Properties

Label 1681.2.a.m.1.13
Level $1681$
Weight $2$
Character 1681.1
Self dual yes
Analytic conductor $13.423$
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] = [1681,2,Mod(1,1681)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1681, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1681.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1681 = 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1681.a (trivial)

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: yes
Analytic conductor: \(13.4228525798\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 41)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 1681.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.706693 q^{2} -0.343106 q^{3} -1.50059 q^{4} -2.37819 q^{5} -0.242471 q^{6} +4.91274 q^{7} -2.47384 q^{8} -2.88228 q^{9} +O(q^{10})\) \(q+0.706693 q^{2} -0.343106 q^{3} -1.50059 q^{4} -2.37819 q^{5} -0.242471 q^{6} +4.91274 q^{7} -2.47384 q^{8} -2.88228 q^{9} -1.68065 q^{10} -1.70460 q^{11} +0.514861 q^{12} -3.97013 q^{13} +3.47180 q^{14} +0.815971 q^{15} +1.25293 q^{16} +2.74718 q^{17} -2.03688 q^{18} +0.965030 q^{19} +3.56867 q^{20} -1.68559 q^{21} -1.20463 q^{22} +6.75640 q^{23} +0.848790 q^{24} +0.655768 q^{25} -2.80566 q^{26} +2.01825 q^{27} -7.37199 q^{28} -0.507132 q^{29} +0.576641 q^{30} -0.590143 q^{31} +5.83311 q^{32} +0.584858 q^{33} +1.94141 q^{34} -11.6834 q^{35} +4.32510 q^{36} +4.25416 q^{37} +0.681980 q^{38} +1.36218 q^{39} +5.88325 q^{40} -1.19120 q^{42} -5.38687 q^{43} +2.55789 q^{44} +6.85459 q^{45} +4.77470 q^{46} +5.02690 q^{47} -0.429888 q^{48} +17.1350 q^{49} +0.463426 q^{50} -0.942574 q^{51} +5.95752 q^{52} +7.62835 q^{53} +1.42628 q^{54} +4.05385 q^{55} -12.1533 q^{56} -0.331108 q^{57} -0.358387 q^{58} +8.42739 q^{59} -1.22443 q^{60} +0.695295 q^{61} -0.417049 q^{62} -14.1599 q^{63} +1.61636 q^{64} +9.44171 q^{65} +0.413315 q^{66} -4.08016 q^{67} -4.12237 q^{68} -2.31816 q^{69} -8.25658 q^{70} +7.27140 q^{71} +7.13029 q^{72} +9.72685 q^{73} +3.00638 q^{74} -0.224998 q^{75} -1.44811 q^{76} -8.37423 q^{77} +0.962641 q^{78} -8.70415 q^{79} -2.97970 q^{80} +7.95436 q^{81} +4.81920 q^{83} +2.52938 q^{84} -6.53329 q^{85} -3.80686 q^{86} +0.174000 q^{87} +4.21689 q^{88} +3.68307 q^{89} +4.84409 q^{90} -19.5042 q^{91} -10.1386 q^{92} +0.202482 q^{93} +3.55247 q^{94} -2.29502 q^{95} -2.00138 q^{96} +4.08190 q^{97} +12.1092 q^{98} +4.91312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9} - 4 q^{10} + 20 q^{16} + 20 q^{18} + 40 q^{20} + 56 q^{21} + 48 q^{23} + 8 q^{25} + 32 q^{31} + 36 q^{32} + 24 q^{33} + 108 q^{36} + 60 q^{39} + 44 q^{40} - 40 q^{42} + 36 q^{43} - 36 q^{45} + 36 q^{46} + 32 q^{49} - 60 q^{50} + 36 q^{51} - 60 q^{57} + 116 q^{59} + 40 q^{61} + 48 q^{62} + 60 q^{64} + 16 q^{66} + 4 q^{72} + 16 q^{73} + 24 q^{74} - 16 q^{77} - 60 q^{78} + 84 q^{80} - 28 q^{81} + 80 q^{83} - 24 q^{84} - 24 q^{86} + 76 q^{87} + 56 q^{90} + 40 q^{91} - 40 q^{92} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.706693 0.499707 0.249854 0.968284i \(-0.419618\pi\)
0.249854 + 0.968284i \(0.419618\pi\)
\(3\) −0.343106 −0.198093 −0.0990463 0.995083i \(-0.531579\pi\)
−0.0990463 + 0.995083i \(0.531579\pi\)
\(4\) −1.50059 −0.750293
\(5\) −2.37819 −1.06356 −0.531779 0.846883i \(-0.678476\pi\)
−0.531779 + 0.846883i \(0.678476\pi\)
\(6\) −0.242471 −0.0989883
\(7\) 4.91274 1.85684 0.928420 0.371531i \(-0.121167\pi\)
0.928420 + 0.371531i \(0.121167\pi\)
\(8\) −2.47384 −0.874634
\(9\) −2.88228 −0.960759
\(10\) −1.68065 −0.531467
\(11\) −1.70460 −0.513955 −0.256977 0.966417i \(-0.582727\pi\)
−0.256977 + 0.966417i \(0.582727\pi\)
\(12\) 0.514861 0.148627
\(13\) −3.97013 −1.10112 −0.550558 0.834797i \(-0.685585\pi\)
−0.550558 + 0.834797i \(0.685585\pi\)
\(14\) 3.47180 0.927877
\(15\) 0.815971 0.210683
\(16\) 1.25293 0.313232
\(17\) 2.74718 0.666288 0.333144 0.942876i \(-0.391890\pi\)
0.333144 + 0.942876i \(0.391890\pi\)
\(18\) −2.03688 −0.480098
\(19\) 0.965030 0.221393 0.110697 0.993854i \(-0.464692\pi\)
0.110697 + 0.993854i \(0.464692\pi\)
\(20\) 3.56867 0.797979
\(21\) −1.68559 −0.367826
\(22\) −1.20463 −0.256827
\(23\) 6.75640 1.40881 0.704403 0.709800i \(-0.251215\pi\)
0.704403 + 0.709800i \(0.251215\pi\)
\(24\) 0.848790 0.173258
\(25\) 0.655768 0.131154
\(26\) −2.80566 −0.550236
\(27\) 2.01825 0.388412
\(28\) −7.37199 −1.39317
\(29\) −0.507132 −0.0941721 −0.0470860 0.998891i \(-0.514993\pi\)
−0.0470860 + 0.998891i \(0.514993\pi\)
\(30\) 0.576641 0.105280
\(31\) −0.590143 −0.105993 −0.0529964 0.998595i \(-0.516877\pi\)
−0.0529964 + 0.998595i \(0.516877\pi\)
\(32\) 5.83311 1.03116
\(33\) 0.584858 0.101811
\(34\) 1.94141 0.332949
\(35\) −11.6834 −1.97486
\(36\) 4.32510 0.720851
\(37\) 4.25416 0.699380 0.349690 0.936866i \(-0.386287\pi\)
0.349690 + 0.936866i \(0.386287\pi\)
\(38\) 0.681980 0.110632
\(39\) 1.36218 0.218123
\(40\) 5.88325 0.930223
\(41\) 0 0
\(42\) −1.19120 −0.183805
\(43\) −5.38687 −0.821490 −0.410745 0.911750i \(-0.634731\pi\)
−0.410745 + 0.911750i \(0.634731\pi\)
\(44\) 2.55789 0.385617
\(45\) 6.85459 1.02182
\(46\) 4.77470 0.703990
\(47\) 5.02690 0.733249 0.366624 0.930369i \(-0.380513\pi\)
0.366624 + 0.930369i \(0.380513\pi\)
\(48\) −0.429888 −0.0620490
\(49\) 17.1350 2.44786
\(50\) 0.463426 0.0655384
