Properties

Label 147.6.a.m.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 \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(6\) 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 \(-0.815343\pi\)
−0.836399 + 0.548121i \(0.815343\pi\)
\(14\) 0 0
\(15\) 213.775 0.245318
\(16\) 1915.32 1.87043
\(17\) 561.757 0.471440 0.235720 0.971821i \(-0.424255\pi\)
0.235720 + 0.971821i \(0.424255\pi\)
\(18\) 828.132 0.602446
\(19\) −1387.58 −0.881807 −0.440903 0.897555i \(-0.645342\pi\)
−0.440903 + 0.897555i \(0.645342\pi\)
\(20\) 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 \(-0.411080\pi\)
0.275730 + 0.961235i \(0.411080\pi\)
\(32\) 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 \(-0.566687\pi\)
−0.207974 + 0.978134i \(0.566687\pi\)
\(42\) 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 \(-0.532864\pi\)
−0.103063 + 0.994675i \(0.532864\pi\)
\(48\) 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 \(-0.328091\pi\)
0.514194 + 0.857674i \(0.328091\pi\)
\(60\) 15504.5 0.556007
\(61\) −21103.5 −0.726157 −0.363078 0.931759i \(-0.618274\pi\)
−0.363078 + 0.931759i \(0.618274\pi\)
\(62\) 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 \(-0.558991\pi\)
−0.184266 + 0.982876i \(0.558991\pi\)
\(74\) −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 \(-0.340943\pi\)
0.479156 + 0.877730i \(0.340943\pi\)
\(84\) 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 \(-0.637330\pi\)
−0.418174 + 0.908367i \(0.637330\pi\)
\(90\) 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 \(-0.611596\pi\)
−0.343451 + 0.939170i \(0.611596\pi\)
\(98\) 0 0
\(99\) 37707.6 0.386670
\(100\) −185728. −1.85728
\(101\) 184623. 1.80087 0.900434 0.434993i \(-0.143249\pi\)
0.900434 + 0.434993i \(0.143249\pi\)
\(102\) 51689.9 0.491932
\(103\) 52043.7 0.483365 0.241683 0.970355i \(-0.422301\pi\)
0.241683 + 0.970355i \(0.422301\pi\)
\(104\) −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 \(-0.831005\pi\)
−0.862345 + 0.506321i \(0.831005\pi\)
\(132\) 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 \(-0.345990\pi\)
0.465178 + 0.885217i \(0.345990\pi\)
\(140\) 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 \(-0.142842\pi\)
0.900990 + 0.433840i \(0.142842\pi\)
\(158\) −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 \(-0.458025\pi\)
0.131486 + 0.991318i \(0.458025\pi\)
\(168\) 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 \(-0.358380\pi\)
0.430379 + 0.902649i \(0.358380\pi\)
\(174\) −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 \(-0.548099\pi\)
−0.150534 + 0.988605i \(0.548099\pi\)
\(182\) 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 \(-0.243330\pi\)
0.721767 + 0.692136i \(0.243330\pi\)
\(200\) −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 \(-0.637128\pi\)
−0.417598 + 0.908632i \(0.637128\pi\)
\(224\) 0 0
\(225\) −207425. −0.273152
\(226\) −988760. −1.28771
\(227\) 1.00223e6 1.29093 0.645467 0.763788i \(-0.276663\pi\)
0.645467 + 0.763788i \(0.276663\pi\)
\(228\) −905734. −1.15389
\(229\) 885861. 1.11629 0.558145 0.829743i \(-0.311513\pi\)
0.558145 + 0.829743i \(0.311513\pi\)
\(230\) 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 \(-0.725854\pi\)
−0.651487 + 0.758660i \(0.725854\pi\)
\(242\) 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 \(-0.443416\pi\)
0.176828 + 0.984242i \(0.443416\pi\)
\(252\) 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 \(-0.498495\pi\)
0.00472811 + 0.999989i \(0.498495\pi\)
\(258\) 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 \(-0.799289\pi\)
−0.807702 + 0.589591i \(0.799289\pi\)
\(270\) 177034. 0.147791
\(271\) 2.29000e6 1.89414 0.947069 0.321029i \(-0.104029\pi\)
0.947069 + 0.321029i \(0.104029\pi\)
\(272\) 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 \(-0.351071\pi\)
