Properties

Label 1120.2.bz.f.271.4
Level $1120$
Weight $2$
Character 1120.271
Analytic conductor $8.943$
Analytic rank $0$
Dimension $24$
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] = [1120,2,Mod(271,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bz (of order \(6\), 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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.4
Character \(\chi\) \(=\) 1120.271
Dual form 1120.2.bz.f.591.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.90624 + 1.10057i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.584379 - 2.58041i) q^{7} +(0.922503 - 1.59782i) q^{9} +O(q^{10})\) \(q+(-1.90624 + 1.10057i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.584379 - 2.58041i) q^{7} +(0.922503 - 1.59782i) q^{9} +(2.90486 + 5.03137i) q^{11} -4.83332 q^{13} +2.20114i q^{15} +(3.78236 - 2.18375i) q^{17} +(-1.63783 - 0.945600i) q^{19} +(3.95388 + 4.27573i) q^{21} +(0.157820 + 0.0911173i) q^{23} +(-0.500000 - 0.866025i) q^{25} -2.54230i q^{27} -4.38898i q^{29} +(-2.03983 - 3.53308i) q^{31} +(-11.0747 - 6.39400i) q^{33} +(-2.52689 - 0.784117i) q^{35} +(-3.69132 - 2.13119i) q^{37} +(9.21347 - 5.31940i) q^{39} +3.44314i q^{41} +2.10796 q^{43} +(-0.922503 - 1.59782i) q^{45} +(0.946816 - 1.63993i) q^{47} +(-6.31700 + 3.01587i) q^{49} +(-4.80673 + 8.32550i) q^{51} +(8.54212 - 4.93180i) q^{53} +5.80972 q^{55} +4.16279 q^{57} +(6.35546 - 3.66932i) q^{59} +(4.89522 - 8.47877i) q^{61} +(-4.66212 - 1.44670i) q^{63} +(-2.41666 + 4.18577i) q^{65} +(-6.04954 - 10.4781i) q^{67} -0.401124 q^{69} -8.80669i q^{71} +(7.34998 - 4.24351i) q^{73} +(1.90624 + 1.10057i) q^{75} +(11.2854 - 10.4360i) q^{77} +(10.5541 + 6.09342i) q^{79} +(5.56549 + 9.63970i) q^{81} -16.9697i q^{83} -4.36749i q^{85} +(4.83037 + 8.36645i) q^{87} +(-5.11692 - 2.95425i) q^{89} +(2.82449 + 12.4719i) q^{91} +(7.77680 + 4.48994i) q^{93} +(-1.63783 + 0.945600i) q^{95} -13.2388i q^{97} +10.7190 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 12 q^{3} + 12 q^{5} + 10 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 12 q^{3} + 12 q^{5} + 10 q^{7} + 12 q^{9} - 8 q^{11} - 20 q^{13} + 6 q^{17} - 18 q^{19} + 26 q^{21} + 18 q^{23} - 12 q^{25} - 6 q^{31} + 12 q^{33} + 8 q^{35} - 18 q^{39} - 32 q^{43} - 12 q^{45} + 8 q^{49} + 22 q^{51} - 30 q^{53} - 16 q^{55} - 44 q^{57} + 18 q^{59} - 22 q^{61} - 12 q^{63} - 10 q^{65} + 8 q^{67} + 12 q^{69} + 30 q^{73} + 12 q^{75} + 32 q^{77} + 6 q^{79} - 4 q^{81} + 14 q^{87} - 60 q^{89} - 18 q^{91} + 18 q^{93} - 18 q^{95} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −1.90624 + 1.10057i −1.10057 + 0.635414i −0.936370 0.351014i \(-0.885837\pi\)
−0.164198 + 0.986427i \(0.552504\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −0.584379 2.58041i −0.220874 0.975302i
\(8\) 0 0
\(9\) 0.922503 1.59782i 0.307501 0.532608i
\(10\) 0 0
\(11\) 2.90486 + 5.03137i 0.875849 + 1.51701i 0.855856 + 0.517214i \(0.173031\pi\)
0.0199929 + 0.999800i \(0.493636\pi\)
\(12\) 0 0
\(13\) −4.83332 −1.34052 −0.670260 0.742126i \(-0.733818\pi\)
−0.670260 + 0.742126i \(0.733818\pi\)
\(14\) 0 0
\(15\) 2.20114i 0.568331i
\(16\) 0 0
\(17\) 3.78236 2.18375i 0.917357 0.529636i 0.0345662 0.999402i \(-0.488995\pi\)
0.882791 + 0.469766i \(0.155662\pi\)
\(18\) 0 0
\(19\) −1.63783 0.945600i −0.375743 0.216936i 0.300221 0.953870i \(-0.402939\pi\)
−0.675965 + 0.736934i \(0.736273\pi\)
\(20\) 0 0
\(21\) 3.95388 + 4.27573i 0.862808 + 0.933041i
\(22\) 0 0
\(23\) 0.157820 + 0.0911173i 0.0329077 + 0.0189993i 0.516364 0.856369i \(-0.327285\pi\)
−0.483456 + 0.875369i \(0.660619\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 2.54230i 0.489266i
\(28\) 0 0
\(29\) 4.38898i 0.815013i −0.913202 0.407506i \(-0.866398\pi\)
0.913202 0.407506i \(-0.133602\pi\)
\(30\) 0 0
\(31\) −2.03983 3.53308i −0.366364 0.634560i 0.622630 0.782516i \(-0.286064\pi\)
−0.988994 + 0.147956i \(0.952731\pi\)
\(32\) 0 0
\(33\) −11.0747 6.39400i −1.92786 1.11305i
\(34\) 0 0
\(35\) −2.52689 0.784117i −0.427122 0.132540i
\(36\) 0 0
\(37\) −3.69132 2.13119i −0.606849 0.350365i 0.164882 0.986313i \(-0.447276\pi\)
−0.771731 + 0.635949i \(0.780609\pi\)
\(38\) 0 0
\(39\) 9.21347 5.31940i 1.47534 0.851785i
\(40\) 0 0
\(41\) 3.44314i 0.537729i 0.963178 + 0.268864i \(0.0866483\pi\)
−0.963178 + 0.268864i \(0.913352\pi\)
\(42\) 0 0
\(43\) 2.10796 0.321461 0.160731 0.986998i \(-0.448615\pi\)
0.160731 + 0.986998i \(0.448615\pi\)
\(44\) 0 0
\(45\) −0.922503 1.59782i −0.137519 0.238189i
\(46\) 0 0
\(47\) 0.946816 1.63993i 0.138107 0.239209i −0.788673 0.614813i \(-0.789231\pi\)
0.926780 + 0.375604i \(0.122565\pi\)
\(48\) 0 0
\(49\) −6.31700 + 3.01587i −0.902429 + 0.430839i
\(50\) 0 0
\(51\) −4.80673 + 8.32550i −0.673076 + 1.16580i
\(52\) 0 0
\(53\) 8.54212 4.93180i 1.17335 0.677434i 0.218884 0.975751i \(-0.429758\pi\)
0.954467 + 0.298317i \(0.0964252\pi\)
\(54\) 0 0
\(55\) 5.80972 0.783383
\(56\) 0 0
\(57\) 4.16279 0.551375
\(58\) 0 0
\(59\) 6.35546 3.66932i 0.827410 0.477705i −0.0255550 0.999673i \(-0.508135\pi\)
0.852965 + 0.521968i \(0.174802\pi\)
\(60\) 0 0
\(61\) 4.89522 8.47877i 0.626769 1.08560i −0.361427 0.932400i \(-0.617710\pi\)
0.988196 0.153195i \(-0.0489562\pi\)
\(62\) 0 0
\(63\) −4.66212 1.44670i −0.587373 0.182267i
\(64\) 0 0
\(65\) −2.41666 + 4.18577i −0.299750 + 0.519181i
