Properties

Label 50.6.b.b.49.2
Level $50$
Weight $6$
Character 50.49
Analytic conductor $8.019$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01919099065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.6.b.b.49.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +O(q^{10})\) \(q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +192.000 q^{11} +96.0000i q^{12} -1106.00i q^{13} +472.000 q^{14} +256.000 q^{16} +762.000i q^{17} +828.000i q^{18} +2740.00 q^{19} -708.000 q^{21} +768.000i q^{22} -1566.00i q^{23} -384.000 q^{24} +4424.00 q^{26} -2700.00i q^{27} +1888.00i q^{28} -5910.00 q^{29} -6868.00 q^{31} +1024.00i q^{32} -1152.00i q^{33} -3048.00 q^{34} -3312.00 q^{36} -5518.00i q^{37} +10960.0i q^{38} -6636.00 q^{39} -378.000 q^{41} -2832.00i q^{42} +2434.00i q^{43} -3072.00 q^{44} +6264.00 q^{46} +13122.0i q^{47} -1536.00i q^{48} +2883.00 q^{49} +4572.00 q^{51} +17696.0i q^{52} +9174.00i q^{53} +10800.0 q^{54} -7552.00 q^{56} -16440.0i q^{57} -23640.0i q^{58} +34980.0 q^{59} -9838.00 q^{61} -27472.0i q^{62} -24426.0i q^{63} -4096.00 q^{64} +4608.00 q^{66} +33722.0i q^{67} -12192.0i q^{68} -9396.00 q^{69} +70212.0 q^{71} -13248.0i q^{72} -21986.0i q^{73} +22072.0 q^{74} -43840.0 q^{76} -22656.0i q^{77} -26544.0i q^{78} -4520.00 q^{79} +34101.0 q^{81} -1512.00i q^{82} +109074. i q^{83} +11328.0 q^{84} -9736.00 q^{86} +35460.0i q^{87} -12288.0i q^{88} -38490.0 q^{89} -130508. q^{91} +25056.0i q^{92} +41208.0i q^{93} -52488.0 q^{94} +6144.00 q^{96} -1918.00i q^{97} +11532.0i q^{98} +39744.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + 48q^{6} + 414q^{9} + O(q^{10}) \) \( 2q - 32q^{4} + 48q^{6} + 414q^{9} + 384q^{11} + 944q^{14} + 512q^{16} + 5480q^{19} - 1416q^{21} - 768q^{24} + 8848q^{26} - 11820q^{29} - 13736q^{31} - 6096q^{34} - 6624q^{36} - 13272q^{39} - 756q^{41} - 6144q^{44} + 12528q^{46} + 5766q^{49} + 9144q^{51} + 21600q^{54} - 15104q^{56} + 69960q^{59} - 19676q^{61} - 8192q^{64} + 9216q^{66} - 18792q^{69} + 140424q^{71} + 44144q^{74} - 87680q^{76} - 9040q^{79} + 68202q^{81} + 22656q^{84} - 19472q^{86} - 76980q^{89} - 261016q^{91} - 104976q^{94} + 12288q^{96} + 79488q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 0.707107i
\(3\) − 6.00000i − 0.384900i −0.981307 0.192450i \(-0.938357\pi\)
0.981307 0.192450i \(-0.0616434\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 24.0000 0.272166
\(7\) − 118.000i − 0.910200i −0.890440 0.455100i \(-0.849603\pi\)
0.890440 0.455100i \(-0.150397\pi\)
\(8\) − 64.0000i − 0.353553i
\(9\) 207.000 0.851852
\(10\) 0 0
\(11\) 192.000 0.478431 0.239216 0.970966i \(-0.423110\pi\)
0.239216 + 0.970966i \(0.423110\pi\)
\(12\) 96.0000i 0.192450i
\(13\) − 1106.00i − 1.81508i −0.419961 0.907542i \(-0.637956\pi\)
0.419961 0.907542i \(-0.362044\pi\)
\(14\) 472.000 0.643609
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 762.000i 0.639488i 0.947504 + 0.319744i \(0.103597\pi\)
−0.947504 + 0.319744i \(0.896403\pi\)
\(18\) 828.000i 0.602350i
\(19\) 2740.00 1.74127 0.870636 0.491928i \(-0.163708\pi\)
0.870636 + 0.491928i \(0.163708\pi\)
\(20\) 0 0
\(21\) −708.000 −0.350336
\(22\) 768.000i 0.338302i
\(23\) − 1566.00i − 0.617266i −0.951181 0.308633i \(-0.900129\pi\)
0.951181 0.308633i \(-0.0998714\pi\)
\(24\) −384.000 −0.136083
\(25\) 0 0
\(26\) 4424.00 1.28346
\(27\) − 2700.00i − 0.712778i
\(28\) 1888.00i 0.455100i
\(29\) −5910.00 −1.30495 −0.652473 0.757812i \(-0.726268\pi\)
−0.652473 + 0.757812i \(0.726268\pi\)
\(30\) 0 0
\(31\) −6868.00 −1.28359 −0.641795 0.766877i \(-0.721810\pi\)
−0.641795 + 0.766877i \(0.721810\pi\)
\(32\) 1024.00i 0.176777i
\(33\) − 1152.00i − 0.184148i
\(34\) −3048.00 −0.452187
\(35\) 0 0
\(36\) −3312.00 −0.425926
\(37\) − 5518.00i − 0.662640i −0.943519 0.331320i \(-0.892506\pi\)
0.943519 0.331320i \(-0.107494\pi\)
\(38\) 10960.0i 1.23127i
\(39\) −6636.00 −0.698626
\(40\) 0 0
\(41\) −378.000 −0.0351182 −0.0175591 0.999846i \(-0.505590\pi\)
−0.0175591 + 0.999846i \(0.505590\pi\)
\(42\) − 2832.00i − 0.247725i
\(43\) 2434.00i 0.200747i 0.994950 + 0.100374i \(0.0320038\pi\)
−0.994950 + 0.100374i \(0.967996\pi\)
\(44\) −3072.00 −0.239216
\(45\) 0 0
\(46\) 6264.00 0.436473
\(47\) 13122.0i 0.866474i 0.901280 + 0.433237i \(0.142629\pi\)
−0.901280 + 0.433237i \(0.857371\pi\)
\(48\) − 1536.00i − 0.0962250i
\(49\) 2883.00 0.171536
\(50\) 0 0
\(51\) 4572.00 0.246139
\(52\) 17696.0i 0.907542i
\(53\) 9174.00i 0.448610i 0.974519 + 0.224305i \(0.0720112\pi\)
−0.974519 + 0.224305i \(0.927989\pi\)
\(54\) 10800.0 0.504010
\(55\) 0 0
\(56\) −7552.00 −0.321804
\(57\) − 16440.0i − 0.670216i
\(58\) − 23640.0i − 0.922736i
\(59\) 34980.0 1.30825 0.654124 0.756388i \(-0.273038\pi\)
0.654124 + 0.756388i \(0.273038\pi\)
\(60\) 0 0
\(61\) −9838.00 −0.338518 −0.169259 0.985572i \(-0.554137\pi\)
−0.169259 + 0.985572i \(0.554137\pi\)
\(62\) − 27472.0i − 0.907635i
\(63\) − 24426.0i − 0.775356i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 4608.00 0.130212
\(67\) 33722.0i 0.917754i 0.888500 + 0.458877i \(0.151748\pi\)
−0.888500 + 0.458877i \(0.848252\pi\)
\(68\) − 12192.0i − 0.319744i
\(69\) −9396.00 −0.237586
\(70\) 0 0
