Properties

Label 531.5.c.d.235.9
Level $531$
Weight $5$
Character 531.235
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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 235.9
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.32

$q$-expansion

\(f(q)\) \(=\) \(q-4.85825i q^{2} -7.60257 q^{4} +12.5087 q^{5} +50.4755 q^{7} -40.7968i q^{8} +O(q^{10})\) \(q-4.85825i q^{2} -7.60257 q^{4} +12.5087 q^{5} +50.4755 q^{7} -40.7968i q^{8} -60.7705i q^{10} -191.226i q^{11} -0.640162i q^{13} -245.222i q^{14} -319.842 q^{16} +549.428 q^{17} +524.497 q^{19} -95.0984 q^{20} -929.025 q^{22} +543.113i q^{23} -468.532 q^{25} -3.11006 q^{26} -383.743 q^{28} +296.169 q^{29} +1078.43i q^{31} +901.123i q^{32} -2669.26i q^{34} +631.384 q^{35} -1168.43i q^{37} -2548.14i q^{38} -510.316i q^{40} +1128.78 q^{41} +1522.74i q^{43} +1453.81i q^{44} +2638.58 q^{46} -103.190i q^{47} +146.777 q^{49} +2276.24i q^{50} +4.86687i q^{52} +2811.88 q^{53} -2392.00i q^{55} -2059.24i q^{56} -1438.86i q^{58} +(2562.55 - 2355.99i) q^{59} -4909.65i q^{61} +5239.30 q^{62} -739.596 q^{64} -8.00760i q^{65} -5854.39i q^{67} -4177.06 q^{68} -3067.42i q^{70} -5828.87 q^{71} +6751.02i q^{73} -5676.53 q^{74} -3987.52 q^{76} -9652.25i q^{77} -9333.98 q^{79} -4000.81 q^{80} -5483.90i q^{82} -1839.42i q^{83} +6872.64 q^{85} +7397.84 q^{86} -7801.43 q^{88} -5734.95i q^{89} -32.3125i q^{91} -4129.05i q^{92} -501.323 q^{94} +6560.79 q^{95} +15272.6i q^{97} -713.079i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.85825i 1.21456i −0.794487 0.607281i \(-0.792260\pi\)
0.794487 0.607281i \(-0.207740\pi\)
\(3\) 0 0
\(4\) −7.60257 −0.475160
\(5\) 12.5087 0.500349 0.250174 0.968201i \(-0.419512\pi\)
0.250174 + 0.968201i \(0.419512\pi\)
\(6\) 0 0
\(7\) 50.4755 1.03011 0.515056 0.857156i \(-0.327771\pi\)
0.515056 + 0.857156i \(0.327771\pi\)
\(8\) 40.7968i 0.637450i
\(9\) 0 0
\(10\) 60.7705i 0.607705i
\(11\) 191.226i 1.58038i −0.612860 0.790192i \(-0.709981\pi\)
0.612860 0.790192i \(-0.290019\pi\)
\(12\) 0 0
\(13\) 0.640162i 0.00378794i −0.999998 0.00189397i \(-0.999397\pi\)
0.999998 0.00189397i \(-0.000602869\pi\)
\(14\) 245.222i 1.25114i
\(15\) 0 0
\(16\) −319.842 −1.24938
\(17\) 549.428 1.90113 0.950567 0.310520i \(-0.100503\pi\)
0.950567 + 0.310520i \(0.100503\pi\)
\(18\) 0 0
\(19\) 524.497 1.45290 0.726450 0.687219i \(-0.241169\pi\)
0.726450 + 0.687219i \(0.241169\pi\)
\(20\) −95.0984 −0.237746
\(21\) 0 0
\(22\) −929.025 −1.91947
\(23\) 543.113i 1.02668i 0.858186 + 0.513339i \(0.171592\pi\)
−0.858186 + 0.513339i \(0.828408\pi\)
\(24\) 0 0
\(25\) −468.532 −0.749651
\(26\) −3.11006 −0.00460068
\(27\) 0 0
\(28\) −383.743 −0.489469
\(29\) 296.169 0.352163 0.176081 0.984376i \(-0.443658\pi\)
0.176081 + 0.984376i \(0.443658\pi\)
\(30\) 0 0
\(31\) 1078.43i 1.12220i 0.827748 + 0.561100i \(0.189622\pi\)
−0.827748 + 0.561100i \(0.810378\pi\)
\(32\) 901.123i 0.880003i
\(33\) 0 0
\(34\) 2669.26i 2.30904i
\(35\) 631.384 0.515416
\(36\) 0 0
\(37\) 1168.43i 0.853493i −0.904371 0.426747i \(-0.859660\pi\)
0.904371 0.426747i \(-0.140340\pi\)
\(38\) 2548.14i 1.76464i
\(39\) 0 0
\(40\) 510.316i 0.318947i
\(41\) 1128.78 0.671494 0.335747 0.941952i \(-0.391011\pi\)
0.335747 + 0.941952i \(0.391011\pi\)
\(42\) 0 0
\(43\) 1522.74i 0.823547i 0.911286 + 0.411773i \(0.135090\pi\)
−0.911286 + 0.411773i \(0.864910\pi\)
\(44\) 1453.81i 0.750936i
\(45\) 0 0
\(46\) 2638.58 1.24696
\(47\) 103.190i 0.0467135i −0.999727 0.0233567i \(-0.992565\pi\)
0.999727 0.0233567i \(-0.00743536\pi\)
\(48\) 0 0
\(49\) 146.777 0.0611316
\(50\) 2276.24i 0.910498i
\(51\) 0 0
\(52\) 4.86687i 0.00179988i
\(53\) 2811.88 1.00103 0.500513 0.865729i \(-0.333145\pi\)
