Properties

Label 450.6.c.f.199.2
Level $450$
Weight $6$
Character 450.199
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.6.c.f.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000i q^{2} -16.0000 q^{4} +118.000i q^{7} -64.0000i q^{8} +O(q^{10})\) \(q+4.00000i q^{2} -16.0000 q^{4} +118.000i q^{7} -64.0000i q^{8} -192.000 q^{11} +1106.00i q^{13} -472.000 q^{14} +256.000 q^{16} +762.000i q^{17} +2740.00 q^{19} -768.000i q^{22} -1566.00i q^{23} -4424.00 q^{26} -1888.00i q^{28} +5910.00 q^{29} -6868.00 q^{31} +1024.00i q^{32} -3048.00 q^{34} +5518.00i q^{37} +10960.0i q^{38} +378.000 q^{41} -2434.00i q^{43} +3072.00 q^{44} +6264.00 q^{46} +13122.0i q^{47} +2883.00 q^{49} -17696.0i q^{52} +9174.00i q^{53} +7552.00 q^{56} +23640.0i q^{58} -34980.0 q^{59} -9838.00 q^{61} -27472.0i q^{62} -4096.00 q^{64} -33722.0i q^{67} -12192.0i q^{68} -70212.0 q^{71} +21986.0i q^{73} -22072.0 q^{74} -43840.0 q^{76} -22656.0i q^{77} -4520.00 q^{79} +1512.00i q^{82} +109074. i q^{83} +9736.00 q^{86} +12288.0i q^{88} +38490.0 q^{89} -130508. q^{91} +25056.0i q^{92} -52488.0 q^{94} +1918.00i q^{97} +11532.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + O(q^{10}) \) \( 2q - 32q^{4} - 384q^{11} - 944q^{14} + 512q^{16} + 5480q^{19} - 8848q^{26} + 11820q^{29} - 13736q^{31} - 6096q^{34} + 756q^{41} + 6144q^{44} + 12528q^{46} + 5766q^{49} + 15104q^{56} - 69960q^{59} - 19676q^{61} - 8192q^{64} - 140424q^{71} - 44144q^{74} - 87680q^{76} - 9040q^{79} + 19472q^{86} + 76980q^{89} - 261016q^{91} - 104976q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 0.707107i
\(3\) 0 0
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 118.000i 0.910200i 0.890440 + 0.455100i \(0.150397\pi\)
−0.890440 + 0.455100i \(0.849603\pi\)
\(8\) − 64.0000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −192.000 −0.478431 −0.239216 0.970966i \(-0.576890\pi\)
−0.239216 + 0.970966i \(0.576890\pi\)
\(12\) 0 0
\(13\) 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 \(0.103597\pi\)
−0.947504 + 0.319744i \(0.896403\pi\)
\(18\) 0 0
\(19\) 2740.00 1.74127 0.870636 0.491928i \(-0.163708\pi\)
0.870636 + 0.491928i \(0.163708\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 768.000i − 0.338302i
\(23\) − 1566.00i − 0.617266i −0.951181 0.308633i \(-0.900129\pi\)
0.951181 0.308633i \(-0.0998714\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4424.00 −1.28346
\(27\) 0 0
\(28\) − 1888.00i − 0.455100i
\(29\) 5910.00 1.30495 0.652473 0.757812i \(-0.273732\pi\)
0.652473 + 0.757812i \(0.273732\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −3048.00 −0.452187
\(35\) 0 0
\(36\) 0 0
\(37\) 5518.00i 0.662640i 0.943519 + 0.331320i \(0.107494\pi\)
−0.943519 + 0.331320i \(0.892506\pi\)
\(38\) 10960.0i 1.23127i
\(39\) 0 0
\(40\) 0 0
\(41\) 378.000 0.0351182 0.0175591 0.999846i \(-0.494410\pi\)
0.0175591 + 0.999846i \(0.494410\pi\)
\(42\) 0 0
\(43\) − 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 \(0.142629\pi\)
−0.901280 + 0.433237i \(0.857371\pi\)
\(48\) 0 0
\(49\) 2883.00 0.171536
\(50\) 0 0
\(51\) 0 0
\(52\) − 17696.0i − 0.907542i
\(53\) 9174.00i 0.448610i 0.974519 + 0.224305i \(0.0720112\pi\)
−0.974519 + 0.224305i \(0.927989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7552.00 0.321804
\(57\) 0 0
\(58\) 23640.0i 0.922736i
\(59\) −34980.0 −1.30825 −0.654124 0.756388i \(-0.726962\pi\)
−0.654124 + 0.756388i \(0.726962\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 33722.0i − 0.917754i −0.888500 0.458877i \(-0.848252\pi\)
0.888500 0.458877i \(-0.151748\pi\)
\(68\) − 12192.0i − 0.319744i
\(69\) 0 0
\(70\) 0 0
\(71\) −70212.0 −1.65297 −0.826486 0.562957i \(-0.809664\pi\)
−0.826486 + 0.562957i \(0.809664\pi\)
\(72\) 0 0
\(73\) 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
\(78\) 0 0