\(71\) 70212.0 1.65297 0.826486 0.562957i \(-0.190336\pi\)
0.826486 + 0.562957i \(0.190336\pi\)
\(72\) − 13248.0i − 0.301175i
\(73\) − 21986.0i − 0.482880i −0.970416 0.241440i \(-0.922380\pi\)
0.970416 0.241440i \(-0.0776197\pi\)
\(74\) 22072.0 0.468557
\(75\) 0 0
\(76\) −43840.0 −0.870636
\(77\) − 22656.0i − 0.435468i
\(78\) − 26544.0i − 0.494003i
\(79\) −4520.00 −0.0814837 −0.0407418 0.999170i \(-0.512972\pi\)
−0.0407418 + 0.999170i \(0.512972\pi\)
\(80\) 0 0
\(81\) 34101.0 0.577503
\(82\) − 1512.00i − 0.0248323i
\(83\) 109074.i 1.73790i 0.494896 + 0.868952i \(0.335206\pi\)
−0.494896 + 0.868952i \(0.664794\pi\)
\(84\) 11328.0 0.175168
\(85\) 0 0
\(86\) −9736.00 −0.141950
\(87\) 35460.0i 0.502274i
\(88\) − 12288.0i − 0.169151i
\(89\) −38490.0 −0.515078 −0.257539 0.966268i \(-0.582912\pi\)
−0.257539 + 0.966268i \(0.582912\pi\)
\(90\) 0 0
\(91\) −130508. −1.65209
\(92\) 25056.0i 0.308633i
\(93\) 41208.0i 0.494054i
\(94\) −52488.0 −0.612689
\(95\) 0 0
\(96\) 6144.00 0.0680414
\(97\) − 1918.00i − 0.0206976i −0.999946 0.0103488i \(-0.996706\pi\)
0.999946 0.0103488i \(-0.00329418\pi\)
\(98\) 11532.0i 0.121294i
\(99\) 39744.0 0.407553
\(100\) 0 0
\(101\) 77622.0 0.757149 0.378575 0.925571i \(-0.376414\pi\)
0.378575 + 0.925571i \(0.376414\pi\)
\(102\) 18288.0i 0.174047i
\(103\) 46714.0i 0.433864i 0.976187 + 0.216932i \(0.0696051\pi\)
−0.976187 + 0.216932i \(0.930395\pi\)
\(104\) −70784.0 −0.641729
\(105\) 0 0
\(106\) −36696.0 −0.317215
\(107\) − 1038.00i − 0.00876472i −0.999990 0.00438236i \(-0.998605\pi\)
0.999990 0.00438236i \(-0.00139495\pi\)
\(108\) 43200.0i 0.356389i
\(109\) −206930. −1.66823 −0.834117 0.551587i \(-0.814023\pi\)
−0.834117 + 0.551587i \(0.814023\pi\)
\(110\) 0 0
\(111\) −33108.0 −0.255050
\(112\) − 30208.0i − 0.227550i
\(113\) − 139386.i − 1.02689i −0.858123 0.513444i \(-0.828369\pi\)
0.858123 0.513444i \(-0.171631\pi\)
\(114\) 65760.0 0.473914
\(115\) 0 0
\(116\) 94560.0 0.652473
\(117\) − 228942.i − 1.54618i
\(118\) 139920.i 0.925070i
\(119\) 89916.0 0.582062
\(120\) 0 0
\(121\) −124187. −0.771104
\(122\) − 39352.0i − 0.239369i
\(123\) 2268.00i 0.0135170i
\(124\) 109888. 0.641795
\(125\) 0 0
\(126\) 97704.0 0.548259
\(127\) 299882.i 1.64984i 0.565252 + 0.824919i \(0.308779\pi\)
−0.565252 + 0.824919i \(0.691221\pi\)
\(128\) − 16384.0i − 0.0883883i
\(129\) 14604.0 0.0772676
\(130\) 0 0
\(131\) 7872.00 0.0400781 0.0200390 0.999799i \(-0.493621\pi\)
0.0200390 + 0.999799i \(0.493621\pi\)
\(132\) 18432.0i 0.0920741i
\(133\) − 323320.i − 1.58491i
\(134\) −134888. −0.648950
\(135\) 0 0
\(136\) 48768.0 0.226093
\(137\) − 164238.i − 0.747605i −0.927508 0.373803i \(-0.878054\pi\)
0.927508 0.373803i \(-0.121946\pi\)
\(138\) − 37584.0i − 0.167998i
\(139\) 282100. 1.23841 0.619207 0.785228i \(-0.287454\pi\)
0.619207 + 0.785228i \(0.287454\pi\)
\(140\) 0 0
\(141\) 78732.0 0.333506
\(142\) 280848.i 1.16883i
\(143\) − 212352.i − 0.868393i
\(144\) 52992.0 0.212963
\(145\) 0 0
\(146\) 87944.0 0.341448
\(147\) − 17298.0i − 0.0660241i
\(148\) 88288.0i 0.331320i
\(149\) 388950. 1.43525 0.717626 0.696429i \(-0.245229\pi\)
0.717626 + 0.696429i \(0.245229\pi\)
\(150\) 0 0
\(151\) −97948.0 −0.349585 −0.174793 0.984605i \(-0.555926\pi\)
−0.174793 + 0.984605i \(0.555926\pi\)
\(152\) − 175360.i − 0.615633i
\(153\) 157734.i 0.544749i
\(154\) 90624.0 0.307923
\(155\) 0 0
\(156\) 106176. 0.349313
\(157\) − 3718.00i − 0.0120382i −0.999982 0.00601908i \(-0.998084\pi\)
0.999982 0.00601908i \(-0.00191594\pi\)
\(158\) − 18080.0i − 0.0576177i
\(159\) 55044.0 0.172670
\(160\) 0 0
\(161\) −184788. −0.561835
\(162\) 136404.i 0.408357i
\(163\) 43234.0i 0.127455i 0.997967 + 0.0637274i \(0.0202988\pi\)
−0.997967 + 0.0637274i \(0.979701\pi\)
\(164\) 6048.00 0.0175591
\(165\) 0 0
\(166\) −436296. −1.22888
\(167\) 186522.i 0.517534i 0.965940 + 0.258767i \(0.0833162\pi\)
−0.965940 + 0.258767i \(0.916684\pi\)
\(168\) 45312.0i 0.123863i
\(169\) −851943. −2.29453
\(170\) 0 0
\(171\) 567180. 1.48331
\(172\) − 38944.0i − 0.100374i
\(173\) 374454.i 0.951225i 0.879655 + 0.475612i \(0.157774\pi\)
−0.879655 + 0.475612i \(0.842226\pi\)
\(174\) −141840. −0.355161
\(175\) 0 0
\(176\) 49152.0 0.119608
\(177\) − 209880.i − 0.503545i
\(178\) − 153960.i − 0.364215i
\(179\) −272100. −0.634740 −0.317370 0.948302i \(-0.602800\pi\)
−0.317370 + 0.948302i \(0.602800\pi\)
\(180\) 0 0
\(181\) −75418.0 −0.171111 −0.0855556 0.996333i \(-0.527267\pi\)
−0.0855556 + 0.996333i \(0.527267\pi\)
\(182\) − 522032.i − 1.16820i
\(183\) 59028.0i 0.130296i
\(184\) −100224. −0.218236
\(185\) 0 0
\(186\) −164832. −0.349349
\(187\) 146304.i 0.305951i
\(188\) − 209952.i − 0.433237i
\(189\) −318600. −0.648771
\(190\) 0 0
\(191\) −356988. −0.708060 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(192\) 24576.0i 0.0481125i
\(193\) 438694.i 0.847751i 0.905720 + 0.423876i \(0.139331\pi\)
−0.905720 + 0.423876i \(0.860669\pi\)
\(194\) 7672.00 0.0146354
\(195\) 0 0
\(196\) −46128.0 −0.0857678
\(197\) − 156798.i − 0.287856i −0.989588 0.143928i \(-0.954027\pi\)
0.989588 0.143928i \(-0.0459733\pi\)
\(198\) 158976.i 0.288183i
\(199\) 162520. 0.290920 0.145460 0.989364i \(-0.453534\pi\)
