Properties

Label 800.2.f.e.49.1
Level $800$
Weight $2$
Character 800.49
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,Mod(49,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.f (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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 800.49
Dual form 800.2.f.e.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.732051 q^{3} -2.73205i q^{7} -2.46410 q^{9} +O(q^{10})\) \(q-0.732051 q^{3} -2.73205i q^{7} -2.46410 q^{9} +2.00000i q^{11} -3.46410 q^{13} +3.46410i q^{17} +7.46410i q^{19} +2.00000i q^{21} -4.19615i q^{23} +4.00000 q^{27} +6.92820i q^{29} -1.46410 q^{31} -1.46410i q^{33} -2.00000 q^{37} +2.53590 q^{39} -5.46410 q^{41} -8.73205 q^{43} +6.73205i q^{47} -0.464102 q^{49} -2.53590i q^{51} -4.53590 q^{53} -5.46410i q^{57} -0.535898i q^{59} +4.92820i q^{61} +6.73205i q^{63} -7.26795 q^{67} +3.07180i q^{69} +1.46410 q^{71} +0.535898i q^{73} +5.46410 q^{77} -14.9282 q^{79} +4.46410 q^{81} +4.73205 q^{83} -5.07180i q^{87} +4.92820 q^{89} +9.46410i q^{91} +1.07180 q^{93} -6.39230i q^{97} -4.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 8 q^{31} - 8 q^{37} + 24 q^{39} - 8 q^{41} - 28 q^{43} + 12 q^{49} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 8 q^{77} - 32 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{89} + 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.73205i − 1.03262i −0.856402 0.516309i \(-0.827306\pi\)
0.856402 0.516309i \(-0.172694\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) 7.46410i 1.71238i 0.516659 + 0.856191i \(0.327175\pi\)
−0.516659 + 0.856191i \(0.672825\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) − 4.19615i − 0.874958i −0.899229 0.437479i \(-0.855871\pi\)
0.899229 0.437479i \(-0.144129\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0 0
\(33\) − 1.46410i − 0.254867i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.53590 0.406069
\(40\) 0 0
\(41\) −5.46410 −0.853349 −0.426675 0.904405i \(-0.640315\pi\)
−0.426675 + 0.904405i \(0.640315\pi\)
\(42\) 0 0
\(43\) −8.73205 −1.33163 −0.665813 0.746119i \(-0.731915\pi\)
−0.665813 + 0.746119i \(0.731915\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.73205i 0.981971i 0.871168 + 0.490985i \(0.163363\pi\)
−0.871168 + 0.490985i \(0.836637\pi\)
\(48\) 0 0
\(49\) −0.464102 −0.0663002
\(50\) 0 0
\(51\) − 2.53590i − 0.355097i
\(52\) 0 0
\(53\) −4.53590 −0.623054 −0.311527 0.950237i \(-0.600840\pi\)
−0.311527 + 0.950237i \(0.600840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.46410i − 0.723738i
\(58\) 0 0
\(59\) − 0.535898i − 0.0697680i −0.999391 0.0348840i \(-0.988894\pi\)
0.999391 0.0348840i \(-0.0111062\pi\)
\(60\) 0 0
\(61\) 4.92820i 0.630992i 0.948927 + 0.315496i \(0.102171\pi\)
−0.948927 + 0.315496i \(0.897829\pi\)
\(62\) 0 0
\(63\) 6.73205i 0.848159i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.26795 −0.887921 −0.443961 0.896046i \(-0.646427\pi\)
−0.443961 + 0.896046i \(0.646427\pi\)
\(68\) 0 0
\(69\) 3.07180i 0.369801i
\(70\) 0 0
\(71\) 1.46410 0.173757 0.0868784 0.996219i \(-0.472311\pi\)
0.0868784 + 0.996219i \(0.472311\pi\)
\(72\) 0 0
\(73\) 0.535898i 0.0627222i 0.999508 + 0.0313611i \(0.00998418\pi\)
−0.999508 + 0.0313611i \(0.990016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.46410 0.622692
\(78\) 0 0
\(79\) −14.9282 −1.67955 −0.839777 0.542931i \(-0.817314\pi\)
−0.839777 + 0.542931i \(0.817314\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) 4.73205 0.519410 0.259705 0.965688i \(-0.416375\pi\)
0.259705 + 0.965688i \(0.416375\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.07180i − 0.543754i
\(88\) 0 0
\(89\) 4.92820 0.522388 0.261194 0.965286i \(-0.415884\pi\)
0.261194 + 0.965286i \(0.415884\pi\)
\(90\) 0 0
\(91\) 9.46410i 0.992107i
\(92\) 0 0
\(93\) 1.07180 0.111140
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.39230i − 0.649040i −0.945879 0.324520i \(-0.894797\pi\)
0.945879 0.324520i \(-0.105203\pi\)
\(98\) 0 0
\(99\) − 4.92820i − 0.495303i
\(100\) 0 0
\(101\) − 10.9282i − 1.08740i −0.839281 0.543698i \(-0.817024\pi\)
0.839281 0.543698i \(-0.182976\pi\)
\(102\) 0 0
\(103\) 1.66025i 0.163590i 0.996649 + 0.0817948i \(0.0260652\pi\)
