Properties

Label 169.4.a.g.1.1
Level $169$
Weight $4$
Character 169.1
Self dual yes
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -18.1771 q^{7} +24.1771 q^{8} -13.4233 q^{9} +O(q^{10})\) \(q-2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -18.1771 q^{7} +24.1771 q^{8} -13.4233 q^{9} +1.43845 q^{10} -64.7386 q^{11} +5.30019 q^{12} +46.5616 q^{14} +2.06913 q^{15} -50.4233 q^{16} -25.5464 q^{17} +34.3845 q^{18} +107.970 q^{19} +0.807764 q^{20} +66.9763 q^{21} +165.831 q^{22} +73.2614 q^{23} -89.0843 q^{24} -124.685 q^{25} +148.946 q^{27} +26.1468 q^{28} +175.909 q^{29} -5.30019 q^{30} +113.093 q^{31} -64.2547 q^{32} +238.540 q^{33} +65.4384 q^{34} +10.2074 q^{35} +19.3087 q^{36} -114.808 q^{37} -276.570 q^{38} -13.5767 q^{40} +69.6458 q^{41} -171.563 q^{42} +438.302 q^{43} +93.1231 q^{44} +7.53789 q^{45} -187.663 q^{46} +31.9479 q^{47} +185.793 q^{48} -12.5937 q^{49} +319.386 q^{50} +94.1298 q^{51} +2.84658 q^{53} -381.533 q^{54} +36.3542 q^{55} -439.469 q^{56} -397.831 q^{57} -450.600 q^{58} -71.6325 q^{59} -2.97633 q^{60} -920.695 q^{61} -289.693 q^{62} +243.996 q^{63} +567.978 q^{64} -611.032 q^{66} +444.280 q^{67} +36.7471 q^{68} -269.943 q^{69} -26.1468 q^{70} +541.719 q^{71} -324.536 q^{72} -764.004 q^{73} +294.086 q^{74} +459.420 q^{75} -155.309 q^{76} +1176.76 q^{77} -421.538 q^{79} +28.3153 q^{80} -186.386 q^{81} -178.401 q^{82} -603.797 q^{83} -96.3419 q^{84} +14.3457 q^{85} -1122.73 q^{86} -648.165 q^{87} -1565.19 q^{88} +1159.88 q^{89} -19.3087 q^{90} -105.383 q^{92} -416.708 q^{93} -81.8362 q^{94} -60.6307 q^{95} +236.757 q^{96} -583.269 q^{97} +32.2595 q^{98} +869.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 7 q^{10} - 80 q^{11} - 43 q^{12} + 89 q^{14} + 33 q^{15} - 39 q^{16} + 19 q^{17} + 110 q^{18} + 84 q^{19} - 19 q^{20} + 303 q^{21} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 335 q^{27} - 125 q^{28} - 44 q^{29} + 43 q^{30} + 86 q^{31} + 123 q^{32} + 106 q^{33} + 135 q^{34} + 107 q^{35} - 250 q^{36} - 209 q^{37} - 314 q^{38} - 89 q^{40} + 230 q^{41} + 197 q^{42} + 287 q^{43} + 178 q^{44} + 180 q^{45} + 4 q^{46} - 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} + 481 q^{51} - 118 q^{53} - 91 q^{54} - 18 q^{55} - 1015 q^{56} - 606 q^{57} - 794 q^{58} + 368 q^{59} - 175 q^{60} - 1058 q^{61} - 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} - 211 q^{68} + 796 q^{69} + 125 q^{70} + 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} - 516 q^{75} - 22 q^{76} + 762 q^{77} - 1008 q^{79} + 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} - 1359 q^{86} - 2558 q^{87} - 1242 q^{88} + 720 q^{89} + 250 q^{90} - 788 q^{92} - 652 q^{93} - 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} + 650 q^{98} + 130 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56155 −0.905646 −0.452823 0.891601i \(-0.649583\pi\)
−0.452823 + 0.891601i \(0.649583\pi\)
\(3\) −3.68466 −0.709113 −0.354556 0.935035i \(-0.615368\pi\)
−0.354556 + 0.935035i \(0.615368\pi\)
\(4\) −1.43845 −0.179806
\(5\) −0.561553 −0.0502268 −0.0251134 0.999685i \(-0.507995\pi\)
−0.0251134 + 0.999685i \(0.507995\pi\)
\(6\) 9.43845 0.642205
\(7\) −18.1771 −0.981470 −0.490735 0.871309i \(-0.663272\pi\)
−0.490735 + 0.871309i \(0.663272\pi\)
\(8\) 24.1771 1.06849
\(9\) −13.4233 −0.497159
\(10\) 1.43845 0.0454877
\(11\) −64.7386 −1.77449 −0.887247 0.461295i \(-0.847385\pi\)
−0.887247 + 0.461295i \(0.847385\pi\)
\(12\) 5.30019 0.127503
\(13\) 0 0
\(14\) 46.5616 0.888864
\(15\) 2.06913 0.0356165
\(16\) −50.4233 −0.787864
\(17\) −25.5464 −0.364465 −0.182233 0.983255i \(-0.558332\pi\)
−0.182233 + 0.983255i \(0.558332\pi\)
\(18\) 34.3845 0.450250
\(19\) 107.970 1.30368 0.651841 0.758356i \(-0.273997\pi\)
0.651841 + 0.758356i \(0.273997\pi\)
\(20\) 0.807764 0.00903108
\(21\) 66.9763 0.695973
\(22\) 165.831 1.60706
\(23\) 73.2614 0.664176 0.332088 0.943248i \(-0.392247\pi\)
0.332088 + 0.943248i \(0.392247\pi\)
\(24\) −89.0843 −0.757677
\(25\) −124.685 −0.997477
\(26\) 0 0
\(27\) 148.946 1.06165
\(28\) 26.1468 0.176474
\(29\) 175.909 1.12640 0.563198 0.826322i \(-0.309571\pi\)
0.563198 + 0.826322i \(0.309571\pi\)
\(30\) −5.30019 −0.0322559
\(31\) 113.093 0.655228 0.327614 0.944812i \(-0.393755\pi\)
0.327614 + 0.944812i \(0.393755\pi\)
\(32\) −64.2547 −0.354961
\(33\) 238.540 1.25832
\(34\) 65.4384 0.330077
\(35\) 10.2074 0.0492961
\(36\) 19.3087 0.0893921
\(37\) −114.808 −0.510116 −0.255058 0.966926i \(-0.582095\pi\)
−0.255058 + 0.966926i \(0.582095\pi\)
\(38\) −276.570 −1.18067
\(39\) 0 0
\(40\) −13.5767 −0.0536666
\(41\) 69.6458 0.265289 0.132645 0.991164i \(-0.457653\pi\)
0.132645 + 0.991164i \(0.457653\pi\)
\(42\) −171.563 −0.630305
\(43\) 438.302 1.55443 0.777214 0.629236i \(-0.216632\pi\)
0.777214 + 0.629236i \(0.216632\pi\)
\(44\) 93.1231 0.319064
\(45\) 7.53789 0.0249707
\(46\) −187.663 −0.601508
\(47\) 31.9479 0.0991506 0.0495753 0.998770i \(-0.484213\pi\)
0.0495753 + 0.998770i \(0.484213\pi\)
\(48\) 185.793 0.558684
\(49\) −12.5937 −0.0367164
\(50\) 319.386 0.903361
\(51\) 94.1298 0.258447
\(52\) 0 0
\(53\) 2.84658 0.00737752 0.00368876 0.999993i \(-0.498826\pi\)
0.00368876 + 0.999993i \(0.498826\pi\)
\(54\) −381.533 −0.961483
\(55\) 36.3542 0.0891272
\(56\) −439.469 −1.04869
