Properties

Label 1323.2.a.bd.1.2
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
Dimension $4$
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] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7168.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.25928\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.25928 q^{2} -0.414214 q^{4} -1.25928 q^{5} +3.04017 q^{8} +O(q^{10})\) \(q-1.25928 q^{2} -0.414214 q^{4} -1.25928 q^{5} +3.04017 q^{8} +1.58579 q^{10} -4.82106 q^{11} +1.17157 q^{13} -3.00000 q^{16} -4.29945 q^{17} +1.58579 q^{19} +0.521611 q^{20} +6.07107 q^{22} +6.60195 q^{23} -3.41421 q^{25} -1.47534 q^{26} -5.03712 q^{29} +10.6569 q^{31} -2.30250 q^{32} +5.41421 q^{34} -5.24264 q^{37} -1.99695 q^{38} -3.82843 q^{40} -0.521611 q^{41} +7.07107 q^{43} +1.99695 q^{44} -8.31371 q^{46} -11.4230 q^{47} +4.29945 q^{50} -0.485281 q^{52} -1.04322 q^{53} +6.07107 q^{55} +6.34315 q^{58} +1.78089 q^{59} +10.4853 q^{61} -13.4200 q^{62} +8.89949 q^{64} -1.47534 q^{65} +5.65685 q^{67} +1.78089 q^{68} +8.81496 q^{71} +8.58579 q^{73} +6.60195 q^{74} -0.656854 q^{76} -13.8995 q^{79} +3.77784 q^{80} +0.656854 q^{82} +14.6792 q^{83} +5.41421 q^{85} -8.90446 q^{86} -14.6569 q^{88} +17.7194 q^{89} -2.73462 q^{92} +14.3848 q^{94} -1.99695 q^{95} +9.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 12 q^{10} + 16 q^{13} - 12 q^{16} + 12 q^{19} - 4 q^{22} - 8 q^{25} + 20 q^{31} + 16 q^{34} - 4 q^{37} - 4 q^{40} + 12 q^{46} + 32 q^{52} - 4 q^{55} + 48 q^{58} + 8 q^{61} - 4 q^{64} + 40 q^{73} + 20 q^{76} - 16 q^{79} - 20 q^{82} + 16 q^{85} - 36 q^{88} - 16 q^{94} + 16 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25928 −0.890446 −0.445223 0.895420i \(-0.646876\pi\)
−0.445223 + 0.895420i \(0.646876\pi\)
\(3\) 0 0
\(4\) −0.414214 −0.207107
\(5\) −1.25928 −0.563167 −0.281584 0.959537i \(-0.590860\pi\)
−0.281584 + 0.959537i \(0.590860\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 3.04017 1.07486
\(9\) 0 0
\(10\) 1.58579 0.501470
\(11\) −4.82106 −1.45360 −0.726802 0.686847i \(-0.758994\pi\)
−0.726802 + 0.686847i \(0.758994\pi\)
\(12\) 0 0
\(13\) 1.17157 0.324936 0.162468 0.986714i \(-0.448055\pi\)
0.162468 + 0.986714i \(0.448055\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.00000 −0.750000
\(17\) −4.29945 −1.04277 −0.521385 0.853322i \(-0.674584\pi\)
−0.521385 + 0.853322i \(0.674584\pi\)
\(18\) 0 0
\(19\) 1.58579 0.363804 0.181902 0.983317i \(-0.441775\pi\)
0.181902 + 0.983317i \(0.441775\pi\)
\(20\) 0.521611 0.116636
\(21\) 0 0
\(22\) 6.07107 1.29436
\(23\) 6.60195 1.37660 0.688301 0.725425i \(-0.258357\pi\)
0.688301 + 0.725425i \(0.258357\pi\)
\(24\) 0 0
\(25\) −3.41421 −0.682843
\(26\) −1.47534 −0.289338
\(27\) 0 0
\(28\) 0 0
\(29\) −5.03712 −0.935370 −0.467685 0.883895i \(-0.654912\pi\)
−0.467685 + 0.883895i \(0.654912\pi\)
\(30\) 0 0
\(31\) 10.6569 1.91403 0.957014 0.290043i \(-0.0936695\pi\)
0.957014 + 0.290043i \(0.0936695\pi\)
\(32\) −2.30250 −0.407029
\(33\) 0 0
\(34\) 5.41421 0.928530
\(35\) 0 0
\(36\) 0 0
\(37\) −5.24264 −0.861885 −0.430942 0.902379i \(-0.641819\pi\)
−0.430942 + 0.902379i \(0.641819\pi\)
\(38\) −1.99695 −0.323948
\(39\) 0 0
\(40\) −3.82843 −0.605327
\(41\) −0.521611 −0.0814619 −0.0407310 0.999170i \(-0.512969\pi\)
−0.0407310 + 0.999170i \(0.512969\pi\)
\(42\) 0 0
\(43\) 7.07107 1.07833 0.539164 0.842201i \(-0.318740\pi\)
0.539164 + 0.842201i \(0.318740\pi\)
\(44\) 1.99695 0.301051
\(45\) 0 0
\(46\) −8.31371 −1.22579
\(47\) −11.4230 −1.66622 −0.833109 0.553109i \(-0.813441\pi\)
−0.833109 + 0.553109i \(0.813441\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.29945 0.608034
\(51\) 0 0
\(52\) −0.485281 −0.0672964
\(53\) −1.04322 −0.143298 −0.0716488 0.997430i \(-0.522826\pi\)
−0.0716488 + 0.997430i \(0.522826\pi\)
\(54\) 0 0
\(55\) 6.07107 0.818623
\(56\) 0 0
\(57\) 0 0
\(58\) 6.34315 0.832896
\(59\) 1.78089 0.231852 0.115926 0.993258i \(-0.463016\pi\)
0.115926 + 0.993258i \(0.463016\pi\)
\(60\) 0 0
\(61\) 10.4853 1.34250 0.671251 0.741230i \(-0.265757\pi\)
0.671251 + 0.741230i \(0.265757\pi\)
\(62\) −13.4200 −1.70434
\(63\) 0 0
\(64\) 8.89949 1.11244
\(65\) −1.47534 −0.182993
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 1.78089 0.215965
\(69\) 0 0
\(70\) 0 0
\(71\) 8.81496 1.04614 0.523072 0.852289i \(-0.324786\pi\)
0.523072 + 0.852289i \(0.324786\pi\)
