Properties

Label 289.4.b.f.288.6
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 288.6
Character \(\chi\) \(=\) 289.288
Dual form 289.4.b.f.288.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.28857 q^{2} +2.85039i q^{3} +10.3918 q^{4} -6.36629i q^{5} -12.2241i q^{6} -29.4308i q^{7} -10.2575 q^{8} +18.8753 q^{9} +O(q^{10})\) \(q-4.28857 q^{2} +2.85039i q^{3} +10.3918 q^{4} -6.36629i q^{5} -12.2241i q^{6} -29.4308i q^{7} -10.2575 q^{8} +18.8753 q^{9} +27.3023i q^{10} -61.3595i q^{11} +29.6207i q^{12} +18.5582 q^{13} +126.216i q^{14} +18.1464 q^{15} -39.1446 q^{16} -80.9480 q^{18} -115.793 q^{19} -66.1574i q^{20} +83.8892 q^{21} +263.144i q^{22} +7.38352i q^{23} -29.2379i q^{24} +84.4703 q^{25} -79.5883 q^{26} +130.762i q^{27} -305.840i q^{28} +164.974i q^{29} -77.8221 q^{30} -127.132i q^{31} +249.934 q^{32} +174.898 q^{33} -187.365 q^{35} +196.149 q^{36} -158.120i q^{37} +496.588 q^{38} +52.8982i q^{39} +65.3022i q^{40} +31.3736i q^{41} -359.764 q^{42} -157.708 q^{43} -637.637i q^{44} -120.166i q^{45} -31.6647i q^{46} -460.704 q^{47} -111.577i q^{48} -523.171 q^{49} -362.257 q^{50} +192.854 q^{52} +166.846 q^{53} -560.783i q^{54} -390.632 q^{55} +301.886i q^{56} -330.056i q^{57} -707.502i q^{58} +343.136 q^{59} +188.574 q^{60} -112.359i q^{61} +545.216i q^{62} -555.515i q^{63} -758.704 q^{64} -118.147i q^{65} -750.063 q^{66} -984.209 q^{67} -21.0459 q^{69} +803.528 q^{70} +524.337i q^{71} -193.613 q^{72} -852.636i q^{73} +678.106i q^{74} +240.773i q^{75} -1203.30 q^{76} -1805.86 q^{77} -226.858i q^{78} -201.021i q^{79} +249.206i q^{80} +136.909 q^{81} -134.548i q^{82} +22.5351 q^{83} +871.761 q^{84} +676.342 q^{86} -470.240 q^{87} +629.395i q^{88} +502.230 q^{89} +515.338i q^{90} -546.184i q^{91} +76.7282i q^{92} +362.377 q^{93} +1975.76 q^{94} +737.175i q^{95} +712.410i q^{96} -680.691i q^{97} +2243.65 q^{98} -1158.18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.28857 −1.51624 −0.758119 0.652116i \(-0.773881\pi\)
−0.758119 + 0.652116i \(0.773881\pi\)
\(3\) 2.85039i 0.548557i 0.961650 + 0.274279i \(0.0884391\pi\)
−0.961650 + 0.274279i \(0.911561\pi\)
\(4\) 10.3918 1.29898
\(5\) − 6.36629i − 0.569418i −0.958614 0.284709i \(-0.908103\pi\)
0.958614 0.284709i \(-0.0918971\pi\)
\(6\) − 12.2241i − 0.831744i
\(7\) − 29.4308i − 1.58911i −0.607190 0.794556i \(-0.707703\pi\)
0.607190 0.794556i \(-0.292297\pi\)
\(8\) −10.2575 −0.453322
\(9\) 18.8753 0.699085
\(10\) 27.3023i 0.863374i
\(11\) − 61.3595i − 1.68187i −0.541136 0.840935i \(-0.682005\pi\)
0.541136 0.840935i \(-0.317995\pi\)
\(12\) 29.6207i 0.712564i
\(13\) 18.5582 0.395933 0.197967 0.980209i \(-0.436566\pi\)
0.197967 + 0.980209i \(0.436566\pi\)
\(14\) 126.216i 2.40947i
\(15\) 18.1464 0.312359
\(16\) −39.1446 −0.611634
\(17\) 0 0
\(18\) −80.9480 −1.05998
\(19\) −115.793 −1.39815 −0.699075 0.715049i \(-0.746404\pi\)
−0.699075 + 0.715049i \(0.746404\pi\)
\(20\) − 66.1574i − 0.739662i
\(21\) 83.8892 0.871720
\(22\) 263.144i 2.55012i
\(23\) 7.38352i 0.0669378i 0.999440 + 0.0334689i \(0.0106555\pi\)
−0.999440 + 0.0334689i \(0.989345\pi\)
\(24\) − 29.2379i − 0.248673i
\(25\) 84.4703 0.675763
\(26\) −79.5883 −0.600329
\(27\) 130.762i 0.932046i
\(28\) − 305.840i − 2.06422i
\(29\) 164.974i 1.05638i 0.849127 + 0.528188i \(0.177128\pi\)
−0.849127 + 0.528188i \(0.822872\pi\)
\(30\) −77.8221 −0.473610
\(31\) − 127.132i − 0.736570i −0.929713 0.368285i \(-0.879945\pi\)
0.929713 0.368285i \(-0.120055\pi\)
\(32\) 249.934 1.38070
\(33\) 174.898 0.922603
\(34\) 0 0
\(35\) −187.365 −0.904870
\(36\) 196.149 0.908096
\(37\) − 158.120i − 0.702559i −0.936271 0.351280i \(-0.885747\pi\)
0.936271 0.351280i \(-0.114253\pi\)
\(38\) 496.588 2.11993
\(39\) 52.8982i 0.217192i
\(40\) 65.3022i 0.258130i
\(41\) 31.3736i 0.119506i 0.998213 + 0.0597528i \(0.0190313\pi\)
−0.998213 + 0.0597528i \(0.980969\pi\)
\(42\) −359.764 −1.32173
\(43\) −157.708 −0.559309 −0.279654 0.960101i \(-0.590220\pi\)
−0.279654 + 0.960101i \(0.590220\pi\)
\(44\) − 637.637i − 2.18471i
\(45\) − 120.166i − 0.398072i
\(46\) − 31.6647i − 0.101494i
\(47\) −460.704 −1.42980 −0.714900 0.699227i \(-0.753528\pi\)
−0.714900 + 0.699227i \(0.753528\pi\)
\(48\) − 111.577i − 0.335517i
\(49\) −523.171 −1.52528
\(50\) −362.257 −1.02462
\(51\) 0 0
\(52\) 192.854 0.514308
\(53\) 166.846 0.432417 0.216208 0.976347i \(-0.430631\pi\)
0.216208 + 0.976347i \(0.430631\pi\)
\(54\) − 560.783i − 1.41320i
\(55\) −390.632 −0.957688
\(56\) 301.886i 0.720379i
\(57\) − 330.056i − 0.766965i
\(58\) − 707.502i − 1.60172i
\(59\) 343.136 0.757161 0.378581 0.925568i \(-0.376412\pi\)
0.378581 + 0.925568i \(0.376412\pi\)
\(60\) 188.574 0.405747
\(61\) − 112.359i − 0.235837i −0.993023 0.117918i \(-0.962378\pi\)
0.993023 0.117918i \(-0.0376221\pi\)
\(62\) 545.216i 1.11682i
\(63\) − 555.515i − 1.11092i
\(64\) −758.704 −1.48184
\(65\) − 118.147i − 0.225452i
\(66\) −750.063 −1.39889
\(67\) −984.209 −1.79463 −0.897316 0.441390i \(-0.854486\pi\)
−0.897316 + 0.441390i \(0.854486\pi\)
\(68\) 0 0
\(69\) −21.0459 −0.0367192
\(70\) 803.528 1.37200
\(71\) 524.337i 0.876442i 0.898867 + 0.438221i \(0.144391\pi\)
−0.898867 + 0.438221i \(0.855609\pi\)
\(72\) −193.613 −0.316910
