Properties

Label 531.5.c.d.235.13
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 235.13
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.28

$q$-expansion

\(f(q)\) \(=\) \(q-3.44422i q^{2} +4.13738 q^{4} +16.2205 q^{5} +92.6602 q^{7} -69.3575i q^{8} +O(q^{10})\) \(q-3.44422i q^{2} +4.13738 q^{4} +16.2205 q^{5} +92.6602 q^{7} -69.3575i q^{8} -55.8670i q^{10} +48.6771i q^{11} -216.742i q^{13} -319.142i q^{14} -172.684 q^{16} -171.979 q^{17} +267.365 q^{19} +67.1105 q^{20} +167.654 q^{22} -734.072i q^{23} -361.894 q^{25} -746.506 q^{26} +383.370 q^{28} -999.352 q^{29} -568.212i q^{31} -514.959i q^{32} +592.334i q^{34} +1503.00 q^{35} +1003.07i q^{37} -920.864i q^{38} -1125.02i q^{40} +2301.23 q^{41} +2157.18i q^{43} +201.396i q^{44} -2528.30 q^{46} +1199.13i q^{47} +6184.90 q^{49} +1246.44i q^{50} -896.743i q^{52} +1101.00 q^{53} +789.569i q^{55} -6426.67i q^{56} +3441.98i q^{58} +(1324.95 - 3218.99i) q^{59} +4030.40i q^{61} -1957.05 q^{62} -4536.57 q^{64} -3515.67i q^{65} -8142.38i q^{67} -711.544 q^{68} -5176.65i q^{70} +2537.07 q^{71} -792.331i q^{73} +3454.80 q^{74} +1106.19 q^{76} +4510.43i q^{77} -3214.46 q^{79} -2801.03 q^{80} -7925.93i q^{82} +10537.1i q^{83} -2789.60 q^{85} +7429.78 q^{86} +3376.12 q^{88} +226.497i q^{89} -20083.3i q^{91} -3037.13i q^{92} +4130.07 q^{94} +4336.81 q^{95} -7280.88i q^{97} -21302.1i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.44422i 0.861054i −0.902578 0.430527i \(-0.858328\pi\)
0.902578 0.430527i \(-0.141672\pi\)
\(3\) 0 0
\(4\) 4.13738 0.258586
\(5\) 16.2205 0.648821 0.324411 0.945916i \(-0.394834\pi\)
0.324411 + 0.945916i \(0.394834\pi\)
\(6\) 0 0
\(7\) 92.6602 1.89102 0.945512 0.325588i \(-0.105562\pi\)
0.945512 + 0.325588i \(0.105562\pi\)
\(8\) 69.3575i 1.08371i
\(9\) 0 0
\(10\) 55.8670i 0.558670i
\(11\) 48.6771i 0.402290i 0.979561 + 0.201145i \(0.0644663\pi\)
−0.979561 + 0.201145i \(0.935534\pi\)
\(12\) 0 0
\(13\) 216.742i 1.28250i −0.767334 0.641248i \(-0.778417\pi\)
0.767334 0.641248i \(-0.221583\pi\)
\(14\) 319.142i 1.62827i
\(15\) 0 0
\(16\) −172.684 −0.674547
\(17\) −171.979 −0.595084 −0.297542 0.954709i \(-0.596167\pi\)
−0.297542 + 0.954709i \(0.596167\pi\)
\(18\) 0 0
\(19\) 267.365 0.740624 0.370312 0.928907i \(-0.379251\pi\)
0.370312 + 0.928907i \(0.379251\pi\)
\(20\) 67.1105 0.167776
\(21\) 0 0
\(22\) 167.654 0.346394
\(23\) 734.072i 1.38766i −0.720139 0.693830i \(-0.755922\pi\)
0.720139 0.693830i \(-0.244078\pi\)
\(24\) 0 0
\(25\) −361.894 −0.579031
\(26\) −746.506 −1.10430
\(27\) 0 0
\(28\) 383.370 0.488993
\(29\) −999.352 −1.18829 −0.594145 0.804358i \(-0.702510\pi\)
−0.594145 + 0.804358i \(0.702510\pi\)
\(30\) 0 0
\(31\) 568.212i 0.591272i −0.955301 0.295636i \(-0.904468\pi\)
0.955301 0.295636i \(-0.0955315\pi\)
\(32\) 514.959i 0.502889i
\(33\) 0 0
\(34\) 592.334i 0.512400i
\(35\) 1503.00 1.22694
\(36\) 0 0
\(37\) 1003.07i 0.732704i 0.930476 + 0.366352i \(0.119393\pi\)
−0.930476 + 0.366352i \(0.880607\pi\)
\(38\) 920.864i 0.637717i
\(39\) 0 0
\(40\) 1125.02i 0.703135i
\(41\) 2301.23 1.36897 0.684483 0.729029i \(-0.260028\pi\)
0.684483 + 0.729029i \(0.260028\pi\)
\(42\) 0 0
\(43\) 2157.18i 1.16667i 0.812231 + 0.583336i \(0.198253\pi\)
−0.812231 + 0.583336i \(0.801747\pi\)
\(44\) 201.396i 0.104027i
\(45\) 0 0
\(46\) −2528.30 −1.19485
\(47\) 1199.13i 0.542840i 0.962461 + 0.271420i \(0.0874932\pi\)
−0.962461 + 0.271420i \(0.912507\pi\)
\(48\) 0 0
\(49\) 6184.90 2.57597
\(50\) 1246.44i 0.498577i
\(51\) 0 0
\(52\) 896.743i 0.331636i
\(53\) 1101.00 0.391956 0.195978 0.980608i \(-0.437212\pi\)
0.195978 + 0.980608i \(0.437212\pi\)
\(54\) 0 0
\(55\) 789.569i 0.261015i
