Properties

Label 2205.2.g.b.2204.22
Level $2205$
Weight $2$
Character 2205.2204
Analytic conductor $17.607$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,2,Mod(2204,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.2204");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.g (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: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2204.22
Character \(\chi\) \(=\) 2205.2204
Dual form 2205.2.g.b.2204.23

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56937 q^{2} +4.60168 q^{4} +(0.884697 - 2.05361i) q^{5} +6.68468 q^{8} +(2.27312 - 5.27649i) q^{10} +5.63712i q^{11} +3.53537 q^{13} +7.97209 q^{16} -5.34283i q^{17} +1.44270i q^{19} +(4.07109 - 9.45005i) q^{20} +14.4839i q^{22} +2.99860 q^{23} +(-3.43462 - 3.63364i) q^{25} +9.08368 q^{26} -2.19404i q^{29} +1.52304i q^{31} +7.11391 q^{32} -13.7277i q^{34} +1.09314i q^{37} +3.70684i q^{38} +(5.91392 - 13.7277i) q^{40} -6.52058 q^{41} -0.486125i q^{43} +25.9402i q^{44} +7.70452 q^{46} +4.25632i q^{47} +(-8.82483 - 9.33618i) q^{50} +16.2686 q^{52} +4.02204 q^{53} +(11.5764 + 4.98714i) q^{55} -5.63731i q^{58} -11.4196 q^{59} -7.00854i q^{61} +3.91326i q^{62} +2.33411 q^{64} +(3.12773 - 7.26027i) q^{65} +2.48980i q^{67} -24.5860i q^{68} -8.13849i q^{71} -3.13802 q^{73} +2.80869i q^{74} +6.63886i q^{76} +12.8050 q^{79} +(7.05288 - 16.3716i) q^{80} -16.7538 q^{82} +6.77241i q^{83} +(-10.9721 - 4.72679i) q^{85} -1.24904i q^{86} +37.6824i q^{88} -10.3303 q^{89} +13.7986 q^{92} +10.9361i q^{94} +(2.96275 + 1.27635i) q^{95} -14.7321 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 48 q^{4} - 24 q^{25} + 48 q^{46} + 48 q^{64} + 120 q^{79} - 72 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56937 1.81682 0.908411 0.418079i \(-0.137297\pi\)
0.908411 + 0.418079i \(0.137297\pi\)
\(3\) 0 0
\(4\) 4.60168 2.30084
\(5\) 0.884697 2.05361i 0.395648 0.918402i
\(6\) 0 0
\(7\) 0 0
\(8\) 6.68468 2.36339
\(9\) 0 0
\(10\) 2.27312 5.27649i 0.718822 1.66857i
\(11\) 5.63712i 1.69966i 0.527060 + 0.849828i \(0.323294\pi\)
−0.527060 + 0.849828i \(0.676706\pi\)
\(12\) 0 0
\(13\) 3.53537 0.980535 0.490267 0.871572i \(-0.336899\pi\)
0.490267 + 0.871572i \(0.336899\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 7.97209 1.99302
\(17\) 5.34283i 1.29583i −0.761714 0.647914i \(-0.775642\pi\)
0.761714 0.647914i \(-0.224358\pi\)
\(18\) 0 0
\(19\) 1.44270i 0.330979i 0.986212 + 0.165489i \(0.0529204\pi\)
−0.986212 + 0.165489i \(0.947080\pi\)
\(20\) 4.07109 9.45005i 0.910323 2.11310i
\(21\) 0 0
\(22\) 14.4839i 3.08797i
\(23\) 2.99860 0.625251 0.312626 0.949876i \(-0.398791\pi\)
0.312626 + 0.949876i \(0.398791\pi\)
\(24\) 0 0
\(25\) −3.43462 3.63364i −0.686925 0.726729i
\(26\) 9.08368 1.78146
\(27\) 0 0
\(28\) 0 0
\(29\) 2.19404i 0.407423i −0.979031 0.203712i \(-0.934699\pi\)
0.979031 0.203712i \(-0.0653005\pi\)
\(30\) 0 0
\(31\) 1.52304i 0.273546i 0.990602 + 0.136773i \(0.0436731\pi\)
−0.990602 + 0.136773i \(0.956327\pi\)
\(32\) 7.11391 1.25757
\(33\) 0 0
\(34\) 13.7277i 2.35429i
\(35\) 0 0
\(36\) 0 0
\(37\) 1.09314i 0.179712i 0.995955 + 0.0898558i \(0.0286406\pi\)
−0.995955 + 0.0898558i \(0.971359\pi\)
\(38\) 3.70684i 0.601329i
\(39\) 0 0
\(40\) 5.91392 13.7277i 0.935073 2.17055i
\(41\) −6.52058 −1.01834 −0.509172 0.860665i \(-0.670048\pi\)
−0.509172 + 0.860665i \(0.670048\pi\)
\(42\) 0 0
\(43\) 0.486125i 0.0741333i −0.999313 0.0370667i \(-0.988199\pi\)
0.999313 0.0370667i \(-0.0118014\pi\)
\(44\) 25.9402i 3.91064i
\(45\) 0 0
\(46\) 7.70452 1.13597
\(47\) 4.25632i 0.620849i 0.950598 + 0.310424i \(0.100471\pi\)
−0.950598 + 0.310424i \(0.899529\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −8.82483 9.33618i −1.24802 1.32034i
\(51\) 0 0
\(52\) 16.2686 2.25605
\(53\) 4.02204 0.552469 0.276235 0.961090i \(-0.410913\pi\)
0.276235 + 0.961090i \(0.410913\pi\)
\(54\) 0 0
\(55\) 11.5764 + 4.98714i 1.56097 + 0.672466i
\(56\) 0 0
\(57\) 0 0
\(58\) 5.63731i 0.740216i
\(59\) −11.4196 −1.48670 −0.743350 0.668903i \(-0.766764\pi\)
−0.743350 + 0.668903i \(0.766764\pi\)
\(60\) 0 0
\(61\) 7.00854i 0.897352i −0.893695 0.448676i \(-0.851896\pi\)
0.893695 0.448676i \(-0.148104\pi\)
\(62\) 3.91326i 0.496984i
\(63\) 0 0
\(64\) 2.33411 0.291764
\(65\) 3.12773 7.26027i 0.387947 0.900525i
\(66\) 0 0
\(67\) 2.48980i 0.304177i 0.988367 + 0.152089i \(0.0485999\pi\)
−0.988367 + 0.152089i \(0.951400\pi\)
\(68\) 24.5860i 2.98149i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.13849i 0.965861i −0.875659 0.482930i \(-0.839572\pi\)
0.875659 0.482930i \(-0.160428\pi\)
\(72\) 0 0
\(73\) −3.13802 −0.367277 −0.183639 0.982994i \(-0.558788\pi\)
−0.183639 + 0.982994i \(0.558788\pi\)
\(74\) 2.80869i 0.326504i
\(75\) 0 0
\(76\) 6.63886i 0.761529i
\(77\) 0 0
\(78\) 0 0
\(79\) 12.8050 1.44068 0.720340 0.693622i \(-0.243986\pi\)
0.720340 + 0.693622i \(0.243986\pi\)