0.500513 + 0.865729i \(0.333145\pi\)
\(54\) 0 0
\(55\) 2392.00i 0.790743i
\(56\) 2059.24i 0.656645i
\(57\) 0 0
\(58\) 1438.86i 0.427723i
\(59\) 2562.55 2355.99i 0.736155 0.676813i
\(60\) 0 0
\(61\) 4909.65i 1.31944i −0.751510 0.659722i \(-0.770674\pi\)
0.751510 0.659722i \(-0.229326\pi\)
\(62\) 5239.30 1.36298
\(63\) 0 0
\(64\) −739.596 −0.180565
\(65\) 8.00760i 0.00189529i
\(66\) 0 0
\(67\) 5854.39i 1.30416i −0.758149 0.652082i \(-0.773896\pi\)
0.758149 0.652082i \(-0.226104\pi\)
\(68\) −4177.06 −0.903343
\(69\) 0 0
\(70\) 3067.42i 0.626004i
\(71\) −5828.87 −1.15629 −0.578146 0.815933i \(-0.696224\pi\)
−0.578146 + 0.815933i \(0.696224\pi\)
\(72\) 0 0
\(73\) 6751.02i 1.26684i 0.773806 + 0.633422i \(0.218350\pi\)
−0.773806 + 0.633422i \(0.781650\pi\)
\(74\) −5676.53 −1.03662
\(75\) 0 0
\(76\) −3987.52 −0.690361
\(77\) 9652.25i 1.62797i
\(78\) 0 0
\(79\) −9333.98 −1.49559 −0.747795 0.663929i \(-0.768888\pi\)
−0.747795 + 0.663929i \(0.768888\pi\)
\(80\) −4000.81 −0.625127
\(81\) 0 0
\(82\) 5483.90i 0.815571i
\(83\) 1839.42i 0.267008i −0.991048 0.133504i \(-0.957377\pi\)
0.991048 0.133504i \(-0.0426229\pi\)
\(84\) 0 0
\(85\) 6872.64 0.951230
\(86\) 7397.84 1.00025
\(87\) 0 0
\(88\) −7801.43 −1.00742
\(89\) 5734.95i 0.724018i −0.932175 0.362009i \(-0.882091\pi\)
0.932175 0.362009i \(-0.117909\pi\)
\(90\) 0 0
\(91\) 32.3125i 0.00390200i
\(92\) 4129.05i 0.487837i
\(93\) 0 0
\(94\) −501.323 −0.0567364
\(95\) 6560.79 0.726957
\(96\) 0 0
\(97\) 15272.6i 1.62319i 0.584223 + 0.811593i \(0.301399\pi\)
−0.584223 + 0.811593i \(0.698601\pi\)
\(98\) 713.079i 0.0742481i
\(99\) 0 0
\(100\) 3562.04 0.356204
\(101\) 13635.2i 1.33665i −0.743868 0.668327i \(-0.767011\pi\)
0.743868 0.668327i \(-0.232989\pi\)
\(102\) 0 0
\(103\) 4773.40i 0.449939i 0.974366 + 0.224969i \(0.0722282\pi\)
−0.974366 + 0.224969i \(0.927772\pi\)
\(104\) −26.1166 −0.00241462
\(105\) 0 0
\(106\) 13660.8i 1.21581i
\(107\) −10911.5 −0.953051 −0.476525 0.879161i \(-0.658104\pi\)
−0.476525 + 0.879161i \(0.658104\pi\)
\(108\) 0 0
\(109\) 1327.72i 0.111752i 0.998438 + 0.0558758i \(0.0177951\pi\)
−0.998438 + 0.0558758i \(0.982205\pi\)
\(110\) −11620.9 −0.960406
\(111\) 0 0
\(112\) −16144.2 −1.28700
\(113\) 7206.57i 0.564380i 0.959359 + 0.282190i \(0.0910608\pi\)
−0.959359 + 0.282190i \(0.908939\pi\)
\(114\) 0 0
\(115\) 6793.65i 0.513697i
\(116\) −2251.64 −0.167334
\(117\) 0 0
\(118\) −11446.0 12449.5i −0.822031 0.894105i
\(119\) 27732.6 1.95838
\(120\) 0 0
\(121\) −21926.5 −1.49761
\(122\) −23852.3 −1.60255
\(123\) 0 0
\(124\) 8198.86i 0.533225i
\(125\) −13678.7 −0.875436
\(126\) 0 0
\(127\) 30152.5 1.86946 0.934729 0.355361i \(-0.115642\pi\)
0.934729 + 0.355361i \(0.115642\pi\)
\(128\) 18011.1i 1.09931i
\(129\) 0 0
\(130\) −38.9029 −0.00230195
\(131\) 30591.4i 1.78261i −0.453404 0.891305i \(-0.649790\pi\)
0.453404 0.891305i \(-0.350210\pi\)
\(132\) 0 0
\(133\) 26474.3 1.49665
\(134\) −28442.1 −1.58399
\(135\) 0 0
\(136\) 22414.9i 1.21188i
\(137\) −28251.4 −1.50521 −0.752607 0.658470i \(-0.771204\pi\)
−0.752607 + 0.658470i \(0.771204\pi\)
\(138\) 0 0
\(139\) 6125.81 0.317054 0.158527 0.987355i \(-0.449325\pi\)
0.158527 + 0.987355i \(0.449325\pi\)
\(140\) −4800.14 −0.244905
\(141\) 0 0
\(142\) 28318.1i 1.40439i
\(143\) −122.416 −0.00598640
\(144\) 0 0
\(145\) 3704.69 0.176204
\(146\) 32798.1 1.53866
\(147\) 0 0
\(148\) 8883.08i 0.405546i
\(149\) 35941.9i 1.61893i 0.587166 + 0.809467i \(0.300244\pi\)
−0.587166 + 0.809467i \(0.699756\pi\)
\(150\) 0 0
\(151\) 3815.72i 0.167349i 0.996493 + 0.0836745i \(0.0266656\pi\)
−0.996493 + 0.0836745i \(0.973334\pi\)
\(152\) 21397.8i 0.926152i
\(153\) 0 0