\(73\) − 852.636i − 1.36703i −0.729934 0.683517i \(-0.760449\pi\)
0.729934 0.683517i \(-0.239551\pi\)
\(74\) 678.106i 1.06525i
\(75\) 240.773i 0.370695i
\(76\) −1203.30 −1.81616
\(77\) −1805.86 −2.67268
\(78\) − 226.858i − 0.329315i
\(79\) − 201.021i − 0.286286i −0.989702 0.143143i \(-0.954279\pi\)
0.989702 0.143143i \(-0.0457209\pi\)
\(80\) 249.206i 0.348276i
\(81\) 136.909 0.187804
\(82\) − 134.548i − 0.181199i
\(83\) 22.5351 0.0298017 0.0149009 0.999889i \(-0.495257\pi\)
0.0149009 + 0.999889i \(0.495257\pi\)
\(84\) 871.761 1.13234
\(85\) 0 0
\(86\) 676.342 0.848045
\(87\) −470.240 −0.579483
\(88\) 629.395i 0.762428i
\(89\) 502.230 0.598161 0.299080 0.954228i \(-0.403320\pi\)
0.299080 + 0.954228i \(0.403320\pi\)
\(90\) 515.338i 0.603572i
\(91\) − 546.184i − 0.629182i
\(92\) 76.7282i 0.0869507i
\(93\) 362.377 0.404051
\(94\) 1975.76 2.16792
\(95\) 737.175i 0.796132i
\(96\) 712.410i 0.757396i
\(97\) − 680.691i − 0.712512i −0.934388 0.356256i \(-0.884053\pi\)
0.934388 0.356256i \(-0.115947\pi\)
\(98\) 2243.65 2.31269
\(99\) − 1158.18i − 1.17577i
\(100\) 877.801 0.877801
\(101\) 1921.92 1.89345 0.946724 0.322045i \(-0.104370\pi\)
0.946724 + 0.322045i \(0.104370\pi\)
\(102\) 0 0
\(103\) −1123.80 −1.07506 −0.537529 0.843245i \(-0.680642\pi\)
−0.537529 + 0.843245i \(0.680642\pi\)
\(104\) −190.361 −0.179485
\(105\) − 534.063i − 0.496373i
\(106\) −715.532 −0.655647
\(107\) 122.786i 0.110937i 0.998460 + 0.0554683i \(0.0176652\pi\)
−0.998460 + 0.0554683i \(0.982335\pi\)
\(108\) 1358.86i 1.21071i
\(109\) 791.497i 0.695520i 0.937584 + 0.347760i \(0.113058\pi\)
−0.937584 + 0.347760i \(0.886942\pi\)
\(110\) 1675.25 1.45208
\(111\) 450.702 0.385394
\(112\) 1152.06i 0.971956i
\(113\) − 73.7910i − 0.0614307i −0.999528 0.0307154i \(-0.990221\pi\)
0.999528 0.0307154i \(-0.00977854\pi\)
\(114\) 1415.47i 1.16290i
\(115\) 47.0056 0.0381156
\(116\) 1714.38i 1.37221i
\(117\) 350.292 0.276791
\(118\) −1471.56 −1.14804
\(119\) 0 0
\(120\) −186.137 −0.141599
\(121\) −2433.98 −1.82869
\(122\) 481.857i 0.357585i
\(123\) −89.4269 −0.0655557
\(124\) − 1321.14i − 0.956788i
\(125\) − 1333.55i − 0.954210i
\(126\) 2382.36i 1.68443i
\(127\) 1475.81 1.03115 0.515577 0.856843i \(-0.327577\pi\)
0.515577 + 0.856843i \(0.327577\pi\)
\(128\) 1254.28 0.866122
\(129\) − 449.529i − 0.306813i
\(130\) 506.682i 0.341838i
\(131\) 1029.79i 0.686821i 0.939185 + 0.343410i \(0.111582\pi\)
−0.939185 + 0.343410i \(0.888418\pi\)
\(132\) 1817.51 1.19844
\(133\) 3407.89i 2.22182i
\(134\) 4220.85 2.72109
\(135\) 832.471 0.530724
\(136\) 0 0
\(137\) −1018.26 −0.635008 −0.317504 0.948257i \(-0.602845\pi\)
−0.317504 + 0.948257i \(0.602845\pi\)
\(138\) 90.2567 0.0556751
\(139\) 1506.59i 0.919332i 0.888092 + 0.459666i \(0.152031\pi\)
−0.888092 + 0.459666i \(0.847969\pi\)
\(140\) −1947.06 −1.17541
\(141\) − 1313.18i − 0.784327i
\(142\) − 2248.66i − 1.32890i
\(143\) − 1138.72i − 0.665908i
\(144\) −738.865 −0.427584
\(145\) 1050.27 0.601520
\(146\) 3656.59i 2.07275i
\(147\) − 1491.24i − 0.836704i
\(148\) − 1643.15i − 0.912609i
\(149\) −424.614 −0.233462 −0.116731 0.993164i \(-0.537241\pi\)
−0.116731 + 0.993164i \(0.537241\pi\)
\(150\) − 1032.57i − 0.562061i
\(151\) −1566.83 −0.844418 −0.422209 0.906499i \(-0.638745\pi\)
−0.422209 + 0.906499i \(0.638745\pi\)
\(152\) 1187.75 0.633811
\(153\) 0 0
\(154\) 7744.54 4.05242
\(155\) −809.362 −0.419416
\(156\) 549.709i 0.282128i
\(157\) −762.291 −0.387500 −0.193750 0.981051i \(-0.562065\pi\)
−0.193750 + 0.981051i \(0.562065\pi\)
\(158\) 862.092i 0.434078i
\(159\) 475.577i 0.237206i
\(160\) − 1591.15i − 0.786199i
\(161\) 217.303 0.106372
\(162\) −587.145 −0.284756
\(163\) − 2253.74i − 1.08299i −0.840705 0.541493i \(-0.817859\pi\)
0.840705 0.541493i \(-0.182141\pi\)
\(164\) 326.029i 0.155235i
\(165\) − 1113.45i − 0.525347i
\(166\) −96.6432 −0.0451865
\(167\) − 1995.81i − 0.924792i −0.886673 0.462396i \(-0.846990\pi\)
0.886673 0.462396i \(-0.153010\pi\)
\(168\) −860.493 −0.395169
\(169\) −1852.59 −0.843237
\(170\) 0 0
\(171\) −2185.63 −0.977425
\(172\) −1638.88 −0.726529
\(173\) 1853.93i 0.814749i 0.913261 + 0.407375i \(0.133556\pi\)
−0.913261 + 0.407375i \(0.866444\pi\)
\(174\) 2016.66 0.878634
\(175\) − 2486.03i − 1.07386i
\(176\) 2401.89i 1.02869i
\(177\) 978.071i 0.415346i
\(178\) −2153.85 −0.906954
\(179\) 1623.39 0.677865 0.338933 0.940811i \(-0.389934\pi\)
0.338933 + 0.940811i \(0.389934\pi\)
\(180\) − 1248.74i − 0.517086i
\(181\) 759.107i 0.311735i 0.987778 + 0.155867i \(0.0498173\pi\)
−0.987778 + 0.155867i \(0.950183\pi\)
\(182\) 2342.35i 0.953990i
\(183\) 320.266 0.129370
\(184\) − 75.7364i − 0.0303444i
\(185\) −1006.64 −0.400050
\(186\) −1554.08 −0.612637
\(187\) 0 0
\(188\) −4787.55 −1.85728
\(189\) 3848.44 1.48113
\(190\) − 3161.42i − 1.20713i
\(191\) −1878.32 −0.711572 −0.355786 0.934567i \(-0.615787\pi\)
−0.355786 + 0.934567i \(0.615787\pi\)
\(192\) − 2162.60i − 0.812876i
\(193\) 497.364i 0.185498i 0.995690 + 0.0927489i \(0.0295654\pi\)
−0.995690 + 0.0927489i \(0.970435\pi\)
\(194\) 2919.19i 1.08034i
\(195\) 336.765 0.123673
\(196\) −5436.70 −1.98130
\(197\) − 1052.05i − 0.380486i −0.981737 0.190243i \(-0.939072\pi\)
