Properties

Label 207.6.c.a.206.26
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 206.26
Character \(\chi\) \(=\) 207.206
Dual form 207.6.c.a.206.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.84990i q^{2} -46.3207 q^{4} +32.1196 q^{5} +32.0928i q^{7} -126.737i q^{8} +O(q^{10})\) \(q+8.84990i q^{2} -46.3207 q^{4} +32.1196 q^{5} +32.0928i q^{7} -126.737i q^{8} +284.255i q^{10} -601.231 q^{11} +865.912 q^{13} -284.018 q^{14} -360.654 q^{16} -831.109 q^{17} +1073.08i q^{19} -1487.80 q^{20} -5320.84i q^{22} +(-2491.41 - 478.774i) q^{23} -2093.33 q^{25} +7663.23i q^{26} -1486.56i q^{28} -1742.95i q^{29} -4617.38 q^{31} -7247.33i q^{32} -7355.23i q^{34} +1030.81i q^{35} -14538.9i q^{37} -9496.69 q^{38} -4070.73i q^{40} +4605.46i q^{41} -16104.7i q^{43} +27849.5 q^{44} +(4237.10 - 22048.7i) q^{46} +6936.30i q^{47} +15777.1 q^{49} -18525.8i q^{50} -40109.6 q^{52} -11493.3 q^{53} -19311.3 q^{55} +4067.35 q^{56} +15425.0 q^{58} +21495.9i q^{59} -3068.46i q^{61} -40863.4i q^{62} +52597.2 q^{64} +27812.7 q^{65} +69034.0i q^{67} +38497.6 q^{68} -9122.54 q^{70} +5054.68i q^{71} +7803.25 q^{73} +128668. q^{74} -49706.0i q^{76} -19295.2i q^{77} -18787.0i q^{79} -11584.0 q^{80} -40757.9 q^{82} -60140.1 q^{83} -26694.9 q^{85} +142525. q^{86} +76198.2i q^{88} -29806.7 q^{89} +27789.6i q^{91} +(115404. + 22177.1i) q^{92} -61385.6 q^{94} +34467.0i q^{95} -126459. i q^{97} +139625. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 600 q^{4}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.84990i 1.56446i 0.622992 + 0.782228i \(0.285917\pi\)
−0.622992 + 0.782228i \(0.714083\pi\)
\(3\) 0 0
\(4\) −46.3207 −1.44752
\(5\) 32.1196 0.574572 0.287286 0.957845i \(-0.407247\pi\)
0.287286 + 0.957845i \(0.407247\pi\)
\(6\) 0 0
\(7\) 32.0928i 0.247550i 0.992310 + 0.123775i \(0.0395001\pi\)
−0.992310 + 0.123775i \(0.960500\pi\)
\(8\) 126.737i 0.700129i
\(9\) 0 0
\(10\) 284.255i 0.898893i
\(11\) −601.231 −1.49817 −0.749083 0.662476i \(-0.769506\pi\)
−0.749083 + 0.662476i \(0.769506\pi\)
\(12\) 0 0
\(13\) 865.912 1.42107 0.710534 0.703662i \(-0.248453\pi\)
0.710534 + 0.703662i \(0.248453\pi\)
\(14\) −284.018 −0.387281
\(15\) 0 0
\(16\) −360.654 −0.352201
\(17\) −831.109 −0.697487 −0.348743 0.937218i \(-0.613391\pi\)
−0.348743 + 0.937218i \(0.613391\pi\)
\(18\) 0 0
\(19\) 1073.08i 0.681946i 0.940073 + 0.340973i \(0.110756\pi\)
−0.940073 + 0.340973i \(0.889244\pi\)
\(20\) −1487.80 −0.831706
\(21\) 0 0
\(22\) 5320.84i 2.34381i
\(23\) −2491.41 478.774i −0.982032 0.188717i
\(24\) 0 0
\(25\) −2093.33 −0.669867
\(26\) 7663.23i 2.22320i
\(27\) 0 0
\(28\) 1486.56i 0.358334i
\(29\) 1742.95i 0.384850i −0.981312 0.192425i \(-0.938365\pi\)
0.981312 0.192425i \(-0.0616352\pi\)
\(30\) 0 0
\(31\) −4617.38 −0.862962 −0.431481 0.902122i \(-0.642009\pi\)
−0.431481 + 0.902122i \(0.642009\pi\)
\(32\) 7247.33i 1.25113i
\(33\) 0 0
\(34\) 7355.23i 1.09119i
\(35\) 1030.81i 0.142235i
\(36\) 0 0
\(37\) 14538.9i 1.74594i −0.487778 0.872968i \(-0.662193\pi\)
0.487778 0.872968i \(-0.337807\pi\)
\(38\) −9496.69 −1.06687
\(39\) 0 0
\(40\) 4070.73i 0.402275i
\(41\) 4605.46i 0.427872i 0.976848 + 0.213936i \(0.0686284\pi\)
−0.976848 + 0.213936i \(0.931372\pi\)
\(42\) 0 0
\(43\) 16104.7i 1.32826i −0.747619 0.664128i \(-0.768803\pi\)
0.747619 0.664128i \(-0.231197\pi\)
\(44\) 27849.5 2.16863
\(45\) 0 0
\(46\) 4237.10 22048.7i 0.295239 1.53635i
\(47\) 6936.30i 0.458019i 0.973424 + 0.229009i \(0.0735486\pi\)
−0.973424 + 0.229009i \(0.926451\pi\)
\(48\) 0 0
\(49\) 15777.1 0.938719
\(50\) 18525.8i 1.04798i
\(51\) 0 0
\(52\) −40109.6 −2.05703
\(53\) −11493.3 −0.562024 −0.281012 0.959704i \(-0.590670\pi\)
−0.281012 + 0.959704i \(0.590670\pi\)
\(54\) 0 0
\(55\) −19311.3 −0.860804
\(56\) 4067.35 0.173317
\(57\) 0 0
\(58\) 15425.0 0.602080
\(59\) 21495.9i 0.803945i 0.915652 + 0.401972i \(0.131675\pi\)
−0.915652 + 0.401972i \(0.868325\pi\)
\(60\) 0 0
\(61\) 3068.46i 0.105583i −0.998606 0.0527917i \(-0.983188\pi\)
0.998606 0.0527917i \(-0.0168119\pi\)
\(62\) 40863.4i 1.35007i
\(63\) 0 0
\(64\) 52597.2 1.60514
\(65\) 27812.7 0.816506
\(66\) 0 0
\(67\) 69034.0i 1.87878i 0.342848 + 0.939391i \(0.388608\pi\)
−0.342848 + 0.939391i \(0.611392\pi\)
\(68\) 38497.6 1.00963
\(69\) 0 0
\(70\) −9122.54 −0.222521
\(71\) 5054.68i 0.119000i 0.998228 + 0.0595001i \(0.0189507\pi\)
−0.998228 + 0.0595001i \(0.981049\pi\)
\(72\) 0 0
\(73\) 7803.25 0.171383 0.0856917 0.996322i \(-0.472690\pi\)
0.0856917 + 0.996322i \(0.472690\pi\)
\(74\) 128668. 2.73144
\(75\) 0 0
\(76\) 49706.0i 0.987132i
\(77\) 19295.2i 0.370871i
\(78\) 0 0
\(79\) 18787.0i 0.338680i −0.985558 0.169340i \(-0.945836\pi\)
0.985558 0.169340i \(-0.0541637\pi\)
