Properties

Label 441.6.a.w.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(-1.74818\) 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)
 
\(f(q)\) \(=\) \(q-2.74818 q^{2} -24.4475 q^{4} -58.3673 q^{5} +155.128 q^{8} +O(q^{10})\) \(q-2.74818 q^{2} -24.4475 q^{4} -58.3673 q^{5} +155.128 q^{8} +160.404 q^{10} -17.4241 q^{11} +889.933 q^{13} +356.000 q^{16} -1026.64 q^{17} -1739.40 q^{19} +1426.93 q^{20} +47.8846 q^{22} -3936.22 q^{23} +281.739 q^{25} -2445.70 q^{26} -5633.53 q^{29} -3096.53 q^{31} -5942.45 q^{32} +2821.38 q^{34} +5026.86 q^{37} +4780.20 q^{38} -9054.40 q^{40} -18367.0 q^{41} -1630.91 q^{43} +425.976 q^{44} +10817.4 q^{46} +9605.23 q^{47} -774.269 q^{50} -21756.6 q^{52} +23256.5 q^{53} +1017.00 q^{55} +15482.0 q^{58} +3603.24 q^{59} +22876.6 q^{61} +8509.83 q^{62} +4938.92 q^{64} -51943.0 q^{65} +47012.8 q^{67} +25098.7 q^{68} +1599.63 q^{71} +5931.35 q^{73} -13814.7 q^{74} +42524.1 q^{76} -88468.9 q^{79} -20778.8 q^{80} +50475.8 q^{82} +95823.9 q^{83} +59921.9 q^{85} +4482.04 q^{86} -2702.96 q^{88} +46507.9 q^{89} +96230.8 q^{92} -26396.9 q^{94} +101524. q^{95} -75981.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74818 −0.485814 −0.242907 0.970050i \(-0.578101\pi\)
−0.242907 + 0.970050i \(0.578101\pi\)
\(3\) 0 0
\(4\) −24.4475 −0.763984
\(5\) −58.3673 −1.04411 −0.522053 0.852913i \(-0.674834\pi\)
−0.522053 + 0.852913i \(0.674834\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 155.128 0.856969
\(9\) 0 0
\(10\) 160.404 0.507242
\(11\) −17.4241 −0.0434179 −0.0217089 0.999764i \(-0.506911\pi\)
−0.0217089 + 0.999764i \(0.506911\pi\)
\(12\) 0 0
\(13\) 889.933 1.46049 0.730246 0.683185i \(-0.239406\pi\)
0.730246 + 0.683185i \(0.239406\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 356.000 0.347656
\(17\) −1026.64 −0.861577 −0.430788 0.902453i \(-0.641764\pi\)
−0.430788 + 0.902453i \(0.641764\pi\)
\(18\) 0 0
\(19\) −1739.40 −1.10539 −0.552696 0.833383i \(-0.686401\pi\)
−0.552696 + 0.833383i \(0.686401\pi\)
\(20\) 1426.93 0.797680
\(21\) 0 0
\(22\) 47.8846 0.0210930
\(23\) −3936.22 −1.55153 −0.775764 0.631023i \(-0.782636\pi\)
−0.775764 + 0.631023i \(0.782636\pi\)
\(24\) 0 0
\(25\) 281.739 0.0901563
\(26\) −2445.70 −0.709528
\(27\) 0 0
\(28\) 0 0
\(29\) −5633.53 −1.24390 −0.621950 0.783057i \(-0.713659\pi\)
−0.621950 + 0.783057i \(0.713659\pi\)
\(30\) 0 0
\(31\) −3096.53 −0.578724 −0.289362 0.957220i \(-0.593443\pi\)
−0.289362 + 0.957220i \(0.593443\pi\)
\(32\) −5942.45 −1.02587
\(33\) 0 0
\(34\) 2821.38 0.418566
\(35\) 0 0
\(36\) 0 0
\(37\) 5026.86 0.603660 0.301830 0.953362i \(-0.402402\pi\)
0.301830 + 0.953362i \(0.402402\pi\)
\(38\) 4780.20 0.537015
\(39\) 0 0
\(40\) −9054.40 −0.894766
\(41\) −18367.0 −1.70639 −0.853195 0.521592i \(-0.825338\pi\)
−0.853195 + 0.521592i \(0.825338\pi\)
\(42\) 0 0
\(43\) −1630.91 −0.134511 −0.0672557 0.997736i \(-0.521424\pi\)
−0.0672557 + 0.997736i \(0.521424\pi\)
\(44\) 425.976 0.0331706
\(45\) 0 0
\(46\) 10817.4 0.753755
\(47\) 9605.23 0.634254 0.317127 0.948383i \(-0.397282\pi\)
0.317127 + 0.948383i \(0.397282\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −774.269 −0.0437992
\(51\) 0 0
\(52\) −21756.6 −1.11579
\(53\) 23256.5 1.13725 0.568624 0.822598i \(-0.307476\pi\)
0.568624 + 0.822598i \(0.307476\pi\)
\(54\) 0 0
\(55\) 1017.00 0.0453328
\(56\) 0 0
\(57\) 0 0
\(58\) 15482.0 0.604304
\(59\) 3603.24 0.134761 0.0673803 0.997727i \(-0.478536\pi\)
0.0673803 + 0.997727i \(0.478536\pi\)
\(60\) 0 0
\(61\) 22876.6 0.787167 0.393584 0.919289i \(-0.371235\pi\)
0.393584 + 0.919289i \(0.371235\pi\)
\(62\) 8509.83 0.281152
\(63\) 0 0
\(64\) 4938.92 0.150724
\(65\) −51943.0 −1.52491
\(66\) 0 0
\(67\) 47012.8 1.27947 0.639733 0.768597i \(-0.279045\pi\)
0.639733 + 0.768597i \(0.279045\pi\)
\(68\) 25098.7 0.658231
\(69\) 0 0
\(70\) 0 0
\(71\) 1599.63 0.0376595 0.0188298 0.999823i \(-0.494006\pi\)
0.0188298 + 0.999823i \(0.494006\pi\)
\(72\) 0 0
\(73\) 5931.35 0.130271 0.0651353 0.997876i \(-0.479252\pi\)
0.0651353 + 0.997876i \(0.479252\pi\)
\(74\) −13814.7 −0.293267
\(75\) 0 0
\(76\) 42524.1 0.844502
\(77\) 0 0
\(78\) 0 0
\(79\) −88468.9 −1.59486 −0.797431 0.603411i \(-0.793808\pi\)
−0.797431 + 0.603411i \(0.793808\pi\)
\(80\) −20778.8 −0.362990
\(81\) 0 0
\(82\) 50475.8 0.828989
\(83\) 95823.9 1.52679 0.763394 0.645933i \(-0.223531\pi\)
0.763394 + 0.645933i \(0.223531\pi\)
\(84\) 0 0
\(85\) 59921.9 0.899577
\(86\) 4482.04 0.0653476
\(87\) 0 0
\(88\) −2702.96 −0.0372078
