Properties

Label 800.6.c.a.449.1
Level $800$
Weight $6$
Character 800.449
Analytic conductor $128.307$
Analytic rank $0$
Dimension $2$
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] = [800,6,Mod(449,800)]
 
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, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.449");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (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: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.6.c.a.449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000i q^{3} +208.000i q^{7} +179.000 q^{9} +O(q^{10})\) \(q-8.00000i q^{3} +208.000i q^{7} +179.000 q^{9} -536.000 q^{11} +694.000i q^{13} +1278.00i q^{17} -1112.00 q^{19} +1664.00 q^{21} +3216.00i q^{23} -3376.00i q^{27} -2918.00 q^{29} -2624.00 q^{31} +4288.00i q^{33} +9458.00i q^{37} +5552.00 q^{39} +170.000 q^{41} -19928.0i q^{43} -32.0000i q^{47} -26457.0 q^{49} +10224.0 q^{51} -22178.0i q^{53} +8896.00i q^{57} -41480.0 q^{59} +15462.0 q^{61} +37232.0i q^{63} +20744.0i q^{67} +25728.0 q^{69} +28592.0 q^{71} -53670.0i q^{73} -111488. i q^{77} +69152.0 q^{79} +16489.0 q^{81} -37800.0i q^{83} +23344.0i q^{87} +126806. q^{89} -144352. q^{91} +20992.0i q^{93} -62290.0i q^{97} -95944.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 358 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 358 q^{9} - 1072 q^{11} - 2224 q^{19} + 3328 q^{21} - 5836 q^{29} - 5248 q^{31} + 11104 q^{39} + 340 q^{41} - 52914 q^{49} + 20448 q^{51} - 82960 q^{59} + 30924 q^{61} + 51456 q^{69} + 57184 q^{71} + 138304 q^{79} + 32978 q^{81} + 253612 q^{89} - 288704 q^{91} - 191888 q^{99}+O(q^{100}) \) 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
\(3\) − 8.00000i − 0.513200i −0.966518 0.256600i \(-0.917398\pi\)
0.966518 0.256600i \(-0.0826023\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 208.000i 1.60442i 0.597042 + 0.802210i \(0.296343\pi\)
−0.597042 + 0.802210i \(0.703657\pi\)
\(8\) 0 0
\(9\) 179.000 0.736626
\(10\) 0 0
\(11\) −536.000 −1.33562 −0.667810 0.744332i \(-0.732768\pi\)
−0.667810 + 0.744332i \(0.732768\pi\)
\(12\) 0 0
\(13\) 694.000i 1.13894i 0.822012 + 0.569470i \(0.192852\pi\)
−0.822012 + 0.569470i \(0.807148\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1278.00i 1.07253i 0.844050 + 0.536264i \(0.180165\pi\)
−0.844050 + 0.536264i \(0.819835\pi\)
\(18\) 0 0
\(19\) −1112.00 −0.706677 −0.353338 0.935496i \(-0.614954\pi\)
−0.353338 + 0.935496i \(0.614954\pi\)
\(20\) 0 0
\(21\) 1664.00 0.823389
\(22\) 0 0
\(23\) 3216.00i 1.26764i 0.773480 + 0.633821i \(0.218514\pi\)
−0.773480 + 0.633821i \(0.781486\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3376.00i − 0.891237i
\(28\) 0 0
\(29\) −2918.00 −0.644303 −0.322152 0.946688i \(-0.604406\pi\)
−0.322152 + 0.946688i \(0.604406\pi\)
\(30\) 0 0
\(31\) −2624.00 −0.490410 −0.245205 0.969471i \(-0.578855\pi\)
−0.245205 + 0.969471i \(0.578855\pi\)
\(32\) 0 0
\(33\) 4288.00i 0.685441i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9458.00i 1.13578i 0.823104 + 0.567891i \(0.192241\pi\)
−0.823104 + 0.567891i \(0.807759\pi\)
\(38\) 0 0
\(39\) 5552.00 0.584505
\(40\) 0 0
\(41\) 170.000 0.0157939 0.00789695 0.999969i \(-0.497486\pi\)
0.00789695 + 0.999969i \(0.497486\pi\)
\(42\) 0 0
\(43\) − 19928.0i − 1.64359i −0.569786 0.821793i \(-0.692974\pi\)
0.569786 0.821793i \(-0.307026\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 32.0000i − 0.00211303i −0.999999 0.00105651i \(-0.999664\pi\)
0.999999 0.00105651i \(-0.000336299\pi\)
\(48\) 0 0
\(49\) −26457.0 −1.57417
\(50\) 0 0
\(51\) 10224.0 0.550422
\(52\) 0 0
\(53\) − 22178.0i − 1.08451i −0.840215 0.542254i \(-0.817571\pi\)
0.840215 0.542254i \(-0.182429\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8896.00i 0.362667i
\(58\) 0 0
\(59\) −41480.0 −1.55135 −0.775673 0.631135i \(-0.782589\pi\)
−0.775673 + 0.631135i \(0.782589\pi\)
\(60\) 0 0
\(61\) 15462.0 0.532036 0.266018 0.963968i \(-0.414292\pi\)
0.266018 + 0.963968i \(0.414292\pi\)
\(62\) 0 0
\(63\) 37232.0i 1.18186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 20744.0i 0.564554i 0.959333 + 0.282277i \(0.0910897\pi\)
−0.959333 + 0.282277i \(0.908910\pi\)
\(68\) 0 0
\(69\) 25728.0 0.650554
\(70\) 0 0
\(71\) 28592.0 0.673130 0.336565 0.941660i \(-0.390735\pi\)
0.336565 + 0.941660i \(0.390735\pi\)
