Properties

Label 1421.4.a.e.1.2
Level $1421$
Weight $4$
Character 1421.1
Self dual yes
Analytic conductor $83.842$
Analytic rank $1$
Dimension $5$
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] = [1421,4,Mod(1,1421)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1421, 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("1421.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1421.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: \(83.8417141182\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.957567\) of defining polynomial
Character \(\chi\) \(=\) 1421.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.84972 q^{2} -4.64574 q^{3} +0.120922 q^{4} -12.8729 q^{5} +13.2391 q^{6} +22.4532 q^{8} -5.41713 q^{9} +O(q^{10})\) \(q-2.84972 q^{2} -4.64574 q^{3} +0.120922 q^{4} -12.8729 q^{5} +13.2391 q^{6} +22.4532 q^{8} -5.41713 q^{9} +36.6841 q^{10} -62.8274 q^{11} -0.561770 q^{12} -22.3936 q^{13} +59.8039 q^{15} -64.9528 q^{16} +57.9808 q^{17} +15.4373 q^{18} -71.3143 q^{19} -1.55661 q^{20} +179.041 q^{22} -49.5307 q^{23} -104.312 q^{24} +40.7104 q^{25} +63.8155 q^{26} +150.601 q^{27} -29.0000 q^{29} -170.425 q^{30} -62.9198 q^{31} +5.47182 q^{32} +291.880 q^{33} -165.229 q^{34} -0.655048 q^{36} +119.123 q^{37} +203.226 q^{38} +104.035 q^{39} -289.037 q^{40} +414.916 q^{41} -348.009 q^{43} -7.59719 q^{44} +69.7340 q^{45} +141.149 q^{46} -553.259 q^{47} +301.753 q^{48} -116.013 q^{50} -269.364 q^{51} -2.70787 q^{52} -107.308 q^{53} -429.172 q^{54} +808.768 q^{55} +331.308 q^{57} +82.6420 q^{58} -136.881 q^{59} +7.23158 q^{60} +579.408 q^{61} +179.304 q^{62} +504.029 q^{64} +288.269 q^{65} -831.776 q^{66} +919.959 q^{67} +7.01113 q^{68} +230.106 q^{69} +781.802 q^{71} -121.632 q^{72} +133.237 q^{73} -339.467 q^{74} -189.130 q^{75} -8.62344 q^{76} -296.470 q^{78} +868.196 q^{79} +836.127 q^{80} -553.392 q^{81} -1182.39 q^{82} +83.3560 q^{83} -746.379 q^{85} +991.730 q^{86} +134.726 q^{87} -1410.68 q^{88} +357.919 q^{89} -198.722 q^{90} -5.98933 q^{92} +292.309 q^{93} +1576.64 q^{94} +918.019 q^{95} -25.4206 q^{96} +187.105 q^{97} +340.344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} + 26 q^{4} - 10 q^{5} - 34 q^{6} - 84 q^{8} + 33 q^{9} + 64 q^{10} + 12 q^{11} + 224 q^{12} - 14 q^{13} - 74 q^{15} + 146 q^{16} - 66 q^{17} - 108 q^{18} - 214 q^{19} - 6 q^{20} - 98 q^{22} + 164 q^{23} - 314 q^{24} + 207 q^{25} - 56 q^{26} - 362 q^{27} - 145 q^{29} - 234 q^{30} - 420 q^{31} - 652 q^{32} + 576 q^{33} - 204 q^{34} - 260 q^{36} + 378 q^{37} + 496 q^{38} - 374 q^{39} + 80 q^{40} + 1158 q^{41} - 204 q^{43} + 784 q^{44} + 1506 q^{45} + 580 q^{46} - 248 q^{47} + 1880 q^{48} + 908 q^{50} + 228 q^{51} - 1482 q^{52} - 554 q^{53} - 918 q^{54} - 546 q^{55} + 44 q^{57} - 440 q^{59} + 636 q^{60} - 618 q^{61} - 1250 q^{62} + 2594 q^{64} - 1656 q^{65} - 2940 q^{66} + 1164 q^{67} - 356 q^{68} + 1968 q^{69} - 692 q^{71} - 2648 q^{72} + 1950 q^{73} - 1832 q^{74} - 3074 q^{75} - 1376 q^{76} - 1302 q^{78} + 272 q^{79} + 890 q^{80} + 1801 q^{81} - 92 q^{82} - 512 q^{83} - 1628 q^{85} + 2446 q^{86} + 232 q^{87} - 6954 q^{88} - 866 q^{89} + 2200 q^{90} + 3468 q^{92} - 40 q^{93} + 5942 q^{94} + 2244 q^{95} - 7386 q^{96} - 1562 q^{97} - 238 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.84972 −1.00753 −0.503765 0.863841i \(-0.668052\pi\)
−0.503765 + 0.863841i \(0.668052\pi\)
\(3\) −4.64574 −0.894072 −0.447036 0.894516i \(-0.647520\pi\)
−0.447036 + 0.894516i \(0.647520\pi\)
\(4\) 0.120922 0.0151152
\(5\) −12.8729 −1.15138 −0.575692 0.817667i \(-0.695267\pi\)
−0.575692 + 0.817667i \(0.695267\pi\)
\(6\) 13.2391 0.900804
\(7\) 0 0
\(8\) 22.4532 0.992300
\(9\) −5.41713 −0.200635
\(10\) 36.6841 1.16005
\(11\) −62.8274 −1.72211 −0.861053 0.508515i \(-0.830195\pi\)
−0.861053 + 0.508515i \(0.830195\pi\)
\(12\) −0.561770 −0.0135141
\(13\) −22.3936 −0.477759 −0.238879 0.971049i \(-0.576780\pi\)
−0.238879 + 0.971049i \(0.576780\pi\)
\(14\) 0 0
\(15\) 59.8039 1.02942
\(16\) −64.9528 −1.01489
\(17\) 57.9808 0.827201 0.413601 0.910458i \(-0.364271\pi\)
0.413601 + 0.910458i \(0.364271\pi\)
\(18\) 15.4373 0.202145
\(19\) −71.3143 −0.861086 −0.430543 0.902570i \(-0.641678\pi\)
−0.430543 + 0.902570i \(0.641678\pi\)
\(20\) −1.55661 −0.0174034
\(21\) 0 0
\(22\) 179.041 1.73507
\(23\) −49.5307 −0.449037 −0.224519 0.974470i \(-0.572081\pi\)
−0.224519 + 0.974470i \(0.572081\pi\)
\(24\) −104.312 −0.887188
\(25\) 40.7104 0.325683
\(26\) 63.8155 0.481356
\(27\) 150.601 1.07345
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) −170.425 −1.03717
\(31\) −62.9198 −0.364540 −0.182270 0.983249i \(-0.558344\pi\)
−0.182270 + 0.983249i \(0.558344\pi\)
\(32\) 5.47182 0.0302278
\(33\) 291.880 1.53969
\(34\) −165.229 −0.833429
\(35\) 0 0
\(36\) −0.655048 −0.00303263
\(37\) 119.123 0.529288 0.264644 0.964346i \(-0.414746\pi\)
0.264644 + 0.964346i \(0.414746\pi\)
\(38\) 203.226 0.867569
\(39\) 104.035 0.427151
\(40\) −289.037 −1.14252
\(41\) 414.916 1.58046 0.790231 0.612810i \(-0.209961\pi\)
0.790231 + 0.612810i \(0.209961\pi\)
