Properties

Label 531.5.c.d.235.21
Level $531$
Weight $5$
Character 531.235
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 235.21
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.20

$q$-expansion

\(f(q)\) \(=\) \(q+0.389117i q^{2} +15.8486 q^{4} +17.9988 q^{5} -47.2538 q^{7} +12.3928i q^{8} +O(q^{10})\) \(q+0.389117i q^{2} +15.8486 q^{4} +17.9988 q^{5} -47.2538 q^{7} +12.3928i q^{8} +7.00365i q^{10} -197.341i q^{11} +176.297i q^{13} -18.3873i q^{14} +248.755 q^{16} -486.600 q^{17} +56.2166 q^{19} +285.256 q^{20} +76.7889 q^{22} -848.854i q^{23} -301.043 q^{25} -68.6003 q^{26} -748.906 q^{28} -275.393 q^{29} -843.954i q^{31} +295.080i q^{32} -189.344i q^{34} -850.512 q^{35} -1855.12i q^{37} +21.8749i q^{38} +223.056i q^{40} -1479.48 q^{41} +2758.38i q^{43} -3127.58i q^{44} +330.304 q^{46} -1437.11i q^{47} -168.080 q^{49} -117.141i q^{50} +2794.06i q^{52} +1119.21 q^{53} -3551.91i q^{55} -585.608i q^{56} -107.160i q^{58} +(1640.41 - 3070.25i) q^{59} +115.162i q^{61} +328.397 q^{62} +3865.26 q^{64} +3173.14i q^{65} +49.5250i q^{67} -7711.92 q^{68} -330.949i q^{70} +2085.72 q^{71} +209.953i q^{73} +721.859 q^{74} +890.954 q^{76} +9325.12i q^{77} -6542.76 q^{79} +4477.30 q^{80} -575.691i q^{82} +6802.27i q^{83} -8758.22 q^{85} -1073.33 q^{86} +2445.62 q^{88} +9119.14i q^{89} -8330.70i q^{91} -13453.1i q^{92} +559.206 q^{94} +1011.83 q^{95} +2946.12i q^{97} -65.4030i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.389117i 0.0972793i 0.998816 + 0.0486397i \(0.0154886\pi\)
−0.998816 + 0.0486397i \(0.984511\pi\)
\(3\) 0 0
\(4\) 15.8486 0.990537
\(5\) 17.9988 0.719952 0.359976 0.932962i \(-0.382785\pi\)
0.359976 + 0.932962i \(0.382785\pi\)
\(6\) 0 0
\(7\) −47.2538 −0.964363 −0.482181 0.876071i \(-0.660155\pi\)
−0.482181 + 0.876071i \(0.660155\pi\)
\(8\) 12.3928i 0.193638i
\(9\) 0 0
\(10\) 7.00365i 0.0700365i
\(11\) 197.341i 1.63092i −0.578814 0.815460i \(-0.696484\pi\)
0.578814 0.815460i \(-0.303516\pi\)
\(12\) 0 0
\(13\) 176.297i 1.04318i 0.853197 + 0.521589i \(0.174661\pi\)
−0.853197 + 0.521589i \(0.825339\pi\)
\(14\) 18.3873i 0.0938126i
\(15\) 0 0
\(16\) 248.755 0.971700
\(17\) −486.600 −1.68374 −0.841868 0.539683i \(-0.818544\pi\)
−0.841868 + 0.539683i \(0.818544\pi\)
\(18\) 0 0
\(19\) 56.2166 0.155725 0.0778624 0.996964i \(-0.475191\pi\)
0.0778624 + 0.996964i \(0.475191\pi\)
\(20\) 285.256 0.713139
\(21\) 0 0
\(22\) 76.7889 0.158655
\(23\) 848.854i 1.60464i −0.596895 0.802319i \(-0.703599\pi\)
0.596895 0.802319i \(-0.296401\pi\)
\(24\) 0 0
\(25\) −301.043 −0.481669
\(26\) −68.6003 −0.101480
\(27\) 0 0
\(28\) −748.906 −0.955237
\(29\) −275.393 −0.327459 −0.163729 0.986505i \(-0.552352\pi\)
−0.163729 + 0.986505i \(0.552352\pi\)
\(30\) 0 0
\(31\) 843.954i 0.878204i −0.898437 0.439102i \(-0.855297\pi\)
0.898437 0.439102i \(-0.144703\pi\)
\(32\) 295.080i 0.288164i
\(33\) 0 0
\(34\) 189.344i 0.163793i
\(35\) −850.512 −0.694295
\(36\) 0 0
\(37\) 1855.12i 1.35509i −0.735481 0.677546i \(-0.763044\pi\)
0.735481 0.677546i \(-0.236956\pi\)
\(38\) 21.8749i 0.0151488i
\(39\) 0 0
\(40\) 223.056i 0.139410i
\(41\) −1479.48 −0.880118 −0.440059 0.897969i \(-0.645042\pi\)
−0.440059 + 0.897969i \(0.645042\pi\)
\(42\) 0 0
\(43\) 2758.38i 1.49182i 0.666045 + 0.745912i \(0.267986\pi\)
−0.666045 + 0.745912i \(0.732014\pi\)
\(44\) 3127.58i 1.61549i
\(45\) 0 0
\(46\) 330.304 0.156098
\(47\) 1437.11i 0.650572i −0.945616 0.325286i \(-0.894539\pi\)
0.945616 0.325286i \(-0.105461\pi\)
\(48\) 0 0
\(49\) −168.080 −0.0700043
\(50\) 117.141i 0.0468564i
\(51\) 0 0
\(52\) 2794.06i 1.03331i
\(53\) 1119.21 0.398439 0.199219 0.979955i \(-0.436159\pi\)
0.199219 + 0.979955i \(0.436159\pi\)
