Properties

Label 289.4.b.f.288.23
Level $289$
Weight $4$
Character 289.288
Analytic conductor $17.052$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(288,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.288");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 288.23
Character \(\chi\) \(=\) 289.288
Dual form 289.4.b.f.288.24

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.35197 q^{2} +1.66219i q^{3} +20.6436 q^{4} -5.96899i q^{5} +8.89600i q^{6} +27.8210i q^{7} +67.6680 q^{8} +24.2371 q^{9} +O(q^{10})\) \(q+5.35197 q^{2} +1.66219i q^{3} +20.6436 q^{4} -5.96899i q^{5} +8.89600i q^{6} +27.8210i q^{7} +67.6680 q^{8} +24.2371 q^{9} -31.9458i q^{10} -18.6361i q^{11} +34.3136i q^{12} -42.5466 q^{13} +148.897i q^{14} +9.92160 q^{15} +197.008 q^{16} +129.716 q^{18} -31.3435 q^{19} -123.221i q^{20} -46.2438 q^{21} -99.7396i q^{22} -60.3201i q^{23} +112.477i q^{24} +89.3712 q^{25} -227.708 q^{26} +85.1659i q^{27} +574.325i q^{28} +117.647i q^{29} +53.1001 q^{30} -228.520i q^{31} +513.039 q^{32} +30.9767 q^{33} +166.063 q^{35} +500.341 q^{36} +99.4898i q^{37} -167.749 q^{38} -70.7206i q^{39} -403.909i q^{40} -270.844i q^{41} -247.496 q^{42} -108.891 q^{43} -384.715i q^{44} -144.671i q^{45} -322.831i q^{46} -250.032 q^{47} +327.466i q^{48} -431.008 q^{49} +478.312 q^{50} -878.313 q^{52} -294.775 q^{53} +455.805i q^{54} -111.238 q^{55} +1882.59i q^{56} -52.0989i q^{57} +629.643i q^{58} -62.0744 q^{59} +204.817 q^{60} -799.234i q^{61} -1223.03i q^{62} +674.301i q^{63} +1169.70 q^{64} +253.960i q^{65} +165.786 q^{66} -645.320 q^{67} +100.264 q^{69} +888.765 q^{70} +1148.97i q^{71} +1640.08 q^{72} -550.710i q^{73} +532.466i q^{74} +148.552i q^{75} -647.041 q^{76} +518.474 q^{77} -378.494i q^{78} +253.834i q^{79} -1175.94i q^{80} +512.840 q^{81} -1449.55i q^{82} -717.541 q^{83} -954.638 q^{84} -582.784 q^{86} -195.552 q^{87} -1261.06i q^{88} -1590.45 q^{89} -774.275i q^{90} -1183.69i q^{91} -1245.22i q^{92} +379.844 q^{93} -1338.16 q^{94} +187.089i q^{95} +852.769i q^{96} +255.084i q^{97} -2306.74 q^{98} -451.684i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.35197 1.89221 0.946103 0.323865i \(-0.104982\pi\)
0.946103 + 0.323865i \(0.104982\pi\)
\(3\) 1.66219i 0.319889i 0.987126 + 0.159944i \(0.0511315\pi\)
−0.987126 + 0.159944i \(0.948868\pi\)
\(4\) 20.6436 2.58045
\(5\) − 5.96899i − 0.533882i −0.963713 0.266941i \(-0.913987\pi\)
0.963713 0.266941i \(-0.0860129\pi\)
\(6\) 8.89600i 0.605296i
\(7\) 27.8210i 1.50219i 0.660193 + 0.751096i \(0.270475\pi\)
−0.660193 + 0.751096i \(0.729525\pi\)
\(8\) 67.6680 2.99053
\(9\) 24.2371 0.897671
\(10\) − 31.9458i − 1.01022i
\(11\) − 18.6361i − 0.510816i −0.966833 0.255408i \(-0.917790\pi\)
0.966833 0.255408i \(-0.0822099\pi\)
\(12\) 34.3136i 0.825456i
\(13\) −42.5466 −0.907715 −0.453857 0.891074i \(-0.649952\pi\)
−0.453857 + 0.891074i \(0.649952\pi\)
\(14\) 148.897i 2.84246i
\(15\) 9.92160 0.170783
\(16\) 197.008 3.07826
\(17\) 0 0
\(18\) 129.716 1.69858
\(19\) −31.3435 −0.378457 −0.189229 0.981933i \(-0.560599\pi\)
−0.189229 + 0.981933i \(0.560599\pi\)
\(20\) − 123.221i − 1.37765i
\(21\) −46.2438 −0.480535
\(22\) − 99.7396i − 0.966570i
\(23\) − 60.3201i − 0.546853i −0.961893 0.273426i \(-0.911843\pi\)
0.961893 0.273426i \(-0.0881570\pi\)
\(24\) 112.477i 0.956638i
\(25\) 89.3712 0.714970
\(26\) −227.708 −1.71758
\(27\) 85.1659i 0.607044i
\(28\) 574.325i 3.87633i
\(29\) 117.647i 0.753327i 0.926350 + 0.376664i \(0.122929\pi\)
−0.926350 + 0.376664i \(0.877071\pi\)
\(30\) 53.1001 0.323157
\(31\) − 228.520i − 1.32398i −0.749512 0.661991i \(-0.769712\pi\)
0.749512 0.661991i \(-0.230288\pi\)
\(32\) 513.039 2.83417
\(33\) 30.9767 0.163405
\(34\) 0 0
\(35\) 166.063 0.801994
\(36\) 500.341 2.31639
\(37\) 99.4898i 0.442055i 0.975268 + 0.221027i \(0.0709410\pi\)
−0.975268 + 0.221027i \(0.929059\pi\)
\(38\) −167.749 −0.716119
\(39\) − 70.7206i − 0.290368i
\(40\) − 403.909i − 1.59659i
\(41\) − 270.844i − 1.03168i −0.856686 0.515838i \(-0.827481\pi\)
0.856686 0.515838i \(-0.172519\pi\)
\(42\) −247.496 −0.909271
\(43\) −108.891 −0.386181 −0.193091 0.981181i \(-0.561851\pi\)
−0.193091 + 0.981181i \(0.561851\pi\)
\(44\) − 384.715i − 1.31813i
\(45\) − 144.671i − 0.479251i
\(46\) − 322.831i − 1.03476i
\(47\) −250.032 −0.775975 −0.387988 0.921665i \(-0.626830\pi\)
−0.387988 + 0.921665i \(0.626830\pi\)
\(48\) 327.466i 0.984700i
\(49\) −431.008 −1.25658
\(50\) 478.312 1.35287
\(51\) 0 0
\(52\) −878.313 −2.34231
\(53\) −294.775 −0.763971 −0.381985 0.924168i \(-0.624760\pi\)
−0.381985 + 0.924168i \(0.624760\pi\)
\(54\) 455.805i 1.14865i
\(55\) −111.238 −0.272716
\(56\) 1882.59i 4.49235i
\(57\) − 52.0989i − 0.121064i
\(58\) 629.643i 1.42545i
\(59\) −62.0744 −0.136973 −0.0684864 0.997652i \(-0.521817\pi\)
−0.0684864 + 0.997652i \(0.521817\pi\)
\(60\) 204.817 0.440697
\(61\) − 799.234i − 1.67757i −0.544466 0.838783i \(-0.683268\pi\)
0.544466 0.838783i \(-0.316732\pi\)
\(62\) − 1223.03i − 2.50525i
\(63\) 674.301i 1.34847i
\(64\) 1169.70 2.28457
\(65\) 253.960i 0.484613i
\(66\) 165.786 0.309195
\(67\) −645.320 −1.17669 −0.588346 0.808609i \(-0.700221\pi\)
−0.588346 + 0.808609i \(0.700221\pi\)
\(68\) 0 0
