Properties

Label 320.2.j.b.47.4
Level $320$
Weight $2$
Character 320.47
Analytic conductor $2.555$
Analytic rank $0$
Dimension $18$
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] = [320,2,Mod(47,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.4
Root \(-1.08900 + 0.902261i\) of defining polynomial
Character \(\chi\) \(=\) 320.47
Dual form 320.2.j.b.143.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.496487i q^{3} +(-0.987189 + 2.00635i) q^{5} +(-1.55426 - 1.55426i) q^{7} +2.75350 q^{9} +O(q^{10})\) \(q-0.496487i q^{3} +(-0.987189 + 2.00635i) q^{5} +(-1.55426 - 1.55426i) q^{7} +2.75350 q^{9} +(4.19607 + 4.19607i) q^{11} +5.09530 q^{13} +(0.996130 + 0.490127i) q^{15} +(0.213542 + 0.213542i) q^{17} +(0.844754 + 0.844754i) q^{19} +(-0.771668 + 0.771668i) q^{21} +(-1.70744 + 1.70744i) q^{23} +(-3.05092 - 3.96130i) q^{25} -2.85654i q^{27} +(-2.24750 + 2.24750i) q^{29} +0.818209i q^{31} +(2.08329 - 2.08329i) q^{33} +(4.65273 - 1.58404i) q^{35} -5.12639 q^{37} -2.52975i q^{39} -3.34727i q^{41} +4.49131 q^{43} +(-2.71822 + 5.52450i) q^{45} +(4.29355 - 4.29355i) q^{47} -2.16858i q^{49} +(0.106021 - 0.106021i) q^{51} +1.00653i q^{53} +(-12.5611 + 4.27649i) q^{55} +(0.419410 - 0.419410i) q^{57} +(-7.65005 + 7.65005i) q^{59} +(-1.90291 - 1.90291i) q^{61} +(-4.27964 - 4.27964i) q^{63} +(-5.03002 + 10.2230i) q^{65} -11.0221 q^{67} +(0.847724 + 0.847724i) q^{69} +10.5331 q^{71} +(-2.70854 - 2.70854i) q^{73} +(-1.96674 + 1.51474i) q^{75} -13.0435i q^{77} -8.32010 q^{79} +6.84226 q^{81} -9.17237i q^{83} +(-0.639248 + 0.217635i) q^{85} +(1.11585 + 1.11585i) q^{87} +4.25101 q^{89} +(-7.91940 - 7.91940i) q^{91} +0.406230 q^{93} +(-2.52881 + 0.860944i) q^{95} +(-7.16000 - 7.16000i) q^{97} +(11.5539 + 11.5539i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{5} - 2 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 4 q^{5} - 2 q^{7} - 10 q^{9} + 2 q^{11} - 20 q^{15} - 6 q^{17} - 2 q^{19} - 16 q^{21} + 2 q^{23} + 6 q^{25} - 14 q^{29} - 8 q^{33} + 6 q^{35} + 8 q^{37} + 44 q^{43} - 4 q^{45} + 38 q^{47} - 8 q^{51} + 6 q^{55} + 24 q^{57} + 10 q^{59} + 14 q^{61} - 6 q^{63} - 12 q^{67} + 32 q^{69} - 24 q^{71} + 14 q^{73} - 64 q^{75} - 16 q^{79} + 2 q^{81} - 10 q^{85} - 24 q^{87} - 12 q^{89} + 16 q^{93} + 34 q^{95} + 18 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 0.496487i 0.286647i −0.989676 0.143324i \(-0.954221\pi\)
0.989676 0.143324i \(-0.0457790\pi\)
\(4\) 0 0
\(5\) −0.987189 + 2.00635i −0.441484 + 0.897269i
\(6\) 0 0
\(7\) −1.55426 1.55426i −0.587453 0.587453i 0.349488 0.936941i \(-0.386356\pi\)
−0.936941 + 0.349488i \(0.886356\pi\)
\(8\) 0 0
\(9\) 2.75350 0.917833
\(10\) 0 0
\(11\) 4.19607 + 4.19607i 1.26516 + 1.26516i 0.948558 + 0.316604i \(0.102543\pi\)
0.316604 + 0.948558i \(0.397457\pi\)
\(12\) 0 0
\(13\) 5.09530 1.41318 0.706591 0.707622i \(-0.250232\pi\)
0.706591 + 0.707622i \(0.250232\pi\)
\(14\) 0 0
\(15\) 0.996130 + 0.490127i 0.257200 + 0.126550i
\(16\) 0 0
\(17\) 0.213542 + 0.213542i 0.0517916 + 0.0517916i 0.732528 0.680737i \(-0.238340\pi\)
−0.680737 + 0.732528i \(0.738340\pi\)
\(18\) 0 0
\(19\) 0.844754 + 0.844754i 0.193800 + 0.193800i 0.797336 0.603536i \(-0.206242\pi\)
−0.603536 + 0.797336i \(0.706242\pi\)
\(20\) 0 0
\(21\) −0.771668 + 0.771668i −0.168392 + 0.168392i
\(22\) 0 0
\(23\) −1.70744 + 1.70744i −0.356027 + 0.356027i −0.862346 0.506319i \(-0.831006\pi\)
0.506319 + 0.862346i \(0.331006\pi\)
\(24\) 0 0
\(25\) −3.05092 3.96130i −0.610183 0.792260i
\(26\) 0 0
\(27\) 2.85654i 0.549741i
\(28\) 0 0
\(29\) −2.24750 + 2.24750i −0.417350 + 0.417350i −0.884289 0.466939i \(-0.845357\pi\)
0.466939 + 0.884289i \(0.345357\pi\)
\(30\) 0 0
\(31\) 0.818209i 0.146955i 0.997297 + 0.0734773i \(0.0234097\pi\)
−0.997297 + 0.0734773i \(0.976590\pi\)
\(32\) 0 0
\(33\) 2.08329 2.08329i 0.362655 0.362655i
\(34\) 0 0
\(35\) 4.65273 1.58404i 0.786455 0.267752i
\(36\) 0 0
\(37\) −5.12639 −0.842774 −0.421387 0.906881i \(-0.638457\pi\)
−0.421387 + 0.906881i \(0.638457\pi\)
\(38\) 0 0
\(39\) 2.52975i 0.405084i
\(40\) 0 0
\(41\) 3.34727i 0.522756i −0.965237 0.261378i \(-0.915823\pi\)
0.965237 0.261378i \(-0.0841769\pi\)
\(42\) 0 0
\(43\) 4.49131 0.684919 0.342460 0.939533i \(-0.388740\pi\)
0.342460 + 0.939533i \(0.388740\pi\)
\(44\) 0 0
\(45\) −2.71822 + 5.52450i −0.405209 + 0.823543i
\(46\) 0 0
\(47\) 4.29355 4.29355i 0.626278 0.626278i −0.320851 0.947130i \(-0.603969\pi\)
0.947130 + 0.320851i \(0.103969\pi\)
\(48\) 0 0
\(49\) 2.16858i 0.309797i
\(50\) 0 0
\(51\) 0.106021 0.106021i 0.0148459 0.0148459i
\(52\) 0 0
\(53\) 1.00653i 0.138258i 0.997608 + 0.0691291i \(0.0220220\pi\)
−0.997608 + 0.0691291i \(0.977978\pi\)
\(54\) 0 0
\(55\) −12.5611 + 4.27649i −1.69374 + 0.576642i
\(56\) 0 0
\(57\) 0.419410 0.419410i 0.0555521 0.0555521i
\(58\) 0 0
\(59\) −7.65005 + 7.65005i −0.995952 + 0.995952i −0.999992 0.00404030i \(-0.998714\pi\)
0.00404030 + 0.999992i \(0.498714\pi\)
\(60\) 0 0
\(61\) −1.90291 1.90291i −0.243643 0.243643i 0.574712 0.818355i \(-0.305114\pi\)
−0.818355 + 0.574712i \(0.805114\pi\)
\(62\) 0 0
\(63\) −4.27964 4.27964i −0.539184 0.539184i
\(64\) 0 0
\(65\) −5.03002 + 10.2230i −0.623897 + 1.26800i
\(66\) 0 0
\(67\) −11.0221 −1.34656 −0.673280 0.739387i \(-0.735115\pi\)
−0.673280 + 0.739387i \(0.735115\pi\)
\(68\) 0 0