\(79\) −4520.00 −0.0814837 −0.0407418 0.999170i \(-0.512972\pi\)
−0.0407418 + 0.999170i \(0.512972\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1512.00i 0.0248323i
\(83\) 109074.i 1.73790i 0.494896 + 0.868952i \(0.335206\pi\)
−0.494896 + 0.868952i \(0.664794\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9736.00 0.141950
\(87\) 0 0
\(88\) 12288.0i 0.169151i
\(89\) 38490.0 0.515078 0.257539 0.966268i \(-0.417088\pi\)
0.257539 + 0.966268i \(0.417088\pi\)
\(90\) 0 0
\(91\) −130508. −1.65209
\(92\) 25056.0i 0.308633i
\(93\) 0 0
\(94\) −52488.0 −0.612689
\(95\) 0 0
\(96\) 0 0
\(97\) 1918.00i 0.0206976i 0.999946 + 0.0103488i \(0.00329418\pi\)
−0.999946 + 0.0103488i \(0.996706\pi\)
\(98\) 11532.0i 0.121294i
\(99\) 0 0
\(100\) 0 0
\(101\) −77622.0 −0.757149 −0.378575 0.925571i \(-0.623586\pi\)
−0.378575 + 0.925571i \(0.623586\pi\)
\(102\) 0 0
\(103\) − 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 \(-0.998605\pi\)
0.999990 0.00438236i \(-0.00139495\pi\)
\(108\) 0 0
\(109\) −206930. −1.66823 −0.834117 0.551587i \(-0.814023\pi\)
−0.834117 + 0.551587i \(0.814023\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 30208.0i 0.227550i
\(113\) − 139386.i − 1.02689i −0.858123 0.513444i \(-0.828369\pi\)
0.858123 0.513444i \(-0.171631\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −94560.0 −0.652473
\(117\) 0 0
\(118\) − 139920.i − 0.925070i
\(119\) −89916.0 −0.582062
\(120\) 0 0
\(121\) −124187. −0.771104
\(122\) − 39352.0i − 0.239369i
\(123\) 0 0
\(124\) 109888. 0.641795
\(125\) 0 0
\(126\) 0 0
\(127\) − 299882.i − 1.64984i −0.565252 0.824919i \(-0.691221\pi\)
0.565252 0.824919i \(-0.308779\pi\)
\(128\) − 16384.0i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −7872.00 −0.0400781 −0.0200390 0.999799i \(-0.506379\pi\)
−0.0200390 + 0.999799i \(0.506379\pi\)
\(132\) 0 0
\(133\) 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 \(-0.121946\pi\)
\(138\) 0 0
\(139\) 282100. 1.23841 0.619207 0.785228i \(-0.287454\pi\)
0.619207 + 0.785228i \(0.287454\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 280848.i − 1.16883i
\(143\) − 212352.i − 0.868393i
\(144\) 0 0
\(145\) 0 0
\(146\) −87944.0 −0.341448
\(147\) 0 0
\(148\) − 88288.0i − 0.331320i
\(149\) −388950. −1.43525 −0.717626 0.696429i \(-0.754771\pi\)
−0.717626 + 0.696429i \(0.754771\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 90624.0 0.307923
\(155\) 0 0
\(156\) 0 0
\(157\) 3718.00i 0.0120382i 0.999982 + 0.00601908i \(0.00191594\pi\)
−0.999982 + 0.00601908i \(0.998084\pi\)
\(158\) − 18080.0i − 0.0576177i
\(159\) 0 0
\(160\) 0 0
\(161\) 184788. 0.561835
\(162\) 0 0
\(163\) − 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 \(0.0833162\pi\)
−0.965940 + 0.258767i \(0.916684\pi\)
\(168\) 0 0
\(169\) −851943. −2.29453
\(170\) 0 0
\(171\) 0 0
\(172\) 38944.0i 0.100374i
\(173\) 374454.i 0.951225i 0.879655 + 0.475612i \(0.157774\pi\)
−0.879655 + 0.475612i \(0.842226\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −49152.0 −0.119608
\(177\) 0 0
\(178\) 153960.i 0.364215i
\(179\) 272100. 0.634740 0.317370 0.948302i \(-0.397200\pi\)
0.317370 + 0.948302i \(0.397200\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −100224. −0.218236
\(185\) 0 0
\(186\) 0 0
\(187\) − 146304.i − 0.305951i
\(188\) − 209952.i − 0.433237i
\(189\) 0 0
\(190\) 0 0
\(191\) 356988. 0.708060 0.354030 0.935234i \(-0.384811\pi\)
0.354030 + 0.935234i \(0.384811\pi\)
\(192\) 0 0
\(193\) − 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 \(-0.954027\pi\)
0.989588 0.143928i \(-0.0459733\pi\)
\(198\) 0 0
\(199\) 162520. 0.290920 0.145460 0.989364i \(-0.453534\pi\)
0.145460 + 0.989364i \(0.453534\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 310488.i − 0.535385i
\(203\) 697380.i 1.18776i
\(204\) 0 0
\(205\) 0 0
\(206\) 186856. 0.306788
