Properties

Label 147.6.a.l.1.4
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $4$
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] = [147,6,Mod(1,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, 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("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-9.22385\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.2239 q^{2} -9.00000 q^{3} +72.5272 q^{4} -23.7528 q^{5} -92.0147 q^{6} +414.344 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.2239 q^{2} -9.00000 q^{3} +72.5272 q^{4} -23.7528 q^{5} -92.0147 q^{6} +414.344 q^{8} +81.0000 q^{9} -242.845 q^{10} +465.526 q^{11} -652.744 q^{12} +1019.30 q^{13} +213.775 q^{15} +1915.32 q^{16} -561.757 q^{17} +828.132 q^{18} +1387.58 q^{19} -1722.72 q^{20} +4759.47 q^{22} +4113.62 q^{23} -3729.09 q^{24} -2560.80 q^{25} +10421.2 q^{26} -729.000 q^{27} -2381.37 q^{29} +2185.61 q^{30} -2950.66 q^{31} +6322.95 q^{32} -4189.73 q^{33} -5743.32 q^{34} +5874.70 q^{36} -9908.95 q^{37} +14186.4 q^{38} -9173.70 q^{39} -9841.83 q^{40} +4477.13 q^{41} +5181.48 q^{43} +33763.3 q^{44} -1923.98 q^{45} +42057.0 q^{46} +3121.59 q^{47} -17237.9 q^{48} -26181.3 q^{50} +5055.81 q^{51} +73926.9 q^{52} +1141.00 q^{53} -7453.19 q^{54} -11057.6 q^{55} -12488.2 q^{57} -24346.8 q^{58} -27497.1 q^{59} +15504.5 q^{60} +21103.5 q^{61} -30167.1 q^{62} +3354.67 q^{64} -24211.2 q^{65} -42835.2 q^{66} -55588.4 q^{67} -40742.6 q^{68} -37022.6 q^{69} -6076.90 q^{71} +33561.8 q^{72} +16779.6 q^{73} -101308. q^{74} +23047.2 q^{75} +100637. q^{76} -93790.5 q^{78} -4845.26 q^{79} -45494.2 q^{80} +6561.00 q^{81} +45773.5 q^{82} -60145.4 q^{83} +13343.3 q^{85} +52974.7 q^{86} +21432.4 q^{87} +192888. q^{88} +62497.4 q^{89} -19670.5 q^{90} +298349. q^{92} +26555.9 q^{93} +31914.7 q^{94} -32958.9 q^{95} -56906.5 q^{96} +63653.8 q^{97} +37707.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9} - 283 q^{10} + 402 q^{11} - 621 q^{12} - 462 q^{13} + 3273 q^{16} - 276 q^{17} + 243 q^{18} - 510 q^{19} - 4719 q^{20} + 1375 q^{22} + 6900 q^{23} - 1107 q^{24} + 2814 q^{25} + 15138 q^{26} - 2916 q^{27} + 540 q^{29} + 2547 q^{30} + 6410 q^{31} + 15519 q^{32} - 3618 q^{33} - 21144 q^{34} + 5589 q^{36} + 15250 q^{37} + 41250 q^{38} + 4158 q^{39} + 8547 q^{40} - 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} + 15060 q^{47} - 29457 q^{48} - 7302 q^{50} + 2484 q^{51} + 47476 q^{52} + 13692 q^{53} - 2187 q^{54} - 73124 q^{55} + 4590 q^{57} + 52309 q^{58} - 34830 q^{59} + 42471 q^{60} + 5364 q^{61} - 16029 q^{62} - 73487 q^{64} + 66864 q^{65} - 12375 q^{66} - 5994 q^{67} + 58272 q^{68} - 62100 q^{69} + 89268 q^{71} + 9963 q^{72} - 59638 q^{73} - 185442 q^{74} - 25326 q^{75} - 21308 q^{76} - 136242 q^{78} - 44062 q^{79} + 33381 q^{80} + 26244 q^{81} - 57596 q^{82} + 208446 q^{83} + 36324 q^{85} - 136968 q^{86} - 4860 q^{87} + 87597 q^{88} + 77520 q^{89} - 22923 q^{90} + 158256 q^{92} - 57690 q^{93} + 73722 q^{94} - 221376 q^{95} - 139671 q^{96} + 188630 q^{97} + 32562 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.2239 1.80734 0.903669 0.428231i \(-0.140863\pi\)
0.903669 + 0.428231i \(0.140863\pi\)
\(3\) −9.00000 −0.577350
\(4\) 72.5272 2.26647
\(5\) −23.7528 −0.424903 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(6\) −92.0147 −1.04347
\(7\) 0 0
\(8\) 414.344 2.28895
\(9\) 81.0000 0.333333
\(10\) −242.845 −0.767944
\(11\) 465.526 1.16001 0.580006 0.814612i \(-0.303050\pi\)
0.580006 + 0.814612i \(0.303050\pi\)
\(12\) −652.744 −1.30855
\(13\) 1019.30 1.67280 0.836399 0.548121i \(-0.184657\pi\)
0.836399 + 0.548121i \(0.184657\pi\)
\(14\) 0 0
\(15\) 213.775 0.245318
\(16\) 1915.32 1.87043
\(17\) −561.757 −0.471440 −0.235720 0.971821i \(-0.575745\pi\)
−0.235720 + 0.971821i \(0.575745\pi\)
\(18\) 828.132 0.602446
\(19\) 1387.58 0.881807 0.440903 0.897555i \(-0.354658\pi\)
0.440903 + 0.897555i \(0.354658\pi\)
\(20\) −1722.72 −0.963032
\(21\) 0 0
\(22\) 4759.47 2.09653
\(23\) 4113.62 1.62145 0.810727 0.585425i \(-0.199072\pi\)
0.810727 + 0.585425i \(0.199072\pi\)
\(24\) −3729.09 −1.32152
\(25\) −2560.80 −0.819457
\(26\) 10421.2 3.02331
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −2381.37 −0.525814 −0.262907 0.964821i \(-0.584681\pi\)
−0.262907 + 0.964821i \(0.584681\pi\)
\(30\) 2185.61 0.443373
\(31\) −2950.66 −0.551460 −0.275730 0.961235i \(-0.588920\pi\)
−0.275730 + 0.961235i \(0.588920\pi\)
\(32\) 6322.95 1.09155
\(33\) −4189.73 −0.669733
\(34\) −5743.32 −0.852051
\(35\) 0 0
\(36\) 5874.70 0.755491
\(37\) −9908.95 −1.18994 −0.594968 0.803750i \(-0.702835\pi\)
−0.594968 + 0.803750i \(0.702835\pi\)
\(38\) 14186.4 1.59372
\(39\) −9173.70 −0.965791
\(40\) −9841.83 −0.972581
\(41\) 4477.13 0.415949 0.207974 0.978134i \(-0.433313\pi\)
0.207974 + 0.978134i \(0.433313\pi\)
\(42\) 0 0
\(43\) 5181.48 0.427349 0.213675 0.976905i \(-0.431457\pi\)
0.213675 + 0.976905i \(0.431457\pi\)
\(44\) 33763.3 2.62914
\(45\) −1923.98 −0.141634
\(46\) 42057.0 2.93052
\(47\) 3121.59 0.206125 0.103063 0.994675i \(-0.467136\pi\)
0.103063 + 0.994675i \(0.467136\pi\)
\(48\) −17237.9 −1.07989
\(49\) 0 0
\(50\) −26181.3 −1.48104
\(51\) 5055.81 0.272186