\(69\) 0.847724 + 0.847724i 0.102054 + 0.102054i
\(70\) 0 0
\(71\) 10.5331 1.25005 0.625027 0.780604i \(-0.285088\pi\)
0.625027 + 0.780604i \(0.285088\pi\)
\(72\) 0 0
\(73\) −2.70854 2.70854i −0.317010 0.317010i 0.530607 0.847618i \(-0.321964\pi\)
−0.847618 + 0.530607i \(0.821964\pi\)
\(74\) 0 0
\(75\) −1.96674 + 1.51474i −0.227099 + 0.174907i
\(76\) 0 0
\(77\) 13.0435i 1.48645i
\(78\) 0 0
\(79\) −8.32010 −0.936085 −0.468042 0.883706i \(-0.655041\pi\)
−0.468042 + 0.883706i \(0.655041\pi\)
\(80\) 0 0
\(81\) 6.84226 0.760252
\(82\) 0 0
\(83\) 9.17237i 1.00680i −0.864054 0.503399i \(-0.832083\pi\)
0.864054 0.503399i \(-0.167917\pi\)
\(84\) 0 0
\(85\) −0.639248 + 0.217635i −0.0693362 + 0.0236058i
\(86\) 0 0
\(87\) 1.11585 + 1.11585i 0.119632 + 0.119632i
\(88\) 0 0
\(89\) 4.25101 0.450606 0.225303 0.974289i \(-0.427663\pi\)
0.225303 + 0.974289i \(0.427663\pi\)
\(90\) 0 0
\(91\) −7.91940 7.91940i −0.830178 0.830178i
\(92\) 0 0
\(93\) 0.406230 0.0421241
\(94\) 0 0
\(95\) −2.52881 + 0.860944i −0.259450 + 0.0883310i
\(96\) 0 0
\(97\) −7.16000 7.16000i −0.726987 0.726987i 0.243031 0.970019i \(-0.421858\pi\)
−0.970019 + 0.243031i \(0.921858\pi\)
\(98\) 0 0
\(99\) 11.5539 + 11.5539i 1.16121 + 1.16121i
\(100\) 0 0
\(101\) 8.38846 8.38846i 0.834683 0.834683i −0.153470 0.988153i \(-0.549045\pi\)
0.988153 + 0.153470i \(0.0490448\pi\)
\(102\) 0 0
\(103\) 5.16478 5.16478i 0.508901 0.508901i −0.405288 0.914189i \(-0.632829\pi\)
0.914189 + 0.405288i \(0.132829\pi\)
\(104\) 0 0
\(105\) −0.786458 2.31002i −0.0767504 0.225435i
\(106\) 0 0
\(107\) 8.97973i 0.868103i −0.900888 0.434052i \(-0.857084\pi\)
0.900888 0.434052i \(-0.142916\pi\)
\(108\) 0 0
\(109\) −10.9081 + 10.9081i −1.04481 + 1.04481i −0.0458592 + 0.998948i \(0.514603\pi\)
−0.998948 + 0.0458592i \(0.985397\pi\)
\(110\) 0 0
\(111\) 2.54519i 0.241579i
\(112\) 0 0
\(113\) −4.29684 + 4.29684i −0.404212 + 0.404212i −0.879715 0.475502i \(-0.842266\pi\)
0.475502 + 0.879715i \(0.342266\pi\)
\(114\) 0 0
\(115\) −1.74017 5.11131i −0.162271 0.476632i
\(116\) 0 0
\(117\) 14.0299 1.29707
\(118\) 0 0
\(119\) 0.663798i 0.0608503i
\(120\) 0 0
\(121\) 24.2140i 2.20127i
\(122\) 0 0
\(123\) −1.66188 −0.149846
\(124\) 0 0
\(125\) 10.9596 2.21067i 0.980257 0.197728i
\(126\) 0 0
\(127\) 0.759686 0.759686i 0.0674112 0.0674112i −0.672597 0.740009i \(-0.734821\pi\)
0.740009 + 0.672597i \(0.234821\pi\)
\(128\) 0 0
\(129\) 2.22988i 0.196330i
\(130\) 0 0
\(131\) −7.59995 + 7.59995i −0.664010 + 0.664010i −0.956323 0.292312i \(-0.905575\pi\)
0.292312 + 0.956323i \(0.405575\pi\)
\(132\) 0 0
\(133\) 2.62593i 0.227697i
\(134\) 0 0
\(135\) 5.73123 + 2.81994i 0.493266 + 0.242702i
\(136\) 0 0
\(137\) −12.7789 + 12.7789i −1.09178 + 1.09178i −0.0964376 + 0.995339i \(0.530745\pi\)
−0.995339 + 0.0964376i \(0.969255\pi\)
\(138\) 0 0
\(139\) 7.74227 7.74227i 0.656691 0.656691i −0.297905 0.954596i \(-0.596288\pi\)
0.954596 + 0.297905i \(0.0962877\pi\)
\(140\) 0 0
\(141\) −2.13169 2.13169i −0.179521 0.179521i
\(142\) 0 0
\(143\) 21.3802 + 21.3802i 1.78790 + 1.78790i
\(144\) 0 0
\(145\) −2.29057 6.72798i −0.190222 0.558728i
\(146\) 0 0
\(147\) −1.07667 −0.0888024
\(148\) 0 0
\(149\) −9.57165 9.57165i −0.784140 0.784140i 0.196386 0.980527i \(-0.437079\pi\)
−0.980527 + 0.196386i \(0.937079\pi\)
\(150\) 0 0
\(151\) 9.68791 0.788391 0.394195 0.919027i \(-0.371023\pi\)
0.394195 + 0.919027i \(0.371023\pi\)
\(152\) 0 0
\(153\) 0.587989 + 0.587989i 0.0475361 + 0.0475361i
\(154\) 0 0
\(155\) −1.64162 0.807726i −0.131858 0.0648781i
\(156\) 0 0
\(157\) 9.97637i 0.796201i −0.917342 0.398101i \(-0.869669\pi\)
0.917342 0.398101i \(-0.130331\pi\)
\(158\) 0 0
\(159\) 0.499732 0.0396313
\(160\) 0 0
\(161\) 5.30761 0.418298
\(162\) 0 0
\(163\) 9.48267i 0.742740i 0.928485 + 0.371370i \(0.121112\pi\)
−0.928485 + 0.371370i \(0.878888\pi\)
\(164\) 0 0
\(165\) 2.12322 + 6.23643i 0.165293 + 0.485506i
\(166\) 0 0
\(167\) 9.43528 + 9.43528i 0.730124 + 0.730124i 0.970644 0.240520i \(-0.0773180\pi\)
−0.240520 + 0.970644i \(0.577318\pi\)
\(168\) 0 0
\(169\) 12.9621 0.997082
\(170\) 0 0
\(171\) 2.32603 + 2.32603i 0.177876 + 0.177876i
\(172\) 0 0
\(173\) 8.94716 0.680240 0.340120 0.940382i \(-0.389532\pi\)
0.340120 + 0.940382i \(0.389532\pi\)
\(174\) 0 0
\(175\) −1.41497 + 10.8988i −0.106962 + 0.823870i
\(176\) 0 0
\(177\) 3.79815 + 3.79815i 0.285487 + 0.285487i
\(178\) 0 0
\(179\) −3.02430 3.02430i −0.226047 0.226047i 0.584992 0.811039i \(-0.301098\pi\)
−0.811039 + 0.584992i \(0.801098\pi\)
\(180\) 0 0
\(181\) −1.54845 + 1.54845i −0.115095 + 0.115095i −0.762309 0.647213i \(-0.775934\pi\)
0.647213 + 0.762309i \(0.275934\pi\)
\(182\) 0 0
\(183\) −0.944773 + 0.944773i −0.0698396 + 0.0698396i
\(184\) 0 0
\(185\) 5.06072 10.2854i 0.372071 0.756195i
\(186\) 0 0
\(187\) 1.79208i 0.131050i
\(188\) 0 0
\(189\) −4.43979 + 4.43979i −0.322947 + 0.322947i
\(190\) 0 0
\(191\) 20.1005i 1.45442i −0.686415 0.727210i \(-0.740817\pi\)
0.686415 0.727210i \(-0.259183\pi\)
\(192\) 0 0
\(193\) 3.82483 3.82483i 0.275317 0.275317i −0.555919 0.831236i \(-0.687634\pi\)
0.831236 + 0.555919i \(0.187634\pi\)
\(194\) 0 0
\(195\) 5.07558 + 2.49734i 0.363470 + 0.178838i
\(196\) 0 0
\(197\) −1.11758 −0.0796246 −0.0398123 0.999207i \(-0.512676\pi\)
−0.0398123 + 0.999207i \(0.512676\pi\)
\(198\) 0 0