\(42\) 0 0
\(43\) −348.009 −1.23421 −0.617103 0.786882i \(-0.711694\pi\)
−0.617103 + 0.786882i \(0.711694\pi\)
\(44\) −7.59719 −0.0260300
\(45\) 69.7340 0.231007
\(46\) 141.149 0.452418
\(47\) −553.259 −1.71705 −0.858523 0.512775i \(-0.828618\pi\)
−0.858523 + 0.512775i \(0.828618\pi\)
\(48\) 301.753 0.907382
\(49\) 0 0
\(50\) −116.013 −0.328136
\(51\) −269.364 −0.739578
\(52\) −2.70787 −0.00722142
\(53\) −107.308 −0.278111 −0.139055 0.990285i \(-0.544407\pi\)
−0.139055 + 0.990285i \(0.544407\pi\)
\(54\) −429.172 −1.08154
\(55\) 808.768 1.98280
\(56\) 0 0
\(57\) 331.308 0.769873
\(58\) 82.6420 0.187093
\(59\) −136.881 −0.302041 −0.151020 0.988531i \(-0.548256\pi\)
−0.151020 + 0.988531i \(0.548256\pi\)
\(60\) 7.23158 0.0155599
\(61\) 579.408 1.21616 0.608078 0.793877i \(-0.291941\pi\)
0.608078 + 0.793877i \(0.291941\pi\)
\(62\) 179.304 0.367285
\(63\) 0 0
\(64\) 504.029 0.984431
\(65\) 288.269 0.550084
\(66\) −831.776 −1.55128
\(67\) 919.959 1.67748 0.838738 0.544535i \(-0.183294\pi\)
0.838738 + 0.544535i \(0.183294\pi\)
\(68\) 7.01113 0.0125033
\(69\) 230.106 0.401472
\(70\) 0 0
\(71\) 781.802 1.30680 0.653400 0.757013i \(-0.273342\pi\)
0.653400 + 0.757013i \(0.273342\pi\)
\(72\) −121.632 −0.199090
\(73\) 133.237 0.213619 0.106810 0.994279i \(-0.465936\pi\)
0.106810 + 0.994279i \(0.465936\pi\)
\(74\) −339.467 −0.533273
\(75\) −189.130 −0.291185
\(76\) −8.62344 −0.0130155
\(77\) 0 0
\(78\) −296.470 −0.430367
\(79\) 868.196 1.23645 0.618225 0.786001i \(-0.287852\pi\)
0.618225 + 0.786001i \(0.287852\pi\)
\(80\) 836.127 1.16852
\(81\) −553.392 −0.759111
\(82\) −1182.39 −1.59236
\(83\) 83.3560 0.110235 0.0551175 0.998480i \(-0.482447\pi\)
0.0551175 + 0.998480i \(0.482447\pi\)
\(84\) 0 0
\(85\) −746.379 −0.952425
\(86\) 991.730 1.24350
\(87\) 134.726 0.166025
\(88\) −1410.68 −1.70885
\(89\) 357.919 0.426284 0.213142 0.977021i \(-0.431630\pi\)
0.213142 + 0.977021i \(0.431630\pi\)
\(90\) −198.722 −0.232747
\(91\) 0 0
\(92\) −5.98933 −0.00678729
\(93\) 292.309 0.325925
\(94\) 1576.64 1.72997
\(95\) 918.019 0.991440
\(96\) −25.4206 −0.0270259
\(97\) 187.105 0.195852 0.0979260 0.995194i \(-0.468779\pi\)
0.0979260 + 0.995194i \(0.468779\pi\)
\(98\) 0 0
\(99\) 340.344 0.345514
\(100\) 4.92277 0.00492277
\(101\) 959.423 0.945209 0.472605 0.881275i \(-0.343314\pi\)
0.472605 + 0.881275i \(0.343314\pi\)
\(102\) 767.612 0.745146
\(103\) −78.8738 −0.0754531 −0.0377265 0.999288i \(-0.512012\pi\)
−0.0377265 + 0.999288i \(0.512012\pi\)
\(104\) −502.808 −0.474080
\(105\) 0 0
\(106\) 305.797 0.280205
\(107\) 713.851 0.644959 0.322479 0.946577i \(-0.395484\pi\)
0.322479 + 0.946577i \(0.395484\pi\)
\(108\) 18.2110 0.0162255
\(109\) 536.561 0.471497 0.235749 0.971814i \(-0.424246\pi\)
0.235749 + 0.971814i \(0.424246\pi\)
\(110\) −2304.76 −1.99773
\(111\) −553.412 −0.473222
\(112\) 0 0
\(113\) 1946.36 1.62034 0.810170 0.586195i \(-0.199375\pi\)
0.810170 + 0.586195i \(0.199375\pi\)
\(114\) −944.135 −0.775670
\(115\) 637.601 0.517014
\(116\) −3.50673 −0.00280682
\(117\) 121.309 0.0958549
\(118\) 390.073 0.304315
\(119\) 0 0
\(120\) 1342.79 1.02149
\(121\) 2616.28 1.96565
\(122\) −1651.15 −1.22531
\(123\) −1927.59 −1.41305
\(124\) −7.60837 −0.00551009
\(125\) 1085.05 0.776397
\(126\) 0 0
\(127\) −1995.14 −1.39402 −0.697009 0.717062i \(-0.745486\pi\)
−0.697009 + 0.717062i \(0.745486\pi\)
\(128\) −1480.12 −1.02207
\(129\) 1616.76 1.10347
\(130\) −821.488 −0.554225
\(131\) −1544.84 −1.03033 −0.515164 0.857092i \(-0.672269\pi\)
−0.515164 + 0.857092i \(0.672269\pi\)
\(132\) 35.2945 0.0232727
\(133\) 0 0
\(134\) −2621.63 −1.69011
\(135\) −1938.67 −1.23596
\(136\) 1301.85 0.820832
\(137\) −1294.93 −0.807543 −0.403771 0.914860i \(-0.632301\pi\)
−0.403771 + 0.914860i \(0.632301\pi\)
\(138\) −655.740 −0.404495
\(139\) −1999.66 −1.22021 −0.610105 0.792320i \(-0.708873\pi\)
−0.610105 + 0.792320i \(0.708873\pi\)
\(140\) 0 0
\(141\) 2570.30 1.53516
\(142\) −2227.92 −1.31664
\(143\) 1406.93 0.822752
\(144\) 351.858 0.203621
\(145\) 373.313 0.213807
\(146\) −379.688 −0.215228
\(147\) 0 0
\(148\) 14.4045 0.00800029
\(149\) 1187.63 0.652984 0.326492 0.945200i \(-0.394133\pi\)
0.326492 + 0.945200i \(0.394133\pi\)
\(150\) 538.968 0.293377
\(151\) −2257.61 −1.21670 −0.608350 0.793669i \(-0.708168\pi\)
−0.608350 + 0.793669i \(0.708168\pi\)
\(152\) −1601.23 −0.854456
\(153\) −314.090 −0.165965
\(154\) 0 0
\(155\) 809.958 0.419725
\(156\) 12.5800 0.00645647
\(157\) −1188.18 −0.603995 −0.301997 0.953309i \(-0.597653\pi\)
−0.301997 + 0.953309i \(0.597653\pi\)
\(158\) −2474.12 −1.24576
\(159\) 498.524 0.248651
\(160\) −70.4379 −0.0348038
\(161\) 0 0
\(162\) 1577.01 0.764827
\(163\) −2452.33 −1.17841 −0.589207 0.807982i \(-0.700560\pi\)
−0.589207 + 0.807982i \(0.700560\pi\)
\(164\) 50.1722 0.0238890
\(165\) −3757.32 −1.77277
\(166\) −237.541 −0.111065
\(167\) 2020.14 0.936067 0.468034 0.883711i \(-0.344963\pi\)
0.468034 + 0.883711i \(0.344963\pi\)
\(168\) 0 0
\(169\) −1695.53 −0.771746
\(170\) 2126.97 0.959597
\(171\) 386.319 0.172764
\(172\) −42.0818 −0.0186553
