19735 0.115381i
\(478\) 0 0
\(479\) 10.9833i 0.501841i −0.968008 0.250921i \(-0.919267\pi\)
0.968008 0.250921i \(-0.0807333\pi\)
\(480\) 0 0
\(481\) −35.2530 −1.60740
\(482\) 0 0
\(483\) 63.8518 2.90536
\(484\) 0 0
\(485\) 38.9923i 1.77055i
\(486\) 0 0
\(487\) −21.4742 −0.973088 −0.486544 0.873656i \(-0.661743\pi\)
−0.486544 + 0.873656i \(0.661743\pi\)
\(488\) 0 0
\(489\) 38.1661i 1.72593i
\(490\) 0 0
\(491\) 1.52728i 0.0689252i 0.999406 + 0.0344626i \(0.0109720\pi\)
−0.999406 + 0.0344626i \(0.989028\pi\)
\(492\) 0 0
\(493\) 46.2999 2.08524
\(494\) 0 0
\(495\) 8.36370i 0.375920i
\(496\) 0 0
\(497\) 46.3105i 2.07731i
\(498\) 0 0
\(499\) 3.54485i 0.158689i 0.996847 + 0.0793446i \(0.0252827\pi\)
−0.996847 + 0.0793446i \(0.974717\pi\)
\(500\) 0 0
\(501\) −48.8551 −2.18269
\(502\) 0 0
\(503\) 28.6581i 1.27780i −0.769289 0.638901i \(-0.779390\pi\)
0.769289 0.638901i \(-0.220610\pi\)
\(504\) 0 0
\(505\) −12.5975 −0.560581
\(506\) 0 0
\(507\) 7.83175i 0.347820i
\(508\) 0 0
\(509\) 7.91923i 0.351014i −0.984478 0.175507i \(-0.943844\pi\)
0.984478 0.175507i \(-0.0561564\pi\)
\(510\) 0 0
\(511\) 16.7684i 0.741791i
\(512\) 0 0
\(513\) −1.56476 −0.0690860
\(514\) 0 0
\(515\) 2.17170 0.0956966
\(516\) 0 0
\(517\) 5.33637 0.234693
\(518\) 0 0
\(519\) 55.2364 2.42461
\(520\) 0 0
\(521\) −15.3321 −0.671712 −0.335856 0.941913i \(-0.609025\pi\)
−0.335856 + 0.941913i \(0.609025\pi\)
\(522\) 0 0
\(523\) −21.8157 −0.953934 −0.476967 0.878921i \(-0.658264\pi\)
−0.476967 + 0.878921i \(0.658264\pi\)
\(524\) 0 0
\(525\) 8.96703i 0.391353i
\(526\) 0 0
\(527\) 20.4093i 0.889042i
\(528\) 0 0
\(529\) −24.5198 −1.06608
\(530\) 0 0
\(531\) 16.5332 0.717478
\(532\) 0 0
\(533\) 21.8531i 0.946564i
\(534\) 0 0
\(535\) 11.1297i 0.481180i
\(536\) 0 0
\(537\) 9.56802 0.412891
\(538\) 0 0
\(539\) −5.67559 −0.244465
\(540\) 0 0
\(541\) 2.96480 0.127467 0.0637335 0.997967i \(-0.479699\pi\)
0.0637335 + 0.997967i \(0.479699\pi\)
\(542\) 0 0
\(543\) 18.2930 0.785027
\(544\) 0 0
\(545\) −39.4725 −1.69082
\(546\) 0 0
\(547\) −24.9037 −1.06481 −0.532403 0.846491i \(-0.678711\pi\)
−0.532403 + 0.846491i \(0.678711\pi\)
\(548\) 0 0
\(549\) 41.5154i 1.77183i
\(550\) 0 0
\(551\) 6.36961i 0.271355i
\(552\) 0 0
\(553\) 29.8748i 1.27041i
\(554\) 0 0
\(555\) 70.1750 2.97876
\(556\) 0 0