\(89\) 46507.9 0.622374 0.311187 0.950349i \(-0.399273\pi\)
0.311187 + 0.950349i \(0.399273\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 96230.8 1.18534
\(93\) 0 0
\(94\) −26396.9 −0.308130
\(95\) 101524. 1.15415
\(96\) 0 0
\(97\) −75981.8 −0.819937 −0.409968 0.912100i \(-0.634460\pi\)
−0.409968 + 0.912100i \(0.634460\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −6887.80 −0.0688780
\(101\) 46157.4 0.450234 0.225117 0.974332i \(-0.427724\pi\)
0.225117 + 0.974332i \(0.427724\pi\)
\(102\) 0 0
\(103\) −81973.6 −0.761344 −0.380672 0.924710i \(-0.624307\pi\)
−0.380672 + 0.924710i \(0.624307\pi\)
\(104\) 138054. 1.25160
\(105\) 0 0
\(106\) −63913.2 −0.552491
\(107\) 2853.67 0.0240960 0.0120480 0.999927i \(-0.496165\pi\)
0.0120480 + 0.999927i \(0.496165\pi\)
\(108\) 0 0
\(109\) −166278. −1.34051 −0.670254 0.742132i \(-0.733815\pi\)
−0.670254 + 0.742132i \(0.733815\pi\)
\(110\) −2794.89 −0.0220234
\(111\) 0 0
\(112\) 0 0
\(113\) −260304. −1.91772 −0.958858 0.283886i \(-0.908376\pi\)
−0.958858 + 0.283886i \(0.908376\pi\)
\(114\) 0 0
\(115\) 229746. 1.61996
\(116\) 137726. 0.950320
\(117\) 0 0
\(118\) −9902.35 −0.0654687
\(119\) 0 0
\(120\) 0 0
\(121\) −160747. −0.998115
\(122\) −62869.1 −0.382417
\(123\) 0 0
\(124\) 75702.5 0.442136
\(125\) 165953. 0.949973
\(126\) 0 0
\(127\) −233743. −1.28596 −0.642982 0.765882i \(-0.722303\pi\)
−0.642982 + 0.765882i \(0.722303\pi\)
\(128\) 176585. 0.952642
\(129\) 0 0
\(130\) 142749. 0.740822
\(131\) 157290. 0.800797 0.400398 0.916341i \(-0.368872\pi\)
0.400398 + 0.916341i \(0.368872\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −129200. −0.621583
\(135\) 0 0
\(136\) −159260. −0.738345
\(137\) 171548. 0.780878 0.390439 0.920629i \(-0.372323\pi\)
0.390439 + 0.920629i \(0.372323\pi\)
\(138\) 0 0
\(139\) 210625. 0.924642 0.462321 0.886713i \(-0.347017\pi\)
0.462321 + 0.886713i \(0.347017\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4396.08 −0.0182955
\(143\) −15506.3 −0.0634114
\(144\) 0 0
\(145\) 328814. 1.29876
\(146\) −16300.4 −0.0632873
\(147\) 0 0
\(148\) −122894. −0.461187
\(149\) −239413. −0.883450 −0.441725 0.897151i \(-0.645633\pi\)
−0.441725 + 0.897151i \(0.645633\pi\)
\(150\) 0 0
\(151\) 217039. 0.774634 0.387317 0.921947i \(-0.373402\pi\)
0.387317 + 0.921947i \(0.373402\pi\)
\(152\) −269830. −0.947287
\(153\) 0 0
\(154\) 0 0
\(155\) 180736. 0.604249
\(156\) 0 0
\(157\) 166904. 0.540404 0.270202 0.962804i \(-0.412909\pi\)
0.270202 + 0.962804i \(0.412909\pi\)
\(158\) 243129. 0.774807
\(159\) 0 0
\(160\) 346844. 1.07111
\(161\) 0 0
\(162\) 0 0
\(163\) 506172. 1.49221 0.746104 0.665830i \(-0.231922\pi\)
0.746104 + 0.665830i \(0.231922\pi\)
\(164\) 449027. 1.30366
\(165\) 0 0
\(166\) −263342. −0.741736
\(167\) 565560. 1.56923 0.784616 0.619982i \(-0.212860\pi\)
0.784616 + 0.619982i \(0.212860\pi\)
\(168\) 0 0
\(169\) 420688. 1.13304
\(170\) −164676. −0.437027
\(171\) 0 0
\(172\) 39871.7 0.102765
\(173\) 659435. 1.67516 0.837581 0.546313i \(-0.183969\pi\)
0.837581 + 0.546313i \(0.183969\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6202.98 −0.0150945
\(177\) 0 0
\(178\) −127812. −0.302358
\(179\) −165290. −0.385580 −0.192790 0.981240i \(-0.561754\pi\)
−0.192790 + 0.981240i \(0.561754\pi\)
\(180\) 0 0
\(181\) 148492. 0.336904 0.168452 0.985710i \(-0.446123\pi\)
0.168452 + 0.985710i \(0.446123\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −610618. −1.32961
\(185\) −293404. −0.630285
\(186\) 0 0
\(187\) 17888.2 0.0374078
\(188\) −234824. −0.484560
\(189\) 0 0
\(190\) −279007. −0.560701
\(191\) −385911. −0.765426 −0.382713 0.923867i \(-0.625010\pi\)
−0.382713 + 0.923867i \(0.625010\pi\)
\(192\) 0 0
\(193\) 496296. 0.959063 0.479531 0.877525i \(-0.340807\pi\)
0.479531 + 0.877525i \(0.340807\pi\)
\(194\) 208812. 0.398337
\(195\) 0 0
\(196\) 0 0
\(197\) −441439. −0.810411 −0.405206 0.914226i \(-0.632800\pi\)
−0.405206 + 0.914226i \(0.632800\pi\)
\(198\) 0 0
\(199\) −75838.8 −0.135756 −0.0678779 0.997694i \(-0.521623\pi\)
−0.0678779 + 0.997694i \(0.521623\pi\)
\(200\) 43705.5 0.0772612
\(201\) 0 0
\(202\) −126849. −0.218730
\(203\) 0 0
\(204\) 0 0
\(205\) 1.07203e6 1.78165
\(206\) 225278. 0.369872
\(207\) 0 0
\(208\) 316816. 0.507749
\(209\) 30307.5 0.0479938
\(210\) 0 0
\(211\) 778704. 1.20411 0.602055 0.798454i \(-0.294349\pi\)
0.602055 + 0.798454i \(0.294349\pi\)
\(212\) −568564. −0.868840
\(213\) 0 0
\(214\) −7842.41 −0.0117062
\(215\) 95191.9 0.140444