0.145460 + 0.989364i \(0.453534\pi\)
\(200\) 0 0
\(201\) 202332. 0.353244
\(202\) 310488.i 0.535385i
\(203\) 697380.i 1.18776i
\(204\) −73152.0 −0.123070
\(205\) 0 0
\(206\) −186856. −0.306788
\(207\) − 324162.i − 0.525819i
\(208\) − 283136.i − 0.453771i
\(209\) 526080. 0.833079
\(210\) 0 0
\(211\) −181648. −0.280882 −0.140441 0.990089i \(-0.544852\pi\)
−0.140441 + 0.990089i \(0.544852\pi\)
\(212\) − 146784.i − 0.224305i
\(213\) − 421272.i − 0.636229i
\(214\) 4152.00 0.00619759
\(215\) 0 0
\(216\) −172800. −0.252005
\(217\) 810424.i 1.16832i
\(218\) − 827720.i − 1.17962i
\(219\) −131916. −0.185861
\(220\) 0 0
\(221\) 842772. 1.16073
\(222\) − 132432.i − 0.180348i
\(223\) 288274.i 0.388189i 0.980983 + 0.194095i \(0.0621769\pi\)
−0.980983 + 0.194095i \(0.937823\pi\)
\(224\) 120832. 0.160902
\(225\) 0 0
\(226\) 557544. 0.726119
\(227\) 1.12552e6i 1.44974i 0.688887 + 0.724869i \(0.258100\pi\)
−0.688887 + 0.724869i \(0.741900\pi\)
\(228\) 263040.i 0.335108i
\(229\) 415810. 0.523970 0.261985 0.965072i \(-0.415623\pi\)
0.261985 + 0.965072i \(0.415623\pi\)
\(230\) 0 0
\(231\) −135936. −0.167612
\(232\) 378240.i 0.461368i
\(233\) − 770586.i − 0.929889i −0.885340 0.464945i \(-0.846074\pi\)
0.885340 0.464945i \(-0.153926\pi\)
\(234\) 915768. 1.09332
\(235\) 0 0
\(236\) −559680. −0.654124
\(237\) 27120.0i 0.0313631i
\(238\) 359664.i 0.411580i
\(239\) 595320. 0.674149 0.337074 0.941478i \(-0.390563\pi\)
0.337074 + 0.941478i \(0.390563\pi\)
\(240\) 0 0
\(241\) 273902. 0.303775 0.151888 0.988398i \(-0.451465\pi\)
0.151888 + 0.988398i \(0.451465\pi\)
\(242\) − 496748.i − 0.545253i
\(243\) − 860706.i − 0.935059i
\(244\) 157408. 0.169259
\(245\) 0 0
\(246\) −9072.00 −0.00955796
\(247\) − 3.03044e6i − 3.16055i
\(248\) 439552.i 0.453817i
\(249\) 654444. 0.668920
\(250\) 0 0
\(251\) 850752. 0.852351 0.426176 0.904640i \(-0.359861\pi\)
0.426176 + 0.904640i \(0.359861\pi\)
\(252\) 390816.i 0.387678i
\(253\) − 300672.i − 0.295319i
\(254\) −1.19953e6 −1.16661
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 825402.i 0.779530i 0.920914 + 0.389765i \(0.127444\pi\)
−0.920914 + 0.389765i \(0.872556\pi\)
\(258\) 58416.0i 0.0546365i
\(259\) −651124. −0.603135
\(260\) 0 0
\(261\) −1.22337e6 −1.11162
\(262\) 31488.0i 0.0283395i
\(263\) − 1.36465e6i − 1.21655i −0.793726 0.608276i \(-0.791861\pi\)
0.793726 0.608276i \(-0.208139\pi\)
\(264\) −73728.0 −0.0651062
\(265\) 0 0
\(266\) 1.29328e6 1.12070
\(267\) 230940.i 0.198254i
\(268\) − 539552.i − 0.458877i
\(269\) 113310. 0.0954745 0.0477373 0.998860i \(-0.484799\pi\)
0.0477373 + 0.998860i \(0.484799\pi\)
\(270\) 0 0
\(271\) −849628. −0.702758 −0.351379 0.936233i \(-0.614287\pi\)
−0.351379 + 0.936233i \(0.614287\pi\)
\(272\) 195072.i 0.159872i
\(273\) 783048.i 0.635890i
\(274\) 656952. 0.528637
\(275\) 0 0
\(276\) 150336. 0.118793
\(277\) 438602.i 0.343456i 0.985144 + 0.171728i \(0.0549350\pi\)
−0.985144 + 0.171728i \(0.945065\pi\)
\(278\) 1.12840e6i 0.875691i
\(279\) −1.42168e6 −1.09343
\(280\) 0 0
\(281\) −1.45670e6 −1.10053 −0.550267 0.834989i \(-0.685474\pi\)
−0.550267 + 0.834989i \(0.685474\pi\)
\(282\) 314928.i 0.235824i
\(283\) 120394.i 0.0893591i 0.999001 + 0.0446795i \(0.0142267\pi\)
−0.999001 + 0.0446795i \(0.985773\pi\)
\(284\) −1.12339e6 −0.826486
\(285\) 0 0
\(286\) 849408. 0.614047
\(287\) 44604.0i 0.0319646i
\(288\) 211968.i 0.150588i
\(289\) 839213. 0.591055
\(290\) 0 0
\(291\) −11508.0 −0.00796650
\(292\) 351776.i 0.241440i
\(293\) 2.64209e6i 1.79796i 0.437993 + 0.898978i \(0.355689\pi\)
−0.437993 + 0.898978i \(0.644311\pi\)
\(294\) 69192.0 0.0466861
\(295\) 0 0
\(296\) −353152. −0.234278
\(297\) − 518400.i − 0.341015i
\(298\) 1.55580e6i 1.01488i
\(299\) −1.73200e6 −1.12039
\(300\) 0 0
\(301\) 287212. 0.182720
\(302\) − 391792.i − 0.247194i
\(303\) − 465732.i − 0.291427i
\(304\) 701440. 0.435318
\(305\) 0 0
\(306\) −630936. −0.385196
\(307\) − 1.44756e6i − 0.876577i −0.898834 0.438288i \(-0.855585\pi\)
0.898834 0.438288i \(-0.144415\pi\)
\(308\) 362496.i 0.217734i
\(309\) 280284. 0.166994
\(310\) 0 0
\(311\) −928068. −0.544100 −0.272050 0.962283i \(-0.587702\pi\)
−0.272050 + 0.962283i \(0.587702\pi\)
\(312\) 424704.i 0.247002i
\(313\) − 2.29563e6i − 1.32446i −0.749299 0.662232i \(-0.769609\pi\)
0.749299 0.662232i \(-0.230391\pi\)
\(314\) 14872.0 0.00851227
\(315\) 0 0
\(316\) 72320.0 0.0407418
\(317\) 2.73652e6i 1.52950i 0.644324 + 0.764752i \(0.277139\pi\)
−0.644324 + 0.764752i \(0.722861\pi\)
\(318\) 220176.i 0.122096i
\(319\) −1.13472e6 −0.624327
\(320\) 0 0
\(321\) −6228.00 −0.00337354
\(322\) − 739152.i − 0.397278i
\(323\) 2.08788e6i 1.11352i
\(324\) −545616. −0.288752
\(325\) 0 0
\(326\) −172936. −0.0901242
\(327\) 1.24158e6i 0.642104i
\(328\) 24192.0i 0.0124162i
\(329\) 1.54840e6 0.788665
\(330\) 0 0
\(331\) 3.81879e6 1.91583 0.957913 0.287059i \(-0.0926776\pi\)
0.957913 + 0.287059i \(0.0926776\pi\)
\(332\) − 1.74518e6i − 0.868952i
\(333\) − 1.14223e6i − 0.564471i
\(334\) −746088. −0.365952
\(335\) 0 0
\(336\) −181248. −0.0875841
\(337\) − 2.21088e6i − 1.06045i −0.847857 0.530225i \(-0.822108\pi\)
0.847857 0.530225i \(-0.177892\pi\)
