Properties

Label 1080.6.a.e.1.3
Level $1080$
Weight $6$
Character 1080.1
Self dual yes
Analytic conductor $173.215$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,6,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(173.214525398\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.15881.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 29x - 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.33626\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-25.0000 q^{5} +85.5605 q^{7} +O(q^{10})\) \(q-25.0000 q^{5} +85.5605 q^{7} +99.8209 q^{11} -546.179 q^{13} +424.929 q^{17} +2122.20 q^{19} +3450.04 q^{23} +625.000 q^{25} -6876.76 q^{29} +2989.87 q^{31} -2139.01 q^{35} -14200.2 q^{37} -423.869 q^{41} -3569.67 q^{43} +928.769 q^{47} -9486.39 q^{49} +33234.4 q^{53} -2495.52 q^{55} +17830.1 q^{59} +30245.0 q^{61} +13654.5 q^{65} +20790.3 q^{67} -19209.9 q^{71} -83174.8 q^{73} +8540.73 q^{77} +42514.1 q^{79} +93576.0 q^{83} -10623.2 q^{85} -72645.6 q^{89} -46731.4 q^{91} -53055.1 q^{95} +16741.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 75 q^{5} - 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 75 q^{5} - 30 q^{7} + 660 q^{11} - 1278 q^{13} - 1731 q^{17} + 1011 q^{19} - 2433 q^{23} + 1875 q^{25} - 3786 q^{29} + 4131 q^{31} + 750 q^{35} - 7122 q^{37} - 17352 q^{41} + 11556 q^{43} + 19548 q^{47} - 7857 q^{49} + 4965 q^{53} - 16500 q^{55} + 32106 q^{59} - 16317 q^{61} + 31950 q^{65} - 51258 q^{67} + 84072 q^{71} - 161892 q^{73} - 142968 q^{77} + 36441 q^{79} + 64581 q^{83} + 43275 q^{85} + 61584 q^{89} - 123588 q^{91} - 25275 q^{95} + 14760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 85.5605 0.659976 0.329988 0.943985i \(-0.392955\pi\)
0.329988 + 0.943985i \(0.392955\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 99.8209 0.248737 0.124368 0.992236i \(-0.460310\pi\)
0.124368 + 0.992236i \(0.460310\pi\)
\(12\) 0 0
\(13\) −546.179 −0.896348 −0.448174 0.893946i \(-0.647926\pi\)
−0.448174 + 0.893946i \(0.647926\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 424.929 0.356610 0.178305 0.983975i \(-0.442939\pi\)
0.178305 + 0.983975i \(0.442939\pi\)
\(18\) 0 0
\(19\) 2122.20 1.34866 0.674331 0.738429i \(-0.264432\pi\)
0.674331 + 0.738429i \(0.264432\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3450.04 1.35989 0.679946 0.733262i \(-0.262003\pi\)
0.679946 + 0.733262i \(0.262003\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6876.76 −1.51841 −0.759205 0.650852i \(-0.774412\pi\)
−0.759205 + 0.650852i \(0.774412\pi\)
\(30\) 0 0
\(31\) 2989.87 0.558790 0.279395 0.960176i \(-0.409866\pi\)
0.279395 + 0.960176i \(0.409866\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2139.01 −0.295150
\(36\) 0 0
\(37\) −14200.2 −1.70526 −0.852629 0.522517i \(-0.824993\pi\)
−0.852629 + 0.522517i \(0.824993\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −423.869 −0.0393797 −0.0196898 0.999806i \(-0.506268\pi\)
−0.0196898 + 0.999806i \(0.506268\pi\)
\(42\) 0 0
\(43\) −3569.67 −0.294413 −0.147207 0.989106i \(-0.547028\pi\)
−0.147207 + 0.989106i \(0.547028\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 928.769 0.0613286 0.0306643 0.999530i \(-0.490238\pi\)
0.0306643 + 0.999530i \(0.490238\pi\)
\(48\) 0 0
\(49\) −9486.39 −0.564431
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 33234.4 1.62517 0.812584 0.582844i \(-0.198060\pi\)
0.812584 + 0.582844i \(0.198060\pi\)
\(54\) 0 0
\(55\) −2495.52 −0.111238
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 17830.1 0.666845 0.333423 0.942778i \(-0.391796\pi\)
0.333423 + 0.942778i \(0.391796\pi\)
\(60\) 0 0
\(61\) 30245.0 1.04071 0.520354 0.853951i \(-0.325800\pi\)
0.520354 + 0.853951i \(0.325800\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13654.5 0.400859
\(66\) 0 0
\(67\) 20790.3 0.565815 0.282908 0.959147i \(-0.408701\pi\)
0.282908 + 0.959147i \(0.408701\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −19209.9 −0.452250 −0.226125 0.974098i \(-0.572606\pi\)
−0.226125 + 0.974098i \(0.572606\pi\)
\(72\) 0 0
