Properties

Label 224.3.n.a.145.9
Level $224$
Weight $3$
Character 224.145
Analytic conductor $6.104$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(17,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.n (of order \(6\), degree \(2\), not 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.9
Character \(\chi\) \(=\) 224.145
Dual form 224.3.n.a.17.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.455431 + 0.788830i) q^{3} +(3.17251 - 5.49495i) q^{5} +(-3.79106 - 5.88455i) q^{7} +(4.08516 - 7.07571i) q^{9} +O(q^{10})\) \(q+(0.455431 + 0.788830i) q^{3} +(3.17251 - 5.49495i) q^{5} +(-3.79106 - 5.88455i) q^{7} +(4.08516 - 7.07571i) q^{9} +(-11.4442 + 6.60732i) q^{11} -19.4243 q^{13} +5.77945 q^{15} +(13.7930 - 7.96338i) q^{17} +(8.22725 - 14.2500i) q^{19} +(2.91534 - 5.67051i) q^{21} +(11.9607 - 20.7166i) q^{23} +(-7.62967 - 13.2150i) q^{25} +15.6398 q^{27} +16.6618i q^{29} +(11.1360 - 6.42939i) q^{31} +(-10.4241 - 6.01837i) q^{33} +(-44.3625 + 2.16288i) q^{35} +(41.1844 + 23.7778i) q^{37} +(-8.84646 - 15.3225i) q^{39} -6.49499i q^{41} -33.2928i q^{43} +(-25.9205 - 44.8956i) q^{45} +(18.9713 + 10.9531i) q^{47} +(-20.2558 + 44.6173i) q^{49} +(12.5635 + 7.25355i) q^{51} +(-32.2028 + 18.5923i) q^{53} +83.8473i q^{55} +14.9878 q^{57} +(27.3428 + 47.3591i) q^{59} +(-5.12340 + 8.87399i) q^{61} +(-57.1245 + 2.78508i) q^{63} +(-61.6240 + 106.736i) q^{65} +(14.8386 - 8.56706i) q^{67} +21.7892 q^{69} -32.0568 q^{71} +(92.8082 - 53.5828i) q^{73} +(6.94958 - 12.0370i) q^{75} +(82.2668 + 42.2953i) q^{77} +(-29.1542 + 50.4965i) q^{79} +(-29.6436 - 51.3443i) q^{81} +36.3441 q^{83} -101.056i q^{85} +(-13.1433 + 7.58829i) q^{87} +(0.929882 + 0.536867i) q^{89} +(73.6388 + 114.303i) q^{91} +(10.1434 + 5.85629i) q^{93} +(-52.2021 - 90.4167i) q^{95} +169.517i q^{97} +107.968i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 4 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 4 q^{7} - 32 q^{9} - 28 q^{15} - 6 q^{17} - 30 q^{23} - 32 q^{25} + 6 q^{31} - 6 q^{33} + 20 q^{39} + 294 q^{47} - 20 q^{49} + 124 q^{57} - 432 q^{63} - 52 q^{65} + 136 q^{71} + 234 q^{73} + 162 q^{79} - 18 q^{81} - 48 q^{87} - 150 q^{89} - 290 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) 0.455431 + 0.788830i 0.151810 + 0.262943i 0.931893 0.362733i \(-0.118156\pi\)
−0.780083 + 0.625677i \(0.784823\pi\)
\(4\) 0 0
\(5\) 3.17251 5.49495i 0.634503 1.09899i −0.352118 0.935956i \(-0.614538\pi\)
0.986620 0.163035i \(-0.0521283\pi\)
\(6\) 0 0
\(7\) −3.79106 5.88455i −0.541579 0.840650i
\(8\) 0 0
\(9\) 4.08516 7.07571i 0.453907 0.786190i
\(10\) 0 0
\(11\) −11.4442 + 6.60732i −1.04038 + 0.600666i −0.919942 0.392054i \(-0.871764\pi\)
−0.120442 + 0.992720i \(0.538431\pi\)
\(12\) 0 0
\(13\) −19.4243 −1.49418 −0.747090 0.664723i \(-0.768550\pi\)
−0.747090 + 0.664723i \(0.768550\pi\)
\(14\) 0 0
\(15\) 5.77945 0.385296
\(16\) 0 0
\(17\) 13.7930 7.96338i 0.811352 0.468434i −0.0360732 0.999349i \(-0.511485\pi\)
0.847425 + 0.530915i \(0.178152\pi\)
\(18\) 0 0
\(19\) 8.22725 14.2500i 0.433013 0.750001i −0.564118 0.825694i \(-0.690784\pi\)
0.997131 + 0.0756934i \(0.0241170\pi\)
\(20\) 0 0
\(21\) 2.91534 5.67051i 0.138826 0.270024i
\(22\) 0 0
\(23\) 11.9607 20.7166i 0.520032 0.900721i −0.479697 0.877434i \(-0.659254\pi\)
0.999729 0.0232870i \(-0.00741316\pi\)
\(24\) 0 0
\(25\) −7.62967 13.2150i −0.305187 0.528599i
\(26\) 0 0
\(27\) 15.6398 0.579252
\(28\) 0 0
\(29\) 16.6618i 0.574544i 0.957849 + 0.287272i \(0.0927483\pi\)
−0.957849 + 0.287272i \(0.907252\pi\)
\(30\) 0 0
\(31\) 11.1360 6.42939i 0.359227 0.207400i −0.309515 0.950895i \(-0.600167\pi\)
0.668741 + 0.743495i \(0.266833\pi\)
\(32\) 0 0
\(33\) −10.4241 6.01837i −0.315882 0.182375i
\(34\) 0 0
\(35\) −44.3625 + 2.16288i −1.26750 + 0.0617964i
\(36\) 0 0
\(37\) 41.1844 + 23.7778i 1.11309 + 0.642644i 0.939628 0.342196i \(-0.111171\pi\)
0.173463 + 0.984840i \(0.444504\pi\)
\(38\) 0 0
\(39\) −8.84646 15.3225i −0.226832 0.392885i
\(40\) 0 0
\(41\) 6.49499i 0.158415i −0.996858 0.0792073i \(-0.974761\pi\)
0.996858 0.0792073i \(-0.0252389\pi\)
\(42\) 0 0
\(43\) 33.2928i 0.774252i −0.922027 0.387126i \(-0.873468\pi\)
0.922027 0.387126i \(-0.126532\pi\)
\(44\) 0 0
\(45\) −25.9205 44.8956i −0.576010 0.997679i
\(46\) 0 0
\(47\) 18.9713 + 10.9531i 0.403645 + 0.233045i 0.688056 0.725658i \(-0.258465\pi\)
−0.284411 + 0.958703i \(0.591798\pi\)
\(48\) 0 0
\(49\) −20.2558 + 44.6173i −0.413383 + 0.910557i
\(50\) 0 0
\(51\) 12.5635 + 7.25355i 0.246343 + 0.142226i
\(52\) 0 0
\(53\) −32.2028 + 18.5923i −0.607601 + 0.350798i −0.772026 0.635591i \(-0.780756\pi\)
0.164425 + 0.986390i \(0.447423\pi\)
\(54\) 0 0
\(55\) 83.8473i 1.52450i
\(56\) 0 0
\(57\) 14.9878 0.262944
\(58\) 0 0
\(59\) 27.3428 + 47.3591i 0.463437 + 0.802696i 0.999129 0.0417169i \(-0.0132827\pi\)
−0.535693 + 0.844413i \(0.679949\pi\)
\(60\) 0 0
\(61\) −5.12340 + 8.87399i −0.0839902 + 0.145475i −0.904960 0.425496i \(-0.860100\pi\)
0.820970 + 0.570971i \(0.193433\pi\)
\(62\) 0 0
\(63\) −57.1245 + 2.78508i −0.906737 + 0.0442076i
\(64\) 0 0
\(65\) −61.6240 + 106.736i −0.948061 + 1.64209i
\(66\) 0 0
\(67\) 14.8386 8.56706i 0.221471 0.127867i −0.385160 0.922850i \(-0.625854\pi\)
0.606631 + 0.794983i \(0.292520\pi\)
\(68\) 0 0
\(69\) 21.7892 0.315785
\(70\) 0 0
\(71\) −32.0568 −0.451505 −0.225752 0.974185i \(-0.572484\pi\)
−0.225752 + 0.974185i \(0.572484\pi\)