\(338\) − 3.40777e6i − 1.62248i
\(339\) −836316. −0.395249
\(340\) 0 0
\(341\) −1.31866e6 −0.614109
\(342\) 2.26872e6i 1.04886i
\(343\) − 2.32342e6i − 1.06633i
\(344\) 155776. 0.0709748
\(345\) 0 0
\(346\) −1.49782e6 −0.672618
\(347\) − 2.32724e6i − 1.03757i −0.854905 0.518785i \(-0.826385\pi\)
0.854905 0.518785i \(-0.173615\pi\)
\(348\) − 567360.i − 0.251137i
\(349\) 311290. 0.136805 0.0684024 0.997658i \(-0.478210\pi\)
0.0684024 + 0.997658i \(0.478210\pi\)
\(350\) 0 0
\(351\) −2.98620e6 −1.29375
\(352\) 196608.i 0.0845755i
\(353\) 3.08657e6i 1.31838i 0.751977 + 0.659189i \(0.229100\pi\)
−0.751977 + 0.659189i \(0.770900\pi\)
\(354\) 839520. 0.356060
\(355\) 0 0
\(356\) 615840. 0.257539
\(357\) − 539496.i − 0.224036i
\(358\) − 1.08840e6i − 0.448829i
\(359\) 3.53076e6 1.44588 0.722940 0.690911i \(-0.242790\pi\)
0.722940 + 0.690911i \(0.242790\pi\)
\(360\) 0 0
\(361\) 5.03150e6 2.03203
\(362\) − 301672.i − 0.120994i
\(363\) 745122.i 0.296798i
\(364\) 2.08813e6 0.826045
\(365\) 0 0
\(366\) −236112. −0.0921330
\(367\) 35762.0i 0.0138598i 0.999976 + 0.00692989i \(0.00220587\pi\)
−0.999976 + 0.00692989i \(0.997794\pi\)
\(368\) − 400896.i − 0.154316i
\(369\) −78246.0 −0.0299155
\(370\) 0 0
\(371\) 1.08253e6 0.408325
\(372\) − 659328.i − 0.247027i
\(373\) 1.71525e6i 0.638346i 0.947696 + 0.319173i \(0.103405\pi\)
−0.947696 + 0.319173i \(0.896595\pi\)
\(374\) −585216. −0.216340
\(375\) 0 0
\(376\) 839808. 0.306345
\(377\) 6.53646e6i 2.36859i
\(378\) − 1.27440e6i − 0.458750i
\(379\) 3.10174e6 1.10919 0.554597 0.832119i \(-0.312873\pi\)
0.554597 + 0.832119i \(0.312873\pi\)
\(380\) 0 0
\(381\) 1.79929e6 0.635023
\(382\) − 1.42795e6i − 0.500674i
\(383\) − 5.31949e6i − 1.85299i −0.376309 0.926494i \(-0.622807\pi\)
0.376309 0.926494i \(-0.377193\pi\)
\(384\) −98304.0 −0.0340207
\(385\) 0 0
\(386\) −1.75478e6 −0.599451
\(387\) 503838.i 0.171007i
\(388\) 30688.0i 0.0103488i
\(389\) −1.16145e6 −0.389158 −0.194579 0.980887i \(-0.562334\pi\)
−0.194579 + 0.980887i \(0.562334\pi\)
\(390\) 0 0
\(391\) 1.19329e6 0.394734
\(392\) − 184512.i − 0.0606470i
\(393\) − 47232.0i − 0.0154261i
\(394\) 627192. 0.203545
\(395\) 0 0
\(396\) −635904. −0.203776
\(397\) 628562.i 0.200157i 0.994980 + 0.100079i \(0.0319095\pi\)
−0.994980 + 0.100079i \(0.968091\pi\)
\(398\) 650080.i 0.205712i
\(399\) −1.93992e6 −0.610031
\(400\) 0 0
\(401\) −2.72432e6 −0.846052 −0.423026 0.906118i \(-0.639032\pi\)
−0.423026 + 0.906118i \(0.639032\pi\)
\(402\) 809328.i 0.249781i
\(403\) 7.59601e6i 2.32982i
\(404\) −1.24195e6 −0.378575
\(405\) 0 0
\(406\) −2.78952e6 −0.839875
\(407\) − 1.05946e6i − 0.317027i
\(408\) − 292608.i − 0.0870233i
\(409\) −1.78019e6 −0.526209 −0.263104 0.964767i \(-0.584746\pi\)
−0.263104 + 0.964767i \(0.584746\pi\)
\(410\) 0 0
\(411\) −985428. −0.287753
\(412\) − 747424.i − 0.216932i
\(413\) − 4.12764e6i − 1.19077i
\(414\) 1.29665e6 0.371810
\(415\) 0 0
\(416\) 1.13254e6 0.320865
\(417\) − 1.69260e6i − 0.476666i
\(418\) 2.10432e6i 0.589076i
\(419\) −650580. −0.181036 −0.0905181 0.995895i \(-0.528852\pi\)
−0.0905181 + 0.995895i \(0.528852\pi\)
\(420\) 0 0
\(421\) −3.54060e6 −0.973579 −0.486790 0.873519i \(-0.661832\pi\)
−0.486790 + 0.873519i \(0.661832\pi\)
\(422\) − 726592.i − 0.198614i
\(423\) 2.71625e6i 0.738107i
\(424\) 587136. 0.158608
\(425\) 0 0
\(426\) 1.68509e6 0.449882
\(427\) 1.16088e6i 0.308119i
\(428\) 16608.0i 0.00438236i
\(429\) −1.27411e6 −0.334245
\(430\) 0 0
\(431\) −548748. −0.142292 −0.0711459 0.997466i \(-0.522666\pi\)
−0.0711459 + 0.997466i \(0.522666\pi\)
\(432\) − 691200.i − 0.178195i
\(433\) 1.49241e6i 0.382534i 0.981538 + 0.191267i \(0.0612596\pi\)
−0.981538 + 0.191267i \(0.938740\pi\)
\(434\) −3.24170e6 −0.826129
\(435\) 0 0
\(436\) 3.31088e6 0.834117
\(437\) − 4.29084e6i − 1.07483i
\(438\) − 527664.i − 0.131423i
\(439\) −4.86212e6 −1.20411 −0.602053 0.798456i \(-0.705650\pi\)
−0.602053 + 0.798456i \(0.705650\pi\)
\(440\) 0 0
\(441\) 596781. 0.146123
\(442\) 3.37109e6i 0.820757i
\(443\) 1.86155e6i 0.450678i 0.974280 + 0.225339i \(0.0723490\pi\)
−0.974280 + 0.225339i \(0.927651\pi\)
\(444\) 529728. 0.127525
\(445\) 0 0
\(446\) −1.15310e6 −0.274491
\(447\) − 2.33370e6i − 0.552429i
\(448\) 483328.i 0.113775i
\(449\) −3.73719e6 −0.874841 −0.437421 0.899257i \(-0.644108\pi\)
−0.437421 + 0.899257i \(0.644108\pi\)
\(450\) 0 0
\(451\) −72576.0 −0.0168016
\(452\) 2.23018e6i 0.513444i
\(453\) 587688.i 0.134555i
\(454\) −4.50209e6 −1.02512
\(455\) 0 0
\(456\) −1.05216e6 −0.236957
\(457\) − 6.48276e6i − 1.45201i −0.687690 0.726005i \(-0.741375\pi\)
0.687690 0.726005i \(-0.258625\pi\)
\(458\) 1.66324e6i 0.370503i
\(459\) 2.05740e6 0.455813
\(460\) 0 0
\(461\) 1.50910e6 0.330724 0.165362 0.986233i \(-0.447121\pi\)
0.165362 + 0.986233i \(0.447121\pi\)
\(462\) − 543744.i − 0.118519i
\(463\) − 8.68401e6i − 1.88264i −0.337513 0.941321i \(-0.609586\pi\)
0.337513 0.941321i \(-0.390414\pi\)
\(464\) −1.51296e6 −0.326236
\(465\) 0 0
\(466\) 3.08234e6 0.657531
\(467\) 6.96412e6i 1.47766i 0.673893 + 0.738829i \(0.264621\pi\)
−0.673893 + 0.738829i \(0.735379\pi\)
\(468\) 3.66307e6i 0.773091i
\(469\) 3.97920e6 0.835340