\(54\) 0 0
\(55\) 3551.91i 1.17418i
\(56\) 585.608i 0.186737i
\(57\) 0 0
\(58\) 107.160i 0.0318550i
\(59\) 1640.41 3070.25i 0.471247 0.882001i
\(60\) 0 0
\(61\) 115.162i 0.0309492i 0.999880 + 0.0154746i \(0.00492592\pi\)
−0.999880 + 0.0154746i \(0.995074\pi\)
\(62\) 328.397 0.0854311
\(63\) 0 0
\(64\) 3865.26 0.943667
\(65\) 3173.14i 0.751038i
\(66\) 0 0
\(67\) 49.5250i 0.0110325i 0.999985 + 0.00551626i \(0.00175589\pi\)
−0.999985 + 0.00551626i \(0.998244\pi\)
\(68\) −7711.92 −1.66780
\(69\) 0 0
\(70\) 330.949i 0.0675406i
\(71\) 2085.72 0.413751 0.206876 0.978367i \(-0.433670\pi\)
0.206876 + 0.978367i \(0.433670\pi\)
\(72\) 0 0
\(73\) 209.953i 0.0393982i 0.999806 + 0.0196991i \(0.00627083\pi\)
−0.999806 + 0.0196991i \(0.993729\pi\)
\(74\) 721.859 0.131822
\(75\) 0 0
\(76\) 890.954 0.154251
\(77\) 9325.12i 1.57280i
\(78\) 0 0
\(79\) −6542.76 −1.04835 −0.524176 0.851610i \(-0.675626\pi\)
−0.524176 + 0.851610i \(0.675626\pi\)
\(80\) 4477.30 0.699577
\(81\) 0 0
\(82\) 575.691i 0.0856173i
\(83\) 6802.27i 0.987410i 0.869629 + 0.493705i \(0.164358\pi\)
−0.869629 + 0.493705i \(0.835642\pi\)
\(84\) 0 0
\(85\) −8758.22 −1.21221
\(86\) −1073.33 −0.145124
\(87\) 0 0
\(88\) 2445.62 0.315808
\(89\) 9119.14i 1.15126i 0.817710 + 0.575630i \(0.195243\pi\)
−0.817710 + 0.575630i \(0.804757\pi\)
\(90\) 0 0
\(91\) 8330.70i 1.00600i
\(92\) 13453.1i 1.58945i
\(93\) 0 0
\(94\) 559.206 0.0632872
\(95\) 1011.83 0.112114
\(96\) 0 0
\(97\) 2946.12i 0.313117i 0.987669 + 0.156559i \(0.0500400\pi\)
−0.987669 + 0.156559i \(0.949960\pi\)
\(98\) 65.4030i 0.00680997i
\(99\) 0 0
\(100\) −4771.11 −0.477111
\(101\) 4577.99i 0.448779i −0.974500 0.224389i \(-0.927961\pi\)
0.974500 0.224389i \(-0.0720388\pi\)
\(102\) 0 0
\(103\) 14158.2i 1.33455i −0.744812 0.667274i \(-0.767461\pi\)
0.744812 0.667274i \(-0.232539\pi\)
\(104\) −2184.82 −0.201999
\(105\) 0 0
\(106\) 435.506i 0.0387598i
\(107\) −2298.00 −0.200717 −0.100358 0.994951i \(-0.531999\pi\)
−0.100358 + 0.994951i \(0.531999\pi\)
\(108\) 0 0
\(109\) 18428.3i 1.55107i −0.631302 0.775537i \(-0.717479\pi\)
0.631302 0.775537i \(-0.282521\pi\)
\(110\) 1382.11 0.114224
\(111\) 0 0
\(112\) −11754.6 −0.937071
\(113\) 15385.4i 1.20490i −0.798156 0.602451i \(-0.794191\pi\)
0.798156 0.602451i \(-0.205809\pi\)
\(114\) 0 0
\(115\) 15278.4i 1.15526i
\(116\) −4364.59 −0.324360
\(117\) 0 0
\(118\) 1194.69 + 638.312i 0.0858005 + 0.0458426i
\(119\) 22993.7 1.62373
\(120\) 0 0
\(121\) −24302.6 −1.65990
\(122\) −44.8115 −0.00301072
\(123\) 0 0
\(124\) 13375.5i 0.869893i
\(125\) −16667.7 −1.06673
\(126\) 0 0
\(127\) 28666.4 1.77732 0.888661 0.458564i \(-0.151636\pi\)
0.888661 + 0.458564i \(0.151636\pi\)
\(128\) 6225.33i 0.379964i
\(129\) 0 0
\(130\) −1234.72 −0.0730605
\(131\) 19845.6i 1.15643i −0.815883 0.578217i \(-0.803749\pi\)
0.815883 0.578217i \(-0.196251\pi\)
\(132\) 0 0
\(133\) −2656.45 −0.150175
\(134\) −19.2710 −0.00107324
\(135\) 0 0
\(136\) 6030.35i 0.326035i
\(137\) −19165.2 −1.02111 −0.510555 0.859845i \(-0.670560\pi\)
−0.510555 + 0.859845i \(0.670560\pi\)
\(138\) 0 0
\(139\) −20252.2 −1.04820 −0.524099 0.851658i \(-0.675598\pi\)
−0.524099 + 0.851658i \(0.675598\pi\)
\(140\) −13479.4 −0.687725
\(141\) 0 0
\(142\) 811.590i 0.0402495i
\(143\) 34790.7 1.70134
\(144\) 0 0
\(145\) −4956.74 −0.235755
\(146\) −81.6964 −0.00383263
\(147\) 0 0
\(148\) 29401.0i 1.34227i
\(149\) 8521.84i 0.383849i −0.981410 0.191925i \(-0.938527\pi\)
0.981410 0.191925i \(-0.0614729\pi\)
\(150\) 0 0
\(151\) 43196.8i 1.89452i −0.320473 0.947258i \(-0.603842\pi\)
0.320473 0.947258i \(-0.396158\pi\)
\(152\) 696.684i 0.0301542i
\(153\) 0 0
\(154\) −3628.57 −0.153001
