Properties

Label 64.12.e.a.17.8
Level $64$
Weight $12$
Character 64.17
Analytic conductor $49.174$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,12,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(49.1739635558\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.8
Character \(\chi\) \(=\) 64.17
Dual form 64.12.e.a.49.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-219.312 - 219.312i) q^{3} +(-4575.66 + 4575.66i) q^{5} +2778.29i q^{7} -80951.6i q^{9} +O(q^{10})\) \(q+(-219.312 - 219.312i) q^{3} +(-4575.66 + 4575.66i) q^{5} +2778.29i q^{7} -80951.6i q^{9} +(-17312.0 + 17312.0i) q^{11} +(-1.16637e6 - 1.16637e6i) q^{13} +2.00699e6 q^{15} -7.63947e6 q^{17} +(-1.62010e6 - 1.62010e6i) q^{19} +(609312. - 609312. i) q^{21} +3.41251e7i q^{23} +6.95479e6i q^{25} +(-5.66041e7 + 5.66041e7i) q^{27} +(-5.22501e7 - 5.22501e7i) q^{29} +1.70466e8 q^{31} +7.59345e6 q^{33} +(-1.27125e7 - 1.27125e7i) q^{35} +(1.36013e8 - 1.36013e8i) q^{37} +5.11599e8i q^{39} -9.43574e8i q^{41} +(1.75831e8 - 1.75831e8i) q^{43} +(3.70407e8 + 3.70407e8i) q^{45} +2.33824e9 q^{47} +1.96961e9 q^{49} +(1.67543e9 + 1.67543e9i) q^{51} +(-2.13123e9 + 2.13123e9i) q^{53} -1.58428e8i q^{55} +7.10614e8i q^{57} +(2.73656e8 - 2.73656e8i) q^{59} +(7.10089e9 + 7.10089e9i) q^{61} +2.24907e8 q^{63} +1.06739e10 q^{65} +(-6.25789e9 - 6.25789e9i) q^{67} +(7.48403e9 - 7.48403e9i) q^{69} -3.05701e9i q^{71} +3.14465e10i q^{73} +(1.52527e9 - 1.52527e9i) q^{75} +(-4.80977e7 - 4.80977e7i) q^{77} +4.76542e10 q^{79} +1.04876e10 q^{81} +(4.66019e10 + 4.66019e10i) q^{83} +(3.49556e10 - 3.49556e10i) q^{85} +2.29182e10i q^{87} +1.17108e10i q^{89} +(3.24052e9 - 3.24052e9i) q^{91} +(-3.73852e10 - 3.73852e10i) q^{93} +1.48261e10 q^{95} +3.59417e10 q^{97} +(1.40143e9 + 1.40143e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 42 q + 2 q^{3} - 2 q^{5} + 540846 q^{11} - 2 q^{13} + 6075004 q^{15} - 4 q^{17} + 11291290 q^{19} + 354292 q^{21} + 66463304 q^{27} + 77673206 q^{29} - 343549808 q^{31} - 4 q^{33} + 434731684 q^{35} - 522762058 q^{37} - 3824193658 q^{43} + 97301954 q^{45} + 4586900144 q^{47} - 8474257474 q^{49} - 7074245796 q^{51} - 2100608058 q^{53} - 955824746 q^{59} + 2150827022 q^{61} - 27758037828 q^{63} - 1884965292 q^{65} + 3186519018 q^{67} - 16193060732 q^{69} - 28890034486 q^{75} - 22711870540 q^{77} - 48011833792 q^{79} - 90656394430 q^{81} - 55713221118 q^{83} - 84575506252 q^{85} + 147369662716 q^{91} - 69689773328 q^{93} - 375702304500 q^{95} - 4 q^{97} + 286271331106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −219.312 219.312i −0.521069 0.521069i 0.396825 0.917894i \(-0.370112\pi\)
−0.917894 + 0.396825i \(0.870112\pi\)
\(4\) 0 0
\(5\) −4575.66 + 4575.66i −0.654815 + 0.654815i −0.954149 0.299333i \(-0.903236\pi\)
0.299333 + 0.954149i \(0.403236\pi\)
\(6\) 0 0
\(7\) 2778.29i 0.0624796i 0.999512 + 0.0312398i \(0.00994555\pi\)
−0.999512 + 0.0312398i \(0.990054\pi\)
\(8\) 0 0
\(9\) 80951.6i 0.456974i
\(10\) 0 0
\(11\) −17312.0 + 17312.0i −0.0324106 + 0.0324106i −0.723126 0.690716i \(-0.757296\pi\)
0.690716 + 0.723126i \(0.257296\pi\)
\(12\) 0 0
\(13\) −1.16637e6 1.16637e6i −0.871263 0.871263i 0.121347 0.992610i \(-0.461278\pi\)
−0.992610 + 0.121347i \(0.961278\pi\)
\(14\) 0 0
\(15\) 2.00699e6 0.682408
\(16\) 0 0
\(17\) −7.63947e6 −1.30495 −0.652475 0.757810i \(-0.726269\pi\)
−0.652475 + 0.757810i \(0.726269\pi\)
\(18\) 0 0
\(19\) −1.62010e6 1.62010e6i −0.150106 0.150106i 0.628060 0.778165i \(-0.283849\pi\)
−0.778165 + 0.628060i \(0.783849\pi\)
\(20\) 0 0
\(21\) 609312. 609312.i 0.0325562 0.0325562i
\(22\) 0 0
\(23\) 3.41251e7i 1.10553i 0.833337 + 0.552765i \(0.186427\pi\)
−0.833337 + 0.552765i \(0.813573\pi\)
\(24\) 0 0
\(25\) 6.95479e6i 0.142434i
\(26\) 0 0
\(27\) −5.66041e7 + 5.66041e7i −0.759184 + 0.759184i
\(28\) 0 0
\(29\) −5.22501e7 5.22501e7i −0.473041 0.473041i 0.429856 0.902897i \(-0.358564\pi\)
−0.902897 + 0.429856i \(0.858564\pi\)
\(30\) 0 0
\(31\) 1.70466e8 1.06942 0.534710 0.845035i \(-0.320421\pi\)
0.534710 + 0.845035i \(0.320421\pi\)
\(32\) 0 0
\(33\) 7.59345e6 0.0337763
\(34\) 0 0
\(35\) −1.27125e7 1.27125e7i −0.0409126 0.0409126i
\(36\) 0 0
\(37\) 1.36013e8 1.36013e8i 0.322455 0.322455i −0.527253 0.849708i \(-0.676778\pi\)
0.849708 + 0.527253i \(0.176778\pi\)
\(38\) 0 0
\(39\) 5.11599e8i 0.907976i
\(40\) 0 0
\(41\) 9.43574e8i 1.27193i −0.771716 0.635967i \(-0.780601\pi\)
0.771716 0.635967i \(-0.219399\pi\)
\(42\) 0 0
\(43\) 1.75831e8 1.75831e8i 0.182397 0.182397i −0.610002 0.792400i \(-0.708831\pi\)
0.792400 + 0.610002i \(0.208831\pi\)
\(44\) 0 0
\(45\) 3.70407e8 + 3.70407e8i 0.299233 + 0.299233i
\(46\) 0 0
\(47\) 2.33824e9 1.48714 0.743569 0.668660i \(-0.233132\pi\)
0.743569 + 0.668660i \(0.233132\pi\)
\(48\) 0 0
\(49\) 1.96961e9 0.996096
\(50\) 0 0
\(51\) 1.67543e9 + 1.67543e9i 0.679969 + 0.679969i
\(52\) 0 0
\(53\) −2.13123e9 + 2.13123e9i −0.700023 + 0.700023i −0.964415 0.264392i \(-0.914829\pi\)
0.264392 + 0.964415i \(0.414829\pi\)
\(54\) 0 0
\(55\) 1.58428e8i 0.0424459i
\(56\) 0 0
\(57\) 7.10614e8i 0.156431i
\(58\) 0 0
\(59\) 2.73656e8 2.73656e8i 0.0498333 0.0498333i −0.681751 0.731584i \(-0.738781\pi\)
0.731584 + 0.681751i \(0.238781\pi\)
\(60\) 0 0
\(61\) 7.10089e9 + 7.10089e9i 1.07646 + 1.07646i 0.996824 + 0.0796382i \(0.0253765\pi\)
0.0796382 + 0.996824i \(0.474624\pi\)
