Properties

Label 160.6.d.a.81.4
Level $160$
Weight $6$
Character 160.81
Analytic conductor $25.661$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,6,Mod(81,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.81");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} + \cdots + 1099511627776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{4}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 81.4
Root \(2.93366 + 2.71913i\) of defining polynomial
Character \(\chi\) \(=\) 160.81
Dual form 160.6.d.a.81.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-18.7876i q^{3} +25.0000i q^{5} +107.536 q^{7} -109.975 q^{9} +O(q^{10})\) \(q-18.7876i q^{3} +25.0000i q^{5} +107.536 q^{7} -109.975 q^{9} -272.206i q^{11} -198.402i q^{13} +469.691 q^{15} +2065.79 q^{17} +1891.04i q^{19} -2020.35i q^{21} +987.677 q^{23} -625.000 q^{25} -2499.22i q^{27} -8015.26i q^{29} -827.342 q^{31} -5114.10 q^{33} +2688.40i q^{35} -9426.30i q^{37} -3727.50 q^{39} -8221.07 q^{41} -9301.63i q^{43} -2749.38i q^{45} -13837.9 q^{47} -5242.97 q^{49} -38811.3i q^{51} +27751.2i q^{53} +6805.14 q^{55} +35528.1 q^{57} -25106.1i q^{59} -26404.6i q^{61} -11826.3 q^{63} +4960.05 q^{65} -38563.9i q^{67} -18556.1i q^{69} +71073.0 q^{71} +18622.0 q^{73} +11742.3i q^{75} -29271.9i q^{77} +75599.4 q^{79} -73678.4 q^{81} +125298. i q^{83} +51644.7i q^{85} -150588. q^{87} +30341.5 q^{89} -21335.4i q^{91} +15543.8i q^{93} -47275.9 q^{95} +15635.2 q^{97} +29935.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 18.7876i − 1.20523i −0.798033 0.602614i \(-0.794126\pi\)
0.798033 0.602614i \(-0.205874\pi\)
\(4\) 0 0
\(5\) 25.0000i 0.447214i
\(6\) 0 0
\(7\) 107.536 0.829487 0.414743 0.909938i \(-0.363871\pi\)
0.414743 + 0.909938i \(0.363871\pi\)
\(8\) 0 0
\(9\) −109.975 −0.452573
\(10\) 0 0
\(11\) − 272.206i − 0.678290i −0.940734 0.339145i \(-0.889862\pi\)
0.940734 0.339145i \(-0.110138\pi\)
\(12\) 0 0
\(13\) − 198.402i − 0.325602i −0.986659 0.162801i \(-0.947947\pi\)
0.986659 0.162801i \(-0.0520529\pi\)
\(14\) 0 0
\(15\) 469.691 0.538994
\(16\) 0 0
\(17\) 2065.79 1.73366 0.866829 0.498606i \(-0.166154\pi\)
0.866829 + 0.498606i \(0.166154\pi\)
\(18\) 0 0
\(19\) 1891.04i 1.20176i 0.799341 + 0.600878i \(0.205182\pi\)
−0.799341 + 0.600878i \(0.794818\pi\)
\(20\) 0 0
\(21\) − 2020.35i − 0.999720i
\(22\) 0 0
\(23\) 987.677 0.389310 0.194655 0.980872i \(-0.437641\pi\)
0.194655 + 0.980872i \(0.437641\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) − 2499.22i − 0.659774i
\(28\) 0 0
\(29\) − 8015.26i − 1.76979i −0.465788 0.884896i \(-0.654229\pi\)
0.465788 0.884896i \(-0.345771\pi\)
\(30\) 0 0
\(31\) −827.342 −0.154625 −0.0773127 0.997007i \(-0.524634\pi\)
−0.0773127 + 0.997007i \(0.524634\pi\)
\(32\) 0 0
\(33\) −5114.10 −0.817494
\(34\) 0 0
\(35\) 2688.40i 0.370958i
\(36\) 0 0
\(37\) − 9426.30i − 1.13198i −0.824414 0.565988i \(-0.808495\pi\)
0.824414 0.565988i \(-0.191505\pi\)
\(38\) 0 0
\(39\) −3727.50 −0.392425
\(40\) 0 0
\(41\) −8221.07 −0.763780 −0.381890 0.924208i \(-0.624727\pi\)
−0.381890 + 0.924208i \(0.624727\pi\)
\(42\) 0 0
\(43\) − 9301.63i − 0.767164i −0.923507 0.383582i \(-0.874690\pi\)
0.923507 0.383582i \(-0.125310\pi\)
\(44\) 0 0
\(45\) − 2749.38i − 0.202397i
\(46\) 0 0
\(47\) −13837.9 −0.913745 −0.456873 0.889532i \(-0.651031\pi\)
−0.456873 + 0.889532i \(0.651031\pi\)
\(48\) 0 0
\(49\) −5242.97 −0.311952
\(50\) 0 0
\(51\) − 38811.3i − 2.08945i
\(52\) 0 0
\(53\) 27751.2i 1.35704i 0.734582 + 0.678520i \(0.237378\pi\)
−0.734582 + 0.678520i \(0.762622\pi\)
\(54\) 0 0
\(55\) 6805.14 0.303340
\(56\) 0 0
\(57\) 35528.1 1.44839
\(58\) 0 0
\(59\) − 25106.1i − 0.938965i −0.882942 0.469482i \(-0.844441\pi\)
0.882942 0.469482i \(-0.155559\pi\)
\(60\) 0 0
\(61\) − 26404.6i − 0.908563i −0.890858 0.454282i \(-0.849896\pi\)
0.890858 0.454282i \(-0.150104\pi\)
\(62\) 0 0
\(63\) −11826.3 −0.375404
\(64\) 0 0
\(65\) 4960.05 0.145614
\(66\) 0 0
\(67\) − 38563.9i − 1.04953i −0.851248 0.524764i \(-0.824154\pi\)
0.851248 0.524764i \(-0.175846\pi\)
\(68\) 0 0
\(69\) − 18556.1i − 0.469207i
\(70\) 0 0
\(71\) 71073.0 1.67324 0.836621 0.547782i \(-0.184528\pi\)
0.836621 + 0.547782i \(0.184528\pi\)
\(72\) 0 0
\(73\) 18622.0 0.408996 0.204498 0.978867i \(-0.434444\pi\)
0.204498 + 0.978867i \(0.434444\pi\)
\(74\) 0 0
\(75\) 11742.3i 0.241045i
\(76\) 0 0
\(77\) − 29271.9i − 0.562632i
\(78\) 0 0
\(79\) 75599.4 1.36286 0.681429 0.731884i \(-0.261359\pi\)
0.681429 + 0.731884i \(0.261359\pi\)