\(155\) 15190.2i 0.632265i
\(156\) 0 0
\(157\) 7274.02i 0.295104i 0.989054 + 0.147552i \(0.0471394\pi\)
−0.989054 + 0.147552i \(0.952861\pi\)
\(158\) 2545.90i 0.101983i
\(159\) 0 0
\(160\) 5311.09i 0.207465i
\(161\) 40111.6i 1.54745i
\(162\) 0 0
\(163\) −15415.8 −0.580219 −0.290109 0.956994i \(-0.593692\pi\)
−0.290109 + 0.956994i \(0.593692\pi\)
\(164\) −23447.6 −0.871789
\(165\) 0 0
\(166\) −2646.88 −0.0960546
\(167\) −37730.3 −1.35288 −0.676438 0.736500i \(-0.736477\pi\)
−0.676438 + 0.736500i \(0.736477\pi\)
\(168\) 0 0
\(169\) −2519.67 −0.0882205
\(170\) 3407.97i 0.117923i
\(171\) 0 0
\(172\) 43716.4i 1.47771i
\(173\) 12104.2i 0.404430i 0.979341 + 0.202215i \(0.0648139\pi\)
−0.979341 + 0.202215i \(0.935186\pi\)
\(174\) 0 0
\(175\) 14225.4 0.464503
\(176\) 49089.7i 1.58476i
\(177\) 0 0
\(178\) −3548.41 −0.111994
\(179\) 288.509i 0.00900436i −0.999990 0.00450218i \(-0.998567\pi\)
0.999990 0.00450218i \(-0.00143309\pi\)
\(180\) 0 0
\(181\) −12771.6 −0.389841 −0.194921 0.980819i \(-0.562445\pi\)
−0.194921 + 0.980819i \(0.562445\pi\)
\(182\) 3241.62 0.0978632
\(183\) 0 0
\(184\) 10519.7 0.310719
\(185\) 33389.9i 0.975601i
\(186\) 0 0
\(187\) 96026.2i 2.74604i
\(188\) 22776.2i 0.644415i
\(189\) 0 0
\(190\) 393.722i 0.0109064i
\(191\) 49064.3i 1.34493i 0.740130 + 0.672464i \(0.234764\pi\)
−0.740130 + 0.672464i \(0.765236\pi\)
\(192\) 0 0
\(193\) 51351.0 1.37859 0.689294 0.724481i \(-0.257921\pi\)
0.689294 + 0.724481i \(0.257921\pi\)
\(194\) −1146.39 −0.0304598
\(195\) 0 0
\(196\) −2663.84 −0.0693418
\(197\) 30154.4 0.776996 0.388498 0.921450i \(-0.372994\pi\)
0.388498 + 0.921450i \(0.372994\pi\)
\(198\) 0 0
\(199\) 67941.6 1.71565 0.857827 0.513939i \(-0.171814\pi\)
0.857827 + 0.513939i \(0.171814\pi\)
\(200\) 3730.78i 0.0932694i
\(201\) 0 0
\(202\) 1781.38 0.0436569
\(203\) 13013.4 0.315789
\(204\) 0 0
\(205\) −26628.8 −0.633643
\(206\) 5509.21 0.129824
\(207\) 0 0
\(208\) 43854.8i 1.01366i
\(209\) 11093.9i 0.253975i
\(210\) 0 0
\(211\) 2303.99i 0.0517507i 0.999665 + 0.0258754i \(0.00823730\pi\)
−0.999665 + 0.0258754i \(0.991763\pi\)
\(212\) 17738.0 0.394668
\(213\) 0 0
\(214\) 894.193i 0.0195256i
\(215\) 49647.6i 1.07404i
\(216\) 0 0
\(217\) 39880.0i 0.846907i
\(218\) 7170.78 0.150888
\(219\) 0 0
\(220\) 56292.7i 1.16307i
\(221\) 85786.1i 1.75644i
\(222\) 0 0
\(223\) 28222.6 0.567527 0.283764 0.958894i \(-0.408417\pi\)
0.283764 + 0.958894i \(0.408417\pi\)
\(224\) 13943.7i 0.277895i
\(225\) 0 0
\(226\) 5986.73 0.117212
\(227\) 63511.2i 1.23253i 0.787538 + 0.616267i \(0.211356\pi\)
−0.787538 + 0.616267i \(0.788644\pi\)
\(228\) 0 0
\(229\) 36930.9i 0.704236i −0.935956 0.352118i \(-0.885462\pi\)
0.935956 0.352118i \(-0.114538\pi\)
\(230\) 5945.07 0.112383
\(231\) 0 0
\(232\) 3412.90i 0.0634085i
\(233\) 53127.8i 0.978610i 0.872113 + 0.489305i \(0.162750\pi\)
−0.872113 + 0.489305i \(0.837250\pi\)
\(234\) 0 0
\(235\) 25866.3i 0.468381i
\(236\) 25998.2 48659.1i 0.466787 0.873655i
\(237\) 0 0
\(238\) 8947.24i 0.157956i
\(239\) 24487.8 0.428700 0.214350 0.976757i \(-0.431237\pi\)
0.214350 + 0.976757i \(0.431237\pi\)
\(240\) 0 0
\(241\) 12746.1 0.219454 0.109727 0.993962i \(-0.465002\pi\)
0.109727 + 0.993962i \(0.465002\pi\)
\(242\) 9456.55i 0.161474i
\(243\) 0 0
\(244\) 1825.16i 0.0306563i
\(245\) −3025.25 −0.0503998
\(246\) 0 0
\(247\) 9910.83i 0.162449i
\(248\) 10459.0 0.170054
\(249\) 0 0
\(250\) 6485.68i 0.103771i
\(251\) −44798.5 −0.711076 −0.355538 0.934662i \(-0.615702\pi\)
−0.355538 + 0.934662i \(0.615702\pi\)
\(252\) 0 0
\(253\) −167514. −2.61704
\(254\) 11154.6i 0.172897i
\(255\) 0 0
\(256\) 59421.8 0.906705
\(257\) −105733. −1.60083 −0.800413 0.599449i \(-0.795386\pi\)
