Properties

Label 387.4.a.h.1.6
Level $387$
Weight $4$
Character 387.1
Self dual yes
Analytic conductor $22.834$
Analytic rank $1$
Dimension $6$
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] = [387,4,Mod(1,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, 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("387.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 387.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: \(22.8337391722\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(4.31455\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.31455 q^{2} +2.98627 q^{4} -8.51910 q^{5} +9.84502 q^{7} -16.6183 q^{8} +O(q^{10})\) \(q+3.31455 q^{2} +2.98627 q^{4} -8.51910 q^{5} +9.84502 q^{7} -16.6183 q^{8} -28.2370 q^{10} -0.597776 q^{11} +34.9021 q^{13} +32.6318 q^{14} -78.9723 q^{16} -91.3552 q^{17} -24.0324 q^{19} -25.4403 q^{20} -1.98136 q^{22} -127.719 q^{23} -52.4250 q^{25} +115.685 q^{26} +29.3999 q^{28} -285.724 q^{29} -192.699 q^{31} -128.812 q^{32} -302.802 q^{34} -83.8707 q^{35} +363.399 q^{37} -79.6565 q^{38} +141.573 q^{40} -191.586 q^{41} -43.0000 q^{43} -1.78512 q^{44} -423.330 q^{46} +458.267 q^{47} -246.076 q^{49} -173.765 q^{50} +104.227 q^{52} +583.068 q^{53} +5.09252 q^{55} -163.607 q^{56} -947.047 q^{58} -416.589 q^{59} -30.9376 q^{61} -638.710 q^{62} +204.825 q^{64} -297.334 q^{65} +16.1466 q^{67} -272.811 q^{68} -277.994 q^{70} +219.461 q^{71} +1031.07 q^{73} +1204.50 q^{74} -71.7670 q^{76} -5.88512 q^{77} +445.438 q^{79} +672.773 q^{80} -635.022 q^{82} -298.511 q^{83} +778.264 q^{85} -142.526 q^{86} +9.93402 q^{88} -846.185 q^{89} +343.612 q^{91} -381.402 q^{92} +1518.95 q^{94} +204.734 q^{95} -259.068 q^{97} -815.631 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.31455 1.17187 0.585936 0.810357i \(-0.300727\pi\)
0.585936 + 0.810357i \(0.300727\pi\)
\(3\) 0 0
\(4\) 2.98627 0.373283
\(5\) −8.51910 −0.761971 −0.380986 0.924581i \(-0.624415\pi\)
−0.380986 + 0.924581i \(0.624415\pi\)
\(6\) 0 0
\(7\) 9.84502 0.531581 0.265791 0.964031i \(-0.414367\pi\)
0.265791 + 0.964031i \(0.414367\pi\)
\(8\) −16.6183 −0.734432
\(9\) 0 0
\(10\) −28.2370 −0.892933
\(11\) −0.597776 −0.0163851 −0.00819256 0.999966i \(-0.502608\pi\)
−0.00819256 + 0.999966i \(0.502608\pi\)
\(12\) 0 0
\(13\) 34.9021 0.744623 0.372311 0.928108i \(-0.378565\pi\)
0.372311 + 0.928108i \(0.378565\pi\)
\(14\) 32.6318 0.622945
\(15\) 0 0
\(16\) −78.9723 −1.23394
\(17\) −91.3552 −1.30335 −0.651673 0.758500i \(-0.725933\pi\)
−0.651673 + 0.758500i \(0.725933\pi\)
\(18\) 0 0
\(19\) −24.0324 −0.290179 −0.145089 0.989419i \(-0.546347\pi\)
−0.145089 + 0.989419i \(0.546347\pi\)
\(20\) −25.4403 −0.284431
\(21\) 0 0
\(22\) −1.98136 −0.0192013
\(23\) −127.719 −1.15788 −0.578938 0.815371i \(-0.696533\pi\)
−0.578938 + 0.815371i \(0.696533\pi\)
\(24\) 0 0
\(25\) −52.4250 −0.419400
\(26\) 115.685 0.872602
\(27\) 0 0
\(28\) 29.3999 0.198430
\(29\) −285.724 −1.82957 −0.914786 0.403939i \(-0.867641\pi\)
−0.914786 + 0.403939i \(0.867641\pi\)
\(30\) 0 0
\(31\) −192.699 −1.11644 −0.558221 0.829692i \(-0.688516\pi\)
−0.558221 + 0.829692i \(0.688516\pi\)
\(32\) −128.812 −0.711591
\(33\) 0 0
\(34\) −302.802 −1.52735
\(35\) −83.8707 −0.405050
\(36\) 0 0
\(37\) 363.399 1.61466 0.807329 0.590101i \(-0.200912\pi\)
0.807329 + 0.590101i \(0.200912\pi\)
\(38\) −79.6565 −0.340053
\(39\) 0 0
\(40\) 141.573 0.559616
\(41\) −191.586 −0.729774 −0.364887 0.931052i \(-0.618892\pi\)
−0.364887 + 0.931052i \(0.618892\pi\)
\(42\) 0 0
\(43\) −43.0000 −0.152499
\(44\) −1.78512 −0.00611629
\(45\) 0 0
\(46\) −423.330 −1.35688
\(47\) 458.267 1.42223 0.711117 0.703073i \(-0.248189\pi\)
0.711117 + 0.703073i \(0.248189\pi\)
\(48\) 0 0
\(49\) −246.076 −0.717422
\(50\) −173.765 −0.491483
\(51\) 0 0
\(52\) 104.227 0.277955
\(53\) 583.068 1.51114 0.755572 0.655066i \(-0.227359\pi\)
0.755572 + 0.655066i \(0.227359\pi\)
\(54\) 0 0
\(55\) 5.09252 0.0124850
\(56\) −163.607 −0.390410
\(57\) 0 0
\(58\) −947.047 −2.14402
\(59\) −416.589 −0.919243 −0.459621 0.888115i \(-0.652015\pi\)
−0.459621 + 0.888115i \(0.652015\pi\)
\(60\) 0 0
\(61\) −30.9376 −0.0649369 −0.0324685 0.999473i \(-0.510337\pi\)
−0.0324685 + 0.999473i \(0.510337\pi\)
\(62\) −638.710 −1.30833
\(63\) 0 0
\(64\) 204.825 0.400049
\(65\) −297.334 −0.567381
\(66\) 0 0
\(67\) 16.1466 0.0294421 0.0147210 0.999892i \(-0.495314\pi\)
0.0147210 + 0.999892i \(0.495314\pi\)
\(68\) −272.811 −0.486517
\(69\) 0 0
\(70\) −277.994 −0.474666
\(71\) 219.461 0.366835 0.183418 0.983035i \(-0.441284\pi\)
