Properties

Label 1089.4.a.u.1.1
Level $1089$
Weight $4$
Character 1089.1
Self dual yes
Analytic conductor $64.253$
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] = [1089,4,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(64.2530799963\)
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 1.1
Root \(-4.42443\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} -31.6977 q^{7} -15.8199 q^{8} +O(q^{10})\) \(q-4.42443 q^{2} +11.5756 q^{4} -2.84886 q^{5} -31.6977 q^{7} -15.8199 q^{8} +12.6046 q^{10} -5.15114 q^{13} +140.244 q^{14} -22.6107 q^{16} +121.942 q^{17} -34.8489 q^{19} -32.9772 q^{20} -116.244 q^{23} -116.884 q^{25} +22.7909 q^{26} -366.919 q^{28} -69.4534 q^{29} +140.605 q^{31} +226.598 q^{32} -539.524 q^{34} +90.3023 q^{35} -420.070 q^{37} +154.186 q^{38} +45.0685 q^{40} -322.058 q^{41} -321.035 q^{43} +514.315 q^{46} +231.408 q^{47} +661.745 q^{49} +517.145 q^{50} -59.6274 q^{52} -4.91916 q^{53} +501.453 q^{56} +307.292 q^{58} -406.443 q^{59} +556.431 q^{61} -622.095 q^{62} -821.683 q^{64} +14.6749 q^{65} +84.7452 q^{67} +1411.55 q^{68} -399.536 q^{70} -49.0808 q^{71} -785.884 q^{73} +1858.57 q^{74} -403.395 q^{76} +383.118 q^{79} +64.4147 q^{80} +1424.92 q^{82} -930.211 q^{83} -347.395 q^{85} +1420.40 q^{86} +732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} -1023.85 q^{94} +99.2794 q^{95} -1171.49 q^{97} -2927.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8} + 104 q^{10} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} - 50 q^{19} + 328 q^{20} - 134 q^{23} + 42 q^{25} - 112 q^{26} - 202 q^{28} - 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 220 q^{35} - 328 q^{37} + 72 q^{38} + 1272 q^{40} - 782 q^{41} - 386 q^{43} + 418 q^{46} - 266 q^{47} + 378 q^{49} + 1379 q^{50} - 592 q^{52} + 522 q^{53} + 1062 q^{56} - 390 q^{58} + 172 q^{59} + 778 q^{61} + 568 q^{62} + 809 q^{64} - 404 q^{65} - 776 q^{67} + 1070 q^{68} + 304 q^{70} - 630 q^{71} - 1296 q^{73} + 2358 q^{74} - 728 q^{76} - 652 q^{79} + 3832 q^{80} - 1070 q^{82} - 324 q^{83} - 616 q^{85} + 1068 q^{86} + 756 q^{89} - 28 q^{91} - 1726 q^{92} - 3722 q^{94} - 156 q^{95} - 452 q^{97} - 4467 q^{98}+O(q^{100}) \) 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 \(-0.785870\pi\)
−0.782136 + 0.623108i \(0.785870\pi\)
\(3\) 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 \(-0.826909\pi\)
−0.855758 + 0.517377i \(0.826909\pi\)
\(8\) −15.8199 −0.699146
\(9\) 0 0
\(10\) 12.6046 0.398591
\(11\) 0 0
\(12\) 0 0
\(13\) −5.15114 −0.109898 −0.0549488 0.998489i \(-0.517500\pi\)
−0.0549488 + 0.998489i \(0.517500\pi\)
\(14\) 140.244 2.67728
\(15\) 0 0
\(16\) −22.6107 −0.353293
\(17\) 121.942 1.73972 0.869861 0.493297i \(-0.164208\pi\)
0.869861 + 0.493297i \(0.164208\pi\)
\(18\) 0 0
\(19\) −34.8489 −0.420783 −0.210391 0.977617i \(-0.567474\pi\)
−0.210391 + 0.977617i \(0.567474\pi\)
\(20\) −32.9772 −0.368696
\(21\) 0 0
\(22\) 0 0
\(23\) −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 \(-0.571378\pi\)
−0.222365 + 0.974963i \(0.571378\pi\)
\(30\) 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 \(-0.710190\pi\)
−0.613378 + 0.789789i \(0.710190\pi\)
\(42\) 0 0
\(43\) −321.035 −1.13854 −0.569272 0.822149i \(-0.692775\pi\)
−0.569272 + 0.822149i \(0.692775\pi\)
\(44\) 0 0
\(45\) 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
\(55\) 0 0
\(56\) 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 \(-0.301501\pi\)