\(154\) −46893.0 −1.97727
\(155\) 13489.8i 0.561491i
\(156\) 0 0
\(157\) 14168.1i 0.574795i −0.957811 0.287398i \(-0.907210\pi\)
0.957811 0.287398i \(-0.0927901\pi\)
\(158\) 45346.8i 1.81649i
\(159\) 0 0
\(160\) 11271.9i 0.440308i
\(161\) 27413.9i 1.05759i
\(162\) 0 0
\(163\) −5436.37 −0.204613 −0.102307 0.994753i \(-0.532622\pi\)
−0.102307 + 0.994753i \(0.532622\pi\)
\(164\) −8581.64 −0.319068
\(165\) 0 0
\(166\) −8936.35 −0.324298
\(167\) −10745.5 −0.385296 −0.192648 0.981268i \(-0.561707\pi\)
−0.192648 + 0.981268i \(0.561707\pi\)
\(168\) 0 0
\(169\) 28560.6 0.999986
\(170\) 33389.0i 1.15533i
\(171\) 0 0
\(172\) 11576.7i 0.391317i
\(173\) 27249.2i 0.910463i 0.890373 + 0.455231i \(0.150443\pi\)
−0.890373 + 0.455231i \(0.849557\pi\)
\(174\) 0 0
\(175\) −23649.4 −0.772225
\(176\) 61162.3i 1.97450i
\(177\) 0 0
\(178\) −27861.8 −0.879365
\(179\) 60313.1i 1.88237i −0.337890 0.941186i \(-0.609713\pi\)
0.337890 0.941186i \(-0.390287\pi\)
\(180\) 0 0
\(181\) −27872.8 −0.850793 −0.425396 0.905007i \(-0.639865\pi\)
−0.425396 + 0.905007i \(0.639865\pi\)
\(182\) −156.982 −0.00473922
\(183\) 0 0
\(184\) 22157.3 0.654456
\(185\) 14615.6i 0.427044i
\(186\) 0 0
\(187\) 105065.i 3.00452i
\(188\) 784.509i 0.0221964i
\(189\) 0 0
\(190\) 31873.9i 0.882934i
\(191\) 32832.5i 0.899989i 0.893031 + 0.449995i \(0.148574\pi\)
−0.893031 + 0.449995i \(0.851426\pi\)
\(192\) 0 0
\(193\) −51056.4 −1.37068 −0.685339 0.728224i \(-0.740346\pi\)
−0.685339 + 0.728224i \(0.740346\pi\)
\(194\) 74197.9 1.97146
\(195\) 0 0
\(196\) −1115.88 −0.0290473
\(197\) −52427.0 −1.35090 −0.675449 0.737407i \(-0.736050\pi\)
−0.675449 + 0.737407i \(0.736050\pi\)
\(198\) 0 0
\(199\) −21283.2 −0.537440 −0.268720 0.963218i \(-0.586601\pi\)
−0.268720 + 0.963218i \(0.586601\pi\)
\(200\) 19114.6i 0.477865i
\(201\) 0 0
\(202\) −66243.2 −1.62345
\(203\) 14949.3 0.362767
\(204\) 0 0
\(205\) 14119.6 0.335981
\(206\) 23190.3 0.546478
\(207\) 0 0
\(208\) 204.751i 0.00473259i
\(209\) 100298.i 2.29614i
\(210\) 0 0
\(211\) 67797.8i 1.52283i 0.648267 + 0.761413i \(0.275494\pi\)
−0.648267 + 0.761413i \(0.724506\pi\)
\(212\) −21377.5 −0.475648
\(213\) 0 0
\(214\) 53010.7i 1.15754i
\(215\) 19047.5i 0.412061i
\(216\) 0 0
\(217\) 54434.5i 1.15599i
\(218\) 6450.40 0.135729
\(219\) 0 0
\(220\) 18185.3i 0.375730i
\(221\) 351.722i 0.00720138i
\(222\) 0 0
\(223\) 45072.8 0.906368 0.453184 0.891417i \(-0.350288\pi\)
0.453184 + 0.891417i \(0.350288\pi\)
\(224\) 45484.6i 0.906502i
\(225\) 0 0
\(226\) 35011.3 0.685474
\(227\) 13186.6i 0.255906i −0.991780 0.127953i \(-0.959159\pi\)
0.991780 0.127953i \(-0.0408407\pi\)
\(228\) 0 0
\(229\) 5098.55i 0.0972244i 0.998818 + 0.0486122i \(0.0154799\pi\)
−0.998818 + 0.0486122i \(0.984520\pi\)
\(230\) 33005.2 0.623917
\(231\) 0 0
\(232\) 12082.7i 0.224486i
\(233\) 45461.0i 0.837390i −0.908127 0.418695i \(-0.862488\pi\)
0.908127 0.418695i \(-0.137512\pi\)
\(234\) 0 0
\(235\) 1290.78i 0.0233730i
\(236\) −19482.0 + 17911.5i −0.349792 + 0.321595i
\(237\) 0 0
\(238\) 134732.i 2.37858i
\(239\) 80213.6 1.40427 0.702137 0.712041i \(-0.252229\pi\)
0.702137 + 0.712041i \(0.252229\pi\)
\(240\) 0 0
\(241\) 27279.2 0.469676 0.234838 0.972035i \(-0.424544\pi\)
0.234838 + 0.972035i \(0.424544\pi\)
\(242\) 106525.i 1.81894i
\(243\) 0 0
\(244\) 37325.9i 0.626947i
\(245\) 1835.99 0.0305871
\(246\) 0 0
\(247\) 335.763i 0.00550350i
\(248\) 43996.7 0.715346
\(249\) 0 0
\(250\) 66454.4i 1.06327i
\(251\) 87290.9 1.38555 0.692774 0.721155i \(-0.256388\pi\)
0.692774 + 0.721155i \(0.256388\pi\)
\(252\) 0 0
\(253\) 103858. 1.62255
\(254\) 146488.i 2.27057i
\(255\) 0 0
\(256\) 75668.9 1.15462