\(56\) 6426.67i 2.04932i
\(57\) 0 0
\(58\) 3441.98i 1.02318i
\(59\) 1324.95 3218.99i 0.380623 0.924730i
\(60\) 0 0
\(61\) 4030.40i 1.08315i 0.840653 + 0.541574i \(0.182172\pi\)
−0.840653 + 0.541574i \(0.817828\pi\)
\(62\) −1957.05 −0.509117
\(63\) 0 0
\(64\) −4536.57 −1.10756
\(65\) 3515.67i 0.832111i
\(66\) 0 0
\(67\) 8142.38i 1.81385i −0.421290 0.906926i \(-0.638423\pi\)
0.421290 0.906926i \(-0.361577\pi\)
\(68\) −711.544 −0.153881
\(69\) 0 0
\(70\) 5176.65i 1.05646i
\(71\) 2537.07 0.503286 0.251643 0.967820i \(-0.419029\pi\)
0.251643 + 0.967820i \(0.419029\pi\)
\(72\) 0 0
\(73\) 792.331i 0.148683i −0.997233 0.0743414i \(-0.976315\pi\)
0.997233 0.0743414i \(-0.0236855\pi\)
\(74\) 3454.80 0.630898
\(75\) 0 0
\(76\) 1106.19 0.191515
\(77\) 4510.43i 0.760740i
\(78\) 0 0
\(79\) −3214.46 −0.515056 −0.257528 0.966271i \(-0.582908\pi\)
−0.257528 + 0.966271i \(0.582908\pi\)
\(80\) −2801.03 −0.437661
\(81\) 0 0
\(82\) 7925.93i 1.17875i
\(83\) 10537.1i 1.52956i 0.644292 + 0.764779i \(0.277152\pi\)
−0.644292 + 0.764779i \(0.722848\pi\)
\(84\) 0 0
\(85\) −2789.60 −0.386103
\(86\) 7429.78 1.00457
\(87\) 0 0
\(88\) 3376.12 0.435966
\(89\) 226.497i 0.0285945i 0.999898 + 0.0142972i \(0.00455111\pi\)
−0.999898 + 0.0142972i \(0.995449\pi\)
\(90\) 0 0
\(91\) 20083.3i 2.42523i
\(92\) 3037.13i 0.358829i
\(93\) 0 0
\(94\) 4130.07 0.467414
\(95\) 4336.81 0.480533
\(96\) 0 0
\(97\) 7280.88i 0.773821i −0.922117 0.386911i \(-0.873542\pi\)
0.922117 0.386911i \(-0.126458\pi\)
\(98\) 21302.1i 2.21805i
\(99\) 0 0
\(100\) −1497.29 −0.149729
\(101\) 11745.9i 1.15145i 0.817643 + 0.575725i \(0.195280\pi\)
−0.817643 + 0.575725i \(0.804720\pi\)
\(102\) 0 0
\(103\) 19029.4i 1.79370i −0.442333 0.896851i \(-0.645849\pi\)
0.442333 0.896851i \(-0.354151\pi\)
\(104\) −15032.7 −1.38985
\(105\) 0 0
\(106\) 3792.10i 0.337495i
\(107\) −6751.89 −0.589737 −0.294868 0.955538i \(-0.595276\pi\)
−0.294868 + 0.955538i \(0.595276\pi\)
\(108\) 0 0
\(109\) 22919.8i 1.92911i 0.263876 + 0.964556i \(0.414999\pi\)
−0.263876 + 0.964556i \(0.585001\pi\)
\(110\) 2719.45 0.224748
\(111\) 0 0
\(112\) −16000.9 −1.27558
\(113\) 1167.57i 0.0914377i −0.998954 0.0457188i \(-0.985442\pi\)
0.998954 0.0457188i \(-0.0145578\pi\)
\(114\) 0 0
\(115\) 11907.0i 0.900343i
\(116\) −4134.70 −0.307276
\(117\) 0 0
\(118\) −11086.9 4563.41i −0.796243 0.327737i
\(119\) −15935.6 −1.12532
\(120\) 0 0
\(121\) 12271.5 0.838163
\(122\) 13881.6 0.932649
\(123\) 0 0
\(124\) 2350.91i 0.152895i
\(125\) −16008.0 −1.02451
\(126\) 0 0
\(127\) −14064.3 −0.871990 −0.435995 0.899949i \(-0.643603\pi\)
−0.435995 + 0.899949i \(0.643603\pi\)
\(128\) 7385.60i 0.450781i
\(129\) 0 0
\(130\) −12108.7 −0.716492
\(131\) 15833.5i 0.922644i 0.887233 + 0.461322i \(0.152625\pi\)
−0.887233 + 0.461322i \(0.847375\pi\)
\(132\) 0 0
\(133\) 24774.1 1.40054
\(134\) −28044.1 −1.56182
\(135\) 0 0
\(136\) 11928.1i 0.644899i
\(137\) 21940.7 1.16899 0.584493 0.811399i \(-0.301293\pi\)
0.584493 + 0.811399i \(0.301293\pi\)
\(138\) 0 0
\(139\) −33025.6 −1.70931 −0.854655 0.519196i \(-0.826231\pi\)
−0.854655 + 0.519196i \(0.826231\pi\)
\(140\) 6218.47 0.317269
\(141\) 0 0
\(142\) 8738.20i 0.433357i
\(143\) 10550.4 0.515936
\(144\) 0 0
\(145\) −16210.0 −0.770988
\(146\) −2728.96 −0.128024
\(147\) 0 0
\(148\) 4150.09i 0.189467i
\(149\) 20821.5i 0.937863i 0.883235 + 0.468932i \(0.155361\pi\)
−0.883235 + 0.468932i \(0.844639\pi\)
\(150\) 0 0
\(151\) 9654.62i 0.423430i 0.977331 + 0.211715i \(0.0679048\pi\)
−0.977331 + 0.211715i \(0.932095\pi\)
\(152\) 18543.8i 0.802622i
\(153\) 0 0
\(154\) 15534.9 0.655038
\(155\) 9216.71i 0.383630i
\(156\) 0 0
