Properties

Label 1200.4.f.m.49.1
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.m.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} +24.0000 q^{11} -74.0000i q^{13} +54.0000i q^{17} -124.000 q^{19} -60.0000 q^{21} -120.000i q^{23} +27.0000i q^{27} +78.0000 q^{29} -200.000 q^{31} -72.0000i q^{33} -70.0000i q^{37} -222.000 q^{39} +330.000 q^{41} +92.0000i q^{43} +24.0000i q^{47} -57.0000 q^{49} +162.000 q^{51} -450.000i q^{53} +372.000i q^{57} +24.0000 q^{59} -322.000 q^{61} +180.000i q^{63} +196.000i q^{67} -360.000 q^{69} +288.000 q^{71} +430.000i q^{73} -480.000i q^{77} -520.000 q^{79} +81.0000 q^{81} +156.000i q^{83} -234.000i q^{87} -1026.00 q^{89} -1480.00 q^{91} +600.000i q^{93} -286.000i q^{97} -216.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 48 q^{11} - 248 q^{19} - 120 q^{21} + 156 q^{29} - 400 q^{31} - 444 q^{39} + 660 q^{41} - 114 q^{49} + 324 q^{51} + 48 q^{59} - 644 q^{61} - 720 q^{69} + 576 q^{71} - 1040 q^{79} + 162 q^{81} - 2052 q^{89} - 2960 q^{91} - 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 20.0000i − 1.07990i −0.841698 0.539949i \(-0.818443\pi\)
0.841698 0.539949i \(-0.181557\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 0 0
\(13\) − 74.0000i − 1.57876i −0.613904 0.789381i \(-0.710402\pi\)
0.613904 0.789381i \(-0.289598\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 54.0000i 0.770407i 0.922832 + 0.385204i \(0.125869\pi\)
−0.922832 + 0.385204i \(0.874131\pi\)
\(18\) 0 0
\(19\) −124.000 −1.49724 −0.748620 0.663000i \(-0.769283\pi\)
−0.748620 + 0.663000i \(0.769283\pi\)
\(20\) 0 0
\(21\) −60.0000 −0.623480
\(22\) 0 0
\(23\) − 120.000i − 1.08790i −0.839117 0.543951i \(-0.816928\pi\)
0.839117 0.543951i \(-0.183072\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 78.0000 0.499456 0.249728 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) 0 0
\(31\) −200.000 −1.15874 −0.579372 0.815063i \(-0.696702\pi\)
−0.579372 + 0.815063i \(0.696702\pi\)
\(32\) 0 0
\(33\) − 72.0000i − 0.379806i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 70.0000i − 0.311025i −0.987834 0.155513i \(-0.950297\pi\)
0.987834 0.155513i \(-0.0497029\pi\)
\(38\) 0 0
\(39\) −222.000 −0.911499
\(40\) 0 0
\(41\) 330.000 1.25701 0.628504 0.777806i \(-0.283668\pi\)
0.628504 + 0.777806i \(0.283668\pi\)
\(42\) 0 0
\(43\) 92.0000i 0.326276i 0.986603 + 0.163138i \(0.0521616\pi\)
−0.986603 + 0.163138i \(0.947838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 24.0000i 0.0744843i 0.999306 + 0.0372421i \(0.0118573\pi\)
−0.999306 + 0.0372421i \(0.988143\pi\)
\(48\) 0 0
\(49\) −57.0000 −0.166181
\(50\) 0 0
\(51\) 162.000 0.444795
\(52\) 0 0
\(53\) − 450.000i − 1.16627i −0.812376 0.583134i \(-0.801826\pi\)
0.812376 0.583134i \(-0.198174\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 372.000i 0.864432i
\(58\) 0 0
\(59\) 24.0000 0.0529582 0.0264791 0.999649i \(-0.491570\pi\)
0.0264791 + 0.999649i \(0.491570\pi\)
\(60\) 0 0
\(61\) −322.000 −0.675867 −0.337933 0.941170i \(-0.609728\pi\)
−0.337933 + 0.941170i \(0.609728\pi\)
\(62\) 0 0
\(63\) 180.000i 0.359966i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 196.000i 0.357391i 0.983904 + 0.178696i \(0.0571877\pi\)
−0.983904 + 0.178696i \(0.942812\pi\)
\(68\) 0 0
\(69\) −360.000 −0.628100
\(70\) 0 0
\(71\) 288.000 0.481399 0.240699 0.970600i \(-0.422623\pi\)
0.240699 + 0.970600i \(0.422623\pi\)
\(72\) 0 0
\(73\) 430.000i 0.689420i 0.938709 + 0.344710i \(0.112023\pi\)
−0.938709 + 0.344710i \(0.887977\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 480.000i − 0.710404i
\(78\) 0 0
\(79\) −520.000 −0.740564 −0.370282 0.928919i \(-0.620739\pi\)
−0.370282 + 0.928919i \(0.620739\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 156.000i 0.206304i 0.994666 + 0.103152i \(0.0328928\pi\)
−0.994666 + 0.103152i \(0.967107\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 234.000i − 0.288361i
\(88\) 0 0
\(89\) −1026.00 −1.22198 −0.610988 0.791640i \(-0.709227\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) −1480.00 −1.70490
\(92\) 0 0
\(93\) 600.000i 0.669001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 286.000i − 0.299370i −0.988734 0.149685i \(-0.952174\pi\)