0.583964 + 0.811779i \(0.301501\pi\)
\(62\) −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 \(-0.716947\pi\)
−0.630005 + 0.776591i \(0.716947\pi\)
\(74\) 1858.57 2.91966
\(75\) 0 0
\(76\) −403.395 −0.608850
\(77\) 0 0
\(78\) 0 0
\(79\) 383.118 0.545622 0.272811 0.962068i \(-0.412047\pi\)
0.272811 + 0.962068i \(0.412047\pi\)
\(80\) 64.4147 0.0900223
\(81\) 0 0
\(82\) 1424.92 1.91898
\(83\) −930.211 −1.23017 −0.615084 0.788462i \(-0.710878\pi\)
−0.615084 + 0.788462i \(0.710878\pi\)
\(84\) 0 0
\(85\) −347.395 −0.443298
\(86\) 1420.40 1.78099
\(87\) 0 0
\(88\) 0 0
\(89\) 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 \(-0.705465\pi\)
−0.601589 + 0.798806i \(0.705465\pi\)
\(102\) 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 \(-0.521878\pi\)
−0.0686765 + 0.997639i \(0.521878\pi\)
\(108\) 0 0
\(109\) −2170.32 −1.90714 −0.953572 0.301164i \(-0.902625\pi\)
−0.953572 + 0.301164i \(0.902625\pi\)
\(110\) 0 0
\(111\) 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
\(121\) 0 0
\(122\) −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 \(-0.387196\pi\)
0.347014 + 0.937860i \(0.387196\pi\)
\(128\) 1822.69 1.25863
\(129\) 0 0
\(130\) −64.9279 −0.0438043
\(131\) 385.814 0.257318 0.128659 0.991689i \(-0.458933\pi\)
0.128659 + 0.991689i \(0.458933\pi\)
\(132\) 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 \(-0.391886\pi\)
0.333156 + 0.942872i \(0.391886\pi\)
\(140\) 1045.30 0.631029
\(141\) 0 0
\(142\) 217.155 0.128333
\(143\) 0 0
\(144\) 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 \(-0.473980\pi\)
0.0816522 + 0.996661i \(0.473980\pi\)
\(150\) 0 0
\(151\) 1887.86 1.01743 0.508716 0.860935i \(-0.330120\pi\)
0.508716 + 0.860935i \(0.330120\pi\)
\(152\) 551.304 0.294189
\(153\) 0 0
\(154\) 0 0
\(155\) −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 \(-0.340893\pi\)
0.479294 + 0.877654i \(0.340893\pi\)
\(168\) 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 \(-0.542372\pi\)
−0.132723 + 0.991153i \(0.542372\pi\)
\(174\) 0 0
\(175\) 3704.96 1.60039
\(176\) 0 0
\(177\) 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
\(187\) 0 0
\(188\) 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 \(-0.410320\pi\)
0.278026 + 0.960574i \(0.410320\pi\)
\(194\) 5183.18 1.91820
\(195\) 0 0
\(196\) 7660.08 2.79157
\(197\) −230.529 −0.0833732 −0.0416866 0.999131i \(-0.513273\pi\)
−0.0416866 + 0.999131i \(0.513273\pi\)
\(198\) 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
\(209\) 0 0
\(210\) 0 0
\(211\) 1051.64 0.343117 0.171558 0.985174i \(-0.445120\pi\)
0.171558 + 0.985174i \(0.445120\pi\)
\(212\) −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
\(220\) 0 0
\(221\) −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 \(-0.540723\pi\)
−0.127587 + 0.991827i \(0.540723\pi\)
\(228\) 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 \(-0.313561\pi\)
0.552796 + 0.833317i \(0.313561\pi\)
\(234\) 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 \(-0.276534\pi\)
0.645777 + 0.763526i \(0.276534\pi\)
\(240\) 0 0
\(241\) −3988.84 −1.06616 −0.533078 0.846066i \(-0.678965\pi\)
−0.533078 + 0.846066i \(0.678965\pi\)
\(242\) 0 0
\(243\) 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
\(253\) 0 0
\(254\) −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 \(-0.150946\pi\)
0.889654 + 0.456636i \(0.150946\pi\)
\(264\) 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 \(-0.495636\pi\)
