Properties

Label 99.4.a.f.1.2
Level $99$
Weight $4$
Character 99.1
Self dual yes
Analytic conductor $5.841$
Analytic rank $0$
Dimension $2$
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] = [99,4,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, 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("99.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.42443\) of defining polynomial
Character \(\chi\) \(=\) 99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} +31.6977 q^{7} +15.8199 q^{8} +O(q^{10})\) \(q+4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} +31.6977 q^{7} +15.8199 q^{8} -12.6046 q^{10} +11.0000 q^{11} +5.15114 q^{13} +140.244 q^{14} -22.6107 q^{16} -121.942 q^{17} +34.8489 q^{19} -32.9772 q^{20} +48.6687 q^{22} -116.244 q^{23} -116.884 q^{25} +22.7909 q^{26} +366.919 q^{28} +69.4534 q^{29} +140.605 q^{31} -226.598 q^{32} -539.524 q^{34} -90.3023 q^{35} -420.070 q^{37} +154.186 q^{38} -45.0685 q^{40} +322.058 q^{41} +321.035 q^{43} +127.331 q^{44} -514.315 q^{46} +231.408 q^{47} +661.745 q^{49} -517.145 q^{50} +59.6274 q^{52} -4.91916 q^{53} -31.3374 q^{55} +501.453 q^{56} +307.292 q^{58} -406.443 q^{59} -556.431 q^{61} +622.095 q^{62} -821.683 q^{64} -14.6749 q^{65} +84.7452 q^{67} -1411.55 q^{68} -399.536 q^{70} -49.0808 q^{71} +785.884 q^{73} -1858.57 q^{74} +403.395 q^{76} +348.675 q^{77} -383.118 q^{79} +64.4147 q^{80} +1424.92 q^{82} +930.211 q^{83} +347.395 q^{85} +1420.40 q^{86} +174.018 q^{88} +732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} +1023.85 q^{94} -99.2794 q^{95} -1171.49 q^{97} +2927.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8} - 104 q^{10} + 22 q^{11} + 30 q^{13} + 182 q^{14} + 201 q^{16} - 106 q^{17} + 50 q^{19} + 328 q^{20} - 11 q^{22} - 134 q^{23} + 42 q^{25} - 112 q^{26} + 202 q^{28} + 198 q^{29} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} - 328 q^{37} + 72 q^{38} - 1272 q^{40} + 782 q^{41} + 386 q^{43} + 363 q^{44} - 418 q^{46} - 266 q^{47} + 378 q^{49} - 1379 q^{50} + 592 q^{52} + 522 q^{53} + 154 q^{55} + 1062 q^{56} - 390 q^{58} + 172 q^{59} - 778 q^{61} - 568 q^{62} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} + 304 q^{70} - 630 q^{71} + 1296 q^{73} - 2358 q^{74} + 728 q^{76} + 264 q^{77} + 652 q^{79} + 3832 q^{80} - 1070 q^{82} + 324 q^{83} + 616 q^{85} + 1068 q^{86} - 627 q^{88} + 756 q^{89} - 28 q^{91} - 1726 q^{92} + 3722 q^{94} + 156 q^{95} - 452 q^{97} + 4467 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.42443 1.56427 0.782136 0.623108i \(-0.214130\pi\)
0.782136 + 0.623108i \(0.214130\pi\)
\(3\) 0 0
\(4\) 11.5756 1.44695
\(5\) −2.84886 −0.254810 −0.127405 0.991851i \(-0.540665\pi\)
−0.127405 + 0.991851i \(0.540665\pi\)
\(6\) 0 0
\(7\) 31.6977 1.71152 0.855758 0.517377i \(-0.173091\pi\)
0.855758 + 0.517377i \(0.173091\pi\)
\(8\) 15.8199 0.699146
\(9\) 0 0
\(10\) −12.6046 −0.398591
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 5.15114 0.109898 0.0549488 0.998489i \(-0.482500\pi\)
0.0549488 + 0.998489i \(0.482500\pi\)
\(14\) 140.244 2.67728
\(15\) 0 0
\(16\) −22.6107 −0.353293
\(17\) −121.942 −1.73972 −0.869861 0.493297i \(-0.835792\pi\)
−0.869861 + 0.493297i \(0.835792\pi\)
\(18\) 0 0
\(19\) 34.8489 0.420783 0.210391 0.977617i \(-0.432526\pi\)
0.210391 + 0.977617i \(0.432526\pi\)
\(20\) −32.9772 −0.368696
\(21\) 0 0
\(22\) 48.6687 0.471646
\(23\) −116.244 −1.05385 −0.526926 0.849911i \(-0.676656\pi\)
−0.526926 + 0.849911i \(0.676656\pi\)
\(24\) 0 0
\(25\) −116.884 −0.935072
\(26\) 22.7909 0.171910
\(27\) 0 0
\(28\) 366.919 2.47647
\(29\) 69.4534 0.444730 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(30\) 0 0
\(31\) 140.605 0.814623 0.407312 0.913289i \(-0.366466\pi\)
0.407312 + 0.913289i \(0.366466\pi\)
\(32\) −226.598 −1.25179
\(33\) 0 0
\(34\) −539.524 −2.72140
\(35\) −90.3023 −0.436111
\(36\) 0 0
\(37\) −420.070 −1.86646 −0.933232 0.359276i \(-0.883024\pi\)
−0.933232 + 0.359276i \(0.883024\pi\)
\(38\) 154.186 0.658219
\(39\) 0 0
\(40\) −45.0685 −0.178149
\(41\) 322.058 1.22676 0.613378 0.789789i \(-0.289810\pi\)
0.613378 + 0.789789i \(0.289810\pi\)
\(42\) 0 0
\(43\) 321.035 1.13854 0.569272 0.822149i \(-0.307225\pi\)
0.569272 + 0.822149i \(0.307225\pi\)
\(44\) 127.331 0.436271
\(45\) 0 0
\(46\) −514.315 −1.64851
\(47\) 231.408 0.718176 0.359088 0.933304i \(-0.383088\pi\)
0.359088 + 0.933304i \(0.383088\pi\)
\(48\) 0 0
\(49\) 661.745 1.92929
\(50\) −517.145 −1.46271
\(51\) 0 0
\(52\) 59.6274 0.159016
\(53\) −4.91916 −0.0127490 −0.00637452 0.999980i \(-0.502029\pi\)
−0.00637452 + 0.999980i \(0.502029\pi\)
\(54\) 0 0
\(55\) −31.3374 −0.0768280
\(56\) 501.453 1.19660