\(80\) −11584.0 −0.202365
\(81\) 0 0
\(82\) −40757.9 −0.669387
\(83\) −60140.1 −0.958227 −0.479114 0.877753i \(-0.659042\pi\)
−0.479114 + 0.877753i \(0.659042\pi\)
\(84\) 0 0
\(85\) −26694.9 −0.400756
\(86\) 142525. 2.07800
\(87\) 0 0
\(88\) 76198.2i 1.04891i
\(89\) −29806.7 −0.398877 −0.199439 0.979910i \(-0.563912\pi\)
−0.199439 + 0.979910i \(0.563912\pi\)
\(90\) 0 0
\(91\) 27789.6i 0.351786i
\(92\) 115404. + 22177.1i 1.42151 + 0.273172i
\(93\) 0 0
\(94\) −61385.6 −0.716550
\(95\) 34467.0i 0.391827i
\(96\) 0 0
\(97\) 126459.i 1.36464i −0.731052 0.682322i \(-0.760970\pi\)
0.731052 0.682322i \(-0.239030\pi\)
\(98\) 139625.i 1.46858i
\(99\) 0 0
\(100\) 96964.8 0.969648
\(101\) 121023.i 1.18050i −0.807222 0.590248i \(-0.799030\pi\)
0.807222 0.590248i \(-0.200970\pi\)
\(102\) 0 0
\(103\) 42716.5i 0.396737i −0.980128 0.198368i \(-0.936436\pi\)
0.980128 0.198368i \(-0.0635642\pi\)
\(104\) 109743.i 0.994932i
\(105\) 0 0
\(106\) 101715.i 0.879262i
\(107\) −83454.0 −0.704673 −0.352337 0.935873i \(-0.614613\pi\)
−0.352337 + 0.935873i \(0.614613\pi\)
\(108\) 0 0
\(109\) 53334.5i 0.429974i −0.976617 0.214987i \(-0.931029\pi\)
0.976617 0.214987i \(-0.0689709\pi\)
\(110\) 170903.i 1.34669i
\(111\) 0 0
\(112\) 11574.4i 0.0871874i
\(113\) −35211.2 −0.259409 −0.129705 0.991553i \(-0.541403\pi\)
−0.129705 + 0.991553i \(0.541403\pi\)
\(114\) 0 0
\(115\) −80022.9 15378.0i −0.564248 0.108431i
\(116\) 80734.9i 0.557079i
\(117\) 0 0
\(118\) −190237. −1.25774
\(119\) 26672.6i 0.172663i
\(120\) 0 0
\(121\) 200428. 1.24450
\(122\) 27155.5 0.165180
\(123\) 0 0
\(124\) 213880. 1.24916
\(125\) −167611. −0.959459
\(126\) 0 0
\(127\) 189969. 1.04514 0.522569 0.852597i \(-0.324974\pi\)
0.522569 + 0.852597i \(0.324974\pi\)
\(128\) 233566.i 1.26004i
\(129\) 0 0
\(130\) 246139.i 1.27739i
\(131\) 197216.i 1.00407i 0.864847 + 0.502036i \(0.167416\pi\)
−0.864847 + 0.502036i \(0.832584\pi\)
\(132\) 0 0
\(133\) −34438.3 −0.168816
\(134\) −610944. −2.93927
\(135\) 0 0
\(136\) 105332.i 0.488331i
\(137\) −52017.2 −0.236780 −0.118390 0.992967i \(-0.537773\pi\)
−0.118390 + 0.992967i \(0.537773\pi\)
\(138\) 0 0
\(139\) −413006. −1.81309 −0.906546 0.422108i \(-0.861290\pi\)
−0.906546 + 0.422108i \(0.861290\pi\)
\(140\) 47747.7i 0.205889i
\(141\) 0 0
\(142\) −44733.4 −0.186171
\(143\) −520613. −2.12900
\(144\) 0 0
\(145\) 55982.9i 0.221124i
\(146\) 69058.0i 0.268122i
\(147\) 0 0
\(148\) 673454.i 2.52728i
\(149\) −498144. −1.83818 −0.919092 0.394042i \(-0.871076\pi\)
−0.919092 + 0.394042i \(0.871076\pi\)
\(150\) 0 0
\(151\) −267038. −0.953083 −0.476541 0.879152i \(-0.658110\pi\)
−0.476541 + 0.879152i \(0.658110\pi\)
\(152\) 135999. 0.477450
\(153\) 0 0
\(154\) 170761. 0.580211
\(155\) −148308. −0.495834
\(156\) 0 0
\(157\) 154389.i 0.499881i 0.968261 + 0.249941i \(0.0804111\pi\)
−0.968261 + 0.249941i \(0.919589\pi\)
\(158\) 166263. 0.529851
\(159\) 0 0
\(160\) 232781.i 0.718866i
\(161\) 15365.2 79956.4i 0.0467169 0.243102i
\(162\) 0 0
\(163\) 314441. 0.926981 0.463490 0.886102i \(-0.346597\pi\)
0.463490 + 0.886102i \(0.346597\pi\)
\(164\) 213328.i 0.619354i
\(165\) 0 0
\(166\) 532233.i 1.49910i
\(167\) 466773.i 1.29513i 0.762009 + 0.647567i \(0.224213\pi\)
−0.762009 + 0.647567i \(0.775787\pi\)
\(168\) 0 0
\(169\) 378510. 1.01944
\(170\) 236247.i 0.626965i
\(171\) 0 0
\(172\) 745982.i 1.92268i
\(173\) 437548.i 1.11150i 0.831348 + 0.555752i \(0.187569\pi\)
−0.831348 + 0.555752i \(0.812431\pi\)
\(174\) 0 0
\(175\) 67181.0i 0.165826i
\(176\) 216837. 0.527656
\(177\) 0 0
\(178\) 263787.i 0.624026i
\(179\) 52237.9i 0.121858i −0.998142 0.0609289i \(-0.980594\pi\)
0.998142 0.0609289i \(-0.0194063\pi\)
\(180\) 0 0
\(181\) 622329.i 1.41196i 0.708230 + 0.705982i \(0.249494\pi\)
−0.708230 + 0.705982i \(0.750506\pi\)
\(182\) −245935. −0.550353
\(183\) 0 0
\(184\) −60678.3 + 315754.i −0.132126 + 0.687549i
\(185\) 466984.i 1.00317i
\(186\) 0 0
\(187\) 499689. 1.04495
\(188\) 321294.i 0.662992i
\(189\) 0 0
\(190\) −305029. −0.612996
\(191\) 814655. 1.61581 0.807905 0.589312i \(-0.200601\pi\)
0.807905 + 0.589312i \(0.200601\pi\)
\(192\) 0 0
\(193\) −396971. −0.767124 −0.383562 0.923515i \(-0.625303\pi\)
−0.383562 + 0.923515i \(0.625303\pi\)
\(194\) 1.11915e6 2.13493
\(195\) 0 0
\(196\) −730804. −1.35882
\(197\) 314623.i 0.577597i 0.957390 + 0.288799i \(0.0932558\pi\)
−0.957390 + 0.288799i \(0.906744\pi\)
\(198\) 0 0
\(199\) 188929.i 0.338194i 0.985599 + 0.169097i \(0.0540851\pi\)
−0.985599 + 0.169097i \(0.945915\pi\)
\(200\) 265303.i 0.468993i
\(201\) 0 0
\(202\) 1.07104e6 1.84683
\(203\) 55936.4 0.0952695
\(204\) 0 0
\(205\) 147925.i 0.245843i
\(206\) 378036. 0.620677
\(207\) 0 0
\(208\) −312294. −0.500502
\(209\) 645172.i 1.02167i