−0.996649 + 0.0817948i \(0.973935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.732051 0.0707700 0.0353850 0.999374i \(-0.488734\pi\)
0.0353850 + 0.999374i \(0.488734\pi\)
\(108\) 0 0
\(109\) − 3.07180i − 0.294225i −0.989120 0.147112i \(-0.953002\pi\)
0.989120 0.147112i \(-0.0469979\pi\)
\(110\) 0 0
\(111\) 1.46410 0.138966
\(112\) 0 0
\(113\) 0.928203i 0.0873180i 0.999046 + 0.0436590i \(0.0139015\pi\)
−0.999046 + 0.0436590i \(0.986098\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.53590 0.789144
\(118\) 0 0
\(119\) 9.46410 0.867573
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 13.2679i 1.17734i 0.808373 + 0.588670i \(0.200348\pi\)
−0.808373 + 0.588670i \(0.799652\pi\)
\(128\) 0 0
\(129\) 6.39230 0.562811
\(130\) 0 0
\(131\) − 7.85641i − 0.686417i −0.939259 0.343209i \(-0.888486\pi\)
0.939259 0.343209i \(-0.111514\pi\)
\(132\) 0 0
\(133\) 20.3923 1.76824
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.92820i 0.762788i 0.924413 + 0.381394i \(0.124556\pi\)
−0.924413 + 0.381394i \(0.875444\pi\)
\(138\) 0 0
\(139\) − 7.46410i − 0.633097i −0.948576 0.316548i \(-0.897476\pi\)
0.948576 0.316548i \(-0.102524\pi\)
\(140\) 0 0
\(141\) − 4.92820i − 0.415030i
\(142\) 0 0
\(143\) − 6.92820i − 0.579365i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.339746 0.0280218
\(148\) 0 0
\(149\) − 19.8564i − 1.62670i −0.581775 0.813350i \(-0.697641\pi\)
0.581775 0.813350i \(-0.302359\pi\)
\(150\) 0 0
\(151\) −8.39230 −0.682956 −0.341478 0.939890i \(-0.610927\pi\)
−0.341478 + 0.939890i \(0.610927\pi\)
\(152\) 0 0
\(153\) − 8.53590i − 0.690086i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −16.9282 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(158\) 0 0
\(159\) 3.32051 0.263333
\(160\) 0 0
\(161\) −11.4641 −0.903498
\(162\) 0 0
\(163\) 10.1962 0.798624 0.399312 0.916815i \(-0.369249\pi\)
0.399312 + 0.916815i \(0.369249\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 20.1962i − 1.56283i −0.624015 0.781413i \(-0.714499\pi\)
0.624015 0.781413i \(-0.285501\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 18.3923i − 1.40649i
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.392305i 0.0294874i
\(178\) 0 0
\(179\) − 15.4641i − 1.15584i −0.816093 0.577921i \(-0.803864\pi\)
0.816093 0.577921i \(-0.196136\pi\)
\(180\) 0 0
\(181\) 16.0000i 1.18927i 0.803996 + 0.594635i \(0.202704\pi\)
−0.803996 + 0.594635i \(0.797296\pi\)
\(182\) 0 0
\(183\) − 3.60770i − 0.266688i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.92820 −0.506640
\(188\) 0 0
\(189\) − 10.9282i − 0.794910i
\(190\) 0 0
\(191\) 19.3205 1.39798 0.698991 0.715130i \(-0.253633\pi\)
0.698991 + 0.715130i \(0.253633\pi\)
\(192\) 0 0
\(193\) 7.46410i 0.537278i 0.963241 + 0.268639i \(0.0865738\pi\)
−0.963241 + 0.268639i \(0.913426\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.5359 −0.893146 −0.446573 0.894747i \(-0.647356\pi\)
−0.446573 + 0.894747i \(0.647356\pi\)
\(198\) 0 0
\(199\) 25.8564 1.83291 0.916456 0.400135i \(-0.131037\pi\)
0.916456 + 0.400135i \(0.131037\pi\)
\(200\) 0 0
\(201\) 5.32051 0.375280
\(202\) 0 0
\(203\) 18.9282 1.32850
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.3397i 0.718662i
\(208\) 0 0
\(209\) −14.9282 −1.03261
\(210\) 0 0
\(211\) 14.7846i 1.01781i 0.860821 + 0.508907i \(0.169950\pi\)
−0.860821 + 0.508907i \(0.830050\pi\)
\(212\) 0 0
\(213\) −1.07180 −0.0734383
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) 0 0
\(219\) − 0.392305i − 0.0265095i
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) 16.1962i 1.08457i 0.840193 + 0.542287i \(0.182442\pi\)
−0.840193 + 0.542287i \(0.817558\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.0526 1.86191 0.930957 0.365129i \(-0.118975\pi\)
0.930957 + 0.365129i \(0.118975\pi\)
\(228\) 0 0
\(229\) 4.00000i 0.264327i 0.991228 + 0.132164i \(0.0421925\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 29.3205i 1.92085i 0.278538 + 0.960425i \(0.410150\pi\)
−0.278538 + 0.960425i \(0.589850\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.9282 0.709863
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −4.39230 −0.282933 −0.141467 0.989943i \(-0.545182\pi\)
−0.141467 + 0.989943i \(0.545182\pi\)
\(242\) 0 0
\(243\) −15.2679 −0.979439
