Properties

Label 4005.2.a.x.1.3
Level $4005$
Weight $2$
Character 4005.1
Self dual yes
Analytic conductor $31.980$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} - 2456 x^{9} - 7002 x^{8} + 6279 x^{7} + 7299 x^{6} - 7119 x^{5} - 3066 x^{4} + \cdots + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.59605\) of defining polynomial
Character \(\chi\) \(=\) 4005.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.59605 q^{2} +0.547372 q^{4} +1.00000 q^{5} +1.65189 q^{7} +2.31847 q^{8} +O(q^{10})\) \(q-1.59605 q^{2} +0.547372 q^{4} +1.00000 q^{5} +1.65189 q^{7} +2.31847 q^{8} -1.59605 q^{10} -0.312701 q^{11} -4.26577 q^{13} -2.63650 q^{14} -4.79513 q^{16} +3.29407 q^{17} +2.57815 q^{19} +0.547372 q^{20} +0.499086 q^{22} -5.06311 q^{23} +1.00000 q^{25} +6.80838 q^{26} +0.904200 q^{28} -5.30438 q^{29} +6.85629 q^{31} +3.01633 q^{32} -5.25750 q^{34} +1.65189 q^{35} +9.86431 q^{37} -4.11485 q^{38} +2.31847 q^{40} -4.30603 q^{41} +1.15263 q^{43} -0.171164 q^{44} +8.08097 q^{46} -8.26666 q^{47} -4.27125 q^{49} -1.59605 q^{50} -2.33496 q^{52} +6.28163 q^{53} -0.312701 q^{55} +3.82986 q^{56} +8.46605 q^{58} +5.37742 q^{59} +11.7981 q^{61} -10.9430 q^{62} +4.77605 q^{64} -4.26577 q^{65} +0.735092 q^{67} +1.80308 q^{68} -2.63650 q^{70} +3.08726 q^{71} +4.86570 q^{73} -15.7439 q^{74} +1.41120 q^{76} -0.516549 q^{77} +10.4024 q^{79} -4.79513 q^{80} +6.87263 q^{82} +6.74042 q^{83} +3.29407 q^{85} -1.83965 q^{86} -0.724987 q^{88} -1.00000 q^{89} -7.04660 q^{91} -2.77140 q^{92} +13.1940 q^{94} +2.57815 q^{95} +4.83568 q^{97} +6.81712 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.59605 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(3\) 0 0
\(4\) 0.547372 0.273686
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.65189 0.624357 0.312179 0.950023i \(-0.398941\pi\)
0.312179 + 0.950023i \(0.398941\pi\)
\(8\) 2.31847 0.819701
\(9\) 0 0
\(10\) −1.59605 −0.504715
\(11\) −0.312701 −0.0942830 −0.0471415 0.998888i \(-0.515011\pi\)
−0.0471415 + 0.998888i \(0.515011\pi\)
\(12\) 0 0
\(13\) −4.26577 −1.18311 −0.591556 0.806264i \(-0.701486\pi\)
−0.591556 + 0.806264i \(0.701486\pi\)
\(14\) −2.63650 −0.704635
\(15\) 0 0
\(16\) −4.79513 −1.19878
\(17\) 3.29407 0.798929 0.399465 0.916749i \(-0.369196\pi\)
0.399465 + 0.916749i \(0.369196\pi\)
\(18\) 0 0
\(19\) 2.57815 0.591467 0.295734 0.955270i \(-0.404436\pi\)
0.295734 + 0.955270i \(0.404436\pi\)
\(20\) 0.547372 0.122396
\(21\) 0 0
\(22\) 0.499086 0.106406
\(23\) −5.06311 −1.05573 −0.527866 0.849328i \(-0.677008\pi\)
−0.527866 + 0.849328i \(0.677008\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.80838 1.33523
\(27\) 0 0
\(28\) 0.904200 0.170878
\(29\) −5.30438 −0.984999 −0.492499 0.870313i \(-0.663917\pi\)
−0.492499 + 0.870313i \(0.663917\pi\)
\(30\) 0 0
\(31\) 6.85629 1.23143 0.615713 0.787971i \(-0.288868\pi\)
0.615713 + 0.787971i \(0.288868\pi\)
\(32\) 3.01633 0.533216
\(33\) 0 0
\(34\) −5.25750 −0.901653
\(35\) 1.65189 0.279221
\(36\) 0 0
\(37\) 9.86431 1.62168 0.810841 0.585266i \(-0.199010\pi\)
0.810841 + 0.585266i \(0.199010\pi\)
\(38\) −4.11485 −0.667516
\(39\) 0 0
\(40\) 2.31847 0.366582
\(41\) −4.30603 −0.672488 −0.336244 0.941775i \(-0.609157\pi\)
−0.336244 + 0.941775i \(0.609157\pi\)
\(42\) 0 0
\(43\) 1.15263 0.175774 0.0878868 0.996130i \(-0.471989\pi\)
0.0878868 + 0.996130i \(0.471989\pi\)
\(44\) −0.171164 −0.0258039
\(45\) 0 0
\(46\) 8.08097 1.19147
\(47\) −8.26666 −1.20582 −0.602908 0.797810i \(-0.705992\pi\)
−0.602908 + 0.797810i \(0.705992\pi\)
\(48\) 0 0
\(49\) −4.27125 −0.610178
\(50\) −1.59605 −0.225715
\(51\) 0 0
\(52\) −2.33496 −0.323801
\(53\) 6.28163 0.862848 0.431424 0.902149i \(-0.358011\pi\)
0.431424 + 0.902149i \(0.358011\pi\)
\(54\) 0 0
\(55\) −0.312701 −0.0421646
\(56\) 3.82986 0.511786
\(57\) 0 0
\(58\) 8.46605 1.11165
\(59\) 5.37742 0.700081 0.350040 0.936735i \(-0.386168\pi\)
0.350040 + 0.936735i \(0.386168\pi\)
\(60\) 0 0
\(61\) 11.7981 1.51059 0.755297 0.655383i \(-0.227493\pi\)
0.755297 + 0.655383i \(0.227493\pi\)
\(62\) −10.9430 −1.38976
\(63\) 0 0
\(64\) 4.77605 0.597006
\(65\) −4.26577 −0.529104
\(66\) 0 0
\(67\) 0.735092 0.0898058 0.0449029 0.998991i \(-0.485702\pi\)
0.0449029 + 0.998991i \(0.485702\pi\)
\(68\) 1.80308 0.218656
\(69\) 0 0
\(70\) −2.63650 −0.315122
\(71\) 3.08726 0.366390 0.183195 0.983077i \(-0.441356\pi\)
0.183195 + 0.983077i \(0.441356\pi\)
\(72\) 0 0
\(73\) 4.86570 0.569487 0.284743 0.958604i \(-0.408092\pi\)