\(157\) 23483.9i 0.952731i −0.879247 0.476366i \(-0.841954\pi\)
0.879247 0.476366i \(-0.158046\pi\)
\(158\) 11071.3i 0.443491i
\(159\) 0 0
\(160\) 8352.90i 0.326285i
\(161\) 68019.2i 2.62410i
\(162\) 0 0
\(163\) 12805.6 0.481977 0.240988 0.970528i \(-0.422528\pi\)
0.240988 + 0.970528i \(0.422528\pi\)
\(164\) 9521.06 0.353995
\(165\) 0 0
\(166\) 36292.2 1.31703
\(167\) 29956.3 1.07413 0.537063 0.843542i \(-0.319534\pi\)
0.537063 + 0.843542i \(0.319534\pi\)
\(168\) 0 0
\(169\) −18416.0 −0.644796
\(170\) 9607.98i 0.332456i
\(171\) 0 0
\(172\) 8925.05i 0.301685i
\(173\) 51496.1i 1.72061i −0.509781 0.860304i \(-0.670274\pi\)
0.509781 0.860304i \(-0.329726\pi\)
\(174\) 0 0
\(175\) −33533.2 −1.09496
\(176\) 8405.76i 0.271364i
\(177\) 0 0
\(178\) 780.104 0.0246214
\(179\) 17057.4i 0.532363i −0.963923 0.266181i \(-0.914238\pi\)
0.963923 0.266181i \(-0.0857620\pi\)
\(180\) 0 0
\(181\) 7526.56 0.229741 0.114871 0.993380i \(-0.463355\pi\)
0.114871 + 0.993380i \(0.463355\pi\)
\(182\) −69171.3 −2.08825
\(183\) 0 0
\(184\) −50913.4 −1.50382
\(185\) 16270.4i 0.475394i
\(186\) 0 0
\(187\) 8371.46i 0.239397i
\(188\) 4961.27i 0.140371i
\(189\) 0 0
\(190\) 14936.9i 0.413765i
\(191\) 34367.0i 0.942053i 0.882119 + 0.471026i \(0.156116\pi\)
−0.882119 + 0.471026i \(0.843884\pi\)
\(192\) 0 0
\(193\) −35591.7 −0.955507 −0.477753 0.878494i \(-0.658549\pi\)
−0.477753 + 0.878494i \(0.658549\pi\)
\(194\) −25076.9 −0.666302
\(195\) 0 0
\(196\) 25589.3 0.666110
\(197\) 25127.2 0.647458 0.323729 0.946150i \(-0.395063\pi\)
0.323729 + 0.946150i \(0.395063\pi\)
\(198\) 0 0
\(199\) 19989.2 0.504765 0.252382 0.967628i \(-0.418786\pi\)
0.252382 + 0.967628i \(0.418786\pi\)
\(200\) 25100.1i 0.627502i
\(201\) 0 0
\(202\) 40455.6 0.991460
\(203\) −92600.1 −2.24709
\(204\) 0 0
\(205\) 37327.2 0.888214
\(206\) −65541.3 −1.54447
\(207\) 0 0
\(208\) 37427.9i 0.865104i
\(209\) 13014.6i 0.297946i
\(210\) 0 0
\(211\) 32244.7i 0.724257i 0.932128 + 0.362129i \(0.117950\pi\)
−0.932128 + 0.362129i \(0.882050\pi\)
\(212\) 4555.27 0.101354
\(213\) 0 0
\(214\) 23255.0i 0.507795i
\(215\) 34990.5i 0.756962i
\(216\) 0 0
\(217\) 52650.6i 1.11811i
\(218\) 78940.7 1.66107
\(219\) 0 0
\(220\) 3266.75i 0.0674948i
\(221\) 37275.1i 0.763193i
\(222\) 0 0
\(223\) −28806.5 −0.579270 −0.289635 0.957137i \(-0.593534\pi\)
−0.289635 + 0.957137i \(0.593534\pi\)
\(224\) 47716.1i 0.950975i
\(225\) 0 0
\(226\) −4021.36 −0.0787328
\(227\) 57367.6i 1.11331i −0.830745 0.556654i \(-0.812085\pi\)
0.830745 0.556654i \(-0.187915\pi\)
\(228\) 0 0
\(229\) 8038.50i 0.153287i −0.997059 0.0766433i \(-0.975580\pi\)
0.997059 0.0766433i \(-0.0244203\pi\)
\(230\) −41010.4 −0.775244
\(231\) 0 0
\(232\) 69312.6i 1.28776i
\(233\) 55821.9i 1.02824i 0.857719 + 0.514118i \(0.171881\pi\)
−0.857719 + 0.514118i \(0.828119\pi\)
\(234\) 0 0
\(235\) 19450.6i 0.352206i
\(236\) 5481.82 13318.2i 0.0984239 0.239122i
\(237\) 0 0
\(238\) 54885.8i 0.968960i
\(239\) 35079.1 0.614120 0.307060 0.951690i \(-0.400655\pi\)
0.307060 + 0.951690i \(0.400655\pi\)
\(240\) 0 0
\(241\) 84946.1 1.46255 0.731273 0.682085i \(-0.238927\pi\)
0.731273 + 0.682085i \(0.238927\pi\)
\(242\) 42265.8i 0.721703i
\(243\) 0 0
\(244\) 16675.3i 0.280087i
\(245\) 100322. 1.67134
\(246\) 0 0
\(247\) 57949.2i 0.949847i
\(248\) −39409.8 −0.640768
\(249\) 0 0
\(250\) 55134.8i 0.882157i
\(251\) 62187.4 0.987086 0.493543 0.869721i \(-0.335702\pi\)
0.493543 + 0.869721i \(0.335702\pi\)
\(252\) 0 0
\(253\) 35732.5 0.558242
\(254\) 48440.5i 0.750830i
\(255\) 0 0
\(256\) −47147.6 −0.719415
\(257\) 11454.6 0.173426 0.0867129 0.996233i \(-0.472364\pi\)
0.0867129 + 0.996233i \(0.472364\pi\)