\(73\) −83174.8 −1.82677 −0.913386 0.407094i \(-0.866542\pi\)
−0.913386 + 0.407094i \(0.866542\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8540.73 0.164160
\(78\) 0 0
\(79\) 42514.1 0.766418 0.383209 0.923662i \(-0.374819\pi\)
0.383209 + 0.923662i \(0.374819\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 93576.0 1.49097 0.745486 0.666522i \(-0.232218\pi\)
0.745486 + 0.666522i \(0.232218\pi\)
\(84\) 0 0
\(85\) −10623.2 −0.159481
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −72645.6 −0.972152 −0.486076 0.873917i \(-0.661572\pi\)
−0.486076 + 0.873917i \(0.661572\pi\)
\(90\) 0 0
\(91\) −46731.4 −0.591569
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −53055.1 −0.603140
\(96\) 0 0
\(97\) 16741.0 0.180655 0.0903277 0.995912i \(-0.471209\pi\)
0.0903277 + 0.995912i \(0.471209\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5922.83 0.0577731 0.0288866 0.999583i \(-0.490804\pi\)
0.0288866 + 0.999583i \(0.490804\pi\)
\(102\) 0 0
\(103\) −142022. −1.31906 −0.659528 0.751680i \(-0.729244\pi\)
−0.659528 + 0.751680i \(0.729244\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3404.42 0.0287464 0.0143732 0.999897i \(-0.495425\pi\)
0.0143732 + 0.999897i \(0.495425\pi\)
\(108\) 0 0
\(109\) 132231. 1.06602 0.533012 0.846107i \(-0.321060\pi\)
0.533012 + 0.846107i \(0.321060\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −212816. −1.56786 −0.783932 0.620847i \(-0.786789\pi\)
−0.783932 + 0.620847i \(0.786789\pi\)
\(114\) 0 0
\(115\) −86250.9 −0.608162
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 36357.1 0.235354
\(120\) 0 0
\(121\) −151087. −0.938130
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) 304653. 1.67608 0.838042 0.545606i \(-0.183700\pi\)
0.838042 + 0.545606i \(0.183700\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 103961. 0.529290 0.264645 0.964346i \(-0.414745\pi\)
0.264645 + 0.964346i \(0.414745\pi\)
\(132\) 0 0
\(133\) 181577. 0.890085
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4202.17 −0.0191281 −0.00956405 0.999954i \(-0.503044\pi\)
−0.00956405 + 0.999954i \(0.503044\pi\)
\(138\) 0 0
\(139\) 154249. 0.677151 0.338575 0.940939i \(-0.390055\pi\)
0.338575 + 0.940939i \(0.390055\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −54520.1 −0.222955
\(144\) 0 0
\(145\) 171919. 0.679054
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 288038. 1.06288 0.531439 0.847096i \(-0.321651\pi\)
0.531439 + 0.847096i \(0.321651\pi\)
\(150\) 0 0
\(151\) 51586.3 0.184116 0.0920581 0.995754i \(-0.470655\pi\)
0.0920581 + 0.995754i \(0.470655\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −74746.9 −0.249899
\(156\) 0 0
\(157\) −247275. −0.800627 −0.400314 0.916378i \(-0.631099\pi\)
−0.400314 + 0.916378i \(0.631099\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 295187. 0.897496
\(162\) 0 0
\(163\) −278761. −0.821794 −0.410897 0.911682i \(-0.634784\pi\)
−0.410897 + 0.911682i \(0.634784\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 489181. 1.35731 0.678654 0.734459i \(-0.262564\pi\)
0.678654 + 0.734459i \(0.262564\pi\)
\(168\) 0 0
\(169\) −72981.4 −0.196560
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 577120. 1.46606 0.733029 0.680198i \(-0.238106\pi\)
0.733029 + 0.680198i \(0.238106\pi\)
\(174\) 0 0
\(175\) 53475.3 0.131995
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 617817. 1.44121 0.720605 0.693346i \(-0.243864\pi\)
0.720605 + 0.693346i \(0.243864\pi\)
\(180\) 0 0
\(181\) −265079. −0.601421 −0.300711 0.953715i \(-0.597224\pi\)
−0.300711 + 0.953715i \(0.597224\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 355005. 0.762615
\(186\) 0 0
\(187\) 42416.8 0.0887021
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 30593.3 0.0606797 0.0303398 0.999540i \(-0.490341\pi\)
0.0303398 + 0.999540i \(0.490341\pi\)
\(192\) 0 0
\(193\) 463235. 0.895176 0.447588 0.894240i \(-0.352283\pi\)
0.447588 + 0.894240i \(0.352283\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −518076. −0.951104 −0.475552 0.879688i \(-0.657752\pi\)
