Properties

Label 84.12.f.b.41.18
Level $84$
Weight $12$
Character 84.41
Analytic conductor $64.541$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,12,Mod(41,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.41");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.f (of order \(2\), degree \(1\), 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: \(64.5408271670\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 41.18
Character \(\chi\) \(=\) 84.41
Dual form 84.12.f.b.41.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(138.135 + 397.575i) q^{3} -8832.29 q^{5} +(16667.3 + 41225.3i) q^{7} +(-138985. + 109838. i) q^{9} +O(q^{10})\) \(q+(138.135 + 397.575i) q^{3} -8832.29 q^{5} +(16667.3 + 41225.3i) q^{7} +(-138985. + 109838. i) q^{9} +674199. i q^{11} +1.36330e6i q^{13} +(-1.22005e6 - 3.51150e6i) q^{15} +7.92109e6 q^{17} +8.91937e6i q^{19} +(-1.40878e7 + 1.23212e7i) q^{21} +2.25397e7i q^{23} +2.91813e7 q^{25} +(-6.28674e7 - 4.00844e7i) q^{27} -8.14998e7i q^{29} +2.35843e8i q^{31} +(-2.68045e8 + 9.31303e7i) q^{33} +(-1.47211e8 - 3.64114e8i) q^{35} -5.33115e8 q^{37} +(-5.42012e8 + 1.88319e8i) q^{39} -9.52106e8 q^{41} +1.43275e9 q^{43} +(1.22755e9 - 9.70120e8i) q^{45} +2.52108e9 q^{47} +(-1.42173e9 + 1.37423e9i) q^{49} +(1.09418e9 + 3.14923e9i) q^{51} -2.77421e9i q^{53} -5.95472e9i q^{55} +(-3.54612e9 + 1.23207e9i) q^{57} +6.78699e9 q^{59} +4.33580e8i q^{61} +(-6.84460e9 - 3.89898e9i) q^{63} -1.20410e10i q^{65} -1.56078e10 q^{67} +(-8.96121e9 + 3.11351e9i) q^{69} -6.69911e9i q^{71} -2.11852e10i q^{73} +(4.03095e9 + 1.16018e10i) q^{75} +(-2.77941e10 + 1.12371e10i) q^{77} -2.95109e10 q^{79} +(7.25237e9 - 3.05315e10i) q^{81} +5.39016e10 q^{83} -6.99614e10 q^{85} +(3.24023e10 - 1.12580e10i) q^{87} -3.45087e10 q^{89} +(-5.62023e10 + 2.27225e10i) q^{91} +(-9.37654e10 + 3.25782e10i) q^{93} -7.87785e10i q^{95} +2.62564e10i q^{97} +(-7.40525e10 - 9.37033e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 138.135 + 397.575i 0.328198 + 0.944609i
\(4\) 0 0
\(5\) −8832.29 −1.26398 −0.631988 0.774979i \(-0.717761\pi\)
−0.631988 + 0.774979i \(0.717761\pi\)
\(6\) 0 0
\(7\) 16667.3 + 41225.3i 0.374823 + 0.927096i
\(8\) 0 0
\(9\) −138985. + 109838.i −0.784572 + 0.620038i
\(10\) 0 0
\(11\) 674199.i 1.26220i 0.775701 + 0.631100i \(0.217396\pi\)
−0.775701 + 0.631100i \(0.782604\pi\)
\(12\) 0 0
\(13\) 1.36330e6i 1.01836i 0.860660 + 0.509180i \(0.170051\pi\)
−0.860660 + 0.509180i \(0.829949\pi\)
\(14\) 0 0
\(15\) −1.22005e6 3.51150e6i −0.414834 1.19396i
\(16\) 0 0
\(17\) 7.92109e6 1.35306 0.676529 0.736416i \(-0.263484\pi\)
0.676529 + 0.736416i \(0.263484\pi\)
\(18\) 0 0
\(19\) 8.91937e6i 0.826398i 0.910641 + 0.413199i \(0.135589\pi\)
−0.910641 + 0.413199i \(0.864411\pi\)
\(20\) 0 0
\(21\) −1.40878e7 + 1.23212e7i −0.752727 + 0.658333i
\(22\) 0 0
\(23\) 2.25397e7i 0.730204i 0.930967 + 0.365102i \(0.118966\pi\)
−0.930967 + 0.365102i \(0.881034\pi\)
\(24\) 0 0
\(25\) 2.91813e7 0.597633
\(26\) 0 0
\(27\) −6.28674e7 4.00844e7i −0.843188 0.537619i
\(28\) 0 0
\(29\) 8.14998e7i 0.737849i −0.929459 0.368925i \(-0.879726\pi\)
0.929459 0.368925i \(-0.120274\pi\)
\(30\) 0 0
\(31\) 2.35843e8i 1.47957i 0.672846 + 0.739783i \(0.265072\pi\)
−0.672846 + 0.739783i \(0.734928\pi\)
\(32\) 0 0
\(33\) −2.68045e8 + 9.31303e7i −1.19229 + 0.414252i
\(34\) 0 0
\(35\) −1.47211e8 3.64114e8i −0.473767 1.17183i
\(36\) 0 0
\(37\) −5.33115e8 −1.26390 −0.631948 0.775011i \(-0.717744\pi\)
−0.631948 + 0.775011i \(0.717744\pi\)
\(38\) 0 0
\(39\) −5.42012e8 + 1.88319e8i −0.961953 + 0.334224i
\(40\) 0 0
\(41\) −9.52106e8 −1.28344 −0.641718 0.766941i \(-0.721778\pi\)
−0.641718 + 0.766941i \(0.721778\pi\)
\(42\) 0 0
\(43\) 1.43275e9 1.48626 0.743130 0.669147i \(-0.233341\pi\)
0.743130 + 0.669147i \(0.233341\pi\)
\(44\) 0 0
\(45\) 1.22755e9 9.70120e8i 0.991680 0.783712i
\(46\) 0 0
\(47\) 2.52108e9 1.60342 0.801711 0.597712i \(-0.203924\pi\)
0.801711 + 0.597712i \(0.203924\pi\)
\(48\) 0 0
\(49\) −1.42173e9 + 1.37423e9i −0.719015 + 0.694995i
\(50\) 0 0
\(51\) 1.09418e9 + 3.14923e9i 0.444071 + 1.27811i
\(52\) 0 0
\(53\) 2.77421e9i 0.911217i −0.890180 0.455608i \(-0.849422\pi\)
0.890180 0.455608i \(-0.150578\pi\)
\(54\) 0 0
\(55\) 5.95472e9i 1.59539i
\(56\) 0 0
\(57\) −3.54612e9 + 1.23207e9i −0.780623 + 0.271222i
\(58\) 0 0
\(59\) 6.78699e9 1.23592 0.617961 0.786209i \(-0.287959\pi\)
0.617961 + 0.786209i \(0.287959\pi\)
\(60\) 0 0
\(61\) 4.33580e8i 0.0657287i 0.999460 + 0.0328644i \(0.0104629\pi\)
−0.999460 + 0.0328644i \(0.989537\pi\)
\(62\) 0 0
\(63\) −6.84460e9 3.89898e9i −0.868911 0.494969i
\(64\) 0 0
\(65\) 1.20410e10i 1.28718i
\(66\) 0 0
\(67\) −1.56078e10 −1.41231 −0.706157 0.708055i \(-0.749573\pi\)
−0.706157 + 0.708055i \(0.749573\pi\)
\(68\) 0 0
\(69\) −8.96121e9 + 3.11351e9i −0.689758 + 0.239652i
\(70\) 0 0
\(71\) 6.69911e9i 0.440653i −0.975426 0.220326i \(-0.929288\pi\)
0.975426 0.220326i \(-0.0707122\pi\)
\(72\) 0 0
\(73\) 2.11852e10i 1.19607i −0.801471 0.598034i \(-0.795949\pi\)
0.801471 0.598034i \(-0.204051\pi\)
\(74\) 0 0
