Properties

Label 207.6.c.a.206.10
Level $207$
Weight $6$
Character 207.206
Analytic conductor $33.199$
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] = [207,6,Mod(206,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.206");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 207.c (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: \(33.1994507013\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 206.10
Character \(\chi\) \(=\) 207.206
Dual form 207.6.c.a.206.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.32542i q^{2} +20.9416 q^{4} -58.3890 q^{5} -22.0963i q^{7} +176.053i q^{8} +O(q^{10})\) \(q+3.32542i q^{2} +20.9416 q^{4} -58.3890 q^{5} -22.0963i q^{7} +176.053i q^{8} -194.168i q^{10} +772.113 q^{11} -840.212 q^{13} +73.4793 q^{14} +84.6829 q^{16} +1076.07 q^{17} -726.287i q^{19} -1222.76 q^{20} +2567.60i q^{22} +(-958.808 + 2348.84i) q^{23} +284.272 q^{25} -2794.05i q^{26} -462.732i q^{28} +8098.76i q^{29} +60.1277 q^{31} +5915.30i q^{32} +3578.39i q^{34} +1290.18i q^{35} -2590.57i q^{37} +2415.21 q^{38} -10279.5i q^{40} -3380.97i q^{41} +15940.7i q^{43} +16169.3 q^{44} +(-7810.85 - 3188.44i) q^{46} +7434.87i q^{47} +16318.8 q^{49} +945.324i q^{50} -17595.4 q^{52} -18754.5 q^{53} -45082.9 q^{55} +3890.11 q^{56} -26931.7 q^{58} +31076.5i q^{59} +40476.3i q^{61} +199.949i q^{62} -16961.0 q^{64} +49059.1 q^{65} -15803.0i q^{67} +22534.7 q^{68} -4290.38 q^{70} +37916.8i q^{71} -51596.6 q^{73} +8614.71 q^{74} -15209.6i q^{76} -17060.8i q^{77} +53278.7i q^{79} -4944.55 q^{80} +11243.1 q^{82} -114127. q^{83} -62830.9 q^{85} -53009.5 q^{86} +135933. i q^{88} +36052.3 q^{89} +18565.6i q^{91} +(-20079.0 + 49188.4i) q^{92} -24724.0 q^{94} +42407.1i q^{95} -134874. i q^{97} +54266.6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 600 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.32542i 0.587856i 0.955828 + 0.293928i \(0.0949626\pi\)
−0.955828 + 0.293928i \(0.905037\pi\)
\(3\) 0 0
\(4\) 20.9416 0.654425
\(5\) −58.3890 −1.04449 −0.522247 0.852794i \(-0.674906\pi\)
−0.522247 + 0.852794i \(0.674906\pi\)
\(6\) 0 0
\(7\) 22.0963i 0.170441i −0.996362 0.0852205i \(-0.972841\pi\)
0.996362 0.0852205i \(-0.0271595\pi\)
\(8\) 176.053i 0.972564i
\(9\) 0 0
\(10\) 194.168i 0.614012i
\(11\) 772.113 1.92397 0.961987 0.273097i \(-0.0880480\pi\)
0.961987 + 0.273097i \(0.0880480\pi\)
\(12\) 0 0
\(13\) −840.212 −1.37889 −0.689447 0.724337i \(-0.742146\pi\)
−0.689447 + 0.724337i \(0.742146\pi\)
\(14\) 73.4793 0.100195
\(15\) 0 0
\(16\) 84.6829 0.0826981
\(17\) 1076.07 0.903067 0.451533 0.892254i \(-0.350877\pi\)
0.451533 + 0.892254i \(0.350877\pi\)
\(18\) 0 0
\(19\) 726.287i 0.461556i −0.973006 0.230778i \(-0.925873\pi\)
0.973006 0.230778i \(-0.0741271\pi\)
\(20\) −1222.76 −0.683543
\(21\) 0 0
\(22\) 2567.60i 1.13102i
\(23\) −958.808 + 2348.84i −0.377931 + 0.925834i
\(24\) 0 0
\(25\) 284.272 0.0909672
\(26\) 2794.05i 0.810590i
\(27\) 0 0
\(28\) 462.732i 0.111541i
\(29\) 8098.76i 1.78823i 0.447836 + 0.894116i \(0.352195\pi\)
−0.447836 + 0.894116i \(0.647805\pi\)
\(30\) 0 0
\(31\) 60.1277 0.0112375 0.00561876 0.999984i \(-0.498211\pi\)
0.00561876 + 0.999984i \(0.498211\pi\)
\(32\) 5915.30i 1.02118i
\(33\) 0 0
\(34\) 3578.39i 0.530873i
\(35\) 1290.18i 0.178025i
\(36\) 0 0
\(37\) 2590.57i 0.311093i −0.987829 0.155547i \(-0.950286\pi\)
0.987829 0.155547i \(-0.0497139\pi\)
\(38\) 2415.21 0.271328
\(39\) 0 0
\(40\) 10279.5i 1.01584i
\(41\) 3380.97i 0.314110i −0.987590 0.157055i \(-0.949800\pi\)
0.987590 0.157055i \(-0.0501999\pi\)
\(42\) 0 0
\(43\) 15940.7i 1.31473i 0.753572 + 0.657365i \(0.228329\pi\)
−0.753572 + 0.657365i \(0.771671\pi\)
\(44\) 16169.3 1.25910
\(45\) 0 0
\(46\) −7810.85 3188.44i −0.544257 0.222169i
\(47\) 7434.87i 0.490940i 0.969404 + 0.245470i \(0.0789423\pi\)
−0.969404 + 0.245470i \(0.921058\pi\)
\(48\) 0 0
\(49\) 16318.8 0.970950
\(50\) 945.324i 0.0534756i
\(51\) 0 0
\(52\) −17595.4 −0.902383
\(53\) −18754.5 −0.917099 −0.458550 0.888669i \(-0.651631\pi\)
−0.458550 + 0.888669i \(0.651631\pi\)
\(54\) 0 0
\(55\) −45082.9 −2.00958
\(56\) 3890.11 0.165765
\(57\) 0 0
\(58\) −26931.7 −1.05122
\(59\) 31076.5i 1.16226i 0.813812 + 0.581128i \(0.197388\pi\)
−0.813812 + 0.581128i \(0.802612\pi\)
\(60\) 0 0
\(61\) 40476.3i 1.39276i 0.717674 + 0.696379i \(0.245207\pi\)
−0.717674 + 0.696379i \(0.754793\pi\)
\(62\) 199.949i 0.00660604i
\(63\) 0 0
\(64\) −16961.0 −0.517608
\(65\) 49059.1 1.44025
\(66\) 0 0
\(67\) 15803.0i 0.430082i −0.976605 0.215041i \(-0.931011\pi\)
0.976605 0.215041i \(-0.0689886\pi\)
\(68\) 22534.7 0.590990
\(69\) 0 0
\(70\) −4290.38 −0.104653
\(71\) 37916.8i 0.892660i 0.894868 + 0.446330i \(0.147269\pi\)
−0.894868 + 0.446330i \(0.852731\pi\)
\(72\) 0 0