\(72\) 0 0
\(73\) 92.8082 53.5828i 1.27135 0.734011i 0.296104 0.955156i \(-0.404313\pi\)
0.975241 + 0.221144i \(0.0709792\pi\)
\(74\) 0 0
\(75\) 6.94958 12.0370i 0.0926611 0.160494i
\(76\) 0 0
\(77\) 82.2668 + 42.2953i 1.06840 + 0.549290i
\(78\) 0 0
\(79\) −29.1542 + 50.4965i −0.369040 + 0.639196i −0.989416 0.145109i \(-0.953647\pi\)
0.620376 + 0.784305i \(0.286980\pi\)
\(80\) 0 0
\(81\) −29.6436 51.3443i −0.365971 0.633880i
\(82\) 0 0
\(83\) 36.3441 0.437880 0.218940 0.975738i \(-0.429740\pi\)
0.218940 + 0.975738i \(0.429740\pi\)
\(84\) 0 0
\(85\) 101.056i 1.18889i
\(86\) 0 0
\(87\) −13.1433 + 7.58829i −0.151072 + 0.0872217i
\(88\) 0 0
\(89\) 0.929882 + 0.536867i 0.0104481 + 0.00603222i 0.505215 0.862994i \(-0.331413\pi\)
−0.494767 + 0.869026i \(0.664747\pi\)
\(90\) 0 0
\(91\) 73.6388 + 114.303i 0.809217 + 1.25608i
\(92\) 0 0
\(93\) 10.1434 + 5.85629i 0.109069 + 0.0629709i
\(94\) 0 0
\(95\) −52.2021 90.4167i −0.549496 0.951755i
\(96\) 0 0
\(97\) 169.517i 1.74760i 0.486286 + 0.873799i \(0.338351\pi\)
−0.486286 + 0.873799i \(0.661649\pi\)
\(98\) 0 0
\(99\) 107.968i 1.09059i
\(100\) 0 0
\(101\) −14.0630 24.3579i −0.139238 0.241167i 0.787971 0.615713i \(-0.211132\pi\)
−0.927208 + 0.374546i \(0.877799\pi\)
\(102\) 0 0
\(103\) 144.029 + 83.1551i 1.39834 + 0.807331i 0.994219 0.107374i \(-0.0342444\pi\)
0.404120 + 0.914706i \(0.367578\pi\)
\(104\) 0 0
\(105\) −21.9102 34.0094i −0.208669 0.323899i
\(106\) 0 0
\(107\) −171.112 98.7918i −1.59918 0.923288i −0.991645 0.128996i \(-0.958825\pi\)
−0.607536 0.794292i \(-0.707842\pi\)
\(108\) 0 0
\(109\) 9.97643 5.75990i 0.0915269 0.0528431i −0.453538 0.891237i \(-0.649838\pi\)
0.545065 + 0.838394i \(0.316505\pi\)
\(110\) 0 0
\(111\) 43.3167i 0.390240i
\(112\) 0 0
\(113\) −14.7908 −0.130892 −0.0654460 0.997856i \(-0.520847\pi\)
−0.0654460 + 0.997856i \(0.520847\pi\)
\(114\) 0 0
\(115\) −75.8911 131.447i −0.659923 1.14302i
\(116\) 0 0
\(117\) −79.3516 + 137.441i −0.678219 + 1.17471i
\(118\) 0 0
\(119\) −99.1509 50.9758i −0.833201 0.428368i
\(120\) 0 0
\(121\) 26.8135 46.4423i 0.221599 0.383821i
\(122\) 0 0
\(123\) 5.12345 2.95802i 0.0416541 0.0240490i
\(124\) 0 0
\(125\) 61.8047 0.494438
\(126\) 0 0
\(127\) 70.2656 0.553272 0.276636 0.960975i \(-0.410780\pi\)
0.276636 + 0.960975i \(0.410780\pi\)
\(128\) 0 0
\(129\) 26.2624 15.1626i 0.203584 0.117540i
\(130\) 0 0
\(131\) 71.0646 123.088i 0.542478 0.939600i −0.456283 0.889835i \(-0.650819\pi\)
0.998761 0.0497649i \(-0.0158472\pi\)
\(132\) 0 0
\(133\) −115.045 + 5.60897i −0.864999 + 0.0421727i
\(134\) 0 0
\(135\) 49.6175 85.9400i 0.367537 0.636593i
\(136\) 0 0
\(137\) −126.537 219.168i −0.923626 1.59977i −0.793756 0.608236i \(-0.791877\pi\)
−0.129870 0.991531i \(-0.541456\pi\)
\(138\) 0 0
\(139\) 49.1909 0.353892 0.176946 0.984221i \(-0.443378\pi\)
0.176946 + 0.984221i \(0.443378\pi\)
\(140\) 0 0
\(141\) 19.9535i 0.141514i
\(142\) 0 0
\(143\) 222.296 128.343i 1.55452 0.897503i
\(144\) 0 0
\(145\) 91.5556 + 52.8597i 0.631418 + 0.364549i
\(146\) 0 0
\(147\) −44.4206 + 4.34174i −0.302181 + 0.0295356i
\(148\) 0 0
\(149\) −36.1077 20.8468i −0.242334 0.139911i 0.373915 0.927463i \(-0.378015\pi\)
−0.616249 + 0.787551i \(0.711348\pi\)
\(150\) 0 0
\(151\) −48.8145 84.5492i −0.323275 0.559928i 0.657887 0.753117i \(-0.271450\pi\)
−0.981162 + 0.193188i \(0.938117\pi\)
\(152\) 0 0
\(153\) 130.127i 0.850503i
\(154\) 0 0
\(155\) 81.5892i 0.526382i
\(156\) 0 0
\(157\) −14.0827 24.3919i −0.0896986 0.155363i 0.817685 0.575666i \(-0.195257\pi\)
−0.907384 + 0.420303i \(0.861924\pi\)
\(158\) 0 0
\(159\) −29.3324 16.9350i −0.184480 0.106510i
\(160\) 0 0
\(161\) −167.251 + 8.15428i −1.03883 + 0.0506477i
\(162\) 0 0
\(163\) 209.952 + 121.216i 1.28805 + 0.743655i 0.978306 0.207165i \(-0.0664237\pi\)
0.309743 + 0.950820i \(0.399757\pi\)
\(164\) 0 0
\(165\) −66.1413 + 38.1867i −0.400856 + 0.231434i
\(166\) 0 0
\(167\) 60.1108i 0.359945i −0.983672 0.179972i \(-0.942399\pi\)
0.983672 0.179972i \(-0.0576008\pi\)
\(168\) 0 0
\(169\) 208.305 1.23257
\(170\) 0 0
\(171\) −67.2193 116.427i −0.393096 0.680861i
\(172\) 0 0
\(173\) 69.6820 120.693i 0.402786 0.697646i −0.591275 0.806470i \(-0.701375\pi\)
0.994061 + 0.108824i \(0.0347085\pi\)
\(174\) 0 0
\(175\) −48.8397 + 94.9959i −0.279084 + 0.542834i
\(176\) 0 0
\(177\) −24.9055 + 43.1376i −0.140709 + 0.243715i
\(178\) 0 0
\(179\) −252.643 + 145.863i −1.41141 + 0.814879i −0.995522 0.0945354i \(-0.969863\pi\)
−0.415891 + 0.909415i \(0.636530\pi\)
\(180\) 0 0
\(181\) 166.844 0.921791 0.460895 0.887455i \(-0.347528\pi\)
0.460895 + 0.887455i \(0.347528\pi\)
\(182\) 0 0
\(183\) −9.33343 −0.0510024
\(184\) 0 0
\(185\) 261.316 150.871i 1.41252 0.815518i
\(186\) 0 0
\(187\) −105.233 + 182.269i −0.562745 + 0.974703i
\(188\) 0 0
\(189\) −59.2914 92.0332i −0.313711 0.486948i
\(190\) 0 0
\(191\) −65.6781 + 113.758i −0.343864 + 0.595590i −0.985147 0.171715i \(-0.945069\pi\)
0.641283 + 0.767305i \(0.278403\pi\)
\(192\) 0 0
\(193\) 40.7196 + 70.5284i 0.210982 + 0.365432i 0.952022 0.306029i \(-0.0990004\pi\)
−0.741040 + 0.671461i \(0.765667\pi\)
\(194\) 0 0
\(195\) −112.262 −0.575702
\(196\) 0 0
\(197\) 2.09549i 0.0106370i 0.999986 + 0.00531851i \(0.00169294\pi\)
−0.999986 + 0.00531851i \(0.998307\pi\)
\(198\) 0 0
\(199\) 109.937 63.4721i 0.552447 0.318955i −0.197662 0.980270i \(-0.563335\pi\)