\(57\) 0 0
\(58\) 307.292 0.695679
\(59\) −406.443 −0.896854 −0.448427 0.893820i \(-0.648016\pi\)
−0.448427 + 0.893820i \(0.648016\pi\)
\(60\) 0 0
\(61\) −556.431 −1.16793 −0.583964 0.811779i \(-0.698499\pi\)
−0.583964 + 0.811779i \(0.698499\pi\)
\(62\) 622.095 1.27429
\(63\) 0 0
\(64\) −821.683 −1.60485
\(65\) −14.6749 −0.0280030
\(66\) 0 0
\(67\) 84.7452 0.154526 0.0772632 0.997011i \(-0.475382\pi\)
0.0772632 + 0.997011i \(0.475382\pi\)
\(68\) −1411.55 −2.51728
\(69\) 0 0
\(70\) −399.536 −0.682196
\(71\) −49.0808 −0.0820398 −0.0410199 0.999158i \(-0.513061\pi\)
−0.0410199 + 0.999158i \(0.513061\pi\)
\(72\) 0 0
\(73\) 785.884 1.26001 0.630005 0.776591i \(-0.283053\pi\)
0.630005 + 0.776591i \(0.283053\pi\)
\(74\) −1858.57 −2.91966
\(75\) 0 0
\(76\) 403.395 0.608850
\(77\) 348.675 0.516041
\(78\) 0 0
\(79\) −383.118 −0.545622 −0.272811 0.962068i \(-0.587953\pi\)
−0.272811 + 0.962068i \(0.587953\pi\)
\(80\) 64.4147 0.0900223
\(81\) 0 0
\(82\) 1424.92 1.91898
\(83\) 930.211 1.23017 0.615084 0.788462i \(-0.289122\pi\)
0.615084 + 0.788462i \(0.289122\pi\)
\(84\) 0 0
\(85\) 347.395 0.443298
\(86\) 1420.40 1.78099
\(87\) 0 0
\(88\) 174.018 0.210800
\(89\) 732.559 0.872484 0.436242 0.899829i \(-0.356309\pi\)
0.436242 + 0.899829i \(0.356309\pi\)
\(90\) 0 0
\(91\) 163.279 0.188092
\(92\) −1345.59 −1.52487
\(93\) 0 0
\(94\) 1023.85 1.12342
\(95\) −99.2794 −0.107220
\(96\) 0 0
\(97\) −1171.49 −1.22626 −0.613128 0.789984i \(-0.710089\pi\)
−0.613128 + 0.789984i \(0.710089\pi\)
\(98\) 2927.84 3.01793
\(99\) 0 0
\(100\) −1353.00 −1.35300
\(101\) 1221.27 1.20318 0.601589 0.798806i \(-0.294535\pi\)
0.601589 + 0.798806i \(0.294535\pi\)
\(102\) 0 0
\(103\) 516.745 0.494334 0.247167 0.968973i \(-0.420500\pi\)
0.247167 + 0.968973i \(0.420500\pi\)
\(104\) 81.4903 0.0768345
\(105\) 0 0
\(106\) −21.7645 −0.0199430
\(107\) 152.025 0.137353 0.0686765 0.997639i \(-0.478122\pi\)
0.0686765 + 0.997639i \(0.478122\pi\)
\(108\) 0 0
\(109\) 2170.32 1.90714 0.953572 0.301164i \(-0.0973752\pi\)
0.953572 + 0.301164i \(0.0973752\pi\)
\(110\) −138.650 −0.120180
\(111\) 0 0
\(112\) −716.708 −0.604666
\(113\) 646.397 0.538123 0.269062 0.963123i \(-0.413286\pi\)
0.269062 + 0.963123i \(0.413286\pi\)
\(114\) 0 0
\(115\) 331.163 0.268532
\(116\) 803.963 0.643501
\(117\) 0 0
\(118\) −1798.28 −1.40292
\(119\) −3865.28 −2.97756
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −2461.89 −1.82696
\(123\) 0 0
\(124\) 1627.58 1.17872
\(125\) 689.093 0.493075
\(126\) 0 0
\(127\) −993.304 −0.694027 −0.347014 0.937860i \(-0.612804\pi\)
−0.347014 + 0.937860i \(0.612804\pi\)
\(128\) −1822.69 −1.25863
\(129\) 0 0
\(130\) −64.9279 −0.0438043
\(131\) −385.814 −0.257318 −0.128659 0.991689i \(-0.541067\pi\)
−0.128659 + 0.991689i \(0.541067\pi\)
\(132\) 0 0
\(133\) 1104.63 0.720177
\(134\) 374.949 0.241721
\(135\) 0 0
\(136\) −1929.11 −1.21632
\(137\) −884.840 −0.551803 −0.275901 0.961186i \(-0.588976\pi\)
−0.275901 + 0.961186i \(0.588976\pi\)
\(138\) 0 0
\(139\) −1091.94 −0.666312 −0.333156 0.942872i \(-0.608114\pi\)
−0.333156 + 0.942872i \(0.608114\pi\)
\(140\) −1045.30 −0.631029
\(141\) 0 0
\(142\) −217.155 −0.128333
\(143\) 56.6626 0.0331354
\(144\) 0 0
\(145\) −197.863 −0.113322
\(146\) 3477.09 1.97100
\(147\) 0 0
\(148\) −4862.55 −2.70067
\(149\) −297.014 −0.163304 −0.0816522 0.996661i \(-0.526020\pi\)
−0.0816522 + 0.996661i \(0.526020\pi\)
\(150\) 0 0
\(151\) −1887.86 −1.01743 −0.508716 0.860935i \(-0.669880\pi\)
−0.508716 + 0.860935i \(0.669880\pi\)
\(152\) 551.304 0.294189
\(153\) 0 0
\(154\) 1542.69 0.807229
\(155\) −400.562 −0.207574
\(156\) 0 0
\(157\) −56.5343 −0.0287384 −0.0143692 0.999897i \(-0.504574\pi\)
−0.0143692 + 0.999897i \(0.504574\pi\)
\(158\) −1695.08 −0.853501
\(159\) 0 0
\(160\) 645.547 0.318968
\(161\) −3684.68 −1.80369
\(162\) 0 0
\(163\) −49.2338 −0.0236582 −0.0118291 0.999930i \(-0.503765\pi\)
−0.0118291 + 0.999930i \(0.503765\pi\)
\(164\) 3728.01 1.77505
\(165\) 0 0
\(166\) 4115.65 1.92432
\(167\) −2068.75 −0.958589 −0.479294 0.877654i \(-0.659107\pi\)
−0.479294 + 0.877654i \(0.659107\pi\)
\(168\) 0 0
\(169\) −2170.47 −0.987923
\(170\) 1537.03 0.693438
\(171\) 0 0
\(172\) 3716.17 1.64741
\(173\) 604.012 0.265446 0.132723 0.991153i \(-0.457628\pi\)
0.132723 + 0.991153i \(0.457628\pi\)
\(174\) 0 0
\(175\) −3704.96 −1.60039
\(176\) −248.718 −0.106522
\(177\) 0 0
\(178\) 3241.15 1.36480
\(179\) 2132.02 0.890251 0.445126 0.895468i \(-0.353159\pi\)
0.445126 + 0.895468i \(0.353159\pi\)
\(180\) 0 0
\(181\) −589.371 −0.242031 −0.121015 0.992651i \(-0.538615\pi\)
−0.121015 + 0.992651i \(0.538615\pi\)
\(182\) 722.418 0.294226