0.450991 + 0.892529i \(0.351071\pi\)
\(284\) −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 \(-0.669108\pi\)
−0.506627 + 0.862165i \(0.669108\pi\)
\(294\) 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 \(-0.289169\pi\)
0.614968 + 0.788552i \(0.289169\pi\)
\(308\) 0 0
\(309\) 468394. 0.279071
\(310\) 716553. 0.423491
\(311\) 289785. 0.169893 0.0849463 0.996386i \(-0.472928\pi\)
0.0849463 + 0.996386i \(0.472928\pi\)
\(312\) −3.80106e6 −2.21064
\(313\) 218211. 0.125897 0.0629484 0.998017i \(-0.479950\pi\)
0.0629484 + 0.998017i \(0.479950\pi\)
\(314\) 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 \(-0.389840\pi\)
0.339212 + 0.940710i \(0.389840\pi\)
\(350\) 0 0
\(351\) −743070. −0.321930
\(352\) 2.94350e6 1.26621
\(353\) −1.67796e6 −0.716710 −0.358355 0.933585i \(-0.616662\pi\)
−0.358355 + 0.933585i \(0.616662\pi\)
\(354\) 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 \(-0.633903\pi\)
−0.408370 + 0.912816i \(0.633903\pi\)
\(368\) 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 \(-0.415347\pi\)
0.262821 + 0.964845i \(0.415347\pi\)
\(384\) −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 \(-0.689455\pi\)
−0.560665 + 0.828042i \(0.689455\pi\)
\(398\) 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 \(-0.279654\pi\)
0.638260 + 0.769821i \(0.279654\pi\)
\(410\) −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 \(-0.567848\pi\)
−0.211541 + 0.977369i \(0.567848\pi\)
\(420\) 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 \(-0.671012\pi\)
−0.511777 + 0.859119i \(0.671012\pi\)
\(434\) 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 \(-0.441111\pi\)
0.183950 + 0.982936i \(0.441111\pi\)
\(440\) 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 \(-0.670287\pi\)
−0.509819 + 0.860282i \(0.670287\pi\)
\(462\) 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 \(-0.680367\pi\)
−0.536799 + 0.843710i \(0.680367\pi\)
\(468\) −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 \(-0.686403\pi\)
−0.552702 + 0.833379i \(0.686403\pi\)
\(480\) 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 \(-0.225844\pi\)
0.758681 + 0.651463i \(0.225844\pi\)
\(504\) 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 \(-0.407714\pi\)
0.285879 + 0.958266i \(0.407714\pi\)
\(510\) 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 \(-0.569472\pi\)
−0.216523 + 0.976278i \(0.569472\pi\)
\(522\) −1.97209e6 −0.316775
\(523\) −5.42355e6 −0.867021 −0.433511 0.901148i \(-0.642725\pi\)
−0.433511 + 0.901148i \(0.642725\pi\)
\(524\) −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 \(-0.491556\pi\)
0.0265244 + 0.999648i \(0.491556\pi\)
\(564\) −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 \(-0.535240\pi\)
−0.110483 + 0.993878i \(0.535240\pi\)
\(578\) −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 \(-0.673352\pi\)
−0.518078 + 0.855333i \(0.673352\pi\)
\(588\) 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 \(-0.141955\pi\)
0.902194 + 0.431329i \(0.141955\pi\)
\(594\) 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 \(-0.243969\pi\)
0.720376 + 0.693583i \(0.243969\pi\)
\(602\) 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 \(-0.770059\pi\)
−0.750233 + 0.661173i \(0.770059\pi\)
\(608\) −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 \(-0.514604\pi\)
−0.0458652 + 0.998948i \(0.514604\pi\)
\(620\) 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 \(-0.690153\pi\)
−0.562481 + 0.826810i \(0.690153\pi\)
\(644\) 0 0
\(645\) 1.10767e6 0.104837
\(646\) −7.96930e6 −0.751344
\(647\) −2.43380e6 −0.228573 −0.114286 0.993448i \(-0.536458\pi\)
−0.114286 + 0.993448i \(0.536458\pi\)
\(648\) 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 \(-0.776685\pi\)
−0.763832 + 0.645415i \(0.776685\pi\)
