Properties

Label 169.4.b.f.168.4
Level $169$
Weight $4$
Character 169.168
Analytic conductor $9.971$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.4.b.f.168.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155i q^{2} -3.68466 q^{3} +1.43845 q^{4} +0.561553i q^{5} -9.43845i q^{6} -18.1771i q^{7} +24.1771i q^{8} -13.4233 q^{9} +O(q^{10})\) \(q+2.56155i q^{2} -3.68466 q^{3} +1.43845 q^{4} +0.561553i q^{5} -9.43845i q^{6} -18.1771i q^{7} +24.1771i q^{8} -13.4233 q^{9} -1.43845 q^{10} -64.7386i q^{11} -5.30019 q^{12} +46.5616 q^{14} -2.06913i q^{15} -50.4233 q^{16} +25.5464 q^{17} -34.3845i q^{18} -107.970i q^{19} +0.807764i q^{20} +66.9763i q^{21} +165.831 q^{22} -73.2614 q^{23} -89.0843i q^{24} +124.685 q^{25} +148.946 q^{27} -26.1468i q^{28} +175.909 q^{29} +5.30019 q^{30} -113.093i q^{31} +64.2547i q^{32} +238.540i q^{33} +65.4384i q^{34} +10.2074 q^{35} -19.3087 q^{36} -114.808i q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458i q^{41} -171.563 q^{42} -438.302 q^{43} -93.1231i q^{44} -7.53789i q^{45} -187.663i q^{46} +31.9479i q^{47} +185.793 q^{48} +12.5937 q^{49} +319.386i q^{50} -94.1298 q^{51} +2.84658 q^{53} +381.533i q^{54} +36.3542 q^{55} +439.469 q^{56} +397.831i q^{57} +450.600i q^{58} -71.6325i q^{59} -2.97633i q^{60} -920.695 q^{61} +289.693 q^{62} +243.996i q^{63} -567.978 q^{64} -611.032 q^{66} -444.280i q^{67} +36.7471 q^{68} +269.943 q^{69} +26.1468i q^{70} -541.719i q^{71} -324.536i q^{72} -764.004i q^{73} +294.086 q^{74} -459.420 q^{75} -155.309i q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153i q^{80} -186.386 q^{81} +178.401 q^{82} +603.797i q^{83} +96.3419i q^{84} +14.3457i q^{85} -1122.73i q^{86} -648.165 q^{87} +1565.19 q^{88} +1159.88i q^{89} +19.3087 q^{90} -105.383 q^{92} +416.708i q^{93} -81.8362 q^{94} +60.6307 q^{95} -236.757i q^{96} +583.269i q^{97} +32.2595i q^{98} +869.006i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155i 0.905646i 0.891601 + 0.452823i \(0.149583\pi\)
−0.891601 + 0.452823i \(0.850417\pi\)
\(3\) −3.68466 −0.709113 −0.354556 0.935035i \(-0.615368\pi\)
−0.354556 + 0.935035i \(0.615368\pi\)
\(4\) 1.43845 0.179806
\(5\) 0.561553i 0.0502268i 0.999685 + 0.0251134i \(0.00799469\pi\)
−0.999685 + 0.0251134i \(0.992005\pi\)
\(6\) − 9.43845i − 0.642205i
\(7\) − 18.1771i − 0.981470i −0.871309 0.490735i \(-0.836728\pi\)
0.871309 0.490735i \(-0.163272\pi\)
\(8\) 24.1771i 1.06849i
\(9\) −13.4233 −0.497159
\(10\) −1.43845 −0.0454877
\(11\) − 64.7386i − 1.77449i −0.461295 0.887247i \(-0.652615\pi\)
0.461295 0.887247i \(-0.347385\pi\)
\(12\) −5.30019 −0.127503
\(13\) 0 0
\(14\) 46.5616 0.888864
\(15\) − 2.06913i − 0.0356165i
\(16\) −50.4233 −0.787864
\(17\) 25.5464 0.364465 0.182233 0.983255i \(-0.441668\pi\)
0.182233 + 0.983255i \(0.441668\pi\)
\(18\) − 34.3845i − 0.450250i
\(19\) − 107.970i − 1.30368i −0.758356 0.651841i \(-0.773997\pi\)
0.758356 0.651841i \(-0.226003\pi\)
\(20\) 0.807764i 0.00903108i
\(21\) 66.9763i 0.695973i
\(22\) 165.831 1.60706
\(23\) −73.2614 −0.664176 −0.332088 0.943248i \(-0.607753\pi\)
−0.332088 + 0.943248i \(0.607753\pi\)
\(24\) − 89.0843i − 0.757677i
\(25\) 124.685 0.997477
\(26\) 0 0
\(27\) 148.946 1.06165
\(28\) − 26.1468i − 0.176474i
\(29\) 175.909 1.12640 0.563198 0.826322i \(-0.309571\pi\)
0.563198 + 0.826322i \(0.309571\pi\)
\(30\) 5.30019 0.0322559
\(31\) − 113.093i − 0.655228i −0.944812 0.327614i \(-0.893755\pi\)
0.944812 0.327614i \(-0.106245\pi\)
\(32\) 64.2547i 0.354961i
\(33\) 238.540i 1.25832i
\(34\) 65.4384i 0.330077i
\(35\) 10.2074 0.0492961
\(36\) −19.3087 −0.0893921
\(37\) − 114.808i − 0.510116i −0.966926 0.255058i \(-0.917905\pi\)
0.966926 0.255058i \(-0.0820945\pi\)
\(38\) 276.570 1.18067
\(39\) 0 0
\(40\) −13.5767 −0.0536666
\(41\) − 69.6458i − 0.265289i −0.991164 0.132645i \(-0.957653\pi\)
0.991164 0.132645i \(-0.0423469\pi\)
\(42\) −171.563 −0.630305
\(43\) −438.302 −1.55443 −0.777214 0.629236i \(-0.783368\pi\)
−0.777214 + 0.629236i \(0.783368\pi\)
\(44\) − 93.1231i − 0.319064i
\(45\) − 7.53789i − 0.0249707i
\(46\) − 187.663i − 0.601508i
\(47\) 31.9479i 0.0991506i 0.998770 + 0.0495753i \(0.0157868\pi\)
−0.998770 + 0.0495753i \(0.984213\pi\)
\(48\) 185.793 0.558684
\(49\) 12.5937 0.0367164
\(50\) 319.386i 0.903361i
\(51\) −94.1298 −0.258447
\(52\) 0 0
\(53\) 2.84658 0.00737752 0.00368876 0.999993i \(-0.498826\pi\)
0.00368876 + 0.999993i \(0.498826\pi\)
\(54\) 381.533i 0.961483i
\(55\) 36.3542 0.0891272
\(56\) 439.469 1.04869
\(57\) 397.831i 0.924457i
\(58\) 450.600i 1.02012i
\(59\) − 71.6325i − 0.158064i −0.996872 0.0790319i \(-0.974817\pi\)
0.996872 0.0790319i \(-0.0251829\pi\)
\(60\) − 2.97633i − 0.00640405i
\(61\) −920.695 −1.93251 −0.966253 0.257593i \(-0.917071\pi\)
−0.966253 + 0.257593i \(0.917071\pi\)
\(62\) 289.693 0.593404
\(63\) 243.996i 0.487947i
\(64\) −567.978 −1.10933
\(65\) 0 0
\(66\) −611.032 −1.13959
\(67\) − 444.280i − 0.810112i −0.914292 0.405056i \(-0.867252\pi\)
0.914292 0.405056i \(-0.132748\pi\)
\(68\) 36.7471 0.0655330
\(69\) 269.943 0.470976
\(70\) 26.1468i 0.0446448i
\(71\) − 541.719i − 0.905496i −0.891639 0.452748i \(-0.850444\pi\)
0.891639 0.452748i \(-0.149556\pi\)
\(72\) − 324.536i − 0.531207i
\(73\) − 764.004i − 1.22493i −0.790498 0.612465i \(-0.790178\pi\)
0.790498 0.612465i \(-0.209822\pi\)
\(74\) 294.086 0.461984