\(66\) 0 0
\(67\) −6.04954 10.4781i −0.739069 1.28011i −0.952915 0.303238i \(-0.901932\pi\)
0.213845 0.976868i \(-0.431401\pi\)
\(68\) 0 0
\(69\) −0.401124 −0.0482896
\(70\) 0 0
\(71\) 8.80669i 1.04516i −0.852590 0.522581i \(-0.824969\pi\)
0.852590 0.522581i \(-0.175031\pi\)
\(72\) 0 0
\(73\) 7.34998 4.24351i 0.860250 0.496666i −0.00384586 0.999993i \(-0.501224\pi\)
0.864096 + 0.503327i \(0.167891\pi\)
\(74\) 0 0
\(75\) 1.90624 + 1.10057i 0.220114 + 0.127083i
\(76\) 0 0
\(77\) 11.2854 10.4360i 1.28610 1.18929i
\(78\) 0 0
\(79\) 10.5541 + 6.09342i 1.18743 + 0.685563i 0.957722 0.287696i \(-0.0928894\pi\)
0.229708 + 0.973259i \(0.426223\pi\)
\(80\) 0 0
\(81\) 5.56549 + 9.63970i 0.618387 + 1.07108i
\(82\) 0 0
\(83\) 16.9697i 1.86267i −0.364163 0.931335i \(-0.618645\pi\)
0.364163 0.931335i \(-0.381355\pi\)
\(84\) 0 0
\(85\) 4.36749i 0.473721i
\(86\) 0 0
\(87\) 4.83037 + 8.36645i 0.517870 + 0.896977i
\(88\) 0 0
\(89\) −5.11692 2.95425i −0.542392 0.313150i 0.203656 0.979043i \(-0.434718\pi\)
−0.746048 + 0.665892i \(0.768051\pi\)
\(90\) 0 0
\(91\) 2.82449 + 12.4719i 0.296087 + 1.30741i
\(92\) 0 0
\(93\) 7.77680 + 4.48994i 0.806417 + 0.465585i
\(94\) 0 0
\(95\) −1.63783 + 0.945600i −0.168038 + 0.0970165i
\(96\) 0 0
\(97\) 13.2388i 1.34420i −0.740461 0.672100i \(-0.765393\pi\)
0.740461 0.672100i \(-0.234607\pi\)
\(98\) 0 0
\(99\) 10.7190 1.07730
\(100\) 0 0
\(101\) 1.23442 + 2.13807i 0.122829 + 0.212746i 0.920882 0.389841i \(-0.127470\pi\)
−0.798053 + 0.602587i \(0.794137\pi\)
\(102\) 0 0
\(103\) 0.920346 1.59409i 0.0906844 0.157070i −0.817115 0.576475i \(-0.804428\pi\)
0.907799 + 0.419405i \(0.137761\pi\)
\(104\) 0 0
\(105\) 5.67983 1.28630i 0.554295 0.125530i
\(106\) 0 0
\(107\) −9.08674 + 15.7387i −0.878448 + 1.52152i −0.0254044 + 0.999677i \(0.508087\pi\)
−0.853044 + 0.521839i \(0.825246\pi\)
\(108\) 0 0
\(109\) −3.40525 + 1.96602i −0.326164 + 0.188311i −0.654137 0.756376i \(-0.726968\pi\)
0.327973 + 0.944687i \(0.393635\pi\)
\(110\) 0 0
\(111\) 9.38206 0.890506
\(112\) 0 0
\(113\) 7.16641 0.674159 0.337079 0.941476i \(-0.390561\pi\)
0.337079 + 0.941476i \(0.390561\pi\)
\(114\) 0 0
\(115\) 0.157820 0.0911173i 0.0147168 0.00849673i
\(116\) 0 0
\(117\) −4.45875 + 7.72278i −0.412212 + 0.713972i
\(118\) 0 0
\(119\) −7.84529 8.48390i −0.719176 0.777717i
\(120\) 0 0
\(121\) −11.3764 + 19.7046i −1.03422 + 1.79133i
\(122\) 0 0
\(123\) −3.78942 6.56346i −0.341680 0.591808i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.88206i 0.255742i −0.991791 0.127871i \(-0.959186\pi\)
0.991791 0.127871i \(-0.0408143\pi\)
\(128\) 0 0
\(129\) −4.01829 + 2.31996i −0.353790 + 0.204261i
\(130\) 0 0
\(131\) −6.92953 4.00076i −0.605436 0.349548i 0.165741 0.986169i \(-0.446998\pi\)
−0.771177 + 0.636621i \(0.780332\pi\)
\(132\) 0 0
\(133\) −1.48292 + 4.77885i −0.128586 + 0.414379i
\(134\) 0 0
\(135\) −2.20170 1.27115i −0.189492 0.109403i
\(136\) 0 0
\(137\) 5.09925 + 8.83216i 0.435658 + 0.754582i 0.997349 0.0727651i \(-0.0231823\pi\)
−0.561691 + 0.827347i \(0.689849\pi\)
\(138\) 0 0
\(139\) 9.93769i 0.842904i −0.906851 0.421452i \(-0.861521\pi\)
0.906851 0.421452i \(-0.138479\pi\)
\(140\) 0 0
\(141\) 4.16814i 0.351021i
\(142\) 0 0
\(143\) −14.0401 24.3182i −1.17409 2.03359i
\(144\) 0 0
\(145\) −3.80097 2.19449i −0.315653 0.182242i
\(146\) 0 0
\(147\) 8.72256 12.7013i 0.719424 1.04758i
\(148\) 0 0
\(149\) −13.6559 7.88422i −1.11873 0.645900i −0.177655 0.984093i \(-0.556851\pi\)
−0.941077 + 0.338192i \(0.890185\pi\)
\(150\) 0 0
\(151\) −5.24155 + 3.02621i −0.426552 + 0.246270i −0.697877 0.716218i \(-0.745872\pi\)
0.271325 + 0.962488i \(0.412538\pi\)
\(152\) 0 0
\(153\) 8.05806i 0.651455i
\(154\) 0 0
\(155\) −4.07965 −0.327686
\(156\) 0 0
\(157\) 10.2074 + 17.6798i 0.814641 + 1.41100i 0.909585 + 0.415518i \(0.136400\pi\)
−0.0949439 + 0.995483i \(0.530267\pi\)
\(158\) 0 0
\(159\) −10.8556 + 18.8024i −0.860902 + 1.49113i
\(160\) 0 0
\(161\) 0.142893 0.460486i 0.0112616 0.0362914i
\(162\) 0 0
\(163\) −1.91267 + 3.31285i −0.149812 + 0.259482i −0.931158 0.364616i \(-0.881200\pi\)
0.781346 + 0.624098i \(0.214534\pi\)
\(164\) 0 0
\(165\) −11.0747 + 6.39400i −0.862167 + 0.497772i
\(166\) 0 0
\(167\) −21.6531 −1.67556 −0.837782 0.546005i \(-0.816148\pi\)
−0.837782 + 0.546005i \(0.816148\pi\)
\(168\) 0 0
\(169\) 10.3609 0.796996
\(170\) 0 0
\(171\) −3.02180 + 1.74464i −0.231083 + 0.133416i
\(172\) 0 0
\(173\) −5.20160 + 9.00944i −0.395470 + 0.684975i −0.993161 0.116752i \(-0.962752\pi\)
0.597691 + 0.801727i \(0.296085\pi\)
\(174\) 0 0
\(175\) −1.94251 + 1.79629i −0.146840 + 0.135787i
\(176\) 0 0
\(177\) −8.07669 + 13.9892i −0.607081 + 1.05150i
\(178\) 0 0
\(179\) −3.80514 6.59069i −0.284409 0.492611i 0.688056 0.725657i \(-0.258464\pi\)
−0.972466 + 0.233046i \(0.925131\pi\)
\(180\) 0 0
\(181\) 11.2225 0.834160 0.417080 0.908870i \(-0.363054\pi\)
0.417080 + 0.908870i \(0.363054\pi\)
\(182\) 0 0
\(183\) 21.5501i 1.59303i
\(184\) 0 0
\(185\) −3.69132 + 2.13119i −0.271391 + 0.156688i
\(186\) 0 0
\(187\) 21.9745 + 12.6870i 1.60693 + 0.927763i
\(188\) 0 0
\(189\) −6.56017 + 1.48567i −0.477182 + 0.108066i
\(190\) 0 0
\(191\) 6.68571 + 3.86000i 0.483761 + 0.279300i 0.721983 0.691911i \(-0.243231\pi\)
−0.238222 + 0.971211i \(0.576564\pi\)
\(192\) 0 0
