Properties

Label 507.4.a.k.1.1
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - 2x^{3} - 13x^{2} + 14x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.29360\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} +3.05006 q^{5} +12.3693 q^{6} +6.68324 q^{7} -4.12311 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} +3.05006 q^{5} +12.3693 q^{6} +6.68324 q^{7} -4.12311 q^{8} +9.00000 q^{9} -12.5757 q^{10} -32.2500 q^{11} -27.0000 q^{12} -27.5557 q^{14} -9.15018 q^{15} -55.0000 q^{16} -28.8586 q^{17} -37.1080 q^{18} +101.194 q^{19} +27.4505 q^{20} -20.0497 q^{21} +132.970 q^{22} -118.990 q^{23} +12.3693 q^{24} -115.697 q^{25} -27.0000 q^{27} +60.1492 q^{28} +160.111 q^{29} +37.7271 q^{30} +38.0705 q^{31} +259.756 q^{32} +96.7499 q^{33} +118.987 q^{34} +20.3843 q^{35} +81.0000 q^{36} +327.568 q^{37} -417.233 q^{38} -12.5757 q^{40} -56.0846 q^{41} +82.6671 q^{42} +127.879 q^{43} -290.250 q^{44} +27.4505 q^{45} +490.608 q^{46} +517.983 q^{47} +165.000 q^{48} -298.334 q^{49} +477.032 q^{50} +86.5757 q^{51} -695.546 q^{53} +111.324 q^{54} -98.3643 q^{55} -27.5557 q^{56} -303.581 q^{57} -660.156 q^{58} -656.523 q^{59} -82.3516 q^{60} +701.304 q^{61} -156.969 q^{62} +60.1492 q^{63} -631.000 q^{64} -398.910 q^{66} +57.1750 q^{67} -259.727 q^{68} +356.970 q^{69} -84.0465 q^{70} -309.226 q^{71} -37.1080 q^{72} +389.711 q^{73} -1350.60 q^{74} +347.091 q^{75} +910.744 q^{76} -215.534 q^{77} +901.820 q^{79} -167.753 q^{80} +81.0000 q^{81} +231.243 q^{82} -687.095 q^{83} -180.448 q^{84} -88.0203 q^{85} -527.257 q^{86} -480.334 q^{87} +132.970 q^{88} -1070.54 q^{89} -113.181 q^{90} -1070.91 q^{92} -114.211 q^{93} -2135.70 q^{94} +308.647 q^{95} -779.267 q^{96} -1754.48 q^{97} +1230.06 q^{98} -290.250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{4} + 36 q^{9} - 136 q^{10} - 108 q^{12} + 204 q^{14} - 220 q^{16} - 144 q^{17} - 68 q^{22} - 276 q^{23} - 120 q^{25} - 108 q^{27} + 12 q^{29} + 408 q^{30} - 804 q^{35} + 324 q^{36} - 612 q^{38} - 136 q^{40} - 612 q^{42} + 940 q^{43} + 660 q^{48} + 692 q^{49} + 432 q^{51} - 2268 q^{53} + 892 q^{55} + 204 q^{56} + 320 q^{61} - 2856 q^{62} - 2524 q^{64} + 204 q^{66} - 1296 q^{68} + 828 q^{69} - 3060 q^{74} + 360 q^{75} - 2976 q^{77} + 8 q^{79} + 324 q^{81} + 68 q^{82} - 36 q^{87} - 68 q^{88} - 1224 q^{90} - 2484 q^{92} - 5372 q^{94} - 108 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.12311 −1.45774 −0.728869 0.684653i \(-0.759954\pi\)
−0.728869 + 0.684653i \(0.759954\pi\)
\(3\) −3.00000 −0.577350
\(4\) 9.00000 1.12500
\(5\) 3.05006 0.272806 0.136403 0.990653i \(-0.456446\pi\)
0.136403 + 0.990653i \(0.456446\pi\)
\(6\) 12.3693 0.841625
\(7\) 6.68324 0.360861 0.180431 0.983588i \(-0.442251\pi\)
0.180431 + 0.983588i \(0.442251\pi\)
\(8\) −4.12311 −0.182217
\(9\) 9.00000 0.333333
\(10\) −12.5757 −0.397679
\(11\) −32.2500 −0.883975 −0.441988 0.897021i \(-0.645727\pi\)
−0.441988 + 0.897021i \(0.645727\pi\)
\(12\) −27.0000 −0.649519
\(13\) 0 0
\(14\) −27.5557 −0.526041
\(15\) −9.15018 −0.157504
\(16\) −55.0000 −0.859375
\(17\) −28.8586 −0.411720 −0.205860 0.978582i \(-0.565999\pi\)
−0.205860 + 0.978582i \(0.565999\pi\)
\(18\) −37.1080 −0.485913
\(19\) 101.194 1.22187 0.610933 0.791682i \(-0.290794\pi\)
0.610933 + 0.791682i \(0.290794\pi\)
\(20\) 27.4505 0.306906
\(21\) −20.0497 −0.208343
\(22\) 132.970 1.28860
\(23\) −118.990 −1.07874 −0.539372 0.842067i \(-0.681338\pi\)
−0.539372 + 0.842067i \(0.681338\pi\)
\(24\) 12.3693 0.105203
\(25\) −115.697 −0.925577
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 60.1492 0.405969
\(29\) 160.111 1.02524 0.512620 0.858616i \(-0.328675\pi\)
0.512620 + 0.858616i \(0.328675\pi\)
\(30\) 37.7271 0.229600
\(31\) 38.0705 0.220570 0.110285 0.993900i \(-0.464824\pi\)
0.110285 + 0.993900i \(0.464824\pi\)
\(32\) 259.756 1.43496
\(33\) 96.7499 0.510363
\(34\) 118.987 0.600179
\(35\) 20.3843 0.0984449
\(36\) 81.0000 0.375000
\(37\) 327.568 1.45545 0.727727 0.685866i \(-0.240577\pi\)
0.727727 + 0.685866i \(0.240577\pi\)
\(38\) −417.233 −1.78116
\(39\) 0 0
\(40\) −12.5757 −0.0497099
\(41\) −56.0846 −0.213633 −0.106816 0.994279i \(-0.534066\pi\)
−0.106816 + 0.994279i \(0.534066\pi\)
\(42\) 82.6671 0.303710
\(43\) 127.879 0.453519 0.226759 0.973951i \(-0.427187\pi\)
0.226759 + 0.973951i \(0.427187\pi\)
\(44\) −290.250 −0.994472
\(45\) 27.4505 0.0909352
\(46\) 490.608 1.57253
\(47\) 517.983 1.60757 0.803783 0.594923i \(-0.202817\pi\)
0.803783 + 0.594923i \(0.202817\pi\)
\(48\) 165.000 0.496160
\(49\) −298.334 −0.869779
\(50\) 477.032 1.34925
\(51\) 86.5757 0.237706
\(52\) 0 0
\(53\) −695.546 −1.80265 −0.901326 0.433141i \(-0.857405\pi\)
−0.901326 + 0.433141i \(0.857405\pi\)
\(54\) 111.324 0.280542
\(55\) −98.3643 −0.241153
\(56\) −27.5557 −0.0657551
\(57\) −303.581 −0.705445
\(58\) −660.156 −1.49453
\(59\) −656.523 −1.44868 −0.724339 0.689444i \(-0.757855\pi\)
−0.724339 + 0.689444i \(0.757855\pi\)
\(60\) −82.3516 −0.177192
\(61\) 701.304 1.47201 0.736007 0.676974i \(-0.236709\pi\)
0.736007 + 0.676974i \(0.236709\pi\)
\(62\) −156.969 −0.321533
\(63\) 60.1492 0.120287