0.988734 0.149685i \(-0.0478260\pi\)
\(98\) 0 0
\(99\) −216.000 −0.219281
\(100\) 0 0
\(101\) −1734.00 −1.70831 −0.854156 0.520017i \(-0.825925\pi\)
−0.854156 + 0.520017i \(0.825925\pi\)
\(102\) 0 0
\(103\) 452.000i 0.432397i 0.976349 + 0.216198i \(0.0693658\pi\)
−0.976349 + 0.216198i \(0.930634\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1404.00i 1.26850i 0.773127 + 0.634251i \(0.218692\pi\)
−0.773127 + 0.634251i \(0.781308\pi\)
\(108\) 0 0
\(109\) 1474.00 1.29526 0.647631 0.761954i \(-0.275760\pi\)
0.647631 + 0.761954i \(0.275760\pi\)
\(110\) 0 0
\(111\) −210.000 −0.179570
\(112\) 0 0
\(113\) − 1086.00i − 0.904091i −0.891995 0.452046i \(-0.850694\pi\)
0.891995 0.452046i \(-0.149306\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 666.000i 0.526254i
\(118\) 0 0
\(119\) 1080.00 0.831962
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) − 990.000i − 0.725734i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1244.00i − 0.869190i −0.900626 0.434595i \(-0.856891\pi\)
0.900626 0.434595i \(-0.143109\pi\)
\(128\) 0 0
\(129\) 276.000 0.188376
\(130\) 0 0
\(131\) −2328.00 −1.55266 −0.776329 0.630327i \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) 0 0
\(133\) 2480.00i 1.61687i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2118.00i 1.32082i 0.750903 + 0.660412i \(0.229618\pi\)
−0.750903 + 0.660412i \(0.770382\pi\)
\(138\) 0 0
\(139\) 2324.00 1.41812 0.709062 0.705147i \(-0.249119\pi\)
0.709062 + 0.705147i \(0.249119\pi\)
\(140\) 0 0
\(141\) 72.0000 0.0430035
\(142\) 0 0
\(143\) − 1776.00i − 1.03858i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 171.000i 0.0959445i
\(148\) 0 0
\(149\) −258.000 −0.141854 −0.0709268 0.997482i \(-0.522596\pi\)
−0.0709268 + 0.997482i \(0.522596\pi\)
\(150\) 0 0
\(151\) 808.000 0.435458 0.217729 0.976009i \(-0.430135\pi\)
0.217729 + 0.976009i \(0.430135\pi\)
\(152\) 0 0
\(153\) − 486.000i − 0.256802i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2378.00i 1.20882i 0.796673 + 0.604411i \(0.206592\pi\)
−0.796673 + 0.604411i \(0.793408\pi\)
\(158\) 0 0
\(159\) −1350.00 −0.673346
\(160\) 0 0
\(161\) −2400.00 −1.17482
\(162\) 0 0
\(163\) − 52.0000i − 0.0249874i −0.999922 0.0124937i \(-0.996023\pi\)
0.999922 0.0124937i \(-0.00397698\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3720.00i 1.72373i 0.507141 + 0.861863i \(0.330702\pi\)
−0.507141 + 0.861863i \(0.669298\pi\)
\(168\) 0 0
\(169\) −3279.00 −1.49249
\(170\) 0 0
\(171\) 1116.00 0.499080
\(172\) 0 0
\(173\) − 426.000i − 0.187215i −0.995609 0.0936075i \(-0.970160\pi\)
0.995609 0.0936075i \(-0.0298399\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 72.0000i − 0.0305754i
\(178\) 0 0
\(179\) −1440.00 −0.601289 −0.300644 0.953736i \(-0.597202\pi\)
−0.300644 + 0.953736i \(0.597202\pi\)
\(180\) 0 0
\(181\) −3130.00 −1.28537 −0.642683 0.766133i \(-0.722179\pi\)
−0.642683 + 0.766133i \(0.722179\pi\)
\(182\) 0 0
\(183\) 966.000i 0.390212i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1296.00i 0.506807i
\(188\) 0 0
\(189\) 540.000 0.207827
\(190\) 0 0
\(191\) −3576.00 −1.35471 −0.677357 0.735655i \(-0.736875\pi\)
−0.677357 + 0.735655i \(0.736875\pi\)
\(192\) 0 0
\(193\) − 2666.00i − 0.994315i −0.867660 0.497158i \(-0.834377\pi\)
0.867660 0.497158i \(-0.165623\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2718.00i − 0.982992i −0.870880 0.491496i \(-0.836450\pi\)
0.870880 0.491496i \(-0.163550\pi\)
\(198\) 0 0
\(199\) −3832.00 −1.36504 −0.682521 0.730866i \(-0.739116\pi\)
−0.682521 + 0.730866i \(0.739116\pi\)
\(200\) 0 0
\(201\) 588.000 0.206340
\(202\) 0 0
\(203\) − 1560.00i − 0.539362i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1080.00i 0.362634i
\(208\) 0 0
\(209\) −2976.00 −0.984948
\(210\) 0 0
\(211\) −1100.00 −0.358896 −0.179448 0.983767i \(-0.557431\pi\)
−0.179448 + 0.983767i \(0.557431\pi\)
\(212\) 0 0
\(213\) − 864.000i − 0.277936i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4000.00i 1.25133i
\(218\) 0 0
\(219\) 1290.00 0.398037
\(220\) 0 0
\(221\) 3996.00 1.21629
\(222\) 0 0
\(223\) 1964.00i 0.589772i 0.955532 + 0.294886i \(0.0952817\pi\)