\(557\) 14.9872i 0.635030i −0.948253 0.317515i \(-0.897152\pi\)
0.948253 0.317515i \(-0.102848\pi\)
\(558\) 0 0
\(559\) 13.1458 0.556007
\(560\) 0 0
\(561\) 17.7279i 0.748471i
\(562\) 0 0
\(563\) 26.6862i 1.12469i 0.826903 + 0.562345i \(0.190101\pi\)
−0.826903 + 0.562345i \(0.809899\pi\)
\(564\) 0 0
\(565\) 2.71949i 0.114410i
\(566\) 0 0
\(567\) −24.5136 −1.02948
\(568\) 0 0
\(569\) 17.8067i 0.746494i −0.927732 0.373247i \(-0.878244\pi\)
0.927732 0.373247i \(-0.121756\pi\)
\(570\) 0 0
\(571\) 15.9075i 0.665708i −0.942978 0.332854i \(-0.891988\pi\)
0.942978 0.332854i \(-0.108012\pi\)
\(572\) 0 0
\(573\) −25.6819 −1.07288
\(574\) 0 0
\(575\) 6.67345i 0.278302i
\(576\) 0 0
\(577\) 42.8577 1.78419 0.892094 0.451849i \(-0.149235\pi\)
0.892094 + 0.451849i \(0.149235\pi\)
\(578\) 0 0
\(579\) 37.4968 1.55831
\(580\) 0 0
\(581\) 48.8295i 2.02579i
\(582\) 0 0
\(583\) −6.87484 0.662486i −0.284727 0.0274374i
\(584\) 0 0
\(585\) 27.8134i 1.14994i
\(586\) 0 0
\(587\) −45.3378 −1.87129 −0.935647 0.352938i \(-0.885183\pi\)
−0.935647 + 0.352938i \(0.885183\pi\)
\(588\) 0 0
\(589\) 2.80776 0.115692
\(590\) 0 0
\(591\) 31.0313i 1.27646i
\(592\) 0 0
\(593\) 16.2727 0.668238 0.334119 0.942531i \(-0.391561\pi\)
0.334119 + 0.942531i \(0.391561\pi\)
\(594\) 0 0
\(595\) 63.9827i 2.62304i
\(596\) 0 0
\(597\) 6.16992i 0.252518i
\(598\) 0 0
\(599\) −17.6092 −0.719494 −0.359747 0.933050i \(-0.617137\pi\)
−0.359747 + 0.933050i \(0.617137\pi\)
\(600\) 0 0
\(601\) 30.9394i 1.26204i −0.775765 0.631022i \(-0.782636\pi\)
0.775765 0.631022i \(-0.217364\pi\)
\(602\) 0 0
\(603\) 31.9223i 1.29997i
\(604\) 0 0
\(605\) 24.6738i 1.00313i
\(606\) 0 0
\(607\) −15.1792 −0.616104 −0.308052 0.951370i \(-0.599677\pi\)
−0.308052 + 0.951370i \(0.599677\pi\)
\(608\) 0 0
\(609\) 58.9996i 2.39078i
\(610\) 0 0
\(611\) 17.7460 0.717928
\(612\) 0 0
\(613\) 35.9950i 1.45383i −0.686730 0.726913i \(-0.740954\pi\)
0.686730 0.726913i \(-0.259046\pi\)
\(614\) 0 0
\(615\) 43.5012i 1.75414i
\(616\) 0 0
\(617\) 0.671258i 0.0270238i −0.999909 0.0135119i \(-0.995699\pi\)
0.999909 0.0135119i \(-0.00430111\pi\)
\(618\) 0 0
\(619\) 31.4141 1.26264 0.631320 0.775523i \(-0.282514\pi\)
0.631320 + 0.775523i \(0.282514\pi\)
\(620\) 0 0
\(621\) −10.7866 −0.432853
\(622\) 0 0
\(623\) −37.6429 −1.50813
\(624\) 0 0
\(625\) −28.9032 −1.15613
\(626\) 0 0