−0.475552 + 0.879688i \(0.657752\pi\)
\(198\) 0 0
\(199\) −53461.6 −0.0956993 −0.0478497 0.998855i \(-0.515237\pi\)
−0.0478497 + 0.998855i \(0.515237\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −588380. −1.00211
\(204\) 0 0
\(205\) 10596.7 0.0176111
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 211840. 0.335462
\(210\) 0 0
\(211\) −318155. −0.491963 −0.245982 0.969275i \(-0.579110\pi\)
−0.245982 + 0.969275i \(0.579110\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 89241.8 0.131666
\(216\) 0 0
\(217\) 255815. 0.368788
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −232087. −0.319647
\(222\) 0 0
\(223\) 282126. 0.379911 0.189955 0.981793i \(-0.439166\pi\)
0.189955 + 0.981793i \(0.439166\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 599160. 0.771753 0.385877 0.922550i \(-0.373899\pi\)
0.385877 + 0.922550i \(0.373899\pi\)
\(228\) 0 0
\(229\) 1.29257e6 1.62879 0.814394 0.580313i \(-0.197070\pi\)
0.814394 + 0.580313i \(0.197070\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 611411. 0.737808 0.368904 0.929467i \(-0.379733\pi\)
0.368904 + 0.929467i \(0.379733\pi\)
\(234\) 0 0
\(235\) −23219.2 −0.0274270
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 960138. 1.08727 0.543637 0.839320i \(-0.317047\pi\)
0.543637 + 0.839320i \(0.317047\pi\)
\(240\) 0 0
\(241\) −345926. −0.383655 −0.191828 0.981429i \(-0.561441\pi\)
−0.191828 + 0.981429i \(0.561441\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 237160. 0.252421
\(246\) 0 0
\(247\) −1.15910e6 −1.20887
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 711996. 0.713335 0.356667 0.934231i \(-0.383913\pi\)
0.356667 + 0.934231i \(0.383913\pi\)
\(252\) 0 0
\(253\) 344386. 0.338255
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 305560. 0.288578 0.144289 0.989536i \(-0.453910\pi\)
0.144289 + 0.989536i \(0.453910\pi\)
\(258\) 0 0
\(259\) −1.21498e6 −1.12543
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.69549e6 1.51149 0.755744 0.654867i \(-0.227275\pi\)
0.755744 + 0.654867i \(0.227275\pi\)
\(264\) 0 0
\(265\) −830860. −0.726797
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −577207. −0.486352 −0.243176 0.969982i \(-0.578189\pi\)
−0.243176 + 0.969982i \(0.578189\pi\)
\(270\) 0 0
\(271\) −1.65995e6 −1.37300 −0.686501 0.727129i \(-0.740854\pi\)
−0.686501 + 0.727129i \(0.740854\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 62388.1 0.0497473
\(276\) 0 0
\(277\) 1.25880e6 0.985732 0.492866 0.870105i \(-0.335949\pi\)
0.492866 + 0.870105i \(0.335949\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.69319e6 1.27921 0.639604 0.768705i \(-0.279099\pi\)
0.639604 + 0.768705i \(0.279099\pi\)
\(282\) 0 0
\(283\) 1.97014e6 1.46228 0.731142 0.682225i \(-0.238988\pi\)
0.731142 + 0.682225i \(0.238988\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −36266.5 −0.0259897
\(288\) 0 0
\(289\) −1.23929e6 −0.872829
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.69945e6 1.83699 0.918494 0.395435i \(-0.129406\pi\)
0.918494 + 0.395435i \(0.129406\pi\)
\(294\) 0 0
\(295\) −445754. −0.298222
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.88434e6 −1.21894
\(300\) 0 0
\(301\) −305423. −0.194306
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −756125. −0.465419
\(306\) 0 0
\(307\) 1.46875e6 0.889410 0.444705 0.895677i \(-0.353308\pi\)
0.444705 + 0.895677i \(0.353308\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 681620. 0.399614 0.199807 0.979835i \(-0.435968\pi\)
0.199807 + 0.979835i \(0.435968\pi\)
\(312\) 0 0
\(313\) −642495. −0.370688 −0.185344 0.982674i \(-0.559340\pi\)
−0.185344 + 0.982674i \(0.559340\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.63094e6 1.47049 0.735245 0.677801i \(-0.237067\pi\)
0.735245 + 0.677801i \(0.237067\pi\)
\(318\) 0 0
\(319\) −686445. −0.377684
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 901786. 0.480947
\(324\) 0 0
\(325\) −341362. −0.179270
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 79465.9 0.0404754
\(330\) 0 0