\(75\) 4.03095e9 + 1.16018e10i 0.196142 + 0.564530i
\(76\) 0 0
\(77\) −2.77941e10 + 1.12371e10i −1.17018 + 0.473102i
\(78\) 0 0
\(79\) −2.95109e10 −1.07903 −0.539515 0.841976i \(-0.681392\pi\)
−0.539515 + 0.841976i \(0.681392\pi\)
\(80\) 0 0
\(81\) 7.25237e9 3.05315e10i 0.231107 0.972928i
\(82\) 0 0
\(83\) 5.39016e10 1.50201 0.751004 0.660297i \(-0.229570\pi\)
0.751004 + 0.660297i \(0.229570\pi\)
\(84\) 0 0
\(85\) −6.99614e10 −1.71023
\(86\) 0 0
\(87\) 3.24023e10 1.12580e10i 0.696979 0.242161i
\(88\) 0 0
\(89\) −3.45087e10 −0.655064 −0.327532 0.944840i \(-0.606217\pi\)
−0.327532 + 0.944840i \(0.606217\pi\)
\(90\) 0 0
\(91\) −5.62023e10 + 2.27225e10i −0.944118 + 0.381705i
\(92\) 0 0
\(93\) −9.37654e10 + 3.25782e10i −1.39761 + 0.485591i
\(94\) 0 0
\(95\) 7.87785e10i 1.04455i
\(96\) 0 0
\(97\) 2.62564e10i 0.310450i 0.987879 + 0.155225i \(0.0496102\pi\)
−0.987879 + 0.155225i \(0.950390\pi\)
\(98\) 0 0
\(99\) −7.40525e10 9.37033e10i −0.782612 0.990287i
\(100\) 0 0
\(101\) −5.61313e10 −0.531419 −0.265710 0.964053i \(-0.585606\pi\)
−0.265710 + 0.964053i \(0.585606\pi\)
\(102\) 0 0
\(103\) 9.75174e10i 0.828853i −0.910083 0.414426i \(-0.863982\pi\)
0.910083 0.414426i \(-0.136018\pi\)
\(104\) 0 0
\(105\) 1.24428e11 1.08824e11i 0.951428 0.832116i
\(106\) 0 0
\(107\) 2.02351e11i 1.39474i 0.716711 + 0.697370i \(0.245647\pi\)
−0.716711 + 0.697370i \(0.754353\pi\)
\(108\) 0 0
\(109\) −6.00804e8 −0.00374014 −0.00187007 0.999998i \(-0.500595\pi\)
−0.00187007 + 0.999998i \(0.500595\pi\)
\(110\) 0 0
\(111\) −7.36417e10 2.11953e11i −0.414808 1.19389i
\(112\) 0 0
\(113\) 2.17953e11i 1.11284i −0.830902 0.556418i \(-0.812175\pi\)
0.830902 0.556418i \(-0.187825\pi\)
\(114\) 0 0
\(115\) 1.99077e11i 0.922960i
\(116\) 0 0
\(117\) −1.49741e11 1.89477e11i −0.631422 0.798977i
\(118\) 0 0
\(119\) 1.32023e11 + 3.26550e11i 0.507157 + 1.25441i
\(120\) 0 0
\(121\) −1.69233e11 −0.593150
\(122\) 0 0
\(123\) −1.31519e11 3.78533e11i −0.421221 1.21234i
\(124\) 0 0
\(125\) 1.73527e11 0.508582
\(126\) 0 0
\(127\) −5.36248e10 −0.144027 −0.0720137 0.997404i \(-0.522943\pi\)
−0.0720137 + 0.997404i \(0.522943\pi\)
\(128\) 0 0
\(129\) 1.97913e11 + 5.69626e11i 0.487787 + 1.40393i
\(130\) 0 0
\(131\) 3.16692e11 0.717207 0.358604 0.933490i \(-0.383253\pi\)
0.358604 + 0.933490i \(0.383253\pi\)
\(132\) 0 0
\(133\) −3.67704e11 + 1.48662e11i −0.766151 + 0.309753i
\(134\) 0 0
\(135\) 5.55263e11 + 3.54037e11i 1.06577 + 0.679537i
\(136\) 0 0
\(137\) 8.47016e11i 1.49944i −0.661756 0.749719i \(-0.730188\pi\)
0.661756 0.749719i \(-0.269812\pi\)
\(138\) 0 0
\(139\) 2.35441e11i 0.384858i 0.981311 + 0.192429i \(0.0616366\pi\)
−0.981311 + 0.192429i \(0.938363\pi\)
\(140\) 0 0
\(141\) 3.48248e11 + 1.00232e12i 0.526240 + 1.51461i
\(142\) 0 0
\(143\) −9.19133e11 −1.28538
\(144\) 0 0
\(145\) 7.19830e11i 0.932623i
\(146\) 0 0
\(147\) −7.42750e11 3.75414e11i −0.892477 0.451092i
\(148\) 0 0
\(149\) 1.08981e12i 1.21570i 0.794051 + 0.607851i \(0.207968\pi\)
−0.794051 + 0.607851i \(0.792032\pi\)
\(150\) 0 0
\(151\) 9.84048e11 1.02010 0.510051 0.860144i \(-0.329627\pi\)
0.510051 + 0.860144i \(0.329627\pi\)
\(152\) 0 0
\(153\) −1.10091e12 + 8.70036e11i −1.06157 + 0.838946i
\(154\) 0 0
\(155\) 2.08304e12i 1.87013i
\(156\) 0 0
\(157\) 4.98975e11i 0.417475i 0.977972 + 0.208738i \(0.0669355\pi\)
−0.977972 + 0.208738i \(0.933065\pi\)
\(158\) 0 0
\(159\) 1.10296e12 3.83214e11i 0.860743 0.299060i
\(160\) 0 0
\(161\) −9.29205e11 + 3.75676e11i −0.676970 + 0.273698i
\(162\) 0 0
\(163\) 2.16163e12 1.47147 0.735733 0.677272i \(-0.236838\pi\)
0.735733 + 0.677272i \(0.236838\pi\)
\(164\) 0 0
\(165\) 2.36745e12 8.22554e11i 1.50702 0.523604i
\(166\) 0 0
\(167\) 1.47075e11 0.0876192 0.0438096 0.999040i \(-0.486050\pi\)
0.0438096 + 0.999040i \(0.486050\pi\)
\(168\) 0 0
\(169\) −6.64150e10 −0.0370586
\(170\) 0 0
\(171\) −9.79684e11 1.23965e12i −0.512398 0.648369i
\(172\) 0 0
\(173\) −6.88452e10 −0.0337769 −0.0168885 0.999857i \(-0.505376\pi\)
−0.0168885 + 0.999857i \(0.505376\pi\)
\(174\) 0 0
\(175\) 4.86374e11 + 1.20301e12i 0.224007 + 0.554063i
\(176\) 0 0
\(177\) 9.37519e11 + 2.69834e12i 0.405627 + 1.16746i
\(178\) 0 0
\(179\) 4.21115e12i 1.71281i 0.516304 + 0.856405i \(0.327307\pi\)
−0.516304 + 0.856405i \(0.672693\pi\)
\(180\) 0 0
\(181\) 5.67474e11i 0.217127i −0.994090 0.108564i \(-0.965375\pi\)
0.994090 0.108564i \(-0.0346251\pi\)
\(182\) 0 0
\(183\) −1.72381e11 + 5.98925e10i −0.0620879 + 0.0215720i
\(184\) 0 0
\(185\) 4.70863e12 1.59753
\(186\) 0 0
\(187\) 5.34039e12i 1.70783i
\(188\) 0 0
\(189\) 6.04660e11 3.25983e12i 0.182378 0.983229i
\(190\) 0 0
\(191\) 4.11700e12i 1.17192i 0.810341 + 0.585959i \(0.199282\pi\)
−0.810341 + 0.585959i \(0.800718\pi\)
\(192\) 0 0
\(193\) 3.29033e12 0.884452 0.442226 0.896904i \(-0.354189\pi\)
0.442226 + 0.896904i \(0.354189\pi\)
\(194\) 0 0
\(195\) 4.78721e12 1.66328e12i 1.21588 0.422451i
\(196\) 0 0
\(197\) 3.20172e12i 0.768810i −0.923165 0.384405i \(-0.874407\pi\)
0.923165 0.384405i \(-0.125593\pi\)
\(198\) 0 0
\(199\) 6.75865e11i 0.153521i 0.997050 + 0.0767605i \(0.0244577\pi\)
