Properties

Label 1521.4.a.z.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.12311 q^{2} +9.00000 q^{4} -3.05006 q^{5} +6.68324 q^{7} +4.12311 q^{8} +O(q^{10})\) \(q+4.12311 q^{2} +9.00000 q^{4} -3.05006 q^{5} +6.68324 q^{7} +4.12311 q^{8} -12.5757 q^{10} +32.2500 q^{11} +27.5557 q^{14} -55.0000 q^{16} +28.8586 q^{17} +101.194 q^{19} -27.4505 q^{20} +132.970 q^{22} +118.990 q^{23} -115.697 q^{25} +60.1492 q^{28} -160.111 q^{29} +38.0705 q^{31} -259.756 q^{32} +118.987 q^{34} -20.3843 q^{35} +327.568 q^{37} +417.233 q^{38} -12.5757 q^{40} +56.0846 q^{41} +127.879 q^{43} +290.250 q^{44} +490.608 q^{46} -517.983 q^{47} -298.334 q^{49} -477.032 q^{50} +695.546 q^{53} -98.3643 q^{55} +27.5557 q^{56} -660.156 q^{58} +656.523 q^{59} +701.304 q^{61} +156.969 q^{62} -631.000 q^{64} +57.1750 q^{67} +259.727 q^{68} -84.0465 q^{70} +309.226 q^{71} +389.711 q^{73} +1350.60 q^{74} +910.744 q^{76} +215.534 q^{77} +901.820 q^{79} +167.753 q^{80} +231.243 q^{82} +687.095 q^{83} -88.0203 q^{85} +527.257 q^{86} +132.970 q^{88} +1070.54 q^{89} +1070.91 q^{92} -2135.70 q^{94} -308.647 q^{95} -1754.48 q^{97} -1230.06 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) 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 \(0.240046\pi\)
\(3\) 0 0
\(4\) 9.00000 1.12500
\(5\) −3.05006 −0.272806 −0.136403 0.990653i \(-0.543554\pi\)
−0.136403 + 0.990653i \(0.543554\pi\)
\(6\) 0 0
\(7\) 6.68324 0.360861 0.180431 0.983588i \(-0.442251\pi\)
0.180431 + 0.983588i \(0.442251\pi\)
\(8\) 4.12311 0.182217
\(9\) 0 0
\(10\) −12.5757 −0.397679
\(11\) 32.2500 0.883975 0.441988 0.897021i \(-0.354273\pi\)
0.441988 + 0.897021i \(0.354273\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 27.5557 0.526041
\(15\) 0 0
\(16\) −55.0000 −0.859375
\(17\) 28.8586 0.411720 0.205860 0.978582i \(-0.434001\pi\)
0.205860 + 0.978582i \(0.434001\pi\)
\(18\) 0 0
\(19\) 101.194 1.22187 0.610933 0.791682i \(-0.290794\pi\)
0.610933 + 0.791682i \(0.290794\pi\)
\(20\) −27.4505 −0.306906
\(21\) 0 0
\(22\) 132.970 1.28860
\(23\) 118.990 1.07874 0.539372 0.842067i \(-0.318662\pi\)
0.539372 + 0.842067i \(0.318662\pi\)
\(24\) 0 0
\(25\) −115.697 −0.925577
\(26\) 0 0
\(27\) 0 0
\(28\) 60.1492 0.405969
\(29\) −160.111 −1.02524 −0.512620 0.858616i \(-0.671325\pi\)
−0.512620 + 0.858616i \(0.671325\pi\)
\(30\) 0 0
\(31\) 38.0705 0.220570 0.110285 0.993900i \(-0.464824\pi\)
0.110285 + 0.993900i \(0.464824\pi\)
\(32\) −259.756 −1.43496
\(33\) 0 0
\(34\) 118.987 0.600179
\(35\) −20.3843 −0.0984449
\(36\) 0 0
\(37\) 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 \(-0.465934\pi\)
0.106816 + 0.994279i \(0.465934\pi\)
\(42\) 0 0
\(43\) 127.879 0.453519 0.226759 0.973951i \(-0.427187\pi\)
0.226759 + 0.973951i \(0.427187\pi\)
\(44\) 290.250 0.994472
\(45\) 0 0
\(46\) 490.608 1.57253
\(47\) −517.983 −1.60757 −0.803783 0.594923i \(-0.797183\pi\)
−0.803783 + 0.594923i \(0.797183\pi\)
\(48\) 0 0
\(49\) −298.334 −0.869779
\(50\) −477.032 −1.34925
\(51\) 0 0
\(52\) 0 0
\(53\) 695.546 1.80265 0.901326 0.433141i \(-0.142595\pi\)
0.901326 + 0.433141i \(0.142595\pi\)
\(54\) 0 0
\(55\) −98.3643 −0.241153
\(56\) 27.5557 0.0657551
\(57\) 0 0
\(58\) −660.156 −1.49453
\(59\) 656.523 1.44868 0.724339 0.689444i \(-0.242145\pi\)
0.724339 + 0.689444i \(0.242145\pi\)
\(60\) 0 0
\(61\) 701.304 1.47201 0.736007 0.676974i \(-0.236709\pi\)
0.736007 + 0.676974i \(0.236709\pi\)
\(62\) 156.969 0.321533
\(63\) 0 0
\(64\) −631.000 −1.23242
\(65\) 0 0
\(66\) 0 0
\(67\) 57.1750 0.104254 0.0521271 0.998640i \(-0.483400\pi\)
0.0521271 + 0.998640i \(0.483400\pi\)
\(68\) 259.727 0.463184
\(69\) 0 0
\(70\) −84.0465 −0.143507
\(71\) 309.226 0.516879 0.258440 0.966027i \(-0.416792\pi\)
0.258440 + 0.966027i \(0.416792\pi\)
\(72\) 0 0
\(73\) 389.711 0.624826 0.312413 0.949946i \(-0.398863\pi\)
0.312413 + 0.949946i \(0.398863\pi\)
\(74\) 1350.60 2.12167
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 231.243 0.311421
\(83\) 687.095 0.908657 0.454328 0.890834i \(-0.349879\pi\)
0.454328 + 0.890834i \(0.349879\pi\)
\(84\) 0 0
\(85\) −88.0203 −0.112319
\(86\) 527.257 0.661111
\(87\) 0 0
\(88\) 132.970 0.161076