\(258\) 0 0
\(259\) 92944.8i 1.38556i
\(260\) 14545.7i 0.215172i
\(261\) 0 0
\(262\) 54533.9 0.794446
\(263\) −39389.7 −0.569471 −0.284735 0.958606i \(-0.591906\pi\)
−0.284735 + 0.958606i \(0.591906\pi\)
\(264\) 0 0
\(265\) 17858.9 0.254310
\(266\) 85327.4i 1.20594i
\(267\) 0 0
\(268\) 33688.1i 0.469037i
\(269\) 13109.8i 0.181172i 0.995889 + 0.0905861i \(0.0288740\pi\)
−0.995889 + 0.0905861i \(0.971126\pi\)
\(270\) 0 0
\(271\) 63942.9 0.870670 0.435335 0.900268i \(-0.356630\pi\)
0.435335 + 0.900268i \(0.356630\pi\)
\(272\) 29698.1 0.401412
\(273\) 0 0
\(274\) 75568.5i 1.00656i
\(275\) 17616.0i 0.232938i
\(276\) 0 0
\(277\) −26097.8 −0.340130 −0.170065 0.985433i \(-0.554398\pi\)
−0.170065 + 0.985433i \(0.554398\pi\)
\(278\) 113747.i 1.47181i
\(279\) 0 0
\(280\) 104244.i 1.32964i
\(281\) 33934.6 0.429765 0.214882 0.976640i \(-0.431063\pi\)
0.214882 + 0.976640i \(0.431063\pi\)
\(282\) 0 0
\(283\) 50184.5i 0.626609i 0.949653 + 0.313305i \(0.101436\pi\)
−0.949653 + 0.313305i \(0.898564\pi\)
\(284\) 10496.8 0.130143
\(285\) 0 0
\(286\) 36337.7i 0.444248i
\(287\) 213232. 2.58875
\(288\) 0 0
\(289\) −53944.1 −0.645875
\(290\) 55830.8i 0.663863i
\(291\) 0 0
\(292\) 3278.17i 0.0384473i
\(293\) 145740. 1.69764 0.848819 0.528684i \(-0.177314\pi\)
0.848819 + 0.528684i \(0.177314\pi\)
\(294\) 0 0
\(295\) 21491.4 52213.7i 0.246957 0.599985i
\(296\) 69570.6 0.794040
\(297\) 0 0
\(298\) 71713.7 0.807551
\(299\) −159104. −1.77967
\(300\) 0 0
\(301\) 199884.i 2.20620i
\(302\) 33252.6 0.364596
\(303\) 0 0
\(304\) −46169.7 −0.499586
\(305\) 65375.2i 0.702770i
\(306\) 0 0
\(307\) −37666.1 −0.399645 −0.199822 0.979832i \(-0.564037\pi\)
−0.199822 + 0.979832i \(0.564037\pi\)
\(308\) 18661.4i 0.196717i
\(309\) 0 0
\(310\) −31744.3 −0.330326
\(311\) 145763. 1.50704 0.753522 0.657423i \(-0.228353\pi\)
0.753522 + 0.657423i \(0.228353\pi\)
\(312\) 0 0
\(313\) 28891.9i 0.294908i 0.989069 + 0.147454i \(0.0471079\pi\)
−0.989069 + 0.147454i \(0.952892\pi\)
\(314\) −80883.5 −0.820353
\(315\) 0 0
\(316\) −13299.5 −0.133186
\(317\) −149119. −1.48393 −0.741967 0.670437i \(-0.766107\pi\)
−0.741967 + 0.670437i \(0.766107\pi\)
\(318\) 0 0
\(319\) 48645.6i 0.478038i
\(320\) −73585.6 −0.718610
\(321\) 0 0
\(322\) −234273. −2.25949
\(323\) −45981.3 −0.440734
\(324\) 0 0
\(325\) 78437.6i 0.742605i
\(326\) 44105.4i 0.415008i
\(327\) 0 0
\(328\) 159608.i 1.48356i
\(329\) 111112.i 1.02652i
\(330\) 0 0
\(331\) 20162.8 0.184033 0.0920164 0.995757i \(-0.470669\pi\)
0.0920164 + 0.995757i \(0.470669\pi\)
\(332\) 43596.1i 0.395523i
\(333\) 0 0
\(334\) 103176.i 0.924880i
\(335\) 132074.i 1.17687i
\(336\) 0 0
\(337\) 56644.5i 0.498767i 0.968405 + 0.249384i \(0.0802280\pi\)
−0.968405 + 0.249384i \(0.919772\pi\)
\(338\) 63428.7i 0.555204i
\(339\) 0 0
\(340\) −11541.6 −0.0998410
\(341\) 27658.9 0.237863
\(342\) 0 0
\(343\) 350617. 2.98020
\(344\) 149616. 1.26433
\(345\) 0 0
\(346\) −177364. −1.48154
\(347\) 98468.3i 0.817782i −0.912583 0.408891i \(-0.865916\pi\)
0.912583 0.408891i \(-0.134084\pi\)
\(348\) 0 0
\(349\) 11046.8i 0.0906953i −0.998971 0.0453476i \(-0.985560\pi\)
0.998971 0.0453476i \(-0.0144396\pi\)
\(350\) 115495.i 0.942820i
\(351\) 0 0
\(352\) 25066.7 0.202307
\(353\) 82041.3i 0.658390i 0.944262 + 0.329195i \(0.106777\pi\)
−0.944262 + 0.329195i \(0.893223\pi\)
\(354\) 0 0
\(355\) 41152.6 0.326543
\(356\) 937.104i 0.00739414i
\(357\) 0 0
\(358\) −58749.5 −0.458393
\(359\) 48438.3 0.375837 0.187919 0.982185i \(-0.439826\pi\)
0.187919 + 0.982185i \(0.439826\pi\)
\(360\) 0 0
\(361\) −58836.8 −0.451476
\(362\) 25923.1i 0.197820i
\(363\) 0 0
\(364\) 83092.3i 0.627131i
