Properties

Label 1560.2.bg.e.841.1
Level $1560$
Weight $2$
Character 1560.841
Analytic conductor $12.457$
Analytic rank $0$
Dimension $4$
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] = [1560,2,Mod(601,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("1560.601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.bg (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: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 841.1
Root \(-1.58114 - 2.73861i\) of defining polynomial
Character \(\chi\) \(=\) 1560.841
Dual form 1560.2.bg.e.601.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} -1.00000 q^{5} +(-2.08114 - 3.60464i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} -1.00000 q^{5} +(-2.08114 - 3.60464i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.00000 - 1.73205i) q^{11} +(3.08114 + 1.87259i) q^{13} +(-0.500000 + 0.866025i) q^{15} +(0.418861 + 0.725489i) q^{17} +(-1.00000 - 1.73205i) q^{19} -4.16228 q^{21} +(1.00000 - 1.73205i) q^{23} +1.00000 q^{25} -1.00000 q^{27} +(3.58114 - 6.20271i) q^{29} -9.32456 q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.08114 + 3.60464i) q^{35} +(-4.16228 + 7.20928i) q^{37} +(3.16228 - 1.73205i) q^{39} +(-2.58114 + 4.47066i) q^{41} +(-5.08114 - 8.80079i) q^{43} +(0.500000 + 0.866025i) q^{45} -7.48683 q^{47} +(-5.16228 + 8.94133i) q^{49} +0.837722 q^{51} -6.32456 q^{53} +(-1.00000 + 1.73205i) q^{55} -2.00000 q^{57} +(3.58114 + 6.20271i) q^{59} +(-6.50000 - 11.2583i) q^{61} +(-2.08114 + 3.60464i) q^{63} +(-3.08114 - 1.87259i) q^{65} +(-5.08114 + 8.80079i) q^{67} +(-1.00000 - 1.73205i) q^{69} +(-2.74342 - 4.75174i) q^{71} -6.16228 q^{73} +(0.500000 - 0.866025i) q^{75} -8.32456 q^{77} +11.0000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(-0.418861 - 0.725489i) q^{85} +(-3.58114 - 6.20271i) q^{87} +(7.74342 - 13.4120i) q^{89} +(0.337722 - 15.0035i) q^{91} +(-4.66228 + 8.07530i) q^{93} +(1.00000 + 1.73205i) q^{95} +(6.40569 + 11.0950i) q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} + 8 q^{17} - 4 q^{19} - 4 q^{21} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 8 q^{29} - 12 q^{31} - 4 q^{33} + 2 q^{35} - 4 q^{37} - 4 q^{41} - 14 q^{43} + 2 q^{45} + 8 q^{47} - 8 q^{49} + 16 q^{51} - 4 q^{55} - 8 q^{57} + 8 q^{59} - 26 q^{61} - 2 q^{63} - 6 q^{65} - 14 q^{67} - 4 q^{69} + 8 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} + 44 q^{79} - 2 q^{81} - 8 q^{85} - 8 q^{87} + 12 q^{89} + 14 q^{91} - 6 q^{93} + 4 q^{95} - 6 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.08114 3.60464i −0.786597 1.36243i −0.928041 0.372479i \(-0.878508\pi\)
0.141444 0.989946i \(-0.454825\pi\)
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) 3.08114 + 1.87259i 0.854554 + 0.519362i
\(14\) 0 0
\(15\) −0.500000 + 0.866025i −0.129099 + 0.223607i
\(16\) 0 0
\(17\) 0.418861 + 0.725489i 0.101589 + 0.175957i 0.912339 0.409435i \(-0.134274\pi\)
−0.810751 + 0.585392i \(0.800941\pi\)
\(18\) 0 0
\(19\) −1.00000 1.73205i −0.229416 0.397360i 0.728219 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) −4.16228 −0.908283
\(22\) 0 0
\(23\) 1.00000 1.73205i 0.208514 0.361158i −0.742732 0.669588i \(-0.766471\pi\)
0.951247 + 0.308431i \(0.0998038\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.58114 6.20271i 0.665001 1.15182i −0.314284 0.949329i \(-0.601765\pi\)
0.979285 0.202486i \(-0.0649021\pi\)
\(30\) 0 0
\(31\) −9.32456 −1.67474 −0.837370 0.546637i \(-0.815908\pi\)
−0.837370 + 0.546637i \(0.815908\pi\)
\(32\) 0 0
\(33\) −1.00000 1.73205i −0.174078 0.301511i
\(34\) 0 0
\(35\) 2.08114 + 3.60464i 0.351777 + 0.609295i
\(36\) 0 0
\(37\) −4.16228 + 7.20928i −0.684274 + 1.18520i 0.289390 + 0.957211i \(0.406548\pi\)
−0.973664 + 0.227986i \(0.926786\pi\)
\(38\) 0 0
\(39\) 3.16228 1.73205i 0.506370 0.277350i
\(40\) 0 0
\(41\) −2.58114 + 4.47066i −0.403106 + 0.698200i −0.994099 0.108476i \(-0.965403\pi\)
0.590993 + 0.806677i \(0.298736\pi\)
\(42\) 0 0
\(43\) −5.08114 8.80079i −0.774866 1.34211i −0.934870 0.354991i \(-0.884484\pi\)
0.160003 0.987117i \(-0.448850\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 0 0
\(47\) −7.48683 −1.09207 −0.546033 0.837763i \(-0.683863\pi\)
−0.546033 + 0.837763i \(0.683863\pi\)
\(48\) 0 0
\(49\) −5.16228 + 8.94133i −0.737468 + 1.27733i
\(50\) 0 0
\(51\) 0.837722 0.117305
\(52\) 0 0
\(53\) −6.32456 −0.868744 −0.434372 0.900733i \(-0.643030\pi\)
−0.434372 + 0.900733i \(0.643030\pi\)
\(54\) 0 0
\(55\) −1.00000 + 1.73205i −0.134840 + 0.233550i
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 3.58114 + 6.20271i 0.466225 + 0.807525i 0.999256 0.0385709i \(-0.0122805\pi\)
−0.533031 + 0.846096i \(0.678947\pi\)
\(60\) 0 0
\(61\) −6.50000 11.2583i −0.832240 1.44148i −0.896258 0.443533i \(-0.853725\pi\)
0.0640184 0.997949i \(-0.479608\pi\)
\(62\) 0 0
\(63\) −2.08114 + 3.60464i −0.262199 + 0.454142i
\(64\) 0 0
\(65\) −3.08114 1.87259i −0.382168 0.232266i
\(66\) 0 0
\(67\) −5.08114 + 8.80079i −0.620760 + 1.07519i 0.368585 + 0.929594i \(0.379842\pi\)
−0.989344 + 0.145593i \(0.953491\pi\)
\(68\) 0 0
\(69\) −1.00000 1.73205i −0.120386 0.208514i