\(210\) 0 0
\(211\) 248477. 0.384220 0.192110 0.981373i \(-0.438467\pi\)
0.192110 + 0.981373i \(0.438467\pi\)
\(212\) 532378. 0.813543
\(213\) 0 0
\(214\) 738560.i 1.10243i
\(215\) 517276.i 0.763179i
\(216\) 0 0
\(217\) 148185.i 0.213626i
\(218\) 472005. 0.672675
\(219\) 0 0
\(220\) 894512. 1.24603
\(221\) −719667. −0.991176
\(222\) 0 0
\(223\) −1.25438e6 −1.68914 −0.844571 0.535443i \(-0.820145\pi\)
−0.844571 + 0.535443i \(0.820145\pi\)
\(224\) 232587. 0.309718
\(225\) 0 0
\(226\) 311616.i 0.405834i
\(227\) −388100. −0.499895 −0.249948 0.968259i \(-0.580413\pi\)
−0.249948 + 0.968259i \(0.580413\pi\)
\(228\) 0 0
\(229\) 1.08193e6i 1.36336i 0.731650 + 0.681680i \(0.238750\pi\)
−0.731650 + 0.681680i \(0.761250\pi\)
\(230\) 136094. 708195.i 0.169636 0.882741i
\(231\) 0 0
\(232\) −220897. −0.269445
\(233\) 1.04198e6i 1.25738i 0.777654 + 0.628692i \(0.216409\pi\)
−0.777654 + 0.628692i \(0.783591\pi\)
\(234\) 0 0
\(235\) 222791.i 0.263165i
\(236\) 995707.i 1.16373i
\(237\) 0 0
\(238\) 236050. 0.270123
\(239\) 947058.i 1.07246i −0.844071 0.536231i \(-0.819848\pi\)
0.844071 0.536231i \(-0.180152\pi\)
\(240\) 0 0
\(241\) 168288.i 0.186642i −0.995636 0.0933210i \(-0.970252\pi\)
0.995636 0.0933210i \(-0.0297483\pi\)
\(242\) 1.77377e6i 1.94697i
\(243\) 0 0
\(244\) 142133.i 0.152834i
\(245\) 506752. 0.539362
\(246\) 0 0
\(247\) 929196.i 0.969092i
\(248\) 585193.i 0.604185i
\(249\) 0 0
\(250\) 1.48334e6i 1.50103i
\(251\) 661022. 0.662264 0.331132 0.943584i \(-0.392569\pi\)
0.331132 + 0.943584i \(0.392569\pi\)
\(252\) 0 0
\(253\) 1.49791e6 + 287854.i 1.47125 + 0.282729i
\(254\) 1.68121e6i 1.63507i
\(255\) 0 0
\(256\) −383921. −0.366135
\(257\) 113921.i 0.107590i −0.998552 0.0537949i \(-0.982868\pi\)
0.998552 0.0537949i \(-0.0171317\pi\)
\(258\) 0 0
\(259\) 466595. 0.432206
\(260\) −1.28830e6 −1.18191
\(261\) 0 0
\(262\) −1.74535e6 −1.57083
\(263\) −1.91611e6 −1.70817 −0.854086 0.520132i \(-0.825883\pi\)
−0.854086 + 0.520132i \(0.825883\pi\)
\(264\) 0 0
\(265\) −369160. −0.322923
\(266\) 304776.i 0.264105i
\(267\) 0 0
\(268\) 3.19771e6i 2.71958i
\(269\) 1.72330e6i 1.45205i 0.687669 + 0.726024i \(0.258634\pi\)
−0.687669 + 0.726024i \(0.741366\pi\)
\(270\) 0 0
\(271\) 156049. 0.129074 0.0645369 0.997915i \(-0.479443\pi\)
0.0645369 + 0.997915i \(0.479443\pi\)
\(272\) 299743. 0.245656
\(273\) 0 0
\(274\) 460347.i 0.370432i
\(275\) 1.25858e6 1.00357
\(276\) 0 0
\(277\) −1.29944e6 −1.01756 −0.508778 0.860898i \(-0.669903\pi\)
−0.508778 + 0.860898i \(0.669903\pi\)
\(278\) 3.65506e6i 2.83650i
\(279\) 0 0
\(280\) 130641. 0.0995831
\(281\) 1.53841e6 1.16227 0.581135 0.813807i \(-0.302609\pi\)
0.581135 + 0.813807i \(0.302609\pi\)
\(282\) 0 0
\(283\) 1.22558e6i 0.909653i 0.890580 + 0.454827i \(0.150299\pi\)
−0.890580 + 0.454827i \(0.849701\pi\)
\(284\) 234136.i 0.172256i
\(285\) 0 0
\(286\) 4.60737e6i 3.33072i
\(287\) −147802. −0.105920
\(288\) 0 0
\(289\) −729114. −0.513513
\(290\) 495443. 0.345939
\(291\) 0 0
\(292\) −361452. −0.248081
\(293\) 38645.9 0.0262987 0.0131493 0.999914i \(-0.495814\pi\)
0.0131493 + 0.999914i \(0.495814\pi\)
\(294\) 0 0
\(295\) 690440.i 0.461924i
\(296\) −1.84262e6 −1.22238
\(297\) 0 0
\(298\) 4.40852e6i 2.87576i
\(299\) −2.15734e6 414576.i −1.39553 0.268180i
\(300\) 0 0
\(301\) 516846. 0.328810
\(302\) 2.36326e6i 1.49106i
\(303\) 0 0
\(304\) 387012.i 0.240182i
\(305\) 98557.4i 0.0606652i
\(306\) 0 0
\(307\) 2.23002e6 1.35040 0.675201 0.737634i \(-0.264057\pi\)
0.675201 + 0.737634i \(0.264057\pi\)
\(308\) 893768.i 0.536844i
\(309\) 0 0
\(310\) 1.31251e6i 0.775710i
\(311\) 2.62823e6i 1.54086i 0.637526 + 0.770429i \(0.279958\pi\)
−0.637526 + 0.770429i \(0.720042\pi\)
\(312\) 0 0
\(313\) 2.55421e6i 1.47366i −0.676081 0.736828i \(-0.736323\pi\)
0.676081 0.736828i \(-0.263677\pi\)
\(314\) −1.36633e6 −0.782042
\(315\) 0 0
\(316\) 870228.i 0.490247i
\(317\) 2.08931e6i 1.16776i −0.811839 0.583881i \(-0.801534\pi\)
0.811839 0.583881i \(-0.198466\pi\)
\(318\) 0 0
\(319\) 1.04792e6i 0.576569i
\(320\) 1.68940e6 0.922269
\(321\) 0 0
\(322\) 707606. + 135981.i 0.380322 + 0.0730865i
\(323\) 891850.i 0.475648i
\(324\) 0 0
\(325\) −1.81264e6 −0.951927
\(326\) 2.78278e6i 1.45022i
\(327\) 0 0
\(328\) 583682. 0.299566
\(329\) −222605. −0.113383
\(330\) 0 0
\(331\) 669432. 0.335843 0.167922 0.985800i \(-0.446294\pi\)
0.167922 + 0.985800i \(0.446294\pi\)
\(332\) 2.78573e6 1.38706
\(333\) 0 0
\(334\) −4.13089e6 −2.02618
\(335\) 2.21734e6i 1.07950i
\(336\) 0 0
\(337\) 2.26733e6i 1.08752i 0.839239 + 0.543762i \(0.183001\pi\)
−0.839239 + 0.543762i \(0.816999\pi\)
\(338\) 3.34977e6i 1.59486i
\(339\) 0 0
\(340\) 1.23652e6 0.580104
\(341\) 2.77611e6 1.29286
\(342\) 0 0