\(470\) 0 0
\(471\) −22308.0 −0.00463349
\(472\) − 2.23872e6i − 0.462535i
\(473\) 467328.i 0.0960437i
\(474\) −108480. −0.0221771
\(475\) 0 0
\(476\) −1.43866e6 −0.291031
\(477\) 1.89902e6i 0.382149i
\(478\) 2.38128e6i 0.476695i
\(479\) 5.51052e6 1.09737 0.548686 0.836029i \(-0.315128\pi\)
0.548686 + 0.836029i \(0.315128\pi\)
\(480\) 0 0
\(481\) −6.10291e6 −1.20275
\(482\) 1.09561e6i 0.214802i
\(483\) 1.10873e6i 0.216251i
\(484\) 1.98699e6 0.385552
\(485\) 0 0
\(486\) 3.44282e6 0.661187
\(487\) 5.51808e6i 1.05430i 0.849771 + 0.527152i \(0.176740\pi\)
−0.849771 + 0.527152i \(0.823260\pi\)
\(488\) 629632.i 0.119684i
\(489\) 259404. 0.0490574
\(490\) 0 0
\(491\) −1.51277e6 −0.283184 −0.141592 0.989925i \(-0.545222\pi\)
−0.141592 + 0.989925i \(0.545222\pi\)
\(492\) − 36288.0i − 0.00675850i
\(493\) − 4.50342e6i − 0.834498i
\(494\) 1.21218e7 2.23485
\(495\) 0 0
\(496\) −1.75821e6 −0.320897
\(497\) − 8.28502e6i − 1.50454i
\(498\) 2.61778e6i 0.472998i
\(499\) 1.93042e6 0.347057 0.173528 0.984829i \(-0.444483\pi\)
0.173528 + 0.984829i \(0.444483\pi\)
\(500\) 0 0
\(501\) 1.11913e6 0.199199
\(502\) 3.40301e6i 0.602703i
\(503\) − 6.73105e6i − 1.18621i −0.805124 0.593106i \(-0.797901\pi\)
0.805124 0.593106i \(-0.202099\pi\)
\(504\) −1.56326e6 −0.274130
\(505\) 0 0
\(506\) 1.20269e6 0.208822
\(507\) 5.11166e6i 0.883165i
\(508\) − 4.79811e6i − 0.824919i
\(509\) 556650. 0.0952331 0.0476165 0.998866i \(-0.484837\pi\)
0.0476165 + 0.998866i \(0.484837\pi\)
\(510\) 0 0
\(511\) −2.59435e6 −0.439517
\(512\) 262144.i 0.0441942i
\(513\) − 7.39800e6i − 1.24114i
\(514\) −3.30161e6 −0.551211
\(515\) 0 0
\(516\) −233664. −0.0386338
\(517\) 2.51942e6i 0.414548i
\(518\) − 2.60450e6i − 0.426481i
\(519\) 2.24672e6 0.366127
\(520\) 0 0
\(521\) 1.01110e7 1.63192 0.815962 0.578106i \(-0.196208\pi\)
0.815962 + 0.578106i \(0.196208\pi\)
\(522\) − 4.89348e6i − 0.786034i
\(523\) 7.03719e6i 1.12498i 0.826804 + 0.562491i \(0.190157\pi\)
−0.826804 + 0.562491i \(0.809843\pi\)
\(524\) −125952. −0.0200390
\(525\) 0 0
\(526\) 5.45858e6 0.860232
\(527\) − 5.23342e6i − 0.820840i
\(528\) − 294912.i − 0.0460371i
\(529\) 3.98399e6 0.618983
\(530\) 0 0
\(531\) 7.24086e6 1.11443
\(532\) 5.17312e6i 0.792453i
\(533\) 418068.i 0.0637425i
\(534\) −923760. −0.140186
\(535\) 0 0
\(536\) 2.15821e6 0.324475
\(537\) 1.63260e6i 0.244312i
\(538\) 453240.i 0.0675107i
\(539\) 553536. 0.0820680
\(540\) 0 0
\(541\) −4.23114e6 −0.621533 −0.310766 0.950486i \(-0.600586\pi\)
−0.310766 + 0.950486i \(0.600586\pi\)
\(542\) − 3.39851e6i − 0.496925i
\(543\) 452508.i 0.0658608i
\(544\) −780288. −0.113047
\(545\) 0 0
\(546\) −3.13219e6 −0.449642
\(547\) 4.44024e6i 0.634510i 0.948340 + 0.317255i \(0.102761\pi\)
−0.948340 + 0.317255i \(0.897239\pi\)
\(548\) 2.62781e6i 0.373803i
\(549\) −2.03647e6 −0.288367
\(550\) 0 0
\(551\) −1.61934e7 −2.27227
\(552\) 601344.i 0.0839992i
\(553\) 533360.i 0.0741665i
\(554\) −1.75441e6 −0.242860
\(555\) 0 0
\(556\) −4.51360e6 −0.619207
\(557\) − 9.01448e6i − 1.23113i −0.788088 0.615563i \(-0.788929\pi\)
0.788088 0.615563i \(-0.211071\pi\)
\(558\) − 5.68670e6i − 0.773170i
\(559\) 2.69200e6 0.364373
\(560\) 0 0
\(561\) 877824. 0.117761
\(562\) − 5.82679e6i − 0.778196i
\(563\) 9.81287e6i 1.30474i 0.757899 + 0.652372i \(0.226226\pi\)
−0.757899 + 0.652372i \(0.773774\pi\)
\(564\) −1.25971e6 −0.166753
\(565\) 0 0
\(566\) −481576. −0.0631864
\(567\) − 4.02392e6i − 0.525644i
\(568\) − 4.49357e6i − 0.584414i
\(569\) −1.33152e7 −1.72412 −0.862061 0.506804i \(-0.830827\pi\)
−0.862061 + 0.506804i \(0.830827\pi\)
\(570\) 0 0
\(571\) 9.95895e6 1.27827 0.639136 0.769094i \(-0.279292\pi\)
0.639136 + 0.769094i \(0.279292\pi\)
\(572\) 3.39763e6i 0.434196i
\(573\) 2.14193e6i 0.272533i
\(574\) −178416. −0.0226024
\(575\) 0 0
\(576\) −847872. −0.106481
\(577\) 4.50372e6i 0.563160i 0.959538 + 0.281580i \(0.0908585\pi\)
−0.959538 + 0.281580i \(0.909141\pi\)
\(578\) 3.35685e6i 0.417939i
\(579\) 2.63216e6 0.326300
\(580\) 0 0
\(581\) 1.28707e7 1.58184
\(582\) − 46032.0i − 0.00563316i
\(583\) 1.76141e6i 0.214629i
\(584\) −1.40710e6 −0.170724
\(585\) 0 0
\(586\) −1.05684e7 −1.27135
\(587\) 625842.i 0.0749669i 0.999297 + 0.0374834i \(0.0119341\pi\)
−0.999297 + 0.0374834i \(0.988066\pi\)
\(588\) 276768.i 0.0330121i
\(589\) −1.88183e7 −2.23508
\(590\) 0 0
\(591\) −940788. −0.110796
\(592\) − 1.41261e6i − 0.165660i
\(593\) 2.50385e6i 0.292397i 0.989255 + 0.146198i \(0.0467038\pi\)
−0.989255 + 0.146198i \(0.953296\pi\)
\(594\) 2.07360e6 0.241134
\(595\) 0 0
\(596\) −6.22320e6 −0.717626
\(597\) − 975120.i − 0.111975i
\(598\) − 6.92798e6i − 0.792235i
\(599\) 756480. 0.0861451 0.0430725 0.999072i \(-0.486285\pi\)
0.0430725 + 0.999072i \(0.486285\pi\)
\(600\) 0 0
\(601\) −1.38565e7 −1.56483 −0.782413 0.622760i \(-0.786011\pi\)
−0.782413 + 0.622760i \(0.786011\pi\)
\(602\) 1.14885e6i 0.129203i
\(603\) 6.98045e6i 0.781791i
\(604\) 1.56717e6 0.174793
\(605\) 0 0
\(606\) 1.86293e6 0.206070
\(607\) 1.13772e7i 1.25333i 0.779291 + 0.626663i \(0.215580\pi\)
−0.779291 + 0.626663i \(0.784420\pi\)
\(608\) 2.80576e6i 0.307816i
\(609\) 4.18428e6 0.457170
\(610\) 0 0
\(611\) 1.45129e7 1.57272