\(193\) −11.7011 20.2669i −0.842265 1.45885i −0.887975 0.459892i \(-0.847888\pi\)
0.0457097 0.998955i \(-0.485445\pi\)
\(194\) 0 0
\(195\) 10.6388i 0.761860i
\(196\) 0 0
\(197\) 20.4272i 1.45538i −0.685907 0.727689i \(-0.740594\pi\)
0.685907 0.727689i \(-0.259406\pi\)
\(198\) 0 0
\(199\) 1.10091 + 1.90683i 0.0780413 + 0.135172i 0.902405 0.430889i \(-0.141800\pi\)
−0.824363 + 0.566061i \(0.808467\pi\)
\(200\) 0 0
\(201\) 23.0638 + 13.3159i 1.62679 + 0.939230i
\(202\) 0 0
\(203\) −11.3253 + 2.56482i −0.794884 + 0.180015i
\(204\) 0 0
\(205\) 2.98185 + 1.72157i 0.208261 + 0.120240i
\(206\) 0 0
\(207\) 0.291179 0.168112i 0.0202383 0.0116846i
\(208\) 0 0
\(209\) 10.9874i 0.760011i
\(210\) 0 0
\(211\) 28.9363 1.99206 0.996028 0.0890425i \(-0.0283807\pi\)
0.996028 + 0.0890425i \(0.0283807\pi\)
\(212\) 0 0
\(213\) 9.69236 + 16.7877i 0.664110 + 1.15027i
\(214\) 0 0
\(215\) 1.05398 1.82555i 0.0718810 0.124501i
\(216\) 0 0
\(217\) −7.92476 + 7.32824i −0.537968 + 0.497473i
\(218\) 0 0
\(219\) −9.34056 + 16.1783i −0.631176 + 1.09323i
\(220\) 0 0
\(221\) −18.2813 + 10.5547i −1.22974 + 0.709989i
\(222\) 0 0
\(223\) −11.0535 −0.740200 −0.370100 0.928992i \(-0.620677\pi\)
−0.370100 + 0.928992i \(0.620677\pi\)
\(224\) 0 0
\(225\) −1.84501 −0.123000
\(226\) 0 0
\(227\) −7.11916 + 4.11025i −0.472515 + 0.272807i −0.717292 0.696773i \(-0.754619\pi\)
0.244777 + 0.969579i \(0.421285\pi\)
\(228\) 0 0
\(229\) −5.03725 + 8.72478i −0.332871 + 0.576549i −0.983074 0.183211i \(-0.941351\pi\)
0.650203 + 0.759761i \(0.274684\pi\)
\(230\) 0 0
\(231\) −10.0273 + 32.3138i −0.659747 + 2.12609i
\(232\) 0 0
\(233\) 3.68557 6.38359i 0.241450 0.418203i −0.719678 0.694308i \(-0.755711\pi\)
0.961127 + 0.276105i \(0.0890438\pi\)
\(234\) 0 0
\(235\) −0.946816 1.63993i −0.0617634 0.106977i
\(236\) 0 0
\(237\) −26.8249 −1.74246
\(238\) 0 0
\(239\) 20.9534i 1.35536i 0.735355 + 0.677682i \(0.237016\pi\)
−0.735355 + 0.677682i \(0.762984\pi\)
\(240\) 0 0
\(241\) −2.42132 + 1.39795i −0.155971 + 0.0900498i −0.575954 0.817482i \(-0.695369\pi\)
0.419983 + 0.907532i \(0.362036\pi\)
\(242\) 0 0
\(243\) −14.6132 8.43695i −0.937439 0.541231i
\(244\) 0 0
\(245\) −0.546681 + 6.97862i −0.0349262 + 0.445848i
\(246\) 0 0
\(247\) 7.91614 + 4.57039i 0.503692 + 0.290807i
\(248\) 0 0
\(249\) 18.6764 + 32.3484i 1.18357 + 2.05000i
\(250\) 0 0
\(251\) 18.7656i 1.18448i 0.805763 + 0.592238i \(0.201756\pi\)
−0.805763 + 0.592238i \(0.798244\pi\)
\(252\) 0 0
\(253\) 1.05873i 0.0665620i
\(254\) 0 0
\(255\) 4.80673 + 8.32550i 0.301009 + 0.521363i
\(256\) 0 0
\(257\) −6.76564 3.90615i −0.422029 0.243659i 0.273916 0.961754i \(-0.411681\pi\)
−0.695945 + 0.718095i \(0.745014\pi\)
\(258\) 0 0
\(259\) −3.34220 + 10.7705i −0.207674 + 0.669248i
\(260\) 0 0
\(261\) −7.01281 4.04885i −0.434082 0.250617i
\(262\) 0 0
\(263\) 5.03985 2.90976i 0.310770 0.179423i −0.336501 0.941683i \(-0.609243\pi\)
0.647271 + 0.762260i \(0.275910\pi\)
\(264\) 0 0
\(265\) 9.86359i 0.605916i
\(266\) 0 0
\(267\) 13.0054 0.795920
\(268\) 0 0
\(269\) −12.0950 20.9492i −0.737445 1.27729i −0.953642 0.300943i \(-0.902699\pi\)
0.216197 0.976350i \(-0.430635\pi\)
\(270\) 0 0
\(271\) 7.83742 13.5748i 0.476089 0.824611i −0.523536 0.852004i \(-0.675387\pi\)
0.999625 + 0.0273932i \(0.00872061\pi\)
\(272\) 0 0
\(273\) −19.1104 20.6660i −1.15661 1.25076i
\(274\) 0 0
\(275\) 2.90486 5.03137i 0.175170 0.303403i
\(276\) 0 0
\(277\) −12.6984 + 7.33142i −0.762972 + 0.440502i −0.830362 0.557225i \(-0.811866\pi\)
0.0673898 + 0.997727i \(0.478533\pi\)
\(278\) 0 0
\(279\) −7.52699 −0.450629
\(280\) 0 0
\(281\) 15.0418 0.897320 0.448660 0.893703i \(-0.351901\pi\)
0.448660 + 0.893703i \(0.351901\pi\)
\(282\) 0 0
\(283\) −4.29409 + 2.47919i −0.255257 + 0.147373i −0.622169 0.782883i \(-0.713748\pi\)
0.366912 + 0.930256i \(0.380415\pi\)
\(284\) 0 0
\(285\) 2.08140 3.60508i 0.123291 0.213547i
\(286\) 0 0
\(287\) 8.88471 2.01210i 0.524448 0.118771i
\(288\) 0 0
\(289\) 1.03750 1.79700i 0.0610294 0.105706i
\(290\) 0 0
\(291\) 14.5702 + 25.2364i 0.854123 + 1.47938i
\(292\) 0 0
\(293\) −2.56812 −0.150031 −0.0750156 0.997182i \(-0.523901\pi\)
−0.0750156 + 0.997182i \(0.523901\pi\)
\(294\) 0 0
\(295\) 7.33865i 0.427273i
\(296\) 0 0
\(297\) 12.7912 7.38503i 0.742223 0.428523i
\(298\) 0 0
\(299\) −0.762793 0.440399i −0.0441135 0.0254689i
\(300\) 0 0
\(301\) −1.23185 5.43941i −0.0710026 0.313522i
\(302\) 0 0
\(303\) −4.70619 2.71712i −0.270363 0.156094i
\(304\) 0 0
\(305\) −4.89522 8.47877i −0.280299 0.485493i
\(306\) 0 0
\(307\) 14.4425i 0.824280i 0.911121 + 0.412140i \(0.135219\pi\)
−0.911121 + 0.412140i \(0.864781\pi\)
\(308\) 0 0
\(309\) 4.05162i 0.230488i
\(310\) 0 0
\(311\) −8.26712 14.3191i −0.468785 0.811960i 0.530578 0.847636i \(-0.321975\pi\)
−0.999363 + 0.0356762i \(0.988641\pi\)
\(312\) 0 0
\(313\) 2.19818 + 1.26912i 0.124248 + 0.0717348i 0.560836 0.827927i \(-0.310480\pi\)
−0.436588 + 0.899662i \(0.643813\pi\)
\(314\) 0 0
\(315\) −3.58394 + 3.31417i −0.201932 + 0.186732i
\(316\) 0 0
\(317\) 14.8432 + 8.56971i 0.833676 + 0.481323i 0.855109 0.518447i \(-0.173490\pi\)
−0.0214339 + 0.999770i \(0.506823\pi\)
\(318\) 0 0
\(319\) 22.0826 12.7494i 1.23639 0.713828i
\(320\) 0 0
\(321\) 40.0023i 2.23271i
\(322\) 0 0
\(323\) −8.25981 −0.459588
\(324\) 0 0
\(325\) 2.41666 + 4.18577i 0.134052 + 0.232185i