\(199\) 25.5830i 1.81353i −0.421635 0.906766i \(-0.638544\pi\)
0.421635 0.906766i \(-0.361456\pi\)
\(200\) 0 0
\(201\) 5.47232i 0.385988i
\(202\) 0 0
\(203\) 6.98637 0.490347
\(204\) 0 0
\(205\) 6.71581 + 3.30439i 0.469053 + 0.230788i
\(206\) 0 0
\(207\) −4.70145 + 4.70145i −0.326773 + 0.326773i
\(208\) 0 0
\(209\) 7.08929i 0.490376i
\(210\) 0 0
\(211\) −0.411613 + 0.411613i −0.0283366 + 0.0283366i −0.721133 0.692797i \(-0.756378\pi\)
0.692797 + 0.721133i \(0.256378\pi\)
\(212\) 0 0
\(213\) 5.22957i 0.358324i
\(214\) 0 0
\(215\) −4.43378 + 9.01117i −0.302381 + 0.614557i
\(216\) 0 0
\(217\) 1.27171 1.27171i 0.0863290 0.0863290i
\(218\) 0 0
\(219\) −1.34475 + 1.34475i −0.0908701 + 0.0908701i
\(220\) 0 0
\(221\) 1.08806 + 1.08806i 0.0731909 + 0.0731909i
\(222\) 0 0
\(223\) −16.7466 16.7466i −1.12143 1.12143i −0.991526 0.129908i \(-0.958532\pi\)
−0.129908 0.991526i \(-0.541468\pi\)
\(224\) 0 0
\(225\) −8.40070 10.9074i −0.560047 0.727163i
\(226\) 0 0
\(227\) −13.7807 −0.914659 −0.457330 0.889297i \(-0.651194\pi\)
−0.457330 + 0.889297i \(0.651194\pi\)
\(228\) 0 0
\(229\) 7.90971 + 7.90971i 0.522688 + 0.522688i 0.918382 0.395694i \(-0.129496\pi\)
−0.395694 + 0.918382i \(0.629496\pi\)
\(230\) 0 0
\(231\) −6.47594 −0.426086
\(232\) 0 0
\(233\) 1.67997 + 1.67997i 0.110058 + 0.110058i 0.759991 0.649933i \(-0.225203\pi\)
−0.649933 + 0.759991i \(0.725203\pi\)
\(234\) 0 0
\(235\) 4.37583 + 12.8529i 0.285448 + 0.838432i
\(236\) 0 0
\(237\) 4.13083i 0.268326i
\(238\) 0 0
\(239\) 11.7685 0.761241 0.380620 0.924731i \(-0.375710\pi\)
0.380620 + 0.924731i \(0.375710\pi\)
\(240\) 0 0
\(241\) −13.2730 −0.854991 −0.427495 0.904018i \(-0.640604\pi\)
−0.427495 + 0.904018i \(0.640604\pi\)
\(242\) 0 0
\(243\) 11.9667i 0.767665i
\(244\) 0 0
\(245\) 4.35094 + 2.14080i 0.277971 + 0.136770i
\(246\) 0 0
\(247\) 4.30427 + 4.30427i 0.273874 + 0.273874i
\(248\) 0 0
\(249\) −4.55396 −0.288596
\(250\) 0 0
\(251\) −10.3795 10.3795i −0.655149 0.655149i 0.299079 0.954228i \(-0.403321\pi\)
−0.954228 + 0.299079i \(0.903321\pi\)
\(252\) 0 0
\(253\) −14.3291 −0.900863
\(254\) 0 0
\(255\) 0.108053 + 0.317378i 0.00676654 + 0.0198750i
\(256\) 0 0
\(257\) 20.4353 + 20.4353i 1.27472 + 1.27472i 0.943582 + 0.331140i \(0.107433\pi\)
0.331140 + 0.943582i \(0.392567\pi\)
\(258\) 0 0
\(259\) 7.96772 + 7.96772i 0.495090 + 0.495090i
\(260\) 0 0
\(261\) −6.18848 + 6.18848i −0.383058 + 0.383058i
\(262\) 0 0
\(263\) −14.0611 + 14.0611i −0.867047 + 0.867047i −0.992144 0.125098i \(-0.960076\pi\)
0.125098 + 0.992144i \(0.460076\pi\)
\(264\) 0 0
\(265\) −2.01946 0.993639i −0.124055 0.0610388i
\(266\) 0 0
\(267\) 2.11057i 0.129165i
\(268\) 0 0
\(269\) 6.61443 6.61443i 0.403289 0.403289i −0.476101 0.879390i \(-0.657950\pi\)
0.879390 + 0.476101i \(0.157950\pi\)
\(270\) 0 0
\(271\) 10.6219i 0.645237i 0.946529 + 0.322619i \(0.104563\pi\)
−0.946529 + 0.322619i \(0.895437\pi\)
\(272\) 0 0
\(273\) −3.93188 + 3.93188i −0.237968 + 0.237968i
\(274\) 0 0
\(275\) 3.82004 29.4237i 0.230357 1.77432i
\(276\) 0 0
\(277\) −8.28511 −0.497804 −0.248902 0.968529i \(-0.580070\pi\)
−0.248902 + 0.968529i \(0.580070\pi\)
\(278\) 0 0
\(279\) 2.25294i 0.134880i
\(280\) 0 0
\(281\) 21.0176i 1.25380i −0.779098 0.626902i \(-0.784323\pi\)
0.779098 0.626902i \(-0.215677\pi\)
\(282\) 0 0
\(283\) −14.4748 −0.860436 −0.430218 0.902725i \(-0.641563\pi\)
−0.430218 + 0.902725i \(0.641563\pi\)
\(284\) 0 0
\(285\) 0.427448 + 1.25552i 0.0253198 + 0.0743706i
\(286\) 0 0
\(287\) −5.20251 + 5.20251i −0.307095 + 0.307095i
\(288\) 0 0
\(289\) 16.9088i 0.994635i
\(290\) 0 0
\(291\) −3.55485 + 3.55485i −0.208389 + 0.208389i
\(292\) 0 0
\(293\) 11.9165i 0.696171i 0.937463 + 0.348086i \(0.113168\pi\)
−0.937463 + 0.348086i \(0.886832\pi\)
\(294\) 0 0
\(295\) −7.79667 22.9008i −0.453940 1.33333i
\(296\) 0 0
\(297\) 11.9862 11.9862i 0.695512 0.695512i
\(298\) 0 0
\(299\) −8.69993 + 8.69993i −0.503130 + 0.503130i
\(300\) 0 0
\(301\) −6.98065 6.98065i −0.402358 0.402358i
\(302\) 0 0
\(303\) −4.16477 4.16477i −0.239260 0.239260i
\(304\) 0 0
\(305\) 5.69645 1.93938i 0.326178 0.111049i
\(306\) 0 0
\(307\) 25.4511 1.45257 0.726287 0.687392i \(-0.241245\pi\)
0.726287 + 0.687392i \(0.241245\pi\)
\(308\) 0 0
\(309\) −2.56425 2.56425i −0.145875 0.145875i
\(310\) 0 0
\(311\) −21.4775 −1.21788 −0.608939 0.793217i \(-0.708404\pi\)
−0.608939 + 0.793217i \(0.708404\pi\)
\(312\) 0 0
\(313\) −18.7965 18.7965i −1.06244 1.06244i −0.997916 0.0645277i \(-0.979446\pi\)
−0.0645277 0.997916i \(-0.520554\pi\)
\(314\) 0 0
\(315\) 12.8113 4.36167i 0.721835 0.245752i
\(316\) 0 0
\(317\) 16.2531i 0.912864i 0.889758 + 0.456432i \(0.150873\pi\)
−0.889758 + 0.456432i \(0.849127\pi\)
\(318\) 0 0
\(319\) −18.8613 −1.05603
\(320\) 0 0
\(321\) −4.45832 −0.248839
\(322\) 0 0
\(323\) 0.360781i 0.0200744i
\(324\) 0 0
\(325\) −15.5453 20.1840i −0.862300 1.11961i
\(326\) 0 0
\(327\) 5.41574 + 5.41574i 0.299491 + 0.299491i
\(328\) 0 0
\(329\) −13.3465 −0.735818
\(330\) 0 0
\(331\) 8.71558 + 8.71558i 0.479052 + 0.479052i 0.904828 0.425777i \(-0.139999\pi\)
−0.425777 + 0.904828i \(0.639999\pi\)
\(332\) 0 0
\(333\) −14.1155 −0.773526
\(334\) 0 0
\(335\) 10.8809 22.1142i 0.594485 1.20823i
\(336\) 0 0
\(337\) 0.0406874 + 0.0406874i 0.00221638 + 0.00221638i 0.708214 0.705998i \(-0.249501\pi\)
−0.705998 + 0.708214i \(0.749501\pi\)
