Properties

Label 160.6.o.a.47.10
Level $160$
Weight $6$
Character 160.47
Analytic conductor $25.661$
Analytic rank $0$
Dimension $56$
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] = [160,6,Mod(47,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.47");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.o (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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.10
Character \(\chi\) \(=\) 160.47
Dual form 160.6.o.a.143.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-10.0657 + 10.0657i) q^{3} +(23.9125 - 50.5291i) q^{5} +(-96.6855 + 96.6855i) q^{7} +40.3635i q^{9} +O(q^{10})\) \(q+(-10.0657 + 10.0657i) q^{3} +(23.9125 - 50.5291i) q^{5} +(-96.6855 + 96.6855i) q^{7} +40.3635i q^{9} +283.587 q^{11} +(-699.519 - 699.519i) q^{13} +(267.915 + 749.307i) q^{15} +(399.025 + 399.025i) q^{17} +338.083i q^{19} -1946.41i q^{21} +(-594.687 - 594.687i) q^{23} +(-1981.38 - 2416.56i) q^{25} +(-2852.25 - 2852.25i) q^{27} +5582.41 q^{29} +144.182i q^{31} +(-2854.50 + 2854.50i) q^{33} +(2573.44 + 7197.43i) q^{35} +(11418.4 - 11418.4i) q^{37} +14082.3 q^{39} +11892.0 q^{41} +(8345.23 - 8345.23i) q^{43} +(2039.53 + 965.191i) q^{45} +(17955.5 - 17955.5i) q^{47} -1889.19i q^{49} -8032.92 q^{51} +(3977.21 + 3977.21i) q^{53} +(6781.27 - 14329.4i) q^{55} +(-3403.04 - 3403.04i) q^{57} +17288.6i q^{59} -6236.61i q^{61} +(-3902.56 - 3902.56i) q^{63} +(-52073.3 + 18618.8i) q^{65} +(-39565.2 - 39565.2i) q^{67} +11971.9 q^{69} -50088.5i q^{71} +(42184.8 - 42184.8i) q^{73} +(44268.3 + 4380.30i) q^{75} +(-27418.7 + 27418.7i) q^{77} -22093.6 q^{79} +47611.5 q^{81} +(-51337.5 + 51337.5i) q^{83} +(29704.0 - 10620.7i) q^{85} +(-56190.8 + 56190.8i) q^{87} +78727.1i q^{89} +135267. q^{91} +(-1451.29 - 1451.29i) q^{93} +(17083.0 + 8084.41i) q^{95} +(-9943.42 - 9943.42i) q^{97} +11446.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{3} + 8 q^{11} - 408 q^{17} - 3120 q^{25} - 968 q^{27} - 976 q^{33} + 4780 q^{35} - 8 q^{41} - 1308 q^{43} - 20872 q^{51} + 968 q^{57} + 17680 q^{65} - 89252 q^{67} - 25184 q^{73} + 127740 q^{75} - 67792 q^{81} + 126444 q^{83} - 329432 q^{91} + 212576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) −10.0657 + 10.0657i −0.645715 + 0.645715i −0.951954 0.306240i \(-0.900929\pi\)
0.306240 + 0.951954i \(0.400929\pi\)
\(4\) 0 0
\(5\) 23.9125 50.5291i 0.427760 0.903892i
\(6\) 0 0
\(7\) −96.6855 + 96.6855i −0.745790 + 0.745790i −0.973686 0.227896i \(-0.926815\pi\)
0.227896 + 0.973686i \(0.426815\pi\)
\(8\) 0 0
\(9\) 40.3635i 0.166105i
\(10\) 0 0
\(11\) 283.587 0.706650 0.353325 0.935501i \(-0.385051\pi\)
0.353325 + 0.935501i \(0.385051\pi\)
\(12\) 0 0
\(13\) −699.519 699.519i −1.14800 1.14800i −0.986946 0.161052i \(-0.948511\pi\)
−0.161052 0.986946i \(-0.551489\pi\)
\(14\) 0 0
\(15\) 267.915 + 749.307i 0.307446 + 0.859868i
\(16\) 0 0
\(17\) 399.025 + 399.025i 0.334871 + 0.334871i 0.854433 0.519562i \(-0.173905\pi\)
−0.519562 + 0.854433i \(0.673905\pi\)
\(18\) 0 0
\(19\) 338.083i 0.214852i 0.994213 + 0.107426i \(0.0342609\pi\)
−0.994213 + 0.107426i \(0.965739\pi\)
\(20\) 0 0
\(21\) 1946.41i 0.963135i
\(22\) 0 0
\(23\) −594.687 594.687i −0.234406 0.234406i 0.580123 0.814529i \(-0.303005\pi\)
−0.814529 + 0.580123i \(0.803005\pi\)
\(24\) 0 0
\(25\) −1981.38 2416.56i −0.634043 0.773298i
\(26\) 0 0
\(27\) −2852.25 2852.25i −0.752971 0.752971i
\(28\) 0 0
\(29\) 5582.41 1.23261 0.616306 0.787507i \(-0.288628\pi\)
0.616306 + 0.787507i \(0.288628\pi\)
\(30\) 0 0
\(31\) 144.182i 0.0269468i 0.999909 + 0.0134734i \(0.00428884\pi\)
−0.999909 + 0.0134734i \(0.995711\pi\)
\(32\) 0 0
\(33\) −2854.50 + 2854.50i −0.456294 + 0.456294i
\(34\) 0 0
\(35\) 2573.44 + 7197.43i 0.355095 + 0.993133i
\(36\) 0 0
\(37\) 11418.4 11418.4i 1.37120 1.37120i 0.512523 0.858674i \(-0.328711\pi\)
0.858674 0.512523i \(-0.171289\pi\)
\(38\) 0 0
\(39\) 14082.3 1.48256
\(40\) 0 0
\(41\) 11892.0 1.10483 0.552413 0.833570i \(-0.313707\pi\)
0.552413 + 0.833570i \(0.313707\pi\)
\(42\) 0 0
\(43\) 8345.23 8345.23i 0.688283 0.688283i −0.273569 0.961852i \(-0.588204\pi\)
0.961852 + 0.273569i \(0.0882042\pi\)
\(44\) 0 0
\(45\) 2039.53 + 965.191i 0.150141 + 0.0710530i
\(46\) 0 0
\(47\) 17955.5 17955.5i 1.18564 1.18564i 0.207383 0.978260i \(-0.433505\pi\)
0.978260 0.207383i \(-0.0664946\pi\)
\(48\) 0 0
\(49\) 1889.19i 0.112405i
\(50\) 0 0
\(51\) −8032.92 −0.432462
\(52\) 0 0
\(53\) 3977.21 + 3977.21i 0.194486 + 0.194486i 0.797631 0.603145i \(-0.206086\pi\)
−0.603145 + 0.797631i \(0.706086\pi\)
\(54\) 0 0
\(55\) 6781.27 14329.4i 0.302276 0.638735i
\(56\) 0 0
\(57\) −3403.04 3403.04i −0.138733 0.138733i
\(58\) 0 0
\(59\) 17288.6i 0.646590i 0.946298 + 0.323295i \(0.104791\pi\)
−0.946298 + 0.323295i \(0.895209\pi\)
\(60\) 0 0
\(61\) 6236.61i 0.214597i −0.994227 0.107299i \(-0.965780\pi\)
0.994227 0.107299i \(-0.0342201\pi\)
\(62\) 0 0
\(63\) −3902.56 3902.56i −0.123879 0.123879i
\(64\) 0 0
\(65\) −52073.3 + 18618.8i −1.52873 + 0.546599i
\(66\) 0 0
\(67\) −39565.2 39565.2i −1.07678 1.07678i −0.996796 0.0799829i \(-0.974513\pi\)
−0.0799829 0.996796i \(-0.525487\pi\)
\(68\) 0 0
\(69\) 11971.9 0.302719
\(70\) 0 0
\(71\) 50088.5i 1.17921i −0.807691 0.589606i \(-0.799283\pi\)