\(64\) −631.000 −1.23242
\(65\) 0 0
\(66\) −398.910 −0.743976
\(67\) 57.1750 0.104254 0.0521271 0.998640i \(-0.483400\pi\)
0.0521271 + 0.998640i \(0.483400\pi\)
\(68\) −259.727 −0.463184
\(69\) 356.970 0.622814
\(70\) −84.0465 −0.143507
\(71\) −309.226 −0.516879 −0.258440 0.966027i \(-0.583208\pi\)
−0.258440 + 0.966027i \(0.583208\pi\)
\(72\) −37.1080 −0.0607391
\(73\) 389.711 0.624826 0.312413 0.949946i \(-0.398863\pi\)
0.312413 + 0.949946i \(0.398863\pi\)
\(74\) −1350.60 −2.12167
\(75\) 347.091 0.534382
\(76\) 910.744 1.37460
\(77\) −215.534 −0.318992
\(78\) 0 0
\(79\) 901.820 1.28434 0.642169 0.766563i \(-0.278035\pi\)
0.642169 + 0.766563i \(0.278035\pi\)
\(80\) −167.753 −0.234442
\(81\) 81.0000 0.111111
\(82\) 231.243 0.311421
\(83\) −687.095 −0.908657 −0.454328 0.890834i \(-0.650121\pi\)
−0.454328 + 0.890834i \(0.650121\pi\)
\(84\) −180.448 −0.234386
\(85\) −88.0203 −0.112319
\(86\) −527.257 −0.661111
\(87\) −480.334 −0.591922
\(88\) 132.970 0.161076
\(89\) −1070.54 −1.27502 −0.637510 0.770442i \(-0.720036\pi\)
−0.637510 + 0.770442i \(0.720036\pi\)
\(90\) −113.181 −0.132560
\(91\) 0 0
\(92\) −1070.91 −1.21359
\(93\) −114.211 −0.127346
\(94\) −2135.70 −2.34341
\(95\) 308.647 0.333332
\(96\) −779.267 −0.828475
\(97\) −1754.48 −1.83650 −0.918251 0.395998i \(-0.870399\pi\)
−0.918251 + 0.395998i \(0.870399\pi\)
\(98\) 1230.06 1.26791
\(99\) −290.250 −0.294658
\(100\) −1041.27 −1.04127
\(101\) −640.840 −0.631346 −0.315673 0.948868i \(-0.602230\pi\)
−0.315673 + 0.948868i \(0.602230\pi\)
\(102\) −356.961 −0.346514
\(103\) −693.153 −0.663091 −0.331546 0.943439i \(-0.607570\pi\)
−0.331546 + 0.943439i \(0.607570\pi\)
\(104\) 0 0
\(105\) −61.1528 −0.0568372
\(106\) 2867.81 2.62779
\(107\) −405.676 −0.366525 −0.183262 0.983064i \(-0.558666\pi\)
−0.183262 + 0.983064i \(0.558666\pi\)
\(108\) −243.000 −0.216506
\(109\) 479.516 0.421370 0.210685 0.977554i \(-0.432431\pi\)
0.210685 + 0.977554i \(0.432431\pi\)
\(110\) 405.566 0.351538
\(111\) −982.704 −0.840307
\(112\) −367.578 −0.310115
\(113\) −1547.20 −1.28804 −0.644021 0.765008i \(-0.722735\pi\)
−0.644021 + 0.765008i \(0.722735\pi\)
\(114\) 1251.70 1.02835
\(115\) −362.926 −0.294288
\(116\) 1441.00 1.15339
\(117\) 0 0
\(118\) 2706.91 2.11179
\(119\) −192.869 −0.148574
\(120\) 37.7271 0.0287000
\(121\) −290.940 −0.218588
\(122\) −2891.55 −2.14581
\(123\) 168.254 0.123341
\(124\) 342.634 0.248141
\(125\) −734.140 −0.525308
\(126\) −248.001 −0.175347
\(127\) −2495.15 −1.74338 −0.871690 0.490059i \(-0.836975\pi\)
−0.871690 + 0.490059i \(0.836975\pi\)
\(128\) 523.634 0.361587
\(129\) −383.636 −0.261839
\(130\) 0 0
\(131\) 43.7571 0.0291838 0.0145919 0.999894i \(-0.495355\pi\)
0.0145919 + 0.999894i \(0.495355\pi\)
\(132\) 870.749 0.574159
\(133\) 676.303 0.440924
\(134\) −235.739 −0.151975
\(135\) −82.3516 −0.0525015
\(136\) 118.987 0.0750224
\(137\) 206.194 0.128586 0.0642932 0.997931i \(-0.479521\pi\)
0.0642932 + 0.997931i \(0.479521\pi\)
\(138\) −1471.83 −0.907899
\(139\) −100.000 −0.0610208 −0.0305104 0.999534i \(-0.509713\pi\)
−0.0305104 + 0.999534i \(0.509713\pi\)
\(140\) 183.459 0.110751
\(141\) −1553.95 −0.928128
\(142\) 1274.97 0.753474
\(143\) 0 0
\(144\) −495.000 −0.286458
\(145\) 488.349 0.279691
\(146\) −1606.82 −0.910832
\(147\) 895.003 0.502167
\(148\) 2948.11 1.63739
\(149\) −380.451 −0.209179 −0.104590 0.994515i \(-0.533353\pi\)
−0.104590 + 0.994515i \(0.533353\pi\)
\(150\) −1431.09 −0.778989
\(151\) 1517.45 0.817805 0.408902 0.912578i \(-0.365912\pi\)
0.408902 + 0.912578i \(0.365912\pi\)
\(152\) −417.233 −0.222645
\(153\) −259.727 −0.137240
\(154\) 888.671 0.465007
\(155\) 116.117 0.0601726
\(156\) 0 0
\(157\) 1450.16 0.737166 0.368583 0.929595i \(-0.379843\pi\)
0.368583 + 0.929595i \(0.379843\pi\)
\(158\) −3718.30 −1.87223
\(159\) 2086.64 1.04076
\(160\) 792.270 0.391465
\(161\) −795.239 −0.389277
\(162\) −333.972 −0.161971
\(163\) −2342.36 −1.12557 −0.562785 0.826603i \(-0.690270\pi\)
−0.562785 + 0.826603i \(0.690270\pi\)
\(164\) −504.762 −0.240337
\(165\) 295.093 0.139230
\(166\) 2832.97 1.32458
\(167\) 40.0731 0.0185686 0.00928428 0.999957i \(-0.497045\pi\)
0.00928428 + 0.999957i \(0.497045\pi\)
\(168\) 82.6671 0.0379637
\(169\) 0 0
\(170\) 362.917 0.163732
\(171\) 910.744 0.407289
\(172\) 1150.91 0.510208
\(173\) 1909.54 0.839189 0.419594 0.907712i \(-0.362172\pi\)
0.419594 + 0.907712i \(0.362172\pi\)
\(174\) 1980.47 0.862868
\(175\) −773.232 −0.334005
\(176\) 1773.75 0.759666
\(177\) 1969.57 0.836395
\(178\) 4413.94 1.85864
\(179\) −509.959 −0.212939 −0.106470 0.994316i \(-0.533955\pi\)
−0.106470 + 0.994316i \(0.533955\pi\)
\(180\) 247.055 0.102302
\(181\) −2136.88 −0.877531 −0.438766 0.898602i \(-0.644584\pi\)
−0.438766 + 0.898602i \(0.644584\pi\)
\(182\) 0 0
\(183\) −2103.91 −0.849867
\(184\) 490.608 0.196566
\(185\) 999.101 0.397056
\(186\) 470.906 0.185637
\(187\) 930.688 0.363950
\(188\) 4661.85 1.80851
\(189\) −180.448 −0.0694478
\(190\) −1272.58 −0.485911
\(191\) 4057.74 1.53721 0.768607 0.639721i \(-0.220950\pi\)
0.768607 + 0.639721i \(0.220950\pi\)
\(192\) 1893.00 0.711539
