Properties

Label 441.6.a.w.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
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] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
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 1.1
Root \(-9.22385\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.2239 q^{2} +72.5272 q^{4} -23.7528 q^{5} -414.344 q^{8} +O(q^{10})\) \(q-10.2239 q^{2} +72.5272 q^{4} -23.7528 q^{5} -414.344 q^{8} +242.845 q^{10} -465.526 q^{11} -1019.30 q^{13} +1915.32 q^{16} -561.757 q^{17} -1387.58 q^{19} -1722.72 q^{20} +4759.47 q^{22} -4113.62 q^{23} -2560.80 q^{25} +10421.2 q^{26} +2381.37 q^{29} +2950.66 q^{31} -6322.95 q^{32} +5743.32 q^{34} -9908.95 q^{37} +14186.4 q^{38} +9841.83 q^{40} +4477.13 q^{41} +5181.48 q^{43} -33763.3 q^{44} +42057.0 q^{46} +3121.59 q^{47} +26181.3 q^{50} -73926.9 q^{52} -1141.00 q^{53} +11057.6 q^{55} -24346.8 q^{58} -27497.1 q^{59} -21103.5 q^{61} -30167.1 q^{62} +3354.67 q^{64} +24211.2 q^{65} -55588.4 q^{67} -40742.6 q^{68} +6076.90 q^{71} -16779.6 q^{73} +101308. q^{74} -100637. q^{76} -4845.26 q^{79} -45494.2 q^{80} -45773.5 q^{82} -60145.4 q^{83} +13343.3 q^{85} -52974.7 q^{86} +192888. q^{88} +62497.4 q^{89} -298349. q^{92} -31914.7 q^{94} +32958.9 q^{95} -63653.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} + 283 q^{10} - 402 q^{11} + 462 q^{13} + 3273 q^{16} - 276 q^{17} + 510 q^{19} - 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} + 15138 q^{26} - 540 q^{29} - 6410 q^{31} - 15519 q^{32} + 21144 q^{34} + 15250 q^{37} + 41250 q^{38} - 8547 q^{40} - 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} + 15060 q^{47} + 7302 q^{50} - 47476 q^{52} - 13692 q^{53} + 73124 q^{55} + 52309 q^{58} - 34830 q^{59} - 5364 q^{61} - 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} + 58272 q^{68} - 89268 q^{71} + 59638 q^{73} + 185442 q^{74} + 21308 q^{76} - 44062 q^{79} + 33381 q^{80} + 57596 q^{82} + 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} + 77520 q^{89} - 158256 q^{92} - 73722 q^{94} + 221376 q^{95} - 188630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.2239 −1.80734 −0.903669 0.428231i \(-0.859137\pi\)
−0.903669 + 0.428231i \(0.859137\pi\)
\(3\) 0 0
\(4\) 72.5272 2.26647
\(5\) −23.7528 −0.424903 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −414.344 −2.28895
\(9\) 0 0
\(10\) 242.845 0.767944
\(11\) −465.526 −1.16001 −0.580006 0.814612i \(-0.696950\pi\)
−0.580006 + 0.814612i \(0.696950\pi\)
\(12\) 0 0
\(13\) −1019.30 −1.67280 −0.836399 0.548121i \(-0.815343\pi\)
−0.836399 + 0.548121i \(0.815343\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1915.32 1.87043
\(17\) −561.757 −0.471440 −0.235720 0.971821i \(-0.575745\pi\)
−0.235720 + 0.971821i \(0.575745\pi\)
\(18\) 0 0
\(19\) −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 \(-0.800928\pi\)
−0.810727 + 0.585425i \(0.800928\pi\)
\(24\) 0 0
\(25\) −2560.80 −0.819457
\(26\) 10421.2 3.02331
\(27\) 0 0
\(28\) 0 0
\(29\) 2381.37 0.525814 0.262907 0.964821i \(-0.415319\pi\)
0.262907 + 0.964821i \(0.415319\pi\)
\(30\) 0 0
\(31\) 2950.66 0.551460 0.275730 0.961235i \(-0.411080\pi\)
0.275730 + 0.961235i \(0.411080\pi\)
\(32\) −6322.95 −1.09155
\(33\) 0 0
\(34\) 5743.32 0.852051
\(35\) 0 0
\(36\) 0 0
\(37\) −9908.95 −1.18994 −0.594968 0.803750i \(-0.702835\pi\)
−0.594968 + 0.803750i \(0.702835\pi\)
\(38\) 14186.4 1.59372
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 42057.0 2.93052
\(47\) 3121.59 0.206125 0.103063 0.994675i \(-0.467136\pi\)
0.103063 + 0.994675i \(0.467136\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 26181.3 1.48104
\(51\) 0 0
\(52\) −73926.9 −3.79135
\(53\) −1141.00 −0.0557950 −0.0278975 0.999611i \(-0.508881\pi\)
−0.0278975 + 0.999611i \(0.508881\pi\)
\(54\) 0 0
\(55\) 11057.6 0.492893
\(56\) 0 0
\(57\) 0 0
