Properties

Label 49.6.a.e.1.1
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
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] = [49,6,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(7.85880717084\)
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 1.1
Root \(-2.54138\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.08276 q^{2} -2.08276 q^{3} -6.16553 q^{4} -41.8276 q^{5} +10.5862 q^{6} +193.986 q^{8} -238.662 q^{9} +O(q^{10})\) \(q-5.08276 q^{2} -2.08276 q^{3} -6.16553 q^{4} -41.8276 q^{5} +10.5862 q^{6} +193.986 q^{8} -238.662 q^{9} +212.600 q^{10} +72.0965 q^{11} +12.8413 q^{12} +632.317 q^{13} +87.1170 q^{15} -788.689 q^{16} +1975.92 q^{17} +1213.06 q^{18} +1864.93 q^{19} +257.889 q^{20} -366.449 q^{22} +413.711 q^{23} -404.027 q^{24} -1375.45 q^{25} -3213.92 q^{26} +1003.19 q^{27} +731.934 q^{29} -442.795 q^{30} -6123.18 q^{31} -2198.84 q^{32} -150.160 q^{33} -10043.2 q^{34} +1471.48 q^{36} +10350.4 q^{37} -9478.97 q^{38} -1316.97 q^{39} -8113.99 q^{40} +3529.84 q^{41} -14515.2 q^{43} -444.513 q^{44} +9982.67 q^{45} -2102.79 q^{46} +21423.3 q^{47} +1642.65 q^{48} +6991.08 q^{50} -4115.38 q^{51} -3898.57 q^{52} +12579.5 q^{53} -5098.97 q^{54} -3015.62 q^{55} -3884.20 q^{57} -3720.25 q^{58} +36133.9 q^{59} -537.122 q^{60} -4024.80 q^{61} +31122.6 q^{62} +36414.2 q^{64} -26448.3 q^{65} +763.227 q^{66} +15565.9 q^{67} -12182.6 q^{68} -861.661 q^{69} +12180.8 q^{71} -46297.2 q^{72} -19589.1 q^{73} -52608.5 q^{74} +2864.74 q^{75} -11498.2 q^{76} +6693.83 q^{78} +36089.8 q^{79} +32989.0 q^{80} +55905.5 q^{81} -17941.3 q^{82} +24572.6 q^{83} -82648.2 q^{85} +73777.5 q^{86} -1524.44 q^{87} +13985.7 q^{88} +70243.3 q^{89} -50739.5 q^{90} -2550.74 q^{92} +12753.1 q^{93} -108890. q^{94} -78005.4 q^{95} +4579.66 q^{96} -105758. q^{97} -17206.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 38 q^{5} + 82 q^{6} + 96 q^{8} - 380 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 38 q^{5} + 82 q^{6} + 96 q^{8} - 380 q^{9} + 778 q^{10} + 424 q^{11} + 196 q^{12} + 924 q^{13} + 892 q^{15} - 2064 q^{16} + 2346 q^{17} + 212 q^{18} + 360 q^{19} + 1708 q^{20} + 2126 q^{22} - 12 q^{23} - 1392 q^{24} + 1872 q^{25} - 1148 q^{26} - 2872 q^{27} - 7052 q^{29} + 5258 q^{30} - 3548 q^{31} - 8096 q^{32} + 3398 q^{33} - 7422 q^{34} - 1096 q^{36} + 11090 q^{37} - 20138 q^{38} + 1624 q^{39} - 15936 q^{40} - 3500 q^{41} - 12680 q^{43} + 5948 q^{44} - 1300 q^{45} - 5118 q^{46} + 22956 q^{47} - 11216 q^{48} + 29992 q^{50} - 384 q^{51} + 1400 q^{52} + 3042 q^{53} - 32546 q^{54} + 25076 q^{55} - 19058 q^{57} - 58852 q^{58} + 65808 q^{59} + 14084 q^{60} + 42486 q^{61} + 49362 q^{62} + 35456 q^{64} - 3164 q^{65} + 25894 q^{66} + 42312 q^{67} - 5460 q^{68} - 5154 q^{69} - 2208 q^{71} - 32448 q^{72} + 50506 q^{73} - 47370 q^{74} + 35608 q^{75} - 38836 q^{76} + 27524 q^{78} + 9004 q^{79} - 68816 q^{80} + 51178 q^{81} - 67732 q^{82} + 104328 q^{83} - 53106 q^{85} + 86776 q^{86} - 80008 q^{87} - 20496 q^{88} + 26666 q^{89} - 130652 q^{90} - 10284 q^{92} + 38718 q^{93} - 98034 q^{94} - 198140 q^{95} - 54880 q^{96} - 209132 q^{97} - 66944 q^{99}+O(q^{100}) \) 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 \(-0.648311\pi\)
−0.449257 + 0.893403i \(0.648311\pi\)
\(3\) −2.08276 −0.133609 −0.0668046 0.997766i \(-0.521280\pi\)
−0.0668046 + 0.997766i \(0.521280\pi\)
\(4\) −6.16553 −0.192673
\(5\) −41.8276 −0.748235 −0.374118 0.927381i \(-0.622054\pi\)
−0.374118 + 0.927381i \(0.622054\pi\)
\(6\) 10.5862 0.120050
\(7\) 0 0
\(8\) 193.986 1.07163
\(9\) −238.662 −0.982149
\(10\) 212.600 0.672300
\(11\) 72.0965 0.179652 0.0898260 0.995957i \(-0.471369\pi\)
0.0898260 + 0.995957i \(0.471369\pi\)
\(12\) 12.8413 0.0257429
\(13\) 632.317 1.03771 0.518856 0.854862i \(-0.326358\pi\)
0.518856 + 0.854862i \(0.326358\pi\)
\(14\) 0 0
\(15\) 87.1170 0.0999712
\(16\) −788.689 −0.770205
\(17\) 1975.92 1.65824 0.829121 0.559069i \(-0.188841\pi\)
0.829121 + 0.559069i \(0.188841\pi\)
\(18\) 1213.06 0.882474
\(19\) 1864.93 1.18516 0.592581 0.805511i \(-0.298109\pi\)
0.592581 + 0.805511i \(0.298109\pi\)
\(20\) 257.889 0.144164
\(21\) 0 0
\(22\) −366.449 −0.161420
\(23\) 413.711 0.163071 0.0815356 0.996670i \(-0.474018\pi\)
0.0815356 + 0.996670i \(0.474018\pi\)
\(24\) −404.027 −0.143180
\(25\) −1375.45 −0.440144
\(26\) −3213.92 −0.932398
\(27\) 1003.19 0.264833
\(28\) 0 0
\(29\) 731.934 0.161613 0.0808066 0.996730i \(-0.474250\pi\)
0.0808066 + 0.996730i \(0.474250\pi\)
\(30\) −442.795 −0.0898255
\(31\) −6123.18 −1.14439 −0.572193 0.820119i \(-0.693907\pi\)
−0.572193 + 0.820119i \(0.693907\pi\)
\(32\) −2198.84 −0.379593
\(33\) −150.160 −0.0240032
\(34\) −10043.2 −1.48995
\(35\) 0 0
\(36\) 1471.48 0.189233
\(37\) 10350.4 1.24295 0.621473 0.783436i \(-0.286535\pi\)