\(69\) 100.264 0.174932
\(70\) 888.765 1.51754
\(71\) 1148.97i 1.92053i 0.279096 + 0.960263i \(0.409965\pi\)
−0.279096 + 0.960263i \(0.590035\pi\)
\(72\) 1640.08 2.68451
\(73\) − 550.710i − 0.882955i −0.897272 0.441477i \(-0.854455\pi\)
0.897272 0.441477i \(-0.145545\pi\)
\(74\) 532.466i 0.836459i
\(75\) 148.552i 0.228711i
\(76\) −647.041 −0.976588
\(77\) 518.474 0.767345
\(78\) − 378.494i − 0.549436i
\(79\) 253.834i 0.361501i 0.983529 + 0.180750i \(0.0578526\pi\)
−0.983529 + 0.180750i \(0.942147\pi\)
\(80\) − 1175.94i − 1.64343i
\(81\) 512.840 0.703484
\(82\) − 1449.55i − 1.95214i
\(83\) −717.541 −0.948920 −0.474460 0.880277i \(-0.657357\pi\)
−0.474460 + 0.880277i \(0.657357\pi\)
\(84\) −954.638 −1.23999
\(85\) 0 0
\(86\) −582.784 −0.730735
\(87\) −195.552 −0.240981
\(88\) − 1261.06i − 1.52761i
\(89\) −1590.45 −1.89424 −0.947121 0.320877i \(-0.896022\pi\)
−0.947121 + 0.320877i \(0.896022\pi\)
\(90\) − 774.275i − 0.906841i
\(91\) − 1183.69i − 1.36356i
\(92\) − 1245.22i − 1.41112i
\(93\) 379.844 0.423527
\(94\) −1338.16 −1.46831
\(95\) 187.089i 0.202052i
\(96\) 852.769i 0.906619i
\(97\) 255.084i 0.267008i 0.991048 + 0.133504i \(0.0426230\pi\)
−0.991048 + 0.133504i \(0.957377\pi\)
\(98\) −2306.74 −2.37771
\(99\) − 451.684i − 0.458545i
\(100\) 1844.94 1.84494
\(101\) 1006.58 0.991663 0.495832 0.868419i \(-0.334863\pi\)
0.495832 + 0.868419i \(0.334863\pi\)
\(102\) 0 0
\(103\) 1631.61 1.56084 0.780422 0.625253i \(-0.215004\pi\)
0.780422 + 0.625253i \(0.215004\pi\)
\(104\) −2879.04 −2.71455
\(105\) 276.029i 0.256549i
\(106\) −1577.63 −1.44559
\(107\) − 42.5550i − 0.0384481i −0.999815 0.0192241i \(-0.993880\pi\)
0.999815 0.0192241i \(-0.00611958\pi\)
\(108\) 1758.13i 1.56644i
\(109\) − 1862.24i − 1.63643i −0.574914 0.818214i \(-0.694964\pi\)
0.574914 0.818214i \(-0.305036\pi\)
\(110\) −595.344 −0.516035
\(111\) −165.371 −0.141408
\(112\) 5480.97i 4.62414i
\(113\) − 13.5364i − 0.0112690i −0.999984 0.00563452i \(-0.998206\pi\)
0.999984 0.00563452i \(-0.00179353\pi\)
\(114\) − 278.832i − 0.229079i
\(115\) −360.050 −0.291955
\(116\) 2428.65i 1.94392i
\(117\) −1031.21 −0.814829
\(118\) −332.220 −0.259181
\(119\) 0 0
\(120\) 671.375 0.510732
\(121\) 983.698 0.739067
\(122\) − 4277.48i − 3.17430i
\(123\) 450.195 0.330022
\(124\) − 4717.47i − 3.41646i
\(125\) − 1279.58i − 0.915592i
\(126\) 3608.84i 2.55159i
\(127\) 1239.07 0.865744 0.432872 0.901455i \(-0.357500\pi\)
0.432872 + 0.901455i \(0.357500\pi\)
\(128\) 2155.89 1.48872
\(129\) − 180.998i − 0.123535i
\(130\) 1359.19i 0.916988i
\(131\) 1310.04i 0.873733i 0.899526 + 0.436866i \(0.143912\pi\)
−0.899526 + 0.436866i \(0.856088\pi\)
\(132\) 639.470 0.421657
\(133\) − 872.007i − 0.568516i
\(134\) −3453.73 −2.22655
\(135\) 508.354 0.324090
\(136\) 0 0
\(137\) −592.057 −0.369218 −0.184609 0.982812i \(-0.559102\pi\)
−0.184609 + 0.982812i \(0.559102\pi\)
\(138\) 536.608 0.331008
\(139\) 1723.70i 1.05182i 0.850541 + 0.525908i \(0.176275\pi\)
−0.850541 + 0.525908i \(0.823725\pi\)
\(140\) 3428.14 2.06950
\(141\) − 415.600i − 0.248226i
\(142\) 6149.24i 3.63403i
\(143\) 792.900i 0.463676i
\(144\) 4774.92 2.76326
\(145\) 702.233 0.402188
\(146\) − 2947.38i − 1.67073i
\(147\) − 716.418i − 0.401967i
\(148\) 2053.82i 1.14070i
\(149\) −230.059 −0.126491 −0.0632456 0.997998i \(-0.520145\pi\)
−0.0632456 + 0.997998i \(0.520145\pi\)
\(150\) 795.046i 0.432768i
\(151\) 1876.54 1.01133 0.505663 0.862731i \(-0.331248\pi\)
0.505663 + 0.862731i \(0.331248\pi\)
\(152\) −2120.95 −1.13179
\(153\) 0 0
\(154\) 2774.85 1.45197
\(155\) −1364.03 −0.706851
\(156\) − 1459.92i − 0.749279i
\(157\) 1076.29 0.547118 0.273559 0.961855i \(-0.411799\pi\)
0.273559 + 0.961855i \(0.411799\pi\)
\(158\) 1358.51i 0.684034i
\(159\) − 489.973i − 0.244386i
\(160\) − 3062.32i − 1.51311i
\(161\) 1678.17 0.821478
\(162\) 2744.70 1.33114
\(163\) 1240.38i 0.596037i 0.954560 + 0.298018i \(0.0963257\pi\)
−0.954560 + 0.298018i \(0.903674\pi\)
\(164\) − 5591.18i − 2.66218i
\(165\) − 184.899i − 0.0872388i
\(166\) −3840.26 −1.79555
\(167\) 2692.11i 1.24743i 0.781650 + 0.623717i \(0.214378\pi\)
−0.781650 + 0.623717i \(0.785622\pi\)
\(168\) −3129.23 −1.43705
\(169\) −386.790 −0.176054
\(170\) 0 0
\(171\) −759.675 −0.339730
\(172\) −2247.91 −0.996520
\(173\) 1819.34i 0.799548i 0.916614 + 0.399774i \(0.130911\pi\)
−0.916614 + 0.399774i \(0.869089\pi\)
\(174\) −1046.59 −0.455986
\(175\) 2486.40i 1.07402i
\(176\) − 3671.46i − 1.57242i
\(177\) − 103.180i − 0.0438161i
\(178\) −8512.05 −3.58430
\(179\) 2087.72 0.871752 0.435876 0.900007i \(-0.356439\pi\)
0.435876 + 0.900007i \(0.356439\pi\)
\(180\) − 2986.53i − 1.23668i
\(181\) 2882.67i 1.18380i 0.806013 + 0.591898i \(0.201621\pi\)
−0.806013 + 0.591898i \(0.798379\pi\)
\(182\) − 6335.06i − 2.58014i
\(183\) 1328.48 0.536635
\(184\) − 4081.74i − 1.63538i
\(185\) 593.853 0.236005
\(186\) 2032.92 0.801401
\(187\) 0 0
\(188\) −5161.54 −2.00236
\(189\) −2369.40 −0.911897
\(190\) 1001.29i 0.382323i
\(191\) −242.916 −0.0920249 −0.0460124 0.998941i \(-0.514651\pi\)
−0.0460124 + 0.998941i \(0.514651\pi\)
\(192\) 1944.27i 0.730810i
\(193\) − 3202.66i − 1.19447i −0.802067 0.597234i \(-0.796266\pi\)
0.802067 0.597234i \(-0.203734\pi\)