\(183\) 0 0
\(184\) −1838.97 −0.736796
\(185\) 1196.72 0.475593
\(186\) 0 0
\(187\) −1341.36 −0.524546
\(188\) 2678.68 1.03916
\(189\) 0 0
\(190\) −439.255 −0.167720
\(191\) 2160.90 0.818624 0.409312 0.912395i \(-0.365769\pi\)
0.409312 + 0.912395i \(0.365769\pi\)
\(192\) 0 0
\(193\) −1490.91 −0.556052 −0.278026 0.960574i \(-0.589680\pi\)
−0.278026 + 0.960574i \(0.589680\pi\)
\(194\) −5183.18 −1.91820
\(195\) 0 0
\(196\) 7660.08 2.79157
\(197\) 230.529 0.0833732 0.0416866 0.999131i \(-0.486727\pi\)
0.0416866 + 0.999131i \(0.486727\pi\)
\(198\) 0 0
\(199\) 22.4007 0.00797963 0.00398982 0.999992i \(-0.498730\pi\)
0.00398982 + 0.999992i \(0.498730\pi\)
\(200\) −1849.09 −0.653752
\(201\) 0 0
\(202\) 5403.43 1.88210
\(203\) 2201.51 0.761163
\(204\) 0 0
\(205\) −917.497 −0.312589
\(206\) 2286.30 0.773273
\(207\) 0 0
\(208\) −116.471 −0.0388260
\(209\) 383.337 0.126871
\(210\) 0 0
\(211\) −1051.64 −0.343117 −0.171558 0.985174i \(-0.554880\pi\)
−0.171558 + 0.985174i \(0.554880\pi\)
\(212\) −56.9421 −0.0184472
\(213\) 0 0
\(214\) 672.622 0.214857
\(215\) −914.583 −0.290112
\(216\) 0 0
\(217\) 4456.84 1.39424
\(218\) 9602.42 2.98329
\(219\) 0 0
\(220\) −362.749 −0.111166
\(221\) −628.141 −0.191191
\(222\) 0 0
\(223\) 3861.80 1.15966 0.579832 0.814736i \(-0.303118\pi\)
0.579832 + 0.814736i \(0.303118\pi\)
\(224\) −7182.65 −2.14246
\(225\) 0 0
\(226\) 2859.94 0.841771
\(227\) 872.721 0.255174 0.127587 0.991827i \(-0.459277\pi\)
0.127587 + 0.991827i \(0.459277\pi\)
\(228\) 0 0
\(229\) 1841.72 0.531459 0.265730 0.964048i \(-0.414387\pi\)
0.265730 + 0.964048i \(0.414387\pi\)
\(230\) 1465.21 0.420057
\(231\) 0 0
\(232\) 1098.74 0.310931
\(233\) −3932.14 −1.10559 −0.552796 0.833317i \(-0.686439\pi\)
−0.552796 + 0.833317i \(0.686439\pi\)
\(234\) 0 0
\(235\) −659.248 −0.182998
\(236\) −4704.81 −1.29770
\(237\) 0 0
\(238\) −17101.7 −4.65772
\(239\) −4772.10 −1.29155 −0.645777 0.763526i \(-0.723466\pi\)
−0.645777 + 0.763526i \(0.723466\pi\)
\(240\) 0 0
\(241\) 3988.84 1.06616 0.533078 0.846066i \(-0.321035\pi\)
0.533078 + 0.846066i \(0.321035\pi\)
\(242\) 535.356 0.142207
\(243\) 0 0
\(244\) −6441.00 −1.68993
\(245\) −1885.22 −0.491601
\(246\) 0 0
\(247\) 179.511 0.0462431
\(248\) 2224.34 0.569540
\(249\) 0 0
\(250\) 3048.84 0.771303
\(251\) 5474.22 1.37661 0.688306 0.725421i \(-0.258355\pi\)
0.688306 + 0.725421i \(0.258355\pi\)
\(252\) 0 0
\(253\) −1278.69 −0.317749
\(254\) −4394.80 −1.08565
\(255\) 0 0
\(256\) −1490.90 −0.363989
\(257\) 6434.01 1.56164 0.780822 0.624754i \(-0.214801\pi\)
0.780822 + 0.624754i \(0.214801\pi\)
\(258\) 0 0
\(259\) −13315.3 −3.19448
\(260\) −169.870 −0.0405188
\(261\) 0 0
\(262\) −1707.01 −0.402516
\(263\) −7589.00 −1.77931 −0.889654 0.456636i \(-0.849054\pi\)
−0.889654 + 0.456636i \(0.849054\pi\)
\(264\) 0 0
\(265\) 14.0140 0.00324858
\(266\) 4887.35 1.12655
\(267\) 0 0
\(268\) 980.974 0.223591
\(269\) −478.178 −0.108383 −0.0541914 0.998531i \(-0.517258\pi\)
−0.0541914 + 0.998531i \(0.517258\pi\)
\(270\) 0 0
\(271\) −122.323 −0.0274192 −0.0137096 0.999906i \(-0.504364\pi\)
−0.0137096 + 0.999906i \(0.504364\pi\)
\(272\) 2757.20 0.614631
\(273\) 0 0
\(274\) −3914.91 −0.863170
\(275\) −1285.72 −0.281935
\(276\) 0 0
\(277\) 8199.41 1.77854 0.889269 0.457385i \(-0.151214\pi\)
0.889269 + 0.457385i \(0.151214\pi\)
\(278\) −4831.22 −1.04229
\(279\) 0 0
\(280\) −1428.57 −0.304905
\(281\) −6943.79 −1.47413 −0.737067 0.675820i \(-0.763790\pi\)
−0.737067 + 0.675820i \(0.763790\pi\)
\(282\) 0 0
\(283\) 1035.14 0.217429 0.108715 0.994073i \(-0.465327\pi\)
0.108715 + 0.994073i \(0.465327\pi\)
\(284\) −568.139 −0.118707
\(285\) 0 0
\(286\) 250.699 0.0518328
\(287\) 10208.5 2.09961
\(288\) 0 0
\(289\) 9956.85 2.02663
\(290\) −875.430 −0.177266
\(291\) 0 0
\(292\) 9097.06 1.82317
\(293\) 6144.81 1.22520 0.612600 0.790393i \(-0.290124\pi\)
0.612600 + 0.790393i \(0.290124\pi\)
\(294\) 0 0
\(295\) 1157.90 0.228527
\(296\) −6645.45 −1.30493
\(297\) 0 0
\(298\) −1314.12 −0.255452
\(299\) −598.791 −0.115816
\(300\) 0 0
\(301\) 10176.1 1.94864
\(302\) −8352.72 −1.59154
\(303\) 0 0
\(304\) −787.958 −0.148659
\(305\) 1585.19 0.297599
\(306\) 0 0
\(307\) −2186.09 −0.406406 −0.203203 0.979137i \(-0.565135\pi\)
−0.203203 + 0.979137i \(0.565135\pi\)
\(308\) 4036.11 0.746684
\(309\) 0 0
\(310\) −1772.26 −0.324702
\(311\) 7484.83 1.36471 0.682357 0.731019i \(-0.260955\pi\)
0.682357 + 0.731019i \(0.260955\pi\)
\(312\) 0 0
\(313\) −6833.33 −1.23400 −0.617001 0.786962i \(-0.711653\pi\)
−0.617001 + 0.786962i \(0.711653\pi\)
\(314\) −250.132 −0.0449546
\(315\) 0 0
\(316\) −4434.81 −0.789485
\(317\) −924.265 −0.163760 −0.0818800 0.996642i \(-0.526092\pi\)