\(173\) −2862.12 −1.25782 −0.628910 0.777478i \(-0.716499\pi\)
−0.628910 + 0.777478i \(0.716499\pi\)
\(174\) −383.933 −0.167275
\(175\) 0 0
\(176\) 4080.81 1.74774
\(177\) 635.914 0.270046
\(178\) −1019.97 −0.429494
\(179\) −232.651 −0.0971460 −0.0485730 0.998820i \(-0.515467\pi\)
−0.0485730 + 0.998820i \(0.515467\pi\)
\(180\) 8.43234 0.00349172
\(181\) 2607.67 1.07086 0.535432 0.844578i \(-0.320149\pi\)
0.535432 + 0.844578i \(0.320149\pi\)
\(182\) 0 0
\(183\) −2691.78 −1.08733
\(184\) −1112.12 −0.445580
\(185\) −1533.45 −0.609413
\(186\) −833.000 −0.328379
\(187\) −3642.78 −1.42453
\(188\) −66.9010 −0.0259535
\(189\) 0 0
\(190\) −2616.10 −0.998905
\(191\) 1528.90 0.579202 0.289601 0.957147i \(-0.406477\pi\)
0.289601 + 0.957147i \(0.406477\pi\)
\(192\) −2341.59 −0.880153
\(193\) 1017.58 0.379518 0.189759 0.981831i \(-0.439229\pi\)
0.189759 + 0.981831i \(0.439229\pi\)
\(194\) −533.197 −0.197327
\(195\) −1339.22 −0.491815
\(196\) 0 0
\(197\) 3290.20 1.18994 0.594968 0.803749i \(-0.297165\pi\)
0.594968 + 0.803749i \(0.297165\pi\)
\(198\) −969.887 −0.348115
\(199\) 29.9190 0.0106578 0.00532891 0.999986i \(-0.498304\pi\)
0.00532891 + 0.999986i \(0.498304\pi\)
\(200\) 914.079 0.323176
\(201\) −4273.89 −1.49979
\(202\) −2734.09 −0.952326
\(203\) 0 0
\(204\) −32.5719 −0.0111789
\(205\) −5341.15 −1.81972
\(206\) 224.769 0.0760212
\(207\) 268.314 0.0900924
\(208\) 1454.53 0.484871
\(209\) 4480.49 1.48288
\(210\) 0 0
\(211\) 2267.20 0.739717 0.369859 0.929088i \(-0.379406\pi\)
0.369859 + 0.929088i \(0.379406\pi\)
\(212\) −12.9758 −0.00420370
\(213\) −3632.05 −1.16837
\(214\) −2034.28 −0.649815
\(215\) 4479.87 1.42105
\(216\) 3381.48 1.06519
\(217\) 0 0
\(218\) −1529.05 −0.475047
\(219\) −618.984 −0.190991
\(220\) 97.7975 0.0299705
\(221\) −1298.40 −0.395203
\(222\) 1577.07 0.476785
\(223\) −4945.03 −1.48495 −0.742474 0.669875i \(-0.766348\pi\)
−0.742474 + 0.669875i \(0.766348\pi\)
\(224\) 0 0
\(225\) −220.534 −0.0653433
\(226\) −5546.59 −1.63254
\(227\) 3559.46 1.04075 0.520374 0.853938i \(-0.325792\pi\)
0.520374 + 0.853938i \(0.325792\pi\)
\(228\) 40.0622 0.0116368
\(229\) −6143.40 −1.77278 −0.886391 0.462937i \(-0.846796\pi\)
−0.886391 + 0.462937i \(0.846796\pi\)
\(230\) −1816.99 −0.520907
\(231\) 0 0
\(232\) −651.143 −0.184266
\(233\) −1087.06 −0.305648 −0.152824 0.988253i \(-0.548837\pi\)
−0.152824 + 0.988253i \(0.548837\pi\)
\(234\) −345.697 −0.0965766
\(235\) 7122.03 1.97698
\(236\) −16.5519 −0.00456540
\(237\) −4033.41 −1.10548
\(238\) 0 0
\(239\) 1079.00 0.292027 0.146013 0.989283i \(-0.453356\pi\)
0.146013 + 0.989283i \(0.453356\pi\)
\(240\) −3884.43 −1.04474
\(241\) 989.224 0.264405 0.132202 0.991223i \(-0.457795\pi\)
0.132202 + 0.991223i \(0.457795\pi\)
\(242\) −7455.68 −1.98045
\(243\) −1495.33 −0.394754
\(244\) 70.0629 0.0183824
\(245\) 0 0
\(246\) 5493.09 1.42369
\(247\) 1596.98 0.411391
\(248\) −1412.75 −0.361733
\(249\) −387.250 −0.0985581
\(250\) −3092.09 −0.782243
\(251\) −900.246 −0.226386 −0.113193 0.993573i \(-0.536108\pi\)
−0.113193 + 0.993573i \(0.536108\pi\)
\(252\) 0 0
\(253\) 3111.88 0.773290
\(254\) 5685.61 1.40451
\(255\) 3467.48 0.851537
\(256\) 185.693 0.0453353
\(257\) 3125.18 0.758534 0.379267 0.925287i \(-0.376176\pi\)
0.379267 + 0.925287i \(0.376176\pi\)
\(258\) −4607.31 −1.11178
\(259\) 0 0
\(260\) 34.8580 0.00831462
\(261\) 157.097 0.0372569
\(262\) 4402.35 1.03808
\(263\) 2814.89 0.659976 0.329988 0.943985i \(-0.392955\pi\)
0.329988 + 0.943985i \(0.392955\pi\)
\(264\) 6553.63 1.52783
\(265\) 1381.36 0.320212
\(266\) 0 0
\(267\) −1662.80 −0.381129
\(268\) 111.243 0.0253554
\(269\) −4409.28 −0.999400 −0.499700 0.866199i \(-0.666556\pi\)
−0.499700 + 0.866199i \(0.666556\pi\)
\(270\) 5524.68 1.24526
\(271\) 4417.97 0.990304 0.495152 0.868806i \(-0.335112\pi\)
0.495152 + 0.868806i \(0.335112\pi\)
\(272\) −3766.01 −0.839515
\(273\) 0 0
\(274\) 3690.19 0.813623
\(275\) −2557.73 −0.560862
\(276\) 27.8248 0.00606833
\(277\) −887.577 −0.192525 −0.0962624 0.995356i \(-0.530689\pi\)
−0.0962624 + 0.995356i \(0.530689\pi\)
\(278\) 5698.49 1.22940
\(279\) 340.845 0.0731393
\(280\) 0 0
\(281\) −4286.58 −0.910021 −0.455010 0.890486i \(-0.650364\pi\)
−0.455010 + 0.890486i \(0.650364\pi\)
\(282\) −7324.64 −1.54672
\(283\) 1709.59 0.359097 0.179549 0.983749i \(-0.442536\pi\)
0.179549 + 0.983749i \(0.442536\pi\)
\(284\) 94.5367 0.0197525
\(285\) −4264.88 −0.886419
\(286\) −4009.36 −0.828946
\(287\) 0 0
\(288\) −29.6416 −0.00606474
\(289\) −1551.22 −0.315738
\(290\) −1063.84 −0.215416
\(291\) −869.241 −0.175106
\(292\) 16.1112 0.00322890
\(293\) −1145.01 −0.228301 −0.114151 0.993463i \(-0.536415\pi\)
−0.114151 + 0.993463i \(0.536415\pi\)
\(294\) 0 0
\(295\) 1762.05 0.347765
\(296\) 2674.68 0.525212
\(297\) −9461.90 −1.84860
\(298\) −3384.42 −0.657900
\(299\) 1109.17 0.214532
\(300\) −22.8699 −0.00440131
\(301\) 0 0
\(302\) 6433.56 1.22586
\(303\) −4457.23 −0.845086
\(304\) 4632.06 0.873905
\(305\) −7458.63 −1.40026
\(306\) 895.069 0.167215
\(307\) −1079.39 −0.200665 −0.100333 0.994954i \(-0.531991\pi\)