0.981737 0.190243i \(-0.0609276\pi\)
\(198\) 4966.92i 1.78275i
\(199\) − 1783.55i − 0.635338i −0.948202 0.317669i \(-0.897100\pi\)
0.948202 0.317669i \(-0.102900\pi\)
\(200\) −866.454 −0.306338
\(201\) − 2805.38i − 0.984458i
\(202\) −8242.29 −2.87092
\(203\) 4855.31 1.67870
\(204\) 0 0
\(205\) 199.733 0.0680487
\(206\) 4819.48 1.63004
\(207\) 139.366i 0.0467952i
\(208\) −726.455 −0.242166
\(209\) 7105.02i 2.35151i
\(210\) 2290.37i 0.752620i
\(211\) − 3756.09i − 1.22550i −0.790278 0.612748i \(-0.790064\pi\)
0.790278 0.612748i \(-0.209936\pi\)
\(212\) 1733.84 0.561700
\(213\) −1494.57 −0.480779
\(214\) − 526.578i − 0.168206i
\(215\) 1004.02i 0.318481i
\(216\) − 1341.30i − 0.422517i
\(217\) −3741.61 −1.17049
\(218\) − 3394.39i − 1.05457i
\(219\) 2430.34 0.749897
\(220\) −4059.38 −1.24402
\(221\) 0 0
\(222\) −1932.87 −0.584349
\(223\) 3356.40 1.00790 0.503949 0.863733i \(-0.331880\pi\)
0.503949 + 0.863733i \(0.331880\pi\)
\(224\) − 7355.76i − 2.19410i
\(225\) 1594.40 0.472415
\(226\) 316.458i 0.0931436i
\(227\) − 1988.64i − 0.581457i −0.956806 0.290728i \(-0.906102\pi\)
0.956806 0.290728i \(-0.0938976\pi\)
\(228\) − 3429.89i − 0.996271i
\(229\) −5203.50 −1.50156 −0.750779 0.660554i \(-0.770322\pi\)
−0.750779 + 0.660554i \(0.770322\pi\)
\(230\) −201.587 −0.0577924
\(231\) − 5147.39i − 1.46612i
\(232\) − 1692.22i − 0.478878i
\(233\) − 5702.70i − 1.60342i −0.597714 0.801709i \(-0.703924\pi\)
0.597714 0.801709i \(-0.296076\pi\)
\(234\) −1502.25 −0.419681
\(235\) 2932.98i 0.814154i
\(236\) 3565.81 0.983536
\(237\) 572.988 0.157045
\(238\) 0 0
\(239\) 1418.04 0.383787 0.191894 0.981416i \(-0.438537\pi\)
0.191894 + 0.981416i \(0.438537\pi\)
\(240\) −710.334 −0.191049
\(241\) 6215.71i 1.66137i 0.556746 + 0.830683i \(0.312050\pi\)
−0.556746 + 0.830683i \(0.687950\pi\)
\(242\) 10438.3 2.77273
\(243\) 3920.83i 1.03507i
\(244\) − 1167.61i − 0.306347i
\(245\) 3330.66i 0.868523i
\(246\) 383.514 0.0993981
\(247\) −2148.92 −0.553573
\(248\) 1304.06i 0.333903i
\(249\) 64.2337i 0.0163480i
\(250\) 5719.02i 1.44681i
\(251\) 1547.45 0.389139 0.194570 0.980889i \(-0.437669\pi\)
0.194570 + 0.980889i \(0.437669\pi\)
\(252\) − 5772.81i − 1.44307i
\(253\) 453.049 0.112581
\(254\) −6329.10 −1.56348
\(255\) 0 0
\(256\) 690.569 0.168596
\(257\) −7251.45 −1.76005 −0.880025 0.474928i \(-0.842474\pi\)
−0.880025 + 0.474928i \(0.842474\pi\)
\(258\) 1927.84i 0.465201i
\(259\) −4653.58 −1.11645
\(260\) − 1227.76i − 0.292857i
\(261\) 3113.93i 0.738496i
\(262\) − 4416.34i − 1.04138i
\(263\) −4635.13 −1.08675 −0.543373 0.839491i \(-0.682853\pi\)
−0.543373 + 0.839491i \(0.682853\pi\)
\(264\) −1794.02 −0.418236
\(265\) − 1062.19i − 0.246226i
\(266\) − 14615.0i − 3.36880i
\(267\) 1431.55i 0.328126i
\(268\) −10227.7 −2.33119
\(269\) 7116.95i 1.61311i 0.591156 + 0.806557i \(0.298672\pi\)
−0.591156 + 0.806557i \(0.701328\pi\)
\(270\) −3570.11 −0.804704
\(271\) 163.713 0.0366970 0.0183485 0.999832i \(-0.494159\pi\)
0.0183485 + 0.999832i \(0.494159\pi\)
\(272\) 0 0
\(273\) 1556.83 0.345143
\(274\) 4366.89 0.962823
\(275\) − 5183.05i − 1.13655i
\(276\) −218.705 −0.0476975
\(277\) − 4174.84i − 0.905567i −0.891620 0.452784i \(-0.850431\pi\)
0.891620 0.452784i \(-0.149569\pi\)
\(278\) − 6461.11i − 1.39393i
\(279\) − 2399.66i − 0.514925i
\(280\) 1921.90 0.410197
\(281\) 6920.09 1.46910 0.734551 0.678553i \(-0.237393\pi\)
0.734551 + 0.678553i \(0.237393\pi\)
\(282\) 5631.68i 1.18923i
\(283\) − 6292.91i − 1.32182i −0.750466 0.660909i \(-0.770171\pi\)
0.750466 0.660909i \(-0.229829\pi\)
\(284\) 5448.82i 1.13848i
\(285\) −2101.23 −0.436724
\(286\) 4883.50i 1.00968i
\(287\) 923.350 0.189908
\(288\) 4717.58 0.965230
\(289\) 0 0
\(290\) −4504.17 −0.912048
\(291\) 1940.23 0.390854
\(292\) − 8860.45i − 1.77575i
\(293\) 2292.96 0.457189 0.228595 0.973522i \(-0.426587\pi\)
0.228595 + 0.973522i \(0.426587\pi\)
\(294\) 6395.29i 1.26864i
\(295\) − 2184.50i − 0.431142i
\(296\) 1621.91i 0.318485i
\(297\) 8023.51 1.56758
\(298\) 1820.99 0.353983
\(299\) 137.025i 0.0265029i
\(300\) 2502.07i 0.481524i
\(301\) 4641.47i 0.888804i
\(302\) 6719.47 1.28034
\(303\) 5478.22i 1.03867i
\(304\) 4532.69 0.855156
\(305\) −715.308 −0.134290
\(306\) 0 0
\(307\) 2300.11 0.427604 0.213802 0.976877i \(-0.431415\pi\)
0.213802 + 0.976877i \(0.431415\pi\)
\(308\) −18766.1 −3.47176
\(309\) − 3203.26i − 0.589731i
\(310\) 3471.01 0.635935
\(311\) 5775.22i 1.05300i 0.850175 + 0.526499i \(0.176496\pi\)
−0.850175 + 0.526499i \(0.823504\pi\)
\(312\) − 542.603i − 0.0984579i
\(313\) − 2799.69i − 0.505585i −0.967521 0.252792i \(-0.918651\pi\)
0.967521 0.252792i \(-0.0813490\pi\)
\(314\) 3269.14 0.587542
\(315\) −3536.57 −0.632581
\(316\) − 2088.97i − 0.371880i
\(317\) − 5140.47i − 0.910781i −0.890292 0.455390i \(-0.849500\pi\)
0.890292 0.455390i \(-0.150500\pi\)
\(318\) − 2039.54i − 0.359660i
\(319\) 10122.7 1.77669
\(320\) 4830.13i 0.843789i
\(321\) −349.989 −0.0608550
\(322\) −931.917 −0.161285
\(323\) 0 0
\(324\) 1422.74 0.243954
\(325\) 1567.62 0.267557
\(326\) 9665.33i 1.64206i
\(327\) −2256.07 −0.381533
\(328\) − 321.815i − 0.0541745i
\(329\) 13558.9i 2.27211i
\(330\) 4775.12i 0.796551i
\(331\) −459.510 −0.0763050 −0.0381525 0.999272i \(-0.512147\pi\)