\(72\) 0 0
\(73\) 8.58579 1.00489 0.502445 0.864609i \(-0.332434\pi\)
0.502445 + 0.864609i \(0.332434\pi\)
\(74\) 6.60195 0.767461
\(75\) 0 0
\(76\) −0.656854 −0.0753463
\(77\) 0 0
\(78\) 0 0
\(79\) −13.8995 −1.56382 −0.781908 0.623394i \(-0.785753\pi\)
−0.781908 + 0.623394i \(0.785753\pi\)
\(80\) 3.77784 0.422375
\(81\) 0 0
\(82\) 0.656854 0.0725374
\(83\) 14.6792 1.61126 0.805628 0.592421i \(-0.201828\pi\)
0.805628 + 0.592421i \(0.201828\pi\)
\(84\) 0 0
\(85\) 5.41421 0.587254
\(86\) −8.90446 −0.960192
\(87\) 0 0
\(88\) −14.6569 −1.56243
\(89\) 17.7194 1.87825 0.939127 0.343570i \(-0.111636\pi\)
0.939127 + 0.343570i \(0.111636\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.73462 −0.285104
\(93\) 0 0
\(94\) 14.3848 1.48368
\(95\) −1.99695 −0.204883
\(96\) 0 0
\(97\) 9.65685 0.980505 0.490252 0.871580i \(-0.336905\pi\)
0.490252 + 0.871580i \(0.336905\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.41421 0.141421
\(101\) 16.4601 1.63784 0.818922 0.573904i \(-0.194572\pi\)
0.818922 + 0.573904i \(0.194572\pi\)
\(102\) 0 0
\(103\) −3.48528 −0.343415 −0.171707 0.985148i \(-0.554928\pi\)
−0.171707 + 0.985148i \(0.554928\pi\)
\(104\) 3.56178 0.349261
\(105\) 0 0
\(106\) 1.31371 0.127599
\(107\) 6.81801 0.659122 0.329561 0.944134i \(-0.393099\pi\)
0.329561 + 0.944134i \(0.393099\pi\)
\(108\) 0 0
\(109\) 2.65685 0.254480 0.127240 0.991872i \(-0.459388\pi\)
0.127240 + 0.991872i \(0.459388\pi\)
\(110\) −7.64518 −0.728939
\(111\) 0 0
\(112\) 0 0
\(113\) −11.4230 −1.07459 −0.537293 0.843395i \(-0.680553\pi\)
−0.537293 + 0.843395i \(0.680553\pi\)
\(114\) 0 0
\(115\) −8.31371 −0.775257
\(116\) 2.08644 0.193721
\(117\) 0 0
\(118\) −2.24264 −0.206452
\(119\) 0 0
\(120\) 0 0
\(121\) 12.2426 1.11297
\(122\) −13.2039 −1.19543
\(123\) 0 0
\(124\) −4.41421 −0.396408
\(125\) 10.5959 0.947722
\(126\) 0 0
\(127\) −4.82843 −0.428454 −0.214227 0.976784i \(-0.568723\pi\)
−0.214227 + 0.976784i \(0.568723\pi\)
\(128\) −6.60195 −0.583536
\(129\) 0 0
\(130\) 1.85786 0.162945
\(131\) 15.4169 1.34698 0.673491 0.739195i \(-0.264794\pi\)
0.673491 + 0.739195i \(0.264794\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.12356 −0.615382
\(135\) 0 0
\(136\) −13.0711 −1.12083
\(137\) 4.60500 0.393432 0.196716 0.980461i \(-0.436972\pi\)
0.196716 + 0.980461i \(0.436972\pi\)
\(138\) 0 0
\(139\) −8.82843 −0.748817 −0.374409 0.927264i \(-0.622154\pi\)
−0.374409 + 0.927264i \(0.622154\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.1005 −0.931534
\(143\) −5.64823 −0.472328
\(144\) 0 0
\(145\) 6.34315 0.526770
\(146\) −10.8119 −0.894800
\(147\) 0 0
\(148\) 2.17157 0.178502
\(149\) −6.81801 −0.558553 −0.279277 0.960211i \(-0.590095\pi\)
−0.279277 + 0.960211i \(0.590095\pi\)
\(150\) 0 0
\(151\) 4.24264 0.345261 0.172631 0.984987i \(-0.444773\pi\)
0.172631 + 0.984987i \(0.444773\pi\)
\(152\) 4.82106 0.391040
\(153\) 0 0
\(154\) 0 0
\(155\) −13.4200 −1.07792
\(156\) 0 0
\(157\) 18.4853 1.47529 0.737643 0.675191i \(-0.235939\pi\)
0.737643 + 0.675191i \(0.235939\pi\)
\(158\) 17.5034 1.39249
\(159\) 0 0
\(160\) 2.89949 0.229225
\(161\) 0 0
\(162\) 0 0
\(163\) −11.6569 −0.913035 −0.456518 0.889714i \(-0.650903\pi\)
−0.456518 + 0.889714i \(0.650903\pi\)
\(164\) 0.216058 0.0168713
\(165\) 0 0
\(166\) −18.4853 −1.43474
\(167\) −11.4230 −0.883939 −0.441970 0.897030i \(-0.645720\pi\)
−0.441970 + 0.897030i \(0.645720\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) −6.81801 −0.522918
\(171\) 0 0
\(172\) −2.92893 −0.223329
\(173\) 13.1144 0.997070 0.498535 0.866869i \(-0.333871\pi\)
0.498535 + 0.866869i \(0.333871\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14.4632 1.09020
\(177\) 0 0
\(178\) −22.3137 −1.67248
\(179\) 17.9355 1.34056 0.670280 0.742108i \(-0.266174\pi\)
0.670280 + 0.742108i \(0.266174\pi\)
\(180\) 0 0
\(181\) 4.10051 0.304788 0.152394 0.988320i \(-0.451302\pi\)
0.152394 + 0.988320i \(0.451302\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 20.0711 1.47966
\(185\) 6.60195 0.485385
\(186\) 0 0
\(187\) 20.7279 1.51578
\(188\) 4.73157 0.345085
\(189\) 0 0
\(190\) 2.51472 0.182437
\(191\) −9.12051 −0.659937 −0.329969 0.943992i \(-0.607038\pi\)
−0.329969 + 0.943992i \(0.607038\pi\)
\(192\) 0 0
\(193\) 23.6569 1.70286 0.851429 0.524470i \(-0.175737\pi\)
0.851429 + 0.524470i \(0.175737\pi\)
\(194\) −12.1607 −0.873086
\(195\) 0 0
