Properties

Label 531.5.c.d.235.38
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 235.38
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.3

$q$-expansion

\(f(q)\) \(=\) \(q+7.65331i q^{2} -42.5731 q^{4} +38.1687 q^{5} -35.1454 q^{7} -203.372i q^{8} +O(q^{10})\) \(q+7.65331i q^{2} -42.5731 q^{4} +38.1687 q^{5} -35.1454 q^{7} -203.372i q^{8} +292.117i q^{10} +147.824i q^{11} -17.0918i q^{13} -268.979i q^{14} +875.302 q^{16} +354.287 q^{17} -647.468 q^{19} -1624.96 q^{20} -1131.34 q^{22} +862.278i q^{23} +831.852 q^{25} +130.809 q^{26} +1496.25 q^{28} -440.875 q^{29} +347.019i q^{31} +3445.00i q^{32} +2711.47i q^{34} -1341.45 q^{35} +972.830i q^{37} -4955.27i q^{38} -7762.47i q^{40} -2783.36 q^{41} -2837.80i q^{43} -6293.33i q^{44} -6599.28 q^{46} -1287.07i q^{47} -1165.80 q^{49} +6366.42i q^{50} +727.650i q^{52} +1103.72 q^{53} +5642.25i q^{55} +7147.61i q^{56} -3374.15i q^{58} +(-1692.03 - 3042.10i) q^{59} -5980.77i q^{61} -2655.84 q^{62} -12360.8 q^{64} -652.371i q^{65} +3352.44i q^{67} -15083.1 q^{68} -10266.6i q^{70} +1396.21 q^{71} -1649.77i q^{73} -7445.37 q^{74} +27564.7 q^{76} -5195.33i q^{77} -8357.83 q^{79} +33409.2 q^{80} -21301.9i q^{82} +5025.77i q^{83} +13522.7 q^{85} +21718.6 q^{86} +30063.3 q^{88} -5292.92i q^{89} +600.697i q^{91} -36709.9i q^{92} +9850.36 q^{94} -24713.0 q^{95} +14051.7i q^{97} -8922.24i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.65331i 1.91333i 0.291195 + 0.956664i \(0.405947\pi\)
−0.291195 + 0.956664i \(0.594053\pi\)
\(3\) 0 0
\(4\) −42.5731 −2.66082
\(5\) 38.1687 1.52675 0.763375 0.645956i \(-0.223541\pi\)
0.763375 + 0.645956i \(0.223541\pi\)
\(6\) 0 0
\(7\) −35.1454 −0.717253 −0.358626 0.933481i \(-0.616755\pi\)
−0.358626 + 0.933481i \(0.616755\pi\)
\(8\) 203.372i 3.17769i
\(9\) 0 0
\(10\) 292.117i 2.92117i
\(11\) 147.824i 1.22169i 0.791752 + 0.610843i \(0.209169\pi\)
−0.791752 + 0.610843i \(0.790831\pi\)
\(12\) 0 0
\(13\) 17.0918i 0.101135i −0.998721 0.0505674i \(-0.983897\pi\)
0.998721 0.0505674i \(-0.0161030\pi\)
\(14\) 268.979i 1.37234i
\(15\) 0 0
\(16\) 875.302 3.41915
\(17\) 354.287 1.22591 0.612953 0.790119i \(-0.289981\pi\)
0.612953 + 0.790119i \(0.289981\pi\)
\(18\) 0 0
\(19\) −647.468 −1.79354 −0.896770 0.442498i \(-0.854092\pi\)
−0.896770 + 0.442498i \(0.854092\pi\)
\(20\) −1624.96 −4.06241
\(21\) 0 0
\(22\) −1131.34 −2.33748
\(23\) 862.278i 1.63001i 0.579451 + 0.815007i \(0.303267\pi\)
−0.579451 + 0.815007i \(0.696733\pi\)
\(24\) 0 0
\(25\) 831.852 1.33096
\(26\) 130.809 0.193504
\(27\) 0 0
\(28\) 1496.25 1.90848
\(29\) −440.875 −0.524227 −0.262113 0.965037i \(-0.584419\pi\)
−0.262113 + 0.965037i \(0.584419\pi\)
\(30\) 0 0
\(31\) 347.019i 0.361101i 0.983566 + 0.180551i \(0.0577880\pi\)
−0.983566 + 0.180551i \(0.942212\pi\)
\(32\) 3445.00i 3.36426i
\(33\) 0 0
\(34\) 2711.47i 2.34556i
\(35\) −1341.45 −1.09507
\(36\) 0 0
\(37\) 972.830i 0.710613i 0.934750 + 0.355307i \(0.115624\pi\)
−0.934750 + 0.355307i \(0.884376\pi\)
\(38\) 4955.27i 3.43163i
\(39\) 0 0
\(40\) 7762.47i 4.85154i
\(41\) −2783.36 −1.65578 −0.827889 0.560892i \(-0.810458\pi\)
−0.827889 + 0.560892i \(0.810458\pi\)
\(42\) 0 0
\(43\) 2837.80i 1.53478i −0.641183 0.767388i \(-0.721556\pi\)
0.641183 0.767388i \(-0.278444\pi\)
\(44\) 6293.33i 3.25069i
\(45\) 0 0
\(46\) −6599.28 −3.11875
\(47\) 1287.07i 0.582649i −0.956624 0.291324i \(-0.905904\pi\)
0.956624 0.291324i \(-0.0940959\pi\)
\(48\) 0 0
\(49\) −1165.80 −0.485548
\(50\) 6366.42i 2.54657i
\(51\) 0 0
\(52\) 727.650i 0.269101i
\(53\) 1103.72 0.392923 0.196462 0.980511i \(-0.437055\pi\)
0.196462 + 0.980511i \(0.437055\pi\)
\(54\) 0 0