\(75\) −459.420 −0.707324
\(76\) − 155.309i − 0.234410i
\(77\) −1176.76 −1.74161
\(78\) 0 0
\(79\) −421.538 −0.600338 −0.300169 0.953886i \(-0.597043\pi\)
−0.300169 + 0.953886i \(0.597043\pi\)
\(80\) − 28.3153i − 0.0395719i
\(81\) −186.386 −0.255674
\(82\) 178.401 0.240258
\(83\) 603.797i 0.798498i 0.916842 + 0.399249i \(0.130729\pi\)
−0.916842 + 0.399249i \(0.869271\pi\)
\(84\) 96.3419i 0.125140i
\(85\) 14.3457i 0.0183059i
\(86\) − 1122.73i − 1.40776i
\(87\) −648.165 −0.798742
\(88\) 1565.19 1.89602
\(89\) 1159.88i 1.38143i 0.723127 + 0.690715i \(0.242704\pi\)
−0.723127 + 0.690715i \(0.757296\pi\)
\(90\) 19.3087 0.0226146
\(91\) 0 0
\(92\) −105.383 −0.119423
\(93\) 416.708i 0.464631i
\(94\) −81.8362 −0.0897953
\(95\) 60.6307 0.0654798
\(96\) − 236.757i − 0.251707i
\(97\) 583.269i 0.610536i 0.952267 + 0.305268i \(0.0987460\pi\)
−0.952267 + 0.305268i \(0.901254\pi\)
\(98\) 32.2595i 0.0332521i
\(99\) 869.006i 0.882206i
\(100\) 179.352 0.179352
\(101\) −921.740 −0.908085 −0.454043 0.890980i \(-0.650019\pi\)
−0.454043 + 0.890980i \(0.650019\pi\)
\(102\) − 241.118i − 0.234061i
\(103\) 930.712 0.890347 0.445174 0.895444i \(-0.353142\pi\)
0.445174 + 0.895444i \(0.353142\pi\)
\(104\) 0 0
\(105\) −37.6107 −0.0349565
\(106\) 7.29168i 0.00668142i
\(107\) 857.383 0.774638 0.387319 0.921946i \(-0.373401\pi\)
0.387319 + 0.921946i \(0.373401\pi\)
\(108\) 214.251 0.190892
\(109\) 671.853i 0.590384i 0.955438 + 0.295192i \(0.0953836\pi\)
−0.955438 + 0.295192i \(0.904616\pi\)
\(110\) 93.1231i 0.0807176i
\(111\) 423.027i 0.361730i
\(112\) 916.548i 0.773265i
\(113\) 641.474 0.534024 0.267012 0.963693i \(-0.413964\pi\)
0.267012 + 0.963693i \(0.413964\pi\)
\(114\) −1019.07 −0.837231
\(115\) − 41.1401i − 0.0333594i
\(116\) 253.036 0.202533
\(117\) 0 0
\(118\) 183.491 0.143150
\(119\) − 464.359i − 0.357712i
\(120\) 50.0255 0.0380557
\(121\) −2860.09 −2.14883
\(122\) − 2358.41i − 1.75017i
\(123\) 256.621i 0.188120i
\(124\) − 162.678i − 0.117814i
\(125\) 140.211i 0.100327i
\(126\) −625.009 −0.441907
\(127\) 553.174 0.386506 0.193253 0.981149i \(-0.438096\pi\)
0.193253 + 0.981149i \(0.438096\pi\)
\(128\) − 940.868i − 0.649702i
\(129\) 1614.99 1.10227
\(130\) 0 0
\(131\) 2056.40 1.37152 0.685758 0.727830i \(-0.259471\pi\)
0.685758 + 0.727830i \(0.259471\pi\)
\(132\) 343.127i 0.226253i
\(133\) −1962.57 −1.27952
\(134\) 1138.05 0.733674
\(135\) 83.6411i 0.0533235i
\(136\) 617.637i 0.389426i
\(137\) 1808.57i 1.12786i 0.825824 + 0.563928i \(0.190710\pi\)
−0.825824 + 0.563928i \(0.809290\pi\)
\(138\) 691.474i 0.426537i
\(139\) 1493.64 0.911428 0.455714 0.890126i \(-0.349384\pi\)
0.455714 + 0.890126i \(0.349384\pi\)
\(140\) 14.6828 0.00886373
\(141\) − 117.717i − 0.0703090i
\(142\) 1387.64 0.820058
\(143\) 0 0
\(144\) 676.847 0.391694
\(145\) 98.7822i 0.0565753i
\(146\) 1957.04 1.10935
\(147\) −46.4036 −0.0260361
\(148\) − 165.145i − 0.0917218i
\(149\) − 2759.02i − 1.51696i −0.651694 0.758482i \(-0.725941\pi\)
0.651694 0.758482i \(-0.274059\pi\)
\(150\) − 1176.83i − 0.640585i
\(151\) 976.355i 0.526190i 0.964770 + 0.263095i \(0.0847432\pi\)
−0.964770 + 0.263095i \(0.915257\pi\)
\(152\) 2610.39 1.39297
\(153\) −342.917 −0.181197
\(154\) − 3014.33i − 1.57728i
\(155\) 63.5076 0.0329100
\(156\) 0 0
\(157\) −564.875 −0.287146 −0.143573 0.989640i \(-0.545859\pi\)
−0.143573 + 0.989640i \(0.545859\pi\)
\(158\) − 1079.79i − 0.543694i
\(159\) −10.4887 −0.00523149
\(160\) −36.0824 −0.0178285
\(161\) 1331.68i 0.651869i
\(162\) − 477.438i − 0.231550i
\(163\) − 1508.53i − 0.724892i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(164\) − 100.182i − 0.0477005i
\(165\) −133.953 −0.0632012
\(166\) −1546.66 −0.723157
\(167\) − 592.521i − 0.274555i −0.990533 0.137277i \(-0.956165\pi\)
0.990533 0.137277i \(-0.0438351\pi\)
\(168\) −1619.29 −0.743638
\(169\) 0 0
\(170\) −36.7471 −0.0165787
\(171\) 1449.31i 0.648137i
\(172\) −630.474 −0.279495
\(173\) 4495.57 1.97568 0.987838 0.155488i \(-0.0496952\pi\)
0.987838 + 0.155488i \(0.0496952\pi\)
\(174\) − 1660.31i − 0.723377i
\(175\) − 2266.40i − 0.978994i
\(176\) 3264.34i 1.39806i
\(177\) 263.941i 0.112085i
\(178\) −2971.10 −1.25109
\(179\) 154.285 0.0644235 0.0322117 0.999481i \(-0.489745\pi\)
0.0322117 + 0.999481i \(0.489745\pi\)
\(180\) − 10.8429i − 0.00448988i
\(181\) −1071.35 −0.439959 −0.219979 0.975505i \(-0.570599\pi\)
−0.219979 + 0.975505i \(0.570599\pi\)
\(182\) 0 0
\(183\) 3392.45 1.37037
\(184\) − 1771.25i − 0.709663i
\(185\) 64.4706 0.0256215
\(186\) −1067.42 −0.420791
\(187\) − 1653.84i − 0.646742i
\(188\) 45.9554i 0.0178279i
\(189\) − 2707.40i − 1.04198i
\(190\) 155.309i 0.0593015i
\(191\) 677.203 0.256548 0.128274 0.991739i \(-0.459056\pi\)
0.128274 + 0.991739i \(0.459056\pi\)
\(192\) 2092.81 0.786642
\(193\) − 1321.68i − 0.492936i −0.969151 0.246468i \(-0.920730\pi\)
0.969151 0.246468i \(-0.0792700\pi\)
\(194\) −1494.07 −0.552929
\(195\) 0 0
\(196\) 18.1154 0.00660183
\(197\) 1267.37i 0.458356i 0.973385 + 0.229178i \(0.0736038\pi\)
−0.973385 + 0.229178i \(0.926396\pi\)
\(198\) −2226.00 −0.798966
\(199\) −2396.24 −0.853593 −0.426796 0.904348i \(-0.640358\pi\)
−0.426796 + 0.904348i \(0.640358\pi\)
\(200\) 3014.51i 1.06579i
\(201\) 1637.02i 0.574460i
\(202\) − 2361.09i − 0.822403i
\(203\) − 3197.51i − 1.10552i
\(204\) −135.401 −0.0464703