\(365\) 12852.0i 0.0964686i
\(366\) 0 0
\(367\) 206944.i 1.53646i 0.640175 + 0.768229i \(0.278862\pi\)
−0.640175 + 0.768229i \(0.721138\pi\)
\(368\) 126762.i 0.936041i
\(369\) 0 0
\(370\) 56038.7 0.409340
\(371\) 102019. 0.741198
\(372\) 0 0
\(373\) 96391.4 0.692821 0.346410 0.938083i \(-0.387401\pi\)
0.346410 + 0.938083i \(0.387401\pi\)
\(374\) −28833.1 −0.206133
\(375\) 0 0
\(376\) 83168.8 0.588281
\(377\) 216601.i 1.52398i
\(378\) 0 0
\(379\) 222227. 1.54710 0.773549 0.633737i \(-0.218480\pi\)
0.773549 + 0.633737i \(0.218480\pi\)
\(380\) 17943.0 0.124259
\(381\) 0 0
\(382\) 118367. 0.811158
\(383\) −251261. −1.71288 −0.856440 0.516246i \(-0.827329\pi\)
−0.856440 + 0.516246i \(0.827329\pi\)
\(384\) 0 0
\(385\) 73161.6i 0.493585i
\(386\) 122585.i 0.822743i
\(387\) 0 0
\(388\) 30123.8i 0.200100i
\(389\) −247468. −1.63538 −0.817691 0.575657i \(-0.804746\pi\)
−0.817691 + 0.575657i \(0.804746\pi\)
\(390\) 0 0
\(391\) 126245.i 0.825774i
\(392\) 428969.i 2.79161i
\(393\) 0 0
\(394\) 86543.4i 0.557496i
\(395\) −52140.3 −0.334179
\(396\) 0 0
\(397\) 15188.5i 0.0963684i 0.998838 + 0.0481842i \(0.0153435\pi\)
−0.998838 + 0.0481842i \(0.984657\pi\)
\(398\) 68847.1i 0.434630i
\(399\) 0 0
\(400\) 62493.4 0.390583
\(401\) 41102.4i 0.255611i 0.991799 + 0.127805i \(0.0407933\pi\)
−0.991799 + 0.127805i \(0.959207\pi\)
\(402\) 0 0
\(403\) −123155. −0.758304
\(404\) 48597.4i 0.297749i
\(405\) 0 0
\(406\) 318935.i 1.93486i
\(407\) −48826.7 −0.294760
\(408\) 0 0
\(409\) 119091.i 0.711925i 0.934500 + 0.355962i \(0.115847\pi\)
−0.934500 + 0.355962i \(0.884153\pi\)
\(410\) 128563.i 0.764800i
\(411\) 0 0
\(412\) 78731.7i 0.463826i
\(413\) 122770. 298272.i 0.719768 1.74869i
\(414\) 0 0
\(415\) 170918.i 0.992411i
\(416\) −111613. −0.644953
\(417\) 0 0
\(418\) 44825.0 0.256547
\(419\) 228690.i 1.30263i 0.758809 + 0.651313i \(0.225782\pi\)
−0.758809 + 0.651313i \(0.774218\pi\)
\(420\) 0 0
\(421\) 282680.i 1.59489i 0.603390 + 0.797447i \(0.293816\pi\)
−0.603390 + 0.797447i \(0.706184\pi\)
\(422\) 111058. 0.623624
\(423\) 0 0
\(424\) 76362.9i 0.424767i
\(425\) 62238.3 0.344572
\(426\) 0 0
\(427\) 373457.i 2.04826i
\(428\) −27935.1 −0.152498
\(429\) 0 0
\(430\) 120515. 0.651785
\(431\) 270326.i 1.45524i 0.685981 + 0.727619i \(0.259373\pi\)
−0.685981 + 0.727619i \(0.740627\pi\)
\(432\) 0 0
\(433\) −201414. −1.07427 −0.537136 0.843496i \(-0.680494\pi\)
−0.537136 + 0.843496i \(0.680494\pi\)
\(434\) −181340. −0.962752
\(435\) 0 0
\(436\) 94827.9i 0.498842i
\(437\) 196265.i 1.02773i
\(438\) 0 0
\(439\) −10876.9 −0.0564388 −0.0282194 0.999602i \(-0.508984\pi\)
−0.0282194 + 0.999602i \(0.508984\pi\)
\(440\) 54762.5 0.282864
\(441\) 0 0
\(442\) 128384. 0.657151
\(443\) 28282.8i 0.144117i −0.997400 0.0720584i \(-0.977043\pi\)
0.997400 0.0720584i \(-0.0229568\pi\)
\(444\) 0 0
\(445\) 3673.90i 0.0185527i
\(446\) 99215.8i 0.498782i
\(447\) 0 0
\(448\) −420360. −2.09443
\(449\) −158321. −0.785320 −0.392660 0.919684i \(-0.628445\pi\)
−0.392660 + 0.919684i \(0.628445\pi\)
\(450\) 0 0
\(451\) 112017.i 0.550721i
\(452\) 4830.67i 0.0236445i
\(453\) 0 0
\(454\) −197586. −0.958618
\(455\) 325762.i 1.57354i
\(456\) 0 0
\(457\) 35833.3i 0.171575i −0.996313 0.0857876i \(-0.972659\pi\)
0.996313 0.0857876i \(-0.0273406\pi\)
\(458\) −27686.3 −0.131988
\(459\) 0 0
\(460\) 49263.9i 0.232816i
\(461\) −307571. −1.44725 −0.723625 0.690193i \(-0.757526\pi\)
−0.723625 + 0.690193i \(0.757526\pi\)
\(462\) 0 0
\(463\) 272379.i 1.27061i −0.772262 0.635304i \(-0.780875\pi\)
0.772262 0.635304i \(-0.219125\pi\)
\(464\) 172572. 0.801558
\(465\) 0 0
\(466\) 192263. 0.885367
