Properties

Label 531.5.c.d.235.2
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 235.2
Character \(\chi\) \(=\) 531.235
Dual form 531.5.c.d.235.39

$q$-expansion

\(f(q)\) \(=\) \(q-7.77798i q^{2} -44.4970 q^{4} -28.5836 q^{5} +72.7001 q^{7} +221.650i q^{8} +O(q^{10})\) \(q-7.77798i q^{2} -44.4970 q^{4} -28.5836 q^{5} +72.7001 q^{7} +221.650i q^{8} +222.323i q^{10} -9.33885i q^{11} +106.784i q^{13} -565.461i q^{14} +1012.03 q^{16} -337.515 q^{17} +546.806 q^{19} +1271.89 q^{20} -72.6374 q^{22} +39.3036i q^{23} +192.023 q^{25} +830.565 q^{26} -3234.94 q^{28} +444.446 q^{29} -1230.69i q^{31} -4325.19i q^{32} +2625.19i q^{34} -2078.03 q^{35} -1943.27i q^{37} -4253.05i q^{38} -6335.54i q^{40} +835.576 q^{41} +2242.91i q^{43} +415.551i q^{44} +305.703 q^{46} -3914.09i q^{47} +2884.31 q^{49} -1493.55i q^{50} -4751.57i q^{52} -3820.28 q^{53} +266.938i q^{55} +16113.9i q^{56} -3456.89i q^{58} +(3000.50 + 1764.76i) q^{59} +2344.07i q^{61} -9572.29 q^{62} -17448.7 q^{64} -3052.27i q^{65} +3650.67i q^{67} +15018.4 q^{68} +16162.9i q^{70} +2630.49 q^{71} +8924.21i q^{73} -15114.7 q^{74} -24331.3 q^{76} -678.935i q^{77} +1497.53 q^{79} -28927.6 q^{80} -6499.09i q^{82} -3985.30i q^{83} +9647.40 q^{85} +17445.3 q^{86} +2069.95 q^{88} -2531.49i q^{89} +7763.21i q^{91} -1748.89i q^{92} -30443.7 q^{94} -15629.7 q^{95} -9210.22i q^{97} -22434.1i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.77798i 1.94450i −0.233953 0.972248i \(-0.575166\pi\)
0.233953 0.972248i \(-0.424834\pi\)
\(3\) 0 0
\(4\) −44.4970 −2.78106
\(5\) −28.5836 −1.14334 −0.571672 0.820482i \(-0.693705\pi\)
−0.571672 + 0.820482i \(0.693705\pi\)
\(6\) 0 0
\(7\) 72.7001 1.48368 0.741838 0.670579i \(-0.233954\pi\)
0.741838 + 0.670579i \(0.233954\pi\)
\(8\) 221.650i 3.46327i
\(9\) 0 0
\(10\) 222.323i 2.22323i
\(11\) 9.33885i 0.0771806i −0.999255 0.0385903i \(-0.987713\pi\)
0.999255 0.0385903i \(-0.0122867\pi\)
\(12\) 0 0
\(13\) 106.784i 0.631858i 0.948783 + 0.315929i \(0.102316\pi\)
−0.948783 + 0.315929i \(0.897684\pi\)
\(14\) 565.461i 2.88500i
\(15\) 0 0
\(16\) 1012.03 3.95326
\(17\) −337.515 −1.16787 −0.583936 0.811800i \(-0.698488\pi\)
−0.583936 + 0.811800i \(0.698488\pi\)
\(18\) 0 0
\(19\) 546.806 1.51470 0.757349 0.653010i \(-0.226494\pi\)
0.757349 + 0.653010i \(0.226494\pi\)
\(20\) 1271.89 3.17972
\(21\) 0 0
\(22\) −72.6374 −0.150077
\(23\) 39.3036i 0.0742979i 0.999310 + 0.0371490i \(0.0118276\pi\)
−0.999310 + 0.0371490i \(0.988172\pi\)
\(24\) 0 0
\(25\) 192.023 0.307237
\(26\) 830.565 1.22865
\(27\) 0 0
\(28\) −3234.94 −4.12620
\(29\) 444.446 0.528473 0.264236 0.964458i \(-0.414880\pi\)
0.264236 + 0.964458i \(0.414880\pi\)
\(30\) 0 0
\(31\) 1230.69i 1.28063i −0.768110 0.640317i \(-0.778803\pi\)
0.768110 0.640317i \(-0.221197\pi\)
\(32\) 4325.19i 4.22382i
\(33\) 0 0
\(34\) 2625.19i 2.27092i
\(35\) −2078.03 −1.69635
\(36\) 0 0
\(37\) 1943.27i 1.41948i −0.704464 0.709740i \(-0.748812\pi\)
0.704464 0.709740i \(-0.251188\pi\)
\(38\) 4253.05i 2.94532i
\(39\) 0 0
\(40\) 6335.54i 3.95971i
\(41\) 835.576 0.497071 0.248535 0.968623i \(-0.420051\pi\)
0.248535 + 0.968623i \(0.420051\pi\)
\(42\) 0 0
\(43\) 2242.91i 1.21304i 0.795069 + 0.606519i \(0.207435\pi\)
−0.795069 + 0.606519i \(0.792565\pi\)
\(44\) 415.551i 0.214644i
\(45\) 0 0
\(46\) 305.703 0.144472
\(47\) 3914.09i 1.77188i −0.463797 0.885942i \(-0.653513\pi\)
0.463797 0.885942i \(-0.346487\pi\)
\(48\) 0 0
\(49\) 2884.31 1.20130
\(50\) 1493.55i 0.597421i
\(51\) 0 0
\(52\) 4751.57i 1.75724i
\(53\) −3820.28 −1.36001 −0.680006 0.733206i \(-0.738023\pi\)
−0.680006 + 0.733206i \(0.738023\pi\)
\(54\) 0 0
\(55\) 266.938i 0.0882440i
