Properties

Label 289.4.b.b.288.3
Level $289$
Weight $4$
Character 289.288
Analytic conductor $17.052$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27793984.1
Defining polynomial: \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 288.3
Root \(-1.93854 - 1.93854i\) of defining polynomial
Character \(\chi\) \(=\) 289.288
Dual form 289.4.b.b.288.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.36122 q^{2} -3.15463i q^{3} -6.14708 q^{4} -3.03171i q^{5} +4.29415i q^{6} -7.94049i q^{7} +19.2573 q^{8} +17.0483 q^{9} +O(q^{10})\) \(q-1.36122 q^{2} -3.15463i q^{3} -6.14708 q^{4} -3.03171i q^{5} +4.29415i q^{6} -7.94049i q^{7} +19.2573 q^{8} +17.0483 q^{9} +4.12682i q^{10} +27.6161i q^{11} +19.3918i q^{12} +58.1117 q^{13} +10.8088i q^{14} -9.56391 q^{15} +22.9632 q^{16} -23.2065 q^{18} -89.1688 q^{19} +18.6361i q^{20} -25.0493 q^{21} -37.5916i q^{22} -115.269i q^{23} -60.7497i q^{24} +115.809 q^{25} -79.1029 q^{26} -138.956i q^{27} +48.8108i q^{28} +128.558i q^{29} +13.0186 q^{30} -273.460i q^{31} -185.316 q^{32} +87.1187 q^{33} -24.0732 q^{35} -104.797 q^{36} +132.351i q^{37} +121.379 q^{38} -183.321i q^{39} -58.3825i q^{40} -470.559i q^{41} +34.0977 q^{42} -352.642 q^{43} -169.758i q^{44} -51.6854i q^{45} +156.907i q^{46} +152.598 q^{47} -72.4403i q^{48} +279.949 q^{49} -157.641 q^{50} -357.217 q^{52} -527.614 q^{53} +189.150i q^{54} +83.7239 q^{55} -152.912i q^{56} +281.295i q^{57} -174.995i q^{58} +292.020 q^{59} +58.7901 q^{60} -53.8962i q^{61} +372.239i q^{62} -135.372i q^{63} +68.5514 q^{64} -176.178i q^{65} -118.588 q^{66} +52.9572 q^{67} -363.632 q^{69} +32.7690 q^{70} -788.400i q^{71} +328.304 q^{72} -295.780i q^{73} -180.159i q^{74} -365.334i q^{75} +548.127 q^{76} +219.285 q^{77} +249.541i q^{78} -720.325i q^{79} -69.6175i q^{80} +21.9487 q^{81} +640.535i q^{82} +116.051 q^{83} +153.980 q^{84} +480.024 q^{86} +405.552 q^{87} +531.812i q^{88} -813.329 q^{89} +70.3553i q^{90} -461.435i q^{91} +708.569i q^{92} -862.664 q^{93} -207.720 q^{94} +270.334i q^{95} +584.605i q^{96} -794.693i q^{97} -381.072 q^{98} +470.808i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9} + 60 q^{13} - 216 q^{15} + 274 q^{16} - 206 q^{18} - 160 q^{19} - 384 q^{21} + 446 q^{25} - 52 q^{26} + 800 q^{30} + 142 q^{32} - 664 q^{33} - 664 q^{35} - 2626 q^{36} + 1448 q^{38} + 2256 q^{42} - 1112 q^{43} + 1280 q^{47} + 538 q^{49} + 1094 q^{50} - 1548 q^{52} - 604 q^{53} + 152 q^{55} - 1272 q^{59} - 2656 q^{60} - 1838 q^{64} - 4936 q^{66} + 2016 q^{67} + 1152 q^{69} + 3008 q^{70} - 1854 q^{72} + 1816 q^{76} + 1008 q^{77} - 1010 q^{81} + 4792 q^{83} - 4080 q^{84} - 2528 q^{86} - 2856 q^{87} - 340 q^{89} - 1264 q^{93} + 4032 q^{94} + 5714 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36122 −0.481264 −0.240632 0.970616i \(-0.577355\pi\)
−0.240632 + 0.970616i \(0.577355\pi\)
\(3\) − 3.15463i − 0.607109i −0.952814 0.303555i \(-0.901827\pi\)
0.952814 0.303555i \(-0.0981735\pi\)
\(4\) −6.14708 −0.768385
\(5\) − 3.03171i − 0.271164i −0.990766 0.135582i \(-0.956710\pi\)
0.990766 0.135582i \(-0.0432904\pi\)
\(6\) 4.29415i 0.292180i
\(7\) − 7.94049i − 0.428746i −0.976752 0.214373i \(-0.931229\pi\)
0.976752 0.214373i \(-0.0687708\pi\)
\(8\) 19.2573 0.851061
\(9\) 17.0483 0.631419
\(10\) 4.12682i 0.130502i
\(11\) 27.6161i 0.756961i 0.925609 + 0.378481i \(0.123553\pi\)
−0.925609 + 0.378481i \(0.876447\pi\)
\(12\) 19.3918i 0.466493i
\(13\) 58.1117 1.23979 0.619896 0.784684i \(-0.287175\pi\)
0.619896 + 0.784684i \(0.287175\pi\)
\(14\) 10.8088i 0.206340i
\(15\) −9.56391 −0.164626
\(16\) 22.9632 0.358799
\(17\) 0 0
\(18\) −23.2065 −0.303879
\(19\) −89.1688 −1.07667 −0.538335 0.842731i \(-0.680946\pi\)
−0.538335 + 0.842731i \(0.680946\pi\)
\(20\) 18.6361i 0.208358i
\(21\) −25.0493 −0.260296
\(22\) − 37.5916i − 0.364298i
\(23\) − 115.269i − 1.04501i −0.852635 0.522507i \(-0.824997\pi\)
0.852635 0.522507i \(-0.175003\pi\)
\(24\) − 60.7497i − 0.516687i
\(25\) 115.809 0.926470
\(26\) −79.1029 −0.596668
\(27\) − 138.956i − 0.990449i
\(28\) 48.8108i 0.329442i
\(29\) 128.558i 0.823191i 0.911367 + 0.411596i \(0.135028\pi\)
−0.911367 + 0.411596i \(0.864972\pi\)
\(30\) 13.0186 0.0792287
\(31\) − 273.460i − 1.58435i −0.610295 0.792174i \(-0.708949\pi\)
0.610295 0.792174i \(-0.291051\pi\)
\(32\) −185.316 −1.02374
\(33\) 87.1187 0.459558
\(34\) 0 0
\(35\) −24.0732 −0.116260
\(36\) −104.797 −0.485172
\(37\) 132.351i 0.588063i 0.955796 + 0.294031i \(0.0949970\pi\)
−0.955796 + 0.294031i \(0.905003\pi\)
\(38\) 121.379 0.518163
\(39\) − 183.321i − 0.752689i
\(40\) − 58.3825i − 0.230777i
\(41\) − 470.559i − 1.79241i −0.443636 0.896207i \(-0.646312\pi\)
0.443636 0.896207i \(-0.353688\pi\)
\(42\) 34.0977 0.125271
\(43\) −352.642 −1.25064 −0.625318 0.780370i \(-0.715031\pi\)
−0.625318 + 0.780370i \(0.715031\pi\)
\(44\) − 169.758i − 0.581637i
\(45\) − 51.6854i − 0.171218i
\(46\) 156.907i 0.502928i
\(47\) 152.598 0.473589 0.236795 0.971560i \(-0.423903\pi\)
0.236795 + 0.971560i \(0.423903\pi\)
\(48\) − 72.4403i − 0.217830i
\(49\) 279.949 0.816177
\(50\) −157.641 −0.445877
\(51\) 0 0
\(52\) −357.217 −0.952637
\(53\) −527.614 −1.36742 −0.683711 0.729753i \(-0.739635\pi\)
−0.683711 + 0.729753i \(0.739635\pi\)
\(54\) 189.150i 0.476668i
\(55\) 83.7239 0.205261
\(56\) − 152.912i − 0.364889i
\(57\) 281.295i 0.653656i
\(58\) − 174.995i − 0.396173i