\(89\) 1070.54 1.27502 0.637510 0.770442i \(-0.279964\pi\)
0.637510 + 0.770442i \(0.279964\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1070.91 1.21359
\(93\) 0 0
\(94\) −2135.70 −2.34341
\(95\) −308.647 −0.333332
\(96\) 0 0
\(97\) −1754.48 −1.83650 −0.918251 0.395998i \(-0.870399\pi\)
−0.918251 + 0.395998i \(0.870399\pi\)
\(98\) −1230.06 −1.26791
\(99\) 0 0
\(100\) −1041.27 −1.04127
\(101\) 640.840 0.631346 0.315673 0.948868i \(-0.397770\pi\)
0.315673 + 0.948868i \(0.397770\pi\)
\(102\) 0 0
\(103\) −693.153 −0.663091 −0.331546 0.943439i \(-0.607570\pi\)
−0.331546 + 0.943439i \(0.607570\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 2867.81 2.62779
\(107\) 405.676 0.366525 0.183262 0.983064i \(-0.441334\pi\)
0.183262 + 0.983064i \(0.441334\pi\)
\(108\) 0 0
\(109\) 479.516 0.421370 0.210685 0.977554i \(-0.432431\pi\)
0.210685 + 0.977554i \(0.432431\pi\)
\(110\) −405.566 −0.351538
\(111\) 0 0
\(112\) −367.578 −0.310115
\(113\) 1547.20 1.28804 0.644021 0.765008i \(-0.277265\pi\)
0.644021 + 0.765008i \(0.277265\pi\)
\(114\) 0 0
\(115\) −362.926 −0.294288
\(116\) −1441.00 −1.15339
\(117\) 0 0
\(118\) 2706.91 2.11179
\(119\) 192.869 0.148574
\(120\) 0 0
\(121\) −290.940 −0.218588
\(122\) 2891.55 2.14581
\(123\) 0 0
\(124\) 342.634 0.248141
\(125\) 734.140 0.525308
\(126\) 0 0
\(127\) −2495.15 −1.74338 −0.871690 0.490059i \(-0.836975\pi\)
−0.871690 + 0.490059i \(0.836975\pi\)
\(128\) −523.634 −0.361587
\(129\) 0 0
\(130\) 0 0
\(131\) −43.7571 −0.0291838 −0.0145919 0.999894i \(-0.504645\pi\)
−0.0145919 + 0.999894i \(0.504645\pi\)
\(132\) 0 0
\(133\) 676.303 0.440924
\(134\) 235.739 0.151975
\(135\) 0 0
\(136\) 118.987 0.0750224
\(137\) −206.194 −0.128586 −0.0642932 0.997931i \(-0.520479\pi\)
−0.0642932 + 0.997931i \(0.520479\pi\)
\(138\) 0 0
\(139\) −100.000 −0.0610208 −0.0305104 0.999534i \(-0.509713\pi\)
−0.0305104 + 0.999534i \(0.509713\pi\)
\(140\) −183.459 −0.110751
\(141\) 0 0
\(142\) 1274.97 0.753474
\(143\) 0 0
\(144\) 0 0
\(145\) 488.349 0.279691
\(146\) 1606.82 0.910832
\(147\) 0 0
\(148\) 2948.11 1.63739
\(149\) 380.451 0.209179 0.104590 0.994515i \(-0.466647\pi\)
0.104590 + 0.994515i \(0.466647\pi\)
\(150\) 0 0
\(151\) 1517.45 0.817805 0.408902 0.912578i \(-0.365912\pi\)
0.408902 + 0.912578i \(0.365912\pi\)
\(152\) 417.233 0.222645
\(153\) 0 0
\(154\) 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
\(159\) 0 0
\(160\) 792.270 0.391465
\(161\) 795.239 0.389277
\(162\) 0 0
\(163\) −2342.36 −1.12557 −0.562785 0.826603i \(-0.690270\pi\)
−0.562785 + 0.826603i \(0.690270\pi\)
\(164\) 504.762 0.240337
\(165\) 0 0
\(166\) 2832.97 1.32458
\(167\) −40.0731 −0.0185686 −0.00928428 0.999957i \(-0.502955\pi\)
−0.00928428 + 0.999957i \(0.502955\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −362.917 −0.163732
\(171\) 0 0
\(172\) 1150.91 0.510208
\(173\) −1909.54 −0.839189 −0.419594 0.907712i \(-0.637828\pi\)
−0.419594 + 0.907712i \(0.637828\pi\)
\(174\) 0 0
\(175\) −773.232 −0.334005
\(176\) −1773.75 −0.759666
\(177\) 0 0
\(178\) 4413.94 1.85864
\(179\) 509.959 0.212939 0.106470 0.994316i \(-0.466045\pi\)
0.106470 + 0.994316i \(0.466045\pi\)
\(180\) 0 0
\(181\) −2136.88 −0.877531 −0.438766 0.898602i \(-0.644584\pi\)
−0.438766 + 0.898602i \(0.644584\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 490.608 0.196566
\(185\) −999.101 −0.397056
\(186\) 0 0
\(187\) 930.688 0.363950
\(188\) −4661.85 −1.80851
\(189\) 0 0
\(190\) −1272.58 −0.485911
\(191\) −4057.74 −1.53721 −0.768607 0.639721i \(-0.779050\pi\)
−0.768607 + 0.639721i \(0.779050\pi\)
\(192\) 0 0
\(193\) −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 \(-0.230079\pi\)
0.749947 + 0.661498i \(0.230079\pi\)
\(198\) 0 0
\(199\) −2404.06 −0.856379 −0.428189 0.903689i \(-0.640848\pi\)
−0.428189 + 0.903689i \(0.640848\pi\)
\(200\) −477.032 −0.168656
\(201\) 0 0
\(202\) 2642.25 0.920337
\(203\) −1070.06 −0.369969
\(204\) 0 0
\(205\) −171.061 −0.0582802
\(206\) −2857.94 −0.966613
\(207\) 0 0
\(208\) 0 0
\(209\) 3263.50 1.08010
\(210\) 0 0
\(211\) 3868.85 1.26229 0.631144 0.775665i \(-0.282586\pi\)
0.631144 + 0.775665i \(0.282586\pi\)
