Properties

Label 531.5.c.d.235.15
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 235.15
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.26

$q$-expansion

\(f(q)\) \(=\) \(q-2.33963i q^{2} +10.5261 q^{4} -30.0439 q^{5} -41.7287 q^{7} -62.0613i q^{8} +O(q^{10})\) \(q-2.33963i q^{2} +10.5261 q^{4} -30.0439 q^{5} -41.7287 q^{7} -62.0613i q^{8} +70.2916i q^{10} -117.370i q^{11} -118.348i q^{13} +97.6298i q^{14} +23.2175 q^{16} -263.868 q^{17} +373.890 q^{19} -316.246 q^{20} -274.603 q^{22} -503.177i q^{23} +277.635 q^{25} -276.891 q^{26} -439.242 q^{28} +549.815 q^{29} +450.839i q^{31} -1047.30i q^{32} +617.355i q^{34} +1253.69 q^{35} +1430.88i q^{37} -874.765i q^{38} +1864.56i q^{40} -1441.61 q^{41} +3202.06i q^{43} -1235.45i q^{44} -1177.25 q^{46} +967.133i q^{47} -659.713 q^{49} -649.564i q^{50} -1245.75i q^{52} -1662.12 q^{53} +3526.26i q^{55} +2589.74i q^{56} -1286.36i q^{58} +(34.3306 + 3480.83i) q^{59} -1712.94i q^{61} +1054.80 q^{62} -2078.82 q^{64} +3555.63i q^{65} +5062.54i q^{67} -2777.51 q^{68} -2933.18i q^{70} -6150.31 q^{71} -4170.58i q^{73} +3347.74 q^{74} +3935.62 q^{76} +4897.71i q^{77} -5990.22 q^{79} -697.544 q^{80} +3372.84i q^{82} -2937.78i q^{83} +7927.64 q^{85} +7491.63 q^{86} -7284.15 q^{88} -3877.47i q^{89} +4938.51i q^{91} -5296.50i q^{92} +2262.73 q^{94} -11233.1 q^{95} +2340.59i q^{97} +1543.48i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.33963i 0.584908i −0.956280 0.292454i \(-0.905528\pi\)
0.956280 0.292454i \(-0.0944718\pi\)
\(3\) 0 0
\(4\) 10.5261 0.657883
\(5\) −30.0439 −1.20176 −0.600878 0.799341i \(-0.705182\pi\)
−0.600878 + 0.799341i \(0.705182\pi\)
\(6\) 0 0
\(7\) −41.7287 −0.851607 −0.425803 0.904816i \(-0.640008\pi\)
−0.425803 + 0.904816i \(0.640008\pi\)
\(8\) 62.0613i 0.969708i
\(9\) 0 0
\(10\) 70.2916i 0.702916i
\(11\) 117.370i 0.970001i −0.874514 0.485000i \(-0.838819\pi\)
0.874514 0.485000i \(-0.161181\pi\)
\(12\) 0 0
\(13\) 118.348i 0.700284i −0.936697 0.350142i \(-0.886133\pi\)
0.936697 0.350142i \(-0.113867\pi\)
\(14\) 97.6298i 0.498111i
\(15\) 0 0
\(16\) 23.2175 0.0906934
\(17\) −263.868 −0.913040 −0.456520 0.889713i \(-0.650904\pi\)
−0.456520 + 0.889713i \(0.650904\pi\)
\(18\) 0 0
\(19\) 373.890 1.03571 0.517854 0.855469i \(-0.326731\pi\)
0.517854 + 0.855469i \(0.326731\pi\)
\(20\) −316.246 −0.790615
\(21\) 0 0
\(22\) −274.603 −0.567361
\(23\) 503.177i 0.951185i −0.879666 0.475592i \(-0.842234\pi\)
0.879666 0.475592i \(-0.157766\pi\)
\(24\) 0 0
\(25\) 277.635 0.444217
\(26\) −276.891 −0.409601
\(27\) 0 0
\(28\) −439.242 −0.560258
\(29\) 549.815 0.653763 0.326882 0.945065i \(-0.394002\pi\)
0.326882 + 0.945065i \(0.394002\pi\)
\(30\) 0 0
\(31\) 450.839i 0.469135i 0.972100 + 0.234568i \(0.0753674\pi\)
−0.972100 + 0.234568i \(0.924633\pi\)
\(32\) 1047.30i 1.02276i
\(33\) 0 0
\(34\) 617.355i 0.534044i
\(35\) 1253.69 1.02342
\(36\) 0 0
\(37\) 1430.88i 1.04520i 0.852577 + 0.522601i \(0.175038\pi\)
−0.852577 + 0.522601i \(0.824962\pi\)
\(38\) 874.765i 0.605793i
\(39\) 0 0
\(40\) 1864.56i 1.16535i
\(41\) −1441.61 −0.857591 −0.428796 0.903401i \(-0.641062\pi\)
−0.428796 + 0.903401i \(0.641062\pi\)
\(42\) 0 0
\(43\) 3202.06i 1.73178i 0.500236 + 0.865889i \(0.333247\pi\)
−0.500236 + 0.865889i \(0.666753\pi\)
\(44\) 1235.45i 0.638147i
\(45\) 0 0
\(46\) −1177.25 −0.556355
\(47\) 967.133i 0.437815i 0.975746 + 0.218907i \(0.0702493\pi\)
−0.975746 + 0.218907i \(0.929751\pi\)
\(48\) 0 0
\(49\) −659.713 −0.274766
\(50\) 649.564i 0.259826i
\(51\) 0 0
\(52\) 1245.75i 0.460705i
\(53\) −1662.12 −0.591713 −0.295856 0.955232i \(-0.595605\pi\)
−0.295856 + 0.955232i \(0.595605\pi\)
\(54\) 0 0