\(331\) 739507. 0.370999 0.185499 0.982644i \(-0.440610\pi\)
0.185499 + 0.982644i \(0.440610\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −519759. −0.253040
\(336\) 0 0
\(337\) 2.86873e6 1.37599 0.687996 0.725715i \(-0.258491\pi\)
0.687996 + 0.725715i \(0.258491\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 298452. 0.138992
\(342\) 0 0
\(343\) −2.24968e6 −1.03249
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 358901. 0.160011 0.0800057 0.996794i \(-0.474506\pi\)
0.0800057 + 0.996794i \(0.474506\pi\)
\(348\) 0 0
\(349\) 2.96201e6 1.30174 0.650869 0.759190i \(-0.274405\pi\)
0.650869 + 0.759190i \(0.274405\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.02769e6 −0.866095 −0.433048 0.901371i \(-0.642562\pi\)
−0.433048 + 0.901371i \(0.642562\pi\)
\(354\) 0 0
\(355\) 480246. 0.202252
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.52540e6 1.85319 0.926596 0.376058i \(-0.122721\pi\)
0.926596 + 0.376058i \(0.122721\pi\)
\(360\) 0 0
\(361\) 2.02765e6 0.818890
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.07937e6 0.816958
\(366\) 0 0
\(367\) −3.58235e6 −1.38836 −0.694182 0.719800i \(-0.744234\pi\)
−0.694182 + 0.719800i \(0.744234\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.84355e6 1.07257
\(372\) 0 0
\(373\) −1.87995e6 −0.699638 −0.349819 0.936817i \(-0.613757\pi\)
−0.349819 + 0.936817i \(0.613757\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.75594e6 1.36102
\(378\) 0 0
\(379\) −1.50759e6 −0.539121 −0.269560 0.962983i \(-0.586878\pi\)
−0.269560 + 0.962983i \(0.586878\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.72039e6 −1.29596 −0.647980 0.761658i \(-0.724386\pi\)
−0.647980 + 0.761658i \(0.724386\pi\)
\(384\) 0 0
\(385\) −213518. −0.0734148
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.42840e6 0.478603 0.239302 0.970945i \(-0.423081\pi\)
0.239302 + 0.970945i \(0.423081\pi\)
\(390\) 0 0
\(391\) 1.46602e6 0.484951
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.06285e6 −0.342752
\(396\) 0 0
\(397\) −2.52884e6 −0.805276 −0.402638 0.915359i \(-0.631907\pi\)
−0.402638 + 0.915359i \(0.631907\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −730346. −0.226813 −0.113406 0.993549i \(-0.536176\pi\)
−0.113406 + 0.993549i \(0.536176\pi\)
\(402\) 0 0
\(403\) −1.63301e6 −0.500871
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.41748e6 −0.424160
\(408\) 0 0
\(409\) −1.51732e6 −0.448506 −0.224253 0.974531i \(-0.571994\pi\)
−0.224253 + 0.974531i \(0.571994\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.52556e6 0.440102
\(414\) 0 0
\(415\) −2.33940e6 −0.666783
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.36661e6 0.380286 0.190143 0.981756i \(-0.439105\pi\)
0.190143 + 0.981756i \(0.439105\pi\)
\(420\) 0 0
\(421\) 3.20736e6 0.881946 0.440973 0.897520i \(-0.354633\pi\)
0.440973 + 0.897520i \(0.354633\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 265581. 0.0713221
\(426\) 0 0
\(427\) 2.58778e6 0.686843
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.51920e6 1.17184 0.585920 0.810369i \(-0.300733\pi\)
0.585920 + 0.810369i \(0.300733\pi\)
\(432\) 0 0
\(433\) 5.08506e6 1.30340 0.651698 0.758478i \(-0.274057\pi\)
0.651698 + 0.758478i \(0.274057\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.32169e6 1.83403
\(438\) 0 0
\(439\) 1.67597e6 0.415054 0.207527 0.978229i \(-0.433459\pi\)
0.207527 + 0.978229i \(0.433459\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.33137e6 1.77491 0.887454 0.460897i \(-0.152472\pi\)
0.887454 + 0.460897i \(0.152472\pi\)
\(444\) 0 0
\(445\) 1.81614e6 0.434759
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.68385e6 0.862354 0.431177 0.902267i \(-0.358098\pi\)
0.431177 + 0.902267i \(0.358098\pi\)
\(450\) 0 0
\(451\) −42311.0 −0.00979518
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.16828e6 0.264557
\(456\) 0 0
\(457\) −810253. −0.181480 −0.0907402 0.995875i \(-0.528923\pi\)
−0.0907402 + 0.995875i \(0.528923\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.76965e6 −1.04528 −0.522642 0.852552i \(-0.675053\pi\)
