Properties

Label 640.2.s.d.223.6
Level $640$
Weight $2$
Character 640.223
Analytic conductor $5.110$
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] = [640,2,Mod(223,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.s (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: \(5.11042572936\)
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 223.6
Root \(-1.08900 - 0.902261i\) of defining polynomial
Character \(\chi\) \(=\) 640.223
Dual form 640.2.s.d.287.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.496487 q^{3} +(2.00635 - 0.987189i) q^{5} +(1.55426 - 1.55426i) q^{7} -2.75350 q^{9} +O(q^{10})\) \(q+0.496487 q^{3} +(2.00635 - 0.987189i) q^{5} +(1.55426 - 1.55426i) q^{7} -2.75350 q^{9} +(4.19607 + 4.19607i) q^{11} -5.09530i q^{13} +(0.996130 - 0.490127i) q^{15} +(0.213542 - 0.213542i) q^{17} +(-0.844754 - 0.844754i) q^{19} +(0.771668 - 0.771668i) q^{21} +(1.70744 + 1.70744i) q^{23} +(3.05092 - 3.96130i) q^{25} -2.85654 q^{27} +(-2.24750 + 2.24750i) q^{29} -0.818209i q^{31} +(2.08329 + 2.08329i) q^{33} +(1.58404 - 4.65273i) q^{35} -5.12639i q^{37} -2.52975i q^{39} -3.34727i q^{41} +4.49131i q^{43} +(-5.52450 + 2.71822i) q^{45} +(4.29355 + 4.29355i) q^{47} +2.16858i q^{49} +(0.106021 - 0.106021i) q^{51} +1.00653 q^{53} +(12.5611 + 4.27649i) q^{55} +(-0.419410 - 0.419410i) q^{57} +(7.65005 - 7.65005i) q^{59} +(1.90291 + 1.90291i) q^{61} +(-4.27964 + 4.27964i) q^{63} +(-5.03002 - 10.2230i) q^{65} +11.0221i q^{67} +(0.847724 + 0.847724i) q^{69} -10.5331 q^{71} +(2.70854 - 2.70854i) q^{73} +(1.51474 - 1.96674i) q^{75} +13.0435 q^{77} -8.32010 q^{79} +6.84226 q^{81} +9.17237 q^{83} +(0.217635 - 0.639248i) q^{85} +(-1.11585 + 1.11585i) q^{87} -4.25101 q^{89} +(-7.91940 - 7.91940i) q^{91} -0.406230i q^{93} +(-2.52881 - 0.860944i) q^{95} +(-7.16000 + 7.16000i) q^{97} +(-11.5539 - 11.5539i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{5} + 2 q^{7} + 10 q^{9} + 2 q^{11} - 20 q^{15} - 6 q^{17} + 2 q^{19} + 16 q^{21} - 2 q^{23} - 6 q^{25} + 24 q^{27} - 14 q^{29} - 8 q^{33} - 2 q^{35} + 14 q^{45} + 38 q^{47} - 8 q^{51} - 12 q^{53} - 6 q^{55} - 24 q^{57} - 10 q^{59} - 14 q^{61} - 6 q^{63} + 32 q^{69} + 24 q^{71} - 14 q^{73} - 16 q^{75} + 44 q^{77} - 16 q^{79} + 2 q^{81} - 40 q^{83} - 14 q^{85} + 24 q^{87} + 12 q^{89} + 34 q^{95} + 18 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.496487 0.286647 0.143324 0.989676i \(-0.454221\pi\)
0.143324 + 0.989676i \(0.454221\pi\)
\(4\) 0 0
\(5\) 2.00635 0.987189i 0.897269 0.441484i
\(6\) 0 0
\(7\) 1.55426 1.55426i 0.587453 0.587453i −0.349488 0.936941i \(-0.613644\pi\)
0.936941 + 0.349488i \(0.113644\pi\)
\(8\) 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
\(13\) 5.09530i 1.41318i −0.707622 0.706591i \(-0.750232\pi\)
0.707622 0.706591i \(-0.249768\pi\)
\(14\) 0 0
\(15\) 0.996130 0.490127i 0.257200 0.126550i
\(16\) 0 0
\(17\) 0.213542 0.213542i 0.0517916 0.0517916i −0.680737 0.732528i \(-0.738340\pi\)
0.732528 + 0.680737i \(0.238340\pi\)
\(18\) 0 0
\(19\) −0.844754 0.844754i −0.193800 0.193800i 0.603536 0.797336i \(-0.293758\pi\)
−0.797336 + 0.603536i \(0.793758\pi\)
\(20\) 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 \(-0.168994\pi\)
−0.506319 + 0.862346i \(0.668994\pi\)
\(24\) 0 0
\(25\) 3.05092 3.96130i 0.610183 0.792260i
\(26\) 0 0
\(27\) −2.85654 −0.549741
\(28\) 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 \(-0.976590\pi\)
0.997297 0.0734773i \(-0.0234097\pi\)
\(32\) 0 0
\(33\) 2.08329 + 2.08329i 0.362655 + 0.362655i
\(34\) 0 0
\(35\) 1.58404 4.65273i 0.267752 0.786455i
\(36\) 0 0
\(37\) 5.12639i 0.842774i −0.906881 0.421387i \(-0.861543\pi\)
0.906881 0.421387i \(-0.138457\pi\)
\(38\) 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
\(43\) 4.49131i 0.684919i 0.939533 + 0.342460i \(0.111260\pi\)
−0.939533 + 0.342460i \(0.888740\pi\)
\(44\) 0 0
\(45\) −5.52450 + 2.71822i −0.823543 + 0.405209i
\(46\) 0 0
\(47\) 4.29355 + 4.29355i 0.626278 + 0.626278i 0.947130 0.320851i \(-0.103969\pi\)