\(55\) 3526.26i 1.16570i
\(56\) 2589.74i 0.825810i
\(57\) 0 0
\(58\) 1286.36i 0.382391i
\(59\) 34.3306 + 3480.83i 0.00986228 + 0.999951i
\(60\) 0 0
\(61\) 1712.94i 0.460343i −0.973150 0.230172i \(-0.926071\pi\)
0.973150 0.230172i \(-0.0739288\pi\)
\(62\) 1054.80 0.274401
\(63\) 0 0
\(64\) −2078.82 −0.507524
\(65\) 3555.63i 0.841570i
\(66\) 0 0
\(67\) 5062.54i 1.12777i 0.825854 + 0.563883i \(0.190693\pi\)
−0.825854 + 0.563883i \(0.809307\pi\)
\(68\) −2777.51 −0.600673
\(69\) 0 0
\(70\) 2933.18i 0.598608i
\(71\) −6150.31 −1.22006 −0.610028 0.792380i \(-0.708842\pi\)
−0.610028 + 0.792380i \(0.708842\pi\)
\(72\) 0 0
\(73\) 4170.58i 0.782619i −0.920259 0.391309i \(-0.872022\pi\)
0.920259 0.391309i \(-0.127978\pi\)
\(74\) 3347.74 0.611347
\(75\) 0 0
\(76\) 3935.62 0.681374
\(77\) 4897.71i 0.826059i
\(78\) 0 0
\(79\) −5990.22 −0.959818 −0.479909 0.877318i \(-0.659330\pi\)
−0.479909 + 0.877318i \(0.659330\pi\)
\(80\) −697.544 −0.108991
\(81\) 0 0
\(82\) 3372.84i 0.501612i
\(83\) 2937.78i 0.426445i −0.977004 0.213222i \(-0.931604\pi\)
0.977004 0.213222i \(-0.0683959\pi\)
\(84\) 0 0
\(85\) 7927.64 1.09725
\(86\) 7491.63 1.01293
\(87\) 0 0
\(88\) −7284.15 −0.940618
\(89\) 3877.47i 0.489518i −0.969584 0.244759i \(-0.921291\pi\)
0.969584 0.244759i \(-0.0787088\pi\)
\(90\) 0 0
\(91\) 4938.51i 0.596367i
\(92\) 5296.50i 0.625769i
\(93\) 0 0
\(94\) 2262.73 0.256081
\(95\) −11233.1 −1.24467
\(96\) 0 0
\(97\) 2340.59i 0.248760i 0.992235 + 0.124380i \(0.0396942\pi\)
−0.992235 + 0.124380i \(0.960306\pi\)
\(98\) 1543.48i 0.160713i
\(99\) 0 0
\(100\) 2922.43 0.292243
\(101\) 18031.3i 1.76761i 0.467860 + 0.883803i \(0.345025\pi\)
−0.467860 + 0.883803i \(0.654975\pi\)
\(102\) 0 0
\(103\) 12277.9i 1.15731i −0.815572 0.578656i \(-0.803577\pi\)
0.815572 0.578656i \(-0.196423\pi\)
\(104\) −7344.83 −0.679071
\(105\) 0 0
\(106\) 3888.75i 0.346097i
\(107\) −7815.43 −0.682630 −0.341315 0.939949i \(-0.610872\pi\)
−0.341315 + 0.939949i \(0.610872\pi\)
\(108\) 0 0
\(109\) 4707.99i 0.396262i −0.980176 0.198131i \(-0.936513\pi\)
0.980176 0.198131i \(-0.0634872\pi\)
\(110\) 8250.13 0.681829
\(111\) 0 0
\(112\) −968.837 −0.0772351
\(113\) 8859.89i 0.693860i 0.937891 + 0.346930i \(0.112776\pi\)
−0.937891 + 0.346930i \(0.887224\pi\)
\(114\) 0 0
\(115\) 15117.4i 1.14309i
\(116\) 5787.42 0.430100
\(117\) 0 0
\(118\) 8143.86 80.3209i 0.584879 0.00576852i
\(119\) 11010.9 0.777551
\(120\) 0 0
\(121\) 865.256 0.0590981
\(122\) −4007.64 −0.269258
\(123\) 0 0
\(124\) 4745.59i 0.308636i
\(125\) 10436.2 0.667916
\(126\) 0 0
\(127\) −7759.81 −0.481109 −0.240554 0.970636i \(-0.577329\pi\)
−0.240554 + 0.970636i \(0.577329\pi\)
\(128\) 11893.2i 0.725901i
\(129\) 0 0
\(130\) 8318.87 0.492241
\(131\) 26570.9i 1.54833i 0.632984 + 0.774165i \(0.281830\pi\)
−0.632984 + 0.774165i \(0.718170\pi\)
\(132\) 0 0
\(133\) −15602.0 −0.882015
\(134\) 11844.5 0.659639
\(135\) 0 0
\(136\) 16376.0i 0.885382i
\(137\) 6370.46 0.339414 0.169707 0.985495i \(-0.445718\pi\)
0.169707 + 0.985495i \(0.445718\pi\)
\(138\) 0 0
\(139\) 26487.6 1.37092 0.685461 0.728109i \(-0.259601\pi\)
0.685461 + 0.728109i \(0.259601\pi\)
\(140\) 13196.5 0.673293
\(141\) 0 0
\(142\) 14389.4i 0.713620i
\(143\) −13890.5 −0.679276
\(144\) 0 0
\(145\) −16518.6 −0.785664
\(146\) −9757.60 −0.457760
\(147\) 0 0
\(148\) 15061.7i 0.687621i
\(149\) 37732.0i 1.69956i 0.527136 + 0.849781i \(0.323266\pi\)
−0.527136 + 0.849781i \(0.676734\pi\)
\(150\) 0 0
\(151\) 19425.6i 0.851961i 0.904732 + 0.425981i \(0.140071\pi\)
−0.904732 + 0.425981i \(0.859929\pi\)
\(152\) 23204.1i 1.00433i
\(153\) 0 0
\(154\) 11458.8 0.483168
\(155\) 13545.0i 0.563786i