\(343\) 1.04571e6i 0.479930i
\(344\) −2.04106e6 −0.929951
\(345\) 0 0
\(346\) −3.87226e6 −1.73890
\(347\) 1.79403e6i 0.799845i −0.916549 0.399922i \(-0.869037\pi\)
0.916549 0.399922i \(-0.130963\pi\)
\(348\) 0 0
\(349\) 3.09620e6 1.36071 0.680355 0.732882i \(-0.261825\pi\)
0.680355 + 0.732882i \(0.261825\pi\)
\(350\) 594545. 0.259427
\(351\) 0 0
\(352\) 4.35732e6i 1.87440i
\(353\) 4.46663e6i 1.90785i −0.300051 0.953923i \(-0.597004\pi\)
0.300051 0.953923i \(-0.402996\pi\)
\(354\) 0 0
\(355\) 162354.i 0.0683742i
\(356\) 1.38067e6 0.577384
\(357\) 0 0
\(358\) 462300. 0.190641
\(359\) 1.24915e6 0.511537 0.255769 0.966738i \(-0.417671\pi\)
0.255769 + 0.966738i \(0.417671\pi\)
\(360\) 0 0
\(361\) 1.32459e6 0.534950
\(362\) −5.50755e6 −2.20896
\(363\) 0 0
\(364\) 1.28723e6i 0.509218i
\(365\) 250637. 0.0984721
\(366\) 0 0
\(367\) 1.81423e6i 0.703117i 0.936166 + 0.351558i \(0.114348\pi\)
−0.936166 + 0.351558i \(0.885652\pi\)
\(368\) 898537. + 172672.i 0.345873 + 0.0664663i
\(369\) 0 0
\(370\) 4.13276e6 1.56941
\(371\) 368853.i 0.139129i
\(372\) 0 0
\(373\) 4.17985e6i 1.55557i 0.628533 + 0.777783i \(0.283656\pi\)
−0.628533 + 0.777783i \(0.716344\pi\)
\(374\) 4.42220e6i 1.63478i
\(375\) 0 0
\(376\) 879085. 0.320672
\(377\) 1.50924e6i 0.546898i
\(378\) 0 0
\(379\) 3.05588e6i 1.09279i 0.837527 + 0.546397i \(0.184001\pi\)
−0.837527 + 0.546397i \(0.815999\pi\)
\(380\) 1.59654e6i 0.567178i
\(381\) 0 0
\(382\) 7.20962e6i 2.52786i
\(383\) 4.23058e6 1.47368 0.736840 0.676068i \(-0.236317\pi\)
0.736840 + 0.676068i \(0.236317\pi\)
\(384\) 0 0
\(385\) 619754.i 0.213092i
\(386\) 3.51316e6i 1.20013i
\(387\) 0 0
\(388\) 5.85766e6i 1.97535i
\(389\) −3.31466e6 −1.11062 −0.555309 0.831644i \(-0.687400\pi\)
−0.555309 + 0.831644i \(0.687400\pi\)
\(390\) 0 0
\(391\) 2.07063e6 + 397913.i 0.684954 + 0.131627i
\(392\) 1.99953e6i 0.657225i
\(393\) 0 0
\(394\) −2.78438e6 −0.903625
\(395\) 603430.i 0.194596i
\(396\) 0 0
\(397\) −1.82036e6 −0.579670 −0.289835 0.957077i \(-0.593601\pi\)
−0.289835 + 0.957077i \(0.593601\pi\)
\(398\) −1.67200e6 −0.529089
\(399\) 0 0
\(400\) 754969. 0.235928
\(401\) 704005. 0.218633 0.109316 0.994007i \(-0.465134\pi\)
0.109316 + 0.994007i \(0.465134\pi\)
\(402\) 0 0
\(403\) −3.99824e6 −1.22633
\(404\) 5.60587e6i 1.70879i
\(405\) 0 0
\(406\) 495031.i 0.149045i
\(407\) 8.74126e6i 2.61570i
\(408\) 0 0
\(409\) 2.24072e6 0.662339 0.331169 0.943571i \(-0.392557\pi\)
0.331169 + 0.943571i \(0.392557\pi\)
\(410\) −1.30913e6 −0.384611
\(411\) 0 0
\(412\) 1.97866e6i 0.574285i
\(413\) −689865. −0.199017
\(414\) 0 0
\(415\) −1.93167e6 −0.550571
\(416\) 6.27555e6i 1.77795i
\(417\) 0 0
\(418\) 5.70971e6 1.59835
\(419\) −3.56788e6 −0.992830 −0.496415 0.868085i \(-0.665351\pi\)
−0.496415 + 0.868085i \(0.665351\pi\)
\(420\) 0 0
\(421\) 5.04487e6i 1.38722i −0.720352 0.693609i \(-0.756020\pi\)
0.720352 0.693609i \(-0.243980\pi\)
\(422\) 2.19899e6i 0.601095i
\(423\) 0 0
\(424\) 1.45663e6i 0.393490i
\(425\) 1.73979e6 0.467223
\(426\) 0 0
\(427\) 98475.4 0.0261371
\(428\) 3.86565e6 1.02003
\(429\) 0 0
\(430\) 4.57784e6 1.19396
\(431\) 905396. 0.234772 0.117386 0.993086i \(-0.462549\pi\)
0.117386 + 0.993086i \(0.462549\pi\)
\(432\) 0 0
\(433\) 477429.i 0.122374i 0.998126 + 0.0611870i \(0.0194886\pi\)
−0.998126 + 0.0611870i \(0.980511\pi\)
\(434\) 1.31142e6 0.334209
\(435\) 0 0
\(436\) 2.47049e6i 0.622397i
\(437\) 513764. 2.67349e6i 0.128695 0.669692i
\(438\) 0 0
\(439\) −2.09936e6 −0.519906 −0.259953 0.965621i \(-0.583707\pi\)
−0.259953 + 0.965621i \(0.583707\pi\)
\(440\) 2.44745e6i 0.602674i
\(441\) 0 0
\(442\) 6.36898e6i 1.55065i
\(443\) 5.15177e6i 1.24723i −0.781731 0.623616i \(-0.785663\pi\)
0.781731 0.623616i \(-0.214337\pi\)
\(444\) 0 0
\(445\) −957379. −0.229184
\(446\) 1.11011e7i 2.64259i
\(447\) 0 0
\(448\) 1.68799e6i 0.397353i
\(449\) 188053.i 0.0440216i −0.999758 0.0220108i \(-0.992993\pi\)
0.999758 0.0220108i \(-0.00700682\pi\)
\(450\) 0 0
\(451\) 2.76895e6i 0.641023i
\(452\) 1.63101e6 0.375501
\(453\) 0 0
\(454\) 3.43464e6i 0.782064i
\(455\) 892588.i 0.202126i
\(456\) 0 0
\(457\) 3.55643e6i 0.796569i 0.917262 + 0.398284i \(0.130394\pi\)
−0.917262 + 0.398284i \(0.869606\pi\)
\(458\) −9.57498e6 −2.13292
\(459\) 0 0
\(460\) 3.70672e6 + 712320.i 0.816761 + 0.156957i
\(461\) 7.65350e6i 1.67729i 0.544680 + 0.838644i \(0.316651\pi\)
−0.544680 + 0.838644i \(0.683349\pi\)
\(462\) 0 0
\(463\) −4.99234e6 −1.08231 −0.541155 0.840923i \(-0.682013\pi\)
−0.541155 + 0.840923i \(0.682013\pi\)
\(464\) 628604.i 0.135545i
\(465\) 0 0
\(466\) −9.22138e6 −1.96712
\(467\) 482188. 0.102311 0.0511557 0.998691i \(-0.483710\pi\)
0.0511557 + 0.998691i \(0.483710\pi\)
\(468\) 0 0
