Properties

Label 336.6.h.b.239.16
Level $336$
Weight $6$
Character 336.239
Analytic conductor $53.889$
Analytic rank $0$
Dimension $40$
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] = [336,6,Mod(239,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.239");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.h (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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 239.16
Character \(\chi\) \(=\) 336.239
Dual form 336.6.h.b.239.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-6.06266 + 14.3612i) q^{3} +44.6235i q^{5} -49.0000i q^{7} +(-169.488 - 174.134i) q^{9} +O(q^{10})\) \(q+(-6.06266 + 14.3612i) q^{3} +44.6235i q^{5} -49.0000i q^{7} +(-169.488 - 174.134i) q^{9} +190.244 q^{11} +542.034 q^{13} +(-640.848 - 270.537i) q^{15} +377.111i q^{17} +214.658i q^{19} +(703.699 + 297.070i) q^{21} +4578.90 q^{23} +1133.74 q^{25} +(3528.33 - 1378.34i) q^{27} -6957.56i q^{29} +7806.37i q^{31} +(-1153.38 + 2732.13i) q^{33} +2186.55 q^{35} +5560.43 q^{37} +(-3286.17 + 7784.27i) q^{39} +3056.15i q^{41} +6656.50i q^{43} +(7770.48 - 7563.17i) q^{45} -5920.56 q^{47} -2401.00 q^{49} +(-5415.77 - 2286.30i) q^{51} -27044.7i q^{53} +8489.34i q^{55} +(-3082.74 - 1301.39i) q^{57} +3587.75 q^{59} -47233.3 q^{61} +(-8532.57 + 8304.93i) q^{63} +24187.5i q^{65} +31211.8i q^{67} +(-27760.3 + 65758.5i) q^{69} +80798.0 q^{71} -29201.7 q^{73} +(-6873.48 + 16281.9i) q^{75} -9321.94i q^{77} +56190.0i q^{79} +(-1596.38 + 59027.4i) q^{81} -51520.4 q^{83} -16828.0 q^{85} +(99918.9 + 42181.3i) q^{87} +77681.5i q^{89} -26559.7i q^{91} +(-112109. - 47327.4i) q^{93} -9578.78 q^{95} +20845.5 q^{97} +(-32244.1 - 33127.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 8 q^{9} - 1048 q^{13} + 980 q^{21} - 43416 q^{25} + 20296 q^{33} - 16192 q^{37} + 56488 q^{45} - 96040 q^{49} + 31088 q^{57} + 173112 q^{61} - 114176 q^{69} - 267488 q^{73} + 64888 q^{81} + 508112 q^{85} - 224544 q^{93} - 276400 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) −6.06266 + 14.3612i −0.388920 + 0.921272i
\(4\) 0 0
\(5\) 44.6235i 0.798250i 0.916897 + 0.399125i \(0.130686\pi\)
−0.916897 + 0.399125i \(0.869314\pi\)
\(6\) 0 0
\(7\) 49.0000i 0.377964i
\(8\) 0 0
\(9\) −169.488 174.134i −0.697483 0.716601i
\(10\) 0 0
\(11\) 190.244 0.474055 0.237027 0.971503i \(-0.423827\pi\)
0.237027 + 0.971503i \(0.423827\pi\)
\(12\) 0 0
\(13\) 542.034 0.889546 0.444773 0.895643i \(-0.353284\pi\)
0.444773 + 0.895643i \(0.353284\pi\)
\(14\) 0 0
\(15\) −640.848 270.537i −0.735405 0.310455i
\(16\) 0 0
\(17\) 377.111i 0.316481i 0.987401 + 0.158240i \(0.0505821\pi\)
−0.987401 + 0.158240i \(0.949418\pi\)
\(18\) 0 0
\(19\) 214.658i 0.136415i 0.997671 + 0.0682075i \(0.0217280\pi\)
−0.997671 + 0.0682075i \(0.978272\pi\)
\(20\) 0 0
\(21\) 703.699 + 297.070i 0.348208 + 0.146998i
\(22\) 0 0
\(23\) 4578.90 1.80485 0.902426 0.430845i \(-0.141785\pi\)
0.902426 + 0.430845i \(0.141785\pi\)
\(24\) 0 0
\(25\) 1133.74 0.362797
\(26\) 0 0
\(27\) 3528.33 1378.34i 0.931449 0.363871i
\(28\) 0 0
\(29\) 6957.56i 1.53625i −0.640300 0.768125i \(-0.721190\pi\)
0.640300 0.768125i \(-0.278810\pi\)
\(30\) 0 0
\(31\) 7806.37i 1.45897i 0.683999 + 0.729483i \(0.260239\pi\)
−0.683999 + 0.729483i \(0.739761\pi\)
\(32\) 0 0
\(33\) −1153.38 + 2732.13i −0.184369 + 0.436733i
\(34\) 0 0
\(35\) 2186.55 0.301710
\(36\) 0 0
\(37\) 5560.43 0.667735 0.333867 0.942620i \(-0.391646\pi\)
0.333867 + 0.942620i \(0.391646\pi\)
\(38\) 0 0
\(39\) −3286.17 + 7784.27i −0.345962 + 0.819514i
\(40\) 0 0
\(41\) 3056.15i 0.283932i 0.989872 + 0.141966i \(0.0453424\pi\)
−0.989872 + 0.141966i \(0.954658\pi\)
\(42\) 0 0
\(43\) 6656.50i 0.549003i 0.961587 + 0.274501i \(0.0885128\pi\)
−0.961587 + 0.274501i \(0.911487\pi\)
\(44\) 0 0
\(45\) 7770.48 7563.17i 0.572027 0.556766i
\(46\) 0 0
\(47\) −5920.56 −0.390947 −0.195474 0.980709i \(-0.562624\pi\)
−0.195474 + 0.980709i \(0.562624\pi\)
\(48\) 0 0
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) −5415.77 2286.30i −0.291565 0.123086i
\(52\) 0 0
\(53\) 27044.7i 1.32249i −0.750171 0.661244i \(-0.770029\pi\)
0.750171 0.661244i \(-0.229971\pi\)
\(54\) 0 0
\(55\) 8489.34i 0.378414i
\(56\) 0 0
\(57\) −3082.74 1301.39i −0.125675 0.0530545i
\(58\) 0 0
\(59\) 3587.75 0.134181 0.0670906 0.997747i \(-0.478628\pi\)
0.0670906 + 0.997747i \(0.478628\pi\)
\(60\) 0 0
\(61\) −47233.3 −1.62526 −0.812632 0.582778i \(-0.801966\pi\)
−0.812632 + 0.582778i \(0.801966\pi\)
\(62\) 0 0
\(63\) −8532.57 + 8304.93i −0.270850 + 0.263624i
\(64\) 0 0
\(65\) 24187.5i 0.710080i
\(66\) 0 0
\(67\) 31211.8i 0.849437i 0.905325 + 0.424719i \(0.139627\pi\)
−0.905325 + 0.424719i \(0.860373\pi\)
\(68\) 0 0
\(69\) −27760.3 + 65758.5i −0.701942 + 1.66276i
\(70\) 0 0
\(71\) 80798.0 1.90219 0.951097 0.308894i \(-0.0999587\pi\)