\(73\) −51596.6 −1.13322 −0.566609 0.823987i \(-0.691745\pi\)
−0.566609 + 0.823987i \(0.691745\pi\)
\(74\) 8614.71 0.182878
\(75\) 0 0
\(76\) 15209.6i 0.302054i
\(77\) 17060.8i 0.327924i
\(78\) 0 0
\(79\) 53278.7i 0.960474i 0.877139 + 0.480237i \(0.159449\pi\)
−0.877139 + 0.480237i \(0.840551\pi\)
\(80\) −4944.55 −0.0863777
\(81\) 0 0
\(82\) 11243.1 0.184651
\(83\) −114127. −1.81842 −0.909209 0.416341i \(-0.863312\pi\)
−0.909209 + 0.416341i \(0.863312\pi\)
\(84\) 0 0
\(85\) −62830.9 −0.943248
\(86\) −53009.5 −0.772872
\(87\) 0 0
\(88\) 135933.i 1.87119i
\(89\) 36052.3 0.482457 0.241228 0.970468i \(-0.422450\pi\)
0.241228 + 0.970468i \(0.422450\pi\)
\(90\) 0 0
\(91\) 18565.6i 0.235020i
\(92\) −20079.0 + 49188.4i −0.247327 + 0.605889i
\(93\) 0 0
\(94\) −24724.0 −0.288602
\(95\) 42407.1i 0.482092i
\(96\) 0 0
\(97\) 134874.i 1.45546i −0.685863 0.727730i \(-0.740575\pi\)
0.685863 0.727730i \(-0.259425\pi\)
\(98\) 54266.6i 0.570779i
\(99\) 0 0
\(100\) 5953.12 0.0595312
\(101\) 104342.i 1.01778i −0.860831 0.508890i \(-0.830056\pi\)
0.860831 0.508890i \(-0.169944\pi\)
\(102\) 0 0
\(103\) 145443.i 1.35082i 0.737441 + 0.675412i \(0.236034\pi\)
−0.737441 + 0.675412i \(0.763966\pi\)
\(104\) 147922.i 1.34106i
\(105\) 0 0
\(106\) 62366.6i 0.539122i
\(107\) 114667. 0.968231 0.484115 0.875004i \(-0.339141\pi\)
0.484115 + 0.875004i \(0.339141\pi\)
\(108\) 0 0
\(109\) 213204.i 1.71882i −0.511291 0.859408i \(-0.670833\pi\)
0.511291 0.859408i \(-0.329167\pi\)
\(110\) 149919.i 1.18134i
\(111\) 0 0
\(112\) 1871.18i 0.0140952i
\(113\) 144078. 1.06146 0.530728 0.847542i \(-0.321919\pi\)
0.530728 + 0.847542i \(0.321919\pi\)
\(114\) 0 0
\(115\) 55983.8 137146.i 0.394746 0.967028i
\(116\) 169601.i 1.17026i
\(117\) 0 0
\(118\) −103342. −0.683239
\(119\) 23777.2i 0.153920i
\(120\) 0 0
\(121\) 435107. 2.70167
\(122\) −134600. −0.818741
\(123\) 0 0
\(124\) 1259.17 0.00735411
\(125\) 165867. 0.949479
\(126\) 0 0
\(127\) 149218. 0.820943 0.410471 0.911873i \(-0.365364\pi\)
0.410471 + 0.911873i \(0.365364\pi\)
\(128\) 132887.i 0.716900i
\(129\) 0 0
\(130\) 163142.i 0.846657i
\(131\) 59735.2i 0.304125i 0.988371 + 0.152062i \(0.0485915\pi\)
−0.988371 + 0.152062i \(0.951409\pi\)
\(132\) 0 0
\(133\) −16048.2 −0.0786680
\(134\) 52551.4 0.252826
\(135\) 0 0
\(136\) 189446.i 0.878290i
\(137\) −51774.6 −0.235676 −0.117838 0.993033i \(-0.537596\pi\)
−0.117838 + 0.993033i \(0.537596\pi\)
\(138\) 0 0
\(139\) −83329.7 −0.365816 −0.182908 0.983130i \(-0.558551\pi\)
−0.182908 + 0.983130i \(0.558551\pi\)
\(140\) 27018.4i 0.116504i
\(141\) 0 0
\(142\) −126089. −0.524756
\(143\) −648739. −2.65295
\(144\) 0 0
\(145\) 472878.i 1.86780i
\(146\) 171580.i 0.666169i
\(147\) 0 0
\(148\) 54250.6i 0.203587i
\(149\) 149607. 0.552062 0.276031 0.961149i \(-0.410981\pi\)
0.276031 + 0.961149i \(0.410981\pi\)
\(150\) 0 0
\(151\) 416254. 1.48565 0.742824 0.669486i \(-0.233486\pi\)
0.742824 + 0.669486i \(0.233486\pi\)
\(152\) 127865. 0.448892
\(153\) 0 0
\(154\) 56734.3 0.192772
\(155\) −3510.79 −0.0117375
\(156\) 0 0
\(157\) 512939.i 1.66080i 0.557169 + 0.830399i \(0.311888\pi\)
−0.557169 + 0.830399i \(0.688112\pi\)
\(158\) −177174. −0.564621
\(159\) 0 0
\(160\) 345388.i 1.06661i
\(161\) 51900.5 + 21186.1i 0.157800 + 0.0644149i
\(162\) 0 0
\(163\) 345214. 1.01770 0.508850 0.860855i \(-0.330071\pi\)
0.508850 + 0.860855i \(0.330071\pi\)
\(164\) 70802.9i 0.205561i
\(165\) 0 0
\(166\) 379520.i 1.06897i
\(167\) 371376.i 1.03044i −0.857058 0.515220i \(-0.827710\pi\)
0.857058 0.515220i \(-0.172290\pi\)
\(168\) 0 0
\(169\) 334664. 0.901346
\(170\) 208939.i 0.554494i
\(171\) 0 0
\(172\) 333824.i 0.860393i
\(173\) 187699.i 0.476811i −0.971166 0.238405i \(-0.923375\pi\)
0.971166 0.238405i \(-0.0766247\pi\)
\(174\) 0 0
\(175\) 6281.36i 0.0155045i
\(176\) 65384.7 0.159109
\(177\) 0 0
\(178\) 119889.i 0.283615i
\(179\) 199147.i 0.464560i −0.972649 0.232280i \(-0.925381\pi\)
0.972649 0.232280i \(-0.0746185\pi\)
\(180\) 0 0
\(181\) 125464.i 0.284657i −0.989819 0.142329i \(-0.954541\pi\)
0.989819 0.142329i \(-0.0454590\pi\)
\(182\) −61738.2 −0.138158
\(183\) 0 0
\(184\) −413519. 168801.i −0.900432 0.367562i
\(185\) 151260.i 0.324935i
\(186\) 0 0
\(187\) 830850. 1.73748
\(188\) 155698.i 0.321284i
\(189\) 0 0
\(190\) −141021. −0.283401
\(191\) −278416. −0.552219 −0.276109 0.961126i \(-0.589045\pi\)
−0.276109 + 0.961126i \(0.589045\pi\)
\(192\) 0 0
\(193\) −500877. −0.967917 −0.483958 0.875091i \(-0.660801\pi\)
−0.483958 + 0.875091i \(0.660801\pi\)
\(194\) 448514. 0.855601
\(195\) 0 0
\(196\) 341741. 0.635414
\(197\) 102101.i 0.187441i −0.995599 0.0937205i \(-0.970124\pi\)
0.995599 0.0937205i \(-0.0298760\pi\)
\(198\) 0 0
\(199\) 600453.i 1.07485i −0.843313 0.537423i \(-0.819398\pi\)