0.807691 0.589606i \(-0.200717\pi\)
\(72\) 0 0
\(73\) 42184.8 42184.8i 0.926508 0.926508i −0.0709703 0.997478i \(-0.522610\pi\)
0.997478 + 0.0709703i \(0.0226096\pi\)
\(74\) 0 0
\(75\) 44268.3 + 4380.30i 0.908741 + 0.0899189i
\(76\) 0 0
\(77\) −27418.7 + 27418.7i −0.527012 + 0.527012i
\(78\) 0 0
\(79\) −22093.6 −0.398290 −0.199145 0.979970i \(-0.563816\pi\)
−0.199145 + 0.979970i \(0.563816\pi\)
\(80\) 0 0
\(81\) 47611.5 0.806304
\(82\) 0 0
\(83\) −51337.5 + 51337.5i −0.817974 + 0.817974i −0.985814 0.167840i \(-0.946321\pi\)
0.167840 + 0.985814i \(0.446321\pi\)
\(84\) 0 0
\(85\) 29704.0 10620.7i 0.445932 0.159443i
\(86\) 0 0
\(87\) −56190.8 + 56190.8i −0.795916 + 0.795916i
\(88\) 0 0
\(89\) 78727.1i 1.05354i 0.850009 + 0.526768i \(0.176596\pi\)
−0.850009 + 0.526768i \(0.823404\pi\)
\(90\) 0 0
\(91\) 135267. 1.71233
\(92\) 0 0
\(93\) −1451.29 1451.29i −0.0173999 0.0173999i
\(94\) 0 0
\(95\) 17083.0 + 8084.41i 0.194203 + 0.0919050i
\(96\) 0 0
\(97\) −9943.42 9943.42i −0.107302 0.107302i 0.651418 0.758719i \(-0.274175\pi\)
−0.758719 + 0.651418i \(0.774175\pi\)
\(98\) 0 0
\(99\) 11446.5i 0.117378i
\(100\) 0 0
\(101\) 34886.7i 0.340296i −0.985419 0.170148i \(-0.945575\pi\)
0.985419 0.170148i \(-0.0544246\pi\)
\(102\) 0 0
\(103\) 74849.2 + 74849.2i 0.695175 + 0.695175i 0.963366 0.268191i \(-0.0864257\pi\)
−0.268191 + 0.963366i \(0.586426\pi\)
\(104\) 0 0
\(105\) −98350.6 46543.7i −0.870570 0.411991i
\(106\) 0 0
\(107\) −117317. 117317.i −0.990608 0.990608i 0.00934815 0.999956i \(-0.497024\pi\)
−0.999956 + 0.00934815i \(0.997024\pi\)
\(108\) 0 0
\(109\) −111201. −0.896486 −0.448243 0.893912i \(-0.647950\pi\)
−0.448243 + 0.893912i \(0.647950\pi\)
\(110\) 0 0
\(111\) 229868.i 1.77080i
\(112\) 0 0
\(113\) 66036.6 66036.6i 0.486506 0.486506i −0.420696 0.907202i \(-0.638214\pi\)
0.907202 + 0.420696i \(0.138214\pi\)
\(114\) 0 0
\(115\) −44269.5 + 15828.6i −0.312148 + 0.111608i
\(116\) 0 0
\(117\) 28235.0 28235.0i 0.190688 0.190688i
\(118\) 0 0
\(119\) −77159.8 −0.499487
\(120\) 0 0
\(121\) −80629.6 −0.500646
\(122\) 0 0
\(123\) −119701. + 119701.i −0.713403 + 0.713403i
\(124\) 0 0
\(125\) −169486. + 42331.7i −0.970196 + 0.242321i
\(126\) 0 0
\(127\) −78416.4 + 78416.4i −0.431418 + 0.431418i −0.889110 0.457693i \(-0.848676\pi\)
0.457693 + 0.889110i \(0.348676\pi\)
\(128\) 0 0
\(129\) 168001.i 0.888870i
\(130\) 0 0
\(131\) 309770. 1.57711 0.788554 0.614966i \(-0.210830\pi\)
0.788554 + 0.614966i \(0.210830\pi\)
\(132\) 0 0
\(133\) −32687.7 32687.7i −0.160234 0.160234i
\(134\) 0 0
\(135\) −212326. + 75917.3i −1.00270 + 0.358514i
\(136\) 0 0
\(137\) 61997.1 + 61997.1i 0.282208 + 0.282208i 0.833989 0.551781i \(-0.186051\pi\)
−0.551781 + 0.833989i \(0.686051\pi\)
\(138\) 0 0
\(139\) 114337.i 0.501938i −0.967995 0.250969i \(-0.919251\pi\)
0.967995 0.250969i \(-0.0807492\pi\)
\(140\) 0 0
\(141\) 361470.i 1.53117i
\(142\) 0 0
\(143\) −198374. 198374.i −0.811232 0.811232i
\(144\) 0 0
\(145\) 133489. 282074.i 0.527262 1.11415i
\(146\) 0 0
\(147\) 19016.0 + 19016.0i 0.0725815 + 0.0725815i
\(148\) 0 0
\(149\) 63562.8 0.234551 0.117276 0.993099i \(-0.462584\pi\)
0.117276 + 0.993099i \(0.462584\pi\)
\(150\) 0 0
\(151\) 397853.i 1.41998i −0.704214 0.709988i \(-0.748700\pi\)
0.704214 0.709988i \(-0.251300\pi\)
\(152\) 0 0
\(153\) −16106.0 + 16106.0i −0.0556237 + 0.0556237i
\(154\) 0 0
\(155\) 7285.39 + 3447.75i 0.0243570 + 0.0115267i
\(156\) 0 0
\(157\) −283821. + 283821.i −0.918958 + 0.918958i −0.996954 0.0779961i \(-0.975148\pi\)
0.0779961 + 0.996954i \(0.475148\pi\)
\(158\) 0 0
\(159\) −80066.8 −0.251165
\(160\) 0 0
\(161\) 114995. 0.349635
\(162\) 0 0
\(163\) −298452. + 298452.i −0.879843 + 0.879843i −0.993518 0.113675i \(-0.963738\pi\)
0.113675 + 0.993518i \(0.463738\pi\)
\(164\) 0 0
\(165\) 75977.1 + 212493.i 0.217256 + 0.607625i
\(166\) 0 0
\(167\) 234469. 234469.i 0.650570 0.650570i −0.302560 0.953130i \(-0.597841\pi\)
0.953130 + 0.302560i \(0.0978413\pi\)
\(168\) 0 0
\(169\) 607360.i 1.63580i
\(170\) 0 0
\(171\) −13646.2 −0.0356879
\(172\) 0 0
\(173\) −131530. 131530.i −0.334126 0.334126i 0.520025 0.854151i \(-0.325923\pi\)
−0.854151 + 0.520025i \(0.825923\pi\)
\(174\) 0 0
\(175\) 425217. + 42074.8i 1.04958 + 0.103855i
\(176\) 0 0
\(177\) −174022. 174022.i −0.417513 0.417513i
\(178\) 0 0
\(179\) 370114.i 0.863383i −0.902021 0.431691i \(-0.857917\pi\)
0.902021 0.431691i \(-0.142083\pi\)
\(180\) 0 0
\(181\) 473727.i 1.07481i −0.843324 0.537405i \(-0.819405\pi\)
0.843324 0.537405i \(-0.180595\pi\)
\(182\) 0 0
\(183\) 62775.9 + 62775.9i 0.138569 + 0.138569i
\(184\) 0 0
\(185\) −303918. 850002.i −0.652871 1.82596i
\(186\) 0 0
\(187\) 113158. + 113158.i 0.236636 + 0.236636i
\(188\) 0 0
\(189\) 551543. 1.12312
\(190\) 0 0
\(191\) 287004.i 0.569252i 0.958639 + 0.284626i \(0.0918695\pi\)
−0.958639 + 0.284626i \(0.908131\pi\)
\(192\) 0 0
\(193\) −391702. + 391702.i −0.756941 + 0.756941i −0.975765 0.218823i \(-0.929778\pi\)
0.218823 + 0.975765i \(0.429778\pi\)
\(194\) 0 0
\(195\) 336743. 711566.i 0.634179 1.34007i
\(196\) 0 0
\(197\) 686198. 686198.i 1.25975 1.25975i 0.308535 0.951213i \(-0.400161\pi\)
0.951213 0.308535i \(-0.0998386\pi\)
\(198\) 0 0
\(199\) 775467. 1.38813 0.694066 0.719912i \(-0.255818\pi\)
