Properties

Label 6016.2.a.o.1.1
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $0$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.10994\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.10994 q^{3} -0.407310 q^{5} +3.05265 q^{7} +6.67175 q^{9} +O(q^{10})\) \(q-3.10994 q^{3} -0.407310 q^{5} +3.05265 q^{7} +6.67175 q^{9} -2.16670 q^{11} -0.550265 q^{13} +1.26671 q^{15} -3.34079 q^{17} +6.78215 q^{19} -9.49357 q^{21} +5.22765 q^{23} -4.83410 q^{25} -11.4189 q^{27} +3.25095 q^{29} +8.03298 q^{31} +6.73833 q^{33} -1.24337 q^{35} -9.82204 q^{37} +1.71129 q^{39} +8.04458 q^{41} +3.91084 q^{43} -2.71747 q^{45} -1.00000 q^{47} +2.31866 q^{49} +10.3897 q^{51} +6.41908 q^{53} +0.882520 q^{55} -21.0921 q^{57} +2.73544 q^{59} +4.63323 q^{61} +20.3665 q^{63} +0.224128 q^{65} -11.2924 q^{67} -16.2577 q^{69} +5.75360 q^{71} -6.94230 q^{73} +15.0338 q^{75} -6.61419 q^{77} +15.0416 q^{79} +15.4970 q^{81} -9.73633 q^{83} +1.36074 q^{85} -10.1103 q^{87} +9.09741 q^{89} -1.67976 q^{91} -24.9821 q^{93} -2.76244 q^{95} -15.1387 q^{97} -14.4557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{3} - 6 q^{5} - 2 q^{7} + 21 q^{9} + 10 q^{11} - 4 q^{13} + 14 q^{15} + 10 q^{17} + 8 q^{19} - 10 q^{21} + 18 q^{23} + 23 q^{25} + 16 q^{27} - 14 q^{29} + 4 q^{31} + 14 q^{33} + 14 q^{35} - 16 q^{37} + 12 q^{39} + 10 q^{41} + 12 q^{43} - 10 q^{45} - 13 q^{47} + 9 q^{49} + 22 q^{51} - 26 q^{53} - 2 q^{55} + 20 q^{57} + 30 q^{59} - 18 q^{61} + 12 q^{63} - 4 q^{65} + 4 q^{67} - 2 q^{69} + 36 q^{71} + 10 q^{73} + 38 q^{75} - 42 q^{77} + 21 q^{81} + 12 q^{83} - 4 q^{85} + 6 q^{87} + 50 q^{89} - 4 q^{91} - 52 q^{93} + 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.10994 −1.79553 −0.897763 0.440478i \(-0.854809\pi\)
−0.897763 + 0.440478i \(0.854809\pi\)
\(4\) 0 0
\(5\) −0.407310 −0.182155 −0.0910773 0.995844i \(-0.529031\pi\)
−0.0910773 + 0.995844i \(0.529031\pi\)
\(6\) 0 0
\(7\) 3.05265 1.15379 0.576896 0.816817i \(-0.304264\pi\)
0.576896 + 0.816817i \(0.304264\pi\)
\(8\) 0 0
\(9\) 6.67175 2.22392
\(10\) 0 0
\(11\) −2.16670 −0.653286 −0.326643 0.945148i \(-0.605917\pi\)
−0.326643 + 0.945148i \(0.605917\pi\)
\(12\) 0 0
\(13\) −0.550265 −0.152616 −0.0763080 0.997084i \(-0.524313\pi\)
−0.0763080 + 0.997084i \(0.524313\pi\)
\(14\) 0 0
\(15\) 1.26671 0.327063
\(16\) 0 0
\(17\) −3.34079 −0.810261 −0.405131 0.914259i \(-0.632774\pi\)
−0.405131 + 0.914259i \(0.632774\pi\)
\(18\) 0 0
\(19\) 6.78215 1.55593 0.777966 0.628306i \(-0.216252\pi\)
0.777966 + 0.628306i \(0.216252\pi\)
\(20\) 0 0
\(21\) −9.49357 −2.07167
\(22\) 0 0
\(23\) 5.22765 1.09004 0.545020 0.838423i \(-0.316522\pi\)
0.545020 + 0.838423i \(0.316522\pi\)
\(24\) 0 0
\(25\) −4.83410 −0.966820
\(26\) 0 0
\(27\) −11.4189 −2.19758
\(28\) 0 0
\(29\) 3.25095 0.603685 0.301843 0.953358i \(-0.402398\pi\)
0.301843 + 0.953358i \(0.402398\pi\)
\(30\) 0 0
\(31\) 8.03298 1.44277 0.721383 0.692536i \(-0.243507\pi\)
0.721383 + 0.692536i \(0.243507\pi\)
\(32\) 0 0
\(33\) 6.73833 1.17299
\(34\) 0 0
\(35\) −1.24337 −0.210169
\(36\) 0 0
\(37\) −9.82204 −1.61473 −0.807366 0.590050i \(-0.799108\pi\)
−0.807366 + 0.590050i \(0.799108\pi\)
\(38\) 0 0
\(39\) 1.71129 0.274026
\(40\) 0 0
\(41\) 8.04458 1.25635 0.628176 0.778071i \(-0.283802\pi\)
0.628176 + 0.778071i \(0.283802\pi\)
\(42\) 0 0
\(43\) 3.91084 0.596398 0.298199 0.954504i \(-0.403614\pi\)
0.298199 + 0.954504i \(0.403614\pi\)
\(44\) 0 0
\(45\) −2.71747 −0.405097
\(46\) 0 0
\(47\) −1.00000 −0.145865
\(48\) 0 0
\(49\) 2.31866 0.331238
\(50\) 0 0
\(51\) 10.3897 1.45485
\(52\) 0 0
\(53\) 6.41908 0.881728 0.440864 0.897574i \(-0.354672\pi\)
0.440864 + 0.897574i \(0.354672\pi\)
\(54\) 0 0
\(55\) 0.882520 0.118999
\(56\) 0 0
\(57\) −21.0921 −2.79372
\(58\) 0 0
\(59\) 2.73544 0.356124 0.178062 0.984019i \(-0.443017\pi\)
0.178062 + 0.984019i \(0.443017\pi\)
\(60\) 0 0
\(61\) 4.63323 0.593224 0.296612 0.954998i \(-0.404143\pi\)
0.296612 + 0.954998i \(0.404143\pi\)
\(62\) 0 0
\(63\) 20.3665 2.56594
\(64\) 0 0
\(65\) 0.224128 0.0277997
\(66\) 0 0
\(67\) −11.2924 −1.37958 −0.689791 0.724009i \(-0.742297\pi\)
−0.689791 + 0.724009i \(0.742297\pi\)
\(68\) 0 0
\(69\) −16.2577 −1.95720
\(70\) 0 0