−0.997050 + 0.0767605i \(0.975542\pi\)
\(200\) 0 0
\(201\) −2.15598e12 6.20528e12i −0.463519 1.33408i
\(202\) 0 0
\(203\) 3.35986e12 1.35838e12i 0.684057 0.276563i
\(204\) 0 0
\(205\) 8.40928e12 1.62223
\(206\) 0 0
\(207\) −2.47571e12 3.13267e12i −0.452754 0.572898i
\(208\) 0 0
\(209\) −6.01343e12 −1.04308
\(210\) 0 0
\(211\) −8.46217e12 −1.39293 −0.696464 0.717592i \(-0.745244\pi\)
−0.696464 + 0.717592i \(0.745244\pi\)
\(212\) 0 0
\(213\) 2.66340e12 9.25380e11i 0.416245 0.144621i
\(214\) 0 0
\(215\) −1.26545e13 −1.87859
\(216\) 0 0
\(217\) −9.72272e12 + 3.93088e12i −1.37170 + 0.554576i
\(218\) 0 0
\(219\) 8.42269e12 2.92641e12i 1.12982 0.392547i
\(220\) 0 0
\(221\) 1.07988e13i 1.37790i
\(222\) 0 0
\(223\) 9.39037e12i 1.14027i 0.821552 + 0.570133i \(0.193108\pi\)
−0.821552 + 0.570133i \(0.806892\pi\)
\(224\) 0 0
\(225\) −4.05575e12 + 3.20521e12i −0.468886 + 0.370555i
\(226\) 0 0
\(227\) −5.62913e12 −0.619867 −0.309934 0.950758i \(-0.600307\pi\)
−0.309934 + 0.950758i \(0.600307\pi\)
\(228\) 0 0
\(229\) 8.24055e10i 0.00864691i 0.999991 + 0.00432346i \(0.00137620\pi\)
−0.999991 + 0.00432346i \(0.998624\pi\)
\(230\) 0 0
\(231\) −8.30691e12 9.49799e12i −0.830948 0.950092i
\(232\) 0 0
\(233\) 1.34554e13i 1.28363i −0.766860 0.641814i \(-0.778182\pi\)
0.766860 0.641814i \(-0.221818\pi\)
\(234\) 0 0
\(235\) −2.22669e13 −2.02668
\(236\) 0 0
\(237\) −4.07648e12 1.17328e13i −0.354135 1.01926i
\(238\) 0 0
\(239\) 4.06540e12i 0.337221i −0.985683 0.168611i \(-0.946072\pi\)
0.985683 0.168611i \(-0.0539281\pi\)
\(240\) 0 0
\(241\) 5.71196e12i 0.452576i 0.974060 + 0.226288i \(0.0726590\pi\)
−0.974060 + 0.226288i \(0.927341\pi\)
\(242\) 0 0
\(243\) 1.31404e13 1.33410e12i 0.994886 0.101008i
\(244\) 0 0
\(245\) 1.25571e13 1.21376e13i 0.908817 0.878456i
\(246\) 0 0
\(247\) −1.21597e13 −0.841571
\(248\) 0 0
\(249\) 7.44569e12 + 2.14299e13i 0.492956 + 1.41881i
\(250\) 0 0
\(251\) 2.51525e13 1.59359 0.796794 0.604251i \(-0.206528\pi\)
0.796794 + 0.604251i \(0.206528\pi\)
\(252\) 0 0
\(253\) −1.51962e13 −0.921664
\(254\) 0 0
\(255\) −9.66411e12 2.78149e13i −0.561294 1.61550i
\(256\) 0 0
\(257\) 1.79007e13 0.995949 0.497974 0.867192i \(-0.334077\pi\)
0.497974 + 0.867192i \(0.334077\pi\)
\(258\) 0 0
\(259\) −8.88560e12 2.19778e13i −0.473738 1.17175i
\(260\) 0 0
\(261\) 8.95176e12 + 1.13272e13i 0.457494 + 0.578896i
\(262\) 0 0
\(263\) 3.77086e12i 0.184792i −0.995722 0.0923960i \(-0.970547\pi\)
0.995722 0.0923960i \(-0.0294526\pi\)
\(264\) 0 0
\(265\) 2.45026e13i 1.15175i
\(266\) 0 0
\(267\) −4.76685e12 1.37198e13i −0.214991 0.618779i
\(268\) 0 0
\(269\) −3.84201e12 −0.166311 −0.0831555 0.996537i \(-0.526500\pi\)
−0.0831555 + 0.996537i \(0.526500\pi\)
\(270\) 0 0
\(271\) 8.19916e12i 0.340752i −0.985379 0.170376i \(-0.945502\pi\)
0.985379 0.170376i \(-0.0544982\pi\)
\(272\) 0 0
\(273\) −1.67974e13 1.92059e13i −0.670420 0.766548i
\(274\) 0 0
\(275\) 1.96740e13i 0.754333i
\(276\) 0 0
\(277\) −5.31900e12 −0.195971 −0.0979854 0.995188i \(-0.531240\pi\)
−0.0979854 + 0.995188i \(0.531240\pi\)
\(278\) 0 0
\(279\) −2.59045e13 3.27786e13i −0.917387 1.16083i
\(280\) 0 0
\(281\) 1.34290e13i 0.457257i −0.973514 0.228628i \(-0.926576\pi\)
0.973514 0.228628i \(-0.0734241\pi\)
\(282\) 0 0
\(283\) 4.84296e13i 1.58593i 0.609264 + 0.792967i \(0.291465\pi\)
−0.609264 + 0.792967i \(0.708535\pi\)
\(284\) 0 0
\(285\) 3.13204e13 1.08820e13i 0.986688 0.342818i
\(286\) 0 0
\(287\) −1.58691e13 3.92509e13i −0.481062 1.18987i
\(288\) 0 0
\(289\) 2.84718e13 0.830763
\(290\) 0 0
\(291\) −1.04389e13 + 3.62693e12i −0.293254 + 0.101889i
\(292\) 0 0
\(293\) −2.71164e13 −0.733601 −0.366801 0.930300i \(-0.619547\pi\)
−0.366801 + 0.930300i \(0.619547\pi\)
\(294\) 0 0
\(295\) −5.99447e13 −1.56217
\(296\) 0 0
\(297\) 2.70248e13 4.23851e13i 0.678582 1.06427i
\(298\) 0 0
\(299\) −3.07282e13 −0.743612
\(300\) 0 0
\(301\) 2.38801e13 + 5.90657e13i 0.557085 + 1.37791i
\(302\) 0 0
\(303\) −7.75368e12 2.23164e13i −0.174411 0.501983i
\(304\) 0 0
\(305\) 3.82951e12i 0.0830795i
\(306\) 0 0
\(307\) 2.27973e13i 0.477115i 0.971128 + 0.238558i \(0.0766746\pi\)
−0.971128 + 0.238558i \(0.923325\pi\)
\(308\) 0 0
\(309\) 3.87705e13 1.34705e13i 0.782942 0.272028i
\(310\) 0 0
\(311\) −1.91597e12 −0.0373429 −0.0186714 0.999826i \(-0.505944\pi\)
−0.0186714 + 0.999826i \(0.505944\pi\)
\(312\) 0 0
\(313\) 6.23953e13i 1.17397i 0.809597 + 0.586987i \(0.199686\pi\)
−0.809597 + 0.586987i \(0.800314\pi\)
\(314\) 0 0
\(315\) 6.04535e13 + 3.44370e13i 1.09828 + 0.625629i
\(316\) 0 0
\(317\) 2.72235e13i 0.477659i 0.971061 + 0.238830i \(0.0767638\pi\)
−0.971061 + 0.238830i \(0.923236\pi\)
\(318\) 0 0
\(319\) 5.49471e13 0.931314
\(320\) 0 0
\(321\) −8.04495e13 + 2.79516e13i −1.31748 + 0.457751i
\(322\) 0 0
\(323\) 7.06512e13i 1.11816i
\(324\) 0 0
\(325\) 3.97827e13i 0.608606i
\(326\) 0 0
\(327\) −8.29919e10 2.38865e11i −0.00122751 0.00353296i
\(328\) 0 0
\(329\) 4.20196e13 + 1.03932e14i 0.601000 + 1.48653i
\(330\) 0 0
\(331\) −6.74470e13 −0.933058 −0.466529 0.884506i \(-0.654496\pi\)
−0.466529 + 0.884506i \(0.654496\pi\)