\(59\) 292.020 0.644368 0.322184 0.946677i \(-0.395583\pi\)
0.322184 + 0.946677i \(0.395583\pi\)
\(60\) 58.7901 0.126496
\(61\) − 53.8962i − 0.113126i −0.998399 0.0565632i \(-0.981986\pi\)
0.998399 0.0565632i \(-0.0180142\pi\)
\(62\) 372.239i 0.762490i
\(63\) − 135.372i − 0.270718i
\(64\) 68.5514 0.133889
\(65\) − 176.178i − 0.336187i
\(66\) −118.588 −0.221169
\(67\) 52.9572 0.0965635 0.0482817 0.998834i \(-0.484625\pi\)
0.0482817 + 0.998834i \(0.484625\pi\)
\(68\) 0 0
\(69\) −363.632 −0.634437
\(70\) 32.7690 0.0559520
\(71\) − 788.400i − 1.31783i −0.752218 0.658915i \(-0.771016\pi\)
0.752218 0.658915i \(-0.228984\pi\)
\(72\) 328.304 0.537375
\(73\) − 295.780i − 0.474224i −0.971482 0.237112i \(-0.923799\pi\)
0.971482 0.237112i \(-0.0762009\pi\)
\(74\) − 180.159i − 0.283014i
\(75\) − 365.334i − 0.562468i
\(76\) 548.127 0.827296
\(77\) 219.285 0.324544
\(78\) 249.541i 0.362242i
\(79\) − 720.325i − 1.02586i −0.858430 0.512930i \(-0.828560\pi\)
0.858430 0.512930i \(-0.171440\pi\)
\(80\) − 69.6175i − 0.0972934i
\(81\) 21.9487 0.0301079
\(82\) 640.535i 0.862625i
\(83\) 116.051 0.153473 0.0767363 0.997051i \(-0.475550\pi\)
0.0767363 + 0.997051i \(0.475550\pi\)
\(84\) 153.980 0.200007
\(85\) 0 0
\(86\) 480.024 0.601887
\(87\) 405.552 0.499767
\(88\) 531.812i 0.644220i
\(89\) −813.329 −0.968682 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(90\) 70.3553i 0.0824011i
\(91\) − 461.435i − 0.531556i
\(92\) 708.569i 0.802972i
\(93\) −862.664 −0.961872
\(94\) −207.720 −0.227922
\(95\) 270.334i 0.291954i
\(96\) 584.605i 0.621521i
\(97\) − 794.693i − 0.831844i −0.909400 0.415922i \(-0.863459\pi\)
0.909400 0.415922i \(-0.136541\pi\)
\(98\) −381.072 −0.392797
\(99\) 470.808i 0.477959i
\(100\) −711.885 −0.711885
\(101\) 265.513 0.261579 0.130790 0.991410i \(-0.458249\pi\)
0.130790 + 0.991410i \(0.458249\pi\)
\(102\) 0 0
\(103\) 523.107 0.500420 0.250210 0.968192i \(-0.419500\pi\)
0.250210 + 0.968192i \(0.419500\pi\)
\(104\) 1119.07 1.05514
\(105\) 75.9421i 0.0705828i
\(106\) 718.199 0.658091
\(107\) 986.039i 0.890878i 0.895312 + 0.445439i \(0.146952\pi\)
−0.895312 + 0.445439i \(0.853048\pi\)
\(108\) 854.174i 0.761046i
\(109\) 1814.39i 1.59438i 0.603732 + 0.797188i \(0.293680\pi\)
−0.603732 + 0.797188i \(0.706320\pi\)
\(110\) −113.967 −0.0987846
\(111\) 417.518 0.357018
\(112\) − 182.339i − 0.153834i
\(113\) − 707.339i − 0.588857i −0.955673 0.294429i \(-0.904871\pi\)
0.955673 0.294429i \(-0.0951293\pi\)
\(114\) − 382.904i − 0.314581i
\(115\) −349.463 −0.283370
\(116\) − 790.253i − 0.632527i
\(117\) 990.706 0.782827
\(118\) −397.503 −0.310112
\(119\) 0 0
\(120\) −184.175 −0.140107
\(121\) 568.350 0.427010
\(122\) 73.3647i 0.0544437i
\(123\) −1484.44 −1.08819
\(124\) 1680.98i 1.21739i
\(125\) − 730.061i − 0.522389i
\(126\) 184.271i 0.130287i
\(127\) −2648.18 −1.85030 −0.925151 0.379600i \(-0.876062\pi\)
−0.925151 + 0.379600i \(0.876062\pi\)
\(128\) 1389.22 0.959302
\(129\) 1112.46i 0.759273i
\(130\) 239.817i 0.161795i
\(131\) 1979.08i 1.31995i 0.751289 + 0.659974i \(0.229433\pi\)
−0.751289 + 0.659974i \(0.770567\pi\)
\(132\) −535.525 −0.353117
\(133\) 708.044i 0.461618i
\(134\) −72.0865 −0.0464726
\(135\) −421.274 −0.268574
\(136\) 0 0
\(137\) 3141.92 1.95936 0.979679 0.200570i \(-0.0642794\pi\)
0.979679 + 0.200570i \(0.0642794\pi\)
\(138\) 494.984 0.305332
\(139\) − 1468.07i − 0.895830i −0.894076 0.447915i \(-0.852167\pi\)
0.894076 0.447915i \(-0.147833\pi\)
\(140\) 147.980 0.0893327
\(141\) − 481.390i − 0.287520i
\(142\) 1073.19i 0.634224i
\(143\) 1604.82i 0.938474i
\(144\) 391.483 0.226553
\(145\) 389.749 0.223220
\(146\) 402.621i 0.228227i
\(147\) − 883.135i − 0.495508i
\(148\) − 813.570i − 0.451858i
\(149\) −286.027 −0.157263 −0.0786316 0.996904i \(-0.525055\pi\)
−0.0786316 + 0.996904i \(0.525055\pi\)
\(150\) 497.300i 0.270696i
\(151\) 669.626 0.360883 0.180442 0.983586i \(-0.442247\pi\)
0.180442 + 0.983586i \(0.442247\pi\)
\(152\) −1717.15 −0.916311
\(153\) 0 0
\(154\) −298.496 −0.156191
\(155\) −829.049 −0.429618
\(156\) 1126.89i 0.578354i
\(157\) 720.809 0.366413 0.183206 0.983074i \(-0.441352\pi\)
0.183206 + 0.983074i \(0.441352\pi\)
\(158\) 980.522i 0.493710i
\(159\) 1664.43i 0.830174i
\(160\) 561.825i 0.277601i
\(161\) −915.294 −0.448045
\(162\) −29.8770 −0.0144899
\(163\) − 676.599i − 0.325125i −0.986698 0.162562i \(-0.948024\pi\)
0.986698 0.162562i \(-0.0519759\pi\)
\(164\) 2892.56i 1.37726i
\(165\) − 264.118i − 0.124616i
\(166\) −157.971 −0.0738609
\(167\) 2835.67i 1.31396i 0.753909 + 0.656979i \(0.228166\pi\)
−0.753909 + 0.656979i \(0.771834\pi\)
\(168\) −482.382 −0.221527
\(169\) 1179.97 0.537083
\(170\) 0 0
\(171\) −1520.18 −0.679829
\(172\) 2167.72 0.960970
\(173\) 177.314i 0.0779243i 0.999241 + 0.0389621i \(0.0124052\pi\)
−0.999241 + 0.0389621i \(0.987595\pi\)
\(174\) −552.046 −0.240520
\(175\) − 919.578i − 0.397220i
\(176\) 634.153i 0.271597i
\(177\) − 921.214i − 0.391202i
\(178\) 1107.12 0.466192
\(179\) 1023.76 0.427483 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(180\) 317.714i 0.131561i
\(181\) − 3450.21i − 1.41686i −0.705779 0.708432i \(-0.749403\pi\)
0.705779 0.708432i \(-0.250597\pi\)
\(182\) 628.116i 0.255819i
\(183\) −170.023 −0.0686800
\(184\) − 2219.78i − 0.889370i
\(185\) 401.248 0.159461
\(186\) 1174.28 0.462915
\(187\) 0 0
\(188\) −938.031 −0.363899
\(189\) −1103.38 −0.424651
\(190\) − 367.984i − 0.140507i
