Properties

Label 819.2.a.m.1.5
Level $819$
Weight $2$
Character 819.1
Self dual yes
Analytic conductor $6.540$
Analytic rank $0$
Dimension $6$
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] = [819,2,Mod(1,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, 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("819.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.199374400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} + 10x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.54574\) of defining polynomial
Character \(\chi\) \(=\) 819.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.15293 q^{2} +2.63509 q^{4} +4.16251 q^{5} -1.00000 q^{7} +1.36730 q^{8} +O(q^{10})\) \(q+2.15293 q^{2} +2.63509 q^{4} +4.16251 q^{5} -1.00000 q^{7} +1.36730 q^{8} +8.96157 q^{10} -0.785627 q^{11} +1.00000 q^{13} -2.15293 q^{14} -2.32648 q^{16} +5.87710 q^{17} -8.59666 q^{19} +10.9686 q^{20} -1.69140 q^{22} +0.928968 q^{23} +12.3265 q^{25} +2.15293 q^{26} -2.63509 q^{28} -5.23482 q^{29} -5.32648 q^{31} -7.74335 q^{32} +12.6530 q^{34} -4.16251 q^{35} -1.27018 q^{37} -18.5080 q^{38} +5.69140 q^{40} -1.07231 q^{41} +8.59666 q^{43} -2.07020 q^{44} +2.00000 q^{46} -1.71459 q^{47} +1.00000 q^{49} +26.5380 q^{50} +2.63509 q^{52} +6.80607 q^{53} -3.27018 q^{55} -1.36730 q^{56} -11.2702 q^{58} -13.7032 q^{59} +10.0000 q^{61} -11.4675 q^{62} -12.0179 q^{64} +4.16251 q^{65} +0.729822 q^{67} +15.4867 q^{68} -8.96157 q^{70} -7.53939 q^{71} -7.32648 q^{73} -2.73460 q^{74} -22.6530 q^{76} +0.785627 q^{77} +10.0563 q^{79} -9.68401 q^{80} -2.30860 q^{82} +10.0396 q^{83} +24.4635 q^{85} +18.5080 q^{86} -1.07419 q^{88} -16.2034 q^{89} -1.00000 q^{91} +2.44791 q^{92} -3.69140 q^{94} -35.7837 q^{95} -10.5967 q^{97} +2.15293 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 12 q^{4} - 6 q^{7} + 8 q^{10} + 6 q^{13} + 28 q^{16} - 2 q^{19} + 28 q^{22} + 32 q^{25} - 12 q^{28} + 10 q^{31} - 8 q^{34} - 4 q^{40} + 2 q^{43} + 12 q^{46} + 6 q^{49} + 12 q^{52} - 12 q^{55} - 60 q^{58} + 60 q^{61} + 8 q^{64} + 12 q^{67} - 8 q^{70} - 2 q^{73} - 52 q^{76} + 26 q^{79} - 52 q^{82} + 40 q^{85} + 108 q^{88} - 6 q^{91} + 16 q^{94} - 14 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.15293 1.52235 0.761174 0.648548i \(-0.224623\pi\)
0.761174 + 0.648548i \(0.224623\pi\)
\(3\) 0 0
\(4\) 2.63509 1.31754
\(5\) 4.16251 1.86153 0.930765 0.365617i \(-0.119142\pi\)
0.930765 + 0.365617i \(0.119142\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.36730 0.483413
\(9\) 0 0
\(10\) 8.96157 2.83390
\(11\) −0.785627 −0.236875 −0.118438 0.992961i \(-0.537789\pi\)
−0.118438 + 0.992961i \(0.537789\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −2.15293 −0.575394
\(15\) 0 0
\(16\) −2.32648 −0.581621
\(17\) 5.87710 1.42541 0.712704 0.701465i \(-0.247470\pi\)
0.712704 + 0.701465i \(0.247470\pi\)
\(18\) 0 0
\(19\) −8.59666 −1.97221 −0.986105 0.166125i \(-0.946875\pi\)
−0.986105 + 0.166125i \(0.946875\pi\)
\(20\) 10.9686 2.45265
\(21\) 0 0
\(22\) −1.69140 −0.360607
\(23\) 0.928968 0.193703 0.0968517 0.995299i \(-0.469123\pi\)
0.0968517 + 0.995299i \(0.469123\pi\)
\(24\) 0 0
\(25\) 12.3265 2.46530
\(26\) 2.15293 0.422223
\(27\) 0 0
\(28\) −2.63509 −0.497985
\(29\) −5.23482 −0.972082 −0.486041 0.873936i \(-0.661559\pi\)
−0.486041 + 0.873936i \(0.661559\pi\)
\(30\) 0 0
\(31\) −5.32648 −0.956665 −0.478332 0.878179i \(-0.658759\pi\)
−0.478332 + 0.878179i \(0.658759\pi\)
\(32\) −7.74335 −1.36884
\(33\) 0 0
\(34\) 12.6530 2.16997
\(35\) −4.16251 −0.703593
\(36\) 0 0
\(37\) −1.27018 −0.208816 −0.104408 0.994535i \(-0.533295\pi\)
−0.104408 + 0.994535i \(0.533295\pi\)
\(38\) −18.5080 −3.00239
\(39\) 0 0
\(40\) 5.69140 0.899889
\(41\) −1.07231 −0.167467 −0.0837334 0.996488i \(-0.526684\pi\)
−0.0837334 + 0.996488i \(0.526684\pi\)
\(42\) 0 0
\(43\) 8.59666 1.31098 0.655489 0.755204i \(-0.272462\pi\)
0.655489 + 0.755204i \(0.272462\pi\)
\(44\) −2.07020 −0.312094
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −1.71459 −0.250099 −0.125050 0.992150i \(-0.539909\pi\)
−0.125050 + 0.992150i \(0.539909\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 26.5380 3.75304
\(51\) 0 0
\(52\) 2.63509 0.365421
\(53\) 6.80607 0.934886 0.467443 0.884023i \(-0.345175\pi\)
0.467443 + 0.884023i \(0.345175\pi\)
\(54\) 0 0
\(55\) −3.27018 −0.440951
\(56\) −1.36730 −0.182713
\(57\) 0 0
\(58\) −11.2702 −1.47985
\(59\) −13.7032 −1.78400 −0.892001 0.452033i \(-0.850699\pi\)