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 25.8564i − 1.64520i
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) − 11.0718i − 0.698846i −0.936965 0.349423i \(-0.886378\pi\)
0.936965 0.349423i \(-0.113622\pi\)
\(252\) 0 0
\(253\) 8.39230 0.527620
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 0 0
\(259\) 5.46410i 0.339523i
\(260\) 0 0
\(261\) − 17.0718i − 1.05672i
\(262\) 0 0
\(263\) − 5.66025i − 0.349026i −0.984655 0.174513i \(-0.944165\pi\)
0.984655 0.174513i \(-0.0558351\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.60770 −0.220787
\(268\) 0 0
\(269\) − 4.92820i − 0.300478i −0.988650 0.150239i \(-0.951996\pi\)
0.988650 0.150239i \(-0.0480043\pi\)
\(270\) 0 0
\(271\) −15.3205 −0.930655 −0.465327 0.885139i \(-0.654063\pi\)
−0.465327 + 0.885139i \(0.654063\pi\)
\(272\) 0 0
\(273\) − 6.92820i − 0.419314i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 3.60770 0.215987
\(280\) 0 0
\(281\) 17.4641 1.04182 0.520910 0.853611i \(-0.325593\pi\)
0.520910 + 0.853611i \(0.325593\pi\)
\(282\) 0 0
\(283\) 7.66025 0.455355 0.227677 0.973737i \(-0.426887\pi\)
0.227677 + 0.973737i \(0.426887\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.9282i 0.881184i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 4.67949i 0.274317i
\(292\) 0 0
\(293\) −11.8564 −0.692659 −0.346329 0.938113i \(-0.612572\pi\)
−0.346329 + 0.938113i \(0.612572\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.00000i 0.464207i
\(298\) 0 0
\(299\) 14.5359i 0.840633i
\(300\) 0 0
\(301\) 23.8564i 1.37506i
\(302\) 0 0
\(303\) 8.00000i 0.459588i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −26.9808 −1.53987 −0.769937 0.638120i \(-0.779712\pi\)
−0.769937 + 0.638120i \(0.779712\pi\)
\(308\) 0 0
\(309\) − 1.21539i − 0.0691411i
\(310\) 0 0
\(311\) 3.32051 0.188289 0.0941444 0.995559i \(-0.469988\pi\)
0.0941444 + 0.995559i \(0.469988\pi\)
\(312\) 0 0
\(313\) − 31.8564i − 1.80063i −0.435238 0.900315i \(-0.643336\pi\)
0.435238 0.900315i \(-0.356664\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.4641 0.868550 0.434275 0.900780i \(-0.357005\pi\)
0.434275 + 0.900780i \(0.357005\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) −0.535898 −0.0299109
\(322\) 0 0
\(323\) −25.8564 −1.43869
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.24871i 0.124354i
\(328\) 0 0
\(329\) 18.3923 1.01400
\(330\) 0 0
\(331\) 14.0000i 0.769510i 0.923019 + 0.384755i \(0.125714\pi\)
−0.923019 + 0.384755i \(0.874286\pi\)
\(332\) 0 0
\(333\) 4.92820 0.270064
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 7.85641i − 0.427966i −0.976837 0.213983i \(-0.931356\pi\)
0.976837 0.213983i \(-0.0686437\pi\)
\(338\) 0 0
\(339\) − 0.679492i − 0.0369049i
\(340\) 0 0
\(341\) − 2.92820i − 0.158571i
\(342\) 0 0
\(343\) − 17.8564i − 0.964155i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.6603 −0.840686 −0.420343 0.907365i \(-0.638090\pi\)
−0.420343 + 0.907365i \(0.638090\pi\)
\(348\) 0 0
\(349\) 28.0000i 1.49881i 0.662114 + 0.749403i \(0.269659\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) −13.8564 −0.739600
\(352\) 0 0
\(353\) − 0.928203i − 0.0494033i −0.999695 0.0247016i \(-0.992136\pi\)
0.999695 0.0247016i \(-0.00786357\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.92820 −0.366679
\(358\) 0 0
\(359\) 5.07180 0.267679 0.133840 0.991003i \(-0.457269\pi\)
0.133840 + 0.991003i \(0.457269\pi\)
\(360\) 0 0
\(361\) −36.7128 −1.93225
\(362\) 0 0
\(363\) −5.12436 −0.268959
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 27.1244i 1.41588i 0.706273 + 0.707940i \(0.250375\pi\)
−0.706273 + 0.707940i \(0.749625\pi\)
\(368\) 0 0
\(369\) 13.4641 0.700913
\(370\) 0 0
\(371\) 12.3923i 0.643376i
\(372\) 0 0
\(373\) −29.7128 −1.53847 −0.769236 0.638965i \(-0.779363\pi\)
−0.769236 + 0.638965i \(0.779363\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 24.0000i − 1.23606i
\(378\) 0 0
\(379\) 12.2487i 0.629174i 0.949229 + 0.314587i \(0.101866\pi\)
−0.949229 + 0.314587i \(0.898134\pi\)
\(380\) 0 0
\(381\) − 9.71281i − 0.497602i
\(382\) 0 0
\(383\) 3.12436i 0.159647i 0.996809 + 0.0798236i \(0.0254357\pi\)
−0.996809 + 0.0798236i \(0.974564\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 21.5167 1.09375
