Properties

Label 736.4.a.e.1.8
Level $736$
Weight $4$
Character 736.1
Self dual yes
Analytic conductor $43.425$
Analytic rank $1$
Dimension $8$
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] = [736,4,Mod(1,736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("736.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 736.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: \(43.4254057642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(9.42564\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.42564 q^{3} +4.36302 q^{5} -9.40153 q^{7} +28.1401 q^{9} -67.5361 q^{11} -33.5730 q^{13} +32.3982 q^{15} +24.0641 q^{17} -71.4057 q^{19} -69.8123 q^{21} +23.0000 q^{23} -105.964 q^{25} +8.46606 q^{27} -96.4102 q^{29} -63.1309 q^{31} -501.498 q^{33} -41.0191 q^{35} +84.0774 q^{37} -249.301 q^{39} -144.498 q^{41} +75.4198 q^{43} +122.776 q^{45} +215.075 q^{47} -254.611 q^{49} +178.691 q^{51} -55.0233 q^{53} -294.661 q^{55} -530.233 q^{57} -139.853 q^{59} +772.119 q^{61} -264.560 q^{63} -146.480 q^{65} -871.806 q^{67} +170.790 q^{69} +148.801 q^{71} +745.997 q^{73} -786.851 q^{75} +634.942 q^{77} -461.010 q^{79} -696.917 q^{81} +345.272 q^{83} +104.992 q^{85} -715.907 q^{87} +1276.79 q^{89} +315.637 q^{91} -468.787 q^{93} -311.545 q^{95} -1024.86 q^{97} -1900.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} + 14 q^{7} + 90 q^{9} - 88 q^{11} - 30 q^{13} - 30 q^{15} + 58 q^{17} - 190 q^{19} - 66 q^{21} + 184 q^{23} + 28 q^{25} - 432 q^{27} + 190 q^{29} + 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 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
\(3\) 7.42564 1.42906 0.714532 0.699602i \(-0.246639\pi\)
0.714532 + 0.699602i \(0.246639\pi\)
\(4\) 0 0
\(5\) 4.36302 0.390241 0.195120 0.980779i \(-0.437490\pi\)
0.195120 + 0.980779i \(0.437490\pi\)
\(6\) 0 0
\(7\) −9.40153 −0.507635 −0.253817 0.967252i \(-0.581686\pi\)
−0.253817 + 0.967252i \(0.581686\pi\)
\(8\) 0 0
\(9\) 28.1401 1.04223
\(10\) 0 0
\(11\) −67.5361 −1.85117 −0.925586 0.378538i \(-0.876427\pi\)
−0.925586 + 0.378538i \(0.876427\pi\)
\(12\) 0 0
\(13\) −33.5730 −0.716267 −0.358133 0.933670i \(-0.616587\pi\)
−0.358133 + 0.933670i \(0.616587\pi\)
\(14\) 0 0
\(15\) 32.3982 0.557679
\(16\) 0 0
\(17\) 24.0641 0.343317 0.171659 0.985156i \(-0.445087\pi\)
0.171659 + 0.985156i \(0.445087\pi\)
\(18\) 0 0
\(19\) −71.4057 −0.862189 −0.431094 0.902307i \(-0.641872\pi\)
−0.431094 + 0.902307i \(0.641872\pi\)
\(20\) 0 0
\(21\) −69.8123 −0.725443
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) −105.964 −0.847712
\(26\) 0 0
\(27\) 8.46606 0.0603442
\(28\) 0 0
\(29\) −96.4102 −0.617342 −0.308671 0.951169i \(-0.599884\pi\)
−0.308671 + 0.951169i \(0.599884\pi\)
\(30\) 0 0
\(31\) −63.1309 −0.365763 −0.182881 0.983135i \(-0.558542\pi\)
−0.182881 + 0.983135i \(0.558542\pi\)
\(32\) 0 0
\(33\) −501.498 −2.64544
\(34\) 0 0
\(35\) −41.0191 −0.198100
\(36\) 0 0
\(37\) 84.0774 0.373574 0.186787 0.982400i \(-0.440193\pi\)
0.186787 + 0.982400i \(0.440193\pi\)
\(38\) 0 0
\(39\) −249.301 −1.02359
\(40\) 0 0
\(41\) −144.498 −0.550411 −0.275206 0.961385i \(-0.588746\pi\)
−0.275206 + 0.961385i \(0.588746\pi\)
\(42\) 0 0
\(43\) 75.4198 0.267475 0.133737 0.991017i \(-0.457302\pi\)
0.133737 + 0.991017i \(0.457302\pi\)
\(44\) 0 0
\(45\) 122.776 0.406719
\(46\) 0 0
\(47\) 215.075 0.667486 0.333743 0.942664i \(-0.391688\pi\)
0.333743 + 0.942664i \(0.391688\pi\)
\(48\) 0 0
\(49\) −254.611 −0.742307
\(50\) 0 0
\(51\) 178.691 0.490623
\(52\) 0 0
\(53\) −55.0233 −0.142604 −0.0713021 0.997455i \(-0.522715\pi\)
−0.0713021 + 0.997455i \(0.522715\pi\)
\(54\) 0 0
\(55\) −294.661 −0.722402
\(56\) 0 0
\(57\) −530.233 −1.23212
\(58\) 0 0
\(59\) −139.853 −0.308598 −0.154299 0.988024i \(-0.549312\pi\)
−0.154299 + 0.988024i \(0.549312\pi\)
\(60\) 0 0
\(61\) 772.119 1.62065 0.810325 0.585981i \(-0.199291\pi\)
0.810325 + 0.585981i \(0.199291\pi\)
\(62\) 0 0
\(63\) −264.560 −0.529070
\(64\) 0 0
\(65\) −146.480 −0.279516
\(66\) 0 0
\(67\) −871.806 −1.58967 −0.794836 0.606824i \(-0.792443\pi\)
−0.794836 + 0.606824i \(0.792443\pi\)
\(68\) 0 0
\(69\) 170.790 0.297981
\(70\) 0 0
\(71\) 148.801 0.248725 0.124362 0.992237i \(-0.460311\pi\)