\(72\) 0 0
\(73\) − 53670.0i − 1.17876i −0.807857 0.589379i \(-0.799373\pi\)
0.807857 0.589379i \(-0.200627\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 111488.i − 2.14290i
\(78\) 0 0
\(79\) 69152.0 1.24663 0.623314 0.781971i \(-0.285786\pi\)
0.623314 + 0.781971i \(0.285786\pi\)
\(80\) 0 0
\(81\) 16489.0 0.279243
\(82\) 0 0
\(83\) − 37800.0i − 0.602277i −0.953580 0.301139i \(-0.902633\pi\)
0.953580 0.301139i \(-0.0973667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23344.0i 0.330657i
\(88\) 0 0
\(89\) 126806. 1.69693 0.848467 0.529249i \(-0.177526\pi\)
0.848467 + 0.529249i \(0.177526\pi\)
\(90\) 0 0
\(91\) −144352. −1.82734
\(92\) 0 0
\(93\) 20992.0i 0.251679i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 62290.0i − 0.672185i −0.941829 0.336093i \(-0.890894\pi\)
0.941829 0.336093i \(-0.109106\pi\)
\(98\) 0 0
\(99\) −95944.0 −0.983852
\(100\) 0 0
\(101\) 6414.00 0.0625641 0.0312821 0.999511i \(-0.490041\pi\)
0.0312821 + 0.999511i \(0.490041\pi\)
\(102\) 0 0
\(103\) − 108432.i − 1.00708i −0.863972 0.503541i \(-0.832030\pi\)
0.863972 0.503541i \(-0.167970\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 103976.i − 0.877958i −0.898497 0.438979i \(-0.855340\pi\)
0.898497 0.438979i \(-0.144660\pi\)
\(108\) 0 0
\(109\) −2486.00 −0.0200417 −0.0100209 0.999950i \(-0.503190\pi\)
−0.0100209 + 0.999950i \(0.503190\pi\)
\(110\) 0 0
\(111\) 75664.0 0.582884
\(112\) 0 0
\(113\) 15794.0i 0.116358i 0.998306 + 0.0581790i \(0.0185294\pi\)
−0.998306 + 0.0581790i \(0.981471\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 124226.i 0.838973i
\(118\) 0 0
\(119\) −265824. −1.72079
\(120\) 0 0
\(121\) 126245. 0.783882
\(122\) 0 0
\(123\) − 1360.00i − 0.00810543i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1024.00i − 0.00563366i −0.999996 0.00281683i \(-0.999103\pi\)
0.999996 0.00281683i \(-0.000896626\pi\)
\(128\) 0 0
\(129\) −159424. −0.843489
\(130\) 0 0
\(131\) −22664.0 −0.115387 −0.0576937 0.998334i \(-0.518375\pi\)
−0.0576937 + 0.998334i \(0.518375\pi\)
\(132\) 0 0
\(133\) − 231296.i − 1.13381i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 53238.0i 0.242337i 0.992632 + 0.121169i \(0.0386642\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(138\) 0 0
\(139\) −19816.0 −0.0869919 −0.0434960 0.999054i \(-0.513850\pi\)
−0.0434960 + 0.999054i \(0.513850\pi\)
\(140\) 0 0
\(141\) −256.000 −0.00108441
\(142\) 0 0
\(143\) − 371984.i − 1.52119i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 211656.i 0.807862i
\(148\) 0 0
\(149\) −452190. −1.66861 −0.834306 0.551302i \(-0.814131\pi\)
−0.834306 + 0.551302i \(0.814131\pi\)
\(150\) 0 0
\(151\) −263280. −0.939670 −0.469835 0.882754i \(-0.655687\pi\)
−0.469835 + 0.882754i \(0.655687\pi\)
\(152\) 0 0
\(153\) 228762.i 0.790051i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 353530.i 1.14466i 0.820023 + 0.572331i \(0.193961\pi\)
−0.820023 + 0.572331i \(0.806039\pi\)
\(158\) 0 0
\(159\) −177424. −0.556570
\(160\) 0 0
\(161\) −668928. −2.03383
\(162\) 0 0
\(163\) − 100936.i − 0.297562i −0.988870 0.148781i \(-0.952465\pi\)
0.988870 0.148781i \(-0.0475349\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 284944.i 0.790621i 0.918548 + 0.395310i \(0.129363\pi\)
−0.918548 + 0.395310i \(0.870637\pi\)
\(168\) 0 0
\(169\) −110343. −0.297186
\(170\) 0 0
\(171\) −199048. −0.520556
\(172\) 0 0
\(173\) 484374.i 1.23045i 0.788350 + 0.615227i \(0.210936\pi\)
−0.788350 + 0.615227i \(0.789064\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 331840.i 0.796151i
\(178\) 0 0
\(179\) −406680. −0.948681 −0.474341 0.880341i \(-0.657313\pi\)
−0.474341 + 0.880341i \(0.657313\pi\)
\(180\) 0 0
\(181\) 570302. 1.29392 0.646962 0.762523i \(-0.276039\pi\)
0.646962 + 0.762523i \(0.276039\pi\)
\(182\) 0 0
\(183\) − 123696.i − 0.273041i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 685008.i − 1.43249i
\(188\) 0 0
\(189\) 702208. 1.42992
\(190\) 0 0
\(191\) 138624. 0.274951 0.137475 0.990505i \(-0.456101\pi\)
0.137475 + 0.990505i \(0.456101\pi\)
\(192\) 0 0
\(193\) 34482.0i 0.0666345i 0.999445 + 0.0333173i \(0.0106072\pi\)
−0.999445 + 0.0333173i \(0.989393\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 643598.i − 1.18154i −0.806839 0.590771i \(-0.798824\pi\)
0.806839 0.590771i \(-0.201176\pi\)
\(198\) 0 0
\(199\) −1.10738e6 −1.98227 −0.991134 0.132865i \(-0.957582\pi\)