−0.955532 + 0.294886i \(0.904718\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 660.000i − 0.192977i −0.995334 0.0964884i \(-0.969239\pi\)
0.995334 0.0964884i \(-0.0307611\pi\)
\(228\) 0 0
\(229\) 1906.00 0.550009 0.275004 0.961443i \(-0.411321\pi\)
0.275004 + 0.961443i \(0.411321\pi\)
\(230\) 0 0
\(231\) −1440.00 −0.410152
\(232\) 0 0
\(233\) 1458.00i 0.409943i 0.978768 + 0.204972i \(0.0657102\pi\)
−0.978768 + 0.204972i \(0.934290\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1560.00i 0.427565i
\(238\) 0 0
\(239\) 1176.00 0.318281 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(240\) 0 0
\(241\) 866.000 0.231469 0.115734 0.993280i \(-0.463078\pi\)
0.115734 + 0.993280i \(0.463078\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9176.00i 2.36379i
\(248\) 0 0
\(249\) 468.000 0.119110
\(250\) 0 0
\(251\) −432.000 −0.108636 −0.0543179 0.998524i \(-0.517298\pi\)
−0.0543179 + 0.998524i \(0.517298\pi\)
\(252\) 0 0
\(253\) − 2880.00i − 0.715668i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2526.00i 0.613103i 0.951854 + 0.306552i \(0.0991752\pi\)
−0.951854 + 0.306552i \(0.900825\pi\)
\(258\) 0 0
\(259\) −1400.00 −0.335876
\(260\) 0 0
\(261\) −702.000 −0.166485
\(262\) 0 0
\(263\) 5448.00i 1.27733i 0.769484 + 0.638666i \(0.220513\pi\)
−0.769484 + 0.638666i \(0.779487\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3078.00i 0.705508i
\(268\) 0 0
\(269\) 2574.00 0.583418 0.291709 0.956507i \(-0.405776\pi\)
0.291709 + 0.956507i \(0.405776\pi\)
\(270\) 0 0
\(271\) 3184.00 0.713706 0.356853 0.934161i \(-0.383850\pi\)
0.356853 + 0.934161i \(0.383850\pi\)
\(272\) 0 0
\(273\) 4440.00i 0.984326i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3962.00i 0.859399i 0.902972 + 0.429699i \(0.141380\pi\)
−0.902972 + 0.429699i \(0.858620\pi\)
\(278\) 0 0
\(279\) 1800.00 0.386248
\(280\) 0 0
\(281\) −8286.00 −1.75908 −0.879540 0.475825i \(-0.842149\pi\)
−0.879540 + 0.475825i \(0.842149\pi\)
\(282\) 0 0
\(283\) − 2716.00i − 0.570493i −0.958454 0.285246i \(-0.907925\pi\)
0.958454 0.285246i \(-0.0920754\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6600.00i − 1.35744i
\(288\) 0 0
\(289\) 1997.00 0.406473
\(290\) 0 0
\(291\) −858.000 −0.172841
\(292\) 0 0
\(293\) − 6018.00i − 1.19992i −0.800032 0.599958i \(-0.795184\pi\)
0.800032 0.599958i \(-0.204816\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 648.000i 0.126602i
\(298\) 0 0
\(299\) −8880.00 −1.71754
\(300\) 0 0
\(301\) 1840.00 0.352345
\(302\) 0 0
\(303\) 5202.00i 0.986294i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 9236.00i − 1.71702i −0.512793 0.858512i \(-0.671389\pi\)
0.512793 0.858512i \(-0.328611\pi\)
\(308\) 0 0
\(309\) 1356.00 0.249644
\(310\) 0 0
\(311\) −1536.00 −0.280060 −0.140030 0.990147i \(-0.544720\pi\)
−0.140030 + 0.990147i \(0.544720\pi\)
\(312\) 0 0
\(313\) 7342.00i 1.32586i 0.748681 + 0.662930i \(0.230687\pi\)
−0.748681 + 0.662930i \(0.769313\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3894.00i − 0.689933i −0.938615 0.344967i \(-0.887890\pi\)
0.938615 0.344967i \(-0.112110\pi\)
\(318\) 0 0
\(319\) 1872.00 0.328564
\(320\) 0 0
\(321\) 4212.00 0.732370
\(322\) 0 0
\(323\) − 6696.00i − 1.15348i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 4422.00i − 0.747820i
\(328\) 0 0
\(329\) 480.000 0.0804354
\(330\) 0 0
\(331\) −3692.00 −0.613084 −0.306542 0.951857i \(-0.599172\pi\)
−0.306542 + 0.951857i \(0.599172\pi\)
\(332\) 0 0
\(333\) 630.000i 0.103675i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 8998.00i − 1.45446i −0.686395 0.727229i \(-0.740808\pi\)
0.686395 0.727229i \(-0.259192\pi\)
\(338\) 0 0
\(339\) −3258.00 −0.521977
\(340\) 0 0
\(341\) −4800.00 −0.762271
\(342\) 0 0
\(343\) − 5720.00i − 0.900440i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 5244.00i − 0.811276i −0.914034 0.405638i \(-0.867049\pi\)
0.914034 0.405638i \(-0.132951\pi\)
\(348\) 0 0
\(349\) −6302.00 −0.966585 −0.483293 0.875459i \(-0.660559\pi\)
−0.483293 + 0.875459i \(0.660559\pi\)
\(350\) 0 0
\(351\) 1998.00 0.303833
\(352\) 0 0
\(353\) − 3414.00i − 0.514756i −0.966311 0.257378i \(-0.917141\pi\)
0.966311 0.257378i \(-0.0828586\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3240.00i − 0.480333i