\(58\) −24346.8 −0.950325
\(59\) −27497.1 −1.02839 −0.514194 0.857674i \(-0.671909\pi\)
−0.514194 + 0.857674i \(0.671909\pi\)
\(60\) 0 0
\(61\) −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
\(66\) 0 0
\(67\) −55588.4 −1.51286 −0.756428 0.654077i \(-0.773057\pi\)
−0.756428 + 0.654077i \(0.773057\pi\)
\(68\) −40742.6 −1.06851
\(69\) 0 0
\(70\) 0 0
\(71\) 6076.90 0.143066 0.0715330 0.997438i \(-0.477211\pi\)
0.0715330 + 0.997438i \(0.477211\pi\)
\(72\) 0 0
\(73\) −16779.6 −0.368532 −0.184266 0.982876i \(-0.558991\pi\)
−0.184266 + 0.982876i \(0.558991\pi\)
\(74\) 101308. 2.15062
\(75\) 0 0
\(76\) −100637. −1.99859
\(77\) 0 0
\(78\) 0 0
\(79\) −4845.26 −0.0873473 −0.0436737 0.999046i \(-0.513906\pi\)
−0.0436737 + 0.999046i \(0.513906\pi\)
\(80\) −45494.2 −0.794752
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) 192888. 2.65520
\(89\) 62497.4 0.836348 0.418174 0.908367i \(-0.362670\pi\)
0.418174 + 0.908367i \(0.362670\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −298349. −3.67498
\(93\) 0 0
\(94\) −31914.7 −0.372539
\(95\) 32958.9 0.374683
\(96\) 0 0
\(97\) −63653.8 −0.686903 −0.343451 0.939170i \(-0.611596\pi\)
−0.343451 + 0.939170i \(0.611596\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −185728. −1.85728
\(101\) −184623. −1.80087 −0.900434 0.434993i \(-0.856751\pi\)
−0.900434 + 0.434993i \(0.856751\pi\)
\(102\) 0 0
\(103\) 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 \(-0.565200\pi\)
−0.203403 + 0.979095i \(0.565200\pi\)
\(108\) 0 0
\(109\) −36435.6 −0.293737 −0.146869 0.989156i \(-0.546919\pi\)
−0.146869 + 0.989156i \(0.546919\pi\)
\(110\) −113051. −0.890824
\(111\) 0 0
\(112\) 0 0
\(113\) 96711.1 0.712492 0.356246 0.934392i \(-0.384056\pi\)
0.356246 + 0.934392i \(0.384056\pi\)
\(114\) 0 0
\(115\) 97710.0 0.688961
\(116\) 172714. 1.19174
\(117\) 0 0
\(118\) 281126. 1.85864
\(119\) 0 0
\(120\) 0 0
\(121\) 55663.5 0.345626
\(122\) 215759. 1.31241
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) −247532. −1.28462
\(131\) 338758. 1.72469 0.862345 0.506321i \(-0.168995\pi\)
0.862345 + 0.506321i \(0.168995\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 568328. 2.73424
\(135\) 0 0
\(136\) 232760. 1.07910
\(137\) 62876.7 0.286213 0.143106 0.989707i \(-0.454291\pi\)
0.143106 + 0.989707i \(0.454291\pi\)
\(138\) 0 0
\(139\) 211927. 0.930356 0.465178 0.885217i \(-0.345990\pi\)
0.465178 + 0.885217i \(0.345990\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −62129.4 −0.258569
\(143\) 474511. 1.94047
\(144\) 0 0
\(145\) −56564.3 −0.223420
\(146\) 171553. 0.666062
\(147\) 0 0
\(148\) −718668. −2.69696
\(149\) −140273. −0.517616 −0.258808 0.965929i \(-0.583330\pi\)
−0.258808 + 0.965929i \(0.583330\pi\)
\(150\) 0 0
\(151\) −163991. −0.585300 −0.292650 0.956220i \(-0.594537\pi\)
−0.292650 + 0.956220i \(0.594537\pi\)
\(152\) 574934. 2.01841
\(153\) 0 0
\(154\) 0 0
\(155\) −70086.4 −0.234317
\(156\) 0 0
\(157\) 556543. 1.80198 0.900990 0.433840i \(-0.142842\pi\)
0.900990 + 0.433840i \(0.142842\pi\)
\(158\) 49537.2 0.157866
\(159\) 0 0
\(160\) 150188. 0.463804
\(161\) 0 0
\(162\) 0 0
\(163\) −19726.6 −0.0581546 −0.0290773 0.999577i \(-0.509257\pi\)
−0.0290773 + 0.999577i \(0.509257\pi\)
\(164\) 324713. 0.942736
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 375798. 0.968576
\(173\) −338841. −0.860757 −0.430379 0.902649i \(-0.641620\pi\)
−0.430379 + 0.902649i \(0.641620\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −891631. −2.16972
\(177\) 0 0
\(178\) −638965. −1.51156
\(179\) −776193. −1.81066 −0.905330 0.424708i \(-0.860377\pi\)
−0.905330 + 0.424708i \(0.860377\pi\)
\(180\) 0 0
\(181\) −132697. −0.301067 −0.150534 0.988605i \(-0.548099\pi\)
−0.150534 + 0.988605i \(0.548099\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.70445e6 3.71142
\(185\) 235366. 0.505607