−0.100333 + 0.994954i \(0.531991\pi\)
\(308\) 0 0
\(309\) 366.427 0.0674605
\(310\) −2308.16 −0.422885
\(311\) −2313.90 −0.421895 −0.210947 0.977497i \(-0.567655\pi\)
−0.210947 + 0.977497i \(0.567655\pi\)
\(312\) 2335.91 0.423862
\(313\) 7653.19 1.38206 0.691029 0.722827i \(-0.257158\pi\)
0.691029 + 0.722827i \(0.257158\pi\)
\(314\) 3385.99 0.608542
\(315\) 0 0
\(316\) 104.984 0.0186892
\(317\) −3657.23 −0.647982 −0.323991 0.946060i \(-0.605025\pi\)
−0.323991 + 0.946060i \(0.605025\pi\)
\(318\) −1420.65 −0.250523
\(319\) 1821.99 0.319787
\(320\) −6488.29 −1.13346
\(321\) −3316.36 −0.576640
\(322\) 0 0
\(323\) −4134.86 −0.712291
\(324\) −66.9170 −0.0114741
\(325\) −911.653 −0.155598
\(326\) 6988.47 1.18729
\(327\) −2492.72 −0.421553
\(328\) 9316.18 1.56829
\(329\) 0 0
\(330\) 10707.3 1.78612
\(331\) 3237.92 0.537681 0.268841 0.963185i \(-0.413360\pi\)
0.268841 + 0.963185i \(0.413360\pi\)
\(332\) 10.0795 0.00166622
\(333\) −645.303 −0.106193
\(334\) −5756.84 −0.943115
\(335\) −11842.5 −1.93142
\(336\) 0 0
\(337\) −7976.89 −1.28940 −0.644702 0.764434i \(-0.723018\pi\)
−0.644702 + 0.764434i \(0.723018\pi\)
\(338\) 4831.78 0.777557
\(339\) −9042.29 −1.44870
\(340\) −90.2533 −0.0143961
\(341\) 3953.09 0.627777
\(342\) −1100.90 −0.174064
\(343\) 0 0
\(344\) −7813.92 −1.22470
\(345\) −2962.13 −0.462248
\(346\) 8156.25 1.26729
\(347\) −8355.93 −1.29271 −0.646354 0.763037i \(-0.723707\pi\)
−0.646354 + 0.763037i \(0.723707\pi\)
\(348\) 16.2913 0.00250950
\(349\) −5544.56 −0.850411 −0.425205 0.905097i \(-0.639798\pi\)
−0.425205 + 0.905097i \(0.639798\pi\)
\(350\) 0 0
\(351\) −3372.51 −0.512852
\(352\) −343.780 −0.0520555
\(353\) 1682.79 0.253727 0.126864 0.991920i \(-0.459509\pi\)
0.126864 + 0.991920i \(0.459509\pi\)
\(354\) −1812.18 −0.272080
\(355\) −10064.0 −1.50463
\(356\) 43.2801 0.00644337
\(357\) 0 0
\(358\) 662.990 0.0978774
\(359\) 7143.13 1.05014 0.525069 0.851059i \(-0.324039\pi\)
0.525069 + 0.851059i \(0.324039\pi\)
\(360\) 1565.75 0.229229
\(361\) −1773.27 −0.258531
\(362\) −7431.13 −1.07893
\(363\) −12154.6 −1.75743
\(364\) 0 0
\(365\) −1715.14 −0.245958
\(366\) 7670.81 1.09552
\(367\) 4456.16 0.633814 0.316907 0.948457i \(-0.397356\pi\)
0.316907 + 0.948457i \(0.397356\pi\)
\(368\) 3217.15 0.455722
\(369\) −2247.65 −0.317095
\(370\) 4369.90 0.614001
\(371\) 0 0
\(372\) 35.3465 0.00492642
\(373\) 2508.28 0.348187 0.174094 0.984729i \(-0.444300\pi\)
0.174094 + 0.984729i \(0.444300\pi\)
\(374\) 10380.9 1.43525
\(375\) −5040.85 −0.694155
\(376\) −12422.4 −1.70383
\(377\) 649.414 0.0887176
\(378\) 0 0
\(379\) −12733.9 −1.72585 −0.862926 0.505331i \(-0.831371\pi\)
−0.862926 + 0.505331i \(0.831371\pi\)
\(380\) 111.008 0.0149858
\(381\) 9268.91 1.24635
\(382\) −4356.95 −0.583563
\(383\) 1027.19 0.137042 0.0685209 0.997650i \(-0.478172\pi\)
0.0685209 + 0.997650i \(0.478172\pi\)
\(384\) 6876.23 0.913806
\(385\) 0 0
\(386\) −2899.82 −0.382376
\(387\) 1885.21 0.247624
\(388\) 22.6250 0.00296034
\(389\) 5153.35 0.671684 0.335842 0.941918i \(-0.390979\pi\)
0.335842 + 0.941918i \(0.390979\pi\)
\(390\) 3816.42 0.495518
\(391\) −2871.83 −0.371444
\(392\) 0 0
\(393\) 7176.90 0.921187
\(394\) −9376.17 −1.19890
\(395\) −11176.2 −1.42363
\(396\) 41.1550 0.00522251
\(397\) −6250.95 −0.790242 −0.395121 0.918629i \(-0.629297\pi\)
−0.395121 + 0.918629i \(0.629297\pi\)
\(398\) −85.2610 −0.0107381
\(399\) 0 0
\(400\) −2644.25 −0.330532
\(401\) 11083.8 1.38029 0.690145 0.723671i \(-0.257547\pi\)
0.690145 + 0.723671i \(0.257547\pi\)
\(402\) 12179.4 1.51108
\(403\) 1409.00 0.174162
\(404\) 116.015 0.0142870
\(405\) 7123.74 0.874028
\(406\) 0 0
\(407\) −7484.17 −0.911490
\(408\) −6048.07 −0.733883
\(409\) 3375.43 0.408079 0.204039 0.978963i \(-0.434593\pi\)
0.204039 + 0.978963i \(0.434593\pi\)
\(410\) 15220.8 1.83342
\(411\) 6015.91 0.722002
\(412\) −9.53754 −0.00114049
\(413\) 0 0
\(414\) −764.621 −0.0907707
\(415\) −1073.03 −0.126923
\(416\) −122.534 −0.0144416
\(417\) 9289.91 1.09096
\(418\) −12768.2 −1.49405
\(419\) −6486.50 −0.756292 −0.378146 0.925746i \(-0.623438\pi\)
−0.378146 + 0.925746i \(0.623438\pi\)
\(420\) 0 0
\(421\) −10938.6 −1.26631 −0.633153 0.774026i \(-0.718240\pi\)
−0.633153 + 0.774026i \(0.718240\pi\)
\(422\) −6460.89 −0.745287
\(423\) 2997.08 0.344499
\(424\) −2409.40 −0.275969
\(425\) 2360.42 0.269406
\(426\) 10350.3 1.17717
\(427\) 0 0
\(428\) 86.3200 0.00974867
\(429\) −6536.23 −0.735600
\(430\) −12766.4 −1.43174
\(431\) 4124.34 0.460934 0.230467 0.973080i \(-0.425975\pi\)
0.230467 + 0.973080i \(0.425975\pi\)
\(432\) −9781.98 −1.08943
\(433\) 1561.41 0.173295 0.0866473 0.996239i \(-0.472385\pi\)
0.0866473 + 0.996239i \(0.472385\pi\)
\(434\) 0 0
\(435\) −1734.31 −0.191158
\(436\) 64.8818 0.00712677
\(437\) 3532.25 0.386660
\(438\) 1763.93 0.192429
\(439\) 15712.7 1.70826 0.854130 0.520060i \(-0.174091\pi\)
0.854130 + 0.520060i \(0.174091\pi\)
\(440\) 18159.4 1.96754
\(441\) 0 0
\(442\) 3700.08 0.398178
\(443\) 12763.5 1.36888 0.684439 0.729070i \(-0.260047\pi\)
0.684439 + 0.729070i \(0.260047\pi\)