\(51\) −0.942574 −0.131987
\(52\) 5.95752 0.826160
\(53\) 7.62835 1.04783 0.523917 0.851769i \(-0.324470\pi\)
0.523917 + 0.851769i \(0.324470\pi\)
\(54\) 1.42628 0.194092
\(55\) 4.05385 0.546620
\(56\) −12.1533 −1.62406
\(57\) −0.331108 −0.0438563
\(58\) −0.358387 −0.0470585
\(59\) 8.42739 1.09715 0.548577 0.836100i \(-0.315170\pi\)
0.548577 + 0.836100i \(0.315170\pi\)
\(60\) −1.22443 −0.158074
\(61\) 0.695295 0.0890234 0.0445117 0.999009i \(-0.485827\pi\)
0.0445117 + 0.999009i \(0.485827\pi\)
\(62\) −0.417049 −0.0529653
\(63\) −14.1599 −1.78398
\(64\) 1.61636 0.202045
\(65\) 9.44171 1.17110
\(66\) 0.413315 0.0508755
\(67\) −4.08016 −0.498471 −0.249235 0.968443i \(-0.580179\pi\)
−0.249235 + 0.968443i \(0.580179\pi\)
\(68\) −4.12237 −0.499911
\(69\) −2.31816 −0.279074
\(70\) −8.25658 −0.986850
\(71\) 7.27140 0.862956 0.431478 0.902123i \(-0.357992\pi\)
0.431478 + 0.902123i \(0.357992\pi\)
\(72\) 7.13029 0.840313
\(73\) 9.72685 1.13844 0.569221 0.822185i \(-0.307245\pi\)
0.569221 + 0.822185i \(0.307245\pi\)
\(74\) 3.00638 0.349485
\(75\) −0.224998 −0.0259806
\(76\) −1.44811 −0.166110
\(77\) −8.37423 −0.954332
\(78\) 0.962641 0.108998
\(79\) −8.70415 −0.979294 −0.489647 0.871921i \(-0.662874\pi\)
−0.489647 + 0.871921i \(0.662874\pi\)
\(80\) −2.97970 −0.333140
\(81\) 7.95436 0.883818
\(82\) 0 0
\(83\) 4.81920 0.528976 0.264488 0.964389i \(-0.414797\pi\)
0.264488 + 0.964389i \(0.414797\pi\)
\(84\) 2.52938 0.275978
\(85\) −6.53329 −0.708635
\(86\) −3.80686 −0.410504
\(87\) 0.174000 0.0186548
\(88\) 4.21689 0.449522
\(89\) 3.68307 0.390405 0.195202 0.980763i \(-0.437464\pi\)
0.195202 + 0.980763i \(0.437464\pi\)
\(90\) 4.84409 0.510612
\(91\) −19.5042 −2.04460
\(92\) −10.1386 −1.05702
\(93\) 0.202482 0.0209964
\(94\) 3.55247 0.366410
\(95\) −2.29502 −0.235464
\(96\) −2.00138 −0.204265
\(97\) 4.08190 0.414454 0.207227 0.978293i \(-0.433556\pi\)
0.207227 + 0.978293i \(0.433556\pi\)
\(98\) 12.1092 1.22321
\(99\) 4.91312 0.493787
\(100\) −0.984036 −0.0984036
\(101\) −10.4203 −1.03686 −0.518428 0.855121i \(-0.673482\pi\)
−0.518428 + 0.855121i \(0.673482\pi\)
\(102\) −0.666110 −0.0659547
\(103\) 0.409816 0.0403804 0.0201902 0.999796i \(-0.493573\pi\)
0.0201902 + 0.999796i \(0.493573\pi\)
\(104\) 9.82146 0.963073
\(105\) 4.00865 0.391204
\(106\) 5.39090 0.523611
\(107\) 13.9793 1.35143 0.675715 0.737163i \(-0.263835\pi\)
0.675715 + 0.737163i \(0.263835\pi\)
\(108\) −3.02855 −0.291423
\(109\) 15.3340 1.46873 0.734366 0.678753i \(-0.237479\pi\)
0.734366 + 0.678753i \(0.237479\pi\)
\(110\) 2.86482 0.273150
\(111\) −1.45963 −0.138542
\(112\) 6.15531 0.581622
\(113\) 8.06399 0.758596 0.379298 0.925274i \(-0.376165\pi\)
0.379298 + 0.925274i \(0.376165\pi\)
\(114\) −0.233992 −0.0219153
\(115\) −16.0680 −1.49835
\(116\) 0.760995 0.0706566
\(117\) 11.4430 1.05791
\(118\) 5.95558 0.548255
\(119\) 13.4962 1.23719
\(120\) −2.01858 −0.184270
\(121\) −8.09435 −0.735850
\(122\) 0.491360 0.0444856
\(123\) 0 0
\(124\) 0.885560 0.0795256
\(125\) 10.3314 0.924068
\(126\) −10.0067 −0.891466
\(127\) −10.7614 −0.954923 −0.477462 0.878653i \(-0.658443\pi\)
−0.477462 + 0.878653i \(0.658443\pi\)
\(128\) −10.5240 −0.930195
\(129\) 1.84827 0.162731
\(130\) 6.67239 0.585207
\(131\) −16.2642 −1.42101 −0.710505 0.703692i \(-0.751534\pi\)
−0.710505 + 0.703692i \(0.751534\pi\)
\(132\) −0.877629 −0.0763878
\(133\) 4.74094 0.411092
\(134\) −2.88342 −0.249089
\(135\) −4.79977 −0.413098
\(136\) −6.79607 −0.582758
\(137\) 17.7338 1.51510 0.757549 0.652779i \(-0.226397\pi\)
0.757549 + 0.652779i \(0.226397\pi\)
\(138\) −1.63823 −0.139455
\(139\) −2.13028 −0.180688 −0.0903438 0.995911i \(-0.528797\pi\)
−0.0903438 + 0.995911i \(0.528797\pi\)
\(140\) 17.5320 1.48172
\(141\) −1.72476 −0.145251
\(142\) 5.13864 0.431225
\(143\) 6.76747 0.565924
\(144\) −3.61129 −0.300941
\(145\) 1.20605 0.100157
\(146\) 6.87389 0.568888
\(147\) −5.87913 −0.484902
\(148\) −6.38373 −0.524740
\(149\) −1.51731 −0.124303 −0.0621513 0.998067i \(-0.519796\pi\)
−0.0621513 + 0.998067i \(0.519796\pi\)
\(150\) −0.159005 −0.0129827
\(151\) 2.09591 0.170563 0.0852815 0.996357i \(-0.472821\pi\)
0.0852815 + 0.996357i \(0.472821\pi\)
\(152\) −2.38733 −0.193638
\(153\) −7.91812 −0.640142
\(154\) −5.91801 −0.476887
\(155\) 1.40347 0.112729
\(156\) −2.04406 −0.163656
\(157\) −6.91424 −0.551816 −0.275908 0.961184i \(-0.588979\pi\)
−0.275908 + 0.961184i \(0.588979\pi\)
\(158\) −6.15116 −0.489360
\(159\) −2.61734 −0.207568
\(160\) −13.8722 −1.09670
\(161\) 33.1924 2.61593
\(162\) 5.62129 0.441650
\(163\) 7.72291 0.604905 0.302453 0.953164i \(-0.402195\pi\)
0.302453 + 0.953164i \(0.402195\pi\)
\(164\) 0 0
\(165\) −1.39090 −0.108281
\(166\) 3.40569 0.264333
\(167\) 18.9794 1.46867 0.734336 0.678786i \(-0.237494\pi\)
0.734336 + 0.678786i \(0.237494\pi\)
\(168\) 4.16988 0.321713
\(169\) 2.76194 0.212457
\(170\) −4.61703 −0.354110
\(171\) −2.78149 −0.212706
\(172\) 8.08346 0.616358
\(173\) 12.9720 0.986242 0.493121 0.869961i \(-0.335856\pi\)
0.493121 + 0.869961i \(0.335856\pi\)
\(174\) 0.122965 0.00932193
\(175\) 3.22162 0.243531
\(176\) −2.13574 −0.160987
\(177\) −2.89149 −0.217338
\(178\) 2.60280 0.195088
\(179\) −0.553509 −0.0413712 −0.0206856 0.999786i \(-0.506585\pi\)
−0.0206856 + 0.999786i \(0.506585\pi\)