\(205\) 39.1098 0.0133246
\(206\) 2384.07i 0.806339i
\(207\) 983.409 0.330201
\(208\) 0 0
\(209\) −6989.81 −2.31337
\(210\) − 96.3419i − 0.0316582i
\(211\) −91.5539 −0.0298712 −0.0149356 0.999888i \(-0.504754\pi\)
−0.0149356 + 0.999888i \(0.504754\pi\)
\(212\) 4.09466 0.00132652
\(213\) 1996.05i 0.642099i
\(214\) 2196.23i 0.701548i
\(215\) − 246.130i − 0.0780740i
\(216\) 3601.08i 1.13436i
\(217\) −2055.70 −0.643087
\(218\) −1720.99 −0.534679
\(219\) 2815.09i 0.868613i
\(220\) 52.2935 0.0160256
\(221\) 0 0
\(222\) −1083.61 −0.327599
\(223\) 1235.42i 0.370985i 0.982646 + 0.185493i \(0.0593880\pi\)
−0.982646 + 0.185493i \(0.940612\pi\)
\(224\) 1167.96 0.348383
\(225\) −1673.68 −0.495905
\(226\) 1643.17i 0.483637i
\(227\) 3301.66i 0.965370i 0.875794 + 0.482685i \(0.160338\pi\)
−0.875794 + 0.482685i \(0.839662\pi\)
\(228\) 572.260i 0.166223i
\(229\) − 211.283i − 0.0609694i −0.999535 0.0304847i \(-0.990295\pi\)
0.999535 0.0304847i \(-0.00970508\pi\)
\(230\) 105.383 0.0302118
\(231\) 4335.96 1.23500
\(232\) 4252.97i 1.20354i
\(233\) 256.724 0.0721827 0.0360913 0.999348i \(-0.488509\pi\)
0.0360913 + 0.999348i \(0.488509\pi\)
\(234\) 0 0
\(235\) −17.9404 −0.00498002
\(236\) − 103.040i − 0.0284208i
\(237\) 1553.22 0.425708
\(238\) 1189.48 0.323960
\(239\) − 3549.62i − 0.960694i −0.877078 0.480347i \(-0.840511\pi\)
0.877078 0.480347i \(-0.159489\pi\)
\(240\) 104.332i 0.0280609i
\(241\) 5030.10i 1.34447i 0.740338 + 0.672235i \(0.234665\pi\)
−0.740338 + 0.672235i \(0.765335\pi\)
\(242\) − 7326.27i − 1.94608i
\(243\) −3334.77 −0.880353
\(244\) −1324.37 −0.347476
\(245\) 7.07204i 0.00184415i
\(246\) −657.349 −0.170370
\(247\) 0 0
\(248\) 2734.25 0.700102
\(249\) − 2224.79i − 0.566226i
\(250\) −359.158 −0.0908606
\(251\) 718.784 0.180754 0.0903770 0.995908i \(-0.471193\pi\)
0.0903770 + 0.995908i \(0.471193\pi\)
\(252\) 350.976i 0.0877357i
\(253\) 4742.84i 1.17858i
\(254\) 1416.98i 0.350038i
\(255\) − 52.8588i − 0.0129810i
\(256\) −2133.74 −0.520933
\(257\) −1280.79 −0.310871 −0.155435 0.987846i \(-0.549678\pi\)
−0.155435 + 0.987846i \(0.549678\pi\)
\(258\) 4136.89i 0.998262i
\(259\) −2086.87 −0.500663
\(260\) 0 0
\(261\) −2361.28 −0.559998
\(262\) 5267.58i 1.24211i
\(263\) 5225.55 1.22517 0.612587 0.790403i \(-0.290129\pi\)
0.612587 + 0.790403i \(0.290129\pi\)
\(264\) −5767.19 −1.34449
\(265\) 1.59851i 0 0.000370549i
\(266\) − 5027.24i − 1.15880i
\(267\) − 4273.77i − 0.979590i
\(268\) − 639.074i − 0.145663i
\(269\) 6443.80 1.46054 0.730270 0.683158i \(-0.239394\pi\)
0.730270 + 0.683158i \(0.239394\pi\)
\(270\) −214.251 −0.0482922
\(271\) 3929.93i 0.880909i 0.897775 + 0.440455i \(0.145183\pi\)
−0.897775 + 0.440455i \(0.854817\pi\)
\(272\) −1288.13 −0.287149
\(273\) 0 0
\(274\) −4632.74 −1.02144
\(275\) − 8071.91i − 1.77002i
\(276\) 388.299 0.0846842
\(277\) 5884.40 1.27639 0.638194 0.769876i \(-0.279682\pi\)
0.638194 + 0.769876i \(0.279682\pi\)
\(278\) 3826.03i 0.825431i
\(279\) 1518.08i 0.325752i
\(280\) 246.785i 0.0526722i
\(281\) − 3529.79i − 0.749358i −0.927155 0.374679i \(-0.877753\pi\)
0.927155 0.374679i \(-0.122247\pi\)
\(282\) 301.538 0.0636750
\(283\) 2611.00 0.548438 0.274219 0.961667i \(-0.411581\pi\)
0.274219 + 0.961667i \(0.411581\pi\)
\(284\) − 779.234i − 0.162813i
\(285\) −223.403 −0.0464325
\(286\) 0 0
\(287\) −1265.96 −0.260373
\(288\) − 862.510i − 0.176472i
\(289\) −4260.38 −0.867165
\(290\) −253.036 −0.0512372
\(291\) − 2149.15i − 0.432939i
\(292\) − 1098.98i − 0.220250i
\(293\) 5491.03i 1.09484i 0.836857 + 0.547422i \(0.184391\pi\)
−0.836857 + 0.547422i \(0.815609\pi\)
\(294\) − 118.865i − 0.0235795i
\(295\) 40.2255 0.00793904
\(296\) 2775.72 0.545052
\(297\) − 9642.56i − 1.88390i
\(298\) 7067.37 1.37383
\(299\) 0 0
\(300\) −660.852 −0.127181
\(301\) 7967.05i 1.52563i
\(302\) −2500.99 −0.476542
\(303\) 3396.30 0.643935
\(304\) 5444.19i 1.02712i
\(305\) − 517.019i − 0.0970637i
\(306\) − 878.399i − 0.164100i
\(307\) 7307.59i 1.35852i 0.733897 + 0.679261i \(0.237700\pi\)
−0.733897 + 0.679261i \(0.762300\pi\)
\(308\) −1692.71 −0.313152
\(309\) −3429.36 −0.631357
\(310\) 162.678i 0.0298048i
\(311\) −7904.92 −1.44131 −0.720654 0.693295i \(-0.756158\pi\)
−0.720654 + 0.693295i \(0.756158\pi\)
\(312\) 0 0
\(313\) 10002.4 1.80629 0.903145 0.429336i \(-0.141252\pi\)
0.903145 + 0.429336i \(0.141252\pi\)
\(314\) − 1446.96i − 0.260053i
\(315\) −137.017 −0.0245080
\(316\) −606.360 −0.107944
\(317\) − 6230.81i − 1.10397i −0.833856 0.551983i \(-0.813871\pi\)
0.833856 0.551983i \(-0.186129\pi\)
\(318\) − 26.8673i − 0.00473788i
\(319\) − 11388.1i − 1.99878i
\(320\) − 318.950i − 0.0557182i
\(321\) −3159.16 −0.549306
\(322\) −3411.16 −0.590362
\(323\) − 2758.24i − 0.475147i
\(324\) −268.107 −0.0459717
\(325\) 0 0
\(326\) 3864.19 0.656495
\(327\) − 2475.55i − 0.418649i
\(328\) 1683.83 0.283458
\(329\) 580.719 0.0973134
\(330\) − 343.127i − 0.0572379i
\(331\) − 4634.51i − 0.769594i −0.923001 0.384797i \(-0.874271\pi\)
0.923001 0.384797i \(-0.125729\pi\)
\(332\) 868.531i 0.143575i
\(333\) 1541.10i 0.253609i
\(334\) 1517.77 0.248649
\(335\) 249.487 0.0406893
\(336\) − 3377.17i − 0.548332i
\(337\) −3029.82 −0.489747 −0.244874 0.969555i \(-0.578746\pi\)
−0.244874 + 0.969555i \(0.578746\pi\)
\(338\) 0 0
\(339\) −2363.61 −0.378684
\(340\) 20.6355i 0.00329151i
\(341\) −7321.47 −1.16270