\(193\) −873.394 −0.325742 −0.162871 0.986647i \(-0.552075\pi\)
−0.162871 + 0.986647i \(0.552075\pi\)
\(194\) 7233.92 2.67714
\(195\) 0 0
\(196\) −2685.01 −0.978502
\(197\) −4147.25 −1.49989 −0.749947 0.661498i \(-0.769921\pi\)
−0.749947 + 0.661498i \(0.769921\pi\)
\(198\) 1196.73 0.429535
\(199\) −2404.06 −0.856379 −0.428189 0.903689i \(-0.640848\pi\)
−0.428189 + 0.903689i \(0.640848\pi\)
\(200\) 477.032 0.168656
\(201\) −171.525 −0.0601912
\(202\) 2642.25 0.920337
\(203\) 1070.06 0.369969
\(204\) 779.181 0.267420
\(205\) −171.061 −0.0582802
\(206\) 2857.94 0.966613
\(207\) −1070.91 −0.359582
\(208\) 0 0
\(209\) −3263.50 −1.08010
\(210\) 252.140 0.0828538
\(211\) 3868.85 1.26229 0.631144 0.775665i \(-0.282586\pi\)
0.631144 + 0.775665i \(0.282586\pi\)
\(212\) −6259.91 −2.02798
\(213\) 927.679 0.298420
\(214\) 1672.64 0.534297
\(215\) 390.037 0.123722
\(216\) 111.324 0.0350677
\(217\) 254.434 0.0795950
\(218\) −1977.09 −0.614246
\(219\) −1169.13 −0.360743
\(220\) −885.279 −0.271298
\(221\) 0 0
\(222\) 4051.79 1.22495
\(223\) −2813.55 −0.844885 −0.422443 0.906390i \(-0.638827\pi\)
−0.422443 + 0.906390i \(0.638827\pi\)
\(224\) 1736.01 0.517822
\(225\) −1041.27 −0.308526
\(226\) 6379.29 1.87763
\(227\) 4518.37 1.32112 0.660561 0.750772i \(-0.270318\pi\)
0.660561 + 0.750772i \(0.270318\pi\)
\(228\) −2732.23 −0.793625
\(229\) 1305.27 0.376658 0.188329 0.982106i \(-0.439693\pi\)
0.188329 + 0.982106i \(0.439693\pi\)
\(230\) 1496.38 0.428994
\(231\) 646.603 0.184170
\(232\) −660.156 −0.186816
\(233\) −3360.55 −0.944879 −0.472440 0.881363i \(-0.656627\pi\)
−0.472440 + 0.881363i \(0.656627\pi\)
\(234\) 0 0
\(235\) 1579.88 0.438553
\(236\) −5908.71 −1.62976
\(237\) −2705.46 −0.741513
\(238\) 795.219 0.216581
\(239\) 4737.17 1.28210 0.641050 0.767499i \(-0.278499\pi\)
0.641050 + 0.767499i \(0.278499\pi\)
\(240\) 503.260 0.135355
\(241\) −4785.28 −1.27903 −0.639516 0.768778i \(-0.720865\pi\)
−0.639516 + 0.768778i \(0.720865\pi\)
\(242\) 1199.58 0.318643
\(243\) −243.000 −0.0641500
\(244\) 6311.74 1.65601
\(245\) −909.937 −0.237281
\(246\) −693.729 −0.179799
\(247\) 0 0
\(248\) −156.969 −0.0401916
\(249\) 2061.29 0.524613
\(250\) 3026.94 0.765762
\(251\) −3273.86 −0.823284 −0.411642 0.911346i \(-0.635045\pi\)
−0.411642 + 0.911346i \(0.635045\pi\)
\(252\) 541.343 0.135323
\(253\) 3837.42 0.953584
\(254\) 10287.8 2.54139
\(255\) 264.061 0.0648476
\(256\) 2889.00 0.705322
\(257\) −6545.81 −1.58878 −0.794390 0.607408i \(-0.792209\pi\)
−0.794390 + 0.607408i \(0.792209\pi\)
\(258\) 1581.77 0.381693
\(259\) 2189.22 0.525217
\(260\) 0 0
\(261\) 1441.00 0.341746
\(262\) −180.415 −0.0425424
\(263\) 88.2014 0.0206796 0.0103398 0.999947i \(-0.496709\pi\)
0.0103398 + 0.999947i \(0.496709\pi\)
\(264\) −398.910 −0.0929970
\(265\) −2121.46 −0.491773
\(266\) −2788.47 −0.642752
\(267\) 3211.61 0.736133
\(268\) 514.575 0.117286
\(269\) −4527.60 −1.02622 −0.513109 0.858324i \(-0.671506\pi\)
−0.513109 + 0.858324i \(0.671506\pi\)
\(270\) 339.544 0.0765334
\(271\) −8321.82 −1.86537 −0.932684 0.360695i \(-0.882540\pi\)
−0.932684 + 0.360695i \(0.882540\pi\)
\(272\) 1587.22 0.353821
\(273\) 0 0
\(274\) −850.160 −0.187445
\(275\) 3731.23 0.818187
\(276\) 3212.73 0.700665
\(277\) −2881.31 −0.624986 −0.312493 0.949920i \(-0.601164\pi\)
−0.312493 + 0.949920i \(0.601164\pi\)
\(278\) 412.311 0.0889523
\(279\) 342.634 0.0735232
\(280\) −84.0465 −0.0179384
\(281\) 2817.99 0.598247 0.299123 0.954214i \(-0.403306\pi\)
0.299123 + 0.954214i \(0.403306\pi\)
\(282\) 6407.10 1.35297
\(283\) 264.601 0.0555792 0.0277896 0.999614i \(-0.491153\pi\)
0.0277896 + 0.999614i \(0.491153\pi\)
\(284\) −2783.04 −0.581489
\(285\) −925.941 −0.192449
\(286\) 0 0
\(287\) −374.827 −0.0770918
\(288\) 2337.80 0.478320
\(289\) −4080.18 −0.830487
\(290\) −2013.52 −0.407716
\(291\) 5263.45 1.06031
\(292\) 3507.40 0.702929
\(293\) −4292.52 −0.855877 −0.427938 0.903808i \(-0.640760\pi\)
−0.427938 + 0.903808i \(0.640760\pi\)
\(294\) −3690.19 −0.732028
\(295\) −2002.43 −0.395208
\(296\) −1350.60 −0.265209
\(297\) 870.749 0.170121
\(298\) 1568.64 0.304929
\(299\) 0 0
\(300\) 3123.82 0.601180
\(301\) 854.643 0.163657
\(302\) −6256.62 −1.19214
\(303\) 1922.52 0.364508
\(304\) −5565.66 −1.05004
\(305\) 2139.02 0.401573
\(306\) 1070.88 0.200060
\(307\) −7026.26 −1.30622 −0.653110 0.757263i \(-0.726536\pi\)
−0.653110 + 0.757263i \(0.726536\pi\)
\(308\) −1939.81 −0.358866
\(309\) 2079.46 0.382836
\(310\) −478.763 −0.0877159
\(311\) −1133.21 −0.206618 −0.103309 0.994649i \(-0.532943\pi\)
−0.103309 + 0.994649i \(0.532943\pi\)
\(312\) 0 0
\(313\) −5285.95 −0.954566 −0.477283 0.878750i \(-0.658378\pi\)
−0.477283 + 0.878750i \(0.658378\pi\)
\(314\) −5979.15 −1.07459
\(315\) 183.459 0.0328150
\(316\) 8116.38 1.44488
\(317\) 4782.16 0.847296 0.423648 0.905827i \(-0.360749\pi\)
0.423648 + 0.905827i \(0.360749\pi\)
\(318\) −8603.43 −1.51716
\(319\) −5163.59 −0.906286
\(320\) −1924.59 −0.336212
\(321\) 1217.03 0.211613
\(322\) 3278.85 0.567464
\(323\) −2920.31 −0.503066
\(324\) 729.000 0.125000
\(325\) 0 0
\(326\) 9657.80 1.64079