0.694066 + 0.719912i \(0.255818\pi\)
\(200\) 0 0
\(201\) 796503. 1.39058
\(202\) 0 0
\(203\) −539738. + 539738.i −0.919270 + 0.919270i
\(204\) 0 0
\(205\) 284367. 600891.i 0.472601 0.998645i
\(206\) 0 0
\(207\) 24003.6 24003.6i 0.0389360 0.0389360i
\(208\) 0 0
\(209\) 95875.8i 0.151825i
\(210\) 0 0
\(211\) 307476. 0.475450 0.237725 0.971333i \(-0.423598\pi\)
0.237725 + 0.971333i \(0.423598\pi\)
\(212\) 0 0
\(213\) 504176. + 504176.i 0.761435 + 0.761435i
\(214\) 0 0
\(215\) −222122. 621233.i −0.327714 0.916554i
\(216\) 0 0
\(217\) −13940.3 13940.3i −0.0200966 0.0200966i
\(218\) 0 0
\(219\) 849240.i 1.19652i
\(220\) 0 0
\(221\) 558251.i 0.768862i
\(222\) 0 0
\(223\) 836770. + 836770.i 1.12679 + 1.12679i 0.990696 + 0.136097i \(0.0434557\pi\)
0.136097 + 0.990696i \(0.456544\pi\)
\(224\) 0 0
\(225\) 97540.6 79975.5i 0.128448 0.105318i
\(226\) 0 0
\(227\) 79173.7 + 79173.7i 0.101980 + 0.101980i 0.756256 0.654276i \(-0.227026\pi\)
−0.654276 + 0.756256i \(0.727026\pi\)
\(228\) 0 0
\(229\) −686727. −0.865357 −0.432679 0.901548i \(-0.642432\pi\)
−0.432679 + 0.901548i \(0.642432\pi\)
\(230\) 0 0
\(231\) 551977.i 0.680599i
\(232\) 0 0
\(233\) −566816. + 566816.i −0.683994 + 0.683994i −0.960898 0.276903i \(-0.910692\pi\)
0.276903 + 0.960898i \(0.410692\pi\)
\(234\) 0 0
\(235\) −477916. 1.33664e6i −0.564523 1.57886i
\(236\) 0 0
\(237\) 222388. 222388.i 0.257182 0.257182i
\(238\) 0 0
\(239\) −764329. −0.865537 −0.432769 0.901505i \(-0.642463\pi\)
−0.432769 + 0.901505i \(0.642463\pi\)
\(240\) 0 0
\(241\) 257789. 0.285905 0.142952 0.989730i \(-0.454340\pi\)
0.142952 + 0.989730i \(0.454340\pi\)
\(242\) 0 0
\(243\) 213854. 213854.i 0.232328 0.232328i
\(244\) 0 0
\(245\) −95459.0 45175.2i −0.101602 0.0480823i
\(246\) 0 0
\(247\) 236495. 236495.i 0.246650 0.246650i
\(248\) 0 0
\(249\) 1.03350e6i 1.05636i
\(250\) 0 0
\(251\) 818014. 0.819552 0.409776 0.912186i \(-0.365607\pi\)
0.409776 + 0.912186i \(0.365607\pi\)
\(252\) 0 0
\(253\) −168645. 168645.i −0.165643 0.165643i
\(254\) 0 0
\(255\) −192087. + 405897.i −0.184990 + 0.390899i
\(256\) 0 0
\(257\) 212976. + 212976.i 0.201140 + 0.201140i 0.800488 0.599348i \(-0.204574\pi\)
−0.599348 + 0.800488i \(0.704574\pi\)
\(258\) 0 0
\(259\) 2.20798e6i 2.04525i
\(260\) 0 0
\(261\) 225325.i 0.204743i
\(262\) 0 0
\(263\) −1.19052e6 1.19052e6i −1.06132 1.06132i −0.997993 0.0633261i \(-0.979829\pi\)
−0.0633261 0.997993i \(-0.520171\pi\)
\(264\) 0 0
\(265\) 296070. 105860.i 0.258988 0.0926012i
\(266\) 0 0
\(267\) −792444. 792444.i −0.680284 0.680284i
\(268\) 0 0
\(269\) 507577. 0.427682 0.213841 0.976868i \(-0.431402\pi\)
0.213841 + 0.976868i \(0.431402\pi\)
\(270\) 0 0
\(271\) 680345.i 0.562737i 0.959600 + 0.281369i \(0.0907884\pi\)
−0.959600 + 0.281369i \(0.909212\pi\)
\(272\) 0 0
\(273\) −1.36155e6 + 1.36155e6i −1.10568 + 1.10568i
\(274\) 0 0
\(275\) −561894. 685303.i −0.448046 0.546451i
\(276\) 0 0
\(277\) −1.29751e6 + 1.29751e6i −1.01604 + 1.01604i −0.0161744 + 0.999869i \(0.505149\pi\)
−0.999869 + 0.0161744i \(0.994851\pi\)
\(278\) 0 0
\(279\) −5819.68 −0.00447599
\(280\) 0 0
\(281\) 1.38793e6 1.04858 0.524289 0.851541i \(-0.324331\pi\)
0.524289 + 0.851541i \(0.324331\pi\)
\(282\) 0 0
\(283\) 506188. 506188.i 0.375704 0.375704i −0.493846 0.869550i \(-0.664409\pi\)
0.869550 + 0.493846i \(0.164409\pi\)
\(284\) 0 0
\(285\) −253328. + 90577.4i −0.184744 + 0.0660553i
\(286\) 0 0
\(287\) −1.14978e6 + 1.14978e6i −0.823969 + 0.823969i
\(288\) 0 0
\(289\) 1.10142e6i 0.775723i
\(290\) 0 0
\(291\) 200175. 0.138573
\(292\) 0 0
\(293\) −358870. 358870.i −0.244212 0.244212i 0.574378 0.818590i \(-0.305244\pi\)
−0.818590 + 0.574378i \(0.805244\pi\)
\(294\) 0 0
\(295\) 873577. + 413413.i 0.584448 + 0.276585i
\(296\) 0 0
\(297\) −808860. 808860.i −0.532087 0.532087i
\(298\) 0 0
\(299\) 831990.i 0.538196i
\(300\) 0 0
\(301\) 1.61373e6i 1.02663i
\(302\) 0 0
\(303\) 351159. + 351159.i 0.219734 + 0.219734i
\(304\) 0 0
\(305\) −315131. 149133.i −0.193973 0.0917961i
\(306\) 0 0
\(307\) −800065. 800065.i −0.484484 0.484484i 0.422076 0.906560i \(-0.361301\pi\)
−0.906560 + 0.422076i \(0.861301\pi\)
\(308\) 0 0
\(309\) −1.50682e6 −0.897770
\(310\) 0 0
\(311\) 1.39145e6i 0.815766i −0.913034 0.407883i \(-0.866267\pi\)
0.913034 0.407883i \(-0.133733\pi\)
\(312\) 0 0
\(313\) −731178. + 731178.i −0.421854 + 0.421854i −0.885842 0.463987i \(-0.846418\pi\)
0.463987 + 0.885842i \(0.346418\pi\)
\(314\) 0 0
\(315\) −290513. + 103873.i −0.164964 + 0.0589829i
\(316\) 0 0
\(317\) −415835. + 415835.i −0.232420 + 0.232420i −0.813702 0.581282i \(-0.802551\pi\)
0.581282 + 0.813702i \(0.302551\pi\)
\(318\) 0 0
\(319\) 1.58310e6 0.871025
\(320\) 0 0
\(321\) 2.36176e6 1.27930
\(322\) 0 0
\(323\) −134903. + 134903.i −0.0719477 + 0.0719477i
\(324\) 0 0
\(325\) −304411. + 3.07644e6i −0.159864 + 1.61562i
\(326\) 0 0
\(327\) 1.11932e6 1.11932e6i 0.578874 0.578874i
\(328\) 0 0
\(329\) 3.47208e6i 1.76848i
\(330\) 0 0
\(331\) 2.46277e6 1.23553 0.617765 0.786363i \(-0.288038\pi\)
0.617765 + 0.786363i \(0.288038\pi\)
\(332\) 0 0
\(333\) 460885. + 460885.i 0.227762 + 0.227762i
\(334\) 0 0
\(335\) −2.94530e6 + 1.05309e6i −1.43390 + 0.512690i
\(336\) 0 0