\(71\) 5.75360 0.682827 0.341414 0.939913i \(-0.389094\pi\)
0.341414 + 0.939913i \(0.389094\pi\)
\(72\) 0 0
\(73\) −6.94230 −0.812535 −0.406267 0.913754i \(-0.633170\pi\)
−0.406267 + 0.913754i \(0.633170\pi\)
\(74\) 0 0
\(75\) 15.0338 1.73595
\(76\) 0 0
\(77\) −6.61419 −0.753757
\(78\) 0 0
\(79\) 15.0416 1.69231 0.846155 0.532937i \(-0.178912\pi\)
0.846155 + 0.532937i \(0.178912\pi\)
\(80\) 0 0
\(81\) 15.4970 1.72189
\(82\) 0 0
\(83\) −9.73633 −1.06870 −0.534350 0.845263i \(-0.679444\pi\)
−0.534350 + 0.845263i \(0.679444\pi\)
\(84\) 0 0
\(85\) 1.36074 0.147593
\(86\) 0 0
\(87\) −10.1103 −1.08393
\(88\) 0 0
\(89\) 9.09741 0.964324 0.482162 0.876082i \(-0.339852\pi\)
0.482162 + 0.876082i \(0.339852\pi\)
\(90\) 0 0
\(91\) −1.67976 −0.176087
\(92\) 0 0
\(93\) −24.9821 −2.59052
\(94\) 0 0
\(95\) −2.76244 −0.283420
\(96\) 0 0
\(97\) −15.1387 −1.53710 −0.768552 0.639787i \(-0.779022\pi\)
−0.768552 + 0.639787i \(0.779022\pi\)
\(98\) 0 0
\(99\) −14.4557 −1.45285
\(100\) 0 0
\(101\) −3.77611 −0.375737 −0.187869 0.982194i \(-0.560158\pi\)
−0.187869 + 0.982194i \(0.560158\pi\)
\(102\) 0 0
\(103\) −4.04840 −0.398901 −0.199451 0.979908i \(-0.563916\pi\)
−0.199451 + 0.979908i \(0.563916\pi\)
\(104\) 0 0
\(105\) 3.86682 0.377363
\(106\) 0 0
\(107\) 14.0615 1.35937 0.679687 0.733503i \(-0.262116\pi\)
0.679687 + 0.733503i \(0.262116\pi\)
\(108\) 0 0
\(109\) −9.62922 −0.922312 −0.461156 0.887319i \(-0.652565\pi\)
−0.461156 + 0.887319i \(0.652565\pi\)
\(110\) 0 0
\(111\) 30.5460 2.89930
\(112\) 0 0
\(113\) −6.99339 −0.657883 −0.328941 0.944350i \(-0.606692\pi\)
−0.328941 + 0.944350i \(0.606692\pi\)
\(114\) 0 0
\(115\) −2.12927 −0.198556
\(116\) 0 0
\(117\) −3.67123 −0.339405
\(118\) 0 0
\(119\) −10.1983 −0.934874
\(120\) 0 0
\(121\) −6.30539 −0.573218
\(122\) 0 0
\(123\) −25.0182 −2.25582
\(124\) 0 0
\(125\) 4.00553 0.358265
\(126\) 0 0
\(127\) 0.458497 0.0406851 0.0203425 0.999793i \(-0.493524\pi\)
0.0203425 + 0.999793i \(0.493524\pi\)
\(128\) 0 0
\(129\) −12.1625 −1.07085
\(130\) 0 0
\(131\) −5.47957 −0.478752 −0.239376 0.970927i \(-0.576943\pi\)
−0.239376 + 0.970927i \(0.576943\pi\)
\(132\) 0 0
\(133\) 20.7035 1.79522
\(134\) 0 0
\(135\) 4.65105 0.400298
\(136\) 0 0
\(137\) 11.6905 0.998790 0.499395 0.866374i \(-0.333556\pi\)
0.499395 + 0.866374i \(0.333556\pi\)
\(138\) 0 0
\(139\) −9.69114 −0.821992 −0.410996 0.911637i \(-0.634819\pi\)
−0.410996 + 0.911637i \(0.634819\pi\)
\(140\) 0 0
\(141\) 3.10994 0.261905
\(142\) 0 0
\(143\) 1.19226 0.0997018
\(144\) 0 0
\(145\) −1.32414 −0.109964
\(146\) 0 0
\(147\) −7.21091 −0.594746
\(148\) 0 0
\(149\) 9.94488 0.814716 0.407358 0.913269i \(-0.366450\pi\)
0.407358 + 0.913269i \(0.366450\pi\)
\(150\) 0 0
\(151\) −7.83567 −0.637657 −0.318829 0.947812i \(-0.603289\pi\)
−0.318829 + 0.947812i \(0.603289\pi\)
\(152\) 0 0
\(153\) −22.2889 −1.80195
\(154\) 0 0
\(155\) −3.27191 −0.262806
\(156\) 0 0
\(157\) 17.9714 1.43427 0.717135 0.696934i \(-0.245453\pi\)
0.717135 + 0.696934i \(0.245453\pi\)
\(158\) 0 0
\(159\) −19.9630 −1.58317
\(160\) 0 0
\(161\) 15.9582 1.25768
\(162\) 0 0
\(163\) −9.23108 −0.723034 −0.361517 0.932366i \(-0.617741\pi\)
−0.361517 + 0.932366i \(0.617741\pi\)
\(164\) 0 0
\(165\) −2.74459 −0.213666
\(166\) 0 0
\(167\) −16.7235 −1.29411 −0.647053 0.762445i \(-0.723999\pi\)
−0.647053 + 0.762445i \(0.723999\pi\)
\(168\) 0 0
\(169\) −12.6972 −0.976708
\(170\) 0 0
\(171\) 45.2488 3.46026
\(172\) 0 0
\(173\) −0.659422 −0.0501349 −0.0250675 0.999686i \(-0.507980\pi\)
−0.0250675 + 0.999686i \(0.507980\pi\)
\(174\) 0 0
\(175\) −14.7568 −1.11551
\(176\) 0 0
\(177\) −8.50707 −0.639430
\(178\) 0 0
\(179\) 0.422467 0.0315767 0.0157883 0.999875i \(-0.494974\pi\)
0.0157883 + 0.999875i \(0.494974\pi\)
\(180\) 0 0
\(181\) −15.4430 −1.14787 −0.573935 0.818901i \(-0.694584\pi\)
−0.573935 + 0.818901i \(0.694584\pi\)
\(182\) 0 0
\(183\) −14.4091 −1.06515
\(184\) 0 0
\(185\) 4.00061 0.294131
\(186\) 0 0
\(187\) 7.23851 0.529332
\(188\) 0 0
\(189\) −34.8580 −2.53555
\(190\) 0 0
\(191\) −6.20080 −0.448674 −0.224337 0.974512i \(-0.572022\pi\)
−0.224337 + 0.974512i \(0.572022\pi\)
\(192\) 0 0
\(193\) −0.914661 −0.0658387 −0.0329194 0.999458i \(-0.510480\pi\)