\(327\) −1438.55 −0.243278
\(328\) 231.243 0.0389276
\(329\) 3461.81 0.580108
\(330\) −1216.70 −0.202961
\(331\) −8669.98 −1.43971 −0.719857 0.694122i \(-0.755793\pi\)
−0.719857 + 0.694122i \(0.755793\pi\)
\(332\) −6183.86 −1.02224
\(333\) 2948.11 0.485152
\(334\) −165.226 −0.0270681
\(335\) 174.387 0.0284412
\(336\) 1102.73 0.179045
\(337\) 8526.59 1.37826 0.689129 0.724639i \(-0.257993\pi\)
0.689129 + 0.724639i \(0.257993\pi\)
\(338\) 0 0
\(339\) 4641.61 0.743651
\(340\) −792.183 −0.126359
\(341\) −1227.77 −0.194978
\(342\) −3755.10 −0.593720
\(343\) −4286.19 −0.674731
\(344\) −527.257 −0.0826389
\(345\) 1088.78 0.169907
\(346\) −7873.23 −1.22332
\(347\) −12581.5 −1.94643 −0.973213 0.229907i \(-0.926158\pi\)
−0.973213 + 0.229907i \(0.926158\pi\)
\(348\) −4323.01 −0.665913
\(349\) 8961.18 1.37444 0.687222 0.726447i \(-0.258830\pi\)
0.687222 + 0.726447i \(0.258830\pi\)
\(350\) 3188.12 0.486892
\(351\) 0 0
\(352\) −8377.11 −1.26847
\(353\) 5357.99 0.807868 0.403934 0.914788i \(-0.367643\pi\)
0.403934 + 0.914788i \(0.367643\pi\)
\(354\) −8120.74 −1.21924
\(355\) −943.159 −0.141007
\(356\) −9634.84 −1.43440
\(357\) 578.606 0.0857790
\(358\) 2102.61 0.310409
\(359\) 2705.40 0.397731 0.198866 0.980027i \(-0.436274\pi\)
0.198866 + 0.980027i \(0.436274\pi\)
\(360\) −113.181 −0.0165700
\(361\) 3381.19 0.492957
\(362\) 8810.59 1.27921
\(363\) 872.820 0.126202
\(364\) 0 0
\(365\) 1188.64 0.170456
\(366\) 8674.65 1.23888
\(367\) 10473.8 1.48972 0.744858 0.667223i \(-0.232517\pi\)
0.744858 + 0.667223i \(0.232517\pi\)
\(368\) 6544.45 0.927046
\(369\) −504.762 −0.0712110
\(370\) −4119.40 −0.578804
\(371\) −4648.50 −0.650507
\(372\) −1027.90 −0.143264
\(373\) −12763.0 −1.77170 −0.885850 0.463973i \(-0.846424\pi\)
−0.885850 + 0.463973i \(0.846424\pi\)
\(374\) −3837.32 −0.530544
\(375\) 2202.42 0.303287
\(376\) −2135.70 −0.292926
\(377\) 0 0
\(378\) 744.004 0.101237
\(379\) 2318.02 0.314166 0.157083 0.987585i \(-0.449791\pi\)
0.157083 + 0.987585i \(0.449791\pi\)
\(380\) 2777.82 0.374998
\(381\) 7485.46 1.00654
\(382\) −16730.5 −2.24086
\(383\) −1983.34 −0.264606 −0.132303 0.991209i \(-0.542237\pi\)
−0.132303 + 0.991209i \(0.542237\pi\)
\(384\) −1570.90 −0.208763
\(385\) −657.392 −0.0870229
\(386\) 3601.10 0.474847
\(387\) 1150.91 0.151173
\(388\) −15790.3 −2.06607
\(389\) −3244.51 −0.422887 −0.211444 0.977390i \(-0.567816\pi\)
−0.211444 + 0.977390i \(0.567816\pi\)
\(390\) 0 0
\(391\) 3433.88 0.444140
\(392\) 1230.06 0.158489
\(393\) −131.271 −0.0168493
\(394\) 17099.5 2.18645
\(395\) 2750.60 0.350374
\(396\) −2612.25 −0.331491
\(397\) −3759.72 −0.475302 −0.237651 0.971351i \(-0.576378\pi\)
−0.237651 + 0.971351i \(0.576378\pi\)
\(398\) 9912.20 1.24838
\(399\) −2028.91 −0.254568
\(400\) 6363.34 0.795418
\(401\) 1997.55 0.248760 0.124380 0.992235i \(-0.460306\pi\)
0.124380 + 0.992235i \(0.460306\pi\)
\(402\) 707.216 0.0877431
\(403\) 0 0
\(404\) −5767.56 −0.710264
\(405\) 247.055 0.0303117
\(406\) −4411.98 −0.539318
\(407\) −10564.1 −1.28659
\(408\) −356.961 −0.0433142
\(409\) 5195.23 0.628087 0.314044 0.949409i \(-0.398316\pi\)
0.314044 + 0.949409i \(0.398316\pi\)
\(410\) 705.304 0.0849573
\(411\) −618.582 −0.0742394
\(412\) −6238.38 −0.745977
\(413\) −4387.70 −0.522772
\(414\) 4415.48 0.524176
\(415\) −2095.68 −0.247887
\(416\) 0 0
\(417\) 300.000 0.0352304
\(418\) 13455.7 1.57450
\(419\) −6822.11 −0.795422 −0.397711 0.917511i \(-0.630195\pi\)
−0.397711 + 0.917511i \(0.630195\pi\)
\(420\) −550.376 −0.0639419
\(421\) −7537.70 −0.872601 −0.436300 0.899801i \(-0.643712\pi\)
−0.436300 + 0.899801i \(0.643712\pi\)
\(422\) −15951.7 −1.84009
\(423\) 4661.85 0.535855
\(424\) 2867.81 0.328474
\(425\) 3338.85 0.381078
\(426\) −3824.92 −0.435019
\(427\) 4686.99 0.531192
\(428\) −3651.08 −0.412340
\(429\) 0 0
\(430\) −1608.16 −0.180355
\(431\) 13404.2 1.49805 0.749023 0.662544i \(-0.230523\pi\)
0.749023 + 0.662544i \(0.230523\pi\)
\(432\) 1485.00 0.165387
\(433\) −17715.9 −1.96622 −0.983110 0.183014i \(-0.941415\pi\)
−0.983110 + 0.183014i \(0.941415\pi\)
\(434\) −1049.06 −0.116029
\(435\) −1465.05 −0.161480
\(436\) 4315.64 0.474041
\(437\) −12041.1 −1.31808
\(438\) 4820.46 0.525869
\(439\) 7163.47 0.778801 0.389401 0.921068i \(-0.372682\pi\)
0.389401 + 0.921068i \(0.372682\pi\)
\(440\) 405.566 0.0439423
\(441\) −2685.01 −0.289926
\(442\) 0 0
\(443\) −10169.2 −1.09064 −0.545321 0.838227i \(-0.683592\pi\)
−0.545321 + 0.838227i \(0.683592\pi\)
\(444\) −8844.33 −0.945346
\(445\) −3265.20 −0.347833
\(446\) 11600.6 1.23162
\(447\) 1141.35 0.120770
\(448\) −4217.13 −0.444733
\(449\) 17142.5 1.80179 0.900895 0.434037i \(-0.142911\pi\)
0.900895 + 0.434037i \(0.142911\pi\)
\(450\) 4293.28 0.449750
\(451\) 1808.73 0.188846
\(452\) −13924.8 −1.44905
\(453\) −4552.36 −0.472160
\(454\) −18629.7 −1.92585
\(455\) 0 0
\(456\) 1251.70 0.128544
\(457\) −14091.1 −1.44235 −0.721177 0.692750i \(-0.756399\pi\)
−0.721177 + 0.692750i \(0.756399\pi\)
\(458\) −5381.76 −0.549068
\(459\) 779.181 0.0792355
\(460\) −3266.34 −0.331074
\(461\) 2922.22 0.295231 0.147616 0.989045i \(-0.452840\pi\)