−0.522642 + 0.852552i \(0.675053\pi\)
\(462\) 0 0
\(463\) 3.70653e6 0.803554 0.401777 0.915738i \(-0.368393\pi\)
0.401777 + 0.915738i \(0.368393\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.33319e6 −0.707241 −0.353621 0.935389i \(-0.615050\pi\)
−0.353621 + 0.935389i \(0.615050\pi\)
\(468\) 0 0
\(469\) 1.77883e6 0.373425
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −356328. −0.0732314
\(474\) 0 0
\(475\) 1.32638e6 0.269732
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.99488e6 0.596404 0.298202 0.954503i \(-0.403613\pi\)
0.298202 + 0.954503i \(0.403613\pi\)
\(480\) 0 0
\(481\) 7.75585e6 1.52850
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −418524. −0.0807916
\(486\) 0 0
\(487\) −7.71593e6 −1.47423 −0.737116 0.675766i \(-0.763813\pi\)
−0.737116 + 0.675766i \(0.763813\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.65091e6 0.496239 0.248120 0.968729i \(-0.420187\pi\)
0.248120 + 0.968729i \(0.420187\pi\)
\(492\) 0 0
\(493\) −2.92214e6 −0.541481
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.64361e6 −0.298474
\(498\) 0 0
\(499\) 3.29296e6 0.592019 0.296010 0.955185i \(-0.404344\pi\)
0.296010 + 0.955185i \(0.404344\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.22340e6 −0.215600 −0.107800 0.994173i \(-0.534381\pi\)
−0.107800 + 0.994173i \(0.534381\pi\)
\(504\) 0 0
\(505\) −148071. −0.0258369
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.10782e6 −0.873858 −0.436929 0.899496i \(-0.643934\pi\)
−0.436929 + 0.899496i \(0.643934\pi\)
\(510\) 0 0
\(511\) −7.11648e6 −1.20563
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.55055e6 0.589899
\(516\) 0 0
\(517\) 92710.6 0.0152547
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −115812. −0.0186922 −0.00934611 0.999956i \(-0.502975\pi\)
−0.00934611 + 0.999956i \(0.502975\pi\)
\(522\) 0 0
\(523\) −6.72323e6 −1.07479 −0.537395 0.843330i \(-0.680592\pi\)
−0.537395 + 0.843330i \(0.680592\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.27048e6 0.199270
\(528\) 0 0
\(529\) 5.46642e6 0.849305
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 231509. 0.0352979
\(534\) 0 0
\(535\) −85110.6 −0.0128558
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −946941. −0.140395
\(540\) 0 0
\(541\) −9.33904e6 −1.37186 −0.685929 0.727669i \(-0.740604\pi\)
−0.685929 + 0.727669i \(0.740604\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.30578e6 −0.476741
\(546\) 0 0
\(547\) −1.29689e6 −0.185325 −0.0926626 0.995698i \(-0.529538\pi\)
−0.0926626 + 0.995698i \(0.529538\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.45939e7 −2.04782
\(552\) 0 0
\(553\) 3.63753e6 0.505818
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.88461e6 −0.393958 −0.196979 0.980408i \(-0.563113\pi\)
−0.196979 + 0.980408i \(0.563113\pi\)
\(558\) 0 0
\(559\) 1.94968e6 0.263897
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.96177e6 1.05862 0.529308 0.848430i \(-0.322452\pi\)
0.529308 + 0.848430i \(0.322452\pi\)
\(564\) 0 0
\(565\) 5.32040e6 0.701170
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.36853e6 0.436174 0.218087 0.975929i \(-0.430018\pi\)
0.218087 + 0.975929i \(0.430018\pi\)
\(570\) 0 0
\(571\) −1.43608e7 −1.84327 −0.921633 0.388063i \(-0.873144\pi\)
−0.921633 + 0.388063i \(0.873144\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.15627e6 0.271978
\(576\) 0 0
\(577\) −5.67748e6 −0.709930 −0.354965 0.934880i \(-0.615507\pi\)
−0.354965 + 0.934880i \(0.615507\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00641e6 0.984006
\(582\) 0 0
\(583\) 3.31749e6 0.404239
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.30937e6 −0.516201 −0.258101 0.966118i \(-0.583097\pi\)
−0.258101 + 0.966118i \(0.583097\pi\)
\(588\) 0 0
\(589\) 6.34513e6 0.753619
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.30041e6 −0.618975 −0.309487 0.950904i \(-0.600157\pi\)
−0.309487 + 0.950904i \(0.600157\pi\)
\(594\) 0 0
\(595\) −908929. −0.105254
