Properties

Label 80.18.a.d.1.2
Level $80$
Weight $18$
Character 80.1
Self dual yes
Analytic conductor $146.578$
Analytic rank $1$
Dimension $2$
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] = [80,18,Mod(1,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(146.577669876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{83281}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 20820 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-143.792\) of defining polynomial
Character \(\chi\) \(=\) 80.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9311.53 q^{3} +390625. q^{5} +2.48655e7 q^{7} -4.24355e7 q^{9} +1.90392e8 q^{11} -1.57377e7 q^{13} +3.63732e9 q^{15} -2.25502e10 q^{17} -1.27984e11 q^{19} +2.31536e11 q^{21} -2.59325e11 q^{23} +1.52588e11 q^{25} -1.59763e12 q^{27} -2.57953e12 q^{29} -7.91253e12 q^{31} +1.77285e12 q^{33} +9.71310e12 q^{35} -2.09888e13 q^{37} -1.46542e11 q^{39} +1.98123e13 q^{41} +4.60777e13 q^{43} -1.65764e13 q^{45} -2.66827e14 q^{47} +3.85664e14 q^{49} -2.09977e14 q^{51} +4.00045e14 q^{53} +7.43721e13 q^{55} -1.19172e15 q^{57} -1.82611e15 q^{59} -1.48236e15 q^{61} -1.05518e15 q^{63} -6.14754e12 q^{65} +2.71410e15 q^{67} -2.41471e15 q^{69} +3.87337e15 q^{71} -2.85786e15 q^{73} +1.42083e15 q^{75} +4.73421e15 q^{77} -2.80988e15 q^{79} -9.39628e15 q^{81} +7.04230e15 q^{83} -8.80866e15 q^{85} -2.40194e16 q^{87} +5.51966e16 q^{89} -3.91327e14 q^{91} -7.36778e16 q^{93} -4.99936e16 q^{95} +6.42662e16 q^{97} -8.07940e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1308 q^{3} + 781250 q^{5} - 603844 q^{7} - 107519094 q^{9} + 471481296 q^{11} - 1541834228 q^{13} + 510937500 q^{15} + 32139900564 q^{17} - 128672529400 q^{19} + 435381246624 q^{21} - 650359859292 q^{23}+ \cdots - 26\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9311.53 0.819390 0.409695 0.912223i \(-0.365635\pi\)
0.409695 + 0.912223i \(0.365635\pi\)
\(4\) 0 0
\(5\) 390625. 0.447214
\(6\) 0 0
\(7\) 2.48655e7 1.63029 0.815144 0.579258i \(-0.196658\pi\)
0.815144 + 0.579258i \(0.196658\pi\)
\(8\) 0 0
\(9\) −4.24355e7 −0.328600
\(10\) 0 0
\(11\) 1.90392e8 0.267801 0.133900 0.990995i \(-0.457250\pi\)
0.133900 + 0.990995i \(0.457250\pi\)
\(12\) 0 0
\(13\) −1.57377e7 −0.00535085 −0.00267543 0.999996i \(-0.500852\pi\)
−0.00267543 + 0.999996i \(0.500852\pi\)
\(14\) 0 0
\(15\) 3.63732e9 0.366442
\(16\) 0 0
\(17\) −2.25502e10 −0.784032 −0.392016 0.919958i \(-0.628222\pi\)
−0.392016 + 0.919958i \(0.628222\pi\)
\(18\) 0 0
\(19\) −1.27984e11 −1.72882 −0.864408 0.502791i \(-0.832306\pi\)
−0.864408 + 0.502791i \(0.832306\pi\)
\(20\) 0 0
\(21\) 2.31536e11 1.33584
\(22\) 0 0
\(23\) −2.59325e11 −0.690490 −0.345245 0.938513i \(-0.612204\pi\)
−0.345245 + 0.938513i \(0.612204\pi\)
\(24\) 0 0
\(25\) 1.52588e11 0.200000
\(26\) 0 0
\(27\) −1.59763e12 −1.08864
\(28\) 0 0
\(29\) −2.57953e12 −0.957542 −0.478771 0.877940i \(-0.658918\pi\)
−0.478771 + 0.877940i \(0.658918\pi\)
\(30\) 0 0
\(31\) −7.91253e12 −1.66625 −0.833126 0.553083i \(-0.813451\pi\)
−0.833126 + 0.553083i \(0.813451\pi\)
\(32\) 0 0
\(33\) 1.77285e12 0.219433
\(34\) 0 0
\(35\) 9.71310e12 0.729087
\(36\) 0 0
\(37\) −2.09888e13 −0.982367 −0.491184 0.871056i \(-0.663436\pi\)
−0.491184 + 0.871056i \(0.663436\pi\)
\(38\) 0 0
\(39\) −1.46542e11 −0.00438444
\(40\) 0 0
\(41\) 1.98123e13 0.387499 0.193750 0.981051i \(-0.437935\pi\)
0.193750 + 0.981051i \(0.437935\pi\)
\(42\) 0 0
\(43\) 4.60777e13 0.601186 0.300593 0.953753i \(-0.402815\pi\)
0.300593 + 0.953753i \(0.402815\pi\)
\(44\) 0 0
\(45\) −1.65764e13 −0.146955
\(46\) 0 0
\(47\) −2.66827e14 −1.63455 −0.817274 0.576250i \(-0.804516\pi\)
−0.817274 + 0.576250i \(0.804516\pi\)
\(48\) 0 0
\(49\) 3.85664e14 1.65784
\(50\) 0 0
\(51\) −2.09977e14 −0.642428
\(52\) 0 0
\(53\) 4.00045e14 0.882599 0.441300 0.897360i \(-0.354518\pi\)
0.441300 + 0.897360i \(0.354518\pi\)
\(54\) 0 0
\(55\) 7.43721e13 0.119764
\(56\) 0 0
\(57\) −1.19172e15 −1.41657
\(58\) 0 0
\(59\) −1.82611e15 −1.61914 −0.809570 0.587024i \(-0.800299\pi\)
−0.809570 + 0.587024i \(0.800299\pi\)
\(60\) 0 0
\(61\) −1.48236e15 −0.990035 −0.495017 0.868883i \(-0.664838\pi\)
−0.495017 + 0.868883i \(0.664838\pi\)
\(62\) 0 0
\(63\) −1.05518e15 −0.535713
\(64\) 0 0
\(65\) −6.14754e12 −0.00239297
\(66\) 0 0
\(67\) 2.71410e15 0.816563 0.408282 0.912856i \(-0.366128\pi\)
0.408282 + 0.912856i \(0.366128\pi\)
\(68\) 0 0
\(69\) −2.41471e15 −0.565780