\(332\) 0 0
\(333\) 7.40948e13 5.85562e13i 0.991618 0.783663i
\(334\) 0 0
\(335\) 1.37853e14 1.78513
\(336\) 0 0
\(337\) 1.46005e14 1.82980 0.914900 0.403681i \(-0.132270\pi\)
0.914900 + 0.403681i \(0.132270\pi\)
\(338\) 0 0
\(339\) 8.66526e13 3.01069e13i 1.05119 0.365231i
\(340\) 0 0
\(341\) −1.59005e14 −1.86751
\(342\) 0 0
\(343\) −8.03495e13 3.57064e13i −0.913831 0.406096i
\(344\) 0 0
\(345\) 7.91480e13 2.74995e13i 0.871836 0.302914i
\(346\) 0 0
\(347\) 1.14206e14i 1.21864i −0.792924 0.609320i \(-0.791442\pi\)
0.792924 0.609320i \(-0.208558\pi\)
\(348\) 0 0
\(349\) 1.28402e14i 1.32750i 0.747956 + 0.663748i \(0.231035\pi\)
−0.747956 + 0.663748i \(0.768965\pi\)
\(350\) 0 0
\(351\) 5.46468e13 8.57068e13i 0.547490 0.858670i
\(352\) 0 0
\(353\) −1.05142e14 −1.02097 −0.510486 0.859886i \(-0.670535\pi\)
−0.510486 + 0.859886i \(0.670535\pi\)
\(354\) 0 0
\(355\) 5.91686e13i 0.556974i
\(356\) 0 0
\(357\) −1.11591e14 + 9.75971e13i −1.01848 + 0.890762i
\(358\) 0 0
\(359\) 7.42230e13i 0.656930i 0.944516 + 0.328465i \(0.106531\pi\)
−0.944516 + 0.328465i \(0.893469\pi\)
\(360\) 0 0
\(361\) 3.69351e13 0.317066
\(362\) 0 0
\(363\) −2.33769e13 6.72826e13i −0.194671 0.560295i
\(364\) 0 0
\(365\) 1.87114e14i 1.51180i
\(366\) 0 0
\(367\) 1.96666e14i 1.54193i −0.636876 0.770966i \(-0.719774\pi\)
0.636876 0.770966i \(-0.280226\pi\)
\(368\) 0 0
\(369\) 1.32328e14 1.04577e14i 1.00695 0.795779i
\(370\) 0 0
\(371\) 1.14368e14 4.62386e13i 0.844785 0.341545i
\(372\) 0 0
\(373\) −1.45118e13 −0.104069 −0.0520347 0.998645i \(-0.516571\pi\)
−0.0520347 + 0.998645i \(0.516571\pi\)
\(374\) 0 0
\(375\) 2.39700e13 + 6.89898e13i 0.166916 + 0.480411i
\(376\) 0 0
\(377\) 1.11108e14 0.751397
\(378\) 0 0
\(379\) 1.98737e14 1.30546 0.652728 0.757592i \(-0.273624\pi\)
0.652728 + 0.757592i \(0.273624\pi\)
\(380\) 0 0
\(381\) −7.40745e12 2.13199e13i −0.0472695 0.136050i
\(382\) 0 0
\(383\) −2.14817e14 −1.33191 −0.665956 0.745991i \(-0.731976\pi\)
−0.665956 + 0.745991i \(0.731976\pi\)
\(384\) 0 0
\(385\) 2.45485e14 9.92493e13i 1.47908 0.597989i
\(386\) 0 0
\(387\) −1.99130e14 + 1.57370e14i −1.16608 + 0.921537i
\(388\) 0 0
\(389\) 2.38756e14i 1.35904i −0.733658 0.679519i \(-0.762189\pi\)
0.733658 0.679519i \(-0.237811\pi\)
\(390\) 0 0
\(391\) 1.78539e14i 0.988008i
\(392\) 0 0
\(393\) 4.37461e13 + 1.25909e14i 0.235386 + 0.677481i
\(394\) 0 0
\(395\) 2.60649e14 1.36387
\(396\) 0 0
\(397\) 2.22230e14i 1.13098i 0.824756 + 0.565489i \(0.191313\pi\)
−0.824756 + 0.565489i \(0.808687\pi\)
\(398\) 0 0
\(399\) −1.09897e14 1.25654e14i −0.544045 0.622052i
\(400\) 0 0
\(401\) 1.83901e14i 0.885707i −0.896594 0.442853i \(-0.853966\pi\)
0.896594 0.442853i \(-0.146034\pi\)
\(402\) 0 0
\(403\) −3.21524e14 −1.50673
\(404\) 0 0
\(405\) −6.40551e13 + 2.69663e14i −0.292113 + 1.22976i
\(406\) 0 0
\(407\) 3.59426e14i 1.59529i
\(408\) 0 0
\(409\) 2.53752e13i 0.109631i −0.998497 0.0548153i \(-0.982543\pi\)
0.998497 0.0548153i \(-0.0174570\pi\)
\(410\) 0 0
\(411\) 3.36752e14 1.17002e14i 1.41638 0.492113i
\(412\) 0 0
\(413\) 1.13121e14 + 2.79796e14i 0.463252 + 1.14582i
\(414\) 0 0
\(415\) −4.76075e14 −1.89850
\(416\) 0 0
\(417\) −9.36055e13 + 3.25226e13i −0.363541 + 0.126310i
\(418\) 0 0
\(419\) −4.54359e14 −1.71879 −0.859394 0.511314i \(-0.829159\pi\)
−0.859394 + 0.511314i \(0.829159\pi\)
\(420\) 0 0
\(421\) −2.91306e14 −1.07349 −0.536745 0.843745i \(-0.680346\pi\)
−0.536745 + 0.843745i \(0.680346\pi\)
\(422\) 0 0
\(423\) −3.50391e14 + 2.76909e14i −1.25800 + 0.994182i
\(424\) 0 0
\(425\) 2.31148e14 0.808632
\(426\) 0 0
\(427\) −1.78745e13 + 7.22662e12i −0.0609368 + 0.0246367i
\(428\) 0 0
\(429\) −1.26964e14 3.65424e14i −0.421858 1.21418i
\(430\) 0 0
\(431\) 2.00039e14i 0.647872i 0.946079 + 0.323936i \(0.105006\pi\)
−0.946079 + 0.323936i \(0.894994\pi\)
\(432\) 0 0
\(433\) 2.71904e14i 0.858483i −0.903190 0.429241i \(-0.858781\pi\)
0.903190 0.429241i \(-0.141219\pi\)
\(434\) 0 0
\(435\) −2.86186e14 + 9.94336e13i −0.880964 + 0.306085i
\(436\) 0 0
\(437\) −2.01040e14 −0.603440
\(438\) 0 0
\(439\) 2.27715e14i 0.666556i −0.942829 0.333278i \(-0.891845\pi\)
0.942829 0.333278i \(-0.108155\pi\)
\(440\) 0 0
\(441\) 4.66556e13 3.47156e14i 0.133196 0.991090i
\(442\) 0 0
\(443\) 6.67383e14i 1.85846i −0.369496 0.929232i \(-0.620470\pi\)
0.369496 0.929232i \(-0.379530\pi\)
\(444\) 0 0
\(445\) 3.04791e14 0.827985
\(446\) 0 0
\(447\) −4.33282e14 + 1.50541e14i −1.14836 + 0.398991i
\(448\) 0 0
\(449\) 3.91260e14i 1.01184i 0.862581 + 0.505918i \(0.168846\pi\)
−0.862581 + 0.505918i \(0.831154\pi\)
\(450\) 0 0
\(451\) 6.41909e14i 1.61995i
\(452\) 0 0
\(453\) 1.35931e14 + 3.91233e14i 0.334795 + 0.963597i
\(454\) 0 0
\(455\) 4.96395e14 2.00692e14i 1.19334 0.482466i
\(456\) 0 0
\(457\) 1.23255e14 0.289244 0.144622 0.989487i \(-0.453803\pi\)
0.144622 + 0.989487i \(0.453803\pi\)
\(458\) 0 0
\(459\) −4.97978e14 3.17512e14i −1.14088 0.727429i
\(460\) 0 0
\(461\) 1.91725e14 0.428868 0.214434 0.976738i \(-0.431209\pi\)
0.214434 + 0.976738i \(0.431209\pi\)
\(462\) 0 0
\(463\) −1.46165e14 −0.319262 −0.159631 0.987177i \(-0.551030\pi\)