\(469\) −2.21550e6 −0.465092
\(470\) −1.97168e6 −0.411710
\(471\) 0 0
\(472\) 2.72433e6 0.562865
\(473\) 9.68266e6i 1.98995i
\(474\) 0 0
\(475\) 2.24632e6i 0.456813i
\(476\) 1.23550e6i 0.249933i
\(477\) 0 0
\(478\) 8.38137e6 1.67782
\(479\) −7.22981e6 −1.43975 −0.719877 0.694102i \(-0.755802\pi\)
−0.719877 + 0.694102i \(0.755802\pi\)
\(480\) 0 0
\(481\) 1.25894e7i 2.48109i
\(482\) 1.48933e6 0.291993
\(483\) 0 0
\(484\) −9.28398e6 −1.80144
\(485\) 4.06180e6i 0.784086i
\(486\) 0 0
\(487\) 7.95633e6 1.52016 0.760082 0.649828i \(-0.225159\pi\)
0.760082 + 0.649828i \(0.225159\pi\)
\(488\) −388887. −0.0739220
\(489\) 0 0
\(490\) 4.48470e6i 0.843808i
\(491\) 4.52236e6i 0.846568i −0.905997 0.423284i \(-0.860877\pi\)
0.905997 0.423284i \(-0.139123\pi\)
\(492\) 0 0
\(493\) 1.44859e6i 0.268427i
\(494\) −8.22329e6 −1.51610
\(495\) 0 0
\(496\) 1.66528e6 0.303936
\(497\) −162219. −0.0294585
\(498\) 0 0
\(499\) 774400. 0.139224 0.0696120 0.997574i \(-0.477824\pi\)
0.0696120 + 0.997574i \(0.477824\pi\)
\(500\) 7.76384e6 1.38884
\(501\) 0 0
\(502\) 5.84997e6i 1.03608i
\(503\) −7.87112e6 −1.38713 −0.693564 0.720395i \(-0.743961\pi\)
−0.693564 + 0.720395i \(0.743961\pi\)
\(504\) 0 0
\(505\) 3.88720e6i 0.678280i
\(506\) −2.54748e6 + 1.32564e7i −0.442317 + 2.30170i
\(507\) 0 0
\(508\) −8.79951e6 −1.51286
\(509\) 457065.i 0.0781957i −0.999235 0.0390979i \(-0.987552\pi\)
0.999235 0.0390979i \(-0.0124484\pi\)
\(510\) 0 0
\(511\) 250429.i 0.0424259i
\(512\) 4.07644e6i 0.687237i
\(513\) 0 0
\(514\) 1.00819e6 0.168319
\(515\) 1.37203e6i 0.227954i
\(516\) 0 0
\(517\) 4.17032e6i 0.686188i
\(518\) 4.12932e6i 0.676168i
\(519\) 0 0
\(520\) 3.52489e6i 0.571660i
\(521\) −7.34583e6 −1.18562 −0.592811 0.805342i \(-0.701982\pi\)
−0.592811 + 0.805342i \(0.701982\pi\)
\(522\) 0 0
\(523\) 1.31316e6i 0.209925i −0.994476 0.104963i \(-0.966528\pi\)
0.994476 0.104963i \(-0.0334723\pi\)
\(524\) 9.13521e6i 1.45342i
\(525\) 0 0
\(526\) 1.69574e7i 2.67236i
\(527\) 3.83755e6 0.601904
\(528\) 0 0
\(529\) 5.97789e6 + 2.38564e6i 0.928772 + 0.370652i
\(530\) 3.26703e6i 0.505200i
\(531\) 0 0
\(532\) 1.59521e6 0.244364
\(533\) 3.98792e6i 0.608036i
\(534\) 0 0
\(535\) −2.68051e6 −0.404886
\(536\) 8.74916e6 1.31539
\(537\) 0 0
\(538\) −1.52511e7 −2.27167
\(539\) −9.48566e6 −1.40636
\(540\) 0 0
\(541\) −1.14053e7 −1.67538 −0.837691 0.546144i \(-0.816095\pi\)
−0.837691 + 0.546144i \(0.816095\pi\)
\(542\) 1.38102e6i 0.201930i
\(543\) 0 0
\(544\) 6.02333e6i 0.872648i
\(545\) 1.71308e6i 0.247051i
\(546\) 0 0
\(547\) 1.26924e7 1.81374 0.906869 0.421413i \(-0.138466\pi\)
0.906869 + 0.421413i \(0.138466\pi\)
\(548\) 2.40947e6 0.342745
\(549\) 0 0
\(550\) 1.11383e7i 1.57004i
\(551\) 1.87034e6 0.262447
\(552\) 0 0
\(553\) 602928. 0.0838403
\(554\) 1.15000e7i 1.59192i
\(555\) 0 0
\(556\) 1.91307e7 2.62449
\(557\) −968825. −0.132314 −0.0661572 0.997809i \(-0.521074\pi\)
−0.0661572 + 0.997809i \(0.521074\pi\)
\(558\) 0 0
\(559\) 1.39453e7i 1.88754i
\(560\) 371765.i 0.0500954i
\(561\) 0 0
\(562\) 1.36148e7i 1.81832i
\(563\) 1.89549e6 0.252028 0.126014 0.992028i \(-0.459782\pi\)
0.126014 + 0.992028i \(0.459782\pi\)
\(564\) 0 0
\(565\) −1.13097e6 −0.149049
\(566\) −1.08463e7 −1.42311
\(567\) 0 0
\(568\) 640615. 0.0833156
\(569\) 1.10647e7 1.43271 0.716354 0.697738i \(-0.245810\pi\)
0.716354 + 0.697738i \(0.245810\pi\)
\(570\) 0 0
\(571\) 5.95073e6i 0.763801i −0.924204 0.381900i \(-0.875270\pi\)
0.924204 0.381900i \(-0.124730\pi\)
\(572\) 2.41152e7 3.08177
\(573\) 0 0
\(574\) 1.30804e6i 0.165707i
\(575\) 5.21535e6 + 1.00223e6i 0.657831 + 0.126415i
\(576\) 0 0
\(577\) −9.66464e6 −1.20850 −0.604249 0.796795i \(-0.706527\pi\)
−0.604249 + 0.796795i \(0.706527\pi\)
\(578\) 6.45259e6i 0.803368i
\(579\) 0 0
\(580\) 2.59317e6i 0.320082i
\(581\) 1.93006e6i 0.237209i
\(582\) 0 0
\(583\) 6.91014e6 0.842006
\(584\) 988961.i 0.119990i
\(585\) 0 0
\(586\) 342012.i 0.0411431i
\(587\) 1.06148e7i 1.27150i −0.771895 0.635749i \(-0.780691\pi\)
0.771895 0.635749i \(-0.219309\pi\)
\(588\) 0 0
\(589\) 4.95484e6i 0.588493i
\(590\) −6.11032e6 −0.722660
\(591\) 0 0
\(592\) 5.24352e6i 0.614920i
\(593\) 4.26629e6i 0.498212i 0.968476 + 0.249106i \(0.0801367\pi\)
−0.968476 + 0.249106i \(0.919863\pi\)
\(594\) 0 0
\(595\) 856713.i 0.0992072i
\(596\) 2.30744e7 2.66081
\(597\) 0 0
\(598\) 3.66895e6 1.90922e7i 0.419555 2.18325i
\(599\) 1.21167e7i 1.37980i −0.723903 0.689902i \(-0.757654\pi\)
0.723903 0.689902i \(-0.242346\pi\)
\(600\) 0 0
\(601\) −1.45959e7 −1.64833 −0.824167 0.566346i \(-0.808357\pi\)
−0.824167 + 0.566346i \(0.808357\pi\)
\(602\) 4.57403e6i 0.514408i
\(603\) 0 0
\(604\) 1.23694e7 1.37961
\(605\) 6.43766e6 0.715056
\(606\) 0 0