\(56\) 16113.9i 5.13838i
\(57\) 0 0
\(58\) 3456.89i 1.02761i
\(59\) 3000.50 + 1764.76i 0.861964 + 0.506970i
\(60\) 0 0
\(61\) 2344.07i 0.629958i 0.949099 + 0.314979i \(0.101997\pi\)
−0.949099 + 0.314979i \(0.898003\pi\)
\(62\) −9572.29 −2.49019
\(63\) 0 0
\(64\) −17448.7 −4.25994
\(65\) 3052.27i 0.722432i
\(66\) 0 0
\(67\) 3650.67i 0.813249i 0.913595 + 0.406624i \(0.133294\pi\)
−0.913595 + 0.406624i \(0.866706\pi\)
\(68\) 15018.4 3.24793
\(69\) 0 0
\(70\) 16162.9i 3.29855i
\(71\) 2630.49 0.521819 0.260909 0.965363i \(-0.415978\pi\)
0.260909 + 0.965363i \(0.415978\pi\)
\(72\) 0 0
\(73\) 8924.21i 1.67465i 0.546705 + 0.837325i \(0.315882\pi\)
−0.546705 + 0.837325i \(0.684118\pi\)
\(74\) −15114.7 −2.76017
\(75\) 0 0
\(76\) −24331.3 −4.21247
\(77\) 678.935i 0.114511i
\(78\) 0 0
\(79\) 1497.53 0.239951 0.119976 0.992777i \(-0.461718\pi\)
0.119976 + 0.992777i \(0.461718\pi\)
\(80\) −28927.6 −4.51993
\(81\) 0 0
\(82\) 6499.09i 0.966552i
\(83\) 3985.30i 0.578503i −0.957253 0.289251i \(-0.906594\pi\)
0.957253 0.289251i \(-0.0934063\pi\)
\(84\) 0 0
\(85\) 9647.40 1.33528
\(86\) 17445.3 2.35875
\(87\) 0 0
\(88\) 2069.95 0.267297
\(89\) 2531.49i 0.319593i −0.987150 0.159796i \(-0.948916\pi\)
0.987150 0.159796i \(-0.0510838\pi\)
\(90\) 0 0
\(91\) 7763.21i 0.937473i
\(92\) 1748.89i 0.206627i
\(93\) 0 0
\(94\) −30443.7 −3.44542
\(95\) −15629.7 −1.73182
\(96\) 0 0
\(97\) 9210.22i 0.978874i −0.872039 0.489437i \(-0.837202\pi\)
0.872039 0.489437i \(-0.162798\pi\)
\(98\) 22434.1i 2.33591i
\(99\) 0 0
\(100\) −8544.45 −0.854445
\(101\) 13290.3i 1.30284i 0.758717 + 0.651420i \(0.225826\pi\)
−0.758717 + 0.651420i \(0.774174\pi\)
\(102\) 0 0
\(103\) 14436.1i 1.36074i −0.732867 0.680372i \(-0.761818\pi\)
0.732867 0.680372i \(-0.238182\pi\)
\(104\) −23668.6 −2.18830
\(105\) 0 0
\(106\) 29714.0i 2.64454i
\(107\) 986.412 0.0861570 0.0430785 0.999072i \(-0.486283\pi\)
0.0430785 + 0.999072i \(0.486283\pi\)
\(108\) 0 0
\(109\) 14934.0i 1.25697i −0.777824 0.628483i \(-0.783676\pi\)
0.777824 0.628483i \(-0.216324\pi\)
\(110\) 2076.24 0.171590
\(111\) 0 0
\(112\) 73575.0 5.86535
\(113\) 19908.8i 1.55915i −0.626308 0.779576i \(-0.715435\pi\)
0.626308 0.779576i \(-0.284565\pi\)
\(114\) 0 0
\(115\) 1123.44i 0.0849481i
\(116\) −19776.5 −1.46972
\(117\) 0 0
\(118\) 13726.3 23337.8i 0.985802 1.67608i
\(119\) −24537.4 −1.73274
\(120\) 0 0
\(121\) 14553.8 0.994043
\(122\) 18232.2 1.22495
\(123\) 0 0
\(124\) 54762.1i 3.56153i
\(125\) 12376.0 0.792067
\(126\) 0 0
\(127\) −15531.0 −0.962922 −0.481461 0.876468i \(-0.659894\pi\)
−0.481461 + 0.876468i \(0.659894\pi\)
\(128\) 66512.8i 4.05962i
\(129\) 0 0
\(130\) −23740.5 −1.40477
\(131\) 1246.21i 0.0726188i −0.999341 0.0363094i \(-0.988440\pi\)
0.999341 0.0363094i \(-0.0115602\pi\)
\(132\) 0 0
\(133\) 39752.9 2.24732
\(134\) 28394.9 1.58136
\(135\) 0 0
\(136\) 74810.0i 4.04466i
\(137\) −16792.4 −0.894689 −0.447344 0.894362i \(-0.647630\pi\)
−0.447344 + 0.894362i \(0.647630\pi\)
\(138\) 0 0
\(139\) −35922.2 −1.85923 −0.929616 0.368530i \(-0.879861\pi\)
−0.929616 + 0.368530i \(0.879861\pi\)
\(140\) 92466.3 4.71767
\(141\) 0 0
\(142\) 20459.9i 1.01467i
\(143\) 997.240 0.0487672
\(144\) 0 0
\(145\) −12703.9 −0.604227
\(146\) 69412.4 3.25635
\(147\) 0 0
\(148\) 86469.7i 3.94767i
\(149\) 25955.6i 1.16912i −0.811351 0.584559i \(-0.801268\pi\)
0.811351 0.584559i \(-0.198732\pi\)
\(150\) 0 0
\(151\) 18565.1i 0.814222i −0.913379 0.407111i \(-0.866536\pi\)
0.913379 0.407111i \(-0.133464\pi\)
\(152\) 121199.i 5.24581i
\(153\) 0 0
\(154\) −5280.75 −0.222666
\(155\) 35177.6i 1.46421i
\(156\) 0 0