\(358\) 0 0
\(359\) 4824.00 0.709195 0.354597 0.935019i \(-0.384618\pi\)
0.354597 + 0.935019i \(0.384618\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 0 0
\(363\) 2265.00i 0.327498i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3508.00i 0.498954i 0.968381 + 0.249477i \(0.0802587\pi\)
−0.968381 + 0.249477i \(0.919741\pi\)
\(368\) 0 0
\(369\) −2970.00 −0.419003
\(370\) 0 0
\(371\) −9000.00 −1.25945
\(372\) 0 0
\(373\) − 10802.0i − 1.49948i −0.661732 0.749740i \(-0.730178\pi\)
0.661732 0.749740i \(-0.269822\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5772.00i − 0.788523i
\(378\) 0 0
\(379\) 1460.00 0.197876 0.0989382 0.995094i \(-0.468455\pi\)
0.0989382 + 0.995094i \(0.468455\pi\)
\(380\) 0 0
\(381\) −3732.00 −0.501827
\(382\) 0 0
\(383\) − 4872.00i − 0.649994i −0.945715 0.324997i \(-0.894637\pi\)
0.945715 0.324997i \(-0.105363\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 828.000i − 0.108759i
\(388\) 0 0
\(389\) 14046.0 1.83075 0.915373 0.402606i \(-0.131896\pi\)
0.915373 + 0.402606i \(0.131896\pi\)
\(390\) 0 0
\(391\) 6480.00 0.838127
\(392\) 0 0
\(393\) 6984.00i 0.896428i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2734.00i − 0.345631i −0.984954 0.172816i \(-0.944714\pi\)
0.984954 0.172816i \(-0.0552864\pi\)
\(398\) 0 0
\(399\) 7440.00 0.933498
\(400\) 0 0
\(401\) −15942.0 −1.98530 −0.992650 0.121019i \(-0.961384\pi\)
−0.992650 + 0.121019i \(0.961384\pi\)
\(402\) 0 0
\(403\) 14800.0i 1.82938i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1680.00i − 0.204606i
\(408\) 0 0
\(409\) −8714.00 −1.05350 −0.526748 0.850022i \(-0.676589\pi\)
−0.526748 + 0.850022i \(0.676589\pi\)
\(410\) 0 0
\(411\) 6354.00 0.762578
\(412\) 0 0
\(413\) − 480.000i − 0.0571895i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 6972.00i − 0.818754i
\(418\) 0 0
\(419\) 11976.0 1.39634 0.698169 0.715933i \(-0.253998\pi\)
0.698169 + 0.715933i \(0.253998\pi\)
\(420\) 0 0
\(421\) 11054.0 1.27967 0.639833 0.768514i \(-0.279004\pi\)
0.639833 + 0.768514i \(0.279004\pi\)
\(422\) 0 0
\(423\) − 216.000i − 0.0248281i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6440.00i 0.729868i
\(428\) 0 0
\(429\) −5328.00 −0.599623
\(430\) 0 0
\(431\) −720.000 −0.0804668 −0.0402334 0.999190i \(-0.512810\pi\)
−0.0402334 + 0.999190i \(0.512810\pi\)
\(432\) 0 0
\(433\) 15622.0i 1.73382i 0.498462 + 0.866912i \(0.333898\pi\)
−0.498462 + 0.866912i \(0.666102\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14880.0i 1.62885i
\(438\) 0 0
\(439\) −9880.00 −1.07414 −0.537069 0.843538i \(-0.680469\pi\)
−0.537069 + 0.843538i \(0.680469\pi\)
\(440\) 0 0
\(441\) 513.000 0.0553936
\(442\) 0 0
\(443\) − 16116.0i − 1.72843i −0.503123 0.864215i \(-0.667816\pi\)
0.503123 0.864215i \(-0.332184\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 774.000i 0.0818992i
\(448\) 0 0
\(449\) −9018.00 −0.947852 −0.473926 0.880565i \(-0.657164\pi\)
−0.473926 + 0.880565i \(0.657164\pi\)
\(450\) 0 0
\(451\) 7920.00 0.826914
\(452\) 0 0
\(453\) − 2424.00i − 0.251412i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3670.00i − 0.375657i −0.982202 0.187829i \(-0.939855\pi\)
0.982202 0.187829i \(-0.0601450\pi\)
\(458\) 0 0
\(459\) −1458.00 −0.148265
\(460\) 0 0
\(461\) 17562.0 1.77428 0.887141 0.461499i \(-0.152688\pi\)
0.887141 + 0.461499i \(0.152688\pi\)
\(462\) 0 0
\(463\) 1172.00i 0.117640i 0.998269 + 0.0588202i \(0.0187338\pi\)
−0.998269 + 0.0588202i \(0.981266\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6876.00i − 0.681335i −0.940184 0.340667i \(-0.889347\pi\)
0.940184 0.340667i \(-0.110653\pi\)
\(468\) 0 0
\(469\) 3920.00 0.385946
\(470\) 0 0
\(471\) 7134.00 0.697914
\(472\) 0 0
\(473\) 2208.00i 0.214638i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4050.00i 0.388756i
\(478\) 0 0
\(479\) 2280.00 0.217486 0.108743 0.994070i \(-0.465317\pi\)
0.108743 + 0.994070i \(0.465317\pi\)
\(480\) 0 0
\(481\) −5180.00 −0.491035
\(482\) 0 0
\(483\) 7200.00i 0.678284i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3076.00i 0.286215i 0.989707 + 0.143108i \(0.0457095\pi\)
−0.989707 + 0.143108i \(0.954290\pi\)
\(488\) 0 0
\(489\) −156.000 −0.0144265
