Properties

Label 4020.2.q.l.841.4
Level $4020$
Weight $2$
Character 4020.841
Analytic conductor $32.100$
Analytic rank $0$
Dimension $22$
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] = [4020,2,Mod(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.q (of order \(3\), degree \(2\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 841.4
Character \(\chi\) \(=\) 4020.841
Dual form 4020.2.q.l.3781.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +(-0.775137 + 1.34258i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +(-0.775137 + 1.34258i) q^{7} +1.00000 q^{9} +(-0.262596 + 0.454829i) q^{11} +(-0.0526231 - 0.0911458i) q^{13} +1.00000 q^{15} +(1.53226 + 2.65395i) q^{17} +(-1.60844 - 2.78590i) q^{19} +(0.775137 - 1.34258i) q^{21} +(-4.09911 - 7.09986i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(-3.20300 + 5.54777i) q^{29} +(1.70073 - 2.94575i) q^{31} +(0.262596 - 0.454829i) q^{33} +(0.775137 - 1.34258i) q^{35} +(-0.396101 - 0.686067i) q^{37} +(0.0526231 + 0.0911458i) q^{39} +(2.52857 - 4.37962i) q^{41} +4.99327 q^{43} -1.00000 q^{45} +(0.703919 - 1.21922i) q^{47} +(2.29832 + 3.98082i) q^{49} +(-1.53226 - 2.65395i) q^{51} -6.53926 q^{53} +(0.262596 - 0.454829i) q^{55} +(1.60844 + 2.78590i) q^{57} +3.21916 q^{59} +(2.81190 + 4.87035i) q^{61} +(-0.775137 + 1.34258i) q^{63} +(0.0526231 + 0.0911458i) q^{65} +(-7.78180 + 2.53843i) q^{67} +(4.09911 + 7.09986i) q^{69} +(2.92291 - 5.06262i) q^{71} +(-4.11127 - 7.12092i) q^{73} -1.00000 q^{75} +(-0.407095 - 0.705110i) q^{77} +(-0.910790 + 1.57753i) q^{79} +1.00000 q^{81} +(8.71382 + 15.0928i) q^{83} +(-1.53226 - 2.65395i) q^{85} +(3.20300 - 5.54777i) q^{87} -8.71903 q^{89} +0.163160 q^{91} +(-1.70073 + 2.94575i) q^{93} +(1.60844 + 2.78590i) q^{95} +(-1.03739 - 1.79682i) q^{97} +(-0.262596 + 0.454829i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 22 q^{3} - 22 q^{5} + q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q - 22 q^{3} - 22 q^{5} + q^{7} + 22 q^{9} - 6 q^{11} - 7 q^{13} + 22 q^{15} + 4 q^{17} + 2 q^{19} - q^{21} + 6 q^{23} + 22 q^{25} - 22 q^{27} + 15 q^{29} - 5 q^{31} + 6 q^{33} - q^{35} + 2 q^{37} + 7 q^{39} - 6 q^{43} - 22 q^{45} - 7 q^{47} - 16 q^{49} - 4 q^{51} + 8 q^{53} + 6 q^{55} - 2 q^{57} - 6 q^{59} + 8 q^{61} + q^{63} + 7 q^{65} - 9 q^{67} - 6 q^{69} + 12 q^{71} - q^{73} - 22 q^{75} + 9 q^{77} - 15 q^{79} + 22 q^{81} - q^{83} - 4 q^{85} - 15 q^{87} + 20 q^{89} + 18 q^{91} + 5 q^{93} - 2 q^{95} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.775137 + 1.34258i −0.292974 + 0.507446i −0.974512 0.224337i \(-0.927978\pi\)
0.681537 + 0.731783i \(0.261312\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.262596 + 0.454829i −0.0791756 + 0.137136i −0.902894 0.429862i \(-0.858562\pi\)
0.823719 + 0.566998i \(0.191895\pi\)
\(12\) 0 0
\(13\) −0.0526231 0.0911458i −0.0145950 0.0252793i 0.858636 0.512586i \(-0.171313\pi\)
−0.873231 + 0.487307i \(0.837979\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.53226 + 2.65395i 0.371627 + 0.643677i 0.989816 0.142353i \(-0.0454668\pi\)
−0.618189 + 0.786029i \(0.712133\pi\)
\(18\) 0 0
\(19\) −1.60844 2.78590i −0.369002 0.639130i 0.620408 0.784279i \(-0.286967\pi\)
−0.989410 + 0.145150i \(0.953634\pi\)
\(20\) 0 0
\(21\) 0.775137 1.34258i 0.169149 0.292974i
\(22\) 0 0
\(23\) −4.09911 7.09986i −0.854723 1.48042i −0.876902 0.480669i \(-0.840394\pi\)
0.0221794 0.999754i \(-0.492939\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.20300 + 5.54777i −0.594783 + 1.03019i 0.398794 + 0.917040i \(0.369429\pi\)
−0.993577 + 0.113154i \(0.963905\pi\)
\(30\) 0 0
\(31\) 1.70073 2.94575i 0.305460 0.529072i −0.671904 0.740638i \(-0.734523\pi\)
0.977364 + 0.211567i \(0.0678565\pi\)
\(32\) 0 0
\(33\) 0.262596 0.454829i 0.0457120 0.0791756i
\(34\) 0 0
\(35\) 0.775137 1.34258i 0.131022 0.226937i
\(36\) 0 0
\(37\) −0.396101 0.686067i −0.0651186 0.112789i 0.831628 0.555333i \(-0.187409\pi\)
−0.896747 + 0.442544i \(0.854076\pi\)
\(38\) 0 0
\(39\) 0.0526231 + 0.0911458i 0.00842643 + 0.0145950i
\(40\) 0 0
\(41\) 2.52857 4.37962i 0.394897 0.683982i −0.598191 0.801353i \(-0.704114\pi\)
0.993088 + 0.117372i \(0.0374470\pi\)
\(42\) 0 0
\(43\) 4.99327 0.761466 0.380733 0.924685i \(-0.375672\pi\)
0.380733 + 0.924685i \(0.375672\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0.703919 1.21922i 0.102677 0.177842i −0.810110 0.586278i \(-0.800592\pi\)
0.912787 + 0.408436i \(0.133926\pi\)
\(48\) 0 0
\(49\) 2.29832 + 3.98082i 0.328332 + 0.568688i
\(50\) 0 0
\(51\) −1.53226 2.65395i −0.214559 0.371627i
\(52\) 0 0
\(53\) −6.53926 −0.898236 −0.449118 0.893472i \(-0.648262\pi\)
−0.449118 + 0.893472i \(0.648262\pi\)
\(54\) 0 0
\(55\) 0.262596 0.454829i 0.0354084 0.0613291i
\(56\) 0 0
\(57\) 1.60844 + 2.78590i 0.213043 + 0.369002i
\(58\) 0 0
\(59\) 3.21916 0.419099 0.209550 0.977798i \(-0.432800\pi\)
0.209550 + 0.977798i \(0.432800\pi\)
\(60\) 0 0
\(61\) 2.81190 + 4.87035i 0.360027 + 0.623584i 0.987965 0.154679i \(-0.0494343\pi\)
−0.627938 + 0.778263i \(0.716101\pi\)
\(62\) 0 0
\(63\) −0.775137 + 1.34258i −0.0976581 + 0.169149i
\(64\) 0 0
\(65\) 0.0526231 + 0.0911458i 0.00652709 + 0.0113052i
\(66\) 0 0