0.843313 0.537423i \(-0.180602\pi\)
\(200\) 50047.0i 0.0884714i
\(201\) 0 0
\(202\) 346979. 0.598308
\(203\) 178953. 0.304788
\(204\) 0 0
\(205\) 197411.i 0.328086i
\(206\) −483657. −0.794090
\(207\) 0 0
\(208\) −71151.6 −0.114032
\(209\) 560775.i 0.888021i
\(210\) 0 0
\(211\) −444157. −0.686800 −0.343400 0.939189i \(-0.611579\pi\)
−0.343400 + 0.939189i \(0.611579\pi\)
\(212\) −392750. −0.600173
\(213\) 0 0
\(214\) 381315.i 0.569180i
\(215\) 930762.i 1.37323i
\(216\) 0 0
\(217\) 1328.60i 0.00191533i
\(218\) 708992. 1.01042
\(219\) 0 0
\(220\) −944108. −1.31512
\(221\) −904131. −1.24523
\(222\) 0 0
\(223\) 144161. 0.194127 0.0970633 0.995278i \(-0.469055\pi\)
0.0970633 + 0.995278i \(0.469055\pi\)
\(224\) 130706. 0.174051
\(225\) 0 0
\(226\) 479120.i 0.623983i
\(227\) 642364. 0.827402 0.413701 0.910413i \(-0.364236\pi\)
0.413701 + 0.910413i \(0.364236\pi\)
\(228\) 0 0
\(229\) 1.06969e6i 1.34794i −0.738758 0.673971i \(-0.764587\pi\)
0.738758 0.673971i \(-0.235413\pi\)
\(230\) 456068. + 186169.i 0.568473 + 0.232054i
\(231\) 0 0
\(232\) −1.42581e6 −1.73917
\(233\) 912914.i 1.10164i −0.834624 0.550820i \(-0.814315\pi\)
0.834624 0.550820i \(-0.185685\pi\)
\(234\) 0 0
\(235\) 434114.i 0.512784i
\(236\) 650792.i 0.760610i
\(237\) 0 0
\(238\) 79069.2 0.0904826
\(239\) 1.29048e6i 1.46136i 0.682720 + 0.730680i \(0.260797\pi\)
−0.682720 + 0.730680i \(0.739203\pi\)
\(240\) 0 0
\(241\) 669866.i 0.742925i 0.928448 + 0.371463i \(0.121144\pi\)
−0.928448 + 0.371463i \(0.878856\pi\)
\(242\) 1.44691e6i 1.58819i
\(243\) 0 0
\(244\) 847638.i 0.911457i
\(245\) −952835. −1.01415
\(246\) 0 0
\(247\) 610235.i 0.636436i
\(248\) 10585.6i 0.0109292i
\(249\) 0 0
\(250\) 551577.i 0.558157i
\(251\) 677038. 0.678310 0.339155 0.940730i \(-0.389859\pi\)
0.339155 + 0.940730i \(0.389859\pi\)
\(252\) 0 0
\(253\) −740308. + 1.81357e6i −0.727129 + 1.78128i
\(254\) 496213.i 0.482596i
\(255\) 0 0
\(256\) −984656. −0.939041
\(257\) 890595.i 0.841099i 0.907270 + 0.420550i \(0.138163\pi\)
−0.907270 + 0.420550i \(0.861837\pi\)
\(258\) 0 0
\(259\) −57241.9 −0.0530230
\(260\) 1.02738e6 0.942533
\(261\) 0 0
\(262\) −198644. −0.178782
\(263\) −2.16840e6 −1.93308 −0.966538 0.256522i \(-0.917423\pi\)
−0.966538 + 0.256522i \(0.917423\pi\)
\(264\) 0 0
\(265\) 1.09506e6 0.957904
\(266\) 53367.1i 0.0462455i
\(267\) 0 0
\(268\) 330940.i 0.281457i
\(269\) 2.10505e6i 1.77371i 0.462051 + 0.886853i \(0.347114\pi\)
−0.462051 + 0.886853i \(0.652886\pi\)
\(270\) 0 0
\(271\) −84020.0 −0.0694959 −0.0347480 0.999396i \(-0.511063\pi\)
−0.0347480 + 0.999396i \(0.511063\pi\)
\(272\) 91125.0 0.0746819
\(273\) 0 0
\(274\) 172172.i 0.138543i
\(275\) 219490. 0.175018
\(276\) 0 0
\(277\) 184609. 0.144562 0.0722808 0.997384i \(-0.476972\pi\)
0.0722808 + 0.997384i \(0.476972\pi\)
\(278\) 277106.i 0.215047i
\(279\) 0 0
\(280\) −227140. −0.173140
\(281\) 1.48283e6 1.12028 0.560140 0.828398i \(-0.310747\pi\)
0.560140 + 0.828398i \(0.310747\pi\)
\(282\) 0 0
\(283\) 402405.i 0.298674i −0.988786 0.149337i \(-0.952286\pi\)
0.988786 0.149337i \(-0.0477139\pi\)
\(284\) 794040.i 0.584180i
\(285\) 0 0
\(286\) 2.15733e6i 1.55955i
\(287\) −74706.8 −0.0535372
\(288\) 0 0
\(289\) −261922. −0.184470
\(290\) 1.57252e6 1.09799
\(291\) 0 0
\(292\) −1.08052e6 −0.741607
\(293\) 947678. 0.644899 0.322450 0.946587i \(-0.395494\pi\)
0.322450 + 0.946587i \(0.395494\pi\)
\(294\) 0 0
\(295\) 1.81452e6i 1.21397i
\(296\) 456076. 0.302558
\(297\) 0 0
\(298\) 497507.i 0.324533i
\(299\) 805602. 1.97352e6i 0.521126 1.27663i
\(300\) 0 0
\(301\) 352230. 0.224084
\(302\) 1.38422e6i 0.873347i
\(303\) 0 0
\(304\) 61504.1i 0.0381698i
\(305\) 2.36337e6i 1.45473i
\(306\) 0 0
\(307\) −2.14928e6 −1.30151 −0.650754 0.759289i \(-0.725547\pi\)
−0.650754 + 0.759289i \(0.725547\pi\)
\(308\) 357281.i 0.214602i
\(309\) 0 0
\(310\) 11674.8i 0.00689996i
\(311\) 825868.i 0.484183i 0.970253 + 0.242091i \(0.0778334\pi\)
−0.970253 + 0.242091i \(0.922167\pi\)
\(312\) 0 0
\(313\) 2.29977e6i 1.32686i −0.748240 0.663428i \(-0.769101\pi\)
0.748240 0.663428i \(-0.230899\pi\)
\(314\) −1.70574e6 −0.976310
\(315\) 0 0
\(316\) 1.11574e6i 0.628559i
\(317\) 1.51861e6i 0.848788i 0.905478 + 0.424394i \(0.139513\pi\)
−0.905478 + 0.424394i \(0.860487\pi\)
\(318\) 0 0
\(319\) 6.25316e6i 3.44051i
\(320\) 990333. 0.540638
\(321\) 0 0
\(322\) −70452.6 + 172591.i −0.0378667 + 0.0927637i
\(323\) 781538.i 0.416816i
\(324\) 0 0
\(325\) −238849. −0.125434
\(326\) 1.14798e6i 0.598261i
\(327\) 0 0
\(328\) 595229. 0.305492
\(329\) 164283. 0.0836764
\(330\) 0 0
\(331\) 2.61384e6 1.31132 0.655662 0.755055i \(-0.272390\pi\)
0.655662 + 0.755055i \(0.272390\pi\)
\(332\) −2.39001e6 −1.19002
\(333\) 0 0
\(334\) 1.23498e6 0.605750
\(335\) 922719.i 0.449218i