−0.892001 + 0.452033i \(0.850699\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −11.4675 −1.45638
\(63\) 0 0
\(64\) −12.0179 −1.50223
\(65\) 4.16251 0.516296
\(66\) 0 0
\(67\) 0.729822 0.0891620 0.0445810 0.999006i \(-0.485805\pi\)
0.0445810 + 0.999006i \(0.485805\pi\)
\(68\) 15.4867 1.87804
\(69\) 0 0
\(70\) −8.96157 −1.07111
\(71\) −7.53939 −0.894761 −0.447381 0.894344i \(-0.647643\pi\)
−0.447381 + 0.894344i \(0.647643\pi\)
\(72\) 0 0
\(73\) −7.32648 −0.857500 −0.428750 0.903423i \(-0.641046\pi\)
−0.428750 + 0.903423i \(0.641046\pi\)
\(74\) −2.73460 −0.317891
\(75\) 0 0
\(76\) −22.6530 −2.59847
\(77\) 0.785627 0.0895305
\(78\) 0 0
\(79\) 10.0563 1.13142 0.565711 0.824603i \(-0.308602\pi\)
0.565711 + 0.824603i \(0.308602\pi\)
\(80\) −9.68401 −1.08271
\(81\) 0 0
\(82\) −2.30860 −0.254943
\(83\) 10.0396 1.10199 0.550995 0.834508i \(-0.314248\pi\)
0.550995 + 0.834508i \(0.314248\pi\)
\(84\) 0 0
\(85\) 24.4635 2.65344
\(86\) 18.5080 1.99577
\(87\) 0 0
\(88\) −1.07419 −0.114509
\(89\) −16.2034 −1.71756 −0.858779 0.512347i \(-0.828776\pi\)
−0.858779 + 0.512347i \(0.828776\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 2.44791 0.255213
\(93\) 0 0
\(94\) −3.69140 −0.380738
\(95\) −35.7837 −3.67133
\(96\) 0 0
\(97\) −10.5967 −1.07593 −0.537964 0.842968i \(-0.680806\pi\)
−0.537964 + 0.842968i \(0.680806\pi\)
\(98\) 2.15293 0.217478
\(99\) 0 0
\(100\) 32.4814 3.24814
\(101\) −11.0596 −1.10047 −0.550237 0.835009i \(-0.685462\pi\)
−0.550237 + 0.835009i \(0.685462\pi\)
\(102\) 0 0
\(103\) 13.9231 1.37189 0.685944 0.727654i \(-0.259389\pi\)
0.685944 + 0.727654i \(0.259389\pi\)
\(104\) 1.36730 0.134075
\(105\) 0 0
\(106\) 14.6530 1.42322
\(107\) 7.73504 0.747775 0.373887 0.927474i \(-0.378025\pi\)
0.373887 + 0.927474i \(0.378025\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −7.04045 −0.671281
\(111\) 0 0
\(112\) 2.32648 0.219832
\(113\) 5.23482 0.492450 0.246225 0.969213i \(-0.420810\pi\)
0.246225 + 0.969213i \(0.420810\pi\)
\(114\) 0 0
\(115\) 3.86684 0.360585
\(116\) −13.7942 −1.28076
\(117\) 0 0
\(118\) −29.5019 −2.71587
\(119\) −5.87710 −0.538753
\(120\) 0 0
\(121\) −10.3828 −0.943890
\(122\) 21.5293 1.94917
\(123\) 0 0
\(124\) −14.0358 −1.26045
\(125\) 30.4966 2.72770
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −10.3869 −0.918082
\(129\) 0 0
\(130\) 8.96157 0.785982
\(131\) −6.46708 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(132\) 0 0
\(133\) 8.59666 0.745425
\(134\) 1.57125 0.135736
\(135\) 0 0
\(136\) 8.03576 0.689061
\(137\) 3.23354 0.276260 0.138130 0.990414i \(-0.455891\pi\)
0.138130 + 0.990414i \(0.455891\pi\)
\(138\) 0 0
\(139\) 5.81053 0.492843 0.246421 0.969163i \(-0.420745\pi\)
0.246421 + 0.969163i \(0.420745\pi\)
\(140\) −10.9686 −0.927014
\(141\) 0 0
\(142\) −16.2318 −1.36214
\(143\) −0.785627 −0.0656974
\(144\) 0 0
\(145\) −21.7900 −1.80956
\(146\) −15.7734 −1.30541
\(147\) 0 0
\(148\) −3.34703 −0.275124
\(149\) −2.94686 −0.241416 −0.120708 0.992688i \(-0.538516\pi\)
−0.120708 + 0.992688i \(0.538516\pi\)
\(150\) 0 0
\(151\) −13.9231 −1.13305 −0.566525 0.824045i \(-0.691712\pi\)
−0.566525 + 0.824045i \(0.691712\pi\)
\(152\) −11.7542 −0.953392
\(153\) 0 0
\(154\) 1.69140 0.136297
\(155\) −22.1715 −1.78086
\(156\) 0 0
\(157\) −19.1933 −1.53179 −0.765897 0.642963i \(-0.777705\pi\)
−0.765897 + 0.642963i \(0.777705\pi\)
\(158\) 21.6505 1.72242
\(159\) 0 0
\(160\) −32.2318 −2.54814
\(161\) −0.928968 −0.0732130
\(162\) 0 0
\(163\) 10.5404 0.825584 0.412792 0.910825i \(-0.364554\pi\)
0.412792 + 0.910825i \(0.364554\pi\)
\(164\) −2.82563 −0.220645
\(165\) 0 0
\(166\) 21.6145 1.67761
\(167\) 2.00128 0.154864 0.0774318 0.996998i \(-0.475328\pi\)
0.0774318 + 0.996998i \(0.475328\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 52.6681 4.03946
\(171\) 0 0
\(172\) 22.6530 1.72727
\(173\) −20.9559 −1.59325 −0.796623 0.604476i \(-0.793383\pi\)
−0.796623 + 0.604476i \(0.793383\pi\)
\(174\) 0 0
\(175\) −12.3265 −0.931795
\(176\) 1.82775 0.137772
\(177\) 0 0
\(178\) −34.8847 −2.61472
\(179\) 11.1119 0.830544 0.415272 0.909697i \(-0.363686\pi\)
0.415272 + 0.909697i \(0.363686\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −2.15293 −0.159585
\(183\) 0 0
\(184\) 1.27018 0.0936387
\(185\) −5.28713 −0.388717
\(186\) 0 0
\(187\) −4.61721 −0.337644
\(188\) −4.51811 −0.329517
\(189\) 0 0
\(190\) −77.0396 −5.58904
\(191\) −5.59042 −0.404509 −0.202254 0.979333i \(-0.564827\pi\)
−0.202254 + 0.979333i \(0.564827\pi\)
\(192\) 0 0
\(193\) 3.92315 0.282394 0.141197 0.989981i \(-0.454905\pi\)
