Properties

Label 531.5.c.d.235.6
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 235.6
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.35

$q$-expansion

\(f(q)\) \(=\) \(q-6.10324i q^{2} -21.2495 q^{4} +12.8499 q^{5} +4.61608 q^{7} +32.0389i q^{8} +O(q^{10})\) \(q-6.10324i q^{2} -21.2495 q^{4} +12.8499 q^{5} +4.61608 q^{7} +32.0389i q^{8} -78.4262i q^{10} +159.523i q^{11} -148.997i q^{13} -28.1731i q^{14} -144.451 q^{16} -154.411 q^{17} +313.342 q^{19} -273.055 q^{20} +973.607 q^{22} -420.147i q^{23} -459.879 q^{25} -909.363 q^{26} -98.0895 q^{28} -1422.43 q^{29} +1100.16i q^{31} +1394.24i q^{32} +942.405i q^{34} +59.3164 q^{35} +186.690i q^{37} -1912.40i q^{38} +411.698i q^{40} -435.578 q^{41} -2365.57i q^{43} -3389.78i q^{44} -2564.25 q^{46} +2367.77i q^{47} -2379.69 q^{49} +2806.75i q^{50} +3166.11i q^{52} -3945.26 q^{53} +2049.86i q^{55} +147.894i q^{56} +8681.45i q^{58} +(-2040.07 + 2820.55i) q^{59} -1734.71i q^{61} +6714.53 q^{62} +6198.17 q^{64} -1914.60i q^{65} +2714.23i q^{67} +3281.15 q^{68} -362.022i q^{70} -5431.15 q^{71} +2072.73i q^{73} +1139.41 q^{74} -6658.35 q^{76} +736.372i q^{77} +10773.3 q^{79} -1856.18 q^{80} +2658.44i q^{82} +10956.5i q^{83} -1984.17 q^{85} -14437.6 q^{86} -5110.95 q^{88} +1731.25i q^{89} -687.782i q^{91} +8927.90i q^{92} +14451.1 q^{94} +4026.42 q^{95} -3594.71i q^{97} +14523.8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.10324i 1.52581i −0.646511 0.762905i \(-0.723773\pi\)
0.646511 0.762905i \(-0.276227\pi\)
\(3\) 0 0
\(4\) −21.2495 −1.32809
\(5\) 12.8499 0.513997 0.256999 0.966412i \(-0.417266\pi\)
0.256999 + 0.966412i \(0.417266\pi\)
\(6\) 0 0
\(7\) 4.61608 0.0942058 0.0471029 0.998890i \(-0.485001\pi\)
0.0471029 + 0.998890i \(0.485001\pi\)
\(8\) 32.0389i 0.500608i
\(9\) 0 0
\(10\) 78.4262i 0.784262i
\(11\) 159.523i 1.31837i 0.751980 + 0.659186i \(0.229099\pi\)
−0.751980 + 0.659186i \(0.770901\pi\)
\(12\) 0 0
\(13\) 148.997i 0.881638i −0.897596 0.440819i \(-0.854688\pi\)
0.897596 0.440819i \(-0.145312\pi\)
\(14\) 28.1731i 0.143740i
\(15\) 0 0
\(16\) −144.451 −0.564261
\(17\) −154.411 −0.534293 −0.267147 0.963656i \(-0.586081\pi\)
−0.267147 + 0.963656i \(0.586081\pi\)
\(18\) 0 0
\(19\) 313.342 0.867982 0.433991 0.900917i \(-0.357105\pi\)
0.433991 + 0.900917i \(0.357105\pi\)
\(20\) −273.055 −0.682636
\(21\) 0 0
\(22\) 973.607 2.01158
\(23\) 420.147i 0.794228i −0.917769 0.397114i \(-0.870012\pi\)
0.917769 0.397114i \(-0.129988\pi\)
\(24\) 0 0
\(25\) −459.879 −0.735807
\(26\) −909.363 −1.34521
\(27\) 0 0
\(28\) −98.0895 −0.125114
\(29\) −1422.43 −1.69136 −0.845680 0.533690i \(-0.820805\pi\)
−0.845680 + 0.533690i \(0.820805\pi\)
\(30\) 0 0
\(31\) 1100.16i 1.14481i 0.819972 + 0.572403i \(0.193989\pi\)
−0.819972 + 0.572403i \(0.806011\pi\)
\(32\) 1394.24i 1.36156i
\(33\) 0 0
\(34\) 942.405i 0.815230i
\(35\) 59.3164 0.0484215
\(36\) 0 0
\(37\) 186.690i 0.136370i 0.997673 + 0.0681848i \(0.0217207\pi\)
−0.997673 + 0.0681848i \(0.978279\pi\)
\(38\) 1912.40i 1.32437i
\(39\) 0 0
\(40\) 411.698i 0.257311i
\(41\) −435.578 −0.259118 −0.129559 0.991572i \(-0.541356\pi\)
−0.129559 + 0.991572i \(0.541356\pi\)
\(42\) 0 0
\(43\) 2365.57i 1.27938i −0.768634 0.639689i \(-0.779063\pi\)
0.768634 0.639689i \(-0.220937\pi\)
\(44\) 3389.78i 1.75092i
\(45\) 0 0
\(46\) −2564.25 −1.21184
\(47\) 2367.77i 1.07187i 0.844258 + 0.535937i \(0.180042\pi\)
−0.844258 + 0.535937i \(0.819958\pi\)
\(48\) 0 0
\(49\) −2379.69 −0.991125
\(50\) 2806.75i 1.12270i
\(51\) 0 0
\(52\) 3166.11i 1.17090i
\(53\) −3945.26 −1.40451 −0.702254 0.711927i \(-0.747823\pi\)
−0.702254 + 0.711927i \(0.747823\pi\)
\(54\) 0 0