−0.0329194 + 0.999458i \(0.510480\pi\)
\(194\) 0 0
\(195\) −0.697026 −0.0499151
\(196\) 0 0
\(197\) 27.3854 1.95113 0.975565 0.219712i \(-0.0705117\pi\)
0.975565 + 0.219712i \(0.0705117\pi\)
\(198\) 0 0
\(199\) −16.5462 −1.17293 −0.586464 0.809975i \(-0.699481\pi\)
−0.586464 + 0.809975i \(0.699481\pi\)
\(200\) 0 0
\(201\) 35.1186 2.47707
\(202\) 0 0
\(203\) 9.92399 0.696528
\(204\) 0 0
\(205\) −3.27664 −0.228850
\(206\) 0 0
\(207\) 34.8776 2.42416
\(208\) 0 0
\(209\) −14.6949 −1.01647
\(210\) 0 0
\(211\) 23.9892 1.65149 0.825743 0.564046i \(-0.190756\pi\)
0.825743 + 0.564046i \(0.190756\pi\)
\(212\) 0 0
\(213\) −17.8934 −1.22603
\(214\) 0 0
\(215\) −1.59292 −0.108637
\(216\) 0 0
\(217\) 24.5219 1.66465
\(218\) 0 0
\(219\) 21.5902 1.45893
\(220\) 0 0
\(221\) 1.83832 0.123659
\(222\) 0 0
\(223\) 15.0889 1.01043 0.505215 0.862993i \(-0.331413\pi\)
0.505215 + 0.862993i \(0.331413\pi\)
\(224\) 0 0
\(225\) −32.2519 −2.15013
\(226\) 0 0
\(227\) 19.4047 1.28794 0.643968 0.765053i \(-0.277287\pi\)
0.643968 + 0.765053i \(0.277287\pi\)
\(228\) 0 0
\(229\) −5.07540 −0.335392 −0.167696 0.985839i \(-0.553633\pi\)
−0.167696 + 0.985839i \(0.553633\pi\)
\(230\) 0 0
\(231\) 20.5697 1.35339
\(232\) 0 0
\(233\) 28.1567 1.84461 0.922304 0.386465i \(-0.126304\pi\)
0.922304 + 0.386465i \(0.126304\pi\)
\(234\) 0 0
\(235\) 0.407310 0.0265700
\(236\) 0 0
\(237\) −46.7785 −3.03859
\(238\) 0 0
\(239\) 7.86942 0.509030 0.254515 0.967069i \(-0.418084\pi\)
0.254515 + 0.967069i \(0.418084\pi\)
\(240\) 0 0
\(241\) 15.8599 1.02162 0.510812 0.859693i \(-0.329345\pi\)
0.510812 + 0.859693i \(0.329345\pi\)
\(242\) 0 0
\(243\) −13.9380 −0.894122
\(244\) 0 0
\(245\) −0.944415 −0.0603365
\(246\) 0 0
\(247\) −3.73198 −0.237460
\(248\) 0 0
\(249\) 30.2794 1.91888
\(250\) 0 0
\(251\) 26.3744 1.66474 0.832369 0.554222i \(-0.186984\pi\)
0.832369 + 0.554222i \(0.186984\pi\)
\(252\) 0 0
\(253\) −11.3268 −0.712108
\(254\) 0 0
\(255\) −4.23182 −0.265007
\(256\) 0 0
\(257\) −21.0078 −1.31043 −0.655217 0.755441i \(-0.727423\pi\)
−0.655217 + 0.755441i \(0.727423\pi\)
\(258\) 0 0
\(259\) −29.9832 −1.86307
\(260\) 0 0
\(261\) 21.6895 1.34255
\(262\) 0 0
\(263\) 4.00628 0.247038 0.123519 0.992342i \(-0.460582\pi\)
0.123519 + 0.992342i \(0.460582\pi\)
\(264\) 0 0
\(265\) −2.61455 −0.160611
\(266\) 0 0
\(267\) −28.2924 −1.73147
\(268\) 0 0
\(269\) 7.17613 0.437536 0.218768 0.975777i \(-0.429796\pi\)
0.218768 + 0.975777i \(0.429796\pi\)
\(270\) 0 0
\(271\) −18.5170 −1.12483 −0.562414 0.826856i \(-0.690127\pi\)
−0.562414 + 0.826856i \(0.690127\pi\)
\(272\) 0 0
\(273\) 5.22397 0.316169
\(274\) 0 0
\(275\) 10.4741 0.631610
\(276\) 0 0
\(277\) −10.5411 −0.633354 −0.316677 0.948533i \(-0.602567\pi\)
−0.316677 + 0.948533i \(0.602567\pi\)
\(278\) 0 0
\(279\) 53.5940 3.20859
\(280\) 0 0
\(281\) 7.60641 0.453760 0.226880 0.973923i \(-0.427147\pi\)
0.226880 + 0.973923i \(0.427147\pi\)
\(282\) 0 0
\(283\) 12.8983 0.766724 0.383362 0.923598i \(-0.374766\pi\)
0.383362 + 0.923598i \(0.374766\pi\)
\(284\) 0 0
\(285\) 8.59102 0.508888
\(286\) 0 0
\(287\) 24.5573 1.44957
\(288\) 0 0
\(289\) −5.83910 −0.343476
\(290\) 0 0
\(291\) 47.0806 2.75991
\(292\) 0 0
\(293\) 1.98565 0.116003 0.0580015 0.998316i \(-0.481527\pi\)
0.0580015 + 0.998316i \(0.481527\pi\)
\(294\) 0 0
\(295\) −1.11417 −0.0648696
\(296\) 0 0
\(297\) 24.7415 1.43564
\(298\) 0 0
\(299\) −2.87659 −0.166357
\(300\) 0 0
\(301\) 11.9384 0.688119
\(302\) 0 0
\(303\) 11.7435 0.674646
\(304\) 0 0
\(305\) −1.88716 −0.108058
\(306\) 0 0
\(307\) −7.55506 −0.431190 −0.215595 0.976483i \(-0.569169\pi\)
−0.215595 + 0.976483i \(0.569169\pi\)
\(308\) 0 0
\(309\) 12.5903 0.716238
\(310\) 0 0
\(311\) 21.5882 1.22415 0.612077 0.790799i \(-0.290334\pi\)
0.612077 + 0.790799i \(0.290334\pi\)
\(312\) 0 0
\(313\) −14.6067 −0.825622 −0.412811 0.910817i \(-0.635453\pi\)
−0.412811 + 0.910817i \(0.635453\pi\)
\(314\) 0 0
\(315\) −8.29548 −0.467397
\(316\) 0 0
\(317\) −8.34814 −0.468878 −0.234439 0.972131i \(-0.575325\pi\)
−0.234439 + 0.972131i \(0.575325\pi\)
\(318\) 0 0
\(319\) −7.04384 −0.394379
\(320\) 0 0
\(321\) −43.7304 −2.44079
\(322\) 0 0
\(323\) −22.6578 −1.26071
