Properties

Label 800.2.d.e.401.4
Level $800$
Weight $2$
Character 800.401
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 800.401
Dual form 800.2.d.e.401.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205i q^{3} +0.732051 q^{7} -4.46410 q^{9} +O(q^{10})\) \(q+2.73205i q^{3} +0.732051 q^{7} -4.46410 q^{9} +2.00000i q^{11} +3.46410i q^{13} -3.46410 q^{17} -0.535898i q^{19} +2.00000i q^{21} -6.19615 q^{23} -4.00000i q^{27} +6.92820i q^{29} +5.46410 q^{31} -5.46410 q^{33} +2.00000i q^{37} -9.46410 q^{39} +1.46410 q^{41} -5.26795i q^{43} +3.26795 q^{47} -6.46410 q^{49} -9.46410i q^{51} -11.4641i q^{53} +1.46410 q^{57} +7.46410i q^{59} -8.92820i q^{61} -3.26795 q^{63} +10.7321i q^{67} -16.9282i q^{69} -5.46410 q^{71} -7.46410 q^{73} +1.46410i q^{77} +1.07180 q^{79} -2.46410 q^{81} +1.26795i q^{83} -18.9282 q^{87} +8.92820 q^{89} +2.53590i q^{91} +14.9282i q^{93} +14.3923 q^{97} -8.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{7} - 4 q^{9} - 4 q^{23} + 8 q^{31} - 8 q^{33} - 24 q^{39} - 8 q^{41} + 20 q^{47} - 12 q^{49} - 8 q^{57} - 20 q^{63} - 8 q^{71} - 16 q^{73} + 32 q^{79} + 4 q^{81} - 48 q^{87} + 8 q^{89} + 16 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 2.73205i 1.57735i 0.614810 + 0.788675i \(0.289233\pi\)
−0.614810 + 0.788675i \(0.710767\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 0 0
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) − 0.535898i − 0.122944i −0.998109 0.0614718i \(-0.980421\pi\)
0.998109 0.0614718i \(-0.0195794\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) −6.19615 −1.29199 −0.645994 0.763343i \(-0.723557\pi\)
−0.645994 + 0.763343i \(0.723557\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) −5.46410 −0.951178
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −9.46410 −1.51547
\(40\) 0 0
\(41\) 1.46410 0.228654 0.114327 0.993443i \(-0.463529\pi\)
0.114327 + 0.993443i \(0.463529\pi\)
\(42\) 0 0
\(43\) − 5.26795i − 0.803355i −0.915781 0.401677i \(-0.868427\pi\)
0.915781 0.401677i \(-0.131573\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.26795 0.476679 0.238340 0.971182i \(-0.423397\pi\)
0.238340 + 0.971182i \(0.423397\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) − 9.46410i − 1.32524i
\(52\) 0 0
\(53\) − 11.4641i − 1.57472i −0.616496 0.787358i \(-0.711449\pi\)
0.616496 0.787358i \(-0.288551\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.46410 0.193925
\(58\) 0 0
\(59\) 7.46410i 0.971743i 0.874030 + 0.485872i \(0.161498\pi\)
−0.874030 + 0.485872i \(0.838502\pi\)
\(60\) 0 0
\(61\) − 8.92820i − 1.14314i −0.820554 0.571570i \(-0.806335\pi\)
0.820554 0.571570i \(-0.193665\pi\)
\(62\) 0 0
\(63\) −3.26795 −0.411723
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.7321i 1.31113i 0.755139 + 0.655564i \(0.227569\pi\)
−0.755139 + 0.655564i \(0.772431\pi\)
\(68\) 0 0
\(69\) − 16.9282i − 2.03792i
\(70\) 0 0
\(71\) −5.46410 −0.648470 −0.324235 0.945977i \(-0.605107\pi\)
−0.324235 + 0.945977i \(0.605107\pi\)
\(72\) 0 0
\(73\) −7.46410 −0.873607 −0.436804 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.46410i 0.166850i
\(78\) 0 0
\(79\) 1.07180 0.120587 0.0602933 0.998181i \(-0.480796\pi\)
0.0602933 + 0.998181i \(0.480796\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 1.26795i 0.139176i 0.997576 + 0.0695878i \(0.0221684\pi\)
−0.997576 + 0.0695878i \(0.977832\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.9282 −2.02932
\(88\) 0 0
\(89\) 8.92820 0.946388 0.473194 0.880958i \(-0.343101\pi\)
0.473194 + 0.880958i \(0.343101\pi\)
\(90\) 0 0
\(91\) 2.53590i 0.265834i
\(92\) 0 0
\(93\) 14.9282i 1.54798i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.3923 1.46132 0.730659 0.682743i \(-0.239213\pi\)
0.730659 + 0.682743i \(0.239213\pi\)
\(98\) 0 0
\(99\) − 8.92820i − 0.897318i
\(100\) 0 0
\(101\) 2.92820i 0.291367i 0.989331 + 0.145684i \(0.0465381\pi\)
−0.989331 + 0.145684i \(0.953462\pi\)
\(102\) 0 0
\(103\) 15.6603 1.54305 0.771525 0.636199i \(-0.219494\pi\)
0.771525 + 0.636199i \(0.219494\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.73205i 0.264117i 0.991242 + 0.132059i \(0.0421587\pi\)
