Properties

Label 189.4.a.l.1.1
Level $189$
Weight $4$
Character 189.1
Self dual yes
Analytic conductor $11.151$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,4,Mod(1,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 189.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: \(11.1513609911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.684742\) of defining polynomial
Character \(\chi\) \(=\) 189.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} -7.00000 q^{7} -54.6288 q^{8} +O(q^{10})\) \(q-5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} -7.00000 q^{7} -54.6288 q^{8} +87.7802 q^{10} +30.3531 q^{11} -89.5603 q^{13} +36.0981 q^{14} +132.967 q^{16} -13.4164 q^{17} -5.37355 q^{19} -316.496 q^{20} -156.527 q^{22} -167.449 q^{23} +164.747 q^{25} +461.852 q^{26} -130.154 q^{28} +135.085 q^{29} -18.9339 q^{31} -248.664 q^{32} +69.1868 q^{34} +119.154 q^{35} +402.560 q^{37} +27.7107 q^{38} +929.889 q^{40} +434.345 q^{41} +53.1868 q^{43} +564.367 q^{44} +863.516 q^{46} +155.218 q^{47} +49.0000 q^{49} -849.581 q^{50} -1665.23 q^{52} +301.707 q^{53} -516.669 q^{55} +382.402 q^{56} -696.615 q^{58} -412.635 q^{59} -571.253 q^{61} +97.6396 q^{62} +218.593 q^{64} +1524.49 q^{65} +820.549 q^{67} -249.456 q^{68} -614.461 q^{70} +8.95825 q^{71} +21.9339 q^{73} -2075.95 q^{74} -99.9124 q^{76} -212.472 q^{77} +619.428 q^{79} -2263.36 q^{80} -2239.87 q^{82} +259.532 q^{83} +228.374 q^{85} -274.278 q^{86} -1658.15 q^{88} +484.329 q^{89} +626.922 q^{91} -3113.45 q^{92} -800.440 q^{94} +91.4683 q^{95} -252.747 q^{97} -252.687 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{4} - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 26 q^{4} - 28 q^{7} + 206 q^{10} - 68 q^{13} + 290 q^{16} + 172 q^{19} - 94 q^{22} + 272 q^{25} - 182 q^{28} + 408 q^{31} + 180 q^{34} + 1320 q^{37} + 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 196 q^{49} - 3952 q^{52} + 352 q^{55} - 1432 q^{58} - 2672 q^{61} + 826 q^{64} + 1444 q^{67} - 1442 q^{70} - 396 q^{73} - 1222 q^{76} + 1220 q^{79} - 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 476 q^{91} - 3492 q^{94} - 624 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.15688 −1.82323 −0.911616 0.411043i \(-0.865164\pi\)
−0.911616 + 0.411043i \(0.865164\pi\)
\(3\) 0 0
\(4\) 18.5934 2.32417
\(5\) −17.0220 −1.52249 −0.761245 0.648464i \(-0.775412\pi\)
−0.761245 + 0.648464i \(0.775412\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −54.6288 −2.41427
\(9\) 0 0
\(10\) 87.7802 2.77585
\(11\) 30.3531 0.831982 0.415991 0.909369i \(-0.363435\pi\)
0.415991 + 0.909369i \(0.363435\pi\)
\(12\) 0 0
\(13\) −89.5603 −1.91074 −0.955368 0.295419i \(-0.904541\pi\)
−0.955368 + 0.295419i \(0.904541\pi\)
\(14\) 36.0981 0.689117
\(15\) 0 0
\(16\) 132.967 2.07761
\(17\) −13.4164 −0.191409 −0.0957046 0.995410i \(-0.530510\pi\)
−0.0957046 + 0.995410i \(0.530510\pi\)
\(18\) 0 0
\(19\) −5.37355 −0.0648830 −0.0324415 0.999474i \(-0.510328\pi\)
−0.0324415 + 0.999474i \(0.510328\pi\)
\(20\) −316.496 −3.53853
\(21\) 0 0
\(22\) −156.527 −1.51690
\(23\) −167.449 −1.51807 −0.759035 0.651050i \(-0.774329\pi\)
−0.759035 + 0.651050i \(0.774329\pi\)
\(24\) 0 0
\(25\) 164.747 1.31798
\(26\) 461.852 3.48371
\(27\) 0 0
\(28\) −130.154 −0.878455
\(29\) 135.085 0.864986 0.432493 0.901637i \(-0.357634\pi\)
0.432493 + 0.901637i \(0.357634\pi\)
\(30\) 0 0
\(31\) −18.9339 −0.109698 −0.0548488 0.998495i \(-0.517468\pi\)
−0.0548488 + 0.998495i \(0.517468\pi\)
\(32\) −248.664 −1.37369
\(33\) 0 0
\(34\) 69.1868 0.348983
\(35\) 119.154 0.575447
\(36\) 0 0
\(37\) 402.560 1.78866 0.894331 0.447406i \(-0.147652\pi\)
0.894331 + 0.447406i \(0.147652\pi\)
\(38\) 27.7107 0.118297
\(39\) 0 0
\(40\) 929.889 3.67571
\(41\) 434.345 1.65447 0.827236 0.561855i \(-0.189912\pi\)
0.827236 + 0.561855i \(0.189912\pi\)
\(42\) 0 0
\(43\) 53.1868 0.188626 0.0943129 0.995543i \(-0.469935\pi\)
0.0943129 + 0.995543i \(0.469935\pi\)
\(44\) 564.367 1.93367
\(45\) 0 0
\(46\) 863.516 2.76779
\(47\) 155.218 0.481720 0.240860 0.970560i \(-0.422570\pi\)
0.240860 + 0.970560i \(0.422570\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −849.581 −2.40298
\(51\) 0 0
\(52\) −1665.23 −4.44088
\(53\) 301.707 0.781938 0.390969 0.920404i \(-0.372140\pi\)
0.390969 + 0.920404i \(0.372140\pi\)
\(54\) 0 0
\(55\) −516.669 −1.26669
\(56\) 382.402 0.912510
\(57\) 0 0
\(58\) −696.615 −1.57707
\(59\) −412.635 −0.910518 −0.455259 0.890359i \(-0.650453\pi\)
−0.455259 + 0.890359i \(0.650453\pi\)
\(60\) 0 0