\(212\) 6259.91 2.02798
\(213\) 0 0
\(214\) 1672.64 0.534297
\(215\) −390.037 −0.123722
\(216\) 0 0
\(217\) 254.434 0.0795950
\(218\) 1977.09 0.614246
\(219\) 0 0
\(220\) −885.279 −0.271298
\(221\) 0 0
\(222\) 0 0
\(223\) −2813.55 −0.844885 −0.422443 0.906390i \(-0.638827\pi\)
−0.422443 + 0.906390i \(0.638827\pi\)
\(224\) −1736.01 −0.517822
\(225\) 0 0
\(226\) 6379.29 1.87763
\(227\) −4518.37 −1.32112 −0.660561 0.750772i \(-0.729682\pi\)
−0.660561 + 0.750772i \(0.729682\pi\)
\(228\) 0 0
\(229\) 1305.27 0.376658 0.188329 0.982106i \(-0.439693\pi\)
0.188329 + 0.982106i \(0.439693\pi\)
\(230\) −1496.38 −0.428994
\(231\) 0 0
\(232\) −660.156 −0.186816
\(233\) 3360.55 0.944879 0.472440 0.881363i \(-0.343373\pi\)
0.472440 + 0.881363i \(0.343373\pi\)
\(234\) 0 0
\(235\) 1579.88 0.438553
\(236\) 5908.71 1.62976
\(237\) 0 0
\(238\) 795.219 0.216581
\(239\) −4737.17 −1.28210 −0.641050 0.767499i \(-0.721501\pi\)
−0.641050 + 0.767499i \(0.721501\pi\)
\(240\) 0 0
\(241\) −4785.28 −1.27903 −0.639516 0.768778i \(-0.720865\pi\)
−0.639516 + 0.768778i \(0.720865\pi\)
\(242\) −1199.58 −0.318643
\(243\) 0 0
\(244\) 6311.74 1.65601
\(245\) 909.937 0.237281
\(246\) 0 0
\(247\) 0 0
\(248\) 156.969 0.0401916
\(249\) 0 0
\(250\) 3026.94 0.765762
\(251\) 3273.86 0.823284 0.411642 0.911346i \(-0.364955\pi\)
0.411642 + 0.911346i \(0.364955\pi\)
\(252\) 0 0
\(253\) 3837.42 0.953584
\(254\) −10287.8 −2.54139
\(255\) 0 0
\(256\) 2889.00 0.705322
\(257\) 6545.81 1.58878 0.794390 0.607408i \(-0.207791\pi\)
0.794390 + 0.607408i \(0.207791\pi\)
\(258\) 0 0
\(259\) 2189.22 0.525217
\(260\) 0 0
\(261\) 0 0
\(262\) −180.415 −0.0425424
\(263\) −88.2014 −0.0206796 −0.0103398 0.999947i \(-0.503291\pi\)
−0.0103398 + 0.999947i \(0.503291\pi\)
\(264\) 0 0
\(265\) −2121.46 −0.491773
\(266\) 2788.47 0.642752
\(267\) 0 0
\(268\) 514.575 0.117286
\(269\) 4527.60 1.02622 0.513109 0.858324i \(-0.328494\pi\)
0.513109 + 0.858324i \(0.328494\pi\)
\(270\) 0 0
\(271\) −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
\(276\) 0 0
\(277\) −2881.31 −0.624986 −0.312493 0.949920i \(-0.601164\pi\)
−0.312493 + 0.949920i \(0.601164\pi\)
\(278\) −412.311 −0.0889523
\(279\) 0 0
\(280\) −84.0465 −0.0179384
\(281\) −2817.99 −0.598247 −0.299123 0.954214i \(-0.596694\pi\)
−0.299123 + 0.954214i \(0.596694\pi\)
\(282\) 0 0
\(283\) 264.601 0.0555792 0.0277896 0.999614i \(-0.491153\pi\)
0.0277896 + 0.999614i \(0.491153\pi\)
\(284\) 2783.04 0.581489
\(285\) 0 0
\(286\) 0 0
\(287\) 374.827 0.0770918
\(288\) 0 0
\(289\) −4080.18 −0.830487
\(290\) 2013.52 0.407716
\(291\) 0 0
\(292\) 3507.40 0.702929
\(293\) 4292.52 0.855877 0.427938 0.903808i \(-0.359240\pi\)
0.427938 + 0.903808i \(0.359240\pi\)
\(294\) 0 0
\(295\) −2002.43 −0.395208
\(296\) 1350.60 0.265209
\(297\) 0 0
\(298\) 1568.64 0.304929
\(299\) 0 0
\(300\) 0 0
\(301\) 854.643 0.163657
\(302\) 6256.62 1.19214
\(303\) 0 0
\(304\) −5565.66 −1.05004
\(305\) −2139.02 −0.401573
\(306\) 0 0
\(307\) −7026.26 −1.30622 −0.653110 0.757263i \(-0.726536\pi\)
−0.653110 + 0.757263i \(0.726536\pi\)
\(308\) 1939.81 0.358866
\(309\) 0 0
\(310\) −478.763 −0.0877159
\(311\) 1133.21 0.206618 0.103309 0.994649i \(-0.467057\pi\)
0.103309 + 0.994649i \(0.467057\pi\)
\(312\) 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
\(315\) 0 0
\(316\) 8116.38 1.44488
\(317\) −4782.16 −0.847296 −0.423648 0.905827i \(-0.639251\pi\)
−0.423648 + 0.905827i \(0.639251\pi\)
\(318\) 0 0
\(319\) −5163.59 −0.906286
\(320\) 1924.59 0.336212
\(321\) 0 0
\(322\) 3278.85 0.567464
\(323\) 2920.31 0.503066
\(324\) 0 0
\(325\) 0 0
\(326\) −9657.80 −1.64079
\(327\) 0 0
\(328\) 231.243 0.0389276
\(329\) −3461.81 −0.580108
\(330\) 0 0
\(331\) −8669.98 −1.43971 −0.719857 0.694122i \(-0.755793\pi\)
−0.719857 + 0.694122i \(0.755793\pi\)
\(332\) 6183.86 1.02224
\(333\) 0 0
\(334\) −165.226 −0.0270681
\(335\) −174.387 −0.0284412
\(336\) 0 0
\(337\) 8526.59 1.37826 0.689129 0.724639i \(-0.257993\pi\)
0.689129 + 0.724639i \(0.257993\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −792.183 −0.126359
\(341\) 1227.77 0.194978
\(342\) 0 0
\(343\) −4286.19 −0.674731