\(70\) 0 0
\(71\) −2.74342 4.75174i −0.325584 0.563927i 0.656047 0.754720i \(-0.272227\pi\)
−0.981630 + 0.190793i \(0.938894\pi\)
\(72\) 0 0
\(73\) −6.16228 −0.721240 −0.360620 0.932713i \(-0.617435\pi\)
−0.360620 + 0.932713i \(0.617435\pi\)
\(74\) 0 0
\(75\) 0.500000 0.866025i 0.0577350 0.100000i
\(76\) 0 0
\(77\) −8.32456 −0.948671
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −0.418861 0.725489i −0.0454319 0.0786903i
\(86\) 0 0
\(87\) −3.58114 6.20271i −0.383938 0.665001i
\(88\) 0 0
\(89\) 7.74342 13.4120i 0.820801 1.42167i −0.0842864 0.996442i \(-0.526861\pi\)
0.905087 0.425227i \(-0.139806\pi\)
\(90\) 0 0
\(91\) 0.337722 15.0035i 0.0354029 1.57279i
\(92\) 0 0
\(93\) −4.66228 + 8.07530i −0.483456 + 0.837370i
\(94\) 0 0
\(95\) 1.00000 + 1.73205i 0.102598 + 0.177705i
\(96\) 0 0
\(97\) 6.40569 + 11.0950i 0.650400 + 1.12653i 0.983026 + 0.183467i \(0.0587319\pi\)
−0.332626 + 0.943059i \(0.607935\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −0.162278 + 0.281073i −0.0161472 + 0.0279678i −0.873986 0.485951i \(-0.838473\pi\)
0.857839 + 0.513919i \(0.171807\pi\)
\(102\) 0 0
\(103\) −10.4868 −1.03330 −0.516649 0.856197i \(-0.672821\pi\)
−0.516649 + 0.856197i \(0.672821\pi\)
\(104\) 0 0
\(105\) 4.16228 0.406197
\(106\) 0 0
\(107\) 9.74342 16.8761i 0.941932 1.63147i 0.180150 0.983639i \(-0.442342\pi\)
0.761781 0.647834i \(-0.224325\pi\)
\(108\) 0 0
\(109\) 7.32456 0.701565 0.350783 0.936457i \(-0.385916\pi\)
0.350783 + 0.936457i \(0.385916\pi\)
\(110\) 0 0
\(111\) 4.16228 + 7.20928i 0.395066 + 0.684274i
\(112\) 0 0
\(113\) 2.16228 + 3.74517i 0.203410 + 0.352316i 0.949625 0.313389i \(-0.101464\pi\)
−0.746215 + 0.665705i \(0.768131\pi\)
\(114\) 0 0
\(115\) −1.00000 + 1.73205i −0.0932505 + 0.161515i
\(116\) 0 0
\(117\) 0.0811388 3.60464i 0.00750129 0.333249i
\(118\) 0 0
\(119\) 1.74342 3.01969i 0.159819 0.276814i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 2.58114 + 4.47066i 0.232733 + 0.403106i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.0811388 + 0.140537i −0.00719991 + 0.0124706i −0.869603 0.493752i \(-0.835625\pi\)
0.862403 + 0.506222i \(0.168958\pi\)
\(128\) 0 0
\(129\) −10.1623 −0.894739
\(130\) 0 0
\(131\) 15.8114 1.38145 0.690724 0.723119i \(-0.257292\pi\)
0.690724 + 0.723119i \(0.257292\pi\)
\(132\) 0 0
\(133\) −4.16228 + 7.20928i −0.360915 + 0.625124i
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −6.41886 11.1178i −0.548400 0.949857i −0.998384 0.0568206i \(-0.981904\pi\)
0.449984 0.893037i \(-0.351430\pi\)
\(138\) 0 0
\(139\) −0.662278 1.14710i −0.0561737 0.0972956i 0.836571 0.547858i \(-0.184557\pi\)
−0.892745 + 0.450563i \(0.851223\pi\)
\(140\) 0 0
\(141\) −3.74342 + 6.48379i −0.315253 + 0.546033i
\(142\) 0 0
\(143\) 6.32456 3.46410i 0.528886 0.289683i
\(144\) 0 0
\(145\) −3.58114 + 6.20271i −0.297397 + 0.515107i
\(146\) 0 0
\(147\) 5.16228 + 8.94133i 0.425777 + 0.737468i
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 4.32456 0.351927 0.175964 0.984397i \(-0.443696\pi\)
0.175964 + 0.984397i \(0.443696\pi\)
\(152\) 0 0
\(153\) 0.418861 0.725489i 0.0338629 0.0586523i
\(154\) 0 0
\(155\) 9.32456 0.748966
\(156\) 0 0
\(157\) −14.8114 −1.18208 −0.591039 0.806643i \(-0.701282\pi\)
−0.591039 + 0.806643i \(0.701282\pi\)
\(158\) 0 0
\(159\) −3.16228 + 5.47723i −0.250785 + 0.434372i
\(160\) 0 0
\(161\) −8.32456 −0.656067
\(162\) 0 0
\(163\) 4.24342 + 7.34981i 0.332370 + 0.575682i 0.982976 0.183734i \(-0.0588184\pi\)
−0.650606 + 0.759415i \(0.725485\pi\)
\(164\) 0 0
\(165\) 1.00000 + 1.73205i 0.0778499 + 0.134840i
\(166\) 0 0
\(167\) 10.3246 17.8827i 0.798938 1.38380i −0.121370 0.992607i \(-0.538729\pi\)
0.920308 0.391194i \(-0.127938\pi\)
\(168\) 0 0
\(169\) 5.98683 + 11.5394i 0.460526 + 0.887646i
\(170\) 0 0
\(171\) −1.00000 + 1.73205i −0.0764719 + 0.132453i
\(172\) 0 0
\(173\) 0.256584 + 0.444416i 0.0195077 + 0.0337883i 0.875614 0.483011i \(-0.160457\pi\)
−0.856107 + 0.516799i \(0.827123\pi\)
\(174\) 0 0
\(175\) −2.08114 3.60464i −0.157319 0.272485i
\(176\) 0 0
\(177\) 7.16228 0.538350
\(178\) 0 0
\(179\) −0.256584 + 0.444416i −0.0191780 + 0.0332172i −0.875455 0.483299i \(-0.839438\pi\)
0.856277 + 0.516517i \(0.172772\pi\)
\(180\) 0 0
\(181\) −24.6491 −1.83215 −0.916077 0.401002i \(-0.868662\pi\)
−0.916077 + 0.401002i \(0.868662\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) 0 0
\(185\) 4.16228 7.20928i 0.306017 0.530037i
\(186\) 0 0
\(187\) 1.67544 0.122521
\(188\) 0 0
\(189\) 2.08114 + 3.60464i 0.151381 + 0.262199i
\(190\) 0 0
\(191\) −5.90569 10.2290i −0.427321 0.740142i 0.569313 0.822121i \(-0.307209\pi\)
−0.996634 + 0.0819791i \(0.973876\pi\)
\(192\) 0 0
\(193\) 0.0811388 0.140537i 0.00584050 0.0101160i −0.863090 0.505049i \(-0.831474\pi\)
0.868931 + 0.494933i \(0.164808\pi\)
\(194\) 0 0
\(195\) −3.16228 + 1.73205i −0.226455 + 0.124035i
\(196\) 0 0
\(197\) −2.83772 + 4.91508i −0.202179 + 0.350185i −0.949230 0.314582i \(-0.898136\pi\)
0.747051 + 0.664767i \(0.231469\pi\)
\(198\) 0 0
\(199\) 3.33772 + 5.78110i 0.236605 + 0.409812i 0.959738 0.280897i \(-0.0906320\pi\)