0.124362 + 0.992237i \(0.460311\pi\)
\(72\) 0 0
\(73\) 745.997 1.19606 0.598030 0.801474i \(-0.295950\pi\)
0.598030 + 0.801474i \(0.295950\pi\)
\(74\) 0 0
\(75\) −786.851 −1.21144
\(76\) 0 0
\(77\) 634.942 0.939719
\(78\) 0 0
\(79\) −461.010 −0.656552 −0.328276 0.944582i \(-0.606468\pi\)
−0.328276 + 0.944582i \(0.606468\pi\)
\(80\) 0 0
\(81\) −696.917 −0.955991
\(82\) 0 0
\(83\) 345.272 0.456609 0.228305 0.973590i \(-0.426682\pi\)
0.228305 + 0.973590i \(0.426682\pi\)
\(84\) 0 0
\(85\) 104.992 0.133976
\(86\) 0 0
\(87\) −715.907 −0.882222
\(88\) 0 0
\(89\) 1276.79 1.52067 0.760337 0.649529i \(-0.225034\pi\)
0.760337 + 0.649529i \(0.225034\pi\)
\(90\) 0 0
\(91\) 315.637 0.363602
\(92\) 0 0
\(93\) −468.787 −0.522699
\(94\) 0 0
\(95\) −311.545 −0.336461
\(96\) 0 0
\(97\) −1024.86 −1.07277 −0.536386 0.843973i \(-0.680211\pi\)
−0.536386 + 0.843973i \(0.680211\pi\)
\(98\) 0 0
\(99\) −1900.47 −1.92934
\(100\) 0 0
\(101\) 126.508 0.124634 0.0623169 0.998056i \(-0.480151\pi\)
0.0623169 + 0.998056i \(0.480151\pi\)
\(102\) 0 0
\(103\) 258.017 0.246827 0.123413 0.992355i \(-0.460616\pi\)
0.123413 + 0.992355i \(0.460616\pi\)
\(104\) 0 0
\(105\) −304.593 −0.283097
\(106\) 0 0
\(107\) −1017.46 −0.919265 −0.459633 0.888109i \(-0.652019\pi\)
−0.459633 + 0.888109i \(0.652019\pi\)
\(108\) 0 0
\(109\) 1861.29 1.63559 0.817797 0.575507i \(-0.195195\pi\)
0.817797 + 0.575507i \(0.195195\pi\)
\(110\) 0 0
\(111\) 624.329 0.533862
\(112\) 0 0
\(113\) −49.7040 −0.0413784 −0.0206892 0.999786i \(-0.506586\pi\)
−0.0206892 + 0.999786i \(0.506586\pi\)
\(114\) 0 0
\(115\) 100.350 0.0813708
\(116\) 0 0
\(117\) −944.748 −0.746512
\(118\) 0 0
\(119\) −226.239 −0.174280
\(120\) 0 0
\(121\) 3230.12 2.42684
\(122\) 0 0
\(123\) −1072.99 −0.786573
\(124\) 0 0
\(125\) −1007.70 −0.721052
\(126\) 0 0
\(127\) 1755.96 1.22690 0.613449 0.789734i \(-0.289782\pi\)
0.613449 + 0.789734i \(0.289782\pi\)
\(128\) 0 0
\(129\) 560.040 0.382239
\(130\) 0 0
\(131\) −377.103 −0.251509 −0.125754 0.992061i \(-0.540135\pi\)
−0.125754 + 0.992061i \(0.540135\pi\)
\(132\) 0 0
\(133\) 671.322 0.437677
\(134\) 0 0
\(135\) 36.9376 0.0235488
\(136\) 0 0
\(137\) 376.686 0.234908 0.117454 0.993078i \(-0.462527\pi\)
0.117454 + 0.993078i \(0.462527\pi\)
\(138\) 0 0
\(139\) −2297.98 −1.40224 −0.701122 0.713041i \(-0.747317\pi\)
−0.701122 + 0.713041i \(0.747317\pi\)
\(140\) 0 0
\(141\) 1597.07 0.953881
\(142\) 0 0
\(143\) 2267.39 1.32593
\(144\) 0 0
\(145\) −420.640 −0.240912
\(146\) 0 0
\(147\) −1890.65 −1.06080
\(148\) 0 0
\(149\) 1191.27 0.654987 0.327493 0.944853i \(-0.393796\pi\)
0.327493 + 0.944853i \(0.393796\pi\)
\(150\) 0 0
\(151\) −2536.35 −1.36692 −0.683462 0.729986i \(-0.739527\pi\)
−0.683462 + 0.729986i \(0.739527\pi\)
\(152\) 0 0
\(153\) 677.166 0.357814
\(154\) 0 0
\(155\) −275.442 −0.142736
\(156\) 0 0
\(157\) 1112.51 0.565529 0.282764 0.959189i \(-0.408749\pi\)
0.282764 + 0.959189i \(0.408749\pi\)
\(158\) 0 0
\(159\) −408.583 −0.203791
\(160\) 0 0
\(161\) −216.235 −0.105849
\(162\) 0 0
\(163\) −1481.99 −0.712135 −0.356068 0.934460i \(-0.615883\pi\)
−0.356068 + 0.934460i \(0.615883\pi\)
\(164\) 0 0
\(165\) −2188.05 −1.03236
\(166\) 0 0
\(167\) −3939.40 −1.82539 −0.912695 0.408642i \(-0.866003\pi\)
−0.912695 + 0.408642i \(0.866003\pi\)
\(168\) 0 0
\(169\) −1069.85 −0.486962
\(170\) 0 0
\(171\) −2009.36 −0.898596
\(172\) 0 0
\(173\) −1332.84 −0.585744 −0.292872 0.956152i \(-0.594611\pi\)
−0.292872 + 0.956152i \(0.594611\pi\)
\(174\) 0 0
\(175\) 996.224 0.430328
\(176\) 0 0
\(177\) −1038.50 −0.441006
\(178\) 0 0
\(179\) −603.340 −0.251931 −0.125966 0.992035i \(-0.540203\pi\)
−0.125966 + 0.992035i \(0.540203\pi\)
\(180\) 0 0
\(181\) −4132.64 −1.69711 −0.848554 0.529109i \(-0.822526\pi\)
−0.848554 + 0.529109i \(0.822526\pi\)
\(182\) 0 0
\(183\) 5733.47 2.31601
\(184\) 0 0
\(185\) 366.832 0.145784
\(186\) 0 0
\(187\) −1625.19 −0.635539
\(188\) 0 0
\(189\) −79.5939 −0.0306328
\(190\) 0 0
\(191\) −2364.21 −0.895647 −0.447823 0.894122i \(-0.647801\pi\)
−0.447823 + 0.894122i \(0.647801\pi\)
\(192\) 0 0
\(193\) −3292.16 −1.22785 −0.613925 0.789365i \(-0.710410\pi\)
−0.613925 + 0.789365i \(0.710410\pi\)
\(194\) 0 0
