Properties

Label 441.6.a.n.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

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: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.54138\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.08276 q^{2} -6.16553 q^{4} -41.8276 q^{5} -193.986 q^{8} +O(q^{10})\) \(q+5.08276 q^{2} -6.16553 q^{4} -41.8276 q^{5} -193.986 q^{8} -212.600 q^{10} -72.0965 q^{11} -632.317 q^{13} -788.689 q^{16} +1975.92 q^{17} -1864.93 q^{19} +257.889 q^{20} -366.449 q^{22} -413.711 q^{23} -1375.45 q^{25} -3213.92 q^{26} -731.934 q^{29} +6123.18 q^{31} +2198.84 q^{32} +10043.2 q^{34} +10350.4 q^{37} -9478.97 q^{38} +8113.99 q^{40} +3529.84 q^{41} -14515.2 q^{43} +444.513 q^{44} -2102.79 q^{46} +21423.3 q^{47} -6991.08 q^{50} +3898.57 q^{52} -12579.5 q^{53} +3015.62 q^{55} -3720.25 q^{58} +36133.9 q^{59} +4024.80 q^{61} +31122.6 q^{62} +36414.2 q^{64} +26448.3 q^{65} +15565.9 q^{67} -12182.6 q^{68} -12180.8 q^{71} +19589.1 q^{73} +52608.5 q^{74} +11498.2 q^{76} +36089.8 q^{79} +32989.0 q^{80} +17941.3 q^{82} +24572.6 q^{83} -82648.2 q^{85} -73777.5 q^{86} +13985.7 q^{88} +70243.3 q^{89} +2550.74 q^{92} +108890. q^{94} +78005.4 q^{95} +105758. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 12 q^{4} + 38 q^{5} - 96 q^{8} - 778 q^{10} - 424 q^{11} - 924 q^{13} - 2064 q^{16} + 2346 q^{17} - 360 q^{19} + 1708 q^{20} + 2126 q^{22} + 12 q^{23} + 1872 q^{25} - 1148 q^{26} + 7052 q^{29} + 3548 q^{31} + 8096 q^{32} + 7422 q^{34} + 11090 q^{37} - 20138 q^{38} + 15936 q^{40} - 3500 q^{41} - 12680 q^{43} - 5948 q^{44} - 5118 q^{46} + 22956 q^{47} - 29992 q^{50} - 1400 q^{52} - 3042 q^{53} - 25076 q^{55} - 58852 q^{58} + 65808 q^{59} - 42486 q^{61} + 49362 q^{62} + 35456 q^{64} + 3164 q^{65} + 42312 q^{67} - 5460 q^{68} + 2208 q^{71} - 50506 q^{73} + 47370 q^{74} + 38836 q^{76} + 9004 q^{79} - 68816 q^{80} + 67732 q^{82} + 104328 q^{83} - 53106 q^{85} - 86776 q^{86} - 20496 q^{88} + 26666 q^{89} + 10284 q^{92} + 98034 q^{94} + 198140 q^{95} + 209132 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.08276 0.898514 0.449257 0.893403i \(-0.351689\pi\)
0.449257 + 0.893403i \(0.351689\pi\)
\(3\) 0 0
\(4\) −6.16553 −0.192673
\(5\) −41.8276 −0.748235 −0.374118 0.927381i \(-0.622054\pi\)
−0.374118 + 0.927381i \(0.622054\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −193.986 −1.07163
\(9\) 0 0
\(10\) −212.600 −0.672300
\(11\) −72.0965 −0.179652 −0.0898260 0.995957i \(-0.528631\pi\)
−0.0898260 + 0.995957i \(0.528631\pi\)
\(12\) 0 0
\(13\) −632.317 −1.03771 −0.518856 0.854862i \(-0.673642\pi\)
−0.518856 + 0.854862i \(0.673642\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −788.689 −0.770205
\(17\) 1975.92 1.65824 0.829121 0.559069i \(-0.188841\pi\)
0.829121 + 0.559069i \(0.188841\pi\)
\(18\) 0 0
\(19\) −1864.93 −1.18516 −0.592581 0.805511i \(-0.701891\pi\)
−0.592581 + 0.805511i \(0.701891\pi\)
\(20\) 257.889 0.144164
\(21\) 0 0
\(22\) −366.449 −0.161420
\(23\) −413.711 −0.163071 −0.0815356 0.996670i \(-0.525982\pi\)
−0.0815356 + 0.996670i \(0.525982\pi\)
\(24\) 0 0
\(25\) −1375.45 −0.440144
\(26\) −3213.92 −0.932398
\(27\) 0 0
\(28\) 0 0
\(29\) −731.934 −0.161613 −0.0808066 0.996730i \(-0.525750\pi\)
−0.0808066 + 0.996730i \(0.525750\pi\)
\(30\) 0 0
\(31\) 6123.18 1.14439 0.572193 0.820119i \(-0.306093\pi\)
0.572193 + 0.820119i \(0.306093\pi\)
\(32\) 2198.84 0.379593
\(33\) 0 0
\(34\) 10043.2 1.48995
\(35\) 0 0
\(36\) 0 0
\(37\) 10350.4 1.24295 0.621473 0.783436i \(-0.286535\pi\)
0.621473 + 0.783436i \(0.286535\pi\)
\(38\) −9478.97 −1.06488
\(39\) 0 0
\(40\) 8113.99 0.801834
\(41\) 3529.84 0.327941 0.163970 0.986465i \(-0.447570\pi\)
0.163970 + 0.986465i \(0.447570\pi\)
\(42\) 0 0
\(43\) −14515.2 −1.19716 −0.598581 0.801062i \(-0.704269\pi\)
−0.598581 + 0.801062i \(0.704269\pi\)
\(44\) 444.513 0.0346140
\(45\) 0 0
\(46\) −2102.79 −0.146522
\(47\) 21423.3 1.41463 0.707314 0.706900i \(-0.249907\pi\)
0.707314 + 0.706900i \(0.249907\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −6991.08 −0.395475
\(51\) 0 0
\(52\) 3898.57 0.199939
\(53\) −12579.5 −0.615138 −0.307569 0.951526i \(-0.599515\pi\)
−0.307569 + 0.951526i \(0.599515\pi\)
\(54\) 0 0
\(55\) 3015.62 0.134422
\(56\) 0 0
\(57\) 0 0
\(58\) −3720.25 −0.145212
\(59\) 36133.9 1.35140 0.675702 0.737175i \(-0.263841\pi\)
0.675702 + 0.737175i \(0.263841\pi\)
\(60\) 0 0
\(61\) 4024.80 0.138490 0.0692451 0.997600i \(-0.477941\pi\)
0.0692451 + 0.997600i \(0.477941\pi\)
\(62\) 31122.6 1.02825
\(63\) 0 0
\(64\) 36414.2 1.11127
\(65\) 26448.3 0.776453
\(66\) 0 0