\(612\) − 2.52374e6i − 0.272375i
\(613\) 7.00161e6i 0.752570i 0.926504 + 0.376285i \(0.122799\pi\)
−0.926504 + 0.376285i \(0.877201\pi\)
\(614\) 5.79023e6 0.619833
\(615\) 0 0
\(616\) −1.44998e6 −0.153961
\(617\) 7.90300e6i 0.835755i 0.908503 + 0.417878i \(0.137226\pi\)
−0.908503 + 0.417878i \(0.862774\pi\)
\(618\) 1.12114e6i 0.118083i
\(619\) −4.02362e6 −0.422076 −0.211038 0.977478i \(-0.567684\pi\)
−0.211038 + 0.977478i \(0.567684\pi\)
\(620\) 0 0
\(621\) −4.22820e6 −0.439974
\(622\) − 3.71227e6i − 0.384737i
\(623\) 4.54182e6i 0.468824i
\(624\) −1.69882e6 −0.174657
\(625\) 0 0
\(626\) 9.18250e6 0.936538
\(627\) − 3.15648e6i − 0.320652i
\(628\) 59488.0i 0.00601908i
\(629\) 4.20472e6 0.423750
\(630\) 0 0
\(631\) −1.00227e7 −1.00210 −0.501049 0.865419i \(-0.667052\pi\)
−0.501049 + 0.865419i \(0.667052\pi\)
\(632\) 289280.i 0.0288088i
\(633\) 1.08989e6i 0.108112i
\(634\) −1.09461e7 −1.08152
\(635\) 0 0
\(636\) −880704. −0.0863351
\(637\) − 3.18860e6i − 0.311352i
\(638\) − 4.53888e6i − 0.441466i
\(639\) 1.45339e7 1.40809
\(640\) 0 0
\(641\) 6.37390e6 0.612718 0.306359 0.951916i \(-0.400889\pi\)
0.306359 + 0.951916i \(0.400889\pi\)
\(642\) − 24912.0i − 0.00238545i
\(643\) − 5.00457e6i − 0.477352i −0.971099 0.238676i \(-0.923287\pi\)
0.971099 0.238676i \(-0.0767134\pi\)
\(644\) 2.95661e6 0.280918
\(645\) 0 0
\(646\) −8.35152e6 −0.787380
\(647\) − 8.71928e6i − 0.818879i −0.912337 0.409440i \(-0.865724\pi\)
0.912337 0.409440i \(-0.134276\pi\)
\(648\) − 2.18246e6i − 0.204178i
\(649\) 6.71616e6 0.625906
\(650\) 0 0
\(651\) 4.86254e6 0.449688
\(652\) − 691744.i − 0.0637274i
\(653\) 1.58477e6i 0.145440i 0.997352 + 0.0727201i \(0.0231680\pi\)
−0.997352 + 0.0727201i \(0.976832\pi\)
\(654\) −4.96632e6 −0.454036
\(655\) 0 0
\(656\) −96768.0 −0.00877955
\(657\) − 4.55110e6i − 0.411342i
\(658\) 6.19358e6i 0.557670i
\(659\) −1.26410e7 −1.13388 −0.566940 0.823759i \(-0.691873\pi\)
−0.566940 + 0.823759i \(0.691873\pi\)
\(660\) 0 0
\(661\) −3.61572e6 −0.321878 −0.160939 0.986964i \(-0.551452\pi\)
−0.160939 + 0.986964i \(0.551452\pi\)
\(662\) 1.52752e7i 1.35469i
\(663\) − 5.05663e6i − 0.446763i
\(664\) 6.98074e6 0.614442
\(665\) 0 0
\(666\) 4.56890e6 0.399141
\(667\) 9.25506e6i 0.805498i
\(668\) − 2.98435e6i − 0.258767i
\(669\) 1.72964e6 0.149414
\(670\) 0 0
\(671\) −1.88890e6 −0.161958
\(672\) − 724992.i − 0.0619313i
\(673\) − 1.11313e7i − 0.947349i −0.880700 0.473675i \(-0.842927\pi\)
0.880700 0.473675i \(-0.157073\pi\)
\(674\) 8.84351e6 0.749851
\(675\) 0 0
\(676\) 1.36311e7 1.14727
\(677\) − 235518.i − 0.0197493i −0.999951 0.00987467i \(-0.996857\pi\)
0.999951 0.00987467i \(-0.00314326\pi\)
\(678\) − 3.34526e6i − 0.279483i
\(679\) −226324. −0.0188389
\(680\) 0 0
\(681\) 6.75313e6 0.558004
\(682\) − 5.27462e6i − 0.434241i
\(683\) − 2.05830e7i − 1.68833i −0.536084 0.844164i \(-0.680097\pi\)
0.536084 0.844164i \(-0.319903\pi\)
\(684\) −9.07488e6 −0.741653
\(685\) 0 0
\(686\) 9.29368e6 0.754011
\(687\) − 2.49486e6i − 0.201676i
\(688\) 623104.i 0.0501868i
\(689\) 1.01464e7 0.814265
\(690\) 0 0
\(691\) −9.54825e6 −0.760727 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(692\) − 5.99126e6i − 0.475612i
\(693\) − 4.68979e6i − 0.370954i
\(694\) 9.30895e6 0.733672
\(695\) 0 0
\(696\) 2.26944e6 0.177581
\(697\) − 288036.i − 0.0224577i
\(698\) 1.24516e6i 0.0967357i
\(699\) −4.62352e6 −0.357915
\(700\) 0 0
\(701\) 1.29304e6 0.0993843 0.0496921 0.998765i \(-0.484176\pi\)
0.0496921 + 0.998765i \(0.484176\pi\)
\(702\) − 1.19448e7i − 0.914821i
\(703\) − 1.51193e7i − 1.15384i
\(704\) −786432. −0.0598039
\(705\) 0 0
\(706\) −1.23463e7 −0.932234
\(707\) − 9.15940e6i − 0.689157i
\(708\) 3.35808e6i 0.251772i
\(709\) 2.12720e7 1.58926 0.794628 0.607097i \(-0.207666\pi\)
0.794628 + 0.607097i \(0.207666\pi\)
\(710\) 0 0
\(711\) −935640. −0.0694120
\(712\) 2.46336e6i 0.182108i
\(713\) 1.07553e7i 0.792316i
\(714\) 2.15798e6 0.158417
\(715\) 0 0
\(716\) 4.35360e6 0.317370
\(717\) − 3.57192e6i − 0.259480i
\(718\) 1.41230e7i 1.02239i
\(719\) −8.31732e6 −0.600014 −0.300007 0.953937i \(-0.596989\pi\)
−0.300007 + 0.953937i \(0.596989\pi\)
\(720\) 0 0
\(721\) 5.51225e6 0.394903
\(722\) 2.01260e7i 1.43686i
\(723\) − 1.64341e6i − 0.116923i
\(724\) 1.20669e6 0.0855556
\(725\) 0 0
\(726\) −2.98049e6 −0.209868
\(727\) − 4.36740e6i − 0.306469i −0.988190 0.153235i \(-0.951031\pi\)
0.988190 0.153235i \(-0.0489690\pi\)
\(728\) 8.35251e6i 0.584102i
\(729\) 3.12231e6 0.217599
\(730\) 0 0
\(731\) −1.85471e6 −0.128375
\(732\) − 944448.i − 0.0651479i
\(733\) 4.05645e6i 0.278860i 0.990232 + 0.139430i \(0.0445271\pi\)
−0.990232 + 0.139430i \(0.955473\pi\)
\(734\) −143048. −0.00980035
\(735\) 0 0
\(736\) 1.60358e6 0.109118
\(737\) 6.47462e6i 0.439082i
\(738\) − 312984.i − 0.0211535i
\(739\) −768260. −0.0517484 −0.0258742 0.999665i \(-0.508237\pi\)
−0.0258742 + 0.999665i \(0.508237\pi\)
\(740\) 0 0
\(741\) −1.81826e7 −1.21650
\(742\) 4.33013e6i 0.288729i
\(743\) − 6.18781e6i − 0.411211i −0.978635 0.205605i \(-0.934084\pi\)
0.978635 0.205605i \(-0.0659164\pi\)
\(744\) 2.63731e6 0.174674
\(745\) 0 0
\(746\) −6.86102e6 −0.451379
\(747\) 2.25783e7i 1.48044i