\(191\) −490.894 −0.185968 −0.0929839 0.995668i \(-0.529640\pi\)
−0.0929839 + 0.995668i \(0.529640\pi\)
\(192\) − 216.254i − 0.0812855i
\(193\) − 3548.80i − 1.32357i −0.749696 0.661783i \(-0.769800\pi\)
0.749696 0.661783i \(-0.230200\pi\)
\(194\) 1081.75i 0.400337i
\(195\) −555.775 −0.204102
\(196\) −1720.87 −0.627138
\(197\) 1363.15i 0.492996i 0.969143 + 0.246498i \(0.0792799\pi\)
−0.969143 + 0.246498i \(0.920720\pi\)
\(198\) − 640.874i − 0.230025i
\(199\) − 3737.46i − 1.33137i −0.746235 0.665683i \(-0.768140\pi\)
0.746235 0.665683i \(-0.231860\pi\)
\(200\) 2230.16 0.788482
\(201\) − 167.060i − 0.0586246i
\(202\) −361.422 −0.125889
\(203\) 1020.81 0.352940
\(204\) 0 0
\(205\) −1426.60 −0.486038
\(206\) −712.064 −0.240834
\(207\) − 1965.15i − 0.659841i
\(208\) 1334.43 0.444836
\(209\) − 2462.50i − 0.814997i
\(210\) − 103.374i − 0.0339690i
\(211\) − 5266.12i − 1.71817i −0.511829 0.859087i \(-0.671032\pi\)
0.511829 0.859087i \(-0.328968\pi\)
\(212\) 3243.28 1.05071
\(213\) −2487.11 −0.800066
\(214\) − 1342.22i − 0.428748i
\(215\) 1069.11i 0.339128i
\(216\) − 2675.92i − 0.842932i
\(217\) −2171.40 −0.679283
\(218\) − 2469.78i − 0.767316i
\(219\) −933.075 −0.287906
\(220\) −514.657 −0.157719
\(221\) 0 0
\(222\) −568.334 −0.171820
\(223\) −704.546 −0.211569 −0.105785 0.994389i \(-0.533735\pi\)
−0.105785 + 0.994389i \(0.533735\pi\)
\(224\) 1471.50i 0.438923i
\(225\) 1974.34 0.584990
\(226\) 962.845i 0.283396i
\(227\) − 2151.26i − 0.629006i −0.949256 0.314503i \(-0.898162\pi\)
0.949256 0.314503i \(-0.101838\pi\)
\(228\) − 1729.14i − 0.502259i
\(229\) 3916.94 1.13030 0.565149 0.824989i \(-0.308819\pi\)
0.565149 + 0.824989i \(0.308819\pi\)
\(230\) 475.696 0.136376
\(231\) − 691.764i − 0.197034i
\(232\) 2475.67i 0.700586i
\(233\) 5192.74i 1.46003i 0.683429 + 0.730017i \(0.260488\pi\)
−0.683429 + 0.730017i \(0.739512\pi\)
\(234\) −1348.57 −0.376747
\(235\) − 462.632i − 0.128420i
\(236\) −1795.07 −0.495123
\(237\) −2272.36 −0.622809
\(238\) 0 0
\(239\) 334.305 0.0904786 0.0452393 0.998976i \(-0.485595\pi\)
0.0452393 + 0.998976i \(0.485595\pi\)
\(240\) −219.618 −0.0590677
\(241\) 1918.45i 0.512773i 0.966574 + 0.256386i \(0.0825320\pi\)
−0.966574 + 0.256386i \(0.917468\pi\)
\(242\) −773.651 −0.205505
\(243\) − 3821.06i − 1.00873i
\(244\) 331.304i 0.0869245i
\(245\) − 848.722i − 0.221318i
\(246\) 2020.65 0.523708
\(247\) −5181.75 −1.33485
\(248\) − 5266.09i − 1.34838i
\(249\) − 366.097i − 0.0931746i
\(250\) 993.775i 0.251407i
\(251\) 7695.71 1.93525 0.967627 0.252385i \(-0.0812148\pi\)
0.967627 + 0.252385i \(0.0812148\pi\)
\(252\) 832.141i 0.208016i
\(253\) 3183.29 0.791035
\(254\) 3604.76 0.890484
\(255\) 0 0
\(256\) −2439.44 −0.595567
\(257\) −5335.10 −1.29492 −0.647460 0.762099i \(-0.724169\pi\)
−0.647460 + 0.762099i \(0.724169\pi\)
\(258\) − 1514.30i − 0.365411i
\(259\) 1050.93 0.252130
\(260\) 1082.98i 0.258321i
\(261\) 2191.69i 0.519778i
\(262\) − 2693.97i − 0.635244i
\(263\) −3934.15 −0.922396 −0.461198 0.887297i \(-0.652580\pi\)
−0.461198 + 0.887297i \(0.652580\pi\)
\(264\) 1677.67 0.391112
\(265\) 1599.57i 0.370795i
\(266\) − 963.804i − 0.222160i
\(267\) 2565.75i 0.588095i
\(268\) −325.532 −0.0741979
\(269\) − 3424.04i − 0.776088i −0.921641 0.388044i \(-0.873151\pi\)
0.921641 0.388044i \(-0.126849\pi\)
\(270\) 573.447 0.129255
\(271\) 549.034 0.123068 0.0615340 0.998105i \(-0.480401\pi\)
0.0615340 + 0.998105i \(0.480401\pi\)
\(272\) 0 0
\(273\) −1455.66 −0.322712
\(274\) −4276.85 −0.942970
\(275\) 3198.19i 0.701302i
\(276\) 2235.27 0.487492
\(277\) − 5203.65i − 1.12873i −0.825527 0.564363i \(-0.809122\pi\)
0.825527 0.564363i \(-0.190878\pi\)
\(278\) 1998.37i 0.431131i
\(279\) − 4662.02i − 1.00039i
\(280\) −463.585 −0.0989447
\(281\) 1986.73 0.421774 0.210887 0.977510i \(-0.432365\pi\)
0.210887 + 0.977510i \(0.432365\pi\)
\(282\) 655.279i 0.138373i
\(283\) 753.696i 0.158313i 0.996862 + 0.0791565i \(0.0252227\pi\)
−0.996862 + 0.0791565i \(0.974777\pi\)
\(284\) 4846.36i 1.01260i
\(285\) 852.803 0.177248
\(286\) − 2184.51i − 0.451654i
\(287\) −3736.47 −0.768490
\(288\) −3159.33 −0.646407
\(289\) 0 0
\(290\) −530.534 −0.107428
\(291\) −2506.96 −0.505020
\(292\) 1818.18i 0.364387i
\(293\) −7202.22 −1.43603 −0.718017 0.696025i \(-0.754950\pi\)
−0.718017 + 0.696025i \(0.754950\pi\)
\(294\) 1202.14i 0.238471i
\(295\) − 885.318i − 0.174729i
\(296\) 2548.72i 0.500477i
\(297\) 3837.43 0.749731
\(298\) 389.345 0.0756852
\(299\) − 6698.50i − 1.29560i
\(300\) 2245.74i 0.432192i
\(301\) 2800.15i 0.536205i
\(302\) −911.509 −0.173680
\(303\) − 837.595i − 0.158807i
\(304\) −2047.60 −0.386308
\(305\) −163.398 −0.0306758
\(306\) 0 0
\(307\) 2425.71 0.450953 0.225477 0.974249i \(-0.427606\pi\)
0.225477 + 0.974249i \(0.427606\pi\)
\(308\) −1347.96 −0.249375
\(309\) − 1650.21i − 0.303809i
\(310\) 1128.52 0.206760
\(311\) 9544.94i 1.74033i 0.492757 + 0.870167i \(0.335989\pi\)
−0.492757 + 0.870167i \(0.664011\pi\)
\(312\) − 3530.27i − 0.640584i
\(313\) 588.379i 0.106253i 0.998588 + 0.0531264i \(0.0169186\pi\)
−0.998588 + 0.0531264i \(0.983081\pi\)
\(314\) −981.180 −0.176341
\(315\) −410.407 −0.0734090
\(316\) 4427.89i 0.788255i
\(317\) 7653.31i 1.35600i 0.735061 + 0.678001i \(0.237154\pi\)
−0.735061 + 0.678001i \(0.762846\pi\)
\(318\) − 2265.65i − 0.399533i
\(319\) −3550.26 −0.623124
\(320\) − 207.828i − 0.0363060i
\(321\) 3110.59 0.540860