\(195\) −1087.71 −0.399447
\(196\) 0 0
\(197\) −3375.84 −1.22091 −0.610453 0.792052i \(-0.709013\pi\)
−0.610453 + 0.792052i \(0.709013\pi\)
\(198\) 0 0
\(199\) 4578.09 1.63082 0.815408 0.578886i \(-0.196512\pi\)
0.815408 + 0.578886i \(0.196512\pi\)
\(200\) 0 0
\(201\) −6473.71 −2.27174
\(202\) 0 0
\(203\) 906.403 0.313384
\(204\) 0 0
\(205\) −630.450 −0.214793
\(206\) 0 0
\(207\) 647.223 0.217319
\(208\) 0 0
\(209\) 4822.46 1.59606
\(210\) 0 0
\(211\) 2634.81 0.859659 0.429829 0.902910i \(-0.358574\pi\)
0.429829 + 0.902910i \(0.358574\pi\)
\(212\) 0 0
\(213\) 1104.94 0.355444
\(214\) 0 0
\(215\) 329.058 0.104380
\(216\) 0 0
\(217\) 593.527 0.185674
\(218\) 0 0
\(219\) 5539.50 1.70925
\(220\) 0 0
\(221\) −807.902 −0.245907
\(222\) 0 0
\(223\) −551.110 −0.165493 −0.0827467 0.996571i \(-0.526369\pi\)
−0.0827467 + 0.996571i \(0.526369\pi\)
\(224\) 0 0
\(225\) −2981.84 −0.883508
\(226\) 0 0
\(227\) −1519.50 −0.444286 −0.222143 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(228\) 0 0
\(229\) 3162.04 0.912461 0.456231 0.889862i \(-0.349199\pi\)
0.456231 + 0.889862i \(0.349199\pi\)
\(230\) 0 0
\(231\) 4714.85 1.34292
\(232\) 0 0
\(233\) 4659.33 1.31005 0.655027 0.755605i \(-0.272657\pi\)
0.655027 + 0.755605i \(0.272657\pi\)
\(234\) 0 0
\(235\) 938.376 0.260480
\(236\) 0 0
\(237\) −3423.29 −0.938256
\(238\) 0 0
\(239\) −1993.51 −0.539536 −0.269768 0.962925i \(-0.586947\pi\)
−0.269768 + 0.962925i \(0.586947\pi\)
\(240\) 0 0
\(241\) 1772.18 0.473677 0.236838 0.971549i \(-0.423889\pi\)
0.236838 + 0.971549i \(0.423889\pi\)
\(242\) 0 0
\(243\) −5403.64 −1.42652
\(244\) 0 0
\(245\) −1110.87 −0.289678
\(246\) 0 0
\(247\) 2397.30 0.617557
\(248\) 0 0
\(249\) 2563.87 0.652524
\(250\) 0 0
\(251\) −1484.40 −0.373285 −0.186643 0.982428i \(-0.559761\pi\)
−0.186643 + 0.982428i \(0.559761\pi\)
\(252\) 0 0
\(253\) −1553.33 −0.385996
\(254\) 0 0
\(255\) 779.633 0.191461
\(256\) 0 0
\(257\) 2791.47 0.677537 0.338769 0.940870i \(-0.389990\pi\)
0.338769 + 0.940870i \(0.389990\pi\)
\(258\) 0 0
\(259\) −790.456 −0.189639
\(260\) 0 0
\(261\) −2712.99 −0.643410
\(262\) 0 0
\(263\) 6898.74 1.61747 0.808735 0.588173i \(-0.200153\pi\)
0.808735 + 0.588173i \(0.200153\pi\)
\(264\) 0 0
\(265\) −240.068 −0.0556500
\(266\) 0 0
\(267\) 9481.01 2.17314
\(268\) 0 0
\(269\) 2861.59 0.648604 0.324302 0.945954i \(-0.394871\pi\)
0.324302 + 0.945954i \(0.394871\pi\)
\(270\) 0 0
\(271\) −2226.19 −0.499009 −0.249504 0.968374i \(-0.580268\pi\)
−0.249504 + 0.968374i \(0.580268\pi\)
\(272\) 0 0
\(273\) 2343.81 0.519611
\(274\) 0 0
\(275\) 7156.39 1.56926
\(276\) 0 0
\(277\) −5283.50 −1.14605 −0.573023 0.819539i \(-0.694230\pi\)
−0.573023 + 0.819539i \(0.694230\pi\)
\(278\) 0 0
\(279\) −1776.51 −0.381208
\(280\) 0 0
\(281\) 7449.27 1.58145 0.790723 0.612174i \(-0.209705\pi\)
0.790723 + 0.612174i \(0.209705\pi\)
\(282\) 0 0
\(283\) −5206.96 −1.09372 −0.546858 0.837225i \(-0.684176\pi\)
−0.546858 + 0.837225i \(0.684176\pi\)
\(284\) 0 0
\(285\) −2313.42 −0.480825
\(286\) 0 0
\(287\) 1358.51 0.279408
\(288\) 0 0
\(289\) −4333.92 −0.882133
\(290\) 0 0
\(291\) −7610.24 −1.53306
\(292\) 0 0
\(293\) −3588.15 −0.715433 −0.357717 0.933830i \(-0.616445\pi\)
−0.357717 + 0.933830i \(0.616445\pi\)
\(294\) 0 0
\(295\) −610.180 −0.120427
\(296\) 0 0
\(297\) −571.764 −0.111708
\(298\) 0 0
\(299\) −772.179 −0.149352
\(300\) 0 0
\(301\) −709.062 −0.135779
\(302\) 0 0
\(303\) 939.402 0.178110
\(304\) 0 0
\(305\) 3368.77 0.632443
\(306\) 0 0
\(307\) −1222.71 −0.227309 −0.113654 0.993520i \(-0.536256\pi\)
−0.113654 + 0.993520i \(0.536256\pi\)
\(308\) 0 0
\(309\) 1915.94 0.352731
\(310\) 0 0
\(311\) −158.959 −0.0289831 −0.0144916 0.999895i \(-0.504613\pi\)
−0.0144916 + 0.999895i \(0.504613\pi\)
\(312\) 0 0
\(313\) 1532.53 0.276754 0.138377 0.990380i \(-0.455811\pi\)
0.138377 + 0.990380i \(0.455811\pi\)
\(314\) 0 0
\(315\) −1154.28 −0.206465
\(316\) 0 0
\(317\) −2444.39 −0.433093 −0.216546 0.976272i \(-0.569479\pi\)
−0.216546 + 0.976272i \(0.569479\pi\)
\(318\) 0 0
\(319\) 6511.16 1.14281
\(320\) 0 0
\(321\) −7555.27 −1.31369
\(322\) 0 0
\(323\) −1718.31 −0.296004
\(324\) 0 0
\(325\) 3557.53 0.607188
\(326\) 0 0
\(327\) 13821.3 2.33737