\(55\) 2049.86i 0.677640i
\(56\) 147.894i 0.0471602i
\(57\) 0 0
\(58\) 8681.45i 2.58069i
\(59\) −2040.07 + 2820.55i −0.586057 + 0.810270i
\(60\) 0 0
\(61\) 1734.71i 0.466194i −0.972453 0.233097i \(-0.925114\pi\)
0.972453 0.233097i \(-0.0748860\pi\)
\(62\) 6714.53 1.74676
\(63\) 0 0
\(64\) 6198.17 1.51322
\(65\) 1914.60i 0.453160i
\(66\) 0 0
\(67\) 2714.23i 0.604640i 0.953206 + 0.302320i \(0.0977611\pi\)
−0.953206 + 0.302320i \(0.902239\pi\)
\(68\) 3281.15 0.709592
\(69\) 0 0
\(70\) 362.022i 0.0738820i
\(71\) −5431.15 −1.07740 −0.538698 0.842499i \(-0.681084\pi\)
−0.538698 + 0.842499i \(0.681084\pi\)
\(72\) 0 0
\(73\) 2072.73i 0.388953i 0.980907 + 0.194476i \(0.0623008\pi\)
−0.980907 + 0.194476i \(0.937699\pi\)
\(74\) 1139.41 0.208074
\(75\) 0 0
\(76\) −6658.35 −1.15276
\(77\) 736.372i 0.124198i
\(78\) 0 0
\(79\) 10773.3 1.72622 0.863108 0.505019i \(-0.168515\pi\)
0.863108 + 0.505019i \(0.168515\pi\)
\(80\) −1856.18 −0.290029
\(81\) 0 0
\(82\) 2658.44i 0.395365i
\(83\) 10956.5i 1.59044i 0.606321 + 0.795220i \(0.292645\pi\)
−0.606321 + 0.795220i \(0.707355\pi\)
\(84\) 0 0
\(85\) −1984.17 −0.274625
\(86\) −14437.6 −1.95209
\(87\) 0 0
\(88\) −5110.95 −0.659988
\(89\) 1731.25i 0.218565i 0.994011 + 0.109283i \(0.0348553\pi\)
−0.994011 + 0.109283i \(0.965145\pi\)
\(90\) 0 0
\(91\) 687.782i 0.0830555i
\(92\) 8927.90i 1.05481i
\(93\) 0 0
\(94\) 14451.1 1.63547
\(95\) 4026.42 0.446140
\(96\) 0 0
\(97\) 3594.71i 0.382050i −0.981585 0.191025i \(-0.938819\pi\)
0.981585 0.191025i \(-0.0611812\pi\)
\(98\) 14523.8i 1.51227i
\(99\) 0 0
\(100\) 9772.20 0.977220
\(101\) 9157.70i 0.897726i 0.893601 + 0.448863i \(0.148171\pi\)
−0.893601 + 0.448863i \(0.851829\pi\)
\(102\) 0 0
\(103\) 2682.66i 0.252866i −0.991975 0.126433i \(-0.959647\pi\)
0.991975 0.126433i \(-0.0403529\pi\)
\(104\) 4773.70 0.441355
\(105\) 0 0
\(106\) 24078.9i 2.14301i
\(107\) 3938.64 0.344016 0.172008 0.985096i \(-0.444974\pi\)
0.172008 + 0.985096i \(0.444974\pi\)
\(108\) 0 0
\(109\) 11260.3i 0.947754i −0.880591 0.473877i \(-0.842854\pi\)
0.880591 0.473877i \(-0.157146\pi\)
\(110\) 12510.8 1.03395
\(111\) 0 0
\(112\) −666.797 −0.0531567
\(113\) 9397.40i 0.735954i 0.929835 + 0.367977i \(0.119950\pi\)
−0.929835 + 0.367977i \(0.880050\pi\)
\(114\) 0 0
\(115\) 5398.85i 0.408231i
\(116\) 30226.0 2.24629
\(117\) 0 0
\(118\) 17214.5 + 12451.0i 1.23632 + 0.894212i
\(119\) −712.773 −0.0503335
\(120\) 0 0
\(121\) −10806.6 −0.738105
\(122\) −10587.3 −0.711323
\(123\) 0 0
\(124\) 23377.8i 1.52041i
\(125\) −13940.6 −0.892200
\(126\) 0 0
\(127\) −9401.70 −0.582906 −0.291453 0.956585i \(-0.594139\pi\)
−0.291453 + 0.956585i \(0.594139\pi\)
\(128\) 15521.0i 0.947328i
\(129\) 0 0
\(130\) −11685.3 −0.691435
\(131\) 28050.5i 1.63455i −0.576249 0.817274i \(-0.695484\pi\)
0.576249 0.817274i \(-0.304516\pi\)
\(132\) 0 0
\(133\) 1446.41 0.0817690
\(134\) 16565.6 0.922566
\(135\) 0 0
\(136\) 4947.16i 0.267472i
\(137\) 10546.1 0.561888 0.280944 0.959724i \(-0.409352\pi\)
0.280944 + 0.959724i \(0.409352\pi\)
\(138\) 0 0
\(139\) 13525.1 0.700020 0.350010 0.936746i \(-0.386178\pi\)
0.350010 + 0.936746i \(0.386178\pi\)
\(140\) −1260.44 −0.0643083
\(141\) 0 0
\(142\) 33147.6i 1.64390i
\(143\) 23768.4 1.16233
\(144\) 0 0
\(145\) −18278.2 −0.869355
\(146\) 12650.4 0.593468
\(147\) 0 0
\(148\) 3967.07i 0.181112i
\(149\) 4534.58i 0.204251i −0.994772 0.102126i \(-0.967436\pi\)
0.994772 0.102126i \(-0.0325644\pi\)
\(150\) 0 0
\(151\) 28293.6i 1.24089i −0.784249 0.620446i \(-0.786952\pi\)
0.784249 0.620446i \(-0.213048\pi\)
\(152\) 10039.1i 0.434519i
\(153\) 0 0
\(154\) 4494.25 0.189503
\(155\) 14137.0i 0.588427i
\(156\) 0 0