\(326\) 0 0
\(327\) 4.32749 7.49543i 0.239311 0.414498i
\(328\) 0 0
\(329\) −4.78499 1.48483i −0.263805 0.0818612i
\(330\) 0 0
\(331\) 11.9027 20.6161i 0.654233 1.13317i −0.327852 0.944729i \(-0.606325\pi\)
0.982085 0.188436i \(-0.0603418\pi\)
\(332\) 0 0
\(333\) −6.81051 + 3.93205i −0.373214 + 0.215475i
\(334\) 0 0
\(335\) −12.0991 −0.661044
\(336\) 0 0
\(337\) 1.22715 0.0668470 0.0334235 0.999441i \(-0.489359\pi\)
0.0334235 + 0.999441i \(0.489359\pi\)
\(338\) 0 0
\(339\) −13.6609 + 7.88712i −0.741958 + 0.428370i
\(340\) 0 0
\(341\) 11.8508 20.5262i 0.641758 1.11156i
\(342\) 0 0
\(343\) 11.4737 + 14.5380i 0.619521 + 0.784980i
\(344\) 0 0
\(345\) −0.200562 + 0.347383i −0.0107979 + 0.0187025i
\(346\) 0 0
\(347\) 15.4834 + 26.8181i 0.831194 + 1.43967i 0.897092 + 0.441844i \(0.145675\pi\)
−0.0658983 + 0.997826i \(0.520991\pi\)
\(348\) 0 0
\(349\) 26.6776 1.42802 0.714010 0.700136i \(-0.246877\pi\)
0.714010 + 0.700136i \(0.246877\pi\)
\(350\) 0 0
\(351\) 12.2877i 0.655871i
\(352\) 0 0
\(353\) −12.1501 + 7.01487i −0.646685 + 0.373364i −0.787185 0.616717i \(-0.788462\pi\)
0.140500 + 0.990081i \(0.455129\pi\)
\(354\) 0 0
\(355\) −7.62681 4.40334i −0.404789 0.233705i
\(356\) 0 0
\(357\) 24.2921 + 7.53807i 1.28568 + 0.398957i
\(358\) 0 0
\(359\) 10.5164 + 6.07165i 0.555035 + 0.320450i 0.751150 0.660131i \(-0.229499\pi\)
−0.196115 + 0.980581i \(0.562833\pi\)
\(360\) 0 0
\(361\) −7.71168 13.3570i −0.405878 0.703001i
\(362\) 0 0
\(363\) 50.0822i 2.62864i
\(364\) 0 0
\(365\) 8.48703i 0.444231i
\(366\) 0 0
\(367\) 5.75583 + 9.96939i 0.300452 + 0.520398i 0.976238 0.216700i \(-0.0695292\pi\)
−0.675787 + 0.737097i \(0.736196\pi\)
\(368\) 0 0
\(369\) 5.50153 + 3.17631i 0.286398 + 0.165352i
\(370\) 0 0
\(371\) −17.7179 19.1601i −0.919866 0.994744i
\(372\) 0 0
\(373\) −20.3241 11.7341i −1.05234 0.607570i −0.129037 0.991640i \(-0.541189\pi\)
−0.923304 + 0.384070i \(0.874522\pi\)
\(374\) 0 0
\(375\) 1.90624 1.10057i 0.0984379 0.0568331i
\(376\) 0 0
\(377\) 21.2133i 1.09254i
\(378\) 0 0
\(379\) 15.5653 0.799536 0.399768 0.916616i \(-0.369091\pi\)
0.399768 + 0.916616i \(0.369091\pi\)
\(380\) 0 0
\(381\) 3.17191 + 5.49391i 0.162502 + 0.281462i
\(382\) 0 0
\(383\) −7.45943 + 12.9201i −0.381159 + 0.660187i −0.991228 0.132161i \(-0.957808\pi\)
0.610069 + 0.792348i \(0.291142\pi\)
\(384\) 0 0
\(385\) −3.39508 14.9915i −0.173029 0.764035i
\(386\) 0 0
\(387\) 1.94460 3.36815i 0.0988498 0.171213i
\(388\) 0 0
\(389\) 10.9524 6.32338i 0.555310 0.320608i −0.195951 0.980614i \(-0.562779\pi\)
0.751261 + 0.660006i \(0.229446\pi\)
\(390\) 0 0
\(391\) 0.795909 0.0402508
\(392\) 0 0
\(393\) 17.6125 0.888431
\(394\) 0 0
\(395\) 10.5541 6.09342i 0.531035 0.306593i
\(396\) 0 0
\(397\) −7.61650 + 13.1922i −0.382261 + 0.662095i −0.991385 0.130979i \(-0.958188\pi\)
0.609124 + 0.793075i \(0.291521\pi\)
\(398\) 0 0
\(399\) −2.43265 10.7417i −0.121785 0.537758i
\(400\) 0 0
\(401\) −7.73594 + 13.3990i −0.386315 + 0.669117i −0.991951 0.126625i \(-0.959585\pi\)
0.605636 + 0.795742i \(0.292919\pi\)
\(402\) 0 0
\(403\) 9.85913 + 17.0765i 0.491118 + 0.850641i
\(404\) 0 0
\(405\) 11.1310 0.553102
\(406\) 0 0
\(407\) 24.7632i 1.22747i
\(408\) 0 0
\(409\) −7.47055 + 4.31313i −0.369395 + 0.213270i −0.673194 0.739466i \(-0.735078\pi\)
0.303799 + 0.952736i \(0.401745\pi\)
\(410\) 0 0
\(411\) −19.4408 11.2241i −0.958944 0.553646i
\(412\) 0 0
\(413\) −13.1823 14.2554i −0.648661 0.701462i
\(414\) 0 0
\(415\) −14.6962 8.48487i −0.721409 0.416506i
\(416\) 0 0
\(417\) 10.9371 + 18.9436i 0.535592 + 0.927673i
\(418\) 0 0
\(419\) 10.6898i 0.522230i 0.965308 + 0.261115i \(0.0840902\pi\)
−0.965308 + 0.261115i \(0.915910\pi\)
\(420\) 0 0
\(421\) 20.4941i 0.998823i 0.866365 + 0.499412i \(0.166450\pi\)
−0.866365 + 0.499412i \(0.833550\pi\)
\(422\) 0 0
\(423\) −1.74688 3.02569i −0.0849363 0.147114i
\(424\) 0 0
\(425\) −3.78236 2.18375i −0.183471 0.105927i
\(426\) 0 0
\(427\) −24.7393 7.67685i −1.19722 0.371509i
\(428\) 0 0
\(429\) 53.5277 + 30.9042i 2.58434 + 1.49207i
\(430\) 0 0
\(431\) 20.9544 12.0980i 1.00934 0.582740i 0.0983389 0.995153i \(-0.468647\pi\)
0.910997 + 0.412412i \(0.135314\pi\)
\(432\) 0 0
\(433\) 20.2949i 0.975311i 0.873036 + 0.487655i \(0.162148\pi\)
−0.873036 + 0.487655i \(0.837852\pi\)
\(434\) 0 0
\(435\) 9.66074 0.463197
\(436\) 0 0
\(437\) −0.172321 0.298469i −0.00824324 0.0142777i
\(438\) 0 0
\(439\) −2.52231 + 4.36878i −0.120384 + 0.208510i −0.919919 0.392109i \(-0.871746\pi\)
0.799535 + 0.600619i \(0.205079\pi\)
\(440\) 0 0
\(441\) −1.00863 + 12.8756i −0.0480300 + 0.613124i
\(442\) 0 0
\(443\) 4.04696 7.00955i 0.192277 0.333034i −0.753727 0.657187i \(-0.771746\pi\)
0.946005 + 0.324153i \(0.105079\pi\)
\(444\) 0 0
\(445\) −5.11692 + 2.95425i −0.242565 + 0.140045i
\(446\) 0 0
\(447\) 34.7085 1.64166
\(448\) 0 0
\(449\) −4.05786 −0.191502 −0.0957511 0.995405i \(-0.530525\pi\)
−0.0957511 + 0.995405i \(0.530525\pi\)
\(450\) 0 0
\(451\) −17.3237 + 10.0019i −0.815742 + 0.470969i
\(452\) 0 0
\(453\) 6.66111 11.5374i 0.312966 0.542074i
\(454\) 0 0
\(455\) 12.2132 + 3.78988i 0.572566 + 0.177673i
\(456\) 0 0
\(457\) −2.70734 + 4.68925i −0.126644 + 0.219354i −0.922374 0.386297i \(-0.873754\pi\)
0.795730 + 0.605651i \(0.207087\pi\)
\(458\) 0 0
\(459\) −5.55174 9.61589i −0.259133 0.448831i