−0.991134 + 0.132865i \(0.957582\pi\)
\(200\) 0 0
\(201\) 165952. 0.289729
\(202\) 0 0
\(203\) − 606944.i − 1.03373i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 575664.i 0.933777i
\(208\) 0 0
\(209\) 596032. 0.943852
\(210\) 0 0
\(211\) 229976. 0.355612 0.177806 0.984066i \(-0.443100\pi\)
0.177806 + 0.984066i \(0.443100\pi\)
\(212\) 0 0
\(213\) − 228736.i − 0.345450i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 545792.i − 0.786824i
\(218\) 0 0
\(219\) −429360. −0.604939
\(220\) 0 0
\(221\) −886932. −1.22155
\(222\) 0 0
\(223\) − 1.08947e6i − 1.46708i −0.679646 0.733540i \(-0.737867\pi\)
0.679646 0.733540i \(-0.262133\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 687048.i 0.884958i 0.896779 + 0.442479i \(0.145901\pi\)
−0.896779 + 0.442479i \(0.854099\pi\)
\(228\) 0 0
\(229\) 699730. 0.881743 0.440871 0.897570i \(-0.354670\pi\)
0.440871 + 0.897570i \(0.354670\pi\)
\(230\) 0 0
\(231\) −891904. −1.09974
\(232\) 0 0
\(233\) 937722.i 1.13158i 0.824550 + 0.565789i \(0.191428\pi\)
−0.824550 + 0.565789i \(0.808572\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 553216.i − 0.639770i
\(238\) 0 0
\(239\) −643488. −0.728695 −0.364347 0.931263i \(-0.618708\pi\)
−0.364347 + 0.931263i \(0.618708\pi\)
\(240\) 0 0
\(241\) 157282. 0.174436 0.0872181 0.996189i \(-0.472202\pi\)
0.0872181 + 0.996189i \(0.472202\pi\)
\(242\) 0 0
\(243\) − 952280.i − 1.03454i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 771728.i − 0.804863i
\(248\) 0 0
\(249\) −302400. −0.309089
\(250\) 0 0
\(251\) −1.58604e6 −1.58902 −0.794511 0.607250i \(-0.792273\pi\)
−0.794511 + 0.607250i \(0.792273\pi\)
\(252\) 0 0
\(253\) − 1.72378e6i − 1.69309i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 654334.i 0.617969i 0.951067 + 0.308984i \(0.0999891\pi\)
−0.951067 + 0.308984i \(0.900011\pi\)
\(258\) 0 0
\(259\) −1.96726e6 −1.82227
\(260\) 0 0
\(261\) −522322. −0.474610
\(262\) 0 0
\(263\) − 330192.i − 0.294359i −0.989110 0.147179i \(-0.952981\pi\)
0.989110 0.147179i \(-0.0470195\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 1.01445e6i − 0.870867i
\(268\) 0 0
\(269\) −1.56956e6 −1.32250 −0.661252 0.750164i \(-0.729974\pi\)
−0.661252 + 0.750164i \(0.729974\pi\)
\(270\) 0 0
\(271\) 957792. 0.792224 0.396112 0.918202i \(-0.370359\pi\)
0.396112 + 0.918202i \(0.370359\pi\)
\(272\) 0 0
\(273\) 1.15482e6i 0.937791i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 565438.i − 0.442778i −0.975186 0.221389i \(-0.928941\pi\)
0.975186 0.221389i \(-0.0710590\pi\)
\(278\) 0 0
\(279\) −469696. −0.361249
\(280\) 0 0
\(281\) −1.34127e6 −1.01333 −0.506664 0.862143i \(-0.669122\pi\)
−0.506664 + 0.862143i \(0.669122\pi\)
\(282\) 0 0
\(283\) − 734264.i − 0.544987i −0.962158 0.272494i \(-0.912152\pi\)
0.962158 0.272494i \(-0.0878483\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 35360.0i 0.0253401i
\(288\) 0 0
\(289\) −213427. −0.150316
\(290\) 0 0
\(291\) −498320. −0.344966
\(292\) 0 0
\(293\) − 1.13320e6i − 0.771149i −0.922677 0.385574i \(-0.874003\pi\)
0.922677 0.385574i \(-0.125997\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.80954e6i 1.19035i
\(298\) 0 0
\(299\) −2.23190e6 −1.44377
\(300\) 0 0
\(301\) 4.14502e6 2.63700
\(302\) 0 0
\(303\) − 51312.0i − 0.0321079i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.91377e6i 1.76445i 0.470829 + 0.882224i \(0.343955\pi\)
−0.470829 + 0.882224i \(0.656045\pi\)
\(308\) 0 0
\(309\) −867456. −0.516834
\(310\) 0 0
\(311\) −1.43813e6 −0.843134 −0.421567 0.906797i \(-0.638520\pi\)
−0.421567 + 0.906797i \(0.638520\pi\)
\(312\) 0 0
\(313\) − 1.37601e6i − 0.793888i −0.917843 0.396944i \(-0.870071\pi\)
0.917843 0.396944i \(-0.129929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.23494e6i 0.690235i 0.938559 + 0.345118i \(0.112161\pi\)
−0.938559 + 0.345118i \(0.887839\pi\)
\(318\) 0 0
\(319\) 1.56405e6 0.860545
\(320\) 0 0
\(321\) −831808. −0.450568
\(322\) 0 0
\(323\) − 1.42114e6i − 0.757930i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 19888.0i 0.0102854i
\(328\) 0 0
\(329\) 6656.00 0.00339019
\(330\) 0 0
\(331\) −1.48930e6 −0.747160 −0.373580 0.927598i \(-0.621870\pi\)
−0.373580 + 0.927598i \(0.621870\pi\)
\(332\) 0 0
\(333\) 1.69298e6i 0.836646i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 838226.i − 0.402056i −0.979586 0.201028i \(-0.935572\pi\)
