Properties

Label 349.2.b.b.348.14
Level $349$
Weight $2$
Character 349.348
Analytic conductor $2.787$
Analytic rank $0$
Dimension $26$
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] = [349,2,Mod(348,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.348");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 349.b (of order \(2\), degree \(1\), 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: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 348.14
Character \(\chi\) \(=\) 349.348
Dual form 349.2.b.b.348.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.719982i q^{2} +1.92112 q^{3} +1.48163 q^{4} +0.227297 q^{5} +1.38317i q^{6} -2.96485i q^{7} +2.50671i q^{8} +0.690697 q^{9} +O(q^{10})\) \(q+0.719982i q^{2} +1.92112 q^{3} +1.48163 q^{4} +0.227297 q^{5} +1.38317i q^{6} -2.96485i q^{7} +2.50671i q^{8} +0.690697 q^{9} +0.163649i q^{10} +3.61680i q^{11} +2.84638 q^{12} -2.92669i q^{13} +2.13464 q^{14} +0.436664 q^{15} +1.15847 q^{16} -1.57443 q^{17} +0.497289i q^{18} +1.16844 q^{19} +0.336769 q^{20} -5.69583i q^{21} -2.60403 q^{22} -5.51662 q^{23} +4.81568i q^{24} -4.94834 q^{25} +2.10716 q^{26} -4.43645 q^{27} -4.39280i q^{28} +7.55239 q^{29} +0.314390i q^{30} +0.298625 q^{31} +5.84749i q^{32} +6.94829i q^{33} -1.13356i q^{34} -0.673900i q^{35} +1.02335 q^{36} -1.62168 q^{37} +0.841253i q^{38} -5.62251i q^{39} +0.569766i q^{40} -2.90221 q^{41} +4.10089 q^{42} +5.41694i q^{43} +5.35874i q^{44} +0.156993 q^{45} -3.97187i q^{46} -2.69771i q^{47} +2.22556 q^{48} -1.79033 q^{49} -3.56271i q^{50} -3.02466 q^{51} -4.33626i q^{52} +6.41377i q^{53} -3.19416i q^{54} +0.822085i q^{55} +7.43201 q^{56} +2.24470 q^{57} +5.43758i q^{58} -5.21837i q^{59} +0.646973 q^{60} +2.01219i q^{61} +0.215004i q^{62} -2.04781i q^{63} -1.89314 q^{64} -0.665226i q^{65} -5.00264 q^{66} +2.81322 q^{67} -2.33271 q^{68} -10.5981 q^{69} +0.485196 q^{70} -6.35775i q^{71} +1.73137i q^{72} -13.8123 q^{73} -1.16758i q^{74} -9.50634 q^{75} +1.73119 q^{76} +10.7233 q^{77} +4.04810 q^{78} +1.13066i q^{79} +0.263317 q^{80} -10.5950 q^{81} -2.08954i q^{82} +6.82981 q^{83} -8.43909i q^{84} -0.357862 q^{85} -3.90010 q^{86} +14.5090 q^{87} -9.06624 q^{88} -0.406795i q^{89} +0.113032i q^{90} -8.67718 q^{91} -8.17358 q^{92} +0.573694 q^{93} +1.94230 q^{94} +0.265582 q^{95} +11.2337i q^{96} -8.19259i q^{97} -1.28900i q^{98} +2.49811i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 2 q^{3} - 36 q^{4} - 12 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 26 q - 2 q^{3} - 36 q^{4} - 12 q^{5} + 32 q^{9} + 12 q^{12} + 4 q^{14} + 12 q^{15} + 20 q^{16} - 14 q^{17} - 4 q^{19} - 2 q^{20} - 12 q^{22} - 18 q^{23} + 18 q^{25} + 22 q^{26} + 4 q^{27} - 18 q^{29} + 10 q^{31} - 54 q^{36} + 30 q^{37} - 16 q^{41} - 44 q^{45} - 74 q^{48} - 22 q^{49} + 32 q^{51} - 38 q^{56} - 16 q^{57} - 78 q^{60} - 96 q^{64} + 104 q^{66} + 72 q^{67} + 36 q^{68} - 40 q^{69} + 86 q^{70} + 72 q^{73} - 38 q^{75} + 96 q^{76} - 28 q^{77} - 30 q^{78} + 30 q^{80} - 6 q^{81} - 8 q^{83} - 22 q^{85} + 60 q^{86} + 32 q^{87} + 110 q^{88} - 12 q^{91} + 14 q^{92} + 84 q^{93} + 22 q^{94} - 10 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.719982i 0.509104i 0.967059 + 0.254552i \(0.0819280\pi\)
−0.967059 + 0.254552i \(0.918072\pi\)
\(3\) 1.92112 1.10916 0.554579 0.832131i \(-0.312879\pi\)
0.554579 + 0.832131i \(0.312879\pi\)
\(4\) 1.48163 0.740813
\(5\) 0.227297 0.101650 0.0508251 0.998708i \(-0.483815\pi\)
0.0508251 + 0.998708i \(0.483815\pi\)
\(6\) 1.38317i 0.564677i
\(7\) 2.96485i 1.12061i −0.828287 0.560304i \(-0.810684\pi\)
0.828287 0.560304i \(-0.189316\pi\)
\(8\) 2.50671i 0.886255i
\(9\) 0.690697 0.230232
\(10\) 0.163649i 0.0517505i
\(11\) 3.61680i 1.09050i 0.838272 + 0.545252i \(0.183566\pi\)
−0.838272 + 0.545252i \(0.816434\pi\)
\(12\) 2.84638 0.821679
\(13\) 2.92669i 0.811717i −0.913936 0.405858i \(-0.866973\pi\)
0.913936 0.405858i \(-0.133027\pi\)
\(14\) 2.13464 0.570506
\(15\) 0.436664 0.112746
\(16\) 1.15847 0.289618
\(17\) −1.57443 −0.381855 −0.190927 0.981604i \(-0.561150\pi\)
−0.190927 + 0.981604i \(0.561150\pi\)
\(18\) 0.497289i 0.117212i
\(19\) 1.16844 0.268058 0.134029 0.990977i \(-0.457209\pi\)
0.134029 + 0.990977i \(0.457209\pi\)
\(20\) 0.336769 0.0753038
\(21\) 5.69583i 1.24293i
\(22\) −2.60403 −0.555180
\(23\) −5.51662 −1.15030 −0.575148 0.818050i \(-0.695055\pi\)
−0.575148 + 0.818050i \(0.695055\pi\)
\(24\) 4.81568i 0.982997i
\(25\) −4.94834 −0.989667
\(26\) 2.10716 0.413248
\(27\) −4.43645 −0.853794
\(28\) 4.39280i 0.830161i
\(29\) 7.55239 1.40244 0.701221 0.712944i \(-0.252638\pi\)
0.701221 + 0.712944i \(0.252638\pi\)
\(30\) 0.314390i 0.0573995i
\(31\) 0.298625 0.0536346 0.0268173 0.999640i \(-0.491463\pi\)
0.0268173 + 0.999640i \(0.491463\pi\)
\(32\) 5.84749i 1.03370i
\(33\) 6.94829i 1.20954i
\(34\) 1.13356i 0.194404i
\(35\) 0.673900i 0.113910i
\(36\) 1.02335 0.170559
\(37\) −1.62168 −0.266603 −0.133301 0.991076i \(-0.542558\pi\)
−0.133301 + 0.991076i \(0.542558\pi\)
\(38\) 0.841253i 0.136469i
\(39\) 5.62251i 0.900322i
\(40\) 0.569766i 0.0900879i
\(41\) −2.90221 −0.453249 −0.226625 0.973982i \(-0.572769\pi\)
−0.226625 + 0.973982i \(0.572769\pi\)
\(42\) 4.10089 0.632781
\(43\) 5.41694i 0.826076i 0.910714 + 0.413038i \(0.135532\pi\)
−0.910714 + 0.413038i \(0.864468\pi\)
\(44\) 5.35874i 0.807860i
\(45\) 0.156993 0.0234031