0.621473 + 0.783436i \(0.286535\pi\)
\(38\) −9478.97 −1.06488
\(39\) −1316.97 −0.138648
\(40\) −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
\(45\) 9982.67 0.734878
\(46\) −2102.79 −0.146522
\(47\) 21423.3 1.41463 0.707314 0.706900i \(-0.249907\pi\)
0.707314 + 0.706900i \(0.249907\pi\)
\(48\) 1642.65 0.102906
\(49\) 0 0
\(50\) 6991.08 0.395475
\(51\) −4115.38 −0.221557
\(52\) −3898.57 −0.199939
\(53\) 12579.5 0.615138 0.307569 0.951526i \(-0.400485\pi\)
0.307569 + 0.951526i \(0.400485\pi\)
\(54\) −5098.97 −0.237957
\(55\) −3015.62 −0.134422
\(56\) 0 0
\(57\) −3884.20 −0.158349
\(58\) −3720.25 −0.145212
\(59\) 36133.9 1.35140 0.675702 0.737175i \(-0.263841\pi\)
0.675702 + 0.737175i \(0.263841\pi\)
\(60\) −537.122 −0.0192617
\(61\) −4024.80 −0.138490 −0.0692451 0.997600i \(-0.522059\pi\)
−0.0692451 + 0.997600i \(0.522059\pi\)
\(62\) 31122.6 1.02825
\(63\) 0 0
\(64\) 36414.2 1.11127
\(65\) −26448.3 −0.776453
\(66\) 763.227 0.0215672
\(67\) 15565.9 0.423632 0.211816 0.977310i \(-0.432062\pi\)
0.211816 + 0.977310i \(0.432062\pi\)
\(68\) −12182.6 −0.319498
\(69\) −861.661 −0.0217878
\(70\) 0 0
\(71\) 12180.8 0.286766 0.143383 0.989667i \(-0.454202\pi\)
0.143383 + 0.989667i \(0.454202\pi\)
\(72\) −46297.2 −1.05250
\(73\) −19589.1 −0.430237 −0.215119 0.976588i \(-0.569014\pi\)
−0.215119 + 0.976588i \(0.569014\pi\)
\(74\) −52608.5 −1.11680
\(75\) 2864.74 0.0588073
\(76\) −11498.2 −0.228348
\(77\) 0 0
\(78\) 6693.83 0.124577
\(79\) 36089.8 0.650604 0.325302 0.945610i \(-0.394534\pi\)
0.325302 + 0.945610i \(0.394534\pi\)
\(80\) 32989.0 0.576294
\(81\) 55905.5 0.946764
\(82\) −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
\(87\) −1524.44 −0.0215930
\(88\) 13985.7 0.192521
\(89\) 70243.3 0.940005 0.470002 0.882665i \(-0.344253\pi\)
0.470002 + 0.882665i \(0.344253\pi\)
\(90\) −50739.5 −0.660298
\(91\) 0 0
\(92\) −2550.74 −0.0314193
\(93\) 12753.1 0.152901
\(94\) −108890. −1.27106
\(95\) −78005.4 −0.886779
\(96\) 4579.66 0.0507172
\(97\) −105758. −1.14126 −0.570630 0.821207i \(-0.693301\pi\)
−0.570630 + 0.821207i \(0.693301\pi\)
\(98\) 0 0
\(99\) −17206.7 −0.176445
\(100\) 8480.37 0.0848037
\(101\) −36461.8 −0.355660 −0.177830 0.984061i \(-0.556908\pi\)
−0.177830 + 0.984061i \(0.556908\pi\)
\(102\) 20917.5 0.199072
\(103\) −64520.1 −0.599242 −0.299621 0.954058i \(-0.596860\pi\)
−0.299621 + 0.954058i \(0.596860\pi\)
\(104\) 122661. 1.11205
\(105\) 0 0
\(106\) −63938.4 −0.552710
\(107\) 66045.6 0.557679 0.278840 0.960338i \(-0.410050\pi\)
0.278840 + 0.960338i \(0.410050\pi\)
\(108\) −6185.18 −0.0510262
\(109\) −37938.0 −0.305850 −0.152925 0.988238i \(-0.548869\pi\)
−0.152925 + 0.988238i \(0.548869\pi\)
\(110\) 15327.7 0.120780
\(111\) −21557.4 −0.166069
\(112\) 0 0
\(113\) 123802. 0.912080 0.456040 0.889959i \(-0.349267\pi\)
0.456040 + 0.889959i \(0.349267\pi\)
\(114\) 19742.4 0.142278
\(115\) −17304.5 −0.122016
\(116\) −4512.76 −0.0311384
\(117\) −150910. −1.01919
\(118\) −183660. −1.21426
\(119\) 0 0
\(120\) 16899.5 0.107132
\(121\) −155853. −0.967725
\(122\) 20457.1 0.124435
\(123\) −7351.81 −0.0438159
\(124\) 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
\(129\) 30231.8 0.159952
\(130\) 134431. 0.697653
\(131\) 147902. 0.753003 0.376501 0.926416i \(-0.377127\pi\)
0.376501 + 0.926416i \(0.377127\pi\)
\(132\) 925.814 0.00462476
\(133\) 0 0
\(134\) −79118.0 −0.380639
\(135\) −41961.0 −0.198158
\(136\) 383302. 1.77703
\(137\) −91157.4 −0.414945 −0.207472 0.978241i \(-0.566524\pi\)
−0.207472 + 0.978241i \(0.566524\pi\)
\(138\) 4379.62 0.0195767
\(139\) 334657. 1.46914 0.734570 0.678533i \(-0.237384\pi\)
0.734570 + 0.678533i \(0.237384\pi\)
\(140\) 0 0
\(141\) −44619.7 −0.189007
\(142\) −61911.9 −0.257664
\(143\) 45587.8 0.186427
\(144\) 188230. 0.756455
\(145\) −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 \(-0.417887\pi\)
0.255115 + 0.966911i \(0.417887\pi\)
\(150\) −14560.8 −0.0528392
\(151\) 111169. 0.396773 0.198386 0.980124i \(-0.436430\pi\)
0.198386 + 0.980124i \(0.436430\pi\)
\(152\) 361770. 1.27006
\(153\) −471578. −1.62864
\(154\) 0 0
\(155\) 256118. 0.856270
\(156\) 8119.79 0.0267137
\(157\) 38148.5 0.123517 0.0617587 0.998091i \(-0.480329\pi\)
0.0617587 + 0.998091i \(0.480329\pi\)
\(158\) −183436. −0.584577
\(159\) −26200.0 −0.0821881
\(160\) 91972.3 0.284025
\(161\) 0 0
\(162\) −284154. −0.850681
\(163\) −212905. −0.627648 −0.313824 0.949481i \(-0.601610\pi\)
−0.313824 + 0.949481i \(0.601610\pi\)
\(164\) −21763.3 −0.0631852
\(165\) 6280.83 0.0179600
\(166\) −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
\(171\) −445087. −1.16400
\(172\) 89494.0 0.230660
\(173\) −712914. −1.81101 −0.905507 0.424331i \(-0.860509\pi\)
−0.905507 + 0.424331i \(0.860509\pi\)
\(174\) 7748.39 0.0194016
\(175\) 0 0