\(337\) −1.44346e6 1.44346e6i −0.692355 0.692355i 0.270395 0.962749i \(-0.412846\pi\)
−0.962749 + 0.270395i \(0.912846\pi\)
\(338\) 0 0
\(339\) 1.32941e6i 0.628289i
\(340\) 0 0
\(341\) 40888.1i 0.0190419i
\(342\) 0 0
\(343\) −1.44234e6 1.44234e6i −0.661959 0.661959i
\(344\) 0 0
\(345\) 286278. 604929.i 0.129491 0.273625i
\(346\) 0 0
\(347\) 966813. + 966813.i 0.431041 + 0.431041i 0.888982 0.457941i \(-0.151413\pi\)
−0.457941 + 0.888982i \(0.651413\pi\)
\(348\) 0 0
\(349\) 1.77134e6 0.778462 0.389231 0.921140i \(-0.372741\pi\)
0.389231 + 0.921140i \(0.372741\pi\)
\(350\) 0 0
\(351\) 3.99041e6i 1.72882i
\(352\) 0 0
\(353\) 1.32730e6 1.32730e6i 0.566932 0.566932i −0.364336 0.931268i \(-0.618704\pi\)
0.931268 + 0.364336i \(0.118704\pi\)
\(354\) 0 0
\(355\) −2.53093e6 1.19774e6i −1.06588 0.504420i
\(356\) 0 0
\(357\) 776668. 776668.i 0.322526 0.322526i
\(358\) 0 0
\(359\) −3.44839e6 −1.41215 −0.706074 0.708138i \(-0.749535\pi\)
−0.706074 + 0.708138i \(0.749535\pi\)
\(360\) 0 0
\(361\) 2.36180e6 0.953839
\(362\) 0 0
\(363\) 811593. 811593.i 0.323275 0.323275i
\(364\) 0 0
\(365\) −1.12282e6 3.14031e6i −0.441141 1.23379i
\(366\) 0 0
\(367\) −2.52705e6 + 2.52705e6i −0.979374 + 0.979374i −0.999792 0.0204173i \(-0.993501\pi\)
0.0204173 + 0.999792i \(0.493501\pi\)
\(368\) 0 0
\(369\) 480001.i 0.183517i
\(370\) 0 0
\(371\) −769077. −0.290092
\(372\) 0 0
\(373\) −268402. 268402.i −0.0998882 0.0998882i 0.655397 0.755285i \(-0.272501\pi\)
−0.755285 + 0.655397i \(0.772501\pi\)
\(374\) 0 0
\(375\) 1.27990e6 2.13210e6i 0.470000 0.782940i
\(376\) 0 0
\(377\) −3.90500e6 3.90500e6i −1.41504 1.41504i
\(378\) 0 0
\(379\) 847168.i 0.302951i −0.988461 0.151475i \(-0.951598\pi\)
0.988461 0.151475i \(-0.0484024\pi\)
\(380\) 0 0
\(381\) 1.57863e6i 0.557145i
\(382\) 0 0
\(383\) 2.32473e6 + 2.32473e6i 0.809795 + 0.809795i 0.984603 0.174807i \(-0.0559303\pi\)
−0.174807 + 0.984603i \(0.555930\pi\)
\(384\) 0 0
\(385\) 729794. + 2.04109e6i 0.250928 + 0.701797i
\(386\) 0 0
\(387\) 336842. + 336842.i 0.114327 + 0.114327i
\(388\) 0 0
\(389\) −3.52032e6 −1.17953 −0.589764 0.807575i \(-0.700779\pi\)
−0.589764 + 0.807575i \(0.700779\pi\)
\(390\) 0 0
\(391\) 474590.i 0.156992i
\(392\) 0 0
\(393\) −3.11805e6 + 3.11805e6i −1.01836 + 1.01836i
\(394\) 0 0
\(395\) −528314. + 1.11637e6i −0.170372 + 0.360011i
\(396\) 0 0
\(397\) 239629. 239629.i 0.0763067 0.0763067i −0.667923 0.744230i \(-0.732817\pi\)
0.744230 + 0.667923i \(0.232817\pi\)
\(398\) 0 0
\(399\) 658050. 0.206931
\(400\) 0 0
\(401\) −1.56689e6 −0.486607 −0.243303 0.969950i \(-0.578231\pi\)
−0.243303 + 0.969950i \(0.578231\pi\)
\(402\) 0 0
\(403\) 100858. 100858.i 0.0309348 0.0309348i
\(404\) 0 0
\(405\) 1.13851e6 2.40577e6i 0.344905 0.728812i
\(406\) 0 0
\(407\) 3.23810e6 3.23810e6i 0.968955 0.968955i
\(408\) 0 0
\(409\) 220244.i 0.0651022i 0.999470 + 0.0325511i \(0.0103632\pi\)
−0.999470 + 0.0325511i \(0.989637\pi\)
\(410\) 0 0
\(411\) −1.24809e6 −0.364452
\(412\) 0 0
\(413\) −1.67156e6 1.67156e6i −0.482220 0.482220i
\(414\) 0 0
\(415\) 1.36643e6 + 3.82165e6i 0.389464 + 1.08926i
\(416\) 0 0
\(417\) 1.15088e6 + 1.15088e6i 0.324109 + 0.324109i
\(418\) 0 0
\(419\) 3.95300e6i 1.10000i −0.835165 0.549999i \(-0.814628\pi\)
0.835165 0.549999i \(-0.185372\pi\)
\(420\) 0 0
\(421\) 1.92105e6i 0.528244i −0.964489 0.264122i \(-0.914918\pi\)
0.964489 0.264122i \(-0.0850821\pi\)
\(422\) 0 0
\(423\) 724748. + 724748.i 0.196941 + 0.196941i
\(424\) 0 0
\(425\) 173644. 1.75489e6i 0.0466324 0.471278i
\(426\) 0 0
\(427\) 602990. + 602990.i 0.160045 + 0.160045i
\(428\) 0 0
\(429\) 3.99355e6 1.04765
\(430\) 0 0
\(431\) 5.12491e6i 1.32890i −0.747331 0.664451i \(-0.768665\pi\)
0.747331 0.664451i \(-0.231335\pi\)
\(432\) 0 0
\(433\) 4.93044e6 4.93044e6i 1.26376 1.26376i 0.314510 0.949254i \(-0.398160\pi\)
0.949254 0.314510i \(-0.101840\pi\)
\(434\) 0 0
\(435\) 1.49561e6 + 4.18294e6i 0.378962 + 1.05988i
\(436\) 0 0
\(437\) 201054. 201054.i 0.0503626 0.0503626i
\(438\) 0 0
\(439\) 2.70879e6 0.670833 0.335417 0.942070i \(-0.391123\pi\)
0.335417 + 0.942070i \(0.391123\pi\)
\(440\) 0 0
\(441\) 76254.2 0.0186710
\(442\) 0 0
\(443\) −940341. + 940341.i −0.227654 + 0.227654i −0.811712 0.584058i \(-0.801464\pi\)
0.584058 + 0.811712i \(0.301464\pi\)
\(444\) 0 0
\(445\) 3.97801e6 + 1.88256e6i 0.952283 + 0.450660i
\(446\) 0 0
\(447\) −639804. + 639804.i −0.151453 + 0.151453i
\(448\) 0 0
\(449\) 2.66405e6i 0.623630i 0.950143 + 0.311815i \(0.100937\pi\)
−0.950143 + 0.311815i \(0.899063\pi\)
\(450\) 0 0
\(451\) 3.37240e6 0.780725
\(452\) 0 0
\(453\) 4.00467e6 + 4.00467e6i 0.916899 + 0.916899i
\(454\) 0 0
\(455\) 3.23457e6 6.83491e6i 0.732466 1.54776i
\(456\) 0 0
\(457\) 4.92890e6 + 4.92890e6i 1.10398 + 1.10398i 0.993926 + 0.110049i \(0.0351007\pi\)
0.110049 + 0.993926i \(0.464899\pi\)
\(458\) 0 0
\(459\) 2.27624e6i 0.504296i
\(460\) 0 0
\(461\) 1.34239e6i 0.294189i −0.989122 0.147095i \(-0.953008\pi\)
0.989122 0.147095i \(-0.0469922\pi\)
\(462\) 0 0
\(463\) 66298.4 + 66298.4i 0.0143731 + 0.0143731i 0.714257 0.699884i \(-0.246765\pi\)
−0.699884 + 0.714257i \(0.746765\pi\)
\(464\) 0 0
\(465\) −108037. + 38628.5i −0.0231706 + 0.00828467i
\(466\) 0 0
\(467\) 493248. + 493248.i 0.104658 + 0.104658i 0.757497 0.652839i \(-0.226422\pi\)
