Properties

Label 531.5.c.d.235.7
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 235.7
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.34

$q$-expansion

\(f(q)\) \(=\) \(q-5.77154i q^{2} -17.3107 q^{4} -34.3452 q^{5} -39.2727 q^{7} +7.56484i q^{8} +O(q^{10})\) \(q-5.77154i q^{2} -17.3107 q^{4} -34.3452 q^{5} -39.2727 q^{7} +7.56484i q^{8} +198.225i q^{10} +56.8541i q^{11} +232.684i q^{13} +226.664i q^{14} -233.311 q^{16} +391.037 q^{17} -352.390 q^{19} +594.541 q^{20} +328.136 q^{22} -173.061i q^{23} +554.595 q^{25} +1342.94 q^{26} +679.838 q^{28} -90.7462 q^{29} -128.035i q^{31} +1467.60i q^{32} -2256.89i q^{34} +1348.83 q^{35} +413.192i q^{37} +2033.83i q^{38} -259.816i q^{40} +761.796 q^{41} -874.013i q^{43} -984.184i q^{44} -998.831 q^{46} +895.574i q^{47} -858.658 q^{49} -3200.87i q^{50} -4027.92i q^{52} -3689.46 q^{53} -1952.67i q^{55} -297.092i q^{56} +523.746i q^{58} +(1547.97 - 3117.88i) q^{59} -1498.06i q^{61} -738.962 q^{62} +4737.35 q^{64} -7991.57i q^{65} +3683.48i q^{67} -6769.13 q^{68} -7784.82i q^{70} +9230.55 q^{71} -4308.55i q^{73} +2384.76 q^{74} +6100.12 q^{76} -2232.81i q^{77} -4707.88 q^{79} +8013.11 q^{80} -4396.74i q^{82} -398.453i q^{83} -13430.3 q^{85} -5044.40 q^{86} -430.092 q^{88} -12419.6i q^{89} -9138.11i q^{91} +2995.82i q^{92} +5168.84 q^{94} +12102.9 q^{95} -397.504i q^{97} +4955.78i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.77154i 1.44289i −0.692474 0.721443i \(-0.743479\pi\)
0.692474 0.721443i \(-0.256521\pi\)
\(3\) 0 0
\(4\) −17.3107 −1.08192
\(5\) −34.3452 −1.37381 −0.686905 0.726748i \(-0.741031\pi\)
−0.686905 + 0.726748i \(0.741031\pi\)
\(6\) 0 0
\(7\) −39.2727 −0.801483 −0.400741 0.916191i \(-0.631247\pi\)
−0.400741 + 0.916191i \(0.631247\pi\)
\(8\) 7.56484i 0.118201i
\(9\) 0 0
\(10\) 198.225i 1.98225i
\(11\) 56.8541i 0.469868i 0.972011 + 0.234934i \(0.0754874\pi\)
−0.972011 + 0.234934i \(0.924513\pi\)
\(12\) 0 0
\(13\) 232.684i 1.37683i 0.725319 + 0.688413i \(0.241692\pi\)
−0.725319 + 0.688413i \(0.758308\pi\)
\(14\) 226.664i 1.15645i
\(15\) 0 0
\(16\) −233.311 −0.911370
\(17\) 391.037 1.35307 0.676535 0.736411i \(-0.263481\pi\)
0.676535 + 0.736411i \(0.263481\pi\)
\(18\) 0 0
\(19\) −352.390 −0.976149 −0.488075 0.872802i \(-0.662301\pi\)
−0.488075 + 0.872802i \(0.662301\pi\)
\(20\) 594.541 1.48635
\(21\) 0 0
\(22\) 328.136 0.677966
\(23\) 173.061i 0.327148i −0.986531 0.163574i \(-0.947698\pi\)
0.986531 0.163574i \(-0.0523023\pi\)
\(24\) 0 0
\(25\) 554.595 0.887352
\(26\) 1342.94 1.98660
\(27\) 0 0
\(28\) 679.838 0.867140
\(29\) −90.7462 −0.107903 −0.0539514 0.998544i \(-0.517182\pi\)
−0.0539514 + 0.998544i \(0.517182\pi\)
\(30\) 0 0
\(31\) 128.035i 0.133232i −0.997779 0.0666158i \(-0.978780\pi\)
0.997779 0.0666158i \(-0.0212202\pi\)
\(32\) 1467.60i 1.43320i
\(33\) 0 0
\(34\) 2256.89i 1.95233i
\(35\) 1348.83 1.10108
\(36\) 0 0
\(37\) 413.192i 0.301821i 0.988547 + 0.150910i \(0.0482204\pi\)
−0.988547 + 0.150910i \(0.951780\pi\)
\(38\) 2033.83i 1.40847i
\(39\) 0 0
\(40\) 259.816i 0.162385i
\(41\) 761.796 0.453180 0.226590 0.973990i \(-0.427242\pi\)
0.226590 + 0.973990i \(0.427242\pi\)
\(42\) 0 0
\(43\) 874.013i 0.472695i −0.971669 0.236347i \(-0.924050\pi\)
0.971669 0.236347i \(-0.0759504\pi\)
\(44\) 984.184i 0.508360i
\(45\) 0 0
\(46\) −998.831 −0.472038
\(47\) 895.574i 0.405420i 0.979239 + 0.202710i \(0.0649749\pi\)
−0.979239 + 0.202710i \(0.935025\pi\)
\(48\) 0 0
\(49\) −858.658 −0.357625
\(50\) 3200.87i 1.28035i
\(51\) 0 0
\(52\) 4027.92i 1.48962i
\(53\) −3689.46 −1.31344 −0.656721 0.754134i \(-0.728057\pi\)
−0.656721 + 0.754134i \(0.728057\pi\)
\(54\) 0 0