\(55\) 5642.25i 1.86521i
\(56\) 7147.61i 2.27921i
\(57\) 0 0
\(58\) 3374.15i 1.00302i
\(59\) −1692.03 3042.10i −0.486076 0.873917i
\(60\) 0 0
\(61\) 5980.77i 1.60730i −0.595100 0.803651i \(-0.702888\pi\)
0.595100 0.803651i \(-0.297112\pi\)
\(62\) −2655.84 −0.690905
\(63\) 0 0
\(64\) −12360.8 −3.01777
\(65\) 652.371i 0.154407i
\(66\) 0 0
\(67\) 3352.44i 0.746812i 0.927668 + 0.373406i \(0.121810\pi\)
−0.927668 + 0.373406i \(0.878190\pi\)
\(68\) −15083.1 −3.26192
\(69\) 0 0
\(70\) 10266.6i 2.09522i
\(71\) 1396.21 0.276971 0.138486 0.990364i \(-0.455777\pi\)
0.138486 + 0.990364i \(0.455777\pi\)
\(72\) 0 0
\(73\) 1649.77i 0.309584i −0.987947 0.154792i \(-0.950529\pi\)
0.987947 0.154792i \(-0.0494707\pi\)
\(74\) −7445.37 −1.35964
\(75\) 0 0
\(76\) 27564.7 4.77229
\(77\) 5195.33i 0.876258i
\(78\) 0 0
\(79\) −8357.83 −1.33918 −0.669591 0.742730i \(-0.733530\pi\)
−0.669591 + 0.742730i \(0.733530\pi\)
\(80\) 33409.2 5.22018
\(81\) 0 0
\(82\) 21301.9i 3.16805i
\(83\) 5025.77i 0.729536i 0.931098 + 0.364768i \(0.118852\pi\)
−0.931098 + 0.364768i \(0.881148\pi\)
\(84\) 0 0
\(85\) 13522.7 1.87165
\(86\) 21718.6 2.93653
\(87\) 0 0
\(88\) 30063.3 3.88214
\(89\) 5292.92i 0.668213i −0.942535 0.334107i \(-0.891565\pi\)
0.942535 0.334107i \(-0.108435\pi\)
\(90\) 0 0
\(91\) 600.697i 0.0725392i
\(92\) 36709.9i 4.33718i
\(93\) 0 0
\(94\) 9850.36 1.11480
\(95\) −24713.0 −2.73828
\(96\) 0 0
\(97\) 14051.7i 1.49343i 0.665142 + 0.746716i \(0.268371\pi\)
−0.665142 + 0.746716i \(0.731629\pi\)
\(98\) 8922.24i 0.929013i
\(99\) 0 0
\(100\) −35414.5 −3.54145
\(101\) 623.626i 0.0611338i −0.999533 0.0305669i \(-0.990269\pi\)
0.999533 0.0305669i \(-0.00973126\pi\)
\(102\) 0 0
\(103\) 8032.20i 0.757112i 0.925578 + 0.378556i \(0.123579\pi\)
−0.925578 + 0.378556i \(0.876421\pi\)
\(104\) −3475.99 −0.321375
\(105\) 0 0
\(106\) 8447.13i 0.751791i
\(107\) −11098.7 −0.969402 −0.484701 0.874680i \(-0.661072\pi\)
−0.484701 + 0.874680i \(0.661072\pi\)
\(108\) 0 0
\(109\) 7809.17i 0.657283i −0.944455 0.328641i \(-0.893409\pi\)
0.944455 0.328641i \(-0.106591\pi\)
\(110\) −43181.9 −3.56875
\(111\) 0 0
\(112\) −30762.8 −2.45239
\(113\) 3130.68i 0.245178i −0.992457 0.122589i \(-0.960880\pi\)
0.992457 0.122589i \(-0.0391198\pi\)
\(114\) 0 0
\(115\) 32912.0i 2.48862i
\(116\) 18769.4 1.39487
\(117\) 0 0
\(118\) 23282.2 12949.6i 1.67209 0.930022i
\(119\) −12451.6 −0.879285
\(120\) 0 0
\(121\) −7210.93 −0.492516
\(122\) 45772.7 3.07530
\(123\) 0 0
\(124\) 14773.7i 0.960827i
\(125\) 7895.26 0.505297
\(126\) 0 0
\(127\) −18711.5 −1.16011 −0.580056 0.814576i \(-0.696969\pi\)
−0.580056 + 0.814576i \(0.696969\pi\)
\(128\) 39481.1i 2.40973i
\(129\) 0 0
\(130\) 4992.80 0.295432
\(131\) 12408.9i 0.723089i 0.932355 + 0.361545i \(0.117751\pi\)
−0.932355 + 0.361545i \(0.882249\pi\)
\(132\) 0 0
\(133\) 22755.5 1.28642
\(134\) −25657.2 −1.42889
\(135\) 0 0
\(136\) 72052.2i 3.89556i
\(137\) 24025.7 1.28007 0.640037 0.768344i \(-0.278919\pi\)
0.640037 + 0.768344i \(0.278919\pi\)
\(138\) 0 0
\(139\) −3014.36 −0.156014 −0.0780072 0.996953i \(-0.524856\pi\)
−0.0780072 + 0.996953i \(0.524856\pi\)
\(140\) 57109.9 2.91377
\(141\) 0 0
\(142\) 10685.6i 0.529937i
\(143\) 2526.57 0.123555
\(144\) 0 0
\(145\) −16827.6 −0.800362
\(146\) 12626.2 0.592335
\(147\) 0 0
\(148\) 41416.4i 1.89082i
\(149\) 10398.6i 0.468383i 0.972190 + 0.234192i \(0.0752443\pi\)
−0.972190 + 0.234192i \(0.924756\pi\)
\(150\) 0 0
\(151\) 35440.0i 1.55432i −0.629305 0.777159i \(-0.716660\pi\)
0.629305 0.777159i \(-0.283340\pi\)
\(152\) 131677.i 5.69932i
\(153\) 0 0
\(154\) 39761.5 1.67657
\(155\) 13245.3i 0.551311i
\(156\) 0 0
