Properties

Label 3808.2.a.j.1.4
Level $3808$
Weight $2$
Character 3808.1
Self dual yes
Analytic conductor $30.407$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3808,2,Mod(1,3808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3808, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3808.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3808.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: \(30.4070330897\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4022000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 14x^{2} - 8x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.350492\) of defining polynomial
Character \(\chi\) \(=\) 3808.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.692013 q^{3} +2.18514 q^{5} -1.00000 q^{7} -2.52112 q^{9} -4.76273 q^{11} +3.00528 q^{13} +1.51215 q^{15} +1.00000 q^{17} -3.29587 q^{19} -0.692013 q^{21} -1.05647 q^{23} -0.225153 q^{25} -3.82068 q^{27} +3.24782 q^{29} +1.23052 q^{31} -3.29587 q^{33} -2.18514 q^{35} +1.17626 q^{37} +2.07969 q^{39} -1.21534 q^{41} -0.149665 q^{43} -5.50900 q^{45} +0.529996 q^{47} +1.00000 q^{49} +0.692013 q^{51} -6.32395 q^{53} -10.4072 q^{55} -2.28079 q^{57} -2.93372 q^{59} -12.4131 q^{61} +2.52112 q^{63} +6.56696 q^{65} -7.61983 q^{67} -0.731091 q^{69} -4.23146 q^{71} +0.412507 q^{73} -0.155809 q^{75} +4.76273 q^{77} -14.5585 q^{79} +4.91939 q^{81} +0.652935 q^{83} +2.18514 q^{85} +2.24753 q^{87} -13.9425 q^{89} -3.00528 q^{91} +0.851538 q^{93} -7.20195 q^{95} +8.93914 q^{97} +12.0074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) 0.692013 0.399534 0.199767 0.979843i \(-0.435982\pi\)
0.199767 + 0.979843i \(0.435982\pi\)
\(4\) 0 0
\(5\) 2.18514 0.977225 0.488613 0.872501i \(-0.337503\pi\)
0.488613 + 0.872501i \(0.337503\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.52112 −0.840373
\(10\) 0 0
\(11\) −4.76273 −1.43602 −0.718009 0.696034i \(-0.754946\pi\)
−0.718009 + 0.696034i \(0.754946\pi\)
\(12\) 0 0
\(13\) 3.00528 0.833514 0.416757 0.909018i \(-0.363167\pi\)
0.416757 + 0.909018i \(0.363167\pi\)
\(14\) 0 0
\(15\) 1.51215 0.390435
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −3.29587 −0.756125 −0.378062 0.925780i \(-0.623410\pi\)
−0.378062 + 0.925780i \(0.623410\pi\)
\(20\) 0 0
\(21\) −0.692013 −0.151010
\(22\) 0 0
\(23\) −1.05647 −0.220289 −0.110145 0.993916i \(-0.535131\pi\)
−0.110145 + 0.993916i \(0.535131\pi\)
\(24\) 0 0
\(25\) −0.225153 −0.0450306
\(26\) 0 0
\(27\) −3.82068 −0.735291
\(28\) 0 0
\(29\) 3.24782 0.603105 0.301553 0.953450i \(-0.402495\pi\)
0.301553 + 0.953450i \(0.402495\pi\)
\(30\) 0 0
\(31\) 1.23052 0.221008 0.110504 0.993876i \(-0.464753\pi\)
0.110504 + 0.993876i \(0.464753\pi\)
\(32\) 0 0
\(33\) −3.29587 −0.573738
\(34\) 0 0
\(35\) −2.18514 −0.369356
\(36\) 0 0
\(37\) 1.17626 0.193377 0.0966884 0.995315i \(-0.469175\pi\)
0.0966884 + 0.995315i \(0.469175\pi\)
\(38\) 0 0
\(39\) 2.07969 0.333017
\(40\) 0 0
\(41\) −1.21534 −0.189805 −0.0949024 0.995487i \(-0.530254\pi\)
−0.0949024 + 0.995487i \(0.530254\pi\)
\(42\) 0 0
\(43\) −0.149665 −0.0228238 −0.0114119 0.999935i \(-0.503633\pi\)
−0.0114119 + 0.999935i \(0.503633\pi\)
\(44\) 0 0
\(45\) −5.50900 −0.821234
\(46\) 0 0
\(47\) 0.529996 0.0773078 0.0386539 0.999253i \(-0.487693\pi\)
0.0386539 + 0.999253i \(0.487693\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.692013 0.0969012
\(52\) 0 0
\(53\) −6.32395 −0.868662 −0.434331 0.900753i \(-0.643015\pi\)
−0.434331 + 0.900753i \(0.643015\pi\)
\(54\) 0 0
\(55\) −10.4072 −1.40331
\(56\) 0 0
\(57\) −2.28079 −0.302097
\(58\) 0 0
\(59\) −2.93372 −0.381938 −0.190969 0.981596i \(-0.561163\pi\)
−0.190969 + 0.981596i \(0.561163\pi\)
\(60\) 0 0
\(61\) −12.4131 −1.58933 −0.794666 0.607048i \(-0.792354\pi\)
−0.794666 + 0.607048i \(0.792354\pi\)
\(62\) 0 0
\(63\) 2.52112 0.317631
\(64\) 0 0
\(65\) 6.56696 0.814531
\(66\) 0 0
\(67\) −7.61983 −0.930910 −0.465455 0.885072i \(-0.654109\pi\)
−0.465455 + 0.885072i \(0.654109\pi\)
\(68\) 0 0
\(69\) −0.731091 −0.0880130
\(70\) 0 0
\(71\) −4.23146 −0.502181 −0.251091 0.967964i \(-0.580789\pi\)
−0.251091 + 0.967964i \(0.580789\pi\)
\(72\) 0 0
\(73\) 0.412507 0.0482803 0.0241402 0.999709i \(-0.492315\pi\)