−0.652839 + 0.757497i \(0.726422\pi\)
\(468\) 0 0
\(469\) 7.65077e6 1.60610
\(470\) 0 0
\(471\) 5.71371e6i 1.18677i
\(472\) 0 0
\(473\) 2.36660e6 2.36660e6i 0.486375 0.486375i
\(474\) 0 0
\(475\) 816996. 669872.i 0.166145 0.136225i
\(476\) 0 0
\(477\) −160534. + 160534.i −0.0323051 + 0.0323051i
\(478\) 0 0
\(479\) 2.01091e6 0.400456 0.200228 0.979749i \(-0.435832\pi\)
0.200228 + 0.979749i \(0.435832\pi\)
\(480\) 0 0
\(481\) −1.59747e7 −3.14826
\(482\) 0 0
\(483\) −1.15751e6 + 1.15751e6i −0.225765 + 0.225765i
\(484\) 0 0
\(485\) −740204. + 264660.i −0.142889 + 0.0510898i
\(486\) 0 0
\(487\) 2.94894e6 2.94894e6i 0.563435 0.563435i −0.366847 0.930281i \(-0.619563\pi\)
0.930281 + 0.366847i \(0.119563\pi\)
\(488\) 0 0
\(489\) 6.00825e6i 1.13625i
\(490\) 0 0
\(491\) −5.11456e6 −0.957425 −0.478712 0.877972i \(-0.658896\pi\)
−0.478712 + 0.877972i \(0.658896\pi\)
\(492\) 0 0
\(493\) 2.22752e6 + 2.22752e6i 0.412766 + 0.412766i
\(494\) 0 0
\(495\) 578384. + 273715.i 0.106097 + 0.0502095i
\(496\) 0 0
\(497\) 4.84283e6 + 4.84283e6i 0.879445 + 0.879445i
\(498\) 0 0
\(499\) 22809.7i 0.00410079i −0.999998 0.00205040i \(-0.999347\pi\)
0.999998 0.00205040i \(-0.000652662\pi\)
\(500\) 0 0
\(501\) 4.72019e6i 0.840165i
\(502\) 0 0
\(503\) −5.18776e6 5.18776e6i −0.914240 0.914240i 0.0823624 0.996602i \(-0.473753\pi\)
−0.996602 + 0.0823624i \(0.973753\pi\)
\(504\) 0 0
\(505\) −1.76280e6 834229.i −0.307591 0.145565i
\(506\) 0 0
\(507\) −6.11350e6 6.11350e6i −1.05626 1.05626i
\(508\) 0 0
\(509\) −1.15335e7 −1.97318 −0.986589 0.163222i \(-0.947811\pi\)
−0.986589 + 0.163222i \(0.947811\pi\)
\(510\) 0 0
\(511\) 8.15733e6i 1.38196i
\(512\) 0 0
\(513\) 964297. 964297.i 0.161777 0.161777i
\(514\) 0 0
\(515\) 5.57190e6 1.99223e6i 0.925732 0.330995i
\(516\) 0 0
\(517\) 5.09195e6 5.09195e6i 0.837834 0.837834i
\(518\) 0 0
\(519\) 2.64789e6 0.431500
\(520\) 0 0
\(521\) 736346. 0.118847 0.0594234 0.998233i \(-0.481074\pi\)
0.0594234 + 0.998233i \(0.481074\pi\)
\(522\) 0 0
\(523\) −5.83896e6 + 5.83896e6i −0.933430 + 0.933430i −0.997918 0.0644887i \(-0.979458\pi\)
0.0644887 + 0.997918i \(0.479458\pi\)
\(524\) 0 0
\(525\) −4.70362e6 + 3.85660e6i −0.744790 + 0.610669i
\(526\) 0 0
\(527\) −57532.1 + 57532.1i −0.00902369 + 0.00902369i
\(528\) 0 0
\(529\) 5.72904e6i 0.890107i
\(530\) 0 0
\(531\) −697827. −0.107402
\(532\) 0 0
\(533\) −8.31866e6 8.31866e6i −1.26834 1.26834i
\(534\) 0 0
\(535\) −8.73328e6 + 3.12258e6i −1.31915 + 0.471661i
\(536\) 0 0
\(537\) 3.72546e6 + 3.72546e6i 0.557499 + 0.557499i
\(538\) 0 0
\(539\) 535749.i 0.0794308i
\(540\) 0 0
\(541\) 5.66802e6i 0.832604i 0.909226 + 0.416302i \(0.136674\pi\)
−0.909226 + 0.416302i \(0.863326\pi\)
\(542\) 0 0
\(543\) 4.76839e6 + 4.76839e6i 0.694021 + 0.694021i
\(544\) 0 0
\(545\) −2.65910e6 + 5.61890e6i −0.383481 + 0.810327i
\(546\) 0 0
\(547\) 3.64899e6 + 3.64899e6i 0.521440 + 0.521440i 0.918006 0.396566i \(-0.129798\pi\)
−0.396566 + 0.918006i \(0.629798\pi\)
\(548\) 0 0
\(549\) 251731. 0.0356456
\(550\) 0 0
\(551\) 1.88732e6i 0.264829i
\(552\) 0 0
\(553\) 2.13613e6 2.13613e6i 0.297040 0.297040i
\(554\) 0 0
\(555\) 1.16150e7 + 5.49671e6i 1.60062 + 0.757479i
\(556\) 0 0
\(557\) −1.61980e6 + 1.61980e6i −0.221219 + 0.221219i −0.809012 0.587793i \(-0.799997\pi\)
0.587793 + 0.809012i \(0.299997\pi\)
\(558\) 0 0
\(559\) −1.16753e7 −1.58030
\(560\) 0 0
\(561\) −2.27803e6 −0.305599
\(562\) 0 0
\(563\) 6.15791e6 6.15791e6i 0.818771 0.818771i −0.167159 0.985930i \(-0.553459\pi\)
0.985930 + 0.167159i \(0.0534592\pi\)
\(564\) 0 0
\(565\) −1.75767e6 4.91587e6i −0.231641 0.647857i
\(566\) 0 0
\(567\) −4.60334e6 + 4.60334e6i −0.601334 + 0.601334i
\(568\) 0 0
\(569\) 4.23310e6i 0.548123i 0.961712 + 0.274061i \(0.0883672\pi\)
−0.961712 + 0.274061i \(0.911633\pi\)
\(570\) 0 0
\(571\) −6.05998e6 −0.777823 −0.388912 0.921275i \(-0.627149\pi\)
−0.388912 + 0.921275i \(0.627149\pi\)
\(572\) 0 0
\(573\) −2.88890e6 2.88890e6i −0.367575 0.367575i
\(574\) 0 0
\(575\) −258791. + 2.61540e6i −0.0326422 + 0.329889i
\(576\) 0 0
\(577\) 6.43951e6 + 6.43951e6i 0.805217 + 0.805217i 0.983906 0.178689i \(-0.0571855\pi\)
−0.178689 + 0.983906i \(0.557185\pi\)
\(578\) 0 0
\(579\) 7.88550e6i 0.977536i
\(580\) 0 0
\(581\) 9.92719e6i 1.22007i
\(582\) 0 0
\(583\) 1.12788e6 + 1.12788e6i 0.137434 + 0.137434i
\(584\) 0 0
\(585\) −751520. 2.10186e6i −0.0907927 0.253930i
\(586\) 0 0
\(587\) 9.09859e6 + 9.09859e6i 1.08988 + 1.08988i 0.995540 + 0.0943400i \(0.0300741\pi\)
0.0943400 + 0.995540i \(0.469926\pi\)
\(588\) 0 0
\(589\) −48745.5 −0.00578956
\(590\) 0 0
\(591\) 1.38141e7i 1.62688i
\(592\) 0 0
\(593\) 3.99751e6 3.99751e6i 0.466823 0.466823i −0.434061 0.900884i \(-0.642920\pi\)
0.900884 + 0.434061i \(0.142920\pi\)
\(594\) 0 0
\(595\) −1.84508e6 + 3.89882e6i −0.213660 + 0.451482i
\(596\) 0 0
\(597\) −7.80561e6 + 7.80561e6i −0.896337 + 0.896337i
\(598\) 0 0
\(599\) −448071. −0.0510246 −0.0255123 0.999675i \(-0.508122\pi\)
−0.0255123 + 0.999675i \(0.508122\pi\)
\(600\) 0 0
\(601\) −3.18693e6 −0.359903 −0.179952 0.983675i \(-0.557594\pi\)
−0.179952 + 0.983675i \(0.557594\pi\)
\(602\) 0 0
\(603\) 1.59699e6 1.59699e6i 0.178858 0.178858i
\(604\) 0 0
\(605\) −1.92806e6 + 4.07414e6i −0.214156 + 0.452531i
\(606\) 0 0