\(67\) −7.78180 + 2.53843i −0.950698 + 0.310118i
\(68\) 0 0
\(69\) 4.09911 + 7.09986i 0.493474 + 0.854723i
\(70\) 0 0
\(71\) 2.92291 5.06262i 0.346885 0.600823i −0.638809 0.769365i \(-0.720573\pi\)
0.985694 + 0.168542i \(0.0539060\pi\)
\(72\) 0 0
\(73\) −4.11127 7.12092i −0.481187 0.833441i 0.518580 0.855029i \(-0.326461\pi\)
−0.999767 + 0.0215885i \(0.993128\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −0.407095 0.705110i −0.0463928 0.0803547i
\(78\) 0 0
\(79\) −0.910790 + 1.57753i −0.102472 + 0.177486i −0.912702 0.408625i \(-0.866008\pi\)
0.810231 + 0.586111i \(0.199342\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.71382 + 15.0928i 0.956466 + 1.65665i 0.730977 + 0.682402i \(0.239064\pi\)
0.225488 + 0.974246i \(0.427602\pi\)
\(84\) 0 0
\(85\) −1.53226 2.65395i −0.166197 0.287861i
\(86\) 0 0
\(87\) 3.20300 5.54777i 0.343398 0.594783i
\(88\) 0 0
\(89\) −8.71903 −0.924215 −0.462108 0.886824i \(-0.652907\pi\)
−0.462108 + 0.886824i \(0.652907\pi\)
\(90\) 0 0
\(91\) 0.163160 0.0171039
\(92\) 0 0
\(93\) −1.70073 + 2.94575i −0.176357 + 0.305460i
\(94\) 0 0
\(95\) 1.60844 + 2.78590i 0.165023 + 0.285827i
\(96\) 0 0
\(97\) −1.03739 1.79682i −0.105331 0.182439i 0.808542 0.588438i \(-0.200257\pi\)
−0.913873 + 0.405999i \(0.866924\pi\)
\(98\) 0 0
\(99\) −0.262596 + 0.454829i −0.0263919 + 0.0457120i
\(100\) 0 0
\(101\) 1.31323 2.27457i 0.130671 0.226328i −0.793265 0.608877i \(-0.791620\pi\)
0.923935 + 0.382549i \(0.124954\pi\)
\(102\) 0 0
\(103\) 3.17348 5.49663i 0.312692 0.541599i −0.666252 0.745727i \(-0.732102\pi\)
0.978944 + 0.204128i \(0.0654358\pi\)
\(104\) 0 0
\(105\) −0.775137 + 1.34258i −0.0756456 + 0.131022i
\(106\) 0 0
\(107\) 10.3184 0.997521 0.498761 0.866740i \(-0.333789\pi\)
0.498761 + 0.866740i \(0.333789\pi\)
\(108\) 0 0
\(109\) 6.42905 0.615792 0.307896 0.951420i \(-0.400375\pi\)
0.307896 + 0.951420i \(0.400375\pi\)
\(110\) 0 0
\(111\) 0.396101 + 0.686067i 0.0375962 + 0.0651186i
\(112\) 0 0
\(113\) −1.66174 + 2.87821i −0.156323 + 0.270759i −0.933540 0.358473i \(-0.883297\pi\)
0.777217 + 0.629233i \(0.216631\pi\)
\(114\) 0 0
\(115\) 4.09911 + 7.09986i 0.382244 + 0.662065i
\(116\) 0 0
\(117\) −0.0526231 0.0911458i −0.00486500 0.00842643i
\(118\) 0 0
\(119\) −4.75084 −0.435508
\(120\) 0 0
\(121\) 5.36209 + 9.28741i 0.487462 + 0.844310i
\(122\) 0 0
\(123\) −2.52857 + 4.37962i −0.227994 + 0.394897i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.8863 18.8556i 0.966001 1.67316i 0.259101 0.965850i \(-0.416574\pi\)
0.706900 0.707313i \(-0.250093\pi\)
\(128\) 0 0
\(129\) −4.99327 −0.439633
\(130\) 0 0
\(131\) 14.8140 1.29431 0.647155 0.762359i \(-0.275959\pi\)
0.647155 + 0.762359i \(0.275959\pi\)
\(132\) 0 0
\(133\) 4.98705 0.432432
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −12.0569 −1.03009 −0.515044 0.857163i \(-0.672225\pi\)
−0.515044 + 0.857163i \(0.672225\pi\)
\(138\) 0 0
\(139\) 22.9507 1.94666 0.973328 0.229418i \(-0.0736821\pi\)
0.973328 + 0.229418i \(0.0736821\pi\)
\(140\) 0 0
\(141\) −0.703919 + 1.21922i −0.0592807 + 0.102677i
\(142\) 0 0
\(143\) 0.0552744 0.00462227
\(144\) 0 0
\(145\) 3.20300 5.54777i 0.265995 0.460717i
\(146\) 0 0
\(147\) −2.29832 3.98082i −0.189563 0.328332i
\(148\) 0 0
\(149\) 4.20473 0.344465 0.172232 0.985056i \(-0.444902\pi\)
0.172232 + 0.985056i \(0.444902\pi\)
\(150\) 0 0
\(151\) 1.97272 + 3.41684i 0.160537 + 0.278059i 0.935062 0.354485i \(-0.115344\pi\)
−0.774524 + 0.632544i \(0.782011\pi\)
\(152\) 0 0
\(153\) 1.53226 + 2.65395i 0.123876 + 0.214559i
\(154\) 0 0
\(155\) −1.70073 + 2.94575i −0.136606 + 0.236608i
\(156\) 0 0
\(157\) 6.28433 + 10.8848i 0.501544 + 0.868700i 0.999998 + 0.00178402i \(0.000567871\pi\)
−0.498454 + 0.866916i \(0.666099\pi\)
\(158\) 0 0
\(159\) 6.53926 0.518597
\(160\) 0 0
\(161\) 12.7095 1.00165
\(162\) 0 0
\(163\) 10.2750 17.7969i 0.804802 1.39396i −0.111623 0.993751i \(-0.535605\pi\)
0.916425 0.400207i \(-0.131062\pi\)
\(164\) 0 0
\(165\) −0.262596 + 0.454829i −0.0204430 + 0.0354084i
\(166\) 0 0
\(167\) −10.3299 + 17.8919i −0.799353 + 1.38452i 0.120685 + 0.992691i \(0.461491\pi\)
−0.920038 + 0.391830i \(0.871842\pi\)
\(168\) 0 0
\(169\) 6.49446 11.2487i 0.499574 0.865288i
\(170\) 0 0
\(171\) −1.60844 2.78590i −0.123001 0.213043i
\(172\) 0 0
\(173\) −0.847605 1.46810i −0.0644422 0.111617i 0.832004 0.554769i \(-0.187193\pi\)
−0.896446 + 0.443152i \(0.853860\pi\)
\(174\) 0 0
\(175\) −0.775137 + 1.34258i −0.0585949 + 0.101489i
\(176\) 0 0
\(177\) −3.21916 −0.241967
\(178\) 0 0
\(179\) 16.2705 1.21612 0.608058 0.793893i \(-0.291949\pi\)
0.608058 + 0.793893i \(0.291949\pi\)
\(180\) 0 0
\(181\) 6.53860 11.3252i 0.486010 0.841794i −0.513861 0.857874i \(-0.671785\pi\)
0.999871 + 0.0160795i \(0.00511848\pi\)
\(182\) 0 0
\(183\) −2.81190 4.87035i −0.207861 0.360027i
\(184\) 0 0
\(185\) 0.396101 + 0.686067i 0.0291219 + 0.0504407i
\(186\) 0 0
\(187\) −1.60946 −0.117695
\(188\) 0 0
\(189\) 0.775137 1.34258i 0.0563829 0.0976581i
\(190\) 0 0
\(191\) −7.38160 12.7853i −0.534114 0.925112i −0.999206 0.0398496i \(-0.987312\pi\)
0.465092 0.885262i \(-0.346021\pi\)
\(192\) 0 0
\(193\) −0.212140 −0.0152702 −0.00763509 0.999971i \(-0.502430\pi\)
−0.00763509 + 0.999971i \(0.502430\pi\)