\(344\) 527.257 0.0826389
\(345\) 0 0
\(346\) −7873.23 −1.22332
\(347\) 12581.5 1.94643 0.973213 0.229907i \(-0.0738421\pi\)
0.973213 + 0.229907i \(0.0738421\pi\)
\(348\) 0 0
\(349\) 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 \(-0.632357\pi\)
−0.403934 + 0.914788i \(0.632357\pi\)
\(354\) 0 0
\(355\) −943.159 −0.141007
\(356\) 9634.84 1.43440
\(357\) 0 0
\(358\) 2102.61 0.310409
\(359\) −2705.40 −0.397731 −0.198866 0.980027i \(-0.563726\pi\)
−0.198866 + 0.980027i \(0.563726\pi\)
\(360\) 0 0
\(361\) 3381.19 0.492957
\(362\) −8810.59 −1.27921
\(363\) 0 0
\(364\) 0 0
\(365\) −1188.64 −0.170456
\(366\) 0 0
\(367\) 10473.8 1.48972 0.744858 0.667223i \(-0.232517\pi\)
0.744858 + 0.667223i \(0.232517\pi\)
\(368\) −6544.45 −0.927046
\(369\) 0 0
\(370\) −4119.40 −0.578804
\(371\) 4648.50 0.650507
\(372\) 0 0
\(373\) −12763.0 −1.77170 −0.885850 0.463973i \(-0.846424\pi\)
−0.885850 + 0.463973i \(0.846424\pi\)
\(374\) 3837.32 0.530544
\(375\) 0 0
\(376\) −2135.70 −0.292926
\(377\) 0 0
\(378\) 0 0
\(379\) 2318.02 0.314166 0.157083 0.987585i \(-0.449791\pi\)
0.157083 + 0.987585i \(0.449791\pi\)
\(380\) −2777.82 −0.374998
\(381\) 0 0
\(382\) −16730.5 −2.24086
\(383\) 1983.34 0.264606 0.132303 0.991209i \(-0.457763\pi\)
0.132303 + 0.991209i \(0.457763\pi\)
\(384\) 0 0
\(385\) −657.392 −0.0870229
\(386\) −3601.10 −0.474847
\(387\) 0 0
\(388\) −15790.3 −2.06607
\(389\) 3244.51 0.422887 0.211444 0.977390i \(-0.432184\pi\)
0.211444 + 0.977390i \(0.432184\pi\)
\(390\) 0 0
\(391\) 3433.88 0.444140
\(392\) −1230.06 −0.158489
\(393\) 0 0
\(394\) 17099.5 2.18645
\(395\) −2750.60 −0.350374
\(396\) 0 0
\(397\) −3759.72 −0.475302 −0.237651 0.971351i \(-0.576378\pi\)
−0.237651 + 0.971351i \(0.576378\pi\)
\(398\) −9912.20 −1.24838
\(399\) 0 0
\(400\) 6363.34 0.795418
\(401\) −1997.55 −0.248760 −0.124380 0.992235i \(-0.539694\pi\)
−0.124380 + 0.992235i \(0.539694\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 5767.56 0.710264
\(405\) 0 0
\(406\) −4411.98 −0.539318
\(407\) 10564.1 1.28659
\(408\) 0 0
\(409\) 5195.23 0.628087 0.314044 0.949409i \(-0.398316\pi\)
0.314044 + 0.949409i \(0.398316\pi\)
\(410\) −705.304 −0.0849573
\(411\) 0 0
\(412\) −6238.38 −0.745977
\(413\) 4387.70 0.522772
\(414\) 0 0
\(415\) −2095.68 −0.247887
\(416\) 0 0
\(417\) 0 0
\(418\) 13455.7 1.57450
\(419\) 6822.11 0.795422 0.397711 0.917511i \(-0.369805\pi\)
0.397711 + 0.917511i \(0.369805\pi\)
\(420\) 0 0
\(421\) −7537.70 −0.872601 −0.436300 0.899801i \(-0.643712\pi\)
−0.436300 + 0.899801i \(0.643712\pi\)
\(422\) 15951.7 1.84009
\(423\) 0 0
\(424\) 2867.81 0.328474
\(425\) −3338.85 −0.381078
\(426\) 0 0
\(427\) 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 \(0.769477\pi\)
\(432\) 0 0
\(433\) −17715.9 −1.96622 −0.983110 0.183014i \(-0.941415\pi\)
−0.983110 + 0.183014i \(0.941415\pi\)
\(434\) 1049.06 0.116029
\(435\) 0 0
\(436\) 4315.64 0.474041
\(437\) 12041.1 1.31808
\(438\) 0 0
\(439\) 7163.47 0.778801 0.389401 0.921068i \(-0.372682\pi\)
0.389401 + 0.921068i \(0.372682\pi\)
\(440\) −405.566 −0.0439423
\(441\) 0 0
\(442\) 0 0
\(443\) 10169.2 1.09064 0.545321 0.838227i \(-0.316408\pi\)
0.545321 + 0.838227i \(0.316408\pi\)
\(444\) 0 0
\(445\) −3265.20 −0.347833
\(446\) −11600.6 −1.23162
\(447\) 0 0
\(448\) −4217.13 −0.444733
\(449\) −17142.5 −1.80179 −0.900895 0.434037i \(-0.857089\pi\)
−0.900895 + 0.434037i \(0.857089\pi\)
\(450\) 0 0
\(451\) 1808.73 0.188846
\(452\) 13924.8 1.44905
\(453\) 0 0
\(454\) −18629.7 −1.92585
\(455\) 0 0
\(456\) 0 0
\(457\) −14091.1 −1.44235 −0.721177 0.692750i \(-0.756399\pi\)
−0.721177 + 0.692750i \(0.756399\pi\)
\(458\) 5381.76 0.549068
\(459\) 0 0
\(460\) −3266.34 −0.331074
\(461\) −2922.22 −0.295231 −0.147616 0.989045i \(-0.547160\pi\)
−0.147616 + 0.989045i \(0.547160\pi\)
\(462\) 0 0
\(463\) 2072.61 0.208040 0.104020 0.994575i \(-0.466829\pi\)
0.104020 + 0.994575i \(0.466829\pi\)
\(464\) 8806.13 0.881065
\(465\) 0 0
\(466\) 13855.9 1.37739
\(467\) 2664.19 0.263992 0.131996 0.991250i \(-0.457861\pi\)
0.131996 + 0.991250i \(0.457861\pi\)
\(468\) 0 0
\(469\) 382.114 0.0376213
\(470\) 6514.01 0.639295