0.183418 + 0.983035i \(0.441284\pi\)
\(72\) 0 0
\(73\) 1031.07 1.65311 0.826557 0.562853i \(-0.190296\pi\)
0.826557 + 0.562853i \(0.190296\pi\)
\(74\) 1204.50 1.89217
\(75\) 0 0
\(76\) −71.7670 −0.108319
\(77\) −5.88512 −0.00871002
\(78\) 0 0
\(79\) 445.438 0.634375 0.317188 0.948363i \(-0.397262\pi\)
0.317188 + 0.948363i \(0.397262\pi\)
\(80\) 672.773 0.940229
\(81\) 0 0
\(82\) −635.022 −0.855201
\(83\) −298.511 −0.394769 −0.197384 0.980326i \(-0.563245\pi\)
−0.197384 + 0.980326i \(0.563245\pi\)
\(84\) 0 0
\(85\) 778.264 0.993112
\(86\) −142.526 −0.178709
\(87\) 0 0
\(88\) 9.93402 0.0120338
\(89\) −846.185 −1.00781 −0.503907 0.863758i \(-0.668105\pi\)
−0.503907 + 0.863758i \(0.668105\pi\)
\(90\) 0 0
\(91\) 343.612 0.395827
\(92\) −381.402 −0.432216
\(93\) 0 0
\(94\) 1518.95 1.66668
\(95\) 204.734 0.221108
\(96\) 0 0
\(97\) −259.068 −0.271179 −0.135590 0.990765i \(-0.543293\pi\)
−0.135590 + 0.990765i \(0.543293\pi\)
\(98\) −815.631 −0.840726
\(99\) 0 0
\(100\) −156.555 −0.156555
\(101\) 1182.01 1.16450 0.582252 0.813009i \(-0.302172\pi\)
0.582252 + 0.813009i \(0.302172\pi\)
\(102\) 0 0
\(103\) 319.565 0.305706 0.152853 0.988249i \(-0.451154\pi\)
0.152853 + 0.988249i \(0.451154\pi\)
\(104\) −580.013 −0.546874
\(105\) 0 0
\(106\) 1932.61 1.77087
\(107\) −1647.14 −1.48818 −0.744089 0.668081i \(-0.767116\pi\)
−0.744089 + 0.668081i \(0.767116\pi\)
\(108\) 0 0
\(109\) −560.538 −0.492567 −0.246283 0.969198i \(-0.579209\pi\)
−0.246283 + 0.969198i \(0.579209\pi\)
\(110\) 16.8794 0.0146308
\(111\) 0 0
\(112\) −777.484 −0.655941
\(113\) 65.6852 0.0546827 0.0273414 0.999626i \(-0.491296\pi\)
0.0273414 + 0.999626i \(0.491296\pi\)
\(114\) 0 0
\(115\) 1088.05 0.882269
\(116\) −853.248 −0.682949
\(117\) 0 0
\(118\) −1380.81 −1.07723
\(119\) −899.393 −0.692834
\(120\) 0 0
\(121\) −1330.64 −0.999732
\(122\) −102.544 −0.0760977
\(123\) 0 0
\(124\) −575.450 −0.416749
\(125\) 1511.50 1.08154
\(126\) 0 0
\(127\) −1906.92 −1.33238 −0.666189 0.745783i \(-0.732076\pi\)
−0.666189 + 0.745783i \(0.732076\pi\)
\(128\) 1709.40 1.18040
\(129\) 0 0
\(130\) −985.530 −0.664898
\(131\) 1536.09 1.02449 0.512246 0.858839i \(-0.328814\pi\)
0.512246 + 0.858839i \(0.328814\pi\)
\(132\) 0 0
\(133\) −236.599 −0.154254
\(134\) 53.5187 0.0345023
\(135\) 0 0
\(136\) 1518.17 0.957218
\(137\) 1695.38 1.05727 0.528634 0.848850i \(-0.322704\pi\)
0.528634 + 0.848850i \(0.322704\pi\)
\(138\) 0 0
\(139\) 947.580 0.578221 0.289110 0.957296i \(-0.406641\pi\)
0.289110 + 0.957296i \(0.406641\pi\)
\(140\) −250.460 −0.151198
\(141\) 0 0
\(142\) 727.417 0.429884
\(143\) −20.8636 −0.0122007
\(144\) 0 0
\(145\) 2434.11 1.39408
\(146\) 3417.53 1.93724
\(147\) 0 0
\(148\) 1085.21 0.602725
\(149\) −3084.10 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(150\) 0 0
\(151\) −2346.83 −1.26478 −0.632391 0.774649i \(-0.717926\pi\)
−0.632391 + 0.774649i \(0.717926\pi\)
\(152\) 399.377 0.213117
\(153\) 0 0
\(154\) −19.5066 −0.0102070
\(155\) 1641.62 0.850697
\(156\) 0 0
\(157\) 1582.73 0.804556 0.402278 0.915517i \(-0.368218\pi\)
0.402278 + 0.915517i \(0.368218\pi\)
\(158\) 1476.43 0.743406
\(159\) 0 0
\(160\) 1097.36 0.542212
\(161\) −1257.39 −0.615505
\(162\) 0 0
\(163\) −820.887 −0.394459 −0.197229 0.980357i \(-0.563194\pi\)
−0.197229 + 0.980357i \(0.563194\pi\)
\(164\) −572.127 −0.272412
\(165\) 0 0
\(166\) −989.430 −0.462618
\(167\) −1690.58 −0.783358 −0.391679 0.920102i \(-0.628106\pi\)
−0.391679 + 0.920102i \(0.628106\pi\)
\(168\) 0 0
\(169\) −978.845 −0.445537
\(170\) 2579.60 1.16380
\(171\) 0 0
\(172\) −128.409 −0.0569252
\(173\) −1390.24 −0.610973 −0.305486 0.952196i \(-0.598819\pi\)
−0.305486 + 0.952196i \(0.598819\pi\)
\(174\) 0 0
\(175\) −516.125 −0.222945
\(176\) 47.2078 0.0202183
\(177\) 0 0
\(178\) −2804.73 −1.18103
\(179\) −693.895 −0.289744 −0.144872 0.989450i \(-0.546277\pi\)
−0.144872 + 0.989450i \(0.546277\pi\)
\(180\) 0 0
\(181\) −2355.50 −0.967309 −0.483655 0.875259i \(-0.660691\pi\)
−0.483655 + 0.875259i \(0.660691\pi\)
\(182\) 1138.92 0.463859
\(183\) 0 0
\(184\) 2122.46 0.850381
\(185\) −3095.83 −1.23032
\(186\) 0 0
\(187\) 54.6100 0.0213555
\(188\) 1368.51 0.530897
\(189\) 0 0
\(190\) 678.602 0.259110
\(191\) 2168.09 0.821349 0.410674 0.911782i \(-0.365293\pi\)
0.410674 + 0.911782i \(0.365293\pi\)
\(192\) 0 0
\(193\) 1305.00 0.486717 0.243358 0.969936i \(-0.421751\pi\)
0.243358 + 0.969936i \(0.421751\pi\)
\(194\) −858.695 −0.317787
\(195\) 0 0
\(196\) −734.847 −0.267802