\(156\) 0 0
\(157\) 40542.8i 1.64480i −0.568907 0.822402i \(-0.692634\pi\)
0.568907 0.822402i \(-0.307366\pi\)
\(158\) 14014.9i 0.561405i
\(159\) 0 0
\(160\) 31465.0i 1.22910i
\(161\) 20996.9i 0.810036i
\(162\) 0 0
\(163\) −6024.65 −0.226755 −0.113377 0.993552i \(-0.536167\pi\)
−0.113377 + 0.993552i \(0.536167\pi\)
\(164\) −15174.6 −0.564195
\(165\) 0 0
\(166\) −6873.32 −0.249431
\(167\) 38348.4 1.37504 0.687519 0.726167i \(-0.258700\pi\)
0.687519 + 0.726167i \(0.258700\pi\)
\(168\) 0 0
\(169\) 14554.8 0.509602
\(170\) 18547.7i 0.641790i
\(171\) 0 0
\(172\) 33705.3i 1.13931i
\(173\) 6642.40i 0.221938i −0.993824 0.110969i \(-0.964604\pi\)
0.993824 0.110969i \(-0.0353955\pi\)
\(174\) 0 0
\(175\) −11585.4 −0.378298
\(176\) 2725.04i 0.0879727i
\(177\) 0 0
\(178\) −9071.85 −0.286323
\(179\) 15505.0i 0.483910i 0.970287 + 0.241955i \(0.0777887\pi\)
−0.970287 + 0.241955i \(0.922211\pi\)
\(180\) 0 0
\(181\) −7207.84 −0.220013 −0.110006 0.993931i \(-0.535087\pi\)
−0.110006 + 0.993931i \(0.535087\pi\)
\(182\) 11554.3 0.348819
\(183\) 0 0
\(184\) −31227.8 −0.922372
\(185\) 42989.3i 1.25608i
\(186\) 0 0
\(187\) 30970.3i 0.885649i
\(188\) 10180.2i 0.288031i
\(189\) 0 0
\(190\) 26281.3i 0.728015i
\(191\) 24422.5i 0.669459i 0.942314 + 0.334729i \(0.108645\pi\)
−0.942314 + 0.334729i \(0.891355\pi\)
\(192\) 0 0
\(193\) −42315.7 −1.13602 −0.568011 0.823021i \(-0.692287\pi\)
−0.568011 + 0.823021i \(0.692287\pi\)
\(194\) 5476.11 0.145502
\(195\) 0 0
\(196\) −6944.23 −0.180764
\(197\) 3081.53 0.0794024 0.0397012 0.999212i \(-0.487359\pi\)
0.0397012 + 0.999212i \(0.487359\pi\)
\(198\) 0 0
\(199\) 71081.9 1.79495 0.897476 0.441064i \(-0.145399\pi\)
0.897476 + 0.441064i \(0.145399\pi\)
\(200\) 17230.4i 0.430761i
\(201\) 0 0
\(202\) 42186.7 1.03389
\(203\) −22943.1 −0.556749
\(204\) 0 0
\(205\) 43311.6 1.03062
\(206\) −28725.8 −0.676920
\(207\) 0 0
\(208\) 2747.74i 0.0635111i
\(209\) 43883.5i 1.00464i
\(210\) 0 0
\(211\) 31417.1i 0.705670i −0.935686 0.352835i \(-0.885218\pi\)
0.935686 0.352835i \(-0.114782\pi\)
\(212\) −17495.7 −0.389278
\(213\) 0 0
\(214\) 18285.2i 0.399275i
\(215\) 96202.3i 2.08117i
\(216\) 0 0
\(217\) 18812.9i 0.399519i
\(218\) −11015.0 −0.231777
\(219\) 0 0
\(220\) 37117.8i 0.766897i
\(221\) 31228.3i 0.639387i
\(222\) 0 0
\(223\) −55061.3 −1.10723 −0.553613 0.832774i \(-0.686751\pi\)
−0.553613 + 0.832774i \(0.686751\pi\)
\(224\) 43702.6i 0.870986i
\(225\) 0 0
\(226\) 20728.9 0.405844
\(227\) 23259.5i 0.451386i −0.974198 0.225693i \(-0.927535\pi\)
0.974198 0.225693i \(-0.0724646\pi\)
\(228\) 0 0
\(229\) 16859.6i 0.321496i −0.986995 0.160748i \(-0.948609\pi\)
0.986995 0.160748i \(-0.0513907\pi\)
\(230\) 35369.1 0.668603
\(231\) 0 0
\(232\) 34122.3i 0.633960i
\(233\) 56400.3i 1.03889i −0.854504 0.519445i \(-0.826139\pi\)
0.854504 0.519445i \(-0.173861\pi\)
\(234\) 0 0
\(235\) 29056.4i 0.526147i
\(236\) 361.368 + 36639.7i 0.00648823 + 0.657851i
\(237\) 0 0
\(238\) 25761.4i 0.454795i
\(239\) −17569.0 −0.307575 −0.153788 0.988104i \(-0.549147\pi\)
−0.153788 + 0.988104i \(0.549147\pi\)
\(240\) 0 0
\(241\) 97271.4 1.67475 0.837377 0.546625i \(-0.184088\pi\)
0.837377 + 0.546625i \(0.184088\pi\)
\(242\) 2024.38i 0.0345670i
\(243\) 0 0
\(244\) 18030.6i 0.302852i
\(245\) 19820.3 0.330202
\(246\) 0 0
\(247\) 44249.2i 0.725289i
\(248\) 27979.7 0.454924
\(249\) 0 0
\(250\) 24416.8i 0.390669i
\(251\) 41339.8 0.656177 0.328089 0.944647i \(-0.393596\pi\)
0.328089 + 0.944647i \(0.393596\pi\)
\(252\) 0 0
\(253\) −59057.9 −0.922650
\(254\) 18155.1i 0.281404i
\(255\) 0 0
\(256\) −61086.7 −0.932109
\(257\) 14776.8 0.223725 0.111863 0.993724i \(-0.464318\pi\)