\(342\) −3712.48 −0.586982
\(343\) − 6463.66i − 1.01751i
\(344\) − 10596.9i − 1.66089i
\(345\) 151.587i 0.0236556i
\(346\) 11515.6i 1.78926i
\(347\) 2841.60 0.439611 0.219805 0.975544i \(-0.429458\pi\)
0.219805 + 0.975544i \(0.429458\pi\)
\(348\) −932.351 −0.143619
\(349\) − 7565.68i − 1.16040i −0.814472 0.580202i \(-0.802973\pi\)
0.814472 0.580202i \(-0.197027\pi\)
\(350\) 5805.51 0.886622
\(351\) 0 0
\(352\) 4159.76 0.629875
\(353\) − 2339.44i − 0.352736i −0.984324 0.176368i \(-0.943565\pi\)
0.984324 0.176368i \(-0.0564348\pi\)
\(354\) −676.100 −0.101509
\(355\) 304.204 0.0454802
\(356\) 1668.43i 0.248389i
\(357\) 1711.00i 0.253658i
\(358\) 395.209i 0.0583449i
\(359\) 2531.68i 0.372192i 0.982532 + 0.186096i \(0.0595835\pi\)
−0.982532 + 0.186096i \(0.940417\pi\)
\(360\) 182.244 0.0266809
\(361\) −4798.45 −0.699585
\(362\) − 2744.31i − 0.398447i
\(363\) 10538.5 1.52376
\(364\) 0 0
\(365\) 429.028 0.0615243
\(366\) 8689.93i 1.24107i
\(367\) 6577.81 0.935583 0.467792 0.883839i \(-0.345050\pi\)
0.467792 + 0.883839i \(0.345050\pi\)
\(368\) 3694.08 0.523280
\(369\) 934.876i 0.131891i
\(370\) 165.145i 0.0232040i
\(371\) − 51.7426i − 0.00724081i
\(372\) 599.413i 0.0835433i
\(373\) 2902.72 0.402942 0.201471 0.979495i \(-0.435428\pi\)
0.201471 + 0.979495i \(0.435428\pi\)
\(374\) 4236.40 0.585719
\(375\) − 516.630i − 0.0711431i
\(376\) −772.407 −0.105941
\(377\) 0 0
\(378\) 6935.16 0.943667
\(379\) − 1865.73i − 0.252866i −0.991975 0.126433i \(-0.959647\pi\)
0.991975 0.126433i \(-0.0403529\pi\)
\(380\) 87.2140 0.0117736
\(381\) −2038.26 −0.274076
\(382\) 1734.69i 0.232342i
\(383\) 10836.0i 1.44567i 0.691019 + 0.722837i \(0.257162\pi\)
−0.691019 + 0.722837i \(0.742838\pi\)
\(384\) 3466.78i 0.460712i
\(385\) − 660.813i − 0.0874757i
\(386\) 3385.55 0.446425
\(387\) 5883.46 0.772798
\(388\) 839.001i 0.109778i
\(389\) 9520.34 1.24088 0.620438 0.784256i \(-0.286955\pi\)
0.620438 + 0.784256i \(0.286955\pi\)
\(390\) 0 0
\(391\) −1871.56 −0.242069
\(392\) 304.480i 0.0392310i
\(393\) −7577.13 −0.972559
\(394\) −3246.43 −0.415108
\(395\) − 236.716i − 0.0301531i
\(396\) 1250.02i 0.158626i
\(397\) 10108.8i 1.27796i 0.769225 + 0.638978i \(0.220642\pi\)
−0.769225 + 0.638978i \(0.779358\pi\)
\(398\) − 6138.10i − 0.773053i
\(399\) 7231.41 0.907327
\(400\) −6287.01 −0.785876
\(401\) − 2084.38i − 0.259573i −0.991542 0.129787i \(-0.958571\pi\)
0.991542 0.129787i \(-0.0414292\pi\)
\(402\) −4193.32 −0.520258
\(403\) 0 0
\(404\) −1325.88 −0.163279
\(405\) − 104.666i − 0.0128417i
\(406\) 8190.60 1.00121
\(407\) −7432.50 −0.905197
\(408\) − 2275.78i − 0.276147i
\(409\) − 9716.53i − 1.17470i −0.809334 0.587349i \(-0.800172\pi\)
0.809334 0.587349i \(-0.199828\pi\)
\(410\) 100.182i 0.0120674i
\(411\) − 6663.95i − 0.799777i
\(412\) 1338.78 0.160090
\(413\) −1302.07 −0.155135
\(414\) 2519.05i 0.299045i
\(415\) −339.064 −0.0401060
\(416\) 0 0
\(417\) −5503.54 −0.646305
\(418\) − 17904.8i − 2.09510i
\(419\) 13381.9 1.56026 0.780129 0.625619i \(-0.215153\pi\)
0.780129 + 0.625619i \(0.215153\pi\)
\(420\) −54.1011 −0.00628539
\(421\) − 9463.37i − 1.09553i −0.836633 0.547763i \(-0.815479\pi\)
0.836633 0.547763i \(-0.184521\pi\)
\(422\) − 234.520i − 0.0270527i
\(423\) − 428.846i − 0.0492936i
\(424\) 68.8221i 0.00788278i
\(425\) 3185.24 0.363546
\(426\) −5112.98 −0.581514
\(427\) 16735.5i 1.89670i
\(428\) 1233.30 0.139285
\(429\) 0 0
\(430\) 630.474 0.0707074
\(431\) 4852.28i 0.542288i 0.962539 + 0.271144i \(0.0874019\pi\)
−0.962539 + 0.271144i \(0.912598\pi\)
\(432\) −7510.35 −0.836439
\(433\) 8208.00 0.910973 0.455486 0.890243i \(-0.349465\pi\)
0.455486 + 0.890243i \(0.349465\pi\)
\(434\) − 5265.78i − 0.582409i
\(435\) − 363.979i − 0.0401183i
\(436\) 966.425i 0.106155i
\(437\) 7910.01i 0.865874i
\(438\) −7211.01 −0.786656
\(439\) 2993.80 0.325481 0.162741 0.986669i \(-0.447967\pi\)
0.162741 + 0.986669i \(0.447967\pi\)
\(440\) 878.938i 0.0952311i
\(441\) −169.049 −0.0182539
\(442\) 0 0
\(443\) 9743.67 1.04500 0.522501 0.852639i \(-0.324999\pi\)
0.522501 + 0.852639i \(0.324999\pi\)
\(444\) 608.503i 0.0650411i
\(445\) −651.335 −0.0693848
\(446\) −3164.59 −0.335981
\(447\) 10166.0i 1.07570i
\(448\) 10324.2i 1.08878i
\(449\) 561.459i 0.0590131i 0.999565 + 0.0295065i \(0.00939359\pi\)
−0.999565 + 0.0295065i \(0.990606\pi\)
\(450\) − 4287.22i − 0.449114i
\(451\) −4508.78 −0.470754
\(452\) 922.726 0.0960207
\(453\) − 3597.54i − 0.373128i
\(454\) −8457.38 −0.874283
\(455\) 0 0
\(456\) −9618.40 −0.987770
\(457\) 13758.4i 1.40830i 0.710054 + 0.704148i \(0.248671\pi\)
−0.710054 + 0.704148i \(0.751329\pi\)
\(458\) 541.213 0.0552166
\(459\) 3805.03 0.386936
\(460\) − 59.1779i − 0.00599823i
\(461\) 12009.2i 1.21329i 0.794974 + 0.606644i \(0.207485\pi\)
−0.794974 + 0.606644i \(0.792515\pi\)
\(462\) 11106.8i 1.11847i
\(463\) − 13635.7i − 1.36870i −0.729156 0.684348i \(-0.760087\pi\)
0.729156 0.684348i \(-0.239913\pi\)
\(464\) −8869.91 −0.887447
\(465\) −234.004 −0.0233369
\(466\) 657.613i 0.0653719i
\(467\) −8821.95 −0.874157 −0.437079 0.899423i \(-0.643987\pi\)
−0.437079 + 0.899423i \(0.643987\pi\)
\(468\) 0 0
\(469\) −8075.72 −0.795100
\(470\) − 45.9554i − 0.00451013i
\(471\) 2081.37 0.203619
\(472\) 1731.87 0.168889
\(473\) 28375.1i 2.75832i
\(474\) 3978.66i 0.385540i
\(475\) − 13462.2i − 1.30039i
\(476\) − 667.956i − 0.0643187i