\(196\) 0 0
\(197\) 3.25623 0.231997 0.115998 0.993249i \(-0.462993\pi\)
0.115998 + 0.993249i \(0.462993\pi\)
\(198\) 0 0
\(199\) −3.92893 −0.278515 −0.139257 0.990256i \(-0.544472\pi\)
−0.139257 + 0.990256i \(0.544472\pi\)
\(200\) −10.3798 −0.733962
\(201\) 0 0
\(202\) −20.7279 −1.45841
\(203\) 0 0
\(204\) 0 0
\(205\) 0.656854 0.0458767
\(206\) 4.38895 0.305792
\(207\) 0 0
\(208\) −3.51472 −0.243702
\(209\) −7.64518 −0.528828
\(210\) 0 0
\(211\) −8.24264 −0.567447 −0.283723 0.958906i \(-0.591570\pi\)
−0.283723 + 0.958906i \(0.591570\pi\)
\(212\) 0.432117 0.0296779
\(213\) 0 0
\(214\) −8.58579 −0.586912
\(215\) −8.90446 −0.607279
\(216\) 0 0
\(217\) 0 0
\(218\) −3.34572 −0.226601
\(219\) 0 0
\(220\) −2.51472 −0.169542
\(221\) −5.03712 −0.338833
\(222\) 0 0
\(223\) −13.7279 −0.919290 −0.459645 0.888103i \(-0.652023\pi\)
−0.459645 + 0.888103i \(0.652023\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.3848 0.956861
\(227\) −7.86123 −0.521768 −0.260884 0.965370i \(-0.584014\pi\)
−0.260884 + 0.965370i \(0.584014\pi\)
\(228\) 0 0
\(229\) 1.51472 0.100095 0.0500477 0.998747i \(-0.484063\pi\)
0.0500477 + 0.998747i \(0.484063\pi\)
\(230\) 10.4693 0.690324
\(231\) 0 0
\(232\) −15.3137 −1.00539
\(233\) 20.0219 1.31168 0.655840 0.754900i \(-0.272315\pi\)
0.655840 + 0.754900i \(0.272315\pi\)
\(234\) 0 0
\(235\) 14.3848 0.938359
\(236\) −0.737669 −0.0480182
\(237\) 0 0
\(238\) 0 0
\(239\) 1.78089 0.115196 0.0575981 0.998340i \(-0.481656\pi\)
0.0575981 + 0.998340i \(0.481656\pi\)
\(240\) 0 0
\(241\) −24.7279 −1.59287 −0.796433 0.604727i \(-0.793282\pi\)
−0.796433 + 0.604727i \(0.793282\pi\)
\(242\) −15.4169 −0.991037
\(243\) 0 0
\(244\) −4.34315 −0.278041
\(245\) 0 0
\(246\) 0 0
\(247\) 1.85786 0.118213
\(248\) 32.3987 2.05732
\(249\) 0 0
\(250\) −13.3431 −0.843895
\(251\) 11.7286 0.740301 0.370150 0.928972i \(-0.379306\pi\)
0.370150 + 0.928972i \(0.379306\pi\)
\(252\) 0 0
\(253\) −31.8284 −2.00104
\(254\) 6.08034 0.381515
\(255\) 0 0
\(256\) −9.48528 −0.592830
\(257\) −10.2903 −0.641891 −0.320946 0.947098i \(-0.604001\pi\)
−0.320946 + 0.947098i \(0.604001\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.611105 0.0378991
\(261\) 0 0
\(262\) −19.4142 −1.19941
\(263\) −6.16984 −0.380448 −0.190224 0.981741i \(-0.560921\pi\)
−0.190224 + 0.981741i \(0.560921\pi\)
\(264\) 0 0
\(265\) 1.31371 0.0807005
\(266\) 0 0
\(267\) 0 0
\(268\) −2.34315 −0.143130
\(269\) 0.521611 0.0318032 0.0159016 0.999874i \(-0.494938\pi\)
0.0159016 + 0.999874i \(0.494938\pi\)
\(270\) 0 0
\(271\) −16.8284 −1.02225 −0.511127 0.859505i \(-0.670772\pi\)
−0.511127 + 0.859505i \(0.670772\pi\)
\(272\) 12.8984 0.782078
\(273\) 0 0
\(274\) −5.79899 −0.350330
\(275\) 16.4601 0.992584
\(276\) 0 0
\(277\) 8.55635 0.514101 0.257051 0.966398i \(-0.417249\pi\)
0.257051 + 0.966398i \(0.417249\pi\)
\(278\) 11.1175 0.666781
\(279\) 0 0
\(280\) 0 0
\(281\) 22.5405 1.34465 0.672326 0.740255i \(-0.265295\pi\)
0.672326 + 0.740255i \(0.265295\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) −3.65128 −0.216663
\(285\) 0 0
\(286\) 7.11270 0.420583
\(287\) 0 0
\(288\) 0 0
\(289\) 1.48528 0.0873695
\(290\) −7.98780 −0.469060
\(291\) 0 0
\(292\) −3.55635 −0.208120
\(293\) −9.94768 −0.581149 −0.290575 0.956852i \(-0.593846\pi\)
−0.290575 + 0.956852i \(0.593846\pi\)
\(294\) 0 0
\(295\) −2.24264 −0.130572
\(296\) −15.9385 −0.926408
\(297\) 0 0
\(298\) 8.58579 0.497361
\(299\) 7.73467 0.447307
\(300\) 0 0
\(301\) 0 0
\(302\) −5.34267 −0.307436
\(303\) 0 0
\(304\) −4.75736 −0.272853
\(305\) −13.2039 −0.756053
\(306\) 0 0
\(307\) −9.92893 −0.566674 −0.283337 0.959020i \(-0.591442\pi\)
−0.283337 + 0.959020i \(0.591442\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 16.8995 0.959827
\(311\) −22.2349 −1.26083 −0.630413 0.776260i \(-0.717115\pi\)
−0.630413 + 0.776260i \(0.717115\pi\)
\(312\) 0 0
\(313\) 24.8284 1.40339 0.701693 0.712480i \(-0.252428\pi\)
0.701693 + 0.712480i \(0.252428\pi\)
\(314\) −23.2781 −1.31366
\(315\) 0 0
\(316\) 5.75736 0.323877
\(317\) 19.4108 1.09022 0.545110 0.838365i \(-0.316488\pi\)
0.545110 + 0.838365i \(0.316488\pi\)
\(318\) 0 0
\(319\) 24.2843 1.35966
\(320\) −11.2070 −0.626488
\(321\) 0 0
\(322\) 0 0
\(323\) −6.81801 −0.379364
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 14.6792 0.813008
\(327\) 0 0
\(328\) −1.58579 −0.0875604