\(157\) 23302.5i 0.945373i 0.881231 + 0.472686i \(0.156716\pi\)
−0.881231 + 0.472686i \(0.843284\pi\)
\(158\) 63965.0i 2.56229i
\(159\) 0 0
\(160\) 131491.i 5.13637i
\(161\) 30305.1i 1.16913i
\(162\) 0 0
\(163\) −8996.62 −0.338613 −0.169307 0.985563i \(-0.554153\pi\)
−0.169307 + 0.985563i \(0.554153\pi\)
\(164\) 118497. 4.40573
\(165\) 0 0
\(166\) −38463.8 −1.39584
\(167\) 441.902 0.0158450 0.00792252 0.999969i \(-0.497478\pi\)
0.00792252 + 0.999969i \(0.497478\pi\)
\(168\) 0 0
\(169\) 28268.9 0.989772
\(170\) 103493.i 3.58108i
\(171\) 0 0
\(172\) 120814.i 4.08376i
\(173\) 2947.42i 0.0984803i 0.998787 + 0.0492402i \(0.0156800\pi\)
−0.998787 + 0.0492402i \(0.984320\pi\)
\(174\) 0 0
\(175\) −29235.8 −0.954637
\(176\) 129391.i 4.17713i
\(177\) 0 0
\(178\) 40508.3 1.27851
\(179\) 17326.2i 0.540750i −0.962755 0.270375i \(-0.912852\pi\)
0.962755 0.270375i \(-0.0871477\pi\)
\(180\) 0 0
\(181\) −9276.44 −0.283155 −0.141578 0.989927i \(-0.545217\pi\)
−0.141578 + 0.989927i \(0.545217\pi\)
\(182\) −4597.32 −0.138791
\(183\) 0 0
\(184\) 175364. 5.17969
\(185\) 37131.7i 1.08493i
\(186\) 0 0
\(187\) 52372.1i 1.49767i
\(188\) 54794.7i 1.55032i
\(189\) 0 0
\(190\) 189136.i 5.23923i
\(191\) 14709.0i 0.403195i −0.979468 0.201598i \(-0.935387\pi\)
0.979468 0.201598i \(-0.0646134\pi\)
\(192\) 0 0
\(193\) −34249.4 −0.919472 −0.459736 0.888056i \(-0.652056\pi\)
−0.459736 + 0.888056i \(0.652056\pi\)
\(194\) −107542. −2.85743
\(195\) 0 0
\(196\) 49631.8 1.29196
\(197\) −7911.51 −0.203858 −0.101929 0.994792i \(-0.532501\pi\)
−0.101929 + 0.994792i \(0.532501\pi\)
\(198\) 0 0
\(199\) −57454.5 −1.45084 −0.725418 0.688309i \(-0.758353\pi\)
−0.725418 + 0.688309i \(0.758353\pi\)
\(200\) 169176.i 4.22939i
\(201\) 0 0
\(202\) 4772.80 0.116969
\(203\) 15494.7 0.376003
\(204\) 0 0
\(205\) −106237. −2.52796
\(206\) −61472.9 −1.44860
\(207\) 0 0
\(208\) 14960.5i 0.345795i
\(209\) 95711.3i 2.19114i
\(210\) 0 0
\(211\) 49004.2i 1.10070i 0.834935 + 0.550349i \(0.185505\pi\)
−0.834935 + 0.550349i \(0.814495\pi\)
\(212\) −46988.9 −1.04550
\(213\) 0 0
\(214\) 84941.6i 1.85478i
\(215\) 108315.i 2.34322i
\(216\) 0 0
\(217\) 12196.1i 0.259001i
\(218\) 59766.0 1.25760
\(219\) 0 0
\(220\) 240208.i 4.96298i
\(221\) 6055.39i 0.123982i
\(222\) 0 0
\(223\) 94322.8 1.89674 0.948368 0.317171i \(-0.102733\pi\)
0.948368 + 0.317171i \(0.102733\pi\)
\(224\) 121076.i 2.41302i
\(225\) 0 0
\(226\) 23960.1 0.469107
\(227\) 73496.8i 1.42632i 0.701002 + 0.713159i \(0.252736\pi\)
−0.701002 + 0.713159i \(0.747264\pi\)
\(228\) 0 0
\(229\) 71268.4i 1.35902i −0.733666 0.679510i \(-0.762192\pi\)
0.733666 0.679510i \(-0.237808\pi\)
\(230\) −251886. −4.76155
\(231\) 0 0
\(232\) 89661.7i 1.66583i
\(233\) 21675.4i 0.399259i 0.979871 + 0.199630i \(0.0639739\pi\)
−0.979871 + 0.199630i \(0.936026\pi\)
\(234\) 0 0
\(235\) 49125.9i 0.889559i
\(236\) 72035.0 + 129512.i 1.29336 + 2.32534i
\(237\) 0 0
\(238\) 95295.6i 1.68236i
\(239\) 556.432 0.00974129 0.00487065 0.999988i \(-0.498450\pi\)
0.00487065 + 0.999988i \(0.498450\pi\)
\(240\) 0 0
\(241\) 34167.1 0.588266 0.294133 0.955764i \(-0.404969\pi\)
0.294133 + 0.955764i \(0.404969\pi\)
\(242\) 55187.5i 0.942345i
\(243\) 0 0
\(244\) 254620.i 4.27675i
\(245\) −44497.2 −0.741310
\(246\) 0 0
\(247\) 11066.4i 0.181389i
\(248\) 70574.0 1.14747
\(249\) 0 0
\(250\) 60424.9i 0.966798i
\(251\) −57018.6 −0.905043 −0.452522 0.891753i \(-0.649475\pi\)
−0.452522 + 0.891753i \(0.649475\pi\)
\(252\) 0 0
\(253\) −127465. −1.99137
\(254\) 143205.i 2.21967i
\(255\) 0 0
\(256\) 104388. 1.59283
\(257\) 3529.50 0.0534375 0.0267188 0.999643i \(-0.491494\pi\)