\(607\) −8.70600e6 −0.959063 −0.479531 0.877525i \(-0.659193\pi\)
−0.479531 + 0.877525i \(0.659193\pi\)
\(608\) 7.77700e6 0.853204
\(609\) 0 0
\(610\) 872223. 0.0949080
\(611\) 6.00622e6i 0.650876i
\(612\) 0 0
\(613\) 1.60427e7i 1.72435i −0.506610 0.862175i \(-0.669102\pi\)
0.506610 0.862175i \(-0.330898\pi\)
\(614\) 1.97355e7i 2.11264i
\(615\) 0 0
\(616\) −2.44542e6 −0.259658
\(617\) 1.35989e7 1.43811 0.719053 0.694955i \(-0.244576\pi\)
0.719053 + 0.694955i \(0.244576\pi\)
\(618\) 0 0
\(619\) 8.39164e6i 0.880279i 0.897929 + 0.440139i \(0.145071\pi\)
−0.897929 + 0.440139i \(0.854929\pi\)
\(620\) 6.86974e6 0.717730
\(621\) 0 0
\(622\) −2.32596e7 −2.41060
\(623\) 956583.i 0.0987421i
\(624\) 0 0
\(625\) 1.15809e6 0.118589
\(626\) 2.26045e7 2.30547
\(627\) 0 0
\(628\) 7.15140e6i 0.723589i
\(629\) 1.20834e7i 1.21777i
\(630\) 0 0
\(631\) 4.63039e6i 0.462961i −0.972840 0.231481i \(-0.925643\pi\)
0.972840 0.231481i \(-0.0743570\pi\)
\(632\) −2.38101e6 −0.237120
\(633\) 0 0
\(634\) 1.84902e7 1.82691
\(635\) 6.10172e6 0.600507
\(636\) 0 0
\(637\) 1.36615e7 1.33398
\(638\) −9.27398e6 −0.902016
\(639\) 0 0
\(640\) 7.50202e6i 0.723983i
\(641\) −7.21313e6 −0.693392 −0.346696 0.937978i \(-0.612696\pi\)
−0.346696 + 0.937978i \(0.612696\pi\)
\(642\) 0 0
\(643\) 1.24779e7i 1.19019i 0.803657 + 0.595093i \(0.202885\pi\)
−0.803657 + 0.595093i \(0.797115\pi\)
\(644\) −711727. + 3.70364e6i −0.0676237 + 0.351895i
\(645\) 0 0
\(646\) 7.89278e6 0.744130
\(647\) 8.39992e6i 0.788887i −0.918920 0.394443i \(-0.870937\pi\)
0.918920 0.394443i \(-0.129063\pi\)
\(648\) 0 0
\(649\) 1.29240e7i 1.20444i
\(650\) 1.60417e7i 1.48925i
\(651\) 0 0
\(652\) −1.45652e7 −1.34183
\(653\) 5.72017e6i 0.524960i −0.964937 0.262480i \(-0.915460\pi\)
0.964937 0.262480i \(-0.0845404\pi\)
\(654\) 0 0
\(655\) 6.33450e6i 0.576912i
\(656\) 1.66098e6i 0.150697i
\(657\) 0 0
\(658\) 1.97004e6i 0.177382i
\(659\) −6.18373e6 −0.554673 −0.277337 0.960773i \(-0.589452\pi\)
−0.277337 + 0.960773i \(0.589452\pi\)
\(660\) 0 0
\(661\) 1.39119e6i 0.123846i 0.998081 + 0.0619231i \(0.0197233\pi\)
−0.998081 + 0.0619231i \(0.980277\pi\)
\(662\) 5.92441e6i 0.525412i
\(663\) 0 0
\(664\) 7.62197e6i 0.670883i
\(665\) −1.10614e6 −0.0969967
\(666\) 0 0
\(667\) −834481. + 4.34241e6i −0.0726276 + 0.377935i
\(668\) 2.16213e7i 1.87473i
\(669\) 0 0
\(670\) −1.96233e7 −1.68882
\(671\) 1.84485e6i 0.158181i
\(672\) 0 0
\(673\) 1.31993e7 1.12335 0.561673 0.827359i \(-0.310158\pi\)
0.561673 + 0.827359i \(0.310158\pi\)
\(674\) −2.00656e7 −1.70138
\(675\) 0 0
\(676\) −1.75328e7 −1.47566
\(677\) 1.85679e7 1.55701 0.778503 0.627640i \(-0.215979\pi\)
0.778503 + 0.627640i \(0.215979\pi\)
\(678\) 0 0
\(679\) 4.05842e6 0.337818
\(680\) 3.38322e6i 0.280581i
\(681\) 0 0
\(682\) 2.45683e7i 2.02262i
\(683\) 1.44272e7i 1.18339i 0.806161 + 0.591696i \(0.201542\pi\)
−0.806161 + 0.591696i \(0.798458\pi\)
\(684\) 0 0
\(685\) −1.67077e6 −0.136047
\(686\) −9.25447e6 −0.750829
\(687\) 0 0
\(688\) 5.80823e6i 0.467813i
\(689\) −9.95218e6 −0.798675
\(690\) 0 0
\(691\) −1.56581e7 −1.24751 −0.623755 0.781620i \(-0.714394\pi\)
−0.623755 + 0.781620i \(0.714394\pi\)
\(692\) 2.02676e7i 1.60893i
\(693\) 0 0
\(694\) 1.58770e7 1.25132
\(695\) −1.32656e7 −1.04175
\(696\) 0 0
\(697\) 3.82764e6i 0.298435i
\(698\) 2.74011e7i 2.12877i
\(699\) 0 0
\(700\) 3.11187e6i 0.240036i
\(701\) −1.87473e6 −0.144093 −0.0720465 0.997401i \(-0.522953\pi\)
−0.0720465 + 0.997401i \(0.522953\pi\)
\(702\) 0 0
\(703\) 1.56015e7 1.19063
\(704\) −3.16231e7 −2.40477
\(705\) 0 0
\(706\) 3.95292e7 2.98474
\(707\) 3.88397e6 0.292232
\(708\) 0 0
\(709\) 4.21563e6i 0.314954i 0.987523 + 0.157477i \(0.0503360\pi\)
−0.987523 + 0.157477i \(0.949664\pi\)
\(710\) −1.43682e6 −0.106968
\(711\) 0 0
\(712\) 3.77761e6i 0.279266i
\(713\) 1.15038e7 + 2.21068e6i 0.847456 + 0.162855i
\(714\) 0 0
\(715\) −1.67219e7 −1.22326
\(716\) 2.41970e6i 0.176392i
\(717\) 0 0
\(718\) 1.10548e7i 0.800278i
\(719\) 3.85693e6i 0.278240i 0.990276 + 0.139120i \(0.0444273\pi\)
−0.990276 + 0.139120i \(0.955573\pi\)
\(720\) 0 0
\(721\) 1.37089e6 0.0982121
\(722\) 1.17225e7i 0.836906i
\(723\) 0 0
\(724\) 2.88267e7i 2.04385i
\(725\) 3.64859e6i 0.257798i
\(726\) 0 0
\(727\) 1.82196e7i 1.27850i 0.768998 + 0.639251i \(0.220756\pi\)
−0.768998 + 0.639251i \(0.779244\pi\)
\(728\) 3.52196e6 0.246295
\(729\) 0 0
\(730\) 2.21811e6i 0.154055i
\(731\) 1.33848e7i 0.926440i
\(732\) 0 0
\(733\) 2.49993e7i 1.71857i −0.511495 0.859286i \(-0.670908\pi\)
0.511495 0.859286i \(-0.329092\pi\)
\(734\) −1.60558e7 −1.09999
\(735\) 0 0
\(736\) −3.46983e6 + 1.80561e7i −0.236110 + 1.22865i
\(737\) 4.15054e7i 2.81473i
\(738\) 0 0
\(739\) 2.49012e6 0.167729 0.0838647 0.996477i \(-0.473274\pi\)