0.141197 + 0.989981i \(0.454905\pi\)
\(194\) −22.8138 −1.63794
\(195\) 0 0
\(196\) 2.63509 0.188221
\(197\) −13.7032 −0.976311 −0.488156 0.872757i \(-0.662330\pi\)
−0.488156 + 0.872757i \(0.662330\pi\)
\(198\) 0 0
\(199\) −9.45964 −0.670576 −0.335288 0.942116i \(-0.608834\pi\)
−0.335288 + 0.942116i \(0.608834\pi\)
\(200\) 16.8540 1.19176
\(201\) 0 0
\(202\) −23.8105 −1.67530
\(203\) 5.23482 0.367412
\(204\) 0 0
\(205\) −4.46350 −0.311745
\(206\) 29.9755 2.08849
\(207\) 0 0
\(208\) −2.32648 −0.161313
\(209\) 6.75377 0.467168
\(210\) 0 0
\(211\) −12.5967 −0.867190 −0.433595 0.901108i \(-0.642755\pi\)
−0.433595 + 0.901108i \(0.642755\pi\)
\(212\) 17.9346 1.23175
\(213\) 0 0
\(214\) 16.6530 1.13837
\(215\) 35.7837 2.44043
\(216\) 0 0
\(217\) 5.32648 0.361585
\(218\) 21.5293 1.45815
\(219\) 0 0
\(220\) −8.61721 −0.580972
\(221\) 5.87710 0.395337
\(222\) 0 0
\(223\) 12.5967 0.843535 0.421767 0.906704i \(-0.361410\pi\)
0.421767 + 0.906704i \(0.361410\pi\)
\(224\) 7.74335 0.517374
\(225\) 0 0
\(226\) 11.2702 0.749681
\(227\) 2.94686 0.195590 0.0977949 0.995207i \(-0.468821\pi\)
0.0977949 + 0.995207i \(0.468821\pi\)
\(228\) 0 0
\(229\) 11.9231 0.787904 0.393952 0.919131i \(-0.371108\pi\)
0.393952 + 0.919131i \(0.371108\pi\)
\(230\) 8.32502 0.548935
\(231\) 0 0
\(232\) −7.15756 −0.469917
\(233\) 16.8844 1.10613 0.553067 0.833137i \(-0.313457\pi\)
0.553067 + 0.833137i \(0.313457\pi\)
\(234\) 0 0
\(235\) −7.13702 −0.465568
\(236\) −36.1091 −2.35050
\(237\) 0 0
\(238\) −12.6530 −0.820170
\(239\) 4.21482 0.272634 0.136317 0.990665i \(-0.456473\pi\)
0.136317 + 0.990665i \(0.456473\pi\)
\(240\) 0 0
\(241\) 21.9795 1.41582 0.707911 0.706302i \(-0.249638\pi\)
0.707911 + 0.706302i \(0.249638\pi\)
\(242\) −22.3534 −1.43693
\(243\) 0 0
\(244\) 26.3509 1.68694
\(245\) 4.16251 0.265933
\(246\) 0 0
\(247\) −8.59666 −0.546993
\(248\) −7.28290 −0.462464
\(249\) 0 0
\(250\) 65.6568 4.15250
\(251\) 28.6909 1.81096 0.905478 0.424394i \(-0.139513\pi\)
0.905478 + 0.424394i \(0.139513\pi\)
\(252\) 0 0
\(253\) −0.729822 −0.0458835
\(254\) 8.61170 0.540346
\(255\) 0 0
\(256\) 1.67352 0.104595
\(257\) −2.44791 −0.152697 −0.0763484 0.997081i \(-0.524326\pi\)
−0.0763484 + 0.997081i \(0.524326\pi\)
\(258\) 0 0
\(259\) 1.27018 0.0789250
\(260\) 10.9686 0.680243
\(261\) 0 0
\(262\) −13.9231 −0.860175
\(263\) 17.5790 1.08397 0.541984 0.840389i \(-0.317673\pi\)
0.541984 + 0.840389i \(0.317673\pi\)
\(264\) 0 0
\(265\) 28.3303 1.74032
\(266\) 18.5080 1.13480
\(267\) 0 0
\(268\) 1.92315 0.117475
\(269\) −16.0601 −0.979199 −0.489600 0.871947i \(-0.662857\pi\)
−0.489600 + 0.871947i \(0.662857\pi\)
\(270\) 0 0
\(271\) −3.38279 −0.205490 −0.102745 0.994708i \(-0.532763\pi\)
−0.102745 + 0.994708i \(0.532763\pi\)
\(272\) −13.6730 −0.829047
\(273\) 0 0
\(274\) 6.96157 0.420564
\(275\) −9.68401 −0.583968
\(276\) 0 0
\(277\) 1.40334 0.0843184 0.0421592 0.999111i \(-0.486576\pi\)
0.0421592 + 0.999111i \(0.486576\pi\)
\(278\) 12.5096 0.750279
\(279\) 0 0
\(280\) −5.69140 −0.340126
\(281\) 30.3532 1.81072 0.905361 0.424644i \(-0.139601\pi\)
0.905361 + 0.424644i \(0.139601\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −19.8670 −1.17889
\(285\) 0 0
\(286\) −1.69140 −0.100014
\(287\) 1.07231 0.0632965
\(288\) 0 0
\(289\) 17.5404 1.03179
\(290\) −46.9122 −2.75478
\(291\) 0 0
\(292\) −19.3059 −1.12979
\(293\) 13.0609 0.763026 0.381513 0.924364i \(-0.375403\pi\)
0.381513 + 0.924364i \(0.375403\pi\)
\(294\) 0 0
\(295\) −57.0396 −3.32097
\(296\) −1.73671 −0.100944
\(297\) 0 0
\(298\) −6.34436 −0.367519
\(299\) 0.928968 0.0537236
\(300\) 0 0
\(301\) −8.59666 −0.495503
\(302\) −29.9755 −1.72490
\(303\) 0 0
\(304\) 20.0000 1.14708
\(305\) 41.6251 2.38345
\(306\) 0 0
\(307\) 17.7900 1.01533 0.507664 0.861555i \(-0.330509\pi\)
0.507664 + 0.861555i \(0.330509\pi\)
\(308\) 2.07020 0.117960
\(309\) 0 0
\(310\) −47.7337 −2.71109
\(311\) 25.5484 1.44872 0.724359 0.689423i \(-0.242136\pi\)
0.724359 + 0.689423i \(0.242136\pi\)
\(312\) 0 0
\(313\) 20.5404 1.16101 0.580505 0.814257i \(-0.302855\pi\)
0.580505 + 0.814257i \(0.302855\pi\)
\(314\) −41.3218 −2.33192
\(315\) 0 0
\(316\) 26.4993 1.49070
\(317\) 0.482333 0.0270905 0.0135453 0.999908i \(-0.495688\pi\)
0.0135453 + 0.999908i \(0.495688\pi\)
\(318\) 0 0
\(319\) 4.11261 0.230262
\(320\) −50.0245 −2.79646
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) −50.5235 −2.81120
\(324\) 0 0
\(325\) 12.3265 0.683750
\(326\) 22.6926 1.25683