\(467\) 217459.i 0.997111i 0.866858 + 0.498556i \(0.166136\pi\)
−0.866858 + 0.498556i \(0.833864\pi\)
\(468\) 0 0
\(469\) 754474.i 3.43004i
\(470\) 66992.0 0.303268
\(471\) 0 0
\(472\) −223261. 91895.2i −1.00214 0.412485i
\(473\) −105005. −0.469341
\(474\) 0 0
\(475\) −96757.9 −0.428844
\(476\) −65931.8 −0.290992
\(477\) 0 0
\(478\) 120820.i 0.528790i
\(479\) 56488.8 0.246202 0.123101 0.992394i \(-0.460716\pi\)
0.123101 + 0.992394i \(0.460716\pi\)
\(480\) 0 0
\(481\) 217408. 0.939691
\(482\) 292573.i 1.25933i
\(483\) 0 0
\(484\) 50772.0 0.216737
\(485\) 118100.i 0.502072i
\(486\) 0 0
\(487\) −374783. −1.58024 −0.790118 0.612954i \(-0.789981\pi\)
−0.790118 + 0.612954i \(0.789981\pi\)
\(488\) 279538. 1.17382
\(489\) 0 0
\(490\) 345532.i 1.43912i
\(491\) 61744.1 0.256114 0.128057 0.991767i \(-0.459126\pi\)
0.128057 + 0.991767i \(0.459126\pi\)
\(492\) 0 0
\(493\) 171868. 0.707133
\(494\) −199590. −0.817870
\(495\) 0 0
\(496\) 98121.2i 0.398841i
\(497\) 235085. 0.951726
\(498\) 0 0
\(499\) −212124. −0.851901 −0.425950 0.904747i \(-0.640060\pi\)
−0.425950 + 0.904747i \(0.640060\pi\)
\(500\) −66231.0 −0.264924
\(501\) 0 0
\(502\) 214187.i 0.849934i
\(503\) 268106.i 1.05967i −0.848101 0.529834i \(-0.822254\pi\)
0.848101 0.529834i \(-0.177746\pi\)
\(504\) 0 0
\(505\) 190525.i 0.747085i
\(506\) 123070.i 0.480676i
\(507\) 0 0
\(508\) −58189.4 −0.225484
\(509\) 241532.i 0.932265i 0.884715 + 0.466133i \(0.154353\pi\)
−0.884715 + 0.466133i \(0.845647\pi\)
\(510\) 0 0
\(511\) 73417.5i 0.281163i
\(512\) 280556.i 1.07024i
\(513\) 0 0
\(514\) 39452.1i 0.149329i
\(515\) 308667.i 1.16379i
\(516\) 0 0
\(517\) −58370.3 −0.218379
\(518\) 320122. 1.19304
\(519\) 0 0
\(520\) −243838. −0.901767
\(521\) −81137.0 −0.298912 −0.149456 0.988768i \(-0.547752\pi\)
−0.149456 + 0.988768i \(0.547752\pi\)
\(522\) 0 0
\(523\) 507362. 1.85488 0.927438 0.373977i \(-0.122006\pi\)
0.927438 + 0.373977i \(0.122006\pi\)
\(524\) 65509.1i 0.238583i
\(525\) 0 0
\(526\) 135667.i 0.490345i
\(527\) 97720.8i 0.351857i
\(528\) 0 0
\(529\) −259020. −0.925598
\(530\) 61509.9i 0.218974i
\(531\) 0 0
\(532\) 102500. 0.362160
\(533\) 498773.i 1.75569i
\(534\) 0 0
\(535\) −109519. −0.382634
\(536\) −564735. −1.96569
\(537\) 0 0
\(538\) 45153.0 0.155999
\(539\) 301063.i 1.03629i
\(540\) 0 0
\(541\) 303310.i 1.03632i −0.855285 0.518158i \(-0.826618\pi\)
0.855285 0.518158i \(-0.173382\pi\)
\(542\) 220233.i 0.749694i
\(543\) 0 0
\(544\) 88562.2i 0.299261i
\(545\) 371771.i 1.25165i
\(546\) 0 0
\(547\) 185234. 0.619077 0.309539 0.950887i \(-0.399825\pi\)
0.309539 + 0.950887i \(0.399825\pi\)
\(548\) 90777.0 0.302284
\(549\) 0 0
\(550\) −60673.2 −0.200573
\(551\) −267192. −0.880076
\(552\) 0 0
\(553\) −297853. −0.973983
\(554\) 89886.5i 0.292870i
\(555\) 0 0
\(556\) −136639. −0.442004
\(557\) −300545. −0.968723 −0.484361 0.874868i \(-0.660948\pi\)
−0.484361 + 0.874868i \(0.660948\pi\)
\(558\) 0 0
\(559\) 467550. 1.49625
\(560\) −259544. −0.827626
\(561\) 0 0
\(562\) 116878.i 0.370050i
\(563\) 110761.i 0.349437i −0.984618 0.174719i \(-0.944098\pi\)
0.984618 0.174719i \(-0.0559016\pi\)
\(564\) 0 0
\(565\) 18938.6i 0.0593267i
\(566\) 172846. 0.539544
\(567\) 0 0
\(568\) 175965.i 0.545417i
\(569\) 85112.7i 0.262887i 0.991324 + 0.131444i \(0.0419612\pi\)
−0.991324 + 0.131444i \(0.958039\pi\)
\(570\) 0 0
\(571\) 419082.i 1.28537i 0.766132 + 0.642683i \(0.222179\pi\)
−0.766132 + 0.642683i \(0.777821\pi\)
\(572\) 43650.9 0.133414
\(573\) 0 0
\(574\) 734418.i 2.22905i
\(575\) 265656.i 0.803497i
\(576\) 0 0
\(577\) −210330. −0.631757 −0.315878 0.948800i \(-0.602299\pi\)
−0.315878 + 0.948800i \(0.602299\pi\)