−0.800413 + 0.599449i \(0.795386\pi\)
\(258\) 0 0
\(259\) 87661.4i 1.30680i
\(260\) 50289.7i 0.743931i
\(261\) 0 0
\(262\) 7722.25 0.112497
\(263\) −9304.75 −0.134522 −0.0672610 0.997735i \(-0.521426\pi\)
−0.0672610 + 0.997735i \(0.521426\pi\)
\(264\) 0 0
\(265\) 20144.5 0.286857
\(266\) 1033.67i 0.0146089i
\(267\) 0 0
\(268\) 784.901i 0.0109281i
\(269\) 6683.88i 0.0923685i −0.998933 0.0461842i \(-0.985294\pi\)
0.998933 0.0461842i \(-0.0147061\pi\)
\(270\) 0 0
\(271\) −137964. −1.87856 −0.939282 0.343145i \(-0.888508\pi\)
−0.939282 + 0.343145i \(0.888508\pi\)
\(272\) −121044. −1.63609
\(273\) 0 0
\(274\) 7457.51i 0.0993329i
\(275\) 59408.2i 0.785563i
\(276\) 0 0
\(277\) 96139.5 1.25298 0.626488 0.779431i \(-0.284492\pi\)
0.626488 + 0.779431i \(0.284492\pi\)
\(278\) 7880.49i 0.101968i
\(279\) 0 0
\(280\) 10540.3i 0.134442i
\(281\) 13544.9 0.171539 0.0857696 0.996315i \(-0.472665\pi\)
0.0857696 + 0.996315i \(0.472665\pi\)
\(282\) 0 0
\(283\) 147321.i 1.83946i 0.392547 + 0.919732i \(0.371594\pi\)
−0.392547 + 0.919732i \(0.628406\pi\)
\(284\) 33055.7 0.409836
\(285\) 0 0
\(286\) 13537.7i 0.165505i
\(287\) 69910.9 0.848753
\(288\) 0 0
\(289\) 153258. 1.83497
\(290\) 1928.76i 0.0229341i
\(291\) 0 0
\(292\) 3327.46i 0.0390254i
\(293\) −21384.4 −0.249094 −0.124547 0.992214i \(-0.539748\pi\)
−0.124547 + 0.992214i \(0.539748\pi\)
\(294\) 0 0
\(295\) 29525.4 55260.8i 0.339275 0.634999i
\(296\) 22990.2 0.262397
\(297\) 0 0
\(298\) 3316.00 0.0373406
\(299\) 149650. 1.67392
\(300\) 0 0
\(301\) 130344.i 1.43866i
\(302\) 16808.6 0.184297
\(303\) 0 0
\(304\) 13984.2 0.151318
\(305\) 2072.78i 0.0222820i
\(306\) 0 0
\(307\) −89703.8 −0.951775 −0.475887 0.879506i \(-0.657873\pi\)
−0.475887 + 0.879506i \(0.657873\pi\)
\(308\) 147790.i 1.55791i
\(309\) 0 0
\(310\) 5910.76 0.0615063
\(311\) 23438.7 0.242333 0.121166 0.992632i \(-0.461337\pi\)
0.121166 + 0.992632i \(0.461337\pi\)
\(312\) 0 0
\(313\) 48687.8i 0.496971i 0.968636 + 0.248486i \(0.0799329\pi\)
−0.968636 + 0.248486i \(0.920067\pi\)
\(314\) −2830.45 −0.0287075
\(315\) 0 0
\(316\) −103694. −1.03843
\(317\) 126653. 1.26036 0.630182 0.776448i \(-0.282980\pi\)
0.630182 + 0.776448i \(0.282980\pi\)
\(318\) 0 0
\(319\) 54346.4i 0.534059i
\(320\) 69570.1 0.679395
\(321\) 0 0
\(322\) −15608.1 −0.150535
\(323\) −27355.0 −0.262199
\(324\) 0 0
\(325\) 53073.0i 0.502466i
\(326\) 5998.57i 0.0564433i
\(327\) 0 0
\(328\) 18334.9i 0.170424i
\(329\) 67909.0i 0.627387i
\(330\) 0 0
\(331\) 177452. 1.61966 0.809831 0.586663i \(-0.199559\pi\)
0.809831 + 0.586663i \(0.199559\pi\)
\(332\) 107806.i 0.978066i
\(333\) 0 0
\(334\) 14681.5i 0.131607i
\(335\) 891.390i 0.00794289i
\(336\) 0 0
\(337\) 88144.0i 0.776127i −0.921633 0.388064i \(-0.873144\pi\)
0.921633 0.388064i \(-0.126856\pi\)
\(338\) 980.446i 0.00858203i
\(339\) 0 0
\(340\) −138805. −1.20074
\(341\) −166547. −1.43228
\(342\) 0 0
\(343\) 121399. 1.03187
\(344\) −34184.2 −0.288874
\(345\) 0 0
\(346\) −4709.95 −0.0393427
\(347\) 79473.9i 0.660033i 0.943975 + 0.330016i \(0.107054\pi\)
−0.943975 + 0.330016i \(0.892946\pi\)
\(348\) 0 0
\(349\) 24382.5i 0.200183i 0.994978 + 0.100091i \(0.0319135\pi\)
−0.994978 + 0.100091i \(0.968086\pi\)
\(350\) 5535.36i 0.0451866i
\(351\) 0 0
\(352\) 58231.5 0.469973
\(353\) 61148.2i 0.490720i −0.969432 0.245360i \(-0.921094\pi\)
0.969432 0.245360i \(-0.0789062\pi\)
\(354\) 0 0
\(355\) 37540.5 0.297881
\(356\) 144525.i 1.14037i
\(357\) 0 0
\(358\) 112.264 0.000875939
\(359\) 241084. 1.87059 0.935297 0.353864i \(-0.115132\pi\)
0.935297 + 0.353864i \(0.115132\pi\)
\(360\) 0 0
\(361\) −127161. −0.975750
\(362\) 4969.64i 0.0379235i
