Properties

Label 387.4.a.h.1.1
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 1.1
Root \(-4.15251\) 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)
 
\(f(q)\) \(=\) \(q-5.15251 q^{2} +18.5484 q^{4} -17.2665 q^{5} -23.3206 q^{7} -54.3507 q^{8} +O(q^{10})\) \(q-5.15251 q^{2} +18.5484 q^{4} -17.2665 q^{5} -23.3206 q^{7} -54.3507 q^{8} +88.9661 q^{10} +60.5580 q^{11} +10.9419 q^{13} +120.160 q^{14} +131.656 q^{16} +3.57473 q^{17} +33.2403 q^{19} -320.267 q^{20} -312.026 q^{22} -63.7158 q^{23} +173.134 q^{25} -56.3783 q^{26} -432.560 q^{28} +89.3510 q^{29} +222.839 q^{31} -243.552 q^{32} -18.4188 q^{34} +402.666 q^{35} -59.6535 q^{37} -171.271 q^{38} +938.449 q^{40} +143.837 q^{41} -43.0000 q^{43} +1123.25 q^{44} +328.296 q^{46} -379.013 q^{47} +200.850 q^{49} -892.073 q^{50} +202.955 q^{52} +150.129 q^{53} -1045.63 q^{55} +1267.49 q^{56} -460.382 q^{58} -207.310 q^{59} -486.557 q^{61} -1148.18 q^{62} +201.659 q^{64} -188.929 q^{65} +1019.41 q^{67} +66.3055 q^{68} -2074.74 q^{70} -13.8437 q^{71} +411.158 q^{73} +307.365 q^{74} +616.555 q^{76} -1412.25 q^{77} -1315.13 q^{79} -2273.24 q^{80} -741.120 q^{82} -813.425 q^{83} -61.7233 q^{85} +221.558 q^{86} -3291.37 q^{88} +350.573 q^{89} -255.171 q^{91} -1181.83 q^{92} +1952.87 q^{94} -573.946 q^{95} -1187.03 q^{97} -1034.88 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.15251 −1.82169 −0.910844 0.412750i \(-0.864568\pi\)
−0.910844 + 0.412750i \(0.864568\pi\)
\(3\) 0 0
\(4\) 18.5484 2.31855
\(5\) −17.2665 −1.54437 −0.772183 0.635400i \(-0.780835\pi\)
−0.772183 + 0.635400i \(0.780835\pi\)
\(6\) 0 0
\(7\) −23.3206 −1.25919 −0.629597 0.776922i \(-0.716780\pi\)
−0.629597 + 0.776922i \(0.716780\pi\)
\(8\) −54.3507 −2.40199
\(9\) 0 0
\(10\) 88.9661 2.81336
\(11\) 60.5580 1.65990 0.829951 0.557836i \(-0.188368\pi\)
0.829951 + 0.557836i \(0.188368\pi\)
\(12\) 0 0
\(13\) 10.9419 0.233441 0.116721 0.993165i \(-0.462762\pi\)
0.116721 + 0.993165i \(0.462762\pi\)
\(14\) 120.160 2.29386
\(15\) 0 0
\(16\) 131.656 2.05712
\(17\) 3.57473 0.0510000 0.0255000 0.999675i \(-0.491882\pi\)
0.0255000 + 0.999675i \(0.491882\pi\)
\(18\) 0 0
\(19\) 33.2403 0.401361 0.200680 0.979657i \(-0.435685\pi\)
0.200680 + 0.979657i \(0.435685\pi\)
\(20\) −320.267 −3.58069
\(21\) 0 0
\(22\) −312.026 −3.02383
\(23\) −63.7158 −0.577637 −0.288819 0.957384i \(-0.593262\pi\)
−0.288819 + 0.957384i \(0.593262\pi\)
\(24\) 0 0
\(25\) 173.134 1.38507
\(26\) −56.3783 −0.425257
\(27\) 0 0
\(28\) −432.560 −2.91950
\(29\) 89.3510 0.572140 0.286070 0.958209i \(-0.407651\pi\)
0.286070 + 0.958209i \(0.407651\pi\)
\(30\) 0 0
\(31\) 222.839 1.29106 0.645532 0.763733i \(-0.276636\pi\)
0.645532 + 0.763733i \(0.276636\pi\)
\(32\) −243.552 −1.34545
\(33\) 0 0
\(34\) −18.4188 −0.0929061
\(35\) 402.666 1.94466
\(36\) 0 0
\(37\) −59.6535 −0.265053 −0.132527 0.991179i \(-0.542309\pi\)
−0.132527 + 0.991179i \(0.542309\pi\)
\(38\) −171.271 −0.731154
\(39\) 0 0
\(40\) 938.449 3.70955
\(41\) 143.837 0.547890 0.273945 0.961745i \(-0.411671\pi\)
0.273945 + 0.961745i \(0.411671\pi\)
\(42\) 0 0
\(43\) −43.0000 −0.152499
\(44\) 1123.25 3.84857
\(45\) 0 0
\(46\) 328.296 1.05228
\(47\) −379.013 −1.17627 −0.588135 0.808763i \(-0.700138\pi\)
−0.588135 + 0.808763i \(0.700138\pi\)
\(48\) 0 0
\(49\) 200.850 0.585569
\(50\) −892.073 −2.52316
\(51\) 0 0
\(52\) 202.955 0.541245
\(53\) 150.129 0.389090 0.194545 0.980894i \(-0.437677\pi\)
0.194545 + 0.980894i \(0.437677\pi\)
\(54\) 0 0
\(55\) −1045.63 −2.56350
\(56\) 1267.49 3.02457
\(57\) 0 0
\(58\) −460.382 −1.04226
\(59\) −207.310 −0.457448 −0.228724 0.973491i \(-0.573455\pi\)
−0.228724 + 0.973491i \(0.573455\pi\)
\(60\) 0 0
\(61\) −486.557 −1.02127 −0.510633 0.859799i \(-0.670589\pi\)
−0.510633 + 0.859799i \(0.670589\pi\)
\(62\) −1148.18 −2.35192
\(63\) 0 0
\(64\) 201.659 0.393866
\(65\) −188.929 −0.360519
\(66\) 0 0
\(67\) 1019.41 1.85882 0.929408 0.369054i \(-0.120318\pi\)
0.929408 + 0.369054i \(0.120318\pi\)
\(68\) 66.3055 0.118246
\(69\) 0 0
\(70\) −2074.74 −3.54256
\(71\) −13.8437 −0.0231400 −0.0115700 0.999933i \(-0.503683\pi\)
−0.0115700 + 0.999933i \(0.503683\pi\)
\(72\) 0 0
\(73\) 411.158 0.659211 0.329605 0.944119i \(-0.393084\pi\)
0.329605 + 0.944119i \(0.393084\pi\)
\(74\) 307.365 0.482844
\(75\) 0 0
\(76\) 616.555 0.930575
\(77\) −1412.25 −2.09014
\(78\) 0 0
\(79\) −1315.13 −1.87296 −0.936481 0.350718i \(-0.885938\pi\)
−0.936481 + 0.350718i \(0.885938\pi\)