\(328\) 0 0
\(329\) −2022.03 −0.338839
\(330\) 0 0
\(331\) 3229.27 0.536245 0.268122 0.963385i \(-0.413597\pi\)
0.268122 + 0.963385i \(0.413597\pi\)
\(332\) 0 0
\(333\) 2365.95 0.389349
\(334\) 0 0
\(335\) −3803.71 −0.620354
\(336\) 0 0
\(337\) 2563.42 0.414357 0.207178 0.978303i \(-0.433572\pi\)
0.207178 + 0.978303i \(0.433572\pi\)
\(338\) 0 0
\(339\) −369.084 −0.0591325
\(340\) 0 0
\(341\) 4263.61 0.677090
\(342\) 0 0
\(343\) 5618.46 0.884456
\(344\) 0 0
\(345\) 745.159 0.116284
\(346\) 0 0
\(347\) −5755.34 −0.890383 −0.445192 0.895435i \(-0.646864\pi\)
−0.445192 + 0.895435i \(0.646864\pi\)
\(348\) 0 0
\(349\) 10312.7 1.58174 0.790868 0.611987i \(-0.209630\pi\)
0.790868 + 0.611987i \(0.209630\pi\)
\(350\) 0 0
\(351\) −284.231 −0.0432226
\(352\) 0 0
\(353\) 5515.25 0.831578 0.415789 0.909461i \(-0.363506\pi\)
0.415789 + 0.909461i \(0.363506\pi\)
\(354\) 0 0
\(355\) 649.223 0.0970625
\(356\) 0 0
\(357\) −1679.97 −0.249057
\(358\) 0 0
\(359\) −563.557 −0.0828507 −0.0414254 0.999142i \(-0.513190\pi\)
−0.0414254 + 0.999142i \(0.513190\pi\)
\(360\) 0 0
\(361\) −1760.23 −0.256631
\(362\) 0 0
\(363\) 23985.7 3.46811
\(364\) 0 0
\(365\) 3254.80 0.466751
\(366\) 0 0
\(367\) 7705.08 1.09592 0.547959 0.836505i \(-0.315405\pi\)
0.547959 + 0.836505i \(0.315405\pi\)
\(368\) 0 0
\(369\) −4066.20 −0.573653
\(370\) 0 0
\(371\) 517.303 0.0723909
\(372\) 0 0
\(373\) −2346.53 −0.325734 −0.162867 0.986648i \(-0.552074\pi\)
−0.162867 + 0.986648i \(0.552074\pi\)
\(374\) 0 0
\(375\) −7482.83 −1.03043
\(376\) 0 0
\(377\) 3236.78 0.442182
\(378\) 0 0
\(379\) −3390.67 −0.459544 −0.229772 0.973244i \(-0.573798\pi\)
−0.229772 + 0.973244i \(0.573798\pi\)
\(380\) 0 0
\(381\) 13039.1 1.75332
\(382\) 0 0
\(383\) 837.622 0.111751 0.0558753 0.998438i \(-0.482205\pi\)
0.0558753 + 0.998438i \(0.482205\pi\)
\(384\) 0 0
\(385\) 2770.27 0.366717
\(386\) 0 0
\(387\) 2122.32 0.278769
\(388\) 0 0
\(389\) 12669.0 1.65126 0.825632 0.564209i \(-0.190819\pi\)
0.825632 + 0.564209i \(0.190819\pi\)
\(390\) 0 0
\(391\) 553.474 0.0715866
\(392\) 0 0
\(393\) −2800.23 −0.359422
\(394\) 0 0
\(395\) −2011.40 −0.256213
\(396\) 0 0
\(397\) 8731.37 1.10382 0.551908 0.833905i \(-0.313900\pi\)
0.551908 + 0.833905i \(0.313900\pi\)
\(398\) 0 0
\(399\) 4985.00 0.625469
\(400\) 0 0
\(401\) −8265.25 −1.02929 −0.514647 0.857402i \(-0.672077\pi\)
−0.514647 + 0.857402i \(0.672077\pi\)
\(402\) 0 0
\(403\) 2119.49 0.261984
\(404\) 0 0
\(405\) −3040.66 −0.373066
\(406\) 0 0
\(407\) −5678.26 −0.691550
\(408\) 0 0
\(409\) 15088.7 1.82418 0.912091 0.409988i \(-0.134467\pi\)
0.912091 + 0.409988i \(0.134467\pi\)
\(410\) 0 0
\(411\) 2797.13 0.335699
\(412\) 0 0
\(413\) 1314.83 0.156655
\(414\) 0 0
\(415\) 1506.43 0.178187
\(416\) 0 0
\(417\) −17064.0 −2.00390
\(418\) 0 0
\(419\) −6923.33 −0.807224 −0.403612 0.914930i \(-0.632245\pi\)
−0.403612 + 0.914930i \(0.632245\pi\)
\(420\) 0 0
\(421\) −3156.36 −0.365396 −0.182698 0.983169i \(-0.558483\pi\)
−0.182698 + 0.983169i \(0.558483\pi\)
\(422\) 0 0
\(423\) 6052.23 0.695672
\(424\) 0 0
\(425\) −2549.93 −0.291034
\(426\) 0 0
\(427\) −7259.09 −0.822698
\(428\) 0 0
\(429\) 16836.8 1.89484
\(430\) 0 0
\(431\) 6355.74 0.710314 0.355157 0.934807i \(-0.384427\pi\)
0.355157 + 0.934807i \(0.384427\pi\)
\(432\) 0 0
\(433\) −5151.76 −0.571773 −0.285886 0.958264i \(-0.592288\pi\)
−0.285886 + 0.958264i \(0.592288\pi\)
\(434\) 0 0
\(435\) −3123.52 −0.344279
\(436\) 0 0
\(437\) −1642.33 −0.179779
\(438\) 0 0
\(439\) 11082.9 1.20492 0.602458 0.798150i \(-0.294188\pi\)
0.602458 + 0.798150i \(0.294188\pi\)
\(440\) 0 0
\(441\) −7164.79 −0.773652
\(442\) 0 0
\(443\) 13618.8 1.46060 0.730301 0.683126i \(-0.239380\pi\)
0.730301 + 0.683126i \(0.239380\pi\)
\(444\) 0 0
\(445\) 5570.68 0.593428
\(446\) 0 0
\(447\) 8845.98 0.936019
\(448\) 0 0
\(449\) −13134.4 −1.38051 −0.690257 0.723564i \(-0.742503\pi\)
−0.690257 + 0.723564i \(0.742503\pi\)
\(450\) 0 0
\(451\) 9758.85 1.01891
\(452\) 0 0
\(453\) −18834.1 −1.95342
\(454\) 0 0
\(455\) 1377.13 0.141892
\(456\) 0 0
\(457\) 12711.1 1.30109 0.650546 0.759467i \(-0.274540\pi\)
0.650546 + 0.759467i \(0.274540\pi\)
\(458\) 0 0
\(459\) 203.728 0.0207172
\(460\) 0 0