\(578\) 185795.i 0.556133i
\(579\) 0 0
\(580\) −67067.0 −0.199367
\(581\) 976372.i 2.89243i
\(582\) 0 0
\(583\) 53593.7i 0.157680i
\(584\) −54954.1 −0.161129
\(585\) 0 0
\(586\) 501962.i 1.46176i
\(587\) 645237.i 1.87259i −0.351213 0.936296i \(-0.614231\pi\)
0.351213 0.936296i \(-0.385769\pi\)
\(588\) 0 0
\(589\) 151920.i 0.437910i
\(590\) −179835. 74021.0i −0.516619 0.212643i
\(591\) 0 0
\(592\) 173215.i 0.494244i
\(593\) 386221. 1.09831 0.549157 0.835719i \(-0.314949\pi\)
0.549157 + 0.835719i \(0.314949\pi\)
\(594\) 0 0
\(595\) −258485. −0.730131
\(596\) 86146.4i 0.242518i
\(597\) 0 0
\(598\) 547989.i 1.53239i
\(599\) 388204. 1.08195 0.540974 0.841039i \(-0.318056\pi\)
0.540974 + 0.841039i \(0.318056\pi\)
\(600\) 0 0
\(601\) 510006.i 1.41197i −0.708225 0.705987i \(-0.750504\pi\)
0.708225 0.705987i \(-0.249496\pi\)
\(602\) 688444. 1.89966
\(603\) 0 0
\(604\) 39944.8i 0.109493i
\(605\) 199051. 0.543818
\(606\) 0 0
\(607\) −560179. −1.52037 −0.760185 0.649707i \(-0.774892\pi\)
−0.760185 + 0.649707i \(0.774892\pi\)
\(608\) 137682.i 0.372452i
\(609\) 0 0
\(610\) 225166. 0.605123
\(611\) 259902. 0.696190
\(612\) 0 0
\(613\) 306896.i 0.816715i 0.912822 + 0.408358i \(0.133898\pi\)
−0.912822 + 0.408358i \(0.866102\pi\)
\(614\) 129730.i 0.344116i
\(615\) 0 0
\(616\) 312832. 0.824422
\(617\) −465889. −1.22380 −0.611902 0.790933i \(-0.709595\pi\)
−0.611902 + 0.790933i \(0.709595\pi\)
\(618\) 0 0
\(619\) 608265. 1.58749 0.793746 0.608250i \(-0.208128\pi\)
0.793746 + 0.608250i \(0.208128\pi\)
\(620\) 38133.0i 0.0992014i
\(621\) 0 0
\(622\) 502039.i 1.29765i
\(623\) 20987.2i 0.0540728i
\(624\) 0 0
\(625\) −33473.7 −0.0856927
\(626\) 99509.8 0.253932
\(627\) 0 0
\(628\) 97161.7i 0.246363i
\(629\) 172508.i 0.436021i
\(630\) 0 0
\(631\) −51223.1 −0.128649 −0.0643245 0.997929i \(-0.520489\pi\)
−0.0643245 + 0.997929i \(0.520489\pi\)
\(632\) 222947.i 0.558171i
\(633\) 0 0
\(634\) 513598.i 1.27775i
\(635\) −228131. −0.565766
\(636\) 0 0
\(637\) 1.34053e6i 3.30367i
\(638\) −167546. −0.411616
\(639\) 0 0
\(640\) 119798.i 0.292476i
\(641\) 194528. 0.473440 0.236720 0.971578i \(-0.423928\pi\)
0.236720 + 0.971578i \(0.423928\pi\)
\(642\) 0 0
\(643\) −127058. −0.307312 −0.153656 0.988124i \(-0.549105\pi\)
−0.153656 + 0.988124i \(0.549105\pi\)
\(644\) 281421.i 0.678555i
\(645\) 0 0
\(646\) 158370.i 0.379495i
\(647\) −298850. −0.713912 −0.356956 0.934121i \(-0.616185\pi\)
−0.356956 + 0.934121i \(0.616185\pi\)
\(648\) 0 0
\(649\) 156691. + 64494.7i 0.372010 + 0.153121i
\(650\) 270156. 0.639423
\(651\) 0 0
\(652\) 52981.8 0.124633
\(653\) 808235. 1.89545 0.947723 0.319094i \(-0.103379\pi\)
0.947723 + 0.319094i \(0.103379\pi\)
\(654\) 0 0
\(655\) 256828.i 0.598631i
\(656\) −397386. −0.923431
\(657\) 0 0
\(658\) 382693. 0.883891
\(659\) 205299.i 0.472732i −0.971664 0.236366i \(-0.924044\pi\)
0.971664 0.236366i \(-0.0759565\pi\)
\(660\) 0 0
\(661\) −218321. −0.499680 −0.249840 0.968287i \(-0.580378\pi\)
−0.249840 + 0.968287i \(0.580378\pi\)
\(662\) 69445.1i 0.158462i
\(663\) 0 0
\(664\) 730829. 1.65760
\(665\) 401849. 0.908699
\(666\) 0 0
\(667\) 733596.i 1.64894i
\(668\) 123940. 0.277754
\(669\) 0 0
\(670\) −454891. −1.01335
\(671\) −196188. −0.435740
\(672\) 0 0
\(673\) 308870.i 0.681939i 0.940074 + 0.340970i \(0.110755\pi\)
−0.940074 + 0.340970i \(0.889245\pi\)
\(674\) 195096. 0.429466
\(675\) 0 0
\(676\) −76194.0 −0.166735
\(677\) 765077. 1.66927 0.834637 0.550800i \(-0.185677\pi\)
0.834637 + 0.550800i \(0.185677\pi\)
\(678\) 0 0
\(679\) 674648.i 1.46331i
\(680\) 193479.i 0.418424i
\(681\) 0 0
\(682\) 95263.3i 0.204813i
\(683\) 32889.7i 0.0705047i −0.999378 0.0352523i \(-0.988777\pi\)