\(627\) −2.43887 −0.0973993
\(628\) 0 0
\(629\) −81.2222 −3.23854
\(630\) 0 0
\(631\) 4.73742i 0.188594i −0.995544 0.0942968i \(-0.969940\pi\)
0.995544 0.0942968i \(-0.0300603\pi\)
\(632\) 0 0
\(633\) 4.97825i 0.197868i
\(634\) 0 0
\(635\) −29.9967 −1.19038
\(636\) 0 0
\(637\) −18.8741 −0.747820
\(638\) 0 0
\(639\) 46.3820i 1.83484i
\(640\) 0 0
\(641\) 42.6838i 1.68591i −0.537986 0.842954i \(-0.680815\pi\)
0.537986 0.842954i \(-0.319185\pi\)
\(642\) 0 0
\(643\) −25.8484 −1.01936 −0.509682 0.860363i \(-0.670237\pi\)
−0.509682 + 0.860363i \(0.670237\pi\)
\(644\) 0 0
\(645\) −26.1682 −1.03037
\(646\) 0 0
\(647\) −41.5257 −1.63254 −0.816272 0.577668i \(-0.803963\pi\)
−0.816272 + 0.577668i \(0.803963\pi\)
\(648\) 0 0
\(649\) 4.34650 0.170615
\(650\) 0 0
\(651\) −26.0074 −1.01931
\(652\) 0 0
\(653\) 9.03265 0.353475 0.176737 0.984258i \(-0.443446\pi\)
0.176737 + 0.984258i \(0.443446\pi\)
\(654\) 0 0
\(655\) 53.1059i 2.07502i
\(656\) 0 0
\(657\) 16.7943i 0.655209i
\(658\) 0 0
\(659\) 27.7419i 1.08067i −0.841450 0.540335i \(-0.818298\pi\)
0.841450 0.540335i \(-0.181702\pi\)
\(660\) 0 0
\(661\) 18.8006 0.731260 0.365630 0.930760i \(-0.380854\pi\)
0.365630 + 0.930760i \(0.380854\pi\)
\(662\) 0 0
\(663\) 58.9539i 2.28958i
\(664\) 0 0
\(665\) −8.80229 −0.341338
\(666\) 0 0
\(667\) 43.9087i 1.70015i
\(668\) 0 0
\(669\) 15.0600i 0.582255i
\(670\) 0 0
\(671\) 10.9142i 0.421338i
\(672\) 0 0
\(673\) −5.60508 −0.216060 −0.108030 0.994148i \(-0.534454\pi\)
−0.108030 + 0.994148i \(0.534454\pi\)
\(674\) 0 0
\(675\) 1.51482i 0.0583055i
\(676\) 0 0
\(677\) 40.8171i 1.56873i −0.620301 0.784364i \(-0.712990\pi\)
0.620301 0.784364i \(-0.287010\pi\)
\(678\) 0 0
\(679\) −57.5095 −2.20701
\(680\) 0 0
\(681\) 7.07965i 0.271293i
\(682\) 0 0
\(683\) −33.9164 −1.29778 −0.648888 0.760884i \(-0.724766\pi\)
−0.648888 + 0.760884i \(0.724766\pi\)
\(684\) 0 0
\(685\) 31.0083 1.18477
\(686\) 0 0
\(687\) 63.1085i 2.40774i
\(688\) 0 0
\(689\) −22.8622 2.20309i −0.870982 0.0839311i
\(690\) 0 0
\(691\) 38.7868i 1.47552i 0.675064 + 0.737759i \(0.264116\pi\)
−0.675064 + 0.737759i \(0.735884\pi\)
\(692\) 0 0
\(693\) 12.3356 0.468589
\(694\) 0 0
\(695\) −23.4071 −0.887883
\(696\) 0 0
\(697\) 50.3492i 1.90711i
\(698\) 0 0
\(699\) −55.9532 −2.11635
\(700\) 0 0
\(701\) 39.1597i 1.47904i 0.673134 + 0.739520i \(0.264948\pi\)