−0.0381525 + 0.999272i \(0.512147\pi\)
\(332\) 234.180 0.0387118
\(333\) − 2984.55i − 0.491148i
\(334\) 8559.16i 1.40221i
\(335\) 6265.76i 1.02190i
\(336\) −3283.81 −0.533174
\(337\) − 6233.46i − 1.00759i −0.863823 0.503796i \(-0.831937\pi\)
0.863823 0.503796i \(-0.168063\pi\)
\(338\) 7944.97 1.27855
\(339\) 210.333 0.0336983
\(340\) 0 0
\(341\) −7800.78 −1.23882
\(342\) 9373.24 1.48201
\(343\) 5302.57i 0.834729i
\(344\) 1617.69 0.253547
\(345\) 133.984i 0.0209086i
\(346\) − 7950.70i − 1.23535i
\(347\) 6370.01i 0.985476i 0.870178 + 0.492738i \(0.164004\pi\)
−0.870178 + 0.492738i \(0.835996\pi\)
\(348\) −4886.65 −0.752735
\(349\) 6494.79 0.996155 0.498078 0.867132i \(-0.334039\pi\)
0.498078 + 0.867132i \(0.334039\pi\)
\(350\) 10661.5i 1.62823i
\(351\) 2426.72i 0.369028i
\(352\) − 15335.8i − 2.32217i
\(353\) 502.630 0.0757856 0.0378928 0.999282i \(-0.487935\pi\)
0.0378928 + 0.999282i \(0.487935\pi\)
\(354\) − 4194.53i − 0.629764i
\(355\) 3338.09 0.499062
\(356\) 5219.09 0.776998
\(357\) 0 0
\(358\) −6962.02 −1.02781
\(359\) 974.431 0.143255 0.0716275 0.997431i \(-0.477181\pi\)
0.0716275 + 0.997431i \(0.477181\pi\)
\(360\) 1232.60i 0.180455i
\(361\) 6549.11 0.954821
\(362\) − 3255.48i − 0.472664i
\(363\) − 6937.80i − 1.00314i
\(364\) − 5675.84i − 0.817294i
\(365\) −5428.13 −0.778415
\(366\) −1373.48 −0.196156
\(367\) 4693.11i 0.667515i 0.942659 + 0.333758i \(0.108317\pi\)
−0.942659 + 0.333758i \(0.891683\pi\)
\(368\) − 289.025i − 0.0409415i
\(369\) 592.186i 0.0835446i
\(370\) 4317.02 0.606571
\(371\) − 4910.42i − 0.687159i
\(372\) 3765.76 0.524853
\(373\) 8651.16 1.20091 0.600456 0.799658i \(-0.294986\pi\)
0.600456 + 0.799658i \(0.294986\pi\)
\(374\) 0 0
\(375\) 3801.13 0.523439
\(376\) 4725.67 0.648159
\(377\) 3061.63i 0.418254i
\(378\) −16504.3 −2.24574
\(379\) − 3490.01i − 0.473007i −0.971631 0.236503i \(-0.923999\pi\)
0.971631 0.236503i \(-0.0760014\pi\)
\(380\) 7660.59i 1.03416i
\(381\) 4206.62i 0.565648i
\(382\) 8055.29 1.07891
\(383\) −3928.41 −0.524105 −0.262052 0.965054i \(-0.584399\pi\)
−0.262052 + 0.965054i \(0.584399\pi\)
\(384\) 3575.18i 0.475118i
\(385\) 11496.6i 1.52187i
\(386\) − 2132.98i − 0.281259i
\(387\) −2976.79 −0.391004
\(388\) − 7073.62i − 0.925537i
\(389\) −6961.15 −0.907312 −0.453656 0.891177i \(-0.649880\pi\)
−0.453656 + 0.891177i \(0.649880\pi\)
\(390\) −1444.24 −0.187518
\(391\) 0 0
\(392\) 5366.43 0.691443
\(393\) −2935.31 −0.376761
\(394\) 4511.81i 0.576908i
\(395\) −1279.76 −0.163017
\(396\) − 12035.6i − 1.52730i
\(397\) 5331.94i 0.674062i 0.941494 + 0.337031i \(0.109423\pi\)
−0.941494 + 0.337031i \(0.890577\pi\)
\(398\) 7648.86i 0.963323i
\(399\) −9713.81 −1.21879
\(400\) −3306.56 −0.413320
\(401\) 546.021i 0.0679975i 0.999422 + 0.0339988i \(0.0108242\pi\)
−0.999422 + 0.0339988i \(0.989176\pi\)
\(402\) 12031.1i 1.49267i
\(403\) − 2359.35i − 0.291632i
\(404\) 19972.3 2.45955
\(405\) − 871.604i − 0.106939i
\(406\) −20822.3 −2.54531
\(407\) −9702.13 −1.18161
\(408\) 0 0
\(409\) 15964.6 1.93007 0.965037 0.262114i \(-0.0844196\pi\)
0.965037 + 0.262114i \(0.0844196\pi\)
\(410\) −856.571 −0.103178
\(411\) − 2902.44i − 0.348338i
\(412\) −11678.3 −1.39648
\(413\) − 10098.8i − 1.20321i
\(414\) − 597.681i − 0.0709527i
\(415\) − 143.465i − 0.0169697i
\(416\) 4638.34 0.546667
\(417\) −4294.36 −0.504306
\(418\) − 30470.4i − 3.56544i
\(419\) 7643.91i 0.891240i 0.895222 + 0.445620i \(0.147017\pi\)
−0.895222 + 0.445620i \(0.852983\pi\)
\(420\) − 5549.89i − 0.644778i
\(421\) −7524.14 −0.871031 −0.435516 0.900181i \(-0.643434\pi\)
−0.435516 + 0.900181i \(0.643434\pi\)
\(422\) 16108.2i 1.85814i
\(423\) −8695.92 −0.999551
\(424\) −1711.43 −0.196024
\(425\) 0 0
\(426\) 6409.55 0.728975
\(427\) −3306.80 −0.374771
\(428\) 1275.97i 0.144104i
\(429\) 3245.80 0.365289
\(430\) − 4305.79i − 0.482892i
\(431\) − 12222.4i − 1.36597i −0.730431 0.682986i \(-0.760681\pi\)
0.730431 0.682986i \(-0.239319\pi\)
\(432\) − 5118.64i − 0.570071i
\(433\) 15420.3 1.71144 0.855719 0.517440i \(-0.173115\pi\)
0.855719 + 0.517440i \(0.173115\pi\)
\(434\) 16046.1 1.77475
\(435\) 2993.68i 0.329968i
\(436\) 8225.10i 0.903465i
\(437\) − 854.963i − 0.0935890i
\(438\) −10422.7 −1.13702
\(439\) − 17832.7i − 1.93874i −0.245607 0.969369i \(-0.578987\pi\)
0.245607 0.969369i \(-0.421013\pi\)
\(440\) 4006.91 0.434141
\(441\) −9875.00 −1.06630
\(442\) 0 0
\(443\) 241.629 0.0259145 0.0129573 0.999916i \(-0.495875\pi\)
0.0129573 + 0.999916i \(0.495875\pi\)
\(444\) 4683.62 0.500618
\(445\) − 3197.35i − 0.340604i
\(446\) −14394.2 −1.52821
\(447\) − 1210.32i − 0.128067i
\(448\) 22329.2i 2.35482i
\(449\) − 14481.4i − 1.52209i −0.648698 0.761046i \(-0.724686\pi\)
0.648698 0.761046i \(-0.275314\pi\)
\(450\) −6837.70 −0.716294
\(451\) 1925.07 0.200993
\(452\) − 766.823i − 0.0797971i
\(453\) − 4466.08i − 0.463211i
\(454\) 8528.42i 0.881627i
\(455\) −3477.16 −0.358268
\(456\) 3385.55i 0.347682i
\(457\) 1814.51 0.185731 0.0928654 0.995679i \(-0.470397\pi\)
0.0928654 + 0.995679i \(0.470397\pi\)
\(458\) 22315.6 2.27672
\(459\) 0 0
\(460\) 488.474 0.0495114
\(461\) 13434.1 1.35724 0.678621 0.734489i \(-0.262578\pi\)
0.678621 + 0.734489i \(0.262578\pi\)