−0.723133 + 0.690709i \(0.757299\pi\)
\(200\) 0 0
\(201\) 5.08114 + 8.80079i 0.358396 + 0.620760i
\(202\) 0 0
\(203\) −29.8114 −2.09235
\(204\) 0 0
\(205\) 2.58114 4.47066i 0.180275 0.312245i
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 11.8246 20.4807i 0.814036 1.40995i −0.0959821 0.995383i \(-0.530599\pi\)
0.910018 0.414569i \(-0.136068\pi\)
\(212\) 0 0
\(213\) −5.48683 −0.375952
\(214\) 0 0
\(215\) 5.08114 + 8.80079i 0.346531 + 0.600209i
\(216\) 0 0
\(217\) 19.4057 + 33.6116i 1.31734 + 2.28171i
\(218\) 0 0
\(219\) −3.08114 + 5.33669i −0.208204 + 0.360620i
\(220\) 0 0
\(221\) −0.0679718 + 3.01969i −0.00457228 + 0.203126i
\(222\) 0 0
\(223\) 2.00000 3.46410i 0.133930 0.231973i −0.791258 0.611482i \(-0.790574\pi\)
0.925188 + 0.379509i \(0.123907\pi\)
\(224\) 0 0
\(225\) −0.500000 0.866025i −0.0333333 0.0577350i
\(226\) 0 0
\(227\) 2.41886 + 4.18959i 0.160545 + 0.278073i 0.935064 0.354478i \(-0.115341\pi\)
−0.774519 + 0.632551i \(0.782008\pi\)
\(228\) 0 0
\(229\) −5.67544 −0.375044 −0.187522 0.982260i \(-0.560046\pi\)
−0.187522 + 0.982260i \(0.560046\pi\)
\(230\) 0 0
\(231\) −4.16228 + 7.20928i −0.273858 + 0.474336i
\(232\) 0 0
\(233\) 20.3246 1.33151 0.665753 0.746172i \(-0.268110\pi\)
0.665753 + 0.746172i \(0.268110\pi\)
\(234\) 0 0
\(235\) 7.48683 0.488387
\(236\) 0 0
\(237\) 5.50000 9.52628i 0.357263 0.618798i
\(238\) 0 0
\(239\) 16.6491 1.07694 0.538471 0.842644i \(-0.319002\pi\)
0.538471 + 0.842644i \(0.319002\pi\)
\(240\) 0 0
\(241\) −2.83772 4.91508i −0.182794 0.316608i 0.760037 0.649880i \(-0.225181\pi\)
−0.942831 + 0.333272i \(0.891847\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 5.16228 8.94133i 0.329806 0.571240i
\(246\) 0 0
\(247\) 0.162278 7.20928i 0.0103255 0.458715i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.48683 16.4317i −0.598804 1.03716i −0.992998 0.118131i \(-0.962310\pi\)
0.394194 0.919027i \(-0.371024\pi\)
\(252\) 0 0
\(253\) −2.00000 3.46410i −0.125739 0.217786i
\(254\) 0 0
\(255\) −0.837722 −0.0524602
\(256\) 0 0
\(257\) 13.0680 22.6344i 0.815158 1.41189i −0.0940569 0.995567i \(-0.529984\pi\)
0.909215 0.416328i \(-0.136683\pi\)
\(258\) 0 0
\(259\) 34.6491 2.15299
\(260\) 0 0
\(261\) −7.16228 −0.443334
\(262\) 0 0
\(263\) −4.74342 + 8.21584i −0.292492 + 0.506610i −0.974398 0.224829i \(-0.927818\pi\)
0.681907 + 0.731439i \(0.261151\pi\)
\(264\) 0 0
\(265\) 6.32456 0.388514
\(266\) 0 0
\(267\) −7.74342 13.4120i −0.473889 0.820801i
\(268\) 0 0
\(269\) 11.7434 + 20.3402i 0.716009 + 1.24016i 0.962569 + 0.271036i \(0.0873663\pi\)
−0.246560 + 0.969127i \(0.579300\pi\)
\(270\) 0 0
\(271\) 4.82456 8.35637i 0.293071 0.507614i −0.681463 0.731852i \(-0.738656\pi\)
0.974534 + 0.224239i \(0.0719895\pi\)
\(272\) 0 0
\(273\) −12.8246 7.79423i −0.776177 0.471728i
\(274\) 0 0
\(275\) 1.00000 1.73205i 0.0603023 0.104447i
\(276\) 0 0
\(277\) −5.83772 10.1112i −0.350755 0.607525i 0.635627 0.771996i \(-0.280742\pi\)
−0.986382 + 0.164471i \(0.947408\pi\)
\(278\) 0 0
\(279\) 4.66228 + 8.07530i 0.279123 + 0.483456i
\(280\) 0 0
\(281\) −3.16228 −0.188646 −0.0943228 0.995542i \(-0.530069\pi\)
−0.0943228 + 0.995542i \(0.530069\pi\)
\(282\) 0 0
\(283\) −12.4057 + 21.4873i −0.737442 + 1.27729i 0.216202 + 0.976349i \(0.430633\pi\)
−0.953644 + 0.300938i \(0.902700\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 21.4868 1.26833
\(288\) 0 0
\(289\) 8.14911 14.1147i 0.479359 0.830275i
\(290\) 0 0
\(291\) 12.8114 0.751017
\(292\) 0 0
\(293\) −13.7434 23.8043i −0.802899 1.39066i −0.917700 0.397273i \(-0.869956\pi\)
0.114801 0.993388i \(-0.463377\pi\)
\(294\) 0 0
\(295\) −3.58114 6.20271i −0.208502 0.361136i
\(296\) 0 0
\(297\) −1.00000 + 1.73205i −0.0580259 + 0.100504i
\(298\) 0 0
\(299\) 6.32456 3.46410i 0.365758 0.200334i
\(300\) 0 0
\(301\) −21.1491 + 36.6313i −1.21901 + 2.11140i
\(302\) 0 0
\(303\) 0.162278 + 0.281073i 0.00932261 + 0.0161472i
\(304\) 0 0
\(305\) 6.50000 + 11.2583i 0.372189 + 0.644650i
\(306\) 0 0
\(307\) −24.1623 −1.37901 −0.689507 0.724279i \(-0.742173\pi\)
−0.689507 + 0.724279i \(0.742173\pi\)
\(308\) 0 0
\(309\) −5.24342 + 9.08186i −0.298288 + 0.516649i
\(310\) 0 0
\(311\) 25.8114 1.46363 0.731815 0.681504i \(-0.238674\pi\)
0.731815 + 0.681504i \(0.238674\pi\)
\(312\) 0 0
\(313\) 18.4868 1.04494 0.522469 0.852658i \(-0.325011\pi\)
0.522469 + 0.852658i \(0.325011\pi\)
\(314\) 0 0
\(315\) 2.08114 3.60464i 0.117259 0.203098i
\(316\) 0 0
\(317\) −2.32456 −0.130560 −0.0652800 0.997867i \(-0.520794\pi\)
−0.0652800 + 0.997867i \(0.520794\pi\)
\(318\) 0 0
\(319\) −7.16228 12.4054i −0.401011 0.694571i
\(320\) 0 0
\(321\) −9.74342 16.8761i −0.543824 0.941932i
\(322\) 0 0
\(323\) 0.837722 1.45098i 0.0466121 0.0807346i
\(324\) 0 0
\(325\) 3.08114 + 1.87259i 0.170911 + 0.103872i
\(326\) 0 0
\(327\) 3.66228 6.34325i 0.202524 0.350783i
\(328\) 0 0
\(329\) 15.5811 + 26.9873i 0.859016 + 1.48786i
\(330\) 0 0
\(331\) 1.33772 + 2.31700i 0.0735279 + 0.127354i 0.900445 0.434970i \(-0.143241\pi\)
−0.826917 + 0.562324i \(0.809908\pi\)
\(332\) 0 0
\(333\) 8.32456 0.456183
\(334\) 0 0