\(363\) 0 0
\(364\) 132030.i 0.996482i
\(365\) 3778.91i 0.0283648i
\(366\) 0 0
\(367\) 237472.i 1.76311i 0.472078 + 0.881557i \(0.343504\pi\)
−0.472078 + 0.881557i \(0.656496\pi\)
\(368\) 211157.i 1.55923i
\(369\) 0 0
\(370\) 12992.6 0.0949058
\(371\) −52887.1 −0.384239
\(372\) 0 0
\(373\) −168379. −1.21023 −0.605117 0.796137i \(-0.706874\pi\)
−0.605117 + 0.796137i \(0.706874\pi\)
\(374\) −37365.5 −0.267133
\(375\) 0 0
\(376\) 17809.9 0.125976
\(377\) 48551.0i 0.341598i
\(378\) 0 0
\(379\) 234352. 1.63152 0.815758 0.578394i \(-0.196320\pi\)
0.815758 + 0.578394i \(0.196320\pi\)
\(380\) 16036.1 0.111053
\(381\) 0 0
\(382\) −19091.8 −0.130834
\(383\) −114382. −0.779762 −0.389881 0.920865i \(-0.627484\pi\)
−0.389881 + 0.920865i \(0.627484\pi\)
\(384\) 0 0
\(385\) 167841.i 1.13234i
\(386\) 19981.6i 0.134108i
\(387\) 0 0
\(388\) 46691.9i 0.310154i
\(389\) 248829. 1.64438 0.822189 0.569215i \(-0.192753\pi\)
0.822189 + 0.569215i \(0.192753\pi\)
\(390\) 0 0
\(391\) 413052.i 2.70179i
\(392\) 2082.99i 0.0135555i
\(393\) 0 0
\(394\) 11733.6i 0.0755857i
\(395\) −117762. −0.754763
\(396\) 0 0
\(397\) 178881.i 1.13497i 0.823385 + 0.567483i \(0.192083\pi\)
−0.823385 + 0.567483i \(0.807917\pi\)
\(398\) 26437.2i 0.166898i
\(399\) 0 0
\(400\) −74886.0 −0.468037
\(401\) 79311.1i 0.493225i 0.969114 + 0.246613i \(0.0793175\pi\)
−0.969114 + 0.246613i \(0.920682\pi\)
\(402\) 0 0
\(403\) 148787. 0.916123
\(404\) 72554.7i 0.444532i
\(405\) 0 0
\(406\) 5063.72i 0.0307198i
\(407\) −366092. −2.21004
\(408\) 0 0
\(409\) 85968.6i 0.513917i 0.966422 + 0.256959i \(0.0827204\pi\)
−0.966422 + 0.256959i \(0.917280\pi\)
\(410\) 10361.7i 0.0616404i
\(411\) 0 0
\(412\) 224388.i 1.32192i
\(413\) −77515.6 + 145081.i −0.454453 + 0.850569i
\(414\) 0 0
\(415\) 122433.i 0.710888i
\(416\) −52021.8 −0.300607
\(417\) 0 0
\(418\) 4316.81 0.0247065
\(419\) 282200.i 1.60742i −0.595023 0.803709i \(-0.702857\pi\)
0.595023 0.803709i \(-0.297143\pi\)
\(420\) 0 0
\(421\) 254214.i 1.43428i −0.696928 0.717141i \(-0.745450\pi\)
0.696928 0.717141i \(-0.254550\pi\)
\(422\) −896.524 −0.00503427
\(423\) 0 0
\(424\) 13870.2i 0.0771529i
\(425\) 146487. 0.811003
\(426\) 0 0
\(427\) 5441.84i 0.0298463i
\(428\) −36420.1 −0.198817
\(429\) 0 0
\(430\) −19318.7 −0.104482
\(431\) 196721.i 1.05900i −0.848310 0.529500i \(-0.822380\pi\)
0.848310 0.529500i \(-0.177620\pi\)
\(432\) 0 0
\(433\) 246611. 1.31533 0.657667 0.753309i \(-0.271543\pi\)
0.657667 + 0.753309i \(0.271543\pi\)
\(434\) −15518.0 −0.0823866
\(435\) 0 0
\(436\) 292063.i 1.53640i
\(437\) 47719.7i 0.249882i
\(438\) 0 0
\(439\) 73628.2 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(440\) 44018.2 0.227367
\(441\) 0 0
\(442\) 33380.9 0.170865
\(443\) 21033.6i 0.107178i 0.998563 + 0.0535890i \(0.0170661\pi\)
−0.998563 + 0.0535890i \(0.982934\pi\)
\(444\) 0 0
\(445\) 164134.i 0.828853i
\(446\) 10981.9i 0.0552087i
\(447\) 0 0
\(448\) −182648. −0.910038
\(449\) −99536.9 −0.493732 −0.246866 0.969050i \(-0.579401\pi\)
−0.246866 + 0.969050i \(0.579401\pi\)
\(450\) 0 0
\(451\) 291962.i 1.43540i
\(452\) 243837.i 1.19350i
\(453\) 0 0
\(454\) −24713.3 −0.119900
\(455\) 149943.i 0.724274i
\(456\) 0 0
\(457\) 282977.i 1.35494i −0.735551 0.677469i \(-0.763077\pi\)
0.735551 0.677469i \(-0.236923\pi\)
\(458\) 14370.4 0.0685077
\(459\) 0 0
\(460\) 242140.i 1.14433i
\(461\) −3519.59 −0.0165611 −0.00828057 0.999966i \(-0.502636\pi\)
−0.00828057 + 0.999966i \(0.502636\pi\)
\(462\) 0 0
\(463\) 295702.i 1.37940i 0.724093 + 0.689702i \(0.242259\pi\)
−0.724093 + 0.689702i \(0.757741\pi\)
\(464\) −68505.4 −0.318192
\(465\) 0 0
\(466\) −20672.9 −0.0951986