\(180\) −10.2859 −0.766666
\(181\) −24.0251 −1.78577 −0.892885 0.450285i \(-0.851322\pi\)
−0.892885 + 0.450285i \(0.851322\pi\)
\(182\) −13.7835 −1.02170
\(183\) −0.238560 −0.0176349
\(184\) −16.7142 −1.23219
\(185\) −10.1172 −0.743830
\(186\) 0.143092 0.0104920
\(187\) −4.68282 −0.342442
\(188\) −7.54329 −0.550151
\(189\) 9.91512 0.721219
\(190\) −1.62187 −0.117663
\(191\) −12.4328 −0.899606 −0.449803 0.893128i \(-0.648506\pi\)
−0.449803 + 0.893128i \(0.648506\pi\)
\(192\) −0.554583 −0.0400236
\(193\) 3.25333 0.234179 0.117090 0.993121i \(-0.462643\pi\)
0.117090 + 0.993121i \(0.462643\pi\)
\(194\) 2.88465 0.207106
\(195\) −3.23951 −0.231986
\(196\) −25.7125 −1.83661
\(197\) 13.5417 0.964807 0.482404 0.875949i \(-0.339764\pi\)
0.482404 + 0.875949i \(0.339764\pi\)
\(198\) 3.47206 0.246749
\(199\) −9.27629 −0.657578 −0.328789 0.944403i \(-0.606641\pi\)
−0.328789 + 0.944403i \(0.606641\pi\)
\(200\) −1.62226 −0.114711
\(201\) 1.39993 0.0987434
\(202\) −7.36393 −0.518124
\(203\) −2.49141 −0.174863
\(204\) 1.41441 0.0990287
\(205\) 0 0
\(206\) 0.289614 0.0201784
\(207\) −19.4738 −1.35352
\(208\) −4.97429 −0.344905
\(209\) −1.64499 −0.113786
\(210\) 2.83288 0.195488
\(211\) −27.2206 −1.87394 −0.936972 0.349405i \(-0.886384\pi\)
−0.936972 + 0.349405i \(0.886384\pi\)
\(212\) −11.4470 −0.786183
\(213\) −2.49486 −0.170945
\(214\) 9.87907 0.675320
\(215\) 12.8110 0.873702
\(216\) −4.99282 −0.339718
\(217\) −2.89922 −0.196812
\(218\) 10.8364 0.733936
\(219\) −3.33735 −0.225517
\(220\) −6.08314 −0.410125
\(221\) −10.9066 −0.733660
\(222\) −1.03151 −0.0692304
\(223\) 9.53126 0.638260 0.319130 0.947711i \(-0.396609\pi\)
0.319130 + 0.947711i \(0.396609\pi\)
\(224\) 28.6566 1.91470
\(225\) −1.89011 −0.126007
\(226\) 5.69876 0.379076
\(227\) −1.52258 −0.101057 −0.0505286 0.998723i \(-0.516091\pi\)
−0.0505286 + 0.998723i \(0.516091\pi\)
\(228\) 0.496856 0.0329051
\(229\) −14.7408 −0.974099 −0.487049 0.873374i \(-0.661927\pi\)
−0.487049 + 0.873374i \(0.661927\pi\)
\(230\) −11.3551 −0.748734
\(231\) 2.87325 0.189046
\(232\) 1.25456 0.0823661
\(233\) −14.8872 −0.975290 −0.487645 0.873042i \(-0.662144\pi\)
−0.487645 + 0.873042i \(0.662144\pi\)
\(234\) 8.08670 0.528644
\(235\) −11.9549 −0.779852
\(236\) −12.6460 −0.823186
\(237\) 2.98645 0.193991
\(238\) 9.53763 0.618233
\(239\) −13.3732 −0.865040 −0.432520 0.901624i \(-0.642375\pi\)
−0.432520 + 0.901624i \(0.642375\pi\)
\(240\) 1.02235 0.0659926
\(241\) −3.99525 −0.257357 −0.128678 0.991686i \(-0.541073\pi\)
−0.128678 + 0.991686i \(0.541073\pi\)
\(242\) −5.72022 −0.367710
\(243\) −8.78393 −0.563490
\(244\) −1.04335 −0.0667936
\(245\) −40.7502 −2.60344
\(246\) 0 0
\(247\) −3.83130 −0.243780
\(248\) 1.45992 0.0927048
\(249\) −1.65350 −0.104786
\(250\) 7.30112 0.461763
\(251\) 27.3187 1.72434 0.862171 0.506618i \(-0.169104\pi\)
0.862171 + 0.506618i \(0.169104\pi\)
\(252\) 21.2481 1.33851
\(253\) −11.5169 −0.724063
\(254\) −7.60503 −0.477182
\(255\) 2.24162 0.140375
\(256\) −10.6699 −0.666870
\(257\) 7.49785 0.467703 0.233851 0.972272i \(-0.424867\pi\)
0.233851 + 0.972272i \(0.424867\pi\)
\(258\) 1.30616 0.0813179
\(259\) 20.8996 1.29864
\(260\) −14.1681 −0.878668
\(261\) 1.46170 0.0904767
\(262\) −11.4938 −0.710089
\(263\) −11.9205 −0.735048 −0.367524 0.930014i \(-0.619794\pi\)
−0.367524 + 0.930014i \(0.619794\pi\)
\(264\) −1.44684 −0.0890470
\(265\) −18.1416 −1.11443
\(266\) 3.35039 0.205426
\(267\) −1.26369 −0.0773363
\(268\) 6.12263 0.373999
\(269\) −21.1828 −1.29154 −0.645769 0.763533i \(-0.723463\pi\)
−0.645769 + 0.763533i \(0.723463\pi\)
\(270\) −3.39196 −0.206428
\(271\) −3.83247 −0.232806 −0.116403 0.993202i \(-0.537136\pi\)
−0.116403 + 0.993202i \(0.537136\pi\)
\(272\) 3.44201 0.208703
\(273\) 6.69202 0.405020
\(274\) 12.5323 0.757105
\(275\) −1.11782 −0.0674070
\(276\) 3.47860 0.209387
\(277\) −1.16792 −0.0701734 −0.0350867 0.999384i \(-0.511171\pi\)
−0.0350867 + 0.999384i \(0.511171\pi\)
\(278\) −1.50545 −0.0902909
\(279\) 1.70096 0.101834
\(280\) 28.9029 1.72728
\(281\) 17.6959 1.05565 0.527826 0.849353i \(-0.323007\pi\)
0.527826 + 0.849353i \(0.323007\pi\)
\(282\) −1.21888 −0.0725830
\(283\) 7.41025 0.440494 0.220247 0.975444i \(-0.429314\pi\)
0.220247 + 0.975444i \(0.429314\pi\)
\(284\) −10.9114 −0.647470
\(285\) 0.787437 0.0466437
\(286\) 4.78252 0.282796
\(287\) 0 0
\(288\) −16.8126 −0.990695
\(289\) −9.45303 −0.556060
\(290\) 0.852310 0.0500494
\(291\) −1.40053 −0.0821003
\(292\) −14.5960 −0.854165
\(293\) −10.3526 −0.604803 −0.302401 0.953181i \(-0.597788\pi\)
−0.302401 + 0.953181i \(0.597788\pi\)
\(294\) −4.15474 −0.242309
\(295\) −20.0419 −1.16689
\(296\) −10.5241 −0.611701
\(297\) −3.44030 −0.199626
\(298\) −1.07227 −0.0621149
\(299\) −26.8238 −1.55126
\(300\) 0.337629 0.0194930
\(301\) −26.4643 −1.52538
\(302\) 1.48117 0.0852315
\(303\) 3.57526 0.205393
\(304\) 1.20911 0.0693474
\(305\) −1.65354 −0.0946814
\(306\) −5.59568 −0.319884
\(307\) −1.37369 −0.0784006 −0.0392003 0.999231i \(-0.512481\pi\)
−0.0392003 + 0.999231i \(0.512481\pi\)
\(308\) 12.5663 0.716029
\(309\) −0.140611 −0.00799906
\(310\) 0.991821 0.0563316
\(311\) 12.1498 0.688953 0.344477 0.938795i \(-0.388056\pi\)
0.344477 + 0.938795i \(0.388056\pi\)
\(312\) −3.36981 −0.190778
