Properties

Label 507.4.a.k.1.4
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 1.4
Root \(4.29360\) 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 \(-0.240046\pi\)
0.728869 + 0.684653i \(0.240046\pi\)
\(3\) −3.00000 −0.577350
\(4\) 9.00000 1.12500
\(5\) −3.05006 −0.272806 −0.136403 0.990653i \(-0.543554\pi\)
−0.136403 + 0.990653i \(0.543554\pi\)
\(6\) −12.3693 −0.841625
\(7\) −6.68324 −0.360861 −0.180431 0.983588i \(-0.557749\pi\)
−0.180431 + 0.983588i \(0.557749\pi\)
\(8\) 4.12311 0.182217
\(9\) 9.00000 0.333333
\(10\) −12.5757 −0.397679
\(11\) 32.2500 0.883975 0.441988 0.897021i \(-0.354273\pi\)
0.441988 + 0.897021i \(0.354273\pi\)
\(12\) −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 \(-0.709206\pi\)
−0.610933 + 0.791682i \(0.709206\pi\)
\(20\) −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 \(-0.535176\pi\)
−0.110285 + 0.993900i \(0.535176\pi\)
\(32\) −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 \(-0.759423\pi\)
−0.727727 + 0.685866i \(0.759423\pi\)
\(38\) −417.233 −1.78116
\(39\) 0 0
\(40\) −12.5757 −0.0497099
\(41\) 56.0846 0.213633 0.106816 0.994279i \(-0.465934\pi\)
0.106816 + 0.994279i \(0.465934\pi\)
\(42\) 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 \(-0.797183\pi\)
−0.803783 + 0.594923i \(0.797183\pi\)
\(48\) 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 \(-0.242145\pi\)
0.724339 + 0.689444i \(0.242145\pi\)
\(60\) 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 \(-0.516600\pi\)
−0.0521271 + 0.998640i \(0.516600\pi\)
\(68\) −259.727 −0.463184
\(69\) 356.970 0.622814
\(70\) 84.0465 0.143507
\(71\) 309.226 0.516879 0.258440 0.966027i \(-0.416792\pi\)
0.258440 + 0.966027i \(0.416792\pi\)
\(72\) 37.1080 0.0607391
\(73\) −389.711 −0.624826 −0.312413 0.949946i \(-0.601137\pi\)
−0.312413 + 0.949946i \(0.601137\pi\)
\(74\) −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 \(-0.349879\pi\)
0.454328 + 0.890834i \(0.349879\pi\)
\(84\) 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 \(-0.279964\pi\)
0.637510 + 0.770442i \(0.279964\pi\)
\(90\) −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 \(-0.129601\pi\)
0.918251 + 0.395998i \(0.129601\pi\)
\(98\) −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 \(-0.567569\pi\)
−0.210685 + 0.977554i \(0.567569\pi\)
\(110\) −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 \(-0.520479\pi\)
−0.0642932 + 0.997931i \(0.520479\pi\)
\(138\) 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 \(-0.466647\pi\)
0.104590 + 0.994515i \(0.466647\pi\)
\(150\) 1431.09 0.778989
\(151\) −1517.45 −0.817805 −0.408902 0.912578i \(-0.634088\pi\)
−0.408902 + 0.912578i \(0.634088\pi\)
\(152\) −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 \(-0.309730\pi\)
0.562785 + 0.826603i \(0.309730\pi\)
\(164\) 504.762 0.240337
\(165\) 295.093 0.139230
\(166\) 2832.97 1.32458
\(167\) −40.0731 −0.0185686 −0.00928428 0.999957i \(-0.502955\pi\)
−0.00928428 + 0.999957i \(0.502955\pi\)
\(168\) 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 \(-0.447925\pi\)
0.162871 + 0.986647i \(0.447925\pi\)
\(194\) 7233.92 2.67714
\(195\) 0 0
\(196\) −2685.01 −0.978502
\(197\) 4147.25 1.49989 0.749947 0.661498i \(-0.230079\pi\)
0.749947 + 0.661498i \(0.230079\pi\)
\(198\) 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 \(-0.361173\pi\)
0.422443 + 0.906390i \(0.361173\pi\)
\(224\) 1736.01 0.517822
\(225\) −1041.27 −0.308526
\(226\) −6379.29 −1.87763
\(227\) −4518.37 −1.32112 −0.660561 0.750772i \(-0.729682\pi\)
−0.660561 + 0.750772i \(0.729682\pi\)
\(228\) 2732.23 0.793625
\(229\) −1305.27 −0.376658 −0.188329 0.982106i \(-0.560307\pi\)
−0.188329 + 0.982106i \(0.560307\pi\)
\(230\) 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 \(-0.721501\pi\)
−0.641050 + 0.767499i \(0.721501\pi\)
\(240\) −503.260 −0.135355
\(241\) 4785.28 1.27903 0.639516 0.768778i \(-0.279135\pi\)
0.639516 + 0.768778i \(0.279135\pi\)
\(242\) −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 \(-0.117460\pi\)