0.979586 0.201028i \(-0.0644282\pi\)
\(338\) 0 0
\(339\) 126352. 0.0597149
\(340\) 0 0
\(341\) 1.40646e6 0.655002
\(342\) 0 0
\(343\) − 2.00720e6i − 0.921203i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 350008.i 0.156047i 0.996952 + 0.0780233i \(0.0248609\pi\)
−0.996952 + 0.0780233i \(0.975139\pi\)
\(348\) 0 0
\(349\) 383642. 0.168602 0.0843010 0.996440i \(-0.473134\pi\)
0.0843010 + 0.996440i \(0.473134\pi\)
\(350\) 0 0
\(351\) 2.34294e6 1.01507
\(352\) 0 0
\(353\) 4.09309e6i 1.74829i 0.485661 + 0.874147i \(0.338579\pi\)
−0.485661 + 0.874147i \(0.661421\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.12659e6i 0.883108i
\(358\) 0 0
\(359\) −3.14430e6 −1.28762 −0.643811 0.765185i \(-0.722648\pi\)
−0.643811 + 0.765185i \(0.722648\pi\)
\(360\) 0 0
\(361\) −1.23955e6 −0.500608
\(362\) 0 0
\(363\) − 1.00996e6i − 0.402288i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.47619e6i 0.572108i 0.958214 + 0.286054i \(0.0923436\pi\)
−0.958214 + 0.286054i \(0.907656\pi\)
\(368\) 0 0
\(369\) 30430.0 0.0116342
\(370\) 0 0
\(371\) 4.61302e6 1.74001
\(372\) 0 0
\(373\) − 3.73981e6i − 1.39180i −0.718138 0.695901i \(-0.755005\pi\)
0.718138 0.695901i \(-0.244995\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.02509e6i − 0.733823i
\(378\) 0 0
\(379\) −1.89966e6 −0.679324 −0.339662 0.940548i \(-0.610313\pi\)
−0.339662 + 0.940548i \(0.610313\pi\)
\(380\) 0 0
\(381\) −8192.00 −0.00289120
\(382\) 0 0
\(383\) 1.74310e6i 0.607192i 0.952801 + 0.303596i \(0.0981874\pi\)
−0.952801 + 0.303596i \(0.901813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 3.56711e6i − 1.21071i
\(388\) 0 0
\(389\) 2.69147e6 0.901812 0.450906 0.892571i \(-0.351101\pi\)
0.450906 + 0.892571i \(0.351101\pi\)
\(390\) 0 0
\(391\) −4.11005e6 −1.35958
\(392\) 0 0
\(393\) 181312.i 0.0592168i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 5.37353e6i − 1.71113i −0.517695 0.855565i \(-0.673210\pi\)
0.517695 0.855565i \(-0.326790\pi\)
\(398\) 0 0
\(399\) −1.85037e6 −0.581870
\(400\) 0 0
\(401\) 156418. 0.0485765 0.0242882 0.999705i \(-0.492268\pi\)
0.0242882 + 0.999705i \(0.492268\pi\)
\(402\) 0 0
\(403\) − 1.82106e6i − 0.558548i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 5.06949e6i − 1.51697i
\(408\) 0 0
\(409\) 306086. 0.0904764 0.0452382 0.998976i \(-0.485595\pi\)
0.0452382 + 0.998976i \(0.485595\pi\)
\(410\) 0 0
\(411\) 425904. 0.124368
\(412\) 0 0
\(413\) − 8.62784e6i − 2.48901i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 158528.i 0.0446443i
\(418\) 0 0
\(419\) 6.70868e6 1.86682 0.933409 0.358814i \(-0.116819\pi\)
0.933409 + 0.358814i \(0.116819\pi\)
\(420\) 0 0
\(421\) 4.02347e6 1.10636 0.553179 0.833063i \(-0.313415\pi\)
0.553179 + 0.833063i \(0.313415\pi\)
\(422\) 0 0
\(423\) − 5728.00i − 0.00155651i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.21610e6i 0.853610i
\(428\) 0 0
\(429\) −2.97587e6 −0.780676
\(430\) 0 0
\(431\) −7.04304e6 −1.82628 −0.913139 0.407648i \(-0.866349\pi\)
−0.913139 + 0.407648i \(0.866349\pi\)
\(432\) 0 0
\(433\) − 1.25142e6i − 0.320763i −0.987055 0.160381i \(-0.948728\pi\)
0.987055 0.160381i \(-0.0512724\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.57619e6i − 0.895813i
\(438\) 0 0
\(439\) 1.25406e6 0.310569 0.155285 0.987870i \(-0.450371\pi\)
0.155285 + 0.987870i \(0.450371\pi\)
\(440\) 0 0
\(441\) −4.73580e6 −1.15957
\(442\) 0 0
\(443\) 5.18081e6i 1.25426i 0.778914 + 0.627131i \(0.215771\pi\)
−0.778914 + 0.627131i \(0.784229\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.61752e6i 0.856332i
\(448\) 0 0
\(449\) 5.73064e6 1.34149 0.670745 0.741688i \(-0.265975\pi\)
0.670745 + 0.741688i \(0.265975\pi\)
\(450\) 0 0
\(451\) −91120.0 −0.0210947
\(452\) 0 0
\(453\) 2.10624e6i 0.482239i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.24153e6i 1.17400i 0.809588 + 0.586999i \(0.199691\pi\)
−0.809588 + 0.586999i \(0.800309\pi\)
\(458\) 0 0
\(459\) 4.31453e6 0.955876
\(460\) 0 0
\(461\) −173994. −0.0381313 −0.0190657 0.999818i \(-0.506069\pi\)
−0.0190657 + 0.999818i \(0.506069\pi\)
\(462\) 0 0
\(463\) 4.01277e6i 0.869945i 0.900444 + 0.434972i \(0.143242\pi\)
−0.900444 + 0.434972i \(0.856758\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 774616.i − 0.164359i −0.996618 0.0821796i \(-0.973812\pi\)
0.996618 0.0821796i \(-0.0261881\pi\)
\(468\) 0 0