\(70\) 0 0
\(71\) 3.87337e15 0.711858 0.355929 0.934513i \(-0.384164\pi\)
0.355929 + 0.934513i \(0.384164\pi\)
\(72\) 0 0
\(73\) −2.85786e15 −0.414759 −0.207380 0.978261i \(-0.566494\pi\)
−0.207380 + 0.978261i \(0.566494\pi\)
\(74\) 0 0
\(75\) 1.42083e15 0.163878
\(76\) 0 0
\(77\) 4.73421e15 0.436593
\(78\) 0 0
\(79\) −2.80988e15 −0.208381 −0.104190 0.994557i \(-0.533225\pi\)
−0.104190 + 0.994557i \(0.533225\pi\)
\(80\) 0 0
\(81\) −9.39628e15 −0.563422
\(82\) 0 0
\(83\) 7.04230e15 0.343203 0.171601 0.985166i \(-0.445106\pi\)
0.171601 + 0.985166i \(0.445106\pi\)
\(84\) 0 0
\(85\) −8.80866e15 −0.350630
\(86\) 0 0
\(87\) −2.40194e16 −0.784600
\(88\) 0 0
\(89\) 5.51966e16 1.48627 0.743135 0.669142i \(-0.233338\pi\)
0.743135 + 0.669142i \(0.233338\pi\)
\(90\) 0 0
\(91\) −3.91327e14 −0.00872343
\(92\) 0 0
\(93\) −7.36778e16 −1.36531
\(94\) 0 0
\(95\) −4.99936e16 −0.773150
\(96\) 0 0
\(97\) 6.42662e16 0.832574 0.416287 0.909233i \(-0.363331\pi\)
0.416287 + 0.909233i \(0.363331\pi\)
\(98\) 0 0
\(99\) −8.07940e15 −0.0879994
\(100\) 0 0
\(101\) −1.56405e17 −1.43721 −0.718603 0.695421i \(-0.755218\pi\)
−0.718603 + 0.695421i \(0.755218\pi\)
\(102\) 0 0
\(103\) 2.33166e17 1.81363 0.906816 0.421527i \(-0.138506\pi\)
0.906816 + 0.421527i \(0.138506\pi\)
\(104\) 0 0
\(105\) 9.04438e16 0.597407
\(106\) 0 0
\(107\) 1.05397e17 0.593017 0.296509 0.955030i \(-0.404178\pi\)
0.296509 + 0.955030i \(0.404178\pi\)
\(108\) 0 0
\(109\) 2.51895e17 1.21086 0.605431 0.795898i \(-0.293001\pi\)
0.605431 + 0.795898i \(0.293001\pi\)
\(110\) 0 0
\(111\) −1.95438e17 −0.804942
\(112\) 0 0
\(113\) 4.09395e17 1.44869 0.724346 0.689436i \(-0.242142\pi\)
0.724346 + 0.689436i \(0.242142\pi\)
\(114\) 0 0
\(115\) −1.01299e17 −0.308796
\(116\) 0 0
\(117\) 6.67838e14 0.00175829
\(118\) 0 0
\(119\) −5.60722e17 −1.27820
\(120\) 0 0
\(121\) −4.69198e17 −0.928283
\(122\) 0 0
\(123\) 1.84482e17 0.317513
\(124\) 0 0
\(125\) 5.96046e16 0.0894427
\(126\) 0 0
\(127\) 6.42410e17 0.842327 0.421164 0.906985i \(-0.361622\pi\)
0.421164 + 0.906985i \(0.361622\pi\)
\(128\) 0 0
\(129\) 4.29054e17 0.492606
\(130\) 0 0
\(131\) −3.57323e17 −0.359961 −0.179980 0.983670i \(-0.557603\pi\)
−0.179980 + 0.983670i \(0.557603\pi\)
\(132\) 0 0
\(133\) −3.18238e18 −2.81847
\(134\) 0 0
\(135\) −6.24075e17 −0.486855
\(136\) 0 0
\(137\) −1.23821e18 −0.852454 −0.426227 0.904616i \(-0.640158\pi\)
−0.426227 + 0.904616i \(0.640158\pi\)
\(138\) 0 0
\(139\) −1.95129e18 −1.18767 −0.593836 0.804586i \(-0.702387\pi\)
−0.593836 + 0.804586i \(0.702387\pi\)
\(140\) 0 0
\(141\) −2.48457e18 −1.33933
\(142\) 0 0
\(143\) −2.99634e15 −0.00143296
\(144\) 0 0
\(145\) −1.00763e18 −0.428226
\(146\) 0 0
\(147\) 3.59112e18 1.35842
\(148\) 0 0
\(149\) 1.86793e18 0.629908 0.314954 0.949107i \(-0.398011\pi\)
0.314954 + 0.949107i \(0.398011\pi\)
\(150\) 0 0
\(151\) −5.18884e18 −1.56231 −0.781154 0.624339i \(-0.785368\pi\)
−0.781154 + 0.624339i \(0.785368\pi\)
\(152\) 0 0
\(153\) 9.56927e17 0.257633
\(154\) 0 0
\(155\) −3.09083e18 −0.745171
\(156\) 0 0
\(157\) 7.78987e18 1.68416 0.842080 0.539353i \(-0.181331\pi\)
0.842080 + 0.539353i \(0.181331\pi\)
\(158\) 0 0
\(159\) 3.72503e18 0.723193
\(160\) 0 0
\(161\) −6.44825e18 −1.12570
\(162\) 0 0
\(163\) −6.16891e18 −0.969647 −0.484824 0.874612i \(-0.661116\pi\)
−0.484824 + 0.874612i \(0.661116\pi\)
\(164\) 0 0
\(165\) 6.92518e17 0.0981336
\(166\) 0 0
\(167\) −1.24726e19 −1.59539 −0.797693 0.603064i \(-0.793946\pi\)
−0.797693 + 0.603064i \(0.793946\pi\)
\(168\) 0 0
\(169\) −8.65017e18 −0.999971
\(170\) 0 0
\(171\) 5.43105e18 0.568089
\(172\) 0 0
\(173\) 1.64494e19 1.55869 0.779344 0.626596i \(-0.215552\pi\)
0.779344 + 0.626596i \(0.215552\pi\)
\(174\) 0 0
\(175\) 3.79418e18 0.326058
\(176\) 0 0
\(177\) −1.70039e19 −1.32671
\(178\) 0 0
\(179\) 1.53067e18 0.108550 0.0542751 0.998526i \(-0.482715\pi\)
0.0542751 + 0.998526i \(0.482715\pi\)
\(180\) 0 0
\(181\) −8.51444e18 −0.549399 −0.274700 0.961530i \(-0.588578\pi\)
−0.274700 + 0.961530i \(0.588578\pi\)
\(182\) 0 0
\(183\) −1.38031e19 −0.811224
\(184\) 0 0
\(185\) −8.19877e18 −0.439328
\(186\) 0 0
\(187\) −4.29338e18 −0.209965
\(188\) 0 0
\(189\) −3.97260e19 −1.77480
\(190\) 0 0
\(191\) −1.05921e19 −0.432710 −0.216355 0.976315i \(-0.569417\pi\)
−0.216355 + 0.976315i \(0.569417\pi\)
\(192\) 0 0
\(193\) −1.55768e19 −0.582425 −0.291213 0.956658i \(-0.594059\pi\)
−0.291213 + 0.956658i \(0.594059\pi\)