0.111863 + 0.993724i \(0.464318\pi\)
\(258\) 0 0
\(259\) 59708.9i 0.890102i
\(260\) 37427.1i 0.553655i
\(261\) 0 0
\(262\) 62166.1 0.905630
\(263\) −46511.4 −0.672431 −0.336216 0.941785i \(-0.609147\pi\)
−0.336216 + 0.941785i \(0.609147\pi\)
\(264\) 0 0
\(265\) 49936.6 0.711094
\(266\) 36502.8i 0.515897i
\(267\) 0 0
\(268\) 53289.0i 0.741939i
\(269\) 52167.3i 0.720931i −0.932773 0.360465i \(-0.882618\pi\)
0.932773 0.360465i \(-0.117382\pi\)
\(270\) 0 0
\(271\) −118272. −1.61044 −0.805220 0.592977i \(-0.797953\pi\)
−0.805220 + 0.592977i \(0.797953\pi\)
\(272\) −6126.37 −0.0828066
\(273\) 0 0
\(274\) 14904.5i 0.198526i
\(275\) 32586.1i 0.430891i
\(276\) 0 0
\(277\) −116769. −1.52184 −0.760919 0.648847i \(-0.775252\pi\)
−0.760919 + 0.648847i \(0.775252\pi\)
\(278\) 61971.2i 0.801863i
\(279\) 0 0
\(280\) 77805.9i 0.992422i
\(281\) −148473. −1.88034 −0.940169 0.340710i \(-0.889333\pi\)
−0.940169 + 0.340710i \(0.889333\pi\)
\(282\) 0 0
\(283\) 30655.6i 0.382769i −0.981515 0.191385i \(-0.938702\pi\)
0.981515 0.191385i \(-0.0612978\pi\)
\(284\) −64738.9 −0.802655
\(285\) 0 0
\(286\) 32498.7i 0.397314i
\(287\) 60156.6 0.730331
\(288\) 0 0
\(289\) −13894.4 −0.166359
\(290\) 38647.4i 0.459541i
\(291\) 0 0
\(292\) 43900.0i 0.514872i
\(293\) −77404.6 −0.901636 −0.450818 0.892616i \(-0.648868\pi\)
−0.450818 + 0.892616i \(0.648868\pi\)
\(294\) 0 0
\(295\) −1031.42 104578.i −0.0118520 1.20170i
\(296\) 88802.5 1.01354
\(297\) 0 0
\(298\) 88278.8 0.994086
\(299\) −59550.0 −0.666100
\(300\) 0 0
\(301\) 133618.i 1.47479i
\(302\) 45448.7 0.498319
\(303\) 0 0
\(304\) 8680.80 0.0939318
\(305\) 51463.3i 0.553220i
\(306\) 0 0
\(307\) −124048. −1.31617 −0.658085 0.752943i \(-0.728633\pi\)
−0.658085 + 0.752943i \(0.728633\pi\)
\(308\) 51553.9i 0.543451i
\(309\) 0 0
\(310\) −31690.2 −0.329762
\(311\) 3890.57 0.0402247 0.0201123 0.999798i \(-0.493598\pi\)
0.0201123 + 0.999798i \(0.493598\pi\)
\(312\) 0 0
\(313\) 18870.8i 0.192620i 0.995351 + 0.0963099i \(0.0307040\pi\)
−0.995351 + 0.0963099i \(0.969296\pi\)
\(314\) −94855.1 −0.962058
\(315\) 0 0
\(316\) −63053.9 −0.631448
\(317\) 31502.4 0.313491 0.156745 0.987639i \(-0.449900\pi\)
0.156745 + 0.987639i \(0.449900\pi\)
\(318\) 0 0
\(319\) 64531.9i 0.634151i
\(320\) 62455.8 0.609920
\(321\) 0 0
\(322\) 49125.1 0.473796
\(323\) −98657.8 −0.945642
\(324\) 0 0
\(325\) 32857.6i 0.311078i
\(326\) 14095.4i 0.132631i
\(327\) 0 0
\(328\) 89468.3i 0.831614i
\(329\) 40357.2i 0.372846i
\(330\) 0 0
\(331\) 4457.32 0.0406834 0.0203417 0.999793i \(-0.493525\pi\)
0.0203417 + 0.999793i \(0.493525\pi\)
\(332\) 30923.4i 0.280551i
\(333\) 0 0
\(334\) 89721.1i 0.804270i
\(335\) 152099.i 1.35530i
\(336\) 0 0
\(337\) 46555.0i 0.409927i −0.978770 0.204964i \(-0.934292\pi\)
0.978770 0.204964i \(-0.0657076\pi\)
\(338\) 34052.7i 0.298070i
\(339\) 0 0
\(340\) 83447.3 0.721863
\(341\) 52915.0 0.455061
\(342\) 0 0
\(343\) 127720. 1.08560
\(344\) 198724. 1.67932
\(345\) 0 0
\(346\) −15540.8 −0.129814
\(347\) 173485.i 1.44080i 0.693560 + 0.720399i \(0.256041\pi\)
−0.693560 + 0.720399i \(0.743959\pi\)
\(348\) 0 0
\(349\) 160924.i 1.32120i 0.750737 + 0.660601i \(0.229699\pi\)
−0.750737 + 0.660601i \(0.770301\pi\)
\(350\) 27105.5i 0.221269i
\(351\) 0 0
\(352\) −122922. −0.992074
\(353\) 109289.i 0.877055i 0.898718 + 0.438527i \(0.144500\pi\)
−0.898718 + 0.438527i \(0.855500\pi\)
\(354\) 0 0
\(355\) 184779. 1.46621
\(356\) 40814.8i 0.322045i
\(357\) 0 0
\(358\) 36275.9 0.283043
\(359\) −127764. −0.991336 −0.495668 0.868512i \(-0.665077\pi\)
−0.495668 + 0.868512i \(0.665077\pi\)
\(360\) 0 0
\(361\) 9472.92 0.0726891
\(362\) 16863.7i 0.128687i
\(363\) 0 0