0.0241402 + 0.999709i \(0.492315\pi\)
\(74\) 0 0
\(75\) −0.155809 −0.0179912
\(76\) 0 0
\(77\) 4.76273 0.542764
\(78\) 0 0
\(79\) −14.5585 −1.63796 −0.818979 0.573823i \(-0.805460\pi\)
−0.818979 + 0.573823i \(0.805460\pi\)
\(80\) 0 0
\(81\) 4.91939 0.546599
\(82\) 0 0
\(83\) 0.652935 0.0716689 0.0358344 0.999358i \(-0.488591\pi\)
0.0358344 + 0.999358i \(0.488591\pi\)
\(84\) 0 0
\(85\) 2.18514 0.237012
\(86\) 0 0
\(87\) 2.24753 0.240961
\(88\) 0 0
\(89\) −13.9425 −1.47790 −0.738952 0.673758i \(-0.764679\pi\)
−0.738952 + 0.673758i \(0.764679\pi\)
\(90\) 0 0
\(91\) −3.00528 −0.315039
\(92\) 0 0
\(93\) 0.851538 0.0883004
\(94\) 0 0
\(95\) −7.20195 −0.738904
\(96\) 0 0
\(97\) 8.93914 0.907632 0.453816 0.891095i \(-0.350062\pi\)
0.453816 + 0.891095i \(0.350062\pi\)
\(98\) 0 0
\(99\) 12.0074 1.20679
\(100\) 0 0
\(101\) −6.45105 −0.641903 −0.320952 0.947096i \(-0.604003\pi\)
−0.320952 + 0.947096i \(0.604003\pi\)
\(102\) 0 0
\(103\) −3.25097 −0.320327 −0.160164 0.987090i \(-0.551202\pi\)
−0.160164 + 0.987090i \(0.551202\pi\)
\(104\) 0 0
\(105\) −1.51215 −0.147570
\(106\) 0 0
\(107\) −5.27557 −0.510009 −0.255005 0.966940i \(-0.582077\pi\)
−0.255005 + 0.966940i \(0.582077\pi\)
\(108\) 0 0
\(109\) −2.49962 −0.239420 −0.119710 0.992809i \(-0.538196\pi\)
−0.119710 + 0.992809i \(0.538196\pi\)
\(110\) 0 0
\(111\) 0.813990 0.0772606
\(112\) 0 0
\(113\) −5.18186 −0.487468 −0.243734 0.969842i \(-0.578372\pi\)
−0.243734 + 0.969842i \(0.578372\pi\)
\(114\) 0 0
\(115\) −2.30854 −0.215272
\(116\) 0 0
\(117\) −7.57666 −0.700462
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 11.6836 1.06215
\(122\) 0 0
\(123\) −0.841033 −0.0758334
\(124\) 0 0
\(125\) −11.4177 −1.02123
\(126\) 0 0
\(127\) 17.8709 1.58578 0.792892 0.609362i \(-0.208574\pi\)
0.792892 + 0.609362i \(0.208574\pi\)
\(128\) 0 0
\(129\) −0.103570 −0.00911887
\(130\) 0 0
\(131\) −17.9932 −1.57207 −0.786035 0.618181i \(-0.787870\pi\)
−0.786035 + 0.618181i \(0.787870\pi\)
\(132\) 0 0
\(133\) 3.29587 0.285788
\(134\) 0 0
\(135\) −8.34874 −0.718545
\(136\) 0 0
\(137\) 3.23571 0.276445 0.138222 0.990401i \(-0.455861\pi\)
0.138222 + 0.990401i \(0.455861\pi\)
\(138\) 0 0
\(139\) −4.44222 −0.376784 −0.188392 0.982094i \(-0.560328\pi\)
−0.188392 + 0.982094i \(0.560328\pi\)
\(140\) 0 0
\(141\) 0.366764 0.0308871
\(142\) 0 0
\(143\) −14.3133 −1.19694
\(144\) 0 0
\(145\) 7.09695 0.589370
\(146\) 0 0
\(147\) 0.692013 0.0570763
\(148\) 0 0
\(149\) 4.81102 0.394134 0.197067 0.980390i \(-0.436858\pi\)
0.197067 + 0.980390i \(0.436858\pi\)
\(150\) 0 0
\(151\) −10.6543 −0.867032 −0.433516 0.901146i \(-0.642727\pi\)
−0.433516 + 0.901146i \(0.642727\pi\)
\(152\) 0 0
\(153\) −2.52112 −0.203820
\(154\) 0 0
\(155\) 2.68887 0.215975
\(156\) 0 0
\(157\) −1.61144 −0.128607 −0.0643035 0.997930i \(-0.520483\pi\)
−0.0643035 + 0.997930i \(0.520483\pi\)
\(158\) 0 0
\(159\) −4.37626 −0.347060
\(160\) 0 0
\(161\) 1.05647 0.0832615
\(162\) 0 0
\(163\) 19.0236 1.49004 0.745022 0.667040i \(-0.232439\pi\)
0.745022 + 0.667040i \(0.232439\pi\)
\(164\) 0 0
\(165\) −7.20195 −0.560671
\(166\) 0 0
\(167\) −15.8539 −1.22681 −0.613404 0.789769i \(-0.710200\pi\)
−0.613404 + 0.789769i \(0.710200\pi\)
\(168\) 0 0
\(169\) −3.96832 −0.305255
\(170\) 0 0
\(171\) 8.30928 0.635427
\(172\) 0 0
\(173\) −6.39955 −0.486549 −0.243275 0.969957i \(-0.578222\pi\)
−0.243275 + 0.969957i \(0.578222\pi\)
\(174\) 0 0
\(175\) 0.225153 0.0170200
\(176\) 0 0
\(177\) −2.03017 −0.152597
\(178\) 0 0
\(179\) 19.7775 1.47824 0.739121 0.673572i \(-0.235241\pi\)
0.739121 + 0.673572i \(0.235241\pi\)
\(180\) 0 0
\(181\) 22.7792 1.69316 0.846581 0.532260i \(-0.178657\pi\)
0.846581 + 0.532260i \(0.178657\pi\)
\(182\) 0 0
\(183\) −8.59001 −0.634991
\(184\) 0 0
\(185\) 2.57031 0.188973
\(186\) 0 0
\(187\) −4.76273 −0.348285
\(188\) 0 0
\(189\) 3.82068 0.277914
\(190\) 0 0
\(191\) −1.40889 −0.101944 −0.0509719 0.998700i \(-0.516232\pi\)
−0.0509719 + 0.998700i \(0.516232\pi\)
\(192\) 0 0
\(193\) 3.65742 0.263267 0.131633 0.991298i \(-0.457978\pi\)
0.131633 + 0.991298i \(0.457978\pi\)
\(194\) 0 0
\(195\) 4.54442 0.325433
\(196\) 0 0
\(197\) 22.6113 1.61099 0.805495 0.592602i \(-0.201899\pi\)
