Properties

Label 3800.2.d.f.3649.2
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.f.3649.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} +1.00000i q^{13} +5.00000i q^{17} -1.00000 q^{19} +3.00000 q^{21} -1.00000i q^{23} +5.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} +2.00000i q^{33} -2.00000i q^{37} -1.00000 q^{39} -8.00000 q^{41} -8.00000i q^{43} +8.00000i q^{47} -2.00000 q^{49} -5.00000 q^{51} +9.00000i q^{53} -1.00000i q^{57} -1.00000 q^{59} +14.0000 q^{61} -6.00000i q^{63} -13.0000i q^{67} +1.00000 q^{69} +10.0000 q^{71} +9.00000i q^{73} -6.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +10.0000i q^{83} +3.00000i q^{87} +12.0000 q^{89} +3.00000 q^{91} +4.00000i q^{93} -14.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} + 4q^{11} - 2q^{19} + 6q^{21} + 6q^{29} + 8q^{31} - 2q^{39} - 16q^{41} - 4q^{49} - 10q^{51} - 2q^{59} + 28q^{61} + 2q^{69} + 20q^{71} + 20q^{79} + 2q^{81} + 24q^{89} + 6q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000i 1.21268i 0.795206 + 0.606339i \(0.207363\pi\)
−0.795206 + 0.606339i \(0.792637\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) 0 0
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) − 6.00000i − 0.755929i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.00000i − 0.683763i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.0000i 1.09764i 0.835940 + 0.548821i \(0.184923\pi\)
−0.835940 + 0.548821i \(0.815077\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.00000i 0.321634i
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.0000i − 1.45010i −0.688694 0.725052i \(-0.741816\pi\)
0.688694 0.725052i \(-0.258184\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 7.00000i − 0.598050i −0.954245 0.299025i \(-0.903339\pi\)
0.954245 0.299025i \(-0.0966615\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 0 0
\(153\) 10.0000i 0.808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 4.00000i − 0.309529i −0.987951 0.154765i \(-0.950538\pi\)
0.987951 0.154765i \(-0.0494619\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.00000i − 0.0751646i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.0000i 0.731272i
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) 23.0000 1.66422 0.832111 0.554609i \(-0.187132\pi\)
0.832111 + 0.554609i \(0.187132\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) 0 0
\(199\) 9.00000 0.637993 0.318997 0.947756i \(-0.396654\pi\)
0.318997 + 0.947756i \(0.396654\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 0 0
\(203\) − 9.00000i − 0.631676i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.00000i − 0.139010i
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 21.0000 1.44570 0.722850 0.691005i \(-0.242832\pi\)
0.722850 + 0.691005i \(0.242832\pi\)
\(212\) 0 0
\(213\) 10.0000i 0.685189i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 12.0000i − 0.814613i
\(218\) 0 0
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.00000i − 0.597351i −0.954355 0.298675i \(-0.903455\pi\)
0.954355 0.298675i \(-0.0965448\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.0000i 0.649570i
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.00000i − 0.0636285i
\(248\) 0 0
\(249\) −10.0000 −0.633724
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) − 2.00000i − 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.00000i 0.249513i 0.992187 + 0.124757i \(0.0398150\pi\)
−0.992187 + 0.124757i \(0.960185\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 32.0000i 1.97320i 0.163144 + 0.986602i \(0.447836\pi\)
−0.163144 + 0.986602i \(0.552164\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 3.00000i 0.181568i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) 7.00000i 0.408944i 0.978872 + 0.204472i \(0.0655478\pi\)
−0.978872 + 0.204472i \(0.934452\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 10.0000i 0.580259i
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) − 14.0000i − 0.804279i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) 13.0000i 0.734803i 0.930062 + 0.367402i \(0.119753\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.00000i − 0.168497i −0.996445 0.0842484i \(-0.973151\pi\)
0.996445 0.0842484i \(-0.0268489\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 0 0
\(323\) − 5.00000i − 0.278207i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.00000i − 0.387101i