\(67\) 15565.9 0.423632 0.211816 0.977310i \(-0.432062\pi\)
0.211816 + 0.977310i \(0.432062\pi\)
\(68\) −12182.6 −0.319498
\(69\) 0 0
\(70\) 0 0
\(71\) −12180.8 −0.286766 −0.143383 0.989667i \(-0.545798\pi\)
−0.143383 + 0.989667i \(0.545798\pi\)
\(72\) 0 0
\(73\) 19589.1 0.430237 0.215119 0.976588i \(-0.430986\pi\)
0.215119 + 0.976588i \(0.430986\pi\)
\(74\) 52608.5 1.11680
\(75\) 0 0
\(76\) 11498.2 0.228348
\(77\) 0 0
\(78\) 0 0
\(79\) 36089.8 0.650604 0.325302 0.945610i \(-0.394534\pi\)
0.325302 + 0.945610i \(0.394534\pi\)
\(80\) 32989.0 0.576294
\(81\) 0 0
\(82\) 17941.3 0.294659
\(83\) 24572.6 0.391522 0.195761 0.980652i \(-0.437282\pi\)
0.195761 + 0.980652i \(0.437282\pi\)
\(84\) 0 0
\(85\) −82648.2 −1.24076
\(86\) −73777.5 −1.07567
\(87\) 0 0
\(88\) 13985.7 0.192521
\(89\) 70243.3 0.940005 0.470002 0.882665i \(-0.344253\pi\)
0.470002 + 0.882665i \(0.344253\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2550.74 0.0314193
\(93\) 0 0
\(94\) 108890. 1.27106
\(95\) 78005.4 0.886779
\(96\) 0 0
\(97\) 105758. 1.14126 0.570630 0.821207i \(-0.306699\pi\)
0.570630 + 0.821207i \(0.306699\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8480.37 0.0848037
\(101\) −36461.8 −0.355660 −0.177830 0.984061i \(-0.556908\pi\)
−0.177830 + 0.984061i \(0.556908\pi\)
\(102\) 0 0
\(103\) 64520.1 0.599242 0.299621 0.954058i \(-0.403140\pi\)
0.299621 + 0.954058i \(0.403140\pi\)
\(104\) 122661. 1.11205
\(105\) 0 0
\(106\) −63938.4 −0.552710
\(107\) −66045.6 −0.557679 −0.278840 0.960338i \(-0.589950\pi\)
−0.278840 + 0.960338i \(0.589950\pi\)
\(108\) 0 0
\(109\) −37938.0 −0.305850 −0.152925 0.988238i \(-0.548869\pi\)
−0.152925 + 0.988238i \(0.548869\pi\)
\(110\) 15327.7 0.120780
\(111\) 0 0
\(112\) 0 0
\(113\) −123802. −0.912080 −0.456040 0.889959i \(-0.650733\pi\)
−0.456040 + 0.889959i \(0.650733\pi\)
\(114\) 0 0
\(115\) 17304.5 0.122016
\(116\) 4512.76 0.0311384
\(117\) 0 0
\(118\) 183660. 1.21426
\(119\) 0 0
\(120\) 0 0
\(121\) −155853. −0.967725
\(122\) 20457.1 0.124435
\(123\) 0 0
\(124\) −37752.6 −0.220492
\(125\) 188243. 1.07757
\(126\) 0 0
\(127\) 128724. 0.708189 0.354095 0.935210i \(-0.384789\pi\)
0.354095 + 0.935210i \(0.384789\pi\)
\(128\) 114722. 0.618902
\(129\) 0 0
\(130\) 134431. 0.697653
\(131\) 147902. 0.753003 0.376501 0.926416i \(-0.377127\pi\)
0.376501 + 0.926416i \(0.377127\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 79118.0 0.380639
\(135\) 0 0
\(136\) −383302. −1.77703
\(137\) 91157.4 0.414945 0.207472 0.978241i \(-0.433476\pi\)
0.207472 + 0.978241i \(0.433476\pi\)
\(138\) 0 0
\(139\) −334657. −1.46914 −0.734570 0.678533i \(-0.762616\pi\)
−0.734570 + 0.678533i \(0.762616\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −61911.9 −0.257664
\(143\) 45587.8 0.186427
\(144\) 0 0
\(145\) 30615.1 0.120925
\(146\) 99566.9 0.386574
\(147\) 0 0
\(148\) −63815.5 −0.239482
\(149\) −138271. −0.510231 −0.255115 0.966911i \(-0.582113\pi\)
−0.255115 + 0.966911i \(0.582113\pi\)
\(150\) 0 0
\(151\) 111169. 0.396773 0.198386 0.980124i \(-0.436430\pi\)
0.198386 + 0.980124i \(0.436430\pi\)
\(152\) 361770. 1.27006
\(153\) 0 0
\(154\) 0 0
\(155\) −256118. −0.856270
\(156\) 0 0
\(157\) −38148.5 −0.123517 −0.0617587 0.998091i \(-0.519671\pi\)
−0.0617587 + 0.998091i \(0.519671\pi\)
\(158\) 183436. 0.584577
\(159\) 0 0
\(160\) −91972.3 −0.284025
\(161\) 0 0
\(162\) 0 0
\(163\) −212905. −0.627648 −0.313824 0.949481i \(-0.601610\pi\)
−0.313824 + 0.949481i \(0.601610\pi\)
\(164\) −21763.3 −0.0631852
\(165\) 0 0
\(166\) 124897. 0.351788
\(167\) 120396. 0.334057 0.167028 0.985952i \(-0.446583\pi\)
0.167028 + 0.985952i \(0.446583\pi\)
\(168\) 0 0
\(169\) 28532.2 0.0768456
\(170\) −420081. −1.11484
\(171\) 0 0
\(172\) 89494.0 0.230660
\(173\) −712914. −1.81101 −0.905507 0.424331i \(-0.860509\pi\)
−0.905507 + 0.424331i \(0.860509\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 56861.7 0.138369
\(177\) 0 0
\(178\) 357030. 0.844607
\(179\) 749738. 1.74895 0.874474 0.485072i \(-0.161207\pi\)
0.874474 + 0.485072i \(0.161207\pi\)
\(180\) 0 0
\(181\) 623718. 1.41511 0.707557 0.706656i \(-0.249797\pi\)
0.707557 + 0.706656i \(0.249797\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 80254.2 0.174752
\(185\) −432932. −0.930016
\(186\) 0 0
\(187\) −142457. −0.297907
\(188\) −132086. −0.272560
\(189\) 0 0
\(190\) 396483. 0.796784
\(191\) −417726. −0.828530 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(192\) 0 0
\(193\) 770700. 1.48933 0.744667 0.667436i \(-0.232608\pi\)
0.744667 + 0.667436i \(0.232608\pi\)
\(194\) 537544. 1.02544
\(195\) 0 0