\(469\) −4.31475e6 −0.905782
\(470\) 0 0
\(471\) 2.82824e6 0.587441
\(472\) 0 0
\(473\) 1.06814e7i 2.19521i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 3.96986e6i − 0.798876i
\(478\) 0 0
\(479\) −2.33530e6 −0.465054 −0.232527 0.972590i \(-0.574699\pi\)
−0.232527 + 0.972590i \(0.574699\pi\)
\(480\) 0 0
\(481\) −6.56385e6 −1.29359
\(482\) 0 0
\(483\) 5.35142e6i 1.04376i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 1.03947e6i − 0.198605i −0.995057 0.0993025i \(-0.968339\pi\)
0.995057 0.0993025i \(-0.0316612\pi\)
\(488\) 0 0
\(489\) −807488. −0.152709
\(490\) 0 0
\(491\) 7.85092e6 1.46966 0.734830 0.678251i \(-0.237262\pi\)
0.734830 + 0.678251i \(0.237262\pi\)
\(492\) 0 0
\(493\) − 3.72920e6i − 0.691033i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.94714e6i 1.07998i
\(498\) 0 0
\(499\) −2.71644e6 −0.488370 −0.244185 0.969729i \(-0.578520\pi\)
−0.244185 + 0.969729i \(0.578520\pi\)
\(500\) 0 0
\(501\) 2.27955e6 0.405747
\(502\) 0 0
\(503\) − 4.62034e6i − 0.814242i −0.913374 0.407121i \(-0.866533\pi\)
0.913374 0.407121i \(-0.133467\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 882744.i 0.152516i
\(508\) 0 0
\(509\) 4.46198e6 0.763366 0.381683 0.924293i \(-0.375345\pi\)
0.381683 + 0.924293i \(0.375345\pi\)
\(510\) 0 0
\(511\) 1.11634e7 1.89122
\(512\) 0 0
\(513\) 3.75411e6i 0.629816i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 17152.0i 0.00282220i
\(518\) 0 0
\(519\) 3.87499e6 0.631470
\(520\) 0 0
\(521\) 3.74375e6 0.604245 0.302122 0.953269i \(-0.402305\pi\)
0.302122 + 0.953269i \(0.402305\pi\)
\(522\) 0 0
\(523\) 9.28433e6i 1.48421i 0.670282 + 0.742107i \(0.266173\pi\)
−0.670282 + 0.742107i \(0.733827\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 3.35347e6i − 0.525979i
\(528\) 0 0
\(529\) −3.90631e6 −0.606915
\(530\) 0 0
\(531\) −7.42492e6 −1.14276
\(532\) 0 0
\(533\) 117980.i 0.0179883i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.25344e6i 0.486863i
\(538\) 0 0
\(539\) 1.41810e7 2.10249
\(540\) 0 0
\(541\) 862150. 0.126645 0.0633227 0.997993i \(-0.479830\pi\)
0.0633227 + 0.997993i \(0.479830\pi\)
\(542\) 0 0
\(543\) − 4.56242e6i − 0.664042i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.75442e6i − 0.250707i −0.992112 0.125353i \(-0.959994\pi\)
0.992112 0.125353i \(-0.0400065\pi\)
\(548\) 0 0
\(549\) 2.76770e6 0.391911
\(550\) 0 0
\(551\) 3.24482e6 0.455314
\(552\) 0 0
\(553\) 1.43836e7i 2.00012i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00292e7i 1.36971i 0.728678 + 0.684856i \(0.240135\pi\)
−0.728678 + 0.684856i \(0.759865\pi\)
\(558\) 0 0
\(559\) 1.38300e7 1.87195
\(560\) 0 0
\(561\) −5.48006e6 −0.735154
\(562\) 0 0
\(563\) − 5.27460e6i − 0.701324i −0.936502 0.350662i \(-0.885957\pi\)
0.936502 0.350662i \(-0.114043\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.42971e6i 0.448023i
\(568\) 0 0
\(569\) −8.36940e6 −1.08371 −0.541856 0.840471i \(-0.682278\pi\)
−0.541856 + 0.840471i \(0.682278\pi\)
\(570\) 0 0
\(571\) 4.02702e6 0.516884 0.258442 0.966027i \(-0.416791\pi\)
0.258442 + 0.966027i \(0.416791\pi\)
\(572\) 0 0
\(573\) − 1.10899e6i − 0.141105i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.37568e6i − 0.297063i −0.988908 0.148532i \(-0.952545\pi\)
0.988908 0.148532i \(-0.0474547\pi\)
\(578\) 0 0
\(579\) 275856. 0.0341968
\(580\) 0 0
\(581\) 7.86240e6 0.966306
\(582\) 0 0
\(583\) 1.18874e7i 1.44849i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.44028e6i 0.412096i 0.978542 + 0.206048i \(0.0660603\pi\)
−0.978542 + 0.206048i \(0.933940\pi\)
\(588\) 0 0
\(589\) 2.91789e6 0.346562
\(590\) 0 0
\(591\) −5.14878e6 −0.606368
\(592\) 0 0
\(593\) − 3.22942e6i − 0.377127i −0.982061 0.188564i \(-0.939617\pi\)
0.982061 0.188564i \(-0.0603832\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.85901e6i 1.01730i
\(598\) 0 0
\(599\) 9.29714e6 1.05872 0.529361 0.848397i \(-0.322432\pi\)
0.529361 + 0.848397i \(0.322432\pi\)
\(600\) 0 0
\(601\) −1.12782e7 −1.27366 −0.636828 0.771006i \(-0.719754\pi\)
−0.636828 + 0.771006i \(0.719754\pi\)
\(602\) 0 0
\(603\) 3.71318e6i 0.415865i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.00115e6i − 0.771255i −0.922655 0.385627i \(-0.873985\pi\)
0.922655 0.385627i \(-0.126015\pi\)
\(608\) 0 0
\(609\) −4.85555e6 −0.530512
\(610\) 0 0
\(611\) 22208.0 0.00240661
\(612\) 0 0
\(613\) − 6.19432e6i − 0.665798i −0.942962 0.332899i \(-0.891973\pi\)