\(490\) 0 0
\(491\) 18912.0 1.73826 0.869131 0.494582i \(-0.164679\pi\)
0.869131 + 0.494582i \(0.164679\pi\)
\(492\) 0 0
\(493\) 4212.00i 0.384785i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5760.00i − 0.519862i
\(498\) 0 0
\(499\) 9956.00 0.893170 0.446585 0.894741i \(-0.352640\pi\)
0.446585 + 0.894741i \(0.352640\pi\)
\(500\) 0 0
\(501\) 11160.0 0.995194
\(502\) 0 0
\(503\) − 10656.0i − 0.944588i −0.881441 0.472294i \(-0.843426\pi\)
0.881441 0.472294i \(-0.156574\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9837.00i 0.861689i
\(508\) 0 0
\(509\) 2766.00 0.240866 0.120433 0.992721i \(-0.461572\pi\)
0.120433 + 0.992721i \(0.461572\pi\)
\(510\) 0 0
\(511\) 8600.00 0.744504
\(512\) 0 0
\(513\) − 3348.00i − 0.288144i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 576.000i 0.0489989i
\(518\) 0 0
\(519\) −1278.00 −0.108089
\(520\) 0 0
\(521\) 10530.0 0.885466 0.442733 0.896654i \(-0.354009\pi\)
0.442733 + 0.896654i \(0.354009\pi\)
\(522\) 0 0
\(523\) 12692.0i 1.06115i 0.847637 + 0.530576i \(0.178024\pi\)
−0.847637 + 0.530576i \(0.821976\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 10800.0i − 0.892705i
\(528\) 0 0
\(529\) −2233.00 −0.183529
\(530\) 0 0
\(531\) −216.000 −0.0176527
\(532\) 0 0
\(533\) − 24420.0i − 1.98452i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4320.00i 0.347154i
\(538\) 0 0
\(539\) −1368.00 −0.109321
\(540\) 0 0
\(541\) 18110.0 1.43920 0.719602 0.694386i \(-0.244324\pi\)
0.719602 + 0.694386i \(0.244324\pi\)
\(542\) 0 0
\(543\) 9390.00i 0.742106i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 3620.00i − 0.282962i −0.989941 0.141481i \(-0.954814\pi\)
0.989941 0.141481i \(-0.0451864\pi\)
\(548\) 0 0
\(549\) 2898.00 0.225289
\(550\) 0 0
\(551\) −9672.00 −0.747806
\(552\) 0 0
\(553\) 10400.0i 0.799734i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 14166.0i − 1.07762i −0.842428 0.538809i \(-0.818875\pi\)
0.842428 0.538809i \(-0.181125\pi\)
\(558\) 0 0
\(559\) 6808.00 0.515112
\(560\) 0 0
\(561\) 3888.00 0.292605
\(562\) 0 0
\(563\) − 13404.0i − 1.00339i −0.865043 0.501697i \(-0.832709\pi\)
0.865043 0.501697i \(-0.167291\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1620.00i − 0.119989i
\(568\) 0 0
\(569\) 18654.0 1.37437 0.687185 0.726483i \(-0.258846\pi\)
0.687185 + 0.726483i \(0.258846\pi\)
\(570\) 0 0
\(571\) 7684.00 0.563162 0.281581 0.959537i \(-0.409141\pi\)
0.281581 + 0.959537i \(0.409141\pi\)
\(572\) 0 0
\(573\) 10728.0i 0.782144i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1726.00i − 0.124531i −0.998060 0.0622654i \(-0.980167\pi\)
0.998060 0.0622654i \(-0.0198325\pi\)
\(578\) 0 0
\(579\) −7998.00 −0.574068
\(580\) 0 0
\(581\) 3120.00 0.222787
\(582\) 0 0
\(583\) − 10800.0i − 0.767222i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 10596.0i − 0.745049i −0.928022 0.372524i \(-0.878492\pi\)
0.928022 0.372524i \(-0.121508\pi\)
\(588\) 0 0
\(589\) 24800.0 1.73492
\(590\) 0 0
\(591\) −8154.00 −0.567531
\(592\) 0 0
\(593\) − 2862.00i − 0.198193i −0.995078 0.0990963i \(-0.968405\pi\)
0.995078 0.0990963i \(-0.0315952\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11496.0i 0.788107i
\(598\) 0 0
\(599\) −23592.0 −1.60925 −0.804627 0.593781i \(-0.797635\pi\)
−0.804627 + 0.593781i \(0.797635\pi\)
\(600\) 0 0
\(601\) −9574.00 −0.649803 −0.324902 0.945748i \(-0.605331\pi\)
−0.324902 + 0.945748i \(0.605331\pi\)
\(602\) 0 0
\(603\) − 1764.00i − 0.119130i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 17444.0i − 1.16644i −0.812314 0.583221i \(-0.801792\pi\)
0.812314 0.583221i \(-0.198208\pi\)
\(608\) 0 0
\(609\) −4680.00 −0.311401
\(610\) 0 0
\(611\) 1776.00 0.117593
\(612\) 0 0
\(613\) 2374.00i 0.156419i 0.996937 + 0.0782096i \(0.0249203\pi\)
−0.996937 + 0.0782096i \(0.975080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12162.0i − 0.793555i −0.917915 0.396778i \(-0.870128\pi\)
0.917915 0.396778i \(-0.129872\pi\)
\(618\) 0 0
\(619\) 8804.00 0.571668 0.285834 0.958279i \(-0.407729\pi\)
0.285834 + 0.958279i \(0.407729\pi\)
\(620\) 0 0
\(621\) 3240.00 0.209367
\(622\) 0 0
\(623\) 20520.0i 1.31961i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8928.00i 0.568660i