0.932684 + 0.360695i \(0.117460\pi\)
\(272\) 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 \(-0.596694\pi\)
−0.299123 + 0.954214i \(0.596694\pi\)
\(282\) 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 \(-0.359240\pi\)
0.427938 + 0.903808i \(0.359240\pi\)
\(294\) 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 \(-0.273464\pi\)
0.653110 + 0.757263i \(0.273464\pi\)
\(308\) −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 \(-0.639251\pi\)
−0.423648 + 0.905827i \(0.639251\pi\)
\(318\) 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 \(-0.244207\pi\)
0.719857 + 0.694122i \(0.244207\pi\)
\(332\) 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 \(-0.741170\pi\)
−0.687222 + 0.726447i \(0.741170\pi\)
\(350\) 3188.12 0.486892
\(351\) 0 0
\(352\) −8377.11 −1.26847
\(353\) −5357.99 −0.807868 −0.403934 0.914788i \(-0.632357\pi\)
−0.403934 + 0.914788i \(0.632357\pi\)
\(354\) −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 \(-0.563726\pi\)
−0.198866 + 0.980027i \(0.563726\pi\)
\(360\) −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 \(-0.550209\pi\)
−0.157083 + 0.987585i \(0.550209\pi\)
\(380\) 2777.82 0.374998
\(381\) 7485.46 1.00654
\(382\) 16730.5 2.24086
\(383\) 1983.34 0.264606 0.132303 0.991209i \(-0.457763\pi\)
0.132303 + 0.991209i \(0.457763\pi\)
\(384\) 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 \(-0.423622\pi\)
0.237651 + 0.971351i \(0.423622\pi\)
\(398\) −9912.20 −1.24838
\(399\) −2028.91 −0.254568
\(400\) 6363.34 0.795418
\(401\) −1997.55 −0.248760 −0.124380 0.992235i \(-0.539694\pi\)
−0.124380 + 0.992235i \(0.539694\pi\)
\(402\) 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 \(-0.601684\pi\)
−0.314044 + 0.949409i \(0.601684\pi\)
\(410\) −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 \(-0.356288\pi\)
0.436300 + 0.899801i \(0.356288\pi\)
\(422\) 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 \(-0.769477\pi\)
−0.749023 + 0.662544i \(0.769477\pi\)
\(432\) 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 \(-0.857089\pi\)
−0.900895 + 0.434037i \(0.857089\pi\)
\(450\) −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 \(-0.243601\pi\)
0.721177 + 0.692750i \(0.243601\pi\)
\(458\) −5381.76 −0.549068
\(459\) 779.181 0.0792355
\(460\) 3266.34 0.331074
\(461\) −2922.22 −0.295231 −0.147616 0.989045i \(-0.547160\pi\)
−0.147616 + 0.989045i \(0.547160\pi\)
\(462\) 2666.01 0.268472
\(463\) −2072.61 −0.208040 −0.104020 0.994575i \(-0.533171\pi\)
−0.104020 + 0.994575i \(0.533171\pi\)
\(464\) −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 \(-0.580101\pi\)
−0.248998 + 0.968504i \(0.580101\pi\)
\(480\) −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 \(-0.692568\pi\)
−0.568737 + 0.822520i \(0.692568\pi\)
\(488\) 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 \(-0.676093\pi\)
−0.525422 + 0.850842i \(0.676093\pi\)
\(500\) 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 \(-0.574535\pi\)
−0.232024 + 0.972710i \(0.574535\pi\)
\(510\) −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 \(-0.566052\pi\)
−0.206022 + 0.978547i \(0.566052\pi\)
\(542\) 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 \(-0.777970\pi\)
−0.766432 + 0.642326i \(0.777970\pi\)
\(558\) −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 \(-0.445928\pi\)
0.169056 + 0.985606i \(0.445928\pi\)
\(578\) −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 \(-0.360250\pi\)
0.425069 + 0.905161i \(0.360250\pi\)
\(588\) 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 \(-0.431853\pi\)
0.212457 + 0.977170i \(0.431853\pi\)
\(594\) −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 \(-0.508437\pi\)
−0.0265030 + 0.999649i \(0.508437\pi\)
\(614\) 28970.0 1.90413
\(615\) 513.184 0.0336481
\(616\) −888.671 −0.0581259
\(617\) 15218.3 0.992973 0.496486 0.868044i \(-0.334623\pi\)
0.496486 + 0.868044i \(0.334623\pi\)
\(618\) 8573.83 0.558074
\(619\) −11462.5 −0.744291 −0.372145 0.928174i \(-0.621378\pi\)
−0.372145 + 0.928174i \(0.621378\pi\)