\(46\) 3.97187i 0.585620i
\(47\) 2.69771i 0.393501i −0.980454 0.196751i \(-0.936961\pi\)
0.980454 0.196751i \(-0.0630390\pi\)
\(48\) 2.22556 0.321232
\(49\) −1.79033 −0.255761
\(50\) 3.56271i 0.503843i
\(51\) −3.02466 −0.423537
\(52\) 4.33626i 0.601330i
\(53\) 6.41377i 0.880999i 0.897753 + 0.440500i \(0.145199\pi\)
−0.897753 + 0.440500i \(0.854801\pi\)
\(54\) 3.19416i 0.434670i
\(55\) 0.822085i 0.110850i
\(56\) 7.43201 0.993144
\(57\) 2.24470 0.297318
\(58\) 5.43758i 0.713989i
\(59\) 5.21837i 0.679374i −0.940539 0.339687i \(-0.889679\pi\)
0.940539 0.339687i \(-0.110321\pi\)
\(60\) 0.646973 0.0835238
\(61\) 2.01219i 0.257634i 0.991668 + 0.128817i \(0.0411180\pi\)
−0.991668 + 0.128817i \(0.958882\pi\)
\(62\) 0.215004i 0.0273056i
\(63\) 2.04781i 0.258000i
\(64\) −1.89314 −0.236643
\(65\) 0.665226i 0.0825111i
\(66\) −5.00264 −0.615783
\(67\) 2.81322 0.343689 0.171845 0.985124i \(-0.445027\pi\)
0.171845 + 0.985124i \(0.445027\pi\)
\(68\) −2.33271 −0.282883
\(69\) −10.5981 −1.27586
\(70\) 0.485196 0.0579920
\(71\) 6.35775i 0.754526i −0.926106 0.377263i \(-0.876865\pi\)
0.926106 0.377263i \(-0.123135\pi\)
\(72\) 1.73137i 0.204044i
\(73\) −13.8123 −1.61661 −0.808304 0.588766i \(-0.799614\pi\)
−0.808304 + 0.588766i \(0.799614\pi\)
\(74\) 1.16758i 0.135728i
\(75\) −9.50634 −1.09770
\(76\) 1.73119 0.198581
\(77\) 10.7233 1.22203
\(78\) 4.04810 0.458357
\(79\) 1.13066i 0.127209i 0.997975 + 0.0636044i \(0.0202596\pi\)
−0.997975 + 0.0636044i \(0.979740\pi\)
\(80\) 0.263317 0.0294397
\(81\) −10.5950 −1.17723
\(82\) 2.08954i 0.230751i
\(83\) 6.82981 0.749669 0.374834 0.927092i \(-0.377700\pi\)
0.374834 + 0.927092i \(0.377700\pi\)
\(84\) 8.43909i 0.920780i
\(85\) −0.357862 −0.0388156
\(86\) −3.90010 −0.420558
\(87\) 14.5090 1.55553
\(88\) −9.06624 −0.966465
\(89\) 0.406795i 0.0431202i −0.999768 0.0215601i \(-0.993137\pi\)
0.999768 0.0215601i \(-0.00686333\pi\)
\(90\) 0.113032i 0.0119146i
\(91\) −8.67718 −0.909616
\(92\) −8.17358 −0.852154
\(93\) 0.573694 0.0594893
\(94\) 1.94230 0.200333
\(95\) 0.265582 0.0272481
\(96\) 11.2337i 1.14654i
\(97\) 8.19259i 0.831831i −0.909403 0.415916i \(-0.863461\pi\)
0.909403 0.415916i \(-0.136539\pi\)
\(98\) 1.28900i 0.130209i
\(99\) 2.49811i 0.251069i
\(100\) −7.33159 −0.733159
\(101\) 13.2009i 1.31354i −0.754091 0.656770i \(-0.771922\pi\)
0.754091 0.656770i \(-0.228078\pi\)
\(102\) 2.17770i 0.215625i
\(103\) 20.2631i 1.99658i −0.0584352 0.998291i \(-0.518611\pi\)
0.0584352 0.998291i \(-0.481389\pi\)
\(104\) 7.33634 0.719388
\(105\) 1.29464i 0.126344i
\(106\) −4.61780 −0.448520
\(107\) 11.9439i 1.15466i −0.816511 0.577331i \(-0.804094\pi\)
0.816511 0.577331i \(-0.195906\pi\)
\(108\) −6.57316 −0.632502
\(109\) 4.55295 0.436094 0.218047 0.975938i \(-0.430031\pi\)
0.218047 + 0.975938i \(0.430031\pi\)
\(110\) −0.591886 −0.0564341
\(111\) −3.11544 −0.295705
\(112\) 3.43469i 0.324548i
\(113\) 6.06748i 0.570780i 0.958411 + 0.285390i \(0.0921232\pi\)
−0.958411 + 0.285390i \(0.907877\pi\)
\(114\) 1.61615i 0.151366i
\(115\) −1.25391 −0.116928
\(116\) 11.1898 1.03895
\(117\) 2.02145i 0.186883i
\(118\) 3.75713 0.345872
\(119\) 4.66794i 0.427909i
\(120\) 1.09459i 0.0999218i
\(121\) −2.08121 −0.189201
\(122\) −1.44874 −0.131163
\(123\) −5.57549 −0.502725
\(124\) 0.442451 0.0397332
\(125\) −2.26122 −0.202250
\(126\) 1.47439 0.131349
\(127\) 19.2080i 1.70443i 0.523190 + 0.852216i \(0.324742\pi\)
−0.523190 + 0.852216i \(0.675258\pi\)
\(128\) 10.3320i 0.913224i
\(129\) 10.4066i 0.916249i
\(130\) 0.478950 0.0420067
\(131\) 16.6511i 1.45481i 0.686208 + 0.727406i \(0.259274\pi\)
−0.686208 + 0.727406i \(0.740726\pi\)
\(132\) 10.2948i 0.896045i
\(133\) 3.46424i 0.300388i
\(134\) 2.02546i 0.174973i
\(135\) −1.00839 −0.0867883
\(136\) 3.94663i 0.338421i
\(137\) 0.220208i 0.0188137i 0.999956 + 0.00940683i \(0.00299433\pi\)
−0.999956 + 0.00940683i \(0.997006\pi\)
\(138\) 7.63043i 0.649545i
\(139\) 16.5090 1.40027 0.700137 0.714009i \(-0.253122\pi\)
0.700137 + 0.714009i \(0.253122\pi\)
\(140\) 0.998469i 0.0843860i
\(141\) 5.18262i 0.436455i
\(142\) 4.57746 0.384132
\(143\) 10.5852 0.885181
\(144\) 0.800152 0.0666793
\(145\) 1.71663 0.142559
\(146\) 9.94460i 0.823021i
\(147\) −3.43944 −0.283680
\(148\) −2.40273 −0.197503
\(149\) 2.83961i 0.232630i 0.993212 + 0.116315i \(0.0371081\pi\)
−0.993212 + 0.116315i \(0.962892\pi\)
\(150\) 6.84439i 0.558842i
\(151\) 2.71918 0.221284 0.110642 0.993860i \(-0.464709\pi\)
0.110642 + 0.993860i \(0.464709\pi\)
\(152\) 2.92893i 0.237567i
\(153\) −1.08745 −0.0879153
\(154\) 7.72054i 0.622139i
\(155\) 0.0678764 0.00545197
\(156\) 8.33046i 0.666971i
\(157\) 8.56005 0.683166 0.341583 0.939852i \(-0.389037\pi\)
0.341583 + 0.939852i \(0.389037\pi\)
\(158\) −0.814052 −0.0647625
\(159\) 12.3216i 0.977168i
\(160\) 1.32912i 0.105076i
\(161\) 16.3560i 1.28903i
\(162\) 7.62822i 0.599330i
\(163\) 12.6066i 0.987427i −0.869625 0.493713i \(-0.835639\pi\)
0.869625 0.493713i \(-0.164361\pi\)
\(164\) −4.30000 −0.335773
\(165\) 1.57932i 0.122950i
\(166\) 4.91734i 0.381659i
\(167\) 21.2390i 1.64353i 0.569829 + 0.821763i \(0.307010\pi\)
−0.569829 + 0.821763i \(0.692990\pi\)
\(168\) 14.2778 1.10155
\(169\) 4.43451 0.341116
\(170\) 0.257654i 0.0197612i
\(171\) 0.807035 0.0617155
\(172\) 8.02589i 0.611968i
\(173\) 9.78902i 0.744245i −0.928184 0.372123i \(-0.878630\pi\)
0.928184 0.372123i \(-0.121370\pi\)