\(444\) −66.9195 −0.00715284
\(445\) −4607.44 −0.490817
\(446\) 14092.0 1.49613
\(447\) −5517.42 −0.583815
\(448\) 0 0
\(449\) 3117.88 0.327710 0.163855 0.986484i \(-0.447607\pi\)
0.163855 + 0.986484i \(0.447607\pi\)
\(450\) 628.460 0.0658353
\(451\) −26068.1 −2.72172
\(452\) 235.357 0.0244918
\(453\) 10488.3 1.08782
\(454\) −10143.5 −1.04858
\(455\) 0 0
\(456\) 7438.91 0.763945
\(457\) −8479.41 −0.867943 −0.433972 0.900927i \(-0.642888\pi\)
−0.433972 + 0.900927i \(0.642888\pi\)
\(458\) 17507.0 1.78613
\(459\) 8732.00 0.887962
\(460\) 77.0997 0.00781477
\(461\) −9253.32 −0.934859 −0.467430 0.884030i \(-0.654820\pi\)
−0.467430 + 0.884030i \(0.654820\pi\)
\(462\) 0 0
\(463\) 521.395 0.0523354 0.0261677 0.999658i \(-0.491670\pi\)
0.0261677 + 0.999658i \(0.491670\pi\)
\(464\) 1883.63 0.188460
\(465\) −3762.85 −0.375265
\(466\) 3097.83 0.307949
\(467\) −4337.35 −0.429783 −0.214891 0.976638i \(-0.568940\pi\)
−0.214891 + 0.976638i \(0.568940\pi\)
\(468\) 14.6689 0.00144887
\(469\) 0 0
\(470\) −20295.8 −1.99186
\(471\) 5519.98 0.540015
\(472\) −3073.42 −0.299715
\(473\) 21864.5 2.12544
\(474\) 11494.1 1.11380
\(475\) −2903.24 −0.280441
\(476\) 0 0
\(477\) 581.300 0.0557986
\(478\) −3074.84 −0.294226
\(479\) 11258.1 1.07390 0.536949 0.843615i \(-0.319577\pi\)
0.536949 + 0.843615i \(0.319577\pi\)
\(480\) 327.236 0.0311171
\(481\) −2667.58 −0.252872
\(482\) −2819.02 −0.266395
\(483\) 0 0
\(484\) 316.365 0.0297112
\(485\) −2408.58 −0.225501
\(486\) 4261.26 0.397726
\(487\) 4353.54 0.405088 0.202544 0.979273i \(-0.435079\pi\)
0.202544 + 0.979273i \(0.435079\pi\)
\(488\) 13009.6 1.20679
\(489\) 11392.9 1.05359
\(490\) 0 0
\(491\) −8458.77 −0.777472 −0.388736 0.921349i \(-0.627088\pi\)
−0.388736 + 0.921349i \(0.627088\pi\)
\(492\) −233.087 −0.0213585
\(493\) −1681.44 −0.153607
\(494\) −4550.96 −0.414489
\(495\) −4381.20 −0.397819
\(496\) 4086.82 0.369967
\(497\) 0 0
\(498\) 1103.56 0.0993002
\(499\) 14850.9 1.33230 0.666149 0.745819i \(-0.267942\pi\)
0.666149 + 0.745819i \(0.267942\pi\)
\(500\) 131.206 0.0117354
\(501\) −9385.04 −0.836912
\(502\) 2565.45 0.228091
\(503\) 1686.45 0.149493 0.0747464 0.997203i \(-0.476185\pi\)
0.0747464 + 0.997203i \(0.476185\pi\)
\(504\) 0 0
\(505\) −12350.5 −1.08830
\(506\) −8868.00 −0.779112
\(507\) 7876.97 0.689997
\(508\) −241.256 −0.0210709
\(509\) 11113.5 0.967773 0.483887 0.875131i \(-0.339225\pi\)
0.483887 + 0.875131i \(0.339225\pi\)
\(510\) −9881.36 −0.857949
\(511\) 0 0
\(512\) 11311.8 0.976395
\(513\) −10740.0 −0.924336
\(514\) −8905.90 −0.764246
\(515\) 1015.33 0.0868754
\(516\) 195.501 0.0166792
\(517\) 34759.8 2.95694
\(518\) 0 0
\(519\) 13296.7 1.12458
\(520\) 6472.57 0.545848
\(521\) 15931.1 1.33964 0.669820 0.742523i \(-0.266371\pi\)
0.669820 + 0.742523i \(0.266371\pi\)
\(522\) −447.682 −0.0375374
\(523\) 7960.43 0.665555 0.332778 0.943005i \(-0.392014\pi\)
0.332778 + 0.943005i \(0.392014\pi\)
\(524\) −186.804 −0.0155736
\(525\) 0 0
\(526\) −8021.67 −0.664945
\(527\) −3648.14 −0.301548
\(528\) −18958.4 −1.56261
\(529\) −9713.71 −0.798366
\(530\) −3936.49 −0.322623
\(531\) 741.503 0.0605998
\(532\) 0 0
\(533\) −9291.45 −0.755079
\(534\) 4738.51 0.383999
\(535\) −9189.30 −0.742594
\(536\) 20656.0 1.66456
\(537\) 1080.83 0.0868555
\(538\) 12565.2 1.00692
\(539\) 0 0
\(540\) −234.427 −0.0186817
\(541\) 13818.9 1.09819 0.549096 0.835760i \(-0.314972\pi\)
0.549096 + 0.835760i \(0.314972\pi\)
\(542\) −12590.0 −0.997760
\(543\) −12114.5 −0.957430
\(544\) 317.261 0.0250045
\(545\) −6907.07 −0.542874
\(546\) 0 0
\(547\) 22093.3 1.72695 0.863474 0.504393i \(-0.168284\pi\)
0.863474 + 0.504393i \(0.168284\pi\)
\(548\) −156.585 −0.0122062
\(549\) −3138.73 −0.244003
\(550\) 7288.82 0.565084
\(551\) 2068.12 0.159900
\(552\) 5166.62 0.398381
\(553\) 0 0
\(554\) 2529.35 0.193974
\(555\) 7124.00 0.544859
\(556\) −241.802 −0.0184437
\(557\) −3110.34 −0.236606 −0.118303 0.992978i \(-0.537745\pi\)
−0.118303 + 0.992978i \(0.537745\pi\)
\(558\) −971.314 −0.0736900
\(559\) 7793.17 0.589653
\(560\) 0 0
\(561\) 16923.4 1.27363
\(562\) 12215.6 0.916873
\(563\) 13284.6 0.994455 0.497227 0.867620i \(-0.334351\pi\)
0.497227 + 0.867620i \(0.334351\pi\)
\(564\) 310.804 0.0232043
\(565\) −25055.2 −1.86563
\(566\) −4871.86 −0.361801
\(567\) 0 0
\(568\) 17553.9 1.29674
\(569\) −6809.18 −0.501680 −0.250840 0.968029i \(-0.580707\pi\)
−0.250840 + 0.968029i \(0.580707\pi\)
\(570\) 12153.7 0.893093
\(571\) −13471.8 −0.987352 −0.493676 0.869646i \(-0.664347\pi\)
−0.493676 + 0.869646i \(0.664347\pi\)
\(572\) 170.128 0.0124361
\(573\) −7102.88 −0.517849
\(574\) 0 0
\(575\) −2016.41 −0.146244
\(576\) −2730.39 −0.197511
\(577\) −5331.06 −0.384636 −0.192318 0.981333i \(-0.561601\pi\)
−0.192318 + 0.981333i \(0.561601\pi\)
\(578\) 4420.56 0.318116
\(579\) −4727.41 −0.339317
\(580\) 45.1416 0.00323173
\(581\) 0 0
\(582\) 2477.10 0.176424
\(583\) 6741.87 0.478936
\(584\) 2991.59 0.211974
\(585\) −1561.59 −0.110366
\(586\) 3262.96 0.230020
\(587\) −3333.96 −0.234425 −0.117212 0.993107i \(-0.537396\pi\)