\(62\) 0 0
\(63\) 2.24907e8 0.0285515
\(64\) 0 0
\(65\) 1.06739e10 1.14103
\(66\) 0 0
\(67\) −6.25789e9 6.25789e9i −0.566261 0.566261i 0.364818 0.931079i \(-0.381131\pi\)
−0.931079 + 0.364818i \(0.881131\pi\)
\(68\) 0 0
\(69\) 7.48403e9 7.48403e9i 0.576057 0.576057i
\(70\) 0 0
\(71\) 3.05701e9i 0.201083i −0.994933 0.100542i \(-0.967942\pi\)
0.994933 0.100542i \(-0.0320576\pi\)
\(72\) 0 0
\(73\) 3.14465e10i 1.77540i 0.460420 + 0.887701i \(0.347699\pi\)
−0.460420 + 0.887701i \(0.652301\pi\)
\(74\) 0 0
\(75\) 1.52527e9 1.52527e9i 0.0742180 0.0742180i
\(76\) 0 0
\(77\) −4.80977e7 4.80977e7i −0.00202500 0.00202500i
\(78\) 0 0
\(79\) 4.76542e10 1.74242 0.871209 0.490912i \(-0.163336\pi\)
0.871209 + 0.490912i \(0.163336\pi\)
\(80\) 0 0
\(81\) 1.04876e10 0.334201
\(82\) 0 0
\(83\) 4.66019e10 + 4.66019e10i 1.29860 + 1.29860i 0.929320 + 0.369276i \(0.120394\pi\)
0.369276 + 0.929320i \(0.379606\pi\)
\(84\) 0 0
\(85\) 3.49556e10 3.49556e10i 0.854501 0.854501i
\(86\) 0 0
\(87\) 2.29182e10i 0.492974i
\(88\) 0 0
\(89\) 1.17108e10i 0.222300i 0.993804 + 0.111150i \(0.0354534\pi\)
−0.993804 + 0.111150i \(0.964547\pi\)
\(90\) 0 0
\(91\) 3.24052e9 3.24052e9i 0.0544361 0.0544361i
\(92\) 0 0
\(93\) −3.73852e10 3.73852e10i −0.557242 0.557242i
\(94\) 0 0
\(95\) 1.48261e10 0.196583
\(96\) 0 0
\(97\) 3.59417e10 0.424966 0.212483 0.977165i \(-0.431845\pi\)
0.212483 + 0.977165i \(0.431845\pi\)
\(98\) 0 0
\(99\) 1.40143e9 + 1.40143e9i 0.0148108 + 0.0148108i
\(100\) 0 0
\(101\) 4.25749e9 4.25749e9i 0.0403075 0.0403075i −0.686666 0.726973i \(-0.740926\pi\)
0.726973 + 0.686666i \(0.240926\pi\)
\(102\) 0 0
\(103\) 8.38400e10i 0.712601i −0.934371 0.356301i \(-0.884038\pi\)
0.934371 0.356301i \(-0.115962\pi\)
\(104\) 0 0
\(105\) 5.57601e9i 0.0426366i
\(106\) 0 0
\(107\) −1.39681e11 + 1.39681e11i −0.962782 + 0.962782i −0.999332 0.0365502i \(-0.988363\pi\)
0.0365502 + 0.999332i \(0.488363\pi\)
\(108\) 0 0
\(109\) −1.03674e11 1.03674e11i −0.645393 0.645393i 0.306483 0.951876i \(-0.400848\pi\)
−0.951876 + 0.306483i \(0.900848\pi\)
\(110\) 0 0
\(111\) −5.96583e10 −0.336043
\(112\) 0 0
\(113\) −4.37338e10 −0.223298 −0.111649 0.993748i \(-0.535613\pi\)
−0.111649 + 0.993748i \(0.535613\pi\)
\(114\) 0 0
\(115\) −1.56145e11 1.56145e11i −0.723917 0.723917i
\(116\) 0 0
\(117\) −9.44197e10 + 9.44197e10i −0.398144 + 0.398144i
\(118\) 0 0
\(119\) 2.12246e10i 0.0815328i
\(120\) 0 0
\(121\) 2.84712e11i 0.997899i
\(122\) 0 0
\(123\) −2.06937e11 + 2.06937e11i −0.662765 + 0.662765i
\(124\) 0 0
\(125\) −2.55244e11 2.55244e11i −0.748083 0.748083i
\(126\) 0 0
\(127\) 6.00265e11 1.61221 0.806107 0.591770i \(-0.201570\pi\)
0.806107 + 0.591770i \(0.201570\pi\)
\(128\) 0 0
\(129\) −7.71236e10 −0.190083
\(130\) 0 0
\(131\) −6.08704e11 6.08704e11i −1.37852 1.37852i −0.847117 0.531407i \(-0.821663\pi\)
−0.531407 0.847117i \(-0.678337\pi\)
\(132\) 0 0
\(133\) 4.50111e9 4.50111e9i 0.00937854 0.00937854i
\(134\) 0 0
\(135\) 5.18002e11i 0.994251i
\(136\) 0 0
\(137\) 3.71507e11i 0.657663i 0.944389 + 0.328832i \(0.106655\pi\)
−0.944389 + 0.328832i \(0.893345\pi\)
\(138\) 0 0
\(139\) −4.19881e11 + 4.19881e11i −0.686348 + 0.686348i −0.961423 0.275075i \(-0.911297\pi\)
0.275075 + 0.961423i \(0.411297\pi\)
\(140\) 0 0
\(141\) −5.12804e11 5.12804e11i −0.774901 0.774901i
\(142\) 0 0
\(143\) 4.03845e10 0.0564763
\(144\) 0 0
\(145\) 4.78158e11 0.619509
\(146\) 0 0
\(147\) −4.31958e11 4.31958e11i −0.519035 0.519035i
\(148\) 0 0
\(149\) 5.05211e11 5.05211e11i 0.563571 0.563571i −0.366749 0.930320i \(-0.619529\pi\)
0.930320 + 0.366749i \(0.119529\pi\)
\(150\) 0 0
\(151\) 8.08446e11i 0.838065i −0.907971 0.419033i \(-0.862369\pi\)
0.907971 0.419033i \(-0.137631\pi\)
\(152\) 0 0
\(153\) 6.18427e11i 0.596328i
\(154\) 0 0
\(155\) −7.79995e11 + 7.79995e11i −0.700273 + 0.700273i
\(156\) 0 0
\(157\) −7.30037e11 7.30037e11i −0.610797 0.610797i 0.332357 0.943154i \(-0.392156\pi\)
−0.943154 + 0.332357i \(0.892156\pi\)
\(158\) 0 0
\(159\) 9.34807e11 0.729521
\(160\) 0 0
\(161\) −9.48093e10 −0.0690730
\(162\) 0 0
\(163\) 9.03848e11 + 9.03848e11i 0.615267 + 0.615267i 0.944314 0.329047i \(-0.106727\pi\)
−0.329047 + 0.944314i \(0.606727\pi\)
\(164\) 0 0
\(165\) −3.47450e10 + 3.47450e10i −0.0221173 + 0.0221173i
\(166\) 0 0
\(167\) 2.22636e12i 1.32634i 0.748469 + 0.663170i \(0.230789\pi\)
−0.748469 + 0.663170i \(0.769211\pi\)
\(168\) 0 0
\(169\) 9.28692e11i 0.518197i
\(170\) 0 0
\(171\) −1.31150e11 + 1.31150e11i −0.0685944 + 0.0685944i
\(172\) 0 0
\(173\) −6.98524e11 6.98524e11i −0.342711 0.342711i 0.514675 0.857385i \(-0.327913\pi\)
−0.857385 + 0.514675i \(0.827913\pi\)
\(174\) 0 0
\(175\) −1.93224e10 −0.00889922
\(176\) 0 0
\(177\) −1.20032e11 −0.0519332
\(178\) 0 0
\(179\) 2.78923e12 + 2.78923e12i 1.13447 + 1.13447i 0.989425 + 0.145044i \(0.0463322\pi\)
0.145044 + 0.989425i \(0.453668\pi\)
\(180\) 0 0
\(181\) 8.65718e11 8.65718e11i 0.331241 0.331241i −0.521817 0.853058i \(-0.674745\pi\)
0.853058 + 0.521817i \(0.174745\pi\)
\(182\) 0 0
\(183\) 3.11462e12i 1.12182i
\(184\) 0 0
\(185\) 1.24469e12i 0.422297i
\(186\) 0 0
\(187\) 1.32254e11 1.32254e11i 0.0422942 0.0422942i
\(188\) 0 0
\(189\) −1.57263e11 1.57263e11i −0.0474335 0.0474335i
\(190\) 0 0
\(191\) 3.97152e12 1.13051 0.565254 0.824917i \(-0.308778\pi\)
0.565254 + 0.824917i \(0.308778\pi\)