\(197\) 4800.27 1.73607 0.868033 0.496507i \(-0.165384\pi\)
0.868033 + 0.496507i \(0.165384\pi\)
\(198\) 0 0
\(199\) −2113.34 −0.752816 −0.376408 0.926454i \(-0.622841\pi\)
−0.376408 + 0.926454i \(0.622841\pi\)
\(200\) 871.213 0.308020
\(201\) 0 0
\(202\) 3917.85 1.36465
\(203\) −2812.96 −0.972566
\(204\) 0 0
\(205\) 1632.14 0.556067
\(206\) 1059.22 0.358248
\(207\) 0 0
\(208\) −2756.30 −0.918822
\(209\) 14.3660 0.00475462
\(210\) 0 0
\(211\) 5564.31 1.81546 0.907732 0.419551i \(-0.137812\pi\)
0.907732 + 0.419551i \(0.137812\pi\)
\(212\) 1741.20 0.564085
\(213\) 0 0
\(214\) −5459.53 −1.74395
\(215\) 366.321 0.116200
\(216\) 0 0
\(217\) −1897.12 −0.593480
\(218\) −1857.93 −0.577225
\(219\) 0 0
\(220\) 15.2076 0.00466044
\(221\) −3188.49 −0.970501
\(222\) 0 0
\(223\) −1709.75 −0.513422 −0.256711 0.966488i \(-0.582639\pi\)
−0.256711 + 0.966488i \(0.582639\pi\)
\(224\) −1268.15 −0.378268
\(225\) 0 0
\(226\) 217.717 0.0640811
\(227\) −347.506 −0.101607 −0.0508035 0.998709i \(-0.516178\pi\)
−0.0508035 + 0.998709i \(0.516178\pi\)
\(228\) 0 0
\(229\) 713.508 0.205895 0.102947 0.994687i \(-0.467173\pi\)
0.102947 + 0.994687i \(0.467173\pi\)
\(230\) 3606.39 1.03391
\(231\) 0 0
\(232\) 4748.24 1.34370
\(233\) 6539.93 1.83882 0.919410 0.393300i \(-0.128667\pi\)
0.919410 + 0.393300i \(0.128667\pi\)
\(234\) 0 0
\(235\) −3904.02 −1.08370
\(236\) −1244.05 −0.343138
\(237\) 0 0
\(238\) −2981.09 −0.811913
\(239\) 2561.41 0.693238 0.346619 0.938006i \(-0.387330\pi\)
0.346619 + 0.938006i \(0.387330\pi\)
\(240\) 0 0
\(241\) −4656.80 −1.24469 −0.622346 0.782742i \(-0.713820\pi\)
−0.622346 + 0.782742i \(0.713820\pi\)
\(242\) −4410.49 −1.17156
\(243\) 0 0
\(244\) −92.3879 −0.0242399
\(245\) 2096.34 0.546655
\(246\) 0 0
\(247\) −838.779 −0.216074
\(248\) 3202.32 0.819950
\(249\) 0 0
\(250\) 5009.95 1.26743
\(251\) −6104.04 −1.53499 −0.767497 0.641053i \(-0.778498\pi\)
−0.767497 + 0.641053i \(0.778498\pi\)
\(252\) 0 0
\(253\) 76.3471 0.0189720
\(254\) −6320.60 −1.56138
\(255\) 0 0
\(256\) 4027.29 0.983225
\(257\) −682.439 −0.165640 −0.0828199 0.996565i \(-0.526393\pi\)
−0.0828199 + 0.996565i \(0.526393\pi\)
\(258\) 0 0
\(259\) 3577.67 0.858322
\(260\) −887.919 −0.211794
\(261\) 0 0
\(262\) 5091.44 1.20057
\(263\) −1671.87 −0.391984 −0.195992 0.980605i \(-0.562793\pi\)
−0.195992 + 0.980605i \(0.562793\pi\)
\(264\) 0 0
\(265\) −4967.22 −1.15145
\(266\) −784.220 −0.180766
\(267\) 0 0
\(268\) 48.2180 0.0109902
\(269\) 261.999 0.0593843 0.0296921 0.999559i \(-0.490547\pi\)
0.0296921 + 0.999559i \(0.490547\pi\)
\(270\) 0 0
\(271\) 4620.04 1.03560 0.517800 0.855502i \(-0.326751\pi\)
0.517800 + 0.855502i \(0.326751\pi\)
\(272\) 7214.53 1.60825
\(273\) 0 0
\(274\) 5619.41 1.23898
\(275\) 31.3384 0.00687192
\(276\) 0 0
\(277\) 5163.93 1.12011 0.560055 0.828456i \(-0.310780\pi\)
0.560055 + 0.828456i \(0.310780\pi\)
\(278\) 3140.81 0.677601
\(279\) 0 0
\(280\) 1393.79 0.297481
\(281\) 5275.67 1.12000 0.560000 0.828493i \(-0.310801\pi\)
0.560000 + 0.828493i \(0.310801\pi\)
\(282\) 0 0
\(283\) −5028.34 −1.05620 −0.528098 0.849183i \(-0.677095\pi\)
−0.528098 + 0.849183i \(0.677095\pi\)
\(284\) 655.371 0.136933
\(285\) 0 0
\(286\) −69.1537 −0.0142977
\(287\) −1886.17 −0.387934
\(288\) 0 0
\(289\) 3432.77 0.698711
\(290\) 8067.99 1.63368
\(291\) 0 0
\(292\) 3079.04 0.617080
\(293\) −8840.69 −1.76273 −0.881363 0.472439i \(-0.843374\pi\)
−0.881363 + 0.472439i \(0.843374\pi\)
\(294\) 0 0
\(295\) 3548.96 0.700436
\(296\) −6039.06 −1.18586
\(297\) 0 0
\(298\) −10222.4 −1.98714
\(299\) −4457.64 −0.862181
\(300\) 0 0
\(301\) −423.336 −0.0810654
\(302\) −7778.68 −1.48216
\(303\) 0 0
\(304\) 1897.89 0.358064
\(305\) 263.560 0.0494801
\(306\) 0 0
\(307\) 6565.15 1.22050 0.610249 0.792209i \(-0.291069\pi\)
0.610249 + 0.792209i \(0.291069\pi\)
\(308\) −17.5745 −0.00325131
\(309\) 0 0
\(310\) 5441.23 0.996908
\(311\) −1250.31 −0.227969 −0.113985 0.993483i \(-0.536361\pi\)
−0.113985 + 0.993483i \(0.536361\pi\)
\(312\) 0 0
\(313\) −2968.85 −0.536132 −0.268066 0.963401i \(-0.586385\pi\)
−0.268066 + 0.963401i \(0.586385\pi\)
\(314\) 5246.03 0.942837
\(315\) 0 0
\(316\) 1330.20 0.236802
\(317\) −1054.32 −0.186803 −0.0934013 0.995629i \(-0.529774\pi\)
−0.0934013 + 0.995629i \(0.529774\pi\)
\(318\) 0 0
\(319\) 170.799 0.0299778
\(320\) −1744.93 −0.304826
\(321\) 0 0
\(322\) −4167.69 −0.721293
\(323\) 2195.48 0.378204
\(324\) 0 0
\(325\) −1829.74 −0.312295
\(326\) −2720.87 −0.462255
\(327\) 0 0
\(328\) 3183.83 0.535969
\(329\) 4511.64 0.756033