0.750108 + 0.661315i \(0.230001\pi\)
\(200\) 0 0
\(201\) 13.5159 + 7.80341i 0.0672433 + 0.0388230i
\(202\) 0 0
\(203\) 98.0469 63.1657i 0.482990 0.311161i
\(204\) 0 0
\(205\) −35.6897 20.6055i −0.174096 0.100514i
\(206\) 0 0
\(207\) −97.7231 169.261i −0.472092 0.817687i
\(208\) 0 0
\(209\) 217.440i 1.04038i
\(210\) 0 0
\(211\) 7.16822i 0.0339726i 0.999856 + 0.0169863i \(0.00540717\pi\)
−0.999856 + 0.0169863i \(0.994593\pi\)
\(212\) 0 0
\(213\) −14.5997 25.2874i −0.0685432 0.118720i
\(214\) 0 0
\(215\) −182.943 105.622i −0.850896 0.491265i
\(216\) 0 0
\(217\) −80.0513 41.1563i −0.368900 0.189660i
\(218\) 0 0
\(219\) 84.5355 + 48.8066i 0.386007 + 0.222861i
\(220\) 0 0
\(221\) −267.920 + 154.683i −1.21231 + 0.699925i
\(222\) 0 0
\(223\) 279.720i 1.25435i 0.778878 + 0.627175i \(0.215789\pi\)
−0.778878 + 0.627175i \(0.784211\pi\)
\(224\) 0 0
\(225\) −124.674 −0.554106
\(226\) 0 0
\(227\) 152.392 + 263.950i 0.671330 + 1.16278i 0.977527 + 0.210809i \(0.0676098\pi\)
−0.306198 + 0.951968i \(0.599057\pi\)
\(228\) 0 0
\(229\) 207.344 359.130i 0.905433 1.56826i 0.0850971 0.996373i \(-0.472880\pi\)
0.820335 0.571883i \(-0.193787\pi\)
\(230\) 0 0
\(231\) 4.10305 + 84.1572i 0.0177621 + 0.364317i
\(232\) 0 0
\(233\) 82.4628 142.830i 0.353918 0.613004i −0.633014 0.774140i \(-0.718183\pi\)
0.986932 + 0.161136i \(0.0515159\pi\)
\(234\) 0 0
\(235\) 120.373 69.4976i 0.512227 0.295735i
\(236\) 0 0
\(237\) −53.1109 −0.224097
\(238\) 0 0
\(239\) 19.1182 0.0799926 0.0399963 0.999200i \(-0.487265\pi\)
0.0399963 + 0.999200i \(0.487265\pi\)
\(240\) 0 0
\(241\) −303.376 + 175.154i −1.25882 + 0.726780i −0.972845 0.231457i \(-0.925651\pi\)
−0.285975 + 0.958237i \(0.592317\pi\)
\(242\) 0 0
\(243\) 97.3804 168.668i 0.400743 0.694106i
\(244\) 0 0
\(245\) 180.908 + 252.854i 0.738401 + 1.03206i
\(246\) 0 0
\(247\) −159.809 + 276.797i −0.647000 + 1.12064i
\(248\) 0 0
\(249\) 16.5522 + 28.6693i 0.0664748 + 0.115138i
\(250\) 0 0
\(251\) 88.3204 0.351874 0.175937 0.984401i \(-0.443704\pi\)
0.175937 + 0.984401i \(0.443704\pi\)
\(252\) 0 0
\(253\) 316.114i 1.24946i
\(254\) 0 0
\(255\) 79.7158 46.0240i 0.312611 0.180486i
\(256\) 0 0
\(257\) 74.5499 + 43.0414i 0.290077 + 0.167476i 0.637977 0.770056i \(-0.279772\pi\)
−0.347899 + 0.937532i \(0.613105\pi\)
\(258\) 0 0
\(259\) −16.2106 332.495i −0.0625893 1.28376i
\(260\) 0 0
\(261\) 117.894 + 68.0661i 0.451701 + 0.260789i
\(262\) 0 0
\(263\) 159.605 + 276.444i 0.606863 + 1.05112i 0.991754 + 0.128156i \(0.0409057\pi\)
−0.384891 + 0.922962i \(0.625761\pi\)
\(264\) 0 0
\(265\) 235.937i 0.890330i
\(266\) 0 0
\(267\) 0.978025i 0.00366302i
\(268\) 0 0
\(269\) 28.7340 + 49.7687i 0.106818 + 0.185014i 0.914479 0.404632i \(-0.132601\pi\)
−0.807662 + 0.589646i \(0.799267\pi\)
\(270\) 0 0
\(271\) −26.7398 15.4382i −0.0986709 0.0569677i 0.449853 0.893103i \(-0.351477\pi\)
−0.548523 + 0.836135i \(0.684810\pi\)
\(272\) 0 0
\(273\) −56.6286 + 110.146i −0.207431 + 0.403465i
\(274\) 0 0
\(275\) 174.631 + 100.823i 0.635023 + 0.366631i
\(276\) 0 0
\(277\) 308.465 178.092i 1.11359 0.642933i 0.173834 0.984775i \(-0.444384\pi\)
0.939757 + 0.341842i \(0.111051\pi\)
\(278\) 0 0
\(279\) 105.060i 0.376561i
\(280\) 0 0
\(281\) −294.160 −1.04683 −0.523416 0.852077i \(-0.675343\pi\)
−0.523416 + 0.852077i \(0.675343\pi\)
\(282\) 0 0
\(283\) −207.501 359.402i −0.733219 1.26997i −0.955501 0.294989i \(-0.904684\pi\)
0.222282 0.974982i \(-0.428649\pi\)
\(284\) 0 0
\(285\) 47.5490 82.3572i 0.166838 0.288973i
\(286\) 0 0
\(287\) −38.2201 + 24.6229i −0.133171 + 0.0857940i
\(288\) 0 0
\(289\) −17.6691 + 30.6037i −0.0611386 + 0.105895i
\(290\) 0 0
\(291\) −133.720 + 77.2034i −0.459520 + 0.265304i
\(292\) 0 0
\(293\) −370.564 −1.26472 −0.632362 0.774673i \(-0.717915\pi\)
−0.632362 + 0.774673i \(0.717915\pi\)
\(294\) 0 0
\(295\) 346.981 1.17621
\(296\) 0 0
\(297\) −178.986 + 103.337i −0.602645 + 0.347937i
\(298\) 0 0
\(299\) −232.329 + 402.406i −0.777021 + 1.34584i
\(300\) 0 0
\(301\) −195.913 + 126.215i −0.650875 + 0.419319i
\(302\) 0 0
\(303\) 12.8095 22.1867i 0.0422755 0.0732233i
\(304\) 0 0
\(305\) 32.5081 + 56.3057i 0.106584 + 0.184609i
\(306\) 0 0
\(307\) 160.327 0.522239 0.261120 0.965306i \(-0.415908\pi\)
0.261120 + 0.965306i \(0.415908\pi\)
\(308\) 0 0
\(309\) 151.486i 0.490245i
\(310\) 0 0
\(311\) −409.490 + 236.419i −1.31669 + 0.760191i −0.983195 0.182561i \(-0.941561\pi\)
−0.333495 + 0.942752i \(0.608228\pi\)
\(312\) 0 0
\(313\) 200.063 + 115.506i 0.639179 + 0.369030i 0.784298 0.620384i \(-0.213023\pi\)
−0.145119 + 0.989414i \(0.546357\pi\)
\(314\) 0 0
\(315\) −165.924 + 322.732i −0.526743 + 1.02455i
\(316\) 0 0
\(317\) −195.132 112.659i −0.615557 0.355392i 0.159580 0.987185i \(-0.448986\pi\)
−0.775137 + 0.631793i \(0.782319\pi\)
\(318\) 0 0
\(319\) −110.090 190.681i −0.345109 0.597746i
\(320\) 0 0
\(321\) 179.972i 0.560659i
\(322\) 0 0
\(323\) 262.067i 0.811353i
\(324\) 0 0
\(325\) 148.201 + 256.692i 0.456004 + 0.789822i
\(326\) 0 0
\(327\) 9.08716 + 5.24648i 0.0277895 + 0.0160443i
\(328\) 0 0
\(329\) −7.46732 153.161i −0.0226970 0.465536i
\(330\) 0 0
\(331\) −17.9257 10.3494i −0.0541561 0.0312671i 0.472677 0.881236i \(-0.343288\pi\)
−0.526834 + 0.849968i \(0.676621\pi\)
\(332\) 0 0
\(333\) 336.490 194.273i 1.01048 0.583401i
\(334\) 0 0
\(335\) 108.716i 0.324527i
\(336\) 0 0
\(337\) 34.9645 0.103752 0.0518762 0.998654i \(-0.483480\pi\)