\(620\) 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 \(-0.545020\pi\)
−0.140963 + 0.990015i \(0.545020\pi\)
\(632\) 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 \(-0.280608\pi\)
0.635951 + 0.771729i \(0.280608\pi\)
\(644\) 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 \(-0.521021\pi\)
−0.0659915 + 0.997820i \(0.521021\pi\)
\(662\) 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 \(-0.270412\pi\)
0.660340 + 0.750967i \(0.270412\pi\)
\(684\) −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 \(-0.612219\pi\)
−0.345288 + 0.938497i \(0.612219\pi\)
\(692\) 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 \(-0.685126\pi\)
−0.549353 + 0.835590i \(0.685126\pi\)
\(710\) −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 \(-0.530866\pi\)
−0.0968160 + 0.995302i \(0.530866\pi\)
\(734\) 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 \(-0.242988\pi\)
0.722511 + 0.691359i \(0.242988\pi\)
\(740\) 8991.91 0.446688
\(741\) 0 0
\(742\) 19166.3 0.948269
\(743\) −34996.7 −1.72800 −0.864000 0.503492i \(-0.832048\pi\)
−0.864000 + 0.503492i \(0.832048\pi\)
\(744\) 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 \(-0.332511\pi\)
0.502236 + 0.864731i \(0.332511\pi\)
\(762\) 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 \(-0.348615\pi\)
0.457864 + 0.889022i \(0.348615\pi\)
\(770\) 2710.50 0.126857
\(771\) 19637.4 0.917283
\(772\) 7860.55 0.366460
\(773\) −29352.0 −1.36574 −0.682870 0.730540i \(-0.739269\pi\)
−0.682870 + 0.730540i \(0.739269\pi\)
\(774\) 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 \(-0.467772\pi\)
0.101076 + 0.994879i \(0.467772\pi\)
\(788\) 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 \(-0.462024\pi\)
0.119024 + 0.992891i \(0.462024\pi\)
\(812\) −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 \(-0.350411\pi\)
0.452839 + 0.891592i \(0.350411\pi\)
\(822\) 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 \(-0.465603\pi\)
0.107851 + 0.994167i \(0.465603\pi\)
\(828\) −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 \(-0.628945\pi\)
−0.394103 + 0.919066i \(0.628945\pi\)
\(840\) −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 \(-0.618168\pi\)
−0.362766 + 0.931880i \(0.618168\pi\)
\(854\) −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 \(-0.625161\pi\)
−0.383152 + 0.923685i \(0.625161\pi\)
\(864\) 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 \(-0.407569\pi\)
0.286316 + 0.958135i \(0.407569\pi\)
\(878\) 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 \(-0.792975\pi\)
−0.795849 + 0.605495i \(0.792975\pi\)
\(930\) −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 \(-0.280778\pi\)
0.635538 + 0.772070i \(0.280778\pi\)
\(942\) −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 \(-0.837590\pi\)
−0.872634 + 0.488375i \(0.837590\pi\)
\(948\) −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 \(-0.263666\pi\)
0.676107 + 0.736803i \(0.263666\pi\)
\(968\) −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 \(-0.515781\pi\)
−0.0495567 + 0.998771i \(0.515781\pi\)
\(978\) −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 \(-0.314376\pi\)
0.550659 + 0.834730i \(0.314376\pi\)
\(984\) −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.1.4 4
3.2 odd 2 1521.4.a.z.1.1 4
13.2 odd 12 39.4.j.b.4.2 4
13.5 odd 4 507.4.b.e.337.1 4
13.7 odd 12 39.4.j.b.10.2 yes 4
13.8 odd 4 507.4.b.e.337.4 4
13.12 even 2 inner 507.4.a.k.1.1 4
39.2 even 12 117.4.q.d.82.1 4
39.20 even 12 117.4.q.d.10.1 4
39.38 odd 2 1521.4.a.z.1.4 4
52.7 even 12 624.4.bv.c.49.1 4
52.15 even 12 624.4.bv.c.433.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.2 4 13.2 odd 12
39.4.j.b.10.2 yes 4 13.7 odd 12
117.4.q.d.10.1 4 39.20 even 12
117.4.q.d.82.1 4 39.2 even 12
507.4.a.k.1.1 4 13.12 even 2 inner
507.4.a.k.1.4 4 1.1 even 1 trivial
507.4.b.e.337.1 4 13.5 odd 4
507.4.b.e.337.4 4 13.8 odd 4
624.4.bv.c.49.1 4 52.7 even 12
624.4.bv.c.433.2 4 52.15 even 12
1521.4.a.z.1.1 4 3.2 odd 2
1521.4.a.z.1.4 4 39.38 odd 2