\(324\) 0 0
\(325\) 2.66003 0.147552
\(326\) 0 0
\(327\) 29.9463 1.65604
\(328\) 0 0
\(329\) −3.05265 −0.168298
\(330\) 0 0
\(331\) 31.8095 1.74841 0.874205 0.485558i \(-0.161383\pi\)
0.874205 + 0.485558i \(0.161383\pi\)
\(332\) 0 0
\(333\) −65.5302 −3.59103
\(334\) 0 0
\(335\) 4.59949 0.251297
\(336\) 0 0
\(337\) −6.01424 −0.327616 −0.163808 0.986492i \(-0.552378\pi\)
−0.163808 + 0.986492i \(0.552378\pi\)
\(338\) 0 0
\(339\) 21.7490 1.18125
\(340\) 0 0
\(341\) −17.4051 −0.942539
\(342\) 0 0
\(343\) −14.2905 −0.771613
\(344\) 0 0
\(345\) 6.62192 0.356512
\(346\) 0 0
\(347\) 12.8677 0.690775 0.345387 0.938460i \(-0.387747\pi\)
0.345387 + 0.938460i \(0.387747\pi\)
\(348\) 0 0
\(349\) 10.9705 0.587237 0.293619 0.955923i \(-0.405140\pi\)
0.293619 + 0.955923i \(0.405140\pi\)
\(350\) 0 0
\(351\) 6.28344 0.335385
\(352\) 0 0
\(353\) 24.9810 1.32961 0.664803 0.747019i \(-0.268516\pi\)
0.664803 + 0.747019i \(0.268516\pi\)
\(354\) 0 0
\(355\) −2.34350 −0.124380
\(356\) 0 0
\(357\) 31.7160 1.67859
\(358\) 0 0
\(359\) −23.2115 −1.22506 −0.612528 0.790449i \(-0.709847\pi\)
−0.612528 + 0.790449i \(0.709847\pi\)
\(360\) 0 0
\(361\) 26.9976 1.42092
\(362\) 0 0
\(363\) 19.6094 1.02923
\(364\) 0 0
\(365\) 2.82767 0.148007
\(366\) 0 0
\(367\) 18.5333 0.967430 0.483715 0.875226i \(-0.339287\pi\)
0.483715 + 0.875226i \(0.339287\pi\)
\(368\) 0 0
\(369\) 53.6714 2.79402
\(370\) 0 0
\(371\) 19.5952 1.01733
\(372\) 0 0
\(373\) 32.7310 1.69475 0.847373 0.530998i \(-0.178183\pi\)
0.847373 + 0.530998i \(0.178183\pi\)
\(374\) 0 0
\(375\) −12.4570 −0.643275
\(376\) 0 0
\(377\) −1.78888 −0.0921320
\(378\) 0 0
\(379\) 18.8297 0.967215 0.483608 0.875285i \(-0.339326\pi\)
0.483608 + 0.875285i \(0.339326\pi\)
\(380\) 0 0
\(381\) −1.42590 −0.0730511
\(382\) 0 0
\(383\) 37.4769 1.91498 0.957489 0.288470i \(-0.0931466\pi\)
0.957489 + 0.288470i \(0.0931466\pi\)
\(384\) 0 0
\(385\) 2.69402 0.137300
\(386\) 0 0
\(387\) 26.0922 1.32634
\(388\) 0 0
\(389\) −9.88039 −0.500956 −0.250478 0.968122i \(-0.580588\pi\)
−0.250478 + 0.968122i \(0.580588\pi\)
\(390\) 0 0
\(391\) −17.4645 −0.883217
\(392\) 0 0
\(393\) 17.0411 0.859612
\(394\) 0 0
\(395\) −6.12659 −0.308262
\(396\) 0 0
\(397\) 8.26429 0.414773 0.207387 0.978259i \(-0.433504\pi\)
0.207387 + 0.978259i \(0.433504\pi\)
\(398\) 0 0
\(399\) −64.3868 −3.22337
\(400\) 0 0
\(401\) 28.0473 1.40061 0.700307 0.713841i \(-0.253046\pi\)
0.700307 + 0.713841i \(0.253046\pi\)
\(402\) 0 0
\(403\) −4.42026 −0.220189
\(404\) 0 0
\(405\) −6.31208 −0.313650
\(406\) 0 0
\(407\) 21.2814 1.05488
\(408\) 0 0
\(409\) 5.98600 0.295989 0.147994 0.988988i \(-0.452718\pi\)
0.147994 + 0.988988i \(0.452718\pi\)
\(410\) 0 0
\(411\) −36.3569 −1.79335
\(412\) 0 0
\(413\) 8.35034 0.410893
\(414\) 0 0
\(415\) 3.96570 0.194669
\(416\) 0 0
\(417\) 30.1389 1.47591
\(418\) 0 0
\(419\) −3.61266 −0.176490 −0.0882449 0.996099i \(-0.528126\pi\)
−0.0882449 + 0.996099i \(0.528126\pi\)
\(420\) 0 0
\(421\) −37.4594 −1.82566 −0.912829 0.408341i \(-0.866107\pi\)
−0.912829 + 0.408341i \(0.866107\pi\)
\(422\) 0 0
\(423\) −6.67175 −0.324392
\(424\) 0 0
\(425\) 16.1497 0.783377
\(426\) 0 0
\(427\) 14.1436 0.684458
\(428\) 0 0
\(429\) −3.70786 −0.179017
\(430\) 0 0
\(431\) −6.49150 −0.312685 −0.156342 0.987703i \(-0.549970\pi\)
−0.156342 + 0.987703i \(0.549970\pi\)
\(432\) 0 0
\(433\) 14.1159 0.678369 0.339184 0.940720i \(-0.389849\pi\)
0.339184 + 0.940720i \(0.389849\pi\)
\(434\) 0 0
\(435\) 4.11801 0.197443
\(436\) 0 0
\(437\) 35.4547 1.69603
\(438\) 0 0
\(439\) 23.0935 1.10219 0.551097 0.834441i \(-0.314209\pi\)
0.551097 + 0.834441i \(0.314209\pi\)
\(440\) 0 0
\(441\) 15.4695 0.736645
\(442\) 0 0
\(443\) 4.41824 0.209917 0.104958 0.994477i \(-0.466529\pi\)
0.104958 + 0.994477i \(0.466529\pi\)
\(444\) 0 0
\(445\) −3.70547 −0.175656
\(446\) 0 0
\(447\) −30.9280 −1.46284
\(448\) 0 0
\(449\) 34.6386 1.63470 0.817348 0.576144i \(-0.195443\pi\)
0.817348 + 0.576144i \(0.195443\pi\)
\(450\) 0 0
\(451\) −17.4302 −0.820758
\(452\) 0 0
\(453\) 24.3685 1.14493
\(454\) 0 0
\(455\) 0.684185 0.0320751
\(456\) 0 0
\(457\) 22.1776 1.03742 0.518712 0.854949i \(-0.326412\pi\)
0.518712 + 0.854949i \(0.326412\pi\)