\(327\) 0 0
\(328\) −1.46617 −0.0809556
\(329\) 1.71459 0.0945287
\(330\) 0 0
\(331\) 14.6530 0.805400 0.402700 0.915332i \(-0.368072\pi\)
0.402700 + 0.915332i \(0.368072\pi\)
\(332\) 26.4553 1.45192
\(333\) 0 0
\(334\) 4.30860 0.235756
\(335\) 3.03789 0.165978
\(336\) 0 0
\(337\) −23.0602 −1.25617 −0.628084 0.778146i \(-0.716160\pi\)
−0.628084 + 0.778146i \(0.716160\pi\)
\(338\) 2.15293 0.117104
\(339\) 0 0
\(340\) 64.4635 3.49602
\(341\) 4.18463 0.226610
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 11.7542 0.633744
\(345\) 0 0
\(346\) −45.1165 −2.42548
\(347\) −14.2021 −0.762410 −0.381205 0.924491i \(-0.624491\pi\)
−0.381205 + 0.924491i \(0.624491\pi\)
\(348\) 0 0
\(349\) 19.6774 1.05331 0.526653 0.850080i \(-0.323447\pi\)
0.526653 + 0.850080i \(0.323447\pi\)
\(350\) −26.5380 −1.41852
\(351\) 0 0
\(352\) 6.08338 0.324245
\(353\) −19.4757 −1.03659 −0.518293 0.855203i \(-0.673432\pi\)
−0.518293 + 0.855203i \(0.673432\pi\)
\(354\) 0 0
\(355\) −31.3828 −1.66563
\(356\) −42.6974 −2.26296
\(357\) 0 0
\(358\) 23.9231 1.26438
\(359\) 22.6182 1.19374 0.596871 0.802337i \(-0.296410\pi\)
0.596871 + 0.802337i \(0.296410\pi\)
\(360\) 0 0
\(361\) 54.9026 2.88961
\(362\) 21.5293 1.13155
\(363\) 0 0
\(364\) −2.63509 −0.138116
\(365\) −30.4966 −1.59626
\(366\) 0 0
\(367\) −15.2702 −0.797097 −0.398548 0.917147i \(-0.630486\pi\)
−0.398548 + 0.917147i \(0.630486\pi\)
\(368\) −2.16123 −0.112662
\(369\) 0 0
\(370\) −11.3828 −0.591763
\(371\) −6.80607 −0.353354
\(372\) 0 0
\(373\) 23.9231 1.23869 0.619347 0.785118i \(-0.287398\pi\)
0.619347 + 0.785118i \(0.287398\pi\)
\(374\) −9.94051 −0.514011
\(375\) 0 0
\(376\) −2.34436 −0.120901
\(377\) −5.23482 −0.269607
\(378\) 0 0
\(379\) 38.0358 1.95377 0.976883 0.213775i \(-0.0685760\pi\)
0.976883 + 0.213775i \(0.0685760\pi\)
\(380\) −94.2932 −4.83714
\(381\) 0 0
\(382\) −12.0358 −0.615803
\(383\) −8.80735 −0.450035 −0.225017 0.974355i \(-0.572244\pi\)
−0.225017 + 0.974355i \(0.572244\pi\)
\(384\) 0 0
\(385\) 3.27018 0.166664
\(386\) 8.44624 0.429902
\(387\) 0 0
\(388\) −27.9231 −1.41758
\(389\) −17.2234 −0.873261 −0.436631 0.899641i \(-0.643828\pi\)
−0.436631 + 0.899641i \(0.643828\pi\)
\(390\) 0 0
\(391\) 5.45964 0.276106
\(392\) 1.36730 0.0690590
\(393\) 0 0
\(394\) −29.5019 −1.48629
\(395\) 41.8595 2.10618
\(396\) 0 0
\(397\) 12.7861 0.641717 0.320859 0.947127i \(-0.396029\pi\)
0.320859 + 0.947127i \(0.396029\pi\)
\(398\) −20.3659 −1.02085
\(399\) 0 0
\(400\) −28.6774 −1.43387
\(401\) 0.482333 0.0240866 0.0120433 0.999927i \(-0.496166\pi\)
0.0120433 + 0.999927i \(0.496166\pi\)
\(402\) 0 0
\(403\) −5.32648 −0.265331
\(404\) −29.1431 −1.44992
\(405\) 0 0
\(406\) 11.2702 0.559330
\(407\) 0.997885 0.0494633
\(408\) 0 0
\(409\) −4.05631 −0.200571 −0.100286 0.994959i \(-0.531976\pi\)
−0.100286 + 0.994959i \(0.531976\pi\)
\(410\) −9.60959 −0.474584
\(411\) 0 0
\(412\) 36.6887 1.80752
\(413\) 13.7032 0.674289
\(414\) 0 0
\(415\) 41.7900 2.05139
\(416\) −7.74335 −0.379649
\(417\) 0 0
\(418\) 14.5404 0.711192
\(419\) −1.46664 −0.0716500 −0.0358250 0.999358i \(-0.511406\pi\)
−0.0358250 + 0.999358i \(0.511406\pi\)
\(420\) 0 0
\(421\) −19.8105 −0.965506 −0.482753 0.875756i \(-0.660363\pi\)
−0.482753 + 0.875756i \(0.660363\pi\)
\(422\) −27.1197 −1.32017
\(423\) 0 0
\(424\) 9.30594 0.451936
\(425\) 72.4440 3.51405
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 20.3825 0.985226
\(429\) 0 0
\(430\) 77.0396 3.71518
\(431\) −30.7611 −1.48171 −0.740856 0.671664i \(-0.765580\pi\)
−0.740856 + 0.671664i \(0.765580\pi\)
\(432\) 0 0
\(433\) −19.1933 −0.922372 −0.461186 0.887303i \(-0.652576\pi\)
−0.461186 + 0.887303i \(0.652576\pi\)
\(434\) 11.4675 0.550459
\(435\) 0 0
\(436\) 26.3509 1.26198
\(437\) −7.98603 −0.382024
\(438\) 0 0
\(439\) 2.18947 0.104498 0.0522488 0.998634i \(-0.483361\pi\)
0.0522488 + 0.998634i \(0.483361\pi\)
\(440\) −4.47131 −0.213161
\(441\) 0 0
\(442\) 12.6530 0.601840
\(443\) 37.8403 1.79785 0.898924 0.438106i \(-0.144350\pi\)
0.898924 + 0.438106i \(0.144350\pi\)
\(444\) 0 0
\(445\) −67.4468 −3.19729
\(446\) 27.1197 1.28415
\(447\) 0 0
\(448\) 12.0179 0.567791
\(449\) −18.7036 −0.882679 −0.441339 0.897340i \(-0.645496\pi\)
−0.441339 + 0.897340i \(0.645496\pi\)
\(450\) 0 0
\(451\) 0.842435 0.0396687
\(452\) 13.7942 0.648825
\(453\) 0 0
\(454\) 6.34436 0.297756
\(455\) −4.16251 −0.195141
\(456\) 0 0
\(457\) 11.5723 0.541327 0.270664 0.962674i \(-0.412757\pi\)
0.270664 + 0.962674i \(0.412757\pi\)
\(458\) 25.6696 1.19946