\(186\) 0 0
\(187\) 261512. 0.546875
\(188\) 226400. 0.467178
\(189\) 0 0
\(190\) −336967. −0.677178
\(191\) −6637.41 −0.0131648 −0.00658242 0.999978i \(-0.502095\pi\)
−0.00658242 + 0.999978i \(0.502095\pi\)
\(192\) 0 0
\(193\) 452590. 0.874604 0.437302 0.899315i \(-0.355934\pi\)
0.437302 + 0.899315i \(0.355934\pi\)
\(194\) 650787. 1.24147
\(195\) 0 0
\(196\) 0 0
\(197\) −816952. −1.49979 −0.749896 0.661556i \(-0.769896\pi\)
−0.749896 + 0.661556i \(0.769896\pi\)
\(198\) 0 0
\(199\) 806417. 1.44353 0.721767 0.692136i \(-0.243330\pi\)
0.721767 + 0.692136i \(0.243330\pi\)
\(200\) 1.06105e6 1.87569
\(201\) 0 0
\(202\) 1.88756e6 3.25478
\(203\) 0 0
\(204\) 0 0
\(205\) −106344. −0.176738
\(206\) −532087. −0.873605
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 492563. 0.735237
\(215\) −123075. −0.181582
\(216\) 0 0
\(217\) 0 0
\(218\) 372512. 0.530883
\(219\) 0 0
\(220\) 801973. 1.11713
\(221\) 572599. 0.788623
\(222\) 0 0
\(223\) −620227. −0.835196 −0.417598 0.908632i \(-0.637128\pi\)
−0.417598 + 0.908632i \(0.637128\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −988760. −1.28771
\(227\) −1.00223e6 −1.29093 −0.645467 0.763788i \(-0.723337\pi\)
−0.645467 + 0.763788i \(0.723337\pi\)
\(228\) 0 0
\(229\) 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 \(-0.617122\pi\)
−0.359704 + 0.933066i \(0.617122\pi\)
\(234\) 0 0
\(235\) −74146.6 −0.0875834
\(236\) −1.99429e6 −2.33081
\(237\) 0 0
\(238\) 0 0
\(239\) −743111. −0.841509 −0.420754 0.907175i \(-0.638235\pi\)
−0.420754 + 0.907175i \(0.638235\pi\)
\(240\) 0 0
\(241\) −1.17484e6 −1.30297 −0.651487 0.758660i \(-0.725854\pi\)
−0.651487 + 0.758660i \(0.725854\pi\)
\(242\) −569095. −0.624664
\(243\) 0 0
\(244\) −1.53058e6 −1.64582
\(245\) 0 0
\(246\) 0 0
\(247\) 1.41436e6 1.47508
\(248\) −1.22259e6 −1.26226
\(249\) 0 0
\(250\) −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
\(255\) 0 0
\(256\) −1.82533e6 −1.74077
\(257\) −10012.7 −0.00945621 −0.00472811 0.999989i \(-0.501505\pi\)
−0.00472811 + 0.999989i \(0.501505\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.75597e6 1.61096
\(261\) 0 0
\(262\) −3.46341e6 −3.11710
\(263\) −1.68180e6 −1.49929 −0.749644 0.661841i \(-0.769775\pi\)
−0.749644 + 0.661841i \(0.769775\pi\)
\(264\) 0 0
\(265\) 27101.9 0.0237075
\(266\) 0 0
\(267\) 0 0
\(268\) −4.03167e6 −3.42885
\(269\) 1.91717e6 1.61540 0.807702 0.589591i \(-0.200711\pi\)
0.807702 + 0.589591i \(0.200711\pi\)
\(270\) 0 0
\(271\) 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
\(276\) 0 0
\(277\) −394802. −0.309157 −0.154579 0.987980i \(-0.549402\pi\)
−0.154579 + 0.987980i \(0.549402\pi\)
\(278\) −2.16671e6 −1.68147
\(279\) 0 0
\(280\) 0 0
\(281\) 1.77699e6 1.34252 0.671259 0.741223i \(-0.265754\pi\)
0.671259 + 0.741223i \(0.265754\pi\)
\(282\) 0 0
\(283\) 1.21524e6 0.901981 0.450991 0.892529i \(-0.351071\pi\)
0.450991 + 0.892529i \(0.351071\pi\)
\(284\) 440741. 0.324255
\(285\) 0 0
\(286\) −4.85133e6 −3.50708
\(287\) 0 0
\(288\) 0 0
\(289\) −1.10429e6 −0.777745
\(290\) 578305. 0.403796
\(291\) 0 0
\(292\) −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
\(297\) 0 0
\(298\) 1.43413e6 0.935508
\(299\) 4.19301e6 2.71236
\(300\) 0 0
\(301\) 0 0
\(302\) 1.67662e6 1.05784
\(303\) 0 0
\(304\) −2.65766e6 −1.64936
\(305\) 501268. 0.308546
\(306\) 0 0
\(307\) 2.03109e6 1.22994 0.614968 0.788552i \(-0.289169\pi\)
0.614968 + 0.788552i \(0.289169\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 716553. 0.423491
\(311\) −289785. −0.169893 −0.0849463 0.996386i \(-0.527072\pi\)
−0.0849463 + 0.996386i \(0.527072\pi\)
\(312\) 0 0
\(313\) 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 \(-0.617456\pi\)
−0.360681 + 0.932689i \(0.617456\pi\)
\(318\) 0 0
\(319\) −1.10859e6 −0.609951
\(320\) −79682.8 −0.0435000
\(321\) 0 0
\(322\) 0 0