0.147616 + 0.989045i \(0.452840\pi\)
\(462\) −2666.01 −0.268472
\(463\) 2072.61 0.208040 0.104020 0.994575i \(-0.466829\pi\)
0.104020 + 0.994575i \(0.466829\pi\)
\(464\) −8806.13 −0.881065
\(465\) −348.352 −0.0347407
\(466\) 13855.9 1.37739
\(467\) −2664.19 −0.263992 −0.131996 0.991250i \(-0.542139\pi\)
−0.131996 + 0.991250i \(0.542139\pi\)
\(468\) 0 0
\(469\) 382.114 0.0376213
\(470\) −6514.01 −0.639295
\(471\) −4350.47 −0.425603
\(472\) 2706.91 0.263974
\(473\) −4124.08 −0.400899
\(474\) 11154.9 1.08093
\(475\) −11707.8 −1.13093
\(476\) −1735.82 −0.167145
\(477\) −6259.91 −0.600884
\(478\) −19531.8 −1.86897
\(479\) 5220.70 0.497995 0.248998 0.968504i \(-0.419899\pi\)
0.248998 + 0.968504i \(0.419899\pi\)
\(480\) −2376.81 −0.226013
\(481\) 0 0
\(482\) 19730.2 1.86449
\(483\) 2385.72 0.224749
\(484\) −2618.46 −0.245911
\(485\) −5351.28 −0.501008
\(486\) 1001.91 0.0935139
\(487\) 12224.6 1.13747 0.568737 0.822520i \(-0.307432\pi\)
0.568737 + 0.822520i \(0.307432\pi\)
\(488\) −2891.55 −0.268226
\(489\) 7027.08 0.649848
\(490\) 3751.77 0.345893
\(491\) 19653.2 1.80639 0.903195 0.429231i \(-0.141216\pi\)
0.903195 + 0.429231i \(0.141216\pi\)
\(492\) 1514.29 0.138759
\(493\) −4620.59 −0.422111
\(494\) 0 0
\(495\) −885.279 −0.0803845
\(496\) −2093.88 −0.189552
\(497\) −2066.63 −0.186522
\(498\) −8498.90 −0.764749
\(499\) 11713.6 1.05084 0.525422 0.850842i \(-0.323907\pi\)
0.525422 + 0.850842i \(0.323907\pi\)
\(500\) −6607.26 −0.590972
\(501\) −120.219 −0.0107206
\(502\) 13498.5 1.20013
\(503\) 13003.3 1.15266 0.576332 0.817216i \(-0.304483\pi\)
0.576332 + 0.817216i \(0.304483\pi\)
\(504\) −248.001 −0.0219184
\(505\) −1954.60 −0.172235
\(506\) −15822.1 −1.39008
\(507\) 0 0
\(508\) −22456.4 −1.96130
\(509\) 5328.93 0.464049 0.232024 0.972710i \(-0.425465\pi\)
0.232024 + 0.972710i \(0.425465\pi\)
\(510\) −1088.75 −0.0945308
\(511\) 2604.54 0.225475
\(512\) −16100.7 −1.38976
\(513\) −2732.23 −0.235148
\(514\) 26989.1 2.31603
\(515\) −2114.16 −0.180895
\(516\) −3452.72 −0.294569
\(517\) −16704.9 −1.42105
\(518\) −9026.37 −0.765629
\(519\) −5728.62 −0.484506
\(520\) 0 0
\(521\) −11700.3 −0.983876 −0.491938 0.870630i \(-0.663711\pi\)
−0.491938 + 0.870630i \(0.663711\pi\)
\(522\) −5941.41 −0.498177
\(523\) −4535.04 −0.379165 −0.189583 0.981865i \(-0.560714\pi\)
−0.189583 + 0.981865i \(0.560714\pi\)
\(524\) 393.814 0.0328318
\(525\) 2319.70 0.192838
\(526\) −363.664 −0.0301454
\(527\) −1098.66 −0.0908128
\(528\) −5321.24 −0.438594
\(529\) 1991.62 0.163690
\(530\) 8746.98 0.716877
\(531\) −5908.71 −0.482893
\(532\) 6086.73 0.496040
\(533\) 0 0
\(534\) −13241.8 −1.07309
\(535\) −1237.33 −0.0999900
\(536\) −235.739 −0.0189969
\(537\) 1529.88 0.122940
\(538\) 18667.8 1.49596
\(539\) 9621.27 0.768863
\(540\) −741.164 −0.0590641
\(541\) 5184.89 0.412044 0.206022 0.978547i \(-0.433948\pi\)
0.206022 + 0.978547i \(0.433948\pi\)
\(542\) 34311.7 2.71922
\(543\) 6410.64 0.506643
\(544\) −7496.18 −0.590801
\(545\) 1462.55 0.114952
\(546\) 0 0
\(547\) 5609.12 0.438443 0.219222 0.975675i \(-0.429648\pi\)
0.219222 + 0.975675i \(0.429648\pi\)
\(548\) 1855.75 0.144660
\(549\) 6311.74 0.490671
\(550\) −15384.2 −1.19270
\(551\) 16202.3 1.25271
\(552\) −1471.83 −0.113487
\(553\) 6027.08 0.463468
\(554\) 11879.9 0.911066
\(555\) −2997.30 −0.229241
\(556\) −900.000 −0.0686484
\(557\) 20150.5 1.53286 0.766432 0.642326i \(-0.222030\pi\)
0.766432 + 0.642326i \(0.222030\pi\)
\(558\) −1412.72 −0.107178
\(559\) 0 0
\(560\) −1121.14 −0.0846011
\(561\) −2792.06 −0.210127
\(562\) −11618.9 −0.872087
\(563\) 16292.2 1.21960 0.609800 0.792556i \(-0.291250\pi\)
0.609800 + 0.792556i \(0.291250\pi\)
\(564\) −13985.5 −1.04414
\(565\) −4719.06 −0.351385
\(566\) −1090.98 −0.0810200
\(567\) 541.343 0.0400957
\(568\) 1274.97 0.0941843
\(569\) 10460.5 0.770700 0.385350 0.922770i \(-0.374081\pi\)
0.385350 + 0.922770i \(0.374081\pi\)
\(570\) 3817.75 0.280541
\(571\) 2225.96 0.163141 0.0815705 0.996668i \(-0.474006\pi\)
0.0815705 + 0.996668i \(0.474006\pi\)
\(572\) 0 0
\(573\) −12173.2 −0.887511
\(574\) 1545.45 0.112380
\(575\) 13766.8 0.998461
\(576\) −5679.00 −0.410807
\(577\) −4686.23 −0.338112 −0.169056 0.985606i \(-0.554072\pi\)
−0.169056 + 0.985606i \(0.554072\pi\)
\(578\) 16823.0 1.21063
\(579\) 2620.18 0.188067
\(580\) 4395.14 0.314652
\(581\) −4592.03 −0.327899
\(582\) −21701.8 −1.54565
\(583\) 22431.3 1.59350
\(584\) −1606.82 −0.113854
\(585\) 0 0
\(586\) 17698.5 1.24764
\(587\) −12090.6 −0.850138 −0.425069 0.905161i \(-0.639750\pi\)
−0.425069 + 0.905161i \(0.639750\pi\)
\(588\) 8055.03 0.564938
\(589\) 3852.50 0.269507
\(590\) 8256.25 0.576109
\(591\) 12441.7 0.865964
\(592\) −18016.2 −1.25078
\(593\) −6135.97 −0.424914 −0.212457 0.977170i \(-0.568147\pi\)
−0.212457 + 0.977170i \(0.568147\pi\)
\(594\) −3590.19 −0.247992
\(595\) −588.261 −0.0405317
\(596\) −3424.06 −0.235327
\(597\) 7212.18 0.494431
\(598\) 0 0
\(599\) −6198.80 −0.422831 −0.211416 0.977396i \(-0.567807\pi\)
−0.211416 + 0.977396i \(0.567807\pi\)
\(600\) −1431.09 −0.0973737
\(601\) 18345.4 1.24513 0.622565 0.782568i \(-0.286091\pi\)