\(80\) 7.05288 16.3716i 0.788536 1.83040i
\(81\) 0 0
\(82\) −16.7538 −1.85015
\(83\) 6.77241i 0.743368i 0.928359 + 0.371684i \(0.121220\pi\)
−0.928359 + 0.371684i \(0.878780\pi\)
\(84\) 0 0
\(85\) −10.9721 4.72679i −1.19009 0.512692i
\(86\) 1.24904i 0.134687i
\(87\) 0 0
\(88\) 37.6824i 4.01695i
\(89\) −10.3303 −1.09501 −0.547503 0.836804i \(-0.684421\pi\)
−0.547503 + 0.836804i \(0.684421\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 13.7986 1.43860
\(93\) 0 0
\(94\) 10.9361i 1.12797i
\(95\) 2.96275 + 1.27635i 0.303972 + 0.130951i
\(96\) 0 0
\(97\) −14.7321 −1.49582 −0.747908 0.663802i \(-0.768942\pi\)
−0.747908 + 0.663802i \(0.768942\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −15.8050 16.7209i −1.58050 1.67209i
\(101\) 14.9337 1.48595 0.742977 0.669317i \(-0.233413\pi\)
0.742977 + 0.669317i \(0.233413\pi\)
\(102\) 0 0
\(103\) 3.91510 0.385767 0.192883 0.981222i \(-0.438216\pi\)
0.192883 + 0.981222i \(0.438216\pi\)
\(104\) 23.6328 2.31739
\(105\) 0 0
\(106\) 10.3341 1.00374
\(107\) −14.8220 −1.43290 −0.716450 0.697638i \(-0.754234\pi\)
−0.716450 + 0.697638i \(0.754234\pi\)
\(108\) 0 0
\(109\) 11.0386 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(110\) 29.7442 + 12.8138i 2.83600 + 1.22175i
\(111\) 0 0
\(112\) 0 0
\(113\) −7.54314 −0.709599 −0.354799 0.934942i \(-0.615451\pi\)
−0.354799 + 0.934942i \(0.615451\pi\)
\(114\) 0 0
\(115\) 2.65285 6.15795i 0.247380 0.574232i
\(116\) 10.0963i 0.937416i
\(117\) 0 0
\(118\) −29.3411 −2.70107
\(119\) 0 0
\(120\) 0 0
\(121\) −20.7771 −1.88883
\(122\) 18.0076i 1.63033i
\(123\) 0 0
\(124\) 7.00854i 0.629385i
\(125\) −10.5007 + 3.83871i −0.939210 + 0.343344i
\(126\) 0 0
\(127\) 3.93715i 0.349366i −0.984625 0.174683i \(-0.944110\pi\)
0.984625 0.174683i \(-0.0558900\pi\)
\(128\) −8.23062 −0.727491
\(129\) 0 0
\(130\) 8.03630 18.6543i 0.704830 1.63609i
\(131\) −3.94799 −0.344937 −0.172469 0.985015i \(-0.555174\pi\)
−0.172469 + 0.985015i \(0.555174\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 6.39722i 0.552635i
\(135\) 0 0
\(136\) 35.7152i 3.06255i
\(137\) 5.66125 0.483673 0.241837 0.970317i \(-0.422250\pi\)
0.241837 + 0.970317i \(0.422250\pi\)
\(138\) 0 0
\(139\) 23.3015i 1.97640i 0.153158 + 0.988202i \(0.451056\pi\)
−0.153158 + 0.988202i \(0.548944\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 20.9108i 1.75480i
\(143\) 19.9293i 1.66657i
\(144\) 0 0
\(145\) −4.50571 1.94106i −0.374179 0.161196i
\(146\) −8.06275 −0.667278
\(147\) 0 0
\(148\) 5.03030i 0.413488i
\(149\) 4.67568i 0.383046i −0.981488 0.191523i \(-0.938657\pi\)
0.981488 0.191523i \(-0.0613427\pi\)
\(150\) 0 0
\(151\) 1.86085 0.151434 0.0757170 0.997129i \(-0.475875\pi\)
0.0757170 + 0.997129i \(0.475875\pi\)
\(152\) 9.64401i 0.782233i
\(153\) 0 0
\(154\) 0 0
\(155\) 3.12773 + 1.34743i 0.251225 + 0.108228i
\(156\) 0 0
\(157\) 6.56796 0.524180 0.262090 0.965043i \(-0.415588\pi\)
0.262090 + 0.965043i \(0.415588\pi\)
\(158\) 32.9009 2.61746
\(159\) 0 0
\(160\) 6.29365 14.6092i 0.497557 1.15496i
\(161\) 0 0
\(162\) 0 0
\(163\) 18.7752i 1.47059i 0.677747 + 0.735295i \(0.262956\pi\)
−0.677747 + 0.735295i \(0.737044\pi\)
\(164\) −30.0056 −2.34304
\(165\) 0 0
\(166\) 17.4008i 1.35057i
\(167\) 4.80581i 0.371884i −0.982561 0.185942i \(-0.940466\pi\)
0.982561 0.185942i \(-0.0595337\pi\)
\(168\) 0 0
\(169\) −0.501166 −0.0385512
\(170\) −28.1914 12.1449i −2.16218 0.931470i
\(171\) 0 0
\(172\) 2.23699i 0.170569i
\(173\) 3.36378i 0.255743i −0.991791 0.127872i \(-0.959185\pi\)
0.991791 0.127872i \(-0.0408145\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 44.9396i 3.38745i
\(177\) 0 0
\(178\) −26.5423 −1.98943
\(179\) 11.1288i 0.831805i 0.909409 + 0.415903i \(0.136534\pi\)
−0.909409 + 0.415903i \(0.863466\pi\)
\(180\) 0 0
\(181\) 2.19549i 0.163190i −0.996666 0.0815949i \(-0.973999\pi\)
0.996666 0.0815949i \(-0.0260014\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 20.0447 1.47771
\(185\) 2.24489 + 0.967100i 0.165048 + 0.0711026i
\(186\) 0 0
\(187\) 30.1182 2.20246
\(188\) 19.5862i 1.42847i
\(189\) 0 0
\(190\) 7.61241 + 3.27943i 0.552262 + 0.237915i
\(191\) 9.95413i 0.720256i −0.932903 0.360128i \(-0.882733\pi\)
0.932903 0.360128i \(-0.117267\pi\)
\(192\) 0 0
\(193\) 24.9494i 1.79590i 0.440102 + 0.897948i \(0.354942\pi\)
−0.440102 + 0.897948i \(0.645058\pi\)
\(194\) −37.8522 −2.71763
\(195\) 0 0
\(196\) 0 0
\(197\) −19.1023 −1.36098 −0.680492 0.732755i \(-0.738234\pi\)
−0.680492 + 0.732755i \(0.738234\pi\)
\(198\) 0 0
\(199\) 1.84327i 0.130666i 0.997864 + 0.0653329i \(0.0208109\pi\)
−0.997864 + 0.0653329i \(0.979189\pi\)
\(200\) −22.9594 24.2898i −1.62347 1.71755i
\(201\) 0 0
\(202\) 38.3701 2.69971
\(203\) 0 0
\(204\) 0 0
\(205\) −5.76873 + 13.3907i −0.402906 + 0.935249i
\(206\) 10.0594 0.700869
\(207\) 0 0
\(208\) 28.1843 1.95423
\(209\) −8.13269 −0.562550
\(210\) 0 0
\(211\) −9.77713 −0.673085 −0.336543 0.941668i \(-0.609258\pi\)