\(194\) 0 0
\(195\) −0.0526231 0.0911458i −0.00376842 0.00652709i
\(196\) 0 0
\(197\) 12.3422 21.3774i 0.879348 1.52307i 0.0272897 0.999628i \(-0.491312\pi\)
0.852058 0.523447i \(-0.175354\pi\)
\(198\) 0 0
\(199\) 5.70515 + 9.88161i 0.404427 + 0.700489i 0.994255 0.107041i \(-0.0341375\pi\)
−0.589827 + 0.807529i \(0.700804\pi\)
\(200\) 0 0
\(201\) 7.78180 2.53843i 0.548886 0.179047i
\(202\) 0 0
\(203\) −4.96554 8.60056i −0.348512 0.603641i
\(204\) 0 0
\(205\) −2.52857 + 4.37962i −0.176603 + 0.305886i
\(206\) 0 0
\(207\) −4.09911 7.09986i −0.284908 0.493474i
\(208\) 0 0
\(209\) 1.68948 0.116864
\(210\) 0 0
\(211\) 11.2335 + 19.4570i 0.773346 + 1.33947i 0.935720 + 0.352745i \(0.114752\pi\)
−0.162374 + 0.986729i \(0.551915\pi\)
\(212\) 0 0
\(213\) −2.92291 + 5.06262i −0.200274 + 0.346885i
\(214\) 0 0
\(215\) −4.99327 −0.340538
\(216\) 0 0
\(217\) 2.63659 + 4.56672i 0.178984 + 0.310009i
\(218\) 0 0
\(219\) 4.11127 + 7.12092i 0.277814 + 0.481187i
\(220\) 0 0
\(221\) 0.161264 0.279318i 0.0108478 0.0187889i
\(222\) 0 0
\(223\) 1.35718 0.0908833 0.0454416 0.998967i \(-0.485530\pi\)
0.0454416 + 0.998967i \(0.485530\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.01431 6.95299i 0.266439 0.461486i −0.701500 0.712669i \(-0.747486\pi\)
0.967940 + 0.251183i \(0.0808195\pi\)
\(228\) 0 0
\(229\) −3.22002 5.57724i −0.212785 0.368555i 0.739800 0.672827i \(-0.234920\pi\)
−0.952585 + 0.304272i \(0.901587\pi\)
\(230\) 0 0
\(231\) 0.407095 + 0.705110i 0.0267849 + 0.0463928i
\(232\) 0 0
\(233\) −13.8495 + 23.9880i −0.907309 + 1.57151i −0.0895227 + 0.995985i \(0.528534\pi\)
−0.817787 + 0.575521i \(0.804799\pi\)
\(234\) 0 0
\(235\) −0.703919 + 1.21922i −0.0459186 + 0.0795334i
\(236\) 0 0
\(237\) 0.910790 1.57753i 0.0591621 0.102472i
\(238\) 0 0
\(239\) 8.63210 14.9512i 0.558364 0.967115i −0.439269 0.898355i \(-0.644763\pi\)
0.997633 0.0687593i \(-0.0219040\pi\)
\(240\) 0 0
\(241\) 10.2489 0.660192 0.330096 0.943947i \(-0.392919\pi\)
0.330096 + 0.943947i \(0.392919\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.29832 3.98082i −0.146835 0.254325i
\(246\) 0 0
\(247\) −0.169282 + 0.293205i −0.0107712 + 0.0186562i
\(248\) 0 0
\(249\) −8.71382 15.0928i −0.552216 0.956466i
\(250\) 0 0
\(251\) −3.09480 5.36035i −0.195342 0.338342i 0.751671 0.659539i \(-0.229248\pi\)
−0.947013 + 0.321197i \(0.895915\pi\)
\(252\) 0 0
\(253\) 4.30563 0.270693
\(254\) 0 0
\(255\) 1.53226 + 2.65395i 0.0959536 + 0.166197i
\(256\) 0 0
\(257\) −7.92981 + 13.7348i −0.494648 + 0.856755i −0.999981 0.00616928i \(-0.998036\pi\)
0.505333 + 0.862924i \(0.331370\pi\)
\(258\) 0 0
\(259\) 1.22813 0.0763123
\(260\) 0 0
\(261\) −3.20300 + 5.54777i −0.198261 + 0.343398i
\(262\) 0 0
\(263\) 18.7400 1.15556 0.577780 0.816193i \(-0.303919\pi\)
0.577780 + 0.816193i \(0.303919\pi\)
\(264\) 0 0
\(265\) 6.53926 0.401703
\(266\) 0 0
\(267\) 8.71903 0.533596
\(268\) 0 0
\(269\) −21.6742 −1.32150 −0.660750 0.750606i \(-0.729762\pi\)
−0.660750 + 0.750606i \(0.729762\pi\)
\(270\) 0 0
\(271\) 30.1382 1.83077 0.915384 0.402582i \(-0.131887\pi\)
0.915384 + 0.402582i \(0.131887\pi\)
\(272\) 0 0
\(273\) −0.163160 −0.00987491
\(274\) 0 0
\(275\) −0.262596 + 0.454829i −0.0158351 + 0.0274272i
\(276\) 0 0
\(277\) 19.8633 1.19347 0.596734 0.802439i \(-0.296465\pi\)
0.596734 + 0.802439i \(0.296465\pi\)
\(278\) 0 0
\(279\) 1.70073 2.94575i 0.101820 0.176357i
\(280\) 0 0
\(281\) 16.1159 + 27.9135i 0.961393 + 1.66518i 0.719009 + 0.695001i \(0.244596\pi\)
0.242384 + 0.970180i \(0.422071\pi\)
\(282\) 0 0
\(283\) −30.5884 −1.81829 −0.909147 0.416476i \(-0.863265\pi\)
−0.909147 + 0.416476i \(0.863265\pi\)
\(284\) 0 0
\(285\) −1.60844 2.78590i −0.0952758 0.165023i
\(286\) 0 0
\(287\) 3.91998 + 6.78961i 0.231389 + 0.400778i
\(288\) 0 0
\(289\) 3.80438 6.58938i 0.223787 0.387610i
\(290\) 0 0
\(291\) 1.03739 + 1.79682i 0.0608130 + 0.105331i
\(292\) 0 0
\(293\) 2.17227 0.126905 0.0634527 0.997985i \(-0.479789\pi\)
0.0634527 + 0.997985i \(0.479789\pi\)
\(294\) 0 0
\(295\) −3.21916 −0.187427
\(296\) 0 0
\(297\) 0.262596 0.454829i 0.0152373 0.0263919i
\(298\) 0 0
\(299\) −0.431415 + 0.747233i −0.0249494 + 0.0432136i
\(300\) 0 0
\(301\) −3.87047 + 6.70385i −0.223090 + 0.386403i
\(302\) 0 0
\(303\) −1.31323 + 2.27457i −0.0754428 + 0.130671i
\(304\) 0 0
\(305\) −2.81190 4.87035i −0.161009 0.278875i
\(306\) 0 0
\(307\) 4.16325 + 7.21095i 0.237609 + 0.411551i 0.960028 0.279905i \(-0.0903030\pi\)
−0.722419 + 0.691456i \(0.756970\pi\)
\(308\) 0 0
\(309\) −3.17348 + 5.49663i −0.180533 + 0.312692i
\(310\) 0 0
\(311\) −10.6902 −0.606187 −0.303094 0.952961i \(-0.598019\pi\)
−0.303094 + 0.952961i \(0.598019\pi\)
\(312\) 0 0
\(313\) 20.7113 1.17067 0.585335 0.810791i \(-0.300963\pi\)
0.585335 + 0.810791i \(0.300963\pi\)
\(314\) 0 0
\(315\) 0.775137 1.34258i 0.0436740 0.0756456i
\(316\) 0 0
\(317\) 10.5331 + 18.2438i 0.591596 + 1.02467i 0.994018 + 0.109220i \(0.0348354\pi\)
−0.402421 + 0.915455i \(0.631831\pi\)
\(318\) 0 0
\(319\) −1.68219 2.91364i −0.0941846 0.163132i
\(320\) 0 0
\(321\) −10.3184 −0.575919
\(322\) 0 0
\(323\) 4.92909 8.53743i 0.274262 0.475035i
\(324\) 0 0
\(325\) −0.0526231 0.0911458i −0.00291900 0.00505586i