\(157\) 46527.1i 1.88759i −0.330537 0.943793i \(-0.607230\pi\)
0.330537 0.943793i \(-0.392770\pi\)
\(158\) 11647.8i 0.466584i
\(159\) 0 0
\(160\) 123630.i 4.82928i
\(161\) 2857.38i 0.110234i
\(162\) 0 0
\(163\) −437.670 −0.0164730 −0.00823648 0.999966i \(-0.502622\pi\)
−0.00823648 + 0.999966i \(0.502622\pi\)
\(164\) −37180.6 −1.38239
\(165\) 0 0
\(166\) −30997.6 −1.12490
\(167\) −49721.0 −1.78282 −0.891408 0.453201i \(-0.850282\pi\)
−0.891408 + 0.453201i \(0.850282\pi\)
\(168\) 0 0
\(169\) 17158.2 0.600755
\(170\) 75037.3i 2.59645i
\(171\) 0 0
\(172\) 99802.8i 3.37354i
\(173\) 30766.7i 1.02799i −0.857793 0.513995i \(-0.828165\pi\)
0.857793 0.513995i \(-0.171835\pi\)
\(174\) 0 0
\(175\) 13960.1 0.455840
\(176\) 9451.23i 0.305115i
\(177\) 0 0
\(178\) −19689.9 −0.621447
\(179\) 20767.6i 0.648156i −0.946030 0.324078i \(-0.894946\pi\)
0.946030 0.324078i \(-0.105054\pi\)
\(180\) 0 0
\(181\) 27789.6 0.848252 0.424126 0.905603i \(-0.360581\pi\)
0.424126 + 0.905603i \(0.360581\pi\)
\(182\) 60382.2 1.82291
\(183\) 0 0
\(184\) −8711.63 −0.257314
\(185\) 55545.6i 1.62295i
\(186\) 0 0
\(187\) 3152.00i 0.0901370i
\(188\) 174165.i 4.92772i
\(189\) 0 0
\(190\) 121568.i 3.36752i
\(191\) 22222.5i 0.609153i −0.952488 0.304576i \(-0.901485\pi\)
0.952488 0.304576i \(-0.0985149\pi\)
\(192\) 0 0
\(193\) 22533.4 0.604941 0.302470 0.953159i \(-0.402189\pi\)
0.302470 + 0.953159i \(0.402189\pi\)
\(194\) −71637.0 −1.90342
\(195\) 0 0
\(196\) −128343. −3.34088
\(197\) −11138.9 −0.287018 −0.143509 0.989649i \(-0.545839\pi\)
−0.143509 + 0.989649i \(0.545839\pi\)
\(198\) 0 0
\(199\) −27923.7 −0.705126 −0.352563 0.935788i \(-0.614690\pi\)
−0.352563 + 0.935788i \(0.614690\pi\)
\(200\) 42561.8i 1.06404i
\(201\) 0 0
\(202\) 103372. 2.53337
\(203\) 32311.3 0.784083
\(204\) 0 0
\(205\) −23883.8 −0.568323
\(206\) −112284. −2.64596
\(207\) 0 0
\(208\) 108069.i 2.49790i
\(209\) 5106.54i 0.116905i
\(210\) 0 0
\(211\) 4163.53i 0.0935182i −0.998906 0.0467591i \(-0.985111\pi\)
0.998906 0.0467591i \(-0.0148893\pi\)
\(212\) 169991. 3.78228
\(213\) 0 0
\(214\) 7672.29i 0.167532i
\(215\) 64110.4i 1.38692i
\(216\) 0 0
\(217\) 89471.3i 1.90005i
\(218\) −116156. −2.44416
\(219\) 0 0
\(220\) 11878.0i 0.245412i
\(221\) 36041.2i 0.737930i
\(222\) 0 0
\(223\) −77429.7 −1.55703 −0.778517 0.627624i \(-0.784028\pi\)
−0.778517 + 0.627624i \(0.784028\pi\)
\(224\) 314442.i 6.26678i
\(225\) 0 0
\(226\) −154850. −3.03176
\(227\) 6994.10i 0.135731i −0.997694 0.0678656i \(-0.978381\pi\)
0.997694 0.0678656i \(-0.0216189\pi\)
\(228\) 0 0
\(229\) 51473.3i 0.981547i −0.871287 0.490773i \(-0.836714\pi\)
0.871287 0.490773i \(-0.163286\pi\)
\(230\) −8738.09 −0.165181
\(231\) 0 0
\(232\) 98511.2i 1.83025i
\(233\) 79635.1i 1.46687i 0.679758 + 0.733437i \(0.262085\pi\)
−0.679758 + 0.733437i \(0.737915\pi\)
\(234\) 0 0
\(235\) 111879.i 2.02587i
\(236\) −133513. 78526.7i −2.39718 1.40992i
\(237\) 0 0
\(238\) 190851.i 3.36931i
\(239\) 27642.4 0.483927 0.241964 0.970285i \(-0.422208\pi\)
0.241964 + 0.970285i \(0.422208\pi\)
\(240\) 0 0
\(241\) 11767.3 0.202602 0.101301 0.994856i \(-0.467700\pi\)
0.101301 + 0.994856i \(0.467700\pi\)
\(242\) 113199.i 1.93291i
\(243\) 0 0
\(244\) 104304.i 1.75195i
\(245\) −82444.0 −1.37349
\(246\) 0 0
\(247\) 58390.2i 0.957075i
\(248\) 272782. 4.43519
\(249\) 0 0
\(250\) 96260.7i 1.54017i
\(251\) 98763.5 1.56765 0.783825 0.620982i \(-0.213266\pi\)
0.783825 + 0.620982i \(0.213266\pi\)
\(252\) 0 0
\(253\) 367.050 0.00573436
\(254\) 120800.i 1.87240i
\(255\) 0 0
\(256\) 238156. 3.63398
\(257\) −36087.2 −0.546370 −0.273185 0.961961i \(-0.588077\pi\)
−0.273185 + 0.961961i \(0.588077\pi\)
\(258\) 0 0