\(461\) 3915.10 0.395541 0.197771 0.980248i \(-0.436630\pi\)
0.197771 + 0.980248i \(0.436630\pi\)
\(462\) 0 0
\(463\) −16351.0 −1.64124 −0.820619 0.571475i \(-0.806371\pi\)
−0.820619 + 0.571475i \(0.806371\pi\)
\(464\) 0 0
\(465\) −2045.33 −0.203978
\(466\) 0 0
\(467\) −7265.82 −0.719961 −0.359981 0.932960i \(-0.617217\pi\)
−0.359981 + 0.932960i \(0.617217\pi\)
\(468\) 0 0
\(469\) 8196.30 0.806973
\(470\) 0 0
\(471\) 8261.10 0.808177
\(472\) 0 0
\(473\) −5093.56 −0.495142
\(474\) 0 0
\(475\) 7566.43 0.730888
\(476\) 0 0
\(477\) −1548.36 −0.148626
\(478\) 0 0
\(479\) −19081.1 −1.82012 −0.910060 0.414476i \(-0.863965\pi\)
−0.910060 + 0.414476i \(0.863965\pi\)
\(480\) 0 0
\(481\) −2822.73 −0.267579
\(482\) 0 0
\(483\) −1605.68 −0.151265
\(484\) 0 0
\(485\) −4471.49 −0.418639
\(486\) 0 0
\(487\) −12350.2 −1.14916 −0.574580 0.818448i \(-0.694835\pi\)
−0.574580 + 0.818448i \(0.694835\pi\)
\(488\) 0 0
\(489\) −11004.7 −1.01769
\(490\) 0 0
\(491\) −11944.1 −1.09782 −0.548910 0.835881i \(-0.684957\pi\)
−0.548910 + 0.835881i \(0.684957\pi\)
\(492\) 0 0
\(493\) −2320.02 −0.211944
\(494\) 0 0
\(495\) −8291.80 −0.752907
\(496\) 0 0
\(497\) −1398.96 −0.126261
\(498\) 0 0
\(499\) −1772.57 −0.159020 −0.0795101 0.996834i \(-0.525336\pi\)
−0.0795101 + 0.996834i \(0.525336\pi\)
\(500\) 0 0
\(501\) −29252.6 −2.60860
\(502\) 0 0
\(503\) −5517.89 −0.489126 −0.244563 0.969633i \(-0.578645\pi\)
−0.244563 + 0.969633i \(0.578645\pi\)
\(504\) 0 0
\(505\) 551.957 0.0486371
\(506\) 0 0
\(507\) −7944.36 −0.695900
\(508\) 0 0
\(509\) −21201.4 −1.84624 −0.923120 0.384512i \(-0.874370\pi\)
−0.923120 + 0.384512i \(0.874370\pi\)
\(510\) 0 0
\(511\) −7013.51 −0.607161
\(512\) 0 0
\(513\) −604.525 −0.0520281
\(514\) 0 0
\(515\) 1125.73 0.0963218
\(516\) 0 0
\(517\) −14525.3 −1.23563
\(518\) 0 0
\(519\) −9897.17 −0.837067
\(520\) 0 0
\(521\) −9203.33 −0.773906 −0.386953 0.922099i \(-0.626472\pi\)
−0.386953 + 0.922099i \(0.626472\pi\)
\(522\) 0 0
\(523\) 21900.1 1.83103 0.915513 0.402289i \(-0.131785\pi\)
0.915513 + 0.402289i \(0.131785\pi\)
\(524\) 0 0
\(525\) 7397.60 0.614967
\(526\) 0 0
\(527\) −1519.19 −0.125573
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −3935.47 −0.321629
\(532\) 0 0
\(533\) 4851.24 0.394241
\(534\) 0 0
\(535\) −4439.19 −0.358735
\(536\) 0 0
\(537\) −4480.18 −0.360026
\(538\) 0 0
\(539\) 17195.4 1.37414
\(540\) 0 0
\(541\) −3490.20 −0.277367 −0.138684 0.990337i \(-0.544287\pi\)
−0.138684 + 0.990337i \(0.544287\pi\)
\(542\) 0 0
\(543\) −30687.5 −2.42528
\(544\) 0 0
\(545\) 8120.87 0.638275
\(546\) 0 0
\(547\) 16506.0 1.29021 0.645106 0.764093i \(-0.276813\pi\)
0.645106 + 0.764093i \(0.276813\pi\)
\(548\) 0 0
\(549\) 21727.5 1.68908
\(550\) 0 0
\(551\) 6884.23 0.532265
\(552\) 0 0
\(553\) 4334.19 0.333289
\(554\) 0 0
\(555\) 2723.96 0.208335
\(556\) 0 0
\(557\) −10101.1 −0.768400 −0.384200 0.923250i \(-0.625523\pi\)
−0.384200 + 0.923250i \(0.625523\pi\)
\(558\) 0 0
\(559\) −2532.07 −0.191583
\(560\) 0 0
\(561\) −12068.1 −0.908227
\(562\) 0 0
\(563\) −1303.81 −0.0976005 −0.0488003 0.998809i \(-0.515540\pi\)
−0.0488003 + 0.998809i \(0.515540\pi\)
\(564\) 0 0
\(565\) −216.860 −0.0161475
\(566\) 0 0
\(567\) 6552.09 0.485294
\(568\) 0 0
\(569\) 1993.03 0.146840 0.0734201 0.997301i \(-0.476609\pi\)
0.0734201 + 0.997301i \(0.476609\pi\)
\(570\) 0 0
\(571\) −1776.69 −0.130214 −0.0651071 0.997878i \(-0.520739\pi\)
−0.0651071 + 0.997878i \(0.520739\pi\)
\(572\) 0 0
\(573\) −17555.8 −1.27994
\(574\) 0 0
\(575\) −2437.17 −0.176760
\(576\) 0 0
\(577\) 9542.03 0.688457 0.344229 0.938886i \(-0.388140\pi\)
0.344229 + 0.938886i \(0.388140\pi\)
\(578\) 0 0
\(579\) −24446.4 −1.75468
\(580\) 0 0
\(581\) −3246.09 −0.231791
\(582\) 0 0
\(583\) 3716.05 0.263985
\(584\) 0 0
\(585\) −4121.95 −0.291319
\(586\) 0 0
\(587\) −8518.89 −0.598999 −0.299499 0.954096i \(-0.596820\pi\)
−0.299499 + 0.954096i \(0.596820\pi\)
\(588\) 0 0
\(589\) 4507.91 0.315357
\(590\) 0 0
\(591\) −25067.8 −1.74475
\(592\) 0 0
\(593\) 11295.1 0.782181 0.391090 0.920352i \(-0.372098\pi\)
0.391090 + 0.920352i \(0.372098\pi\)
\(594\) 0 0
\(595\) −987.086 −0.0680110
\(596\) 0 0
\(597\) 33995.3 2.33054
\(598\) 0 0
