Properties

Label 819.2.a.i.1.3
Level $819$
Weight $2$
Character 819.1
Self dual yes
Analytic conductor $6.540$
Analytic rank $1$
Dimension $3$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.81361\) 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)
 
\(f(q)\) \(=\) \(q+1.81361 q^{2} +1.28917 q^{4} -2.81361 q^{5} -1.00000 q^{7} -1.28917 q^{8} +O(q^{10})\) \(q+1.81361 q^{2} +1.28917 q^{4} -2.81361 q^{5} -1.00000 q^{7} -1.28917 q^{8} -5.10278 q^{10} -3.10278 q^{11} +1.00000 q^{13} -1.81361 q^{14} -4.91638 q^{16} +0.524438 q^{17} +0.813607 q^{19} -3.62721 q^{20} -5.62721 q^{22} -7.33804 q^{23} +2.91638 q^{25} +1.81361 q^{26} -1.28917 q^{28} -8.28917 q^{29} +1.39194 q^{31} -6.33804 q^{32} +0.951124 q^{34} +2.81361 q^{35} -6.15165 q^{37} +1.47556 q^{38} +3.62721 q^{40} +4.20555 q^{41} +6.75971 q^{43} -4.00000 q^{44} -13.3083 q^{46} +5.97028 q^{47} +1.00000 q^{49} +5.28917 q^{50} +1.28917 q^{52} +2.49472 q^{53} +8.72999 q^{55} +1.28917 q^{56} -15.0333 q^{58} +4.47054 q^{59} -2.00000 q^{61} +2.52444 q^{62} -1.66196 q^{64} -2.81361 q^{65} +10.0383 q^{67} +0.676089 q^{68} +5.10278 q^{70} +8.72999 q^{71} -2.34307 q^{73} -11.1567 q^{74} +1.04888 q^{76} +3.10278 q^{77} -13.5436 q^{79} +13.8328 q^{80} +7.62721 q^{82} -16.4791 q^{83} -1.47556 q^{85} +12.2594 q^{86} +4.00000 q^{88} +10.6464 q^{89} -1.00000 q^{91} -9.45998 q^{92} +10.8277 q^{94} -2.28917 q^{95} -1.18639 q^{97} +1.81361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8} - 8 q^{10} - 2 q^{11} + 3 q^{13} + q^{14} - q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} - q^{26} - 3 q^{28} - 24 q^{29} - 4 q^{31} - 7 q^{32} + 14 q^{34} + 2 q^{35} + 10 q^{38} - 2 q^{40} - 2 q^{41} + 10 q^{43} - 12 q^{44} - 18 q^{46} + 8 q^{47} + 3 q^{49} + 15 q^{50} + 3 q^{52} - 8 q^{53} + 6 q^{55} + 3 q^{56} + 12 q^{58} + 4 q^{59} - 6 q^{61} + 2 q^{62} - 17 q^{64} - 2 q^{65} - 12 q^{67} - 22 q^{68} + 8 q^{70} + 6 q^{71} - 10 q^{73} - 30 q^{74} - 8 q^{76} + 2 q^{77} - 14 q^{79} + 14 q^{80} + 10 q^{82} + 12 q^{83} - 10 q^{85} + 26 q^{86} + 12 q^{88} - 2 q^{89} - 3 q^{91} + 12 q^{92} - 10 q^{94} - 6 q^{95} - 10 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.81361 1.28241 0.641207 0.767368i \(-0.278434\pi\)
0.641207 + 0.767368i \(0.278434\pi\)
\(3\) 0 0
\(4\) 1.28917 0.644584
\(5\) −2.81361 −1.25828 −0.629142 0.777291i \(-0.716593\pi\)
−0.629142 + 0.777291i \(0.716593\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.28917 −0.455790
\(9\) 0 0
\(10\) −5.10278 −1.61364
\(11\) −3.10278 −0.935522 −0.467761 0.883855i \(-0.654939\pi\)
−0.467761 + 0.883855i \(0.654939\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.81361 −0.484707
\(15\) 0 0
\(16\) −4.91638 −1.22910
\(17\) 0.524438 0.127195 0.0635974 0.997976i \(-0.479743\pi\)
0.0635974 + 0.997976i \(0.479743\pi\)
\(18\) 0 0
\(19\) 0.813607 0.186654 0.0933271 0.995636i \(-0.470250\pi\)
0.0933271 + 0.995636i \(0.470250\pi\)
\(20\) −3.62721 −0.811069
\(21\) 0 0
\(22\) −5.62721 −1.19973
\(23\) −7.33804 −1.53009 −0.765044 0.643978i \(-0.777283\pi\)
−0.765044 + 0.643978i \(0.777283\pi\)
\(24\) 0 0
\(25\) 2.91638 0.583276
\(26\) 1.81361 0.355677
\(27\) 0 0
\(28\) −1.28917 −0.243630
\(29\) −8.28917 −1.53926 −0.769630 0.638490i \(-0.779559\pi\)
−0.769630 + 0.638490i \(0.779559\pi\)
\(30\) 0 0
\(31\) 1.39194 0.250000 0.125000 0.992157i \(-0.460107\pi\)
0.125000 + 0.992157i \(0.460107\pi\)
\(32\) −6.33804 −1.12042
\(33\) 0 0
\(34\) 0.951124 0.163116
\(35\) 2.81361 0.475586
\(36\) 0 0
\(37\) −6.15165 −1.01133 −0.505663 0.862731i \(-0.668752\pi\)
−0.505663 + 0.862731i \(0.668752\pi\)
\(38\) 1.47556 0.239368
\(39\) 0 0
\(40\) 3.62721 0.573513
\(41\) 4.20555 0.656797 0.328398 0.944539i \(-0.393491\pi\)
0.328398 + 0.944539i \(0.393491\pi\)
\(42\) 0 0
\(43\) 6.75971 1.03085 0.515423 0.856936i \(-0.327635\pi\)
0.515423 + 0.856936i \(0.327635\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −13.3083 −1.96221
\(47\) 5.97028 0.870855 0.435427 0.900224i \(-0.356597\pi\)
0.435427 + 0.900224i \(0.356597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.28917 0.748001
\(51\) 0 0
\(52\) 1.28917 0.178776
\(53\) 2.49472 0.342676 0.171338 0.985212i \(-0.445191\pi\)
0.171338 + 0.985212i \(0.445191\pi\)
\(54\) 0 0
\(55\) 8.72999 1.17715
\(56\) 1.28917 0.172272
\(57\) 0 0
\(58\) −15.0333 −1.97397
\(59\) 4.47054 0.582015 0.291007 0.956721i \(-0.406010\pi\)
0.291007 + 0.956721i \(0.406010\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.52444 0.320604
\(63\) 0 0
\(64\) −1.66196 −0.207744
\(65\) −2.81361 −0.348985
\(66\) 0 0
\(67\) 10.0383 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(68\) 0.676089 0.0819878