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 0 0
\(333\) − 4.00000i − 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 12.0000i − 0.653682i −0.945079 0.326841i \(-0.894016\pi\)
0.945079 0.326841i \(-0.105984\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) − 15.0000i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) − 7.00000i − 0.372572i −0.982496 0.186286i \(-0.940355\pi\)
0.982496 0.186286i \(-0.0596452\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 15.0000i 0.793884i
\(358\) 0 0
\(359\) −17.0000 −0.897226 −0.448613 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) −16.0000 −0.832927
\(370\) 0 0
\(371\) 27.0000 1.40177
\(372\) 0 0
\(373\) − 13.0000i − 0.673114i −0.941663 0.336557i \(-0.890737\pi\)
0.941663 0.336557i \(-0.109263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000i 0.154508i
\(378\) 0 0
\(379\) −9.00000 −0.462299 −0.231149 0.972918i \(-0.574249\pi\)
−0.231149 + 0.972918i \(0.574249\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) − 6.00000i − 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 16.0000i − 0.813326i
\(388\) 0 0
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.00000i 0.200754i 0.994949 + 0.100377i \(0.0320049\pi\)
−0.994949 + 0.100377i \(0.967995\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.00000i − 0.198273i
\(408\) 0 0
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 0 0
\(411\) 7.00000 0.345285
\(412\) 0 0
\(413\) 3.00000i 0.147620i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) 0 0
\(423\) 16.0000i 0.777947i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 42.0000i − 2.03252i
\(428\) 0 0
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.00000i 0.0478365i
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 6.00000i 0.285069i 0.989790 + 0.142534i \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.00000i 0.378387i
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 0 0
\(453\) 22.0000i 1.03365i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1.00000i − 0.0467780i −0.999726 0.0233890i \(-0.992554\pi\)
0.999726 0.0233890i \(-0.00744563\pi\)
\(458\) 0 0
\(459\) −25.0000 −1.16690
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) 0 0
\(469\) −39.0000 −1.80085
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 18.0000i 0.824163i
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) − 3.00000i − 0.136505i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 26.0000i − 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 15.0000i 0.675566i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 30.0000i − 1.34568i
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) 0 0
\(503\) − 33.0000i − 1.47140i −0.677309 0.735699i \(-0.736854\pi\)
0.677309 0.735699i \(-0.263146\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000i 0.532939i
\(508\) 0 0
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 27.0000 1.19441
\(512\) 0 0
\(513\) − 5.00000i − 0.220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) − 5.00000i − 0.218635i −0.994007 0.109317i \(-0.965134\pi\)
0.994007 0.109317i \(-0.0348665\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.0000i 0.871214i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) − 8.00000i − 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 0 0
\(543\) − 14.0000i − 0.600798i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) 28.0000 1.19501
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 0 0
\(553\) − 30.0000i − 1.27573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.0000i − 1.01691i −0.861088 0.508456i \(-0.830216\pi\)
0.861088 0.508456i \(-0.169784\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −10.0000 −0.422200
\(562\) 0 0
\(563\) − 44.0000i − 1.85438i −0.374593 0.927189i \(-0.622217\pi\)
0.374593 0.927189i \(-0.377783\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.00000i − 0.125988i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 23.0000i 0.960839i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.0000i 1.20729i 0.797255 + 0.603643i \(0.206285\pi\)
−0.797255 + 0.603643i \(0.793715\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 30.0000 1.24461
\(582\) 0 0
\(583\) 18.0000i 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 44.0000i 1.81607i 0.418890 + 0.908037i \(0.362419\pi\)
−0.418890 + 0.908037i \(0.637581\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −8.00000 −0.329076
