Properties

Label 50.6.b.b.49.1
Level $50$
Weight $6$
Character 50.49
Analytic conductor $8.019$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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

$q$-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 \(0.0616434\pi\)
−0.981307 + 0.192450i \(0.938357\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 24.0000 0.272166
\(7\) 118.000i 0.910200i 0.890440 + 0.455100i \(0.150397\pi\)
−0.890440 + 0.455100i \(0.849603\pi\)
\(8\) 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 \(0.362044\pi\)
−0.419961 + 0.907542i \(0.637956\pi\)
\(14\) 472.000 0.643609
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 762.000i − 0.639488i −0.947504 0.319744i \(-0.896403\pi\)
0.947504 0.319744i \(-0.103597\pi\)
\(18\) − 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 \(0.0998714\pi\)
−0.951181 + 0.308633i \(0.900129\pi\)
\(24\) −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 \(0.107494\pi\)
−0.943519 + 0.331320i \(0.892506\pi\)
\(38\) − 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 \(-0.967996\pi\)
0.994950 0.100374i \(-0.0320038\pi\)
\(44\) −3072.00 −0.239216
\(45\) 0 0
\(46\) 6264.00 0.436473
\(47\) − 13122.0i − 0.866474i −0.901280 0.433237i \(-0.857371\pi\)
0.901280 0.433237i \(-0.142629\pi\)
\(48\) 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 \(-0.927989\pi\)
0.974519 0.224305i \(-0.0720112\pi\)
\(54\) 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 \(-0.848252\pi\)
0.888500 0.458877i \(-0.151748\pi\)
\(68\) 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 \(0.0776197\pi\)
−0.970416 + 0.241440i \(0.922380\pi\)
\(74\) 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 \(-0.664794\pi\)
0.494896 0.868952i \(-0.335206\pi\)
\(84\) 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 \(0.00329418\pi\)
−0.999946 + 0.0103488i \(0.996706\pi\)
\(98\) − 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 \(-0.930395\pi\)
0.976187 0.216932i \(-0.0696051\pi\)
\(104\) −70784.0 −0.641729
\(105\) 0 0
\(106\) −36696.0 −0.317215
\(107\) 1038.00i 0.00876472i 0.999990 + 0.00438236i \(0.00139495\pi\)
−0.999990 + 0.00438236i \(0.998605\pi\)
\(108\) − 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 \(0.171631\pi\)
−0.858123 + 0.513444i \(0.828369\pi\)
\(114\) 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 \(-0.691221\pi\)
0.565252 0.824919i \(-0.308779\pi\)
\(128\) 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 \(0.121946\pi\)
−0.927508 + 0.373803i \(0.878054\pi\)
\(138\) 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 \(0.00191594\pi\)
−0.999982 + 0.00601908i \(0.998084\pi\)
\(158\) 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 \(-0.979701\pi\)
0.997967 0.0637274i \(-0.0202988\pi\)
\(164\) 6048.00 0.0175591
\(165\) 0 0
\(166\) −436296. −1.22888
\(167\) − 186522.i − 0.517534i −0.965940 0.258767i \(-0.916684\pi\)
0.965940 0.258767i \(-0.0833162\pi\)
\(168\) − 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 \(-0.842226\pi\)
0.879655 0.475612i \(-0.157774\pi\)
\(174\) −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 \(-0.860669\pi\)
0.905720 0.423876i \(-0.139331\pi\)
\(194\) 7672.00 0.0146354
\(195\) 0 0
\(196\) −46128.0 −0.0857678
\(197\) 156798.i 0.287856i 0.989588 + 0.143928i \(0.0459733\pi\)
−0.989588 + 0.143928i \(0.954027\pi\)
\(198\) − 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 \(-0.937823\pi\)
0.980983 0.194095i \(-0.0621769\pi\)
\(224\) 120832. 0.160902
\(225\) 0 0
\(226\) 557544. 0.726119
\(227\) − 1.12552e6i − 1.44974i −0.688887 0.724869i \(-0.741900\pi\)
0.688887 0.724869i \(-0.258100\pi\)
\(228\) − 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 \(0.153926\pi\)
−0.885340 + 0.464945i \(0.846074\pi\)
\(234\) 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 \(-0.872556\pi\)
0.920914 0.389765i \(-0.127444\pi\)
\(258\) − 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 \(0.208139\pi\)
−0.793726 + 0.608276i \(0.791861\pi\)
\(264\) −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 \(-0.945065\pi\)
0.985144 0.171728i \(-0.0549350\pi\)