0.0518762 + 0.998654i \(0.483480\pi\)
\(338\) 0 0
\(339\) −6.73619 11.6674i −0.0198708 0.0344172i
\(340\) 0 0
\(341\) −84.9621 + 147.159i −0.249156 + 0.431550i
\(342\) 0 0
\(343\) 339.343 49.9505i 0.989339 0.145628i
\(344\) 0 0
\(345\) 69.1264 119.730i 0.200366 0.347045i
\(346\) 0 0
\(347\) 379.958 219.369i 1.09498 0.632188i 0.160083 0.987104i \(-0.448824\pi\)
0.934898 + 0.354916i \(0.115491\pi\)
\(348\) 0 0
\(349\) 435.121 1.24677 0.623383 0.781917i \(-0.285758\pi\)
0.623383 + 0.781917i \(0.285758\pi\)
\(350\) 0 0
\(351\) −303.793 −0.865507
\(352\) 0 0
\(353\) 243.447 140.554i 0.689653 0.398171i −0.113829 0.993500i \(-0.536312\pi\)
0.803482 + 0.595329i \(0.202978\pi\)
\(354\) 0 0
\(355\) −101.701 + 176.151i −0.286481 + 0.496200i
\(356\) 0 0
\(357\) −4.94514 101.429i −0.0138519 0.284115i
\(358\) 0 0
\(359\) 131.965 228.570i 0.367590 0.636685i −0.621598 0.783336i \(-0.713516\pi\)
0.989188 + 0.146652i \(0.0468497\pi\)
\(360\) 0 0
\(361\) 45.1247 + 78.1583i 0.124999 + 0.216505i
\(362\) 0 0
\(363\) 48.8468 0.134564
\(364\) 0 0
\(365\) 679.969i 1.86293i
\(366\) 0 0
\(367\) 134.181 77.4694i 0.365615 0.211088i −0.305926 0.952055i \(-0.598966\pi\)
0.671541 + 0.740967i \(0.265633\pi\)
\(368\) 0 0
\(369\) −45.9567 26.5331i −0.124544 0.0719055i
\(370\) 0 0
\(371\) 231.490 + 119.015i 0.623962 + 0.320794i
\(372\) 0 0
\(373\) −506.505 292.431i −1.35792 0.783997i −0.368579 0.929597i \(-0.620155\pi\)
−0.989344 + 0.145600i \(0.953489\pi\)
\(374\) 0 0
\(375\) 28.1478 + 48.7534i 0.0750608 + 0.130009i
\(376\) 0 0
\(377\) 323.644i 0.858472i
\(378\) 0 0
\(379\) 128.176i 0.338195i −0.985599 0.169098i \(-0.945915\pi\)
0.985599 0.169098i \(-0.0540853\pi\)
\(380\) 0 0
\(381\) 32.0011 + 55.4276i 0.0839925 + 0.145479i
\(382\) 0 0
\(383\) 216.437 + 124.960i 0.565110 + 0.326266i 0.755194 0.655502i \(-0.227543\pi\)
−0.190084 + 0.981768i \(0.560876\pi\)
\(384\) 0 0
\(385\) 493.403 317.870i 1.28157 0.825636i
\(386\) 0 0
\(387\) −235.571 136.007i −0.608709 0.351439i
\(388\) 0 0
\(389\) −187.428 + 108.212i −0.481821 + 0.278179i −0.721175 0.692753i \(-0.756398\pi\)
0.239354 + 0.970932i \(0.423064\pi\)
\(390\) 0 0
\(391\) 380.991i 0.974402i
\(392\) 0 0
\(393\) 129.460 0.329415
\(394\) 0 0
\(395\) 184.984 + 320.401i 0.468314 + 0.811143i
\(396\) 0 0
\(397\) −349.941 + 606.116i −0.881463 + 1.52674i −0.0317493 + 0.999496i \(0.510108\pi\)
−0.849714 + 0.527244i \(0.823226\pi\)
\(398\) 0 0
\(399\) −56.8196 88.1964i −0.142405 0.221044i
\(400\) 0 0
\(401\) 90.4903 156.734i 0.225662 0.390858i −0.730856 0.682532i \(-0.760879\pi\)
0.956518 + 0.291674i \(0.0942123\pi\)
\(402\) 0 0
\(403\) −216.310 + 124.887i −0.536749 + 0.309892i
\(404\) 0 0
\(405\) −376.179 −0.928837
\(406\) 0 0
\(407\) −628.431 −1.54406
\(408\) 0 0
\(409\) −310.767 + 179.421i −0.759821 + 0.438683i −0.829232 0.558905i \(-0.811222\pi\)
0.0694104 + 0.997588i \(0.477888\pi\)
\(410\) 0 0
\(411\) 115.258 199.632i 0.280432 0.485723i
\(412\) 0 0
\(413\) 175.029 340.441i 0.423798 0.824312i
\(414\) 0 0
\(415\) 115.302 199.709i 0.277836 0.481226i
\(416\) 0 0
\(417\) 22.4031 + 38.8033i 0.0537245 + 0.0930535i
\(418\) 0 0
\(419\) −780.890 −1.86370 −0.931849 0.362846i \(-0.881805\pi\)
−0.931849 + 0.362846i \(0.881805\pi\)
\(420\) 0 0
\(421\) 114.961i 0.273068i −0.990635 0.136534i \(-0.956404\pi\)
0.990635 0.136534i \(-0.0435962\pi\)
\(422\) 0 0
\(423\) 155.002 89.4904i 0.366435 0.211561i
\(424\) 0 0
\(425\) −210.472 121.516i −0.495228 0.285920i
\(426\) 0 0
\(427\) 71.6425 3.49290i 0.167781 0.00818010i
\(428\) 0 0
\(429\) 202.482 + 116.903i 0.471985 + 0.272501i
\(430\) 0 0
\(431\) 154.856 + 268.219i 0.359295 + 0.622317i 0.987843 0.155453i \(-0.0496838\pi\)
−0.628548 + 0.777771i \(0.716351\pi\)
\(432\) 0 0
\(433\) 595.775i 1.37592i −0.725747 0.687962i \(-0.758506\pi\)
0.725747 0.687962i \(-0.241494\pi\)
\(434\) 0 0
\(435\) 96.2958i 0.221370i
\(436\) 0 0
\(437\) −196.808 340.881i −0.450361 0.780048i
\(438\) 0 0
\(439\) −698.796 403.450i −1.59179 0.919020i −0.993000 0.118113i \(-0.962315\pi\)
−0.598789 0.800907i \(-0.704351\pi\)
\(440\) 0 0
\(441\) 232.951 + 325.593i 0.528233 + 0.738306i
\(442\) 0 0
\(443\) −385.214 222.403i −0.869557 0.502039i −0.00235617 0.999997i \(-0.500750\pi\)
−0.867201 + 0.497958i \(0.834083\pi\)
\(444\) 0 0
\(445\) 5.90012 3.40644i 0.0132587 0.00765491i
\(446\) 0 0
\(447\) 37.9772i 0.0849601i
\(448\) 0 0
\(449\) 262.420 0.584455 0.292228 0.956349i \(-0.405604\pi\)
0.292228 + 0.956349i \(0.405604\pi\)
\(450\) 0 0
\(451\) 42.9145 + 74.3302i 0.0951542 + 0.164812i
\(452\) 0 0
\(453\) 44.4633 77.0127i 0.0981530 0.170006i
\(454\) 0 0
\(455\) 861.712 42.0124i 1.89387 0.0923350i
\(456\) 0 0
\(457\) −194.738 + 337.296i −0.426122 + 0.738065i −0.996524 0.0833004i \(-0.973454\pi\)
0.570403 + 0.821365i \(0.306787\pi\)
\(458\) 0 0
\(459\) 215.720 124.546i 0.469978 0.271342i
\(460\) 0 0
\(461\) 158.714 0.344283 0.172141 0.985072i \(-0.444931\pi\)
0.172141 + 0.985072i \(0.444931\pi\)
\(462\) 0 0
\(463\) 528.844 1.14221 0.571106 0.820877i \(-0.306515\pi\)
0.571106 + 0.820877i \(0.306515\pi\)
\(464\) 0 0
\(465\) 64.3601 37.1583i 0.138409 0.0799103i
\(466\) 0 0
\(467\) 218.449 378.365i 0.467771 0.810203i −0.531551 0.847026i \(-0.678391\pi\)
0.999322 + 0.0368236i \(0.0117240\pi\)
\(468\) 0 0
\(469\) −106.667 54.8401i −0.227435 0.116930i
\(470\) 0 0
\(471\) 12.8274 22.2177i 0.0272344 0.0471713i