\(607\) −5.74029e6 + 5.74029e6i −0.632357 + 0.632357i −0.948659 0.316302i \(-0.897559\pi\)
0.316302 + 0.948659i \(0.397559\pi\)
\(608\) 0 0
\(609\) 1.08657e7i 1.18717i
\(610\) 0 0
\(611\) −2.51205e7 −2.72223
\(612\) 0 0
\(613\) 4.84512e6 + 4.84512e6i 0.520779 + 0.520779i 0.917807 0.397028i \(-0.129958\pi\)
−0.397028 + 0.917807i \(0.629958\pi\)
\(614\) 0 0
\(615\) 3.18604e6 + 8.91074e6i 0.339674 + 0.950005i
\(616\) 0 0
\(617\) −7.13817e6 7.13817e6i −0.754874 0.754874i 0.220511 0.975384i \(-0.429228\pi\)
−0.975384 + 0.220511i \(0.929228\pi\)
\(618\) 0 0
\(619\) 8.04378e6i 0.843789i −0.906645 0.421894i \(-0.861365\pi\)
0.906645 0.421894i \(-0.138635\pi\)
\(620\) 0 0
\(621\) 3.39239e6i 0.353002i
\(622\) 0 0
\(623\) −7.61178e6 7.61178e6i −0.785717 0.785717i
\(624\) 0 0
\(625\) −1.91386e6 + 9.57625e6i −0.195979 + 0.980608i
\(626\) 0 0
\(627\) −965057. 965057.i −0.0980357 0.0980357i
\(628\) 0 0
\(629\) 9.11242e6 0.918348
\(630\) 0 0
\(631\) 1.34627e7i 1.34604i −0.739622 0.673022i \(-0.764996\pi\)
0.739622 0.673022i \(-0.235004\pi\)
\(632\) 0 0
\(633\) −3.09496e6 + 3.09496e6i −0.307005 + 0.307005i
\(634\) 0 0
\(635\) 2.08718e6 + 5.83745e6i 0.205412 + 0.574498i
\(636\) 0 0
\(637\) −1.32152e6 + 1.32152e6i −0.129041 + 0.129041i
\(638\) 0 0
\(639\) 2.02175e6 0.195873
\(640\) 0 0
\(641\) −2.67079e6 −0.256741 −0.128370 0.991726i \(-0.540975\pi\)
−0.128370 + 0.991726i \(0.540975\pi\)
\(642\) 0 0
\(643\) 4.33127e6 4.33127e6i 0.413131 0.413131i −0.469697 0.882828i \(-0.655637\pi\)
0.882828 + 0.469697i \(0.155637\pi\)
\(644\) 0 0
\(645\) 8.48895e6 + 4.01733e6i 0.803442 + 0.380223i
\(646\) 0 0
\(647\) −2.01146e6 + 2.01146e6i −0.188908 + 0.188908i −0.795224 0.606316i \(-0.792647\pi\)
0.606316 + 0.795224i \(0.292647\pi\)
\(648\) 0 0
\(649\) 4.90281e6i 0.456913i
\(650\) 0 0
\(651\) 280638. 0.0259534
\(652\) 0 0
\(653\) 3.78586e6 + 3.78586e6i 0.347441 + 0.347441i 0.859156 0.511714i \(-0.170989\pi\)
−0.511714 + 0.859156i \(0.670989\pi\)
\(654\) 0 0
\(655\) 7.40738e6 1.56524e7i 0.674623 1.42554i
\(656\) 0 0
\(657\) 1.70273e6 + 1.70273e6i 0.153897 + 0.153897i
\(658\) 0 0
\(659\) 3.51951e6i 0.315696i 0.987463 + 0.157848i \(0.0504556\pi\)
−0.987463 + 0.157848i \(0.949544\pi\)
\(660\) 0 0
\(661\) 1.09747e7i 0.976991i 0.872566 + 0.488496i \(0.162454\pi\)
−0.872566 + 0.488496i \(0.837546\pi\)
\(662\) 0 0
\(663\) 5.61918e6 + 5.61918e6i 0.496466 + 0.496466i
\(664\) 0 0
\(665\) −2.43333e6 + 870037.i −0.213376 + 0.0762928i
\(666\) 0 0
\(667\) −3.31979e6 3.31979e6i −0.288932 0.288932i
\(668\) 0 0
\(669\) −1.68453e7 −1.45517
\(670\) 0 0
\(671\) 1.76862e6i 0.151645i
\(672\) 0 0
\(673\) 1.14860e7 1.14860e7i 0.977532 0.977532i −0.0222213 0.999753i \(-0.507074\pi\)
0.999753 + 0.0222213i \(0.00707384\pi\)
\(674\) 0 0
\(675\) −1.24122e6 + 1.25440e7i −0.104855 + 1.05969i
\(676\) 0 0
\(677\) −1.56904e6 + 1.56904e6i −0.131571 + 0.131571i −0.769826 0.638254i \(-0.779657\pi\)
0.638254 + 0.769826i \(0.279657\pi\)
\(678\) 0 0
\(679\) 1.92277e6 0.160049
\(680\) 0 0
\(681\) −1.59388e6 −0.131700
\(682\) 0 0
\(683\) 8.69137e6 8.69137e6i 0.712913 0.712913i −0.254231 0.967144i \(-0.581822\pi\)
0.967144 + 0.254231i \(0.0818223\pi\)
\(684\) 0 0
\(685\) 4.61516e6 1.65015e6i 0.375803 0.134369i
\(686\) 0 0
\(687\) 6.91239e6 6.91239e6i 0.558774 0.558774i
\(688\) 0 0
\(689\) 5.56427e6i 0.446539i
\(690\) 0 0
\(691\) 1.40891e7 1.12250 0.561252 0.827645i \(-0.310320\pi\)
0.561252 + 0.827645i \(0.310320\pi\)
\(692\) 0 0
\(693\) −1.10671e6 1.10671e6i −0.0875392 0.0875392i
\(694\) 0 0
\(695\) −5.77735e6 2.73409e6i −0.453698 0.214709i
\(696\) 0 0
\(697\) 4.74519e6 + 4.74519e6i 0.369974 + 0.369974i
\(698\) 0 0
\(699\) 1.14108e7i 0.883331i
\(700\) 0 0
\(701\) 1.15723e7i 0.889457i −0.895665 0.444729i \(-0.853300\pi\)
0.895665 0.444729i \(-0.146700\pi\)
\(702\) 0 0
\(703\) 3.86035e6 + 3.86035e6i 0.294604 + 0.294604i
\(704\) 0 0
\(705\) 1.82648e7 + 8.64365e6i 1.38402 + 0.654975i
\(706\) 0 0
\(707\) 3.37304e6 + 3.37304e6i 0.253789 + 0.253789i
\(708\) 0 0
\(709\) −9.95975e6 −0.744103 −0.372051 0.928212i \(-0.621345\pi\)
−0.372051 + 0.928212i \(0.621345\pi\)
\(710\) 0 0
\(711\) 891775.i 0.0661578i
\(712\) 0 0
\(713\) 85743.2 85743.2i 0.00631649 0.00631649i
\(714\) 0 0
\(715\) −1.47673e7 + 5.28005e6i −1.08028 + 0.386254i
\(716\) 0 0
\(717\) 7.69351e6 7.69351e6i 0.558890 0.558890i
\(718\) 0 0
\(719\) −3.73997e6 −0.269803 −0.134901 0.990859i \(-0.543072\pi\)
−0.134901 + 0.990859i \(0.543072\pi\)
\(720\) 0 0
\(721\) −1.44737e7 −1.03691
\(722\) 0 0
\(723\) −2.59482e6 + 2.59482e6i −0.184613 + 0.184613i
\(724\) 0 0
\(725\) −1.10609e7 1.34902e7i −0.781529 0.953176i
\(726\) 0 0
\(727\) 4.69069e6 4.69069e6i 0.329155 0.329155i −0.523110 0.852265i \(-0.675228\pi\)
0.852265 + 0.523110i \(0.175228\pi\)
\(728\) 0 0
\(729\) 1.58748e7i 1.10634i
\(730\) 0 0
\(731\) 6.65991e6 0.460972
\(732\) 0 0
\(733\) 5.58662e6 + 5.58662e6i 0.384051 + 0.384051i 0.872559 0.488508i \(-0.162459\pi\)
−0.488508 + 0.872559i \(0.662459\pi\)
\(734\) 0 0
\(735\) 1.41558e6 506142.i 0.0966533 0.0345584i
\(736\) 0 0
\(737\) −1.12202e7 1.12202e7i −0.760905 0.760905i
\(738\) 0 0
\(739\) 1.60299e7i 1.07974i 0.841749 + 0.539870i \(0.181527\pi\)
−0.841749 + 0.539870i \(0.818473\pi\)