−0.336543 + 0.941668i \(0.609258\pi\)
\(212\) 18.5081 1.27114
\(213\) 0 0
\(214\) −38.0833 −2.60332
\(215\) −0.998310 0.430073i −0.0680842 0.0293307i
\(216\) 0 0
\(217\) 0 0
\(218\) 28.3624 1.92094
\(219\) 0 0
\(220\) 53.2711 + 22.9492i 3.59154 + 1.54724i
\(221\) 18.8889i 1.27060i
\(222\) 0 0
\(223\) 7.84782 0.525529 0.262765 0.964860i \(-0.415366\pi\)
0.262765 + 0.964860i \(0.415366\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −19.3811 −1.28921
\(227\) 0.549482i 0.0364704i 0.999834 + 0.0182352i \(0.00580476\pi\)
−0.999834 + 0.0182352i \(0.994195\pi\)
\(228\) 0 0
\(229\) 21.4582i 1.41800i 0.705210 + 0.708998i \(0.250853\pi\)
−0.705210 + 0.708998i \(0.749147\pi\)
\(230\) 6.81617 15.8221i 0.449445 1.04328i
\(231\) 0 0
\(232\) 14.6665i 0.962902i
\(233\) 18.7664 1.22943 0.614713 0.788751i \(-0.289272\pi\)
0.614713 + 0.788751i \(0.289272\pi\)
\(234\) 0 0
\(235\) 8.74083 + 3.76556i 0.570189 + 0.245638i
\(236\) −52.5491 −3.42066
\(237\) 0 0
\(238\) 0 0
\(239\) 7.35866i 0.475992i −0.971266 0.237996i \(-0.923509\pi\)
0.971266 0.237996i \(-0.0764905\pi\)
\(240\) 0 0
\(241\) 27.5358i 1.77374i 0.462022 + 0.886868i \(0.347124\pi\)
−0.462022 + 0.886868i \(0.652876\pi\)
\(242\) −53.3842 −3.43167
\(243\) 0 0
\(244\) 32.2511i 2.06466i
\(245\) 0 0
\(246\) 0 0
\(247\) 5.10049i 0.324536i
\(248\) 10.1810i 0.646497i
\(249\) 0 0
\(250\) −26.9802 + 9.86307i −1.70638 + 0.623795i
\(251\) −22.4204 −1.41516 −0.707582 0.706631i \(-0.750214\pi\)
−0.707582 + 0.706631i \(0.750214\pi\)
\(252\) 0 0
\(253\) 16.9035i 1.06271i
\(254\) 10.1160i 0.634735i
\(255\) 0 0
\(256\) −25.8158 −1.61349
\(257\) 17.1275i 1.06838i −0.845364 0.534191i \(-0.820616\pi\)
0.845364 0.534191i \(-0.179384\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 14.3928 33.4094i 0.892604 2.07196i
\(261\) 0 0
\(262\) −10.1439 −0.626689
\(263\) 1.61764 0.0997481 0.0498741 0.998756i \(-0.484118\pi\)
0.0498741 + 0.998756i \(0.484118\pi\)
\(264\) 0 0
\(265\) 3.55828 8.25969i 0.218583 0.507389i
\(266\) 0 0
\(267\) 0 0
\(268\) 11.4572i 0.699863i
\(269\) −11.6810 −0.712201 −0.356101 0.934448i \(-0.615894\pi\)
−0.356101 + 0.934448i \(0.615894\pi\)
\(270\) 0 0
\(271\) 5.83773i 0.354617i −0.984155 0.177308i \(-0.943261\pi\)
0.984155 0.177308i \(-0.0567390\pi\)
\(272\) 42.5936i 2.58261i
\(273\) 0 0
\(274\) 14.5459 0.878748
\(275\) 20.4833 19.3614i 1.23519 1.16754i
\(276\) 0 0
\(277\) 16.6000i 0.997395i −0.866776 0.498698i \(-0.833812\pi\)
0.866776 0.498698i \(-0.166188\pi\)
\(278\) 59.8701i 3.59077i
\(279\) 0 0
\(280\) 0 0
\(281\) 20.4255i 1.21848i −0.792984 0.609242i \(-0.791474\pi\)
0.792984 0.609242i \(-0.208526\pi\)
\(282\) 0 0
\(283\) −6.63286 −0.394283 −0.197141 0.980375i \(-0.563166\pi\)
−0.197141 + 0.980375i \(0.563166\pi\)
\(284\) 37.4507i 2.22229i
\(285\) 0 0
\(286\) 51.2058i 3.02786i
\(287\) 0 0
\(288\) 0 0
\(289\) −11.5459 −0.679168
\(290\) −11.5768 4.98731i −0.679816 0.292865i
\(291\) 0 0
\(292\) −14.4402 −0.845047
\(293\) 16.3716i 0.956437i 0.878241 + 0.478219i \(0.158717\pi\)
−0.878241 + 0.478219i \(0.841283\pi\)
\(294\) 0 0
\(295\) −10.1028 + 23.4513i −0.588210 + 1.36539i
\(296\) 7.30732i 0.424729i
\(297\) 0 0
\(298\) 12.0136i 0.695927i
\(299\) 10.6012 0.613081
\(300\) 0 0
\(301\) 0 0
\(302\) 4.78122 0.275129
\(303\) 0 0
\(304\) 11.5014i 0.659648i
\(305\) −14.3928 6.20043i −0.824130 0.355036i
\(306\) 0 0
\(307\) 25.5071 1.45576 0.727882 0.685702i \(-0.240505\pi\)
0.727882 + 0.685702i \(0.240505\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.03630 + 3.46205i 0.456431 + 0.196631i
\(311\) −31.0360 −1.75989 −0.879946 0.475073i \(-0.842422\pi\)
−0.879946 + 0.475073i \(0.842422\pi\)
\(312\) 0 0
\(313\) 15.0469 0.850501 0.425251 0.905076i \(-0.360186\pi\)
0.425251 + 0.905076i \(0.360186\pi\)
\(314\) 16.8755 0.952341
\(315\) 0 0
\(316\) 58.9247 3.31477
\(317\) −15.5812 −0.875129 −0.437564 0.899187i \(-0.644159\pi\)
−0.437564 + 0.899187i \(0.644159\pi\)
\(318\) 0 0
\(319\) 12.3681 0.692480
\(320\) 2.06498 4.79335i 0.115436 0.267956i
\(321\) 0 0
\(322\) 0 0
\(323\) 7.70812 0.428891
\(324\) 0 0
\(325\) −12.1427 12.8463i −0.673554 0.712583i
\(326\) 48.2406i 2.67180i
\(327\) 0 0
\(328\) −43.5880 −2.40675
\(329\) 0 0
\(330\) 0 0
\(331\) 13.6682 0.751273 0.375637 0.926767i \(-0.377424\pi\)
0.375637 + 0.926767i \(0.377424\pi\)
\(332\) 31.1644i 1.71037i
\(333\) 0 0
\(334\) 12.3479i 0.675647i
\(335\) 5.11307 + 2.20271i 0.279357 + 0.120347i
\(336\) 0 0
\(337\) 22.8443i 1.24441i −0.782855 0.622204i \(-0.786237\pi\)
0.782855 0.622204i \(-0.213763\pi\)
\(338\) −1.28768 −0.0700406
\(339\) 0 0
\(340\) −50.4900 21.7512i −2.73821 1.17962i
\(341\) −8.58556 −0.464934
\(342\) 0 0
\(343\) 0 0
\(344\) 3.24959i 0.175206i
\(345\) 0 0
\(346\) 8.64280i 0.464640i
\(347\) 19.6248 1.05352 0.526758 0.850015i \(-0.323407\pi\)
0.526758 + 0.850015i \(0.323407\pi\)
\(348\) 0 0