−0.991242 + 0.132059i \(0.957841\pi\)
\(108\) 0 0
\(109\) 16.9282i 1.62143i 0.585443 + 0.810714i \(0.300921\pi\)
−0.585443 + 0.810714i \(0.699079\pi\)
\(110\) 0 0
\(111\) −5.46410 −0.518630
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 15.4641i − 1.42966i
\(118\) 0 0
\(119\) −2.53590 −0.232465
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 4.00000i 0.360668i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.7321 1.48473 0.742365 0.669996i \(-0.233704\pi\)
0.742365 + 0.669996i \(0.233704\pi\)
\(128\) 0 0
\(129\) 14.3923 1.26717
\(130\) 0 0
\(131\) 19.8564i 1.73486i 0.497557 + 0.867431i \(0.334230\pi\)
−0.497557 + 0.867431i \(0.665770\pi\)
\(132\) 0 0
\(133\) − 0.392305i − 0.0340171i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.92820 −0.421045 −0.210522 0.977589i \(-0.567516\pi\)
−0.210522 + 0.977589i \(0.567516\pi\)
\(138\) 0 0
\(139\) 0.535898i 0.0454543i 0.999742 + 0.0227272i \(0.00723490\pi\)
−0.999742 + 0.0227272i \(0.992765\pi\)
\(140\) 0 0
\(141\) 8.92820i 0.751890i
\(142\) 0 0
\(143\) −6.92820 −0.579365
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 17.6603i − 1.45659i
\(148\) 0 0
\(149\) − 7.85641i − 0.643622i −0.946804 0.321811i \(-0.895708\pi\)
0.946804 0.321811i \(-0.104292\pi\)
\(150\) 0 0
\(151\) 12.3923 1.00847 0.504236 0.863566i \(-0.331774\pi\)
0.504236 + 0.863566i \(0.331774\pi\)
\(152\) 0 0
\(153\) 15.4641 1.25020
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.07180i 0.245156i 0.992459 + 0.122578i \(0.0391162\pi\)
−0.992459 + 0.122578i \(0.960884\pi\)
\(158\) 0 0
\(159\) 31.3205 2.48388
\(160\) 0 0
\(161\) −4.53590 −0.357479
\(162\) 0 0
\(163\) − 0.196152i − 0.0153638i −0.999970 0.00768192i \(-0.997555\pi\)
0.999970 0.00768192i \(-0.00244526\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.80385 −0.758645 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.39230i 0.182944i
\(172\) 0 0
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20.3923 −1.53278
\(178\) 0 0
\(179\) 8.53590i 0.638003i 0.947754 + 0.319002i \(0.103348\pi\)
−0.947754 + 0.319002i \(0.896652\pi\)
\(180\) 0 0
\(181\) 16.0000i 1.18927i 0.803996 + 0.594635i \(0.202704\pi\)
−0.803996 + 0.594635i \(0.797296\pi\)
\(182\) 0 0
\(183\) 24.3923 1.80313
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.92820i − 0.506640i
\(188\) 0 0
\(189\) − 2.92820i − 0.212995i
\(190\) 0 0
\(191\) −15.3205 −1.10855 −0.554277 0.832333i \(-0.687005\pi\)
−0.554277 + 0.832333i \(0.687005\pi\)
\(192\) 0 0
\(193\) −0.535898 −0.0385748 −0.0192874 0.999814i \(-0.506140\pi\)
−0.0192874 + 0.999814i \(0.506140\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.4641i 1.38676i 0.720572 + 0.693380i \(0.243879\pi\)
−0.720572 + 0.693380i \(0.756121\pi\)
\(198\) 0 0
\(199\) 1.85641 0.131597 0.0657986 0.997833i \(-0.479041\pi\)
0.0657986 + 0.997833i \(0.479041\pi\)
\(200\) 0 0
\(201\) −29.3205 −2.06811
\(202\) 0 0
\(203\) 5.07180i 0.355970i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 27.6603 1.92252
\(208\) 0 0
\(209\) 1.07180 0.0741377
\(210\) 0 0
\(211\) − 26.7846i − 1.84393i −0.387275 0.921964i \(-0.626584\pi\)
0.387275 0.921964i \(-0.373416\pi\)
\(212\) 0 0
\(213\) − 14.9282i − 1.02286i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) − 20.3923i − 1.37798i
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) −5.80385 −0.388654 −0.194327 0.980937i \(-0.562252\pi\)
−0.194327 + 0.980937i \(0.562252\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.0526i 0.667212i 0.942713 + 0.333606i \(0.108265\pi\)
−0.942713 + 0.333606i \(0.891735\pi\)
\(228\) 0 0
\(229\) − 4.00000i − 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 5.32051 0.348558 0.174279 0.984696i \(-0.444241\pi\)
0.174279 + 0.984696i \(0.444241\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.92820i 0.190207i
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 16.3923 1.05592 0.527961 0.849269i \(-0.322957\pi\)
0.527961 + 0.849269i \(0.322957\pi\)
\(242\) 0 0
\(243\) − 18.7321i − 1.20166i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.85641 0.118120
