Properties

Label 4800.2.f.b.3649.2
Level $4800$
Weight $2$
Character 4800.3649
Analytic conductor $38.328$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.3281929702\)
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 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.b.3649.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +5.00000i q^{13} -6.00000i q^{17} +5.00000 q^{19} +1.00000 q^{21} +6.00000i q^{23} -1.00000i q^{27} -6.00000 q^{29} -1.00000 q^{31} -6.00000i q^{33} -2.00000i q^{37} -5.00000 q^{39} -1.00000i q^{43} +6.00000i q^{47} +6.00000 q^{49} +6.00000 q^{51} -12.0000i q^{53} +5.00000i q^{57} -6.00000 q^{59} +13.0000 q^{61} +1.00000i q^{63} -11.0000i q^{67} -6.00000 q^{69} -2.00000i q^{73} +6.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -6.00000i q^{87} +5.00000 q^{91} -1.00000i q^{93} -7.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 12q^{11} + 10q^{19} + 2q^{21} - 12q^{29} - 2q^{31} - 10q^{39} + 12q^{49} + 12q^{51} - 12q^{59} + 26q^{61} - 12q^{69} - 16q^{79} + 2q^{81} + 10q^{91} + 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) − 6.00000i − 1.04447i
\(34\) 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
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) − 12.0000i − 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000i 0.662266i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.0000i − 1.34386i −0.740613 0.671932i \(-0.765465\pi\)
0.740613 0.671932i \(-0.234535\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 0 0
\(93\) − 1.00000i − 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.00000i − 0.710742i −0.934725 0.355371i \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 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
\(113\) − 12.0000i − 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 5.00000i − 0.462250i
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) − 5.00000i − 0.433555i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) − 30.0000i − 2.50873i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 23.0000i − 1.83560i −0.397043 0.917800i \(-0.629964\pi\)
0.397043 0.917800i \(-0.370036\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.00000i − 0.450988i
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 13.0000i 0.960988i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 36.0000i 2.63258i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) − 11.0000i − 0.791797i −0.918294 0.395899i \(-0.870433\pi\)
0.918294 0.395899i \(-0.129567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.00000i − 0.417029i
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.00000i 0.0678844i
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) 25.0000i 1.67412i 0.547108 + 0.837062i \(0.315729\pi\)
−0.547108 + 0.837062i \(0.684271\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 25.0000i 1.59071i
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) − 36.0000i − 2.26330i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 5.00000i 0.302614i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.00000i 0.420589i 0.977638 + 0.210295i \(0.0674423\pi\)
−0.977638 + 0.210295i \(0.932558\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 17.0000i 1.01055i 0.862960 + 0.505273i \(0.168608\pi\)
−0.862960 + 0.505273i \(0.831392\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) 0 0
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 6.00000i 0.348155i
\(298\) 0 0
\(299\) −30.0000 −1.73494
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 17.0000i − 0.970241i −0.874447 0.485121i \(-0.838776\pi\)
0.874447 0.485121i \(-0.161224\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) − 11.0000i − 0.621757i −0.950450 0.310878i \(-0.899377\pi\)
0.950450 0.310878i \(-0.100623\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 24.0000i − 1.34797i −0.738743 0.673987i \(-0.764580\pi\)
0.738743 0.673987i \(-0.235420\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 30.0000i − 1.66924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.00000i − 0.387101i
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 17.0000i 0.926049i 0.886345 + 0.463025i \(0.153236\pi\)
−0.886345 + 0.463025i \(0.846764\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 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
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.00000i − 0.317554i
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 31.0000i − 1.61819i −0.587680 0.809093i \(-0.699959\pi\)
0.587680 0.809093i \(-0.300041\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) − 1.00000i − 0.0517780i −0.999665 0.0258890i \(-0.991758\pi\)
0.999665 0.0258890i \(-0.00824165\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 30.0000i − 1.54508i
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.00000i 0.0508329i
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) − 12.0000i − 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13.0000i 0.652451i 0.945292 + 0.326226i \(0.105777\pi\)
−0.945292 + 0.326226i \(0.894223\pi\)
\(398\) 0 0
\(399\) 5.00000 0.250313
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) − 5.00000i − 0.249068i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 6.00000i 0.295241i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 13.0000i − 0.629114i
\(428\) 0 0
\(429\) 30.0000 1.44841
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) − 11.0000i − 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.0000i 1.43509i
\(438\) 0 0
\(439\) −23.0000 −1.09773 −0.548865 0.835911i \(-0.684940\pi\)
−0.548865 + 0.835911i \(0.684940\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 18.0000i 0.851371i
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.00000i 0.234920i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) 0 0
\(469\) −11.0000 −0.507933
\(470\) 0 0
\(471\) 23.0000 1.05978
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.00000i − 0.317200i −0.987343 0.158600i \(-0.949302\pi\)
0.987343 0.158600i \(-0.0506981\pi\)
\(488\) 0 0
\(489\) −5.00000 −0.226108
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 36.0000i 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 29.0000 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) − 5.00000i − 0.220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 36.0000i − 1.58328i
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 17.0000i 0.743358i 0.928361 + 0.371679i \(0.121218\pi\)
−0.928361 + 0.371679i \(0.878782\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000i 0.261364i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 6.00000i − 0.258919i