\(52\) 73926.9 3.79135
\(53\) 1141.00 0.0557950 0.0278975 0.999611i \(-0.491119\pi\)
0.0278975 + 0.999611i \(0.491119\pi\)
\(54\) −7453.19 −0.347823
\(55\) −11057.6 −0.492893
\(56\) 0 0
\(57\) −12488.2 −0.509111
\(58\) −24346.8 −0.950325
\(59\) −27497.1 −1.02839 −0.514194 0.857674i \(-0.671909\pi\)
−0.514194 + 0.857674i \(0.671909\pi\)
\(60\) 15504.5 0.556007
\(61\) 21103.5 0.726157 0.363078 0.931759i \(-0.381726\pi\)
0.363078 + 0.931759i \(0.381726\pi\)
\(62\) −30167.1 −0.996676
\(63\) 0 0
\(64\) 3354.67 0.102376
\(65\) −24211.2 −0.710778
\(66\) −42835.2 −1.21043
\(67\) −55588.4 −1.51286 −0.756428 0.654077i \(-0.773057\pi\)
−0.756428 + 0.654077i \(0.773057\pi\)
\(68\) −40742.6 −1.06851
\(69\) −37022.6 −0.936146
\(70\) 0 0
\(71\) −6076.90 −0.143066 −0.0715330 0.997438i \(-0.522789\pi\)
−0.0715330 + 0.997438i \(0.522789\pi\)
\(72\) 33561.8 0.762982
\(73\) 16779.6 0.368532 0.184266 0.982876i \(-0.441009\pi\)
0.184266 + 0.982876i \(0.441009\pi\)
\(74\) −101308. −2.15062
\(75\) 23047.2 0.473114
\(76\) 100637. 1.99859
\(77\) 0 0
\(78\) −93790.5 −1.74551
\(79\) −4845.26 −0.0873473 −0.0436737 0.999046i \(-0.513906\pi\)
−0.0436737 + 0.999046i \(0.513906\pi\)
\(80\) −45494.2 −0.794752
\(81\) 6561.00 0.111111
\(82\) 45773.5 0.751760
\(83\) −60145.4 −0.958313 −0.479156 0.877730i \(-0.659057\pi\)
−0.479156 + 0.877730i \(0.659057\pi\)
\(84\) 0 0
\(85\) 13343.3 0.200316
\(86\) 52974.7 0.772365
\(87\) 21432.4 0.303579
\(88\) 192888. 2.65520
\(89\) 62497.4 0.836348 0.418174 0.908367i \(-0.362670\pi\)
0.418174 + 0.908367i \(0.362670\pi\)
\(90\) −19670.5 −0.255981
\(91\) 0 0
\(92\) 298349. 3.67498
\(93\) 26555.9 0.318386
\(94\) 31914.7 0.372539
\(95\) −32958.9 −0.374683
\(96\) −56906.5 −0.630208
\(97\) 63653.8 0.686903 0.343451 0.939170i \(-0.388404\pi\)
0.343451 + 0.939170i \(0.388404\pi\)
\(98\) 0 0
\(99\) 37707.6 0.386670
\(100\) −185728. −1.85728
\(101\) −184623. −1.80087 −0.900434 0.434993i \(-0.856751\pi\)
−0.900434 + 0.434993i \(0.856751\pi\)
\(102\) 51689.9 0.491932
\(103\) −52043.7 −0.483365 −0.241683 0.970355i \(-0.577699\pi\)
−0.241683 + 0.970355i \(0.577699\pi\)
\(104\) 422341. 3.82895
\(105\) 0 0
\(106\) 11665.4 0.100840
\(107\) 48177.8 0.406806 0.203403 0.979095i \(-0.434800\pi\)
0.203403 + 0.979095i \(0.434800\pi\)
\(108\) −52872.3 −0.436183
\(109\) −36435.6 −0.293737 −0.146869 0.989156i \(-0.546919\pi\)
−0.146869 + 0.989156i \(0.546919\pi\)
\(110\) −113051. −0.890824
\(111\) 89180.6 0.687010
\(112\) 0 0
\(113\) −96711.1 −0.712492 −0.356246 0.934392i \(-0.615944\pi\)
−0.356246 + 0.934392i \(0.615944\pi\)
\(114\) −127678. −0.920137
\(115\) −97710.0 −0.688961
\(116\) −172714. −1.19174
\(117\) 82563.3 0.557599
\(118\) −281126. −1.85864
\(119\) 0 0
\(120\) 88576.5 0.561520
\(121\) 55663.5 0.345626
\(122\) 215759. 1.31241
\(123\) −40294.1 −0.240148
\(124\) −214003. −1.24987
\(125\) 135054. 0.773093
\(126\) 0 0
\(127\) 23322.9 0.128314 0.0641568 0.997940i \(-0.479564\pi\)
0.0641568 + 0.997940i \(0.479564\pi\)
\(128\) −168037. −0.906524
\(129\) −46633.4 −0.246730
\(130\) −247532. −1.28462
\(131\) 338758. 1.72469 0.862345 0.506321i \(-0.168995\pi\)
0.862345 + 0.506321i \(0.168995\pi\)
\(132\) −303870. −1.51793
\(133\) 0 0
\(134\) −568328. −2.73424
\(135\) 17315.8 0.0817727
\(136\) −232760. −1.07910
\(137\) −62876.7 −0.286213 −0.143106 0.989707i \(-0.545709\pi\)
−0.143106 + 0.989707i \(0.545709\pi\)
\(138\) −378513. −1.69193
\(139\) −211927. −0.930356 −0.465178 0.885217i \(-0.654010\pi\)
−0.465178 + 0.885217i \(0.654010\pi\)
\(140\) 0 0
\(141\) −28094.3 −0.119007
\(142\) −62129.4 −0.258569
\(143\) 474511. 1.94047
\(144\) 155141. 0.623477
\(145\) 56564.3 0.223420
\(146\) 171553. 0.666062
\(147\) 0 0
\(148\) −718668. −2.69696
\(149\) 140273. 0.517616 0.258808 0.965929i \(-0.416670\pi\)
0.258808 + 0.965929i \(0.416670\pi\)
\(150\) 235632. 0.855077
\(151\) −163991. −0.585300 −0.292650 0.956220i \(-0.594537\pi\)
−0.292650 + 0.956220i \(0.594537\pi\)
\(152\) 574934. 2.01841
\(153\) −45502.3 −0.157147
\(154\) 0 0
\(155\) 70086.4 0.234317
\(156\) −665342. −2.18894
\(157\) −556543. −1.80198 −0.900990 0.433840i \(-0.857158\pi\)
−0.900990 + 0.433840i \(0.857158\pi\)
\(158\) −49537.2 −0.157866
\(159\) −10269.0 −0.0322132
\(160\) −150188. −0.463804
\(161\) 0 0
\(162\) 67078.7 0.200815
\(163\) −19726.6 −0.0581546 −0.0290773 0.999577i \(-0.509257\pi\)
−0.0290773 + 0.999577i \(0.509257\pi\)
\(164\) 324713. 0.942736
\(165\) 99518.0 0.284572
\(166\) −614918. −1.73200
\(167\) −94776.2 −0.262971 −0.131486 0.991318i \(-0.541975\pi\)
−0.131486 + 0.991318i \(0.541975\pi\)
\(168\) 0 0
\(169\) 667679. 1.79825
\(170\) 136420. 0.362039
\(171\) 112394. 0.293936
\(172\) 375798. 0.968576
\(173\) −338841. −0.860757 −0.430379 0.902649i \(-0.641620\pi\)
−0.430379 + 0.902649i \(0.641620\pi\)
\(174\) 219121. 0.548670
\(175\) 0 0
\(176\) 891631. 2.16972
\(177\) 247474. 0.593740
\(178\) 638965. 1.51156
\(179\) 776193. 1.81066 0.905330 0.424708i \(-0.139623\pi\)
0.905330 + 0.424708i \(0.139623\pi\)
\(180\) −139541. −0.321011
\(181\) 132697. 0.301067 0.150534 0.988605i \(-0.451901\pi\)
0.150534 + 0.988605i \(0.451901\pi\)
\(182\) 0 0
\(183\) −189932. −0.419247