\(57\) −397.831 −0.924457
\(58\) −450.600 −1.02012
\(59\) −71.6325 −0.158064 −0.0790319 0.996872i \(-0.525183\pi\)
−0.0790319 + 0.996872i \(0.525183\pi\)
\(60\) −2.97633 −0.00640405
\(61\) −920.695 −1.93251 −0.966253 0.257593i \(-0.917071\pi\)
−0.966253 + 0.257593i \(0.917071\pi\)
\(62\) −289.693 −0.593404
\(63\) 243.996 0.487947
\(64\) 567.978 1.10933
\(65\) 0 0
\(66\) −611.032 −1.13959
\(67\) 444.280 0.810112 0.405056 0.914292i \(-0.367252\pi\)
0.405056 + 0.914292i \(0.367252\pi\)
\(68\) 36.7471 0.0655330
\(69\) −269.943 −0.470976
\(70\) −26.1468 −0.0446448
\(71\) 541.719 0.905496 0.452748 0.891639i \(-0.350444\pi\)
0.452748 + 0.891639i \(0.350444\pi\)
\(72\) −324.536 −0.531207
\(73\) −764.004 −1.22493 −0.612465 0.790498i \(-0.709822\pi\)
−0.612465 + 0.790498i \(0.709822\pi\)
\(74\) 294.086 0.461984
\(75\) 459.420 0.707324
\(76\) −155.309 −0.234410
\(77\) 1176.76 1.74161
\(78\) 0 0
\(79\) −421.538 −0.600338 −0.300169 0.953886i \(-0.597043\pi\)
−0.300169 + 0.953886i \(0.597043\pi\)
\(80\) 28.3153 0.0395719
\(81\) −186.386 −0.255674
\(82\) −178.401 −0.240258
\(83\) −603.797 −0.798498 −0.399249 0.916842i \(-0.630729\pi\)
−0.399249 + 0.916842i \(0.630729\pi\)
\(84\) −96.3419 −0.125140
\(85\) 14.3457 0.0183059
\(86\) −1122.73 −1.40776
\(87\) −648.165 −0.798742
\(88\) −1565.19 −1.89602
\(89\) 1159.88 1.38143 0.690715 0.723127i \(-0.257296\pi\)
0.690715 + 0.723127i \(0.257296\pi\)
\(90\) −19.3087 −0.0226146
\(91\) 0 0
\(92\) −105.383 −0.119423
\(93\) −416.708 −0.464631
\(94\) −81.8362 −0.0897953
\(95\) −60.6307 −0.0654798
\(96\) 236.757 0.251707
\(97\) −583.269 −0.610536 −0.305268 0.952267i \(-0.598746\pi\)
−0.305268 + 0.952267i \(0.598746\pi\)
\(98\) 32.2595 0.0332521
\(99\) 869.006 0.882206
\(100\) 179.352 0.179352
\(101\) 921.740 0.908085 0.454043 0.890980i \(-0.349981\pi\)
0.454043 + 0.890980i \(0.349981\pi\)
\(102\) −241.118 −0.234061
\(103\) −930.712 −0.890347 −0.445174 0.895444i \(-0.646858\pi\)
−0.445174 + 0.895444i \(0.646858\pi\)
\(104\) 0 0
\(105\) −37.6107 −0.0349565
\(106\) −7.29168 −0.00668142
\(107\) 857.383 0.774638 0.387319 0.921946i \(-0.373401\pi\)
0.387319 + 0.921946i \(0.373401\pi\)
\(108\) −214.251 −0.190892
\(109\) −671.853 −0.590384 −0.295192 0.955438i \(-0.595384\pi\)
−0.295192 + 0.955438i \(0.595384\pi\)
\(110\) −93.1231 −0.0807176
\(111\) 423.027 0.361730
\(112\) 916.548 0.773265
\(113\) 641.474 0.534024 0.267012 0.963693i \(-0.413964\pi\)
0.267012 + 0.963693i \(0.413964\pi\)
\(114\) 1019.07 0.837231
\(115\) −41.1401 −0.0333594
\(116\) −253.036 −0.202533
\(117\) 0 0
\(118\) 183.491 0.143150
\(119\) 464.359 0.357712
\(120\) 50.0255 0.0380557
\(121\) 2860.09 2.14883
\(122\) 2358.41 1.75017
\(123\) −256.621 −0.188120
\(124\) −162.678 −0.117814
\(125\) 140.211 0.100327
\(126\) −625.009 −0.441907
\(127\) −553.174 −0.386506 −0.193253 0.981149i \(-0.561904\pi\)
−0.193253 + 0.981149i \(0.561904\pi\)
\(128\) −940.868 −0.649702
\(129\) −1614.99 −1.10227
\(130\) 0 0
\(131\) 2056.40 1.37152 0.685758 0.727830i \(-0.259471\pi\)
0.685758 + 0.727830i \(0.259471\pi\)
\(132\) −343.127 −0.226253
\(133\) −1962.57 −1.27952
\(134\) −1138.05 −0.733674
\(135\) −83.6411 −0.0533235
\(136\) −617.637 −0.389426
\(137\) 1808.57 1.12786 0.563928 0.825824i \(-0.309290\pi\)
0.563928 + 0.825824i \(0.309290\pi\)
\(138\) 691.474 0.426537
\(139\) 1493.64 0.911428 0.455714 0.890126i \(-0.349384\pi\)
0.455714 + 0.890126i \(0.349384\pi\)
\(140\) −14.6828 −0.00886373
\(141\) −117.717 −0.0703090
\(142\) −1387.64 −0.820058
\(143\) 0 0
\(144\) 676.847 0.391694
\(145\) −98.7822 −0.0565753
\(146\) 1957.04 1.10935
\(147\) 46.4036 0.0260361
\(148\) 165.145 0.0917218
\(149\) 2759.02 1.51696 0.758482 0.651694i \(-0.225941\pi\)
0.758482 + 0.651694i \(0.225941\pi\)
\(150\) −1176.83 −0.640585
\(151\) 976.355 0.526190 0.263095 0.964770i \(-0.415257\pi\)
0.263095 + 0.964770i \(0.415257\pi\)
\(152\) 2610.39 1.39297
\(153\) 342.917 0.181197
\(154\) −3014.33 −1.57728
\(155\) −63.5076 −0.0329100
\(156\) 0 0
\(157\) −564.875 −0.287146 −0.143573 0.989640i \(-0.545859\pi\)
−0.143573 + 0.989640i \(0.545859\pi\)
\(158\) 1079.79 0.543694
\(159\) −10.4887 −0.00523149
\(160\) 36.0824 0.0178285
\(161\) −1331.68 −0.651869
\(162\) 477.438 0.231550
\(163\) −1508.53 −0.724892 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(164\) −100.182 −0.0477005
\(165\) −133.953 −0.0632012
\(166\) 1546.66 0.723157
\(167\) −592.521 −0.274555 −0.137277 0.990533i \(-0.543835\pi\)
−0.137277 + 0.990533i \(0.543835\pi\)
\(168\) 1619.29 0.743638
\(169\) 0 0
\(170\) −36.7471 −0.0165787
\(171\) −1449.31 −0.648137
\(172\) −630.474 −0.279495
\(173\) −4495.57 −1.97568 −0.987838 0.155488i \(-0.950305\pi\)
−0.987838 + 0.155488i \(0.950305\pi\)
\(174\) 1660.31 0.723377
\(175\) 2266.40 0.978994
\(176\) 3264.34 1.39806
\(177\) 263.941 0.112085
\(178\) −2971.10 −1.25109
\(179\) −154.285 −0.0644235 −0.0322117 0.999481i \(-0.510255\pi\)
−0.0322117 + 0.999481i \(0.510255\pi\)
\(180\) −10.8429 −0.00448988
\(181\) 1071.35 0.439959 0.219979 0.975505i \(-0.429401\pi\)
0.219979 + 0.975505i \(0.429401\pi\)
\(182\) 0 0
\(183\) 3392.45 1.37037
\(184\) 1771.25 0.709663
\(185\) 64.4706 0.0256215
\(186\) 1067.42 0.420791
\(187\) 1653.84 0.646742
\(188\) −45.9554 −0.0178279