0.951097 + 0.308894i \(0.0999587\pi\)
\(72\) 0 0
\(73\) −29201.7 −0.641359 −0.320679 0.947188i \(-0.603911\pi\)
−0.320679 + 0.947188i \(0.603911\pi\)
\(74\) 0 0
\(75\) −6873.48 + 16281.9i −0.141099 + 0.334235i
\(76\) 0 0
\(77\) 9321.94i 0.179176i
\(78\) 0 0
\(79\) 56190.0i 1.01296i 0.862252 + 0.506479i \(0.169053\pi\)
−0.862252 + 0.506479i \(0.830947\pi\)
\(80\) 0 0
\(81\) −1596.38 + 59027.4i −0.0270348 + 0.999634i
\(82\) 0 0
\(83\) −51520.4 −0.820888 −0.410444 0.911886i \(-0.634626\pi\)
−0.410444 + 0.911886i \(0.634626\pi\)
\(84\) 0 0
\(85\) −16828.0 −0.252631
\(86\) 0 0
\(87\) 99918.9 + 42181.3i 1.41530 + 0.597478i
\(88\) 0 0
\(89\) 77681.5i 1.03954i 0.854305 + 0.519772i \(0.173983\pi\)
−0.854305 + 0.519772i \(0.826017\pi\)
\(90\) 0 0
\(91\) 26559.7i 0.336217i
\(92\) 0 0
\(93\) −112109. 47327.4i −1.34410 0.567420i
\(94\) 0 0
\(95\) −9578.78 −0.108893
\(96\) 0 0
\(97\) 20845.5 0.224948 0.112474 0.993655i \(-0.464122\pi\)
0.112474 + 0.993655i \(0.464122\pi\)
\(98\) 0 0
\(99\) −32244.1 33127.9i −0.330645 0.339708i
\(100\) 0 0
\(101\) 111322.i 1.08587i 0.839774 + 0.542937i \(0.182688\pi\)
−0.839774 + 0.542937i \(0.817312\pi\)
\(102\) 0 0
\(103\) 70195.7i 0.651955i 0.945378 + 0.325977i \(0.105693\pi\)
−0.945378 + 0.325977i \(0.894307\pi\)
\(104\) 0 0
\(105\) −13256.3 + 31401.5i −0.117341 + 0.277957i
\(106\) 0 0
\(107\) 97869.1 0.826392 0.413196 0.910642i \(-0.364412\pi\)
0.413196 + 0.910642i \(0.364412\pi\)
\(108\) 0 0
\(109\) 1440.79 0.0116154 0.00580772 0.999983i \(-0.498151\pi\)
0.00580772 + 0.999983i \(0.498151\pi\)
\(110\) 0 0
\(111\) −33711.0 + 79854.5i −0.259695 + 0.615165i
\(112\) 0 0
\(113\) 212764.i 1.56748i −0.621089 0.783740i \(-0.713309\pi\)
0.621089 0.783740i \(-0.286691\pi\)
\(114\) 0 0
\(115\) 204327.i 1.44072i
\(116\) 0 0
\(117\) −91868.5 94386.7i −0.620443 0.637450i
\(118\) 0 0
\(119\) 18478.5 0.119619
\(120\) 0 0
\(121\) −124858. −0.775272
\(122\) 0 0
\(123\) −43890.0 18528.4i −0.261579 0.110427i
\(124\) 0 0
\(125\) 190040.i 1.08785i
\(126\) 0 0
\(127\) 239639.i 1.31841i 0.751965 + 0.659203i \(0.229106\pi\)
−0.751965 + 0.659203i \(0.770894\pi\)
\(128\) 0 0
\(129\) −95595.3 40356.0i −0.505781 0.213518i
\(130\) 0 0
\(131\) 157566. 0.802204 0.401102 0.916033i \(-0.368627\pi\)
0.401102 + 0.916033i \(0.368627\pi\)
\(132\) 0 0
\(133\) 10518.2 0.0515600
\(134\) 0 0
\(135\) 61506.5 + 157446.i 0.290460 + 0.743529i
\(136\) 0 0
\(137\) 90882.5i 0.413694i 0.978373 + 0.206847i \(0.0663202\pi\)
−0.978373 + 0.206847i \(0.933680\pi\)
\(138\) 0 0
\(139\) 108809.i 0.477668i 0.971060 + 0.238834i \(0.0767652\pi\)
−0.971060 + 0.238834i \(0.923235\pi\)
\(140\) 0 0
\(141\) 35894.3 85026.4i 0.152047 0.360169i
\(142\) 0 0
\(143\) 103119. 0.421693
\(144\) 0 0
\(145\) 310471. 1.22631
\(146\) 0 0
\(147\) 14556.4 34481.3i 0.0555599 0.131610i
\(148\) 0 0
\(149\) 24511.2i 0.0904479i −0.998977 0.0452240i \(-0.985600\pi\)
0.998977 0.0452240i \(-0.0144001\pi\)
\(150\) 0 0
\(151\) 47092.6i 0.168078i 0.996462 + 0.0840389i \(0.0267820\pi\)
−0.996462 + 0.0840389i \(0.973218\pi\)
\(152\) 0 0
\(153\) 65668.0 63916.0i 0.226791 0.220740i
\(154\) 0 0
\(155\) −348348. −1.16462
\(156\) 0 0
\(157\) −45035.0 −0.145815 −0.0729073 0.997339i \(-0.523228\pi\)
−0.0729073 + 0.997339i \(0.523228\pi\)
\(158\) 0 0
\(159\) 388394. + 163963.i 1.21837 + 0.514342i
\(160\) 0 0
\(161\) 224366.i 0.682170i
\(162\) 0 0
\(163\) 72731.6i 0.214415i −0.994237 0.107207i \(-0.965809\pi\)
0.994237 0.107207i \(-0.0341908\pi\)
\(164\) 0 0
\(165\) −121917. 51468.0i −0.348622 0.147173i
\(166\) 0 0
\(167\) −172434. −0.478444 −0.239222 0.970965i \(-0.576892\pi\)
−0.239222 + 0.970965i \(0.576892\pi\)
\(168\) 0 0
\(169\) −77491.8 −0.208708
\(170\) 0 0
\(171\) 37379.2 36382.0i 0.0977552 0.0951471i
\(172\) 0 0
\(173\) 230047.i 0.584388i −0.956359 0.292194i \(-0.905615\pi\)
0.956359 0.292194i \(-0.0943853\pi\)
\(174\) 0 0
\(175\) 55553.3i 0.137124i
\(176\) 0 0
\(177\) −21751.3 + 51524.4i −0.0521857 + 0.123617i
\(178\) 0 0
\(179\) 35064.1 0.0817956 0.0408978 0.999163i \(-0.486978\pi\)
0.0408978 + 0.999163i \(0.486978\pi\)
\(180\) 0 0
\(181\) 396840. 0.900366 0.450183 0.892936i \(-0.351359\pi\)
0.450183 + 0.892936i \(0.351359\pi\)
\(182\) 0 0
\(183\) 286359. 678327.i 0.632097 1.49731i
\(184\) 0 0
\(185\) 248126.i 0.533019i
\(186\) 0 0
\(187\) 71743.1i 0.150029i
\(188\) 0 0
\(189\) −67538.8 172888.i −0.137530 0.352055i
\(190\) 0 0
\(191\) 437830. 0.868404 0.434202 0.900815i \(-0.357030\pi\)
0.434202 + 0.900815i \(0.357030\pi\)
\(192\) 0 0
\(193\) 708524. 1.36918 0.684591 0.728927i \(-0.259981\pi\)
0.684591 + 0.728927i \(0.259981\pi\)
\(194\) 0 0
\(195\) −347361. 146640.i −0.654177 0.276164i
\(196\) 0 0
\(197\) 530365.i 0.973664i 0.873496 + 0.486832i \(0.161848\pi\)
−0.873496 + 0.486832i \(0.838152\pi\)
\(198\) 0 0