0.284743 + 0.958604i \(0.408092\pi\)
\(74\) −15.7439 −1.83019
\(75\) 0 0
\(76\) 1.41120 0.161876
\(77\) −0.516549 −0.0588663
\(78\) 0 0
\(79\) 10.4024 1.17036 0.585182 0.810902i \(-0.301023\pi\)
0.585182 + 0.810902i \(0.301023\pi\)
\(80\) −4.79513 −0.536112
\(81\) 0 0
\(82\) 6.87263 0.758955
\(83\) 6.74042 0.739858 0.369929 0.929060i \(-0.379382\pi\)
0.369929 + 0.929060i \(0.379382\pi\)
\(84\) 0 0
\(85\) 3.29407 0.357292
\(86\) −1.83965 −0.198374
\(87\) 0 0
\(88\) −0.724987 −0.0772839
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −7.04660 −0.738685
\(92\) −2.77140 −0.288939
\(93\) 0 0
\(94\) 13.1940 1.36086
\(95\) 2.57815 0.264512
\(96\) 0 0
\(97\) 4.83568 0.490989 0.245495 0.969398i \(-0.421050\pi\)
0.245495 + 0.969398i \(0.421050\pi\)
\(98\) 6.81712 0.688633
\(99\) 0 0
\(100\) 0.547372 0.0547372
\(101\) −17.9368 −1.78478 −0.892389 0.451268i \(-0.850972\pi\)
−0.892389 + 0.451268i \(0.850972\pi\)
\(102\) 0 0
\(103\) −12.1543 −1.19759 −0.598797 0.800901i \(-0.704354\pi\)
−0.598797 + 0.800901i \(0.704354\pi\)
\(104\) −9.89004 −0.969799
\(105\) 0 0
\(106\) −10.0258 −0.973790
\(107\) 2.47111 0.238891 0.119446 0.992841i \(-0.461888\pi\)
0.119446 + 0.992841i \(0.461888\pi\)
\(108\) 0 0
\(109\) 10.6997 1.02485 0.512425 0.858732i \(-0.328747\pi\)
0.512425 + 0.858732i \(0.328747\pi\)
\(110\) 0.499086 0.0475860
\(111\) 0 0
\(112\) −7.92104 −0.748468
\(113\) 17.1698 1.61520 0.807599 0.589732i \(-0.200767\pi\)
0.807599 + 0.589732i \(0.200767\pi\)
\(114\) 0 0
\(115\) −5.06311 −0.472138
\(116\) −2.90347 −0.269580
\(117\) 0 0
\(118\) −8.58263 −0.790095
\(119\) 5.44145 0.498817
\(120\) 0 0
\(121\) −10.9022 −0.991111
\(122\) −18.8304 −1.70482
\(123\) 0 0
\(124\) 3.75294 0.337024
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.530525 −0.0470765 −0.0235382 0.999723i \(-0.507493\pi\)
−0.0235382 + 0.999723i \(0.507493\pi\)
\(128\) −13.6555 −1.20698
\(129\) 0 0
\(130\) 6.80838 0.597134
\(131\) −3.01167 −0.263131 −0.131565 0.991308i \(-0.542000\pi\)
−0.131565 + 0.991308i \(0.542000\pi\)
\(132\) 0 0
\(133\) 4.25882 0.369287
\(134\) −1.17324 −0.101353
\(135\) 0 0
\(136\) 7.63719 0.654883
\(137\) −11.0835 −0.946925 −0.473463 0.880814i \(-0.656996\pi\)
−0.473463 + 0.880814i \(0.656996\pi\)
\(138\) 0 0
\(139\) 5.13255 0.435337 0.217669 0.976023i \(-0.430155\pi\)
0.217669 + 0.976023i \(0.430155\pi\)
\(140\) 0.904200 0.0764188
\(141\) 0 0
\(142\) −4.92742 −0.413500
\(143\) 1.33391 0.111547
\(144\) 0 0
\(145\) −5.30438 −0.440505
\(146\) −7.76589 −0.642710
\(147\) 0 0
\(148\) 5.39944 0.443831
\(149\) −1.19416 −0.0978294 −0.0489147 0.998803i \(-0.515576\pi\)
−0.0489147 + 0.998803i \(0.515576\pi\)
\(150\) 0 0
\(151\) −1.29569 −0.105442 −0.0527208 0.998609i \(-0.516789\pi\)
−0.0527208 + 0.998609i \(0.516789\pi\)
\(152\) 5.97734 0.484826
\(153\) 0 0
\(154\) 0.824438 0.0664351
\(155\) 6.85629 0.550710
\(156\) 0 0
\(157\) 19.1427 1.52775 0.763876 0.645363i \(-0.223294\pi\)
0.763876 + 0.645363i \(0.223294\pi\)
\(158\) −16.6028 −1.32084
\(159\) 0 0
\(160\) 3.01633 0.238462
\(161\) −8.36373 −0.659154
\(162\) 0 0
\(163\) −21.1277 −1.65485 −0.827424 0.561578i \(-0.810195\pi\)
−0.827424 + 0.561578i \(0.810195\pi\)
\(164\) −2.35700 −0.184050
\(165\) 0 0
\(166\) −10.7580 −0.834986
\(167\) 11.5765 0.895813 0.447906 0.894080i \(-0.352170\pi\)
0.447906 + 0.894080i \(0.352170\pi\)
\(168\) 0 0
\(169\) 5.19681 0.399754
\(170\) −5.25750 −0.403232
\(171\) 0 0
\(172\) 0.630914 0.0481068
\(173\) −16.3647 −1.24419 −0.622093 0.782943i \(-0.713718\pi\)
−0.622093 + 0.782943i \(0.713718\pi\)
\(174\) 0 0
\(175\) 1.65189 0.124871
\(176\) 1.49944 0.113025
\(177\) 0 0
\(178\) 1.59605 0.119629
\(179\) −14.5100 −1.08452 −0.542262 0.840209i \(-0.682432\pi\)
−0.542262 + 0.840209i \(0.682432\pi\)
\(180\) 0 0
\(181\) −20.6306 −1.53346 −0.766731 0.641968i \(-0.778118\pi\)
−0.766731 + 0.641968i \(0.778118\pi\)
\(182\) 11.2467 0.833662
\(183\) 0 0
\(184\) −11.7387 −0.865385
\(185\) 9.86431 0.725238
\(186\) 0 0
\(187\) −1.03006 −0.0753254
\(188\) −4.52494 −0.330015
\(189\) 0 0
\(190\) −4.11485 −0.298522
\(191\) 19.7013 1.42554 0.712768 0.701400i \(-0.247441\pi\)
0.712768 + 0.701400i \(0.247441\pi\)
\(192\) 0 0
\(193\) 18.5596 1.33595 0.667975 0.744184i \(-0.267161\pi\)
0.667975 + 0.744184i \(0.267161\pi\)
\(194\) −7.71799 −0.554119
\(195\) 0 0
\(196\) −2.33796 −0.166997
\(197\) 13.1125 0.934229 0.467115 0.884197i \(-0.345294\pi\)
0.467115 + 0.884197i \(0.345294\pi\)
\(198\) 0 0