\(55\) 1952.67i 0.645509i
\(56\) 297.092i 0.0947358i
\(57\) 0 0
\(58\) 523.746i 0.155691i
\(59\) 1547.97 3117.88i 0.444691 0.895684i
\(60\) 0 0
\(61\) 1498.06i 0.402596i −0.979530 0.201298i \(-0.935484\pi\)
0.979530 0.201298i \(-0.0645159\pi\)
\(62\) −738.962 −0.192238
\(63\) 0 0
\(64\) 4737.35 1.15658
\(65\) 7991.57i 1.89150i
\(66\) 0 0
\(67\) 3683.48i 0.820558i 0.911960 + 0.410279i \(0.134569\pi\)
−0.911960 + 0.410279i \(0.865431\pi\)
\(68\) −6769.13 −1.46391
\(69\) 0 0
\(70\) 7784.82i 1.58874i
\(71\) 9230.55 1.83109 0.915547 0.402211i \(-0.131758\pi\)
0.915547 + 0.402211i \(0.131758\pi\)
\(72\) 0 0
\(73\) 4308.55i 0.808510i −0.914646 0.404255i \(-0.867531\pi\)
0.914646 0.404255i \(-0.132469\pi\)
\(74\) 2384.76 0.435493
\(75\) 0 0
\(76\) 6100.12 1.05612
\(77\) 2232.81i 0.376591i
\(78\) 0 0
\(79\) −4707.88 −0.754346 −0.377173 0.926143i \(-0.623104\pi\)
−0.377173 + 0.926143i \(0.623104\pi\)
\(80\) 8013.11 1.25205
\(81\) 0 0
\(82\) 4396.74i 0.653887i
\(83\) 398.453i 0.0578390i −0.999582 0.0289195i \(-0.990793\pi\)
0.999582 0.0289195i \(-0.00920665\pi\)
\(84\) 0 0
\(85\) −13430.3 −1.85886
\(86\) −5044.40 −0.682045
\(87\) 0 0
\(88\) −430.092 −0.0555387
\(89\) 12419.6i 1.56794i −0.620802 0.783968i \(-0.713193\pi\)
0.620802 0.783968i \(-0.286807\pi\)
\(90\) 0 0
\(91\) 9138.11i 1.10350i
\(92\) 2995.82i 0.353948i
\(93\) 0 0
\(94\) 5168.84 0.584975
\(95\) 12102.9 1.34104
\(96\) 0 0
\(97\) 397.504i 0.0422472i −0.999777 0.0211236i \(-0.993276\pi\)
0.999777 0.0211236i \(-0.00672435\pi\)
\(98\) 4955.78i 0.516012i
\(99\) 0 0
\(100\) −9600.44 −0.960044
\(101\) 7090.10i 0.695040i −0.937673 0.347520i \(-0.887024\pi\)
0.937673 0.347520i \(-0.112976\pi\)
\(102\) 0 0
\(103\) 6266.89i 0.590714i −0.955387 0.295357i \(-0.904561\pi\)
0.955387 0.295357i \(-0.0954387\pi\)
\(104\) −1760.22 −0.162742
\(105\) 0 0
\(106\) 21293.9i 1.89515i
\(107\) −1508.55 −0.131763 −0.0658814 0.997827i \(-0.520986\pi\)
−0.0658814 + 0.997827i \(0.520986\pi\)
\(108\) 0 0
\(109\) 16261.8i 1.36872i 0.729144 + 0.684361i \(0.239919\pi\)
−0.729144 + 0.684361i \(0.760081\pi\)
\(110\) −11269.9 −0.931396
\(111\) 0 0
\(112\) 9162.73 0.730447
\(113\) 22671.8i 1.77553i −0.460293 0.887767i \(-0.652256\pi\)
0.460293 0.887767i \(-0.347744\pi\)
\(114\) 0 0
\(115\) 5943.83i 0.449439i
\(116\) 1570.88 0.116742
\(117\) 0 0
\(118\) −17995.0 8934.17i −1.29237 0.641638i
\(119\) −15357.1 −1.08446
\(120\) 0 0
\(121\) 11408.6 0.779224
\(122\) −8646.11 −0.580900
\(123\) 0 0
\(124\) 2216.39i 0.144146i
\(125\) 2418.07 0.154757
\(126\) 0 0
\(127\) 7430.12 0.460668 0.230334 0.973112i \(-0.426018\pi\)
0.230334 + 0.973112i \(0.426018\pi\)
\(128\) 3860.20i 0.235608i
\(129\) 0 0
\(130\) −46123.7 −2.72921
\(131\) 30063.5i 1.75185i 0.482446 + 0.875926i \(0.339748\pi\)
−0.482446 + 0.875926i \(0.660252\pi\)
\(132\) 0 0
\(133\) 13839.3 0.782367
\(134\) 21259.4 1.18397
\(135\) 0 0
\(136\) 2958.14i 0.159934i
\(137\) −20907.4 −1.11393 −0.556967 0.830535i \(-0.688035\pi\)
−0.556967 + 0.830535i \(0.688035\pi\)
\(138\) 0 0
\(139\) 22666.8 1.17317 0.586583 0.809889i \(-0.300473\pi\)
0.586583 + 0.809889i \(0.300473\pi\)
\(140\) −23349.2 −1.19129
\(141\) 0 0
\(142\) 53274.5i 2.64206i
\(143\) −13229.0 −0.646927
\(144\) 0 0
\(145\) 3116.70 0.148238
\(146\) −24867.0 −1.16659
\(147\) 0 0
\(148\) 7152.65i 0.326546i
\(149\) 14748.1i 0.664299i 0.943227 + 0.332150i \(0.107774\pi\)
−0.943227 + 0.332150i \(0.892226\pi\)
\(150\) 0 0
\(151\) 1352.46i 0.0593158i 0.999560 + 0.0296579i \(0.00944179\pi\)
−0.999560 + 0.0296579i \(0.990558\pi\)
\(152\) 2665.77i 0.115382i
\(153\) 0 0
\(154\) −12886.8 −0.543378
\(155\) 4397.41i 0.183035i