\(323\) 779481. 0.415719
\(324\) 0 0
\(325\) 2.61023e6 1.37079
\(326\) 201682. 0.105105
\(327\) 0 0
\(328\) −1.85507e6 −0.952084
\(329\) 0 0
\(330\) 0 0
\(331\) −3.48561e6 −1.74867 −0.874336 0.485321i \(-0.838703\pi\)
−0.874336 + 0.485321i \(0.838703\pi\)
\(332\) −4.36218e6 −2.17199
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 967752. 0.454012
\(341\) −1.37361e6 −0.639700
\(342\) 0 0
\(343\) 0 0
\(344\) −2.14692e6 −0.978180
\(345\) 0 0
\(346\) 3.46426e6 1.55568
\(347\) 1.50601e6 0.671437 0.335719 0.941962i \(-0.391021\pi\)
0.335719 + 0.941962i \(0.391021\pi\)
\(348\) 0 0
\(349\) 1.54370e6 0.678423 0.339212 0.940710i \(-0.389840\pi\)
0.339212 + 0.940710i \(0.389840\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.94350e6 1.26621
\(353\) 1.67796e6 0.716710 0.358355 0.933585i \(-0.383338\pi\)
0.358355 + 0.933585i \(0.383338\pi\)
\(354\) 0 0
\(355\) −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 \(-0.328967\pi\)
0.511832 + 0.859086i \(0.328967\pi\)
\(360\) 0 0
\(361\) −550727. −0.222417
\(362\) 1.35667e6 0.544130
\(363\) 0 0
\(364\) 0 0
\(365\) 398564. 0.156591
\(366\) 0 0
\(367\) −2.10741e6 −0.816740 −0.408370 0.912816i \(-0.633903\pi\)
−0.408370 + 0.912816i \(0.633903\pi\)
\(368\) −7.87890e6 −3.03281
\(369\) 0 0
\(370\) −2.40634e6 −0.913804
\(371\) 0 0
\(372\) 0 0
\(373\) 2.73225e6 1.01683 0.508414 0.861113i \(-0.330232\pi\)
0.508414 + 0.861113i \(0.330232\pi\)
\(374\) −2.67366e6 −0.988389
\(375\) 0 0
\(376\) −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
\(381\) 0 0
\(382\) 67859.9 0.0237933
\(383\) −1.50899e6 −0.525643 −0.262821 0.964845i \(-0.584653\pi\)
−0.262821 + 0.964845i \(0.584653\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.62721e6 −1.58071
\(387\) 0 0
\(388\) −4.61663e6 −1.55685
\(389\) −4.50065e6 −1.50800 −0.754000 0.656874i \(-0.771878\pi\)
−0.754000 + 0.656874i \(0.771878\pi\)
\(390\) 0 0
\(391\) 2.31085e6 0.764417
\(392\) 0 0
\(393\) 0 0
\(394\) 8.35240e6 2.71063
\(395\) 115089. 0.0371142
\(396\) 0 0
\(397\) −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 \(-0.455015\pi\)
0.140853 + 0.990030i \(0.455015\pi\)
\(402\) 0 0
\(403\) −3.00760e6 −0.922482
\(404\) −1.33902e7 −4.08162
\(405\) 0 0
\(406\) 0 0
\(407\) 4.61287e6 1.38034
\(408\) 0 0
\(409\) 4.31853e6 1.27652 0.638260 0.769821i \(-0.279654\pi\)
0.638260 + 0.769821i \(0.279654\pi\)
\(410\) 1.08725e6 0.319425
\(411\) 0 0
\(412\) 3.77458e6 1.09553
\(413\) 0 0
\(414\) 0 0
\(415\) 1.42862e6 0.407190
\(416\) 6.44498e6 1.82595
\(417\) 0 0
\(418\) −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
\(423\) 0 0
\(424\) 472765. 0.127712
\(425\) 1.43855e6 0.386325
\(426\) 0 0
\(427\) 0 0
\(428\) −3.49420e6 −0.922016
\(429\) 0 0
\(430\) 1.25830e6 0.328180
\(431\) 6.84785e6 1.77567 0.887833 0.460166i \(-0.152210\pi\)
0.887833 + 0.460166i \(0.152210\pi\)
\(432\) 0 0
\(433\) −3.99328e6 −1.02355 −0.511777 0.859119i \(-0.671012\pi\)
−0.511777 + 0.859119i \(0.671012\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.64257e6 −0.665748
\(437\) 5.70797e6 1.42981
\(438\) 0 0
\(439\) 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 \(-0.261750\pi\)
0.680528 + 0.732722i \(0.261750\pi\)
\(444\) 0 0
\(445\) −1.48449e6 −0.355367
\(446\) 6.34111e6 1.50948
\(447\) 0 0
\(448\) 0 0
\(449\) −1.94883e6 −0.456202 −0.228101 0.973637i \(-0.573252\pi\)
−0.228101 + 0.973637i \(0.573252\pi\)
\(450\) 0 0
\(451\) −2.08422e6 −0.482505
\(452\) 7.01418e6 1.61484
\(453\) 0 0
\(454\) 1.02467e7 2.33315
\(455\) 0 0
\(456\) 0 0
\(457\) −2.67312e6 −0.598726 −0.299363 0.954139i \(-0.596774\pi\)
−0.299363 + 0.954139i \(0.596774\pi\)
\(458\) −9.05691e6 −2.01751
\(459\) 0 0
\(460\) 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
\(465\) 0 0
\(466\) 6.09509e6 1.30021
\(467\) 5.05980e6 1.07360 0.536799 0.843710i \(-0.319633\pi\)