−0.673134 + 0.739520i \(0.735052\pi\)
\(702\) 0 0
\(703\) 11.1740i 0.421434i
\(704\) 0 0
\(705\) −35.3255 −1.33044
\(706\) 0 0
\(707\) 18.5800i 0.698771i
\(708\) 0 0
\(709\) 3.27467i 0.122983i −0.998108 0.0614913i \(-0.980414\pi\)
0.998108 0.0614913i \(-0.0195857\pi\)
\(710\) 0 0
\(711\) 29.9210i 1.12212i
\(712\) 0 0
\(713\) 19.3552 0.724859
\(714\) 0 0
\(715\) 7.31202i 0.273454i
\(716\) 0 0
\(717\) −46.7084 −1.74436
\(718\) 0 0
\(719\) 22.2113i 0.828341i −0.910199 0.414170i \(-0.864072\pi\)
0.910199 0.414170i \(-0.135928\pi\)
\(720\) 0 0
\(721\) 3.20303i 0.119287i
\(722\) 0 0
\(723\) 61.9317i 2.30327i
\(724\) 0 0
\(725\) 6.16632 0.229011
\(726\) 0 0
\(727\) −10.1898 −0.377920 −0.188960 0.981985i \(-0.560512\pi\)
−0.188960 + 0.981985i \(0.560512\pi\)
\(728\) 0 0
\(729\) 36.6193 1.35627
\(730\) 0 0
\(731\) 30.2876 1.12023
\(732\) 0 0
\(733\) 20.3091 0.750132 0.375066 0.926998i \(-0.377620\pi\)
0.375066 + 0.926998i \(0.377620\pi\)
\(734\) 0 0
\(735\) 37.5711 1.38583
\(736\) 0 0
\(737\) 8.39222i 0.309131i
\(738\) 0 0
\(739\) 24.9263i 0.916929i −0.888713 0.458464i \(-0.848400\pi\)
0.888713 0.458464i \(-0.151600\pi\)
\(740\) 0 0
\(741\) −8.11046 −0.297945
\(742\) 0 0
\(743\) 17.0233 0.624523 0.312262 0.949996i \(-0.398913\pi\)
0.312262 + 0.949996i \(0.398913\pi\)
\(744\) 0 0
\(745\) 41.7681i 1.53027i
\(746\) 0 0
\(747\) 48.9050i 1.78934i
\(748\) 0 0
\(749\) 16.4152 0.599797
\(750\) 0 0
\(751\) 3.83424 0.139914 0.0699568 0.997550i \(-0.477714\pi\)
0.0699568 + 0.997550i \(0.477714\pi\)
\(752\) 0 0
\(753\) 31.3955 1.14412
\(754\) 0 0
\(755\) −33.8006 −1.23013
\(756\) 0 0
\(757\) −31.0335 −1.12793 −0.563965 0.825798i \(-0.690725\pi\)
−0.563965 + 0.825798i \(0.690725\pi\)
\(758\) 0 0
\(759\) −16.8123 −0.610248
\(760\) 0 0
\(761\) 10.3119i 0.373806i −0.982378 0.186903i \(-0.940155\pi\)
0.982378 0.186903i \(-0.0598450\pi\)
\(762\) 0 0
\(763\) 58.2177i 2.10762i
\(764\) 0 0
\(765\) 64.0816i 2.31687i
\(766\) 0 0
\(767\) 14.4542 0.521912
\(768\) 0 0
\(769\) 20.9935i 0.757044i 0.925592 + 0.378522i \(0.123568\pi\)
−0.925592 + 0.378522i \(0.876432\pi\)
\(770\) 0 0
\(771\) −24.3167 −0.875744
\(772\) 0 0
\(773\) 9.05916i 0.325835i −0.986640 0.162918i \(-0.947909\pi\)
0.986640 0.162918i \(-0.0520905\pi\)
\(774\) 0 0
\(775\) 2.71815i 0.0976388i
\(776\) 0 0