−0.117212 + 0.993107i \(0.537396\pi\)
\(588\) 0 0
\(589\) 4487.09 0.313900
\(590\) −5021.36 −0.350383
\(591\) −15285.4 −1.06389
\(592\) −7737.34 −0.537167
\(593\) 23405.3 1.62081 0.810404 0.585872i \(-0.199248\pi\)
0.810404 + 0.585872i \(0.199248\pi\)
\(594\) 26963.8 1.86252
\(595\) 0 0
\(596\) 143.610 0.00986998
\(597\) −138.996 −0.00952886
\(598\) −3160.83 −0.216147
\(599\) −10352.1 −0.706133 −0.353067 0.935598i \(-0.614861\pi\)
−0.353067 + 0.935598i \(0.614861\pi\)
\(600\) −4246.57 −0.288943
\(601\) −15171.1 −1.02969 −0.514843 0.857285i \(-0.672150\pi\)
−0.514843 + 0.857285i \(0.672150\pi\)
\(602\) 0 0
\(603\) −4983.54 −0.336560
\(604\) −272.994 −0.0183907
\(605\) −33679.0 −2.26322
\(606\) 12701.9 0.851449
\(607\) 20823.0 1.39239 0.696193 0.717854i \(-0.254876\pi\)
0.696193 + 0.717854i \(0.254876\pi\)
\(608\) −390.219 −0.0260287
\(609\) 0 0
\(610\) 21255.0 1.41081
\(611\) 12389.5 0.820334
\(612\) −37.9802 −0.00250859
\(613\) −19071.6 −1.25660 −0.628299 0.777972i \(-0.716249\pi\)
−0.628299 + 0.777972i \(0.716249\pi\)
\(614\) 3075.97 0.202176
\(615\) 24813.6 1.62696
\(616\) 0 0
\(617\) 15200.4 0.991806 0.495903 0.868378i \(-0.334837\pi\)
0.495903 + 0.868378i \(0.334837\pi\)
\(618\) −1044.22 −0.0679684
\(619\) −4358.76 −0.283026 −0.141513 0.989936i \(-0.545197\pi\)
−0.141513 + 0.989936i \(0.545197\pi\)
\(620\) 97.9414 0.00634423
\(621\) −7459.39 −0.482021
\(622\) 6593.98 0.425071
\(623\) 0 0
\(624\) −6757.34 −0.433510
\(625\) −19056.5 −1.21961
\(626\) −21809.5 −1.39246
\(627\) −20815.2 −1.32580
\(628\) −143.677 −0.00912950
\(629\) 6906.83 0.437827
\(630\) 0 0
\(631\) −2580.66 −0.162812 −0.0814062 0.996681i \(-0.525941\pi\)
−0.0814062 + 0.996681i \(0.525941\pi\)
\(632\) 19493.8 1.22693
\(633\) −10532.8 −0.661361
\(634\) 10422.1 0.652861
\(635\) 25683.2 1.60505
\(636\) 60.2823 0.00375841
\(637\) 0 0
\(638\) −5192.18 −0.322195
\(639\) −4235.12 −0.262189
\(640\) 19053.3 1.17680
\(641\) 19858.9 1.22368 0.611840 0.790982i \(-0.290430\pi\)
0.611840 + 0.790982i \(0.290430\pi\)
\(642\) 9450.72 0.580981
\(643\) −17371.9 −1.06545 −0.532723 0.846290i \(-0.678831\pi\)
−0.532723 + 0.846290i \(0.678831\pi\)
\(644\) 0 0
\(645\) −20812.3 −1.27052
\(646\) 11783.2 0.717654
\(647\) −3275.07 −0.199005 −0.0995024 0.995037i \(-0.531725\pi\)
−0.0995024 + 0.995037i \(0.531725\pi\)
\(648\) −12425.4 −0.753266
\(649\) 8599.88 0.520146
\(650\) 2597.96 0.156770
\(651\) 0 0
\(652\) −296.540 −0.0178120
\(653\) 20726.5 1.24210 0.621049 0.783772i \(-0.286707\pi\)
0.621049 + 0.783772i \(0.286707\pi\)
\(654\) 7103.56 0.424727
\(655\) 19886.4 1.18630
\(656\) −26949.9 −1.60399
\(657\) −721.762 −0.0428594
\(658\) 0 0
\(659\) −18404.6 −1.08792 −0.543960 0.839111i \(-0.683076\pi\)
−0.543960 + 0.839111i \(0.683076\pi\)
\(660\) −454.341 −0.0267958
\(661\) 7146.10 0.420501 0.210250 0.977648i \(-0.432572\pi\)
0.210250 + 0.977648i \(0.432572\pi\)
\(662\) −9227.19 −0.541729
\(663\) 6032.02 0.353340
\(664\) 1871.61 0.109386
\(665\) 0 0
\(666\) 1838.94 0.106993
\(667\) 1436.39 0.0833841
\(668\) 244.279 0.0141488
\(669\) 22973.3 1.32765
\(670\) 33747.9 1.94596
\(671\) −36402.7 −2.09435
\(672\) 0 0
\(673\) −26819.0 −1.53610 −0.768051 0.640389i \(-0.778773\pi\)
−0.768051 + 0.640389i \(0.778773\pi\)
\(674\) 22731.9 1.29911
\(675\) 6131.05 0.349606
\(676\) −205.026 −0.0116651
\(677\) 20093.1 1.14068 0.570340 0.821408i \(-0.306811\pi\)
0.570340 + 0.821408i \(0.306811\pi\)
\(678\) 25768.0 1.45961
\(679\) 0 0
\(680\) −16758.6 −0.945092
\(681\) −16536.3 −0.930505
\(682\) −11265.2 −0.632503
\(683\) −6876.22 −0.385229 −0.192614 0.981275i \(-0.561697\pi\)
−0.192614 + 0.981275i \(0.561697\pi\)
\(684\) 46.7143 0.00261135
\(685\) 16669.5 0.929791
\(686\) 0 0
\(687\) 28540.6 1.58500
\(688\) 22604.1 1.25258
\(689\) 2403.01 0.132870
\(690\) 8441.24 0.465728
\(691\) −17332.7 −0.954224 −0.477112 0.878842i \(-0.658316\pi\)
−0.477112 + 0.878842i \(0.658316\pi\)
\(692\) −346.092 −0.0190122
\(693\) 0 0
\(694\) 23812.1 1.30244
\(695\) 25741.4 1.40493
\(696\) 3025.04 0.164747
\(697\) 24057.1 1.30736
\(698\) 15800.5 0.856814
\(699\) 5050.21 0.273271
\(700\) 0 0
\(701\) 11127.5 0.599545 0.299772 0.954011i \(-0.403089\pi\)
0.299772 + 0.954011i \(0.403089\pi\)
\(702\) 9610.71 0.516714
\(703\) −8495.15 −0.455762
\(704\) −31666.8 −1.69530
\(705\) −33087.1 −1.76756
\(706\) −4795.48 −0.255638
\(707\) 0 0
\(708\) 76.8957 0.00408180
\(709\) 17432.0 0.923374 0.461687 0.887043i \(-0.347244\pi\)
0.461687 + 0.887043i \(0.347244\pi\)
\(710\) 28679.7 1.51596
\(711\) −4703.13 −0.248075
\(712\) 8036.42 0.423002
\(713\) 3116.46 0.163692
\(714\) 0 0
\(715\) −18111.2 −0.947302
\(716\) −28.1325 −0.00146838
\(717\) −5012.73 −0.261093
\(718\) −20355.9 −1.05805
\(719\) −21082.7 −1.09353 −0.546767 0.837285i \(-0.684142\pi\)
−0.546767 + 0.837285i \(0.684142\pi\)
\(720\) −4529.41 −0.234446
\(721\) 0 0
\(722\) 5053.32 0.260478
\(723\) −4595.68 −0.236397
\(724\) 315.323 0.0161863
\(725\) −1180.60 −0.0604779
\(726\) 34637.1 1.77067
\(727\) 25839.1 1.31818 0.659091 0.752063i \(-0.270941\pi\)
0.659091 + 0.752063i \(0.270941\pi\)