\(199\) 22.3651 1.58542 0.792711 0.609598i \(-0.208669\pi\)
0.792711 + 0.609598i \(0.208669\pi\)
\(200\) 2.31847 0.163940
\(201\) 0 0
\(202\) 28.6280 2.01426
\(203\) −8.76227 −0.614991
\(204\) 0 0
\(205\) −4.30603 −0.300746
\(206\) 19.3988 1.35158
\(207\) 0 0
\(208\) 20.4549 1.41829
\(209\) −0.806189 −0.0557653
\(210\) 0 0
\(211\) −17.5062 −1.20518 −0.602589 0.798051i \(-0.705864\pi\)
−0.602589 + 0.798051i \(0.705864\pi\)
\(212\) 3.43838 0.236149
\(213\) 0 0
\(214\) −3.94401 −0.269607
\(215\) 1.15263 0.0786084
\(216\) 0 0
\(217\) 11.3259 0.768850
\(218\) −17.0773 −1.15662
\(219\) 0 0
\(220\) −0.171164 −0.0115399
\(221\) −14.0517 −0.945223
\(222\) 0 0
\(223\) 22.4098 1.50067 0.750336 0.661057i \(-0.229892\pi\)
0.750336 + 0.661057i \(0.229892\pi\)
\(224\) 4.98265 0.332917
\(225\) 0 0
\(226\) −27.4038 −1.82288
\(227\) 27.4458 1.82164 0.910821 0.412801i \(-0.135449\pi\)
0.910821 + 0.412801i \(0.135449\pi\)
\(228\) 0 0
\(229\) 26.7472 1.76750 0.883751 0.467958i \(-0.155010\pi\)
0.883751 + 0.467958i \(0.155010\pi\)
\(230\) 8.08097 0.532844
\(231\) 0 0
\(232\) −12.2980 −0.807405
\(233\) 3.78149 0.247734 0.123867 0.992299i \(-0.460470\pi\)
0.123867 + 0.992299i \(0.460470\pi\)
\(234\) 0 0
\(235\) −8.26666 −0.539258
\(236\) 2.94345 0.191602
\(237\) 0 0
\(238\) −8.68483 −0.562954
\(239\) −20.3619 −1.31710 −0.658550 0.752537i \(-0.728830\pi\)
−0.658550 + 0.752537i \(0.728830\pi\)
\(240\) 0 0
\(241\) −2.45826 −0.158350 −0.0791752 0.996861i \(-0.525229\pi\)
−0.0791752 + 0.996861i \(0.525229\pi\)
\(242\) 17.4005 1.11854
\(243\) 0 0
\(244\) 6.45795 0.413428
\(245\) −4.27125 −0.272880
\(246\) 0 0
\(247\) −10.9978 −0.699772
\(248\) 15.8961 1.00940
\(249\) 0 0
\(250\) −1.59605 −0.100943
\(251\) −20.7248 −1.30814 −0.654070 0.756434i \(-0.726940\pi\)
−0.654070 + 0.756434i \(0.726940\pi\)
\(252\) 0 0
\(253\) 1.58324 0.0995376
\(254\) 0.846743 0.0531294
\(255\) 0 0
\(256\) 12.2427 0.765168
\(257\) 11.2294 0.700473 0.350237 0.936661i \(-0.386101\pi\)
0.350237 + 0.936661i \(0.386101\pi\)
\(258\) 0 0
\(259\) 16.2948 1.01251
\(260\) −2.33496 −0.144808
\(261\) 0 0
\(262\) 4.80677 0.296963
\(263\) 2.14224 0.132096 0.0660480 0.997816i \(-0.478961\pi\)
0.0660480 + 0.997816i \(0.478961\pi\)
\(264\) 0 0
\(265\) 6.28163 0.385877
\(266\) −6.79729 −0.416768
\(267\) 0 0
\(268\) 0.402369 0.0245786
\(269\) 12.2972 0.749774 0.374887 0.927070i \(-0.377681\pi\)
0.374887 + 0.927070i \(0.377681\pi\)
\(270\) 0 0
\(271\) 6.67702 0.405600 0.202800 0.979220i \(-0.434996\pi\)
0.202800 + 0.979220i \(0.434996\pi\)
\(272\) −15.7955 −0.957742
\(273\) 0 0
\(274\) 17.6898 1.06868
\(275\) −0.312701 −0.0188566
\(276\) 0 0
\(277\) 9.65816 0.580303 0.290151 0.956981i \(-0.406294\pi\)
0.290151 + 0.956981i \(0.406294\pi\)
\(278\) −8.19180 −0.491312
\(279\) 0 0
\(280\) 3.82986 0.228878
\(281\) 19.0745 1.13789 0.568944 0.822376i \(-0.307352\pi\)
0.568944 + 0.822376i \(0.307352\pi\)
\(282\) 0 0
\(283\) 3.01850 0.179431 0.0897156 0.995967i \(-0.471404\pi\)
0.0897156 + 0.995967i \(0.471404\pi\)
\(284\) 1.68988 0.100276
\(285\) 0 0
\(286\) −2.12899 −0.125890
\(287\) −7.11310 −0.419873
\(288\) 0 0
\(289\) −6.14910 −0.361712
\(290\) 8.46605 0.497144
\(291\) 0 0
\(292\) 2.66334 0.155860
\(293\) −22.7079 −1.32661 −0.663305 0.748349i \(-0.730847\pi\)
−0.663305 + 0.748349i \(0.730847\pi\)
\(294\) 0 0
\(295\) 5.37742 0.313086
\(296\) 22.8701 1.32930
\(297\) 0 0
\(298\) 1.90594 0.110408
\(299\) 21.5981 1.24905
\(300\) 0 0
\(301\) 1.90401 0.109746
\(302\) 2.06798 0.118999
\(303\) 0 0
\(304\) −12.3625 −0.709040
\(305\) 11.7981 0.675558
\(306\) 0 0
\(307\) 22.4280 1.28003 0.640016 0.768362i \(-0.278928\pi\)
0.640016 + 0.768362i \(0.278928\pi\)
\(308\) −0.282744 −0.0161109
\(309\) 0 0
\(310\) −10.9430 −0.621519
\(311\) 2.80828 0.159243 0.0796213 0.996825i \(-0.474629\pi\)
0.0796213 + 0.996825i \(0.474629\pi\)
\(312\) 0 0
\(313\) 24.0478 1.35926 0.679631 0.733554i \(-0.262140\pi\)
0.679631 + 0.733554i \(0.262140\pi\)
\(314\) −30.5527 −1.72419
\(315\) 0 0
\(316\) 5.69399 0.320312
\(317\) −0.567561 −0.0318774 −0.0159387 0.999873i \(-0.505074\pi\)
−0.0159387 + 0.999873i \(0.505074\pi\)
\(318\) 0 0
\(319\) 1.65869 0.0928686
\(320\) 4.77605 0.266989
\(321\) 0 0
\(322\) 13.3489 0.743906
\(323\) 8.49259 0.472540
\(324\) 0 0
\(325\) −4.26577 −0.236622
\(326\) 33.7208 1.86762
\(327\) 0 0
\(328\) −9.98337 −0.551239
\(329\) −13.6557 −0.752860
\(330\) 0 0
\(331\) 21.8342 1.20012 0.600059 0.799956i \(-0.295144\pi\)