\(196\) 0 0
\(197\) 479193. 0.879721 0.439861 0.898066i \(-0.355028\pi\)
0.439861 + 0.898066i \(0.355028\pi\)
\(198\) 0 0
\(199\) −428686. −0.767373 −0.383687 0.923463i \(-0.625346\pi\)
−0.383687 + 0.923463i \(0.625346\pi\)
\(200\) 266818. 0.471673
\(201\) 0 0
\(202\) −185327. −0.319565
\(203\) 0 0
\(204\) 0 0
\(205\) −147645. −0.245377
\(206\) 327940. 0.538427
\(207\) 0 0
\(208\) 498702. 0.799250
\(209\) 134455. 0.212917
\(210\) 0 0
\(211\) −588544. −0.910066 −0.455033 0.890475i \(-0.650373\pi\)
−0.455033 + 0.890475i \(0.650373\pi\)
\(212\) 77559.0 0.118520
\(213\) 0 0
\(214\) −335694. −0.501083
\(215\) 607138. 0.895759
\(216\) 0 0
\(217\) 0 0
\(218\) −192830. −0.274811
\(219\) 0 0
\(220\) −18592.9 −0.0258994
\(221\) −1.24941e6 −1.72078
\(222\) 0 0
\(223\) −363249. −0.489151 −0.244575 0.969630i \(-0.578649\pi\)
−0.244575 + 0.969630i \(0.578649\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −629258. −0.819517
\(227\) 843041. 1.08589 0.542943 0.839770i \(-0.317310\pi\)
0.542943 + 0.839770i \(0.317310\pi\)
\(228\) 0 0
\(229\) −568666. −0.716587 −0.358293 0.933609i \(-0.616641\pi\)
−0.358293 + 0.933609i \(0.616641\pi\)
\(230\) 87954.8 0.109633
\(231\) 0 0
\(232\) 141985. 0.173190
\(233\) 1.05651e6 1.27492 0.637461 0.770482i \(-0.279985\pi\)
0.637461 + 0.770482i \(0.279985\pi\)
\(234\) 0 0
\(235\) −896086. −1.05847
\(236\) −222785. −0.260379
\(237\) 0 0
\(238\) 0 0
\(239\) −853715. −0.966759 −0.483379 0.875411i \(-0.660591\pi\)
−0.483379 + 0.875411i \(0.660591\pi\)
\(240\) 0 0
\(241\) −388888. −0.431302 −0.215651 0.976470i \(-0.569187\pi\)
−0.215651 + 0.976470i \(0.569187\pi\)
\(242\) −792164. −0.869515
\(243\) 0 0
\(244\) −24815.0 −0.0266833
\(245\) 0 0
\(246\) 0 0
\(247\) 1.17922e6 1.22986
\(248\) −1.18781e6 −1.22636
\(249\) 0 0
\(250\) 956795. 0.968209
\(251\) 839328. 0.840906 0.420453 0.907314i \(-0.361871\pi\)
0.420453 + 0.907314i \(0.361871\pi\)
\(252\) 0 0
\(253\) 29827.1 0.0292961
\(254\) 654272. 0.636318
\(255\) 0 0
\(256\) −582151. −0.555182
\(257\) −291986. −0.275759 −0.137879 0.990449i \(-0.544029\pi\)
−0.137879 + 0.990449i \(0.544029\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −163068. −0.149601
\(261\) 0 0
\(262\) 751752. 0.676584
\(263\) 288495. 0.257187 0.128594 0.991697i \(-0.458954\pi\)
0.128594 + 0.991697i \(0.458954\pi\)
\(264\) 0 0
\(265\) 526169. 0.460268
\(266\) 0 0
\(267\) 0 0
\(268\) −95972.2 −0.0816222
\(269\) 259370. 0.218544 0.109272 0.994012i \(-0.465148\pi\)
0.109272 + 0.994012i \(0.465148\pi\)
\(270\) 0 0
\(271\) −2.19551e6 −1.81599 −0.907994 0.418984i \(-0.862386\pi\)
−0.907994 + 0.418984i \(0.862386\pi\)
\(272\) −1.55839e6 −1.27719
\(273\) 0 0
\(274\) 463331. 0.372834
\(275\) 99165.1 0.0790728
\(276\) 0 0
\(277\) 126991. 0.0994426 0.0497213 0.998763i \(-0.484167\pi\)
0.0497213 + 0.998763i \(0.484167\pi\)
\(278\) −1.70098e6 −1.32004
\(279\) 0 0
\(280\) 0 0
\(281\) 2.22759e6 1.68294 0.841472 0.540301i \(-0.181690\pi\)
0.841472 + 0.540301i \(0.181690\pi\)
\(282\) 0 0
\(283\) 1.18895e6 0.882463 0.441231 0.897393i \(-0.354542\pi\)
0.441231 + 0.897393i \(0.354542\pi\)
\(284\) 75100.7 0.0552520
\(285\) 0 0
\(286\) 231712. 0.167507
\(287\) 0 0
\(288\) 0 0
\(289\) 2.48442e6 1.74977
\(290\) 155609. 0.108653
\(291\) 0 0
\(292\) −120777. −0.0828949
\(293\) −1.83223e6 −1.24684 −0.623421 0.781886i \(-0.714258\pi\)
−0.623421 + 0.781886i \(0.714258\pi\)
\(294\) 0 0
\(295\) −1.51140e6 −1.01117
\(296\) −2.00783e6 −1.33198
\(297\) 0 0
\(298\) −702800. −0.458449
\(299\) 261596. 0.169221
\(300\) 0 0
\(301\) 0 0
\(302\) 565047. 0.356506
\(303\) 0 0
\(304\) 1.47085e6 0.912817
\(305\) −168348. −0.103623
\(306\) 0 0
\(307\) −717638. −0.434569 −0.217285 0.976108i \(-0.569720\pi\)
−0.217285 + 0.976108i \(0.569720\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.30179e6 −0.769370
\(311\) 856892. 0.502372 0.251186 0.967939i \(-0.419179\pi\)
0.251186 + 0.967939i \(0.419179\pi\)
\(312\) 0 0
\(313\) 1.61699e6 0.932924 0.466462 0.884541i \(-0.345528\pi\)
0.466462 + 0.884541i \(0.345528\pi\)
\(314\) −193900. −0.110982
\(315\) 0 0
\(316\) −222512. −0.125354
\(317\) 2.26559e6 1.26629 0.633145 0.774033i \(-0.281764\pi\)
0.633145 + 0.774033i \(0.281764\pi\)
\(318\) 0 0
\(319\) 52769.8 0.0290341
\(320\) −1.52312e6 −0.831495
\(321\) 0 0
\(322\) 0 0
\(323\) −3.68495e6 −1.96528
\(324\) 0 0
\(325\) 869721. 0.456743
\(326\) −1.08214e6 −0.563950
\(327\) 0 0
\(328\) −684740. −0.351432
\(329\) 0 0
\(330\) 0 0
\(331\) 709650. 0.356020 0.178010 0.984029i \(-0.443034\pi\)
0.178010 + 0.984029i \(0.443034\pi\)
\(332\) −151503. −0.0754355