\(61\) −571.253 −1.19904 −0.599520 0.800360i \(-0.704642\pi\)
−0.599520 + 0.800360i \(0.704642\pi\)
\(62\) 97.6396 0.200004
\(63\) 0 0
\(64\) 218.593 0.426940
\(65\) 1524.49 2.90908
\(66\) 0 0
\(67\) 820.549 1.49621 0.748104 0.663581i \(-0.230964\pi\)
0.748104 + 0.663581i \(0.230964\pi\)
\(68\) −249.456 −0.444868
\(69\) 0 0
\(70\) −614.461 −1.04917
\(71\) 8.95825 0.0149739 0.00748696 0.999972i \(-0.497617\pi\)
0.00748696 + 0.999972i \(0.497617\pi\)
\(72\) 0 0
\(73\) 21.9339 0.0351666 0.0175833 0.999845i \(-0.494403\pi\)
0.0175833 + 0.999845i \(0.494403\pi\)
\(74\) −2075.95 −3.26115
\(75\) 0 0
\(76\) −99.9124 −0.150799
\(77\) −212.472 −0.314460
\(78\) 0 0
\(79\) 619.428 0.882166 0.441083 0.897466i \(-0.354594\pi\)
0.441083 + 0.897466i \(0.354594\pi\)
\(80\) −2263.36 −3.16314
\(81\) 0 0
\(82\) −2239.87 −3.01649
\(83\) 259.532 0.343221 0.171610 0.985165i \(-0.445103\pi\)
0.171610 + 0.985165i \(0.445103\pi\)
\(84\) 0 0
\(85\) 228.374 0.291419
\(86\) −274.278 −0.343908
\(87\) 0 0
\(88\) −1658.15 −2.00863
\(89\) 484.329 0.576840 0.288420 0.957504i \(-0.406870\pi\)
0.288420 + 0.957504i \(0.406870\pi\)
\(90\) 0 0
\(91\) 626.922 0.722190
\(92\) −3113.45 −3.52826
\(93\) 0 0
\(94\) −800.440 −0.878288
\(95\) 91.4683 0.0987837
\(96\) 0 0
\(97\) −252.747 −0.264563 −0.132281 0.991212i \(-0.542230\pi\)
−0.132281 + 0.991212i \(0.542230\pi\)
\(98\) −252.687 −0.260462
\(99\) 0 0
\(100\) 3063.21 3.06321
\(101\) −629.378 −0.620054 −0.310027 0.950728i \(-0.600338\pi\)
−0.310027 + 0.950728i \(0.600338\pi\)
\(102\) 0 0
\(103\) 1172.54 1.12168 0.560842 0.827923i \(-0.310477\pi\)
0.560842 + 0.827923i \(0.310477\pi\)
\(104\) 4892.57 4.61304
\(105\) 0 0
\(106\) −1555.87 −1.42565
\(107\) −1002.87 −0.906087 −0.453044 0.891488i \(-0.649662\pi\)
−0.453044 + 0.891488i \(0.649662\pi\)
\(108\) 0 0
\(109\) 542.802 0.476981 0.238491 0.971145i \(-0.423347\pi\)
0.238491 + 0.971145i \(0.423347\pi\)
\(110\) 2664.40 2.30946
\(111\) 0 0
\(112\) −930.769 −0.785262
\(113\) −1388.74 −1.15612 −0.578062 0.815993i \(-0.696191\pi\)
−0.578062 + 0.815993i \(0.696191\pi\)
\(114\) 0 0
\(115\) 2850.32 2.31125
\(116\) 2511.68 2.01038
\(117\) 0 0
\(118\) 2127.91 1.66009
\(119\) 93.9149 0.0723459
\(120\) 0 0
\(121\) −409.689 −0.307805
\(122\) 2945.88 2.18613
\(123\) 0 0
\(124\) −352.045 −0.254956
\(125\) −676.573 −0.484117
\(126\) 0 0
\(127\) −2330.00 −1.62798 −0.813991 0.580877i \(-0.802710\pi\)
−0.813991 + 0.580877i \(0.802710\pi\)
\(128\) 862.051 0.595276
\(129\) 0 0
\(130\) −7861.62 −5.30392
\(131\) 226.817 0.151276 0.0756379 0.997135i \(-0.475901\pi\)
0.0756379 + 0.997135i \(0.475901\pi\)
\(132\) 0 0
\(133\) 37.6148 0.0245235
\(134\) −4231.47 −2.72793
\(135\) 0 0
\(136\) 732.922 0.462114
\(137\) 1900.83 1.18540 0.592698 0.805425i \(-0.298063\pi\)
0.592698 + 0.805425i \(0.298063\pi\)
\(138\) 0 0
\(139\) 1958.99 1.19539 0.597695 0.801724i \(-0.296083\pi\)
0.597695 + 0.801724i \(0.296083\pi\)
\(140\) 2215.47 1.33744
\(141\) 0 0
\(142\) −46.1966 −0.0273009
\(143\) −2718.43 −1.58970
\(144\) 0 0
\(145\) −2299.40 −1.31693
\(146\) −113.110 −0.0641169
\(147\) 0 0
\(148\) 7484.96 4.15716
\(149\) −1518.37 −0.834831 −0.417416 0.908716i \(-0.637064\pi\)
−0.417416 + 0.908716i \(0.637064\pi\)
\(150\) 0 0
\(151\) −1815.18 −0.978262 −0.489131 0.872210i \(-0.662686\pi\)
−0.489131 + 0.872210i \(0.662686\pi\)
\(152\) 293.550 0.156645
\(153\) 0 0
\(154\) 1095.69 0.573333
\(155\) 322.291 0.167013
\(156\) 0 0
\(157\) −1858.79 −0.944889 −0.472445 0.881360i \(-0.656628\pi\)
−0.472445 + 0.881360i \(0.656628\pi\)
\(158\) −3194.31 −1.60839
\(159\) 0 0
\(160\) 4232.75 2.09142
\(161\) 1172.15 0.573776
\(162\) 0 0
\(163\) 870.089 0.418102 0.209051 0.977905i \(-0.432962\pi\)
0.209051 + 0.977905i \(0.432962\pi\)
\(164\) 8075.95 3.84528
\(165\) 0 0
\(166\) −1338.37 −0.625771
\(167\) −1749.13 −0.810488 −0.405244 0.914209i \(-0.632813\pi\)
−0.405244 + 0.914209i \(0.632813\pi\)
\(168\) 0 0
\(169\) 5824.05 2.65091
\(170\) −1177.69 −0.531324
\(171\) 0 0
\(172\) 988.922 0.438399
\(173\) −39.3374 −0.0172877 −0.00864383 0.999963i \(-0.502751\pi\)
−0.00864383 + 0.999963i \(0.502751\pi\)
\(174\) 0 0
\(175\) −1153.23 −0.498148
\(176\) 4035.96 1.72853
\(177\) 0 0
\(178\) −2497.62 −1.05171
\(179\) 2650.58 1.10678 0.553391 0.832922i \(-0.313334\pi\)
0.553391 + 0.832922i \(0.313334\pi\)
\(180\) 0 0
\(181\) 2719.71 1.11688 0.558438 0.829546i \(-0.311401\pi\)
0.558438 + 0.829546i \(0.311401\pi\)
\(182\) −3232.96 −1.31672
\(183\) 0 0
\(184\) 9147.55 3.66504
\(185\) −6852.37 −2.72322
\(186\) 0 0
\(187\) −407.230 −0.159249