\(69\) 0 0
\(70\) 5.10278 0.609898
\(71\) 8.72999 1.03606 0.518029 0.855363i \(-0.326666\pi\)
0.518029 + 0.855363i \(0.326666\pi\)
\(72\) 0 0
\(73\) −2.34307 −0.274235 −0.137118 0.990555i \(-0.543784\pi\)
−0.137118 + 0.990555i \(0.543784\pi\)
\(74\) −11.1567 −1.29694
\(75\) 0 0
\(76\) 1.04888 0.120314
\(77\) 3.10278 0.353594
\(78\) 0 0
\(79\) −13.5436 −1.52377 −0.761887 0.647710i \(-0.775727\pi\)
−0.761887 + 0.647710i \(0.775727\pi\)
\(80\) 13.8328 1.54655
\(81\) 0 0
\(82\) 7.62721 0.842285
\(83\) −16.4791 −1.80882 −0.904410 0.426665i \(-0.859688\pi\)
−0.904410 + 0.426665i \(0.859688\pi\)
\(84\) 0 0
\(85\) −1.47556 −0.160047
\(86\) 12.2594 1.32197
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 10.6464 1.12851 0.564256 0.825600i \(-0.309163\pi\)
0.564256 + 0.825600i \(0.309163\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −9.45998 −0.986271
\(93\) 0 0
\(94\) 10.8277 1.11680
\(95\) −2.28917 −0.234864
\(96\) 0 0
\(97\) −1.18639 −0.120460 −0.0602300 0.998185i \(-0.519183\pi\)
−0.0602300 + 0.998185i \(0.519183\pi\)
\(98\) 1.81361 0.183202
\(99\) 0 0
\(100\) 3.75971 0.375971
\(101\) −13.1028 −1.30377 −0.651887 0.758316i \(-0.726023\pi\)
−0.651887 + 0.758316i \(0.726023\pi\)
\(102\) 0 0
\(103\) 4.41110 0.434639 0.217319 0.976101i \(-0.430269\pi\)
0.217319 + 0.976101i \(0.430269\pi\)
\(104\) −1.28917 −0.126413
\(105\) 0 0
\(106\) 4.52444 0.439452
\(107\) −0.578337 −0.0559100 −0.0279550 0.999609i \(-0.508900\pi\)
−0.0279550 + 0.999609i \(0.508900\pi\)
\(108\) 0 0
\(109\) 5.57331 0.533827 0.266913 0.963721i \(-0.413996\pi\)
0.266913 + 0.963721i \(0.413996\pi\)
\(110\) 15.8328 1.50959
\(111\) 0 0
\(112\) 4.91638 0.464554
\(113\) −5.44584 −0.512302 −0.256151 0.966637i \(-0.582454\pi\)
−0.256151 + 0.966637i \(0.582454\pi\)
\(114\) 0 0
\(115\) 20.6464 1.92528
\(116\) −10.6861 −0.992183
\(117\) 0 0
\(118\) 8.10780 0.746383
\(119\) −0.524438 −0.0480751
\(120\) 0 0
\(121\) −1.37279 −0.124799
\(122\) −3.62721 −0.328392
\(123\) 0 0
\(124\) 1.79445 0.161146
\(125\) 5.86248 0.524356
\(126\) 0 0
\(127\) −12.8816 −1.14306 −0.571530 0.820581i \(-0.693650\pi\)
−0.571530 + 0.820581i \(0.693650\pi\)
\(128\) 9.66196 0.854004
\(129\) 0 0
\(130\) −5.10278 −0.447543
\(131\) −9.04888 −0.790604 −0.395302 0.918551i \(-0.629360\pi\)
−0.395302 + 0.918551i \(0.629360\pi\)
\(132\) 0 0
\(133\) −0.813607 −0.0705486
\(134\) 18.2056 1.57272
\(135\) 0 0
\(136\) −0.676089 −0.0579741
\(137\) 6.25945 0.534781 0.267390 0.963588i \(-0.413839\pi\)
0.267390 + 0.963588i \(0.413839\pi\)
\(138\) 0 0
\(139\) −11.5733 −0.981636 −0.490818 0.871262i \(-0.663302\pi\)
−0.490818 + 0.871262i \(0.663302\pi\)
\(140\) 3.62721 0.306555
\(141\) 0 0
\(142\) 15.8328 1.32866
\(143\) −3.10278 −0.259467
\(144\) 0 0
\(145\) 23.3225 1.93682
\(146\) −4.24940 −0.351683
\(147\) 0 0
\(148\) −7.93051 −0.651884
\(149\) −8.52444 −0.698349 −0.349175 0.937058i \(-0.613538\pi\)
−0.349175 + 0.937058i \(0.613538\pi\)
\(150\) 0 0
\(151\) −11.9844 −0.975278 −0.487639 0.873045i \(-0.662142\pi\)
−0.487639 + 0.873045i \(0.662142\pi\)
\(152\) −1.04888 −0.0850751
\(153\) 0 0
\(154\) 5.62721 0.453454
\(155\) −3.91638 −0.314571
\(156\) 0 0
\(157\) −12.8277 −1.02377 −0.511883 0.859055i \(-0.671052\pi\)
−0.511883 + 0.859055i \(0.671052\pi\)
\(158\) −24.5628 −1.95411
\(159\) 0 0
\(160\) 17.8328 1.40980
\(161\) 7.33804 0.578319
\(162\) 0 0
\(163\) −13.4600 −1.05427 −0.527133 0.849783i \(-0.676733\pi\)
−0.527133 + 0.849783i \(0.676733\pi\)
\(164\) 5.42166 0.423361
\(165\) 0 0
\(166\) −29.8867 −2.31965
\(167\) 2.02972 0.157064 0.0785322 0.996912i \(-0.474977\pi\)
0.0785322 + 0.996912i \(0.474977\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −2.67609 −0.205247
\(171\) 0 0
\(172\) 8.71440 0.664467
\(173\) −20.2978 −1.54321 −0.771605 0.636102i \(-0.780546\pi\)
−0.771605 + 0.636102i \(0.780546\pi\)
\(174\) 0 0
\(175\) −2.91638 −0.220458
\(176\) 15.2544 1.14985
\(177\) 0 0
\(178\) 19.3083 1.44722
\(179\) 11.0036 0.822445 0.411223 0.911535i \(-0.365102\pi\)
0.411223 + 0.911535i \(0.365102\pi\)
\(180\) 0 0
\(181\) 0.691675 0.0514118 0.0257059 0.999670i \(-0.491817\pi\)
0.0257059 + 0.999670i \(0.491817\pi\)
\(182\) −1.81361 −0.134433
\(183\) 0 0
\(184\) 9.45998 0.697399
\(185\) 17.3083 1.27253
\(186\) 0 0
\(187\) −1.62721 −0.118994
\(188\) 7.69670 0.561339
\(189\) 0 0
\(190\) −4.15165 −0.301192
\(191\) −7.83276 −0.566759 −0.283379 0.959008i \(-0.591456\pi\)
−0.283379 + 0.959008i \(0.591456\pi\)
\(192\) 0 0
\(193\) −12.2056 −0.878575 −0.439287 0.898347i \(-0.644769\pi\)
−0.439287 + 0.898347i \(0.644769\pi\)
\(194\) −2.15165 −0.154480