\(259\) 141276.i 2.10605i
\(260\) 135817.i 2.00913i
\(261\) 0 0
\(262\) −9693.01 −0.141207
\(263\) 16587.3 0.239808 0.119904 0.992786i \(-0.461741\pi\)
0.119904 + 0.992786i \(0.461741\pi\)
\(264\) 0 0
\(265\) 109197. 1.55496
\(266\) 309197.i 4.36991i
\(267\) 0 0
\(268\) 162444.i 2.26170i
\(269\) 47175.2i 0.651943i 0.945380 + 0.325971i \(0.105691\pi\)
−0.945380 + 0.325971i \(0.894309\pi\)
\(270\) 0 0
\(271\) 37006.4 0.503893 0.251946 0.967741i \(-0.418929\pi\)
0.251946 + 0.967741i \(0.418929\pi\)
\(272\) −341577. −4.61690
\(273\) 0 0
\(274\) 130611.i 1.73972i
\(275\) 1793.27i 0.0237127i
\(276\) 0 0
\(277\) −37499.2 −0.488722 −0.244361 0.969684i \(-0.578578\pi\)
−0.244361 + 0.969684i \(0.578578\pi\)
\(278\) 279402.i 3.61527i
\(279\) 0 0
\(280\) 460595.i 5.87494i
\(281\) 8246.03 0.104432 0.0522158 0.998636i \(-0.483372\pi\)
0.0522158 + 0.998636i \(0.483372\pi\)
\(282\) 0 0
\(283\) 3380.85i 0.0422137i 0.999777 + 0.0211069i \(0.00671902\pi\)
−0.999777 + 0.0211069i \(0.993281\pi\)
\(284\) −117049. −1.45121
\(285\) 0 0
\(286\) 7756.52i 0.0948276i
\(287\) 60746.5 0.737492
\(288\) 0 0
\(289\) 30395.4 0.363925
\(290\) 98810.4i 1.17492i
\(291\) 0 0
\(292\) 397101.i 4.65731i
\(293\) 146875. 1.71085 0.855427 0.517923i \(-0.173295\pi\)
0.855427 + 0.517923i \(0.173295\pi\)
\(294\) 0 0
\(295\) −85765.0 50443.3i −0.985521 0.579642i
\(296\) 430724. 4.91605
\(297\) 0 0
\(298\) −201882. −2.27334
\(299\) −4197.00 −0.0469458
\(300\) 0 0
\(301\) 163060.i 1.79976i
\(302\) −144399. −1.58325
\(303\) 0 0
\(304\) 553386. 5.98799
\(305\) 67002.1i 0.720259i
\(306\) 0 0
\(307\) 92101.3 0.977213 0.488606 0.872504i \(-0.337505\pi\)
0.488606 + 0.872504i \(0.337505\pi\)
\(308\) 30210.6i 0.318462i
\(309\) 0 0
\(310\) 273611. 2.84714
\(311\) 32978.0 0.340960 0.170480 0.985361i \(-0.445468\pi\)
0.170480 + 0.985361i \(0.445468\pi\)
\(312\) 0 0
\(313\) 40714.0i 0.415581i −0.978173 0.207790i \(-0.933373\pi\)
0.978173 0.207790i \(-0.0666272\pi\)
\(314\) −361887. −3.67040
\(315\) 0 0
\(316\) −66635.9 −0.667320
\(317\) −5097.47 −0.0507266 −0.0253633 0.999678i \(-0.508074\pi\)
−0.0253633 + 0.999678i \(0.508074\pi\)
\(318\) 0 0
\(319\) 4150.61i 0.0407878i
\(320\) 498748. 4.87058
\(321\) 0 0
\(322\) 22224.6 0.214350
\(323\) −184555. −1.76897
\(324\) 0 0
\(325\) 20505.0i 0.194130i
\(326\) 3404.19i 0.0320316i
\(327\) 0 0
\(328\) 185205.i 1.72149i
\(329\) 284555.i 2.62890i
\(330\) 0 0
\(331\) −18523.1 −0.169067 −0.0845334 0.996421i \(-0.526940\pi\)
−0.0845334 + 0.996421i \(0.526940\pi\)
\(332\) 177334.i 1.60885i
\(333\) 0 0
\(334\) 386729.i 3.46668i
\(335\) 104349.i 0.929823i
\(336\) 0 0
\(337\) 137124.i 1.20741i −0.797209 0.603704i \(-0.793691\pi\)
0.797209 0.603704i \(-0.206309\pi\)
\(338\) 133456.i 1.16817i
\(339\) 0 0
\(340\) −429281. −3.71350
\(341\) −11493.2 −0.0988401
\(342\) 0 0
\(343\) 35136.7 0.298657
\(344\) −497140. −4.20108
\(345\) 0 0
\(346\) −239303. −1.99892
\(347\) 92388.3i 0.767287i −0.923481 0.383643i \(-0.874669\pi\)
0.923481 0.383643i \(-0.125331\pi\)
\(348\) 0 0
\(349\) 192179.i 1.57781i −0.614513 0.788907i \(-0.710648\pi\)
0.614513 0.788907i \(-0.289352\pi\)
\(350\) 108581.i 0.886379i
\(351\) 0 0
\(352\) −40392.3 −0.325997
\(353\) 133742.i 1.07329i −0.843808 0.536645i \(-0.819692\pi\)
0.843808 0.536645i \(-0.180308\pi\)
\(354\) 0 0
\(355\) −75188.9 −0.596619
\(356\) 112644.i 0.888808i
\(357\) 0 0
\(358\) −161530. −1.26034
\(359\) 63441.2 0.492246 0.246123 0.969239i \(-0.420843\pi\)
0.246123 + 0.969239i \(0.420843\pi\)
\(360\) 0 0
\(361\) 168676. 1.29431
\(362\) 216147.i 1.64942i
\(363\) 0 0
\(364\) 345440.i 2.60717i
\(365\) 255086.i 1.91470i