\(777\) 103.501i 3.71307i
\(778\) 0 0
\(779\) −6.92669 −0.248175
\(780\) 0 0
\(781\) 12.1936i 0.436322i
\(782\) 0 0
\(783\) 9.96695i 0.356190i
\(784\) 0 0
\(785\) 46.6519 1.66508
\(786\) 0 0
\(787\) 23.4232i 0.834948i −0.908689 0.417474i \(-0.862916\pi\)
0.908689 0.417474i \(-0.137084\pi\)
\(788\) 0 0
\(789\) −18.4236 −0.655899
\(790\) 0 0
\(791\) −4.01096 −0.142613
\(792\) 0 0
\(793\) 36.2951i 1.28888i
\(794\) 0 0
\(795\) 45.5099 + 4.38551i 1.61407 + 0.155538i
\(796\) 0 0
\(797\) 55.5691i 1.96836i 0.177176 + 0.984179i \(0.443304\pi\)
−0.177176 + 0.984179i \(0.556696\pi\)
\(798\) 0 0
\(799\) 40.8866 1.44646
\(800\) 0 0
\(801\) 37.7010 1.33210
\(802\) 0 0
\(803\) 4.41515i 0.155807i
\(804\) 0 0
\(805\) −60.6783 −2.13863
\(806\) 0 0
\(807\) 22.0258i 0.775346i
\(808\) 0 0
\(809\) 30.6779i 1.07858i 0.842121 + 0.539289i \(0.181307\pi\)
−0.842121 + 0.539289i \(0.818693\pi\)
\(810\) 0 0
\(811\) 21.7290 0.763010 0.381505 0.924367i \(-0.375406\pi\)
0.381505 + 0.924367i \(0.375406\pi\)
\(812\) 0 0
\(813\) 45.2325i 1.58637i
\(814\) 0 0
\(815\) 36.2691i 1.27045i
\(816\) 0 0
\(817\) 4.16676i 0.145776i
\(818\) 0 0
\(819\) 41.0218 1.43342
\(820\) 0 0
\(821\) 6.69953i 0.233815i 0.993143 + 0.116908i \(0.0372982\pi\)
−0.993143 + 0.116908i \(0.962702\pi\)
\(822\) 0 0
\(823\) −33.4208 −1.16497 −0.582487 0.812840i \(-0.697920\pi\)
−0.582487 + 0.812840i \(0.697920\pi\)
\(824\) 0 0
\(825\) 2.36103i 0.0822006i
\(826\) 0 0
\(827\) 16.1652i 0.562119i −0.959690 0.281059i \(-0.909314\pi\)
0.959690 0.281059i \(-0.0906858\pi\)
\(828\) 0 0
\(829\) 22.0835i 0.766992i 0.923543 + 0.383496i \(0.125280\pi\)
−0.923543 + 0.383496i \(0.874720\pi\)
\(830\) 0 0
\(831\) 39.8563 1.38260
\(832\) 0 0
\(833\) −43.4856 −1.50669
\(834\) 0 0
\(835\) 46.4269 1.60667
\(836\) 0 0
\(837\) 4.39349 0.151861
\(838\) 0 0
\(839\) 36.4795 1.25941 0.629706 0.776833i \(-0.283175\pi\)
0.629706 + 0.776833i \(0.283175\pi\)
\(840\) 0 0
\(841\) 11.5720 0.399034
\(842\) 0 0
\(843\) 68.8852i 2.37253i
\(844\) 0 0
\(845\) 7.44250i 0.256030i
\(846\) 0 0
\(847\) −36.3913 −1.25042
\(848\) 0 0
\(849\) −32.4545 −1.11384
\(850\) 0 0
\(851\) 77.0274i 2.64046i
\(852\) 0 0
\(853\) 25.6039i 0.876661i 0.898814 + 0.438330i \(0.144430\pi\)
−0.898814 + 0.438330i \(0.855570\pi\)
\(854\) 0 0
\(855\) 8.81589 0.301497
\(856\) 0 0