\(460\) 0 0
\(461\) −18.9095 −0.880702 −0.440351 0.897826i \(-0.645146\pi\)
−0.440351 + 0.897826i \(0.645146\pi\)
\(462\) 0 0
\(463\) 0.304782i 0.0141644i 0.999975 + 0.00708220i \(0.00225435\pi\)
−0.999975 + 0.00708220i \(0.997746\pi\)
\(464\) 0 0
\(465\) 7.77680 4.48994i 0.360641 0.208216i
\(466\) 0 0
\(467\) 3.35368 + 1.93625i 0.155190 + 0.0895989i 0.575584 0.817743i \(-0.304775\pi\)
−0.420394 + 0.907342i \(0.638108\pi\)
\(468\) 0 0
\(469\) −23.5026 + 21.7335i −1.08525 + 1.00356i
\(470\) 0 0
\(471\) −38.9156 22.4680i −1.79314 1.03527i
\(472\) 0 0
\(473\) 6.12334 + 10.6059i 0.281552 + 0.487662i
\(474\) 0 0
\(475\) 1.89120i 0.0867742i
\(476\) 0 0
\(477\) 18.1984i 0.833247i
\(478\) 0 0
\(479\) −16.4744 28.5344i −0.752732 1.30377i −0.946494 0.322722i \(-0.895402\pi\)
0.193761 0.981049i \(-0.437931\pi\)
\(480\) 0 0
\(481\) 17.8413 + 10.3007i 0.813494 + 0.469671i
\(482\) 0 0
\(483\) 0.234408 + 1.03506i 0.0106659 + 0.0470970i
\(484\) 0 0
\(485\) −11.4652 6.61941i −0.520606 0.300572i
\(486\) 0 0
\(487\) −35.8923 + 20.7224i −1.62643 + 0.939022i −0.641289 + 0.767300i \(0.721600\pi\)
−0.985145 + 0.171722i \(0.945067\pi\)
\(488\) 0 0
\(489\) 8.42012i 0.380771i
\(490\) 0 0
\(491\) 25.4339 1.14782 0.573908 0.818920i \(-0.305427\pi\)
0.573908 + 0.818920i \(0.305427\pi\)
\(492\) 0 0
\(493\) −9.58441 16.6007i −0.431660 0.747658i
\(494\) 0 0
\(495\) 5.35949 9.28291i 0.240891 0.417236i
\(496\) 0 0
\(497\) −22.7248 + 5.14644i −1.01935 + 0.230849i
\(498\) 0 0
\(499\) −3.25059 + 5.63019i −0.145516 + 0.252042i −0.929565 0.368657i \(-0.879818\pi\)
0.784049 + 0.620699i \(0.213151\pi\)
\(500\) 0 0
\(501\) 41.2759 23.8307i 1.84407 1.06468i
\(502\) 0 0
\(503\) 23.4094 1.04378 0.521888 0.853014i \(-0.325228\pi\)
0.521888 + 0.853014i \(0.325228\pi\)
\(504\) 0 0
\(505\) 2.46883 0.109862
\(506\) 0 0
\(507\) −19.7505 + 11.4029i −0.877149 + 0.506422i
\(508\) 0 0
\(509\) 8.44031 14.6190i 0.374110 0.647978i −0.616083 0.787681i \(-0.711281\pi\)
0.990193 + 0.139703i \(0.0446148\pi\)
\(510\) 0 0
\(511\) −15.2452 16.4861i −0.674406 0.729303i
\(512\) 0 0
\(513\) −2.40400 + 4.16385i −0.106139 + 0.183838i
\(514\) 0 0
\(515\) −0.920346 1.59409i −0.0405553 0.0702438i
\(516\) 0 0
\(517\) 11.0015 0.483844
\(518\) 0 0
\(519\) 22.8989i 1.00515i
\(520\) 0 0
\(521\) 21.2245 12.2540i 0.929864 0.536857i 0.0430955 0.999071i \(-0.486278\pi\)
0.886769 + 0.462214i \(0.152945\pi\)
\(522\) 0 0
\(523\) 25.2087 + 14.5543i 1.10230 + 0.636414i 0.936824 0.349800i \(-0.113751\pi\)
0.165476 + 0.986214i \(0.447084\pi\)
\(524\) 0 0
\(525\) 1.72595 5.56203i 0.0753266 0.242747i
\(526\) 0 0
\(527\) −15.4307 8.90893i −0.672173 0.388079i
\(528\) 0 0
\(529\) −11.4834 19.8898i −0.499278 0.864775i
\(530\) 0 0
\(531\) 13.5399i 0.587580i
\(532\) 0 0
\(533\) 16.6418i 0.720837i
\(534\) 0 0
\(535\) 9.08674 + 15.7387i 0.392854 + 0.680443i
\(536\) 0 0
\(537\) 14.5070 + 8.37563i 0.626024 + 0.361435i
\(538\) 0 0
\(539\) −33.5240 23.0225i −1.44398 0.991649i
\(540\) 0 0
\(541\) −3.98939 2.30327i −0.171517 0.0990254i 0.411784 0.911282i \(-0.364906\pi\)
−0.583301 + 0.812256i \(0.698239\pi\)
\(542\) 0 0
\(543\) −21.3927 + 12.3511i −0.918050 + 0.530036i
\(544\) 0 0
\(545\) 3.93204i 0.168430i
\(546\) 0 0
\(547\) −4.27444 −0.182762 −0.0913808 0.995816i \(-0.529128\pi\)
−0.0913808 + 0.995816i \(0.529128\pi\)
\(548\) 0 0
\(549\) −9.03171 15.6434i −0.385464 0.667644i
\(550\) 0 0
\(551\) −4.15022 + 7.18839i −0.176805 + 0.306236i
\(552\) 0 0
\(553\) 9.55590 30.7948i 0.406358 1.30953i
\(554\) 0 0
\(555\) 4.69103 8.12511i 0.199123 0.344892i
\(556\) 0 0
\(557\) −16.1482 + 9.32317i −0.684221 + 0.395035i −0.801443 0.598070i \(-0.795934\pi\)
0.117222 + 0.993106i \(0.462601\pi\)
\(558\) 0 0
\(559\) −10.1885 −0.430926
\(560\) 0 0
\(561\) −55.8515 −2.35805
\(562\) 0 0
\(563\) 19.1853 11.0766i 0.808565 0.466825i −0.0378926 0.999282i \(-0.512064\pi\)
0.846457 + 0.532457i \(0.178731\pi\)
\(564\) 0 0
\(565\) 3.58320 6.20629i 0.150746 0.261101i
\(566\) 0 0
\(567\) 21.6220 19.9945i 0.908039 0.839688i
\(568\) 0 0
\(569\) −0.619323 + 1.07270i −0.0259634 + 0.0449699i −0.878715 0.477346i \(-0.841599\pi\)
0.852752 + 0.522316i \(0.174932\pi\)
\(570\) 0 0
\(571\) −10.3953 18.0051i −0.435028 0.753490i 0.562270 0.826954i \(-0.309928\pi\)
−0.997298 + 0.0734633i \(0.976595\pi\)
\(572\) 0 0
\(573\) −16.9928 −0.709883
\(574\) 0 0
\(575\) 0.182235i 0.00759971i
\(576\) 0 0
\(577\) −29.9908 + 17.3152i −1.24853 + 0.720842i −0.970817 0.239821i \(-0.922911\pi\)
−0.277717 + 0.960663i \(0.589578\pi\)
\(578\) 0 0
\(579\) 44.6103 + 25.7558i 1.85394 + 1.07037i
\(580\) 0 0
\(581\) −43.7888 + 9.91675i −1.81667 + 0.411416i
\(582\) 0 0
\(583\) 49.6274 + 28.6524i 2.05536 + 1.18666i
\(584\) 0 0
\(585\) 4.45875 + 7.72278i 0.184347 + 0.319298i
\(586\) 0 0
\(587\) 22.4911i 0.928306i −0.885755 0.464153i \(-0.846359\pi\)
0.885755 0.464153i \(-0.153641\pi\)
\(588\) 0 0
\(589\) 7.71544i 0.317909i
\(590\) 0 0
\(591\) 22.4815 + 38.9392i 0.924767 + 1.60174i
\(592\) 0 0
\(593\) −3.70525 2.13923i −0.152156 0.0878474i 0.421989 0.906601i \(-0.361332\pi\)
−0.574145 + 0.818754i \(0.694665\pi\)
\(594\) 0 0
\(595\) −11.2699 + 2.55227i −0.462021 + 0.104633i
\(596\) 0 0
\(597\) −4.19719 2.42325i −0.171780 0.0991771i
\(598\) 0 0
\(599\) 5.50041 3.17566i 0.224741 0.129754i −0.383403 0.923581i \(-0.625248\pi\)