\(194\) 0 0
\(195\) −5.72431e16 −0.00196078
\(196\) 0 0
\(197\) 8.47320e18 0.266124 0.133062 0.991108i \(-0.457519\pi\)
0.133062 + 0.991108i \(0.457519\pi\)
\(198\) 0 0
\(199\) −9.48640e18 −0.273433 −0.136716 0.990610i \(-0.543655\pi\)
−0.136716 + 0.990610i \(0.543655\pi\)
\(200\) 0 0
\(201\) 2.52724e19 0.669083
\(202\) 0 0
\(203\) −6.41414e19 −1.56107
\(204\) 0 0
\(205\) 7.73916e18 0.173295
\(206\) 0 0
\(207\) 1.10046e19 0.226895
\(208\) 0 0
\(209\) −2.43671e19 −0.462978
\(210\) 0 0
\(211\) 5.06102e19 0.886824 0.443412 0.896318i \(-0.353768\pi\)
0.443412 + 0.896318i \(0.353768\pi\)
\(212\) 0 0
\(213\) 3.60671e19 0.583289
\(214\) 0 0
\(215\) 1.79991e19 0.268858
\(216\) 0 0
\(217\) −1.96749e20 −2.71647
\(218\) 0 0
\(219\) −2.66110e19 −0.339849
\(220\) 0 0
\(221\) 3.54888e17 0.00419524
\(222\) 0 0
\(223\) −1.13017e20 −1.23752 −0.618759 0.785581i \(-0.712364\pi\)
−0.618759 + 0.785581i \(0.712364\pi\)
\(224\) 0 0
\(225\) −6.47514e18 −0.0657201
\(226\) 0 0
\(227\) −1.34057e20 −1.26203 −0.631016 0.775770i \(-0.717362\pi\)
−0.631016 + 0.775770i \(0.717362\pi\)
\(228\) 0 0
\(229\) −1.53092e20 −1.33768 −0.668840 0.743406i \(-0.733209\pi\)
−0.668840 + 0.743406i \(0.733209\pi\)
\(230\) 0 0
\(231\) 4.40828e19 0.357740
\(232\) 0 0
\(233\) 1.54095e20 1.16215 0.581076 0.813849i \(-0.302632\pi\)
0.581076 + 0.813849i \(0.302632\pi\)
\(234\) 0 0
\(235\) −1.04229e20 −0.730992
\(236\) 0 0
\(237\) −2.61643e19 −0.170745
\(238\) 0 0
\(239\) −1.10626e20 −0.672161 −0.336081 0.941833i \(-0.609101\pi\)
−0.336081 + 0.941833i \(0.609101\pi\)
\(240\) 0 0
\(241\) −1.68590e20 −0.954303 −0.477152 0.878821i \(-0.658331\pi\)
−0.477152 + 0.878821i \(0.658331\pi\)
\(242\) 0 0
\(243\) 1.18825e20 0.626980
\(244\) 0 0
\(245\) 1.50650e20 0.741408
\(246\) 0 0
\(247\) 2.01417e18 0.00925064
\(248\) 0 0
\(249\) 6.55746e19 0.281217
\(250\) 0 0
\(251\) 8.71684e19 0.349247 0.174623 0.984635i \(-0.444129\pi\)
0.174623 + 0.984635i \(0.444129\pi\)
\(252\) 0 0
\(253\) −4.93735e19 −0.184914
\(254\) 0 0
\(255\) −8.20221e19 −0.287303
\(256\) 0 0
\(257\) 2.57989e20 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(258\) 0 0
\(259\) −5.21899e20 −1.60154
\(260\) 0 0
\(261\) 1.09464e20 0.314648
\(262\) 0 0
\(263\) 4.12838e20 1.11213 0.556065 0.831139i \(-0.312311\pi\)
0.556065 + 0.831139i \(0.312311\pi\)
\(264\) 0 0
\(265\) 1.56267e20 0.394710
\(266\) 0 0
\(267\) 5.13965e20 1.21783
\(268\) 0 0
\(269\) 3.61622e20 0.804194 0.402097 0.915597i \(-0.368282\pi\)
0.402097 + 0.915597i \(0.368282\pi\)
\(270\) 0 0
\(271\) 6.61685e20 1.38170 0.690848 0.723000i \(-0.257238\pi\)
0.690848 + 0.723000i \(0.257238\pi\)
\(272\) 0 0
\(273\) −3.64385e18 −0.00714789
\(274\) 0 0
\(275\) 2.90516e19 0.0535602
\(276\) 0 0
\(277\) 2.43700e20 0.422452 0.211226 0.977437i \(-0.432254\pi\)
0.211226 + 0.977437i \(0.432254\pi\)
\(278\) 0 0
\(279\) 3.35772e20 0.547531
\(280\) 0 0
\(281\) −9.47052e20 −1.45335 −0.726675 0.686981i \(-0.758935\pi\)
−0.726675 + 0.686981i \(0.758935\pi\)
\(282\) 0 0
\(283\) 3.67504e20 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(284\) 0 0
\(285\) −4.65517e20 −0.633511
\(286\) 0 0
\(287\) 4.92642e20 0.631736
\(288\) 0 0
\(289\) −3.18730e20 −0.385293
\(290\) 0 0
\(291\) 5.98417e20 0.682203
\(292\) 0 0
\(293\) 1.20904e21 1.30037 0.650183 0.759778i \(-0.274692\pi\)
0.650183 + 0.759778i \(0.274692\pi\)
\(294\) 0 0
\(295\) −7.13323e20 −0.724101
\(296\) 0 0
\(297\) −3.04177e20 −0.291539
\(298\) 0 0
\(299\) 4.08118e18 0.00369471
\(300\) 0 0
\(301\) 1.14575e21 0.980106
\(302\) 0 0
\(303\) −1.45637e21 −1.17763
\(304\) 0 0
\(305\) −5.79047e20 −0.442757
\(306\) 0 0
\(307\) 4.76931e20 0.344969 0.172484 0.985012i \(-0.444821\pi\)
0.172484 + 0.985012i \(0.444821\pi\)
\(308\) 0 0
\(309\) 2.17114e21 1.48607
\(310\) 0 0
\(311\) −3.50418e20 −0.227051 −0.113525 0.993535i \(-0.536214\pi\)
−0.113525 + 0.993535i \(0.536214\pi\)
\(312\) 0 0
\(313\) 2.04751e21 1.25631 0.628157 0.778087i \(-0.283810\pi\)
0.628157 + 0.778087i \(0.283810\pi\)
\(314\) 0 0
\(315\) −4.12180e20 −0.239578
\(316\) 0 0
\(317\) 1.31663e21 0.725205 0.362602 0.931944i \(-0.381888\pi\)
0.362602 + 0.931944i \(0.381888\pi\)
\(318\) 0 0
\(319\) −4.91123e20 −0.256430
\(320\) 0 0
\(321\) 9.81411e20 0.485912
\(322\) 0 0
\(323\) 2.88605e21 1.35545
\(324\) 0 0
\(325\) −2.40138e18 −0.00107017
\(326\) 0 0
\(327\) 2.34553e21 0.992167
\(328\) 0 0
\(329\) −6.63479e21 −2.66478
\(330\) 0 0