−0.159631 + 0.987177i \(0.551030\pi\)
\(464\) 0 0
\(465\) 8.28164e14 2.87740e14i 1.76655 0.613775i
\(466\) 0 0
\(467\) −5.34962e14 −1.11450 −0.557250 0.830345i \(-0.688143\pi\)
−0.557250 + 0.830345i \(0.688143\pi\)
\(468\) 0 0
\(469\) −2.60141e14 6.43438e14i −0.529368 1.30935i
\(470\) 0 0
\(471\) −1.98380e14 + 6.89258e13i −0.394351 + 0.137015i
\(472\) 0 0
\(473\) 9.65960e14i 1.87596i
\(474\) 0 0
\(475\) 2.60279e14i 0.493883i
\(476\) 0 0
\(477\) 3.04713e14 + 3.85572e14i 0.564989 + 0.714915i
\(478\) 0 0
\(479\) 2.74443e14 0.497287 0.248643 0.968595i \(-0.420015\pi\)
0.248643 + 0.968595i \(0.420015\pi\)
\(480\) 0 0
\(481\) 7.26794e14i 1.28710i
\(482\) 0 0
\(483\) −2.77715e14 3.17535e14i −0.480717 0.549645i
\(484\) 0 0
\(485\) 2.31905e14i 0.392401i
\(486\) 0 0
\(487\) −5.12082e14 −0.847092 −0.423546 0.905875i \(-0.639215\pi\)
−0.423546 + 0.905875i \(0.639215\pi\)
\(488\) 0 0
\(489\) 2.98597e14 + 8.59411e14i 0.482932 + 1.38996i
\(490\) 0 0
\(491\) 2.16539e14i 0.342443i −0.985233 0.171221i \(-0.945229\pi\)
0.985233 0.171221i \(-0.0547713\pi\)
\(492\) 0 0
\(493\) 6.45568e14i 0.998352i
\(494\) 0 0
\(495\) 6.54054e14 + 8.27615e14i 0.989202 + 1.25170i
\(496\) 0 0
\(497\) 2.76173e14 1.11656e14i 0.408527 0.165167i
\(498\) 0 0
\(499\) 2.26752e14 0.328094 0.164047 0.986453i \(-0.447545\pi\)
0.164047 + 0.986453i \(0.447545\pi\)
\(500\) 0 0
\(501\) 2.03162e13 + 5.84735e13i 0.0287565 + 0.0827659i
\(502\) 0 0
\(503\) 5.99152e14 0.829685 0.414843 0.909893i \(-0.363837\pi\)
0.414843 + 0.909893i \(0.363837\pi\)
\(504\) 0 0
\(505\) 4.95768e14 0.671701
\(506\) 0 0
\(507\) −9.17422e12 2.64049e13i −0.0121626 0.0350059i
\(508\) 0 0
\(509\) −1.37350e15 −1.78188 −0.890942 0.454117i \(-0.849955\pi\)
−0.890942 + 0.454117i \(0.849955\pi\)
\(510\) 0 0
\(511\) 8.73365e14 3.53100e14i 1.10887 0.448314i
\(512\) 0 0
\(513\) 3.57527e14 5.60737e14i 0.444287 0.696809i
\(514\) 0 0
\(515\) 8.61303e14i 1.04765i
\(516\) 0 0
\(517\) 1.69971e15i 2.02384i
\(518\) 0 0
\(519\) −9.50991e12 2.73711e13i −0.0110855 0.0319060i
\(520\) 0 0
\(521\) −4.11317e14 −0.469428 −0.234714 0.972064i \(-0.575415\pi\)
−0.234714 + 0.972064i \(0.575415\pi\)
\(522\) 0 0
\(523\) 6.15252e13i 0.0687534i 0.999409 + 0.0343767i \(0.0109446\pi\)
−0.999409 + 0.0343767i \(0.989055\pi\)
\(524\) 0 0
\(525\) −4.11101e14 + 3.59547e14i −0.449855 + 0.393441i
\(526\) 0 0
\(527\) 1.86814e15i 2.00194i
\(528\) 0 0
\(529\) 4.44773e14 0.466802
\(530\) 0 0
\(531\) −9.43286e14 + 7.45468e14i −0.969669 + 0.766318i
\(532\) 0 0
\(533\) 1.29800e15i 1.30700i
\(534\) 0 0
\(535\) 1.78722e15i 1.76292i
\(536\) 0 0
\(537\) −1.67425e15 + 5.81707e14i −1.61794 + 0.562141i
\(538\) 0 0
\(539\) −9.26506e14 9.58527e14i −0.877223 0.907541i
\(540\) 0 0
\(541\) 1.90399e15 1.76637 0.883183 0.469029i \(-0.155396\pi\)
0.883183 + 0.469029i \(0.155396\pi\)
\(542\) 0 0
\(543\) 2.25614e14 7.83879e13i 0.205100 0.0712607i
\(544\) 0 0
\(545\) 5.30648e12 0.00472744
\(546\) 0 0
\(547\) 1.33498e15 1.16559 0.582794 0.812620i \(-0.301959\pi\)
0.582794 + 0.812620i \(0.301959\pi\)
\(548\) 0 0
\(549\) −4.76235e13 6.02609e13i −0.0407543 0.0515689i
\(550\) 0 0
\(551\) 7.26927e14 0.609757
\(552\) 0 0
\(553\) −4.91867e14 1.21660e15i −0.404445 1.00036i
\(554\) 0 0
\(555\) 6.50425e14 + 1.87203e15i 0.524307 + 1.50904i
\(556\) 0 0
\(557\) 1.32763e15i 1.04924i −0.851337 0.524619i \(-0.824208\pi\)
0.851337 0.524619i \(-0.175792\pi\)
\(558\) 0 0
\(559\) 1.95327e15i 1.51355i
\(560\) 0 0
\(561\) −2.12321e15 + 7.37694e14i −1.61323 + 0.560506i
\(562\) 0 0
\(563\) 1.06094e15 0.790486 0.395243 0.918577i \(-0.370660\pi\)
0.395243 + 0.918577i \(0.370660\pi\)
\(564\) 0 0
\(565\) 1.92502e15i 1.40660i
\(566\) 0 0
\(567\) 1.37955e15 2.09898e14i 0.988622 0.150418i
\(568\) 0 0
\(569\) 1.33450e15i 0.937994i −0.883200 0.468997i \(-0.844616\pi\)
0.883200 0.468997i \(-0.155384\pi\)
\(570\) 0 0
\(571\) 1.90913e15 1.31624 0.658122 0.752912i \(-0.271351\pi\)
0.658122 + 0.752912i \(0.271351\pi\)
\(572\) 0 0
\(573\) −1.63682e15 + 5.68700e14i −1.10700 + 0.384621i
\(574\) 0 0
\(575\) 6.57737e14i 0.436394i
\(576\) 0 0
\(577\) 9.11507e13i 0.0593325i −0.999560 0.0296663i \(-0.990556\pi\)
0.999560 0.0296663i \(-0.00944445\pi\)
\(578\) 0 0
\(579\) 4.54509e14 + 1.30815e15i 0.290276 + 0.835462i
\(580\) 0 0
\(581\) 8.98396e14 + 2.22211e15i 0.562988 + 1.39251i
\(582\) 0 0
\(583\) 1.87037e15 1.15014
\(584\) 0 0
\(585\) 1.32256e15 + 1.67352e15i 0.798102 + 1.00989i
\(586\) 0 0
\(587\) −2.05609e15 −1.21768 −0.608840 0.793293i \(-0.708365\pi\)
−0.608840 + 0.793293i \(0.708365\pi\)
\(588\) 0 0
\(589\) −2.10357e15 −1.22271
\(590\) 0 0
\(591\) 1.27292e15 4.42268e14i 0.726224 0.252322i
\(592\) 0 0
\(593\) 2.10809e15 1.18056 0.590280 0.807198i \(-0.299017\pi\)
0.590280 + 0.807198i \(0.299017\pi\)
\(594\) 0 0
\(595\) −1.16607e15 2.88418e15i −0.641034 1.58555i
\(596\) 0 0
\(597\) −2.68707e14 + 9.33604e13i −0.145017 + 0.0503853i
\(598\) 0 0
\(599\) 1.09829e15i 0.581930i −0.956734 0.290965i \(-0.906024\pi\)
0.956734 0.290965i \(-0.0939763\pi\)
\(600\) 0 0