\(195\) 0 0
\(196\) 1.28917 0.0920835
\(197\) 18.8222 1.34103 0.670513 0.741898i \(-0.266074\pi\)
0.670513 + 0.741898i \(0.266074\pi\)
\(198\) 0 0
\(199\) 21.6116 1.53201 0.766004 0.642836i \(-0.222242\pi\)
0.766004 + 0.642836i \(0.222242\pi\)
\(200\) −3.75971 −0.265851
\(201\) 0 0
\(202\) −23.7633 −1.67198
\(203\) 8.28917 0.581786
\(204\) 0 0
\(205\) −11.8328 −0.826436
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −4.91638 −0.340890
\(209\) −2.52444 −0.174619
\(210\) 0 0
\(211\) −17.3764 −1.19624 −0.598119 0.801407i \(-0.704085\pi\)
−0.598119 + 0.801407i \(0.704085\pi\)
\(212\) 3.21611 0.220884
\(213\) 0 0
\(214\) −1.04888 −0.0716997
\(215\) −19.0192 −1.29710
\(216\) 0 0
\(217\) −1.39194 −0.0944913
\(218\) 10.1078 0.684586
\(219\) 0 0
\(220\) 11.2544 0.758773
\(221\) 0.524438 0.0352775
\(222\) 0 0
\(223\) −10.5486 −0.706388 −0.353194 0.935550i \(-0.614904\pi\)
−0.353194 + 0.935550i \(0.614904\pi\)
\(224\) 6.33804 0.423478
\(225\) 0 0
\(226\) −9.87662 −0.656983
\(227\) 6.95112 0.461362 0.230681 0.973029i \(-0.425905\pi\)
0.230681 + 0.973029i \(0.425905\pi\)
\(228\) 0 0
\(229\) −21.0872 −1.39348 −0.696740 0.717323i \(-0.745367\pi\)
−0.696740 + 0.717323i \(0.745367\pi\)
\(230\) 37.4444 2.46901
\(231\) 0 0
\(232\) 10.6861 0.701579
\(233\) −6.08362 −0.398551 −0.199276 0.979943i \(-0.563859\pi\)
−0.199276 + 0.979943i \(0.563859\pi\)
\(234\) 0 0
\(235\) −16.7980 −1.09578
\(236\) 5.76328 0.375157
\(237\) 0 0
\(238\) −0.951124 −0.0616522
\(239\) 14.2056 0.918881 0.459440 0.888209i \(-0.348050\pi\)
0.459440 + 0.888209i \(0.348050\pi\)
\(240\) 0 0
\(241\) 8.44082 0.543721 0.271860 0.962337i \(-0.412361\pi\)
0.271860 + 0.962337i \(0.412361\pi\)
\(242\) −2.48970 −0.160044
\(243\) 0 0
\(244\) −2.57834 −0.165061
\(245\) −2.81361 −0.179755
\(246\) 0 0
\(247\) 0.813607 0.0517685
\(248\) −1.79445 −0.113948
\(249\) 0 0
\(250\) 10.6322 0.672442
\(251\) 23.9844 1.51388 0.756941 0.653483i \(-0.226693\pi\)
0.756941 + 0.653483i \(0.226693\pi\)
\(252\) 0 0
\(253\) 22.7683 1.43143
\(254\) −23.3622 −1.46588
\(255\) 0 0
\(256\) 20.8469 1.30293
\(257\) −15.6116 −0.973827 −0.486913 0.873450i \(-0.661877\pi\)
−0.486913 + 0.873450i \(0.661877\pi\)
\(258\) 0 0
\(259\) 6.15165 0.382245
\(260\) −3.62721 −0.224950
\(261\) 0 0
\(262\) −16.4111 −1.01388
\(263\) 15.1708 0.935472 0.467736 0.883868i \(-0.345070\pi\)
0.467736 + 0.883868i \(0.345070\pi\)
\(264\) 0 0
\(265\) −7.01916 −0.431183
\(266\) −1.47556 −0.0904725
\(267\) 0 0
\(268\) 12.9411 0.790502
\(269\) −23.2927 −1.42018 −0.710092 0.704109i \(-0.751347\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(270\) 0 0
\(271\) 12.7456 0.774238 0.387119 0.922030i \(-0.373470\pi\)
0.387119 + 0.922030i \(0.373470\pi\)
\(272\) −2.57834 −0.156335
\(273\) 0 0
\(274\) 11.3522 0.685810
\(275\) −9.04888 −0.545668
\(276\) 0 0
\(277\) 8.12193 0.488000 0.244000 0.969775i \(-0.421540\pi\)
0.244000 + 0.969775i \(0.421540\pi\)
\(278\) −20.9894 −1.25886
\(279\) 0 0
\(280\) −3.62721 −0.216767
\(281\) −19.0333 −1.13543 −0.567715 0.823225i \(-0.692173\pi\)
−0.567715 + 0.823225i \(0.692173\pi\)
\(282\) 0 0
\(283\) 11.1466 0.662598 0.331299 0.943526i \(-0.392513\pi\)
0.331299 + 0.943526i \(0.392513\pi\)
\(284\) 11.2544 0.667827
\(285\) 0 0
\(286\) −5.62721 −0.332744
\(287\) −4.20555 −0.248246
\(288\) 0 0
\(289\) −16.7250 −0.983821
\(290\) 42.2978 2.48381
\(291\) 0 0
\(292\) −3.02061 −0.176768
\(293\) 14.1758 0.828161 0.414080 0.910240i \(-0.364103\pi\)
0.414080 + 0.910240i \(0.364103\pi\)
\(294\) 0 0
\(295\) −12.5783 −0.732339
\(296\) 7.93051 0.460952
\(297\) 0 0
\(298\) −15.4600 −0.895572
\(299\) −7.33804 −0.424370
\(300\) 0 0
\(301\) −6.75971 −0.389623
\(302\) −21.7350 −1.25071
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 5.62721 0.322213
\(306\) 0 0
\(307\) 13.5592 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) −7.10278 −0.403411
\(311\) −0.426686 −0.0241952 −0.0120976 0.999927i \(-0.503851\pi\)
−0.0120976 + 0.999927i \(0.503851\pi\)
\(312\) 0 0
\(313\) −18.1517 −1.02599 −0.512996 0.858391i \(-0.671464\pi\)
−0.512996 + 0.858391i \(0.671464\pi\)
\(314\) −23.2645 −1.31289
\(315\) 0 0
\(316\) −17.4600 −0.982200
\(317\) −9.42166 −0.529173 −0.264587 0.964362i \(-0.585236\pi\)
−0.264587 + 0.964362i \(0.585236\pi\)
\(318\) 0 0
\(319\) 25.7194 1.44001
\(320\) 4.67609 0.261401
\(321\) 0 0
\(322\) 13.3083 0.741644
\(323\) 0.426686 0.0237415
\(324\) 0 0
\(325\) 2.91638 0.161772
\(326\) −24.4111 −1.35201
\(327\) 0 0
\(328\) −5.42166 −0.299361
\(329\) −5.97028 −0.329152
\(330\) 0 0
\(331\) −17.4005 −0.956420 −0.478210 0.878246i \(-0.658714\pi\)