\(349\) 6.14050i 0.328693i −0.986403 0.164347i \(-0.947448\pi\)
0.986403 0.164347i \(-0.0525516\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 40.1020i 2.13744i
\(353\) 24.0041i 1.27761i −0.769369 0.638805i \(-0.779429\pi\)
0.769369 0.638805i \(-0.220571\pi\)
\(354\) 0 0
\(355\) −16.7133 7.20009i −0.887049 0.382141i
\(356\) −47.5366 −2.51943
\(357\) 0 0
\(358\) 28.5940i 1.51124i
\(359\) 16.9508i 0.894631i −0.894376 0.447315i \(-0.852380\pi\)
0.894376 0.447315i \(-0.147620\pi\)
\(360\) 0 0
\(361\) 16.9186 0.890453
\(362\) 5.64104i 0.296487i
\(363\) 0 0
\(364\) 0 0
\(365\) −2.77620 + 6.44427i −0.145313 + 0.337308i
\(366\) 0 0
\(367\) 35.0036 1.82717 0.913587 0.406642i \(-0.133300\pi\)
0.913587 + 0.406642i \(0.133300\pi\)
\(368\) 23.9051 1.24614
\(369\) 0 0
\(370\) 5.76796 + 2.48484i 0.299862 + 0.129181i
\(371\) 0 0
\(372\) 0 0
\(373\) 5.99153i 0.310229i 0.987896 + 0.155115i \(0.0495747\pi\)
−0.987896 + 0.155115i \(0.950425\pi\)
\(374\) 77.3849 4.00148
\(375\) 0 0
\(376\) 28.4522i 1.46731i
\(377\) 7.75675i 0.399493i
\(378\) 0 0
\(379\) −3.12842 −0.160696 −0.0803481 0.996767i \(-0.525603\pi\)
−0.0803481 + 0.996767i \(0.525603\pi\)
\(380\) 13.6336 + 5.87337i 0.699390 + 0.301298i
\(381\) 0 0
\(382\) 25.5759i 1.30858i
\(383\) 16.9659i 0.866917i −0.901173 0.433459i \(-0.857293\pi\)
0.901173 0.433459i \(-0.142707\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 64.1043i 3.26282i
\(387\) 0 0
\(388\) −67.7923 −3.44163
\(389\) 0.449473i 0.0227892i −0.999935 0.0113946i \(-0.996373\pi\)
0.999935 0.0113946i \(-0.00362709\pi\)
\(390\) 0 0
\(391\) 16.0210i 0.810218i
\(392\) 0 0
\(393\) 0 0
\(394\) −49.0810 −2.47267
\(395\) 11.3286 26.2965i 0.570002 1.32312i
\(396\) 0 0
\(397\) 22.7326 1.14092 0.570459 0.821326i \(-0.306765\pi\)
0.570459 + 0.821326i \(0.306765\pi\)
\(398\) 4.73604i 0.237396i
\(399\) 0 0
\(400\) −27.3811 28.9677i −1.36906 1.44839i
\(401\) 9.71458i 0.485123i −0.970136 0.242562i \(-0.922012\pi\)
0.970136 0.242562i \(-0.0779876\pi\)
\(402\) 0 0
\(403\) 5.38451i 0.268221i
\(404\) 68.7199 3.41894
\(405\) 0 0
\(406\) 0 0
\(407\) −6.16218 −0.305448
\(408\) 0 0
\(409\) 4.06026i 0.200767i 0.994949 + 0.100383i \(0.0320069\pi\)
−0.994949 + 0.100383i \(0.967993\pi\)
\(410\) −14.8220 + 34.4058i −0.732008 + 1.69918i
\(411\) 0 0
\(412\) 18.0161 0.887587
\(413\) 0 0
\(414\) 0 0
\(415\) 13.9079 + 5.99153i 0.682711 + 0.294112i
\(416\) 25.1503 1.23310
\(417\) 0 0
\(418\) −20.8959 −1.02205
\(419\) −20.9314 −1.02257 −0.511283 0.859412i \(-0.670830\pi\)
−0.511283 + 0.859412i \(0.670830\pi\)
\(420\) 0 0
\(421\) −19.7795 −0.963992 −0.481996 0.876173i \(-0.660088\pi\)
−0.481996 + 0.876173i \(0.660088\pi\)
\(422\) −25.1211 −1.22288
\(423\) 0 0
\(424\) 26.8860 1.30570
\(425\) −19.4139 + 18.3506i −0.941715 + 0.890136i
\(426\) 0 0
\(427\) 0 0
\(428\) −68.2062 −3.29687
\(429\) 0 0
\(430\) −2.56503 1.10502i −0.123697 0.0532887i
\(431\) 10.6596i 0.513454i 0.966484 + 0.256727i \(0.0826442\pi\)
−0.966484 + 0.256727i \(0.917356\pi\)
\(432\) 0 0
\(433\) −21.2351 −1.02050 −0.510248 0.860028i \(-0.670446\pi\)
−0.510248 + 0.860028i \(0.670446\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 50.7963 2.43270
\(437\) 4.32609i 0.206945i
\(438\) 0 0
\(439\) 21.4436i 1.02345i −0.859150 0.511725i \(-0.829007\pi\)
0.859150 0.511725i \(-0.170993\pi\)
\(440\) 77.3849 + 33.3375i 3.68918 + 1.58930i
\(441\) 0 0
\(442\) 48.5326i 2.30846i
\(443\) 0.170970 0.00812304 0.00406152 0.999992i \(-0.498707\pi\)
0.00406152 + 0.999992i \(0.498707\pi\)
\(444\) 0 0
\(445\) −9.13915 + 21.2143i −0.433237 + 1.00566i
\(446\) 20.1640 0.954792
\(447\) 0 0
\(448\) 0 0
\(449\) 9.38752i 0.443025i −0.975158 0.221512i \(-0.928901\pi\)
0.975158 0.221512i \(-0.0710993\pi\)
\(450\) 0 0
\(451\) 36.7573i 1.73083i
\(452\) −34.7111 −1.63267
\(453\) 0 0
\(454\) 1.41182i 0.0662602i
\(455\) 0 0
\(456\) 0 0
\(457\) 16.8925i 0.790196i −0.918639 0.395098i \(-0.870711\pi\)
0.918639 0.395098i \(-0.129289\pi\)
\(458\) 55.1341i 2.57625i
\(459\) 0 0
\(460\) 12.2076 28.3369i 0.569181 1.32122i
\(461\) 31.0360 1.44549 0.722746 0.691113i \(-0.242880\pi\)
0.722746 + 0.691113i \(0.242880\pi\)
\(462\) 0 0
\(463\) 11.9434i 0.555055i −0.960718 0.277528i \(-0.910485\pi\)
0.960718 0.277528i \(-0.0895150\pi\)
\(464\) 17.4911i 0.812004i
\(465\) 0 0
\(466\) 48.2178 2.23365
\(467\) 34.5606i 1.59928i −0.600483 0.799638i \(-0.705025\pi\)
0.600483 0.799638i \(-0.294975\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 22.4584 + 9.67512i 1.03593 + 0.446280i
\(471\) 0 0
\(472\) −76.3362 −3.51366
\(473\) 2.74034 0.126001
\(474\) 0 0
\(475\) 5.24227 4.95514i 0.240532 0.227357i
\(476\) 0 0
\(477\) 0 0
\(478\) 18.9071i 0.864793i
\(479\) 2.97230 0.135808 0.0679040 0.997692i \(-0.478369\pi\)
0.0679040 + 0.997692i \(0.478369\pi\)
\(480\) 0 0
\(481\) 3.86467i 0.176214i
\(482\) 70.7498i 3.22256i
\(483\) 0 0
\(484\) −95.6097 −4.34589