\(331\) −3.85862e21 −1.47195 −0.735977 0.677007i \(-0.763277\pi\)
−0.735977 + 0.677007i \(0.763277\pi\)
\(332\) 0 0
\(333\) 8.90672e20 0.322806
\(334\) 0 0
\(335\) 1.06019e21 0.365178
\(336\) 0 0
\(337\) 4.69149e21 1.53623 0.768115 0.640312i \(-0.221195\pi\)
0.768115 + 0.640312i \(0.221195\pi\)
\(338\) 0 0
\(339\) 3.81210e21 1.18704
\(340\) 0 0
\(341\) −1.50649e21 −0.446224
\(342\) 0 0
\(343\) 3.80526e21 1.07247
\(344\) 0 0
\(345\) −9.43246e20 −0.253025
\(346\) 0 0
\(347\) −2.51637e21 −0.642648 −0.321324 0.946969i \(-0.604128\pi\)
−0.321324 + 0.946969i \(0.604128\pi\)
\(348\) 0 0
\(349\) −2.20222e21 −0.535606 −0.267803 0.963474i \(-0.586298\pi\)
−0.267803 + 0.963474i \(0.586298\pi\)
\(350\) 0 0
\(351\) 2.51431e19 0.00582516
\(352\) 0 0
\(353\) 1.30214e21 0.287456 0.143728 0.989617i \(-0.454091\pi\)
0.143728 + 0.989617i \(0.454091\pi\)
\(354\) 0 0
\(355\) 1.51304e21 0.318353
\(356\) 0 0
\(357\) −5.22118e21 −1.04734
\(358\) 0 0
\(359\) −2.07162e21 −0.396285 −0.198143 0.980173i \(-0.563491\pi\)
−0.198143 + 0.980173i \(0.563491\pi\)
\(360\) 0 0
\(361\) 1.08994e22 1.98880
\(362\) 0 0
\(363\) −4.36895e21 −0.760625
\(364\) 0 0
\(365\) −1.11635e21 −0.185486
\(366\) 0 0
\(367\) −4.69137e21 −0.744111 −0.372056 0.928210i \(-0.621347\pi\)
−0.372056 + 0.928210i \(0.621347\pi\)
\(368\) 0 0
\(369\) −8.40743e20 −0.127332
\(370\) 0 0
\(371\) 9.94732e21 1.43889
\(372\) 0 0
\(373\) −7.77652e20 −0.107463 −0.0537317 0.998555i \(-0.517112\pi\)
−0.0537317 + 0.998555i \(0.517112\pi\)
\(374\) 0 0
\(375\) 5.55011e20 0.0732885
\(376\) 0 0
\(377\) 4.05959e19 0.00512367
\(378\) 0 0
\(379\) −3.50054e20 −0.0422379 −0.0211189 0.999777i \(-0.506723\pi\)
−0.0211189 + 0.999777i \(0.506723\pi\)
\(380\) 0 0
\(381\) 5.98183e21 0.690194
\(382\) 0 0
\(383\) 1.02315e22 1.12914 0.564570 0.825385i \(-0.309042\pi\)
0.564570 + 0.825385i \(0.309042\pi\)
\(384\) 0 0
\(385\) 1.84930e21 0.195250
\(386\) 0 0
\(387\) −1.95533e21 −0.197550
\(388\) 0 0
\(389\) −4.13104e21 −0.399473 −0.199737 0.979850i \(-0.564009\pi\)
−0.199737 + 0.979850i \(0.564009\pi\)
\(390\) 0 0
\(391\) 5.84781e21 0.541366
\(392\) 0 0
\(393\) −3.32723e21 −0.294948
\(394\) 0 0
\(395\) −1.09761e21 −0.0931907
\(396\) 0 0
\(397\) −7.65334e21 −0.622489 −0.311244 0.950330i \(-0.600746\pi\)
−0.311244 + 0.950330i \(0.600746\pi\)
\(398\) 0 0
\(399\) −2.96328e22 −2.30942
\(400\) 0 0
\(401\) −4.08273e20 −0.0304947 −0.0152473 0.999884i \(-0.504854\pi\)
−0.0152473 + 0.999884i \(0.504854\pi\)
\(402\) 0 0
\(403\) 1.24525e20 0.00891588
\(404\) 0 0
\(405\) −3.67042e21 −0.251970
\(406\) 0 0
\(407\) −3.99612e21 −0.263079
\(408\) 0 0
\(409\) 2.00923e22 1.26877 0.634383 0.773019i \(-0.281254\pi\)
0.634383 + 0.773019i \(0.281254\pi\)
\(410\) 0 0
\(411\) −1.15297e22 −0.698493
\(412\) 0 0
\(413\) −4.54071e22 −2.63966
\(414\) 0 0
\(415\) 2.75090e21 0.153485
\(416\) 0 0
\(417\) −1.81695e22 −0.973167
\(418\) 0 0
\(419\) −3.31114e22 −1.70278 −0.851390 0.524533i \(-0.824240\pi\)
−0.851390 + 0.524533i \(0.824240\pi\)
\(420\) 0 0
\(421\) −2.31325e22 −1.14242 −0.571208 0.820805i \(-0.693525\pi\)
−0.571208 + 0.820805i \(0.693525\pi\)
\(422\) 0 0
\(423\) 1.13229e22 0.537113
\(424\) 0 0
\(425\) −3.44088e21 −0.156806
\(426\) 0 0
\(427\) −3.68597e22 −1.61404
\(428\) 0 0
\(429\) −2.79005e19 −0.00117416
\(430\) 0 0
\(431\) −4.44596e22 −1.79849 −0.899247 0.437440i \(-0.855885\pi\)
−0.899247 + 0.437440i \(0.855885\pi\)
\(432\) 0 0
\(433\) 2.85304e22 1.10959 0.554793 0.831989i \(-0.312798\pi\)
0.554793 + 0.831989i \(0.312798\pi\)
\(434\) 0 0
\(435\) −9.38257e21 −0.350884
\(436\) 0 0
\(437\) 3.31893e22 1.19373
\(438\) 0 0
\(439\) 4.28578e22 1.48279 0.741397 0.671066i \(-0.234163\pi\)
0.741397 + 0.671066i \(0.234163\pi\)
\(440\) 0 0
\(441\) −1.63658e22 −0.544767
\(442\) 0 0
\(443\) 1.40464e22 0.449919 0.224959 0.974368i \(-0.427775\pi\)
0.224959 + 0.974368i \(0.427775\pi\)
\(444\) 0 0
\(445\) 2.15612e22 0.664680
\(446\) 0 0
\(447\) 1.73933e22 0.516140
\(448\) 0 0
\(449\) −3.88401e22 −1.10965 −0.554826 0.831966i \(-0.687215\pi\)
−0.554826 + 0.831966i \(0.687215\pi\)
\(450\) 0 0
\(451\) 3.77210e21 0.103773
\(452\) 0 0
\(453\) −4.83161e22 −1.28014
\(454\) 0 0
\(455\) −1.52862e20 −0.00390124
\(456\) 0 0
\(457\) −2.21160e22 −0.543774 −0.271887 0.962329i \(-0.587648\pi\)
−0.271887 + 0.962329i \(0.587648\pi\)
\(458\) 0 0
\(459\) 3.60269e22 0.853530
\(460\) 0 0
\(461\) 1.69257e22 0.386446 0.193223 0.981155i \(-0.438106\pi\)