\(184\) 1.70445e6 3.71142
\(185\) 235366. 0.505607
\(186\) 271504. 0.575431
\(187\) −261512. −0.546875
\(188\) 226400. 0.467178
\(189\) 0 0
\(190\) −336967. −0.677178
\(191\) 6637.41 0.0131648 0.00658242 0.999978i \(-0.497905\pi\)
0.00658242 + 0.999978i \(0.497905\pi\)
\(192\) −30192.0 −0.0591070
\(193\) 452590. 0.874604 0.437302 0.899315i \(-0.355934\pi\)
0.437302 + 0.899315i \(0.355934\pi\)
\(194\) 650787. 1.24147
\(195\) 217901. 0.410368
\(196\) 0 0
\(197\) 816952. 1.49979 0.749896 0.661556i \(-0.230104\pi\)
0.749896 + 0.661556i \(0.230104\pi\)
\(198\) 385517. 0.698845
\(199\) −806417. −1.44353 −0.721767 0.692136i \(-0.756670\pi\)
−0.721767 + 0.692136i \(0.756670\pi\)
\(200\) −1.06105e6 −1.87569
\(201\) 500296. 0.873447
\(202\) −1.88756e6 −3.25478
\(203\) 0 0
\(204\) 366684. 0.616902
\(205\) −106344. −0.176738
\(206\) −532087. −0.873605
\(207\) 333203. 0.540484
\(208\) 1.95229e6 3.12885
\(209\) 645954. 1.02291
\(210\) 0 0
\(211\) 68773.9 0.106345 0.0531726 0.998585i \(-0.483067\pi\)
0.0531726 + 0.998585i \(0.483067\pi\)
\(212\) 82753.3 0.126458
\(213\) 54692.1 0.0825992
\(214\) 492563. 0.735237
\(215\) −123075. −0.181582
\(216\) −302057. −0.440508
\(217\) 0 0
\(218\) −372512. −0.530883
\(219\) −151017. −0.212772
\(220\) −801973. −1.11713
\(221\) −572599. −0.788623
\(222\) 911769. 1.24166
\(223\) 620227. 0.835196 0.417598 0.908632i \(-0.362872\pi\)
0.417598 + 0.908632i \(0.362872\pi\)
\(224\) 0 0
\(225\) −207425. −0.273152
\(226\) −988760. −1.28771
\(227\) −1.00223e6 −1.29093 −0.645467 0.763788i \(-0.723337\pi\)
−0.645467 + 0.763788i \(0.723337\pi\)
\(228\) −905734. −1.15389
\(229\) −885861. −1.11629 −0.558145 0.829743i \(-0.688487\pi\)
−0.558145 + 0.829743i \(0.688487\pi\)
\(230\) −998973. −1.24519
\(231\) 0 0
\(232\) −986707. −1.20356
\(233\) 596163. 0.719408 0.359704 0.933066i \(-0.382878\pi\)
0.359704 + 0.933066i \(0.382878\pi\)
\(234\) 844115. 1.00777
\(235\) −74146.6 −0.0875834
\(236\) −1.99429e6 −2.33081
\(237\) 43607.4 0.0504300
\(238\) 0 0
\(239\) 743111. 0.841509 0.420754 0.907175i \(-0.361765\pi\)
0.420754 + 0.907175i \(0.361765\pi\)
\(240\) 409448. 0.458850
\(241\) 1.17484e6 1.30297 0.651487 0.758660i \(-0.274146\pi\)
0.651487 + 0.758660i \(0.274146\pi\)
\(242\) 569095. 0.624664
\(243\) −59049.0 −0.0641500
\(244\) 1.53058e6 1.64582
\(245\) 0 0
\(246\) −411961. −0.434029
\(247\) 1.41436e6 1.47508
\(248\) −1.22259e6 −1.26226
\(249\) 541309. 0.553282
\(250\) 1.38077e6 1.39724
\(251\) −352992. −0.353655 −0.176828 0.984242i \(-0.556584\pi\)
−0.176828 + 0.984242i \(0.556584\pi\)
\(252\) 0 0
\(253\) 1.91500e6 1.88090
\(254\) 238450. 0.231906
\(255\) −120090. −0.115653
\(256\) −1.82533e6 −1.74077
\(257\) −10012.7 −0.00945621 −0.00472811 0.999989i \(-0.501505\pi\)
−0.00472811 + 0.999989i \(0.501505\pi\)
\(258\) −476773. −0.445925
\(259\) 0 0
\(260\) −1.75597e6 −1.61096
\(261\) −192891. −0.175271
\(262\) 3.46341e6 3.11710
\(263\) 1.68180e6 1.49929 0.749644 0.661841i \(-0.230225\pi\)
0.749644 + 0.661841i \(0.230225\pi\)
\(264\) −1.73599e6 −1.53298
\(265\) −27101.9 −0.0237075
\(266\) 0 0
\(267\) −562477. −0.482866
\(268\) −4.03167e6 −3.42885
\(269\) 1.91717e6 1.61540 0.807702 0.589591i \(-0.200711\pi\)
0.807702 + 0.589591i \(0.200711\pi\)
\(270\) 177034. 0.147791
\(271\) −2.29000e6 −1.89414 −0.947069 0.321029i \(-0.895971\pi\)
−0.947069 + 0.321029i \(0.895971\pi\)
\(272\) −1.07594e6 −0.881795
\(273\) 0 0
\(274\) −642843. −0.517283
\(275\) −1.19212e6 −0.950580
\(276\) −2.68514e6 −2.12175
\(277\) −394802. −0.309157 −0.154579 0.987980i \(-0.549402\pi\)
−0.154579 + 0.987980i \(0.549402\pi\)
\(278\) −2.16671e6 −1.68147
\(279\) −239003. −0.183820
\(280\) 0 0
\(281\) −1.77699e6 −1.34252 −0.671259 0.741223i \(-0.734246\pi\)
−0.671259 + 0.741223i \(0.734246\pi\)
\(282\) −287232. −0.215085
\(283\) −1.21524e6 −0.901981 −0.450991 0.892529i \(-0.648929\pi\)
−0.450991 + 0.892529i \(0.648929\pi\)
\(284\) −440741. −0.324255
\(285\) 296630. 0.216323
\(286\) 4.85133e6 3.50708
\(287\) 0 0
\(288\) 512159. 0.363851
\(289\) −1.10429e6 −0.777745
\(290\) 578305. 0.403796
\(291\) −572884. −0.396583
\(292\) 1.21698e6 0.835268
\(293\) 1.48897e6 1.01325 0.506627 0.862165i \(-0.330892\pi\)
0.506627 + 0.862165i \(0.330892\pi\)
\(294\) 0 0
\(295\) 653133. 0.436965
\(296\) −4.10571e6 −2.72370
\(297\) −339368. −0.223244
\(298\) 1.43413e6 0.935508
\(299\) 4.19301e6 2.71236
\(300\) 1.67155e6 1.07230
\(301\) 0 0
\(302\) −1.67662e6 −1.05784
\(303\) 1.66161e6 1.03973
\(304\) 2.65766e6 1.64936
\(305\) −501268. −0.308546
\(306\) −465209. −0.284017
\(307\) −2.03109e6 −1.22994 −0.614968 0.788552i \(-0.710831\pi\)
−0.614968 + 0.788552i \(0.710831\pi\)
\(308\) 0 0
\(309\) 468394. 0.279071
\(310\) 716553. 0.423491
\(311\) −289785. −0.169893 −0.0849463 0.996386i \(-0.527072\pi\)
−0.0849463 + 0.996386i \(0.527072\pi\)
\(312\) −3.80106e6 −2.21064
\(313\) −218211. −0.125897 −0.0629484 0.998017i \(-0.520050\pi\)
−0.0629484 + 0.998017i \(0.520050\pi\)
\(314\) −5.69002e6 −3.25679
\(315\) 0 0
\(316\) −351413. −0.197970
\(317\) 1.29063e6 0.721363 0.360681 0.932689i \(-0.382544\pi\)
0.360681 + 0.932689i \(0.382544\pi\)
\(318\) −104988. −0.0582202