\(364\) 51983.4i 0.392339i
\(365\) 125300.i 0.940516i
\(366\) 0 0
\(367\) 131014.i 0.972718i −0.873759 0.486359i \(-0.838325\pi\)
0.873759 0.486359i \(-0.161675\pi\)
\(368\) 11682.5i 0.0862662i
\(369\) 0 0
\(370\) −100579. −0.734690
\(371\) 69358.2 0.503907
\(372\) 0 0
\(373\) 162049. 1.16474 0.582369 0.812924i \(-0.302126\pi\)
0.582369 + 0.812924i \(0.302126\pi\)
\(374\) 72459.0 0.518023
\(375\) 0 0
\(376\) 60021.6 0.424553
\(377\) 65069.5i 0.457820i
\(378\) 0 0
\(379\) −170314. −1.18569 −0.592847 0.805315i \(-0.701996\pi\)
−0.592847 + 0.805315i \(0.701996\pi\)
\(380\) −118241. −0.818845
\(381\) 0 0
\(382\) 57139.7 0.391572
\(383\) −197908. −1.34916 −0.674582 0.738200i \(-0.735676\pi\)
−0.674582 + 0.738200i \(0.735676\pi\)
\(384\) 0 0
\(385\) 147146.i 0.992722i
\(386\) 99003.0i 0.664468i
\(387\) 0 0
\(388\) 24637.3i 0.163655i
\(389\) 130696. 0.863699 0.431849 0.901946i \(-0.357861\pi\)
0.431849 + 0.901946i \(0.357861\pi\)
\(390\) 0 0
\(391\) 132772.i 0.868470i
\(392\) 40942.7i 0.266443i
\(393\) 0 0
\(394\) 7209.64i 0.0464431i
\(395\) 179970. 1.15347
\(396\) 0 0
\(397\) 35127.3i 0.222876i 0.993771 + 0.111438i \(0.0355457\pi\)
−0.993771 + 0.111438i \(0.964454\pi\)
\(398\) 166305.i 1.04988i
\(399\) 0 0
\(400\) 6446.00 0.0402875
\(401\) 177132.i 1.10156i −0.834650 0.550780i \(-0.814330\pi\)
0.834650 0.550780i \(-0.185670\pi\)
\(402\) 0 0
\(403\) 53355.9 0.328528
\(404\) 189800.i 1.16288i
\(405\) 0 0
\(406\) 53678.3i 0.325647i
\(407\) 167943. 1.01385
\(408\) 0 0
\(409\) 194928.i 1.16527i −0.812733 0.582636i \(-0.802021\pi\)
0.812733 0.582636i \(-0.197979\pi\)
\(410\) 101333.i 0.602815i
\(411\) 0 0
\(412\) 129239.i 0.761376i
\(413\) −1432.57 145251.i −0.00839878 0.851565i
\(414\) 0 0
\(415\) 88262.3i 0.512483i
\(416\) −123946. −0.716219
\(417\) 0 0
\(418\) −102671. −0.587620
\(419\) 86241.5i 0.491234i −0.969367 0.245617i \(-0.921009\pi\)
0.969367 0.245617i \(-0.0789906\pi\)
\(420\) 0 0
\(421\) 51752.7i 0.291990i −0.989285 0.145995i \(-0.953362\pi\)
0.989285 0.145995i \(-0.0466384\pi\)
\(422\) −73504.5 −0.412752
\(423\) 0 0
\(424\) 103153.i 0.573789i
\(425\) −73259.2 −0.405587
\(426\) 0 0
\(427\) 71478.7i 0.392031i
\(428\) −82266.2 −0.449091
\(429\) 0 0
\(430\) −225078. −1.21729
\(431\) 184933.i 0.995545i −0.867308 0.497773i \(-0.834151\pi\)
0.867308 0.497773i \(-0.165849\pi\)
\(432\) 0 0
\(433\) −201660. −1.07558 −0.537790 0.843079i \(-0.680741\pi\)
−0.537790 + 0.843079i \(0.680741\pi\)
\(434\) −44015.3 −0.233681
\(435\) 0 0
\(436\) 49556.9i 0.260694i
\(437\) 188133.i 0.985149i
\(438\) 0 0
\(439\) 264954. 1.37480 0.687402 0.726277i \(-0.258751\pi\)
0.687402 + 0.726277i \(0.258751\pi\)
\(440\) 218844. 1.13039
\(441\) 0 0
\(442\) 73062.7 0.373982
\(443\) 377029.i 1.92117i −0.277978 0.960587i \(-0.589664\pi\)
0.277978 0.960587i \(-0.410336\pi\)
\(444\) 0 0
\(445\) 116494.i 0.588281i
\(446\) 128823.i 0.647625i
\(447\) 0 0
\(448\) 86746.5 0.432211
\(449\) −108089. −0.536155 −0.268078 0.963397i \(-0.586388\pi\)
−0.268078 + 0.963397i \(0.586388\pi\)
\(450\) 0 0
\(451\) 169202.i 0.831864i
\(452\) 93260.4i 0.456479i
\(453\) 0 0
\(454\) −54418.5 −0.264019
\(455\) 148372.i 0.716687i
\(456\) 0 0
\(457\) 265515.i 1.27132i 0.771968 + 0.635662i \(0.219273\pi\)
−0.771968 + 0.635662i \(0.780727\pi\)
\(458\) −39445.2 −0.188046
\(459\) 0 0
\(460\) 159128.i 0.752021i
\(461\) −216820. −1.02023 −0.510114 0.860107i \(-0.670397\pi\)
−0.510114 + 0.860107i \(0.670397\pi\)
\(462\) 0 0
\(463\) 245917.i 1.14716i 0.819148 + 0.573582i \(0.194447\pi\)
−0.819148 + 0.573582i \(0.805553\pi\)
\(464\) 12765.3 0.0592920
\(465\) 0 0
\(466\) −131956. −0.607654
