Properties

Label 4020.2.q.l.841.8
Level $4020$
Weight $2$
Character 4020.841
Analytic conductor $32.100$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 841.8
Character \(\chi\) \(=\) 4020.841
Dual form 4020.2.q.l.3781.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +(0.637931 - 1.10493i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +(0.637931 - 1.10493i) q^{7} +1.00000 q^{9} +(-0.198096 + 0.343112i) q^{11} +(1.57788 + 2.73297i) q^{13} +1.00000 q^{15} +(-0.405163 - 0.701763i) q^{17} +(2.24701 + 3.89194i) q^{19} +(-0.637931 + 1.10493i) q^{21} +(-0.399310 - 0.691625i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(2.44753 - 4.23925i) q^{29} +(-0.134765 + 0.233420i) q^{31} +(0.198096 - 0.343112i) q^{33} +(-0.637931 + 1.10493i) q^{35} +(-2.34684 - 4.06485i) q^{37} +(-1.57788 - 2.73297i) q^{39} +(-3.47144 + 6.01270i) q^{41} -2.23162 q^{43} -1.00000 q^{45} +(1.85168 - 3.20720i) q^{47} +(2.68609 + 4.65244i) q^{49} +(0.405163 + 0.701763i) q^{51} -13.4945 q^{53} +(0.198096 - 0.343112i) q^{55} +(-2.24701 - 3.89194i) q^{57} +4.24630 q^{59} +(-1.93403 - 3.34985i) q^{61} +(0.637931 - 1.10493i) q^{63} +(-1.57788 - 2.73297i) q^{65} +(-0.366493 + 8.17714i) q^{67} +(0.399310 + 0.691625i) q^{69} +(2.11699 - 3.66673i) q^{71} +(5.97631 + 10.3513i) q^{73} -1.00000 q^{75} +(0.252743 + 0.437764i) q^{77} +(-0.255601 + 0.442715i) q^{79} +1.00000 q^{81} +(1.92890 + 3.34095i) q^{83} +(0.405163 + 0.701763i) q^{85} +(-2.44753 + 4.23925i) q^{87} -0.151145 q^{89} +4.02632 q^{91} +(0.134765 - 0.233420i) q^{93} +(-2.24701 - 3.89194i) q^{95} +(3.70238 + 6.41271i) q^{97} +(-0.198096 + 0.343112i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 22 q^{3} - 22 q^{5} + q^{7} + 22 q^{9}+O(q^{10}) \) 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
\(7\) 0.637931 1.10493i 0.241115 0.417624i −0.719917 0.694060i \(-0.755820\pi\)
0.961032 + 0.276436i \(0.0891535\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.198096 + 0.343112i −0.0597282 + 0.103452i −0.894343 0.447381i \(-0.852357\pi\)
0.834615 + 0.550833i \(0.185690\pi\)
\(12\) 0 0
\(13\) 1.57788 + 2.73297i 0.437625 + 0.757990i 0.997506 0.0705839i \(-0.0224863\pi\)
−0.559880 + 0.828573i \(0.689153\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −0.405163 0.701763i −0.0982665 0.170203i 0.812701 0.582681i \(-0.197996\pi\)
−0.910967 + 0.412479i \(0.864663\pi\)
\(18\) 0 0
\(19\) 2.24701 + 3.89194i 0.515500 + 0.892871i 0.999838 + 0.0179908i \(0.00572696\pi\)
−0.484339 + 0.874881i \(0.660940\pi\)
\(20\) 0 0
\(21\) −0.637931 + 1.10493i −0.139208 + 0.241115i
\(22\) 0 0
\(23\) −0.399310 0.691625i −0.0832619 0.144214i 0.821387 0.570371i \(-0.193200\pi\)
−0.904649 + 0.426157i \(0.859867\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.44753 4.23925i 0.454496 0.787209i −0.544163 0.838979i \(-0.683153\pi\)
0.998659 + 0.0517698i \(0.0164862\pi\)
\(30\) 0 0
\(31\) −0.134765 + 0.233420i −0.0242045 + 0.0419235i −0.877874 0.478892i \(-0.841039\pi\)
0.853669 + 0.520815i \(0.174372\pi\)
\(32\) 0 0
\(33\) 0.198096 0.343112i 0.0344841 0.0597282i
\(34\) 0 0
\(35\) −0.637931 + 1.10493i −0.107830 + 0.186767i
\(36\) 0 0
\(37\) −2.34684 4.06485i −0.385819 0.668258i 0.606064 0.795416i \(-0.292748\pi\)
−0.991882 + 0.127159i \(0.959414\pi\)
\(38\) 0 0
\(39\) −1.57788 2.73297i −0.252663 0.437625i
\(40\) 0 0
\(41\) −3.47144 + 6.01270i −0.542147 + 0.939026i 0.456633 + 0.889655i \(0.349055\pi\)
−0.998780 + 0.0493714i \(0.984278\pi\)
\(42\) 0 0
\(43\) −2.23162 −0.340320 −0.170160 0.985416i \(-0.554428\pi\)
−0.170160 + 0.985416i \(0.554428\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.85168 3.20720i 0.270095 0.467818i −0.698791 0.715326i \(-0.746278\pi\)
0.968886 + 0.247508i \(0.0796117\pi\)
\(48\) 0 0
\(49\) 2.68609 + 4.65244i 0.383727 + 0.664635i
\(50\) 0 0
\(51\) 0.405163 + 0.701763i 0.0567342 + 0.0982665i
\(52\) 0 0
\(53\) −13.4945 −1.85362 −0.926809 0.375532i \(-0.877460\pi\)
−0.926809 + 0.375532i \(0.877460\pi\)
\(54\) 0 0
\(55\) 0.198096 0.343112i 0.0267113 0.0462653i
\(56\) 0 0
\(57\) −2.24701 3.89194i −0.297624 0.515500i
\(58\) 0 0
\(59\) 4.24630 0.552822 0.276411 0.961040i \(-0.410855\pi\)
0.276411 + 0.961040i \(0.410855\pi\)
\(60\) 0 0
\(61\) −1.93403 3.34985i −0.247628 0.428904i 0.715239 0.698880i \(-0.246318\pi\)
−0.962867 + 0.269976i \(0.912984\pi\)
\(62\) 0 0
\(63\) 0.637931 1.10493i 0.0803717 0.139208i
\(64\) 0 0
\(65\) −1.57788 2.73297i −0.195712 0.338983i
\(66\) 0 0
\(67\) −0.366493 + 8.17714i −0.0447743 + 0.998997i
\(68\) 0 0
\(69\) 0.399310 + 0.691625i 0.0480713 + 0.0832619i
\(70\) 0 0
\(71\) 2.11699 3.66673i 0.251240 0.435161i −0.712627 0.701543i \(-0.752495\pi\)
0.963868 + 0.266382i \(0.0858283\pi\)
\(72\) 0 0
\(73\) 5.97631 + 10.3513i 0.699475 + 1.21153i 0.968649 + 0.248434i \(0.0799159\pi\)
−0.269174 + 0.963091i \(0.586751\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0.252743 + 0.437764i 0.0288027 + 0.0498878i
\(78\) 0 0
\(79\) −0.255601 + 0.442715i −0.0287574 + 0.0498093i −0.880046 0.474889i \(-0.842488\pi\)