\(388\) 0 0
\(389\) 34.7846i 1.76365i 0.471577 + 0.881825i \(0.343685\pi\)
−0.471577 + 0.881825i \(0.656315\pi\)
\(390\) 0 0
\(391\) 14.5359 0.735112
\(392\) 0 0
\(393\) 5.75129i 0.290114i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.2487 −0.815499 −0.407750 0.913094i \(-0.633686\pi\)
−0.407750 + 0.913094i \(0.633686\pi\)
\(398\) 0 0
\(399\) −14.9282 −0.747345
\(400\) 0 0
\(401\) 19.8564 0.991582 0.495791 0.868442i \(-0.334878\pi\)
0.495791 + 0.868442i \(0.334878\pi\)
\(402\) 0 0
\(403\) 5.07180 0.252644
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.00000i − 0.198273i
\(408\) 0 0
\(409\) −23.3205 −1.15312 −0.576562 0.817053i \(-0.695606\pi\)
−0.576562 + 0.817053i \(0.695606\pi\)
\(410\) 0 0
\(411\) − 6.53590i − 0.322392i
\(412\) 0 0
\(413\) −1.46410 −0.0720437
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.46410i 0.267578i
\(418\) 0 0
\(419\) − 2.39230i − 0.116872i −0.998291 0.0584359i \(-0.981389\pi\)
0.998291 0.0584359i \(-0.0186113\pi\)
\(420\) 0 0
\(421\) 27.8564i 1.35764i 0.734306 + 0.678819i \(0.237508\pi\)
−0.734306 + 0.678819i \(0.762492\pi\)
\(422\) 0 0
\(423\) − 16.5885i − 0.806558i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 13.4641 0.651574
\(428\) 0 0
\(429\) 5.07180i 0.244869i
\(430\) 0 0
\(431\) 14.5359 0.700170 0.350085 0.936718i \(-0.386153\pi\)
0.350085 + 0.936718i \(0.386153\pi\)
\(432\) 0 0
\(433\) − 12.5359i − 0.602437i −0.953555 0.301218i \(-0.902607\pi\)
0.953555 0.301218i \(-0.0973933\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.3205 1.49826
\(438\) 0 0
\(439\) −0.784610 −0.0374474 −0.0187237 0.999825i \(-0.505960\pi\)
−0.0187237 + 0.999825i \(0.505960\pi\)
\(440\) 0 0
\(441\) 1.14359 0.0544568
\(442\) 0 0
\(443\) −30.9808 −1.47194 −0.735970 0.677014i \(-0.763274\pi\)
−0.735970 + 0.677014i \(0.763274\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.5359i 0.687524i
\(448\) 0 0
\(449\) −11.3205 −0.534248 −0.267124 0.963662i \(-0.586073\pi\)
−0.267124 + 0.963662i \(0.586073\pi\)
\(450\) 0 0
\(451\) − 10.9282i − 0.514589i
\(452\) 0 0
\(453\) 6.14359 0.288651
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 14.7846i − 0.691595i −0.938309 0.345797i \(-0.887608\pi\)
0.938309 0.345797i \(-0.112392\pi\)
\(458\) 0 0
\(459\) 13.8564i 0.646762i
\(460\) 0 0
\(461\) 2.92820i 0.136380i 0.997672 + 0.0681900i \(0.0217224\pi\)
−0.997672 + 0.0681900i \(0.978278\pi\)
\(462\) 0 0
\(463\) 14.7321i 0.684656i 0.939580 + 0.342328i \(0.111215\pi\)
−0.939580 + 0.342328i \(0.888785\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.33975 0.385917 0.192959 0.981207i \(-0.438192\pi\)
0.192959 + 0.981207i \(0.438192\pi\)
\(468\) 0 0
\(469\) 19.8564i 0.916884i
\(470\) 0 0
\(471\) 12.3923 0.571007
\(472\) 0 0
\(473\) − 17.4641i − 0.803000i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.1769 0.511756
\(478\) 0 0
\(479\) −21.8564 −0.998645 −0.499322 0.866416i \(-0.666418\pi\)
−0.499322 + 0.866416i \(0.666418\pi\)
\(480\) 0 0
\(481\) 6.92820 0.315899
\(482\) 0 0
\(483\) 8.39230 0.381863
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 24.5885i − 1.11421i −0.830442 0.557105i \(-0.811912\pi\)
0.830442 0.557105i \(-0.188088\pi\)
\(488\) 0 0
\(489\) −7.46410 −0.337538
\(490\) 0 0
\(491\) 3.07180i 0.138628i 0.997595 + 0.0693141i \(0.0220811\pi\)
−0.997595 + 0.0693141i \(0.977919\pi\)
\(492\) 0 0
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 4.00000i − 0.179425i
\(498\) 0 0
\(499\) 24.5359i 1.09838i 0.835698 + 0.549189i \(0.185063\pi\)
−0.835698 + 0.549189i \(0.814937\pi\)
\(500\) 0 0
\(501\) 14.7846i 0.660528i
\(502\) 0 0
\(503\) 17.6603i 0.787432i 0.919232 + 0.393716i \(0.128811\pi\)
−0.919232 + 0.393716i \(0.871189\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.732051 0.0325115
\(508\) 0 0
\(509\) − 25.8564i − 1.14607i −0.819533 0.573033i \(-0.805767\pi\)
0.819533 0.573033i \(-0.194233\pi\)
\(510\) 0 0
\(511\) 1.46410 0.0647680
\(512\) 0 0
\(513\) 29.8564i 1.31819i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −13.4641 −0.592151
\(518\) 0 0
\(519\) 1.46410 0.0642669
\(520\) 0 0
\(521\) −16.1436 −0.707264 −0.353632 0.935385i \(-0.615053\pi\)
−0.353632 + 0.935385i \(0.615053\pi\)
\(522\) 0 0