\(467\) 198167.i 0.908651i 0.890836 + 0.454326i \(0.150120\pi\)
−0.890836 + 0.454326i \(0.849880\pi\)
\(468\) 0 0
\(469\) 211254.i 0.960414i
\(470\) −67981.3 −0.307747
\(471\) 0 0
\(472\) 216025. 2130.60i 0.969661 0.00956353i
\(473\) 375826. 1.67983
\(474\) 0 0
\(475\) 103805. 0.460078
\(476\) 115902. 0.511538
\(477\) 0 0
\(478\) 41105.0i 0.179903i
\(479\) 419986. 1.83048 0.915238 0.402915i \(-0.132003\pi\)
0.915238 + 0.402915i \(0.132003\pi\)
\(480\) 0 0
\(481\) 169342. 0.731939
\(482\) 227579.i 0.979577i
\(483\) 0 0
\(484\) 9107.80 0.0388797
\(485\) 70320.3i 0.298949i
\(486\) 0 0
\(487\) −336768. −1.41995 −0.709975 0.704227i \(-0.751294\pi\)
−0.709975 + 0.704227i \(0.751294\pi\)
\(488\) −106307. −0.446399
\(489\) 0 0
\(490\) 46372.3i 0.193137i
\(491\) −279399. −1.15894 −0.579471 0.814993i \(-0.696741\pi\)
−0.579471 + 0.814993i \(0.696741\pi\)
\(492\) 0 0
\(493\) −145079. −0.596912
\(494\) −103527. −0.424227
\(495\) 0 0
\(496\) 10467.4i 0.0425474i
\(497\) 256644. 1.03901
\(498\) 0 0
\(499\) 23800.2 0.0955826 0.0477913 0.998857i \(-0.484782\pi\)
0.0477913 + 0.998857i \(0.484782\pi\)
\(500\) 109853. 0.439410
\(501\) 0 0
\(502\) 96719.9i 0.383803i
\(503\) 25529.1i 0.100902i 0.998727 + 0.0504509i \(0.0160659\pi\)
−0.998727 + 0.0504509i \(0.983934\pi\)
\(504\) 0 0
\(505\) 541732.i 2.12423i
\(506\) 138174.i 0.539665i
\(507\) 0 0
\(508\) −81680.7 −0.316513
\(509\) 6537.03i 0.0252316i 0.999920 + 0.0126158i \(0.00401584\pi\)
−0.999920 + 0.0126158i \(0.995984\pi\)
\(510\) 0 0
\(511\) 174033.i 0.666483i
\(512\) 47370.3i 0.180703i
\(513\) 0 0
\(514\) 34572.3i 0.130859i
\(515\) 368877.i 1.39081i
\(516\) 0 0
\(517\) 113513. 0.424681
\(518\) −139697. −0.520627
\(519\) 0 0
\(520\) 220667. 0.816078
\(521\) 307294. 1.13208 0.566042 0.824376i \(-0.308474\pi\)
0.566042 + 0.824376i \(0.308474\pi\)
\(522\) 0 0
\(523\) −92199.8 −0.337075 −0.168537 0.985695i \(-0.553904\pi\)
−0.168537 + 0.985695i \(0.553904\pi\)
\(524\) 279689.i 1.01862i
\(525\) 0 0
\(526\) 108819.i 0.393310i
\(527\) 118962.i 0.428339i
\(528\) 0 0
\(529\) 26654.1 0.0952472
\(530\) 116833.i 0.415924i
\(531\) 0 0
\(532\) −164228. −0.580263
\(533\) 170612.i 0.600557i
\(534\) 0 0
\(535\) 234806. 0.820354
\(536\) 314188. 1.09360
\(537\) 0 0
\(538\) −122052. −0.421678
\(539\) 77430.6i 0.266523i
\(540\) 0 0
\(541\) 234603.i 0.801564i −0.916173 0.400782i \(-0.868738\pi\)
0.916173 0.400782i \(-0.131262\pi\)
\(542\) 276713.i 0.941958i
\(543\) 0 0
\(544\) 276350.i 0.933816i
\(545\) 141446.i 0.476210i
\(546\) 0 0
\(547\) −246372. −0.823411 −0.411706 0.911317i \(-0.635067\pi\)
−0.411706 + 0.911317i \(0.635067\pi\)
\(548\) 67056.3 0.223295
\(549\) 0 0
\(550\) −76239.4 −0.252031
\(551\) 205570. 0.677107
\(552\) 0 0
\(553\) 249964. 0.817387
\(554\) 273197.i 0.890134i
\(555\) 0 0
\(556\) 278812. 0.901907
\(557\) −567300. −1.82853 −0.914266 0.405115i \(-0.867231\pi\)
−0.914266 + 0.405115i \(0.867231\pi\)
\(558\) 0 0
\(559\) 378957. 1.21274
\(560\) 29107.6 0.0928177
\(561\) 0 0
\(562\) 347373.i 1.09982i
\(563\) 327570.i 1.03344i 0.856153 + 0.516722i \(0.172848\pi\)
−0.856153 + 0.516722i \(0.827152\pi\)
\(564\) 0 0
\(565\) 266186.i 0.833850i
\(566\) −71722.8 −0.223885
\(567\) 0 0
\(568\) 381696.i 1.18310i
\(569\) 379301.i 1.17154i 0.810476 + 0.585772i \(0.199209\pi\)
−0.810476 + 0.585772i \(0.800791\pi\)
\(570\) 0 0
\(571\) 580504.i 1.78046i 0.455509 + 0.890231i \(0.349457\pi\)
−0.455509 + 0.890231i \(0.650543\pi\)
\(572\) −146213. −0.446884
\(573\) 0 0
\(574\) 140744.i 0.427176i
\(575\) 139700.i 0.422532i
\(576\) 0 0
\(577\) −34030.7 −0.102216 −0.0511080 0.998693i \(-0.516275\pi\)
−0.0511080 + 0.998693i \(0.516275\pi\)