\(156\) 0 0
\(157\) 43867.4i 1.77968i 0.456270 + 0.889841i \(0.349185\pi\)
−0.456270 + 0.889841i \(0.650815\pi\)
\(158\) 27171.7i 1.08844i
\(159\) 0 0
\(160\) 50405.1i 1.96895i
\(161\) 6796.58i 0.262204i
\(162\) 0 0
\(163\) 24914.8 0.937741 0.468870 0.883267i \(-0.344661\pi\)
0.468870 + 0.883267i \(0.344661\pi\)
\(164\) −13187.2 −0.490304
\(165\) 0 0
\(166\) −2299.69 −0.0834551
\(167\) 15384.8 0.551644 0.275822 0.961209i \(-0.411050\pi\)
0.275822 + 0.961209i \(0.411050\pi\)
\(168\) 0 0
\(169\) −25580.7 −0.895651
\(170\) 77513.4i 2.68212i
\(171\) 0 0
\(172\) 15129.8i 0.511418i
\(173\) 54918.5i 1.83496i −0.397784 0.917479i \(-0.630221\pi\)
0.397784 0.917479i \(-0.369779\pi\)
\(174\) 0 0
\(175\) −21780.4 −0.711198
\(176\) 13264.7i 0.428224i
\(177\) 0 0
\(178\) −71680.3 −2.26235
\(179\) 50544.7i 1.57750i −0.614713 0.788751i \(-0.710728\pi\)
0.614713 0.788751i \(-0.289272\pi\)
\(180\) 0 0
\(181\) 34588.1 1.05577 0.527885 0.849316i \(-0.322985\pi\)
0.527885 + 0.849316i \(0.322985\pi\)
\(182\) −52741.0 −1.59223
\(183\) 0 0
\(184\) 1309.18 0.0386691
\(185\) 14191.2i 0.414644i
\(186\) 0 0
\(187\) 22232.0i 0.635765i
\(188\) 15503.0i 0.438632i
\(189\) 0 0
\(190\) 69852.5i 1.93497i
\(191\) 29175.3i 0.799739i 0.916572 + 0.399869i \(0.130945\pi\)
−0.916572 + 0.399869i \(0.869055\pi\)
\(192\) 0 0
\(193\) 31941.0 0.857500 0.428750 0.903423i \(-0.358954\pi\)
0.428750 + 0.903423i \(0.358954\pi\)
\(194\) −2294.21 −0.0609579
\(195\) 0 0
\(196\) 14864.0 0.386922
\(197\) 4250.44 0.109522 0.0547610 0.998499i \(-0.482560\pi\)
0.0547610 + 0.998499i \(0.482560\pi\)
\(198\) 0 0
\(199\) 28562.3 0.721251 0.360625 0.932711i \(-0.382563\pi\)
0.360625 + 0.932711i \(0.382563\pi\)
\(200\) 4195.43i 0.104886i
\(201\) 0 0
\(202\) −40920.8 −1.00286
\(203\) 3563.84 0.0864822
\(204\) 0 0
\(205\) −26164.1 −0.622583
\(206\) −36169.6 −0.852334
\(207\) 0 0
\(208\) 54287.6i 1.25480i
\(209\) 20034.8i 0.458662i
\(210\) 0 0
\(211\) 26481.8i 0.594816i −0.954750 0.297408i \(-0.903878\pi\)
0.954750 0.297408i \(-0.0961222\pi\)
\(212\) 63867.2 1.42104
\(213\) 0 0
\(214\) 8706.68i 0.190119i
\(215\) 30018.2i 0.649393i
\(216\) 0 0
\(217\) 5028.29i 0.106783i
\(218\) 93855.5 1.97491
\(219\) 0 0
\(220\) 33802.0i 0.698389i
\(221\) 90988.0i 1.86294i
\(222\) 0 0
\(223\) −20499.0 −0.412215 −0.206108 0.978529i \(-0.566080\pi\)
−0.206108 + 0.978529i \(0.566080\pi\)
\(224\) 57636.6i 1.14869i
\(225\) 0 0
\(226\) −130851. −2.56189
\(227\) 78399.4i 1.52146i −0.649067 0.760731i \(-0.724841\pi\)
0.649067 0.760731i \(-0.275159\pi\)
\(228\) 0 0
\(229\) 31210.2i 0.595149i 0.954699 + 0.297574i \(0.0961776\pi\)
−0.954699 + 0.297574i \(0.903822\pi\)
\(230\) 34305.1 0.648490
\(231\) 0 0
\(232\) 686.481i 0.0127542i
\(233\) 91009.4i 1.67639i 0.545372 + 0.838194i \(0.316388\pi\)
−0.545372 + 0.838194i \(0.683612\pi\)
\(234\) 0 0
\(235\) 30758.7i 0.556970i
\(236\) −26796.5 + 53972.7i −0.481120 + 0.969058i
\(237\) 0 0
\(238\) 88634.0i 1.56476i
\(239\) 96173.1 1.68367 0.841837 0.539732i \(-0.181474\pi\)
0.841837 + 0.539732i \(0.181474\pi\)
\(240\) 0 0
\(241\) −69208.9 −1.19159 −0.595796 0.803136i \(-0.703163\pi\)
−0.595796 + 0.803136i \(0.703163\pi\)
\(242\) 65845.3i 1.12433i
\(243\) 0 0
\(244\) 25932.5i 0.435576i
\(245\) 29490.8 0.491309
\(246\) 0 0
\(247\) 81995.4i 1.34399i
\(248\) 968.568 0.0157481
\(249\) 0 0
\(250\) 13956.0i 0.223296i
\(251\) −94134.4 −1.49417 −0.747087 0.664726i \(-0.768548\pi\)
−0.747087 + 0.664726i \(0.768548\pi\)
\(252\) 0 0
\(253\) 9839.24 0.153717
\(254\) 42883.3i 0.664692i
\(255\) 0 0
\(256\) 53518.2 0.816623
\(257\) −11680.1 −0.176841 −0.0884203 0.996083i \(-0.528182\pi\)