−0.0818800 + 0.996642i \(0.526092\pi\)
\(318\) 0 0
\(319\) 763.988 0.134091
\(320\) 2340.86 0.408931
\(321\) 0 0
\(322\) −16302.6 −2.82145
\(323\) −4249.54 −0.732046
\(324\) 0 0
\(325\) −602.086 −0.102762
\(326\) −217.831 −0.0370078
\(327\) 0 0
\(328\) 5094.91 0.857681
\(329\) 7335.10 1.22917
\(330\) 0 0
\(331\) −9820.46 −1.63076 −0.815380 0.578927i \(-0.803472\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(332\) 10767.7 1.77999
\(333\) 0 0
\(334\) −9153.02 −1.49949
\(335\) −241.427 −0.0393748
\(336\) 0 0
\(337\) 600.808 0.0971161 0.0485580 0.998820i \(-0.484537\pi\)
0.0485580 + 0.998820i \(0.484537\pi\)
\(338\) −9603.07 −1.54538
\(339\) 0 0
\(340\) 4021.30 0.641428
\(341\) 1546.65 0.245618
\(342\) 0 0
\(343\) 10103.5 1.59049
\(344\) 5078.73 0.796008
\(345\) 0 0
\(346\) 2672.41 0.415230
\(347\) 3143.41 0.486303 0.243152 0.969988i \(-0.421819\pi\)
0.243152 + 0.969988i \(0.421819\pi\)
\(348\) 0 0
\(349\) 720.663 0.110533 0.0552667 0.998472i \(-0.482399\pi\)
0.0552667 + 0.998472i \(0.482399\pi\)
\(350\) −16392.3 −2.50345
\(351\) 0 0
\(352\) −2492.58 −0.377429
\(353\) −1207.12 −0.182007 −0.0910034 0.995851i \(-0.529007\pi\)
−0.0910034 + 0.995851i \(0.529007\pi\)
\(354\) 0 0
\(355\) 139.824 0.0209045
\(356\) 8479.79 1.26244
\(357\) 0 0
\(358\) 9432.99 1.39260
\(359\) −8748.31 −1.28612 −0.643062 0.765814i \(-0.722336\pi\)
−0.643062 + 0.765814i \(0.722336\pi\)
\(360\) 0 0
\(361\) −5644.56 −0.822942
\(362\) −2607.63 −0.378602
\(363\) 0 0
\(364\) 1890.05 0.272158
\(365\) −2238.87 −0.321063
\(366\) 0 0
\(367\) −6730.45 −0.957293 −0.478647 0.878008i \(-0.658872\pi\)
−0.478647 + 0.878008i \(0.658872\pi\)
\(368\) 2628.37 0.372318
\(369\) 0 0
\(370\) 5294.81 0.743956
\(371\) −155.926 −0.0218202
\(372\) 0 0
\(373\) −227.394 −0.0315657 −0.0157828 0.999875i \(-0.505024\pi\)
−0.0157828 + 0.999875i \(0.505024\pi\)
\(374\) −5934.76 −0.820533
\(375\) 0 0
\(376\) 3660.84 0.502110
\(377\) 357.764 0.0488748
\(378\) 0 0
\(379\) 11356.2 1.53913 0.769565 0.638568i \(-0.220473\pi\)
0.769565 + 0.638568i \(0.220473\pi\)
\(380\) −1149.22 −0.155141
\(381\) 0 0
\(382\) 9560.74 1.28055
\(383\) −10753.6 −1.43468 −0.717338 0.696725i \(-0.754640\pi\)
−0.717338 + 0.696725i \(0.754640\pi\)
\(384\) 0 0
\(385\) −993.325 −0.131492
\(386\) −6596.43 −0.869817
\(387\) 0 0
\(388\) −13560.7 −1.77433
\(389\) 11727.1 1.52850 0.764252 0.644918i \(-0.223109\pi\)
0.764252 + 0.644918i \(0.223109\pi\)
\(390\) 0 0
\(391\) 14175.1 1.83341
\(392\) 10468.7 1.34885
\(393\) 0 0
\(394\) 1019.96 0.130418
\(395\) 1091.45 0.139030
\(396\) 0 0
\(397\) −359.905 −0.0454990 −0.0227495 0.999741i \(-0.507242\pi\)
−0.0227495 + 0.999741i \(0.507242\pi\)
\(398\) 99.1105 0.0124823
\(399\) 0 0
\(400\) 2642.83 0.330354
\(401\) 4066.71 0.506438 0.253219 0.967409i \(-0.418511\pi\)
0.253219 + 0.967409i \(0.418511\pi\)
\(402\) 0 0
\(403\) 724.274 0.0895252
\(404\) 14136.9 1.74093
\(405\) 0 0
\(406\) 9740.45 1.19067
\(407\) −4620.77 −0.562760
\(408\) 0 0
\(409\) −13488.8 −1.63076 −0.815379 0.578927i \(-0.803472\pi\)
−0.815379 + 0.578927i \(0.803472\pi\)
\(410\) −4059.40 −0.488975
\(411\) 0 0
\(412\) 5981.62 0.715275
\(413\) −12883.3 −1.53498
\(414\) 0 0
\(415\) −2650.04 −0.313459
\(416\) −1167.24 −0.137569
\(417\) 0 0
\(418\) 1696.05 0.198460
\(419\) 7040.12 0.820841 0.410420 0.911896i \(-0.365382\pi\)
0.410420 + 0.911896i \(0.365382\pi\)
\(420\) 0 0
\(421\) 9171.74 1.06177 0.530883 0.847445i \(-0.321860\pi\)
0.530883 + 0.847445i \(0.321860\pi\)
\(422\) −4652.89 −0.536728
\(423\) 0 0
\(424\) −77.8204 −0.00891343
\(425\) 14253.1 1.62677
\(426\) 0 0
\(427\) −17637.6 −1.99893
\(428\) 1759.77 0.198742
\(429\) 0 0
\(430\) −4046.51 −0.453814
\(431\) −992.995 −0.110976 −0.0554882 0.998459i \(-0.517672\pi\)
−0.0554882 + 0.998459i \(0.517672\pi\)
\(432\) 0 0
\(433\) 3790.21 0.420660 0.210330 0.977630i \(-0.432546\pi\)
0.210330 + 0.977630i \(0.432546\pi\)
\(434\) 19719.0 2.18097
\(435\) 0 0
\(436\) 25122.7 2.75954
\(437\) −4050.98 −0.443443
\(438\) 0 0
\(439\) −5136.97 −0.558483 −0.279242 0.960221i \(-0.590083\pi\)
−0.279242 + 0.960221i \(0.590083\pi\)
\(440\) −495.754 −0.0537139
\(441\) 0 0
\(442\) −2779.16 −0.299075
\(443\) −10676.8 −1.14508 −0.572541 0.819876i \(-0.694042\pi\)
−0.572541 + 0.819876i \(0.694042\pi\)
\(444\) 0 0
\(445\) −2086.96 −0.222317
\(446\) 17086.2 1.81403
\(447\) 0 0
\(448\) −26045.5 −2.74672
\(449\) −10529.9 −1.10676 −0.553379 0.832929i \(-0.686662\pi\)
−0.553379 + 0.832929i \(0.686662\pi\)
\(450\) 0 0
\(451\) 3542.64 0.369881
\(452\) 7482.42 0.778636
\(453\) 0 0
\(454\) 3861.29 0.399162
\(455\) −465.160 −0.0479275
\(456\) 0 0