0.805495 + 0.592602i \(0.201899\pi\)
\(198\) 0 0
\(199\) −13.3510 −0.946428 −0.473214 0.880948i \(-0.656906\pi\)
−0.473214 + 0.880948i \(0.656906\pi\)
\(200\) 0 0
\(201\) −5.27302 −0.371930
\(202\) 0 0
\(203\) −3.24782 −0.227952
\(204\) 0 0
\(205\) −2.65570 −0.185482
\(206\) 0 0
\(207\) 2.66349 0.185125
\(208\) 0 0
\(209\) 15.6973 1.08581
\(210\) 0 0
\(211\) 11.5748 0.796840 0.398420 0.917203i \(-0.369559\pi\)
0.398420 + 0.917203i \(0.369559\pi\)
\(212\) 0 0
\(213\) −2.92822 −0.200638
\(214\) 0 0
\(215\) −0.327040 −0.0223040
\(216\) 0 0
\(217\) −1.23052 −0.0835334
\(218\) 0 0
\(219\) 0.285460 0.0192896
\(220\) 0 0
\(221\) 3.00528 0.202157
\(222\) 0 0
\(223\) −13.2790 −0.889230 −0.444615 0.895722i \(-0.646659\pi\)
−0.444615 + 0.895722i \(0.646659\pi\)
\(224\) 0 0
\(225\) 0.567637 0.0378425
\(226\) 0 0
\(227\) 23.5760 1.56480 0.782398 0.622778i \(-0.213996\pi\)
0.782398 + 0.622778i \(0.213996\pi\)
\(228\) 0 0
\(229\) −17.1700 −1.13463 −0.567314 0.823501i \(-0.692017\pi\)
−0.567314 + 0.823501i \(0.692017\pi\)
\(230\) 0 0
\(231\) 3.29587 0.216852
\(232\) 0 0
\(233\) 21.5395 1.41110 0.705551 0.708659i \(-0.250700\pi\)
0.705551 + 0.708659i \(0.250700\pi\)
\(234\) 0 0
\(235\) 1.15812 0.0755472
\(236\) 0 0
\(237\) −10.0747 −0.654420
\(238\) 0 0
\(239\) −13.9134 −0.899985 −0.449992 0.893032i \(-0.648573\pi\)
−0.449992 + 0.893032i \(0.648573\pi\)
\(240\) 0 0
\(241\) 7.28389 0.469197 0.234598 0.972092i \(-0.424622\pi\)
0.234598 + 0.972092i \(0.424622\pi\)
\(242\) 0 0
\(243\) 14.8663 0.953676
\(244\) 0 0
\(245\) 2.18514 0.139604
\(246\) 0 0
\(247\) −9.90500 −0.630240
\(248\) 0 0
\(249\) 0.451839 0.0286341
\(250\) 0 0
\(251\) −4.83826 −0.305388 −0.152694 0.988273i \(-0.548795\pi\)
−0.152694 + 0.988273i \(0.548795\pi\)
\(252\) 0 0
\(253\) 5.03168 0.316339
\(254\) 0 0
\(255\) 1.51215 0.0946943
\(256\) 0 0
\(257\) 4.24023 0.264498 0.132249 0.991217i \(-0.457780\pi\)
0.132249 + 0.991217i \(0.457780\pi\)
\(258\) 0 0
\(259\) −1.17626 −0.0730895
\(260\) 0 0
\(261\) −8.18814 −0.506833
\(262\) 0 0
\(263\) 3.93053 0.242367 0.121183 0.992630i \(-0.461331\pi\)
0.121183 + 0.992630i \(0.461331\pi\)
\(264\) 0 0
\(265\) −13.8187 −0.848878
\(266\) 0 0
\(267\) −9.64840 −0.590473
\(268\) 0 0
\(269\) 30.6455 1.86849 0.934245 0.356633i \(-0.116075\pi\)
0.934245 + 0.356633i \(0.116075\pi\)
\(270\) 0 0
\(271\) −0.342060 −0.0207787 −0.0103893 0.999946i \(-0.503307\pi\)
−0.0103893 + 0.999946i \(0.503307\pi\)
\(272\) 0 0
\(273\) −2.07969 −0.125869
\(274\) 0 0
\(275\) 1.07234 0.0646647
\(276\) 0 0
\(277\) 17.5416 1.05398 0.526988 0.849873i \(-0.323321\pi\)
0.526988 + 0.849873i \(0.323321\pi\)
\(278\) 0 0
\(279\) −3.10229 −0.185730
\(280\) 0 0
\(281\) −29.5940 −1.76543 −0.882715 0.469909i \(-0.844287\pi\)
−0.882715 + 0.469909i \(0.844287\pi\)
\(282\) 0 0
\(283\) −19.9864 −1.18807 −0.594035 0.804439i \(-0.702466\pi\)
−0.594035 + 0.804439i \(0.702466\pi\)
\(284\) 0 0
\(285\) −4.98384 −0.295217
\(286\) 0 0
\(287\) 1.21534 0.0717394
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.18600 0.362630
\(292\) 0 0
\(293\) 2.83826 0.165813 0.0829065 0.996557i \(-0.473580\pi\)
0.0829065 + 0.996557i \(0.473580\pi\)
\(294\) 0 0
\(295\) −6.41060 −0.373239
\(296\) 0 0
\(297\) 18.1969 1.05589
\(298\) 0 0
\(299\) −3.17499 −0.183614
\(300\) 0 0
\(301\) 0.149665 0.00862657
\(302\) 0 0
\(303\) −4.46421 −0.256462
\(304\) 0 0
\(305\) −27.1243 −1.55313
\(306\) 0 0
\(307\) 10.9778 0.626536 0.313268 0.949665i \(-0.398576\pi\)
0.313268 + 0.949665i \(0.398576\pi\)
\(308\) 0 0
\(309\) −2.24971 −0.127982
\(310\) 0 0
\(311\) 19.8668 1.12655 0.563273 0.826271i \(-0.309542\pi\)
0.563273 + 0.826271i \(0.309542\pi\)
\(312\) 0 0
\(313\) −16.9055 −0.955553 −0.477776 0.878481i \(-0.658557\pi\)
−0.477776 + 0.878481i \(0.658557\pi\)
\(314\) 0 0
\(315\) 5.50900 0.310397
\(316\) 0 0
\(317\) 6.80503 0.382209 0.191104 0.981570i \(-0.438793\pi\)
0.191104 + 0.981570i \(0.438793\pi\)
\(318\) 0 0
\(319\) −15.4685 −0.866070
\(320\) 0 0
\(321\) −3.65077 −0.203766
\(322\) 0 0
\(323\) −3.29587 −0.183387
\(324\) 0 0
\(325\) −0.676647 −0.0375336
\(326\) 0 0
\(327\) −1.72977 −0.0956564
\(328\) 0 0
\(329\) −0.529996 −0.0292196
\(330\) 0 0