\(257\) 1669.20 0.0252721 0.0126361 0.999920i \(-0.495978\pi\)
0.0126361 + 0.999920i \(0.495978\pi\)
\(258\) 0 0
\(259\) 58977.2i 0.879194i
\(260\) 60.8783i 0.000900567i
\(261\) 0 0
\(262\) −148620. −2.16509
\(263\) −99916.5 −1.44453 −0.722264 0.691618i \(-0.756898\pi\)
−0.722264 + 0.691618i \(0.756898\pi\)
\(264\) 0 0
\(265\) 35173.1 0.500862
\(266\) 128618.i 1.81777i
\(267\) 0 0
\(268\) 44508.4i 0.619687i
\(269\) 24594.1i 0.339881i 0.985454 + 0.169940i \(0.0543575\pi\)
−0.985454 + 0.169940i \(0.945642\pi\)
\(270\) 0 0
\(271\) 12306.3 0.167567 0.0837836 0.996484i \(-0.473300\pi\)
0.0837836 + 0.996484i \(0.473300\pi\)
\(272\) −175730. −2.37524
\(273\) 0 0
\(274\) 137252.i 1.82818i
\(275\) 89595.7i 1.18474i
\(276\) 0 0
\(277\) 30853.9 0.402115 0.201058 0.979579i \(-0.435562\pi\)
0.201058 + 0.979579i \(0.435562\pi\)
\(278\) 29760.7i 0.385082i
\(279\) 0 0
\(280\) 25758.5i 0.328552i
\(281\) −6311.40 −0.0799306 −0.0399653 0.999201i \(-0.512725\pi\)
−0.0399653 + 0.999201i \(0.512725\pi\)
\(282\) 0 0
\(283\) 37474.9i 0.467916i −0.972247 0.233958i \(-0.924832\pi\)
0.972247 0.233958i \(-0.0751678\pi\)
\(284\) 44314.3 0.549424
\(285\) 0 0
\(286\) 594.726i 0.00727085i
\(287\) 56975.9 0.691715
\(288\) 0 0
\(289\) 218350. 2.61431
\(290\) 17998.3i 0.214011i
\(291\) 0 0
\(292\) 51325.0i 0.601954i
\(293\) −13284.3 −0.154740 −0.0773700 0.997002i \(-0.524652\pi\)
−0.0773700 + 0.997002i \(0.524652\pi\)
\(294\) 0 0
\(295\) 32054.3 29470.4i 0.368334 0.338643i
\(296\) −47668.3 −0.544059
\(297\) 0 0
\(298\) 174615. 1.96629
\(299\) 347.680 0.00388899
\(300\) 0 0
\(301\) 76861.0i 0.848346i
\(302\) 18537.7 0.203256
\(303\) 0 0
\(304\) −167756. −1.81523
\(305\) 61413.4i 0.660182i
\(306\) 0 0
\(307\) 27275.0 0.289393 0.144697 0.989476i \(-0.453779\pi\)
0.144697 + 0.989476i \(0.453779\pi\)
\(308\) 73381.9i 0.773548i
\(309\) 0 0
\(310\) 65536.9 0.681966
\(311\) −74737.1 −0.772708 −0.386354 0.922351i \(-0.626266\pi\)
−0.386354 + 0.922351i \(0.626266\pi\)
\(312\) 0 0
\(313\) 147610.i 1.50670i 0.657618 + 0.753352i \(0.271564\pi\)
−0.657618 + 0.753352i \(0.728436\pi\)
\(314\) −68832.3 −0.698124
\(315\) 0 0
\(316\) 70962.2 0.710645
\(317\) 75632.0 0.752640 0.376320 0.926490i \(-0.377189\pi\)
0.376320 + 0.926490i \(0.377189\pi\)
\(318\) 0 0
\(319\) 56635.3i 0.556552i
\(320\) −9251.40 −0.0903457
\(321\) 0 0
\(322\) 133183. 1.28451
\(323\) 288173. 2.76216
\(324\) 0 0
\(325\) 299.936i 0.00283963i
\(326\) 26411.2i 0.248515i
\(327\) 0 0
\(328\) 46050.7i 0.428044i
\(329\) 5208.57i 0.0481201i
\(330\) 0 0
\(331\) 43603.6 0.397985 0.198992 0.980001i \(-0.436233\pi\)
0.198992 + 0.980001i \(0.436233\pi\)
\(332\) 13984.3i 0.126872i
\(333\) 0 0
\(334\) 52204.3i 0.467965i
\(335\) 73230.9i 0.652537i
\(336\) 0 0
\(337\) 111943.i 0.985682i 0.870119 + 0.492841i \(0.164042\pi\)
−0.870119 + 0.492841i \(0.835958\pi\)
\(338\) 138754.i 1.21454i
\(339\) 0 0
\(340\) −52249.7 −0.451987
\(341\) 206225. 1.77351
\(342\) 0 0
\(343\) −113783. −0.967140
\(344\) 62122.9 0.524970
\(345\) 0 0
\(346\) 132384. 1.10581
\(347\) 43440.9i 0.360778i 0.983595 + 0.180389i \(0.0577357\pi\)
−0.983595 + 0.180389i \(0.942264\pi\)
\(348\) 0 0
\(349\) 63411.1i 0.520612i −0.965526 0.260306i \(-0.916177\pi\)
0.965526 0.260306i \(-0.0838235\pi\)
\(350\) 114895.i 0.937915i
\(351\) 0 0
\(352\) 172318. 1.39074
\(353\) 152287.i 1.22212i −0.791585 0.611059i \(-0.790744\pi\)
0.791585 0.611059i \(-0.209256\pi\)
\(354\) 0 0
\(355\) −72911.7 −0.578549
\(356\) 43600.3i 0.344025i
\(357\) 0 0
\(358\) −293016. −2.28626
\(359\) −38536.4 −0.299008 −0.149504 0.988761i \(-0.547768\pi\)
−0.149504 + 0.988761i \(0.547768\pi\)
\(360\) 0 0
\(361\) 144776. 1.11092
\(362\) 135413.i 1.03334i