\(189\) −2707.40 −1.04198
\(190\) 155.309 0.0593015
\(191\) 677.203 0.256548 0.128274 0.991739i \(-0.459056\pi\)
0.128274 + 0.991739i \(0.459056\pi\)
\(192\) −2092.81 −0.786642
\(193\) −1321.68 −0.492936 −0.246468 0.969151i \(-0.579270\pi\)
−0.246468 + 0.969151i \(0.579270\pi\)
\(194\) 1494.07 0.552929
\(195\) 0 0
\(196\) 18.1154 0.00660183
\(197\) −1267.37 −0.458356 −0.229178 0.973385i \(-0.573604\pi\)
−0.229178 + 0.973385i \(0.573604\pi\)
\(198\) −2226.00 −0.798966
\(199\) 2396.24 0.853593 0.426796 0.904348i \(-0.359642\pi\)
0.426796 + 0.904348i \(0.359642\pi\)
\(200\) −3014.51 −1.06579
\(201\) −1637.02 −0.574460
\(202\) −2361.09 −0.822403
\(203\) −3197.51 −1.10552
\(204\) −135.401 −0.0464703
\(205\) −39.1098 −0.0133246
\(206\) 2384.07 0.806339
\(207\) −983.409 −0.330201
\(208\) 0 0
\(209\) −6989.81 −2.31337
\(210\) 96.3419 0.0316582
\(211\) −91.5539 −0.0298712 −0.0149356 0.999888i \(-0.504754\pi\)
−0.0149356 + 0.999888i \(0.504754\pi\)
\(212\) −4.09466 −0.00132652
\(213\) −1996.05 −0.642099
\(214\) −2196.23 −0.701548
\(215\) −246.130 −0.0780740
\(216\) 3601.08 1.13436
\(217\) −2055.70 −0.643087
\(218\) 1720.99 0.534679
\(219\) 2815.09 0.868613
\(220\) −52.2935 −0.0160256
\(221\) 0 0
\(222\) −1083.61 −0.327599
\(223\) −1235.42 −0.370985 −0.185493 0.982646i \(-0.559388\pi\)
−0.185493 + 0.982646i \(0.559388\pi\)
\(224\) 1167.96 0.348383
\(225\) 1673.68 0.495905
\(226\) −1643.17 −0.483637
\(227\) −3301.66 −0.965370 −0.482685 0.875794i \(-0.660338\pi\)
−0.482685 + 0.875794i \(0.660338\pi\)
\(228\) 572.260 0.166223
\(229\) −211.283 −0.0609694 −0.0304847 0.999535i \(-0.509705\pi\)
−0.0304847 + 0.999535i \(0.509705\pi\)
\(230\) 105.383 0.0302118
\(231\) −4335.96 −1.23500
\(232\) 4252.97 1.20354
\(233\) −256.724 −0.0721827 −0.0360913 0.999348i \(-0.511491\pi\)
−0.0360913 + 0.999348i \(0.511491\pi\)
\(234\) 0 0
\(235\) −17.9404 −0.00498002
\(236\) 103.040 0.0284208
\(237\) 1553.22 0.425708
\(238\) −1189.48 −0.323960
\(239\) 3549.62 0.960694 0.480347 0.877078i \(-0.340511\pi\)
0.480347 + 0.877078i \(0.340511\pi\)
\(240\) −104.332 −0.0280609
\(241\) 5030.10 1.34447 0.672235 0.740338i \(-0.265335\pi\)
0.672235 + 0.740338i \(0.265335\pi\)
\(242\) −7326.27 −1.94608
\(243\) −3334.77 −0.880353
\(244\) 1324.37 0.347476
\(245\) 7.07204 0.00184415
\(246\) 657.349 0.170370
\(247\) 0 0
\(248\) 2734.25 0.700102
\(249\) 2224.79 0.566226
\(250\) −359.158 −0.0908606
\(251\) −718.784 −0.180754 −0.0903770 0.995908i \(-0.528807\pi\)
−0.0903770 + 0.995908i \(0.528807\pi\)
\(252\) −350.976 −0.0877357
\(253\) −4742.84 −1.17858
\(254\) 1416.98 0.350038
\(255\) −52.8588 −0.0129810
\(256\) −2133.74 −0.520933
\(257\) 1280.79 0.310871 0.155435 0.987846i \(-0.450322\pi\)
0.155435 + 0.987846i \(0.450322\pi\)
\(258\) 4136.89 0.998262
\(259\) 2086.87 0.500663
\(260\) 0 0
\(261\) −2361.28 −0.559998
\(262\) −5267.58 −1.24211
\(263\) 5225.55 1.22517 0.612587 0.790403i \(-0.290129\pi\)
0.612587 + 0.790403i \(0.290129\pi\)
\(264\) 5767.19 1.34449
\(265\) −1.59851 −0.000370549 0
\(266\) 5027.24 1.15880
\(267\) −4273.77 −0.979590
\(268\) −639.074 −0.145663
\(269\) 6443.80 1.46054 0.730270 0.683158i \(-0.239394\pi\)
0.730270 + 0.683158i \(0.239394\pi\)
\(270\) 214.251 0.0482922
\(271\) 3929.93 0.880909 0.440455 0.897775i \(-0.354817\pi\)
0.440455 + 0.897775i \(0.354817\pi\)
\(272\) 1288.13 0.287149
\(273\) 0 0
\(274\) −4632.74 −1.02144
\(275\) 8071.91 1.77002
\(276\) 388.299 0.0846842
\(277\) −5884.40 −1.27639 −0.638194 0.769876i \(-0.720318\pi\)
−0.638194 + 0.769876i \(0.720318\pi\)
\(278\) −3826.03 −0.825431
\(279\) −1518.08 −0.325752
\(280\) 246.785 0.0526722
\(281\) −3529.79 −0.749358 −0.374679 0.927155i \(-0.622247\pi\)
−0.374679 + 0.927155i \(0.622247\pi\)
\(282\) 301.538 0.0636750
\(283\) −2611.00 −0.548438 −0.274219 0.961667i \(-0.588419\pi\)
−0.274219 + 0.961667i \(0.588419\pi\)
\(284\) −779.234 −0.162813
\(285\) 223.403 0.0464325
\(286\) 0 0
\(287\) −1265.96 −0.260373
\(288\) 862.510 0.176472
\(289\) −4260.38 −0.867165
\(290\) 253.036 0.0512372
\(291\) 2149.15 0.432939
\(292\) 1098.98 0.220250
\(293\) 5491.03 1.09484 0.547422 0.836857i \(-0.315609\pi\)
0.547422 + 0.836857i \(0.315609\pi\)
\(294\) −118.865 −0.0235795
\(295\) 40.2255 0.00793904
\(296\) −2775.72 −0.545052
\(297\) −9642.56 −1.88390
\(298\) −7067.37 −1.37383
\(299\) 0 0
\(300\) −660.852 −0.127181
\(301\) −7967.05 −1.52563
\(302\) −2500.99 −0.476542
\(303\) −3396.30 −0.643935
\(304\) −5444.19 −1.02712
\(305\) 517.019 0.0970637
\(306\) −878.399 −0.164100
\(307\) 7307.59 1.35852 0.679261 0.733897i \(-0.262300\pi\)
0.679261 + 0.733897i \(0.262300\pi\)
\(308\) −1692.71 −0.313152
\(309\) 3429.36 0.631357
\(310\) 162.678 0.0298048
\(311\) 7904.92 1.44131 0.720654 0.693295i \(-0.243842\pi\)
0.720654 + 0.693295i \(0.243842\pi\)
\(312\) 0 0
\(313\) 10002.4 1.80629 0.903145 0.429336i \(-0.141252\pi\)
0.903145 + 0.429336i \(0.141252\pi\)
\(314\) 1446.96 0.260053
\(315\) −137.017 −0.0245080
\(316\) 606.360 0.107944
\(317\) 6230.81 1.10397 0.551983 0.833856i \(-0.313871\pi\)
0.551983 + 0.833856i \(0.313871\pi\)
\(318\) 26.8673 0.00473788
\(319\) −11388.1 −1.99878
\(320\) −318.950 −0.0557182
\(321\) −3159.16 −0.549306
\(322\) 3411.16 0.590362
\(323\) −2758.24 −0.475147
\(324\) 268.107 0.0459717