\(740\) 0 0
\(741\) 4.76098e6i 0.318531i
\(742\) 0 0
\(743\) 269940. + 269940.i 0.0179389 + 0.0179389i 0.716019 0.698080i \(-0.245962\pi\)
−0.698080 + 0.716019i \(0.745962\pi\)
\(744\) 0 0
\(745\) 1.51995e6 3.21177e6i 0.100332 0.212009i
\(746\) 0 0
\(747\) −2.07216e6 2.07216e6i −0.135869 0.135869i
\(748\) 0 0
\(749\) 2.26857e7 1.47757
\(750\) 0 0
\(751\) 1.18503e7i 0.766707i −0.923602 0.383353i \(-0.874769\pi\)
0.923602 0.383353i \(-0.125231\pi\)
\(752\) 0 0
\(753\) −8.23388e6 + 8.23388e6i −0.529197 + 0.529197i
\(754\) 0 0
\(755\) −2.01032e7 9.51367e6i −1.28350 0.607408i
\(756\) 0 0
\(757\) 1.72173e6 1.72173e6i 0.109201 0.109201i −0.650395 0.759596i \(-0.725397\pi\)
0.759596 + 0.650395i \(0.225397\pi\)
\(758\) 0 0
\(759\) 3.39507e6 0.213916
\(760\) 0 0
\(761\) 1.62361e7 1.01629 0.508147 0.861271i \(-0.330331\pi\)
0.508147 + 0.861271i \(0.330331\pi\)
\(762\) 0 0
\(763\) 1.07516e7 1.07516e7i 0.668590 0.668590i
\(764\) 0 0
\(765\) 428688. + 1.19896e6i 0.0264842 + 0.0740714i
\(766\) 0 0
\(767\) 1.20937e7 1.20937e7i 0.742284 0.742284i
\(768\) 0 0
\(769\) 1.91198e7i 1.16592i −0.812502 0.582958i \(-0.801895\pi\)
0.812502 0.582958i \(-0.198105\pi\)
\(770\) 0 0
\(771\) −4.28751e6 −0.259758
\(772\) 0 0
\(773\) −1.13423e6 1.13423e6i −0.0682737 0.0682737i 0.672145 0.740419i \(-0.265373\pi\)
−0.740419 + 0.672145i \(0.765373\pi\)
\(774\) 0 0
\(775\) 348424. 285680.i 0.0208379 0.0170854i
\(776\) 0 0
\(777\) −2.22249e7 2.22249e7i −1.32065 1.32065i
\(778\) 0 0
\(779\) 4.02047e6i 0.237374i
\(780\) 0 0
\(781\) 1.42044e7i 0.833290i
\(782\) 0 0
\(783\) −1.59224e7 1.59224e7i −0.928121 0.928121i
\(784\) 0 0
\(785\) 7.55436e6 + 2.11281e7i 0.437546 + 1.22373i
\(786\) 0 0
\(787\) −5.19263e6 5.19263e6i −0.298848 0.298848i 0.541715 0.840563i \(-0.317775\pi\)
−0.840563 + 0.541715i \(0.817775\pi\)
\(788\) 0 0
\(789\) 2.39668e7 1.37062
\(790\) 0 0
\(791\) 1.27696e7i 0.725663i
\(792\) 0 0
\(793\) −4.36263e6 + 4.36263e6i −0.246357 + 0.246357i
\(794\) 0 0
\(795\) −1.91460e6 + 4.04570e6i −0.107438 + 0.227026i
\(796\) 0 0
\(797\) 1.09058e7 1.09058e7i 0.608154 0.608154i −0.334309 0.942463i \(-0.608503\pi\)
0.942463 + 0.334309i \(0.108503\pi\)
\(798\) 0 0
\(799\) 1.43294e7 0.794075
\(800\) 0 0
\(801\) −3.17770e6 −0.174997
\(802\) 0 0
\(803\) 1.19631e7 1.19631e7i 0.654717 0.654717i
\(804\) 0 0
\(805\) 2.74983e6 5.81061e6i 0.149560 0.316033i
\(806\) 0 0
\(807\) −5.10912e6 + 5.10912e6i −0.276161 + 0.276161i
\(808\) 0 0
\(809\) 6.21861e6i 0.334058i −0.985952 0.167029i \(-0.946583\pi\)
0.985952 0.167029i \(-0.0534174\pi\)
\(810\) 0 0
\(811\) 7.23689e6 0.386367 0.193183 0.981163i \(-0.438119\pi\)
0.193183 + 0.981163i \(0.438119\pi\)
\(812\) 0 0
\(813\) −6.84814e6 6.84814e6i −0.363368 0.363368i
\(814\) 0 0
\(815\) 7.94378e6 + 2.22172e7i 0.418922 + 1.17164i
\(816\) 0 0
\(817\) 2.82138e6 + 2.82138e6i 0.147879 + 0.147879i
\(818\) 0 0
\(819\) 5.45983e6i 0.284426i
\(820\) 0 0
\(821\) 9.66191e6i 0.500271i 0.968211 + 0.250135i \(0.0804751\pi\)
−0.968211 + 0.250135i \(0.919525\pi\)
\(822\) 0 0
\(823\) 1.66712e7 + 1.66712e7i 0.857961 + 0.857961i 0.991098 0.133137i \(-0.0425050\pi\)
−0.133137 + 0.991098i \(0.542505\pi\)
\(824\) 0 0
\(825\) 1.25539e7 + 1.24220e6i 0.642161 + 0.0635411i
\(826\) 0 0
\(827\) 1.31048e7 + 1.31048e7i 0.666297 + 0.666297i 0.956857 0.290560i \(-0.0938415\pi\)
−0.290560 + 0.956857i \(0.593841\pi\)
\(828\) 0 0
\(829\) −7.86602e6 −0.397529 −0.198764 0.980047i \(-0.563693\pi\)
−0.198764 + 0.980047i \(0.563693\pi\)
\(830\) 0 0
\(831\) 2.61207e7i 1.31215i
\(832\) 0 0
\(833\) 753833. 753833.i 0.0376411 0.0376411i
\(834\) 0 0
\(835\) −6.24077e6 1.74542e7i −0.309758 0.866333i
\(836\) 0 0
\(837\) 411243. 411243.i 0.0202901 0.0202901i
\(838\) 0 0
\(839\) 7.46126e6 0.365938 0.182969 0.983119i \(-0.441429\pi\)
0.182969 + 0.983119i \(0.441429\pi\)
\(840\) 0 0
\(841\) 1.06521e7 0.519333
\(842\) 0 0
\(843\) −1.39704e7 + 1.39704e7i −0.677082 + 0.677082i
\(844\) 0 0
\(845\) 3.06894e7 + 1.45235e7i 1.47859 + 0.699729i
\(846\) 0 0
\(847\) 7.79572e6 7.79572e6i 0.373377 0.373377i
\(848\) 0 0
\(849\) 1.01903e7i 0.485195i
\(850\) 0 0
\(851\) −1.35807e7 −0.642834
\(852\) 0 0
\(853\) −1.59287e7 1.59287e7i −0.749562 0.749562i 0.224835 0.974397i \(-0.427816\pi\)
−0.974397 + 0.224835i \(0.927816\pi\)
\(854\) 0 0
\(855\) −326315. + 689530.i −0.0152659 + 0.0322581i
\(856\) 0 0
\(857\) 4.57402e6 + 4.57402e6i 0.212738 + 0.212738i 0.805430 0.592691i \(-0.201935\pi\)
−0.592691 + 0.805430i \(0.701935\pi\)
\(858\) 0 0
\(859\) 1.72334e7i 0.796872i 0.917196 + 0.398436i \(0.130447\pi\)
−0.917196 + 0.398436i \(0.869553\pi\)
\(860\) 0 0
\(861\) 2.31467e7i 1.06410i
\(862\) 0 0
\(863\) −1.12321e6 1.12321e6i −0.0513373 0.0513373i 0.680972 0.732309i \(-0.261557\pi\)
−0.732309 + 0.680972i \(0.761557\pi\)
\(864\) 0 0
\(865\) −9.79132e6 + 3.50089e6i −0.444940 + 0.159088i
\(866\) 0 0
\(867\) 1.10865e7 + 1.10865e7i 0.500896 + 0.500896i
\(868\) 0 0
\(869\) −6.26546e6 −0.281451
\(870\) 0 0
\(871\) 5.53532e7i 2.47228i
\(872\) 0 0
\(873\) 401351. 401351.i 0.0178233 0.0178233i
\(874\) 0 0
\(875\) 1.22940e7 2.04797e7i 0.542842 0.904283i
\(876\) 0 0
\(877\) 5.11686e6 5.11686e6i 0.224649 0.224649i −0.585804 0.810453i \(-0.699221\pi\)
0.810453 + 0.585804i \(0.199221\pi\)