\(333\) 0 0
\(334\) 611943. 0.300155
\(335\) −651086. −0.316976
\(336\) 0 0
\(337\) 603572. 0.289504 0.144752 0.989468i \(-0.453762\pi\)
0.144752 + 0.989468i \(0.453762\pi\)
\(338\) 145023. 0.0690468
\(339\) 0 0
\(340\) 509570. 0.239060
\(341\) −441459. −0.205591
\(342\) 0 0
\(343\) 0 0
\(344\) 2.81576e6 1.28292
\(345\) 0 0
\(346\) −3.62357e6 −1.62722
\(347\) −1.75731e6 −0.783474 −0.391737 0.920077i \(-0.628126\pi\)
−0.391737 + 0.920077i \(0.628126\pi\)
\(348\) 0 0
\(349\) 391875. 0.172220 0.0861102 0.996286i \(-0.472556\pi\)
0.0861102 + 0.996286i \(0.472556\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −158529. −0.0681948
\(353\) −492407. −0.210323 −0.105162 0.994455i \(-0.533536\pi\)
−0.105162 + 0.994455i \(0.533536\pi\)
\(354\) 0 0
\(355\) 509492. 0.214569
\(356\) −433087. −0.181113
\(357\) 0 0
\(358\) 3.81074e6 1.57145
\(359\) 3.77032e6 1.54398 0.771991 0.635634i \(-0.219261\pi\)
0.771991 + 0.635634i \(0.219261\pi\)
\(360\) 0 0
\(361\) 1.00185e6 0.404607
\(362\) 3.17021e6 1.27150
\(363\) 0 0
\(364\) 0 0
\(365\) −819367. −0.321919
\(366\) 0 0
\(367\) −2.19768e6 −0.851726 −0.425863 0.904788i \(-0.640030\pi\)
−0.425863 + 0.904788i \(0.640030\pi\)
\(368\) 326289. 0.125598
\(369\) 0 0
\(370\) −2.20049e6 −0.835632
\(371\) 0 0
\(372\) 0 0
\(373\) −1.65636e6 −0.616427 −0.308213 0.951317i \(-0.599731\pi\)
−0.308213 + 0.951317i \(0.599731\pi\)
\(374\) −724076. −0.267673
\(375\) 0 0
\(376\) −4.15583e6 −1.51596
\(377\) 462814. 0.167708
\(378\) 0 0
\(379\) −2.82050e6 −1.00862 −0.504310 0.863523i \(-0.668253\pi\)
−0.504310 + 0.863523i \(0.668253\pi\)
\(380\) −480944. −0.170858
\(381\) 0 0
\(382\) −2.12320e6 −0.744446
\(383\) −3.24845e6 −1.13156 −0.565781 0.824555i \(-0.691425\pi\)
−0.565781 + 0.824555i \(0.691425\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.91729e6 1.33819
\(387\) 0 0
\(388\) −652055. −0.219890
\(389\) −4.65348e6 −1.55921 −0.779604 0.626273i \(-0.784580\pi\)
−0.779604 + 0.626273i \(0.784580\pi\)
\(390\) 0 0
\(391\) −817461. −0.270411
\(392\) 0 0
\(393\) 0 0
\(394\) 2.43563e6 0.790442
\(395\) −1.50955e6 −0.486805
\(396\) 0 0
\(397\) −1.16361e6 −0.370536 −0.185268 0.982688i \(-0.559315\pi\)
−0.185268 + 0.982688i \(0.559315\pi\)
\(398\) −2.17891e6 −0.689495
\(399\) 0 0
\(400\) 1.08480e6 0.339001
\(401\) −322380. −0.100117 −0.0500584 0.998746i \(-0.515941\pi\)
−0.0500584 + 0.998746i \(0.515941\pi\)
\(402\) 0 0
\(403\) −3.87179e6 −1.18754
\(404\) 224806. 0.0685259
\(405\) 0 0
\(406\) 0 0
\(407\) −746226. −0.223298
\(408\) 0 0
\(409\) 1.38690e6 0.409956 0.204978 0.978767i \(-0.434288\pi\)
0.204978 + 0.978767i \(0.434288\pi\)
\(410\) −750443. −0.220474
\(411\) 0 0
\(412\) −397800. −0.115457
\(413\) 0 0
\(414\) 0 0
\(415\) −1.02781e6 −0.292950
\(416\) −1.39036e6 −0.393909
\(417\) 0 0
\(418\) 683400. 0.191309
\(419\) −4.90871e6 −1.36594 −0.682971 0.730446i \(-0.739312\pi\)
−0.682971 + 0.730446i \(0.739312\pi\)
\(420\) 0 0
\(421\) 2.43924e6 0.670733 0.335367 0.942088i \(-0.391140\pi\)
0.335367 + 0.942088i \(0.391140\pi\)
\(422\) −2.99143e6 −0.817707
\(423\) 0 0
\(424\) 2.44024e6 0.659202
\(425\) −2.71779e6 −0.729865
\(426\) 0 0
\(427\) 0 0
\(428\) 407206. 0.107450
\(429\) 0 0
\(430\) 3.08594e6 0.804852
\(431\) 5.22752e6 1.35551 0.677755 0.735288i \(-0.262953\pi\)
0.677755 + 0.735288i \(0.262953\pi\)
\(432\) 0 0
\(433\) −2.63022e6 −0.674174 −0.337087 0.941473i \(-0.609442\pi\)
−0.337087 + 0.941473i \(0.609442\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 233908. 0.0589289
\(437\) 771539. 0.193266
\(438\) 0 0
\(439\) 2.55412e6 0.632527 0.316264 0.948671i \(-0.397572\pi\)
0.316264 + 0.948671i \(0.397572\pi\)
\(440\) −584990. −0.144051
\(441\) 0 0
\(442\) −6.35046e6 −1.54614
\(443\) −3.83900e6 −0.929414 −0.464707 0.885465i \(-0.653840\pi\)
−0.464707 + 0.885465i \(0.653840\pi\)
\(444\) 0 0
\(445\) −2.93811e6 −0.703345
\(446\) −1.84631e6 −0.439509
\(447\) 0 0
\(448\) 0 0
\(449\) −1.49369e6 −0.349658 −0.174829 0.984599i \(-0.555937\pi\)
−0.174829 + 0.984599i \(0.555937\pi\)
\(450\) 0 0
\(451\) −254489. −0.0589152
\(452\) 763307. 0.175733
\(453\) 0 0
\(454\) 4.28498e6 0.975684
\(455\) 0 0
\(456\) 0 0
\(457\) −2.16221e6 −0.484293 −0.242146 0.970240i \(-0.577851\pi\)
−0.242146 + 0.970240i \(0.577851\pi\)
\(458\) −2.89040e6 −0.643863
\(459\) 0 0
\(460\) −106692. −0.0235091
\(461\) 6.11949e6 1.34111 0.670553 0.741862i \(-0.266057\pi\)
0.670553 + 0.741862i \(0.266057\pi\)
\(462\) 0 0
\(463\) 3.93615e6 0.853335 0.426667 0.904409i \(-0.359687\pi\)
0.426667 + 0.904409i \(0.359687\pi\)
\(464\) 577268. 0.124475
\(465\) 0 0
\(466\) 5.36999e6 1.14554