\(80\) −2273.24 −3.17695
\(81\) 0 0
\(82\) −741.120 −0.998085
\(83\) −813.425 −1.07572 −0.537861 0.843033i \(-0.680768\pi\)
−0.537861 + 0.843033i \(0.680768\pi\)
\(84\) 0 0
\(85\) −61.7233 −0.0787627
\(86\) 221.558 0.277805
\(87\) 0 0
\(88\) −3291.37 −3.98706
\(89\) 350.573 0.417535 0.208768 0.977965i \(-0.433055\pi\)
0.208768 + 0.977965i \(0.433055\pi\)
\(90\) 0 0
\(91\) −255.171 −0.293948
\(92\) −1181.83 −1.33928
\(93\) 0 0
\(94\) 1952.87 2.14280
\(95\) −573.946 −0.619848
\(96\) 0 0
\(97\) −1187.03 −1.24252 −0.621262 0.783603i \(-0.713380\pi\)
−0.621262 + 0.783603i \(0.713380\pi\)
\(98\) −1034.88 −1.06672
\(99\) 0 0
\(100\) 3211.35 3.21135
\(101\) −469.014 −0.462066 −0.231033 0.972946i \(-0.574210\pi\)
−0.231033 + 0.972946i \(0.574210\pi\)
\(102\) 0 0
\(103\) −1299.82 −1.24345 −0.621725 0.783235i \(-0.713568\pi\)
−0.621725 + 0.783235i \(0.713568\pi\)
\(104\) −594.700 −0.560722
\(105\) 0 0
\(106\) −773.541 −0.708801
\(107\) 1480.92 1.33800 0.669000 0.743263i \(-0.266723\pi\)
0.669000 + 0.743263i \(0.266723\pi\)
\(108\) 0 0
\(109\) 350.586 0.308073 0.154037 0.988065i \(-0.450773\pi\)
0.154037 + 0.988065i \(0.450773\pi\)
\(110\) 5387.61 4.66990
\(111\) 0 0
\(112\) −3070.29 −2.59031
\(113\) −785.106 −0.653598 −0.326799 0.945094i \(-0.605970\pi\)
−0.326799 + 0.945094i \(0.605970\pi\)
\(114\) 0 0
\(115\) 1100.15 0.892084
\(116\) 1657.32 1.32653
\(117\) 0 0
\(118\) 1068.17 0.833329
\(119\) −83.3649 −0.0642189
\(120\) 0 0
\(121\) 2336.27 1.75528
\(122\) 2506.99 1.86043
\(123\) 0 0
\(124\) 4133.30 2.99340
\(125\) −831.100 −0.594687
\(126\) 0 0
\(127\) −731.562 −0.511147 −0.255573 0.966790i \(-0.582264\pi\)
−0.255573 + 0.966790i \(0.582264\pi\)
\(128\) 909.364 0.627947
\(129\) 0 0
\(130\) 973.458 0.656753
\(131\) −2462.56 −1.64240 −0.821202 0.570638i \(-0.806696\pi\)
−0.821202 + 0.570638i \(0.806696\pi\)
\(132\) 0 0
\(133\) −775.184 −0.505391
\(134\) −5252.52 −3.38618
\(135\) 0 0
\(136\) −194.289 −0.122501
\(137\) −2384.64 −1.48710 −0.743552 0.668678i \(-0.766861\pi\)
−0.743552 + 0.668678i \(0.766861\pi\)
\(138\) 0 0
\(139\) −3086.87 −1.88363 −0.941816 0.336129i \(-0.890882\pi\)
−0.941816 + 0.336129i \(0.890882\pi\)
\(140\) 7468.81 4.50878
\(141\) 0 0
\(142\) 71.3297 0.0421539
\(143\) 662.619 0.387490
\(144\) 0 0
\(145\) −1542.78 −0.883594
\(146\) −2118.50 −1.20088
\(147\) 0 0
\(148\) −1106.48 −0.614539
\(149\) 1445.58 0.794808 0.397404 0.917644i \(-0.369911\pi\)
0.397404 + 0.917644i \(0.369911\pi\)
\(150\) 0 0
\(151\) −825.878 −0.445093 −0.222546 0.974922i \(-0.571437\pi\)
−0.222546 + 0.974922i \(0.571437\pi\)
\(152\) −1806.64 −0.964063
\(153\) 0 0
\(154\) 7276.63 3.80758
\(155\) −3847.65 −1.99388
\(156\) 0 0
\(157\) 1187.82 0.603809 0.301904 0.953338i \(-0.402378\pi\)
0.301904 + 0.953338i \(0.402378\pi\)
\(158\) 6776.24 3.41195
\(159\) 0 0
\(160\) 4205.30 2.07786
\(161\) 1485.89 0.727357
\(162\) 0 0
\(163\) 1514.60 0.727809 0.363904 0.931436i \(-0.381443\pi\)
0.363904 + 0.931436i \(0.381443\pi\)
\(164\) 2667.94 1.27031
\(165\) 0 0
\(166\) 4191.18 1.95963
\(167\) 3354.48 1.55435 0.777177 0.629282i \(-0.216651\pi\)
0.777177 + 0.629282i \(0.216651\pi\)
\(168\) 0 0
\(169\) −2077.27 −0.945505
\(170\) 318.030 0.143481
\(171\) 0 0
\(172\) −797.581 −0.353575
\(173\) −1654.64 −0.727166 −0.363583 0.931562i \(-0.618447\pi\)
−0.363583 + 0.931562i \(0.618447\pi\)
\(174\) 0 0
\(175\) −4037.58 −1.74407
\(176\) 7972.81 3.41462
\(177\) 0 0
\(178\) −1806.33 −0.760619
\(179\) 1334.05 0.557049 0.278524 0.960429i \(-0.410155\pi\)
0.278524 + 0.960429i \(0.410155\pi\)
\(180\) 0 0
\(181\) 2771.58 1.13817 0.569087 0.822277i \(-0.307297\pi\)
0.569087 + 0.822277i \(0.307297\pi\)
\(182\) 1314.77 0.535481
\(183\) 0 0
\(184\) 3463.00 1.38748
\(185\) 1030.01 0.409339
\(186\) 0 0
\(187\) 216.479 0.0846550
\(188\) −7030.08 −2.72724
\(189\) 0 0
\(190\) 2957.26 1.12917
\(191\) −81.5135 −0.0308802 −0.0154401 0.999881i \(-0.504915\pi\)
−0.0154401 + 0.999881i \(0.504915\pi\)
\(192\) 0 0
\(193\) 3305.29 1.23275 0.616373 0.787454i \(-0.288601\pi\)
0.616373 + 0.787454i \(0.288601\pi\)
\(194\) 6116.20 2.26349
\(195\) 0 0
\(196\) 3725.45 1.35767
\(197\) −2241.99 −0.810840 −0.405420 0.914131i \(-0.632875\pi\)
−0.405420 + 0.914131i \(0.632875\pi\)
\(198\) 0 0
\(199\) −4074.29 −1.45135 −0.725675 0.688037i \(-0.758472\pi\)
−0.725675 + 0.688037i \(0.758472\pi\)
\(200\) −9409.94 −3.32691
\(201\) 0 0
\(202\) 2416.60 0.841740
\(203\) −2083.72 −0.720435