\(216\) 0 0
\(217\) 0 0
\(218\) 456963. 0.651238
\(219\) 0 0
\(220\) −24863.0 −0.0346336
\(221\) −913637. −1.25833
\(222\) 0 0
\(223\) 738085. 0.993903 0.496951 0.867778i \(-0.334453\pi\)
0.496951 + 0.867778i \(0.334453\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 715362. 0.931654
\(227\) 544115. 0.700852 0.350426 0.936590i \(-0.386037\pi\)
0.350426 + 0.936590i \(0.386037\pi\)
\(228\) 0 0
\(229\) −233562. −0.294316 −0.147158 0.989113i \(-0.547013\pi\)
−0.147158 + 0.989113i \(0.547013\pi\)
\(230\) −631385. −0.787000
\(231\) 0 0
\(232\) −873917. −1.06598
\(233\) 618100. 0.745880 0.372940 0.927856i \(-0.378350\pi\)
0.372940 + 0.927856i \(0.378350\pi\)
\(234\) 0 0
\(235\) −560631. −0.662228
\(236\) −88090.2 −0.102955
\(237\) 0 0
\(238\) 0 0
\(239\) −937500. −1.06164 −0.530819 0.847485i \(-0.678116\pi\)
−0.530819 + 0.847485i \(0.678116\pi\)
\(240\) 0 0
\(241\) 932036. 1.03369 0.516845 0.856079i \(-0.327106\pi\)
0.516845 + 0.856079i \(0.327106\pi\)
\(242\) 441763. 0.484899
\(243\) 0 0
\(244\) −559276. −0.601383
\(245\) 0 0
\(246\) 0 0
\(247\) −1.54795e6 −1.61442
\(248\) −480359. −0.495948
\(249\) 0 0
\(250\) −456070. −0.461510
\(251\) −214975. −0.215379 −0.107690 0.994185i \(-0.534345\pi\)
−0.107690 + 0.994185i \(0.534345\pi\)
\(252\) 0 0
\(253\) 68585.1 0.0673641
\(254\) 642367. 0.624740
\(255\) 0 0
\(256\) −643334. −0.613531
\(257\) −79187.0 −0.0747862 −0.0373931 0.999301i \(-0.511905\pi\)
−0.0373931 + 0.999301i \(0.511905\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.26988e6 1.16501
\(261\) 0 0
\(262\) −432261. −0.389039
\(263\) 432828. 0.385856 0.192928 0.981213i \(-0.438202\pi\)
0.192928 + 0.981213i \(0.438202\pi\)
\(264\) 0 0
\(265\) −1.35742e6 −1.18741
\(266\) 0 0
\(267\) 0 0
\(268\) −1.14934e6 −0.977492
\(269\) 4691.86 0.00395334 0.00197667 0.999998i \(-0.499371\pi\)
0.00197667 + 0.999998i \(0.499371\pi\)
\(270\) 0 0
\(271\) 105264. 0.0870678 0.0435339 0.999052i \(-0.486138\pi\)
0.0435339 + 0.999052i \(0.486138\pi\)
\(272\) −365482. −0.299533
\(273\) 0 0
\(274\) −471444. −0.379362
\(275\) −4909.04 −0.00391440
\(276\) 0 0
\(277\) −763117. −0.597574 −0.298787 0.954320i \(-0.596582\pi\)
−0.298787 + 0.954320i \(0.596582\pi\)
\(278\) −578837. −0.449205
\(279\) 0 0
\(280\) 0 0
\(281\) 729540. 0.551167 0.275584 0.961277i \(-0.411129\pi\)
0.275584 + 0.961277i \(0.411129\pi\)
\(282\) 0 0
\(283\) −1.19043e6 −0.883563 −0.441781 0.897123i \(-0.645653\pi\)
−0.441781 + 0.897123i \(0.645653\pi\)
\(284\) −39107.0 −0.0287713
\(285\) 0 0
\(286\) 42614.1 0.0308062
\(287\) 0 0
\(288\) 0 0
\(289\) −365877. −0.257686
\(290\) −903639. −0.630958
\(291\) 0 0
\(292\) −145007. −0.0995246
\(293\) 1.02503e6 0.697537 0.348769 0.937209i \(-0.386600\pi\)
0.348769 + 0.937209i \(0.386600\pi\)
\(294\) 0 0
\(295\) −210311. −0.140704
\(296\) 779807. 0.517318
\(297\) 0 0
\(298\) 657950. 0.429193
\(299\) −3.50297e6 −2.26599
\(300\) 0 0
\(301\) 0 0
\(302\) −596464. −0.376328
\(303\) 0 0
\(304\) −619228. −0.384297
\(305\) −1.33525e6 −0.821886
\(306\) 0 0
\(307\) −709845. −0.429850 −0.214925 0.976631i \(-0.568951\pi\)
−0.214925 + 0.976631i \(0.568951\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −496696. −0.293553
\(311\) −1.56763e6 −0.919058 −0.459529 0.888163i \(-0.651982\pi\)
−0.459529 + 0.888163i \(0.651982\pi\)
\(312\) 0 0
\(313\) 372952. 0.215175 0.107588 0.994196i \(-0.465687\pi\)
0.107588 + 0.994196i \(0.465687\pi\)
\(314\) −458684. −0.262536
\(315\) 0 0
\(316\) 2.16284e6 1.21845
\(317\) 3.02816e6 1.69251 0.846253 0.532781i \(-0.178853\pi\)
0.846253 + 0.532781i \(0.178853\pi\)
\(318\) 0 0
\(319\) 98159.1 0.0540075
\(320\) −288271. −0.157372
\(321\) 0 0
\(322\) 0 0
\(323\) 1.78573e6 0.952380
\(324\) 0 0
\(325\) 250728. 0.131673
\(326\) −1.39105e6 −0.724936
\(327\) 0 0
\(328\) −2.84923e6 −1.46232
\(329\) 0 0
\(330\) 0 0
\(331\) 1.06690e6 0.535244 0.267622 0.963524i \(-0.413762\pi\)
0.267622 + 0.963524i \(0.413762\pi\)
\(332\) −2.34266e6 −1.16644
\(333\) 0 0
\(334\) −1.55426e6 −0.762356
\(335\) −2.74401e6 −1.33590
\(336\) 0 0
\(337\) 1.55734e6 0.746981 0.373490 0.927634i \(-0.378161\pi\)
0.373490 + 0.927634i \(0.378161\pi\)
\(338\) −1.15613e6 −0.550445
\(339\) 0 0
\(340\) −1.46494e6 −0.687263
\(341\) 53954.3 0.0251270
\(342\) 0 0
\(343\) 0 0
\(344\) −253000. −0.115272
\(345\) 0 0
\(346\) −1.81225e6 −0.813818
\(347\) 2.23496e6 0.996428 0.498214 0.867054i \(-0.333989\pi\)
0.498214 + 0.867054i \(0.333989\pi\)
\(348\) 0 0