\(467\) 5.47044e6 1.16073 0.580363 0.814358i \(-0.302911\pi\)
0.580363 + 0.814358i \(0.302911\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.55459e6 −0.951054
\(471\) 0 0
\(472\) −7.00949e6 −1.44821
\(473\) 1.04650e6 0.215073
\(474\) 0 0
\(475\) 2.56511e6 0.521641
\(476\) 0 0
\(477\) 0 0
\(478\) −4.33923e6 −0.868646
\(479\) 6.66289e6 1.32686 0.663428 0.748240i \(-0.269101\pi\)
0.663428 + 0.748240i \(0.269101\pi\)
\(480\) 0 0
\(481\) −6.54473e6 −1.28982
\(482\) −1.97663e6 −0.387531
\(483\) 0 0
\(484\) 960916. 0.186454
\(485\) −4.42362e6 −0.853931
\(486\) 0 0
\(487\) 9.53693e6 1.82216 0.911079 0.412232i \(-0.135251\pi\)
0.911079 + 0.412232i \(0.135251\pi\)
\(488\) −780755. −0.148411
\(489\) 0 0
\(490\) 0 0
\(491\) 8.19294e6 1.53369 0.766843 0.641835i \(-0.221827\pi\)
0.766843 + 0.641835i \(0.221827\pi\)
\(492\) 0 0
\(493\) −1.44625e6 −0.267994
\(494\) 5.99372e6 1.10504
\(495\) 0 0
\(496\) −4.82928e6 −0.881411
\(497\) 0 0
\(498\) 0 0
\(499\) −4.31437e6 −0.775650 −0.387825 0.921733i \(-0.626774\pi\)
−0.387825 + 0.921733i \(0.626774\pi\)
\(500\) −1.16062e6 −0.207618
\(501\) 0 0
\(502\) 4.26610e6 0.755566
\(503\) 1.04015e7 1.83306 0.916529 0.399968i \(-0.130979\pi\)
0.916529 + 0.399968i \(0.130979\pi\)
\(504\) 0 0
\(505\) 1.52511e6 0.266117
\(506\) 151604. 0.0263229
\(507\) 0 0
\(508\) −793649. −0.136449
\(509\) 3.09396e6 0.529322 0.264661 0.964342i \(-0.414740\pi\)
0.264661 + 0.964342i \(0.414740\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6.63004e6 −1.11774
\(513\) 0 0
\(514\) −1.48410e6 −0.247773
\(515\) −2.69872e6 −0.448374
\(516\) 0 0
\(517\) −1.54455e6 −0.254141
\(518\) 0 0
\(519\) 0 0
\(520\) −5.13061e6 −0.832072
\(521\) 7.60175e6 1.22693 0.613464 0.789723i \(-0.289776\pi\)
0.613464 + 0.789723i \(0.289776\pi\)
\(522\) 0 0
\(523\) 4.75669e6 0.760415 0.380208 0.924901i \(-0.375853\pi\)
0.380208 + 0.924901i \(0.375853\pi\)
\(524\) −911895. −0.145083
\(525\) 0 0
\(526\) 1.46635e6 0.231086
\(527\) 1.20989e7 1.89767
\(528\) 0 0
\(529\) −6.26519e6 −0.973408
\(530\) 2.67439e6 0.413557
\(531\) 0 0
\(532\) 0 0
\(533\) −2.23198e6 −0.340308
\(534\) 0 0
\(535\) 2.76253e6 0.417275
\(536\) −3.01958e6 −0.453978
\(537\) 0 0
\(538\) 1.31832e6 0.196365
\(539\) 0 0
\(540\) 0 0
\(541\) 1.10052e7 1.61661 0.808305 0.588764i \(-0.200385\pi\)
0.808305 + 0.588764i \(0.200385\pi\)
\(542\) −1.11593e7 −1.63169
\(543\) 0 0
\(544\) 4.34474e6 0.629458
\(545\) 1.58686e6 0.228848
\(546\) 0 0
\(547\) −4.46311e6 −0.637778 −0.318889 0.947792i \(-0.603310\pi\)
−0.318889 + 0.947792i \(0.603310\pi\)
\(548\) −562033. −0.0799485
\(549\) 0 0
\(550\) 504032. 0.0710480
\(551\) 1.36500e6 0.191538
\(552\) 0 0
\(553\) 0 0
\(554\) 645463. 0.0893505
\(555\) 0 0
\(556\) 2.06334e6 0.283063
\(557\) 6.45222e6 0.881194 0.440597 0.897705i \(-0.354767\pi\)
0.440597 + 0.897705i \(0.354767\pi\)
\(558\) 0 0
\(559\) 9.17823e6 1.24231
\(560\) 0 0
\(561\) 0 0
\(562\) 1.13223e7 1.51215
\(563\) −1.74748e6 −0.232349 −0.116175 0.993229i \(-0.537063\pi\)
−0.116175 + 0.993229i \(0.537063\pi\)
\(564\) 0 0
\(565\) 5.17836e6 0.682450
\(566\) 6.04313e6 0.792905
\(567\) 0 0
\(568\) 2.36290e6 0.307308
\(569\) −512789. −0.0663985 −0.0331992 0.999449i \(-0.510570\pi\)
−0.0331992 + 0.999449i \(0.510570\pi\)
\(570\) 0 0
\(571\) 5.22364e6 0.670475 0.335238 0.942134i \(-0.391183\pi\)
0.335238 + 0.942134i \(0.391183\pi\)
\(572\) −281073. −0.0359194
\(573\) 0 0
\(574\) 0 0
\(575\) 569038. 0.0717748
\(576\) 0 0
\(577\) 6.63973e6 0.830254 0.415127 0.909763i \(-0.363737\pi\)
0.415127 + 0.909763i \(0.363737\pi\)
\(578\) 1.26277e7 1.57219
\(579\) 0 0
\(580\) −188758. −0.0232989
\(581\) 0 0
\(582\) 0 0
\(583\) 906935. 0.110511
\(584\) −3.80002e6 −0.461056
\(585\) 0 0
\(586\) −9.31280e6 −1.12030
\(587\) −774096. −0.0927256 −0.0463628 0.998925i \(-0.514763\pi\)
−0.0463628 + 0.998925i \(0.514763\pi\)
\(588\) 0 0
\(589\) −1.14193e7 −1.35628
\(590\) −7.68207e6 −0.908549
\(591\) 0 0
\(592\) −8.16324e6 −0.957322
\(593\) 1.43756e7 1.67876 0.839379 0.543546i \(-0.182919\pi\)
0.839379 + 0.543546i \(0.182919\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 852515. 0.0983075
\(597\) 0 0
\(598\) 1.32963e6 0.152047
\(599\) −1.20835e7 −1.37602 −0.688010 0.725701i \(-0.741516\pi\)
−0.688010 + 0.725701i \(0.741516\pi\)
\(600\) 0 0
\(601\) 5.75607e6 0.650040 0.325020 0.945707i \(-0.394629\pi\)
0.325020 + 0.945707i \(0.394629\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −685416. −0.0764473
\(605\) 6.51897e6 0.724086
\(606\) 0 0
\(607\) 4.20121e6 0.462810 0.231405 0.972858i \(-0.425668\pi\)