\(157\) 8264.64i 0.335293i 0.985847 + 0.167647i \(0.0536167\pi\)
−0.985847 + 0.167647i \(0.946383\pi\)
\(158\) 65752.1i 2.63388i
\(159\) 0 0
\(160\) 17915.9i 0.699839i
\(161\) 1939.43i 0.0748209i
\(162\) 0 0
\(163\) −22072.4 −0.830759 −0.415379 0.909648i \(-0.636351\pi\)
−0.415379 + 0.909648i \(0.636351\pi\)
\(164\) 9255.81 0.344133
\(165\) 0 0
\(166\) 66870.4 2.42671
\(167\) 13873.2 0.497445 0.248723 0.968575i \(-0.419989\pi\)
0.248723 + 0.968575i \(0.419989\pi\)
\(168\) 0 0
\(169\) 6360.93 0.222714
\(170\) 12109.8i 0.419026i
\(171\) 0 0
\(172\) 50267.2i 1.69913i
\(173\) 26589.0i 0.888402i 0.895927 + 0.444201i \(0.146512\pi\)
−0.895927 + 0.444201i \(0.853488\pi\)
\(174\) 0 0
\(175\) −2122.84 −0.0693173
\(176\) 23043.2i 0.743906i
\(177\) 0 0
\(178\) 10566.3 0.333489
\(179\) 23765.9i 0.741735i −0.928686 0.370867i \(-0.879060\pi\)
0.928686 0.370867i \(-0.120940\pi\)
\(180\) 0 0
\(181\) −25062.8 −0.765019 −0.382510 0.923951i \(-0.624940\pi\)
−0.382510 + 0.923951i \(0.624940\pi\)
\(182\) −4197.70 −0.126727
\(183\) 0 0
\(184\) 13461.0 0.397597
\(185\) 2398.95i 0.0700936i
\(186\) 0 0
\(187\) 24632.1i 0.704397i
\(188\) 50313.9i 1.42355i
\(189\) 0 0
\(190\) 24574.2i 0.680725i
\(191\) 8426.68i 0.230988i −0.993308 0.115494i \(-0.963155\pi\)
0.993308 0.115494i \(-0.0368451\pi\)
\(192\) 0 0
\(193\) 5162.04 0.138582 0.0692910 0.997596i \(-0.477926\pi\)
0.0692910 + 0.997596i \(0.477926\pi\)
\(194\) −21939.4 −0.582936
\(195\) 0 0
\(196\) 50567.3 1.31631
\(197\) −13261.3 −0.341707 −0.170853 0.985296i \(-0.554652\pi\)
−0.170853 + 0.985296i \(0.554652\pi\)
\(198\) 0 0
\(199\) −19128.2 −0.483022 −0.241511 0.970398i \(-0.577643\pi\)
−0.241511 + 0.970398i \(0.577643\pi\)
\(200\) 14734.0i 0.368351i
\(201\) 0 0
\(202\) 55891.6 1.36976
\(203\) −6566.08 −0.159336
\(204\) 0 0
\(205\) −5597.15 −0.133186
\(206\) −16372.9 −0.385825
\(207\) 0 0
\(208\) 21522.7i 0.497474i
\(209\) 49985.2i 1.14432i
\(210\) 0 0
\(211\) 88561.9i 1.98922i 0.103705 + 0.994608i \(0.466930\pi\)
−0.103705 + 0.994608i \(0.533070\pi\)
\(212\) 83834.9 1.86532
\(213\) 0 0
\(214\) 24038.5i 0.524903i
\(215\) 30397.4i 0.657597i
\(216\) 0 0
\(217\) 5078.43i 0.107847i
\(218\) −68724.1 −1.44609
\(219\) 0 0
\(220\) 43558.5i 0.899969i
\(221\) 23006.7i 0.471053i
\(222\) 0 0
\(223\) 19289.2 0.387885 0.193943 0.981013i \(-0.437872\pi\)
0.193943 + 0.981013i \(0.437872\pi\)
\(224\) 6435.93i 0.128267i
\(225\) 0 0
\(226\) 57354.6 1.12293
\(227\) 27309.5i 0.529983i 0.964251 + 0.264991i \(0.0853691\pi\)
−0.964251 + 0.264991i \(0.914631\pi\)
\(228\) 0 0
\(229\) 7085.68i 0.135117i 0.997715 + 0.0675586i \(0.0215210\pi\)
−0.997715 + 0.0675586i \(0.978479\pi\)
\(230\) −32950.5 −0.622882
\(231\) 0 0
\(232\) 45573.3i 0.846709i
\(233\) 59816.5i 1.10182i 0.834566 + 0.550908i \(0.185719\pi\)
−0.834566 + 0.550908i \(0.814281\pi\)
\(234\) 0 0
\(235\) 30425.7i 0.550940i
\(236\) 43350.4 59935.2i 0.778339 1.07611i
\(237\) 0 0
\(238\) 4350.22i 0.0767994i
\(239\) −78985.4 −1.38277 −0.691387 0.722485i \(-0.743000\pi\)
−0.691387 + 0.722485i \(0.743000\pi\)
\(240\) 0 0
\(241\) −50419.4 −0.868088 −0.434044 0.900892i \(-0.642914\pi\)
−0.434044 + 0.900892i \(0.642914\pi\)
\(242\) 65955.2i 1.12621i
\(243\) 0 0
\(244\) 36861.7i 0.619149i
\(245\) −30578.9 −0.509436
\(246\) 0 0
\(247\) 46686.9i 0.765246i
\(248\) −35247.9 −0.573099
\(249\) 0 0
\(250\) 85082.9i 1.36133i
\(251\) −107267. −1.70262 −0.851312 0.524660i \(-0.824193\pi\)
−0.851312 + 0.524660i \(0.824193\pi\)
\(252\) 0 0
\(253\) 67023.0 1.04709
\(254\) 57380.8i 0.889404i
\(255\) 0 0
\(256\) 4442.15 0.0677818
\(257\) −52186.5 −0.790118 −0.395059 0.918656i \(-0.629276\pi\)