\(857\) 31.2444 1.06729 0.533644 0.845709i \(-0.320822\pi\)
0.533644 + 0.845709i \(0.320822\pi\)
\(858\) 0 0
\(859\) −46.4493 −1.58483 −0.792415 0.609983i \(-0.791176\pi\)
−0.792415 + 0.609983i \(0.791176\pi\)
\(860\) 0 0
\(861\) 64.1596 2.18655
\(862\) 0 0
\(863\) 22.2303 0.756728 0.378364 0.925657i \(-0.376487\pi\)
0.378364 + 0.925657i \(0.376487\pi\)
\(864\) 0 0
\(865\) −52.4911 −1.78475
\(866\) 0 0
\(867\) 92.1262i 3.12877i
\(868\) 0 0
\(869\) 7.86609i 0.266839i
\(870\) 0 0
\(871\) 27.9083i 0.945635i
\(872\) 0 0
\(873\) 57.5984 1.94941
\(874\) 0 0
\(875\) 35.4901i 1.19978i
\(876\) 0 0
\(877\) −31.8793 −1.07649 −0.538244 0.842789i \(-0.680912\pi\)
−0.538244 + 0.842789i \(0.680912\pi\)
\(878\) 0 0
\(879\) 14.9981i 0.505872i
\(880\) 0 0
\(881\) 1.55656i 0.0524418i 0.999656 + 0.0262209i \(0.00834732\pi\)
−0.999656 + 0.0262209i \(0.991653\pi\)
\(882\) 0 0
\(883\) 1.37840i 0.0463869i 0.999731 + 0.0231934i \(0.00738336\pi\)
−0.999731 + 0.0231934i \(0.992617\pi\)
\(884\) 0 0
\(885\) −28.7728 −0.967187
\(886\) 0 0
\(887\) 7.54983i 0.253499i −0.991935 0.126749i \(-0.959546\pi\)
0.991935 0.126749i \(-0.0404544\pi\)
\(888\) 0 0
\(889\) 44.2420i 1.48383i
\(890\) 0 0
\(891\) 6.45448 0.216233
\(892\) 0 0
\(893\) 5.62489i 0.188230i
\(894\) 0 0
\(895\) −9.09247 −0.303928
\(896\) 0 0
\(897\) −55.9091 −1.86675
\(898\) 0 0
\(899\) 17.8844i 0.596477i
\(900\) 0 0
\(901\) −52.6742 5.07589i −1.75483 0.169102i
\(902\) 0 0
\(903\) 38.5953i 1.28437i
\(904\) 0 0
\(905\) −17.3838 −0.577857
\(906\) 0 0
\(907\) −14.3016 −0.474876 −0.237438 0.971403i \(-0.576308\pi\)
−0.237438 + 0.971403i \(0.576308\pi\)
\(908\) 0 0
\(909\) 18.6087i 0.617211i
\(910\) 0 0
\(911\) −16.3908 −0.543051 −0.271526 0.962431i \(-0.587528\pi\)
−0.271526 + 0.962431i \(0.587528\pi\)
\(912\) 0 0
\(913\) 12.8569i 0.425501i
\(914\) 0 0
\(915\) 72.2495i 2.38850i
\(916\) 0 0
\(917\) −78.3256 −2.58654
\(918\) 0 0
\(919\) 8.06505i 0.266042i −0.991113 0.133021i \(-0.957532\pi\)
0.991113 0.133021i \(-0.0424677\pi\)
\(920\) 0 0
\(921\) 79.9310i 2.63382i
\(922\) 0 0
\(923\) 40.5498i 1.33471i
\(924\) 0 0
\(925\) −10.8173 −0.355672
\(926\) 0 0
\(927\) 3.20798i 0.105364i
\(928\) 0 0
\(929\) −15.4599 −0.507223 −0.253611 0.967306i \(-0.581618\pi\)
−0.253611 + 0.967306i \(0.581618\pi\)
\(930\) 0 0
\(931\) 5.98244i 0.196067i
\(932\) 0 0