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) − 24.9282i − 1.57345i −0.617301 0.786727i \(-0.711774\pi\)
0.617301 0.786727i \(-0.288226\pi\)
\(252\) 0 0
\(253\) − 12.3923i − 0.779098i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 1.46410i 0.0909748i
\(260\) 0 0
\(261\) − 30.9282i − 1.91441i
\(262\) 0 0
\(263\) −11.6603 −0.719002 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 24.3923i 1.49278i
\(268\) 0 0
\(269\) − 8.92820i − 0.544362i −0.962246 0.272181i \(-0.912255\pi\)
0.962246 0.272181i \(-0.0877450\pi\)
\(270\) 0 0
\(271\) 19.3205 1.17364 0.586819 0.809718i \(-0.300380\pi\)
0.586819 + 0.809718i \(0.300380\pi\)
\(272\) 0 0
\(273\) −6.92820 −0.419314
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −24.3923 −1.46033
\(280\) 0 0
\(281\) 10.5359 0.628519 0.314260 0.949337i \(-0.398244\pi\)
0.314260 + 0.949337i \(0.398244\pi\)
\(282\) 0 0
\(283\) − 9.66025i − 0.574242i −0.957894 0.287121i \(-0.907302\pi\)
0.957894 0.287121i \(-0.0926983\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.07180 0.0632662
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 39.3205i 2.30501i
\(292\) 0 0
\(293\) 15.8564i 0.926341i 0.886269 + 0.463171i \(0.153288\pi\)
−0.886269 + 0.463171i \(0.846712\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 0 0
\(299\) − 21.4641i − 1.24130i
\(300\) 0 0
\(301\) − 3.85641i − 0.222280i
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 24.9808i − 1.42573i −0.701303 0.712864i \(-0.747398\pi\)
0.701303 0.712864i \(-0.252602\pi\)
\(308\) 0 0
\(309\) 42.7846i 2.43393i
\(310\) 0 0
\(311\) −31.3205 −1.77602 −0.888012 0.459821i \(-0.847914\pi\)
−0.888012 + 0.459821i \(0.847914\pi\)
\(312\) 0 0
\(313\) 4.14359 0.234210 0.117105 0.993120i \(-0.462639\pi\)
0.117105 + 0.993120i \(0.462639\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 8.53590i − 0.479424i −0.970844 0.239712i \(-0.922947\pi\)
0.970844 0.239712i \(-0.0770530\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) −7.46410 −0.416606
\(322\) 0 0
\(323\) 1.85641i 0.103293i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −46.2487 −2.55756
\(328\) 0 0
\(329\) 2.39230 0.131892
\(330\) 0 0
\(331\) 14.0000i 0.769510i 0.923019 + 0.384755i \(0.125714\pi\)
−0.923019 + 0.384755i \(0.874286\pi\)
\(332\) 0 0
\(333\) − 8.92820i − 0.489263i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.8564 1.08165 0.540824 0.841136i \(-0.318113\pi\)
0.540824 + 0.841136i \(0.318113\pi\)
\(338\) 0 0
\(339\) 35.3205i 1.91835i
\(340\) 0 0
\(341\) 10.9282i 0.591795i
\(342\) 0 0
\(343\) −9.85641 −0.532196
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1.66025i − 0.0891271i −0.999007 0.0445636i \(-0.985810\pi\)
0.999007 0.0445636i \(-0.0141897\pi\)
\(348\) 0 0
\(349\) − 28.0000i − 1.49881i −0.662114 0.749403i \(-0.730341\pi\)
0.662114 0.749403i \(-0.269659\pi\)
\(350\) 0 0
\(351\) 13.8564 0.739600
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.92820i − 0.366679i
\(358\) 0 0
\(359\) −18.9282 −0.998992 −0.499496 0.866316i \(-0.666482\pi\)
−0.499496 + 0.866316i \(0.666482\pi\)
\(360\) 0 0
\(361\) 18.7128 0.984885
\(362\) 0 0
\(363\) 19.1244i 1.00377i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.87564 0.150107 0.0750537 0.997179i \(-0.476087\pi\)
0.0750537 + 0.997179i \(0.476087\pi\)
\(368\) 0 0
\(369\) −6.53590 −0.340245
\(370\) 0 0
\(371\) − 8.39230i − 0.435707i
\(372\) 0 0
\(373\) 25.7128i 1.33136i 0.746238 + 0.665679i \(0.231858\pi\)
−0.746238 + 0.665679i \(0.768142\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 36.2487i 1.86197i 0.365056 + 0.930986i \(0.381050\pi\)
−0.365056 + 0.930986i \(0.618950\pi\)
\(380\) 0 0
\(381\) 45.7128i 2.34194i
\(382\) 0 0
\(383\) 21.1244 1.07940 0.539702 0.841856i \(-0.318537\pi\)
0.539702 + 0.841856i \(0.318537\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 23.5167i 1.19542i
\(388\) 0 0
\(389\) 6.78461i 0.343993i 0.985098 + 0.171997i \(0.0550218\pi\)
−0.985098 + 0.171997i \(0.944978\pi\)
\(390\) 0 0
\(391\) 21.4641 1.08549
\(392\) 0 0
\(393\) −54.2487 −2.73649
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 32.2487i − 1.61852i −0.587453 0.809258i \(-0.699869\pi\)
0.587453 0.809258i \(-0.300131\pi\)