\(331\) 23.6993 1.30263 0.651315 0.758808i \(-0.274218\pi\)
0.651315 + 0.758808i \(0.274218\pi\)
\(332\) 0 0
\(333\) −2.96550 −0.162509
\(334\) 0 0
\(335\) −16.6504 −0.909709
\(336\) 0 0
\(337\) 1.09300 0.0595395 0.0297697 0.999557i \(-0.490523\pi\)
0.0297697 + 0.999557i \(0.490523\pi\)
\(338\) 0 0
\(339\) −3.58591 −0.194760
\(340\) 0 0
\(341\) −5.86065 −0.317372
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.59754 −0.0860086
\(346\) 0 0
\(347\) −21.6284 −1.16107 −0.580537 0.814234i \(-0.697157\pi\)
−0.580537 + 0.814234i \(0.697157\pi\)
\(348\) 0 0
\(349\) 16.8211 0.900412 0.450206 0.892925i \(-0.351350\pi\)
0.450206 + 0.892925i \(0.351350\pi\)
\(350\) 0 0
\(351\) −11.4822 −0.612875
\(352\) 0 0
\(353\) 11.2360 0.598030 0.299015 0.954248i \(-0.403342\pi\)
0.299015 + 0.954248i \(0.403342\pi\)
\(354\) 0 0
\(355\) −9.24633 −0.490744
\(356\) 0 0
\(357\) −0.692013 −0.0366252
\(358\) 0 0
\(359\) −16.5430 −0.873106 −0.436553 0.899679i \(-0.643801\pi\)
−0.436553 + 0.899679i \(0.643801\pi\)
\(360\) 0 0
\(361\) −8.13723 −0.428275
\(362\) 0 0
\(363\) 8.08521 0.424363
\(364\) 0 0
\(365\) 0.901387 0.0471808
\(366\) 0 0
\(367\) −23.9945 −1.25250 −0.626252 0.779621i \(-0.715412\pi\)
−0.626252 + 0.779621i \(0.715412\pi\)
\(368\) 0 0
\(369\) 3.06402 0.159507
\(370\) 0 0
\(371\) 6.32395 0.328323
\(372\) 0 0
\(373\) −5.52152 −0.285893 −0.142947 0.989730i \(-0.545658\pi\)
−0.142947 + 0.989730i \(0.545658\pi\)
\(374\) 0 0
\(375\) −7.90120 −0.408016
\(376\) 0 0
\(377\) 9.76060 0.502696
\(378\) 0 0
\(379\) −14.6513 −0.752585 −0.376293 0.926501i \(-0.622801\pi\)
−0.376293 + 0.926501i \(0.622801\pi\)
\(380\) 0 0
\(381\) 12.3669 0.633574
\(382\) 0 0
\(383\) 6.08107 0.310728 0.155364 0.987857i \(-0.450345\pi\)
0.155364 + 0.987857i \(0.450345\pi\)
\(384\) 0 0
\(385\) 10.4072 0.530402
\(386\) 0 0
\(387\) 0.377324 0.0191805
\(388\) 0 0
\(389\) −11.7879 −0.597671 −0.298835 0.954305i \(-0.596598\pi\)
−0.298835 + 0.954305i \(0.596598\pi\)
\(390\) 0 0
\(391\) −1.05647 −0.0534280
\(392\) 0 0
\(393\) −12.4515 −0.628095
\(394\) 0 0
\(395\) −31.8124 −1.60065
\(396\) 0 0
\(397\) −22.8898 −1.14881 −0.574404 0.818572i \(-0.694766\pi\)
−0.574404 + 0.818572i \(0.694766\pi\)
\(398\) 0 0
\(399\) 2.28079 0.114182
\(400\) 0 0
\(401\) −36.9705 −1.84622 −0.923109 0.384539i \(-0.874360\pi\)
−0.923109 + 0.384539i \(0.874360\pi\)
\(402\) 0 0
\(403\) 3.69806 0.184214
\(404\) 0 0
\(405\) 10.7496 0.534150
\(406\) 0 0
\(407\) −5.60223 −0.277692
\(408\) 0 0
\(409\) −7.15953 −0.354016 −0.177008 0.984209i \(-0.556642\pi\)
−0.177008 + 0.984209i \(0.556642\pi\)
\(410\) 0 0
\(411\) 2.23915 0.110449
\(412\) 0 0
\(413\) 2.93372 0.144359
\(414\) 0 0
\(415\) 1.42676 0.0700367
\(416\) 0 0
\(417\) −3.07407 −0.150538
\(418\) 0 0
\(419\) −29.6690 −1.44943 −0.724713 0.689051i \(-0.758028\pi\)
−0.724713 + 0.689051i \(0.758028\pi\)
\(420\) 0 0
\(421\) 24.4657 1.19239 0.596193 0.802841i \(-0.296679\pi\)
0.596193 + 0.802841i \(0.296679\pi\)
\(422\) 0 0
\(423\) −1.33618 −0.0649674
\(424\) 0 0
\(425\) −0.225153 −0.0109215
\(426\) 0 0
\(427\) 12.4131 0.600711
\(428\) 0 0
\(429\) −9.90500 −0.478218
\(430\) 0 0
\(431\) −2.50038 −0.120439 −0.0602195 0.998185i \(-0.519180\pi\)
−0.0602195 + 0.998185i \(0.519180\pi\)
\(432\) 0 0
\(433\) 17.8990 0.860169 0.430084 0.902789i \(-0.358484\pi\)
0.430084 + 0.902789i \(0.358484\pi\)
\(434\) 0 0
\(435\) 4.91118 0.235473
\(436\) 0 0
\(437\) 3.48199 0.166566
\(438\) 0 0
\(439\) −9.78957 −0.467231 −0.233615 0.972329i \(-0.575056\pi\)
−0.233615 + 0.972329i \(0.575056\pi\)
\(440\) 0 0
\(441\) −2.52112 −0.120053
\(442\) 0 0
\(443\) 16.7723 0.796877 0.398438 0.917195i \(-0.369552\pi\)
0.398438 + 0.917195i \(0.369552\pi\)
\(444\) 0 0
\(445\) −30.4664 −1.44425
\(446\) 0 0
\(447\) 3.32929 0.157470
\(448\) 0 0
\(449\) −9.46455 −0.446660 −0.223330 0.974743i \(-0.571693\pi\)
−0.223330 + 0.974743i \(0.571693\pi\)
\(450\) 0 0
\(451\) 5.78835 0.272563
\(452\) 0 0
\(453\) −7.37289 −0.346409
\(454\) 0 0
\(455\) −6.56696 −0.307864
\(456\) 0 0
\(457\) 25.7452 1.20431 0.602154 0.798380i \(-0.294309\pi\)
0.602154 + 0.798380i \(0.294309\pi\)
\(458\) 0 0
\(459\) −3.82068 −0.178334
\(460\) 0 0