\(80\) 0 0
\(81\) −73678.4 −1.24775
\(82\) 0 0
\(83\) 125298.i 1.99641i 0.0599304 + 0.998203i \(0.480912\pi\)
−0.0599304 + 0.998203i \(0.519088\pi\)
\(84\) 0 0
\(85\) 51644.7i 0.775315i
\(86\) 0 0
\(87\) −150588. −2.13300
\(88\) 0 0
\(89\) 30341.5 0.406034 0.203017 0.979175i \(-0.434925\pi\)
0.203017 + 0.979175i \(0.434925\pi\)
\(90\) 0 0
\(91\) − 21335.4i − 0.270083i
\(92\) 0 0
\(93\) 15543.8i 0.186359i
\(94\) 0 0
\(95\) −47275.9 −0.537441
\(96\) 0 0
\(97\) 15635.2 0.168723 0.0843617 0.996435i \(-0.473115\pi\)
0.0843617 + 0.996435i \(0.473115\pi\)
\(98\) 0 0
\(99\) 29935.9i 0.306976i
\(100\) 0 0
\(101\) − 102692.i − 1.00169i −0.865538 0.500843i \(-0.833024\pi\)
0.865538 0.500843i \(-0.166976\pi\)
\(102\) 0 0
\(103\) 35981.8 0.334187 0.167093 0.985941i \(-0.446562\pi\)
0.167093 + 0.985941i \(0.446562\pi\)
\(104\) 0 0
\(105\) 50508.8 0.447088
\(106\) 0 0
\(107\) 94984.6i 0.802035i 0.916070 + 0.401018i \(0.131343\pi\)
−0.916070 + 0.401018i \(0.868657\pi\)
\(108\) 0 0
\(109\) 173158.i 1.39597i 0.716113 + 0.697984i \(0.245919\pi\)
−0.716113 + 0.697984i \(0.754081\pi\)
\(110\) 0 0
\(111\) −177098. −1.36429
\(112\) 0 0
\(113\) −237780. −1.75178 −0.875891 0.482510i \(-0.839725\pi\)
−0.875891 + 0.482510i \(0.839725\pi\)
\(114\) 0 0
\(115\) 24691.9i 0.174105i
\(116\) 0 0
\(117\) 21819.3i 0.147359i
\(118\) 0 0
\(119\) 222147. 1.43805
\(120\) 0 0
\(121\) 86955.1 0.539923
\(122\) 0 0
\(123\) 154454.i 0.920529i
\(124\) 0 0
\(125\) − 15625.0i − 0.0894427i
\(126\) 0 0
\(127\) 63282.9 0.348158 0.174079 0.984732i \(-0.444305\pi\)
0.174079 + 0.984732i \(0.444305\pi\)
\(128\) 0 0
\(129\) −174756. −0.924607
\(130\) 0 0
\(131\) 180082.i 0.916835i 0.888737 + 0.458417i \(0.151584\pi\)
−0.888737 + 0.458417i \(0.848416\pi\)
\(132\) 0 0
\(133\) 203355.i 0.996840i
\(134\) 0 0
\(135\) 62480.5 0.295060
\(136\) 0 0
\(137\) 365739. 1.66483 0.832414 0.554155i \(-0.186958\pi\)
0.832414 + 0.554155i \(0.186958\pi\)
\(138\) 0 0
\(139\) − 111886.i − 0.491180i −0.969374 0.245590i \(-0.921018\pi\)
0.969374 0.245590i \(-0.0789817\pi\)
\(140\) 0 0
\(141\) 259981.i 1.10127i
\(142\) 0 0
\(143\) −54006.1 −0.220853
\(144\) 0 0
\(145\) 200381. 0.791475
\(146\) 0 0
\(147\) 98503.1i 0.375973i
\(148\) 0 0
\(149\) − 136480.i − 0.503621i −0.967777 0.251811i \(-0.918974\pi\)
0.967777 0.251811i \(-0.0810260\pi\)
\(150\) 0 0
\(151\) −186354. −0.665115 −0.332557 0.943083i \(-0.607912\pi\)
−0.332557 + 0.943083i \(0.607912\pi\)
\(152\) 0 0
\(153\) −227186. −0.784607
\(154\) 0 0
\(155\) − 20683.6i − 0.0691506i
\(156\) 0 0
\(157\) 74689.6i 0.241830i 0.992663 + 0.120915i \(0.0385829\pi\)
−0.992663 + 0.120915i \(0.961417\pi\)
\(158\) 0 0
\(159\) 521380. 1.63554
\(160\) 0 0
\(161\) 106211. 0.322927
\(162\) 0 0
\(163\) − 548779.i − 1.61781i −0.587937 0.808907i \(-0.700060\pi\)
0.587937 0.808907i \(-0.299940\pi\)
\(164\) 0 0
\(165\) − 127852.i − 0.365594i
\(166\) 0 0
\(167\) −224312. −0.622387 −0.311194 0.950347i \(-0.600729\pi\)
−0.311194 + 0.950347i \(0.600729\pi\)
\(168\) 0 0
\(169\) 331930. 0.893983
\(170\) 0 0
\(171\) − 207967.i − 0.543882i
\(172\) 0 0
\(173\) − 165260.i − 0.419809i −0.977722 0.209905i \(-0.932685\pi\)
0.977722 0.209905i \(-0.0673154\pi\)
\(174\) 0 0
\(175\) −67210.1 −0.165897
\(176\) 0 0
\(177\) −471684. −1.13167
\(178\) 0 0
\(179\) 431975.i 1.00769i 0.863795 + 0.503844i \(0.168081\pi\)
−0.863795 + 0.503844i \(0.831919\pi\)
\(180\) 0 0
\(181\) 216944.i 0.492210i 0.969243 + 0.246105i \(0.0791508\pi\)
−0.969243 + 0.246105i \(0.920849\pi\)
\(182\) 0 0
\(183\) −496080. −1.09503
\(184\) 0 0
\(185\) 235658. 0.506235
\(186\) 0 0
\(187\) − 562319.i − 1.17592i
\(188\) 0 0
\(189\) − 268756.i − 0.547274i
\(190\) 0 0
\(191\) 34566.0 0.0685592 0.0342796 0.999412i \(-0.489086\pi\)
0.0342796 + 0.999412i \(0.489086\pi\)
\(192\) 0 0
\(193\) −473601. −0.915208 −0.457604 0.889156i \(-0.651292\pi\)
−0.457604 + 0.889156i \(0.651292\pi\)
\(194\) 0 0
\(195\) − 93187.6i − 0.175498i
\(196\) 0 0
\(197\) 394784.i 0.724760i 0.932030 + 0.362380i \(0.118036\pi\)
−0.932030 + 0.362380i \(0.881964\pi\)
\(198\) 0 0
\(199\) 477089. 0.854017 0.427009 0.904248i \(-0.359567\pi\)
0.427009 + 0.904248i \(0.359567\pi\)
\(200\) 0 0
\(201\) −724524. −1.26492
\(202\) 0 0
\(203\) − 861930.i − 1.46802i
\(204\) 0 0
\(205\) − 205527.i − 0.341573i
\(206\) 0 0
\(207\) −108620. −0.176191
\(208\) 0 0
\(209\) 514751. 0.815139
\(210\) 0 0
\(211\) 435894.i 0.674023i 0.941500 + 0.337012i \(0.109416\pi\)