−0.395059 + 0.918656i \(0.629276\pi\)
\(258\) 0 0
\(259\) 861.776i 0.0128468i
\(260\) 40684.3i 0.601838i
\(261\) 0 0
\(262\) −171199. −2.49401
\(263\) 109997. 1.59026 0.795132 0.606436i \(-0.207401\pi\)
0.795132 + 0.606436i \(0.207401\pi\)
\(264\) 0 0
\(265\) −50696.3 −0.721913
\(266\) 8827.79i 0.124764i
\(267\) 0 0
\(268\) 57676.0i 0.803019i
\(269\) 60277.7i 0.833014i 0.909133 + 0.416507i \(0.136746\pi\)
−0.909133 + 0.416507i \(0.863254\pi\)
\(270\) 0 0
\(271\) −68787.2 −0.936633 −0.468316 0.883561i \(-0.655139\pi\)
−0.468316 + 0.883561i \(0.655139\pi\)
\(272\) 22304.8 0.301481
\(273\) 0 0
\(274\) 64365.2i 0.857334i
\(275\) 73361.3i 0.970067i
\(276\) 0 0
\(277\) −136750. −1.78225 −0.891126 0.453755i \(-0.850084\pi\)
−0.891126 + 0.453755i \(0.850084\pi\)
\(278\) 82546.8i 1.06810i
\(279\) 0 0
\(280\) 1900.43i 0.0242402i
\(281\) 128919. 1.63270 0.816349 0.577559i \(-0.195995\pi\)
0.816349 + 0.577559i \(0.195995\pi\)
\(282\) 0 0
\(283\) 35381.9i 0.441782i 0.975298 + 0.220891i \(0.0708965\pi\)
−0.975298 + 0.220891i \(0.929103\pi\)
\(284\) 115409. 1.43088
\(285\) 0 0
\(286\) 145064.i 1.77349i
\(287\) −2010.66 −0.0244105
\(288\) 0 0
\(289\) −59678.3 −0.714531
\(290\) 111556.i 1.32647i
\(291\) 0 0
\(292\) 44044.5i 0.516566i
\(293\) 27782.7 0.323622 0.161811 0.986822i \(-0.448266\pi\)
0.161811 + 0.986822i \(0.448266\pi\)
\(294\) 0 0
\(295\) −26214.7 + 36243.8i −0.301232 + 0.416476i
\(296\) −5981.34 −0.0682677
\(297\) 0 0
\(298\) −27675.6 −0.311648
\(299\) −62600.5 −0.700222
\(300\) 0 0
\(301\) 10919.7i 0.120525i
\(302\) −172682. −1.89336
\(303\) 0 0
\(304\) −45262.4 −0.489768
\(305\) 22290.9i 0.239622i
\(306\) 0 0
\(307\) −87467.0 −0.928042 −0.464021 0.885824i \(-0.653594\pi\)
−0.464021 + 0.885824i \(0.653594\pi\)
\(308\) 15647.5i 0.164947i
\(309\) 0 0
\(310\) 86281.2 0.897827
\(311\) −122891. −1.27057 −0.635284 0.772279i \(-0.719117\pi\)
−0.635284 + 0.772279i \(0.719117\pi\)
\(312\) 0 0
\(313\) 13347.4i 0.136242i −0.997677 0.0681208i \(-0.978300\pi\)
0.997677 0.0681208i \(-0.0217003\pi\)
\(314\) 50441.0 0.511593
\(315\) 0 0
\(316\) −228928. −2.29258
\(317\) 1111.83 0.0110642 0.00553210 0.999985i \(-0.498239\pi\)
0.00553210 + 0.999985i \(0.498239\pi\)
\(318\) 0 0
\(319\) 226911.i 2.22984i
\(320\) 79646.0 0.777793
\(321\) 0 0
\(322\) −11836.8 −0.114162
\(323\) −48383.3 −0.463757
\(324\) 0 0
\(325\) 68520.6i 0.648716i
\(326\) 134713.i 1.26758i
\(327\) 0 0
\(328\) 13955.5i 0.129717i
\(329\) 10929.8i 0.100977i
\(330\) 0 0
\(331\) 141657. 1.29295 0.646476 0.762935i \(-0.276242\pi\)
0.646476 + 0.762935i \(0.276242\pi\)
\(332\) 232821.i 2.11225i
\(333\) 0 0
\(334\) 84671.7i 0.759006i
\(335\) 34877.7i 0.310783i
\(336\) 0 0
\(337\) 163985.i 1.44392i −0.691935 0.721960i \(-0.743242\pi\)
0.691935 0.721960i \(-0.256758\pi\)
\(338\) 38822.2i 0.339819i
\(339\) 0 0
\(340\) 42162.6 0.364728
\(341\) −175501. −1.50928
\(342\) 0 0
\(343\) −22068.1 −0.187576
\(344\) 75790.3 0.640467
\(345\) 0 0
\(346\) 162279. 1.35553
\(347\) 163886.i 1.36108i −0.732710 0.680541i \(-0.761745\pi\)
0.732710 0.680541i \(-0.238255\pi\)
\(348\) 0 0
\(349\) 63926.5i 0.524844i −0.964953 0.262422i \(-0.915479\pi\)
0.964953 0.262422i \(-0.0845212\pi\)
\(350\) 12956.2i 0.105765i
\(351\) 0 0
\(352\) −222413. −1.79505
\(353\) 87690.5i 0.703725i −0.936052 0.351863i \(-0.885548\pi\)
0.936052 0.351863i \(-0.114452\pi\)
\(354\) 0 0
\(355\) −69789.9 −0.553779
\(356\) 36788.3i 0.290275i
\(357\) 0 0
\(358\) −145049. −1.13175
\(359\) −119280. −0.925504 −0.462752 0.886488i \(-0.653138\pi\)
−0.462752 + 0.886488i \(0.653138\pi\)
\(360\) 0 0
\(361\) −32138.1 −0.246607
\(362\) 152964.i 1.16727i