\(319\) −1.10859e6 −0.609951
\(320\) −79682.8 −0.0435000
\(321\) −433600. −0.234870
\(322\) 0 0
\(323\) −779481. −0.415719
\(324\) 475851. 0.251830
\(325\) −2.61023e6 −1.37079
\(326\) −201682. −0.105105
\(327\) 327920. 0.169589
\(328\) 1.85507e6 0.952084
\(329\) 0 0
\(330\) 1.01746e6 0.514318
\(331\) −3.48561e6 −1.74867 −0.874336 0.485321i \(-0.838703\pi\)
−0.874336 + 0.485321i \(0.838703\pi\)
\(332\) −4.36218e6 −2.17199
\(333\) −802625. −0.396645
\(334\) −968978. −0.475278
\(335\) 1.32038e6 0.642817
\(336\) 0 0
\(337\) 249198. 0.119528 0.0597641 0.998213i \(-0.480965\pi\)
0.0597641 + 0.998213i \(0.480965\pi\)
\(338\) 6.82625e6 3.25005
\(339\) 870400. 0.411358
\(340\) 967752. 0.454012
\(341\) −1.37361e6 −0.639700
\(342\) 1.14910e6 0.531241
\(343\) 0 0
\(344\) 2.14692e6 0.978180
\(345\) 879390. 0.397772
\(346\) −3.46426e6 −1.55568
\(347\) −1.50601e6 −0.671437 −0.335719 0.941962i \(-0.608979\pi\)
−0.335719 + 0.941962i \(0.608979\pi\)
\(348\) 1.55443e6 0.688054
\(349\) −1.54370e6 −0.678423 −0.339212 0.940710i \(-0.610160\pi\)
−0.339212 + 0.940710i \(0.610160\pi\)
\(350\) 0 0
\(351\) −743070. −0.321930
\(352\) 2.94350e6 1.26621
\(353\) 1.67796e6 0.716710 0.358355 0.933585i \(-0.383338\pi\)
0.358355 + 0.933585i \(0.383338\pi\)
\(354\) 2.53014e6 1.07309
\(355\) 144344. 0.0607892
\(356\) 4.53276e6 1.89556
\(357\) 0 0
\(358\) 7.93568e6 3.27248
\(359\) −2.49973e6 −1.02366 −0.511832 0.859086i \(-0.671033\pi\)
−0.511832 + 0.859086i \(0.671033\pi\)
\(360\) −797188. −0.324194
\(361\) −550727. −0.222417
\(362\) 1.35667e6 0.544130
\(363\) −500971. −0.199547
\(364\) 0 0
\(365\) −398564. −0.156591
\(366\) −1.94183e6 −0.757721
\(367\) 2.10741e6 0.816740 0.408370 0.912816i \(-0.366097\pi\)
0.408370 + 0.912816i \(0.366097\pi\)
\(368\) 7.87890e6 3.03281
\(369\) 362647. 0.138650
\(370\) 2.40634e6 0.913804
\(371\) 0 0
\(372\) 1.92602e6 0.721613
\(373\) 2.73225e6 1.01683 0.508414 0.861113i \(-0.330232\pi\)
0.508414 + 0.861113i \(0.330232\pi\)
\(374\) −2.67366e6 −0.988389
\(375\) −1.21548e6 −0.446346
\(376\) 1.29341e6 0.471810
\(377\) −2.42733e6 −0.879581
\(378\) 0 0
\(379\) −2.04295e6 −0.730567 −0.365283 0.930896i \(-0.619028\pi\)
−0.365283 + 0.930896i \(0.619028\pi\)
\(380\) −2.39041e6 −0.849208
\(381\) −209906. −0.0740819
\(382\) 67859.9 0.0237933
\(383\) −1.50899e6 −0.525643 −0.262821 0.964845i \(-0.584653\pi\)
−0.262821 + 0.964845i \(0.584653\pi\)
\(384\) 1.51233e6 0.523382
\(385\) 0 0
\(386\) 4.62721e6 1.58071
\(387\) 419700. 0.142450
\(388\) 4.61663e6 1.55685
\(389\) 4.50065e6 1.50800 0.754000 0.656874i \(-0.228122\pi\)
0.754000 + 0.656874i \(0.228122\pi\)
\(390\) 2.22779e6 0.741673
\(391\) −2.31085e6 −0.764417
\(392\) 0 0
\(393\) −3.04882e6 −0.995751
\(394\) 8.35240e6 2.71063
\(395\) 115089. 0.0371142
\(396\) 2.73483e6 0.876378
\(397\) 3.52136e6 1.12133 0.560665 0.828042i \(-0.310545\pi\)
0.560665 + 0.828042i \(0.310545\pi\)
\(398\) −8.24469e6 −2.60895
\(399\) 0 0
\(400\) −4.90476e6 −1.53274
\(401\) −907107. −0.281707 −0.140853 0.990030i \(-0.544985\pi\)
−0.140853 + 0.990030i \(0.544985\pi\)
\(402\) 5.11495e6 1.57862
\(403\) −3.00760e6 −0.922482
\(404\) −1.33902e7 −4.08162
\(405\) −155842. −0.0472115
\(406\) 0 0
\(407\) −4.61287e6 −1.38034
\(408\) 2.09484e6 0.623019
\(409\) −4.31853e6 −1.27652 −0.638260 0.769821i \(-0.720346\pi\)
−0.638260 + 0.769821i \(0.720346\pi\)
\(410\) −1.08725e6 −0.319425
\(411\) 565891. 0.165245
\(412\) −3.77458e6 −1.09553
\(413\) 0 0
\(414\) 3.40662e6 0.976838
\(415\) 1.42862e6 0.407190
\(416\) 6.44498e6 1.82595
\(417\) 1.90734e6 0.537141
\(418\) 6.60413e6 1.84874
\(419\) 1.52041e6 0.423083 0.211541 0.977369i \(-0.432152\pi\)
0.211541 + 0.977369i \(0.432152\pi\)
\(420\) 0 0
\(421\) −4.42050e6 −1.21553 −0.607766 0.794116i \(-0.707934\pi\)
−0.607766 + 0.794116i \(0.707934\pi\)
\(422\) 703134. 0.192202
\(423\) 252849. 0.0687085
\(424\) 472765. 0.127712
\(425\) 1.43855e6 0.386325
\(426\) 559164. 0.149285
\(427\) 0 0
\(428\) 3.49420e6 0.922016
\(429\) −4.27060e6 −1.12033
\(430\) −1.25830e6 −0.328180
\(431\) −6.84785e6 −1.77567 −0.887833 0.460166i \(-0.847790\pi\)
−0.887833 + 0.460166i \(0.847790\pi\)
\(432\) −1.39627e6 −0.359964
\(433\) 3.99328e6 1.02355 0.511777 0.859119i \(-0.328988\pi\)
0.511777 + 0.859119i \(0.328988\pi\)
\(434\) 0 0
\(435\) −509079. −0.128992
\(436\) −2.64257e6 −0.665748
\(437\) 5.70797e6 1.42981
\(438\) −1.54397e6 −0.384551
\(439\) −1.48557e6 −0.367901 −0.183950 0.982936i \(-0.558889\pi\)
−0.183950 + 0.982936i \(0.558889\pi\)
\(440\) −4.58163e6 −1.12821
\(441\) 0 0
\(442\) −5.85416e6 −1.42531
\(443\) −5.62193e6 −1.36106 −0.680528 0.732722i \(-0.738250\pi\)
−0.680528 + 0.732722i \(0.738250\pi\)
\(444\) 6.46801e6 1.55709
\(445\) −1.48449e6 −0.355367
\(446\) 6.34111e6 1.50948
\(447\) −1.26246e6 −0.298846
\(448\) 0 0
\(449\) 1.94883e6 0.456202 0.228101 0.973637i \(-0.426748\pi\)
0.228101 + 0.973637i \(0.426748\pi\)
\(450\) −2.12068e6 −0.493679
\(451\) 2.08422e6 0.482505
\(452\) −7.01418e6 −1.61484
\(453\) 1.47592e6 0.337923
\(454\) −1.02467e7 −2.33315
\(455\) 0 0
\(456\) −5.17441e6 −1.16533
\(457\) −2.67312e6 −0.598726 −0.299363 0.954139i \(-0.596774\pi\)
−0.299363 + 0.954139i \(0.596774\pi\)