−0.0884203 + 0.996083i \(0.528182\pi\)
\(258\) 0 0
\(259\) 16227.2i 0.241904i
\(260\) 138340.i 2.04645i
\(261\) 0 0
\(262\) 173513. 2.52772
\(263\) 67551.5 0.976616 0.488308 0.872671i \(-0.337614\pi\)
0.488308 + 0.872671i \(0.337614\pi\)
\(264\) 0 0
\(265\) 126715. 1.80442
\(266\) 79874.1i 1.12887i
\(267\) 0 0
\(268\) 63763.7i 0.887778i
\(269\) 8085.17i 0.111734i −0.998438 0.0558669i \(-0.982208\pi\)
0.998438 0.0558669i \(-0.0177923\pi\)
\(270\) 0 0
\(271\) −3043.53 −0.0414419 −0.0207209 0.999785i \(-0.506596\pi\)
−0.0207209 + 0.999785i \(0.506596\pi\)
\(272\) −91233.1 −1.23315
\(273\) 0 0
\(274\) 120668.i 1.60728i
\(275\) 31531.0i 0.416939i
\(276\) 0 0
\(277\) 110726. 1.44308 0.721540 0.692373i \(-0.243435\pi\)
0.721540 + 0.692373i \(0.243435\pi\)
\(278\) 130822.i 1.69275i
\(279\) 0 0
\(280\) 10203.7i 0.130149i
\(281\) 132826. 1.68217 0.841085 0.540903i \(-0.181917\pi\)
0.841085 + 0.540903i \(0.181917\pi\)
\(282\) 0 0
\(283\) 136650.i 1.70622i 0.521728 + 0.853112i \(0.325287\pi\)
−0.521728 + 0.853112i \(0.674713\pi\)
\(284\) −159787. −1.98110
\(285\) 0 0
\(286\) 76351.8i 0.933442i
\(287\) −29917.7 −0.363216
\(288\) 0 0
\(289\) 69389.1 0.830798
\(290\) 17988.2i 0.213890i
\(291\) 0 0
\(292\) 74584.1i 0.874743i
\(293\) 138078. 1.60838 0.804189 0.594374i \(-0.202600\pi\)
0.804189 + 0.594374i \(0.202600\pi\)
\(294\) 0 0
\(295\) −53165.4 + 107084.i −0.610921 + 1.23050i
\(296\) −3125.74 −0.0356754
\(297\) 0 0
\(298\) 85119.4 0.958508
\(299\) 40268.6 0.450426
\(300\) 0 0
\(301\) 34324.8i 0.378857i
\(302\) 7805.78 0.0855859
\(303\) 0 0
\(304\) 82216.3 0.889633
\(305\) 51451.2i 0.553090i
\(306\) 0 0
\(307\) 88705.4 0.941181 0.470591 0.882352i \(-0.344041\pi\)
0.470591 + 0.882352i \(0.344041\pi\)
\(308\) 38651.5i 0.407442i
\(309\) 0 0
\(310\) 25379.8 0.264098
\(311\) 120340. 1.24420 0.622101 0.782937i \(-0.286279\pi\)
0.622101 + 0.782937i \(0.286279\pi\)
\(312\) 0 0
\(313\) 153842.i 1.57031i −0.619299 0.785155i \(-0.712583\pi\)
0.619299 0.785155i \(-0.287417\pi\)
\(314\) 253183. 2.56788
\(315\) 0 0
\(316\) 81496.7 0.816142
\(317\) −13221.7 −0.131574 −0.0657868 0.997834i \(-0.520956\pi\)
−0.0657868 + 0.997834i \(0.520956\pi\)
\(318\) 0 0
\(319\) 5159.29i 0.0507001i
\(320\) −162705. −1.58892
\(321\) 0 0
\(322\) 39226.8 0.378330
\(323\) −137798. −1.32080
\(324\) 0 0
\(325\) 129045.i 1.22173i
\(326\) 143797.i 1.35305i
\(327\) 0 0
\(328\) 5762.87i 0.0535662i
\(329\) 35171.6i 0.324938i
\(330\) 0 0
\(331\) 56327.8 0.514123 0.257061 0.966395i \(-0.417246\pi\)
0.257061 + 0.966395i \(0.417246\pi\)
\(332\) 6897.50i 0.0625771i
\(333\) 0 0
\(334\) 88794.1i 0.795960i
\(335\) 126510.i 1.12729i
\(336\) 0 0
\(337\) 115339.i 1.01558i 0.861481 + 0.507791i \(0.169538\pi\)
−0.861481 + 0.507791i \(0.830462\pi\)
\(338\) 147640.i 1.29232i
\(339\) 0 0
\(340\) 232487. 2.01114
\(341\) 7279.34 0.0626013
\(342\) 0 0
\(343\) 128015. 1.08811
\(344\) 6611.77 0.0558729
\(345\) 0 0
\(346\) −316964. −2.64763
\(347\) 9584.04i 0.0795957i 0.999208 + 0.0397979i \(0.0126714\pi\)
−0.999208 + 0.0397979i \(0.987329\pi\)
\(348\) 0 0
\(349\) 148678.i 1.22066i 0.792147 + 0.610330i \(0.208963\pi\)
−0.792147 + 0.610330i \(0.791037\pi\)
\(350\) 125707.i 1.02618i
\(351\) 0 0
\(352\) −83439.0 −0.673416
\(353\) 12045.3i 0.0966652i −0.998831 0.0483326i \(-0.984609\pi\)
0.998831 0.0483326i \(-0.0153907\pi\)
\(354\) 0 0
\(355\) −317025. −2.51557
\(356\) 214992.i 1.69638i
\(357\) 0 0
\(358\) −291721. −2.27615
\(359\) 11159.4 0.0865872 0.0432936 0.999062i \(-0.486215\pi\)
0.0432936 + 0.999062i \(0.486215\pi\)
\(360\) 0 0
\(361\) −6142.36 −0.0471326
\(362\) 199627.i 1.52336i
\(363\) 0 0