\(363\) 0 0
\(364\) 245.658i 0.00185408i
\(365\) 84446.6i 0.633864i
\(366\) 0 0
\(367\) 127968.i 0.950102i 0.879958 + 0.475051i \(0.157570\pi\)
−0.879958 + 0.475051i \(0.842430\pi\)
\(368\) 173710.i 1.28271i
\(369\) 0 0
\(370\) −71006.1 −0.518672
\(371\) 141931. 1.03117
\(372\) 0 0
\(373\) −181481. −1.30441 −0.652203 0.758044i \(-0.726155\pi\)
−0.652203 + 0.758044i \(0.726155\pi\)
\(374\) −510432. −3.64918
\(375\) 0 0
\(376\) −4209.83 −0.0297775
\(377\) 189.596i 0.00133397i
\(378\) 0 0
\(379\) −52087.4 −0.362622 −0.181311 0.983426i \(-0.558034\pi\)
−0.181311 + 0.983426i \(0.558034\pi\)
\(380\) −49878.8 −0.345421
\(381\) 0 0
\(382\) 159508. 1.09309
\(383\) 129457. 0.882527 0.441263 0.897378i \(-0.354530\pi\)
0.441263 + 0.897378i \(0.354530\pi\)
\(384\) 0 0
\(385\) 120737.i 0.814554i
\(386\) 248045.i 1.66477i
\(387\) 0 0
\(388\) 116111.i 0.771274i
\(389\) 239714. 1.58414 0.792070 0.610430i \(-0.209003\pi\)
0.792070 + 0.610430i \(0.209003\pi\)
\(390\) 0 0
\(391\) 298401.i 1.95185i
\(392\) 5988.03i 0.0389684i
\(393\) 0 0
\(394\) 254703.i 1.64075i
\(395\) −116756. −0.748317
\(396\) 0 0
\(397\) 63125.7i 0.400521i −0.979743 0.200260i \(-0.935821\pi\)
0.979743 0.200260i \(-0.0641788\pi\)
\(398\) 103399.i 0.652755i
\(399\) 0 0
\(400\) 149856. 0.936601
\(401\) 5633.87i 0.0350363i −0.999847 0.0175181i \(-0.994424\pi\)
0.999847 0.0175181i \(-0.00557648\pi\)
\(402\) 0 0
\(403\) 690.372 0.00425082
\(404\) 103663.i 0.635125i
\(405\) 0 0
\(406\) 72627.2i 0.440603i
\(407\) −223435. −1.34885
\(408\) 0 0
\(409\) 208211.i 1.24468i 0.782747 + 0.622340i \(0.213818\pi\)
−0.782747 + 0.622340i \(0.786182\pi\)
\(410\) 68596.6i 0.408070i
\(411\) 0 0
\(412\) 36290.1i 0.213793i
\(413\) 129346. 118920.i 0.758322 0.697194i
\(414\) 0 0
\(415\) 23008.8i 0.133597i
\(416\) 576.864 0.00333340
\(417\) 0 0
\(418\) −487271. −2.78880
\(419\) 29129.4i 0.165922i −0.996553 0.0829610i \(-0.973562\pi\)
0.996553 0.0829610i \(-0.0264377\pi\)
\(420\) 0 0
\(421\) 297451.i 1.67823i 0.543956 + 0.839113i \(0.316926\pi\)
−0.543956 + 0.839113i \(0.683074\pi\)
\(422\) 329378. 1.84957
\(423\) 0 0
\(424\) 114716.i 0.638105i
\(425\) −257424. −1.42519
\(426\) 0 0
\(427\) 247817.i 1.35917i
\(428\) 82955.2 0.452852
\(429\) 0 0
\(430\) 92537.5 0.500473
\(431\) 340194.i 1.83135i 0.401917 + 0.915676i \(0.368344\pi\)
−0.401917 + 0.915676i \(0.631656\pi\)
\(432\) 0 0
\(433\) −19705.6 −0.105103 −0.0525514 0.998618i \(-0.516735\pi\)
−0.0525514 + 0.998618i \(0.516735\pi\)
\(434\) 264456. 1.40402
\(435\) 0 0
\(436\) 10094.1i 0.0530999i
\(437\) 284861.i 1.49166i
\(438\) 0 0
\(439\) 29208.1 0.151557 0.0757783 0.997125i \(-0.475856\pi\)
0.0757783 + 0.997125i \(0.475856\pi\)
\(440\) −97585.9 −0.504059
\(441\) 0 0
\(442\) −1708.75 −0.00874652
\(443\) 136923.i 0.697698i 0.937179 + 0.348849i \(0.113427\pi\)
−0.937179 + 0.348849i \(0.886573\pi\)
\(444\) 0 0
\(445\) 71736.9i 0.362262i
\(446\) 218975.i 1.10084i
\(447\) 0 0
\(448\) −37331.5 −0.186003
\(449\) −324943. −1.61181 −0.805906 0.592043i \(-0.798322\pi\)
−0.805906 + 0.592043i \(0.798322\pi\)
\(450\) 0 0
\(451\) 215853.i 1.06122i
\(452\) 54788.4i 0.268171i
\(453\) 0 0
\(454\) −64063.8 −0.310814
\(455\) 404.188i 0.00195236i
\(456\) 0 0
\(457\) 142683.i 0.683189i −0.939847 0.341595i \(-0.889033\pi\)
0.939847 0.341595i \(-0.110967\pi\)
\(458\) 24770.0 0.118085
\(459\) 0 0
\(460\) 51649.1i 0.244089i
\(461\) −107011. −0.503533 −0.251767 0.967788i \(-0.581012\pi\)
−0.251767 + 0.967788i \(0.581012\pi\)
\(462\) 0 0
\(463\) 344520.i 1.60714i 0.595213 + 0.803568i \(0.297068\pi\)
−0.595213 + 0.803568i \(0.702932\pi\)
\(464\) −94727.2 −0.439986
\(465\) 0 0
\(466\) −220861. −1.01706