0.851289 + 0.524698i \(0.175822\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.92890 + 3.34095i 0.211724 + 0.366717i 0.952254 0.305306i \(-0.0987589\pi\)
−0.740530 + 0.672023i \(0.765426\pi\)
\(84\) 0 0
\(85\) 0.405163 + 0.701763i 0.0439461 + 0.0761169i
\(86\) 0 0
\(87\) −2.44753 + 4.23925i −0.262403 + 0.454496i
\(88\) 0 0
\(89\) −0.151145 −0.0160214 −0.00801068 0.999968i \(-0.502550\pi\)
−0.00801068 + 0.999968i \(0.502550\pi\)
\(90\) 0 0
\(91\) 4.02632 0.422073
\(92\) 0 0
\(93\) 0.134765 0.233420i 0.0139745 0.0242045i
\(94\) 0 0
\(95\) −2.24701 3.89194i −0.230538 0.399304i
\(96\) 0 0
\(97\) 3.70238 + 6.41271i 0.375920 + 0.651112i 0.990464 0.137770i \(-0.0439934\pi\)
−0.614544 + 0.788882i \(0.710660\pi\)
\(98\) 0 0
\(99\) −0.198096 + 0.343112i −0.0199094 + 0.0344841i
\(100\) 0 0
\(101\) −2.40349 + 4.16297i −0.239156 + 0.414231i −0.960472 0.278375i \(-0.910204\pi\)
0.721316 + 0.692606i \(0.243537\pi\)
\(102\) 0 0
\(103\) 5.28897 9.16076i 0.521138 0.902637i −0.478560 0.878055i \(-0.658841\pi\)
0.999698 0.0245820i \(-0.00782549\pi\)
\(104\) 0 0
\(105\) 0.637931 1.10493i 0.0622557 0.107830i
\(106\) 0 0
\(107\) 10.9381 1.05743 0.528713 0.848800i \(-0.322675\pi\)
0.528713 + 0.848800i \(0.322675\pi\)
\(108\) 0 0
\(109\) −11.5903 −1.11015 −0.555075 0.831800i \(-0.687311\pi\)
−0.555075 + 0.831800i \(0.687311\pi\)
\(110\) 0 0
\(111\) 2.34684 + 4.06485i 0.222753 + 0.385819i
\(112\) 0 0
\(113\) 5.17900 8.97029i 0.487199 0.843854i −0.512693 0.858572i \(-0.671352\pi\)
0.999892 + 0.0147186i \(0.00468525\pi\)
\(114\) 0 0
\(115\) 0.399310 + 0.691625i 0.0372358 + 0.0644944i
\(116\) 0 0
\(117\) 1.57788 + 2.73297i 0.145875 + 0.252663i
\(118\) 0 0
\(119\) −1.03386 −0.0947742
\(120\) 0 0
\(121\) 5.42152 + 9.39034i 0.492865 + 0.853667i
\(122\) 0 0
\(123\) 3.47144 6.01270i 0.313009 0.542147i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.69595 + 11.5977i −0.594169 + 1.02913i 0.399494 + 0.916736i \(0.369186\pi\)
−0.993664 + 0.112395i \(0.964148\pi\)
\(128\) 0 0
\(129\) 2.23162 0.196484
\(130\) 0 0
\(131\) −3.97325 −0.347145 −0.173572 0.984821i \(-0.555531\pi\)
−0.173572 + 0.984821i \(0.555531\pi\)
\(132\) 0 0
\(133\) 5.73375 0.497179
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.9590 0.936293 0.468146 0.883651i \(-0.344922\pi\)
0.468146 + 0.883651i \(0.344922\pi\)
\(138\) 0 0
\(139\) 6.91210 0.586276 0.293138 0.956070i \(-0.405300\pi\)
0.293138 + 0.956070i \(0.405300\pi\)
\(140\) 0 0
\(141\) −1.85168 + 3.20720i −0.155939 + 0.270095i
\(142\) 0 0
\(143\) −1.25029 −0.104554
\(144\) 0 0
\(145\) −2.44753 + 4.23925i −0.203257 + 0.352051i
\(146\) 0 0
\(147\) −2.68609 4.65244i −0.221545 0.383727i
\(148\) 0 0
\(149\) 15.4625 1.26674 0.633370 0.773849i \(-0.281671\pi\)
0.633370 + 0.773849i \(0.281671\pi\)
\(150\) 0 0
\(151\) 6.39067 + 11.0690i 0.520065 + 0.900779i 0.999728 + 0.0233265i \(0.00742572\pi\)
−0.479663 + 0.877453i \(0.659241\pi\)
\(152\) 0 0
\(153\) −0.405163 0.701763i −0.0327555 0.0567342i
\(154\) 0 0
\(155\) 0.134765 0.233420i 0.0108246 0.0187488i
\(156\) 0 0
\(157\) −2.24885 3.89513i −0.179478 0.310865i 0.762224 0.647314i \(-0.224108\pi\)
−0.941702 + 0.336448i \(0.890774\pi\)
\(158\) 0 0
\(159\) 13.4945 1.07019
\(160\) 0 0
\(161\) −1.01893 −0.0803028
\(162\) 0 0
\(163\) −0.839614 + 1.45425i −0.0657636 + 0.113906i −0.897032 0.441965i \(-0.854282\pi\)
0.831269 + 0.555871i \(0.187615\pi\)
\(164\) 0 0
\(165\) −0.198096 + 0.343112i −0.0154218 + 0.0267113i
\(166\) 0 0
\(167\) 6.61904 11.4645i 0.512197 0.887151i −0.487703 0.873010i \(-0.662165\pi\)
0.999900 0.0141416i \(-0.00450156\pi\)
\(168\) 0 0
\(169\) 1.52058 2.63373i 0.116968 0.202594i
\(170\) 0 0
\(171\) 2.24701 + 3.89194i 0.171833 + 0.297624i
\(172\) 0 0
\(173\) 9.85480 + 17.0690i 0.749246 + 1.29773i 0.948184 + 0.317721i \(0.102917\pi\)
−0.198938 + 0.980012i \(0.563749\pi\)
\(174\) 0 0
\(175\) 0.637931 1.10493i 0.0482230 0.0835247i
\(176\) 0 0
\(177\) −4.24630 −0.319172
\(178\) 0 0
\(179\) −15.8857 −1.18735 −0.593676 0.804704i \(-0.702324\pi\)
−0.593676 + 0.804704i \(0.702324\pi\)
\(180\) 0 0
\(181\) −12.4536 + 21.5702i −0.925666 + 1.60330i −0.135179 + 0.990821i \(0.543161\pi\)
−0.790487 + 0.612479i \(0.790172\pi\)
\(182\) 0 0
\(183\) 1.93403 + 3.34985i 0.142968 + 0.247628i
\(184\) 0 0
\(185\) 2.34684 + 4.06485i 0.172543 + 0.298854i
\(186\) 0 0
\(187\) 0.321045 0.0234771
\(188\) 0 0
\(189\) −0.637931 + 1.10493i −0.0464026 + 0.0803717i
\(190\) 0 0
\(191\) 11.7379 + 20.3306i 0.849324 + 1.47107i 0.881812 + 0.471601i \(0.156324\pi\)
−0.0324880 + 0.999472i \(0.510343\pi\)
\(192\) 0 0
\(193\) −22.6977 −1.63382 −0.816909 0.576767i \(-0.804314\pi\)
−0.816909 + 0.576767i \(0.804314\pi\)
\(194\) 0 0
\(195\) 1.57788 + 2.73297i 0.112994 + 0.195712i
\(196\) 0 0
\(197\) −6.66683 + 11.5473i −0.474992 + 0.822710i −0.999590 0.0286400i \(-0.990882\pi\)
0.524598 + 0.851350i \(0.324216\pi\)
\(198\) 0 0
\(199\) 9.65718 + 16.7267i 0.684579 + 1.18573i 0.973569 + 0.228394i \(0.0733473\pi\)
−0.288990 + 0.957332i \(0.593319\pi\)
\(200\) 0 0
\(201\) 0.366493 8.17714i 0.0258504 0.576771i
\(202\) 0 0
\(203\) −3.12271 5.40870i −0.219172 0.379616i
\(204\) 0 0
\(205\) 3.47144 6.01270i 0.242456 0.419945i