\(601\) 7.20536e14i 0.374840i −0.982280 0.187420i \(-0.939987\pi\)
0.982280 0.187420i \(-0.0600126\pi\)
\(602\) 0 0
\(603\) 2.16925e15 1.71433e15i 1.10806 0.875688i
\(604\) 0 0
\(605\) 1.49471e15 0.749727
\(606\) 0 0
\(607\) 3.41473e15i 1.68197i −0.541058 0.840985i \(-0.681976\pi\)
0.541058 0.840985i \(-0.318024\pi\)
\(608\) 0 0
\(609\) 1.00417e15 + 1.14815e15i 0.485750 + 0.555399i
\(610\) 0 0
\(611\) 3.43697e15i 1.63286i
\(612\) 0 0
\(613\) 2.09343e15 0.976847 0.488424 0.872607i \(-0.337572\pi\)
0.488424 + 0.872607i \(0.337572\pi\)
\(614\) 0 0
\(615\) 1.16161e15 + 3.34332e15i 0.532413 + 1.53237i
\(616\) 0 0
\(617\) 2.31713e14i 0.104324i −0.998639 0.0521618i \(-0.983389\pi\)
0.998639 0.0521618i \(-0.0166112\pi\)
\(618\) 0 0
\(619\) 7.77570e14i 0.343907i −0.985105 0.171954i \(-0.944992\pi\)
0.985105 0.171954i \(-0.0550079\pi\)
\(620\) 0 0
\(621\) 9.03488e14 1.41701e15i 0.392571 0.615700i
\(622\) 0 0
\(623\) −5.75168e14 1.42263e15i −0.245533 0.607307i
\(624\) 0 0
\(625\) −2.95751e15 −1.24047
\(626\) 0 0
\(627\) −8.30664e14 2.39079e15i −0.342337 0.985303i
\(628\) 0 0
\(629\) −4.22285e15 −1.71012
\(630\) 0 0
\(631\) −1.49304e15 −0.594168 −0.297084 0.954851i \(-0.596014\pi\)
−0.297084 + 0.954851i \(0.596014\pi\)
\(632\) 0 0
\(633\) −1.16892e15 3.36435e15i −0.457156 1.31577i
\(634\) 0 0
\(635\) 4.73630e14 0.182047
\(636\) 0 0
\(637\) −1.87348e15 1.93823e15i −0.707755 0.732217i
\(638\) 0 0
\(639\) 7.35816e14 + 9.31074e14i 0.273221 + 0.345724i
\(640\) 0 0
\(641\) 4.52879e15i 1.65296i −0.562965 0.826481i \(-0.690339\pi\)
0.562965 0.826481i \(-0.309661\pi\)
\(642\) 0 0
\(643\) 4.92986e15i 1.76878i 0.466748 + 0.884391i \(0.345426\pi\)
−0.466748 + 0.884391i \(0.654574\pi\)
\(644\) 0 0
\(645\) −1.74802e15 5.03111e15i −0.616551 1.77454i
\(646\) 0 0
\(647\) −2.40609e15 −0.834332 −0.417166 0.908830i \(-0.636977\pi\)
−0.417166 + 0.908830i \(0.636977\pi\)
\(648\) 0 0
\(649\) 4.57578e15i 1.55998i
\(650\) 0 0
\(651\) −2.90586e15 3.32252e15i −0.974047 1.11371i
\(652\) 0 0
\(653\) 3.48746e15i 1.14944i 0.818350 + 0.574720i \(0.194889\pi\)
−0.818350 + 0.574720i \(0.805111\pi\)
\(654\) 0 0
\(655\) −2.79712e15 −0.906532
\(656\) 0 0
\(657\) 2.32693e15 + 2.94441e15i 0.741607 + 0.938402i
\(658\) 0 0
\(659\) 4.71622e14i 0.147817i 0.997265 + 0.0739085i \(0.0235473\pi\)
−0.997265 + 0.0739085i \(0.976453\pi\)
\(660\) 0 0
\(661\) 4.67965e15i 1.44246i −0.692694 0.721232i \(-0.743576\pi\)
0.692694 0.721232i \(-0.256424\pi\)
\(662\) 0 0
\(663\) −4.29333e15 + 1.49169e15i −1.30158 + 0.452224i
\(664\) 0 0
\(665\) 3.24767e15 1.31303e15i 0.968395 0.391521i
\(666\) 0 0
\(667\) 1.83698e15 0.538781
\(668\) 0 0
\(669\) −3.73338e15 + 1.29714e15i −1.07711 + 0.374233i
\(670\) 0 0
\(671\) −2.92319e14 −0.0829628
\(672\) 0 0
\(673\) −3.26519e15 −0.911645 −0.455823 0.890071i \(-0.650655\pi\)
−0.455823 + 0.890071i \(0.650655\pi\)
\(674\) 0 0
\(675\) −1.83455e15 1.16971e15i −0.503917 0.321299i
\(676\) 0 0
\(677\) 1.18223e15 0.319495 0.159747 0.987158i \(-0.448932\pi\)
0.159747 + 0.987158i \(0.448932\pi\)
\(678\) 0 0
\(679\) −1.08243e15 + 4.37625e14i −0.287817 + 0.116364i
\(680\) 0 0
\(681\) −7.77578e14 2.23800e15i −0.203439 0.585532i
\(682\) 0 0
\(683\) 3.58812e15i 0.923746i 0.886946 + 0.461873i \(0.152822\pi\)
−0.886946 + 0.461873i \(0.847178\pi\)
\(684\) 0 0
\(685\) 7.48110e15i 1.89525i
\(686\) 0 0
\(687\) −3.27624e13 + 1.13831e13i −0.00816795 + 0.00283790i
\(688\) 0 0
\(689\) 3.78206e15 0.927947
\(690\) 0 0
\(691\) 4.35230e15i 1.05097i 0.850804 + 0.525484i \(0.176116\pi\)
−0.850804 + 0.525484i \(0.823884\pi\)
\(692\) 0 0
\(693\) 2.62869e15 4.61462e15i 0.624750 1.09674i
\(694\) 0 0
\(695\) 2.07949e15i 0.486451i
\(696\) 0 0
\(697\) −7.54172e15 −1.73656
\(698\) 0 0
\(699\) 5.34953e15 1.85866e15i 1.21253 0.421284i
\(700\) 0 0
\(701\) 2.76253e15i 0.616393i 0.951323 + 0.308197i \(0.0997255\pi\)
−0.951323 + 0.308197i \(0.900275\pi\)
\(702\) 0 0
\(703\) 4.75505e15i 1.04448i
\(704\) 0 0
\(705\) −3.07583e15 8.85275e15i −0.665154 1.91442i
\(706\) 0 0
\(707\) −9.35558e14 2.31403e15i −0.199188 0.492677i
\(708\) 0 0
\(709\) −6.99109e15 −1.46552 −0.732758 0.680489i \(-0.761767\pi\)
−0.732758 + 0.680489i \(0.761767\pi\)
\(710\) 0 0
\(711\) 4.10156e15 3.24141e15i 0.846576 0.669039i
\(712\) 0 0
\(713\) −5.31583e15 −1.08039
\(714\) 0 0
\(715\) 8.11805e15 1.62468
\(716\) 0 0
\(717\) 1.61630e15 5.61573e14i 0.318542 0.110675i
\(718\) 0 0
\(719\) −1.40034e15 −0.271784 −0.135892 0.990724i \(-0.543390\pi\)
−0.135892 + 0.990724i \(0.543390\pi\)
\(720\) 0 0
\(721\) 4.02019e15 1.62535e15i 0.768426 0.310673i
\(722\) 0 0
\(723\) −2.27093e15 + 7.89019e14i −0.427507 + 0.148534i
\(724\) 0 0
\(725\) 2.37827e15i 0.440963i
\(726\) 0 0
\(727\) 8.90831e15i 1.62688i 0.581647 + 0.813441i \(0.302408\pi\)
−0.581647 + 0.813441i \(0.697592\pi\)
\(728\) 0 0
\(729\) 2.34555e15 + 5.04000e15i 0.421932 + 0.906627i
\(730\) 0 0
\(731\) 1.13490e16 2.01099
\(732\) 0 0
\(733\) 1.12028e14i 0.0195549i 0.999952 + 0.00977746i \(0.00311231\pi\)
−0.999952 + 0.00977746i \(0.996888\pi\)
\(734\) 0 0
\(735\) 6.56019e15 + 3.31577e15i 1.12807 + 0.570169i