\(398\) 0 0
\(399\) 1.07180 0.0536570
\(400\) 0 0
\(401\) −7.85641 −0.392330 −0.196165 0.980571i \(-0.562849\pi\)
−0.196165 + 0.980571i \(0.562849\pi\)
\(402\) 0 0
\(403\) 18.9282i 0.942881i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −11.3205 −0.559763 −0.279882 0.960035i \(-0.590295\pi\)
−0.279882 + 0.960035i \(0.590295\pi\)
\(410\) 0 0
\(411\) − 13.4641i − 0.664135i
\(412\) 0 0
\(413\) 5.46410i 0.268871i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.46410 −0.0716974
\(418\) 0 0
\(419\) − 18.3923i − 0.898523i −0.893400 0.449261i \(-0.851687\pi\)
0.893400 0.449261i \(-0.148313\pi\)
\(420\) 0 0
\(421\) 0.143594i 0.00699832i 0.999994 + 0.00349916i \(0.00111382\pi\)
−0.999994 + 0.00349916i \(0.998886\pi\)
\(422\) 0 0
\(423\) −14.5885 −0.709315
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.53590i − 0.316294i
\(428\) 0 0
\(429\) − 18.9282i − 0.913862i
\(430\) 0 0
\(431\) 21.4641 1.03389 0.516945 0.856019i \(-0.327069\pi\)
0.516945 + 0.856019i \(0.327069\pi\)
\(432\) 0 0
\(433\) 19.4641 0.935385 0.467693 0.883891i \(-0.345085\pi\)
0.467693 + 0.883891i \(0.345085\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.32051i 0.158841i
\(438\) 0 0
\(439\) −40.7846 −1.94654 −0.973272 0.229657i \(-0.926240\pi\)
−0.973272 + 0.229657i \(0.926240\pi\)
\(440\) 0 0
\(441\) 28.8564 1.37411
\(442\) 0 0
\(443\) 20.9808i 0.996826i 0.866940 + 0.498413i \(0.166084\pi\)
−0.866940 + 0.498413i \(0.833916\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 21.4641 1.01522
\(448\) 0 0
\(449\) −23.3205 −1.10056 −0.550281 0.834979i \(-0.685480\pi\)
−0.550281 + 0.834979i \(0.685480\pi\)
\(450\) 0 0
\(451\) 2.92820i 0.137884i
\(452\) 0 0
\(453\) 33.8564i 1.59071i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.7846 1.25293 0.626466 0.779449i \(-0.284501\pi\)
0.626466 + 0.779449i \(0.284501\pi\)
\(458\) 0 0
\(459\) 13.8564i 0.646762i
\(460\) 0 0
\(461\) − 10.9282i − 0.508977i −0.967076 0.254489i \(-0.918093\pi\)
0.967076 0.254489i \(-0.0819071\pi\)
\(462\) 0 0
\(463\) −11.2679 −0.523666 −0.261833 0.965113i \(-0.584327\pi\)
−0.261833 + 0.965113i \(0.584327\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 25.6603i − 1.18741i −0.804681 0.593707i \(-0.797664\pi\)
0.804681 0.593707i \(-0.202336\pi\)
\(468\) 0 0
\(469\) 7.85641i 0.362775i
\(470\) 0 0
\(471\) −8.39230 −0.386697
\(472\) 0 0
\(473\) 10.5359 0.484441
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 51.1769i 2.34323i
\(478\) 0 0
\(479\) −5.85641 −0.267586 −0.133793 0.991009i \(-0.542716\pi\)
−0.133793 + 0.991009i \(0.542716\pi\)
\(480\) 0 0
\(481\) −6.92820 −0.315899
\(482\) 0 0
\(483\) − 12.3923i − 0.563869i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.58846 0.298551 0.149276 0.988796i \(-0.452306\pi\)
0.149276 + 0.988796i \(0.452306\pi\)
\(488\) 0 0
\(489\) 0.535898 0.0242342
\(490\) 0 0
\(491\) 16.9282i 0.763959i 0.924171 + 0.381980i \(0.124758\pi\)
−0.924171 + 0.381980i \(0.875242\pi\)
\(492\) 0 0
\(493\) − 24.0000i − 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) − 31.4641i − 1.40853i −0.709939 0.704263i \(-0.751277\pi\)
0.709939 0.704263i \(-0.248723\pi\)
\(500\) 0 0
\(501\) − 26.7846i − 1.19665i
\(502\) 0 0
\(503\) −0.339746 −0.0151485 −0.00757426 0.999971i \(-0.502411\pi\)
−0.00757426 + 0.999971i \(0.502411\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.73205i 0.121335i
\(508\) 0 0
\(509\) − 1.85641i − 0.0822838i −0.999153 0.0411419i \(-0.986900\pi\)
0.999153 0.0411419i \(-0.0130996\pi\)
\(510\) 0 0
\(511\) −5.46410 −0.241718
\(512\) 0 0
\(513\) −2.14359 −0.0946420
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.53590i 0.287448i
\(518\) 0 0
\(519\) 5.46410 0.239847
\(520\) 0 0
\(521\) −43.8564 −1.92138 −0.960692 0.277616i \(-0.910456\pi\)
−0.960692 + 0.277616i \(0.910456\pi\)
\(522\) 0 0
\(523\) − 11.8038i − 0.516146i −0.966125 0.258073i \(-0.916912\pi\)
0.966125 0.258073i \(-0.0830875\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.9282 −0.824525
\(528\) 0 0
\(529\) 15.3923 0.669231
\(530\) 0 0
\(531\) − 33.3205i − 1.44599i
\(532\) 0 0