\(322\) 1245.92 0.215628
\(323\) 0 0
\(324\) −134.920 −0.0231345
\(325\) 6729.85 1.14863
\(326\) 921.001i 0.156471i
\(327\) 5723.73 0.967960
\(328\) − 9061.70i − 1.52545i
\(329\) − 1211.70i − 0.203050i
\(330\) 359.523i 0.0599730i
\(331\) −752.266 −0.124919 −0.0624597 0.998047i \(-0.519894\pi\)
−0.0624597 + 0.998047i \(0.519894\pi\)
\(332\) −713.373 −0.117926
\(333\) 2256.36i 0.371314i
\(334\) − 3859.98i − 0.632361i
\(335\) − 160.551i − 0.0261845i
\(336\) −575.211 −0.0933939
\(337\) 1968.57i 0.318204i 0.987262 + 0.159102i \(0.0508598\pi\)
−0.987262 + 0.159102i \(0.949140\pi\)
\(338\) −1606.20 −0.258479
\(339\) −2231.39 −0.357501
\(340\) 0 0
\(341\) 7551.89 1.19929
\(342\) 2069.30 0.327178
\(343\) − 4946.52i − 0.778678i
\(344\) −6790.93 −1.06437
\(345\) 1102.43i 0.172037i
\(346\) − 241.363i − 0.0375022i
\(347\) 3983.10i 0.616207i 0.951353 + 0.308104i \(0.0996944\pi\)
−0.951353 + 0.308104i \(0.900306\pi\)
\(348\) −2492.96 −0.384013
\(349\) −1495.61 −0.229393 −0.114697 0.993401i \(-0.536590\pi\)
−0.114697 + 0.993401i \(0.536590\pi\)
\(350\) 1251.75i 0.191168i
\(351\) − 8074.98i − 1.22795i
\(352\) − 5117.72i − 0.774930i
\(353\) 6482.49 0.977417 0.488708 0.872447i \(-0.337468\pi\)
0.488708 + 0.872447i \(0.337468\pi\)
\(354\) 1253.98i 0.188272i
\(355\) −2390.20 −0.357348
\(356\) 4999.59 0.744320
\(357\) 0 0
\(358\) −1393.56 −0.205732
\(359\) 4943.42 0.726751 0.363376 0.931643i \(-0.381624\pi\)
0.363376 + 0.931643i \(0.381624\pi\)
\(360\) − 995.322i − 0.145717i
\(361\) 1092.08 0.159218
\(362\) 4696.50i 0.681886i
\(363\) − 1792.94i − 0.259242i
\(364\) 2836.48i 0.408439i
\(365\) −896.717 −0.128593
\(366\) 231.439 0.0330533
\(367\) − 14.8871i − 0.00211743i −0.999999 0.00105872i \(-0.999663\pi\)
0.999999 0.00105872i \(-0.000337000\pi\)
\(368\) − 2646.95i − 0.374950i
\(369\) − 8022.23i − 1.13176i
\(370\) −546.188 −0.0767431
\(371\) 4189.51i 0.586276i
\(372\) 5302.86 0.739088
\(373\) 1923.18 0.266966 0.133483 0.991051i \(-0.457384\pi\)
0.133483 + 0.991051i \(0.457384\pi\)
\(374\) 0 0
\(375\) −2303.07 −0.317147
\(376\) 2938.63 0.403053
\(377\) 7470.70i 1.02059i
\(378\) 1501.94 0.204369
\(379\) 9592.87i 1.30014i 0.759875 + 0.650069i \(0.225260\pi\)
−0.759875 + 0.650069i \(0.774740\pi\)
\(380\) − 1661.76i − 0.224333i
\(381\) 8354.04i 1.12333i
\(382\) 668.215 0.0894996
\(383\) 9083.77 1.21190 0.605951 0.795502i \(-0.292793\pi\)
0.605951 + 0.795502i \(0.292793\pi\)
\(384\) − 4382.47i − 0.582401i
\(385\) − 664.809i − 0.0880046i
\(386\) 4830.70i 0.636985i
\(387\) −6011.95 −0.789675
\(388\) 4885.04i 0.639176i
\(389\) 1143.78 0.149079 0.0745396 0.997218i \(-0.476251\pi\)
0.0745396 + 0.997218i \(0.476251\pi\)
\(390\) 756.533 0.0982271
\(391\) 0 0
\(392\) 5391.06 0.694616
\(393\) 6243.27 0.801352
\(394\) − 1855.55i − 0.237262i
\(395\) −2183.81 −0.278176
\(396\) − 2894.09i − 0.367256i
\(397\) 10604.5i 1.34061i 0.742084 + 0.670307i \(0.233838\pi\)
−0.742084 + 0.670307i \(0.766162\pi\)
\(398\) 5087.51i 0.640739i
\(399\) 2233.62 0.280252
\(400\) 2659.33 0.332417
\(401\) 13785.4i 1.71674i 0.513035 + 0.858368i \(0.328521\pi\)
−0.513035 + 0.858368i \(0.671479\pi\)
\(402\) 227.406i 0.0282139i
\(403\) − 15891.2i − 1.96426i
\(404\) −1632.13 −0.200993
\(405\) − 66.5420i − 0.00816419i
\(406\) −1389.55 −0.169857
\(407\) −3655.01 −0.445141
\(408\) 0 0
\(409\) −9505.94 −1.14924 −0.574619 0.818421i \(-0.694850\pi\)
−0.574619 + 0.818421i \(0.694850\pi\)
\(410\) 1941.91 0.233913
\(411\) − 9911.59i − 1.18954i
\(412\) −3215.58 −0.384515
\(413\) − 2318.78i − 0.276270i
\(414\) 2675.00i 0.317558i
\(415\) − 351.832i − 0.0416162i
\(416\) −10769.1 −1.26922
\(417\) −4631.23 −0.543866
\(418\) 3352.00i 0.392229i
\(419\) 9680.86i 1.12874i 0.825523 + 0.564369i \(0.190880\pi\)
−0.825523 + 0.564369i \(0.809120\pi\)
\(420\) − 466.822i − 0.0542347i
\(421\) −12360.3 −1.43089 −0.715444 0.698671i \(-0.753775\pi\)
−0.715444 + 0.698671i \(0.753775\pi\)
\(422\) 7168.36i 0.826897i
\(423\) 2601.54 0.299033
\(424\) −10160.4 −1.16376
\(425\) 0 0
\(426\) 3385.51 0.385043
\(427\) −427.962 −0.0485025
\(428\) − 6061.25i − 0.684537i
\(429\) 5062.61 0.569756
\(430\) − 1455.29i − 0.163210i
\(431\) 2970.58i 0.331990i 0.986127 + 0.165995i \(0.0530835\pi\)
−0.986127 + 0.165995i \(0.946916\pi\)
\(432\) − 3190.87i − 0.355372i
\(433\) −6131.50 −0.680510 −0.340255 0.940333i \(-0.610513\pi\)
−0.340255 + 0.940333i \(0.610513\pi\)
\(434\) 2955.76 0.326915
\(435\) − 1229.51i − 0.135519i
\(436\) − 11153.2i − 1.22509i
\(437\) 10278.4i 1.12513i
\(438\) 1270.12 0.138559
\(439\) 2544.91i 0.276679i 0.990385 + 0.138339i \(0.0441765\pi\)
−0.990385 + 0.138339i \(0.955824\pi\)
\(440\) 1612.30 0.174689
\(441\) 4772.65 0.515349
\(442\) 0 0
\(443\) 8529.82 0.914817 0.457408 0.889257i \(-0.348778\pi\)
0.457408 + 0.889257i \(0.348778\pi\)
\(444\) −2566.51 −0.274327
\(445\) 2465.77i 0.262672i
\(446\) 959.043 0.101821
\(447\) 902.308i 0.0954759i
\(448\) − 544.331i − 0.0574046i
\(449\) 8855.74i 0.930798i 0.885101 + 0.465399i \(0.154089\pi\)
−0.885101 + 0.465399i \(0.845911\pi\)
\(450\) −2687.52 −0.281535
\(451\) 12995.0 1.35679
\(452\) 4348.07i 0.452469i
\(453\) − 2112.42i − 0.219095i
\(454\) 2928.35i 0.302718i
\(455\) −1398.94 −0.144139
\(456\) 5416.98i 0.556301i
\(457\) 7154.78 0.732356 0.366178 0.930545i \(-0.380666\pi\)
0.366178 + 0.930545i \(0.380666\pi\)
\(458\) −5331.82 −0.543973
\(459\) 0 0