\(461\) −15.2232 −0.709017 −0.354508 0.935053i \(-0.615352\pi\)
−0.354508 + 0.935053i \(0.615352\pi\)
\(462\) 0 0
\(463\) −25.1295 −1.16787 −0.583935 0.811801i \(-0.698488\pi\)
−0.583935 + 0.811801i \(0.698488\pi\)
\(464\) 0 0
\(465\) 1.86073 0.0862894
\(466\) 0 0
\(467\) 22.0858 1.02201 0.511005 0.859578i \(-0.329273\pi\)
0.511005 + 0.859578i \(0.329273\pi\)
\(468\) 0 0
\(469\) 7.61983 0.351851
\(470\) 0 0
\(471\) −1.11514 −0.0513828
\(472\) 0 0
\(473\) 0.712816 0.0327753
\(474\) 0 0
\(475\) 0.742075 0.0340487
\(476\) 0 0
\(477\) 15.9434 0.730000
\(478\) 0 0
\(479\) −11.3333 −0.517830 −0.258915 0.965900i \(-0.583365\pi\)
−0.258915 + 0.965900i \(0.583365\pi\)
\(480\) 0 0
\(481\) 3.53500 0.161182
\(482\) 0 0
\(483\) 0.731091 0.0332658
\(484\) 0 0
\(485\) 19.5333 0.886961
\(486\) 0 0
\(487\) 7.65208 0.346749 0.173374 0.984856i \(-0.444533\pi\)
0.173374 + 0.984856i \(0.444533\pi\)
\(488\) 0 0
\(489\) 13.1646 0.595323
\(490\) 0 0
\(491\) −6.72345 −0.303425 −0.151713 0.988425i \(-0.548479\pi\)
−0.151713 + 0.988425i \(0.548479\pi\)
\(492\) 0 0
\(493\) 3.24782 0.146275
\(494\) 0 0
\(495\) 26.2379 1.17931
\(496\) 0 0
\(497\) 4.23146 0.189807
\(498\) 0 0
\(499\) 31.7247 1.42019 0.710096 0.704105i \(-0.248652\pi\)
0.710096 + 0.704105i \(0.248652\pi\)
\(500\) 0 0
\(501\) −10.9711 −0.490151
\(502\) 0 0
\(503\) 14.6228 0.651997 0.325998 0.945370i \(-0.394300\pi\)
0.325998 + 0.945370i \(0.394300\pi\)
\(504\) 0 0
\(505\) −14.0965 −0.627284
\(506\) 0 0
\(507\) −2.74613 −0.121960
\(508\) 0 0
\(509\) 15.0973 0.669178 0.334589 0.942364i \(-0.391403\pi\)
0.334589 + 0.942364i \(0.391403\pi\)
\(510\) 0 0
\(511\) −0.412507 −0.0182483
\(512\) 0 0
\(513\) 12.5925 0.555972
\(514\) 0 0
\(515\) −7.10382 −0.313032
\(516\) 0 0
\(517\) −2.52423 −0.111015
\(518\) 0 0
\(519\) −4.42857 −0.194393
\(520\) 0 0
\(521\) 24.6836 1.08141 0.540703 0.841213i \(-0.318158\pi\)
0.540703 + 0.841213i \(0.318158\pi\)
\(522\) 0 0
\(523\) −1.24305 −0.0543547 −0.0271774 0.999631i \(-0.508652\pi\)
−0.0271774 + 0.999631i \(0.508652\pi\)
\(524\) 0 0
\(525\) 0.155809 0.00680005
\(526\) 0 0
\(527\) 1.23052 0.0536024
\(528\) 0 0
\(529\) −21.8839 −0.951473
\(530\) 0 0
\(531\) 7.39625 0.320970
\(532\) 0 0
\(533\) −3.65244 −0.158205
\(534\) 0 0
\(535\) −11.5279 −0.498394
\(536\) 0 0
\(537\) 13.6863 0.590608
\(538\) 0 0
\(539\) −4.76273 −0.205145
\(540\) 0 0
\(541\) −21.7406 −0.934703 −0.467352 0.884072i \(-0.654792\pi\)
−0.467352 + 0.884072i \(0.654792\pi\)
\(542\) 0 0
\(543\) 15.7635 0.676476
\(544\) 0 0
\(545\) −5.46202 −0.233967
\(546\) 0 0
\(547\) 30.2901 1.29511 0.647556 0.762018i \(-0.275791\pi\)
0.647556 + 0.762018i \(0.275791\pi\)
\(548\) 0 0
\(549\) 31.2948 1.33563
\(550\) 0 0
\(551\) −10.7044 −0.456023
\(552\) 0 0
\(553\) 14.5585 0.619090
\(554\) 0 0
\(555\) 1.77868 0.0755010
\(556\) 0 0
\(557\) −6.56186 −0.278035 −0.139018 0.990290i \(-0.544394\pi\)
−0.139018 + 0.990290i \(0.544394\pi\)
\(558\) 0 0
\(559\) −0.449786 −0.0190239
\(560\) 0 0
\(561\) −3.29587 −0.139152
\(562\) 0 0
\(563\) 24.1250 1.01675 0.508373 0.861137i \(-0.330247\pi\)
0.508373 + 0.861137i \(0.330247\pi\)
\(564\) 0 0
\(565\) −11.3231 −0.476366
\(566\) 0 0
\(567\) −4.91939 −0.206595
\(568\) 0 0
\(569\) −20.3512 −0.853166 −0.426583 0.904448i \(-0.640283\pi\)
−0.426583 + 0.904448i \(0.640283\pi\)
\(570\) 0 0
\(571\) 29.3914 1.22999 0.614997 0.788530i \(-0.289157\pi\)
0.614997 + 0.788530i \(0.289157\pi\)
\(572\) 0 0
\(573\) −0.974972 −0.0407300
\(574\) 0 0
\(575\) 0.237867 0.00991976
\(576\) 0 0
\(577\) 13.7482 0.572344 0.286172 0.958178i \(-0.407617\pi\)
0.286172 + 0.958178i \(0.407617\pi\)
\(578\) 0 0
\(579\) 2.53098 0.105184
\(580\) 0 0
\(581\) −0.652935 −0.0270883
\(582\) 0 0
\(583\) 30.1193 1.24741
\(584\) 0 0
\(585\) −16.5561 −0.684509
\(586\) 0 0
\(587\) 2.94284 0.121464 0.0607320 0.998154i \(-0.480657\pi\)
0.0607320 + 0.998154i \(0.480657\pi\)
\(588\) 0 0
\(589\) −4.05565 −0.167110
\(590\) 0 0
\(591\) 15.6473 0.643645
\(592\) 0 0
\(593\) 8.89968 0.365466 0.182733 0.983163i \(-0.441506\pi\)
0.182733 + 0.983163i \(0.441506\pi\)
\(594\) 0 0
\(595\) −2.18514 −0.0895821
\(596\) 0 0
\(597\) −9.23907 −0.378130
\(598\) 0 0
\(599\) −31.6963 −1.29508 −0.647538 0.762033i \(-0.724201\pi\)