−0.941500 + 0.337012i \(0.890584\pi\)
\(212\) 0 0
\(213\) − 1.33529e6i − 2.01664i
\(214\) 0 0
\(215\) 232541. 0.343086
\(216\) 0 0
\(217\) −88969.2 −0.128260
\(218\) 0 0
\(219\) − 349863.i − 0.492933i
\(220\) 0 0
\(221\) − 409856.i − 0.564483i
\(222\) 0 0
\(223\) −1.14075e6 −1.53613 −0.768067 0.640369i \(-0.778781\pi\)
−0.768067 + 0.640369i \(0.778781\pi\)
\(224\) 0 0
\(225\) 68734.6 0.0905147
\(226\) 0 0
\(227\) 1.08895e6i 1.40262i 0.712854 + 0.701312i \(0.247402\pi\)
−0.712854 + 0.701312i \(0.752598\pi\)
\(228\) 0 0
\(229\) − 474386.i − 0.597783i −0.954287 0.298891i \(-0.903383\pi\)
0.954287 0.298891i \(-0.0966168\pi\)
\(230\) 0 0
\(231\) −549951. −0.678100
\(232\) 0 0
\(233\) −271057. −0.327093 −0.163547 0.986536i \(-0.552293\pi\)
−0.163547 + 0.986536i \(0.552293\pi\)
\(234\) 0 0
\(235\) − 345947.i − 0.408639i
\(236\) 0 0
\(237\) − 1.42033e6i − 1.64255i
\(238\) 0 0
\(239\) 824404. 0.933567 0.466784 0.884372i \(-0.345413\pi\)
0.466784 + 0.884372i \(0.345413\pi\)
\(240\) 0 0
\(241\) −86717.2 −0.0961750 −0.0480875 0.998843i \(-0.515313\pi\)
−0.0480875 + 0.998843i \(0.515313\pi\)
\(242\) 0 0
\(243\) 776933.i 0.844050i
\(244\) 0 0
\(245\) − 131074.i − 0.139509i
\(246\) 0 0
\(247\) 375186. 0.391294
\(248\) 0 0
\(249\) 2.35405e6 2.40612
\(250\) 0 0
\(251\) 411977.i 0.412751i 0.978473 + 0.206376i \(0.0661669\pi\)
−0.978473 + 0.206376i \(0.933833\pi\)
\(252\) 0 0
\(253\) − 268851.i − 0.264065i
\(254\) 0 0
\(255\) 970282. 0.934431
\(256\) 0 0
\(257\) −88057.5 −0.0831637 −0.0415818 0.999135i \(-0.513240\pi\)
−0.0415818 + 0.999135i \(0.513240\pi\)
\(258\) 0 0
\(259\) − 1.01367e6i − 0.938958i
\(260\) 0 0
\(261\) 881480.i 0.800961i
\(262\) 0 0
\(263\) 126284. 0.112579 0.0562896 0.998414i \(-0.482073\pi\)
0.0562896 + 0.998414i \(0.482073\pi\)
\(264\) 0 0
\(265\) −693781. −0.606886
\(266\) 0 0
\(267\) − 570045.i − 0.489363i
\(268\) 0 0
\(269\) 1.88178e6i 1.58558i 0.609495 + 0.792790i \(0.291372\pi\)
−0.609495 + 0.792790i \(0.708628\pi\)
\(270\) 0 0
\(271\) 1.17095e6 0.968532 0.484266 0.874921i \(-0.339087\pi\)
0.484266 + 0.874921i \(0.339087\pi\)
\(272\) 0 0
\(273\) −400841. −0.325511
\(274\) 0 0
\(275\) 170128.i 0.135658i
\(276\) 0 0
\(277\) 1.11420e6i 0.872496i 0.899827 + 0.436248i \(0.143693\pi\)
−0.899827 + 0.436248i \(0.856307\pi\)
\(278\) 0 0
\(279\) 90987.2 0.0699793
\(280\) 0 0
\(281\) −1.87861e6 −1.41929 −0.709646 0.704558i \(-0.751145\pi\)
−0.709646 + 0.704558i \(0.751145\pi\)
\(282\) 0 0
\(283\) − 3072.89i − 0.00228077i −0.999999 0.00114038i \(-0.999637\pi\)
0.999999 0.00114038i \(-0.000362996\pi\)
\(284\) 0 0
\(285\) 888203.i 0.647739i
\(286\) 0 0
\(287\) −884062. −0.633546
\(288\) 0 0
\(289\) 2.84762e6 2.00557
\(290\) 0 0
\(291\) − 293749.i − 0.203350i
\(292\) 0 0
\(293\) 2.45968e6i 1.67382i 0.547339 + 0.836911i \(0.315641\pi\)
−0.547339 + 0.836911i \(0.684359\pi\)
\(294\) 0 0
\(295\) 627653. 0.419918
\(296\) 0 0
\(297\) −680301. −0.447518
\(298\) 0 0
\(299\) − 195957.i − 0.126760i
\(300\) 0 0
\(301\) − 1.00026e6i − 0.636352i
\(302\) 0 0
\(303\) −1.92933e6 −1.20726
\(304\) 0 0
\(305\) 660115. 0.406322
\(306\) 0 0
\(307\) − 257478.i − 0.155917i −0.996957 0.0779585i \(-0.975160\pi\)
0.996957 0.0779585i \(-0.0248402\pi\)
\(308\) 0 0
\(309\) − 676012.i − 0.402771i
\(310\) 0 0
\(311\) −2.43192e6 −1.42576 −0.712882 0.701284i \(-0.752610\pi\)
−0.712882 + 0.701284i \(0.752610\pi\)
\(312\) 0 0
\(313\) −2.62542e6 −1.51474 −0.757369 0.652987i \(-0.773515\pi\)
−0.757369 + 0.652987i \(0.773515\pi\)
\(314\) 0 0
\(315\) − 295658.i − 0.167886i
\(316\) 0 0
\(317\) 2.54754e6i 1.42388i 0.702240 + 0.711940i \(0.252183\pi\)
−0.702240 + 0.711940i \(0.747817\pi\)
\(318\) 0 0
\(319\) −2.18180e6 −1.20043
\(320\) 0 0
\(321\) 1.78454e6 0.966635
\(322\) 0 0
\(323\) 3.90648e6i 2.08343i
\(324\) 0 0
\(325\) 124001.i 0.0651205i
\(326\) 0 0
\(327\) 3.25322e6 1.68246
\(328\) 0 0
\(329\) −1.48807e6 −0.757940
\(330\) 0 0
\(331\) − 2.11710e6i − 1.06211i −0.847336 0.531057i \(-0.821795\pi\)
0.847336 0.531057i \(-0.178205\pi\)
\(332\) 0 0
\(333\) 1.03666e6i 0.512302i
\(334\) 0 0
\(335\) 964097. 0.469363
\(336\) 0 0
\(337\) −1.56954e6 −0.752832 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(338\) 0 0
\(339\) 4.46733e6i 2.11129i
\(340\) 0 0
\(341\) 225207.i 0.104881i
\(342\) 0 0
\(343\) −2.37117e6 −1.08825
\(344\) 0 0
\(345\) 463903. 0.209836
\(346\) 0 0
\(347\) − 428538.i − 0.191058i −0.995427 0.0955292i \(-0.969546\pi\)
0.995427 0.0955292i \(-0.0304543\pi\)