\(485\) −13.0334 + 30.2539i −0.591817 + 1.37376i
\(486\) 0 0
\(487\) 22.7124i 1.02920i −0.857431 0.514598i \(-0.827941\pi\)
0.857431 0.514598i \(-0.172059\pi\)
\(488\) 46.8499i 2.12079i
\(489\) 0 0
\(490\) 0 0
\(491\) 13.0036i 0.586846i 0.955983 + 0.293423i \(0.0947944\pi\)
−0.955983 + 0.293423i \(0.905206\pi\)
\(492\) 0 0
\(493\) −11.7224 −0.527950
\(494\) 13.1051i 0.589624i
\(495\) 0 0
\(496\) 12.1418i 0.545184i
\(497\) 0 0
\(498\) 0 0
\(499\) −21.6077 −0.967295 −0.483648 0.875263i \(-0.660688\pi\)
−0.483648 + 0.875263i \(0.660688\pi\)
\(500\) −48.3208 + 17.6645i −2.16097 + 0.789980i
\(501\) 0 0
\(502\) −57.6064 −2.57110
\(503\) 32.4125i 1.44520i −0.691265 0.722601i \(-0.742946\pi\)
0.691265 0.722601i \(-0.257054\pi\)
\(504\) 0 0
\(505\) 13.2118 30.6679i 0.587915 1.36470i
\(506\) 43.4313i 1.93076i
\(507\) 0 0
\(508\) 18.1175i 0.803835i
\(509\) 36.5604 1.62051 0.810255 0.586077i \(-0.199329\pi\)
0.810255 + 0.586077i \(0.199329\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −49.8691 −2.20392
\(513\) 0 0
\(514\) 44.0068i 1.94106i
\(515\) 3.46368 8.04010i 0.152628 0.354289i
\(516\) 0 0
\(517\) −23.9934 −1.05523
\(518\) 0 0
\(519\) 0 0
\(520\) 20.9079 48.5326i 0.916871 2.12830i
\(521\) 18.8722 0.826804 0.413402 0.910549i \(-0.364340\pi\)
0.413402 + 0.910549i \(0.364340\pi\)
\(522\) 0 0
\(523\) −28.2248 −1.23418 −0.617092 0.786891i \(-0.711690\pi\)
−0.617092 + 0.786891i \(0.711690\pi\)
\(524\) −18.1674 −0.793645
\(525\) 0 0
\(526\) 4.15633 0.181225
\(527\) 8.13735 0.354468
\(528\) 0 0
\(529\) −14.0084 −0.609061
\(530\) 9.14255 21.2222i 0.397127 0.921835i
\(531\) 0 0
\(532\) 0 0
\(533\) −23.0527 −0.998521
\(534\) 0 0
\(535\) −13.1130 + 30.4387i −0.566924 + 1.31598i
\(536\) 16.6435i 0.718890i
\(537\) 0 0
\(538\) −30.0128 −1.29394
\(539\) 0 0
\(540\) 0 0
\(541\) 28.4537 1.22332 0.611661 0.791120i \(-0.290502\pi\)
0.611661 + 0.791120i \(0.290502\pi\)
\(542\) 14.9993i 0.644275i
\(543\) 0 0
\(544\) 38.0084i 1.62960i
\(545\) 9.76584 22.6690i 0.418323 0.971035i
\(546\) 0 0
\(547\) 21.9338i 0.937820i 0.883246 + 0.468910i \(0.155353\pi\)
−0.883246 + 0.468910i \(0.844647\pi\)
\(548\) 26.0513 1.11285
\(549\) 0 0
\(550\) 52.6292 49.7466i 2.24412 2.12120i
\(551\) 3.16535 0.134848
\(552\) 0 0
\(553\) 0 0
\(554\) 42.6515i 1.81209i
\(555\) 0 0
\(556\) 107.226i 4.54739i
\(557\) 24.4336 1.03529 0.517643 0.855597i \(-0.326810\pi\)
0.517643 + 0.855597i \(0.326810\pi\)
\(558\) 0 0
\(559\) 1.71863i 0.0726903i
\(560\) 0 0
\(561\) 0 0
\(562\) 52.4808i 2.21377i
\(563\) 21.3140i 0.898279i 0.893462 + 0.449139i \(0.148269\pi\)
−0.893462 + 0.449139i \(0.851731\pi\)
\(564\) 0 0
\(565\) −6.67339 + 15.4907i −0.280752 + 0.651697i
\(566\) −17.0423 −0.716341
\(567\) 0 0
\(568\) 54.4032i 2.28271i
\(569\) 27.9099i 1.17004i 0.811018 + 0.585022i \(0.198914\pi\)
−0.811018 + 0.585022i \(0.801086\pi\)
\(570\) 0 0
\(571\) −46.3872 −1.94124 −0.970622 0.240609i \(-0.922653\pi\)
−0.970622 + 0.240609i \(0.922653\pi\)
\(572\) 91.7083i 3.83451i
\(573\) 0 0
\(574\) 0 0
\(575\) −10.2991 10.8958i −0.429501 0.454388i
\(576\) 0 0
\(577\) 32.4333 1.35022 0.675108 0.737719i \(-0.264097\pi\)
0.675108 + 0.737719i \(0.264097\pi\)
\(578\) −29.6656 −1.23393
\(579\) 0 0
\(580\) −20.7338 8.93214i −0.860925 0.370887i
\(581\) 0 0
\(582\) 0 0
\(583\) 22.6727i 0.939007i
\(584\) −20.9767 −0.868021
\(585\) 0 0
\(586\) 42.0647i 1.73768i
\(587\) 0.790380i 0.0326225i 0.999867 + 0.0163112i \(0.00519226\pi\)
−0.999867 + 0.0163112i \(0.994808\pi\)
\(588\) 0 0
\(589\) −2.19729 −0.0905379
\(590\) −25.9580 + 60.2552i −1.06867 + 2.48067i
\(591\) 0 0
\(592\) 8.71464i 0.358170i
\(593\) 20.2848i 0.832998i 0.909136 + 0.416499i \(0.136743\pi\)
−0.909136 + 0.416499i \(0.863257\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 21.5160i 0.881328i
\(597\) 0 0
\(598\) 27.2383 1.11386
\(599\) 10.1357i 0.414135i 0.978327 + 0.207068i \(0.0663920\pi\)
−0.978327 + 0.207068i \(0.933608\pi\)
\(600\) 0 0
\(601\) 35.6657i 1.45483i 0.686196 + 0.727417i \(0.259279\pi\)
−0.686196 + 0.727417i \(0.740721\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.56305 0.348425
\(605\) −18.3815 + 42.6681i −0.747312 + 1.73471i
\(606\) 0 0
\(607\) 26.2665 1.06613 0.533063 0.846076i \(-0.321041\pi\)
0.533063 + 0.846076i \(0.321041\pi\)
\(608\) 10.2633i 0.416230i
\(609\) 0 0
\(610\) −36.9805 15.9312i −1.49730 0.645036i
\(611\) 15.0477i 0.608764i
\(612\) 0 0
\(613\) 25.8819i 1.04536i 0.852528 + 0.522681i \(0.175068\pi\)
−0.852528 + 0.522681i \(0.824932\pi\)
\(614\) 65.5371 2.64486
\(615\) 0 0
\(616\) 0 0
\(617\) 16.1754 0.651198 0.325599 0.945508i \(-0.394434\pi\)
0.325599 + 0.945508i \(0.394434\pi\)
\(618\) 0 0
\(619\) 6.10962i 0.245566i 0.992434 + 0.122783i \(0.0391820\pi\)
−0.992434 + 0.122783i \(0.960818\pi\)
\(620\) 14.3928 + 6.20043i 0.578029 + 0.249015i
\(621\) 0 0
\(622\) −79.7432 −3.19741
\(623\) 0 0
\(624\) 0 0