−0.478210 + 0.878246i \(0.658714\pi\)
\(332\) −21.2444 −1.16594
\(333\) 0 0
\(334\) 3.68111 0.201421
\(335\) −28.2439 −1.54313
\(336\) 0 0
\(337\) 22.0524 1.20127 0.600637 0.799522i \(-0.294914\pi\)
0.600637 + 0.799522i \(0.294914\pi\)
\(338\) 1.81361 0.0986472
\(339\) 0 0
\(340\) −1.90225 −0.103164
\(341\) −4.31889 −0.233881
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −8.71440 −0.469849
\(345\) 0 0
\(346\) −36.8122 −1.97903
\(347\) −25.3522 −1.36098 −0.680488 0.732759i \(-0.738232\pi\)
−0.680488 + 0.732759i \(0.738232\pi\)
\(348\) 0 0
\(349\) −5.70529 −0.305397 −0.152699 0.988273i \(-0.548796\pi\)
−0.152699 + 0.988273i \(0.548796\pi\)
\(350\) −5.28917 −0.282718
\(351\) 0 0
\(352\) 19.6655 1.04818
\(353\) 28.6761 1.52627 0.763137 0.646237i \(-0.223658\pi\)
0.763137 + 0.646237i \(0.223658\pi\)
\(354\) 0 0
\(355\) −24.5628 −1.30366
\(356\) 13.7250 0.727422
\(357\) 0 0
\(358\) 19.9561 1.05472
\(359\) −11.0433 −0.582845 −0.291423 0.956594i \(-0.594129\pi\)
−0.291423 + 0.956594i \(0.594129\pi\)
\(360\) 0 0
\(361\) −18.3380 −0.965160
\(362\) 1.25443 0.0659312
\(363\) 0 0
\(364\) −1.28917 −0.0675708
\(365\) 6.59247 0.345066
\(366\) 0 0
\(367\) 27.3466 1.42748 0.713741 0.700409i \(-0.246999\pi\)
0.713741 + 0.700409i \(0.246999\pi\)
\(368\) 36.0766 1.88062
\(369\) 0 0
\(370\) 31.3905 1.63191
\(371\) −2.49472 −0.129519
\(372\) 0 0
\(373\) 16.1461 0.836014 0.418007 0.908444i \(-0.362729\pi\)
0.418007 + 0.908444i \(0.362729\pi\)
\(374\) −2.95112 −0.152599
\(375\) 0 0
\(376\) −7.69670 −0.396927
\(377\) −8.28917 −0.426914
\(378\) 0 0
\(379\) 26.1305 1.34223 0.671117 0.741351i \(-0.265815\pi\)
0.671117 + 0.741351i \(0.265815\pi\)
\(380\) −2.95112 −0.151389
\(381\) 0 0
\(382\) −14.2056 −0.726819
\(383\) 21.0489 1.07555 0.537774 0.843089i \(-0.319265\pi\)
0.537774 + 0.843089i \(0.319265\pi\)
\(384\) 0 0
\(385\) −8.72999 −0.444921
\(386\) −22.1361 −1.12670
\(387\) 0 0
\(388\) −1.52946 −0.0776466
\(389\) 21.6061 1.09547 0.547736 0.836651i \(-0.315490\pi\)
0.547736 + 0.836651i \(0.315490\pi\)
\(390\) 0 0
\(391\) −3.84835 −0.194619
\(392\) −1.28917 −0.0651128
\(393\) 0 0
\(394\) 34.1361 1.71975
\(395\) 38.1063 1.91734
\(396\) 0 0
\(397\) −27.6952 −1.38998 −0.694992 0.719017i \(-0.744592\pi\)
−0.694992 + 0.719017i \(0.744592\pi\)
\(398\) 39.1950 1.96467
\(399\) 0 0
\(400\) −14.3380 −0.716902
\(401\) 2.57834 0.128756 0.0643780 0.997926i \(-0.479494\pi\)
0.0643780 + 0.997926i \(0.479494\pi\)
\(402\) 0 0
\(403\) 1.39194 0.0693376
\(404\) −16.8917 −0.840393
\(405\) 0 0
\(406\) 15.0333 0.746090
\(407\) 19.0872 0.946117
\(408\) 0 0
\(409\) 15.1169 0.747483 0.373742 0.927533i \(-0.378075\pi\)
0.373742 + 0.927533i \(0.378075\pi\)
\(410\) −21.4600 −1.05983
\(411\) 0 0
\(412\) 5.68665 0.280161
\(413\) −4.47054 −0.219981
\(414\) 0 0
\(415\) 46.3658 2.27601
\(416\) −6.33804 −0.310748
\(417\) 0 0
\(418\) −4.57834 −0.223934
\(419\) 9.99446 0.488261 0.244131 0.969742i \(-0.421497\pi\)
0.244131 + 0.969742i \(0.421497\pi\)
\(420\) 0 0
\(421\) −25.9250 −1.26351 −0.631753 0.775170i \(-0.717664\pi\)
−0.631753 + 0.775170i \(0.717664\pi\)
\(422\) −31.5139 −1.53407
\(423\) 0 0
\(424\) −3.21611 −0.156188
\(425\) 1.52946 0.0741898
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −0.745574 −0.0360387
\(429\) 0 0
\(430\) −34.4933 −1.66341
\(431\) −30.6761 −1.47762 −0.738808 0.673916i \(-0.764611\pi\)
−0.738808 + 0.673916i \(0.764611\pi\)
\(432\) 0 0
\(433\) −3.51941 −0.169132 −0.0845661 0.996418i \(-0.526950\pi\)
−0.0845661 + 0.996418i \(0.526950\pi\)
\(434\) −2.52444 −0.121177
\(435\) 0 0
\(436\) 7.18494 0.344096
\(437\) −5.97028 −0.285597
\(438\) 0 0
\(439\) 32.3517 1.54406 0.772030 0.635586i \(-0.219241\pi\)
0.772030 + 0.635586i \(0.219241\pi\)
\(440\) −11.2544 −0.536534
\(441\) 0 0
\(442\) 0.951124 0.0452404
\(443\) −15.4458 −0.733854 −0.366927 0.930250i \(-0.619590\pi\)
−0.366927 + 0.930250i \(0.619590\pi\)
\(444\) 0 0
\(445\) −29.9547 −1.41999
\(446\) −19.1310 −0.905881
\(447\) 0 0
\(448\) 1.66196 0.0785200
\(449\) 14.4705 0.682907 0.341453 0.939899i \(-0.389081\pi\)
0.341453 + 0.939899i \(0.389081\pi\)
\(450\) 0 0
\(451\) −13.0489 −0.614448
\(452\) −7.02061 −0.330222
\(453\) 0 0
\(454\) 12.6066 0.591657
\(455\) 2.81361 0.131904
\(456\) 0 0
\(457\) −34.6705 −1.62182 −0.810910 0.585171i \(-0.801027\pi\)
−0.810910 + 0.585171i \(0.801027\pi\)
\(458\) −38.2439 −1.78702
\(459\) 0 0
\(460\) 26.6167 1.24101
\(461\) −12.5400 −0.584047 −0.292024 0.956411i \(-0.594329\pi\)
−0.292024 + 0.956411i \(0.594329\pi\)
\(462\) 0 0
\(463\) 12.1517 0.564735 0.282368 0.959306i \(-0.408880\pi\)
0.282368 + 0.959306i \(0.408880\pi\)