\(338\) 0 0
\(339\) 2.13333 + 2.13333i 0.115866 + 0.115866i
\(340\) 0 0
\(341\) −3.43326 + 3.43326i −0.185921 + 0.185921i
\(342\) 0 0
\(343\) −14.2503 + 14.2503i −0.769445 + 0.769445i
\(344\) 0 0
\(345\) −2.53770 + 0.863971i −0.136625 + 0.0465146i
\(346\) 0 0
\(347\) 35.7094i 1.91698i 0.285124 + 0.958491i \(0.407965\pi\)
−0.285124 + 0.958491i \(0.592035\pi\)
\(348\) 0 0
\(349\) 0.274452 0.274452i 0.0146911 0.0146911i −0.699723 0.714414i \(-0.746693\pi\)
0.714414 + 0.699723i \(0.246693\pi\)
\(350\) 0 0
\(351\) 14.5549i 0.776884i
\(352\) 0 0
\(353\) −15.6215 + 15.6215i −0.831446 + 0.831446i −0.987715 0.156268i \(-0.950054\pi\)
0.156268 + 0.987715i \(0.450054\pi\)
\(354\) 0 0
\(355\) −10.3982 + 21.1332i −0.551879 + 1.12163i
\(356\) 0 0
\(357\) −0.329567 −0.0174426
\(358\) 0 0
\(359\) 0.768787i 0.0405750i 0.999794 + 0.0202875i \(0.00645816\pi\)
−0.999794 + 0.0202875i \(0.993542\pi\)
\(360\) 0 0
\(361\) 17.5728i 0.924883i
\(362\) 0 0
\(363\) 12.0219 0.630988
\(364\) 0 0
\(365\) 8.10812 2.76045i 0.424399 0.144488i
\(366\) 0 0
\(367\) 13.7849 13.7849i 0.719568 0.719568i −0.248949 0.968517i \(-0.580085\pi\)
0.968517 + 0.248949i \(0.0800852\pi\)
\(368\) 0 0
\(369\) 9.21671i 0.479803i
\(370\) 0 0
\(371\) 1.56441 1.56441i 0.0812202 0.0812202i
\(372\) 0 0
\(373\) 21.4003i 1.10806i 0.832496 + 0.554031i \(0.186911\pi\)
−0.832496 + 0.554031i \(0.813089\pi\)
\(374\) 0 0
\(375\) −1.09757 5.44131i −0.0566782 0.280988i
\(376\) 0 0
\(377\) −11.4517 + 11.4517i −0.589791 + 0.589791i
\(378\) 0 0
\(379\) 11.3922 11.3922i 0.585180 0.585180i −0.351142 0.936322i \(-0.614207\pi\)
0.936322 + 0.351142i \(0.114207\pi\)
\(380\) 0 0
\(381\) −0.377174 0.377174i −0.0193232 0.0193232i
\(382\) 0 0
\(383\) 4.42635 + 4.42635i 0.226176 + 0.226176i 0.811093 0.584917i \(-0.198873\pi\)
−0.584917 + 0.811093i \(0.698873\pi\)
\(384\) 0 0
\(385\) 26.1699 + 12.8764i 1.33374 + 0.656243i
\(386\) 0 0
\(387\) 12.3668 0.628642
\(388\) 0 0
\(389\) −12.3502 12.3502i −0.626180 0.626180i 0.320924 0.947105i \(-0.396006\pi\)
−0.947105 + 0.320924i \(0.896006\pi\)
\(390\) 0 0
\(391\) −0.729222 −0.0368784
\(392\) 0 0
\(393\) 3.77328 + 3.77328i 0.190337 + 0.190337i
\(394\) 0 0
\(395\) 8.21351 16.6931i 0.413267 0.839920i
\(396\) 0 0
\(397\) 17.9832i 0.902551i 0.892385 + 0.451275i \(0.149031\pi\)
−0.892385 + 0.451275i \(0.850969\pi\)
\(398\) 0 0
\(399\) −1.30374 −0.0652686
\(400\) 0 0
\(401\) 9.06570 0.452720 0.226360 0.974044i \(-0.427317\pi\)
0.226360 + 0.974044i \(0.427317\pi\)
\(402\) 0 0
\(403\) 4.16902i 0.207674i
\(404\) 0 0
\(405\) −6.75461 + 13.7280i −0.335639 + 0.682150i
\(406\) 0 0
\(407\) −21.5107 21.5107i −1.06625 1.06625i
\(408\) 0 0
\(409\) 30.0616 1.48645 0.743226 0.669040i \(-0.233295\pi\)
0.743226 + 0.669040i \(0.233295\pi\)
\(410\) 0 0
\(411\) 6.34457 + 6.34457i 0.312955 + 0.312955i
\(412\) 0 0
\(413\) 23.7803 1.17015
\(414\) 0 0
\(415\) 18.4030 + 9.05486i 0.903369 + 0.444485i
\(416\) 0 0
\(417\) −3.84394 3.84394i −0.188239 0.188239i
\(418\) 0 0
\(419\) −15.3986 15.3986i −0.752271 0.752271i 0.222631 0.974903i \(-0.428535\pi\)
−0.974903 + 0.222631i \(0.928535\pi\)
\(420\) 0 0
\(421\) −3.86468 + 3.86468i −0.188353 + 0.188353i −0.794984 0.606631i \(-0.792521\pi\)
0.606631 + 0.794984i \(0.292521\pi\)
\(422\) 0 0
\(423\) 11.8223 11.8223i 0.574819 0.574819i
\(424\) 0 0
\(425\) 0.194406 1.49740i 0.00943006 0.0726348i
\(426\) 0 0
\(427\) 5.91523i 0.286258i
\(428\) 0 0
\(429\) 10.6150 10.6150i 0.512497 0.512497i
\(430\) 0 0
\(431\) 27.2692i 1.31351i 0.754103 + 0.656756i \(0.228072\pi\)
−0.754103 + 0.656756i \(0.771928\pi\)
\(432\) 0 0
\(433\) 19.1435 19.1435i 0.919978 0.919978i −0.0770497 0.997027i \(-0.524550\pi\)
0.997027 + 0.0770497i \(0.0245500\pi\)
\(434\) 0 0
\(435\) −3.34036 + 1.13724i −0.160158 + 0.0545265i
\(436\) 0 0
\(437\) −2.88474 −0.137996
\(438\) 0 0
\(439\) 30.1995i 1.44134i 0.693276 + 0.720672i \(0.256167\pi\)
−0.693276 + 0.720672i \(0.743833\pi\)
\(440\) 0 0
\(441\) 5.97118i 0.284342i
\(442\) 0 0
\(443\) −27.7051 −1.31631 −0.658153 0.752884i \(-0.728662\pi\)
−0.658153 + 0.752884i \(0.728662\pi\)
\(444\) 0 0
\(445\) −4.19655 + 8.52903i −0.198935 + 0.404315i
\(446\) 0 0
\(447\) −4.75220 + 4.75220i −0.224772 + 0.224772i
\(448\) 0 0
\(449\) 9.78315i 0.461695i 0.972990 + 0.230848i \(0.0741499\pi\)
−0.972990 + 0.230848i \(0.925850\pi\)
\(450\) 0 0
\(451\) 14.0454 14.0454i 0.661371 0.661371i
\(452\) 0 0
\(453\) 4.80992i 0.225990i
\(454\) 0 0
\(455\) 23.7071 8.07118i 1.11140 0.378383i
\(456\) 0 0
\(457\) 0.557108 0.557108i 0.0260604 0.0260604i −0.693957 0.720017i \(-0.744134\pi\)
0.720017 + 0.693957i \(0.244134\pi\)
\(458\) 0 0
\(459\) 0.609992 0.609992i 0.0284720 0.0284720i
\(460\) 0 0
\(461\) −12.5791 12.5791i −0.585865 0.585865i 0.350644 0.936509i \(-0.385963\pi\)
−0.936509 + 0.350644i \(0.885963\pi\)
\(462\) 0 0
\(463\) −3.29549 3.29549i −0.153154 0.153154i 0.626371 0.779525i \(-0.284540\pi\)
−0.779525 + 0.626371i \(0.784540\pi\)
\(464\) 0 0
\(465\) −0.401026 + 0.815042i −0.0185971 + 0.0377967i
\(466\) 0 0
\(467\) −10.1995 −0.471979 −0.235989 0.971756i \(-0.575833\pi\)
−0.235989 + 0.971756i \(0.575833\pi\)
\(468\) 0 0
\(469\) 17.1311 + 17.1311i 0.791042 + 0.791042i
\(470\) 0 0
\(471\) −4.95314 −0.228229
\(472\) 0 0
\(473\) 18.8459 + 18.8459i 0.866534 + 0.866534i
\(474\) 0 0