\(364\) 158187.i 1.19390i
\(365\) 147978.i 1.11074i
\(366\) 0 0
\(367\) 24075.5i 0.178749i −0.995998 0.0893744i \(-0.971513\pi\)
0.995998 0.0893744i \(-0.0284868\pi\)
\(368\) 40377.1i 0.298153i
\(369\) 0 0
\(370\) −81905.0 −0.598284
\(371\) 144895. 1.05270
\(372\) 0 0
\(373\) −87079.2 −0.625888 −0.312944 0.949772i \(-0.601315\pi\)
−0.312944 + 0.949772i \(0.601315\pi\)
\(374\) 128313. 0.917336
\(375\) 0 0
\(376\) −6774.87 −0.0479210
\(377\) 21115.2i 0.148563i
\(378\) 0 0
\(379\) 201608. 1.40355 0.701776 0.712398i \(-0.252391\pi\)
0.701776 + 0.712398i \(0.252391\pi\)
\(380\) −209510. −1.45090
\(381\) 0 0
\(382\) 168386. 1.15393
\(383\) −181721. −1.23882 −0.619408 0.785069i \(-0.712627\pi\)
−0.619408 + 0.785069i \(0.712627\pi\)
\(384\) 0 0
\(385\) 76686.4i 0.517365i
\(386\) 184349.i 1.23727i
\(387\) 0 0
\(388\) 6881.08i 0.0457081i
\(389\) 133518. 0.882350 0.441175 0.897421i \(-0.354562\pi\)
0.441175 + 0.897421i \(0.354562\pi\)
\(390\) 0 0
\(391\) 67673.4i 0.442654i
\(392\) 6495.61i 0.0422715i
\(393\) 0 0
\(394\) 24531.6i 0.158028i
\(395\) 161693. 1.03633
\(396\) 0 0
\(397\) 27629.4i 0.175304i 0.996151 + 0.0876518i \(0.0279363\pi\)
−0.996151 + 0.0876518i \(0.972064\pi\)
\(398\) 164848.i 1.04068i
\(399\) 0 0
\(400\) −129393. −0.808706
\(401\) 208735.i 1.29809i −0.760748 0.649047i \(-0.775168\pi\)
0.760748 0.649047i \(-0.224832\pi\)
\(402\) 0 0
\(403\) 29791.8 0.183437
\(404\) 122735.i 0.751977i
\(405\) 0 0
\(406\) 20568.9i 0.124784i
\(407\) −23491.7 −0.141816
\(408\) 0 0
\(409\) 278718.i 1.66616i −0.553150 0.833082i \(-0.686574\pi\)
0.553150 0.833082i \(-0.313426\pi\)
\(410\) 151007.i 0.898316i
\(411\) 0 0
\(412\) 108484.i 0.639106i
\(413\) −60792.9 + 122447.i −0.356412 + 0.717875i
\(414\) 0 0
\(415\) 13685.0i 0.0794598i
\(416\) −341486. −1.97327
\(417\) 0 0
\(418\) −115632. −0.661796
\(419\) 15318.9i 0.0872570i −0.999048 0.0436285i \(-0.986108\pi\)
0.999048 0.0436285i \(-0.0138918\pi\)
\(420\) 0 0
\(421\) 278756.i 1.57275i −0.617747 0.786377i \(-0.711954\pi\)
0.617747 0.786377i \(-0.288046\pi\)
\(422\) −152841. −0.858252
\(423\) 0 0
\(424\) 27910.2i 0.155250i
\(425\) 216867. 1.20065
\(426\) 0 0
\(427\) 58832.8i 0.322674i
\(428\) 26114.1 0.142557
\(429\) 0 0
\(430\) 173251. 0.937000
\(431\) 202239.i 1.08870i −0.838857 0.544352i \(-0.816776\pi\)
0.838857 0.544352i \(-0.183224\pi\)
\(432\) 0 0
\(433\) −88674.2 −0.472957 −0.236478 0.971637i \(-0.575993\pi\)
−0.236478 + 0.971637i \(0.575993\pi\)
\(434\) 29021.0 0.154075
\(435\) 0 0
\(436\) 281503.i 1.48085i
\(437\) 60985.1i 0.319345i
\(438\) 0 0
\(439\) −374655. −1.94403 −0.972013 0.234928i \(-0.924515\pi\)
−0.972013 + 0.234928i \(0.924515\pi\)
\(440\) 14771.6 0.0762997
\(441\) 0 0
\(442\) 525141. 2.68801
\(443\) 19857.5i 0.101185i 0.998719 + 0.0505925i \(0.0161110\pi\)
−0.998719 + 0.0505925i \(0.983889\pi\)
\(444\) 0 0
\(445\) 426555.i 2.15404i
\(446\) 118311.i 0.594779i
\(447\) 0 0
\(448\) −186048. −0.926978
\(449\) −71307.0 −0.353703 −0.176852 0.984238i \(-0.556591\pi\)
−0.176852 + 0.984238i \(0.556591\pi\)
\(450\) 0 0
\(451\) 43311.2i 0.212935i
\(452\) 392465.i 1.92099i
\(453\) 0 0
\(454\) −452486. −2.19530
\(455\) 313850.i 1.51600i
\(456\) 0 0
\(457\) 134049.i 0.641846i −0.947105 0.320923i \(-0.896007\pi\)
0.947105 0.320923i \(-0.103993\pi\)
\(458\) 180131. 0.858732
\(459\) 0 0
\(460\) 102892.i 0.486257i
\(461\) −281430. −1.32425 −0.662123 0.749395i \(-0.730344\pi\)
−0.662123 + 0.749395i \(0.730344\pi\)
\(462\) 0 0
\(463\) 323834.i 1.51064i −0.655356 0.755320i \(-0.727482\pi\)
0.655356 0.755320i \(-0.272518\pi\)
\(464\) 21172.0 0.0983393
\(465\) 0 0
\(466\) 525265. 2.41884