\(192\) 0 0
\(193\) 5.02696e12 1.35126 0.675632 0.737239i \(-0.263871\pi\)
0.675632 + 0.737239i \(0.263871\pi\)
\(194\) 0 0
\(195\) −2.34090e12 2.34090e12i −0.594557 0.594557i
\(196\) 0 0
\(197\) 2.21725e12 2.21725e12i 0.532415 0.532415i −0.388876 0.921290i \(-0.627136\pi\)
0.921290 + 0.388876i \(0.127136\pi\)
\(198\) 0 0
\(199\) 1.69752e12i 0.385588i 0.981239 + 0.192794i \(0.0617548\pi\)
−0.981239 + 0.192794i \(0.938245\pi\)
\(200\) 0 0
\(201\) 2.74486e12i 0.590122i
\(202\) 0 0
\(203\) 1.45166e11 1.45166e11i 0.0295554 0.0295554i
\(204\) 0 0
\(205\) 4.31747e12 + 4.31747e12i 0.832882 + 0.832882i
\(206\) 0 0
\(207\) 2.76248e12 0.505198
\(208\) 0 0
\(209\) 5.60943e10 0.00973003
\(210\) 0 0
\(211\) −1.20649e12 1.20649e12i −0.198596 0.198596i 0.600802 0.799398i \(-0.294848\pi\)
−0.799398 + 0.600802i \(0.794848\pi\)
\(212\) 0 0
\(213\) −6.70439e11 + 6.70439e11i −0.104778 + 0.104778i
\(214\) 0 0
\(215\) 1.60908e12i 0.238873i
\(216\) 0 0
\(217\) 4.73604e11i 0.0668170i
\(218\) 0 0
\(219\) 6.89660e12 6.89660e12i 0.925108 0.925108i
\(220\) 0 0
\(221\) 8.91047e12 + 8.91047e12i 1.13695 + 1.13695i
\(222\) 0 0
\(223\) −3.81312e12 −0.463025 −0.231512 0.972832i \(-0.574367\pi\)
−0.231512 + 0.972832i \(0.574367\pi\)
\(224\) 0 0
\(225\) 5.63001e11 0.0650887
\(226\) 0 0
\(227\) −2.20437e12 2.20437e12i −0.242740 0.242740i 0.575243 0.817983i \(-0.304908\pi\)
−0.817983 + 0.575243i \(0.804908\pi\)
\(228\) 0 0
\(229\) −6.95781e12 + 6.95781e12i −0.730092 + 0.730092i −0.970638 0.240546i \(-0.922673\pi\)
0.240546 + 0.970638i \(0.422673\pi\)
\(230\) 0 0
\(231\) 2.10968e10i 0.00211033i
\(232\) 0 0
\(233\) 1.59857e13i 1.52502i −0.646977 0.762510i \(-0.723967\pi\)
0.646977 0.762510i \(-0.276033\pi\)
\(234\) 0 0
\(235\) −1.06990e13 + 1.06990e13i −0.973800 + 0.973800i
\(236\) 0 0
\(237\) −1.04511e13 1.04511e13i −0.907920 0.907920i
\(238\) 0 0
\(239\) −1.48081e13 −1.22831 −0.614157 0.789184i \(-0.710504\pi\)
−0.614157 + 0.789184i \(0.710504\pi\)
\(240\) 0 0
\(241\) −1.00810e13 −0.798750 −0.399375 0.916788i \(-0.630773\pi\)
−0.399375 + 0.916788i \(0.630773\pi\)
\(242\) 0 0
\(243\) 7.72719e12 + 7.72719e12i 0.585042 + 0.585042i
\(244\) 0 0
\(245\) −9.01226e12 + 9.01226e12i −0.652259 + 0.652259i
\(246\) 0 0
\(247\) 3.77928e12i 0.261563i
\(248\) 0 0
\(249\) 2.04407e13i 1.35332i
\(250\) 0 0
\(251\) 2.01824e13 2.01824e13i 1.27870 1.27870i 0.337302 0.941397i \(-0.390486\pi\)
0.941397 0.337302i \(-0.109514\pi\)
\(252\) 0 0
\(253\) −5.90772e11 5.90772e11i −0.0358309 0.0358309i
\(254\) 0 0
\(255\) −1.53324e13 −0.890509
\(256\) 0 0
\(257\) −2.83439e13 −1.57699 −0.788493 0.615044i \(-0.789138\pi\)
−0.788493 + 0.615044i \(0.789138\pi\)
\(258\) 0 0
\(259\) 3.77882e11 + 3.77882e11i 0.0201469 + 0.0201469i
\(260\) 0 0
\(261\) −4.22973e12 + 4.22973e12i −0.216167 + 0.216167i
\(262\) 0 0
\(263\) 1.43628e13i 0.703855i 0.936027 + 0.351928i \(0.114474\pi\)
−0.936027 + 0.351928i \(0.885526\pi\)
\(264\) 0 0
\(265\) 1.95035e13i 0.916772i
\(266\) 0 0
\(267\) 2.56831e12 2.56831e12i 0.115834 0.115834i
\(268\) 0 0
\(269\) 1.53045e13 + 1.53045e13i 0.662492 + 0.662492i 0.955967 0.293475i \(-0.0948116\pi\)
−0.293475 + 0.955967i \(0.594812\pi\)
\(270\) 0 0
\(271\) −3.43046e13 −1.42568 −0.712838 0.701329i \(-0.752590\pi\)
−0.712838 + 0.701329i \(0.752590\pi\)
\(272\) 0 0
\(273\) −1.42137e12 −0.0567300
\(274\) 0 0
\(275\) −1.20401e11 1.20401e11i −0.00461637 0.00461637i
\(276\) 0 0
\(277\) 3.09049e13 3.09049e13i 1.13864 1.13864i 0.149951 0.988693i \(-0.452088\pi\)
0.988693 0.149951i \(-0.0479117\pi\)
\(278\) 0 0
\(279\) 1.37995e13i 0.488697i
\(280\) 0 0
\(281\) 2.94675e13i 1.00336i 0.865052 + 0.501682i \(0.167285\pi\)
−0.865052 + 0.501682i \(0.832715\pi\)
\(282\) 0 0
\(283\) −7.63667e12 + 7.63667e12i −0.250080 + 0.250080i −0.821003 0.570923i \(-0.806585\pi\)
0.570923 + 0.821003i \(0.306585\pi\)
\(284\) 0 0
\(285\) −3.25153e12 3.25153e12i −0.102433 0.102433i
\(286\) 0 0
\(287\) 2.62152e12 0.0794699
\(288\) 0 0
\(289\) 2.40896e13 0.702896
\(290\) 0 0
\(291\) −7.88244e12 7.88244e12i −0.221437 0.221437i
\(292\) 0 0
\(293\) −1.88996e13 + 1.88996e13i −0.511307 + 0.511307i −0.914927 0.403620i \(-0.867752\pi\)
0.403620 + 0.914927i \(0.367752\pi\)
\(294\) 0 0
\(295\) 2.50432e12i 0.0652632i
\(296\) 0 0
\(297\) 1.95986e12i 0.0492112i
\(298\) 0 0
\(299\) 3.98025e13 3.98025e13i 0.963206 0.963206i
\(300\) 0 0
\(301\) 4.88509e11 + 4.88509e11i 0.0113961 + 0.0113961i
\(302\) 0 0
\(303\) −1.86744e12 −0.0420060
\(304\) 0 0
\(305\) −6.49825e13 −1.40977
\(306\) 0 0
\(307\) 8.05621e12 + 8.05621e12i 0.168605 + 0.168605i 0.786366 0.617761i \(-0.211960\pi\)
−0.617761 + 0.786366i \(0.711960\pi\)
\(308\) 0 0
\(309\) −1.83871e13 + 1.83871e13i −0.371315 + 0.371315i
\(310\) 0 0
\(311\) 2.29206e13i 0.446729i −0.974735 0.223365i \(-0.928296\pi\)
0.974735 0.223365i \(-0.0717041\pi\)
\(312\) 0 0
\(313\) 1.16826e13i 0.219809i −0.993942 0.109904i \(-0.964946\pi\)
0.993942 0.109904i \(-0.0350544\pi\)
\(314\) 0 0
\(315\) −1.02910e12 + 1.02910e12i −0.0186960 + 0.0186960i
\(316\) 0 0
\(317\) 4.67319e13 + 4.67319e13i 0.819950 + 0.819950i 0.986100 0.166150i \(-0.0531337\pi\)
−0.166150 + 0.986100i \(0.553134\pi\)
\(318\) 0 0
\(319\) 1.80911e12 0.0306631
\(320\) 0 0
\(321\) 6.12676e13 1.00335
\(322\) 0 0
\(323\) 1.23767e13 + 1.23767e13i 0.195880 + 0.195880i
\(324\) 0 0