\(462\) 22075.0i 2.22299i
\(463\) −16593.9 −1.66562 −0.832810 0.553559i \(-0.813269\pi\)
−0.832810 + 0.553559i \(0.813269\pi\)
\(464\) − 6457.84i − 0.646116i
\(465\) − 2307.00i − 0.230074i
\(466\) 24456.4i 2.43116i
\(467\) 3944.12 0.390818 0.195409 0.980722i \(-0.437397\pi\)
0.195409 + 0.980722i \(0.437397\pi\)
\(468\) 3640.17 0.359545
\(469\) 28966.0i 2.85187i
\(470\) − 12578.3i − 1.23445i
\(471\) − 2172.83i − 0.212566i
\(472\) −3519.72 −0.343238
\(473\) 9676.89i 0.940685i
\(474\) −2457.30 −0.238117
\(475\) −9781.11 −0.944817
\(476\) 0 0
\(477\) 3149.27 0.302296
\(478\) −6081.35 −0.581913
\(479\) − 11162.8i − 1.06480i −0.846492 0.532401i \(-0.821290\pi\)
0.846492 0.532401i \(-0.178710\pi\)
\(480\) 4535.41 0.431275
\(481\) − 2934.42i − 0.278166i
\(482\) − 26656.5i − 2.51903i
\(483\) 619.397i 0.0583510i
\(484\) −25293.5 −2.37543
\(485\) −4333.48 −0.405717
\(486\) − 16814.7i − 1.56941i
\(487\) 9984.01i 0.928991i 0.885575 + 0.464496i \(0.153764\pi\)
−0.885575 + 0.464496i \(0.846236\pi\)
\(488\) 1152.52i 0.106910i
\(489\) 6424.04 0.594080
\(490\) − 14283.8i − 1.31689i
\(491\) 19639.7 1.80514 0.902572 0.430540i \(-0.141677\pi\)
0.902572 + 0.430540i \(0.141677\pi\)
\(492\) −929.309 −0.0851554
\(493\) 0 0
\(494\) 9215.80 0.839349
\(495\) −7373.30 −0.669505
\(496\) 4976.55i 0.450511i
\(497\) 15431.7 1.39277
\(498\) − 275.470i − 0.0247874i
\(499\) 5764.32i 0.517127i 0.965994 + 0.258564i \(0.0832491\pi\)
−0.965994 + 0.258564i \(0.916751\pi\)
\(500\) − 13858.0i − 1.23950i
\(501\) 5688.83 0.507302
\(502\) −6636.33 −0.590027
\(503\) − 1747.34i − 0.154890i −0.996997 0.0774452i \(-0.975324\pi\)
0.996997 0.0774452i \(-0.0246763\pi\)
\(504\) 5698.19i 0.503606i
\(505\) − 12235.5i − 1.07816i
\(506\) −1942.93 −0.170699
\(507\) − 5280.61i − 0.462564i
\(508\) 15336.3 1.33945
\(509\) −9110.89 −0.793385 −0.396693 0.917952i \(-0.629842\pi\)
−0.396693 + 0.917952i \(0.629842\pi\)
\(510\) 0 0
\(511\) −25093.8 −2.17237
\(512\) −12995.8 −1.12175
\(513\) − 15141.4i − 1.30314i
\(514\) 31098.3 2.66865
\(515\) 7154.42i 0.612158i
\(516\) − 4671.43i − 0.398543i
\(517\) 28268.5i 2.40474i
\(518\) 19957.2 1.69280
\(519\) −5284.42 −0.446937
\(520\) 1211.89i 0.102202i
\(521\) − 9231.61i − 0.776285i −0.921599 0.388142i \(-0.873117\pi\)
0.921599 0.388142i \(-0.126883\pi\)
\(522\) − 13354.3i − 1.11974i
\(523\) 14927.5 1.24806 0.624028 0.781402i \(-0.285495\pi\)
0.624028 + 0.781402i \(0.285495\pi\)
\(524\) 10701.4i 0.892165i
\(525\) 7086.14 0.589076
\(526\) 19878.1 1.64777
\(527\) 0 0
\(528\) −6846.32 −0.564295
\(529\) 12112.5 0.995519
\(530\) 4555.28i 0.373338i
\(531\) 6476.79 0.529320
\(532\) 35414.2i 2.88609i
\(533\) 582.239i 0.0473162i
\(534\) − 6139.31i − 0.497516i
\(535\) 781.694 0.0631693
\(536\) 10095.5 0.813545
\(537\) 4627.29i 0.371848i
\(538\) − 30521.5i − 2.44587i
\(539\) 32101.5i 2.56532i
\(540\) 8650.90 0.689399
\(541\) 9005.10i 0.715637i 0.933791 + 0.357818i \(0.116479\pi\)
−0.933791 + 0.357818i \(0.883521\pi\)
\(542\) −702.096 −0.0556413
\(543\) −2163.75 −0.171004
\(544\) 0 0
\(545\) 5038.90 0.396042
\(546\) −6676.59 −0.523318
\(547\) − 6774.33i − 0.529524i −0.964314 0.264762i \(-0.914707\pi\)
0.964314 0.264762i \(-0.0852933\pi\)
\(548\) −10581.6 −0.824861
\(549\) − 2120.80i − 0.164870i
\(550\) 22227.9i 1.72327i
\(551\) − 19102.9i − 1.47697i
\(552\) 215.878 0.0166456
\(553\) −5916.20 −0.454941
\(554\) 17904.1i 1.37306i
\(555\) − 2869.30i − 0.219450i
\(556\) 15656.2i 1.19419i
\(557\) −8558.56 −0.651055 −0.325528 0.945533i \(-0.605542\pi\)
−0.325528 + 0.945533i \(0.605542\pi\)
\(558\) 10291.1i 0.780748i
\(559\) −2926.79 −0.221449
\(560\) 7334.32 0.553450
\(561\) 0 0
\(562\) −29677.3 −2.22751
\(563\) −13915.1 −1.04165 −0.520827 0.853662i \(-0.674376\pi\)
−0.520827 + 0.853662i \(0.674376\pi\)
\(564\) − 13646.4i − 1.01882i
\(565\) −469.775 −0.0349798
\(566\) 26987.6i 2.00419i
\(567\) − 4029.35i − 0.298442i
\(568\) − 5378.39i − 0.397310i
\(569\) 12378.8 0.912031 0.456015 0.889972i \(-0.349276\pi\)
0.456015 + 0.889972i \(0.349276\pi\)
\(570\) 9011.29 0.662178
\(571\) 785.845i 0.0575947i 0.999585 + 0.0287974i \(0.00916775\pi\)
−0.999585 + 0.0287974i \(0.990832\pi\)
\(572\) − 11833.4i − 0.865000i
\(573\) − 5353.93i − 0.390338i
\(574\) −3959.85 −0.287946
\(575\) 623.688i 0.0452341i
\(576\) −14320.8 −1.03593
\(577\) 19662.8 1.41867 0.709334 0.704873i \(-0.248996\pi\)
0.709334 + 0.704873i \(0.248996\pi\)
\(578\) 0 0
\(579\) −1417.68 −0.101756
\(580\) 10914.2 0.781361
\(581\) − 663.224i − 0.0473583i
\(582\) −8320.82 −0.592627
\(583\) − 10237.6i − 0.727269i
\(584\) 8745.92i 0.619707i
\(585\) − 2230.06i − 0.157610i
\(586\) −9833.54 −0.693208
\(587\) 10024.7 0.704875 0.352438 0.935835i \(-0.385353\pi\)
0.352438 + 0.935835i \(0.385353\pi\)
\(588\) − 15496.7i − 1.08686i
\(589\) 14721.1i 1.02983i
\(590\) 9368.40i 0.653713i
\(591\) 2998.76 0.208719
\(592\) 6189.52i 0.429709i
\(593\) −770.054 −0.0533260 −0.0266630 0.999644i \(-0.508488\pi\)
−0.0266630 + 0.999644i \(0.508488\pi\)
\(594\) −34409.4 −2.37682
\(595\) 0 0
\(596\) −4412.52 −0.303261
\(597\) 5083.80 0.348519
\(598\) − 587.641i − 0.0401847i
\(599\) 20011.7 1.36503 0.682517 0.730870i \(-0.260885\pi\)