\(457\) −14072.5 −1.44045 −0.720225 0.693741i \(-0.755961\pi\)
−0.720225 + 0.693741i \(0.755961\pi\)
\(458\) 8148.55 0.831347
\(459\) 0 0
\(460\) 3833.41 0.388551
\(461\) 30.8173 0.00311346 0.00155673 0.999999i \(-0.499504\pi\)
0.00155673 + 0.999999i \(0.499504\pi\)
\(462\) 0 0
\(463\) 17591.3 1.76573 0.882867 0.469622i \(-0.155610\pi\)
0.882867 + 0.469622i \(0.155610\pi\)
\(464\) −1570.39 −0.157120
\(465\) 0 0
\(466\) −17397.5 −1.72945
\(467\) −13273.1 −1.31522 −0.657609 0.753360i \(-0.728432\pi\)
−0.657609 + 0.753360i \(0.728432\pi\)
\(468\) 0 0
\(469\) 2686.23 0.264474
\(470\) −2916.79 −0.286259
\(471\) 0 0
\(472\) −6429.87 −0.627031
\(473\) 3531.39 0.343284
\(474\) 0 0
\(475\) −4073.27 −0.393462
\(476\) −44742.9 −4.30837
\(477\) 0 0
\(478\) −21113.8 −2.02034
\(479\) −2496.68 −0.238155 −0.119077 0.992885i \(-0.537994\pi\)
−0.119077 + 0.992885i \(0.537994\pi\)
\(480\) 0 0
\(481\) −2163.84 −0.205120
\(482\) 17648.3 1.66776
\(483\) 0 0
\(484\) 1400.64 0.131541
\(485\) 3337.41 0.312462
\(486\) 0 0
\(487\) −3464.42 −0.322357 −0.161178 0.986925i \(-0.551529\pi\)
−0.161178 + 0.986925i \(0.551529\pi\)
\(488\) −8802.65 −0.816552
\(489\) 0 0
\(490\) −8341.01 −0.768997
\(491\) 16224.6 1.49125 0.745625 0.666366i \(-0.232151\pi\)
0.745625 + 0.666366i \(0.232151\pi\)
\(492\) 0 0
\(493\) −8469.29 −0.773707
\(494\) 794.236 0.0723367
\(495\) 0 0
\(496\) −3179.17 −0.287800
\(497\) −1555.75 −0.140412
\(498\) 0 0
\(499\) 9993.81 0.896562 0.448281 0.893893i \(-0.352036\pi\)
0.448281 + 0.893893i \(0.352036\pi\)
\(500\) 7976.65 0.713453
\(501\) 0 0
\(502\) 24220.3 2.15340
\(503\) 15334.8 1.35933 0.679667 0.733520i \(-0.262124\pi\)
0.679667 + 0.733520i \(0.262124\pi\)
\(504\) 0 0
\(505\) −3479.23 −0.306581
\(506\) −5657.46 −0.497045
\(507\) 0 0
\(508\) −11498.1 −1.00422
\(509\) 7291.23 0.634927 0.317464 0.948270i \(-0.397169\pi\)
0.317464 + 0.948270i \(0.397169\pi\)
\(510\) 0 0
\(511\) 24910.7 2.15653
\(512\) 7985.14 0.689251
\(513\) 0 0
\(514\) 28466.8 2.44283
\(515\) −1472.13 −0.125961
\(516\) 0 0
\(517\) 2545.49 0.216538
\(518\) −58912.5 −4.99704
\(519\) 0 0
\(520\) −232.154 −0.0195782
\(521\) −16794.3 −1.41223 −0.706114 0.708098i \(-0.749553\pi\)
−0.706114 + 0.708098i \(0.749553\pi\)
\(522\) 0 0
\(523\) −21009.4 −1.75655 −0.878275 0.478157i \(-0.841305\pi\)
−0.878275 + 0.478157i \(0.841305\pi\)
\(524\) −4466.01 −0.372326
\(525\) 0 0
\(526\) −33577.0 −2.78332
\(527\) −17145.6 −1.41722
\(528\) 0 0
\(529\) 1345.73 0.110605
\(530\) 62.0039 0.00508166
\(531\) 0 0
\(532\) 12786.7 1.04206
\(533\) 1658.97 0.134818
\(534\) 0 0
\(535\) −433.097 −0.0349989
\(536\) 1340.66 0.108036
\(537\) 0 0
\(538\) −2115.66 −0.169540
\(539\) 7279.20 0.581702
\(540\) 0 0
\(541\) −16802.8 −1.33532 −0.667662 0.744464i \(-0.732705\pi\)
−0.667662 + 0.744464i \(0.732705\pi\)
\(542\) −541.211 −0.0428911
\(543\) 0 0
\(544\) 27631.9 2.17777
\(545\) −6182.93 −0.485959
\(546\) 0 0
\(547\) 16784.5 1.31198 0.655990 0.754770i \(-0.272251\pi\)
0.655990 + 0.754770i \(0.272251\pi\)
\(548\) −10242.5 −0.798429
\(549\) 0 0
\(550\) −5688.59 −0.441023
\(551\) 2420.37 0.187135
\(552\) 0 0
\(553\) −12144.0 −0.933840
\(554\) 36277.7 2.78212
\(555\) 0 0
\(556\) −12639.9 −0.964117
\(557\) −18127.0 −1.37893 −0.689467 0.724317i \(-0.742155\pi\)
−0.689467 + 0.724317i \(0.742155\pi\)
\(558\) 0 0
\(559\) 1653.70 0.125123
\(560\) 2041.80 0.154075
\(561\) 0 0
\(562\) −30722.3 −2.30595
\(563\) −2090.88 −0.156518 −0.0782592 0.996933i \(-0.524936\pi\)
−0.0782592 + 0.996933i \(0.524936\pi\)
\(564\) 0 0
\(565\) −1841.49 −0.137119
\(566\) 4579.89 0.340119
\(567\) 0 0
\(568\) −776.452 −0.0573578
\(569\) −6249.23 −0.460424 −0.230212 0.973140i \(-0.573942\pi\)
−0.230212 + 0.973140i \(0.573942\pi\)
\(570\) 0 0
\(571\) 6048.79 0.443317 0.221659 0.975124i \(-0.428853\pi\)
0.221659 + 0.975124i \(0.428853\pi\)
\(572\) 655.902 0.0479451
\(573\) 0 0
\(574\) 45166.8 3.28437
\(575\) 13587.1 0.985428
\(576\) 0 0
\(577\) −15729.1 −1.13486 −0.567429 0.823423i \(-0.692062\pi\)
−0.567429 + 0.823423i \(0.692062\pi\)
\(578\) 44053.4 3.17021
\(579\) 0 0
\(580\) −2290.38 −0.163970
\(581\) 29485.6 2.10545
\(582\) 0 0
\(583\) −54.1108 −0.00384398
\(584\) 12432.6 0.880931
\(585\) 0 0
\(586\) 27187.3 1.91655
\(587\) −15620.5 −1.09835 −0.549173 0.835709i \(-0.685057\pi\)
−0.549173 + 0.835709i \(0.685057\pi\)
\(588\) 0 0
\(589\) 4899.91 0.342780
\(590\) 5123.04 0.357478
\(591\) 0 0
\(592\) 9498.09 0.659407
\(593\) 493.541 0.0341776 0.0170888 0.999854i \(-0.494560\pi\)
0.0170888 + 0.999854i \(0.494560\pi\)
\(594\) 0 0
\(595\) 11011.6 0.758711
\(596\) −3438.11 −0.236293
\(597\) 0 0
\(598\) −2649.31 −0.181168