0.231405 + 0.972858i \(0.425668\pi\)
\(608\) −4.10067e6 −0.449879
\(609\) 0 0
\(610\) −855671. −0.0931070
\(611\) −1.35463e7 −1.46798
\(612\) 0 0
\(613\) −2.64543e6 −0.284344 −0.142172 0.989842i \(-0.545409\pi\)
−0.142172 + 0.989842i \(0.545409\pi\)
\(614\) −3.64758e6 −0.390467
\(615\) 0 0
\(616\) 0 0
\(617\) −6.43533e6 −0.680546 −0.340273 0.940327i \(-0.610520\pi\)
−0.340273 + 0.940327i \(0.610520\pi\)
\(618\) 0 0
\(619\) 1.41177e7 1.48094 0.740469 0.672090i \(-0.234603\pi\)
0.740469 + 0.672090i \(0.234603\pi\)
\(620\) 1.57910e6 0.164980
\(621\) 0 0
\(622\) 4.35538e6 0.451388
\(623\) 0 0
\(624\) 0 0
\(625\) −3.57548e6 −0.366129
\(626\) 8.21877e6 0.838246
\(627\) 0 0
\(628\) 235206. 0.0237984
\(629\) 2.04516e7 2.06111
\(630\) 0 0
\(631\) −4.70856e6 −0.470777 −0.235388 0.971901i \(-0.575636\pi\)
−0.235388 + 0.971901i \(0.575636\pi\)
\(632\) −7.00092e6 −0.697208
\(633\) 0 0
\(634\) 1.15155e7 1.13778
\(635\) −5.38421e6 −0.529892
\(636\) 0 0
\(637\) 0 0
\(638\) 268217. 0.0260876
\(639\) 0 0
\(640\) −4.79855e6 −0.463085
\(641\) 1.04174e7 1.00141 0.500707 0.865617i \(-0.333073\pi\)
0.500707 + 0.865617i \(0.333073\pi\)
\(642\) 0 0
\(643\) 1.27284e7 1.21407 0.607037 0.794674i \(-0.292358\pi\)
0.607037 + 0.794674i \(0.292358\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.87297e7 −1.76584
\(647\) 1.61348e7 1.51531 0.757657 0.652653i \(-0.226344\pi\)
0.757657 + 0.652653i \(0.226344\pi\)
\(648\) 0 0
\(649\) −2.60513e6 −0.242783
\(650\) 4.42058e6 0.410390
\(651\) 0 0
\(652\) 1.31267e6 0.120931
\(653\) 1.50295e7 1.37931 0.689654 0.724139i \(-0.257763\pi\)
0.689654 + 0.724139i \(0.257763\pi\)
\(654\) 0 0
\(655\) −6.18640e6 −0.563423
\(656\) −2.78395e6 −0.252581
\(657\) 0 0
\(658\) 0 0
\(659\) −1.67927e7 −1.50628 −0.753140 0.657860i \(-0.771462\pi\)
−0.753140 + 0.657860i \(0.771462\pi\)
\(660\) 0 0
\(661\) 1.08540e7 0.966246 0.483123 0.875552i \(-0.339502\pi\)
0.483123 + 0.875552i \(0.339502\pi\)
\(662\) 3.60698e6 0.319889
\(663\) 0 0
\(664\) −4.76675e6 −0.419567
\(665\) 0 0
\(666\) 0 0
\(667\) 302809. 0.0263544
\(668\) −742303. −0.0643636
\(669\) 0 0
\(670\) −3.30932e6 −0.284807
\(671\) −290174. −0.0248801
\(672\) 0 0
\(673\) 1.23697e7 1.05274 0.526371 0.850255i \(-0.323552\pi\)
0.526371 + 0.850255i \(0.323552\pi\)
\(674\) 3.06781e6 0.260123
\(675\) 0 0
\(676\) −175916. −0.0148060
\(677\) 1.00501e6 0.0842746 0.0421373 0.999112i \(-0.486583\pi\)
0.0421373 + 0.999112i \(0.486583\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.60326e7 1.32963
\(681\) 0 0
\(682\) −2.24383e6 −0.184727
\(683\) −1.87019e6 −0.153403 −0.0767014 0.997054i \(-0.524439\pi\)
−0.0767014 + 0.997054i \(0.524439\pi\)
\(684\) 0 0
\(685\) −3.81290e6 −0.310476
\(686\) 0 0
\(687\) 0 0
\(688\) 1.14480e7 0.922060
\(689\) 7.95421e6 0.638336
\(690\) 0 0
\(691\) 1.93867e7 1.54457 0.772286 0.635275i \(-0.219113\pi\)
0.772286 + 0.635275i \(0.219113\pi\)
\(692\) 4.39549e6 0.348933
\(693\) 0 0
\(694\) −8.93199e6 −0.703963
\(695\) 1.39979e7 1.09926
\(696\) 0 0
\(697\) 6.97469e6 0.543805
\(698\) 1.99181e6 0.154742
\(699\) 0 0
\(700\) 0 0
\(701\) −1.17488e7 −0.903024 −0.451512 0.892265i \(-0.649115\pi\)
−0.451512 + 0.892265i \(0.649115\pi\)
\(702\) 0 0
\(703\) −1.93027e7 −1.47309
\(704\) −2.62534e6 −0.199643
\(705\) 0 0
\(706\) −2.50279e6 −0.188978
\(707\) 0 0
\(708\) 0 0
\(709\) 1.67948e7 1.25475 0.627377 0.778716i \(-0.284129\pi\)
0.627377 + 0.778716i \(0.284129\pi\)
\(710\) 2.58963e6 0.192793
\(711\) 0 0
\(712\) −1.36262e7 −1.00734
\(713\) −2.53322e6 −0.186616
\(714\) 0 0
\(715\) −1.90683e6 −0.139491
\(716\) −4.62253e6 −0.336974
\(717\) 0 0
\(718\) 1.91636e7 1.38729
\(719\) 1.65130e7 1.19126 0.595628 0.803261i \(-0.296903\pi\)
0.595628 + 0.803261i \(0.296903\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5.09215e6 0.363545
\(723\) 0 0
\(724\) −3.84555e6 −0.272654
\(725\) 1.00674e6 0.0711331
\(726\) 0 0
\(727\) −1.25756e6 −0.0882453 −0.0441227 0.999026i \(-0.514049\pi\)
−0.0441227 + 0.999026i \(0.514049\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.16465e6 −0.289248
\(731\) −2.86810e7 −1.98518
\(732\) 0 0
\(733\) −1.98332e6 −0.136343 −0.0681716 0.997674i \(-0.521717\pi\)
−0.0681716 + 0.997674i \(0.521717\pi\)
\(734\) −1.11703e7 −0.765288
\(735\) 0 0
\(736\) −909684. −0.0619007
\(737\) −1.12225e6 −0.0761063
\(738\) 0 0
\(739\) −2.38807e7 −1.60856 −0.804278 0.594253i \(-0.797448\pi\)
−0.804278 + 0.594253i \(0.797448\pi\)
\(740\) 2.66925e6 0.179189
\(741\) 0 0
\(742\) 0 0
\(743\) 1.90819e7 1.26809 0.634043 0.773298i \(-0.281394\pi\)
0.634043 + 0.773298i \(0.281394\pi\)
\(744\) 0 0