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 4152.00 0.00619759
\(215\) 0 0
\(216\) 0 0
\(217\) − 810424.i − 1.16832i
\(218\) − 827720.i − 1.17962i
\(219\) 0 0
\(220\) 0 0
\(221\) −842772. −1.16073
\(222\) 0 0
\(223\) − 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 \(0.258100\pi\)
−0.688887 + 0.724869i \(0.741900\pi\)
\(228\) 0 0
\(229\) 415810. 0.523970 0.261985 0.965072i \(-0.415623\pi\)
0.261985 + 0.965072i \(0.415623\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 378240.i − 0.461368i
\(233\) − 770586.i − 0.929889i −0.885340 0.464945i \(-0.846074\pi\)
0.885340 0.464945i \(-0.153926\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 559680. 0.654124
\(237\) 0 0
\(238\) − 359664.i − 0.411580i
\(239\) −595320. −0.674149 −0.337074 0.941478i \(-0.609437\pi\)
−0.337074 + 0.941478i \(0.609437\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 157408. 0.169259
\(245\) 0 0
\(246\) 0 0
\(247\) 3.03044e6i 3.16055i
\(248\) 439552.i 0.453817i
\(249\) 0 0
\(250\) 0 0
\(251\) −850752. −0.852351 −0.426176 0.904640i \(-0.640139\pi\)
−0.426176 + 0.904640i \(0.640139\pi\)
\(252\) 0 0
\(253\) 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 \(0.872556\pi\)
\(258\) 0 0
\(259\) −651124. −0.603135
\(260\) 0 0
\(261\) 0 0
\(262\) − 31488.0i − 0.0283395i
\(263\) − 1.36465e6i − 1.21655i −0.793726 0.608276i \(-0.791861\pi\)
0.793726 0.608276i \(-0.208139\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.29328e6 −1.12070
\(267\) 0 0
\(268\) 539552.i 0.458877i
\(269\) −113310. −0.0954745 −0.0477373 0.998860i \(-0.515201\pi\)
−0.0477373 + 0.998860i \(0.515201\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 656952. 0.528637
\(275\) 0 0
\(276\) 0 0
\(277\) − 438602.i − 0.343456i −0.985144 0.171728i \(-0.945065\pi\)
0.985144 0.171728i \(-0.0549350\pi\)
\(278\) 1.12840e6i 0.875691i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.45670e6 1.10053 0.550267 0.834989i \(-0.314526\pi\)
0.550267 + 0.834989i \(0.314526\pi\)
\(282\) 0 0
\(283\) − 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
\(288\) 0 0
\(289\) 839213. 0.591055
\(290\) 0 0
\(291\) 0 0
\(292\) − 351776.i − 0.241440i
\(293\) 2.64209e6i 1.79796i 0.437993 + 0.898978i \(0.355689\pi\)
−0.437993 + 0.898978i \(0.644311\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 353152. 0.234278
\(297\) 0 0
\(298\) − 1.55580e6i − 1.01488i
\(299\) 1.73200e6 1.12039
\(300\) 0 0
\(301\) 287212. 0.182720
\(302\) − 391792.i − 0.247194i
\(303\) 0 0
\(304\) 701440. 0.435318
\(305\) 0 0
\(306\) 0 0
\(307\) 1.44756e6i 0.876577i 0.898834 + 0.438288i \(0.144415\pi\)
−0.898834 + 0.438288i \(0.855585\pi\)
\(308\) 362496.i 0.217734i
\(309\) 0 0
\(310\) 0 0
\(311\) 928068. 0.544100 0.272050 0.962283i \(-0.412298\pi\)
0.272050 + 0.962283i \(0.412298\pi\)
\(312\) 0 0
\(313\) 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 \(0.277139\pi\)
−0.644324 + 0.764752i \(0.722861\pi\)
\(318\) 0 0
\(319\) −1.13472e6 −0.624327
\(320\) 0 0
\(321\) 0 0
\(322\) 739152.i 0.397278i
\(323\) 2.08788e6i 1.11352i
\(324\) 0 0
\(325\) 0 0
\(326\) 172936. 0.0901242
\(327\) 0 0
\(328\) − 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
\(333\) 0 0
\(334\) −746088. −0.365952
\(335\) 0 0
\(336\) 0 0
\(337\) 2.21088e6i 1.06045i 0.847857 + 0.530225i \(0.177892\pi\)
−0.847857 + 0.530225i \(0.822108\pi\)
\(338\) − 3.40777e6i − 1.62248i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.31866e6 0.614109
\(342\) 0 0
\(343\) 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 \(-0.173615\pi\)
\(348\) 0 0
\(349\) 311290. 0.136805 0.0684024 0.997658i \(-0.478210\pi\)
0.0684024 + 0.997658i \(0.478210\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 196608.i − 0.0845755i
\(353\) 3.08657e6i 1.31838i 0.751977 + 0.659189i \(0.229100\pi\)
