Properties

Label 13.4.a.b.1.2
Level $13$
Weight $4$
Character 13.1
Self dual yes
Analytic conductor $0.767$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.767024830075\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 13.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} +0.561553 q^{5} -9.43845 q^{6} +18.1771 q^{7} -24.1771 q^{8} -13.4233 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} +0.561553 q^{5} -9.43845 q^{6} +18.1771 q^{7} -24.1771 q^{8} -13.4233 q^{9} +1.43845 q^{10} +64.7386 q^{11} +5.30019 q^{12} -13.0000 q^{13} +46.5616 q^{14} -2.06913 q^{15} -50.4233 q^{16} -25.5464 q^{17} -34.3845 q^{18} -107.970 q^{19} -0.807764 q^{20} -66.9763 q^{21} +165.831 q^{22} +73.2614 q^{23} +89.0843 q^{24} -124.685 q^{25} -33.3002 q^{26} +148.946 q^{27} -26.1468 q^{28} +175.909 q^{29} -5.30019 q^{30} -113.093 q^{31} +64.2547 q^{32} -238.540 q^{33} -65.4384 q^{34} +10.2074 q^{35} +19.3087 q^{36} +114.808 q^{37} -276.570 q^{38} +47.9006 q^{39} -13.5767 q^{40} -69.6458 q^{41} -171.563 q^{42} +438.302 q^{43} -93.1231 q^{44} -7.53789 q^{45} +187.663 q^{46} -31.9479 q^{47} +185.793 q^{48} -12.5937 q^{49} -319.386 q^{50} +94.1298 q^{51} +18.6998 q^{52} +2.84658 q^{53} +381.533 q^{54} +36.3542 q^{55} -439.469 q^{56} +397.831 q^{57} +450.600 q^{58} +71.6325 q^{59} +2.97633 q^{60} -920.695 q^{61} -289.693 q^{62} -243.996 q^{63} +567.978 q^{64} -7.30019 q^{65} -611.032 q^{66} -444.280 q^{67} +36.7471 q^{68} -269.943 q^{69} +26.1468 q^{70} -541.719 q^{71} +324.536 q^{72} +764.004 q^{73} +294.086 q^{74} +459.420 q^{75} +155.309 q^{76} +1176.76 q^{77} +122.700 q^{78} -421.538 q^{79} -28.3153 q^{80} -186.386 q^{81} -178.401 q^{82} +603.797 q^{83} +96.3419 q^{84} -14.3457 q^{85} +1122.73 q^{86} -648.165 q^{87} -1565.19 q^{88} -1159.88 q^{89} -19.3087 q^{90} -236.302 q^{91} -105.383 q^{92} +416.708 q^{93} -81.8362 q^{94} -60.6307 q^{95} -236.757 q^{96} +583.269 q^{97} -32.2595 q^{98} -869.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{3} - 7 q^{4} - 3 q^{5} - 23 q^{6} - 9 q^{7} - 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{3} - 7 q^{4} - 3 q^{5} - 23 q^{6} - 9 q^{7} - 3 q^{8} + 35 q^{9} + 7 q^{10} + 80 q^{11} - 43 q^{12} - 26 q^{13} + 89 q^{14} - 33 q^{15} - 39 q^{16} + 19 q^{17} - 110 q^{18} - 84 q^{19} + 19 q^{20} - 303 q^{21} + 142 q^{22} + 196 q^{23} + 273 q^{24} - 237 q^{25} - 13 q^{26} + 335 q^{27} + 125 q^{28} - 44 q^{29} + 43 q^{30} - 86 q^{31} - 123 q^{32} - 106 q^{33} - 135 q^{34} + 107 q^{35} - 250 q^{36} + 209 q^{37} - 314 q^{38} - 65 q^{39} - 89 q^{40} - 230 q^{41} + 197 q^{42} + 287 q^{43} - 178 q^{44} - 180 q^{45} - 4 q^{46} + 435 q^{47} + 285 q^{48} + 383 q^{49} - 144 q^{50} + 481 q^{51} + 91 q^{52} - 118 q^{53} + 91 q^{54} - 18 q^{55} - 1015 q^{56} + 606 q^{57} + 794 q^{58} - 368 q^{59} + 175 q^{60} - 1058 q^{61} - 332 q^{62} - 1560 q^{63} + 769 q^{64} + 39 q^{65} - 818 q^{66} + 68 q^{67} - 211 q^{68} + 796 q^{69} - 125 q^{70} - 131 q^{71} + 1350 q^{72} + 456 q^{73} + 147 q^{74} - 516 q^{75} + 22 q^{76} + 762 q^{77} + 299 q^{78} - 1008 q^{79} - 69 q^{80} + 122 q^{81} + 72 q^{82} + 1958 q^{83} + 1409 q^{84} - 173 q^{85} + 1359 q^{86} - 2558 q^{87} - 1242 q^{88} - 720 q^{89} + 250 q^{90} + 117 q^{91} - 788 q^{92} + 652 q^{93} - 811 q^{94} - 146 q^{95} - 1863 q^{96} - 928 q^{97} - 650 q^{98} - 130 q^{99}+O(q^{100}) \) 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 \(-0.350417\pi\)