\(176\) −56861.7 −0.138369
\(177\) −75258.4 −0.180560
\(178\) −357030. −0.844607
\(179\) −749738. −1.74895 −0.874474 0.485072i \(-0.838793\pi\)
−0.874474 + 0.485072i \(0.838793\pi\)
\(180\) −61548.4 −0.141591
\(181\) −623718. −1.41511 −0.707557 0.706656i \(-0.750203\pi\)
−0.707557 + 0.706656i \(0.750203\pi\)
\(182\) 0 0
\(183\) 8382.69 0.0185036
\(184\) 80254.2 0.174752
\(185\) −432932. −0.930016
\(186\) −64821.1 −0.137383
\(187\) 142457. 0.297907
\(188\) −132086. −0.272560
\(189\) 0 0
\(190\) 396483. 0.796784
\(191\) 417726. 0.828530 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(192\) −75842.2 −0.148477
\(193\) 770700. 1.48933 0.744667 0.667436i \(-0.232608\pi\)
0.744667 + 0.667436i \(0.232608\pi\)
\(194\) 537544. 1.02544
\(195\) 55085.6 0.103741
\(196\) 0 0
\(197\) −479193. −0.879721 −0.439861 0.898066i \(-0.644972\pi\)
−0.439861 + 0.898066i \(0.644972\pi\)
\(198\) 87457.5 0.158538
\(199\) 428686. 0.767373 0.383687 0.923463i \(-0.374654\pi\)
0.383687 + 0.923463i \(0.374654\pi\)
\(200\) −266818. −0.471673
\(201\) −32420.2 −0.0566011
\(202\) 185327. 0.319565
\(203\) 0 0
\(204\) 25373.5 0.0426879
\(205\) −147645. −0.245377
\(206\) 327940. 0.538427
\(207\) −98737.0 −0.160160
\(208\) −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
\(213\) −25369.6 −0.0383147
\(214\) −335694. −0.501083
\(215\) 607138. 0.895759
\(216\) 194605. 0.283804
\(217\) 0 0
\(218\) 192830. 0.274811
\(219\) 40799.5 0.0574837
\(220\) 18592.9 0.0258994
\(221\) 1.24941e6 1.72078
\(222\) 109571. 0.149215
\(223\) 363249. 0.489151 0.244575 0.969630i \(-0.421351\pi\)
0.244575 + 0.969630i \(0.421351\pi\)
\(224\) 0 0
\(225\) 328268. 0.432287
\(226\) −629258. −0.819517
\(227\) 843041. 1.08589 0.542943 0.839770i \(-0.317310\pi\)
0.542943 + 0.839770i \(0.317310\pi\)
\(228\) 23948.1 0.0305094
\(229\) 568666. 0.716587 0.358293 0.933609i \(-0.383359\pi\)
0.358293 + 0.933609i \(0.383359\pi\)
\(230\) 87954.8 0.109633
\(231\) 0 0
\(232\) 141985. 0.173190
\(233\) −1.05651e6 −1.27492 −0.637461 0.770482i \(-0.720015\pi\)
−0.637461 + 0.770482i \(0.720015\pi\)
\(234\) 767041. 0.915754
\(235\) −896086. −1.05847
\(236\) −222785. −0.260379
\(237\) −75166.5 −0.0869267
\(238\) 0 0
\(239\) 853715. 0.966759 0.483379 0.875411i \(-0.339409\pi\)
0.483379 + 0.875411i \(0.339409\pi\)
\(240\) −68708.3 −0.0769983
\(241\) 388888. 0.431302 0.215651 0.976470i \(-0.430813\pi\)
0.215651 + 0.976470i \(0.430813\pi\)
\(242\) 792164. 0.869515
\(243\) −360212. −0.391330
\(244\) 24815.0 0.0266833
\(245\) 0 0
\(246\) 37367.5 0.0393692
\(247\) 1.17922e6 1.22986
\(248\) −1.18781e6 −1.22636
\(249\) −51178.9 −0.0523109
\(250\) −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
\(255\) 172137. 0.165776
\(256\) −582151. −0.555182
\(257\) −291986. −0.275759 −0.137879 0.990449i \(-0.544029\pi\)
−0.137879 + 0.990449i \(0.544029\pi\)
\(258\) −153661. −0.143719
\(259\) 0 0
\(260\) 163068. 0.149601
\(261\) −174685. −0.158728
\(262\) −751752. −0.676584
\(263\) −288495. −0.257187 −0.128594 0.991697i \(-0.541046\pi\)
−0.128594 + 0.991697i \(0.541046\pi\)
\(264\) −29128.9 −0.0257226
\(265\) −526169. −0.460268
\(266\) 0 0
\(267\) −146300. −0.125593
\(268\) −95972.2 −0.0816222
\(269\) 259370. 0.218544 0.109272 0.994012i \(-0.465148\pi\)
0.109272 + 0.994012i \(0.465148\pi\)
\(270\) 213278. 0.178047
\(271\) 2.19551e6 1.81599 0.907994 0.418984i \(-0.137614\pi\)
0.907994 + 0.418984i \(0.137614\pi\)
\(272\) −1.55839e6 −1.27719
\(273\) 0 0
\(274\) 463331. 0.372834
\(275\) −99165.1 −0.0790728
\(276\) 5312.59 0.00419792
\(277\) 126991. 0.0994426 0.0497213 0.998763i \(-0.484167\pi\)
0.0497213 + 0.998763i \(0.484167\pi\)
\(278\) −1.70098e6 −1.32004
\(279\) 1.46137e6 1.12396
\(280\) 0 0
\(281\) −2.22759e6 −1.68294 −0.841472 0.540301i \(-0.818310\pi\)
−0.841472 + 0.540301i \(0.818310\pi\)
\(282\) 226791. 0.169826
\(283\) −1.18895e6 −0.882463 −0.441231 0.897393i \(-0.645458\pi\)
−0.441231 + 0.897393i \(0.645458\pi\)
\(284\) −75100.7 −0.0552520
\(285\) 162467. 0.118482
\(286\) −231712. −0.167507
\(287\) 0 0
\(288\) 524780. 0.372817
\(289\) 2.48442e6 1.74977
\(290\) 155609. 0.108653
\(291\) 220269. 0.152483
\(292\) 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
\(297\) 72326.3 0.0475779
\(298\) −702800. −0.458449
\(299\) 261596. 0.169221
\(300\) −17662.6 −0.0113306
\(301\) 0 0
\(302\) −565047. −0.356506
\(303\) 75941.3 0.0475194
\(304\) −1.47085e6 −0.912817
\(305\) 168348. 0.103623
\(306\) 2.39692e6 1.46336
\(307\) 717638. 0.434569 0.217285 0.976108i \(-0.430280\pi\)
0.217285 + 0.976108i \(0.430280\pi\)
\(308\) 0 0
\(309\) 134380. 0.0800642
\(310\) −1.30179e6 −0.769370
\(311\) 856892. 0.502372 0.251186 0.967939i \(-0.419179\pi\)
0.251186 + 0.967939i \(0.419179\pi\)
\(312\) −255474. −0.148580