\(625\) −1.40672 + 24.9604i −0.0562687 + 0.998416i
\(626\) 38.6611 1.54521
\(627\) 0 0
\(628\) 30.2236 1.20605
\(629\) 5.84048 0.232875
\(630\) 0 0
\(631\) 1.89949 0.0756174 0.0378087 0.999285i \(-0.487962\pi\)
0.0378087 + 0.999285i \(0.487962\pi\)
\(632\) 85.5976 3.40489
\(633\) 0 0
\(634\) −40.0340 −1.58995
\(635\) −8.08537 3.48318i −0.320858 0.138226i
\(636\) 0 0
\(637\) 0 0
\(638\) 31.7782 1.25811
\(639\) 0 0
\(640\) −7.28160 + 16.9025i −0.287831 + 0.668129i
\(641\) 11.3453i 0.448113i −0.974576 0.224057i \(-0.928070\pi\)
0.974576 0.224057i \(-0.0719300\pi\)
\(642\) 0 0
\(643\) 18.2255 0.718742 0.359371 0.933195i \(-0.382991\pi\)
0.359371 + 0.933195i \(0.382991\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 19.8050 0.779219
\(647\) 24.5411i 0.964812i −0.875948 0.482406i \(-0.839763\pi\)
0.875948 0.482406i \(-0.160237\pi\)
\(648\) 0 0
\(649\) 64.3734i 2.52688i
\(650\) −31.1990 33.0069i −1.22373 1.29464i
\(651\) 0 0
\(652\) 86.3976i 3.38359i
\(653\) −8.75313 −0.342536 −0.171268 0.985224i \(-0.554786\pi\)
−0.171268 + 0.985224i \(0.554786\pi\)
\(654\) 0 0
\(655\) −3.49277 + 8.10762i −0.136474 + 0.316791i
\(656\) −51.9827 −2.02958
\(657\) 0 0
\(658\) 0 0
\(659\) 2.51324i 0.0979020i 0.998801 + 0.0489510i \(0.0155878\pi\)
−0.998801 + 0.0489510i \(0.984412\pi\)
\(660\) 0 0
\(661\) 39.8826i 1.55125i −0.631192 0.775626i \(-0.717434\pi\)
0.631192 0.775626i \(-0.282566\pi\)
\(662\) 35.1188 1.36493
\(663\) 0 0
\(664\) 45.2714i 1.75687i
\(665\) 0 0
\(666\) 0 0
\(667\) 6.57906i 0.254742i
\(668\) 22.1148i 0.855646i
\(669\) 0 0
\(670\) 13.1374 + 5.65960i 0.507541 + 0.218649i
\(671\) 39.5080 1.52519
\(672\) 0 0
\(673\) 36.0634i 1.39014i −0.718941 0.695071i \(-0.755373\pi\)
0.718941 0.695071i \(-0.244627\pi\)
\(674\) 58.6955i 2.26087i
\(675\) 0 0
\(676\) −2.30620 −0.0887001
\(677\) 7.90377i 0.303767i −0.988398 0.151883i \(-0.951466\pi\)
0.988398 0.151883i \(-0.0485338\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −73.3450 31.5971i −2.81265 1.21169i
\(681\) 0 0
\(682\) −22.0595 −0.844702
\(683\) 25.2861 0.967545 0.483772 0.875194i \(-0.339266\pi\)
0.483772 + 0.875194i \(0.339266\pi\)
\(684\) 0 0
\(685\) 5.00849 11.6260i 0.191365 0.444206i
\(686\) 0 0
\(687\) 0 0
\(688\) 3.87543i 0.147749i
\(689\) 14.2194 0.541715
\(690\) 0 0
\(691\) 15.1704i 0.577111i 0.957463 + 0.288555i \(0.0931749\pi\)
−0.957463 + 0.288555i \(0.906825\pi\)
\(692\) 15.4790i 0.588424i
\(693\) 0 0
\(694\) 50.4235 1.91405
\(695\) 47.8521 + 20.6147i 1.81513 + 0.781961i
\(696\) 0 0
\(697\) 34.8384i 1.31960i
\(698\) 15.7772i 0.597177i
\(699\) 0 0
\(700\) 0 0
\(701\) 34.6815i 1.30990i 0.755671 + 0.654951i \(0.227311\pi\)
−0.755671 + 0.654951i \(0.772689\pi\)
\(702\) 0 0
\(703\) −1.57708 −0.0594807
\(704\) 13.1577i 0.495898i
\(705\) 0 0
\(706\) 61.6755i 2.32119i
\(707\) 0 0
\(708\) 0 0
\(709\) 34.4621 1.29425 0.647126 0.762383i \(-0.275971\pi\)
0.647126 + 0.762383i \(0.275971\pi\)
\(710\) −42.9426 18.4997i −1.61161 0.694282i
\(711\) 0 0
\(712\) −69.0545 −2.58793
\(713\) 4.56699i 0.171035i
\(714\) 0 0
\(715\) 40.9270 + 17.6314i 1.53058 + 0.659376i
\(716\) 51.2112i 1.91385i
\(717\) 0 0
\(718\) 43.5530i 1.62538i
\(719\) −26.9650 −1.00562 −0.502812 0.864396i \(-0.667701\pi\)
−0.502812 + 0.864396i \(0.667701\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 43.4702 1.61779
\(723\) 0 0
\(724\) 10.1030i 0.375473i
\(725\) −7.97237 + 7.53571i −0.296086 + 0.279869i
\(726\) 0 0
\(727\) −10.4196 −0.386442 −0.193221 0.981155i \(-0.561894\pi\)
−0.193221 + 0.981155i \(0.561894\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7.13308 + 16.5577i −0.264007 + 0.612829i
\(731\) −2.59728 −0.0960640
\(732\) 0 0
\(733\) 22.7082 0.838748 0.419374 0.907814i \(-0.362250\pi\)
0.419374 + 0.907814i \(0.362250\pi\)
\(734\) 89.9374 3.31965
\(735\) 0 0
\(736\) 21.3318 0.786300
\(737\) −14.0353 −0.516996
\(738\) 0 0
\(739\) 29.7795 1.09546 0.547728 0.836657i \(-0.315493\pi\)
0.547728 + 0.836657i \(0.315493\pi\)
\(740\) 10.3303 + 4.45029i 0.379748 + 0.163596i
\(741\) 0 0
\(742\) 0 0
\(743\) 34.8044 1.27685 0.638424 0.769684i \(-0.279586\pi\)
0.638424 + 0.769684i \(0.279586\pi\)
\(744\) 0 0
\(745\) −9.60201 4.13656i −0.351791 0.151552i
\(746\) 15.3945i 0.563632i
\(747\) 0 0
\(748\) 138.594 5.06751
\(749\) 0 0
\(750\) 0 0
\(751\) 7.49044 0.273330 0.136665 0.990617i \(-0.456362\pi\)
0.136665 + 0.990617i \(0.456362\pi\)
\(752\) 33.9318i 1.23737i
\(753\) 0 0
\(754\) 19.9300i 0.725807i
\(755\) 1.64629 3.82146i 0.0599146 0.139077i
\(756\) 0 0
\(757\) 2.66139i 0.0967300i −0.998830 0.0483650i \(-0.984599\pi\)
0.998830 0.0483650i \(-0.0154011\pi\)
\(758\) −8.03808 −0.291956
\(759\) 0 0
\(760\) 19.8050 + 8.53203i 0.718404 + 0.309489i
\(761\) 5.56959 0.201897 0.100949 0.994892i \(-0.467812\pi\)
0.100949 + 0.994892i \(0.467812\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 45.8057i 1.65719i
\(765\) 0 0