0.0267188 + 0.999643i \(0.491494\pi\)
\(258\) 0 0
\(259\) 34190.5i 0.509689i
\(260\) 27773.5i 0.410850i
\(261\) 0 0
\(262\) −94969.4 −1.38351
\(263\) −39797.7 −0.575369 −0.287684 0.957725i \(-0.592885\pi\)
−0.287684 + 0.957725i \(0.592885\pi\)
\(264\) 0 0
\(265\) 42127.7 0.599896
\(266\) 174155.i 2.46134i
\(267\) 0 0
\(268\) 142724.i 1.98713i
\(269\) 99994.5i 1.38188i −0.722910 0.690942i \(-0.757196\pi\)
0.722910 0.690942i \(-0.242804\pi\)
\(270\) 0 0
\(271\) −70921.8 −0.965698 −0.482849 0.875704i \(-0.660398\pi\)
−0.482849 + 0.875704i \(0.660398\pi\)
\(272\) 310108. 4.19156
\(273\) 0 0
\(274\) 183876.i 2.44920i
\(275\) 122968.i 1.62602i
\(276\) 0 0
\(277\) −27566.1 −0.359265 −0.179633 0.983734i \(-0.557491\pi\)
−0.179633 + 0.983734i \(0.557491\pi\)
\(278\) 23069.8i 0.298507i
\(279\) 0 0
\(280\) 272815.i 3.47978i
\(281\) −116989. −1.48160 −0.740802 0.671723i \(-0.765554\pi\)
−0.740802 + 0.671723i \(0.765554\pi\)
\(282\) 0 0
\(283\) 124680.i 1.55676i −0.627791 0.778382i \(-0.716041\pi\)
0.627791 0.778382i \(-0.283959\pi\)
\(284\) −59441.1 −0.736971
\(285\) 0 0
\(286\) 19336.6i 0.236401i
\(287\) 97822.4 1.18761
\(288\) 0 0
\(289\) 41998.2 0.502846
\(290\) 128787.i 1.53136i
\(291\) 0 0
\(292\) 70236.0i 0.823747i
\(293\) 117066. 1.36362 0.681812 0.731528i \(-0.261192\pi\)
0.681812 + 0.731528i \(0.261192\pi\)
\(294\) 0 0
\(295\) −64582.6 116113.i −0.742116 1.33425i
\(296\) 197847. 2.25811
\(297\) 0 0
\(298\) −79583.5 −0.896170
\(299\) 14737.8 0.164851
\(300\) 0 0
\(301\) 99735.6i 1.10082i
\(302\) 271233. 2.97392
\(303\) 0 0
\(304\) −566730. −6.13238
\(305\) 228278.i 2.45395i
\(306\) 0 0
\(307\) −74586.4 −0.791376 −0.395688 0.918385i \(-0.629494\pi\)
−0.395688 + 0.918385i \(0.629494\pi\)
\(308\) 221182.i 2.33157i
\(309\) 0 0
\(310\) −101370. −1.05484
\(311\) −164539. −1.70117 −0.850587 0.525834i \(-0.823753\pi\)
−0.850587 + 0.525834i \(0.823753\pi\)
\(312\) 0 0
\(313\) 135789.i 1.38604i 0.720919 + 0.693019i \(0.243720\pi\)
−0.720919 + 0.693019i \(0.756280\pi\)
\(314\) −178341. −1.80881
\(315\) 0 0
\(316\) 355819. 3.56332
\(317\) 92280.2 0.918311 0.459156 0.888356i \(-0.348152\pi\)
0.459156 + 0.888356i \(0.348152\pi\)
\(318\) 0 0
\(319\) 65171.8i 0.640440i
\(320\) −471796. −4.60738
\(321\) 0 0
\(322\) 231934. 2.23693
\(323\) −229389. −2.19871
\(324\) 0 0
\(325\) 14217.8i 0.134607i
\(326\) 68853.9i 0.647878i
\(327\) 0 0
\(328\) 566059.i 5.26156i
\(329\) 45234.6i 0.417907i
\(330\) 0 0
\(331\) 10830.5 0.0988536 0.0494268 0.998778i \(-0.484261\pi\)
0.0494268 + 0.998778i \(0.484261\pi\)
\(332\) 213963.i 1.94116i
\(333\) 0 0
\(334\) 3382.02i 0.0303167i
\(335\) 127958.i 1.14019i
\(336\) 0 0
\(337\) 124414.i 1.09549i −0.836645 0.547746i \(-0.815486\pi\)
0.836645 0.547746i \(-0.184514\pi\)
\(338\) 216350.i 1.89376i
\(339\) 0 0
\(340\) −575703. −4.98013
\(341\) −51297.7 −0.441153
\(342\) 0 0
\(343\) 125357. 1.06551
\(344\) −577130. −4.87705
\(345\) 0 0
\(346\) −22557.5 −0.188425
\(347\) 226048.i 1.87733i 0.344829 + 0.938665i \(0.387937\pi\)
−0.344829 + 0.938665i \(0.612063\pi\)
\(348\) 0 0
\(349\) 223773.i 1.83720i 0.395185 + 0.918602i \(0.370681\pi\)
−0.395185 + 0.918602i \(0.629319\pi\)
\(350\) 223750.i 1.82653i
\(351\) 0 0
\(352\) −509253. −4.11006
\(353\) 132923.i 1.06672i −0.845889 0.533358i \(-0.820930\pi\)
0.845889 0.533358i \(-0.179070\pi\)
\(354\) 0 0
\(355\) 53291.6 0.422866
\(356\) 225336.i 1.77800i
\(357\) 0 0
\(358\) 132603. 1.03463
\(359\) −52427.0 −0.406786 −0.203393 0.979097i \(-0.565197\pi\)
−0.203393 + 0.979097i \(0.565197\pi\)
\(360\) 0 0
\(361\) 288893. 2.21678
\(362\) 70995.5i 0.541768i
\(363\) 0 0