\(467\) 77026.5i 0.353189i −0.984284 0.176594i \(-0.943492\pi\)
0.984284 0.176594i \(-0.0565081\pi\)
\(468\) 0 0
\(469\) 2340.24i 0.0106394i
\(470\) 10065.0 0.0455638
\(471\) 0 0
\(472\) 38049.1 + 20329.3i 0.170789 + 0.0912513i
\(473\) 544342. 2.43304
\(474\) 0 0
\(475\) −16923.6 −0.0750077
\(476\) 364417. 1.60837
\(477\) 0 0
\(478\) 9528.62i 0.0417037i
\(479\) −76085.5 −0.331612 −0.165806 0.986158i \(-0.553023\pi\)
−0.165806 + 0.986158i \(0.553023\pi\)
\(480\) 0 0
\(481\) 327052. 1.41360
\(482\) 4959.73i 0.0213483i
\(483\) 0 0
\(484\) −385161. −1.64419
\(485\) 53026.7i 0.225430i
\(486\) 0 0
\(487\) 174246. 0.734690 0.367345 0.930085i \(-0.380267\pi\)
0.367345 + 0.930085i \(0.380267\pi\)
\(488\) −1427.18 −0.00599295
\(489\) 0 0
\(490\) 1177.18i 0.00490285i
\(491\) −328811. −1.36390 −0.681951 0.731398i \(-0.738868\pi\)
−0.681951 + 0.731398i \(0.738868\pi\)
\(492\) 0 0
\(493\) 134006. 0.551354
\(494\) −3856.48 −0.0158029
\(495\) 0 0
\(496\) 209938.i 0.853351i
\(497\) −98558.2 −0.399006
\(498\) 0 0
\(499\) −289611. −1.16309 −0.581546 0.813513i \(-0.697552\pi\)
−0.581546 + 0.813513i \(0.697552\pi\)
\(500\) −264159. −1.05664
\(501\) 0 0
\(502\) 17431.9i 0.0691730i
\(503\) 453397.i 1.79202i 0.444036 + 0.896009i \(0.353546\pi\)
−0.444036 + 0.896009i \(0.646454\pi\)
\(504\) 0 0
\(505\) 82398.4i 0.323099i
\(506\) 65182.6i 0.254584i
\(507\) 0 0
\(508\) 454323. 1.76050
\(509\) 427793.i 1.65119i 0.564260 + 0.825597i \(0.309161\pi\)
−0.564260 + 0.825597i \(0.690839\pi\)
\(510\) 0 0
\(511\) 9921.08i 0.0379942i
\(512\) 122727.i 0.468167i
\(513\) 0 0
\(514\) 41142.5i 0.155727i
\(515\) 254831.i 0.960811i
\(516\) 0 0
\(517\) −283602. −1.06103
\(518\) −34110.6 −0.127125
\(519\) 0 0
\(520\) −39324.2 −0.145430
\(521\) 142162. 0.523730 0.261865 0.965105i \(-0.415663\pi\)
0.261865 + 0.965105i \(0.415663\pi\)
\(522\) 0 0
\(523\) −197294. −0.721289 −0.360645 0.932703i \(-0.617443\pi\)
−0.360645 + 0.932703i \(0.617443\pi\)
\(524\) 314524.i 1.14549i
\(525\) 0 0
\(526\) 3620.64i 0.0130862i
\(527\) 410668.i 1.47866i
\(528\) 0 0
\(529\) −440712. −1.57487
\(530\) 7838.58i 0.0279052i
\(531\) 0 0
\(532\) −42101.0 −0.148754
\(533\) 260828.i 0.918120i
\(534\) 0 0
\(535\) −41361.3 −0.144506
\(536\) −613.755 −0.00213632
\(537\) 0 0
\(538\) 2600.81 0.00898554
\(539\) 33169.2i 0.114171i
\(540\) 0 0
\(541\) 225666.i 0.771029i −0.922702 0.385515i \(-0.874024\pi\)
0.922702 0.385515i \(-0.125976\pi\)
\(542\) 53684.1i 0.182746i
\(543\) 0 0
\(544\) 143586.i 0.485193i
\(545\) 331688.i 1.11670i
\(546\) 0 0
\(547\) 648.964 0.00216893 0.00108447 0.999999i \(-0.499655\pi\)
0.00108447 + 0.999999i \(0.499655\pi\)
\(548\) −303741. −1.01145
\(549\) 0 0
\(550\) −23116.8 −0.0764190
\(551\) −15481.7 −0.0509935
\(552\) 0 0
\(553\) 309170. 1.01099
\(554\) 37409.6i 0.121889i
\(555\) 0 0
\(556\) −320969. −1.03828
\(557\) −179490. −0.578536 −0.289268 0.957248i \(-0.593412\pi\)
−0.289268 + 0.957248i \(0.593412\pi\)
\(558\) 0 0
\(559\) −486295. −1.55624
\(560\) −211569. −0.674647
\(561\) 0 0
\(562\) 5270.56i 0.0166872i
\(563\) 56816.3i 0.179249i −0.995976 0.0896243i \(-0.971433\pi\)
0.995976 0.0896243i \(-0.0285666\pi\)
\(564\) 0 0
\(565\) 276919.i 0.867472i
\(566\) −57325.1 −0.178942
\(567\) 0 0
\(568\) 25848.0i 0.0801180i
\(569\) 361962.i 1.11799i 0.829170 + 0.558996i \(0.188813\pi\)
−0.829170 + 0.558996i \(0.811187\pi\)
\(570\) 0 0
\(571\) 496112.i 1.52163i −0.648972 0.760813i \(-0.724801\pi\)
0.648972 0.760813i \(-0.275199\pi\)
\(572\) 551383. 1.68524
\(573\) 0 0
\(574\) 27203.6i 0.0825661i
\(575\) 255541.i 0.772904i
\(576\) 0 0
\(577\) −9847.40 −0.0295781 −0.0147890 0.999891i \(-0.504708\pi\)
−0.0147890 + 0.999891i \(0.504708\pi\)