0.536799 + 0.843710i \(0.319633\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 758064. 0.158293
\(471\) 0 0
\(472\) 1.13932e7 2.35392
\(473\) −2.41212e6 −0.495730
\(474\) 0 0
\(475\) 3.55331e6 0.722603
\(476\) 0 0
\(477\) 0 0
\(478\) 7.59745e6 1.52089
\(479\) 5.55085e6 1.10540 0.552702 0.833379i \(-0.313597\pi\)
0.552702 + 0.833379i \(0.313597\pi\)
\(480\) 0 0
\(481\) 1.01002e7 1.99052
\(482\) 1.20114e7 2.35491
\(483\) 0 0
\(484\) 4.03711e6 0.783353
\(485\) 1.51196e6 0.291867
\(486\) 0 0
\(487\) −6.87810e6 −1.31415 −0.657077 0.753824i \(-0.728207\pi\)
−0.657077 + 0.753824i \(0.728207\pi\)
\(488\) 8.74411e6 1.66213
\(489\) 0 0
\(490\) 0 0
\(491\) −7.44459e6 −1.39360 −0.696798 0.717267i \(-0.745393\pi\)
−0.696798 + 0.717267i \(0.745393\pi\)
\(492\) 0 0
\(493\) −1.33775e6 −0.247890
\(494\) −1.44602e7 −2.66598
\(495\) 0 0
\(496\) 5.65145e6 1.03147
\(497\) 0 0
\(498\) 0 0
\(499\) 2.15329e6 0.387126 0.193563 0.981088i \(-0.437996\pi\)
0.193563 + 0.981088i \(0.437996\pi\)
\(500\) 9.79507e6 1.75220
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 1.69154e6 0.290820
\(509\) −3.34201e6 −0.571759 −0.285879 0.958266i \(-0.592286\pi\)
−0.285879 + 0.958266i \(0.592286\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.32848e7 2.23964
\(513\) 0 0
\(514\) 102368. 0.0170906
\(515\) −1.23619e6 −0.205383
\(516\) 0 0
\(517\) −1.45318e6 −0.239108
\(518\) 0 0
\(519\) 0 0
\(520\) −1.00318e7 −1.62693
\(521\) 2.68305e6 0.433046 0.216523 0.976278i \(-0.430528\pi\)
0.216523 + 0.976278i \(0.430528\pi\)
\(522\) 0 0
\(523\) −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
\(528\) 0 0
\(529\) 1.04855e7 1.62911
\(530\) −277086. −0.0428474
\(531\) 0 0
\(532\) 0 0
\(533\) −4.56353e6 −0.695798
\(534\) 0 0
\(535\) 1.14436e6 0.172853
\(536\) 2.30327e7 3.46285
\(537\) 0 0
\(538\) −1.96009e7 −2.91958
\(539\) 0 0
\(540\) 0 0
\(541\) 2.18456e6 0.320900 0.160450 0.987044i \(-0.448705\pi\)
0.160450 + 0.987044i \(0.448705\pi\)
\(542\) −2.34126e7 −3.42335
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) −1.21881e7 −1.71802
\(551\) −3.30434e6 −0.463667
\(552\) 0 0
\(553\) 0 0
\(554\) 4.03639e6 0.558752
\(555\) 0 0
\(556\) 1.53705e7 2.10863
\(557\) −5.47656e6 −0.747946 −0.373973 0.927440i \(-0.622005\pi\)
−0.373973 + 0.927440i \(0.622005\pi\)
\(558\) 0 0
\(559\) −5.28149e6 −0.714869
\(560\) 0 0
\(561\) 0 0
\(562\) −1.81677e7 −2.42638
\(563\) −398976. −0.0530488 −0.0265244 0.999648i \(-0.508444\pi\)
−0.0265244 + 0.999648i \(0.508444\pi\)
\(564\) 0 0
\(565\) −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 \(-0.411675\pi\)
0.273934 + 0.961749i \(0.411675\pi\)
\(570\) 0 0
\(571\) 8.75102e6 1.12323 0.561615 0.827399i \(-0.310180\pi\)
0.561615 + 0.827399i \(0.310180\pi\)
\(572\) 3.44149e7 4.39801
\(573\) 0 0
\(574\) 0 0
\(575\) 1.05342e7 1.32871
\(576\) 0 0
\(577\) −1.76712e6 −0.220966 −0.110483 0.993878i \(-0.535240\pi\)
−0.110483 + 0.993878i \(0.535240\pi\)
\(578\) 1.12901e7 1.40565
\(579\) 0 0
\(580\) −4.10245e6 −0.506376
\(581\) 0 0
\(582\) 0 0
\(583\) 531164. 0.0647228
\(584\) 6.95254e6 0.843551
\(585\) 0 0
\(586\) −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
\(591\) 0 0
\(592\) −1.89788e7 −2.22569
\(593\) −1.54514e7 −1.80439 −0.902194 0.431329i \(-0.858045\pi\)
−0.902194 + 0.431329i \(0.858045\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.01736e7 −1.17316
\(597\) 0 0
\(598\) −4.28687e7 −4.90216
\(599\) 3.84801e6 0.438196 0.219098 0.975703i \(-0.429688\pi\)
0.219098 + 0.975703i \(0.429688\pi\)
\(600\) 0 0
\(601\) 1.27578e7 1.44075 0.720376 0.693583i \(-0.243969\pi\)
0.720376 + 0.693583i \(0.243969\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.18938e7 −1.32657
\(605\) −1.32216e6 −0.146858
\(606\) 0 0
\(607\) −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
\(612\) 0 0