\(599\) −7880.18 −0.537522 −0.268761 0.963207i \(-0.586614\pi\)
−0.268761 + 0.963207i \(0.586614\pi\)
\(600\) 0 0
\(601\) 21198.1 1.43875 0.719375 0.694622i \(-0.244428\pi\)
0.719375 + 0.694622i \(0.244428\pi\)
\(602\) 0 0
\(603\) −24532.7 −1.65680
\(604\) 0 0
\(605\) 14093.1 0.947050
\(606\) 0 0
\(607\) 13439.0 0.898638 0.449319 0.893371i \(-0.351667\pi\)
0.449319 + 0.893371i \(0.351667\pi\)
\(608\) 0 0
\(609\) 6730.62 0.447846
\(610\) 0 0
\(611\) −7220.70 −0.478098
\(612\) 0 0
\(613\) −24209.4 −1.59512 −0.797561 0.603238i \(-0.793877\pi\)
−0.797561 + 0.603238i \(0.793877\pi\)
\(614\) 0 0
\(615\) −4681.49 −0.306953
\(616\) 0 0
\(617\) −21713.2 −1.41676 −0.708379 0.705832i \(-0.750573\pi\)
−0.708379 + 0.705832i \(0.750573\pi\)
\(618\) 0 0
\(619\) −15809.5 −1.02656 −0.513279 0.858222i \(-0.671569\pi\)
−0.513279 + 0.858222i \(0.671569\pi\)
\(620\) 0 0
\(621\) 194.719 0.0125826
\(622\) 0 0
\(623\) −12003.8 −0.771947
\(624\) 0 0
\(625\) 8848.88 0.566328
\(626\) 0 0
\(627\) 35809.8 2.28087
\(628\) 0 0
\(629\) 2023.25 0.128254
\(630\) 0 0
\(631\) 29674.4 1.87214 0.936068 0.351818i \(-0.114436\pi\)
0.936068 + 0.351818i \(0.114436\pi\)
\(632\) 0 0
\(633\) 19565.2 1.22851
\(634\) 0 0
\(635\) 7661.28 0.478785
\(636\) 0 0
\(637\) 8548.06 0.531690
\(638\) 0 0
\(639\) 4187.28 0.259227
\(640\) 0 0
\(641\) −6257.34 −0.385569 −0.192785 0.981241i \(-0.561752\pi\)
−0.192785 + 0.981241i \(0.561752\pi\)
\(642\) 0 0
\(643\) −1466.42 −0.0899377 −0.0449688 0.998988i \(-0.514319\pi\)
−0.0449688 + 0.998988i \(0.514319\pi\)
\(644\) 0 0
\(645\) 2443.47 0.149165
\(646\) 0 0
\(647\) −25202.9 −1.53142 −0.765711 0.643185i \(-0.777613\pi\)
−0.765711 + 0.643185i \(0.777613\pi\)
\(648\) 0 0
\(649\) 9445.10 0.571267
\(650\) 0 0
\(651\) 4407.32 0.265340
\(652\) 0 0
\(653\) −27800.2 −1.66601 −0.833006 0.553263i \(-0.813382\pi\)
−0.833006 + 0.553263i \(0.813382\pi\)
\(654\) 0 0
\(655\) −1645.31 −0.0981489
\(656\) 0 0
\(657\) 20992.4 1.24656
\(658\) 0 0
\(659\) −1065.63 −0.0629912 −0.0314956 0.999504i \(-0.510027\pi\)
−0.0314956 + 0.999504i \(0.510027\pi\)
\(660\) 0 0
\(661\) −14149.4 −0.832599 −0.416300 0.909227i \(-0.636673\pi\)
−0.416300 + 0.909227i \(0.636673\pi\)
\(662\) 0 0
\(663\) −5999.19 −0.351417
\(664\) 0 0
\(665\) 2928.99 0.170799
\(666\) 0 0
\(667\) −2217.43 −0.128725
\(668\) 0 0
\(669\) −4092.34 −0.236501
\(670\) 0 0
\(671\) −52145.9 −3.00010
\(672\) 0 0
\(673\) 1643.03 0.0941073 0.0470536 0.998892i \(-0.485017\pi\)
0.0470536 + 0.998892i \(0.485017\pi\)
\(674\) 0 0
\(675\) −897.098 −0.0511545
\(676\) 0 0
\(677\) 30414.5 1.72662 0.863311 0.504672i \(-0.168387\pi\)
0.863311 + 0.504672i \(0.168387\pi\)
\(678\) 0 0
\(679\) 9635.25 0.544576
\(680\) 0 0
\(681\) −11283.3 −0.634914
\(682\) 0 0
\(683\) −18182.8 −1.01866 −0.509331 0.860571i \(-0.670107\pi\)
−0.509331 + 0.860571i \(0.670107\pi\)
\(684\) 0 0
\(685\) 1643.49 0.0916708
\(686\) 0 0
\(687\) 23480.2 1.30397
\(688\) 0 0
\(689\) 1847.29 0.102143
\(690\) 0 0
\(691\) −15330.4 −0.843989 −0.421995 0.906598i \(-0.638670\pi\)
−0.421995 + 0.906598i \(0.638670\pi\)
\(692\) 0 0
\(693\) 17867.3 0.979400
\(694\) 0 0
\(695\) −10026.1 −0.547212
\(696\) 0 0
\(697\) −3477.22 −0.188966
\(698\) 0 0
\(699\) 34598.5 1.87215
\(700\) 0 0
\(701\) −1245.98 −0.0671328 −0.0335664 0.999436i \(-0.510687\pi\)
−0.0335664 + 0.999436i \(0.510687\pi\)
\(702\) 0 0
\(703\) −6003.61 −0.322092
\(704\) 0 0
\(705\) 6968.04 0.372243
\(706\) 0 0
\(707\) −1189.37 −0.0632684
\(708\) 0 0
\(709\) −10282.8 −0.544683 −0.272341 0.962201i \(-0.587798\pi\)
−0.272341 + 0.962201i \(0.587798\pi\)
\(710\) 0 0
\(711\) −12972.9 −0.684276
\(712\) 0 0
\(713\) −1452.01 −0.0762668
\(714\) 0 0
\(715\) 9892.66 0.517433
\(716\) 0 0
\(717\) −14803.1 −0.771032
\(718\) 0 0
\(719\) −14800.1 −0.767666 −0.383833 0.923403i \(-0.625396\pi\)
−0.383833 + 0.923403i \(0.625396\pi\)
\(720\) 0 0
\(721\) −2425.75 −0.125298
\(722\) 0 0
\(723\) 13159.6 0.676915
\(724\) 0 0
\(725\) 10216.0 0.523328
\(726\) 0 0
\(727\) −24168.5 −1.23296 −0.616478 0.787372i \(-0.711441\pi\)
−0.616478 + 0.787372i \(0.711441\pi\)
\(728\) 0 0
\(729\) −21308.7 −1.08259
\(730\) 0 0
\(731\) 1814.91 0.0918287
\(732\) 0 0