\(188\) 2886.03 1.11960
\(189\) 0 0
\(190\) −471.691 −0.180106
\(191\) −4339.00 −1.64376 −0.821881 0.569659i \(-0.807075\pi\)
−0.821881 + 0.569659i \(0.807075\pi\)
\(192\) 0 0
\(193\) 977.689 0.364640 0.182320 0.983239i \(-0.441639\pi\)
0.182320 + 0.983239i \(0.441639\pi\)
\(194\) 1303.39 0.482359
\(195\) 0 0
\(196\) 911.076 0.332025
\(197\) 1556.32 0.562859 0.281430 0.959582i \(-0.409191\pi\)
0.281430 + 0.959582i \(0.409191\pi\)
\(198\) 0 0
\(199\) 1723.21 0.613846 0.306923 0.951734i \(-0.400701\pi\)
0.306923 + 0.951734i \(0.400701\pi\)
\(200\) −8999.94 −3.18196
\(201\) 0 0
\(202\) 3245.62 1.13050
\(203\) −945.592 −0.326934
\(204\) 0 0
\(205\) −7393.41 −2.51892
\(206\) −6046.63 −2.04509
\(207\) 0 0
\(208\) −11908.6 −3.96976
\(209\) −163.104 −0.0539815
\(210\) 0 0
\(211\) 1295.19 0.422581 0.211290 0.977423i \(-0.432233\pi\)
0.211290 + 0.977423i \(0.432233\pi\)
\(212\) 5609.76 1.81736
\(213\) 0 0
\(214\) 5171.69 1.65201
\(215\) −905.343 −0.287181
\(216\) 0 0
\(217\) 132.537 0.0414618
\(218\) −2799.16 −0.869647
\(219\) 0 0
\(220\) −9606.63 −2.94400
\(221\) 1201.58 0.365732
\(222\) 0 0
\(223\) −483.611 −0.145224 −0.0726121 0.997360i \(-0.523134\pi\)
−0.0726121 + 0.997360i \(0.523134\pi\)
\(224\) 1740.65 0.519205
\(225\) 0 0
\(226\) 7161.58 2.10788
\(227\) 4019.86 1.17536 0.587681 0.809092i \(-0.300041\pi\)
0.587681 + 0.809092i \(0.300041\pi\)
\(228\) 0 0
\(229\) 5070.37 1.46314 0.731570 0.681766i \(-0.238788\pi\)
0.731570 + 0.681766i \(0.238788\pi\)
\(230\) −14698.7 −4.21394
\(231\) 0 0
\(232\) −7379.51 −2.08831
\(233\) 4418.26 1.24227 0.621137 0.783702i \(-0.286671\pi\)
0.621137 + 0.783702i \(0.286671\pi\)
\(234\) 0 0
\(235\) −2642.11 −0.733415
\(236\) −7672.29 −2.11620
\(237\) 0 0
\(238\) −484.307 −0.131903
\(239\) 3228.15 0.873690 0.436845 0.899537i \(-0.356096\pi\)
0.436845 + 0.899537i \(0.356096\pi\)
\(240\) 0 0
\(241\) −1537.14 −0.410854 −0.205427 0.978672i \(-0.565858\pi\)
−0.205427 + 0.978672i \(0.565858\pi\)
\(242\) 2112.72 0.561200
\(243\) 0 0
\(244\) −10621.5 −2.78678
\(245\) −834.076 −0.217499
\(246\) 0 0
\(247\) 481.257 0.123974
\(248\) 1034.33 0.264840
\(249\) 0 0
\(250\) 3489.01 0.882657
\(251\) −178.384 −0.0448587 −0.0224293 0.999748i \(-0.507140\pi\)
−0.0224293 + 0.999748i \(0.507140\pi\)
\(252\) 0 0
\(253\) −5082.61 −1.26301
\(254\) 12015.5 2.96819
\(255\) 0 0
\(256\) −6194.24 −1.51227
\(257\) −3271.92 −0.794152 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(258\) 0 0
\(259\) −2817.92 −0.676051
\(260\) 28345.5 6.76120
\(261\) 0 0
\(262\) −1169.67 −0.275811
\(263\) 4738.45 1.11097 0.555486 0.831526i \(-0.312532\pi\)
0.555486 + 0.831526i \(0.312532\pi\)
\(264\) 0 0
\(265\) −5135.65 −1.19049
\(266\) −193.975 −0.0447119
\(267\) 0 0
\(268\) 15256.8 3.47745
\(269\) −418.456 −0.0948466 −0.0474233 0.998875i \(-0.515101\pi\)
−0.0474233 + 0.998875i \(0.515101\pi\)
\(270\) 0 0
\(271\) 5263.95 1.17993 0.589967 0.807428i \(-0.299141\pi\)
0.589967 + 0.807428i \(0.299141\pi\)
\(272\) −1783.94 −0.397673
\(273\) 0 0
\(274\) −9802.37 −2.16125
\(275\) 5000.59 1.09653
\(276\) 0 0
\(277\) 2855.61 0.619412 0.309706 0.950832i \(-0.399769\pi\)
0.309706 + 0.950832i \(0.399769\pi\)
\(278\) −10102.3 −2.17947
\(279\) 0 0
\(280\) −6509.22 −1.38929
\(281\) −8589.72 −1.82356 −0.911779 0.410682i \(-0.865291\pi\)
−0.911779 + 0.410682i \(0.865291\pi\)
\(282\) 0 0
\(283\) 3681.18 0.773228 0.386614 0.922242i \(-0.373645\pi\)
0.386614 + 0.922242i \(0.373645\pi\)
\(284\) 166.564 0.0348020
\(285\) 0 0
\(286\) 14018.6 2.89839
\(287\) −3040.42 −0.625332
\(288\) 0 0
\(289\) −4733.00 −0.963363
\(290\) 11857.7 2.40107
\(291\) 0 0
\(292\) 407.825 0.0817334
\(293\) −1958.68 −0.390536 −0.195268 0.980750i \(-0.562558\pi\)
−0.195268 + 0.980750i \(0.562558\pi\)
\(294\) 0 0
\(295\) 7023.86 1.38625
\(296\) −21991.4 −4.31832
\(297\) 0 0
\(298\) 7830.06 1.52209
\(299\) 14996.8 2.90063
\(300\) 0 0
\(301\) −372.307 −0.0712938
\(302\) 9360.68 1.78360
\(303\) 0 0
\(304\) −714.504 −0.134801
\(305\) 9723.84 1.82553
\(306\) 0 0
\(307\) −5672.32 −1.05452 −0.527258 0.849706i \(-0.676780\pi\)
−0.527258 + 0.849706i \(0.676780\pi\)
\(308\) −3950.57 −0.730859
\(309\) 0 0
\(310\) −1662.02 −0.304504
\(311\) −1271.71 −0.231872 −0.115936 0.993257i \(-0.536987\pi\)
−0.115936 + 0.993257i \(0.536987\pi\)
\(312\) 0 0
\(313\) −4868.38 −0.879160 −0.439580 0.898204i \(-0.644873\pi\)
−0.439580 + 0.898204i \(0.644873\pi\)
\(314\) 9585.55 1.72275
\(315\) 0 0
\(316\) 11517.3 2.05031
\(317\) −3224.18 −0.571256 −0.285628 0.958341i \(-0.592202\pi\)