0.622565 + 0.782568i \(0.286091\pi\)
\(602\) −3523.79 −0.238569
\(603\) 514.575 0.0347514
\(604\) 13657.1 0.920030
\(605\) −887.384 −0.0596319
\(606\) −7926.75 −0.531357
\(607\) 10388.1 0.694631 0.347315 0.937748i \(-0.387093\pi\)
0.347315 + 0.937748i \(0.387093\pi\)
\(608\) 26285.7 1.75333
\(609\) −3210.19 −0.213602
\(610\) −8819.40 −0.585389
\(611\) 0 0
\(612\) −2337.54 −0.154395
\(613\) 804.480 0.0530060 0.0265030 0.999649i \(-0.491563\pi\)
0.0265030 + 0.999649i \(0.491563\pi\)
\(614\) 28970.0 1.90413
\(615\) 513.184 0.0336481
\(616\) 888.671 0.0581259
\(617\) −15218.3 −0.992973 −0.496486 0.868044i \(-0.665377\pi\)
−0.496486 + 0.868044i \(0.665377\pi\)
\(618\) −8573.83 −0.558074
\(619\) 11462.5 0.744291 0.372145 0.928174i \(-0.378622\pi\)
0.372145 + 0.928174i \(0.378622\pi\)
\(620\) 1045.05 0.0676942
\(621\) 3212.73 0.207605
\(622\) 4672.33 0.301195
\(623\) −7154.66 −0.460105
\(624\) 0 0
\(625\) 12223.0 0.782270
\(626\) 21794.5 1.39151
\(627\) 9790.49 0.623596
\(628\) 13051.4 0.829311
\(629\) −9453.14 −0.599239
\(630\) −756.419 −0.0478356
\(631\) 4468.68 0.281926 0.140963 0.990015i \(-0.454980\pi\)
0.140963 + 0.990015i \(0.454980\pi\)
\(632\) −3718.30 −0.234028
\(633\) −11606.6 −0.728783
\(634\) −19717.3 −1.23514
\(635\) −7610.37 −0.475603
\(636\) 18779.7 1.17086
\(637\) 0 0
\(638\) 21290.0 1.32113
\(639\) −2783.04 −0.172293
\(640\) 1597.12 0.0986430
\(641\) 6142.36 0.378484 0.189242 0.981930i \(-0.439397\pi\)
0.189242 + 0.981930i \(0.439397\pi\)
\(642\) −5017.93 −0.308477
\(643\) −20738.2 −1.27190 −0.635951 0.771729i \(-0.719392\pi\)
−0.635951 + 0.771729i \(0.719392\pi\)
\(644\) −7157.15 −0.437937
\(645\) −1170.11 −0.0714312
\(646\) 12040.7 0.733339
\(647\) 852.757 0.0518166 0.0259083 0.999664i \(-0.491752\pi\)
0.0259083 + 0.999664i \(0.491752\pi\)
\(648\) −333.972 −0.0202464
\(649\) 21172.8 1.28060
\(650\) 0 0
\(651\) −763.303 −0.0459542
\(652\) −21081.3 −1.26627
\(653\) −7345.75 −0.440217 −0.220108 0.975475i \(-0.570641\pi\)
−0.220108 + 0.975475i \(0.570641\pi\)
\(654\) 5931.28 0.354635
\(655\) 133.462 0.00796151
\(656\) 3084.65 0.183591
\(657\) 3507.40 0.208275
\(658\) −14273.4 −0.845645
\(659\) 12540.7 0.741297 0.370648 0.928773i \(-0.379136\pi\)
0.370648 + 0.928773i \(0.379136\pi\)
\(660\) 2655.84 0.156634
\(661\) 2242.95 0.131983 0.0659915 0.997820i \(-0.478979\pi\)
0.0659915 + 0.997820i \(0.478979\pi\)
\(662\) 35747.3 2.09873
\(663\) 0 0
\(664\) 2832.97 0.165573
\(665\) 2062.76 0.120287
\(666\) −12155.4 −0.707224
\(667\) −19051.7 −1.10597
\(668\) 360.658 0.0208896
\(669\) 8440.66 0.487795
\(670\) −719.016 −0.0414597
\(671\) −22617.0 −1.30122
\(672\) −5208.03 −0.298964
\(673\) −4776.46 −0.273579 −0.136790 0.990600i \(-0.543678\pi\)
−0.136790 + 0.990600i \(0.543678\pi\)
\(674\) −35156.0 −2.00914
\(675\) 3123.82 0.178127
\(676\) 0 0
\(677\) −7933.57 −0.450387 −0.225193 0.974314i \(-0.572301\pi\)
−0.225193 + 0.974314i \(0.572301\pi\)
\(678\) −19137.9 −1.08405
\(679\) −11725.6 −0.662723
\(680\) 362.917 0.0204665
\(681\) −13555.1 −0.762750
\(682\) 5062.23 0.284227
\(683\) −23573.8 −1.32068 −0.660340 0.750967i \(-0.729588\pi\)
−0.660340 + 0.750967i \(0.729588\pi\)
\(684\) 8196.70 0.458200
\(685\) 628.904 0.0350791
\(686\) 17672.4 0.983581
\(687\) −3915.81 −0.217463
\(688\) −7033.32 −0.389743
\(689\) 0 0
\(690\) −4489.15 −0.247680
\(691\) 12543.8 0.690575 0.345288 0.938497i \(-0.387781\pi\)
0.345288 + 0.938497i \(0.387781\pi\)
\(692\) 17185.9 0.944087
\(693\) −1939.81 −0.106331
\(694\) 51874.8 2.83738
\(695\) −305.006 −0.0166468
\(696\) 1980.47 0.107858
\(697\) 1618.52 0.0879568
\(698\) −36947.9 −2.00358
\(699\) 10081.6 0.545526
\(700\) −6959.09 −0.375755
\(701\) −581.786 −0.0313463 −0.0156731 0.999877i \(-0.504989\pi\)
−0.0156731 + 0.999877i \(0.504989\pi\)
\(702\) 0 0
\(703\) 33147.9 1.77837
\(704\) 20349.7 1.08943
\(705\) −4739.64 −0.253199
\(706\) −22091.6 −1.17766
\(707\) −4282.89 −0.227828
\(708\) 17726.1 0.940944
\(709\) 20742.0 1.09871 0.549353 0.835590i \(-0.314874\pi\)
0.549353 + 0.835590i \(0.314874\pi\)
\(710\) 3888.74 0.205552
\(711\) 8116.38 0.428113
\(712\) 4413.94 0.232331
\(713\) −4530.01 −0.237938
\(714\) −2385.66 −0.125043
\(715\) 0 0
\(716\) −4589.63 −0.239556
\(717\) −14211.5 −0.740221
\(718\) −11154.6 −0.579788
\(719\) 25350.2 1.31489 0.657443 0.753504i \(-0.271638\pi\)
0.657443 + 0.753504i \(0.271638\pi\)
\(720\) −1509.78 −0.0781474
\(721\) −4632.51 −0.239284
\(722\) −13941.0 −0.718602
\(723\) 14355.8 0.738449
\(724\) −19231.9 −0.987223
\(725\) −18524.4 −0.948938
\(726\) −3598.73 −0.183969
\(727\) 33428.2 1.70534 0.852672 0.522447i \(-0.174981\pi\)
0.852672 + 0.522447i \(0.174981\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −4900.90 −0.248480
\(731\) −3690.39 −0.186722
\(732\) −18935.2 −0.956101
\(733\) 3842.67 0.193632 0.0968160 0.995302i \(-0.469134\pi\)
0.0968160 + 0.995302i \(0.469134\pi\)
\(734\) −43184.4 −2.17162
\(735\) 2729.81 0.136994
\(736\) −30908.3 −1.54796
\(737\) −1843.89 −0.0921582
\(738\) 2081.19 0.103807
\(739\) −29029.6 −1.44502 −0.722511 0.691359i \(-0.757012\pi\)