\(194\) 1365.20i 0.505235i
\(195\) −422.130 −0.155022
\(196\) −8897.54 −3.24254
\(197\) 3624.82i 1.31095i 0.755215 + 0.655477i \(0.227532\pi\)
−0.755215 + 0.655477i \(0.772468\pi\)
\(198\) − 2417.40i − 0.867662i
\(199\) − 65.9115i − 0.0234791i −0.999931 0.0117395i \(-0.996263\pi\)
0.999931 0.0117395i \(-0.00373690\pi\)
\(200\) 6047.57 2.13814
\(201\) − 1072.65i − 0.376411i
\(202\) 5387.16 1.87643
\(203\) −3273.06 −1.13164
\(204\) 0 0
\(205\) −1616.66 −0.550793
\(206\) 8732.31 2.95344
\(207\) − 1461.99i − 0.490894i
\(208\) −8382.03 −2.79418
\(209\) 584.119i 0.193322i
\(210\) 1477.30i 0.485444i
\(211\) − 3523.72i − 1.14968i −0.818265 0.574841i \(-0.805064\pi\)
0.818265 0.574841i \(-0.194936\pi\)
\(212\) −6085.21 −1.97139
\(213\) −1909.80 −0.614355
\(214\) − 227.753i − 0.0727518i
\(215\) 649.972i 0.206175i
\(216\) 5763.01i 1.81538i
\(217\) 6357.66 1.98888
\(218\) − 9966.68i − 3.09646i
\(219\) 915.385 0.282447
\(220\) −2296.36 −0.703729
\(221\) 0 0
\(222\) −885.061 −0.267574
\(223\) 935.115 0.280807 0.140404 0.990094i \(-0.455160\pi\)
0.140404 + 0.990094i \(0.455160\pi\)
\(224\) 14273.3i 4.25747i
\(225\) 2166.10 0.641808
\(226\) − 72.4466i − 0.0213233i
\(227\) 4987.57i 1.45831i 0.684349 + 0.729155i \(0.260087\pi\)
−0.684349 + 0.729155i \(0.739913\pi\)
\(228\) − 1075.51i − 0.312400i
\(229\) 181.488 0.0523714 0.0261857 0.999657i \(-0.491664\pi\)
0.0261857 + 0.999657i \(0.491664\pi\)
\(230\) −1926.98 −0.552439
\(231\) 861.803i 0.245465i
\(232\) 7960.94i 2.25285i
\(233\) − 2290.78i − 0.644095i −0.946723 0.322048i \(-0.895629\pi\)
0.946723 0.322048i \(-0.104371\pi\)
\(234\) −5518.98 −1.54183
\(235\) 1492.43i 0.414280i
\(236\) −1281.44 −0.353451
\(237\) −421.921 −0.115640
\(238\) 0 0
\(239\) 809.874 0.219190 0.109595 0.993976i \(-0.465045\pi\)
0.109595 + 0.993976i \(0.465045\pi\)
\(240\) 1954.64 0.525714
\(241\) − 5634.28i − 1.50596i −0.658044 0.752979i \(-0.728616\pi\)
0.658044 0.752979i \(-0.271384\pi\)
\(242\) 5264.72 1.39847
\(243\) 3151.92i 0.832081i
\(244\) − 16499.1i − 4.32887i
\(245\) 2572.68i 0.670867i
\(246\) 2409.43 0.624469
\(247\) 1333.56 0.343531
\(248\) − 15463.5i − 3.95941i
\(249\) − 1192.69i − 0.303549i
\(250\) − 6848.27i − 1.73249i
\(251\) 3676.86 0.924627 0.462314 0.886717i \(-0.347019\pi\)
0.462314 + 0.886717i \(0.347019\pi\)
\(252\) 13920.0i 3.47967i
\(253\) −1124.13 −0.279341
\(254\) 6631.45 1.63817
\(255\) 0 0
\(256\) 2180.66 0.532389
\(257\) 180.020 0.0436938 0.0218469 0.999761i \(-0.493045\pi\)
0.0218469 + 0.999761i \(0.493045\pi\)
\(258\) − 968.698i − 0.233754i
\(259\) −2767.91 −0.664051
\(260\) 5242.64i 1.25052i
\(261\) 2851.42i 0.676240i
\(262\) 7011.31i 1.65328i
\(263\) 2035.20 0.477171 0.238586 0.971121i \(-0.423316\pi\)
0.238586 + 0.971121i \(0.423316\pi\)
\(264\) 2096.13 0.488666
\(265\) 1759.51i 0.407871i
\(266\) − 4666.95i − 1.07575i
\(267\) − 2643.64i − 0.605947i
\(268\) −13321.7 −3.03639
\(269\) 2036.45i 0.461578i 0.973004 + 0.230789i \(0.0741308\pi\)
−0.973004 + 0.230789i \(0.925869\pi\)
\(270\) 2720.70 0.613245
\(271\) 5126.86 1.14920 0.574602 0.818433i \(-0.305157\pi\)
0.574602 + 0.818433i \(0.305157\pi\)
\(272\) 0 0
\(273\) 1967.52 0.436189
\(274\) −3168.67 −0.698637
\(275\) − 1665.53i − 0.365218i
\(276\) 2069.80 0.451403
\(277\) 4012.79i 0.870415i 0.900330 + 0.435208i \(0.143325\pi\)
−0.900330 + 0.435208i \(0.856675\pi\)
\(278\) 9225.20i 1.99025i
\(279\) − 5538.67i − 1.18850i
\(280\) 11237.2 2.39839
\(281\) 6603.25 1.40184 0.700920 0.713240i \(-0.252773\pi\)
0.700920 + 0.713240i \(0.252773\pi\)
\(282\) − 2224.28i − 0.469695i
\(283\) − 108.788i − 0.0228508i −0.999935 0.0114254i \(-0.996363\pi\)
0.999935 0.0114254i \(-0.00363689\pi\)
\(284\) 23718.8i 4.95582i
\(285\) −310.977 −0.0646341
\(286\) 4243.58i 0.877370i
\(287\) 7535.15 1.54978
\(288\) 12434.6 2.54415
\(289\) 0 0
\(290\) 3758.33 0.761023
\(291\) −423.998 −0.0854131
\(292\) − 11368.6i − 2.27842i
\(293\) 9133.49 1.82111 0.910553 0.413393i \(-0.135656\pi\)
0.910553 + 0.413393i \(0.135656\pi\)
\(294\) − 3834.25i − 0.760605i
\(295\) 370.521i 0.0731274i
\(296\) 6732.27i 1.32198i
\(297\) 1587.16 0.310088
\(298\) −1231.27 −0.239348
\(299\) 2566.41i 0.496387i
\(300\) 3066.65i 0.590176i
\(301\) − 3029.47i − 0.580119i
\(302\) 10043.2 1.91364
\(303\) 1673.12i 0.317222i
\(304\) −6174.93 −1.16499
\(305\) −4770.62 −0.895622
\(306\) 0 0
\(307\) −4469.59 −0.830921 −0.415461 0.909611i \(-0.636380\pi\)
−0.415461 + 0.909611i \(0.636380\pi\)
\(308\) 10703.1 1.98009
\(309\) 2712.04i 0.499297i
\(310\) −7300.27 −1.33751
\(311\) − 9396.83i − 1.71333i −0.515874 0.856664i \(-0.672533\pi\)
0.515874 0.856664i \(-0.327467\pi\)
\(312\) − 4785.52i − 0.868355i
\(313\) 10276.8i 1.85583i 0.372786 + 0.927917i \(0.378403\pi\)
−0.372786 + 0.927917i \(0.621597\pi\)
\(314\) 5760.29 1.03526
\(315\) 4024.89 0.719927
\(316\) 5240.04i 0.932833i
\(317\) 7547.44i 1.33724i 0.743602 + 0.668622i \(0.233116\pi\)
−0.743602 + 0.668622i \(0.766884\pi\)
\(318\) − 2622.32i − 0.462429i
\(319\) 2192.48 0.384812
\(320\) − 6981.93i − 1.21969i
\(321\) 70.7346 0.0122991
\(322\) 8981.49 1.55441
\(323\) 0 0
\(324\) 10586.9 1.81530
\(325\) −3802.44 −0.648989
\(326\) 6638.47i 1.12782i
\(327\) 3095.41 0.523475