\(349\) −1.72982e6 −0.760218 −0.380109 0.924942i \(-0.624114\pi\)
−0.380109 + 0.924942i \(0.624114\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 103542. 0.0445409
\(353\) 2.36574e6 1.01048 0.505242 0.862978i \(-0.331403\pi\)
0.505242 + 0.862978i \(0.331403\pi\)
\(354\) 0 0
\(355\) −93366.3 −0.0393205
\(356\) −1.13700e6 −0.475484
\(357\) 0 0
\(358\) 454247. 0.187320
\(359\) −51028.7 −0.0208967 −0.0104484 0.999945i \(-0.503326\pi\)
−0.0104484 + 0.999945i \(0.503326\pi\)
\(360\) 0 0
\(361\) 549424. 0.221891
\(362\) −408083. −0.163673
\(363\) 0 0
\(364\) 0 0
\(365\) −346197. −0.136016
\(366\) 0 0
\(367\) 3.76823e6 1.46040 0.730200 0.683233i \(-0.239427\pi\)
0.730200 + 0.683233i \(0.239427\pi\)
\(368\) −1.40130e6 −0.539399
\(369\) 0 0
\(370\) 806328. 0.306201
\(371\) 0 0
\(372\) 0 0
\(373\) 4.63440e6 1.72473 0.862366 0.506285i \(-0.168982\pi\)
0.862366 + 0.506285i \(0.168982\pi\)
\(374\) −49160.0 −0.0181733
\(375\) 0 0
\(376\) 1.49004e6 0.543536
\(377\) −5.01346e6 −1.81670
\(378\) 0 0
\(379\) −4.17169e6 −1.49181 −0.745905 0.666052i \(-0.767983\pi\)
−0.745905 + 0.666052i \(0.767983\pi\)
\(380\) −2.48201e6 −0.881749
\(381\) 0 0
\(382\) 1.06055e6 0.371855
\(383\) 4.17172e6 1.45318 0.726588 0.687074i \(-0.241105\pi\)
0.726588 + 0.687074i \(0.241105\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.36391e6 −0.465927
\(387\) 0 0
\(388\) 1.85757e6 0.626419
\(389\) −2.75394e6 −0.922743 −0.461371 0.887207i \(-0.652642\pi\)
−0.461371 + 0.887207i \(0.652642\pi\)
\(390\) 0 0
\(391\) 4.04106e6 1.33676
\(392\) 0 0
\(393\) 0 0
\(394\) 1.21316e6 0.393709
\(395\) 5.16369e6 1.66520
\(396\) 0 0
\(397\) −2.53204e6 −0.806297 −0.403148 0.915135i \(-0.632084\pi\)
−0.403148 + 0.915135i \(0.632084\pi\)
\(398\) 208419. 0.0659522
\(399\) 0 0
\(400\) 100299. 0.0313434
\(401\) 2.15747e6 0.670014 0.335007 0.942216i \(-0.391261\pi\)
0.335007 + 0.942216i \(0.391261\pi\)
\(402\) 0 0
\(403\) −2.75571e6 −0.845221
\(404\) −1.12843e6 −0.343972
\(405\) 0 0
\(406\) 0 0
\(407\) −87588.5 −0.0262096
\(408\) 0 0
\(409\) −4.32804e6 −1.27933 −0.639665 0.768653i \(-0.720927\pi\)
−0.639665 + 0.768653i \(0.720927\pi\)
\(410\) −2.94614e6 −0.865552
\(411\) 0 0
\(412\) 2.00405e6 0.581655
\(413\) 0 0
\(414\) 0 0
\(415\) −5.59298e6 −1.59413
\(416\) −5.28838e6 −1.49827
\(417\) 0 0
\(418\) −83290.6 −0.0233161
\(419\) 1.51129e6 0.420544 0.210272 0.977643i \(-0.432565\pi\)
0.210272 + 0.977643i \(0.432565\pi\)
\(420\) 0 0
\(421\) 1.11586e6 0.306835 0.153418 0.988161i \(-0.450972\pi\)
0.153418 + 0.988161i \(0.450972\pi\)
\(422\) −2.14002e6 −0.584974
\(423\) 0 0
\(424\) 3.60774e6 0.974586
\(425\) −289243. −0.0776766
\(426\) 0 0
\(427\) 0 0
\(428\) −69765.2 −0.0184090
\(429\) 0 0
\(430\) −261605. −0.0682298
\(431\) −6.21345e6 −1.61116 −0.805581 0.592485i \(-0.798147\pi\)
−0.805581 + 0.592485i \(0.798147\pi\)
\(432\) 0 0
\(433\) 3.24118e6 0.830775 0.415388 0.909644i \(-0.363646\pi\)
0.415388 + 0.909644i \(0.363646\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.06509e6 1.02413
\(437\) 6.84667e6 1.71505
\(438\) 0 0
\(439\) 2.30497e6 0.570825 0.285413 0.958405i \(-0.407869\pi\)
0.285413 + 0.958405i \(0.407869\pi\)
\(440\) 157765. 0.0388488
\(441\) 0 0
\(442\) 2.51084e6 0.611313
\(443\) −1.53266e6 −0.371052 −0.185526 0.982639i \(-0.559399\pi\)
−0.185526 + 0.982639i \(0.559399\pi\)
\(444\) 0 0
\(445\) −2.71454e6 −0.649824
\(446\) −2.02839e6 −0.482852
\(447\) 0 0
\(448\) 0 0
\(449\) 3.55718e6 0.832702 0.416351 0.909204i \(-0.363309\pi\)
0.416351 + 0.909204i \(0.363309\pi\)
\(450\) 0 0
\(451\) 320028. 0.0740878
\(452\) 6.36378e6 1.46511
\(453\) 0 0
\(454\) −1.49533e6 −0.340484
\(455\) 0 0
\(456\) 0 0
\(457\) 2.51883e6 0.564167 0.282083 0.959390i \(-0.408974\pi\)
0.282083 + 0.959390i \(0.408974\pi\)
\(458\) 641872. 0.142983
\(459\) 0 0
\(460\) −5.61673e6 −1.23762
\(461\) 6.63271e6 1.45358 0.726789 0.686861i \(-0.241012\pi\)
0.726789 + 0.686861i \(0.241012\pi\)
\(462\) 0 0
\(463\) −4.40432e6 −0.954830 −0.477415 0.878678i \(-0.658426\pi\)
−0.477415 + 0.878678i \(0.658426\pi\)
\(464\) −2.00554e6 −0.432450
\(465\) 0 0
\(466\) −1.69865e6 −0.362359
\(467\) −245845. −0.0521637 −0.0260819 0.999660i \(-0.508303\pi\)
−0.0260819 + 0.999660i \(0.508303\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.54072e6 0.321720
\(471\) 0 0
\(472\) 558963. 0.115486
\(473\) 28417.2 0.00584020
\(474\) 0 0
\(475\) −490057. −0.0996581
\(476\) 0 0
\(477\) 0 0
\(478\) 2.57642e6 0.515759