\(458\) 0 0
\(459\) 38.1483 1.78061
\(460\) 0 0
\(461\) −22.1451 −1.03140 −0.515699 0.856770i \(-0.672468\pi\)
−0.515699 + 0.856770i \(0.672468\pi\)
\(462\) 0 0
\(463\) −33.6623 −1.56442 −0.782210 0.623015i \(-0.785908\pi\)
−0.782210 + 0.623015i \(0.785908\pi\)
\(464\) 0 0
\(465\) 10.1755 0.471876
\(466\) 0 0
\(467\) 27.6298 1.27855 0.639276 0.768977i \(-0.279234\pi\)
0.639276 + 0.768977i \(0.279234\pi\)
\(468\) 0 0
\(469\) −34.4716 −1.59175
\(470\) 0 0
\(471\) −55.8899 −2.57527
\(472\) 0 0
\(473\) −8.47364 −0.389618
\(474\) 0 0
\(475\) −32.7856 −1.50431
\(476\) 0 0
\(477\) 42.8265 1.96089
\(478\) 0 0
\(479\) 19.8209 0.905640 0.452820 0.891602i \(-0.350418\pi\)
0.452820 + 0.891602i \(0.350418\pi\)
\(480\) 0 0
\(481\) 5.40472 0.246434
\(482\) 0 0
\(483\) −49.6290 −2.25820
\(484\) 0 0
\(485\) 6.16615 0.279990
\(486\) 0 0
\(487\) −4.49345 −0.203618 −0.101809 0.994804i \(-0.532463\pi\)
−0.101809 + 0.994804i \(0.532463\pi\)
\(488\) 0 0
\(489\) 28.7081 1.29823
\(490\) 0 0
\(491\) 10.6023 0.478475 0.239238 0.970961i \(-0.423103\pi\)
0.239238 + 0.970961i \(0.423103\pi\)
\(492\) 0 0
\(493\) −10.8607 −0.489143
\(494\) 0 0
\(495\) 5.88795 0.264644
\(496\) 0 0
\(497\) 17.5637 0.787841
\(498\) 0 0
\(499\) −19.4794 −0.872016 −0.436008 0.899943i \(-0.643608\pi\)
−0.436008 + 0.899943i \(0.643608\pi\)
\(500\) 0 0
\(501\) 52.0092 2.32360
\(502\) 0 0
\(503\) −7.10841 −0.316948 −0.158474 0.987363i \(-0.550657\pi\)
−0.158474 + 0.987363i \(0.550657\pi\)
\(504\) 0 0
\(505\) 1.53805 0.0684422
\(506\) 0 0
\(507\) 39.4876 1.75371
\(508\) 0 0
\(509\) −4.46106 −0.197733 −0.0988666 0.995101i \(-0.531522\pi\)
−0.0988666 + 0.995101i \(0.531522\pi\)
\(510\) 0 0
\(511\) −21.1924 −0.937497
\(512\) 0 0
\(513\) −77.4449 −3.41928
\(514\) 0 0
\(515\) 1.64896 0.0726617
\(516\) 0 0
\(517\) 2.16670 0.0952915
\(518\) 0 0
\(519\) 2.05077 0.0900186
\(520\) 0 0
\(521\) −29.6999 −1.30118 −0.650589 0.759430i \(-0.725478\pi\)
−0.650589 + 0.759430i \(0.725478\pi\)
\(522\) 0 0
\(523\) 39.9686 1.74771 0.873853 0.486191i \(-0.161614\pi\)
0.873853 + 0.486191i \(0.161614\pi\)
\(524\) 0 0
\(525\) 45.8928 2.00293
\(526\) 0 0
\(527\) −26.8365 −1.16902
\(528\) 0 0
\(529\) 4.32829 0.188186
\(530\) 0 0
\(531\) 18.2502 0.791990
\(532\) 0 0
\(533\) −4.42665 −0.191739
\(534\) 0 0
\(535\) −5.72738 −0.247616
\(536\) 0 0
\(537\) −1.31385 −0.0566968
\(538\) 0 0
\(539\) −5.02386 −0.216393
\(540\) 0 0
\(541\) 13.3928 0.575801 0.287901 0.957660i \(-0.407043\pi\)
0.287901 + 0.957660i \(0.407043\pi\)
\(542\) 0 0
\(543\) 48.0269 2.06103
\(544\) 0 0
\(545\) 3.92208 0.168003
\(546\) 0 0
\(547\) 28.7590 1.22965 0.614824 0.788665i \(-0.289227\pi\)
0.614824 + 0.788665i \(0.289227\pi\)
\(548\) 0 0
\(549\) 30.9117 1.31928
\(550\) 0 0
\(551\) 22.0484 0.939294
\(552\) 0 0
\(553\) 45.9167 1.95258
\(554\) 0 0
\(555\) −12.4417 −0.528120
\(556\) 0 0
\(557\) −19.6279 −0.831659 −0.415830 0.909442i \(-0.636509\pi\)
−0.415830 + 0.909442i \(0.636509\pi\)
\(558\) 0 0
\(559\) −2.15200 −0.0910198
\(560\) 0 0
\(561\) −22.5114 −0.950430
\(562\) 0 0
\(563\) −29.6491 −1.24956 −0.624780 0.780801i \(-0.714812\pi\)
−0.624780 + 0.780801i \(0.714812\pi\)
\(564\) 0 0
\(565\) 2.84848 0.119836
\(566\) 0 0
\(567\) 47.3069 1.98670
\(568\) 0 0
\(569\) 44.2613 1.85553 0.927765 0.373164i \(-0.121727\pi\)
0.927765 + 0.373164i \(0.121727\pi\)
\(570\) 0 0
\(571\) 28.6496 1.19895 0.599475 0.800393i \(-0.295376\pi\)
0.599475 + 0.800393i \(0.295376\pi\)
\(572\) 0 0
\(573\) 19.2841 0.805607
\(574\) 0 0
\(575\) −25.2710 −1.05387
\(576\) 0 0
\(577\) 5.20852 0.216833 0.108417 0.994106i \(-0.465422\pi\)
0.108417 + 0.994106i \(0.465422\pi\)
\(578\) 0 0
\(579\) 2.84454 0.118215
\(580\) 0 0
\(581\) −29.7216 −1.23306
\(582\) 0 0
\(583\) −13.9082 −0.576020
\(584\) 0 0
\(585\) 1.49533 0.0618242
\(586\) 0 0
\(587\) 6.64376 0.274217 0.137109 0.990556i \(-0.456219\pi\)
0.137109 + 0.990556i \(0.456219\pi\)
\(588\) 0 0
\(589\) 54.4809 2.24485
\(590\) 0 0
\(591\) −85.1671 −3.50331
\(592\) 0 0
\(593\) −0.225118 −0.00924447 −0.00462223 0.999989i \(-0.501471\pi\)
−0.00462223 + 0.999989i \(0.501471\pi\)
\(594\) 0 0
\(595\) 4.15386 0.170291
\(596\) 0 0
\(597\) 51.4577 2.10603
\(598\) 0 0