\(326\) 0 0
\(327\) −6.42905 −0.355527
\(328\) 0 0
\(329\) 1.09127 + 1.89013i 0.0601635 + 0.104206i
\(330\) 0 0
\(331\) 16.6168 28.7812i 0.913344 1.58196i 0.104036 0.994574i \(-0.466824\pi\)
0.809308 0.587385i \(-0.199842\pi\)
\(332\) 0 0
\(333\) −0.396101 0.686067i −0.0217062 0.0375962i
\(334\) 0 0
\(335\) 7.78180 2.53843i 0.425165 0.138689i
\(336\) 0 0
\(337\) 0.722325 + 1.25110i 0.0393476 + 0.0681520i 0.885029 0.465537i \(-0.154139\pi\)
−0.845681 + 0.533689i \(0.820805\pi\)
\(338\) 0 0
\(339\) 1.66174 2.87821i 0.0902531 0.156323i
\(340\) 0 0
\(341\) 0.893208 + 1.54708i 0.0483699 + 0.0837791i
\(342\) 0 0
\(343\) −17.9780 −0.970720
\(344\) 0 0
\(345\) −4.09911 7.09986i −0.220688 0.382244i
\(346\) 0 0
\(347\) 10.9030 18.8845i 0.585301 1.01377i −0.409536 0.912294i \(-0.634310\pi\)
0.994838 0.101478i \(-0.0323571\pi\)
\(348\) 0 0
\(349\) 31.3545 1.67837 0.839184 0.543847i \(-0.183033\pi\)
0.839184 + 0.543847i \(0.183033\pi\)
\(350\) 0 0
\(351\) 0.0526231 + 0.0911458i 0.00280881 + 0.00486500i
\(352\) 0 0
\(353\) −15.3212 26.5372i −0.815467 1.41243i −0.908992 0.416814i \(-0.863147\pi\)
0.0935249 0.995617i \(-0.470187\pi\)
\(354\) 0 0
\(355\) −2.92291 + 5.06262i −0.155132 + 0.268696i
\(356\) 0 0
\(357\) 4.75084 0.251441
\(358\) 0 0
\(359\) −35.5977 −1.87877 −0.939387 0.342860i \(-0.888604\pi\)
−0.939387 + 0.342860i \(0.888604\pi\)
\(360\) 0 0
\(361\) 4.32583 7.49257i 0.227675 0.394346i
\(362\) 0 0
\(363\) −5.36209 9.28741i −0.281437 0.487462i
\(364\) 0 0
\(365\) 4.11127 + 7.12092i 0.215194 + 0.372726i
\(366\) 0 0
\(367\) 1.77266 3.07033i 0.0925319 0.160270i −0.816044 0.577990i \(-0.803837\pi\)
0.908576 + 0.417720i \(0.137171\pi\)
\(368\) 0 0
\(369\) 2.52857 4.37962i 0.131632 0.227994i
\(370\) 0 0
\(371\) 5.06882 8.77945i 0.263160 0.455807i
\(372\) 0 0
\(373\) −16.5747 + 28.7082i −0.858204 + 1.48645i 0.0154362 + 0.999881i \(0.495086\pi\)
−0.873640 + 0.486572i \(0.838247\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0.674208 0.0347235
\(378\) 0 0
\(379\) 5.35411 + 9.27360i 0.275022 + 0.476353i 0.970141 0.242543i \(-0.0779814\pi\)
−0.695118 + 0.718895i \(0.744648\pi\)
\(380\) 0 0
\(381\) −10.8863 + 18.8556i −0.557721 + 0.966001i
\(382\) 0 0
\(383\) −12.3745 21.4332i −0.632307 1.09519i −0.987079 0.160235i \(-0.948775\pi\)
0.354772 0.934953i \(-0.384559\pi\)
\(384\) 0 0
\(385\) 0.407095 + 0.705110i 0.0207475 + 0.0359357i
\(386\) 0 0
\(387\) 4.99327 0.253822
\(388\) 0 0
\(389\) 10.2179 + 17.6979i 0.518068 + 0.897319i 0.999780 + 0.0209898i \(0.00668174\pi\)
−0.481712 + 0.876329i \(0.659985\pi\)
\(390\) 0 0
\(391\) 12.5618 21.7576i 0.635276 1.10033i
\(392\) 0 0
\(393\) −14.8140 −0.747270
\(394\) 0 0
\(395\) 0.910790 1.57753i 0.0458268 0.0793743i
\(396\) 0 0
\(397\) 15.3340 0.769592 0.384796 0.923002i \(-0.374272\pi\)
0.384796 + 0.923002i \(0.374272\pi\)
\(398\) 0 0
\(399\) −4.98705 −0.249665
\(400\) 0 0
\(401\) 26.8598 1.34132 0.670658 0.741767i \(-0.266012\pi\)
0.670658 + 0.741767i \(0.266012\pi\)
\(402\) 0 0
\(403\) −0.357990 −0.0178328
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0.416058 0.0206232
\(408\) 0 0
\(409\) 2.48655 4.30683i 0.122952 0.212959i −0.797979 0.602686i \(-0.794097\pi\)
0.920931 + 0.389727i \(0.127431\pi\)
\(410\) 0 0
\(411\) 12.0569 0.594722
\(412\) 0 0
\(413\) −2.49529 + 4.32197i −0.122785 + 0.212670i
\(414\) 0 0
\(415\) −8.71382 15.0928i −0.427745 0.740875i
\(416\) 0 0
\(417\) −22.9507 −1.12390
\(418\) 0 0
\(419\) −17.0807 29.5846i −0.834447 1.44530i −0.894480 0.447108i \(-0.852454\pi\)
0.0600336 0.998196i \(-0.480879\pi\)
\(420\) 0 0
\(421\) −4.93960 8.55564i −0.240741 0.416976i 0.720184 0.693783i \(-0.244057\pi\)
−0.960926 + 0.276806i \(0.910724\pi\)
\(422\) 0 0
\(423\) 0.703919 1.21922i 0.0342257 0.0592807i
\(424\) 0 0
\(425\) 1.53226 + 2.65395i 0.0743254 + 0.128735i
\(426\) 0 0
\(427\) −8.71843 −0.421914
\(428\) 0 0
\(429\) −0.0552744 −0.00266867
\(430\) 0 0
\(431\) 3.23898 5.61007i 0.156016 0.270228i −0.777413 0.628991i \(-0.783468\pi\)
0.933429 + 0.358763i \(0.116802\pi\)
\(432\) 0 0
\(433\) −15.1428 + 26.2281i −0.727716 + 1.26044i 0.230130 + 0.973160i \(0.426085\pi\)
−0.957846 + 0.287282i \(0.907248\pi\)
\(434\) 0 0
\(435\) −3.20300 + 5.54777i −0.153572 + 0.265995i
\(436\) 0 0
\(437\) −13.1863 + 22.8394i −0.630788 + 1.09256i
\(438\) 0 0
\(439\) 7.74874 + 13.4212i 0.369827 + 0.640560i 0.989538 0.144270i \(-0.0460835\pi\)
−0.619711 + 0.784830i \(0.712750\pi\)
\(440\) 0 0
\(441\) 2.29832 + 3.98082i 0.109444 + 0.189563i
\(442\) 0 0
\(443\) −0.827526 + 1.43332i −0.0393169 + 0.0680989i −0.885014 0.465564i \(-0.845852\pi\)
0.845697 + 0.533663i \(0.179185\pi\)
\(444\) 0 0
\(445\) 8.71903 0.413322
\(446\) 0 0
\(447\) −4.20473 −0.198877
\(448\) 0 0
\(449\) −1.79139 + 3.10278i −0.0845410 + 0.146429i −0.905196 0.424995i \(-0.860276\pi\)
0.820655 + 0.571425i \(0.193609\pi\)
\(450\) 0 0
\(451\) 1.32799 + 2.30014i 0.0625324 + 0.108309i
\(452\) 0 0
\(453\) −1.97272 3.41684i −0.0926863 0.160537i
\(454\) 0 0
\(455\) −0.163160 −0.00764908
\(456\) 0 0
\(457\) 4.72189 8.17855i 0.220881 0.382577i −0.734195 0.678939i \(-0.762440\pi\)
0.955076 + 0.296362i \(0.0957735\pi\)
\(458\) 0 0