−0.320851 + 0.947130i \(0.603969\pi\)
\(48\) 0 0
\(49\) 2.16858i 0.309797i
\(50\) 0 0
\(51\) 0.106021 0.106021i 0.0148459 0.0148459i
\(52\) 0 0
\(53\) 1.00653 0.138258 0.0691291 0.997608i \(-0.477978\pi\)
0.0691291 + 0.997608i \(0.477978\pi\)
\(54\) 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 −0.00404030 0.999992i \(-0.501286\pi\)
0.999992 + 0.00404030i \(0.00128607\pi\)
\(60\) 0 0
\(61\) 1.90291 + 1.90291i 0.243643 + 0.243643i 0.818355 0.574712i \(-0.194886\pi\)
−0.574712 + 0.818355i \(0.694886\pi\)
\(62\) 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
\(67\) 11.0221i 1.34656i 0.739387 + 0.673280i \(0.235115\pi\)
−0.739387 + 0.673280i \(0.764885\pi\)
\(68\) 0 0
\(69\) 0.847724 + 0.847724i 0.102054 + 0.102054i
\(70\) 0 0
\(71\) −10.5331 −1.25005 −0.625027 0.780604i \(-0.714912\pi\)
−0.625027 + 0.780604i \(0.714912\pi\)
\(72\) 0 0
\(73\) 2.70854 2.70854i 0.317010 0.317010i −0.530607 0.847618i \(-0.678036\pi\)
0.847618 + 0.530607i \(0.178036\pi\)
\(74\) 0 0
\(75\) 1.51474 1.96674i 0.174907 0.227099i
\(76\) 0 0
\(77\) 13.0435 1.48645
\(78\) 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
\(83\) 9.17237 1.00680 0.503399 0.864054i \(-0.332083\pi\)
0.503399 + 0.864054i \(0.332083\pi\)
\(84\) 0 0
\(85\) 0.217635 0.639248i 0.0236058 0.0693362i
\(86\) 0 0
\(87\) −1.11585 + 1.11585i −0.119632 + 0.119632i
\(88\) 0 0
\(89\) −4.25101 −0.450606 −0.225303 0.974289i \(-0.572337\pi\)
−0.225303 + 0.974289i \(0.572337\pi\)
\(90\) 0 0
\(91\) −7.91940 7.91940i −0.830178 0.830178i
\(92\) 0 0
\(93\) 0.406230i 0.0421241i
\(94\) 0 0
\(95\) −2.52881 0.860944i −0.259450 0.0883310i
\(96\) 0 0
\(97\) −7.16000 + 7.16000i −0.726987 + 0.726987i −0.970019 0.243031i \(-0.921858\pi\)
0.243031 + 0.970019i \(0.421858\pi\)
\(98\) 0 0
\(99\) −11.5539 11.5539i −1.16121 1.16121i
\(100\) 0 0
\(101\) −8.38846 + 8.38846i −0.834683 + 0.834683i −0.988153 0.153470i \(-0.950955\pi\)
0.153470 + 0.988153i \(0.450955\pi\)
\(102\) 0 0
\(103\) −5.16478 5.16478i −0.508901 0.508901i 0.405288 0.914189i \(-0.367171\pi\)
−0.914189 + 0.405288i \(0.867171\pi\)
\(104\) 0 0
\(105\) 0.786458 2.31002i 0.0767504 0.225435i
\(106\) 0 0
\(107\) −8.97973 −0.868103 −0.434052 0.900888i \(-0.642916\pi\)
−0.434052 + 0.900888i \(0.642916\pi\)
\(108\) 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 0.475502 0.879715i \(-0.342266\pi\)
−0.879715 + 0.475502i \(0.842266\pi\)
\(114\) 0 0
\(115\) 5.11131 + 1.74017i 0.476632 + 0.162271i
\(116\) 0 0
\(117\) 14.0299i 1.29707i
\(118\) 0 0
\(119\) 0.663798i 0.0608503i
\(120\) 0 0
\(121\) 24.2140i 2.20127i
\(122\) 0 0
\(123\) 1.66188i 0.149846i
\(124\) 0 0
\(125\) 2.21067 10.9596i 0.197728 0.980257i
\(126\) 0 0
\(127\) 0.759686 + 0.759686i 0.0674112 + 0.0674112i 0.740009 0.672597i \(-0.234821\pi\)
−0.672597 + 0.740009i \(0.734821\pi\)
\(128\) 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
\(133\) −2.62593 −0.227697
\(134\) 0 0
\(135\) −5.73123 + 2.81994i −0.493266 + 0.242702i
\(136\) 0 0
\(137\) 12.7789 + 12.7789i 1.09178 + 1.09178i 0.995339 + 0.0964376i \(0.0307448\pi\)
0.0964376 + 0.995339i \(0.469255\pi\)
\(138\) 0 0
\(139\) −7.74227 + 7.74227i −0.656691 + 0.656691i −0.954596 0.297905i \(-0.903712\pi\)
0.297905 + 0.954596i \(0.403712\pi\)
\(140\) 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
\(147\) 1.07667i 0.0888024i
\(148\) 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 \(-0.628977\pi\)
−0.394195 + 0.919027i \(0.628977\pi\)
\(152\) 0 0
\(153\) −0.587989 + 0.587989i −0.0475361 + 0.0475361i
\(154\) 0 0
\(155\) −0.807726 1.64162i −0.0648781 0.131858i
\(156\) 0 0
\(157\) 9.97637 0.796201 0.398101 0.917342i \(-0.369669\pi\)
0.398101 + 0.917342i \(0.369669\pi\)
\(158\) 0 0
\(159\) 0.499732 0.0396313
\(160\) 0 0
\(161\) 5.30761 0.418298
\(162\) 0 0
\(163\) −9.48267 −0.742740 −0.371370 0.928485i \(-0.621112\pi\)
−0.371370 + 0.928485i \(0.621112\pi\)
\(164\) 0 0
\(165\) 6.23643 + 2.12322i 0.485506 + 0.165293i
\(166\) 0 0