\(199\) 1.00530e6i 1.79954i 0.436361 + 0.899772i \(0.356267\pi\)
−0.436361 + 0.899772i \(0.643733\pi\)
\(200\) 0 0
\(201\) −448238. 189226.i −0.782562 0.330363i
\(202\) 0 0
\(203\) −340920. −0.580648
\(204\) 0 0
\(205\) −136376. −0.226649
\(206\) 0 0
\(207\) −776070. 797343.i −1.25885 1.29336i
\(208\) 0 0
\(209\) 40837.2i 0.0646682i
\(210\) 0 0
\(211\) 1.00443e6i 1.55316i −0.630020 0.776579i \(-0.716953\pi\)
0.630020 0.776579i \(-0.283047\pi\)
\(212\) 0 0
\(213\) −489850. + 1.16036e6i −0.739800 + 1.75244i
\(214\) 0 0
\(215\) −297036. −0.438241
\(216\) 0 0
\(217\) 382512. 0.551437
\(218\) 0 0
\(219\) 177040. 419372.i 0.249437 0.590866i
\(220\) 0 0
\(221\) 204407.i 0.281524i
\(222\) 0 0
\(223\) 1.03170e6i 1.38928i −0.719358 0.694639i \(-0.755564\pi\)
0.719358 0.694639i \(-0.244436\pi\)
\(224\) 0 0
\(225\) −192156. 197423.i −0.253045 0.259981i
\(226\) 0 0
\(227\) 906383. 1.16747 0.583737 0.811943i \(-0.301590\pi\)
0.583737 + 0.811943i \(0.301590\pi\)
\(228\) 0 0
\(229\) 132754. 0.167286 0.0836429 0.996496i \(-0.473345\pi\)
0.0836429 + 0.996496i \(0.473345\pi\)
\(230\) 0 0
\(231\) 133874. + 56515.7i 0.165070 + 0.0696850i
\(232\) 0 0
\(233\) 653204.i 0.788241i −0.919059 0.394121i \(-0.871049\pi\)
0.919059 0.394121i \(-0.128951\pi\)
\(234\) 0 0
\(235\) 264196.i 0.312074i
\(236\) 0 0
\(237\) −806956. 340661.i −0.933209 0.393959i
\(238\) 0 0
\(239\) −1.30676e6 −1.47979 −0.739896 0.672721i \(-0.765125\pi\)
−0.739896 + 0.672721i \(0.765125\pi\)
\(240\) 0 0
\(241\) 503326. 0.558222 0.279111 0.960259i \(-0.409960\pi\)
0.279111 + 0.960259i \(0.409960\pi\)
\(242\) 0 0
\(243\) −838027. 380789.i −0.910421 0.413684i
\(244\) 0 0
\(245\) 107141.i 0.114036i
\(246\) 0 0
\(247\) 116352.i 0.121347i
\(248\) 0 0
\(249\) 312351. 739895.i 0.319260 0.756261i
\(250\) 0 0
\(251\) −975300. −0.977134 −0.488567 0.872526i \(-0.662480\pi\)
−0.488567 + 0.872526i \(0.662480\pi\)
\(252\) 0 0
\(253\) 871106. 0.855598
\(254\) 0 0
\(255\) 102023. 241671.i 0.0982531 0.232742i
\(256\) 0 0
\(257\) 403880.i 0.381434i 0.981645 + 0.190717i \(0.0610812\pi\)
−0.981645 + 0.190717i \(0.938919\pi\)
\(258\) 0 0
\(259\) 272461.i 0.252380i
\(260\) 0 0
\(261\) −1.21155e6 + 1.17923e6i −1.10088 + 1.07151i
\(262\) 0 0
\(263\) −894632. −0.797545 −0.398772 0.917050i \(-0.630564\pi\)
−0.398772 + 0.917050i \(0.630564\pi\)
\(264\) 0 0
\(265\) 1.20683e6 1.05568
\(266\) 0 0
\(267\) −1.11560e6 470956.i −0.957702 0.404299i
\(268\) 0 0
\(269\) 1.52378e6i 1.28393i −0.766733 0.641967i \(-0.778119\pi\)
0.766733 0.641967i \(-0.221881\pi\)
\(270\) 0 0
\(271\) 384963.i 0.318417i −0.987245 0.159209i \(-0.949106\pi\)
0.987245 0.159209i \(-0.0508942\pi\)
\(272\) 0 0
\(273\) 381429. + 161022.i 0.309747 + 0.130761i
\(274\) 0 0
\(275\) 215687. 0.171986
\(276\) 0 0
\(277\) 901496. 0.705934 0.352967 0.935636i \(-0.385173\pi\)
0.352967 + 0.935636i \(0.385173\pi\)
\(278\) 0 0
\(279\) 1.35936e6 1.32309e6i 1.04550 1.01760i
\(280\) 0 0
\(281\) 872264.i 0.658995i −0.944156 0.329498i \(-0.893121\pi\)
0.944156 0.329498i \(-0.106879\pi\)
\(282\) 0 0
\(283\) 377286.i 0.280030i 0.990149 + 0.140015i \(0.0447150\pi\)
−0.990149 + 0.140015i \(0.955285\pi\)
\(284\) 0 0
\(285\) 58072.8 137563.i 0.0423507 0.100320i
\(286\) 0 0
\(287\) 149751. 0.107316
\(288\) 0 0
\(289\) 1.27764e6 0.899840
\(290\) 0 0
\(291\) −126379. + 299367.i −0.0874869 + 0.207239i
\(292\) 0 0
\(293\) 1.08830e6i 0.740591i −0.928914 0.370295i \(-0.879256\pi\)
0.928914 0.370295i \(-0.120744\pi\)
\(294\) 0 0
\(295\) 160098.i 0.107110i
\(296\) 0 0
\(297\) 671241. 262221.i 0.441558 0.172495i
\(298\) 0 0
\(299\) 2.48192e6 1.60550
\(300\) 0 0
\(301\) 326168. 0.207504
\(302\) 0 0
\(303\) −1.59872e6 674910.i −1.00038 0.422317i
\(304\) 0 0
\(305\) 2.10772e6i 1.29737i
\(306\) 0 0
\(307\) 209669.i 0.126966i 0.997983 + 0.0634832i \(0.0202209\pi\)
−0.997983 + 0.0634832i \(0.979779\pi\)
\(308\) 0 0
\(309\) −1.00809e6 425572.i −0.600628 0.253558i
\(310\) 0 0
\(311\) 1.46997e6 0.861803 0.430902 0.902399i \(-0.358196\pi\)
0.430902 + 0.902399i \(0.358196\pi\)
\(312\) 0 0
\(313\) 2.38412e6 1.37552 0.687761 0.725937i \(-0.258594\pi\)
0.687761 + 0.725937i \(0.258594\pi\)
\(314\) 0 0
\(315\) −370595. 380753.i −0.210438 0.216206i
\(316\) 0 0
\(317\) 2.81250e6i 1.57197i 0.618244 + 0.785986i \(0.287844\pi\)
−0.618244 + 0.785986i \(0.712156\pi\)
\(318\) 0 0
\(319\) 1.32363e6i 0.728266i
\(320\) 0 0
\(321\) −593347. + 1.40552e6i −0.321400 + 0.761332i
\(322\) 0 0
\(323\) −80949.8 −0.0431727
\(324\) 0 0
\(325\) 614526. 0.322725
\(326\) 0 0
\(327\) −8735.03 + 20691.5i −0.00451747 + 0.0107010i
\(328\) 0 0
\(329\) 290107.i 0.147764i
\(330\) 0 0
\(331\) 947216.i 0.475203i 0.971363 + 0.237601i \(0.0763612\pi\)
−0.971363 + 0.237601i \(0.923639\pi\)
\(332\) 0 0
\(333\) −942428. 968261.i −0.465734 0.478500i
\(334\) 0 0
\(335\) −1.39278e6 −0.678063