−0.647538 + 0.762033i \(0.724201\pi\)
\(600\) 0 0
\(601\) −7.02587 −0.286591 −0.143296 0.989680i \(-0.545770\pi\)
−0.143296 + 0.989680i \(0.545770\pi\)
\(602\) 0 0
\(603\) 19.2105 0.782311
\(604\) 0 0
\(605\) 25.5303 1.03796
\(606\) 0 0
\(607\) 34.4722 1.39918 0.699591 0.714543i \(-0.253365\pi\)
0.699591 + 0.714543i \(0.253365\pi\)
\(608\) 0 0
\(609\) −2.24753 −0.0910747
\(610\) 0 0
\(611\) 1.59278 0.0644371
\(612\) 0 0
\(613\) −1.02437 −0.0413741 −0.0206870 0.999786i \(-0.506585\pi\)
−0.0206870 + 0.999786i \(0.506585\pi\)
\(614\) 0 0
\(615\) −1.83778 −0.0741063
\(616\) 0 0
\(617\) 15.9782 0.643259 0.321629 0.946866i \(-0.395770\pi\)
0.321629 + 0.946866i \(0.395770\pi\)
\(618\) 0 0
\(619\) −0.157524 −0.00633143 −0.00316572 0.999995i \(-0.501008\pi\)
−0.00316572 + 0.999995i \(0.501008\pi\)
\(620\) 0 0
\(621\) 4.03644 0.161977
\(622\) 0 0
\(623\) 13.9425 0.558595
\(624\) 0 0
\(625\) −23.8235 −0.952942
\(626\) 0 0
\(627\) 10.8628 0.433817
\(628\) 0 0
\(629\) 1.17626 0.0469008
\(630\) 0 0
\(631\) 26.0474 1.03693 0.518465 0.855099i \(-0.326504\pi\)
0.518465 + 0.855099i \(0.326504\pi\)
\(632\) 0 0
\(633\) 8.00989 0.318365
\(634\) 0 0
\(635\) 39.0504 1.54967
\(636\) 0 0
\(637\) 3.00528 0.119073
\(638\) 0 0
\(639\) 10.6680 0.422020
\(640\) 0 0
\(641\) −34.9914 −1.38208 −0.691039 0.722818i \(-0.742847\pi\)
−0.691039 + 0.722818i \(0.742847\pi\)
\(642\) 0 0
\(643\) 19.8670 0.783478 0.391739 0.920076i \(-0.371874\pi\)
0.391739 + 0.920076i \(0.371874\pi\)
\(644\) 0 0
\(645\) −0.226316 −0.00891119
\(646\) 0 0
\(647\) 25.5462 1.00433 0.502163 0.864773i \(-0.332538\pi\)
0.502163 + 0.864773i \(0.332538\pi\)
\(648\) 0 0
\(649\) 13.9725 0.548469
\(650\) 0 0
\(651\) −0.851538 −0.0333744
\(652\) 0 0
\(653\) −20.5504 −0.804200 −0.402100 0.915596i \(-0.631720\pi\)
−0.402100 + 0.915596i \(0.631720\pi\)
\(654\) 0 0
\(655\) −39.3176 −1.53627
\(656\) 0 0
\(657\) −1.03998 −0.0405735
\(658\) 0 0
\(659\) −14.4156 −0.561551 −0.280776 0.959773i \(-0.590592\pi\)
−0.280776 + 0.959773i \(0.590592\pi\)
\(660\) 0 0
\(661\) −14.9184 −0.580260 −0.290130 0.956987i \(-0.593699\pi\)
−0.290130 + 0.956987i \(0.593699\pi\)
\(662\) 0 0
\(663\) 2.07969 0.0807685
\(664\) 0 0
\(665\) 7.20195 0.279280
\(666\) 0 0
\(667\) −3.43123 −0.132858
\(668\) 0 0
\(669\) −9.18926 −0.355277
\(670\) 0 0
\(671\) 59.1201 2.28231
\(672\) 0 0
\(673\) 0.899084 0.0346571 0.0173286 0.999850i \(-0.494484\pi\)
0.0173286 + 0.999850i \(0.494484\pi\)
\(674\) 0 0
\(675\) 0.860239 0.0331106
\(676\) 0 0
\(677\) −19.7181 −0.757827 −0.378914 0.925432i \(-0.623702\pi\)
−0.378914 + 0.925432i \(0.623702\pi\)
\(678\) 0 0
\(679\) −8.93914 −0.343053
\(680\) 0 0
\(681\) 16.3149 0.625189
\(682\) 0 0
\(683\) 5.09474 0.194945 0.0974724 0.995238i \(-0.468924\pi\)
0.0974724 + 0.995238i \(0.468924\pi\)
\(684\) 0 0
\(685\) 7.07048 0.270149
\(686\) 0 0
\(687\) −11.8819 −0.453322
\(688\) 0 0
\(689\) −19.0052 −0.724042
\(690\) 0 0
\(691\) 2.77065 0.105400 0.0527002 0.998610i \(-0.483217\pi\)
0.0527002 + 0.998610i \(0.483217\pi\)
\(692\) 0 0
\(693\) −12.0074 −0.456124
\(694\) 0 0
\(695\) −9.70687 −0.368203
\(696\) 0 0
\(697\) −1.21534 −0.0460344
\(698\) 0 0
\(699\) 14.9056 0.563783
\(700\) 0 0
\(701\) 42.4680 1.60399 0.801997 0.597328i \(-0.203771\pi\)
0.801997 + 0.597328i \(0.203771\pi\)
\(702\) 0 0
\(703\) −3.87682 −0.146217
\(704\) 0 0
\(705\) 0.801431 0.0301836
\(706\) 0 0
\(707\) 6.45105 0.242617
\(708\) 0 0
\(709\) −46.2542 −1.73711 −0.868556 0.495591i \(-0.834952\pi\)
−0.868556 + 0.495591i \(0.834952\pi\)
\(710\) 0 0
\(711\) 36.7037 1.37650
\(712\) 0 0
\(713\) −1.30001 −0.0486858
\(714\) 0 0
\(715\) −31.2766 −1.16968
\(716\) 0 0
\(717\) −9.62827 −0.359574
\(718\) 0 0
\(719\) 29.5694 1.10275 0.551376 0.834257i \(-0.314103\pi\)
0.551376 + 0.834257i \(0.314103\pi\)
\(720\) 0 0
\(721\) 3.25097 0.121072
\(722\) 0 0
\(723\) 5.04055 0.187460
\(724\) 0 0
\(725\) −0.731257 −0.0271582
\(726\) 0 0
\(727\) 6.58624 0.244270 0.122135 0.992513i \(-0.461026\pi\)
0.122135 + 0.992513i \(0.461026\pi\)
\(728\) 0 0
\(729\) −4.47048 −0.165573
\(730\) 0 0
\(731\) −0.149665 −0.00553558
\(732\) 0 0
\(733\) −17.3643 −0.641364 −0.320682 0.947187i \(-0.603912\pi\)