\(328\) − 18327.5i − 3.08526i
\(329\) − 6956.13i − 1.16566i
\(330\) − 989.576i − 0.165074i
\(331\) −6285.90 −1.04382 −0.521910 0.853000i \(-0.674780\pi\)
−0.521910 + 0.853000i \(0.674780\pi\)
\(332\) −14812.6 −2.44864
\(333\) 2411.35i 0.396820i
\(334\) 14408.1i 2.36040i
\(335\) 3851.91i 0.628215i
\(336\) −9110.43 −1.47921
\(337\) 464.322i 0.0750541i 0.999296 + 0.0375271i \(0.0119480\pi\)
−0.999296 + 0.0375271i \(0.988052\pi\)
\(338\) −2070.09 −0.333130
\(339\) 22.5002 0.00360484
\(340\) 0 0
\(341\) −4258.71 −0.676312
\(342\) −4065.76 −0.642839
\(343\) − 2448.47i − 0.385437i
\(344\) −7368.47 −1.15489
\(345\) − 598.472i − 0.0933932i
\(346\) 9737.05i 1.51291i
\(347\) − 7445.29i − 1.15183i −0.817510 0.575914i \(-0.804646\pi\)
0.817510 0.575914i \(-0.195354\pi\)
\(348\) −4036.89 −0.621839
\(349\) −6395.90 −0.980987 −0.490494 0.871445i \(-0.663184\pi\)
−0.490494 + 0.871445i \(0.663184\pi\)
\(350\) 13307.1i 2.03227i
\(351\) − 3623.52i − 0.551023i
\(352\) − 9561.02i − 1.44774i
\(353\) −8031.82 −1.21102 −0.605511 0.795837i \(-0.707031\pi\)
−0.605511 + 0.795837i \(0.707031\pi\)
\(354\) − 552.214i − 0.0829091i
\(355\) 6858.17 1.02534
\(356\) −32832.6 −4.88799
\(357\) 0 0
\(358\) 11173.4 1.64953
\(359\) 2730.88 0.401477 0.200738 0.979645i \(-0.435666\pi\)
0.200738 + 0.979645i \(0.435666\pi\)
\(360\) − 9789.60i − 1.43321i
\(361\) −5876.59 −0.856770
\(362\) 15428.0i 2.23999i
\(363\) 1635.09i 0.236419i
\(364\) − 24435.5i − 3.51860i
\(365\) −3287.18 −0.471394
\(366\) 7109.99 1.01542
\(367\) − 2802.02i − 0.398541i −0.979945 0.199270i \(-0.936143\pi\)
0.979945 0.199270i \(-0.0638572\pi\)
\(368\) − 11883.6i − 1.68335i
\(369\) − 6564.48i − 0.926106i
\(370\) 3178.28 0.446570
\(371\) − 8200.93i − 1.14763i
\(372\) 7841.35 1.09289
\(373\) 8987.58 1.24761 0.623806 0.781579i \(-0.285585\pi\)
0.623806 + 0.781579i \(0.285585\pi\)
\(374\) 0 0
\(375\) 2126.91 0.292888
\(376\) −16919.1 −2.32058
\(377\) − 5005.47i − 0.683807i
\(378\) −12681.0 −1.72550
\(379\) 7786.77i 1.05536i 0.849445 + 0.527678i \(0.176937\pi\)
−0.849445 + 0.527678i \(0.823063\pi\)
\(380\) 3862.18i 0.521383i
\(381\) 2059.57i 0.276942i
\(382\) −1300.08 −0.174130
\(383\) −7557.45 −1.00827 −0.504135 0.863625i \(-0.668189\pi\)
−0.504135 + 0.863625i \(0.668189\pi\)
\(384\) 3583.51i 0.476225i
\(385\) − 3094.76i − 0.409672i
\(386\) − 17140.5i − 2.26018i
\(387\) −2639.22 −0.346664
\(388\) 5265.84i 0.689001i
\(389\) −10883.8 −1.41859 −0.709296 0.704911i \(-0.750987\pi\)
−0.709296 + 0.704911i \(0.750987\pi\)
\(390\) −2259.23 −0.293334
\(391\) 0 0
\(392\) −29165.4 −3.75785
\(393\) −2177.54 −0.279497
\(394\) 19399.9i 2.48060i
\(395\) 1515.13 0.192999
\(396\) − 9324.37i − 1.18325i
\(397\) − 2103.72i − 0.265952i −0.991119 0.132976i \(-0.957547\pi\)
0.991119 0.132976i \(-0.0424533\pi\)
\(398\) − 352.756i − 0.0444273i
\(399\) 1449.44 0.181862
\(400\) 17606.9 2.20086
\(401\) − 10029.6i − 1.24901i −0.781021 0.624505i \(-0.785301\pi\)
0.781021 0.624505i \(-0.214699\pi\)
\(402\) − 5740.77i − 0.712247i
\(403\) 9722.75i 1.20180i
\(404\) 20779.3 2.55893
\(405\) − 3061.13i − 0.375578i
\(406\) −17517.3 −2.14130
\(407\) 1854.10 0.225809
\(408\) 0 0
\(409\) −515.825 −0.0623616 −0.0311808 0.999514i \(-0.509927\pi\)
−0.0311808 + 0.999514i \(0.509927\pi\)
\(410\) −8652.33 −1.04222
\(411\) − 984.113i − 0.118109i
\(412\) 33682.2 4.02767
\(413\) − 1726.97i − 0.205760i
\(414\) − 7824.50i − 0.928873i
\(415\) 4282.99i 0.506611i
\(416\) −21828.1 −2.57262
\(417\) −2865.12 −0.336465
\(418\) 3126.18i 0.365805i
\(419\) − 9370.98i − 1.09261i −0.837587 0.546304i \(-0.816034\pi\)
0.837587 0.546304i \(-0.183966\pi\)
\(420\) 5698.22i 0.662011i
\(421\) −283.696 −0.0328421 −0.0164210 0.999865i \(-0.505227\pi\)
−0.0164210 + 0.999865i \(0.505227\pi\)
\(422\) − 18858.8i − 2.17544i
\(423\) −6060.04 −0.696571
\(424\) −19946.8 −2.28468
\(425\) 0 0
\(426\) −10221.2 −1.16249
\(427\) 22235.5 2.52003
\(428\) − 878.487i − 0.0992133i
\(429\) −1317.95 −0.148325
\(430\) 3478.63i 0.390126i
\(431\) 8243.79i 0.921321i 0.887576 + 0.460661i \(0.152387\pi\)
−0.887576 + 0.460661i \(0.847613\pi\)
\(432\) 16778.4i 1.86864i
\(433\) 1340.39 0.148765 0.0743824 0.997230i \(-0.476301\pi\)
0.0743824 + 0.997230i \(0.476301\pi\)
\(434\) 34026.0 3.76336
\(435\) 1167.25i 0.128656i
\(436\) − 38443.4i − 4.22272i
\(437\) 1890.64i 0.206960i
\(438\) 4899.11 0.534449
\(439\) 4717.90i 0.512923i 0.966554 + 0.256461i \(0.0825566\pi\)
−0.966554 + 0.256461i \(0.917443\pi\)
\(440\) −7527.27 −0.815565
\(441\) −10446.4 −1.12800
\(442\) 0 0
\(443\) 14002.0 1.50171 0.750855 0.660467i \(-0.229642\pi\)
0.750855 + 0.660467i \(0.229642\pi\)
\(444\) −3413.85 −0.364897
\(445\) 9493.38i 1.01130i
\(446\) 5004.71 0.531345
\(447\) − 382.403i − 0.0404632i
\(448\) 32542.3i 3.43187i
\(449\) 4574.27i 0.480787i 0.970676 + 0.240393i \(0.0772764\pi\)
−0.970676 + 0.240393i \(0.922724\pi\)
\(450\) 11592.9 1.21443
\(451\) −5047.46 −0.526997
\(452\) − 279.440i − 0.0290791i
\(453\) 3119.16i 0.323512i
\(454\) 26693.3i 2.75942i
\(455\) −7065.42 −0.727982
\(456\) − 3525.43i − 0.362046i
\(457\) 1535.79 0.157202 0.0786009 0.996906i \(-0.474955\pi\)
0.0786009 + 0.996906i \(0.474955\pi\)
\(458\) 971.318 0.0990976