\(336\) 0 0
\(337\) 3.63228e6 1.74223 0.871113 0.491083i \(-0.163399\pi\)
0.871113 + 0.491083i \(0.163399\pi\)
\(338\) 0 0
\(339\) 3.05555e6 + 1.28992e6i 1.44408 + 0.609624i
\(340\) 0 0
\(341\) 1.48511e6i 0.691630i
\(342\) 0 0
\(343\) 117649.i 0.0539949i
\(344\) 0 0
\(345\) −2.93438e6 1.23876e6i −1.32730 0.560325i
\(346\) 0 0
\(347\) 2.46255e6 1.09790 0.548948 0.835857i \(-0.315029\pi\)
0.548948 + 0.835857i \(0.315029\pi\)
\(348\) 0 0
\(349\) −2.32760e6 −1.02293 −0.511463 0.859305i \(-0.670896\pi\)
−0.511463 + 0.859305i \(0.670896\pi\)
\(350\) 0 0
\(351\) 1.91247e6 747109.i 0.828567 0.323680i
\(352\) 0 0
\(353\) 1.40361e6i 0.599529i 0.954013 + 0.299764i \(0.0969081\pi\)
−0.954013 + 0.299764i \(0.903092\pi\)
\(354\) 0 0
\(355\) 3.60549e6i 1.51843i
\(356\) 0 0
\(357\) −112029. + 265373.i −0.0465220 + 0.110201i
\(358\) 0 0
\(359\) −1.82377e6 −0.746851 −0.373425 0.927660i \(-0.621817\pi\)
−0.373425 + 0.927660i \(0.621817\pi\)
\(360\) 0 0
\(361\) 2.43002e6 0.981391
\(362\) 0 0
\(363\) 756973. 1.79312e6i 0.301519 0.714236i
\(364\) 0 0
\(365\) 1.30308e6i 0.511965i
\(366\) 0 0
\(367\) 2.60865e6i 1.01100i 0.862827 + 0.505499i \(0.168692\pi\)
−0.862827 + 0.505499i \(0.831308\pi\)
\(368\) 0 0
\(369\) 532180. 517982.i 0.203466 0.198038i
\(370\) 0 0
\(371\) −1.32519e6 −0.499854
\(372\) 0 0
\(373\) −4.80916e6 −1.78977 −0.894884 0.446299i \(-0.852742\pi\)
−0.894884 + 0.446299i \(0.852742\pi\)
\(374\) 0 0
\(375\) −2.72920e6 1.15215e6i −1.00221 0.423087i
\(376\) 0 0
\(377\) 3.77124e6i 1.36657i
\(378\) 0 0
\(379\) 121899.i 0.0435914i 0.999762 + 0.0217957i \(0.00693834\pi\)
−0.999762 + 0.0217957i \(0.993062\pi\)
\(380\) 0 0
\(381\) −3.44151e6 1.45285e6i −1.21461 0.512754i
\(382\) 0 0
\(383\) −3.67894e6 −1.28152 −0.640760 0.767741i \(-0.721381\pi\)
−0.640760 + 0.767741i \(0.721381\pi\)
\(384\) 0 0
\(385\) 415978. 0.143027
\(386\) 0 0
\(387\) 1.15912e6 1.12820e6i 0.393416 0.382920i
\(388\) 0 0
\(389\) 2.10472e6i 0.705212i 0.935772 + 0.352606i \(0.114704\pi\)
−0.935772 + 0.352606i \(0.885296\pi\)
\(390\) 0 0
\(391\) 1.72676e6i 0.571201i
\(392\) 0 0
\(393\) −955270. + 2.26284e6i −0.311993 + 0.739048i
\(394\) 0 0
\(395\) −2.50740e6 −0.808593
\(396\) 0 0
\(397\) 671819. 0.213932 0.106966 0.994263i \(-0.465886\pi\)
0.106966 + 0.994263i \(0.465886\pi\)
\(398\) 0 0
\(399\) −63768.4 + 151054.i −0.0200527 + 0.0475008i
\(400\) 0 0
\(401\) 5.47259e6i 1.69954i 0.527151 + 0.849771i \(0.323260\pi\)
−0.527151 + 0.849771i \(0.676740\pi\)
\(402\) 0 0
\(403\) 4.23132e6i 1.29782i
\(404\) 0 0
\(405\) −2.63401e6 71236.1i −0.797958 0.0215805i
\(406\) 0 0
\(407\) 1.05784e6 0.316543
\(408\) 0 0
\(409\) 4.97372e6 1.47019 0.735094 0.677965i \(-0.237138\pi\)
0.735094 + 0.677965i \(0.237138\pi\)
\(410\) 0 0
\(411\) −1.30518e6 550990.i −0.381124 0.160894i
\(412\) 0 0
\(413\) 175800.i 0.0507158i
\(414\) 0 0
\(415\) 2.29902e6i 0.655274i
\(416\) 0 0
\(417\) −1.56262e6 659670.i −0.440062 0.185775i
\(418\) 0 0
\(419\) −147486. −0.0410408 −0.0205204 0.999789i \(-0.506532\pi\)
−0.0205204 + 0.999789i \(0.506532\pi\)
\(420\) 0 0
\(421\) −3.87541e6 −1.06565 −0.532823 0.846227i \(-0.678869\pi\)
−0.532823 + 0.846227i \(0.678869\pi\)
\(422\) 0 0
\(423\) 1.00347e6 + 1.03097e6i 0.272679 + 0.280153i
\(424\) 0 0
\(425\) 427547.i 0.114818i
\(426\) 0 0
\(427\) 2.31443e6i 0.614292i
\(428\) 0 0
\(429\) −625173. + 1.48091e6i −0.164005 + 0.388494i
\(430\) 0 0
\(431\) 7.53594e6 1.95409 0.977045 0.213035i \(-0.0683348\pi\)
0.977045 + 0.213035i \(0.0683348\pi\)
\(432\) 0 0
\(433\) −4.26964e6 −1.09439 −0.547194 0.837006i \(-0.684304\pi\)
−0.547194 + 0.837006i \(0.684304\pi\)
\(434\) 0 0
\(435\) −1.88228e6 + 4.45874e6i −0.476937 + 1.12977i
\(436\) 0 0
\(437\) 982895.i 0.246209i
\(438\) 0 0
\(439\) 5.60539e6i 1.38818i 0.719890 + 0.694088i \(0.244192\pi\)
−0.719890 + 0.694088i \(0.755808\pi\)
\(440\) 0 0
\(441\) 406942. + 418096.i 0.0996404 + 0.102372i
\(442\) 0 0
\(443\) 209607. 0.0507454 0.0253727 0.999678i \(-0.491923\pi\)
0.0253727 + 0.999678i \(0.491923\pi\)
\(444\) 0 0
\(445\) −3.46642e6 −0.829815
\(446\) 0 0
\(447\) 352010. + 148603.i 0.0833271 + 0.0351770i
\(448\) 0 0
\(449\) 2.52872e6i 0.591951i 0.955196 + 0.295975i \(0.0956447\pi\)
−0.955196 + 0.295975i \(0.904355\pi\)
\(450\) 0 0
\(451\) 581413.i 0.134599i
\(452\) 0 0
\(453\) −676307. 285506.i −0.154845 0.0653688i
\(454\) 0 0
\(455\) 1.18519e6 0.268385
\(456\) 0 0
\(457\) −3.30478e6 −0.740206 −0.370103 0.928991i \(-0.620678\pi\)
−0.370103 + 0.928991i \(0.620678\pi\)
\(458\) 0 0
\(459\) 519789. + 1.33057e6i 0.115158 + 0.294786i
\(460\) 0 0
\(461\) 6.34695e6i 1.39095i 0.718549 + 0.695476i \(0.244807\pi\)
−0.718549 + 0.695476i \(0.755193\pi\)
\(462\) 0 0
\(463\) 383220.i 0.0830799i −0.999137 0.0415399i \(-0.986774\pi\)
0.999137 0.0415399i \(-0.0132264\pi\)
\(464\) 0 0