\(459\) −1.53226 2.65395i −0.0715196 0.123876i
\(460\) 0 0
\(461\) 9.43538 0.439450 0.219725 0.975562i \(-0.429484\pi\)
0.219725 + 0.975562i \(0.429484\pi\)
\(462\) 0 0
\(463\) 4.77207 + 8.26546i 0.221777 + 0.384129i 0.955348 0.295484i \(-0.0954811\pi\)
−0.733571 + 0.679613i \(0.762148\pi\)
\(464\) 0 0
\(465\) 1.70073 2.94575i 0.0788694 0.136606i
\(466\) 0 0
\(467\) −2.02364 3.50505i −0.0936430 0.162194i 0.815398 0.578900i \(-0.196518\pi\)
−0.909041 + 0.416706i \(0.863185\pi\)
\(468\) 0 0
\(469\) 2.62393 12.4153i 0.121162 0.573285i
\(470\) 0 0
\(471\) −6.28433 10.8848i −0.289567 0.501544i
\(472\) 0 0
\(473\) −1.31121 + 2.27108i −0.0602895 + 0.104425i
\(474\) 0 0
\(475\) −1.60844 2.78590i −0.0738003 0.127826i
\(476\) 0 0
\(477\) −6.53926 −0.299412
\(478\) 0 0
\(479\) −11.3259 19.6170i −0.517494 0.896325i −0.999794 0.0203189i \(-0.993532\pi\)
0.482300 0.876006i \(-0.339802\pi\)
\(480\) 0 0
\(481\) −0.0416881 + 0.0722059i −0.00190081 + 0.00329231i
\(482\) 0 0
\(483\) −12.7095 −0.578301
\(484\) 0 0
\(485\) 1.03739 + 1.79682i 0.0471055 + 0.0815892i
\(486\) 0 0
\(487\) 1.43999 + 2.49413i 0.0652520 + 0.113020i 0.896806 0.442425i \(-0.145882\pi\)
−0.831554 + 0.555444i \(0.812548\pi\)
\(488\) 0 0
\(489\) −10.2750 + 17.7969i −0.464653 + 0.804802i
\(490\) 0 0
\(491\) 13.6498 0.616008 0.308004 0.951385i \(-0.400339\pi\)
0.308004 + 0.951385i \(0.400339\pi\)
\(492\) 0 0
\(493\) −19.6313 −0.884149
\(494\) 0 0
\(495\) 0.262596 0.454829i 0.0118028 0.0204430i
\(496\) 0 0
\(497\) 4.53131 + 7.84846i 0.203257 + 0.352051i
\(498\) 0 0
\(499\) 2.10697 + 3.64938i 0.0943209 + 0.163369i 0.909325 0.416087i \(-0.136599\pi\)
−0.815004 + 0.579455i \(0.803265\pi\)
\(500\) 0 0
\(501\) 10.3299 17.8919i 0.461507 0.799353i
\(502\) 0 0
\(503\) −1.69612 + 2.93776i −0.0756262 + 0.130988i −0.901358 0.433074i \(-0.857429\pi\)
0.825732 + 0.564062i \(0.190762\pi\)
\(504\) 0 0
\(505\) −1.31323 + 2.27457i −0.0584378 + 0.101217i
\(506\) 0 0
\(507\) −6.49446 + 11.2487i −0.288429 + 0.499574i
\(508\) 0 0
\(509\) −14.0321 −0.621963 −0.310982 0.950416i \(-0.600658\pi\)
−0.310982 + 0.950416i \(0.600658\pi\)
\(510\) 0 0
\(511\) 12.7472 0.563902
\(512\) 0 0
\(513\) 1.60844 + 2.78590i 0.0710144 + 0.123001i
\(514\) 0 0
\(515\) −3.17348 + 5.49663i −0.139840 + 0.242210i
\(516\) 0 0
\(517\) 0.369692 + 0.640326i 0.0162590 + 0.0281615i
\(518\) 0 0
\(519\) 0.847605 + 1.46810i 0.0372057 + 0.0644422i
\(520\) 0 0
\(521\) 11.0180 0.482709 0.241355 0.970437i \(-0.422408\pi\)
0.241355 + 0.970437i \(0.422408\pi\)
\(522\) 0 0
\(523\) 13.0711 + 22.6398i 0.571559 + 0.989970i 0.996406 + 0.0847044i \(0.0269946\pi\)
−0.424847 + 0.905265i \(0.639672\pi\)
\(524\) 0 0
\(525\) 0.775137 1.34258i 0.0338298 0.0585949i
\(526\) 0 0
\(527\) 10.4238 0.454068
\(528\) 0 0
\(529\) −22.1053 + 38.2876i −0.961102 + 1.66468i
\(530\) 0 0
\(531\) 3.21916 0.139700
\(532\) 0 0
\(533\) −0.532245 −0.0230541
\(534\) 0 0
\(535\) −10.3184 −0.446105
\(536\) 0 0
\(537\) −16.2705 −0.702125
\(538\) 0 0
\(539\) −2.41412 −0.103984
\(540\) 0 0
\(541\) 22.0989 0.950105 0.475052 0.879957i \(-0.342429\pi\)
0.475052 + 0.879957i \(0.342429\pi\)
\(542\) 0 0
\(543\) −6.53860 + 11.3252i −0.280598 + 0.486010i
\(544\) 0 0
\(545\) −6.42905 −0.275390
\(546\) 0 0
\(547\) −15.6090 + 27.0356i −0.667393 + 1.15596i 0.311238 + 0.950332i \(0.399257\pi\)
−0.978631 + 0.205626i \(0.934077\pi\)
\(548\) 0 0
\(549\) 2.81190 + 4.87035i 0.120009 + 0.207861i
\(550\) 0 0
\(551\) 20.6074 0.877904
\(552\) 0 0
\(553\) −1.41197 2.44561i −0.0600432 0.103998i
\(554\) 0 0
\(555\) −0.396101 0.686067i −0.0168136 0.0291219i
\(556\) 0 0
\(557\) 4.40764 7.63426i 0.186758 0.323474i −0.757410 0.652940i \(-0.773535\pi\)
0.944167 + 0.329466i \(0.106869\pi\)
\(558\) 0 0
\(559\) −0.262761 0.455116i −0.0111136 0.0192493i
\(560\) 0 0
\(561\) 1.60946 0.0679513
\(562\) 0 0
\(563\) −26.4203 −1.11348 −0.556741 0.830686i \(-0.687948\pi\)
−0.556741 + 0.830686i \(0.687948\pi\)
\(564\) 0 0
\(565\) 1.66174 2.87821i 0.0699098 0.121087i
\(566\) 0 0
\(567\) −0.775137 + 1.34258i −0.0325527 + 0.0563829i
\(568\) 0 0
\(569\) −22.2680 + 38.5694i −0.933524 + 1.61691i −0.156280 + 0.987713i \(0.549950\pi\)
−0.777245 + 0.629199i \(0.783383\pi\)
\(570\) 0 0
\(571\) −7.98314 + 13.8272i −0.334084 + 0.578651i −0.983308 0.181946i \(-0.941760\pi\)
0.649224 + 0.760597i \(0.275094\pi\)
\(572\) 0 0
\(573\) 7.38160 + 12.7853i 0.308371 + 0.534114i
\(574\) 0 0
\(575\) −4.09911 7.09986i −0.170945 0.296085i
\(576\) 0 0
\(577\) 2.46657 4.27223i 0.102685 0.177855i −0.810105 0.586285i \(-0.800590\pi\)
0.912790 + 0.408429i \(0.133923\pi\)
\(578\) 0 0
\(579\) 0.212140 0.00881624
\(580\) 0 0
\(581\) −27.0176 −1.12088
\(582\) 0 0
\(583\) 1.71718 2.97424i 0.0711183 0.123181i
\(584\) 0 0
\(585\) 0.0526231 + 0.0911458i 0.00217570 + 0.00376842i
\(586\) 0 0
\(587\) 10.9315 + 18.9338i 0.451189 + 0.781483i 0.998460 0.0554725i \(-0.0176665\pi\)
−0.547271 + 0.836956i \(0.684333\pi\)
\(588\) 0 0
\(589\) −10.9421 −0.450861
\(590\) 0 0
\(591\) −12.3422 + 21.3774i −0.507692 + 0.879348i
\(592\) 0 0
\(593\) −5.47607 9.48483i −0.224875 0.389495i 0.731407 0.681941i \(-0.238864\pi\)
−0.956282 + 0.292446i \(0.905531\pi\)
\(594\) 0 0
\(595\) 4.75084 0.194765