\(364\) 25573.6i 0.193014i
\(365\) 62969.7i 0.472657i
\(366\) 0 0
\(367\) 30282.9i 0.224835i 0.993661 + 0.112418i \(0.0358595\pi\)
−0.993661 + 0.112418i \(0.964141\pi\)
\(368\) 754753.i 5.57326i
\(369\) 0 0
\(370\) −284180. −2.07582
\(371\) −38790.7 −0.281826
\(372\) 0 0
\(373\) 47523.7 0.341580 0.170790 0.985307i \(-0.445368\pi\)
0.170790 + 0.985307i \(0.445368\pi\)
\(374\) −400820. −2.86554
\(375\) 0 0
\(376\) −261755. −1.85148
\(377\) 7535.33i 0.0530175i
\(378\) 0 0
\(379\) 185993. 1.29485 0.647425 0.762130i \(-0.275846\pi\)
0.647425 + 0.762130i \(0.275846\pi\)
\(380\) 1.05211e6 7.28609
\(381\) 0 0
\(382\) 112572. 0.771445
\(383\) 141142. 0.962188 0.481094 0.876669i \(-0.340240\pi\)
0.481094 + 0.876669i \(0.340240\pi\)
\(384\) 0 0
\(385\) 198299.i 1.33783i
\(386\) 262121.i 1.75925i
\(387\) 0 0
\(388\) 598225.i 3.97376i
\(389\) −58679.8 −0.387784 −0.193892 0.981023i \(-0.562111\pi\)
−0.193892 + 0.981023i \(0.562111\pi\)
\(390\) 0 0
\(391\) 305494.i 1.99825i
\(392\) 237092.i 1.54292i
\(393\) 0 0
\(394\) 60549.3i 0.390046i
\(395\) −319008. −2.04459
\(396\) 0 0
\(397\) 277665.i 1.76173i 0.473365 + 0.880866i \(0.343039\pi\)
−0.473365 + 0.880866i \(0.656961\pi\)
\(398\) 439717.i 2.77592i
\(399\) 0 0
\(400\) 728121. 4.55076
\(401\) 59794.3i 0.371853i 0.982564 + 0.185927i \(0.0595286\pi\)
−0.982564 + 0.185927i \(0.940471\pi\)
\(402\) 0 0
\(403\) 5931.16 0.0365199
\(404\) 26549.7i 0.162666i
\(405\) 0 0
\(406\) 118586.i 0.719417i
\(407\) −143808. −0.868146
\(408\) 0 0
\(409\) 92288.3i 0.551696i −0.961201 0.275848i \(-0.911041\pi\)
0.961201 0.275848i \(-0.0889587\pi\)
\(410\) 813068.i 4.83681i
\(411\) 0 0
\(412\) 341956.i 2.01454i
\(413\) 59467.0 + 106916.i 0.348639 + 0.626819i
\(414\) 0 0
\(415\) 191827.i 1.11382i
\(416\) 58881.1 0.340243
\(417\) 0 0
\(418\) 732508. 4.19237
\(419\) 100145.i 0.570429i 0.958464 + 0.285214i \(0.0920648\pi\)
−0.958464 + 0.285214i \(0.907935\pi\)
\(420\) 0 0
\(421\) 102049.i 0.575762i 0.957666 + 0.287881i \(0.0929507\pi\)
−0.957666 + 0.287881i \(0.907049\pi\)
\(422\) −375044. −2.10600
\(423\) 0 0
\(424\) 224467.i 1.24859i
\(425\) 294714. 1.63164
\(426\) 0 0
\(427\) 210197.i 1.15284i
\(428\) 472506. 2.57941
\(429\) 0 0
\(430\) 828970. 4.48334
\(431\) 259464.i 1.39676i 0.715725 + 0.698382i \(0.246096\pi\)
−0.715725 + 0.698382i \(0.753904\pi\)
\(432\) 0 0
\(433\) −332067. −1.77113 −0.885565 0.464516i \(-0.846228\pi\)
−0.885565 + 0.464516i \(0.846228\pi\)
\(434\) 93340.5 0.495554
\(435\) 0 0
\(436\) 332461.i 1.74891i
\(437\) 558297.i 2.92350i
\(438\) 0 0
\(439\) 70862.7 0.367696 0.183848 0.982955i \(-0.441145\pi\)
0.183848 + 0.982955i \(0.441145\pi\)
\(440\) 1.14748e6 5.92706
\(441\) 0 0
\(442\) 46343.8 0.237218
\(443\) 85593.5i 0.436147i 0.975932 + 0.218074i \(0.0699773\pi\)
−0.975932 + 0.218074i \(0.930023\pi\)
\(444\) 0 0
\(445\) 202024.i 1.02019i
\(446\) 721882.i 3.62908i
\(447\) 0 0
\(448\) 434425. 2.16451
\(449\) −70020.7 −0.347323 −0.173662 0.984805i \(-0.555560\pi\)
−0.173662 + 0.984805i \(0.555560\pi\)
\(450\) 0 0
\(451\) 411448.i 2.02284i
\(452\) 133283.i 0.652376i
\(453\) 0 0
\(454\) −562494. −2.72901
\(455\) 22927.8i 0.110749i
\(456\) 0 0
\(457\) 97704.0i 0.467821i −0.972258 0.233911i \(-0.924848\pi\)
0.972258 0.233911i \(-0.0751523\pi\)
\(458\) 545439. 2.60025
\(459\) 0 0
\(460\) 1.40117e6i 6.62178i
\(461\) −149231. −0.702194 −0.351097 0.936339i \(-0.614191\pi\)
−0.351097 + 0.936339i \(0.614191\pi\)
\(462\) 0 0
\(463\) 350096.i 1.63315i 0.577242 + 0.816573i \(0.304129\pi\)
−0.577242 + 0.816573i \(0.695871\pi\)
\(464\) −385898. −1.79241
\(465\) 0 0
\(466\) −165888. −0.763914