\(878\) 0 0
\(879\) 7.22455e6 0.315383
\(880\) 0 0
\(881\) 3.96052e6 0.171915 0.0859573 0.996299i \(-0.472605\pi\)
0.0859573 + 0.996299i \(0.472605\pi\)
\(882\) 0 0
\(883\) 1.32476e7 1.32476e7i 0.571790 0.571790i −0.360838 0.932628i \(-0.617509\pi\)
0.932628 + 0.360838i \(0.117509\pi\)
\(884\) 0 0
\(885\) −1.29545e7 + 4.63187e6i −0.555982 + 0.198792i
\(886\) 0 0
\(887\) −2.58340e7 + 2.58340e7i −1.10251 + 1.10251i −0.108401 + 0.994107i \(0.534573\pi\)
−0.994107 + 0.108401i \(0.965427\pi\)
\(888\) 0 0
\(889\) 1.51635e7i 0.643494i
\(890\) 0 0
\(891\) 1.35020e7 0.569775
\(892\) 0 0
\(893\) 6.07046e6 + 6.07046e6i 0.254738 + 0.254738i
\(894\) 0 0
\(895\) −1.87016e7 8.85036e6i −0.780405 0.369321i
\(896\) 0 0
\(897\) −8.37456e6 8.37456e6i −0.347521 0.347521i
\(898\) 0 0
\(899\) 804882.i 0.0332149i
\(900\) 0 0
\(901\) 3.17401e6i 0.130256i
\(902\) 0 0
\(903\) −1.62433e7 1.62433e7i −0.662910 0.662910i
\(904\) 0 0
\(905\) −2.39370e7 1.13280e7i −0.971513 0.459761i
\(906\) 0 0
\(907\) −1.97512e7 1.97512e7i −0.797214 0.797214i 0.185441 0.982655i \(-0.440629\pi\)
−0.982655 + 0.185441i \(0.940629\pi\)
\(908\) 0 0
\(909\) 1.40815e6 0.0565248
\(910\) 0 0
\(911\) 1.40853e7i 0.562304i −0.959663 0.281152i \(-0.909284\pi\)
0.959663 0.281152i \(-0.0907165\pi\)
\(912\) 0 0
\(913\) −1.45586e7 + 1.45586e7i −0.578021 + 0.578021i
\(914\) 0 0
\(915\) 4.67314e6 1.67088e6i 0.184525 0.0659771i
\(916\) 0 0
\(917\) −2.99503e7 + 2.99503e7i −1.17619 + 1.17619i
\(918\) 0 0
\(919\) −2.47354e7 −0.966116 −0.483058 0.875588i \(-0.660474\pi\)
−0.483058 + 0.875588i \(0.660474\pi\)
\(920\) 0 0
\(921\) 1.61064e7 0.625677
\(922\) 0 0
\(923\) −3.50378e7 + 3.50378e7i −1.35373 + 1.35373i
\(924\) 0 0
\(925\) −5.02173e7 4.96895e6i −1.92974 0.190946i
\(926\) 0 0
\(927\) −3.02117e6 + 3.02117e6i −0.115472 + 0.115472i
\(928\) 0 0
\(929\) 1.69900e7i 0.645884i 0.946419 + 0.322942i \(0.104672\pi\)
−0.946419 + 0.322942i \(0.895328\pi\)
\(930\) 0 0
\(931\) 638702. 0.0241504
\(932\) 0 0
\(933\) 1.40059e7 + 1.40059e7i 0.526752 + 0.526752i
\(934\) 0 0
\(935\) 8.42367e6 3.01189e6i 0.315117 0.112670i
\(936\) 0 0
\(937\) 2.16683e7 + 2.16683e7i 0.806260 + 0.806260i 0.984066 0.177806i \(-0.0568998\pi\)
−0.177806 + 0.984066i \(0.556900\pi\)
\(938\) 0 0
\(939\) 1.47196e7i 0.544795i
\(940\) 0 0
\(941\) 5.14995e7i 1.89596i 0.318331 + 0.947980i \(0.396878\pi\)
−0.318331 + 0.947980i \(0.603122\pi\)
\(942\) 0 0
\(943\) −7.07200e6 7.07200e6i −0.258978 0.258978i
\(944\) 0 0
\(945\) 1.31888e7 2.78690e7i 0.480424 1.01518i
\(946\) 0 0
\(947\) 1.45885e7 + 1.45885e7i 0.528610 + 0.528610i 0.920158 0.391548i \(-0.128060\pi\)
−0.391548 + 0.920158i \(0.628060\pi\)
\(948\) 0 0
\(949\) −5.90182e7 −2.12726
\(950\) 0 0
\(951\) 8.37134e6i 0.300154i
\(952\) 0 0
\(953\) −1.66848e7 + 1.66848e7i −0.595097 + 0.595097i −0.939004 0.343907i \(-0.888250\pi\)
0.343907 + 0.939004i \(0.388250\pi\)
\(954\) 0 0
\(955\) 1.45021e7 + 6.86299e6i 0.514543 + 0.243503i
\(956\) 0 0
\(957\) −1.59350e7 + 1.59350e7i −0.562434 + 0.562434i
\(958\) 0 0
\(959\) −1.19884e7 −0.420936
\(960\) 0 0
\(961\) 2.86084e7 0.999274
\(962\) 0 0
\(963\) 4.73532e6 4.73532e6i 0.164545 0.164545i
\(964\) 0 0
\(965\) 1.04258e7 + 2.91589e7i 0.360404 + 1.00798i
\(966\) 0 0
\(967\) 7.09186e6 7.09186e6i 0.243890 0.243890i −0.574567 0.818457i \(-0.694830\pi\)
0.818457 + 0.574567i \(0.194830\pi\)
\(968\) 0 0
\(969\) 2.71579e6i 0.0929153i
\(970\) 0 0
\(971\) −2.60523e7 −0.886742 −0.443371 0.896338i \(-0.646218\pi\)
−0.443371 + 0.896338i \(0.646218\pi\)
\(972\) 0 0
\(973\) 1.10547e7 + 1.10547e7i 0.374340 + 0.374340i
\(974\) 0 0
\(975\) −2.79024e7 3.40306e7i −0.940006 1.14646i
\(976\) 0 0
\(977\) −2.40458e6 2.40458e6i −0.0805939 0.0805939i 0.665661 0.746255i \(-0.268150\pi\)
−0.746255 + 0.665661i \(0.768150\pi\)
\(978\) 0 0
\(979\) 2.23260e7i 0.744481i
\(980\) 0 0
\(981\) 4.48847e6i 0.148911i
\(982\) 0 0
\(983\) 2.99200e7 + 2.99200e7i 0.987591 + 0.987591i 0.999924 0.0123328i \(-0.00392576\pi\)
−0.0123328 + 0.999924i \(0.503926\pi\)
\(984\) 0 0
\(985\) −1.82643e7 5.10817e7i −0.599807 1.67755i
\(986\) 0 0
\(987\) −3.49489e7 3.49489e7i −1.14193 1.14193i
\(988\) 0 0
\(989\) −9.92561e6 −0.322676
\(990\) 0 0
\(991\) 3.78149e7i 1.22315i −0.791187 0.611574i \(-0.790537\pi\)
0.791187 0.611574i \(-0.209463\pi\)
\(992\) 0 0
\(993\) −2.47895e7 + 2.47895e7i −0.797800 + 0.797800i
\(994\) 0 0
\(995\) 1.85434e7 3.91837e7i 0.593787 1.25472i
\(996\) 0 0
\(997\) −1.28781e6 + 1.28781e6i −0.0410312 + 0.0410312i −0.727325 0.686294i \(-0.759237\pi\)
0.686294 + 0.727325i \(0.259237\pi\)
\(998\) 0 0
\(999\) −6.51361e7 −2.06494
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 160.6.o.a.47.10 56
4.3 odd 2 40.6.k.a.27.9 yes 56
5.3 odd 4 inner 160.6.o.a.143.9 56
8.3 odd 2 inner 160.6.o.a.47.9 56
8.5 even 2 40.6.k.a.27.7 yes 56
20.3 even 4 40.6.k.a.3.7 56
40.3 even 4 inner 160.6.o.a.143.10 56
40.13 odd 4 40.6.k.a.3.9 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.k.a.3.7 56 20.3 even 4
40.6.k.a.3.9 yes 56 40.13 odd 4
40.6.k.a.27.7 yes 56 8.5 even 2
40.6.k.a.27.9 yes 56 4.3 odd 2
160.6.o.a.47.9 56 8.3 odd 2 inner
160.6.o.a.47.10 56 1.1 even 1 trivial
160.6.o.a.143.9 56 5.3 odd 4 inner
160.6.o.a.143.10 56 40.3 even 4 inner