\(599\) 12455.1 0.849585 0.424793 0.905291i \(-0.360347\pi\)
0.424793 + 0.905291i \(0.360347\pi\)
\(600\) 0 0
\(601\) 12454.8 0.845329 0.422664 0.906286i \(-0.361095\pi\)
0.422664 + 0.906286i \(0.361095\pi\)
\(602\) 45023.3 3.04820
\(603\) 0 0
\(604\) −21853.1 −1.47217
\(605\) −344.712 −0.0231645
\(606\) 0 0
\(607\) −4243.19 −0.283733 −0.141867 0.989886i \(-0.545310\pi\)
−0.141867 + 0.989886i \(0.545310\pi\)
\(608\) −7896.70 −0.526732
\(609\) 0 0
\(610\) 7013.57 0.465526
\(611\) 1192.01 0.0789259
\(612\) 0 0
\(613\) 5733.14 0.377748 0.188874 0.982001i \(-0.439516\pi\)
0.188874 + 0.982001i \(0.439516\pi\)
\(614\) −9672.18 −0.635729
\(615\) 0 0
\(616\) 5515.99 0.360788
\(617\) −15642.1 −1.02063 −0.510314 0.859988i \(-0.670471\pi\)
−0.510314 + 0.859988i \(0.670471\pi\)
\(618\) 0 0
\(619\) −7467.40 −0.484879 −0.242440 0.970167i \(-0.577948\pi\)
−0.242440 + 0.970167i \(0.577948\pi\)
\(620\) −4636.74 −0.300348
\(621\) 0 0
\(622\) 33116.1 2.13478
\(623\) 23220.4 1.49327
\(624\) 0 0
\(625\) 12647.4 0.809432
\(626\) −30233.6 −1.93031
\(627\) 0 0
\(628\) −654.416 −0.0415829
\(629\) 51224.2 3.24713
\(630\) 0 0
\(631\) −1486.38 −0.0937745 −0.0468872 0.998900i \(-0.514930\pi\)
−0.0468872 + 0.998900i \(0.514930\pi\)
\(632\) −6060.87 −0.381469
\(633\) 0 0
\(634\) −4089.35 −0.256165
\(635\) 2829.78 0.176845
\(636\) 0 0
\(637\) 3408.74 0.212024
\(638\) 3380.21 0.209755
\(639\) 0 0
\(640\) 5192.58 0.320711
\(641\) −12386.0 −0.763211 −0.381606 0.924325i \(-0.624629\pi\)
−0.381606 + 0.924325i \(0.624629\pi\)
\(642\) 0 0
\(643\) −14458.1 −0.886737 −0.443369 0.896339i \(-0.646217\pi\)
−0.443369 + 0.896339i \(0.646217\pi\)
\(644\) −42652.3 −2.60984
\(645\) 0 0
\(646\) −18801.8 −1.14512
\(647\) −15792.8 −0.959625 −0.479813 0.877371i \(-0.659295\pi\)
−0.479813 + 0.877371i \(0.659295\pi\)
\(648\) 0 0
\(649\) −4470.87 −0.270412
\(650\) −2663.89 −0.160748
\(651\) 0 0
\(652\) −569.909 −0.0342321
\(653\) 3179.93 0.190567 0.0952837 0.995450i \(-0.469624\pi\)
0.0952837 + 0.995450i \(0.469624\pi\)
\(654\) 0 0
\(655\) 1099.13 0.0655672
\(656\) −7281.96 −0.433404
\(657\) 0 0
\(658\) 32453.6 1.92276
\(659\) −11593.5 −0.685308 −0.342654 0.939462i \(-0.611326\pi\)
−0.342654 + 0.939462i \(0.611326\pi\)
\(660\) 0 0
\(661\) 3233.88 0.190293 0.0951464 0.995463i \(-0.469668\pi\)
0.0951464 + 0.995463i \(0.469668\pi\)
\(662\) −43449.9 −2.55095
\(663\) 0 0
\(664\) 14715.8 0.860066
\(665\) −3146.93 −0.183508
\(666\) 0 0
\(667\) −8073.56 −0.468680
\(668\) −23946.9 −1.38703
\(669\) 0 0
\(670\) −1068.18 −0.0615929
\(671\) −6120.74 −0.352144
\(672\) 0 0
\(673\) −5495.72 −0.314776 −0.157388 0.987537i \(-0.550307\pi\)
−0.157388 + 0.987537i \(0.550307\pi\)
\(674\) 2658.23 0.151916
\(675\) 0 0
\(676\) −25124.4 −1.42947
\(677\) −33836.7 −1.92090 −0.960451 0.278448i \(-0.910180\pi\)
−0.960451 + 0.278448i \(0.910180\pi\)
\(678\) 0 0
\(679\) −37133.6 −2.09876
\(680\) 5495.75 0.309930
\(681\) 0 0
\(682\) 6843.04 0.384214
\(683\) 21080.3 1.18099 0.590493 0.807043i \(-0.298933\pi\)
0.590493 + 0.807043i \(0.298933\pi\)
\(684\) 0 0
\(685\) 2520.78 0.140605
\(686\) 44702.2 2.48796
\(687\) 0 0
\(688\) −7258.84 −0.402239
\(689\) −25.3393 −0.00140109
\(690\) 0 0
\(691\) 11811.3 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(692\) 6991.79 0.384087
\(693\) 0 0
\(694\) 13907.8 0.760711
\(695\) 3110.79 0.169783
\(696\) 0 0
\(697\) −39272.4 −2.13422
\(698\) 3188.52 0.172904
\(699\) 0 0
\(700\) −42887.0 −2.31568
\(701\) −4244.99 −0.228718 −0.114359 0.993440i \(-0.536481\pi\)
−0.114359 + 0.993440i \(0.536481\pi\)
\(702\) 0 0
\(703\) −14639.0 −0.785376
\(704\) −9038.51 −0.483880
\(705\) 0 0
\(706\) −5340.81 −0.284708
\(707\) 38711.5 2.05926
\(708\) 0 0
\(709\) −898.822 −0.0476107 −0.0238053 0.999717i \(-0.507578\pi\)
−0.0238053 + 0.999717i \(0.507578\pi\)
\(710\) 618.643 0.0327004
\(711\) 0 0
\(712\) 11589.0 0.609993
\(713\) −16344.5 −0.858493
\(714\) 0 0
\(715\) −161.424 −0.00844322
\(716\) 24679.4 1.28815
\(717\) 0 0
\(718\) −38706.3 −2.01185
\(719\) −10741.8 −0.557165 −0.278582 0.960412i \(-0.589865\pi\)
−0.278582 + 0.960412i \(0.589865\pi\)
\(720\) 0 0
\(721\) 16379.6 0.846061
\(722\) −24973.9 −1.28730
\(723\) 0 0
\(724\) −6822.30 −0.350206
\(725\) −8117.99 −0.415855
\(726\) 0 0
\(727\) 16794.2 0.856758 0.428379 0.903599i \(-0.359085\pi\)
0.428379 + 0.903599i \(0.359085\pi\)
\(728\) 2583.06 0.131503
\(729\) 0 0
\(730\) −9905.73 −0.502229
\(731\) −39147.7 −1.98075
\(732\) 0 0
\(733\) 8659.40 0.436347 0.218173 0.975910i \(-0.429990\pi\)
0.218173 + 0.975910i \(0.429990\pi\)
\(734\) −29778.4 −1.49747
\(735\) 0 0
\(736\) 26340.8 1.31920
\(737\) 932.197 0.0465915