\(204\) 0 0
\(205\) −2483.56 −0.846143
\(206\) 6697.36 2.26518
\(207\) 0 0
\(208\) 1440.56 0.480217
\(209\) 2012.97 0.666220
\(210\) 0 0
\(211\) 4267.62 1.39239 0.696196 0.717851i \(-0.254874\pi\)
0.696196 + 0.717851i \(0.254874\pi\)
\(212\) 2784.65 0.902125
\(213\) 0 0
\(214\) −7630.46 −2.43742
\(215\) 742.461 0.235514
\(216\) 0 0
\(217\) −5196.73 −1.62570
\(218\) −1806.40 −0.561214
\(219\) 0 0
\(220\) −19394.7 −5.94360
\(221\) 39.1143 0.0119055
\(222\) 0 0
\(223\) −889.370 −0.267070 −0.133535 0.991044i \(-0.542633\pi\)
−0.133535 + 0.991044i \(0.542633\pi\)
\(224\) 5679.78 1.69418
\(225\) 0 0
\(226\) 4045.27 1.19065
\(227\) −232.389 −0.0679479 −0.0339739 0.999423i \(-0.510816\pi\)
−0.0339739 + 0.999423i \(0.510816\pi\)
\(228\) 0 0
\(229\) 851.806 0.245803 0.122902 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(230\) −5668.54 −1.62510
\(231\) 0 0
\(232\) −4856.29 −1.37427
\(233\) 2083.62 0.585848 0.292924 0.956136i \(-0.405372\pi\)
0.292924 + 0.956136i \(0.405372\pi\)
\(234\) 0 0
\(235\) 6544.24 1.81659
\(236\) −3845.27 −1.06062
\(237\) 0 0
\(238\) 429.539 0.116987
\(239\) −986.983 −0.267124 −0.133562 0.991040i \(-0.542642\pi\)
−0.133562 + 0.991040i \(0.542642\pi\)
\(240\) 0 0
\(241\) 14.8772 0.00397644 0.00198822 0.999998i \(-0.499367\pi\)
0.00198822 + 0.999998i \(0.499367\pi\)
\(242\) −12037.7 −3.19757
\(243\) 0 0
\(244\) −9024.85 −2.36786
\(245\) −3467.99 −0.904333
\(246\) 0 0
\(247\) 363.712 0.0936942
\(248\) −12111.4 −3.10112
\(249\) 0 0
\(250\) 4282.26 1.08333
\(251\) 3027.27 0.761274 0.380637 0.924725i \(-0.375705\pi\)
0.380637 + 0.924725i \(0.375705\pi\)
\(252\) 0 0
\(253\) −3858.50 −0.958822
\(254\) 3769.38 0.931150
\(255\) 0 0
\(256\) −6298.79 −1.53779
\(257\) −5759.92 −1.39803 −0.699016 0.715106i \(-0.746378\pi\)
−0.699016 + 0.715106i \(0.746378\pi\)
\(258\) 0 0
\(259\) 1391.15 0.333753
\(260\) −3504.32 −0.835881
\(261\) 0 0
\(262\) 12688.4 2.99195
\(263\) −2545.09 −0.596719 −0.298359 0.954454i \(-0.596439\pi\)
−0.298359 + 0.954454i \(0.596439\pi\)
\(264\) 0 0
\(265\) −2592.21 −0.600898
\(266\) 3994.15 0.920665
\(267\) 0 0
\(268\) 18908.4 4.30976
\(269\) −6757.89 −1.53173 −0.765866 0.643001i \(-0.777689\pi\)
−0.765866 + 0.643001i \(0.777689\pi\)
\(270\) 0 0
\(271\) 2485.54 0.557143 0.278571 0.960415i \(-0.410139\pi\)
0.278571 + 0.960415i \(0.410139\pi\)
\(272\) 470.634 0.104913
\(273\) 0 0
\(274\) 12286.9 2.70904
\(275\) 10484.6 2.29908
\(276\) 0 0
\(277\) −2090.19 −0.453385 −0.226692 0.973966i \(-0.572791\pi\)
−0.226692 + 0.973966i \(0.572791\pi\)
\(278\) 15905.1 3.43139
\(279\) 0 0
\(280\) −21885.2 −4.67104
\(281\) 6049.22 1.28422 0.642111 0.766612i \(-0.278059\pi\)
0.642111 + 0.766612i \(0.278059\pi\)
\(282\) 0 0
\(283\) −650.183 −0.136570 −0.0682851 0.997666i \(-0.521753\pi\)
−0.0682851 + 0.997666i \(0.521753\pi\)
\(284\) −256.778 −0.0536513
\(285\) 0 0
\(286\) −3414.16 −0.705885
\(287\) −3354.35 −0.689900
\(288\) 0 0
\(289\) −4900.22 −0.997399
\(290\) 7949.21 1.60963
\(291\) 0 0
\(292\) 7626.31 1.52841
\(293\) 2172.20 0.433110 0.216555 0.976270i \(-0.430518\pi\)
0.216555 + 0.976270i \(0.430518\pi\)
\(294\) 0 0
\(295\) 3579.53 0.706468
\(296\) 3242.21 0.636654
\(297\) 0 0
\(298\) −7448.36 −1.44789
\(299\) −697.171 −0.134844
\(300\) 0 0
\(301\) 1002.79 0.192025
\(302\) 4255.35 0.810820
\(303\) 0 0
\(304\) 4376.28 0.825648
\(305\) 8401.16 1.57721
\(306\) 0 0
\(307\) 1257.47 0.233771 0.116885 0.993145i \(-0.462709\pi\)
0.116885 + 0.993145i \(0.462709\pi\)
\(308\) −26194.9 −4.84609
\(309\) 0 0
\(310\) 19825.1 3.63222
\(311\) 1316.02 0.239950 0.119975 0.992777i \(-0.461719\pi\)
0.119975 + 0.992777i \(0.461719\pi\)
\(312\) 0 0
\(313\) 3137.09 0.566514 0.283257 0.959044i \(-0.408585\pi\)
0.283257 + 0.959044i \(0.408585\pi\)
\(314\) −6120.24 −1.09995
\(315\) 0 0
\(316\) −24393.6 −4.34256
\(317\) −571.518 −0.101261 −0.0506303 0.998717i \(-0.516123\pi\)
−0.0506303 + 0.998717i \(0.516123\pi\)
\(318\) 0 0
\(319\) 5410.92 0.949696
\(320\) −3481.96 −0.608273
\(321\) 0 0
\(322\) −7656.07 −1.32502
\(323\) 118.825 0.0204694
\(324\) 0 0
\(325\) 1894.41 0.323332
\(326\) −7804.01 −1.32584
\(327\) 0 0
\(328\) −7817.62 −1.31602
\(329\) 8838.80 1.48115
\(330\) 0 0
\(331\) −2257.30 −0.374841 −0.187420 0.982280i \(-0.560013\pi\)
−0.187420 + 0.982280i \(0.560013\pi\)
\(332\) −15087.7 −2.49412
\(333\) 0 0
\(334\) −17284.0 −2.83155
\(335\) −17601.7 −2.87069
\(336\) 0 0