\(592\) 0 0
\(593\) 2.00000i 0.0821302i 0.999156 + 0.0410651i \(0.0130751\pi\)
−0.999156 + 0.0410651i \(0.986925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9.00000i 0.368345i
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) − 26.0000i − 1.05880i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 42.0000i − 1.70473i −0.522949 0.852364i \(-0.675168\pi\)
0.522949 0.852364i \(-0.324832\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) − 36.0000i − 1.44231i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 2.00000i − 0.0798723i
\(628\) 0 0
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 21.0000i 0.834675i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) 20.0000 0.791188
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) − 46.0000i − 1.81406i −0.421063 0.907031i \(-0.638343\pi\)
0.421063 0.907031i \(-0.361657\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000i 0.353827i 0.984226 + 0.176913i \(0.0566112\pi\)
−0.984226 + 0.176913i \(0.943389\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000i 0.702247i
\(658\) 0 0
\(659\) −19.0000 −0.740135 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) 0 0
\(663\) − 5.00000i − 0.194184i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.00000i − 0.116160i
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) − 12.0000i − 0.462566i −0.972887 0.231283i \(-0.925708\pi\)
0.972887 0.231283i \(-0.0742923\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 27.0000i − 1.03769i −0.854867 0.518847i \(-0.826361\pi\)
0.854867 0.518847i \(-0.173639\pi\)
\(678\) 0 0
\(679\) −42.0000 −1.61181
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 2.00000i − 0.0763048i
\(688\) 0 0
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 0 0
\(693\) − 12.0000i − 0.455842i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 40.0000i − 1.51511i
\(698\) 0 0
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 44.0000 1.66186 0.830929 0.556379i \(-0.187810\pi\)
0.830929 + 0.556379i \(0.187810\pi\)
\(702\) 0 0
\(703\) 2.00000i 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 42.0000i 1.57957i
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) − 4.00000i − 0.149801i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 9.00000i 0.336111i
\(718\) 0 0
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.00000i − 0.259616i −0.991539 0.129808i \(-0.958564\pi\)
0.991539 0.129808i \(-0.0414360\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 26.0000i − 0.957722i
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) − 28.0000i − 1.02722i −0.858024 0.513610i \(-0.828308\pi\)
0.858024 0.513610i \(-0.171692\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 20.0000i 0.731762i
\(748\) 0 0
\(749\) −45.0000 −1.64426
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) − 18.0000i − 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000i 0.799604i 0.916602 + 0.399802i \(0.130921\pi\)
−0.916602 + 0.399802i \(0.869079\pi\)
\(758\) 0 0
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −37.0000 −1.34125 −0.670624 0.741797i \(-0.733974\pi\)
−0.670624 + 0.741797i \(0.733974\pi\)
\(762\) 0 0
\(763\) 21.0000i 0.760251i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.00000i − 0.0361079i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 0 0
\(773\) − 49.0000i − 1.76241i −0.472737 0.881204i \(-0.656734\pi\)
0.472737 0.881204i \(-0.343266\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 6.00000i − 0.215249i
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 0 0
\(783\) 15.0000i 0.536056i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000i 0.249523i 0.992187 + 0.124762i \(0.0398166\pi\)
−0.992187 + 0.124762i \(0.960183\pi\)
\(788\) 0 0
\(789\) −32.0000 −1.13923
\(790\) 0 0
\(791\) −54.0000 −1.92002
\(792\) 0 0
\(793\) 14.0000i 0.497155i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.00000i 0.106265i 0.998587 + 0.0531327i \(0.0169206\pi\)
−0.998587 + 0.0531327i \(0.983079\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) 18.0000i 0.635206i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 10.0000i − 0.352017i
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 27.0000 0.948098 0.474049 0.880498i \(-0.342792\pi\)
0.474049 + 0.880498i \(0.342792\pi\)
\(812\) 0 0
\(813\) − 25.0000i − 0.876788i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.00000i 0.279885i