\(348\) 0 0
\(349\) − 522898.i − 0.229802i −0.993377 0.114901i \(-0.963345\pi\)
0.993377 0.114901i \(-0.0366551\pi\)
\(350\) 0 0
\(351\) −495850. −0.214824
\(352\) 0 0
\(353\) 1.89389e6 0.808943 0.404472 0.914550i \(-0.367455\pi\)
0.404472 + 0.914550i \(0.367455\pi\)
\(354\) 0 0
\(355\) 1.77682e6i 0.748297i
\(356\) 0 0
\(357\) − 4.17361e6i − 1.73317i
\(358\) 0 0
\(359\) 2.57094e6 1.05282 0.526412 0.850229i \(-0.323537\pi\)
0.526412 + 0.850229i \(0.323537\pi\)
\(360\) 0 0
\(361\) −1.09992e6 −0.444216
\(362\) 0 0
\(363\) − 1.63368e6i − 0.650730i
\(364\) 0 0
\(365\) 465550.i 0.182909i
\(366\) 0 0
\(367\) 2.47263e6 0.958282 0.479141 0.877738i \(-0.340948\pi\)
0.479141 + 0.877738i \(0.340948\pi\)
\(368\) 0 0
\(369\) 904114. 0.345667
\(370\) 0 0
\(371\) 2.98426e6i 1.12565i
\(372\) 0 0
\(373\) 2.62533e6i 0.977038i 0.872553 + 0.488519i \(0.162463\pi\)
−0.872553 + 0.488519i \(0.837537\pi\)
\(374\) 0 0
\(375\) −293557. −0.107799
\(376\) 0 0
\(377\) −1.59024e6 −0.576249
\(378\) 0 0
\(379\) − 2.14283e6i − 0.766284i −0.923689 0.383142i \(-0.874842\pi\)
0.923689 0.383142i \(-0.125158\pi\)
\(380\) 0 0
\(381\) − 1.18894e6i − 0.419610i
\(382\) 0 0
\(383\) 641783. 0.223559 0.111779 0.993733i \(-0.464345\pi\)
0.111779 + 0.993733i \(0.464345\pi\)
\(384\) 0 0
\(385\) 731799. 0.251617
\(386\) 0 0
\(387\) 1.02295e6i 0.347198i
\(388\) 0 0
\(389\) − 2.36321e6i − 0.791823i −0.918289 0.395911i \(-0.870429\pi\)
0.918289 0.395911i \(-0.129571\pi\)
\(390\) 0 0
\(391\) 2.04033e6 0.674930
\(392\) 0 0
\(393\) 3.38331e6 1.10499
\(394\) 0 0
\(395\) 1.88998e6i 0.609489i
\(396\) 0 0
\(397\) − 403888.i − 0.128613i −0.997930 0.0643065i \(-0.979516\pi\)
0.997930 0.0643065i \(-0.0204835\pi\)
\(398\) 0 0
\(399\) 3.82056e6 1.20142
\(400\) 0 0
\(401\) 2.28335e6 0.709108 0.354554 0.935036i \(-0.384633\pi\)
0.354554 + 0.935036i \(0.384633\pi\)
\(402\) 0 0
\(403\) 164146.i 0.0503464i
\(404\) 0 0
\(405\) − 1.84196e6i − 0.558011i
\(406\) 0 0
\(407\) −2.56589e6 −0.767807
\(408\) 0 0
\(409\) 3.27964e6 0.969435 0.484717 0.874671i \(-0.338922\pi\)
0.484717 + 0.874671i \(0.338922\pi\)
\(410\) 0 0
\(411\) − 6.87136e6i − 2.00650i
\(412\) 0 0
\(413\) − 2.69981e6i − 0.778859i
\(414\) 0 0
\(415\) −3.13245e6 −0.892820
\(416\) 0 0
\(417\) −2.10208e6 −0.591983
\(418\) 0 0
\(419\) − 604670.i − 0.168261i −0.996455 0.0841305i \(-0.973189\pi\)
0.996455 0.0841305i \(-0.0268113\pi\)
\(420\) 0 0
\(421\) − 3.46346e6i − 0.952367i −0.879346 0.476184i \(-0.842020\pi\)
0.879346 0.476184i \(-0.157980\pi\)
\(422\) 0 0
\(423\) 1.52183e6 0.413537
\(424\) 0 0
\(425\) −1.29112e6 −0.346732
\(426\) 0 0
\(427\) − 2.83945e6i − 0.753641i
\(428\) 0 0
\(429\) 1.01465e6i 0.266178i
\(430\) 0 0
\(431\) −5.00345e6 −1.29741 −0.648704 0.761041i \(-0.724689\pi\)
−0.648704 + 0.761041i \(0.724689\pi\)
\(432\) 0 0
\(433\) −4.11259e6 −1.05413 −0.527067 0.849824i \(-0.676708\pi\)
−0.527067 + 0.849824i \(0.676708\pi\)
\(434\) 0 0
\(435\) − 3.76469e6i − 0.953908i
\(436\) 0 0
\(437\) 1.86773e6i 0.467855i
\(438\) 0 0
\(439\) −4.20313e6 −1.04091 −0.520454 0.853890i \(-0.674237\pi\)
−0.520454 + 0.853890i \(0.674237\pi\)
\(440\) 0 0
\(441\) 576598. 0.141181
\(442\) 0 0
\(443\) − 1.32165e6i − 0.319969i −0.987120 0.159984i \(-0.948856\pi\)
0.987120 0.159984i \(-0.0511443\pi\)
\(444\) 0 0
\(445\) 758538.i 0.181584i
\(446\) 0 0
\(447\) −2.56414e6 −0.606978
\(448\) 0 0
\(449\) −5.52202e6 −1.29265 −0.646327 0.763061i \(-0.723696\pi\)
−0.646327 + 0.763061i \(0.723696\pi\)
\(450\) 0 0
\(451\) 2.23782e6i 0.518064i
\(452\) 0 0
\(453\) 3.50115e6i 0.801615i
\(454\) 0 0
\(455\) 533385. 0.120785
\(456\) 0 0
\(457\) 3.46838e6 0.776849 0.388424 0.921481i \(-0.373019\pi\)
0.388424 + 0.921481i \(0.373019\pi\)
\(458\) 0 0
\(459\) − 5.16286e6i − 1.14382i
\(460\) 0 0
\(461\) 2.42345e6i 0.531107i 0.964096 + 0.265554i \(0.0855547\pi\)
−0.964096 + 0.265554i \(0.914445\pi\)
\(462\) 0 0
\(463\) 6.58501e6 1.42759 0.713796 0.700354i \(-0.246974\pi\)
0.713796 + 0.700354i \(0.246974\pi\)
\(464\) 0 0
\(465\) −388595. −0.0833422
\(466\) 0 0
\(467\) 1.15697e6i 0.245488i 0.992438 + 0.122744i \(0.0391694\pi\)
−0.992438 + 0.122744i \(0.960831\pi\)
\(468\) 0 0
\(469\) − 4.14701e6i − 0.870569i
\(470\) 0 0
\(471\) 1.40324e6 0.291461
\(472\) 0 0
\(473\) −2.53196e6 −0.520359
\(474\) 0 0
\(475\) − 1.18190e6i − 0.240351i
\(476\) 0 0
\(477\) − 3.05195e6i − 0.614160i
\(478\) 0 0
\(479\) 380517. 0.0757767 0.0378883 0.999282i \(-0.487937\pi\)