\(337\) −9222.66 −1.49077 −0.745386 0.666633i \(-0.767735\pi\)
−0.745386 + 0.666633i \(0.767735\pi\)
\(338\) 10703.2 1.72242
\(339\) 0 0
\(340\) −1144.87 −0.182615
\(341\) 13494.7 2.14304
\(342\) 0 0
\(343\) 3315.02 0.521849
\(344\) 2337.08 0.366299
\(345\) 0 0
\(346\) 8525.54 1.32467
\(347\) −8840.48 −1.36767 −0.683835 0.729636i \(-0.739689\pi\)
−0.683835 + 0.729636i \(0.739689\pi\)
\(348\) 0 0
\(349\) 1431.64 0.219581 0.109791 0.993955i \(-0.464982\pi\)
0.109791 + 0.993955i \(0.464982\pi\)
\(350\) 20803.7 3.17715
\(351\) 0 0
\(352\) −14749.0 −2.23331
\(353\) 8308.06 1.25267 0.626336 0.779553i \(-0.284554\pi\)
0.626336 + 0.779553i \(0.284554\pi\)
\(354\) 0 0
\(355\) 239.032 0.0357367
\(356\) 6502.56 0.968076
\(357\) 0 0
\(358\) −6873.72 −1.01477
\(359\) −11642.2 −1.71157 −0.855786 0.517330i \(-0.826926\pi\)
−0.855786 + 0.517330i \(0.826926\pi\)
\(360\) 0 0
\(361\) −5754.08 −0.838909
\(362\) −14280.6 −2.07340
\(363\) 0 0
\(364\) −4733.02 −0.681532
\(365\) −7099.27 −1.01806
\(366\) 0 0
\(367\) 4373.93 0.622119 0.311059 0.950391i \(-0.399316\pi\)
0.311059 + 0.950391i \(0.399316\pi\)
\(368\) −8388.55 −1.18827
\(369\) 0 0
\(370\) −5307.14 −0.745689
\(371\) −3501.09 −0.489940
\(372\) 0 0
\(373\) 7970.63 1.10644 0.553222 0.833034i \(-0.313398\pi\)
0.553222 + 0.833034i \(0.313398\pi\)
\(374\) −1115.41 −0.154215
\(375\) 0 0
\(376\) 20599.6 2.82538
\(377\) 977.669 0.133561
\(378\) 0 0
\(379\) −2020.56 −0.273850 −0.136925 0.990581i \(-0.543722\pi\)
−0.136925 + 0.990581i \(0.543722\pi\)
\(380\) −10645.8 −1.43715
\(381\) 0 0
\(382\) 420.000 0.0562540
\(383\) −11625.0 −1.55094 −0.775468 0.631386i \(-0.782486\pi\)
−0.775468 + 0.631386i \(0.782486\pi\)
\(384\) 0 0
\(385\) 24384.7 3.22794
\(386\) −17030.5 −2.24568
\(387\) 0 0
\(388\) −22017.6 −2.88086
\(389\) −6212.26 −0.809702 −0.404851 0.914383i \(-0.632677\pi\)
−0.404851 + 0.914383i \(0.632677\pi\)
\(390\) 0 0
\(391\) −227.767 −0.0294595
\(392\) −10916.4 −1.40653
\(393\) 0 0
\(394\) 11551.9 1.47710
\(395\) 22707.8 2.89254
\(396\) 0 0
\(397\) −1647.75 −0.208307 −0.104154 0.994561i \(-0.533213\pi\)
−0.104154 + 0.994561i \(0.533213\pi\)
\(398\) 20992.8 2.64391
\(399\) 0 0
\(400\) 22794.0 2.84925
\(401\) −6893.37 −0.858450 −0.429225 0.903198i \(-0.641213\pi\)
−0.429225 + 0.903198i \(0.641213\pi\)
\(402\) 0 0
\(403\) 2438.28 0.301388
\(404\) −8699.46 −1.07132
\(405\) 0 0
\(406\) 10736.4 1.31241
\(407\) −3612.49 −0.439962
\(408\) 0 0
\(409\) −2962.23 −0.358125 −0.179062 0.983838i \(-0.557306\pi\)
−0.179062 + 0.983838i \(0.557306\pi\)
\(410\) 12796.6 1.54141
\(411\) 0 0
\(412\) −24109.6 −2.88300
\(413\) 4834.59 0.576016
\(414\) 0 0
\(415\) 14045.0 1.66131
\(416\) −2664.92 −0.314083
\(417\) 0 0
\(418\) −10371.8 −1.21365
\(419\) 5520.25 0.643633 0.321816 0.946802i \(-0.395707\pi\)
0.321816 + 0.946802i \(0.395707\pi\)
\(420\) 0 0
\(421\) 11017.9 1.27548 0.637742 0.770250i \(-0.279868\pi\)
0.637742 + 0.770250i \(0.279868\pi\)
\(422\) −21989.0 −2.53651
\(423\) 0 0
\(424\) −8159.61 −0.934589
\(425\) 618.906 0.0706385
\(426\) 0 0
\(427\) 11346.8 1.28597
\(428\) 27468.7 3.10222
\(429\) 0 0
\(430\) −3825.54 −0.429033
\(431\) −8825.55 −0.986338 −0.493169 0.869934i \(-0.664162\pi\)
−0.493169 + 0.869934i \(0.664162\pi\)
\(432\) 0 0
\(433\) 4570.88 0.507303 0.253652 0.967296i \(-0.418368\pi\)
0.253652 + 0.967296i \(0.418368\pi\)
\(434\) 26776.2 2.96152
\(435\) 0 0
\(436\) 6502.80 0.714283
\(437\) −2117.93 −0.231841
\(438\) 0 0
\(439\) −3059.73 −0.332649 −0.166325 0.986071i \(-0.553190\pi\)
−0.166325 + 0.986071i \(0.553190\pi\)
\(440\) 56830.6 6.15749
\(441\) 0 0
\(442\) −201.537 −0.0216881
\(443\) 2764.55 0.296496 0.148248 0.988950i \(-0.452637\pi\)
0.148248 + 0.988950i \(0.452637\pi\)
\(444\) 0 0
\(445\) −6053.18 −0.644828
\(446\) 4582.49 0.486518
\(447\) 0 0
\(448\) −4702.82 −0.495954
\(449\) −7567.33 −0.795377 −0.397688 0.917521i \(-0.630188\pi\)
−0.397688 + 0.917521i \(0.630188\pi\)
\(450\) 0 0
\(451\) 8710.46 0.909444
\(452\) −14562.4 −1.51540
\(453\) 0 0
\(454\) 1197.38 0.123780
\(455\) 4405.93 0.453963
\(456\) 0 0
\(457\) −8079.05 −0.826963 −0.413482 0.910512i \(-0.635687\pi\)
−0.413482 + 0.910512i \(0.635687\pi\)
\(458\) −4388.94 −0.447777
\(459\) 0 0
\(460\) 20406.0 2.06834
\(461\) −14466.4 −1.46154 −0.730770 0.682624i \(-0.760839\pi\)
−0.730770 + 0.682624i \(0.760839\pi\)
\(462\) 0 0
\(463\) −7966.79 −0.799672 −0.399836 0.916587i \(-0.630933\pi\)
−0.399836 + 0.916587i \(0.630933\pi\)
\(464\) 11763.6 1.17696