\(471\) 0 0
\(472\) 2706.91 0.263974
\(473\) 4124.08 0.400899
\(474\) 0 0
\(475\) −11707.8 −1.13093
\(476\) 1735.82 0.167145
\(477\) 0 0
\(478\) −19531.8 −1.86897
\(479\) −5220.70 −0.497995 −0.248998 0.968504i \(-0.580101\pi\)
−0.248998 + 0.968504i \(0.580101\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −19730.2 −1.86449
\(483\) 0 0
\(484\) −2618.46 −0.245911
\(485\) 5351.28 0.501008
\(486\) 0 0
\(487\) 12224.6 1.13747 0.568737 0.822520i \(-0.307432\pi\)
0.568737 + 0.822520i \(0.307432\pi\)
\(488\) 2891.55 0.268226
\(489\) 0 0
\(490\) 3751.77 0.345893
\(491\) −19653.2 −1.80639 −0.903195 0.429231i \(-0.858784\pi\)
−0.903195 + 0.429231i \(0.858784\pi\)
\(492\) 0 0
\(493\) −4620.59 −0.422111
\(494\) 0 0
\(495\) 0 0
\(496\) −2093.88 −0.189552
\(497\) 2066.63 0.186522
\(498\) 0 0
\(499\) 11713.6 1.05084 0.525422 0.850842i \(-0.323907\pi\)
0.525422 + 0.850842i \(0.323907\pi\)
\(500\) 6607.26 0.590972
\(501\) 0 0
\(502\) 13498.5 1.20013
\(503\) −13003.3 −1.15266 −0.576332 0.817216i \(-0.695517\pi\)
−0.576332 + 0.817216i \(0.695517\pi\)
\(504\) 0 0
\(505\) −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 \(0.574535\pi\)
\(510\) 0 0
\(511\) 2604.54 0.225475
\(512\) 16100.7 1.38976
\(513\) 0 0
\(514\) 26989.1 2.31603
\(515\) 2114.16 0.180895
\(516\) 0 0
\(517\) −16704.9 −1.42105
\(518\) 9026.37 0.765629
\(519\) 0 0
\(520\) 0 0
\(521\) 11700.3 0.983876 0.491938 0.870630i \(-0.336289\pi\)
0.491938 + 0.870630i \(0.336289\pi\)
\(522\) 0 0
\(523\) −4535.04 −0.379165 −0.189583 0.981865i \(-0.560714\pi\)
−0.189583 + 0.981865i \(0.560714\pi\)
\(524\) −393.814 −0.0328318
\(525\) 0 0
\(526\) −363.664 −0.0301454
\(527\) 1098.66 0.0908128
\(528\) 0 0
\(529\) 1991.62 0.163690
\(530\) −8746.98 −0.716877
\(531\) 0 0
\(532\) 6086.73 0.496040
\(533\) 0 0
\(534\) 0 0
\(535\) −1237.33 −0.0999900
\(536\) 235.739 0.0189969
\(537\) 0 0
\(538\) 18667.8 1.49596
\(539\) −9621.27 −0.768863
\(540\) 0 0
\(541\) 5184.89 0.412044 0.206022 0.978547i \(-0.433948\pi\)
0.206022 + 0.978547i \(0.433948\pi\)
\(542\) −34311.7 −2.71922
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) −15384.2 −1.19270
\(551\) −16202.3 −1.25271
\(552\) 0 0
\(553\) 6027.08 0.463468
\(554\) −11879.9 −0.911066
\(555\) 0 0
\(556\) −900.000 −0.0686484
\(557\) −20150.5 −1.53286 −0.766432 0.642326i \(-0.777970\pi\)
−0.766432 + 0.642326i \(0.777970\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1121.14 0.0846011
\(561\) 0 0
\(562\) −11618.9 −0.872087
\(563\) −16292.2 −1.21960 −0.609800 0.792556i \(-0.708750\pi\)
−0.609800 + 0.792556i \(0.708750\pi\)
\(564\) 0 0
\(565\) −4719.06 −0.351385
\(566\) 1090.98 0.0810200
\(567\) 0 0
\(568\) 1274.97 0.0941843
\(569\) −10460.5 −0.770700 −0.385350 0.922770i \(-0.625919\pi\)
−0.385350 + 0.922770i \(0.625919\pi\)
\(570\) 0 0
\(571\) 2225.96 0.163141 0.0815705 0.996668i \(-0.474006\pi\)
0.0815705 + 0.996668i \(0.474006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1545.45 0.112380
\(575\) −13766.8 −0.998461
\(576\) 0 0
\(577\) −4686.23 −0.338112 −0.169056 0.985606i \(-0.554072\pi\)
−0.169056 + 0.985606i \(0.554072\pi\)
\(578\) −16823.0 −1.21063
\(579\) 0 0
\(580\) 4395.14 0.314652
\(581\) 4592.03 0.327899
\(582\) 0 0
\(583\) 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 \(0.360250\pi\)
\(588\) 0 0
\(589\) 3852.50 0.269507
\(590\) −8256.25 −0.576109
\(591\) 0 0
\(592\) −18016.2 −1.25078
\(593\) 6135.97 0.424914 0.212457 0.977170i \(-0.431853\pi\)
0.212457 + 0.977170i \(0.431853\pi\)
\(594\) 0 0
\(595\) −588.261 −0.0405317
\(596\) 3424.06 0.235327
\(597\) 0 0
\(598\) 0 0
\(599\) 6198.80 0.422831 0.211416 0.977396i \(-0.432193\pi\)
0.211416 + 0.977396i \(0.432193\pi\)
\(600\) 0 0
\(601\) 18345.4 1.24513 0.622565 0.782568i \(-0.286091\pi\)
0.622565 + 0.782568i \(0.286091\pi\)
\(602\) 3523.79 0.238569
\(603\) 0 0
\(604\) 13657.1 0.920030
\(605\) 887.384 0.0596319
\(606\) 0 0
\(607\) 10388.1 0.694631 0.347315 0.937748i \(-0.387093\pi\)
0.347315 + 0.937748i \(0.387093\pi\)
\(608\) −26285.7 −1.75333
\(609\) 0 0
\(610\) −8819.40 −0.585389
\(611\) 0 0
\(612\) 0 0
\(613\) 804.480 0.0530060 0.0265030 0.999649i \(-0.491563\pi\)