0.0378883 + 0.999282i \(0.487937\pi\)
\(480\) 0 0
\(481\) −1.87020e6 −0.368574
\(482\) 0 0
\(483\) − 1.99545e6i − 0.389201i
\(484\) 0 0
\(485\) 390881.i 0.0754554i
\(486\) 0 0
\(487\) −2.67715e6 −0.511505 −0.255753 0.966742i \(-0.582323\pi\)
−0.255753 + 0.966742i \(0.582323\pi\)
\(488\) 0 0
\(489\) −1.03103e7 −1.94983
\(490\) 0 0
\(491\) 5.10695e6i 0.956000i 0.878360 + 0.478000i \(0.158638\pi\)
−0.878360 + 0.478000i \(0.841362\pi\)
\(492\) 0 0
\(493\) − 1.65578e7i − 3.06822i
\(494\) 0 0
\(495\) −748397. −0.137284
\(496\) 0 0
\(497\) 7.64292e6 1.38793
\(498\) 0 0
\(499\) − 259376.i − 0.0466313i −0.999728 0.0233157i \(-0.992578\pi\)
0.999728 0.0233157i \(-0.00742228\pi\)
\(500\) 0 0
\(501\) 4.21429e6i 0.750118i
\(502\) 0 0
\(503\) 324388. 0.0571669 0.0285835 0.999591i \(-0.490900\pi\)
0.0285835 + 0.999591i \(0.490900\pi\)
\(504\) 0 0
\(505\) 2.56729e6 0.447967
\(506\) 0 0
\(507\) − 6.23617e6i − 1.07745i
\(508\) 0 0
\(509\) 1.13575e7i 1.94307i 0.236890 + 0.971537i \(0.423872\pi\)
−0.236890 + 0.971537i \(0.576128\pi\)
\(510\) 0 0
\(511\) 2.00254e6 0.339257
\(512\) 0 0
\(513\) 4.72612e6 0.792887
\(514\) 0 0
\(515\) 899544.i 0.149453i
\(516\) 0 0
\(517\) 3.76675e6i 0.619784i
\(518\) 0 0
\(519\) −3.10484e6 −0.505966
\(520\) 0 0
\(521\) 9.15918e6 1.47830 0.739149 0.673541i \(-0.235228\pi\)
0.739149 + 0.673541i \(0.235228\pi\)
\(522\) 0 0
\(523\) − 7.08351e6i − 1.13238i −0.824273 0.566192i \(-0.808416\pi\)
0.824273 0.566192i \(-0.191584\pi\)
\(524\) 0 0
\(525\) 1.26272e6i 0.199944i
\(526\) 0 0
\(527\) −1.70911e6 −0.268068
\(528\) 0 0
\(529\) −5.46084e6 −0.848438
\(530\) 0 0
\(531\) 2.76105e6i 0.424950i
\(532\) 0 0
\(533\) 1.63108e6i 0.248689i
\(534\) 0 0
\(535\) −2.37461e6 −0.358681
\(536\) 0 0
\(537\) 8.11579e6 1.21449
\(538\) 0 0
\(539\) 1.42717e6i 0.211594i
\(540\) 0 0
\(541\) − 2.53136e6i − 0.371843i −0.982565 0.185922i \(-0.940473\pi\)
0.982565 0.185922i \(-0.0595271\pi\)
\(542\) 0 0
\(543\) 4.07586e6 0.593225
\(544\) 0 0
\(545\) −4.32894e6 −0.624296
\(546\) 0 0
\(547\) 1.30020e7i 1.85798i 0.370108 + 0.928989i \(0.379321\pi\)
−0.370108 + 0.928989i \(0.620679\pi\)
\(548\) 0 0
\(549\) 2.90386e6i 0.411192i
\(550\) 0 0
\(551\) 1.51571e7 2.12686
\(552\) 0 0
\(553\) 8.12967e6 1.13047
\(554\) 0 0
\(555\) − 4.42745e6i − 0.610128i
\(556\) 0 0
\(557\) 5.70438e6i 0.779059i 0.921014 + 0.389530i \(0.127362\pi\)
−0.921014 + 0.389530i \(0.872638\pi\)
\(558\) 0 0
\(559\) −1.84546e6 −0.249790
\(560\) 0 0
\(561\) −1.05646e7 −1.41725
\(562\) 0 0
\(563\) 1.98939e6i 0.264514i 0.991215 + 0.132257i \(0.0422224\pi\)
−0.991215 + 0.132257i \(0.957778\pi\)
\(564\) 0 0
\(565\) − 5.94451e6i − 0.783420i
\(566\) 0 0
\(567\) −7.92310e6 −1.03499
\(568\) 0 0
\(569\) 1.30277e6 0.168689 0.0843447 0.996437i \(-0.473120\pi\)
0.0843447 + 0.996437i \(0.473120\pi\)
\(570\) 0 0
\(571\) 2.90628e6i 0.373033i 0.982452 + 0.186517i \(0.0597199\pi\)
−0.982452 + 0.186517i \(0.940280\pi\)
\(572\) 0 0
\(573\) − 649413.i − 0.0826294i
\(574\) 0 0
\(575\) −617298. −0.0778619
\(576\) 0 0
\(577\) 4.01853e6 0.502490 0.251245 0.967923i \(-0.419160\pi\)
0.251245 + 0.967923i \(0.419160\pi\)
\(578\) 0 0
\(579\) 8.89785e6i 1.10303i
\(580\) 0 0
\(581\) 1.34741e7i 1.65599i
\(582\) 0 0
\(583\) 7.55404e6 0.920466
\(584\) 0 0
\(585\) −545483. −0.0659009
\(586\) 0 0
\(587\) 1.29654e7i 1.55307i 0.630074 + 0.776535i \(0.283025\pi\)
−0.630074 + 0.776535i \(0.716975\pi\)
\(588\) 0 0
\(589\) − 1.56454e6i − 0.185822i
\(590\) 0 0
\(591\) 7.41706e6 0.873501
\(592\) 0 0
\(593\) −107000. −0.0124953 −0.00624767 0.999980i \(-0.501989\pi\)
−0.00624767 + 0.999980i \(0.501989\pi\)
\(594\) 0 0
\(595\) 5.55367e6i 0.643114i
\(596\) 0 0
\(597\) − 8.96337e6i − 1.02928i
\(598\) 0 0
\(599\) −1.06394e7 −1.21157 −0.605785 0.795629i \(-0.707141\pi\)
−0.605785 + 0.795629i \(0.707141\pi\)
\(600\) 0 0
\(601\) −6.42705e6 −0.725814 −0.362907 0.931825i \(-0.618216\pi\)
−0.362907 + 0.931825i \(0.618216\pi\)
\(602\) 0 0
\(603\) 4.24108e6i 0.474988i
\(604\) 0 0
\(605\) 2.17388e6i 0.241461i
\(606\) 0 0
\(607\) −1.96807e6 −0.216805 −0.108403 0.994107i \(-0.534574\pi\)
−0.108403 + 0.994107i \(0.534574\pi\)
\(608\) 0 0
\(609\) −1.61936e7 −1.76930
\(610\) 0 0
\(611\) 2.74546e6i 0.297518i
\(612\) 0 0
\(613\) 8.12485e6i 0.873301i 0.899631 + 0.436651i \(0.143835\pi\)
−0.899631 + 0.436651i \(0.856165\pi\)
\(614\) 0 0
\(615\) −3.86136e6 −0.411673
\(616\) 0 0
\(617\) 1.40750e6 0.148846 0.0744229 0.997227i \(-0.476289\pi\)
0.0744229 + 0.997227i \(0.476289\pi\)