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −479201. −0.0545696 −0.0272848 0.999628i \(-0.508686\pi\)
−0.0272848 + 0.999628i \(0.508686\pi\)
\(600\) 0 0
\(601\) 6.07294e6 0.685825 0.342912 0.939367i \(-0.388587\pi\)
0.342912 + 0.939367i \(0.388587\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.77717e6 0.419545
\(606\) 0 0
\(607\) −7.45020e6 −0.820722 −0.410361 0.911923i \(-0.634597\pi\)
−0.410361 + 0.911923i \(0.634597\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −507274. −0.0549718
\(612\) 0 0
\(613\) 1.15156e7 1.23776 0.618878 0.785487i \(-0.287587\pi\)
0.618878 + 0.785487i \(0.287587\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00826e7 −1.06625 −0.533126 0.846036i \(-0.678983\pi\)
−0.533126 + 0.846036i \(0.678983\pi\)
\(618\) 0 0
\(619\) −1.39730e7 −1.46576 −0.732882 0.680355i \(-0.761825\pi\)
−0.732882 + 0.680355i \(0.761825\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.21559e6 −0.641597
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.03407e6 −0.608113
\(630\) 0 0
\(631\) 6.09736e6 0.609633 0.304817 0.952411i \(-0.401405\pi\)
0.304817 + 0.952411i \(0.401405\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.61632e6 −0.749568
\(636\) 0 0
\(637\) 5.18127e6 0.505927
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −548849. −0.0527604 −0.0263802 0.999652i \(-0.508398\pi\)
−0.0263802 + 0.999652i \(0.508398\pi\)
\(642\) 0 0
\(643\) −1.29095e7 −1.23135 −0.615677 0.787999i \(-0.711117\pi\)
−0.615677 + 0.787999i \(0.711117\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.71727e6 0.161279 0.0806397 0.996743i \(-0.474304\pi\)
0.0806397 + 0.996743i \(0.474304\pi\)
\(648\) 0 0
\(649\) 1.77982e6 0.165869
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.15426e7 −1.05931 −0.529653 0.848214i \(-0.677678\pi\)
−0.529653 + 0.848214i \(0.677678\pi\)
\(654\) 0 0
\(655\) −2.59904e6 −0.236706
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.09831e7 −0.985169 −0.492585 0.870265i \(-0.663948\pi\)
−0.492585 + 0.870265i \(0.663948\pi\)
\(660\) 0 0
\(661\) 1.79023e7 1.59370 0.796849 0.604178i \(-0.206499\pi\)
0.796849 + 0.604178i \(0.206499\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.53942e6 −0.398058
\(666\) 0 0
\(667\) −2.37251e7 −2.06487
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.01908e6 0.258862
\(672\) 0 0
\(673\) 1.51352e7 1.28810 0.644050 0.764983i \(-0.277253\pi\)
0.644050 + 0.764983i \(0.277253\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.74374e6 −0.146221 −0.0731105 0.997324i \(-0.523293\pi\)
−0.0731105 + 0.997324i \(0.523293\pi\)
\(678\) 0 0
\(679\) 1.43237e6 0.119228
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3.91354e6 −0.321010 −0.160505 0.987035i \(-0.551312\pi\)
−0.160505 + 0.987035i \(0.551312\pi\)
\(684\) 0 0
\(685\) 105054. 0.00855434
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.81519e7 −1.45672
\(690\) 0 0
\(691\) 2.09646e7 1.67029 0.835143 0.550032i \(-0.185385\pi\)
0.835143 + 0.550032i \(0.185385\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.85622e6 −0.302831
\(696\) 0 0
\(697\) −180114. −0.0140432
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.37397e7 1.05604 0.528022 0.849231i \(-0.322934\pi\)
0.528022 + 0.849231i \(0.322934\pi\)
\(702\) 0 0
\(703\) −3.01357e7 −2.29982
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 506761. 0.0381289
\(708\) 0 0
\(709\) 8.09118e6 0.604500 0.302250 0.953229i \(-0.402262\pi\)
0.302250 + 0.953229i \(0.402262\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.03152e7 0.759894
\(714\) 0 0
\(715\) 1.36300e6 0.0997084
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.50696e6 0.613694 0.306847 0.951759i \(-0.400726\pi\)
0.306847 + 0.951759i \(0.400726\pi\)
\(720\) 0 0
\(721\) −1.21515e7 −0.870545
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.29798e6 −0.303682
\(726\) 0 0
\(727\) −1.35919e7 −0.953771 −0.476885 0.878966i \(-0.658234\pi\)
−0.476885 + 0.878966i \(0.658234\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.51686e6 −0.104991
\(732\) 0 0