\(325\) 8.11188e12 8.11188e12i 0.124097 0.124097i
\(326\) 0 0
\(327\) 4.54739e13i 0.672588i
\(328\) 0 0
\(329\) 6.49631e12i 0.0929157i
\(330\) 0 0
\(331\) −2.48513e13 + 2.48513e13i −0.343791 + 0.343791i −0.857790 0.514000i \(-0.828163\pi\)
0.514000 + 0.857790i \(0.328163\pi\)
\(332\) 0 0
\(333\) −1.10104e13 1.10104e13i −0.147354 0.147354i
\(334\) 0 0
\(335\) 5.72680e13 0.741593
\(336\) 0 0
\(337\) 1.14158e14 1.43068 0.715342 0.698775i \(-0.246271\pi\)
0.715342 + 0.698775i \(0.246271\pi\)
\(338\) 0 0
\(339\) 9.59133e12 + 9.59133e12i 0.116354 + 0.116354i
\(340\) 0 0
\(341\) −2.95111e12 + 2.95111e12i −0.0346606 + 0.0346606i
\(342\) 0 0
\(343\) 1.09657e13i 0.124715i
\(344\) 0 0
\(345\) 6.84888e13i 0.754422i
\(346\) 0 0
\(347\) −6.86725e13 + 6.86725e13i −0.732776 + 0.732776i −0.971169 0.238393i \(-0.923379\pi\)
0.238393 + 0.971169i \(0.423379\pi\)
\(348\) 0 0
\(349\) 5.60520e13 + 5.60520e13i 0.579497 + 0.579497i 0.934765 0.355268i \(-0.115610\pi\)
−0.355268 + 0.934765i \(0.615610\pi\)
\(350\) 0 0
\(351\) 1.32043e14 1.32290
\(352\) 0 0
\(353\) 5.46239e13 0.530423 0.265211 0.964190i \(-0.414558\pi\)
0.265211 + 0.964190i \(0.414558\pi\)
\(354\) 0 0
\(355\) 1.39878e13 + 1.39878e13i 0.131672 + 0.131672i
\(356\) 0 0
\(357\) −4.65482e12 + 4.65482e12i −0.0424842 + 0.0424842i
\(358\) 0 0
\(359\) 3.93500e13i 0.348278i −0.984721 0.174139i \(-0.944286\pi\)
0.984721 0.174139i \(-0.0557141\pi\)
\(360\) 0 0
\(361\) 1.11241e14i 0.954937i
\(362\) 0 0
\(363\) 6.24408e13 6.24408e13i 0.519974 0.519974i
\(364\) 0 0
\(365\) −1.43889e14 1.43889e14i −1.16256 1.16256i
\(366\) 0 0
\(367\) 1.36804e14 1.07260 0.536299 0.844028i \(-0.319822\pi\)
0.536299 + 0.844028i \(0.319822\pi\)
\(368\) 0 0
\(369\) −7.63838e13 −0.581241
\(370\) 0 0
\(371\) −5.92116e12 5.92116e12i −0.0437372 0.0437372i
\(372\) 0 0
\(373\) 1.67793e14 1.67793e14i 1.20331 1.20331i 0.230152 0.973155i \(-0.426078\pi\)
0.973155 0.230152i \(-0.0739224\pi\)
\(374\) 0 0
\(375\) 1.11956e14i 0.779606i
\(376\) 0 0
\(377\) 1.21886e14i 0.824285i
\(378\) 0 0
\(379\) −8.77288e13 + 8.77288e13i −0.576271 + 0.576271i −0.933874 0.357603i \(-0.883594\pi\)
0.357603 + 0.933874i \(0.383594\pi\)
\(380\) 0 0
\(381\) −1.31645e14 1.31645e14i −0.840075 0.840075i
\(382\) 0 0
\(383\) 6.00616e13 0.372395 0.186197 0.982512i \(-0.440384\pi\)
0.186197 + 0.982512i \(0.440384\pi\)
\(384\) 0 0
\(385\) 4.40157e11 0.00265200
\(386\) 0 0
\(387\) −1.42338e13 1.42338e13i −0.0833508 0.0833508i
\(388\) 0 0
\(389\) −3.16819e13 + 3.16819e13i −0.180339 + 0.180339i −0.791503 0.611165i \(-0.790701\pi\)
0.611165 + 0.791503i \(0.290701\pi\)
\(390\) 0 0
\(391\) 2.60697e14i 1.44266i
\(392\) 0 0
\(393\) 2.66992e14i 1.43661i
\(394\) 0 0
\(395\) −2.18050e14 + 2.18050e14i −1.14096 + 1.14096i
\(396\) 0 0
\(397\) −9.07399e13 9.07399e13i −0.461796 0.461796i 0.437448 0.899244i \(-0.355883\pi\)
−0.899244 + 0.437448i \(0.855883\pi\)
\(398\) 0 0
\(399\) −1.97429e12 −0.00977373
\(400\) 0 0
\(401\) −1.18853e14 −0.572424 −0.286212 0.958166i \(-0.592396\pi\)
−0.286212 + 0.958166i \(0.592396\pi\)
\(402\) 0 0
\(403\) −1.98827e14 1.98827e14i −0.931746 0.931746i
\(404\) 0 0
\(405\) −4.79876e13 + 4.79876e13i −0.218840 + 0.218840i
\(406\) 0 0
\(407\) 4.70929e12i 0.0209019i
\(408\) 0 0
\(409\) 2.44879e14i 1.05797i 0.848631 + 0.528986i \(0.177428\pi\)
−0.848631 + 0.528986i \(0.822572\pi\)
\(410\) 0 0
\(411\) 8.14758e13 8.14758e13i 0.342688 0.342688i
\(412\) 0 0
\(413\) 7.60297e11 + 7.60297e11i 0.00311356 + 0.00311356i
\(414\) 0 0
\(415\) −4.26469e14 −1.70068
\(416\) 0 0
\(417\) 1.84170e14 0.715270
\(418\) 0 0
\(419\) −2.30256e13 2.30256e13i −0.0871033 0.0871033i 0.662213 0.749316i \(-0.269618\pi\)
−0.749316 + 0.662213i \(0.769618\pi\)
\(420\) 0 0
\(421\) −2.71463e13 + 2.71463e13i −0.100036 + 0.100036i −0.755354 0.655317i \(-0.772535\pi\)
0.655317 + 0.755354i \(0.272535\pi\)
\(422\) 0 0
\(423\) 1.89284e14i 0.679583i
\(424\) 0 0
\(425\) 5.31309e13i 0.185869i
\(426\) 0 0
\(427\) −1.97283e13 + 1.97283e13i −0.0672569 + 0.0672569i
\(428\) 0 0
\(429\) −8.85680e12 8.85680e12i −0.0294281 0.0294281i
\(430\) 0 0
\(431\) −4.22363e13 −0.136792 −0.0683961 0.997658i \(-0.521788\pi\)
−0.0683961 + 0.997658i \(0.521788\pi\)
\(432\) 0 0
\(433\) −3.22774e14 −1.01909 −0.509547 0.860443i \(-0.670187\pi\)
−0.509547 + 0.860443i \(0.670187\pi\)
\(434\) 0 0
\(435\) −1.04866e14 1.04866e14i −0.322807 0.322807i
\(436\) 0 0
\(437\) 5.52860e13 5.52860e13i 0.165946 0.165946i
\(438\) 0 0
\(439\) 1.45795e14i 0.426764i −0.976969 0.213382i \(-0.931552\pi\)
0.976969 0.213382i \(-0.0684479\pi\)
\(440\) 0 0
\(441\) 1.59443e14i 0.455190i
\(442\) 0 0
\(443\) −2.05734e14 + 2.05734e14i −0.572909 + 0.572909i −0.932940 0.360031i \(-0.882766\pi\)
0.360031 + 0.932940i \(0.382766\pi\)
\(444\) 0 0
\(445\) −5.35844e13 5.35844e13i −0.145566 0.145566i
\(446\) 0 0
\(447\) −2.21598e14 −0.587319
\(448\) 0 0
\(449\) 1.85617e13 0.0480024 0.0240012 0.999712i \(-0.492359\pi\)
0.0240012 + 0.999712i \(0.492359\pi\)
\(450\) 0 0
\(451\) 1.63351e13 + 1.63351e13i 0.0412241 + 0.0412241i
\(452\) 0 0
\(453\) −1.77302e14 + 1.77302e14i −0.436690 + 0.436690i
\(454\) 0 0
\(455\) 2.96551e13i 0.0712912i
\(456\) 0 0
\(457\) 5.36809e14i 1.25974i −0.776701 0.629869i \(-0.783109\pi\)
0.776701 0.629869i \(-0.216891\pi\)
\(458\) 0 0
\(459\) 4.32425e14 4.32425e14i 0.990698 0.990698i