\(467\) 411702.i 1.88777i 0.330272 + 0.943886i \(0.392859\pi\)
−0.330272 + 0.943886i \(0.607141\pi\)
\(468\) 0 0
\(469\) 144660.i 0.657663i
\(470\) −177525. −0.803645
\(471\) 0 0
\(472\) 23586.2 + 11710.1i 0.105870 + 0.0525628i
\(473\) 49691.2 0.222104
\(474\) 0 0
\(475\) −195434. −0.866188
\(476\) 265842. 1.17330
\(477\) 0 0
\(478\) 555067.i 2.42935i
\(479\) −84811.3 −0.369643 −0.184822 0.982772i \(-0.559171\pi\)
−0.184822 + 0.982772i \(0.559171\pi\)
\(480\) 0 0
\(481\) −96143.1 −0.415554
\(482\) 399442.i 1.71933i
\(483\) 0 0
\(484\) −197491. −0.843058
\(485\) 13652.4i 0.0580396i
\(486\) 0 0
\(487\) 249337. 1.05131 0.525653 0.850699i \(-0.323821\pi\)
0.525653 + 0.850699i \(0.323821\pi\)
\(488\) 11332.6 0.0475871
\(489\) 0 0
\(490\) 170207.i 0.708902i
\(491\) −50328.2 −0.208760 −0.104380 0.994537i \(-0.533286\pi\)
−0.104380 + 0.994537i \(0.533286\pi\)
\(492\) 0 0
\(493\) −35485.1 −0.146000
\(494\) −473240. −1.93922
\(495\) 0 0
\(496\) 29872.0i 0.121423i
\(497\) −362508. −1.46759
\(498\) 0 0
\(499\) −23067.0 −0.0926380 −0.0463190 0.998927i \(-0.514749\pi\)
−0.0463190 + 0.998927i \(0.514749\pi\)
\(500\) −41858.5 −0.167434
\(501\) 0 0
\(502\) 543301.i 2.15592i
\(503\) 410578.i 1.62278i 0.584505 + 0.811390i \(0.301289\pi\)
−0.584505 + 0.811390i \(0.698711\pi\)
\(504\) 0 0
\(505\) 243511.i 0.954852i
\(506\) 56787.6i 0.221795i
\(507\) 0 0
\(508\) −128621. −0.498406
\(509\) 19719.0i 0.0761115i −0.999276 0.0380557i \(-0.987884\pi\)
0.999276 0.0380557i \(-0.0121164\pi\)
\(510\) 0 0
\(511\) 169208.i 0.648007i
\(512\) 370646.i 1.41390i
\(513\) 0 0
\(514\) 67412.5i 0.255161i
\(515\) 215238.i 0.811529i
\(516\) 0 0
\(517\) −50917.0 −0.190494
\(518\) −93655.8 −0.349040
\(519\) 0 0
\(520\) 60455.0 0.223576
\(521\) −420195. −1.54802 −0.774008 0.633175i \(-0.781751\pi\)
−0.774008 + 0.633175i \(0.781751\pi\)
\(522\) 0 0
\(523\) −523604. −1.91425 −0.957126 0.289670i \(-0.906454\pi\)
−0.957126 + 0.289670i \(0.906454\pi\)
\(524\) 520421.i 1.89536i
\(525\) 0 0
\(526\) 389877.i 1.40915i
\(527\) 50066.6i 0.180272i
\(528\) 0 0
\(529\) 249891. 0.892974
\(530\) 731343.i 2.60357i
\(531\) 0 0
\(532\) −239568. −0.846458
\(533\) 177257.i 0.623950i
\(534\) 0 0
\(535\) 51811.6 0.181017
\(536\) −27865.0 −0.0969905
\(537\) 0 0
\(538\) −46663.9 −0.161219
\(539\) 48818.2i 0.168037i
\(540\) 0 0
\(541\) 62563.9i 0.213761i −0.994272 0.106881i \(-0.965914\pi\)
0.994272 0.106881i \(-0.0340863\pi\)
\(542\) 17565.9i 0.0597959i
\(543\) 0 0
\(544\) 573886.i 1.93922i
\(545\) 558515.i 1.88036i
\(546\) 0 0
\(547\) 28370.0 0.0948166 0.0474083 0.998876i \(-0.484904\pi\)
0.0474083 + 0.998876i \(0.484904\pi\)
\(548\) 361923. 1.20519
\(549\) 0 0
\(550\) 181982. 0.601595
\(551\) 31978.0 0.105329
\(552\) 0 0
\(553\) 184891. 0.604596
\(554\) 639060.i 2.08220i
\(555\) 0 0
\(556\) −392378. −1.26927
\(557\) 85305.1 0.274957 0.137478 0.990505i \(-0.456100\pi\)
0.137478 + 0.990505i \(0.456100\pi\)
\(558\) 0 0
\(559\) 203369. 0.650819
\(560\) −314696. −1.00350
\(561\) 0 0
\(562\) 766610.i 2.42718i
\(563\) 625430.i 1.97316i −0.163282 0.986580i \(-0.552208\pi\)
0.163282 0.986580i \(-0.447792\pi\)
\(564\) 0 0
\(565\) 778668.i 2.43925i
\(566\) 788680. 2.46189
\(567\) 0 0
\(568\) 69827.6i 0.216437i
\(569\) 182688.i 0.564269i 0.959375 + 0.282134i \(0.0910424\pi\)
−0.959375 + 0.282134i \(0.908958\pi\)
\(570\) 0 0
\(571\) 366777.i 1.12494i 0.826817 + 0.562471i \(0.190149\pi\)
−0.826817 + 0.562471i \(0.809851\pi\)
\(572\) 229004. 0.699923
\(573\) 0 0
\(574\) 172672.i 0.524079i
\(575\) 95979.0i 0.290296i
\(576\) 0 0
\(577\) 485860. 1.45935 0.729675 0.683794i \(-0.239671\pi\)
0.729675 + 0.683794i \(0.239671\pi\)