\(329\) 0 0
\(330\) 0 0
\(331\) −2.97056 −0.163277 −0.0816384 0.996662i \(-0.526015\pi\)
−0.0816384 + 0.996662i \(0.526015\pi\)
\(332\) −6.08034 −0.333702
\(333\) 0 0
\(334\) 14.3848 0.787100
\(335\) −7.12356 −0.389202
\(336\) 0 0
\(337\) −28.7990 −1.56878 −0.784390 0.620267i \(-0.787024\pi\)
−0.784390 + 0.620267i \(0.787024\pi\)
\(338\) 14.6422 0.796429
\(339\) 0 0
\(340\) −2.24264 −0.121624
\(341\) −51.3774 −2.78224
\(342\) 0 0
\(343\) 0 0
\(344\) 21.4973 1.15905
\(345\) 0 0
\(346\) −16.5147 −0.887837
\(347\) −11.6391 −0.624818 −0.312409 0.949948i \(-0.601136\pi\)
−0.312409 + 0.949948i \(0.601136\pi\)
\(348\) 0 0
\(349\) −1.07107 −0.0573329 −0.0286665 0.999589i \(-0.509126\pi\)
−0.0286665 + 0.999589i \(0.509126\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 11.1005 0.591659
\(353\) −28.4048 −1.51183 −0.755916 0.654668i \(-0.772808\pi\)
−0.755916 + 0.654668i \(0.772808\pi\)
\(354\) 0 0
\(355\) −11.1005 −0.589154
\(356\) −7.33962 −0.388999
\(357\) 0 0
\(358\) −22.5858 −1.19370
\(359\) −26.5344 −1.40043 −0.700215 0.713932i \(-0.746913\pi\)
−0.700215 + 0.713932i \(0.746913\pi\)
\(360\) 0 0
\(361\) −16.4853 −0.867646
\(362\) −5.16368 −0.271397
\(363\) 0 0
\(364\) 0 0
\(365\) −10.8119 −0.565921
\(366\) 0 0
\(367\) −7.38478 −0.385482 −0.192741 0.981250i \(-0.561738\pi\)
−0.192741 + 0.981250i \(0.561738\pi\)
\(368\) −19.8059 −1.03245
\(369\) 0 0
\(370\) −8.31371 −0.432209
\(371\) 0 0
\(372\) 0 0
\(373\) 13.1421 0.680474 0.340237 0.940340i \(-0.389493\pi\)
0.340237 + 0.940340i \(0.389493\pi\)
\(374\) −26.1023 −1.34972
\(375\) 0 0
\(376\) −34.7279 −1.79096
\(377\) −5.90135 −0.303935
\(378\) 0 0
\(379\) −1.89949 −0.0975705 −0.0487853 0.998809i \(-0.515535\pi\)
−0.0487853 + 0.998809i \(0.515535\pi\)
\(380\) 0.827164 0.0424326
\(381\) 0 0
\(382\) 11.4853 0.587638
\(383\) 6.81801 0.348384 0.174192 0.984712i \(-0.444269\pi\)
0.174192 + 0.984712i \(0.444269\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −29.7906 −1.51630
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) −11.1175 −0.563678 −0.281839 0.959462i \(-0.590944\pi\)
−0.281839 + 0.959462i \(0.590944\pi\)
\(390\) 0 0
\(391\) −28.3848 −1.43548
\(392\) 0 0
\(393\) 0 0
\(394\) −4.10051 −0.206580
\(395\) 17.5034 0.880690
\(396\) 0 0
\(397\) 8.92893 0.448130 0.224065 0.974574i \(-0.428067\pi\)
0.224065 + 0.974574i \(0.428067\pi\)
\(398\) 4.94763 0.248002
\(399\) 0 0
\(400\) 10.2426 0.512132
\(401\) −21.8028 −1.08878 −0.544390 0.838832i \(-0.683239\pi\)
−0.544390 + 0.838832i \(0.683239\pi\)
\(402\) 0 0
\(403\) 12.4853 0.621936
\(404\) −6.81801 −0.339209
\(405\) 0 0
\(406\) 0 0
\(407\) 25.2751 1.25284
\(408\) 0 0
\(409\) −9.07107 −0.448535 −0.224268 0.974528i \(-0.571999\pi\)
−0.224268 + 0.974528i \(0.571999\pi\)
\(410\) −0.827164 −0.0408507
\(411\) 0 0
\(412\) 1.44365 0.0711236
\(413\) 0 0
\(414\) 0 0
\(415\) −18.4853 −0.907407
\(416\) −2.69755 −0.132258
\(417\) 0 0
\(418\) 9.62742 0.470892
\(419\) 17.5034 0.855095 0.427547 0.903993i \(-0.359378\pi\)
0.427547 + 0.903993i \(0.359378\pi\)
\(420\) 0 0
\(421\) 22.0711 1.07568 0.537839 0.843048i \(-0.319241\pi\)
0.537839 + 0.843048i \(0.319241\pi\)
\(422\) 10.3798 0.505280
\(423\) 0 0
\(424\) −3.17157 −0.154025
\(425\) 14.6792 0.712048
\(426\) 0 0
\(427\) 0 0
\(428\) −2.82411 −0.136509
\(429\) 0 0
\(430\) 11.2132 0.540749
\(431\) 17.7194 0.853514 0.426757 0.904366i \(-0.359656\pi\)
0.426757 + 0.904366i \(0.359656\pi\)
\(432\) 0 0
\(433\) 24.9706 1.20001 0.600004 0.799997i \(-0.295166\pi\)
0.600004 + 0.799997i \(0.295166\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.10051 −0.0527046
\(437\) 10.4693 0.500814
\(438\) 0 0
\(439\) 19.7990 0.944954 0.472477 0.881343i \(-0.343360\pi\)
0.472477 + 0.881343i \(0.343360\pi\)
\(440\) 18.4571 0.879907
\(441\) 0 0
\(442\) 6.34315 0.301713
\(443\) 14.4632 0.687167 0.343583 0.939122i \(-0.388359\pi\)
0.343583 + 0.939122i \(0.388359\pi\)
\(444\) 0 0
\(445\) −22.3137 −1.05777
\(446\) 17.2873 0.818577
\(447\) 0 0
\(448\) 0 0
\(449\) 9.94768 0.469460 0.234730 0.972061i \(-0.424579\pi\)
0.234730 + 0.972061i \(0.424579\pi\)
\(450\) 0 0
\(451\) 2.51472 0.118413
\(452\) 4.73157 0.222554
\(453\) 0 0
\(454\) 9.89949 0.464606
\(455\) 0 0
\(456\) 0 0
\(457\) 19.9706 0.934184 0.467092 0.884209i \(-0.345302\pi\)
0.467092 + 0.884209i \(0.345302\pi\)
\(458\) −1.90746 −0.0891295
\(459\) 0 0