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) − 11.0000i − 0.383436i −0.981450 0.191718i \(-0.938594\pi\)
0.981450 0.191718i \(-0.0614059\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 33.0000i − 1.14752i −0.819023 0.573761i \(-0.805484\pi\)
0.819023 0.573761i \(-0.194516\pi\)
\(828\) 0 0
\(829\) −39.0000 −1.35453 −0.677263 0.735741i \(-0.736834\pi\)
−0.677263 + 0.735741i \(0.736834\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 10.0000i − 0.346479i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 20.0000i 0.691301i
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 21.0000i 0.721569i
\(848\) 0 0
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.0000i 1.50301i 0.659727 + 0.751506i \(0.270672\pi\)
−0.659727 + 0.751506i \(0.729328\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) 54.0000i 1.83818i 0.394046 + 0.919091i \(0.371075\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 8.00000i − 0.271694i
\(868\) 0 0
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 13.0000 0.440488
\(872\) 0 0
\(873\) − 28.0000i − 0.947656i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 37.0000i 1.24940i 0.780864 + 0.624701i \(0.214779\pi\)
−0.780864 + 0.624701i \(0.785221\pi\)
\(878\) 0 0
\(879\) −7.00000 −0.236104
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) 46.0000i 1.54802i 0.633171 + 0.774012i \(0.281753\pi\)
−0.633171 + 0.774012i \(0.718247\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.0000i 0.604381i 0.953248 + 0.302190i \(0.0977178\pi\)
−0.953248 + 0.302190i \(0.902282\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) − 8.00000i − 0.267710i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.00000i 0.0333890i
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −45.0000 −1.49917
\(902\) 0 0
\(903\) − 24.0000i − 0.798670i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.00000i 0.166022i 0.996549 + 0.0830111i \(0.0264537\pi\)
−0.996549 + 0.0830111i \(0.973546\pi\)
\(908\) 0 0
\(909\) −28.0000 −0.928701
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 20.0000i 0.661903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 43.0000 1.41844 0.709220 0.704988i \(-0.249047\pi\)
0.709220 + 0.704988i \(0.249047\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) 10.0000i 0.329154i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.0000i 0.394132i
\(928\) 0 0
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) − 9.00000i − 0.294647i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 49.0000i − 1.60076i −0.599493 0.800380i \(-0.704631\pi\)
0.599493 0.800380i \(-0.295369\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.0000i 1.03986i 0.854209 + 0.519930i \(0.174042\pi\)
−0.854209 + 0.519930i \(0.825958\pi\)
\(948\) 0 0
\(949\) −9.00000 −0.292152
\(950\) 0 0
\(951\) 3.00000 0.0972817
\(952\) 0 0
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.00000i 0.193952i
\(958\) 0 0
\(959\) −21.0000 −0.678125
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 30.0000i − 0.966736i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) − 36.0000i − 1.15411i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) 10.0000i 0.318950i 0.987202 + 0.159475i \(0.0509802\pi\)
−0.987202 + 0.159475i \(0.949020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 0 0
\(993\) − 25.0000i − 0.793351i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.0000i 0.506725i 0.967371 + 0.253363i \(0.0815366\pi\)
−0.967371 + 0.253363i \(0.918463\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 3800.2.d.f.3649.2 2
5.2 odd 4 152.2.a.b.1.1 1
5.3 odd 4 3800.2.a.d.1.1 1
5.4 even 2 inner 3800.2.d.f.3649.1 2
15.2 even 4 1368.2.a.g.1.1 1
20.3 even 4 7600.2.a.o.1.1 1
20.7 even 4 304.2.a.b.1.1 1
35.27 even 4 7448.2.a.g.1.1 1
40.27 even 4 1216.2.a.l.1.1 1
40.37 odd 4 1216.2.a.f.1.1 1
60.47 odd 4 2736.2.a.k.1.1 1
95.37 even 4 2888.2.a.b.1.1 1
380.227 odd 4 5776.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.b.1.1 1 5.2 odd 4
304.2.a.b.1.1 1 20.7 even 4
1216.2.a.f.1.1 1 40.37 odd 4
1216.2.a.l.1.1 1 40.27 even 4
1368.2.a.g.1.1 1 15.2 even 4
2736.2.a.k.1.1 1 60.47 odd 4
2888.2.a.b.1.1 1 95.37 even 4
3800.2.a.d.1.1 1 5.3 odd 4
3800.2.d.f.3649.1 2 5.4 even 2 inner
3800.2.d.f.3649.2 2 1.1 even 1 trivial
5776.2.a.l.1.1 1 380.227 odd 4
7448.2.a.g.1.1 1 35.27 even 4
7600.2.a.o.1.1 1 20.3 even 4