\(460\) 0 0
\(461\) −1.02395e14 1.02395e14i −0.229046 0.229046i 0.583248 0.812294i \(-0.301782\pi\)
−0.812294 + 0.583248i \(0.801782\pi\)
\(462\) 0 0
\(463\) 6.64099e14 1.45057 0.725283 0.688451i \(-0.241709\pi\)
0.725283 + 0.688451i \(0.241709\pi\)
\(464\) 0 0
\(465\) 3.42124e14 0.729781
\(466\) 0 0
\(467\) 3.58885e14 + 3.58885e14i 0.747675 + 0.747675i 0.974042 0.226367i \(-0.0726849\pi\)
−0.226367 + 0.974042i \(0.572685\pi\)
\(468\) 0 0
\(469\) 1.73862e13 1.73862e13i 0.0353798 0.0353798i
\(470\) 0 0
\(471\) 3.20212e14i 0.636535i
\(472\) 0 0
\(473\) 6.08796e12i 0.0118232i
\(474\) 0 0
\(475\) 1.12675e13 1.12675e13i 0.0213802 0.0213802i
\(476\) 0 0
\(477\) 1.72526e14 + 1.72526e14i 0.319892 + 0.319892i
\(478\) 0 0
\(479\) 1.53948e14 0.278951 0.139475 0.990226i \(-0.455458\pi\)
0.139475 + 0.990226i \(0.455458\pi\)
\(480\) 0 0
\(481\) −3.17283e14 −0.561886
\(482\) 0 0
\(483\) 2.07928e13 + 2.07928e13i 0.0359918 + 0.0359918i
\(484\) 0 0
\(485\) −1.64457e14 + 1.64457e14i −0.278274 + 0.278274i
\(486\) 0 0
\(487\) 9.15975e14i 1.51522i 0.652710 + 0.757608i \(0.273632\pi\)
−0.652710 + 0.757608i \(0.726368\pi\)
\(488\) 0 0
\(489\) 3.96449e14i 0.641193i
\(490\) 0 0
\(491\) −1.86596e14 + 1.86596e14i −0.295089 + 0.295089i −0.839087 0.543998i \(-0.816910\pi\)
0.543998 + 0.839087i \(0.316910\pi\)
\(492\) 0 0
\(493\) 3.99163e14 + 3.99163e14i 0.617295 + 0.617295i
\(494\) 0 0
\(495\) −1.28250e13 −0.0193967
\(496\) 0 0
\(497\) 8.49326e12 0.0125636
\(498\) 0 0
\(499\) −1.06833e14 1.06833e14i −0.154580 0.154580i 0.625580 0.780160i \(-0.284862\pi\)
−0.780160 + 0.625580i \(0.784862\pi\)
\(500\) 0 0
\(501\) 4.88267e14 4.88267e14i 0.691114 0.691114i
\(502\) 0 0
\(503\) 1.16223e14i 0.160941i 0.996757 + 0.0804706i \(0.0256423\pi\)
−0.996757 + 0.0804706i \(0.974358\pi\)
\(504\) 0 0
\(505\) 3.89616e13i 0.0527879i
\(506\) 0 0
\(507\) 2.03673e14 2.03673e14i 0.270017 0.270017i
\(508\) 0 0
\(509\) 8.43081e14 + 8.43081e14i 1.09376 + 1.09376i 0.995124 + 0.0986346i \(0.0314475\pi\)
0.0986346 + 0.995124i \(0.468552\pi\)
\(510\) 0 0
\(511\) −8.73675e13 −0.110926
\(512\) 0 0
\(513\) 1.83409e14 0.227916
\(514\) 0 0
\(515\) 3.83624e14 + 3.83624e14i 0.466622 + 0.466622i
\(516\) 0 0
\(517\) −4.04796e13 + 4.04796e13i −0.0481990 + 0.0481990i
\(518\) 0 0
\(519\) 3.06389e14i 0.357152i
\(520\) 0 0
\(521\) 2.87101e14i 0.327662i 0.986488 + 0.163831i \(0.0523852\pi\)
−0.986488 + 0.163831i \(0.947615\pi\)
\(522\) 0 0
\(523\) 1.16485e15 1.16485e15i 1.30169 1.30169i 0.374446 0.927249i \(-0.377833\pi\)
0.927249 0.374446i \(-0.122167\pi\)
\(524\) 0 0
\(525\) 4.23764e12 + 4.23764e12i 0.00463711 + 0.00463711i
\(526\) 0 0
\(527\) −1.30227e15 −1.39554
\(528\) 0 0
\(529\) −2.11709e14 −0.222195
\(530\) 0 0
\(531\) −2.21529e13 2.21529e13i −0.0227725 0.0227725i
\(532\) 0 0
\(533\) −1.10056e15 + 1.10056e15i −1.10819 + 1.10819i
\(534\) 0 0
\(535\) 1.27827e15i 1.26089i
\(536\) 0 0
\(537\) 1.22342e15i 1.18227i
\(538\) 0 0
\(539\) −3.40978e13 + 3.40978e13i −0.0322841 + 0.0322841i
\(540\) 0 0
\(541\) 6.21814e14 + 6.21814e14i 0.576866 + 0.576866i 0.934039 0.357172i \(-0.116259\pi\)
−0.357172 + 0.934039i \(0.616259\pi\)
\(542\) 0 0
\(543\) −3.79725e14 −0.345199
\(544\) 0 0
\(545\) 9.48754e14 0.845226
\(546\) 0 0
\(547\) 7.07617e14 + 7.07617e14i 0.617829 + 0.617829i 0.944974 0.327145i \(-0.106087\pi\)
−0.327145 + 0.944974i \(0.606087\pi\)
\(548\) 0 0
\(549\) 5.74828e14 5.74828e14i 0.491915 0.491915i
\(550\) 0 0
\(551\) 1.69301e14i 0.142012i
\(552\) 0 0
\(553\) 1.32397e14i 0.108866i
\(554\) 0 0
\(555\) 2.72976e14 2.72976e14i 0.220046 0.220046i
\(556\) 0 0
\(557\) 8.59075e14 + 8.59075e14i 0.678934 + 0.678934i 0.959759 0.280825i \(-0.0906081\pi\)
−0.280825 + 0.959759i \(0.590608\pi\)
\(558\) 0 0
\(559\) −4.10169e14 −0.317832
\(560\) 0 0
\(561\) −5.80099e13 −0.0440764
\(562\) 0 0
\(563\) 9.71752e14 + 9.71752e14i 0.724034 + 0.724034i 0.969424 0.245390i \(-0.0789160\pi\)
−0.245390 + 0.969424i \(0.578916\pi\)
\(564\) 0 0
\(565\) 2.00111e14 2.00111e14i 0.146219 0.146219i
\(566\) 0 0
\(567\) 2.91375e13i 0.0208807i
\(568\) 0 0
\(569\) 1.36146e15i 0.956949i 0.878102 + 0.478474i \(0.158810\pi\)
−0.878102 + 0.478474i \(0.841190\pi\)
\(570\) 0 0
\(571\) 1.36626e15 1.36626e15i 0.941966 0.941966i −0.0564400 0.998406i \(-0.517975\pi\)
0.998406 + 0.0564400i \(0.0179750\pi\)
\(572\) 0 0
\(573\) −8.71002e14 8.71002e14i −0.589072 0.589072i
\(574\) 0 0
\(575\) −2.37333e14 −0.157465
\(576\) 0 0
\(577\) −1.37482e15 −0.894911 −0.447456 0.894306i \(-0.647670\pi\)
−0.447456 + 0.894306i \(0.647670\pi\)
\(578\) 0 0
\(579\) −1.10247e15 1.10247e15i −0.704102 0.704102i
\(580\) 0 0
\(581\) −1.29474e14 + 1.29474e14i −0.0811357 + 0.0811357i
\(582\) 0 0
\(583\) 7.37915e13i 0.0453763i
\(584\) 0 0
\(585\) 8.64065e14i 0.521422i
\(586\) 0 0
\(587\) −1.27134e15 + 1.27134e15i −0.752925 + 0.752925i −0.975024 0.222099i \(-0.928709\pi\)
0.222099 + 0.975024i \(0.428709\pi\)
\(588\) 0 0
\(589\) −2.76172e14 2.76172e14i −0.160526 0.160526i
\(590\) 0 0
\(591\) −9.72538e14 −0.554850
\(592\) 0 0
\(593\) −1.66805e15 −0.934131 −0.467066 0.884223i \(-0.654689\pi\)
−0.467066 + 0.884223i \(0.654689\pi\)
\(594\) 0 0
\(595\) 9.71168e13 + 9.71168e13i 0.0533889 + 0.0533889i
\(596\) 0 0
\(597\) 3.72286e14 3.72286e14i 0.200918 0.200918i
\(598\) 0 0
\(599\) 1.85853e15i 0.984741i 0.870386 + 0.492370i \(0.163869\pi\)