\(330\) 0 0
\(331\) −2881.73 −0.478532 −0.239266 0.970954i \(-0.576907\pi\)
−0.239266 + 0.970954i \(0.576907\pi\)
\(332\) −891.432 −0.147361
\(333\) 0 0
\(334\) −5603.51 −0.917995
\(335\) −137.554 −0.0224340
\(336\) 0 0
\(337\) −6147.77 −0.993740 −0.496870 0.867825i \(-0.665517\pi\)
−0.496870 + 0.867825i \(0.665517\pi\)
\(338\) −3244.43 −0.522112
\(339\) 0 0
\(340\) 2324.10 0.370712
\(341\) 115.191 0.0182930
\(342\) 0 0
\(343\) −5799.46 −0.912949
\(344\) 714.586 0.112000
\(345\) 0 0
\(346\) −4608.04 −0.715982
\(347\) −1404.42 −0.217271 −0.108636 0.994082i \(-0.534648\pi\)
−0.108636 + 0.994082i \(0.534648\pi\)
\(348\) 0 0
\(349\) −4472.14 −0.685926 −0.342963 0.939349i \(-0.611430\pi\)
−0.342963 + 0.939349i \(0.611430\pi\)
\(350\) −1710.72 −0.261263
\(351\) 0 0
\(352\) 77.0007 0.0116595
\(353\) −2146.48 −0.323642 −0.161821 0.986820i \(-0.551737\pi\)
−0.161821 + 0.986820i \(0.551737\pi\)
\(354\) 0 0
\(355\) −1869.61 −0.279518
\(356\) −2526.93 −0.376200
\(357\) 0 0
\(358\) −2299.95 −0.339543
\(359\) −1960.49 −0.288219 −0.144109 0.989562i \(-0.546032\pi\)
−0.144109 + 0.989562i \(0.546032\pi\)
\(360\) 0 0
\(361\) −6281.45 −0.915796
\(362\) −7807.43 −1.13356
\(363\) 0 0
\(364\) 1026.12 0.147756
\(365\) −8783.77 −1.25963
\(366\) 0 0
\(367\) 13485.2 1.91804 0.959021 0.283337i \(-0.0914414\pi\)
0.959021 + 0.283337i \(0.0914414\pi\)
\(368\) 10086.2 1.42875
\(369\) 0 0
\(370\) −10261.3 −1.44178
\(371\) 5740.32 0.803295
\(372\) 0 0
\(373\) −12034.5 −1.67057 −0.835285 0.549817i \(-0.814697\pi\)
−0.835285 + 0.549817i \(0.814697\pi\)
\(374\) 181.008 0.0250259
\(375\) 0 0
\(376\) −7615.60 −1.04453
\(377\) −9972.36 −1.36234
\(378\) 0 0
\(379\) −10185.1 −1.38041 −0.690204 0.723615i \(-0.742479\pi\)
−0.690204 + 0.723615i \(0.742479\pi\)
\(380\) 611.390 0.0825359
\(381\) 0 0
\(382\) 7186.26 0.962515
\(383\) −13508.3 −1.80219 −0.901097 0.433618i \(-0.857237\pi\)
−0.901097 + 0.433618i \(0.857237\pi\)
\(384\) 0 0
\(385\) 50.1359 0.00663679
\(386\) 4325.51 0.570369
\(387\) 0 0
\(388\) −773.646 −0.101227
\(389\) 6651.52 0.866954 0.433477 0.901165i \(-0.357286\pi\)
0.433477 + 0.901165i \(0.357286\pi\)
\(390\) 0 0
\(391\) 11667.7 1.50911
\(392\) 4089.35 0.526897
\(393\) 0 0
\(394\) 15910.7 2.03445
\(395\) −3794.73 −0.483376
\(396\) 0 0
\(397\) −8308.29 −1.05033 −0.525165 0.851000i \(-0.675996\pi\)
−0.525165 + 0.851000i \(0.675996\pi\)
\(398\) −7004.77 −0.882204
\(399\) 0 0
\(400\) 4140.12 0.517515
\(401\) −2963.57 −0.369061 −0.184530 0.982827i \(-0.559076\pi\)
−0.184530 + 0.982827i \(0.559076\pi\)
\(402\) 0 0
\(403\) −6725.59 −0.831328
\(404\) 3529.81 0.434690
\(405\) 0 0
\(406\) −9323.70 −1.13972
\(407\) −217.231 −0.0264564
\(408\) 0 0
\(409\) −9951.28 −1.20308 −0.601540 0.798843i \(-0.705446\pi\)
−0.601540 + 0.798843i \(0.705446\pi\)
\(410\) 5409.82 0.651639
\(411\) 0 0
\(412\) 954.308 0.114115
\(413\) −4101.33 −0.488652
\(414\) 0 0
\(415\) 2543.04 0.300802
\(416\) −4495.80 −0.529867
\(417\) 0 0
\(418\) 47.6168 0.00557180
\(419\) −5281.70 −0.615818 −0.307909 0.951416i \(-0.599629\pi\)
−0.307909 + 0.951416i \(0.599629\pi\)
\(420\) 0 0
\(421\) −1668.54 −0.193159 −0.0965795 0.995325i \(-0.530790\pi\)
−0.0965795 + 0.995325i \(0.530790\pi\)
\(422\) 18443.2 2.12749
\(423\) 0 0
\(424\) −9689.60 −1.10983
\(425\) 4789.29 0.546623
\(426\) 0 0
\(427\) −304.581 −0.0345192
\(428\) −4918.80 −0.555512
\(429\) 0 0
\(430\) 1214.19 0.136171
\(431\) 10348.7 1.15657 0.578284 0.815835i \(-0.303722\pi\)
0.578284 + 0.815835i \(0.303722\pi\)
\(432\) 0 0
\(433\) 2968.27 0.329436 0.164718 0.986341i \(-0.447329\pi\)
0.164718 + 0.986341i \(0.447329\pi\)
\(434\) −6288.12 −0.695482
\(435\) 0 0
\(436\) −1673.91 −0.183867
\(437\) 3069.38 0.335991
\(438\) 0 0
\(439\) −9224.53 −1.00288 −0.501438 0.865194i \(-0.667196\pi\)
−0.501438 + 0.865194i \(0.667196\pi\)
\(440\) −84.6289 −0.00916937
\(441\) 0 0
\(442\) −10568.4 −1.13730
\(443\) 10115.5 1.08488 0.542438 0.840096i \(-0.317501\pi\)
0.542438 + 0.840096i \(0.317501\pi\)
\(444\) 0 0
\(445\) 7208.73 0.767925
\(446\) −5667.05 −0.601665
\(447\) 0 0
\(448\) 2016.51 0.212659
\(449\) −14386.0 −1.51207 −0.756034 0.654532i \(-0.772866\pi\)
−0.756034 + 0.654532i \(0.772866\pi\)
\(450\) 0 0
\(451\) 114.526 0.0119574
\(452\) 196.154 0.0204121
\(453\) 0 0
\(454\) −1151.83 −0.119070
\(455\) −2927.26 −0.301609
\(456\) 0 0
\(457\) −10031.8 −1.02684 −0.513421 0.858137i \(-0.671622\pi\)
−0.513421 + 0.858137i \(0.671622\pi\)
\(458\) 2364.96 0.241282
\(459\) 0 0
\(460\) 3249.20 0.329336