\(475\) 0.769051 5.92360i 0.0352865 0.271793i
\(476\) 0 0
\(477\) 2.77149i 0.126898i
\(478\) 0 0
\(479\) 5.65795 0.258518 0.129259 0.991611i \(-0.458740\pi\)
0.129259 + 0.991611i \(0.458740\pi\)
\(480\) 0 0
\(481\) −26.1205 −1.19099
\(482\) 0 0
\(483\) 2.63516i 0.119904i
\(484\) 0 0
\(485\) 21.4338 7.29722i 0.973257 0.331350i
\(486\) 0 0
\(487\) 19.7470 + 19.7470i 0.894823 + 0.894823i 0.994972 0.100149i \(-0.0319321\pi\)
−0.100149 + 0.994972i \(0.531932\pi\)
\(488\) 0 0
\(489\) 4.70802 0.212904
\(490\) 0 0
\(491\) 4.21405 + 4.21405i 0.190177 + 0.190177i 0.795773 0.605595i \(-0.207065\pi\)
−0.605595 + 0.795773i \(0.707065\pi\)
\(492\) 0 0
\(493\) −0.959871 −0.0432304
\(494\) 0 0
\(495\) −34.5870 + 11.7753i −1.55457 + 0.529261i
\(496\) 0 0
\(497\) −16.3712 16.3712i −0.734348 0.734348i
\(498\) 0 0
\(499\) −16.8862 16.8862i −0.755928 0.755928i 0.219650 0.975579i \(-0.429508\pi\)
−0.975579 + 0.219650i \(0.929508\pi\)
\(500\) 0 0
\(501\) 4.68450 4.68450i 0.209288 0.209288i
\(502\) 0 0
\(503\) 20.3714 20.3714i 0.908317 0.908317i −0.0878190 0.996136i \(-0.527990\pi\)
0.996136 + 0.0878190i \(0.0279897\pi\)
\(504\) 0 0
\(505\) 8.54923 + 25.1112i 0.380436 + 1.11744i
\(506\) 0 0
\(507\) 6.43550i 0.285811i
\(508\) 0 0
\(509\) 20.6309 20.6309i 0.914448 0.914448i −0.0821701 0.996618i \(-0.526185\pi\)
0.996618 + 0.0821701i \(0.0261851\pi\)
\(510\) 0 0
\(511\) 8.41952i 0.372458i
\(512\) 0 0
\(513\) 2.41307 2.41307i 0.106540 0.106540i
\(514\) 0 0
\(515\) 5.26376 + 15.4610i 0.231949 + 0.681293i
\(516\) 0 0
\(517\) 36.0320 1.58469
\(518\) 0 0
\(519\) 4.44215i 0.194989i
\(520\) 0 0
\(521\) 19.0433i 0.834300i 0.908838 + 0.417150i \(0.136971\pi\)
−0.908838 + 0.417150i \(0.863029\pi\)
\(522\) 0 0
\(523\) 19.1782 0.838603 0.419301 0.907847i \(-0.362275\pi\)
0.419301 + 0.907847i \(0.362275\pi\)
\(524\) 0 0
\(525\) 5.41111 + 0.702515i 0.236160 + 0.0306603i
\(526\) 0 0
\(527\) −0.174722 + 0.174722i −0.00761101 + 0.00761101i
\(528\) 0 0
\(529\) 17.1693i 0.746490i
\(530\) 0 0
\(531\) −21.0644 + 21.0644i −0.914118 + 0.914118i
\(532\) 0 0
\(533\) 17.0553i 0.738749i
\(534\) 0 0
\(535\) 18.0165 + 8.86469i 0.778922 + 0.383254i
\(536\) 0 0
\(537\) −1.50153 + 1.50153i −0.0647957 + 0.0647957i
\(538\) 0 0
\(539\) 9.09950 9.09950i 0.391943 0.391943i
\(540\) 0 0
\(541\) 14.5231 + 14.5231i 0.624398 + 0.624398i 0.946653 0.322255i \(-0.104441\pi\)
−0.322255 + 0.946653i \(0.604441\pi\)
\(542\) 0 0
\(543\) 0.768787 + 0.768787i 0.0329918 + 0.0329918i
\(544\) 0 0
\(545\) −11.1172 32.6539i −0.476207 1.39874i
\(546\) 0 0
\(547\) 9.97058 0.426311 0.213156 0.977018i \(-0.431626\pi\)
0.213156 + 0.977018i \(0.431626\pi\)
\(548\) 0 0
\(549\) −5.23967 5.23967i −0.223624 0.223624i
\(550\) 0 0
\(551\) −3.79716 −0.161765
\(552\) 0 0
\(553\) 12.9316 + 12.9316i 0.549906 + 0.549906i
\(554\) 0 0
\(555\) −5.10655 2.51258i −0.216761 0.106653i
\(556\) 0 0
\(557\) 11.4424i 0.484831i 0.970173 + 0.242416i \(0.0779397\pi\)
−0.970173 + 0.242416i \(0.922060\pi\)
\(558\) 0 0
\(559\) 22.8846 0.967915
\(560\) 0 0
\(561\) 0.889743 0.0375650
\(562\) 0 0
\(563\) 47.0585i 1.98328i 0.129034 + 0.991640i \(0.458812\pi\)
−0.129034 + 0.991640i \(0.541188\pi\)
\(564\) 0 0
\(565\) −4.37919 12.8628i −0.184234 0.541141i
\(566\) 0 0
\(567\) −10.6346 10.6346i −0.446612 0.446612i
\(568\) 0 0
\(569\) −41.4684 −1.73845 −0.869224 0.494419i \(-0.835381\pi\)
−0.869224 + 0.494419i \(0.835381\pi\)
\(570\) 0 0
\(571\) −16.1745 16.1745i −0.676881 0.676881i 0.282412 0.959293i \(-0.408865\pi\)
−0.959293 + 0.282412i \(0.908865\pi\)
\(572\) 0 0
\(573\) −9.97963 −0.416905
\(574\) 0 0
\(575\) 11.9730 + 1.55443i 0.499307 + 0.0648242i
\(576\) 0 0
\(577\) 20.0316 + 20.0316i 0.833926 + 0.833926i 0.988051 0.154125i \(-0.0492560\pi\)
−0.154125 + 0.988051i \(0.549256\pi\)
\(578\) 0 0
\(579\) −1.89898 1.89898i −0.0789189 0.0789189i
\(580\) 0 0
\(581\) −14.2562 + 14.2562i −0.591447 + 0.591447i
\(582\) 0 0
\(583\) −4.22349 + 4.22349i −0.174919 + 0.174919i
\(584\) 0 0
\(585\) −13.8502 + 28.1490i −0.572634 + 1.16382i
\(586\) 0 0
\(587\) 29.1190i 1.20187i −0.799298 0.600935i \(-0.794795\pi\)
0.799298 0.600935i \(-0.205205\pi\)
\(588\) 0 0
\(589\) −0.691185 + 0.691185i −0.0284798 + 0.0284798i
\(590\) 0 0
\(591\) 0.554866i 0.0228242i
\(592\) 0 0
\(593\) −10.3431 + 10.3431i −0.424740 + 0.424740i −0.886832 0.462092i \(-0.847099\pi\)
0.462092 + 0.886832i \(0.347099\pi\)
\(594\) 0 0
\(595\) 1.33181 + 0.655294i 0.0545991 + 0.0268644i
\(596\) 0 0
\(597\) −12.7016 −0.519843
\(598\) 0 0
\(599\) 2.59479i 0.106020i 0.998594 + 0.0530101i \(0.0168816\pi\)
−0.998594 + 0.0530101i \(0.983118\pi\)
\(600\) 0 0
\(601\) 14.4092i 0.587765i −0.955842 0.293882i \(-0.905053\pi\)
0.955842 0.293882i \(-0.0949474\pi\)
\(602\) 0 0
\(603\) −30.3493 −1.23592
\(604\) 0 0
\(605\) −48.5818 23.9038i −1.97513 0.971826i
\(606\) 0 0
\(607\) −11.8502 + 11.8502i −0.480985 + 0.480985i −0.905446 0.424461i \(-0.860464\pi\)
0.424461 + 0.905446i \(0.360464\pi\)
\(608\) 0 0
\(609\) 3.46864i 0.140557i
\(610\) 0 0
\(611\) 21.8769 21.8769i 0.885045 0.885045i
\(612\) 0 0
\(613\) 16.8256i 0.679579i −0.940502 0.339789i \(-0.889644\pi\)
0.940502 0.339789i \(-0.110356\pi\)
\(614\) 0 0
\(615\) 1.64059 3.33431i 0.0661548 0.134453i
\(616\) 0 0
\(617\) 22.4849 22.4849i 0.905209 0.905209i −0.0906720 0.995881i \(-0.528902\pi\)