\(736\) 0 0
\(737\) 1.05228e16i 1.78262i
\(738\) 0 0
\(739\) 5.21696e15 0.870709 0.435355 0.900259i \(-0.356623\pi\)
0.435355 + 0.900259i \(0.356623\pi\)
\(740\) 0 0
\(741\) −1.67968e15 4.83441e15i −0.276202 0.794956i
\(742\) 0 0
\(743\) 7.45447e14i 0.120775i 0.998175 + 0.0603877i \(0.0192337\pi\)
−0.998175 + 0.0603877i \(0.980766\pi\)
\(744\) 0 0
\(745\) 9.62554e15i 1.53662i
\(746\) 0 0
\(747\) −7.49150e15 + 5.92044e15i −1.17843 + 0.931302i
\(748\) 0 0
\(749\) −8.34196e15 + 3.37264e15i −1.29306 + 0.522781i
\(750\) 0 0
\(751\) −1.48803e15 −0.227296 −0.113648 0.993521i \(-0.536254\pi\)
−0.113648 + 0.993521i \(0.536254\pi\)
\(752\) 0 0
\(753\) 3.47444e15 + 1.00000e16i 0.523013 + 1.50532i
\(754\) 0 0
\(755\) −8.69140e15 −1.28938
\(756\) 0 0
\(757\) −1.24719e16 −1.82350 −0.911751 0.410744i \(-0.865269\pi\)
−0.911751 + 0.410744i \(0.865269\pi\)
\(758\) 0 0
\(759\) −2.09913e15 6.04164e15i −0.302488 0.870612i
\(760\) 0 0
\(761\) 3.09265e15 0.439253 0.219627 0.975584i \(-0.429516\pi\)
0.219627 + 0.975584i \(0.429516\pi\)
\(762\) 0 0
\(763\) −1.00138e13 2.47683e13i −0.00140189 0.00346747i
\(764\) 0 0
\(765\) 9.72356e15 7.68441e15i 1.34180 1.06041i
\(766\) 0 0
\(767\) 9.25267e15i 1.25861i
\(768\) 0 0
\(769\) 1.06465e16i 1.42762i 0.700338 + 0.713811i \(0.253033\pi\)
−0.700338 + 0.713811i \(0.746967\pi\)
\(770\) 0 0
\(771\) 2.47270e15 + 7.11686e15i 0.326868 + 0.940782i
\(772\) 0 0
\(773\) 5.28652e15 0.688942 0.344471 0.938797i \(-0.388058\pi\)
0.344471 + 0.938797i \(0.388058\pi\)
\(774\) 0 0
\(775\) 6.88222e15i 0.884238i
\(776\) 0 0
\(777\) 7.51043e15 6.56860e15i 0.951369 0.832064i
\(778\) 0 0
\(779\) 8.49219e15i 1.06063i
\(780\) 0 0
\(781\) 4.51654e15 0.556192
\(782\) 0 0
\(783\) −3.26687e15 + 5.12368e15i −0.396681 + 0.622146i
\(784\) 0 0
\(785\) 4.40710e15i 0.527679i
\(786\) 0 0
\(787\) 8.57270e15i 1.01218i 0.862482 + 0.506088i \(0.168909\pi\)
−0.862482 + 0.506088i \(0.831091\pi\)
\(788\) 0 0
\(789\) 1.49920e15 5.20886e14i 0.174556 0.0606484i
\(790\) 0 0
\(791\) 8.98518e15 3.63269e15i 1.03171 0.417117i
\(792\) 0 0
\(793\) −5.91098e14 −0.0669355
\(794\) 0 0
\(795\) −9.74162e15 + 3.38466e15i −1.08796 + 0.378004i
\(796\) 0 0
\(797\) 3.31753e15 0.365422 0.182711 0.983167i \(-0.441513\pi\)
0.182711 + 0.983167i \(0.441513\pi\)
\(798\) 0 0
\(799\) 1.99697e16 2.16952
\(800\) 0 0
\(801\) 4.79618e15 3.79036e15i 0.513945 0.406164i
\(802\) 0 0
\(803\) 1.42830e16 1.50968
\(804\) 0 0
\(805\) 8.20701e15 3.31808e15i 0.855673 0.345947i
\(806\) 0 0
\(807\) −5.30715e14 1.52749e15i −0.0545829 0.157099i
\(808\) 0 0
\(809\) 1.00470e16i 1.01934i 0.860370 + 0.509670i \(0.170232\pi\)
−0.860370 + 0.509670i \(0.829768\pi\)
\(810\) 0 0
\(811\) 6.54645e15i 0.655226i 0.944812 + 0.327613i \(0.106244\pi\)
−0.944812 + 0.327613i \(0.893756\pi\)
\(812\) 0 0
\(813\) 3.25978e15 1.13259e15i 0.321877 0.111834i
\(814\) 0 0
\(815\) −1.90922e16 −1.85990
\(816\) 0 0
\(817\) 1.27792e16i 1.22824i
\(818\) 0 0
\(819\) 5.31547e15 9.33121e15i 0.504057 0.884864i
\(820\) 0 0
\(821\) 1.46809e16i 1.37361i 0.726840 + 0.686806i \(0.240988\pi\)
−0.726840 + 0.686806i \(0.759012\pi\)
\(822\) 0 0
\(823\) −8.36776e15 −0.772521 −0.386260 0.922390i \(-0.626233\pi\)
−0.386260 + 0.922390i \(0.626233\pi\)
\(824\) 0 0
\(825\) −7.82189e15 + 2.71766e15i −0.712549 + 0.247571i
\(826\) 0 0
\(827\) 1.51787e16i 1.36444i 0.731146 + 0.682221i \(0.238986\pi\)
−0.731146 + 0.682221i \(0.761014\pi\)
\(828\) 0 0
\(829\) 1.18299e16i 1.04937i 0.851295 + 0.524687i \(0.175818\pi\)
−0.851295 + 0.524687i \(0.824182\pi\)
\(830\) 0 0
\(831\) −7.34739e14 2.11470e15i −0.0643172 0.185116i
\(832\) 0 0
\(833\) −1.12616e16 + 1.08854e16i −0.972868 + 0.940367i
\(834\) 0 0
\(835\) −1.29901e15 −0.110749
\(836\) 0 0
\(837\) 9.45363e15 1.48268e16i 0.795442 1.24755i
\(838\) 0 0
\(839\) −1.20775e16 −1.00296 −0.501482 0.865168i \(-0.667212\pi\)
−0.501482 + 0.865168i \(0.667212\pi\)
\(840\) 0 0
\(841\) 5.55829e15 0.455578
\(842\) 0 0
\(843\) 5.33905e15 1.85502e15i 0.431929 0.150071i
\(844\) 0 0
\(845\) 5.86597e14 0.0468412
\(846\) 0 0
\(847\) −2.82065e15 6.97667e15i −0.222326 0.549907i
\(848\) 0 0
\(849\) −1.92544e16 + 6.68981e15i −1.49809 + 0.520501i
\(850\) 0 0
\(851\) 1.20162e16i 0.922903i
\(852\) 0 0
\(853\) 2.32449e16i 1.76241i 0.472732 + 0.881206i \(0.343268\pi\)
−0.472732 + 0.881206i \(0.656732\pi\)
\(854\) 0 0
\(855\) 8.65286e15 + 1.09490e16i 0.647658 + 0.819522i
\(856\) 0 0
\(857\) −1.35819e16 −1.00362 −0.501808 0.864979i \(-0.667331\pi\)
−0.501808 + 0.864979i \(0.667331\pi\)
\(858\) 0 0
\(859\) 1.98996e15i 0.145172i −0.997362 0.0725858i \(-0.976875\pi\)
0.997362 0.0725858i \(-0.0231251\pi\)
\(860\) 0 0
\(861\) 1.34131e16 1.17311e16i 0.966077 0.844928i
\(862\) 0 0
\(863\) 2.28322e16i 1.62364i −0.583909 0.811819i \(-0.698478\pi\)
0.583909 0.811819i \(-0.301522\pi\)
\(864\) 0 0
\(865\) 6.08061e14 0.0426932
\(866\) 0 0
\(867\) 3.93295e15 + 1.13197e16i 0.272655 + 0.784746i
\(868\) 0 0
\(869\) 1.98962e16i 1.36195i
\(870\) 0 0
\(871\) 2.12781e16i 1.43825i
\(872\) 0 0
\(873\) −2.88395e15 3.64924e15i −0.192491 0.243570i