\(313\) −1.61699e6 −0.932924 −0.466462 0.884541i \(-0.654472\pi\)
−0.466462 + 0.884541i \(0.654472\pi\)
\(314\) −193900. −0.110982
\(315\) 0 0
\(316\) −222512. −0.125354
\(317\) −2.26559e6 −1.26629 −0.633145 0.774033i \(-0.718236\pi\)
−0.633145 + 0.774033i \(0.718236\pi\)
\(318\) 133169. 0.0738472
\(319\) 52769.8 0.0290341
\(320\) −1.52312e6 −0.831495
\(321\) −137557. −0.0745111
\(322\) 0 0
\(323\) 3.68495e6 1.96528
\(324\) −344687. −0.182416
\(325\) −869721. −0.456743
\(326\) 1.08214e6 0.563950
\(327\) 79015.9 0.0408644
\(328\) 684740. 0.351432
\(329\) 0 0
\(330\) −31924.0 −0.0161373
\(331\) 709650. 0.356020 0.178010 0.984029i \(-0.443034\pi\)
0.178010 + 0.984029i \(0.443034\pi\)
\(332\) −151503. −0.0754355
\(333\) −2.47024e6 −1.22076
\(334\) −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
\(339\) −257851. −0.121862
\(340\) 509570. 0.239060
\(341\) −441459. −0.205591
\(342\) 2.26227e6 1.04587
\(343\) 0 0
\(344\) −2.81576e6 −1.28292
\(345\) 36041.2 0.0163024
\(346\) 3.62357e6 1.62722
\(347\) 1.75731e6 0.783474 0.391737 0.920077i \(-0.371874\pi\)
0.391737 + 0.920077i \(0.371874\pi\)
\(348\) 9399.00 0.00416038
\(349\) −391875. −0.172220 −0.0861102 0.996286i \(-0.527444\pi\)
−0.0861102 + 0.996286i \(0.527444\pi\)
\(350\) 0 0
\(351\) 634333. 0.274821
\(352\) −158529. −0.0681948
\(353\) −492407. −0.210323 −0.105162 0.994455i \(-0.533536\pi\)
−0.105162 + 0.994455i \(0.533536\pi\)
\(354\) 382521. 0.162236
\(355\) −509492. −0.214569
\(356\) −433087. −0.181113
\(357\) 0 0
\(358\) 3.81074e6 1.57145
\(359\) −3.77032e6 −1.54398 −0.771991 0.635634i \(-0.780739\pi\)
−0.771991 + 0.635634i \(0.780739\pi\)
\(360\) 1.93650e6 0.787520
\(361\) 1.00185e6 0.404607
\(362\) 3.17021e6 1.27150
\(363\) 324605. 0.129297
\(364\) 0 0
\(365\) 819367. 0.321919
\(366\) −42607.2 −0.0166257
\(367\) 2.19768e6 0.851726 0.425863 0.904788i \(-0.359970\pi\)
0.425863 + 0.904788i \(0.359970\pi\)
\(368\) −326289. −0.125598
\(369\) −842439. −0.322086
\(370\) 2.20049e6 0.835632
\(371\) 0 0
\(372\) −78629.7 −0.0294598
\(373\) −1.65636e6 −0.616427 −0.308213 0.951317i \(-0.599731\pi\)
−0.308213 + 0.951317i \(0.599731\pi\)
\(374\) −724076. −0.267673
\(375\) −392066. −0.143973
\(376\) 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
\(381\) −268101. −0.0946206
\(382\) −2.12320e6 −0.744446
\(383\) −3.24845e6 −1.13156 −0.565781 0.824555i \(-0.691425\pi\)
−0.565781 + 0.824555i \(0.691425\pi\)
\(384\) 238939. 0.0826911
\(385\) 0 0
\(386\) −3.91729e6 −1.33819
\(387\) 3.46424e6 1.17579
\(388\) 652055. 0.219890
\(389\) 4.65348e6 1.55921 0.779604 0.626273i \(-0.215420\pi\)
0.779604 + 0.626273i \(0.215420\pi\)
\(390\) −279987. −0.0932130
\(391\) 817461. 0.270411
\(392\) 0 0
\(393\) −308045. −0.100608
\(394\) 2.43563e6 0.790442
\(395\) −1.50955e6 −0.486805
\(396\) 106088. 0.0339961
\(397\) 1.16361e6 0.370536 0.185268 0.982688i \(-0.440685\pi\)
0.185268 + 0.982688i \(0.440685\pi\)
\(398\) −2.17891e6 −0.689495
\(399\) 0 0
\(400\) 1.08480e6 0.339001
\(401\) 322380. 0.100117 0.0500584 0.998746i \(-0.484059\pi\)
0.0500584 + 0.998746i \(0.484059\pi\)
\(402\) 164784. 0.0508569
\(403\) −3.87179e6 −1.18754
\(404\) 224806. 0.0685259
\(405\) −2.33839e6 −0.708403
\(406\) 0 0
\(407\) 746226. 0.223298
\(408\) −798328. −0.237427
\(409\) −1.38690e6 −0.409956 −0.204978 0.978767i \(-0.565712\pi\)
−0.204978 + 0.978767i \(0.565712\pi\)
\(410\) 750443. 0.220474
\(411\) 189859. 0.0554405
\(412\) 397800. 0.115457
\(413\) 0 0
\(414\) 501857. 0.143906
\(415\) −1.02781e6 −0.292950
\(416\) −1.39036e6 −0.393909
\(417\) −697011. −0.196291
\(418\) −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
\(423\) −5.11293e6 −1.38937
\(424\) 2.44024e6 0.659202
\(425\) −2.71779e6 −0.729865
\(426\) 128948. 0.0344263
\(427\) 0 0
\(428\) −407206. −0.107450
\(429\) −94948.7 −0.0249084
\(430\) −3.08594e6 −0.804852
\(431\) −5.22752e6 −1.35551 −0.677755 0.735288i \(-0.737047\pi\)
−0.677755 + 0.735288i \(0.737047\pi\)
\(432\) −791204. −0.203976
\(433\) 2.63022e6 0.674174 0.337087 0.941473i \(-0.390558\pi\)
0.337087 + 0.941473i \(0.390558\pi\)
\(434\) 0 0
\(435\) 63763.9 0.0161567
\(436\) 233908. 0.0589289
\(437\) 771539. 0.193266
\(438\) −207374. −0.0516499
\(439\) −2.55412e6 −0.632527 −0.316264 0.948671i \(-0.602428\pi\)
−0.316264 + 0.948671i \(0.602428\pi\)
\(440\) −584990. −0.144051
\(441\) 0 0
\(442\) −6.35046e6 −1.54614
\(443\) 3.83900e6 0.929414 0.464707 0.885465i \(-0.346160\pi\)
0.464707 + 0.885465i \(0.346160\pi\)
\(444\) 132913. 0.0319970
\(445\) −2.93811e6 −0.703345
\(446\) −1.84631e6 −0.439509
\(447\) −287986. −0.0681715
\(448\) 0 0
\(449\) 1.49369e6 0.349658 0.174829 0.984599i \(-0.444063\pi\)
0.174829 + 0.984599i \(0.444063\pi\)
\(450\) −1.66851e6 −0.388416
\(451\) 254489. 0.0589152
\(452\) −763307. −0.175733