\(335\) 5.08114 8.80079i 0.277612 0.480839i
\(336\) 0 0
\(337\) −1.83772 −0.100107 −0.0500536 0.998747i \(-0.515939\pi\)
−0.0500536 + 0.998747i \(0.515939\pi\)
\(338\) 0 0
\(339\) 4.32456 0.234878
\(340\) 0 0
\(341\) −9.32456 + 16.1506i −0.504953 + 0.874604i
\(342\) 0 0
\(343\) 13.8377 0.747167
\(344\) 0 0
\(345\) 1.00000 + 1.73205i 0.0538382 + 0.0932505i
\(346\) 0 0
\(347\) 13.1623 + 22.7977i 0.706588 + 1.22385i 0.966115 + 0.258111i \(0.0830999\pi\)
−0.259527 + 0.965736i \(0.583567\pi\)
\(348\) 0 0
\(349\) 0.662278 1.14710i 0.0354509 0.0614028i −0.847756 0.530387i \(-0.822047\pi\)
0.883206 + 0.468984i \(0.155380\pi\)
\(350\) 0 0
\(351\) −3.08114 1.87259i −0.164459 0.0999513i
\(352\) 0 0
\(353\) −1.83772 + 3.18303i −0.0978121 + 0.169416i −0.910779 0.412895i \(-0.864518\pi\)
0.812967 + 0.582310i \(0.197851\pi\)
\(354\) 0 0
\(355\) 2.74342 + 4.75174i 0.145605 + 0.252196i
\(356\) 0 0
\(357\) −1.74342 3.01969i −0.0922714 0.159819i
\(358\) 0 0
\(359\) −12.5132 −0.660420 −0.330210 0.943908i \(-0.607120\pi\)
−0.330210 + 0.943908i \(0.607120\pi\)
\(360\) 0 0
\(361\) 7.50000 12.9904i 0.394737 0.683704i
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 6.16228 0.322548
\(366\) 0 0
\(367\) 15.0811 26.1213i 0.787229 1.36352i −0.140429 0.990091i \(-0.544848\pi\)
0.927658 0.373430i \(-0.121818\pi\)
\(368\) 0 0
\(369\) 5.16228 0.268737
\(370\) 0 0
\(371\) 13.1623 + 22.7977i 0.683351 + 1.18360i
\(372\) 0 0
\(373\) −15.7302 27.2456i −0.814481 1.41072i −0.909700 0.415267i \(-0.863688\pi\)
0.0952182 0.995456i \(-0.469645\pi\)
\(374\) 0 0
\(375\) −0.500000 + 0.866025i −0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) 22.6491 12.4054i 1.16649 0.638912i
\(378\) 0 0
\(379\) 1.98683 3.44130i 0.102057 0.176767i −0.810475 0.585773i \(-0.800791\pi\)
0.912532 + 0.409006i \(0.134124\pi\)
\(380\) 0 0
\(381\) 0.0811388 + 0.140537i 0.00415687 + 0.00719991i
\(382\) 0 0
\(383\) −4.74342 8.21584i −0.242377 0.419810i 0.719014 0.694996i \(-0.244594\pi\)
−0.961391 + 0.275186i \(0.911261\pi\)
\(384\) 0 0
\(385\) 8.32456 0.424259
\(386\) 0 0
\(387\) −5.08114 + 8.80079i −0.258289 + 0.447369i
\(388\) 0 0
\(389\) 30.9737 1.57043 0.785214 0.619225i \(-0.212553\pi\)
0.785214 + 0.619225i \(0.212553\pi\)
\(390\) 0 0
\(391\) 1.67544 0.0847309
\(392\) 0 0
\(393\) 7.90569 13.6931i 0.398790 0.690724i
\(394\) 0 0
\(395\) −11.0000 −0.553470
\(396\) 0 0
\(397\) −5.75658 9.97070i −0.288915 0.500415i 0.684636 0.728885i \(-0.259961\pi\)
−0.973551 + 0.228470i \(0.926628\pi\)
\(398\) 0 0
\(399\) 4.16228 + 7.20928i 0.208375 + 0.360915i
\(400\) 0 0
\(401\) 7.48683 12.9676i 0.373875 0.647570i −0.616283 0.787525i \(-0.711362\pi\)
0.990158 + 0.139955i \(0.0446957\pi\)
\(402\) 0 0
\(403\) −28.7302 17.4610i −1.43116 0.869797i
\(404\) 0 0
\(405\) 0.500000 0.866025i 0.0248452 0.0430331i
\(406\) 0 0
\(407\) 8.32456 + 14.4186i 0.412633 + 0.714701i
\(408\) 0 0
\(409\) 10.8246 + 18.7487i 0.535240 + 0.927063i 0.999152 + 0.0411812i \(0.0131121\pi\)
−0.463912 + 0.885881i \(0.653555\pi\)
\(410\) 0 0
\(411\) −12.8377 −0.633238
\(412\) 0 0
\(413\) 14.9057 25.8174i 0.733461 1.27039i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.32456 −0.0648638
\(418\) 0 0
\(419\) −18.9057 + 32.7456i −0.923604 + 1.59973i −0.129812 + 0.991539i \(0.541437\pi\)
−0.793792 + 0.608190i \(0.791896\pi\)
\(420\) 0 0
\(421\) 9.32456 0.454451 0.227226 0.973842i \(-0.427035\pi\)
0.227226 + 0.973842i \(0.427035\pi\)
\(422\) 0 0
\(423\) 3.74342 + 6.48379i 0.182011 + 0.315253i
\(424\) 0 0
\(425\) 0.418861 + 0.725489i 0.0203178 + 0.0351914i
\(426\) 0 0
\(427\) −27.0548 + 46.8603i −1.30927 + 2.26773i
\(428\) 0 0
\(429\) 0.162278 7.20928i 0.00783484 0.348067i
\(430\) 0 0
\(431\) −14.3246 + 24.8109i −0.689990 + 1.19510i 0.281851 + 0.959458i \(0.409052\pi\)
−0.971841 + 0.235639i \(0.924282\pi\)
\(432\) 0 0
\(433\) 13.2434 + 22.9383i 0.636438 + 1.10234i 0.986209 + 0.165508i \(0.0529262\pi\)
−0.349771 + 0.936835i \(0.613740\pi\)
\(434\) 0 0
\(435\) 3.58114 + 6.20271i 0.171702 + 0.297397i
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 16.3114 28.2522i 0.778500 1.34840i −0.154306 0.988023i \(-0.549314\pi\)
0.932806 0.360379i \(-0.117352\pi\)
\(440\) 0 0
\(441\) 10.3246 0.491645
\(442\) 0 0
\(443\) −6.13594 −0.291527 −0.145764 0.989319i \(-0.546564\pi\)
−0.145764 + 0.989319i \(0.546564\pi\)
\(444\) 0 0
\(445\) −7.74342 + 13.4120i −0.367073 + 0.635789i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.16228 3.74517i −0.102044 0.176746i 0.810483 0.585763i \(-0.199205\pi\)
−0.912527 + 0.409017i \(0.865872\pi\)
\(450\) 0 0
\(451\) 5.16228 + 8.94133i 0.243082 + 0.421031i
\(452\) 0 0
\(453\) 2.16228 3.74517i 0.101593 0.175964i
\(454\) 0 0
\(455\) −0.337722 + 15.0035i −0.0158327 + 0.703375i
\(456\) 0 0
\(457\) −7.91886 + 13.7159i −0.370429 + 0.641601i −0.989631 0.143630i \(-0.954123\pi\)
0.619203 + 0.785231i \(0.287456\pi\)
\(458\) 0 0
\(459\) −0.418861 0.725489i −0.0195508 0.0338629i
\(460\) 0 0
\(461\) −5.41886 9.38574i −0.252382 0.437138i 0.711799 0.702383i \(-0.247880\pi\)
−0.964181 + 0.265245i \(0.914547\pi\)
\(462\) 0 0