0.0265030 + 0.999649i \(0.491563\pi\)
\(614\) −28970.0 −1.90413
\(615\) 0 0
\(616\) 888.671 0.0581259
\(617\) 15218.3 0.992973 0.496486 0.868044i \(-0.334623\pi\)
0.496486 + 0.868044i \(0.334623\pi\)
\(618\) 0 0
\(619\) 11462.5 0.744291 0.372145 0.928174i \(-0.378622\pi\)
0.372145 + 0.928174i \(0.378622\pi\)
\(620\) −1045.05 −0.0676942
\(621\) 0 0
\(622\) 4672.33 0.301195
\(623\) 7154.66 0.460105
\(624\) 0 0
\(625\) 12223.0 0.782270
\(626\) −21794.5 −1.39151
\(627\) 0 0
\(628\) 13051.4 0.829311
\(629\) 9453.14 0.599239
\(630\) 0 0
\(631\) 4468.68 0.281926 0.140963 0.990015i \(-0.454980\pi\)
0.140963 + 0.990015i \(0.454980\pi\)
\(632\) 3718.30 0.234028
\(633\) 0 0
\(634\) −19717.3 −1.23514
\(635\) 7610.37 0.475603
\(636\) 0 0
\(637\) 0 0
\(638\) −21290.0 −1.32113
\(639\) 0 0
\(640\) 1597.12 0.0986430
\(641\) −6142.36 −0.378484 −0.189242 0.981930i \(-0.560603\pi\)
−0.189242 + 0.981930i \(0.560603\pi\)
\(642\) 0 0
\(643\) −20738.2 −1.27190 −0.635951 0.771729i \(-0.719392\pi\)
−0.635951 + 0.771729i \(0.719392\pi\)
\(644\) 7157.15 0.437937
\(645\) 0 0
\(646\) 12040.7 0.733339
\(647\) −852.757 −0.0518166 −0.0259083 0.999664i \(-0.508248\pi\)
−0.0259083 + 0.999664i \(0.508248\pi\)
\(648\) 0 0
\(649\) 21172.8 1.28060
\(650\) 0 0
\(651\) 0 0
\(652\) −21081.3 −1.26627
\(653\) 7345.75 0.440217 0.220108 0.975475i \(-0.429359\pi\)
0.220108 + 0.975475i \(0.429359\pi\)
\(654\) 0 0
\(655\) 133.462 0.00796151
\(656\) −3084.65 −0.183591
\(657\) 0 0
\(658\) −14273.4 −0.845645
\(659\) −12540.7 −0.741297 −0.370648 0.928773i \(-0.620864\pi\)
−0.370648 + 0.928773i \(0.620864\pi\)
\(660\) 0 0
\(661\) 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
\(666\) 0 0
\(667\) −19051.7 −1.10597
\(668\) −360.658 −0.0208896
\(669\) 0 0
\(670\) −719.016 −0.0414597
\(671\) 22617.0 1.30122
\(672\) 0 0
\(673\) −4776.46 −0.273579 −0.136790 0.990600i \(-0.543678\pi\)
−0.136790 + 0.990600i \(0.543678\pi\)
\(674\) 35156.0 2.00914
\(675\) 0 0
\(676\) 0 0
\(677\) 7933.57 0.450387 0.225193 0.974314i \(-0.427699\pi\)
0.225193 + 0.974314i \(0.427699\pi\)
\(678\) 0 0
\(679\) −11725.6 −0.662723
\(680\) −362.917 −0.0204665
\(681\) 0 0
\(682\) 5062.23 0.284227
\(683\) 23573.8 1.32068 0.660340 0.750967i \(-0.270412\pi\)
0.660340 + 0.750967i \(0.270412\pi\)
\(684\) 0 0
\(685\) 628.904 0.0350791
\(686\) −17672.4 −0.983581
\(687\) 0 0
\(688\) −7033.32 −0.389743
\(689\) 0 0
\(690\) 0 0
\(691\) 12543.8 0.690575 0.345288 0.938497i \(-0.387781\pi\)
0.345288 + 0.938497i \(0.387781\pi\)
\(692\) −17185.9 −0.944087
\(693\) 0 0
\(694\) 51874.8 2.83738
\(695\) 305.006 0.0166468
\(696\) 0 0
\(697\) 1618.52 0.0879568
\(698\) 36947.9 2.00358
\(699\) 0 0
\(700\) −6959.09 −0.375755
\(701\) 581.786 0.0313463 0.0156731 0.999877i \(-0.495011\pi\)
0.0156731 + 0.999877i \(0.495011\pi\)
\(702\) 0 0
\(703\) 33147.9 1.77837
\(704\) −20349.7 −1.08943
\(705\) 0 0
\(706\) −22091.6 −1.17766
\(707\) 4282.89 0.227828
\(708\) 0 0
\(709\) 20742.0 1.09871 0.549353 0.835590i \(-0.314874\pi\)
0.549353 + 0.835590i \(0.314874\pi\)
\(710\) −3888.74 −0.205552
\(711\) 0 0
\(712\) 4413.94 0.232331
\(713\) 4530.01 0.237938
\(714\) 0 0
\(715\) 0 0
\(716\) 4589.63 0.239556
\(717\) 0 0
\(718\) −11154.6 −0.579788
\(719\) −25350.2 −1.31489 −0.657443 0.753504i \(-0.728362\pi\)
−0.657443 + 0.753504i \(0.728362\pi\)
\(720\) 0 0
\(721\) −4632.51 −0.239284
\(722\) 13941.0 0.718602
\(723\) 0 0
\(724\) −19231.9 −0.987223
\(725\) 18524.4 0.948938
\(726\) 0 0
\(727\) 33428.2 1.70534 0.852672 0.522447i \(-0.174981\pi\)
0.852672 + 0.522447i \(0.174981\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4900.90 −0.248480
\(731\) 3690.39 0.186722
\(732\) 0 0
\(733\) 3842.67 0.193632 0.0968160 0.995302i \(-0.469134\pi\)
0.0968160 + 0.995302i \(0.469134\pi\)
\(734\) 43184.4 2.17162
\(735\) 0 0
\(736\) −30908.3 −1.54796
\(737\) 1843.89 0.0921582
\(738\) 0 0
\(739\) −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 \(-0.832048\pi\)
−0.864000 + 0.503492i \(0.832048\pi\)
\(744\) 0 0
\(745\) −1160.40 −0.0570653
\(746\) −52623.2 −2.58267
\(747\) 0 0
\(748\) 8376.19 0.409444
\(749\) 2711.23 0.132265