\(766\) 43.5917i 1.57503i
\(767\) −40.3724 −1.45776
\(768\) 0 0
\(769\) 11.5002i 0.414709i 0.978266 + 0.207355i \(0.0664854\pi\)
−0.978266 + 0.207355i \(0.933515\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 114.809i 4.13207i
\(773\) 25.3743i 0.912650i −0.889813 0.456325i \(-0.849165\pi\)
0.889813 0.456325i \(-0.150835\pi\)
\(774\) 0 0
\(775\) 5.53418 5.23107i 0.198794 0.187906i
\(776\) −98.4793 −3.53520
\(777\) 0 0
\(778\) 1.15486i 0.0414039i
\(779\) 9.40726i 0.337050i
\(780\) 0 0
\(781\) 45.8776 1.64163
\(782\) 41.1640i 1.47202i
\(783\) 0 0
\(784\) 0 0
\(785\) 5.81065 13.4880i 0.207391 0.481408i
\(786\) 0 0
\(787\) 23.9934 0.855273 0.427636 0.903951i \(-0.359346\pi\)
0.427636 + 0.903951i \(0.359346\pi\)
\(788\) −87.9028 −3.13141
\(789\) 0 0
\(790\) 29.1073 67.5656i 1.03559 2.40388i
\(791\) 0 0
\(792\) 0 0
\(793\) 24.7778i 0.879885i
\(794\) 58.4086 2.07284
\(795\) 0 0
\(796\) 8.48212i 0.300641i
\(797\) 28.1188i 0.996020i 0.867171 + 0.498010i \(0.165936\pi\)
−0.867171 + 0.498010i \(0.834064\pi\)
\(798\) 0 0
\(799\) 22.7408 0.804513
\(800\) −24.4336 25.8494i −0.863859 0.913915i
\(801\) 0 0
\(802\) 24.9604i 0.881382i
\(803\) 17.6894i 0.624245i
\(804\) 0 0
\(805\) 0 0
\(806\) 13.8348i 0.487310i
\(807\) 0 0
\(808\) 99.8268 3.51189
\(809\) 28.0566i 0.986417i 0.869911 + 0.493209i \(0.164176\pi\)
−0.869911 + 0.493209i \(0.835824\pi\)
\(810\) 0 0
\(811\) 43.4980i 1.52742i −0.645559 0.763710i \(-0.723376\pi\)
0.645559 0.763710i \(-0.276624\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −15.8329 −0.554944
\(815\) 38.5570 + 16.6104i 1.35059 + 0.581836i
\(816\) 0 0
\(817\) 0.701333 0.0245365
\(818\) 10.4323i 0.364757i
\(819\) 0 0
\(820\) −26.5459 + 61.6198i −0.927022 + 2.15186i
\(821\) 36.4741i 1.27295i −0.771295 0.636477i \(-0.780391\pi\)
0.771295 0.636477i \(-0.219609\pi\)
\(822\) 0 0
\(823\) 46.1045i 1.60710i −0.595235 0.803551i \(-0.702941\pi\)
0.595235 0.803551i \(-0.297059\pi\)
\(824\) 26.1712 0.911718
\(825\) 0 0
\(826\) 0 0
\(827\) 20.1533 0.700800 0.350400 0.936600i \(-0.386046\pi\)
0.350400 + 0.936600i \(0.386046\pi\)
\(828\) 0 0
\(829\) 7.08776i 0.246168i 0.992396 + 0.123084i \(0.0392785\pi\)
−0.992396 + 0.123084i \(0.960722\pi\)
\(830\) 35.7345 + 15.3945i 1.24036 + 0.534350i
\(831\) 0 0
\(832\) 8.25194 0.286085
\(833\) 0 0
\(834\) 0 0
\(835\) −9.86925 4.25168i −0.341539 0.147135i
\(836\) −37.4240 −1.29434
\(837\) 0 0
\(838\) −53.7806 −1.85782
\(839\) −33.0805 −1.14206 −0.571032 0.820928i \(-0.693457\pi\)
−0.571032 + 0.820928i \(0.693457\pi\)
\(840\) 0 0
\(841\) 24.1862 0.834006
\(842\) −50.8208 −1.75140
\(843\) 0 0
\(844\) −44.9912 −1.54866
\(845\) −0.443379 + 1.02920i −0.0152527 + 0.0354055i
\(846\) 0 0
\(847\) 0 0
\(848\) 32.0640 1.10108
\(849\) 0 0
\(850\) −49.8817 + 47.1496i −1.71093 + 1.61722i
\(851\) 3.27790i 0.112365i
\(852\) 0 0
\(853\) 38.1187 1.30516 0.652580 0.757719i \(-0.273686\pi\)
0.652580 + 0.757719i \(0.273686\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −99.0806 −3.38651
\(857\) 7.03613i 0.240350i −0.992753 0.120175i \(-0.961654\pi\)
0.992753 0.120175i \(-0.0383455\pi\)
\(858\) 0 0
\(859\) 25.6740i 0.875984i −0.898979 0.437992i \(-0.855690\pi\)
0.898979 0.437992i \(-0.144310\pi\)
\(860\) −4.59390 1.97906i −0.156651 0.0674853i
\(861\) 0 0
\(862\) 27.3885i 0.932855i
\(863\) 50.9858 1.73558 0.867788 0.496934i \(-0.165541\pi\)
0.867788 + 0.496934i \(0.165541\pi\)
\(864\) 0 0
\(865\) −6.90788 2.97592i −0.234875 0.101184i
\(866\) −54.5610 −1.85406
\(867\) 0 0
\(868\) 0 0
\(869\) 72.1835i 2.44866i
\(870\) 0 0
\(871\) 8.80235i 0.298256i
\(872\) 73.7898 2.49884
\(873\) 0 0
\(874\) 11.1153i 0.375982i
\(875\) 0 0
\(876\) 0 0
\(877\) 38.5532i 1.30185i −0.759142 0.650925i \(-0.774381\pi\)
0.759142 0.650925i \(-0.225619\pi\)
\(878\) 55.0967i 1.85942i
\(879\) 0 0
\(880\) 92.2885 + 39.7580i 3.11104 + 1.34024i
\(881\) 34.4764 1.16154 0.580770 0.814068i \(-0.302752\pi\)
0.580770 + 0.814068i \(0.302752\pi\)
\(882\) 0 0
\(883\) 25.5869i 0.861068i 0.902574 + 0.430534i \(0.141675\pi\)
−0.902574 + 0.430534i \(0.858325\pi\)
\(884\) 86.9206i 2.92346i
\(885\) 0 0
\(886\) 0.439286 0.0147581
\(887\) 23.7956i 0.798979i 0.916738 + 0.399489i \(0.130813\pi\)
−0.916738 + 0.399489i \(0.869187\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −23.4819 + 54.5075i −0.787114 + 1.82710i
\(891\) 0 0
\(892\) 36.1132 1.20916
\(893\) −6.14061 −0.205488
\(894\) 0 0
\(895\) 22.8542 + 9.84561i 0.763932 + 0.329102i
\(896\) 0 0
\(897\) 0 0
\(898\) 24.1201i 0.804897i
\(899\) 3.34161 0.111449
\(900\) 0 0
\(901\) 21.4891i 0.715904i
\(902\) 94.4432i 3.14461i
\(903\) 0 0
\(904\) −50.4235 −1.67706
\(905\) −4.50869 1.94235i −0.149874 0.0645658i
\(906\) 0 0
\(907\) 38.7755i 1.28752i 0.765227 + 0.643760i \(0.222626\pi\)
−0.765227 + 0.643760i \(0.777374\pi\)
\(908\) 2.52854i 0.0839125i
\(909\) 0 0
\(910\) 0 0
\(911\) 31.4438i 1.04178i 0.853624 + 0.520889i \(0.174400\pi\)