−0.751977 + 0.659189i \(0.770900\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −615840. −0.257539
\(357\) 0 0
\(358\) 1.08840e6i 0.448829i
\(359\) −3.53076e6 −1.44588 −0.722940 0.690911i \(-0.757210\pi\)
−0.722940 + 0.690911i \(0.757210\pi\)
\(360\) 0 0
\(361\) 5.03150e6 2.03203
\(362\) − 301672.i − 0.120994i
\(363\) 0 0
\(364\) 2.08813e6 0.826045
\(365\) 0 0
\(366\) 0 0
\(367\) − 35762.0i − 0.0138598i −0.999976 0.00692989i \(-0.997794\pi\)
0.999976 0.00692989i \(-0.00220587\pi\)
\(368\) − 400896.i − 0.154316i
\(369\) 0 0
\(370\) 0 0
\(371\) −1.08253e6 −0.408325
\(372\) 0 0
\(373\) − 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
\(378\) 0 0
\(379\) 3.10174e6 1.10919 0.554597 0.832119i \(-0.312873\pi\)
0.554597 + 0.832119i \(0.312873\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.42795e6i 0.500674i
\(383\) − 5.31949e6i − 1.85299i −0.376309 0.926494i \(-0.622807\pi\)
0.376309 0.926494i \(-0.377193\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.75478e6 0.599451
\(387\) 0 0
\(388\) − 30688.0i − 0.0103488i
\(389\) 1.16145e6 0.389158 0.194579 0.980887i \(-0.437666\pi\)
0.194579 + 0.980887i \(0.437666\pi\)
\(390\) 0 0
\(391\) 1.19329e6 0.394734
\(392\) − 184512.i − 0.0606470i
\(393\) 0 0
\(394\) 627192. 0.203545
\(395\) 0 0
\(396\) 0 0
\(397\) − 628562.i − 0.200157i −0.994980 0.100079i \(-0.968091\pi\)
0.994980 0.100079i \(-0.0319095\pi\)
\(398\) 650080.i 0.205712i
\(399\) 0 0
\(400\) 0 0
\(401\) 2.72432e6 0.846052 0.423026 0.906118i \(-0.360968\pi\)
0.423026 + 0.906118i \(0.360968\pi\)
\(402\) 0 0
\(403\) − 7.59601e6i − 2.32982i
\(404\) 1.24195e6 0.378575
\(405\) 0 0
\(406\) −2.78952e6 −0.839875
\(407\) − 1.05946e6i − 0.317027i
\(408\) 0 0
\(409\) −1.78019e6 −0.526209 −0.263104 0.964767i \(-0.584746\pi\)
−0.263104 + 0.964767i \(0.584746\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 747424.i 0.216932i
\(413\) − 4.12764e6i − 1.19077i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.13254e6 −0.320865
\(417\) 0 0
\(418\) − 2.10432e6i − 0.589076i
\(419\) 650580. 0.181036 0.0905181 0.995895i \(-0.471148\pi\)
0.0905181 + 0.995895i \(0.471148\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 587136. 0.158608
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.16088e6i − 0.308119i
\(428\) 16608.0i 0.00438236i
\(429\) 0 0
\(430\) 0 0
\(431\) 548748. 0.142292 0.0711459 0.997466i \(-0.477334\pi\)
0.0711459 + 0.997466i \(0.477334\pi\)
\(432\) 0 0
\(433\) − 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
\(438\) 0 0
\(439\) −4.86212e6 −1.20411 −0.602053 0.798456i \(-0.705650\pi\)
−0.602053 + 0.798456i \(0.705650\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.37109e6i − 0.820757i
\(443\) 1.86155e6i 0.450678i 0.974280 + 0.225339i \(0.0723490\pi\)
−0.974280 + 0.225339i \(0.927651\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.15310e6 0.274491
\(447\) 0 0
\(448\) − 483328.i − 0.113775i
\(449\) 3.73719e6 0.874841 0.437421 0.899257i \(-0.355892\pi\)
0.437421 + 0.899257i \(0.355892\pi\)
\(450\) 0 0
\(451\) −72576.0 −0.0168016
\(452\) 2.23018e6i 0.513444i
\(453\) 0 0
\(454\) −4.50209e6 −1.02512
\(455\) 0 0
\(456\) 0 0
\(457\) 6.48276e6i 1.45201i 0.687690 + 0.726005i \(0.258625\pi\)
−0.687690 + 0.726005i \(0.741375\pi\)
\(458\) 1.66324e6i 0.370503i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.50910e6 −0.330724 −0.165362 0.986233i \(-0.552879\pi\)
−0.165362 + 0.986233i \(0.552879\pi\)
\(462\) 0 0
\(463\) 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 \(0.264621\pi\)
−0.673893 + 0.738829i \(0.735379\pi\)
\(468\) 0 0
\(469\) 3.97920e6 0.835340
\(470\) 0 0
\(471\) 0 0
\(472\) 2.23872e6i 0.462535i
\(473\) 467328.i 0.0960437i
\(474\) 0 0
\(475\) 0 0
\(476\) 1.43866e6 0.291031
\(477\) 0 0
\(478\) − 2.38128e6i − 0.476695i
\(479\) −5.51052e6 −1.09737 −0.548686 0.836029i \(-0.684872\pi\)