\(599\) −17.4269 −0.712045 −0.356023 0.934477i \(-0.615867\pi\)
−0.356023 + 0.934477i \(0.615867\pi\)
\(600\) 0 0
\(601\) −34.2190 −1.39582 −0.697910 0.716185i \(-0.745887\pi\)
−0.697910 + 0.716185i \(0.745887\pi\)
\(602\) 0 0
\(603\) −75.3398 −3.06807
\(604\) 0 0
\(605\) 2.56825 0.104414
\(606\) 0 0
\(607\) −29.8314 −1.21082 −0.605410 0.795914i \(-0.706991\pi\)
−0.605410 + 0.795914i \(0.706991\pi\)
\(608\) 0 0
\(609\) −30.8631 −1.25063
\(610\) 0 0
\(611\) 0.550265 0.0222613
\(612\) 0 0
\(613\) 39.6737 1.60241 0.801203 0.598393i \(-0.204194\pi\)
0.801203 + 0.598393i \(0.204194\pi\)
\(614\) 0 0
\(615\) 10.1902 0.410907
\(616\) 0 0
\(617\) −25.4005 −1.02259 −0.511294 0.859406i \(-0.670834\pi\)
−0.511294 + 0.859406i \(0.670834\pi\)
\(618\) 0 0
\(619\) 7.43419 0.298805 0.149403 0.988776i \(-0.452265\pi\)
0.149403 + 0.988776i \(0.452265\pi\)
\(620\) 0 0
\(621\) −59.6942 −2.39544
\(622\) 0 0
\(623\) 27.7712 1.11263
\(624\) 0 0
\(625\) 22.5390 0.901560
\(626\) 0 0
\(627\) 45.7004 1.82510
\(628\) 0 0
\(629\) 32.8134 1.30836
\(630\) 0 0
\(631\) 32.0345 1.27527 0.637636 0.770338i \(-0.279912\pi\)
0.637636 + 0.770338i \(0.279912\pi\)
\(632\) 0 0
\(633\) −74.6051 −2.96529
\(634\) 0 0
\(635\) −0.186751 −0.00741097
\(636\) 0 0
\(637\) −1.27588 −0.0505521
\(638\) 0 0
\(639\) 38.3866 1.51855
\(640\) 0 0
\(641\) 17.2147 0.679938 0.339969 0.940437i \(-0.389583\pi\)
0.339969 + 0.940437i \(0.389583\pi\)
\(642\) 0 0
\(643\) −14.1099 −0.556439 −0.278219 0.960518i \(-0.589744\pi\)
−0.278219 + 0.960518i \(0.589744\pi\)
\(644\) 0 0
\(645\) 4.95391 0.195060
\(646\) 0 0
\(647\) 16.4499 0.646714 0.323357 0.946277i \(-0.395189\pi\)
0.323357 + 0.946277i \(0.395189\pi\)
\(648\) 0 0
\(649\) −5.92689 −0.232651
\(650\) 0 0
\(651\) −76.2616 −2.98893
\(652\) 0 0
\(653\) −27.6510 −1.08207 −0.541035 0.841000i \(-0.681967\pi\)
−0.541035 + 0.841000i \(0.681967\pi\)
\(654\) 0 0
\(655\) 2.23188 0.0872068
\(656\) 0 0
\(657\) −46.3173 −1.80701
\(658\) 0 0
\(659\) 7.95658 0.309945 0.154972 0.987919i \(-0.450471\pi\)
0.154972 + 0.987919i \(0.450471\pi\)
\(660\) 0 0
\(661\) −45.3923 −1.76555 −0.882777 0.469792i \(-0.844329\pi\)
−0.882777 + 0.469792i \(0.844329\pi\)
\(662\) 0 0
\(663\) −5.71707 −0.222033
\(664\) 0 0
\(665\) −8.43275 −0.327008
\(666\) 0 0
\(667\) 16.9948 0.658041
\(668\) 0 0
\(669\) −46.9258 −1.81426
\(670\) 0 0
\(671\) −10.0388 −0.387545
\(672\) 0 0
\(673\) −8.24632 −0.317872 −0.158936 0.987289i \(-0.550806\pi\)
−0.158936 + 0.987289i \(0.550806\pi\)
\(674\) 0 0
\(675\) 55.2003 2.12466
\(676\) 0 0
\(677\) 9.09676 0.349617 0.174808 0.984602i \(-0.444069\pi\)
0.174808 + 0.984602i \(0.444069\pi\)
\(678\) 0 0
\(679\) −46.2132 −1.77350
\(680\) 0 0
\(681\) −60.3475 −2.31252
\(682\) 0 0
\(683\) −26.9031 −1.02942 −0.514710 0.857364i \(-0.672100\pi\)
−0.514710 + 0.857364i \(0.672100\pi\)
\(684\) 0 0
\(685\) −4.76167 −0.181934
\(686\) 0 0
\(687\) 15.7842 0.602205
\(688\) 0 0
\(689\) −3.53219 −0.134566
\(690\) 0 0
\(691\) −13.1874 −0.501673 −0.250836 0.968029i \(-0.580706\pi\)
−0.250836 + 0.968029i \(0.580706\pi\)
\(692\) 0 0
\(693\) −44.1282 −1.67629
\(694\) 0 0
\(695\) 3.94730 0.149730
\(696\) 0 0
\(697\) −26.8753 −1.01797
\(698\) 0 0
\(699\) −87.5658 −3.31204
\(700\) 0 0
\(701\) −6.53505 −0.246825 −0.123413 0.992355i \(-0.539384\pi\)
−0.123413 + 0.992355i \(0.539384\pi\)
\(702\) 0 0
\(703\) −66.6145 −2.51241
\(704\) 0 0
\(705\) −1.26671 −0.0477071
\(706\) 0 0
\(707\) −11.5271 −0.433523
\(708\) 0 0
\(709\) 43.6864 1.64068 0.820339 0.571877i \(-0.193785\pi\)
0.820339 + 0.571877i \(0.193785\pi\)
\(710\) 0 0
\(711\) 100.354 3.76356
\(712\) 0 0
\(713\) 41.9936 1.57267
\(714\) 0 0
\(715\) −0.485620 −0.0181611
\(716\) 0 0
\(717\) −24.4734 −0.913977
\(718\) 0 0
\(719\) 5.47228 0.204082 0.102041 0.994780i \(-0.467463\pi\)
0.102041 + 0.994780i \(0.467463\pi\)
\(720\) 0 0
\(721\) −12.3584 −0.460249
\(722\) 0 0
\(723\) −49.3233 −1.83435
\(724\) 0 0
\(725\) −15.7154 −0.583655
\(726\) 0 0
\(727\) 4.64274 0.172190 0.0860948 0.996287i \(-0.472561\pi\)
0.0860948 + 0.996287i \(0.472561\pi\)
\(728\) 0 0
\(729\) −3.14464 −0.116468
\(730\) 0 0
\(731\) −13.0653 −0.483238
\(732\) 0 0