−0.853624 + 0.520889i \(0.825600\pi\)
\(912\) 0 0
\(913\) −38.1769 −1.26347
\(914\) 43.4030i 1.43564i
\(915\) 0 0
\(916\) 98.7437i 3.26258i
\(917\) 0 0
\(918\) 0 0
\(919\) 27.9744 0.922791 0.461396 0.887195i \(-0.347349\pi\)
0.461396 + 0.887195i \(0.347349\pi\)
\(920\) 17.7335 41.1640i 0.584655 1.35714i
\(921\) 0 0
\(922\) 79.7432 2.62620
\(923\) 28.7726i 0.947060i
\(924\) 0 0
\(925\) 3.97209 3.75454i 0.130602 0.123448i
\(926\) 30.6870i 1.00844i
\(927\) 0 0
\(928\) 15.6082i 0.512365i
\(929\) −22.4904 −0.737885 −0.368943 0.929452i \(-0.620280\pi\)
−0.368943 + 0.929452i \(0.620280\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 86.3568 2.82871
\(933\) 0 0
\(934\) 88.7992i 2.90560i
\(935\) 26.6455 61.8510i 0.871400 2.02274i
\(936\) 0 0
\(937\) −32.0994 −1.04864 −0.524321 0.851521i \(-0.675681\pi\)
−0.524321 + 0.851521i \(0.675681\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 40.2225 + 17.3279i 1.31191 + 0.565173i
\(941\) −10.2182 −0.333105 −0.166552 0.986033i \(-0.553264\pi\)
−0.166552 + 0.986033i \(0.553264\pi\)
\(942\) 0 0
\(943\) −19.5526 −0.636721
\(944\) −91.0378 −2.96303
\(945\) 0 0
\(946\) 7.04097 0.228922
\(947\) 10.2991 0.334675 0.167337 0.985900i \(-0.446483\pi\)
0.167337 + 0.985900i \(0.446483\pi\)
\(948\) 0 0
\(949\) −11.0941 −0.360128
\(950\) 13.4693 12.7316i 0.437003 0.413068i
\(951\) 0 0
\(952\) 0 0
\(953\) 0.930159 0.0301308 0.0150654 0.999887i \(-0.495204\pi\)
0.0150654 + 0.999887i \(0.495204\pi\)
\(954\) 0 0
\(955\) −20.4419 8.80639i −0.661484 0.284968i
\(956\) 33.8622i 1.09518i
\(957\) 0 0
\(958\) 7.63696 0.246739
\(959\) 0 0
\(960\) 0 0
\(961\) 28.6803 0.925173
\(962\) 9.92977i 0.320149i
\(963\) 0 0
\(964\) 126.711i 4.08108i
\(965\) 51.2363 + 22.0726i 1.64935 + 0.710543i
\(966\) 0 0
\(967\) 4.22117i 0.135744i −0.997694 0.0678719i \(-0.978379\pi\)
0.997694 0.0678719i \(-0.0216209\pi\)
\(968\) −138.889 −4.46405
\(969\) 0 0
\(970\) −33.4877 + 77.7337i −1.07523 + 2.49588i
\(971\) 4.57870 0.146938 0.0734688 0.997298i \(-0.476593\pi\)
0.0734688 + 0.997298i \(0.476593\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 58.3566i 1.86987i
\(975\) 0 0
\(976\) 55.8727i 1.78844i
\(977\) −45.5111 −1.45603 −0.728015 0.685562i \(-0.759557\pi\)
−0.728015 + 0.685562i \(0.759557\pi\)
\(978\) 0 0
\(979\) 58.2329i 1.86113i
\(980\) 0 0
\(981\) 0 0
\(982\) 33.4112i 1.06619i
\(983\) 10.5166i 0.335428i 0.985836 + 0.167714i \(0.0536385\pi\)
−0.985836 + 0.167714i \(0.946362\pi\)
\(984\) 0 0
\(985\) −16.8998 + 39.2287i −0.538471 + 1.24993i
\(986\) −30.1192 −0.959192
\(987\) 0 0
\(988\) 23.4708i 0.746706i
\(989\) 1.45769i 0.0463520i
\(990\) 0 0
\(991\) −27.9079 −0.886523 −0.443262 0.896392i \(-0.646179\pi\)
−0.443262 + 0.896392i \(0.646179\pi\)
\(992\) 10.8348i 0.344004i
\(993\) 0 0
\(994\) 0 0
\(995\) 3.78535 + 1.63073i 0.120004 + 0.0516977i
\(996\) 0 0
\(997\) −38.9134 −1.23240 −0.616201 0.787589i \(-0.711329\pi\)
−0.616201 + 0.787589i \(0.711329\pi\)
\(998\) −55.5184 −1.75740
\(999\) 0 0
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 2205.2.g.b.2204.22 24
3.2 odd 2 inner 2205.2.g.b.2204.4 24
5.4 even 2 inner 2205.2.g.b.2204.2 24
7.4 even 3 315.2.bb.b.89.1 24
7.5 odd 6 315.2.bb.b.269.2 yes 24
7.6 odd 2 inner 2205.2.g.b.2204.21 24
15.14 odd 2 inner 2205.2.g.b.2204.24 24
21.5 even 6 315.2.bb.b.269.11 yes 24
21.11 odd 6 315.2.bb.b.89.12 yes 24
21.20 even 2 inner 2205.2.g.b.2204.3 24
35.4 even 6 315.2.bb.b.89.11 yes 24
35.12 even 12 1575.2.bk.i.1151.12 24
35.18 odd 12 1575.2.bk.i.26.11 24
35.19 odd 6 315.2.bb.b.269.12 yes 24
35.32 odd 12 1575.2.bk.i.26.1 24
35.33 even 12 1575.2.bk.i.1151.2 24
35.34 odd 2 inner 2205.2.g.b.2204.1 24
105.32 even 12 1575.2.bk.i.26.12 24
105.47 odd 12 1575.2.bk.i.1151.1 24
105.53 even 12 1575.2.bk.i.26.2 24
105.68 odd 12 1575.2.bk.i.1151.11 24
105.74 odd 6 315.2.bb.b.89.2 yes 24
105.89 even 6 315.2.bb.b.269.1 yes 24
105.104 even 2 inner 2205.2.g.b.2204.23 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.bb.b.89.1 24 7.4 even 3
315.2.bb.b.89.2 yes 24 105.74 odd 6
315.2.bb.b.89.11 yes 24 35.4 even 6
315.2.bb.b.89.12 yes 24 21.11 odd 6
315.2.bb.b.269.1 yes 24 105.89 even 6
315.2.bb.b.269.2 yes 24 7.5 odd 6
315.2.bb.b.269.11 yes 24 21.5 even 6
315.2.bb.b.269.12 yes 24 35.19 odd 6
1575.2.bk.i.26.1 24 35.32 odd 12
1575.2.bk.i.26.2 24 105.53 even 12
1575.2.bk.i.26.11 24 35.18 odd 12
1575.2.bk.i.26.12 24 105.32 even 12
1575.2.bk.i.1151.1 24 105.47 odd 12
1575.2.bk.i.1151.2 24 35.33 even 12
1575.2.bk.i.1151.11 24 105.68 odd 12
1575.2.bk.i.1151.12 24 35.12 even 12
2205.2.g.b.2204.1 24 35.34 odd 2 inner
2205.2.g.b.2204.2 24 5.4 even 2 inner
2205.2.g.b.2204.3 24 21.20 even 2 inner
2205.2.g.b.2204.4 24 3.2 odd 2 inner
2205.2.g.b.2204.21 24 7.6 odd 2 inner
2205.2.g.b.2204.22 24 1.1 even 1 trivial
2205.2.g.b.2204.23 24 105.104 even 2 inner
2205.2.g.b.2204.24 24 15.14 odd 2 inner