\(363\) 0 0
\(364\) 14615.0i 0.110305i
\(365\) 26634.4i 0.199921i
\(366\) 0 0
\(367\) 56563.1i 0.419953i 0.977706 + 0.209977i \(0.0673388\pi\)
−0.977706 + 0.209977i \(0.932661\pi\)
\(368\) 60690.5i 0.448152i
\(369\) 0 0
\(370\) 14641.4 0.106949
\(371\) −18211.7 −0.132313
\(372\) 0 0
\(373\) 123782. 0.889691 0.444846 0.895607i \(-0.353259\pi\)
0.444846 + 0.895607i \(0.353259\pi\)
\(374\) −150335. −1.07478
\(375\) 0 0
\(376\) −75860.8 −0.536589
\(377\) 211938.i 1.49117i
\(378\) 0 0
\(379\) 237594. 1.65408 0.827042 0.562140i \(-0.190022\pi\)
0.827042 + 0.562140i \(0.190022\pi\)
\(380\) −85559.3 −0.592516
\(381\) 0 0
\(382\) −51430.0 −0.352444
\(383\) −271890. −1.85351 −0.926757 0.375661i \(-0.877416\pi\)
−0.926757 + 0.375661i \(0.877416\pi\)
\(384\) 0 0
\(385\) 9462.33i 0.0638376i
\(386\) 31505.2i 0.211450i
\(387\) 0 0
\(388\) 76385.8i 0.507398i
\(389\) 76616.7 0.506319 0.253160 0.967425i \(-0.418530\pi\)
0.253160 + 0.967425i \(0.418530\pi\)
\(390\) 0 0
\(391\) 64875.1i 0.424351i
\(392\) 76242.8i 0.496166i
\(393\) 0 0
\(394\) 80936.8i 0.521379i
\(395\) 138436. 0.887270
\(396\) 0 0
\(397\) 2682.43i 0.0170195i 0.999964 + 0.00850977i \(0.00270878\pi\)
−0.999964 + 0.00850977i \(0.997291\pi\)
\(398\) 116744.i 0.737000i
\(399\) 0 0
\(400\) 66429.9 0.415187
\(401\) 208279.i 1.29526i −0.761955 0.647630i \(-0.775760\pi\)
0.761955 0.647630i \(-0.224240\pi\)
\(402\) 0 0
\(403\) 163920. 1.00931
\(404\) 194597.i 1.19226i
\(405\) 0 0
\(406\) 40074.3i 0.243116i
\(407\) −29781.3 −0.179786
\(408\) 0 0
\(409\) 306294.i 1.83102i −0.402299 0.915508i \(-0.631789\pi\)
0.402299 0.915508i \(-0.368211\pi\)
\(410\) 34160.7i 0.203217i
\(411\) 0 0
\(412\) 57005.1i 0.335830i
\(413\) −9417.11 + 13019.9i −0.0552100 + 0.0763321i
\(414\) 0 0
\(415\) 140791.i 0.817482i
\(416\) 207737. 1.20041
\(417\) 0 0
\(418\) 305071. 1.74602
\(419\) 214239.i 1.22031i 0.792282 + 0.610155i \(0.208893\pi\)
−0.792282 + 0.610155i \(0.791107\pi\)
\(420\) 0 0
\(421\) 211231.i 1.19177i −0.803069 0.595886i \(-0.796801\pi\)
0.803069 0.595886i \(-0.203199\pi\)
\(422\) 540514. 3.03516
\(423\) 0 0
\(424\) 126402.i 0.703108i
\(425\) 71010.3 0.393137
\(426\) 0 0
\(427\) 8007.56i 0.0439182i
\(428\) −83694.2 −0.456886
\(429\) 0 0
\(430\) −185523. −1.00337
\(431\) 77388.2i 0.416601i −0.978065 0.208300i \(-0.933207\pi\)
0.978065 0.208300i \(-0.0667932\pi\)
\(432\) 0 0
\(433\) 246940. 1.31709 0.658546 0.752541i \(-0.271172\pi\)
0.658546 + 0.752541i \(0.271172\pi\)
\(434\) 30994.8 0.164555
\(435\) 0 0
\(436\) 239275.i 1.25871i
\(437\) 131649.i 0.689375i
\(438\) 0 0
\(439\) −43951.4 −0.228057 −0.114028 0.993477i \(-0.536376\pi\)
−0.114028 + 0.993477i \(0.536376\pi\)
\(440\) −65675.3 −0.339232
\(441\) 0 0
\(442\) 140415. 0.718738
\(443\) 204422.i 1.04165i −0.853665 0.520823i \(-0.825625\pi\)
0.853665 0.520823i \(-0.174375\pi\)
\(444\) 0 0
\(445\) 22246.5i 0.112342i
\(446\) 117726.i 0.591839i
\(447\) 0 0
\(448\) 28611.3 0.142554
\(449\) −37083.0 −0.183943 −0.0919713 0.995762i \(-0.529317\pi\)
−0.0919713 + 0.995762i \(0.529317\pi\)
\(450\) 0 0
\(451\) 69484.7i 0.341614i
\(452\) 199690.i 0.977416i
\(453\) 0 0
\(454\) 166676. 0.808653
\(455\) 8837.95i 0.0426903i
\(456\) 0 0
\(457\) 29076.9i 0.139225i −0.997574 0.0696123i \(-0.977824\pi\)
0.997574 0.0696123i \(-0.0221762\pi\)
\(458\) 43245.6 0.206163
\(459\) 0 0
\(460\) 114723.i 0.542169i
\(461\) 254103. 1.19566 0.597831 0.801622i \(-0.296029\pi\)
0.597831 + 0.801622i \(0.296029\pi\)
\(462\) 0 0
\(463\) 386615.i 1.80350i 0.432256 + 0.901751i \(0.357718\pi\)
−0.432256 + 0.901751i \(0.642282\pi\)
\(464\) 205472. 0.954369
\(465\) 0 0
\(466\) 365074. 1.68116