\(472\) 0 0
\(473\) 219.977 + 381.011i 0.465067 + 0.805519i
\(474\) 0 0
\(475\) −251.085 −0.528600
\(476\) 0 0
\(477\) 303.811i 0.636920i
\(478\) 0 0
\(479\) −472.737 + 272.935i −0.986925 + 0.569802i −0.904354 0.426783i \(-0.859647\pi\)
−0.0825716 + 0.996585i \(0.526313\pi\)
\(480\) 0 0
\(481\) −799.980 461.869i −1.66316 0.960226i
\(482\) 0 0
\(483\) −82.6039 128.219i −0.171023 0.265464i
\(484\) 0 0
\(485\) 931.488 + 537.795i 1.92059 + 1.10886i
\(486\) 0 0
\(487\) 324.115 + 561.384i 0.665534 + 1.15274i 0.979140 + 0.203185i \(0.0651294\pi\)
−0.313606 + 0.949553i \(0.601537\pi\)
\(488\) 0 0
\(489\) 220.822i 0.451579i
\(490\) 0 0
\(491\) 732.074i 1.49098i 0.666514 + 0.745492i \(0.267786\pi\)
−0.666514 + 0.745492i \(0.732214\pi\)
\(492\) 0 0
\(493\) 132.684 + 229.815i 0.269136 + 0.466157i
\(494\) 0 0
\(495\) 593.279 + 342.530i 1.19854 + 0.691980i
\(496\) 0 0
\(497\) 121.529 + 188.640i 0.244526 + 0.379557i
\(498\) 0 0
\(499\) −23.1264 13.3520i −0.0463454 0.0267575i 0.476648 0.879094i \(-0.341852\pi\)
−0.522994 + 0.852337i \(0.675185\pi\)
\(500\) 0 0
\(501\) 47.4172 27.3763i 0.0946451 0.0546434i
\(502\) 0 0
\(503\) 616.414i 1.22548i −0.790286 0.612738i \(-0.790068\pi\)
0.790286 0.612738i \(-0.209932\pi\)
\(504\) 0 0
\(505\) −178.460 −0.353387
\(506\) 0 0
\(507\) 94.8687 + 164.317i 0.187118 + 0.324097i
\(508\) 0 0
\(509\) −66.3763 + 114.967i −0.130405 + 0.225869i −0.923833 0.382796i \(-0.874961\pi\)
0.793428 + 0.608665i \(0.208295\pi\)
\(510\) 0 0
\(511\) −667.152 342.999i −1.30558 0.671230i
\(512\) 0 0
\(513\) 128.673 222.868i 0.250824 0.434440i
\(514\) 0 0
\(515\) 913.867 527.621i 1.77450 1.02451i
\(516\) 0 0
\(517\) −289.483 −0.559928
\(518\) 0 0
\(519\) 126.942 0.244589
\(520\) 0 0
\(521\) −585.480 + 338.027i −1.12376 + 0.648804i −0.942359 0.334605i \(-0.891397\pi\)
−0.181403 + 0.983409i \(0.558064\pi\)
\(522\) 0 0
\(523\) 186.224 322.550i 0.356069 0.616730i −0.631231 0.775595i \(-0.717450\pi\)
0.987300 + 0.158865i \(0.0507833\pi\)
\(524\) 0 0
\(525\) −97.1787 + 4.73791i −0.185102 + 0.00902459i
\(526\) 0 0
\(527\) 102.399 177.361i 0.194306 0.336548i
\(528\) 0 0
\(529\) −21.6179 37.4433i −0.0408656 0.0707813i
\(530\) 0 0
\(531\) 446.799 0.841429
\(532\) 0 0
\(533\) 126.161i 0.236700i
\(534\) 0 0
\(535\) −1085.71 + 626.836i −2.02937 + 1.17166i
\(536\) 0 0
\(537\) −230.123 132.862i −0.428534 0.247414i
\(538\) 0 0
\(539\) −62.9892 644.447i −0.116863 1.19563i
\(540\) 0 0
\(541\) −60.3373 34.8357i −0.111529 0.0643914i 0.443198 0.896424i \(-0.353844\pi\)
−0.554727 + 0.832032i \(0.687177\pi\)
\(542\) 0 0
\(543\) 75.9860 + 131.612i 0.139937 + 0.242379i
\(544\) 0 0
\(545\) 73.0934i 0.134116i
\(546\) 0 0
\(547\) 466.463i 0.852765i 0.904543 + 0.426383i \(0.140212\pi\)
−0.904543 + 0.426383i \(0.859788\pi\)
\(548\) 0 0
\(549\) 41.8599 + 72.5034i 0.0762475 + 0.132065i
\(550\) 0 0
\(551\) 237.430 + 137.081i 0.430908 + 0.248785i
\(552\) 0 0
\(553\) 407.674 19.8760i 0.737204 0.0359421i
\(554\) 0 0
\(555\) 238.023 + 137.423i 0.428870 + 0.247608i
\(556\) 0 0
\(557\) −118.835 + 68.6094i −0.213348 + 0.123177i −0.602866 0.797842i \(-0.705975\pi\)
0.389518 + 0.921019i \(0.372642\pi\)
\(558\) 0 0
\(559\) 646.692i 1.15687i
\(560\) 0 0
\(561\) −191.706 −0.341722
\(562\) 0 0
\(563\) 84.5632 + 146.468i 0.150201 + 0.260156i 0.931301 0.364250i \(-0.118675\pi\)
−0.781100 + 0.624406i \(0.785341\pi\)
\(564\) 0 0
\(565\) −46.9240 + 81.2747i −0.0830513 + 0.143849i
\(566\) 0 0
\(567\) −189.757 + 369.088i −0.334669 + 0.650949i
\(568\) 0 0
\(569\) −372.466 + 645.129i −0.654597 + 1.13379i 0.327398 + 0.944887i \(0.393828\pi\)
−0.981995 + 0.188908i \(0.939505\pi\)
\(570\) 0 0
\(571\) −767.828 + 443.306i −1.34471 + 0.776367i −0.987494 0.157655i \(-0.949606\pi\)
−0.357213 + 0.934023i \(0.616273\pi\)
\(572\) 0 0
\(573\) −119.647 −0.208809
\(574\) 0 0
\(575\) −365.026 −0.634827
\(576\) 0 0
\(577\) −207.900 + 120.031i −0.360311 + 0.208026i −0.669217 0.743067i \(-0.733370\pi\)
0.308906 + 0.951093i \(0.400037\pi\)
\(578\) 0 0
\(579\) −37.0900 + 64.2417i −0.0640586 + 0.110953i
\(580\) 0 0
\(581\) −137.782 213.868i −0.237147 0.368104i
\(582\) 0 0
\(583\) 245.691 425.549i 0.421425 0.729930i
\(584\) 0 0
\(585\) 503.488 + 872.067i 0.860663 + 1.49071i
\(586\) 0 0
\(587\) −190.873 −0.325168 −0.162584 0.986695i \(-0.551983\pi\)
−0.162584 + 0.986695i \(0.551983\pi\)
\(588\) 0 0
\(589\) 211.585i 0.359227i
\(590\) 0 0
\(591\) −1.65299 + 0.954353i −0.00279693 + 0.00161481i
\(592\) 0 0
\(593\) 637.548 + 368.089i 1.07512 + 0.620723i 0.929577 0.368629i \(-0.120173\pi\)
0.145547 + 0.989351i \(0.453506\pi\)
\(594\) 0 0
\(595\) −594.667 + 383.108i −0.999441 + 0.643879i
\(596\) 0 0
\(597\) 100.137 + 57.8144i 0.167734 + 0.0968415i
\(598\) 0 0
\(599\) 558.330 + 967.057i 0.932104 + 1.61445i 0.779718 + 0.626131i \(0.215362\pi\)
0.152386 + 0.988321i \(0.451304\pi\)
\(600\) 0 0
\(601\) 183.100i 0.304659i 0.988330 + 0.152329i \(0.0486774\pi\)
−0.988330 + 0.152329i \(0.951323\pi\)
\(602\) 0 0
\(603\) 139.991i 0.232158i
\(604\) 0 0
\(605\) −170.132 294.678i −0.281210 0.487071i
\(606\) 0 0
\(607\) −394.026 227.491i −0.649136 0.374779i 0.138989 0.990294i \(-0.455615\pi\)
−0.788125 + 0.615515i \(0.788948\pi\)
\(608\) 0 0
\(609\) 94.4807 + 48.5748i 0.155141 + 0.0797615i
\(610\) 0 0
\(611\) −368.505 212.757i −0.603118 0.348211i
\(612\) 0 0
\(613\) 232.853 134.438i 0.379859 0.219312i −0.297898 0.954598i \(-0.596286\pi\)