\(613\) 1.53474e7 1.64962 0.824812 0.565407i \(-0.191281\pi\)
0.824812 + 0.565407i \(0.191281\pi\)
\(614\) −2.07655e7 −2.22291
\(615\) 0 0
\(616\) 0 0
\(617\) −1.18485e7 −1.25300 −0.626500 0.779421i \(-0.715513\pi\)
−0.626500 + 0.779421i \(0.715513\pi\)
\(618\) 0 0
\(619\) −874460. −0.0917304 −0.0458652 0.998948i \(-0.514604\pi\)
−0.0458652 + 0.998948i \(0.514604\pi\)
\(620\) −5.08317e6 −0.531074
\(621\) 0 0
\(622\) 2.96272e6 0.307054
\(623\) 0 0
\(624\) 0 0
\(625\) 4.79460e6 0.490967
\(626\) −2.23095e6 −0.227538
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 1.31952e7 1.30375
\(635\) −553984. −0.0545209
\(636\) 0 0
\(637\) 0 0
\(638\) 1.13341e7 1.10239
\(639\) 0 0
\(640\) −3.99135e6 −0.385185
\(641\) −812551. −0.0781098 −0.0390549 0.999237i \(-0.512435\pi\)
−0.0390549 + 0.999237i \(0.512435\pi\)
\(642\) 0 0
\(643\) −1.17941e7 −1.12496 −0.562481 0.826810i \(-0.690153\pi\)
−0.562481 + 0.826810i \(0.690153\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −7.96930e6 −0.751344
\(647\) 2.43380e6 0.228573 0.114286 0.993448i \(-0.463542\pi\)
0.114286 + 0.993448i \(0.463542\pi\)
\(648\) 0 0
\(649\) 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 \(-0.356963\pi\)
0.434393 + 0.900723i \(0.356963\pi\)
\(654\) 0 0
\(655\) −8.04646e6 −0.732827
\(656\) 8.57513e6 0.778003
\(657\) 0 0
\(658\) 0 0
\(659\) −1.40662e7 −1.26172 −0.630860 0.775896i \(-0.717298\pi\)
−0.630860 + 0.775896i \(0.717298\pi\)
\(660\) 0 0
\(661\) −1.71606e7 −1.52766 −0.763832 0.645415i \(-0.776685\pi\)
−0.763832 + 0.645415i \(0.776685\pi\)
\(662\) 3.56363e7 3.16044
\(663\) 0 0
\(664\) 2.49209e7 2.19353
\(665\) 0 0
\(666\) 0 0
\(667\) −9.79606e6 −0.852583
\(668\) −6.87385e6 −0.596017
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 4.84249e7 4.07570
\(677\) −1.39274e7 −1.16788 −0.583938 0.811798i \(-0.698489\pi\)
−0.583938 + 0.811798i \(0.698489\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5.52872e6 −0.458513
\(681\) 0 0
\(682\) 1.40436e7 1.15615
\(683\) −1.83101e7 −1.50189 −0.750945 0.660365i \(-0.770402\pi\)
−0.750945 + 0.660365i \(0.770402\pi\)
\(684\) 0 0
\(685\) −1.49350e6 −0.121613
\(686\) 0 0
\(687\) 0 0
\(688\) 9.92420e6 0.799327
\(689\) 1.16302e6 0.0933337
\(690\) 0 0
\(691\) −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
\(696\) 0 0
\(697\) −2.51506e6 −0.196095
\(698\) −1.57826e7 −1.22614
\(699\) 0 0
\(700\) 0 0
\(701\) −8.22993e6 −0.632559 −0.316279 0.948666i \(-0.602434\pi\)
−0.316279 + 0.948666i \(0.602434\pi\)
\(702\) 0 0
\(703\) 1.37494e7 1.04929
\(704\) −1.56169e6 −0.118758
\(705\) 0 0
\(706\) −1.71552e7 −1.29534
\(707\) 0 0
\(708\) 0 0
\(709\) 2.46367e7 1.84063 0.920314 0.391180i \(-0.127933\pi\)
0.920314 + 0.391180i \(0.127933\pi\)
\(710\) 1.47575e6 0.109867
\(711\) 0 0
\(712\) −2.58954e7 −1.91436
\(713\) −1.21379e7 −0.894167
\(714\) 0 0
\(715\) −1.12710e7 −0.824510
\(716\) −5.62951e7 −4.10382
\(717\) 0 0
\(718\) −2.55569e7 −1.85011
\(719\) 1.88254e7 1.35807 0.679034 0.734107i \(-0.262399\pi\)
0.679034 + 0.734107i \(0.262399\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5.63055e6 0.401983
\(723\) 0 0
\(724\) −9.62411e6 −0.682361
\(725\) −6.09823e6 −0.430882
\(726\) 0 0
\(727\) 6.77607e6 0.475491 0.237745 0.971328i \(-0.423592\pi\)
0.237745 + 0.971328i \(0.423592\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.07486e6 −0.283012
\(731\) −2.91073e6 −0.201469
\(732\) 0 0
\(733\) −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
\(738\) 0 0
\(739\) −1.76534e7 −1.18910 −0.594548 0.804060i \(-0.702669\pi\)
−0.594548 + 0.804060i \(0.702669\pi\)
\(740\) 1.70704e7 1.14595
\(741\) 0 0
\(742\) 0 0
\(743\) 1.36977e7 0.910281 0.455141 0.890420i \(-0.349589\pi\)
0.455141 + 0.890420i \(0.349589\pi\)
\(744\) 0 0
\(745\) 3.33188e6 0.219937
\(746\) −2.79341e7 −1.83775
\(747\) 0 0
\(748\) 1.89668e7 1.23948
\(749\) 0 0
\(750\) 0 0