\(523\) −22.1962 −0.970570 −0.485285 0.874356i \(-0.661284\pi\)
−0.485285 + 0.874356i \(0.661284\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5.07180i − 0.220931i
\(528\) 0 0
\(529\) 5.39230 0.234448
\(530\) 0 0
\(531\) 1.32051i 0.0573052i
\(532\) 0 0
\(533\) 18.9282 0.819871
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 11.3205i 0.488516i
\(538\) 0 0
\(539\) − 0.928203i − 0.0399805i
\(540\) 0 0
\(541\) − 13.0718i − 0.562000i −0.959708 0.281000i \(-0.909334\pi\)
0.959708 0.281000i \(-0.0906662\pi\)
\(542\) 0 0
\(543\) − 11.7128i − 0.502645i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −36.7321 −1.57055 −0.785275 0.619148i \(-0.787478\pi\)
−0.785275 + 0.619148i \(0.787478\pi\)
\(548\) 0 0
\(549\) − 12.1436i − 0.518276i
\(550\) 0 0
\(551\) −51.7128 −2.20304
\(552\) 0 0
\(553\) 40.7846i 1.73434i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.7846 −1.13490 −0.567450 0.823408i \(-0.692070\pi\)
−0.567450 + 0.823408i \(0.692070\pi\)
\(558\) 0 0
\(559\) 30.2487 1.27938
\(560\) 0 0
\(561\) 5.07180 0.214131
\(562\) 0 0
\(563\) 16.0526 0.676535 0.338267 0.941050i \(-0.390159\pi\)
0.338267 + 0.941050i \(0.390159\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 12.1962i − 0.512190i
\(568\) 0 0
\(569\) 6.53590 0.273999 0.137000 0.990571i \(-0.456254\pi\)
0.137000 + 0.990571i \(0.456254\pi\)
\(570\) 0 0
\(571\) 34.7846i 1.45569i 0.685741 + 0.727845i \(0.259478\pi\)
−0.685741 + 0.727845i \(0.740522\pi\)
\(572\) 0 0
\(573\) −14.1436 −0.590857
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 43.5692i 1.81381i 0.421335 + 0.906905i \(0.361562\pi\)
−0.421335 + 0.906905i \(0.638438\pi\)
\(578\) 0 0
\(579\) − 5.46410i − 0.227080i
\(580\) 0 0
\(581\) − 12.9282i − 0.536352i
\(582\) 0 0
\(583\) − 9.07180i − 0.375715i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.1962 0.585938 0.292969 0.956122i \(-0.405357\pi\)
0.292969 + 0.956122i \(0.405357\pi\)
\(588\) 0 0
\(589\) − 10.9282i − 0.450289i
\(590\) 0 0
\(591\) 9.17691 0.377488
\(592\) 0 0
\(593\) − 36.6410i − 1.50467i −0.658783 0.752333i \(-0.728928\pi\)
0.658783 0.752333i \(-0.271072\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.9282 −0.774680
\(598\) 0 0
\(599\) 34.6410 1.41539 0.707697 0.706516i \(-0.249734\pi\)
0.707697 + 0.706516i \(0.249734\pi\)
\(600\) 0 0
\(601\) 25.4641 1.03870 0.519351 0.854561i \(-0.326174\pi\)
0.519351 + 0.854561i \(0.326174\pi\)
\(602\) 0 0
\(603\) 17.9090 0.729309
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 20.9808i − 0.851583i −0.904821 0.425791i \(-0.859996\pi\)
0.904821 0.425791i \(-0.140004\pi\)
\(608\) 0 0
\(609\) −13.8564 −0.561490
\(610\) 0 0
\(611\) − 23.3205i − 0.943447i
\(612\) 0 0
\(613\) −5.60770 −0.226493 −0.113246 0.993567i \(-0.536125\pi\)
−0.113246 + 0.993567i \(0.536125\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.4641i 1.10566i 0.833293 + 0.552832i \(0.186453\pi\)
−0.833293 + 0.552832i \(0.813547\pi\)
\(618\) 0 0
\(619\) − 33.3205i − 1.33926i −0.742693 0.669632i \(-0.766452\pi\)
0.742693 0.669632i \(-0.233548\pi\)
\(620\) 0 0
\(621\) − 16.7846i − 0.673543i
\(622\) 0 0
\(623\) − 13.4641i − 0.539428i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 10.9282 0.436430
\(628\) 0 0
\(629\) − 6.92820i − 0.276246i
\(630\) 0 0
\(631\) −11.3205 −0.450662 −0.225331 0.974282i \(-0.572346\pi\)
−0.225331 + 0.974282i \(0.572346\pi\)
\(632\) 0 0
\(633\) − 10.8231i − 0.430179i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.60770 0.0636992
\(638\) 0 0
\(639\) −3.60770 −0.142718
\(640\) 0 0
\(641\) −20.3923 −0.805448 −0.402724 0.915322i \(-0.631936\pi\)
−0.402724 + 0.915322i \(0.631936\pi\)
\(642\) 0 0
\(643\) 14.8756 0.586638 0.293319 0.956015i \(-0.405240\pi\)
0.293319 + 0.956015i \(0.405240\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.2679i 0.521617i 0.965391 + 0.260808i \(0.0839891\pi\)
−0.965391 + 0.260808i \(0.916011\pi\)
\(648\) 0 0
\(649\) 1.07180 0.0420717
\(650\) 0 0
\(651\) − 2.92820i − 0.114765i
\(652\) 0 0
\(653\) 36.2487 1.41852 0.709261 0.704946i \(-0.249029\pi\)
0.709261 + 0.704946i \(0.249029\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.32051i − 0.0515179i
\(658\) 0 0
\(659\) 17.3205i 0.674711i 0.941377 + 0.337356i \(0.109532\pi\)