\(465\) 0 0
\(466\) −10735.9 −1.06723
\(467\) −6737.29 −0.667590 −0.333795 0.942646i \(-0.608329\pi\)
−0.333795 + 0.942646i \(0.608329\pi\)
\(468\) 0 0
\(469\) −23773.2 −2.34061
\(470\) −33719.3 −3.30927
\(471\) 0 0
\(472\) 11267.4 1.09878
\(473\) −2603.99 −0.253133
\(474\) 0 0
\(475\) 5755.02 0.555912
\(476\) −1546.28 −0.148895
\(477\) 0 0
\(478\) 5085.44 0.486616
\(479\) 18115.7 1.72803 0.864014 0.503468i \(-0.167943\pi\)
0.864014 + 0.503468i \(0.167943\pi\)
\(480\) 0 0
\(481\) −652.722 −0.0618743
\(482\) −76.6548 −0.00724384
\(483\) 0 0
\(484\) 43334.1 4.06969
\(485\) 20496.0 1.91891
\(486\) 0 0
\(487\) −14966.4 −1.39260 −0.696298 0.717753i \(-0.745171\pi\)
−0.696298 + 0.717753i \(0.745171\pi\)
\(488\) 26444.7 2.45307
\(489\) 0 0
\(490\) 17868.8 1.64741
\(491\) 15713.8 1.44430 0.722151 0.691736i \(-0.243154\pi\)
0.722151 + 0.691736i \(0.243154\pi\)
\(492\) 0 0
\(493\) 319.406 0.0291791
\(494\) −1874.03 −0.170682
\(495\) 0 0
\(496\) 29338.0 2.65588
\(497\) 322.843 0.0291378
\(498\) 0 0
\(499\) −15391.5 −1.38080 −0.690398 0.723430i \(-0.742565\pi\)
−0.690398 + 0.723430i \(0.742565\pi\)
\(500\) −15415.6 −1.37881
\(501\) 0 0
\(502\) −15598.1 −1.38680
\(503\) 12150.9 1.07710 0.538548 0.842595i \(-0.318973\pi\)
0.538548 + 0.842595i \(0.318973\pi\)
\(504\) 0 0
\(505\) 8098.25 0.713599
\(506\) 19881.0 1.74667
\(507\) 0 0
\(508\) −13569.3 −1.18512
\(509\) −16646.6 −1.44960 −0.724799 0.688960i \(-0.758067\pi\)
−0.724799 + 0.688960i \(0.758067\pi\)
\(510\) 0 0
\(511\) −9588.44 −0.830074
\(512\) 25179.7 2.17343
\(513\) 0 0
\(514\) 29678.1 2.54678
\(515\) 22443.5 1.92034
\(516\) 0 0
\(517\) −22952.3 −1.95249
\(518\) −7167.94 −0.607995
\(519\) 0 0
\(520\) 10268.4 0.865961
\(521\) −16675.9 −1.40227 −0.701135 0.713028i \(-0.747323\pi\)
−0.701135 + 0.713028i \(0.747323\pi\)
\(522\) 0 0
\(523\) 14326.5 1.19781 0.598903 0.800821i \(-0.295603\pi\)
0.598903 + 0.800821i \(0.295603\pi\)
\(524\) −45676.5 −3.80799
\(525\) 0 0
\(526\) 13113.6 1.08704
\(527\) 796.588 0.0658443
\(528\) 0 0
\(529\) −8107.30 −0.666335
\(530\) 13356.4 1.09465
\(531\) 0 0
\(532\) −14378.4 −1.17177
\(533\) 1573.84 0.127900
\(534\) 0 0
\(535\) −25570.4 −2.06636
\(536\) −55405.6 −4.46485
\(537\) 0 0
\(538\) 34820.1 2.79034
\(539\) 12163.1 0.971987
\(540\) 0 0
\(541\) −8159.73 −0.648455 −0.324228 0.945979i \(-0.605104\pi\)
−0.324228 + 0.945979i \(0.605104\pi\)
\(542\) −12806.8 −1.01494
\(543\) 0 0
\(544\) −870.633 −0.0686178
\(545\) −6053.40 −0.475778
\(546\) 0 0
\(547\) −6089.67 −0.476006 −0.238003 0.971264i \(-0.576493\pi\)
−0.238003 + 0.971264i \(0.576493\pi\)
\(548\) −44231.2 −3.44793
\(549\) 0 0
\(550\) −54022.2 −4.18821
\(551\) 2970.06 0.229635
\(552\) 0 0
\(553\) 30669.7 2.35842
\(554\) 10769.8 0.825926
\(555\) 0 0
\(556\) −57256.5 −4.36729
\(557\) −4160.66 −0.316504 −0.158252 0.987399i \(-0.550586\pi\)
−0.158252 + 0.987399i \(0.550586\pi\)
\(558\) 0 0
\(559\) −470.501 −0.0355995
\(560\) 53013.3 4.00039
\(561\) 0 0
\(562\) −31168.7 −2.33945
\(563\) −7731.24 −0.578744 −0.289372 0.957217i \(-0.593446\pi\)
−0.289372 + 0.957217i \(0.593446\pi\)
\(564\) 0 0
\(565\) 13556.1 1.00939
\(566\) 3350.08 0.248788
\(567\) 0 0
\(568\) 752.414 0.0555820
\(569\) −9134.32 −0.672989 −0.336494 0.941685i \(-0.609241\pi\)
−0.336494 + 0.941685i \(0.609241\pi\)
\(570\) 0 0
\(571\) 10421.5 0.763797 0.381898 0.924204i \(-0.375270\pi\)
0.381898 + 0.924204i \(0.375270\pi\)
\(572\) 12290.5 0.898414
\(573\) 0 0
\(574\) 17283.4 1.25678
\(575\) −11031.3 −0.800067
\(576\) 0 0
\(577\) 806.571 0.0581941 0.0290971 0.999577i \(-0.490737\pi\)
0.0290971 + 0.999577i \(0.490737\pi\)
\(578\) 25248.5 1.81695
\(579\) 0 0
\(580\) −28616.1 −2.04866
\(581\) 18969.6 1.35454
\(582\) 0 0
\(583\) 9091.50 0.645852
\(584\) −22346.7 −1.58341
\(585\) 0 0
\(586\) −11192.3 −0.788992
\(587\) −11514.2 −0.809614 −0.404807 0.914402i \(-0.632661\pi\)
−0.404807 + 0.914402i \(0.632661\pi\)
\(588\) 0 0
\(589\) 7407.23 0.518183
\(590\) −18443.6 −1.28696
\(591\) 0 0
\(592\) −7853.72 −0.545246
\(593\) −2702.41 −0.187141 −0.0935706 0.995613i \(-0.529828\pi\)
−0.0935706 + 0.995613i \(0.529828\pi\)
\(594\) 0 0
\(595\) 1439.42 0.0991775
\(596\) 26813.1 1.84280
\(597\) 0 0
\(598\) 3592.18 0.245644
\(599\) 27360.1 1.86628 0.933142 0.359507i \(-0.117055\pi\)
0.933142 + 0.359507i \(0.117055\pi\)
\(600\) 0 0
\(601\) −11506.8 −0.780985 −0.390492 0.920606i \(-0.627695\pi\)
−0.390492 + 0.920606i \(0.627695\pi\)