0.0137096 + 0.999906i \(0.495636\pi\)
\(272\) −2757.20 −0.614631
\(273\) 0 0
\(274\) 3914.91 0.863170
\(275\) 0 0
\(276\) 0 0
\(277\) −8199.41 −1.77854 −0.889269 0.457385i \(-0.848786\pi\)
−0.889269 + 0.457385i \(0.848786\pi\)
\(278\) −4831.22 −1.04229
\(279\) 0 0
\(280\) −1428.57 −0.304905
\(281\) 6943.79 1.47413 0.737067 0.675820i \(-0.236210\pi\)
0.737067 + 0.675820i \(0.236210\pi\)
\(282\) 0 0
\(283\) −1035.14 −0.217429 −0.108715 0.994073i \(-0.534673\pi\)
−0.108715 + 0.994073i \(0.534673\pi\)
\(284\) −568.139 −0.118707
\(285\) 0 0
\(286\) 0 0
\(287\) 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 \(-0.709876\pi\)
−0.612600 + 0.790393i \(0.709876\pi\)
\(294\) 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 \(-0.434865\pi\)
0.203203 + 0.979137i \(0.434865\pi\)
\(308\) 0 0
\(309\) 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
\(319\) 0 0
\(320\) 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 \(-0.515463\pi\)
−0.0485580 + 0.998820i \(0.515463\pi\)
\(338\) 9603.07 1.54538
\(339\) 0 0
\(340\) −4021.30 −0.641428
\(341\) 0 0
\(342\) 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 \(-0.578181\pi\)
−0.243152 + 0.969988i \(0.578181\pi\)
\(348\) 0 0
\(349\) −720.663 −0.110533 −0.0552667 0.998472i \(-0.517601\pi\)
−0.0552667 + 0.998472i \(0.517601\pi\)
\(350\) −16392.3 −2.50345
\(351\) 0 0
\(352\) 0 0
\(353\) −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 \(-0.277664\pi\)
0.643062 + 0.765814i \(0.277664\pi\)
\(360\) 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 \(-0.494976\pi\)
0.0157828 + 0.999875i \(0.494976\pi\)
\(374\) 0 0
\(375\) 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
\(385\) 0 0
\(386\) −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
\(407\) 0 0
\(408\) 0 0
\(409\) 13488.8 1.63076 0.815379 0.578927i \(-0.196528\pi\)
0.815379 + 0.578927i \(0.196528\pi\)
\(410\) −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
\(418\) 0 0
\(419\) 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 \(-0.482328\pi\)
0.0554882 + 0.998459i \(0.482328\pi\)
\(432\) 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 \(-0.409917\pi\)
0.279242 + 0.960221i \(0.409917\pi\)
\(440\) 0 0
\(441\) 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
\(451\) 0 0
\(452\) 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 \(-0.244039\pi\)
0.720225 + 0.693741i \(0.244039\pi\)
\(458\) −8148.55 −0.831347
\(459\) 0 0
\(460\) 3833.41 0.388551
\(461\) −30.8173 −0.00311346 −0.00155673 0.999999i \(-0.500496\pi\)
−0.00155673 + 0.999999i \(0.500496\pi\)
\(462\) 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
\(473\) 0 0
\(474\) 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 \(-0.462006\pi\)
0.119077 + 0.992885i \(0.462006\pi\)
\(480\) 0 0
\(481\) 2163.84 0.205120
\(482\) 17648.3 1.66776
\(483\) 0 0
\(484\) 0 0
\(485\) 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 \(-0.767849\pi\)
−0.745625 + 0.666366i \(0.767849\pi\)
\(492\) 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 \(-0.737876\pi\)
−0.679667 + 0.733520i \(0.737876\pi\)
\(504\) 0 0
\(505\) 3479.23 0.306581
\(506\) 0 0
\(507\) 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
\(517\) 0 0
\(518\) −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 \(-0.158695\pi\)
0.878275 + 0.478157i \(0.158695\pi\)
\(524\) 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