\(733\) −14.4222 −0.532696 −0.266348 0.963877i \(-0.585817\pi\)
−0.266348 + 0.963877i \(0.585817\pi\)
\(734\) 0 0
\(735\) 2.93708 0.108336
\(736\) 0 0
\(737\) 24.4672 0.901261
\(738\) 0 0
\(739\) 36.1648 1.33034 0.665172 0.746690i \(-0.268358\pi\)
0.665172 + 0.746690i \(0.268358\pi\)
\(740\) 0 0
\(741\) 11.6062 0.426366
\(742\) 0 0
\(743\) −45.9267 −1.68489 −0.842444 0.538784i \(-0.818884\pi\)
−0.842444 + 0.538784i \(0.818884\pi\)
\(744\) 0 0
\(745\) −4.05065 −0.148404
\(746\) 0 0
\(747\) −64.9583 −2.37670
\(748\) 0 0
\(749\) 42.9247 1.56844
\(750\) 0 0
\(751\) 30.5643 1.11531 0.557653 0.830074i \(-0.311702\pi\)
0.557653 + 0.830074i \(0.311702\pi\)
\(752\) 0 0
\(753\) −82.0229 −2.98908
\(754\) 0 0
\(755\) 3.19154 0.116152
\(756\) 0 0
\(757\) 1.46143 0.0531164 0.0265582 0.999647i \(-0.491545\pi\)
0.0265582 + 0.999647i \(0.491545\pi\)
\(758\) 0 0
\(759\) 35.2256 1.27861
\(760\) 0 0
\(761\) −26.1612 −0.948343 −0.474171 0.880433i \(-0.657252\pi\)
−0.474171 + 0.880433i \(0.657252\pi\)
\(762\) 0 0
\(763\) −29.3946 −1.06416
\(764\) 0 0
\(765\) 9.07851 0.328234
\(766\) 0 0
\(767\) −1.50522 −0.0543502
\(768\) 0 0
\(769\) −34.7026 −1.25141 −0.625704 0.780061i \(-0.715188\pi\)
−0.625704 + 0.780061i \(0.715188\pi\)
\(770\) 0 0
\(771\) 65.3332 2.35292
\(772\) 0 0
\(773\) −35.1723 −1.26506 −0.632530 0.774536i \(-0.717983\pi\)
−0.632530 + 0.774536i \(0.717983\pi\)
\(774\) 0 0
\(775\) −38.8322 −1.39489
\(776\) 0 0
\(777\) 93.2462 3.34519
\(778\) 0 0
\(779\) 54.5596 1.95480
\(780\) 0 0
\(781\) −12.4664 −0.446081
\(782\) 0 0
\(783\) −37.1223 −1.32664
\(784\) 0 0
\(785\) −7.31991 −0.261259
\(786\) 0 0
\(787\) 1.57031 0.0559756 0.0279878 0.999608i \(-0.491090\pi\)
0.0279878 + 0.999608i \(0.491090\pi\)
\(788\) 0 0
\(789\) −12.4593 −0.443563
\(790\) 0 0
\(791\) −21.3484 −0.759060
\(792\) 0 0
\(793\) −2.54950 −0.0905354
\(794\) 0 0
\(795\) 8.13111 0.288381
\(796\) 0 0
\(797\) 11.3579 0.402318 0.201159 0.979559i \(-0.435529\pi\)
0.201159 + 0.979559i \(0.435529\pi\)
\(798\) 0 0
\(799\) 3.34079 0.118189
\(800\) 0 0
\(801\) 60.6957 2.14458
\(802\) 0 0
\(803\) 15.0419 0.530818
\(804\) 0 0
\(805\) −6.49992 −0.229092
\(806\) 0 0
\(807\) −22.3174 −0.785608
\(808\) 0 0
\(809\) 35.4551 1.24653 0.623267 0.782009i \(-0.285805\pi\)
0.623267 + 0.782009i \(0.285805\pi\)
\(810\) 0 0
\(811\) −1.60023 −0.0561917 −0.0280959 0.999605i \(-0.508944\pi\)
−0.0280959 + 0.999605i \(0.508944\pi\)
\(812\) 0 0
\(813\) 57.5868 2.01966
\(814\) 0 0
\(815\) 3.75991 0.131704
\(816\) 0 0
\(817\) 26.5239 0.927954
\(818\) 0 0
\(819\) −11.2070 −0.391603
\(820\) 0 0
\(821\) 9.29436 0.324375 0.162188 0.986760i \(-0.448145\pi\)
0.162188 + 0.986760i \(0.448145\pi\)
\(822\) 0 0
\(823\) 17.5978 0.613421 0.306711 0.951803i \(-0.400772\pi\)
0.306711 + 0.951803i \(0.400772\pi\)
\(824\) 0 0
\(825\) −32.5737 −1.13407
\(826\) 0 0
\(827\) −49.0737 −1.70646 −0.853230 0.521535i \(-0.825359\pi\)
−0.853230 + 0.521535i \(0.825359\pi\)
\(828\) 0 0
\(829\) 7.75210 0.269241 0.134621 0.990897i \(-0.457018\pi\)
0.134621 + 0.990897i \(0.457018\pi\)
\(830\) 0 0
\(831\) 32.7823 1.13720
\(832\) 0 0
\(833\) −7.74618 −0.268389
\(834\) 0 0
\(835\) 6.81166 0.235727
\(836\) 0 0
\(837\) −91.7281 −3.17059
\(838\) 0 0
\(839\) −39.9387 −1.37884 −0.689419 0.724363i \(-0.742134\pi\)
−0.689419 + 0.724363i \(0.742134\pi\)
\(840\) 0 0
\(841\) −18.4314 −0.635564
\(842\) 0 0
\(843\) −23.6555 −0.814739
\(844\) 0 0
\(845\) 5.17170 0.177912
\(846\) 0 0
\(847\) −19.2481 −0.661374
\(848\) 0 0
\(849\) −40.1130 −1.37667
\(850\) 0 0
\(851\) −51.3461 −1.76012
\(852\) 0 0
\(853\) 12.4661 0.426832 0.213416 0.976961i \(-0.431541\pi\)
0.213416 + 0.976961i \(0.431541\pi\)
\(854\) 0 0
\(855\) −18.4303 −0.630303
\(856\) 0 0
\(857\) 15.2044 0.519373 0.259686 0.965693i \(-0.416381\pi\)
0.259686 + 0.965693i \(0.416381\pi\)
\(858\) 0 0
\(859\) 41.5280 1.41692 0.708459 0.705752i \(-0.249391\pi\)
0.708459 + 0.705752i \(0.249391\pi\)
\(860\) 0 0
\(861\) −76.3718 −2.60274
\(862\) 0 0
\(863\) −8.60696 −0.292984 −0.146492 0.989212i \(-0.546798\pi\)
−0.146492 + 0.989212i \(0.546798\pi\)
\(864\) 0 0
\(865\) 0.268589 0.00913231
\(866\) 0 0
\(867\) 18.1593 0.616721
\(868\) 0 0
\(869\) −32.5907 −1.10556