−0.941377 + 0.337356i \(0.890468\pi\)
\(660\) 0 0
\(661\) − 35.8564i − 1.39465i −0.716754 0.697326i \(-0.754373\pi\)
0.716754 0.697326i \(-0.245627\pi\)
\(662\) 0 0
\(663\) 8.78461i 0.341166i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 29.0718 1.12566
\(668\) 0 0
\(669\) − 11.8564i − 0.458395i
\(670\) 0 0
\(671\) −9.85641 −0.380502
\(672\) 0 0
\(673\) 19.4641i 0.750286i 0.926967 + 0.375143i \(0.122406\pi\)
−0.926967 + 0.375143i \(0.877594\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 38.3923 1.47554 0.737768 0.675054i \(-0.235880\pi\)
0.737768 + 0.675054i \(0.235880\pi\)
\(678\) 0 0
\(679\) −17.4641 −0.670211
\(680\) 0 0
\(681\) −20.5359 −0.786937
\(682\) 0 0
\(683\) 34.9808 1.33850 0.669251 0.743037i \(-0.266615\pi\)
0.669251 + 0.743037i \(0.266615\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 2.92820i − 0.111718i
\(688\) 0 0
\(689\) 15.7128 0.598610
\(690\) 0 0
\(691\) − 18.0000i − 0.684752i −0.939563 0.342376i \(-0.888768\pi\)
0.939563 0.342376i \(-0.111232\pi\)
\(692\) 0 0
\(693\) −13.4641 −0.511459
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 18.9282i − 0.716957i
\(698\) 0 0
\(699\) − 21.4641i − 0.811847i
\(700\) 0 0
\(701\) 32.9282i 1.24368i 0.783144 + 0.621841i \(0.213615\pi\)
−0.783144 + 0.621841i \(0.786385\pi\)
\(702\) 0 0
\(703\) − 14.9282i − 0.563028i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.8564 −1.12287
\(708\) 0 0
\(709\) − 28.7846i − 1.08103i −0.841335 0.540514i \(-0.818230\pi\)
0.841335 0.540514i \(-0.181770\pi\)
\(710\) 0 0
\(711\) 36.7846 1.37953
\(712\) 0 0
\(713\) 6.14359i 0.230079i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 14.6410 0.546779
\(718\) 0 0
\(719\) 25.8564 0.964281 0.482141 0.876094i \(-0.339859\pi\)
0.482141 + 0.876094i \(0.339859\pi\)
\(720\) 0 0
\(721\) 4.53590 0.168926
\(722\) 0 0
\(723\) 3.21539 0.119582
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 14.0526i 0.521181i 0.965449 + 0.260590i \(0.0839172\pi\)
−0.965449 + 0.260590i \(0.916083\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) − 30.2487i − 1.11879i
\(732\) 0 0
\(733\) 48.9282 1.80720 0.903602 0.428373i \(-0.140913\pi\)
0.903602 + 0.428373i \(0.140913\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 14.5359i − 0.535437i
\(738\) 0 0
\(739\) − 5.32051i − 0.195718i −0.995200 0.0978590i \(-0.968801\pi\)
0.995200 0.0978590i \(-0.0311994\pi\)
\(740\) 0 0
\(741\) 18.9282i 0.695345i
\(742\) 0 0
\(743\) 40.9808i 1.50344i 0.659483 + 0.751719i \(0.270775\pi\)
−0.659483 + 0.751719i \(0.729225\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.6603 −0.426626
\(748\) 0 0
\(749\) − 2.00000i − 0.0730784i
\(750\) 0 0
\(751\) 22.2487 0.811867 0.405934 0.913903i \(-0.366946\pi\)
0.405934 + 0.913903i \(0.366946\pi\)
\(752\) 0 0
\(753\) 8.10512i 0.295367i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −32.9282 −1.19680 −0.598398 0.801199i \(-0.704196\pi\)
−0.598398 + 0.801199i \(0.704196\pi\)
\(758\) 0 0
\(759\) −6.14359 −0.222998
\(760\) 0 0
\(761\) 49.7128 1.80209 0.901044 0.433728i \(-0.142802\pi\)
0.901044 + 0.433728i \(0.142802\pi\)
\(762\) 0 0
\(763\) −8.39230 −0.303822
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.85641i 0.0670310i
\(768\) 0 0
\(769\) 0.928203 0.0334719 0.0167359 0.999860i \(-0.494673\pi\)
0.0167359 + 0.999860i \(0.494673\pi\)
\(770\) 0 0
\(771\) − 1.46410i − 0.0527283i
\(772\) 0 0
\(773\) 1.60770 0.0578248 0.0289124 0.999582i \(-0.490796\pi\)
0.0289124 + 0.999582i \(0.490796\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 4.00000i − 0.143499i
\(778\) 0 0
\(779\) − 40.7846i − 1.46126i
\(780\) 0 0
\(781\) 2.92820i 0.104779i
\(782\) 0 0
\(783\) 27.7128i 0.990375i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 14.5885 0.520022 0.260011 0.965606i \(-0.416274\pi\)
0.260011 + 0.965606i \(0.416274\pi\)
\(788\) 0 0
\(789\) 4.14359i 0.147516i
\(790\) 0 0
\(791\) 2.53590 0.0901662
\(792\) 0 0
\(793\) − 17.0718i − 0.606237i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.1051 −0.924691 −0.462345 0.886700i \(-0.652992\pi\)
−0.462345 + 0.886700i \(0.652992\pi\)
\(798\) 0 0
\(799\) −23.3205 −0.825020
\(800\) 0 0
\(801\) −12.1436 −0.429073
\(802\) 0 0
\(803\) −1.07180 −0.0378229