0.995881 + 0.0906720i \(0.0289015\pi\)
\(618\) 0 0
\(619\) −14.1269 + 14.1269i −0.567809 + 0.567809i −0.931514 0.363705i \(-0.881512\pi\)
0.363705 + 0.931514i \(0.381512\pi\)
\(620\) 0 0
\(621\) 4.87738 + 4.87738i 0.195723 + 0.195723i
\(622\) 0 0
\(623\) −6.60715 6.60715i −0.264710 0.264710i
\(624\) 0 0
\(625\) −6.38382 + 24.1712i −0.255353 + 0.966848i
\(626\) 0 0
\(627\) 3.51974 0.140565
\(628\) 0 0
\(629\) −1.09470 1.09470i −0.0436486 0.0436486i
\(630\) 0 0
\(631\) 33.9235 1.35047 0.675236 0.737601i \(-0.264042\pi\)
0.675236 + 0.737601i \(0.264042\pi\)
\(632\) 0 0
\(633\) 0.204361 + 0.204361i 0.00812261 + 0.00812261i
\(634\) 0 0
\(635\) 0.774246 + 2.27415i 0.0307250 + 0.0902470i
\(636\) 0 0
\(637\) 11.0496i 0.437799i
\(638\) 0 0
\(639\) 29.0030 1.14734
\(640\) 0 0
\(641\) 18.8495 0.744509 0.372254 0.928131i \(-0.378585\pi\)
0.372254 + 0.928131i \(0.378585\pi\)
\(642\) 0 0
\(643\) 16.4916i 0.650364i −0.945652 0.325182i \(-0.894574\pi\)
0.945652 0.325182i \(-0.105426\pi\)
\(644\) 0 0
\(645\) 4.47393 + 2.20131i 0.176161 + 0.0866766i
\(646\) 0 0
\(647\) 0.316870 + 0.316870i 0.0124574 + 0.0124574i 0.713308 0.700851i \(-0.247196\pi\)
−0.700851 + 0.713308i \(0.747196\pi\)
\(648\) 0 0
\(649\) −64.2002 −2.52008
\(650\) 0 0
\(651\) −0.631386 0.631386i −0.0247460 0.0247460i
\(652\) 0 0
\(653\) −17.0751 −0.668200 −0.334100 0.942538i \(-0.608432\pi\)
−0.334100 + 0.942538i \(0.608432\pi\)
\(654\) 0 0
\(655\) −7.74560 22.7508i −0.302646 0.888946i
\(656\) 0 0
\(657\) −7.45796 7.45796i −0.290963 0.290963i
\(658\) 0 0
\(659\) −7.42245 7.42245i −0.289138 0.289138i 0.547601 0.836739i \(-0.315541\pi\)
−0.836739 + 0.547601i \(0.815541\pi\)
\(660\) 0 0
\(661\) 31.7614 31.7614i 1.23538 1.23538i 0.273507 0.961870i \(-0.411816\pi\)
0.961870 0.273507i \(-0.0881837\pi\)
\(662\) 0 0
\(663\) 0.540209 0.540209i 0.0209800 0.0209800i
\(664\) 0 0
\(665\) 5.26854 + 2.59229i 0.204305 + 0.100525i
\(666\) 0 0
\(667\) 7.67495i 0.297175i
\(668\) 0 0
\(669\) −8.31446 + 8.31446i −0.321456 + 0.321456i
\(670\) 0 0
\(671\) 15.9695i 0.616496i
\(672\) 0 0
\(673\) 4.14672 4.14672i 0.159844 0.159844i −0.622653 0.782498i \(-0.713945\pi\)
0.782498 + 0.622653i \(0.213945\pi\)
\(674\) 0 0
\(675\) −11.3156 + 8.71507i −0.435538 + 0.335443i
\(676\) 0 0
\(677\) −25.2618 −0.970890 −0.485445 0.874267i \(-0.661342\pi\)
−0.485445 + 0.874267i \(0.661342\pi\)
\(678\) 0 0
\(679\) 22.2569i 0.854143i
\(680\) 0 0
\(681\) 6.84196i 0.262184i
\(682\) 0 0
\(683\) 8.20306 0.313881 0.156941 0.987608i \(-0.449837\pi\)
0.156941 + 0.987608i \(0.449837\pi\)
\(684\) 0 0
\(685\) −13.0238 38.2542i −0.497615 1.46162i
\(686\) 0 0
\(687\) 3.92707 3.92707i 0.149827 0.149827i
\(688\) 0 0
\(689\) 5.12859i 0.195384i
\(690\) 0 0
\(691\) 7.89158 7.89158i 0.300210 0.300210i −0.540886 0.841096i \(-0.681911\pi\)
0.841096 + 0.540886i \(0.181911\pi\)
\(692\) 0 0
\(693\) 35.9153i 1.36431i
\(694\) 0 0
\(695\) 7.89066 + 23.1768i 0.299310 + 0.879147i
\(696\) 0 0
\(697\) 0.714783 0.714783i 0.0270744 0.0270744i
\(698\) 0 0
\(699\) 0.834083 0.834083i 0.0315479 0.0315479i
\(700\) 0 0
\(701\) 1.50228 + 1.50228i 0.0567405 + 0.0567405i 0.734908 0.678167i \(-0.237225\pi\)
−0.678167 + 0.734908i \(0.737225\pi\)
\(702\) 0 0
\(703\) −4.33054 4.33054i −0.163329 0.163329i
\(704\) 0 0
\(705\) 6.38131 2.17255i 0.240334 0.0818228i
\(706\) 0 0
\(707\) −26.0756 −0.980675
\(708\) 0 0
\(709\) 36.0738 + 36.0738i 1.35478 + 1.35478i 0.880228 + 0.474551i \(0.157390\pi\)
0.474551 + 0.880228i \(0.342610\pi\)
\(710\) 0 0
\(711\) −22.9094 −0.859170
\(712\) 0 0
\(713\) −1.39704 1.39704i −0.0523197 0.0523197i
\(714\) 0 0
\(715\) −64.0026 + 21.7900i −2.39356 + 0.814899i
\(716\) 0 0
\(717\) 5.84291i 0.218207i
\(718\) 0 0
\(719\) −35.0340 −1.30655 −0.653274 0.757121i \(-0.726605\pi\)
−0.653274 + 0.757121i \(0.726605\pi\)
\(720\) 0 0
\(721\) −16.0548 −0.597911
\(722\) 0 0
\(723\) 6.58989i 0.245081i
\(724\) 0 0
\(725\) 15.7599 + 2.04609i 0.585310 + 0.0759898i
\(726\) 0 0
\(727\) −25.4241 25.4241i −0.942928 0.942928i 0.0555295 0.998457i \(-0.482315\pi\)
−0.998457 + 0.0555295i \(0.982315\pi\)
\(728\) 0 0
\(729\) 14.5855 0.540203
\(730\) 0 0
\(731\) 0.959085 + 0.959085i 0.0354731 + 0.0354731i
\(732\) 0 0
\(733\) −7.37554 −0.272422 −0.136211 0.990680i \(-0.543492\pi\)
−0.136211 + 0.990680i \(0.543492\pi\)
\(734\) 0 0
\(735\) 1.06288 2.16019i 0.0392049 0.0796797i
\(736\) 0 0
\(737\) −46.2494 46.2494i −1.70362 1.70362i
\(738\) 0 0
\(739\) 5.55025 + 5.55025i 0.204169 + 0.204169i 0.801784 0.597614i \(-0.203885\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(740\) 0 0
\(741\) 2.13702 2.13702i 0.0785053 0.0785053i
\(742\) 0 0
\(743\) 6.78835 6.78835i 0.249040 0.249040i −0.571536 0.820577i \(-0.693652\pi\)
0.820577 + 0.571536i \(0.193652\pi\)
\(744\) 0 0
\(745\) 28.6532 9.75510i 1.04977 0.357399i
\(746\) 0 0
\(747\) 25.2561i 0.924073i
\(748\) 0 0
\(749\) −13.9568 + 13.9568i −0.509970 + 0.509970i
\(750\) 0 0
\(751\) 3.93385i 0.143548i 0.997421 + 0.0717742i \(0.0228661\pi\)
−0.997421 + 0.0717742i \(0.977134\pi\)
\(752\) 0 0
\(753\) −5.15330 + 5.15330i −0.187797 + 0.187797i
\(754\) 0 0
\(755\) −9.56379 + 19.4374i −0.348062 + 0.707398i
\(756\) 0 0
\(757\) −21.8327 −0.793525 −0.396762 0.917921i \(-0.629866\pi\)
−0.396762 + 0.917921i \(0.629866\pi\)
\(758\) 0 0
\(759\) 7.11421i 0.258230i