\(167\) −9.43528 + 9.43528i −0.730124 + 0.730124i −0.970644 0.240520i \(-0.922682\pi\)
0.240520 + 0.970644i \(0.422682\pi\)
\(168\) 0 0
\(169\) −12.9621 −0.997082
\(170\) 0 0
\(171\) 2.32603 + 2.32603i 0.177876 + 0.177876i
\(172\) 0 0
\(173\) 8.94716i 0.680240i −0.940382 0.340120i \(-0.889532\pi\)
0.940382 0.340120i \(-0.110468\pi\)
\(174\) 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 0.811039 0.584992i \(-0.198902\pi\)
−0.584992 + 0.811039i \(0.698902\pi\)
\(180\) 0 0
\(181\) 1.54845 1.54845i 0.115095 0.115095i −0.647213 0.762309i \(-0.724066\pi\)
0.762309 + 0.647213i \(0.224066\pi\)
\(182\) 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
\(187\) 1.79208 0.131050
\(188\) 0 0
\(189\) −4.43979 + 4.43979i −0.322947 + 0.322947i
\(190\) 0 0
\(191\) 20.1005i 1.45442i 0.686415 + 0.727210i \(0.259183\pi\)
−0.686415 + 0.727210i \(0.740817\pi\)
\(192\) 0 0
\(193\) 3.82483 + 3.82483i 0.275317 + 0.275317i 0.831236 0.555919i \(-0.187634\pi\)
−0.555919 + 0.831236i \(0.687634\pi\)
\(194\) 0 0
\(195\) −2.49734 5.07558i −0.178838 0.363470i
\(196\) 0 0
\(197\) 1.11758i 0.0796246i −0.999207 0.0398123i \(-0.987324\pi\)
0.999207 0.0398123i \(-0.0126760\pi\)
\(198\) 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
\(203\) 6.98637i 0.490347i
\(204\) 0 0
\(205\) −3.30439 6.71581i −0.230788 0.469053i
\(206\) 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
\(213\) −5.22957 −0.358324
\(214\) 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 −0.129908 + 0.991526i \(0.541468\pi\)
−0.991526 + 0.129908i \(0.958532\pi\)
\(224\) 0 0
\(225\) −8.40070 + 10.9074i −0.560047 + 0.727163i
\(226\) 0 0
\(227\) 13.7807i 0.914659i 0.889297 + 0.457330i \(0.151194\pi\)
−0.889297 + 0.457330i \(0.848806\pi\)
\(228\) 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 \(-0.774797\pi\)
0.649933 + 0.759991i \(0.274797\pi\)
\(234\) 0 0
\(235\) 12.8529 + 4.37583i 0.838432 + 0.285448i
\(236\) 0 0
\(237\) −4.13083 −0.268326
\(238\) 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
\(243\) 11.9667 0.767665
\(244\) 0 0
\(245\) 2.14080 + 4.35094i 0.136770 + 0.277971i
\(246\) 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
\(253\) 14.3291i 0.900863i
\(254\) 0 0
\(255\) 0.108053 0.317378i 0.00676654 0.0198750i
\(256\) 0 0
\(257\) 20.4353 20.4353i 1.27472 1.27472i 0.331140 0.943582i \(-0.392567\pi\)
0.943582 0.331140i \(-0.107433\pi\)
\(258\) 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 \(-0.0399244\pi\)
−0.125098 + 0.992144i \(0.539924\pi\)
\(264\) 0 0
\(265\) 2.01946 0.993639i 0.124055 0.0610388i
\(266\) 0 0
\(267\) −2.11057 −0.129165
\(268\) 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 \(-0.895437\pi\)
0.946529 0.322619i \(-0.104563\pi\)
\(272\) 0 0
\(273\) −3.93188 3.93188i −0.237968 0.237968i
\(274\) 0 0
\(275\) 29.4237 3.82004i 1.77432 0.230357i
\(276\) 0 0
\(277\) 8.28511i 0.497804i −0.968529 0.248902i \(-0.919930\pi\)
0.968529 0.248902i \(-0.0800697\pi\)
\(278\) 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
\(283\) 14.4748i 0.860436i −0.902725 0.430218i \(-0.858437\pi\)
0.902725 0.430218i \(-0.141563\pi\)
\(284\) 0 0
\(285\) −1.25552 0.427448i −0.0743706 0.0253198i
\(286\) 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
\(293\) 11.9165 0.696171 0.348086 0.937463i \(-0.386832\pi\)
0.348086 + 0.937463i \(0.386832\pi\)
\(294\) 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
\(307\) 25.4511i 1.45257i −0.687392 0.726287i \(-0.741245\pi\)
0.687392 0.726287i \(-0.258755\pi\)
\(308\) 0 0
\(309\) −2.56425 2.56425i −0.145875 0.145875i
\(310\) 0 0
\(311\) 21.4775 1.21788 0.608939 0.793217i \(-0.291596\pi\)
0.608939 + 0.793217i \(0.291596\pi\)
\(312\) 0 0
\(313\) 18.7965 18.7965i 1.06244 1.06244i 0.0645277 0.997916i \(-0.479446\pi\)
0.997916 0.0645277i \(-0.0205541\pi\)
\(314\) 0 0
\(315\) −4.36167 + 12.8113i −0.245752 + 0.721835i
\(316\) 0 0
\(317\) −16.2531 −0.912864 −0.456432 0.889758i \(-0.650873\pi\)
−0.456432 + 0.889758i \(0.650873\pi\)
\(318\) 0 0
\(319\) −18.8613 −1.05603