\(733\) 1.21454e7 0.834932 0.417466 0.908692i \(-0.362918\pi\)
0.417466 + 0.908692i \(0.362918\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.07531e6 0.140739
\(738\) 0 0
\(739\) −4.50729e6 −0.303602 −0.151801 0.988411i \(-0.548507\pi\)
−0.151801 + 0.988411i \(0.548507\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.82273e6 0.519860 0.259930 0.965628i \(-0.416301\pi\)
0.259930 + 0.965628i \(0.416301\pi\)
\(744\) 0 0
\(745\) −7.20094e6 −0.475334
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 291284. 0.0189720
\(750\) 0 0
\(751\) 1.62764e7 1.05307 0.526537 0.850152i \(-0.323490\pi\)
0.526537 + 0.850152i \(0.323490\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.28966e6 −0.0823393
\(756\) 0 0
\(757\) 5.05461e6 0.320588 0.160294 0.987069i \(-0.448756\pi\)
0.160294 + 0.987069i \(0.448756\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −786485. −0.0492299 −0.0246149 0.999697i \(-0.507836\pi\)
−0.0246149 + 0.999697i \(0.507836\pi\)
\(762\) 0 0
\(763\) 1.13138e7 0.703551
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.73845e6 −0.597725
\(768\) 0 0
\(769\) 1.44258e7 0.879679 0.439840 0.898076i \(-0.355035\pi\)
0.439840 + 0.898076i \(0.355035\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.81060e7 −1.08987 −0.544935 0.838478i \(-0.683446\pi\)
−0.544935 + 0.838478i \(0.683446\pi\)
\(774\) 0 0
\(775\) 1.86867e6 0.111758
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −899537. −0.0531099
\(780\) 0 0
\(781\) −1.91755e6 −0.112491
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.18187e6 0.358051
\(786\) 0 0
\(787\) 1.86788e7 1.07501 0.537504 0.843261i \(-0.319367\pi\)
0.537504 + 0.843261i \(0.319367\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.82087e7 −1.03475
\(792\) 0 0
\(793\) −1.65192e7 −0.932836
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.99446e6 0.334275 0.167138 0.985934i \(-0.446548\pi\)
0.167138 + 0.985934i \(0.446548\pi\)
\(798\) 0 0
\(799\) 394661. 0.0218704
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −8.30259e6 −0.454386
\(804\) 0 0
\(805\) −7.37968e6 −0.401373
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.85790e7 −0.998047 −0.499023 0.866589i \(-0.666308\pi\)
−0.499023 + 0.866589i \(0.666308\pi\)
\(810\) 0 0
\(811\) −2.31663e7 −1.23682 −0.618408 0.785857i \(-0.712222\pi\)
−0.618408 + 0.785857i \(0.712222\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.96902e6 0.367517
\(816\) 0 0
\(817\) −7.57558e6 −0.397064
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.02639e7 −0.531440 −0.265720 0.964050i \(-0.585610\pi\)
−0.265720 + 0.964050i \(0.585610\pi\)
\(822\) 0 0
\(823\) 3.46900e7 1.78527 0.892637 0.450776i \(-0.148853\pi\)
0.892637 + 0.450776i \(0.148853\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.50380e7 −0.764585 −0.382293 0.924041i \(-0.624865\pi\)
−0.382293 + 0.924041i \(0.624865\pi\)
\(828\) 0 0
\(829\) −1.38937e7 −0.702155 −0.351078 0.936346i \(-0.614185\pi\)
−0.351078 + 0.936346i \(0.614185\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.03104e6 −0.201282
\(834\) 0 0
\(835\) −1.22295e7 −0.607006
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.79014e7 0.877973 0.438986 0.898494i \(-0.355338\pi\)
0.438986 + 0.898494i \(0.355338\pi\)
\(840\) 0 0
\(841\) 2.67787e7 1.30557
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.82454e6 0.0879044
\(846\) 0 0
\(847\) −1.29271e7 −0.619144
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.89912e7 −2.31897
\(852\) 0 0
\(853\) 1.07670e7 0.506664 0.253332 0.967379i \(-0.418473\pi\)
0.253332 + 0.967379i \(0.418473\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.76117e7 −0.819122 −0.409561 0.912283i \(-0.634318\pi\)
−0.409561 + 0.912283i \(0.634318\pi\)
\(858\) 0 0
\(859\) 1.95268e7 0.902917 0.451458 0.892292i \(-0.350904\pi\)
0.451458 + 0.892292i \(0.350904\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.78308e7 −1.72910 −0.864548 0.502550i \(-0.832395\pi\)
−0.864548 + 0.502550i \(0.832395\pi\)
\(864\) 0 0
\(865\) −1.44280e7 −0.655641