0.600059 + 0.799956i \(0.295144\pi\)
\(332\) 3.68952 0.202489
\(333\) 0 0
\(334\) −18.4766 −1.01099
\(335\) 0.735092 0.0401624
\(336\) 0 0
\(337\) 26.8567 1.46298 0.731490 0.681852i \(-0.238825\pi\)
0.731490 + 0.681852i \(0.238825\pi\)
\(338\) −8.29436 −0.451153
\(339\) 0 0
\(340\) 1.80308 0.0977858
\(341\) −2.14397 −0.116102
\(342\) 0 0
\(343\) −18.6189 −1.00533
\(344\) 2.67232 0.144082
\(345\) 0 0
\(346\) 26.1189 1.40416
\(347\) 19.1660 1.02889 0.514443 0.857525i \(-0.327999\pi\)
0.514443 + 0.857525i \(0.327999\pi\)
\(348\) 0 0
\(349\) −16.8496 −0.901937 −0.450968 0.892540i \(-0.648921\pi\)
−0.450968 + 0.892540i \(0.648921\pi\)
\(350\) −2.63650 −0.140927
\(351\) 0 0
\(352\) −0.943209 −0.0502732
\(353\) 20.7985 1.10699 0.553497 0.832851i \(-0.313293\pi\)
0.553497 + 0.832851i \(0.313293\pi\)
\(354\) 0 0
\(355\) 3.08726 0.163855
\(356\) −0.547372 −0.0290106
\(357\) 0 0
\(358\) 23.1586 1.22397
\(359\) 24.1169 1.27284 0.636419 0.771343i \(-0.280415\pi\)
0.636419 + 0.771343i \(0.280415\pi\)
\(360\) 0 0
\(361\) −12.3532 −0.650167
\(362\) 32.9275 1.73063
\(363\) 0 0
\(364\) −3.85711 −0.202167
\(365\) 4.86570 0.254682
\(366\) 0 0
\(367\) 1.78328 0.0930864 0.0465432 0.998916i \(-0.485179\pi\)
0.0465432 + 0.998916i \(0.485179\pi\)
\(368\) 24.2783 1.26559
\(369\) 0 0
\(370\) −15.7439 −0.818487
\(371\) 10.3766 0.538725
\(372\) 0 0
\(373\) 4.80838 0.248968 0.124484 0.992222i \(-0.460272\pi\)
0.124484 + 0.992222i \(0.460272\pi\)
\(374\) 1.64403 0.0850105
\(375\) 0 0
\(376\) −19.1660 −0.988410
\(377\) 22.6273 1.16536
\(378\) 0 0
\(379\) 14.9770 0.769319 0.384659 0.923059i \(-0.374319\pi\)
0.384659 + 0.923059i \(0.374319\pi\)
\(380\) 1.41120 0.0723932
\(381\) 0 0
\(382\) −31.4442 −1.60883
\(383\) 16.5874 0.847579 0.423789 0.905761i \(-0.360700\pi\)
0.423789 + 0.905761i \(0.360700\pi\)
\(384\) 0 0
\(385\) −0.516549 −0.0263258
\(386\) −29.6220 −1.50772
\(387\) 0 0
\(388\) 2.64692 0.134377
\(389\) 4.03093 0.204376 0.102188 0.994765i \(-0.467416\pi\)
0.102188 + 0.994765i \(0.467416\pi\)
\(390\) 0 0
\(391\) −16.6782 −0.843455
\(392\) −9.90274 −0.500164
\(393\) 0 0
\(394\) −20.9282 −1.05435
\(395\) 10.4024 0.523402
\(396\) 0 0
\(397\) 3.69116 0.185254 0.0926270 0.995701i \(-0.470474\pi\)
0.0926270 + 0.995701i \(0.470474\pi\)
\(398\) −35.6958 −1.78927
\(399\) 0 0
\(400\) −4.79513 −0.239756
\(401\) −20.7374 −1.03558 −0.517788 0.855509i \(-0.673244\pi\)
−0.517788 + 0.855509i \(0.673244\pi\)
\(402\) 0 0
\(403\) −29.2474 −1.45691
\(404\) −9.81809 −0.488468
\(405\) 0 0
\(406\) 13.9850 0.694065
\(407\) −3.08458 −0.152897
\(408\) 0 0
\(409\) 3.59295 0.177660 0.0888300 0.996047i \(-0.471687\pi\)
0.0888300 + 0.996047i \(0.471687\pi\)
\(410\) 6.87263 0.339415
\(411\) 0 0
\(412\) −6.65289 −0.327765
\(413\) 8.88293 0.437101
\(414\) 0 0
\(415\) 6.74042 0.330874
\(416\) −12.8670 −0.630854
\(417\) 0 0
\(418\) 1.28672 0.0629354
\(419\) −17.8945 −0.874204 −0.437102 0.899412i \(-0.643995\pi\)
−0.437102 + 0.899412i \(0.643995\pi\)
\(420\) 0 0
\(421\) −15.0678 −0.734362 −0.367181 0.930150i \(-0.619677\pi\)
−0.367181 + 0.930150i \(0.619677\pi\)
\(422\) 27.9408 1.36014
\(423\) 0 0
\(424\) 14.5637 0.707278
\(425\) 3.29407 0.159786
\(426\) 0 0
\(427\) 19.4892 0.943150
\(428\) 1.35262 0.0653811
\(429\) 0 0
\(430\) −1.83965 −0.0887156
\(431\) 21.4719 1.03426 0.517132 0.855905i \(-0.326999\pi\)
0.517132 + 0.855905i \(0.326999\pi\)
\(432\) 0 0
\(433\) −2.12381 −0.102064 −0.0510319 0.998697i \(-0.516251\pi\)
−0.0510319 + 0.998697i \(0.516251\pi\)
\(434\) −18.0766 −0.867706
\(435\) 0 0
\(436\) 5.85674 0.280487
\(437\) −13.0534 −0.624431
\(438\) 0 0
\(439\) −38.5896 −1.84178 −0.920891 0.389819i \(-0.872538\pi\)
−0.920891 + 0.389819i \(0.872538\pi\)
\(440\) −0.724987 −0.0345624
\(441\) 0 0
\(442\) 22.4273 1.06676
\(443\) −18.5685 −0.882217 −0.441108 0.897454i \(-0.645415\pi\)
−0.441108 + 0.897454i \(0.645415\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) −35.7671 −1.69362
\(447\) 0 0
\(448\) 7.88953 0.372745
\(449\) 36.0323 1.70047 0.850236 0.526402i \(-0.176459\pi\)
0.850236 + 0.526402i \(0.176459\pi\)
\(450\) 0 0
\(451\) 1.34650 0.0634042
\(452\) 9.39826 0.442057
\(453\) 0 0
\(454\) −43.8049 −2.05586
\(455\) −7.04660 −0.330350
\(456\) 0 0
\(457\) 10.4574 0.489175 0.244588 0.969627i \(-0.421347\pi\)
0.244588 + 0.969627i \(0.421347\pi\)
\(458\) −42.6898 −1.99476
\(459\) 0 0
\(460\) −2.77140 −0.129217
\(461\) 0.185494 0.00863930 0.00431965 0.999991i \(-0.498625\pi\)