0.452823 + 0.891601i \(0.350417\pi\)
\(3\) −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 \(-0.492005\pi\)
0.0251134 + 0.999685i \(0.492005\pi\)
\(6\) −9.43845 −0.642205
\(7\) 18.1771 0.981470 0.490735 0.871309i \(-0.336728\pi\)
0.490735 + 0.871309i \(0.336728\pi\)
\(8\) −24.1771 −1.06849
\(9\) −13.4233 −0.497159
\(10\) 1.43845 0.0454877
\(11\) 64.7386 1.77449 0.887247 0.461295i \(-0.152615\pi\)
0.887247 + 0.461295i \(0.152615\pi\)
\(12\) 5.30019 0.127503
\(13\) −13.0000 −0.277350
\(14\) 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 \(-0.726003\pi\)
−0.651841 + 0.758356i \(0.726003\pi\)
\(20\) −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
\(26\) −33.3002 −0.251181
\(27\) 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 \(-0.606245\pi\)
−0.327614 + 0.944812i \(0.606245\pi\)
\(32\) 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 \(-0.417905\pi\)
0.255058 + 0.966926i \(0.417905\pi\)
\(38\) −276.570 −1.18067
\(39\) 47.9006 0.196673
\(40\) −13.5767 −0.0536666
\(41\) −69.6458 −0.265289 −0.132645 0.991164i \(-0.542347\pi\)
−0.132645 + 0.991164i \(0.542347\pi\)
\(42\) −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 \(-0.515787\pi\)
−0.0495753 + 0.998770i \(0.515787\pi\)
\(48\) 185.793 0.558684
\(49\) −12.5937 −0.0367164
\(50\) −319.386 −0.903361
\(51\) 94.1298 0.258447
\(52\) 18.6998 0.0498692
\(53\) 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 \(-0.474817\pi\)
0.0790319 + 0.996872i \(0.474817\pi\)
\(60\) 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
\(65\) −7.30019 −0.0139304
\(66\) −611.032 −1.13959
\(67\) −444.280 −0.810112 −0.405056 0.914292i \(-0.632748\pi\)
−0.405056 + 0.914292i \(0.632748\pi\)
\(68\) 36.7471 0.0655330
\(69\) −269.943 −0.470976
\(70\) 26.1468 0.0446448
\(71\) −541.719 −0.905496 −0.452748 0.891639i \(-0.649556\pi\)
−0.452748 + 0.891639i \(0.649556\pi\)
\(72\) 324.536 0.531207
\(73\) 764.004 1.22493 0.612465 0.790498i \(-0.290178\pi\)
0.612465 + 0.790498i \(0.290178\pi\)
\(74\) 294.086 0.461984
\(75\) 459.420 0.707324
\(76\) 155.309 0.234410
\(77\) 1176.76 1.74161
\(78\) 122.700 0.178116
\(79\) −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 \(-0.369271\pi\)
0.399249 + 0.916842i \(0.369271\pi\)
\(84\) 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 \(-0.742704\pi\)
−0.690715 + 0.723127i \(0.742704\pi\)
\(90\) −19.3087 −0.0226146
\(91\) −236.302 −0.272211
\(92\) −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 \(-0.401254\pi\)
0.305268 + 0.952267i \(0.401254\pi\)
\(98\) −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 \(0.646858\pi\)
\(104\) 314.302 0.296345
\(105\) −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 \(-0.404616\pi\)
0.295192 + 0.955438i \(0.404616\pi\)
\(110\) 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
\(117\) 174.503 0.137887
\(118\) 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
\(130\) −18.6998 −0.0126160
\(131\) 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 \(-0.690710\pi\)
−0.563928 + 0.825824i \(0.690710\pi\)
\(138\) −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
\(143\) −841.602 −0.492156
\(144\) 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 \(-0.774059\pi\)