0.608143 + 0.793827i \(0.291915\pi\)
\(600\) 0 0
\(601\) 3.72971i 0.152138i −0.997103 0.0760691i \(-0.975763\pi\)
0.997103 0.0760691i \(-0.0242369\pi\)
\(602\) 0 0
\(603\) −22.3229 −0.909059
\(604\) 0 0
\(605\) 11.3764 + 19.7046i 0.462518 + 0.801105i
\(606\) 0 0
\(607\) −9.66175 + 16.7346i −0.392158 + 0.679238i −0.992734 0.120330i \(-0.961605\pi\)
0.600576 + 0.799568i \(0.294938\pi\)
\(608\) 0 0
\(609\) 18.7661 17.3535i 0.760440 0.703199i
\(610\) 0 0
\(611\) −4.57626 + 7.92631i −0.185136 + 0.320664i
\(612\) 0 0
\(613\) 26.3430 15.2092i 1.06399 0.614292i 0.137453 0.990508i \(-0.456108\pi\)
0.926532 + 0.376216i \(0.122775\pi\)
\(614\) 0 0
\(615\) −7.57883 −0.305608
\(616\) 0 0
\(617\) 9.32522 0.375419 0.187710 0.982225i \(-0.439894\pi\)
0.187710 + 0.982225i \(0.439894\pi\)
\(618\) 0 0
\(619\) 9.29185 5.36465i 0.373471 0.215624i −0.301503 0.953465i \(-0.597488\pi\)
0.674974 + 0.737842i \(0.264155\pi\)
\(620\) 0 0
\(621\) 0.231647 0.401225i 0.00929569 0.0161006i
\(622\) 0 0
\(623\) −4.63296 + 14.9301i −0.185616 + 0.598163i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 12.0923 + 20.9445i 0.482921 + 0.836444i
\(628\) 0 0
\(629\) −18.6159 −0.742264
\(630\) 0 0
\(631\) 7.84644i 0.312362i −0.987728 0.156181i \(-0.950082\pi\)
0.987728 0.156181i \(-0.0499183\pi\)
\(632\) 0 0
\(633\) −55.1595 + 31.8464i −2.19239 + 1.26578i
\(634\) 0 0
\(635\) −2.49594 1.44103i −0.0990484 0.0571856i
\(636\) 0 0
\(637\) 30.5321 14.5767i 1.20972 0.577548i
\(638\) 0 0
\(639\) −14.0715 8.12420i −0.556661 0.321388i
\(640\) 0 0
\(641\) 6.79977 + 11.7775i 0.268575 + 0.465185i 0.968494 0.249037i \(-0.0801141\pi\)
−0.699919 + 0.714222i \(0.746781\pi\)
\(642\) 0 0
\(643\) 5.68857i 0.224335i −0.993689 0.112168i \(-0.964221\pi\)
0.993689 0.112168i \(-0.0357794\pi\)
\(644\) 0 0
\(645\) 4.63992i 0.182697i
\(646\) 0 0
\(647\) −0.225415 0.390431i −0.00886200 0.0153494i 0.861560 0.507655i \(-0.169488\pi\)
−0.870422 + 0.492306i \(0.836154\pi\)
\(648\) 0 0
\(649\) 36.9234 + 21.3178i 1.44937 + 0.836795i
\(650\) 0 0
\(651\) 7.04127 22.6911i 0.275969 0.889336i
\(652\) 0 0
\(653\) 3.69894 + 2.13559i 0.144751 + 0.0835720i 0.570626 0.821210i \(-0.306700\pi\)
−0.425875 + 0.904782i \(0.640034\pi\)
\(654\) 0 0
\(655\) −6.92953 + 4.00076i −0.270759 + 0.156323i
\(656\) 0 0
\(657\) 15.6586i 0.610901i
\(658\) 0 0
\(659\) −15.5436 −0.605493 −0.302746 0.953071i \(-0.597903\pi\)
−0.302746 + 0.953071i \(0.597903\pi\)
\(660\) 0 0
\(661\) 14.4897 + 25.0969i 0.563583 + 0.976155i 0.997180 + 0.0750479i \(0.0239110\pi\)
−0.433597 + 0.901107i \(0.642756\pi\)
\(662\) 0 0
\(663\) 23.2324 40.2398i 0.902273 1.56278i
\(664\) 0 0
\(665\) 3.39715 + 3.67367i 0.131736 + 0.142459i
\(666\) 0 0
\(667\) 0.399912 0.692668i 0.0154846 0.0268202i
\(668\) 0 0
\(669\) 21.0707 12.1652i 0.814641 0.470333i
\(670\) 0 0
\(671\) 56.8797 2.19582
\(672\) 0 0
\(673\) 32.3539 1.24715 0.623576 0.781763i \(-0.285679\pi\)
0.623576 + 0.781763i \(0.285679\pi\)
\(674\) 0 0
\(675\) −2.20170 + 1.27115i −0.0847433 + 0.0489266i
\(676\) 0 0
\(677\) 22.4527 38.8892i 0.862927 1.49463i −0.00616405 0.999981i \(-0.501962\pi\)
0.869091 0.494652i \(-0.164705\pi\)
\(678\) 0 0
\(679\) −34.1616 + 7.73649i −1.31100 + 0.296899i
\(680\) 0 0
\(681\) 9.04722 15.6703i 0.346690 0.600485i
\(682\) 0 0
\(683\) −5.62256 9.73856i −0.215141 0.372636i 0.738175 0.674609i \(-0.235688\pi\)
−0.953316 + 0.301974i \(0.902355\pi\)
\(684\) 0 0
\(685\) 10.1985 0.389665
\(686\) 0 0
\(687\) 22.1754i 0.846043i
\(688\) 0 0
\(689\) −41.2868 + 23.8369i −1.57290 + 0.908115i
\(690\) 0 0
\(691\) 8.53812 + 4.92949i 0.324805 + 0.187527i 0.653533 0.756898i \(-0.273286\pi\)
−0.328727 + 0.944425i \(0.606620\pi\)
\(692\) 0 0
\(693\) −6.26394 27.6593i −0.237948 1.05069i
\(694\) 0 0
\(695\) −8.60629 4.96884i −0.326455 0.188479i
\(696\) 0 0
\(697\) 7.51896 + 13.0232i 0.284801 + 0.493289i
\(698\) 0 0
\(699\) 16.2249i 0.613681i
\(700\) 0 0
\(701\) 6.00307i 0.226733i −0.993553 0.113366i \(-0.963837\pi\)
0.993553 0.113366i \(-0.0361634\pi\)
\(702\) 0 0
\(703\) 4.03050 + 6.98103i 0.152013 + 0.263294i
\(704\) 0 0
\(705\) 3.60972 + 2.08407i 0.135950 + 0.0784907i
\(706\) 0 0
\(707\) 4.79573 4.43474i 0.180362 0.166785i
\(708\) 0 0
\(709\) 10.2323 + 5.90764i 0.384283 + 0.221866i 0.679680 0.733508i \(-0.262119\pi\)
−0.295397 + 0.955375i \(0.595452\pi\)
\(710\) 0 0
\(711\) 19.4724 11.2424i 0.730272 0.421623i
\(712\) 0 0
\(713\) 0.743454i 0.0278426i
\(714\) 0 0
\(715\) −28.0802 −1.05014
\(716\) 0 0
\(717\) −23.0607 39.9423i −0.861217 1.49167i
\(718\) 0 0
\(719\) 3.06796 5.31386i 0.114415 0.198173i −0.803130 0.595803i \(-0.796834\pi\)
0.917546 + 0.397630i \(0.130167\pi\)
\(720\) 0 0
\(721\) −4.65122 1.44332i −0.173221 0.0537520i
\(722\) 0 0
\(723\) 3.07708 5.32966i 0.114438 0.198212i
\(724\) 0 0
\(725\) −3.80097 + 2.19449i −0.141164 + 0.0815013i
\(726\) 0 0
\(727\) −14.8986 −0.552558 −0.276279 0.961077i \(-0.589101\pi\)
−0.276279 + 0.961077i \(0.589101\pi\)
\(728\) 0 0
\(729\) 3.74887 0.138847
\(730\) 0 0
\(731\) 7.97308 4.60326i 0.294895 0.170258i
\(732\) 0 0
\(733\) −23.7602 + 41.1539i −0.877604 + 1.52006i −0.0236421 + 0.999720i \(0.507526\pi\)
−0.853962 + 0.520335i \(0.825807\pi\)
\(734\) 0 0
\(735\) −6.63835 13.9046i −0.244859 0.512879i
\(736\) 0 0
\(737\) 35.1462 60.8750i 1.29463 2.24236i