\(533\) 5.07180i 0.219684i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −23.3205 −1.00635
\(538\) 0 0
\(539\) − 12.9282i − 0.556857i
\(540\) 0 0
\(541\) − 26.9282i − 1.15773i −0.815422 0.578867i \(-0.803495\pi\)
0.815422 0.578867i \(-0.196505\pi\)
\(542\) 0 0
\(543\) −43.7128 −1.87590
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 33.2679i 1.42243i 0.702972 + 0.711217i \(0.251856\pi\)
−0.702972 + 0.711217i \(0.748144\pi\)
\(548\) 0 0
\(549\) 39.8564i 1.70103i
\(550\) 0 0
\(551\) 3.71281 0.158171
\(552\) 0 0
\(553\) 0.784610 0.0333650
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 14.7846i − 0.626444i −0.949680 0.313222i \(-0.898592\pi\)
0.949680 0.313222i \(-0.101408\pi\)
\(558\) 0 0
\(559\) 18.2487 0.771838
\(560\) 0 0
\(561\) 18.9282 0.799149
\(562\) 0 0
\(563\) − 22.0526i − 0.929405i −0.885467 0.464702i \(-0.846161\pi\)
0.885467 0.464702i \(-0.153839\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.80385 −0.0757545
\(568\) 0 0
\(569\) −13.4641 −0.564445 −0.282222 0.959349i \(-0.591072\pi\)
−0.282222 + 0.959349i \(0.591072\pi\)
\(570\) 0 0
\(571\) − 6.78461i − 0.283927i −0.989872 0.141964i \(-0.954658\pi\)
0.989872 0.141964i \(-0.0453416\pi\)
\(572\) 0 0
\(573\) − 41.8564i − 1.74858i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −39.5692 −1.64729 −0.823644 0.567107i \(-0.808063\pi\)
−0.823644 + 0.567107i \(0.808063\pi\)
\(578\) 0 0
\(579\) − 1.46410i − 0.0608460i
\(580\) 0 0
\(581\) 0.928203i 0.0385084i
\(582\) 0 0
\(583\) 22.9282 0.949589
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 3.80385i − 0.157002i −0.996914 0.0785008i \(-0.974987\pi\)
0.996914 0.0785008i \(-0.0250133\pi\)
\(588\) 0 0
\(589\) − 2.92820i − 0.120655i
\(590\) 0 0
\(591\) −53.1769 −2.18741
\(592\) 0 0
\(593\) −32.6410 −1.34041 −0.670203 0.742178i \(-0.733793\pi\)
−0.670203 + 0.742178i \(0.733793\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.07180i 0.207575i
\(598\) 0 0
\(599\) 34.6410 1.41539 0.707697 0.706516i \(-0.249734\pi\)
0.707697 + 0.706516i \(0.249734\pi\)
\(600\) 0 0
\(601\) 18.5359 0.756095 0.378048 0.925786i \(-0.376596\pi\)
0.378048 + 0.925786i \(0.376596\pi\)
\(602\) 0 0
\(603\) − 47.9090i − 1.95100i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 30.9808 1.25747 0.628735 0.777619i \(-0.283573\pi\)
0.628735 + 0.777619i \(0.283573\pi\)
\(608\) 0 0
\(609\) −13.8564 −0.561490
\(610\) 0 0
\(611\) 11.3205i 0.457979i
\(612\) 0 0
\(613\) − 26.3923i − 1.06598i −0.846123 0.532988i \(-0.821069\pi\)
0.846123 0.532988i \(-0.178931\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.5359 0.826744 0.413372 0.910562i \(-0.364351\pi\)
0.413372 + 0.910562i \(0.364351\pi\)
\(618\) 0 0
\(619\) − 1.32051i − 0.0530757i −0.999648 0.0265379i \(-0.991552\pi\)
0.999648 0.0265379i \(-0.00844825\pi\)
\(620\) 0 0
\(621\) 24.7846i 0.994572i
\(622\) 0 0
\(623\) 6.53590 0.261855
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.92820i 0.116941i
\(628\) 0 0
\(629\) − 6.92820i − 0.276246i
\(630\) 0 0
\(631\) 23.3205 0.928375 0.464187 0.885737i \(-0.346346\pi\)
0.464187 + 0.885737i \(0.346346\pi\)
\(632\) 0 0
\(633\) 73.1769 2.90852
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 22.3923i − 0.887215i
\(638\) 0 0
\(639\) 24.3923 0.964945
\(640\) 0 0
\(641\) 0.392305 0.0154951 0.00774755 0.999970i \(-0.497534\pi\)
0.00774755 + 0.999970i \(0.497534\pi\)
\(642\) 0 0
\(643\) 39.1244i 1.54291i 0.636281 + 0.771457i \(0.280472\pi\)
−0.636281 + 0.771457i \(0.719528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.7321 0.657805 0.328902 0.944364i \(-0.393321\pi\)
0.328902 + 0.944364i \(0.393321\pi\)
\(648\) 0 0
\(649\) −14.9282 −0.585983
\(650\) 0 0
\(651\) 10.9282i 0.428310i
\(652\) 0 0
\(653\) − 12.2487i − 0.479329i −0.970856 0.239665i \(-0.922963\pi\)
0.970856 0.239665i \(-0.0770375\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 33.3205 1.29996
\(658\) 0 0
\(659\) 17.3205i 0.674711i 0.941377 + 0.337356i \(0.109532\pi\)
−0.941377 + 0.337356i \(0.890468\pi\)
\(660\) 0 0
\(661\) − 8.14359i − 0.316749i −0.987379 0.158375i \(-0.949375\pi\)
0.987379 0.158375i \(-0.0506253\pi\)
\(662\) 0 0
\(663\) 32.7846 1.27325