\(728\) 0 0
\(729\) 21888.5 1.11205
\(730\) 4887.67 0.247810
\(731\) −20177.9 −1.02094
\(732\) −325.494 −0.0164352
\(733\) −1278.54 −0.0644256 −0.0322128 0.999481i \(-0.510255\pi\)
−0.0322128 + 0.999481i \(0.510255\pi\)
\(734\) −12698.8 −0.638586
\(735\) 0 0
\(736\) −271.023 −0.0135734
\(737\) −57798.6 −2.88879
\(738\) 6405.19 0.319483
\(739\) 4224.54 0.210287 0.105144 0.994457i \(-0.466470\pi\)
0.105144 + 0.994457i \(0.466470\pi\)
\(740\) −185.427 −0.00921140
\(741\) −7419.17 −0.367814
\(742\) 0 0
\(743\) −17992.3 −0.888390 −0.444195 0.895930i \(-0.646510\pi\)
−0.444195 + 0.895930i \(0.646510\pi\)
\(744\) 6563.27 0.323416
\(745\) −15288.2 −0.751835
\(746\) −7147.91 −0.350809
\(747\) −451.550 −0.0221170
\(748\) −440.491 −0.0215320
\(749\) 0 0
\(750\) 14365.0 0.699381
\(751\) −10082.4 −0.489895 −0.244948 0.969536i \(-0.578771\pi\)
−0.244948 + 0.969536i \(0.578771\pi\)
\(752\) 35935.7 1.74261
\(753\) 4182.30 0.202406
\(754\) −1850.65 −0.0893856
\(755\) 29061.9 1.40089
\(756\) 0 0
\(757\) 10806.5 0.518851 0.259425 0.965763i \(-0.416467\pi\)
0.259425 + 0.965763i \(0.416467\pi\)
\(758\) 36288.2 1.73885
\(759\) −14457.0 −0.691377
\(760\) 20612.5 0.983806
\(761\) 30710.4 1.46288 0.731439 0.681907i \(-0.238849\pi\)
0.731439 + 0.681907i \(0.238849\pi\)
\(762\) −26413.8 −1.25574
\(763\) 0 0
\(764\) 184.877 0.00875475
\(765\) 4043.23 0.191089
\(766\) −2927.21 −0.138074
\(767\) 3065.26 0.144303
\(768\) −862.682 −0.0405330
\(769\) −10757.1 −0.504436 −0.252218 0.967670i \(-0.581160\pi\)
−0.252218 + 0.967670i \(0.581160\pi\)
\(770\) 0 0
\(771\) −14518.8 −0.678185
\(772\) 123.047 0.00573649
\(773\) 18077.2 0.841125 0.420563 0.907263i \(-0.361833\pi\)
0.420563 + 0.907263i \(0.361833\pi\)
\(774\) −5372.33 −0.249489
\(775\) −2561.49 −0.118725
\(776\) 4201.10 0.194344
\(777\) 0 0
\(778\) −14685.6 −0.676742
\(779\) −29589.4 −1.36091
\(780\) −161.941 −0.00743387
\(781\) −49118.6 −2.25045
\(782\) 8183.92 0.374241
\(783\) −4367.44 −0.199335
\(784\) 0 0
\(785\) 15295.3 0.695430
\(786\) −20452.2 −0.928123
\(787\) 31543.7 1.42873 0.714366 0.699773i \(-0.246715\pi\)
0.714366 + 0.699773i \(0.246715\pi\)
\(788\) 397.857 0.0179861
\(789\) −13077.3 −0.590066
\(790\) 31848.9 1.43435
\(791\) 0 0
\(792\) 7641.81 0.342854
\(793\) −12975.0 −0.581030
\(794\) 17813.5 0.796192
\(795\) −6417.42 −0.286293
\(796\) 3.61786 0.000161095 0
\(797\) −18280.9 −0.812477 −0.406239 0.913767i \(-0.633160\pi\)
−0.406239 + 0.913767i \(0.633160\pi\)
\(798\) 0 0
\(799\) −32078.4 −1.42034
\(800\) 222.760 0.00984470
\(801\) −1938.89 −0.0855273
\(802\) −31585.6 −1.39068
\(803\) −8370.93 −0.367875
\(804\) −516.805 −0.0226695
\(805\) 0 0
\(806\) −4015.26 −0.175473
\(807\) 20484.4 0.893536
\(808\) 21542.1 0.937932
\(809\) 21776.3 0.946372 0.473186 0.880963i \(-0.343104\pi\)
0.473186 + 0.880963i \(0.343104\pi\)
\(810\) −20300.7 −0.880609
\(811\) −17035.7 −0.737612 −0.368806 0.929506i \(-0.620233\pi\)
−0.368806 + 0.929506i \(0.620233\pi\)
\(812\) 0 0
\(813\) −20524.7 −0.885404
\(814\) 21327.8 0.918353
\(815\) 31568.5 1.35681
\(816\) 17495.9 0.750588
\(817\) 24818.0 1.06276
\(818\) −9619.03 −0.411151
\(819\) 0 0
\(820\) −645.860 −0.0275054
\(821\) −45120.7 −1.91805 −0.959027 0.283314i \(-0.908566\pi\)
−0.959027 + 0.283314i \(0.908566\pi\)
\(822\) −17143.7 −0.727438
\(823\) 22261.7 0.942883 0.471442 0.881897i \(-0.343734\pi\)
0.471442 + 0.881897i \(0.343734\pi\)
\(824\) −1770.97 −0.0748721
\(825\) 11882.5 0.501451
\(826\) 0 0
\(827\) 10280.4 0.432266 0.216133 0.976364i \(-0.430655\pi\)
0.216133 + 0.976364i \(0.430655\pi\)
\(828\) 32.4450 0.00136176
\(829\) −28509.2 −1.19441 −0.597206 0.802088i \(-0.703722\pi\)
−0.597206 + 0.802088i \(0.703722\pi\)
\(830\) 3057.84 0.127878
\(831\) 4123.45 0.172131
\(832\) −11287.0 −0.470321
\(833\) 0 0
\(834\) −26473.7 −1.09917
\(835\) −26005.0 −1.07777
\(836\) 541.788 0.0224140
\(837\) −9475.82 −0.391317
\(838\) 18484.7 0.761986
\(839\) −4746.97 −0.195332 −0.0976661 0.995219i \(-0.531138\pi\)
−0.0976661 + 0.995219i \(0.531138\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 31172.0 1.27584
\(843\) 19914.3 0.813625
\(844\) 274.153 0.0111810
\(845\) 21826.3 0.888576
\(846\) −8540.85 −0.347093
\(847\) 0 0
\(848\) 6969.94 0.282251
\(849\) −7942.30 −0.321059
\(850\) −6726.56 −0.271434
\(851\) −5900.22 −0.237670
\(852\) −439.193 −0.0176602
\(853\) −35313.3 −1.41747 −0.708736 0.705473i \(-0.750734\pi\)
−0.708736 + 0.705473i \(0.750734\pi\)
\(854\) 0 0
\(855\) −4973.03 −0.198917
\(856\) 16028.2 0.639993
\(857\) −32142.8 −1.28119 −0.640594 0.767880i \(-0.721312\pi\)
−0.640594 + 0.767880i \(0.721312\pi\)
\(858\) 18626.4 0.741138
\(859\) −37568.2 −1.49221 −0.746107 0.665826i \(-0.768079\pi\)
−0.746107 + 0.665826i \(0.768079\pi\)
\(860\) 541.713 0.0214794
\(861\) 0 0
\(862\) −11753.2 −0.464404
\(863\) 3416.95 0.134779 0.0673896 0.997727i \(-0.478533\pi\)
0.0673896 + 0.997727i \(0.478533\pi\)
\(864\) 824.064 0.0324482
\(865\) 36843.6 1.44823
\(866\) −4449.59 −0.174599
\(867\) 7206.57 0.282293
\(868\) 0 0
\(869\) −54546.5 −2.12930