\(206\) 0 0
\(207\) −0.399310 0.691625i −0.0277540 0.0480713i
\(208\) 0 0
\(209\) −1.78050 −0.123159
\(210\) 0 0
\(211\) 7.90244 + 13.6874i 0.544027 + 0.942282i 0.998667 + 0.0516068i \(0.0164342\pi\)
−0.454641 + 0.890675i \(0.650232\pi\)
\(212\) 0 0
\(213\) −2.11699 + 3.66673i −0.145054 + 0.251240i
\(214\) 0 0
\(215\) 2.23162 0.152196
\(216\) 0 0
\(217\) 0.171942 + 0.297812i 0.0116722 + 0.0202168i
\(218\) 0 0
\(219\) −5.97631 10.3513i −0.403842 0.699475i
\(220\) 0 0
\(221\) 1.27860 2.21460i 0.0860078 0.148970i
\(222\) 0 0
\(223\) 0.134772 0.00902497 0.00451248 0.999990i \(-0.498564\pi\)
0.00451248 + 0.999990i \(0.498564\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 8.05133 13.9453i 0.534386 0.925583i −0.464807 0.885412i \(-0.653876\pi\)
0.999193 0.0401711i \(-0.0127903\pi\)
\(228\) 0 0
\(229\) 6.14349 + 10.6408i 0.405973 + 0.703167i 0.994434 0.105359i \(-0.0335992\pi\)
−0.588461 + 0.808526i \(0.700266\pi\)
\(230\) 0 0
\(231\) −0.252743 0.437764i −0.0166293 0.0288027i
\(232\) 0 0
\(233\) −2.41572 + 4.18416i −0.158259 + 0.274113i −0.934241 0.356642i \(-0.883922\pi\)
0.775982 + 0.630755i \(0.217255\pi\)
\(234\) 0 0
\(235\) −1.85168 + 3.20720i −0.120790 + 0.209214i
\(236\) 0 0
\(237\) 0.255601 0.442715i 0.0166031 0.0287574i
\(238\) 0 0
\(239\) −1.22293 + 2.11817i −0.0791045 + 0.137013i −0.902864 0.429927i \(-0.858539\pi\)
0.823759 + 0.566940i \(0.191873\pi\)
\(240\) 0 0
\(241\) 18.1372 1.16832 0.584159 0.811639i \(-0.301425\pi\)
0.584159 + 0.811639i \(0.301425\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.68609 4.65244i −0.171608 0.297234i
\(246\) 0 0
\(247\) −7.09103 + 12.2820i −0.451191 + 0.781487i
\(248\) 0 0
\(249\) −1.92890 3.34095i −0.122239 0.211724i
\(250\) 0 0
\(251\) 10.8009 + 18.7078i 0.681748 + 1.18082i 0.974447 + 0.224618i \(0.0721135\pi\)
−0.292698 + 0.956205i \(0.594553\pi\)
\(252\) 0 0
\(253\) 0.316407 0.0198923
\(254\) 0 0
\(255\) −0.405163 0.701763i −0.0253723 0.0439461i
\(256\) 0 0
\(257\) −13.2554 + 22.9591i −0.826851 + 1.43215i 0.0736456 + 0.997284i \(0.476537\pi\)
−0.900497 + 0.434863i \(0.856797\pi\)
\(258\) 0 0
\(259\) −5.98849 −0.372107
\(260\) 0 0
\(261\) 2.44753 4.23925i 0.151499 0.262403i
\(262\) 0 0
\(263\) −1.98177 −0.122201 −0.0611005 0.998132i \(-0.519461\pi\)
−0.0611005 + 0.998132i \(0.519461\pi\)
\(264\) 0 0
\(265\) 13.4945 0.828963
\(266\) 0 0
\(267\) 0.151145 0.00924993
\(268\) 0 0
\(269\) −19.6325 −1.19701 −0.598507 0.801117i \(-0.704239\pi\)
−0.598507 + 0.801117i \(0.704239\pi\)
\(270\) 0 0
\(271\) 16.1989 0.984012 0.492006 0.870592i \(-0.336264\pi\)
0.492006 + 0.870592i \(0.336264\pi\)
\(272\) 0 0
\(273\) −4.02632 −0.243684
\(274\) 0 0
\(275\) −0.198096 + 0.343112i −0.0119456 + 0.0206905i
\(276\) 0 0
\(277\) 20.6486 1.24066 0.620328 0.784343i \(-0.287000\pi\)
0.620328 + 0.784343i \(0.287000\pi\)
\(278\) 0 0
\(279\) −0.134765 + 0.233420i −0.00806818 + 0.0139745i
\(280\) 0 0
\(281\) 3.36886 + 5.83504i 0.200969 + 0.348089i 0.948841 0.315754i \(-0.102257\pi\)
−0.747872 + 0.663843i \(0.768924\pi\)
\(282\) 0 0
\(283\) 16.5873 0.986015 0.493007 0.870025i \(-0.335898\pi\)
0.493007 + 0.870025i \(0.335898\pi\)
\(284\) 0 0
\(285\) 2.24701 + 3.89194i 0.133101 + 0.230538i
\(286\) 0 0
\(287\) 4.42907 + 7.67138i 0.261440 + 0.452827i
\(288\) 0 0
\(289\) 8.17169 14.1538i 0.480687 0.832575i
\(290\) 0 0
\(291\) −3.70238 6.41271i −0.217037 0.375920i
\(292\) 0 0
\(293\) −30.5791 −1.78645 −0.893225 0.449611i \(-0.851563\pi\)
−0.893225 + 0.449611i \(0.851563\pi\)
\(294\) 0 0
\(295\) −4.24630 −0.247229
\(296\) 0 0
\(297\) 0.198096 0.343112i 0.0114947 0.0199094i
\(298\) 0 0
\(299\) 1.26013 2.18260i 0.0728750 0.126223i
\(300\) 0 0
\(301\) −1.42362 + 2.46579i −0.0820562 + 0.142126i
\(302\) 0 0
\(303\) 2.40349 4.16297i 0.138077 0.239156i
\(304\) 0 0
\(305\) 1.93403 + 3.34985i 0.110743 + 0.191812i
\(306\) 0 0
\(307\) 9.72786 + 16.8491i 0.555198 + 0.961631i 0.997888 + 0.0649567i \(0.0206909\pi\)
−0.442690 + 0.896675i \(0.645976\pi\)
\(308\) 0 0
\(309\) −5.28897 + 9.16076i −0.300879 + 0.521138i
\(310\) 0 0
\(311\) 31.9314 1.81066 0.905331 0.424706i \(-0.139623\pi\)
0.905331 + 0.424706i \(0.139623\pi\)
\(312\) 0 0
\(313\) −24.0643 −1.36019 −0.680097 0.733122i \(-0.738062\pi\)
−0.680097 + 0.733122i \(0.738062\pi\)
\(314\) 0 0
\(315\) −0.637931 + 1.10493i −0.0359433 + 0.0622557i
\(316\) 0 0
\(317\) −15.6368 27.0837i −0.878248 1.52117i −0.853262 0.521483i \(-0.825379\pi\)
−0.0249863 0.999688i \(-0.507954\pi\)
\(318\) 0 0
\(319\) 0.969693 + 1.67956i 0.0542924 + 0.0940372i
\(320\) 0 0
\(321\) −10.9381 −0.610506
\(322\) 0 0
\(323\) 1.82081 3.15374i 0.101313 0.175479i
\(324\) 0 0
\(325\) 1.57788 + 2.73297i 0.0875251 + 0.151598i
\(326\) 0 0
\(327\) 11.5903 0.640946
\(328\) 0 0
\(329\) −2.36248 4.09194i −0.130248 0.225596i
\(330\) 0 0
\(331\) −13.6369 + 23.6198i −0.749551 + 1.29826i 0.198487 + 0.980103i \(0.436397\pi\)
−0.948038 + 0.318157i \(0.896936\pi\)
\(332\) 0 0
\(333\) −2.34684 4.06485i −0.128606 0.222753i
\(334\) 0 0
\(335\) 0.366493 8.17714i 0.0200237 0.446765i
\(336\) 0 0