\(366\) 0 0
\(367\) 62757.7i 0.465946i −0.972483 0.232973i \(-0.925155\pi\)
0.972483 0.232973i \(-0.0748453\pi\)
\(368\) 39776.6i 0.293719i
\(369\) 0 0
\(370\) 432033. 3.15583
\(371\) −277735. −2.01782
\(372\) 0 0
\(373\) −102419. −0.736148 −0.368074 0.929797i \(-0.619983\pi\)
−0.368074 + 0.929797i \(0.619983\pi\)
\(374\) 24516.2 0.175271
\(375\) 0 0
\(376\) 867556. 6.13652
\(377\) 47459.7i 0.333920i
\(378\) 0 0
\(379\) −115222. −0.802151 −0.401076 0.916045i \(-0.631364\pi\)
−0.401076 + 0.916045i \(0.631364\pi\)
\(380\) 695475. 4.81631
\(381\) 0 0
\(382\) −172846. −1.18450
\(383\) −68700.2 −0.468339 −0.234170 0.972196i \(-0.575237\pi\)
−0.234170 + 0.972196i \(0.575237\pi\)
\(384\) 0 0
\(385\) 19406.4i 0.130925i
\(386\) 175265.i 1.17631i
\(387\) 0 0
\(388\) 409828.i 2.72231i
\(389\) −1971.29 −0.0130272 −0.00651360 0.999979i \(-0.502073\pi\)
−0.00651360 + 0.999979i \(0.502073\pi\)
\(390\) 0 0
\(391\) 13265.6i 0.0867705i
\(392\) 639306.i 4.16041i
\(393\) 0 0
\(394\) 86638.0i 0.558105i
\(395\) −42805.0 −0.274347
\(396\) 0 0
\(397\) 42440.9i 0.269280i 0.990895 + 0.134640i \(0.0429877\pi\)
−0.990895 + 0.134640i \(0.957012\pi\)
\(398\) 217190.i 1.37111i
\(399\) 0 0
\(400\) 194334. 1.21459
\(401\) 103710.i 0.644959i 0.946576 + 0.322480i \(0.104516\pi\)
−0.946576 + 0.322480i \(0.895484\pi\)
\(402\) 0 0
\(403\) 131418. 0.809180
\(404\) 591378.i 3.62328i
\(405\) 0 0
\(406\) 251316.i 1.52465i
\(407\) −18147.9 −0.109556
\(408\) 0 0
\(409\) 29250.4i 0.174858i 0.996171 + 0.0874289i \(0.0278651\pi\)
−0.996171 + 0.0874289i \(0.972135\pi\)
\(410\) 185768.i 1.10510i
\(411\) 0 0
\(412\) 642365.i 3.78432i
\(413\) 218136. + 128299.i 1.27887 + 0.752180i
\(414\) 0 0
\(415\) 113914.i 0.661428i
\(416\) 461861. 2.66885
\(417\) 0 0
\(418\) −39718.6 −0.227322
\(419\) 54737.0i 0.311784i −0.987774 0.155892i \(-0.950175\pi\)
0.987774 0.155892i \(-0.0498251\pi\)
\(420\) 0 0
\(421\) 135111.i 0.762303i 0.924513 + 0.381151i \(0.124472\pi\)
−0.924513 + 0.381151i \(0.875528\pi\)
\(422\) −32383.8 −0.181846
\(423\) 0 0
\(424\) 846762.i 4.71010i
\(425\) −64810.6 −0.358813
\(426\) 0 0
\(427\) 170414.i 0.934653i
\(428\) −43892.4 −0.239608
\(429\) 0 0
\(430\) −498650. −2.69686
\(431\) 332324.i 1.78899i 0.447081 + 0.894493i \(0.352463\pi\)
−0.447081 + 0.894493i \(0.647537\pi\)
\(432\) 0 0
\(433\) 245001. 1.30675 0.653375 0.757035i \(-0.273353\pi\)
0.653375 + 0.757035i \(0.273353\pi\)
\(434\) −695907. −3.69463
\(435\) 0 0
\(436\) 664519.i 3.49570i
\(437\) 21491.5i 0.112539i
\(438\) 0 0
\(439\) −181499. −0.941771 −0.470885 0.882194i \(-0.656066\pi\)
−0.470885 + 0.882194i \(0.656066\pi\)
\(440\) −59166.7 −0.305613
\(441\) 0 0
\(442\) −280328. −1.43490
\(443\) 57457.9i 0.292781i −0.989227 0.146390i \(-0.953234\pi\)
0.989227 0.146390i \(-0.0467655\pi\)
\(444\) 0 0
\(445\) 72359.3i 0.365405i
\(446\) 602247.i 3.02765i
\(447\) 0 0
\(448\) −1.26852e6 −6.32038
\(449\) −26596.2 −0.131925 −0.0659624 0.997822i \(-0.521012\pi\)
−0.0659624 + 0.997822i \(0.521012\pi\)
\(450\) 0 0
\(451\) 7803.31i 0.0383642i
\(452\) 885883.i 4.33610i
\(453\) 0 0
\(454\) −54400.0 −0.263929
\(455\) 221901.i 1.07185i
\(456\) 0 0
\(457\) 300518.i 1.43892i −0.694532 0.719462i \(-0.744388\pi\)
0.694532 0.719462i \(-0.255612\pi\)
\(458\) −400358. −1.90861
\(459\) 0 0
\(460\) 49989.7i 0.236246i
\(461\) 140059. 0.659038 0.329519 0.944149i \(-0.393113\pi\)
0.329519 + 0.944149i \(0.393113\pi\)
\(462\) 0 0
\(463\) 127338.i 0.594014i 0.954875 + 0.297007i \(0.0959885\pi\)
−0.954875 + 0.297007i \(0.904012\pi\)
\(464\) 449794. 2.08919
\(465\) 0 0
\(466\) 619400. 2.85233
\(467\) 27447.8i 0.125856i 0.998018 + 0.0629280i \(0.0200439\pi\)