\(467\) 344576.i 1.57998i −0.613122 0.789988i \(-0.710087\pi\)
0.613122 0.789988i \(-0.289913\pi\)
\(468\) 0 0
\(469\) 12529.1i 0.0569606i
\(470\) 185695. 0.840629
\(471\) 0 0
\(472\) −90367.4 65361.5i −0.405628 0.293385i
\(473\) 377363. 1.68670
\(474\) 0 0
\(475\) −144099. −0.638667
\(476\) 15146.1 0.0668476
\(477\) 0 0
\(478\) 482067.i 2.10985i
\(479\) −304897. −1.32887 −0.664435 0.747346i \(-0.731328\pi\)
−0.664435 + 0.747346i \(0.731328\pi\)
\(480\) 0 0
\(481\) 27816.2 0.120229
\(482\) 307722.i 1.32454i
\(483\) 0 0
\(484\) 229635. 0.980273
\(485\) 46191.8i 0.196373i
\(486\) 0 0
\(487\) −327375. −1.38034 −0.690171 0.723646i \(-0.742465\pi\)
−0.690171 + 0.723646i \(0.742465\pi\)
\(488\) 55578.2 0.233381
\(489\) 0 0
\(490\) 186630.i 0.777301i
\(491\) 330527. 1.37102 0.685511 0.728063i \(-0.259579\pi\)
0.685511 + 0.728063i \(0.259579\pi\)
\(492\) 0 0
\(493\) 219639. 0.903683
\(494\) −284941. −1.16762
\(495\) 0 0
\(496\) 158919.i 0.645969i
\(497\) −25070.7 −0.101497
\(498\) 0 0
\(499\) −207100. −0.831724 −0.415862 0.909428i \(-0.636520\pi\)
−0.415862 + 0.909428i \(0.636520\pi\)
\(500\) 296231. 1.18492
\(501\) 0 0
\(502\) 654676.i 2.59788i
\(503\) 187431.i 0.740807i 0.928871 + 0.370403i \(0.120781\pi\)
−0.928871 + 0.370403i \(0.879219\pi\)
\(504\) 0 0
\(505\) 117676.i 0.461429i
\(506\) 409058.i 1.59766i
\(507\) 0 0
\(508\) 199781. 0.774154
\(509\) 27177.2i 0.104899i 0.998624 + 0.0524493i \(0.0167028\pi\)
−0.998624 + 0.0524493i \(0.983297\pi\)
\(510\) 0 0
\(511\) 9567.90i 0.0366416i
\(512\) 275448.i 1.05075i
\(513\) 0 0
\(514\) 318507.i 1.20557i
\(515\) 34471.9i 0.129972i
\(516\) 0 0
\(517\) −377714. −1.41313
\(518\) 5259.63 0.0196018
\(519\) 0 0
\(520\) 61341.7 0.226855
\(521\) −492814. −1.81555 −0.907774 0.419460i \(-0.862219\pi\)
−0.907774 + 0.419460i \(0.862219\pi\)
\(522\) 0 0
\(523\) −154182. −0.563678 −0.281839 0.959462i \(-0.590944\pi\)
−0.281839 + 0.959462i \(0.590944\pi\)
\(524\) 596059.i 2.17083i
\(525\) 0 0
\(526\) 671338.i 2.42644i
\(527\) 169876.i 0.611662i
\(528\) 0 0
\(529\) 103318. 0.369202
\(530\) 309412.i 1.10150i
\(531\) 0 0
\(532\) −30735.5 −0.108597
\(533\) 64899.8i 0.228449i
\(534\) 0 0
\(535\) 50611.3 0.176823
\(536\) −86961.0 −0.302688
\(537\) 0 0
\(538\) 367889. 1.27102
\(539\) 379616.i 1.30667i
\(540\) 0 0
\(541\) 175670.i 0.600210i −0.953906 0.300105i \(-0.902978\pi\)
0.953906 0.300105i \(-0.0970217\pi\)
\(542\) 419825.i 1.42912i
\(543\) 0 0
\(544\) 215286.i 0.727474i
\(545\) 144694.i 0.487143i
\(546\) 0 0
\(547\) 22187.9 0.0741552 0.0370776 0.999312i \(-0.488195\pi\)
0.0370776 + 0.999312i \(0.488195\pi\)
\(548\) −224099. −0.746240
\(549\) 0 0
\(550\) −447742. −1.48014
\(551\) −445708. −1.46807
\(552\) 0 0
\(553\) 49730.5 0.162620
\(554\) 834620.i 2.71938i
\(555\) 0 0
\(556\) −287401. −0.929692
\(557\) 475276. 1.53192 0.765959 0.642890i \(-0.222265\pi\)
0.765959 + 0.642890i \(0.222265\pi\)
\(558\) 0 0
\(559\) −352463. −1.12795
\(560\) −8568.30 −0.0273224
\(561\) 0 0
\(562\) 786826.i 2.49118i
\(563\) 28507.6i 0.0899382i −0.998988 0.0449691i \(-0.985681\pi\)
0.998988 0.0449691i \(-0.0143189\pi\)
\(564\) 0 0
\(565\) 120756.i 0.378278i
\(566\) 215944. 0.674075
\(567\) 0 0
\(568\) 174008.i 0.539353i
\(569\) 112438.i 0.347287i −0.984809 0.173644i \(-0.944446\pi\)
0.984809 0.173644i \(-0.0555541\pi\)
\(570\) 0 0
\(571\) 428720.i 1.31493i −0.753486 0.657464i \(-0.771629\pi\)
0.753486 0.657464i \(-0.228371\pi\)
\(572\) −505067. −1.54368
\(573\) 0 0
\(574\) 12271.6i 0.0372457i
\(575\) 193217.i 0.584398i
\(576\) 0 0
\(577\) −75213.5 −0.225915 −0.112957 0.993600i \(-0.536032\pi\)
−0.112957 + 0.993600i \(0.536032\pi\)