\(751\) 1.16089e7 0.751090 0.375545 0.926804i \(-0.377456\pi\)
0.375545 + 0.926804i \(0.377456\pi\)
\(752\) 5.97885e6 0.385543
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) −1.36563e7 −0.857629
\(761\) 3.02125e7 1.89115 0.945574 0.325406i \(-0.105501\pi\)
0.945574 + 0.325406i \(0.105501\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −481393. −0.0298378
\(765\) 0 0
\(766\) 1.54277e7 0.950014
\(767\) 2.80278e7 1.72028
\(768\) 0 0
\(769\) −2.58756e7 −1.57788 −0.788940 0.614470i \(-0.789370\pi\)
−0.788940 + 0.614470i \(0.789370\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.28251e7 1.98227
\(773\) −2.55409e7 −1.53740 −0.768701 0.639608i \(-0.779097\pi\)
−0.768701 + 0.639608i \(0.779097\pi\)
\(774\) 0 0
\(775\) −7.55605e6 −0.451898
\(776\) 2.63746e7 1.57228
\(777\) 0 0
\(778\) 4.60140e7 2.72547
\(779\) −6.21236e6 −0.366786
\(780\) 0 0
\(781\) −2.82896e6 −0.165958
\(782\) −2.36258e7 −1.38156
\(783\) 0 0
\(784\) 0 0
\(785\) −1.32195e7 −0.765667
\(786\) 0 0
\(787\) −8.81179e6 −0.507139 −0.253570 0.967317i \(-0.581605\pi\)
−0.253570 + 0.967317i \(0.581605\pi\)
\(788\) −5.92512e7 −3.39924
\(789\) 0 0
\(790\) −1.17665e6 −0.0670779
\(791\) 0 0
\(792\) 0 0
\(793\) 2.15108e7 1.21471
\(794\) 3.60018e7 2.02662
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) −9.27413e6 −0.509140
\(803\) 7.81136e6 0.427501
\(804\) 0 0
\(805\) 0 0
\(806\) 3.07493e7 1.66724
\(807\) 0 0
\(808\) 7.64973e7 4.12209
\(809\) −1.37021e7 −0.736067 −0.368034 0.929813i \(-0.619969\pi\)
−0.368034 + 0.929813i \(0.619969\pi\)
\(810\) 0 0
\(811\) 3.30799e7 1.76609 0.883043 0.469292i \(-0.155491\pi\)
0.883043 + 0.469292i \(0.155491\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −4.71613e7 −2.49474
\(815\) 468563. 0.0247101
\(816\) 0 0
\(817\) −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 \(-0.625451\pi\)
−0.383991 + 0.923337i \(0.625451\pi\)
\(822\) 0 0
\(823\) 1.08722e7 0.559525 0.279762 0.960069i \(-0.409744\pi\)
0.279762 + 0.960069i \(0.409744\pi\)
\(824\) −2.15640e7 −1.10640
\(825\) 0 0
\(826\) 0 0
\(827\) 1.34267e7 0.682662 0.341331 0.939943i \(-0.389122\pi\)
0.341331 + 0.939943i \(0.389122\pi\)
\(828\) 0 0
\(829\) −1.46754e7 −0.741658 −0.370829 0.928701i \(-0.620926\pi\)
−0.370829 + 0.928701i \(0.620926\pi\)
\(830\) −1.46060e7 −0.735931
\(831\) 0 0
\(832\) −3.41941e6 −0.171255
\(833\) 0 0
\(834\) 0 0
\(835\) 2.25120e6 0.111737
\(836\) 4.68492e7 2.31839
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) 4.98798e6 0.241028
\(845\) −1.58593e7 −0.764084
\(846\) 0 0
\(847\) 0 0
\(848\) −2.18537e6 −0.104361
\(849\) 0 0
\(850\) −1.47075e7 −0.698219
\(851\) 4.07616e7 1.92942
\(852\) 0 0
\(853\) 3.43515e6 0.161649 0.0808244 0.996728i \(-0.474245\pi\)
0.0808244 + 0.996728i \(0.474245\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.99622e7 0.931158
\(857\) 6.40626e6 0.297956 0.148978 0.988840i \(-0.452402\pi\)
0.148978 + 0.988840i \(0.452402\pi\)
\(858\) 0 0
\(859\) 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 \(-0.505956\pi\)
−0.0187114 + 0.999825i \(0.505956\pi\)
\(864\) 0 0
\(865\) 8.04843e6 0.365739
\(866\) 4.08267e7 1.84991
\(867\) 0 0
\(868\) 0 0
\(869\) 2.25560e6 0.101324
\(870\) 0 0
\(871\) 5.66613e7 2.53070
\(872\) 1.50968e7 0.672350
\(873\) 0 0
\(874\) −5.83574e7 −2.58415
\(875\) 0 0
\(876\) 0 0
\(877\) −3.33962e7 −1.46622 −0.733108 0.680113i \(-0.761931\pi\)
−0.733108 + 0.680113i \(0.761931\pi\)
\(878\) −1.51882e7 −0.664921
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) −5.74778e7 −2.45989
\(887\) 4.75528e6 0.202940 0.101470 0.994839i \(-0.467645\pi\)
0.101470 + 0.994839i \(0.467645\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.51772e7 0.642269
\(891\) 0 0
\(892\) −4.49833e7 −1.89295
\(893\) −4.33145e6 −0.181763
\(894\) 0 0
\(895\) 1.84368e7 0.769356
\(896\) 0 0