0.942962 0.332899i \(-0.108027\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.62407e6i 0.277500i 0.990327 + 0.138750i \(0.0443084\pi\)
−0.990327 + 0.138750i \(0.955692\pi\)
\(618\) 0 0
\(619\) 2.83721e6 0.297622 0.148811 0.988866i \(-0.452455\pi\)
0.148811 + 0.988866i \(0.452455\pi\)
\(620\) 0 0
\(621\) 1.08572e7 1.12977
\(622\) 0 0
\(623\) 2.63756e7i 2.72259i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4.76826e6i − 0.484385i
\(628\) 0 0
\(629\) −1.20873e7 −1.21816
\(630\) 0 0
\(631\) 1.29656e7 1.29634 0.648170 0.761496i \(-0.275535\pi\)
0.648170 + 0.761496i \(0.275535\pi\)
\(632\) 0 0
\(633\) − 1.83981e6i − 0.182500i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.83612e7i − 1.79288i
\(638\) 0 0
\(639\) 5.11797e6 0.495844
\(640\) 0 0
\(641\) −1.16798e7 −1.12276 −0.561382 0.827557i \(-0.689730\pi\)
−0.561382 + 0.827557i \(0.689730\pi\)
\(642\) 0 0
\(643\) 7.02732e6i 0.670289i 0.942167 + 0.335145i \(0.108785\pi\)
−0.942167 + 0.335145i \(0.891215\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.72821e6i 0.913634i 0.889561 + 0.456817i \(0.151011\pi\)
−0.889561 + 0.456817i \(0.848989\pi\)
\(648\) 0 0
\(649\) 2.22333e7 2.07201
\(650\) 0 0
\(651\) −4.36634e6 −0.403798
\(652\) 0 0
\(653\) − 9.81425e6i − 0.900688i −0.892855 0.450344i \(-0.851301\pi\)
0.892855 0.450344i \(-0.148699\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 9.60693e6i − 0.868303i
\(658\) 0 0
\(659\) −1.46652e7 −1.31545 −0.657724 0.753259i \(-0.728481\pi\)
−0.657724 + 0.753259i \(0.728481\pi\)
\(660\) 0 0
\(661\) −1.41836e7 −1.26265 −0.631327 0.775517i \(-0.717489\pi\)
−0.631327 + 0.775517i \(0.717489\pi\)
\(662\) 0 0
\(663\) 7.09546e6i 0.626897i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.38429e6i − 0.816745i
\(668\) 0 0
\(669\) −8.71578e6 −0.752906
\(670\) 0 0
\(671\) −8.28763e6 −0.710598
\(672\) 0 0
\(673\) 5.49941e6i 0.468035i 0.972232 + 0.234018i \(0.0751873\pi\)
−0.972232 + 0.234018i \(0.924813\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.77375e7i − 1.48737i −0.668528 0.743687i \(-0.733075\pi\)
0.668528 0.743687i \(-0.266925\pi\)
\(678\) 0 0
\(679\) 1.29563e7 1.07847
\(680\) 0 0
\(681\) 5.49638e6 0.454160
\(682\) 0 0
\(683\) − 9.39670e6i − 0.770768i −0.922756 0.385384i \(-0.874069\pi\)
0.922756 0.385384i \(-0.125931\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 5.59784e6i − 0.452510i
\(688\) 0 0
\(689\) 1.53915e7 1.23519
\(690\) 0 0
\(691\) −1.34767e7 −1.07371 −0.536857 0.843673i \(-0.680389\pi\)
−0.536857 + 0.843673i \(0.680389\pi\)
\(692\) 0 0
\(693\) − 1.99564e7i − 1.57851i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 217260.i 0.0169394i
\(698\) 0 0
\(699\) 7.50178e6 0.580726
\(700\) 0 0
\(701\) 2.15594e7 1.65707 0.828536 0.559935i \(-0.189174\pi\)
0.828536 + 0.559935i \(0.189174\pi\)
\(702\) 0 0
\(703\) − 1.05173e7i − 0.802631i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.33411e6i 0.100379i
\(708\) 0 0
\(709\) 6.38165e6 0.476779 0.238390 0.971170i \(-0.423380\pi\)
0.238390 + 0.971170i \(0.423380\pi\)
\(710\) 0 0
\(711\) 1.23782e7 0.918298
\(712\) 0 0
\(713\) − 8.43878e6i − 0.621664i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.14790e6i 0.373966i
\(718\) 0 0
\(719\) 1.63566e7 1.17997 0.589986 0.807413i \(-0.299133\pi\)
0.589986 + 0.807413i \(0.299133\pi\)
\(720\) 0 0
\(721\) 2.25539e7 1.61578
\(722\) 0 0
\(723\) − 1.25826e6i − 0.0895207i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.13130e7i 1.49558i 0.663937 + 0.747788i \(0.268884\pi\)
−0.663937 + 0.747788i \(0.731116\pi\)
\(728\) 0 0
\(729\) −3.61141e6 −0.251686
\(730\) 0 0
\(731\) 2.54680e7 1.76279
\(732\) 0 0
\(733\) 1.21571e7i 0.835737i 0.908507 + 0.417869i \(0.137223\pi\)
−0.908507 + 0.417869i \(0.862777\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.11188e7i − 0.754030i
\(738\) 0 0
\(739\) 1.92337e7 1.29555 0.647773 0.761834i \(-0.275701\pi\)
0.647773 + 0.761834i \(0.275701\pi\)
\(740\) 0 0
\(741\) −6.17382e6 −0.413056
\(742\) 0 0
\(743\) 1.66565e6i 0.110691i 0.998467 + 0.0553454i \(0.0176260\pi\)
−0.998467 + 0.0553454i \(0.982374\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.76620e6i − 0.443653i
\(748\) 0 0
\(749\) 2.16270e7 1.40861
\(750\) 0 0
\(751\) 9.81290e6 0.634888 0.317444 0.948277i \(-0.397175\pi\)
0.317444 + 0.948277i \(0.397175\pi\)
\(752\) 0 0