\(745\) 5.78356e6 0.381773
\(746\) −8.41886e6 −0.553868
\(747\) 0 0
\(748\) 878323. 0.0573985
\(749\) 0 0
\(750\) 0 0
\(751\) −3.75805e6 −0.243144 −0.121572 0.992583i \(-0.538794\pi\)
−0.121572 + 0.992583i \(0.538794\pi\)
\(752\) −1.68963e7 −1.08955
\(753\) 0 0
\(754\) 2.35238e6 0.150688
\(755\) −4.64994e6 −0.296880
\(756\) 0 0
\(757\) 1.69904e7 1.07761 0.538807 0.842429i \(-0.318875\pi\)
0.538807 + 0.842429i \(0.318875\pi\)
\(758\) −1.43359e7 −0.906259
\(759\) 0 0
\(760\) −1.51320e7 −0.950302
\(761\) −2.23998e7 −1.40211 −0.701056 0.713106i \(-0.747288\pi\)
−0.701056 + 0.713106i \(0.747288\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.57550e6 0.159635
\(765\) 0 0
\(766\) −1.65111e7 −1.01672
\(767\) −2.28481e7 −1.40237
\(768\) 0 0
\(769\) −1.87866e7 −1.14560 −0.572799 0.819696i \(-0.694142\pi\)
−0.572799 + 0.819696i \(0.694142\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.75177e6 −0.286954
\(773\) −9.30837e6 −0.560306 −0.280153 0.959955i \(-0.590385\pi\)
−0.280153 + 0.959955i \(0.590385\pi\)
\(774\) 0 0
\(775\) −8.42212e6 −0.503694
\(776\) −2.05156e7 −1.22301
\(777\) 0 0
\(778\) −2.36525e7 −1.40097
\(779\) −6.58288e6 −0.388662
\(780\) 0 0
\(781\) 878189. 0.0515182
\(782\) −4.15496e6 −0.242969
\(783\) 0 0
\(784\) 0 0
\(785\) 1.59566e6 0.0924201
\(786\) 0 0
\(787\) 1.73427e7 0.998111 0.499056 0.866570i \(-0.333680\pi\)
0.499056 + 0.866570i \(0.333680\pi\)
\(788\) −2.95448e6 −0.169498
\(789\) 0 0
\(790\) −7.67268e6 −0.437401
\(791\) 0 0
\(792\) 0 0
\(793\) −2.54495e6 −0.143713
\(794\) −5.91434e6 −0.332932
\(795\) 0 0
\(796\) 2.64307e6 0.147852
\(797\) −3.10445e7 −1.73117 −0.865584 0.500764i \(-0.833052\pi\)
−0.865584 + 0.500764i \(0.833052\pi\)
\(798\) 0 0
\(799\) 4.23309e7 2.34580
\(800\) −3.02439e6 −0.167076
\(801\) 0 0
\(802\) −1.63858e6 −0.0899563
\(803\) −1.41231e6 −0.0772930
\(804\) 0 0
\(805\) 0 0
\(806\) −1.96794e7 −1.06702
\(807\) 0 0
\(808\) 7.07309e6 0.381137
\(809\) 2.47038e7 1.32707 0.663533 0.748147i \(-0.269056\pi\)
0.663533 + 0.748147i \(0.269056\pi\)
\(810\) 0 0
\(811\) 8.42005e6 0.449534 0.224767 0.974413i \(-0.427838\pi\)
0.224767 + 0.974413i \(0.427838\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.79289e6 −0.200636
\(815\) 8.90529e6 0.469628
\(816\) 0 0
\(817\) 2.70698e7 1.41883
\(818\) 7.04930e6 0.368352
\(819\) 0 0
\(820\) 910307. 0.0472774
\(821\) 2.58827e7 1.34014 0.670071 0.742297i \(-0.266264\pi\)
0.670071 + 0.742297i \(0.266264\pi\)
\(822\) 0 0
\(823\) 1.72004e7 0.885195 0.442597 0.896720i \(-0.354057\pi\)
0.442597 + 0.896720i \(0.354057\pi\)
\(824\) −1.25160e7 −0.642167
\(825\) 0 0
\(826\) 0 0
\(827\) 2.40337e7 1.22196 0.610979 0.791647i \(-0.290776\pi\)
0.610979 + 0.791647i \(0.290776\pi\)
\(828\) 0 0
\(829\) −3.24736e7 −1.64113 −0.820567 0.571550i \(-0.806342\pi\)
−0.820567 + 0.571550i \(0.806342\pi\)
\(830\) −5.22413e6 −0.263220
\(831\) 0 0
\(832\) −2.30254e7 −1.15318
\(833\) 0 0
\(834\) 0 0
\(835\) −5.03587e6 −0.249953
\(836\) −828983. −0.0410232
\(837\) 0 0
\(838\) −2.49498e7 −1.22732
\(839\) 1.24404e7 0.610139 0.305069 0.952330i \(-0.401320\pi\)
0.305069 + 0.952330i \(0.401320\pi\)
\(840\) 0 0
\(841\) −1.99754e7 −0.973881
\(842\) 1.23981e7 0.602663
\(843\) 0 0
\(844\) 3.62868e6 0.175345
\(845\) −1.19344e6 −0.0574986
\(846\) 0 0
\(847\) 0 0
\(848\) 9.92129e6 0.473782
\(849\) 0 0
\(850\) −1.38139e7 −0.655794
\(851\) −4.28206e6 −0.202689
\(852\) 0 0
\(853\) −999355. −0.0470270 −0.0235135 0.999724i \(-0.507485\pi\)
−0.0235135 + 0.999724i \(0.507485\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.28119e7 0.597628
\(857\) 2.64465e7 1.23003 0.615016 0.788514i \(-0.289149\pi\)
0.615016 + 0.788514i \(0.289149\pi\)
\(858\) 0 0
\(859\) −2.86716e7 −1.32577 −0.662887 0.748719i \(-0.730669\pi\)
−0.662887 + 0.748719i \(0.730669\pi\)
\(860\) −3.74332e6 −0.172588
\(861\) 0 0
\(862\) 2.65703e7 1.21794
\(863\) −4.08173e6 −0.186560 −0.0932798 0.995640i \(-0.529735\pi\)
−0.0932798 + 0.995640i \(0.529735\pi\)
\(864\) 0 0
\(865\) 2.98195e7 1.35506
\(866\) −1.33688e7 −0.605755
\(867\) 0 0
\(868\) 0 0
\(869\) −2.60195e6 −0.116882
\(870\) 0 0
\(871\) −9.84261e6 −0.439608
\(872\) 7.35946e6 0.327759
\(873\) 0 0
\(874\) 3.92155e6 0.173652
\(875\) 0 0
\(876\) 0 0
\(877\) −2.82405e7 −1.23986 −0.619931 0.784657i \(-0.712839\pi\)
−0.619931 + 0.784657i \(0.712839\pi\)
\(878\) 1.29820e7 0.568335
\(879\) 0 0
\(880\) −2.37839e6 −0.103532
\(881\) −1.61480e7 −0.700936 −0.350468 0.936575i \(-0.613978\pi\)
−0.350468 + 0.936575i \(0.613978\pi\)
\(882\) 0 0
\(883\) −3.86021e7 −1.66613 −0.833065 0.553174i \(-0.813416\pi\)