\(748\) − 2.34086e6i − 0.152976i
\(749\) −122484. −0.00797765
\(750\) 0 0
\(751\) 1.81698e7 1.17557 0.587787 0.809016i \(-0.299999\pi\)
0.587787 + 0.809016i \(0.299999\pi\)
\(752\) 3.35923e6i 0.216618i
\(753\) − 5.10451e6i − 0.328070i
\(754\) −2.61458e7 −1.67484
\(755\) 0 0
\(756\) 5.09760e6 0.324385
\(757\) 1.93494e7i 1.22724i 0.789603 + 0.613618i \(0.210286\pi\)
−0.789603 + 0.613618i \(0.789714\pi\)
\(758\) 1.24070e7i 0.784318i
\(759\) −1.80403e6 −0.113668
\(760\) 0 0
\(761\) −3.01992e7 −1.89031 −0.945155 0.326621i \(-0.894090\pi\)
−0.945155 + 0.326621i \(0.894090\pi\)
\(762\) 7.19717e6i 0.449029i
\(763\) 2.44177e7i 1.51843i
\(764\) 5.71181e6 0.354030
\(765\) 0 0
\(766\) 2.12779e7 1.31026
\(767\) − 3.86879e7i − 2.37458i
\(768\) − 393216.i − 0.0240563i
\(769\) −2.15854e7 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(770\) 0 0
\(771\) 4.95241e6 0.300041
\(772\) − 7.01910e6i − 0.423876i
\(773\) − 3.90895e6i − 0.235294i −0.993055 0.117647i \(-0.962465\pi\)
0.993055 0.117647i \(-0.0375351\pi\)
\(774\) −2.01535e6 −0.120920
\(775\) 0 0
\(776\) −122752. −0.00731769
\(777\) 3.90674e6i 0.232147i
\(778\) − 4.64580e6i − 0.275177i
\(779\) −1.03572e6 −0.0611503
\(780\) 0 0
\(781\) 1.34807e7 0.790833
\(782\) 4.77317e6i 0.279119i
\(783\) 1.59570e7i 0.930137i
\(784\) 738048. 0.0428839
\(785\) 0 0
\(786\) 188928. 0.0109079
\(787\) − 2.65082e7i − 1.52561i −0.646628 0.762806i \(-0.723821\pi\)
0.646628 0.762806i \(-0.276179\pi\)
\(788\) 2.50877e6i 0.143928i
\(789\) −8.18788e6 −0.468251
\(790\) 0 0
\(791\) −1.64475e7 −0.934674
\(792\) − 2.54362e6i − 0.144092i
\(793\) 1.08808e7i 0.614439i
\(794\) −2.51425e6 −0.141533
\(795\) 0 0
\(796\) −2.60032e6 −0.145460
\(797\) 1.07940e7i 0.601919i 0.953637 + 0.300960i \(0.0973070\pi\)
−0.953637 + 0.300960i \(0.902693\pi\)
\(798\) − 7.75968e6i − 0.431357i
\(799\) −9.99896e6 −0.554100
\(800\) 0 0
\(801\) −7.96743e6 −0.438770
\(802\) − 1.08973e7i − 0.598249i
\(803\) − 4.22131e6i − 0.231025i
\(804\) −3.23731e6 −0.176622
\(805\) 0 0
\(806\) −3.03840e7 −1.64743
\(807\) − 679860.i − 0.0367482i
\(808\) − 4.96781e6i − 0.267693i
\(809\) 1.11446e7 0.598675 0.299338 0.954147i \(-0.403234\pi\)
0.299338 + 0.954147i \(0.403234\pi\)
\(810\) 0 0
\(811\) −1.14866e7 −0.613253 −0.306626 0.951830i \(-0.599200\pi\)
−0.306626 + 0.951830i \(0.599200\pi\)
\(812\) − 1.11581e7i − 0.593881i
\(813\) 5.09777e6i 0.270492i
\(814\) 4.23782e6 0.224172
\(815\) 0 0
\(816\) 1.17043e6 0.0615348
\(817\) 6.66916e6i 0.349555i
\(818\) − 7.12076e6i − 0.372086i
\(819\) −2.70152e7 −1.40734
\(820\) 0 0
\(821\) 3.04347e7 1.57584 0.787918 0.615781i \(-0.211159\pi\)
0.787918 + 0.615781i \(0.211159\pi\)
\(822\) − 3.94171e6i − 0.203472i
\(823\) − 4.09773e6i − 0.210884i −0.994425 0.105442i \(-0.966374\pi\)
0.994425 0.105442i \(-0.0336257\pi\)
\(824\) 2.98970e6 0.153394
\(825\) 0 0
\(826\) 1.65106e7 0.841999
\(827\) − 1.70652e7i − 0.867654i −0.900996 0.433827i \(-0.857163\pi\)
0.900996 0.433827i \(-0.142837\pi\)
\(828\) 5.18659e6i 0.262909i
\(829\) 2.47617e7 1.25139 0.625697 0.780066i \(-0.284815\pi\)
0.625697 + 0.780066i \(0.284815\pi\)
\(830\) 0 0
\(831\) 2.63161e6 0.132196
\(832\) 4.53018e6i 0.226886i
\(833\) 2.19685e6i 0.109695i
\(834\) 6.77040e6 0.337054
\(835\) 0 0
\(836\) −8.41728e6 −0.416539
\(837\) 1.85436e7i 0.914914i
\(838\) − 2.60232e6i − 0.128012i
\(839\) −3.16529e7 −1.55242 −0.776208 0.630476i \(-0.782860\pi\)
−0.776208 + 0.630476i \(0.782860\pi\)
\(840\) 0 0
\(841\) 1.44170e7 0.702884
\(842\) − 1.41624e7i − 0.688425i
\(843\) 8.74019e6i 0.423596i
\(844\) 2.90637e6 0.140441
\(845\) 0 0
\(846\) −1.08650e7 −0.521921
\(847\) 1.46541e7i 0.701859i
\(848\) 2.34854e6i 0.112153i
\(849\) 722364. 0.0343943
\(850\) 0 0
\(851\) −8.64119e6 −0.409025
\(852\) 6.74035e6i 0.318115i
\(853\) − 2.82671e7i − 1.33017i −0.746765 0.665087i \(-0.768394\pi\)
0.746765 0.665087i \(-0.231606\pi\)
\(854\) −4.64354e6 −0.217873
\(855\) 0 0
\(856\) −66432.0 −0.00309880
\(857\) 2.60870e7i 1.21331i 0.794966 + 0.606655i \(0.207489\pi\)
−0.794966 + 0.606655i \(0.792511\pi\)
\(858\) − 5.09645e6i − 0.236347i
\(859\) 3.38111e7 1.56342 0.781710 0.623642i \(-0.214348\pi\)
0.781710 + 0.623642i \(0.214348\pi\)
\(860\) 0 0
\(861\) 267624. 0.0123032
\(862\) − 2.19499e6i − 0.100615i
\(863\) − 2.22817e7i − 1.01841i −0.860646 0.509204i \(-0.829940\pi\)
0.860646 0.509204i \(-0.170060\pi\)
\(864\) 2.76480e6 0.126003
\(865\) 0 0
\(866\) −5.96966e6 −0.270492
\(867\) − 5.03528e6i − 0.227497i
\(868\) − 1.29668e7i − 0.584162i
\(869\) −867840. −0.0389843
\(870\) 0 0
\(871\) 3.72965e7 1.66580
\(872\) 1.32435e7i 0.589810i
\(873\) − 397026.i − 0.0176313i
\(874\) 1.71634e7 0.760018
\(875\) 0 0
\(876\) 2.11066e6 0.0929303
\(877\) − 3.46748e7i − 1.52235i −0.648545 0.761177i \(-0.724622\pi\)
0.648545 0.761177i \(-0.275378\pi\)
\(878\) − 1.94485e7i − 0.851431i
\(879\) 1.58526e7 0.692034
\(880\) 0 0
\(881\) 1.42603e7 0.618998 0.309499 0.950900i \(-0.399839\pi\)
0.309499 + 0.950900i \(0.399839\pi\)
\(882\) 2.38712e6i 0.103325i
\(883\) 3.75177e7i 1.61933i 0.586895 + 0.809663i \(0.300350\pi\)
−0.586895 + 0.809663i \(0.699650\pi\)
\(884\) −1.34844e7 −0.580363