\(463\) −16.4868 −0.766208 −0.383104 0.923705i \(-0.625145\pi\)
−0.383104 + 0.923705i \(0.625145\pi\)
\(464\) 0 0
\(465\) 4.66228 8.07530i 0.216208 0.374483i
\(466\) 0 0
\(467\) −6.18861 −0.286375 −0.143187 0.989696i \(-0.545735\pi\)
−0.143187 + 0.989696i \(0.545735\pi\)
\(468\) 0 0
\(469\) 42.2982 1.95315
\(470\) 0 0
\(471\) −7.40569 + 12.8270i −0.341236 + 0.591039i
\(472\) 0 0
\(473\) −20.3246 −0.934524
\(474\) 0 0
\(475\) −1.00000 1.73205i −0.0458831 0.0794719i
\(476\) 0 0
\(477\) 3.16228 + 5.47723i 0.144791 + 0.250785i
\(478\) 0 0
\(479\) 14.5811 25.2553i 0.666229 1.15394i −0.312721 0.949845i \(-0.601241\pi\)
0.978950 0.204098i \(-0.0654261\pi\)
\(480\) 0 0
\(481\) −26.3246 + 14.4186i −1.20030 + 0.657429i
\(482\) 0 0
\(483\) −4.16228 + 7.20928i −0.189390 + 0.328033i
\(484\) 0 0
\(485\) −6.40569 11.0950i −0.290868 0.503797i
\(486\) 0 0
\(487\) 7.00000 + 12.1244i 0.317200 + 0.549407i 0.979903 0.199476i \(-0.0639239\pi\)
−0.662702 + 0.748883i \(0.730591\pi\)
\(488\) 0 0
\(489\) 8.48683 0.383788
\(490\) 0 0
\(491\) 16.7434 29.0004i 0.755620 1.30877i −0.189446 0.981891i \(-0.560669\pi\)
0.945066 0.326881i \(-0.105998\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) −11.4189 + 19.7780i −0.512206 + 0.887167i
\(498\) 0 0
\(499\) 14.6491 0.655784 0.327892 0.944715i \(-0.393662\pi\)
0.327892 + 0.944715i \(0.393662\pi\)
\(500\) 0 0
\(501\) −10.3246 17.8827i −0.461267 0.798938i
\(502\) 0 0
\(503\) 5.64911 + 9.78455i 0.251881 + 0.436271i 0.964044 0.265743i \(-0.0856174\pi\)
−0.712162 + 0.702015i \(0.752284\pi\)
\(504\) 0 0
\(505\) 0.162278 0.281073i 0.00722126 0.0125076i
\(506\) 0 0
\(507\) 12.9868 + 0.584952i 0.576766 + 0.0259786i
\(508\) 0 0
\(509\) 5.32456 9.22240i 0.236007 0.408776i −0.723558 0.690264i \(-0.757495\pi\)
0.959565 + 0.281488i \(0.0908279\pi\)
\(510\) 0 0
\(511\) 12.8246 + 22.2128i 0.567325 + 0.982636i
\(512\) 0 0
\(513\) 1.00000 + 1.73205i 0.0441511 + 0.0764719i
\(514\) 0 0
\(515\) 10.4868 0.462105
\(516\) 0 0
\(517\) −7.48683 + 12.9676i −0.329271 + 0.570313i
\(518\) 0 0
\(519\) 0.513167 0.0225255
\(520\) 0 0
\(521\) 12.1359 0.531685 0.265843 0.964016i \(-0.414350\pi\)
0.265843 + 0.964016i \(0.414350\pi\)
\(522\) 0 0
\(523\) −5.48683 + 9.50347i −0.239922 + 0.415558i −0.960692 0.277617i \(-0.910455\pi\)
0.720769 + 0.693175i \(0.243789\pi\)
\(524\) 0 0
\(525\) −4.16228 −0.181657
\(526\) 0 0
\(527\) −3.90569 6.76486i −0.170135 0.294682i
\(528\) 0 0
\(529\) 9.50000 + 16.4545i 0.413043 + 0.715412i
\(530\) 0 0
\(531\) 3.58114 6.20271i 0.155408 0.269175i
\(532\) 0 0
\(533\) −16.3246 + 8.94133i −0.707095 + 0.387292i
\(534\) 0 0
\(535\) −9.74342 + 16.8761i −0.421245 + 0.729617i
\(536\) 0 0
\(537\) 0.256584 + 0.444416i 0.0110724 + 0.0191780i
\(538\) 0 0
\(539\) 10.3246 + 17.8827i 0.444710 + 0.770260i
\(540\) 0 0
\(541\) 36.6228 1.57454 0.787268 0.616611i \(-0.211495\pi\)
0.787268 + 0.616611i \(0.211495\pi\)
\(542\) 0 0
\(543\) −12.3246 + 21.3468i −0.528897 + 0.916077i
\(544\) 0 0
\(545\) −7.32456 −0.313749
\(546\) 0 0
\(547\) −25.1359 −1.07474 −0.537368 0.843348i \(-0.680581\pi\)
−0.537368 + 0.843348i \(0.680581\pi\)
\(548\) 0 0
\(549\) −6.50000 + 11.2583i −0.277413 + 0.480494i
\(550\) 0 0
\(551\) −14.3246 −0.610247
\(552\) 0 0
\(553\) −22.8925 39.6510i −0.973489 1.68613i
\(554\) 0 0
\(555\) −4.16228 7.20928i −0.176679 0.306017i
\(556\) 0 0
\(557\) −7.48683 + 12.9676i −0.317227 + 0.549454i −0.979908 0.199448i \(-0.936085\pi\)
0.662681 + 0.748902i \(0.269418\pi\)
\(558\) 0 0
\(559\) 0.824555 36.6313i 0.0348750 1.54934i
\(560\) 0 0
\(561\) 0.837722 1.45098i 0.0353687 0.0612603i
\(562\) 0 0
\(563\) 9.00000 + 15.5885i 0.379305 + 0.656975i 0.990961 0.134148i \(-0.0428299\pi\)
−0.611656 + 0.791123i \(0.709497\pi\)
\(564\) 0 0
\(565\) −2.16228 3.74517i −0.0909677 0.157561i
\(566\) 0 0
\(567\) 4.16228 0.174799
\(568\) 0 0
\(569\) 19.5811 33.9155i 0.820884 1.42181i −0.0841407 0.996454i \(-0.526815\pi\)
0.905025 0.425359i \(-0.139852\pi\)
\(570\) 0 0
\(571\) −16.9737 −0.710326 −0.355163 0.934804i \(-0.615575\pi\)
−0.355163 + 0.934804i \(0.615575\pi\)
\(572\) 0 0
\(573\) −11.8114 −0.493428
\(574\) 0 0
\(575\) 1.00000 1.73205i 0.0417029 0.0722315i
\(576\) 0 0
\(577\) 21.6228 0.900168 0.450084 0.892986i \(-0.351394\pi\)
0.450084 + 0.892986i \(0.351394\pi\)
\(578\) 0 0
\(579\) −0.0811388 0.140537i −0.00337201 0.00584050i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.32456 + 10.9545i −0.261936 + 0.453687i
\(584\) 0 0
\(585\) −0.0811388 + 3.60464i −0.00335468 + 0.149033i
\(586\) 0 0
\(587\) 16.3925 28.3927i 0.676592 1.17189i −0.299409 0.954125i \(-0.596789\pi\)
0.976001 0.217767i \(-0.0698772\pi\)
\(588\) 0 0
\(589\) 9.32456 + 16.1506i 0.384212 + 0.665474i
\(590\) 0 0
\(591\) 2.83772 + 4.91508i 0.116728 + 0.202179i
\(592\) 0 0
\(593\) 32.3246 1.32741 0.663705 0.747994i \(-0.268983\pi\)
0.663705 + 0.747994i \(0.268983\pi\)
\(594\) 0 0
\(595\) −1.74342 + 3.01969i −0.0714731 + 0.123795i
\(596\) 0 0
\(597\) 6.67544 0.273208
\(598\) 0 0
\(599\) −33.4868 −1.36823 −0.684117 0.729372i \(-0.739812\pi\)
−0.684117 + 0.729372i \(0.739812\pi\)