\(738\) 0 0
\(739\) 16705.7 0.831567 0.415783 0.909464i \(-0.363507\pi\)
0.415783 + 0.909464i \(0.363507\pi\)
\(740\) 13852.7 0.688157
\(741\) 0 0
\(742\) −689.884 −0.0341327
\(743\) −1292.12 −0.0637996 −0.0318998 0.999491i \(-0.510156\pi\)
−0.0318998 + 0.999491i \(0.510156\pi\)
\(744\) 0 0
\(745\) 846.151 0.0416115
\(746\) −1006.09 −0.0493773
\(747\) 0 0
\(748\) −15527.0 −0.758990
\(749\) 4818.83 0.235082
\(750\) 0 0
\(751\) −14980.4 −0.727886 −0.363943 0.931421i \(-0.618570\pi\)
−0.363943 + 0.931421i \(0.618570\pi\)
\(752\) −5232.30 −0.253726
\(753\) 0 0
\(754\) 1582.90 0.0764535
\(755\) 5378.25 0.259251
\(756\) 0 0
\(757\) 3003.41 0.144202 0.0721010 0.997397i \(-0.477030\pi\)
0.0721010 + 0.997397i \(0.477030\pi\)
\(758\) 50244.8 2.40762
\(759\) 0 0
\(760\) −1570.59 −0.0749621
\(761\) 20375.0 0.970555 0.485277 0.874360i \(-0.338719\pi\)
0.485277 + 0.874360i \(0.338719\pi\)
\(762\) 0 0
\(763\) 68794.1 3.26411
\(764\) 25013.6 1.18450
\(765\) 0 0
\(766\) −47578.4 −2.24422
\(767\) −2093.65 −0.0985621
\(768\) 0 0
\(769\) −12372.4 −0.580184 −0.290092 0.956999i \(-0.593686\pi\)
−0.290092 + 0.956999i \(0.593686\pi\)
\(770\) −4394.90 −0.205690
\(771\) 0 0
\(772\) −17258.1 −0.804578
\(773\) −21023.6 −0.978225 −0.489113 0.872221i \(-0.662679\pi\)
−0.489113 + 0.872221i \(0.662679\pi\)
\(774\) 0 0
\(775\) −16434.4 −0.761732
\(776\) −18532.8 −0.857331
\(777\) 0 0
\(778\) 51885.7 2.39099
\(779\) 11223.4 0.516198
\(780\) 0 0
\(781\) −539.889 −0.0247359
\(782\) 62716.6 2.86795
\(783\) 0 0
\(784\) −14962.5 −0.681602
\(785\) 161.058 0.00732281
\(786\) 0 0
\(787\) 30286.2 1.37177 0.685886 0.727709i \(-0.259415\pi\)
0.685886 + 0.727709i \(0.259415\pi\)
\(788\) 2668.51 0.120637
\(789\) 0 0
\(790\) 4829.03 0.217480
\(791\) 20489.3 0.921007
\(792\) 0 0
\(793\) −2866.25 −0.128353
\(794\) −1592.37 −0.0711729
\(795\) 0 0
\(796\) 259.301 0.0115461
\(797\) −32337.8 −1.43722 −0.718610 0.695413i \(-0.755221\pi\)
−0.718610 + 0.695413i \(0.755221\pi\)
\(798\) 0 0
\(799\) −28218.3 −1.24943
\(800\) 26485.7 1.17052
\(801\) 0 0
\(802\) 17992.9 0.792207
\(803\) 8644.72 0.379907
\(804\) 0 0
\(805\) 10497.1 0.459596
\(806\) 3204.50 0.140042
\(807\) 0 0
\(808\) 19320.3 0.841197
\(809\) 891.707 0.0387525 0.0193762 0.999812i \(-0.493832\pi\)
0.0193762 + 0.999812i \(0.493832\pi\)
\(810\) 0 0
\(811\) −10114.9 −0.437957 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(812\) 25483.8 1.10136
\(813\) 0 0
\(814\) −20444.3 −0.880309
\(815\) 140.260 0.00602833
\(816\) 0 0
\(817\) 11187.7 0.479080
\(818\) −59680.4 −2.55095
\(819\) 0 0
\(820\) −10620.6 −0.452300
\(821\) −10833.5 −0.460525 −0.230262 0.973129i \(-0.573958\pi\)
−0.230262 + 0.973129i \(0.573958\pi\)
\(822\) 0 0
\(823\) 31958.5 1.35359 0.676794 0.736173i \(-0.263369\pi\)
0.676794 + 0.736173i \(0.263369\pi\)
\(824\) 8174.84 0.345612
\(825\) 0 0
\(826\) −57001.3 −2.40112
\(827\) −34847.3 −1.46525 −0.732624 0.680634i \(-0.761704\pi\)
−0.732624 + 0.680634i \(0.761704\pi\)
\(828\) 0 0
\(829\) 6537.91 0.273910 0.136955 0.990577i \(-0.456268\pi\)
0.136955 + 0.990577i \(0.456268\pi\)
\(830\) −11724.9 −0.490334
\(831\) 0 0
\(832\) −4232.60 −0.176369
\(833\) −80694.5 −3.35642
\(834\) 0 0
\(835\) 5893.56 0.244258
\(836\) 4437.35 0.183575
\(837\) 0 0
\(838\) 31148.5 1.28402
\(839\) −2710.34 −0.111527 −0.0557635 0.998444i \(-0.517759\pi\)
−0.0557635 + 0.998444i \(0.517759\pi\)
\(840\) 0 0
\(841\) −19565.2 −0.802215
\(842\) 40579.7 1.66089
\(843\) 0 0
\(844\) −12173.3 −0.496471
\(845\) 6183.35 0.251732
\(846\) 0 0
\(847\) 3835.42 0.155592
\(848\) 111.226 0.00450414
\(849\) 0 0
\(850\) 63061.7 2.54470
\(851\) 48830.8 1.96698
\(852\) 0 0
\(853\) 9759.32 0.391738 0.195869 0.980630i \(-0.437247\pi\)
0.195869 + 0.980630i \(0.437247\pi\)
\(854\) −78036.2 −3.12687
\(855\) 0 0
\(856\) 2405.01 0.0960298
\(857\) 13649.8 0.544072 0.272036 0.962287i \(-0.412303\pi\)
0.272036 + 0.962287i \(0.412303\pi\)
\(858\) 0 0
\(859\) 7796.42 0.309674 0.154837 0.987940i \(-0.450515\pi\)
0.154837 + 0.987940i \(0.450515\pi\)
\(860\) −10586.8 −0.419776
\(861\) 0 0
\(862\) −4393.43 −0.173597
\(863\) −7183.57 −0.283350 −0.141675 0.989913i \(-0.545249\pi\)
−0.141675 + 0.989913i \(0.545249\pi\)
\(864\) 0 0
\(865\) −1720.75 −0.0676383
\(866\) 16769.5 0.658026
\(867\) 0 0
\(868\) 51590.5 2.01739
\(869\) −4214.30 −0.164511
\(870\) 0 0
\(871\) 436.534 0.0169821
\(872\) 34334.1 1.33337
\(873\) 0 0
\(874\) −17923.3 −0.693666
\(875\) 21842.7 0.843905
\(876\) 0 0
\(877\) 17063.1 0.656991 0.328495 0.944506i \(-0.393458\pi\)
0.328495 + 0.944506i \(0.393458\pi\)
\(878\) −22728.2 −0.873620
\(879\) 0 0
\(880\) 708.562 0.0271428