\(578\) 400482.i 1.19875i
\(579\) 0 0
\(580\) −53952.3 −0.160381
\(581\) 15648.3i 0.0463570i
\(582\) 0 0
\(583\) 209761.i 0.617145i
\(584\) 32593.5 0.0955664
\(585\) 0 0
\(586\) 796921.i 2.32071i
\(587\) 466181.i 1.35294i 0.736470 + 0.676470i \(0.236491\pi\)
−0.736470 + 0.676470i \(0.763509\pi\)
\(588\) 0 0
\(589\) 45118.4i 0.130054i
\(590\) 618041. + 306846.i 1.77547 + 0.881489i
\(591\) 0 0
\(592\) 96402.1i 0.275070i
\(593\) 263489. 0.749294 0.374647 0.927167i \(-0.377764\pi\)
0.374647 + 0.927167i \(0.377764\pi\)
\(594\) 0 0
\(595\) 527442. 1.48984
\(596\) 255300.i 0.718719i
\(597\) 0 0
\(598\) 232412.i 0.649914i
\(599\) 182847. 0.509606 0.254803 0.966993i \(-0.417989\pi\)
0.254803 + 0.966993i \(0.417989\pi\)
\(600\) 0 0
\(601\) 529477.i 1.46588i 0.680294 + 0.732939i \(0.261852\pi\)
−0.680294 + 0.732939i \(0.738148\pi\)
\(602\) 198107. 0.546647
\(603\) 0 0
\(604\) 23412.0i 0.0641749i
\(605\) −391832. −1.07051
\(606\) 0 0
\(607\) −161986. −0.439642 −0.219821 0.975540i \(-0.570547\pi\)
−0.219821 + 0.975540i \(0.570547\pi\)
\(608\) 517167.i 1.39902i
\(609\) 0 0
\(610\) 296953. 0.798046
\(611\) −208385. −0.558193
\(612\) 0 0
\(613\) 97048.9i 0.258268i −0.991627 0.129134i \(-0.958780\pi\)
0.991627 0.129134i \(-0.0412197\pi\)
\(614\) 511967.i 1.35802i
\(615\) 0 0
\(616\) 16890.9 0.0445134
\(617\) 17414.4 0.0457444 0.0228722 0.999738i \(-0.492719\pi\)
0.0228722 + 0.999738i \(0.492719\pi\)
\(618\) 0 0
\(619\) −216510. −0.565061 −0.282531 0.959258i \(-0.591174\pi\)
−0.282531 + 0.959258i \(0.591174\pi\)
\(620\) 76122.3i 0.198029i
\(621\) 0 0
\(622\) 694550.i 1.79524i
\(623\) 487751.i 1.25667i
\(624\) 0 0
\(625\) −429671. −1.09996
\(626\) −887904. −2.26578
\(627\) 0 0
\(628\) 759376.i 1.92547i
\(629\) 161574.i 0.408384i
\(630\) 0 0
\(631\) −218714. −0.549310 −0.274655 0.961543i \(-0.588564\pi\)
−0.274655 + 0.961543i \(0.588564\pi\)
\(632\) 35614.3i 0.0891643i
\(633\) 0 0
\(634\) 76309.6i 0.189846i
\(635\) −255189. −0.632871
\(636\) 0 0
\(637\) 199796.i 0.492388i
\(638\) −29777.1 −0.0731544
\(639\) 0 0
\(640\) 132580.i 0.323681i
\(641\) 210985. 0.513495 0.256747 0.966479i \(-0.417349\pi\)
0.256747 + 0.966479i \(0.417349\pi\)
\(642\) 0 0
\(643\) 192225. 0.464932 0.232466 0.972605i \(-0.425321\pi\)
0.232466 + 0.972605i \(0.425321\pi\)
\(644\) 117654.i 0.283683i
\(645\) 0 0
\(646\) 795305.i 1.90576i
\(647\) 671147. 1.60328 0.801639 0.597808i \(-0.203962\pi\)
0.801639 + 0.597808i \(0.203962\pi\)
\(648\) 0 0
\(649\) 177264. + 88008.3i 0.420853 + 0.208946i
\(650\) 744790. 1.76282
\(651\) 0 0
\(652\) −431294. −1.01456
\(653\) −330138. −0.774227 −0.387114 0.922032i \(-0.626528\pi\)
−0.387114 + 0.922032i \(0.626528\pi\)
\(654\) 0 0
\(655\) 1.03254e6i 2.40671i
\(656\) −177735. −0.413014
\(657\) 0 0
\(658\) −202994. −0.468848
\(659\) 45073.9i 0.103790i −0.998653 0.0518949i \(-0.983474\pi\)
0.998653 0.0518949i \(-0.0165261\pi\)
\(660\) 0 0
\(661\) −304683. −0.697341 −0.348670 0.937245i \(-0.613367\pi\)
−0.348670 + 0.937245i \(0.613367\pi\)
\(662\) 325098.i 0.741821i
\(663\) 0 0
\(664\) 3014.23 0.00683661
\(665\) −475314. −1.07482
\(666\) 0 0
\(667\) 15704.7i 0.0353002i
\(668\) −266322. −0.596835
\(669\) 0 0
\(670\) −730159. −1.62655
\(671\) 85170.7 0.189167
\(672\) 0 0
\(673\) 188450.i 0.416070i 0.978121 + 0.208035i \(0.0667068\pi\)
−0.978121 + 0.208035i \(0.933293\pi\)
\(674\) 665681. 1.46537
\(675\) 0 0
\(676\) 442820. 0.969022
\(677\) 254006. 0.554199 0.277100 0.960841i \(-0.410627\pi\)
0.277100 + 0.960841i \(0.410627\pi\)
\(678\) 0 0
\(679\) 15611.0i 0.0338604i
\(680\) 101598.i 0.219719i
\(681\) 0 0
\(682\) 42013.0i 0.0903265i