\(465\) 2.11191e6 5.00270e6i 0.452943 1.07293i
\(466\) 0 0
\(467\) 8.95851e6 1.90083 0.950415 0.310983i \(-0.100658\pi\)
0.950415 + 0.310983i \(0.100658\pi\)
\(468\) 0 0
\(469\) 1.52938e6 0.321057
\(470\) 0 0
\(471\) 273032. 646757.i 0.0567101 0.134335i
\(472\) 0 0
\(473\) 1.26636e6i 0.260257i
\(474\) 0 0
\(475\) 243366.i 0.0494909i
\(476\) 0 0
\(477\) −4.70940e6 + 4.58376e6i −0.947697 + 0.922413i
\(478\) 0 0
\(479\) −5.76160e6 −1.14737 −0.573686 0.819075i \(-0.694487\pi\)
−0.573686 + 0.819075i \(0.694487\pi\)
\(480\) 0 0
\(481\) 3.01394e6 0.593981
\(482\) 0 0
\(483\) 3.22217e6 + 1.36025e6i 0.628464 + 0.265309i
\(484\) 0 0
\(485\) 930200.i 0.179565i
\(486\) 0 0
\(487\) 6.10617e6i 1.16667i 0.812233 + 0.583333i \(0.198252\pi\)
−0.812233 + 0.583333i \(0.801748\pi\)
\(488\) 0 0
\(489\) 1.04451e6 + 440947.i 0.197534 + 0.0833900i
\(490\) 0 0
\(491\) −4.14096e6 −0.775171 −0.387585 0.921834i \(-0.626691\pi\)
−0.387585 + 0.921834i \(0.626691\pi\)
\(492\) 0 0
\(493\) 2.62378e6 0.486194
\(494\) 0 0
\(495\) 1.47828e6 1.43884e6i 0.271172 0.263937i
\(496\) 0 0
\(497\) 3.95910e6i 0.718961i
\(498\) 0 0
\(499\) 5.59199e6i 1.00535i −0.864477 0.502673i \(-0.832350\pi\)
0.864477 0.502673i \(-0.167650\pi\)
\(500\) 0 0
\(501\) 1.04541e6 2.47636e6i 0.186076 0.440777i
\(502\) 0 0
\(503\) 7.65132e6 1.34839 0.674197 0.738552i \(-0.264490\pi\)
0.674197 + 0.738552i \(0.264490\pi\)
\(504\) 0 0
\(505\) −4.96760e6 −0.866798
\(506\) 0 0
\(507\) 469806. 1.11288e6i 0.0811706 0.192277i
\(508\) 0 0
\(509\) 7.91287e6i 1.35375i −0.736097 0.676877i \(-0.763333\pi\)
0.736097 0.676877i \(-0.236667\pi\)
\(510\) 0 0
\(511\) 1.43088e6i 0.242411i
\(512\) 0 0
\(513\) 295871. + 757382.i 0.0496375 + 0.127064i
\(514\) 0 0
\(515\) −3.13238e6 −0.520423
\(516\) 0 0
\(517\) −1.12635e6 −0.185330
\(518\) 0 0
\(519\) 3.30375e6 + 1.39470e6i 0.538380 + 0.227280i
\(520\) 0 0
\(521\) 1.06748e7i 1.72292i −0.507826 0.861460i \(-0.669551\pi\)
0.507826 0.861460i \(-0.330449\pi\)
\(522\) 0 0
\(523\) 3.59838e6i 0.575244i −0.957744 0.287622i \(-0.907135\pi\)
0.957744 0.287622i \(-0.0928647\pi\)
\(524\) 0 0
\(525\) 797812. + 336801.i 0.126329 + 0.0533304i
\(526\) 0 0
\(527\) −2.94387e6 −0.461735
\(528\) 0 0
\(529\) 1.45300e7 2.25749
\(530\) 0 0
\(531\) −608082. 624749.i −0.0935892 0.0961545i
\(532\) 0 0
\(533\) 1.65654e6i 0.252571i
\(534\) 0 0
\(535\) 4.36727e6i 0.659668i
\(536\) 0 0
\(537\) −212582. + 503563.i −0.0318119 + 0.0753560i
\(538\) 0 0
\(539\) −456775. −0.0677221
\(540\) 0 0
\(541\) −1.15956e7 −1.70333 −0.851666 0.524086i \(-0.824407\pi\)
−0.851666 + 0.524086i \(0.824407\pi\)
\(542\) 0 0
\(543\) −2.40590e6 + 5.69910e6i −0.350170 + 0.829482i
\(544\) 0 0
\(545\) 64293.3i 0.00927202i
\(546\) 0 0
\(547\) 550260.i 0.0786321i −0.999227 0.0393160i \(-0.987482\pi\)
0.999227 0.0393160i \(-0.0125179\pi\)
\(548\) 0 0
\(549\) 8.00550e6 + 8.22493e6i 1.13359 + 1.16467i
\(550\) 0 0
\(551\) 1.49349e6 0.209568
\(552\) 0 0
\(553\) 2.75331e6 0.382862
\(554\) 0 0
\(555\) −3.56339e6 1.50430e6i −0.491056 0.207302i
\(556\) 0 0
\(557\) 5.20802e6i 0.711270i 0.934625 + 0.355635i \(0.115735\pi\)
−0.934625 + 0.355635i \(0.884265\pi\)
\(558\) 0 0
\(559\) 3.60805e6i 0.488363i
\(560\) 0 0
\(561\) −1.03032e6 434954.i −0.138218 0.0583493i
\(562\) 0 0
\(563\) 3.38957e6 0.450685 0.225343 0.974280i \(-0.427650\pi\)
0.225343 + 0.974280i \(0.427650\pi\)
\(564\) 0 0
\(565\) 9.49428e6 1.25124
\(566\) 0 0
\(567\) 2.89234e6 + 78222.6i 0.377826 + 0.0102182i
\(568\) 0 0
\(569\) 1.22684e7i 1.58857i −0.607547 0.794284i \(-0.707846\pi\)
0.607547 0.794284i \(-0.292154\pi\)
\(570\) 0 0
\(571\) 1.40136e6i 0.179871i −0.995948 0.0899354i \(-0.971334\pi\)
0.995948 0.0899354i \(-0.0286661\pi\)
\(572\) 0 0
\(573\) −2.65441e6 + 6.28777e6i −0.337740 + 0.800036i
\(574\) 0 0
\(575\) 5.19128e6 0.654795
\(576\) 0 0
\(577\) 1.11633e7 1.39590 0.697950 0.716146i \(-0.254096\pi\)
0.697950 + 0.716146i \(0.254096\pi\)
\(578\) 0 0
\(579\) −4.29554e6 + 1.01753e7i −0.532502 + 1.26139i
\(580\) 0 0
\(581\) 2.52450e6i 0.310267i
\(582\) 0 0
\(583\) 5.14508e6i 0.626932i
\(584\) 0 0
\(585\) 4.21187e6 4.09950e6i 0.508844 0.495269i
\(586\) 0 0
\(587\) −1.39598e7 −1.67218 −0.836091 0.548591i \(-0.815164\pi\)
−0.836091 + 0.548591i \(0.815164\pi\)
\(588\) 0 0
\(589\) −1.67570e6 −0.199025
\(590\) 0 0
\(591\) −7.61667e6 3.21542e6i −0.897009 0.378677i
\(592\) 0 0
\(593\) 7.08900e6i 0.827843i −0.910313 0.413922i \(-0.864159\pi\)
0.910313 0.413922i \(-0.135841\pi\)
\(594\) 0 0
\(595\) 824574.i 0.0954855i
\(596\) 0 0
\(597\) −1.44373e7 6.09478e6i −1.65787 0.699878i
\(598\) 0 0
\(599\) 1.17685e7 1.34016 0.670079 0.742290i \(-0.266260\pi\)
0.670079 + 0.742290i \(0.266260\pi\)
\(600\) 0 0
\(601\) −1.00504e7 −1.13501 −0.567504 0.823371i \(-0.692091\pi\)
−0.567504 + 0.823371i \(0.692091\pi\)
\(602\) 0 0