0.193223 + 0.981155i \(0.438106\pi\)
\(462\) 0 0
\(463\) −5.00810e22 −1.10213 −0.551067 0.834461i \(-0.685779\pi\)
−0.551067 + 0.834461i \(0.685779\pi\)
\(464\) 0 0
\(465\) −2.87804e22 −0.610585
\(466\) 0 0
\(467\) 6.87611e22 1.40653 0.703266 0.710927i \(-0.251724\pi\)
0.703266 + 0.710927i \(0.251724\pi\)
\(468\) 0 0
\(469\) 6.74875e22 1.33123
\(470\) 0 0
\(471\) 7.25357e22 1.37998
\(472\) 0 0
\(473\) 8.77284e21 0.160998
\(474\) 0 0
\(475\) −1.95287e22 −0.345763
\(476\) 0 0
\(477\) −1.69761e22 −0.290022
\(478\) 0 0
\(479\) 3.44291e22 0.567641 0.283821 0.958877i \(-0.408398\pi\)
0.283821 + 0.958877i \(0.408398\pi\)
\(480\) 0 0
\(481\) 3.30316e20 0.00525650
\(482\) 0 0
\(483\) −6.00431e22 −0.922385
\(484\) 0 0
\(485\) 2.51040e22 0.372338
\(486\) 0 0
\(487\) −8.83844e20 −0.0126584 −0.00632921 0.999980i \(-0.502015\pi\)
−0.00632921 + 0.999980i \(0.502015\pi\)
\(488\) 0 0
\(489\) −5.74420e22 −0.794519
\(490\) 0 0
\(491\) −1.04515e23 −1.39632 −0.698158 0.715943i \(-0.745997\pi\)
−0.698158 + 0.715943i \(0.745997\pi\)
\(492\) 0 0
\(493\) 5.81688e22 0.750744
\(494\) 0 0
\(495\) −3.15602e21 −0.0393545
\(496\) 0 0
\(497\) 9.63135e22 1.16053
\(498\) 0 0
\(499\) 5.12067e22 0.596310 0.298155 0.954518i \(-0.403629\pi\)
0.298155 + 0.954518i \(0.403629\pi\)
\(500\) 0 0
\(501\) −1.16139e23 −1.30724
\(502\) 0 0
\(503\) −5.06660e22 −0.551301 −0.275650 0.961258i \(-0.588893\pi\)
−0.275650 + 0.961258i \(0.588893\pi\)
\(504\) 0 0
\(505\) −6.10957e22 −0.642738
\(506\) 0 0
\(507\) −8.05463e22 −0.819366
\(508\) 0 0
\(509\) 1.68111e23 1.65385 0.826923 0.562316i \(-0.190089\pi\)
0.826923 + 0.562316i \(0.190089\pi\)
\(510\) 0 0
\(511\) −7.10621e22 −0.676177
\(512\) 0 0
\(513\) 2.04471e23 1.88206
\(514\) 0 0
\(515\) 9.10805e22 0.811081
\(516\) 0 0
\(517\) −5.08018e22 −0.437733
\(518\) 0 0
\(519\) 1.53170e23 1.27717
\(520\) 0 0
\(521\) 1.13516e23 0.916085 0.458042 0.888930i \(-0.348551\pi\)
0.458042 + 0.888930i \(0.348551\pi\)
\(522\) 0 0
\(523\) −3.13139e22 −0.244609 −0.122305 0.992493i \(-0.539028\pi\)
−0.122305 + 0.992493i \(0.539028\pi\)
\(524\) 0 0
\(525\) 3.53296e22 0.267168
\(526\) 0 0
\(527\) 1.78429e23 1.30640
\(528\) 0 0
\(529\) −7.38008e22 −0.523224
\(530\) 0 0
\(531\) 7.74918e22 0.532050
\(532\) 0 0
\(533\) −3.11800e20 −0.00207345
\(534\) 0 0
\(535\) 4.11708e22 0.265205
\(536\) 0 0
\(537\) 1.42529e22 0.0889449
\(538\) 0 0
\(539\) 7.34275e22 0.443971
\(540\) 0 0
\(541\) −1.32903e23 −0.778680 −0.389340 0.921094i \(-0.627297\pi\)
−0.389340 + 0.921094i \(0.627297\pi\)
\(542\) 0 0
\(543\) −7.92825e22 −0.450172
\(544\) 0 0
\(545\) 9.83965e22 0.541514
\(546\) 0 0
\(547\) −3.86923e22 −0.206411 −0.103205 0.994660i \(-0.532910\pi\)
−0.103205 + 0.994660i \(0.532910\pi\)
\(548\) 0 0
\(549\) 6.29047e22 0.325326
\(550\) 0 0
\(551\) 3.30138e23 1.65541
\(552\) 0 0
\(553\) −6.98692e22 −0.339721
\(554\) 0 0
\(555\) −7.63431e22 −0.359981
\(556\) 0 0
\(557\) −1.53384e23 −0.701475 −0.350737 0.936474i \(-0.614069\pi\)
−0.350737 + 0.936474i \(0.614069\pi\)
\(558\) 0 0
\(559\) −7.25157e20 −0.00321686
\(560\) 0 0
\(561\) −3.99780e22 −0.172043
\(562\) 0 0
\(563\) 1.19252e23 0.497904 0.248952 0.968516i \(-0.419914\pi\)
0.248952 + 0.968516i \(0.419914\pi\)
\(564\) 0 0
\(565\) 1.59920e23 0.647875
\(566\) 0 0
\(567\) −2.33644e23 −0.918540
\(568\) 0 0
\(569\) −1.07363e23 −0.409638 −0.204819 0.978800i \(-0.565661\pi\)
−0.204819 + 0.978800i \(0.565661\pi\)
\(570\) 0 0
\(571\) 2.25644e23 0.835634 0.417817 0.908531i \(-0.362795\pi\)
0.417817 + 0.908531i \(0.362795\pi\)
\(572\) 0 0
\(573\) −9.86284e22 −0.354558
\(574\) 0 0
\(575\) −3.95698e22 −0.138098
\(576\) 0 0
\(577\) −1.29594e23 −0.439127 −0.219563 0.975598i \(-0.570463\pi\)
−0.219563 + 0.975598i \(0.570463\pi\)
\(578\) 0 0
\(579\) −1.45044e23 −0.477233
\(580\) 0 0
\(581\) 1.75111e23 0.559519
\(582\) 0 0
\(583\) 7.61655e22 0.236361
\(584\) 0 0
\(585\) 2.60874e20 0.000786332 0
\(586\) 0 0
\(587\) −3.21537e23 −0.941471 −0.470735 0.882274i \(-0.656011\pi\)
−0.470735 + 0.882274i \(0.656011\pi\)
\(588\) 0 0
\(589\) 1.01267e24 2.88064
\(590\) 0 0
\(591\) 7.88985e22 0.218060
\(592\) 0 0
\(593\) −8.23836e22 −0.221246 −0.110623 0.993862i \(-0.535285\pi\)
−0.110623 + 0.993862i \(0.535285\pi\)
\(594\) 0 0
\(595\) −2.19032e23 −0.571628
\(596\) 0 0
\(597\) −8.83330e22 −0.224048
\(598\) 0 0
\(599\) 7.29015e22 0.179725 0.0898625 0.995954i \(-0.471357\pi\)