\(459\) 0 0
\(460\) 10.1895 0.475086
\(461\) −23.0095 −1.07166 −0.535829 0.844327i \(-0.680001\pi\)
−0.535829 + 0.844327i \(0.680001\pi\)
\(462\) 0 0
\(463\) −4.72982 −0.219813 −0.109907 0.993942i \(-0.535055\pi\)
−0.109907 + 0.993942i \(0.535055\pi\)
\(464\) 12.1787 0.565383
\(465\) 0 0
\(466\) 36.3509 1.68392
\(467\) 5.18251 0.239818 0.119909 0.992785i \(-0.461740\pi\)
0.119909 + 0.992785i \(0.461740\pi\)
\(468\) 0 0
\(469\) −0.729822 −0.0337001
\(470\) −15.3655 −0.708756
\(471\) 0 0
\(472\) −18.7363 −0.862410
\(473\) −6.75377 −0.310538
\(474\) 0 0
\(475\) −105.967 −4.86208
\(476\) −15.4867 −0.709831
\(477\) 0 0
\(478\) 9.07419 0.415044
\(479\) −18.0779 −0.826003 −0.413001 0.910730i \(-0.635520\pi\)
−0.413001 + 0.910730i \(0.635520\pi\)
\(480\) 0 0
\(481\) −1.27018 −0.0579151
\(482\) 47.3201 2.15537
\(483\) 0 0
\(484\) −27.3596 −1.24362
\(485\) −44.1087 −2.00287
\(486\) 0 0
\(487\) 1.81053 0.0820431 0.0410215 0.999158i \(-0.486939\pi\)
0.0410215 + 0.999158i \(0.486939\pi\)
\(488\) 13.6730 0.618947
\(489\) 0 0
\(490\) 8.96157 0.404843
\(491\) 9.48836 0.428204 0.214102 0.976811i \(-0.431318\pi\)
0.214102 + 0.976811i \(0.431318\pi\)
\(492\) 0 0
\(493\) −30.7656 −1.38561
\(494\) −18.5080 −0.832713
\(495\) 0 0
\(496\) 12.3920 0.556416
\(497\) 7.53939 0.338188
\(498\) 0 0
\(499\) −38.3866 −1.71842 −0.859211 0.511621i \(-0.829045\pi\)
−0.859211 + 0.511621i \(0.829045\pi\)
\(500\) 80.3611 3.59386
\(501\) 0 0
\(502\) 61.7694 2.75691
\(503\) −2.85582 −0.127335 −0.0636674 0.997971i \(-0.520280\pi\)
−0.0636674 + 0.997971i \(0.520280\pi\)
\(504\) 0 0
\(505\) −46.0358 −2.04856
\(506\) −1.57125 −0.0698507
\(507\) 0 0
\(508\) 10.5404 0.467653
\(509\) −26.6730 −1.18226 −0.591131 0.806576i \(-0.701318\pi\)
−0.591131 + 0.806576i \(0.701318\pi\)
\(510\) 0 0
\(511\) 7.32648 0.324105
\(512\) 24.3768 1.07731
\(513\) 0 0
\(514\) −5.27018 −0.232458
\(515\) 57.9552 2.55381
\(516\) 0 0
\(517\) 1.34703 0.0592424
\(518\) 2.73460 0.120151
\(519\) 0 0
\(520\) 5.69140 0.249584
\(521\) −0.0166098 −0.000727689 0 −0.000363845 1.00000i \(-0.500116\pi\)
−0.000363845 1.00000i \(0.500116\pi\)
\(522\) 0 0
\(523\) −16.7298 −0.731544 −0.365772 0.930704i \(-0.619195\pi\)
−0.365772 + 0.930704i \(0.619195\pi\)
\(524\) −17.0413 −0.744454
\(525\) 0 0
\(526\) 37.8463 1.65018
\(527\) −31.3043 −1.36364
\(528\) 0 0
\(529\) −22.1370 −0.962479
\(530\) 60.9931 2.64937
\(531\) 0 0
\(532\) 22.6530 0.982131
\(533\) −1.07231 −0.0464469
\(534\) 0 0
\(535\) 32.1972 1.39201
\(536\) 0.997885 0.0431021
\(537\) 0 0
\(538\) −34.5761 −1.49068
\(539\) −0.785627 −0.0338393
\(540\) 0 0
\(541\) 33.2702 1.43040 0.715198 0.698922i \(-0.246336\pi\)
0.715198 + 0.698922i \(0.246336\pi\)
\(542\) −7.28290 −0.312827
\(543\) 0 0
\(544\) −45.5085 −1.95116
\(545\) 41.6251 1.78302
\(546\) 0 0
\(547\) 29.6774 1.26891 0.634456 0.772959i \(-0.281224\pi\)
0.634456 + 0.772959i \(0.281224\pi\)
\(548\) 8.52067 0.363985
\(549\) 0 0
\(550\) −20.8490 −0.889003
\(551\) 45.0020 1.91715
\(552\) 0 0
\(553\) −10.0563 −0.427638
\(554\) 3.02128 0.128362
\(555\) 0 0
\(556\) 15.3113 0.649343
\(557\) 30.4578 1.29054 0.645270 0.763955i \(-0.276745\pi\)
0.645270 + 0.763955i \(0.276745\pi\)
\(558\) 0 0
\(559\) 8.59666 0.363600
\(560\) 9.68401 0.409224
\(561\) 0 0
\(562\) 65.3482 2.75655
\(563\) −9.18507 −0.387105 −0.193552 0.981090i \(-0.562001\pi\)
−0.193552 + 0.981090i \(0.562001\pi\)
\(564\) 0 0
\(565\) 21.7900 0.916712
\(566\) −8.61170 −0.361977
\(567\) 0 0
\(568\) −10.3086 −0.432539
\(569\) −33.6391 −1.41022 −0.705112 0.709096i \(-0.749103\pi\)
−0.705112 + 0.709096i \(0.749103\pi\)
\(570\) 0 0
\(571\) 14.0563 0.588238 0.294119 0.955769i \(-0.404974\pi\)
0.294119 + 0.955769i \(0.404974\pi\)
\(572\) −2.07020 −0.0865592
\(573\) 0 0
\(574\) 2.30860 0.0963593
\(575\) 11.4509 0.477536
\(576\) 0 0
\(577\) −31.9231 −1.32898 −0.664489 0.747298i \(-0.731351\pi\)
−0.664489 + 0.747298i \(0.731351\pi\)
\(578\) 37.7631 1.57074
\(579\) 0 0
\(580\) −57.4186 −2.38418
\(581\) −10.0396 −0.416513
\(582\) 0 0
\(583\) −5.34703 −0.221451
\(584\) −10.0175 −0.414527
\(585\) 0 0
\(586\) 28.1191 1.16159
\(587\) −4.75249 −0.196156 −0.0980781 0.995179i \(-0.531269\pi\)
−0.0980781 + 0.995179i \(0.531269\pi\)
\(588\) 0 0
\(589\) 45.7900 1.88674
\(590\) −122.802 −5.05568
\(591\) 0 0
\(592\) 2.95505 0.121452
\(593\) 12.8788 0.528870 0.264435 0.964404i \(-0.414815\pi\)
0.264435 + 0.964404i \(0.414815\pi\)
\(594\) 0 0
\(595\) −24.4635 −1.00291
\(596\) −7.76523 −0.318076