\(464\) 40.7527 1.89190
\(465\) 0 0
\(466\) −11.0333 −0.511107
\(467\) 37.0333 1.71370 0.856848 0.515569i \(-0.172419\pi\)
0.856848 + 0.515569i \(0.172419\pi\)
\(468\) 0 0
\(469\) −10.0383 −0.463526
\(470\) −30.4650 −1.40525
\(471\) 0 0
\(472\) −5.76328 −0.265276
\(473\) −20.9739 −0.964379
\(474\) 0 0
\(475\) 2.37279 0.108871
\(476\) −0.676089 −0.0309885
\(477\) 0 0
\(478\) 25.7633 1.17838
\(479\) −12.0086 −0.548687 −0.274343 0.961632i \(-0.588460\pi\)
−0.274343 + 0.961632i \(0.588460\pi\)
\(480\) 0 0
\(481\) −6.15165 −0.280491
\(482\) 15.3083 0.697275
\(483\) 0 0
\(484\) −1.76975 −0.0804434
\(485\) 3.33804 0.151573
\(486\) 0 0
\(487\) 11.1184 0.503821 0.251911 0.967751i \(-0.418941\pi\)
0.251911 + 0.967751i \(0.418941\pi\)
\(488\) 2.57834 0.116716
\(489\) 0 0
\(490\) −5.10278 −0.230520
\(491\) −0.0594386 −0.00268243 −0.00134121 0.999999i \(-0.500427\pi\)
−0.00134121 + 0.999999i \(0.500427\pi\)
\(492\) 0 0
\(493\) −4.34715 −0.195786
\(494\) 1.47556 0.0663887
\(495\) 0 0
\(496\) −6.84333 −0.307274
\(497\) −8.72999 −0.391593
\(498\) 0 0
\(499\) 10.2978 0.460991 0.230496 0.973073i \(-0.425965\pi\)
0.230496 + 0.973073i \(0.425965\pi\)
\(500\) 7.55773 0.337992
\(501\) 0 0
\(502\) 43.4983 1.94142
\(503\) 9.32391 0.415733 0.207866 0.978157i \(-0.433348\pi\)
0.207866 + 0.978157i \(0.433348\pi\)
\(504\) 0 0
\(505\) 36.8661 1.64052
\(506\) 41.2927 1.83569
\(507\) 0 0
\(508\) −16.6066 −0.736799
\(509\) −39.6952 −1.75946 −0.879730 0.475473i \(-0.842277\pi\)
−0.879730 + 0.475473i \(0.842277\pi\)
\(510\) 0 0
\(511\) 2.34307 0.103651
\(512\) 18.4842 0.816892
\(513\) 0 0
\(514\) −28.3133 −1.24885
\(515\) −12.4111 −0.546898
\(516\) 0 0
\(517\) −18.5244 −0.814704
\(518\) 11.1567 0.490196
\(519\) 0 0
\(520\) 3.62721 0.159064
\(521\) −22.3627 −0.979729 −0.489865 0.871798i \(-0.662954\pi\)
−0.489865 + 0.871798i \(0.662954\pi\)
\(522\) 0 0
\(523\) −20.6550 −0.903178 −0.451589 0.892226i \(-0.649143\pi\)
−0.451589 + 0.892226i \(0.649143\pi\)
\(524\) −11.6655 −0.509611
\(525\) 0 0
\(526\) 27.5139 1.19966
\(527\) 0.729988 0.0317988
\(528\) 0 0
\(529\) 30.8469 1.34117
\(530\) −12.7300 −0.552955
\(531\) 0 0
\(532\) −1.04888 −0.0454745
\(533\) 4.20555 0.182163
\(534\) 0 0
\(535\) 1.62721 0.0703506
\(536\) −12.9411 −0.558969
\(537\) 0 0
\(538\) −42.2439 −1.82126
\(539\) −3.10278 −0.133646
\(540\) 0 0
\(541\) 5.62167 0.241695 0.120847 0.992671i \(-0.461439\pi\)
0.120847 + 0.992671i \(0.461439\pi\)
\(542\) 23.1155 0.992894
\(543\) 0 0
\(544\) −3.32391 −0.142512
\(545\) −15.6811 −0.671705
\(546\) 0 0
\(547\) −10.3970 −0.444542 −0.222271 0.974985i \(-0.571347\pi\)
−0.222271 + 0.974985i \(0.571347\pi\)
\(548\) 8.06949 0.344711
\(549\) 0 0
\(550\) −16.4111 −0.699772
\(551\) −6.74412 −0.287309
\(552\) 0 0
\(553\) 13.5436 0.575932
\(554\) 14.7300 0.625817
\(555\) 0 0
\(556\) −14.9200 −0.632747
\(557\) −14.6550 −0.620951 −0.310475 0.950581i \(-0.600488\pi\)
−0.310475 + 0.950581i \(0.600488\pi\)
\(558\) 0 0
\(559\) 6.75971 0.285905
\(560\) −13.8328 −0.584541
\(561\) 0 0
\(562\) −34.5189 −1.45609
\(563\) 24.7456 1.04290 0.521451 0.853281i \(-0.325391\pi\)
0.521451 + 0.853281i \(0.325391\pi\)
\(564\) 0 0
\(565\) 15.3225 0.644621
\(566\) 20.2156 0.849725
\(567\) 0 0
\(568\) −11.2544 −0.472225
\(569\) 20.5330 0.860789 0.430395 0.902641i \(-0.358374\pi\)
0.430395 + 0.902641i \(0.358374\pi\)
\(570\) 0 0
\(571\) −41.8953 −1.75326 −0.876631 0.481163i \(-0.840214\pi\)
−0.876631 + 0.481163i \(0.840214\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) −7.62721 −0.318354
\(575\) −21.4005 −0.892464
\(576\) 0 0
\(577\) −20.1744 −0.839870 −0.419935 0.907554i \(-0.637947\pi\)
−0.419935 + 0.907554i \(0.637947\pi\)
\(578\) −30.3325 −1.26167
\(579\) 0 0
\(580\) 30.0666 1.24845
\(581\) 16.4791 0.683670
\(582\) 0 0
\(583\) −7.74055 −0.320581
\(584\) 3.02061 0.124994
\(585\) 0 0
\(586\) 25.7094 1.06204
\(587\) 18.7441 0.773653 0.386826 0.922153i \(-0.373571\pi\)
0.386826 + 0.922153i \(0.373571\pi\)
\(588\) 0 0
\(589\) 1.13249 0.0466636
\(590\) −22.8122 −0.939162
\(591\) 0 0
\(592\) 30.2439 1.24302
\(593\) 2.98084 0.122409 0.0612043 0.998125i \(-0.480506\pi\)
0.0612043 + 0.998125i \(0.480506\pi\)
\(594\) 0 0
\(595\) 1.47556 0.0604921
\(596\) −10.9894 −0.450145
\(597\) 0 0
\(598\) −13.3083 −0.544218
\(599\) −7.47411 −0.305384 −0.152692 0.988274i \(-0.548794\pi\)
−0.152692 + 0.988274i \(0.548794\pi\)
\(600\) 0 0
\(601\) 21.4700 0.875780 0.437890 0.899028i \(-0.355726\pi\)
0.437890 + 0.899028i \(0.355726\pi\)
\(602\) −12.2594 −0.499658
\(603\) 0 0
\(604\) −15.4499 −0.628649
\(605\) 3.86248 0.157032
\(606\) 0 0