−0.285628 + 0.958341i \(0.592202\pi\)
\(318\) 0 0
\(319\) 4100.24 0.719653
\(320\) −3720.89 −0.650012
\(321\) 0 0
\(322\) −6044.61 −1.04613
\(323\) 72.0937 0.0124192
\(324\) 0 0
\(325\) −14754.8 −2.51831
\(326\) −4486.94 −0.762297
\(327\) 0 0
\(328\) −23727.8 −3.99435
\(329\) −1086.53 −0.182073
\(330\) 0 0
\(331\) 5309.27 0.881643 0.440821 0.897595i \(-0.354687\pi\)
0.440821 + 0.897595i \(0.354687\pi\)
\(332\) 4825.57 0.797704
\(333\) 0 0
\(334\) 9020.04 1.47771
\(335\) −13967.3 −2.27796
\(336\) 0 0
\(337\) −2304.33 −0.372478 −0.186239 0.982505i \(-0.559630\pi\)
−0.186239 + 0.982505i \(0.559630\pi\)
\(338\) −30033.9 −4.83322
\(339\) 0 0
\(340\) 4246.24 0.677308
\(341\) −574.702 −0.0912664
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −2905.53 −0.455394
\(345\) 0 0
\(346\) 202.858 0.0315194
\(347\) 5196.20 0.803881 0.401940 0.915666i \(-0.368336\pi\)
0.401940 + 0.915666i \(0.368336\pi\)
\(348\) 0 0
\(349\) −9030.60 −1.38509 −0.692546 0.721374i \(-0.743511\pi\)
−0.692546 + 0.721374i \(0.743511\pi\)
\(350\) 5947.06 0.908240
\(351\) 0 0
\(352\) −7547.72 −1.14288
\(353\) 8885.51 1.33974 0.669869 0.742479i \(-0.266350\pi\)
0.669869 + 0.742479i \(0.266350\pi\)
\(354\) 0 0
\(355\) −152.487 −0.0227976
\(356\) 9005.31 1.34068
\(357\) 0 0
\(358\) −13668.7 −2.01792
\(359\) 37.2983 0.00548337 0.00274169 0.999996i \(-0.499127\pi\)
0.00274169 + 0.999996i \(0.499127\pi\)
\(360\) 0 0
\(361\) −6830.12 −0.995790
\(362\) −14025.2 −2.03632
\(363\) 0 0
\(364\) 11656.6 1.67850
\(365\) −373.357 −0.0535409
\(366\) 0 0
\(367\) 3335.79 0.474460 0.237230 0.971454i \(-0.423760\pi\)
0.237230 + 0.971454i \(0.423760\pi\)
\(368\) −22265.2 −3.15395
\(369\) 0 0
\(370\) 35336.8 4.96506
\(371\) −2111.95 −0.295545
\(372\) 0 0
\(373\) 3935.81 0.546349 0.273175 0.961964i \(-0.411926\pi\)
0.273175 + 0.961964i \(0.411926\pi\)
\(374\) 2100.03 0.290348
\(375\) 0 0
\(376\) −8479.37 −1.16301
\(377\) −12098.2 −1.65276
\(378\) 0 0
\(379\) 2308.00 0.312808 0.156404 0.987693i \(-0.450010\pi\)
0.156404 + 0.987693i \(0.450010\pi\)
\(380\) 1700.71 0.229590
\(381\) 0 0
\(382\) 22375.7 2.99696
\(383\) 2931.62 0.391119 0.195559 0.980692i \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(384\) 0 0
\(385\) 3616.69 0.478762
\(386\) −5041.82 −0.664824
\(387\) 0 0
\(388\) −4699.42 −0.614890
\(389\) 7610.12 0.991898 0.495949 0.868352i \(-0.334820\pi\)
0.495949 + 0.868352i \(0.334820\pi\)
\(390\) 0 0
\(391\) 2246.57 0.290572
\(392\) −2676.81 −0.344896
\(393\) 0 0
\(394\) −8025.76 −1.02622
\(395\) −10543.9 −1.34309
\(396\) 0 0
\(397\) −4269.74 −0.539778 −0.269889 0.962891i \(-0.586987\pi\)
−0.269889 + 0.962891i \(0.586987\pi\)
\(398\) −8886.41 −1.11918
\(399\) 0 0
\(400\) 21905.9 2.73824
\(401\) 11128.8 1.38590 0.692948 0.720988i \(-0.256312\pi\)
0.692948 + 0.720988i \(0.256312\pi\)
\(402\) 0 0
\(403\) 1695.72 0.209603
\(404\) −11702.3 −1.44111
\(405\) 0 0
\(406\) 4876.30 0.596076
\(407\) 12219.0 1.48814
\(408\) 0 0
\(409\) 7220.39 0.872922 0.436461 0.899723i \(-0.356232\pi\)
0.436461 + 0.899723i \(0.356232\pi\)
\(410\) 38126.9 4.59257
\(411\) 0 0
\(412\) 21801.4 2.60699
\(413\) 2888.45 0.344143
\(414\) 0 0
\(415\) −4417.74 −0.522550
\(416\) 22270.4 2.62475
\(417\) 0 0
\(418\) 841.106 0.0984207
\(419\) 16631.2 1.93911 0.969556 0.244870i \(-0.0787454\pi\)
0.969556 + 0.244870i \(0.0787454\pi\)
\(420\) 0 0
\(421\) −654.498 −0.0757679 −0.0378839 0.999282i \(-0.512062\pi\)
−0.0378839 + 0.999282i \(0.512062\pi\)
\(422\) −6679.14 −0.770463
\(423\) 0 0
\(424\) −16481.9 −1.88781
\(425\) −2210.31 −0.252273
\(426\) 0 0
\(427\) 3998.77 0.453195
\(428\) −18646.8 −2.10590
\(429\) 0 0
\(430\) 4668.74 0.523597
\(431\) 3049.91 0.340856 0.170428 0.985370i \(-0.445485\pi\)
0.170428 + 0.985370i \(0.445485\pi\)
\(432\) 0 0
\(433\) −1599.41 −0.177512 −0.0887562 0.996053i \(-0.528289\pi\)
−0.0887562 + 0.996053i \(0.528289\pi\)
\(434\) −683.477 −0.0755944
\(435\) 0 0
\(436\) 10092.5 1.10859
\(437\) 899.797 0.0984968
\(438\) 0 0
\(439\) −15893.9 −1.72796 −0.863979 0.503528i \(-0.832035\pi\)
−0.863979 + 0.503528i \(0.832035\pi\)
\(440\) 28225.0 3.05813
\(441\) 0 0
\(442\) −6196.39 −0.666815
\(443\) 6160.22 0.660679 0.330339 0.943862i \(-0.392837\pi\)
0.330339 + 0.943862i \(0.392837\pi\)
\(444\) 0 0
\(445\) −8244.23 −0.878233
\(446\) 2493.92 0.264777
\(447\) 0 0
\(448\) −1530.15 −0.161368
\(449\) −8742.47 −0.918892 −0.459446 0.888206i \(-0.651952\pi\)
−0.459446 + 0.888206i \(0.651952\pi\)
\(450\) 0 0
\(451\) 13183.7 1.37649
\(452\) −25821.4 −2.68703
\(453\) 0 0