−0.548686 + 0.836029i \(0.684872\pi\)
\(480\) 0 0
\(481\) −6.10291e6 −1.20275
\(482\) 1.09561e6i 0.214802i
\(483\) 0 0
\(484\) 1.98699e6 0.385552
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.51808e6i − 1.05430i −0.849771 0.527152i \(-0.823260\pi\)
0.849771 0.527152i \(-0.176740\pi\)
\(488\) 629632.i 0.119684i
\(489\) 0 0
\(490\) 0 0
\(491\) 1.51277e6 0.283184 0.141592 0.989925i \(-0.454778\pi\)
0.141592 + 0.989925i \(0.454778\pi\)
\(492\) 0 0
\(493\) 4.50342e6i 0.834498i
\(494\) −1.21218e7 −2.23485
\(495\) 0 0
\(496\) −1.75821e6 −0.320897
\(497\) − 8.28502e6i − 1.50454i
\(498\) 0 0
\(499\) 1.93042e6 0.347057 0.173528 0.984829i \(-0.444483\pi\)
0.173528 + 0.984829i \(0.444483\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 3.40301e6i − 0.602703i
\(503\) − 6.73105e6i − 1.18621i −0.805124 0.593106i \(-0.797901\pi\)
0.805124 0.593106i \(-0.202099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.20269e6 −0.208822
\(507\) 0 0
\(508\) 4.79811e6i 0.824919i
\(509\) −556650. −0.0952331 −0.0476165 0.998866i \(-0.515163\pi\)
−0.0476165 + 0.998866i \(0.515163\pi\)
\(510\) 0 0
\(511\) −2.59435e6 −0.439517
\(512\) 262144.i 0.0441942i
\(513\) 0 0
\(514\) −3.30161e6 −0.551211
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.51942e6i − 0.414548i
\(518\) − 2.60450e6i − 0.426481i
\(519\) 0 0
\(520\) 0 0
\(521\) −1.01110e7 −1.63192 −0.815962 0.578106i \(-0.803792\pi\)
−0.815962 + 0.578106i \(0.803792\pi\)
\(522\) 0 0
\(523\) − 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
\(528\) 0 0
\(529\) 3.98399e6 0.618983
\(530\) 0 0
\(531\) 0 0
\(532\) − 5.17312e6i − 0.792453i
\(533\) 418068.i 0.0637425i
\(534\) 0 0
\(535\) 0 0
\(536\) −2.15821e6 −0.324475
\(537\) 0 0
\(538\) − 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
\(543\) 0 0
\(544\) −780288. −0.113047
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.44024e6i − 0.634510i −0.948340 0.317255i \(-0.897239\pi\)
0.948340 0.317255i \(-0.102761\pi\)
\(548\) 2.62781e6i 0.373803i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.61934e7 2.27227
\(552\) 0 0
\(553\) − 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 \(-0.211071\pi\)
\(558\) 0 0
\(559\) 2.69200e6 0.364373
\(560\) 0 0
\(561\) 0 0
\(562\) 5.82679e6i 0.778196i
\(563\) 9.81287e6i 1.30474i 0.757899 + 0.652372i \(0.226226\pi\)
−0.757899 + 0.652372i \(0.773774\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 481576. 0.0631864
\(567\) 0 0
\(568\) 4.49357e6i 0.584414i
\(569\) 1.33152e7 1.72412 0.862061 0.506804i \(-0.169173\pi\)
0.862061 + 0.506804i \(0.169173\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −178416. −0.0226024
\(575\) 0 0
\(576\) 0 0
\(577\) − 4.50372e6i − 0.563160i −0.959538 0.281580i \(-0.909141\pi\)
0.959538 0.281580i \(-0.0908585\pi\)
\(578\) 3.35685e6i 0.417939i
\(579\) 0 0
\(580\) 0 0
\(581\) −1.28707e7 −1.58184
\(582\) 0 0
\(583\) − 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 \(0.988066\pi\)
\(588\) 0 0
\(589\) −1.88183e7 −2.23508
\(590\) 0 0
\(591\) 0 0
\(592\) 1.41261e6i 0.165660i
\(593\) 2.50385e6i 0.292397i 0.989255 + 0.146198i \(0.0467038\pi\)
−0.989255 + 0.146198i \(0.953296\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.22320e6 0.717626
\(597\) 0 0
\(598\) 6.92798e6i 0.792235i
\(599\) −756480. −0.0861451 −0.0430725 0.999072i \(-0.513715\pi\)
−0.0430725 + 0.999072i \(0.513715\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 1.56717e6 0.174793
\(605\) 0 0
\(606\) 0 0
\(607\) − 1.13772e7i − 1.25333i −0.779291 0.626663i \(-0.784420\pi\)
0.779291 0.626663i \(-0.215580\pi\)
\(608\) 2.80576e6i 0.307816i
\(609\) 0 0
\(610\) 0 0
\(611\) −1.45129e7 −1.57272
\(612\) 0 0
\(613\) − 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 \(0.137226\pi\)
−0.908503 + 0.417878i \(0.862774\pi\)