\(607\) −22.9044 −0.929660 −0.464830 0.885400i \(-0.653884\pi\)
−0.464830 + 0.885400i \(0.653884\pi\)
\(608\) −5.15667 −0.209131
\(609\) 0 0
\(610\) 10.2056 0.413211
\(611\) 5.97028 0.241532
\(612\) 0 0
\(613\) −20.1461 −0.813694 −0.406847 0.913496i \(-0.633372\pi\)
−0.406847 + 0.913496i \(0.633372\pi\)
\(614\) 24.5910 0.992413
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 13.7844 0.554939 0.277470 0.960734i \(-0.410504\pi\)
0.277470 + 0.960734i \(0.410504\pi\)
\(618\) 0 0
\(619\) −19.6655 −0.790424 −0.395212 0.918590i \(-0.629329\pi\)
−0.395212 + 0.918590i \(0.629329\pi\)
\(620\) −5.04888 −0.202768
\(621\) 0 0
\(622\) −0.773841 −0.0310282
\(623\) −10.6464 −0.426538
\(624\) 0 0
\(625\) −31.0766 −1.24307
\(626\) −32.9200 −1.31575
\(627\) 0 0
\(628\) −16.5371 −0.659903
\(629\) −3.22616 −0.128635
\(630\) 0 0
\(631\) −16.1672 −0.643608 −0.321804 0.946806i \(-0.604289\pi\)
−0.321804 + 0.946806i \(0.604289\pi\)
\(632\) 17.4600 0.694521
\(633\) 0 0
\(634\) −17.0872 −0.678619
\(635\) 36.2439 1.43829
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 46.6449 1.84669
\(639\) 0 0
\(640\) −27.1849 −1.07458
\(641\) 29.0036 1.14557 0.572786 0.819705i \(-0.305863\pi\)
0.572786 + 0.819705i \(0.305863\pi\)
\(642\) 0 0
\(643\) −39.2233 −1.54681 −0.773407 0.633910i \(-0.781449\pi\)
−0.773407 + 0.633910i \(0.781449\pi\)
\(644\) 9.45998 0.372775
\(645\) 0 0
\(646\) 0.773841 0.0304464
\(647\) 11.9844 0.471156 0.235578 0.971855i \(-0.424302\pi\)
0.235578 + 0.971855i \(0.424302\pi\)
\(648\) 0 0
\(649\) −13.8711 −0.544487
\(650\) 5.28917 0.207458
\(651\) 0 0
\(652\) −17.3522 −0.679564
\(653\) −45.3311 −1.77394 −0.886971 0.461826i \(-0.847194\pi\)
−0.886971 + 0.461826i \(0.847194\pi\)
\(654\) 0 0
\(655\) 25.4600 0.994804
\(656\) −20.6761 −0.807266
\(657\) 0 0
\(658\) −10.8277 −0.422109
\(659\) −6.12193 −0.238477 −0.119238 0.992866i \(-0.538045\pi\)
−0.119238 + 0.992866i \(0.538045\pi\)
\(660\) 0 0
\(661\) −27.5280 −1.07072 −0.535358 0.844625i \(-0.679823\pi\)
−0.535358 + 0.844625i \(0.679823\pi\)
\(662\) −31.5577 −1.22653
\(663\) 0 0
\(664\) 21.2444 0.824442
\(665\) 2.28917 0.0887701
\(666\) 0 0
\(667\) 60.8263 2.35520
\(668\) 2.61665 0.101241
\(669\) 0 0
\(670\) −51.2233 −1.97893
\(671\) 6.20555 0.239563
\(672\) 0 0
\(673\) 27.9547 1.07757 0.538787 0.842442i \(-0.318883\pi\)
0.538787 + 0.842442i \(0.318883\pi\)
\(674\) 39.9945 1.54053
\(675\) 0 0
\(676\) 1.28917 0.0495834
\(677\) 12.6605 0.486583 0.243291 0.969953i \(-0.421773\pi\)
0.243291 + 0.969953i \(0.421773\pi\)
\(678\) 0 0
\(679\) 1.18639 0.0455296
\(680\) 1.90225 0.0729479
\(681\) 0 0
\(682\) −7.83276 −0.299932
\(683\) 28.3033 1.08300 0.541498 0.840702i \(-0.317857\pi\)
0.541498 + 0.840702i \(0.317857\pi\)
\(684\) 0 0
\(685\) −17.6116 −0.672906
\(686\) −1.81361 −0.0692438
\(687\) 0 0
\(688\) −33.2333 −1.26701
\(689\) 2.49472 0.0950412
\(690\) 0 0
\(691\) 12.2353 0.465452 0.232726 0.972542i \(-0.425236\pi\)
0.232726 + 0.972542i \(0.425236\pi\)
\(692\) −26.1672 −0.994729
\(693\) 0 0
\(694\) −45.9789 −1.74533
\(695\) 32.5628 1.23518
\(696\) 0 0
\(697\) 2.20555 0.0835412
\(698\) −10.3472 −0.391646
\(699\) 0 0
\(700\) −3.75971 −0.142104
\(701\) −51.0419 −1.92783 −0.963913 0.266219i \(-0.914226\pi\)
−0.963913 + 0.266219i \(0.914226\pi\)
\(702\) 0 0
\(703\) −5.00502 −0.188768
\(704\) 5.15667 0.194349
\(705\) 0 0
\(706\) 52.0071 1.95731
\(707\) 13.1028 0.492781
\(708\) 0 0
\(709\) 42.5910 1.59954 0.799770 0.600307i \(-0.204955\pi\)
0.799770 + 0.600307i \(0.204955\pi\)
\(710\) −44.5472 −1.67183
\(711\) 0 0
\(712\) −13.7250 −0.514365
\(713\) −10.2141 −0.382523
\(714\) 0 0
\(715\) 8.72999 0.326483
\(716\) 14.1855 0.530135
\(717\) 0 0
\(718\) −20.0283 −0.747448
\(719\) 42.4933 1.58473 0.792366 0.610046i \(-0.208849\pi\)
0.792366 + 0.610046i \(0.208849\pi\)
\(720\) 0 0
\(721\) −4.41110 −0.164278
\(722\) −33.2580 −1.23773
\(723\) 0 0
\(724\) 0.891685 0.0331392
\(725\) −24.1744 −0.897814
\(726\) 0 0
\(727\) −3.75614 −0.139307 −0.0696537 0.997571i \(-0.522189\pi\)
−0.0696537 + 0.997571i \(0.522189\pi\)
\(728\) 1.28917 0.0477798
\(729\) 0 0
\(730\) 11.9561 0.442517
\(731\) 3.54505 0.131118
\(732\) 0 0
\(733\) 45.7819 1.69099 0.845497 0.533980i \(-0.179304\pi\)
0.845497 + 0.533980i \(0.179304\pi\)
\(734\) 49.5960 1.83062
\(735\) 0 0
\(736\) 46.5089 1.71434
\(737\) −31.1466 −1.14730
\(738\) 0 0
\(739\) −14.0539 −0.516981 −0.258491 0.966014i \(-0.583225\pi\)
−0.258491 + 0.966014i \(0.583225\pi\)
\(740\) 22.3133 0.820255
\(741\) 0 0
\(742\) −4.52444 −0.166097
\(743\) −4.74557 −0.174098 −0.0870491 0.996204i \(-0.527744\pi\)