−0.870386 + 0.492370i \(0.836131\pi\)
\(600\) 0 0
\(601\) 2.26696e15i 1.17932i −0.807650 0.589662i \(-0.799261\pi\)
0.807650 0.589662i \(-0.200739\pi\)
\(602\) 0 0
\(603\) −5.06586e14 + 5.06586e14i −0.258767 + 0.258767i
\(604\) 0 0
\(605\) −1.30275e15 1.30275e15i −0.653440 0.653440i
\(606\) 0 0
\(607\) 1.39375e15 0.686508 0.343254 0.939243i \(-0.388471\pi\)
0.343254 + 0.939243i \(0.388471\pi\)
\(608\) 0 0
\(609\) −6.36733e13 −0.0308008
\(610\) 0 0
\(611\) −2.72726e15 2.72726e15i −1.29569 1.29569i
\(612\) 0 0
\(613\) −4.24343e13 + 4.24343e13i −0.0198009 + 0.0198009i −0.716938 0.697137i \(-0.754457\pi\)
0.697137 + 0.716938i \(0.254457\pi\)
\(614\) 0 0
\(615\) 1.89375e15i 0.867978i
\(616\) 0 0
\(617\) 2.15894e15i 0.972013i 0.873955 + 0.486007i \(0.161547\pi\)
−0.873955 + 0.486007i \(0.838453\pi\)
\(618\) 0 0
\(619\) −1.84800e15 + 1.84800e15i −0.817339 + 0.817339i −0.985722 0.168382i \(-0.946146\pi\)
0.168382 + 0.985722i \(0.446146\pi\)
\(620\) 0 0
\(621\) −1.93162e15 1.93162e15i −0.839300 0.839300i
\(622\) 0 0
\(623\) −3.25359e13 −0.0138892
\(624\) 0 0
\(625\) 1.99623e15 0.837278
\(626\) 0 0
\(627\) −1.23021e13 1.23021e13i −0.00507002 0.00507002i
\(628\) 0 0
\(629\) −1.03906e15 + 1.03906e15i −0.420788 + 0.420788i
\(630\) 0 0
\(631\) 1.80781e15i 0.719435i −0.933061 0.359718i \(-0.882873\pi\)
0.933061 0.359718i \(-0.117127\pi\)
\(632\) 0 0
\(633\) 5.29195e14i 0.206964i
\(634\) 0 0
\(635\) −2.74661e15 + 2.74661e15i −1.05570 + 1.05570i
\(636\) 0 0
\(637\) −2.29730e15 2.29730e15i −0.867861 0.867861i
\(638\) 0 0
\(639\) −2.47470e14 −0.0918898
\(640\) 0 0
\(641\) −3.37793e15 −1.23291 −0.616456 0.787389i \(-0.711432\pi\)
−0.616456 + 0.787389i \(0.711432\pi\)
\(642\) 0 0
\(643\) −1.45344e15 1.45344e15i −0.521480 0.521480i 0.396538 0.918018i \(-0.370211\pi\)
−0.918018 + 0.396538i \(0.870211\pi\)
\(644\) 0 0
\(645\) 3.52891e14 3.52891e14i 0.124469 0.124469i
\(646\) 0 0
\(647\) 4.51422e15i 1.56534i 0.622436 + 0.782670i \(0.286143\pi\)
−0.622436 + 0.782670i \(0.713857\pi\)
\(648\) 0 0
\(649\) 9.47507e12i 0.00323025i
\(650\) 0 0
\(651\) 1.03867e14 1.03867e14i 0.0348163 0.0348163i
\(652\) 0 0
\(653\) 1.89288e15 + 1.89288e15i 0.623880 + 0.623880i 0.946521 0.322641i \(-0.104571\pi\)
−0.322641 + 0.946521i \(0.604571\pi\)
\(654\) 0 0
\(655\) 5.57045e15 1.80536
\(656\) 0 0
\(657\) 2.54565e15 0.811313
\(658\) 0 0
\(659\) 2.29856e15 + 2.29856e15i 0.720420 + 0.720420i 0.968691 0.248270i \(-0.0798621\pi\)
−0.248270 + 0.968691i \(0.579862\pi\)
\(660\) 0 0
\(661\) −9.03518e13 + 9.03518e13i −0.0278502 + 0.0278502i −0.720895 0.693045i \(-0.756269\pi\)
0.693045 + 0.720895i \(0.256269\pi\)
\(662\) 0 0
\(663\) 3.90834e15i 1.18486i
\(664\) 0 0
\(665\) 4.11911e13i 0.0122824i
\(666\) 0 0
\(667\) 1.78304e15 1.78304e15i 0.522960 0.522960i
\(668\) 0 0
\(669\) 8.36263e14 + 8.36263e14i 0.241268 + 0.241268i
\(670\) 0 0
\(671\) −2.45861e14 −0.0697776
\(672\) 0 0
\(673\) 5.82094e14 0.162521 0.0812606 0.996693i \(-0.474105\pi\)
0.0812606 + 0.996693i \(0.474105\pi\)
\(674\) 0 0
\(675\) −3.93670e14 3.93670e14i −0.108134 0.108134i
\(676\) 0 0
\(677\) −5.51457e14 + 5.51457e14i −0.149030 + 0.149030i −0.777685 0.628655i \(-0.783606\pi\)
0.628655 + 0.777685i \(0.283606\pi\)
\(678\) 0 0
\(679\) 9.98564e13i 0.0265517i
\(680\) 0 0
\(681\) 9.66887e14i 0.252969i
\(682\) 0 0
\(683\) 1.35222e15 1.35222e15i 0.348123 0.348123i −0.511287 0.859410i \(-0.670831\pi\)
0.859410 + 0.511287i \(0.170831\pi\)
\(684\) 0 0
\(685\) −1.69989e15 1.69989e15i −0.430648 0.430648i
\(686\) 0 0
\(687\) 3.05186e15 0.760856
\(688\) 0 0
\(689\) 4.97161e15 1.21981
\(690\) 0 0
\(691\) −1.10782e15 1.10782e15i −0.267510 0.267510i 0.560586 0.828096i \(-0.310576\pi\)
−0.828096 + 0.560586i \(0.810576\pi\)
\(692\) 0 0
\(693\) −3.89358e12 + 3.89358e12i −0.000925373 + 0.000925373i
\(694\) 0 0
\(695\) 3.84246e15i 0.898863i
\(696\) 0 0
\(697\) 7.20840e15i 1.65981i
\(698\) 0 0
\(699\) −3.50587e15 + 3.50587e15i −0.794641 + 0.794641i
\(700\) 0 0
\(701\) 2.35478e14 + 2.35478e14i 0.0525413 + 0.0525413i 0.732889 0.680348i \(-0.238171\pi\)
−0.680348 + 0.732889i \(0.738171\pi\)
\(702\) 0 0
\(703\) −4.40708e14 −0.0968047
\(704\) 0 0
\(705\) 4.69284e15 1.01483
\(706\) 0 0
\(707\) 1.18285e13 + 1.18285e13i 0.00251840 + 0.00251840i
\(708\) 0 0
\(709\) 6.09313e15 6.09313e15i 1.27728 1.27728i 0.335097 0.942184i \(-0.391231\pi\)
0.942184 0.335097i \(-0.108769\pi\)
\(710\) 0 0
\(711\) 3.85768e15i 0.796240i
\(712\) 0 0
\(713\) 5.81716e15i 1.18228i
\(714\) 0 0
\(715\) −1.84786e14 + 1.84786e14i −0.0369815 + 0.0369815i
\(716\) 0 0
\(717\) 3.24758e15 + 3.24758e15i 0.640037 + 0.640037i
\(718\) 0 0
\(719\) −2.52319e15 −0.489713 −0.244856 0.969559i \(-0.578741\pi\)
−0.244856 + 0.969559i \(0.578741\pi\)
\(720\) 0 0
\(721\) 2.32932e14 0.0445230
\(722\) 0 0
\(723\) 2.21089e15 + 2.21089e15i 0.416204 + 0.416204i
\(724\) 0 0
\(725\) 3.63389e14 3.63389e14i 0.0673771 0.0673771i
\(726\) 0 0
\(727\) 5.17879e14i 0.0945777i 0.998881 + 0.0472888i \(0.0150581\pi\)
−0.998881 + 0.0472888i \(0.984942\pi\)
\(728\) 0 0
\(729\) 5.24717e15i 0.943896i
\(730\) 0 0
\(731\) −1.34325e15 + 1.34325e15i −0.238019 + 0.238019i
\(732\) 0 0
\(733\) −2.12556e15 2.12556e15i −0.371023 0.371023i 0.496827 0.867850i \(-0.334499\pi\)
−0.867850 + 0.496827i \(0.834499\pi\)
\(734\) 0 0