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.60770i 0.126997i
\(808\) 0 0
\(809\) 3.85641 0.135584 0.0677920 0.997699i \(-0.478405\pi\)
0.0677920 + 0.997699i \(0.478405\pi\)
\(810\) 0 0
\(811\) 15.0718i 0.529242i 0.964352 + 0.264621i \(0.0852469\pi\)
−0.964352 + 0.264621i \(0.914753\pi\)
\(812\) 0 0
\(813\) 11.2154 0.393341
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 65.1769i − 2.28025i
\(818\) 0 0
\(819\) − 23.3205i − 0.814885i
\(820\) 0 0
\(821\) − 6.78461i − 0.236785i −0.992967 0.118392i \(-0.962226\pi\)
0.992967 0.118392i \(-0.0377740\pi\)
\(822\) 0 0
\(823\) − 15.1244i − 0.527202i −0.964632 0.263601i \(-0.915090\pi\)
0.964632 0.263601i \(-0.0849102\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.12436 0.0390977 0.0195488 0.999809i \(-0.493777\pi\)
0.0195488 + 0.999809i \(0.493777\pi\)
\(828\) 0 0
\(829\) − 15.0718i − 0.523465i −0.965140 0.261733i \(-0.915706\pi\)
0.965140 0.261733i \(-0.0842938\pi\)
\(830\) 0 0
\(831\) −1.46410 −0.0507891
\(832\) 0 0
\(833\) − 1.60770i − 0.0557033i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.85641 −0.202427
\(838\) 0 0
\(839\) −16.7846 −0.579469 −0.289735 0.957107i \(-0.593567\pi\)
−0.289735 + 0.957107i \(0.593567\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) −12.7846 −0.440325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 19.1244i − 0.657121i
\(848\) 0 0
\(849\) −5.60770 −0.192456
\(850\) 0 0
\(851\) 8.39230i 0.287685i
\(852\) 0 0
\(853\) 42.3923 1.45148 0.725742 0.687967i \(-0.241496\pi\)
0.725742 + 0.687967i \(0.241496\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 7.85641i − 0.268370i −0.990956 0.134185i \(-0.957158\pi\)
0.990956 0.134185i \(-0.0428416\pi\)
\(858\) 0 0
\(859\) − 20.2487i − 0.690877i −0.938441 0.345439i \(-0.887730\pi\)
0.938441 0.345439i \(-0.112270\pi\)
\(860\) 0 0
\(861\) − 10.9282i − 0.372432i
\(862\) 0 0
\(863\) − 30.3397i − 1.03278i −0.856354 0.516388i \(-0.827276\pi\)
0.856354 0.516388i \(-0.172724\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.66025 −0.124309
\(868\) 0 0
\(869\) − 29.8564i − 1.01281i
\(870\) 0 0
\(871\) 25.1769 0.853087
\(872\) 0 0
\(873\) 15.7513i 0.533100i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 53.7128 1.81375 0.906876 0.421397i \(-0.138460\pi\)
0.906876 + 0.421397i \(0.138460\pi\)
\(878\) 0 0
\(879\) 8.67949 0.292752
\(880\) 0 0
\(881\) 2.53590 0.0854366 0.0427183 0.999087i \(-0.486398\pi\)
0.0427183 + 0.999087i \(0.486398\pi\)
\(882\) 0 0
\(883\) 37.9090 1.27574 0.637869 0.770145i \(-0.279816\pi\)
0.637869 + 0.770145i \(0.279816\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 51.9090i 1.74293i 0.490455 + 0.871466i \(0.336830\pi\)
−0.490455 + 0.871466i \(0.663170\pi\)
\(888\) 0 0
\(889\) 36.2487 1.21574
\(890\) 0 0
\(891\) 8.92820i 0.299106i
\(892\) 0 0
\(893\) −50.2487 −1.68151
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 10.6410i − 0.355293i
\(898\) 0 0
\(899\) − 10.1436i − 0.338308i
\(900\) 0 0
\(901\) − 15.7128i − 0.523470i
\(902\) 0 0
\(903\) − 17.4641i − 0.581169i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −29.1244 −0.967058 −0.483529 0.875328i \(-0.660645\pi\)
−0.483529 + 0.875328i \(0.660645\pi\)
\(908\) 0 0
\(909\) 26.9282i 0.893152i
\(910\) 0 0
\(911\) −13.1769 −0.436571 −0.218285 0.975885i \(-0.570046\pi\)
−0.218285 + 0.975885i \(0.570046\pi\)
\(912\) 0 0
\(913\) 9.46410i 0.313216i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −21.4641 −0.708807
\(918\) 0 0
\(919\) −25.0718 −0.827042 −0.413521 0.910495i \(-0.635701\pi\)
−0.413521 + 0.910495i \(0.635701\pi\)
\(920\) 0 0
\(921\) 19.7513 0.650827
\(922\) 0 0
\(923\) −5.07180 −0.166940
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.09103i − 0.134367i
\(928\) 0 0
\(929\) 10.5359 0.345672 0.172836 0.984951i \(-0.444707\pi\)
0.172836 + 0.984951i \(0.444707\pi\)
\(930\) 0 0
\(931\) − 3.46410i − 0.113531i
\(932\) 0 0
\(933\) −2.43078 −0.0795802
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44.2487i 1.44554i 0.691087 + 0.722771i \(0.257132\pi\)
−0.691087 + 0.722771i \(0.742868\pi\)
\(938\) 0 0
\(939\) 23.3205i 0.761036i
\(940\) 0 0
\(941\) − 32.0000i − 1.04317i −0.853199 0.521585i \(-0.825341\pi\)
0.853199 0.521585i \(-0.174659\pi\)