\(600\) 0 0
\(601\) −18.6491 + 32.3012i −0.760713 + 1.31759i 0.181770 + 0.983341i \(0.441817\pi\)
−0.942483 + 0.334253i \(0.891516\pi\)
\(602\) 0 0
\(603\) 10.1623 0.413840
\(604\) 0 0
\(605\) −3.50000 6.06218i −0.142295 0.246463i
\(606\) 0 0
\(607\) −0.675445 1.16990i −0.0274155 0.0474850i 0.851992 0.523555i \(-0.175394\pi\)
−0.879408 + 0.476070i \(0.842061\pi\)
\(608\) 0 0
\(609\) −14.9057 + 25.8174i −0.604009 + 1.04617i
\(610\) 0 0
\(611\) −23.0680 14.0197i −0.933230 0.567178i
\(612\) 0 0
\(613\) 13.7302 23.7815i 0.554560 0.960525i −0.443378 0.896335i \(-0.646220\pi\)
0.997938 0.0641906i \(-0.0204466\pi\)
\(614\) 0 0
\(615\) −2.58114 4.47066i −0.104082 0.180275i
\(616\) 0 0
\(617\) −6.48683 11.2355i −0.261150 0.452325i 0.705398 0.708812i \(-0.250768\pi\)
−0.966548 + 0.256487i \(0.917435\pi\)
\(618\) 0 0
\(619\) −18.6754 −0.750629 −0.375315 0.926897i \(-0.622465\pi\)
−0.375315 + 0.926897i \(0.622465\pi\)
\(620\) 0 0
\(621\) −1.00000 + 1.73205i −0.0401286 + 0.0695048i
\(622\) 0 0
\(623\) −64.4605 −2.58256
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.00000 + 3.46410i −0.0798723 + 0.138343i
\(628\) 0 0
\(629\) −6.97367 −0.278058
\(630\) 0 0
\(631\) −0.662278 1.14710i −0.0263649 0.0456653i 0.852542 0.522659i \(-0.175060\pi\)
−0.878907 + 0.476994i \(0.841727\pi\)
\(632\) 0 0
\(633\) −11.8246 20.4807i −0.469984 0.814036i
\(634\) 0 0
\(635\) 0.0811388 0.140537i 0.00321990 0.00557702i
\(636\) 0 0
\(637\) −32.6491 + 17.8827i −1.29360 + 0.708537i
\(638\) 0 0
\(639\) −2.74342 + 4.75174i −0.108528 + 0.187976i
\(640\) 0 0
\(641\) −1.83772 3.18303i −0.0725857 0.125722i 0.827448 0.561542i \(-0.189792\pi\)
−0.900034 + 0.435820i \(0.856458\pi\)
\(642\) 0 0
\(643\) 0.756584 + 1.31044i 0.0298367 + 0.0516788i 0.880558 0.473938i \(-0.157168\pi\)
−0.850721 + 0.525617i \(0.823835\pi\)
\(644\) 0 0
\(645\) 10.1623 0.400139
\(646\) 0 0
\(647\) 6.67544 11.5622i 0.262439 0.454557i −0.704451 0.709753i \(-0.748807\pi\)
0.966889 + 0.255196i \(0.0821399\pi\)
\(648\) 0 0
\(649\) 14.3246 0.562288
\(650\) 0 0
\(651\) 38.8114 1.52114
\(652\) 0 0
\(653\) 16.7434 29.0004i 0.655221 1.13488i −0.326618 0.945156i \(-0.605909\pi\)
0.981838 0.189719i \(-0.0607576\pi\)
\(654\) 0 0
\(655\) −15.8114 −0.617802
\(656\) 0 0
\(657\) 3.08114 + 5.33669i 0.120207 + 0.208204i
\(658\) 0 0
\(659\) 1.48683 + 2.57527i 0.0579188 + 0.100318i 0.893531 0.449002i \(-0.148220\pi\)
−0.835612 + 0.549320i \(0.814887\pi\)
\(660\) 0 0
\(661\) 13.6623 23.6638i 0.531401 0.920414i −0.467927 0.883767i \(-0.654999\pi\)
0.999328 0.0366466i \(-0.0116676\pi\)
\(662\) 0 0
\(663\) 2.58114 + 1.56871i 0.100243 + 0.0609236i
\(664\) 0 0
\(665\) 4.16228 7.20928i 0.161406 0.279564i
\(666\) 0 0
\(667\) −7.16228 12.4054i −0.277324 0.480340i
\(668\) 0 0
\(669\) −2.00000 3.46410i −0.0773245 0.133930i
\(670\) 0 0
\(671\) −26.0000 −1.00372
\(672\) 0 0
\(673\) 14.9189 25.8402i 0.575080 0.996067i −0.420953 0.907082i \(-0.638304\pi\)
0.996033 0.0889851i \(-0.0283624\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −29.9473 −1.15097 −0.575485 0.817813i \(-0.695187\pi\)
−0.575485 + 0.817813i \(0.695187\pi\)
\(678\) 0 0
\(679\) 26.6623 46.1804i 1.02320 1.77224i
\(680\) 0 0
\(681\) 4.83772 0.185382
\(682\) 0 0
\(683\) −19.7434 34.1966i −0.755461 1.30850i −0.945145 0.326651i \(-0.894080\pi\)
0.189684 0.981845i \(-0.439254\pi\)
\(684\) 0 0
\(685\) 6.41886 + 11.1178i 0.245252 + 0.424789i
\(686\) 0 0
\(687\) −2.83772 + 4.91508i −0.108266 + 0.187522i
\(688\) 0 0
\(689\) −19.4868 11.8433i −0.742389 0.451193i
\(690\) 0 0
\(691\) −4.82456 + 8.35637i −0.183535 + 0.317891i −0.943082 0.332561i \(-0.892087\pi\)
0.759547 + 0.650452i \(0.225421\pi\)
\(692\) 0 0
\(693\) 4.16228 + 7.20928i 0.158112 + 0.273858i
\(694\) 0 0
\(695\) 0.662278 + 1.14710i 0.0251216 + 0.0435119i
\(696\) 0 0
\(697\) −4.32456 −0.163804
\(698\) 0 0
\(699\) 10.1623 17.6016i 0.384373 0.665753i
\(700\) 0 0
\(701\) −18.3246 −0.692109 −0.346054 0.938214i \(-0.612479\pi\)
−0.346054 + 0.938214i \(0.612479\pi\)
\(702\) 0 0
\(703\) 16.6491 0.627933
\(704\) 0 0
\(705\) 3.74342 6.48379i 0.140985 0.244194i
\(706\) 0 0
\(707\) 1.35089 0.0508054
\(708\) 0 0
\(709\) −12.3114 21.3240i −0.462364 0.800838i 0.536714 0.843764i \(-0.319665\pi\)
−0.999078 + 0.0429263i \(0.986332\pi\)
\(710\) 0 0
\(711\) −5.50000 9.52628i −0.206266 0.357263i
\(712\) 0 0
\(713\) −9.32456 + 16.1506i −0.349207 + 0.604845i
\(714\) 0 0
\(715\) −6.32456 + 3.46410i −0.236525 + 0.129550i
\(716\) 0 0
\(717\) 8.32456 14.4186i 0.310886 0.538471i
\(718\) 0 0
\(719\) 13.0680 + 22.6344i 0.487353 + 0.844120i 0.999894 0.0145423i \(-0.00462913\pi\)
−0.512541 + 0.858663i \(0.671296\pi\)
\(720\) 0 0
\(721\) 21.8246 + 37.8012i 0.812789 + 1.40779i
\(722\) 0 0
\(723\) −5.67544 −0.211072
\(724\) 0 0
\(725\) 3.58114 6.20271i 0.133000 0.230363i
\(726\) 0 0
\(727\) −40.1623 −1.48954 −0.744768 0.667323i \(-0.767440\pi\)
−0.744768 + 0.667323i \(0.767440\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.25658 7.37262i 0.157435 0.272686i
\(732\) 0 0
\(733\) 35.8377 1.32370 0.661848 0.749638i \(-0.269772\pi\)