\(738\) 0 0
\(739\) −1.95274 3.38225i −0.0718328 0.124418i 0.827872 0.560917i \(-0.189551\pi\)
−0.899705 + 0.436499i \(0.856218\pi\)
\(740\) 0 0
\(741\) −20.1201 −0.739130
\(742\) 0 0
\(743\) 48.0392i 1.76239i −0.472755 0.881194i \(-0.656741\pi\)
0.472755 0.881194i \(-0.343259\pi\)
\(744\) 0 0
\(745\) −13.6559 + 7.88422i −0.500312 + 0.288855i
\(746\) 0 0
\(747\) −27.1146 15.6546i −0.992073 0.572773i
\(748\) 0 0
\(749\) 45.9223 + 14.2501i 1.67797 + 0.520688i
\(750\) 0 0
\(751\) −0.139857 0.0807464i −0.00510345 0.00294648i 0.497446 0.867495i \(-0.334271\pi\)
−0.502550 + 0.864548i \(0.667605\pi\)
\(752\) 0 0
\(753\) −20.6529 35.7718i −0.752633 1.30360i
\(754\) 0 0
\(755\) 6.05243i 0.220270i
\(756\) 0 0
\(757\) 17.0175i 0.618513i 0.950979 + 0.309257i \(0.100080\pi\)
−0.950979 + 0.309257i \(0.899920\pi\)
\(758\) 0 0
\(759\) −1.16521 2.01820i −0.0422944 0.0732560i
\(760\) 0 0
\(761\) 23.5345 + 13.5876i 0.853125 + 0.492552i 0.861704 0.507411i \(-0.169398\pi\)
−0.00857924 + 0.999963i \(0.502731\pi\)
\(762\) 0 0
\(763\) 7.06309 + 7.63803i 0.255701 + 0.276515i
\(764\) 0 0
\(765\) −6.97848 4.02903i −0.252308 0.145670i
\(766\) 0 0
\(767\) −30.7179 + 17.7350i −1.10916 + 0.640374i
\(768\) 0 0
\(769\) 16.2516i 0.586049i −0.956105 0.293025i \(-0.905338\pi\)
0.956105 0.293025i \(-0.0946618\pi\)
\(770\) 0 0
\(771\) 17.1959 0.619296
\(772\) 0 0
\(773\) 2.07903 + 3.60098i 0.0747774 + 0.129518i 0.900989 0.433841i \(-0.142842\pi\)
−0.826212 + 0.563359i \(0.809509\pi\)
\(774\) 0 0
\(775\) −2.03983 + 3.53308i −0.0732727 + 0.126912i
\(776\) 0 0
\(777\) −5.48268 24.2095i −0.196690 0.868513i
\(778\) 0 0
\(779\) 3.25584 5.63928i 0.116652 0.202048i
\(780\) 0 0
\(781\) 44.3097 25.5822i 1.58552 0.915403i
\(782\) 0 0
\(783\) −11.1581 −0.398758
\(784\) 0 0
\(785\) 20.4149 0.728637
\(786\) 0 0
\(787\) −0.346588 + 0.200103i −0.0123545 + 0.00713289i −0.506165 0.862437i \(-0.668937\pi\)
0.493810 + 0.869570i \(0.335604\pi\)
\(788\) 0 0
\(789\) −6.40478 + 11.0934i −0.228016 + 0.394936i
\(790\) 0 0
\(791\) −4.18790 18.4922i −0.148904 0.657509i
\(792\) 0 0
\(793\) −23.6601 + 40.9806i −0.840196 + 1.45526i
\(794\) 0 0
\(795\) 10.8556 + 18.8024i 0.385007 + 0.666852i
\(796\) 0 0
\(797\) −4.32443 −0.153179 −0.0765895 0.997063i \(-0.524403\pi\)
−0.0765895 + 0.997063i \(0.524403\pi\)
\(798\) 0 0
\(799\) 8.27042i 0.292586i
\(800\) 0 0
\(801\) −9.44075 + 5.45062i −0.333572 + 0.192588i
\(802\) 0 0
\(803\) 42.7014 + 24.6536i 1.50690 + 0.870008i
\(804\) 0 0
\(805\) −0.327346 0.353992i −0.0115374 0.0124766i
\(806\) 0 0
\(807\) 46.1120 + 26.6228i 1.62322 + 0.937165i
\(808\) 0 0
\(809\) −18.2699 31.6444i −0.642335 1.11256i −0.984910 0.173066i \(-0.944633\pi\)
0.342576 0.939490i \(-0.388701\pi\)
\(810\) 0 0
\(811\) 17.6473i 0.619680i 0.950789 + 0.309840i \(0.100275\pi\)
−0.950789 + 0.309840i \(0.899725\pi\)
\(812\) 0 0
\(813\) 34.5025i 1.21005i
\(814\) 0 0
\(815\) 1.91267 + 3.31285i 0.0669981 + 0.116044i
\(816\) 0 0
\(817\) −3.45248 1.99329i −0.120787 0.0697364i
\(818\) 0 0
\(819\) 22.5335 + 6.99236i 0.787385 + 0.244333i
\(820\) 0 0
\(821\) −37.1429 21.4445i −1.29630 0.748418i −0.316535 0.948581i \(-0.602520\pi\)
−0.979763 + 0.200163i \(0.935853\pi\)
\(822\) 0 0
\(823\) 31.2832 18.0614i 1.09046 0.629580i 0.156764 0.987636i \(-0.449894\pi\)
0.933700 + 0.358056i \(0.116560\pi\)
\(824\) 0 0
\(825\) 12.7880i 0.445221i
\(826\) 0 0
\(827\) 7.53954 0.262176 0.131088 0.991371i \(-0.458153\pi\)
0.131088 + 0.991371i \(0.458153\pi\)
\(828\) 0 0
\(829\) −28.0133 48.5204i −0.972941 1.68518i −0.686570 0.727063i \(-0.740885\pi\)
−0.286370 0.958119i \(-0.592449\pi\)
\(830\) 0 0
\(831\) 16.1375 27.9509i 0.559802 0.969606i
\(832\) 0 0
\(833\) −17.3073 + 25.2018i −0.599662 + 0.873192i
\(834\) 0 0
\(835\) −10.8265 + 18.7521i −0.374667 + 0.648943i
\(836\) 0 0
\(837\) −8.98215 + 5.18585i −0.310469 + 0.179249i
\(838\) 0 0
\(839\) −33.7497 −1.16517 −0.582584 0.812770i \(-0.697958\pi\)
−0.582584 + 0.812770i \(0.697958\pi\)
\(840\) 0 0
\(841\) 9.73688 0.335755
\(842\) 0 0
\(843\) −28.6733 + 16.5546i −0.987562 + 0.570169i
\(844\) 0 0
\(845\) 5.18047 8.97285i 0.178214 0.308675i
\(846\) 0 0
\(847\) 57.4940 + 17.8409i 1.97552 + 0.613021i
\(848\) 0 0
\(849\) 5.45705 9.45188i 0.187285 0.324388i
\(850\) 0 0
\(851\) −0.388376 0.672687i −0.0133133 0.0230594i
\(852\) 0 0
\(853\) 53.4433 1.82987 0.914933 0.403607i \(-0.132244\pi\)
0.914933 + 0.403607i \(0.132244\pi\)
\(854\) 0 0
\(855\) 3.48928i 0.119331i
\(856\) 0 0
\(857\) 30.9412 17.8639i 1.05693 0.610220i 0.132350 0.991203i \(-0.457748\pi\)
0.924582 + 0.380983i \(0.124414\pi\)
\(858\) 0 0
\(859\) −4.24262 2.44948i −0.144756 0.0835751i 0.425873 0.904783i \(-0.359967\pi\)
−0.570629 + 0.821208i \(0.693300\pi\)
\(860\) 0 0
\(861\) −14.7220 + 13.6138i −0.501723 + 0.463957i
\(862\) 0 0
\(863\) −27.0242 15.6024i −0.919914 0.531113i −0.0363065 0.999341i \(-0.511559\pi\)
−0.883608 + 0.468228i \(0.844893\pi\)
\(864\) 0 0
\(865\) 5.20160 + 9.00944i 0.176860 + 0.306330i
\(866\) 0 0
\(867\) 4.56736i 0.155116i
\(868\) 0 0
\(869\) 70.8021i 2.40180i
\(870\) 0 0
\(871\) 29.2394 + 50.6441i 0.990738 + 1.71601i
\(872\) 0 0
\(873\) −21.1533 12.2129i −0.715931 0.413343i
\(874\) 0 0
\(875\) 0.584379 + 2.58041i 0.0197556 + 0.0872337i
\(876\) 0 0