−0.0870491 + 0.996204i \(0.527744\pi\)
\(744\) 0 0
\(745\) 23.9844 0.878721
\(746\) 29.2827 1.07212
\(747\) 0 0
\(748\) −2.09775 −0.0767014
\(749\) 0.578337 0.0211320
\(750\) 0 0
\(751\) 36.1008 1.31734 0.658669 0.752433i \(-0.271120\pi\)
0.658669 + 0.752433i \(0.271120\pi\)
\(752\) −29.3522 −1.07036
\(753\) 0 0
\(754\) −15.0333 −0.547480
\(755\) 33.7194 1.22718
\(756\) 0 0
\(757\) −1.03474 −0.0376084 −0.0188042 0.999823i \(-0.505986\pi\)
−0.0188042 + 0.999823i \(0.505986\pi\)
\(758\) 47.3905 1.72130
\(759\) 0 0
\(760\) 2.95112 0.107049
\(761\) −29.8414 −1.08175 −0.540874 0.841104i \(-0.681906\pi\)
−0.540874 + 0.841104i \(0.681906\pi\)
\(762\) 0 0
\(763\) −5.57331 −0.201768
\(764\) −10.0978 −0.365324
\(765\) 0 0
\(766\) 38.1744 1.37930
\(767\) 4.47054 0.161422
\(768\) 0 0
\(769\) 23.6358 0.852329 0.426164 0.904646i \(-0.359864\pi\)
0.426164 + 0.904646i \(0.359864\pi\)
\(770\) −15.8328 −0.570573
\(771\) 0 0
\(772\) −15.7350 −0.566315
\(773\) −13.0278 −0.468576 −0.234288 0.972167i \(-0.575276\pi\)
−0.234288 + 0.972167i \(0.575276\pi\)
\(774\) 0 0
\(775\) 4.05944 0.145819
\(776\) 1.52946 0.0549045
\(777\) 0 0
\(778\) 39.1849 1.40485
\(779\) 3.42166 0.122594
\(780\) 0 0
\(781\) −27.0872 −0.969256
\(782\) −6.97939 −0.249583
\(783\) 0 0
\(784\) −4.91638 −0.175585
\(785\) 36.0922 1.28819
\(786\) 0 0
\(787\) 46.2141 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(788\) 24.2650 0.864404
\(789\) 0 0
\(790\) 69.1099 2.45882
\(791\) 5.44584 0.193632
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −50.2283 −1.78253
\(795\) 0 0
\(796\) 27.8610 0.987508
\(797\) −53.1155 −1.88145 −0.940723 0.339176i \(-0.889852\pi\)
−0.940723 + 0.339176i \(0.889852\pi\)
\(798\) 0 0
\(799\) 3.13104 0.110768
\(800\) −18.4842 −0.653514
\(801\) 0 0
\(802\) 4.67609 0.165118
\(803\) 7.27001 0.256553
\(804\) 0 0
\(805\) −20.6464 −0.727689
\(806\) 2.52444 0.0889195
\(807\) 0 0
\(808\) 16.8917 0.594247
\(809\) 54.4635 1.91484 0.957418 0.288705i \(-0.0932246\pi\)
0.957418 + 0.288705i \(0.0932246\pi\)
\(810\) 0 0
\(811\) 38.0978 1.33779 0.668897 0.743356i \(-0.266767\pi\)
0.668897 + 0.743356i \(0.266767\pi\)
\(812\) 10.6861 0.375010
\(813\) 0 0
\(814\) 34.6167 1.21331
\(815\) 37.8711 1.32657
\(816\) 0 0
\(817\) 5.49974 0.192412
\(818\) 27.4161 0.958582
\(819\) 0 0
\(820\) −15.2544 −0.532708
\(821\) 2.30330 0.0803858 0.0401929 0.999192i \(-0.487203\pi\)
0.0401929 + 0.999192i \(0.487203\pi\)
\(822\) 0 0
\(823\) 23.6172 0.823243 0.411621 0.911355i \(-0.364963\pi\)
0.411621 + 0.911355i \(0.364963\pi\)
\(824\) −5.68665 −0.198104
\(825\) 0 0
\(826\) −8.10780 −0.282106
\(827\) 48.1643 1.67484 0.837419 0.546562i \(-0.184064\pi\)
0.837419 + 0.546562i \(0.184064\pi\)
\(828\) 0 0
\(829\) 13.0716 0.453996 0.226998 0.973895i \(-0.427109\pi\)
0.226998 + 0.973895i \(0.427109\pi\)
\(830\) 84.0893 2.91878
\(831\) 0 0
\(832\) −1.66196 −0.0576179
\(833\) 0.524438 0.0181707
\(834\) 0 0
\(835\) −5.71083 −0.197631
\(836\) −3.25443 −0.112557
\(837\) 0 0
\(838\) 18.1260 0.626153
\(839\) 17.6756 0.610229 0.305114 0.952316i \(-0.401305\pi\)
0.305114 + 0.952316i \(0.401305\pi\)
\(840\) 0 0
\(841\) 39.7103 1.36932
\(842\) −47.0177 −1.62034
\(843\) 0 0
\(844\) −22.4011 −0.771076
\(845\) −2.81361 −0.0967910
\(846\) 0 0
\(847\) 1.37279 0.0471695
\(848\) −12.2650 −0.421181
\(849\) 0 0
\(850\) 2.77384 0.0951420
\(851\) 45.1411 1.54742
\(852\) 0 0
\(853\) −5.48970 −0.187964 −0.0939818 0.995574i \(-0.529960\pi\)
−0.0939818 + 0.995574i \(0.529960\pi\)
\(854\) 3.62721 0.124121
\(855\) 0 0
\(856\) 0.745574 0.0254832
\(857\) 11.0489 0.377422 0.188711 0.982033i \(-0.439569\pi\)
0.188711 + 0.982033i \(0.439569\pi\)
\(858\) 0 0
\(859\) 45.2616 1.54430 0.772152 0.635437i \(-0.219180\pi\)
0.772152 + 0.635437i \(0.219180\pi\)
\(860\) −24.5189 −0.836087
\(861\) 0 0
\(862\) −55.6344 −1.89491
\(863\) 5.90225 0.200915 0.100457 0.994941i \(-0.467969\pi\)
0.100457 + 0.994941i \(0.467969\pi\)
\(864\) 0 0
\(865\) 57.1099 1.94180
\(866\) −6.38283 −0.216898
\(867\) 0 0
\(868\) −1.79445 −0.0609076
\(869\) 42.0227 1.42552
\(870\) 0 0
\(871\) 10.0383 0.340135
\(872\) −7.18494 −0.243313
\(873\) 0 0
\(874\) −10.8277 −0.366254
\(875\) −5.86248 −0.198188
\(876\) 0 0
\(877\) −4.90727 −0.165707 −0.0828534 0.996562i \(-0.526403\pi\)
−0.0828534 + 0.996562i \(0.526403\pi\)
\(878\) 58.6732 1.98012
\(879\) 0 0
\(880\) −42.9200 −1.44683
\(881\) −44.2822 −1.49190 −0.745952 0.665999i \(-0.768005\pi\)
−0.745952 + 0.665999i \(0.768005\pi\)
\(882\) 0 0
\(883\) −58.8605 −1.98081 −0.990407 0.138181i \(-0.955874\pi\)
−0.990407 + 0.138181i \(0.955874\pi\)
\(884\) 0.676089 0.0227393
\(885\) 0 0