\(603\) 5.43503e6 5.29003e6i 0.608708 0.592468i
\(604\) 0 0
\(605\) 5.57162e6i 0.618861i
\(606\) 0 0
\(607\) 3.19698e6i 0.352183i −0.984374 0.176091i \(-0.943655\pi\)
0.984374 0.176091i \(-0.0563454\pi\)
\(608\) 0 0
\(609\) 2.06688e6 4.89603e6i 0.225825 0.534935i
\(610\) 0 0
\(611\) −3.20915e6 −0.347766
\(612\) 0 0
\(613\) −1.41787e7 −1.52400 −0.761999 0.647578i \(-0.775782\pi\)
−0.761999 + 0.647578i \(0.775782\pi\)
\(614\) 0 0
\(615\) 826802. 1.95853e6i 0.0881482 0.208805i
\(616\) 0 0
\(617\) 7.12714e6i 0.753707i 0.926273 + 0.376853i \(0.122994\pi\)
−0.926273 + 0.376853i \(0.877006\pi\)
\(618\) 0 0
\(619\) 1.78191e7i 1.86922i −0.355678 0.934609i \(-0.615750\pi\)
0.355678 0.934609i \(-0.384250\pi\)
\(620\) 0 0
\(621\) 1.61558e7 6.31129e6i 1.68113 0.656733i
\(622\) 0 0
\(623\) 3.80639e6 0.392910
\(624\) 0 0
\(625\) −4.93732e6 −0.505581
\(626\) 0 0
\(627\) −586472. 247582.i −0.0595769 0.0251507i
\(628\) 0 0
\(629\) 2.09690e6i 0.211325i
\(630\) 0 0
\(631\) 5.00865e6i 0.500780i −0.968145 0.250390i \(-0.919441\pi\)
0.968145 0.250390i \(-0.0805589\pi\)
\(632\) 0 0
\(633\) 1.44249e7 + 6.08954e6i 1.43088 + 0.604054i
\(634\) 0 0
\(635\) −1.06936e7 −1.05242
\(636\) 0 0
\(637\) −1.30142e6 −0.127078
\(638\) 0 0
\(639\) −1.36943e7 1.40697e7i −1.32675 1.36311i
\(640\) 0 0
\(641\) 5.66542e6i 0.544611i 0.962211 + 0.272306i \(0.0877862\pi\)
−0.962211 + 0.272306i \(0.912214\pi\)
\(642\) 0 0
\(643\) 1.14723e7i 1.09427i −0.837046 0.547133i \(-0.815719\pi\)
0.837046 0.547133i \(-0.184281\pi\)
\(644\) 0 0
\(645\) 1.80083e6 4.26580e6i 0.170441 0.403739i
\(646\) 0 0
\(647\) −1.08061e7 −1.01486 −0.507432 0.861692i \(-0.669405\pi\)
−0.507432 + 0.861692i \(0.669405\pi\)
\(648\) 0 0
\(649\) 682546. 0.0636093
\(650\) 0 0
\(651\) −2.31904e6 + 5.49334e6i −0.214465 + 0.508024i
\(652\) 0 0
\(653\) 1.87285e7i 1.71878i 0.511318 + 0.859392i \(0.329157\pi\)
−0.511318 + 0.859392i \(0.670843\pi\)
\(654\) 0 0
\(655\) 7.03116e6i 0.640360i
\(656\) 0 0
\(657\) 4.94935e6 + 5.08501e6i 0.447337 + 0.459599i
\(658\) 0 0
\(659\) 6.06692e6 0.544195 0.272098 0.962270i \(-0.412283\pi\)
0.272098 + 0.962270i \(0.412283\pi\)
\(660\) 0 0
\(661\) 8.66576e6 0.771442 0.385721 0.922616i \(-0.373953\pi\)
0.385721 + 0.922616i \(0.373953\pi\)
\(662\) 0 0
\(663\) −2.93554e6 1.23925e6i −0.259360 0.109490i
\(664\) 0 0
\(665\) 469360.i 0.0411578i
\(666\) 0 0
\(667\) 3.18580e7i 2.77270i
\(668\) 0 0
\(669\) 1.48164e7 + 6.25482e6i 1.27990 + 0.540318i
\(670\) 0 0
\(671\) −8.98583e6 −0.770464
\(672\) 0 0
\(673\) −1.47083e7 −1.25177 −0.625886 0.779915i \(-0.715262\pi\)
−0.625886 + 0.779915i \(0.715262\pi\)
\(674\) 0 0
\(675\) 4.00021e6 1.56268e6i 0.337927 0.132011i
\(676\) 0 0
\(677\) 8.29853e6i 0.695872i −0.937518 0.347936i \(-0.886883\pi\)
0.937518 0.347936i \(-0.113117\pi\)
\(678\) 0 0
\(679\) 1.02143e6i 0.0850225i
\(680\) 0 0
\(681\) −5.49509e6 + 1.30167e7i −0.454053 + 1.07556i
\(682\) 0 0
\(683\) −1.26991e7 −1.04165 −0.520826 0.853663i \(-0.674376\pi\)
−0.520826 + 0.853663i \(0.674376\pi\)
\(684\) 0 0
\(685\) −4.05550e6 −0.330231
\(686\) 0 0
\(687\) −804842. + 1.90651e6i −0.0650607 + 0.154116i
\(688\) 0 0
\(689\) 1.46591e7i 1.17641i
\(690\) 0 0
\(691\) 9.85072e6i 0.784825i −0.919789 0.392413i \(-0.871641\pi\)
0.919789 0.392413i \(-0.128359\pi\)
\(692\) 0 0
\(693\) −1.62327e6 + 1.57996e6i −0.128398 + 0.124972i
\(694\) 0 0
\(695\) −4.85543e6 −0.381299
\(696\) 0 0
\(697\) −1.15251e6 −0.0898591
\(698\) 0 0
\(699\) 9.38080e6 + 3.96015e6i 0.726184 + 0.306563i
\(700\) 0 0
\(701\) 1.12185e7i 0.862262i −0.902289 0.431131i \(-0.858115\pi\)
0.902289 0.431131i \(-0.141885\pi\)
\(702\) 0 0
\(703\) 1.19359e6i 0.0910891i
\(704\) 0 0
\(705\) 3.79418e6 + 1.60173e6i 0.287505 + 0.121372i
\(706\) 0 0
\(707\) 5.45480e6 0.410422
\(708\) 0 0
\(709\) −2.40771e7 −1.79882 −0.899412 0.437101i \(-0.856005\pi\)
−0.899412 + 0.437101i \(0.856005\pi\)
\(710\) 0 0
\(711\) 9.78460e6 9.52355e6i 0.725887 0.706521i
\(712\) 0 0
\(713\) 3.57446e7i 2.63322i
\(714\) 0 0
\(715\) 4.60151e6i 0.336617i
\(716\) 0 0
\(717\) 7.92243e6 1.87666e7i 0.575520 1.36329i
\(718\) 0 0
\(719\) 1.64759e7 1.18858 0.594289 0.804252i \(-0.297434\pi\)
0.594289 + 0.804252i \(0.297434\pi\)
\(720\) 0 0
\(721\) 3.43959e6 0.246416
\(722\) 0 0
\(723\) −3.05149e6 + 7.22837e6i −0.217103 + 0.514274i
\(724\) 0 0
\(725\) 7.88807e6i 0.557347i
\(726\) 0 0
\(727\) 9.00670e6i 0.632018i −0.948756 0.316009i \(-0.897657\pi\)
0.948756 0.316009i \(-0.102343\pi\)
\(728\) 0 0
\(729\) 1.05493e7 9.72648e6i 0.735196 0.677855i
\(730\) 0 0
\(731\) −2.51024e6 −0.173749
\(732\) 0 0
\(733\) 1.72333e7 1.18470 0.592348 0.805682i \(-0.298201\pi\)
0.592348 + 0.805682i \(0.298201\pi\)
\(734\) 0 0
\(735\) 1.53868e6 + 649560.i 0.105058 + 0.0443507i
\(736\) 0 0
\(737\) 5.93784e6i 0.402680i
\(738\) 0 0