0.682517 + 0.730870i \(0.260885\pi\)
\(600\) − 2469.73i − 0.168044i
\(601\) − 16303.2i − 1.10652i −0.833008 0.553261i \(-0.813383\pi\)
0.833008 0.553261i \(-0.186617\pi\)
\(602\) − 19905.3i − 1.34764i
\(603\) −18577.2 −1.25460
\(604\) −16282.2 −1.09688
\(605\) 15495.5i 1.04129i
\(606\) − 23493.7i − 1.57486i
\(607\) 26777.6i 1.79056i 0.445506 + 0.895279i \(0.353024\pi\)
−0.445506 + 0.895279i \(0.646976\pi\)
\(608\) −28940.7 −1.93043
\(609\) 13839.5i 0.920864i
\(610\) 3067.65 0.203615
\(611\) −8549.85 −0.566105
\(612\) 0 0
\(613\) 1385.13 0.0912639 0.0456320 0.998958i \(-0.485470\pi\)
0.0456320 + 0.998958i \(0.485470\pi\)
\(614\) −9864.19 −0.648349
\(615\) 569.318i 0.0373286i
\(616\) 18523.6 1.21158
\(617\) − 818.646i − 0.0534156i −0.999643 0.0267078i \(-0.991498\pi\)
0.999643 0.0267078i \(-0.00850237\pi\)
\(618\) 13737.4i 0.894173i
\(619\) 9468.86i 0.614839i 0.951574 + 0.307420i \(0.0994655\pi\)
−0.951574 + 0.307420i \(0.900534\pi\)
\(620\) −8410.75 −0.544813
\(621\) −965.486 −0.0623891
\(622\) − 24767.4i − 1.59660i
\(623\) − 14781.0i − 0.950545i
\(624\) − 2070.68i − 0.132842i
\(625\) 2069.03 0.132418
\(626\) 12006.7i 0.766587i
\(627\) −20252.1 −1.28994
\(628\) −7921.60 −0.503354
\(629\) 0 0
\(630\) 15166.8 0.959143
\(631\) −11291.5 −0.712374 −0.356187 0.934415i \(-0.615923\pi\)
−0.356187 + 0.934415i \(0.615923\pi\)
\(632\) 2061.97i 0.129780i
\(633\) 10706.3 0.672255
\(634\) 22045.3i 1.38096i
\(635\) − 9395.41i − 0.587158i
\(636\) 4942.11i 0.308125i
\(637\) −9709.13 −0.603909
\(638\) −43412.0 −2.69388
\(639\) 9897.02i 0.612707i
\(640\) − 7985.10i − 0.493186i
\(641\) − 27604.9i − 1.70098i −0.525993 0.850489i \(-0.676306\pi\)
0.525993 0.850489i \(-0.323694\pi\)
\(642\) 1500.95 0.0922707
\(643\) − 30899.1i − 1.89509i −0.319625 0.947544i \(-0.603557\pi\)
0.319625 0.947544i \(-0.396443\pi\)
\(644\) 2258.17 0.138175
\(645\) −2861.84 −0.174705
\(646\) 0 0
\(647\) 26023.7 1.58129 0.790647 0.612272i \(-0.209744\pi\)
0.790647 + 0.612272i \(0.209744\pi\)
\(648\) −1404.35 −0.0851357
\(649\) − 21054.7i − 1.27345i
\(650\) −6722.85 −0.405680
\(651\) − 10665.0i − 0.642082i
\(652\) − 23420.5i − 1.40677i
\(653\) − 18446.4i − 1.10546i −0.833362 0.552728i \(-0.813587\pi\)
0.833362 0.552728i \(-0.186413\pi\)
\(654\) 9675.33 0.578494
\(655\) 6555.97 0.391088
\(656\) − 1228.11i − 0.0730938i
\(657\) − 16093.8i − 0.955673i
\(658\) − 58148.2i − 3.44506i
\(659\) −987.581 −0.0583774 −0.0291887 0.999574i \(-0.509292\pi\)
−0.0291887 + 0.999574i \(0.509292\pi\)
\(660\) − 11570.8i − 0.682414i
\(661\) −4126.78 −0.242834 −0.121417 0.992602i \(-0.538744\pi\)
−0.121417 + 0.992602i \(0.538744\pi\)
\(662\) 1970.64 0.115697
\(663\) 0 0
\(664\) −231.153 −0.0135098
\(665\) 21695.6 1.26514
\(666\) 12799.5i 0.744698i
\(667\) −1218.09 −0.0707115
\(668\) − 20740.1i − 1.20128i
\(669\) 9567.05i 0.552890i
\(670\) − 26871.2i − 1.54944i
\(671\) −6894.26 −0.396647
\(672\) 20966.8 1.20359
\(673\) − 7788.28i − 0.446086i −0.974809 0.223043i \(-0.928401\pi\)
0.974809 0.223043i \(-0.0715991\pi\)
\(674\) 26732.6i 1.52775i
\(675\) 11045.5i 0.629842i
\(676\) −19251.8 −1.09535
\(677\) 849.049i 0.0482003i 0.999710 + 0.0241001i \(0.00767206\pi\)
−0.999710 + 0.0241001i \(0.992328\pi\)
\(678\) −902.027 −0.0510946
\(679\) −20033.3 −1.13226
\(680\) 0 0
\(681\) 5668.39 0.318962
\(682\) 33454.2 1.87834
\(683\) 18449.2i 1.03359i 0.856110 + 0.516793i \(0.172874\pi\)
−0.856110 + 0.516793i \(0.827126\pi\)
\(684\) −22712.7 −1.26965
\(685\) 6482.56i 0.361585i
\(686\) − 22740.4i − 1.26565i
\(687\) − 14832.0i − 0.823691i
\(688\) 6173.42 0.342092
\(689\) 3096.37 0.171208
\(690\) − 574.601i − 0.0317024i
\(691\) 33194.1i 1.82744i 0.406339 + 0.913722i \(0.366805\pi\)
−0.406339 + 0.913722i \(0.633195\pi\)
\(692\) 19265.7i 1.05834i
\(693\) −34086.1 −1.86843
\(694\) − 27318.2i − 1.49422i
\(695\) 9591.38 0.523484
\(696\) 4823.49 0.262692
\(697\) 0 0
\(698\) −27853.4 −1.51041
\(699\) 16254.9 0.879567
\(700\) − 25834.4i − 1.39492i
\(701\) −1102.73 −0.0594146 −0.0297073 0.999559i \(-0.509458\pi\)
−0.0297073 + 0.999559i \(0.509458\pi\)
\(702\) − 10407.2i − 0.559534i
\(703\) 18309.2i 0.982282i
\(704\) 46553.7i 2.49227i
\(705\) −8360.12 −0.446610
\(706\) −2155.57 −0.114909
\(707\) − 56563.7i − 3.00890i
\(708\) 10163.9i 0.539526i
\(709\) − 5881.49i − 0.311543i −0.987793 0.155771i \(-0.950214\pi\)
0.987793 0.155771i \(-0.0497863\pi\)
\(710\) −14315.6 −0.756697
\(711\) − 3794.33i − 0.200138i
\(712\) −5151.63 −0.271159
\(713\) 938.685 0.0493044
\(714\) 0 0
\(715\) −7249.45 −0.379180
\(716\) 16870.0 0.880532
\(717\) 4041.95i 0.210529i
\(718\) −4178.92 −0.217209
\(719\) 13392.6i 0.694661i 0.937743 + 0.347330i \(0.112912\pi\)
−0.937743 + 0.347330i \(0.887088\pi\)
\(720\) 4703.83i 0.243474i
\(721\) 33074.2i 1.70839i
\(722\) −28086.3 −1.44774
\(723\) −17717.2 −0.911355
\(724\) 7888.51i 0.404937i
\(725\) 13935.4i 0.713859i
\(726\) 29753.2i 1.52100i
\(727\) 30483.7 1.55513 0.777564 0.628804i \(-0.216455\pi\)
0.777564 + 0.628804i \(0.216455\pi\)
\(728\) 5602.48i 0.285222i
\(729\) −7479.33 −0.379989
\(730\) 23278.9 1.18026
\(731\) 0 0
\(732\) 3328.14 0.168049
\(733\) −23656.0 −1.19203 −0.596013 0.802975i \(-0.703249\pi\)