\(174\) 10.4462i 0.791927i
\(175\) 14.6711i 1.10903i
\(176\) 4.18995i 0.315829i
\(177\) 10.0251i 0.753533i
\(178\) 0.292885 0.0219527
\(179\) 9.70344i 0.725269i 0.931931 + 0.362635i \(0.118123\pi\)
−0.931931 + 0.362635i \(0.881877\pi\)
\(180\) 0.232605 0.0173374
\(181\) 20.2057 1.50188 0.750940 0.660371i \(-0.229601\pi\)
0.750940 + 0.660371i \(0.229601\pi\)
\(182\) 6.24741i 0.463089i
\(183\) 3.86565i 0.285757i
\(184\) 13.8286i 1.01945i
\(185\) −0.368603 −0.0271002
\(186\) 0.413049i 0.0302862i
\(187\) 5.69438i 0.416415i
\(188\) 3.99700i 0.291511i
\(189\) 13.1534i 0.956769i
\(190\) 0.191214i 0.0138721i
\(191\) 2.75632 0.199441 0.0997203 0.995016i \(-0.468205\pi\)
0.0997203 + 0.995016i \(0.468205\pi\)
\(192\) −3.63695 −0.262475
\(193\) 20.5656i 1.48035i −0.672416 0.740173i \(-0.734744\pi\)
0.672416 0.740173i \(-0.265256\pi\)
\(194\) 5.89851 0.423488
\(195\) 1.27798i 0.0915179i
\(196\) −2.65260 −0.189472
\(197\) 22.5289i 1.60512i 0.596571 + 0.802560i \(0.296529\pi\)
−0.596571 + 0.802560i \(0.703471\pi\)
\(198\) −1.79859 −0.127820
\(199\) 15.8099i 1.12074i 0.828244 + 0.560368i \(0.189340\pi\)
−0.828244 + 0.560368i \(0.810660\pi\)
\(200\) 12.4040i 0.877097i
\(201\) 5.40452 0.381206
\(202\) 9.50442 0.668728
\(203\) 22.3917i 1.57159i
\(204\) −4.48142 −0.313762
\(205\) −0.659663 −0.0460729
\(206\) 14.5891 1.01647
\(207\) −3.81031 −0.264835
\(208\) 3.39048i 0.235087i
\(209\) 4.22600i 0.292318i
\(210\) 0.932119 0.0643223
\(211\) 14.9620i 1.03003i 0.857182 + 0.515014i \(0.172213\pi\)
−0.857182 + 0.515014i \(0.827787\pi\)
\(212\) 9.50281i 0.652656i
\(213\) 12.2140i 0.836889i
\(214\) 8.59939 0.587842
\(215\) 1.23125i 0.0839708i
\(216\) 11.1209i 0.756679i
\(217\) 0.885378i 0.0601034i
\(218\) 3.27804i 0.222017i
\(219\) −26.5351 −1.79307
\(220\) 1.21802i 0.0821191i
\(221\) 4.60786i 0.309958i
\(222\) 2.24306i 0.150544i
\(223\) −20.3450 −1.36240 −0.681200 0.732097i \(-0.738542\pi\)
−0.681200 + 0.732097i \(0.738542\pi\)
\(224\) 17.3369 1.15837
\(225\) −3.41780 −0.227853
\(226\) −4.36847 −0.290586
\(227\) 7.36520 0.488846 0.244423 0.969669i \(-0.421401\pi\)
0.244423 + 0.969669i \(0.421401\pi\)
\(228\) 3.32581 0.220257
\(229\) 16.7515i 1.10697i −0.832860 0.553484i \(-0.813298\pi\)
0.832860 0.553484i \(-0.186702\pi\)
\(230\) 0.902792i 0.0595283i
\(231\) 20.6006 1.35542
\(232\) 18.9316i 1.24292i
\(233\) −22.3820 −1.46629 −0.733147 0.680070i \(-0.761949\pi\)
−0.733147 + 0.680070i \(0.761949\pi\)
\(234\) 1.45541 0.0951430
\(235\) 0.613180i 0.0399995i
\(236\) 7.73168i 0.503289i
\(237\) 2.17212i 0.141095i
\(238\) −3.36083 −0.217850
\(239\) 23.2227 1.50215 0.751077 0.660215i \(-0.229535\pi\)
0.751077 + 0.660215i \(0.229535\pi\)
\(240\) 0.505862 0.0326533
\(241\) −21.4469 −1.38151 −0.690757 0.723087i \(-0.742723\pi\)
−0.690757 + 0.723087i \(0.742723\pi\)
\(242\) 1.49843i 0.0963227i
\(243\) −7.04497 −0.451935
\(244\) 2.98131i 0.190859i
\(245\) −0.406936 −0.0259982
\(246\) 4.01425i 0.255939i
\(247\) 3.41965i 0.217587i
\(248\) 0.748565i 0.0475339i
\(249\) 13.1209 0.831501
\(250\) 1.62804i 0.102966i
\(251\) 3.80427i 0.240123i 0.992766 + 0.120062i \(0.0383092\pi\)
−0.992766 + 0.120062i \(0.961691\pi\)
\(252\) 3.03409i 0.191130i
\(253\) 19.9525i 1.25440i
\(254\) −13.8294 −0.867733
\(255\) −0.687496 −0.0430526
\(256\) −11.2251 −0.701569
\(257\) 5.78099 0.360608 0.180304 0.983611i \(-0.442292\pi\)
0.180304 + 0.983611i \(0.442292\pi\)
\(258\) −7.49255 −0.466466
\(259\) 4.80804i 0.298757i
\(260\) 0.985616i 0.0611253i
\(261\) 5.21641 0.322887
\(262\) −11.9885 −0.740650
\(263\) 28.3586 1.74867 0.874334 0.485325i \(-0.161299\pi\)
0.874334 + 0.485325i \(0.161299\pi\)
\(264\) −17.4173 −1.07196
\(265\) 1.45783i 0.0895537i
\(266\) 2.49419 0.152928
\(267\) 0.781502i 0.0478271i
\(268\) 4.16814 0.254609
\(269\) −14.7707 −0.900588 −0.450294 0.892880i \(-0.648681\pi\)
−0.450294 + 0.892880i \(0.648681\pi\)
\(270\) 0.726022i 0.0441843i
\(271\) 12.3384 0.749502 0.374751 0.927125i \(-0.377728\pi\)
0.374751 + 0.927125i \(0.377728\pi\)
\(272\) −1.82393 −0.110592
\(273\) −16.6699 −1.00891
\(274\) −0.158546 −0.00957810
\(275\) 17.8971i 1.07924i
\(276\) −15.7024 −0.945174
\(277\) 13.3371i 0.801350i 0.916220 + 0.400675i \(0.131224\pi\)
−0.916220 + 0.400675i \(0.868776\pi\)
\(278\) 11.8862i 0.712885i
\(279\) 0.206259 0.0123484
\(280\) 1.68927 0.100953
\(281\) −6.82202 −0.406967 −0.203484 0.979078i \(-0.565226\pi\)
−0.203484 + 0.979078i \(0.565226\pi\)
\(282\) 3.73139 0.222201
\(283\) 14.9414 0.888177 0.444088 0.895983i \(-0.353528\pi\)
0.444088 + 0.895983i \(0.353528\pi\)
\(284\) 9.41981i 0.558963i
\(285\) 0.510214 0.0302225
\(286\) 7.62116i 0.450649i
\(287\) 8.60462i 0.507915i
\(288\) 4.03884i 0.237991i
\(289\) −14.5212 −0.854187
\(290\) 1.23594i 0.0725771i
\(291\) 15.7389i 0.922633i
\(292\) −20.4647 −1.19760
\(293\) −17.2715 −1.00901 −0.504506 0.863408i \(-0.668325\pi\)
−0.504506 + 0.863408i \(0.668325\pi\)
\(294\) 2.47633i 0.144423i
\(295\) 1.18612i 0.0690585i
\(296\) 4.06508i 0.236278i
\(297\) 16.0457i 0.931067i
\(298\) −2.04446 −0.118433
\(299\) 16.1454i 0.933714i
\(300\) −14.0848 −0.813189
\(301\) 16.0604 0.925707
\(302\) 1.95776i 0.112656i
\(303\) 25.3605i 1.45692i
\(304\) 1.35360 0.0776343
\(305\) 0.457364i 0.0261886i
\(306\) 0.782945i 0.0447580i
\(307\) 26.9626 1.53884 0.769418 0.638746i \(-0.220547\pi\)
0.769418 + 0.638746i \(0.220547\pi\)