0.0898625 + 0.995954i \(0.471357\pi\)
\(600\) 0 0
\(601\) 3.79880e23 0.910361 0.455180 0.890399i \(-0.349575\pi\)
0.455180 + 0.890399i \(0.349575\pi\)
\(602\) 0 0
\(603\) −1.15174e23 −0.268323
\(604\) 0 0
\(605\) −1.83280e23 −0.415141
\(606\) 0 0
\(607\) 3.76245e23 0.828641 0.414321 0.910131i \(-0.364019\pi\)
0.414321 + 0.910131i \(0.364019\pi\)
\(608\) 0 0
\(609\) −5.97255e23 −1.27912
\(610\) 0 0
\(611\) 4.19924e21 0.00874623
\(612\) 0 0
\(613\) −4.60370e23 −0.932595 −0.466298 0.884628i \(-0.654412\pi\)
−0.466298 + 0.884628i \(0.654412\pi\)
\(614\) 0 0
\(615\) 7.20635e22 0.141996
\(616\) 0 0
\(617\) −3.36863e23 −0.645698 −0.322849 0.946451i \(-0.604641\pi\)
−0.322849 + 0.946451i \(0.604641\pi\)
\(618\) 0 0
\(619\) −1.49640e23 −0.279047 −0.139523 0.990219i \(-0.544557\pi\)
−0.139523 + 0.990219i \(0.544557\pi\)
\(620\) 0 0
\(621\) 4.14306e23 0.751696
\(622\) 0 0
\(623\) 1.37249e24 2.42305
\(624\) 0 0
\(625\) 2.32831e22 0.0400000
\(626\) 0 0
\(627\) −2.26895e23 −0.379360
\(628\) 0 0
\(629\) 4.73302e23 0.770208
\(630\) 0 0
\(631\) 6.17252e21 0.00977717 0.00488858 0.999988i \(-0.498444\pi\)
0.00488858 + 0.999988i \(0.498444\pi\)
\(632\) 0 0
\(633\) 4.71259e23 0.726654
\(634\) 0 0
\(635\) 2.50942e23 0.376700
\(636\) 0 0
\(637\) −6.06947e21 −0.00887086
\(638\) 0 0
\(639\) −1.64369e23 −0.233917
\(640\) 0 0
\(641\) −1.59514e23 −0.221057 −0.110529 0.993873i \(-0.535254\pi\)
−0.110529 + 0.993873i \(0.535254\pi\)
\(642\) 0 0
\(643\) −1.37854e24 −1.86049 −0.930244 0.366941i \(-0.880405\pi\)
−0.930244 + 0.366941i \(0.880405\pi\)
\(644\) 0 0
\(645\) 1.67599e23 0.220300
\(646\) 0 0
\(647\) −3.15056e23 −0.403369 −0.201684 0.979451i \(-0.564641\pi\)
−0.201684 + 0.979451i \(0.564641\pi\)
\(648\) 0 0
\(649\) −3.47677e23 −0.433607
\(650\) 0 0
\(651\) −1.83204e24 −2.22585
\(652\) 0 0
\(653\) −2.27963e23 −0.269838 −0.134919 0.990857i \(-0.543077\pi\)
−0.134919 + 0.990857i \(0.543077\pi\)
\(654\) 0 0
\(655\) −1.39579e23 −0.160979
\(656\) 0 0
\(657\) 1.21275e23 0.136290
\(658\) 0 0
\(659\) −1.18670e24 −1.29961 −0.649806 0.760100i \(-0.725150\pi\)
−0.649806 + 0.760100i \(0.725150\pi\)
\(660\) 0 0
\(661\) −1.50237e24 −1.60348 −0.801741 0.597672i \(-0.796092\pi\)
−0.801741 + 0.597672i \(0.796092\pi\)
\(662\) 0 0
\(663\) 3.30455e21 0.00343754
\(664\) 0 0
\(665\) −1.24312e24 −1.26046
\(666\) 0 0
\(667\) 6.68936e23 0.661173
\(668\) 0 0
\(669\) −1.05236e24 −1.01401
\(670\) 0 0
\(671\) −2.82230e23 −0.265132
\(672\) 0 0
\(673\) 2.53432e23 0.232131 0.116065 0.993242i \(-0.462972\pi\)
0.116065 + 0.993242i \(0.462972\pi\)
\(674\) 0 0
\(675\) −2.43779e23 −0.217728
\(676\) 0 0
\(677\) 4.60323e23 0.400921 0.200460 0.979702i \(-0.435756\pi\)
0.200460 + 0.979702i \(0.435756\pi\)
\(678\) 0 0
\(679\) 1.59801e24 1.35734
\(680\) 0 0
\(681\) −1.24828e24 −1.03410
\(682\) 0 0
\(683\) 6.27345e23 0.506910 0.253455 0.967347i \(-0.418433\pi\)
0.253455 + 0.967347i \(0.418433\pi\)
\(684\) 0 0
\(685\) −4.83678e23 −0.381229
\(686\) 0 0
\(687\) −1.42553e24 −1.09608
\(688\) 0 0
\(689\) −6.29579e21 −0.00472266
\(690\) 0 0
\(691\) −4.11645e23 −0.301272 −0.150636 0.988589i \(-0.548132\pi\)
−0.150636 + 0.988589i \(0.548132\pi\)
\(692\) 0 0
\(693\) −2.00899e23 −0.143464
\(694\) 0 0
\(695\) −7.62224e23 −0.531144
\(696\) 0 0
\(697\) −4.46770e23 −0.303812
\(698\) 0 0
\(699\) 1.43486e24 0.952255
\(700\) 0 0
\(701\) 2.88292e23 0.186737 0.0933683 0.995632i \(-0.470237\pi\)
0.0933683 + 0.995632i \(0.470237\pi\)
\(702\) 0 0
\(703\) 2.68623e24 1.69833
\(704\) 0 0
\(705\) −9.70534e23 −0.598967
\(706\) 0 0
\(707\) −3.88909e24 −2.34306
\(708\) 0 0
\(709\) −2.85869e24 −1.68141 −0.840707 0.541491i \(-0.817860\pi\)
−0.840707 + 0.541491i \(0.817860\pi\)
\(710\) 0 0
\(711\) 1.19239e23 0.0684740
\(712\) 0 0
\(713\) 2.05191e24 1.15053
\(714\) 0 0
\(715\) −1.17045e21 −0.000640841 0
\(716\) 0 0
\(717\) −1.03009e24 −0.550762
\(718\) 0 0
\(719\) 1.21983e24 0.636946 0.318473 0.947932i \(-0.396830\pi\)
0.318473 + 0.947932i \(0.396830\pi\)
\(720\) 0 0
\(721\) 5.79780e24 2.95674
\(722\) 0 0
\(723\) −1.56983e24 −0.781947
\(724\) 0 0
\(725\) −3.93605e23 −0.191508
\(726\) 0 0
\(727\) 2.28060e24 1.08394 0.541972 0.840396i \(-0.317678\pi\)
0.541972 + 0.840396i \(0.317678\pi\)
\(728\) 0 0
\(729\) 2.31988e24 1.07716
\(730\) 0 0
\(731\) −1.03906e24 −0.471349
\(732\) 0 0
\(733\) −8.50533e23 −0.376970 −0.188485 0.982076i \(-0.560358\pi\)
−0.188485 + 0.982076i \(0.560358\pi\)
\(734\) 0 0