\(320\) 0 0
\(321\) −4.45832 −0.248839
\(322\) 0 0
\(323\) −0.360781 −0.0200744
\(324\) 0 0
\(325\) −20.1840 15.5453i −1.11961 0.862300i
\(326\) 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
\(333\) 14.1155i 0.773526i
\(334\) 0 0
\(335\) 10.8809 + 22.1142i 0.594485 + 1.20823i
\(336\) 0 0
\(337\) 0.0406874 0.0406874i 0.00221638 0.00221638i −0.705998 0.708214i \(-0.749501\pi\)
0.708214 + 0.705998i \(0.249501\pi\)
\(338\) 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
\(347\) 35.7094 1.91698 0.958491 0.285124i \(-0.0920348\pi\)
0.958491 + 0.285124i \(0.0920348\pi\)
\(348\) 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 0.156268 0.987715i \(-0.450054\pi\)
−0.987715 + 0.156268i \(0.950054\pi\)
\(354\) 0 0
\(355\) −21.1332 + 10.3982i −1.12163 + 0.551879i
\(356\) 0 0
\(357\) 0.329567i 0.0174426i
\(358\) 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
\(363\) 12.0219i 0.630988i
\(364\) 0 0
\(365\) 2.76045 8.10812i 0.144488 0.424399i
\(366\) 0 0
\(367\) 13.7849 + 13.7849i 0.719568 + 0.719568i 0.968517 0.248949i \(-0.0800852\pi\)
−0.248949 + 0.968517i \(0.580085\pi\)
\(368\) 0 0
\(369\) 9.21671i 0.479803i
\(370\) 0 0
\(371\) 1.56441 1.56441i 0.0812202 0.0812202i
\(372\) 0 0
\(373\) 21.4003 1.10806 0.554031 0.832496i \(-0.313089\pi\)
0.554031 + 0.832496i \(0.313089\pi\)
\(374\) 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 −0.936322 0.351142i \(-0.885793\pi\)
0.351142 + 0.936322i \(0.385793\pi\)
\(380\) 0 0
\(381\) 0.377174 + 0.377174i 0.0193232 + 0.0193232i
\(382\) 0 0
\(383\) 4.42635 4.42635i 0.226176 0.226176i −0.584917 0.811093i \(-0.698873\pi\)
0.811093 + 0.584917i \(0.198873\pi\)
\(384\) 0 0
\(385\) 26.1699 12.8764i 1.33374 0.656243i
\(386\) 0 0
\(387\) 12.3668i 0.628642i
\(388\) 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
\(395\) −16.6931 + 8.21351i −0.839920 + 0.413267i
\(396\) 0 0
\(397\) −17.9832 −0.902551 −0.451275 0.892385i \(-0.649031\pi\)
−0.451275 + 0.892385i \(0.649031\pi\)
\(398\) 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
\(403\) −4.16902 −0.207674
\(404\) 0 0
\(405\) 13.7280 6.75461i 0.682150 0.335639i
\(406\) 0 0
\(407\) 21.5107 21.5107i 1.06625 1.06625i
\(408\) 0 0
\(409\) −30.0616 −1.48645 −0.743226 0.669040i \(-0.766705\pi\)
−0.743226 + 0.669040i \(0.766705\pi\)
\(410\) 0 0
\(411\) 6.34457 + 6.34457i 0.312955 + 0.312955i
\(412\) 0 0
\(413\) 23.7803i 1.17015i
\(414\) 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 0.974903 0.222631i \(-0.0714646\pi\)
−0.222631 + 0.974903i \(0.571465\pi\)
\(420\) 0 0
\(421\) 3.86468 3.86468i 0.188353 0.188353i −0.606631 0.794984i \(-0.707479\pi\)
0.794984 + 0.606631i \(0.207479\pi\)
\(422\) 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
\(427\) 5.91523 0.286258
\(428\) 0 0
\(429\) 10.6150 10.6150i 0.512497 0.512497i
\(430\) 0 0
\(431\) 27.2692i 1.31351i −0.754103 0.656756i \(-0.771928\pi\)
0.754103 0.656756i \(-0.228072\pi\)
\(432\) 0 0
\(433\) 19.1435 + 19.1435i 0.919978 + 0.919978i 0.997027 0.0770497i \(-0.0245500\pi\)
−0.0770497 + 0.997027i \(0.524550\pi\)
\(434\) 0 0
\(435\) −1.13724 + 3.34036i −0.0545265 + 0.160158i
\(436\) 0 0
\(437\) 2.88474i 0.137996i
\(438\) 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
\(443\) 27.7051i 1.31631i −0.752884 0.658153i \(-0.771338\pi\)
0.752884 0.658153i \(-0.228662\pi\)
\(444\) 0 0
\(445\) −8.52903 + 4.19655i −0.404315 + 0.198935i
\(446\) 0 0
\(447\) −4.75220 4.75220i −0.224772 0.224772i
\(448\) 0 0
\(449\) 9.78315i 0.461695i −0.972990 0.230848i \(-0.925850\pi\)
0.972990 0.230848i \(-0.0741499\pi\)
\(450\) 0 0
\(451\) 14.0454 14.0454i 0.661371 0.661371i
\(452\) 0 0
\(453\) −4.80992 −0.225990
\(454\) 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 \(-0.255866\pi\)
−0.720017 + 0.693957i \(0.755866\pi\)
\(458\) 0 0
\(459\) −0.609992 + 0.609992i −0.0284720 + 0.0284720i
\(460\) 0 0
\(461\) 12.5791 + 12.5791i 0.585865 + 0.585865i 0.936509 0.350644i \(-0.114037\pi\)
−0.350644 + 0.936509i \(0.614037\pi\)