\(461\) 124.788 0.0126072 0.00630362 0.999980i \(-0.497993\pi\)
0.00630362 + 0.999980i \(0.497993\pi\)
\(462\) 0 0
\(463\) 15438.9 1.54969 0.774847 0.632148i \(-0.217827\pi\)
0.774847 + 0.632148i \(0.217827\pi\)
\(464\) 22564.3 2.25759
\(465\) 0 0
\(466\) 21677.0 2.15486
\(467\) 12014.7 1.19052 0.595259 0.803534i \(-0.297049\pi\)
0.595259 + 0.803534i \(0.297049\pi\)
\(468\) 0 0
\(469\) 158.963 0.0156508
\(470\) −12940.1 −1.26996
\(471\) 0 0
\(472\) 6923.00 0.675121
\(473\) 25.7044 0.00249871
\(474\) 0 0
\(475\) 1259.90 0.121701
\(476\) −2685.83 −0.258623
\(477\) 0 0
\(478\) 8489.93 0.812386
\(479\) −6241.39 −0.595358 −0.297679 0.954666i \(-0.596212\pi\)
−0.297679 + 0.954666i \(0.596212\pi\)
\(480\) 0 0
\(481\) 12683.4 1.20231
\(482\) −15435.2 −1.45862
\(483\) 0 0
\(484\) −3973.65 −0.373183
\(485\) 2207.03 0.206631
\(486\) 0 0
\(487\) 924.180 0.0859930 0.0429965 0.999075i \(-0.486310\pi\)
0.0429965 + 0.999075i \(0.486310\pi\)
\(488\) 514.130 0.0476917
\(489\) 0 0
\(490\) 6948.44 0.640609
\(491\) 7044.15 0.647451 0.323725 0.946151i \(-0.395065\pi\)
0.323725 + 0.946151i \(0.395065\pi\)
\(492\) 0 0
\(493\) 26102.3 2.38457
\(494\) −2780.18 −0.253211
\(495\) 0 0
\(496\) 15217.9 1.37763
\(497\) 2160.60 0.195003
\(498\) 0 0
\(499\) −3981.76 −0.357210 −0.178605 0.983921i \(-0.557158\pi\)
−0.178605 + 0.983921i \(0.557158\pi\)
\(500\) 4513.74 0.403722
\(501\) 0 0
\(502\) −20232.2 −1.79882
\(503\) 11650.2 1.03272 0.516358 0.856373i \(-0.327288\pi\)
0.516358 + 0.856373i \(0.327288\pi\)
\(504\) 0 0
\(505\) −10069.7 −0.887318
\(506\) 253.057 0.0222327
\(507\) 0 0
\(508\) −5694.58 −0.497354
\(509\) −11408.7 −0.993481 −0.496740 0.867899i \(-0.665470\pi\)
−0.496740 + 0.867899i \(0.665470\pi\)
\(510\) 0 0
\(511\) 10150.9 0.878764
\(512\) −326.513 −0.0281836
\(513\) 0 0
\(514\) −2261.98 −0.194108
\(515\) −2722.41 −0.232939
\(516\) 0 0
\(517\) −273.941 −0.0233035
\(518\) 11858.4 1.00584
\(519\) 0 0
\(520\) 4941.19 0.416703
\(521\) −15272.4 −1.28425 −0.642127 0.766598i \(-0.721948\pi\)
−0.642127 + 0.766598i \(0.721948\pi\)
\(522\) 0 0
\(523\) −20831.4 −1.74167 −0.870835 0.491575i \(-0.836421\pi\)
−0.870835 + 0.491575i \(0.836421\pi\)
\(524\) 4587.16 0.382426
\(525\) 0 0
\(526\) −5541.50 −0.459355
\(527\) 17604.0 1.45511
\(528\) 0 0
\(529\) 4145.03 0.340678
\(530\) −16464.1 −1.34935
\(531\) 0 0
\(532\) −706.548 −0.0575803
\(533\) −6686.75 −0.543406
\(534\) 0 0
\(535\) 14032.1 1.13395
\(536\) −268.328 −0.0216232
\(537\) 0 0
\(538\) 868.410 0.0695907
\(539\) 147.098 0.0117550
\(540\) 0 0
\(541\) −1411.76 −0.112192 −0.0560962 0.998425i \(-0.517865\pi\)
−0.0560962 + 0.998425i \(0.517865\pi\)
\(542\) 15313.4 1.21359
\(543\) 0 0
\(544\) 11767.6 0.927450
\(545\) 4775.27 0.375322
\(546\) 0 0
\(547\) −328.281 −0.0256605 −0.0128302 0.999918i \(-0.504084\pi\)
−0.0128302 + 0.999918i \(0.504084\pi\)
\(548\) 5062.84 0.394661
\(549\) 0 0
\(550\) 103.873 0.00805301
\(551\) 6866.62 0.530903
\(552\) 0 0
\(553\) 4385.34 0.337222
\(554\) 17116.1 1.31262
\(555\) 0 0
\(556\) 2829.73 0.215840
\(557\) 1908.49 0.145180 0.0725901 0.997362i \(-0.476874\pi\)
0.0725901 + 0.997362i \(0.476874\pi\)
\(558\) 0 0
\(559\) −1500.79 −0.113554
\(560\) 6623.46 0.499808
\(561\) 0 0
\(562\) 17486.5 1.31250
\(563\) −3413.23 −0.255507 −0.127753 0.991806i \(-0.540777\pi\)
−0.127753 + 0.991806i \(0.540777\pi\)
\(564\) 0 0
\(565\) −559.579 −0.0416667
\(566\) −16666.7 −1.23773
\(567\) 0 0
\(568\) −3647.07 −0.269415
\(569\) 15405.8 1.13505 0.567527 0.823355i \(-0.307900\pi\)
0.567527 + 0.823355i \(0.307900\pi\)
\(570\) 0 0
\(571\) 18819.2 1.37926 0.689632 0.724160i \(-0.257772\pi\)
0.689632 + 0.724160i \(0.257772\pi\)
\(572\) −62.3044 −0.00455433
\(573\) 0 0
\(574\) −6251.81 −0.454609
\(575\) 6695.64 0.485613
\(576\) 0 0
\(577\) 8098.53 0.584309 0.292154 0.956371i \(-0.405628\pi\)
0.292154 + 0.956371i \(0.405628\pi\)
\(578\) 11378.1 0.818799
\(579\) 0 0
\(580\) 7268.90 0.520387
\(581\) −2938.84 −0.209852
\(582\) 0 0
\(583\) −348.545 −0.0247603
\(584\) −17134.6 −1.21410
\(585\) 0 0
\(586\) −29303.0 −2.06569
\(587\) 3631.44 0.255342 0.127671 0.991817i \(-0.459250\pi\)
0.127671 + 0.991817i \(0.459250\pi\)
\(588\) 0 0
\(589\) 4631.00 0.323968
\(590\) 11763.2 0.820822
\(591\) 0 0
\(592\) −28698.4 −1.99240
\(593\) 14298.2 0.990147 0.495073 0.868851i \(-0.335141\pi\)
0.495073 + 0.868851i \(0.335141\pi\)
\(594\) 0 0
\(595\) 7662.02 0.527920
\(596\) −9209.95 −0.632977
\(597\) 0 0
\(598\) −14775.1 −1.01037
\(599\) 27823.6 1.89790 0.948949 0.315430i \(-0.102149\pi\)