\(628\) 0 0
\(629\) 3780.00 0.239616
\(630\) 0 0
\(631\) 12688.0 0.800478 0.400239 0.916411i \(-0.368927\pi\)
0.400239 + 0.916411i \(0.368927\pi\)
\(632\) 0 0
\(633\) 3300.00i 0.207209i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4218.00i 0.262360i
\(638\) 0 0
\(639\) −2592.00 −0.160466
\(640\) 0 0
\(641\) −9150.00 −0.563812 −0.281906 0.959442i \(-0.590967\pi\)
−0.281906 + 0.959442i \(0.590967\pi\)
\(642\) 0 0
\(643\) 25292.0i 1.55120i 0.631227 + 0.775598i \(0.282552\pi\)
−0.631227 + 0.775598i \(0.717448\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2736.00i 0.166249i 0.996539 + 0.0831246i \(0.0264900\pi\)
−0.996539 + 0.0831246i \(0.973510\pi\)
\(648\) 0 0
\(649\) 576.000 0.0348382
\(650\) 0 0
\(651\) 12000.0 0.722453
\(652\) 0 0
\(653\) − 22218.0i − 1.33148i −0.746183 0.665741i \(-0.768116\pi\)
0.746183 0.665741i \(-0.231884\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 3870.00i − 0.229807i
\(658\) 0 0
\(659\) 14520.0 0.858299 0.429149 0.903234i \(-0.358813\pi\)
0.429149 + 0.903234i \(0.358813\pi\)
\(660\) 0 0
\(661\) −10618.0 −0.624799 −0.312400 0.949951i \(-0.601133\pi\)
−0.312400 + 0.949951i \(0.601133\pi\)
\(662\) 0 0
\(663\) − 11988.0i − 0.702225i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 9360.00i − 0.543359i
\(668\) 0 0
\(669\) 5892.00 0.340505
\(670\) 0 0
\(671\) −7728.00 −0.444614
\(672\) 0 0
\(673\) − 1370.00i − 0.0784690i −0.999230 0.0392345i \(-0.987508\pi\)
0.999230 0.0392345i \(-0.0124919\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 13758.0i − 0.781038i −0.920595 0.390519i \(-0.872296\pi\)
0.920595 0.390519i \(-0.127704\pi\)
\(678\) 0 0
\(679\) −5720.00 −0.323289
\(680\) 0 0
\(681\) −1980.00 −0.111415
\(682\) 0 0
\(683\) 11988.0i 0.671608i 0.941932 + 0.335804i \(0.109008\pi\)
−0.941932 + 0.335804i \(0.890992\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 5718.00i − 0.317548i
\(688\) 0 0
\(689\) −33300.0 −1.84126
\(690\) 0 0
\(691\) −32996.0 −1.81654 −0.908268 0.418388i \(-0.862595\pi\)
−0.908268 + 0.418388i \(0.862595\pi\)
\(692\) 0 0
\(693\) 4320.00i 0.236801i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 17820.0i 0.968408i
\(698\) 0 0
\(699\) 4374.00 0.236681
\(700\) 0 0
\(701\) −25902.0 −1.39558 −0.697792 0.716300i \(-0.745834\pi\)
−0.697792 + 0.716300i \(0.745834\pi\)
\(702\) 0 0
\(703\) 8680.00i 0.465679i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 34680.0i 1.84480i
\(708\) 0 0
\(709\) 27394.0 1.45106 0.725531 0.688189i \(-0.241594\pi\)
0.725531 + 0.688189i \(0.241594\pi\)
\(710\) 0 0
\(711\) 4680.00 0.246855
\(712\) 0 0
\(713\) 24000.0i 1.26060i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3528.00i − 0.183760i
\(718\) 0 0
\(719\) 34848.0 1.80753 0.903763 0.428033i \(-0.140793\pi\)
0.903763 + 0.428033i \(0.140793\pi\)
\(720\) 0 0
\(721\) 9040.00 0.466945
\(722\) 0 0
\(723\) − 2598.00i − 0.133639i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 28028.0i − 1.42985i −0.699201 0.714925i \(-0.746461\pi\)
0.699201 0.714925i \(-0.253539\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −4968.00 −0.251365
\(732\) 0 0
\(733\) − 18002.0i − 0.907120i −0.891226 0.453560i \(-0.850154\pi\)
0.891226 0.453560i \(-0.149846\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4704.00i 0.235107i
\(738\) 0 0
\(739\) 15284.0 0.760800 0.380400 0.924822i \(-0.375786\pi\)
0.380400 + 0.924822i \(0.375786\pi\)
\(740\) 0 0
\(741\) 27528.0 1.36473
\(742\) 0 0
\(743\) − 18768.0i − 0.926691i −0.886178 0.463345i \(-0.846649\pi\)
0.886178 0.463345i \(-0.153351\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 1404.00i − 0.0687680i
\(748\) 0 0
\(749\) 28080.0 1.36985
\(750\) 0 0
\(751\) −8696.00 −0.422532 −0.211266 0.977429i \(-0.567759\pi\)
−0.211266 + 0.977429i \(0.567759\pi\)
\(752\) 0 0
\(753\) 1296.00i 0.0627209i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 38662.0i − 1.85627i −0.372247 0.928134i \(-0.621413\pi\)
0.372247 0.928134i \(-0.378587\pi\)
\(758\) 0 0
\(759\) −8640.00 −0.413191
\(760\) 0 0
\(761\) 23874.0 1.13723 0.568615 0.822604i \(-0.307479\pi\)
0.568615 + 0.822604i \(0.307479\pi\)
\(762\) 0 0
\(763\) − 29480.0i − 1.39875i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1776.00i − 0.0836084i