0.00431965 + 0.999991i \(0.498625\pi\)
\(462\) 0 0
\(463\) −7.19924 −0.334577 −0.167288 0.985908i \(-0.553501\pi\)
−0.167288 + 0.985908i \(0.553501\pi\)
\(464\) 25.4352 1.18080
\(465\) 0 0
\(466\) −6.03544 −0.279587
\(467\) 18.1602 0.840354 0.420177 0.907442i \(-0.361968\pi\)
0.420177 + 0.907442i \(0.361968\pi\)
\(468\) 0 0
\(469\) 1.21429 0.0560709
\(470\) 13.1940 0.608594
\(471\) 0 0
\(472\) 12.4674 0.573857
\(473\) −0.360427 −0.0165725
\(474\) 0 0
\(475\) 2.57815 0.118293
\(476\) 2.97850 0.136519
\(477\) 0 0
\(478\) 32.4985 1.48645
\(479\) −14.5331 −0.664035 −0.332018 0.943273i \(-0.607729\pi\)
−0.332018 + 0.943273i \(0.607729\pi\)
\(480\) 0 0
\(481\) −42.0789 −1.91863
\(482\) 3.92350 0.178710
\(483\) 0 0
\(484\) −5.96756 −0.271253
\(485\) 4.83568 0.219577
\(486\) 0 0
\(487\) 22.5099 1.02002 0.510011 0.860168i \(-0.329641\pi\)
0.510011 + 0.860168i \(0.329641\pi\)
\(488\) 27.3535 1.23824
\(489\) 0 0
\(490\) 6.81712 0.307966
\(491\) −14.0158 −0.632524 −0.316262 0.948672i \(-0.602428\pi\)
−0.316262 + 0.948672i \(0.602428\pi\)
\(492\) 0 0
\(493\) −17.4730 −0.786944
\(494\) 17.5530 0.789746
\(495\) 0 0
\(496\) −32.8768 −1.47621
\(497\) 5.09983 0.228758
\(498\) 0 0
\(499\) 21.7705 0.974583 0.487292 0.873239i \(-0.337985\pi\)
0.487292 + 0.873239i \(0.337985\pi\)
\(500\) 0.547372 0.0244792
\(501\) 0 0
\(502\) 33.0779 1.47634
\(503\) 21.0760 0.939731 0.469865 0.882738i \(-0.344302\pi\)
0.469865 + 0.882738i \(0.344302\pi\)
\(504\) 0 0
\(505\) −17.9368 −0.798177
\(506\) −2.52693 −0.112336
\(507\) 0 0
\(508\) −0.290394 −0.0128842
\(509\) −13.1967 −0.584934 −0.292467 0.956276i \(-0.594476\pi\)
−0.292467 + 0.956276i \(0.594476\pi\)
\(510\) 0 0
\(511\) 8.03761 0.355563
\(512\) 7.77101 0.343433
\(513\) 0 0
\(514\) −17.9227 −0.790538
\(515\) −12.1543 −0.535581
\(516\) 0 0
\(517\) 2.58500 0.113688
\(518\) −26.0073 −1.14269
\(519\) 0 0
\(520\) −9.89004 −0.433707
\(521\) −31.6444 −1.38637 −0.693183 0.720762i \(-0.743792\pi\)
−0.693183 + 0.720762i \(0.743792\pi\)
\(522\) 0 0
\(523\) 19.5164 0.853393 0.426696 0.904395i \(-0.359677\pi\)
0.426696 + 0.904395i \(0.359677\pi\)
\(524\) −1.64850 −0.0720151
\(525\) 0 0
\(526\) −3.41911 −0.149080
\(527\) 22.5851 0.983822
\(528\) 0 0
\(529\) 2.63511 0.114570
\(530\) −10.0258 −0.435492
\(531\) 0 0
\(532\) 2.33116 0.101069
\(533\) 18.3685 0.795629
\(534\) 0 0
\(535\) 2.47111 0.106835
\(536\) 1.70429 0.0736139
\(537\) 0 0
\(538\) −19.6269 −0.846178
\(539\) 1.33562 0.0575294
\(540\) 0 0
\(541\) −16.8616 −0.724938 −0.362469 0.931996i \(-0.618066\pi\)
−0.362469 + 0.931996i \(0.618066\pi\)
\(542\) −10.6569 −0.457751
\(543\) 0 0
\(544\) 9.93599 0.426002
\(545\) 10.6997 0.458327
\(546\) 0 0
\(547\) −3.59443 −0.153687 −0.0768433 0.997043i \(-0.524484\pi\)
−0.0768433 + 0.997043i \(0.524484\pi\)
\(548\) −6.06678 −0.259160
\(549\) 0 0
\(550\) 0.499086 0.0212811
\(551\) −13.6755 −0.582594
\(552\) 0 0
\(553\) 17.1837 0.730725
\(554\) −15.4149 −0.654916
\(555\) 0 0
\(556\) 2.80941 0.119146
\(557\) −0.0100663 −0.000426524 0 −0.000213262 1.00000i \(-0.500068\pi\)
−0.000213262 1.00000i \(0.500068\pi\)
\(558\) 0 0
\(559\) −4.91684 −0.207960
\(560\) −7.92104 −0.334725
\(561\) 0 0
\(562\) −30.4438 −1.28419
\(563\) 37.8978 1.59720 0.798601 0.601861i \(-0.205574\pi\)
0.798601 + 0.601861i \(0.205574\pi\)
\(564\) 0 0
\(565\) 17.1698 0.722339
\(566\) −4.81768 −0.202502
\(567\) 0 0
\(568\) 7.15771 0.300331
\(569\) 23.1594 0.970891 0.485446 0.874267i \(-0.338657\pi\)
0.485446 + 0.874267i \(0.338657\pi\)
\(570\) 0 0
\(571\) −24.5298 −1.02654 −0.513271 0.858227i \(-0.671566\pi\)
−0.513271 + 0.858227i \(0.671566\pi\)
\(572\) 0.730146 0.0305289
\(573\) 0 0
\(574\) 11.3528 0.473859
\(575\) −5.06311 −0.211146
\(576\) 0 0
\(577\) −11.7052 −0.487292 −0.243646 0.969864i \(-0.578343\pi\)
−0.243646 + 0.969864i \(0.578343\pi\)
\(578\) 9.81427 0.408220
\(579\) 0 0
\(580\) −2.90347 −0.120560
\(581\) 11.1345 0.461935
\(582\) 0 0
\(583\) −1.96427 −0.0813519
\(584\) 11.2810 0.466809
\(585\) 0 0
\(586\) 36.2429 1.49718
\(587\) 6.12814 0.252936 0.126468 0.991971i \(-0.459636\pi\)
0.126468 + 0.991971i \(0.459636\pi\)
\(588\) 0 0
\(589\) 17.6765 0.728348
\(590\) −8.58263 −0.353341
\(591\) 0 0
\(592\) −47.3006 −1.94404
\(593\) 18.5032 0.759837 0.379918 0.925020i \(-0.375952\pi\)
0.379918 + 0.925020i \(0.375952\pi\)
\(594\) 0 0
\(595\) 5.44145 0.223078
\(596\) −0.653649 −0.0267745
\(597\) 0 0
\(598\) −34.4716 −1.40965