−0.833065 + 0.553174i \(0.813416\pi\)
\(884\) 7.70328e6 0.331547
\(885\) 0 0
\(886\) −1.95127e7 −0.835091
\(887\) −7.29088e6 −0.311151 −0.155575 0.987824i \(-0.549723\pi\)
−0.155575 + 0.987824i \(0.549723\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.49337e7 −0.631965
\(891\) 0 0
\(892\) 2.23962e6 0.0942460
\(893\) −3.99529e7 −1.67656
\(894\) 0 0
\(895\) −3.13598e7 −1.30862
\(896\) 0 0
\(897\) 0 0
\(898\) −7.59205e6 −0.314173
\(899\) −4.48176e6 −0.184948
\(900\) 0 0
\(901\) −2.48561e7 −1.02005
\(902\) −1.29351e6 −0.0529361
\(903\) 0 0
\(904\) 2.40160e7 0.977415
\(905\) −2.60886e7 −1.05884
\(906\) 0 0
\(907\) −3.73335e7 −1.50689 −0.753443 0.657514i \(-0.771608\pi\)
−0.753443 + 0.657514i \(0.771608\pi\)
\(908\) −5.19779e6 −0.209221
\(909\) 0 0
\(910\) 0 0
\(911\) 2475.17 9.88120e−5 0 4.94060e−5 1.00000i \(-0.499984\pi\)
4.94060e−5 1.00000i \(0.499984\pi\)
\(912\) 0 0
\(913\) −1.77160e6 −0.0703377
\(914\) −1.09900e7 −0.435144
\(915\) 0 0
\(916\) 3.50613e6 0.138067
\(917\) 0 0
\(918\) 0 0
\(919\) −4.48238e6 −0.175073 −0.0875366 0.996161i \(-0.527899\pi\)
−0.0875366 + 0.996161i \(0.527899\pi\)
\(920\) −3.35684e6 −0.130756
\(921\) 0 0
\(922\) 3.11039e7 1.20500
\(923\) 7.70210e6 0.297581
\(924\) 0 0
\(925\) −1.42364e7 −0.547075
\(926\) 2.00065e7 0.766733
\(927\) 0 0
\(928\) −1.60941e6 −0.0613473
\(929\) −2.12859e7 −0.809193 −0.404596 0.914495i \(-0.632588\pi\)
−0.404596 + 0.914495i \(0.632588\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.51394e6 −0.245643
\(933\) 0 0
\(934\) 2.78049e7 1.04293
\(935\) 5.95865e6 0.222904
\(936\) 0 0
\(937\) 6.79757e6 0.252932 0.126466 0.991971i \(-0.459636\pi\)
0.126466 + 0.991971i \(0.459636\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 5.52484e6 0.203939
\(941\) −4.90883e7 −1.80719 −0.903595 0.428388i \(-0.859082\pi\)
−0.903595 + 0.428388i \(0.859082\pi\)
\(942\) 0 0
\(943\) −1.46033e6 −0.0534776
\(944\) −2.84985e7 −1.04086
\(945\) 0 0
\(946\) 5.31910e6 0.193246
\(947\) −2.45484e7 −0.889505 −0.444753 0.895653i \(-0.646708\pi\)
−0.444753 + 0.895653i \(0.646708\pi\)
\(948\) 0 0
\(949\) −1.23865e7 −0.446462
\(950\) 1.30378e7 0.468702
\(951\) 0 0
\(952\) 0 0
\(953\) 513120. 0.0183015 0.00915075 0.999958i \(-0.497087\pi\)
0.00915075 + 0.999958i \(0.497087\pi\)
\(954\) 0 0
\(955\) 1.74725e7 0.619935
\(956\) 5.26360e6 0.186268
\(957\) 0 0
\(958\) 3.38659e7 1.19220
\(959\) 0 0
\(960\) 0 0
\(961\) 8.86412e6 0.309619
\(962\) −3.32653e7 −1.15892
\(963\) 0 0
\(964\) 2.39770e6 0.0831002
\(965\) −3.22366e7 −1.11437
\(966\) 0 0
\(967\) 3.34818e7 1.15144 0.575722 0.817645i \(-0.304721\pi\)
0.575722 + 0.817645i \(0.304721\pi\)
\(968\) 3.02334e7 1.03705
\(969\) 0 0
\(970\) −2.24842e7 −0.767269
\(971\) −4.76036e6 −0.162029 −0.0810143 0.996713i \(-0.525816\pi\)
−0.0810143 + 0.996713i \(0.525816\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 4.84739e7 1.63723
\(975\) 0 0
\(976\) −3.17431e6 −0.106666
\(977\) −2.87338e7 −0.963067 −0.481534 0.876428i \(-0.659920\pi\)
−0.481534 + 0.876428i \(0.659920\pi\)
\(978\) 0 0
\(979\) −5.06430e6 −0.168874
\(980\) 0 0
\(981\) 0 0
\(982\) 4.16428e7 1.37804
\(983\) 4.97072e7 1.64072 0.820362 0.571845i \(-0.193772\pi\)
0.820362 + 0.571845i \(0.193772\pi\)
\(984\) 0 0
\(985\) −2.00435e7 −0.658239
\(986\) −7.35092e6 −0.240796
\(987\) 0 0
\(988\) −7.27054e6 −0.236960
\(989\) 6.00511e6 0.195223
\(990\) 0 0
\(991\) −2.91066e6 −0.0941471 −0.0470736 0.998891i \(-0.514990\pi\)
−0.0470736 + 0.998891i \(0.514990\pi\)
\(992\) 1.34639e7 0.434401
\(993\) 0 0
\(994\) 0 0
\(995\) 1.79309e7 0.574176
\(996\) 0 0
\(997\) 1.43353e7 0.456740 0.228370 0.973574i \(-0.426660\pi\)
0.228370 + 0.973574i \(0.426660\pi\)
\(998\) −2.19289e7 −0.696933
\(999\) 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 441.6.a.n.1.2 2
3.2 odd 2 49.6.a.d.1.1 2
7.2 even 3 63.6.e.d.46.1 4
7.4 even 3 63.6.e.d.37.1 4
7.6 odd 2 441.6.a.m.1.2 2
12.11 even 2 784.6.a.ba.1.1 2
21.2 odd 6 7.6.c.a.4.2 yes 4
21.5 even 6 49.6.c.f.18.2 4
21.11 odd 6 7.6.c.a.2.2 4
21.17 even 6 49.6.c.f.30.2 4
21.20 even 2 49.6.a.e.1.1 2
84.11 even 6 112.6.i.c.65.2 4
84.23 even 6 112.6.i.c.81.2 4
84.83 odd 2 784.6.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.2 4 21.11 odd 6
7.6.c.a.4.2 yes 4 21.2 odd 6
49.6.a.d.1.1 2 3.2 odd 2
49.6.a.e.1.1 2 21.20 even 2
49.6.c.f.18.2 4 21.5 even 6
49.6.c.f.30.2 4 21.17 even 6
63.6.e.d.37.1 4 7.4 even 3
63.6.e.d.46.1 4 7.2 even 3
112.6.i.c.65.2 4 84.11 even 6
112.6.i.c.81.2 4 84.23 even 6
441.6.a.m.1.2 2 7.6 odd 2
441.6.a.n.1.2 2 1.1 even 1 trivial
784.6.a.t.1.2 2 84.83 odd 2
784.6.a.ba.1.1 2 12.11 even 2