\(870\) 0 0
\(871\) 6.21378 0.210546
\(872\) 0 0
\(873\) −101.002 −3.41839
\(874\) 0 0
\(875\) 12.2275 0.413364
\(876\) 0 0
\(877\) −1.78883 −0.0604044 −0.0302022 0.999544i \(-0.509615\pi\)
−0.0302022 + 0.999544i \(0.509615\pi\)
\(878\) 0 0
\(879\) −6.17526 −0.208286
\(880\) 0 0
\(881\) −5.30452 −0.178714 −0.0893570 0.996000i \(-0.528481\pi\)
−0.0893570 + 0.996000i \(0.528481\pi\)
\(882\) 0 0
\(883\) −4.44640 −0.149633 −0.0748167 0.997197i \(-0.523837\pi\)
−0.0748167 + 0.997197i \(0.523837\pi\)
\(884\) 0 0
\(885\) 3.46501 0.116475
\(886\) 0 0
\(887\) −32.9646 −1.10684 −0.553421 0.832902i \(-0.686678\pi\)
−0.553421 + 0.832902i \(0.686678\pi\)
\(888\) 0 0
\(889\) 1.39963 0.0469421
\(890\) 0 0
\(891\) −33.5774 −1.12489
\(892\) 0 0
\(893\) −6.78215 −0.226956
\(894\) 0 0
\(895\) −0.172075 −0.00575183
\(896\) 0 0
\(897\) 8.94603 0.298699
\(898\) 0 0
\(899\) 26.1148 0.870977
\(900\) 0 0
\(901\) −21.4448 −0.714430
\(902\) 0 0
\(903\) −37.1278 −1.23554
\(904\) 0 0
\(905\) 6.29009 0.209090
\(906\) 0 0
\(907\) −39.7603 −1.32022 −0.660110 0.751169i \(-0.729490\pi\)
−0.660110 + 0.751169i \(0.729490\pi\)
\(908\) 0 0
\(909\) −25.1933 −0.835608
\(910\) 0 0
\(911\) 23.2653 0.770813 0.385407 0.922747i \(-0.374061\pi\)
0.385407 + 0.922747i \(0.374061\pi\)
\(912\) 0 0
\(913\) 21.0957 0.698167
\(914\) 0 0
\(915\) 5.86896 0.194022
\(916\) 0 0
\(917\) −16.7272 −0.552380
\(918\) 0 0
\(919\) 11.7907 0.388939 0.194469 0.980909i \(-0.437702\pi\)
0.194469 + 0.980909i \(0.437702\pi\)
\(920\) 0 0
\(921\) 23.4958 0.774213
\(922\) 0 0
\(923\) −3.16600 −0.104210
\(924\) 0 0
\(925\) 47.4807 1.56116
\(926\) 0 0
\(927\) −27.0099 −0.887123
\(928\) 0 0
\(929\) 20.8905 0.685397 0.342698 0.939446i \(-0.388659\pi\)
0.342698 + 0.939446i \(0.388659\pi\)
\(930\) 0 0
\(931\) 15.7255 0.515383
\(932\) 0 0
\(933\) −67.1380 −2.19800
\(934\) 0 0
\(935\) −2.94832 −0.0964203
\(936\) 0 0
\(937\) 32.9342 1.07591 0.537957 0.842972i \(-0.319196\pi\)
0.537957 + 0.842972i \(0.319196\pi\)
\(938\) 0 0
\(939\) 45.4262 1.48243
\(940\) 0 0
\(941\) 45.1275 1.47111 0.735556 0.677463i \(-0.236921\pi\)
0.735556 + 0.677463i \(0.236921\pi\)
\(942\) 0 0
\(943\) 42.0542 1.36947
\(944\) 0 0
\(945\) 14.1980 0.461861
\(946\) 0 0
\(947\) −30.4571 −0.989721 −0.494861 0.868972i \(-0.664781\pi\)
−0.494861 + 0.868972i \(0.664781\pi\)
\(948\) 0 0
\(949\) 3.82010 0.124006
\(950\) 0 0
\(951\) 25.9622 0.841883
\(952\) 0 0
\(953\) 8.75634 0.283646 0.141823 0.989892i \(-0.454704\pi\)
0.141823 + 0.989892i \(0.454704\pi\)
\(954\) 0 0
\(955\) 2.52565 0.0817281
\(956\) 0 0
\(957\) 21.9059 0.708118
\(958\) 0 0
\(959\) 35.6871 1.15240
\(960\) 0 0
\(961\) 33.5288 1.08157
\(962\) 0 0
\(963\) 93.8146 3.02313
\(964\) 0 0
\(965\) 0.372550 0.0119928
\(966\) 0 0
\(967\) 47.9641 1.54242 0.771210 0.636581i \(-0.219652\pi\)
0.771210 + 0.636581i \(0.219652\pi\)
\(968\) 0 0
\(969\) 70.4644 2.26364
\(970\) 0 0
\(971\) −56.6897 −1.81926 −0.909629 0.415422i \(-0.863634\pi\)
−0.909629 + 0.415422i \(0.863634\pi\)
\(972\) 0 0
\(973\) −29.5836 −0.948408
\(974\) 0 0
\(975\) −8.27255 −0.264934
\(976\) 0 0
\(977\) −9.49520 −0.303778 −0.151889 0.988398i \(-0.548536\pi\)
−0.151889 + 0.988398i \(0.548536\pi\)
\(978\) 0 0
\(979\) −19.7114 −0.629979
\(980\) 0 0
\(981\) −64.2437 −2.05114
\(982\) 0 0
\(983\) −2.08823 −0.0666042 −0.0333021 0.999445i \(-0.510602\pi\)
−0.0333021 + 0.999445i \(0.510602\pi\)
\(984\) 0 0
\(985\) −11.1543 −0.355407
\(986\) 0 0
\(987\) 9.49357 0.302184
\(988\) 0 0
\(989\) 20.4445 0.650097
\(990\) 0 0
\(991\) 1.54116 0.0489567 0.0244784 0.999700i \(-0.492208\pi\)
0.0244784 + 0.999700i \(0.492208\pi\)
\(992\) 0 0
\(993\) −98.9258 −3.13932
\(994\) 0 0
\(995\) 6.73943 0.213654
\(996\) 0 0
\(997\) 37.2343 1.17922 0.589611 0.807687i \(-0.299281\pi\)
0.589611 + 0.807687i \(0.299281\pi\)
\(998\) 0 0
\(999\) 112.157 3.54850
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 6016.2.a.o.1.1 yes 13
4.3 odd 2 6016.2.a.m.1.13 13
8.3 odd 2 6016.2.a.p.1.1 yes 13
8.5 even 2 6016.2.a.n.1.13 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.13 13 4.3 odd 2
6016.2.a.n.1.13 yes 13 8.5 even 2
6016.2.a.o.1.1 yes 13 1.1 even 1 trivial
6016.2.a.p.1.1 yes 13 8.3 odd 2