\(578\) 364231.i 1.09024i
\(579\) 0 0
\(580\) 388402. 1.15458
\(581\) 50576.3i 0.149829i
\(582\) 0 0
\(583\) 629360.i 1.85166i
\(584\) −66408.0 −0.194713
\(585\) 0 0
\(586\) 169564.i 0.493786i
\(587\) 134395.i 0.390039i −0.980799 0.195019i \(-0.937523\pi\)
0.980799 0.195019i \(-0.0624770\pi\)
\(588\) 0 0
\(589\) 344725.i 0.993671i
\(590\) 221205. + 159994.i 0.635463 + 0.459622i
\(591\) 0 0
\(592\) 26967.5i 0.0769480i
\(593\) 351119. 0.998493 0.499247 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(594\) 0 0
\(595\) −9159.08 −0.0258713
\(596\) 96357.6i 0.271265i
\(597\) 0 0
\(598\) 382066.i 1.06840i
\(599\) −31310.5 −0.0872641 −0.0436321 0.999048i \(-0.513893\pi\)
−0.0436321 + 0.999048i \(0.513893\pi\)
\(600\) 0 0
\(601\) 583406.i 1.61518i −0.589742 0.807592i \(-0.700771\pi\)
0.589742 0.807592i \(-0.299229\pi\)
\(602\) −66645.3 −0.183898
\(603\) 0 0
\(604\) 601224.i 1.64802i
\(605\) −138864. −0.379384
\(606\) 0 0
\(607\) 470041. 1.27573 0.637864 0.770149i \(-0.279818\pi\)
0.637864 + 0.770149i \(0.279818\pi\)
\(608\) 436873.i 1.18181i
\(609\) 0 0
\(610\) −136046. −0.365618
\(611\) 352790. 0.945005
\(612\) 0 0
\(613\) 463701.i 1.23400i 0.786961 + 0.617002i \(0.211653\pi\)
−0.786961 + 0.617002i \(0.788347\pi\)
\(614\) 533832.i 1.41602i
\(615\) 0 0
\(616\) −23592.6 −0.0621747
\(617\) −23997.3 −0.0630365 −0.0315183 0.999503i \(-0.510034\pi\)
−0.0315183 + 0.999503i \(0.510034\pi\)
\(618\) 0 0
\(619\) 479797. 1.25221 0.626104 0.779739i \(-0.284648\pi\)
0.626104 + 0.779739i \(0.284648\pi\)
\(620\) 300403.i 0.781486i
\(621\) 0 0
\(622\) 750031.i 1.93864i
\(623\) 7991.61i 0.0205901i
\(624\) 0 0
\(625\) 108289. 0.277219
\(626\) −81462.6 −0.207879
\(627\) 0 0
\(628\) 175619.i 0.445301i
\(629\) 28826.9i 0.0728613i
\(630\) 0 0
\(631\) −359273. −0.902331 −0.451166 0.892440i \(-0.648992\pi\)
−0.451166 + 0.892440i \(0.648992\pi\)
\(632\) 345166.i 0.864158i
\(633\) 0 0
\(634\) 6785.76i 0.0168818i
\(635\) −120811. −0.299612
\(636\) 0 0
\(637\) 354567.i 0.873814i
\(638\) −1.38489e6 −3.40232
\(639\) 0 0
\(640\) 199444.i 0.486924i
\(641\) 107842. 0.262465 0.131232 0.991352i \(-0.458107\pi\)
0.131232 + 0.991352i \(0.458107\pi\)
\(642\) 0 0
\(643\) 727011. 1.75840 0.879202 0.476448i \(-0.158076\pi\)
0.879202 + 0.476448i \(0.158076\pi\)
\(644\) 41212.0i 0.0993691i
\(645\) 0 0
\(646\) 295295.i 0.707605i
\(647\) 413352. 0.987443 0.493721 0.869620i \(-0.335636\pi\)
0.493721 + 0.869620i \(0.335636\pi\)
\(648\) 0 0
\(649\) −449942. 325437.i −1.06824 0.772642i
\(650\) 418197. 0.989816
\(651\) 0 0
\(652\) 469028. 1.10333
\(653\) 189327. 0.444002 0.222001 0.975046i \(-0.428741\pi\)
0.222001 + 0.975046i \(0.428741\pi\)
\(654\) 0 0
\(655\) 360447.i 0.840153i
\(656\) 62919.6 0.146210
\(657\) 0 0
\(658\) 66707.3 0.154071
\(659\) 496507.i 1.14328i −0.820503 0.571642i \(-0.806307\pi\)
0.820503 0.571642i \(-0.193693\pi\)
\(660\) 0 0
\(661\) −542654. −1.24199 −0.620997 0.783813i \(-0.713272\pi\)
−0.620997 + 0.783813i \(0.713272\pi\)
\(662\) 864566.i 1.97280i
\(663\) 0 0
\(664\) −351036. −0.796188
\(665\) 18586.3 0.0420290
\(666\) 0 0
\(667\) 597631.i 1.34333i
\(668\) −294799. −0.660654
\(669\) 0 0
\(670\) 212867. 0.474196
\(671\) 276726. 0.614617
\(672\) 0 0
\(673\) 243660.i 0.537964i 0.963145 + 0.268982i \(0.0866873\pi\)
−0.963145 + 0.268982i \(0.913313\pi\)
\(674\) −1.00084e6 −2.20315
\(675\) 0 0
\(676\) −135166. −0.295785
\(677\) −430256. −0.938750 −0.469375 0.882999i \(-0.655521\pi\)
−0.469375 + 0.882999i \(0.655521\pi\)
\(678\) 0 0
\(679\) 16593.5i 0.0359913i
\(680\) 63570.6i 0.137480i
\(681\) 0 0
\(682\) 1.07112e6i 2.30287i