\(336\) 0 0
\(337\) 3.30404e6i 1.58479i −0.610010 0.792394i \(-0.708835\pi\)
0.610010 0.792394i \(-0.291165\pi\)
\(338\) 1.11290e6i 0.529862i
\(339\) 0 0
\(340\) −1.31578e6 −0.617285
\(341\) 46425.3 0.0216207
\(342\) 0 0
\(343\) 731956.i 0.335931i
\(344\) −2.80641e6 −1.27866
\(345\) 0 0
\(346\) 624176. 0.280296
\(347\) 1.71268e6i 0.763579i −0.924249 0.381789i \(-0.875308\pi\)
0.924249 0.381789i \(-0.124692\pi\)
\(348\) 0 0
\(349\) −16530.5 −0.00726477 −0.00363238 0.999993i \(-0.501156\pi\)
−0.00363238 + 0.999993i \(0.501156\pi\)
\(350\) 20888.1 0.00911443
\(351\) 0 0
\(352\) 4.56728e6i 1.96472i
\(353\) 2.05582e6i 0.878107i −0.898461 0.439053i \(-0.855314\pi\)
0.898461 0.439053i \(-0.144686\pi\)
\(354\) 0 0
\(355\) 2.21393e6i 0.932378i
\(356\) 754994. 0.315732
\(357\) 0 0
\(358\) 662247. 0.273094
\(359\) −2.10257e6 −0.861020 −0.430510 0.902586i \(-0.641666\pi\)
−0.430510 + 0.902586i \(0.641666\pi\)
\(360\) 0 0
\(361\) 1.94861e6 0.786966
\(362\) 417219. 0.167337
\(363\) 0 0
\(364\) 388793.i 0.153803i
\(365\) 3.01267e6 1.18364
\(366\) 0 0
\(367\) 907054.i 0.351535i −0.984432 0.175767i \(-0.943759\pi\)
0.984432 0.175767i \(-0.0562406\pi\)
\(368\) −81194.6 + 198906.i −0.0312542 + 0.0765647i
\(369\) 0 0
\(370\) −503004. −0.191015
\(371\) 414405.i 0.156311i
\(372\) 0 0
\(373\) 16128.8i 0.00600246i −0.999995 0.00300123i \(-0.999045\pi\)
0.999995 0.00300123i \(-0.000955323\pi\)
\(374\) 2.76292e6i 1.02139i
\(375\) 0 0
\(376\) −1.30893e6 −0.477471
\(377\) 6.80468e6i 2.46578i
\(378\) 0 0
\(379\) 1.72620e6i 0.617295i −0.951176 0.308648i \(-0.900124\pi\)
0.951176 0.308648i \(-0.0998764\pi\)
\(380\) 888074.i 0.315493i
\(381\) 0 0
\(382\) 925849.i 0.324625i
\(383\) 795331. 0.277046 0.138523 0.990359i \(-0.455765\pi\)
0.138523 + 0.990359i \(0.455765\pi\)
\(384\) 0 0
\(385\) 996164.i 0.342514i
\(386\) 1.66562e6i 0.568995i
\(387\) 0 0
\(388\) 2.82449e6i 0.952491i
\(389\) −4.97973e6 −1.66852 −0.834261 0.551369i \(-0.814106\pi\)
−0.834261 + 0.551369i \(0.814106\pi\)
\(390\) 0 0
\(391\) −1.03175e6 + 2.52752e6i −0.341297 + 0.836090i
\(392\) 2.87296e6i 0.944311i
\(393\) 0 0
\(394\) 339528. 0.110188
\(395\) 3.11089e6i 1.00321i
\(396\) 0 0
\(397\) −1.14563e6 −0.364811 −0.182405 0.983223i \(-0.558388\pi\)
−0.182405 + 0.983223i \(0.558388\pi\)
\(398\) 1.99675e6 0.631854
\(399\) 0 0
\(400\) 24073.0 0.00752281
\(401\) −1.82077e6 −0.565450 −0.282725 0.959201i \(-0.591238\pi\)
−0.282725 + 0.959201i \(0.591238\pi\)
\(402\) 0 0
\(403\) −50520.0 −0.0154953
\(404\) 2.18508e6i 0.666062i
\(405\) 0 0
\(406\) 595091.i 0.179171i
\(407\) 2.00021e6i 0.598535i
\(408\) 0 0
\(409\) −876882. −0.259199 −0.129599 0.991566i \(-0.541369\pi\)
−0.129599 + 0.991566i \(0.541369\pi\)
\(410\) −656474. −0.192867
\(411\) 0 0
\(412\) 3.04580e6i 0.884013i
\(413\) 686675. 0.198096
\(414\) 0 0
\(415\) 6.66376e6 1.89933
\(416\) 4.97010e6i 1.40810i
\(417\) 0 0
\(418\) 1.86481e6 0.522028
\(419\) −719232. −0.200140 −0.100070 0.994980i \(-0.531907\pi\)
−0.100070 + 0.994980i \(0.531907\pi\)
\(420\) 0 0
\(421\) 1.40118e6i 0.385292i −0.981268 0.192646i \(-0.938293\pi\)
0.981268 0.192646i \(-0.0617069\pi\)
\(422\) 1.47701e6i 0.403739i
\(423\) 0 0
\(424\) 3.30179e6i 0.891937i
\(425\) 305898. 0.0821494
\(426\) 0 0
\(427\) 894375. 0.237383
\(428\) 2.40131e6 0.633635
\(429\) 0 0
\(430\) 3.09517e6 0.807260
\(431\) −5.32487e6 −1.38075 −0.690377 0.723450i \(-0.742555\pi\)
−0.690377 + 0.723450i \(0.742555\pi\)
\(432\) 0 0
\(433\) 421470.i 0.108031i 0.998540 + 0.0540153i \(0.0172020\pi\)
−0.998540 + 0.0540153i \(0.982798\pi\)
\(434\) 4418.14 0.00112594
\(435\) 0 0
\(436\) 4.46484e6i 1.12484i
\(437\) 1.70593e6 + 696370.i 0.427324 + 0.174436i
\(438\) 0 0
\(439\) 6.14641e6 1.52216 0.761080 0.648658i \(-0.224670\pi\)
0.761080 + 0.648658i \(0.224670\pi\)
\(440\) 7.93697e6i 1.95444i
\(441\) 0 0
\(442\) 3.00661e6i 0.732017i
\(443\) 7.40665e6i 1.79313i 0.442908 + 0.896567i \(0.353947\pi\)
−0.442908 + 0.896567i \(0.646053\pi\)
\(444\) 0 0
\(445\) −2.10506e6 −0.503923
\(446\) 479394.i 0.114118i
\(447\) 0 0
\(448\) 374774.i 0.0882216i
\(449\) 690558.i 0.161653i 0.996728 + 0.0808266i \(0.0257560\pi\)
−0.996728 + 0.0808266i \(0.974244\pi\)
\(450\) 0 0
\(451\) 2.61049e6i 0.604339i
\(452\) 3.01723e6 0.694644
\(453\) 0 0
\(454\) 2.13613e6i 0.486393i
\(455\) 1.08402e6i 0.245477i
\(456\) 0 0
\(457\) 755259.i 0.169163i −0.996417 0.0845815i \(-0.973045\pi\)
0.996417 0.0845815i \(-0.0269553\pi\)
\(458\) 3.55718e6 0.792395
\(459\) 0 0
\(460\) 1.17239e6 2.87206e6i 0.258332 0.632848i
\(461\) 7.11457e6i 1.55918i −0.626290 0.779590i \(-0.715427\pi\)
0.626290 0.779590i \(-0.284573\pi\)
\(462\) 0 0
\(463\) 4.59086e6 0.995272 0.497636 0.867386i \(-0.334202\pi\)
0.497636 + 0.867386i \(0.334202\pi\)