\(618\) 0 0
\(619\) −4.02362e6 −0.422076 −0.211038 0.977478i \(-0.567684\pi\)
−0.211038 + 0.977478i \(0.567684\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.71227e6i 0.384737i
\(623\) 4.54182e6i 0.468824i
\(624\) 0 0
\(625\) 0 0
\(626\) −9.18250e6 −0.936538
\(627\) 0 0
\(628\) − 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
\(633\) 0 0
\(634\) −1.09461e7 −1.08152
\(635\) 0 0
\(636\) 0 0
\(637\) 3.18860e6i 0.311352i
\(638\) − 4.53888e6i − 0.441466i
\(639\) 0 0
\(640\) 0 0
\(641\) −6.37390e6 −0.612718 −0.306359 0.951916i \(-0.599111\pi\)
−0.306359 + 0.951916i \(0.599111\pi\)
\(642\) 0 0
\(643\) 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 \(-0.865724\pi\)
0.912337 0.409440i \(-0.134276\pi\)
\(648\) 0 0
\(649\) 6.71616e6 0.625906
\(650\) 0 0
\(651\) 0 0
\(652\) 691744.i 0.0637274i
\(653\) 1.58477e6i 0.145440i 0.997352 + 0.0727201i \(0.0231680\pi\)
−0.997352 + 0.0727201i \(0.976832\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 96768.0 0.00877955
\(657\) 0 0
\(658\) − 6.19358e6i − 0.557670i
\(659\) 1.26410e7 1.13388 0.566940 0.823759i \(-0.308127\pi\)
0.566940 + 0.823759i \(0.308127\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 6.98074e6 0.614442
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.25506e6i − 0.805498i
\(668\) − 2.98435e6i − 0.258767i
\(669\) 0 0
\(670\) 0 0
\(671\) 1.88890e6 0.161958
\(672\) 0 0
\(673\) 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 \(-0.996857\pi\)
0.999951 0.00987467i \(-0.00314326\pi\)
\(678\) 0 0
\(679\) −226324. −0.0188389
\(680\) 0 0
\(681\) 0 0
\(682\) 5.27462e6i 0.434241i
\(683\) − 2.05830e7i − 1.68833i −0.536084 0.844164i \(-0.680097\pi\)
0.536084 0.844164i \(-0.319903\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9.29368e6 −0.754011
\(687\) 0 0
\(688\) − 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
\(693\) 0 0
\(694\) 9.30895e6 0.733672
\(695\) 0 0
\(696\) 0 0
\(697\) 288036.i 0.0224577i
\(698\) 1.24516e6i 0.0967357i
\(699\) 0 0
\(700\) 0 0
\(701\) −1.29304e6 −0.0993843 −0.0496921 0.998765i \(-0.515824\pi\)
−0.0496921 + 0.998765i \(0.515824\pi\)
\(702\) 0 0
\(703\) 1.51193e7i 1.15384i
\(704\) 786432. 0.0598039
\(705\) 0 0
\(706\) −1.23463e7 −0.932234
\(707\) − 9.15940e6i − 0.689157i
\(708\) 0 0
\(709\) 2.12720e7 1.58926 0.794628 0.607097i \(-0.207666\pi\)
0.794628 + 0.607097i \(0.207666\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 2.46336e6i − 0.182108i
\(713\) 1.07553e7i 0.792316i
\(714\) 0 0
\(715\) 0 0
\(716\) −4.35360e6 −0.317370
\(717\) 0 0
\(718\) − 1.41230e7i − 1.02239i
\(719\) 8.31732e6 0.600014 0.300007 0.953937i \(-0.403011\pi\)
0.300007 + 0.953937i \(0.403011\pi\)
\(720\) 0 0
\(721\) 5.51225e6 0.394903
\(722\) 2.01260e7i 1.43686i
\(723\) 0 0
\(724\) 1.20669e6 0.0855556
\(725\) 0 0
\(726\) 0 0
\(727\) 4.36740e6i 0.306469i 0.988190 + 0.153235i \(0.0489690\pi\)
−0.988190 + 0.153235i \(0.951031\pi\)
\(728\) 8.35251e6i 0.584102i
\(729\) 0 0
\(730\) 0 0
\(731\) 1.85471e6 0.128375
\(732\) 0 0
\(733\) − 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
\(738\) 0 0
\(739\) −768260. −0.0517484 −0.0258742 0.999665i \(-0.508237\pi\)
−0.0258742 + 0.999665i \(0.508237\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 4.33013e6i − 0.288729i
\(743\) − 6.18781e6i − 0.411211i −0.978635 0.205605i \(-0.934084\pi\)
0.978635 0.205605i \(-0.0659164\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.86102e6 0.451379
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) −2.61458e7 −1.67484
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.93494e7i − 1.22724i −0.789603 0.613618i \(-0.789714\pi\)
0.789603 0.613618i \(-0.210286\pi\)
\(758\) 1.24070e7i 0.784318i
\(759\) 0 0
\(760\) 0 0
\(761\) 3.01992e7 1.89031 0.945155 0.326621i \(-0.105910\pi\)