\(897\) 0 0
\(898\) 1.99245e7 0.824512
\(899\) 7.02661e6 0.289966
\(900\) 0 0
\(901\) 640963. 0.0263040
\(902\) 2.13087e7 0.872050
\(903\) 0 0
\(904\) −4.00716e7 −1.63086
\(905\) 3.15192e6 0.127924
\(906\) 0 0
\(907\) −1.90902e7 −0.770537 −0.385268 0.922805i \(-0.625891\pi\)
−0.385268 + 0.922805i \(0.625891\pi\)
\(908\) −7.26891e7 −2.92587
\(909\) 0 0
\(910\) 0 0
\(911\) 1.22591e6 0.0489399 0.0244700 0.999701i \(-0.492210\pi\)
0.0244700 + 0.999701i \(0.492210\pi\)
\(912\) 0 0
\(913\) 2.79993e7 1.11165
\(914\) 2.73296e7 1.08210
\(915\) 0 0
\(916\) 6.42490e7 2.53004
\(917\) 0 0
\(918\) 0 0
\(919\) −1.69227e7 −0.660970 −0.330485 0.943811i \(-0.607212\pi\)
−0.330485 + 0.943811i \(0.607212\pi\)
\(920\) −4.04855e7 −1.57699
\(921\) 0 0
\(922\) −4.75677e7 −1.84283
\(923\) −6.19419e6 −0.239321
\(924\) 0 0
\(925\) 2.53749e7 0.975101
\(926\) 5.30195e7 2.03193
\(927\) 0 0
\(928\) −1.50573e7 −0.573954
\(929\) −6.77240e6 −0.257456 −0.128728 0.991680i \(-0.541089\pi\)
−0.128728 + 0.991680i \(0.541089\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −4.32380e7 −1.63052
\(933\) 0 0
\(934\) −5.17307e7 −1.94035
\(935\) −6.21166e6 −0.232369
\(936\) 0 0
\(937\) −1.41035e7 −0.524779 −0.262389 0.964962i \(-0.584511\pi\)
−0.262389 + 0.964962i \(0.584511\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −5.37765e6 −0.198505
\(941\) 5.62524e6 0.207094 0.103547 0.994625i \(-0.466981\pi\)
0.103547 + 0.994625i \(0.466981\pi\)
\(942\) 0 0
\(943\) −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 \(-0.658829\pi\)
−0.478526 + 0.878073i \(0.658829\pi\)
\(948\) 0 0
\(949\) 1.71035e7 0.616480
\(950\) −3.63286e7 −1.30599
\(951\) 0 0
\(952\) 0 0
\(953\) −4.21824e7 −1.50452 −0.752262 0.658864i \(-0.771037\pi\)
−0.752262 + 0.658864i \(0.771037\pi\)
\(954\) 0 0
\(955\) 157657. 0.00559378
\(956\) −5.38957e7 −1.90726
\(957\) 0 0
\(958\) −5.67511e7 −1.99784
\(959\) 0 0
\(960\) 0 0
\(961\) −1.99228e7 −0.695892
\(962\) −1.03263e8 −3.59755
\(963\) 0 0
\(964\) −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
\(969\) 0 0
\(970\) −1.54580e7 −0.527503
\(971\) −1.64551e7 −0.560084 −0.280042 0.959988i \(-0.590348\pi\)
−0.280042 + 0.959988i \(0.590348\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 7.03207e7 2.37512
\(975\) 0 0
\(976\) −4.04200e7 −1.35823
\(977\) −1.71176e7 −0.573729 −0.286865 0.957971i \(-0.592613\pi\)
−0.286865 + 0.957971i \(0.592613\pi\)
\(978\) 0 0
\(979\) −2.90942e7 −0.970174
\(980\) 0 0
\(981\) 0 0
\(982\) 7.61124e7 2.51870
\(983\) 2.33803e7 0.771730 0.385865 0.922555i \(-0.373903\pi\)
0.385865 + 0.922555i \(0.373903\pi\)
\(984\) 0 0
\(985\) 1.94049e7 0.637266
\(986\) 1.36770e7 0.448021
\(987\) 0 0
\(988\) 1.02579e8 3.34324
\(989\) −2.13146e7 −0.692927
\(990\) 0 0
\(991\) −4.70955e6 −0.152333 −0.0761667 0.997095i \(-0.524268\pi\)
−0.0761667 + 0.997095i \(0.524268\pi\)
\(992\) −1.86568e7 −0.601948
\(993\) 0 0
\(994\) 0 0
\(995\) −1.91547e7 −0.613362
\(996\) 0 0
\(997\) −4.55428e7 −1.45105 −0.725524 0.688197i \(-0.758403\pi\)
−0.725524 + 0.688197i \(0.758403\pi\)
\(998\) −2.20150e7 −0.699667
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.6.a.w.1.1 4
3.2 odd 2 147.6.a.m.1.4 4
7.2 even 3 63.6.e.e.46.4 8
7.4 even 3 63.6.e.e.37.4 8
7.6 odd 2 441.6.a.v.1.1 4
21.2 odd 6 21.6.e.c.4.1 8
21.5 even 6 147.6.e.o.67.1 8
21.11 odd 6 21.6.e.c.16.1 yes 8
21.17 even 6 147.6.e.o.79.1 8
21.20 even 2 147.6.a.l.1.4 4
84.11 even 6 336.6.q.j.289.2 8
84.23 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 21.2 odd 6
21.6.e.c.16.1 yes 8 21.11 odd 6
63.6.e.e.37.4 8 7.4 even 3
63.6.e.e.46.4 8 7.2 even 3
147.6.a.l.1.4 4 21.20 even 2
147.6.a.m.1.4 4 3.2 odd 2
147.6.e.o.67.1 8 21.5 even 6
147.6.e.o.79.1 8 21.17 even 6
336.6.q.j.193.2 8 84.23 even 6
336.6.q.j.289.2 8 84.11 even 6
441.6.a.v.1.1 4 7.6 odd 2
441.6.a.w.1.1 4 1.1 even 1 trivial