0.661848 + 0.749638i \(0.269772\pi\)
\(734\) 0 0
\(735\) −5.16228 8.94133i −0.190413 0.329806i
\(736\) 0 0
\(737\) 10.1623 + 17.6016i 0.374332 + 0.648363i
\(738\) 0 0
\(739\) 3.67544 6.36606i 0.135203 0.234179i −0.790472 0.612499i \(-0.790165\pi\)
0.925675 + 0.378319i \(0.123498\pi\)
\(740\) 0 0
\(741\) −6.16228 3.74517i −0.226377 0.137582i
\(742\) 0 0
\(743\) −24.7171 + 42.8112i −0.906782 + 1.57059i −0.0882744 + 0.996096i \(0.528135\pi\)
−0.818507 + 0.574496i \(0.805198\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −81.1096 −2.96368
\(750\) 0 0
\(751\) 7.64911 13.2486i 0.279120 0.483450i −0.692046 0.721853i \(-0.743291\pi\)
0.971166 + 0.238403i \(0.0766239\pi\)
\(752\) 0 0
\(753\) −18.9737 −0.691439
\(754\) 0 0
\(755\) −4.32456 −0.157387
\(756\) 0 0
\(757\) −13.8114 + 23.9220i −0.501983 + 0.869461i 0.498014 + 0.867169i \(0.334063\pi\)
−0.999997 + 0.00229180i \(0.999270\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 2.64911 + 4.58839i 0.0960302 + 0.166329i 0.910038 0.414525i \(-0.136052\pi\)
−0.814008 + 0.580854i \(0.802719\pi\)
\(762\) 0 0
\(763\) −15.2434 26.4024i −0.551849 0.955830i
\(764\) 0 0
\(765\) −0.418861 + 0.725489i −0.0151440 + 0.0262301i
\(766\) 0 0
\(767\) −0.581139 + 25.8174i −0.0209837 + 0.932213i
\(768\) 0 0
\(769\) 25.4868 44.1445i 0.919079 1.59189i 0.118261 0.992983i \(-0.462268\pi\)
0.800817 0.598908i \(-0.204399\pi\)
\(770\) 0 0
\(771\) −13.0680 22.6344i −0.470632 0.815158i
\(772\) 0 0
\(773\) 4.16228 + 7.20928i 0.149707 + 0.259300i 0.931119 0.364715i \(-0.118834\pi\)
−0.781412 + 0.624015i \(0.785500\pi\)
\(774\) 0 0
\(775\) −9.32456 −0.334948
\(776\) 0 0
\(777\) 17.3246 30.0070i 0.621515 1.07650i
\(778\) 0 0
\(779\) 10.3246 0.369916
\(780\) 0 0
\(781\) −10.9737 −0.392669
\(782\) 0 0
\(783\) −3.58114 + 6.20271i −0.127979 + 0.221667i
\(784\) 0 0
\(785\) 14.8114 0.528641
\(786\) 0 0
\(787\) −12.0811 20.9251i −0.430646 0.745901i 0.566283 0.824211i \(-0.308381\pi\)
−0.996929 + 0.0783100i \(0.975048\pi\)
\(788\) 0 0
\(789\) 4.74342 + 8.21584i 0.168870 + 0.292492i
\(790\) 0 0
\(791\) 9.00000 15.5885i 0.320003 0.554262i
\(792\) 0 0
\(793\) 1.05480 46.8603i 0.0374572 1.66406i
\(794\) 0 0
\(795\) 3.16228 5.47723i 0.112154 0.194257i
\(796\) 0 0
\(797\) 11.5811 + 20.0591i 0.410225 + 0.710531i 0.994914 0.100727i \(-0.0321168\pi\)
−0.584689 + 0.811257i \(0.698784\pi\)
\(798\) 0 0
\(799\) −3.13594 5.43161i −0.110942 0.192157i
\(800\) 0 0
\(801\) −15.4868 −0.547200
\(802\) 0 0
\(803\) −6.16228 + 10.6734i −0.217462 + 0.376655i
\(804\) 0 0
\(805\) 8.32456 0.293402
\(806\) 0 0
\(807\) 23.4868 0.826776
\(808\) 0 0
\(809\) −11.1623 + 19.3336i −0.392445 + 0.679734i −0.992771 0.120021i \(-0.961704\pi\)
0.600327 + 0.799755i \(0.295037\pi\)
\(810\) 0 0
\(811\) −3.97367 −0.139534 −0.0697671 0.997563i \(-0.522226\pi\)
−0.0697671 + 0.997563i \(0.522226\pi\)
\(812\) 0 0
\(813\) −4.82456 8.35637i −0.169205 0.293071i
\(814\) 0 0
\(815\) −4.24342 7.34981i −0.148640 0.257453i
\(816\) 0 0
\(817\) −10.1623 + 17.6016i −0.355533 + 0.615801i
\(818\) 0 0
\(819\) −13.1623 + 7.20928i −0.459927 + 0.251913i
\(820\) 0 0
\(821\) −21.0000 + 36.3731i −0.732905 + 1.26943i 0.222731 + 0.974880i \(0.428503\pi\)
−0.955636 + 0.294549i \(0.904831\pi\)
\(822\) 0 0
\(823\) 9.16228 + 15.8695i 0.319377 + 0.553177i 0.980358 0.197225i \(-0.0631931\pi\)
−0.660981 + 0.750402i \(0.729860\pi\)
\(824\) 0 0
\(825\) −1.00000 1.73205i −0.0348155 0.0603023i
\(826\) 0 0
\(827\) −28.4605 −0.989669 −0.494834 0.868987i \(-0.664771\pi\)
−0.494834 + 0.868987i \(0.664771\pi\)
\(828\) 0 0
\(829\) 4.82456 8.35637i 0.167564 0.290229i −0.769999 0.638045i \(-0.779743\pi\)
0.937563 + 0.347816i \(0.113077\pi\)
\(830\) 0 0
\(831\) −11.6754 −0.405017
\(832\) 0 0
\(833\) −8.64911 −0.299674
\(834\) 0 0
\(835\) −10.3246 + 17.8827i −0.357296 + 0.618855i
\(836\) 0 0
\(837\) 9.32456 0.322304
\(838\) 0 0
\(839\) 26.4868 + 45.8765i 0.914427 + 1.58383i 0.807738 + 0.589542i \(0.200692\pi\)
0.106689 + 0.994292i \(0.465975\pi\)
\(840\) 0 0
\(841\) −11.1491 19.3108i −0.384452 0.665891i
\(842\) 0 0
\(843\) −1.58114 + 2.73861i −0.0544573 + 0.0943228i
\(844\) 0 0
\(845\) −5.98683 11.5394i −0.205953 0.396968i
\(846\) 0 0
\(847\) 14.5680 25.2325i 0.500561 0.866998i
\(848\) 0 0
\(849\) 12.4057 + 21.4873i 0.425762 + 0.737442i
\(850\) 0 0
\(851\) 8.32456 + 14.4186i 0.285362 + 0.494262i
\(852\) 0 0
\(853\) −31.7851 −1.08830 −0.544150 0.838988i \(-0.683148\pi\)
−0.544150 + 0.838988i \(0.683148\pi\)
\(854\) 0 0
\(855\) 1.00000 1.73205i 0.0341993 0.0592349i
\(856\) 0 0
\(857\) 3.48683 0.119108 0.0595540 0.998225i \(-0.481032\pi\)
0.0595540 + 0.998225i \(0.481032\pi\)
\(858\) 0 0
\(859\) −42.9473 −1.46534 −0.732672 0.680582i \(-0.761727\pi\)
−0.732672 + 0.680582i \(0.761727\pi\)
\(860\) 0 0
\(861\) 10.7434 18.6081i 0.366135 0.634164i
\(862\) 0 0
\(863\) −11.6754 −0.397437 −0.198718 0.980057i \(-0.563678\pi\)
−0.198718 + 0.980057i \(0.563678\pi\)
\(864\) 0 0
\(865\) −0.256584 0.444416i −0.00872410 0.0151106i
\(866\) 0 0
\(867\) −8.14911 14.1147i −0.276758 0.479359i
\(868\) 0 0