\(683\) 163345.i 0.350158i −0.984554 0.175079i \(-0.943982\pi\)
0.984554 0.175079i \(-0.0560181\pi\)
\(684\) 0 0
\(685\) 135516. 0.288809
\(686\) 134687.i 0.286205i
\(687\) 0 0
\(688\) 341709.i 0.721903i
\(689\) 587832.i 1.23827i
\(690\) 0 0
\(691\) 529874.i 1.10973i 0.831941 + 0.554864i \(0.187230\pi\)
−0.831941 + 0.554864i \(0.812770\pi\)
\(692\) 565002.i 1.17988i
\(693\) 0 0
\(694\) −1.00024e6 −2.07675
\(695\) 173796. 0.359808
\(696\) 0 0
\(697\) 67257.9 0.138445
\(698\) −390159. −0.800812
\(699\) 0 0
\(700\) 45109.3 0.0920598
\(701\) 182341.i 0.371064i 0.982638 + 0.185532i \(0.0594008\pi\)
−0.982638 + 0.185532i \(0.940599\pi\)
\(702\) 0 0
\(703\) 58497.7i 0.118366i
\(704\) 988750.i 1.99499i
\(705\) 0 0
\(706\) −535196. −1.07375
\(707\) 42272.7i 0.0845710i
\(708\) 0 0
\(709\) −860411. −1.71164 −0.855822 0.517271i \(-0.826948\pi\)
−0.855822 + 0.517271i \(0.826948\pi\)
\(710\) 425945.i 0.844960i
\(711\) 0 0
\(712\) −55467.5 −0.109415
\(713\) 462228. 0.909237
\(714\) 0 0
\(715\) 305423. 0.597433
\(716\) 505014.i 0.985093i
\(717\) 0 0
\(718\) 727993.i 1.41214i
\(719\) 198523.i 0.384019i 0.981393 + 0.192010i \(0.0615005\pi\)
−0.981393 + 0.192010i \(0.938499\pi\)
\(720\) 0 0
\(721\) 12383.4i 0.0238215i
\(722\) 196146.i 0.376275i
\(723\) 0 0
\(724\) 532572. 1.01602
\(725\) 654148. 1.24452
\(726\) 0 0
\(727\) −649.946 −0.00122973 −0.000614863 1.00000i \(-0.500196\pi\)
−0.000614863 1.00000i \(0.500196\pi\)
\(728\) 22035.8 0.0415783
\(729\) 0 0
\(730\) 162556. 0.305041
\(731\) 365269.i 0.683563i
\(732\) 0 0
\(733\) 27948.5 0.0520176 0.0260088 0.999662i \(-0.491720\pi\)
0.0260088 + 0.999662i \(0.491720\pi\)
\(734\) 345218. 0.640769
\(735\) 0 0
\(736\) 585785. 1.08139
\(737\) −432982. −0.797141
\(738\) 0 0
\(739\) 406137.i 0.743675i −0.928298 0.371838i \(-0.878728\pi\)
0.928298 0.371838i \(-0.121272\pi\)
\(740\) 50976.5i 0.0930908i
\(741\) 0 0
\(742\) 111150.i 0.201884i
\(743\) 426473. 0.772528 0.386264 0.922388i \(-0.373765\pi\)
0.386264 + 0.922388i \(0.373765\pi\)
\(744\) 0 0
\(745\) 58269.1i 0.104985i
\(746\) 755470.i 1.35750i
\(747\) 0 0
\(748\) 523419.i 0.935506i
\(749\) 18181.1 0.0324083
\(750\) 0 0
\(751\) 625542.i 1.10911i −0.832146 0.554557i \(-0.812888\pi\)
0.832146 0.554557i \(-0.187112\pi\)
\(752\) 342026.i 0.604816i
\(753\) 0 0
\(754\) 1.29351e6 2.27524
\(755\) 363570.i 0.637815i
\(756\) 0 0
\(757\) −42832.4 −0.0747448 −0.0373724 0.999301i \(-0.511899\pi\)
−0.0373724 + 0.999301i \(0.511899\pi\)
\(758\) 1.45009e6i 2.52382i
\(759\) 0 0
\(760\) 129002.i 0.223342i
\(761\) −297397. −0.513532 −0.256766 0.966474i \(-0.582657\pi\)
−0.256766 + 0.966474i \(0.582657\pi\)
\(762\) 0 0
\(763\) 51978.3i 0.0892839i
\(764\) 179063.i 0.306774i
\(765\) 0 0
\(766\) 1.65941e6i 2.82811i
\(767\) 420253. + 303963.i 0.714365 + 0.516691i
\(768\) 0 0
\(769\) 211683.i 0.357960i −0.983853 0.178980i \(-0.942720\pi\)
0.983853 0.178980i \(-0.0572797\pi\)
\(770\) 57750.8 0.0974040
\(771\) 0 0
\(772\) −109691. −0.184050
\(773\) 869866.i 1.45577i 0.685698 + 0.727886i \(0.259497\pi\)
−0.685698 + 0.727886i \(0.740503\pi\)
\(774\) 0 0
\(775\) 505940.i 0.842356i
\(776\) 115171. 0.191257
\(777\) 0 0
\(778\) 467610.i 0.772546i
\(779\) −136485. −0.224910
\(780\) 0 0
\(781\) 866394.i 1.42041i
\(782\) 395948. 0.647478
\(783\) 0 0
\(784\) 343748. 0.559253
\(785\) 106200.i 0.172340i
\(786\) 0 0
\(787\) −281012. −0.453707 −0.226853 0.973929i \(-0.572844\pi\)
−0.226853 + 0.973929i \(0.572844\pi\)
\(788\) 281796. 0.453818
\(789\) 0 0
\(790\) 844910.i 1.35381i
\(791\) 43379.2i 0.0693312i
\(792\) 0 0
\(793\) −258466. −0.411015
\(794\) 16371.5 0.0259686
\(795\) 0 0
\(796\) 406464. 0.641499
\(797\) 515861.i