\(464\) 685826.i 0.147883i
\(465\) 0 0
\(466\) 3.03582e6 0.647606
\(467\) −2.05341e6 −0.435696 −0.217848 0.975983i \(-0.569904\pi\)
−0.217848 + 0.975983i \(0.569904\pi\)
\(468\) 0 0
\(469\) −349187. −0.0733037
\(470\) 1.44361e6 0.301443
\(471\) 0 0
\(472\) −5.47110e6 −1.13037
\(473\) 1.23080e7i 2.52951i
\(474\) 0 0
\(475\) 206463.i 0.0419864i
\(476\) 497934.i 0.100729i
\(477\) 0 0
\(478\) −4.29139e6 −0.859069
\(479\) 3.38495e6 0.674084 0.337042 0.941490i \(-0.390574\pi\)
0.337042 + 0.941490i \(0.390574\pi\)
\(480\) 0 0
\(481\) 2.17662e6i 0.428964i
\(482\) −2.22758e6 −0.436733
\(483\) 0 0
\(484\) 9.11184e6 1.76804
\(485\) 7.87518e6i 1.52022i
\(486\) 0 0
\(487\) −2.06385e6 −0.394326 −0.197163 0.980371i \(-0.563173\pi\)
−0.197163 + 0.980371i \(0.563173\pi\)
\(488\) −7.12596e6 −1.35455
\(489\) 0 0
\(490\) 3.16857e6i 0.596175i
\(491\) 971280.i 0.181820i −0.995859 0.0909098i \(-0.971022\pi\)
0.995859 0.0909098i \(-0.0289775\pi\)
\(492\) 0 0
\(493\) 8.71487e6i 1.61489i
\(494\) −2.02929e6 −0.374133
\(495\) 0 0
\(496\) 5091.78 0.000929321
\(497\) 837821. 0.152146
\(498\) 0 0
\(499\) −4.36873e6 −0.785423 −0.392712 0.919662i \(-0.628463\pi\)
−0.392712 + 0.919662i \(0.628463\pi\)
\(500\) 3.47353e6 0.621363
\(501\) 0 0
\(502\) 2.25143e6i 0.398749i
\(503\) 678984. 0.119657 0.0598287 0.998209i \(-0.480945\pi\)
0.0598287 + 0.998209i \(0.480945\pi\)
\(504\) 0 0
\(505\) 6.09240e6i 1.06307i
\(506\) −6.03086e6 2.46183e6i −1.04714 0.427447i
\(507\) 0 0
\(508\) 3.12487e6 0.537246
\(509\) 6.95416e6i 1.18974i 0.803824 + 0.594868i \(0.202796\pi\)
−0.803824 + 0.594868i \(0.797204\pi\)
\(510\) 0 0
\(511\) 1.14009e6i 0.193147i
\(512\) 978002.i 0.164879i
\(513\) 0 0
\(514\) −2.96160e6 −0.494445
\(515\) 8.49225e6i 1.41093i
\(516\) 0 0
\(517\) 5.74056e6i 0.944556i
\(518\) 190353.i 0.0311699i
\(519\) 0 0
\(520\) 8.63700e6i 1.40073i
\(521\) −5.69845e6 −0.919735 −0.459867 0.887988i \(-0.652103\pi\)
−0.459867 + 0.887988i \(0.652103\pi\)
\(522\) 0 0
\(523\) 9.59760e6i 1.53429i −0.641471 0.767147i \(-0.721676\pi\)
0.641471 0.767147i \(-0.278324\pi\)
\(524\) 1.25095e6i 0.199027i
\(525\) 0 0
\(526\) 7.21081e6i 1.13637i
\(527\) 64701.8 0.0101482
\(528\) 0 0
\(529\) −4.59772e6 4.50417e6i −0.714337 0.699802i
\(530\) 3.64152e6i 0.563110i
\(531\) 0 0
\(532\) −336076. −0.0514824
\(533\) 2.84073e6i 0.433124i
\(534\) 0 0
\(535\) −6.69529e6 −1.01131
\(536\) 2.78216e6 0.418282
\(537\) 0 0
\(538\) −7.00017e6 −1.04268
\(539\) 1.25999e7 1.86808
\(540\) 0 0
\(541\) 2.57197e6 0.377810 0.188905 0.981995i \(-0.439506\pi\)
0.188905 + 0.981995i \(0.439506\pi\)
\(542\) 279401.i 0.0408536i
\(543\) 0 0
\(544\) 6.36530e6i 0.922192i
\(545\) 1.24488e7i 1.79529i
\(546\) 0 0
\(547\) −2.19546e6 −0.313731 −0.156866 0.987620i \(-0.550139\pi\)
−0.156866 + 0.987620i \(0.550139\pi\)
\(548\) −1.08424e6 −0.154232
\(549\) 0 0
\(550\) 729896.i 0.102886i
\(551\) 5.88202e6 0.825368
\(552\) 0 0
\(553\) 1.17726e6 0.163704
\(554\) 613901.i 0.0849814i
\(555\) 0 0
\(556\) −1.74506e6 −0.239399
\(557\) 2.05279e6 0.280353 0.140177 0.990127i \(-0.455233\pi\)
0.140177 + 0.990127i \(0.455233\pi\)
\(558\) 0 0
\(559\) 1.33936e7i 1.81287i
\(560\) 109256.i 0.0147223i
\(561\) 0 0
\(562\) 4.93103e6i 0.658563i
\(563\) 8.39159e6 1.11577 0.557883 0.829920i \(-0.311614\pi\)
0.557883 + 0.829920i \(0.311614\pi\)
\(564\) 0 0
\(565\) −8.41257e6 −1.10868
\(566\) 1.33816e6 0.175577
\(567\) 0 0
\(568\) −6.67537e6 −0.868169
\(569\) 1.21065e7 1.56760 0.783802 0.621011i \(-0.213278\pi\)
0.783802 + 0.621011i \(0.213278\pi\)
\(570\) 0 0
\(571\) 8.53237e6i 1.09517i 0.836752 + 0.547583i \(0.184452\pi\)
−0.836752 + 0.547583i \(0.815548\pi\)
\(572\) −1.35856e7 −1.73616
\(573\) 0 0
\(574\) 248431.i 0.0314721i
\(575\) −272563. + 667709.i −0.0343793 + 0.0842205i
\(576\) 0 0
\(577\) −4.03720e6 −0.504825 −0.252413 0.967620i \(-0.581224\pi\)
−0.252413 + 0.967620i \(0.581224\pi\)
\(578\) 870998.i 0.108442i
\(579\) 0 0
\(580\) 9.90284e6i 1.22233i
\(581\) 2.52178e6i 0.309933i
\(582\) 0 0
\(583\) −1.44806e7 −1.76447
\(584\) 9.08372e6i 1.10213i
\(585\) 0 0
\(586\) 3.15142e6i 0.379108i
\(587\) 1.87167e6i 0.224200i −0.993697 0.112100i \(-0.964242\pi\)
0.993697 0.112100i \(-0.0357577\pi\)
\(588\) 0 0
\(589\) 43669.9i 0.00518674i
\(590\) 6.03404e6 0.713639
\(591\) 0 0
\(592\) 219377.i 0.0257268i
\(593\) 8.93054e6i 1.04290i 0.853283 + 0.521448i \(0.174608\pi\)
−0.853283 + 0.521448i \(0.825392\pi\)
\(594\) 0 0
\(595\) 1.38833e6i 0.160768i
\(596\) 3.13302e6 0.361283
\(597\) 0 0
\(598\) 6.56278e6 + 2.67896e6i 0.750472 + 0.306347i
\(599\) 4.76499e6i 0.542619i 0.962492 + 0.271309i \(0.0874567\pi\)
−0.962492 + 0.271309i \(0.912543\pi\)
\(600\) 0 0
\(601\) −1.06072e7 −1.19789 −0.598944 0.800791i \(-0.704413\pi\)
−0.598944 + 0.800791i \(0.704413\pi\)