−0.596013 + 0.802975i \(0.703249\pi\)
\(734\) − 20126.7i − 1.01211i
\(735\) −9493.67 −0.476434
\(736\) 1845.39i 0.0924214i
\(737\) 60390.5i 3.01834i
\(738\) − 2539.63i − 0.126674i
\(739\) 27884.4 1.38801 0.694007 0.719968i \(-0.255843\pi\)
0.694007 + 0.719968i \(0.255843\pi\)
\(740\) −10460.8 −0.519656
\(741\) − 6125.26i − 0.303667i
\(742\) 21058.7i 1.04190i
\(743\) − 3391.10i − 0.167439i −0.996489 0.0837196i \(-0.973320\pi\)
0.996489 0.0837196i \(-0.0266800\pi\)
\(744\) −3717.08 −0.183165
\(745\) 2703.22i 0.132937i
\(746\) −37101.1 −1.82087
\(747\) 425.356 0.0208339
\(748\) 0 0
\(749\) 3613.70 0.176291
\(750\) −16301.4 −0.793658
\(751\) 19843.5i 0.964181i 0.876121 + 0.482091i \(0.160122\pi\)
−0.876121 + 0.482091i \(0.839878\pi\)
\(752\) 18034.1 0.874514
\(753\) 4410.82i 0.213465i
\(754\) − 13130.0i − 0.634173i
\(755\) 9974.92i 0.480827i
\(756\) 39992.3 1.92395
\(757\) 23915.1 1.14823 0.574114 0.818775i \(-0.305347\pi\)
0.574114 + 0.818775i \(0.305347\pi\)
\(758\) 14967.1i 0.717191i
\(759\) 1291.36i 0.0617570i
\(760\) − 7561.57i − 0.360904i
\(761\) 17162.3 0.817520 0.408760 0.912642i \(-0.365961\pi\)
0.408760 + 0.912642i \(0.365961\pi\)
\(762\) − 18040.4i − 0.857656i
\(763\) 23294.4 1.10526
\(764\) −19519.1 −0.924317
\(765\) 0 0
\(766\) 16847.2 0.794668
\(767\) 6368.00 0.299785
\(768\) 1968.39i 0.0924845i
\(769\) −18387.3 −0.862240 −0.431120 0.902295i \(-0.641881\pi\)
−0.431120 + 0.902295i \(0.641881\pi\)
\(770\) − 49304.0i − 2.30752i
\(771\) − 20669.4i − 0.965488i
\(772\) 5168.52i 0.240957i
\(773\) −4428.78 −0.206070 −0.103035 0.994678i \(-0.532855\pi\)
−0.103035 + 0.994678i \(0.532855\pi\)
\(774\) 12766.2 0.592855
\(775\) − 10738.9i − 0.497746i
\(776\) 6982.18i 0.322997i
\(777\) − 13264.5i − 0.612435i
\(778\) 29853.4 1.37570
\(779\) − 3632.86i − 0.167087i
\(780\) 3499.61 0.160649
\(781\) 32173.1 1.47406
\(782\) 0 0
\(783\) −21572.4 −0.984591
\(784\) 20479.3 0.932913
\(785\) 4852.97i 0.220650i
\(786\) 12588.3 0.571259
\(787\) 17665.8i 0.800151i 0.916482 + 0.400075i \(0.131016\pi\)
−0.916482 + 0.400075i \(0.868984\pi\)
\(788\) − 10932.8i − 0.494243i
\(789\) − 13211.9i − 0.596143i
\(790\) 5488.33 0.247172
\(791\) −2171.73 −0.0976203
\(792\) 11880.0i 0.533002i
\(793\) − 2085.18i − 0.0933756i
\(794\) − 22866.4i − 1.02204i
\(795\) 3027.66 0.135069
\(796\) − 18534.3i − 0.825290i
\(797\) −12688.4 −0.563921 −0.281960 0.959426i \(-0.590985\pi\)
−0.281960 + 0.959426i \(0.590985\pi\)
\(798\) 41658.3 1.84798
\(799\) 0 0
\(800\) 21112.0 0.933029
\(801\) 9479.74 0.418165
\(802\) − 2341.65i − 0.103100i
\(803\) −52317.3 −2.29918
\(804\) − 29153.0i − 1.27879i
\(805\) − 1383.41i − 0.0605700i
\(806\) 10118.3i 0.442184i
\(807\) −20286.1 −0.884886
\(808\) −19714.1 −0.858342
\(809\) 28633.8i 1.24439i 0.782863 + 0.622194i \(0.213758\pi\)
−0.782863 + 0.622194i \(0.786242\pi\)
\(810\) 3737.94i 0.162145i
\(811\) − 2305.93i − 0.0998425i −0.998753 0.0499213i \(-0.984103\pi\)
0.998753 0.0499213i \(-0.0158970\pi\)
\(812\) 50455.6 2.18060
\(813\) 466.646i 0.0201304i
\(814\) 41608.3 1.79161
\(815\) −14348.0 −0.616672
\(816\) 0 0
\(817\) 18261.6 0.781997
\(818\) −68465.4 −2.92645
\(819\) − 10309.4i − 0.439852i
\(820\) 2075.60 0.0883938
\(821\) 28803.5i 1.22442i 0.790695 + 0.612210i \(0.209719\pi\)
−0.790695 + 0.612210i \(0.790281\pi\)
\(822\) 12447.3i 0.528164i
\(823\) − 14597.0i − 0.618248i −0.951022 0.309124i \(-0.899964\pi\)
0.951022 0.309124i \(-0.100036\pi\)
\(824\) 11527.3 0.487347
\(825\) 14773.7 0.623460
\(826\) 43309.3i 1.82436i
\(827\) − 32943.4i − 1.38519i −0.721326 0.692596i \(-0.756467\pi\)
0.721326 0.692596i \(-0.243533\pi\)
\(828\) 1448.27i 0.0607859i
\(829\) −35091.9 −1.47020 −0.735098 0.677961i \(-0.762864\pi\)
−0.735098 + 0.677961i \(0.762864\pi\)
\(830\) 615.259i 0.0257300i
\(831\) 11899.9 0.496756
\(832\) −14080.2 −0.586711
\(833\) 0 0
\(834\) 18416.7 0.764648
\(835\) −12705.9 −0.526594
\(836\) 73834.1i 3.05455i
\(837\) 16624.1 0.686517
\(838\) − 32781.5i − 1.35133i
\(839\) − 19365.1i − 0.796848i −0.917201 0.398424i \(-0.869557\pi\)
0.917201 0.398424i \(-0.130443\pi\)
\(840\) 5478.15i 0.225017i
\(841\) −2827.42 −0.115930
\(842\) 32267.8 1.32069
\(843\) 19724.9i 0.805887i
\(844\) − 39032.6i − 1.59189i
\(845\) 11794.1i 0.480155i
\(846\) 37293.0 1.51556
\(847\) 71634.1i 2.90599i
\(848\) −6531.13 −0.264481
\(849\) 17937.2 0.725094
\(850\) 0 0
\(851\) 1167.48 0.0470278
\(852\) −15531.3 −0.624521
\(853\) 37350.6i 1.49925i 0.661863 + 0.749625i \(0.269766\pi\)
−0.661863 + 0.749625i \(0.730234\pi\)
\(854\) 14181.4 0.568242
\(855\) 13914.4i 0.556564i
\(856\) − 1259.48i − 0.0502899i
\(857\) 14189.2i 0.565569i 0.959183 + 0.282785i \(0.0912582\pi\)
−0.959183 + 0.282785i \(0.908742\pi\)
\(858\) −13919.9 −0.553865
\(859\) −24014.5 −0.953859 −0.476930 0.878942i \(-0.658250\pi\)
−0.476930 + 0.878942i \(0.658250\pi\)
\(860\) 10433.6i 0.413699i
\(861\) 2631.90i 0.104175i
\(862\) 52416.8i 2.07114i
\(863\) −8498.07 −0.335200 −0.167600 0.985855i \(-0.553602\pi\)
−0.167600 + 0.985855i \(0.553602\pi\)
\(864\) 32682.0i 1.28688i
\(865\) 11802.7 0.463933
\(866\) −66131.1 −2.59495
\(867\) 0 0
\(868\) −38882.1 −1.52044
\(869\) −12334.5 −0.481497