\(454\) −20729.9 −2.14296
\(455\) −10671.4 −1.09953
\(456\) 0 0
\(457\) −580.112 −0.0593796 −0.0296898 0.999559i \(-0.509452\pi\)
−0.0296898 + 0.999559i \(0.509452\pi\)
\(458\) −26147.3 −2.66764
\(459\) 0 0
\(460\) 52997.0 5.37174
\(461\) −1957.05 −0.197720 −0.0988599 0.995101i \(-0.531520\pi\)
−0.0988599 + 0.995101i \(0.531520\pi\)
\(462\) 0 0
\(463\) 8111.80 0.814227 0.407114 0.913378i \(-0.366535\pi\)
0.407114 + 0.913378i \(0.366535\pi\)
\(464\) 17961.8 1.79710
\(465\) 0 0
\(466\) −22784.4 −2.26495
\(467\) −1240.29 −0.122899 −0.0614493 0.998110i \(-0.519572\pi\)
−0.0614493 + 0.998110i \(0.519572\pi\)
\(468\) 0 0
\(469\) −5743.84 −0.565514
\(470\) 13625.1 1.33718
\(471\) 0 0
\(472\) 22541.8 2.19824
\(473\) 1614.38 0.156933
\(474\) 0 0
\(475\) −885.276 −0.0855142
\(476\) 1746.20 0.168144
\(477\) 0 0
\(478\) −16647.2 −1.59294
\(479\) −18605.0 −1.77470 −0.887352 0.461092i \(-0.847458\pi\)
−0.887352 + 0.461092i \(0.847458\pi\)
\(480\) 0 0
\(481\) −36053.4 −3.41766
\(482\) 7926.84 0.749083
\(483\) 0 0
\(484\) −7617.51 −0.715393
\(485\) 4302.25 0.402794
\(486\) 0 0
\(487\) 874.431 0.0813640 0.0406820 0.999172i \(-0.487047\pi\)
0.0406820 + 0.999172i \(0.487047\pi\)
\(488\) 31206.9 2.89481
\(489\) 0 0
\(490\) 4301.23 0.396550
\(491\) −5849.95 −0.537688 −0.268844 0.963184i \(-0.586642\pi\)
−0.268844 + 0.963184i \(0.586642\pi\)
\(492\) 0 0
\(493\) −1812.35 −0.165566
\(494\) −2481.78 −0.226034
\(495\) 0 0
\(496\) −2517.58 −0.227908
\(497\) −62.7077 −0.00565961
\(498\) 0 0
\(499\) −1095.76 −0.0983029 −0.0491514 0.998791i \(-0.515652\pi\)
−0.0491514 + 0.998791i \(0.515652\pi\)
\(500\) −12579.8 −1.12517
\(501\) 0 0
\(502\) 919.907 0.0817878
\(503\) −4988.25 −0.442177 −0.221088 0.975254i \(-0.570961\pi\)
−0.221088 + 0.975254i \(0.570961\pi\)
\(504\) 0 0
\(505\) 10713.2 0.944025
\(506\) 26210.4 2.30275
\(507\) 0 0
\(508\) −43322.5 −3.78371
\(509\) 16831.7 1.46572 0.732859 0.680381i \(-0.238186\pi\)
0.732859 + 0.680381i \(0.238186\pi\)
\(510\) 0 0
\(511\) −153.537 −0.0132917
\(512\) 25046.5 2.16193
\(513\) 0 0
\(514\) 16872.9 1.44792
\(515\) −19958.9 −1.70775
\(516\) 0 0
\(517\) 4711.35 0.400783
\(518\) 14531.7 1.23260
\(519\) 0 0
\(520\) −83281.2 −7.02331
\(521\) 12826.2 1.07855 0.539277 0.842129i \(-0.318698\pi\)
0.539277 + 0.842129i \(0.318698\pi\)
\(522\) 0 0
\(523\) −3377.82 −0.282412 −0.141206 0.989980i \(-0.545098\pi\)
−0.141206 + 0.989980i \(0.545098\pi\)
\(524\) 4217.30 0.351591
\(525\) 0 0
\(526\) −24435.6 −2.02556
\(527\) 254.024 0.0209971
\(528\) 0 0
\(529\) 15872.3 1.30453
\(530\) 26483.9 2.17054
\(531\) 0 0
\(532\) 699.387 0.0569968
\(533\) −38900.1 −3.16126
\(534\) 0 0
\(535\) 17070.8 1.37951
\(536\) −44825.6 −3.61226
\(537\) 0 0
\(538\) 2157.93 0.172927
\(539\) 1487.30 0.118855
\(540\) 0 0
\(541\) 4866.90 0.386774 0.193387 0.981123i \(-0.438053\pi\)
0.193387 + 0.981123i \(0.438053\pi\)
\(542\) −27145.5 −2.15129
\(543\) 0 0
\(544\) 3336.18 0.262936
\(545\) −9239.55 −0.726199
\(546\) 0 0
\(547\) 6870.41 0.537034 0.268517 0.963275i \(-0.413467\pi\)
0.268517 + 0.963275i \(0.413467\pi\)
\(548\) 35342.9 2.75506
\(549\) 0 0
\(550\) −25787.4 −1.99923
\(551\) −725.883 −0.0561228
\(552\) 0 0
\(553\) −4336.00 −0.333427
\(554\) −14726.0 −1.12933
\(555\) 0 0
\(556\) 36424.2 2.77829
\(557\) 19795.9 1.50589 0.752943 0.658086i \(-0.228634\pi\)
0.752943 + 0.658086i \(0.228634\pi\)
\(558\) 0 0
\(559\) −4763.42 −0.360414
\(560\) 15843.5 1.19555
\(561\) 0 0
\(562\) 44296.1 3.32477
\(563\) −6981.58 −0.522626 −0.261313 0.965254i \(-0.584156\pi\)
−0.261313 + 0.965254i \(0.584156\pi\)
\(564\) 0 0
\(565\) 23639.1 1.76019
\(566\) −18983.4 −1.40977
\(567\) 0 0
\(568\) −489.378 −0.0361512
\(569\) 3412.25 0.251404 0.125702 0.992068i \(-0.459882\pi\)
0.125702 + 0.992068i \(0.459882\pi\)
\(570\) 0 0
\(571\) 20647.3 1.51325 0.756624 0.653850i \(-0.226847\pi\)
0.756624 + 0.653850i \(0.226847\pi\)
\(572\) −50544.9 −3.69473
\(573\) 0 0
\(574\) 15679.1 1.14012
\(575\) −27586.8 −2.00078
\(576\) 0 0
\(577\) 12938.7 0.933530 0.466765 0.884381i \(-0.345419\pi\)
0.466765 + 0.884381i \(0.345419\pi\)
\(578\) 24407.5 1.75643
\(579\) 0 0
\(580\) −42753.7 −3.06078
\(581\) −1816.72 −0.129725
\(582\) 0 0
\(583\) 9157.75 0.650558
\(584\) −1198.22 −0.0849019
\(585\) 0 0
\(586\) 10100.7 0.712038
\(587\) −14032.4 −0.986676 −0.493338 0.869838i \(-0.664223\pi\)
−0.493338 + 0.869838i \(0.664223\pi\)
\(588\) 0 0
\(589\) 101.742 0.00711750
\(590\) −36221.2 −2.52746
\(591\) 0 0
\(592\) 53527.2 3.71614
\(593\) −23344.9 −1.61663 −0.808313 0.588752i \(-0.799619\pi\)