\(869\) 11.0000 19.0526i 0.373149 0.646314i
\(870\) 0 0
\(871\) −32.1359 + 17.6016i −1.08888 + 0.596407i
\(872\) 0 0
\(873\) 6.40569 11.0950i 0.216800 0.375508i
\(874\) 0 0
\(875\) 2.08114 + 3.60464i 0.0703553 + 0.121859i
\(876\) 0 0
\(877\) −12.1623 21.0657i −0.410691 0.711338i 0.584275 0.811556i \(-0.301379\pi\)
−0.994965 + 0.100219i \(0.968046\pi\)
\(878\) 0 0
\(879\) −27.4868 −0.927108
\(880\) 0 0
\(881\) −5.16228 + 8.94133i −0.173922 + 0.301241i −0.939788 0.341759i \(-0.888977\pi\)
0.765866 + 0.643000i \(0.222311\pi\)
\(882\) 0 0
\(883\) 5.46050 0.183760 0.0918802 0.995770i \(-0.470712\pi\)
0.0918802 + 0.995770i \(0.470712\pi\)
\(884\) 0 0
\(885\) −7.16228 −0.240757
\(886\) 0 0
\(887\) −19.2302 + 33.3078i −0.645689 + 1.11837i 0.338453 + 0.940983i \(0.390096\pi\)
−0.984142 + 0.177382i \(0.943237\pi\)
\(888\) 0 0
\(889\) 0.675445 0.0226537
\(890\) 0 0
\(891\) 1.00000 + 1.73205i 0.0335013 + 0.0580259i
\(892\) 0 0
\(893\) 7.48683 + 12.9676i 0.250537 + 0.433943i
\(894\) 0 0
\(895\) 0.256584 0.444416i 0.00857664 0.0148552i
\(896\) 0 0
\(897\) 0.162278 7.20928i 0.00541829 0.240711i
\(898\) 0 0
\(899\) −33.3925 + 57.8376i −1.11370 + 1.92899i
\(900\) 0 0
\(901\) −2.64911 4.58839i −0.0882547 0.152862i
\(902\) 0 0
\(903\) 21.1491 + 36.6313i 0.703798 + 1.21901i
\(904\) 0 0
\(905\) 24.6491 0.819364
\(906\) 0 0
\(907\) −7.64911 + 13.2486i −0.253985 + 0.439914i −0.964619 0.263647i \(-0.915075\pi\)
0.710635 + 0.703561i \(0.248408\pi\)
\(908\) 0 0
\(909\) 0.324555 0.0107648
\(910\) 0 0
\(911\) −6.97367 −0.231048 −0.115524 0.993305i \(-0.536855\pi\)
−0.115524 + 0.993305i \(0.536855\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 13.0000 0.429767
\(916\) 0 0
\(917\) −32.9057 56.9943i −1.08664 1.88212i
\(918\) 0 0
\(919\) 14.1623 + 24.5298i 0.467170 + 0.809163i 0.999297 0.0375022i \(-0.0119401\pi\)
−0.532126 + 0.846665i \(0.678607\pi\)
\(920\) 0 0
\(921\) −12.0811 + 20.9251i −0.398087 + 0.689507i
\(922\) 0 0
\(923\) 0.445195 19.7780i 0.0146538 0.651002i
\(924\) 0 0
\(925\) −4.16228 + 7.20928i −0.136855 + 0.237040i
\(926\) 0 0
\(927\) 5.24342 + 9.08186i 0.172216 + 0.298288i
\(928\) 0 0
\(929\) −20.3246 35.2032i −0.666827 1.15498i −0.978786 0.204883i \(-0.934319\pi\)
0.311959 0.950095i \(-0.399015\pi\)
\(930\) 0 0
\(931\) 20.6491 0.676747
\(932\) 0 0
\(933\) 12.9057 22.3533i 0.422513 0.731815i
\(934\) 0 0
\(935\) −1.67544 −0.0547929
\(936\) 0 0
\(937\) 20.9737 0.685180 0.342590 0.939485i \(-0.388696\pi\)
0.342590 + 0.939485i \(0.388696\pi\)
\(938\) 0 0
\(939\) 9.24342 16.0101i 0.301647 0.522469i
\(940\) 0 0
\(941\) −40.4605 −1.31897 −0.659487 0.751716i \(-0.729227\pi\)
−0.659487 + 0.751716i \(0.729227\pi\)
\(942\) 0 0
\(943\) 5.16228 + 8.94133i 0.168107 + 0.291170i
\(944\) 0 0
\(945\) −2.08114 3.60464i −0.0676995 0.117259i
\(946\) 0 0
\(947\) 20.2302 35.0398i 0.657395 1.13864i −0.323893 0.946094i \(-0.604992\pi\)
0.981288 0.192547i \(-0.0616749\pi\)
\(948\) 0 0
\(949\) −18.9868 11.5394i −0.616339 0.374585i
\(950\) 0 0
\(951\) −1.16228 + 2.01312i −0.0376894 + 0.0652800i
\(952\) 0 0
\(953\) 21.8377 + 37.8240i 0.707393 + 1.22524i 0.965821 + 0.259210i \(0.0834623\pi\)
−0.258428 + 0.966031i \(0.583204\pi\)
\(954\) 0 0
\(955\) 5.90569 + 10.2290i 0.191104 + 0.331001i
\(956\) 0 0
\(957\) −14.3246 −0.463047
\(958\) 0 0
\(959\) −26.7171 + 46.2753i −0.862740 + 1.49431i
\(960\) 0 0
\(961\) 55.9473 1.80475
\(962\) 0 0
\(963\) −19.4868 −0.627954
\(964\) 0 0
\(965\) −0.0811388 + 0.140537i −0.00261195 + 0.00452403i
\(966\) 0 0
\(967\) −30.9737 −0.996046 −0.498023 0.867164i \(-0.665940\pi\)
−0.498023 + 0.867164i \(0.665940\pi\)
\(968\) 0 0
\(969\) −0.837722 1.45098i −0.0269115 0.0466121i
\(970\) 0 0
\(971\) −13.8114 23.9220i −0.443229 0.767694i 0.554698 0.832052i \(-0.312834\pi\)
−0.997927 + 0.0643571i \(0.979500\pi\)
\(972\) 0 0
\(973\) −2.75658 + 4.77454i −0.0883720 + 0.153065i
\(974\) 0 0
\(975\) 3.16228 1.73205i 0.101274 0.0554700i
\(976\) 0 0
\(977\) −11.4868 + 19.8958i −0.367496 + 0.636522i −0.989173 0.146751i \(-0.953118\pi\)
0.621677 + 0.783274i \(0.286452\pi\)
\(978\) 0 0
\(979\) −15.4868 26.8240i −0.494961 0.857298i
\(980\) 0 0
\(981\) −3.66228 6.34325i −0.116928 0.202524i
\(982\) 0 0
\(983\) 9.62278 0.306919 0.153459 0.988155i \(-0.450959\pi\)
0.153459 + 0.988155i \(0.450959\pi\)
\(984\) 0 0
\(985\) 2.83772 4.91508i 0.0904174 0.156607i
\(986\) 0 0
\(987\) 31.1623 0.991906
\(988\) 0 0
\(989\) −20.3246 −0.646283
\(990\) 0 0
\(991\) 22.8114 39.5105i 0.724628 1.25509i −0.234499 0.972116i \(-0.575345\pi\)
0.959127 0.282976i \(-0.0913216\pi\)
\(992\) 0 0
\(993\) 2.67544 0.0849027
\(994\) 0 0
\(995\) −3.33772 5.78110i −0.105813 0.183273i
\(996\) 0 0
\(997\) −6.75658 11.7027i −0.213983 0.370630i 0.738974 0.673734i \(-0.235310\pi\)
−0.952958 + 0.303104i \(0.901977\pi\)
\(998\) 0 0
\(999\) 4.16228 7.20928i 0.131689 0.228091i
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 1560.2.bg.e.841.1 yes 4
13.3 even 3 inner 1560.2.bg.e.601.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.bg.e.601.1 4 13.3 even 3 inner
1560.2.bg.e.841.1 yes 4 1.1 even 1 trivial