−0.758482 + 0.651694i \(0.774059\pi\)
\(150\) 1176.83 0.640585
\(151\) −976.355 −0.526190 −0.263095 0.964770i \(-0.584743\pi\)
−0.263095 + 0.964770i \(0.584743\pi\)
\(152\) 2610.39 1.39297
\(153\) 342.917 0.181197
\(154\) 3014.33 1.57728
\(155\) −63.5076 −0.0329100
\(156\) −68.9024 −0.0353629
\(157\) −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 \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(164\) 100.182 0.0477005
\(165\) −133.953 −0.0632012
\(166\) 1546.66 0.723157
\(167\) 592.521 0.274555 0.137277 0.990533i \(-0.456165\pi\)
0.137277 + 0.990533i \(0.456165\pi\)
\(168\) 1619.29 0.743638
\(169\) 169.000 0.0769231
\(170\) −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 \(0.429401\pi\)
\(182\) −605.300 −0.246527
\(183\) 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 \(-0.420730\pi\)
0.246468 + 0.969151i \(0.420730\pi\)
\(194\) 1494.07 0.552929
\(195\) 26.8987 0.00987823
\(196\) 18.1154 0.00660183
\(197\) 1267.37 0.458356 0.229178 0.973385i \(-0.426396\pi\)
0.229178 + 0.973385i \(0.426396\pi\)
\(198\) −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
\(208\) 655.503 0.218514
\(209\) −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
\(221\) 332.103 0.101085
\(222\) −1083.61 −0.327599
\(223\) 1235.42 0.370985 0.185493 0.982646i \(-0.440612\pi\)
0.185493 + 0.982646i \(0.440612\pi\)
\(224\) 1167.96 0.348383
\(225\) 1673.68 0.495905
\(226\) 1643.17 0.483637
\(227\) 3301.66 0.965370 0.482685 0.875794i \(-0.339662\pi\)
0.482685 + 0.875794i \(0.339662\pi\)
\(228\) −572.260 −0.166223
\(229\) 211.283 0.0609694 0.0304847 0.999535i \(-0.490295\pi\)
0.0304847 + 0.999535i \(0.490295\pi\)
\(230\) 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 \(0.511491\pi\)
\(234\) 446.998 0.124877
\(235\) −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 \(-0.659489\pi\)
−0.480347 + 0.877078i \(0.659489\pi\)
\(240\) 104.332 0.0280609
\(241\) −5030.10 −1.34447 −0.672235 0.740338i \(-0.734665\pi\)
−0.672235 + 0.740338i \(0.734665\pi\)
\(242\) 7326.27 1.94608
\(243\) −3334.77 −0.880353
\(244\) 1324.37 0.347476
\(245\) −7.07204 −0.00184415
\(246\) 657.349 0.170370
\(247\) 1403.61 0.361576
\(248\) 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
\(260\) 10.5009 0.00250477
\(261\) −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 \(-0.645183\pi\)
−0.440455 + 0.897775i \(0.645183\pi\)
\(272\) 1288.13 0.287149
\(273\) 870.692 0.193028
\(274\) −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 \(-0.377753\pi\)
0.374679 + 0.927155i \(0.377753\pi\)
\(282\) 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
\(286\) −2155.81 −0.445719
\(287\) −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 \(-0.684391\pi\)
−0.547422 + 0.836857i \(0.684391\pi\)
\(294\) 118.865 0.0235795
\(295\) 40.2255 0.00793904
\(296\) −2775.72 −0.545052
\(297\) 9642.56 1.88390
\(298\) −7067.37 −1.37383
\(299\) −952.398 −0.184209
\(300\) −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 \(-0.737700\pi\)
−0.679261 + 0.733897i \(0.737700\pi\)
\(308\) −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 \(0.243842\pi\)
\(312\) −1158.10 −0.210142
\(313\) 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 \(-0.686129\pi\)
−0.551983 + 0.833856i \(0.686129\pi\)
\(318\) −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