\(325\) 0 0
\(326\) 3864.19 0.656495
\(327\) 2475.55 0.418649
\(328\) 1683.83 0.283458
\(329\) −580.719 −0.0973134
\(330\) 343.127 0.0572379
\(331\) 4634.51 0.769594 0.384797 0.923001i \(-0.374271\pi\)
0.384797 + 0.923001i \(0.374271\pi\)
\(332\) 868.531 0.143575
\(333\) 1541.10 0.253609
\(334\) 1517.77 0.248649
\(335\) −249.487 −0.0406893
\(336\) −3377.17 −0.548332
\(337\) 3029.82 0.489747 0.244874 0.969555i \(-0.421254\pi\)
0.244874 + 0.969555i \(0.421254\pi\)
\(338\) 0 0
\(339\) −2363.61 −0.378684
\(340\) −20.6355 −0.00329151
\(341\) −7321.47 −1.16270
\(342\) 3712.48 0.586982
\(343\) 6463.66 1.01751
\(344\) 10596.9 1.66089
\(345\) 151.587 0.0236556
\(346\) 11515.6 1.78926
\(347\) 2841.60 0.439611 0.219805 0.975544i \(-0.429458\pi\)
0.219805 + 0.975544i \(0.429458\pi\)
\(348\) 932.351 0.143619
\(349\) −7565.68 −1.16040 −0.580202 0.814472i \(-0.697027\pi\)
−0.580202 + 0.814472i \(0.697027\pi\)
\(350\) −5805.51 −0.886622
\(351\) 0 0
\(352\) 4159.76 0.629875
\(353\) 2339.44 0.352736 0.176368 0.984324i \(-0.443565\pi\)
0.176368 + 0.984324i \(0.443565\pi\)
\(354\) −676.100 −0.101509
\(355\) −304.204 −0.0454802
\(356\) −1668.43 −0.248389
\(357\) −1711.00 −0.253658
\(358\) 395.209 0.0583449
\(359\) 2531.68 0.372192 0.186096 0.982532i \(-0.440417\pi\)
0.186096 + 0.982532i \(0.440417\pi\)
\(360\) 182.244 0.0266809
\(361\) 4798.45 0.699585
\(362\) −2744.31 −0.398447
\(363\) −10538.5 −1.52376
\(364\) 0 0
\(365\) 429.028 0.0615243
\(366\) −8689.93 −1.24107
\(367\) 6577.81 0.935583 0.467792 0.883839i \(-0.345050\pi\)
0.467792 + 0.883839i \(0.345050\pi\)
\(368\) −3694.08 −0.523280
\(369\) −934.876 −0.131891
\(370\) −165.145 −0.0232040
\(371\) −51.7426 −0.00724081
\(372\) 599.413 0.0835433
\(373\) 2902.72 0.402942 0.201471 0.979495i \(-0.435428\pi\)
0.201471 + 0.979495i \(0.435428\pi\)
\(374\) −4236.40 −0.585719
\(375\) −516.630 −0.0711431
\(376\) 772.407 0.105941
\(377\) 0 0
\(378\) 6935.16 0.943667
\(379\) 1865.73 0.252866 0.126433 0.991975i \(-0.459647\pi\)
0.126433 + 0.991975i \(0.459647\pi\)
\(380\) 87.2140 0.0117736
\(381\) 2038.26 0.274076
\(382\) −1734.69 −0.232342
\(383\) −10836.0 −1.44567 −0.722837 0.691019i \(-0.757162\pi\)
−0.722837 + 0.691019i \(0.757162\pi\)
\(384\) 3466.78 0.460712
\(385\) −660.813 −0.0874757
\(386\) 3385.55 0.446425
\(387\) −5883.46 −0.772798
\(388\) 839.001 0.109778
\(389\) −9520.34 −1.24088 −0.620438 0.784256i \(-0.713045\pi\)
−0.620438 + 0.784256i \(0.713045\pi\)
\(390\) 0 0
\(391\) −1871.56 −0.242069
\(392\) −304.480 −0.0392310
\(393\) −7577.13 −0.972559
\(394\) 3246.43 0.415108
\(395\) 236.716 0.0301531
\(396\) −1250.02 −0.158626
\(397\) 10108.8 1.27796 0.638978 0.769225i \(-0.279358\pi\)
0.638978 + 0.769225i \(0.279358\pi\)
\(398\) −6138.10 −0.773053
\(399\) 7231.41 0.907327
\(400\) 6287.01 0.785876
\(401\) −2084.38 −0.259573 −0.129787 0.991542i \(-0.541429\pi\)
−0.129787 + 0.991542i \(0.541429\pi\)
\(402\) 4193.32 0.520258
\(403\) 0 0
\(404\) −1325.88 −0.163279
\(405\) 104.666 0.0128417
\(406\) 8190.60 1.00121
\(407\) 7432.50 0.905197
\(408\) 2275.78 0.276147
\(409\) 9716.53 1.17470 0.587349 0.809334i \(-0.300172\pi\)
0.587349 + 0.809334i \(0.300172\pi\)
\(410\) 100.182 0.0120674
\(411\) −6663.95 −0.799777
\(412\) 1338.78 0.160090
\(413\) 1302.07 0.155135
\(414\) 2519.05 0.299045
\(415\) 339.064 0.0401060
\(416\) 0 0
\(417\) −5503.54 −0.646305
\(418\) 17904.8 2.09510
\(419\) 13381.9 1.56026 0.780129 0.625619i \(-0.215153\pi\)
0.780129 + 0.625619i \(0.215153\pi\)
\(420\) 54.1011 0.00628539
\(421\) 9463.37 1.09553 0.547763 0.836633i \(-0.315479\pi\)
0.547763 + 0.836633i \(0.315479\pi\)
\(422\) 234.520 0.0270527
\(423\) −428.846 −0.0492936
\(424\) 68.8221 0.00788278
\(425\) 3185.24 0.363546
\(426\) 5112.98 0.581514
\(427\) 16735.5 1.89670
\(428\) −1233.30 −0.139285
\(429\) 0 0
\(430\) 630.474 0.0707074
\(431\) −4852.28 −0.542288 −0.271144 0.962539i \(-0.587402\pi\)
−0.271144 + 0.962539i \(0.587402\pi\)
\(432\) −7510.35 −0.836439
\(433\) −8208.00 −0.910973 −0.455486 0.890243i \(-0.650535\pi\)
−0.455486 + 0.890243i \(0.650535\pi\)
\(434\) 5265.78 0.582409
\(435\) 363.979 0.0401183
\(436\) 966.425 0.106155
\(437\) 7910.01 0.865874
\(438\) −7211.01 −0.786656
\(439\) −2993.80 −0.325481 −0.162741 0.986669i \(-0.552033\pi\)
−0.162741 + 0.986669i \(0.552033\pi\)
\(440\) 878.938 0.0952311
\(441\) 169.049 0.0182539
\(442\) 0 0
\(443\) 9743.67 1.04500 0.522501 0.852639i \(-0.324999\pi\)
0.522501 + 0.852639i \(0.324999\pi\)
\(444\) −608.503 −0.0650411
\(445\) −651.335 −0.0693848
\(446\) 3164.59 0.335981
\(447\) −10166.0 −1.07570
\(448\) −10324.2 −1.08878
\(449\) 561.459 0.0590131 0.0295065 0.999565i \(-0.490606\pi\)
0.0295065 + 0.999565i \(0.490606\pi\)
\(450\) −4287.22 −0.449114
\(451\) −4508.78 −0.470754
\(452\) −922.726 −0.0960207
\(453\) −3597.54 −0.373128
\(454\) 8457.38 0.874283
\(455\) 0 0
\(456\) −9618.40 −0.987770
\(457\) −13758.4 −1.40830 −0.704148 0.710054i \(-0.748671\pi\)
−0.704148 + 0.710054i \(0.748671\pi\)
\(458\) 541.213 0.0552166
\(459\) −3805.03 −0.386936
\(460\) 59.1779 0.00599823
\(461\) −12009.2 −1.21329 −0.606644 0.794974i \(-0.707485\pi\)
−0.606644 + 0.794974i \(0.707485\pi\)
\(462\) 11106.8 1.11847
\(463\) −13635.7 −1.36870 −0.684348 0.729156i \(-0.739913\pi\)