\(602\) 1.17131e6i 0.131729i
\(603\) 0 0
\(604\) 8.71703e6 0.972246
\(605\) −2.54055e7 −2.82188
\(606\) 0 0
\(607\) 1.37303e7 1.51255 0.756275 0.654254i \(-0.227017\pi\)
0.756275 + 0.654254i \(0.227017\pi\)
\(608\) 4.29620e6 0.471331
\(609\) 0 0
\(610\) 7.85918e6 0.855170
\(611\) 6.24687e6i 0.676954i
\(612\) 0 0
\(613\) 1.36411e7i 1.46622i 0.680110 + 0.733110i \(0.261932\pi\)
−0.680110 + 0.733110i \(0.738068\pi\)
\(614\) 7.14724e6i 0.765099i
\(615\) 0 0
\(616\) 3.00361e6 0.318927
\(617\) 5.85548e6 0.619227 0.309613 0.950863i \(-0.399800\pi\)
0.309613 + 0.950863i \(0.399800\pi\)
\(618\) 0 0
\(619\) 1.00952e7i 1.05899i −0.848314 0.529493i \(-0.822382\pi\)
0.848314 0.529493i \(-0.177618\pi\)
\(620\) −73521.7 −0.00768132
\(621\) 0 0
\(622\) −2.74635e6 −0.284630
\(623\) 796622.i 0.0822304i
\(624\) 0 0
\(625\) −1.05732e7 −1.08269
\(626\) 7.64769e6 0.780000
\(627\) 0 0
\(628\) 1.07418e7i 1.08687i
\(629\) 2.78764e6i 0.280938i
\(630\) 0 0
\(631\) 1.41421e7i 1.41397i 0.707230 + 0.706984i \(0.249945\pi\)
−0.707230 + 0.706984i \(0.750055\pi\)
\(632\) −9.37986e6 −0.934123
\(633\) 0 0
\(634\) −5.05002e6 −0.498965
\(635\) −8.71271e6 −0.857470
\(636\) 0 0
\(637\) −1.37112e7 −1.33884
\(638\) −2.07943e7 −2.02252
\(639\) 0 0
\(640\) 7.75915e6i 0.748797i
\(641\) −2.31120e6 −0.222173 −0.111087 0.993811i \(-0.535433\pi\)
−0.111087 + 0.993811i \(0.535433\pi\)
\(642\) 0 0
\(643\) 2.43098e6i 0.231875i 0.993257 + 0.115937i \(0.0369872\pi\)
−0.993257 + 0.115937i \(0.963013\pi\)
\(644\) 1.08688e6 + 443671.i 0.103268 + 0.0421547i
\(645\) 0 0
\(646\) 2.59894e6 0.245028
\(647\) 1.42011e7i 1.33371i −0.745189 0.666853i \(-0.767641\pi\)
0.745189 0.666853i \(-0.232359\pi\)
\(648\) 0 0
\(649\) 2.39945e7i 2.23615i
\(650\) 794272.i 0.0737371i
\(651\) 0 0
\(652\) 7.22934e6 0.666008
\(653\) 688639.i 0.0631987i 0.999501 + 0.0315994i \(0.0100601\pi\)
−0.999501 + 0.0315994i \(0.989940\pi\)
\(654\) 0 0
\(655\) 3.48788e6i 0.317657i
\(656\) 286310.i 0.0259763i
\(657\) 0 0
\(658\) 546309.i 0.0491896i
\(659\) 7.43942e6 0.667307 0.333654 0.942696i \(-0.391718\pi\)
0.333654 + 0.942696i \(0.391718\pi\)
\(660\) 0 0
\(661\) 1.35233e7i 1.20387i −0.798546 0.601933i \(-0.794397\pi\)
0.798546 0.601933i \(-0.205603\pi\)
\(662\) 8.69212e6i 0.770869i
\(663\) 0 0
\(664\) 2.00924e7i 1.76853i
\(665\) 937040. 0.0821683
\(666\) 0 0
\(667\) −1.90227e7 7.76516e6i −1.65560 0.675827i
\(668\) 7.77721e6i 0.674346i
\(669\) 0 0
\(670\) −3.06842e6 −0.264076
\(671\) 3.12522e7i 2.67963i
\(672\) 0 0
\(673\) 4.98340e6 0.424119 0.212060 0.977257i \(-0.431983\pi\)
0.212060 + 0.977257i \(0.431983\pi\)
\(674\) 1.09873e7 0.931626
\(675\) 0 0
\(676\) 7.00840e6 0.589864
\(677\) 4.62140e6 0.387527 0.193763 0.981048i \(-0.437931\pi\)
0.193763 + 0.981048i \(0.437931\pi\)
\(678\) 0 0
\(679\) −2.98023e6 −0.248070
\(680\) 1.10616e7i 0.917368i
\(681\) 0 0
\(682\) 154384.i 0.0127098i
\(683\) 5.51387e6i 0.452277i 0.974095 + 0.226139i \(0.0726102\pi\)
−0.974095 + 0.226139i \(0.927390\pi\)
\(684\) 0 0
\(685\) 3.02306e6 0.246162
\(686\) 2.43406e6 0.197479
\(687\) 0 0
\(688\) 1.34991e6i 0.108726i
\(689\) 1.57578e7 1.26458
\(690\) 0 0
\(691\) 861740. 0.0686564 0.0343282 0.999411i \(-0.489071\pi\)
0.0343282 + 0.999411i \(0.489071\pi\)
\(692\) 3.93071e6i 0.312037i
\(693\) 0 0
\(694\) 5.69539e6 0.448874
\(695\) 4.86554e6 0.382093
\(696\) 0 0
\(697\) 3.63817e6i 0.283662i
\(698\) 54970.7i 0.00427064i
\(699\) 0 0
\(700\) 131542.i 0.0101466i
\(701\) −1.36300e7 −1.04762 −0.523808 0.851837i \(-0.675489\pi\)
−0.523808 + 0.851837i \(0.675489\pi\)
\(702\) 0 0
\(703\) −1.88149e6 −0.143587
\(704\) −1.30958e7 −0.995863
\(705\) 0 0
\(706\) 6.83644e6 0.516200
\(707\) −2.30556e6 −0.173472
\(708\) 0 0
\(709\) 3.42866e6i 0.256158i −0.991764 0.128079i \(-0.959119\pi\)
0.991764 0.128079i \(-0.0408812\pi\)
\(710\) 7.36222e6 0.548104
\(711\) 0 0
\(712\) 6.34711e6i 0.469220i
\(713\) −57650.9 + 141230.i −0.00424700 + 0.0104041i
\(714\) 0 0
\(715\) 3.78792e7 2.77099
\(716\) 4.17046e6i 0.304020i
\(717\) 0 0
\(718\) 6.99190e6i 0.506156i
\(719\) 1.80809e7i 1.30436i −0.758062 0.652182i \(-0.773854\pi\)
0.758062 0.652182i \(-0.226146\pi\)
\(720\) 0 0
\(721\) 3.21374e6 0.230236
\(722\) 6.47992e6i 0.462623i
\(723\) 0 0
\(724\) 2.62742e6i 0.186287i
\(725\) 2.30225e6i 0.162670i
\(726\) 0 0
\(727\) 3.45058e6i 0.242134i 0.992644 + 0.121067i \(0.0386316\pi\)
−0.992644 + 0.121067i \(0.961368\pi\)
\(728\) −3.26852e6 −0.228572
\(729\) 0 0
\(730\) 1.00184e7i 0.695809i
\(731\) 1.71534e7i 1.18729i
\(732\) 0 0
\(733\) 1.11140e7i 0.764031i 0.924156 + 0.382016i \(0.124770\pi\)
−0.924156 + 0.382016i \(0.875230\pi\)
\(734\) 3.01633e6 0.206652
\(735\) 0 0
\(736\) −1.38941e7 5.67164e6i −0.945441 0.385935i
\(737\) 1.22017e7i 0.827467i