\(479\) 40351.1 0.00803557 0.00401779 0.999992i \(-0.498721\pi\)
0.00401779 + 0.999992i \(0.498721\pi\)
\(480\) 0 0
\(481\) 4.47357e6 0.881640
\(482\) −2.56140e6 −0.502181
\(483\) 0 0
\(484\) 3.92987e6 0.762544
\(485\) 4.43485e6 0.856101
\(486\) 0 0
\(487\) −3.30968e6 −0.632359 −0.316180 0.948699i \(-0.602400\pi\)
−0.316180 + 0.948699i \(0.602400\pi\)
\(488\) 3.54880e6 0.674578
\(489\) 0 0
\(490\) 0 0
\(491\) 1.97959e6 0.370570 0.185285 0.982685i \(-0.440679\pi\)
0.185285 + 0.982685i \(0.440679\pi\)
\(492\) 0 0
\(493\) 5.78358e6 1.07171
\(494\) 4.25405e6 0.784306
\(495\) 0 0
\(496\) −1.10237e6 −0.201197
\(497\) 0 0
\(498\) 0 0
\(499\) −3.16995e6 −0.569904 −0.284952 0.958542i \(-0.591978\pi\)
−0.284952 + 0.958542i \(0.591978\pi\)
\(500\) −4.05715e6 −0.725764
\(501\) 0 0
\(502\) 590791. 0.104634
\(503\) 4.01273e6 0.707164 0.353582 0.935403i \(-0.384963\pi\)
0.353582 + 0.935403i \(0.384963\pi\)
\(504\) 0 0
\(505\) −2.69408e6 −0.470092
\(506\) −188484. −0.0327264
\(507\) 0 0
\(508\) 5.71442e6 0.982456
\(509\) −4.11257e6 −0.703590 −0.351795 0.936077i \(-0.614429\pi\)
−0.351795 + 0.936077i \(0.614429\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.88273e6 −0.654580
\(513\) 0 0
\(514\) 217620. 0.0363322
\(515\) 4.78457e6 0.794923
\(516\) 0 0
\(517\) −167363. −0.0275380
\(518\) 0 0
\(519\) 0 0
\(520\) −8.05781e6 −1.30680
\(521\) −8.55515e6 −1.38081 −0.690404 0.723424i \(-0.742567\pi\)
−0.690404 + 0.723424i \(0.742567\pi\)
\(522\) 0 0
\(523\) −1.79364e6 −0.286735 −0.143368 0.989670i \(-0.545793\pi\)
−0.143368 + 0.989670i \(0.545793\pi\)
\(524\) −3.84534e6 −0.611796
\(525\) 0 0
\(526\) −1.18949e6 −0.187455
\(527\) 3.17901e6 0.498615
\(528\) 0 0
\(529\) 9.05749e6 1.40724
\(530\) 3.73044e6 0.576859
\(531\) 0 0
\(532\) 0 0
\(533\) −1.63454e7 −2.49217
\(534\) 0 0
\(535\) −166561. −0.0251588
\(536\) 7.29299e6 1.09646
\(537\) 0 0
\(538\) −12894.1 −0.00192059
\(539\) 0 0
\(540\) 0 0
\(541\) −357561. −0.0525239 −0.0262619 0.999655i \(-0.508360\pi\)
−0.0262619 + 0.999655i \(0.508360\pi\)
\(542\) −289285. −0.0422988
\(543\) 0 0
\(544\) 6.10073e6 0.883862
\(545\) 9.70522e6 1.39963
\(546\) 0 0
\(547\) −3.79404e6 −0.542167 −0.271084 0.962556i \(-0.587382\pi\)
−0.271084 + 0.962556i \(0.587382\pi\)
\(548\) −4.19391e6 −0.596578
\(549\) 0 0
\(550\) 13490.9 0.00190167
\(551\) 9.79897e6 1.37500
\(552\) 0 0
\(553\) 0 0
\(554\) 2.09718e6 0.290310
\(555\) 0 0
\(556\) −5.14926e6 −0.706412
\(557\) 4.89599e6 0.668655 0.334328 0.942457i \(-0.391491\pi\)
0.334328 + 0.942457i \(0.391491\pi\)
\(558\) 0 0
\(559\) −1.45140e6 −0.196453
\(560\) 0 0
\(561\) 0 0
\(562\) −2.00491e6 −0.267765
\(563\) 4.33167e6 0.575949 0.287974 0.957638i \(-0.407018\pi\)
0.287974 + 0.957638i \(0.407018\pi\)
\(564\) 0 0
\(565\) 1.51932e7 2.00230
\(566\) 3.27151e6 0.429248
\(567\) 0 0
\(568\) 248148. 0.0322730
\(569\) 2.19696e6 0.284473 0.142236 0.989833i \(-0.454571\pi\)
0.142236 + 0.989833i \(0.454571\pi\)
\(570\) 0 0
\(571\) −7.47692e6 −0.959693 −0.479846 0.877353i \(-0.659308\pi\)
−0.479846 + 0.877353i \(0.659308\pi\)
\(572\) 379090. 0.0484454
\(573\) 0 0
\(574\) 0 0
\(575\) −1.10899e6 −0.139880
\(576\) 0 0
\(577\) −1.36622e6 −0.170837 −0.0854183 0.996345i \(-0.527223\pi\)
−0.0854183 + 0.996345i \(0.527223\pi\)
\(578\) 1.00550e6 0.125187
\(579\) 0 0
\(580\) −8.03867e6 −0.992234
\(581\) 0 0
\(582\) 0 0
\(583\) −405224. −0.0493769
\(584\) 920118. 0.111638
\(585\) 0 0
\(586\) −2.81697e6 −0.338874
\(587\) 9.27217e6 1.11067 0.555336 0.831626i \(-0.312590\pi\)
0.555336 + 0.831626i \(0.312590\pi\)
\(588\) 0 0
\(589\) 5.38612e6 0.639717
\(590\) 577973. 0.0683562
\(591\) 0 0
\(592\) 1.78956e6 0.209866
\(593\) −9.21045e6 −1.07558 −0.537792 0.843078i \(-0.680741\pi\)
−0.537792 + 0.843078i \(0.680741\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 5.85305e6 0.674942
\(597\) 0 0
\(598\) 9.62681e6 1.10085
\(599\) 1.36916e7 1.55915 0.779575 0.626309i \(-0.215435\pi\)
0.779575 + 0.626309i \(0.215435\pi\)
\(600\) 0 0
\(601\) 1.61113e6 0.181946 0.0909732 0.995853i \(-0.471002\pi\)
0.0909732 + 0.995853i \(0.471002\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −5.30607e6 −0.591808
\(605\) 9.38239e6 1.04214
\(606\) 0 0
\(607\) −1.40656e7 −1.54948 −0.774742 0.632278i \(-0.782120\pi\)
−0.774742 + 0.632278i \(0.782120\pi\)
\(608\) 1.03363e7 1.13398
\(609\) 0 0
\(610\) 3.66950e6 0.399284
\(611\) 8.54802e6 0.926323
\(612\) 0 0
\(613\) 1.35968e7 1.46146 0.730729 0.682667i \(-0.239180\pi\)
0.730729 + 0.682667i \(0.239180\pi\)