\(870\) 4942.31 0.192598
\(871\) −20601.2 −0.801429
\(872\) 12047.5 0.467867
\(873\) −1013.57 −0.0392947
\(874\) −10065.9 −0.389571
\(875\) 0 0
\(876\) −74.8485 −0.00288687
\(877\) −11891.0 −0.457847 −0.228923 0.973444i \(-0.573521\pi\)
−0.228923 + 0.973444i \(0.573521\pi\)
\(878\) −44776.8 −1.72112
\(879\) 5319.42 0.204118
\(880\) −52531.7 −2.01232
\(881\) −20042.7 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(882\) 0 0
\(883\) −18042.8 −0.687641 −0.343821 0.939035i \(-0.611721\pi\)
−0.343821 + 0.939035i \(0.611721\pi\)
\(884\) −157.004 −0.00597356
\(885\) −8186.02 −0.310927
\(886\) −36372.5 −1.37919
\(887\) −2247.91 −0.0850929 −0.0425464 0.999094i \(-0.513547\pi\)
−0.0425464 + 0.999094i \(0.513547\pi\)
\(888\) −12425.9 −0.469578
\(889\) 0 0
\(890\) 13129.9 0.494512
\(891\) 34768.2 1.30727
\(892\) −597.960 −0.0224453
\(893\) 39455.3 1.47852
\(894\) 15723.1 0.588211
\(895\) 2994.88 0.111852
\(896\) 0 0
\(897\) −5152.91 −0.191807
\(898\) −8885.09 −0.330177
\(899\) 1824.68 0.0676934
\(900\) −26.6673 −0.000987677 0
\(901\) −6221.80 −0.230053
\(902\) 74286.7 2.74222
\(903\) 0 0
\(904\) 43702.1 1.60786
\(905\) −33568.1 −1.23298
\(906\) −29888.6 −1.09601
\(907\) −1798.34 −0.0658354 −0.0329177 0.999458i \(-0.510480\pi\)
−0.0329177 + 0.999458i \(0.510480\pi\)
\(908\) 430.416 0.0157311
\(909\) −5197.32 −0.189642
\(910\) 0 0
\(911\) 4602.02 0.167367 0.0836837 0.996492i \(-0.473331\pi\)
0.0836837 + 0.996492i \(0.473331\pi\)
\(912\) −21519.3 −0.781334
\(913\) −5237.04 −0.189836
\(914\) 24164.0 0.874478
\(915\) 34650.8 1.25194
\(916\) −742.869 −0.0267960
\(917\) 0 0
\(918\) −24883.8 −0.894648
\(919\) 20622.0 0.740215 0.370107 0.928989i \(-0.379321\pi\)
0.370107 + 0.928989i \(0.379321\pi\)
\(920\) 14316.2 0.513033
\(921\) 5014.58 0.179409
\(922\) 26369.4 0.941898
\(923\) −17507.4 −0.624335
\(924\) 0 0
\(925\) 4849.53 0.172380
\(926\) −1485.83 −0.0527294
\(927\) 427.270 0.0151385
\(928\) −158.683 −0.00561316
\(929\) 24061.8 0.849777 0.424889 0.905246i \(-0.360313\pi\)
0.424889 + 0.905246i \(0.360313\pi\)
\(930\) 10723.1 0.378090
\(931\) 0 0
\(932\) −131.449 −0.00461992
\(933\) 10749.8 0.377204
\(934\) 12360.2 0.433019
\(935\) 46893.0 1.64018
\(936\) 2723.78 0.0951169
\(937\) 6581.96 0.229481 0.114740 0.993396i \(-0.463396\pi\)
0.114740 + 0.993396i \(0.463396\pi\)
\(938\) 0 0
\(939\) −35554.7 −1.23566
\(940\) 861.207 0.0298824
\(941\) −23579.7 −0.816873 −0.408436 0.912787i \(-0.633926\pi\)
−0.408436 + 0.912787i \(0.633926\pi\)
\(942\) −15730.4 −0.544081
\(943\) −20551.0 −0.709686
\(944\) 8890.80 0.306537
\(945\) 0 0
\(946\) −62307.8 −2.14144
\(947\) −36368.6 −1.24796 −0.623982 0.781439i \(-0.714486\pi\)
−0.623982 + 0.781439i \(0.714486\pi\)
\(948\) −487.726 −0.0167095
\(949\) −2983.65 −0.102058
\(950\) 8273.42 0.282553
\(951\) 16990.5 0.579343
\(952\) 0 0
\(953\) 44404.3 1.50933 0.754667 0.656108i \(-0.227798\pi\)
0.754667 + 0.656108i \(0.227798\pi\)
\(954\) −1656.55 −0.0562187
\(955\) −19681.4 −0.666884
\(956\) 130.474 0.00441404
\(957\) −8464.51 −0.285913
\(958\) −32082.5 −1.08198
\(959\) 0 0
\(960\) 30142.9 1.01339
\(961\) −25832.1 −0.867111
\(962\) 7601.88 0.254776
\(963\) −3867.02 −0.129401
\(964\) 119.619 0.00399653
\(965\) −13099.2 −0.436971
\(966\) 0 0
\(967\) 21928.3 0.729230 0.364615 0.931158i \(-0.381201\pi\)
0.364615 + 0.931158i \(0.381201\pi\)
\(968\) 58743.9 1.95052
\(969\) 19209.5 0.636840
\(970\) 6863.77 0.227198
\(971\) 6352.95 0.209965 0.104982 0.994474i \(-0.466521\pi\)
0.104982 + 0.994474i \(0.466521\pi\)
\(972\) −180.817 −0.00596678
\(973\) 0 0
\(974\) −12406.4 −0.408138
\(975\) 4235.30 0.139116
\(976\) −37634.1 −1.23426
\(977\) 12792.8 0.418913 0.209456 0.977818i \(-0.432831\pi\)
0.209456 + 0.977818i \(0.432831\pi\)
\(978\) −32466.6 −1.06152
\(979\) −22487.1 −0.734107
\(980\) 0 0
\(981\) −2906.62 −0.0945986
\(982\) 24105.1 0.783326
\(983\) −7536.73 −0.244542 −0.122271 0.992497i \(-0.539018\pi\)
−0.122271 + 0.992497i \(0.539018\pi\)
\(984\) −43280.5 −1.40217
\(985\) −42354.3 −1.37007
\(986\) 4791.65 0.154764
\(987\) 0 0
\(988\) 193.110 0.00621826
\(989\) 17237.1 0.554205
\(990\) 12485.2 0.400814
\(991\) −28088.3 −0.900357 −0.450179 0.892938i \(-0.648640\pi\)
−0.450179 + 0.892938i \(0.648640\pi\)
\(992\) −344.286 −0.0110192
\(993\) −15042.5 −0.480726
\(994\) 0 0
\(995\) −385.144 −0.0122712
\(996\) −46.8269 −0.00148973
\(997\) 44951.9 1.42793 0.713963 0.700184i \(-0.246899\pi\)
0.713963 + 0.700184i \(0.246899\pi\)
\(998\) −42320.9 −1.34233
\(999\) 17940.0 0.568166
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 1421.4.a.e.1.2 5
7.6 odd 2 29.4.a.b.1.2 5
21.20 even 2 261.4.a.f.1.4 5
28.27 even 2 464.4.a.l.1.2 5
35.34 odd 2 725.4.a.c.1.4 5
56.13 odd 2 1856.4.a.y.1.2 5
56.27 even 2 1856.4.a.bb.1.4 5
203.202 odd 2 841.4.a.b.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.2 5 7.6 odd 2
261.4.a.f.1.4 5 21.20 even 2
464.4.a.l.1.2 5 28.27 even 2
725.4.a.c.1.4 5 35.34 odd 2
841.4.a.b.1.4 5 203.202 odd 2
1421.4.a.e.1.2 5 1.1 even 1 trivial
1856.4.a.y.1.2 5 56.13 odd 2
1856.4.a.bb.1.4 5 56.27 even 2