0.677757 + 0.735286i \(0.262952\pi\)
\(614\) 0 0
\(615\) 37.5375i 0.0610366i
\(616\) 0 0
\(617\) −184.934 −0.299731 −0.149866 0.988706i \(-0.547884\pi\)
−0.149866 + 0.988706i \(0.547884\pi\)
\(618\) 0 0
\(619\) 496.809 + 860.498i 0.802599 + 1.39014i 0.917900 + 0.396812i \(0.129884\pi\)
−0.115301 + 0.993331i \(0.536783\pi\)
\(620\) 0 0
\(621\) 187.064 324.004i 0.301229 0.521745i
\(622\) 0 0
\(623\) −0.366012 7.50723i −0.000587499 0.0120501i
\(624\) 0 0
\(625\) 386.818 669.988i 0.618909 1.07198i
\(626\) 0 0
\(627\) −171.524 + 99.0292i −0.273562 + 0.157941i
\(628\) 0 0
\(629\) 757.408 1.20415
\(630\) 0 0
\(631\) −805.857 −1.27711 −0.638555 0.769576i \(-0.720468\pi\)
−0.638555 + 0.769576i \(0.720468\pi\)
\(632\) 0 0
\(633\) −5.65451 + 3.26463i −0.00893287 + 0.00515740i
\(634\) 0 0
\(635\) 222.918 386.106i 0.351053 0.608041i
\(636\) 0 0
\(637\) 393.455 866.662i 0.617669 1.36054i
\(638\) 0 0
\(639\) −130.957 + 226.825i −0.204941 + 0.354969i
\(640\) 0 0
\(641\) −2.75221 4.76696i −0.00429361 0.00743676i 0.863871 0.503714i \(-0.168033\pi\)
−0.868164 + 0.496277i \(0.834700\pi\)
\(642\) 0 0
\(643\) 1024.08 1.59266 0.796331 0.604861i \(-0.206771\pi\)
0.796331 + 0.604861i \(0.206771\pi\)
\(644\) 0 0
\(645\) 192.414i 0.298317i
\(646\) 0 0
\(647\) −395.404 + 228.287i −0.611134 + 0.352839i −0.773409 0.633907i \(-0.781450\pi\)
0.162275 + 0.986746i \(0.448117\pi\)
\(648\) 0 0
\(649\) −625.834 361.325i −0.964304 0.556741i
\(650\) 0 0
\(651\) −3.99255 81.8908i −0.00613295 0.125792i
\(652\) 0 0
\(653\) −24.4603 14.1222i −0.0374584 0.0216266i 0.481154 0.876636i \(-0.340218\pi\)
−0.518612 + 0.855010i \(0.673551\pi\)
\(654\) 0 0
\(655\) −450.907 780.994i −0.688407 1.19236i
\(656\) 0 0
\(657\) 875.579i 1.33269i
\(658\) 0 0
\(659\) 132.188i 0.200589i −0.994958 0.100295i \(-0.968021\pi\)
0.994958 0.100295i \(-0.0319785\pi\)
\(660\) 0 0
\(661\) 346.924 + 600.889i 0.524847 + 0.909061i 0.999581 + 0.0289321i \(0.00921065\pi\)
−0.474735 + 0.880129i \(0.657456\pi\)
\(662\) 0 0
\(663\) −244.038 140.895i −0.368082 0.212512i
\(664\) 0 0
\(665\) −334.160 + 649.960i −0.502497 + 0.977384i
\(666\) 0 0
\(667\) 345.175 + 199.287i 0.517504 + 0.298781i
\(668\) 0 0
\(669\) −220.652 + 127.393i −0.329823 + 0.190424i
\(670\) 0 0
\(671\) 135.408i 0.201800i
\(672\) 0 0
\(673\) 532.137 0.790694 0.395347 0.918532i \(-0.370624\pi\)
0.395347 + 0.918532i \(0.370624\pi\)
\(674\) 0 0
\(675\) −119.327 206.680i −0.176780 0.306192i
\(676\) 0 0
\(677\) −143.115 + 247.883i −0.211396 + 0.366149i −0.952152 0.305626i \(-0.901134\pi\)
0.740756 + 0.671775i \(0.234468\pi\)
\(678\) 0 0
\(679\) 997.531 642.649i 1.46912 0.946464i
\(680\) 0 0
\(681\) −138.808 + 240.423i −0.203830 + 0.353043i
\(682\) 0 0
\(683\) −387.838 + 223.918i −0.567844 + 0.327845i −0.756288 0.654239i \(-0.772989\pi\)
0.188443 + 0.982084i \(0.439656\pi\)
\(684\) 0 0
\(685\) −1605.76 −2.34417
\(686\) 0 0
\(687\) 377.724 0.549817
\(688\) 0 0
\(689\) 625.519 361.143i 0.907865 0.524156i
\(690\) 0 0
\(691\) −510.366 + 883.980i −0.738591 + 1.27928i 0.214539 + 0.976715i \(0.431175\pi\)
−0.953130 + 0.302561i \(0.902158\pi\)
\(692\) 0 0
\(693\) 635.343 409.313i 0.916801 0.590639i
\(694\) 0 0
\(695\) 156.059 270.302i 0.224545 0.388924i
\(696\) 0 0
\(697\) −51.7221 89.5854i −0.0742068 0.128530i
\(698\) 0 0
\(699\) 150.225 0.214914
\(700\) 0 0
\(701\) 1311.02i 1.87021i 0.354369 + 0.935106i \(0.384696\pi\)
−0.354369 + 0.935106i \(0.615304\pi\)
\(702\) 0 0
\(703\) 677.669 391.252i 0.963967 0.556546i
\(704\) 0 0
\(705\) 109.644 + 63.3028i 0.155523 + 0.0897912i
\(706\) 0 0
\(707\) −90.0213 + 175.096i −0.127329 + 0.247661i
\(708\) 0 0
\(709\) −465.495 268.754i −0.656552 0.379061i 0.134410 0.990926i \(-0.457086\pi\)
−0.790962 + 0.611865i \(0.790419\pi\)
\(710\) 0 0
\(711\) 238.199 + 412.573i 0.335020 + 0.580271i
\(712\) 0 0
\(713\) 307.601i 0.431417i
\(714\) 0 0
\(715\) 1628.68i 2.27787i
\(716\) 0 0
\(717\) 8.70704 + 15.0810i 0.0121437 + 0.0210335i
\(718\) 0 0
\(719\) 233.275 + 134.681i 0.324443 + 0.187318i 0.653371 0.757037i \(-0.273354\pi\)
−0.328928 + 0.944355i \(0.606687\pi\)
\(720\) 0 0
\(721\) −56.6914 1162.79i −0.0786288 1.61275i
\(722\) 0 0
\(723\) −276.334 159.541i −0.382204 0.220666i
\(724\) 0 0
\(725\) 220.185 127.124i 0.303703 0.175343i
\(726\) 0 0
\(727\) 460.316i 0.633172i −0.948564 0.316586i \(-0.897463\pi\)
0.948564 0.316586i \(-0.102537\pi\)
\(728\) 0 0
\(729\) −356.185 −0.488594
\(730\) 0 0
\(731\) −265.124 459.208i −0.362686 0.628191i
\(732\) 0 0
\(733\) −33.3410 + 57.7484i −0.0454857 + 0.0787836i −0.887872 0.460091i \(-0.847817\pi\)
0.842386 + 0.538874i \(0.181150\pi\)
\(734\) 0 0
\(735\) −117.067 + 257.863i −0.159275 + 0.350834i
\(736\) 0 0
\(737\) −113.211 + 196.087i −0.153610 + 0.266061i
\(738\) 0 0
\(739\) 808.772 466.944i 1.09441 0.631860i 0.159665 0.987171i \(-0.448958\pi\)
0.934748 + 0.355311i \(0.115625\pi\)
\(740\) 0 0
\(741\) −291.128 −0.392885
\(742\) 0 0
\(743\) 1198.23 1.61269 0.806345 0.591446i \(-0.201443\pi\)
0.806345 + 0.591446i \(0.201443\pi\)
\(744\) 0 0
\(745\) −229.104 + 132.273i −0.307523 + 0.177548i
\(746\) 0 0
\(747\) 148.471 257.160i 0.198757 0.344257i
\(748\) 0 0
\(749\) 67.3518 + 1381.44i 0.0899222 + 1.84438i
\(750\) 0 0
\(751\) −84.2993 + 146.011i −0.112249 + 0.194422i −0.916677 0.399629i \(-0.869139\pi\)
0.804427 + 0.594051i \(0.202472\pi\)