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 42.9282i − 1.66219i
\(668\) 0 0
\(669\) − 15.8564i − 0.613044i
\(670\) 0 0
\(671\) 17.8564 0.689339
\(672\) 0 0
\(673\) −12.5359 −0.483223 −0.241612 0.970373i \(-0.577676\pi\)
−0.241612 + 0.970373i \(0.577676\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 17.6077i − 0.676719i −0.941017 0.338359i \(-0.890128\pi\)
0.941017 0.338359i \(-0.109872\pi\)
\(678\) 0 0
\(679\) 10.5359 0.404331
\(680\) 0 0
\(681\) −27.4641 −1.05243
\(682\) 0 0
\(683\) − 16.9808i − 0.649751i −0.945757 0.324875i \(-0.894678\pi\)
0.945757 0.324875i \(-0.105322\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.9282 0.416937
\(688\) 0 0
\(689\) 39.7128 1.51294
\(690\) 0 0
\(691\) − 18.0000i − 0.684752i −0.939563 0.342376i \(-0.888768\pi\)
0.939563 0.342376i \(-0.111232\pi\)
\(692\) 0 0
\(693\) − 6.53590i − 0.248278i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −5.07180 −0.192108
\(698\) 0 0
\(699\) 14.5359i 0.549798i
\(700\) 0 0
\(701\) 19.0718i 0.720332i 0.932888 + 0.360166i \(0.117280\pi\)
−0.932888 + 0.360166i \(0.882720\pi\)
\(702\) 0 0
\(703\) 1.07180 0.0404236
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.14359i 0.0806181i
\(708\) 0 0
\(709\) − 12.7846i − 0.480136i −0.970756 0.240068i \(-0.922830\pi\)
0.970756 0.240068i \(-0.0771698\pi\)
\(710\) 0 0
\(711\) −4.78461 −0.179437
\(712\) 0 0
\(713\) −33.8564 −1.26793
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 54.6410i 2.04061i
\(718\) 0 0
\(719\) 1.85641 0.0692323 0.0346161 0.999401i \(-0.488979\pi\)
0.0346161 + 0.999401i \(0.488979\pi\)
\(720\) 0 0
\(721\) 11.4641 0.426945
\(722\) 0 0
\(723\) 44.7846i 1.66556i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −24.0526 −0.892060 −0.446030 0.895018i \(-0.647163\pi\)
−0.446030 + 0.895018i \(0.647163\pi\)
\(728\) 0 0
\(729\) 43.7846 1.62165
\(730\) 0 0
\(731\) 18.2487i 0.674953i
\(732\) 0 0
\(733\) 35.0718i 1.29541i 0.761893 + 0.647703i \(0.224270\pi\)
−0.761893 + 0.647703i \(0.775730\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.4641 −0.790640
\(738\) 0 0
\(739\) − 29.3205i − 1.07857i −0.842123 0.539286i \(-0.818694\pi\)
0.842123 0.539286i \(-0.181306\pi\)
\(740\) 0 0
\(741\) 5.07180i 0.186317i
\(742\) 0 0
\(743\) 10.9808 0.402845 0.201423 0.979504i \(-0.435444\pi\)
0.201423 + 0.979504i \(0.435444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 5.66025i − 0.207098i
\(748\) 0 0
\(749\) 2.00000i 0.0730784i
\(750\) 0 0
\(751\) −26.2487 −0.957829 −0.478915 0.877862i \(-0.658970\pi\)
−0.478915 + 0.877862i \(0.658970\pi\)
\(752\) 0 0
\(753\) 68.1051 2.48189
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19.0718i 0.693176i 0.938017 + 0.346588i \(0.112660\pi\)
−0.938017 + 0.346588i \(0.887340\pi\)
\(758\) 0 0
\(759\) 33.8564 1.22891
\(760\) 0 0
\(761\) −5.71281 −0.207089 −0.103545 0.994625i \(-0.533018\pi\)
−0.103545 + 0.994625i \(0.533018\pi\)
\(762\) 0 0
\(763\) 12.3923i 0.448632i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25.8564 −0.933621
\(768\) 0 0
\(769\) 12.9282 0.466203 0.233101 0.972452i \(-0.425113\pi\)
0.233101 + 0.972452i \(0.425113\pi\)
\(770\) 0 0
\(771\) 5.46410i 0.196785i
\(772\) 0 0
\(773\) 22.3923i 0.805395i 0.915333 + 0.402698i \(0.131927\pi\)
−0.915333 + 0.402698i \(0.868073\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) − 0.784610i − 0.0281116i
\(780\) 0 0
\(781\) − 10.9282i − 0.391042i
\(782\) 0 0
\(783\) 27.7128 0.990375
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.5885i 0.591315i 0.955294 + 0.295657i \(0.0955387\pi\)
−0.955294 + 0.295657i \(0.904461\pi\)
\(788\) 0 0
\(789\) − 31.8564i − 1.13412i
\(790\) 0 0
\(791\) 9.46410 0.336505
\(792\) 0 0
\(793\) 30.9282 1.09829
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 50.1051i − 1.77481i −0.460986 0.887407i \(-0.652504\pi\)
0.460986 0.887407i \(-0.347496\pi\)
\(798\) 0 0
\(799\) −11.3205 −0.400491
\(800\) 0 0
\(801\) −39.8564 −1.40826
\(802\) 0 0
\(803\) − 14.9282i − 0.526805i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.3923 0.858650
\(808\) 0 0
\(809\) 23.8564 0.838747 0.419373 0.907814i \(-0.362250\pi\)