\(933\) 21.3642i 0.699432i
\(934\) 0 0
\(935\) 16.8468i 0.550948i
\(936\) 0 0
\(937\) −28.4639 −0.929874 −0.464937 0.885344i \(-0.653923\pi\)
−0.464937 + 0.885344i \(0.653923\pi\)
\(938\) 0 0
\(939\) 3.84353 0.125429
\(940\) 0 0
\(941\) 50.6180 1.65010 0.825051 0.565059i \(-0.191147\pi\)
0.825051 + 0.565059i \(0.191147\pi\)
\(942\) 0 0
\(943\) −47.7489 −1.55492
\(944\) 0 0
\(945\) −13.7735 −0.448052
\(946\) 0 0
\(947\) −22.7460 −0.739145 −0.369573 0.929202i \(-0.620496\pi\)
−0.369573 + 0.929202i \(0.620496\pi\)
\(948\) 0 0
\(949\) 14.6826i 0.476616i
\(950\) 0 0
\(951\) 25.9321i 0.840906i
\(952\) 0 0
\(953\) −18.5790 −0.601834 −0.300917 0.953650i \(-0.597293\pi\)
−0.300917 + 0.953650i \(0.597293\pi\)
\(954\) 0 0
\(955\) 24.4055 0.789744
\(956\) 0 0
\(957\) 15.5347i 0.502165i
\(958\) 0 0
\(959\) 45.7340i 1.47683i
\(960\) 0 0
\(961\) 23.1165 0.745692
\(962\) 0 0
\(963\) −16.4405 −0.529788
\(964\) 0 0
\(965\) −35.6331 −1.14707
\(966\) 0 0
\(967\) −3.93218 −0.126450 −0.0632252 0.997999i \(-0.520139\pi\)
−0.0632252 + 0.997999i \(0.520139\pi\)
\(968\) 0 0
\(969\) −18.6863 −0.600292
\(970\) 0 0
\(971\) 42.5985 1.36705 0.683526 0.729926i \(-0.260446\pi\)
0.683526 + 0.729926i \(0.260446\pi\)
\(972\) 0 0
\(973\) 34.5230i 1.10676i
\(974\) 0 0
\(975\) 7.85160i 0.251452i
\(976\) 0 0
\(977\) 16.4273i 0.525557i 0.964856 + 0.262779i \(0.0846388\pi\)
−0.964856 + 0.262779i \(0.915361\pi\)
\(978\) 0 0
\(979\) 9.91143 0.316771
\(980\) 0 0
\(981\) 58.3077i 1.86162i
\(982\) 0 0
\(983\) 5.71022 0.182128 0.0910638 0.995845i \(-0.470973\pi\)
0.0910638 + 0.995845i \(0.470973\pi\)
\(984\) 0 0
\(985\) 29.4890i 0.939598i
\(986\) 0 0
\(987\) 52.1014i 1.65841i
\(988\) 0 0
\(989\) 28.7234i 0.913351i
\(990\) 0 0
\(991\) 5.61664 0.178418 0.0892092 0.996013i \(-0.471566\pi\)
0.0892092 + 0.996013i \(0.471566\pi\)
\(992\) 0 0
\(993\) 53.8504i 1.70889i
\(994\) 0 0
\(995\) 5.86326i 0.185878i
\(996\) 0 0
\(997\) −39.0475 −1.23665 −0.618323 0.785924i \(-0.712188\pi\)
−0.618323 + 0.785924i \(0.712188\pi\)
\(998\) 0 0
\(999\) 17.4846i 0.553189i
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 4028.2.c.a.3497.10 82
53.52 even 2 inner 4028.2.c.a.3497.73 yes 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.c.a.3497.10 82 1.1 even 1 trivial
4028.2.c.a.3497.73 yes 82 53.52 even 2 inner