0.948949 + 0.315430i \(0.102149\pi\)
\(600\) 0 0
\(601\) 3557.15 0.241430 0.120715 0.992687i \(-0.461481\pi\)
0.120715 + 0.992687i \(0.461481\pi\)
\(602\) −1403.17 −0.0949982
\(603\) 0 0
\(604\) −7008.25 −0.472122
\(605\) 11335.9 0.761767
\(606\) 0 0
\(607\) 19170.9 1.28192 0.640958 0.767576i \(-0.278537\pi\)
0.640958 + 0.767576i \(0.278537\pi\)
\(608\) 3095.65 0.206489
\(609\) 0 0
\(610\) 873.585 0.0579843
\(611\) 15994.5 1.05903
\(612\) 0 0
\(613\) 4287.83 0.282518 0.141259 0.989973i \(-0.454885\pi\)
0.141259 + 0.989973i \(0.454885\pi\)
\(614\) 21760.6 1.43027
\(615\) 0 0
\(616\) 97.8006 0.00639692
\(617\) 10569.5 0.689649 0.344824 0.938667i \(-0.387938\pi\)
0.344824 + 0.938667i \(0.387938\pi\)
\(618\) 0 0
\(619\) 19947.5 1.29525 0.647624 0.761960i \(-0.275763\pi\)
0.647624 + 0.761960i \(0.275763\pi\)
\(620\) 4902.31 0.317551
\(621\) 0 0
\(622\) −4144.21 −0.267150
\(623\) −8330.71 −0.535735
\(624\) 0 0
\(625\) −6323.50 −0.404704
\(626\) −9840.42 −0.628278
\(627\) 0 0
\(628\) 4726.44 0.300328
\(629\) −33198.3 −2.10446
\(630\) 0 0
\(631\) 8854.29 0.558611 0.279306 0.960202i \(-0.409896\pi\)
0.279306 + 0.960202i \(0.409896\pi\)
\(632\) −7402.41 −0.465905
\(633\) 0 0
\(634\) −3494.60 −0.218909
\(635\) 16245.3 1.01523
\(636\) 0 0
\(637\) −8588.55 −0.534208
\(638\) 566.122 0.0351301
\(639\) 0 0
\(640\) −14562.5 −0.899429
\(641\) −10022.5 −0.617571 −0.308786 0.951132i \(-0.599923\pi\)
−0.308786 + 0.951132i \(0.599923\pi\)
\(642\) 0 0
\(643\) −5806.52 −0.356123 −0.178061 0.984019i \(-0.556983\pi\)
−0.178061 + 0.984019i \(0.556983\pi\)
\(644\) −3754.91 −0.229758
\(645\) 0 0
\(646\) 7277.04 0.443206
\(647\) −8583.93 −0.521591 −0.260795 0.965394i \(-0.583985\pi\)
−0.260795 + 0.965394i \(0.583985\pi\)
\(648\) 0 0
\(649\) 249.027 0.0150619
\(650\) −6064.77 −0.365969
\(651\) 0 0
\(652\) −2451.39 −0.147245
\(653\) −17048.8 −1.02170 −0.510849 0.859670i \(-0.670669\pi\)
−0.510849 + 0.859670i \(0.670669\pi\)
\(654\) 0 0
\(655\) −13086.1 −0.780633
\(656\) 15130.0 0.900499
\(657\) 0 0
\(658\) 14954.1 0.885974
\(659\) 2694.69 0.159287 0.0796437 0.996823i \(-0.474622\pi\)
0.0796437 + 0.996823i \(0.474622\pi\)
\(660\) 0 0
\(661\) −29666.8 −1.74570 −0.872848 0.487992i \(-0.837730\pi\)
−0.872848 + 0.487992i \(0.837730\pi\)
\(662\) −9551.64 −0.560778
\(663\) 0 0
\(664\) 4960.74 0.289931
\(665\) 2015.61 0.117537
\(666\) 0 0
\(667\) 36492.2 2.11842
\(668\) −5048.52 −0.292415
\(669\) 0 0
\(670\) −455.931 −0.0262898
\(671\) 18.4938 0.00106400
\(672\) 0 0
\(673\) 17932.8 1.02713 0.513564 0.858051i \(-0.328325\pi\)
0.513564 + 0.858051i \(0.328325\pi\)
\(674\) −20377.1 −1.16454
\(675\) 0 0
\(676\) −2923.09 −0.166312
\(677\) −2440.74 −0.138560 −0.0692801 0.997597i \(-0.522070\pi\)
−0.0692801 + 0.997597i \(0.522070\pi\)
\(678\) 0 0
\(679\) −2550.53 −0.144154
\(680\) −12933.4 −0.729373
\(681\) 0 0
\(682\) 381.806 0.0214371
\(683\) 12722.2 0.712737 0.356369 0.934345i \(-0.384015\pi\)
0.356369 + 0.934345i \(0.384015\pi\)
\(684\) 0 0
\(685\) −14443.1 −0.805608
\(686\) −19222.6 −1.06986
\(687\) 0 0
\(688\) 3395.81 0.188175
\(689\) 20350.3 1.12523
\(690\) 0 0
\(691\) −5584.80 −0.307461 −0.153731 0.988113i \(-0.549129\pi\)
−0.153731 + 0.988113i \(0.549129\pi\)
\(692\) −4151.64 −0.228066
\(693\) 0 0
\(694\) −4655.02 −0.254614
\(695\) −8072.53 −0.440588
\(696\) 0 0
\(697\) 17502.4 0.951148
\(698\) −14823.1 −0.803817
\(699\) 0 0
\(700\) −1541.29 −0.0832217
\(701\) −19891.2 −1.07173 −0.535864 0.844304i \(-0.680014\pi\)
−0.535864 + 0.844304i \(0.680014\pi\)
\(702\) 0 0
\(703\) −8733.33 −0.468540
\(704\) −122.440 −0.00655486
\(705\) 0 0
\(706\) −7114.62 −0.379266
\(707\) 11637.0 0.619028
\(708\) 0 0
\(709\) −14613.5 −0.774078 −0.387039 0.922063i \(-0.626502\pi\)
−0.387039 + 0.922063i \(0.626502\pi\)
\(710\) −6196.94 −0.327559
\(711\) 0 0
\(712\) 14062.1 0.740170
\(713\) 24611.2 1.29270
\(714\) 0 0
\(715\) 177.739 0.00929661
\(716\) −2072.16 −0.108157
\(717\) 0 0
\(718\) −6498.14 −0.337755
\(719\) 11027.7 0.571995 0.285998 0.958230i \(-0.407675\pi\)
0.285998 + 0.958230i \(0.407675\pi\)
\(720\) 0 0
\(721\) 3146.13 0.162508
\(722\) −20820.2 −1.07320
\(723\) 0 0
\(724\) −7034.15 −0.361080
\(725\) 14979.1 0.767322
\(726\) 0 0
\(727\) 1650.84 0.0842178 0.0421089 0.999113i \(-0.486592\pi\)
0.0421089 + 0.999113i \(0.486592\pi\)
\(728\) −5710.24 −0.290708
\(729\) 0 0
\(730\) −29114.3 −1.47612
\(731\) 3928.27 0.198758
\(732\) 0 0
\(733\) −28314.8 −1.42678 −0.713391 0.700767i \(-0.752841\pi\)
−0.713391 + 0.700767i \(0.752841\pi\)