0.0838647 + 0.996477i \(0.473274\pi\)
\(740\) 2.16310e7i 1.45210i
\(741\) 0 0
\(742\) 3.26431e6 0.217661
\(743\) −1.05540e7 −0.701367 −0.350684 0.936494i \(-0.614051\pi\)
−0.350684 + 0.936494i \(0.614051\pi\)
\(744\) 0 0
\(745\) −1.60002e7 −1.05617
\(746\) −3.69912e7 −2.43362
\(747\) 0 0
\(748\) −2.31460e7 −1.51259
\(749\) 2.67828e6i 0.174442i
\(750\) 0 0
\(751\) 1.14752e7i 0.742440i 0.928545 + 0.371220i \(0.121060\pi\)
−0.928545 + 0.371220i \(0.878940\pi\)
\(752\) 2.50160e6i 0.161315i
\(753\) 0 0
\(754\) 1.33567e7 0.855598
\(755\) −8.57714e6 −0.547615
\(756\) 0 0
\(757\) 2.88533e7i 1.83002i −0.403433 0.915009i \(-0.632183\pi\)
0.403433 0.915009i \(-0.367817\pi\)
\(758\) −2.70442e7 −1.70963
\(759\) 0 0
\(760\) 4.36824e6 0.274329
\(761\) 1.15175e7i 0.720934i −0.932772 0.360467i \(-0.882617\pi\)
0.932772 0.360467i \(-0.117383\pi\)
\(762\) 0 0
\(763\) 1.71166e6 0.106440
\(764\) −3.77354e7 −2.33892
\(765\) 0 0
\(766\) 3.74402e7i 2.30551i
\(767\) 1.86136e7i 1.14246i
\(768\) 0 0
\(769\) 7.23163e6i 0.440981i 0.975389 + 0.220491i \(0.0707659\pi\)
−0.975389 + 0.220491i \(0.929234\pi\)
\(770\) 5.48476e6 0.333373
\(771\) 0 0
\(772\) 1.83880e7 1.11043
\(773\) 2.66964e7 1.60696 0.803478 0.595335i \(-0.202981\pi\)
0.803478 + 0.595335i \(0.202981\pi\)
\(774\) 0 0
\(775\) 9.66572e6 0.578070
\(776\) −1.60270e7 −0.955427
\(777\) 0 0
\(778\) 2.93344e7i 1.73751i
\(779\) −4.94205e6 −0.291785
\(780\) 0 0
\(781\) 3.03903e6i 0.178282i
\(782\) −3.52149e6 + 1.83249e7i −0.205925 + 1.07158i
\(783\) 0 0
\(784\) −5.69006e6 −0.330618
\(785\) 4.95890e6i 0.287218i
\(786\) 0 0
\(787\) 1.40550e7i 0.808897i 0.914561 + 0.404449i \(0.132537\pi\)
−0.914561 + 0.404449i \(0.867463\pi\)
\(788\) 1.45736e7i 0.836085i
\(789\) 0 0
\(790\) 5.34030e6 0.304437
\(791\) 1.13003e6i 0.0642167i
\(792\) 0 0
\(793\) 2.65701e6i 0.150041i
\(794\) 1.61100e7i 0.906869i
\(795\) 0 0
\(796\) 8.75132e6i 0.489543i
\(797\) 2.43756e7 1.35928 0.679642 0.733544i \(-0.262135\pi\)
0.679642 + 0.733544i \(0.262135\pi\)
\(798\) 0 0
\(799\) 5.76482e6i 0.319462i
\(800\) 1.51711e7i 0.838092i
\(801\) 0 0
\(802\) 6.23037e6i 0.342041i
\(803\) −4.69156e6 −0.256761
\(804\) 0 0
\(805\) 493523. 2.56816e6i 0.0268422 0.139680i
\(806\) 3.53841e7i 1.91854i
\(807\) 0 0
\(808\) −1.53381e7 −0.826499
\(809\) 8.73191e6i 0.469070i 0.972108 + 0.234535i \(0.0753568\pi\)
−0.972108 + 0.234535i \(0.924643\pi\)
\(810\) 0 0
\(811\) 9.83211e6 0.524922 0.262461 0.964943i \(-0.415466\pi\)
0.262461 + 0.964943i \(0.415466\pi\)
\(812\) −2.59101e6 −0.137905
\(813\) 0 0
\(814\) −7.73593e7 −4.09215
\(815\) 1.00997e7 0.532617
\(816\) 0 0
\(817\) 1.72817e7 0.905798
\(818\) 1.98302e7i 1.03620i
\(819\) 0 0
\(820\) 6.85201e6i 0.355864i
\(821\) 1.77459e7i 0.918841i −0.888219 0.459421i \(-0.848057\pi\)
0.888219 0.459421i \(-0.151943\pi\)
\(822\) 0 0
\(823\) 6.14010e6 0.315992 0.157996 0.987440i \(-0.449497\pi\)
0.157996 + 0.987440i \(0.449497\pi\)
\(824\) −5.41375e6 −0.277767
\(825\) 0 0
\(826\) 6.10524e6i 0.311353i
\(827\) 2.46889e7 1.25527 0.627637 0.778506i \(-0.284022\pi\)
0.627637 + 0.778506i \(0.284022\pi\)
\(828\) 0 0
\(829\) −1.63192e7 −0.824729 −0.412364 0.911019i \(-0.635297\pi\)
−0.412364 + 0.911019i \(0.635297\pi\)
\(830\) 1.70951e7i 0.861343i
\(831\) 0 0
\(832\) 4.55446e7 2.28102
\(833\) −1.31125e7 −0.654744
\(834\) 0 0
\(835\) 1.49925e7i 0.744147i
\(836\) 2.98848e7i 1.47889i
\(837\) 0 0
\(838\) 3.15754e7i 1.55324i
\(839\) 9.90074e6 0.485582 0.242791 0.970079i \(-0.421937\pi\)
0.242791 + 0.970079i \(0.421937\pi\)
\(840\) 0 0
\(841\) 1.74733e7 0.851891
\(842\) 4.46466e7 2.17024
\(843\) 0 0
\(844\) −1.15096e7 −0.556167
\(845\) 1.21576e7 0.585740
\(846\) 0 0
\(847\) 6.43231e6i 0.308076i
\(848\) 4.14511e6 0.197946
\(849\) 0 0
\(850\) 1.53970e7i 0.730950i
\(851\) −6.96086e6 + 3.62224e7i −0.329487 + 1.71456i
\(852\) 0 0
\(853\) −2.20295e7 −1.03665 −0.518325 0.855184i \(-0.673444\pi\)
−0.518325 + 0.855184i \(0.673444\pi\)
\(854\) 871498.i 0.0408904i
\(855\) 0 0
\(856\) 1.05767e7i 0.493362i
\(857\) 5.35124e6i 0.248887i −0.992227 0.124443i \(-0.960285\pi\)
0.992227 0.124443i \(-0.0397145\pi\)
\(858\) 0 0
\(859\) −2.27154e7 −1.05036 −0.525180 0.850991i \(-0.676002\pi\)
−0.525180 + 0.850991i \(0.676002\pi\)
\(860\) 2.39606e7i 1.10472i
\(861\) 0 0
\(862\) 8.01267e6i 0.367290i
\(863\) 2.50118e7i 1.14319i −0.820537 0.571594i \(-0.806325\pi\)
0.820537 0.571594i \(-0.193675\pi\)
\(864\) 0 0
\(865\) 1.40539e7i 0.638639i
\(866\) −4.22520e6 −0.191449
\(867\) 0 0
\(868\) 6.86403e6i 0.309229i
\(869\) 1.12953e7i 0.507399i
\(870\) 0 0
\(871\) 5.97774e7i 2.66988i
\(872\) −6.75945e6 −0.301037
\(873\) 0 0
\(874\) 2.36601e7 + 4.54676e6i 1.04770 + 0.201337i
\(875\) 5.37910e6i 0.237514i