\(760\) 0 0
\(761\) 4.27291i 0.154893i 0.996997 + 0.0774464i \(0.0246767\pi\)
−0.996997 + 0.0774464i \(0.975323\pi\)
\(762\) 0 0
\(763\) 33.9080 1.22755
\(764\) 0 0
\(765\) −1.76017 + 0.599258i −0.0636391 + 0.0216662i
\(766\) 0 0
\(767\) −38.9793 + 38.9793i −1.40746 + 1.40746i
\(768\) 0 0
\(769\) 26.1800i 0.944074i 0.881579 + 0.472037i \(0.156481\pi\)
−0.881579 + 0.472037i \(0.843519\pi\)
\(770\) 0 0
\(771\) 10.1459 10.1459i 0.365395 0.365395i
\(772\) 0 0
\(773\) 15.0077i 0.539791i 0.962890 + 0.269895i \(0.0869891\pi\)
−0.962890 + 0.269895i \(0.913011\pi\)
\(774\) 0 0
\(775\) 3.24117 2.49629i 0.116426 0.0896693i
\(776\) 0 0
\(777\) 3.95587 3.95587i 0.141916 0.141916i
\(778\) 0 0
\(779\) 2.82762 2.82762i 0.101310 0.101310i
\(780\) 0 0
\(781\) 44.1977 + 44.1977i 1.58152 + 1.58152i
\(782\) 0 0
\(783\) 6.42007 + 6.42007i 0.229434 + 0.229434i
\(784\) 0 0
\(785\) 20.0161 + 9.84856i 0.714407 + 0.351510i
\(786\) 0 0
\(787\) 42.9223 1.53001 0.765007 0.644022i \(-0.222736\pi\)
0.765007 + 0.644022i \(0.222736\pi\)
\(788\) 0 0
\(789\) 6.98118 + 6.98118i 0.248536 + 0.248536i
\(790\) 0 0
\(791\) 13.3568 0.474912
\(792\) 0 0
\(793\) −9.69591 9.69591i −0.344312 0.344312i
\(794\) 0 0
\(795\) −0.493329 + 1.00264i −0.0174966 + 0.0355599i
\(796\) 0 0
\(797\) 0.280831i 0.00994753i 0.999988 + 0.00497377i \(0.00158321\pi\)
−0.999988 + 0.00497377i \(0.998417\pi\)
\(798\) 0 0
\(799\) 1.83371 0.0648719
\(800\) 0 0
\(801\) 11.7052 0.413581
\(802\) 0 0
\(803\) 22.7304i 0.802139i
\(804\) 0 0
\(805\) −5.23961 + 10.6489i −0.184672 + 0.375326i
\(806\) 0 0
\(807\) −3.28398 3.28398i −0.115602 0.115602i
\(808\) 0 0
\(809\) 16.5787 0.582876 0.291438 0.956590i \(-0.405866\pi\)
0.291438 + 0.956590i \(0.405866\pi\)
\(810\) 0 0
\(811\) −7.25384 7.25384i −0.254717 0.254717i 0.568184 0.822901i \(-0.307646\pi\)
−0.822901 + 0.568184i \(0.807646\pi\)
\(812\) 0 0
\(813\) 5.27366 0.184955
\(814\) 0 0
\(815\) −19.0256 9.36118i −0.666437 0.327908i
\(816\) 0 0
\(817\) 3.79405 + 3.79405i 0.132737 + 0.132737i
\(818\) 0 0
\(819\) −21.8061 21.8061i −0.761965 0.761965i
\(820\) 0 0
\(821\) −15.3525 + 15.3525i −0.535806 + 0.535806i −0.922294 0.386489i \(-0.873688\pi\)
0.386489 + 0.922294i \(0.373688\pi\)
\(822\) 0 0
\(823\) 26.7794 26.7794i 0.933472 0.933472i −0.0644492 0.997921i \(-0.520529\pi\)
0.997921 + 0.0644492i \(0.0205290\pi\)
\(824\) 0 0
\(825\) −14.6085 1.89660i −0.508603 0.0660311i
\(826\) 0 0
\(827\) 39.4186i 1.37072i 0.728205 + 0.685359i \(0.240355\pi\)
−0.728205 + 0.685359i \(0.759645\pi\)
\(828\) 0 0
\(829\) 20.7102 20.7102i 0.719296 0.719296i −0.249165 0.968461i \(-0.580156\pi\)
0.968461 + 0.249165i \(0.0801561\pi\)
\(830\) 0 0
\(831\) 4.11345i 0.142694i
\(832\) 0 0
\(833\) 0.463083 0.463083i 0.0160449 0.0160449i
\(834\) 0 0
\(835\) −28.2449 + 9.61612i −0.977456 + 0.332779i
\(836\) 0 0
\(837\) 2.33725 0.0807870
\(838\) 0 0
\(839\) 31.8706i 1.10029i 0.835068 + 0.550147i \(0.185428\pi\)
−0.835068 + 0.550147i \(0.814572\pi\)
\(840\) 0 0
\(841\) 18.8975i 0.651638i
\(842\) 0 0
\(843\) −10.4350 −0.359399
\(844\) 0 0
\(845\) −12.7960 + 26.0065i −0.440196 + 0.894651i
\(846\) 0 0
\(847\) 37.6347 37.6347i 1.29314 1.29314i
\(848\) 0 0
\(849\) 7.18654i 0.246642i
\(850\) 0 0
\(851\) 8.75302 8.75302i 0.300050 0.300050i
\(852\) 0 0
\(853\) 26.5538i 0.909185i 0.890700 + 0.454592i \(0.150215\pi\)
−0.890700 + 0.454592i \(0.849785\pi\)
\(854\) 0 0
\(855\) −6.96307 + 2.37061i −0.238132 + 0.0810731i
\(856\) 0 0
\(857\) 20.7249 20.7249i 0.707951 0.707951i −0.258153 0.966104i \(-0.583114\pi\)
0.966104 + 0.258153i \(0.0831140\pi\)
\(858\) 0 0
\(859\) 35.9248 35.9248i 1.22574 1.22574i 0.260176 0.965561i \(-0.416219\pi\)
0.965561 0.260176i \(-0.0837807\pi\)
\(860\) 0 0
\(861\) 2.58298 + 2.58298i 0.0880278 + 0.0880278i
\(862\) 0 0
\(863\) 9.19232 + 9.19232i 0.312910 + 0.312910i 0.846036 0.533126i \(-0.178983\pi\)
−0.533126 + 0.846036i \(0.678983\pi\)
\(864\) 0 0
\(865\) −8.83254 + 17.9512i −0.300315 + 0.610358i
\(866\) 0 0
\(867\) −8.39500 −0.285109
\(868\) 0 0
\(869\) −34.9117 34.9117i −1.18430 1.18430i
\(870\) 0 0
\(871\) −56.1608 −1.90293
\(872\) 0 0
\(873\) −19.7151 19.7151i −0.667253 0.667253i
\(874\) 0 0
\(875\) −20.4700 13.5981i −0.692011 0.459699i
\(876\) 0 0
\(877\) 17.9106i 0.604799i −0.953181 0.302399i \(-0.902212\pi\)
0.953181 0.302399i \(-0.0977877\pi\)
\(878\) 0 0
\(879\) 5.91641 0.199556
\(880\) 0 0
\(881\) 6.01537 0.202663 0.101332 0.994853i \(-0.467690\pi\)
0.101332 + 0.994853i \(0.467690\pi\)
\(882\) 0 0
\(883\) 19.8374i 0.667580i 0.942647 + 0.333790i \(0.108328\pi\)
−0.942647 + 0.333790i \(0.891672\pi\)
\(884\) 0 0
\(885\) −11.3699 + 3.87095i −0.382196 + 0.130120i
\(886\) 0 0
\(887\) −14.3740 14.3740i −0.482632 0.482632i 0.423339 0.905971i \(-0.360858\pi\)
−0.905971 + 0.423339i \(0.860858\pi\)
\(888\) 0 0
\(889\) −2.36149 −0.0792019
\(890\) 0 0
\(891\) 28.7106 + 28.7106i 0.961842 + 0.961842i
\(892\) 0 0
\(893\) 7.25398 0.242745
\(894\) 0 0
\(895\) 9.05337 3.08226i 0.302621 0.103029i
\(896\) 0 0
\(897\) 4.31941 + 4.31941i 0.144221 + 0.144221i
\(898\) 0 0
\(899\) −1.83892 1.83892i −0.0613315 0.0613315i
\(900\) 0 0
\(901\) −0.214938 + 0.214938i −0.00716061 + 0.00716061i
\(902\) 0 0
\(903\) −3.46580 + 3.46580i −0.115335 + 0.115335i
\(904\) 0 0
\(905\) −1.57813 4.63536i −0.0524588 0.154084i