\(460\) 2148.17 0.217737
\(461\) 7263.06 0.733784 0.366892 0.930264i \(-0.380422\pi\)
0.366892 + 0.930264i \(0.380422\pi\)
\(462\) 941.645i 0.0948253i
\(463\) 352.898 0.0354224 0.0177112 0.999843i \(-0.494362\pi\)
0.0177112 + 0.999843i \(0.494362\pi\)
\(464\) 2952.09i 0.295360i
\(465\) 2615.34i 0.260825i
\(466\) − 7068.47i − 0.702662i
\(467\) −1483.02 −0.146951 −0.0734753 0.997297i \(-0.523409\pi\)
−0.0734753 + 0.997297i \(0.523409\pi\)
\(468\) −6089.94 −0.601512
\(469\) − 420.506i − 0.0414012i
\(470\) 629.745i 0.0618042i
\(471\) − 2273.89i − 0.222452i
\(472\) 5623.51 0.548396
\(473\) − 9738.60i − 0.946683i
\(474\) 3093.19 0.299736
\(475\) −10326.5 −0.997502
\(476\) 0 0
\(477\) −8994.92 −0.863415
\(478\) −455.063 −0.0435441
\(479\) 9990.10i 0.952942i 0.879190 + 0.476471i \(0.158084\pi\)
−0.879190 + 0.476471i \(0.841916\pi\)
\(480\) 1772.35 0.168534
\(481\) 7691.13i 0.729075i
\(482\) − 2611.44i − 0.246779i
\(483\) 2887.42i 0.272012i
\(484\) −3493.69 −0.328108
\(485\) −2409.27 −0.225566
\(486\) 5201.30i 0.485465i
\(487\) − 1129.88i − 0.105133i −0.998617 0.0525663i \(-0.983260\pi\)
0.998617 0.0525663i \(-0.0167401\pi\)
\(488\) − 1037.90i − 0.0962774i
\(489\) −2134.42 −0.197386
\(490\) 1155.30i 0.106512i
\(491\) −18774.9 −1.72566 −0.862832 0.505491i \(-0.831311\pi\)
−0.862832 + 0.505491i \(0.831311\pi\)
\(492\) 9124.97 0.836149
\(493\) 0 0
\(494\) 7053.51 0.642414
\(495\) 1427.35 0.129605
\(496\) − 6279.49i − 0.568463i
\(497\) −6260.28 −0.565014
\(498\) 498.339i 0.0448416i
\(499\) 17329.1i 1.55462i 0.629118 + 0.777310i \(0.283416\pi\)
−0.629118 + 0.777310i \(0.716584\pi\)
\(500\) 4487.74i 0.401396i
\(501\) 8945.50 0.797716
\(502\) −10475.6 −0.931369
\(503\) − 20837.0i − 1.84707i −0.383518 0.923533i \(-0.625288\pi\)
0.383518 0.923533i \(-0.374712\pi\)
\(504\) − 2606.90i − 0.230398i
\(505\) − 804.957i − 0.0709309i
\(506\) −4333.16 −0.380697
\(507\) − 3722.37i − 0.326068i
\(508\) 16278.6 1.42174
\(509\) 11835.0 1.03060 0.515301 0.857009i \(-0.327680\pi\)
0.515301 + 0.857009i \(0.327680\pi\)
\(510\) 0 0
\(511\) −2348.63 −0.203322
\(512\) −7793.12 −0.672676
\(513\) 12390.6i 1.06639i
\(514\) 7262.26 0.623199
\(515\) − 1585.91i − 0.135696i
\(516\) − 6838.35i − 0.583414i
\(517\) 4214.16i 0.358489i
\(518\) −1430.55 −0.121341
\(519\) 559.359 0.0473086
\(520\) − 3392.71i − 0.286115i
\(521\) 7686.37i 0.646346i 0.946340 + 0.323173i \(0.104750\pi\)
−0.946340 + 0.323173i \(0.895250\pi\)
\(522\) − 2983.37i − 0.250151i
\(523\) 11476.4 0.959518 0.479759 0.877400i \(-0.340724\pi\)
0.479759 + 0.877400i \(0.340724\pi\)
\(524\) − 12165.6i − 1.01423i
\(525\) −2900.93 −0.241156
\(526\) 5355.25 0.443916
\(527\) 0 0
\(528\) 2000.52 0.164889
\(529\) −1120.01 −0.0920535
\(530\) − 2177.37i − 0.178451i
\(531\) 4978.44 0.406866
\(532\) − 4352.40i − 0.354700i
\(533\) − 27345.0i − 2.22222i
\(534\) − 3492.56i − 0.283029i
\(535\) 2989.38 0.241574
\(536\) 1019.81 0.0821814
\(537\) − 3229.59i − 0.259529i
\(538\) 4660.88i 0.373504i
\(539\) 7731.09i 0.617814i
\(540\) 2589.60 0.206368
\(541\) 546.481i 0.0434289i 0.999764 + 0.0217145i \(0.00691247\pi\)
−0.999764 + 0.0217145i \(0.993088\pi\)
\(542\) −747.357 −0.0592283
\(543\) −10884.1 −0.860191
\(544\) 0 0
\(545\) 5500.69 0.432337
\(546\) 1981.47 0.155310
\(547\) − 8397.33i − 0.656388i −0.944610 0.328194i \(-0.893560\pi\)
0.944610 0.328194i \(-0.106440\pi\)
\(548\) −19313.6 −1.50554
\(549\) − 918.839i − 0.0714301i
\(550\) − 4353.44i − 0.337512i
\(551\) − 11463.3i − 0.886305i
\(552\) −7002.58 −0.539945
\(553\) −5719.73 −0.439833
\(554\) 7083.32i 0.543215i
\(555\) − 1265.79i − 0.0968105i
\(556\) 9024.36i 0.688342i
\(557\) −4881.65 −0.371350 −0.185675 0.982611i \(-0.559447\pi\)
−0.185675 + 0.982611i \(0.559447\pi\)
\(558\) 6346.04i 0.481451i
\(559\) −20492.6 −1.55053
\(560\) −552.797 −0.0417142
\(561\) 0 0
\(562\) −2704.38 −0.202985
\(563\) −7198.57 −0.538870 −0.269435 0.963019i \(-0.586837\pi\)
−0.269435 + 0.963019i \(0.586837\pi\)
\(564\) 2959.14i 0.220926i
\(565\) −2144.44 −0.159677
\(566\) − 1025.95i − 0.0761904i
\(567\) − 174.283i − 0.0129087i
\(568\) − 15182.5i − 1.12155i
\(569\) 23946.9 1.76433 0.882167 0.470937i \(-0.156084\pi\)
0.882167 + 0.470937i \(0.156084\pi\)
\(570\) −1160.85 −0.0853032
\(571\) 1593.15i 0.116763i 0.998294 + 0.0583813i \(0.0185939\pi\)
−0.998294 + 0.0583813i \(0.981406\pi\)
\(572\) − 9864.95i − 0.721109i
\(573\) 1548.59i 0.112903i
\(574\) 5086.16 0.369847
\(575\) − 13349.2i − 0.968174i
\(576\) 1168.69 0.0845403
\(577\) 12937.4 0.933435 0.466717 0.884406i \(-0.345436\pi\)
0.466717 + 0.884406i \(0.345436\pi\)
\(578\) 0 0
\(579\) −11195.2 −0.803549
\(580\) −2395.82 −0.171519
\(581\) − 921.499i − 0.0658007i
\(582\) 3412.53 0.243048
\(583\) − 14570.6i − 1.03508i
\(584\) − 5695.92i − 0.403594i
\(585\) − 3003.53i − 0.212275i
\(586\) 9803.82 0.691112
\(587\) 12899.2 0.906998 0.453499 0.891257i \(-0.350176\pi\)
0.453499 + 0.891257i \(0.350176\pi\)
\(588\) 5428.70i 0.380741i
\(589\) 24384.1i 1.70582i
\(590\) 1205.11i 0.0840911i
\(591\) 4300.23 0.299302
\(592\) 3039.19i 0.210997i
\(593\) −4357.13 −0.301730 −0.150865 0.988554i \(-0.548206\pi\)
−0.150865 + 0.988554i \(0.548206\pi\)
\(594\) −5223.59 −0.360819
\(595\) 0 0
\(596\) 1758.23 0.120839
\(597\) −11790.3 −0.808284
\(598\) 9118.14i 0.623526i
\(599\) 13726.8 0.936328 0.468164 0.883642i \(-0.344916\pi\)
0.468164 + 0.883642i \(0.344916\pi\)