\(733\) −37052.5 −1.86707 −0.933537 0.358481i \(-0.883295\pi\)
−0.933537 + 0.358481i \(0.883295\pi\)
\(734\) 0 0
\(735\) −8248.95 −0.413969
\(736\) 0 0
\(737\) 58878.3 2.94276
\(738\) 0 0
\(739\) 2133.67 0.106209 0.0531045 0.998589i \(-0.483088\pi\)
0.0531045 + 0.998589i \(0.483088\pi\)
\(740\) 0 0
\(741\) 17801.5 0.882529
\(742\) 0 0
\(743\) −26389.0 −1.30299 −0.651494 0.758654i \(-0.725857\pi\)
−0.651494 + 0.758654i \(0.725857\pi\)
\(744\) 0 0
\(745\) 5197.56 0.255602
\(746\) 0 0
\(747\) 9716.00 0.475890
\(748\) 0 0
\(749\) 9565.66 0.466651
\(750\) 0 0
\(751\) 10857.9 0.527579 0.263789 0.964580i \(-0.415028\pi\)
0.263789 + 0.964580i \(0.415028\pi\)
\(752\) 0 0
\(753\) −11022.6 −0.533449
\(754\) 0 0
\(755\) −11066.2 −0.533430
\(756\) 0 0
\(757\) 4597.95 0.220760 0.110380 0.993889i \(-0.464793\pi\)
0.110380 + 0.993889i \(0.464793\pi\)
\(758\) 0 0
\(759\) −11534.5 −0.551613
\(760\) 0 0
\(761\) 1093.90 0.0521077 0.0260538 0.999661i \(-0.491706\pi\)
0.0260538 + 0.999661i \(0.491706\pi\)
\(762\) 0 0
\(763\) −17499.0 −0.830284
\(764\) 0 0
\(765\) 2954.49 0.139634
\(766\) 0 0
\(767\) 4695.27 0.221038
\(768\) 0 0
\(769\) 32217.7 1.51079 0.755397 0.655267i \(-0.227444\pi\)
0.755397 + 0.655267i \(0.227444\pi\)
\(770\) 0 0
\(771\) 20728.4 0.968244
\(772\) 0 0
\(773\) −2841.25 −0.132203 −0.0661014 0.997813i \(-0.521056\pi\)
−0.0661014 + 0.997813i \(0.521056\pi\)
\(774\) 0 0
\(775\) 6689.61 0.310062
\(776\) 0 0
\(777\) −5869.64 −0.271007
\(778\) 0 0
\(779\) 10318.0 0.474558
\(780\) 0 0
\(781\) −10049.4 −0.460432
\(782\) 0 0
\(783\) −816.214 −0.0372530
\(784\) 0 0
\(785\) 4853.91 0.220692
\(786\) 0 0
\(787\) −14015.7 −0.634822 −0.317411 0.948288i \(-0.602814\pi\)
−0.317411 + 0.948288i \(0.602814\pi\)
\(788\) 0 0
\(789\) 51227.6 2.31147
\(790\) 0 0
\(791\) 467.294 0.0210051
\(792\) 0 0
\(793\) −25922.3 −1.16082
\(794\) 0 0
\(795\) −1782.66 −0.0795274
\(796\) 0 0
\(797\) −33612.0 −1.49385 −0.746925 0.664908i \(-0.768471\pi\)
−0.746925 + 0.664908i \(0.768471\pi\)
\(798\) 0 0
\(799\) 5175.57 0.229160
\(800\) 0 0
\(801\) 35929.1 1.58489
\(802\) 0 0
\(803\) −50381.7 −2.21411
\(804\) 0 0
\(805\) −943.439 −0.0413066
\(806\) 0 0
\(807\) 21249.2 0.926897
\(808\) 0 0
\(809\) −377.089 −0.0163878 −0.00819390 0.999966i \(-0.502608\pi\)
−0.00819390 + 0.999966i \(0.502608\pi\)
\(810\) 0 0
\(811\) 24291.0 1.05175 0.525877 0.850561i \(-0.323737\pi\)
0.525877 + 0.850561i \(0.323737\pi\)
\(812\) 0 0
\(813\) −16530.9 −0.713116
\(814\) 0 0
\(815\) −6465.93 −0.277904
\(816\) 0 0
\(817\) −5385.40 −0.230614
\(818\) 0 0
\(819\) 8882.07 0.378956
\(820\) 0 0
\(821\) 5538.66 0.235445 0.117723 0.993047i \(-0.462441\pi\)
0.117723 + 0.993047i \(0.462441\pi\)
\(822\) 0 0
\(823\) 41353.5 1.75151 0.875756 0.482755i \(-0.160364\pi\)
0.875756 + 0.482755i \(0.160364\pi\)
\(824\) 0 0
\(825\) 53140.8 2.24258
\(826\) 0 0
\(827\) −3467.59 −0.145804 −0.0729019 0.997339i \(-0.523226\pi\)
−0.0729019 + 0.997339i \(0.523226\pi\)
\(828\) 0 0
\(829\) −5524.89 −0.231468 −0.115734 0.993280i \(-0.536922\pi\)
−0.115734 + 0.993280i \(0.536922\pi\)
\(830\) 0 0
\(831\) −39233.4 −1.63778
\(832\) 0 0
\(833\) −6126.98 −0.254847
\(834\) 0 0
\(835\) −17187.7 −0.712341
\(836\) 0 0
\(837\) −534.470 −0.0220717
\(838\) 0 0
\(839\) 42684.4 1.75641 0.878206 0.478283i \(-0.158741\pi\)
0.878206 + 0.478283i \(0.158741\pi\)
\(840\) 0 0
\(841\) −15094.1 −0.618889
\(842\) 0 0
\(843\) 55315.6 2.25999
\(844\) 0 0
\(845\) −4667.80 −0.190032
\(846\) 0 0
\(847\) −30368.1 −1.23195
\(848\) 0 0
\(849\) −38665.0 −1.56299
\(850\) 0 0
\(851\) 1933.78 0.0778956
\(852\) 0 0
\(853\) −39711.0 −1.59400 −0.796998 0.603981i \(-0.793580\pi\)
−0.796998 + 0.603981i \(0.793580\pi\)
\(854\) 0 0
\(855\) −8766.90 −0.350669
\(856\) 0 0
\(857\) 17088.5 0.681136 0.340568 0.940220i \(-0.389381\pi\)
0.340568 + 0.940220i \(0.389381\pi\)
\(858\) 0 0
\(859\) 34573.9 1.37328 0.686639 0.726999i \(-0.259085\pi\)
0.686639 + 0.726999i \(0.259085\pi\)
\(860\) 0 0
\(861\) 10087.8 0.399292
\(862\) 0 0
\(863\) −34786.0 −1.37211 −0.686054 0.727551i \(-0.740658\pi\)
−0.686054 + 0.727551i \(0.740658\pi\)
\(864\) 0 0
\(865\) −5815.20 −0.228581
\(866\) 0 0
\(867\) −32182.1 −1.26063
\(868\) 0 0
\(869\) 31134.8 1.21539