\(906\) 0 0
\(907\) 39.0417i 1.29636i 0.761487 + 0.648180i \(0.224469\pi\)
−0.761487 + 0.648180i \(0.775531\pi\)
\(908\) 0 0
\(909\) 23.0976 23.0976i 0.766100 0.766100i
\(910\) 0 0
\(911\) 14.0166i 0.464392i −0.972669 0.232196i \(-0.925409\pi\)
0.972669 0.232196i \(-0.0745911\pi\)
\(912\) 0 0
\(913\) 38.4879 38.4879i 1.27376 1.27376i
\(914\) 0 0
\(915\) −0.962880 2.82822i −0.0318318 0.0934980i
\(916\) 0 0
\(917\) 23.6245 0.780150
\(918\) 0 0
\(919\) 8.15149i 0.268893i 0.990921 + 0.134446i \(0.0429256\pi\)
−0.990921 + 0.134446i \(0.957074\pi\)
\(920\) 0 0
\(921\) 12.6362i 0.416376i
\(922\) 0 0
\(923\) 53.6695 1.76655
\(924\) 0 0
\(925\) 15.6402 + 20.3072i 0.514247 + 0.667696i
\(926\) 0 0
\(927\) 14.2212 14.2212i 0.467086 0.467086i
\(928\) 0 0
\(929\) 13.4779i 0.442196i −0.975252 0.221098i \(-0.929036\pi\)
0.975252 0.221098i \(-0.0709641\pi\)
\(930\) 0 0
\(931\) 1.83192 1.83192i 0.0600386 0.0600386i
\(932\) 0 0
\(933\) 10.6633i 0.349101i
\(934\) 0 0
\(935\) −3.59554 1.76912i −0.117587 0.0578563i
\(936\) 0 0
\(937\) 15.8564 15.8564i 0.518005 0.518005i −0.398963 0.916967i \(-0.630630\pi\)
0.916967 + 0.398963i \(0.130630\pi\)
\(938\) 0 0
\(939\) −9.33225 + 9.33225i −0.304546 + 0.304546i
\(940\) 0 0
\(941\) 15.7073 + 15.7073i 0.512044 + 0.512044i 0.915152 0.403108i \(-0.132070\pi\)
−0.403108 + 0.915152i \(0.632070\pi\)
\(942\) 0 0
\(943\) 5.71527 + 5.71527i 0.186115 + 0.186115i
\(944\) 0 0
\(945\) −4.52489 13.2907i −0.147195 0.432347i
\(946\) 0 0
\(947\) 33.6925 1.09486 0.547430 0.836852i \(-0.315606\pi\)
0.547430 + 0.836852i \(0.315606\pi\)
\(948\) 0 0
\(949\) −13.8008 13.8008i −0.447993 0.447993i
\(950\) 0 0
\(951\) 8.06945 0.261670
\(952\) 0 0
\(953\) 33.5702 + 33.5702i 1.08745 + 1.08745i 0.995791 + 0.0916550i \(0.0292157\pi\)
0.0916550 + 0.995791i \(0.470784\pi\)
\(954\) 0 0
\(955\) 40.3287 + 19.8430i 1.30501 + 0.642103i
\(956\) 0 0
\(957\) 9.36440i 0.302708i
\(958\) 0 0
\(959\) 39.7234 1.28274
\(960\) 0 0
\(961\) 30.3305 0.978404
\(962\) 0 0
\(963\) 24.7257i 0.796774i
\(964\) 0 0
\(965\) 3.89813 + 11.4498i 0.125485 + 0.368582i
\(966\) 0 0
\(967\) −28.6436 28.6436i −0.921115 0.921115i 0.0759933 0.997108i \(-0.475787\pi\)
−0.997108 + 0.0759933i \(0.975787\pi\)
\(968\) 0 0
\(969\) 0.179123 0.00575427
\(970\) 0 0
\(971\) 35.7115 + 35.7115i 1.14604 + 1.14604i 0.987325 + 0.158713i \(0.0507345\pi\)
0.158713 + 0.987325i \(0.449266\pi\)
\(972\) 0 0
\(973\) −24.0669 −0.771551
\(974\) 0 0
\(975\) −10.0211 + 7.71806i −0.320932 + 0.247176i
\(976\) 0 0
\(977\) 7.12822 + 7.12822i 0.228052 + 0.228052i 0.811879 0.583826i \(-0.198445\pi\)
−0.583826 + 0.811879i \(0.698445\pi\)
\(978\) 0 0
\(979\) 17.8375 + 17.8375i 0.570090 + 0.570090i
\(980\) 0 0
\(981\) −30.0355 + 30.0355i −0.958959 + 0.958959i
\(982\) 0 0
\(983\) 23.9941 23.9941i 0.765292 0.765292i −0.211982 0.977274i \(-0.567992\pi\)
0.977274 + 0.211982i \(0.0679918\pi\)
\(984\) 0 0
\(985\) 1.10327 2.24227i 0.0351530 0.0714447i
\(986\) 0 0
\(987\) 6.62639i 0.210920i
\(988\) 0 0
\(989\) −7.66866 + 7.66866i −0.243849 + 0.243849i
\(990\) 0 0
\(991\) 40.6040i 1.28983i 0.764255 + 0.644914i \(0.223107\pi\)
−0.764255 + 0.644914i \(0.776893\pi\)
\(992\) 0 0
\(993\) 4.32718 4.32718i 0.137319 0.137319i
\(994\) 0 0
\(995\) 51.3286 + 25.2553i 1.62723 + 0.800645i
\(996\) 0 0
\(997\) −54.9087 −1.73898 −0.869488 0.493953i \(-0.835551\pi\)
−0.869488 + 0.493953i \(0.835551\pi\)
\(998\) 0 0
\(999\) 14.6437i 0.463308i
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 320.2.j.b.47.4 18
4.3 odd 2 80.2.j.b.67.6 yes 18
5.2 odd 4 1600.2.s.d.943.4 18
5.3 odd 4 320.2.s.b.303.6 18
5.4 even 2 1600.2.j.d.1007.6 18
8.3 odd 2 640.2.j.d.607.4 18
8.5 even 2 640.2.j.c.607.6 18
12.11 even 2 720.2.bd.g.307.4 18
16.3 odd 4 640.2.s.c.287.4 18
16.5 even 4 80.2.s.b.27.9 yes 18
16.11 odd 4 320.2.s.b.207.6 18
16.13 even 4 640.2.s.d.287.6 18
20.3 even 4 80.2.s.b.3.9 yes 18
20.7 even 4 400.2.s.d.243.1 18
20.19 odd 2 400.2.j.d.307.4 18
40.3 even 4 640.2.s.d.223.6 18
40.13 odd 4 640.2.s.c.223.4 18
48.5 odd 4 720.2.z.g.667.1 18
60.23 odd 4 720.2.z.g.163.1 18
80.3 even 4 640.2.j.c.543.4 18
80.13 odd 4 640.2.j.d.543.6 18
80.27 even 4 1600.2.j.d.143.4 18
80.37 odd 4 400.2.j.d.43.4 18
80.43 even 4 inner 320.2.j.b.143.6 18
80.53 odd 4 80.2.j.b.43.6 18
80.59 odd 4 1600.2.s.d.207.4 18
80.69 even 4 400.2.s.d.107.1 18
240.53 even 4 720.2.bd.g.523.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.6 18 80.53 odd 4
80.2.j.b.67.6 yes 18 4.3 odd 2
80.2.s.b.3.9 yes 18 20.3 even 4
80.2.s.b.27.9 yes 18 16.5 even 4
320.2.j.b.47.4 18 1.1 even 1 trivial
320.2.j.b.143.6 18 80.43 even 4 inner
320.2.s.b.207.6 18 16.11 odd 4
320.2.s.b.303.6 18 5.3 odd 4
400.2.j.d.43.4 18 80.37 odd 4
400.2.j.d.307.4 18 20.19 odd 2
400.2.s.d.107.1 18 80.69 even 4
400.2.s.d.243.1 18 20.7 even 4
640.2.j.c.543.4 18 80.3 even 4
640.2.j.c.607.6 18 8.5 even 2
640.2.j.d.543.6 18 80.13 odd 4
640.2.j.d.607.4 18 8.3 odd 2
640.2.s.c.223.4 18 40.13 odd 4
640.2.s.c.287.4 18 16.3 odd 4
640.2.s.d.223.6 18 40.3 even 4
640.2.s.d.287.6 18 16.13 even 4
720.2.z.g.163.1 18 60.23 odd 4
720.2.z.g.667.1 18 48.5 odd 4
720.2.bd.g.307.4 18 12.11 even 2
720.2.bd.g.523.4 18 240.53 even 4
1600.2.j.d.143.4 18 80.27 even 4
1600.2.j.d.1007.6 18 5.4 even 2
1600.2.s.d.207.4 18 80.59 odd 4
1600.2.s.d.943.4 18 5.2 odd 4