\(874\) 0 0
\(875\) 2.89222e15 + 7.15369e15i 0.190628 + 0.471504i
\(876\) 0 0
\(877\) −1.18440e15 −0.0770907 −0.0385454 0.999257i \(-0.512272\pi\)
−0.0385454 + 0.999257i \(0.512272\pi\)
\(878\) 0 0
\(879\) −3.74572e15 1.07808e16i −0.240767 0.692966i
\(880\) 0 0
\(881\) 2.02832e16 1.28756 0.643781 0.765209i \(-0.277365\pi\)
0.643781 + 0.765209i \(0.277365\pi\)
\(882\) 0 0
\(883\) 2.64816e15 0.166020 0.0830099 0.996549i \(-0.473547\pi\)
0.0830099 + 0.996549i \(0.473547\pi\)
\(884\) 0 0
\(885\) −8.28044e15 2.38325e16i −0.512702 1.47564i
\(886\) 0 0
\(887\) −1.18280e16 −0.723322 −0.361661 0.932310i \(-0.617790\pi\)
−0.361661 + 0.932310i \(0.617790\pi\)
\(888\) 0 0
\(889\) −8.93782e14 2.21070e15i −0.0539849 0.133527i
\(890\) 0 0
\(891\) 2.05843e16 + 4.88954e15i 1.22803 + 0.291703i
\(892\) 0 0
\(893\) 2.24864e16i 1.32506i
\(894\) 0 0
\(895\) 3.71941e16i 2.16495i
\(896\) 0 0
\(897\) −4.24464e15 1.22168e16i −0.244052 0.702422i
\(898\) 0 0
\(899\) 1.92212e16 1.09170
\(900\) 0 0
\(901\) 2.19748e16i 1.23293i
\(902\) 0 0
\(903\) −2.01844e16 + 1.76532e16i −1.11875 + 0.978453i
\(904\) 0 0
\(905\) 5.01210e15i 0.274443i
\(906\) 0 0
\(907\) 2.85504e15 0.154444 0.0772221 0.997014i \(-0.475395\pi\)
0.0772221 + 0.997014i \(0.475395\pi\)
\(908\) 0 0
\(909\) 7.80138e15 6.16533e15i 0.416937 0.329500i
\(910\) 0 0
\(911\) 2.07192e16i 1.09401i 0.837130 + 0.547005i \(0.184232\pi\)
−0.837130 + 0.547005i \(0.815768\pi\)
\(912\) 0 0
\(913\) 3.63404e16i 1.89584i
\(914\) 0 0
\(915\) 1.52252e15 5.28988e14i 0.0784776 0.0272665i
\(916\) 0 0
\(917\) 5.27841e15 + 1.30557e16i 0.268826 + 0.664920i
\(918\) 0 0
\(919\) −1.86579e16 −0.938918 −0.469459 0.882954i \(-0.655551\pi\)
−0.469459 + 0.882954i \(0.655551\pi\)
\(920\) 0 0
\(921\) −9.06365e15 + 3.14911e15i −0.450687 + 0.156588i
\(922\) 0 0
\(923\) 9.13287e15 0.448743
\(924\) 0 0
\(925\) −1.55570e16 −0.755346
\(926\) 0 0
\(927\) 1.07111e16 + 1.35534e16i 0.513920 + 0.650295i
\(928\) 0 0
\(929\) −2.49449e15 −0.118276 −0.0591379 0.998250i \(-0.518835\pi\)
−0.0591379 + 0.998250i \(0.518835\pi\)
\(930\) 0 0
\(931\) −1.22573e16 1.26809e16i −0.574342 0.594193i
\(932\) 0 0
\(933\) −2.64663e14 7.61743e14i −0.0122559 0.0352744i
\(934\) 0 0
\(935\) 4.71679e16i 2.15865i
\(936\) 0 0
\(937\) 7.25594e15i 0.328190i 0.986445 + 0.164095i \(0.0524704\pi\)
−0.986445 + 0.164095i \(0.947530\pi\)
\(938\) 0 0
\(939\) −2.48068e16 + 8.61896e15i −1.10895 + 0.385296i
\(940\) 0 0
\(941\) 2.08328e16 0.920461 0.460230 0.887800i \(-0.347767\pi\)
0.460230 + 0.887800i \(0.347767\pi\)
\(942\) 0 0
\(943\) 2.14602e16i 0.937170i
\(944\) 0 0
\(945\) −5.34054e15 + 2.87917e16i −0.230521 + 1.24278i
\(946\) 0 0
\(947\) 1.74940e16i 0.746386i 0.927754 + 0.373193i \(0.121737\pi\)
−0.927754 + 0.373193i \(0.878263\pi\)
\(948\) 0 0
\(949\) 2.88816e16 1.21803
\(950\) 0 0
\(951\) −1.08234e16 + 3.76051e15i −0.451201 + 0.156767i
\(952\) 0 0
\(953\) 4.74959e16i 1.95725i −0.205661 0.978623i \(-0.565935\pi\)
0.205661 0.978623i \(-0.434065\pi\)
\(954\) 0 0
\(955\) 3.63625e16i 1.48127i
\(956\) 0 0
\(957\) 7.59010e15 + 2.18456e16i 0.305655 + 0.879727i
\(958\) 0 0
\(959\) 3.49185e16 1.41175e16i 1.39012 0.562025i
\(960\) 0 0
\(961\) −3.02136e16 −1.18912
\(962\) 0 0
\(963\) −2.22257e16 2.81236e16i −0.864792 1.09427i
\(964\) 0 0
\(965\) −2.90612e16 −1.11793
\(966\) 0 0
\(967\) −4.24033e16 −1.61270 −0.806352 0.591436i \(-0.798561\pi\)
−0.806352 + 0.591436i \(0.798561\pi\)
\(968\) 0 0
\(969\) −2.80891e16 + 9.75938e15i −1.05623 + 0.366979i
\(970\) 0 0
\(971\) −3.76749e16 −1.40070 −0.700352 0.713798i \(-0.746974\pi\)
−0.700352 + 0.713798i \(0.746974\pi\)
\(972\) 0 0
\(973\) −9.70613e15 + 3.92417e15i −0.356801 + 0.144254i
\(974\) 0 0
\(975\) −1.58166e16 + 5.49538e15i −0.574895 + 0.199743i
\(976\) 0 0
\(977\) 3.16313e16i 1.13683i −0.822740 0.568417i \(-0.807556\pi\)
0.822740 0.568417i \(-0.192444\pi\)
\(978\) 0 0
\(979\) 2.32657e16i 0.826822i
\(980\) 0 0
\(981\) 8.35025e13 6.59910e13i 0.00293441 0.00231902i
\(982\) 0 0
\(983\) 2.31947e15 0.0806018 0.0403009 0.999188i \(-0.487168\pi\)
0.0403009 + 0.999188i \(0.487168\pi\)
\(984\) 0 0
\(985\) 2.82785e16i 0.971756i
\(986\) 0 0
\(987\) −3.55164e16 + 3.10626e16i −1.20694 + 1.05558i
\(988\) 0 0
\(989\) 3.22938e16i 1.08527i
\(990\) 0 0
\(991\) −2.66064e16 −0.884262 −0.442131 0.896951i \(-0.645777\pi\)
−0.442131 + 0.896951i \(0.645777\pi\)
\(992\) 0 0
\(993\) −9.31678e15 2.68152e16i −0.306228 0.881375i
\(994\) 0 0
\(995\) 5.96944e15i 0.194047i
\(996\) 0 0
\(997\) 3.50447e16i 1.12668i −0.826226 0.563339i \(-0.809517\pi\)
0.826226 0.563339i \(-0.190483\pi\)
\(998\) 0 0
\(999\) 3.35155e16 + 2.13696e16i 1.06570 + 0.679494i
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 84.12.f.b.41.18 yes 28
3.2 odd 2 inner 84.12.f.b.41.12 yes 28
7.6 odd 2 inner 84.12.f.b.41.11 28
21.20 even 2 inner 84.12.f.b.41.17 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.12.f.b.41.11 28 7.6 odd 2 inner
84.12.f.b.41.12 yes 28 3.2 odd 2 inner
84.12.f.b.41.17 yes 28 21.20 even 2 inner
84.12.f.b.41.18 yes 28 1.1 even 1 trivial