\(458\) −9.05691e6 −2.01751
\(459\) 409521. 0.0907286
\(460\) −7.08663e6 −1.56151
\(461\) 4.65262e6 1.01964 0.509819 0.860282i \(-0.329713\pi\)
0.509819 + 0.860282i \(0.329713\pi\)
\(462\) 0 0
\(463\) −5.18586e6 −1.12426 −0.562132 0.827047i \(-0.690019\pi\)
−0.562132 + 0.827047i \(0.690019\pi\)
\(464\) −4.56109e6 −0.983499
\(465\) −630777. −0.135283
\(466\) 6.09509e6 1.30021
\(467\) 5.05980e6 1.07360 0.536799 0.843710i \(-0.319633\pi\)
0.536799 + 0.843710i \(0.319633\pi\)
\(468\) 5.98808e6 1.26378
\(469\) 0 0
\(470\) −758064. −0.158293
\(471\) 5.00889e6 1.04037
\(472\) −1.13932e7 −2.35392
\(473\) 2.41212e6 0.495730
\(474\) 445835. 0.0911441
\(475\) −3.55331e6 −0.722603
\(476\) 0 0
\(477\) 92420.8 0.0185983
\(478\) 7.59745e6 1.52089
\(479\) 5.55085e6 1.10540 0.552702 0.833379i \(-0.313597\pi\)
0.552702 + 0.833379i \(0.313597\pi\)
\(480\) 1.35169e6 0.267778
\(481\) −1.01002e7 −1.99052
\(482\) 1.20114e7 2.35491
\(483\) 0 0
\(484\) 4.03711e6 0.783353
\(485\) −1.51196e6 −0.291867
\(486\) −603708. −0.115941
\(487\) −6.87810e6 −1.31415 −0.657077 0.753824i \(-0.728207\pi\)
−0.657077 + 0.753824i \(0.728207\pi\)
\(488\) 8.74411e6 1.66213
\(489\) 177540. 0.0335756
\(490\) 0 0
\(491\) 7.44459e6 1.39360 0.696798 0.717267i \(-0.254607\pi\)
0.696798 + 0.717267i \(0.254607\pi\)
\(492\) −2.92242e6 −0.544289
\(493\) 1.33775e6 0.247890
\(494\) 1.44602e7 2.66598
\(495\) −895662. −0.164298
\(496\) −5.65145e6 −1.03147
\(497\) 0 0
\(498\) 5.53426e6 0.999968
\(499\) 2.15329e6 0.387126 0.193563 0.981088i \(-0.437996\pi\)
0.193563 + 0.981088i \(0.437996\pi\)
\(500\) 9.79507e6 1.75220
\(501\) 852986. 0.151826
\(502\) −3.60894e6 −0.639175
\(503\) −8.61012e6 −1.51736 −0.758681 0.651463i \(-0.774156\pi\)
−0.758681 + 0.651463i \(0.774156\pi\)
\(504\) 0 0
\(505\) 4.38531e6 0.765195
\(506\) 1.95786e7 3.39943
\(507\) −6.00911e6 −1.03822
\(508\) 1.69154e6 0.290820
\(509\) −3.34201e6 −0.571759 −0.285879 0.958266i \(-0.592286\pi\)
−0.285879 + 0.958266i \(0.592286\pi\)
\(510\) −1.22778e6 −0.209024
\(511\) 0 0
\(512\) −1.32848e7 −2.23964
\(513\) −1.01154e6 −0.169704
\(514\) −102368. −0.0170906
\(515\) 1.23619e6 0.205383
\(516\) −3.38219e6 −0.559208
\(517\) 1.45318e6 0.239108
\(518\) 0 0
\(519\) 3.04957e6 0.496958
\(520\) −1.00318e7 −1.62693
\(521\) 2.68305e6 0.433046 0.216523 0.976278i \(-0.430528\pi\)
0.216523 + 0.976278i \(0.430528\pi\)
\(522\) −1.97209e6 −0.316775
\(523\) 5.42355e6 0.867021 0.433511 0.901148i \(-0.357275\pi\)
0.433511 + 0.901148i \(0.357275\pi\)
\(524\) 2.45692e7 3.90897
\(525\) 0 0
\(526\) 1.71945e7 2.70972
\(527\) 1.65755e6 0.259980
\(528\) −8.02468e6 −1.25269
\(529\) 1.04855e7 1.62911
\(530\) −277086. −0.0428474
\(531\) −2.22726e6 −0.342796
\(532\) 0 0
\(533\) 4.56353e6 0.695798
\(534\) −5.75068e6 −0.872702
\(535\) −1.14436e6 −0.172853
\(536\) −2.30327e7 −3.46285
\(537\) −6.98574e6 −1.04539
\(538\) 1.96009e7 2.91958
\(539\) 0 0
\(540\) 1.25587e6 0.185336
\(541\) 2.18456e6 0.320900 0.160450 0.987044i \(-0.448705\pi\)
0.160450 + 0.987044i \(0.448705\pi\)
\(542\) −2.34126e7 −3.42335
\(543\) −1.19427e6 −0.173821
\(544\) −3.55196e6 −0.514601
\(545\) 865447. 0.124810
\(546\) 0 0
\(547\) −691437. −0.0988062 −0.0494031 0.998779i \(-0.515732\pi\)
−0.0494031 + 0.998779i \(0.515732\pi\)
\(548\) −4.56027e6 −0.648693
\(549\) 1.70939e6 0.242052
\(550\) −1.21881e7 −1.71802
\(551\) −3.30434e6 −0.463667
\(552\) −1.53401e7 −2.14279
\(553\) 0 0
\(554\) −4.03639e6 −0.558752
\(555\) −2.11829e6 −0.291913
\(556\) −1.53705e7 −2.10863
\(557\) 5.47656e6 0.747946 0.373973 0.927440i \(-0.377995\pi\)
0.373973 + 0.927440i \(0.377995\pi\)
\(558\) −2.44353e6 −0.332225
\(559\) 5.28149e6 0.714869
\(560\) 0 0
\(561\) 2.35361e6 0.315739
\(562\) −1.81677e7 −2.42638
\(563\) −398976. −0.0530488 −0.0265244 0.999648i \(-0.508444\pi\)
−0.0265244 + 0.999648i \(0.508444\pi\)
\(564\) −2.03760e6 −0.269725
\(565\) 2.29716e6 0.302740
\(566\) −1.24245e7 −1.63019
\(567\) 0 0
\(568\) −2.51793e6 −0.327471
\(569\) −4.23113e6 −0.547868 −0.273934 0.961749i \(-0.588325\pi\)
−0.273934 + 0.961749i \(0.588325\pi\)
\(570\) 3.03270e6 0.390969
\(571\) 8.75102e6 1.12323 0.561615 0.827399i \(-0.310180\pi\)
0.561615 + 0.827399i \(0.310180\pi\)
\(572\) 3.44149e7 4.39801
\(573\) −59736.7 −0.00760072
\(574\) 0 0
\(575\) −1.05342e7 −1.32871
\(576\) 271728. 0.0341254
\(577\) 1.76712e6 0.220966 0.110483 0.993878i \(-0.464760\pi\)
0.110483 + 0.993878i \(0.464760\pi\)
\(578\) −1.12901e7 −1.40565
\(579\) −4.07331e6 −0.504953
\(580\) 4.10245e6 0.506376
\(581\) 0 0
\(582\) −5.85709e6 −0.716761
\(583\) 531164. 0.0647228
\(584\) 6.95254e6 0.843551
\(585\) −1.96111e6 −0.236926
\(586\) 1.52231e7 1.83129
\(587\) 8.65009e6 1.03616 0.518078 0.855333i \(-0.326648\pi\)
0.518078 + 0.855333i \(0.326648\pi\)
\(588\) 0 0
\(589\) −4.09426e6 −0.486281
\(590\) 6.67754e6 0.789744
\(591\) −7.35257e6 −0.865905
\(592\) −1.89788e7 −2.22569
\(593\) −1.54514e7 −1.80439 −0.902194 0.431329i \(-0.858045\pi\)
−0.902194 + 0.431329i \(0.858045\pi\)
\(594\) −3.46965e6 −0.403478
\(595\) 0 0
\(596\) 1.01736e7 1.17316
\(597\) 7.25775e6 0.833425
\(598\) 4.28687e7 4.90216
\(599\) −3.84801e6 −0.438196 −0.219098 0.975703i \(-0.570312\pi\)