\(477\) −38.2105 −0.00366780
\(478\) 9092.54 0.870049
\(479\) 14620.0i 1.39459i 0.716786 + 0.697293i \(0.245612\pi\)
−0.716786 + 0.697293i \(0.754388\pi\)
\(480\) 132.951 0.0126424
\(481\) 0 0
\(482\) −12884.9 −1.21761
\(483\) − 4906.78i − 0.462249i
\(484\) −4114.09 −0.386372
\(485\) −327.536 −0.0306653
\(486\) − 8542.20i − 0.797288i
\(487\) − 9798.86i − 0.911763i −0.890040 0.455882i \(-0.849324\pi\)
0.890040 0.455882i \(-0.150676\pi\)
\(488\) − 22259.7i − 2.06486i
\(489\) 5558.43i 0.514030i
\(490\) −18.1154 −0.00167014
\(491\) 10836.1 0.995977 0.497989 0.867184i \(-0.334072\pi\)
0.497989 + 0.867184i \(0.334072\pi\)
\(492\) 369.136i 0.0338251i
\(493\) 4493.84 0.410532
\(494\) 0 0
\(495\) −487.993 −0.0443104
\(496\) 5702.51i 0.516230i
\(497\) −9846.86 −0.888717
\(498\) 5698.91 0.512800
\(499\) 2589.96i 0.232349i 0.993229 + 0.116175i \(0.0370633\pi\)
−0.993229 + 0.116175i \(0.962937\pi\)
\(500\) 201.686i 0.0180394i
\(501\) 2183.24i 0.194690i
\(502\) 1841.20i 0.163699i
\(503\) −17067.5 −1.51292 −0.756462 0.654038i \(-0.773074\pi\)
−0.756462 + 0.654038i \(0.773074\pi\)
\(504\) −5899.12 −0.521364
\(505\) − 517.606i − 0.0456102i
\(506\) −12149.0 −1.06737
\(507\) 0 0
\(508\) 795.712 0.0694961
\(509\) − 1012.89i − 0.0882038i −0.999027 0.0441019i \(-0.985957\pi\)
0.999027 0.0441019i \(-0.0140426\pi\)
\(510\) 135.401 0.0117562
\(511\) −13887.4 −1.20223
\(512\) − 12992.6i − 1.12148i
\(513\) − 16081.7i − 1.38406i
\(514\) − 3280.82i − 0.281539i
\(515\) 522.644i 0.0447193i
\(516\) 2323.08 0.198194
\(517\) 2068.26 0.175942
\(518\) − 5345.63i − 0.453424i
\(519\) −16564.6 −1.40098
\(520\) 0 0
\(521\) −14367.7 −1.20818 −0.604089 0.796917i \(-0.706463\pi\)
−0.604089 + 0.796917i \(0.706463\pi\)
\(522\) − 6048.54i − 0.507160i
\(523\) −16219.9 −1.35611 −0.678057 0.735010i \(-0.737178\pi\)
−0.678057 + 0.735010i \(0.737178\pi\)
\(524\) 2958.02 0.246607
\(525\) 8350.92i 0.694217i
\(526\) 13385.5i 1.10957i
\(527\) − 2889.11i − 0.238808i
\(528\) − 12028.0i − 0.991382i
\(529\) −6799.77 −0.558870
\(530\) −4.09466 −0.000335586 0
\(531\) 961.545i 0.0785828i
\(532\) −2823.06 −0.230066
\(533\) 0 0
\(534\) 10947.5 0.887161
\(535\) 481.466i 0.0389076i
\(536\) 10741.4 0.865593
\(537\) −568.488 −0.0456835
\(538\) 16506.1i 1.32273i
\(539\) − 815.301i − 0.0651530i
\(540\) 120.313i 0.00958788i
\(541\) − 17592.2i − 1.39806i −0.715094 0.699029i \(-0.753616\pi\)
0.715094 0.699029i \(-0.246384\pi\)
\(542\) −10066.7 −0.797792
\(543\) 3947.55 0.311980
\(544\) 1641.48i 0.129371i
\(545\) −377.281 −0.0296531
\(546\) 0 0
\(547\) 10504.6 0.821103 0.410552 0.911837i \(-0.365336\pi\)
0.410552 + 0.911837i \(0.365336\pi\)
\(548\) 2601.53i 0.202795i
\(549\) 12358.8 0.960763
\(550\) 20676.6 1.60301
\(551\) − 18992.8i − 1.46846i
\(552\) 6526.44i 0.503231i
\(553\) 7662.33i 0.589214i
\(554\) 15073.2i 1.15596i
\(555\) −237.552 −0.0181685
\(556\) 2148.52 0.163880
\(557\) 507.558i 0.0386102i 0.999814 + 0.0193051i \(0.00614539\pi\)
−0.999814 + 0.0193051i \(0.993855\pi\)
\(558\) −3888.64 −0.295016
\(559\) 0 0
\(560\) −514.690 −0.0388386
\(561\) 6093.83i 0.458613i
\(562\) 9041.75 0.678653
\(563\) 3443.14 0.257746 0.128873 0.991661i \(-0.458864\pi\)
0.128873 + 0.991661i \(0.458864\pi\)
\(564\) − 169.330i − 0.0126420i
\(565\) 360.221i 0.0268223i
\(566\) 6688.21i 0.496690i
\(567\) 3387.96i 0.250936i
\(568\) 13097.2 0.967509
\(569\) −23972.2 −1.76620 −0.883098 0.469189i \(-0.844546\pi\)
−0.883098 + 0.469189i \(0.844546\pi\)
\(570\) − 572.260i − 0.0420514i
\(571\) 7458.32 0.546622 0.273311 0.961926i \(-0.411881\pi\)
0.273311 + 0.961926i \(0.411881\pi\)
\(572\) 0 0
\(573\) −2495.26 −0.181922
\(574\) − 3242.82i − 0.235806i
\(575\) −9134.57 −0.662501
\(576\) 7624.14 0.551515
\(577\) 5669.57i 0.409059i 0.978860 + 0.204530i \(0.0655665\pi\)
−0.978860 + 0.204530i \(0.934434\pi\)
\(578\) − 10913.2i − 0.785344i
\(579\) 4869.94i 0.349547i
\(580\) 142.093i 0.0101726i
\(581\) 10975.3 0.783702
\(582\) 5505.15 0.392089
\(583\) − 184.284i − 0.0130914i
\(584\) 18471.4 1.30882
\(585\) 0 0
\(586\) −14065.6 −0.991541
\(587\) 1017.39i 0.0715371i 0.999360 + 0.0357685i \(0.0113879\pi\)
−0.999360 + 0.0357685i \(0.988612\pi\)
\(588\) −66.7491 −0.00468144
\(589\) −12210.6 −0.854208
\(590\) 103.040i 0.00718996i
\(591\) − 4669.81i − 0.325026i
\(592\) 5788.99i 0.401902i
\(593\) 10198.2i 0.706221i 0.935582 + 0.353111i \(0.114876\pi\)
−0.935582 + 0.353111i \(0.885124\pi\)
\(594\) 24699.9 1.70615
\(595\) 260.762 0.0179667
\(596\) − 3968.70i − 0.272759i
\(597\) 8829.33 0.605294
\(598\) 0 0
\(599\) 12516.3 0.853763 0.426881 0.904308i \(-0.359612\pi\)
0.426881 + 0.904308i \(0.359612\pi\)
\(600\) − 11107.4i − 0.755766i
\(601\) 9627.46 0.653431 0.326716 0.945123i \(-0.394058\pi\)
0.326716 + 0.945123i \(0.394058\pi\)
\(602\) −20408.0 −1.38168
\(603\) 5963.70i 0.402754i
\(604\) 1404.44i 0.0946120i
\(605\) − 1606.09i − 0.107929i
\(606\) 8699.80i 0.583177i
\(607\) 6667.20 0.445821 0.222910 0.974839i \(-0.428444\pi\)
0.222910 + 0.974839i \(0.428444\pi\)
\(608\) 6937.56 0.462755
\(609\) 11781.7i 0.783942i
\(610\) 1324.37 0.0879053
\(611\) 0 0
\(612\) −493.268 −0.0325803
\(613\) − 23085.4i − 1.52106i −0.649302 0.760530i \(-0.724939\pi\)
0.649302 0.760530i \(-0.275061\pi\)
\(614\) −18718.8 −1.23034
\(615\) −144.106 −0.00944866
\(616\) − 28450.6i − 1.86089i