\(618\) 0 0
\(619\) 5.01827e6i 0.526414i 0.964739 + 0.263207i \(0.0847802\pi\)
−0.964739 + 0.263207i \(0.915220\pi\)
\(620\) 0 0
\(621\) − 2.46842e6i − 0.256856i
\(622\) 0 0
\(623\) 3.26281e6 0.336800
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 9.67095e6i − 0.982427i
\(628\) 0 0
\(629\) − 1.94727e7i − 1.96246i
\(630\) 0 0
\(631\) −5.59458e6 −0.559363 −0.279682 0.960093i \(-0.590229\pi\)
−0.279682 + 0.960093i \(0.590229\pi\)
\(632\) 0 0
\(633\) 8.18942e6 0.812351
\(634\) 0 0
\(635\) 1.58207e6i 0.155701i
\(636\) 0 0
\(637\) 1.04022e6i 0.101572i
\(638\) 0 0
\(639\) −7.81627e6 −0.757265
\(640\) 0 0
\(641\) 3.70806e6 0.356453 0.178226 0.983990i \(-0.442964\pi\)
0.178226 + 0.983990i \(0.442964\pi\)
\(642\) 0 0
\(643\) 1.36989e6i 0.130665i 0.997864 + 0.0653325i \(0.0208108\pi\)
−0.997864 + 0.0653325i \(0.979189\pi\)
\(644\) 0 0
\(645\) − 4.36889e6i − 0.413497i
\(646\) 0 0
\(647\) 1.38641e7 1.30206 0.651028 0.759053i \(-0.274338\pi\)
0.651028 + 0.759053i \(0.274338\pi\)
\(648\) 0 0
\(649\) −6.83402e6 −0.636890
\(650\) 0 0
\(651\) 1.67152e6i 0.154582i
\(652\) 0 0
\(653\) 1.00652e7i 0.923714i 0.886954 + 0.461857i \(0.152817\pi\)
−0.886954 + 0.461857i \(0.847183\pi\)
\(654\) 0 0
\(655\) −4.50204e6 −0.410021
\(656\) 0 0
\(657\) −2.04796e6 −0.185101
\(658\) 0 0
\(659\) − 2.94962e6i − 0.264578i −0.991211 0.132289i \(-0.957767\pi\)
0.991211 0.132289i \(-0.0422326\pi\)
\(660\) 0 0
\(661\) 2.46213e6i 0.219184i 0.993977 + 0.109592i \(0.0349544\pi\)
−0.993977 + 0.109592i \(0.965046\pi\)
\(662\) 0 0
\(663\) −7.70023e6 −0.680331
\(664\) 0 0
\(665\) −5.08387e6 −0.445801
\(666\) 0 0
\(667\) − 7.91648e6i − 0.688997i
\(668\) 0 0
\(669\) 2.14320e7i 1.85139i
\(670\) 0 0
\(671\) −7.18748e6 −0.616269
\(672\) 0 0
\(673\) 8.46100e6 0.720085 0.360043 0.932936i \(-0.382762\pi\)
0.360043 + 0.932936i \(0.382762\pi\)
\(674\) 0 0
\(675\) 1.56201e6i 0.131955i
\(676\) 0 0
\(677\) 2.01968e7i 1.69360i 0.531913 + 0.846799i \(0.321473\pi\)
−0.531913 + 0.846799i \(0.678527\pi\)
\(678\) 0 0
\(679\) 1.68135e6 0.139954
\(680\) 0 0
\(681\) 2.04587e7 1.69048
\(682\) 0 0
\(683\) − 5.52541e6i − 0.453224i −0.973985 0.226612i \(-0.927235\pi\)
0.973985 0.226612i \(-0.0727649\pi\)
\(684\) 0 0
\(685\) 9.14346e6i 0.744534i
\(686\) 0 0
\(687\) −8.91259e6 −0.720464
\(688\) 0 0
\(689\) 5.50590e6 0.441855
\(690\) 0 0
\(691\) − 1.66737e7i − 1.32843i −0.747543 0.664214i \(-0.768766\pi\)
0.747543 0.664214i \(-0.231234\pi\)
\(692\) 0 0
\(693\) 3.21919e6i 0.254632i
\(694\) 0 0
\(695\) 2.79716e6 0.219662
\(696\) 0 0
\(697\) −1.69830e7 −1.32413
\(698\) 0 0
\(699\) 5.09253e6i 0.394222i
\(700\) 0 0
\(701\) − 1.49989e7i − 1.15283i −0.817157 0.576415i \(-0.804451\pi\)
0.817157 0.576415i \(-0.195549\pi\)
\(702\) 0 0
\(703\) 1.78255e7 1.36036
\(704\) 0 0
\(705\) −6.49953e6 −0.492503
\(706\) 0 0
\(707\) − 1.10431e7i − 0.830884i
\(708\) 0 0
\(709\) − 1.84405e7i − 1.37771i −0.724900 0.688854i \(-0.758114\pi\)
0.724900 0.688854i \(-0.241886\pi\)
\(710\) 0 0
\(711\) −8.31407e6 −0.616793
\(712\) 0 0
\(713\) −817147. −0.0601972
\(714\) 0 0
\(715\) − 1.35015e6i − 0.0987684i
\(716\) 0 0
\(717\) − 1.54886e7i − 1.12516i
\(718\) 0 0
\(719\) 334083. 0.0241009 0.0120504 0.999927i \(-0.496164\pi\)
0.0120504 + 0.999927i \(0.496164\pi\)
\(720\) 0 0
\(721\) 3.86934e6 0.277204
\(722\) 0 0
\(723\) 1.62921e6i 0.115913i
\(724\) 0 0
\(725\) 5.00953e6i 0.353959i
\(726\) 0 0
\(727\) 2.11438e7 1.48370 0.741851 0.670564i \(-0.233948\pi\)
0.741851 + 0.670564i \(0.233948\pi\)
\(728\) 0 0
\(729\) −3.30712e6 −0.230479
\(730\) 0 0
\(731\) − 1.92152e7i − 1.33000i
\(732\) 0 0
\(733\) − 1.48236e6i − 0.101904i −0.998701 0.0509521i \(-0.983774\pi\)
0.998701 0.0509521i \(-0.0162256\pi\)
\(734\) 0 0
\(735\) −2.46258e6 −0.168140
\(736\) 0 0
\(737\) −1.04973e7 −0.711884
\(738\) 0 0
\(739\) − 1.69772e7i − 1.14355i −0.820411 0.571775i \(-0.806255\pi\)
0.820411 0.571775i \(-0.193745\pi\)
\(740\) 0 0
\(741\) − 7.04885e6i − 0.471599i
\(742\) 0 0
\(743\) 2.01294e7 1.33770 0.668849 0.743399i \(-0.266787\pi\)
0.668849 + 0.743399i \(0.266787\pi\)
\(744\) 0 0
\(745\) 3.41201e6 0.225226
\(746\) 0 0
\(747\) − 1.37797e7i − 0.903520i
\(748\) 0 0
\(749\) 1.02143e7i 0.665278i
\(750\) 0 0
\(751\) 1.16155e7 0.751514 0.375757 0.926718i \(-0.377383\pi\)
0.375757 + 0.926718i \(0.377383\pi\)
\(752\) 0 0
\(753\) 7.74007e6 0.497459
\(754\) 0 0
\(755\) − 4.65885e6i − 0.297448i
\(756\) 0 0
\(757\) − 9.97313e6i − 0.632545i −0.948668 0.316273i \(-0.897569\pi\)