\(596\) 0 0
\(597\) −5.70515 9.88161i −0.233496 0.404427i
\(598\) 0 0
\(599\) −10.9642 + 18.9906i −0.447985 + 0.775933i −0.998255 0.0590539i \(-0.981192\pi\)
0.550270 + 0.834987i \(0.314525\pi\)
\(600\) 0 0
\(601\) −15.1117 26.1742i −0.616417 1.06767i −0.990134 0.140123i \(-0.955250\pi\)
0.373717 0.927543i \(-0.378083\pi\)
\(602\) 0 0
\(603\) −7.78180 + 2.53843i −0.316899 + 0.103373i
\(604\) 0 0
\(605\) −5.36209 9.28741i −0.218000 0.377587i
\(606\) 0 0
\(607\) 22.5252 39.0148i 0.914271 1.58356i 0.106305 0.994334i \(-0.466098\pi\)
0.807966 0.589229i \(-0.200569\pi\)
\(608\) 0 0
\(609\) 4.96554 + 8.60056i 0.201214 + 0.348512i
\(610\) 0 0
\(611\) −0.148169 −0.00599430
\(612\) 0 0
\(613\) −7.17046 12.4196i −0.289612 0.501623i 0.684105 0.729384i \(-0.260193\pi\)
−0.973717 + 0.227760i \(0.926860\pi\)
\(614\) 0 0
\(615\) 2.52857 4.37962i 0.101962 0.176603i
\(616\) 0 0
\(617\) −43.9933 −1.77110 −0.885551 0.464542i \(-0.846219\pi\)
−0.885551 + 0.464542i \(0.846219\pi\)
\(618\) 0 0
\(619\) 4.82126 + 8.35066i 0.193783 + 0.335641i 0.946501 0.322702i \(-0.104591\pi\)
−0.752718 + 0.658343i \(0.771258\pi\)
\(620\) 0 0
\(621\) 4.09911 + 7.09986i 0.164491 + 0.284908i
\(622\) 0 0
\(623\) 6.75844 11.7060i 0.270771 0.468990i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.68948 −0.0674713
\(628\) 0 0
\(629\) 1.21386 2.10246i 0.0483996 0.0838306i
\(630\) 0 0
\(631\) −3.47163 6.01304i −0.138203 0.239375i 0.788613 0.614890i \(-0.210799\pi\)
−0.926817 + 0.375514i \(0.877466\pi\)
\(632\) 0 0
\(633\) −11.2335 19.4570i −0.446491 0.773346i
\(634\) 0 0
\(635\) −10.8863 + 18.8556i −0.432009 + 0.748261i
\(636\) 0 0
\(637\) 0.241890 0.418965i 0.00958402 0.0166000i
\(638\) 0 0
\(639\) 2.92291 5.06262i 0.115628 0.200274i
\(640\) 0 0
\(641\) −12.5610 + 21.7563i −0.496131 + 0.859324i −0.999990 0.00446177i \(-0.998580\pi\)
0.503859 + 0.863786i \(0.331913\pi\)
\(642\) 0 0
\(643\) 5.90010 0.232677 0.116339 0.993210i \(-0.462884\pi\)
0.116339 + 0.993210i \(0.462884\pi\)
\(644\) 0 0
\(645\) 4.99327 0.196610
\(646\) 0 0
\(647\) −18.9350 32.7964i −0.744413 1.28936i −0.950468 0.310821i \(-0.899396\pi\)
0.206055 0.978540i \(-0.433937\pi\)
\(648\) 0 0
\(649\) −0.845338 + 1.46417i −0.0331824 + 0.0574736i
\(650\) 0 0
\(651\) −2.63659 4.56672i −0.103336 0.178984i
\(652\) 0 0
\(653\) 7.30803 + 12.6579i 0.285985 + 0.495341i 0.972848 0.231446i \(-0.0743458\pi\)
−0.686862 + 0.726788i \(0.741012\pi\)
\(654\) 0 0
\(655\) −14.8140 −0.578833
\(656\) 0 0
\(657\) −4.11127 7.12092i −0.160396 0.277814i
\(658\) 0 0
\(659\) 12.9618 22.4506i 0.504922 0.874550i −0.495062 0.868858i \(-0.664855\pi\)
0.999984 0.00569253i \(-0.00181200\pi\)
\(660\) 0 0
\(661\) 26.1006 1.01519 0.507597 0.861595i \(-0.330534\pi\)
0.507597 + 0.861595i \(0.330534\pi\)
\(662\) 0 0
\(663\) −0.161264 + 0.279318i −0.00626298 + 0.0108478i
\(664\) 0 0
\(665\) −4.98705 −0.193389
\(666\) 0 0
\(667\) 52.5178 2.03350
\(668\) 0 0
\(669\) −1.35718 −0.0524715
\(670\) 0 0
\(671\) −2.95357 −0.114021
\(672\) 0 0
\(673\) −5.13781 −0.198048 −0.0990240 0.995085i \(-0.531572\pi\)
−0.0990240 + 0.995085i \(0.531572\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 24.0258 41.6138i 0.923385 1.59935i 0.129246 0.991613i \(-0.458744\pi\)
0.794138 0.607737i \(-0.207922\pi\)
\(678\) 0 0
\(679\) 3.21648 0.123437
\(680\) 0 0
\(681\) −4.01431 + 6.95299i −0.153829 + 0.266439i
\(682\) 0 0
\(683\) 4.21652 + 7.30323i 0.161341 + 0.279450i 0.935350 0.353724i \(-0.115085\pi\)
−0.774009 + 0.633174i \(0.781752\pi\)
\(684\) 0 0
\(685\) 12.0569 0.460670
\(686\) 0 0
\(687\) 3.22002 + 5.57724i 0.122852 + 0.212785i
\(688\) 0 0
\(689\) 0.344116 + 0.596026i 0.0131098 + 0.0227068i
\(690\) 0 0
\(691\) 3.27042 5.66453i 0.124413 0.215489i −0.797091 0.603860i \(-0.793629\pi\)
0.921503 + 0.388371i \(0.126962\pi\)
\(692\) 0 0
\(693\) −0.407095 0.705110i −0.0154643 0.0267849i
\(694\) 0 0
\(695\) −22.9507 −0.870571
\(696\) 0 0
\(697\) 15.4977 0.587017
\(698\) 0 0
\(699\) 13.8495 23.9880i 0.523835 0.907309i
\(700\) 0 0
\(701\) −5.06428 + 8.77159i −0.191275 + 0.331298i −0.945673 0.325119i \(-0.894596\pi\)
0.754398 + 0.656417i \(0.227929\pi\)
\(702\) 0 0
\(703\) −1.27421 + 2.20700i −0.0480578 + 0.0832385i
\(704\) 0 0
\(705\) 0.703919 1.21922i 0.0265111 0.0459186i
\(706\) 0 0
\(707\) 2.03586 + 3.52621i 0.0765664 + 0.132617i
\(708\) 0 0
\(709\) −12.9515 22.4327i −0.486405 0.842479i 0.513473 0.858106i \(-0.328359\pi\)
−0.999878 + 0.0156274i \(0.995025\pi\)
\(710\) 0 0
\(711\) −0.910790 + 1.57753i −0.0341573 + 0.0591621i
\(712\) 0 0
\(713\) −27.8859 −1.04433
\(714\) 0 0
\(715\) −0.0552744 −0.00206714
\(716\) 0 0
\(717\) −8.63210 + 14.9512i −0.322372 + 0.558364i
\(718\) 0 0
\(719\) −16.5293 28.6296i −0.616440 1.06770i −0.990130 0.140151i \(-0.955241\pi\)
0.373691 0.927553i \(-0.378092\pi\)
\(720\) 0 0
\(721\) 4.91976 + 8.52128i 0.183222 + 0.317349i
\(722\) 0 0
\(723\) −10.2489 −0.381162
\(724\) 0 0
\(725\) −3.20300 + 5.54777i −0.118957 + 0.206039i
\(726\) 0 0
\(727\) −12.0841 20.9303i −0.448175 0.776262i 0.550092 0.835104i \(-0.314593\pi\)
−0.998267 + 0.0588418i \(0.981259\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.65097 + 13.2519i 0.282981 + 0.490138i