0.945155 + 0.326621i \(0.105910\pi\)
\(762\) 0 0
\(763\) − 2.44177e7i − 1.51843i
\(764\) −5.71181e6 −0.354030
\(765\) 0 0
\(766\) 2.12779e7 1.31026
\(767\) − 3.86879e7i − 2.37458i
\(768\) 0 0
\(769\) −2.15854e7 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.01910e6i 0.423876i
\(773\) − 3.90895e6i − 0.235294i −0.993055 0.117647i \(-0.962465\pi\)
0.993055 0.117647i \(-0.0375351\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 122752. 0.00731769
\(777\) 0 0
\(778\) 4.64580e6i 0.275177i
\(779\) 1.03572e6 0.0611503
\(780\) 0 0
\(781\) 1.34807e7 0.790833
\(782\) 4.77317e6i 0.279119i
\(783\) 0 0
\(784\) 738048. 0.0428839
\(785\) 0 0
\(786\) 0 0
\(787\) 2.65082e7i 1.52561i 0.646628 + 0.762806i \(0.276179\pi\)
−0.646628 + 0.762806i \(0.723821\pi\)
\(788\) 2.50877e6i 0.143928i
\(789\) 0 0
\(790\) 0 0
\(791\) 1.64475e7 0.934674
\(792\) 0 0
\(793\) − 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 \(0.902693\pi\)
\(798\) 0 0
\(799\) −9.99896e6 −0.554100
\(800\) 0 0
\(801\) 0 0
\(802\) 1.08973e7i 0.598249i
\(803\) − 4.22131e6i − 0.231025i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.03840e7 1.64743
\(807\) 0 0
\(808\) 4.96781e6i 0.267693i
\(809\) −1.11446e7 −0.598675 −0.299338 0.954147i \(-0.596766\pi\)
−0.299338 + 0.954147i \(0.596766\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 4.23782e6 0.224172
\(815\) 0 0
\(816\) 0 0
\(817\) − 6.66916e6i − 0.349555i
\(818\) − 7.12076e6i − 0.372086i
\(819\) 0 0
\(820\) 0 0
\(821\) −3.04347e7 −1.57584 −0.787918 0.615781i \(-0.788841\pi\)
−0.787918 + 0.615781i \(0.788841\pi\)
\(822\) 0 0
\(823\) 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 \(-0.857163\pi\)
0.900996 0.433827i \(-0.142837\pi\)
\(828\) 0 0
\(829\) 2.47617e7 1.25139 0.625697 0.780066i \(-0.284815\pi\)
0.625697 + 0.780066i \(0.284815\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 4.53018e6i − 0.226886i
\(833\) 2.19685e6i 0.109695i
\(834\) 0 0
\(835\) 0 0
\(836\) 8.41728e6 0.416539
\(837\) 0 0
\(838\) 2.60232e6i 0.128012i
\(839\) 3.16529e7 1.55242 0.776208 0.630476i \(-0.217140\pi\)
0.776208 + 0.630476i \(0.217140\pi\)
\(840\) 0 0
\(841\) 1.44170e7 0.702884
\(842\) − 1.41624e7i − 0.688425i
\(843\) 0 0
\(844\) 2.90637e6 0.140441
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.46541e7i − 0.701859i
\(848\) 2.34854e6i 0.112153i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.64119e6 0.409025
\(852\) 0 0
\(853\) 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 \(0.207489\pi\)
−0.794966 + 0.606655i \(0.792511\pi\)
\(858\) 0 0
\(859\) 3.38111e7 1.56342 0.781710 0.623642i \(-0.214348\pi\)
0.781710 + 0.623642i \(0.214348\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.19499e6i 0.100615i
\(863\) − 2.22817e7i − 1.01841i −0.860646 0.509204i \(-0.829940\pi\)
0.860646 0.509204i \(-0.170060\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 5.96966e6 0.270492
\(867\) 0 0
\(868\) 1.29668e7i 0.584162i
\(869\) 867840. 0.0389843
\(870\) 0 0
\(871\) 3.72965e7 1.66580
\(872\) 1.32435e7i 0.589810i
\(873\) 0 0
\(874\) 1.71634e7 0.760018
\(875\) 0 0
\(876\) 0 0
\(877\) 3.46748e7i 1.52235i 0.648545 + 0.761177i \(0.275378\pi\)
−0.648545 + 0.761177i \(0.724622\pi\)
\(878\) − 1.94485e7i − 0.851431i
\(879\) 0 0
\(880\) 0 0
\(881\) −1.42603e7 −0.618998 −0.309499 0.950900i \(-0.600161\pi\)
−0.309499 + 0.950900i \(0.600161\pi\)
\(882\) 0 0
\(883\) − 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 \(0.335800\pi\)
−0.493275 + 0.869873i \(0.664200\pi\)
\(888\) 0 0
\(889\) 3.53861e7 1.50168
\(890\) 0 0
\(891\) 0 0
\(892\) 4.61238e6i 0.194095i
\(893\) 3.59543e7i 1.50877i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.93331e6 0.0804511
\(897\) 0 0
\(898\) 1.49488e7i 0.618606i
\(899\) −4.05899e7 −1.67501
\(900\) 0 0
\(901\) −6.99059e6 −0.286881
\(902\) − 290304.i − 0.0118806i