\(308\) 15.8879 0.905295
\(309\) 38.9278i 2.21453i
\(310\) 0.0488698i 0.00277562i
\(311\) 13.8831i 0.787240i 0.919273 + 0.393620i \(0.128777\pi\)
−0.919273 + 0.393620i \(0.871223\pi\)
\(312\) 14.0940 0.797915
\(313\) 12.4262 0.702371 0.351185 0.936306i \(-0.385779\pi\)
0.351185 + 0.936306i \(0.385779\pi\)
\(314\) 6.16308i 0.347802i
\(315\) 0.465461i 0.0262257i
\(316\) 1.67521i 0.0942380i
\(317\) 32.8113i 1.84287i −0.388534 0.921434i \(-0.627019\pi\)
0.388534 0.921434i \(-0.372981\pi\)
\(318\) −8.87133 −0.497480
\(319\) 27.3154i 1.52937i
\(320\) −0.430305 −0.0240548
\(321\) 22.9457i 1.28070i
\(322\) −11.7760 −0.656250
\(323\) −1.83962 −0.102359
\(324\) −15.6979 −0.872104
\(325\) 14.4822i 0.803329i
\(326\) 9.07653 0.502703
\(327\) 8.74676 0.483697
\(328\) 7.27500i 0.401694i
\(329\) −7.99830 −0.440961
\(330\) −1.13708 −0.0625944
\(331\) 14.6238i 0.803797i −0.915684 0.401898i \(-0.868350\pi\)
0.915684 0.401898i \(-0.131650\pi\)
\(332\) 10.1192 0.555365
\(333\) −1.12009 −0.0613805
\(334\) −15.2917 −0.836726
\(335\) 0.639435 0.0349361
\(336\) 6.59845i 0.359975i
\(337\) 15.0233 0.818370 0.409185 0.912451i \(-0.365813\pi\)
0.409185 + 0.912451i \(0.365813\pi\)
\(338\) 3.19277i 0.173664i
\(339\) 11.6563i 0.633086i
\(340\) −0.530218 −0.0287551
\(341\) 1.08007i 0.0584888i
\(342\) 0.581050i 0.0314196i
\(343\) 15.4459i 0.833999i
\(344\) −13.5787 −0.732114
\(345\) −2.40891 −0.129691
\(346\) 7.04791 0.378898
\(347\) 8.69561i 0.466805i 0.972380 + 0.233402i \(0.0749859\pi\)
−0.972380 + 0.233402i \(0.925014\pi\)
\(348\) 21.4970 1.15236
\(349\) −16.4065 + 8.93451i −0.878222 + 0.478253i
\(350\) −10.5629 −0.564611
\(351\) 12.9841i 0.693039i
\(352\) −21.1492 −1.12725
\(353\) −34.4982 −1.83616 −0.918078 0.396401i \(-0.870259\pi\)
−0.918078 + 0.396401i \(0.870259\pi\)
\(354\) 7.21789 0.383627
\(355\) 1.44509i 0.0766977i
\(356\) 0.602719i 0.0319440i
\(357\) 8.96767i 0.474619i
\(358\) −6.98630 −0.369237
\(359\) 18.6713i 0.985436i −0.870189 0.492718i \(-0.836003\pi\)
0.870189 0.492718i \(-0.163997\pi\)
\(360\) 0.393535i 0.0207411i
\(361\) −17.6348 −0.928145
\(362\) 14.5477i 0.764612i
\(363\) −3.99824 −0.209853
\(364\) −12.8563 −0.673855
\(365\) −3.13949 −0.164328
\(366\) −2.78320 −0.145480
\(367\) 17.5618i 0.916717i 0.888767 + 0.458358i \(0.151562\pi\)
−0.888767 + 0.458358i \(0.848438\pi\)
\(368\) −6.39085 −0.333146
\(369\) −2.00455 −0.104353
\(370\) 0.265387i 0.0137968i
\(371\) 19.0159 0.987254
\(372\) 0.850000 0.0440704
\(373\) 26.0464i 1.34863i 0.738443 + 0.674316i \(0.235562\pi\)
−0.738443 + 0.674316i \(0.764438\pi\)
\(374\) 4.09985 0.211998
\(375\) −4.34408 −0.224327
\(376\) 6.76237 0.348742
\(377\) 22.1035i 1.13839i
\(378\) −9.47020 −0.487094
\(379\) 21.4445i 1.10153i 0.834660 + 0.550765i \(0.185664\pi\)
−0.834660 + 0.550765i \(0.814336\pi\)
\(380\) 0.393493 0.0201858
\(381\) 36.9008i 1.89048i
\(382\) 1.98450i 0.101536i
\(383\) 11.0196i 0.563074i 0.959550 + 0.281537i \(0.0908442\pi\)
−0.959550 + 0.281537i \(0.909156\pi\)
\(384\) 19.8489i 1.01291i
\(385\) 2.43736 0.124219
\(386\) 14.8069 0.753650
\(387\) 3.74146i 0.190189i
\(388\) 12.1384i 0.616232i
\(389\) 4.80635i 0.243691i 0.992549 + 0.121846i \(0.0388813\pi\)
−0.992549 + 0.121846i \(0.961119\pi\)
\(390\) 0.920120 0.0465921
\(391\) 8.68553 0.439246
\(392\) 4.48783i 0.226670i
\(393\) 31.9887i 1.61362i
\(394\) −16.2204 −0.817173
\(395\) 0.256994i 0.0129308i
\(396\) 3.70126i 0.185995i
\(397\) −4.67721 −0.234742 −0.117371 0.993088i \(-0.537447\pi\)
−0.117371 + 0.993088i \(0.537447\pi\)
\(398\) −11.3828 −0.570571
\(399\) 6.65521i 0.333177i
\(400\) −5.73250 −0.286625
\(401\) 10.4723i 0.522962i −0.965209 0.261481i \(-0.915789\pi\)
0.965209 0.261481i \(-0.0842108\pi\)
\(402\) 3.89116i 0.194073i
\(403\) 0.873981i 0.0435361i
\(404\) 19.5588i 0.973088i
\(405\) −2.40821 −0.119665
\(406\) 16.1216 0.800101
\(407\) 5.86529i 0.290732i
\(408\) 7.58194i 0.375362i
\(409\) 10.3473 0.511643 0.255822 0.966724i \(-0.417654\pi\)
0.255822 + 0.966724i \(0.417654\pi\)
\(410\) 0.474945i 0.0234559i
\(411\) 0.423046i 0.0208673i
\(412\) 30.0223i 1.47909i
\(413\) −15.4717 −0.761312
\(414\) 2.74335i 0.134829i
\(415\) 1.55239 0.0762040
\(416\) 17.1138 0.839072
\(417\) 31.7157 1.55313
\(418\) −3.04264 −0.148820
\(419\) 17.4860 0.854245 0.427123 0.904194i \(-0.359527\pi\)
0.427123 + 0.904194i \(0.359527\pi\)
\(420\) 1.91818i 0.0935974i
\(421\) 3.35844i 0.163680i −0.996645 0.0818401i \(-0.973920\pi\)
0.996645 0.0818401i \(-0.0260797\pi\)
\(422\) −10.7724 −0.524391
\(423\) 1.86330i 0.0905967i
\(424\) −16.0774 −0.780790
\(425\) 7.79080 0.377909
\(426\) 8.79384 0.426063
\(427\) 5.96583 0.288707
\(428\) 17.6964i 0.855388i
\(429\) 20.3355 0.981806
\(430\) −0.886479 −0.0427498
\(431\) 15.5444i 0.748748i −0.927278 0.374374i \(-0.877858\pi\)
0.927278 0.374374i \(-0.122142\pi\)
\(432\) −5.13949 −0.247274
\(433\) 4.89134i 0.235063i −0.993069 0.117531i \(-0.962502\pi\)
0.993069 0.117531i \(-0.0374980\pi\)
\(434\) 0.637456 0.0305988
\(435\) 3.29785 0.158120
\(436\) 6.74577 0.323064
\(437\) −6.44582 −0.308346
\(438\) 19.1048i 0.912861i
\(439\) 1.35921i 0.0648714i −0.999474 0.0324357i \(-0.989674\pi\)
0.999474 0.0324357i \(-0.0103264\pi\)
\(440\) −2.06073 −0.0982413
\(441\) −1.23657 −0.0588845
\(442\) −3.31757 −0.157801
\(443\) −16.0173 −0.761005 −0.380503 0.924780i \(-0.624249\pi\)
−0.380503 + 0.924780i \(0.624249\pi\)