\(600\) − 7035.35i − 0.478695i
\(601\) 2531.41i 0.171811i 0.996303 + 0.0859056i \(0.0273783\pi\)
−0.996303 + 0.0859056i \(0.972622\pi\)
\(602\) − 3811.62i − 0.258057i
\(603\) 902.830 0.0609720
\(604\) −4116.24 −0.277297
\(605\) − 1723.07i − 0.115790i
\(606\) 1140.15i 0.0764283i
\(607\) − 185.004i − 0.0123708i −0.999981 0.00618540i \(-0.998031\pi\)
0.999981 0.00618540i \(-0.00196889\pi\)
\(608\) 16524.4 1.10223
\(609\) − 3220.28i − 0.214273i
\(610\) 222.420 0.0147632
\(611\) 8867.73 0.587152
\(612\) 0 0
\(613\) −17706.9 −1.16668 −0.583339 0.812228i \(-0.698254\pi\)
−0.583339 + 0.812228i \(0.698254\pi\)
\(614\) −3301.93 −0.217028
\(615\) 4500.39i 0.295078i
\(616\) 4222.84 0.276207
\(617\) 6183.89i 0.403491i 0.979438 + 0.201746i \(0.0646614\pi\)
−0.979438 + 0.201746i \(0.935339\pi\)
\(618\) 2246.30i 0.146213i
\(619\) − 1247.51i − 0.0810046i −0.999179 0.0405023i \(-0.987104\pi\)
0.999179 0.0405023i \(-0.0128958\pi\)
\(620\) 5096.23 0.330112
\(621\) −16017.4 −1.03503
\(622\) − 12992.8i − 0.837561i
\(623\) 6458.23i 0.415318i
\(624\) − 4209.63i − 0.270064i
\(625\) 12262.8 0.784817
\(626\) − 800.914i − 0.0511357i
\(627\) −7768.27 −0.494792
\(628\) −4430.87 −0.281546
\(629\) 0 0
\(630\) 558.655 0.0353292
\(631\) −24053.3 −1.51750 −0.758752 0.651379i \(-0.774191\pi\)
−0.758752 + 0.651379i \(0.774191\pi\)
\(632\) − 13871.5i − 0.873069i
\(633\) −16612.7 −1.04312
\(634\) − 10417.8i − 0.652596i
\(635\) 8028.51i 0.501735i
\(636\) − 10231.4i − 0.637893i
\(637\) 16268.3 1.01189
\(638\) 4832.69 0.299887
\(639\) − 13440.9i − 0.832102i
\(640\) − 4211.70i − 0.260128i
\(641\) 21286.8i 1.31167i 0.754905 + 0.655834i \(0.227683\pi\)
−0.754905 + 0.655834i \(0.772317\pi\)
\(642\) −4234.20 −0.260297
\(643\) 1789.41i 0.109747i 0.998493 + 0.0548736i \(0.0174756\pi\)
−0.998493 + 0.0548736i \(0.982524\pi\)
\(644\) 5626.38 0.344271
\(645\) 3372.64 0.205888
\(646\) 0 0
\(647\) −4378.61 −0.266060 −0.133030 0.991112i \(-0.542471\pi\)
−0.133030 + 0.991112i \(0.542471\pi\)
\(648\) 422.672 0.0256237
\(649\) 8064.45i 0.487762i
\(650\) −9160.81 −0.552795
\(651\) 6849.97i 0.412399i
\(652\) 4159.11i 0.249821i
\(653\) − 7665.15i − 0.459358i −0.973266 0.229679i \(-0.926232\pi\)
0.973266 0.229679i \(-0.0737676\pi\)
\(654\) −7791.26 −0.465845
\(655\) 5999.99 0.357922
\(656\) − 10805.5i − 0.643117i
\(657\) − 5042.54i − 0.299434i
\(658\) 1649.39i 0.0977205i
\(659\) −4710.22 −0.278428 −0.139214 0.990262i \(-0.544458\pi\)
−0.139214 + 0.990262i \(0.544458\pi\)
\(660\) 1623.55i 0.0957527i
\(661\) 31266.6 1.83983 0.919916 0.392116i \(-0.128257\pi\)
0.919916 + 0.392116i \(0.128257\pi\)
\(662\) 1024.00 0.0601192
\(663\) 0 0
\(664\) 2234.82 0.130614
\(665\) 2146.58 0.125174
\(666\) − 3071.40i − 0.178700i
\(667\) 14818.7 0.860246
\(668\) − 17431.1i − 1.00962i
\(669\) 2222.58i 0.128446i
\(670\) 218.545i 0.0126017i
\(671\) 1488.40 0.0856322
\(672\) 4642.05 0.266474
\(673\) 11723.0i 0.671454i 0.941959 + 0.335727i \(0.108982\pi\)
−0.941959 + 0.335727i \(0.891018\pi\)
\(674\) − 2679.65i − 0.153140i
\(675\) − 16092.3i − 0.917621i
\(676\) −7253.37 −0.412686
\(677\) 289.531i 0.0164366i 0.999966 + 0.00821829i \(0.00261599\pi\)
−0.999966 + 0.00821829i \(0.997384\pi\)
\(678\) 3037.42 0.172052
\(679\) −6310.25 −0.356650
\(680\) 0 0
\(681\) −6786.45 −0.381875
\(682\) −10279.8 −0.577175
\(683\) − 1720.10i − 0.0963660i −0.998839 0.0481830i \(-0.984657\pi\)
0.998839 0.0481830i \(-0.0153431\pi\)
\(684\) 9344.64 0.522370
\(685\) − 9525.37i − 0.531308i
\(686\) 6733.30i 0.374750i
\(687\) − 12356.5i − 0.686215i
\(688\) −8097.77 −0.448728
\(689\) −30660.5 −1.69532
\(690\) − 1500.65i − 0.0827951i
\(691\) − 16777.7i − 0.923665i −0.886967 0.461832i \(-0.847192\pi\)
0.886967 0.461832i \(-0.152808\pi\)
\(692\) − 1089.96i − 0.0598758i
\(693\) 3738.44 0.204923
\(694\) − 5421.88i − 0.296559i
\(695\) −4450.77 −0.242917
\(696\) 7809.84 0.425332
\(697\) 0 0
\(698\) 2035.86 0.110399
\(699\) 16381.2 0.886400
\(700\) 5652.71i 0.305218i
\(701\) 23981.1 1.29209 0.646043 0.763301i \(-0.276423\pi\)
0.646043 + 0.763301i \(0.276423\pi\)
\(702\) 10991.8i 0.590969i
\(703\) − 11801.6i − 0.633150i
\(704\) 1893.12i 0.101349i
\(705\) −1459.43 −0.0779652
\(706\) −8824.10 −0.470396
\(707\) − 2108.30i − 0.112151i
\(708\) 5662.78i 0.300593i
\(709\) 7709.28i 0.408361i 0.978933 + 0.204181i \(0.0654530\pi\)
−0.978933 + 0.204181i \(0.934547\pi\)
\(710\) 3253.59 0.171979
\(711\) − 12280.3i − 0.647747i
\(712\) −15662.5 −0.824407
\(713\) −31521.5 −1.65567
\(714\) 0 0
\(715\) 4865.34 0.254480
\(716\) −6293.13 −0.328471
\(717\) − 1054.61i − 0.0549304i
\(718\) −6729.09 −0.349760
\(719\) 11976.5i 0.621209i 0.950539 + 0.310605i \(0.100532\pi\)
−0.950539 + 0.310605i \(0.899468\pi\)
\(720\) − 1186.86i − 0.0614329i
\(721\) − 4153.72i − 0.214553i
\(722\) −1486.56 −0.0766260
\(723\) 6052.00 0.311309
\(724\) 21208.7i 1.08870i
\(725\) 14888.1i 0.762662i
\(726\) 2440.58i 0.124764i
\(727\) −18597.3 −0.948745 −0.474372 0.880324i \(-0.657325\pi\)
−0.474372 + 0.880324i \(0.657325\pi\)
\(728\) − 8886.00i − 0.452386i
\(729\) −11461.4 −0.582300
\(730\) 1220.63 0.0618870
\(731\) 0 0
\(732\) 1045.14 0.0527727
\(733\) 23569.5 1.18767 0.593833 0.804588i \(-0.297614\pi\)
0.593833 + 0.804588i \(0.297614\pi\)
\(734\) 20.2646i 0.00101905i
\(735\) −2677.41 −0.134364
\(736\) 21361.3i 1.06982i
\(737\) 1462.47i 0.0730948i
\(738\) 10920.0i 0.544678i