0.948668 0.316273i \(-0.102431\pi\)
\(758\) 0 0
\(759\) −5.05108e6 −0.318258
\(760\) 0 0
\(761\) 2.00566e7 1.25544 0.627719 0.778440i \(-0.283989\pi\)
0.627719 + 0.778440i \(0.283989\pi\)
\(762\) 0 0
\(763\) 1.86207e7i 1.15794i
\(764\) 0 0
\(765\) − 5.67964e6i − 0.350887i
\(766\) 0 0
\(767\) −4.98110e6 −0.305729
\(768\) 0 0
\(769\) 2.63971e7 1.60968 0.804842 0.593489i \(-0.202250\pi\)
0.804842 + 0.593489i \(0.202250\pi\)
\(770\) 0 0
\(771\) 1.65439e6i 0.100231i
\(772\) 0 0
\(773\) − 902367.i − 0.0543169i −0.999631 0.0271584i \(-0.991354\pi\)
0.999631 0.0271584i \(-0.00864586\pi\)
\(774\) 0 0
\(775\) 517089. 0.0309251
\(776\) 0 0
\(777\) −1.90444e7 −1.13166
\(778\) 0 0
\(779\) − 1.55463e7i − 0.917877i
\(780\) 0 0
\(781\) − 1.93465e7i − 1.13494i
\(782\) 0 0
\(783\) −2.00319e7 −1.16766
\(784\) 0 0
\(785\) −1.86724e6 −0.108150
\(786\) 0 0
\(787\) 1.59032e7i 0.915269i 0.889140 + 0.457635i \(0.151303\pi\)
−0.889140 + 0.457635i \(0.848697\pi\)
\(788\) 0 0
\(789\) − 2.37258e6i − 0.135684i
\(790\) 0 0
\(791\) −2.55700e7 −1.45308
\(792\) 0 0
\(793\) −5.23873e6 −0.295830
\(794\) 0 0
\(795\) 1.30345e7i 0.731436i
\(796\) 0 0
\(797\) − 2.52099e7i − 1.40580i −0.711287 0.702902i \(-0.751887\pi\)
0.711287 0.702902i \(-0.248113\pi\)
\(798\) 0 0
\(799\) −2.85861e7 −1.58412
\(800\) 0 0
\(801\) −3.33682e6 −0.183760
\(802\) 0 0
\(803\) − 5.06901e6i − 0.277418i
\(804\) 0 0
\(805\) 2.65527e6i 0.144417i
\(806\) 0 0
\(807\) 3.53542e7 1.91099
\(808\) 0 0
\(809\) −1.43630e7 −0.771570 −0.385785 0.922589i \(-0.626069\pi\)
−0.385785 + 0.922589i \(0.626069\pi\)
\(810\) 0 0
\(811\) 1.49512e7i 0.798222i 0.916903 + 0.399111i \(0.130681\pi\)
−0.916903 + 0.399111i \(0.869319\pi\)
\(812\) 0 0
\(813\) − 2.19993e7i − 1.16730i
\(814\) 0 0
\(815\) 1.37195e7 0.723508
\(816\) 0 0
\(817\) 1.75897e7 0.921943
\(818\) 0 0
\(819\) 2.34637e6i 0.122232i
\(820\) 0 0
\(821\) − 2.49701e7i − 1.29289i −0.762960 0.646446i \(-0.776254\pi\)
0.762960 0.646446i \(-0.223746\pi\)
\(822\) 0 0
\(823\) −2.50946e7 −1.29146 −0.645729 0.763567i \(-0.723446\pi\)
−0.645729 + 0.763567i \(0.723446\pi\)
\(824\) 0 0
\(825\) 3.19631e6 0.163499
\(826\) 0 0
\(827\) 2.47367e7i 1.25770i 0.777526 + 0.628850i \(0.216474\pi\)
−0.777526 + 0.628850i \(0.783526\pi\)
\(828\) 0 0
\(829\) 1.02078e7i 0.515878i 0.966161 + 0.257939i \(0.0830434\pi\)
−0.966161 + 0.257939i \(0.916957\pi\)
\(830\) 0 0
\(831\) 2.09332e7 1.05156
\(832\) 0 0
\(833\) −1.08309e7 −0.540818
\(834\) 0 0
\(835\) − 5.60779e6i − 0.278340i
\(836\) 0 0
\(837\) 2.06771e6i 0.102018i
\(838\) 0 0
\(839\) −2.64451e7 −1.29700 −0.648501 0.761214i \(-0.724604\pi\)
−0.648501 + 0.761214i \(0.724604\pi\)
\(840\) 0 0
\(841\) −4.37332e7 −2.13217
\(842\) 0 0
\(843\) 3.52947e7i 1.71057i
\(844\) 0 0
\(845\) 8.29824e6i 0.399801i
\(846\) 0 0
\(847\) 9.35082e6 0.447859
\(848\) 0 0
\(849\) −57732.4 −0.00274885
\(850\) 0 0
\(851\) − 9.31014e6i − 0.440689i
\(852\) 0 0
\(853\) 2.30070e7i 1.08265i 0.840814 + 0.541324i \(0.182077\pi\)
−0.840814 + 0.541324i \(0.817923\pi\)
\(854\) 0 0
\(855\) 5.19919e6 0.243232
\(856\) 0 0
\(857\) −1.81332e7 −0.843379 −0.421689 0.906740i \(-0.638563\pi\)
−0.421689 + 0.906740i \(0.638563\pi\)
\(858\) 0 0
\(859\) − 1.83891e6i − 0.0850310i −0.999096 0.0425155i \(-0.986463\pi\)
0.999096 0.0425155i \(-0.0135372\pi\)
\(860\) 0 0
\(861\) 1.66094e7i 0.763567i
\(862\) 0 0
\(863\) −1.14230e7 −0.522097 −0.261049 0.965326i \(-0.584068\pi\)
−0.261049 + 0.965326i \(0.584068\pi\)
\(864\) 0 0
\(865\) 4.13150e6 0.187744
\(866\) 0 0
\(867\) − 5.35001e7i − 2.41717i
\(868\) 0 0
\(869\) − 2.05786e7i − 0.924413i
\(870\) 0 0
\(871\) −7.65115e6 −0.341729
\(872\) 0 0
\(873\) −1.71949e6 −0.0763597
\(874\) 0 0
\(875\) − 1.68025e6i − 0.0741915i
\(876\) 0 0
\(877\) 2.12365e7i 0.932362i 0.884689 + 0.466181i \(0.154370\pi\)
−0.884689 + 0.466181i \(0.845630\pi\)
\(878\) 0 0
\(879\) 4.62116e7 2.01734
\(880\) 0 0
\(881\) 1.35067e7 0.586287 0.293143 0.956069i \(-0.405299\pi\)
0.293143 + 0.956069i \(0.405299\pi\)
\(882\) 0 0
\(883\) − 2.44466e7i − 1.05515i −0.849507 0.527577i \(-0.823100\pi\)
0.849507 0.527577i \(-0.176900\pi\)
\(884\) 0 0
\(885\) − 1.17921e7i − 0.506096i
\(886\) 0 0
\(887\) −8.02242e6 −0.342371 −0.171185 0.985239i \(-0.554760\pi\)
−0.171185 + 0.985239i \(0.554760\pi\)
\(888\) 0 0
\(889\) 6.80520e6 0.288793
\(890\) 0 0
\(891\) 2.00557e7i 0.846337i
\(892\) 0 0
\(893\) − 2.61680e7i − 1.09810i
\(894\) 0 0
\(895\) −1.07994e7 −0.450652