\(735\) 1.40278e24 0.607503
\(736\) 0 0
\(737\) 5.16744e23 0.218676
\(738\) 0 0
\(739\) −2.14385e24 −0.886578 −0.443289 0.896379i \(-0.646188\pi\)
−0.443289 + 0.896379i \(0.646188\pi\)
\(740\) 0 0
\(741\) 1.87550e22 0.00757988
\(742\) 0 0
\(743\) −2.23446e24 −0.882606 −0.441303 0.897358i \(-0.645484\pi\)
−0.441303 + 0.897358i \(0.645484\pi\)
\(744\) 0 0
\(745\) 7.29660e23 0.281703
\(746\) 0 0
\(747\) −2.98843e23 −0.112777
\(748\) 0 0
\(749\) 2.62076e24 0.966789
\(750\) 0 0
\(751\) 1.71798e24 0.619555 0.309778 0.950809i \(-0.399745\pi\)
0.309778 + 0.950809i \(0.399745\pi\)
\(752\) 0 0
\(753\) 8.11671e23 0.286169
\(754\) 0 0
\(755\) −2.02689e24 −0.698685
\(756\) 0 0
\(757\) −1.09729e24 −0.369832 −0.184916 0.982754i \(-0.559201\pi\)
−0.184916 + 0.982754i \(0.559201\pi\)
\(758\) 0 0
\(759\) −4.59743e23 −0.151516
\(760\) 0 0
\(761\) −1.70967e24 −0.550990 −0.275495 0.961302i \(-0.588842\pi\)
−0.275495 + 0.961302i \(0.588842\pi\)
\(762\) 0 0
\(763\) 6.26350e24 1.97405
\(764\) 0 0
\(765\) 3.73800e23 0.115217
\(766\) 0 0
\(767\) 2.87388e22 0.00866378
\(768\) 0 0
\(769\) 1.22119e24 0.360090 0.180045 0.983658i \(-0.442376\pi\)
0.180045 + 0.983658i \(0.442376\pi\)
\(770\) 0 0
\(771\) 2.40228e24 0.692884
\(772\) 0 0
\(773\) 2.59069e24 0.730952 0.365476 0.930821i \(-0.380906\pi\)
0.365476 + 0.930821i \(0.380906\pi\)
\(774\) 0 0
\(775\) −1.20736e24 −0.333251
\(776\) 0 0
\(777\) −4.85968e24 −1.31229
\(778\) 0 0
\(779\) −2.53564e24 −0.669915
\(780\) 0 0
\(781\) 7.37461e23 0.190636
\(782\) 0 0
\(783\) 4.12114e24 1.04242
\(784\) 0 0
\(785\) 3.04292e24 0.753179
\(786\) 0 0
\(787\) −2.64413e24 −0.640467 −0.320234 0.947339i \(-0.603761\pi\)
−0.320234 + 0.947339i \(0.603761\pi\)
\(788\) 0 0
\(789\) 3.84415e24 0.911268
\(790\) 0 0
\(791\) 1.01798e25 2.36179
\(792\) 0 0
\(793\) 2.33290e22 0.00529753
\(794\) 0 0
\(795\) 1.45509e24 0.323422
\(796\) 0 0
\(797\) −6.24909e24 −1.35963 −0.679815 0.733383i \(-0.737940\pi\)
−0.679815 + 0.733383i \(0.737940\pi\)
\(798\) 0 0
\(799\) 6.01699e24 1.28154
\(800\) 0 0
\(801\) −2.34230e24 −0.488388
\(802\) 0 0
\(803\) −5.44114e23 −0.111073
\(804\) 0 0
\(805\) −2.51885e24 −0.503427
\(806\) 0 0
\(807\) 3.36726e24 0.658948
\(808\) 0 0
\(809\) 5.76985e24 1.10561 0.552805 0.833311i \(-0.313557\pi\)
0.552805 + 0.833311i \(0.313557\pi\)
\(810\) 0 0
\(811\) −3.03433e24 −0.569359 −0.284679 0.958623i \(-0.591887\pi\)
−0.284679 + 0.958623i \(0.591887\pi\)
\(812\) 0 0
\(813\) 6.16130e24 1.13215
\(814\) 0 0
\(815\) −2.40973e24 −0.433639
\(816\) 0 0
\(817\) −5.89719e24 −1.03934
\(818\) 0 0
\(819\) 1.66061e22 0.00286652
\(820\) 0 0
\(821\) 4.15299e24 0.702172 0.351086 0.936343i \(-0.385812\pi\)
0.351086 + 0.936343i \(0.385812\pi\)
\(822\) 0 0
\(823\) −9.22848e23 −0.152838 −0.0764191 0.997076i \(-0.524349\pi\)
−0.0764191 + 0.997076i \(0.524349\pi\)
\(824\) 0 0
\(825\) 2.70515e23 0.0438867
\(826\) 0 0
\(827\) −7.67636e24 −1.22000 −0.609998 0.792403i \(-0.708830\pi\)
−0.609998 + 0.792403i \(0.708830\pi\)
\(828\) 0 0
\(829\) 8.23009e23 0.128142 0.0640709 0.997945i \(-0.479592\pi\)
0.0640709 + 0.997945i \(0.479592\pi\)
\(830\) 0 0
\(831\) 2.26922e24 0.346153
\(832\) 0 0
\(833\) −8.69679e24 −1.29980
\(834\) 0 0
\(835\) −4.87209e24 −0.713478
\(836\) 0 0
\(837\) 1.26413e25 1.81395
\(838\) 0 0
\(839\) 9.57083e24 1.34578 0.672888 0.739744i \(-0.265053\pi\)
0.672888 + 0.739744i \(0.265053\pi\)
\(840\) 0 0
\(841\) −6.03171e23 −0.0831140
\(842\) 0 0
\(843\) −8.81851e24 −1.19086
\(844\) 0 0
\(845\) −3.37897e24 −0.447201
\(846\) 0 0
\(847\) −1.16669e25 −1.51337
\(848\) 0 0
\(849\) 3.42203e24 0.435079
\(850\) 0 0
\(851\) 5.44292e24 0.678314
\(852\) 0 0
\(853\) 2.48597e24 0.303690 0.151845 0.988404i \(-0.451479\pi\)
0.151845 + 0.988404i \(0.451479\pi\)
\(854\) 0 0
\(855\) 2.12150e24 0.254057
\(856\) 0 0
\(857\) −2.10495e24 −0.247118 −0.123559 0.992337i \(-0.539431\pi\)
−0.123559 + 0.992337i \(0.539431\pi\)
\(858\) 0 0
\(859\) 7.58565e23 0.0873074 0.0436537 0.999047i \(-0.486100\pi\)
0.0436537 + 0.999047i \(0.486100\pi\)
\(860\) 0 0
\(861\) 4.58725e24 0.517638
\(862\) 0 0
\(863\) 5.83299e24 0.645356 0.322678 0.946509i \(-0.395417\pi\)
0.322678 + 0.946509i \(0.395417\pi\)
\(864\) 0 0
\(865\) 6.42556e24 0.697067
\(866\) 0 0
\(867\) −2.96787e24 −0.315706
\(868\) 0 0
\(869\) −5.34980e23 −0.0558046
\(870\) 0 0
\(871\) −4.27137e22 −0.00436931
\(872\) 0 0
\(873\) −2.72717e24 −0.273584
\(874\) 0 0