\(467\) 370985.i 1.70107i 0.525917 + 0.850536i \(0.323722\pi\)
−0.525917 + 0.850536i \(0.676278\pi\)
\(468\) 0 0
\(469\) 117823.i 0.535653i
\(470\) 375976. 1.70202
\(471\) 0 0
\(472\) −618680. + 344112.i −2.77704 + 1.54460i
\(473\) 419495. 1.87501
\(474\) 0 0
\(475\) −538597. −2.38713
\(476\) 530102. 2.33962
\(477\) 0 0
\(478\) 4258.55i 0.0186383i
\(479\) 64617.2 0.281629 0.140814 0.990036i \(-0.455028\pi\)
0.140814 + 0.990036i \(0.455028\pi\)
\(480\) 0 0
\(481\) 16627.4 0.0718677
\(482\) 261491.i 1.12555i
\(483\) 0 0
\(484\) 306992. 1.31050
\(485\) 536336.i 2.28010i
\(486\) 0 0
\(487\) 177831. 0.749808 0.374904 0.927064i \(-0.377676\pi\)
0.374904 + 0.927064i \(0.377676\pi\)
\(488\) −1.21632e6 −5.10752
\(489\) 0 0
\(490\) 340550.i 1.41837i
\(491\) 146359. 0.607094 0.303547 0.952817i \(-0.401829\pi\)
0.303547 + 0.952817i \(0.401829\pi\)
\(492\) 0 0
\(493\) −156196. −0.642653
\(494\) −84694.3 −0.347057
\(495\) 0 0
\(496\) 303746.i 1.23466i
\(497\) −49070.4 −0.198658
\(498\) 0 0
\(499\) 297990. 1.19674 0.598372 0.801218i \(-0.295815\pi\)
0.598372 + 0.801218i \(0.295815\pi\)
\(500\) −336126. −1.34450
\(501\) 0 0
\(502\) 436381.i 1.73164i
\(503\) 379938.i 1.50168i −0.660486 0.750839i \(-0.729650\pi\)
0.660486 0.750839i \(-0.270350\pi\)
\(504\) 0 0
\(505\) 23803.0i 0.0933359i
\(506\) 975532.i 3.81013i
\(507\) 0 0
\(508\) 796605. 3.08685
\(509\) 68834.8i 0.265688i 0.991137 + 0.132844i \(0.0424110\pi\)
−0.991137 + 0.132844i \(0.957589\pi\)
\(510\) 0 0
\(511\) 57981.9i 0.222050i
\(512\) 167216.i 0.637880i
\(513\) 0 0
\(514\) 27012.3i 0.102243i
\(515\) 306579.i 1.15592i
\(516\) 0 0
\(517\) 190260. 0.711814
\(518\) 261670. 0.975203
\(519\) 0 0
\(520\) −132674. −0.490659
\(521\) −450862. −1.66099 −0.830497 0.557023i \(-0.811943\pi\)
−0.830497 + 0.557023i \(0.811943\pi\)
\(522\) 0 0
\(523\) −157180. −0.574639 −0.287319 0.957835i \(-0.592764\pi\)
−0.287319 + 0.957835i \(0.592764\pi\)
\(524\) 528287.i 1.92401i
\(525\) 0 0
\(526\) 304584.i 1.10087i
\(527\) 122944.i 0.442677i
\(528\) 0 0
\(529\) −463682. −1.65695
\(530\) 322416.i 1.14780i
\(531\) 0 0
\(532\) −968773. −3.42294
\(533\) 47572.6i 0.167457i
\(534\) 0 0
\(535\) −423623. −1.48003
\(536\) 681793. 2.37314
\(537\) 0 0
\(538\) 765289. 2.64400
\(539\) 172333.i 0.593187i
\(540\) 0 0
\(541\) 272034.i 0.929455i 0.885454 + 0.464727i \(0.153848\pi\)
−0.885454 + 0.464727i \(0.846152\pi\)
\(542\) 542787.i 1.84770i
\(543\) 0 0
\(544\) 1.22052e6i 4.12426i
\(545\) 298066.i 1.00351i
\(546\) 0 0
\(547\) 351369. 1.17433 0.587163 0.809469i \(-0.300245\pi\)
0.587163 + 0.809469i \(0.300245\pi\)
\(548\) −1.02285e6 −3.40605
\(549\) 0 0
\(550\) −941109. −3.11110
\(551\) 285452. 0.940221
\(552\) 0 0
\(553\) 293739. 0.960532
\(554\) 210972.i 0.687392i
\(555\) 0 0
\(556\) 128331. 0.415127
\(557\) −88233.8 −0.284397 −0.142198 0.989838i \(-0.545417\pi\)
−0.142198 + 0.989838i \(0.545417\pi\)
\(558\) 0 0
\(559\) −48503.0 −0.155219
\(560\) −1.17418e6 −3.74419
\(561\) 0 0
\(562\) 895352.i 2.83479i
\(563\) 203437.i 0.641819i 0.947110 + 0.320910i \(0.103989\pi\)
−0.947110 + 0.320910i \(0.896011\pi\)
\(564\) 0 0
\(565\) 119494.i 0.374326i
\(566\) 954212. 2.97860
\(567\) 0 0
\(568\) 283951.i 0.880130i
\(569\) 20875.8i 0.0644791i 0.999480 + 0.0322395i \(0.0102639\pi\)
−0.999480 + 0.0322395i \(0.989736\pi\)
\(570\) 0 0
\(571\) 308480.i 0.946139i 0.881025 + 0.473069i \(0.156854\pi\)
−0.881025 + 0.473069i \(0.843146\pi\)
\(572\) −107564. −0.328757
\(573\) 0 0
\(574\) 748665.i 2.27229i
\(575\) 717287.i 2.16949i
\(576\) 0 0
\(577\) −332414. −0.998454 −0.499227 0.866471i \(-0.666383\pi\)
−0.499227 + 0.866471i \(0.666383\pi\)