−0.684348 + 0.729156i \(0.739913\pi\)
\(464\) −8869.91 −0.887447
\(465\) 234.004 0.0233369
\(466\) 657.613 0.0653719
\(467\) 8821.95 0.874157 0.437079 0.899423i \(-0.356013\pi\)
0.437079 + 0.899423i \(0.356013\pi\)
\(468\) 0 0
\(469\) −8075.72 −0.795100
\(470\) 45.9554 0.00451013
\(471\) 2081.37 0.203619
\(472\) −1731.87 −0.168889
\(473\) −28375.1 −2.75832
\(474\) −3978.66 −0.385540
\(475\) −13462.2 −1.30039
\(476\) −667.956 −0.0643187
\(477\) −38.2105 −0.00366780
\(478\) −9092.54 −0.870049
\(479\) 14620.0 1.39459 0.697293 0.716786i \(-0.254388\pi\)
0.697293 + 0.716786i \(0.254388\pi\)
\(480\) −132.951 −0.0126424
\(481\) 0 0
\(482\) −12884.9 −1.21761
\(483\) 4906.78 0.462249
\(484\) −4114.09 −0.386372
\(485\) 327.536 0.0306653
\(486\) 8542.20 0.797288
\(487\) 9798.86 0.911763 0.455882 0.890040i \(-0.349324\pi\)
0.455882 + 0.890040i \(0.349324\pi\)
\(488\) −22259.7 −2.06486
\(489\) 5558.43 0.514030
\(490\) −18.1154 −0.00167014
\(491\) −10836.1 −0.995977 −0.497989 0.867184i \(-0.665928\pi\)
−0.497989 + 0.867184i \(0.665928\pi\)
\(492\) 369.136 0.0338251
\(493\) −4493.84 −0.410532
\(494\) 0 0
\(495\) −487.993 −0.0443104
\(496\) −5702.51 −0.516230
\(497\) −9846.86 −0.888717
\(498\) −5698.91 −0.512800
\(499\) −2589.96 −0.232349 −0.116175 0.993229i \(-0.537063\pi\)
−0.116175 + 0.993229i \(0.537063\pi\)
\(500\) −201.686 −0.0180394
\(501\) 2183.24 0.194690
\(502\) 1841.20 0.163699
\(503\) −17067.5 −1.51292 −0.756462 0.654038i \(-0.773074\pi\)
−0.756462 + 0.654038i \(0.773074\pi\)
\(504\) 5899.12 0.521364
\(505\) −517.606 −0.0456102
\(506\) 12149.0 1.06737
\(507\) 0 0
\(508\) 795.712 0.0694961
\(509\) 1012.89 0.0882038 0.0441019 0.999027i \(-0.485957\pi\)
0.0441019 + 0.999027i \(0.485957\pi\)
\(510\) 135.401 0.0117562
\(511\) 13887.4 1.20223
\(512\) 12992.6 1.12148
\(513\) 16081.7 1.38406
\(514\) −3280.82 −0.281539
\(515\) 522.644 0.0447193
\(516\) 2323.08 0.198194
\(517\) −2068.26 −0.175942
\(518\) −5345.63 −0.453424
\(519\) 16564.6 1.40098
\(520\) 0 0
\(521\) −14367.7 −1.20818 −0.604089 0.796917i \(-0.706463\pi\)
−0.604089 + 0.796917i \(0.706463\pi\)
\(522\) 6048.54 0.507160
\(523\) −16219.9 −1.35611 −0.678057 0.735010i \(-0.737178\pi\)
−0.678057 + 0.735010i \(0.737178\pi\)
\(524\) −2958.02 −0.246607
\(525\) −8350.92 −0.694217
\(526\) −13385.5 −1.10957
\(527\) −2889.11 −0.238808
\(528\) −12028.0 −0.991382
\(529\) −6799.77 −0.558870
\(530\) 4.09466 0.000335586 0
\(531\) 961.545 0.0785828
\(532\) 2823.06 0.230066
\(533\) 0 0
\(534\) 10947.5 0.887161
\(535\) −481.466 −0.0389076
\(536\) 10741.4 0.865593
\(537\) 568.488 0.0456835
\(538\) −16506.1 −1.32273
\(539\) 815.301 0.0651530
\(540\) 120.313 0.00958788
\(541\) −17592.2 −1.39806 −0.699029 0.715094i \(-0.746384\pi\)
−0.699029 + 0.715094i \(0.746384\pi\)
\(542\) −10066.7 −0.797792
\(543\) −3947.55 −0.311980
\(544\) 1641.48 0.129371
\(545\) 377.281 0.0296531
\(546\) 0 0
\(547\) 10504.6 0.821103 0.410552 0.911837i \(-0.365336\pi\)
0.410552 + 0.911837i \(0.365336\pi\)
\(548\) −2601.53 −0.202795
\(549\) 12358.8 0.960763
\(550\) −20676.6 −1.60301
\(551\) 18992.8 1.46846
\(552\) −6526.44 −0.503231
\(553\) 7662.33 0.589214
\(554\) 15073.2 1.15596
\(555\) −237.552 −0.0181685
\(556\) −2148.52 −0.163880
\(557\) 507.558 0.0386102 0.0193051 0.999814i \(-0.493855\pi\)
0.0193051 + 0.999814i \(0.493855\pi\)
\(558\) 3888.64 0.295016
\(559\) 0 0
\(560\) −514.690 −0.0388386
\(561\) −6093.83 −0.458613
\(562\) 9041.75 0.678653
\(563\) −3443.14 −0.257746 −0.128873 0.991661i \(-0.541136\pi\)
−0.128873 + 0.991661i \(0.541136\pi\)
\(564\) 169.330 0.0126420
\(565\) −360.221 −0.0268223
\(566\) 6688.21 0.496690
\(567\) 3387.96 0.250936
\(568\) 13097.2 0.967509
\(569\) 23972.2 1.76620 0.883098 0.469189i \(-0.155454\pi\)
0.883098 + 0.469189i \(0.155454\pi\)
\(570\) −572.260 −0.0420514
\(571\) −7458.32 −0.546622 −0.273311 0.961926i \(-0.588119\pi\)
−0.273311 + 0.961926i \(0.588119\pi\)
\(572\) 0 0
\(573\) −2495.26 −0.181922
\(574\) 3242.82 0.235806
\(575\) −9134.57 −0.662501
\(576\) −7624.14 −0.551515
\(577\) −5669.57 −0.409059 −0.204530 0.978860i \(-0.565566\pi\)
−0.204530 + 0.978860i \(0.565566\pi\)
\(578\) 10913.2 0.785344
\(579\) 4869.94 0.349547
\(580\) 142.093 0.0101726
\(581\) 10975.3 0.783702
\(582\) −5505.15 −0.392089
\(583\) −184.284 −0.0130914
\(584\) −18471.4 −1.30882
\(585\) 0 0
\(586\) −14065.6 −0.991541
\(587\) −1017.39 −0.0715371 −0.0357685 0.999360i \(-0.511388\pi\)
−0.0357685 + 0.999360i \(0.511388\pi\)
\(588\) −66.7491 −0.00468144
\(589\) 12210.6 0.854208
\(590\) −103.040 −0.00718996
\(591\) 4669.81 0.325026
\(592\) 5788.99 0.401902
\(593\) 10198.2 0.706221 0.353111 0.935582i \(-0.385124\pi\)
0.353111 + 0.935582i \(0.385124\pi\)
\(594\) 24699.9 1.70615
\(595\) −260.762 −0.0179667
\(596\) −3968.70 −0.272759
\(597\) −8829.33 −0.605294
\(598\) 0 0
\(599\) 12516.3 0.853763 0.426881 0.904308i \(-0.359612\pi\)
0.426881 + 0.904308i \(0.359612\pi\)
\(600\) 11107.4 0.755766
\(601\) 9627.46 0.653431 0.326716 0.945123i \(-0.394058\pi\)
0.326716 + 0.945123i \(0.394058\pi\)
\(602\) 20408.0 1.38168
\(603\) −5963.70 −0.402754
\(604\) −1404.44 −0.0946120
\(605\) −1606.09 −0.107929
\(606\) 8699.80 0.583177
\(607\) 6667.20 0.445821 0.222910 0.974839i \(-0.428444\pi\)