\(460\) 3.44365 0.160561
\(461\) −30.3122 −1.41178 −0.705890 0.708321i \(-0.749453\pi\)
−0.705890 + 0.708321i \(0.749453\pi\)
\(462\) 0 0
\(463\) 11.8995 0.553016 0.276508 0.961012i \(-0.410823\pi\)
0.276508 + 0.961012i \(0.410823\pi\)
\(464\) 15.1114 0.701527
\(465\) 0 0
\(466\) −25.2132 −1.16798
\(467\) −2.82411 −0.130684 −0.0653422 0.997863i \(-0.520814\pi\)
−0.0653422 + 0.997863i \(0.520814\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.1145 −0.835558
\(471\) 0 0
\(472\) 5.41421 0.249209
\(473\) −34.0901 −1.56746
\(474\) 0 0
\(475\) −5.41421 −0.248421
\(476\) 0 0
\(477\) 0 0
\(478\) −2.24264 −0.102576
\(479\) 26.4078 1.20660 0.603302 0.797513i \(-0.293851\pi\)
0.603302 + 0.797513i \(0.293851\pi\)
\(480\) 0 0
\(481\) −6.14214 −0.280057
\(482\) 31.1394 1.41836
\(483\) 0 0
\(484\) −5.07107 −0.230503
\(485\) −12.1607 −0.552188
\(486\) 0 0
\(487\) 32.5858 1.47660 0.738301 0.674471i \(-0.235628\pi\)
0.738301 + 0.674471i \(0.235628\pi\)
\(488\) 31.8771 1.44301
\(489\) 0 0
\(490\) 0 0
\(491\) 11.2070 0.505763 0.252881 0.967497i \(-0.418622\pi\)
0.252881 + 0.967497i \(0.418622\pi\)
\(492\) 0 0
\(493\) 21.6569 0.975376
\(494\) −2.33957 −0.105262
\(495\) 0 0
\(496\) −31.9706 −1.43552
\(497\) 0 0
\(498\) 0 0
\(499\) −41.5563 −1.86032 −0.930159 0.367157i \(-0.880331\pi\)
−0.930159 + 0.367157i \(0.880331\pi\)
\(500\) −4.38895 −0.196280
\(501\) 0 0
\(502\) −14.7696 −0.659197
\(503\) 18.2410 0.813327 0.406664 0.913578i \(-0.366692\pi\)
0.406664 + 0.913578i \(0.366692\pi\)
\(504\) 0 0
\(505\) −20.7279 −0.922380
\(506\) 40.0809 1.78181
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −8.90446 −0.394683 −0.197342 0.980335i \(-0.563231\pi\)
−0.197342 + 0.980335i \(0.563231\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 25.1485 1.11142
\(513\) 0 0
\(514\) 12.9584 0.571569
\(515\) 4.38895 0.193400
\(516\) 0 0
\(517\) 55.0711 2.42202
\(518\) 0 0
\(519\) 0 0
\(520\) −4.48528 −0.196693
\(521\) −10.4693 −0.458668 −0.229334 0.973348i \(-0.573655\pi\)
−0.229334 + 0.973348i \(0.573655\pi\)
\(522\) 0 0
\(523\) −29.7696 −1.30173 −0.650866 0.759193i \(-0.725594\pi\)
−0.650866 + 0.759193i \(0.725594\pi\)
\(524\) −6.38589 −0.278969
\(525\) 0 0
\(526\) 7.76955 0.338769
\(527\) −45.8186 −1.99589
\(528\) 0 0
\(529\) 20.5858 0.895034
\(530\) −1.65433 −0.0718594
\(531\) 0 0
\(532\) 0 0
\(533\) −0.611105 −0.0264699
\(534\) 0 0
\(535\) −8.58579 −0.371196
\(536\) 17.1978 0.742832
\(537\) 0 0
\(538\) −0.656854 −0.0283190
\(539\) 0 0
\(540\) 0 0
\(541\) 15.3848 0.661443 0.330722 0.943728i \(-0.392708\pi\)
0.330722 + 0.943728i \(0.392708\pi\)
\(542\) 21.1917 0.910262
\(543\) 0 0
\(544\) 9.89949 0.424437
\(545\) −3.34572 −0.143315
\(546\) 0 0
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) −1.90746 −0.0814824
\(549\) 0 0
\(550\) −20.7279 −0.883842
\(551\) −7.98780 −0.340292
\(552\) 0 0
\(553\) 0 0
\(554\) −10.7748 −0.457779
\(555\) 0 0
\(556\) 3.65685 0.155085
\(557\) 39.7383 1.68377 0.841883 0.539661i \(-0.181448\pi\)
0.841883 + 0.539661i \(0.181448\pi\)
\(558\) 0 0
\(559\) 8.28427 0.350387
\(560\) 0 0
\(561\) 0 0
\(562\) −28.3848 −1.19734
\(563\) 40.4760 1.70586 0.852929 0.522027i \(-0.174824\pi\)
0.852929 + 0.522027i \(0.174824\pi\)
\(564\) 0 0
\(565\) 14.3848 0.605172
\(566\) −10.6853 −0.449139
\(567\) 0 0
\(568\) 26.7990 1.12446
\(569\) −24.6269 −1.03241 −0.516207 0.856464i \(-0.672657\pi\)
−0.516207 + 0.856464i \(0.672657\pi\)
\(570\) 0 0
\(571\) −32.0416 −1.34090 −0.670450 0.741954i \(-0.733899\pi\)
−0.670450 + 0.741954i \(0.733899\pi\)
\(572\) 2.33957 0.0978224
\(573\) 0 0
\(574\) 0 0
\(575\) −22.5405 −0.940003
\(576\) 0 0
\(577\) 41.3553 1.72165 0.860823 0.508905i \(-0.169950\pi\)
0.860823 + 0.508905i \(0.169950\pi\)
\(578\) −1.87039 −0.0777978
\(579\) 0 0
\(580\) −2.62742 −0.109098
\(581\) 0 0
\(582\) 0 0
\(583\) 5.02944 0.208298
\(584\) 26.1023 1.08012
\(585\) 0 0
\(586\) 12.5269 0.517482
\(587\) −35.1333 −1.45011 −0.725053 0.688693i \(-0.758185\pi\)
−0.725053 + 0.688693i \(0.758185\pi\)
\(588\) 0 0
\(589\) 16.8995 0.696332
\(590\) 2.82411 0.116267
\(591\) 0 0
\(592\) 15.7279 0.646414
\(593\) −1.87039 −0.0768075 −0.0384038 0.999262i \(-0.512227\pi\)
−0.0384038 + 0.999262i \(0.512227\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.82411 0.115680
\(597\) 0 0
\(598\) −9.74012 −0.398303
\(599\) 44.5593 1.82065 0.910323 0.413899i \(-0.135833\pi\)