\(313\) 30.1141 1.70215 0.851076 0.525042i \(-0.175951\pi\)
0.851076 + 0.525042i \(0.175951\pi\)
\(314\) −4.88624 −0.275747
\(315\) 33.6748 1.89736
\(316\) 13.0613 0.734757
\(317\) −8.47148 −0.475806 −0.237903 0.971289i \(-0.576460\pi\)
−0.237903 + 0.971289i \(0.576460\pi\)
\(318\) −1.84965 −0.103723
\(319\) 0.864455 0.0484002
\(320\) −3.84400 −0.214886
\(321\) −4.79639 −0.267708
\(322\) 23.4568 1.30720
\(323\) 2.65111 0.147512
\(324\) −11.9362 −0.663122
\(325\) −2.60348 −0.144415
\(326\) 5.45773 0.302275
\(327\) −5.26120 −0.290945
\(328\) 0 0
\(329\) 24.6959 1.36153
\(330\) −0.982939 −0.0541090
\(331\) −10.4458 −0.574155 −0.287077 0.957907i \(-0.592684\pi\)
−0.287077 + 0.957907i \(0.592684\pi\)
\(332\) −7.23162 −0.396887
\(333\) −12.2617 −0.671936
\(334\) 13.4126 0.733906
\(335\) 9.70338 0.530152
\(336\) −2.11193 −0.115215
\(337\) 22.8689 1.24575 0.622875 0.782321i \(-0.285964\pi\)
0.622875 + 0.782321i \(0.285964\pi\)
\(338\) 1.95184 0.106166
\(339\) −2.76681 −0.150272
\(340\) 9.80377 0.531684
\(341\) 1.00595 0.0544755
\(342\) −1.96566 −0.106290
\(343\) 49.7906 2.68844
\(344\) 13.3262 0.718503
\(345\) 5.51302 0.296811
\(346\) 9.16721 0.492832
\(347\) −1.67186 −0.0897500 −0.0448750 0.998993i \(-0.514289\pi\)
−0.0448750 + 0.998993i \(0.514289\pi\)
\(348\) −0.261102 −0.0139966
\(349\) 34.5435 1.84907 0.924536 0.381095i \(-0.124453\pi\)
0.924536 + 0.381095i \(0.124453\pi\)
\(350\) 2.27669 0.121694
\(351\) −8.01271 −0.427687
\(352\) −9.94310 −0.529969
\(353\) −25.6054 −1.36284 −0.681418 0.731894i \(-0.738636\pi\)
−0.681418 + 0.731894i \(0.738636\pi\)
\(354\) −2.04340 −0.108605
\(355\) −17.2927 −0.917803
\(356\) −5.52677 −0.292918
\(357\) −4.63062 −0.245078
\(358\) −0.391160 −0.0206735
\(359\) 26.4143 1.39409 0.697047 0.717025i \(-0.254497\pi\)
0.697047 + 0.717025i \(0.254497\pi\)
\(360\) −16.9572 −0.893720
\(361\) −18.0687 −0.950985
\(362\) −16.9783 −0.892362
\(363\) 2.77723 0.145767
\(364\) 29.2677 1.53405
\(365\) −23.1323 −1.21080
\(366\) −0.168589 −0.00881227
\(367\) 20.0908 1.04873 0.524366 0.851493i \(-0.324302\pi\)
0.524366 + 0.851493i \(0.324302\pi\)
\(368\) 8.46528 0.441283
\(369\) 0 0
\(370\) −7.14974 −0.371697
\(371\) 37.4761 1.94566
\(372\) −0.303841 −0.0157534
\(373\) 7.68309 0.397815 0.198908 0.980018i \(-0.436261\pi\)
0.198908 + 0.980018i \(0.436261\pi\)
\(374\) −3.30932 −0.171121
\(375\) −3.54477 −0.183051
\(376\) −12.4357 −0.641324
\(377\) 2.01338 0.103694
\(378\) 7.00694 0.360398
\(379\) 23.3163 1.19768 0.598838 0.800870i \(-0.295629\pi\)
0.598838 + 0.800870i \(0.295629\pi\)
\(380\) 3.44388 0.176667
\(381\) 3.69232 0.189163
\(382\) −8.78617 −0.449540
\(383\) −16.2989 −0.832837 −0.416418 0.909173i \(-0.636715\pi\)
−0.416418 + 0.909173i \(0.636715\pi\)
\(384\) 3.61084 0.184265
\(385\) 19.9155 1.01499
\(386\) 2.29910 0.117021
\(387\) 15.5265 0.789254
\(388\) −6.12524 −0.310962
\(389\) −0.290902 −0.0147493 −0.00737466 0.999973i \(-0.502347\pi\)
−0.00737466 + 0.999973i \(0.502347\pi\)
\(390\) −2.28934 −0.115925
\(391\) 18.5610 0.938671
\(392\) −42.3892 −2.14098
\(393\) 5.58036 0.281492
\(394\) 9.56983 0.482121
\(395\) 20.7001 1.04153
\(396\) −7.37255 −0.370485
\(397\) 28.2999 1.42033 0.710166 0.704035i \(-0.248620\pi\)
0.710166 + 0.704035i \(0.248620\pi\)
\(398\) −6.55548 −0.328597
\(399\) −1.62665 −0.0814343
\(400\) 0.821630 0.0410815
\(401\) 6.77610 0.338382 0.169191 0.985583i \(-0.445885\pi\)
0.169191 + 0.985583i \(0.445885\pi\)
\(402\) 0.989319 0.0493428
\(403\) 2.34294 0.116710
\(404\) 15.6365 0.777945
\(405\) −18.9169 −0.939991
\(406\) −1.76066 −0.0873801
\(407\) −7.25162 −0.359450
\(408\) 2.33177 0.115440
\(409\) −16.9059 −0.835945 −0.417972 0.908460i \(-0.637259\pi\)
−0.417972 + 0.908460i \(0.637259\pi\)
\(410\) 0 0
\(411\) −6.08457 −0.300130
\(412\) −0.614965 −0.0302971
\(413\) 41.4016 2.03724
\(414\) −13.7620 −0.676365
\(415\) −11.4610 −0.562596
\(416\) −23.1582 −1.13542
\(417\) 0.730911 0.0357929
\(418\) −1.16250 −0.0568597
\(419\) 12.2920 0.600504 0.300252 0.953860i \(-0.402929\pi\)
0.300252 + 0.953860i \(0.402929\pi\)
\(420\) −6.01533 −0.293518
\(421\) −29.1328 −1.41985 −0.709924 0.704278i \(-0.751271\pi\)
−0.709924 + 0.704278i \(0.751271\pi\)
\(422\) −19.2366 −0.936423
\(423\) −14.4889 −0.704476
\(424\) −18.8713 −0.916472
\(425\) 1.80151 0.0873861
\(426\) −1.76310 −0.0854225
\(427\) 3.41580 0.165302
\(428\) −20.9772 −1.01397
\(429\) −2.32196 −0.112105
\(430\) 9.05342 0.436595
\(431\) −8.60832 −0.414648 −0.207324 0.978272i \(-0.566475\pi\)
−0.207324 + 0.978272i \(0.566475\pi\)
\(432\) 2.52872 0.121663
\(433\) 11.8393 0.568961 0.284481 0.958682i \(-0.408179\pi\)
0.284481 + 0.958682i \(0.408179\pi\)
\(434\) −2.04885 −0.0983482
\(435\) −0.413805 −0.0198404
\(436\) −23.0100 −1.10198
\(437\) 6.52013 0.311900
\(438\) −2.35848 −0.112692
\(439\) 30.4135 1.45156 0.725778 0.687929i \(-0.241480\pi\)
0.725778 + 0.687929i \(0.241480\pi\)
\(440\) −10.0286 −0.478093
\(441\) −49.3878 −2.35180
\(442\) −7.70765 −0.366615
\(443\) 13.3534 0.634438 0.317219 0.948352i \(-0.397251\pi\)
0.317219 + 0.948352i \(0.397251\pi\)
\(444\) 2.19030 0.103947
\(445\) −8.75903 −0.415218
\(446\) 6.73567 0.318943
\(447\) 0.520598 0.0246234
\(448\) 7.94075 0.375165
\(449\) 25.1502 1.18691 0.593456 0.804866i \(-0.297763\pi\)