\(325\) 1620.90 0.276650
\(326\) 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 \(-0.625729\pi\)
−0.384797 + 0.923001i \(0.625729\pi\)
\(332\) −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 \(0.421254\pi\)
\(338\) 432.902 0.0696651
\(339\) −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 \(-0.302973\pi\)
0.580202 + 0.814472i \(0.302973\pi\)
\(350\) −5805.51 −0.886622
\(351\) −1936.30 −0.294450
\(352\) 4159.76 0.629875
\(353\) −2339.44 −0.352736 −0.176368 0.984324i \(-0.556435\pi\)
−0.176368 + 0.984324i \(0.556435\pi\)
\(354\) −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 \(-0.559583\pi\)
−0.186096 + 0.982532i \(0.559583\pi\)
\(360\) 182.244 0.0266809
\(361\) 4798.45 0.699585
\(362\) 2744.31 0.398447
\(363\) −10538.5 −1.52376
\(364\) 339.908 0.0489451
\(365\) 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
\(377\) −2286.82 −0.312406
\(378\) 6935.16 0.943667
\(379\) −1865.73 −0.252866 −0.126433 0.991975i \(-0.540353\pi\)
−0.126433 + 0.991975i \(0.540353\pi\)
\(380\) 87.2140 0.0117736
\(381\) 2038.26 0.274076
\(382\) 1734.69 0.232342
\(383\) 10836.0 1.44567 0.722837 0.691019i \(-0.242838\pi\)
0.722837 + 0.691019i \(0.242838\pi\)
\(384\) −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 \(0.713045\pi\)
\(390\) 68.9024 0.00894618
\(391\) −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 \(-0.720642\pi\)
−0.638978 + 0.769225i \(0.720642\pi\)
\(398\) 6138.10 0.773053
\(399\) 7231.41 0.907327
\(400\) 6287.01 0.785876
\(401\) 2084.38 0.259573 0.129787 0.991542i \(-0.458571\pi\)
0.129787 + 0.991542i \(0.458571\pi\)
\(402\) 4193.32 0.520258
\(403\) 1470.21 0.181728
\(404\) −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 \(-0.699828\pi\)
−0.587349 + 0.809334i \(0.699828\pi\)
\(410\) −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
\(416\) −835.311 −0.0984483
\(417\) −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 \(-0.684521\pi\)
−0.547763 + 0.836633i \(0.684521\pi\)
\(422\) −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
\(429\) 3101.02 0.348994
\(430\) 630.474 0.0707074
\(431\) 4852.28 0.542288 0.271144 0.962539i \(-0.412598\pi\)
0.271144 + 0.962539i \(0.412598\pi\)
\(432\) −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
\(442\) 850.700 0.0915468
\(443\) 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 \(-0.509394\pi\)
−0.0295065 + 0.999565i \(0.509394\pi\)
\(450\) 4287.22 0.449114
\(451\) −4508.78 −0.470754
\(452\) −922.726 −0.0960207
\(453\) 3597.54 0.373128
\(454\) 8457.38 0.874283
\(455\) −132.696 −0.0136723
\(456\) −9618.40 −0.987770
\(457\) 13758.4 1.40830 0.704148 0.710054i \(-0.251329\pi\)
0.704148 + 0.710054i \(0.251329\pi\)
\(458\) 541.213 0.0552166
\(459\) −3805.03 −0.386936
\(460\) −59.1779 −0.00599823
\(461\) 12009.2 1.21329 0.606644 0.794974i \(-0.292515\pi\)
0.606644 + 0.794974i \(0.292515\pi\)
\(462\) −11106.8 −1.11847
\(463\) 13635.7 1.36870 0.684348 0.729156i \(-0.260087\pi\)
0.684348 + 0.729156i \(0.260087\pi\)
\(464\) −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 \(0.356013\pi\)
\(468\) −251.013 −0.0247929
\(469\) −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 \(-0.745612\pi\)
−0.697293 + 0.716786i \(0.745612\pi\)
\(480\) −132.951 −0.0126424
\(481\) −1492.50 −0.141481
\(482\) −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 \(-0.650676\pi\)