\(875\) 1.48210e24 0.145817
\(876\) 0 0
\(877\) −1.53835e25 −1.48443 −0.742214 0.670163i \(-0.766224\pi\)
−0.742214 + 0.670163i \(0.766224\pi\)
\(878\) 0 0
\(879\) 1.12580e25 1.06551
\(880\) 0 0
\(881\) −1.57727e25 −1.46424 −0.732118 0.681178i \(-0.761468\pi\)
−0.732118 + 0.681178i \(0.761468\pi\)
\(882\) 0 0
\(883\) −1.75208e25 −1.59547 −0.797734 0.603009i \(-0.793968\pi\)
−0.797734 + 0.603009i \(0.793968\pi\)
\(884\) 0 0
\(885\) −6.64213e24 −0.593321
\(886\) 0 0
\(887\) 1.05988e25 0.928764 0.464382 0.885635i \(-0.346276\pi\)
0.464382 + 0.885635i \(0.346276\pi\)
\(888\) 0 0
\(889\) 1.59739e25 1.37324
\(890\) 0 0
\(891\) −1.78898e24 −0.150885
\(892\) 0 0
\(893\) 3.41494e25 2.82583
\(894\) 0 0
\(895\) 5.97918e23 0.0485451
\(896\) 0 0
\(897\) 3.80020e22 0.00302741
\(898\) 0 0
\(899\) 2.04106e25 1.59551
\(900\) 0 0
\(901\) −9.02107e24 −0.691986
\(902\) 0 0
\(903\) 1.06687e25 0.803089
\(904\) 0 0
\(905\) −3.32595e24 −0.245699
\(906\) 0 0
\(907\) −6.72762e24 −0.487753 −0.243876 0.969806i \(-0.578419\pi\)
−0.243876 + 0.969806i \(0.578419\pi\)
\(908\) 0 0
\(909\) 6.63712e24 0.472266
\(910\) 0 0
\(911\) −1.90734e25 −1.33205 −0.666027 0.745928i \(-0.732006\pi\)
−0.666027 + 0.745928i \(0.732006\pi\)
\(912\) 0 0
\(913\) 1.34080e24 0.0919100
\(914\) 0 0
\(915\) −5.39182e24 −0.362791
\(916\) 0 0
\(917\) −8.88503e24 −0.586840
\(918\) 0 0
\(919\) 1.73304e25 1.12364 0.561818 0.827261i \(-0.310102\pi\)
0.561818 + 0.827261i \(0.310102\pi\)
\(920\) 0 0
\(921\) 4.44096e24 0.282664
\(922\) 0 0
\(923\) −6.09580e22 −0.00380905
\(924\) 0 0
\(925\) −3.20264e24 −0.196473
\(926\) 0 0
\(927\) −9.89452e24 −0.595960
\(928\) 0 0
\(929\) −6.38514e24 −0.377605 −0.188802 0.982015i \(-0.560461\pi\)
−0.188802 + 0.982015i \(0.560461\pi\)
\(930\) 0 0
\(931\) −4.93587e25 −2.86610
\(932\) 0 0
\(933\) −3.26293e24 −0.186043
\(934\) 0 0
\(935\) −1.67710e24 −0.0938990
\(936\) 0 0
\(937\) −2.87864e25 −1.58271 −0.791354 0.611358i \(-0.790624\pi\)
−0.791354 + 0.611358i \(0.790624\pi\)
\(938\) 0 0
\(939\) 1.90654e25 1.02941
\(940\) 0 0
\(941\) −2.31234e25 −1.22614 −0.613071 0.790028i \(-0.710066\pi\)
−0.613071 + 0.790028i \(0.710066\pi\)
\(942\) 0 0
\(943\) −5.13781e24 −0.267564
\(944\) 0 0
\(945\) −1.55180e25 −0.793714
\(946\) 0 0
\(947\) −1.60999e25 −0.808816 −0.404408 0.914579i \(-0.632522\pi\)
−0.404408 + 0.914579i \(0.632522\pi\)
\(948\) 0 0
\(949\) 4.49761e22 0.00221932
\(950\) 0 0
\(951\) 1.22599e25 0.594225
\(952\) 0 0
\(953\) −1.12344e25 −0.534883 −0.267441 0.963574i \(-0.586178\pi\)
−0.267441 + 0.963574i \(0.586178\pi\)
\(954\) 0 0
\(955\) −4.13753e24 −0.193514
\(956\) 0 0
\(957\) −4.57311e24 −0.210117
\(958\) 0 0
\(959\) −3.07889e25 −1.38975
\(960\) 0 0
\(961\) 4.00580e25 1.77640
\(962\) 0 0
\(963\) −4.47259e24 −0.194866
\(964\) 0 0
\(965\) −6.08468e24 −0.260469
\(966\) 0 0
\(967\) −2.40854e24 −0.101305 −0.0506524 0.998716i \(-0.516130\pi\)
−0.0506524 + 0.998716i \(0.516130\pi\)
\(968\) 0 0
\(969\) 2.68736e25 1.11064
\(970\) 0 0
\(971\) 2.55647e25 1.03819 0.519095 0.854716i \(-0.326269\pi\)
0.519095 + 0.854716i \(0.326269\pi\)
\(972\) 0 0
\(973\) −4.85200e25 −1.93625
\(974\) 0 0
\(975\) −2.23606e22 −0.000876887 0
\(976\) 0 0
\(977\) −2.30301e25 −0.887547 −0.443774 0.896139i \(-0.646361\pi\)
−0.443774 + 0.896139i \(0.646361\pi\)
\(978\) 0 0
\(979\) 1.05090e25 0.398024
\(980\) 0 0
\(981\) −1.06893e25 −0.397889
\(982\) 0 0
\(983\) 2.04718e25 0.748949 0.374474 0.927237i \(-0.377823\pi\)
0.374474 + 0.927237i \(0.377823\pi\)
\(984\) 0 0
\(985\) 3.30984e24 0.119014
\(986\) 0 0
\(987\) −6.17801e25 −2.18350
\(988\) 0 0
\(989\) −1.19491e25 −0.415113
\(990\) 0 0
\(991\) −1.20622e25 −0.411907 −0.205954 0.978562i \(-0.566030\pi\)
−0.205954 + 0.978562i \(0.566030\pi\)
\(992\) 0 0
\(993\) −3.59297e25 −1.20610
\(994\) 0 0
\(995\) −3.70563e24 −0.122283
\(996\) 0 0
\(997\) −1.73789e23 −0.00563786 −0.00281893 0.999996i \(-0.500897\pi\)
−0.00281893 + 0.999996i \(0.500897\pi\)
\(998\) 0 0
\(999\) 3.35325e25 1.06945
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 80.18.a.d.1.2 2
4.3 odd 2 10.18.a.c.1.1 2
12.11 even 2 90.18.a.k.1.1 2
20.3 even 4 50.18.b.d.49.3 4
20.7 even 4 50.18.b.d.49.2 4
20.19 odd 2 50.18.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.18.a.c.1.1 2 4.3 odd 2
50.18.a.f.1.2 2 20.19 odd 2
50.18.b.d.49.2 4 20.7 even 4
50.18.b.d.49.3 4 20.3 even 4
80.18.a.d.1.2 2 1.1 even 1 trivial
90.18.a.k.1.1 2 12.11 even 2