\(614\) 1.95078e6 0.208828
\(615\) 0 0
\(616\) 0 0
\(617\) 5.74287e6 0.607318 0.303659 0.952781i \(-0.401792\pi\)
0.303659 + 0.952781i \(0.401792\pi\)
\(618\) 0 0
\(619\) −6.02598e6 −0.632122 −0.316061 0.948739i \(-0.602360\pi\)
−0.316061 + 0.948739i \(0.602360\pi\)
\(620\) −4.41855e6 −0.461637
\(621\) 0 0
\(622\) 4.30814e6 0.446492
\(623\) 0 0
\(624\) 0 0
\(625\) −1.05667e7 −1.08203
\(626\) −1.02494e6 −0.104535
\(627\) 0 0
\(628\) −4.08039e6 −0.412860
\(629\) −5.16075e6 −0.520099
\(630\) 0 0
\(631\) −6.90670e6 −0.690554 −0.345277 0.938501i \(-0.612215\pi\)
−0.345277 + 0.938501i \(0.612215\pi\)
\(632\) −1.37240e7 −1.36675
\(633\) 0 0
\(634\) −8.32193e6 −0.822244
\(635\) 1.36429e7 1.34268
\(636\) 0 0
\(637\) 0 0
\(638\) −269759. −0.0262376
\(639\) 0 0
\(640\) −1.03068e7 −0.994658
\(641\) −1.64966e7 −1.58580 −0.792900 0.609352i \(-0.791430\pi\)
−0.792900 + 0.609352i \(0.791430\pi\)
\(642\) 0 0
\(643\) 1.70171e7 1.62315 0.811576 0.584247i \(-0.198610\pi\)
0.811576 + 0.584247i \(0.198610\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.90752e6 −0.462680
\(647\) −3.48573e6 −0.327366 −0.163683 0.986513i \(-0.552337\pi\)
−0.163683 + 0.986513i \(0.552337\pi\)
\(648\) 0 0
\(649\) −62783.2 −0.00585102
\(650\) −689047. −0.0639684
\(651\) 0 0
\(652\) −1.23746e7 −1.14002
\(653\) −1.54449e7 −1.41743 −0.708716 0.705494i \(-0.750725\pi\)
−0.708716 + 0.705494i \(0.750725\pi\)
\(654\) 0 0
\(655\) −9.18058e6 −0.836116
\(656\) −6.53865e6 −0.593238
\(657\) 0 0
\(658\) 0 0
\(659\) 3.11193e6 0.279136 0.139568 0.990212i \(-0.455429\pi\)
0.139568 + 0.990212i \(0.455429\pi\)
\(660\) 0 0
\(661\) 8.17220e6 0.727504 0.363752 0.931496i \(-0.381496\pi\)
0.363752 + 0.931496i \(0.381496\pi\)
\(662\) −2.93202e6 −0.260029
\(663\) 0 0
\(664\) 1.48650e7 1.30841
\(665\) 0 0
\(666\) 0 0
\(667\) 2.21748e7 1.92995
\(668\) −1.38265e7 −1.19887
\(669\) 0 0
\(670\) 7.54103e6 0.648998
\(671\) −398604. −0.0341771
\(672\) 0 0
\(673\) 1.60182e7 1.36325 0.681627 0.731700i \(-0.261273\pi\)
0.681627 + 0.731700i \(0.261273\pi\)
\(674\) −4.27986e6 −0.362894
\(675\) 0 0
\(676\) −1.02848e7 −0.865621
\(677\) −1.02437e6 −0.0858984 −0.0429492 0.999077i \(-0.513675\pi\)
−0.0429492 + 0.999077i \(0.513675\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 9.29556e6 0.770910
\(681\) 0 0
\(682\) −148276. −0.0122070
\(683\) 5.39377e6 0.442426 0.221213 0.975226i \(-0.428998\pi\)
0.221213 + 0.975226i \(0.428998\pi\)
\(684\) 0 0
\(685\) −1.00128e7 −0.815319
\(686\) 0 0
\(687\) 0 0
\(688\) −580605. −0.0467638
\(689\) 2.06968e7 1.66094
\(690\) 0 0
\(691\) −905653. −0.0721550 −0.0360775 0.999349i \(-0.511486\pi\)
−0.0360775 + 0.999349i \(0.511486\pi\)
\(692\) −1.61215e7 −1.27980
\(693\) 0 0
\(694\) −6.14207e6 −0.484079
\(695\) −1.22936e7 −0.965424
\(696\) 0 0
\(697\) 1.88562e7 1.47019
\(698\) 4.75387e6 0.369325
\(699\) 0 0
\(700\) 0 0
\(701\) −1.12573e7 −0.865246 −0.432623 0.901575i \(-0.642412\pi\)
−0.432623 + 0.901575i \(0.642412\pi\)
\(702\) 0 0
\(703\) −8.74374e6 −0.667281
\(704\) −86056.2 −0.00654411
\(705\) 0 0
\(706\) −6.50147e6 −0.490908
\(707\) 0 0
\(708\) 0 0
\(709\) 4.35509e6 0.325373 0.162687 0.986678i \(-0.447984\pi\)
0.162687 + 0.986678i \(0.447984\pi\)
\(710\) 256587. 0.0191025
\(711\) 0 0
\(712\) 7.21467e6 0.533355
\(713\) 1.21886e7 0.897907
\(714\) 0 0
\(715\) 905060. 0.0662082
\(716\) 4.04093e6 0.294577
\(717\) 0 0
\(718\) 140236. 0.0101519
\(719\) 1.41444e7 1.02038 0.510191 0.860061i \(-0.329575\pi\)
0.510191 + 0.860061i \(0.329575\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.50992e6 −0.107798
\(723\) 0 0
\(724\) −3.63025e6 −0.257389
\(725\) −1.58718e6 −0.112145
\(726\) 0 0
\(727\) −6.26406e6 −0.439561 −0.219781 0.975549i \(-0.570534\pi\)
−0.219781 + 0.975549i \(0.570534\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 951411. 0.0660786
\(731\) 1.67435e6 0.115892
\(732\) 0 0
\(733\) −2.04177e7 −1.40361 −0.701806 0.712368i \(-0.747623\pi\)
−0.701806 + 0.712368i \(0.747623\pi\)
\(734\) −1.03558e7 −0.709484
\(735\) 0 0
\(736\) 2.33908e7 1.59166
\(737\) −819155. −0.0555517
\(738\) 0 0
\(739\) 1.48451e7 0.999937 0.499969 0.866043i \(-0.333345\pi\)
0.499969 + 0.866043i \(0.333345\pi\)
\(740\) 7.17300e6 0.481528
\(741\) 0 0
\(742\) 0 0
\(743\) −2.36601e7 −1.57234 −0.786168 0.618013i \(-0.787938\pi\)
−0.786168 + 0.618013i \(0.787938\pi\)
\(744\) 0 0
\(745\) 1.39739e7 0.922415
\(746\) −1.27362e7 −0.837900
\(747\) 0 0
\(748\) −437322. −0.0285790
\(749\) 0 0
\(750\) 0 0