\(459\) 0 0
\(460\) −7432.72 −0.753374
\(461\) −14514.7 −1.46642 −0.733209 0.680003i \(-0.761978\pi\)
−0.733209 + 0.680003i \(0.761978\pi\)
\(462\) 4612.34i 0.464471i
\(463\) 145.386 0.0145932 0.00729659 0.999973i \(-0.497677\pi\)
0.00729659 + 0.999973i \(0.497677\pi\)
\(464\) 23177.4i 2.31894i
\(465\) − 2267.29i − 0.226114i
\(466\) − 12260.2i − 1.21876i
\(467\) −6038.32 −0.598330 −0.299165 0.954201i \(-0.596708\pi\)
−0.299165 + 0.954201i \(0.596708\pi\)
\(468\) −21287.8 −2.10262
\(469\) − 17953.4i − 1.76762i
\(470\) 7987.46i 0.783903i
\(471\) 1789.01i 0.175017i
\(472\) −4200.45 −0.409622
\(473\) 2029.31i 0.197268i
\(474\) −2258.11 −0.218815
\(475\) −2801.20 −0.270585
\(476\) 0 0
\(477\) −7144.50 −0.685795
\(478\) 4334.42 0.414753
\(479\) 9142.21i 0.872063i 0.899931 + 0.436031i \(0.143616\pi\)
−0.899931 + 0.436031i \(0.856384\pi\)
\(480\) 5090.17 0.484028
\(481\) − 4232.95i − 0.401260i
\(482\) − 30154.5i − 2.84958i
\(483\) 2789.43i 0.262782i
\(484\) 20307.0 1.90712
\(485\) 1522.59 0.142551
\(486\) 16869.0i 1.57447i
\(487\) − 4376.57i − 0.407231i −0.979051 0.203615i \(-0.934731\pi\)
0.979051 0.203615i \(-0.0652692\pi\)
\(488\) − 54082.6i − 5.01681i
\(489\) −2061.75 −0.190666
\(490\) 13768.9i 1.26942i
\(491\) −9490.30 −0.872284 −0.436142 0.899878i \(-0.643655\pi\)
−0.436142 + 0.899878i \(0.643655\pi\)
\(492\) 9293.62 0.851603
\(493\) 0 0
\(494\) 7137.16 0.650032
\(495\) −2696.10 −0.244809
\(496\) − 45020.4i − 4.07556i
\(497\) −31965.4 −2.88500
\(498\) − 6383.24i − 0.574377i
\(499\) 2331.32i 0.209147i 0.994517 + 0.104573i \(0.0333477\pi\)
−0.994517 + 0.104573i \(0.966652\pi\)
\(500\) − 26415.1i − 2.36264i
\(501\) −4474.80 −0.399041
\(502\) 19678.4 1.74959
\(503\) − 4372.95i − 0.387635i −0.981038 0.193817i \(-0.937913\pi\)
0.981038 0.193817i \(-0.0620869\pi\)
\(504\) 45628.6i 4.03266i
\(505\) − 6008.23i − 0.529431i
\(506\) −6016.30 −0.528572
\(507\) − 642.919i − 0.0563176i
\(508\) 25578.8 2.23401
\(509\) 4983.33 0.433954 0.216977 0.976177i \(-0.430380\pi\)
0.216977 + 0.976177i \(0.430380\pi\)
\(510\) 0 0
\(511\) 15321.3 1.32637
\(512\) −5576.30 −0.481328
\(513\) − 2669.40i − 0.229740i
\(514\) 963.460 0.0826778
\(515\) − 9739.03i − 0.833307i
\(516\) − 3736.46i − 0.318776i
\(517\) 4659.60i 0.396381i
\(518\) −14813.7 −1.25652
\(519\) −3024.09 −0.255767
\(520\) 17185.0i 1.44925i
\(521\) 6480.02i 0.544903i 0.962169 + 0.272452i \(0.0878345\pi\)
−0.962169 + 0.272452i \(0.912166\pi\)
\(522\) 15260.7i 1.27959i
\(523\) −15597.0 −1.30403 −0.652017 0.758204i \(-0.726077\pi\)
−0.652017 + 0.758204i \(0.726077\pi\)
\(524\) 27044.0i 2.25462i
\(525\) −4132.87 −0.343568
\(526\) 10892.4 0.902907
\(527\) 0 0
\(528\) 6102.67 0.503001
\(529\) 8528.48 0.700952
\(530\) 9416.83i 0.771775i
\(531\) −1504.50 −0.122957
\(532\) − 18001.3i − 1.46702i
\(533\) 11523.5i 0.936468i
\(534\) − 14148.7i − 1.14658i
\(535\) −254.010 −0.0205268
\(536\) −43667.5 −3.51894
\(537\) 3470.19i 0.278864i
\(538\) 10899.0i 0.873402i
\(539\) 8032.29i 0.641883i
\(540\) 10494.2 0.836297
\(541\) − 10629.6i − 0.844739i −0.906424 0.422370i \(-0.861198\pi\)
0.906424 0.422370i \(-0.138802\pi\)
\(542\) 27438.8 2.17453
\(543\) −4791.55 −0.378683
\(544\) 0 0
\(545\) −11115.7 −0.873660
\(546\) 10530.1 0.825359
\(547\) 12199.5i 0.953588i 0.879015 + 0.476794i \(0.158201\pi\)
−0.879015 + 0.476794i \(0.841799\pi\)
\(548\) −12222.2 −0.952747
\(549\) − 19371.1i − 1.50590i
\(550\) − 8913.85i − 0.691068i
\(551\) − 3687.46i − 0.285102i
\(552\) 6784.64 0.523140
\(553\) −7061.91 −0.543044
\(554\) 21476.3i 1.64701i
\(555\) 987.098i 0.0754954i
\(556\) 35583.4i 2.71416i
\(557\) −17588.3 −1.33796 −0.668978 0.743282i \(-0.733268\pi\)
−0.668978 + 0.743282i \(0.733268\pi\)
\(558\) − 29642.8i − 2.24889i
\(559\) 4632.96 0.350542
\(560\) 32715.8 2.46874
\(561\) 0 0
\(562\) 35340.4 2.65257
\(563\) 4001.08 0.299513 0.149756 0.988723i \(-0.452151\pi\)
0.149756 + 0.988723i \(0.452151\pi\)
\(564\) − 8579.48i − 0.640534i
\(565\) −80.7988 −0.00601634
\(566\) − 582.229i − 0.0432384i
\(567\) 14267.7i 1.05677i
\(568\) 77748.3i 5.74339i
\(569\) 8055.37 0.593495 0.296747 0.954956i \(-0.404098\pi\)
0.296747 + 0.954956i \(0.404098\pi\)
\(570\) −1664.34 −0.122301
\(571\) − 12525.5i − 0.917998i −0.888437 0.458999i \(-0.848208\pi\)
0.888437 0.458999i \(-0.151792\pi\)
\(572\) 16368.3i 1.19649i
\(573\) − 403.772i − 0.0294378i
\(574\) 40327.9 2.93250
\(575\) − 5390.88i − 0.390983i
\(576\) 28350.2 2.05080
\(577\) 7783.77 0.561599 0.280799 0.959766i \(-0.409400\pi\)
0.280799 + 0.959766i \(0.409400\pi\)
\(578\) 0 0
\(579\) 5323.43 0.382097
\(580\) 14496.6 1.03782
\(581\) − 19962.7i − 1.42546i
\(582\) −2269.22 −0.161619
\(583\) 5493.44i 0.390249i
\(584\) − 37265.4i − 2.64050i
\(585\) 6155.25i 0.435023i
\(586\) 48882.1 3.44591
\(587\) −17282.7 −1.21522 −0.607608 0.794237i \(-0.707871\pi\)
−0.607608 + 0.794237i \(0.707871\pi\)
\(588\) − 14789.4i − 1.03725i
\(589\) 7162.62i 0.501070i
\(590\) 1983.02i 0.138372i
\(591\) −6025.15 −0.419360
\(592\) 19600.3i 1.36076i
\(593\) −22285.6 −1.54327 −0.771636 0.636065i \(-0.780561\pi\)
−0.771636 + 0.636065i \(0.780561\pi\)
\(594\) 8494.41 0.586751
\(595\) 0 0
\(596\) −4749.25 −0.326404
\(597\) 109.557 0.00751070