0.999378 0.0352523i \(-0.0112235\pi\)
\(684\) 0 0
\(685\) 355890. 0.758463
\(686\) 1.20760e6i 2.56611i
\(687\) 0 0
\(688\) 372510.i 0.786975i
\(689\) 238634.i 0.502682i
\(690\) 0 0
\(691\) 498934.i 1.04493i 0.852661 + 0.522465i \(0.174987\pi\)
−0.852661 + 0.522465i \(0.825013\pi\)
\(692\) 213059.i 0.444925i
\(693\) 0 0
\(694\) −339146. −0.704154
\(695\) −535693. −1.10904
\(696\) 0 0
\(697\) −395764. −0.814650
\(698\) −38047.5 −0.0780935
\(699\) 0 0
\(700\) −138739. −0.283142
\(701\) 70692.6i 0.143859i −0.997410 0.0719297i \(-0.977084\pi\)
0.997410 0.0719297i \(-0.0229157\pi\)
\(702\) 0 0
\(703\) 268187.i 0.542658i
\(704\) 220827.i 0.445561i
\(705\) 0 0
\(706\) 282568. 0.566909
\(707\) 1.08838e6i 2.17742i
\(708\) 0 0
\(709\) −845715. −1.68241 −0.841204 0.540717i \(-0.818153\pi\)
−0.841204 + 0.540717i \(0.818153\pi\)
\(710\) 141738.i 0.281171i
\(711\) 0 0
\(712\) 15709.3 0.0309881
\(713\) −417108. −0.820484
\(714\) 0 0
\(715\) 171133. 0.334750
\(716\) 70573.0i 0.137662i
\(717\) 0 0
\(718\) 166832.i 0.323616i
\(719\) 541985.i 1.04841i −0.851593 0.524203i \(-0.824363\pi\)
0.851593 0.524203i \(-0.175637\pi\)
\(720\) 0 0
\(721\) 1.76327e6i 3.39193i
\(722\) 202647.i 0.388745i
\(723\) 0 0
\(724\) 31140.2 0.0594080
\(725\) 361660. 0.688057
\(726\) 0 0
\(727\) 324956. 0.614832 0.307416 0.951575i \(-0.400536\pi\)
0.307416 + 0.951575i \(0.400536\pi\)
\(728\) −1.39293e6 −2.62825
\(729\) 0 0
\(730\) −44265.2 −0.0830647
\(731\) 370990.i 0.694268i
\(732\) 0 0
\(733\) −228228. −0.424777 −0.212389 0.977185i \(-0.568124\pi\)
−0.212389 + 0.977185i \(0.568124\pi\)
\(734\) 712760. 1.32297
\(735\) 0 0
\(736\) −378016. −0.697839
\(737\) 396348. 0.729695
\(738\) 0 0
\(739\) 153824.i 0.281667i −0.990033 0.140834i \(-0.955022\pi\)
0.990033 0.140834i \(-0.0449782\pi\)
\(740\) 67316.7i 0.122930i
\(741\) 0 0
\(742\) 351376.i 0.638212i
\(743\) −40762.4 −0.0738384 −0.0369192 0.999318i \(-0.511754\pi\)
−0.0369192 + 0.999318i \(0.511754\pi\)
\(744\) 0 0
\(745\) 337736.i 0.608506i
\(746\) 331993.i 0.596556i
\(747\) 0 0
\(748\) 34635.9i 0.0619047i
\(749\) −625632. −1.11521
\(750\) 0 0
\(751\) 947341.i 1.67968i 0.542834 + 0.839840i \(0.317351\pi\)
−0.542834 + 0.839840i \(0.682649\pi\)
\(752\) 207071.i 0.366171i
\(753\) 0 0
\(754\) 746022. 1.31223
\(755\) 156603.i 0.274730i
\(756\) 0 0
\(757\) −384807. −0.671508 −0.335754 0.941950i \(-0.608991\pi\)
−0.335754 + 0.941950i \(0.608991\pi\)
\(758\) 765397.i 1.33213i
\(759\) 0 0
\(760\) 300790.i 0.520758i
\(761\) −621369. −1.07295 −0.536476 0.843916i \(-0.680245\pi\)
−0.536476 + 0.843916i \(0.680245\pi\)
\(762\) 0 0
\(763\) 2.12375e6i 3.64800i
\(764\) 142189.i 0.243602i
\(765\) 0 0
\(766\) 865396.i 1.47488i
\(767\) −697689. 287172.i −1.18596 0.488148i
\(768\) 0 0
\(769\) 140425.i 0.237460i −0.992927 0.118730i \(-0.962118\pi\)
0.992927 0.118730i \(-0.0378823\pi\)
\(770\) 251984. 0.425003
\(771\) 0 0
\(772\) −147256. −0.247081
\(773\) 233110.i 0.390124i 0.980791 + 0.195062i \(0.0624908\pi\)
−0.980791 + 0.195062i \(0.937509\pi\)
\(774\) 0 0
\(775\) 205633.i 0.342365i
\(776\) −504984. −0.838598
\(777\) 0 0
\(778\) 852332.i 1.40815i
\(779\) 615269. 1.01389
\(780\) 0 0
\(781\) 123497.i 0.202467i
\(782\) 434816. 0.711036
\(783\) 0 0
\(784\) −1.06803e6 −1.73761
\(785\) 380921.i 0.618153i
\(786\) 0 0
\(787\) 95032.6 0.153435 0.0767173 0.997053i \(-0.475556\pi\)
0.0767173 + 0.997053i \(0.475556\pi\)
\(788\) 103961. 0.167424
\(789\) 0 0
\(790\) 179583.i 0.287746i
\(791\) 108187.i 0.172911i
\(792\) 0 0
\(793\) 873555. 1.38913
\(794\) 52312.6 0.0829784
\(795\) 0 0
\(796\) 82702.9 0.130525
\(797\) 1.06273e6i