\(752\) 0 0
\(753\) 40.2239 + 69.6698i 0.0534182 + 0.0925230i
\(754\) 0 0
\(755\) −619.458 −0.820474
\(756\) 0 0
\(757\) 209.207i 0.276364i −0.990407 0.138182i \(-0.955874\pi\)
0.990407 0.138182i \(-0.0441259\pi\)
\(758\) 0 0
\(759\) −249.360 + 143.968i −0.328538 + 0.189681i
\(760\) 0 0
\(761\) 479.127 + 276.624i 0.629602 + 0.363501i 0.780598 0.625033i \(-0.214915\pi\)
−0.150996 + 0.988534i \(0.548248\pi\)
\(762\) 0 0
\(763\) −71.7156 36.8707i −0.0939916 0.0483233i
\(764\) 0 0
\(765\) −715.041 412.829i −0.934695 0.539646i
\(766\) 0 0
\(767\) −531.115 919.919i −0.692458 1.19937i
\(768\) 0 0
\(769\) 219.524i 0.285467i −0.989761 0.142734i \(-0.954411\pi\)
0.989761 0.142734i \(-0.0455892\pi\)
\(770\) 0 0
\(771\) 78.4096i 0.101699i
\(772\) 0 0
\(773\) −333.337 577.357i −0.431225 0.746904i 0.565754 0.824574i \(-0.308585\pi\)
−0.996979 + 0.0776701i \(0.975252\pi\)
\(774\) 0 0
\(775\) −169.928 98.1082i −0.219262 0.126591i
\(776\) 0 0
\(777\) 254.899 164.216i 0.328055 0.211346i
\(778\) 0 0
\(779\) −92.5538 53.4359i −0.118811 0.0685956i
\(780\) 0 0
\(781\) 366.866 211.810i 0.469738 0.271204i
\(782\) 0 0
\(783\) 260.587i 0.332806i
\(784\) 0 0
\(785\) −178.710 −0.227656
\(786\) 0 0
\(787\) −459.932 796.626i −0.584412 1.01223i −0.994948 0.100387i \(-0.967992\pi\)
0.410536 0.911844i \(-0.365341\pi\)
\(788\) 0 0
\(789\) −145.378 + 251.803i −0.184256 + 0.319141i
\(790\) 0 0
\(791\) 56.0727 + 87.0371i 0.0708884 + 0.110034i
\(792\) 0 0
\(793\) 99.5187 172.371i 0.125496 0.217366i
\(794\) 0 0
\(795\) −186.115 + 107.453i −0.234106 + 0.135161i
\(796\) 0 0
\(797\) 1016.13 1.27494 0.637470 0.770476i \(-0.279981\pi\)
0.637470 + 0.770476i \(0.279981\pi\)
\(798\) 0 0
\(799\) 348.895 0.436664
\(800\) 0 0
\(801\) 7.59744 4.38638i 0.00948494 0.00547613i
\(802\) 0 0
\(803\) −708.078 + 1226.43i −0.881791 + 1.52731i
\(804\) 0 0
\(805\) −485.800 + 944.909i −0.603478 + 1.17380i
\(806\) 0 0
\(807\) −26.1727 + 45.3325i −0.0324321 + 0.0561741i
\(808\) 0 0
\(809\) −565.950 980.254i −0.699567 1.21169i −0.968617 0.248560i \(-0.920043\pi\)
0.269049 0.963126i \(-0.413291\pi\)
\(810\) 0 0
\(811\) −481.066 −0.593176 −0.296588 0.955006i \(-0.595849\pi\)
−0.296588 + 0.955006i \(0.595849\pi\)
\(812\) 0 0
\(813\) 28.1242i 0.0345931i
\(814\) 0 0
\(815\) 1332.15 769.117i 1.63454 0.943702i
\(816\) 0 0
\(817\) −474.423 273.908i −0.580690 0.335261i
\(818\) 0 0
\(819\) 1109.60 54.0983i 1.35483 0.0660541i
\(820\) 0 0
\(821\) −630.185 363.838i −0.767582 0.443164i 0.0644292 0.997922i \(-0.479477\pi\)
−0.832011 + 0.554758i \(0.812811\pi\)
\(822\) 0 0
\(823\) −313.323 542.692i −0.380709 0.659407i 0.610455 0.792051i \(-0.290987\pi\)
−0.991164 + 0.132644i \(0.957653\pi\)
\(824\) 0 0
\(825\) 183.673i 0.222633i
\(826\) 0 0
\(827\) 1468.52i 1.77572i 0.460116 + 0.887859i \(0.347808\pi\)
−0.460116 + 0.887859i \(0.652192\pi\)
\(828\) 0 0
\(829\) −409.352 709.019i −0.493790 0.855270i 0.506184 0.862425i \(-0.331056\pi\)
−0.999974 + 0.00715566i \(0.997722\pi\)
\(830\) 0 0
\(831\) 280.969 + 162.218i 0.338110 + 0.195208i
\(832\) 0 0
\(833\) 75.9169 + 776.710i 0.0911367 + 0.932425i
\(834\) 0 0
\(835\) −330.306 190.702i −0.395576 0.228386i
\(836\) 0 0
\(837\) 174.165 100.554i 0.208083 0.120137i
\(838\) 0 0
\(839\) 1108.84i 1.32162i −0.750555 0.660808i \(-0.770214\pi\)
0.750555 0.660808i \(-0.229786\pi\)
\(840\) 0 0
\(841\) 563.386 0.669900
\(842\) 0 0
\(843\) −133.970 232.042i −0.158920 0.275258i
\(844\) 0 0
\(845\) 660.850 1144.63i 0.782072 1.35459i
\(846\) 0 0
\(847\) −374.943 + 18.2802i −0.442672 + 0.0215823i
\(848\) 0 0
\(849\) 189.005 327.366i 0.222621 0.385590i
\(850\) 0 0
\(851\) 985.191 568.800i 1.15769 0.668390i
\(852\) 0 0
\(853\) 610.400 0.715592 0.357796 0.933800i \(-0.383528\pi\)
0.357796 + 0.933800i \(0.383528\pi\)
\(854\) 0 0
\(855\) −853.017 −0.997680
\(856\) 0 0
\(857\) 384.614 222.057i 0.448791 0.259110i −0.258529 0.966004i \(-0.583238\pi\)
0.707319 + 0.706894i \(0.249904\pi\)
\(858\) 0 0
\(859\) −40.7547 + 70.5892i −0.0474443 + 0.0821760i −0.888772 0.458349i \(-0.848441\pi\)
0.841328 + 0.540525i \(0.181774\pi\)
\(860\) 0 0
\(861\) −36.8299 18.9351i −0.0427757 0.0219920i
\(862\) 0 0
\(863\) −525.730 + 910.592i −0.609189 + 1.05515i 0.382185 + 0.924086i \(0.375172\pi\)
−0.991374 + 0.131061i \(0.958162\pi\)
\(864\) 0 0
\(865\) −442.134 765.799i −0.511138 0.885317i
\(866\) 0 0
\(867\) −32.1882 −0.0371259
\(868\) 0 0
\(869\) 770.524i 0.886679i
\(870\) 0 0
\(871\) −288.230 + 166.409i −0.330918 + 0.191056i
\(872\) 0 0
\(873\) 1199.45 + 692.505i 1.37395 + 0.793248i
\(874\) 0 0
\(875\) −234.305 363.693i −0.267777 0.415649i
\(876\) 0 0
\(877\) 1350.68 + 779.814i 1.54011 + 0.889183i 0.998831 + 0.0483410i \(0.0153934\pi\)
0.541280 + 0.840842i \(0.317940\pi\)
\(878\) 0 0
\(879\) −168.767 292.312i −0.191998 0.332551i
\(880\) 0 0
\(881\) 1515.22i 1.71989i 0.510389 + 0.859944i \(0.329501\pi\)
−0.510389 + 0.859944i \(0.670499\pi\)
\(882\) 0 0
\(883\) 763.828i 0.865037i −0.901625 0.432519i \(-0.857625\pi\)
0.901625 0.432519i \(-0.142375\pi\)
\(884\) 0 0
\(885\) 158.026 + 273.709i 0.178561 + 0.309276i
\(886\) 0 0
\(887\) −496.554 286.686i −0.559813 0.323208i 0.193258 0.981148i \(-0.438095\pi\)
−0.753070 + 0.657940i \(0.771428\pi\)
\(888\) 0 0
\(889\) −266.381 413.481i −0.299641 0.465108i
\(890\) 0 0
\(891\) 678.496 + 391.730i 0.761500 + 0.439652i
\(892\) 0 0