0.593456 + 0.804866i \(0.297763\pi\)
\(450\) −1.33572 −0.0629666
\(451\) 0 0
\(452\) −12.1007 −0.569169
\(453\) −0.719121 −0.0337873
\(454\) −1.07600 −0.0504990
\(455\) 46.3847 2.17455
\(456\) 0.819108 0.0383582
\(457\) 32.1746 1.50506 0.752531 0.658556i \(-0.228833\pi\)
0.752531 + 0.658556i \(0.228833\pi\)
\(458\) −10.4172 −0.486764
\(459\) 5.54448 0.258794
\(460\) 24.1114 1.12420
\(461\) 29.4423 1.37126 0.685632 0.727948i \(-0.259526\pi\)
0.685632 + 0.727948i \(0.259526\pi\)
\(462\) 2.03051 0.0944677
\(463\) 20.5209 0.953687 0.476843 0.878988i \(-0.341781\pi\)
0.476843 + 0.878988i \(0.341781\pi\)
\(464\) −0.635400 −0.0294977
\(465\) −0.481539 −0.0223308
\(466\) −10.5206 −0.487359
\(467\) −24.6933 −1.14267 −0.571334 0.820718i \(-0.693574\pi\)
−0.571334 + 0.820718i \(0.693574\pi\)
\(468\) −17.1712 −0.793741
\(469\) −20.0448 −0.925581
\(470\) −8.44844 −0.389698
\(471\) 2.37232 0.109311
\(472\) −20.8480 −0.959607
\(473\) 9.18244 0.422209
\(474\) 2.11050 0.0969386
\(475\) 0.632836 0.0290365
\(476\) −20.2521 −0.928255
\(477\) −21.9870 −1.00672
\(478\) −9.45073 −0.432266
\(479\) −29.0111 −1.32555 −0.662776 0.748817i \(-0.730622\pi\)
−0.662776 + 0.748817i \(0.730622\pi\)
\(480\) 4.75965 0.217247
\(481\) −16.8896 −0.770098
\(482\) −2.82341 −0.128603
\(483\) −11.3885 −0.518196
\(484\) 12.1463 0.552103
\(485\) −9.70752 −0.440796
\(486\) −6.20754 −0.281580
\(487\) −26.3805 −1.19541 −0.597707 0.801715i \(-0.703921\pi\)
−0.597707 + 0.801715i \(0.703921\pi\)
\(488\) −1.72005 −0.0778629
\(489\) −2.64978 −0.119827
\(490\) −28.7979 −1.30096
\(491\) −17.4748 −0.788628 −0.394314 0.918976i \(-0.629018\pi\)
−0.394314 + 0.918976i \(0.629018\pi\)
\(492\) 0 0
\(493\) −1.39318 −0.0627457
\(494\) −2.70755 −0.121818
\(495\) −11.6843 −0.525171
\(496\) −0.739406 −0.0332003
\(497\) 35.7225 1.60237
\(498\) −1.16852 −0.0523624
\(499\) −6.32000 −0.282922 −0.141461 0.989944i \(-0.545180\pi\)
−0.141461 + 0.989944i \(0.545180\pi\)
\(500\) −15.5031 −0.693321
\(501\) −6.51196 −0.290933
\(502\) 19.3059 0.861666
\(503\) −11.0972 −0.494798 −0.247399 0.968914i \(-0.579576\pi\)
−0.247399 + 0.968914i \(0.579576\pi\)
\(504\) 35.0292 1.56033
\(505\) 24.7813 1.10275
\(506\) −8.13892 −0.361819
\(507\) −0.947639 −0.0420861
\(508\) 16.1485 0.716472
\(509\) −21.0225 −0.931808 −0.465904 0.884835i \(-0.654271\pi\)
−0.465904 + 0.884835i \(0.654271\pi\)
\(510\) 1.58413 0.0701466
\(511\) 47.7855 2.11391
\(512\) 13.5076 0.596955
\(513\) 1.94767 0.0859917
\(514\) 5.29867 0.233714
\(515\) −0.974619 −0.0429469
\(516\) −2.77349 −0.122096
\(517\) −8.56883 −0.376857
\(518\) 14.7696 0.648938
\(519\) −4.45077 −0.195367
\(520\) −23.3573 −1.02428
\(521\) 9.55125 0.418448 0.209224 0.977868i \(-0.432906\pi\)
0.209224 + 0.977868i \(0.432906\pi\)
\(522\) 1.03297 0.0452119
\(523\) −9.97758 −0.436289 −0.218145 0.975916i \(-0.570000\pi\)
−0.218145 + 0.975916i \(0.570000\pi\)
\(524\) 24.4058 1.06617
\(525\) −1.10536 −0.0482418
\(526\) −8.42411 −0.367309
\(527\) −1.62123 −0.0706217
\(528\) 0.732785 0.0318904
\(529\) 22.6489 0.984735
\(530\) −12.8206 −0.556890
\(531\) −24.2901 −1.05410
\(532\) −7.11419 −0.308439
\(533\) 0 0
\(534\) −0.893038 −0.0386455
\(535\) −33.2454 −1.43732
\(536\) 10.0937 0.435979
\(537\) 0.189912 0.00819532
\(538\) −14.9697 −0.645390
\(539\) −29.2083 −1.25809
\(540\) 7.20246 0.309945
\(541\) −4.87015 −0.209384 −0.104692 0.994505i \(-0.533386\pi\)
−0.104692 + 0.994505i \(0.533386\pi\)
\(542\) −2.70838 −0.116335
\(543\) 8.24316 0.353748
\(544\) 16.0246 0.687048
\(545\) −36.4672 −1.56208
\(546\) 4.72920 0.202391
\(547\) 20.1046 0.859611 0.429806 0.902921i \(-0.358582\pi\)
0.429806 + 0.902921i \(0.358582\pi\)
\(548\) −26.6110 −1.13677
\(549\) −2.00403 −0.0855300
\(550\) −0.789955 −0.0336838
\(551\) −0.489398 −0.0208491
\(552\) 5.73476 0.244088
\(553\) −42.7612 −1.81839
\(554\) −0.825359 −0.0350661
\(555\) 3.47127 0.147347
\(556\) 3.19666 0.135569
\(557\) 20.3234 0.861130 0.430565 0.902560i \(-0.358314\pi\)
0.430565 + 0.902560i \(0.358314\pi\)
\(558\) 1.20205 0.0508869
\(559\) 21.3866 0.904556
\(560\) −14.6385 −0.618588
\(561\) 1.60671 0.0678352
\(562\) 12.5056 0.527517
\(563\) 16.0734 0.677415 0.338707 0.940892i \(-0.390010\pi\)
0.338707 + 0.940892i \(0.390010\pi\)
\(564\) 2.58815 0.108981
\(565\) −19.1777 −0.806811
\(566\) 5.23677 0.220118
\(567\) 39.0777 1.64111
\(568\) −17.9883 −0.754771
\(569\) 14.3261 0.600580 0.300290 0.953848i \(-0.402916\pi\)
0.300290 + 0.953848i \(0.402916\pi\)
\(570\) 0.556476 0.0233082
\(571\) 8.19798 0.343075 0.171537 0.985178i \(-0.445127\pi\)
0.171537 + 0.985178i \(0.445127\pi\)
\(572\) −10.1552 −0.424609
\(573\) 4.26578 0.178205
\(574\) 0 0
\(575\) 4.43063 0.184770
\(576\) −4.65880 −0.194117
\(577\) −4.94814 −0.205994 −0.102997 0.994682i \(-0.532843\pi\)
−0.102997 + 0.994682i \(0.532843\pi\)
\(578\) −6.68038 −0.277867
\(579\) −1.11624 −0.0463892
\(580\) −1.80979 −0.0751474
\(581\) 23.6755 0.982224
\(582\) −0.989742 −0.0410261
\(583\) −13.0033 −0.538540
\(584\) −24.0627 −0.995720
\(585\) −27.2136 −1.12515
\(586\) −7.31608 −0.302224
\(587\) −10.1283 −0.418038 −0.209019 0.977912i \(-0.567027\pi\)
−0.209019 + 0.977912i \(0.567027\pi\)
\(588\) 8.82214 0.363819
\(589\) −0.569506 −0.0234661
\(590\) −14.1635 −0.583101
\(591\) −4.64625 −0.191121
\(592\) 5.33016 0.219068