\(602\) −5166.87 −0.349810
\(603\) 0 0
\(604\) −15318.7 −1.03197
\(605\) −40339.4 −2.71079
\(606\) 0 0
\(607\) 8189.12 0.547588 0.273794 0.961788i \(-0.411721\pi\)
0.273794 + 0.961788i \(0.411721\pi\)
\(608\) −8095.75 −0.540010
\(609\) 0 0
\(610\) −43287.1 −2.87318
\(611\) −4147.12 −0.274590
\(612\) 0 0
\(613\) −24210.1 −1.59517 −0.797583 0.603209i \(-0.793889\pi\)
−0.797583 + 0.603209i \(0.793889\pi\)
\(614\) −6479.14 −0.425858
\(615\) 0 0
\(616\) 76756.8 5.02048
\(617\) 8779.45 0.572848 0.286424 0.958103i \(-0.407533\pi\)
0.286424 + 0.958103i \(0.407533\pi\)
\(618\) 0 0
\(619\) −6089.56 −0.395412 −0.197706 0.980261i \(-0.563349\pi\)
−0.197706 + 0.980261i \(0.563349\pi\)
\(620\) −71367.8 −4.62290
\(621\) 0 0
\(622\) −6780.79 −0.437114
\(623\) −8175.57 −0.525758
\(624\) 0 0
\(625\) −7291.47 −0.466654
\(626\) −16163.9 −1.03201
\(627\) 0 0
\(628\) 22032.1 1.39996
\(629\) −213.245 −0.0135177
\(630\) 0 0
\(631\) 13768.3 0.868634 0.434317 0.900760i \(-0.356990\pi\)
0.434317 + 0.900760i \(0.356990\pi\)
\(632\) 71478.5 4.49883
\(633\) 0 0
\(634\) 2944.75 0.184465
\(635\) 12631.5 0.789398
\(636\) 0 0
\(637\) 2197.68 0.136696
\(638\) −27879.8 −1.73005
\(639\) 0 0
\(640\) −15701.6 −0.969780
\(641\) −393.612 −0.0242539 −0.0121269 0.999926i \(-0.503860\pi\)
−0.0121269 + 0.999926i \(0.503860\pi\)
\(642\) 0 0
\(643\) −29944.9 −1.83656 −0.918281 0.395929i \(-0.870423\pi\)
−0.918281 + 0.395929i \(0.870423\pi\)
\(644\) 27560.9 1.68641
\(645\) 0 0
\(646\) −612.249 −0.0372889
\(647\) 8372.05 0.508716 0.254358 0.967110i \(-0.418136\pi\)
0.254358 + 0.967110i \(0.418136\pi\)
\(648\) 0 0
\(649\) −12554.3 −0.759320
\(650\) −9760.97 −0.589010
\(651\) 0 0
\(652\) 28093.4 1.68746
\(653\) −7203.34 −0.431682 −0.215841 0.976429i \(-0.569249\pi\)
−0.215841 + 0.976429i \(0.569249\pi\)
\(654\) 0 0
\(655\) 42519.9 2.53647
\(656\) 18936.9 1.12708
\(657\) 0 0
\(658\) −45542.0 −2.69820
\(659\) 28054.8 1.65836 0.829182 0.558979i \(-0.188807\pi\)
0.829182 + 0.558979i \(0.188807\pi\)
\(660\) 0 0
\(661\) −11582.1 −0.681531 −0.340766 0.940148i \(-0.610686\pi\)
−0.340766 + 0.940148i \(0.610686\pi\)
\(662\) 11630.8 0.682843
\(663\) 0 0
\(664\) 44210.2 2.58387
\(665\) 13384.8 0.780509
\(666\) 0 0
\(667\) −5693.07 −0.330489
\(668\) 62220.1 3.60385
\(669\) 0 0
\(670\) 90692.9 5.22951
\(671\) −29464.9 −1.69520
\(672\) 0 0
\(673\) 9582.65 0.548862 0.274431 0.961607i \(-0.411511\pi\)
0.274431 + 0.961607i \(0.411511\pi\)
\(674\) 47519.9 2.71572
\(675\) 0 0
\(676\) −38530.1 −2.19220
\(677\) −8247.63 −0.468216 −0.234108 0.972211i \(-0.575217\pi\)
−0.234108 + 0.972211i \(0.575217\pi\)
\(678\) 0 0
\(679\) 27682.3 1.56458
\(680\) 3354.70 0.189187
\(681\) 0 0
\(682\) −69531.5 −3.90395
\(683\) −23171.1 −1.29812 −0.649061 0.760736i \(-0.724838\pi\)
−0.649061 + 0.760736i \(0.724838\pi\)
\(684\) 0 0
\(685\) 41174.4 2.29663
\(686\) −17080.7 −0.950647
\(687\) 0 0
\(688\) −5661.20 −0.313708
\(689\) 1642.69 0.0908297
\(690\) 0 0
\(691\) 24019.8 1.32237 0.661185 0.750223i \(-0.270054\pi\)
0.661185 + 0.750223i \(0.270054\pi\)
\(692\) −30690.8 −1.68597
\(693\) 0 0
\(694\) 45550.7 2.49147
\(695\) 53299.6 2.90902
\(696\) 0 0
\(697\) 514.177 0.0279424
\(698\) −7376.54 −0.400009
\(699\) 0 0
\(700\) −74890.6 −4.04371
\(701\) 756.339 0.0407511 0.0203755 0.999792i \(-0.493514\pi\)
0.0203755 + 0.999792i \(0.493514\pi\)
\(702\) 0 0
\(703\) −1982.90 −0.106382
\(704\) 12212.1 0.653779
\(705\) 0 0
\(706\) −42807.4 −2.28198
\(707\) 10937.7 0.581830
\(708\) 0 0
\(709\) −26580.1 −1.40795 −0.703975 0.710224i \(-0.748594\pi\)
−0.703975 + 0.710224i \(0.748594\pi\)
\(710\) −1231.62 −0.0651011
\(711\) 0 0
\(712\) −19053.9 −1.00291
\(713\) −14198.3 −0.745767
\(714\) 0 0
\(715\) −11441.1 −0.598426
\(716\) 24744.5 1.29154
\(717\) 0 0
\(718\) 59986.8 3.11795
\(719\) −14125.2 −0.732656 −0.366328 0.930486i \(-0.619385\pi\)
−0.366328 + 0.930486i \(0.619385\pi\)
\(720\) 0 0
\(721\) 30312.7 1.56575
\(722\) 29648.0 1.52823
\(723\) 0 0
\(724\) 51408.3 2.63891
\(725\) 15469.6 0.792453
\(726\) 0 0
\(727\) −19206.2 −0.979805 −0.489903 0.871777i \(-0.662968\pi\)
−0.489903 + 0.871777i \(0.662968\pi\)
\(728\) 13868.8 0.706058
\(729\) 0 0
\(730\) 36579.1 1.85459
\(731\) −153.713 −0.00777742
\(732\) 0 0
\(733\) −29831.4 −1.50320 −0.751602 0.659617i \(-0.770718\pi\)
−0.751602 + 0.659617i \(0.770718\pi\)
\(734\) −22536.8 −1.13331
\(735\) 0 0
\(736\) 15518.1 0.777181
\(737\) 61733.4 3.08545
\(738\) 0 0