−0.808313 + 0.588752i \(0.799619\pi\)
\(594\) 0 0
\(595\) −1598.61 −0.110146
\(596\) −28231.7 −1.94029
\(597\) 0 0
\(598\) −77336.7 −5.28852
\(599\) −21813.4 −1.48793 −0.743966 0.668217i \(-0.767058\pi\)
−0.743966 + 0.668217i \(0.767058\pi\)
\(600\) 0 0
\(601\) 1010.28 0.0685690 0.0342845 0.999412i \(-0.489085\pi\)
0.0342845 + 0.999412i \(0.489085\pi\)
\(602\) 1919.94 0.129985
\(603\) 0 0
\(604\) −33750.4 −2.27365
\(605\) 6973.71 0.468631
\(606\) 0 0
\(607\) −1763.19 −0.117901 −0.0589504 0.998261i \(-0.518775\pi\)
−0.0589504 + 0.998261i \(0.518775\pi\)
\(608\) 1336.21 0.0891288
\(609\) 0 0
\(610\) −50144.7 −3.32836
\(611\) −13901.4 −0.920440
\(612\) 0 0
\(613\) 264.802 0.0174474 0.00872368 0.999962i \(-0.497223\pi\)
0.00872368 + 0.999962i \(0.497223\pi\)
\(614\) 29251.4 1.92263
\(615\) 0 0
\(616\) 11607.1 0.759192
\(617\) 17930.8 1.16996 0.584980 0.811048i \(-0.301102\pi\)
0.584980 + 0.811048i \(0.301102\pi\)
\(618\) 0 0
\(619\) −19581.1 −1.27146 −0.635729 0.771912i \(-0.719300\pi\)
−0.635729 + 0.771912i \(0.719300\pi\)
\(620\) 5992.49 0.388168
\(621\) 0 0
\(622\) 6558.07 0.422757
\(623\) −3390.30 −0.218025
\(624\) 0 0
\(625\) −9076.78 −0.580914
\(626\) 25105.6 1.60291
\(627\) 0 0
\(628\) −34561.2 −2.19609
\(629\) −5400.91 −0.342366
\(630\) 0 0
\(631\) 13595.9 0.857754 0.428877 0.903363i \(-0.358909\pi\)
0.428877 + 0.903363i \(0.358909\pi\)
\(632\) −33838.6 −2.12979
\(633\) 0 0
\(634\) 16626.7 1.04153
\(635\) 39661.1 2.47859
\(636\) 0 0
\(637\) −4388.46 −0.272962
\(638\) −21144.4 −1.31209
\(639\) 0 0
\(640\) −14673.8 −0.906301
\(641\) 9626.58 0.593178 0.296589 0.955005i \(-0.404151\pi\)
0.296589 + 0.955005i \(0.404151\pi\)
\(642\) 0 0
\(643\) 26571.5 1.62967 0.814834 0.579695i \(-0.196828\pi\)
0.814834 + 0.579695i \(0.196828\pi\)
\(644\) 21794.1 1.33356
\(645\) 0 0
\(646\) −371.778 −0.0226431
\(647\) 29729.0 1.80644 0.903220 0.429179i \(-0.141197\pi\)
0.903220 + 0.429179i \(0.141197\pi\)
\(648\) 0 0
\(649\) −12524.8 −0.757535
\(650\) 76088.7 4.59145
\(651\) 0 0
\(652\) 16177.9 0.971742
\(653\) 16796.1 1.00656 0.503278 0.864125i \(-0.332127\pi\)
0.503278 + 0.864125i \(0.332127\pi\)
\(654\) 0 0
\(655\) −3860.88 −0.230316
\(656\) 57753.6 3.43734
\(657\) 0 0
\(658\) 5603.08 0.331962
\(659\) 2078.96 0.122891 0.0614453 0.998110i \(-0.480429\pi\)
0.0614453 + 0.998110i \(0.480429\pi\)
\(660\) 0 0
\(661\) −6600.40 −0.388390 −0.194195 0.980963i \(-0.562209\pi\)
−0.194195 + 0.980963i \(0.562209\pi\)
\(662\) −27379.2 −1.60744
\(663\) 0 0
\(664\) −14177.9 −0.828629
\(665\) −640.278 −0.0373367
\(666\) 0 0
\(667\) −22619.8 −1.31311
\(668\) −32522.2 −1.88372
\(669\) 0 0
\(670\) 72027.9 4.15325
\(671\) −17339.3 −0.997580
\(672\) 0 0
\(673\) −21644.8 −1.23974 −0.619871 0.784704i \(-0.712815\pi\)
−0.619871 + 0.784704i \(0.712815\pi\)
\(674\) 11883.2 0.679113
\(675\) 0 0
\(676\) 108289. 6.16118
\(677\) −349.464 −0.0198390 −0.00991949 0.999951i \(-0.503158\pi\)
−0.00991949 + 0.999951i \(0.503158\pi\)
\(678\) 0 0
\(679\) 1769.23 0.0999953
\(680\) −12475.8 −0.703565
\(681\) 0 0
\(682\) 2963.67 0.166400
\(683\) 14807.9 0.829589 0.414795 0.909915i \(-0.363853\pi\)
0.414795 + 0.909915i \(0.363853\pi\)
\(684\) 0 0
\(685\) −32355.9 −1.80475
\(686\) 1768.81 0.0984453
\(687\) 0 0
\(688\) 7072.08 0.391890
\(689\) −27021.0 −1.49408
\(690\) 0 0
\(691\) 21679.4 1.19352 0.596760 0.802420i \(-0.296454\pi\)
0.596760 + 0.802420i \(0.296454\pi\)
\(692\) −731.415 −0.0401795
\(693\) 0 0
\(694\) −26796.2 −1.46566
\(695\) −33345.8 −1.81997
\(696\) 0 0
\(697\) −5827.35 −0.316681
\(698\) 46569.7 2.52534
\(699\) 0 0
\(700\) −21442.4 −1.15778
\(701\) −5065.92 −0.272949 −0.136474 0.990644i \(-0.543577\pi\)
−0.136474 + 0.990644i \(0.543577\pi\)
\(702\) 0 0
\(703\) −2163.18 −0.116054
\(704\) 6634.99 0.355207
\(705\) 0 0
\(706\) −45821.5 −2.44265
\(707\) 4405.64 0.234358
\(708\) 0 0
\(709\) −16981.8 −0.899527 −0.449764 0.893148i \(-0.648492\pi\)
−0.449764 + 0.893148i \(0.648492\pi\)
\(710\) 786.356 0.0415654
\(711\) 0 0
\(712\) −26458.3 −1.39265
\(713\) 3170.46 0.166528
\(714\) 0 0
\(715\) 46273.1 2.42030
\(716\) 49283.3 2.57235
\(717\) 0 0
\(718\) −192.343 −0.00999746
\(719\) 31450.6 1.63130 0.815652 0.578543i \(-0.196378\pi\)
0.815652 + 0.578543i \(0.196378\pi\)
\(720\) 0 0
\(721\) −8207.76 −0.423957
\(722\) 35222.1 1.81556
\(723\) 0 0
\(724\) 50568.7 2.59581
\(725\) 22254.8 1.14003
\(726\) 0 0
\(727\) 14930.2 0.761665 0.380833 0.924644i \(-0.375637\pi\)
0.380833 + 0.924644i \(0.375637\pi\)
\(728\) −34248.0 −1.74357