\(453\) −231539. −0.0530126
\(454\) −4.28498e6 −0.975684
\(455\) 0 0
\(456\) −753481. −0.169692
\(457\) −2.16221e6 −0.484293 −0.242146 0.970240i \(-0.577851\pi\)
−0.242146 + 0.970240i \(0.577851\pi\)
\(458\) −2.89040e6 −0.643863
\(459\) 1.98222e6 0.439158
\(460\) 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
\(465\) −533433. −0.114406
\(466\) 5.36999e6 1.14554
\(467\) 5.47044e6 1.16073 0.580363 0.814358i \(-0.302911\pi\)
0.580363 + 0.814358i \(0.302911\pi\)
\(468\) 930441. 0.196369
\(469\) 0 0
\(470\) 4.55459e6 0.951054
\(471\) −79454.3 −0.0165031
\(472\) 7.00949e6 1.44821
\(473\) −1.04650e6 −0.215073
\(474\) 382053. 0.0781049
\(475\) −2.56511e6 −0.521641
\(476\) 0 0
\(477\) −3.00224e6 −0.604157
\(478\) −4.33923e6 −0.868646
\(479\) 6.66289e6 1.32686 0.663428 0.748240i \(-0.269101\pi\)
0.663428 + 0.748240i \(0.269101\pi\)
\(480\) −191556. −0.0379484
\(481\) 6.54473e6 1.28982
\(482\) −1.97663e6 −0.387531
\(483\) 0 0
\(484\) 960916. 0.186454
\(485\) 4.42362e6 0.853931
\(486\) 1.83087e6 0.351615
\(487\) 9.53693e6 1.82216 0.911079 0.412232i \(-0.135251\pi\)
0.911079 + 0.412232i \(0.135251\pi\)
\(488\) −780755. −0.148411
\(489\) 443430. 0.0838595
\(490\) 0 0
\(491\) −8.19294e6 −1.53369 −0.766843 0.641835i \(-0.778173\pi\)
−0.766843 + 0.641835i \(0.778173\pi\)
\(492\) 45327.8 0.00844213
\(493\) 1.44625e6 0.267994
\(494\) −5.99372e6 −1.10504
\(495\) 719715. 0.132022
\(496\) 4.82928e6 0.881411
\(497\) 0 0
\(498\) 260130. 0.0470021
\(499\) −4.31437e6 −0.775650 −0.387825 0.921733i \(-0.626774\pi\)
−0.387825 + 0.921733i \(0.626774\pi\)
\(500\) −1.16062e6 −0.207618
\(501\) −250756. −0.0446331
\(502\) −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
\(507\) −59425.9 −0.0102673
\(508\) −793649. −0.136449
\(509\) 3.09396e6 0.529322 0.264661 0.964342i \(-0.414740\pi\)
0.264661 + 0.964342i \(0.414740\pi\)
\(510\) −874930. −0.148952
\(511\) 0 0
\(512\) 6.63004e6 1.11774
\(513\) 1.87087e6 0.313870
\(514\) 1.48410e6 0.247773
\(515\) 2.69872e6 0.448374
\(516\) −186395. −0.0308184
\(517\) 1.54455e6 0.254141
\(518\) 0 0
\(519\) 1.48483e6 0.241968
\(520\) −5.13061e6 −0.832072
\(521\) 7.60175e6 1.22693 0.613464 0.789723i \(-0.289776\pi\)
0.613464 + 0.789723i \(0.289776\pi\)
\(522\) 887882. 0.142619
\(523\) −4.75669e6 −0.760415 −0.380208 0.924901i \(-0.624147\pi\)
−0.380208 + 0.924901i \(0.624147\pi\)
\(524\) −911895. −0.145083
\(525\) 0 0
\(526\) 1.46635e6 0.231086
\(527\) −1.20989e7 −1.89767
\(528\) 118429. 0.0184874
\(529\) −6.26519e6 −0.973408
\(530\) 2.67439e6 0.413557
\(531\) −8.62380e6 −1.32728
\(532\) 0 0
\(533\) 2.23198e6 0.340308
\(534\) 743609. 0.112847
\(535\) −2.76253e6 −0.417275
\(536\) 3.01958e6 0.453978
\(537\) 1.56153e6 0.233676
\(538\) −1.31832e6 −0.196365
\(539\) 0 0
\(540\) 258711. 0.0381796
\(541\) 1.10052e7 1.61661 0.808305 0.588764i \(-0.200385\pi\)
0.808305 + 0.588764i \(0.200385\pi\)
\(542\) −1.11593e7 −1.63169
\(543\) 1.29906e6 0.189072
\(544\) −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
\(549\) 960566. 0.136018
\(550\) 504032. 0.0710480
\(551\) 1.36500e6 0.191538
\(552\) −167150. −0.0233485
\(553\) 0 0
\(554\) −645463. −0.0893505
\(555\) 901694. 0.124259
\(556\) −2.06334e6 −0.283063
\(557\) −6.45222e6 −0.881194 −0.440597 0.897705i \(-0.645233\pi\)
−0.440597 + 0.897705i \(0.645233\pi\)
\(558\) −7.42780e6 −1.00989
\(559\) −9.17823e6 −1.24231
\(560\) 0 0
\(561\) −296704. −0.0398031
\(562\) 1.13223e7 1.51215
\(563\) −1.74748e6 −0.232349 −0.116175 0.993229i \(-0.537063\pi\)
−0.116175 + 0.993229i \(0.537063\pi\)
\(564\) 275104. 0.0364165
\(565\) −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 \(-0.489430\pi\)
0.0331992 + 0.999449i \(0.489430\pi\)
\(570\) −825780. −0.106458
\(571\) 5.22364e6 0.670475 0.335238 0.942134i \(-0.391183\pi\)
0.335238 + 0.942134i \(0.391183\pi\)
\(572\) −281073. −0.0359194
\(573\) −870024. −0.110699
\(574\) 0 0
\(575\) −569038. −0.0717748
\(576\) −8.69070e6 −1.09144
\(577\) −6.63973e6 −0.830254 −0.415127 0.909763i \(-0.636263\pi\)
−0.415127 + 0.909763i \(0.636263\pi\)
\(578\) −1.26277e7 −1.57219
\(579\) −1.60519e6 −0.198989
\(580\) 188758. 0.0232989
\(581\) 0 0
\(582\) −1.11958e6 −0.137008
\(583\) 906935. 0.110511
\(584\) −3.80002e6 −0.461056
\(585\) 6.31221e6 0.762592
\(586\) 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
\(591\) 998046. 0.117539
\(592\) −8.16324e6 −0.957322
\(593\) 1.43756e7 1.67876 0.839379 0.543546i \(-0.182919\pi\)
0.839379 + 0.543546i \(0.182919\pi\)
\(594\) −367617. −0.0427494
\(595\) 0 0
\(596\) −852515. −0.0983075
\(597\) −892851. −0.102528
\(598\) −1.32963e6 −0.152047
\(599\) 1.20835e7 1.37602 0.688010 0.725701i \(-0.258484\pi\)
0.688010 + 0.725701i \(0.258484\pi\)
\(600\) 555719. 0.0630199
\(601\) −5.75607e6 −0.650040 −0.325020 0.945707i \(-0.605371\pi\)