\(337\) 2.41117 + 4.17628i 0.131345 + 0.227496i 0.924195 0.381920i \(-0.124737\pi\)
−0.792850 + 0.609417i \(0.791404\pi\)
\(338\) 0 0
\(339\) −5.17900 + 8.97029i −0.281285 + 0.487199i
\(340\) 0 0
\(341\) −0.0533929 0.0924792i −0.00289139 0.00500803i
\(342\) 0 0
\(343\) 15.7852 0.852320
\(344\) 0 0
\(345\) −0.399310 0.691625i −0.0214981 0.0372358i
\(346\) 0 0
\(347\) −7.67706 + 13.2971i −0.412126 + 0.713824i −0.995122 0.0986516i \(-0.968547\pi\)
0.582996 + 0.812475i \(0.301880\pi\)
\(348\) 0 0
\(349\) −6.66432 −0.356733 −0.178367 0.983964i \(-0.557081\pi\)
−0.178367 + 0.983964i \(0.557081\pi\)
\(350\) 0 0
\(351\) −1.57788 2.73297i −0.0842211 0.145875i
\(352\) 0 0
\(353\) −5.15671 8.93168i −0.274464 0.475386i 0.695536 0.718491i \(-0.255167\pi\)
−0.970000 + 0.243106i \(0.921834\pi\)
\(354\) 0 0
\(355\) −2.11699 + 3.66673i −0.112358 + 0.194610i
\(356\) 0 0
\(357\) 1.03386 0.0547179
\(358\) 0 0
\(359\) −0.396073 −0.0209039 −0.0104520 0.999945i \(-0.503327\pi\)
−0.0104520 + 0.999945i \(0.503327\pi\)
\(360\) 0 0
\(361\) −0.598113 + 1.03596i −0.0314796 + 0.0545243i
\(362\) 0 0
\(363\) −5.42152 9.39034i −0.284556 0.492865i
\(364\) 0 0
\(365\) −5.97631 10.3513i −0.312815 0.541811i
\(366\) 0 0
\(367\) 4.40357 7.62721i 0.229865 0.398137i −0.727903 0.685680i \(-0.759505\pi\)
0.957768 + 0.287543i \(0.0928383\pi\)
\(368\) 0 0
\(369\) −3.47144 + 6.01270i −0.180716 + 0.313009i
\(370\) 0 0
\(371\) −8.60859 + 14.9105i −0.446936 + 0.774115i
\(372\) 0 0
\(373\) 4.77879 8.27711i 0.247436 0.428572i −0.715377 0.698738i \(-0.753745\pi\)
0.962814 + 0.270166i \(0.0870786\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 15.4477 0.795595
\(378\) 0 0
\(379\) 10.9734 + 19.0065i 0.563665 + 0.976296i 0.997173 + 0.0751465i \(0.0239424\pi\)
−0.433507 + 0.901150i \(0.642724\pi\)
\(380\) 0 0
\(381\) 6.69595 11.5977i 0.343044 0.594169i
\(382\) 0 0
\(383\) 1.34372 + 2.32738i 0.0686606 + 0.118924i 0.898312 0.439358i \(-0.144794\pi\)
−0.829651 + 0.558282i \(0.811461\pi\)
\(384\) 0 0
\(385\) −0.252743 0.437764i −0.0128810 0.0223105i
\(386\) 0 0
\(387\) −2.23162 −0.113440
\(388\) 0 0
\(389\) 1.41142 + 2.44465i 0.0715617 + 0.123948i 0.899586 0.436744i \(-0.143868\pi\)
−0.828024 + 0.560692i \(0.810535\pi\)
\(390\) 0 0
\(391\) −0.323571 + 0.560442i −0.0163637 + 0.0283428i
\(392\) 0 0
\(393\) 3.97325 0.200424
\(394\) 0 0
\(395\) 0.255601 0.442715i 0.0128607 0.0222754i
\(396\) 0 0
\(397\) 13.2818 0.666594 0.333297 0.942822i \(-0.391839\pi\)
0.333297 + 0.942822i \(0.391839\pi\)
\(398\) 0 0
\(399\) −5.73375 −0.287046
\(400\) 0 0
\(401\) −15.5196 −0.775014 −0.387507 0.921867i \(-0.626664\pi\)
−0.387507 + 0.921867i \(0.626664\pi\)
\(402\) 0 0
\(403\) −0.850574 −0.0423701
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 1.85960 0.0921770
\(408\) 0 0
\(409\) −4.25265 + 7.36581i −0.210280 + 0.364216i −0.951802 0.306713i \(-0.900771\pi\)
0.741522 + 0.670929i \(0.234104\pi\)
\(410\) 0 0
\(411\) −10.9590 −0.540569
\(412\) 0 0
\(413\) 2.70885 4.69186i 0.133294 0.230871i
\(414\) 0 0
\(415\) −1.92890 3.34095i −0.0946859 0.164001i
\(416\) 0 0
\(417\) −6.91210 −0.338487
\(418\) 0 0
\(419\) −1.42591 2.46975i −0.0696604 0.120655i 0.829091 0.559113i \(-0.188858\pi\)
−0.898752 + 0.438458i \(0.855525\pi\)
\(420\) 0 0
\(421\) 17.0879 + 29.5972i 0.832816 + 1.44248i 0.895797 + 0.444464i \(0.146606\pi\)
−0.0629811 + 0.998015i \(0.520061\pi\)
\(422\) 0 0
\(423\) 1.85168 3.20720i 0.0900316 0.155939i
\(424\) 0 0
\(425\) −0.405163 0.701763i −0.0196533 0.0340405i
\(426\) 0 0
\(427\) −4.93512 −0.238827
\(428\) 0 0
\(429\) 1.25029 0.0603645
\(430\) 0 0
\(431\) 2.40057 4.15791i 0.115631 0.200279i −0.802401 0.596786i \(-0.796444\pi\)
0.918032 + 0.396506i \(0.129778\pi\)
\(432\) 0 0
\(433\) 11.8358 20.5002i 0.568791 0.985175i −0.427895 0.903828i \(-0.640745\pi\)
0.996686 0.0813463i \(-0.0259220\pi\)
\(434\) 0 0
\(435\) 2.44753 4.23925i 0.117350 0.203257i
\(436\) 0 0
\(437\) 1.79451 3.10818i 0.0858429 0.148684i
\(438\) 0 0
\(439\) 8.82558 + 15.2864i 0.421222 + 0.729578i 0.996059 0.0886896i \(-0.0282679\pi\)
−0.574837 + 0.818268i \(0.694935\pi\)
\(440\) 0 0
\(441\) 2.68609 + 4.65244i 0.127909 + 0.221545i
\(442\) 0 0
\(443\) 10.1982 17.6638i 0.484531 0.839233i −0.515311 0.857003i \(-0.672323\pi\)
0.999842 + 0.0177704i \(0.00565680\pi\)
\(444\) 0 0
\(445\) 0.151145 0.00716497
\(446\) 0 0
\(447\) −15.4625 −0.731353
\(448\) 0 0
\(449\) 5.86508 10.1586i 0.276790 0.479415i −0.693795 0.720172i \(-0.744063\pi\)
0.970585 + 0.240758i \(0.0773960\pi\)
\(450\) 0 0
\(451\) −1.37535 2.38218i −0.0647629 0.112173i
\(452\) 0 0
\(453\) −6.39067 11.0690i −0.300260 0.520065i
\(454\) 0 0
\(455\) −4.02632 −0.188757
\(456\) 0 0
\(457\) −9.21088 + 15.9537i −0.430867 + 0.746283i −0.996948 0.0780661i \(-0.975125\pi\)
0.566081 + 0.824349i \(0.308459\pi\)
\(458\) 0 0
\(459\) 0.405163 + 0.701763i 0.0189114 + 0.0327555i
\(460\) 0 0
\(461\) 5.69453 0.265221 0.132610 0.991168i \(-0.457664\pi\)
0.132610 + 0.991168i \(0.457664\pi\)
\(462\) 0 0
\(463\) −5.37110 9.30301i −0.249616 0.432348i 0.713803 0.700346i \(-0.246971\pi\)
−0.963419 + 0.267999i \(0.913638\pi\)
\(464\) 0 0