\(729\) 0 0
\(730\) 1925.36 0.0976174
\(731\) −713.575 −0.0361047
\(732\) 0 0
\(733\) 26799.5 1.35043 0.675213 0.737623i \(-0.264052\pi\)
0.675213 + 0.737623i \(0.264052\pi\)
\(734\) −17202.3 −0.865050
\(735\) 0 0
\(736\) 41638.6 2.08535
\(737\) 24906.2 1.24482
\(738\) 0 0
\(739\) 22511.2 1.12055 0.560275 0.828307i \(-0.310695\pi\)
0.560275 + 0.828307i \(0.310695\pi\)
\(740\) −127409. −6.32924
\(741\) 0 0
\(742\) 10891.1 0.538846
\(743\) 23783.0 1.17431 0.587155 0.809474i \(-0.300248\pi\)
0.587155 + 0.809474i \(0.300248\pi\)
\(744\) 0 0
\(745\) 25845.7 1.27102
\(746\) −20296.5 −0.996121
\(747\) 0 0
\(748\) −7571.78 −0.370123
\(749\) 7020.11 0.342469
\(750\) 0 0
\(751\) −6729.39 −0.326976 −0.163488 0.986545i \(-0.552274\pi\)
−0.163488 + 0.986545i \(0.552274\pi\)
\(752\) 20638.8 1.00083
\(753\) 0 0
\(754\) 62389.0 3.01336
\(755\) 30898.0 1.48939
\(756\) 0 0
\(757\) 34646.1 1.66345 0.831726 0.555187i \(-0.187353\pi\)
0.831726 + 0.555187i \(0.187353\pi\)
\(758\) −11902.1 −0.570321
\(759\) 0 0
\(760\) −4996.80 −0.238491
\(761\) −24096.3 −1.14782 −0.573908 0.818919i \(-0.694573\pi\)
−0.573908 + 0.818919i \(0.694573\pi\)
\(762\) 0 0
\(763\) −3799.61 −0.180282
\(764\) −80676.6 −3.82039
\(765\) 0 0
\(766\) −15118.0 −0.713100
\(767\) 36955.8 1.73976
\(768\) 0 0
\(769\) −28099.8 −1.31769 −0.658846 0.752278i \(-0.728955\pi\)
−0.658846 + 0.752278i \(0.728955\pi\)
\(770\) −18650.8 −0.872894
\(771\) 0 0
\(772\) 18178.5 0.847487
\(773\) −9455.98 −0.439984 −0.219992 0.975502i \(-0.570603\pi\)
−0.219992 + 0.975502i \(0.570603\pi\)
\(774\) 0 0
\(775\) −3119.30 −0.144579
\(776\) 13807.3 0.638727
\(777\) 0 0
\(778\) −39244.5 −1.80846
\(779\) −2333.98 −0.107347
\(780\) 0 0
\(781\) 271.911 0.0124580
\(782\) −11585.3 −0.529781
\(783\) 0 0
\(784\) 6515.38 0.296801
\(785\) 31640.2 1.43858
\(786\) 0 0
\(787\) −16846.0 −0.763017 −0.381509 0.924365i \(-0.624595\pi\)
−0.381509 + 0.924365i \(0.624595\pi\)
\(788\) 28937.3 1.30818
\(789\) 0 0
\(790\) 54373.5 2.44876
\(791\) 9721.20 0.436974
\(792\) 0 0
\(793\) 51161.6 2.29105
\(794\) 22018.5 0.984140
\(795\) 0 0
\(796\) 32040.4 1.42669
\(797\) 35955.5 1.59801 0.799003 0.601327i \(-0.205361\pi\)
0.799003 + 0.601327i \(0.205361\pi\)
\(798\) 0 0
\(799\) −2082.47 −0.0922057
\(800\) −40966.6 −1.81049
\(801\) 0 0
\(802\) −57389.7 −2.52681
\(803\) 665.761 0.0292580
\(804\) 0 0
\(805\) −19952.2 −0.873569
\(806\) −8744.64 −0.382155
\(807\) 0 0
\(808\) 34382.1 1.49698
\(809\) −13680.6 −0.594543 −0.297272 0.954793i \(-0.596077\pi\)
−0.297272 + 0.954793i \(0.596077\pi\)
\(810\) 0 0
\(811\) 10215.1 0.442294 0.221147 0.975241i \(-0.429020\pi\)
0.221147 + 0.975241i \(0.429020\pi\)
\(812\) −17581.8 −0.759851
\(813\) 0 0
\(814\) −63011.7 −2.71322
\(815\) −14810.6 −0.636557
\(816\) 0 0
\(817\) −285.802 −0.0122386
\(818\) −37234.6 −1.59154
\(819\) 0 0
\(820\) −137469. −5.85440
\(821\) −14255.6 −0.605996 −0.302998 0.952991i \(-0.597988\pi\)
−0.302998 + 0.952991i \(0.597988\pi\)
\(822\) 0 0
\(823\) −31349.9 −1.32781 −0.663905 0.747817i \(-0.731102\pi\)
−0.663905 + 0.747817i \(0.731102\pi\)
\(824\) −64054.3 −2.70805
\(825\) 0 0
\(826\) −14895.4 −0.627453
\(827\) 6114.87 0.257116 0.128558 0.991702i \(-0.458965\pi\)
0.128558 + 0.991702i \(0.458965\pi\)
\(828\) 0 0
\(829\) 32418.4 1.35819 0.679093 0.734052i \(-0.262373\pi\)
0.679093 + 0.734052i \(0.262373\pi\)
\(830\) 22781.7 0.952730
\(831\) 0 0
\(832\) −19577.3 −0.815770
\(833\) −657.404 −0.0273442
\(834\) 0 0
\(835\) 29773.6 1.23396
\(836\) −3032.65 −0.125462
\(837\) 0 0
\(838\) −85765.1 −3.53545
\(839\) 44683.5 1.83867 0.919336 0.393474i \(-0.128727\pi\)
0.919336 + 0.393474i \(0.128727\pi\)
\(840\) 0 0
\(841\) −6141.15 −0.251800
\(842\) 3375.17 0.138142
\(843\) 0 0
\(844\) 24082.0 0.982151
\(845\) −99136.8 −4.03599
\(846\) 0 0
\(847\) 2867.82 0.116339
\(848\) 40117.1 1.62456
\(849\) 0 0
\(850\) 11398.3 0.459952
\(851\) −67408.4 −2.71531
\(852\) 0 0
\(853\) −9965.17 −0.400001 −0.200001 0.979796i \(-0.564094\pi\)
−0.200001 + 0.979796i \(0.564094\pi\)
\(854\) −20621.2 −0.826279
\(855\) 0 0
\(856\) 54785.7 2.18754
\(857\) −8639.89 −0.344379 −0.172190 0.985064i \(-0.555084\pi\)
−0.172190 + 0.985064i \(0.555084\pi\)
\(858\) 0 0
\(859\) 15551.5 0.617707 0.308853 0.951110i \(-0.400055\pi\)
0.308853 + 0.951110i \(0.400055\pi\)
\(860\) −16833.4 −0.667458
\(861\) 0 0
\(862\) −15728.0 −0.621460
\(863\) 47746.4 1.88332 0.941661 0.336563i \(-0.109265\pi\)
0.941661 + 0.336563i \(0.109265\pi\)
\(864\) 0 0
\(865\) 669.599 0.0263203
\(866\) 8247.97 0.323646