\(877\) −4.65217 2.68593i −0.157092 0.0906974i 0.419393 0.907805i \(-0.362243\pi\)
−0.576485 + 0.817107i \(0.695576\pi\)
\(878\) 0 0
\(879\) 4.89545 2.82639i 0.165120 0.0953318i
\(880\) 0 0
\(881\) 12.9255i 0.435471i 0.976008 + 0.217736i \(0.0698670\pi\)
−0.976008 + 0.217736i \(0.930133\pi\)
\(882\) 0 0
\(883\) −42.5290 −1.43122 −0.715608 0.698502i \(-0.753850\pi\)
−0.715608 + 0.698502i \(0.753850\pi\)
\(884\) 0 0
\(885\) 8.07669 + 13.9892i 0.271495 + 0.470243i
\(886\) 0 0
\(887\) −19.2391 + 33.3231i −0.645987 + 1.11888i 0.338086 + 0.941115i \(0.390220\pi\)
−0.984073 + 0.177766i \(0.943113\pi\)
\(888\) 0 0
\(889\) −7.43690 + 1.68422i −0.249426 + 0.0564868i
\(890\) 0 0
\(891\) −32.3339 + 56.0040i −1.08323 + 1.87620i
\(892\) 0 0
\(893\) −3.10144 + 1.79062i −0.103786 + 0.0599207i
\(894\) 0 0
\(895\) −7.61028 −0.254383
\(896\) 0 0
\(897\) 1.93876 0.0647332
\(898\) 0 0
\(899\) −15.5066 + 8.95275i −0.517175 + 0.298591i
\(900\) 0 0
\(901\) 21.5396 37.3077i 0.717588 1.24290i
\(902\) 0 0
\(903\) 8.33464 + 9.01308i 0.277359 + 0.299937i
\(904\) 0 0
\(905\) 5.61124 9.71894i 0.186524 0.323069i
\(906\) 0 0
\(907\) −29.8056 51.6248i −0.989679 1.71417i −0.618941 0.785438i \(-0.712438\pi\)
−0.370738 0.928737i \(-0.620895\pi\)
\(908\) 0 0
\(909\) 4.55501 0.151080
\(910\) 0 0
\(911\) 40.9457i 1.35659i 0.734790 + 0.678295i \(0.237281\pi\)
−0.734790 + 0.678295i \(0.762719\pi\)
\(912\) 0 0
\(913\) 85.3810 49.2947i 2.82570 1.63142i
\(914\) 0 0
\(915\) 18.6629 + 10.7751i 0.616978 + 0.356212i
\(916\) 0 0
\(917\) −6.27413 + 20.2190i −0.207190 + 0.667689i
\(918\) 0 0
\(919\) 22.8509 + 13.1930i 0.753781 + 0.435196i 0.827058 0.562116i \(-0.190013\pi\)
−0.0732776 + 0.997312i \(0.523346\pi\)
\(920\) 0 0
\(921\) −15.8950 27.5310i −0.523759 0.907177i
\(922\) 0 0
\(923\) 42.5655i 1.40106i
\(924\) 0 0
\(925\) 4.26237i 0.140146i
\(926\) 0 0
\(927\) −1.69805 2.94110i −0.0557711 0.0965984i
\(928\) 0 0
\(929\) −38.7083 22.3482i −1.26998 0.733222i −0.294994 0.955499i \(-0.595318\pi\)
−0.974984 + 0.222277i \(0.928651\pi\)
\(930\) 0 0
\(931\) 13.1980 + 1.03388i 0.432546 + 0.0338842i
\(932\) 0 0
\(933\) 31.5182 + 18.1971i 1.03186 + 0.595745i
\(934\) 0 0
\(935\) 21.9745 12.6870i 0.718642 0.414908i
\(936\) 0 0
\(937\) 14.4033i 0.470535i −0.971931 0.235268i \(-0.924403\pi\)
0.971931 0.235268i \(-0.0755967\pi\)
\(938\) 0 0
\(939\) −5.58701 −0.182325
\(940\) 0 0
\(941\) 5.58349 + 9.67088i 0.182016 + 0.315262i 0.942567 0.334017i \(-0.108404\pi\)
−0.760551 + 0.649279i \(0.775071\pi\)
\(942\) 0 0
\(943\) −0.313730 + 0.543396i −0.0102165 + 0.0176954i
\(944\) 0 0
\(945\) −1.99346 + 6.42410i −0.0648472 + 0.208976i
\(946\) 0 0
\(947\) 10.3303 17.8927i 0.335691 0.581434i −0.647926 0.761703i \(-0.724364\pi\)
0.983617 + 0.180269i \(0.0576968\pi\)
\(948\) 0 0
\(949\) −35.5248 + 20.5102i −1.15318 + 0.665791i
\(950\) 0 0
\(951\) −37.7262 −1.22336
\(952\) 0 0
\(953\) 29.6708 0.961132 0.480566 0.876959i \(-0.340431\pi\)
0.480566 + 0.876959i \(0.340431\pi\)
\(954\) 0 0
\(955\) 6.68571 3.86000i 0.216344 0.124907i
\(956\) 0 0
\(957\) −28.0631 + 48.6067i −0.907152 + 1.57123i
\(958\) 0 0
\(959\) 19.8107 18.3195i 0.639720 0.591566i
\(960\) 0 0
\(961\) 7.17822 12.4330i 0.231555 0.401066i
\(962\) 0 0
\(963\) 16.7651 + 29.0380i 0.540248 + 0.935736i
\(964\) 0 0
\(965\) −23.4022 −0.753345
\(966\) 0 0
\(967\) 50.5419i 1.62532i −0.582740 0.812658i \(-0.698020\pi\)
0.582740 0.812658i \(-0.301980\pi\)
\(968\) 0 0
\(969\) 15.7452 9.09048i 0.505808 0.292028i
\(970\) 0 0
\(971\) −30.5599 17.6438i −0.980715 0.566216i −0.0782293 0.996935i \(-0.524927\pi\)
−0.902486 + 0.430719i \(0.858260\pi\)
\(972\) 0 0
\(973\) −25.6433 + 5.80737i −0.822086 + 0.186176i
\(974\) 0 0
\(975\) −9.21347 5.31940i −0.295067 0.170357i
\(976\) 0 0
\(977\) −13.3142 23.0609i −0.425960 0.737784i 0.570550 0.821263i \(-0.306730\pi\)
−0.996510 + 0.0834788i \(0.973397\pi\)
\(978\) 0 0
\(979\) 34.3268i 1.09709i
\(980\) 0 0
\(981\) 7.25465i 0.231623i
\(982\) 0 0
\(983\) 9.60664 + 16.6392i 0.306404 + 0.530708i 0.977573 0.210597i \(-0.0675408\pi\)
−0.671169 + 0.741305i \(0.734207\pi\)
\(984\) 0 0
\(985\) −17.6905 10.2136i −0.563666 0.325432i
\(986\) 0 0
\(987\) 10.7555 2.43577i 0.342351 0.0775315i
\(988\) 0 0
\(989\) 0.332678 + 0.192072i 0.0105786 + 0.00610753i
\(990\) 0 0
\(991\) −52.2605 + 30.1726i −1.66011 + 0.958465i −0.687450 + 0.726232i \(0.741270\pi\)
−0.972660 + 0.232233i \(0.925397\pi\)
\(992\) 0 0
\(993\) 52.3991i 1.66283i
\(994\) 0 0
\(995\) 2.20182 0.0698023
\(996\) 0 0
\(997\) 8.59147 + 14.8809i 0.272095 + 0.471282i 0.969398 0.245495i \(-0.0789503\pi\)
−0.697303 + 0.716776i \(0.745617\pi\)
\(998\) 0 0
\(999\) −5.41811 + 9.38444i −0.171421 + 0.296911i
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 1120.2.bz.f.271.4 24
4.3 odd 2 280.2.bj.e.131.11 24
7.3 odd 6 1120.2.bz.e.591.4 24
8.3 odd 2 1120.2.bz.e.271.4 24
8.5 even 2 280.2.bj.f.131.2 yes 24
28.3 even 6 280.2.bj.f.171.2 yes 24
56.3 even 6 inner 1120.2.bz.f.591.4 24
56.45 odd 6 280.2.bj.e.171.11 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.bj.e.131.11 24 4.3 odd 2
280.2.bj.e.171.11 yes 24 56.45 odd 6
280.2.bj.f.131.2 yes 24 8.5 even 2
280.2.bj.f.171.2 yes 24 28.3 even 6
1120.2.bz.e.271.4 24 8.3 odd 2
1120.2.bz.e.591.4 24 7.3 odd 6
1120.2.bz.f.271.4 24 1.1 even 1 trivial
1120.2.bz.f.591.4 24 56.3 even 6 inner