0.910323 + 0.413899i \(0.135833\pi\)
\(600\) 0 0
\(601\) −18.6274 −0.759828 −0.379914 0.925022i \(-0.624046\pi\)
−0.379914 + 0.925022i \(0.624046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.75736 −0.0715059
\(605\) −15.4169 −0.626787
\(606\) 0 0
\(607\) 29.3137 1.18981 0.594903 0.803797i \(-0.297190\pi\)
0.594903 + 0.803797i \(0.297190\pi\)
\(608\) −3.65128 −0.148079
\(609\) 0 0
\(610\) 16.6274 0.673224
\(611\) −13.3829 −0.541414
\(612\) 0 0
\(613\) −39.6274 −1.60054 −0.800268 0.599642i \(-0.795310\pi\)
−0.800268 + 0.599642i \(0.795310\pi\)
\(614\) 12.5033 0.504592
\(615\) 0 0
\(616\) 0 0
\(617\) −11.8551 −0.477270 −0.238635 0.971109i \(-0.576700\pi\)
−0.238635 + 0.971109i \(0.576700\pi\)
\(618\) 0 0
\(619\) 16.4558 0.661416 0.330708 0.943733i \(-0.392712\pi\)
0.330708 + 0.943733i \(0.392712\pi\)
\(620\) 5.55873 0.223244
\(621\) 0 0
\(622\) 28.0000 1.12270
\(623\) 0 0
\(624\) 0 0
\(625\) 3.72792 0.149117
\(626\) −31.2659 −1.24964
\(627\) 0 0
\(628\) −7.65685 −0.305542
\(629\) 22.5405 0.898748
\(630\) 0 0
\(631\) 29.3553 1.16862 0.584309 0.811531i \(-0.301366\pi\)
0.584309 + 0.811531i \(0.301366\pi\)
\(632\) −42.2568 −1.68089
\(633\) 0 0
\(634\) −24.4437 −0.970781
\(635\) 6.08034 0.241291
\(636\) 0 0
\(637\) 0 0
\(638\) −30.5807 −1.21070
\(639\) 0 0
\(640\) 8.31371 0.328628
\(641\) 45.3865 1.79266 0.896330 0.443388i \(-0.146224\pi\)
0.896330 + 0.443388i \(0.146224\pi\)
\(642\) 0 0
\(643\) −2.75736 −0.108740 −0.0543698 0.998521i \(-0.517315\pi\)
−0.0543698 + 0.998521i \(0.517315\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.58579 0.337803
\(647\) −6.81801 −0.268044 −0.134022 0.990978i \(-0.542789\pi\)
−0.134022 + 0.990978i \(0.542789\pi\)
\(648\) 0 0
\(649\) −8.58579 −0.337022
\(650\) 5.03712 0.197572
\(651\) 0 0
\(652\) 4.82843 0.189096
\(653\) 37.6518 1.47343 0.736715 0.676203i \(-0.236376\pi\)
0.736715 + 0.676203i \(0.236376\pi\)
\(654\) 0 0
\(655\) −19.4142 −0.758576
\(656\) 1.56483 0.0610965
\(657\) 0 0
\(658\) 0 0
\(659\) −13.8521 −0.539600 −0.269800 0.962916i \(-0.586958\pi\)
−0.269800 + 0.962916i \(0.586958\pi\)
\(660\) 0 0
\(661\) 1.85786 0.0722625 0.0361313 0.999347i \(-0.488497\pi\)
0.0361313 + 0.999347i \(0.488497\pi\)
\(662\) 3.74077 0.145389
\(663\) 0 0
\(664\) 44.6274 1.73188
\(665\) 0 0
\(666\) 0 0
\(667\) −33.2548 −1.28763
\(668\) 4.73157 0.183070
\(669\) 0 0
\(670\) 8.97056 0.346563
\(671\) −50.5502 −1.95147
\(672\) 0 0
\(673\) 15.1716 0.584821 0.292411 0.956293i \(-0.405543\pi\)
0.292411 + 0.956293i \(0.405543\pi\)
\(674\) 36.2660 1.39691
\(675\) 0 0
\(676\) 4.81623 0.185240
\(677\) 23.4942 0.902956 0.451478 0.892282i \(-0.350897\pi\)
0.451478 + 0.892282i \(0.350897\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 16.4601 0.631217
\(681\) 0 0
\(682\) 64.6985 2.47743
\(683\) −13.8521 −0.530035 −0.265018 0.964244i \(-0.585378\pi\)
−0.265018 + 0.964244i \(0.585378\pi\)
\(684\) 0 0
\(685\) −5.79899 −0.221568
\(686\) 0 0
\(687\) 0 0
\(688\) −21.2132 −0.808746
\(689\) −1.22221 −0.0465625
\(690\) 0 0
\(691\) −44.8284 −1.70535 −0.852677 0.522439i \(-0.825022\pi\)
−0.852677 + 0.522439i \(0.825022\pi\)
\(692\) −5.43217 −0.206500
\(693\) 0 0
\(694\) 14.6569 0.556367
\(695\) 11.1175 0.421709
\(696\) 0 0
\(697\) 2.24264 0.0849461
\(698\) 1.34877 0.0510519
\(699\) 0 0
\(700\) 0 0
\(701\) −2.51856 −0.0951247 −0.0475624 0.998868i \(-0.515145\pi\)
−0.0475624 + 0.998868i \(0.515145\pi\)
\(702\) 0 0
\(703\) −8.31371 −0.313557
\(704\) −42.9050 −1.61704
\(705\) 0 0
\(706\) 35.7696 1.34620
\(707\) 0 0
\(708\) 0 0
\(709\) 31.3431 1.17712 0.588558 0.808455i \(-0.299696\pi\)
0.588558 + 0.808455i \(0.299696\pi\)
\(710\) 13.9786 0.524609
\(711\) 0 0
\(712\) 53.8701 2.01887
\(713\) 70.3561 2.63485
\(714\) 0 0
\(715\) 7.11270 0.266000
\(716\) −7.42912 −0.277639
\(717\) 0 0
\(718\) 33.4142 1.24701
\(719\) 29.0529 1.08349 0.541746 0.840542i \(-0.317764\pi\)
0.541746 + 0.840542i \(0.317764\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 20.7596 0.772592
\(723\) 0 0
\(724\) −1.69848 −0.0631237
\(725\) 17.1978 0.638710
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 13.6152 0.503922
\(731\) −30.4017 −1.12445
\(732\) 0 0
\(733\) 24.2426 0.895422 0.447711 0.894178i \(-0.352239\pi\)
0.447711 + 0.894178i \(0.352239\pi\)
\(734\) 9.29950 0.343251
\(735\) 0 0
\(736\) −15.2010 −0.560317