\(867\) 0 0
\(868\) 2464.31 0.0963643
\(869\) 18801.6 0.733946
\(870\) 0 0
\(871\) −73488.6 −2.85886
\(872\) −29652.6 −1.15156
\(873\) 0 0
\(874\) −4640.14 −0.179583
\(875\) 4736.01 0.182979
\(876\) 0 0
\(877\) −37958.8 −1.46155 −0.730773 0.682620i \(-0.760840\pi\)
−0.730773 + 0.682620i \(0.760840\pi\)
\(878\) 81962.8 3.15047
\(879\) 0 0
\(880\) −68699.9 −2.63168
\(881\) −32323.6 −1.23611 −0.618053 0.786136i \(-0.712078\pi\)
−0.618053 + 0.786136i \(0.712078\pi\)
\(882\) 0 0
\(883\) 27760.0 1.05798 0.528991 0.848628i \(-0.322571\pi\)
0.528991 + 0.848628i \(0.322571\pi\)
\(884\) 22341.4 0.850026
\(885\) 0 0
\(886\) −31767.5 −1.20457
\(887\) 32759.6 1.24009 0.620044 0.784567i \(-0.287115\pi\)
0.620044 + 0.784567i \(0.287115\pi\)
\(888\) 0 0
\(889\) 16310.0 0.615319
\(890\) 42514.5 1.60122
\(891\) 0 0
\(892\) −8991.97 −0.337526
\(893\) −834.071 −0.0312554
\(894\) 0 0
\(895\) −45118.1 −1.68506
\(896\) −6034.36 −0.224993
\(897\) 0 0
\(898\) 45083.8 1.67535
\(899\) −2557.67 −0.0948868
\(900\) 0 0
\(901\) −4047.83 −0.149670
\(902\) −67986.9 −2.50966
\(903\) 0 0
\(904\) 75865.3 2.79120
\(905\) −46294.8 −1.70043
\(906\) 0 0
\(907\) −42682.6 −1.56257 −0.781286 0.624174i \(-0.785436\pi\)
−0.781286 + 0.624174i \(0.785436\pi\)
\(908\) 74742.8 2.73175
\(909\) 0 0
\(910\) 55031.3 2.00469
\(911\) 50130.0 1.82314 0.911570 0.411144i \(-0.134871\pi\)
0.911570 + 0.411144i \(0.134871\pi\)
\(912\) 0 0
\(913\) 7877.60 0.285554
\(914\) 2991.57 0.108263
\(915\) 0 0
\(916\) 94275.3 3.40059
\(917\) −1587.72 −0.0571769
\(918\) 0 0
\(919\) −33784.7 −1.21268 −0.606341 0.795205i \(-0.707363\pi\)
−0.606341 + 0.795205i \(0.707363\pi\)
\(920\) −155709. −5.57998
\(921\) 0 0
\(922\) 10092.3 0.360489
\(923\) −802.303 −0.0286112
\(924\) 0 0
\(925\) 66320.6 2.35742
\(926\) −41831.5 −1.48452
\(927\) 0 0
\(928\) −33590.7 −1.18822
\(929\) 50854.8 1.79601 0.898004 0.439987i \(-0.145017\pi\)
0.898004 + 0.439987i \(0.145017\pi\)
\(930\) 0 0
\(931\) −263.304 −0.00926899
\(932\) 82150.4 2.88726
\(933\) 0 0
\(934\) 6396.01 0.224073
\(935\) 6931.85 0.242455
\(936\) 0 0
\(937\) −3684.00 −0.128443 −0.0642215 0.997936i \(-0.520456\pi\)
−0.0642215 + 0.997936i \(0.520456\pi\)
\(938\) 29620.3 1.03106
\(939\) 0 0
\(940\) −49125.8 −1.70458
\(941\) 28660.5 0.992886 0.496443 0.868069i \(-0.334639\pi\)
0.496443 + 0.868069i \(0.334639\pi\)
\(942\) 0 0
\(943\) −72730.8 −2.51160
\(944\) −54866.9 −1.89170
\(945\) 0 0
\(946\) −8325.18 −0.286126
\(947\) −31873.0 −1.09370 −0.546849 0.837231i \(-0.684173\pi\)
−0.546849 + 0.837231i \(0.684173\pi\)
\(948\) 0 0
\(949\) −1964.40 −0.0671942
\(950\) 4565.26 0.155912
\(951\) 0 0
\(952\) −5130.46 −0.174663
\(953\) 3310.08 0.112512 0.0562561 0.998416i \(-0.482084\pi\)
0.0562561 + 0.998416i \(0.482084\pi\)
\(954\) 0 0
\(955\) 73858.2 2.50261
\(956\) 60022.3 2.03061
\(957\) 0 0
\(958\) 95943.6 3.23570
\(959\) −13305.8 −0.448037
\(960\) 0 0
\(961\) −29432.5 −0.987966
\(962\) 185923. 6.23119
\(963\) 0 0
\(964\) −28580.6 −0.954897
\(965\) −16642.2 −0.555161
\(966\) 0 0
\(967\) −168.413 −0.00560061 −0.00280031 0.999996i \(-0.500891\pi\)
−0.00280031 + 0.999996i \(0.500891\pi\)
\(968\) 22380.8 0.743127
\(969\) 0 0
\(970\) −22186.2 −0.734387
\(971\) −37669.9 −1.24499 −0.622495 0.782624i \(-0.713881\pi\)
−0.622495 + 0.782624i \(0.713881\pi\)
\(972\) 0 0
\(973\) −13712.9 −0.451815
\(974\) −4509.33 −0.148345
\(975\) 0 0
\(976\) −75957.7 −2.49114
\(977\) −26688.7 −0.873948 −0.436974 0.899474i \(-0.643950\pi\)
−0.436974 + 0.899474i \(0.643950\pi\)
\(978\) 0 0
\(979\) 14700.9 0.479921
\(980\) −15508.3 −0.505504
\(981\) 0 0
\(982\) 30167.5 0.980329
\(983\) 1102.29 0.0357655 0.0178828 0.999840i \(-0.494307\pi\)
0.0178828 + 0.999840i \(0.494307\pi\)
\(984\) 0 0
\(985\) −26491.6 −0.856948
\(986\) 9346.07 0.301866
\(987\) 0 0
\(988\) 8948.19 0.288137
\(989\) −8906.09 −0.286347
\(990\) 0 0
\(991\) 16523.6 0.529656 0.264828 0.964296i \(-0.414685\pi\)
0.264828 + 0.964296i \(0.414685\pi\)
\(992\) 4708.17 0.150690
\(993\) 0 0
\(994\) 323.376 0.0103188
\(995\) −29332.5 −0.934575
\(996\) 0 0
\(997\) 17994.5 0.571605 0.285802 0.958289i \(-0.407740\pi\)
0.285802 + 0.958289i \(0.407740\pi\)
\(998\) 5650.72 0.179229
\(999\) 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 189.4.a.l.1.1 4
3.2 odd 2 inner 189.4.a.l.1.4 yes 4
7.6 odd 2 1323.4.a.ba.1.1 4
21.20 even 2 1323.4.a.ba.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.l.1.1 4 1.1 even 1 trivial
189.4.a.l.1.4 yes 4 3.2 odd 2 inner
1323.4.a.ba.1.1 4 7.6 odd 2
1323.4.a.ba.1.4 4 21.20 even 2