\(599\) −45.8000 −1.87134 −0.935668 0.352882i \(-0.885202\pi\)
−0.935668 + 0.352882i \(0.885202\pi\)
\(600\) 0 0
\(601\) −35.5468 −1.44998 −0.724991 0.688758i \(-0.758156\pi\)
−0.724991 + 0.688758i \(0.758156\pi\)
\(602\) −3.03890 −0.123856
\(603\) 0 0
\(604\) −0.709224 −0.0288579
\(605\) −10.9022 −0.443238
\(606\) 0 0
\(607\) −32.3881 −1.31459 −0.657297 0.753632i \(-0.728300\pi\)
−0.657297 + 0.753632i \(0.728300\pi\)
\(608\) 7.77653 0.315380
\(609\) 0 0
\(610\) −18.8304 −0.762419
\(611\) 35.2637 1.42662
\(612\) 0 0
\(613\) 9.42107 0.380513 0.190257 0.981734i \(-0.439068\pi\)
0.190257 + 0.981734i \(0.439068\pi\)
\(614\) −35.7961 −1.44461
\(615\) 0 0
\(616\) −1.19760 −0.0482528
\(617\) 32.5296 1.30959 0.654796 0.755805i \(-0.272754\pi\)
0.654796 + 0.755805i \(0.272754\pi\)
\(618\) 0 0
\(619\) 3.60286 0.144811 0.0724056 0.997375i \(-0.476932\pi\)
0.0724056 + 0.997375i \(0.476932\pi\)
\(620\) 3.75294 0.150722
\(621\) 0 0
\(622\) −4.48214 −0.179718
\(623\) −1.65189 −0.0661817
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −38.3815 −1.53403
\(627\) 0 0
\(628\) 10.4782 0.418124
\(629\) 32.4937 1.29561
\(630\) 0 0
\(631\) 6.77736 0.269802 0.134901 0.990859i \(-0.456928\pi\)
0.134901 + 0.990859i \(0.456928\pi\)
\(632\) 24.1176 0.959348
\(633\) 0 0
\(634\) 0.905855 0.0359761
\(635\) −0.530525 −0.0210532
\(636\) 0 0
\(637\) 18.2202 0.721909
\(638\) −2.64734 −0.104809
\(639\) 0 0
\(640\) −13.6555 −0.539780
\(641\) 12.3607 0.488220 0.244110 0.969748i \(-0.421504\pi\)
0.244110 + 0.969748i \(0.421504\pi\)
\(642\) 0 0
\(643\) −13.7782 −0.543360 −0.271680 0.962388i \(-0.587579\pi\)
−0.271680 + 0.962388i \(0.587579\pi\)
\(644\) −4.57807 −0.180401
\(645\) 0 0
\(646\) −13.5546 −0.533298
\(647\) 15.0963 0.593499 0.296749 0.954955i \(-0.404097\pi\)
0.296749 + 0.954955i \(0.404097\pi\)
\(648\) 0 0
\(649\) −1.68153 −0.0660057
\(650\) 6.80838 0.267047
\(651\) 0 0
\(652\) −11.5647 −0.452908
\(653\) −8.93166 −0.349523 −0.174761 0.984611i \(-0.555915\pi\)
−0.174761 + 0.984611i \(0.555915\pi\)
\(654\) 0 0
\(655\) −3.01167 −0.117676
\(656\) 20.6479 0.806167
\(657\) 0 0
\(658\) 21.7951 0.849661
\(659\) 23.2632 0.906205 0.453103 0.891458i \(-0.350317\pi\)
0.453103 + 0.891458i \(0.350317\pi\)
\(660\) 0 0
\(661\) 32.4474 1.26206 0.631028 0.775760i \(-0.282633\pi\)
0.631028 + 0.775760i \(0.282633\pi\)
\(662\) −34.8485 −1.35443
\(663\) 0 0
\(664\) 15.6274 0.606462
\(665\) 4.25882 0.165150
\(666\) 0 0
\(667\) 26.8567 1.03989
\(668\) 6.33662 0.245171
\(669\) 0 0
\(670\) −1.17324 −0.0453263
\(671\) −3.68928 −0.142423
\(672\) 0 0
\(673\) 9.30803 0.358798 0.179399 0.983776i \(-0.442585\pi\)
0.179399 + 0.983776i \(0.442585\pi\)
\(674\) −42.8647 −1.65109
\(675\) 0 0
\(676\) 2.84458 0.109407
\(677\) −13.4553 −0.517127 −0.258564 0.965994i \(-0.583249\pi\)
−0.258564 + 0.965994i \(0.583249\pi\)
\(678\) 0 0
\(679\) 7.98803 0.306553
\(680\) 7.63719 0.292873
\(681\) 0 0
\(682\) 3.42188 0.131031
\(683\) −5.46981 −0.209296 −0.104648 0.994509i \(-0.533372\pi\)
−0.104648 + 0.994509i \(0.533372\pi\)
\(684\) 0 0
\(685\) −11.0835 −0.423478
\(686\) 29.7167 1.13459
\(687\) 0 0
\(688\) −5.52698 −0.210714
\(689\) −26.7960 −1.02085
\(690\) 0 0
\(691\) 43.5511 1.65676 0.828382 0.560164i \(-0.189262\pi\)
0.828382 + 0.560164i \(0.189262\pi\)
\(692\) −8.95758 −0.340516
\(693\) 0 0
\(694\) −30.5899 −1.16118
\(695\) 5.13255 0.194689
\(696\) 0 0
\(697\) −14.1843 −0.537270
\(698\) 26.8927 1.01790
\(699\) 0 0
\(700\) 0.904200 0.0341755
\(701\) 2.98694 0.112815 0.0564075 0.998408i \(-0.482035\pi\)
0.0564075 + 0.998408i \(0.482035\pi\)
\(702\) 0 0
\(703\) 25.4316 0.959172
\(704\) −1.49348 −0.0562875
\(705\) 0 0
\(706\) −33.1954 −1.24933
\(707\) −29.6297 −1.11434
\(708\) 0 0
\(709\) 19.1866 0.720568 0.360284 0.932843i \(-0.382680\pi\)
0.360284 + 0.932843i \(0.382680\pi\)
\(710\) −4.92742 −0.184923
\(711\) 0 0
\(712\) −2.31847 −0.0868882
\(713\) −34.7142 −1.30006
\(714\) 0 0
\(715\) 1.33391 0.0498855
\(716\) −7.94234 −0.296819
\(717\) 0 0
\(718\) −38.4917 −1.43650
\(719\) 43.3314 1.61599 0.807994 0.589190i \(-0.200553\pi\)
0.807994 + 0.589190i \(0.200553\pi\)
\(720\) 0 0
\(721\) −20.0775 −0.747727
\(722\) 19.7163 0.733763
\(723\) 0 0
\(724\) −11.2926 −0.419687
\(725\) −5.30438 −0.197000
\(726\) 0 0
\(727\) 38.6380 1.43300 0.716502 0.697585i \(-0.245742\pi\)
0.716502 + 0.697585i \(0.245742\pi\)
\(728\) −16.3373 −0.605501
\(729\) 0 0
\(730\) −7.76589 −0.287428
\(731\) 3.79683 0.140431
\(732\) 0 0