\(896\) 0 0
\(897\) −3.68157e6 −0.152775
\(898\) 0 0
\(899\) 6.63136e6i 0.273655i
\(900\) 0 0
\(901\) 5.73281e7i 2.35264i
\(902\) 0 0
\(903\) −1.87926e7 −0.766949
\(904\) 0 0
\(905\) −5.42359e6 −0.220123
\(906\) 0 0
\(907\) 2.02868e7i 0.818831i 0.912348 + 0.409416i \(0.134267\pi\)
−0.912348 + 0.409416i \(0.865733\pi\)
\(908\) 0 0
\(909\) 1.12935e7i 0.453336i
\(910\) 0 0
\(911\) −1.72893e7 −0.690209 −0.345104 0.938564i \(-0.612156\pi\)
−0.345104 + 0.938564i \(0.612156\pi\)
\(912\) 0 0
\(913\) 3.41068e7 1.35414
\(914\) 0 0
\(915\) − 1.24020e7i − 0.489710i
\(916\) 0 0
\(917\) 1.93653e7i 0.760502i
\(918\) 0 0
\(919\) 2.14060e7 0.836079 0.418039 0.908429i \(-0.362717\pi\)
0.418039 + 0.908429i \(0.362717\pi\)
\(920\) 0 0
\(921\) −4.83740e6 −0.187916
\(922\) 0 0
\(923\) − 1.41010e7i − 0.544812i
\(924\) 0 0
\(925\) 5.89144e6i 0.226395i
\(926\) 0 0
\(927\) −3.95710e6 −0.151244
\(928\) 0 0
\(929\) 4.14158e7 1.57444 0.787221 0.616670i \(-0.211519\pi\)
0.787221 + 0.616670i \(0.211519\pi\)
\(930\) 0 0
\(931\) − 9.91466e6i − 0.374890i
\(932\) 0 0
\(933\) 4.56900e7i 1.71837i
\(934\) 0 0
\(935\) 1.40580e7 0.525889
\(936\) 0 0
\(937\) 1.39335e7 0.518455 0.259228 0.965816i \(-0.416532\pi\)
0.259228 + 0.965816i \(0.416532\pi\)
\(938\) 0 0
\(939\) 4.93254e7i 1.82560i
\(940\) 0 0
\(941\) 1.51311e7i 0.557051i 0.960429 + 0.278526i \(0.0898457\pi\)
−0.960429 + 0.278526i \(0.910154\pi\)
\(942\) 0 0
\(943\) −8.11975e6 −0.297347
\(944\) 0 0
\(945\) 6.71891e6 0.244748
\(946\) 0 0
\(947\) 2.72196e7i 0.986296i 0.869945 + 0.493148i \(0.164154\pi\)
−0.869945 + 0.493148i \(0.835846\pi\)
\(948\) 0 0
\(949\) − 3.69464e6i − 0.133170i
\(950\) 0 0
\(951\) 4.78623e7 1.71610
\(952\) 0 0
\(953\) −2.79494e7 −0.996873 −0.498437 0.866926i \(-0.666092\pi\)
−0.498437 + 0.866926i \(0.666092\pi\)
\(954\) 0 0
\(955\) 864150.i 0.0306606i
\(956\) 0 0
\(957\) 4.09908e7i 1.44679i
\(958\) 0 0
\(959\) 3.93301e7 1.38095
\(960\) 0 0
\(961\) −2.79447e7 −0.976091
\(962\) 0 0
\(963\) − 1.04460e7i − 0.362980i
\(964\) 0 0
\(965\) − 1.18400e7i − 0.409293i
\(966\) 0 0
\(967\) −3.73167e7 −1.28333 −0.641663 0.766987i \(-0.721755\pi\)
−0.641663 + 0.766987i \(0.721755\pi\)
\(968\) 0 0
\(969\) 7.33936e7 2.51101
\(970\) 0 0
\(971\) 1.77774e7i 0.605091i 0.953135 + 0.302545i \(0.0978364\pi\)
−0.953135 + 0.302545i \(0.902164\pi\)
\(972\) 0 0
\(973\) − 1.20318e7i − 0.407427i
\(974\) 0 0
\(975\) 2.32969e6 0.0784850
\(976\) 0 0
\(977\) 1.67378e7 0.561000 0.280500 0.959854i \(-0.409500\pi\)
0.280500 + 0.959854i \(0.409500\pi\)
\(978\) 0 0
\(979\) − 8.25913e6i − 0.275409i
\(980\) 0 0
\(981\) − 1.90431e7i − 0.631778i
\(982\) 0 0
\(983\) 2.99351e7 0.988091 0.494045 0.869436i \(-0.335518\pi\)
0.494045 + 0.869436i \(0.335518\pi\)
\(984\) 0 0
\(985\) −9.86961e6 −0.324123
\(986\) 0 0
\(987\) 2.79574e7i 0.913490i
\(988\) 0 0
\(989\) − 9.18701e6i − 0.298664i
\(990\) 0 0
\(991\) −1.30589e7 −0.422398 −0.211199 0.977443i \(-0.567737\pi\)
−0.211199 + 0.977443i \(0.567737\pi\)
\(992\) 0 0
\(993\) −3.97753e7 −1.28009
\(994\) 0 0
\(995\) 1.19272e7i 0.381928i
\(996\) 0 0
\(997\) 7.99417e6i 0.254704i 0.991858 + 0.127352i \(0.0406477\pi\)
−0.991858 + 0.127352i \(0.959352\pi\)
\(998\) 0 0
\(999\) −2.35584e7 −0.746848
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 160.6.d.a.81.4 20
4.3 odd 2 40.6.d.a.21.20 yes 20
5.2 odd 4 800.6.f.c.49.4 20
5.3 odd 4 800.6.f.b.49.17 20
5.4 even 2 800.6.d.c.401.17 20
8.3 odd 2 40.6.d.a.21.19 20
8.5 even 2 inner 160.6.d.a.81.17 20
12.11 even 2 360.6.k.b.181.1 20
20.3 even 4 200.6.f.c.149.9 20
20.7 even 4 200.6.f.b.149.12 20
20.19 odd 2 200.6.d.b.101.1 20
24.11 even 2 360.6.k.b.181.2 20
40.3 even 4 200.6.f.b.149.11 20
40.13 odd 4 800.6.f.c.49.3 20
40.19 odd 2 200.6.d.b.101.2 20
40.27 even 4 200.6.f.c.149.10 20
40.29 even 2 800.6.d.c.401.4 20
40.37 odd 4 800.6.f.b.49.18 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.d.a.21.19 20 8.3 odd 2
40.6.d.a.21.20 yes 20 4.3 odd 2
160.6.d.a.81.4 20 1.1 even 1 trivial
160.6.d.a.81.17 20 8.5 even 2 inner
200.6.d.b.101.1 20 20.19 odd 2
200.6.d.b.101.2 20 40.19 odd 2
200.6.f.b.149.11 20 40.3 even 4
200.6.f.b.149.12 20 20.7 even 4
200.6.f.c.149.9 20 20.3 even 4
200.6.f.c.149.10 20 40.27 even 4
360.6.k.b.181.1 20 12.11 even 2
360.6.k.b.181.2 20 24.11 even 2
800.6.d.c.401.4 20 40.29 even 2
800.6.d.c.401.17 20 5.4 even 2
800.6.f.b.49.17 20 5.3 odd 4
800.6.f.b.49.18 20 40.37 odd 4
800.6.f.c.49.3 20 40.13 odd 4
800.6.f.c.49.4 20 5.2 odd 4