\(683\) 537333.i 1.15187i −0.817497 0.575933i \(-0.804639\pi\)
0.817497 0.575933i \(-0.195361\pi\)
\(684\) 0 0
\(685\) 718071. 1.53033
\(686\) 738847.i 1.57002i
\(687\) 0 0
\(688\) 203917.i 0.430800i
\(689\) 858477.i 1.80838i
\(690\) 0 0
\(691\) 70666.8i 0.147999i 0.997258 + 0.0739996i \(0.0235763\pi\)
−0.997258 + 0.0739996i \(0.976424\pi\)
\(692\) 950678.i 1.98528i
\(693\) 0 0
\(694\) 55314.7 0.114848
\(695\) −778495. −1.61171
\(696\) 0 0
\(697\) 297890. 0.613184
\(698\) 858099. 1.76127
\(699\) 0 0
\(700\) 377035. 0.769459
\(701\) 792788.i 1.61332i 0.591015 + 0.806661i \(0.298727\pi\)
−0.591015 + 0.806661i \(0.701273\pi\)
\(702\) 0 0
\(703\) 145605.i 0.294622i
\(704\) 269337.i 0.543440i
\(705\) 0 0
\(706\) −69520.3 −0.139477
\(707\) 278447.i 0.557063i
\(708\) 0 0
\(709\) −867302. −1.72535 −0.862677 0.505756i \(-0.831214\pi\)
−0.862677 + 0.505756i \(0.831214\pi\)
\(710\) 1.82972e6i 3.62969i
\(711\) 0 0
\(712\) 93952.4 0.185331
\(713\) −22158.0 −0.0435864
\(714\) 0 0
\(715\) 454353. 0.888754
\(716\) 874965.i 1.70673i
\(717\) 0 0
\(718\) 64407.2i 0.124935i
\(719\) 267439.i 0.517329i 0.965967 + 0.258665i \(0.0832825\pi\)
−0.965967 + 0.258665i \(0.916718\pi\)
\(720\) 0 0
\(721\) 246117.i 0.473448i
\(722\) 35450.9i 0.0680069i
\(723\) 0 0
\(724\) −598745. −1.14226
\(725\) −50327.4 −0.0957477
\(726\) 0 0
\(727\) 717136. 1.35685 0.678426 0.734668i \(-0.262662\pi\)
0.678426 + 0.734668i \(0.262662\pi\)
\(728\) 69128.4 0.130435
\(729\) 0 0
\(730\) 854062. 1.60267
\(731\) 341772.i 0.639589i
\(732\) 0 0
\(733\) −443873. −0.826134 −0.413067 0.910701i \(-0.635543\pi\)
−0.413067 + 0.910701i \(0.635543\pi\)
\(734\) −138953. −0.257914
\(735\) 0 0
\(736\) 253985. 0.468870
\(737\) −209421. −0.385554
\(738\) 0 0
\(739\) 177524.i 0.325063i −0.986703 0.162532i \(-0.948034\pi\)
0.986703 0.162532i \(-0.0519659\pi\)
\(740\) 245660.i 0.448611i
\(741\) 0 0
\(742\) 836267.i 1.51893i
\(743\) 266645. 0.483009 0.241505 0.970400i \(-0.422359\pi\)
0.241505 + 0.970400i \(0.422359\pi\)
\(744\) 0 0
\(745\) 506527.i 0.912621i
\(746\) 502581.i 0.903085i
\(747\) 0 0
\(748\) 384853.i 0.687846i
\(749\) 59244.9 0.105606
\(750\) 0 0
\(751\) 438213.i 0.776972i 0.921455 + 0.388486i \(0.127002\pi\)
−0.921455 + 0.388486i \(0.872998\pi\)
\(752\) 208947.i 0.369488i
\(753\) 0 0
\(754\) −121867. −0.214360
\(755\) 46450.5i 0.0814886i
\(756\) 0 0
\(757\) 999861. 1.74481 0.872404 0.488785i \(-0.162560\pi\)
0.872404 + 0.488785i \(0.162560\pi\)
\(758\) 1.16359e6i 2.02516i
\(759\) 0 0
\(760\) 91556.7i 0.158512i
\(761\) 326900. 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(762\) 0 0
\(763\) 638643.i 1.09701i
\(764\) 505045.i 0.865253i
\(765\) 0 0
\(766\) 1.04881e6i 1.78747i
\(767\) 725479. + 360187.i 1.23320 + 0.612262i
\(768\) 0 0
\(769\) 235002.i 0.397392i 0.980061 + 0.198696i \(0.0636707\pi\)
−0.980061 + 0.198696i \(0.936329\pi\)
\(770\) 442599. 0.746498
\(771\) 0 0
\(772\) −552922. −0.927746
\(773\) 210457.i 0.352213i 0.984371 + 0.176106i \(0.0563503\pi\)
−0.984371 + 0.176106i \(0.943650\pi\)
\(774\) 0 0
\(775\) 71007.9i 0.118223i
\(776\) 3007.05 0.00499365
\(777\) 0 0
\(778\) 770606.i 1.27313i
\(779\) −268449. −0.442371
\(780\) 0 0
\(781\) 524794.i 0.860373i
\(782\) −390580. −0.638700
\(783\) 0 0
\(784\) 200334. 0.325929
\(785\) 1.50664e6i 2.44494i
\(786\) 0 0
\(787\) −11734.5 −0.0189459 −0.00947293 0.999955i \(-0.503015\pi\)
−0.00947293 + 0.999955i \(0.503015\pi\)
\(788\) −73578.1 −0.118494
\(789\) 0 0
\(790\) 933219.i 1.49530i
\(791\) 890382.i 1.42306i
\(792\) 0 0
\(793\) 348574. 0.554305
\(794\) 159464. 0.252943
\(795\) 0 0
\(796\) −494433. −0.780335
\(797\) 592344.i