\(278\) − 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 \(-0.985773\pi\)
0.999001 0.0446795i \(-0.0142267\pi\)
\(284\) −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 \(-0.644311\pi\)
0.437993 0.898978i \(-0.355689\pi\)
\(294\) 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 \(0.144415\pi\)
−0.898834 + 0.438288i \(0.855585\pi\)
\(308\) − 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 \(0.230391\pi\)
−0.749299 + 0.662232i \(0.769609\pi\)
\(314\) 14872.0 0.00851227
\(315\) 0 0
\(316\) 72320.0 0.0407418
\(317\) − 2.73652e6i − 1.52950i −0.644324 0.764752i \(-0.722861\pi\)
0.644324 0.764752i \(-0.277139\pi\)
\(318\) − 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 \(0.177892\pi\)
−0.847857 + 0.530225i \(0.822108\pi\)
\(338\) 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 \(0.173615\pi\)
−0.854905 + 0.518785i \(0.826385\pi\)
\(348\) 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 \(-0.770900\pi\)
0.751977 0.659189i \(-0.229100\pi\)
\(354\) 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 \(-0.997794\pi\)
0.999976 0.00692989i \(-0.00220587\pi\)
\(368\) 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 \(-0.896595\pi\)
0.947696 0.319173i \(-0.103405\pi\)
\(374\) −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 \(0.377193\pi\)
−0.376309 + 0.926494i \(0.622807\pi\)
\(384\) −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 \(-0.968091\pi\)
0.994980 0.100079i \(-0.0319095\pi\)
\(398\) − 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 \(-0.938740\pi\)
0.981538 0.191267i \(-0.0612596\pi\)
\(434\) −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 \(-0.927651\pi\)
0.974280 0.225339i \(-0.0723490\pi\)
\(444\) 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 \(0.258625\pi\)
−0.687690 + 0.726005i \(0.741375\pi\)
\(458\) − 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 \(0.390414\pi\)
−0.337513 + 0.941321i \(0.609586\pi\)
\(464\) −1.51296e6 −0.326236
\(465\) 0 0
\(466\) 3.08234e6 0.657531
\(467\) − 6.96412e6i − 1.47766i −0.673893 0.738829i \(-0.735379\pi\)
0.673893 0.738829i \(-0.264621\pi\)
\(468\) − 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 \(-0.823260\pi\)
0.849771 0.527152i \(-0.176740\pi\)
\(488\) − 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 \(0.202099\pi\)
−0.805124 + 0.593106i \(0.797901\pi\)
\(504\) −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 \(-0.809843\pi\)
0.826804 0.562491i \(-0.190157\pi\)
\(524\) −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 \(-0.897239\pi\)
0.948340 0.317255i \(-0.102761\pi\)
\(548\) − 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 \(0.211071\pi\)
−0.788088 + 0.615563i \(0.788929\pi\)
\(558\) 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 \(-0.773774\pi\)
0.757899 0.652372i \(-0.226226\pi\)
\(564\) −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 \(-0.909141\pi\)
0.959538 0.281580i \(-0.0908585\pi\)
\(578\) − 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 \(-0.988066\pi\)
0.999297 0.0374834i \(-0.0119341\pi\)
\(588\) − 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 \(-0.953296\pi\)
0.989255 0.146198i \(-0.0467038\pi\)
\(594\) 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 \(-0.784420\pi\)
0.779291 0.626663i \(-0.215580\pi\)
\(608\) − 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 \(-0.877201\pi\)
0.926504 0.376285i \(-0.122799\pi\)
\(614\) 5.79023e6 0.619833
\(615\) 0 0
\(616\) −1.44998e6 −0.153961
\(617\) − 7.90300e6i − 0.835755i −0.908503 0.417878i \(-0.862774\pi\)
0.908503 0.417878i \(-0.137226\pi\)
\(618\) − 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 \(0.0767134\pi\)
−0.971099 + 0.238676i \(0.923287\pi\)
\(644\) 2.95661e6 0.280918
\(645\) 0 0
\(646\) −8.35152e6 −0.787380
\(647\) 8.71928e6i 0.818879i 0.912337 + 0.409440i \(0.134276\pi\)
−0.912337 + 0.409440i \(0.865724\pi\)
\(648\) 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 \(-0.976832\pi\)
0.997352 0.0727201i \(-0.0231680\pi\)
\(654\) −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 \(0.157073\pi\)
−0.880700 + 0.473675i \(0.842927\pi\)
\(674\) 8.84351e6 0.749851