\(739\) −10149.1 −0.505199 −0.252599 0.967571i \(-0.581285\pi\)
−0.252599 + 0.967571i \(0.581285\pi\)
\(740\) −2466.50 −0.122528
\(741\) 16346.5i 0.810397i
\(742\) − 5702.85i − 0.282154i
\(743\) − 27758.0i − 1.37058i −0.728269 0.685291i \(-0.759675\pi\)
0.728269 0.685291i \(-0.240325\pi\)
\(744\) −16612.6 −0.818611
\(745\) 867.148i 0.0426441i
\(746\) −2617.87 −0.128481
\(747\) 1978.47 0.0969054
\(748\) 0 0
\(749\) 7829.63 0.381960
\(750\) 3134.99 0.152632
\(751\) 815.225i 0.0396112i 0.999804 + 0.0198056i \(0.00630473\pi\)
−0.999804 + 0.0198056i \(0.993695\pi\)
\(752\) 3504.13 0.169924
\(753\) − 24277.1i − 1.17491i
\(754\) − 10169.3i − 0.491172i
\(755\) − 2030.11i − 0.0978585i
\(756\) 6782.56 0.326295
\(757\) 13239.4 0.635659 0.317829 0.948148i \(-0.397046\pi\)
0.317829 + 0.948148i \(0.397046\pi\)
\(758\) − 13058.0i − 0.625710i
\(759\) − 10042.1i − 0.480244i
\(760\) 5205.90i 0.248471i
\(761\) −11028.2 −0.525324 −0.262662 0.964888i \(-0.584600\pi\)
−0.262662 + 0.964888i \(0.584600\pi\)
\(762\) − 11371.7i − 0.540621i
\(763\) 14407.1 0.683582
\(764\) 3017.56 0.142895
\(765\) 0 0
\(766\) −12365.0 −0.583246
\(767\) 16969.8 0.798882
\(768\) 7695.55i 0.361574i
\(769\) −18921.2 −0.887277 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(770\) 904.952i 0.0423535i
\(771\) 16830.3i 0.786158i
\(772\) 21814.7i 1.01701i
\(773\) −38728.6 −1.80203 −0.901016 0.433786i \(-0.857177\pi\)
−0.901016 + 0.433786i \(0.857177\pi\)
\(774\) 8183.59 0.380043
\(775\) − 31669.0i − 1.46785i
\(776\) − 15303.6i − 0.707949i
\(777\) − 3315.29i − 0.153070i
\(778\) −1556.93 −0.0717465
\(779\) 41959.2i 1.92984i
\(780\) 3416.39 0.156829
\(781\) 21772.5 0.997545
\(782\) 0 0
\(783\) 17863.9 0.815329
\(784\) 6428.50 0.292844
\(785\) − 2185.28i − 0.0993579i
\(786\) −8498.47 −0.385662
\(787\) 20587.3i 0.932477i 0.884659 + 0.466239i \(0.154391\pi\)
−0.884659 + 0.466239i \(0.845609\pi\)
\(788\) − 8379.37i − 0.378811i
\(789\) 12410.8i 0.559995i
\(790\) 2972.66 0.133876
\(791\) −5616.62 −0.252470
\(792\) 9066.49i 0.406772i
\(793\) − 3132.00i − 0.140253i
\(794\) − 14435.0i − 0.645190i
\(795\) 5046.05 0.225113
\(796\) 22974.5i 1.02300i
\(797\) 15871.4 0.705385 0.352693 0.935739i \(-0.385266\pi\)
0.352693 + 0.935739i \(0.385266\pi\)
\(798\) −3040.45 −0.134876
\(799\) 0 0
\(800\) −21461.3 −0.948463
\(801\) −13865.9 −0.611644
\(802\) − 18765.0i − 0.826204i
\(803\) 8168.28 0.358969
\(804\) 1026.93i 0.0450462i
\(805\) 2774.90i 0.121494i
\(806\) 21631.4i 0.945329i
\(807\) −10801.6 −0.471170
\(808\) 5113.06 0.222620
\(809\) 39667.1i 1.72388i 0.507007 + 0.861942i \(0.330752\pi\)
−0.507007 + 0.861942i \(0.669248\pi\)
\(810\) 90.5783i 0.00392913i
\(811\) − 8003.87i − 0.346552i −0.984873 0.173276i \(-0.944565\pi\)
0.984873 0.173276i \(-0.0554353\pi\)
\(812\) −6275.00 −0.271194
\(813\) − 1732.00i − 0.0747157i
\(814\) 4975.28 0.214230
\(815\) −2051.25 −0.0881621
\(816\) 0 0
\(817\) 31444.7 1.34652
\(818\) 12939.7 0.553088
\(819\) − 7866.69i − 0.335634i
\(820\) 8769.40 0.373464
\(821\) 13279.1i 0.564489i 0.959343 + 0.282244i \(0.0910789\pi\)
−0.959343 + 0.282244i \(0.908921\pi\)
\(822\) 13491.9i 0.572486i
\(823\) 28934.0i 1.22549i 0.790281 + 0.612745i \(0.209935\pi\)
−0.790281 + 0.612745i \(0.790065\pi\)
\(824\) 10073.6 0.425887
\(825\) 10089.1 0.425767
\(826\) 3156.37i 0.132959i
\(827\) − 13679.6i − 0.575193i −0.957752 0.287597i \(-0.907144\pi\)
0.957752 0.287597i \(-0.0928563\pi\)
\(828\) 12079.9i 0.507012i
\(829\) 16514.5 0.691886 0.345943 0.938256i \(-0.387559\pi\)
0.345943 + 0.938256i \(0.387559\pi\)
\(830\) 478.921i 0.0200284i
\(831\) −16415.6 −0.685260
\(832\) 3983.64 0.165995
\(833\) 0 0
\(834\) 6304.13 0.261744
\(835\) 8596.93 0.356298
\(836\) 15137.1i 0.626231i
\(837\) −37998.9 −1.56922
\(838\) − 13177.8i − 0.543221i
\(839\) − 87.9839i − 0.00362043i −0.999998 0.00181022i \(-0.999424\pi\)
0.999998 0.00181022i \(-0.000576210\pi\)
\(840\) 1462.44i 0.0600702i
\(841\) 7861.94 0.322356
\(842\) 16825.1 0.688635
\(843\) − 6267.41i − 0.256063i
\(844\) 32371.3i 1.32022i
\(845\) − 3577.33i − 0.145638i
\(846\) −3541.27 −0.143914
\(847\) − 4512.98i − 0.183079i
\(848\) −12115.7 −0.490630
\(849\) 2377.63 0.0961133
\(850\) 0 0
\(851\) 15256.0 0.614534
\(852\) 15288.5 0.614758
\(853\) − 8162.96i − 0.327660i −0.986489 0.163830i \(-0.947615\pi\)
0.986489 0.163830i \(-0.0523849\pi\)
\(854\) 582.551 0.0233425
\(855\) 4608.73i 0.184345i
\(856\) 18988.4i 0.758191i
\(857\) 18724.9i 0.746361i 0.927759 + 0.373181i \(0.121733\pi\)
−0.927759 + 0.373181i \(0.878267\pi\)
\(858\) −6891.34 −0.274203
\(859\) 46422.5 1.84391 0.921953 0.387301i \(-0.126593\pi\)
0.921953 + 0.387301i \(0.126593\pi\)
\(860\) − 6571.88i − 0.260580i
\(861\) 11787.2i 0.466557i
\(862\) − 4043.61i − 0.159775i
\(863\) 29112.3 1.14831 0.574157 0.818746i \(-0.305330\pi\)
0.574157 + 0.818746i \(0.305330\pi\)
\(864\) 25750.8i 1.01396i
\(865\) 537.563 0.0211303
\(866\) 8346.33 0.327505
\(867\) 0 0
\(868\) 13347.8 0.521950
\(869\) 19892.6 0.776536
\(870\) 1673.64i 0.0652204i
\(871\) 3077.43 0.119719
\(872\) 34940.2i 1.35691i
\(873\) − 13548.2i − 0.525241i
\(874\) − 13991.2i − 0.541487i
\(875\) −5797.04 −0.223972
\(876\) 5735.69 0.221222
\(877\) 39163.0i 1.50791i 0.656924 + 0.753957i \(0.271857\pi\)
−0.656924 + 0.753957i \(0.728143\pi\)
\(878\) − 3464.19i − 0.133156i
\(879\) 22720.3i 0.871830i
\(880\) 1922.57 0.0736473