0.222910 + 0.974839i \(0.428444\pi\)
\(608\) −6937.56 −0.462755
\(609\) 11781.7 0.783942
\(610\) −1324.37 −0.0879053
\(611\) 0 0
\(612\) −493.268 −0.0325803
\(613\) 23085.4 1.52106 0.760530 0.649302i \(-0.224939\pi\)
0.760530 + 0.649302i \(0.224939\pi\)
\(614\) −18718.8 −1.23034
\(615\) 144.106 0.00944866
\(616\) 28450.6 1.86089
\(617\) −3049.24 −0.198959 −0.0994796 0.995040i \(-0.531718\pi\)
−0.0994796 + 0.995040i \(0.531718\pi\)
\(618\) −8784.48 −0.571786
\(619\) −7296.58 −0.473787 −0.236894 0.971536i \(-0.576129\pi\)
−0.236894 + 0.971536i \(0.576129\pi\)
\(620\) 91.3523 0.00591741
\(621\) 10912.0 0.705126
\(622\) −20248.9 −1.30531
\(623\) −21083.3 −1.35583
\(624\) 0 0
\(625\) 15506.8 0.992438
\(626\) −25621.7 −1.63586
\(627\) 25755.1 1.64044
\(628\) 812.543 0.0516305
\(629\) 2932.92 0.185920
\(630\) 350.976 0.0221956
\(631\) 23829.5 1.50339 0.751694 0.659512i \(-0.229237\pi\)
0.751694 + 0.659512i \(0.229237\pi\)
\(632\) −10191.6 −0.641453
\(633\) 337.345 0.0211821
\(634\) −15960.5 −0.999801
\(635\) 310.637 0.0194130
\(636\) 15.0874 0.000940653 0
\(637\) 0 0
\(638\) 29171.3 1.81019
\(639\) −7271.65 −0.450175
\(640\) 528.347 0.0326324
\(641\) 13405.3 0.826016 0.413008 0.910727i \(-0.364478\pi\)
0.413008 + 0.910727i \(0.364478\pi\)
\(642\) 8092.36 0.497477
\(643\) −5251.51 −0.322083 −0.161042 0.986948i \(-0.551485\pi\)
−0.161042 + 0.986948i \(0.551485\pi\)
\(644\) 1915.55 0.117210
\(645\) 906.904 0.0553633
\(646\) 7065.37 0.430315
\(647\) 21611.4 1.31319 0.656595 0.754244i \(-0.271996\pi\)
0.656595 + 0.754244i \(0.271996\pi\)
\(648\) −4506.28 −0.273184
\(649\) 4637.39 0.280483
\(650\) 0 0
\(651\) 7574.54 0.456021
\(652\) 2169.94 0.130340
\(653\) −21595.8 −1.29420 −0.647099 0.762406i \(-0.724018\pi\)
−0.647099 + 0.762406i \(0.724018\pi\)
\(654\) −6341.25 −0.379148
\(655\) −1154.78 −0.0688869
\(656\) −3511.77 −0.209012
\(657\) 10255.4 0.608985
\(658\) 1487.54 0.0881314
\(659\) −16642.6 −0.983768 −0.491884 0.870661i \(-0.663692\pi\)
−0.491884 + 0.870661i \(0.663692\pi\)
\(660\) 192.684 0.0113640
\(661\) −26981.1 −1.58766 −0.793831 0.608139i \(-0.791916\pi\)
−0.793831 + 0.608139i \(0.791916\pi\)
\(662\) −11871.5 −0.696980
\(663\) 0 0
\(664\) −14598.1 −0.853185
\(665\) 1102.09 0.0642664
\(666\) −3947.60 −0.229680
\(667\) 12887.3 0.748126
\(668\) 852.310 0.0493665
\(669\) 4552.09 0.263070
\(670\) 639.074 0.0368501
\(671\) 59604.5 3.42922
\(672\) −4303.55 −0.247043
\(673\) 11149.2 0.638591 0.319296 0.947655i \(-0.396554\pi\)
0.319296 + 0.947655i \(0.396554\pi\)
\(674\) −7761.04 −0.443537
\(675\) −18571.3 −1.05898
\(676\) 0 0
\(677\) 3314.33 0.188154 0.0940769 0.995565i \(-0.470010\pi\)
0.0940769 + 0.995565i \(0.470010\pi\)
\(678\) 6054.51 0.342953
\(679\) 10602.1 0.599223
\(680\) 346.836 0.0195596
\(681\) 12165.5 0.684556
\(682\) 18754.3 1.05299
\(683\) −24505.2 −1.37287 −0.686433 0.727193i \(-0.740824\pi\)
−0.686433 + 0.727193i \(0.740824\pi\)
\(684\) 2084.75 0.116539
\(685\) −1015.61 −0.0566486
\(686\) −16557.0 −0.921500
\(687\) 778.506 0.0432342
\(688\) −22100.6 −1.22468
\(689\) 0 0
\(690\) −388.299 −0.0214236
\(691\) 21752.8 1.19756 0.598782 0.800912i \(-0.295652\pi\)
0.598782 + 0.800912i \(0.295652\pi\)
\(692\) 6466.64 0.355238
\(693\) −15796.0 −0.865858
\(694\) −7278.90 −0.398132
\(695\) −838.755 −0.0457781
\(696\) −15670.7 −0.853445
\(697\) −1779.20 −0.0966887
\(698\) 19379.9 1.05092
\(699\) 945.941 0.0511857
\(700\) −3260.10 −0.176029
\(701\) 34250.9 1.84542 0.922709 0.385496i \(-0.125970\pi\)
0.922709 + 0.385496i \(0.125970\pi\)
\(702\) 0 0
\(703\) −12395.8 −0.665028
\(704\) −36770.1 −1.96850
\(705\) 66.1043 0.00353140
\(706\) −5992.59 −0.319454
\(707\) −16754.6 −0.891259
\(708\) −379.666 −0.0201536
\(709\) 5527.11 0.292771 0.146386 0.989228i \(-0.453236\pi\)
0.146386 + 0.989228i \(0.453236\pi\)
\(710\) 779.234 0.0411889
\(711\) 5658.43 0.298464
\(712\) 28042.6 1.47604
\(713\) 8285.33 0.435187
\(714\) 4382.83 0.229724
\(715\) 0 0
\(716\) 221.931 0.0115837
\(717\) −13079.1 −0.681241
\(718\) −6485.02 −0.337074
\(719\) −3777.78 −0.195949 −0.0979745 0.995189i \(-0.531236\pi\)
−0.0979745 + 0.995189i \(0.531236\pi\)
\(720\) −380.085 −0.0196735
\(721\) 16917.6 0.873849
\(722\) −12291.5 −0.633576
\(723\) −18534.2 −0.953380
\(724\) −1541.08 −0.0791072
\(725\) −21933.2 −1.12355
\(726\) 26994.8 1.37999
\(727\) 19076.8 0.973204 0.486602 0.873624i \(-0.338236\pi\)
0.486602 + 0.873624i \(0.338236\pi\)
\(728\) 0 0
\(729\) 17319.9 0.879944
\(730\) −1098.98 −0.0557192
\(731\) −11197.0 −0.566535
\(732\) −4879.86 −0.246400
\(733\) −7997.30 −0.402984 −0.201492 0.979490i \(-0.564579\pi\)
−0.201492 + 0.979490i \(0.564579\pi\)
\(734\) −16849.4 −0.847307
\(735\) −26.0581 −0.00130771
\(736\) −4707.39 −0.235756
\(737\) −28762.1 −1.43754
\(738\) 2394.74 0.119446
\(739\) −28983.6 −1.44273 −0.721367 0.692553i \(-0.756486\pi\)
−0.721367 + 0.692553i \(0.756486\pi\)
\(740\) −92.7376 −0.00460689
\(741\) 0 0
\(742\) 132.541 0.00655761
\(743\) 19145.4 0.945324 0.472662 0.881244i \(-0.343293\pi\)
0.472662 + 0.881244i \(0.343293\pi\)
\(744\) −10074.8 −0.496451
\(745\) −1549.33 −0.0761923
\(746\) −7435.47 −0.364922
\(747\) 8104.95 0.396981
\(748\) −2378.96 −0.116288
\(749\) −15584.7 −0.760284
\(750\) 1323.38 0.0644304