\(462\) 0 0
\(463\) −3.29549 + 3.29549i −0.153154 + 0.153154i −0.779525 0.626371i \(-0.784540\pi\)
0.626371 + 0.779525i \(0.284540\pi\)
\(464\) 0 0
\(465\) −0.401026 0.815042i −0.0185971 0.0377967i
\(466\) 0 0
\(467\) 10.1995i 0.471979i 0.971756 + 0.235989i \(0.0758331\pi\)
−0.971756 + 0.235989i \(0.924167\pi\)
\(468\) 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
\(475\) −5.92360 + 0.769051i −0.271793 + 0.0352865i
\(476\) 0 0
\(477\) −2.77149 −0.126898
\(478\) 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
\(483\) 2.63516 0.119904
\(484\) 0 0
\(485\) −7.29722 + 21.4338i −0.331350 + 0.973257i
\(486\) 0 0
\(487\) −19.7470 + 19.7470i −0.894823 + 0.894823i −0.994972 0.100149i \(-0.968068\pi\)
0.100149 + 0.994972i \(0.468068\pi\)
\(488\) 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
\(493\) 0.959871i 0.0432304i
\(494\) 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 0.975579 0.219650i \(-0.0704917\pi\)
−0.219650 + 0.975579i \(0.570492\pi\)
\(500\) 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 \(-0.472010\pi\)
−0.996136 + 0.0878190i \(0.972010\pi\)
\(504\) 0 0
\(505\) −8.54923 + 25.1112i −0.380436 + 1.11744i
\(506\) 0 0
\(507\) −6.43550 −0.285811
\(508\) 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
\(515\) −15.4610 5.26376i −0.681293 0.231949i
\(516\) 0 0
\(517\) 36.0320i 1.58469i
\(518\) 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
\(523\) 19.1782i 0.838603i 0.907847 + 0.419301i \(0.137725\pi\)
−0.907847 + 0.419301i \(0.862275\pi\)
\(524\) 0 0
\(525\) −0.702515 5.41111i −0.0306603 0.236160i
\(526\) 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
\(533\) −17.0553 −0.738749
\(534\) 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 0.322255 0.946653i \(-0.395559\pi\)
−0.946653 + 0.322255i \(0.895559\pi\)
\(542\) 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
\(547\) 9.97058i 0.426311i −0.977018 0.213156i \(-0.931626\pi\)
0.977018 0.213156i \(-0.0683742\pi\)
\(548\) 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
\(555\) −2.51258 5.10655i −0.106653 0.216761i
\(556\) 0 0
\(557\) −11.4424 −0.484831 −0.242416 0.970173i \(-0.577940\pi\)
−0.242416 + 0.970173i \(0.577940\pi\)
\(558\) 0 0
\(559\) 22.8846 0.967915
\(560\) 0 0
\(561\) 0.889743 0.0375650
\(562\) 0 0
\(563\) −47.0585 −1.98328 −0.991640 0.129034i \(-0.958812\pi\)
−0.991640 + 0.129034i \(0.958812\pi\)
\(564\) 0 0
\(565\) −12.8628 4.37919i −0.541141 0.184234i
\(566\) 0 0
\(567\) 10.6346 10.6346i 0.446612 0.446612i
\(568\) 0 0
\(569\) 41.4684 1.73845 0.869224 0.494419i \(-0.164619\pi\)
0.869224 + 0.494419i \(0.164619\pi\)
\(570\) 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
\(573\) 9.97963i 0.416905i
\(574\) 0 0
\(575\) 11.9730 1.55443i 0.499307 0.0648242i
\(576\) 0 0
\(577\) 20.0316 20.0316i 0.833926 0.833926i −0.154125 0.988051i \(-0.549256\pi\)
0.988051 + 0.154125i \(0.0492560\pi\)
\(578\) 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
\(587\) −29.1190 −1.20187 −0.600935 0.799298i \(-0.705205\pi\)
−0.600935 + 0.799298i \(0.705205\pi\)
\(588\) 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 0.462092 0.886832i \(-0.347099\pi\)
−0.886832 + 0.462092i \(0.847099\pi\)
\(594\) 0 0
\(595\) −0.655294 1.33181i −0.0268644 0.0545991i
\(596\) 0 0
\(597\) 12.7016i 0.519843i
\(598\) 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
\(603\) 30.3493i 1.23592i
\(604\) 0 0
\(605\) 23.9038 + 48.5818i 0.971826 + 1.97513i
\(606\) 0 0
\(607\) −11.8502 11.8502i −0.480985 0.480985i 0.424461 0.905446i \(-0.360464\pi\)
−0.905446 + 0.424461i \(0.860464\pi\)
\(608\) 0 0
\(609\) 3.46864i 0.140557i
\(610\) 0 0
\(611\) 21.8769 21.8769i 0.885045 0.885045i
\(612\) 0 0
\(613\) −16.8256 −0.679579 −0.339789 0.940502i \(-0.610356\pi\)
−0.339789 + 0.940502i \(0.610356\pi\)
\(614\) 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 \(-0.471098\pi\)
−0.995881 + 0.0906720i \(0.971098\pi\)
\(618\) 0 0
\(619\) 14.1269 14.1269i 0.567809 0.567809i −0.363705 0.931514i \(-0.618488\pi\)