\(598\) 13735.4i 0.939266i
\(599\) 14494.3 0.988681 0.494340 0.869268i \(-0.335410\pi\)
0.494340 + 0.869268i \(0.335410\pi\)
\(600\) 10052.2i 0.683967i
\(601\) − 13389.8i − 0.908790i −0.890800 0.454395i \(-0.849856\pi\)
0.890800 0.454395i \(-0.150144\pi\)
\(602\) − 16213.6i − 1.09770i
\(603\) −15640.7 −1.05628
\(604\) 38738.4 2.60967
\(605\) − 5871.68i − 0.394575i
\(606\) 8954.49i 0.600250i
\(607\) 5081.09i 0.339761i 0.985465 + 0.169881i \(0.0543382\pi\)
−0.985465 + 0.169881i \(0.945662\pi\)
\(608\) −16080.4 −1.07261
\(609\) − 5440.45i − 0.362000i
\(610\) −25532.2 −1.69470
\(611\) 10638.0 0.704365
\(612\) 0 0
\(613\) −29310.1 −1.93120 −0.965598 0.260040i \(-0.916264\pi\)
−0.965598 + 0.260040i \(0.916264\pi\)
\(614\) −23921.1 −1.57227
\(615\) − 2687.20i − 0.176193i
\(616\) 35084.1 2.29477
\(617\) − 9815.75i − 0.640466i −0.947339 0.320233i \(-0.896239\pi\)
0.947339 0.320233i \(-0.103761\pi\)
\(618\) 14514.8i 0.944773i
\(619\) − 10696.9i − 0.694579i −0.937758 0.347290i \(-0.887102\pi\)
0.937758 0.347290i \(-0.112898\pi\)
\(620\) −28158.5 −1.82399
\(621\) 5137.22 0.331964
\(622\) − 50291.5i − 3.24197i
\(623\) − 44247.9i − 2.84552i
\(624\) − 13932.5i − 0.893827i
\(625\) 3533.62 0.226151
\(626\) 55000.8i 3.51162i
\(627\) −970.917 −0.0618416
\(628\) 22218.5 1.41181
\(629\) 0 0
\(630\) 21541.1 1.36225
\(631\) 6104.87 0.385152 0.192576 0.981282i \(-0.438316\pi\)
0.192576 + 0.981282i \(0.438316\pi\)
\(632\) 17176.4i 1.08108i
\(633\) 5857.10 0.367771
\(634\) 40393.7i 2.53034i
\(635\) − 7395.98i − 0.462205i
\(636\) − 10114.8i − 0.630625i
\(637\) 18337.9 1.14062
\(638\) 11734.1 0.728144
\(639\) 27847.7i 1.72400i
\(640\) − 12868.5i − 0.794800i
\(641\) − 3753.93i − 0.231312i −0.993289 0.115656i \(-0.963103\pi\)
0.993289 0.115656i \(-0.0368971\pi\)
\(642\) 378.569 0.0232725
\(643\) 22480.9i 1.37879i 0.724386 + 0.689394i \(0.242123\pi\)
−0.724386 + 0.689394i \(0.757877\pi\)
\(644\) 34643.3 2.11978
\(645\) −1080.38 −0.0659532
\(646\) 0 0
\(647\) 20385.1 1.23867 0.619337 0.785126i \(-0.287402\pi\)
0.619337 + 0.785126i \(0.287402\pi\)
\(648\) 34702.9 2.10379
\(649\) 1156.82i 0.0699680i
\(650\) −20350.5 −1.22802
\(651\) 10567.7i 0.636220i
\(652\) 25605.9i 1.53804i
\(653\) − 7944.24i − 0.476083i −0.971255 0.238041i \(-0.923495\pi\)
0.971255 0.238041i \(-0.0765054\pi\)
\(654\) 16566.5 0.990524
\(655\) 7819.63 0.466470
\(656\) − 53358.5i − 3.17576i
\(657\) − 13347.6i − 0.792603i
\(658\) − 37229.0i − 2.20568i
\(659\) −672.385 −0.0397457 −0.0198728 0.999803i \(-0.506326\pi\)
−0.0198728 + 0.999803i \(0.506326\pi\)
\(660\) − 3816.98i − 0.225115i
\(661\) 7967.64 0.468843 0.234421 0.972135i \(-0.424680\pi\)
0.234421 + 0.972135i \(0.424680\pi\)
\(662\) −33642.0 −1.97512
\(663\) 0 0
\(664\) −48554.5 −2.83777
\(665\) −5205.00 −0.303520
\(666\) 12905.4i 0.750865i
\(667\) 7096.48 0.411959
\(668\) 55574.7i 3.21894i
\(669\) 1554.34i 0.0898271i
\(670\) 20615.3i 1.18871i
\(671\) −14894.6 −0.856928
\(672\) −23724.9 −1.36192
\(673\) − 20798.2i − 1.19125i −0.803262 0.595626i \(-0.796904\pi\)
0.803262 0.595626i \(-0.203096\pi\)
\(674\) 2485.04i 0.142018i
\(675\) 7611.38i 0.434018i
\(676\) −7984.72 −0.454297
\(677\) − 19180.9i − 1.08890i −0.838794 0.544449i \(-0.816739\pi\)
0.838794 0.544449i \(-0.183261\pi\)
\(678\) 120.420 0.00682110
\(679\) −7096.68 −0.401098
\(680\) 0 0
\(681\) −8290.29 −0.466497
\(682\) −22792.5 −1.27972
\(683\) − 7081.83i − 0.396748i −0.980126 0.198374i \(-0.936434\pi\)
0.980126 0.198374i \(-0.0635661\pi\)
\(684\) −15682.4 −0.876655
\(685\) 3533.98i 0.197119i
\(686\) − 13104.1i − 0.729326i
\(687\) 301.668i 0.0167530i
\(688\) −21452.5 −1.18876
\(689\) 12541.7 0.693468
\(690\) − 3203.00i − 0.176719i
\(691\) 27665.8i 1.52309i 0.648110 + 0.761547i \(0.275560\pi\)
−0.648110 + 0.761547i \(0.724440\pi\)
\(692\) 37557.7i 2.06319i
\(693\) 12566.3 0.688823
\(694\) − 39847.0i − 2.17950i
\(695\) 10288.8 0.561546
\(696\) −13232.6 −0.720662
\(697\) 0 0
\(698\) −34230.7 −1.85623
\(699\) 3807.72 0.206039
\(700\) 51328.1i 2.77146i
\(701\) −7813.26 −0.420974 −0.210487 0.977597i \(-0.567505\pi\)
−0.210487 + 0.977597i \(0.567505\pi\)
\(702\) − 19393.0i − 1.04265i
\(703\) − 3118.36i − 0.167299i
\(704\) − 21798.6i − 1.16700i
\(705\) −2480.71 −0.132523
\(706\) −42986.0 −2.29150
\(707\) 28003.9i 1.48967i
\(708\) − 2129.99i − 0.113065i
\(709\) − 21810.2i − 1.15529i −0.816288 0.577645i \(-0.803972\pi\)
0.816288 0.577645i \(-0.196028\pi\)
\(710\) 36704.7 1.94015
\(711\) 6152.20i 0.324509i
\(712\) −107623. −5.66479
\(713\) −13784.4 −0.724023
\(714\) 0 0
\(715\) 4732.81 0.247548
\(716\) 43098.0 2.24951
\(717\) 1346.17i 0.0701164i
\(718\) 14615.6 0.759677
\(719\) − 2333.31i − 0.121026i −0.998167 0.0605131i \(-0.980726\pi\)
0.998167 0.0605131i \(-0.0192737\pi\)
\(720\) − 28501.4i − 1.47526i
\(721\) 45392.9i 2.34469i
\(722\) −31451.3 −1.62119
\(723\) 9365.26 0.481740
\(724\) 59508.6i 3.05472i
\(725\) 10514.3i 0.538606i
\(726\) 8750.97i 0.447354i
\(727\) 545.182 0.0278125 0.0139063 0.999903i \(-0.495573\pi\)
0.0139063 + 0.999903i \(0.495573\pi\)
\(728\) − 80097.8i − 4.07778i
\(729\) 8607.59 0.437311
\(730\) −17592.9 −0.891975
\(731\) 0 0
\(732\) 27424.6 1.38476