\(735\) 3.95299e15 0.679744
\(736\) 0 0
\(737\) 2.16673e14 0.0367057
\(738\) 0 0
\(739\) 5.41609e14 + 5.41609e14i 0.0903945 + 0.0903945i 0.750858 0.660464i \(-0.229640\pi\)
−0.660464 + 0.750858i \(0.729640\pi\)
\(740\) 0 0
\(741\) 8.28842e14 8.28842e14i 0.136292 0.136292i
\(742\) 0 0
\(743\) 1.22006e16i 1.97670i 0.152185 + 0.988352i \(0.451369\pi\)
−0.152185 + 0.988352i \(0.548631\pi\)
\(744\) 0 0
\(745\) 4.62335e15i 0.738070i
\(746\) 0 0
\(747\) 3.77250e15 3.77250e15i 0.593424 0.593424i
\(748\) 0 0
\(749\) −3.88075e14 3.88075e14i −0.0601542 0.0601542i
\(750\) 0 0
\(751\) 4.45402e15 0.680351 0.340175 0.940362i \(-0.389514\pi\)
0.340175 + 0.940362i \(0.389514\pi\)
\(752\) 0 0
\(753\) −8.85250e15 −1.33258
\(754\) 0 0
\(755\) 3.69918e15 + 3.69918e15i 0.548778 + 0.548778i
\(756\) 0 0
\(757\) −1.55016e15 + 1.55016e15i −0.226646 + 0.226646i −0.811290 0.584644i \(-0.801234\pi\)
0.584644 + 0.811290i \(0.301234\pi\)
\(758\) 0 0
\(759\) 2.59127e14i 0.0373407i
\(760\) 0 0
\(761\) 7.65104e14i 0.108669i −0.998523 0.0543344i \(-0.982696\pi\)
0.998523 0.0543344i \(-0.0173037\pi\)
\(762\) 0 0
\(763\) 2.88036e14 2.88036e14i 0.0403239 0.0403239i
\(764\) 0 0
\(765\) −2.82971e15 2.82971e15i −0.390485 0.390485i
\(766\) 0 0
\(767\) −6.38371e14 −0.0868358
\(768\) 0 0
\(769\) −5.01633e15 −0.672653 −0.336327 0.941745i \(-0.609185\pi\)
−0.336327 + 0.941745i \(0.609185\pi\)
\(770\) 0 0
\(771\) 6.21616e15 + 6.21616e15i 0.821718 + 0.821718i
\(772\) 0 0
\(773\) −2.87468e15 + 2.87468e15i −0.374630 + 0.374630i −0.869160 0.494530i \(-0.835340\pi\)
0.494530 + 0.869160i \(0.335340\pi\)
\(774\) 0 0
\(775\) 1.18556e15i 0.152322i
\(776\) 0 0
\(777\) 1.65748e14i 0.0209958i
\(778\) 0 0
\(779\) −1.52868e15 + 1.52868e15i −0.190924 + 0.190924i
\(780\) 0 0
\(781\) 5.29229e13 + 5.29229e13i 0.00651723 + 0.00651723i
\(782\) 0 0
\(783\) 5.91514e15 0.718250
\(784\) 0 0
\(785\) 6.68081e15 0.799918
\(786\) 0 0
\(787\) −8.90637e15 8.90637e15i −1.05157 1.05157i −0.998596 0.0529775i \(-0.983129\pi\)
−0.0529775 0.998596i \(-0.516871\pi\)
\(788\) 0 0
\(789\) 3.14994e15 3.14994e15i 0.366757 0.366757i
\(790\) 0 0
\(791\) 1.21505e14i 0.0139516i
\(792\) 0 0
\(793\) 1.65646e16i 1.87576i
\(794\) 0 0
\(795\) −4.27736e15 + 4.27736e15i −0.477701 + 0.477701i
\(796\) 0 0
\(797\) −1.42605e13 1.42605e13i −0.00157077 0.00157077i 0.706321 0.707892i \(-0.250354\pi\)
−0.707892 + 0.706321i \(0.750354\pi\)
\(798\) 0 0
\(799\) −1.78629e16 −1.94064
\(800\) 0 0
\(801\) 9.48004e14 0.101585
\(802\) 0 0
\(803\) −5.44402e14 5.44402e14i −0.0575419 0.0575419i
\(804\) 0 0
\(805\) 4.33815e14 4.33815e14i 0.0452301 0.0452301i
\(806\) 0 0
\(807\) 6.71291e15i 0.690409i
\(808\) 0 0
\(809\) 1.34639e16i 1.36601i −0.730414 0.683005i \(-0.760673\pi\)
0.730414 0.683005i \(-0.239327\pi\)
\(810\) 0 0
\(811\) 1.18874e16 1.18874e16i 1.18979 1.18979i 0.212668 0.977125i \(-0.431785\pi\)
0.977125 0.212668i \(-0.0682151\pi\)
\(812\) 0 0
\(813\) 7.52340e15 + 7.52340e15i 0.742876 + 0.742876i
\(814\) 0 0
\(815\) −8.27140e15 −0.805772
\(816\) 0 0
\(817\) −5.69727e14 −0.0547577
\(818\) 0 0
\(819\) −2.62325e14 2.62325e14i −0.0248759 0.0248759i
\(820\) 0 0
\(821\) −1.13620e16 + 1.13620e16i −1.06309 + 1.06309i −0.0652148 + 0.997871i \(0.520773\pi\)
−0.997871 + 0.0652148i \(0.979227\pi\)
\(822\) 0 0
\(823\) 1.24911e15i 0.115319i 0.998336 + 0.0576595i \(0.0183638\pi\)
−0.998336 + 0.0576595i \(0.981636\pi\)
\(824\) 0 0
\(825\) 5.28108e13i 0.00481090i
\(826\) 0 0
\(827\) 4.94453e15 4.94453e15i 0.444472 0.444472i −0.449040 0.893512i \(-0.648234\pi\)
0.893512 + 0.449040i \(0.148234\pi\)
\(828\) 0 0
\(829\) 1.22877e16 + 1.22877e16i 1.08999 + 1.08999i 0.995529 + 0.0944611i \(0.0301128\pi\)
0.0944611 + 0.995529i \(0.469887\pi\)
\(830\) 0 0
\(831\) −1.35556e16 −1.18663
\(832\) 0 0
\(833\) −1.50468e16 −1.29986
\(834\) 0 0
\(835\) −1.01871e16 1.01871e16i −0.868507 0.868507i
\(836\) 0 0
\(837\) −9.64907e15 + 9.64907e15i −0.811887 + 0.811887i
\(838\) 0 0
\(839\) 1.84580e15i 0.153283i −0.997059 0.0766417i \(-0.975580\pi\)
0.997059 0.0766417i \(-0.0244198\pi\)
\(840\) 0 0
\(841\) 6.74035e15i 0.552465i
\(842\) 0 0
\(843\) 6.46258e15 6.46258e15i 0.522822 0.522822i
\(844\) 0 0
\(845\) −4.24938e15 4.24938e15i −0.339323 0.339323i
\(846\) 0 0
\(847\) −7.91013e14 −0.0623483
\(848\) 0 0
\(849\) 3.34963e15 0.260618
\(850\) 0 0
\(851\) 4.64144e15 + 4.64144e15i 0.356484 + 0.356484i
\(852\) 0 0
\(853\) −3.94698e14 + 3.94698e14i −0.0299257 + 0.0299257i −0.721911 0.691986i \(-0.756736\pi\)
0.691986 + 0.721911i \(0.256736\pi\)
\(854\) 0 0
\(855\) 1.20019e15i 0.0898333i
\(856\) 0 0
\(857\) 6.04705e15i 0.446837i 0.974723 + 0.223418i \(0.0717216\pi\)
−0.974723 + 0.223418i \(0.928278\pi\)
\(858\) 0 0
\(859\) −1.24768e16 + 1.24768e16i −0.910209 + 0.910209i −0.996288 0.0860789i \(-0.972566\pi\)
0.0860789 + 0.996288i \(0.472566\pi\)
\(860\) 0 0
\(861\) −5.74931e14 5.74931e14i −0.0414093 0.0414093i
\(862\) 0 0
\(863\) −7.09979e15 −0.504878 −0.252439 0.967613i \(-0.581233\pi\)
−0.252439 + 0.967613i \(0.581233\pi\)
\(864\) 0 0
\(865\) 6.39241e15 0.448824
\(866\) 0 0
\(867\) −5.28313e15 5.28313e15i −0.366257 0.366257i
\(868\) 0 0
\(869\) −8.24989e14 + 8.24989e14i −0.0564728 + 0.0564728i
\(870\) 0 0
\(871\) 1.45981e16i 0.986724i
\(872\) 0 0
\(873\) 2.90954e15i 0.194198i
\(874\) 0 0
\(875\) 7.09141e14 7.09141e14i 0.0467399 0.0467399i