−0.219098 + 0.975703i \(0.570312\pi\)
\(600\) 9.54948e6 1.08293
\(601\) −1.27578e7 −1.44075 −0.720376 0.693583i \(-0.756031\pi\)
−0.720376 + 0.693583i \(0.756031\pi\)
\(602\) 0 0
\(603\) −4.50266e6 −0.504285
\(604\) −1.18938e7 −1.32657
\(605\) −1.32216e6 −0.146858
\(606\) 1.69880e7 1.87915
\(607\) 1.36207e7 1.50047 0.750233 0.661173i \(-0.229941\pi\)
0.750233 + 0.661173i \(0.229941\pi\)
\(608\) 8.77359e6 0.962539
\(609\) 0 0
\(610\) −5.12489e6 −0.557648
\(611\) 3.18184e6 0.344806
\(612\) −3.30015e6 −0.356168
\(613\) 1.53474e7 1.64962 0.824812 0.565407i \(-0.191281\pi\)
0.824812 + 0.565407i \(0.191281\pi\)
\(614\) −2.07655e7 −2.22291
\(615\) 957099. 0.102040
\(616\) 0 0
\(617\) 1.18485e7 1.25300 0.626500 0.779421i \(-0.284487\pi\)
0.626500 + 0.779421i \(0.284487\pi\)
\(618\) 4.78879e6 0.504376
\(619\) 874460. 0.0917304 0.0458652 0.998948i \(-0.485396\pi\)
0.0458652 + 0.998948i \(0.485396\pi\)
\(620\) 5.08317e6 0.531074
\(621\) −2.99883e6 −0.312049
\(622\) −2.96272e6 −0.307054
\(623\) 0 0
\(624\) −1.75706e7 −1.80644
\(625\) 4.79460e6 0.490967
\(626\) −2.23095e6 −0.227538
\(627\) −5.81358e6 −0.590575
\(628\) −4.03645e7 −4.08414
\(629\) 5.56642e6 0.560983
\(630\) 0 0
\(631\) −4.43859e6 −0.443784 −0.221892 0.975071i \(-0.571223\pi\)
−0.221892 + 0.975071i \(0.571223\pi\)
\(632\) −2.00760e6 −0.199933
\(633\) −618965. −0.0613984
\(634\) 1.31952e7 1.30375
\(635\) −553984. −0.0545209
\(636\) −744780. −0.0730104
\(637\) 0 0
\(638\) −1.13341e7 −1.10239
\(639\) −492229. −0.0476887
\(640\) 3.99135e6 0.385185
\(641\) 812551. 0.0781098 0.0390549 0.999237i \(-0.487565\pi\)
0.0390549 + 0.999237i \(0.487565\pi\)
\(642\) −4.43306e6 −0.424489
\(643\) 1.17941e7 1.12496 0.562481 0.826810i \(-0.309847\pi\)
0.562481 + 0.826810i \(0.309847\pi\)
\(644\) 0 0
\(645\) 1.10767e6 0.104837
\(646\) −7.96930e6 −0.751344
\(647\) 2.43380e6 0.228573 0.114286 0.993448i \(-0.463542\pi\)
0.114286 + 0.993448i \(0.463542\pi\)
\(648\) 2.71851e6 0.254327
\(649\) −1.28006e7 −1.19294
\(650\) −2.66866e7 −2.47748
\(651\) 0 0
\(652\) −1.43072e6 −0.131806
\(653\) −9.46664e6 −0.868786 −0.434393 0.900723i \(-0.643037\pi\)
−0.434393 + 0.900723i \(0.643037\pi\)
\(654\) 3.35261e6 0.306506
\(655\) −8.04646e6 −0.732827
\(656\) 8.57513e6 0.778003
\(657\) 1.35915e6 0.122844
\(658\) 0 0
\(659\) 1.40662e7 1.26172 0.630860 0.775896i \(-0.282702\pi\)
0.630860 + 0.775896i \(0.282702\pi\)
\(660\) 7.21776e6 0.644974
\(661\) 1.71606e7 1.52766 0.763832 0.645415i \(-0.223315\pi\)
0.763832 + 0.645415i \(0.223315\pi\)
\(662\) −3.56363e7 −3.16044
\(663\) 5.15339e6 0.455312
\(664\) −2.49209e7 −2.19353
\(665\) 0 0
\(666\) −8.20592e6 −0.716872
\(667\) −9.79606e6 −0.852583
\(668\) −6.87385e6 −0.596017
\(669\) −5.58204e6 −0.482201
\(670\) 1.34994e7 1.16179
\(671\) 9.82424e6 0.842350
\(672\) 0 0
\(673\) 5.87113e6 0.499671 0.249836 0.968288i \(-0.419623\pi\)
0.249836 + 0.968288i \(0.419623\pi\)
\(674\) 2.54777e6 0.216028
\(675\) 1.86683e6 0.157705
\(676\) 4.84249e7 4.07570
\(677\) −1.39274e7 −1.16788 −0.583938 0.811798i \(-0.698489\pi\)
−0.583938 + 0.811798i \(0.698489\pi\)
\(678\) 8.89884e6 0.743462
\(679\) 0 0
\(680\) 5.52872e6 0.458513
\(681\) 9.02009e6 0.745321
\(682\) −1.40436e7 −1.15615
\(683\) 1.83101e7 1.50189 0.750945 0.660365i \(-0.229598\pi\)
0.750945 + 0.660365i \(0.229598\pi\)
\(684\) 8.15160e6 0.666197
\(685\) 1.49350e6 0.121613
\(686\) 0 0
\(687\) 7.97275e6 0.644490
\(688\) 9.92420e6 0.799327
\(689\) 1.16302e6 0.0933337
\(690\) 8.99076e6 0.718908
\(691\) 1.76717e7 1.40793 0.703967 0.710233i \(-0.251410\pi\)
0.703967 + 0.710233i \(0.251410\pi\)
\(692\) −2.45752e7 −1.95088
\(693\) 0 0
\(694\) −1.53973e7 −1.21351
\(695\) 5.03386e6 0.395311
\(696\) 8.88036e6 0.694877
\(697\) −2.51506e6 −0.196095
\(698\) −1.57826e7 −1.22614
\(699\) −5.36547e6 −0.415351
\(700\) 0 0
\(701\) 8.22993e6 0.632559 0.316279 0.948666i \(-0.397566\pi\)
0.316279 + 0.948666i \(0.397566\pi\)
\(702\) −7.59703e6 −0.581837
\(703\) −1.37494e7 −1.04929
\(704\) 1.56169e6 0.118758
\(705\) 667320. 0.0505663
\(706\) 1.71552e7 1.29534
\(707\) 0 0
\(708\) 1.79486e7 1.34570
\(709\) 2.46367e7 1.84063 0.920314 0.391180i \(-0.127933\pi\)
0.920314 + 0.391180i \(0.127933\pi\)
\(710\) 1.47575e6 0.109867
\(711\) −392466. −0.0291158
\(712\) 2.58954e7 1.91436
\(713\) −1.21379e7 −0.894167
\(714\) 0 0
\(715\) −1.12710e7 −0.824510
\(716\) 5.62951e7 4.10382
\(717\) −6.68800e6 −0.485845
\(718\) −2.55569e7 −1.85011
\(719\) 1.88254e7 1.35807 0.679034 0.734107i \(-0.262399\pi\)
0.679034 + 0.734107i \(0.262399\pi\)
\(720\) −3.68503e6 −0.264917
\(721\) 0 0
\(722\) −5.63055e6 −0.401983
\(723\) −1.05735e7 −0.752272
\(724\) 9.62411e6 0.682361
\(725\) 6.09823e6 0.430882
\(726\) −5.12186e6 −0.360650
\(727\) −6.77607e6 −0.475491 −0.237745 0.971328i \(-0.576408\pi\)
−0.237745 + 0.971328i \(0.576408\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −4.07486e6 −0.283012
\(731\) −2.91073e6 −0.201469
\(732\) −1.37752e7 −0.950212
\(733\) 1.72227e7 1.18397 0.591986 0.805948i \(-0.298344\pi\)
0.591986 + 0.805948i \(0.298344\pi\)
\(734\) 2.15459e7 1.47613
\(735\) 0 0
\(736\) 2.60102e7 1.76990
\(737\) −2.58779e7 −1.75493
\(738\) 3.70765e6 0.250587