\(465\) −0.134765 + 0.233420i −0.00624959 + 0.0108246i
\(466\) 0 0
\(467\) −11.7396 20.3336i −0.543243 0.940925i −0.998715 0.0506749i \(-0.983863\pi\)
0.455472 0.890250i \(-0.349471\pi\)
\(468\) 0 0
\(469\) 8.80136 + 5.62140i 0.406409 + 0.259572i
\(470\) 0 0
\(471\) 2.24885 + 3.89513i 0.103622 + 0.179478i
\(472\) 0 0
\(473\) 0.442076 0.765698i 0.0203267 0.0352068i
\(474\) 0 0
\(475\) 2.24701 + 3.89194i 0.103100 + 0.178574i
\(476\) 0 0
\(477\) −13.4945 −0.617873
\(478\) 0 0
\(479\) 13.8767 + 24.0351i 0.634041 + 1.09819i 0.986718 + 0.162446i \(0.0519382\pi\)
−0.352677 + 0.935745i \(0.614728\pi\)
\(480\) 0 0
\(481\) 7.40608 12.8277i 0.337688 0.584893i
\(482\) 0 0
\(483\) 1.01893 0.0463628
\(484\) 0 0
\(485\) −3.70238 6.41271i −0.168117 0.291186i
\(486\) 0 0
\(487\) −7.89159 13.6686i −0.357602 0.619385i 0.629958 0.776629i \(-0.283072\pi\)
−0.987560 + 0.157245i \(0.949739\pi\)
\(488\) 0 0
\(489\) 0.839614 1.45425i 0.0379686 0.0657636i
\(490\) 0 0
\(491\) 19.8770 0.897038 0.448519 0.893773i \(-0.351952\pi\)
0.448519 + 0.893773i \(0.351952\pi\)
\(492\) 0 0
\(493\) −3.96660 −0.178647
\(494\) 0 0
\(495\) 0.198096 0.343112i 0.00890375 0.0154218i
\(496\) 0 0
\(497\) −2.70098 4.67824i −0.121156 0.209848i
\(498\) 0 0
\(499\) −22.0933 38.2668i −0.989034 1.71306i −0.622420 0.782684i \(-0.713850\pi\)
−0.366614 0.930373i \(-0.619483\pi\)
\(500\) 0 0
\(501\) −6.61904 + 11.4645i −0.295717 + 0.512197i
\(502\) 0 0
\(503\) 9.32110 16.1446i 0.415607 0.719853i −0.579885 0.814699i \(-0.696902\pi\)
0.995492 + 0.0948457i \(0.0302358\pi\)
\(504\) 0 0
\(505\) 2.40349 4.16297i 0.106954 0.185250i
\(506\) 0 0
\(507\) −1.52058 + 2.63373i −0.0675314 + 0.116968i
\(508\) 0 0
\(509\) −39.9221 −1.76952 −0.884759 0.466050i \(-0.845677\pi\)
−0.884759 + 0.466050i \(0.845677\pi\)
\(510\) 0 0
\(511\) 15.2499 0.674616
\(512\) 0 0
\(513\) −2.24701 3.89194i −0.0992079 0.171833i
\(514\) 0 0
\(515\) −5.28897 + 9.16076i −0.233060 + 0.403671i
\(516\) 0 0
\(517\) 0.733619 + 1.27067i 0.0322645 + 0.0558838i
\(518\) 0 0
\(519\) −9.85480 17.0690i −0.432578 0.749246i
\(520\) 0 0
\(521\) 0.778849 0.0341220 0.0170610 0.999854i \(-0.494569\pi\)
0.0170610 + 0.999854i \(0.494569\pi\)
\(522\) 0 0
\(523\) −3.69671 6.40289i −0.161646 0.279979i 0.773813 0.633414i \(-0.218347\pi\)
−0.935459 + 0.353435i \(0.885014\pi\)
\(524\) 0 0
\(525\) −0.637931 + 1.10493i −0.0278416 + 0.0482230i
\(526\) 0 0
\(527\) 0.218408 0.00951398
\(528\) 0 0
\(529\) 11.1811 19.3662i 0.486135 0.842010i
\(530\) 0 0
\(531\) 4.24630 0.184274
\(532\) 0 0
\(533\) −21.9100 −0.949030
\(534\) 0 0
\(535\) −10.9381 −0.472896
\(536\) 0 0
\(537\) 15.8857 0.685519
\(538\) 0 0
\(539\) −2.12841 −0.0916773
\(540\) 0 0
\(541\) −6.57101 −0.282510 −0.141255 0.989973i \(-0.545114\pi\)
−0.141255 + 0.989973i \(0.545114\pi\)
\(542\) 0 0
\(543\) 12.4536 21.5702i 0.534433 0.925666i
\(544\) 0 0
\(545\) 11.5903 0.496474
\(546\) 0 0
\(547\) −0.930680 + 1.61199i −0.0397930 + 0.0689235i −0.885236 0.465142i \(-0.846003\pi\)
0.845443 + 0.534066i \(0.179337\pi\)
\(548\) 0 0
\(549\) −1.93403 3.34985i −0.0825426 0.142968i
\(550\) 0 0
\(551\) 21.9985 0.937169
\(552\) 0 0
\(553\) 0.326112 + 0.564842i 0.0138677 + 0.0240195i
\(554\) 0 0
\(555\) −2.34684 4.06485i −0.0996180 0.172543i
\(556\) 0 0
\(557\) −16.8596 + 29.2017i −0.714364 + 1.23731i 0.248841 + 0.968544i \(0.419950\pi\)
−0.963204 + 0.268770i \(0.913383\pi\)
\(558\) 0 0
\(559\) −3.52124 6.09896i −0.148933 0.257959i
\(560\) 0 0
\(561\) −0.321045 −0.0135545
\(562\) 0 0
\(563\) 4.40249 0.185543 0.0927714 0.995687i \(-0.470427\pi\)
0.0927714 + 0.995687i \(0.470427\pi\)
\(564\) 0 0
\(565\) −5.17900 + 8.97029i −0.217882 + 0.377383i
\(566\) 0 0
\(567\) 0.637931 1.10493i 0.0267906 0.0464026i
\(568\) 0 0
\(569\) 11.3850 19.7194i 0.477285 0.826681i −0.522377 0.852715i \(-0.674954\pi\)
0.999661 + 0.0260339i \(0.00828778\pi\)
\(570\) 0 0
\(571\) −6.29626 + 10.9054i −0.263490 + 0.456378i −0.967167 0.254142i \(-0.918207\pi\)
0.703677 + 0.710520i \(0.251540\pi\)
\(572\) 0 0
\(573\) −11.7379 20.3306i −0.490358 0.849324i
\(574\) 0 0
\(575\) −0.399310 0.691625i −0.0166524 0.0288428i
\(576\) 0 0
\(577\) −1.85892 + 3.21974i −0.0773878 + 0.134040i −0.902122 0.431481i \(-0.857991\pi\)
0.824734 + 0.565520i \(0.191325\pi\)
\(578\) 0 0
\(579\) 22.6977 0.943285
\(580\) 0 0
\(581\) 4.92201 0.204199
\(582\) 0 0
\(583\) 2.67322 4.63015i 0.110713 0.191761i
\(584\) 0 0
\(585\) −1.57788 2.73297i −0.0652374 0.112994i
\(586\) 0 0
\(587\) −1.32677 2.29803i −0.0547616 0.0948499i 0.837345 0.546675i \(-0.184107\pi\)
−0.892107 + 0.451825i \(0.850773\pi\)
\(588\) 0 0
\(589\) −1.21128 −0.0499097
\(590\) 0 0
\(591\) 6.66683 11.5473i 0.274237 0.474992i
\(592\) 0 0
\(593\) −14.3424 24.8418i −0.588972 1.02013i −0.994367 0.105988i \(-0.966200\pi\)
0.405396 0.914141i \(-0.367134\pi\)
\(594\) 0 0
\(595\) 1.03386 0.0423843
\(596\) 0 0
\(597\) −9.65718 16.7267i −0.395242 0.684579i
\(598\) 0 0
\(599\) 15.9479 27.6226i 0.651614 1.12863i −0.331117 0.943590i \(-0.607425\pi\)
0.982731 0.185039i \(-0.0592413\pi\)
\(600\) 0 0
\(601\) −6.43018 11.1374i −0.262293 0.454304i 0.704558 0.709646i \(-0.251145\pi\)