\(881\) 32174.9 1.23042 0.615210 0.788363i \(-0.289071\pi\)
0.615210 + 0.788363i \(0.289071\pi\)
\(882\) 0 0
\(883\) 2843.68 0.108378 0.0541889 0.998531i \(-0.482743\pi\)
0.0541889 + 0.998531i \(0.482743\pi\)
\(884\) −7271.09 −0.276644
\(885\) 0 0
\(886\) −47238.9 −1.79122
\(887\) 31417.8 1.18930 0.594649 0.803985i \(-0.297291\pi\)
0.594649 + 0.803985i \(0.297291\pi\)
\(888\) 0 0
\(889\) −31485.5 −1.18784
\(890\) −9233.59 −0.347765
\(891\) 0 0
\(892\) 44702.5 1.67797
\(893\) 8064.30 0.302196
\(894\) 0 0
\(895\) −6073.83 −0.226845
\(896\) −57775.1 −2.15416
\(897\) 0 0
\(898\) −46588.6 −1.73127
\(899\) 9765.47 0.362288
\(900\) 0 0
\(901\) 599.852 0.0221798
\(902\) 15674.1 0.578594
\(903\) 0 0
\(904\) 10225.9 0.376227
\(905\) 1679.03 0.0616718
\(906\) 0 0
\(907\) 12253.1 0.448573 0.224287 0.974523i \(-0.427995\pi\)
0.224287 + 0.974523i \(0.427995\pi\)
\(908\) 10102.2 0.369223
\(909\) 0 0
\(910\) −2058.07 −0.0749717
\(911\) 48422.4 1.76104 0.880518 0.474012i \(-0.157195\pi\)
0.880518 + 0.474012i \(0.157195\pi\)
\(912\) 0 0
\(913\) 10232.3 0.370909
\(914\) −62262.9 −2.25326
\(915\) 0 0
\(916\) 21318.9 0.768993
\(917\) −12229.4 −0.440404
\(918\) 0 0
\(919\) 5546.18 0.199077 0.0995385 0.995034i \(-0.468263\pi\)
0.0995385 + 0.995034i \(0.468263\pi\)
\(920\) 5238.96 0.187743
\(921\) 0 0
\(922\) 136.349 0.00487030
\(923\) −252.822 −0.00901598
\(924\) 0 0
\(925\) 49099.5 1.74528
\(926\) 77831.3 2.76209
\(927\) 0 0
\(928\) −15738.0 −0.556709
\(929\) 35684.5 1.26025 0.630125 0.776494i \(-0.283004\pi\)
0.630125 + 0.776494i \(0.283004\pi\)
\(930\) 0 0
\(931\) 23061.1 0.811811
\(932\) −45516.7 −1.59973
\(933\) 0 0
\(934\) −58726.0 −2.05736
\(935\) 3821.35 0.133659
\(936\) 0 0
\(937\) −48903.6 −1.70503 −0.852514 0.522705i \(-0.824923\pi\)
−0.852514 + 0.522705i \(0.824923\pi\)
\(938\) 11885.0 0.413710
\(939\) 0 0
\(940\) −7631.17 −0.264789
\(941\) 23741.9 0.822490 0.411245 0.911525i \(-0.365094\pi\)
0.411245 + 0.911525i \(0.365094\pi\)
\(942\) 0 0
\(943\) −37437.4 −1.29282
\(944\) 9189.97 0.316852
\(945\) 0 0
\(946\) 15624.4 0.536989
\(947\) −37612.4 −1.29064 −0.645321 0.763911i \(-0.723276\pi\)
−0.645321 + 0.763911i \(0.723276\pi\)
\(948\) 0 0
\(949\) 4048.20 0.138472
\(950\) −18021.9 −0.615482
\(951\) 0 0
\(952\) −61148.2 −2.08175
\(953\) 48294.3 1.64156 0.820779 0.571246i \(-0.193540\pi\)
0.820779 + 0.571246i \(0.193540\pi\)
\(954\) 0 0
\(955\) −6156.09 −0.208593
\(956\) −55239.8 −1.86881
\(957\) 0 0
\(958\) −11046.4 −0.372539
\(959\) −28047.4 −0.944419
\(960\) 0 0
\(961\) −10021.4 −0.336389
\(962\) −9573.76 −0.320863
\(963\) 0 0
\(964\) 46173.1 1.54267
\(965\) 4247.39 0.141687
\(966\) 0 0
\(967\) 1840.92 0.0612204 0.0306102 0.999531i \(-0.490255\pi\)
0.0306102 + 0.999531i \(0.490255\pi\)
\(968\) 1914.20 0.0635587
\(969\) 0 0
\(970\) 14766.1 0.488775
\(971\) −31461.8 −1.03981 −0.519906 0.854223i \(-0.674033\pi\)
−0.519906 + 0.854223i \(0.674033\pi\)
\(972\) 0 0
\(973\) −34612.1 −1.14040
\(974\) −15328.1 −0.504254
\(975\) 0 0
\(976\) 12581.3 0.412620
\(977\) 7040.11 0.230535 0.115268 0.993334i \(-0.463227\pi\)
0.115268 + 0.993334i \(0.463227\pi\)
\(978\) 0 0
\(979\) 8058.15 0.263064
\(980\) −21822.5 −0.711320
\(981\) 0 0
\(982\) 71784.4 2.33272
\(983\) 24610.9 0.798541 0.399270 0.916833i \(-0.369263\pi\)
0.399270 + 0.916833i \(0.369263\pi\)
\(984\) 0 0
\(985\) −656.744 −0.0212443
\(986\) −37471.8 −1.21029
\(987\) 0 0
\(988\) 2077.95 0.0669112
\(989\) −37318.5 −1.19986
\(990\) 0 0
\(991\) −40003.3 −1.28229 −0.641144 0.767421i \(-0.721540\pi\)
−0.641144 + 0.767421i \(0.721540\pi\)
\(992\) −31860.8 −1.01974
\(993\) 0 0
\(994\) −6883.31 −0.219643
\(995\) −63.8165 −0.00203329
\(996\) 0 0
\(997\) −7342.61 −0.233242 −0.116621 0.993176i \(-0.537206\pi\)
−0.116621 + 0.993176i \(0.537206\pi\)
\(998\) 44216.9 1.40247
\(999\) 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 99.4.a.f.1.2 2
3.2 odd 2 33.4.a.c.1.1 2
4.3 odd 2 1584.4.a.bj.1.1 2
5.4 even 2 2475.4.a.p.1.1 2
11.10 odd 2 1089.4.a.u.1.1 2
12.11 even 2 528.4.a.p.1.2 2
15.2 even 4 825.4.c.h.199.2 4
15.8 even 4 825.4.c.h.199.3 4
15.14 odd 2 825.4.a.l.1.2 2
21.20 even 2 1617.4.a.k.1.1 2
24.5 odd 2 2112.4.a.bn.1.1 2
24.11 even 2 2112.4.a.bg.1.1 2
33.32 even 2 363.4.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 3.2 odd 2
99.4.a.f.1.2 2 1.1 even 1 trivial
363.4.a.i.1.2 2 33.32 even 2
528.4.a.p.1.2 2 12.11 even 2
825.4.a.l.1.2 2 15.14 odd 2
825.4.c.h.199.2 4 15.2 even 4
825.4.c.h.199.3 4 15.8 even 4
1089.4.a.u.1.1 2 11.10 odd 2
1584.4.a.bj.1.1 2 4.3 odd 2
1617.4.a.k.1.1 2 21.20 even 2
2112.4.a.bg.1.1 2 24.11 even 2
2112.4.a.bn.1.1 2 24.5 odd 2
2475.4.a.p.1.1 2 5.4 even 2