\(739\) 11720.8 0.583431 0.291715 0.956505i \(-0.405774\pi\)
0.291715 + 0.956505i \(0.405774\pi\)
\(740\) 19105.0 0.949073
\(741\) 0 0
\(742\) 18039.4 0.892518
\(743\) −4257.57 −0.210222 −0.105111 0.994460i \(-0.533520\pi\)
−0.105111 + 0.994460i \(0.533520\pi\)
\(744\) 0 0
\(745\) −24960.1 −1.22747
\(746\) −41068.8 −2.01560
\(747\) 0 0
\(748\) 4015.33 0.196277
\(749\) −34535.9 −1.68480
\(750\) 0 0
\(751\) −16414.4 −0.797564 −0.398782 0.917046i \(-0.630567\pi\)
−0.398782 + 0.917046i \(0.630567\pi\)
\(752\) −49899.2 −2.41973
\(753\) 0 0
\(754\) −5037.45 −0.243307
\(755\) 14260.1 0.687386
\(756\) 0 0
\(757\) −22813.5 −1.09534 −0.547668 0.836696i \(-0.684484\pi\)
−0.547668 + 0.836696i \(0.684484\pi\)
\(758\) 10410.9 0.498869
\(759\) 0 0
\(760\) 31194.4 1.48887
\(761\) −1695.44 −0.0807617 −0.0403808 0.999184i \(-0.512857\pi\)
−0.0403808 + 0.999184i \(0.512857\pi\)
\(762\) 0 0
\(763\) −8175.86 −0.387924
\(764\) −1511.95 −0.0715972
\(765\) 0 0
\(766\) 59897.9 2.82532
\(767\) −2268.36 −0.106787
\(768\) 0 0
\(769\) 21604.9 1.01313 0.506563 0.862203i \(-0.330916\pi\)
0.506563 + 0.862203i \(0.330916\pi\)
\(770\) −125642. −5.88030
\(771\) 0 0
\(772\) 61307.8 2.85818
\(773\) −34435.4 −1.60227 −0.801136 0.598482i \(-0.795771\pi\)
−0.801136 + 0.598482i \(0.795771\pi\)
\(774\) 0 0
\(775\) 38580.9 1.78821
\(776\) 64516.1 2.98453
\(777\) 0 0
\(778\) 32008.8 1.47503
\(779\) 4781.18 0.219902
\(780\) 0 0
\(781\) −838.345 −0.0384102
\(782\) 1173.57 0.0536660
\(783\) 0 0
\(784\) 26443.1 1.20459
\(785\) −20509.5 −0.932502
\(786\) 0 0
\(787\) −12352.5 −0.559490 −0.279745 0.960074i \(-0.590250\pi\)
−0.279745 + 0.960074i \(0.590250\pi\)
\(788\) −41585.4 −1.87997
\(789\) 0 0
\(790\) −117002. −5.26931
\(791\) 18309.1 0.823006
\(792\) 0 0
\(793\) −5323.85 −0.238406
\(794\) 8490.03 0.379471
\(795\) 0 0
\(796\) −75571.5 −3.36503
\(797\) −29811.8 −1.32495 −0.662477 0.749083i \(-0.730495\pi\)
−0.662477 + 0.749083i \(0.730495\pi\)
\(798\) 0 0
\(799\) −1354.87 −0.0599897
\(800\) −42167.0 −1.86354
\(801\) 0 0
\(802\) 35518.2 1.56383
\(803\) 24898.9 1.09423
\(804\) 0 0
\(805\) −25656.2 −1.12331
\(806\) −12563.3 −0.549035
\(807\) 0 0
\(808\) 25491.3 1.10988
\(809\) 1273.45 0.0553424 0.0276712 0.999617i \(-0.491191\pi\)
0.0276712 + 0.999617i \(0.491191\pi\)
\(810\) 0 0
\(811\) 37825.5 1.63777 0.818886 0.573956i \(-0.194592\pi\)
0.818886 + 0.573956i \(0.194592\pi\)
\(812\) −38649.6 −1.67036
\(813\) 0 0
\(814\) 18613.4 0.801475
\(815\) −26152.0 −1.12400
\(816\) 0 0
\(817\) −1429.33 −0.0612070
\(818\) 15262.9 0.652391
\(819\) 0 0
\(820\) −46066.1 −1.96182
\(821\) −189.193 −0.00804247 −0.00402124 0.999992i \(-0.501280\pi\)
−0.00402124 + 0.999992i \(0.501280\pi\)
\(822\) 0 0
\(823\) 26388.5 1.11767 0.558837 0.829278i \(-0.311248\pi\)
0.558837 + 0.829278i \(0.311248\pi\)
\(824\) 70646.4 2.98675
\(825\) 0 0
\(826\) −24910.3 −1.04932
\(827\) 37247.2 1.56616 0.783079 0.621923i \(-0.213648\pi\)
0.783079 + 0.621923i \(0.213648\pi\)
\(828\) 0 0
\(829\) −18429.8 −0.772128 −0.386064 0.922472i \(-0.626166\pi\)
−0.386064 + 0.922472i \(0.626166\pi\)
\(830\) −72367.2 −3.02639
\(831\) 0 0
\(832\) 2206.54 0.0919445
\(833\) 717.985 0.0298640
\(834\) 0 0
\(835\) −57920.2 −2.40049
\(836\) 37337.3 1.54466
\(837\) 0 0
\(838\) −28443.2 −1.17250
\(839\) −24344.3 −1.00174 −0.500870 0.865523i \(-0.666986\pi\)
−0.500870 + 0.865523i \(0.666986\pi\)
\(840\) 0 0
\(841\) −16405.4 −0.672656
\(842\) −56769.8 −2.32354
\(843\) 0 0
\(844\) 79157.5 3.22833
\(845\) 35867.4 1.46021
\(846\) 0 0
\(847\) −54483.3 −2.21023
\(848\) 19765.3 0.800406
\(849\) 0 0
\(850\) −3188.92 −0.128681
\(851\) 3800.87 0.153105
\(852\) 0 0
\(853\) 14146.8 0.567852 0.283926 0.958846i \(-0.408363\pi\)
0.283926 + 0.958846i \(0.408363\pi\)
\(854\) −58464.5 −2.34264
\(855\) 0 0
\(856\) −80489.1 −3.21386
\(857\) −8682.05 −0.346060 −0.173030 0.984917i \(-0.555356\pi\)
−0.173030 + 0.984917i \(0.555356\pi\)
\(858\) 0 0
\(859\) −25428.8 −1.01003 −0.505017 0.863110i \(-0.668514\pi\)
−0.505017 + 0.863110i \(0.668514\pi\)
\(860\) 13771.5 0.546050
\(861\) 0 0
\(862\) 45473.7 1.79680
\(863\) 39552.6 1.56012 0.780062 0.625703i \(-0.215188\pi\)
0.780062 + 0.625703i \(0.215188\pi\)
\(864\) 0 0
\(865\) 28569.9 1.12301
\(866\) −23551.5 −0.924148
\(867\) 0 0
\(868\) −96391.0 −3.76927
\(869\) −79641.9 −3.10894
\(870\) 0 0
\(871\) 11154.3 0.433924
\(872\) −19054.6 −0.739988
\(873\) 0 0
\(874\) 10912.7 0.422342