\(467\) 12985.4i 0.0595419i −0.999557 0.0297709i \(-0.990522\pi\)
0.999557 0.0297709i \(-0.00947778\pi\)
\(468\) 0 0
\(469\) 295503.i 1.34343i
\(470\) −6270.91 −0.0283880
\(471\) 0 0
\(472\) −96116.8 104544.i −0.431435 0.469262i
\(473\) 291188. 1.30152
\(474\) 0 0
\(475\) −245744. −1.08917
\(476\) −210839. −0.930545
\(477\) 0 0
\(478\) 389697.i 1.70558i
\(479\) −148631. −0.647798 −0.323899 0.946092i \(-0.604994\pi\)
−0.323899 + 0.946092i \(0.604994\pi\)
\(480\) 0 0
\(481\) −747.985 −0.00323298
\(482\) 132529.i 0.570450i
\(483\) 0 0
\(484\) 166698. 0.711606
\(485\) 191040.i 0.812159i
\(486\) 0 0
\(487\) 174834. 0.737169 0.368584 0.929594i \(-0.379843\pi\)
0.368584 + 0.929594i \(0.379843\pi\)
\(488\) −200298. −0.841079
\(489\) 0 0
\(490\) 8919.71i 0.0371500i
\(491\) 289993. 1.20288 0.601442 0.798916i \(-0.294593\pi\)
0.601442 + 0.798916i \(0.294593\pi\)
\(492\) 0 0
\(493\) 162723. 0.669508
\(494\) −1631.22 −0.00668434
\(495\) 0 0
\(496\) 344928.i 1.40206i
\(497\) −294215. −1.19111
\(498\) 0 0
\(499\) −337039. −1.35356 −0.676782 0.736183i \(-0.736626\pi\)
−0.676782 + 0.736183i \(0.736626\pi\)
\(500\) 103993. 0.415972
\(501\) 0 0
\(502\) 424081.i 1.68283i
\(503\) 424514.i 1.67786i 0.544238 + 0.838931i \(0.316819\pi\)
−0.544238 + 0.838931i \(0.683181\pi\)
\(504\) 0 0
\(505\) 170559.i 0.668793i
\(506\) 504566.i 1.97068i
\(507\) 0 0
\(508\) −229236. −0.888292
\(509\) 166873.i 0.644095i −0.946724 0.322047i \(-0.895629\pi\)
0.946724 0.322047i \(-0.104371\pi\)
\(510\) 0 0
\(511\) 340761.i 1.30499i
\(512\) 79440.3i 0.303041i
\(513\) 0 0
\(514\) 8109.37i 0.0306945i
\(515\) 59709.1i 0.225126i
\(516\) 0 0
\(517\) −19732.7 −0.0738252
\(518\) −286526. −1.06784
\(519\) 0 0
\(520\) −326.685 −0.00120815
\(521\) −234510. −0.863943 −0.431972 0.901887i \(-0.642182\pi\)
−0.431972 + 0.901887i \(0.642182\pi\)
\(522\) 0 0
\(523\) 133011. 0.486277 0.243138 0.969992i \(-0.421823\pi\)
0.243138 + 0.969992i \(0.421823\pi\)
\(524\) 232573.i 0.847026i
\(525\) 0 0
\(526\) 485419.i 1.75447i
\(527\) 592521.i 2.13345i
\(528\) 0 0
\(529\) −15130.5 −0.0540683
\(530\) 170879.i 0.608328i
\(531\) 0 0
\(532\) −201272. −0.711149
\(533\) 722.603i 0.00254358i
\(534\) 0 0
\(535\) −136489. −0.476858
\(536\) −238840. −0.831339
\(537\) 0 0
\(538\) 119484. 0.412806
\(539\) 28067.6i 0.0966114i
\(540\) 0 0
\(541\) 37399.8i 0.127784i 0.997957 + 0.0638918i \(0.0203512\pi\)
−0.997957 + 0.0638918i \(0.979649\pi\)
\(542\) 59787.0i 0.203521i
\(543\) 0 0
\(544\) 495102.i 1.67300i
\(545\) 16608.1i 0.0559148i
\(546\) 0 0
\(547\) −3623.89 −0.0121116 −0.00605579 0.999982i \(-0.501928\pi\)
−0.00605579 + 0.999982i \(0.501928\pi\)
\(548\) 214783. 0.715218
\(549\) 0 0
\(550\) 435278. 1.43894
\(551\) 155340. 0.511657
\(552\) 0 0
\(553\) −471138. −1.54063
\(554\) 149896.i 0.488394i
\(555\) 0 0
\(556\) −46571.8 −0.150652
\(557\) −223715. −0.721082 −0.360541 0.932743i \(-0.617408\pi\)
−0.360541 + 0.932743i \(0.617408\pi\)
\(558\) 0 0
\(559\) 974.798 0.00311954
\(560\) −201943. −0.643951
\(561\) 0 0
\(562\) 30662.3i 0.0970807i
\(563\) 74625.8i 0.235436i 0.993047 + 0.117718i \(0.0375578\pi\)
−0.993047 + 0.117718i \(0.962442\pi\)
\(564\) 0 0
\(565\) 90144.9i 0.282387i
\(566\) −182062. −0.568313
\(567\) 0 0
\(568\) 237799.i 0.737079i
\(569\) 319733.i 0.987558i −0.869588 0.493779i \(-0.835615\pi\)
0.869588 0.493779i \(-0.164385\pi\)
\(570\) 0 0
\(571\) 25255.1i 0.0774598i −0.999250 0.0387299i \(-0.987669\pi\)
0.999250 0.0387299i \(-0.0123312\pi\)
\(572\) 930.674 0.00284450
\(573\) 0 0
\(574\) 276803.i 0.840130i
\(575\) 254466.i 0.769650i
\(576\) 0 0
\(577\) 442259. 1.32839 0.664194 0.747560i \(-0.268775\pi\)