−0.320682 + 0.947187i \(0.603912\pi\)
\(734\) 0 0
\(735\) 1.51215 0.0557764
\(736\) 0 0
\(737\) 36.2912 1.33680
\(738\) 0 0
\(739\) 26.6329 0.979707 0.489853 0.871805i \(-0.337050\pi\)
0.489853 + 0.871805i \(0.337050\pi\)
\(740\) 0 0
\(741\) −6.85439 −0.251802
\(742\) 0 0
\(743\) 38.7890 1.42303 0.711515 0.702671i \(-0.248009\pi\)
0.711515 + 0.702671i \(0.248009\pi\)
\(744\) 0 0
\(745\) 10.5128 0.385158
\(746\) 0 0
\(747\) −1.64613 −0.0602286
\(748\) 0 0
\(749\) 5.27557 0.192765
\(750\) 0 0
\(751\) 3.97854 0.145179 0.0725894 0.997362i \(-0.476874\pi\)
0.0725894 + 0.997362i \(0.476874\pi\)
\(752\) 0 0
\(753\) −3.34814 −0.122013
\(754\) 0 0
\(755\) −23.2811 −0.847286
\(756\) 0 0
\(757\) 5.41672 0.196874 0.0984369 0.995143i \(-0.468616\pi\)
0.0984369 + 0.995143i \(0.468616\pi\)
\(758\) 0 0
\(759\) 3.48199 0.126388
\(760\) 0 0
\(761\) −7.41577 −0.268821 −0.134411 0.990926i \(-0.542914\pi\)
−0.134411 + 0.990926i \(0.542914\pi\)
\(762\) 0 0
\(763\) 2.49962 0.0904922
\(764\) 0 0
\(765\) −5.50900 −0.199178
\(766\) 0 0
\(767\) −8.81664 −0.318350
\(768\) 0 0
\(769\) −14.6536 −0.528423 −0.264211 0.964465i \(-0.585112\pi\)
−0.264211 + 0.964465i \(0.585112\pi\)
\(770\) 0 0
\(771\) 2.93429 0.105676
\(772\) 0 0
\(773\) −31.6813 −1.13950 −0.569748 0.821819i \(-0.692959\pi\)
−0.569748 + 0.821819i \(0.692959\pi\)
\(774\) 0 0
\(775\) −0.277056 −0.00995214
\(776\) 0 0
\(777\) −0.813990 −0.0292017
\(778\) 0 0
\(779\) 4.00561 0.143516
\(780\) 0 0
\(781\) 20.1533 0.721141
\(782\) 0 0
\(783\) −12.4089 −0.443458
\(784\) 0 0
\(785\) −3.52123 −0.125678
\(786\) 0 0
\(787\) 40.5020 1.44374 0.721870 0.692028i \(-0.243283\pi\)
0.721870 + 0.692028i \(0.243283\pi\)
\(788\) 0 0
\(789\) 2.71998 0.0968338
\(790\) 0 0
\(791\) 5.18186 0.184246
\(792\) 0 0
\(793\) −37.3047 −1.32473
\(794\) 0 0
\(795\) −9.56275 −0.339156
\(796\) 0 0
\(797\) 21.5399 0.762981 0.381491 0.924373i \(-0.375411\pi\)
0.381491 + 0.924373i \(0.375411\pi\)
\(798\) 0 0
\(799\) 0.529996 0.0187499
\(800\) 0 0
\(801\) 35.1507 1.24199
\(802\) 0 0
\(803\) −1.96466 −0.0693314
\(804\) 0 0
\(805\) 2.30854 0.0813653
\(806\) 0 0
\(807\) 21.2071 0.746525
\(808\) 0 0
\(809\) −47.9994 −1.68757 −0.843784 0.536683i \(-0.819677\pi\)
−0.843784 + 0.536683i \(0.819677\pi\)
\(810\) 0 0
\(811\) 13.5842 0.477007 0.238503 0.971142i \(-0.423343\pi\)
0.238503 + 0.971142i \(0.423343\pi\)
\(812\) 0 0
\(813\) −0.236710 −0.00830179
\(814\) 0 0
\(815\) 41.5693 1.45611
\(816\) 0 0
\(817\) 0.493278 0.0172576
\(818\) 0 0
\(819\) 7.57666 0.264750
\(820\) 0 0
\(821\) −45.6149 −1.59197 −0.795986 0.605315i \(-0.793047\pi\)
−0.795986 + 0.605315i \(0.793047\pi\)
\(822\) 0 0
\(823\) −8.84854 −0.308441 −0.154220 0.988036i \(-0.549287\pi\)
−0.154220 + 0.988036i \(0.549287\pi\)
\(824\) 0 0
\(825\) 0.742075 0.0258357
\(826\) 0 0
\(827\) −14.2215 −0.494531 −0.247265 0.968948i \(-0.579532\pi\)
−0.247265 + 0.968948i \(0.579532\pi\)
\(828\) 0 0
\(829\) −3.10407 −0.107809 −0.0539043 0.998546i \(-0.517167\pi\)
−0.0539043 + 0.998546i \(0.517167\pi\)
\(830\) 0 0
\(831\) 12.1390 0.421099
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −34.6429 −1.19887
\(836\) 0 0
\(837\) −4.70144 −0.162506
\(838\) 0 0
\(839\) −45.6207 −1.57500 −0.787501 0.616313i \(-0.788626\pi\)
−0.787501 + 0.616313i \(0.788626\pi\)
\(840\) 0 0
\(841\) −18.4517 −0.636264
\(842\) 0 0
\(843\) −20.4794 −0.705349
\(844\) 0 0
\(845\) −8.67133 −0.298303
\(846\) 0 0
\(847\) −11.6836 −0.401453
\(848\) 0 0
\(849\) −13.8309 −0.474674
\(850\) 0 0
\(851\) −1.24269 −0.0425988
\(852\) 0 0
\(853\) 34.9002 1.19496 0.597481 0.801883i \(-0.296168\pi\)
0.597481 + 0.801883i \(0.296168\pi\)
\(854\) 0 0
\(855\) 18.1570 0.620955
\(856\) 0 0
\(857\) −1.60518 −0.0548319 −0.0274160 0.999624i \(-0.508728\pi\)
−0.0274160 + 0.999624i \(0.508728\pi\)
\(858\) 0 0
\(859\) 48.0128 1.63818 0.819088 0.573667i \(-0.194480\pi\)
0.819088 + 0.573667i \(0.194480\pi\)
\(860\) 0 0
\(861\) 0.841033 0.0286623
\(862\) 0 0
\(863\) 21.0701 0.717236 0.358618 0.933484i \(-0.383248\pi\)
0.358618 + 0.933484i \(0.383248\pi\)
\(864\) 0 0
\(865\) −13.9839 −0.475468
\(866\) 0 0
\(867\) 0.692013 0.0235020
\(868\) 0 0
\(869\) 69.3382 2.35214
\(870\) 0 0