\(875\) 19381.8 0.748826
\(876\) 0 0
\(877\) 1955.60 0.0752975 0.0376487 0.999291i \(-0.488013\pi\)
0.0376487 + 0.999291i \(0.488013\pi\)
\(878\) 15765.3 0.605983
\(879\) 0 0
\(880\) −137663. −5.27343
\(881\) −3037.09 −0.116143 −0.0580716 0.998312i \(-0.518495\pi\)
−0.0580716 + 0.998312i \(0.518495\pi\)
\(882\) 0 0
\(883\) 15702.3 0.598443 0.299222 0.954184i \(-0.403273\pi\)
0.299222 + 0.954184i \(0.403273\pi\)
\(884\) 725.508 0.0276035
\(885\) 0 0
\(886\) −14244.4 −0.540123
\(887\) −2964.83 −0.112231 −0.0561157 0.998424i \(-0.517872\pi\)
−0.0561157 + 0.998424i \(0.517872\pi\)
\(888\) 0 0
\(889\) 17060.5 0.643633
\(890\) 31189.1 1.17467
\(891\) 0 0
\(892\) −16496.4 −0.619215
\(893\) −12598.5 −0.472109
\(894\) 0 0
\(895\) −23034.5 −0.860287
\(896\) −21206.9 −0.790707
\(897\) 0 0
\(898\) 38990.8 1.44893
\(899\) 19910.9 0.738670
\(900\) 0 0
\(901\) 536.670 0.0198436
\(902\) −44880.7 −1.65672
\(903\) 0 0
\(904\) 42671.1 1.56993
\(905\) −47855.5 −1.75776
\(906\) 0 0
\(907\) 47065.4 1.72302 0.861511 0.507738i \(-0.169518\pi\)
0.861511 + 0.507738i \(0.169518\pi\)
\(908\) −4310.43 −0.157540
\(909\) 0 0
\(910\) −22701.6 −0.826979
\(911\) 40218.3 1.46267 0.731335 0.682018i \(-0.238898\pi\)
0.731335 + 0.682018i \(0.238898\pi\)
\(912\) 0 0
\(913\) −49259.4 −1.78559
\(914\) 41627.4 1.50647
\(915\) 0 0
\(916\) 15799.6 0.569906
\(917\) 57428.4 2.06810
\(918\) 0 0
\(919\) −13015.3 −0.467175 −0.233587 0.972336i \(-0.575047\pi\)
−0.233587 + 0.972336i \(0.575047\pi\)
\(920\) −59794.0 −2.14277
\(921\) 0 0
\(922\) 74538.6 2.66247
\(923\) −151.476 −0.00540184
\(924\) 0 0
\(925\) −10328.0 −0.367117
\(926\) 41049.0 1.45675
\(927\) 0 0
\(928\) −21761.6 −0.769784
\(929\) 17365.8 0.613298 0.306649 0.951823i \(-0.400792\pi\)
0.306649 + 0.951823i \(0.400792\pi\)
\(930\) 0 0
\(931\) 6676.32 0.235024
\(932\) 38647.8 1.35832
\(933\) 0 0
\(934\) 34714.0 1.21614
\(935\) −3737.84 −0.130738
\(936\) 0 0
\(937\) −41655.6 −1.45232 −0.726162 0.687523i \(-0.758698\pi\)
−0.726162 + 0.687523i \(0.758698\pi\)
\(938\) 122492. 4.26386
\(939\) 0 0
\(940\) 121385. 4.21186
\(941\) 11377.5 0.394149 0.197075 0.980388i \(-0.436856\pi\)
0.197075 + 0.980388i \(0.436856\pi\)
\(942\) 0 0
\(943\) −9164.66 −0.316482
\(944\) −27293.5 −0.941027
\(945\) 0 0
\(946\) 13417.1 0.461129
\(947\) −39835.0 −1.36691 −0.683455 0.729992i \(-0.739524\pi\)
−0.683455 + 0.729992i \(0.739524\pi\)
\(948\) 0 0
\(949\) 4498.84 0.153887
\(950\) −29652.8 −1.01270
\(951\) 0 0
\(952\) 4530.94 0.154253
\(953\) −16468.3 −0.559769 −0.279885 0.960034i \(-0.590296\pi\)
−0.279885 + 0.960034i \(0.590296\pi\)
\(954\) 0 0
\(955\) 1407.46 0.0476903
\(956\) −18306.9 −0.619340
\(957\) 0 0
\(958\) −93341.2 −3.14793
\(959\) 55611.2 1.87255
\(960\) 0 0
\(961\) 19866.1 0.666848
\(962\) 3363.16 0.112716
\(963\) 0 0
\(964\) 275.948 0.00921958
\(965\) −57070.9 −1.90381
\(966\) 0 0
\(967\) 55135.3 1.83354 0.916770 0.399416i \(-0.130787\pi\)
0.916770 + 0.399416i \(0.130787\pi\)
\(968\) −126978. −4.21615
\(969\) 0 0
\(970\) −105606. −3.49566
\(971\) 13040.1 0.430975 0.215488 0.976507i \(-0.430866\pi\)
0.215488 + 0.976507i \(0.430866\pi\)
\(972\) 0 0
\(973\) 71987.6 2.37186
\(974\) 77114.7 2.53687
\(975\) 0 0
\(976\) −64058.0 −2.10087
\(977\) 45144.9 1.47831 0.739157 0.673534i \(-0.235224\pi\)
0.739157 + 0.673534i \(0.235224\pi\)
\(978\) 0 0
\(979\) 21230.0 0.693068
\(980\) −64325.6 −2.09674
\(981\) 0 0
\(982\) −80965.4 −2.63107
\(983\) 26573.6 0.862225 0.431113 0.902298i \(-0.358121\pi\)
0.431113 + 0.902298i \(0.358121\pi\)
\(984\) 0 0
\(985\) 38711.5 1.25223
\(986\) −1645.74 −0.0531553
\(987\) 0 0
\(988\) 6746.28 0.217235
\(989\) 2739.78 0.0880889
\(990\) 0 0
\(991\) 20658.0 0.662183 0.331092 0.943599i \(-0.392583\pi\)
0.331092 + 0.943599i \(0.392583\pi\)
\(992\) −54272.8 −1.73706
\(993\) 0 0
\(994\) −1663.45 −0.0530800
\(995\) 70348.9 2.24142
\(996\) 0 0
\(997\) −27284.2 −0.866701 −0.433350 0.901226i \(-0.642669\pi\)
−0.433350 + 0.901226i \(0.642669\pi\)
\(998\) 79304.8 2.51538
\(999\) 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.1.1 6
3.2 odd 2 43.4.a.b.1.6 6
12.11 even 2 688.4.a.i.1.5 6
15.14 odd 2 1075.4.a.b.1.1 6
21.20 even 2 2107.4.a.c.1.6 6
129.128 even 2 1849.4.a.c.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.a.b.1.6 6 3.2 odd 2
387.4.a.h.1.1 6 1.1 even 1 trivial
688.4.a.i.1.5 6 12.11 even 2
1075.4.a.b.1.1 6 15.14 odd 2
1849.4.a.c.1.1 6 129.128 even 2
2107.4.a.c.1.6 6 21.20 even 2