\(444\) −4.61592 −0.219062
\(445\) 0.0924632i 0.00438318i
\(446\) 14.6480i 0.693603i
\(447\) 5.45522i 0.258023i
\(448\) 5.61289i 0.265184i
\(449\) −6.76249 −0.319141 −0.159571 0.987187i \(-0.551011\pi\)
−0.159571 + 0.987187i \(0.551011\pi\)
\(450\) 2.46075i 0.116001i
\(451\) 10.4967i 0.494271i
\(452\) 8.98973i 0.422842i
\(453\) 5.22386 0.245439
\(454\) 5.30281i 0.248873i
\(455\) −1.97229 −0.0924626
\(456\) 5.62682i 0.263500i
\(457\) −2.63584 −0.123300 −0.0616498 0.998098i \(-0.519636\pi\)
−0.0616498 + 0.998098i \(0.519636\pi\)
\(458\) 12.0607 0.563561
\(459\) 6.98486 0.326026
\(460\) −1.85783 −0.0866216
\(461\) 10.5486i 0.491296i 0.969359 + 0.245648i \(0.0790008\pi\)
−0.969359 + 0.245648i \(0.920999\pi\)
\(462\) 14.8321i 0.690051i
\(463\) 6.55200i 0.304497i −0.988342 0.152249i \(-0.951349\pi\)
0.988342 0.152249i \(-0.0486514\pi\)
\(464\) 8.74922 0.406172
\(465\) 0.130399 0.00604709
\(466\) 16.1146i 0.746496i
\(467\) 10.0296 0.464116 0.232058 0.972702i \(-0.425454\pi\)
0.232058 + 0.972702i \(0.425454\pi\)
\(468\) 2.99504i 0.138446i
\(469\) 8.34076i 0.385141i
\(470\) 0.441478 0.0203639
\(471\) 16.4449 0.757739
\(472\) 13.0809 0.602098
\(473\) −19.5920 −0.900840
\(474\) −1.56389 −0.0718318
\(475\) −5.78182 −0.265288
\(476\) 6.91615i 0.317001i
\(477\) 4.42997i 0.202834i
\(478\) 16.7199i 0.764752i
\(479\) 21.8304 0.997458 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(480\) 2.55339i 0.116546i
\(481\) 4.74615i 0.216406i
\(482\) 15.4413i 0.703334i
\(483\) 31.4217i 1.42974i
\(484\) −3.08357 −0.140162
\(485\) 1.86215i 0.0845558i
\(486\) 5.07225i 0.230082i
\(487\) 26.4902i 1.20038i −0.799856 0.600192i \(-0.795091\pi\)
0.799856 0.600192i \(-0.204909\pi\)
\(488\) −5.04397 −0.228330
\(489\) 24.2188i 1.09521i
\(490\) 0.292986i 0.0132358i
\(491\) 7.03673 0.317563 0.158782 0.987314i \(-0.449243\pi\)
0.158782 + 0.987314i \(0.449243\pi\)
\(492\) −8.26080 −0.372426
\(493\) −11.8907 −0.535530
\(494\) 2.46208 0.110774
\(495\) 0.567812i 0.0255212i
\(496\) 0.345948 0.0155335
\(497\) −18.8498 −0.845527
\(498\) 9.44679i 0.423321i
\(499\) 18.9065i 0.846370i −0.906043 0.423185i \(-0.860912\pi\)
0.906043 0.423185i \(-0.139088\pi\)
\(500\) −3.35029 −0.149829
\(501\) 40.8027i 1.82293i
\(502\) −2.73900 −0.122248
\(503\) 23.0817i 1.02916i −0.857442 0.514581i \(-0.827948\pi\)
0.857442 0.514581i \(-0.172052\pi\)
\(504\) 5.13326 0.228654
\(505\) 3.00052i 0.133522i
\(506\) 14.3654 0.638621
\(507\) 8.51922 0.378352
\(508\) 28.4590i 1.26267i
\(509\) 16.8514i 0.746925i 0.927645 + 0.373463i \(0.121830\pi\)
−0.927645 + 0.373463i \(0.878170\pi\)
\(510\) 0.494984i 0.0219183i
\(511\) 40.9514i 1.81158i
\(512\) 12.5820i 0.556053i
\(513\) −5.18371 −0.228866
\(514\) 4.16221i 0.183587i
\(515\) 4.60573i 0.202953i
\(516\) 15.4187i 0.678770i
\(517\) 9.75706 0.429115
\(518\) −3.46170 −0.152098
\(519\) 18.8059i 0.825486i
\(520\) 1.66753 0.0731259
\(521\) 13.5854i 0.595189i −0.954692 0.297595i \(-0.903816\pi\)
0.954692 0.297595i \(-0.0961844\pi\)
\(522\) 3.75572i 0.164383i
\(523\) 25.0563i 1.09563i −0.836598 0.547817i \(-0.815459\pi\)
0.836598 0.547817i \(-0.184541\pi\)
\(524\) 24.6707i 1.07774i
\(525\) 28.1849i 1.23009i
\(526\) 20.4177i 0.890254i
\(527\) −0.470163 −0.0204806
\(528\) 8.04939i 0.350305i
\(529\) 7.43314 0.323180
\(530\) −1.04961 −0.0455921
\(531\) 3.60431i 0.156414i
\(532\) 5.13271i 0.222531i
\(533\) 8.49386i 0.367910i
\(534\) 0.562667 0.0243490
\(535\) 2.71481i 0.117371i
\(536\) 7.05191i 0.304596i
\(537\) 18.6415i 0.804439i
\(538\) 10.6347i 0.458493i
\(539\) 6.47526i 0.278909i
\(540\) −1.49406 −0.0642939
\(541\) 9.12575 0.392347 0.196173 0.980569i \(-0.437148\pi\)
0.196173 + 0.980569i \(0.437148\pi\)
\(542\) 8.88340i 0.381574i
\(543\) 38.8176 1.66582
\(544\) 9.20645i 0.394723i
\(545\) 1.03487 0.0443290
\(546\) 12.0020i 0.513639i
\(547\) 17.3348 0.741182 0.370591 0.928796i \(-0.379155\pi\)
0.370591 + 0.928796i \(0.379155\pi\)
\(548\) 0.326266i 0.0139374i
\(549\) 1.38981i 0.0593157i
\(550\) 12.8856 0.549444
\(551\) 8.82448 0.375936
\(552\) 26.5663i 1.13074i
\(553\) 3.35223 0.142551
\(554\) −9.60248 −0.407970
\(555\) −0.708130 −0.0300584
\(556\) 24.4601 1.03734
\(557\) 4.76111i 0.201735i −0.994900 0.100867i \(-0.967838\pi\)
0.994900 0.100867i \(-0.0321618\pi\)
\(558\) 0.148503i 0.00628662i
\(559\) 15.8537 0.670540
\(560\) 0.780694i 0.0329903i
\(561\) 10.9396i 0.461870i
\(562\) 4.91173i 0.207189i
\(563\) −38.3244 −1.61518 −0.807591 0.589743i \(-0.799229\pi\)
−0.807591 + 0.589743i \(0.799229\pi\)
\(564\) 7.67871i 0.323332i
\(565\) 1.37912i 0.0580199i
\(566\) 10.7576i 0.452174i
\(567\) 31.4127i 1.31921i
\(568\) 15.9370 0.668702
\(569\) 26.9667i 1.13050i −0.824919 0.565251i \(-0.808779\pi\)
0.824919 0.565251i \(-0.191221\pi\)
\(570\) 0.367345i 0.0153864i
\(571\) 44.8884i 1.87852i 0.343207 + 0.939260i \(0.388487\pi\)
−0.343207 + 0.939260i \(0.611513\pi\)
\(572\) 15.6833 0.655754
\(573\) 5.29523 0.221211
\(574\) −6.19517 −0.258581
\(575\) 27.2981 1.13841
\(576\) −1.30759 −0.0544828
\(577\) 6.68314 0.278223 0.139111 0.990277i \(-0.455575\pi\)
0.139111 + 0.990277i \(0.455575\pi\)
\(578\) 10.4550i 0.434870i
\(579\) 39.5090i 1.64194i
\(580\) 2.54341 0.105609
\(581\) 20.2494i 0.840085i
\(582\) 11.3317 0.469716
\(583\) −23.1973 −0.960734
\(584\) 34.6234i 1.43273i
\(585\) 0.459469i 0.0189967i
\(586\) 12.4352i 0.513692i