\(739\) 1.11962e6i 0.0754154i −0.999289 0.0377077i \(-0.987994\pi\)
0.999289 0.0377077i \(-0.0120056\pi\)
\(740\) 0 0
\(741\) −1.67095e6 705401.i −0.111794 0.0471944i
\(742\) 0 0
\(743\) −2.11818e6 −0.140764 −0.0703818 0.997520i \(-0.522422\pi\)
−0.0703818 + 0.997520i \(0.522422\pi\)
\(744\) 0 0
\(745\) 1.09378e6 0.0722001
\(746\) 0 0
\(747\) 8.73211e6 + 8.97146e6i 0.572556 + 0.588250i
\(748\) 0 0
\(749\) 4.79559e6i 0.312347i
\(750\) 0 0
\(751\) 2.63493e7i 1.70479i −0.522902 0.852393i \(-0.675150\pi\)
0.522902 0.852393i \(-0.324850\pi\)
\(752\) 0 0
\(753\) 5.91291e6 1.40065e7i 0.380026 0.900206i
\(754\) 0 0
\(755\) −2.10144e6 −0.134168
\(756\) 0 0
\(757\) 4.02559e6 0.255323 0.127662 0.991818i \(-0.459253\pi\)
0.127662 + 0.991818i \(0.459253\pi\)
\(758\) 0 0
\(759\) −5.28122e6 + 1.25101e7i −0.332759 + 0.788238i
\(760\) 0 0
\(761\) 2.11368e7i 1.32305i 0.749922 + 0.661527i \(0.230091\pi\)
−0.749922 + 0.661527i \(0.769909\pi\)
\(762\) 0 0
\(763\) 70598.9i 0.00439022i
\(764\) 0 0
\(765\) 2.85216e6 + 2.93034e6i 0.176206 + 0.181036i
\(766\) 0 0
\(767\) 1.94468e6 0.119360
\(768\) 0 0
\(769\) −1.28275e7 −0.782217 −0.391108 0.920345i \(-0.627908\pi\)
−0.391108 + 0.920345i \(0.627908\pi\)
\(770\) 0 0
\(771\) −5.80020e6 2.44858e6i −0.351404 0.148347i
\(772\) 0 0
\(773\) 1.82599e6i 0.109913i −0.998489 0.0549566i \(-0.982498\pi\)
0.998489 0.0549566i \(-0.0175021\pi\)
\(774\) 0 0
\(775\) 8.85040e6i 0.529308i
\(776\) 0 0
\(777\) 3.91287e6 + 1.65184e6i 0.232511 + 0.0981556i
\(778\) 0 0
\(779\) −656025. −0.0387326
\(780\) 0 0
\(781\) 1.53713e7 0.901743
\(782\) 0 0
\(783\) −9.58990e6 2.45485e7i −0.558997 1.43094i
\(784\) 0 0
\(785\) 2.00962e6i 0.116396i
\(786\) 0 0
\(787\) 2.74454e7i 1.57955i 0.613398 + 0.789774i \(0.289802\pi\)
−0.613398 + 0.789774i \(0.710198\pi\)
\(788\) 0 0
\(789\) 5.42385e6 1.28480e7i 0.310181 0.734755i
\(790\) 0 0
\(791\) −1.04254e7 −0.592452
\(792\) 0 0
\(793\) −2.56021e7 −1.44575
\(794\) 0 0
\(795\) −7.31659e6 + 1.73315e7i −0.410573 + 0.972565i
\(796\) 0 0
\(797\) 2.81269e7i 1.56847i −0.620465 0.784234i \(-0.713056\pi\)
0.620465 0.784234i \(-0.286944\pi\)
\(798\) 0 0
\(799\) 2.23271e6i 0.123727i
\(800\) 0 0
\(801\) 1.35270e7 1.31661e7i 0.744938 0.725064i
\(802\) 0 0
\(803\) −5.55544e6 −0.304039
\(804\) 0 0
\(805\) 1.00120e7 0.544542
\(806\) 0 0
\(807\) 2.18834e7 + 9.23817e6i 1.18285 + 0.499347i
\(808\) 0 0
\(809\) 2.60102e7i 1.39724i −0.715491 0.698622i \(-0.753797\pi\)
0.715491 0.698622i \(-0.246203\pi\)
\(810\) 0 0
\(811\) 2.50958e6i 0.133983i 0.997754 + 0.0669913i \(0.0213400\pi\)
−0.997754 + 0.0669913i \(0.978660\pi\)
\(812\) 0 0
\(813\) 5.52854e6 + 2.33390e6i 0.293349 + 0.123839i
\(814\) 0 0
\(815\) 3.24554e6 0.171156
\(816\) 0 0
\(817\) −1.42887e6 −0.0748922
\(818\) 0 0
\(819\) −4.62495e6 + 4.50156e6i −0.240933 + 0.234506i
\(820\) 0 0
\(821\) 5.92935e6i 0.307007i 0.988148 + 0.153504i \(0.0490557\pi\)
−0.988148 + 0.153504i \(0.950944\pi\)
\(822\) 0 0
\(823\) 8.73575e6i 0.449574i 0.974408 + 0.224787i \(0.0721686\pi\)
−0.974408 + 0.224787i \(0.927831\pi\)
\(824\) 0 0
\(825\) −1.30764e6 + 3.09752e6i −0.0668886 + 0.158445i
\(826\) 0 0
\(827\) −6.46971e6 −0.328943 −0.164472 0.986382i \(-0.552592\pi\)
−0.164472 + 0.986382i \(0.552592\pi\)
\(828\) 0 0
\(829\) 1.72888e7 0.873731 0.436865 0.899527i \(-0.356089\pi\)
0.436865 + 0.899527i \(0.356089\pi\)
\(830\) 0 0
\(831\) −5.46546e6 + 1.29466e7i −0.274552 + 0.650357i
\(832\) 0 0
\(833\) 905445.i 0.0452115i
\(834\) 0 0
\(835\) 7.69461e6i 0.381918i
\(836\) 0 0
\(837\) 1.07599e7 + 2.75434e7i 0.530875 + 1.35895i
\(838\) 0 0
\(839\) 4.19248e6 0.205621 0.102810 0.994701i \(-0.467217\pi\)
0.102810 + 0.994701i \(0.467217\pi\)
\(840\) 0 0
\(841\) −2.78965e7 −1.36006
\(842\) 0 0
\(843\) 1.25268e7 + 5.28824e6i 0.607113 + 0.256296i
\(844\) 0 0
\(845\) 3.45796e6i 0.166601i
\(846\) 0 0
\(847\) 6.11806e6i 0.293025i
\(848\) 0 0
\(849\) −5.41827e6 2.28735e6i −0.257983 0.108909i
\(850\) 0 0
\(851\) 2.54606e7 1.20516
\(852\) 0 0
\(853\) −3.78255e7 −1.77997 −0.889985 0.455990i \(-0.849285\pi\)
−0.889985 + 0.455990i \(0.849285\pi\)
\(854\) 0 0
\(855\) 1.62349e6 + 1.66799e6i 0.0759512 + 0.0780331i
\(856\) 0 0
\(857\) 1.24919e7i 0.580998i 0.956875 + 0.290499i \(0.0938213\pi\)
−0.956875 + 0.290499i \(0.906179\pi\)
\(858\) 0 0
\(859\) 5.06556e6i 0.234231i 0.993118 + 0.117116i \(0.0373648\pi\)
−0.993118 + 0.117116i \(0.962635\pi\)
\(860\) 0 0
\(861\) −907891. + 2.15061e6i −0.0417374 + 0.0988675i
\(862\) 0 0
\(863\) −2.86177e7 −1.30800 −0.653999 0.756495i \(-0.726910\pi\)
−0.653999 + 0.756495i \(0.726910\pi\)
\(864\) 0 0
\(865\) 1.02655e7 0.466488
\(866\) 0 0
\(867\) −7.74592e6 + 1.83485e7i −0.349965 + 0.828997i
\(868\) 0 0
\(869\) 1.06898e7i 0.480197i
\(870\) 0 0
\(871\) 1.69178e7i 0.755613i
\(872\) 0 0
\(873\) −3.53307e6 3.62991e6i −0.156898 0.161198i
\(874\) 0 0