0.419373 + 0.907814i \(0.362250\pi\)
\(810\) 0 0
\(811\) 28.9282i 1.01581i 0.861414 + 0.507903i \(0.169579\pi\)
−0.861414 + 0.507903i \(0.830421\pi\)
\(812\) 0 0
\(813\) 52.7846i 1.85124i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.82309 −0.0987673
\(818\) 0 0
\(819\) − 11.3205i − 0.395571i
\(820\) 0 0
\(821\) 34.7846i 1.21399i 0.794705 + 0.606996i \(0.207625\pi\)
−0.794705 + 0.606996i \(0.792375\pi\)
\(822\) 0 0
\(823\) −9.12436 −0.318055 −0.159028 0.987274i \(-0.550836\pi\)
−0.159028 + 0.987274i \(0.550836\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.1244i 0.804113i 0.915615 + 0.402056i \(0.131704\pi\)
−0.915615 + 0.402056i \(0.868296\pi\)
\(828\) 0 0
\(829\) 28.9282i 1.00472i 0.864659 + 0.502359i \(0.167534\pi\)
−0.864659 + 0.502359i \(0.832466\pi\)
\(830\) 0 0
\(831\) 5.46410 0.189548
\(832\) 0 0
\(833\) 22.3923 0.775847
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 21.8564i − 0.755468i
\(838\) 0 0
\(839\) −24.7846 −0.855660 −0.427830 0.903859i \(-0.640722\pi\)
−0.427830 + 0.903859i \(0.640722\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 28.7846i 0.991395i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.12436 0.176075
\(848\) 0 0
\(849\) 26.3923 0.905782
\(850\) 0 0
\(851\) − 12.3923i − 0.424803i
\(852\) 0 0
\(853\) 21.6077i 0.739833i 0.929065 + 0.369917i \(0.120614\pi\)
−0.929065 + 0.369917i \(0.879386\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.8564 0.678282 0.339141 0.940736i \(-0.389864\pi\)
0.339141 + 0.940736i \(0.389864\pi\)
\(858\) 0 0
\(859\) − 28.2487i − 0.963834i −0.876217 0.481917i \(-0.839941\pi\)
0.876217 0.481917i \(-0.160059\pi\)
\(860\) 0 0
\(861\) 2.92820i 0.0997929i
\(862\) 0 0
\(863\) 47.6603 1.62237 0.811187 0.584787i \(-0.198822\pi\)
0.811187 + 0.584787i \(0.198822\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.6603i − 0.463927i
\(868\) 0 0
\(869\) 2.14359i 0.0727164i
\(870\) 0 0
\(871\) −37.1769 −1.25969
\(872\) 0 0
\(873\) −64.2487 −2.17449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.71281i 0.0578376i 0.999582 + 0.0289188i \(0.00920642\pi\)
−0.999582 + 0.0289188i \(0.990794\pi\)
\(878\) 0 0
\(879\) −43.3205 −1.46116
\(880\) 0 0
\(881\) 9.46410 0.318854 0.159427 0.987210i \(-0.449035\pi\)
0.159427 + 0.987210i \(0.449035\pi\)
\(882\) 0 0
\(883\) − 27.9090i − 0.939211i −0.882876 0.469606i \(-0.844396\pi\)
0.882876 0.469606i \(-0.155604\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.9090 −0.467017 −0.233509 0.972355i \(-0.575021\pi\)
−0.233509 + 0.972355i \(0.575021\pi\)
\(888\) 0 0
\(889\) 12.2487 0.410809
\(890\) 0 0
\(891\) − 4.92820i − 0.165101i
\(892\) 0 0
\(893\) − 1.75129i − 0.0586046i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 58.6410 1.95797
\(898\) 0 0
\(899\) 37.8564i 1.26258i
\(900\) 0 0
\(901\) 39.7128i 1.32303i
\(902\) 0 0
\(903\) 10.5359 0.350613
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.87564i 0.161893i 0.996718 + 0.0809466i \(0.0257943\pi\)
−0.996718 + 0.0809466i \(0.974206\pi\)
\(908\) 0 0
\(909\) − 13.0718i − 0.433564i
\(910\) 0 0
\(911\) 49.1769 1.62930 0.814652 0.579950i \(-0.196928\pi\)
0.814652 + 0.579950i \(0.196928\pi\)
\(912\) 0 0
\(913\) −2.53590 −0.0839260
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 14.5359i 0.480018i
\(918\) 0 0
\(919\) 38.9282 1.28412 0.642061 0.766653i \(-0.278079\pi\)
0.642061 + 0.766653i \(0.278079\pi\)
\(920\) 0 0
\(921\) 68.2487 2.24887
\(922\) 0 0
\(923\) − 18.9282i − 0.623029i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −69.9090 −2.29611
\(928\) 0 0
\(929\) −17.4641 −0.572979 −0.286489 0.958083i \(-0.592488\pi\)
−0.286489 + 0.958083i \(0.592488\pi\)
\(930\) 0 0
\(931\) 3.46410i 0.113531i
\(932\) 0 0
\(933\) − 85.5692i − 2.80141i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.24871 −0.138799 −0.0693997 0.997589i \(-0.522108\pi\)
−0.0693997 + 0.997589i \(0.522108\pi\)
\(938\) 0 0
\(939\) 11.3205i 0.369431i
\(940\) 0 0
\(941\) − 32.0000i − 1.04317i −0.853199 0.521585i \(-0.825341\pi\)
0.853199 0.521585i \(-0.174659\pi\)
\(942\) 0 0
\(943\) −9.07180 −0.295418