\(750\) 0 0
\(751\) 10454.1 0.507957 0.253979 0.967210i \(-0.418261\pi\)
0.253979 + 0.967210i \(0.418261\pi\)
\(752\) 28489.1 1.38150
\(753\) 0 0
\(754\) 0 0
\(755\) −4628.32 −0.223102
\(756\) 0 0
\(757\) −28130.4 −1.35062 −0.675308 0.737536i \(-0.735989\pi\)
−0.675308 + 0.737536i \(0.735989\pi\)
\(758\) 9557.45 0.457971
\(759\) 0 0
\(760\) −1272.58 −0.0607388
\(761\) 21087.0 1.00447 0.502236 0.864731i \(-0.332511\pi\)
0.502236 + 0.864731i \(0.332511\pi\)
\(762\) 0 0
\(763\) 3204.72 0.152056
\(764\) −36519.7 −1.72937
\(765\) 0 0
\(766\) 8177.54 0.385727
\(767\) 0 0
\(768\) 0 0
\(769\) −19527.9 −0.915728 −0.457864 0.889022i \(-0.651385\pi\)
−0.457864 + 0.889022i \(0.651385\pi\)
\(770\) −2710.50 −0.126857
\(771\) 0 0
\(772\) −7860.55 −0.366460
\(773\) −29352.0 −1.36574 −0.682870 0.730540i \(-0.739269\pi\)
−0.682870 + 0.730540i \(0.739269\pi\)
\(774\) 0 0
\(775\) −4404.64 −0.204154
\(776\) −7233.92 −0.334643
\(777\) 0 0
\(778\) 13377.5 0.616459
\(779\) 5675.42 0.261031
\(780\) 0 0
\(781\) 9972.54 0.456908
\(782\) 14158.3 0.647440
\(783\) 0 0
\(784\) 16408.4 0.747467
\(785\) −4423.06 −0.201103
\(786\) 0 0
\(787\) −4463.12 −0.202151 −0.101076 0.994879i \(-0.532228\pi\)
−0.101076 + 0.994879i \(0.532228\pi\)
\(788\) 37325.2 1.68738
\(789\) 0 0
\(790\) −11341.0 −0.510754
\(791\) 10340.3 0.464804
\(792\) 0 0
\(793\) 0 0
\(794\) −15501.7 −0.692866
\(795\) 0 0
\(796\) −21636.6 −0.963426
\(797\) 34785.5 1.54600 0.773002 0.634404i \(-0.218754\pi\)
0.773002 + 0.634404i \(0.218754\pi\)
\(798\) 0 0
\(799\) −14948.2 −0.661866
\(800\) 30053.0 1.32817
\(801\) 0 0
\(802\) −8236.09 −0.362627
\(803\) 12568.2 0.552330
\(804\) 0 0
\(805\) −2425.53 −0.106197
\(806\) 0 0
\(807\) 0 0
\(808\) 2642.25 0.115042
\(809\) 10620.0 0.461530 0.230765 0.973010i \(-0.425877\pi\)
0.230765 + 0.973010i \(0.425877\pi\)
\(810\) 0 0
\(811\) −5497.87 −0.238047 −0.119024 0.992891i \(-0.537976\pi\)
−0.119024 + 0.992891i \(0.537976\pi\)
\(812\) −9630.57 −0.416215
\(813\) 0 0
\(814\) 43556.7 1.87551
\(815\) 7144.34 0.307062
\(816\) 0 0
\(817\) 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 \(0.350411\pi\)
\(822\) 0 0
\(823\) 17342.6 0.734537 0.367268 0.930115i \(-0.380293\pi\)
0.367268 + 0.930115i \(0.380293\pi\)
\(824\) −2857.94 −0.120827
\(825\) 0 0
\(826\) 18091.0 0.762064
\(827\) 5129.96 0.215703 0.107851 0.994167i \(-0.465603\pi\)
0.107851 + 0.994167i \(0.465603\pi\)
\(828\) 0 0
\(829\) −8471.81 −0.354931 −0.177466 0.984127i \(-0.556790\pi\)
−0.177466 + 0.984127i \(0.556790\pi\)
\(830\) −8640.72 −0.361354
\(831\) 0 0
\(832\) 0 0
\(833\) −8609.50 −0.358105
\(834\) 0 0
\(835\) 122.225 0.00506561
\(836\) 29371.5 1.21511
\(837\) 0 0
\(838\) 28128.3 1.15952
\(839\) −19155.0 −0.788207 −0.394103 0.919066i \(-0.628945\pi\)
−0.394103 + 0.919066i \(0.628945\pi\)
\(840\) 0 0
\(841\) 1246.67 0.0511160
\(842\) −31078.7 −1.27202
\(843\) 0 0
\(844\) 34819.7 1.42007
\(845\) 0 0
\(846\) 0 0
\(847\) −1944.42 −0.0788797
\(848\) −38255.0 −1.54915
\(849\) 0 0
\(850\) −13766.4 −0.555512
\(851\) 38977.3 1.57006
\(852\) 0 0
\(853\) 18075.1 0.725532 0.362766 0.931880i \(-0.381832\pi\)
0.362766 + 0.931880i \(0.381832\pi\)
\(854\) 19324.9 0.774339
\(855\) 0 0
\(856\) 1672.64 0.0667871
\(857\) 21054.6 0.839219 0.419609 0.907705i \(-0.362167\pi\)
0.419609 + 0.907705i \(0.362167\pi\)
\(858\) 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
\(861\) 0 0
\(862\) −55267.0 −2.18376
\(863\) −19427.5 −0.766304 −0.383152 0.923685i \(-0.625161\pi\)
−0.383152 + 0.923685i \(0.625161\pi\)
\(864\) 0 0
\(865\) 5824.21 0.228935
\(866\) −73044.7 −2.86623
\(867\) 0 0
\(868\) 2289.91 0.0895444
\(869\) 29083.7 1.13532
\(870\) 0 0
\(871\) 0 0
\(872\) 1977.09 0.0767808
\(873\) 0 0
\(874\) 49646.5 1.92142
\(875\) 4906.44 0.189563
\(876\) 0 0
\(877\) −14872.2 −0.572632 −0.286316 0.958135i \(-0.592431\pi\)
−0.286316 + 0.958135i \(0.592431\pi\)
\(878\) 29535.7 1.13529
\(879\) 0 0
\(880\) 5410.04 0.207241
\(881\) 12940.6 0.494870 0.247435 0.968905i \(-0.420412\pi\)
0.247435 + 0.968905i \(0.420412\pi\)
\(882\) 0 0
\(883\) −25585.5 −0.975108 −0.487554 0.873093i \(-0.662111\pi\)