\(942\) 0 0
\(943\) 22.9282i 0.746645i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −21.1244 −0.686449 −0.343225 0.939253i \(-0.611519\pi\)
−0.343225 + 0.939253i \(0.611519\pi\)
\(948\) 0 0
\(949\) − 1.85641i − 0.0602615i
\(950\) 0 0
\(951\) −11.3205 −0.367093
\(952\) 0 0
\(953\) − 58.7846i − 1.90422i −0.305755 0.952110i \(-0.598909\pi\)
0.305755 0.952110i \(-0.401091\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.1436 0.327896
\(958\) 0 0
\(959\) 24.3923 0.787669
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) −1.80385 −0.0581282
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 33.6603i 1.08244i 0.840881 + 0.541220i \(0.182038\pi\)
−0.840881 + 0.541220i \(0.817962\pi\)
\(968\) 0 0
\(969\) 18.9282 0.608061
\(970\) 0 0
\(971\) − 23.0718i − 0.740409i −0.928950 0.370205i \(-0.879288\pi\)
0.928950 0.370205i \(-0.120712\pi\)
\(972\) 0 0
\(973\) −20.3923 −0.653747
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31.4641i 1.00663i 0.864104 + 0.503313i \(0.167886\pi\)
−0.864104 + 0.503313i \(0.832114\pi\)
\(978\) 0 0
\(979\) 9.85641i 0.315012i
\(980\) 0 0
\(981\) 7.56922i 0.241667i
\(982\) 0 0
\(983\) 45.2679i 1.44382i 0.691985 + 0.721912i \(0.256736\pi\)
−0.691985 + 0.721912i \(0.743264\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −13.4641 −0.428567
\(988\) 0 0
\(989\) 36.6410i 1.16512i
\(990\) 0 0
\(991\) −34.5359 −1.09707 −0.548534 0.836128i \(-0.684814\pi\)
−0.548534 + 0.836128i \(0.684814\pi\)
\(992\) 0 0
\(993\) − 10.2487i − 0.325233i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 51.1769 1.62079 0.810395 0.585884i \(-0.199253\pi\)
0.810395 + 0.585884i \(0.199253\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 800.2.f.e.49.1 4
3.2 odd 2 7200.2.d.n.2449.1 4
4.3 odd 2 200.2.f.e.149.3 4
5.2 odd 4 160.2.d.a.81.3 4
5.3 odd 4 800.2.d.e.401.2 4
5.4 even 2 800.2.f.c.49.4 4
8.3 odd 2 200.2.f.c.149.1 4
8.5 even 2 800.2.f.c.49.3 4
12.11 even 2 1800.2.d.l.1549.2 4
15.2 even 4 1440.2.k.e.721.4 4
15.8 even 4 7200.2.k.j.3601.1 4
15.14 odd 2 7200.2.d.o.2449.4 4
20.3 even 4 200.2.d.f.101.1 4
20.7 even 4 40.2.d.a.21.4 yes 4
20.19 odd 2 200.2.f.c.149.2 4
24.5 odd 2 7200.2.d.o.2449.1 4
24.11 even 2 1800.2.d.p.1549.4 4
40.3 even 4 200.2.d.f.101.2 4
40.13 odd 4 800.2.d.e.401.3 4
40.19 odd 2 200.2.f.e.149.4 4
40.27 even 4 40.2.d.a.21.3 4
40.29 even 2 inner 800.2.f.e.49.2 4
40.37 odd 4 160.2.d.a.81.2 4
60.23 odd 4 1800.2.k.j.901.4 4
60.47 odd 4 360.2.k.e.181.1 4
60.59 even 2 1800.2.d.p.1549.3 4
80.3 even 4 6400.2.a.ce.1.1 2
80.13 odd 4 6400.2.a.be.1.2 2
80.27 even 4 1280.2.a.o.1.1 2
80.37 odd 4 1280.2.a.d.1.2 2
80.43 even 4 6400.2.a.z.1.2 2
80.53 odd 4 6400.2.a.cj.1.1 2
80.67 even 4 1280.2.a.a.1.2 2
80.77 odd 4 1280.2.a.n.1.1 2
120.29 odd 2 7200.2.d.n.2449.4 4
120.53 even 4 7200.2.k.j.3601.2 4
120.59 even 2 1800.2.d.l.1549.1 4
120.77 even 4 1440.2.k.e.721.2 4
120.83 odd 4 1800.2.k.j.901.3 4
120.107 odd 4 360.2.k.e.181.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.3 4 40.27 even 4
40.2.d.a.21.4 yes 4 20.7 even 4
160.2.d.a.81.2 4 40.37 odd 4
160.2.d.a.81.3 4 5.2 odd 4
200.2.d.f.101.1 4 20.3 even 4
200.2.d.f.101.2 4 40.3 even 4
200.2.f.c.149.1 4 8.3 odd 2
200.2.f.c.149.2 4 20.19 odd 2
200.2.f.e.149.3 4 4.3 odd 2
200.2.f.e.149.4 4 40.19 odd 2
360.2.k.e.181.1 4 60.47 odd 4
360.2.k.e.181.2 4 120.107 odd 4
800.2.d.e.401.2 4 5.3 odd 4
800.2.d.e.401.3 4 40.13 odd 4
800.2.f.c.49.3 4 8.5 even 2
800.2.f.c.49.4 4 5.4 even 2
800.2.f.e.49.1 4 1.1 even 1 trivial
800.2.f.e.49.2 4 40.29 even 2 inner
1280.2.a.a.1.2 2 80.67 even 4
1280.2.a.d.1.2 2 80.37 odd 4
1280.2.a.n.1.1 2 80.77 odd 4
1280.2.a.o.1.1 2 80.27 even 4
1440.2.k.e.721.2 4 120.77 even 4
1440.2.k.e.721.4 4 15.2 even 4
1800.2.d.l.1549.1 4 120.59 even 2
1800.2.d.l.1549.2 4 12.11 even 2
1800.2.d.p.1549.3 4 60.59 even 2
1800.2.d.p.1549.4 4 24.11 even 2
1800.2.k.j.901.3 4 120.83 odd 4
1800.2.k.j.901.4 4 60.23 odd 4
6400.2.a.z.1.2 2 80.43 even 4
6400.2.a.be.1.2 2 80.13 odd 4
6400.2.a.ce.1.1 2 80.3 even 4
6400.2.a.cj.1.1 2 80.53 odd 4
7200.2.d.n.2449.1 4 3.2 odd 2
7200.2.d.n.2449.4 4 120.29 odd 2
7200.2.d.o.2449.1 4 24.5 odd 2
7200.2.d.o.2449.4 4 15.14 odd 2
7200.2.k.j.3601.1 4 15.8 even 4
7200.2.k.j.3601.2 4 120.53 even 4