\(587\) −17.9833 −0.742250 −0.371125 0.928583i \(-0.621028\pi\)
−0.371125 + 0.928583i \(0.621028\pi\)
\(588\) −5.09596 −0.210154
\(589\) 0.348924 0.0143772
\(590\) 0.853983 0.0351579
\(591\) 43.2808i 1.78033i
\(592\) −1.87867 −0.0772129
\(593\) 33.6245i 1.38079i −0.723432 0.690396i \(-0.757436\pi\)
0.723432 0.690396i \(-0.242564\pi\)
\(594\) 11.5526 0.474010
\(595\) 1.06101i 0.0434971i
\(596\) 4.20724i 0.172335i
\(597\) 30.3727i 1.24307i
\(598\) −11.6244 −0.475357
\(599\) 21.5246i 0.879473i 0.898127 + 0.439736i \(0.144928\pi\)
−0.898127 + 0.439736i \(0.855072\pi\)
\(600\) 23.8296i 0.972840i
\(601\) 2.31490i 0.0944267i −0.998885 0.0472134i \(-0.984966\pi\)
0.998885 0.0472134i \(-0.0150341\pi\)
\(602\) 11.5632i 0.471281i
\(603\) 1.94308 0.0791283
\(604\) 4.02881 0.163930
\(605\) −0.473051 −0.0192323
\(606\) 18.2591 0.741726
\(607\) 22.3647 0.907754 0.453877 0.891064i \(-0.350041\pi\)
0.453877 + 0.891064i \(0.350041\pi\)
\(608\) 6.83242i 0.277091i
\(609\) 43.0171i 1.74314i
\(610\) −0.329293 −0.0133327
\(611\) −7.89535 −0.319412
\(612\) −1.61120 −0.0651288
\(613\) 10.9512 0.442314 0.221157 0.975238i \(-0.429017\pi\)
0.221157 + 0.975238i \(0.429017\pi\)
\(614\) 19.4126i 0.783427i
\(615\) −1.26729 −0.0511021
\(616\) 26.8800i 1.08303i
\(617\) −35.1527 −1.41519 −0.707597 0.706616i \(-0.750221\pi\)
−0.707597 + 0.706616i \(0.750221\pi\)
\(618\) 28.0273 1.12742
\(619\) 35.8209i 1.43976i −0.694097 0.719882i \(-0.744196\pi\)
0.694097 0.719882i \(-0.255804\pi\)
\(620\) 0.100568 0.00403889
\(621\) 24.4742 0.982116
\(622\) −9.99560 −0.400787
\(623\) −1.20609 −0.0483208
\(624\) 6.51351i 0.260749i
\(625\) 24.2277 0.969109
\(626\) 8.94664i 0.357580i
\(627\) 8.11864i 0.324227i
\(628\) 12.6828 0.506099
\(629\) 2.55322 0.101804
\(630\) 0.335123 0.0133516
\(631\) −35.9978 −1.43305 −0.716525 0.697561i \(-0.754268\pi\)
−0.716525 + 0.697561i \(0.754268\pi\)
\(632\) −2.83422 −0.112739
\(633\) 28.7438i 1.14246i
\(634\) 23.6236 0.938211
\(635\) 4.36591i 0.173256i
\(636\) 18.2560i 0.723899i
\(637\) 5.23973i 0.207606i
\(638\) −19.6666 −0.778608
\(639\) 4.39127i 0.173716i
\(640\) 2.34842i 0.0928294i
\(641\) 1.58340 0.0625407 0.0312704 0.999511i \(-0.490045\pi\)
0.0312704 + 0.999511i \(0.490045\pi\)
\(642\) 16.5205 0.652010
\(643\) 16.2454i 0.640655i −0.947307 0.320328i \(-0.896207\pi\)
0.947307 0.320328i \(-0.103793\pi\)
\(644\) 24.2334i 0.954931i
\(645\) 2.36538i 0.0931369i
\(646\) 1.32449i 0.0521114i
\(647\) −30.9792 −1.21792 −0.608958 0.793202i \(-0.708412\pi\)
−0.608958 + 0.793202i \(0.708412\pi\)
\(648\) 26.5586i 1.04332i
\(649\) 18.8738 0.740860
\(650\) −10.4269 −0.408978
\(651\) 1.70092i 0.0666641i
\(652\) 18.6783i 0.731499i
\(653\) −29.1806 −1.14192 −0.570962 0.820976i \(-0.693430\pi\)
−0.570962 + 0.820976i \(0.693430\pi\)
\(654\) 6.29750i 0.246252i
\(655\) 3.78473i 0.147882i
\(656\) −3.36213 −0.131269
\(657\) −9.54011 −0.372195
\(658\) 5.75863i 0.224495i
\(659\) 11.2198i 0.437063i 0.975830 + 0.218531i \(0.0701266\pi\)
−0.975830 + 0.218531i \(0.929873\pi\)
\(660\) 2.33997i 0.0910831i
\(661\) 3.44171 0.133867 0.0669336 0.997757i \(-0.478678\pi\)
0.0669336 + 0.997757i \(0.478678\pi\)
\(662\) 10.5289 0.409216
\(663\) 8.85224i 0.343792i
\(664\) 17.1203i 0.664398i
\(665\) 0.787410i 0.0305344i
\(666\) 0.806444i 0.0312491i
\(667\) −41.6637 −1.61322
\(668\) 31.4683i 1.21755i
\(669\) −39.0851 −1.51112
\(670\) 0.460381i 0.0177861i
\(671\) −7.27767 −0.280951
\(672\) 33.3063 1.28482
\(673\) −23.3596 −0.900448 −0.450224 0.892916i \(-0.648656\pi\)
−0.450224 + 0.892916i \(0.648656\pi\)
\(674\) 10.8165i 0.416635i
\(675\) 21.9530 0.844972
\(676\) 6.57029 0.252703
\(677\) 36.8091i 1.41469i −0.706869 0.707345i \(-0.749893\pi\)
0.706869 0.707345i \(-0.250107\pi\)
\(678\) −8.39235 −0.322306
\(679\) −24.2898 −0.932156
\(680\) 0.897056i 0.0344005i
\(681\) 14.1494 0.542207
\(682\) −0.777627 −0.0297769
\(683\) −39.4358 −1.50897 −0.754485 0.656318i \(-0.772113\pi\)
−0.754485 + 0.656318i \(0.772113\pi\)
\(684\) 1.19572 0.0457197
\(685\) 0.0500526i 0.00191241i
\(686\) 11.1208 0.424592
\(687\) 32.1815i 1.22780i
\(688\) 6.27537i 0.239246i
\(689\) 18.7711 0.715122
\(690\) 1.73437i 0.0660264i
\(691\) 45.5476i 1.73271i 0.499427 + 0.866356i \(0.333544\pi\)
−0.499427 + 0.866356i \(0.666456\pi\)
\(692\) 14.5037i 0.551347i
\(693\) 7.40651 0.281350
\(694\) −6.26068 −0.237652
\(695\) 3.75244 0.142338
\(696\) 36.3699i 1.37860i
\(697\) 4.56932 0.173075
\(698\) −6.43268 11.8124i −0.243481 0.447106i
\(699\) −42.9985 −1.62635
\(700\) 21.7370i 0.821583i
\(701\) −40.1238 −1.51546 −0.757728 0.652571i \(-0.773690\pi\)
−0.757728 + 0.652571i \(0.773690\pi\)
\(702\) −9.34830 −0.352829
\(703\) −1.89483 −0.0714649
\(704\) 6.84711i 0.258060i
\(705\) 1.17799i 0.0443657i
\(706\) 24.8381i 0.934794i
\(707\) −39.1387 −1.47196
\(708\) 14.8535i 0.558227i
\(709\) 42.2100i 1.58523i −0.609722 0.792615i \(-0.708719\pi\)
0.609722 0.792615i \(-0.291281\pi\)
\(710\) 1.04044 0.0390471
\(711\) 0.780940i 0.0292876i
\(712\) 1.01972 0.0382155
\(713\) −1.64740 −0.0616957
\(714\) −6.45656 −0.241631
\(715\) 2.40599 0.0899788
\(716\) 14.3769i 0.537289i
\(717\) 44.6136 1.66613
\(718\) 13.4430 0.501689
\(719\) 36.2471i 1.35179i 0.736998 + 0.675894i \(0.236242\pi\)
−0.736998 + 0.675894i \(0.763758\pi\)
\(720\) 0.181872 0.00677796
\(721\) −60.0770 −2.23739
\(722\) 12.6967i 0.472522i
\(723\) −41.2020 −1.53232