\(733\) 2753.41 0.138744 0.0693721 0.997591i \(-0.477900\pi\)
0.0693721 + 0.997591i \(0.477900\pi\)
\(734\) − 14996.4i − 0.754122i
\(735\) −4276.29 −0.214603
\(736\) − 30946.6i − 1.54987i
\(737\) 12026.2i 0.601074i
\(738\) − 35132.9i − 1.75238i
\(739\) −25776.3 −1.28308 −0.641540 0.767089i \(-0.721704\pi\)
−0.641540 + 0.767089i \(0.721704\pi\)
\(740\) 12259.2 0.608999
\(741\) 2216.63i 0.109892i
\(742\) − 43891.1i − 2.17156i
\(743\) − 17597.0i − 0.868872i −0.900703 0.434436i \(-0.856948\pi\)
0.900703 0.434436i \(-0.143052\pi\)
\(744\) 25703.3 1.26657
\(745\) 1373.22i 0.0675314i
\(746\) 48101.3 2.36074
\(747\) −17391.1 −0.851818
\(748\) 0 0
\(749\) 1183.92 0.0577565
\(750\) 11383.1 0.554204
\(751\) 1515.51i 0.0736373i 0.999322 + 0.0368186i \(0.0117224\pi\)
−0.999322 + 0.0368186i \(0.988278\pi\)
\(752\) −49258.3 −2.38865
\(753\) 6111.65i 0.295778i
\(754\) − 26789.1i − 1.29390i
\(755\) − 11201.0i − 0.539929i
\(756\) −48912.9 −2.35310
\(757\) −19464.1 −0.934523 −0.467261 0.884119i \(-0.654759\pi\)
−0.467261 + 0.884119i \(0.654759\pi\)
\(758\) 41674.6i 1.99695i
\(759\) − 1868.52i − 0.0893583i
\(760\) 12659.9i 0.604242i
\(761\) 11937.8 0.568652 0.284326 0.958728i \(-0.408230\pi\)
0.284326 + 0.958728i \(0.408230\pi\)
\(762\) 11022.7i 0.524031i
\(763\) 51809.5 2.45823
\(764\) −5014.65 −0.237465
\(765\) 0 0
\(766\) −40447.2 −1.90786
\(767\) 2641.05 0.124332
\(768\) 3624.68i 0.170305i
\(769\) 26884.3 1.26069 0.630345 0.776315i \(-0.282913\pi\)
0.630345 + 0.776315i \(0.282913\pi\)
\(770\) − 16563.1i − 0.775184i
\(771\) 299.227i 0.0139772i
\(772\) − 66114.3i − 3.08226i
\(773\) 12551.4 0.584013 0.292006 0.956416i \(-0.405677\pi\)
0.292006 + 0.956416i \(0.405677\pi\)
\(774\) −14125.0 −0.655959
\(775\) − 20423.1i − 0.946607i
\(776\) 17261.0i 0.798497i
\(777\) − 4600.79i − 0.212423i
\(778\) −58249.9 −2.68427
\(779\) 8489.19i 0.390445i
\(780\) −8714.27 −0.400027
\(781\) 21412.2 0.981036
\(782\) 0 0
\(783\) −10019.5 −0.457303
\(784\) −84912.2 −3.86808
\(785\) − 6424.38i − 0.292097i
\(786\) −11654.1 −0.528867
\(787\) 17781.7i 0.805400i 0.915332 + 0.402700i \(0.131928\pi\)
−0.915332 + 0.402700i \(0.868072\pi\)
\(788\) 74829.3i 3.38285i
\(789\) 3382.90i 0.152642i
\(790\) 8108.94 0.365194
\(791\) 376.597 0.0169283
\(792\) − 30564.6i − 1.37129i
\(793\) 34004.7i 1.52275i
\(794\) − 11259.1i − 0.503235i
\(795\) −2924.64 −0.130473
\(796\) − 1360.65i − 0.0605865i
\(797\) −36529.5 −1.62352 −0.811758 0.583995i \(-0.801489\pi\)
−0.811758 + 0.583995i \(0.801489\pi\)
\(798\) 7757.37 0.344120
\(799\) 0 0
\(800\) 45850.9 2.02634
\(801\) −38548.0 −1.70041
\(802\) − 53677.9i − 2.36338i
\(803\) −10263.1 −0.451028
\(804\) − 22143.2i − 0.971308i
\(805\) − 10017.0i − 0.438573i
\(806\) 52035.9i 2.27405i
\(807\) −3384.97 −0.147654
\(808\) 68112.9 2.96560
\(809\) 19732.6i 0.857555i 0.903410 + 0.428777i \(0.141055\pi\)
−0.903410 + 0.428777i \(0.858945\pi\)
\(810\) − 16383.1i − 0.710671i
\(811\) − 14943.7i − 0.647035i −0.946222 0.323517i \(-0.895135\pi\)
0.946222 0.323517i \(-0.104865\pi\)
\(812\) −67567.6 −2.92014
\(813\) 8521.82i 0.367618i
\(814\) 9923.07 0.427277
\(815\) 7403.80 0.318213
\(816\) 0 0
\(817\) 3413.04 0.146153
\(818\) −2760.68 −0.118001
\(819\) − 28689.2i − 1.22403i
\(820\) −33373.7 −1.42129
\(821\) 31002.2i 1.31789i 0.752193 + 0.658943i \(0.228996\pi\)
−0.752193 + 0.658943i \(0.771004\pi\)
\(822\) − 5266.94i − 0.223486i
\(823\) − 27393.7i − 1.16025i −0.814528 0.580124i \(-0.803004\pi\)
0.814528 0.580124i \(-0.196996\pi\)
\(824\) 110408. 4.66775
\(825\) 2768.42 0.116829
\(826\) − 9242.70i − 0.389340i
\(827\) − 674.496i − 0.0283610i −0.999899 0.0141805i \(-0.995486\pi\)
0.999899 0.0141805i \(-0.00451394\pi\)
\(828\) − 30180.6i − 1.26673i
\(829\) 42790.2 1.79272 0.896359 0.443328i \(-0.146202\pi\)
0.896359 + 0.443328i \(0.146202\pi\)
\(830\) 22922.4i 0.958613i
\(831\) −6670.02 −0.278436
\(832\) −49766.8 −2.07374
\(833\) 0 0
\(834\) −15334.1 −0.636661
\(835\) 16069.2 0.665983
\(836\) 12058.3i 0.498857i
\(837\) 19462.1 0.803715
\(838\) − 50153.2i − 2.06744i
\(839\) 25582.3i 1.05268i 0.850274 + 0.526340i \(0.176436\pi\)
−0.850274 + 0.526340i \(0.823564\pi\)
\(840\) 18678.3i 0.767218i
\(841\) 10548.2 0.432498
\(842\) −1518.33 −0.0621440
\(843\) 10975.9i 0.448433i
\(844\) − 72742.2i − 2.96669i
\(845\) 2308.74i 0.0939919i
\(846\) −32433.2 −1.31806
\(847\) 27367.4i 1.11022i
\(848\) −58073.2 −2.35170
\(849\) 180.826 0.00730971
\(850\) 0 0
\(851\) 6001.24 0.241739
\(852\) −39425.2 −1.58531
\(853\) − 26776.3i − 1.07480i −0.843328 0.537400i \(-0.819407\pi\)
0.843328 0.537400i \(-0.180593\pi\)
\(854\) 119004. 4.76841
\(855\) 4534.49i 0.181376i
\(856\) − 2879.61i − 0.114980i
\(857\) − 2239.80i − 0.0892767i −0.999003 0.0446383i \(-0.985786\pi\)
0.999003 0.0446383i \(-0.0142135\pi\)
\(858\) −7053.64 −0.280661
\(859\) 13841.5 0.549784 0.274892 0.961475i \(-0.411358\pi\)
0.274892 + 0.961475i \(0.411358\pi\)
\(860\) 13417.7i 0.532024i
\(861\) 12524.9i 0.495756i
\(862\) 44120.5i 1.74333i
\(863\) 41377.2 1.63209 0.816046 0.577987i \(-0.196161\pi\)
0.816046 + 0.577987i \(0.196161\pi\)
\(864\) 43693.4i 1.72046i
\(865\) 10859.6 0.426864
\(866\) 7173.74 0.281494
\(867\) 0 0
\(868\) 131245. 5.13219