\(751\) 2.06573e7 1.33652 0.668258 0.743929i \(-0.267040\pi\)
0.668258 + 0.743929i \(0.267040\pi\)
\(752\) 3.41946e6 0.220503
\(753\) 0 0
\(754\) 1.37779e7 0.882581
\(755\) −1.26680e7 −0.808799
\(756\) 0 0
\(757\) −1.26697e7 −0.803573 −0.401787 0.915733i \(-0.631611\pi\)
−0.401787 + 0.915733i \(0.631611\pi\)
\(758\) 1.14646e7 0.724743
\(759\) 0 0
\(760\) 1.57492e7 0.989067
\(761\) −1.62752e7 −1.01874 −0.509372 0.860546i \(-0.670122\pi\)
−0.509372 + 0.860546i \(0.670122\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 9.43455e6 0.584774
\(765\) 0 0
\(766\) −1.14646e7 −0.705974
\(767\) 3.20664e6 0.196817
\(768\) 0 0
\(769\) 1.60471e7 0.978547 0.489273 0.872130i \(-0.337262\pi\)
0.489273 + 0.872130i \(0.337262\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.21332e7 −0.732709
\(773\) −2.15677e6 −0.129824 −0.0649121 0.997891i \(-0.520677\pi\)
−0.0649121 + 0.997891i \(0.520677\pi\)
\(774\) 0 0
\(775\) −872413. −0.0521756
\(776\) −1.17869e7 −0.702661
\(777\) 0 0
\(778\) 7.56833e6 0.448282
\(779\) 3.19476e7 1.88623
\(780\) 0 0
\(781\) −27872.2 −0.00163510
\(782\) −1.11056e7 −0.649418
\(783\) 0 0
\(784\) 0 0
\(785\) −9.74175e6 −0.564239
\(786\) 0 0
\(787\) 1.40101e7 0.806312 0.403156 0.915131i \(-0.367913\pi\)
0.403156 + 0.915131i \(0.367913\pi\)
\(788\) 1.07921e7 0.619142
\(789\) 0 0
\(790\) −1.41908e7 −0.808980
\(791\) 0 0
\(792\) 0 0
\(793\) 2.03587e7 1.14965
\(794\) 6.95852e6 0.391711
\(795\) 0 0
\(796\) 1.85407e6 0.103715
\(797\) −4.73289e6 −0.263925 −0.131963 0.991255i \(-0.542128\pi\)
−0.131963 + 0.991255i \(0.542128\pi\)
\(798\) 0 0
\(799\) −9.86107e6 −0.546459
\(800\) −1.67422e6 −0.0924883
\(801\) 0 0
\(802\) −5.92912e6 −0.325502
\(803\) −103348. −0.00565607
\(804\) 0 0
\(805\) 0 0
\(806\) 7.57318e6 0.410621
\(807\) 0 0
\(808\) 7.16031e6 0.385836
\(809\) −8.10892e6 −0.435604 −0.217802 0.975993i \(-0.569889\pi\)
−0.217802 + 0.975993i \(0.569889\pi\)
\(810\) 0 0
\(811\) −9.13444e6 −0.487674 −0.243837 0.969816i \(-0.578406\pi\)
−0.243837 + 0.969816i \(0.578406\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 240709. 0.0127330
\(815\) −2.95439e7 −1.55802
\(816\) 0 0
\(817\) 2.83681e6 0.148688
\(818\) 1.18942e7 0.621517
\(819\) 0 0
\(820\) −2.62085e7 −1.36115
\(821\) −1.70155e6 −0.0881023 −0.0440511 0.999029i \(-0.514026\pi\)
−0.0440511 + 0.999029i \(0.514026\pi\)
\(822\) 0 0
\(823\) −2.49207e7 −1.28251 −0.641256 0.767327i \(-0.721586\pi\)
−0.641256 + 0.767327i \(0.721586\pi\)
\(824\) −1.27164e7 −0.652448
\(825\) 0 0
\(826\) 0 0
\(827\) 1.91232e7 0.972292 0.486146 0.873878i \(-0.338402\pi\)
0.486146 + 0.873878i \(0.338402\pi\)
\(828\) 0 0
\(829\) −2.20267e7 −1.11318 −0.556588 0.830789i \(-0.687890\pi\)
−0.556588 + 0.830789i \(0.687890\pi\)
\(830\) 1.53705e7 0.774450
\(831\) 0 0
\(832\) 4.39531e6 0.220131
\(833\) 0 0
\(834\) 0 0
\(835\) −3.30102e7 −1.63844
\(836\) −740943. −0.0366665
\(837\) 0 0
\(838\) −4.15329e6 −0.204306
\(839\) −4.28039e6 −0.209932 −0.104966 0.994476i \(-0.533473\pi\)
−0.104966 + 0.994476i \(0.533473\pi\)
\(840\) 0 0
\(841\) 1.12255e7 0.547286
\(842\) −3.06659e6 −0.149065
\(843\) 0 0
\(844\) −1.90374e7 −0.919922
\(845\) −2.45544e7 −1.18301
\(846\) 0 0
\(847\) 0 0
\(848\) 8.27933e6 0.395372
\(849\) 0 0
\(850\) 794892. 0.0377364
\(851\) −1.97868e7 −0.936596
\(852\) 0 0
\(853\) −1.72415e7 −0.811341 −0.405670 0.914019i \(-0.632962\pi\)
−0.405670 + 0.914019i \(0.632962\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 442685. 0.0206495
\(857\) −2.08340e7 −0.968991 −0.484495 0.874794i \(-0.660997\pi\)
−0.484495 + 0.874794i \(0.660997\pi\)
\(858\) 0 0
\(859\) −1.58356e7 −0.732238 −0.366119 0.930568i \(-0.619314\pi\)
−0.366119 + 0.930568i \(0.619314\pi\)
\(860\) −2.32720e6 −0.107297
\(861\) 0 0
\(862\) 1.70757e7 0.782726
\(863\) 1.59282e7 0.728012 0.364006 0.931397i \(-0.381409\pi\)
0.364006 + 0.931397i \(0.381409\pi\)
\(864\) 0 0
\(865\) −3.84894e7 −1.74905
\(866\) −8.90735e6 −0.403603
\(867\) 0 0
\(868\) 0 0
\(869\) 1.54149e6 0.0692455
\(870\) 0 0
\(871\) 4.18382e7 1.86865
\(872\) −2.57944e7 −1.14877
\(873\) 0 0
\(874\) −1.88159e7 −0.833195
\(875\) 0 0
\(876\) 0 0
\(877\) 5.20734e6 0.228621 0.114311 0.993445i \(-0.463534\pi\)
0.114311 + 0.993445i \(0.463534\pi\)
\(878\) −6.33446e6 −0.277315
\(879\) 0 0
\(880\) 362051. 0.0157603
\(881\) 932829. 0.0404913 0.0202457 0.999795i \(-0.493555\pi\)
0.0202457 + 0.999795i \(0.493555\pi\)
\(882\) 0 0
\(883\) 1.33789e7 0.577457 0.288728 0.957411i \(-0.406768\pi\)