\(593\) −15.1299 −0.621310 −0.310655 0.950523i \(-0.600548\pi\)
−0.310655 + 0.950523i \(0.600548\pi\)
\(594\) −2.43123 −0.0997546
\(595\) −32.0964 −1.31582
\(596\) 2.27685 0.0932633
\(597\) 3.18275 0.130261
\(598\) −18.9562 −0.775175
\(599\) 3.27805 0.133937 0.0669687 0.997755i \(-0.478667\pi\)
0.0669687 + 0.997755i \(0.478667\pi\)
\(600\) 0.556609 0.0227235
\(601\) 7.14693 0.291529 0.145765 0.989319i \(-0.453436\pi\)
0.145765 + 0.989319i \(0.453436\pi\)
\(602\) −18.7021 −0.762241
\(603\) 11.7602 0.478910
\(604\) −3.14510 −0.127972
\(605\) 19.2499 0.782619
\(606\) 2.52661 0.102637
\(607\) 33.9444 1.37776 0.688880 0.724876i \(-0.258103\pi\)
0.688880 + 0.724876i \(0.258103\pi\)
\(608\) 5.62913 0.228291
\(609\) 0.854818 0.0346390
\(610\) −1.16854 −0.0473130
\(611\) −19.9575 −0.807392
\(612\) 11.8818 0.480294
\(613\) 15.4196 0.622792 0.311396 0.950280i \(-0.399203\pi\)
0.311396 + 0.950280i \(0.399203\pi\)
\(614\) −0.970776 −0.0391773
\(615\) 0 0
\(616\) 20.7165 0.834691
\(617\) 0.726595 0.0292516 0.0146258 0.999893i \(-0.495344\pi\)
0.0146258 + 0.999893i \(0.495344\pi\)
\(618\) −0.0993685 −0.00399719
\(619\) −24.5798 −0.987946 −0.493973 0.869477i \(-0.664456\pi\)
−0.493973 + 0.869477i \(0.664456\pi\)
\(620\) −2.10603 −0.0845800
\(621\) 13.6361 0.547197
\(622\) 8.58619 0.344275
\(623\) 18.0940 0.724920
\(624\) 1.70671 0.0683231
\(625\) −27.8488 −1.11395
\(626\) 21.2814 0.850578
\(627\) 0.564406 0.0225402
\(628\) 10.3754 0.414024
\(629\) 11.6869 0.465988
\(630\) 23.7977 0.948125
\(631\) 13.6232 0.542333 0.271166 0.962533i \(-0.412591\pi\)
0.271166 + 0.962533i \(0.412591\pi\)
\(632\) 21.5327 0.856523
\(633\) 9.33956 0.371214
\(634\) −5.98674 −0.237764
\(635\) 25.5927 1.01562
\(636\) 3.92754 0.155737
\(637\) −68.0282 −2.69538
\(638\) 0.610904 0.0241859
\(639\) −20.9582 −0.829093
\(640\) 25.0279 0.989315
\(641\) −9.49502 −0.375031 −0.187515 0.982262i \(-0.560043\pi\)
−0.187515 + 0.982262i \(0.560043\pi\)
\(642\) −3.38957 −0.133776
\(643\) 39.6479 1.56356 0.781781 0.623554i \(-0.214312\pi\)
0.781781 + 0.623554i \(0.214312\pi\)
\(644\) −49.8081 −1.96271
\(645\) −4.39553 −0.173074
\(646\) 1.87352 0.0737126
\(647\) 40.5997 1.59614 0.798070 0.602564i \(-0.205854\pi\)
0.798070 + 0.602564i \(0.205854\pi\)
\(648\) −19.6778 −0.773017
\(649\) −14.3653 −0.563887
\(650\) −1.83986 −0.0721654
\(651\) 0.994740 0.0389869
\(652\) −11.5889 −0.453856
\(653\) 9.76983 0.382323 0.191161 0.981559i \(-0.438775\pi\)
0.191161 + 0.981559i \(0.438775\pi\)
\(654\) −3.71805 −0.145387
\(655\) 38.6793 1.51133
\(656\) 0 0
\(657\) −28.0355 −1.09377
\(658\) 17.4524 0.680364
\(659\) −39.0664 −1.52181 −0.760905 0.648863i \(-0.775245\pi\)
−0.760905 + 0.648863i \(0.775245\pi\)
\(660\) 2.08717 0.0812428
\(661\) −37.2590 −1.44921 −0.724603 0.689167i \(-0.757977\pi\)
−0.724603 + 0.689167i \(0.757977\pi\)
\(662\) −7.38199 −0.286909
\(663\) 3.74214 0.145333
\(664\) −11.9219 −0.462660
\(665\) −11.2748 −0.437220
\(666\) −8.66523 −0.335771
\(667\) −3.42639 −0.132670
\(668\) −28.4802 −1.10193
\(669\) −3.27024 −0.126435
\(670\) 6.85730 0.264921
\(671\) −1.18520 −0.0457540
\(672\) −9.83225 −0.379287
\(673\) 32.0460 1.23528 0.617641 0.786460i \(-0.288088\pi\)
0.617641 + 0.786460i \(0.288088\pi\)
\(674\) 16.1613 0.622510
\(675\) 1.32350 0.0509416
\(676\) −4.14453 −0.159405
\(677\) 36.6269 1.40769 0.703843 0.710356i \(-0.251466\pi\)
0.703843 + 0.710356i \(0.251466\pi\)
\(678\) −1.95528 −0.0750922
\(679\) 20.0533 0.769576
\(680\) 16.1623 0.619796
\(681\) 0.522407 0.0200187
\(682\) 0.710901 0.0272218
\(683\) 45.5625 1.74340 0.871700 0.490040i \(-0.163018\pi\)
0.871700 + 0.490040i \(0.163018\pi\)
\(684\) 4.17386 0.159591
\(685\) −42.1742 −1.61139
\(686\) 35.1867 1.34343
\(687\) 5.05766 0.192962
\(688\) −6.74936 −0.257317
\(689\) −30.2856 −1.15379
\(690\) 3.89601 0.148319
\(691\) 12.4918 0.475211 0.237606 0.971362i \(-0.423637\pi\)
0.237606 + 0.971362i \(0.423637\pi\)
\(692\) −19.4656 −0.739970
\(693\) 24.1369 0.916884
\(694\) −1.18149 −0.0448487
\(695\) 5.06619 0.192172
\(696\) −0.430449 −0.0163161
\(697\) 0 0
\(698\) 24.4116 0.923994
\(699\) 5.10788 0.193198
\(700\) −4.83431 −0.182720
\(701\) −33.2743 −1.25675 −0.628377 0.777909i \(-0.716280\pi\)
−0.628377 + 0.777909i \(0.716280\pi\)
\(702\) −5.66252 −0.213718
\(703\) 4.10539 0.154838
\(704\) −2.75524 −0.103842
\(705\) 4.10180 0.154483
\(706\) −18.0951 −0.681019
\(707\) −51.1921 −1.92528
\(708\) 4.33893 0.163067
\(709\) 27.8535 1.04606 0.523030 0.852314i \(-0.324802\pi\)
0.523030 + 0.852314i \(0.324802\pi\)
\(710\) −12.2206 −0.458633
\(711\) 25.0878 0.940866
\(712\) −9.11133 −0.341461
\(713\) −3.98724 −0.149323
\(714\) −3.27242 −0.122467
\(715\) −16.0943 −0.601893
\(716\) 0.830587 0.0310405
\(717\) 4.58843 0.171358
\(718\) 18.6668 0.696639
\(719\) 29.5641 1.10255 0.551277 0.834322i \(-0.314141\pi\)
0.551277 + 0.834322i \(0.314141\pi\)
\(720\) 8.58831 0.320068
\(721\) 2.01332 0.0749800
\(722\) −12.7690 −0.475214
\(723\) 1.37080 0.0509805
\(724\) 36.0517 1.33985
\(725\) −0.332561 −0.0123510
\(726\) 1.96264 0.0728406
\(727\) 14.5061 0.538000 0.269000 0.963140i \(-0.413307\pi\)
0.269000 + 0.963140i \(0.413307\pi\)
\(728\) 48.2503 1.78827
\(729\) −20.8493 −0.772195
\(730\) −16.3474 −0.605044
\(731\) −14.7987 −0.547349