\(578\) 59635.5i 0.178504i
\(579\) 0 0
\(580\) −78557.4 −0.233524
\(581\) 321433.i 0.952222i
\(582\) 0 0
\(583\) 220867.i 0.649821i
\(584\) −2601.92 −0.00762900
\(585\) 0 0
\(586\) 8321.06i 0.0242317i
\(587\) 550647.i 1.59807i −0.601282 0.799037i \(-0.705343\pi\)
0.601282 0.799037i \(-0.294657\pi\)
\(588\) 0 0
\(589\) 47444.3i 0.136758i
\(590\) 21502.9 + 11488.9i 0.0617723 + 0.0330045i
\(591\) 0 0
\(592\) 461471.i 1.31674i
\(593\) 455801. 1.29618 0.648090 0.761563i \(-0.275568\pi\)
0.648090 + 0.761563i \(0.275568\pi\)
\(594\) 0 0
\(595\) 413859. 1.16901
\(596\) 135059.i 0.380217i
\(597\) 0 0
\(598\) 58231.6i 0.162838i
\(599\) −556678. −1.55149 −0.775747 0.631044i \(-0.782627\pi\)
−0.775747 + 0.631044i \(0.782627\pi\)
\(600\) 0 0
\(601\) 157038.i 0.434766i 0.976086 + 0.217383i \(0.0697520\pi\)
−0.976086 + 0.217383i \(0.930248\pi\)
\(602\) 50719.1 0.139952
\(603\) 0 0
\(604\) 684609.i 1.87659i
\(605\) −437417. −1.19505
\(606\) 0 0
\(607\) −550937. −1.49529 −0.747644 0.664100i \(-0.768815\pi\)
−0.747644 + 0.664100i \(0.768815\pi\)
\(608\) 16588.4i 0.0448743i
\(609\) 0 0
\(610\) −806.554 −0.00216757
\(611\) 253359. 0.678662
\(612\) 0 0
\(613\) 582881.i 1.55117i −0.631244 0.775585i \(-0.717455\pi\)
0.631244 0.775585i \(-0.282545\pi\)
\(614\) 34905.3i 0.0925880i
\(615\) 0 0
\(616\) −115565. −0.304554
\(617\) 498475. 1.30940 0.654701 0.755888i \(-0.272795\pi\)
0.654701 + 0.755888i \(0.272795\pi\)
\(618\) 0 0
\(619\) 469002. 1.22403 0.612017 0.790844i \(-0.290358\pi\)
0.612017 + 0.790844i \(0.290358\pi\)
\(620\) 240743.i 0.626282i
\(621\) 0 0
\(622\) 9120.40i 0.0235740i
\(623\) 430914.i 1.11023i
\(624\) 0 0
\(625\) −111846. −0.286327
\(626\) −18945.3 −0.0483450
\(627\) 0 0
\(628\) 115283.i 0.292311i
\(629\) 902701.i 2.28162i
\(630\) 0 0
\(631\) −55781.7 −0.140098 −0.0700492 0.997544i \(-0.522316\pi\)
−0.0700492 + 0.997544i \(0.522316\pi\)
\(632\) 81083.4i 0.203001i
\(633\) 0 0
\(634\) 49282.7i 0.122607i
\(635\) 515962. 1.27959
\(636\) 0 0
\(637\) 29632.1i 0.0730269i
\(638\) −21147.1 −0.0519529
\(639\) 0 0
\(640\) 112048.i 0.273556i
\(641\) 77645.7 0.188974 0.0944869 0.995526i \(-0.469879\pi\)
0.0944869 + 0.995526i \(0.469879\pi\)
\(642\) 0 0
\(643\) 295599. 0.714960 0.357480 0.933921i \(-0.383636\pi\)
0.357480 + 0.933921i \(0.383636\pi\)
\(644\) 635711.i 1.53281i
\(645\) 0 0
\(646\) 10644.3i 0.0255066i
\(647\) 392444. 0.937496 0.468748 0.883332i \(-0.344705\pi\)
0.468748 + 0.883332i \(0.344705\pi\)
\(648\) 0 0
\(649\) −605886. 323721.i −1.43847 0.768565i
\(650\) 20651.6 0.0488796
\(651\) 0 0
\(652\) −244319. −0.574728
\(653\) 225024. 0.527720 0.263860 0.964561i \(-0.415004\pi\)
0.263860 + 0.964561i \(0.415004\pi\)
\(654\) 0 0
\(655\) 357197.i 0.832577i
\(656\) −368028. −0.855210
\(657\) 0 0
\(658\) −26424.6 −0.0610318
\(659\) 490926.i 1.13044i 0.824942 + 0.565218i \(0.191208\pi\)
−0.824942 + 0.565218i \(0.808792\pi\)
\(660\) 0 0
\(661\) −209902. −0.480412 −0.240206 0.970722i \(-0.577215\pi\)
−0.240206 + 0.970722i \(0.577215\pi\)
\(662\) 69049.6i 0.157560i
\(663\) 0 0
\(664\) −84299.4 −0.191200
\(665\) −47812.9 −0.108119
\(666\) 0 0
\(667\) 233768.i 0.525453i
\(668\) −597973. −1.34007
\(669\) 0 0
\(670\) −346.855 −0.000772679
\(671\) 22726.2 0.0504757
\(672\) 0 0
\(673\) 535935.i 1.18327i 0.806208 + 0.591633i \(0.201516\pi\)
−0.806208 + 0.591633i \(0.798484\pi\)
\(674\) 34298.4 0.0755012
\(675\) 0 0
\(676\) −39933.1 −0.0873857
\(677\) −589822. −1.28690 −0.643448 0.765489i \(-0.722497\pi\)
−0.643448 + 0.765489i \(0.722497\pi\)
\(678\) 0 0
\(679\) 139215.i 0.301959i
\(680\) 108539.i 0.234730i
\(681\) 0 0
\(682\) 64806.3i 0.139331i