\(885\) 0 0
\(886\) −7.44622e6 −0.318677
\(887\) 4.07657e7i 1.73975i 0.493275 + 0.869873i \(0.335800\pi\)
−0.493275 + 0.869873i \(0.664200\pi\)
\(888\) 2.11891e6i 0.0901738i
\(889\) 3.53861e7 1.50168
\(890\) 0 0
\(891\) 6.54739e6 0.276296
\(892\) − 4.61238e6i − 0.194095i
\(893\) 3.59543e7i 1.50877i
\(894\) 9.33480e6 0.390626
\(895\) 0 0
\(896\) −1.93331e6 −0.0804511
\(897\) 1.03920e7i 0.431238i
\(898\) − 1.49488e7i − 0.618606i
\(899\) 4.05899e7 1.67501
\(900\) 0 0
\(901\) −6.99059e6 −0.286881
\(902\) − 290304.i − 0.0118806i
\(903\) − 1.72327e6i − 0.0703290i
\(904\) −8.92070e6 −0.363060
\(905\) 0 0
\(906\) −2.35075e6 −0.0951451
\(907\) − 3.57116e7i − 1.44142i −0.693235 0.720712i \(-0.743815\pi\)
0.693235 0.720712i \(-0.256185\pi\)
\(908\) − 1.80084e7i − 0.724869i
\(909\) 1.60678e7 0.644979
\(910\) 0 0
\(911\) −2.11389e7 −0.843893 −0.421947 0.906621i \(-0.638653\pi\)
−0.421947 + 0.906621i \(0.638653\pi\)
\(912\) − 4.20864e6i − 0.167554i
\(913\) 2.09422e7i 0.831468i
\(914\) 2.59310e7 1.02673
\(915\) 0 0
\(916\) −6.65296e6 −0.261985
\(917\) − 928896.i − 0.0364791i
\(918\) 8.22960e6i 0.322309i
\(919\) −1.85996e7 −0.726465 −0.363233 0.931698i \(-0.618327\pi\)
−0.363233 + 0.931698i \(0.618327\pi\)
\(920\) 0 0
\(921\) −8.68535e6 −0.337395
\(922\) 6.03641e6i 0.233857i
\(923\) − 7.76545e7i − 3.00028i
\(924\) 2.17498e6 0.0838059
\(925\) 0 0
\(926\) 3.47360e7 1.33123
\(927\) 9.66980e6i 0.369588i
\(928\) − 6.05184e6i − 0.230684i
\(929\) −4.45110e7 −1.69211 −0.846055 0.533096i \(-0.821028\pi\)
−0.846055 + 0.533096i \(0.821028\pi\)
\(930\) 0 0
\(931\) 7.89942e6 0.298690
\(932\) 1.23294e7i 0.464945i
\(933\) 5.56841e6i 0.209424i
\(934\) −2.78565e7 −1.04486
\(935\) 0 0
\(936\) −1.46523e7 −0.546658
\(937\) − 2.19419e7i − 0.816441i −0.912883 0.408221i \(-0.866149\pi\)
0.912883 0.408221i \(-0.133851\pi\)
\(938\) 1.59168e7i 0.590675i
\(939\) −1.37738e7 −0.509787
\(940\) 0 0
\(941\) −7.77722e6 −0.286319 −0.143160 0.989700i \(-0.545726\pi\)
−0.143160 + 0.989700i \(0.545726\pi\)
\(942\) − 89232.0i − 0.00327637i
\(943\) 591948.i 0.0216773i
\(944\) 8.95488e6 0.327062
\(945\) 0 0
\(946\) −1.86931e6 −0.0679132
\(947\) 3.17199e7i 1.14936i 0.818378 + 0.574681i \(0.194874\pi\)
−0.818378 + 0.574681i \(0.805126\pi\)
\(948\) − 433920.i − 0.0156815i
\(949\) −2.43165e7 −0.876468
\(950\) 0 0
\(951\) 1.64191e7 0.588707
\(952\) − 5.75462e6i − 0.205790i
\(953\) 5.60285e6i 0.199838i 0.994996 + 0.0999188i \(0.0318583\pi\)
−0.994996 + 0.0999188i \(0.968142\pi\)
\(954\) −7.59607e6 −0.270220
\(955\) 0 0
\(956\) −9.52512e6 −0.337074
\(957\) 6.80832e6i 0.240304i
\(958\) 2.20421e7i 0.775959i
\(959\) −1.93801e7 −0.680470
\(960\) 0 0
\(961\) 1.85403e7 0.647601
\(962\) − 2.44116e7i − 0.850470i
\(963\) − 214866.i − 0.00746624i
\(964\) −4.38243e6 −0.151888
\(965\) 0 0
\(966\) −4.43491e6 −0.152912
\(967\) − 2.03532e7i − 0.699949i −0.936759 0.349975i \(-0.886190\pi\)
0.936759 0.349975i \(-0.113810\pi\)
\(968\) 7.94797e6i 0.272626i
\(969\) 1.25273e7 0.428595
\(970\) 0 0
\(971\) −2.34306e7 −0.797510 −0.398755 0.917057i \(-0.630558\pi\)
−0.398755 + 0.917057i \(0.630558\pi\)
\(972\) 1.37713e7i 0.467530i
\(973\) − 3.32878e7i − 1.12721i
\(974\) −2.20723e7 −0.745505
\(975\) 0 0
\(976\) −2.51853e6 −0.0846296
\(977\) − 4.30412e7i − 1.44261i −0.692619 0.721303i \(-0.743543\pi\)
0.692619 0.721303i \(-0.256457\pi\)
\(978\) 1.03762e6i 0.0346888i
\(979\) −7.39008e6 −0.246429
\(980\) 0 0
\(981\) −4.28345e7 −1.42109
\(982\) − 6.05107e6i − 0.200241i
\(983\) 4.75003e7i 1.56788i 0.620837 + 0.783940i \(0.286793\pi\)
−0.620837 + 0.783940i \(0.713207\pi\)
\(984\) 145152. 0.00477898
\(985\) 0 0
\(986\) 1.80137e7 0.590079
\(987\) − 9.29038e6i − 0.303557i
\(988\) 4.84870e7i 1.58028i
\(989\) 3.81164e6 0.123914
\(990\) 0 0
\(991\) 2.09231e7 0.676770 0.338385 0.941008i \(-0.390119\pi\)
0.338385 + 0.941008i \(0.390119\pi\)
\(992\) − 7.03283e6i − 0.226909i
\(993\) − 2.29128e7i − 0.737402i
\(994\) 3.31401e7 1.06387
\(995\) 0 0
\(996\) −1.04711e7 −0.334460
\(997\) 2.96332e7i 0.944148i 0.881559 + 0.472074i \(0.156495\pi\)
−0.881559 + 0.472074i \(0.843505\pi\)
\(998\) 7.72168e6i 0.245406i
\(999\) −1.48986e7 −0.472315
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 50.6.b.b.49.2 2
3.2 odd 2 450.6.c.f.199.1 2
4.3 odd 2 400.6.c.i.49.2 2
5.2 odd 4 50.6.a.b.1.1 1
5.3 odd 4 10.6.a.c.1.1 1
5.4 even 2 inner 50.6.b.b.49.1 2
15.2 even 4 450.6.a.u.1.1 1
15.8 even 4 90.6.a.b.1.1 1
15.14 odd 2 450.6.c.f.199.2 2
20.3 even 4 80.6.a.c.1.1 1
20.7 even 4 400.6.a.i.1.1 1
20.19 odd 2 400.6.c.i.49.1 2
35.13 even 4 490.6.a.k.1.1 1
40.3 even 4 320.6.a.k.1.1 1
40.13 odd 4 320.6.a.f.1.1 1
60.23 odd 4 720.6.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 5.3 odd 4
50.6.a.b.1.1 1 5.2 odd 4
50.6.b.b.49.1 2 5.4 even 2 inner
50.6.b.b.49.2 2 1.1 even 1 trivial
80.6.a.c.1.1 1 20.3 even 4
90.6.a.b.1.1 1 15.8 even 4
320.6.a.f.1.1 1 40.13 odd 4
320.6.a.k.1.1 1 40.3 even 4
400.6.a.i.1.1 1 20.7 even 4
400.6.c.i.49.1 2 20.19 odd 2
400.6.c.i.49.2 2 4.3 odd 2
450.6.a.u.1.1 1 15.2 even 4
450.6.c.f.199.1 2 3.2 odd 2
450.6.c.f.199.2 2 15.14 odd 2
490.6.a.k.1.1 1 35.13 even 4
720.6.a.v.1.1 1 60.23 odd 4