\(539\) 0 0
\(540\) 0 0
\(541\) 16802.8 1.33532 0.667662 0.744464i \(-0.267295\pi\)
0.667662 + 0.744464i \(0.267295\pi\)
\(542\) −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 \(-0.727749\pi\)
−0.655990 + 0.754770i \(0.727749\pi\)
\(548\) −10242.5 −0.798429
\(549\) 0 0
\(550\) 0 0
\(551\) 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 \(-0.257845\pi\)
0.689467 + 0.724317i \(0.257845\pi\)
\(558\) 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 \(-0.475064\pi\)
0.0782592 + 0.996933i \(0.475064\pi\)
\(564\) 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 \(-0.426058\pi\)
0.230212 + 0.973140i \(0.426058\pi\)
\(570\) 0 0
\(571\) −6048.79 −0.443317 −0.221659 0.975124i \(-0.571147\pi\)
−0.221659 + 0.975124i \(0.571147\pi\)
\(572\) 0 0
\(573\) 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
\(583\) 0 0
\(584\) 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 \(-0.505440\pi\)
−0.0170888 + 0.999854i \(0.505440\pi\)
\(594\) 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 \(-0.638905\pi\)
−0.422664 + 0.906286i \(0.638905\pi\)
\(602\) −45023.3 −3.04820
\(603\) 0 0
\(604\) 21853.1 1.47217
\(605\) 0 0
\(606\) 0 0
\(607\) 4243.19 0.283733 0.141867 0.989886i \(-0.454690\pi\)
0.141867 + 0.989886i \(0.454690\pi\)
\(608\) −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 \(-0.560484\pi\)
−0.188874 + 0.982001i \(0.560484\pi\)
\(614\) −9672.18 −0.635729
\(615\) 0 0
\(616\) 0 0
\(617\) −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
\(638\) 0 0
\(639\) 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
\(649\) 0 0
\(650\) −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 \(-0.388674\pi\)
0.342654 + 0.939462i \(0.388674\pi\)
\(660\) 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
\(671\) 0 0
\(672\) 0 0
\(673\) 5495.72 0.314776 0.157388 0.987537i \(-0.449693\pi\)
0.157388 + 0.987537i \(0.449693\pi\)
\(674\) 2658.23 0.151916
\(675\) 0 0
\(676\) −25124.4 −1.42947
\(677\) 33836.7 1.92090 0.960451 0.278448i \(-0.0898200\pi\)
0.960451 + 0.278448i \(0.0898200\pi\)
\(678\) 0 0
\(679\) 37133.6 2.09876
\(680\) 5495.75 0.309930
\(681\) 0 0
\(682\) 0 0
\(683\) 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 \(-0.463519\pi\)
0.114359 + 0.993440i \(0.463519\pi\)
\(702\) 0 0
\(703\) 14639.0 0.785376
\(704\) 0 0
\(705\) 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
\(715\) 0 0
\(716\) 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 \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(734\) 29778.4 1.49747
\(735\) 0 0
\(736\) −26340.8 −1.31920
\(737\) 0 0
\(738\) 0 0
\(739\) −16705.7 −0.831567 −0.415783 0.909464i \(-0.636493\pi\)
−0.415783 + 0.909464i \(0.636493\pi\)
\(740\) 13852.7 0.688157
\(741\) 0 0
\(742\) −689.884 −0.0341327
\(743\) 1292.12 0.0637996 0.0318998 0.999491i \(-0.489844\pi\)
0.0318998 + 0.999491i \(0.489844\pi\)
\(744\) 0 0
\(745\) −846.151 −0.0416115
\(746\) −1006.09 −0.0493773
\(747\) 0 0
\(748\) 0 0
\(749\) 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 \(-0.661281\pi\)
−0.485277 + 0.874360i \(0.661281\pi\)
\(762\) 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 \(-0.406314\pi\)
0.290092 + 0.956999i \(0.406314\pi\)
\(770\) 0 0
\(771\) 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
\(781\) 0 0
\(782\) 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 \(-0.740585\pi\)
−0.685886 + 0.727709i \(0.740585\pi\)