\(578\) 32507.8i 0.0973044i
\(579\) 0 0
\(580\) −173877. −0.516875
\(581\) 122590.i 0.363163i
\(582\) 0 0
\(583\) 195083.i 0.573962i
\(584\) −258831. −0.758912
\(585\) 0 0
\(586\) 181098.i 0.527374i
\(587\) 360445.i 1.04607i 0.852310 + 0.523037i \(0.175201\pi\)
−0.852310 + 0.523037i \(0.824799\pi\)
\(588\) 0 0
\(589\) 168564.i 0.485886i
\(590\) −244673. + 2413.15i −0.702882 + 0.00693235i
\(591\) 0 0
\(592\) 33221.5i 0.0947930i
\(593\) −203460. −0.578589 −0.289294 0.957240i \(-0.593421\pi\)
−0.289294 + 0.957240i \(0.593421\pi\)
\(594\) 0 0
\(595\) −330810. −0.934426
\(596\) 397172.i 1.11811i
\(597\) 0 0
\(598\) 139325.i 0.389607i
\(599\) −68798.6 −0.191746 −0.0958729 0.995394i \(-0.530564\pi\)
−0.0958729 + 0.995394i \(0.530564\pi\)
\(600\) 0 0
\(601\) 549707.i 1.52189i 0.648818 + 0.760943i \(0.275263\pi\)
−0.648818 + 0.760943i \(0.724737\pi\)
\(602\) −312616. −0.862618
\(603\) 0 0
\(604\) 204476.i 0.560491i
\(605\) −25995.7 −0.0710215
\(606\) 0 0
\(607\) −350392. −0.950991 −0.475495 0.879718i \(-0.657731\pi\)
−0.475495 + 0.879718i \(0.657731\pi\)
\(608\) 391576.i 1.05928i
\(609\) 0 0
\(610\) 120405. 0.323583
\(611\) 114458. 0.306595
\(612\) 0 0
\(613\) 450003.i 1.19755i −0.800917 0.598776i \(-0.795654\pi\)
0.800917 0.598776i \(-0.204346\pi\)
\(614\) 290226.i 0.769838i
\(615\) 0 0
\(616\) 303958. 0.801037
\(617\) −165810. −0.435552 −0.217776 0.975999i \(-0.569880\pi\)
−0.217776 + 0.975999i \(0.569880\pi\)
\(618\) 0 0
\(619\) −286451. −0.747599 −0.373799 0.927510i \(-0.621945\pi\)
−0.373799 + 0.927510i \(0.621945\pi\)
\(620\) 142576.i 0.370905i
\(621\) 0 0
\(622\) 9102.49i 0.0235277i
\(623\) 161802.i 0.416877i
\(624\) 0 0
\(625\) −487066. −1.24689
\(626\) 44150.6 0.112665
\(627\) 0 0
\(628\) 426758.i 1.08209i
\(629\) 377565.i 0.954312i
\(630\) 0 0
\(631\) 94171.7 0.236517 0.118258 0.992983i \(-0.462269\pi\)
0.118258 + 0.992983i \(0.462269\pi\)
\(632\) 371761.i 0.930743i
\(633\) 0 0
\(634\) 73703.9i 0.183363i
\(635\) 233135. 0.578175
\(636\) 0 0
\(637\) 78075.7i 0.192414i
\(638\) −150981. −0.370920
\(639\) 0 0
\(640\) 357317.i 0.872356i
\(641\) −739487. −1.79976 −0.899880 0.436137i \(-0.856346\pi\)
−0.899880 + 0.436137i \(0.856346\pi\)
\(642\) 0 0
\(643\) 446727. 1.08049 0.540244 0.841508i \(-0.318332\pi\)
0.540244 + 0.841508i \(0.318332\pi\)
\(644\) 221016.i 0.532909i
\(645\) 0 0
\(646\) 230823.i 0.553113i
\(647\) 370357. 0.884732 0.442366 0.896835i \(-0.354139\pi\)
0.442366 + 0.896835i \(0.354139\pi\)
\(648\) 0 0
\(649\) 408546. 4029.38i 0.969954 0.00956642i
\(650\) −76874.6 −0.181952
\(651\) 0 0
\(652\) −63416.2 −0.149178
\(653\) 339613. 0.796449 0.398224 0.917288i \(-0.369627\pi\)
0.398224 + 0.917288i \(0.369627\pi\)
\(654\) 0 0
\(655\) 798293.i 1.86071i
\(656\) −33470.6 −0.0777779
\(657\) 0 0
\(658\) −94421.0 −0.218081
\(659\) 387109.i 0.891378i 0.895188 + 0.445689i \(0.147041\pi\)
−0.895188 + 0.445689i \(0.852959\pi\)
\(660\) 0 0
\(661\) −367793. −0.841784 −0.420892 0.907111i \(-0.638283\pi\)
−0.420892 + 0.907111i \(0.638283\pi\)
\(662\) 10428.5i 0.0237961i
\(663\) 0 0
\(664\) −182322. −0.413527
\(665\) 468744. 1.05997
\(666\) 0 0
\(667\) 276654.i 0.621850i
\(668\) 403660. 0.904614
\(669\) 0 0
\(670\) −355854. −0.792725
\(671\) −201048. −0.446533
\(672\) 0 0
\(673\) 356274.i 0.786600i −0.919410 0.393300i \(-0.871333\pi\)
0.919410 0.393300i \(-0.128667\pi\)
\(674\) −108922. −0.239770
\(675\) 0 0
\(676\) 153205. 0.335259
\(677\) −427232. −0.932152 −0.466076 0.884745i \(-0.654333\pi\)
−0.466076 + 0.884745i \(0.654333\pi\)
\(678\) 0 0
\(679\) 97669.7i 0.211846i
\(680\) 492000.i 1.06401i
\(681\) 0 0
\(682\) 123802.i 0.266169i