\(732\) 0 0
\(733\) −8.15456 + 14.1241i −0.301196 + 0.521686i −0.976407 0.215938i \(-0.930719\pi\)
0.675211 + 0.737624i \(0.264052\pi\)
\(734\) 0 0
\(735\) 2.29832 + 3.98082i 0.0847750 + 0.146835i
\(736\) 0 0
\(737\) 0.888916 4.20597i 0.0327436 0.154929i
\(738\) 0 0
\(739\) −18.7995 32.5617i −0.691552 1.19780i −0.971329 0.237738i \(-0.923594\pi\)
0.279778 0.960065i \(-0.409739\pi\)
\(740\) 0 0
\(741\) 0.169282 0.293205i 0.00621874 0.0107712i
\(742\) 0 0
\(743\) −7.81340 13.5332i −0.286646 0.496485i 0.686361 0.727261i \(-0.259207\pi\)
−0.973007 + 0.230776i \(0.925874\pi\)
\(744\) 0 0
\(745\) −4.20473 −0.154049
\(746\) 0 0
\(747\) 8.71382 + 15.0928i 0.318822 + 0.552216i
\(748\) 0 0
\(749\) −7.99820 + 13.8533i −0.292248 + 0.506188i
\(750\) 0 0
\(751\) 6.95650 0.253846 0.126923 0.991913i \(-0.459490\pi\)
0.126923 + 0.991913i \(0.459490\pi\)
\(752\) 0 0
\(753\) 3.09480 + 5.36035i 0.112781 + 0.195342i
\(754\) 0 0
\(755\) −1.97272 3.41684i −0.0717945 0.124352i
\(756\) 0 0
\(757\) −11.2090 + 19.4146i −0.407398 + 0.705634i −0.994597 0.103808i \(-0.966897\pi\)
0.587199 + 0.809442i \(0.300230\pi\)
\(758\) 0 0
\(759\) −4.30563 −0.156284
\(760\) 0 0
\(761\) −7.91663 −0.286978 −0.143489 0.989652i \(-0.545832\pi\)
−0.143489 + 0.989652i \(0.545832\pi\)
\(762\) 0 0
\(763\) −4.98340 + 8.63150i −0.180411 + 0.312481i
\(764\) 0 0
\(765\) −1.53226 2.65395i −0.0553989 0.0959536i
\(766\) 0 0
\(767\) −0.169402 0.293413i −0.00611676 0.0105945i
\(768\) 0 0
\(769\) 7.32536 12.6879i 0.264159 0.457537i −0.703184 0.711008i \(-0.748239\pi\)
0.967343 + 0.253471i \(0.0815723\pi\)
\(770\) 0 0
\(771\) 7.92981 13.7348i 0.285585 0.494648i
\(772\) 0 0
\(773\) −23.9940 + 41.5588i −0.863004 + 1.49477i 0.00601145 + 0.999982i \(0.498086\pi\)
−0.869015 + 0.494785i \(0.835247\pi\)
\(774\) 0 0
\(775\) 1.70073 2.94575i 0.0610920 0.105814i
\(776\) 0 0
\(777\) −1.22813 −0.0440589
\(778\) 0 0
\(779\) −16.2682 −0.582871
\(780\) 0 0
\(781\) 1.53509 + 2.65885i 0.0549297 + 0.0951410i
\(782\) 0 0
\(783\) 3.20300 5.54777i 0.114466 0.198261i
\(784\) 0 0
\(785\) −6.28433 10.8848i −0.224297 0.388494i
\(786\) 0 0
\(787\) −14.2603 24.6996i −0.508326 0.880447i −0.999954 0.00964096i \(-0.996931\pi\)
0.491627 0.870806i \(-0.336402\pi\)
\(788\) 0 0
\(789\) −18.7400 −0.667163
\(790\) 0 0
\(791\) −2.57615 4.46202i −0.0915972 0.158651i
\(792\) 0 0
\(793\) 0.295941 0.512585i 0.0105092 0.0182024i
\(794\) 0 0
\(795\) −6.53926 −0.231924
\(796\) 0 0
\(797\) 4.24546 7.35336i 0.150382 0.260469i −0.780986 0.624549i \(-0.785283\pi\)
0.931368 + 0.364079i \(0.118616\pi\)
\(798\) 0 0
\(799\) 4.31434 0.152630
\(800\) 0 0
\(801\) −8.71903 −0.308072
\(802\) 0 0
\(803\) 4.31840 0.152393
\(804\) 0 0
\(805\) −12.7095 −0.447950
\(806\) 0 0
\(807\) 21.6742 0.762968
\(808\) 0 0
\(809\) −4.78307 −0.168164 −0.0840818 0.996459i \(-0.526796\pi\)
−0.0840818 + 0.996459i \(0.526796\pi\)
\(810\) 0 0
\(811\) −24.2422 + 41.9887i −0.851259 + 1.47442i 0.0288136 + 0.999585i \(0.490827\pi\)
−0.880073 + 0.474839i \(0.842506\pi\)
\(812\) 0 0
\(813\) −30.1382 −1.05699
\(814\) 0 0
\(815\) −10.2750 + 17.7969i −0.359918 + 0.623397i
\(816\) 0 0
\(817\) −8.03138 13.9108i −0.280982 0.486676i
\(818\) 0 0
\(819\) 0.163160 0.00570128
\(820\) 0 0
\(821\) 16.3825 + 28.3753i 0.571754 + 0.990306i 0.996386 + 0.0849404i \(0.0270700\pi\)
−0.424633 + 0.905366i \(0.639597\pi\)
\(822\) 0 0
\(823\) −11.2708 19.5217i −0.392876 0.680482i 0.599951 0.800037i \(-0.295187\pi\)
−0.992828 + 0.119555i \(0.961853\pi\)
\(824\) 0 0
\(825\) 0.262596 0.454829i 0.00914241 0.0158351i
\(826\) 0 0
\(827\) 23.7232 + 41.0897i 0.824936 + 1.42883i 0.901968 + 0.431802i \(0.142122\pi\)
−0.0770328 + 0.997029i \(0.524545\pi\)
\(828\) 0 0
\(829\) 39.4233 1.36923 0.684613 0.728906i \(-0.259971\pi\)
0.684613 + 0.728906i \(0.259971\pi\)
\(830\) 0 0
\(831\) −19.8633 −0.689049
\(832\) 0 0
\(833\) −7.04325 + 12.1993i −0.244034 + 0.422679i
\(834\) 0 0
\(835\) 10.3299 17.8919i 0.357482 0.619176i
\(836\) 0 0
\(837\) −1.70073 + 2.94575i −0.0587858 + 0.101820i
\(838\) 0 0
\(839\) −8.81831 + 15.2738i −0.304442 + 0.527309i −0.977137 0.212611i \(-0.931803\pi\)
0.672695 + 0.739920i \(0.265137\pi\)
\(840\) 0 0
\(841\) −6.01848 10.4243i −0.207534 0.359459i
\(842\) 0 0
\(843\) −16.1159 27.9135i −0.555060 0.961393i
\(844\) 0 0
\(845\) −6.49446 + 11.2487i −0.223416 + 0.386968i
\(846\) 0 0
\(847\) −16.6254 −0.571256
\(848\) 0 0
\(849\) 30.5884 1.04979
\(850\) 0 0
\(851\) −3.24732 + 5.62452i −0.111317 + 0.192806i
\(852\) 0 0
\(853\) −15.6954 27.1852i −0.537400 0.930805i −0.999043 0.0437386i \(-0.986073\pi\)
0.461643 0.887066i \(-0.347260\pi\)
\(854\) 0 0
\(855\) 1.60844 + 2.78590i 0.0550075 + 0.0952758i
\(856\) 0 0
\(857\) 33.2972 1.13741 0.568706 0.822541i \(-0.307444\pi\)
0.568706 + 0.822541i \(0.307444\pi\)
\(858\) 0 0
\(859\) 0.331699 0.574519i 0.0113174 0.0196023i −0.860311 0.509769i \(-0.829731\pi\)
0.871629 + 0.490167i \(0.163064\pi\)
\(860\) 0 0
\(861\) −3.91998 6.78961i −0.133593 0.231389i
\(862\) 0 0
\(863\) −12.9776 −0.441763 −0.220882 0.975301i \(-0.570893\pi\)
−0.220882 + 0.975301i \(0.570893\pi\)
\(864\) 0 0
\(865\) 0.847605 + 1.46810i 0.0288194 + 0.0499167i
\(866\) 0 0