0.931514 + 0.363705i \(0.118488\pi\)
\(620\) 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
\(627\) 3.51974i 0.140565i
\(628\) 0 0
\(629\) −1.09470 1.09470i −0.0436486 0.0436486i
\(630\) 0 0
\(631\) −33.9235 −1.35047 −0.675236 0.737601i \(-0.735958\pi\)
−0.675236 + 0.737601i \(0.735958\pi\)
\(632\) 0 0
\(633\) −0.204361 + 0.204361i −0.00812261 + 0.00812261i
\(634\) 0 0
\(635\) 2.27415 + 0.774246i 0.0902470 + 0.0307250i
\(636\) 0 0
\(637\) 11.0496 0.437799
\(638\) 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
\(643\) 16.4916 0.650364 0.325182 0.945652i \(-0.394574\pi\)
0.325182 + 0.945652i \(0.394574\pi\)
\(644\) 0 0
\(645\) 2.20131 + 4.47393i 0.0866766 + 0.176161i
\(646\) 0 0
\(647\) −0.316870 + 0.316870i −0.0124574 + 0.0124574i −0.713308 0.700851i \(-0.752804\pi\)
0.700851 + 0.713308i \(0.252804\pi\)
\(648\) 0 0
\(649\) 64.2002 2.52008
\(650\) 0 0
\(651\) −0.631386 0.631386i −0.0247460 0.0247460i
\(652\) 0 0
\(653\) 17.0751i 0.668200i 0.942538 + 0.334100i \(0.108432\pi\)
−0.942538 + 0.334100i \(0.891568\pi\)
\(654\) 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 0.836739 0.547601i \(-0.184459\pi\)
−0.547601 + 0.836739i \(0.684459\pi\)
\(660\) 0 0
\(661\) −31.7614 + 31.7614i −1.23538 + 1.23538i −0.273507 + 0.961870i \(0.588184\pi\)
−0.961870 + 0.273507i \(0.911816\pi\)
\(662\) 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
\(667\) −7.67495 −0.297175
\(668\) 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 0.782498 0.622653i \(-0.213945\pi\)
−0.622653 + 0.782498i \(0.713945\pi\)
\(674\) 0 0
\(675\) −8.71507 + 11.3156i −0.335443 + 0.435538i
\(676\) 0 0
\(677\) 25.2618i 0.970890i −0.874267 0.485445i \(-0.838658\pi\)
0.874267 0.485445i \(-0.161342\pi\)
\(678\) 0 0
\(679\) 22.2569i 0.854143i
\(680\) 0 0
\(681\) 6.84196i 0.262184i
\(682\) 0 0
\(683\) 8.20306i 0.313881i 0.987608 + 0.156941i \(0.0501631\pi\)
−0.987608 + 0.156941i \(0.949837\pi\)
\(684\) 0 0
\(685\) 38.2542 + 13.0238i 1.46162 + 0.497615i
\(686\) 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
\(693\) −35.9153 −1.36431
\(694\) 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 0.678167 0.734908i \(-0.262775\pi\)
−0.734908 + 0.678167i \(0.762775\pi\)
\(702\) 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
\(707\) 26.0756i 0.980675i
\(708\) 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
\(715\) 21.7900 64.0026i 0.814899 2.39356i
\(716\) 0 0
\(717\) 5.84291 0.218207
\(718\) 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
\(723\) −6.58989 −0.245081
\(724\) 0 0
\(725\) 2.04609 + 15.7599i 0.0759898 + 0.585310i
\(726\) 0 0
\(727\) 25.4241 25.4241i 0.942928 0.942928i −0.0555295 0.998457i \(-0.517685\pi\)
0.998457 + 0.0555295i \(0.0176847\pi\)
\(728\) 0 0
\(729\) −14.5855 −0.540203
\(730\) 0 0
\(731\) 0.959085 + 0.959085i 0.0354731 + 0.0354731i
\(732\) 0 0
\(733\) 7.37554i 0.272422i 0.990680 + 0.136211i \(0.0434925\pi\)
−0.990680 + 0.136211i \(0.956508\pi\)
\(734\) 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 0.597614 0.801784i \(-0.296115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(740\) 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 \(-0.306348\pi\)
−0.820577 + 0.571536i \(0.806348\pi\)
\(744\) 0 0
\(745\) −28.6532 9.75510i −1.04977 0.357399i
\(746\) 0 0
\(747\) −25.2561 −0.924073
\(748\) 0 0
\(749\) −13.9568 + 13.9568i −0.509970 + 0.509970i
\(750\) 0 0
\(751\) 3.93385i 0.143548i −0.997421 0.0717742i \(-0.977134\pi\)
0.997421 0.0717742i \(-0.0228661\pi\)
\(752\) 0 0
\(753\) −5.15330 5.15330i −0.187797 0.187797i
\(754\) 0 0
\(755\) −19.4374 + 9.56379i −0.707398 + 0.348062i
\(756\) 0 0
\(757\) 21.8327i 0.793525i −0.917921 0.396762i \(-0.870134\pi\)
0.917921 0.396762i \(-0.129866\pi\)
\(758\) 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
\(763\) 33.9080i 1.22755i
\(764\) 0 0
\(765\) −0.599258 + 1.76017i −0.0216662 + 0.0636391i
\(766\) 0 0
\(767\) −38.9793 38.9793i −1.40746 1.40746i
\(768\) 0 0