\(733\) 26.6371 0.983865 0.491932 0.870633i \(-0.336291\pi\)
0.491932 + 0.870633i \(0.336291\pi\)
\(734\) −2.84620 −0.105055
\(735\) 0 0
\(736\) −15.2720 −0.562933
\(737\) −0.229864 −0.00846716
\(738\) 0 0
\(739\) 16.8083 0.618304 0.309152 0.951013i \(-0.399955\pi\)
0.309152 + 0.951013i \(0.399955\pi\)
\(740\) 5.39944 0.198487
\(741\) 0 0
\(742\) −16.5615 −0.607993
\(743\) 30.2622 1.11021 0.555107 0.831779i \(-0.312677\pi\)
0.555107 + 0.831779i \(0.312677\pi\)
\(744\) 0 0
\(745\) −1.19416 −0.0437506
\(746\) −7.67440 −0.280980
\(747\) 0 0
\(748\) −0.563826 −0.0206155
\(749\) 4.08201 0.149153
\(750\) 0 0
\(751\) −23.5893 −0.860787 −0.430394 0.902641i \(-0.641625\pi\)
−0.430394 + 0.902641i \(0.641625\pi\)
\(752\) 39.6397 1.44551
\(753\) 0 0
\(754\) −36.1142 −1.31520
\(755\) −1.29569 −0.0471550
\(756\) 0 0
\(757\) −45.5701 −1.65627 −0.828137 0.560526i \(-0.810599\pi\)
−0.828137 + 0.560526i \(0.810599\pi\)
\(758\) −23.9041 −0.868235
\(759\) 0 0
\(760\) 5.97734 0.216821
\(761\) −34.3822 −1.24636 −0.623178 0.782080i \(-0.714159\pi\)
−0.623178 + 0.782080i \(0.714159\pi\)
\(762\) 0 0
\(763\) 17.6748 0.639872
\(764\) 10.7839 0.390149
\(765\) 0 0
\(766\) −26.4744 −0.956558
\(767\) −22.9389 −0.828274
\(768\) 0 0
\(769\) −5.77880 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(770\) 0.824438 0.0297107
\(771\) 0 0
\(772\) 10.1590 0.365631
\(773\) −7.43120 −0.267282 −0.133641 0.991030i \(-0.542667\pi\)
−0.133641 + 0.991030i \(0.542667\pi\)
\(774\) 0 0
\(775\) 6.85629 0.246285
\(776\) 11.2114 0.402465
\(777\) 0 0
\(778\) −6.43356 −0.230654
\(779\) −11.1016 −0.397755
\(780\) 0 0
\(781\) −0.965390 −0.0345444
\(782\) 26.6193 0.951904
\(783\) 0 0
\(784\) 20.4812 0.731470
\(785\) 19.1427 0.683232
\(786\) 0 0
\(787\) −6.45412 −0.230064 −0.115032 0.993362i \(-0.536697\pi\)
−0.115032 + 0.993362i \(0.536697\pi\)
\(788\) 7.17743 0.255685
\(789\) 0 0
\(790\) −16.6028 −0.590700
\(791\) 28.3627 1.00846
\(792\) 0 0
\(793\) −50.3280 −1.78720
\(794\) −5.89127 −0.209073
\(795\) 0 0
\(796\) 12.2420 0.433907
\(797\) 18.2999 0.648216 0.324108 0.946020i \(-0.394936\pi\)
0.324108 + 0.946020i \(0.394936\pi\)
\(798\) 0 0
\(799\) −27.2310 −0.963362
\(800\) 3.01633 0.106643
\(801\) 0 0
\(802\) 33.0979 1.16873
\(803\) −1.52151 −0.0536929
\(804\) 0 0
\(805\) −8.36373 −0.294783
\(806\) 46.6802 1.64424
\(807\) 0 0
\(808\) −41.5858 −1.46298
\(809\) −48.0281 −1.68858 −0.844288 0.535889i \(-0.819977\pi\)
−0.844288 + 0.535889i \(0.819977\pi\)
\(810\) 0 0
\(811\) 20.2349 0.710543 0.355272 0.934763i \(-0.384388\pi\)
0.355272 + 0.934763i \(0.384388\pi\)
\(812\) −4.79622 −0.168314
\(813\) 0 0
\(814\) 4.92314 0.172556
\(815\) −21.1277 −0.740071
\(816\) 0 0
\(817\) 2.97164 0.103964
\(818\) −5.73452 −0.200503
\(819\) 0 0
\(820\) −2.35700 −0.0823099
\(821\) 18.6121 0.649566 0.324783 0.945788i \(-0.394709\pi\)
0.324783 + 0.945788i \(0.394709\pi\)
\(822\) 0 0
\(823\) −39.6408 −1.38179 −0.690896 0.722954i \(-0.742784\pi\)
−0.690896 + 0.722954i \(0.742784\pi\)
\(824\) −28.1792 −0.981670
\(825\) 0 0
\(826\) −14.1776 −0.493302
\(827\) 46.0444 1.60112 0.800561 0.599251i \(-0.204535\pi\)
0.800561 + 0.599251i \(0.204535\pi\)
\(828\) 0 0
\(829\) −22.8796 −0.794641 −0.397320 0.917680i \(-0.630060\pi\)
−0.397320 + 0.917680i \(0.630060\pi\)
\(830\) −10.7580 −0.373417
\(831\) 0 0
\(832\) −20.3735 −0.706326
\(833\) −14.0698 −0.487489
\(834\) 0 0
\(835\) 11.5765 0.400620
\(836\) −0.441285 −0.0152622
\(837\) 0 0
\(838\) 28.5605 0.986607
\(839\) −51.4718 −1.77700 −0.888502 0.458872i \(-0.848254\pi\)
−0.888502 + 0.458872i \(0.848254\pi\)
\(840\) 0 0
\(841\) −0.863551 −0.0297776
\(842\) 24.0490 0.828784
\(843\) 0 0
\(844\) −9.58242 −0.329840
\(845\) 5.19681 0.178776
\(846\) 0 0
\(847\) −18.0093 −0.618807
\(848\) −30.1212 −1.03437
\(849\) 0 0
\(850\) −5.25750 −0.180331
\(851\) −49.9441 −1.71206
\(852\) 0 0
\(853\) −52.3073 −1.79097 −0.895485 0.445093i \(-0.853171\pi\)
−0.895485 + 0.445093i \(0.853171\pi\)
\(854\) −31.1058 −1.06442
\(855\) 0 0
\(856\) 5.72918 0.195819
\(857\) −44.2539 −1.51169 −0.755843 0.654753i \(-0.772773\pi\)
−0.755843 + 0.654753i \(0.772773\pi\)
\(858\) 0 0
\(859\) 19.9864 0.681928 0.340964 0.940076i \(-0.389247\pi\)
0.340964 + 0.940076i \(0.389247\pi\)
\(860\) 0.630914 0.0215140
\(861\) 0 0
\(862\) −34.2702 −1.16725
\(863\) −43.4641 −1.47953 −0.739767 0.672863i \(-0.765064\pi\)
−0.739767 + 0.672863i \(0.765064\pi\)
\(864\) 0 0
\(865\) −16.3647 −0.556417
\(866\) 3.38970 0.115187
\(867\) 0 0