\(876\) 0 0
\(877\) 1.33025e7 0.584030 0.292015 0.956414i \(-0.405674\pi\)
0.292015 + 0.956414i \(0.405674\pi\)
\(878\) 1.85791e7i 0.813370i
\(879\) 0 0
\(880\) 6.96469e6 0.303176
\(881\) 2.81119e7 1.22025 0.610127 0.792304i \(-0.291119\pi\)
0.610127 + 0.792304i \(0.291119\pi\)
\(882\) 0 0
\(883\) −3.69156e7 −1.59334 −0.796668 0.604417i \(-0.793406\pi\)
−0.796668 + 0.604417i \(0.793406\pi\)
\(884\) 3.33355e7 1.43475
\(885\) 0 0
\(886\) 4.55927e7 1.95124
\(887\) 2.10720e7i 0.899283i −0.893209 0.449641i \(-0.851552\pi\)
0.893209 0.449641i \(-0.148448\pi\)
\(888\) 0 0
\(889\) 6.09665e6i 0.258724i
\(890\) 8.47271e6i 0.358548i
\(891\) 0 0
\(892\) 5.81037e7 2.44507
\(893\) −7.44323e6 −0.312344
\(894\) 0 0
\(895\) 1.67786e6i 0.0700161i
\(896\) −7.49578e6 −0.311923
\(897\) 0 0
\(898\) 1.66425e6 0.0688698
\(899\) 8.04789e6i 0.332111i
\(900\) 0 0
\(901\) 9.55219e6 0.392004
\(902\) 2.45049e7 1.00285
\(903\) 0 0
\(904\) 4.46257e6i 0.181620i
\(905\) 1.99889e7i 0.811275i
\(906\) 0 0
\(907\) 4.67560e6i 0.188721i 0.995538 + 0.0943603i \(0.0300806\pi\)
−0.995538 + 0.0943603i \(0.969919\pi\)
\(908\) 1.79771e7 0.723609
\(909\) 0 0
\(910\) −7.89931e6 −0.316217
\(911\) −2.67198e7 −1.06669 −0.533343 0.845899i \(-0.679065\pi\)
−0.533343 + 0.845899i \(0.679065\pi\)
\(912\) 0 0
\(913\) 3.61581e7 1.43558
\(914\) −3.14740e7 −1.24620
\(915\) 0 0
\(916\) 5.01158e7i 1.97349i
\(917\) −6.32923e6 −0.248558
\(918\) 0 0
\(919\) 1.67106e6i 0.0652685i 0.999467 + 0.0326343i \(0.0103897\pi\)
−0.999467 + 0.0326343i \(0.989610\pi\)
\(920\) −1.94896e6 + 1.01419e7i −0.0759160 + 0.395046i
\(921\) 0 0
\(922\) −6.77327e7 −2.62404
\(923\) 4.37691e6i 0.169108i
\(924\) 0 0
\(925\) 3.04348e7i 1.16954i
\(926\) 4.41817e7i 1.69323i
\(927\) 0 0
\(928\) −1.26318e7 −0.481498
\(929\) 3.81445e7i 1.45008i −0.688705 0.725041i \(-0.741821\pi\)
0.688705 0.725041i \(-0.258179\pi\)
\(930\) 0 0
\(931\) 1.69301e7i 0.640155i
\(932\) 4.82651e7i 1.82009i
\(933\) 0 0
\(934\) 4.26732e6i 0.160062i
\(935\) 1.60498e7 0.600399
\(936\) 0 0
\(937\) 5.11671e6i 0.190389i 0.995459 + 0.0951944i \(0.0303473\pi\)
−0.995459 + 0.0951944i \(0.969653\pi\)
\(938\) 1.96069e7i 0.727616i
\(939\) 0 0
\(940\) 1.03198e7i 0.380937i
\(941\) 1.94139e7 0.714725 0.357362 0.933966i \(-0.383676\pi\)
0.357362 + 0.933966i \(0.383676\pi\)
\(942\) 0 0
\(943\) 2.20498e6 1.14741e7i 0.0807467 0.420184i
\(944\) 7.75259e6i 0.283150i
\(945\) 0 0
\(946\) −8.56905e7 −3.11319
\(947\) 1.61762e7i 0.586140i 0.956091 + 0.293070i \(0.0946768\pi\)
−0.956091 + 0.293070i \(0.905323\pi\)
\(948\) 0 0
\(949\) 6.75693e6 0.243548
\(950\) 1.98797e7 0.714664
\(951\) 0 0
\(952\) −3.38041e6 −0.120886
\(953\) −1.69946e7 −0.606148 −0.303074 0.952967i \(-0.598013\pi\)
−0.303074 + 0.952967i \(0.598013\pi\)
\(954\) 0 0
\(955\) 2.61664e7 0.928399
\(956\) 4.38684e7i 1.55241i
\(957\) 0 0
\(958\) 6.39831e7i 2.25243i
\(959\) 1.66938e6i 0.0586150i
\(960\) 0 0
\(961\) −7.30894e6 −0.255297
\(962\) 1.11415e8 3.88156
\(963\) 0 0
\(964\) 7.79520e6i 0.270168i
\(965\) −1.27505e7 −0.440768
\(966\) 0 0
\(967\) −9.47024e6 −0.325683 −0.162841 0.986652i \(-0.552066\pi\)
−0.162841 + 0.986652i \(0.552066\pi\)
\(968\) 2.54017e7i 0.871312i
\(969\) 0 0
\(970\) 3.59465e7 1.22667
\(971\) 2.74957e7 0.935874 0.467937 0.883762i \(-0.344997\pi\)
0.467937 + 0.883762i \(0.344997\pi\)
\(972\) 0 0
\(973\) 1.32545e7i 0.448831i
\(974\) 7.04127e7i 2.37823i
\(975\) 0 0
\(976\) 1.10665e6i 0.0371866i
\(977\) 3.33362e7 1.11732 0.558662 0.829395i \(-0.311315\pi\)
0.558662 + 0.829395i \(0.311315\pi\)
\(978\) 0 0
\(979\) 1.79207e7 0.597585
\(980\) −2.34731e7 −0.780738
\(981\) 0 0
\(982\) 4.00225e7 1.32442
\(983\) −3.35806e7 −1.10842 −0.554210 0.832377i \(-0.686979\pi\)
−0.554210 + 0.832377i \(0.686979\pi\)
\(984\) 0 0
\(985\) 1.01056e7i 0.331871i
\(986\) −1.28198e7 −0.419943
\(987\) 0 0
\(988\) 4.30410e7i 1.40278i
\(989\) −7.71051e6 + 4.01234e7i −0.250664 + 1.30439i
\(990\) 0 0
\(991\) 1.73879e7 0.562422 0.281211 0.959646i \(-0.409264\pi\)
0.281211 + 0.959646i \(0.409264\pi\)
\(992\) 3.34637e7i 1.07968i
\(993\) 0 0
\(994\) 1.43562e6i 0.0460866i
\(995\) 6.06831e6i 0.194317i
\(996\) 0 0
\(997\) −4.16006e7 −1.32544 −0.662722 0.748865i \(-0.730599\pi\)
−0.662722 + 0.748865i \(0.730599\pi\)
\(998\) 6.85336e6i 0.217810i
\(999\) 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.206.26 yes 40
3.2 odd 2 inner 207.6.c.a.206.15 40
23.22 odd 2 inner 207.6.c.a.206.16 yes 40
69.68 even 2 inner 207.6.c.a.206.25 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
207.6.c.a.206.15 40 3.2 odd 2 inner
207.6.c.a.206.16 yes 40 23.22 odd 2 inner
207.6.c.a.206.25 yes 40 69.68 even 2 inner
207.6.c.a.206.26 yes 40 1.1 even 1 trivial