\(769\) 26.1800i 0.944074i −0.881579 0.472037i \(-0.843519\pi\)
0.881579 0.472037i \(-0.156481\pi\)
\(770\) 0 0
\(771\) 10.1459 10.1459i 0.365395 0.365395i
\(772\) 0 0
\(773\) 15.0077 0.539791 0.269895 0.962890i \(-0.413011\pi\)
0.269895 + 0.962890i \(0.413011\pi\)
\(774\) 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
\(787\) 42.9223i 1.53001i −0.644022 0.765007i \(-0.722736\pi\)
0.644022 0.765007i \(-0.277264\pi\)
\(788\) 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
\(795\) 1.00264 0.493329i 0.0355599 0.0174966i
\(796\) 0 0
\(797\) −0.280831 −0.00994753 −0.00497377 0.999988i \(-0.501583\pi\)
−0.00497377 + 0.999988i \(0.501583\pi\)
\(798\) 0 0
\(799\) 1.83371 0.0648719
\(800\) 0 0
\(801\) 11.7052 0.413581
\(802\) 0 0
\(803\) 22.7304 0.802139
\(804\) 0 0
\(805\) 10.6489 5.23961i 0.375326 0.184672i
\(806\) 0 0
\(807\) 3.28398 3.28398i 0.115602 0.115602i
\(808\) 0 0
\(809\) −16.5787 −0.582876 −0.291438 0.956590i \(-0.594134\pi\)
−0.291438 + 0.956590i \(0.594134\pi\)
\(810\) 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
\(813\) 5.27366i 0.184955i
\(814\) 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 −0.386489 0.922294i \(-0.626312\pi\)
0.922294 + 0.386489i \(0.126312\pi\)
\(822\) 0 0
\(823\) −26.7794 26.7794i −0.933472 0.933472i 0.0644492 0.997921i \(-0.479471\pi\)
−0.997921 + 0.0644492i \(0.979471\pi\)
\(824\) 0 0
\(825\) 14.6085 1.89660i 0.508603 0.0660311i
\(826\) 0 0
\(827\) 39.4186 1.37072 0.685359 0.728205i \(-0.259645\pi\)
0.685359 + 0.728205i \(0.259645\pi\)
\(828\) 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
\(835\) −9.61612 + 28.2449i −0.332779 + 0.977456i
\(836\) 0 0
\(837\) 2.33725i 0.0807870i
\(838\) 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
\(843\) 10.4350i 0.359399i
\(844\) 0 0
\(845\) −26.0065 + 12.7960i −0.894651 + 0.440196i
\(846\) 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
\(853\) 26.5538 0.909185 0.454592 0.890700i \(-0.349785\pi\)
0.454592 + 0.890700i \(0.349785\pi\)
\(854\) 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 \(-0.416886\pi\)
−0.966104 + 0.258153i \(0.916886\pi\)
\(858\) 0 0
\(859\) −35.9248 + 35.9248i −1.22574 + 1.22574i −0.260176 + 0.965561i \(0.583781\pi\)
−0.965561 + 0.260176i \(0.916219\pi\)
\(860\) 0 0
\(861\) −2.58298 2.58298i −0.0880278 0.0880278i
\(862\) 0 0
\(863\) 9.19232 9.19232i 0.312910 0.312910i −0.533126 0.846036i \(-0.678983\pi\)
0.846036 + 0.533126i \(0.178983\pi\)
\(864\) 0 0
\(865\) −8.83254 17.9512i −0.300315 0.610358i
\(866\) 0 0
\(867\) 8.39500i 0.285109i
\(868\) 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
\(875\) −13.5981 20.4700i −0.459699 0.692011i
\(876\) 0 0
\(877\) 17.9106 0.604799 0.302399 0.953181i \(-0.402212\pi\)
0.302399 + 0.953181i \(0.402212\pi\)
\(878\) 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
\(883\) −19.8374 −0.667580 −0.333790 0.942647i \(-0.608328\pi\)
−0.333790 + 0.942647i \(0.608328\pi\)
\(884\) 0 0
\(885\) 3.87095 11.3699i 0.130120 0.382196i
\(886\) 0 0
\(887\) 14.3740 14.3740i 0.482632 0.482632i −0.423339 0.905971i \(-0.639142\pi\)
0.905971 + 0.423339i \(0.139142\pi\)
\(888\) 0 0
\(889\) 2.36149 0.0792019
\(890\) 0 0
\(891\) 28.7106 + 28.7106i 0.961842 + 0.961842i
\(892\) 0 0
\(893\) 7.25398i 0.242745i
\(894\) 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
\(907\) 39.0417 1.29636 0.648180 0.761487i \(-0.275531\pi\)
0.648180 + 0.761487i \(0.275531\pi\)
\(908\) 0 0
\(909\) 23.0976 23.0976i 0.766100 0.766100i
\(910\) 0 0
\(911\) 14.0166i 0.464392i 0.972669 + 0.232196i \(0.0745911\pi\)
−0.972669 + 0.232196i \(0.925409\pi\)
\(912\) 0 0
\(913\) 38.4879 + 38.4879i 1.27376 + 1.27376i
\(914\) 0 0
\(915\) 2.82822 + 0.962880i 0.0934980 + 0.0318318i
\(916\) 0 0
\(917\) 23.6245i 0.780150i
\(918\) 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
\(923\) 53.6695i 1.76655i
\(924\) 0 0
\(925\) −20.3072 15.6402i −0.667696 0.514247i
\(926\) 0 0
\(927\) 14.2212 + 14.2212i 0.467086 + 0.467086i
\(928\) 0 0