\(662\) −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 \(-0.301511\pi\)
0.583938 + 0.811798i \(0.301511\pi\)
\(678\) −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 \(-0.748590\pi\)
−0.703967 + 0.710233i \(0.748590\pi\)
\(692\) 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 \(-0.737601\pi\)
−0.679034 + 0.734107i \(0.737601\pi\)
\(720\) 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 \(-0.423592\pi\)
0.237745 + 0.971328i \(0.423592\pi\)
\(728\) 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 \(-0.701656\pi\)
−0.591986 + 0.805948i \(0.701656\pi\)
\(734\) −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 \(-0.894499\pi\)
−0.945574 + 0.325406i \(0.894499\pi\)
\(762\) 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 \(-0.789370\pi\)
−0.788940 + 0.614470i \(0.789370\pi\)
\(770\) 0 0
\(771\) 90114.1 0.00545955
\(772\) 3.28251e7 1.98227
\(773\) 2.55409e7 1.53740 0.768701 0.639608i \(-0.220903\pi\)
0.768701 + 0.639608i \(0.220903\pi\)
\(774\) 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 \(-0.581605\pi\)
−0.253570 + 0.967317i \(0.581605\pi\)
\(788\) 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 \(-0.696517\pi\)
−0.578899 + 0.815399i \(0.696517\pi\)
\(798\) 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 \(-0.155491\pi\)
0.883043 + 0.469292i \(0.155491\pi\)
\(812\) 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 \(-0.620926\pi\)
−0.370829 + 0.928701i \(0.620926\pi\)
\(830\) 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 \(-0.262683\pi\)
0.678379 + 0.734712i \(0.262683\pi\)
\(840\) 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 \(-0.474245\pi\)
0.0808244 + 0.996728i \(0.474245\pi\)
\(854\) 0 0
\(855\) −2.66967e6 −0.124894
\(856\) 1.99622e7 0.931158
\(857\) −6.40626e6 −0.297956 −0.148978 0.988840i \(-0.547598\pi\)
−0.148978 + 0.988840i \(0.547598\pi\)
\(858\) −4.36619e7 −2.02481
\(859\) 1.81739e7 0.840360 0.420180 0.907441i \(-0.361967\pi\)
0.420180 + 0.907441i \(0.361967\pi\)
\(860\) 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 \(-0.586967\pi\)
−0.269829 + 0.962908i \(0.586967\pi\)
\(882\) 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 \(-0.532355\pi\)
−0.101470 + 0.994839i \(0.532355\pi\)
\(888\) −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 \(-0.458911\pi\)
0.128728 + 0.991680i \(0.458911\pi\)
\(930\) 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 \(-0.584511\pi\)
−0.262389 + 0.964962i \(0.584511\pi\)
\(938\) 0 0
\(939\) 1.96389e6 0.0726866
\(940\) −5.37765e6 −0.198505
\(941\) −5.62524e6 −0.207094 −0.103547 0.994625i \(-0.533019\pi\)
−0.103547 + 0.994625i \(0.533019\pi\)
\(942\) 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 \(-0.409652\pi\)
0.280042 + 0.959988i \(0.409652\pi\)
\(972\) 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 \(-0.626097\pi\)
−0.385865 + 0.922555i \(0.626097\pi\)
\(984\) −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 \(-0.758403\pi\)
−0.725524 + 0.688197i \(0.758403\pi\)
\(998\) 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.m.1.4 4
3.2 odd 2 441.6.a.w.1.1 4
7.2 even 3 21.6.e.c.4.1 8
7.3 odd 6 147.6.e.o.79.1 8
7.4 even 3 21.6.e.c.16.1 yes 8
7.5 odd 6 147.6.e.o.67.1 8
7.6 odd 2 147.6.a.l.1.4 4
21.2 odd 6 63.6.e.e.46.4 8
21.11 odd 6 63.6.e.e.37.4 8
21.20 even 2 441.6.a.v.1.1 4
28.11 odd 6 336.6.q.j.289.2 8
28.23 odd 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 7.2 even 3
21.6.e.c.16.1 yes 8 7.4 even 3
63.6.e.e.37.4 8 21.11 odd 6
63.6.e.e.46.4 8 21.2 odd 6
147.6.a.l.1.4 4 7.6 odd 2
147.6.a.m.1.4 4 1.1 even 1 trivial
147.6.e.o.67.1 8 7.5 odd 6
147.6.e.o.79.1 8 7.3 odd 6
336.6.q.j.193.2 8 28.23 odd 6
336.6.q.j.289.2 8 28.11 odd 6
441.6.a.v.1.1 4 21.20 even 2
441.6.a.w.1.1 4 3.2 odd 2