\(578\) 321425.i 0.962110i
\(579\) 0 0
\(580\) 716405. 2.12962
\(581\) 176633.i 0.523262i
\(582\) 0 0
\(583\) 163157.i 0.480029i
\(584\) −335518. −0.983763
\(585\) 0 0
\(586\) 895940.i 2.60906i
\(587\) 398675.i 1.15703i 0.815673 + 0.578513i \(0.196367\pi\)
−0.815673 + 0.578513i \(0.803633\pi\)
\(588\) 0 0
\(589\) 224683.i 0.647650i
\(590\) 888650. 494271.i 2.55286 1.41991i
\(591\) 0 0
\(592\) 851520.i 2.42969i
\(593\) −150216. −0.427177 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(594\) 0 0
\(595\) −475260. −1.34245
\(596\) 442700.i 1.24628i
\(597\) 0 0
\(598\) 112793.i 0.315414i
\(599\) 79272.8 0.220938 0.110469 0.993880i \(-0.464765\pi\)
0.110469 + 0.993880i \(0.464765\pi\)
\(600\) 0 0
\(601\) 510334.i 1.41288i 0.707772 + 0.706441i \(0.249700\pi\)
−0.707772 + 0.706441i \(0.750300\pi\)
\(602\) −763307. −2.10623
\(603\) 0 0
\(604\) 1.50879e6i 4.13576i
\(605\) −275232. −0.751949
\(606\) 0 0
\(607\) −108872. −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(608\) 2.23052e6i 6.03392i
\(609\) 0 0
\(610\) 1.74709e6 4.69521
\(611\) −21998.3 −0.0589260
\(612\) 0 0
\(613\) 71190.4i 0.189453i 0.995503 + 0.0947263i \(0.0301976\pi\)
−0.995503 + 0.0947263i \(0.969802\pi\)
\(614\) 570833.i 1.51416i
\(615\) 0 0
\(616\) −1.05659e6 −2.78448
\(617\) 463811. 1.21835 0.609173 0.793037i \(-0.291502\pi\)
0.609173 + 0.793037i \(0.291502\pi\)
\(618\) 0 0
\(619\) 192587. 0.502627 0.251313 0.967906i \(-0.419138\pi\)
0.251313 + 0.967906i \(0.419138\pi\)
\(620\) 563892.i 1.46694i
\(621\) 0 0
\(622\) 1.25927e6i 3.25490i
\(623\) 186022.i 0.479278i
\(624\) 0 0
\(625\) −218555. −0.559501
\(626\) −1.03923e6 −2.65194
\(627\) 0 0
\(628\) 992060.i 2.51547i
\(629\) 344661.i 0.871145i
\(630\) 0 0
\(631\) −13953.1 −0.0350438 −0.0175219 0.999846i \(-0.505578\pi\)
−0.0175219 + 0.999846i \(0.505578\pi\)
\(632\) 1.69975e6i 4.25551i
\(633\) 0 0
\(634\) 706249.i 1.75703i
\(635\) −714192. −1.77120
\(636\) 0 0
\(637\) 19925.6i 0.0491058i
\(638\) 498780. 1.22537
\(639\) 0 0
\(640\) 1.50694e6i 3.67906i
\(641\) 416645. 1.01403 0.507014 0.861938i \(-0.330749\pi\)
0.507014 + 0.861938i \(0.330749\pi\)
\(642\) 0 0
\(643\) 79601.2 0.192530 0.0962648 0.995356i \(-0.469310\pi\)
0.0962648 + 0.995356i \(0.469310\pi\)
\(644\) 1.29018e6i 3.11085i
\(645\) 0 0
\(646\) 1.75559e6i 4.20685i
\(647\) 681530. 1.62808 0.814041 0.580807i \(-0.197263\pi\)
0.814041 + 0.580807i \(0.197263\pi\)
\(648\) 0 0
\(649\) 449696. 250123.i 1.06765 0.593832i
\(650\) 108813. 0.257546
\(651\) 0 0
\(652\) 383014. 0.900990
\(653\) 383016. 0.898237 0.449118 0.893472i \(-0.351738\pi\)
0.449118 + 0.893472i \(0.351738\pi\)
\(654\) 0 0
\(655\) 473633.i 1.10398i
\(656\) −2.43628e6 −5.66135
\(657\) 0 0
\(658\) −346195. −0.799592
\(659\) 527534.i 1.21473i 0.794423 + 0.607365i \(0.207773\pi\)
−0.794423 + 0.607365i \(0.792227\pi\)
\(660\) 0 0
\(661\) 282774. 0.647197 0.323599 0.946194i \(-0.395107\pi\)
0.323599 + 0.946194i \(0.395107\pi\)
\(662\) 82889.2i 0.189139i
\(663\) 0 0
\(664\) 1.02210e6 2.31824
\(665\) 868549. 1.96404
\(666\) 0 0
\(667\) 380156.i 0.854497i
\(668\) −18813.2 −0.0421608
\(669\) 0 0
\(670\) −979304. −2.18156
\(671\) 884102. 1.96362
\(672\) 0 0
\(673\) 709243.i 1.56590i 0.622083 + 0.782952i \(0.286287\pi\)
−0.622083 + 0.782952i \(0.713713\pi\)
\(674\) 952178. 2.09604
\(675\) 0 0
\(676\) −1.20349e6 −2.63361
\(677\) 419269. 0.914777 0.457389 0.889267i \(-0.348785\pi\)
0.457389 + 0.889267i \(0.348785\pi\)
\(678\) 0 0
\(679\) 493853.i 1.07117i
\(680\) 2.75014e6i 5.94754i
\(681\) 0 0
\(682\) 392597.i 0.844069i
\(683\) 341293.i 0.731620i −0.930690 0.365810i \(-0.880792\pi\)