\(768\) 0 0
\(769\) −23618.0 −1.10753 −0.553763 0.832675i \(-0.686808\pi\)
−0.553763 + 0.832675i \(0.686808\pi\)
\(770\) 0 0
\(771\) 7578.00 0.353975
\(772\) 0 0
\(773\) − 11538.0i − 0.536860i −0.963299 0.268430i \(-0.913495\pi\)
0.963299 0.268430i \(-0.0865049\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 4200.00i 0.193918i
\(778\) 0 0
\(779\) −40920.0 −1.88204
\(780\) 0 0
\(781\) 6912.00 0.316685
\(782\) 0 0
\(783\) 2106.00i 0.0961204i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 14884.0i 0.674152i 0.941478 + 0.337076i \(0.109438\pi\)
−0.941478 + 0.337076i \(0.890562\pi\)
\(788\) 0 0
\(789\) 16344.0 0.737467
\(790\) 0 0
\(791\) −21720.0 −0.976327
\(792\) 0 0
\(793\) 23828.0i 1.06703i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 11334.0i − 0.503728i −0.967763 0.251864i \(-0.918957\pi\)
0.967763 0.251864i \(-0.0810435\pi\)
\(798\) 0 0
\(799\) −1296.00 −0.0573832
\(800\) 0 0
\(801\) 9234.00 0.407325
\(802\) 0 0
\(803\) 10320.0i 0.453530i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 7722.00i − 0.336837i
\(808\) 0 0
\(809\) −44730.0 −1.94391 −0.971955 0.235167i \(-0.924436\pi\)
−0.971955 + 0.235167i \(0.924436\pi\)
\(810\) 0 0
\(811\) 42748.0 1.85091 0.925453 0.378862i \(-0.123684\pi\)
0.925453 + 0.378862i \(0.123684\pi\)
\(812\) 0 0
\(813\) − 9552.00i − 0.412058i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 11408.0i − 0.488513i
\(818\) 0 0
\(819\) 13320.0 0.568301
\(820\) 0 0
\(821\) −31686.0 −1.34695 −0.673477 0.739208i \(-0.735200\pi\)
−0.673477 + 0.739208i \(0.735200\pi\)
\(822\) 0 0
\(823\) 11036.0i 0.467425i 0.972306 + 0.233713i \(0.0750875\pi\)
−0.972306 + 0.233713i \(0.924913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 25884.0i − 1.08836i −0.838968 0.544181i \(-0.816841\pi\)
0.838968 0.544181i \(-0.183159\pi\)
\(828\) 0 0
\(829\) −15950.0 −0.668234 −0.334117 0.942532i \(-0.608438\pi\)
−0.334117 + 0.942532i \(0.608438\pi\)
\(830\) 0 0
\(831\) 11886.0 0.496174
\(832\) 0 0
\(833\) − 3078.00i − 0.128027i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 5400.00i − 0.223000i
\(838\) 0 0
\(839\) 13800.0 0.567853 0.283927 0.958846i \(-0.408363\pi\)
0.283927 + 0.958846i \(0.408363\pi\)
\(840\) 0 0
\(841\) −18305.0 −0.750543
\(842\) 0 0
\(843\) 24858.0i 1.01560i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 15100.0i 0.612565i
\(848\) 0 0
\(849\) −8148.00 −0.329374
\(850\) 0 0
\(851\) −8400.00 −0.338365
\(852\) 0 0
\(853\) 27862.0i 1.11838i 0.829040 + 0.559189i \(0.188887\pi\)
−0.829040 + 0.559189i \(0.811113\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 7314.00i − 0.291530i −0.989319 0.145765i \(-0.953436\pi\)
0.989319 0.145765i \(-0.0465644\pi\)
\(858\) 0 0
\(859\) −28780.0 −1.14314 −0.571572 0.820552i \(-0.693666\pi\)
−0.571572 + 0.820552i \(0.693666\pi\)
\(860\) 0 0
\(861\) −19800.0 −0.783719
\(862\) 0 0
\(863\) − 32688.0i − 1.28935i −0.764455 0.644677i \(-0.776992\pi\)
0.764455 0.644677i \(-0.223008\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 5991.00i − 0.234677i
\(868\) 0 0
\(869\) −12480.0 −0.487175
\(870\) 0 0
\(871\) 14504.0 0.564236
\(872\) 0 0
\(873\) 2574.00i 0.0997900i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 36650.0i 1.41115i 0.708633 + 0.705577i \(0.249312\pi\)
−0.708633 + 0.705577i \(0.750688\pi\)
\(878\) 0 0
\(879\) −18054.0 −0.692772
\(880\) 0 0
\(881\) −2646.00 −0.101187 −0.0505936 0.998719i \(-0.516111\pi\)
−0.0505936 + 0.998719i \(0.516111\pi\)
\(882\) 0 0
\(883\) 10892.0i 0.415113i 0.978223 + 0.207557i \(0.0665511\pi\)
−0.978223 + 0.207557i \(0.933449\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 43464.0i 1.64530i 0.568550 + 0.822648i \(0.307504\pi\)
−0.568550 + 0.822648i \(0.692496\pi\)
\(888\) 0 0
\(889\) −24880.0 −0.938637
\(890\) 0 0
\(891\) 1944.00 0.0730937
\(892\) 0 0
\(893\) − 2976.00i − 0.111521i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 26640.0i 0.991621i
\(898\) 0 0
\(899\) −15600.0 −0.578742
\(900\) 0 0
\(901\) 24300.0 0.898502
\(902\) 0 0
\(903\) − 5520.00i − 0.203426i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14884.0i 0.544890i 0.962171 + 0.272445i \(0.0878323\pi\)