\(737\) −27.2720 −1.00458
\(738\) 0 0
\(739\) 1.55635 0.0572512 0.0286256 0.999590i \(-0.490887\pi\)
0.0286256 + 0.999590i \(0.490887\pi\)
\(740\) −2.73462 −0.100527
\(741\) 0 0
\(742\) 0 0
\(743\) −10.2903 −0.377514 −0.188757 0.982024i \(-0.560446\pi\)
−0.188757 + 0.982024i \(0.560446\pi\)
\(744\) 0 0
\(745\) 8.58579 0.314559
\(746\) −16.5496 −0.605925
\(747\) 0 0
\(748\) −8.58579 −0.313927
\(749\) 0 0
\(750\) 0 0
\(751\) −31.8995 −1.16403 −0.582015 0.813178i \(-0.697735\pi\)
−0.582015 + 0.813178i \(0.697735\pi\)
\(752\) 34.2690 1.24966
\(753\) 0 0
\(754\) 7.43146 0.270638
\(755\) −5.34267 −0.194440
\(756\) 0 0
\(757\) 1.65685 0.0602194 0.0301097 0.999547i \(-0.490414\pi\)
0.0301097 + 0.999547i \(0.490414\pi\)
\(758\) 2.39200 0.0868812
\(759\) 0 0
\(760\) −6.07107 −0.220221
\(761\) 0.305553 0.0110763 0.00553814 0.999985i \(-0.498237\pi\)
0.00553814 + 0.999985i \(0.498237\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 3.77784 0.136677
\(765\) 0 0
\(766\) −8.58579 −0.310217
\(767\) 2.08644 0.0753371
\(768\) 0 0
\(769\) 22.9706 0.828340 0.414170 0.910200i \(-0.364072\pi\)
0.414170 + 0.910200i \(0.364072\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9.79899 −0.352673
\(773\) −6.72852 −0.242008 −0.121004 0.992652i \(-0.538611\pi\)
−0.121004 + 0.992652i \(0.538611\pi\)
\(774\) 0 0
\(775\) −36.3848 −1.30698
\(776\) 29.3585 1.05391
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) −0.827164 −0.0296362
\(780\) 0 0
\(781\) −42.4975 −1.52068
\(782\) 35.7444 1.27822
\(783\) 0 0
\(784\) 0 0
\(785\) −23.2781 −0.830833
\(786\) 0 0
\(787\) −17.6569 −0.629399 −0.314699 0.949191i \(-0.601904\pi\)
−0.314699 + 0.949191i \(0.601904\pi\)
\(788\) −1.34877 −0.0480481
\(789\) 0 0
\(790\) −22.0416 −0.784206
\(791\) 0 0
\(792\) 0 0
\(793\) 12.2843 0.436227
\(794\) −11.2440 −0.399036
\(795\) 0 0
\(796\) 1.62742 0.0576823
\(797\) −3.16674 −0.112172 −0.0560858 0.998426i \(-0.517862\pi\)
−0.0560858 + 0.998426i \(0.517862\pi\)
\(798\) 0 0
\(799\) 49.1127 1.73748
\(800\) 7.86123 0.277937
\(801\) 0 0
\(802\) 27.4558 0.969500
\(803\) −41.3926 −1.46071
\(804\) 0 0
\(805\) 0 0
\(806\) −15.7225 −0.553800
\(807\) 0 0
\(808\) 50.0416 1.76046
\(809\) −29.9696 −1.05367 −0.526837 0.849966i \(-0.676622\pi\)
−0.526837 + 0.849966i \(0.676622\pi\)
\(810\) 0 0
\(811\) 37.6274 1.32128 0.660639 0.750704i \(-0.270286\pi\)
0.660639 + 0.750704i \(0.270286\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −31.8284 −1.11559
\(815\) 14.6792 0.514192
\(816\) 0 0
\(817\) 11.2132 0.392300
\(818\) 11.4230 0.399396
\(819\) 0 0
\(820\) −0.272078 −0.00950137
\(821\) −42.2568 −1.47477 −0.737387 0.675471i \(-0.763941\pi\)
−0.737387 + 0.675471i \(0.763941\pi\)
\(822\) 0 0
\(823\) −50.7279 −1.76826 −0.884132 0.467237i \(-0.845249\pi\)
−0.884132 + 0.467237i \(0.845249\pi\)
\(824\) −10.5959 −0.369124
\(825\) 0 0
\(826\) 0 0
\(827\) 18.1515 0.631191 0.315595 0.948894i \(-0.397796\pi\)
0.315595 + 0.948894i \(0.397796\pi\)
\(828\) 0 0
\(829\) 39.2132 1.36193 0.680965 0.732316i \(-0.261560\pi\)
0.680965 + 0.732316i \(0.261560\pi\)
\(830\) 23.2781 0.807996
\(831\) 0 0
\(832\) 10.4264 0.361471
\(833\) 0 0
\(834\) 0 0
\(835\) 14.3848 0.497806
\(836\) 3.16674 0.109524
\(837\) 0 0
\(838\) −22.0416 −0.761415
\(839\) −12.1607 −0.419833 −0.209917 0.977719i \(-0.567319\pi\)
−0.209917 + 0.977719i \(0.567319\pi\)
\(840\) 0 0
\(841\) −3.62742 −0.125083
\(842\) −27.7937 −0.957833
\(843\) 0 0
\(844\) 3.41421 0.117522
\(845\) 14.6422 0.503706
\(846\) 0 0
\(847\) 0 0
\(848\) 3.12967 0.107473
\(849\) 0 0
\(850\) −18.4853 −0.634040
\(851\) −34.6117 −1.18647
\(852\) 0 0
\(853\) 38.8284 1.32946 0.664730 0.747084i \(-0.268547\pi\)
0.664730 + 0.747084i \(0.268547\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 20.7279 0.708466
\(857\) 33.0098 1.12759 0.563796 0.825914i \(-0.309340\pi\)
0.563796 + 0.825914i \(0.309340\pi\)
\(858\) 0 0
\(859\) −6.27208 −0.214001 −0.107000 0.994259i \(-0.534125\pi\)
−0.107000 + 0.994259i \(0.534125\pi\)
\(860\) 3.68835 0.125772
\(861\) 0 0
\(862\) −22.3137 −0.760008
\(863\) 40.3494 1.37351 0.686755 0.726889i \(-0.259035\pi\)
0.686755 + 0.726889i \(0.259035\pi\)
\(864\) 0 0
\(865\) −16.5147 −0.561517
\(866\) −31.4449 −1.06854
\(867\) 0 0
\(868\) 0 0
\(869\) 67.0103 2.27317
\(870\) 0 0
\(871\) 6.62742 0.224561
\(872\) 8.07729 0.273532
\(873\) 0 0