\(683\) 569992.i 1.22188i −0.791678 0.610939i \(-0.790792\pi\)
0.791678 0.610939i \(-0.209208\pi\)
\(684\) 0 0
\(685\) −344951. −0.735150
\(686\) 47238.4i 0.100380i
\(687\) 0 0
\(688\) 686161.i 1.44960i
\(689\) 197314.i 0.415642i
\(690\) 0 0
\(691\) 151727.i 0.317765i 0.987297 + 0.158883i \(0.0507891\pi\)
−0.987297 + 0.158883i \(0.949211\pi\)
\(692\) 191834.i 0.400602i
\(693\) 0 0
\(694\) −30924.7 −0.0642076
\(695\) −364516. −0.754652
\(696\) 0 0
\(697\) 719914. 1.48189
\(698\) −9487.65 −0.0194737
\(699\) 0 0
\(700\) 225453. 0.460108
\(701\) 288172.i 0.586429i −0.956047 0.293214i \(-0.905275\pi\)
0.956047 0.293214i \(-0.0947250\pi\)
\(702\) 0 0
\(703\) 104289.i 0.211021i
\(704\) 762776.i 1.53905i
\(705\) 0 0
\(706\) 23793.8 0.0477369
\(707\) 216328.i 0.432786i
\(708\) 0 0
\(709\) −17367.9 −0.0345506 −0.0172753 0.999851i \(-0.505499\pi\)
−0.0172753 + 0.999851i \(0.505499\pi\)
\(710\) 14607.7i 0.0289777i
\(711\) 0 0
\(712\) −113012. −0.222928
\(713\) −716394. −1.40920
\(714\) 0 0
\(715\) 626191. 1.22488
\(716\) 4572.46i 0.00891915i
\(717\) 0 0
\(718\) 93810.0i 0.181970i
\(719\) 524931.i 1.01542i 0.861529 + 0.507708i \(0.169507\pi\)
−0.861529 + 0.507708i \(0.830493\pi\)
\(720\) 0 0
\(721\) 669029.i 1.28699i
\(722\) 49480.4i 0.0949203i
\(723\) 0 0
\(724\) −202412. −0.386152
\(725\) 82905.1 0.157727
\(726\) 0 0
\(727\) 416443. 0.787928 0.393964 0.919126i \(-0.371103\pi\)
0.393964 + 0.919126i \(0.371103\pi\)
\(728\) 103241. 0.194800
\(729\) 0 0
\(730\) −1470.44 −0.00275931
\(731\) 1.34223e6i 2.51184i
\(732\) 0 0
\(733\) −22984.6 −0.0427788 −0.0213894 0.999771i \(-0.506809\pi\)
−0.0213894 + 0.999771i \(0.506809\pi\)
\(734\) −92404.5 −0.171515
\(735\) 0 0
\(736\) 250480. 0.462400
\(737\) 9773.32 0.0179931
\(738\) 0 0
\(739\) 339929.i 0.622442i 0.950337 + 0.311221i \(0.100738\pi\)
−0.950337 + 0.311221i \(0.899262\pi\)
\(740\) 529184.i 0.966369i
\(741\) 0 0
\(742\) 20579.3i 0.0373786i
\(743\) 28658.0 0.0519121 0.0259560 0.999663i \(-0.491737\pi\)
0.0259560 + 0.999663i \(0.491737\pi\)
\(744\) 0 0
\(745\) 153383.i 0.276353i
\(746\) 65519.0i 0.117731i
\(747\) 0 0
\(748\) 1.52188e6i 2.72005i
\(749\) 108589. 0.193564
\(750\) 0 0
\(751\) 398692.i 0.706899i 0.935454 + 0.353449i \(0.114991\pi\)
−0.935454 + 0.353449i \(0.885009\pi\)
\(752\) 357489.i 0.632161i
\(753\) 0 0
\(754\) 18892.0 0.0332304
\(755\) 777492.i 1.36396i
\(756\) 0 0
\(757\) 55848.3 0.0974581 0.0487290 0.998812i \(-0.484483\pi\)
0.0487290 + 0.998812i \(0.484483\pi\)
\(758\) 91190.6i 0.158713i
\(759\) 0 0
\(760\) 12539.5i 0.0217096i
\(761\) 31452.4 0.0543106 0.0271553 0.999631i \(-0.491355\pi\)
0.0271553 + 0.999631i \(0.491355\pi\)
\(762\) 0 0
\(763\) 870808.i 1.49580i
\(764\) 777600.i 1.33220i
\(765\) 0 0
\(766\) 44508.2i 0.0758547i
\(767\) 541276. + 289199.i 0.920085 + 0.491594i
\(768\) 0 0
\(769\) 1.10780e6i 1.87330i −0.350263 0.936651i \(-0.613908\pi\)
0.350263 0.936651i \(-0.386092\pi\)
\(770\) −65309.9 −0.110153
\(771\) 0 0
\(772\) 813842. 1.36554
\(773\) 392220.i 0.656403i 0.944608 + 0.328201i \(0.106442\pi\)
−0.944608 + 0.328201i \(0.893558\pi\)
\(774\) 0 0
\(775\) 254066.i 0.423003i
\(776\) −36510.8 −0.0606314
\(777\) 0 0
\(778\) 96823.6i 0.159964i
\(779\) −83171.3 −0.137056
\(780\) 0 0
\(781\) 411599.i 0.674795i
\(782\) −160726. −0.262828
\(783\) 0 0
\(784\) −41810.8 −0.0680232
\(785\) 130924.i 0.212461i
\(786\) 0 0
\(787\) 208960. 0.337376 0.168688 0.985670i \(-0.446047\pi\)
0.168688 + 0.985670i \(0.446047\pi\)
\(788\) 477905. 0.769643
\(789\) 0 0
\(790\) 45823.2i 0.0734228i
\(791\) 727018.i 1.16196i
\(792\) 0 0
\(793\) −20302.7 −0.0322855
\(794\) −69605.7 −0.110409
\(795\) 0 0
\(796\) 1.07678e6 1.69942
\(797\)