−0.455882 + 0.890040i \(0.650676\pi\)
\(488\) 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
\(494\) 3595.41 0.327460
\(495\) −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 \(-0.462937\pi\)
0.116175 + 0.993229i \(0.462937\pi\)
\(500\) 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
\(507\) −622.707 −0.0545471
\(508\) 795.712 0.0694961
\(509\) −1012.89 −0.0882038 −0.0441019 0.999027i \(-0.514043\pi\)
−0.0441019 + 0.999027i \(0.514043\pi\)
\(510\) 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
\(520\) 176.497 0.0148845
\(521\) −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
\(533\) 905.396 0.0735780
\(534\) 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 \(-0.253616\pi\)
0.699029 + 0.715094i \(0.253616\pi\)
\(542\) −10066.7 −0.797792
\(543\) −3947.55 −0.311980
\(544\) −1641.48 −0.129371
\(545\) 377.281 0.0296531
\(546\) 2230.32 0.174815
\(547\) 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 \(-0.506145\pi\)
−0.0193051 + 0.999814i \(0.506145\pi\)
\(558\) 3888.64 0.295016
\(559\) −5697.93 −0.431121
\(560\) −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 \(0.588119\pi\)
\(572\) 1210.60 0.0884926
\(573\) −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 \(-0.434434\pi\)
0.204530 + 0.978860i \(0.434434\pi\)
\(578\) −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
\(585\) 97.9925 0.00692563
\(586\) −14065.6 −0.991541
\(587\) 1017.39 0.0715371 0.0357685 0.999360i \(-0.488612\pi\)
0.0357685 + 0.999360i \(0.488612\pi\)
\(588\) −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 \(-0.614876\pi\)
−0.353111 + 0.935582i \(0.614876\pi\)
\(594\) 24699.9 1.70615
\(595\) −260.762 −0.0179667
\(596\) 3968.70 0.272759
\(597\) −8829.33 −0.605294
\(598\) −2439.62 −0.166828
\(599\) 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
\(611\) 415.323 0.0274994
\(612\) −493.268 −0.0325803
\(613\) −23085.4 −1.52106 −0.760530 0.649302i \(-0.775061\pi\)
−0.760530 + 0.649302i \(0.775061\pi\)
\(614\) −18718.8 −1.23034
\(615\) 144.106 0.00944866
\(616\) −28450.6 −1.86089
\(617\) 3049.24 0.198959 0.0994796 0.995040i \(-0.468282\pi\)
0.0994796 + 0.995040i \(0.468282\pi\)
\(618\) 8784.48 0.571786
\(619\) 7296.58 0.473787 0.236894 0.971536i \(-0.423871\pi\)
0.236894 + 0.971536i \(0.423871\pi\)
\(620\) 91.3523 0.00591741
\(621\) 10912.0 0.705126
\(622\) 20248.9 1.30531
\(623\) −21083.3 −1.35583
\(624\) −2415.30 −0.154951
\(625\) 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 \(-0.770763\pi\)
−0.751694 + 0.659512i \(0.770763\pi\)
\(632\) 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
\(637\) 163.718 0.0101833
\(638\) 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 \(-0.448515\pi\)
0.161042 + 0.986948i \(0.448515\pi\)
\(644\) −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
\(650\) 4152.02 0.250547
\(651\) 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 \(-0.208084\pi\)
0.793831 + 0.608139i \(0.208084\pi\)
\(662\) −11871.5 −0.696980
\(663\) −1223.69 −0.0716803
\(664\) −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
\(676\) −243.098 −0.0138312
\(677\) 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 \(-0.259176\pi\)
0.686433 + 0.727193i \(0.259176\pi\)
\(684\) −2084.75 −0.116539
\(685\) −1015.61 −0.0566486