\(675\) 0 0
\(676\) 1.36311e7 1.14727
\(677\) 235518.i 0.0197493i 0.999951 + 0.00987467i \(0.00314326\pi\)
−0.999951 + 0.00987467i \(0.996857\pi\)
\(678\) 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 \(0.319903\pi\)
−0.536084 + 0.844164i \(0.680097\pi\)
\(684\) −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 \(0.0489690\pi\)
−0.988190 + 0.153235i \(0.951031\pi\)
\(728\) − 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 \(-0.955473\pi\)
0.990232 0.139430i \(-0.0445271\pi\)
\(734\) −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 \(0.0659164\pi\)
−0.978635 + 0.205605i \(0.934084\pi\)
\(744\) 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 \(-0.789714\pi\)
0.789603 0.613618i \(-0.210286\pi\)
\(758\) − 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 \(0.0375351\pi\)
−0.993055 + 0.117647i \(0.962465\pi\)
\(774\) −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 \(0.276179\pi\)
−0.646628 + 0.762806i \(0.723821\pi\)
\(788\) − 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 \(-0.902693\pi\)
0.953637 0.300960i \(-0.0973070\pi\)
\(798\) 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 \(0.0336257\pi\)
−0.994425 + 0.105442i \(0.966374\pi\)
\(824\) 2.98970e6 0.153394
\(825\) 0 0
\(826\) 1.65106e7 0.841999
\(827\) 1.70652e7i 0.867654i 0.900996 + 0.433827i \(0.142837\pi\)
−0.900996 + 0.433827i \(0.857163\pi\)
\(828\) − 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 \(0.231606\pi\)
−0.746765 + 0.665087i \(0.768394\pi\)
\(854\) −4.64354e6 −0.217873
\(855\) 0 0
\(856\) −66432.0 −0.00309880
\(857\) − 2.60870e7i − 1.21331i −0.794966 0.606655i \(-0.792511\pi\)
0.794966 0.606655i \(-0.207489\pi\)
\(858\) 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 \(0.170060\pi\)
−0.860646 + 0.509204i \(0.829940\pi\)
\(864\) 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 \(0.275378\pi\)
−0.648545 + 0.761177i \(0.724622\pi\)
\(878\) 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 \(-0.699650\pi\)
0.586895 0.809663i \(-0.300350\pi\)
\(884\) −1.34844e7 −0.580363
\(885\) 0 0
\(886\) −7.44622e6 −0.318677
\(887\) − 4.07657e7i − 1.73975i −0.493275 0.869873i \(-0.664200\pi\)
0.493275 0.869873i \(-0.335800\pi\)
\(888\) − 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 \(0.256185\pi\)
−0.693235 + 0.720712i \(0.743815\pi\)
\(908\) 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 \(0.133851\pi\)
−0.912883 + 0.408221i \(0.866149\pi\)
\(938\) − 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 \(-0.805126\pi\)
0.818378 0.574681i \(-0.194874\pi\)
\(948\) 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 \(-0.968142\pi\)
0.994996 0.0999188i \(-0.0318583\pi\)
\(954\) −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 \(0.113810\pi\)
−0.936759 + 0.349975i \(0.886190\pi\)
\(968\) − 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 \(0.256457\pi\)
−0.692619 + 0.721303i \(0.743543\pi\)
\(978\) − 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 \(-0.713207\pi\)
0.620837 0.783940i \(-0.286793\pi\)
\(984\) 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 \(-0.843505\pi\)
0.881559 0.472074i \(-0.156495\pi\)
\(998\) − 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.49.1 2
3.2 odd 2 450.6.c.f.199.2 2
4.3 odd 2 400.6.c.i.49.1 2
5.2 odd 4 10.6.a.c.1.1 1
5.3 odd 4 50.6.a.b.1.1 1
5.4 even 2 inner 50.6.b.b.49.2 2
15.2 even 4 90.6.a.b.1.1 1
15.8 even 4 450.6.a.u.1.1 1
15.14 odd 2 450.6.c.f.199.1 2
20.3 even 4 400.6.a.i.1.1 1
20.7 even 4 80.6.a.c.1.1 1
20.19 odd 2 400.6.c.i.49.2 2
35.27 even 4 490.6.a.k.1.1 1
40.27 even 4 320.6.a.k.1.1 1
40.37 odd 4 320.6.a.f.1.1 1
60.47 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 5.2 odd 4
50.6.a.b.1.1 1 5.3 odd 4
50.6.b.b.49.1 2 1.1 even 1 trivial
50.6.b.b.49.2 2 5.4 even 2 inner
80.6.a.c.1.1 1 20.7 even 4
90.6.a.b.1.1 1 15.2 even 4
320.6.a.f.1.1 1 40.37 odd 4
320.6.a.k.1.1 1 40.27 even 4
400.6.a.i.1.1 1 20.3 even 4
400.6.c.i.49.1 2 4.3 odd 2
400.6.c.i.49.2 2 20.19 odd 2
450.6.a.u.1.1 1 15.8 even 4
450.6.c.f.199.1 2 15.14 odd 2
450.6.c.f.199.2 2 3.2 odd 2
490.6.a.k.1.1 1 35.27 even 4
720.6.a.v.1.1 1 60.47 odd 4