\(738\) 0 0
\(739\) 2.00762e7 1.35229 0.676146 0.736768i \(-0.263649\pi\)
0.676146 + 0.736768i \(0.263649\pi\)
\(740\) 3.16764e6i 0.212646i
\(741\) 0 0
\(742\) −1.37807e6 −0.0918885
\(743\) 9.79383e6 0.650849 0.325425 0.945568i \(-0.394493\pi\)
0.325425 + 0.945568i \(0.394493\pi\)
\(744\) 0 0
\(745\) −8.73543e6 −0.576625
\(746\) 53634.9 0.00352858
\(747\) 0 0
\(748\) 1.73993e7 1.13705
\(749\) 2.53371e6i 0.165026i
\(750\) 0 0
\(751\) 1.58594e7i 1.02609i −0.858361 0.513046i \(-0.828517\pi\)
0.858361 0.513046i \(-0.171483\pi\)
\(752\) 629606.i 0.0405998i
\(753\) 0 0
\(754\) 2.26284e7 1.44952
\(755\) −2.43046e7 −1.55175
\(756\) 0 0
\(757\) 1.68578e7i 1.06921i −0.845103 0.534604i \(-0.820461\pi\)
0.845103 0.534604i \(-0.179539\pi\)
\(758\) 5.74033e6 0.362881
\(759\) 0 0
\(760\) −7.46590e6 −0.468865
\(761\) 9.41728e6i 0.589473i 0.955579 + 0.294736i \(0.0952318\pi\)
−0.955579 + 0.294736i \(0.904768\pi\)
\(762\) 0 0
\(763\) −4.71102e6 −0.292957
\(764\) −5.83048e6 −0.361386
\(765\) 0 0
\(766\) 2.64481e6i 0.162863i
\(767\) 2.61108e7i 1.60263i
\(768\) 0 0
\(769\) 2.37459e7i 1.44802i −0.689791 0.724008i \(-0.742298\pi\)
0.689791 0.724008i \(-0.257702\pi\)
\(770\) −3.31266e6 −0.201349
\(771\) 0 0
\(772\) −1.04892e7 −0.633429
\(773\) 2.90678e7 1.74970 0.874849 0.484397i \(-0.160961\pi\)
0.874849 + 0.484397i \(0.160961\pi\)
\(774\) 0 0
\(775\) 17092.6 0.00102224
\(776\) 2.37450e7 1.41553
\(777\) 0 0
\(778\) 1.65597e7i 0.980851i
\(779\) −2.45555e6 −0.144979
\(780\) 0 0
\(781\) 2.92761e7i 1.71745i
\(782\) −8.40506e6 3.43099e6i −0.491500 0.200633i
\(783\) 0 0
\(784\) 1.38192e6 0.0802957
\(785\) 2.99500e7i 1.73469i
\(786\) 0 0
\(787\) 2.29567e7i 1.32122i 0.750731 + 0.660608i \(0.229701\pi\)
−0.750731 + 0.660608i \(0.770299\pi\)
\(788\) 2.13816e6i 0.122666i
\(789\) 0 0
\(790\) 1.03450e7 0.589743
\(791\) 3.18359e6i 0.180916i
\(792\) 0 0
\(793\) 3.40087e7i 1.92047i
\(794\) 3.80969e6i 0.214456i
\(795\) 0 0
\(796\) 1.25744e7i 0.703406i
\(797\) 1.46871e7 0.819010 0.409505 0.912308i \(-0.365701\pi\)
0.409505 + 0.912308i \(0.365701\pi\)
\(798\) 0 0
\(799\) 8.00047e6i 0.443352i
\(800\) 1.68156e6i 0.0928937i
\(801\) 0 0
\(802\) 6.05482e6i 0.332403i
\(803\) −3.98384e7 −2.18028
\(804\) 0 0
\(805\) −3.03042e6 1.23703e6i −0.164821 0.0672810i
\(806\) 168000.i 0.00910902i
\(807\) 0 0
\(808\) 1.83696e7 0.989857
\(809\) 1.93541e7i 1.03968i 0.854262 + 0.519842i \(0.174009\pi\)
−0.854262 + 0.519842i \(0.825991\pi\)
\(810\) 0 0
\(811\) 1.63564e7 0.873242 0.436621 0.899646i \(-0.356175\pi\)
0.436621 + 0.899646i \(0.356175\pi\)
\(812\) 3.74756e6 0.199461
\(813\) 0 0
\(814\) 6.65152e6 0.351852
\(815\) −2.01567e7 −1.06298
\(816\) 0 0
\(817\) 1.15775e7 0.606821
\(818\) 2.91599e6i 0.152371i
\(819\) 0 0
\(820\) 4.13411e6i 0.214708i
\(821\) 9.18663e6i 0.475662i −0.971307 0.237831i \(-0.923564\pi\)
0.971307 0.237831i \(-0.0764364\pi\)
\(822\) 0 0
\(823\) 2.37028e7 1.21983 0.609915 0.792467i \(-0.291204\pi\)
0.609915 + 0.792467i \(0.291204\pi\)
\(824\) −2.56056e7 −1.31376
\(825\) 0 0
\(826\) 2.28348e6i 0.116452i
\(827\) −2.32953e7 −1.18441 −0.592207 0.805786i \(-0.701743\pi\)
−0.592207 + 0.805786i \(0.701743\pi\)
\(828\) 0 0
\(829\) 2.92385e7 1.47764 0.738821 0.673902i \(-0.235383\pi\)
0.738821 + 0.673902i \(0.235383\pi\)
\(830\) 2.21598e7i 1.11653i
\(831\) 0 0
\(832\) 1.42508e7 0.713726
\(833\) 1.75602e7 0.876833
\(834\) 0 0
\(835\) 2.16843e7i 1.07629i
\(836\) 1.17435e7i 0.581144i
\(837\) 0 0
\(838\) 2.39174e6i 0.117653i
\(839\) −1.62630e7 −0.797617 −0.398809 0.917034i \(-0.630576\pi\)
−0.398809 + 0.917034i \(0.630576\pi\)
\(840\) 0 0
\(841\) −4.50788e7 −2.19777
\(842\) 4.65952e6 0.226496
\(843\) 0 0
\(844\) −9.30136e6 −0.449459
\(845\) −1.95407e7 −0.941451
\(846\) 0 0
\(847\) 9.61425e6i 0.460476i
\(848\) −1.58819e6 −0.0758424
\(849\) 0 0
\(850\) 1.01724e6i 0.0482920i
\(851\) 6.08481e6 + 2.48386e6i 0.288020 + 0.117572i
\(852\) 0 0
\(853\) −898065. −0.0422606 −0.0211303 0.999777i \(-0.506726\pi\)
−0.0211303 + 0.999777i \(0.506726\pi\)
\(854\) 2.97417e6i 0.139547i
\(855\) 0 0
\(856\) 2.01874e7i 0.941666i
\(857\) 9.28556e6i 0.431873i 0.976407 + 0.215937i \(0.0692804\pi\)
−0.976407 + 0.215937i \(0.930720\pi\)
\(858\) 0 0
\(859\) 1.69543e7 0.783963 0.391982 0.919973i \(-0.371790\pi\)
0.391982 + 0.919973i \(0.371790\pi\)
\(860\) 1.94917e7i 0.898675i
\(861\) 0 0
\(862\) 1.77074e7i 0.811684i
\(863\) 3.28624e7i 1.50201i −0.660297 0.751005i \(-0.729569\pi\)
0.660297 0.751005i \(-0.270431\pi\)
\(864\) 0 0
\(865\) 1.09595e7i 0.498026i
\(866\) −1.40156e6 −0.0635064
\(867\) 0 0
\(868\) 27823.0i 0.00125344i
\(869\) 4.11372e7i 1.84793i
\(870\) 0 0
\(871\) 1.32778e7i 0.593038i
\(872\) 3.75352e7 1.67166
\(873\) 0 0
\(874\) −2.31572e6 + 5.67292e6i −0.102543 + 0.251205i