\(686\) −16557.0 −0.921500
\(687\) −778.506 −0.0432342
\(688\) −22100.6 −1.22468
\(689\) −37.0056 −0.00204616
\(690\) −388.299 −0.0214236
\(691\) −21752.8 −1.19756 −0.598782 0.800912i \(-0.704348\pi\)
−0.598782 + 0.800912i \(0.704348\pi\)
\(692\) 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 \(0.125970\pi\)
\(702\) −4959.93 −0.266667
\(703\) −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 \(-0.546764\pi\)
−0.146386 + 0.989228i \(0.546764\pi\)
\(710\) −779.234 −0.0411889
\(711\) 5658.43 0.298464
\(712\) 28042.6 1.47604
\(713\) −8285.33 −0.435187
\(714\) 4382.83 0.229724
\(715\) −472.604 −0.0247194
\(716\) 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 \(0.338236\pi\)
\(728\) 5713.09 0.290853
\(729\) 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 \(-0.435421\pi\)
0.201492 + 0.979490i \(0.435421\pi\)
\(734\) 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 \(-0.243514\pi\)
0.721367 + 0.692553i \(0.243514\pi\)
\(740\) −92.7376 −0.00460689
\(741\) −5171.81 −0.256398
\(742\) 132.541 0.00655761
\(743\) −19145.4 −0.945324 −0.472662 0.881244i \(-0.656707\pi\)
−0.472662 + 0.881244i \(0.656707\pi\)
\(744\) −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
\(754\) −5857.80 −0.282929
\(755\) −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 \(-0.517773\pi\)
−0.0558053 + 0.998442i \(0.517773\pi\)
\(762\) 5221.11 0.248216
\(763\) 12212.3 0.579444
\(764\) −974.121 −0.0461289
\(765\) 192.566 0.00910096
\(766\) 27756.9 1.30927
\(767\) −931.223 −0.0438390
\(768\) 7862.11 0.369400
\(769\) −7100.18 −0.332950 −0.166475 0.986046i \(-0.553239\pi\)
−0.166475 + 0.986046i \(0.553239\pi\)
\(770\) 1692.71 0.0792219
\(771\) −4719.29 −0.220442
\(772\) −1901.17 −0.0886328
\(773\) 12270.4 0.570940 0.285470 0.958388i \(-0.407850\pi\)
0.285470 + 0.958388i \(0.407850\pi\)
\(774\) −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
\(780\) −38.6924 −0.00177616
\(781\) −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 \(-0.475285\pi\)
0.0775663 + 0.996987i \(0.475285\pi\)
\(788\) −1823.04 −0.0824151
\(789\) −19254.3 −0.868787
\(790\) −606.360 −0.0273080
\(791\) 11660.1 0.524129
\(792\) 21010.0 0.942624
\(793\) 11969.0 0.535981
\(794\) −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
\(806\) 3766.01 0.164581
\(807\) −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 \(-0.410600\pi\)
0.277180 + 0.960818i \(0.410600\pi\)
\(812\) −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
\(819\) 3171.95 0.135332
\(820\) 56.2574 0.00239585
\(821\) 19335.1 0.821923 0.410962 0.911653i \(-0.365193\pi\)
0.410962 + 0.911653i \(0.365193\pi\)
\(822\) 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 \(-0.546943\pi\)
−0.146941 + 0.989145i \(0.546943\pi\)
\(828\) 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
\(832\) −7383.72 −0.307673
\(833\) 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 \(-0.526477\pi\)
−0.0830841 + 0.996543i \(0.526477\pi\)
\(840\) 909.318 0.0373505
\(841\) 6555.00 0.268769
\(842\) −24240.9 −0.992159
\(843\) −13006.1 −0.531380
\(844\) 131.695 0.00537102
\(845\) 94.9024 0.00386360
\(846\) 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 \(-0.447931\pi\)
0.162850 + 0.986651i \(0.447931\pi\)
\(854\) −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 \(0.646581\pi\)
\(858\) 7943.42 0.316065