\(597\) 0 0
\(598\) 2.00000 0.0817861
\(599\) −27.9440 −1.14176 −0.570881 0.821033i \(-0.693398\pi\)
−0.570881 + 0.821033i \(0.693398\pi\)
\(600\) 0 0
\(601\) −13.6211 −0.555615 −0.277807 0.960637i \(-0.589608\pi\)
−0.277807 + 0.960637i \(0.589608\pi\)
\(602\) −18.5080 −0.754329
\(603\) 0 0
\(604\) −36.6887 −1.49284
\(605\) −43.2185 −1.75708
\(606\) 0 0
\(607\) 19.1165 0.775914 0.387957 0.921678i \(-0.373181\pi\)
0.387957 + 0.921678i \(0.373181\pi\)
\(608\) 66.5669 2.69965
\(609\) 0 0
\(610\) 89.6157 3.62843
\(611\) −1.71459 −0.0693651
\(612\) 0 0
\(613\) −6.11261 −0.246886 −0.123443 0.992352i \(-0.539394\pi\)
−0.123443 + 0.992352i \(0.539394\pi\)
\(614\) 38.3005 1.54568
\(615\) 0 0
\(616\) 1.07419 0.0432802
\(617\) −7.80947 −0.314397 −0.157199 0.987567i \(-0.550246\pi\)
−0.157199 + 0.987567i \(0.550246\pi\)
\(618\) 0 0
\(619\) 20.4635 0.822498 0.411249 0.911523i \(-0.365093\pi\)
0.411249 + 0.911523i \(0.365093\pi\)
\(620\) −58.4240 −2.34636
\(621\) 0 0
\(622\) 55.0039 2.20545
\(623\) 16.2034 0.649176
\(624\) 0 0
\(625\) 65.3098 2.61239
\(626\) 44.2219 1.76746
\(627\) 0 0
\(628\) −50.5761 −2.01821
\(629\) −7.46497 −0.297648
\(630\) 0 0
\(631\) 22.2740 0.886715 0.443358 0.896345i \(-0.353787\pi\)
0.443358 + 0.896345i \(0.353787\pi\)
\(632\) 13.7500 0.546945
\(633\) 0 0
\(634\) 1.03843 0.0412412
\(635\) 16.6500 0.660737
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 8.85415 0.350539
\(639\) 0 0
\(640\) −43.2356 −1.70904
\(641\) 3.76818 0.148834 0.0744171 0.997227i \(-0.476290\pi\)
0.0744171 + 0.997227i \(0.476290\pi\)
\(642\) 0 0
\(643\) −29.4596 −1.16177 −0.580887 0.813984i \(-0.697294\pi\)
−0.580887 + 0.813984i \(0.697294\pi\)
\(644\) −2.44791 −0.0964613
\(645\) 0 0
\(646\) −108.773 −4.27963
\(647\) −1.85794 −0.0730430 −0.0365215 0.999333i \(-0.511628\pi\)
−0.0365215 + 0.999333i \(0.511628\pi\)
\(648\) 0 0
\(649\) 10.7656 0.422586
\(650\) 26.5380 1.04091
\(651\) 0 0
\(652\) 27.7748 1.08774
\(653\) −42.9097 −1.67918 −0.839592 0.543217i \(-0.817206\pi\)
−0.839592 + 0.543217i \(0.817206\pi\)
\(654\) 0 0
\(655\) −26.9193 −1.05182
\(656\) 2.49471 0.0974022
\(657\) 0 0
\(658\) 3.69140 0.143906
\(659\) −11.1119 −0.432859 −0.216430 0.976298i \(-0.569441\pi\)
−0.216430 + 0.976298i \(0.569441\pi\)
\(660\) 0 0
\(661\) 35.9026 1.39645 0.698225 0.715879i \(-0.253974\pi\)
0.698225 + 0.715879i \(0.253974\pi\)
\(662\) 31.5468 1.22610
\(663\) 0 0
\(664\) 13.7272 0.532717
\(665\) 35.7837 1.38763
\(666\) 0 0
\(667\) −4.86298 −0.188295
\(668\) 5.27355 0.204040
\(669\) 0 0
\(670\) 6.54036 0.252676
\(671\) −7.85627 −0.303288
\(672\) 0 0
\(673\) 4.05631 0.156359 0.0781796 0.996939i \(-0.475089\pi\)
0.0781796 + 0.996939i \(0.475089\pi\)
\(674\) −49.6468 −1.91232
\(675\) 0 0
\(676\) 2.63509 0.101350
\(677\) −41.0351 −1.57711 −0.788554 0.614966i \(-0.789170\pi\)
−0.788554 + 0.614966i \(0.789170\pi\)
\(678\) 0 0
\(679\) 10.5967 0.406663
\(680\) 33.4489 1.28271
\(681\) 0 0
\(682\) 9.00919 0.344980
\(683\) −33.3745 −1.27704 −0.638520 0.769605i \(-0.720453\pi\)
−0.638520 + 0.769605i \(0.720453\pi\)
\(684\) 0 0
\(685\) 13.4596 0.514267
\(686\) −2.15293 −0.0821991
\(687\) 0 0
\(688\) −20.0000 −0.762493
\(689\) 6.80607 0.259291
\(690\) 0 0
\(691\) −23.8668 −0.907937 −0.453969 0.891018i \(-0.649992\pi\)
−0.453969 + 0.891018i \(0.649992\pi\)
\(692\) −55.2206 −2.09917
\(693\) 0 0
\(694\) −30.5761 −1.16065
\(695\) 24.1864 0.917442
\(696\) 0 0
\(697\) −6.30208 −0.238708
\(698\) 42.3639 1.60350
\(699\) 0 0
\(700\) −32.4814 −1.22768
\(701\) 22.2762 0.841359 0.420679 0.907209i \(-0.361792\pi\)
0.420679 + 0.907209i \(0.361792\pi\)
\(702\) 0 0
\(703\) 10.9193 0.411829
\(704\) 9.44157 0.355842
\(705\) 0 0
\(706\) −41.9297 −1.57804
\(707\) 11.0596 0.415940
\(708\) 0 0
\(709\) −24.3866 −0.915860 −0.457930 0.888988i \(-0.651409\pi\)
−0.457930 + 0.888988i \(0.651409\pi\)
\(710\) −67.5648 −2.53566
\(711\) 0 0
\(712\) −22.1549 −0.830290
\(713\) −4.94814 −0.185309
\(714\) 0 0
\(715\) −3.27018 −0.122298
\(716\) 29.2809 1.09428
\(717\) 0 0
\(718\) 48.6953 1.81729
\(719\) 20.6526 0.770212 0.385106 0.922872i \(-0.374165\pi\)
0.385106 + 0.922872i \(0.374165\pi\)
\(720\) 0 0
\(721\) −13.9231 −0.518525
\(722\) 118.201 4.39899
\(723\) 0 0
\(724\) 26.3509 0.979323
\(725\) −64.5269 −2.39647
\(726\) 0 0
\(727\) −29.1933 −1.08272 −0.541360 0.840791i \(-0.682090\pi\)
−0.541360 + 0.840791i \(0.682090\pi\)
\(728\) −1.36730 −0.0506755
\(729\) 0 0
\(730\) −65.6568 −2.43007
\(731\) 50.5235 1.86868