−0.325020 + 0.945707i \(0.605371\pi\)
\(602\) 0 0
\(603\) −3.71500e6 −0.416069
\(604\) −685416. −0.0764473
\(605\) 6.51897e6 0.724086
\(606\) −385991. −0.0426969
\(607\) −4.20121e6 −0.462810 −0.231405 0.972858i \(-0.574332\pi\)
−0.231405 + 0.972858i \(0.574332\pi\)
\(608\) −4.10067e6 −0.449879
\(609\) 0 0
\(610\) −855671. −0.0931070
\(611\) 1.35463e7 1.46798
\(612\) 2.90753e6 0.313795
\(613\) −2.64543e6 −0.284344 −0.142172 0.989842i \(-0.545409\pi\)
−0.142172 + 0.989842i \(0.545409\pi\)
\(614\) −3.64758e6 −0.390467
\(615\) 307509. 0.0327846
\(616\) 0 0
\(617\) 6.43533e6 0.680546 0.340273 0.940327i \(-0.389480\pi\)
0.340273 + 0.940327i \(0.389480\pi\)
\(618\) −683022. −0.0719388
\(619\) −1.41177e7 −1.48094 −0.740469 0.672090i \(-0.765397\pi\)
−0.740469 + 0.672090i \(0.765397\pi\)
\(620\) −1.57910e6 −0.164980
\(621\) 415029. 0.0431867
\(622\) −4.35538e6 −0.451388
\(623\) 0 0
\(624\) 1.03868e6 0.106787
\(625\) −3.57548e6 −0.366129
\(626\) 8.21877e6 0.838246
\(627\) −280037. −0.0284476
\(628\) −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
\(633\) 1.22580e6 0.121593
\(634\) 1.15155e7 1.13778
\(635\) −5.38421e6 −0.529892
\(636\) 161537. 0.0158354
\(637\) 0 0
\(638\) −268217. −0.0260876
\(639\) −2.90708e6 −0.281647
\(640\) 4.79855e6 0.463085
\(641\) −1.04174e7 −1.00141 −0.500707 0.865617i \(-0.666927\pi\)
−0.500707 + 0.865617i \(0.666927\pi\)
\(642\) 699171. 0.0669493
\(643\) −1.27284e7 −1.21407 −0.607037 0.794674i \(-0.707642\pi\)
−0.607037 + 0.794674i \(0.707642\pi\)
\(644\) 0 0
\(645\) −1.26452e6 −0.119682
\(646\) −1.87297e7 −1.76584
\(647\) 1.61348e7 1.51531 0.757657 0.652653i \(-0.226344\pi\)
0.757657 + 0.652653i \(0.226344\pi\)
\(648\) 1.08449e7 1.01458
\(649\) 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 \(-0.742237\pi\)
−0.689654 + 0.724139i \(0.742237\pi\)
\(654\) −401619. −0.0367172
\(655\) −6.18640e6 −0.563423
\(656\) −2.78395e6 −0.252581
\(657\) 4.67518e6 0.422557
\(658\) 0 0
\(659\) 1.67927e7 1.50628 0.753140 0.657860i \(-0.228538\pi\)
0.753140 + 0.657860i \(0.228538\pi\)
\(660\) −38724.6 −0.00346041
\(661\) −1.08540e7 −0.966246 −0.483123 0.875552i \(-0.660498\pi\)
−0.483123 + 0.875552i \(0.660498\pi\)
\(662\) −3.60698e6 −0.319889
\(663\) −2.60223e6 −0.229912
\(664\) 4.76675e6 0.419567
\(665\) 0 0
\(666\) 1.25557e7 1.09687
\(667\) 302809. 0.0263544
\(668\) −742303. −0.0643636
\(669\) −756562. −0.0653551
\(670\) 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
\(675\) −1.37983e6 −0.116565
\(676\) −175916. −0.0148060
\(677\) 1.00501e6 0.0842746 0.0421373 0.999112i \(-0.486583\pi\)
0.0421373 + 0.999112i \(0.486583\pi\)
\(678\) 1.31060e6 0.109495
\(679\) 0 0
\(680\) −1.60326e7 −1.32963
\(681\) −1.75586e6 −0.145084
\(682\) 2.24383e6 0.184727
\(683\) 1.87019e6 0.153403 0.0767014 0.997054i \(-0.475561\pi\)
0.0767014 + 0.997054i \(0.475561\pi\)
\(684\) 2.74419e6 0.224272
\(685\) 3.81290e6 0.310476
\(686\) 0 0
\(687\) −1.18440e6 −0.0957426
\(688\) 1.14480e7 0.922060
\(689\) 7.95421e6 0.638336
\(690\) −183189. −0.0146479
\(691\) −1.93867e7 −1.54457 −0.772286 0.635275i \(-0.780887\pi\)
−0.772286 + 0.635275i \(0.780887\pi\)
\(692\) 4.39549e6 0.348933
\(693\) 0 0
\(694\) −8.93199e6 −0.703963
\(695\) −1.39979e7 −1.09926
\(696\) −295721. −0.0231398
\(697\) 6.97469e6 0.543805
\(698\) 1.99181e6 0.154742
\(699\) 2.20046e6 0.170342
\(700\) 0 0
\(701\) 1.17488e7 0.903024 0.451512 0.892265i \(-0.350885\pi\)
0.451512 + 0.892265i \(0.350885\pi\)
\(702\) −3.22416e6 −0.246930
\(703\) 1.93027e7 1.47309
\(704\) 2.62534e6 0.199643
\(705\) 1.86634e6 0.141422
\(706\) 2.50279e6 0.188978
\(707\) 0 0
\(708\) 464008. 0.0347890
\(709\) 1.67948e7 1.25475 0.627377 0.778716i \(-0.284129\pi\)
0.627377 + 0.778716i \(0.284129\pi\)
\(710\) 2.58963e6 0.192793
\(711\) −8.61326e6 −0.638990
\(712\) 1.36262e7 1.00734
\(713\) −2.53322e6 −0.186616
\(714\) 0 0
\(715\) −1.90683e6 −0.139491
\(716\) 4.62253e6 0.336974
\(717\) −1.77809e6 −0.129168
\(718\) 1.91636e7 1.38729
\(719\) 1.65130e7 1.19126 0.595628 0.803261i \(-0.296903\pi\)
0.595628 + 0.803261i \(0.296903\pi\)
\(720\) −7.87323e6 −0.566007
\(721\) 0 0
\(722\) −5.09215e6 −0.363545
\(723\) −809961. −0.0576260
\(724\) 3.84555e6 0.272654
\(725\) −1.00674e6 −0.0711331
\(726\) −1.64989e6 −0.116175
\(727\) 1.25756e6 0.0882453 0.0441227 0.999026i \(-0.485951\pi\)
0.0441227 + 0.999026i \(0.485951\pi\)
\(728\) 0 0
\(729\) −1.28348e7 −0.894479
\(730\) −4.16465e6 −0.289248
\(731\) −2.86810e7 −1.98518
\(732\) −51683.7 −0.00356513
\(733\) 1.98332e6 0.136343 0.0681716 0.997674i \(-0.478283\pi\)
0.0681716 + 0.997674i \(0.478283\pi\)
\(734\) −1.11703e7 −0.765288
\(735\) 0 0
\(736\) −909684. −0.0619007
\(737\) 1.12225e6 0.0761063
\(738\) 4.28191e6 0.289399
\(739\) −2.38807e7 −1.60856 −0.804278 0.594253i \(-0.797448\pi\)