\(903\) 0 0
\(904\) −8.92070e6 −0.363060
\(905\) 0 0
\(906\) 0 0
\(907\) 3.57116e7i 1.44142i 0.693235 + 0.720712i \(0.256185\pi\)
−0.693235 + 0.720712i \(0.743815\pi\)
\(908\) − 1.80084e7i − 0.724869i
\(909\) 0 0
\(910\) 0 0
\(911\) 2.11389e7 0.843893 0.421947 0.906621i \(-0.361347\pi\)
0.421947 + 0.906621i \(0.361347\pi\)
\(912\) 0 0
\(913\) − 2.09422e7i − 0.831468i
\(914\) −2.59310e7 −1.02673
\(915\) 0 0
\(916\) −6.65296e6 −0.261985
\(917\) − 928896.i − 0.0364791i
\(918\) 0 0
\(919\) −1.85996e7 −0.726465 −0.363233 0.931698i \(-0.618327\pi\)
−0.363233 + 0.931698i \(0.618327\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 6.03641e6i − 0.233857i
\(923\) − 7.76545e7i − 3.00028i
\(924\) 0 0
\(925\) 0 0
\(926\) −3.47360e7 −1.33123
\(927\) 0 0
\(928\) 6.05184e6i 0.230684i
\(929\) 4.45110e7 1.69211 0.846055 0.533096i \(-0.178972\pi\)
0.846055 + 0.533096i \(0.178972\pi\)
\(930\) 0 0
\(931\) 7.89942e6 0.298690
\(932\) 1.23294e7i 0.464945i
\(933\) 0 0
\(934\) −2.78565e7 −1.04486
\(935\) 0 0
\(936\) 0 0
\(937\) 2.19419e7i 0.816441i 0.912883 + 0.408221i \(0.133851\pi\)
−0.912883 + 0.408221i \(0.866149\pi\)
\(938\) 1.59168e7i 0.590675i
\(939\) 0 0
\(940\) 0 0
\(941\) 7.77722e6 0.286319 0.143160 0.989700i \(-0.454274\pi\)
0.143160 + 0.989700i \(0.454274\pi\)
\(942\) 0 0
\(943\) − 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 \(0.805126\pi\)
\(948\) 0 0
\(949\) −2.43165e7 −0.876468
\(950\) 0 0
\(951\) 0 0
\(952\) 5.75462e6i 0.205790i
\(953\) 5.60285e6i 0.199838i 0.994996 + 0.0999188i \(0.0318583\pi\)
−0.994996 + 0.0999188i \(0.968142\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9.52512e6 0.337074
\(957\) 0 0
\(958\) − 2.20421e7i − 0.775959i
\(959\) 1.93801e7 0.680470
\(960\) 0 0
\(961\) 1.85403e7 0.647601
\(962\) − 2.44116e7i − 0.850470i
\(963\) 0 0
\(964\) −4.38243e6 −0.151888
\(965\) 0 0
\(966\) 0 0
\(967\) 2.03532e7i 0.699949i 0.936759 + 0.349975i \(0.113810\pi\)
−0.936759 + 0.349975i \(0.886190\pi\)
\(968\) 7.94797e6i 0.272626i
\(969\) 0 0
\(970\) 0 0
\(971\) 2.34306e7 0.797510 0.398755 0.917057i \(-0.369442\pi\)
0.398755 + 0.917057i \(0.369442\pi\)
\(972\) 0 0
\(973\) 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 \(-0.256457\pi\)
\(978\) 0 0
\(979\) −7.39008e6 −0.246429
\(980\) 0 0
\(981\) 0 0
\(982\) 6.05107e6i 0.200241i
\(983\) 4.75003e7i 1.56788i 0.620837 + 0.783940i \(0.286793\pi\)
−0.620837 + 0.783940i \(0.713207\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.80137e7 −0.590079
\(987\) 0 0
\(988\) − 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
\(993\) 0 0
\(994\) 3.31401e7 1.06387
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.96332e7i − 0.944148i −0.881559 0.472074i \(-0.843505\pi\)
0.881559 0.472074i \(-0.156495\pi\)
\(998\) 7.72168e6i 0.245406i
\(999\) 0 0
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 450.6.c.f.199.2 2
3.2 odd 2 50.6.b.b.49.1 2
5.2 odd 4 90.6.a.b.1.1 1
5.3 odd 4 450.6.a.u.1.1 1
5.4 even 2 inner 450.6.c.f.199.1 2
12.11 even 2 400.6.c.i.49.1 2
15.2 even 4 10.6.a.c.1.1 1
15.8 even 4 50.6.a.b.1.1 1
15.14 odd 2 50.6.b.b.49.2 2
20.7 even 4 720.6.a.v.1.1 1
60.23 odd 4 400.6.a.i.1.1 1
60.47 odd 4 80.6.a.c.1.1 1
60.59 even 2 400.6.c.i.49.2 2
105.62 odd 4 490.6.a.k.1.1 1
120.77 even 4 320.6.a.f.1.1 1
120.107 odd 4 320.6.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 15.2 even 4
50.6.a.b.1.1 1 15.8 even 4
50.6.b.b.49.1 2 3.2 odd 2
50.6.b.b.49.2 2 15.14 odd 2
80.6.a.c.1.1 1 60.47 odd 4
90.6.a.b.1.1 1 5.2 odd 4
320.6.a.f.1.1 1 120.77 even 4
320.6.a.k.1.1 1 120.107 odd 4
400.6.a.i.1.1 1 60.23 odd 4
400.6.c.i.49.1 2 12.11 even 2
400.6.c.i.49.2 2 60.59 even 2
450.6.a.u.1.1 1 5.3 odd 4
450.6.c.f.199.1 2 5.4 even 2 inner
450.6.c.f.199.2 2 1.1 even 1 trivial
490.6.a.k.1.1 1 105.62 odd 4
720.6.a.v.1.1 1 20.7 even 4