\(870\) 0 0
\(871\) 29269.1 1.13863
\(872\) 0 0
\(873\) −28839.7 −1.11807
\(874\) 0 0
\(875\) 9473.93 0.366031
\(876\) 0 0
\(877\) −10814.7 −0.416406 −0.208203 0.978086i \(-0.566761\pi\)
−0.208203 + 0.978086i \(0.566761\pi\)
\(878\) 0 0
\(879\) −26644.3 −1.02240
\(880\) 0 0
\(881\) 10411.1 0.398135 0.199068 0.979986i \(-0.436209\pi\)
0.199068 + 0.979986i \(0.436209\pi\)
\(882\) 0 0
\(883\) −28277.5 −1.07771 −0.538853 0.842400i \(-0.681142\pi\)
−0.538853 + 0.842400i \(0.681142\pi\)
\(884\) 0 0
\(885\) −4530.98 −0.172099
\(886\) 0 0
\(887\) 16488.4 0.624154 0.312077 0.950057i \(-0.398975\pi\)
0.312077 + 0.950057i \(0.398975\pi\)
\(888\) 0 0
\(889\) −16508.7 −0.622816
\(890\) 0 0
\(891\) 47067.0 1.76970
\(892\) 0 0
\(893\) −15357.6 −0.575499
\(894\) 0 0
\(895\) −2632.38 −0.0983139
\(896\) 0 0
\(897\) −5733.92 −0.213434
\(898\) 0 0
\(899\) 6086.46 0.225801
\(900\) 0 0
\(901\) −1324.08 −0.0489585
\(902\) 0 0
\(903\) −5265.23 −0.194038
\(904\) 0 0
\(905\) −18030.8 −0.662280
\(906\) 0 0
\(907\) 42168.8 1.54376 0.771880 0.635768i \(-0.219316\pi\)
0.771880 + 0.635768i \(0.219316\pi\)
\(908\) 0 0
\(909\) 3559.95 0.129897
\(910\) 0 0
\(911\) 20373.7 0.740954 0.370477 0.928842i \(-0.379194\pi\)
0.370477 + 0.928842i \(0.379194\pi\)
\(912\) 0 0
\(913\) −23318.3 −0.845262
\(914\) 0 0
\(915\) 25015.3 0.903803
\(916\) 0 0
\(917\) 3545.34 0.127675
\(918\) 0 0
\(919\) 36823.5 1.32176 0.660879 0.750492i \(-0.270183\pi\)
0.660879 + 0.750492i \(0.270183\pi\)
\(920\) 0 0
\(921\) −9079.41 −0.324839
\(922\) 0 0
\(923\) −4995.70 −0.178153
\(924\) 0 0
\(925\) −8909.19 −0.316683
\(926\) 0 0
\(927\) 7260.62 0.257249
\(928\) 0 0
\(929\) 10251.0 0.362028 0.181014 0.983480i \(-0.442062\pi\)
0.181014 + 0.983480i \(0.442062\pi\)
\(930\) 0 0
\(931\) 18180.7 0.640009
\(932\) 0 0
\(933\) −1180.37 −0.0414187
\(934\) 0 0
\(935\) −7090.75 −0.248013
\(936\) 0 0
\(937\) −28272.1 −0.985708 −0.492854 0.870112i \(-0.664046\pi\)
−0.492854 + 0.870112i \(0.664046\pi\)
\(938\) 0 0
\(939\) 11380.0 0.395499
\(940\) 0 0
\(941\) −43672.5 −1.51295 −0.756473 0.654025i \(-0.773079\pi\)
−0.756473 + 0.654025i \(0.773079\pi\)
\(942\) 0 0
\(943\) −3323.46 −0.114769
\(944\) 0 0
\(945\) −347.270 −0.0119542
\(946\) 0 0
\(947\) −43064.2 −1.47772 −0.738858 0.673861i \(-0.764635\pi\)
−0.738858 + 0.673861i \(0.764635\pi\)
\(948\) 0 0
\(949\) −25045.3 −0.856698
\(950\) 0 0
\(951\) −18151.1 −0.618918
\(952\) 0 0
\(953\) −11297.0 −0.383994 −0.191997 0.981396i \(-0.561496\pi\)
−0.191997 + 0.981396i \(0.561496\pi\)
\(954\) 0 0
\(955\) −10315.1 −0.349518
\(956\) 0 0
\(957\) 48349.5 1.63314
\(958\) 0 0
\(959\) −3541.42 −0.119248
\(960\) 0 0
\(961\) −25805.5 −0.866218
\(962\) 0 0
\(963\) −28631.4 −0.958082
\(964\) 0 0
\(965\) −14363.8 −0.479157
\(966\) 0 0
\(967\) 25167.0 0.836937 0.418468 0.908231i \(-0.362567\pi\)
0.418468 + 0.908231i \(0.362567\pi\)
\(968\) 0 0
\(969\) −12759.6 −0.423009
\(970\) 0 0
\(971\) 2387.35 0.0789018 0.0394509 0.999222i \(-0.487439\pi\)
0.0394509 + 0.999222i \(0.487439\pi\)
\(972\) 0 0
\(973\) 21604.5 0.711828
\(974\) 0 0
\(975\) 26416.9 0.867711
\(976\) 0 0
\(977\) −50653.2 −1.65869 −0.829345 0.558736i \(-0.811286\pi\)
−0.829345 + 0.558736i \(0.811286\pi\)
\(978\) 0 0
\(979\) −86229.7 −2.81503
\(980\) 0 0
\(981\) 52377.0 1.70466
\(982\) 0 0
\(983\) 3927.47 0.127433 0.0637166 0.997968i \(-0.479705\pi\)
0.0637166 + 0.997968i \(0.479705\pi\)
\(984\) 0 0
\(985\) −14728.9 −0.476447
\(986\) 0 0
\(987\) −15014.9 −0.484223
\(988\) 0 0
\(989\) 1734.66 0.0557723
\(990\) 0 0
\(991\) 41415.4 1.32755 0.663776 0.747931i \(-0.268953\pi\)
0.663776 + 0.747931i \(0.268953\pi\)
\(992\) 0 0
\(993\) 23979.4 0.766328
\(994\) 0 0
\(995\) 19974.3 0.636411
\(996\) 0 0
\(997\) 54644.4 1.73581 0.867907 0.496727i \(-0.165465\pi\)
0.867907 + 0.496727i \(0.165465\pi\)
\(998\) 0 0
\(999\) 711.805 0.0225430
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 736.4.a.e.1.8 8
4.3 odd 2 736.4.a.f.1.1 yes 8
8.3 odd 2 1472.4.a.bg.1.8 8
8.5 even 2 1472.4.a.bh.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.4.a.e.1.8 8 1.1 even 1 trivial
736.4.a.f.1.1 yes 8 4.3 odd 2
1472.4.a.bg.1.8 8 8.3 odd 2
1472.4.a.bh.1.1 8 8.5 even 2