−0.804278 + 0.594253i \(0.797448\pi\)
\(740\) 2.66925e6 0.179189
\(741\) −2.45604e6 −0.164320
\(742\) 0 0
\(743\) −1.90819e7 −1.26809 −0.634043 0.773298i \(-0.718606\pi\)
−0.634043 + 0.773298i \(0.718606\pi\)
\(744\) 2.47393e6 0.163853
\(745\) −5.78356e6 −0.381773
\(746\) 8.41886e6 0.553868
\(747\) −5.86455e6 −0.384532
\(748\) −878323. −0.0573985
\(749\) 0 0
\(750\) 1.99278e6 0.129362
\(751\) −3.75805e6 −0.243144 −0.121572 0.992583i \(-0.538794\pi\)
−0.121572 + 0.992583i \(0.538794\pi\)
\(752\) −1.68963e7 −1.08955
\(753\) −1.74812e6 −0.112353
\(754\) −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
\(759\) −62122.7 −0.00391423
\(760\) −1.51320e7 −0.950302
\(761\) −2.23998e7 −1.40211 −0.701056 0.713106i \(-0.747288\pi\)
−0.701056 + 0.713106i \(0.747288\pi\)
\(762\) 1.36269e6 0.0850180
\(763\) 0 0
\(764\) −2.57550e6 −0.159635
\(765\) 1.97250e7 1.21861
\(766\) 1.65111e7 1.01672
\(767\) 2.28481e7 1.40237
\(768\) 1.21248e6 0.0741775
\(769\) 1.87866e7 1.14560 0.572799 0.819696i \(-0.305858\pi\)
0.572799 + 0.819696i \(0.305858\pi\)
\(770\) 0 0
\(771\) 608138. 0.0368439
\(772\) −4.75177e6 −0.286954
\(773\) −9.30837e6 −0.560306 −0.280153 0.959955i \(-0.590385\pi\)
−0.280153 + 0.959955i \(0.590385\pi\)
\(774\) −1.76079e7 −1.05646
\(775\) 8.42212e6 0.503694
\(776\) −2.05156e7 −1.22301
\(777\) 0 0
\(778\) −2.36525e7 −1.40097
\(779\) 6.58288e6 0.388662
\(780\) −339632. −0.0199881
\(781\) 878189. 0.0515182
\(782\) −4.15496e6 −0.242969
\(783\) 734267. 0.0428006
\(784\) 0 0
\(785\) −1.59566e6 −0.0924201
\(786\) 1.56572e6 0.0903979
\(787\) −1.73427e7 −0.998111 −0.499056 0.866570i \(-0.666320\pi\)
−0.499056 + 0.866570i \(0.666320\pi\)
\(788\) 2.95448e6 0.169498
\(789\) 600867. 0.0343626
\(790\) 7.67268e6 0.437401
\(791\) 0 0
\(792\) −3.33786e6 −0.189084
\(793\) −2.54495e6 −0.143713
\(794\) −5.91434e6 −0.332932
\(795\) 1.09589e6 0.0614960
\(796\) −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
\(801\) −1.67644e7 −0.923224
\(802\) −1.63858e6 −0.0899563
\(803\) −1.41231e6 −0.0772930
\(804\) 199887. 0.0109055
\(805\) 0 0
\(806\) 1.96794e7 1.06702
\(807\) −540206. −0.0291995
\(808\) −7.07309e6 −0.381137
\(809\) −2.47038e7 −1.32707 −0.663533 0.748147i \(-0.730944\pi\)
−0.663533 + 0.748147i \(0.730944\pi\)
\(810\) 1.18855e7 0.636510
\(811\) −8.42005e6 −0.449534 −0.224767 0.974413i \(-0.572162\pi\)
−0.224767 + 0.974413i \(0.572162\pi\)
\(812\) 0 0
\(813\) −4.57273e6 −0.242633
\(814\) −3.79289e6 −0.200636
\(815\) 8.90529e6 0.469628
\(816\) 3.24576e6 0.170644
\(817\) −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 \(-0.733736\pi\)
−0.670071 + 0.742297i \(0.733736\pi\)
\(822\) −965009. −0.0498141
\(823\) 1.72004e7 0.885195 0.442597 0.896720i \(-0.354057\pi\)
0.442597 + 0.896720i \(0.354057\pi\)
\(824\) −1.25160e7 −0.642167
\(825\) 206537. 0.0105649
\(826\) 0 0
\(827\) −2.40337e7 −1.22196 −0.610979 0.791647i \(-0.709224\pi\)
−0.610979 + 0.791647i \(0.709224\pi\)
\(828\) 608766. 0.0308585
\(829\) 3.24736e7 1.64113 0.820567 0.571550i \(-0.193658\pi\)
0.820567 + 0.571550i \(0.193658\pi\)
\(830\) 5.22413e6 0.263220
\(831\) −264491. −0.0132865
\(832\) 2.30254e7 1.15318
\(833\) 0 0
\(834\) 3.54274e6 0.176370
\(835\) −5.03587e6 −0.249953
\(836\) −828983. −0.0410232
\(837\) −6.14269e6 −0.303072
\(838\) 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
\(843\) 4.63954e6 0.224857
\(844\) 3.62868e6 0.175345
\(845\) −1.19344e6 −0.0574986
\(846\) 2.59878e7 1.24837
\(847\) 0 0
\(848\) −9.92129e6 −0.473782
\(849\) 2.47629e6 0.117905
\(850\) 1.38139e7 0.655794
\(851\) 4.28206e6 0.202689
\(852\) 156417. 0.00738219
\(853\) 999355. 0.0470270 0.0235135 0.999724i \(-0.492515\pi\)
0.0235135 + 0.999724i \(0.492515\pi\)
\(854\) 0 0
\(855\) 1.86169e7 0.870949
\(856\) 1.28119e7 0.597628
\(857\) 2.64465e7 1.23003 0.615016 0.788514i \(-0.289149\pi\)
0.615016 + 0.788514i \(0.289149\pi\)
\(858\) 482601. 0.0223805
\(859\) 2.86716e7 1.32577 0.662887 0.748719i \(-0.269331\pi\)
0.662887 + 0.748719i \(0.269331\pi\)
\(860\) −3.74332e6 −0.172588
\(861\) 0 0
\(862\) 2.65703e7 1.21794
\(863\) 4.08173e6 0.186560 0.0932798 0.995640i \(-0.470265\pi\)
0.0932798 + 0.995640i \(0.470265\pi\)
\(864\) −2.20585e6 −0.100529
\(865\) 2.98195e7 1.35506
\(866\) −1.33688e7 −0.605755
\(867\) −5.17446e6 −0.233785
\(868\) 0 0
\(869\) 2.60195e6 0.116882
\(870\) −324097. −0.0145170
\(871\) 9.84261e6 0.439608
\(872\) −7.35946e6 −0.327759
\(873\) 2.52405e7 1.12089
\(874\) −3.92155e6 −0.173652
\(875\) 0 0
\(876\) −251550. −0.0110755
\(877\) −2.82405e7 −1.23986 −0.619931 0.784657i \(-0.712839\pi\)
−0.619931 + 0.784657i \(0.712839\pi\)
\(878\) 1.29820e7 0.568335
\(879\) 3.81610e6 0.166590
\(880\) 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
\(885\) 3.14788e6 0.135102