\(886\) −28.0127 −0.941104
\(887\) 10.1289 0.340096 0.170048 0.985436i \(-0.445608\pi\)
0.170048 + 0.985436i \(0.445608\pi\)
\(888\) 0 0
\(889\) 12.8816 0.432036
\(890\) −54.3260 −1.82101
\(891\) 0 0
\(892\) −13.5989 −0.455326
\(893\) 4.85746 0.162549
\(894\) 0 0
\(895\) −30.9597 −1.03487
\(896\) −9.66196 −0.322783
\(897\) 0 0
\(898\) 26.2439 0.875769
\(899\) −11.5381 −0.384816
\(900\) 0 0
\(901\) 1.30833 0.0435866
\(902\) −23.6655 −0.787976
\(903\) 0 0
\(904\) 7.02061 0.233502
\(905\) −1.94610 −0.0646906
\(906\) 0 0
\(907\) 37.9547 1.26026 0.630132 0.776488i \(-0.283001\pi\)
0.630132 + 0.776488i \(0.283001\pi\)
\(908\) 8.96117 0.297387
\(909\) 0 0
\(910\) 5.10278 0.169155
\(911\) −5.57477 −0.184700 −0.0923501 0.995727i \(-0.529438\pi\)
−0.0923501 + 0.995727i \(0.529438\pi\)
\(912\) 0 0
\(913\) 51.1310 1.69219
\(914\) −62.8787 −2.07984
\(915\) 0 0
\(916\) −27.1849 −0.898216
\(917\) 9.04888 0.298820
\(918\) 0 0
\(919\) 15.7844 0.520679 0.260340 0.965517i \(-0.416165\pi\)
0.260340 + 0.965517i \(0.416165\pi\)
\(920\) −26.6167 −0.877525
\(921\) 0 0
\(922\) −22.7427 −0.748990
\(923\) 8.72999 0.287351
\(924\) 0 0
\(925\) −17.9406 −0.589882
\(926\) 22.0383 0.724224
\(927\) 0 0
\(928\) 52.5371 1.72462
\(929\) 45.2630 1.48503 0.742516 0.669829i \(-0.233632\pi\)
0.742516 + 0.669829i \(0.233632\pi\)
\(930\) 0 0
\(931\) 0.813607 0.0266649
\(932\) −7.84281 −0.256900
\(933\) 0 0
\(934\) 67.1638 2.19767
\(935\) 4.57834 0.149728
\(936\) 0 0
\(937\) 53.6188 1.75165 0.875824 0.482630i \(-0.160318\pi\)
0.875824 + 0.482630i \(0.160318\pi\)
\(938\) −18.2056 −0.594432
\(939\) 0 0
\(940\) −21.6555 −0.706324
\(941\) −20.7753 −0.677255 −0.338628 0.940920i \(-0.609963\pi\)
−0.338628 + 0.940920i \(0.609963\pi\)
\(942\) 0 0
\(943\) −30.8605 −1.00496
\(944\) −21.9789 −0.715351
\(945\) 0 0
\(946\) −38.0383 −1.23673
\(947\) 10.8605 0.352919 0.176460 0.984308i \(-0.443535\pi\)
0.176460 + 0.984308i \(0.443535\pi\)
\(948\) 0 0
\(949\) −2.34307 −0.0760592
\(950\) 4.30330 0.139618
\(951\) 0 0
\(952\) 0.676089 0.0219122
\(953\) 25.7180 0.833087 0.416543 0.909116i \(-0.363241\pi\)
0.416543 + 0.909116i \(0.363241\pi\)
\(954\) 0 0
\(955\) 22.0383 0.713143
\(956\) 18.3133 0.592296
\(957\) 0 0
\(958\) −21.7789 −0.703643
\(959\) −6.25945 −0.202128
\(960\) 0 0
\(961\) −29.0625 −0.937500
\(962\) −11.1567 −0.359706
\(963\) 0 0
\(964\) 10.8816 0.350474
\(965\) 34.3416 1.10550
\(966\) 0 0
\(967\) 33.5038 1.07741 0.538705 0.842494i \(-0.318914\pi\)
0.538705 + 0.842494i \(0.318914\pi\)
\(968\) 1.76975 0.0568820
\(969\) 0 0
\(970\) 6.05390 0.194379
\(971\) −2.03831 −0.0654126 −0.0327063 0.999465i \(-0.510413\pi\)
−0.0327063 + 0.999465i \(0.510413\pi\)
\(972\) 0 0
\(973\) 11.5733 0.371023
\(974\) 20.1643 0.646107
\(975\) 0 0
\(976\) 9.83276 0.314739
\(977\) −15.1411 −0.484406 −0.242203 0.970226i \(-0.577870\pi\)
−0.242203 + 0.970226i \(0.577870\pi\)
\(978\) 0 0
\(979\) −33.0333 −1.05575
\(980\) −3.62721 −0.115867
\(981\) 0 0
\(982\) −0.107798 −0.00343998
\(983\) −49.3124 −1.57282 −0.786411 0.617704i \(-0.788063\pi\)
−0.786411 + 0.617704i \(0.788063\pi\)
\(984\) 0 0
\(985\) −52.9583 −1.68739
\(986\) −7.88403 −0.251079
\(987\) 0 0
\(988\) 1.04888 0.0333692
\(989\) −49.6030 −1.57728
\(990\) 0 0
\(991\) 5.43171 0.172544 0.0862720 0.996272i \(-0.472505\pi\)
0.0862720 + 0.996272i \(0.472505\pi\)
\(992\) −8.82220 −0.280105
\(993\) 0 0
\(994\) −15.8328 −0.502185
\(995\) −60.8066 −1.92770
\(996\) 0 0
\(997\) −53.6061 −1.69772 −0.848861 0.528616i \(-0.822711\pi\)
−0.848861 + 0.528616i \(0.822711\pi\)
\(998\) 18.6761 0.591181
\(999\) 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.i.1.3 3
3.2 odd 2 91.2.a.d.1.1 3
7.6 odd 2 5733.2.a.x.1.3 3
12.11 even 2 1456.2.a.t.1.3 3
15.14 odd 2 2275.2.a.m.1.3 3
21.2 odd 6 637.2.e.j.508.3 6
21.5 even 6 637.2.e.i.508.3 6
21.11 odd 6 637.2.e.j.79.3 6
21.17 even 6 637.2.e.i.79.3 6
21.20 even 2 637.2.a.j.1.1 3
24.5 odd 2 5824.2.a.by.1.3 3
24.11 even 2 5824.2.a.bs.1.1 3
39.5 even 4 1183.2.c.f.337.5 6
39.8 even 4 1183.2.c.f.337.2 6
39.38 odd 2 1183.2.a.i.1.3 3
273.272 even 2 8281.2.a.bg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 3.2 odd 2
637.2.a.j.1.1 3 21.20 even 2
637.2.e.i.79.3 6 21.17 even 6
637.2.e.i.508.3 6 21.5 even 6
637.2.e.j.79.3 6 21.11 odd 6
637.2.e.j.508.3 6 21.2 odd 6
819.2.a.i.1.3 3 1.1 even 1 trivial
1183.2.a.i.1.3 3 39.38 odd 2
1183.2.c.f.337.2 6 39.8 even 4
1183.2.c.f.337.5 6 39.5 even 4
1456.2.a.t.1.3 3 12.11 even 2
2275.2.a.m.1.3 3 15.14 odd 2
5733.2.a.x.1.3 3 7.6 odd 2
5824.2.a.bs.1.1 3 24.11 even 2
5824.2.a.by.1.3 3 24.5 odd 2
8281.2.a.bg.1.3 3 273.272 even 2