\(751\) −25516.9 −1.23985 −0.619923 0.784663i \(-0.712836\pi\)
−0.619923 + 0.784663i \(0.712836\pi\)
\(752\) −1610.92 −0.0781172
\(753\) 2648.47 0.128175
\(754\) 0 0
\(755\) −548.275 −0.0264288
\(756\) 3894.46 0.187355
\(757\) −17230.6 −0.827289 −0.413645 0.910438i \(-0.635744\pi\)
−0.413645 + 0.910438i \(0.635744\pi\)
\(758\) −4779.17 −0.229007
\(759\) 17475.7 0.835744
\(760\) −1465.87 −0.0699642
\(761\) 2343.06 0.111611 0.0558053 0.998442i \(-0.482227\pi\)
0.0558053 + 0.998442i \(0.482227\pi\)
\(762\) −5221.11 −0.248216
\(763\) 12212.3 0.579444
\(764\) −974.121 −0.0461289
\(765\) −192.566 −0.00910096
\(766\) 27756.9 1.30927
\(767\) 0 0
\(768\) 7862.11 0.369400
\(769\) 7100.18 0.332950 0.166475 0.986046i \(-0.446761\pi\)
0.166475 + 0.986046i \(0.446761\pi\)
\(770\) 1692.71 0.0792219
\(771\) −4719.29 −0.220442
\(772\) 1901.17 0.0886328
\(773\) −12270.4 −0.570940 −0.285470 0.958388i \(-0.592150\pi\)
−0.285470 + 0.958388i \(0.592150\pi\)
\(774\) 15070.8 0.699881
\(775\) −14100.9 −0.653575
\(776\) −14101.7 −0.652349
\(777\) −7689.40 −0.355027
\(778\) 24386.9 1.12379
\(779\) 7519.64 0.345852
\(780\) 0 0
\(781\) −35070.1 −1.60680
\(782\) 4794.11 0.219229
\(783\) 26201.0 1.19584
\(784\) 635.017 0.0289275
\(785\) 317.207 0.0144224
\(786\) 19409.2 0.880794
\(787\) −3425.04 −0.155133 −0.0775663 0.996987i \(-0.524715\pi\)
−0.0775663 + 0.996987i \(0.524715\pi\)
\(788\) 1823.04 0.0824151
\(789\) −19254.3 −0.868787
\(790\) −606.360 −0.0273080
\(791\) −11660.1 −0.524129
\(792\) 21010.0 0.942624
\(793\) 0 0
\(794\) −25894.3 −1.15737
\(795\) 5.88995 0.000262761 0
\(796\) −3446.87 −0.153481
\(797\) −11781.1 −0.523600 −0.261800 0.965122i \(-0.584316\pi\)
−0.261800 + 0.965122i \(0.584316\pi\)
\(798\) −18523.6 −0.821717
\(799\) −816.154 −0.0361370
\(800\) 8011.58 0.354065
\(801\) −15569.4 −0.686790
\(802\) 5339.24 0.235081
\(803\) 49460.6 2.17363
\(804\) 2354.77 0.103291
\(805\) 747.807 0.0327413
\(806\) 0 0
\(807\) −23743.2 −1.03569
\(808\) 22285.0 0.970276
\(809\) 18910.1 0.821810 0.410905 0.911678i \(-0.365213\pi\)
0.410905 + 0.911678i \(0.365213\pi\)
\(810\) −268.107 −0.0116300
\(811\) −12803.3 −0.554359 −0.277180 0.960818i \(-0.589400\pi\)
−0.277180 + 0.960818i \(0.589400\pi\)
\(812\) 4599.45 0.198780
\(813\) −14480.5 −0.624664
\(814\) −19038.7 −0.819788
\(815\) 847.121 0.0364090
\(816\) −4746.33 −0.203621
\(817\) 47323.3 2.02648
\(818\) −24889.4 −1.06386
\(819\) 0 0
\(820\) 56.2574 0.00239585
\(821\) −19335.1 −0.821923 −0.410962 0.911653i \(-0.634807\pi\)
−0.410962 + 0.911653i \(0.634807\pi\)
\(822\) 17070.1 0.724315
\(823\) −2125.90 −0.0900417 −0.0450209 0.998986i \(-0.514335\pi\)
−0.0450209 + 0.998986i \(0.514335\pi\)
\(824\) −22501.9 −0.951324
\(825\) −29742.2 −1.25514
\(826\) −3335.32 −0.140497
\(827\) 6989.24 0.293881 0.146941 0.989145i \(-0.453057\pi\)
0.146941 + 0.989145i \(0.453057\pi\)
\(828\) 1414.58 0.0593721
\(829\) −32649.7 −1.36788 −0.683938 0.729540i \(-0.739734\pi\)
−0.683938 + 0.729540i \(0.739734\pi\)
\(830\) −868.531 −0.0363219
\(831\) 21682.0 0.905103
\(832\) 0 0
\(833\) 321.724 0.0133819
\(834\) 14097.6 0.585324
\(835\) 332.732 0.0137900
\(836\) 10054.5 0.415958
\(837\) 16844.7 0.695626
\(838\) −34278.4 −1.41304
\(839\) 4038.23 0.166168 0.0830841 0.996543i \(-0.473523\pi\)
0.0830841 + 0.996543i \(0.473523\pi\)
\(840\) −909.318 −0.0373505
\(841\) 6555.00 0.268769
\(842\) −24240.9 −0.992159
\(843\) 13006.1 0.531380
\(844\) 131.695 0.00537102
\(845\) 0 0
\(846\) 1098.51 0.0446426
\(847\) −51988.1 −2.10901
\(848\) −143.534 −0.00581248
\(849\) 9620.64 0.388904
\(850\) −8159.17 −0.329244
\(851\) −8410.97 −0.338807
\(852\) 2871.21 0.115453
\(853\) −8114.12 −0.325700 −0.162850 0.986651i \(-0.552069\pi\)
−0.162850 + 0.986651i \(0.552069\pi\)
\(854\) −42869.0 −1.71774
\(855\) 813.863 0.0325538
\(856\) 20729.0 0.827690
\(857\) −22298.1 −0.888786 −0.444393 0.895832i \(-0.646581\pi\)
−0.444393 + 0.895832i \(0.646581\pi\)
\(858\) 0 0
\(859\) 33550.5 1.33263 0.666315 0.745670i \(-0.267870\pi\)
0.666315 + 0.745670i \(0.267870\pi\)
\(860\) 354.045 0.0140382
\(861\) 4664.62 0.184634
\(862\) 12429.4 0.491121
\(863\) 14120.5 0.556972 0.278486 0.960440i \(-0.410167\pi\)
0.278486 + 0.960440i \(0.410167\pi\)
\(864\) −9570.49 −0.376846
\(865\) 2524.50 0.0992319
\(866\) 21025.2 0.825018
\(867\) 15698.1 0.614918
\(868\) 2957.01 0.115631
\(869\) 27289.8 1.06530
\(870\) −932.351 −0.0363329
\(871\) 0 0
\(872\) −16243.4 −0.630817
\(873\) 7829.39 0.303533
\(874\) −20261.9 −0.784175
\(875\) −2548.63 −0.0984679
\(876\) −4049.36 −0.156182
\(877\) 1941.69 0.0747619 0.0373809 0.999301i \(-0.488099\pi\)
0.0373809 + 0.999301i \(0.488099\pi\)
\(878\) 7668.78 0.294771
\(879\) −20232.6 −0.776368
\(880\) −1833.10 −0.0702201
\(881\) −790.231 −0.0302197 −0.0151099 0.999886i \(-0.504810\pi\)
−0.0151099 + 0.999886i \(0.504810\pi\)
\(882\) −433.029 −0.0165316
\(883\) −36638.6 −1.39636 −0.698180 0.715922i \(-0.746007\pi\)
−0.698180 + 0.715922i \(0.746007\pi\)
\(884\) 0 0
\(885\) −148.217 −0.00562968
\(886\) −24958.9 −0.946401
\(887\) −40686.3 −1.54015 −0.770075 0.637954i \(-0.779781\pi\)
−0.770075 + 0.637954i \(0.779781\pi\)
\(888\) 10227.6 0.386503
\(889\) 10055.1 0.379344
\(890\) 1668.43 0.0628381
\(891\) 12066.4 0.453692