−0.722511 + 0.691359i \(0.757012\pi\)
\(740\) 8991.91 0.446688
\(741\) 0 0
\(742\) 19166.3 0.948269
\(743\) 34996.7 1.72800 0.864000 0.503492i \(-0.167952\pi\)
0.864000 + 0.503492i \(0.167952\pi\)
\(744\) 470.906 0.0232046
\(745\) −1160.40 −0.0570653
\(746\) 52623.2 2.58267
\(747\) −6183.86 −0.302886
\(748\) 8376.19 0.409444
\(749\) −2711.23 −0.132265
\(750\) −9080.82 −0.442113
\(751\) 10454.1 0.507957 0.253979 0.967210i \(-0.418261\pi\)
0.253979 + 0.967210i \(0.418261\pi\)
\(752\) −28489.1 −1.38150
\(753\) 9821.58 0.475323
\(754\) 0 0
\(755\) 4628.32 0.223102
\(756\) −1624.03 −0.0781287
\(757\) −28130.4 −1.35062 −0.675308 0.737536i \(-0.735989\pi\)
−0.675308 + 0.737536i \(0.735989\pi\)
\(758\) −9557.45 −0.457971
\(759\) −11512.3 −0.550552
\(760\) −1272.58 −0.0607388
\(761\) −21087.0 −1.00447 −0.502236 0.864731i \(-0.667489\pi\)
−0.502236 + 0.864731i \(0.667489\pi\)
\(762\) −30863.4 −1.46727
\(763\) 3204.72 0.152056
\(764\) 36519.7 1.72937
\(765\) −792.183 −0.0374398
\(766\) 8177.54 0.385727
\(767\) 0 0
\(768\) −8667.00 −0.407218
\(769\) −19527.9 −0.915728 −0.457864 0.889022i \(-0.651385\pi\)
−0.457864 + 0.889022i \(0.651385\pi\)
\(770\) 2710.50 0.126857
\(771\) 19637.4 0.917283
\(772\) −7860.55 −0.366460
\(773\) 29352.0 1.36574 0.682870 0.730540i \(-0.260731\pi\)
0.682870 + 0.730540i \(0.260731\pi\)
\(774\) −4745.31 −0.220370
\(775\) −4404.64 −0.204154
\(776\) 7233.92 0.334643
\(777\) −6567.65 −0.303234
\(778\) 13377.5 0.616459
\(779\) −5675.42 −0.261031
\(780\) 0 0
\(781\) 9972.54 0.456908
\(782\) −14158.3 −0.647440
\(783\) −4323.01 −0.197307
\(784\) 16408.4 0.747467
\(785\) 4423.06 0.201103
\(786\) 541.246 0.0245618
\(787\) −4463.12 −0.202151 −0.101076 0.994879i \(-0.532228\pi\)
−0.101076 + 0.994879i \(0.532228\pi\)
\(788\) −37325.2 −1.68738
\(789\) −264.604 −0.0119394
\(790\) −11341.0 −0.510754
\(791\) −10340.3 −0.464804
\(792\) 1196.73 0.0536919
\(793\) 0 0
\(794\) 15501.7 0.692866
\(795\) 6364.37 0.283926
\(796\) −21636.6 −0.963426
\(797\) −34785.5 −1.54600 −0.773002 0.634404i \(-0.781246\pi\)
−0.773002 + 0.634404i \(0.781246\pi\)
\(798\) 8365.40 0.371093
\(799\) −14948.2 −0.661866
\(800\) −30053.0 −1.32817
\(801\) −9634.84 −0.425007
\(802\) −8236.09 −0.362627
\(803\) −12568.2 −0.552330
\(804\) −1543.72 −0.0677152
\(805\) −2425.53 −0.106197
\(806\) 0 0
\(807\) 13582.8 0.592487
\(808\) 2642.25 0.115042
\(809\) −10620.0 −0.461530 −0.230765 0.973010i \(-0.574123\pi\)
−0.230765 + 0.973010i \(0.574123\pi\)
\(810\) −1018.63 −0.0441866
\(811\) −5497.87 −0.238047 −0.119024 0.992891i \(-0.537976\pi\)
−0.119024 + 0.992891i \(0.537976\pi\)
\(812\) 9630.57 0.416215
\(813\) 24965.5 1.07697
\(814\) 43556.7 1.87551
\(815\) −7144.34 −0.307062
\(816\) −4761.66 −0.204279
\(817\) 12940.5 0.554139
\(818\) −21420.5 −0.915587
\(819\) 0 0
\(820\) −1539.55 −0.0655653
\(821\) −21305.3 −0.905678 −0.452839 0.891592i \(-0.649589\pi\)
−0.452839 + 0.891592i \(0.649589\pi\)
\(822\) 2550.48 0.108222
\(823\) 17342.6 0.734537 0.367268 0.930115i \(-0.380293\pi\)
0.367268 + 0.930115i \(0.380293\pi\)
\(824\) 2857.94 0.120827
\(825\) −11193.7 −0.472381
\(826\) 18091.0 0.762064
\(827\) −5129.96 −0.215703 −0.107851 0.994167i \(-0.534397\pi\)
−0.107851 + 0.994167i \(0.534397\pi\)
\(828\) −9638.19 −0.404529
\(829\) −8471.81 −0.354931 −0.177466 0.984127i \(-0.556790\pi\)
−0.177466 + 0.984127i \(0.556790\pi\)
\(830\) 8640.72 0.361354
\(831\) 8643.93 0.360836
\(832\) 0 0
\(833\) 8609.50 0.358105
\(834\) −1236.93 −0.0513566
\(835\) 122.225 0.00506561
\(836\) −29371.5 −1.21511
\(837\) −1027.90 −0.0424486
\(838\) 28128.3 1.15952
\(839\) 19155.0 0.788207 0.394103 0.919066i \(-0.371055\pi\)
0.394103 + 0.919066i \(0.371055\pi\)
\(840\) 252.140 0.0103567
\(841\) 1246.67 0.0511160
\(842\) 31078.7 1.27202
\(843\) −8453.98 −0.345398
\(844\) 34819.7 1.42007
\(845\) 0 0
\(846\) −19221.3 −0.781136
\(847\) −1944.42 −0.0788797
\(848\) 38255.0 1.54915
\(849\) −793.804 −0.0320887
\(850\) −13766.4 −0.555512
\(851\) −38977.3 −1.57006
\(852\) 8349.11 0.335723
\(853\) 18075.1 0.725532 0.362766 0.931880i \(-0.381832\pi\)
0.362766 + 0.931880i \(0.381832\pi\)
\(854\) −19324.9 −0.774339
\(855\) 2777.82 0.111111
\(856\) 1672.64 0.0667871
\(857\) −21054.6 −0.839219 −0.419609 0.907705i \(-0.637833\pi\)
−0.419609 + 0.907705i \(0.637833\pi\)
\(858\) 0 0
\(859\) 920.322 0.0365553 0.0182776 0.999833i \(-0.494182\pi\)
0.0182776 + 0.999833i \(0.494182\pi\)
\(860\) 3510.33 0.139188
\(861\) 1124.48 0.0445090
\(862\) −55267.0 −2.18376
\(863\) 19427.5 0.766304 0.383152 0.923685i \(-0.374839\pi\)
0.383152 + 0.923685i \(0.374839\pi\)
\(864\) −7013.40 −0.276158
\(865\) 5824.21 0.228935
\(866\) 73044.7 2.86623
\(867\) 12240.5 0.479482
\(868\) 2289.91 0.0895444
\(869\) −29083.7 −1.13532
\(870\) 6040.55 0.235395
\(871\) 0 0
\(872\) −1977.09 −0.0767808
\(873\) −15790.3 −0.612168
\(874\) 49646.5 1.92142
\(875\) −4906.44 −0.189563
\(876\) −10522.2 −0.405836
\(877\) −14872.2 −0.572632 −0.286316 0.958135i \(-0.592431\pi\)
−0.286316 + 0.958135i \(0.592431\pi\)
\(878\) −29535.7 −1.13529
\(879\) 12877.6 0.494141
\(880\) 5410.04 0.207241