\(869\) 4730.46 0.184660
\(870\) 6247.07i 0.243443i
\(871\) 27456.2 1.06810
\(872\) − 126014.i − 4.89379i
\(873\) 6182.49i 0.239686i
\(874\) 10118.7i 0.391612i
\(875\) 35599.2 1.37540
\(876\) 18896.8 0.728840
\(877\) 12605.0i 0.485337i 0.970109 + 0.242668i \(0.0780227\pi\)
−0.970109 + 0.242668i \(0.921977\pi\)
\(878\) 25250.1i 0.970556i
\(879\) 15181.6i 0.582552i
\(880\) −21914.9 −0.839489
\(881\) − 20007.8i − 0.765132i −0.923928 0.382566i \(-0.875040\pi\)
0.923928 0.382566i \(-0.124960\pi\)
\(882\) −55908.8 −2.13441
\(883\) 43309.8 1.65061 0.825307 0.564685i \(-0.191002\pi\)
0.825307 + 0.564685i \(0.191002\pi\)
\(884\) 0 0
\(885\) −615.877 −0.0233926
\(886\) 74938.5 2.84154
\(887\) 39472.3i 1.49420i 0.664714 + 0.747098i \(0.268553\pi\)
−0.664714 + 0.747098i \(0.731447\pi\)
\(888\) −11190.3 −0.422886
\(889\) 34472.1i 1.30051i
\(890\) 50808.3i 1.91359i
\(891\) − 9557.31i − 0.359351i
\(892\) 19304.1 0.724607
\(893\) 7836.86 0.293673
\(894\) − 2046.61i − 0.0765647i
\(895\) − 12461.6i − 0.465413i
\(896\) 59979.1i 2.23634i
\(897\) −4265.87 −0.158789
\(898\) 24481.4i 0.909748i
\(899\) 26884.7 0.997392
\(900\) 44716.1 1.65615
\(901\) 0 0
\(902\) −27013.9 −0.997187
\(903\) 5035.56 0.185574
\(904\) − 915.984i − 0.0337004i
\(905\) 17206.6 0.632008
\(906\) 16693.7i 0.612152i
\(907\) 8879.42i 0.325068i 0.986703 + 0.162534i \(0.0519667\pi\)
−0.986703 + 0.162534i \(0.948033\pi\)
\(908\) 102961.i 3.76309i
\(909\) 24396.5 0.890187
\(910\) −37813.9 −1.37749
\(911\) − 9321.10i − 0.338992i −0.985531 0.169496i \(-0.945786\pi\)
0.985531 0.169496i \(-0.0542140\pi\)
\(912\) − 10263.9i − 0.372667i
\(913\) 13372.1i 0.484724i
\(914\) 8219.50 0.297458
\(915\) − 7929.68i − 0.286500i
\(916\) 3746.56 0.135142
\(917\) −36446.7 −1.31251
\(918\) 0 0
\(919\) −8783.98 −0.315296 −0.157648 0.987495i \(-0.550391\pi\)
−0.157648 + 0.987495i \(0.550391\pi\)
\(920\) −24363.9 −0.873101
\(921\) − 7429.31i − 0.265803i
\(922\) −77682.5 −2.77477
\(923\) − 48884.6i − 1.74329i
\(924\) 17790.7i 0.633410i
\(925\) 8891.52i 0.316056i
\(926\) 778.100 0.0276133
\(927\) 39545.4 1.40112
\(928\) 60357.5i 2.13506i
\(929\) 24381.1i 0.861054i 0.902578 + 0.430527i \(0.141672\pi\)
−0.902578 + 0.430527i \(0.858328\pi\)
\(930\) − 12134.4i − 0.427854i
\(931\) 13509.3 0.475563
\(932\) − 47289.9i − 1.66205i
\(933\) 15619.3 0.548075
\(934\) −32316.9 −1.13216
\(935\) 0 0
\(936\) −69779.7 −2.43677
\(937\) −23120.9 −0.806113 −0.403056 0.915175i \(-0.632052\pi\)
−0.403056 + 0.915175i \(0.632052\pi\)
\(938\) − 96086.3i − 3.34470i
\(939\) −17081.9 −0.593661
\(940\) 30809.2i 1.06903i
\(941\) − 46768.5i − 1.62020i −0.586291 0.810101i \(-0.699412\pi\)
0.586291 0.810101i \(-0.300588\pi\)
\(942\) 9574.70i 0.331168i
\(943\) −16337.3 −0.564175
\(944\) −12229.2 −0.421638
\(945\) 14142.9i 0.486846i
\(946\) 10860.8i 0.373271i
\(947\) 38093.5i 1.30715i 0.756861 + 0.653576i \(0.226732\pi\)
−0.756861 + 0.653576i \(0.773268\pi\)
\(948\) −8709.95 −0.298403
\(949\) 23430.8i 0.801471i
\(950\) −14992.0 −0.512003
\(951\) −12545.3 −0.427770
\(952\) 0 0
\(953\) −31247.7 −1.06213 −0.531066 0.847330i \(-0.678208\pi\)
−0.531066 + 0.847330i \(0.678208\pi\)
\(954\) −38237.1 −1.29767
\(955\) 1449.96i 0.0491305i
\(956\) 16718.7 0.565608
\(957\) 3644.31i 0.123097i
\(958\) 48928.8i 1.65012i
\(959\) − 16471.6i − 0.554637i
\(960\) 11605.3 0.390166
\(961\) −22430.5 −0.752928
\(962\) − 22654.6i − 0.759266i
\(963\) − 1031.41i − 0.0345138i
\(964\) − 116312.i − 3.88604i
\(965\) −19116.6 −0.637705
\(966\) 14929.0i 0.497238i
\(967\) 48567.3 1.61512 0.807558 0.589788i \(-0.200789\pi\)
0.807558 + 0.589788i \(0.200789\pi\)
\(968\) 66564.8 2.21020
\(969\) 0 0
\(970\) 8148.86 0.269736
\(971\) −16286.5 −0.538267 −0.269133 0.963103i \(-0.586737\pi\)
−0.269133 + 0.963103i \(0.586737\pi\)
\(972\) 65066.9i 2.14714i
\(973\) −47955.1 −1.58003
\(974\) − 23423.3i − 0.770565i
\(975\) − 6320.38i − 0.207604i
\(976\) − 157456.i − 5.16398i
\(977\) −12003.7 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(978\) −11034.4 −0.360779
\(979\) 29639.7i 0.967610i
\(980\) 53109.3i 1.73114i
\(981\) − 45135.4i − 1.46897i
\(982\) −50791.8 −1.65054
\(983\) 10008.2i 0.324731i 0.986731 + 0.162365i \(0.0519124\pi\)
−0.986731 + 0.162365i \(0.948088\pi\)
\(984\) 30463.8 0.986940
\(985\) 21636.5 0.699895
\(986\) 0 0
\(987\) 11562.4 0.372883
\(988\) 27529.4 0.886464
\(989\) 6568.35i 0.211184i
\(990\) −14429.4 −0.463229
\(991\) 13384.7i 0.429040i 0.976720 + 0.214520i \(0.0688187\pi\)
−0.976720 + 0.214520i \(0.931181\pi\)
\(992\) − 117240.i − 3.75239i
\(993\) − 10448.4i − 0.333907i
\(994\) −171078. −5.45902
\(995\) −393.425 −0.0125351
\(996\) − 24621.4i − 0.783292i
\(997\) − 21615.7i − 0.686635i −0.939219 0.343317i \(-0.888449\pi\)
0.939219 0.343317i \(-0.111551\pi\)
\(998\) 12477.1i 0.395748i
\(999\) −8473.14 −0.268347
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 289.4.b.f.288.23 24
17.4 even 4 289.4.a.i.1.1 yes 12
17.13 even 4 289.4.a.h.1.1 12
17.16 even 2 inner 289.4.b.f.288.24 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
289.4.a.h.1.1 12 17.13 even 4
289.4.a.i.1.1 yes 12 17.4 even 4
289.4.b.f.288.23 24 1.1 even 1 trivial
289.4.b.f.288.24 24 17.16 even 2 inner