\(753\) 1.26883e7i 0.815486i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.92753e7i 1.22254i 0.791423 + 0.611269i \(0.209341\pi\)
−0.791423 + 0.611269i \(0.790659\pi\)
\(758\) 0 0
\(759\) −1.37902e7 −0.868893
\(760\) 0 0
\(761\) −1.17863e7 −0.737762 −0.368881 0.929477i \(-0.620259\pi\)
−0.368881 + 0.929477i \(0.620259\pi\)
\(762\) 0 0
\(763\) − 517088.i − 0.0321553i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 2.87871e7i − 1.76689i
\(768\) 0 0
\(769\) −1.22941e7 −0.749690 −0.374845 0.927087i \(-0.622304\pi\)
−0.374845 + 0.927087i \(0.622304\pi\)
\(770\) 0 0
\(771\) 5.23467e6 0.317142
\(772\) 0 0
\(773\) 2.57086e6i 0.154750i 0.997002 + 0.0773749i \(0.0246538\pi\)
−0.997002 + 0.0773749i \(0.975346\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.57381e7i 0.935190i
\(778\) 0 0
\(779\) −189040. −0.0111612
\(780\) 0 0
\(781\) −1.53253e7 −0.899046
\(782\) 0 0
\(783\) 9.85117e6i 0.574227i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 5.54594e6i − 0.319182i −0.987183 0.159591i \(-0.948982\pi\)
0.987183 0.159591i \(-0.0510176\pi\)
\(788\) 0 0
\(789\) −2.64154e6 −0.151065
\(790\) 0 0
\(791\) −3.28515e6 −0.186687
\(792\) 0 0
\(793\) 1.07306e7i 0.605958i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 2.09610e7i − 1.16887i −0.811440 0.584436i \(-0.801316\pi\)
0.811440 0.584436i \(-0.198684\pi\)
\(798\) 0 0
\(799\) 40896.0 0.00226628
\(800\) 0 0
\(801\) 2.26983e7 1.25000
\(802\) 0 0
\(803\) 2.87671e7i 1.57437i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.25565e7i 0.678709i
\(808\) 0 0
\(809\) −2.70297e7 −1.45201 −0.726005 0.687690i \(-0.758625\pi\)
−0.726005 + 0.687690i \(0.758625\pi\)
\(810\) 0 0
\(811\) −2.13052e6 −0.113745 −0.0568727 0.998381i \(-0.518113\pi\)
−0.0568727 + 0.998381i \(0.518113\pi\)
\(812\) 0 0
\(813\) − 7.66234e6i − 0.406570i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.21599e7i 1.16148i
\(818\) 0 0
\(819\) −2.58390e7 −1.34607
\(820\) 0 0
\(821\) 1.58060e7 0.818396 0.409198 0.912446i \(-0.365809\pi\)
0.409198 + 0.912446i \(0.365809\pi\)
\(822\) 0 0
\(823\) − 2.28848e7i − 1.17773i −0.808230 0.588867i \(-0.799574\pi\)
0.808230 0.588867i \(-0.200426\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.55336e7i 1.29822i 0.760695 + 0.649109i \(0.224858\pi\)
−0.760695 + 0.649109i \(0.775142\pi\)
\(828\) 0 0
\(829\) −8.31786e6 −0.420364 −0.210182 0.977662i \(-0.567406\pi\)
−0.210182 + 0.977662i \(0.567406\pi\)
\(830\) 0 0
\(831\) −4.52350e6 −0.227234
\(832\) 0 0
\(833\) − 3.38120e7i − 1.68834i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.85862e6i 0.437072i
\(838\) 0 0
\(839\) −3.66261e7 −1.79633 −0.898164 0.439660i \(-0.855099\pi\)
−0.898164 + 0.439660i \(0.855099\pi\)
\(840\) 0 0
\(841\) −1.19964e7 −0.584873
\(842\) 0 0
\(843\) 1.07302e7i 0.520041i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.62590e7i 1.25768i
\(848\) 0 0
\(849\) −5.87411e6 −0.279687
\(850\) 0 0
\(851\) −3.04169e7 −1.43976
\(852\) 0 0
\(853\) 1.74802e7i 0.822571i 0.911507 + 0.411286i \(0.134920\pi\)
−0.911507 + 0.411286i \(0.865080\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 9.31062e6i − 0.433038i −0.976278 0.216519i \(-0.930530\pi\)
0.976278 0.216519i \(-0.0694704\pi\)
\(858\) 0 0
\(859\) 3.49525e7 1.61620 0.808101 0.589045i \(-0.200496\pi\)
0.808101 + 0.589045i \(0.200496\pi\)
\(860\) 0 0
\(861\) 282880. 0.0130045
\(862\) 0 0
\(863\) − 2.02349e7i − 0.924858i −0.886656 0.462429i \(-0.846978\pi\)
0.886656 0.462429i \(-0.153022\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.70742e6i 0.0771421i
\(868\) 0 0
\(869\) −3.70655e7 −1.66502
\(870\) 0 0
\(871\) −1.43963e7 −0.642994
\(872\) 0 0
\(873\) − 1.11499e7i − 0.495149i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.98342e7i 0.870797i 0.900238 + 0.435398i \(0.143392\pi\)
−0.900238 + 0.435398i \(0.856608\pi\)
\(878\) 0 0
\(879\) −9.06562e6 −0.395754
\(880\) 0 0
\(881\) −3.81023e7 −1.65391 −0.826954 0.562270i \(-0.809928\pi\)
−0.826954 + 0.562270i \(0.809928\pi\)
\(882\) 0 0
\(883\) − 2.41560e7i − 1.04261i −0.853369 0.521307i \(-0.825445\pi\)
0.853369 0.521307i \(-0.174555\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.02368e7i − 0.863638i −0.901960 0.431819i \(-0.857872\pi\)
0.901960 0.431819i \(-0.142128\pi\)
\(888\) 0 0
\(889\) 212992. 0.00903876
\(890\) 0 0
\(891\) −8.83810e6 −0.372962
\(892\) 0 0