0.664194 + 0.747560i \(0.268775\pi\)
\(578\) 1.06080e6i 3.17524i
\(579\) 0 0
\(580\) −28165.2 −0.0837252
\(581\) 92845.6i 0.275048i
\(582\) 0 0
\(583\) 537706.i 1.58201i
\(584\) 275420. 0.807551
\(585\) 0 0
\(586\) 64538.3i 0.187941i
\(587\) 122543.i 0.355641i 0.984063 + 0.177821i \(0.0569047\pi\)
−0.984063 + 0.177821i \(0.943095\pi\)
\(588\) 0 0
\(589\) 565635.i 1.63044i
\(590\) −143174. 155728.i −0.411302 0.447365i
\(591\) 0 0
\(592\) 373714.i 1.06634i
\(593\) 531802. 1.51231 0.756154 0.654393i \(-0.227076\pi\)
0.756154 + 0.654393i \(0.227076\pi\)
\(594\) 0 0
\(595\) 346900. 0.979874
\(596\) 273251.i 0.769253i
\(597\) 0 0
\(598\) 1689.11i 0.00472342i
\(599\) −55605.2 −0.154975 −0.0774875 0.996993i \(-0.524690\pi\)
−0.0774875 + 0.996993i \(0.524690\pi\)
\(600\) 0 0
\(601\) 351895.i 0.974236i 0.873336 + 0.487118i \(0.161952\pi\)
−0.873336 + 0.487118i \(0.838048\pi\)
\(602\) 373410. 1.03037
\(603\) 0 0
\(604\) 29009.3i 0.0795176i
\(605\) −274273. −0.749329
\(606\) 0 0
\(607\) 130700. 0.354731 0.177365 0.984145i \(-0.443243\pi\)
0.177365 + 0.984145i \(0.443243\pi\)
\(608\) 472636.i 1.27856i
\(609\) 0 0
\(610\) −298362. −0.801832
\(611\) −66.0583 −0.000176948
\(612\) 0 0
\(613\) 509749.i 1.35655i −0.734808 0.678275i \(-0.762728\pi\)
0.734808 0.678275i \(-0.237272\pi\)
\(614\) 132509.i 0.351486i
\(615\) 0 0
\(616\) −393781. −1.03775
\(617\) 77698.9 0.204101 0.102050 0.994779i \(-0.467460\pi\)
0.102050 + 0.994779i \(0.467460\pi\)
\(618\) 0 0
\(619\) 221586. 0.578309 0.289155 0.957282i \(-0.406626\pi\)
0.289155 + 0.957282i \(0.406626\pi\)
\(620\) 102557.i 0.266798i
\(621\) 0 0
\(622\) 363091.i 0.938502i
\(623\) 289475.i 0.745820i
\(624\) 0 0
\(625\) 121730. 0.311628
\(626\) 717127. 1.82998
\(627\) 0 0
\(628\) 107714.i 0.273120i
\(629\) 641969.i 1.62260i
\(630\) 0 0
\(631\) 2526.90 0.00634642 0.00317321 0.999995i \(-0.498990\pi\)
0.00317321 + 0.999995i \(0.498990\pi\)
\(632\) 380797.i 0.953365i
\(633\) 0 0
\(634\) 367439.i 0.914127i
\(635\) 377169. 0.935381
\(636\) 0 0
\(637\) 93.9610i 0.000231563i
\(638\) −275148. −0.675967
\(639\) 0 0
\(640\) 225296.i 0.550039i
\(641\) −359210. −0.874244 −0.437122 0.899402i \(-0.644002\pi\)
−0.437122 + 0.899402i \(0.644002\pi\)
\(642\) 0 0
\(643\) 512961. 1.24069 0.620344 0.784330i \(-0.286993\pi\)
0.620344 + 0.784330i \(0.286993\pi\)
\(644\) 208416.i 0.502527i
\(645\) 0 0
\(646\) 1.40002e6i 3.35481i
\(647\) 621636. 1.48500 0.742502 0.669844i \(-0.233639\pi\)
0.742502 + 0.669844i \(0.233639\pi\)
\(648\) 0 0
\(649\) −450527. 490028.i −1.06962 1.16341i
\(650\) 1457.16 0.00344891
\(651\) 0 0
\(652\) 41330.3 0.0972241
\(653\) −194566. −0.456291 −0.228145 0.973627i \(-0.573266\pi\)
−0.228145 + 0.973627i \(0.573266\pi\)
\(654\) 0 0
\(655\) 382659.i 0.891927i
\(656\) −361032. −0.838954
\(657\) 0 0
\(658\) −25304.5 −0.0584449
\(659\) 481512.i 1.10876i −0.832265 0.554378i \(-0.812956\pi\)
0.832265 0.554378i \(-0.187044\pi\)
\(660\) 0 0
\(661\) 340031. 0.778244 0.389122 0.921186i \(-0.372778\pi\)
0.389122 + 0.921186i \(0.372778\pi\)
\(662\) 211837.i 0.483377i
\(663\) 0 0
\(664\) −75042.4 −0.170204
\(665\) 331159. 0.748847
\(666\) 0 0
\(667\) 160853.i 0.361558i
\(668\) 81693.4 0.183077
\(669\) 0 0
\(670\) −355774. −0.792546
\(671\) −938855. −2.08523
\(672\) 0 0
\(673\) 413955.i 0.913951i −0.889479 0.456976i \(-0.848933\pi\)
0.889479 0.456976i \(-0.151067\pi\)
\(674\) 543846. 1.19717
\(675\) 0 0
\(676\) −217134. −0.475153
\(677\) 374555. 0.817219 0.408609 0.912709i \(-0.366014\pi\)
0.408609 + 0.912709i \(0.366014\pi\)
\(678\) 0 0
\(679\) 770890.i 1.67206i
\(680\) 280382.i 0.606362i
\(681\) 0 0
\(682\) 1.00189e6i 2.15403i