\(867\) −3.80438 + 6.58938i −0.129203 + 0.223787i
\(868\) 0 0
\(869\) −0.478339 0.828507i −0.0162265 0.0281052i
\(870\) 0 0
\(871\) 0.640869 + 0.575699i 0.0217150 + 0.0195068i
\(872\) 0 0
\(873\) −1.03739 1.79682i −0.0351104 0.0608130i
\(874\) 0 0
\(875\) 0.775137 1.34258i 0.0262044 0.0453874i
\(876\) 0 0
\(877\) −20.6267 35.7265i −0.696515 1.20640i −0.969667 0.244428i \(-0.921400\pi\)
0.273153 0.961971i \(-0.411934\pi\)
\(878\) 0 0
\(879\) −2.17227 −0.0732688
\(880\) 0 0
\(881\) −0.741941 1.28508i −0.0249966 0.0432954i 0.853256 0.521491i \(-0.174624\pi\)
−0.878253 + 0.478196i \(0.841291\pi\)
\(882\) 0 0
\(883\) −11.9427 + 20.6854i −0.401905 + 0.696120i −0.993956 0.109781i \(-0.964985\pi\)
0.592051 + 0.805901i \(0.298319\pi\)
\(884\) 0 0
\(885\) 3.21916 0.108211
\(886\) 0 0
\(887\) −1.24802 2.16163i −0.0419044 0.0725805i 0.844313 0.535851i \(-0.180009\pi\)
−0.886217 + 0.463271i \(0.846676\pi\)
\(888\) 0 0
\(889\) 16.8767 + 29.2313i 0.566027 + 0.980388i
\(890\) 0 0
\(891\) −0.262596 + 0.454829i −0.00879729 + 0.0152373i
\(892\) 0 0
\(893\) −4.52885 −0.151552
\(894\) 0 0
\(895\) −16.2705 −0.543864
\(896\) 0 0
\(897\) 0.431415 0.747233i 0.0144045 0.0249494i
\(898\) 0 0
\(899\) 10.8949 + 18.8705i 0.363365 + 0.629366i
\(900\) 0 0
\(901\) −10.0198 17.3548i −0.333809 0.578173i
\(902\) 0 0
\(903\) 3.87047 6.70385i 0.128801 0.223090i
\(904\) 0 0
\(905\) −6.53860 + 11.3252i −0.217350 + 0.376462i
\(906\) 0 0
\(907\) −15.0591 + 26.0831i −0.500028 + 0.866074i 0.499972 + 0.866042i \(0.333344\pi\)
−1.00000 3.22262e-5i \(0.999990\pi\)
\(908\) 0 0
\(909\) 1.31323 2.27457i 0.0435569 0.0754428i
\(910\) 0 0
\(911\) 18.5596 0.614907 0.307454 0.951563i \(-0.400523\pi\)
0.307454 + 0.951563i \(0.400523\pi\)
\(912\) 0 0
\(913\) −9.15284 −0.302915
\(914\) 0 0
\(915\) 2.81190 + 4.87035i 0.0929585 + 0.161009i
\(916\) 0 0
\(917\) −11.4829 + 19.8890i −0.379199 + 0.656792i
\(918\) 0 0
\(919\) 4.49216 + 7.78064i 0.148183 + 0.256660i 0.930556 0.366150i \(-0.119324\pi\)
−0.782373 + 0.622810i \(0.785991\pi\)
\(920\) 0 0
\(921\) −4.16325 7.21095i −0.137184 0.237609i
\(922\) 0 0
\(923\) −0.615249 −0.0202512
\(924\) 0 0
\(925\) −0.396101 0.686067i −0.0130237 0.0225577i
\(926\) 0 0
\(927\) 3.17348 5.49663i 0.104231 0.180533i
\(928\) 0 0
\(929\) −33.5743 −1.10154 −0.550769 0.834658i \(-0.685666\pi\)
−0.550769 + 0.834658i \(0.685666\pi\)
\(930\) 0 0
\(931\) 7.39344 12.8058i 0.242310 0.419694i
\(932\) 0 0
\(933\) 10.6902 0.349982
\(934\) 0 0
\(935\) 1.60946 0.0526348
\(936\) 0 0
\(937\) 7.89342 0.257867 0.128933 0.991653i \(-0.458845\pi\)
0.128933 + 0.991653i \(0.458845\pi\)
\(938\) 0 0
\(939\) −20.7113 −0.675887
\(940\) 0 0
\(941\) −22.8789 −0.745830 −0.372915 0.927866i \(-0.621642\pi\)
−0.372915 + 0.927866i \(0.621642\pi\)
\(942\) 0 0
\(943\) −41.4596 −1.35011
\(944\) 0 0
\(945\) −0.775137 + 1.34258i −0.0252152 + 0.0436740i
\(946\) 0 0
\(947\) −32.4028 −1.05295 −0.526475 0.850191i \(-0.676487\pi\)
−0.526475 + 0.850191i \(0.676487\pi\)
\(948\) 0 0
\(949\) −0.432695 + 0.749449i −0.0140459 + 0.0243282i
\(950\) 0 0
\(951\) −10.5331 18.2438i −0.341558 0.591596i
\(952\) 0 0
\(953\) 45.3592 1.46933 0.734664 0.678432i \(-0.237340\pi\)
0.734664 + 0.678432i \(0.237340\pi\)
\(954\) 0 0
\(955\) 7.38160 + 12.7853i 0.238863 + 0.413723i
\(956\) 0 0
\(957\) 1.68219 + 2.91364i 0.0543775 + 0.0941846i
\(958\) 0 0
\(959\) 9.34574 16.1873i 0.301790 0.522715i
\(960\) 0 0
\(961\) 9.71505 + 16.8270i 0.313389 + 0.542805i
\(962\) 0 0
\(963\) 10.3184 0.332507
\(964\) 0 0
\(965\) 0.212140 0.00682903
\(966\) 0 0
\(967\) −12.6946 + 21.9877i −0.408231 + 0.707077i −0.994692 0.102901i \(-0.967187\pi\)
0.586461 + 0.809978i \(0.300521\pi\)
\(968\) 0 0
\(969\) −4.92909 + 8.53743i −0.158345 + 0.274262i
\(970\) 0 0
\(971\) 26.9116 46.6122i 0.863634 1.49586i −0.00476379 0.999989i \(-0.501516\pi\)
0.868397 0.495869i \(-0.165150\pi\)
\(972\) 0 0
\(973\) −17.7900 + 30.8131i −0.570320 + 0.987824i
\(974\) 0 0
\(975\) 0.0526231 + 0.0911458i 0.00168529 + 0.00291900i
\(976\) 0 0
\(977\) −4.21136 7.29429i −0.134733 0.233365i 0.790762 0.612124i \(-0.209684\pi\)
−0.925496 + 0.378758i \(0.876351\pi\)
\(978\) 0 0
\(979\) 2.28958 3.96567i 0.0731753 0.126743i
\(980\) 0 0
\(981\) 6.42905 0.205264
\(982\) 0 0
\(983\) 1.92197 0.0613013 0.0306507 0.999530i \(-0.490242\pi\)
0.0306507 + 0.999530i \(0.490242\pi\)
\(984\) 0 0
\(985\) −12.3422 + 21.3774i −0.393256 + 0.681140i
\(986\) 0 0
\(987\) −1.09127 1.89013i −0.0347354 0.0601635i
\(988\) 0 0
\(989\) −20.4679 35.4515i −0.650843 1.12729i
\(990\) 0 0
\(991\) 18.2784 0.580633 0.290317 0.956931i \(-0.406239\pi\)
0.290317 + 0.956931i \(0.406239\pi\)
\(992\) 0 0
\(993\) −16.6168 + 28.7812i −0.527319 + 0.913344i
\(994\) 0 0
\(995\) −5.70515 9.88161i −0.180865 0.313268i
\(996\) 0 0
\(997\) 19.7744 0.626260 0.313130 0.949710i \(-0.398622\pi\)
0.313130 + 0.949710i \(0.398622\pi\)
\(998\) 0 0
\(999\) 0.396101 + 0.686067i 0.0125321 + 0.0217062i
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 4020.2.q.l.841.4 22
67.29 even 3 inner 4020.2.q.l.3781.4 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.l.841.4 22 1.1 even 1 trivial
4020.2.q.l.3781.4 yes 22 67.29 even 3 inner