\(617\) 3049.24i 0.198959i 0.995040 + 0.0994796i \(0.0317178\pi\)
−0.995040 + 0.0994796i \(0.968282\pi\)
\(618\) − 8784.48i − 0.571786i
\(619\) − 7296.58i − 0.473787i −0.971536 0.236894i \(-0.923871\pi\)
0.971536 0.236894i \(-0.0761293\pi\)
\(620\) 91.3523 0.00591741
\(621\) −10912.0 −0.705126
\(622\) − 20248.9i − 1.30531i
\(623\) 21083.3 1.35583
\(624\) 0 0
\(625\) 15506.8 0.992438
\(626\) 25621.7i 1.63586i
\(627\) 25755.1 1.64044
\(628\) −812.543 −0.0516305
\(629\) − 2932.92i − 0.185920i
\(630\) − 350.976i − 0.0221956i
\(631\) 23829.5i 1.50339i 0.659512 + 0.751694i \(0.270763\pi\)
−0.659512 + 0.751694i \(0.729237\pi\)
\(632\) − 10191.6i − 0.641453i
\(633\) 337.345 0.0211821
\(634\) 15960.5 0.999801
\(635\) 310.637i 0.0194130i
\(636\) −15.0874 −0.000940653 0
\(637\) 0 0
\(638\) 29171.3 1.81019
\(639\) 7271.65i 0.450175i
\(640\) 528.347 0.0326324
\(641\) −13405.3 −0.826016 −0.413008 0.910727i \(-0.635522\pi\)
−0.413008 + 0.910727i \(0.635522\pi\)
\(642\) − 8092.36i − 0.497477i
\(643\) 5251.51i 0.322083i 0.986948 + 0.161042i \(0.0514853\pi\)
−0.986948 + 0.161042i \(0.948515\pi\)
\(644\) 1915.55i 0.117210i
\(645\) 906.904i 0.0553633i
\(646\) 7065.37 0.430315
\(647\) −21611.4 −1.31319 −0.656595 0.754244i \(-0.728004\pi\)
−0.656595 + 0.754244i \(0.728004\pi\)
\(648\) − 4506.28i − 0.273184i
\(649\) −4637.39 −0.280483
\(650\) 0 0
\(651\) 7574.54 0.456021
\(652\) − 2169.94i − 0.130340i
\(653\) −21595.8 −1.29420 −0.647099 0.762406i \(-0.724018\pi\)
−0.647099 + 0.762406i \(0.724018\pi\)
\(654\) 6341.25 0.379148
\(655\) 1154.78i 0.0688869i
\(656\) 3511.77i 0.209012i
\(657\) 10255.4i 0.608985i
\(658\) 1487.54i 0.0881314i
\(659\) −16642.6 −0.983768 −0.491884 0.870661i \(-0.663692\pi\)
−0.491884 + 0.870661i \(0.663692\pi\)
\(660\) −192.684 −0.0113640
\(661\) − 26981.1i − 1.58766i −0.608139 0.793831i \(-0.708084\pi\)
0.608139 0.793831i \(-0.291916\pi\)
\(662\) 11871.5 0.696980
\(663\) 0 0
\(664\) −14598.1 −0.853185
\(665\) − 1102.09i − 0.0642664i
\(666\) −3947.60 −0.229680
\(667\) −12887.3 −0.748126
\(668\) − 852.310i − 0.0493665i
\(669\) − 4552.09i − 0.263070i
\(670\) 639.074i 0.0368501i
\(671\) 59604.5i 3.42922i
\(672\) −4303.55 −0.247043
\(673\) −11149.2 −0.638591 −0.319296 0.947655i \(-0.603446\pi\)
−0.319296 + 0.947655i \(0.603446\pi\)
\(674\) − 7761.04i − 0.443537i
\(675\) 18571.3 1.05898
\(676\) 0 0
\(677\) 3314.33 0.188154 0.0940769 0.995565i \(-0.470010\pi\)
0.0940769 + 0.995565i \(0.470010\pi\)
\(678\) − 6054.51i − 0.342953i
\(679\) 10602.1 0.599223
\(680\) −346.836 −0.0195596
\(681\) − 12165.5i − 0.684556i
\(682\) − 18754.3i − 1.05299i
\(683\) − 24505.2i − 1.37287i −0.727193 0.686433i \(-0.759176\pi\)
0.727193 0.686433i \(-0.240824\pi\)
\(684\) 2084.75i 0.116539i
\(685\) −1015.61 −0.0566486
\(686\) 16557.0 0.921500
\(687\) 778.506i 0.0432342i
\(688\) 22100.6 1.22468
\(689\) 0 0
\(690\) −388.299 −0.0214236
\(691\) − 21752.8i − 1.19756i −0.800912 0.598782i \(-0.795652\pi\)
0.800912 0.598782i \(-0.204348\pi\)
\(692\) 6466.64 0.355238
\(693\) 15796.0 0.865858
\(694\) 7278.90i 0.398132i
\(695\) 838.755i 0.0457781i
\(696\) − 15670.7i − 0.853445i
\(697\) − 1779.20i − 0.0966887i
\(698\) 19379.9 1.05092
\(699\) −945.941 −0.0511857
\(700\) − 3260.10i − 0.176029i
\(701\) −34250.9 −1.84542 −0.922709 0.385496i \(-0.874030\pi\)
−0.922709 + 0.385496i \(0.874030\pi\)
\(702\) 0 0
\(703\) −12395.8 −0.665028
\(704\) 36770.1i 1.96850i
\(705\) 66.1043 0.00353140
\(706\) 5992.59 0.319454
\(707\) 16754.6i 0.891259i
\(708\) 379.666i 0.0201536i
\(709\) 5527.11i 0.292771i 0.989228 + 0.146386i \(0.0467641\pi\)
−0.989228 + 0.146386i \(0.953236\pi\)
\(710\) 779.234i 0.0411889i
\(711\) 5658.43 0.298464
\(712\) −28042.6 −1.47604
\(713\) 8285.33i 0.435187i
\(714\) −4382.83 −0.229724
\(715\) 0 0
\(716\) 221.931 0.0115837
\(717\) 13079.1i 0.681241i
\(718\) −6485.02 −0.337074
\(719\) 3777.78 0.195949 0.0979745 0.995189i \(-0.468764\pi\)
0.0979745 + 0.995189i \(0.468764\pi\)
\(720\) 380.085i 0.0196735i
\(721\) − 16917.6i − 0.873849i
\(722\) − 12291.5i − 0.633576i
\(723\) − 18534.2i − 0.953380i
\(724\) −1541.08 −0.0791072
\(725\) 21933.2 1.12355
\(726\) 26994.8i 1.37999i
\(727\) −19076.8 −0.973204 −0.486602 0.873624i \(-0.661764\pi\)
−0.486602 + 0.873624i \(0.661764\pi\)
\(728\) 0 0
\(729\) 17319.9 0.879944
\(730\) 1098.98i 0.0557192i
\(731\) −11197.0 −0.566535
\(732\) 4879.86 0.246400
\(733\) 7997.30i 0.402984i 0.979490 + 0.201492i \(0.0645790\pi\)
−0.979490 + 0.201492i \(0.935421\pi\)
\(734\) 16849.4i 0.847307i
\(735\) − 26.0581i − 0.00130771i
\(736\) − 4707.39i − 0.235756i
\(737\) −28762.1 −1.43754
\(738\) −2394.74 −0.119446
\(739\) − 28983.6i − 1.44273i −0.692553 0.721367i \(-0.743514\pi\)
0.692553 0.721367i \(-0.256486\pi\)
\(740\) 92.7376 0.00460689
\(741\) 0 0
\(742\) 132.541 0.00655761
\(743\) − 19145.4i − 0.945324i −0.881244 0.472662i \(-0.843293\pi\)
0.881244 0.472662i \(-0.156707\pi\)
\(744\) −10074.8 −0.496451
\(745\) 1549.33 0.0761923
\(746\) 7435.47i 0.364922i
\(747\) − 8104.95i − 0.396981i
\(748\) − 2378.96i − 0.116288i
\(749\) − 15584.7i − 0.760284i
\(750\) 1323.38 0.0644304
\(751\) 25516.9 1.23985 0.619923 0.784663i \(-0.287164\pi\)
0.619923 + 0.784663i \(0.287164\pi\)
\(752\) − 1610.92i − 0.0781172i
\(753\) −2648.47 −0.128175
\(754\) 0 0
\(755\) −548.275 −0.0264288
\(756\) − 3894.46i − 0.187355i