\(871\) −22.8997 −0.775926
\(872\) 0 0
\(873\) −22.5366 −0.762749
\(874\) 0 0
\(875\) 11.4177 0.385989
\(876\) 0 0
\(877\) −8.61011 −0.290743 −0.145371 0.989377i \(-0.546438\pi\)
−0.145371 + 0.989377i \(0.546438\pi\)
\(878\) 0 0
\(879\) 1.96411 0.0662479
\(880\) 0 0
\(881\) 11.1694 0.376305 0.188153 0.982140i \(-0.439750\pi\)
0.188153 + 0.982140i \(0.439750\pi\)
\(882\) 0 0
\(883\) 33.6407 1.13210 0.566051 0.824371i \(-0.308471\pi\)
0.566051 + 0.824371i \(0.308471\pi\)
\(884\) 0 0
\(885\) −4.43621 −0.149122
\(886\) 0 0
\(887\) −51.2232 −1.71991 −0.859953 0.510374i \(-0.829507\pi\)
−0.859953 + 0.510374i \(0.829507\pi\)
\(888\) 0 0
\(889\) −17.8709 −0.599370
\(890\) 0 0
\(891\) −23.4297 −0.784926
\(892\) 0 0
\(893\) −1.74680 −0.0584543
\(894\) 0 0
\(895\) 43.2167 1.44458
\(896\) 0 0
\(897\) −2.19713 −0.0733601
\(898\) 0 0
\(899\) 3.99652 0.133291
\(900\) 0 0
\(901\) −6.32395 −0.210681
\(902\) 0 0
\(903\) 0.103570 0.00344661
\(904\) 0 0
\(905\) 49.7757 1.65460
\(906\) 0 0
\(907\) −6.43221 −0.213578 −0.106789 0.994282i \(-0.534057\pi\)
−0.106789 + 0.994282i \(0.534057\pi\)
\(908\) 0 0
\(909\) 16.2639 0.539438
\(910\) 0 0
\(911\) 48.8018 1.61688 0.808438 0.588581i \(-0.200313\pi\)
0.808438 + 0.588581i \(0.200313\pi\)
\(912\) 0 0
\(913\) −3.10975 −0.102918
\(914\) 0 0
\(915\) −18.7704 −0.620530
\(916\) 0 0
\(917\) 17.9932 0.594187
\(918\) 0 0
\(919\) 38.2201 1.26076 0.630382 0.776285i \(-0.282898\pi\)
0.630382 + 0.776285i \(0.282898\pi\)
\(920\) 0 0
\(921\) 7.59677 0.250322
\(922\) 0 0
\(923\) −12.7167 −0.418575
\(924\) 0 0
\(925\) −0.264840 −0.00870787
\(926\) 0 0
\(927\) 8.19607 0.269194
\(928\) 0 0
\(929\) 22.9110 0.751686 0.375843 0.926683i \(-0.377353\pi\)
0.375843 + 0.926683i \(0.377353\pi\)
\(930\) 0 0
\(931\) −3.29587 −0.108018
\(932\) 0 0
\(933\) 13.7481 0.450093
\(934\) 0 0
\(935\) −10.4072 −0.340353
\(936\) 0 0
\(937\) 33.7022 1.10100 0.550501 0.834835i \(-0.314437\pi\)
0.550501 + 0.834835i \(0.314437\pi\)
\(938\) 0 0
\(939\) −11.6988 −0.381776
\(940\) 0 0
\(941\) 18.0639 0.588865 0.294433 0.955672i \(-0.404869\pi\)
0.294433 + 0.955672i \(0.404869\pi\)
\(942\) 0 0
\(943\) 1.28397 0.0418119
\(944\) 0 0
\(945\) 8.34874 0.271585
\(946\) 0 0
\(947\) −46.2560 −1.50312 −0.751559 0.659666i \(-0.770698\pi\)
−0.751559 + 0.659666i \(0.770698\pi\)
\(948\) 0 0
\(949\) 1.23970 0.0402423
\(950\) 0 0
\(951\) 4.70917 0.152705
\(952\) 0 0
\(953\) −19.9327 −0.645683 −0.322842 0.946453i \(-0.604638\pi\)
−0.322842 + 0.946453i \(0.604638\pi\)
\(954\) 0 0
\(955\) −3.07863 −0.0996221
\(956\) 0 0
\(957\) −10.7044 −0.346024
\(958\) 0 0
\(959\) −3.23571 −0.104486
\(960\) 0 0
\(961\) −29.4858 −0.951155
\(962\) 0 0
\(963\) 13.3003 0.428598
\(964\) 0 0
\(965\) 7.99198 0.257271
\(966\) 0 0
\(967\) 10.0746 0.323978 0.161989 0.986793i \(-0.448209\pi\)
0.161989 + 0.986793i \(0.448209\pi\)
\(968\) 0 0
\(969\) −2.28079 −0.0732694
\(970\) 0 0
\(971\) −25.4973 −0.818248 −0.409124 0.912479i \(-0.634166\pi\)
−0.409124 + 0.912479i \(0.634166\pi\)
\(972\) 0 0
\(973\) 4.44222 0.142411
\(974\) 0 0
\(975\) −0.468248 −0.0149959
\(976\) 0 0
\(977\) −33.9875 −1.08736 −0.543678 0.839294i \(-0.682969\pi\)
−0.543678 + 0.839294i \(0.682969\pi\)
\(978\) 0 0
\(979\) 66.4045 2.12230
\(980\) 0 0
\(981\) 6.30183 0.201202
\(982\) 0 0
\(983\) −33.4135 −1.06572 −0.532862 0.846202i \(-0.678884\pi\)
−0.532862 + 0.846202i \(0.678884\pi\)
\(984\) 0 0
\(985\) 49.4090 1.57430
\(986\) 0 0
\(987\) −0.366764 −0.0116742
\(988\) 0 0
\(989\) 0.158117 0.00502783
\(990\) 0 0
\(991\) 29.9427 0.951160 0.475580 0.879672i \(-0.342238\pi\)
0.475580 + 0.879672i \(0.342238\pi\)
\(992\) 0 0
\(993\) 16.4002 0.520445
\(994\) 0 0
\(995\) −29.1739 −0.924873
\(996\) 0 0
\(997\) 5.35532 0.169605 0.0848023 0.996398i \(-0.472974\pi\)
0.0848023 + 0.996398i \(0.472974\pi\)
\(998\) 0 0
\(999\) −4.49414 −0.142188
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 3808.2.a.j.1.4 6
4.3 odd 2 3808.2.a.n.1.3 yes 6
8.3 odd 2 7616.2.a.bw.1.4 6
8.5 even 2 7616.2.a.ca.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3808.2.a.j.1.4 6 1.1 even 1 trivial
3808.2.a.n.1.3 yes 6 4.3 odd 2
7616.2.a.bw.1.4 6 8.3 odd 2
7616.2.a.ca.1.3 6 8.5 even 2