0.288728 + 0.957411i \(0.406768\pi\)
\(884\) 2.23361e7 0.961341
\(885\) 0 0
\(886\) 4.21202e6 0.180263
\(887\) 6.96900e6 0.297414 0.148707 0.988881i \(-0.452489\pi\)
0.148707 + 0.988881i \(0.452489\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.46004e6 0.315694
\(891\) 0 0
\(892\) −1.80443e7 −0.759326
\(893\) −1.67074e7 −0.701099
\(894\) 0 0
\(895\) 9.64753e6 0.402586
\(896\) 0 0
\(897\) 0 0
\(898\) −9.77577e6 −0.404539
\(899\) 1.74444e7 0.719874
\(900\) 0 0
\(901\) −2.38760e7 −0.979826
\(902\) −879496. −0.0359929
\(903\) 0 0
\(904\) −4.03804e7 −1.64342
\(905\) −8.66706e6 −0.351763
\(906\) 0 0
\(907\) −3.76563e7 −1.51992 −0.759959 0.649972i \(-0.774781\pi\)
−0.759959 + 0.649972i \(0.774781\pi\)
\(908\) −1.33023e7 −0.535440
\(909\) 0 0
\(910\) 0 0
\(911\) 3.09942e7 1.23733 0.618663 0.785657i \(-0.287675\pi\)
0.618663 + 0.785657i \(0.287675\pi\)
\(912\) 0 0
\(913\) −1.66965e6 −0.0662899
\(914\) −6.92219e6 −0.274080
\(915\) 0 0
\(916\) 5.71002e6 0.224853
\(917\) 0 0
\(918\) 0 0
\(919\) 716014. 0.0279662 0.0139831 0.999902i \(-0.495549\pi\)
0.0139831 + 0.999902i \(0.495549\pi\)
\(920\) 3.56401e7 1.38826
\(921\) 0 0
\(922\) −1.82279e7 −0.706169
\(923\) 1.42357e6 0.0550014
\(924\) 0 0
\(925\) 1.41626e6 0.0544238
\(926\) 1.21039e7 0.463870
\(927\) 0 0
\(928\) 3.34769e7 1.27607
\(929\) −4.17630e7 −1.58764 −0.793820 0.608153i \(-0.791911\pi\)
−0.793820 + 0.608153i \(0.791911\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.51110e7 −0.569840
\(933\) 0 0
\(934\) 675626. 0.0253419
\(935\) −1.04409e6 −0.0390577
\(936\) 0 0
\(937\) −2.17917e7 −0.810854 −0.405427 0.914127i \(-0.632877\pi\)
−0.405427 + 0.914127i \(0.632877\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.37060e7 0.505932
\(941\) 2.15218e6 0.0792329 0.0396164 0.999215i \(-0.487386\pi\)
0.0396164 + 0.999215i \(0.487386\pi\)
\(942\) 0 0
\(943\) 7.22965e7 2.64751
\(944\) 1.28275e6 0.0468504
\(945\) 0 0
\(946\) −78095.5 −0.00283725
\(947\) 2.64249e7 0.957501 0.478750 0.877951i \(-0.341090\pi\)
0.478750 + 0.877951i \(0.341090\pi\)
\(948\) 0 0
\(949\) 5.27850e6 0.190259
\(950\) 1.34677e6 0.0484153
\(951\) 0 0
\(952\) 0 0
\(953\) −9.07051e6 −0.323519 −0.161759 0.986830i \(-0.551717\pi\)
−0.161759 + 0.986830i \(0.551717\pi\)
\(954\) 0 0
\(955\) 2.25246e7 0.799186
\(956\) 2.29195e7 0.811075
\(957\) 0 0
\(958\) −110892. −0.00390380
\(959\) 0 0
\(960\) 0 0
\(961\) −1.90406e7 −0.665079
\(962\) −1.22942e7 −0.428314
\(963\) 0 0
\(964\) −2.27860e7 −0.789723
\(965\) −2.89674e7 −1.00136
\(966\) 0 0
\(967\) 1.80108e7 0.619395 0.309698 0.950835i \(-0.399772\pi\)
0.309698 + 0.950835i \(0.399772\pi\)
\(968\) −2.49364e7 −0.855354
\(969\) 0 0
\(970\) −1.21878e7 −0.415906
\(971\) 4.67631e7 1.59168 0.795838 0.605509i \(-0.207030\pi\)
0.795838 + 0.605509i \(0.207030\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 9.09561e6 0.307209
\(975\) 0 0
\(976\) 8.14408e6 0.273664
\(977\) −4.32433e6 −0.144938 −0.0724690 0.997371i \(-0.523088\pi\)
−0.0724690 + 0.997371i \(0.523088\pi\)
\(978\) 0 0
\(979\) −810357. −0.0270221
\(980\) 0 0
\(981\) 0 0
\(982\) −5.44026e6 −0.180028
\(983\) −2.08733e7 −0.688982 −0.344491 0.938790i \(-0.611949\pi\)
−0.344491 + 0.938790i \(0.611949\pi\)
\(984\) 0 0
\(985\) 2.57656e7 0.846155
\(986\) −1.58943e7 −0.520655
\(987\) 0 0
\(988\) 3.78436e7 1.23339
\(989\) 6.41963e6 0.208698
\(990\) 0 0
\(991\) 4.12209e7 1.33332 0.666658 0.745364i \(-0.267724\pi\)
0.666658 + 0.745364i \(0.267724\pi\)
\(992\) 1.84010e7 0.593693
\(993\) 0 0
\(994\) 0 0
\(995\) 4.42650e6 0.141743
\(996\) 0 0
\(997\) −5.34615e7 −1.70335 −0.851674 0.524073i \(-0.824412\pi\)
−0.851674 + 0.524073i \(0.824412\pi\)
\(998\) 8.71160e6 0.276867
\(999\) 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.1.2 4
3.2 odd 2 147.6.a.m.1.3 4
7.2 even 3 63.6.e.e.46.3 8
7.4 even 3 63.6.e.e.37.3 8
7.6 odd 2 441.6.a.v.1.2 4
21.2 odd 6 21.6.e.c.4.2 8
21.5 even 6 147.6.e.o.67.2 8
21.11 odd 6 21.6.e.c.16.2 yes 8
21.17 even 6 147.6.e.o.79.2 8
21.20 even 2 147.6.a.l.1.3 4
84.11 even 6 336.6.q.j.289.1 8
84.23 even 6 336.6.q.j.193.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.2 8 21.2 odd 6
21.6.e.c.16.2 yes 8 21.11 odd 6
63.6.e.e.37.3 8 7.4 even 3
63.6.e.e.46.3 8 7.2 even 3
147.6.a.l.1.3 4 21.20 even 2
147.6.a.m.1.3 4 3.2 odd 2
147.6.e.o.67.2 8 21.5 even 6
147.6.e.o.79.2 8 21.17 even 6
336.6.q.j.193.1 8 84.23 even 6
336.6.q.j.289.1 8 84.11 even 6
441.6.a.v.1.2 4 7.6 odd 2
441.6.a.w.1.2 4 1.1 even 1 trivial