\(739\) −1.76534e7 −1.18910 −0.594548 0.804060i \(-0.702669\pi\)
−0.594548 + 0.804060i \(0.702669\pi\)
\(740\) 1.70704e7 1.14595
\(741\) −1.27292e7 −0.851641
\(742\) 0 0
\(743\) −1.36977e7 −0.910281 −0.455141 0.890420i \(-0.650411\pi\)
−0.455141 + 0.890420i \(0.650411\pi\)
\(744\) 1.10033e7 0.728768
\(745\) −3.33188e6 −0.219937
\(746\) 2.79341e7 1.83775
\(747\) −4.87178e6 −0.319438
\(748\) −1.89668e7 −1.23948
\(749\) 0 0
\(750\) −1.24269e7 −0.806698
\(751\) 1.16089e7 0.751090 0.375545 0.926804i \(-0.377456\pi\)
0.375545 + 0.926804i \(0.377456\pi\)
\(752\) 5.97885e6 0.385543
\(753\) 3.17693e6 0.204183
\(754\) −2.48167e7 −1.58970
\(755\) 3.89526e6 0.248696
\(756\) 0 0
\(757\) 6.25226e6 0.396550 0.198275 0.980146i \(-0.436466\pi\)
0.198275 + 0.980146i \(0.436466\pi\)
\(758\) −2.08868e7 −1.32038
\(759\) −1.72350e7 −1.08594
\(760\) −1.36563e7 −0.857629
\(761\) 3.02125e7 1.89115 0.945574 0.325406i \(-0.105501\pi\)
0.945574 + 0.325406i \(0.105501\pi\)
\(762\) −2.14605e6 −0.133891
\(763\) 0 0
\(764\) 481393. 0.0298378
\(765\) 1.08081e6 0.0667721
\(766\) −1.54277e7 −0.950014
\(767\) −2.80278e7 −1.72028
\(768\) 1.64280e7 1.00504
\(769\) 2.58756e7 1.57788 0.788940 0.614470i \(-0.210630\pi\)
0.788940 + 0.614470i \(0.210630\pi\)
\(770\) 0 0
\(771\) 90114.1 0.00545955
\(772\) 3.28251e7 1.98227
\(773\) −2.55409e7 −1.53740 −0.768701 0.639608i \(-0.779097\pi\)
−0.768701 + 0.639608i \(0.779097\pi\)
\(774\) 4.29095e6 0.257455
\(775\) 7.55605e6 0.451898
\(776\) 2.63746e7 1.57228
\(777\) 0 0
\(778\) 4.60140e7 2.72547
\(779\) 6.21236e6 0.366786
\(780\) 1.58038e7 0.930087
\(781\) −2.82896e6 −0.165958
\(782\) −2.36258e7 −1.38156
\(783\) 1.73602e6 0.101193
\(784\) 0 0
\(785\) 1.32195e7 0.765667
\(786\) −3.11707e7 −1.79966
\(787\) 8.81179e6 0.507139 0.253570 0.967317i \(-0.418395\pi\)
0.253570 + 0.967317i \(0.418395\pi\)
\(788\) 5.92512e7 3.39924
\(789\) −1.51362e7 −0.865614
\(790\) 1.17665e6 0.0670779
\(791\) 0 0
\(792\) 1.56239e7 0.885068
\(793\) 2.15108e7 1.21471
\(794\) 3.60018e7 2.02662
\(795\) 243917. 0.0136875
\(796\) −5.84871e7 −3.27173
\(797\) 2.07624e7 1.15780 0.578899 0.815399i \(-0.303483\pi\)
0.578899 + 0.815399i \(0.303483\pi\)
\(798\) 0 0
\(799\) −1.75358e6 −0.0971757
\(800\) −1.61918e7 −0.894481
\(801\) 5.06229e6 0.278783
\(802\) −9.27413e6 −0.509140
\(803\) 7.81136e6 0.427501
\(804\) 3.62850e7 1.97965
\(805\) 0 0
\(806\) −3.07493e7 −1.66724
\(807\) −1.72546e7 −0.932653
\(808\) −7.64973e7 −4.12209
\(809\) 1.37021e7 0.736067 0.368034 0.929813i \(-0.380031\pi\)
0.368034 + 0.929813i \(0.380031\pi\)
\(810\) −1.59331e6 −0.0853271
\(811\) −3.30799e7 −1.76609 −0.883043 0.469292i \(-0.844509\pi\)
−0.883043 + 0.469292i \(0.844509\pi\)
\(812\) 0 0
\(813\) 2.06100e7 1.09358
\(814\) −4.71613e7 −2.49474
\(815\) 468563. 0.0247101
\(816\) 9.68350e6 0.509104
\(817\) 7.18971e6 0.376840
\(818\) −4.41520e7 −2.30711
\(819\) 0 0
\(820\) −7.71285e6 −0.400572
\(821\) 1.48323e7 0.767982 0.383991 0.923337i \(-0.374549\pi\)
0.383991 + 0.923337i \(0.374549\pi\)
\(822\) 5.78558e6 0.298653
\(823\) 1.08722e7 0.559525 0.279762 0.960069i \(-0.409744\pi\)
0.279762 + 0.960069i \(0.409744\pi\)
\(824\) −2.15640e7 −1.10640
\(825\) 1.07291e7 0.548817
\(826\) 0 0
\(827\) −1.34267e7 −0.682662 −0.341331 0.939943i \(-0.610878\pi\)
−0.341331 + 0.939943i \(0.610878\pi\)
\(828\) 2.41663e7 1.22499
\(829\) 1.46754e7 0.741658 0.370829 0.928701i \(-0.379074\pi\)
0.370829 + 0.928701i \(0.379074\pi\)
\(830\) 1.46060e7 0.735931
\(831\) 3.55322e6 0.178492
\(832\) 3.41941e6 0.171255
\(833\) 0 0
\(834\) 1.95004e7 0.970796
\(835\) 2.25120e6 0.111737
\(836\) 4.68492e7 2.31839
\(837\) 2.15103e6 0.106129
\(838\) 1.55444e7 0.764654
\(839\) −2.76635e7 −1.35676 −0.678379 0.734712i \(-0.737317\pi\)
−0.678379 + 0.734712i \(0.737317\pi\)
\(840\) 0 0
\(841\) −1.48402e7 −0.723519
\(842\) −4.51946e7 −2.19688
\(843\) 1.59929e7 0.775103
\(844\) 4.98798e6 0.241028
\(845\) −1.58593e7 −0.764084
\(846\) 2.58509e6 0.124180
\(847\) 0 0
\(848\) 2.18537e6 0.104361
\(849\) 1.09372e7 0.520759
\(850\) 1.47075e7 0.698219
\(851\) −4.07616e7 −1.92942
\(852\) 3.96667e6 0.187209
\(853\) −3.43515e6 −0.161649 −0.0808244 0.996728i \(-0.525755\pi\)
−0.0808244 + 0.996728i \(0.525755\pi\)
\(854\) 0 0
\(855\) −2.66967e6 −0.124894
\(856\) 1.99622e7 0.931158
\(857\) 6.40626e6 0.297956 0.148978 0.988840i \(-0.452402\pi\)
0.148978 + 0.988840i \(0.452402\pi\)
\(858\) −4.36619e7 −2.02481
\(859\) −1.81739e7 −0.840360 −0.420180 0.907441i \(-0.638033\pi\)
−0.420180 + 0.907441i \(0.638033\pi\)
\(860\) −8.92627e6 −0.411551
\(861\) 0 0
\(862\) −7.00114e7 −3.20923
\(863\) 818774. 0.0374229 0.0187114 0.999825i \(-0.494044\pi\)
0.0187114 + 0.999825i \(0.494044\pi\)
\(864\) −4.60943e6 −0.210069
\(865\) 8.04843e6 0.365739
\(866\) 4.08267e7 1.84991
\(867\) 9.93858e6 0.449031
\(868\) 0 0
\(869\) −2.25560e6 −0.101324
\(870\) −5.20475e6 −0.233132
\(871\) −5.66613e7 −2.53070
\(872\) −1.50968e7 −0.672350
\(873\) 5.15596e6 0.228968
\(874\) 5.83574e7 2.58415
\(875\) 0 0
\(876\) −1.09528e7 −0.482242
\(877\) −3.33962e7 −1.46622 −0.733108 0.680113i \(-0.761931\pi\)
−0.733108 + 0.680113i \(0.761931\pi\)
\(878\) −1.51882e7 −0.664921