−0.966851 + 0.255342i \(0.917812\pi\)
\(602\) 0 0
\(603\) −0.366493 + 8.17714i −0.0149248 + 0.332999i
\(604\) 0 0
\(605\) −5.42152 9.39034i −0.220416 0.381772i
\(606\) 0 0
\(607\) 14.2787 24.7314i 0.579553 1.00381i −0.415978 0.909375i \(-0.636561\pi\)
0.995531 0.0944402i \(-0.0301061\pi\)
\(608\) 0 0
\(609\) 3.12271 + 5.40870i 0.126539 + 0.219172i
\(610\) 0 0
\(611\) 11.6869 0.472801
\(612\) 0 0
\(613\) 6.60763 + 11.4447i 0.266880 + 0.462249i 0.968054 0.250740i \(-0.0806741\pi\)
−0.701175 + 0.712989i \(0.747341\pi\)
\(614\) 0 0
\(615\) −3.47144 + 6.01270i −0.139982 + 0.242456i
\(616\) 0 0
\(617\) −16.2276 −0.653300 −0.326650 0.945145i \(-0.605920\pi\)
−0.326650 + 0.945145i \(0.605920\pi\)
\(618\) 0 0
\(619\) 19.7548 + 34.2164i 0.794014 + 1.37527i 0.923463 + 0.383687i \(0.125346\pi\)
−0.129449 + 0.991586i \(0.541321\pi\)
\(620\) 0 0
\(621\) 0.399310 + 0.691625i 0.0160238 + 0.0277540i
\(622\) 0 0
\(623\) −0.0964201 + 0.167005i −0.00386299 + 0.00669090i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.78050 0.0711061
\(628\) 0 0
\(629\) −1.90171 + 3.29386i −0.0758261 + 0.131335i
\(630\) 0 0
\(631\) 10.9545 + 18.9738i 0.436092 + 0.755334i 0.997384 0.0722841i \(-0.0230288\pi\)
−0.561292 + 0.827618i \(0.689695\pi\)
\(632\) 0 0
\(633\) −7.90244 13.6874i −0.314094 0.544027i
\(634\) 0 0
\(635\) 6.69595 11.5977i 0.265721 0.460241i
\(636\) 0 0
\(637\) −8.47666 + 14.6820i −0.335857 + 0.581722i
\(638\) 0 0
\(639\) 2.11699 3.66673i 0.0837467 0.145054i
\(640\) 0 0
\(641\) −5.05891 + 8.76229i −0.199815 + 0.346090i −0.948468 0.316872i \(-0.897367\pi\)
0.748653 + 0.662962i \(0.230701\pi\)
\(642\) 0 0
\(643\) −20.0113 −0.789167 −0.394584 0.918860i \(-0.629111\pi\)
−0.394584 + 0.918860i \(0.629111\pi\)
\(644\) 0 0
\(645\) −2.23162 −0.0878701
\(646\) 0 0
\(647\) −2.21176 3.83089i −0.0869534 0.150608i 0.819269 0.573410i \(-0.194380\pi\)
−0.906222 + 0.422802i \(0.861046\pi\)
\(648\) 0 0
\(649\) −0.841176 + 1.45696i −0.0330190 + 0.0571907i
\(650\) 0 0
\(651\) −0.171942 0.297812i −0.00673893 0.0116722i
\(652\) 0 0
\(653\) −0.209793 0.363372i −0.00820984 0.0142199i 0.861891 0.507093i \(-0.169280\pi\)
−0.870101 + 0.492873i \(0.835947\pi\)
\(654\) 0 0
\(655\) 3.97325 0.155248
\(656\) 0 0
\(657\) 5.97631 + 10.3513i 0.233158 + 0.403842i
\(658\) 0 0
\(659\) −0.936555 + 1.62216i −0.0364830 + 0.0631904i −0.883690 0.468072i \(-0.844949\pi\)
0.847207 + 0.531262i \(0.178282\pi\)
\(660\) 0 0
\(661\) −14.6664 −0.570457 −0.285228 0.958460i \(-0.592069\pi\)
−0.285228 + 0.958460i \(0.592069\pi\)
\(662\) 0 0
\(663\) −1.27860 + 2.21460i −0.0496567 + 0.0860078i
\(664\) 0 0
\(665\) −5.73375 −0.222345
\(666\) 0 0
\(667\) −3.90930 −0.151369
\(668\) 0 0
\(669\) −0.134772 −0.00521057
\(670\) 0 0
\(671\) 1.53250 0.0591614
\(672\) 0 0
\(673\) −26.6890 −1.02878 −0.514392 0.857555i \(-0.671982\pi\)
−0.514392 + 0.857555i \(0.671982\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 7.99612 13.8497i 0.307316 0.532287i −0.670458 0.741947i \(-0.733903\pi\)
0.977774 + 0.209661i \(0.0672359\pi\)
\(678\) 0 0
\(679\) 9.44745 0.362560
\(680\) 0 0
\(681\) −8.05133 + 13.9453i −0.308528 + 0.534386i
\(682\) 0 0
\(683\) 7.17371 + 12.4252i 0.274495 + 0.475438i 0.970007 0.243075i \(-0.0781562\pi\)
−0.695513 + 0.718514i \(0.744823\pi\)
\(684\) 0 0
\(685\) −10.9590 −0.418723
\(686\) 0 0
\(687\) −6.14349 10.6408i −0.234389 0.405973i
\(688\) 0 0
\(689\) −21.2928 36.8802i −0.811191 1.40502i
\(690\) 0 0
\(691\) 0.996519 1.72602i 0.0379094 0.0656609i −0.846448 0.532471i \(-0.821264\pi\)
0.884358 + 0.466810i \(0.154597\pi\)
\(692\) 0 0
\(693\) 0.252743 + 0.437764i 0.00960091 + 0.0166293i
\(694\) 0 0
\(695\) −6.91210 −0.262191
\(696\) 0 0
\(697\) 5.62599 0.213100
\(698\) 0 0
\(699\) 2.41572 4.18416i 0.0913711 0.158259i
\(700\) 0 0
\(701\) −22.6284 + 39.1935i −0.854662 + 1.48032i 0.0222959 + 0.999751i \(0.492902\pi\)
−0.876958 + 0.480567i \(0.840431\pi\)
\(702\) 0 0
\(703\) 10.5468 18.2675i 0.397779 0.688973i
\(704\) 0 0
\(705\) 1.85168 3.20720i 0.0697381 0.120790i
\(706\) 0 0
\(707\) 3.06652 + 5.31137i 0.115328 + 0.199755i
\(708\) 0 0
\(709\) 6.66494 + 11.5440i 0.250307 + 0.433544i 0.963610 0.267311i \(-0.0861351\pi\)
−0.713303 + 0.700855i \(0.752802\pi\)
\(710\) 0 0
\(711\) −0.255601 + 0.442715i −0.00958580 + 0.0166031i
\(712\) 0 0
\(713\) 0.215252 0.00806126
\(714\) 0 0
\(715\) 1.25029 0.0467581
\(716\) 0 0
\(717\) 1.22293 2.11817i 0.0456710 0.0791045i
\(718\) 0 0
\(719\) −13.7027 23.7338i −0.511025 0.885121i −0.999918 0.0127778i \(-0.995933\pi\)
0.488893 0.872344i \(-0.337401\pi\)
\(720\) 0 0
\(721\) −6.74799 11.6879i −0.251308 0.435279i
\(722\) 0 0
\(723\) −18.1372 −0.674529
\(724\) 0 0
\(725\) 2.44753 4.23925i 0.0908991 0.157442i
\(726\) 0 0
\(727\) −10.3913 17.9982i −0.385391 0.667517i 0.606432 0.795135i \(-0.292600\pi\)
−0.991823 + 0.127618i \(0.959267\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.904172 + 1.56607i 0.0334420 + 0.0579233i
\(732\) 0 0
\(733\) 19.7811 34.2619i 0.730631 1.26549i −0.225983 0.974131i \(-0.572559\pi\)
0.956614 0.291359i \(-0.0941074\pi\)