\(881\) 35073.2i 1.34125i 0.741795 + 0.670627i \(0.233975\pi\)
−0.741795 + 0.670627i \(0.766025\pi\)
\(882\) −6496.63 −0.248019
\(883\) −48775.7 −1.85893 −0.929463 0.368915i \(-0.879729\pi\)
−0.929463 + 0.368915i \(0.879729\pi\)
\(884\) 0 0
\(885\) −2792.85 −0.106080
\(886\) −11611.0 −0.440269
\(887\) − 13296.0i − 0.503309i −0.967817 0.251654i \(-0.919025\pi\)
0.967817 0.251654i \(-0.0809746\pi\)
\(888\) 8040.27 0.303844
\(889\) 21027.9i 0.793309i
\(890\) − 3356.46i − 0.126415i
\(891\) 606.137i 0.0227905i
\(892\) 4330.90 0.162566
\(893\) −13607.0 −0.509899
\(894\) − 1228.24i − 0.0459492i
\(895\) − 3103.74i − 0.115918i
\(896\) − 11031.1i − 0.411297i
\(897\) −21131.3 −0.786570
\(898\) − 12054.6i − 0.447960i
\(899\) 35155.3 1.30422
\(900\) −12136.4 −0.449498
\(901\) 0 0
\(902\) −17689.1 −0.652974
\(903\) 8833.44 0.325535
\(904\) − 13621.4i − 0.501153i
\(905\) −10460.0 −0.384202
\(906\) 2875.47i 0.105443i
\(907\) − 11675.0i − 0.427410i −0.976898 0.213705i \(-0.931447\pi\)
0.976898 0.213705i \(-0.0685532\pi\)
\(908\) 13224.0i 0.483319i
\(909\) 4526.54 0.165166
\(910\) 1904.26 0.0693689
\(911\) 18552.9i 0.674738i 0.941372 + 0.337369i \(0.109537\pi\)
−0.941372 + 0.337369i \(0.890463\pi\)
\(912\) 6459.41i 0.234531i
\(913\) 3204.87i 0.116173i
\(914\) −9739.24 −0.352457
\(915\) 515.459i 0.0186235i
\(916\) −24077.7 −0.868504
\(917\) 15714.9 0.565922
\(918\) 0 0
\(919\) 33956.8 1.21886 0.609429 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(920\) −6729.71 −0.241165
\(921\) − 7652.23i − 0.273778i
\(922\) −9886.63 −0.353144
\(923\) − 45815.3i − 1.63383i
\(924\) 4252.33i 0.151398i
\(925\) 15327.4i 0.544823i
\(926\) −480.372 −0.0170475
\(927\) 8918.08 0.315974
\(928\) − 23823.8i − 0.842732i
\(929\) − 23695.3i − 0.836832i −0.908256 0.418416i \(-0.862585\pi\)
0.908256 0.418416i \(-0.137415\pi\)
\(930\) − 3560.06i − 0.125526i
\(931\) −24962.7 −0.878753
\(932\) − 31920.2i − 1.12187i
\(933\) 30110.8 1.05657
\(934\) 2018.72 0.0707221
\(935\) 0 0
\(936\) 19078.3 0.666234
\(937\) −7990.62 −0.278593 −0.139297 0.990251i \(-0.544484\pi\)
−0.139297 + 0.990251i \(0.544484\pi\)
\(938\) 572.402i 0.0199249i
\(939\) 1856.12 0.0645071
\(940\) 2843.83i 0.0986762i
\(941\) 24385.9i 0.844799i 0.906410 + 0.422400i \(0.138812\pi\)
−0.906410 + 0.422400i \(0.861188\pi\)
\(942\) 3095.26i 0.107058i
\(943\) −54241.0 −1.87310
\(944\) 6705.69 0.231199
\(945\) 3345.12i 0.115150i
\(946\) 13256.4i 0.455605i
\(947\) 1174.62i 0.0403064i 0.999797 + 0.0201532i \(0.00641539\pi\)
−0.999797 + 0.0201532i \(0.993585\pi\)
\(948\) 13968.4 0.478557
\(949\) − 17188.3i − 0.587939i
\(950\) 14056.7 0.480063
\(951\) 24143.4 0.823241
\(952\) 0 0
\(953\) −33546.9 −1.14029 −0.570143 0.821546i \(-0.693112\pi\)
−0.570143 + 0.821546i \(0.693112\pi\)
\(954\) 12244.1 0.415531
\(955\) 1488.25i 0.0504277i
\(956\) −2055.00 −0.0695224
\(957\) 11199.8i 0.378304i
\(958\) − 13598.7i − 0.458617i
\(959\) − 24948.4i − 0.840067i
\(960\) −655.620 −0.0220417
\(961\) −44989.1 −1.51016
\(962\) − 10469.3i − 0.350878i
\(963\) 16810.3i 0.562517i
\(964\) − 11792.9i − 0.394007i
\(965\) −10758.9 −0.358903
\(966\) − 3930.41i − 0.130910i
\(967\) −24766.8 −0.823625 −0.411813 0.911269i \(-0.635104\pi\)
−0.411813 + 0.911269i \(0.635104\pi\)
\(968\) 10944.9 0.363411
\(969\) 0 0
\(970\) 3279.56 0.108557
\(971\) −42324.3 −1.39882 −0.699409 0.714721i \(-0.746553\pi\)
−0.699409 + 0.714721i \(0.746553\pi\)
\(972\) 23488.3i 0.775091i
\(973\) −11657.2 −0.384083
\(974\) 1538.01i 0.0505966i
\(975\) − 21230.2i − 0.697344i
\(976\) − 1237.63i − 0.0405896i
\(977\) 11320.4 0.370698 0.185349 0.982673i \(-0.440658\pi\)
0.185349 + 0.982673i \(0.440658\pi\)
\(978\) 2905.42 0.0949949
\(979\) − 22461.0i − 0.733254i
\(980\) 5217.16i 0.170057i
\(981\) 30932.2i 1.00672i
\(982\) 25556.8 0.830500
\(983\) 11311.9i 0.367032i 0.983017 + 0.183516i \(0.0587478\pi\)
−0.983017 + 0.183516i \(0.941252\pi\)
\(984\) −28586.3 −0.926116
\(985\) 4132.66 0.133683
\(986\) 0 0
\(987\) −3822.47 −0.123273
\(988\) 31852.6 1.02568
\(989\) 40648.8i 1.30693i
\(990\) −1942.94 −0.0623744
\(991\) 29405.5i 0.942580i 0.881978 + 0.471290i \(0.156212\pi\)
−0.881978 + 0.471290i \(0.843788\pi\)
\(992\) 50676.5i 1.62196i
\(993\) 2373.12i 0.0758397i
\(994\) 8521.63 0.271921
\(995\) −11330.9 −0.361018
\(996\) 2250.43i 0.0715939i
\(997\) − 54905.9i − 1.74412i −0.489398 0.872060i \(-0.662784\pi\)
0.489398 0.872060i \(-0.337216\pi\)
\(998\) − 23588.7i − 0.748183i
\(999\) 18390.9 0.582446
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.b.288.3 6
17.4 even 4 289.4.a.b.1.2 3
17.13 even 4 17.4.a.b.1.2 3
17.16 even 2 inner 289.4.b.b.288.4 6
51.47 odd 4 153.4.a.g.1.2 3
68.47 odd 4 272.4.a.h.1.2 3
85.13 odd 4 425.4.b.f.324.3 6
85.47 odd 4 425.4.b.f.324.4 6
85.64 even 4 425.4.a.g.1.2 3
119.13 odd 4 833.4.a.d.1.2 3
136.13 even 4 1088.4.a.v.1.2 3
136.115 odd 4 1088.4.a.x.1.2 3
187.98 odd 4 2057.4.a.e.1.2 3
204.47 even 4 2448.4.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.2 3 17.13 even 4
153.4.a.g.1.2 3 51.47 odd 4
272.4.a.h.1.2 3 68.47 odd 4
289.4.a.b.1.2 3 17.4 even 4
289.4.b.b.288.3 6 1.1 even 1 trivial
289.4.b.b.288.4 6 17.16 even 2 inner
425.4.a.g.1.2 3 85.64 even 4
425.4.b.f.324.3 6 85.13 odd 4
425.4.b.f.324.4 6 85.47 odd 4
833.4.a.d.1.2 3 119.13 odd 4
1088.4.a.v.1.2 3 136.13 even 4
1088.4.a.x.1.2 3 136.115 odd 4
2057.4.a.e.1.2 3 187.98 odd 4
2448.4.a.bi.1.1 3 204.47 even 4