\(683\) 816072.i 1.74939i −0.484673 0.874695i \(-0.661061\pi\)
0.484673 0.874695i \(-0.338939\pi\)
\(684\) 0 0
\(685\) −191393. −0.407893
\(686\) 298817.i 0.634975i
\(687\) 0 0
\(688\) 74343.8i 0.157061i
\(689\) 196709.i 0.414367i
\(690\) 0 0
\(691\) 101566.i 0.212712i 0.994328 + 0.106356i \(0.0339183\pi\)
−0.994328 + 0.106356i \(0.966082\pi\)
\(692\) 69918.7i 0.146010i
\(693\) 0 0
\(694\) 405891. 0.842733
\(695\) −795790. −1.64751
\(696\) 0 0
\(697\) 380396. 0.783015
\(698\) 376502. 0.772782
\(699\) 0 0
\(700\) −121949. −0.248876
\(701\) 265473.i 0.540236i 0.962827 + 0.270118i \(0.0870628\pi\)
−0.962827 + 0.270118i \(0.912937\pi\)
\(702\) 0 0
\(703\) 534993.i 1.08252i
\(704\) 243991.i 0.492299i
\(705\) 0 0
\(706\) 255696. 0.512996
\(707\) 752425.i 1.50530i
\(708\) 0 0
\(709\) −380605. −0.757150 −0.378575 0.925571i \(-0.623586\pi\)
−0.378575 + 0.925571i \(0.623586\pi\)
\(710\) 432315.i 0.857597i
\(711\) 0 0
\(712\) −240641. −0.474689
\(713\) 226852. 0.446234
\(714\) 0 0
\(715\) 417325. 0.816324
\(716\) 163207.i 0.318356i
\(717\) 0 0
\(718\) 298921.i 0.579840i
\(719\) 356832.i 0.690249i −0.938557 0.345124i \(-0.887837\pi\)
0.938557 0.345124i \(-0.112163\pi\)
\(720\) 0 0
\(721\) 512342.i 0.985575i
\(722\) 22163.1i 0.0425164i
\(723\) 0 0
\(724\) −75870.6 −0.144743
\(725\) 152648. 0.290413
\(726\) 0 0
\(727\) 84925.0 0.160682 0.0803409 0.996767i \(-0.474399\pi\)
0.0803409 + 0.996767i \(0.474399\pi\)
\(728\) 306491. 0.578302
\(729\) 0 0
\(730\) 293156. 0.550115
\(731\) 844922.i 1.58118i
\(732\) 0 0
\(733\) −359849. −0.669750 −0.334875 0.942263i \(-0.608694\pi\)
−0.334875 + 0.942263i \(0.608694\pi\)
\(734\) −306525. −0.568950
\(735\) 0 0
\(736\) −526978. −0.972830
\(737\) 594191. 1.09393
\(738\) 0 0
\(739\) 813633.i 1.48984i −0.667154 0.744920i \(-0.732488\pi\)
0.667154 0.744920i \(-0.267512\pi\)
\(740\) 452511.i 0.826353i
\(741\) 0 0
\(742\) 162273.i 0.294739i
\(743\) 407788. 0.738680 0.369340 0.929294i \(-0.379584\pi\)
0.369340 + 0.929294i \(0.379584\pi\)
\(744\) 0 0
\(745\) 1.13361e6i 2.04246i
\(746\) 379135.i 0.681265i
\(747\) 0 0
\(748\) 325997.i 0.582654i
\(749\) 326128. 0.581332
\(750\) 0 0
\(751\) 726139.i 1.28748i 0.765245 + 0.643739i \(0.222618\pi\)
−0.765245 + 0.643739i \(0.777382\pi\)
\(752\) 22454.4i 0.0397069i
\(753\) 0 0
\(754\) −152239. −0.267782
\(755\) 583620.i 1.02385i
\(756\) 0 0
\(757\) 226073. 0.394508 0.197254 0.980352i \(-0.436798\pi\)
0.197254 + 0.980352i \(0.436798\pi\)
\(758\) 398472.i 0.693521i
\(759\) 0 0
\(760\) 697142.i 1.20696i
\(761\) 32895.1 0.0568018 0.0284009 0.999597i \(-0.490958\pi\)
0.0284009 + 0.999597i \(0.490958\pi\)
\(762\) 0 0
\(763\) 196458.i 0.337459i
\(764\) 257075.i 0.440426i
\(765\) 0 0
\(766\) 463031.i 0.789137i
\(767\) 411949. 4062.96i 0.700250 0.00690639i
\(768\) 0 0
\(769\) 954320.i 1.61377i −0.590710 0.806884i \(-0.701152\pi\)
0.590710 0.806884i \(-0.298848\pi\)
\(770\) −344268. −0.580650
\(771\) 0 0
\(772\) −445420. −0.747370
\(773\) 430986.i 0.721280i −0.932705 0.360640i \(-0.882558\pi\)
0.932705 0.360640i \(-0.117442\pi\)
\(774\) 0 0
\(775\) 125169.i 0.208398i
\(776\) 145260. 0.241225
\(777\) 0 0
\(778\) 305780.i 0.505184i
\(779\) −539004. −0.888213
\(780\) 0 0
\(781\) 721862.i 1.18346i
\(782\) 310639. 0.507974
\(783\) 0 0
\(784\) −15316.9 −0.0249195
\(785\) 1.21806e6i 1.97665i
\(786\) 0 0
\(787\) 449468. 0.725687 0.362844 0.931850i \(-0.381806\pi\)
0.362844 + 0.931850i \(0.381806\pi\)
\(788\) 32436.6 0.0522375
\(789\) 0 0
\(790\) 421062.i 0.674671i
\(791\) 369712.i 0.590895i
\(792\) 0 0
\(793\) −202723. −0.322371
\(794\) 82184.9 0.130362
\(795\) 0 0
\(796\) 748217. 1.18087
\(797\) 300325.i