\(757\) −17230.6 −0.827289 −0.413645 0.910438i \(-0.635744\pi\)
−0.413645 + 0.910438i \(0.635744\pi\)
\(758\) 4779.17 0.229007
\(759\) − 17475.7i − 0.835744i
\(760\) 1465.87i 0.0699642i
\(761\) 2343.06i 0.111611i 0.998442 + 0.0558053i \(0.0177726\pi\)
−0.998442 + 0.0558053i \(0.982227\pi\)
\(762\) − 5221.11i − 0.248216i
\(763\) 12212.3 0.579444
\(764\) 974.121 0.0461289
\(765\) − 192.566i − 0.00910096i
\(766\) −27756.9 −1.30927
\(767\) 0 0
\(768\) 7862.11 0.369400
\(769\) − 7100.18i − 0.332950i −0.986046 0.166475i \(-0.946761\pi\)
0.986046 0.166475i \(-0.0532386\pi\)
\(770\) 1692.71 0.0792219
\(771\) 4719.29 0.220442
\(772\) − 1901.17i − 0.0886328i
\(773\) 12270.4i 0.570940i 0.958388 + 0.285470i \(0.0921498\pi\)
−0.958388 + 0.285470i \(0.907850\pi\)
\(774\) 15070.8i 0.699881i
\(775\) − 14100.9i − 0.653575i
\(776\) −14101.7 −0.652349
\(777\) 7689.40 0.355027
\(778\) 24386.9i 1.12379i
\(779\) −7519.64 −0.345852
\(780\) 0 0
\(781\) −35070.1 −1.60680
\(782\) − 4794.11i − 0.219229i
\(783\) 26201.0 1.19584
\(784\) −635.017 −0.0289275
\(785\) − 317.207i − 0.0144224i
\(786\) − 19409.2i − 0.880794i
\(787\) − 3425.04i − 0.155133i −0.996987 0.0775663i \(-0.975285\pi\)
0.996987 0.0775663i \(-0.0247150\pi\)
\(788\) 1823.04i 0.0824151i
\(789\) −19254.3 −0.868787
\(790\) 606.360 0.0273080
\(791\) − 11660.1i − 0.524129i
\(792\) −21010.0 −0.942624
\(793\) 0 0
\(794\) −25894.3 −1.15737
\(795\) − 5.88995i 0 0.000262761i
\(796\) −3446.87 −0.153481
\(797\) 11781.1 0.523600 0.261800 0.965122i \(-0.415684\pi\)
0.261800 + 0.965122i \(0.415684\pi\)
\(798\) 18523.6i 0.821717i
\(799\) 816.154i 0.0361370i
\(800\) 8011.58i 0.354065i
\(801\) − 15569.4i − 0.686790i
\(802\) 5339.24 0.235081
\(803\) −49460.6 −2.17363
\(804\) 2354.77i 0.103291i
\(805\) −747.807 −0.0327413
\(806\) 0 0
\(807\) −23743.2 −1.03569
\(808\) − 22285.0i − 0.970276i
\(809\) 18910.1 0.821810 0.410905 0.911678i \(-0.365213\pi\)
0.410905 + 0.911678i \(0.365213\pi\)
\(810\) 268.107 0.0116300
\(811\) 12803.3i 0.554359i 0.960818 + 0.277180i \(0.0893997\pi\)
−0.960818 + 0.277180i \(0.910600\pi\)
\(812\) − 4599.45i − 0.198780i
\(813\) − 14480.5i − 0.624664i
\(814\) − 19038.7i − 0.819788i
\(815\) 847.121 0.0364090
\(816\) 4746.33 0.203621
\(817\) 47323.3i 2.02648i
\(818\) 24889.4 1.06386
\(819\) 0 0
\(820\) 56.2574 0.00239585
\(821\) 19335.1i 0.821923i 0.911653 + 0.410962i \(0.134807\pi\)
−0.911653 + 0.410962i \(0.865193\pi\)
\(822\) 17070.1 0.724315
\(823\) 2125.90 0.0900417 0.0450209 0.998986i \(-0.485665\pi\)
0.0450209 + 0.998986i \(0.485665\pi\)
\(824\) 22501.9i 0.951324i
\(825\) 29742.2i 1.25514i
\(826\) − 3335.32i − 0.140497i
\(827\) 6989.24i 0.293881i 0.989145 + 0.146941i \(0.0469426\pi\)
−0.989145 + 0.146941i \(0.953057\pi\)
\(828\) 1414.58 0.0593721
\(829\) 32649.7 1.36788 0.683938 0.729540i \(-0.260266\pi\)
0.683938 + 0.729540i \(0.260266\pi\)
\(830\) − 868.531i − 0.0363219i
\(831\) −21682.0 −0.905103
\(832\) 0 0
\(833\) 321.724 0.0133819
\(834\) − 14097.6i − 0.585324i
\(835\) 332.732 0.0137900
\(836\) −10054.5 −0.415958
\(837\) − 16844.7i − 0.695626i
\(838\) 34278.4i 1.41304i
\(839\) 4038.23i 0.166168i 0.996543 + 0.0830841i \(0.0264770\pi\)
−0.996543 + 0.0830841i \(0.973523\pi\)
\(840\) − 909.318i − 0.0373505i
\(841\) 6555.00 0.268769
\(842\) 24240.9 0.992159
\(843\) 13006.1i 0.531380i
\(844\) −131.695 −0.00537102
\(845\) 0 0
\(846\) 1098.51 0.0446426
\(847\) 51988.1i 2.10901i
\(848\) −143.534 −0.00581248
\(849\) −9620.64 −0.388904
\(850\) 8159.17i 0.329244i
\(851\) 8410.97i 0.338807i
\(852\) 2871.21i 0.115453i
\(853\) − 8114.12i − 0.325700i −0.986651 0.162850i \(-0.947931\pi\)
0.986651 0.162850i \(-0.0520687\pi\)
\(854\) −42869.0 −1.71774
\(855\) −813.863 −0.0325538
\(856\) 20729.0i 0.827690i
\(857\) 22298.1 0.888786 0.444393 0.895832i \(-0.353419\pi\)
0.444393 + 0.895832i \(0.353419\pi\)
\(858\) 0 0
\(859\) 33550.5 1.33263 0.666315 0.745670i \(-0.267870\pi\)
0.666315 + 0.745670i \(0.267870\pi\)
\(860\) − 354.045i − 0.0140382i
\(861\) 4664.62 0.184634
\(862\) −12429.4 −0.491121
\(863\) − 14120.5i − 0.556972i −0.960440 0.278486i \(-0.910167\pi\)
0.960440 0.278486i \(-0.0898326\pi\)
\(864\) 9570.49i 0.376846i
\(865\) 2524.50i 0.0992319i
\(866\) 21025.2i 0.825018i
\(867\) 15698.1 0.614918
\(868\) −2957.01 −0.115631
\(869\) 27289.8i 1.06530i
\(870\) 932.351 0.0363329
\(871\) 0 0
\(872\) −16243.4 −0.630817
\(873\) − 7829.39i − 0.303533i
\(874\) −20261.9 −0.784175
\(875\) 2548.63 0.0984679
\(876\) 4049.36i 0.156182i
\(877\) − 1941.69i − 0.0747619i −0.999301 0.0373809i \(-0.988099\pi\)
0.999301 0.0373809i \(-0.0119015\pi\)
\(878\) 7668.78i 0.294771i
\(879\) − 20232.6i − 0.776368i
\(880\) −1833.10 −0.0702201
\(881\) 790.231 0.0302197 0.0151099 0.999886i \(-0.495190\pi\)
0.0151099 + 0.999886i \(0.495190\pi\)
\(882\) − 433.029i − 0.0165316i
\(883\) 36638.6 1.39636 0.698180 0.715922i \(-0.253993\pi\)
0.698180 + 0.715922i \(0.253993\pi\)
\(884\) 0 0
\(885\) −148.217 −0.00562968
\(886\) 24958.9i 0.946401i
\(887\) −40686.3 −1.54015 −0.770075 0.637954i \(-0.779781\pi\)
−0.770075 + 0.637954i \(0.779781\pi\)
\(888\) −10227.6 −0.386503
\(889\) − 10055.1i − 0.379344i
\(890\) − 1668.43i − 0.0628381i
\(891\) 12066.4i 0.453692i
\(892\) 1777.08i 0.0667053i
\(893\) 3449.40 0.129261
\(894\) −26040.9 −0.974202
\(895\) 86.6392i 0.00323579i