\(875\) 9.31196e6 0.411170
\(876\) 0 0
\(877\) −2.65453e7 −1.16544 −0.582719 0.812674i \(-0.698011\pi\)
−0.582719 + 0.812674i \(0.698011\pi\)
\(878\) 0 0
\(879\) 1.56292e7 + 6.59797e6i 0.682285 + 0.288030i
\(880\) 0 0
\(881\) 2.02490e7i 0.878951i −0.898255 0.439476i \(-0.855164\pi\)
0.898255 0.439476i \(-0.144836\pi\)
\(882\) 0 0
\(883\) 3.11513e7i 1.34454i −0.740306 0.672271i \(-0.765319\pi\)
0.740306 0.672271i \(-0.234681\pi\)
\(884\) 0 0
\(885\) −2.29920e6 970619.i −0.0986776 0.0416573i
\(886\) 0 0
\(887\) −2.00955e7 −0.857608 −0.428804 0.903398i \(-0.641065\pi\)
−0.428804 + 0.903398i \(0.641065\pi\)
\(888\) 0 0
\(889\) 1.17423e7 0.498310
\(890\) 0 0
\(891\) −303701. + 1.12296e7i −0.0128160 + 0.473881i
\(892\) 0 0
\(893\) 1.27089e6i 0.0533311i
\(894\) 0 0
\(895\) 1.56468e6i 0.0652934i
\(896\) 0 0
\(897\) −1.50470e7 + 3.56434e7i −0.624410 + 1.47910i
\(898\) 0 0
\(899\) 5.43133e7 2.24134
\(900\) 0 0
\(901\) 1.01989e7 0.418542
\(902\) 0 0
\(903\) −1.97745e6 + 4.68417e6i −0.0807022 + 0.191167i
\(904\) 0 0
\(905\) 1.77084e7i 0.718717i
\(906\) 0 0
\(907\) 2.84073e6i 0.114660i −0.998355 0.0573299i \(-0.981741\pi\)
0.998355 0.0573299i \(-0.0182587\pi\)
\(908\) 0 0
\(909\) 1.93850e7 1.88679e7i 0.778138 0.757378i
\(910\) 0 0
\(911\) 6.98893e6 0.279007 0.139503 0.990222i \(-0.455449\pi\)
0.139503 + 0.990222i \(0.455449\pi\)
\(912\) 0 0
\(913\) −9.80143e6 −0.389146
\(914\) 0 0
\(915\) 3.02694e7 + 1.27784e7i 1.19523 + 0.504571i
\(916\) 0 0
\(917\) 7.72075e6i 0.303205i
\(918\) 0 0
\(919\) 1.42936e7i 0.558281i 0.960250 + 0.279141i \(0.0900496\pi\)
−0.960250 + 0.279141i \(0.909950\pi\)
\(920\) 0 0
\(921\) −3.01110e6 1.27115e6i −0.116971 0.0493797i
\(922\) 0 0
\(923\) 4.37953e7 1.69209
\(924\) 0 0
\(925\) 6.30408e6 0.242252
\(926\) 0 0
\(927\) 1.22235e7 1.18974e7i 0.467192 0.454727i
\(928\) 0 0
\(929\) 2.82939e6i 0.107561i 0.998553 + 0.0537804i \(0.0171271\pi\)
−0.998553 + 0.0537804i \(0.982873\pi\)
\(930\) 0 0
\(931\) 515393.i 0.0194879i
\(932\) 0 0
\(933\) −8.91194e6 + 2.11106e7i −0.335172 + 0.793955i
\(934\) 0 0
\(935\) −3.20143e6 −0.119761
\(936\) 0 0
\(937\) −4.90222e7 −1.82408 −0.912041 0.410100i \(-0.865494\pi\)
−0.912041 + 0.410100i \(0.865494\pi\)
\(938\) 0 0
\(939\) −1.44541e7 + 3.42388e7i −0.534967 + 1.26723i
\(940\) 0 0
\(941\) 2.87482e7i 1.05837i 0.848507 + 0.529184i \(0.177502\pi\)
−0.848507 + 0.529184i \(0.822498\pi\)
\(942\) 0 0
\(943\) 1.39938e7i 0.512456i
\(944\) 0 0
\(945\) 7.71487e6 3.01382e6i 0.281028 0.109784i
\(946\) 0 0
\(947\) 3.26394e6 0.118268 0.0591339 0.998250i \(-0.481166\pi\)
0.0591339 + 0.998250i \(0.481166\pi\)
\(948\) 0 0
\(949\) −1.58283e7 −0.570518
\(950\) 0 0
\(951\) −4.03909e7 1.70512e7i −1.44821 0.611371i
\(952\) 0 0
\(953\) 3.40785e7i 1.21548i 0.794135 + 0.607741i \(0.207924\pi\)
−0.794135 + 0.607741i \(0.792076\pi\)
\(954\) 0 0
\(955\) 1.95375e7i 0.693204i
\(956\) 0 0
\(957\) 1.90089e7 + 8.02472e6i 0.670931 + 0.283237i
\(958\) 0 0
\(959\) 4.45324e6 0.156362
\(960\) 0 0
\(961\) −3.23103e7 −1.12858
\(962\) 0 0
\(963\) −1.65877e7 1.70424e7i −0.576395 0.592194i
\(964\) 0 0
\(965\) 3.16168e7i 1.09295i
\(966\) 0 0
\(967\) 213581.i 0.00734506i −0.999993 0.00367253i \(-0.998831\pi\)
0.999993 0.00367253i \(-0.00116901\pi\)
\(968\) 0 0
\(969\) 490771. 1.16254e6i 0.0167907 0.0397738i
\(970\) 0 0
\(971\) 7.63347e6 0.259821 0.129910 0.991526i \(-0.458531\pi\)
0.129910 + 0.991526i \(0.458531\pi\)
\(972\) 0 0
\(973\) 5.33162e6 0.180542
\(974\) 0 0
\(975\) −3.72566e6 + 8.82534e6i −0.125514 + 0.297317i
\(976\) 0 0
\(977\) 2.42456e7i 0.812638i −0.913731 0.406319i \(-0.866812\pi\)
0.913731 0.406319i \(-0.133188\pi\)
\(978\) 0 0
\(979\) 1.47784e7i 0.492800i
\(980\) 0 0
\(981\) −244198. 250891.i −0.00810157 0.00832363i
\(982\) 0 0
\(983\) 2.16068e7 0.713192 0.356596 0.934259i \(-0.383937\pi\)
0.356596 + 0.934259i \(0.383937\pi\)
\(984\) 0 0
\(985\) −2.36667e7 −0.777227
\(986\) 0 0
\(987\) −4.16629e6 1.75882e6i −0.136131 0.0574684i
\(988\) 0 0
\(989\) 3.04794e7i 0.990868i
\(990\) 0 0
\(991\) 1.65466e7i 0.535211i 0.963529 + 0.267606i \(0.0862324\pi\)
−0.963529 + 0.267606i \(0.913768\pi\)
\(992\) 0 0
\(993\) −1.36032e7 5.74265e6i −0.437791 0.184816i
\(994\) 0 0
\(995\) −4.48600e7 −1.43649
\(996\) 0 0
\(997\) −1.55649e7 −0.495915 −0.247957 0.968771i \(-0.579759\pi\)
−0.247957 + 0.968771i \(0.579759\pi\)
\(998\) 0 0
\(999\) 1.96190e7 7.66417e6i 0.621961 0.242969i
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 336.6.h.b.239.16 yes 40
3.2 odd 2 inner 336.6.h.b.239.26 yes 40
4.3 odd 2 inner 336.6.h.b.239.25 yes 40
12.11 even 2 inner 336.6.h.b.239.15 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.6.h.b.239.15 40 12.11 even 2 inner
336.6.h.b.239.16 yes 40 1.1 even 1 trivial
336.6.h.b.239.25 yes 40 4.3 odd 2 inner
336.6.h.b.239.26 yes 40 3.2 odd 2 inner