\(724\) 29.9373 1.11261
\(725\) −37.3717 −1.38795
\(726\) 2.87866i 0.106837i
\(727\) 47.7872 1.77233 0.886165 0.463370i \(-0.153360\pi\)
0.886165 + 0.463370i \(0.153360\pi\)
\(728\) 21.7511i 0.806151i
\(729\) 18.2509 0.675958
\(730\) 2.26037i 0.0836602i
\(731\) 8.52859i 0.315441i
\(732\) 5.72745i 0.211693i
\(733\) 44.7698i 1.65361i 0.562488 + 0.826805i \(0.309844\pi\)
−0.562488 + 0.826805i \(0.690156\pi\)
\(734\) −12.6441 −0.466704
\(735\) −0.781772 −0.0288361
\(736\) 32.2584i 1.18906i
\(737\) 10.1748i 0.374795i
\(738\) 1.44324i 0.0531263i
\(739\) −43.8350 −1.61250 −0.806249 0.591576i \(-0.798506\pi\)
−0.806249 + 0.591576i \(0.798506\pi\)
\(740\) −0.546132 −0.0200762
\(741\) 6.56955i 0.241338i
\(742\) 13.6911i 0.502615i
\(743\) −20.1179 −0.738054 −0.369027 0.929419i \(-0.620309\pi\)
−0.369027 + 0.929419i \(0.620309\pi\)
\(744\) 1.43808i 0.0527227i
\(745\) 0.645433i 0.0236468i
\(746\) −18.7529 −0.686594
\(747\) 4.71732 0.172598
\(748\) 8.43695i 0.308485i
\(749\) −35.4119 −1.29392
\(750\) 3.12766i 0.114206i
\(751\) 15.1066i 0.551246i −0.961266 0.275623i \(-0.911116\pi\)
0.961266 0.275623i \(-0.0888842\pi\)
\(752\) 3.12522i 0.113965i
\(753\) 7.30845i 0.266335i
\(754\) 15.9141 0.579557
\(755\) 0.618060 0.0224935
\(756\) 19.4884i 0.708787i
\(757\) 42.9205i 1.55997i −0.625796 0.779986i \(-0.715226\pi\)
0.625796 0.779986i \(-0.284774\pi\)
\(758\) −15.4397 −0.560794
\(759\) 38.3311i 1.39133i
\(760\) 0.665735i 0.0241488i
\(761\) 32.6493i 1.18354i −0.806108 0.591768i \(-0.798430\pi\)
0.806108 0.591768i \(-0.201570\pi\)
\(762\) −26.5679 −0.962453
\(763\) 13.4988i 0.488690i
\(764\) 4.08384 0.147748
\(765\) −0.247174 −0.00893660
\(766\) −7.93389 −0.286663
\(767\) −15.2725 −0.551459
\(768\) −21.5648 −0.778151
\(769\) 3.08282i 0.111169i −0.998454 0.0555847i \(-0.982298\pi\)
0.998454 0.0555847i \(-0.0177023\pi\)
\(770\) 1.75485i 0.0632405i
\(771\) 11.1060 0.399972
\(772\) 30.4706i 1.09666i
\(773\) 24.6690 0.887282 0.443641 0.896205i \(-0.353687\pi\)
0.443641 + 0.896205i \(0.353687\pi\)
\(774\) −2.69378 −0.0968261
\(775\) −1.47770 −0.0530804
\(776\) 20.5364 0.737214
\(777\) 9.23682i 0.331369i
\(778\) −3.46048 −0.124064
\(779\) −3.39105 −0.121497
\(780\) 1.89349i 0.0677977i
\(781\) 22.9947 0.822814
\(782\) 6.25342i 0.223622i
\(783\) −33.5057 −1.19740
\(784\) −2.07405 −0.0740730
\(785\) 1.94567 0.0694439
\(786\) −23.0313 −0.821498
\(787\) 9.44552i 0.336696i −0.985728 0.168348i \(-0.946157\pi\)
0.985728 0.168348i \(-0.0538433\pi\)
\(788\) 33.3795i 1.18909i
\(789\) 54.4803 1.93955
\(790\) −0.185031 −0.00658312
\(791\) 17.9892 0.639621
\(792\) −6.26202 −0.222511
\(793\) 5.88904 0.209126
\(794\) 3.36750i 0.119508i
\(795\) 2.80066i 0.0993292i
\(796\) 23.4244i 0.830256i
\(797\) 25.6341i 0.908006i −0.891000 0.454003i \(-0.849996\pi\)
0.891000 0.454003i \(-0.150004\pi\)
\(798\) 4.79163 0.169622
\(799\) 4.24735i 0.150260i
\(800\) 28.9354i 1.02302i
\(801\) 0.280972i 0.00992766i
\(802\) 7.53986 0.266242
\(803\) 49.9563i 1.76292i
\(804\) 8.00749 0.282402
\(805\) 3.71765i 0.131030i
\(806\) 0.629250 0.0221644
\(807\) −28.3764 −0.998895
\(808\) 33.0908 1.16413
\(809\) −25.8850 −0.910068 −0.455034 0.890474i \(-0.650373\pi\)
−0.455034 + 0.890474i \(0.650373\pi\)
\(810\) 1.73387i 0.0609220i
\(811\) 2.52277i 0.0885864i −0.999019 0.0442932i \(-0.985896\pi\)
0.999019 0.0442932i \(-0.0141036\pi\)
\(812\) 33.1761i 1.16425i
\(813\) 23.7035 0.831317
\(814\) 4.22290 0.148013
\(815\) 2.86544i 0.100372i
\(816\) −3.50398 −0.122664
\(817\) 6.32935i 0.221436i
\(818\) 7.44990i 0.260480i
\(819\) −5.99330 −0.209423
\(820\) −0.977374 −0.0341314
\(821\) −32.4133 −1.13123 −0.565616 0.824669i \(-0.691362\pi\)
−0.565616 + 0.824669i \(0.691362\pi\)
\(822\) −0.304585 −0.0106236
\(823\) 20.7188 0.722213 0.361107 0.932525i \(-0.382399\pi\)
0.361107 + 0.932525i \(0.382399\pi\)
\(824\) 50.7936 1.76948
\(825\) 34.3825i 1.19704i
\(826\) 11.1393i 0.387587i
\(827\) 18.2348i 0.634085i −0.948411 0.317043i \(-0.897310\pi\)
0.948411 0.317043i \(-0.102690\pi\)
\(828\) −5.64546 −0.196193
\(829\) 38.0099i 1.32014i −0.751204 0.660070i \(-0.770527\pi\)
0.751204 0.660070i \(-0.229473\pi\)
\(830\) 1.11769i 0.0387957i
\(831\) 25.6222i 0.888824i
\(832\) 5.54064i 0.192087i
\(833\) 2.81875 0.0976638
\(834\) 22.8347i 0.790702i
\(835\) 4.82756i 0.167065i
\(836\) 6.26135i 0.216553i
\(837\) −1.32483 −0.0457929
\(838\) 12.5896i 0.434899i
\(839\) 18.7178i 0.646210i 0.946363 + 0.323105i \(0.104727\pi\)
−0.946363 + 0.323105i \(0.895273\pi\)
\(840\) 3.24529 0.111973
\(841\) 28.0385 0.966846
\(842\) 2.41801 0.0833302
\(843\) −13.1059 −0.451391
\(844\) 22.1681i 0.763058i
\(845\) 1.00795 0.0346745
\(846\) 1.34154 0.0461231
\(847\) 6.17046i 0.212020i
\(848\) 7.43017i 0.255153i
\(849\) 28.7043 0.985128
\(850\) 5.60923i 0.192395i
\(851\) 8.94621 0.306672
\(852\) 18.0966i 0.619978i
\(853\) −22.7361 −0.778468 −0.389234 0.921139i \(-0.627260\pi\)
−0.389234 + 0.921139i \(0.627260\pi\)
\(854\) 4.29529i 0.146982i
\(855\) 0.183436 0.00627339
\(856\) 29.9399 1.02332
\(857\) 37.7022i 1.28788i 0.765074 + 0.643942i \(0.222702\pi\)
−0.765074 + 0.643942i \(0.777298\pi\)
\(858\) 14.6412i 0.499841i
\(859\) 7.14352i 0.243734i 0.992546 + 0.121867i \(0.0388881\pi\)
−0.992546 + 0.121867i \(0.961112\pi\)
\(860\) 1.82426i 0.0622067i
\(861\) 16.5305i 0.563358i
\(862\) 11.1917 0.381190