\(732\) 0.357980 0.0132313
\(733\) 44.6474 1.64909 0.824544 0.565798i \(-0.191432\pi\)
0.824544 + 0.565798i \(0.191432\pi\)
\(734\) 14.1980 0.524059
\(735\) 13.9817 0.515721
\(736\) 39.4108 1.45270
\(737\) 6.95502 0.256191
\(738\) 0 0
\(739\) −2.65655 −0.0977228 −0.0488614 0.998806i \(-0.515559\pi\)
−0.0488614 + 0.998806i \(0.515559\pi\)
\(740\) 15.1817 0.558090
\(741\) 1.31454 0.0482909
\(742\) 26.4841 0.972261
\(743\) 30.6558 1.12465 0.562326 0.826916i \(-0.309907\pi\)
0.562326 + 0.826916i \(0.309907\pi\)
\(744\) −0.500907 −0.0183641
\(745\) 3.60844 0.132203
\(746\) 5.42958 0.198791
\(747\) −13.8903 −0.508219
\(748\) 7.02698 0.256932
\(749\) 68.6767 2.50939
\(750\) −2.50506 −0.0914719
\(751\) −50.6405 −1.84790 −0.923949 0.382515i \(-0.875058\pi\)
−0.923949 + 0.382515i \(0.875058\pi\)
\(752\) 6.29835 0.229677
\(753\) −9.37322 −0.341579
\(754\) 1.42284 0.0518168
\(755\) −4.98447 −0.181403
\(756\) −14.8785 −0.541125
\(757\) −28.9856 −1.05350 −0.526750 0.850020i \(-0.676590\pi\)
−0.526750 + 0.850020i \(0.676590\pi\)
\(758\) 16.4774 0.598487
\(759\) 3.95153 0.143431
\(760\) 5.67751 0.205945
\(761\) 38.9868 1.41327 0.706636 0.707577i \(-0.250212\pi\)
0.706636 + 0.707577i \(0.250212\pi\)
\(762\) 2.60933 0.0945262
\(763\) 75.3320 2.72720
\(764\) 18.6565 0.674968
\(765\) 18.8308 0.680828
\(766\) −11.5183 −0.416174
\(767\) −33.4579 −1.20809
\(768\) 3.66092 0.132102
\(769\) 2.64416 0.0953507 0.0476753 0.998863i \(-0.484819\pi\)
0.0476753 + 0.998863i \(0.484819\pi\)
\(770\) 14.0741 0.507196
\(771\) −2.57256 −0.0926485
\(772\) −4.88189 −0.175703
\(773\) −20.9380 −0.753089 −0.376544 0.926399i \(-0.622888\pi\)
−0.376544 + 0.926399i \(0.622888\pi\)
\(774\) 10.9724 0.394396
\(775\) −0.386997 −0.0139013
\(776\) −10.0980 −0.362496
\(777\) −7.17078 −0.257250
\(778\) −0.205578 −0.00737034
\(779\) 0 0
\(780\) 4.86116 0.174058
\(781\) −12.3948 −0.443521
\(782\) 13.1169 0.469060
\(783\) −1.02352 −0.0365776
\(784\) 21.4689 0.766748
\(785\) 16.4433 0.586888
\(786\) 3.94360 0.140663
\(787\) 22.3722 0.797482 0.398741 0.917064i \(-0.369447\pi\)
0.398741 + 0.917064i \(0.369447\pi\)
\(788\) −20.3205 −0.723888
\(789\) 4.08999 0.145608
\(790\) 14.6286 0.520462
\(791\) 39.6163 1.40859
\(792\) −12.1543 −0.431883
\(793\) −2.76041 −0.0980251
\(794\) 19.9993 0.709750
\(795\) 6.22451 0.220761
\(796\) 13.9199 0.493376
\(797\) 20.1097 0.712321 0.356160 0.934425i \(-0.384086\pi\)
0.356160 + 0.934425i \(0.384086\pi\)
\(798\) −1.14954 −0.0406933
\(799\) 13.8098 0.488555
\(800\) 3.82517 0.135240
\(801\) −10.6156 −0.375085
\(802\) 4.78862 0.169092
\(803\) −16.5804 −0.585108
\(804\) −2.10071 −0.0740864
\(805\) −78.9377 −2.78219
\(806\) 1.65574 0.0583210
\(807\) 7.26795 0.255844
\(808\) 25.7781 0.906869
\(809\) −9.18730 −0.323008 −0.161504 0.986872i \(-0.551635\pi\)
−0.161504 + 0.986872i \(0.551635\pi\)
\(810\) −13.3685 −0.469720
\(811\) 18.4060 0.646324 0.323162 0.946344i \(-0.395254\pi\)
0.323162 + 0.946344i \(0.395254\pi\)
\(812\) 3.73857 0.131198
\(813\) 1.31495 0.0461172
\(814\) −5.12467 −0.179620
\(815\) −18.3665 −0.643351
\(816\) −1.18098 −0.0413425
\(817\) −5.19849 −0.181872
\(818\) −11.9473 −0.417728
\(819\) 56.2166 1.96437
\(820\) 0 0
\(821\) 18.0353 0.629436 0.314718 0.949185i \(-0.398090\pi\)
0.314718 + 0.949185i \(0.398090\pi\)
\(822\) −4.29992 −0.149977
\(823\) −44.4796 −1.55046 −0.775230 0.631679i \(-0.782366\pi\)
−0.775230 + 0.631679i \(0.782366\pi\)
\(824\) −1.01382 −0.0353181
\(825\) 0.383531 0.0133528
\(826\) 29.2582 1.01802
\(827\) −44.9128 −1.56177 −0.780886 0.624674i \(-0.785232\pi\)
−0.780886 + 0.624674i \(0.785232\pi\)
\(828\) 29.2221 1.01554
\(829\) −45.0253 −1.56379 −0.781896 0.623409i \(-0.785747\pi\)
−0.781896 + 0.623409i \(0.785747\pi\)
\(830\) −8.09937 −0.281133
\(831\) 0.400720 0.0139008
\(832\) −6.41716 −0.222475
\(833\) 47.0729 1.63098
\(834\) 0.516530 0.0178860
\(835\) −45.1366 −1.56202
\(836\) 2.46844 0.0853729
\(837\) −1.19105 −0.0411688
\(838\) 8.68667 0.300076
\(839\) −10.9413 −0.377737 −0.188869 0.982002i \(-0.560482\pi\)
−0.188869 + 0.982002i \(0.560482\pi\)
\(840\) −9.91676 −0.342161
\(841\) −28.7428 −0.991132
\(842\) −20.5880 −0.709508
\(843\) −6.07159 −0.209117
\(844\) 40.8468 1.40601
\(845\) −6.56841 −0.225960
\(846\) −10.2392 −0.352031
\(847\) −39.7654 −1.36636
\(848\) 9.55778 0.328216
\(849\) −2.54251 −0.0872585
\(850\) 1.27311 0.0436674
\(851\) 28.7428 0.985290
\(852\) 3.74376 0.128259
\(853\) −16.7236 −0.572607 −0.286303 0.958139i \(-0.592427\pi\)
−0.286303 + 0.958139i \(0.592427\pi\)
\(854\) 2.41392 0.0826027
\(855\) 6.61489 0.226224
\(856\) −34.5825 −1.18201
\(857\) 11.1225 0.379937 0.189968 0.981790i \(-0.439161\pi\)
0.189968 + 0.981790i \(0.439161\pi\)
\(858\) −1.64091 −0.0560199
\(859\) −47.3689 −1.61620 −0.808102 0.589042i \(-0.799505\pi\)
−0.808102 + 0.589042i \(0.799505\pi\)
\(860\) −19.2240 −0.655532
\(861\) 0 0
\(862\) −6.08343 −0.207203
\(863\) 14.0929 0.479729 0.239864 0.970806i \(-0.422897\pi\)
0.239864 + 0.970806i \(0.422897\pi\)
\(864\) 11.7727 0.400514
\(865\) −30.8498 −1.04892
\(866\) 8.36675 0.284314
\(867\) 3.24339 0.110151
\(868\) 4.35052 0.147666
\(869\) 14.8371 0.503313
\(870\) −0.292433 −0.00991441
\(871\) 16.1988 0.548874
\(872\) −37.9339 −1.28460
\(873\) −11.7652 −0.398191