\(788\) −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
\(803\) 0 0
\(804\) 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 \(-0.506168\pi\)
−0.0193762 + 0.999812i \(0.506168\pi\)
\(810\) 0 0
\(811\) 10114.9 0.437957 0.218978 0.975730i \(-0.429728\pi\)
0.218978 + 0.975730i \(0.429728\pi\)
\(812\) 25483.8 1.10136
\(813\) 0 0
\(814\) 0 0
\(815\) 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 \(-0.426042\pi\)
0.230262 + 0.973129i \(0.426042\pi\)
\(822\) 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 \(-0.238296\pi\)
0.732624 + 0.680634i \(0.238296\pi\)
\(828\) 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
\(836\) 0 0
\(837\) 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
\(847\) 0 0
\(848\) 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 \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(854\) 78036.2 3.12687
\(855\) 0 0
\(856\) 2405.01 0.0960298
\(857\) −13649.8 −0.544072 −0.272036 0.962287i \(-0.587697\pi\)
−0.272036 + 0.962287i \(0.587697\pi\)
\(858\) 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
\(869\) 0 0
\(870\) 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 \(-0.606542\pi\)
−0.328495 + 0.944506i \(0.606542\pi\)
\(878\) −22728.2 −0.873620
\(879\) 0 0
\(880\) 0 0
\(881\) 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 \(-0.702709\pi\)
−0.594649 + 0.803985i \(0.702709\pi\)
\(888\) 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
\(902\) 0 0
\(903\) 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
\(913\) 0 0
\(914\) −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 \(-0.531737\pi\)
−0.0995385 + 0.995034i \(0.531737\pi\)
\(920\) −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
\(935\) 0 0
\(936\) 0 0
\(937\) 48903.6 1.70503 0.852514 0.522705i \(-0.175077\pi\)
0.852514 + 0.522705i \(0.175077\pi\)
\(938\) 11885.0 0.413710
\(939\) 0 0
\(940\) −7631.17 −0.264789
\(941\) −23741.9 −0.822490 −0.411245 0.911525i \(-0.634906\pi\)
−0.411245 + 0.911525i \(0.634906\pi\)
\(942\) 0 0
\(943\) 37437.4 1.29282
\(944\) 9189.97 0.316852
\(945\) 0 0
\(946\) 0 0
\(947\) −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 \(-0.806460\pi\)
−0.820779 + 0.571246i \(0.806460\pi\)
\(954\) 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 \(-0.509745\pi\)
−0.0306102 + 0.999531i \(0.509745\pi\)
\(968\) 0 0
\(969\) 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
\(979\) 0 0
\(980\) −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 \(-0.462794\pi\)
0.116621 + 0.993176i \(0.462794\pi\)
\(998\) −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 1089.4.a.u.1.1 2
3.2 odd 2 363.4.a.i.1.2 2
11.10 odd 2 99.4.a.f.1.2 2
33.32 even 2 33.4.a.c.1.1 2
44.43 even 2 1584.4.a.bj.1.1 2
55.54 odd 2 2475.4.a.p.1.1 2
132.131 odd 2 528.4.a.p.1.2 2
165.32 odd 4 825.4.c.h.199.2 4
165.98 odd 4 825.4.c.h.199.3 4
165.164 even 2 825.4.a.l.1.2 2
231.230 odd 2 1617.4.a.k.1.1 2
264.131 odd 2 2112.4.a.bg.1.1 2
264.197 even 2 2112.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 33.32 even 2
99.4.a.f.1.2 2 11.10 odd 2
363.4.a.i.1.2 2 3.2 odd 2
528.4.a.p.1.2 2 132.131 odd 2
825.4.a.l.1.2 2 165.164 even 2
825.4.c.h.199.2 4 165.32 odd 4
825.4.c.h.199.3 4 165.98 odd 4
1089.4.a.u.1.1 2 1.1 even 1 trivial
1584.4.a.bj.1.1 2 44.43 even 2
1617.4.a.k.1.1 2 231.230 odd 2
2112.4.a.bg.1.1 2 264.131 odd 2
2112.4.a.bn.1.1 2 264.197 even 2
2475.4.a.p.1.1 2 55.54 odd 2