\(734\) 0 0
\(735\) 2.68609 + 4.65244i 0.0990779 + 0.171608i
\(736\) 0 0
\(737\) −2.73308 1.74561i −0.100674 0.0643003i
\(738\) 0 0
\(739\) 22.4427 + 38.8719i 0.825567 + 1.42992i 0.901485 + 0.432810i \(0.142478\pi\)
−0.0759177 + 0.997114i \(0.524189\pi\)
\(740\) 0 0
\(741\) 7.09103 12.2820i 0.260496 0.451191i
\(742\) 0 0
\(743\) −13.9220 24.1137i −0.510750 0.884644i −0.999922 0.0124575i \(-0.996035\pi\)
0.489173 0.872187i \(-0.337299\pi\)
\(744\) 0 0
\(745\) −15.4625 −0.566504
\(746\) 0 0
\(747\) 1.92890 + 3.34095i 0.0705747 + 0.122239i
\(748\) 0 0
\(749\) 6.97776 12.0858i 0.254962 0.441607i
\(750\) 0 0
\(751\) −17.8633 −0.651840 −0.325920 0.945397i \(-0.605674\pi\)
−0.325920 + 0.945397i \(0.605674\pi\)
\(752\) 0 0
\(753\) −10.8009 18.7078i −0.393608 0.681748i
\(754\) 0 0
\(755\) −6.39067 11.0690i −0.232580 0.402841i
\(756\) 0 0
\(757\) −7.36267 + 12.7525i −0.267601 + 0.463498i −0.968242 0.250016i \(-0.919564\pi\)
0.700641 + 0.713514i \(0.252898\pi\)
\(758\) 0 0
\(759\) −0.316407 −0.0114848
\(760\) 0 0
\(761\) −27.1522 −0.984265 −0.492132 0.870520i \(-0.663782\pi\)
−0.492132 + 0.870520i \(0.663782\pi\)
\(762\) 0 0
\(763\) −7.39381 + 12.8065i −0.267674 + 0.463625i
\(764\) 0 0
\(765\) 0.405163 + 0.701763i 0.0146487 + 0.0253723i
\(766\) 0 0
\(767\) 6.70016 + 11.6050i 0.241929 + 0.419033i
\(768\) 0 0
\(769\) −14.9693 + 25.9275i −0.539806 + 0.934971i 0.459108 + 0.888380i \(0.348169\pi\)
−0.998914 + 0.0465909i \(0.985164\pi\)
\(770\) 0 0
\(771\) 13.2554 22.9591i 0.477383 0.826851i
\(772\) 0 0
\(773\) 1.92267 3.33017i 0.0691537 0.119778i −0.829375 0.558692i \(-0.811303\pi\)
0.898529 + 0.438914i \(0.144637\pi\)
\(774\) 0 0
\(775\) −0.134765 + 0.233420i −0.00484091 + 0.00838470i
\(776\) 0 0
\(777\) 5.98849 0.214836
\(778\) 0 0
\(779\) −31.2014 −1.11791
\(780\) 0 0
\(781\) 0.838733 + 1.45273i 0.0300122 + 0.0519827i
\(782\) 0 0
\(783\) −2.44753 + 4.23925i −0.0874677 + 0.151499i
\(784\) 0 0
\(785\) 2.24885 + 3.89513i 0.0802650 + 0.139023i
\(786\) 0 0
\(787\) 18.3853 + 31.8442i 0.655364 + 1.13512i 0.981802 + 0.189905i \(0.0608181\pi\)
−0.326438 + 0.945218i \(0.605849\pi\)
\(788\) 0 0
\(789\) 1.98177 0.0705527
\(790\) 0 0
\(791\) −6.60768 11.4448i −0.234942 0.406932i
\(792\) 0 0
\(793\) 6.10335 10.5713i 0.216736 0.375399i
\(794\) 0 0
\(795\) −13.4945 −0.478602
\(796\) 0 0
\(797\) 14.6157 25.3152i 0.517716 0.896711i −0.482072 0.876132i \(-0.660116\pi\)
0.999788 0.0205793i \(-0.00655105\pi\)
\(798\) 0 0
\(799\) −3.00092 −0.106165
\(800\) 0 0
\(801\) −0.151145 −0.00534045
\(802\) 0 0
\(803\) −4.73553 −0.167113
\(804\) 0 0
\(805\) 1.01893 0.0359125
\(806\) 0 0
\(807\) 19.6325 0.691097
\(808\) 0 0
\(809\) −20.9794 −0.737596 −0.368798 0.929510i \(-0.620231\pi\)
−0.368798 + 0.929510i \(0.620231\pi\)
\(810\) 0 0
\(811\) 20.5242 35.5490i 0.720703 1.24829i −0.240015 0.970769i \(-0.577152\pi\)
0.960718 0.277525i \(-0.0895142\pi\)
\(812\) 0 0
\(813\) −16.1989 −0.568120
\(814\) 0 0
\(815\) 0.839614 1.45425i 0.0294104 0.0509403i
\(816\) 0 0
\(817\) −5.01448 8.68534i −0.175435 0.303862i
\(818\) 0 0
\(819\) 4.02632 0.140691
\(820\) 0 0
\(821\) −8.81163 15.2622i −0.307528 0.532654i 0.670293 0.742096i \(-0.266168\pi\)
−0.977821 + 0.209443i \(0.932835\pi\)
\(822\) 0 0
\(823\) −18.7257 32.4338i −0.652736 1.13057i −0.982456 0.186493i \(-0.940288\pi\)
0.329720 0.944079i \(-0.393046\pi\)
\(824\) 0 0
\(825\) 0.198096 0.343112i 0.00689682 0.0119456i
\(826\) 0 0
\(827\) 26.9726 + 46.7180i 0.937930 + 1.62454i 0.769323 + 0.638859i \(0.220593\pi\)
0.168607 + 0.985683i \(0.446073\pi\)
\(828\) 0 0
\(829\) −18.4348 −0.640268 −0.320134 0.947372i \(-0.603728\pi\)
−0.320134 + 0.947372i \(0.603728\pi\)
\(830\) 0 0
\(831\) −20.6486 −0.716293
\(832\) 0 0
\(833\) 2.17661 3.77000i 0.0754150 0.130623i
\(834\) 0 0
\(835\) −6.61904 + 11.4645i −0.229061 + 0.396746i
\(836\) 0 0
\(837\) 0.134765 0.233420i 0.00465817 0.00806818i
\(838\) 0 0
\(839\) 3.01582 5.22355i 0.104118 0.180337i −0.809260 0.587451i \(-0.800131\pi\)
0.913377 + 0.407114i \(0.133465\pi\)
\(840\) 0 0
\(841\) 2.51916 + 4.36331i 0.0868675 + 0.150459i
\(842\) 0 0
\(843\) −3.36886 5.83504i −0.116030 0.200969i
\(844\) 0 0
\(845\) −1.52058 + 2.63373i −0.0523096 + 0.0906029i
\(846\) 0 0
\(847\) 13.8342 0.475349
\(848\) 0 0
\(849\) −16.5873 −0.569276
\(850\) 0 0
\(851\) −1.87424 + 3.24627i −0.0642480 + 0.111281i
\(852\) 0 0
\(853\) −26.8343 46.4784i −0.918789 1.59139i −0.801257 0.598320i \(-0.795835\pi\)
−0.117531 0.993069i \(-0.537498\pi\)
\(854\) 0 0
\(855\) −2.24701 3.89194i −0.0768461 0.133101i
\(856\) 0 0
\(857\) 38.1692 1.30383 0.651917 0.758290i \(-0.273965\pi\)
0.651917 + 0.758290i \(0.273965\pi\)
\(858\) 0 0
\(859\) 4.85685 8.41231i 0.165714 0.287024i −0.771195 0.636599i \(-0.780341\pi\)
0.936908 + 0.349575i \(0.113674\pi\)
\(860\) 0 0
\(861\) −4.42907 7.67138i −0.150942 0.261440i
\(862\) 0 0
\(863\) 13.1358 0.447148 0.223574 0.974687i \(-0.428228\pi\)
0.223574 + 0.974687i \(0.428228\pi\)
\(864\) 0 0
\(865\) −9.85480 17.0690i −0.335073 0.580364i
\(866\) 0 0
\(867\) −8.17169 + 14.1538i −0.277525 + 0.480687i
\(868\) 0 0