\(859\) 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 \(-0.589833\pi\)
−0.278486 + 0.960440i \(0.589833\pi\)
\(864\) 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
\(871\) 5775.64 0.224685
\(872\) −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 \(-0.511901\pi\)
−0.0373809 + 0.999301i \(0.511901\pi\)
\(878\) −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 \(0.746007\pi\)
\(884\) −477.713 −0.0181756
\(885\) −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
\(897\) 3509.26 0.130625
\(898\) −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
\(910\) −339.908 −0.0123822
\(911\) −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
\(923\) 7042.34 0.251139
\(924\) 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 \(-0.0567330\pi\)
0.984159 + 0.177290i \(0.0567330\pi\)
\(930\) 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
\(936\) −4218.97 −0.147330
\(937\) −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 \(-0.451160\pi\)
0.152834 + 0.988252i \(0.451160\pi\)
\(942\) 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 \(-0.338680\pi\)
0.485383 + 0.874301i \(0.338680\pi\)
\(948\) −2234.23 −0.0765447
\(949\) −9932.05 −0.339734
\(950\) 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
\(962\) −3823.12 −0.128131
\(963\) −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 \(-0.646231\pi\)
−0.443410 + 0.896319i \(0.646231\pi\)
\(968\) −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
\(975\) −5972.46 −0.196176
\(976\) 46424.5 1.52255
\(977\) 778.759 0.0255012 0.0127506 0.999919i \(-0.495941\pi\)
0.0127506 + 0.999919i \(0.495941\pi\)
\(978\) −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 \(-0.468977\pi\)
0.0973059 + 0.995255i \(0.468977\pi\)
\(984\) −6204.35 −0.201004
\(985\) 711.693 0.0230218
\(986\) −11511.2 −0.371797
\(987\) 2139.75 0.0690062
\(988\) −2019.01 −0.0650135
\(989\) 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 13.4.a.b.1.2 2
3.2 odd 2 117.4.a.d.1.1 2
4.3 odd 2 208.4.a.h.1.2 2
5.2 odd 4 325.4.b.e.274.4 4
5.3 odd 4 325.4.b.e.274.1 4
5.4 even 2 325.4.a.f.1.1 2
7.6 odd 2 637.4.a.b.1.2 2
8.3 odd 2 832.4.a.z.1.1 2
8.5 even 2 832.4.a.s.1.2 2
11.10 odd 2 1573.4.a.b.1.1 2
12.11 even 2 1872.4.a.bb.1.1 2
13.2 odd 12 169.4.e.f.147.4 8
13.3 even 3 169.4.c.g.22.1 4
13.4 even 6 169.4.c.j.146.2 4
13.5 odd 4 169.4.b.f.168.1 4
13.6 odd 12 169.4.e.f.23.1 8
13.7 odd 12 169.4.e.f.23.4 8
13.8 odd 4 169.4.b.f.168.4 4
13.9 even 3 169.4.c.g.146.1 4
13.10 even 6 169.4.c.j.22.2 4
13.11 odd 12 169.4.e.f.147.1 8
13.12 even 2 169.4.a.g.1.1 2
39.38 odd 2 1521.4.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.b.1.2 2 1.1 even 1 trivial
117.4.a.d.1.1 2 3.2 odd 2
169.4.a.g.1.1 2 13.12 even 2
169.4.b.f.168.1 4 13.5 odd 4
169.4.b.f.168.4 4 13.8 odd 4
169.4.c.g.22.1 4 13.3 even 3
169.4.c.g.146.1 4 13.9 even 3
169.4.c.j.22.2 4 13.10 even 6
169.4.c.j.146.2 4 13.4 even 6
169.4.e.f.23.1 8 13.6 odd 12
169.4.e.f.23.4 8 13.7 odd 12
169.4.e.f.147.1 8 13.11 odd 12
169.4.e.f.147.4 8 13.2 odd 12
208.4.a.h.1.2 2 4.3 odd 2
325.4.a.f.1.1 2 5.4 even 2
325.4.b.e.274.1 4 5.3 odd 4
325.4.b.e.274.4 4 5.2 odd 4
637.4.a.b.1.2 2 7.6 odd 2
832.4.a.s.1.2 2 8.5 even 2
832.4.a.z.1.1 2 8.3 odd 2
1521.4.a.r.1.2 2 39.38 odd 2
1573.4.a.b.1.1 2 11.10 odd 2
1872.4.a.bb.1.1 2 12.11 even 2