\(732\) 0 0
\(733\) −4.78613 −0.176780 −0.0883899 0.996086i \(-0.528172\pi\)
−0.0883899 + 0.996086i \(0.528172\pi\)
\(734\) −32.8756 −1.21346
\(735\) 0 0
\(736\) −7.19332 −0.265149
\(737\) −0.573368 −0.0211203
\(738\) 0 0
\(739\) −44.5761 −1.63976 −0.819879 0.572536i \(-0.805960\pi\)
−0.819879 + 0.572536i \(0.805960\pi\)
\(740\) −13.9320 −0.512152
\(741\) 0 0
\(742\) −14.6530 −0.537927
\(743\) 5.39477 0.197915 0.0989575 0.995092i \(-0.468449\pi\)
0.0989575 + 0.995092i \(0.468449\pi\)
\(744\) 0 0
\(745\) −12.2663 −0.449403
\(746\) 51.5048 1.88572
\(747\) 0 0
\(748\) −12.1668 −0.444861
\(749\) −7.73504 −0.282632
\(750\) 0 0
\(751\) −45.7900 −1.67090 −0.835450 0.549566i \(-0.814793\pi\)
−0.835450 + 0.549566i \(0.814793\pi\)
\(752\) 3.98898 0.145463
\(753\) 0 0
\(754\) −11.2702 −0.410436
\(755\) −57.9552 −2.10921
\(756\) 0 0
\(757\) 48.2535 1.75380 0.876901 0.480670i \(-0.159607\pi\)
0.876901 + 0.480670i \(0.159607\pi\)
\(758\) 81.8882 2.97431
\(759\) 0 0
\(760\) −48.9270 −1.77477
\(761\) 30.4221 1.10280 0.551401 0.834240i \(-0.314094\pi\)
0.551401 + 0.834240i \(0.314094\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) −14.7313 −0.532958
\(765\) 0 0
\(766\) −18.9616 −0.685109
\(767\) −13.7032 −0.494793
\(768\) 0 0
\(769\) 45.2496 1.63174 0.815872 0.578233i \(-0.196258\pi\)
0.815872 + 0.578233i \(0.196258\pi\)
\(770\) 7.04045 0.253720
\(771\) 0 0
\(772\) 10.3378 0.372067
\(773\) 24.1894 0.870033 0.435017 0.900422i \(-0.356742\pi\)
0.435017 + 0.900422i \(0.356742\pi\)
\(774\) 0 0
\(775\) −65.6568 −2.35846
\(776\) −14.4888 −0.520118
\(777\) 0 0
\(778\) −37.0807 −1.32941
\(779\) 9.21829 0.330280
\(780\) 0 0
\(781\) 5.92315 0.211947
\(782\) 11.7542 0.420330
\(783\) 0 0
\(784\) −2.32648 −0.0830887
\(785\) −79.8924 −2.85148
\(786\) 0 0
\(787\) −17.0602 −0.608129 −0.304065 0.952651i \(-0.598344\pi\)
−0.304065 + 0.952651i \(0.598344\pi\)
\(788\) −36.1091 −1.28633
\(789\) 0 0
\(790\) 90.1203 3.20634
\(791\) −5.23482 −0.186129
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 27.5276 0.976917
\(795\) 0 0
\(796\) −24.9270 −0.883514
\(797\) −44.7510 −1.58516 −0.792581 0.609767i \(-0.791263\pi\)
−0.792581 + 0.609767i \(0.791263\pi\)
\(798\) 0 0
\(799\) −10.0769 −0.356493
\(800\) −95.4482 −3.37460
\(801\) 0 0
\(802\) 1.03843 0.0366681
\(803\) 5.75588 0.203121
\(804\) 0 0
\(805\) −3.86684 −0.136288
\(806\) −11.4675 −0.403926
\(807\) 0 0
\(808\) −15.1218 −0.531983
\(809\) −32.4591 −1.14120 −0.570601 0.821228i \(-0.693290\pi\)
−0.570601 + 0.821228i \(0.693290\pi\)
\(810\) 0 0
\(811\) 20.4635 0.718571 0.359285 0.933228i \(-0.383020\pi\)
0.359285 + 0.933228i \(0.383020\pi\)
\(812\) 13.7942 0.484082
\(813\) 0 0
\(814\) 2.14837 0.0753004
\(815\) 43.8743 1.53685
\(816\) 0 0
\(817\) −73.9026 −2.58552
\(818\) −8.73293 −0.305340
\(819\) 0 0
\(820\) −11.7617 −0.410737
\(821\) 30.9266 1.07935 0.539673 0.841875i \(-0.318548\pi\)
0.539673 + 0.841875i \(0.318548\pi\)
\(822\) 0 0
\(823\) 11.8463 0.412936 0.206468 0.978453i \(-0.433803\pi\)
0.206468 + 0.978453i \(0.433803\pi\)
\(824\) 19.0371 0.663189
\(825\) 0 0
\(826\) 29.5019 1.02650
\(827\) −55.1296 −1.91704 −0.958522 0.285019i \(-0.908000\pi\)
−0.958522 + 0.285019i \(0.908000\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 89.9707 3.12293
\(831\) 0 0
\(832\) −12.0179 −0.416645
\(833\) 5.87710 0.203630
\(834\) 0 0
\(835\) 8.33034 0.288283
\(836\) 17.7968 0.615514
\(837\) 0 0
\(838\) −3.15756 −0.109076
\(839\) −14.2765 −0.492881 −0.246441 0.969158i \(-0.579261\pi\)
−0.246441 + 0.969158i \(0.579261\pi\)
\(840\) 0 0
\(841\) −1.59666 −0.0550573
\(842\) −42.6506 −1.46984
\(843\) 0 0
\(844\) −33.1933 −1.14256
\(845\) 4.16251 0.143195
\(846\) 0 0
\(847\) 10.3828 0.356757
\(848\) −15.8342 −0.543749
\(849\) 0 0
\(850\) 155.967 5.34961
\(851\) −1.17996 −0.0404483
\(852\) 0 0
\(853\) 16.6735 0.570890 0.285445 0.958395i \(-0.407859\pi\)
0.285445 + 0.958395i \(0.407859\pi\)
\(854\) −21.5293 −0.736716
\(855\) 0 0
\(856\) 10.5761 0.361484
\(857\) −56.1139 −1.91681 −0.958407 0.285404i \(-0.907872\pi\)
−0.958407 + 0.285404i \(0.907872\pi\)
\(858\) 0 0
\(859\) −43.9589 −1.49986 −0.749929 0.661518i \(-0.769913\pi\)
−0.749929 + 0.661518i \(0.769913\pi\)
\(860\) 94.2932 3.21537
\(861\) 0 0
\(862\) −66.2264 −2.25568
\(863\) 14.3978 0.490106 0.245053 0.969510i \(-0.421195\pi\)
0.245053 + 0.969510i \(0.421195\pi\)
\(864\) 0 0
\(865\) −87.2291 −2.96588
\(866\) −41.3218 −1.40417
\(867\) 0 0
\(868\) 14.0358 0.476405
\(869\) −7.90050 −0.268006