\(868\) 6.19946 0.210423
\(869\) −3.25285 −0.110345
\(870\) 0 0
\(871\) −3.13573 −0.106250
\(872\) 24.8070 0.840071
\(873\) 0 0
\(874\) 20.8339 0.704718
\(875\) 1.65189 0.0558442
\(876\) 0 0
\(877\) −0.561222 −0.0189511 −0.00947555 0.999955i \(-0.503016\pi\)
−0.00947555 + 0.999955i \(0.503016\pi\)
\(878\) 61.5909 2.07859
\(879\) 0 0
\(880\) 1.49944 0.0505462
\(881\) −38.9725 −1.31302 −0.656509 0.754318i \(-0.727968\pi\)
−0.656509 + 0.754318i \(0.727968\pi\)
\(882\) 0 0
\(883\) 45.0122 1.51478 0.757390 0.652963i \(-0.226474\pi\)
0.757390 + 0.652963i \(0.226474\pi\)
\(884\) −7.69153 −0.258694
\(885\) 0 0
\(886\) 29.6362 0.995649
\(887\) −50.6113 −1.69936 −0.849681 0.527297i \(-0.823206\pi\)
−0.849681 + 0.527297i \(0.823206\pi\)
\(888\) 0 0
\(889\) −0.876371 −0.0293925
\(890\) 1.59605 0.0534997
\(891\) 0 0
\(892\) 12.2665 0.410712
\(893\) −21.3127 −0.713201
\(894\) 0 0
\(895\) −14.5100 −0.485014
\(896\) −22.5574 −0.753589
\(897\) 0 0
\(898\) −57.5094 −1.91911
\(899\) −36.3684 −1.21295
\(900\) 0 0
\(901\) 20.6921 0.689354
\(902\) −2.14908 −0.0715565
\(903\) 0 0
\(904\) 39.8076 1.32398
\(905\) −20.6306 −0.685785
\(906\) 0 0
\(907\) −18.0299 −0.598672 −0.299336 0.954148i \(-0.596765\pi\)
−0.299336 + 0.954148i \(0.596765\pi\)
\(908\) 15.0231 0.498558
\(909\) 0 0
\(910\) 11.2467 0.372825
\(911\) 12.8735 0.426517 0.213258 0.976996i \(-0.431592\pi\)
0.213258 + 0.976996i \(0.431592\pi\)
\(912\) 0 0
\(913\) −2.10774 −0.0697560
\(914\) −16.6905 −0.552072
\(915\) 0 0
\(916\) 14.6406 0.483740
\(917\) −4.97496 −0.164288
\(918\) 0 0
\(919\) 7.98015 0.263241 0.131621 0.991300i \(-0.457982\pi\)
0.131621 + 0.991300i \(0.457982\pi\)
\(920\) −11.7387 −0.387012
\(921\) 0 0
\(922\) −0.296057 −0.00975011
\(923\) −13.1695 −0.433481
\(924\) 0 0
\(925\) 9.86431 0.324336
\(926\) 11.4903 0.377596
\(927\) 0 0
\(928\) −15.9997 −0.525217
\(929\) 59.0076 1.93598 0.967989 0.250994i \(-0.0807575\pi\)
0.967989 + 0.250994i \(0.0807575\pi\)
\(930\) 0 0
\(931\) −11.0119 −0.360900
\(932\) 2.06988 0.0678012
\(933\) 0 0
\(934\) −28.9846 −0.948404
\(935\) −1.03006 −0.0336866
\(936\) 0 0
\(937\) −1.28011 −0.0418194 −0.0209097 0.999781i \(-0.506656\pi\)
−0.0209097 + 0.999781i \(0.506656\pi\)
\(938\) −1.93807 −0.0632803
\(939\) 0 0
\(940\) −4.52494 −0.147587
\(941\) 17.8453 0.581739 0.290870 0.956763i \(-0.406055\pi\)
0.290870 + 0.956763i \(0.406055\pi\)
\(942\) 0 0
\(943\) 21.8019 0.709967
\(944\) −25.7854 −0.839244
\(945\) 0 0
\(946\) 0.575260 0.0187033
\(947\) −37.6144 −1.22230 −0.611152 0.791513i \(-0.709293\pi\)
−0.611152 + 0.791513i \(0.709293\pi\)
\(948\) 0 0
\(949\) −20.7560 −0.673767
\(950\) −4.11485 −0.133503
\(951\) 0 0
\(952\) 12.6158 0.408881
\(953\) −31.4840 −1.01987 −0.509933 0.860214i \(-0.670330\pi\)
−0.509933 + 0.860214i \(0.670330\pi\)
\(954\) 0 0
\(955\) 19.7013 0.637519
\(956\) −11.1455 −0.360471
\(957\) 0 0
\(958\) 23.1956 0.749415
\(959\) −18.3087 −0.591220
\(960\) 0 0
\(961\) 16.0087 0.516410
\(962\) 67.1599 2.16532
\(963\) 0 0
\(964\) −1.34558 −0.0433382
\(965\) 18.5596 0.597455
\(966\) 0 0
\(967\) 17.5286 0.563681 0.281841 0.959461i \(-0.409055\pi\)
0.281841 + 0.959461i \(0.409055\pi\)
\(968\) −25.2764 −0.812415
\(969\) 0 0
\(970\) −7.71799 −0.247810
\(971\) −37.3269 −1.19788 −0.598939 0.800795i \(-0.704411\pi\)
−0.598939 + 0.800795i \(0.704411\pi\)
\(972\) 0 0
\(973\) 8.47843 0.271806
\(974\) −35.9269 −1.15117
\(975\) 0 0
\(976\) −56.5734 −1.81087
\(977\) 11.3776 0.364001 0.182000 0.983298i \(-0.441743\pi\)
0.182000 + 0.983298i \(0.441743\pi\)
\(978\) 0 0
\(979\) 0.312701 0.00999398
\(980\) −2.33796 −0.0746834
\(981\) 0 0
\(982\) 22.3699 0.713852
\(983\) −17.9011 −0.570958 −0.285479 0.958385i \(-0.592153\pi\)
−0.285479 + 0.958385i \(0.592153\pi\)
\(984\) 0 0
\(985\) 13.1125 0.417800
\(986\) 27.8878 0.888127
\(987\) 0 0
\(988\) −6.01987 −0.191518
\(989\) −5.83587 −0.185570
\(990\) 0 0
\(991\) −41.1988 −1.30872 −0.654362 0.756181i \(-0.727063\pi\)
−0.654362 + 0.756181i \(0.727063\pi\)
\(992\) 20.6808 0.656616
\(993\) 0 0
\(994\) −8.13957 −0.258171
\(995\) 22.3651 0.709022
\(996\) 0 0
\(997\) −38.0200 −1.20411 −0.602053 0.798456i \(-0.705650\pi\)
−0.602053 + 0.798456i \(0.705650\pi\)
\(998\) −34.7468 −1.09989
\(999\) 0 0
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 4005.2.a.x.1.3 yes 17
3.2 odd 2 4005.2.a.w.1.15 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4005.2.a.w.1.15 17 3.2 odd 2
4005.2.a.x.1.3 yes 17 1.1 even 1 trivial