\(870\) − 12838.6i − 0.500310i
\(871\) −18265.2 −0.710554
\(872\) − 8118.78i − 0.315294i
\(873\) − 12848.2i − 0.498106i
\(874\) 3666.57i 0.141903i
\(875\) −39247.4 −1.51635
\(876\) 25255.7 0.974100
\(877\) 16013.2i 0.616566i 0.951295 + 0.308283i \(0.0997544\pi\)
−0.951295 + 0.308283i \(0.900246\pi\)
\(878\) 76476.6i 2.93959i
\(879\) 6535.84i 0.250794i
\(880\) 15291.1 0.585755
\(881\) 16824.6i 0.643399i 0.946842 + 0.321699i \(0.104254\pi\)
−0.946842 + 0.321699i \(0.895746\pi\)
\(882\) 42349.6 1.61676
\(883\) 11327.3 0.431703 0.215851 0.976426i \(-0.430747\pi\)
0.215851 + 0.976426i \(0.430747\pi\)
\(884\) 0 0
\(885\) 6226.69 0.236506
\(886\) −1036.24 −0.0392926
\(887\) 11077.4i 0.419327i 0.977774 + 0.209663i \(0.0672368\pi\)
−0.977774 + 0.209663i \(0.932763\pi\)
\(888\) −4623.08 −0.174708
\(889\) − 43434.1i − 1.63862i
\(890\) 13712.0i 0.516436i
\(891\) − 8400.68i − 0.315862i
\(892\) 34879.2 1.30924
\(893\) 53346.5 1.99907
\(894\) 5190.52i 0.194180i
\(895\) − 10335.0i − 0.385989i
\(896\) − 36914.4i − 1.37637i
\(897\) −390.575 −0.0145384
\(898\) 62104.5i 2.30785i
\(899\) 20973.5 0.778095
\(900\) 16568.7 0.613657
\(901\) 0 0
\(902\) −8255.78 −0.304753
\(903\) −13230.0 −0.487560
\(904\) 756.911i 0.0278479i
\(905\) 4832.70 0.177508
\(906\) 19153.1i 0.702339i
\(907\) 1761.65i 0.0644924i 0.999480 + 0.0322462i \(0.0102661\pi\)
−0.999480 + 0.0322462i \(0.989734\pi\)
\(908\) − 20665.6i − 0.755299i
\(909\) 36276.8 1.32368
\(910\) 14912.1 0.543220
\(911\) − 44996.2i − 1.63643i −0.574909 0.818217i \(-0.694963\pi\)
0.574909 0.818217i \(-0.305037\pi\)
\(912\) 12919.9i 0.469102i
\(913\) − 1382.74i − 0.0501227i
\(914\) −7781.63 −0.281612
\(915\) − 2038.90i − 0.0736657i
\(916\) −54073.8 −1.95049
\(917\) 30307.6 1.09144
\(918\) 0 0
\(919\) 34461.8 1.23699 0.618493 0.785790i \(-0.287743\pi\)
0.618493 + 0.785790i \(0.287743\pi\)
\(920\) −482.160 −0.0172786
\(921\) 6556.21i 0.234565i
\(922\) −57613.1 −2.05790
\(923\) 9730.78i 0.347012i
\(924\) − 53490.8i − 1.90446i
\(925\) − 13356.4i − 0.474763i
\(926\) 71163.9 2.52548
\(927\) −21212.0 −0.751557
\(928\) 41232.7i 1.45854i
\(929\) − 37878.7i − 1.33774i −0.743379 0.668870i \(-0.766778\pi\)
0.743379 0.668870i \(-0.233222\pi\)
\(930\) 9893.71i 0.348847i
\(931\) 60579.8 2.13257
\(932\) − 59261.5i − 2.08281i
\(933\) −16461.6 −0.577630
\(934\) −16914.6 −0.592573
\(935\) 0 0
\(936\) −3593.12 −0.125475
\(937\) 599.505 0.0209018 0.0104509 0.999945i \(-0.496673\pi\)
0.0104509 + 0.999945i \(0.496673\pi\)
\(938\) − 124223.i − 4.32412i
\(939\) 7980.22 0.277342
\(940\) 30479.0i 1.05757i
\(941\) 10118.0i 0.350519i 0.984522 + 0.175260i \(0.0560765\pi\)
−0.984522 + 0.175260i \(0.943924\pi\)
\(942\) 9318.32i 0.322301i
\(943\) −231.647 −0.00799945
\(944\) −13431.9 −0.463106
\(945\) − 24500.3i − 0.843380i
\(946\) − 41500.0i − 1.42630i
\(947\) − 39320.2i − 1.34924i −0.738163 0.674622i \(-0.764307\pi\)
0.738163 0.674622i \(-0.235693\pi\)
\(948\) 5954.39 0.203997
\(949\) − 15823.4i − 0.541254i
\(950\) 41947.0 1.43257
\(951\) 14652.3 0.499616
\(952\) 0 0
\(953\) −41203.8 −1.40055 −0.700274 0.713874i \(-0.746939\pi\)
−0.700274 + 0.713874i \(0.746939\pi\)
\(954\) −13505.9 −0.458353
\(955\) 11957.9i 0.405182i
\(956\) 14736.0 0.498531
\(957\) 28853.7i 0.974615i
\(958\) 47872.3i 1.61449i
\(959\) 29968.3i 1.00910i
\(960\) −13767.7 −0.462867
\(961\) 13628.3 0.457465
\(962\) 12584.5i 0.421766i
\(963\) 2317.63i 0.0775540i
\(964\) 64592.6i 2.15808i
\(965\) 3166.37 0.105626
\(966\) − 2656.33i − 0.0884740i
\(967\) −9403.83 −0.312727 −0.156363 0.987700i \(-0.549977\pi\)
−0.156363 + 0.987700i \(0.549977\pi\)
\(968\) 24966.6 0.828984
\(969\) 0 0
\(970\) 18584.4 0.615164
\(971\) 44819.8 1.48129 0.740646 0.671895i \(-0.234519\pi\)
0.740646 + 0.671895i \(0.234519\pi\)
\(972\) 40744.6i 1.34453i
\(973\) 44340.1 1.46092
\(974\) − 42817.1i − 1.40857i
\(975\) 4468.33i 0.146770i
\(976\) 4398.23i 0.144246i
\(977\) −13700.4 −0.448632 −0.224316 0.974516i \(-0.572015\pi\)
−0.224316 + 0.974516i \(0.572015\pi\)
\(978\) −27549.9 −0.900766
\(979\) − 30816.6i − 1.00603i
\(980\) 34611.6i 1.12819i
\(981\) 14939.7i 0.486227i
\(982\) −84226.0 −2.73703
\(983\) − 34126.9i − 1.10730i −0.832748 0.553652i \(-0.813234\pi\)
0.832748 0.553652i \(-0.186766\pi\)
\(984\) 917.297 0.0297178
\(985\) −6697.69 −0.216656
\(986\) 0 0
\(987\) −38648.1 −1.24638
\(988\) −22331.2 −0.719080
\(989\) − 1164.44i − 0.0374389i
\(990\) 31620.9 1.01513
\(991\) − 38645.7i − 1.23877i −0.785087 0.619386i \(-0.787382\pi\)
0.785087 0.619386i \(-0.212618\pi\)
\(992\) − 31774.8i − 1.01699i
\(993\) − 1309.78i − 0.0418577i
\(994\) −66179.7 −2.11176
\(995\) −11354.6 −0.361773
\(996\) 667.505i 0.0212356i
\(997\) 13993.3i 0.444505i 0.974989 + 0.222252i \(0.0713409\pi\)
−0.974989 + 0.222252i \(0.928659\pi\)
\(998\) − 24720.7i − 0.784088i
\(999\) 20676.1 0.654817
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.288.6 24
17.4 even 4 289.4.a.i.1.10 yes 12
17.13 even 4 289.4.a.h.1.10 12
17.16 even 2 inner 289.4.b.f.288.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
289.4.a.h.1.10 12 17.13 even 4
289.4.a.i.1.10 yes 12 17.4 even 4
289.4.b.f.288.5 24 17.16 even 2 inner
289.4.b.f.288.6 24 1.1 even 1 trivial