\(870\) 0 0
\(871\) 0.729822 0.0247291
\(872\) 13.6730 0.463026
\(873\) 0 0
\(874\) −17.1933 −0.581573
\(875\) −30.4966 −1.03097
\(876\) 0 0
\(877\) 39.1933 1.32346 0.661732 0.749740i \(-0.269822\pi\)
0.661732 + 0.749740i \(0.269822\pi\)
\(878\) 4.71376 0.159082
\(879\) 0 0
\(880\) 7.60802 0.256466
\(881\) 16.1647 0.544602 0.272301 0.962212i \(-0.412215\pi\)
0.272301 + 0.962212i \(0.412215\pi\)
\(882\) 0 0
\(883\) −18.3866 −0.618760 −0.309380 0.950938i \(-0.600121\pi\)
−0.309380 + 0.950938i \(0.600121\pi\)
\(884\) 15.4867 0.520874
\(885\) 0 0
\(886\) 81.4674 2.73695
\(887\) 41.3384 1.38801 0.694004 0.719971i \(-0.255845\pi\)
0.694004 + 0.719971i \(0.255845\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) −145.208 −4.86738
\(891\) 0 0
\(892\) 33.1933 1.11139
\(893\) 14.7398 0.493248
\(894\) 0 0
\(895\) 46.2535 1.54608
\(896\) 10.3869 0.347002
\(897\) 0 0
\(898\) −40.2675 −1.34374
\(899\) 27.8832 0.929956
\(900\) 0 0
\(901\) 40.0000 1.33259
\(902\) 1.81370 0.0603896
\(903\) 0 0
\(904\) 7.15756 0.238057
\(905\) 41.6251 1.38366
\(906\) 0 0
\(907\) 41.6363 1.38251 0.691255 0.722611i \(-0.257058\pi\)
0.691255 + 0.722611i \(0.257058\pi\)
\(908\) 7.76523 0.257698
\(909\) 0 0
\(910\) −8.96157 −0.297073
\(911\) −4.93153 −0.163389 −0.0816944 0.996657i \(-0.526033\pi\)
−0.0816944 + 0.996657i \(0.526033\pi\)
\(912\) 0 0
\(913\) −7.88739 −0.261034
\(914\) 24.9142 0.824089
\(915\) 0 0
\(916\) 31.4186 1.03810
\(917\) 6.46708 0.213562
\(918\) 0 0
\(919\) −6.38665 −0.210676 −0.105338 0.994436i \(-0.533592\pi\)
−0.105338 + 0.994436i \(0.533592\pi\)
\(920\) 5.28713 0.174311
\(921\) 0 0
\(922\) −49.5377 −1.63144
\(923\) −7.53939 −0.248162
\(924\) 0 0
\(925\) −15.6568 −0.514793
\(926\) −10.1830 −0.334633
\(927\) 0 0
\(928\) 40.5350 1.33063
\(929\) −36.7514 −1.20577 −0.602887 0.797827i \(-0.705983\pi\)
−0.602887 + 0.797827i \(0.705983\pi\)
\(930\) 0 0
\(931\) −8.59666 −0.281744
\(932\) 44.4919 1.45738
\(933\) 0 0
\(934\) 11.1576 0.365087
\(935\) −19.2192 −0.628534
\(936\) 0 0
\(937\) −8.27404 −0.270301 −0.135150 0.990825i \(-0.543152\pi\)
−0.135150 + 0.990825i \(0.543152\pi\)
\(938\) −1.57125 −0.0513032
\(939\) 0 0
\(940\) −18.8067 −0.613406
\(941\) −56.5439 −1.84328 −0.921640 0.388047i \(-0.873150\pi\)
−0.921640 + 0.388047i \(0.873150\pi\)
\(942\) 0 0
\(943\) −0.996143 −0.0324389
\(944\) 31.8802 1.03761
\(945\) 0 0
\(946\) −14.5404 −0.472748
\(947\) −10.9686 −0.356431 −0.178216 0.983991i \(-0.557032\pi\)
−0.178216 + 0.983991i \(0.557032\pi\)
\(948\) 0 0
\(949\) −7.32648 −0.237828
\(950\) −228.138 −7.40178
\(951\) 0 0
\(952\) −8.03576 −0.260440
\(953\) 11.4152 0.369775 0.184888 0.982760i \(-0.440808\pi\)
0.184888 + 0.982760i \(0.440808\pi\)
\(954\) 0 0
\(955\) −23.2702 −0.753005
\(956\) 11.1064 0.359207
\(957\) 0 0
\(958\) −38.9205 −1.25746
\(959\) −3.23354 −0.104416
\(960\) 0 0
\(961\) −2.62856 −0.0847924
\(962\) −2.73460 −0.0881670
\(963\) 0 0
\(964\) 57.9178 1.86541
\(965\) 16.3301 0.525686
\(966\) 0 0
\(967\) −2.65297 −0.0853137 −0.0426569 0.999090i \(-0.513582\pi\)
−0.0426569 + 0.999090i \(0.513582\pi\)
\(968\) −14.1964 −0.456289
\(969\) 0 0
\(970\) −94.9628 −3.04907
\(971\) −39.4805 −1.26699 −0.633494 0.773747i \(-0.718380\pi\)
−0.633494 + 0.773747i \(0.718380\pi\)
\(972\) 0 0
\(973\) −5.81053 −0.186277
\(974\) 3.89794 0.124898
\(975\) 0 0
\(976\) −23.2648 −0.744689
\(977\) 16.4544 0.526423 0.263211 0.964738i \(-0.415218\pi\)
0.263211 + 0.964738i \(0.415218\pi\)
\(978\) 0 0
\(979\) 12.7298 0.406847
\(980\) 10.9686 0.350379
\(981\) 0 0
\(982\) 20.4277 0.651875
\(983\) 19.6492 0.626712 0.313356 0.949636i \(-0.398547\pi\)
0.313356 + 0.949636i \(0.398547\pi\)
\(984\) 0 0
\(985\) −57.0396 −1.81743
\(986\) −66.2360 −2.10938
\(987\) 0 0
\(988\) −22.6530 −0.720687
\(989\) 7.98603 0.253941
\(990\) 0 0
\(991\) −22.5404 −0.716018 −0.358009 0.933718i \(-0.616544\pi\)
−0.358009 + 0.933718i \(0.616544\pi\)
\(992\) 41.2448 1.30952
\(993\) 0 0
\(994\) 16.2318 0.514840
\(995\) −39.3759 −1.24830
\(996\) 0 0
\(997\) 8.42774 0.266909 0.133455 0.991055i \(-0.457393\pi\)
0.133455 + 0.991055i \(0.457393\pi\)
\(998\) −82.6436 −2.61604
\(999\) 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 819.2.a.m.1.5 yes 6
3.2 odd 2 inner 819.2.a.m.1.2 6
7.6 odd 2 5733.2.a.bv.1.5 6
21.20 even 2 5733.2.a.bv.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
819.2.a.m.1.2 6 3.2 odd 2 inner
819.2.a.m.1.5 yes 6 1.1 even 1 trivial
5733.2.a.bv.1.2 6 21.20 even 2
5733.2.a.bv.1.5 6 7.6 odd 2