\(881\) −12940.6 −0.494870 −0.247435 0.968905i \(-0.579588\pi\)
−0.247435 + 0.968905i \(0.579588\pi\)
\(882\) 11070.6 0.422637
\(883\) −25585.5 −0.975108 −0.487554 0.873093i \(-0.662111\pi\)
−0.487554 + 0.873093i \(0.662111\pi\)
\(884\) 0 0
\(885\) 6007.30 0.228173
\(886\) 41928.8 1.58987
\(887\) 3716.46 0.140684 0.0703418 0.997523i \(-0.477591\pi\)
0.0703418 + 0.997523i \(0.477591\pi\)
\(888\) 4051.79 0.153118
\(889\) −16675.7 −0.629118
\(890\) 13462.8 0.507049
\(891\) −2612.25 −0.0982195
\(892\) −25322.0 −0.950496
\(893\) 52416.7 1.96423
\(894\) −4705.92 −0.176051
\(895\) −1555.40 −0.0580910
\(896\) 3499.58 0.130483
\(897\) 0 0
\(898\) −70680.3 −2.62654
\(899\) 6095.52 0.226137
\(900\) −9371.47 −0.347091
\(901\) 20072.5 0.742187
\(902\) −7457.57 −0.275288
\(903\) −2563.93 −0.0944876
\(904\) 6379.29 0.234703
\(905\) −6517.61 −0.239395
\(906\) 18769.8 0.688285
\(907\) −12960.4 −0.474469 −0.237235 0.971452i \(-0.576241\pi\)
−0.237235 + 0.971452i \(0.576241\pi\)
\(908\) 40665.3 1.48626
\(909\) −5767.56 −0.210449
\(910\) 0 0
\(911\) −36607.1 −1.33134 −0.665668 0.746248i \(-0.731853\pi\)
−0.665668 + 0.746248i \(0.731853\pi\)
\(912\) 16697.0 0.606242
\(913\) 22158.8 0.803230
\(914\) 58099.3 2.10258
\(915\) −6417.06 −0.231849
\(916\) 11747.4 0.423740
\(917\) 292.440 0.0105313
\(918\) −3212.65 −0.115505
\(919\) 20356.3 0.730676 0.365338 0.930875i \(-0.380953\pi\)
0.365338 + 0.930875i \(0.380953\pi\)
\(920\) 1496.38 0.0536243
\(921\) 21078.8 0.754147
\(922\) −12048.6 −0.430370
\(923\) 0 0
\(924\) 5819.43 0.207192
\(925\) −37898.7 −1.34714
\(926\) −8545.61 −0.303268
\(927\) −6238.38 −0.221030
\(928\) 41589.8 1.47118
\(929\) 45069.7 1.59170 0.795849 0.605495i \(-0.207025\pi\)
0.795849 + 0.605495i \(0.207025\pi\)
\(930\) 1436.29 0.0506428
\(931\) −30189.6 −1.06275
\(932\) −30244.9 −1.06299
\(933\) 3399.62 0.119291
\(934\) 10984.7 0.384831
\(935\) 2838.65 0.0992876
\(936\) 0 0
\(937\) −6771.10 −0.236075 −0.118037 0.993009i \(-0.537660\pi\)
−0.118037 + 0.993009i \(0.537660\pi\)
\(938\) −1575.50 −0.0548420
\(939\) 15857.8 0.551119
\(940\) 14218.9 0.493372
\(941\) −36690.7 −1.27108 −0.635538 0.772070i \(-0.719222\pi\)
−0.635538 + 0.772070i \(0.719222\pi\)
\(942\) 17937.4 0.620417
\(943\) 6673.51 0.230455
\(944\) 36108.8 1.24496
\(945\) −550.376 −0.0189457
\(946\) 17004.0 0.584406
\(947\) 50861.2 1.74527 0.872634 0.488375i \(-0.162410\pi\)
0.872634 + 0.488375i \(0.162410\pi\)
\(948\) −24349.1 −0.834202
\(949\) 0 0
\(950\) 48272.6 1.64860
\(951\) −14346.5 −0.489187
\(952\) 795.219 0.0270727
\(953\) 11855.6 0.402980 0.201490 0.979491i \(-0.435422\pi\)
0.201490 + 0.979491i \(0.435422\pi\)
\(954\) 25810.3 0.875931
\(955\) 12376.4 0.419361
\(956\) 42634.5 1.44236
\(957\) 15490.8 0.523245
\(958\) −21525.5 −0.725947
\(959\) 1378.04 0.0464019
\(960\) 5773.76 0.194112
\(961\) −28341.6 −0.951349
\(962\) 0 0
\(963\) −3651.08 −0.122175
\(964\) −43067.5 −1.43891
\(965\) −2663.90 −0.0888643
\(966\) −9836.56 −0.327625
\(967\) −40661.7 −1.35221 −0.676107 0.736803i \(-0.736334\pi\)
−0.676107 + 0.736803i \(0.736334\pi\)
\(968\) 1199.58 0.0398304
\(969\) 8760.93 0.290445
\(970\) 22063.9 0.730339
\(971\) 57318.3 1.89437 0.947184 0.320690i \(-0.103915\pi\)
0.947184 + 0.320690i \(0.103915\pi\)
\(972\) −2187.00 −0.0721688
\(973\) −668.324 −0.0220200
\(974\) −50403.3 −1.65814
\(975\) 0 0
\(976\) −38571.7 −1.26501
\(977\) 3026.73 0.0991134 0.0495567 0.998771i \(-0.484219\pi\)
0.0495567 + 0.998771i \(0.484219\pi\)
\(978\) −28973.4 −0.947308
\(979\) 34524.8 1.12709
\(980\) −8189.43 −0.266941
\(981\) 4315.64 0.140457
\(982\) −81032.3 −2.63324
\(983\) −33942.5 −1.10132 −0.550659 0.834730i \(-0.685624\pi\)
−0.550659 + 0.834730i \(0.685624\pi\)
\(984\) −693.729 −0.0224749
\(985\) −12649.3 −0.409179
\(986\) 19051.2 0.615327
\(987\) −10385.4 −0.334925
\(988\) 0 0
\(989\) −15216.3 −0.489231
\(990\) 3650.10 0.117179
\(991\) 21637.5 0.693580 0.346790 0.937943i \(-0.387272\pi\)
0.346790 + 0.937943i \(0.387272\pi\)
\(992\) 9889.02 0.316509
\(993\) 26009.9 0.831219
\(994\) 8520.95 0.271900
\(995\) −7332.53 −0.233625
\(996\) 18551.6 0.590190
\(997\) 19624.6 0.623388 0.311694 0.950182i \(-0.399104\pi\)
0.311694 + 0.950182i \(0.399104\pi\)
\(998\) −48296.3 −1.53186
\(999\) −8844.33 −0.280102
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 507.4.a.k.1.1 4
3.2 odd 2 1521.4.a.z.1.4 4
13.5 odd 4 507.4.b.e.337.4 4
13.6 odd 12 39.4.j.b.10.2 yes 4
13.8 odd 4 507.4.b.e.337.1 4
13.11 odd 12 39.4.j.b.4.2 4
13.12 even 2 inner 507.4.a.k.1.4 4
39.11 even 12 117.4.q.d.82.1 4
39.32 even 12 117.4.q.d.10.1 4
39.38 odd 2 1521.4.a.z.1.1 4
52.11 even 12 624.4.bv.c.433.2 4
52.19 even 12 624.4.bv.c.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.2 4 13.11 odd 12
39.4.j.b.10.2 yes 4 13.6 odd 12
117.4.q.d.10.1 4 39.32 even 12
117.4.q.d.82.1 4 39.11 even 12
507.4.a.k.1.1 4 1.1 even 1 trivial
507.4.a.k.1.4 4 13.12 even 2 inner
507.4.b.e.337.1 4 13.8 odd 4
507.4.b.e.337.4 4 13.5 odd 4
624.4.bv.c.49.1 4 52.19 even 12
624.4.bv.c.433.2 4 52.11 even 12
1521.4.a.z.1.1 4 39.38 odd 2
1521.4.a.z.1.4 4 3.2 odd 2