\(734\) 44697.4 2.24770
\(735\) 0 0
\(736\) 16451.7 0.823935
\(737\) −9.65204 −0.000482412 0
\(738\) 0 0
\(739\) −28963.4 −1.44172 −0.720862 0.693078i \(-0.756254\pi\)
−0.720862 + 0.693078i \(0.756254\pi\)
\(740\) −9244.97 −0.459259
\(741\) 0 0
\(742\) 19026.6 0.941359
\(743\) 15510.4 0.765844 0.382922 0.923781i \(-0.374918\pi\)
0.382922 + 0.923781i \(0.374918\pi\)
\(744\) 0 0
\(745\) 26273.8 1.29208
\(746\) −39889.0 −1.95769
\(747\) 0 0
\(748\) 163.080 0.00797165
\(749\) −16216.1 −0.791087
\(750\) 0 0
\(751\) −2901.63 −0.140988 −0.0704941 0.997512i \(-0.522458\pi\)
−0.0704941 + 0.997512i \(0.522458\pi\)
\(752\) −36190.4 −1.75496
\(753\) 0 0
\(754\) −33053.9 −1.59649
\(755\) 19992.8 0.963727
\(756\) 0 0
\(757\) 9260.00 0.444598 0.222299 0.974979i \(-0.428644\pi\)
0.222299 + 0.974979i \(0.428644\pi\)
\(758\) −33759.2 −1.61766
\(759\) 0 0
\(760\) −3402.33 −0.162389
\(761\) 6418.29 0.305733 0.152866 0.988247i \(-0.451150\pi\)
0.152866 + 0.988247i \(0.451150\pi\)
\(762\) 0 0
\(763\) −5518.50 −0.261839
\(764\) 6474.50 0.306596
\(765\) 0 0
\(766\) −44773.9 −2.11194
\(767\) −14539.8 −0.684489
\(768\) 0 0
\(769\) 5335.88 0.250217 0.125108 0.992143i \(-0.460072\pi\)
0.125108 + 0.992143i \(0.460072\pi\)
\(770\) 166.178 0.00777746
\(771\) 0 0
\(772\) 3897.09 0.181683
\(773\) −20062.9 −0.933520 −0.466760 0.884384i \(-0.654579\pi\)
−0.466760 + 0.884384i \(0.654579\pi\)
\(774\) 0 0
\(775\) 10102.2 0.468236
\(776\) 4305.27 0.199162
\(777\) 0 0
\(778\) 22046.8 1.01596
\(779\) 4604.26 0.211765
\(780\) 0 0
\(781\) −131.189 −0.00601064
\(782\) 38673.4 1.76849
\(783\) 0 0
\(784\) 19433.2 0.885257
\(785\) −13483.4 −0.613049
\(786\) 0 0
\(787\) 17182.2 0.778248 0.389124 0.921185i \(-0.372778\pi\)
0.389124 + 0.921185i \(0.372778\pi\)
\(788\) 14334.9 0.648045
\(789\) 0 0
\(790\) −12577.8 −0.566454
\(791\) 646.672 0.0290683
\(792\) 0 0
\(793\) −1079.79 −0.0483535
\(794\) −27538.3 −1.23085
\(795\) 0 0
\(796\) −6310.99 −0.281014
\(797\) 3342.90 0.148572 0.0742859 0.997237i \(-0.476332\pi\)
0.0742859 + 0.997237i \(0.476332\pi\)
\(798\) 0 0
\(799\) −41865.0 −1.85366
\(800\) 6752.95 0.298441
\(801\) 0 0
\(802\) −9822.90 −0.432492
\(803\) −616.348 −0.0270865
\(804\) 0 0
\(805\) 10711.8 0.468997
\(806\) −22292.3 −0.974210
\(807\) 0 0
\(808\) −19643.1 −0.855248
\(809\) 10782.5 0.468596 0.234298 0.972165i \(-0.424721\pi\)
0.234298 + 0.972165i \(0.424721\pi\)
\(810\) 0 0
\(811\) −22440.1 −0.971612 −0.485806 0.874067i \(-0.661474\pi\)
−0.485806 + 0.874067i \(0.661474\pi\)
\(812\) −8400.24 −0.363043
\(813\) 0 0
\(814\) −720.024 −0.0310035
\(815\) 6993.21 0.300566
\(816\) 0 0
\(817\) 1033.39 0.0442519
\(818\) −32984.1 −1.40985
\(819\) 0 0
\(820\) 4874.01 0.207570
\(821\) 10938.7 0.464996 0.232498 0.972597i \(-0.425310\pi\)
0.232498 + 0.972597i \(0.425310\pi\)
\(822\) 0 0
\(823\) −37928.5 −1.60645 −0.803223 0.595678i \(-0.796883\pi\)
−0.803223 + 0.595678i \(0.796883\pi\)
\(824\) −5310.63 −0.224520
\(825\) 0 0
\(826\) −13594.1 −0.572637
\(827\) −36038.1 −1.51532 −0.757660 0.652650i \(-0.773657\pi\)
−0.757660 + 0.652650i \(0.773657\pi\)
\(828\) 0 0
\(829\) 29221.5 1.22425 0.612127 0.790760i \(-0.290314\pi\)
0.612127 + 0.790760i \(0.290314\pi\)
\(830\) 8429.05 0.352502
\(831\) 0 0
\(832\) 7148.82 0.297886
\(833\) 22480.3 0.935048
\(834\) 0 0
\(835\) 14402.2 0.596896
\(836\) 42.9006 0.00177482
\(837\) 0 0
\(838\) −17506.5 −0.721660
\(839\) 6285.44 0.258638 0.129319 0.991603i \(-0.458721\pi\)
0.129319 + 0.991603i \(0.458721\pi\)
\(840\) 0 0
\(841\) 57249.1 2.34733
\(842\) −5530.48 −0.226357
\(843\) 0 0
\(844\) 16616.5 0.677682
\(845\) 8338.87 0.339486
\(846\) 0 0
\(847\) −13100.2 −0.531438
\(848\) −46046.3 −1.86466
\(849\) 0 0
\(850\) 15874.4 0.640572
\(851\) −46412.8 −1.86958
\(852\) 0 0
\(853\) −35129.3 −1.41009 −0.705044 0.709164i \(-0.749073\pi\)
−0.705044 + 0.709164i \(0.749073\pi\)
\(854\) −1009.55 −0.0404521
\(855\) 0 0
\(856\) 27372.6 1.09296
\(857\) −32983.7 −1.31470 −0.657352 0.753584i \(-0.728324\pi\)
−0.657352 + 0.753584i \(0.728324\pi\)
\(858\) 0 0
\(859\) −35152.9 −1.39628 −0.698139 0.715962i \(-0.745988\pi\)
−0.698139 + 0.715962i \(0.745988\pi\)
\(860\) 1093.93 0.0433753
\(861\) 0 0
\(862\) 34301.4 1.35535
\(863\) −24925.6 −0.983173 −0.491587 0.870829i \(-0.663583\pi\)
−0.491587 + 0.870829i \(0.663583\pi\)
\(864\) 0 0
\(865\) 11843.6 0.465544
\(866\) 9838.48 0.386057
\(867\) 0 0
\(868\) −5665.31 −0.221536
\(869\) −266.272 −0.0103943
\(870\) 0 0
\(871\) 563.549 0.0219232
\(872\) 9315.17 0.361756