\(893\) 35584.0i 0.00149323i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.78552e7i 0.740942i
\(898\) 0 0
\(899\) 7.65683e6 0.315973
\(900\) 0 0
\(901\) 2.83435e7 1.16316
\(902\) 0 0
\(903\) − 3.31602e7i − 1.35331i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1.97976e7i − 0.799088i −0.916714 0.399544i \(-0.869169\pi\)
0.916714 0.399544i \(-0.130831\pi\)
\(908\) 0 0
\(909\) 1.14811e6 0.0460863
\(910\) 0 0
\(911\) −2.13242e7 −0.851288 −0.425644 0.904891i \(-0.639952\pi\)
−0.425644 + 0.904891i \(0.639952\pi\)
\(912\) 0 0
\(913\) 2.02608e7i 0.804414i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.71411e6i − 0.185130i
\(918\) 0 0
\(919\) −3.49941e7 −1.36680 −0.683401 0.730043i \(-0.739500\pi\)
−0.683401 + 0.730043i \(0.739500\pi\)
\(920\) 0 0
\(921\) 2.33101e7 0.905515
\(922\) 0 0
\(923\) 1.98428e7i 0.766655i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 1.94093e7i − 0.741842i
\(928\) 0 0
\(929\) 2.88107e7 1.09525 0.547627 0.836722i \(-0.315531\pi\)
0.547627 + 0.836722i \(0.315531\pi\)
\(930\) 0 0
\(931\) 2.94202e7 1.11243
\(932\) 0 0
\(933\) 1.15050e7i 0.432697i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.28854e7i − 0.479457i −0.970840 0.239729i \(-0.922941\pi\)
0.970840 0.239729i \(-0.0770585\pi\)
\(938\) 0 0
\(939\) −1.10080e7 −0.407424
\(940\) 0 0
\(941\) −5.27615e7 −1.94242 −0.971210 0.238227i \(-0.923434\pi\)
−0.971210 + 0.238227i \(0.923434\pi\)
\(942\) 0 0
\(943\) 546720.i 0.0200210i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 5.53961e6i − 0.200726i −0.994951 0.100363i \(-0.968000\pi\)
0.994951 0.100363i \(-0.0320004\pi\)
\(948\) 0 0
\(949\) 3.72470e7 1.34253
\(950\) 0 0
\(951\) 9.87950e6 0.354229
\(952\) 0 0
\(953\) − 228102.i − 0.00813574i −0.999992 0.00406787i \(-0.998705\pi\)
0.999992 0.00406787i \(-0.00129485\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1.25124e7i − 0.441632i
\(958\) 0 0
\(959\) −1.10735e7 −0.388811
\(960\) 0 0
\(961\) −2.17438e7 −0.759498
\(962\) 0 0
\(963\) − 1.86117e7i − 0.646726i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 7.36709e6i − 0.253355i −0.991944 0.126678i \(-0.959569\pi\)
0.991944 0.126678i \(-0.0404313\pi\)
\(968\) 0 0
\(969\) −1.13691e7 −0.388970
\(970\) 0 0
\(971\) −1.91161e7 −0.650654 −0.325327 0.945602i \(-0.605474\pi\)
−0.325327 + 0.945602i \(0.605474\pi\)
\(972\) 0 0
\(973\) − 4.12173e6i − 0.139572i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 5.57040e7i − 1.86702i −0.358546 0.933512i \(-0.616727\pi\)
0.358546 0.933512i \(-0.383273\pi\)
\(978\) 0 0
\(979\) −6.79680e7 −2.26646
\(980\) 0 0
\(981\) −444994. −0.0147632
\(982\) 0 0
\(983\) − 1.55469e7i − 0.513167i −0.966522 0.256584i \(-0.917403\pi\)
0.966522 0.256584i \(-0.0825969\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 53248.0i − 0.00173984i
\(988\) 0 0
\(989\) 6.40884e7 2.08348
\(990\) 0 0
\(991\) 2.36890e7 0.766237 0.383118 0.923699i \(-0.374850\pi\)
0.383118 + 0.923699i \(0.374850\pi\)
\(992\) 0 0
\(993\) 1.19144e7i 0.383442i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 3.71720e6i − 0.118434i −0.998245 0.0592172i \(-0.981140\pi\)
0.998245 0.0592172i \(-0.0188605\pi\)
\(998\) 0 0
\(999\) 3.19302e7 1.01225
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 800.6.c.a.449.1 2
4.3 odd 2 800.6.c.b.449.2 2
5.2 odd 4 32.6.a.a.1.1 1
5.3 odd 4 800.6.a.e.1.1 1
5.4 even 2 inner 800.6.c.a.449.2 2
15.2 even 4 288.6.a.d.1.1 1
20.3 even 4 800.6.a.a.1.1 1
20.7 even 4 32.6.a.c.1.1 yes 1
20.19 odd 2 800.6.c.b.449.1 2
40.27 even 4 64.6.a.c.1.1 1
40.37 odd 4 64.6.a.e.1.1 1
60.47 odd 4 288.6.a.e.1.1 1
80.27 even 4 256.6.b.b.129.1 2
80.37 odd 4 256.6.b.h.129.2 2
80.67 even 4 256.6.b.b.129.2 2
80.77 odd 4 256.6.b.h.129.1 2
120.77 even 4 576.6.a.u.1.1 1
120.107 odd 4 576.6.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.6.a.a.1.1 1 5.2 odd 4
32.6.a.c.1.1 yes 1 20.7 even 4
64.6.a.c.1.1 1 40.27 even 4
64.6.a.e.1.1 1 40.37 odd 4
256.6.b.b.129.1 2 80.27 even 4
256.6.b.b.129.2 2 80.67 even 4
256.6.b.h.129.1 2 80.77 odd 4
256.6.b.h.129.2 2 80.37 odd 4
288.6.a.d.1.1 1 15.2 even 4
288.6.a.e.1.1 1 60.47 odd 4
576.6.a.u.1.1 1 120.77 even 4
576.6.a.v.1.1 1 120.107 odd 4
800.6.a.a.1.1 1 20.3 even 4
800.6.a.e.1.1 1 5.3 odd 4
800.6.c.a.449.1 2 1.1 even 1 trivial
800.6.c.a.449.2 2 5.4 even 2 inner
800.6.c.b.449.1 2 20.19 odd 2
800.6.c.b.449.2 2 4.3 odd 2