−0.998018 + 0.0629280i \(0.979956\pi\)
\(468\) 0 0
\(469\) 265404.i 1.20660i
\(470\) 870192. 3.93930
\(471\) 0 0
\(472\) −391159. + 665058.i −1.75578 + 2.98522i
\(473\) 20946.2 0.0936230
\(474\) 0 0
\(475\) 104999. 0.465371
\(476\) 1.09184e6 4.81887
\(477\) 0 0
\(478\) 215002.i 0.940995i
\(479\) 403672. 1.75937 0.879687 0.475554i \(-0.157752\pi\)
0.879687 + 0.475554i \(0.157752\pi\)
\(480\) 0 0
\(481\) 207510. 0.896910
\(482\) 91526.0i 0.393959i
\(483\) 0 0
\(484\) −647600. −2.76450
\(485\) 263261.i 1.11919i
\(486\) 0 0
\(487\) 160164. 0.675316 0.337658 0.941269i \(-0.390365\pi\)
0.337658 + 0.941269i \(0.390365\pi\)
\(488\) −519563. −2.18172
\(489\) 0 0
\(490\) 641248.i 2.67075i
\(491\) 27802.8 0.115326 0.0576628 0.998336i \(-0.481635\pi\)
0.0576628 + 0.998336i \(0.481635\pi\)
\(492\) 0 0
\(493\) −150007. −0.617189
\(494\) 454158. 1.86103
\(495\) 0 0
\(496\) 1.24550e6i 5.06268i
\(497\) 191237. 0.774210
\(498\) 0 0
\(499\) 136913. 0.549848 0.274924 0.961466i \(-0.411347\pi\)
0.274924 + 0.961466i \(0.411347\pi\)
\(500\) −550697. −2.20279
\(501\) 0 0
\(502\) 768181.i 3.04829i
\(503\) 45212.1i 0.178698i −0.996000 0.0893489i \(-0.971521\pi\)
0.996000 0.0893489i \(-0.0284786\pi\)
\(504\) 0 0
\(505\) 379884.i 1.48960i
\(506\) 2854.91i 0.0111504i
\(507\) 0 0
\(508\) 691082. 2.67795
\(509\) 420229.i 1.62200i −0.585048 0.810999i \(-0.698924\pi\)
0.585048 0.810999i \(-0.301076\pi\)
\(510\) 0 0
\(511\) 648791.i 2.48464i
\(512\) 788170.i 3.00663i
\(513\) 0 0
\(514\) 280686.i 1.06241i
\(515\) 412637.i 1.55580i
\(516\) 0 0
\(517\) −36553.1 −0.136755
\(518\) −1.09884e6 −4.09520
\(519\) 0 0
\(520\) 676535. 2.50198
\(521\) −176873. −0.651608 −0.325804 0.945437i \(-0.605635\pi\)
−0.325804 + 0.945437i \(0.605635\pi\)
\(522\) 0 0
\(523\) −361972. −1.32334 −0.661670 0.749795i \(-0.730152\pi\)
−0.661670 + 0.749795i \(0.730152\pi\)
\(524\) 55452.7i 0.201958i
\(525\) 0 0
\(526\) 129015.i 0.466305i
\(527\) 415376.i 1.49562i
\(528\) 0 0
\(529\) 278296. 0.994480
\(530\) 849335.i 3.02362i
\(531\) 0 0
\(532\) −1.76889e6 −6.24995
\(533\) 89226.1i 0.314078i
\(534\) 0 0
\(535\) −28195.2 −0.0985071
\(536\) −809170. −2.81650
\(537\) 0 0
\(538\) 366928. 1.26770
\(539\) 26936.1i 0.0927166i
\(540\) 0 0
\(541\) 276519.i 0.944779i −0.881390 0.472390i \(-0.843392\pi\)
0.881390 0.472390i \(-0.156608\pi\)
\(542\) 287835.i 0.979818i
\(543\) 0 0
\(544\) 1.45982e6i 4.93288i
\(545\) 426868.i 1.43714i
\(546\) 0 0
\(547\) −424332. −1.41818 −0.709090 0.705118i \(-0.750894\pi\)
−0.709090 + 0.705118i \(0.750894\pi\)
\(548\) 747213. 2.48819
\(549\) 0 0
\(550\) −13948.1 −0.0461093
\(551\) 243026. 0.800477
\(552\) 0 0
\(553\) 108871. 0.356010
\(554\) 291668.i 0.950318i
\(555\) 0 0
\(556\) 1.59843e6 5.17064
\(557\) 408914. 1.31802 0.659010 0.752134i \(-0.270976\pi\)
0.659010 + 0.752134i \(0.270976\pi\)
\(558\) 0 0
\(559\) −239507. −0.766468
\(560\) −2.10304e6 −6.70612
\(561\) 0 0
\(562\) 64137.5i 0.203067i
\(563\) 165648.i 0.522600i 0.965258 + 0.261300i \(0.0841512\pi\)
−0.965258 + 0.261300i \(0.915849\pi\)
\(564\) 0 0
\(565\) 569065.i 1.78265i
\(566\) 26296.2 0.0820844
\(567\) 0 0
\(568\) 583047.i 1.80720i
\(569\) 413247.i 1.27639i 0.769873 + 0.638197i \(0.220320\pi\)
−0.769873 + 0.638197i \(0.779680\pi\)
\(570\) 0 0
\(571\) 531700.i 1.63078i −0.578914 0.815388i \(-0.696523\pi\)
0.578914 0.815388i \(-0.303477\pi\)
\(572\) −44374.2 −0.135625
\(573\) 0 0
\(574\) 472485.i 1.43405i
\(575\) 7547.20i 0.0228271i
\(576\) 0 0
\(577\) −94373.0 −0.283463 −0.141731 0.989905i \(-0.545267\pi\)
−0.141731 + 0.989905i \(0.545267\pi\)
\(578\) 236415.i 0.707651i
\(579\) 0 0