\(896\) −17102.2 −0.637663
\(897\) 0 0
\(898\) −1438.21 −0.0534449
\(899\) − 19894.0i − 0.738046i
\(900\) −2407.50 −0.0891666
\(901\) 72.7200 0.00268885
\(902\) − 11549.5i − 0.426336i
\(903\) − 29355.9i − 1.08184i
\(904\) 15509.0i 0.570598i
\(905\) − 601.618i − 0.0220977i
\(906\) 9215.28 0.337922
\(907\) 10464.4 0.383093 0.191547 0.981484i \(-0.438650\pi\)
0.191547 + 0.981484i \(0.438650\pi\)
\(908\) 4749.26i 0.173579i
\(909\) 12372.8 0.451463
\(910\) 0 0
\(911\) −35611.5 −1.29513 −0.647563 0.762011i \(-0.724212\pi\)
−0.647563 + 0.762011i \(0.724212\pi\)
\(912\) − 20060.0i − 0.728346i
\(913\) 39089.0 1.41693
\(914\) −35242.9 −1.27542
\(915\) 1905.04i 0.0688291i
\(916\) − 303.920i − 0.0109627i
\(917\) − 37379.4i − 1.34610i
\(918\) 9746.80i 0.350427i
\(919\) 1077.25 0.0386674 0.0193337 0.999813i \(-0.493846\pi\)
0.0193337 + 0.999813i \(0.493846\pi\)
\(920\) 994.648 0.0356441
\(921\) − 26926.0i − 0.963346i
\(922\) −30762.3 −1.09881
\(923\) 0 0
\(924\) 6237.04 0.222060
\(925\) − 14314.8i − 0.508829i
\(926\) 34928.6 1.23955
\(927\) −12493.2 −0.442644
\(928\) 11303.0i 0.399826i
\(929\) 55733.8i 1.96832i 0.177290 + 0.984159i \(0.443267\pi\)
−0.177290 + 0.984159i \(0.556733\pi\)
\(930\) − 599.413i − 0.0211350i
\(931\) − 1359.74i − 0.0478665i
\(932\) 369.284 0.0129789
\(933\) 29126.9 1.02205
\(934\) − 22597.9i − 0.791677i
\(935\) 928.718 0.0324838
\(936\) 0 0
\(937\) −3198.60 −0.111519 −0.0557596 0.998444i \(-0.517758\pi\)
−0.0557596 + 0.998444i \(0.517758\pi\)
\(938\) − 20686.4i − 0.720079i
\(939\) −36855.4 −1.28086
\(940\) −25.8064 −0.000895437 0
\(941\) 8823.35i 0.305667i 0.988252 + 0.152834i \(0.0488399\pi\)
−0.988252 + 0.152834i \(0.951160\pi\)
\(942\) 5331.54i 0.184407i
\(943\) 5102.35i 0.176199i
\(944\) 3611.95i 0.124533i
\(945\) 1520.35 0.0523354
\(946\) −72684.3 −2.49806
\(947\) − 28290.4i − 0.970766i −0.874301 0.485383i \(-0.838680\pi\)
0.874301 0.485383i \(-0.161320\pi\)
\(948\) 2234.23 0.0765447
\(949\) 0 0
\(950\) 34484.0 1.17769
\(951\) 22958.4i 0.782836i
\(952\) 11226.8 0.382210
\(953\) 12399.0 0.421452 0.210726 0.977545i \(-0.432417\pi\)
0.210726 + 0.977545i \(0.432417\pi\)
\(954\) − 97.8783i − 0.00332173i
\(955\) 380.285i 0.0128856i
\(956\) − 5105.94i − 0.172739i
\(957\) 41961.3i 1.41736i
\(958\) −37450.0 −1.26300
\(959\) 32874.5 1.10696
\(960\) 1175.22i 0.0395105i
\(961\) 17001.0 0.570676
\(962\) 0 0
\(963\) −11508.9 −0.385118
\(964\) 7235.53i 0.241744i
\(965\) 742.193 0.0247586
\(966\) 12569.0 0.418634
\(967\) − 26667.1i − 0.886820i −0.896319 0.443410i \(-0.853769\pi\)
0.896319 0.443410i \(-0.146231\pi\)
\(968\) − 69148.6i − 2.29599i
\(969\) 10163.2i 0.336933i
\(970\) − 839.001i − 0.0277719i
\(971\) 49420.7 1.63335 0.816676 0.577096i \(-0.195814\pi\)
0.816676 + 0.577096i \(0.195814\pi\)
\(972\) −4796.89 −0.158293
\(973\) − 27149.9i − 0.894539i
\(974\) 25100.3 0.825735
\(975\) 0 0
\(976\) 46424.5 1.52255
\(977\) 778.759i 0.0255012i 0.999919 + 0.0127506i \(0.00405876\pi\)
−0.999919 + 0.0127506i \(0.995941\pi\)
\(978\) −14238.2 −0.465529
\(979\) 75089.2 2.45134
\(980\) 10.1728i 0 0.000331589i
\(981\) − 9018.48i − 0.293515i
\(982\) 27757.2i 0.902003i
\(983\) − 5997.90i − 0.194612i −0.995255 0.0973059i \(-0.968977\pi\)
0.995255 0.0973059i \(-0.0310225\pi\)
\(984\) −6204.35 −0.201004
\(985\) −711.693 −0.0230218
\(986\) 11511.2i 0.371797i
\(987\) −2139.75 −0.0690062
\(988\) 0 0
\(989\) 32110.6 1.03241
\(990\) − 1250.02i − 0.0401295i
\(991\) 8974.94 0.287688 0.143844 0.989600i \(-0.454054\pi\)
0.143844 + 0.989600i \(0.454054\pi\)
\(992\) 7266.75 0.232580
\(993\) 17076.6i 0.545729i
\(994\) − 25223.3i − 0.804863i
\(995\) − 1345.62i − 0.0428732i
\(996\) − 3200.24i − 0.101811i
\(997\) 28530.2 0.906280 0.453140 0.891439i \(-0.350304\pi\)
0.453140 + 0.891439i \(0.350304\pi\)
\(998\) −6634.31 −0.210426
\(999\) − 17100.2i − 0.541567i
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 169.4.b.f.168.4 4
13.2 odd 12 169.4.c.g.22.1 4
13.3 even 3 169.4.e.f.147.1 8
13.4 even 6 169.4.e.f.23.1 8
13.5 odd 4 13.4.a.b.1.2 2
13.6 odd 12 169.4.c.g.146.1 4
13.7 odd 12 169.4.c.j.146.2 4
13.8 odd 4 169.4.a.g.1.1 2
13.9 even 3 169.4.e.f.23.4 8
13.10 even 6 169.4.e.f.147.4 8
13.11 odd 12 169.4.c.j.22.2 4
13.12 even 2 inner 169.4.b.f.168.1 4
39.5 even 4 117.4.a.d.1.1 2
39.8 even 4 1521.4.a.r.1.2 2
52.31 even 4 208.4.a.h.1.2 2
65.18 even 4 325.4.b.e.274.1 4
65.44 odd 4 325.4.a.f.1.1 2
65.57 even 4 325.4.b.e.274.4 4
91.83 even 4 637.4.a.b.1.2 2
104.5 odd 4 832.4.a.s.1.2 2
104.83 even 4 832.4.a.z.1.1 2
143.109 even 4 1573.4.a.b.1.1 2
156.83 odd 4 1872.4.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.2 2 13.5 odd 4
117.4.a.d.1.1 2 39.5 even 4
169.4.a.g.1.1 2 13.8 odd 4
169.4.b.f.168.1 4 13.12 even 2 inner
169.4.b.f.168.4 4 1.1 even 1 trivial
169.4.c.g.22.1 4 13.2 odd 12
169.4.c.g.146.1 4 13.6 odd 12
169.4.c.j.22.2 4 13.11 odd 12
169.4.c.j.146.2 4 13.7 odd 12
169.4.e.f.23.1 8 13.4 even 6
169.4.e.f.23.4 8 13.9 even 3
169.4.e.f.147.1 8 13.3 even 3
169.4.e.f.147.4 8 13.10 even 6
208.4.a.h.1.2 2 52.31 even 4
325.4.a.f.1.1 2 65.44 odd 4
325.4.b.e.274.1 4 65.18 even 4
325.4.b.e.274.4 4 65.57 even 4
637.4.a.b.1.2 2 91.83 even 4
832.4.a.s.1.2 2 104.5 odd 4
832.4.a.z.1.1 2 104.83 even 4
1521.4.a.r.1.2 2 39.8 even 4
1573.4.a.b.1.1 2 143.109 even 4
1872.4.a.bb.1.1 2 156.83 odd 4