\(893\) 312.163 180.228i 0.349567 0.201823i
\(894\) 0 0
\(895\) 1851.01i 2.06817i
\(896\) 0 0
\(897\) −423.240 −0.471840
\(898\) 0 0
\(899\) 107.125 + 185.546i 0.119160 + 0.206391i
\(900\) 0 0
\(901\) −296.115 + 512.887i −0.328652 + 0.569242i
\(902\) 0 0
\(903\) −188.787 97.0600i −0.209067 0.107486i
\(904\) 0 0
\(905\) 529.315 916.800i 0.584878 1.01304i
\(906\) 0 0
\(907\) −885.036 + 510.976i −0.975784 + 0.563369i −0.900995 0.433830i \(-0.857162\pi\)
−0.0747894 + 0.997199i \(0.523828\pi\)
\(908\) 0 0
\(909\) −229.799 −0.252804
\(910\) 0 0
\(911\) −630.111 −0.691669 −0.345835 0.938295i \(-0.612404\pi\)
−0.345835 + 0.938295i \(0.612404\pi\)
\(912\) 0 0
\(913\) −415.930 + 240.137i −0.455564 + 0.263020i
\(914\) 0 0
\(915\) −29.6104 + 51.2868i −0.0323611 + 0.0560511i
\(916\) 0 0
\(917\) −993.725 + 48.4486i −1.08367 + 0.0528339i
\(918\) 0 0
\(919\) −421.489 + 730.041i −0.458639 + 0.794386i −0.998889 0.0471182i \(-0.984996\pi\)
0.540250 + 0.841504i \(0.318330\pi\)
\(920\) 0 0
\(921\) 73.0182 + 126.471i 0.0792814 + 0.137319i
\(922\) 0 0
\(923\) 622.683 0.674630
\(924\) 0 0
\(925\) 725.668i 0.784506i
\(926\) 0 0
\(927\) 1176.76 679.405i 1.26943 0.732907i
\(928\) 0 0
\(929\) 670.867 + 387.325i 0.722139 + 0.416927i 0.815540 0.578701i \(-0.196440\pi\)
−0.0934003 + 0.995629i \(0.529774\pi\)
\(930\) 0 0
\(931\) 469.148 + 655.723i 0.503918 + 0.704321i
\(932\) 0 0
\(933\) −372.990 215.346i −0.399774 0.230810i
\(934\) 0 0
\(935\) 667.708 + 1156.50i 0.714126 + 1.23690i
\(936\) 0 0
\(937\) 1586.27i 1.69293i 0.532447 + 0.846463i \(0.321273\pi\)
−0.532447 + 0.846463i \(0.678727\pi\)
\(938\) 0 0
\(939\) 210.421i 0.224090i
\(940\) 0 0
\(941\) 410.023 + 710.181i 0.435731 + 0.754708i 0.997355 0.0726842i \(-0.0231565\pi\)
−0.561624 + 0.827393i \(0.689823\pi\)
\(942\) 0 0
\(943\) −134.554 77.6849i −0.142687 0.0823805i
\(944\) 0 0
\(945\) −693.821 + 33.8270i −0.734202 + 0.0357957i
\(946\) 0 0
\(947\) 551.949 + 318.668i 0.582839 + 0.336502i 0.762261 0.647270i \(-0.224089\pi\)
−0.179422 + 0.983772i \(0.557423\pi\)
\(948\) 0 0
\(949\) −1802.74 + 1040.81i −1.89962 + 1.09675i
\(950\) 0 0
\(951\) 205.234i 0.215809i
\(952\) 0 0
\(953\) −350.626 −0.367918 −0.183959 0.982934i \(-0.558891\pi\)
−0.183959 + 0.982934i \(0.558891\pi\)
\(954\) 0 0
\(955\) 416.729 + 721.796i 0.436365 + 0.755807i
\(956\) 0 0
\(957\) 100.277 173.684i 0.104782 0.181488i
\(958\) 0 0
\(959\) −809.997 + 1575.49i −0.844627 + 1.64285i
\(960\) 0 0
\(961\) −397.826 + 689.055i −0.413971 + 0.717019i
\(962\) 0 0
\(963\) −1398.04 + 807.161i −1.45176 + 0.838174i
\(964\) 0 0
\(965\) 516.734 0.535475
\(966\) 0 0
\(967\) 649.816 0.671992 0.335996 0.941863i \(-0.390927\pi\)
0.335996 + 0.941863i \(0.390927\pi\)
\(968\) 0 0
\(969\) 206.726 119.354i 0.213340 0.123172i
\(970\) 0 0
\(971\) 485.305 840.573i 0.499799 0.865677i −0.500201 0.865909i \(-0.666741\pi\)
1.00000 0.000232071i \(7.38706e-5\pi\)
\(972\) 0 0
\(973\) −186.486 289.466i −0.191660 0.297499i
\(974\) 0 0
\(975\) −134.991 + 233.811i −0.138452 + 0.239807i
\(976\) 0 0
\(977\) 300.437 + 520.373i 0.307510 + 0.532623i 0.977817 0.209461i \(-0.0671709\pi\)
−0.670307 + 0.742084i \(0.733838\pi\)
\(978\) 0 0
\(979\) −14.1890 −0.0144934
\(980\) 0 0
\(981\) 94.1205i 0.0959434i
\(982\) 0 0
\(983\) −1098.66 + 634.311i −1.11766 + 0.645281i −0.940802 0.338955i \(-0.889926\pi\)
−0.176857 + 0.984236i \(0.556593\pi\)
\(984\) 0 0
\(985\) 11.5146 + 6.64797i 0.0116900 + 0.00674921i
\(986\) 0 0
\(987\) 117.417 75.6449i 0.118964 0.0766413i
\(988\) 0 0
\(989\) −689.714 398.207i −0.697385 0.402636i
\(990\) 0 0
\(991\) −774.555 1341.57i −0.781590 1.35375i −0.931015 0.364980i \(-0.881076\pi\)
0.149426 0.988773i \(-0.452258\pi\)
\(992\) 0 0
\(993\) 18.8538i 0.0189867i
\(994\) 0 0
\(995\) 805.464i 0.809512i
\(996\) 0 0
\(997\) 470.469 + 814.876i 0.471885 + 0.817328i 0.999483 0.0321661i \(-0.0102406\pi\)
−0.527598 + 0.849494i \(0.676907\pi\)
\(998\) 0 0
\(999\) 644.116 + 371.881i 0.644761 + 0.372253i
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 224.3.n.a.145.9 28
4.3 odd 2 56.3.j.a.5.4 28
7.2 even 3 1568.3.h.a.881.12 28
7.3 odd 6 inner 224.3.n.a.17.6 28
7.5 odd 6 1568.3.h.a.881.18 28
8.3 odd 2 56.3.j.a.5.6 yes 28
8.5 even 2 inner 224.3.n.a.145.6 28
28.3 even 6 56.3.j.a.45.6 yes 28
28.11 odd 6 392.3.j.e.325.6 28
28.19 even 6 392.3.h.a.293.27 28
28.23 odd 6 392.3.h.a.293.28 28
28.27 even 2 392.3.j.e.117.4 28
56.3 even 6 56.3.j.a.45.4 yes 28
56.5 odd 6 1568.3.h.a.881.11 28
56.11 odd 6 392.3.j.e.325.4 28
56.19 even 6 392.3.h.a.293.26 28
56.27 even 2 392.3.j.e.117.6 28
56.37 even 6 1568.3.h.a.881.17 28
56.45 odd 6 inner 224.3.n.a.17.9 28
56.51 odd 6 392.3.h.a.293.25 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.j.a.5.4 28 4.3 odd 2
56.3.j.a.5.6 yes 28 8.3 odd 2
56.3.j.a.45.4 yes 28 56.3 even 6
56.3.j.a.45.6 yes 28 28.3 even 6
224.3.n.a.17.6 28 7.3 odd 6 inner
224.3.n.a.17.9 28 56.45 odd 6 inner
224.3.n.a.145.6 28 8.5 even 2 inner
224.3.n.a.145.9 28 1.1 even 1 trivial
392.3.h.a.293.25 28 56.51 odd 6
392.3.h.a.293.26 28 56.19 even 6
392.3.h.a.293.27 28 28.19 even 6
392.3.h.a.293.28 28 28.23 odd 6
392.3.j.e.117.4 28 28.27 even 2
392.3.j.e.117.6 28 56.27 even 2
392.3.j.e.325.4 28 56.11 odd 6
392.3.j.e.325.6 28 28.11 odd 6
1568.3.h.a.881.11 28 56.5 odd 6
1568.3.h.a.881.12 28 7.2 even 3
1568.3.h.a.881.17 28 56.37 even 6
1568.3.h.a.881.18 28 7.5 odd 6