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.24380e6 0.190636
\(870\) 0 0
\(871\) −1.13553e7 −0.507168
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.33688e6 −0.0590301
\(876\) 0 0
\(877\) 4.51942e6 0.198419 0.0992096 0.995067i \(-0.468369\pi\)
0.0992096 + 0.995067i \(0.468369\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.48399e7 −1.07823 −0.539114 0.842233i \(-0.681241\pi\)
−0.539114 + 0.842233i \(0.681241\pi\)
\(882\) 0 0
\(883\) −4.40390e6 −0.190080 −0.0950399 0.995473i \(-0.530298\pi\)
−0.0950399 + 0.995473i \(0.530298\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.67390e7 −1.56790 −0.783951 0.620823i \(-0.786799\pi\)
−0.783951 + 0.620823i \(0.786799\pi\)
\(888\) 0 0
\(889\) 2.60663e7 1.10618
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.97104e6 0.0827116
\(894\) 0 0
\(895\) −1.54454e7 −0.644528
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.05607e7 −0.848473
\(900\) 0 0
\(901\) 1.41223e7 0.579552
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.62697e6 0.268964
\(906\) 0 0
\(907\) −3.61944e6 −0.146091 −0.0730454 0.997329i \(-0.523272\pi\)
−0.0730454 + 0.997329i \(0.523272\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.06846e7 −0.426543 −0.213271 0.976993i \(-0.568412\pi\)
−0.213271 + 0.976993i \(0.568412\pi\)
\(912\) 0 0
\(913\) 9.34084e6 0.370859
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.89499e6 0.349319
\(918\) 0 0
\(919\) 423307. 0.0165336 0.00826679 0.999966i \(-0.497369\pi\)
0.00826679 + 0.999966i \(0.497369\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.04920e7 0.405373
\(924\) 0 0
\(925\) −8.87512e6 −0.341052
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.51666e6 0.0956720 0.0478360 0.998855i \(-0.484768\pi\)
0.0478360 + 0.998855i \(0.484768\pi\)
\(930\) 0 0
\(931\) −2.01321e7 −0.761227
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.06042e6 −0.0396688
\(936\) 0 0
\(937\) −1.39054e7 −0.517408 −0.258704 0.965957i \(-0.583295\pi\)
−0.258704 + 0.965957i \(0.583295\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.15658e7 −0.425795 −0.212897 0.977075i \(-0.568290\pi\)
−0.212897 + 0.977075i \(0.568290\pi\)
\(942\) 0 0
\(943\) −1.46237e6 −0.0535521
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.37017e7 0.858826 0.429413 0.903108i \(-0.358721\pi\)
0.429413 + 0.903108i \(0.358721\pi\)
\(948\) 0 0
\(949\) 4.54283e7 1.63742
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.67313e6 −0.0596757 −0.0298378 0.999555i \(-0.509499\pi\)
−0.0298378 + 0.999555i \(0.509499\pi\)
\(954\) 0 0
\(955\) −764833. −0.0271368
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −359540. −0.0126241
\(960\) 0 0
\(961\) −1.96898e7 −0.687753
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.15809e7 −0.400335
\(966\) 0 0
\(967\) −2.69820e7 −0.927915 −0.463957 0.885858i \(-0.653571\pi\)
−0.463957 + 0.885858i \(0.653571\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.59915e7 1.22504 0.612522 0.790454i \(-0.290155\pi\)
0.612522 + 0.790454i \(0.290155\pi\)
\(972\) 0 0
\(973\) 1.31976e7 0.446903
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.65489e7 −1.56017 −0.780087 0.625671i \(-0.784825\pi\)
−0.780087 + 0.625671i \(0.784825\pi\)
\(978\) 0 0
\(979\) −7.25155e6 −0.241810
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.41850e7 −1.12837 −0.564186 0.825648i \(-0.690810\pi\)
−0.564186 + 0.825648i \(0.690810\pi\)
\(984\) 0 0
\(985\) 1.29519e7 0.425346
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.23155e7 −0.400370
\(990\) 0 0
\(991\) −4.37070e7 −1.41373 −0.706866 0.707347i \(-0.749892\pi\)
−0.706866 + 0.707347i \(0.749892\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.33654e6 0.0427980
\(996\) 0 0
\(997\) −1.73468e7 −0.552689 −0.276344 0.961059i \(-0.589123\pi\)
−0.276344 + 0.961059i \(0.589123\pi\)
\(998\) 0 0
\(999\) 0 0
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 1080.6.a.e.1.3 3
3.2 odd 2 1080.6.a.g.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.6.a.e.1.3 3 1.1 even 1 trivial
1080.6.a.g.1.3 yes 3 3.2 odd 2