\(580\) 565284. 1.68039
\(581\) 289732.i 0.858310i
\(582\) 0 0
\(583\) 35677.0i 0.104967i
\(584\) −1.97805e6 −5.79977
\(585\) 0 0
\(586\) 1.14239e6i 3.32675i
\(587\) 16716.5i 0.0485142i 0.999706 + 0.0242571i \(0.00772203\pi\)
−0.999706 + 0.0242571i \(0.992278\pi\)
\(588\) 0 0
\(589\) 672949.i 1.93978i
\(590\) −392347. + 667079.i −1.12711 + 1.91634i
\(591\) 0 0
\(592\) 1.96665e6i 5.61157i
\(593\) 406858. 1.15700 0.578500 0.815683i \(-0.303638\pi\)
0.578500 + 0.815683i \(0.303638\pi\)
\(594\) 0 0
\(595\) 701367. 1.98112
\(596\) 1.15495e6i 3.25139i
\(597\) 0 0
\(598\) 32644.2i 0.0912858i
\(599\) 216469. 0.603313 0.301656 0.953417i \(-0.402460\pi\)
0.301656 + 0.953417i \(0.402460\pi\)
\(600\) 0 0
\(601\) 114850.i 0.317966i 0.987281 + 0.158983i \(0.0508214\pi\)
−0.987281 + 0.158983i \(0.949179\pi\)
\(602\) 1.26828e6 3.49962
\(603\) 0 0
\(604\) 826091.i 2.26440i
\(605\) −416000. −1.13653
\(606\) 0 0
\(607\) 460090. 1.24872 0.624360 0.781136i \(-0.285360\pi\)
0.624360 + 0.781136i \(0.285360\pi\)
\(608\) 2.36504e6i 6.39781i
\(609\) 0 0
\(610\) −521141. −1.40054
\(611\) 417962. 1.11958
\(612\) 0 0
\(613\) 54204.8i 0.144250i 0.997396 + 0.0721252i \(0.0229781\pi\)
−0.997396 + 0.0721252i \(0.977022\pi\)
\(614\) 716363.i 1.90019i
\(615\) 0 0
\(616\) 150486. 0.396583
\(617\) 446671. 1.17332 0.586662 0.809832i \(-0.300442\pi\)
0.586662 + 0.809832i \(0.300442\pi\)
\(618\) 0 0
\(619\) 608019. 1.58685 0.793425 0.608668i \(-0.208296\pi\)
0.793425 + 0.608668i \(0.208296\pi\)
\(620\) 1.56530e6i 4.07205i
\(621\) 0 0
\(622\) 256503.i 0.662996i
\(623\) 184040.i 0.474172i
\(624\) 0 0
\(625\) −473767. −1.21284
\(626\) −316673. −0.808095
\(627\) 0 0
\(628\) 2.07032e6i 5.24950i
\(629\) 655882.i 1.65777i
\(630\) 0 0
\(631\) −89180.6 −0.223981 −0.111991 0.993709i \(-0.535723\pi\)
−0.111991 + 0.993709i \(0.535723\pi\)
\(632\) 331928.i 0.831016i
\(633\) 0 0
\(634\) 39648.0i 0.0986377i
\(635\) 443931. 1.10095
\(636\) 0 0
\(637\) 307998.i 0.759048i
\(638\) −32283.4 −0.0793118
\(639\) 0 0
\(640\) 1.90118e6i 4.64155i
\(641\) −496660. −1.20877 −0.604384 0.796693i \(-0.706581\pi\)
−0.604384 + 0.796693i \(0.706581\pi\)
\(642\) 0 0
\(643\) −477619. −1.15521 −0.577603 0.816318i \(-0.696012\pi\)
−0.577603 + 0.816318i \(0.696012\pi\)
\(644\) 127145.i 0.306568i
\(645\) 0 0
\(646\) 1.43547e6i 3.43976i
\(647\) −255266. −0.609796 −0.304898 0.952385i \(-0.598622\pi\)
−0.304898 + 0.952385i \(0.598622\pi\)
\(648\) 0 0
\(649\) 16480.9 28021.2i 0.0391282 0.0665268i
\(650\) 159487. 0.377485
\(651\) 0 0
\(652\) 19475.0 0.0458124
\(653\) −12720.0 −0.0298305 −0.0149152 0.999889i \(-0.504748\pi\)
−0.0149152 + 0.999889i \(0.504748\pi\)
\(654\) 0 0
\(655\) 35621.2i 0.0830283i
\(656\) 845631. 1.96505
\(657\) 0 0
\(658\) −2.21326e6 −5.11189
\(659\) 732103.i 1.68578i −0.538084 0.842891i \(-0.680852\pi\)
0.538084 0.842891i \(-0.319148\pi\)
\(660\) 0 0
\(661\) 601813. 1.37740 0.688698 0.725049i \(-0.258183\pi\)
0.688698 + 0.725049i \(0.258183\pi\)
\(662\) 144073.i 0.328750i
\(663\) 0 0
\(664\) 883341. 2.00351
\(665\) −1.13628e6 −2.56946
\(666\) 0 0
\(667\) 17468.3i 0.0392644i
\(668\) 2.21244e6 4.95813
\(669\) 0 0
\(670\) −811628. −1.80804
\(671\) 21890.9 0.0486205
\(672\) 0 0
\(673\) 177803.i 0.392563i −0.980548 0.196282i \(-0.937113\pi\)
0.980548 0.196282i \(-0.0628867\pi\)
\(674\) −1.06655e6 −2.34780
\(675\) 0 0
\(676\) −763488. −1.67074
\(677\) −238228. −0.519774 −0.259887 0.965639i \(-0.583685\pi\)
−0.259887 + 0.965639i \(0.583685\pi\)
\(678\) 0 0
\(679\) 669585.i 1.45233i
\(680\) 2.13834e6i 4.62444i
\(681\) 0 0
\(682\) 89394.1i 0.192194i
\(683\) 660787.i 1.41651i −0.705956 0.708256i \(-0.749482\pi\)