0.930690 0.365810i \(-0.119208\pi\)
\(684\) 0 0
\(685\) 917030. 1.95435
\(686\) 959393.i 2.03868i
\(687\) 0 0
\(688\) 2.48393e6i 5.24762i
\(689\) 18864.6i 0.0397382i
\(690\) 0 0
\(691\) 57041.7i 0.119464i −0.998214 0.0597319i \(-0.980975\pi\)
0.998214 0.0597319i \(-0.0190246\pi\)
\(692\) 125481.i 0.262039i
\(693\) 0 0
\(694\) −1.73001e6 −3.59195
\(695\) −115054. −0.238195
\(696\) 0 0
\(697\) −986109. −2.02983
\(698\) −1.71261e6 −3.51517
\(699\) 0 0
\(700\) 1.24466e6 2.54012
\(701\) 292521.i 0.595279i −0.954678 0.297640i \(-0.903801\pi\)
0.954678 0.297640i \(-0.0961994\pi\)
\(702\) 0 0
\(703\) 629876.i 1.27451i
\(704\) 1.82722e6i 3.68677i
\(705\) 0 0
\(706\) 1.01730e6 2.04098
\(707\) 21917.6i 0.0438484i
\(708\) 0 0
\(709\) 265410. 0.527988 0.263994 0.964524i \(-0.414960\pi\)
0.263994 + 0.964524i \(0.414960\pi\)
\(710\) 407857.i 0.809080i
\(711\) 0 0
\(712\) −1.07643e6 −2.12338
\(713\) −299226. −0.588601
\(714\) 0 0
\(715\) 96436.1 0.188637
\(716\) 737630.i 1.43884i
\(717\) 0 0
\(718\) 401240.i 0.778315i
\(719\) 843328.i 1.63132i −0.578532 0.815659i \(-0.696374\pi\)
0.578532 0.815659i \(-0.303626\pi\)
\(720\) 0 0
\(721\) 282295.i 0.543041i
\(722\) 2.21099e6i 4.24143i
\(723\) 0 0
\(724\) 394927. 0.753425
\(725\) −366742. −0.697726
\(726\) 0 0
\(727\) 361987. 0.684895 0.342448 0.939537i \(-0.388744\pi\)
0.342448 + 0.939537i \(0.388744\pi\)
\(728\) 122165. 0.230507
\(729\) 0 0
\(730\) 481927. 0.904347
\(731\) 1.00540e6i 1.88149i
\(732\) 0 0
\(733\) 364992. 0.679322 0.339661 0.940548i \(-0.389688\pi\)
0.339661 + 0.940548i \(0.389688\pi\)
\(734\) −231764. −0.430184
\(735\) 0 0
\(736\) −2.97054e6 −5.48379
\(737\) −495571. −0.912369
\(738\) 0 0
\(739\) 397416.i 0.727707i −0.931456 0.363854i \(-0.881461\pi\)
0.931456 0.363854i \(-0.118539\pi\)
\(740\) 1.58081e6i 2.88680i
\(741\) 0 0
\(742\) 296878.i 0.539224i
\(743\) −208889. −0.378389 −0.189195 0.981940i \(-0.560588\pi\)
−0.189195 + 0.981940i \(0.560588\pi\)
\(744\) 0 0
\(745\) 396900.i 0.715104i
\(746\) 363714.i 0.653555i
\(747\) 0 0
\(748\) 2.22965e6i 3.98504i
\(749\) 390068. 0.695306
\(750\) 0 0
\(751\) 1.12660e6i 1.99751i 0.0498694 + 0.998756i \(0.484120\pi\)
−0.0498694 + 0.998756i \(0.515880\pi\)
\(752\) 1.12658e6i 1.99216i
\(753\) 0 0
\(754\) −57670.2 −0.101440
\(755\) 1.35270e6i 2.37305i
\(756\) 0 0
\(757\) −287250. −0.501266 −0.250633 0.968082i \(-0.580639\pi\)
−0.250633 + 0.968082i \(0.580639\pi\)
\(758\) 1.42347e6i 2.47747i
\(759\) 0 0
\(760\) 5.02595e6i 8.70143i
\(761\) 39472.2 0.0681587 0.0340794 0.999419i \(-0.489150\pi\)
0.0340794 + 0.999419i \(0.489150\pi\)
\(762\) 0 0
\(763\) 274456.i 0.471438i
\(764\) 626207.i 1.07283i
\(765\) 0 0
\(766\) 1.08021e6i 1.84098i
\(767\) −51994.9 + 28919.8i −0.0883833 + 0.0491591i
\(768\) 0 0
\(769\) 696042.i 1.17702i 0.808491 + 0.588509i \(0.200285\pi\)
−0.808491 + 0.588509i \(0.799715\pi\)
\(770\) 1.51765e6 2.55970
\(771\) 0 0
\(772\) 1.45810e6 2.44655
\(773\) 245846.i 0.411438i −0.978611 0.205719i \(-0.934047\pi\)
0.978611 0.205719i \(-0.0659533\pi\)
\(774\) 0 0
\(775\) 288668.i 0.480613i
\(776\) 2.85773e6 4.74567
\(777\) 0 0
\(778\) 449095.i 0.741957i
\(779\) 1.80214e6 2.96970
\(780\) 0 0
\(781\) 206394.i 0.338372i
\(782\) −2.33804e6 −3.82330
\(783\) 0 0
\(784\) −1.02043e6 −1.66016
\(785\) 889426.i 1.44335i
\(786\) 0 0
\(787\) 437284. 0.706016 0.353008 0.935620i \(-0.385159\pi\)
0.353008 + 0.935620i \(0.385159\pi\)
\(788\) 336818. 0.542429
\(789\) 0 0
\(790\) 2.44146e6i 3.91198i
\(791\) 110029.i 0.175855i
\(792\) 0 0
\(793\) −102222. −0.162554
\(794\) −2.12506e6 −3.37077
\(795\) 0 0
\(796\) 2.44602e6 3.86041
\(797\) 377368.i 0.594085i