\(892\) 1777.08 0.0667053
\(893\) 3449.40 0.129261
\(894\) 26040.9 0.974202
\(895\) 86.6392 0.00323579
\(896\) 17102.2 0.637663
\(897\) 0 0
\(898\) −1438.21 −0.0534449
\(899\) 19894.0 0.738046
\(900\) −2407.50 −0.0891666
\(901\) −72.7200 −0.00268885
\(902\) 11549.5 0.426336
\(903\) 29355.9 1.08184
\(904\) 15509.0 0.570598
\(905\) −601.618 −0.0220977
\(906\) 9215.28 0.337922
\(907\) −10464.4 −0.383093 −0.191547 0.981484i \(-0.561350\pi\)
−0.191547 + 0.981484i \(0.561350\pi\)
\(908\) 4749.26 0.173579
\(909\) −12372.8 −0.451463
\(910\) 0 0
\(911\) −35611.5 −1.29513 −0.647563 0.762011i \(-0.724212\pi\)
−0.647563 + 0.762011i \(0.724212\pi\)
\(912\) 20060.0 0.728346
\(913\) 39089.0 1.41693
\(914\) 35242.9 1.27542
\(915\) −1905.04 −0.0688291
\(916\) 303.920 0.0109627
\(917\) −37379.4 −1.34610
\(918\) 9746.80 0.350427
\(919\) 1077.25 0.0386674 0.0193337 0.999813i \(-0.493846\pi\)
0.0193337 + 0.999813i \(0.493846\pi\)
\(920\) −994.648 −0.0356441
\(921\) −26926.0 −0.963346
\(922\) 30762.3 1.09881
\(923\) 0 0
\(924\) 6237.04 0.222060
\(925\) 14314.8 0.508829
\(926\) 34928.6 1.23955
\(927\) 12493.2 0.442644
\(928\) −11303.0 −0.399826
\(929\) −55733.8 −1.96832 −0.984159 0.177290i \(-0.943267\pi\)
−0.984159 + 0.177290i \(0.943267\pi\)
\(930\) −599.413 −0.0211350
\(931\) −1359.74 −0.0478665
\(932\) 369.284 0.0129789
\(933\) −29126.9 −1.02205
\(934\) −22597.9 −0.791677
\(935\) −928.718 −0.0324838
\(936\) 0 0
\(937\) −3198.60 −0.111519 −0.0557596 0.998444i \(-0.517758\pi\)
−0.0557596 + 0.998444i \(0.517758\pi\)
\(938\) 20686.4 0.720079
\(939\) −36855.4 −1.28086
\(940\) 25.8064 0.000895437 0
\(941\) −8823.35 −0.305667 −0.152834 0.988252i \(-0.548840\pi\)
−0.152834 + 0.988252i \(0.548840\pi\)
\(942\) −5331.54 −0.184407
\(943\) 5102.35 0.176199
\(944\) 3611.95 0.124533
\(945\) 1520.35 0.0523354
\(946\) 72684.3 2.49806
\(947\) −28290.4 −0.970766 −0.485383 0.874301i \(-0.661320\pi\)
−0.485383 + 0.874301i \(0.661320\pi\)
\(948\) −2234.23 −0.0765447
\(949\) 0 0
\(950\) 34484.0 1.17769
\(951\) −22958.4 −0.782836
\(952\) 11226.8 0.382210
\(953\) −12399.0 −0.421452 −0.210726 0.977545i \(-0.567583\pi\)
−0.210726 + 0.977545i \(0.567583\pi\)
\(954\) 97.8783 0.00332173
\(955\) −380.285 −0.0128856
\(956\) −5105.94 −0.172739
\(957\) 41961.3 1.41736
\(958\) −37450.0 −1.26300
\(959\) −32874.5 −1.10696
\(960\) 1175.22 0.0395105
\(961\) −17001.0 −0.570676
\(962\) 0 0
\(963\) −11508.9 −0.385118
\(964\) −7235.53 −0.241744
\(965\) 742.193 0.0247586
\(966\) −12569.0 −0.418634
\(967\) 26667.1 0.886820 0.443410 0.896319i \(-0.353769\pi\)
0.443410 + 0.896319i \(0.353769\pi\)
\(968\) 69148.6 2.29599
\(969\) 10163.2 0.336933
\(970\) −839.001 −0.0277719
\(971\) 49420.7 1.63335 0.816676 0.577096i \(-0.195814\pi\)
0.816676 + 0.577096i \(0.195814\pi\)
\(972\) 4796.89 0.158293
\(973\) −27149.9 −0.894539
\(974\) −25100.3 −0.825735
\(975\) 0 0
\(976\) 46424.5 1.52255
\(977\) −778.759 −0.0255012 −0.0127506 0.999919i \(-0.504059\pi\)
−0.0127506 + 0.999919i \(0.504059\pi\)
\(978\) −14238.2 −0.465529
\(979\) −75089.2 −2.45134
\(980\) −10.1728 −0.000331589 0
\(981\) 9018.48 0.293515
\(982\) 27757.2 0.902003
\(983\) −5997.90 −0.194612 −0.0973059 0.995255i \(-0.531023\pi\)
−0.0973059 + 0.995255i \(0.531023\pi\)
\(984\) −6204.35 −0.201004
\(985\) 711.693 0.0230218
\(986\) 11511.2 0.371797
\(987\) 2139.75 0.0690062
\(988\) 0 0
\(989\) 32110.6 1.03241
\(990\) 1250.02 0.0401295
\(991\) 8974.94 0.287688 0.143844 0.989600i \(-0.454054\pi\)
0.143844 + 0.989600i \(0.454054\pi\)
\(992\) −7266.75 −0.232580
\(993\) −17076.6 −0.545729
\(994\) 25223.3 0.804863
\(995\) −1345.62 −0.0428732
\(996\) −3200.24 −0.101811
\(997\) 28530.2 0.906280 0.453140 0.891439i \(-0.350304\pi\)
0.453140 + 0.891439i \(0.350304\pi\)
\(998\) 6634.31 0.210426
\(999\) −17100.2 −0.541567
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.a.g.1.1 2
3.2 odd 2 1521.4.a.r.1.2 2
13.2 odd 12 169.4.e.f.147.1 8
13.3 even 3 169.4.c.j.22.2 4
13.4 even 6 169.4.c.g.146.1 4
13.5 odd 4 169.4.b.f.168.4 4
13.6 odd 12 169.4.e.f.23.4 8
13.7 odd 12 169.4.e.f.23.1 8
13.8 odd 4 169.4.b.f.168.1 4
13.9 even 3 169.4.c.j.146.2 4
13.10 even 6 169.4.c.g.22.1 4
13.11 odd 12 169.4.e.f.147.4 8
13.12 even 2 13.4.a.b.1.2 2
39.38 odd 2 117.4.a.d.1.1 2
52.51 odd 2 208.4.a.h.1.2 2
65.12 odd 4 325.4.b.e.274.4 4
65.38 odd 4 325.4.b.e.274.1 4
65.64 even 2 325.4.a.f.1.1 2
91.90 odd 2 637.4.a.b.1.2 2
104.51 odd 2 832.4.a.z.1.1 2
104.77 even 2 832.4.a.s.1.2 2
143.142 odd 2 1573.4.a.b.1.1 2
156.155 even 2 1872.4.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.2 2 13.12 even 2
117.4.a.d.1.1 2 39.38 odd 2
169.4.a.g.1.1 2 1.1 even 1 trivial
169.4.b.f.168.1 4 13.8 odd 4
169.4.b.f.168.4 4 13.5 odd 4
169.4.c.g.22.1 4 13.10 even 6
169.4.c.g.146.1 4 13.4 even 6
169.4.c.j.22.2 4 13.3 even 3
169.4.c.j.146.2 4 13.9 even 3
169.4.e.f.23.1 8 13.7 odd 12
169.4.e.f.23.4 8 13.6 odd 12
169.4.e.f.147.1 8 13.2 odd 12
169.4.e.f.147.4 8 13.11 odd 12
208.4.a.h.1.2 2 52.51 odd 2
325.4.a.f.1.1 2 65.64 even 2
325.4.b.e.274.1 4 65.38 odd 4
325.4.b.e.274.4 4 65.12 odd 4
637.4.a.b.1.2 2 91.90 odd 2
832.4.a.s.1.2 2 104.77 even 2
832.4.a.z.1.1 2 104.51 odd 2
1521.4.a.r.1.2 2 3.2 odd 2
1573.4.a.b.1.1 2 143.142 odd 2
1872.4.a.bb.1.1 2 156.155 even 2