0.705956 0.708256i \(-0.250518\pi\)
\(684\) 0 0
\(685\) 479988. 1.02294
\(686\) 273293.i 0.580737i
\(687\) 0 0
\(688\) 2.26990e6i 4.79545i
\(689\) 407945.i 0.859335i
\(690\) 0 0
\(691\) 252072.i 0.527919i 0.964534 + 0.263960i \(0.0850286\pi\)
−0.964534 + 0.263960i \(0.914971\pi\)
\(692\) 1.36903e6i 2.85891i
\(693\) 0 0
\(694\) −718594. −1.49199
\(695\) 1.02679e6 2.12574
\(696\) 0 0
\(697\) −282019. −0.580515
\(698\) −1.49477e6 −3.06805
\(699\) 0 0
\(700\) −621183. −1.26772
\(701\) 601181.i 1.22340i 0.791089 + 0.611701i \(0.209514\pi\)
−0.791089 + 0.611701i \(0.790486\pi\)
\(702\) 0 0
\(703\) 1.06259e6i 2.15008i
\(704\) 162951.i 0.328785i
\(705\) 0 0
\(706\) −1.04024e6 −2.08701
\(707\) 966205.i 1.93299i
\(708\) 0 0
\(709\) 769466. 1.53072 0.765362 0.643600i \(-0.222560\pi\)
0.765362 + 0.643600i \(0.222560\pi\)
\(710\) 584818.i 1.16012i
\(711\) 0 0
\(712\) 561105. 1.10684
\(713\) 48370.6 0.0951485
\(714\) 0 0
\(715\) −28504.7 −0.0557577
\(716\) 924096.i 1.80256i
\(717\) 0 0
\(718\) 493444.i 0.957171i
\(719\) 412760.i 0.798435i −0.916856 0.399218i \(-0.869282\pi\)
0.916856 0.399218i \(-0.130718\pi\)
\(720\) 0 0
\(721\) 1.04951e6i 2.01890i
\(722\) 1.31196e6i 2.51678i
\(723\) 0 0
\(724\) −1.23655e6 −2.35904
\(725\) 85343.8 0.162366
\(726\) 0 0
\(727\) −352870. −0.667645 −0.333822 0.942636i \(-0.608339\pi\)
−0.333822 + 0.942636i \(0.608339\pi\)
\(728\) −1.72071e6 −3.24673
\(729\) 0 0
\(730\) −1.98406e6 −3.72313
\(731\) 757015.i 1.41667i
\(732\) 0 0
\(733\) 566886. 1.05509 0.527543 0.849528i \(-0.323113\pi\)
0.527543 + 0.849528i \(0.323113\pi\)
\(734\) −488129. −0.906029
\(735\) 0 0
\(736\) 169996. 0.313821
\(737\) 34093.1 0.0627670
\(738\) 0 0
\(739\) 110960.i 0.203179i −0.994826 0.101589i \(-0.967607\pi\)
0.994826 0.101589i \(-0.0323928\pi\)
\(740\) 2.47162e6i 4.51354i
\(741\) 0 0
\(742\) 2.16022e6i 3.92364i
\(743\) 565178. 1.02378 0.511891 0.859050i \(-0.328945\pi\)
0.511891 + 0.859050i \(0.328945\pi\)
\(744\) 0 0
\(745\) 741904.i 1.33670i
\(746\) 796617.i 1.43144i
\(747\) 0 0
\(748\) 140255.i 0.250677i
\(749\) 71712.3 0.127829
\(750\) 0 0
\(751\) 217676.i 0.385950i −0.981204 0.192975i \(-0.938186\pi\)
0.981204 0.192975i \(-0.0618136\pi\)
\(752\) 3.96119e6i 7.00471i
\(753\) 0 0
\(754\) 369141. 0.649306
\(755\) 530657.i 0.930936i
\(756\) 0 0
\(757\) 169980. 0.296623 0.148312 0.988941i \(-0.452616\pi\)
0.148312 + 0.988941i \(0.452616\pi\)
\(758\) 896194.i 1.55978i
\(759\) 0 0
\(760\) 3.46431e6i 5.99777i
\(761\) −880285. −1.52004 −0.760019 0.649901i \(-0.774810\pi\)
−0.760019 + 0.649901i \(0.774810\pi\)
\(762\) 0 0
\(763\) 1.08570e6i 1.86493i
\(764\) 988836.i 1.69409i
\(765\) 0 0
\(766\) 534349.i 0.910684i
\(767\) −188449. + 320405.i −0.320333 + 0.544639i
\(768\) 0 0
\(769\) 619749.i 1.04801i 0.851717 + 0.524003i \(0.175562\pi\)
−0.851717 + 0.524003i \(0.824438\pi\)
\(770\) 150943. 0.254584
\(771\) 0 0
\(772\) −1.00267e6 −1.68238
\(773\) 1.11076e6i 1.85892i 0.368917 + 0.929462i \(0.379729\pi\)
−0.368917 + 0.929462i \(0.620271\pi\)
\(774\) 0 0
\(775\) 236321.i 0.393458i
\(776\) 2.04144e6 3.39011
\(777\) 0 0
\(778\) 15332.7i 0.0253313i
\(779\) 456898. 0.752912
\(780\) 0 0
\(781\) 24565.7i 0.0402743i
\(782\) −103179. −0.168725
\(783\) 0 0
\(784\) 2.91902e6 4.74903
\(785\) 1.32991e6i 2.15816i
\(786\) 0 0
\(787\) 243033. 0.392389 0.196194 0.980565i \(-0.437142\pi\)
0.196194 + 0.980565i \(0.437142\pi\)
\(788\) 495647. 0.798216
\(789\) 0 0
\(790\) 332936.i 0.533466i
\(791\) 1.44737e6i 2.31328i
\(792\) 0 0
\(793\) −250310. −0.398044
\(794\) 330105. 0.523613
\(795\) 0 0
\(796\) 1.24252e6 1.96100
\(797\) 419997.i