\(886\) −1.95127e7 −0.835091
\(887\) −7.29088e6 −0.311151 −0.155575 0.987824i \(-0.549723\pi\)
−0.155575 + 0.987824i \(0.549723\pi\)
\(888\) −4.18184e6 −0.177965
\(889\) 0 0
\(890\) 1.49337e7 0.631965
\(891\) 4.03059e6 0.170088
\(892\) −2.23962e6 −0.0942460
\(893\) 3.99529e7 1.67656
\(894\) 1.46377e6 0.0612531
\(895\) 3.13598e7 1.30862
\(896\) 0 0
\(897\) −544843. −0.0226095
\(898\) −7.59205e6 −0.314173
\(899\) −4.48176e6 −0.184948
\(900\) −2.02394e6 −0.0832898
\(901\) 2.48561e7 1.02005
\(902\) −1.29351e6 −0.0529361
\(903\) 0 0
\(904\) 2.40160e7 0.977415
\(905\) 2.60886e7 1.05884
\(906\) 1.17686e6 0.0476325
\(907\) −3.73335e7 −1.50689 −0.753443 0.657514i \(-0.771608\pi\)
−0.753443 + 0.657514i \(0.771608\pi\)
\(908\) −5.19779e6 −0.209221
\(909\) 8.70205e6 0.349311
\(910\) 0 0
\(911\) −2475.17 −9.88120e−5 0 −4.94060e−5 1.00000i \(-0.500016\pi\)
−4.94060e−5 1.00000i \(0.500016\pi\)
\(912\) 3.06342e6 0.121961
\(913\) 1.77160e6 0.0703377
\(914\) 1.09900e7 0.435144
\(915\) −350628. −0.0138450
\(916\) −3.50613e6 −0.138067
\(917\) 0 0
\(918\) −1.00752e7 −0.394590
\(919\) −4.48238e6 −0.175073 −0.0875366 0.996161i \(-0.527899\pi\)
−0.0875366 + 0.996161i \(0.527899\pi\)
\(920\) −3.35684e6 −0.130756
\(921\) −1.49467e6 −0.0580625
\(922\) −3.11039e7 −1.20500
\(923\) 7.70210e6 0.297581
\(924\) 0 0
\(925\) −1.42364e7 −0.547075
\(926\) −2.00065e7 −0.766733
\(927\) 1.53985e7 0.588544
\(928\) −1.60941e6 −0.0613473
\(929\) −2.12859e7 −0.809193 −0.404596 0.914495i \(-0.632588\pi\)
−0.404596 + 0.914495i \(0.632588\pi\)
\(930\) 2.71131e6 0.102795
\(931\) 0 0
\(932\) 6.51394e6 0.245643
\(933\) −1.78470e6 −0.0671215
\(934\) −2.78049e7 −1.04293
\(935\) −5.95865e6 −0.222904
\(936\) −2.92745e7 −1.09219
\(937\) −6.79757e6 −0.252932 −0.126466 0.991971i \(-0.540364\pi\)
−0.126466 + 0.991971i \(0.540364\pi\)
\(938\) 0 0
\(939\) 3.36781e6 0.124647
\(940\) 5.52484e6 0.203939
\(941\) −4.90883e7 −1.80719 −0.903595 0.428388i \(-0.859082\pi\)
−0.903595 + 0.428388i \(0.859082\pi\)
\(942\) 403847. 0.0148283
\(943\) 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 \(-0.353292\pi\)
0.444753 + 0.895653i \(0.353292\pi\)
\(948\) 463441. 0.0167484
\(949\) −1.23865e7 −0.446462
\(950\) 1.30378e7 0.468702
\(951\) 4.71869e6 0.169188
\(952\) 0 0
\(953\) −513120. −0.0183015 −0.00915075 0.999958i \(-0.502913\pi\)
−0.00915075 + 0.999958i \(0.502913\pi\)
\(954\) 1.52597e7 0.542843
\(955\) −1.74725e7 −0.619935
\(956\) −5.26360e6 −0.186268
\(957\) −109907. −0.00387923
\(958\) −3.38659e7 −1.19220
\(959\) 0 0
\(960\) 3.17230e6 0.111095
\(961\) 8.86412e6 0.309619
\(962\) −3.32653e7 −1.15892
\(963\) −1.57626e7 −0.547724
\(964\) −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
\(969\) −7.67488e6 −0.262580
\(970\) −2.24842e7 −0.767269
\(971\) −4.76036e6 −0.162029 −0.0810143 0.996713i \(-0.525816\pi\)
−0.0810143 + 0.996713i \(0.525816\pi\)
\(972\) 2.22090e6 0.0753986
\(973\) 0 0
\(974\) −4.84739e7 −1.63723
\(975\) 1.81142e6 0.0610250
\(976\) 3.17431e6 0.106666
\(977\) 2.87338e7 0.963067 0.481534 0.876428i \(-0.340080\pi\)
0.481534 + 0.876428i \(0.340080\pi\)
\(978\) −2.25385e6 −0.0753490
\(979\) 5.06430e6 0.168874
\(980\) 0 0
\(981\) 9.05437e6 0.300390
\(982\) 4.16428e7 1.37804
\(983\) 4.97072e7 1.64072 0.820362 0.571845i \(-0.193772\pi\)
0.820362 + 0.571845i \(0.193772\pi\)
\(984\) −1.42615e6 −0.0469546
\(985\) 2.00435e7 0.658239
\(986\) −7.35092e6 −0.240796
\(987\) 0 0
\(988\) −7.27054e6 −0.236960
\(989\) −6.00511e6 −0.195223
\(990\) −3.65814e6 −0.118624
\(991\) −2.91066e6 −0.0941471 −0.0470736 0.998891i \(-0.514990\pi\)
−0.0470736 + 0.998891i \(0.514990\pi\)
\(992\) 1.34639e7 0.434401
\(993\) −1.47803e6 −0.0475676
\(994\) 0 0
\(995\) −1.79309e7 −0.574176
\(996\) 315545. 0.0100789
\(997\) −1.43353e7 −0.456740 −0.228370 0.973574i \(-0.573340\pi\)
−0.228370 + 0.973574i \(0.573340\pi\)
\(998\) 2.19289e7 0.696933
\(999\) 1.03834e7 0.329174
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 49.6.a.e.1.1 2
3.2 odd 2 441.6.a.m.1.2 2
4.3 odd 2 784.6.a.t.1.2 2
7.2 even 3 49.6.c.f.18.2 4
7.3 odd 6 7.6.c.a.2.2 4
7.4 even 3 49.6.c.f.30.2 4
7.5 odd 6 7.6.c.a.4.2 yes 4
7.6 odd 2 49.6.a.d.1.1 2
21.5 even 6 63.6.e.d.46.1 4
21.17 even 6 63.6.e.d.37.1 4
21.20 even 2 441.6.a.n.1.2 2
28.3 even 6 112.6.i.c.65.2 4
28.19 even 6 112.6.i.c.81.2 4
28.27 even 2 784.6.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.2 4 7.3 odd 6
7.6.c.a.4.2 yes 4 7.5 odd 6
49.6.a.d.1.1 2 7.6 odd 2
49.6.a.e.1.1 2 1.1 even 1 trivial
49.6.c.f.18.2 4 7.2 even 3
49.6.c.f.30.2 4 7.4 even 3
63.6.e.d.37.1 4 21.17 even 6
63.6.e.d.46.1 4 21.5 even 6
112.6.i.c.65.2 4 28.3 even 6
112.6.i.c.81.2 4 28.19 even 6
441.6.a.m.1.2 2 3.2 odd 2
441.6.a.n.1.2 2 21.20 even 2
784.6.a.t.1.2 2 4.3 odd 2
784.6.a.ba.1.1 2 28.27 even 2