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.12436i − 0.101528i −0.998711 0.0507640i \(-0.983834\pi\)
0.998711 0.0507640i \(-0.0161656\pi\)
\(948\) 0 0
\(949\) − 25.8564i − 0.839334i
\(950\) 0 0
\(951\) 23.3205 0.756219
\(952\) 0 0
\(953\) 17.2154 0.557661 0.278831 0.960340i \(-0.410053\pi\)
0.278831 + 0.960340i \(0.410053\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 37.8564i − 1.22372i
\(958\) 0 0
\(959\) −3.60770 −0.116499
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) − 12.1962i − 0.393016i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.3397 0.525451 0.262725 0.964871i \(-0.415379\pi\)
0.262725 + 0.964871i \(0.415379\pi\)
\(968\) 0 0
\(969\) −5.07180 −0.162930
\(970\) 0 0
\(971\) − 36.9282i − 1.18508i −0.805540 0.592541i \(-0.798125\pi\)
0.805540 0.592541i \(-0.201875\pi\)
\(972\) 0 0
\(973\) 0.392305i 0.0125767i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.5359 0.784973 0.392486 0.919758i \(-0.371615\pi\)
0.392486 + 0.919758i \(0.371615\pi\)
\(978\) 0 0
\(979\) 17.8564i 0.570693i
\(980\) 0 0
\(981\) − 75.5692i − 2.41274i
\(982\) 0 0
\(983\) −48.7321 −1.55431 −0.777156 0.629309i \(-0.783338\pi\)
−0.777156 + 0.629309i \(0.783338\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.53590i 0.208040i
\(988\) 0 0
\(989\) 32.6410i 1.03792i
\(990\) 0 0
\(991\) −41.4641 −1.31715 −0.658575 0.752515i \(-0.728841\pi\)
−0.658575 + 0.752515i \(0.728841\pi\)
\(992\) 0 0
\(993\) −38.2487 −1.21379
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 11.1769i 0.353976i 0.984213 + 0.176988i \(0.0566354\pi\)
−0.984213 + 0.176988i \(0.943365\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 800.2.d.e.401.4 4
3.2 odd 2 7200.2.k.j.3601.3 4
4.3 odd 2 200.2.d.f.101.4 4
5.2 odd 4 800.2.f.e.49.4 4
5.3 odd 4 800.2.f.c.49.1 4
5.4 even 2 160.2.d.a.81.1 4
8.3 odd 2 200.2.d.f.101.3 4
8.5 even 2 inner 800.2.d.e.401.1 4
12.11 even 2 1800.2.k.j.901.1 4
15.2 even 4 7200.2.d.n.2449.3 4
15.8 even 4 7200.2.d.o.2449.2 4
15.14 odd 2 1440.2.k.e.721.3 4
16.3 odd 4 6400.2.a.ce.1.2 2
16.5 even 4 6400.2.a.cj.1.2 2
16.11 odd 4 6400.2.a.z.1.1 2
16.13 even 4 6400.2.a.be.1.1 2
20.3 even 4 200.2.f.c.149.3 4
20.7 even 4 200.2.f.e.149.2 4
20.19 odd 2 40.2.d.a.21.1 4
24.5 odd 2 7200.2.k.j.3601.4 4
24.11 even 2 1800.2.k.j.901.2 4
40.3 even 4 200.2.f.e.149.1 4
40.13 odd 4 800.2.f.e.49.3 4
40.19 odd 2 40.2.d.a.21.2 yes 4
40.27 even 4 200.2.f.c.149.4 4
40.29 even 2 160.2.d.a.81.4 4
40.37 odd 4 800.2.f.c.49.2 4
60.23 odd 4 1800.2.d.p.1549.2 4
60.47 odd 4 1800.2.d.l.1549.3 4
60.59 even 2 360.2.k.e.181.4 4
80.19 odd 4 1280.2.a.a.1.1 2
80.29 even 4 1280.2.a.n.1.2 2
80.59 odd 4 1280.2.a.o.1.2 2
80.69 even 4 1280.2.a.d.1.1 2
120.29 odd 2 1440.2.k.e.721.1 4
120.53 even 4 7200.2.d.n.2449.2 4
120.59 even 2 360.2.k.e.181.3 4
120.77 even 4 7200.2.d.o.2449.3 4
120.83 odd 4 1800.2.d.l.1549.4 4
120.107 odd 4 1800.2.d.p.1549.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 20.19 odd 2
40.2.d.a.21.2 yes 4 40.19 odd 2
160.2.d.a.81.1 4 5.4 even 2
160.2.d.a.81.4 4 40.29 even 2
200.2.d.f.101.3 4 8.3 odd 2
200.2.d.f.101.4 4 4.3 odd 2
200.2.f.c.149.3 4 20.3 even 4
200.2.f.c.149.4 4 40.27 even 4
200.2.f.e.149.1 4 40.3 even 4
200.2.f.e.149.2 4 20.7 even 4
360.2.k.e.181.3 4 120.59 even 2
360.2.k.e.181.4 4 60.59 even 2
800.2.d.e.401.1 4 8.5 even 2 inner
800.2.d.e.401.4 4 1.1 even 1 trivial
800.2.f.c.49.1 4 5.3 odd 4
800.2.f.c.49.2 4 40.37 odd 4
800.2.f.e.49.3 4 40.13 odd 4
800.2.f.e.49.4 4 5.2 odd 4
1280.2.a.a.1.1 2 80.19 odd 4
1280.2.a.d.1.1 2 80.69 even 4
1280.2.a.n.1.2 2 80.29 even 4
1280.2.a.o.1.2 2 80.59 odd 4
1440.2.k.e.721.1 4 120.29 odd 2
1440.2.k.e.721.3 4 15.14 odd 2
1800.2.d.l.1549.3 4 60.47 odd 4
1800.2.d.l.1549.4 4 120.83 odd 4
1800.2.d.p.1549.1 4 120.107 odd 4
1800.2.d.p.1549.2 4 60.23 odd 4
1800.2.k.j.901.1 4 12.11 even 2
1800.2.k.j.901.2 4 24.11 even 2
6400.2.a.z.1.1 2 16.11 odd 4
6400.2.a.be.1.1 2 16.13 even 4
6400.2.a.ce.1.2 2 16.3 odd 4
6400.2.a.cj.1.2 2 16.5 even 4
7200.2.d.n.2449.2 4 120.53 even 4
7200.2.d.n.2449.3 4 15.2 even 4
7200.2.d.o.2449.2 4 15.8 even 4
7200.2.d.o.2449.3 4 120.77 even 4
7200.2.k.j.3601.3 4 3.2 odd 2
7200.2.k.j.3601.4 4 24.5 odd 2