\(874\) −13.1838 −0.445948
\(875\) 0 0
\(876\) 0 0
\(877\) −21.1716 −0.714913 −0.357457 0.933930i \(-0.616356\pi\)
−0.357457 + 0.933930i \(0.616356\pi\)
\(878\) −24.9325 −0.841430
\(879\) 0 0
\(880\) −18.2132 −0.613967
\(881\) −52.1150 −1.75580 −0.877900 0.478844i \(-0.841056\pi\)
−0.877900 + 0.478844i \(0.841056\pi\)
\(882\) 0 0
\(883\) 0.627417 0.0211143 0.0105571 0.999944i \(-0.496639\pi\)
0.0105571 + 0.999944i \(0.496639\pi\)
\(884\) 2.08644 0.0701747
\(885\) 0 0
\(886\) −18.2132 −0.611885
\(887\) −36.3031 −1.21894 −0.609469 0.792810i \(-0.708617\pi\)
−0.609469 + 0.792810i \(0.708617\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 28.0992 0.941888
\(891\) 0 0
\(892\) 5.68629 0.190391
\(893\) −18.1145 −0.606177
\(894\) 0 0
\(895\) −22.5858 −0.754960
\(896\) 0 0
\(897\) 0 0
\(898\) −12.5269 −0.418028
\(899\) −53.6799 −1.79032
\(900\) 0 0
\(901\) 4.48528 0.149426
\(902\) −3.16674 −0.105441
\(903\) 0 0
\(904\) −34.7279 −1.15503
\(905\) −5.16368 −0.171647
\(906\) 0 0
\(907\) 50.7279 1.68439 0.842197 0.539171i \(-0.181262\pi\)
0.842197 + 0.539171i \(0.181262\pi\)
\(908\) 3.25623 0.108062
\(909\) 0 0
\(910\) 0 0
\(911\) 0.916658 0.0303702 0.0151851 0.999885i \(-0.495166\pi\)
0.0151851 + 0.999885i \(0.495166\pi\)
\(912\) 0 0
\(913\) −70.7696 −2.34213
\(914\) −25.1485 −0.831840
\(915\) 0 0
\(916\) −0.627417 −0.0207304
\(917\) 0 0
\(918\) 0 0
\(919\) −13.1716 −0.434490 −0.217245 0.976117i \(-0.569707\pi\)
−0.217245 + 0.976117i \(0.569707\pi\)
\(920\) −25.2751 −0.833295
\(921\) 0 0
\(922\) 38.1716 1.25711
\(923\) 10.3274 0.339929
\(924\) 0 0
\(925\) 17.8995 0.588532
\(926\) −14.9848 −0.492431
\(927\) 0 0
\(928\) 11.5980 0.380722
\(929\) −38.6951 −1.26954 −0.634772 0.772700i \(-0.718906\pi\)
−0.634772 + 0.772700i \(0.718906\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8.29335 −0.271658
\(933\) 0 0
\(934\) 3.55635 0.116367
\(935\) −26.1023 −0.853635
\(936\) 0 0
\(937\) −31.2548 −1.02105 −0.510525 0.859863i \(-0.670549\pi\)
−0.510525 + 0.859863i \(0.670549\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −5.95837 −0.194341
\(941\) −10.2903 −0.335454 −0.167727 0.985833i \(-0.553643\pi\)
−0.167727 + 0.985833i \(0.553643\pi\)
\(942\) 0 0
\(943\) −3.44365 −0.112141
\(944\) −5.34267 −0.173889
\(945\) 0 0
\(946\) 42.9289 1.39574
\(947\) 6.16984 0.200493 0.100246 0.994963i \(-0.468037\pi\)
0.100246 + 0.994963i \(0.468037\pi\)
\(948\) 0 0
\(949\) 10.0589 0.326525
\(950\) 6.81801 0.221206
\(951\) 0 0
\(952\) 0 0
\(953\) 31.2659 1.01280 0.506402 0.862298i \(-0.330975\pi\)
0.506402 + 0.862298i \(0.330975\pi\)
\(954\) 0 0
\(955\) 11.4853 0.371655
\(956\) −0.737669 −0.0238579
\(957\) 0 0
\(958\) −33.2548 −1.07441
\(959\) 0 0
\(960\) 0 0
\(961\) 82.5685 2.66350
\(962\) 7.73467 0.249376
\(963\) 0 0
\(964\) 10.2426 0.329893
\(965\) −29.7906 −0.958994
\(966\) 0 0
\(967\) −17.8995 −0.575609 −0.287804 0.957689i \(-0.592925\pi\)
−0.287804 + 0.957689i \(0.592925\pi\)
\(968\) 37.2197 1.19629
\(969\) 0 0
\(970\) 15.3137 0.491694
\(971\) −0.432117 −0.0138673 −0.00693364 0.999976i \(-0.502207\pi\)
−0.00693364 + 0.999976i \(0.502207\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −41.0346 −1.31483
\(975\) 0 0
\(976\) −31.4558 −1.00688
\(977\) 18.0620 0.577856 0.288928 0.957351i \(-0.406701\pi\)
0.288928 + 0.957351i \(0.406701\pi\)
\(978\) 0 0
\(979\) −85.4264 −2.73024
\(980\) 0 0
\(981\) 0 0
\(982\) −14.1127 −0.450354
\(983\) 5.77479 0.184187 0.0920936 0.995750i \(-0.470644\pi\)
0.0920936 + 0.995750i \(0.470644\pi\)
\(984\) 0 0
\(985\) −4.10051 −0.130653
\(986\) −27.2720 −0.868519
\(987\) 0 0
\(988\) −0.769553 −0.0244827
\(989\) 46.6829 1.48443
\(990\) 0 0
\(991\) −6.97056 −0.221427 −0.110714 0.993852i \(-0.535314\pi\)
−0.110714 + 0.993852i \(0.535314\pi\)
\(992\) −24.5374 −0.779064
\(993\) 0 0
\(994\) 0 0
\(995\) 4.94763 0.156850
\(996\) 0 0
\(997\) −0.870058 −0.0275550 −0.0137775 0.999905i \(-0.504386\pi\)
−0.0137775 + 0.999905i \(0.504386\pi\)
\(998\) 52.3311 1.65651
\(999\) 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 1323.2.a.bd.1.2 yes 4
3.2 odd 2 inner 1323.2.a.bd.1.3 yes 4
7.6 odd 2 1323.2.a.bc.1.2 4
21.20 even 2 1323.2.a.bc.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.bc.1.2 4 7.6 odd 2
1323.2.a.bc.1.3 yes 4 21.20 even 2
1323.2.a.bd.1.2 yes 4 1.1 even 1 trivial
1323.2.a.bd.1.3 yes 4 3.2 odd 2 inner