−0.487554 + 0.873093i \(0.662111\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 41928.8 1.58987
\(887\) −3716.46 −0.140684 −0.0703418 0.997523i \(-0.522409\pi\)
−0.0703418 + 0.997523i \(0.522409\pi\)
\(888\) 0 0
\(889\) −16675.7 −0.629118
\(890\) −13462.8 −0.507049
\(891\) 0 0
\(892\) −25322.0 −0.950496
\(893\) −52416.7 −1.96423
\(894\) 0 0
\(895\) −1555.40 −0.0580910
\(896\) −3499.58 −0.130483
\(897\) 0 0
\(898\) −70680.3 −2.62654
\(899\) −6095.52 −0.226137
\(900\) 0 0
\(901\) 20072.5 0.742187
\(902\) 7457.57 0.275288
\(903\) 0 0
\(904\) 6379.29 0.234703
\(905\) 6517.61 0.239395
\(906\) 0 0
\(907\) −12960.4 −0.474469 −0.237235 0.971452i \(-0.576241\pi\)
−0.237235 + 0.971452i \(0.576241\pi\)
\(908\) −40665.3 −1.48626
\(909\) 0 0
\(910\) 0 0
\(911\) 36607.1 1.33134 0.665668 0.746248i \(-0.268147\pi\)
0.665668 + 0.746248i \(0.268147\pi\)
\(912\) 0 0
\(913\) 22158.8 0.803230
\(914\) −58099.3 −2.10258
\(915\) 0 0
\(916\) 11747.4 0.423740
\(917\) −292.440 −0.0105313
\(918\) 0 0
\(919\) 20356.3 0.730676 0.365338 0.930875i \(-0.380953\pi\)
0.365338 + 0.930875i \(0.380953\pi\)
\(920\) −1496.38 −0.0536243
\(921\) 0 0
\(922\) −12048.6 −0.430370
\(923\) 0 0
\(924\) 0 0
\(925\) −37898.7 −1.34714
\(926\) 8545.61 0.303268
\(927\) 0 0
\(928\) 41589.8 1.47118
\(929\) −45069.7 −1.59170 −0.795849 0.605495i \(-0.792975\pi\)
−0.795849 + 0.605495i \(0.792975\pi\)
\(930\) 0 0
\(931\) −30189.6 −1.06275
\(932\) 30244.9 1.06299
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 14218.9 0.493372
\(941\) 36690.7 1.27108 0.635538 0.772070i \(-0.280778\pi\)
0.635538 + 0.772070i \(0.280778\pi\)
\(942\) 0 0
\(943\) 6673.51 0.230455
\(944\) −36108.8 −1.24496
\(945\) 0 0
\(946\) 17004.0 0.584406
\(947\) −50861.2 −1.74527 −0.872634 0.488375i \(-0.837590\pi\)
−0.872634 + 0.488375i \(0.837590\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −48272.6 −1.64860
\(951\) 0 0
\(952\) 795.219 0.0270727
\(953\) −11855.6 −0.402980 −0.201490 0.979491i \(-0.564578\pi\)
−0.201490 + 0.979491i \(0.564578\pi\)
\(954\) 0 0
\(955\) 12376.4 0.419361
\(956\) −42634.5 −1.44236
\(957\) 0 0
\(958\) −21525.5 −0.725947
\(959\) −1378.04 −0.0464019
\(960\) 0 0
\(961\) −28341.6 −0.951349
\(962\) 0 0
\(963\) 0 0
\(964\) −43067.5 −1.43891
\(965\) 2663.90 0.0888643
\(966\) 0 0
\(967\) −40661.7 −1.35221 −0.676107 0.736803i \(-0.736334\pi\)
−0.676107 + 0.736803i \(0.736334\pi\)
\(968\) −1199.58 −0.0398304
\(969\) 0 0
\(970\) 22063.9 0.730339
\(971\) −57318.3 −1.89437 −0.947184 0.320690i \(-0.896085\pi\)
−0.947184 + 0.320690i \(0.896085\pi\)
\(972\) 0 0
\(973\) −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 \(0.515781\pi\)
\(978\) 0 0
\(979\) 34524.8 1.12709
\(980\) 8189.43 0.266941
\(981\) 0 0
\(982\) −81032.3 −2.63324
\(983\) 33942.5 1.10132 0.550659 0.834730i \(-0.314376\pi\)
0.550659 + 0.834730i \(0.314376\pi\)
\(984\) 0 0
\(985\) −12649.3 −0.409179
\(986\) −19051.2 −0.615327
\(987\) 0 0
\(988\) 0 0
\(989\) 15216.3 0.489231
\(990\) 0 0
\(991\) 21637.5 0.693580 0.346790 0.937943i \(-0.387272\pi\)
0.346790 + 0.937943i \(0.387272\pi\)
\(992\) −9889.02 −0.316509
\(993\) 0 0
\(994\) 8520.95 0.271900
\(995\) 7332.53 0.233625
\(996\) 0 0
\(997\) 19624.6 0.623388 0.311694 0.950182i \(-0.399104\pi\)
0.311694 + 0.950182i \(0.399104\pi\)
\(998\) 48296.3 1.53186
\(999\) 0 0
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 1521.4.a.z.1.4 4
3.2 odd 2 507.4.a.k.1.1 4
13.6 odd 12 117.4.q.d.10.1 4
13.11 odd 12 117.4.q.d.82.1 4
13.12 even 2 inner 1521.4.a.z.1.1 4
39.5 even 4 507.4.b.e.337.4 4
39.8 even 4 507.4.b.e.337.1 4
39.11 even 12 39.4.j.b.4.2 4
39.32 even 12 39.4.j.b.10.2 yes 4
39.38 odd 2 507.4.a.k.1.4 4
156.11 odd 12 624.4.bv.c.433.2 4
156.71 odd 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 39.11 even 12
39.4.j.b.10.2 yes 4 39.32 even 12
117.4.q.d.10.1 4 13.6 odd 12
117.4.q.d.82.1 4 13.11 odd 12
507.4.a.k.1.1 4 3.2 odd 2
507.4.a.k.1.4 4 39.38 odd 2
507.4.b.e.337.1 4 39.8 even 4
507.4.b.e.337.4 4 39.5 even 4
624.4.bv.c.49.1 4 156.71 odd 12
624.4.bv.c.433.2 4 156.11 odd 12
1521.4.a.z.1.1 4 13.12 even 2 inner
1521.4.a.z.1.4 4 1.1 even 1 trivial