\(875\) 3.66505e6i 0.161830i
\(876\) 0 0
\(877\) −1.51246e7 −0.664024 −0.332012 0.943275i \(-0.607727\pi\)
−0.332012 + 0.943275i \(0.607727\pi\)
\(878\) 2.04394e7i 0.894810i
\(879\) 0 0
\(880\) −3.81775e6 −0.166188
\(881\) 4.00600e7 1.73889 0.869444 0.494032i \(-0.164477\pi\)
0.869444 + 0.494032i \(0.164477\pi\)
\(882\) 0 0
\(883\) −2.17336e7 −0.938057 −0.469028 0.883183i \(-0.655396\pi\)
−0.469028 + 0.883183i \(0.655396\pi\)
\(884\) −1.89340e7 −0.814912
\(885\) 0 0
\(886\) −2.46302e7 −1.05410
\(887\) 2.68081e7i 1.14408i −0.820225 0.572042i \(-0.806152\pi\)
0.820225 0.572042i \(-0.193848\pi\)
\(888\) 0 0
\(889\) 3.29717e6i 0.139922i
\(890\) 7.00019e6i 0.296234i
\(891\) 0 0
\(892\) 3.01896e6 0.127041
\(893\) 5.39985e6 0.226596
\(894\) 0 0
\(895\) 1.16280e7i 0.485230i
\(896\) 2.93631e6 0.122189
\(897\) 0 0
\(898\) −2.29639e6 −0.0950288
\(899\) 486960.i 0.0200953i
\(900\) 0 0
\(901\) −2.01813e7 −0.828202
\(902\) 8.68096e6 0.355264
\(903\) 0 0
\(904\) 2.53654e7i 1.03233i
\(905\) 7.32571e6i 0.297323i
\(906\) 0 0
\(907\) 2.44022e6i 0.0984942i −0.998787 0.0492471i \(-0.984318\pi\)
0.998787 0.0492471i \(-0.0156822\pi\)
\(908\) 1.34521e7 0.541473
\(909\) 0 0
\(910\) 3.60483e6 0.144305
\(911\) 1.03581e7 0.413506 0.206753 0.978393i \(-0.433710\pi\)
0.206753 + 0.978393i \(0.433710\pi\)
\(912\) 0 0
\(913\) −8.81190e7 −3.49859
\(914\) 2.51155e6 0.0994434
\(915\) 0 0
\(916\) 2.24011e7i 0.882127i
\(917\) 1.31993e6 0.0518354
\(918\) 0 0
\(919\) 4.51155e7i 1.76213i −0.472999 0.881063i \(-0.656828\pi\)
0.472999 0.881063i \(-0.343172\pi\)
\(920\) 2.41450e7 + 9.85611e6i 0.940496 + 0.383916i
\(921\) 0 0
\(922\) 2.36589e7 0.916574
\(923\) 3.18582e7i 1.23088i
\(924\) 0 0
\(925\) 736426.i 0.0282992i
\(926\) 1.52665e7i 0.585077i
\(927\) 0 0
\(928\) −4.79066e7 −1.82610
\(929\) 2.31231e7i 0.879038i −0.898233 0.439519i \(-0.855149\pi\)
0.898233 0.439519i \(-0.144851\pi\)
\(930\) 0 0
\(931\) 1.18521e7i 0.448148i
\(932\) 1.91179e7i 0.720942i
\(933\) 0 0
\(934\) 6.82844e6i 0.256126i
\(935\) −4.85125e7 −1.81478
\(936\) 0 0
\(937\) 4.04682e7i 1.50579i 0.658139 + 0.752896i \(0.271344\pi\)
−0.658139 + 0.752896i \(0.728656\pi\)
\(938\) 1.16119e6i 0.0430920i
\(939\) 0 0
\(940\) 9.09106e6i 0.335579i
\(941\) −3.52131e7 −1.29637 −0.648186 0.761482i \(-0.724472\pi\)
−0.648186 + 0.761482i \(0.724472\pi\)
\(942\) 0 0
\(943\) 7.94134e6 + 3.24170e6i 0.290813 + 0.118712i
\(944\) 2.63165e6i 0.0961164i
\(945\) 0 0
\(946\) −4.09293e7 −1.48698
\(947\) 2.02835e7i 0.734966i 0.930030 + 0.367483i \(0.119780\pi\)
−0.930030 + 0.367483i \(0.880220\pi\)
\(948\) 0 0
\(949\) 4.33521e7 1.56259
\(950\) 686576. 0.0246820
\(951\) 0 0
\(952\) 4.18605e6 0.149697
\(953\) 2.40451e7 0.857618 0.428809 0.903395i \(-0.358933\pi\)
0.428809 + 0.903395i \(0.358933\pi\)
\(954\) 0 0
\(955\) 1.62564e7 0.576789
\(956\) 2.70248e7i 0.956351i
\(957\) 0 0
\(958\) 1.12564e7i 0.396264i
\(959\) 1.14403e6i 0.0401688i
\(960\) 0 0
\(961\) −2.86255e7 −0.999874
\(962\) −7.23818e6 −0.252169
\(963\) 0 0
\(964\) 1.40281e7i 0.486189i
\(965\) 2.92457e7 1.01098
\(966\) 0 0
\(967\) −8.51724e6 −0.292909 −0.146454 0.989217i \(-0.546786\pi\)
−0.146454 + 0.989217i \(0.546786\pi\)
\(968\) 7.66018e7i 2.62755i
\(969\) 0 0
\(970\) −2.61883e7 −0.893670
\(971\) 2.04367e7 0.695604 0.347802 0.937568i \(-0.386928\pi\)
0.347802 + 0.937568i \(0.386928\pi\)
\(972\) 0 0
\(973\) 1.84128e6i 0.0623501i
\(974\) 6.86316e6i 0.231807i
\(975\) 0 0
\(976\) 3.42765e6i 0.115179i
\(977\) 2.06610e7 0.692491 0.346245 0.938144i \(-0.387456\pi\)
0.346245 + 0.938144i \(0.387456\pi\)
\(978\) 0 0
\(979\) 2.78365e7 0.928233
\(980\) −1.99539e7 −0.663686
\(981\) 0 0
\(982\) 3.22991e6 0.106884
\(983\) 2.99382e7 0.988192 0.494096 0.869407i \(-0.335499\pi\)
0.494096 + 0.869407i \(0.335499\pi\)
\(984\) 0 0
\(985\) 5.96158e6i 0.195781i
\(986\) −2.89806e7 −0.949324
\(987\) 0 0
\(988\) 1.27793e7i 0.416500i
\(989\) −3.74421e7 1.52841e7i −1.21722 0.496877i
\(990\) 0 0
\(991\) 5.61416e7 1.81594 0.907968 0.419040i \(-0.137633\pi\)
0.907968 + 0.419040i \(0.137633\pi\)
\(992\) 355673.i 0.0114755i
\(993\) 0 0
\(994\) 2.78610e6i 0.0894399i
\(995\) 3.50598e7i 1.12267i
\(996\) 0 0
\(997\) −4.14679e7 −1.32122 −0.660609 0.750730i \(-0.729702\pi\)
−0.660609 + 0.750730i \(0.729702\pi\)
\(998\) 1.45278e7i 0.461716i
\(999\) 0 0
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 207.6.c.a.206.10 yes 40
3.2 odd 2 inner 207.6.c.a.206.31 yes 40
23.22 odd 2 inner 207.6.c.a.206.32 yes 40
69.68 even 2 inner 207.6.c.a.206.9 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
207.6.c.a.206.9 40 69.68 even 2 inner
207.6.c.a.206.10 yes 40 1.1 even 1 trivial
207.6.c.a.206.31 yes 40 3.2 odd 2 inner
207.6.c.a.206.32 yes 40 23.22 odd 2 inner