\(879\) −1.34008e7 −0.585002
\(880\) −2.11788e7 −0.921921
\(881\) 1.24325e7 0.539658 0.269829 0.962908i \(-0.413033\pi\)
0.269829 + 0.962908i \(0.413033\pi\)
\(882\) 0 0
\(883\) −2.33298e7 −1.00695 −0.503476 0.864009i \(-0.667946\pi\)
−0.503476 + 0.864009i \(0.667946\pi\)
\(884\) −4.15290e7 −1.78739
\(885\) −5.87820e6 −0.252282
\(886\) −5.74778e7 −2.45989
\(887\) 4.75528e6 0.202940 0.101470 0.994839i \(-0.467645\pi\)
0.101470 + 0.994839i \(0.467645\pi\)
\(888\) 3.69514e7 1.57253
\(889\) 0 0
\(890\) −1.51772e7 −0.642269
\(891\) 3.05432e6 0.128890
\(892\) 4.49833e7 1.89295
\(893\) 4.33145e6 0.181763
\(894\) −1.29072e7 −0.540116
\(895\) −1.84368e7 −0.769356
\(896\) 0 0
\(897\) −3.77371e7 −1.56598
\(898\) 1.99245e7 0.824512
\(899\) 7.02661e6 0.289966
\(900\) −1.50440e7 −0.619093
\(901\) −640963. −0.0263040
\(902\) 2.13087e7 0.872050
\(903\) 0 0
\(904\) −4.00716e7 −1.63086
\(905\) −3.15192e6 −0.127924
\(906\) 1.50896e7 0.610742
\(907\) −1.90902e7 −0.770537 −0.385268 0.922805i \(-0.625891\pi\)
−0.385268 + 0.922805i \(0.625891\pi\)
\(908\) −7.26891e7 −2.92587
\(909\) −1.49544e7 −0.600289
\(910\) 0 0
\(911\) −1.22591e6 −0.0489399 −0.0244700 0.999701i \(-0.507790\pi\)
−0.0244700 + 0.999701i \(0.507790\pi\)
\(912\) −2.39189e7 −0.952257
\(913\) −2.79993e7 −1.11165
\(914\) −2.73296e7 −1.08210
\(915\) 4.51141e6 0.178139
\(916\) −6.42490e7 −2.53004
\(917\) 0 0
\(918\) 4.18688e6 0.163977
\(919\) −1.69227e7 −0.660970 −0.330485 0.943811i \(-0.607212\pi\)
−0.330485 + 0.943811i \(0.607212\pi\)
\(920\) −4.04855e7 −1.57699
\(921\) 1.82798e7 0.710103
\(922\) 4.75677e7 1.84283
\(923\) −6.19419e6 −0.239321
\(924\) 0 0
\(925\) 2.53749e7 0.975101
\(926\) −5.30195e7 −2.03193
\(927\) −4.21554e6 −0.161122
\(928\) −1.50573e7 −0.573954
\(929\) −6.77240e6 −0.257456 −0.128728 0.991680i \(-0.541089\pi\)
−0.128728 + 0.991680i \(0.541089\pi\)
\(930\) −6.44897e6 −0.244502
\(931\) 0 0
\(932\) 4.32380e7 1.63052
\(933\) 2.60806e6 0.0980875
\(934\) 5.17307e7 1.94035
\(935\) 6.21166e6 0.232369
\(936\) 3.42096e7 1.27632
\(937\) 1.41035e7 0.524779 0.262389 0.964962i \(-0.415489\pi\)
0.262389 + 0.964962i \(0.415489\pi\)
\(938\) 0 0
\(939\) 1.96389e6 0.0726866
\(940\) −5.37765e6 −0.198505
\(941\) 5.62524e6 0.207094 0.103547 0.994625i \(-0.466981\pi\)
0.103547 + 0.994625i \(0.466981\pi\)
\(942\) 5.12102e7 1.88031
\(943\) 1.84172e7 0.674441
\(944\) −5.26657e7 −1.92353
\(945\) 0 0
\(946\) 2.46611e7 0.895952
\(947\) 2.64126e7 0.957052 0.478526 0.878073i \(-0.341171\pi\)
0.478526 + 0.878073i \(0.341171\pi\)
\(948\) 3.16272e6 0.114298
\(949\) 1.71035e7 0.616480
\(950\) −3.63286e7 −1.30599
\(951\) −1.16157e7 −0.416479
\(952\) 0 0
\(953\) 4.21824e7 1.50452 0.752262 0.658864i \(-0.228963\pi\)
0.752262 + 0.658864i \(0.228963\pi\)
\(954\) 944896. 0.0336135
\(955\) −157657. −0.00559378
\(956\) 5.38957e7 1.90726
\(957\) 9.97732e6 0.352155
\(958\) 5.67511e7 1.99784
\(959\) 0 0
\(960\) 717145. 0.0251148
\(961\) −1.99228e7 −0.695892
\(962\) −1.03263e8 −3.59755
\(963\) 3.90240e6 0.135602
\(964\) 8.52077e7 2.95315
\(965\) −1.07503e7 −0.371622
\(966\) 0 0
\(967\) 4.01501e7 1.38077 0.690384 0.723443i \(-0.257441\pi\)
0.690384 + 0.723443i \(0.257441\pi\)
\(968\) 2.30638e7 0.791120
\(969\) 7.01533e6 0.240015
\(970\) −1.54580e7 −0.527503
\(971\) −1.64551e7 −0.560084 −0.280042 0.959988i \(-0.590348\pi\)
−0.280042 + 0.959988i \(0.590348\pi\)
\(972\) −4.28266e6 −0.145394
\(973\) 0 0
\(974\) −7.03207e7 −2.37512
\(975\) 2.34920e7 0.791424
\(976\) 4.04200e7 1.35823
\(977\) 1.71176e7 0.573729 0.286865 0.957971i \(-0.407387\pi\)
0.286865 + 0.957971i \(0.407387\pi\)
\(978\) 1.81514e6 0.0606825
\(979\) 2.90942e7 0.970174
\(980\) 0 0
\(981\) −2.95128e6 −0.0979125
\(982\) 7.61124e7 2.51870
\(983\) 2.33803e7 0.771730 0.385865 0.922555i \(-0.373903\pi\)
0.385865 + 0.922555i \(0.373903\pi\)
\(984\) −1.66956e7 −0.549686
\(985\) −1.94049e7 −0.637266
\(986\) 1.36770e7 0.448021
\(987\) 0 0
\(988\) 1.02579e8 3.34324
\(989\) 2.13146e7 0.692927
\(990\) −9.15712e6 −0.296941
\(991\) −4.70955e6 −0.152333 −0.0761667 0.997095i \(-0.524268\pi\)
−0.0761667 + 0.997095i \(0.524268\pi\)
\(992\) −1.86568e7 −0.601948
\(993\) 3.13705e7 1.00960
\(994\) 0 0
\(995\) 1.91547e7 0.613362
\(996\) 3.92596e7 1.25400
\(997\) 4.55428e7 1.45105 0.725524 0.688197i \(-0.241597\pi\)
0.725524 + 0.688197i \(0.241597\pi\)
\(998\) 2.20150e7 0.699667
\(999\) 7.22363e6 0.229003
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 147.6.a.l.1.4 4
3.2 odd 2 441.6.a.v.1.1 4
7.2 even 3 147.6.e.o.67.1 8
7.3 odd 6 21.6.e.c.16.1 yes 8
7.4 even 3 147.6.e.o.79.1 8
7.5 odd 6 21.6.e.c.4.1 8
7.6 odd 2 147.6.a.m.1.4 4
21.5 even 6 63.6.e.e.46.4 8
21.17 even 6 63.6.e.e.37.4 8
21.20 even 2 441.6.a.w.1.1 4
28.3 even 6 336.6.q.j.289.2 8
28.19 even 6 336.6.q.j.193.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.1 8 7.5 odd 6
21.6.e.c.16.1 yes 8 7.3 odd 6
63.6.e.e.37.4 8 21.17 even 6
63.6.e.e.46.4 8 21.5 even 6
147.6.a.l.1.4 4 1.1 even 1 trivial
147.6.a.m.1.4 4 7.6 odd 2
147.6.e.o.67.1 8 7.2 even 3
147.6.e.o.79.1 8 7.4 even 3
336.6.q.j.193.2 8 28.19 even 6
336.6.q.j.289.2 8 28.3 even 6
441.6.a.v.1.1 4 3.2 odd 2
441.6.a.w.1.1 4 21.20 even 2