\(683\) 648839.i 1.39090i 0.718576 + 0.695449i \(0.244794\pi\)
−0.718576 + 0.695449i \(0.755206\pi\)
\(684\) 0 0
\(685\) −353388. −0.753132
\(686\) 552786.i 1.17465i
\(687\) 0 0
\(688\) 487036.i 1.02893i
\(689\) 1800.06i 0.00379183i
\(690\) 0 0
\(691\) 622050.i 1.30277i 0.758746 + 0.651387i \(0.225813\pi\)
−0.758746 + 0.651387i \(0.774187\pi\)
\(692\) 207164.i 0.432616i
\(693\) 0 0
\(694\) 211047. 0.438187
\(695\) 76626.0 0.158638
\(696\) 0 0
\(697\) 620184. 1.27660
\(698\) −308067. −0.632316
\(699\) 0 0
\(700\) 179796. 0.366931
\(701\) 98359.5i 0.200161i 0.994979 + 0.100081i \(0.0319101\pi\)
−0.994979 + 0.100081i \(0.968090\pi\)
\(702\) 0 0
\(703\) 612839.i 1.24004i
\(704\) 141430.i 0.285363i
\(705\) 0 0
\(706\) −739847. −1.48434
\(707\) 688244.i 1.37690i
\(708\) 0 0
\(709\) −772872. −1.53750 −0.768750 0.639549i \(-0.779121\pi\)
−0.768750 + 0.639549i \(0.779121\pi\)
\(710\) 354223.i 0.702684i
\(711\) 0 0
\(712\) −233968. −0.461526
\(713\) −585711. −1.15214
\(714\) 0 0
\(715\) −1531.27 −0.00299529
\(716\) 458534.i 0.894428i
\(717\) 0 0
\(718\) 187220.i 0.363164i
\(719\) 273060.i 0.528202i 0.964495 + 0.264101i \(0.0850752\pi\)
−0.964495 + 0.264101i \(0.914925\pi\)
\(720\) 0 0
\(721\) 240940.i 0.463487i
\(722\) 703358.i 1.34928i
\(723\) 0 0
\(724\) 211905. 0.404263
\(725\) −138764. −0.263999
\(726\) 0 0
\(727\) 31372.5 0.0593581 0.0296791 0.999559i \(-0.490551\pi\)
0.0296791 + 0.999559i \(0.490551\pi\)
\(728\) −1318.25 −0.00248733
\(729\) 0 0
\(730\) 410262. 0.769867
\(731\) 836634.i 1.56567i
\(732\) 0 0
\(733\) 67831.8 0.126248 0.0631241 0.998006i \(-0.479894\pi\)
0.0631241 + 0.998006i \(0.479894\pi\)
\(734\) 621701. 1.15396
\(735\) 0 0
\(736\) −489411. −0.903480
\(737\) −1.11951e6 −2.06108
\(738\) 0 0
\(739\) 788958.i 1.44466i 0.691549 + 0.722329i \(0.256929\pi\)
−0.691549 + 0.722329i \(0.743071\pi\)
\(740\) 111116.i 0.202914i
\(741\) 0 0
\(742\) 689537.i 1.25242i
\(743\) −50905.9 −0.0922127 −0.0461064 0.998937i \(-0.514681\pi\)
−0.0461064 + 0.998937i \(0.514681\pi\)
\(744\) 0 0
\(745\) 449588.i 0.810031i
\(746\) 881678.i 1.58428i
\(747\) 0 0
\(748\) 798764.i 1.42763i
\(749\) −550762. −0.981749
\(750\) 0 0
\(751\) 473502.i 0.839541i −0.907630 0.419770i \(-0.862111\pi\)
0.907630 0.419770i \(-0.137889\pi\)
\(752\) 33004.5i 0.0583630i
\(753\) 0 0
\(754\) −921.103 −0.00162019
\(755\) 47729.8i 0.0837329i
\(756\) 0 0
\(757\) −5901.48 −0.0102984 −0.00514919 0.999987i \(-0.501639\pi\)
−0.00514919 + 0.999987i \(0.501639\pi\)
\(758\) 253053.i 0.440427i
\(759\) 0 0
\(760\) 267659.i 0.463399i
\(761\) −559827. −0.966684 −0.483342 0.875431i \(-0.660577\pi\)
−0.483342 + 0.875431i \(0.660577\pi\)
\(762\) 0 0
\(763\) 67017.4i 0.115117i
\(764\) 249611.i 0.427639i
\(765\) 0 0
\(766\) 628934.i 1.07188i
\(767\) −1508.21 1640.45i −0.00256373 0.00278851i
\(768\) 0 0
\(769\) 752520.i 1.27252i −0.771473 0.636262i \(-0.780480\pi\)
0.771473 0.636262i \(-0.219520\pi\)
\(770\) −586572. −0.989327
\(771\) 0 0
\(772\) 388160. 0.651292
\(773\) 293795.i 0.491684i 0.969310 + 0.245842i \(0.0790644\pi\)
−0.969310 + 0.245842i \(0.920936\pi\)
\(774\) 0 0
\(775\) 505281.i 0.841258i
\(776\) 623072. 1.03470
\(777\) 0 0
\(778\) 1.16459e6i 1.92404i
\(779\) 592043. 0.975614
\(780\) 0 0
\(781\) 1.11463e6i 1.82738i
\(782\) 1.44971e6 2.37065
\(783\) 0 0
\(784\) −46945.5 −0.0763768
\(785\) 177225.i 0.287598i
\(786\) 0 0
\(787\) 642350. 1.03710 0.518552 0.855046i \(-0.326471\pi\)
0.518552 + 0.855046i \(0.326471\pi\)
\(788\) 398579. 0.641893
\(789\) 0 0
\(790\) 567230.i 0.908877i
\(791\) 363755.i 0.581375i
\(792\) 0 0
\(793\) −3142.97 −0.00499797
\(794\) −306680. −0.486457
\(795\) 0 0
\(796\) 161807. 0.255370
\(797\)