\(876\) 0 0
\(877\) 1.22409e15 + 1.22409e15i 0.0796738 + 0.0796738i 0.745821 0.666147i \(-0.232058\pi\)
−0.666147 + 0.745821i \(0.732058\pi\)
\(878\) 0 0
\(879\) 8.28983e15 0.532852
\(880\) 0 0
\(881\) −2.00411e16 −1.27219 −0.636097 0.771609i \(-0.719452\pi\)
−0.636097 + 0.771609i \(0.719452\pi\)
\(882\) 0 0
\(883\) 9.72509e15 + 9.72509e15i 0.609691 + 0.609691i 0.942865 0.333174i \(-0.108120\pi\)
−0.333174 + 0.942865i \(0.608120\pi\)
\(884\) 0 0
\(885\) 5.49227e14 5.49227e14i 0.0340066 0.0340066i
\(886\) 0 0
\(887\) 1.20718e16i 0.738230i −0.929384 0.369115i \(-0.879661\pi\)
0.929384 0.369115i \(-0.120339\pi\)
\(888\) 0 0
\(889\) 1.66771e15i 0.100730i
\(890\) 0 0
\(891\) −1.81561e14 + 1.81561e14i −0.0108317 + 0.0108317i
\(892\) 0 0
\(893\) −3.78818e15 3.78818e15i −0.223228 0.223228i
\(894\) 0 0
\(895\) −2.55251e16 −1.48573
\(896\) 0 0
\(897\) −1.74583e16 −1.00379
\(898\) 0 0
\(899\) −8.90687e15 8.90687e15i −0.505879 0.505879i
\(900\) 0 0
\(901\) 1.62814e16 1.62814e16i 0.913496 0.913496i
\(902\) 0 0
\(903\) 2.14272e14i 0.0118763i
\(904\) 0 0
\(905\) 7.92246e15i 0.433803i
\(906\) 0 0
\(907\) 2.15151e16 2.15151e16i 1.16387 1.16387i 0.180243 0.983622i \(-0.442312\pi\)
0.983622 0.180243i \(-0.0576884\pi\)
\(908\) 0 0
\(909\) −3.44650e14 3.44650e14i −0.0184195 0.0184195i
\(910\) 0 0
\(911\) 2.28388e16 1.20593 0.602964 0.797768i \(-0.293986\pi\)
0.602964 + 0.797768i \(0.293986\pi\)
\(912\) 0 0
\(913\) −1.61354e15 −0.0841765
\(914\) 0 0
\(915\) 1.42514e16 + 1.42514e16i 0.734586 + 0.734586i
\(916\) 0 0
\(917\) 1.69116e15 1.69116e15i 0.0861296 0.0861296i
\(918\) 0 0
\(919\) 2.19185e16i 1.10300i 0.834175 + 0.551500i \(0.185944\pi\)
−0.834175 + 0.551500i \(0.814056\pi\)
\(920\) 0 0
\(921\) 3.53365e15i 0.175709i
\(922\) 0 0
\(923\) −3.56561e15 + 3.56561e15i −0.175196 + 0.175196i
\(924\) 0 0
\(925\) 9.45938e14 + 9.45938e14i 0.0459286 + 0.0459286i
\(926\) 0 0
\(927\) −6.78698e15 −0.325640
\(928\) 0 0
\(929\) 4.13223e16 1.95929 0.979644 0.200741i \(-0.0643350\pi\)
0.979644 + 0.200741i \(0.0643350\pi\)
\(930\) 0 0
\(931\) −3.19096e15 3.19096e15i −0.149520 0.149520i
\(932\) 0 0
\(933\) −5.02677e15 + 5.02677e15i −0.232777 + 0.232777i
\(934\) 0 0
\(935\) 1.21030e15i 0.0553898i
\(936\) 0 0
\(937\) 9.03330e15i 0.408582i −0.978910 0.204291i \(-0.934511\pi\)
0.978910 0.204291i \(-0.0654888\pi\)
\(938\) 0 0
\(939\) −2.56213e15 + 2.56213e15i −0.114535 + 0.114535i
\(940\) 0 0
\(941\) 3.44069e15 + 3.44069e15i 0.152021 + 0.152021i 0.779020 0.626999i \(-0.215717\pi\)
−0.626999 + 0.779020i \(0.715717\pi\)
\(942\) 0 0
\(943\) 3.21995e16 1.40616
\(944\) 0 0
\(945\) 1.43916e15 0.0621204
\(946\) 0 0
\(947\) 2.83365e16 + 2.83365e16i 1.20899 + 1.20899i 0.971356 + 0.237630i \(0.0763705\pi\)
0.237630 + 0.971356i \(0.423629\pi\)
\(948\) 0 0
\(949\) 3.66784e16 3.66784e16i 1.54684 1.54684i
\(950\) 0 0
\(951\) 2.04977e16i 0.854501i
\(952\) 0 0
\(953\) 2.18190e16i 0.899133i 0.893247 + 0.449566i \(0.148422\pi\)
−0.893247 + 0.449566i \(0.851578\pi\)
\(954\) 0 0
\(955\) −1.81723e16 + 1.81723e16i −0.740273 + 0.740273i
\(956\) 0 0
\(957\) −3.96759e14 3.96759e14i −0.0159776 0.0159776i
\(958\) 0 0
\(959\) −1.03215e15 −0.0410905
\(960\) 0 0
\(961\) 3.65018e15 0.143660
\(962\) 0 0
\(963\) 1.13074e16 + 1.13074e16i 0.439966 + 0.439966i
\(964\) 0 0
\(965\) −2.30017e16 + 2.30017e16i −0.884829 + 0.884829i
\(966\) 0 0
\(967\) 3.21064e16i 1.22109i 0.791983 + 0.610543i \(0.209049\pi\)
−0.791983 + 0.610543i \(0.790951\pi\)
\(968\) 0 0
\(969\) 5.42872e15i 0.204134i
\(970\) 0 0
\(971\) 7.64302e15 7.64302e15i 0.284158 0.284158i −0.550607 0.834765i \(-0.685604\pi\)
0.834765 + 0.550607i \(0.185604\pi\)
\(972\) 0 0
\(973\) −1.16655e15 1.16655e15i −0.0428828 0.0428828i
\(974\) 0 0
\(975\) −3.55806e15 −0.129327
\(976\) 0 0
\(977\) 4.77134e16 1.71483 0.857413 0.514628i \(-0.172070\pi\)
0.857413 + 0.514628i \(0.172070\pi\)
\(978\) 0 0
\(979\) −2.02736e14 2.02736e14i −0.00720488 0.00720488i
\(980\) 0 0
\(981\) −8.39257e15 + 8.39257e15i −0.294928 + 0.294928i
\(982\) 0 0
\(983\) 5.15731e16i 1.79217i −0.443884 0.896084i \(-0.646400\pi\)
0.443884 0.896084i \(-0.353600\pi\)
\(984\) 0 0
\(985\) 2.02907e16i 0.697266i
\(986\) 0 0
\(987\) 1.42472e15 1.42472e15i 0.0484155 0.0484155i
\(988\) 0 0
\(989\) 6.00024e15 + 6.00024e15i 0.201646 + 0.201646i
\(990\) 0 0
\(991\) 5.04756e16 1.67755 0.838776 0.544476i \(-0.183272\pi\)
0.838776 + 0.544476i \(0.183272\pi\)
\(992\) 0 0
\(993\) 1.09004e16 0.358278
\(994\) 0 0
\(995\) −7.76728e15 7.76728e15i −0.252489 0.252489i
\(996\) 0 0
\(997\) 5.06141e15 5.06141e15i 0.162723 0.162723i −0.621049 0.783772i \(-0.713293\pi\)
0.783772 + 0.621049i \(0.213293\pi\)
\(998\) 0 0
\(999\) 1.53977e16i 0.489606i
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 64.12.e.a.17.8 42
4.3 odd 2 16.12.e.a.13.6 yes 42
8.3 odd 2 128.12.e.b.33.8 42
8.5 even 2 128.12.e.a.33.14 42
16.3 odd 4 128.12.e.b.97.8 42
16.5 even 4 inner 64.12.e.a.49.8 42
16.11 odd 4 16.12.e.a.5.6 42
16.13 even 4 128.12.e.a.97.14 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.12.e.a.5.6 42 16.11 odd 4
16.12.e.a.13.6 yes 42 4.3 odd 2
64.12.e.a.17.8 42 1.1 even 1 trivial
64.12.e.a.49.8 42 16.5 even 4 inner
128.12.e.a.33.14 42 8.5 even 2
128.12.e.a.97.14 42 16.13 even 4
128.12.e.b.33.8 42 8.3 odd 2
128.12.e.b.97.8 42 16.3 odd 4