\(863\) 6.29022i 0.214121i 0.994252 + 0.107061i \(0.0341439\pi\)
−0.994252 + 0.107061i \(0.965856\pi\)
\(864\) 25.9421i 0.882567i
\(865\) 2.22501i 0.0756527i
\(866\) 3.52167 0.119671
\(867\) −27.8969 −0.947428
\(868\) 1.31180i 0.0445254i
\(869\) −4.08935 −0.138722
\(870\) 2.37439i 0.0804995i
\(871\) 8.23340i 0.278978i
\(872\) 11.4129i 0.386490i
\(873\) 5.65859i 0.191514i
\(874\) 4.64087i 0.156980i
\(875\) 6.70419i 0.226643i
\(876\) −39.3151 −1.32833
\(877\) 36.4226i 1.22990i 0.788565 + 0.614951i \(0.210824\pi\)
−0.788565 + 0.614951i \(0.789176\pi\)
\(878\) 0.978604 0.0330263
\(879\) −33.1806 −1.11915
\(880\) 0.952362i 0.0321041i
\(881\) 40.4785i 1.36376i 0.731466 + 0.681878i \(0.238837\pi\)
−0.731466 + 0.681878i \(0.761163\pi\)
\(882\) 0.890311i 0.0299783i
\(883\) −5.99811 −0.201852 −0.100926 0.994894i \(-0.532181\pi\)
−0.100926 + 0.994894i \(0.532181\pi\)
\(884\) 6.82712i 0.229621i
\(885\) 2.27867i 0.0765968i
\(886\) 11.5322i 0.387431i
\(887\) 1.36885i 0.0459616i 0.999736 + 0.0229808i \(0.00731566\pi\)
−0.999736 + 0.0229808i \(0.992684\pi\)
\(888\) 7.80950i 0.262070i
\(889\) 56.9487 1.91000
\(890\) 0.0665718 0.00223149
\(891\) 38.3200i 1.28377i
\(892\) −30.1437 −1.00928
\(893\) 3.15210i 0.105481i
\(894\) −3.92766 −0.131360
\(895\) 2.20556i 0.0737238i
\(896\) 30.6327 1.02337
\(897\) 31.0173i 1.03564i
\(898\) 4.86886i 0.162476i
\(899\) 2.25533 0.0752195
\(900\) −5.06390 −0.168797
\(901\) 10.0980i 0.336414i
\(902\) 7.55744 0.251635
\(903\) 30.8540 1.02676
\(904\) −15.2094 −0.505857
\(905\) 4.59269 0.152666
\(906\) 3.76109i 0.124954i
\(907\) 40.1750i 1.33399i 0.745063 + 0.666994i \(0.232419\pi\)
−0.745063 + 0.666994i \(0.767581\pi\)
\(908\) 10.9125 0.362143
\(909\) 9.11783i 0.302419i
\(910\) 1.42002i 0.0470731i
\(911\) 45.9632i 1.52283i −0.648265 0.761415i \(-0.724505\pi\)
0.648265 0.761415i \(-0.275495\pi\)
\(912\) 2.60043 0.0861087
\(913\) 24.7020i 0.817517i
\(914\) 1.89776i 0.0627723i
\(915\) 0.878650i 0.0290473i
\(916\) 24.8194i 0.820056i
\(917\) 49.3679 1.63027
\(918\) 5.02897i 0.165981i
\(919\) 32.3909i 1.06848i 0.845334 + 0.534238i \(0.179401\pi\)
−0.845334 + 0.534238i \(0.820599\pi\)
\(920\) 3.14318i 0.103628i
\(921\) 51.7983 1.70681
\(922\) −7.59478 −0.250121
\(923\) −18.6071 −0.612461
\(924\) 30.5225 1.00412
\(925\) 8.02463 0.263848
\(926\) 4.71732 0.155021
\(927\) 13.9957i 0.459677i
\(928\) 44.1625i 1.44971i
\(929\) −12.7665 −0.418854 −0.209427 0.977824i \(-0.567160\pi\)
−0.209427 + 0.977824i \(0.567160\pi\)
\(930\) 0.0938846i 0.00307860i
\(931\) −2.09189 −0.0685588
\(932\) −33.1618 −1.08625
\(933\) 26.6711i 0.873174i
\(934\) 7.22115i 0.236283i
\(935\) 1.29431i 0.0423286i
\(936\) 5.06719 0.165626
\(937\) −19.4848 −0.636541 −0.318270 0.948000i \(-0.603102\pi\)
−0.318270 + 0.948000i \(0.603102\pi\)
\(938\) 6.00520 0.196077
\(939\) 23.8722 0.779041
\(940\) 0.908504i 0.0296321i
\(941\) 21.6860 0.706945 0.353472 0.935445i \(-0.385001\pi\)
0.353472 + 0.935445i \(0.385001\pi\)
\(942\) 11.8400i 0.385768i
\(943\) 16.0104 0.521371
\(944\) 6.04533i 0.196759i
\(945\) 2.98972i 0.0972557i
\(946\) 14.1059i 0.458621i
\(947\) −50.6863 −1.64708 −0.823541 0.567257i \(-0.808005\pi\)
−0.823541 + 0.567257i \(0.808005\pi\)
\(948\) 3.21828i 0.104525i
\(949\) 40.4243i 1.31223i
\(950\) 4.16280i 0.135059i
\(951\) 63.0345i 2.04403i
\(952\) −11.7012 −0.379237
\(953\) −0.0601424 −0.00194820 −0.000974102 1.00000i \(-0.500310\pi\)
−0.000974102 1.00000i \(0.500310\pi\)
\(954\) −3.18950 −0.103264
\(955\) 0.626503 0.0202732
\(956\) 34.4074 1.11282
\(957\) 52.4762i 1.69631i
\(958\) 15.7175i 0.507809i
\(959\) 0.652884 0.0210827
\(960\) −0.826667 −0.0266806
\(961\) −30.9108 −0.997123
\(962\) −3.41714 −0.110173
\(963\) 8.24962i 0.265840i
\(964\) −31.7763 −1.02344
\(965\) 4.67450i 0.150477i
\(966\) −22.6231 −0.727885
\(967\) 35.3304 1.13615 0.568074 0.822978i \(-0.307689\pi\)
0.568074 + 0.822978i \(0.307689\pi\)
\(968\) 5.21697i 0.167680i
\(969\) −3.53413 −0.113532
\(970\) 1.34071 0.0430477
\(971\) 4.65749 0.149466 0.0747329 0.997204i \(-0.476190\pi\)
0.0747329 + 0.997204i \(0.476190\pi\)
\(972\) −10.4380 −0.334799
\(973\) 48.9466i 1.56916i
\(974\) 19.0724 0.611120
\(975\) 27.8221i 0.891019i
\(976\) 2.33106i 0.0746155i
\(977\) 42.8303 1.37026 0.685131 0.728420i \(-0.259745\pi\)
0.685131 + 0.728420i \(0.259745\pi\)
\(978\) 17.4371 0.557577
\(979\) 1.47129 0.0470228
\(980\) −0.602927 −0.0192598
\(981\) 3.14471 0.100403
\(982\) 5.06631i 0.161673i
\(983\) 20.5074 0.654084 0.327042 0.945010i \(-0.393948\pi\)
0.327042 + 0.945010i \(0.393948\pi\)
\(984\) 13.9761i 0.445543i
\(985\) 5.12075i 0.163161i
\(986\) 8.56107i 0.272640i
\(987\) −15.3657 −0.489095
\(988\) 5.06664i 0.161191i
\(989\) 29.8832i 0.950232i
\(990\) −0.408814 −0.0129930
\(991\) 44.6137 1.41720 0.708600 0.705611i \(-0.249327\pi\)
0.708600 + 0.705611i \(0.249327\pi\)
\(992\) 1.74621i 0.0554421i
\(993\) 28.0941i 0.891538i
\(994\) 13.5715i 0.430461i
\(995\) 3.59354i 0.113923i
\(996\) 19.4402 0.615987
\(997\) 9.03393i 0.286107i −0.989715 0.143054i \(-0.954308\pi\)
0.989715 0.143054i \(-0.0456922\pi\)
\(998\) 13.6123 0.430890
\(999\) 7.19450 0.227624
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 349.2.b.b.348.14 yes 26
349.348 even 2 inner 349.2.b.b.348.13 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
349.2.b.b.348.13 26 349.348 even 2 inner
349.2.b.b.348.14 yes 26 1.1 even 1 trivial