\(929\) 13.4779i 0.442196i 0.975252 + 0.221098i \(0.0709641\pi\)
−0.975252 + 0.221098i \(0.929036\pi\)
\(930\) 0 0
\(931\) 1.83192 1.83192i 0.0600386 0.0600386i
\(932\) 0 0
\(933\) 10.6633 0.349101
\(934\) 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 \(-0.369370\pi\)
−0.916967 + 0.398963i \(0.869370\pi\)
\(938\) 0 0
\(939\) 9.33225 9.33225i 0.304546 0.304546i
\(940\) 0 0
\(941\) −15.7073 15.7073i −0.512044 0.512044i 0.403108 0.915152i \(-0.367930\pi\)
−0.915152 + 0.403108i \(0.867930\pi\)
\(942\) 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
\(947\) 33.6925i 1.09486i −0.836852 0.547430i \(-0.815606\pi\)
0.836852 0.547430i \(-0.184394\pi\)
\(948\) 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 −0.0916550 + 0.995791i \(0.529216\pi\)
−0.995791 + 0.0916550i \(0.970784\pi\)
\(954\) 0 0
\(955\) 19.8430 + 40.3287i 0.642103 + 1.30501i
\(956\) 0 0
\(957\) −9.36440 −0.302708
\(958\) 0 0
\(959\) 39.7234 1.28274
\(960\) 0 0
\(961\) 30.3305 0.978404
\(962\) 0 0
\(963\) 24.7257 0.796774
\(964\) 0 0
\(965\) 11.4498 + 3.89813i 0.368582 + 0.125485i
\(966\) 0 0
\(967\) 28.6436 28.6436i 0.921115 0.921115i −0.0759933 0.997108i \(-0.524213\pi\)
0.997108 + 0.0759933i \(0.0242128\pi\)
\(968\) 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
\(973\) 24.0669i 0.771551i
\(974\) 0 0
\(975\) −10.0211 7.71806i −0.320932 0.247176i
\(976\) 0 0
\(977\) 7.12822 7.12822i 0.228052 0.228052i −0.583826 0.811879i \(-0.698445\pi\)
0.811879 + 0.583826i \(0.198445\pi\)
\(978\) 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 \(-0.432008\pi\)
−0.977274 + 0.211982i \(0.932008\pi\)
\(984\) 0 0
\(985\) −1.10327 2.24227i −0.0351530 0.0714447i
\(986\) 0 0
\(987\) 6.62639 0.210920
\(988\) 0 0
\(989\) −7.66866 + 7.66866i −0.243849 + 0.243849i
\(990\) 0 0
\(991\) 40.6040i 1.28983i −0.764255 0.644914i \(-0.776893\pi\)
0.764255 0.644914i \(-0.223107\pi\)
\(992\) 0 0
\(993\) 4.32718 + 4.32718i 0.137319 + 0.137319i
\(994\) 0 0
\(995\) −25.2553 51.3286i −0.800645 1.62723i
\(996\) 0 0
\(997\) 54.9087i 1.73898i −0.493953 0.869488i \(-0.664449\pi\)
0.493953 0.869488i \(-0.335551\pi\)
\(998\) 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 640.2.s.d.223.6 18
4.3 odd 2 640.2.s.c.223.4 18
5.2 odd 4 640.2.j.d.607.4 18
8.3 odd 2 320.2.s.b.303.6 18
8.5 even 2 80.2.s.b.3.9 yes 18
16.3 odd 4 80.2.j.b.43.6 18
16.5 even 4 640.2.j.c.543.4 18
16.11 odd 4 640.2.j.d.543.6 18
16.13 even 4 320.2.j.b.143.6 18
20.7 even 4 640.2.j.c.607.6 18
24.5 odd 2 720.2.z.g.163.1 18
40.3 even 4 1600.2.j.d.1007.6 18
40.13 odd 4 400.2.j.d.307.4 18
40.19 odd 2 1600.2.s.d.943.4 18
40.27 even 4 320.2.j.b.47.4 18
40.29 even 2 400.2.s.d.243.1 18
40.37 odd 4 80.2.j.b.67.6 yes 18
48.35 even 4 720.2.bd.g.523.4 18
80.3 even 4 400.2.s.d.107.1 18
80.13 odd 4 1600.2.s.d.207.4 18
80.19 odd 4 400.2.j.d.43.4 18
80.27 even 4 inner 640.2.s.d.287.6 18
80.29 even 4 1600.2.j.d.143.4 18
80.37 odd 4 640.2.s.c.287.4 18
80.67 even 4 80.2.s.b.27.9 yes 18
80.77 odd 4 320.2.s.b.207.6 18
120.77 even 4 720.2.bd.g.307.4 18
240.227 odd 4 720.2.z.g.667.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.6 18 16.3 odd 4
80.2.j.b.67.6 yes 18 40.37 odd 4
80.2.s.b.3.9 yes 18 8.5 even 2
80.2.s.b.27.9 yes 18 80.67 even 4
320.2.j.b.47.4 18 40.27 even 4
320.2.j.b.143.6 18 16.13 even 4
320.2.s.b.207.6 18 80.77 odd 4
320.2.s.b.303.6 18 8.3 odd 2
400.2.j.d.43.4 18 80.19 odd 4
400.2.j.d.307.4 18 40.13 odd 4
400.2.s.d.107.1 18 80.3 even 4
400.2.s.d.243.1 18 40.29 even 2
640.2.j.c.543.4 18 16.5 even 4
640.2.j.c.607.6 18 20.7 even 4
640.2.j.d.543.6 18 16.11 odd 4
640.2.j.d.607.4 18 5.2 odd 4
640.2.s.c.223.4 18 4.3 odd 2
640.2.s.c.287.4 18 80.37 odd 4
640.2.s.d.223.6 18 1.1 even 1 trivial
640.2.s.d.287.6 18 80.27 even 4 inner
720.2.z.g.163.1 18 24.5 odd 2
720.2.z.g.667.1 18 240.227 odd 4
720.2.bd.g.307.4 18 120.77 even 4
720.2.bd.g.523.4 18 48.35 even 4
1600.2.j.d.143.4 18 80.29 even 4
1600.2.j.d.1007.6 18 40.3 even 4
1600.2.s.d.207.4 18 80.13 odd 4
1600.2.s.d.943.4 18 40.19 odd 2