−0.962171 + 0.272445i \(0.912168\pi\)
\(908\) 0 0
\(909\) 15606.0 0.569437
\(910\) 0 0
\(911\) 1248.00 0.0453876 0.0226938 0.999742i \(-0.492776\pi\)
0.0226938 + 0.999742i \(0.492776\pi\)
\(912\) 0 0
\(913\) 3744.00i 0.135716i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 46560.0i 1.67671i
\(918\) 0 0
\(919\) −6640.00 −0.238339 −0.119169 0.992874i \(-0.538023\pi\)
−0.119169 + 0.992874i \(0.538023\pi\)
\(920\) 0 0
\(921\) −27708.0 −0.991324
\(922\) 0 0
\(923\) − 21312.0i − 0.760014i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4068.00i − 0.144132i
\(928\) 0 0
\(929\) −29946.0 −1.05758 −0.528792 0.848751i \(-0.677355\pi\)
−0.528792 + 0.848751i \(0.677355\pi\)
\(930\) 0 0
\(931\) 7068.00 0.248812
\(932\) 0 0
\(933\) 4608.00i 0.161693i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 45002.0i 1.56900i 0.620130 + 0.784499i \(0.287080\pi\)
−0.620130 + 0.784499i \(0.712920\pi\)
\(938\) 0 0
\(939\) 22026.0 0.765486
\(940\) 0 0
\(941\) 6090.00 0.210976 0.105488 0.994421i \(-0.466360\pi\)
0.105488 + 0.994421i \(0.466360\pi\)
\(942\) 0 0
\(943\) − 39600.0i − 1.36750i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 56388.0i − 1.93491i −0.253035 0.967457i \(-0.581429\pi\)
0.253035 0.967457i \(-0.418571\pi\)
\(948\) 0 0
\(949\) 31820.0 1.08843
\(950\) 0 0
\(951\) −11682.0 −0.398333
\(952\) 0 0
\(953\) − 10854.0i − 0.368936i −0.982839 0.184468i \(-0.940944\pi\)
0.982839 0.184468i \(-0.0590561\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 5616.00i − 0.189696i
\(958\) 0 0
\(959\) 42360.0 1.42636
\(960\) 0 0
\(961\) 10209.0 0.342687
\(962\) 0 0
\(963\) − 12636.0i − 0.422834i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42316.0i 1.40723i 0.710582 + 0.703615i \(0.248432\pi\)
−0.710582 + 0.703615i \(0.751568\pi\)
\(968\) 0 0
\(969\) −20088.0 −0.665964
\(970\) 0 0
\(971\) −24480.0 −0.809063 −0.404532 0.914524i \(-0.632565\pi\)
−0.404532 + 0.914524i \(0.632565\pi\)
\(972\) 0 0
\(973\) − 46480.0i − 1.53143i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 6906.00i − 0.226144i −0.993587 0.113072i \(-0.963931\pi\)
0.993587 0.113072i \(-0.0360690\pi\)
\(978\) 0 0
\(979\) −24624.0 −0.803868
\(980\) 0 0
\(981\) −13266.0 −0.431754
\(982\) 0 0
\(983\) 6960.00i 0.225829i 0.993605 + 0.112914i \(0.0360186\pi\)
−0.993605 + 0.112914i \(0.963981\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 1440.00i − 0.0464394i
\(988\) 0 0
\(989\) 11040.0 0.354956
\(990\) 0 0
\(991\) −47792.0 −1.53195 −0.765975 0.642870i \(-0.777744\pi\)
−0.765975 + 0.642870i \(0.777744\pi\)
\(992\) 0 0
\(993\) 11076.0i 0.353964i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 9938.00i 0.315687i 0.987464 + 0.157843i \(0.0504541\pi\)
−0.987464 + 0.157843i \(0.949546\pi\)
\(998\) 0 0
\(999\) 1890.00 0.0598568
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 1200.4.f.m.49.1 2
4.3 odd 2 75.4.b.a.49.2 2
5.2 odd 4 1200.4.a.o.1.1 1
5.3 odd 4 240.4.a.f.1.1 1
5.4 even 2 inner 1200.4.f.m.49.2 2
12.11 even 2 225.4.b.d.199.1 2
15.8 even 4 720.4.a.r.1.1 1
20.3 even 4 15.4.a.b.1.1 1
20.7 even 4 75.4.a.a.1.1 1
20.19 odd 2 75.4.b.a.49.1 2
40.3 even 4 960.4.a.bi.1.1 1
40.13 odd 4 960.4.a.l.1.1 1
60.23 odd 4 45.4.a.b.1.1 1
60.47 odd 4 225.4.a.g.1.1 1
60.59 even 2 225.4.b.d.199.2 2
140.83 odd 4 735.4.a.i.1.1 1
180.23 odd 12 405.4.e.k.136.1 2
180.43 even 12 405.4.e.d.271.1 2
180.83 odd 12 405.4.e.k.271.1 2
180.103 even 12 405.4.e.d.136.1 2
220.43 odd 4 1815.4.a.a.1.1 1
420.83 even 4 2205.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 20.3 even 4
45.4.a.b.1.1 1 60.23 odd 4
75.4.a.a.1.1 1 20.7 even 4
75.4.b.a.49.1 2 20.19 odd 2
75.4.b.a.49.2 2 4.3 odd 2
225.4.a.g.1.1 1 60.47 odd 4
225.4.b.d.199.1 2 12.11 even 2
225.4.b.d.199.2 2 60.59 even 2
240.4.a.f.1.1 1 5.3 odd 4
405.4.e.d.136.1 2 180.103 even 12
405.4.e.d.271.1 2 180.43 even 12
405.4.e.k.136.1 2 180.23 odd 12
405.4.e.k.271.1 2 180.83 odd 12
720.4.a.r.1.1 1 15.8 even 4
735.4.a.i.1.1 1 140.83 odd 4
960.4.a.l.1.1 1 40.13 odd 4
960.4.a.bi.1.1 1 40.3 even 4
1200.4.a.o.1.1 1 5.2 odd 4
1200.4.f.m.49.1 2 1.1 even 1 trivial
1200.4.f.m.49.2 2 5.4 even 2 inner
1815.4.a.a.1.1 1 220.43 odd 4
2205.4.a.c.1.1 1 420.83 even 4