\(873\) 0 0
\(874\) 10173.6 0.393739
\(875\) 14880.8 0.574927
\(876\) 0 0
\(877\) 2057.63 0.0792261 0.0396131 0.999215i \(-0.487387\pi\)
0.0396131 + 0.999215i \(0.487387\pi\)
\(878\) −30575.2 −1.17524
\(879\) 0 0
\(880\) −402.168 −0.0154058
\(881\) −17912.5 −0.685004 −0.342502 0.939517i \(-0.611274\pi\)
−0.342502 + 0.939517i \(0.611274\pi\)
\(882\) 0 0
\(883\) −27215.2 −1.03722 −0.518610 0.855011i \(-0.673550\pi\)
−0.518610 + 0.855011i \(0.673550\pi\)
\(884\) −9521.67 −0.362272
\(885\) 0 0
\(886\) 33528.2 1.27133
\(887\) −25313.0 −0.958206 −0.479103 0.877759i \(-0.659038\pi\)
−0.479103 + 0.877759i \(0.659038\pi\)
\(888\) 0 0
\(889\) −18773.7 −0.708267
\(890\) 23893.7 0.899910
\(891\) 0 0
\(892\) −5105.76 −0.191652
\(893\) −11013.2 −0.412703
\(894\) 0 0
\(895\) 5911.36 0.220777
\(896\) 16829.1 0.627477
\(897\) 0 0
\(898\) −47683.3 −1.77195
\(899\) 55058.6 2.04261
\(900\) 0 0
\(901\) −53266.3 −1.96954
\(902\) 379.601 0.0140126
\(903\) 0 0
\(904\) −1091.58 −0.0401607
\(905\) 20066.7 0.737062
\(906\) 0 0
\(907\) 45998.9 1.68398 0.841989 0.539495i \(-0.181385\pi\)
0.841989 + 0.539495i \(0.181385\pi\)
\(908\) −1037.75 −0.0379282
\(909\) 0 0
\(910\) −9702.57 −0.353447
\(911\) 46592.7 1.69449 0.847247 0.531199i \(-0.178258\pi\)
0.847247 + 0.531199i \(0.178258\pi\)
\(912\) 0 0
\(913\) 178.443 0.00646833
\(914\) −33250.9 −1.20333
\(915\) 0 0
\(916\) 2130.72 0.0768571
\(917\) 15122.8 0.544601
\(918\) 0 0
\(919\) 33168.1 1.19055 0.595274 0.803523i \(-0.297043\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(920\) −18081.5 −0.647966
\(921\) 0 0
\(922\) 413.615 0.0147741
\(923\) 7659.66 0.273154
\(924\) 0 0
\(925\) −19051.2 −0.677188
\(926\) 51173.2 1.81604
\(927\) 0 0
\(928\) 36804.6 1.30191
\(929\) −19909.9 −0.703148 −0.351574 0.936160i \(-0.614353\pi\)
−0.351574 + 0.936160i \(0.614353\pi\)
\(930\) 0 0
\(931\) 5913.78 0.208181
\(932\) 19530.0 0.686401
\(933\) 0 0
\(934\) 39823.2 1.39514
\(935\) −465.228 −0.0162723
\(936\) 0 0
\(937\) 26262.2 0.915634 0.457817 0.889046i \(-0.348631\pi\)
0.457817 + 0.889046i \(0.348631\pi\)
\(938\) 526.893 0.0183408
\(939\) 0 0
\(940\) −11658.4 −0.404528
\(941\) −42227.8 −1.46290 −0.731449 0.681896i \(-0.761155\pi\)
−0.731449 + 0.681896i \(0.761155\pi\)
\(942\) 0 0
\(943\) 24469.1 0.844988
\(944\) 32899.0 1.13429
\(945\) 0 0
\(946\) 85.1986 0.00292817
\(947\) 35227.0 1.20879 0.604394 0.796685i \(-0.293415\pi\)
0.604394 + 0.796685i \(0.293415\pi\)
\(948\) 0 0
\(949\) 35986.4 1.23095
\(950\) 4175.99 0.142618
\(951\) 0 0
\(952\) 14946.4 0.508839
\(953\) −49209.9 −1.67268 −0.836340 0.548211i \(-0.815309\pi\)
−0.836340 + 0.548211i \(0.815309\pi\)
\(954\) 0 0
\(955\) −18470.2 −0.625844
\(956\) 7649.06 0.258774
\(957\) 0 0
\(958\) −20687.4 −0.697683
\(959\) 16691.0 0.562024
\(960\) 0 0
\(961\) 7341.79 0.246443
\(962\) 42039.7 1.40895
\(963\) 0 0
\(964\) −13906.4 −0.464623
\(965\) −11117.5 −0.370864
\(966\) 0 0
\(967\) 13452.5 0.447366 0.223683 0.974662i \(-0.428192\pi\)
0.223683 + 0.974662i \(0.428192\pi\)
\(968\) 22113.0 0.734234
\(969\) 0 0
\(970\) 7315.30 0.242145
\(971\) −37322.8 −1.23352 −0.616758 0.787152i \(-0.711554\pi\)
−0.616758 + 0.787152i \(0.711554\pi\)
\(972\) 0 0
\(973\) 9328.95 0.307371
\(974\) 3063.24 0.100773
\(975\) 0 0
\(976\) 2443.21 0.0801285
\(977\) 45153.4 1.47859 0.739297 0.673380i \(-0.235158\pi\)
0.739297 + 0.673380i \(0.235158\pi\)
\(978\) 0 0
\(979\) 505.829 0.0165132
\(980\) 6260.24 0.204057
\(981\) 0 0
\(982\) 23348.2 0.758729
\(983\) 15802.7 0.512745 0.256372 0.966578i \(-0.417473\pi\)
0.256372 + 0.966578i \(0.417473\pi\)
\(984\) 0 0
\(985\) −40894.0 −1.32283
\(986\) 86517.6 2.79440
\(987\) 0 0
\(988\) −2504.82 −0.0806568
\(989\) 5491.90 0.176575
\(990\) 0 0
\(991\) 27570.0 0.883744 0.441872 0.897078i \(-0.354315\pi\)
0.441872 + 0.897078i \(0.354315\pi\)
\(992\) 24821.9 0.794451
\(993\) 0 0
\(994\) 7161.43 0.228518
\(995\) 18003.7 0.573624
\(996\) 0 0
\(997\) −14969.8 −0.475526 −0.237763 0.971323i \(-0.576414\pi\)
−0.237763 + 0.971323i \(0.576414\pi\)
\(998\) −13197.7 −0.418605
\(999\) 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 387.4.a.h.1.6 6
3.2 odd 2 43.4.a.b.1.1 6
12.11 even 2 688.4.a.i.1.6 6
15.14 odd 2 1075.4.a.b.1.6 6
21.20 even 2 2107.4.a.c.1.1 6
129.128 even 2 1849.4.a.c.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.a.b.1.1 6 3.2 odd 2
387.4.a.h.1.6 6 1.1 even 1 trivial
688.4.a.i.1.6 6 12.11 even 2
1075.4.a.b.1.6 6 15.14 odd 2
1849.4.a.c.1.6 6 129.128 even 2
2107.4.a.c.1.1 6 21.20 even 2