\(869\) −0.101267 0.175400i −0.00343526 0.00595004i
\(870\) 0 0
\(871\) −22.9262 + 11.9009i −0.776824 + 0.403248i
\(872\) 0 0
\(873\) 3.70238 + 6.41271i 0.125307 + 0.217037i
\(874\) 0 0
\(875\) −0.637931 + 1.10493i −0.0215660 + 0.0373534i
\(876\) 0 0
\(877\) 24.6683 + 42.7268i 0.832990 + 1.44278i 0.895657 + 0.444746i \(0.146706\pi\)
−0.0626674 + 0.998034i \(0.519961\pi\)
\(878\) 0 0
\(879\) 30.5791 1.03141
\(880\) 0 0
\(881\) −4.81030 8.33168i −0.162063 0.280701i 0.773545 0.633741i \(-0.218481\pi\)
−0.935608 + 0.353039i \(0.885148\pi\)
\(882\) 0 0
\(883\) −16.5403 + 28.6487i −0.556627 + 0.964106i 0.441148 + 0.897434i \(0.354571\pi\)
−0.997775 + 0.0666719i \(0.978762\pi\)
\(884\) 0 0
\(885\) 4.24630 0.142738
\(886\) 0 0
\(887\) −16.4481 28.4890i −0.552274 0.956567i −0.998110 0.0614522i \(-0.980427\pi\)
0.445836 0.895115i \(-0.352907\pi\)
\(888\) 0 0
\(889\) 8.54310 + 14.7971i 0.286526 + 0.496278i
\(890\) 0 0
\(891\) −0.198096 + 0.343112i −0.00663647 + 0.0114947i
\(892\) 0 0
\(893\) 16.6429 0.556935
\(894\) 0 0
\(895\) 15.8857 0.531000
\(896\) 0 0
\(897\) −1.26013 + 2.18260i −0.0420744 + 0.0728750i
\(898\) 0 0
\(899\) 0.659685 + 1.14261i 0.0220017 + 0.0381081i
\(900\) 0 0
\(901\) 5.46749 + 9.46998i 0.182149 + 0.315491i
\(902\) 0 0
\(903\) 1.42362 2.46579i 0.0473752 0.0820562i
\(904\) 0 0
\(905\) 12.4536 21.5702i 0.413970 0.717018i
\(906\) 0 0
\(907\) 8.23573 14.2647i 0.273463 0.473651i −0.696283 0.717767i \(-0.745164\pi\)
0.969746 + 0.244116i \(0.0784976\pi\)
\(908\) 0 0
\(909\) −2.40349 + 4.16297i −0.0797187 + 0.138077i
\(910\) 0 0
\(911\) 6.94871 0.230221 0.115110 0.993353i \(-0.463278\pi\)
0.115110 + 0.993353i \(0.463278\pi\)
\(912\) 0 0
\(913\) −1.52843 −0.0505836
\(914\) 0 0
\(915\) −1.93403 3.34985i −0.0639372 0.110743i
\(916\) 0 0
\(917\) −2.53466 + 4.39016i −0.0837018 + 0.144976i
\(918\) 0 0
\(919\) −16.7034 28.9311i −0.550994 0.954349i −0.998203 0.0599197i \(-0.980916\pi\)
0.447210 0.894429i \(-0.352418\pi\)
\(920\) 0 0
\(921\) −9.72786 16.8491i −0.320544 0.555198i
\(922\) 0 0
\(923\) 13.3614 0.439796
\(924\) 0 0
\(925\) −2.34684 4.06485i −0.0771637 0.133652i
\(926\) 0 0
\(927\) 5.28897 9.16076i 0.173713 0.300879i
\(928\) 0 0
\(929\) −4.16111 −0.136522 −0.0682609 0.997668i \(-0.521745\pi\)
−0.0682609 + 0.997668i \(0.521745\pi\)
\(930\) 0 0
\(931\) −12.0713 + 20.9082i −0.395622 + 0.685238i
\(932\) 0 0
\(933\) −31.9314 −1.04539
\(934\) 0 0
\(935\) −0.321045 −0.0104993
\(936\) 0 0
\(937\) 3.23307 0.105620 0.0528099 0.998605i \(-0.483182\pi\)
0.0528099 + 0.998605i \(0.483182\pi\)
\(938\) 0 0
\(939\) 24.0643 0.785308
\(940\) 0 0
\(941\) −30.5570 −0.996131 −0.498065 0.867139i \(-0.665956\pi\)
−0.498065 + 0.867139i \(0.665956\pi\)
\(942\) 0 0
\(943\) 5.54471 0.180561
\(944\) 0 0
\(945\) 0.637931 1.10493i 0.0207519 0.0359433i
\(946\) 0 0
\(947\) −1.08738 −0.0353350 −0.0176675 0.999844i \(-0.505624\pi\)
−0.0176675 + 0.999844i \(0.505624\pi\)
\(948\) 0 0
\(949\) −18.8598 + 32.6662i −0.612216 + 1.06039i
\(950\) 0 0
\(951\) 15.6368 + 27.0837i 0.507057 + 0.878248i
\(952\) 0 0
\(953\) −36.4069 −1.17933 −0.589667 0.807646i \(-0.700741\pi\)
−0.589667 + 0.807646i \(0.700741\pi\)
\(954\) 0 0
\(955\) −11.7379 20.3306i −0.379829 0.657884i
\(956\) 0 0
\(957\) −0.969693 1.67956i −0.0313457 0.0542924i
\(958\) 0 0
\(959\) 6.99110 12.1089i 0.225754 0.391018i
\(960\) 0 0
\(961\) 15.4637 + 26.7839i 0.498828 + 0.863996i
\(962\) 0 0
\(963\) 10.9381 0.352476
\(964\) 0 0
\(965\) 22.6977 0.730665
\(966\) 0 0
\(967\) −2.18753 + 3.78891i −0.0703463 + 0.121843i −0.899053 0.437840i \(-0.855744\pi\)
0.828707 + 0.559683i \(0.189077\pi\)
\(968\) 0 0
\(969\) −1.82081 + 3.15374i −0.0584929 + 0.101313i
\(970\) 0 0
\(971\) 12.7527 22.0883i 0.409253 0.708847i −0.585553 0.810634i \(-0.699123\pi\)
0.994806 + 0.101787i \(0.0324560\pi\)
\(972\) 0 0
\(973\) 4.40944 7.63737i 0.141360 0.244843i
\(974\) 0 0
\(975\) −1.57788 2.73297i −0.0505326 0.0875251i
\(976\) 0 0
\(977\) −6.37763 11.0464i −0.204038 0.353405i 0.745788 0.666184i \(-0.232073\pi\)
−0.949826 + 0.312779i \(0.898740\pi\)
\(978\) 0 0
\(979\) 0.0299412 0.0518598i 0.000956926 0.00165745i
\(980\) 0 0
\(981\) −11.5903 −0.370050
\(982\) 0 0
\(983\) 51.9035 1.65546 0.827732 0.561123i \(-0.189631\pi\)
0.827732 + 0.561123i \(0.189631\pi\)
\(984\) 0 0
\(985\) 6.66683 11.5473i 0.212423 0.367927i
\(986\) 0 0
\(987\) 2.36248 + 4.09194i 0.0751986 + 0.130248i
\(988\) 0 0
\(989\) 0.891110 + 1.54345i 0.0283356 + 0.0490788i
\(990\) 0 0
\(991\) 17.9247 0.569397 0.284699 0.958617i \(-0.408106\pi\)
0.284699 + 0.958617i \(0.408106\pi\)
\(992\) 0 0
\(993\) 13.6369 23.6198i 0.432753 0.749551i
\(994\) 0 0
\(995\) −9.65718 16.7267i −0.306153 0.530273i
\(996\) 0 0
\(997\) −6.78152 −0.214773 −0.107386 0.994217i \(-0.534248\pi\)
−0.107386 + 0.994217i \(0.534248\pi\)
\(998\) 0 0
\(999\) 2.34684 + 4.06485i 0.0742508 + 0.128606i
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.841.8 22
67.29 even 3 inner 4020.2.q.l.3781.8 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.l.841.8 22 1.1 even 1 trivial
4020.2.q.l.3781.8 yes 22 67.29 even 3 inner