\(874\) 4.60773 0.155859
\(875\) 50.7554 1.71585
\(876\) 5.00797 0.169204
\(877\) −35.3772 −1.19460 −0.597301 0.802017i \(-0.703760\pi\)
−0.597301 + 0.802017i \(0.703760\pi\)
\(878\) 21.4930 0.725353
\(879\) 3.55203 0.119807
\(880\) 5.07918 0.171219
\(881\) −21.6923 −0.730832 −0.365416 0.930844i \(-0.619073\pi\)
−0.365416 + 0.930844i \(0.619073\pi\)
\(882\) −34.9020 −1.17521
\(883\) −3.22329 −0.108472 −0.0542362 0.998528i \(-0.517272\pi\)
−0.0542362 + 0.998528i \(0.517272\pi\)
\(884\) 16.3664 0.550460
\(885\) 6.87651 0.231151
\(886\) 9.43673 0.317033
\(887\) −14.4724 −0.485935 −0.242967 0.970034i \(-0.578121\pi\)
−0.242967 + 0.970034i \(0.578121\pi\)
\(888\) 3.61089 0.121173
\(889\) −52.8681 −1.77314
\(890\) −6.18994 −0.207487
\(891\) −13.5590 −0.454242
\(892\) −14.3025 −0.478882
\(893\) 4.85111 0.162336
\(894\) 0.367902 0.0123045
\(895\) 1.31635 0.0440006
\(896\) −51.7014 −1.72722
\(897\) 9.20341 0.307293
\(898\) 17.7735 0.593108
\(899\) 0.299280 0.00998156
\(900\) 2.83627 0.0945422
\(901\) 20.9564 0.698160
\(902\) 0 0
\(903\) 9.08007 0.302166
\(904\) −19.9490 −0.663494
\(905\) 57.1361 1.89927
\(906\) −0.508198 −0.0168837
\(907\) 42.1901 1.40090 0.700450 0.713702i \(-0.252983\pi\)
0.700450 + 0.713702i \(0.252983\pi\)
\(908\) 2.28476 0.0758225
\(909\) 30.0341 0.996168
\(910\) 32.7797 1.08664
\(911\) 46.8754 1.55305 0.776525 0.630087i \(-0.216981\pi\)
0.776525 + 0.630087i \(0.216981\pi\)
\(912\) −0.414855 −0.0137372
\(913\) −8.21479 −0.271870
\(914\) 22.7375 0.752090
\(915\) 0.567340 0.0187557
\(916\) 22.1198 0.730859
\(917\) −79.9018 −2.63859
\(918\) 3.91824 0.129321
\(919\) −26.0204 −0.858333 −0.429167 0.903225i \(-0.641193\pi\)
−0.429167 + 0.903225i \(0.641193\pi\)
\(920\) 39.7495 1.31050
\(921\) 0.471322 0.0155306
\(922\) 20.8066 0.685230
\(923\) −28.8684 −0.950215
\(924\) −4.31156 −0.141840
\(925\) 2.78974 0.0917262
\(926\) 14.5020 0.476564
\(927\) −1.18120 −0.0387959
\(928\) −2.95816 −0.0971063
\(929\) −36.2974 −1.19088 −0.595440 0.803400i \(-0.703022\pi\)
−0.595440 + 0.803400i \(0.703022\pi\)
\(930\) −0.340300 −0.0111589
\(931\) 16.5358 0.541939
\(932\) 22.3394 0.731753
\(933\) −4.16868 −0.136477
\(934\) −17.4505 −0.570999
\(935\) 11.1366 0.364207
\(936\) −28.3082 −0.925282
\(937\) 11.1948 0.365719 0.182859 0.983139i \(-0.441465\pi\)
0.182859 + 0.983139i \(0.441465\pi\)
\(938\) −14.1655 −0.462519
\(939\) −10.3324 −0.337184
\(940\) 17.9394 0.585117
\(941\) −15.4599 −0.503977 −0.251989 0.967730i \(-0.581085\pi\)
−0.251989 + 0.967730i \(0.581085\pi\)
\(942\) 1.67650 0.0546234
\(943\) 0 0
\(944\) 10.5589 0.343664
\(945\) −23.5800 −0.767058
\(946\) 6.48916 0.210981
\(947\) 37.9174 1.23215 0.616074 0.787688i \(-0.288722\pi\)
0.616074 + 0.787688i \(0.288722\pi\)
\(948\) −4.48143 −0.145550
\(949\) −38.6169 −1.25356
\(950\) 0.447221 0.0145098
\(951\) 2.90662 0.0942536
\(952\) −33.3873 −1.08209
\(953\) −35.0022 −1.13383 −0.566916 0.823775i \(-0.691864\pi\)
−0.566916 + 0.823775i \(0.691864\pi\)
\(954\) −15.5381 −0.503064
\(955\) 29.5675 0.956783
\(956\) 20.0676 0.649033
\(957\) −0.296600 −0.00958772
\(958\) −20.5020 −0.662388
\(959\) 87.1213 2.81329
\(960\) 1.31890 0.0425674
\(961\) −30.6517 −0.988766
\(962\) −11.9357 −0.384824
\(963\) −40.2923 −1.29840
\(964\) 5.99522 0.193093
\(965\) −7.73701 −0.249063
\(966\) −8.04819 −0.258946
\(967\) 19.0146 0.611468 0.305734 0.952117i \(-0.401098\pi\)
0.305734 + 0.952117i \(0.401098\pi\)
\(968\) 20.0241 0.643600
\(969\) −0.909612 −0.0292210
\(970\) −6.86023 −0.220269
\(971\) −37.8905 −1.21596 −0.607982 0.793951i \(-0.708021\pi\)
−0.607982 + 0.793951i \(0.708021\pi\)
\(972\) 13.1810 0.422782
\(973\) −10.4655 −0.335508
\(974\) −18.6429 −0.597356
\(975\) 0.893272 0.0286076
\(976\) 0.871155 0.0278850
\(977\) 22.0118 0.704221 0.352110 0.935958i \(-0.385464\pi\)
0.352110 + 0.935958i \(0.385464\pi\)
\(978\) −1.87258 −0.0598785
\(979\) −6.27815 −0.200651
\(980\) 61.1492 1.95334
\(981\) −44.1969 −1.41110
\(982\) −12.3493 −0.394083
\(983\) 12.0990 0.385900 0.192950 0.981209i \(-0.438195\pi\)
0.192950 + 0.981209i \(0.438195\pi\)
\(984\) 0 0
\(985\) −32.2047 −1.02613
\(986\) −0.984551 −0.0313545
\(987\) −8.47331 −0.269708
\(988\) 5.74919 0.182906
\(989\) −36.3958 −1.15732
\(990\) −8.25721 −0.262431
\(991\) 37.5585 1.19309 0.596543 0.802581i \(-0.296541\pi\)
0.596543 + 0.802581i \(0.296541\pi\)
\(992\) −3.44237 −0.109295
\(993\) 3.58403 0.113736
\(994\) 25.2448 0.800717
\(995\) 22.0607 0.699372
\(996\) 2.48122 0.0786204
\(997\) 7.69301 0.243640 0.121820 0.992552i \(-0.461127\pi\)
0.121820 + 0.992552i \(0.461127\pi\)
\(998\) −4.46630 −0.141378
\(999\) 8.58595 0.271647
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 1681.2.a.m.1.13 24
41.34 odd 40 41.2.g.a.8.2 24
41.35 odd 40 41.2.g.a.36.2 yes 24
41.40 even 2 inner 1681.2.a.m.1.14 24
123.35 even 40 369.2.u.a.118.2 24
123.116 even 40 369.2.u.a.172.2 24
164.35 even 40 656.2.bs.d.241.2 24
164.75 even 40 656.2.bs.d.49.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
41.2.g.a.8.2 24 41.34 odd 40
41.2.g.a.36.2 yes 24 41.35 odd 40
369.2.u.a.118.2 24 123.35 even 40
369.2.u.a.172.2 24 123.116 even 40
656.2.bs.d.49.2 24 164.75 even 40
656.2.bs.d.241.2 24 164.35 even 40
1681.2.a.m.1.13 24 1.1 even 1 trivial
1681.2.a.m.1.14 24 41.40 even 2 inner