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) − 5.00000i − 0.214571i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) −30.0000 −1.27804
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 0 0
\(563\) − 30.0000i − 1.26435i −0.774826 0.632175i \(-0.782163\pi\)
0.774826 0.632175i \(-0.217837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −5.00000 −0.209243 −0.104622 0.994512i \(-0.533363\pi\)
−0.104622 + 0.994512i \(0.533363\pi\)
\(572\) 0 0
\(573\) − 6.00000i − 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 37.0000i − 1.54033i −0.637845 0.770165i \(-0.720174\pi\)
0.637845 0.770165i \(-0.279826\pi\)
\(578\) 0 0
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 72.0000i 2.98194i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) − 24.0000i − 0.985562i −0.870153 0.492781i \(-0.835980\pi\)
0.870153 0.492781i \(-0.164020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000i 0.286491i
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) 11.0000i 0.447955i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.0000i 1.44931i 0.689114 + 0.724653i \(0.258000\pi\)
−0.689114 + 0.724653i \(0.742000\pi\)
\(618\) 0 0
\(619\) −31.0000 −1.24600 −0.622998 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 30.0000i − 1.19808i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 0 0
\(633\) − 5.00000i − 0.198732i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 30.0000i 1.18864i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 0 0
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) 0 0
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 30.0000i 1.16510i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 36.0000i − 1.39393i
\(668\) 0 0
\(669\) −25.0000 −0.966556
\(670\) 0 0
\(671\) −78.0000 −3.01116
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 19.0000i − 0.724895i
\(688\) 0 0
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) − 10.0000i − 0.377157i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) 29.0000 1.08912 0.544559 0.838723i \(-0.316697\pi\)
0.544559 + 0.838723i \(0.316697\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) − 6.00000i − 0.224702i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) − 7.00000i − 0.260333i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.0000i 0.853023i 0.904482 + 0.426511i \(0.140258\pi\)
−0.904482 + 0.426511i \(0.859742\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 66.0000i 2.43114i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −25.0000 −0.918398
\(742\) 0 0
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19.0000i 0.690567i 0.938498 + 0.345283i \(0.112217\pi\)
−0.938498 + 0.345283i \(0.887783\pi\)
\(758\) 0 0
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 30.0000i − 1.08324i
\(768\) 0 0
\(769\) 13.0000 0.468792 0.234396 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.00000i − 0.0717496i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 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
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 65.0000i 2.30822i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.0000i 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000i 0.211210i
\(808\) 0 0
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) − 16.0000i − 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 5.00000i − 0.174928i
\(818\) 0 0
\(819\) −5.00000 −0.174714
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 13.0000i 0.453152i 0.973994 + 0.226576i \(0.0727531\pi\)
−0.973994 + 0.226576i \(0.927247\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −7.00000 −0.242827
\(832\) 0 0
\(833\) − 36.0000i − 1.24733i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000i 0.0345651i
\(838\) 0 0
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 25.0000i − 0.859010i
\(848\) 0 0
\(849\) −17.0000 −0.583438
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 23.0000i 0.787505i 0.919216 + 0.393753i \(0.128823\pi\)
−0.919216 + 0.393753i \(0.871177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 12.0000i − 0.409912i −0.978771 0.204956i \(-0.934295\pi\)
0.978771 0.204956i \(-0.0657052\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 48.0000i − 1.63394i −0.576681 0.816970i \(-0.695652\pi\)
0.576681 0.816970i \(-0.304348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 55.0000 1.86360
\(872\) 0 0
\(873\) 7.00000i 0.236914i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 49.0000i 1.65461i 0.561751 + 0.827306i \(0.310128\pi\)
−0.561751 + 0.827306i \(0.689872\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 23.0000i 0.774012i 0.922077 + 0.387006i \(0.126491\pi\)
−0.922077 + 0.387006i \(0.873509\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 6.00000i − 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 0 0
\(893\) 30.0000i 1.00391i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 30.0000i − 1.00167i
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −72.0000 −2.39867
\(902\) 0 0
\(903\) − 1.00000i − 0.0332779i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 36.0000i 1.19143i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −53.0000 −1.74831 −0.874154 0.485648i \(-0.838584\pi\)
−0.874154 + 0.485648i \(0.838584\pi\)
\(920\) 0 0
\(921\) 17.0000 0.560169
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 30.0000 0.983210
\(932\) 0 0
\(933\) − 6.00000i − 0.196431i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 47.0000i 1.53542i 0.640796 + 0.767712i \(0.278605\pi\)
−0.640796 + 0.767712i \(0.721395\pi\)
\(938\) 0 0
\(939\) 11.0000 0.358971
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 42.0000i − 1.36482i −0.730971 0.682408i \(-0.760933\pi\)
0.730971 0.682408i \(-0.239067\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 36.0000i 1.16371i
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 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
\(969\) 30.0000 0.963739
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 7.00000 0.223493
\(982\) 0 0
\(983\) − 12.0000i − 0.382741i −0.981518 0.191370i \(-0.938707\pi\)
0.981518 0.191370i \(-0.0612931\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −37.0000 −1.17534 −0.587672 0.809099i \(-0.699955\pi\)
−0.587672 + 0.809099i \(0.699955\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 26.0000i − 0.823428i −0.911313 0.411714i \(-0.864930\pi\)
0.911313 0.411714i \(-0.135070\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 4800.2.f.b.3649.2 2
4.3 odd 2 4800.2.f.bi.3649.1 2
5.2 odd 4 4800.2.a.ce.1.1 1
5.3 odd 4 4800.2.a.o.1.1 1
5.4 even 2 inner 4800.2.f.b.3649.1 2
8.3 odd 2 1200.2.f.a.49.2 2
8.5 even 2 300.2.d.a.49.1 2
20.3 even 4 4800.2.a.cf.1.1 1
20.7 even 4 4800.2.a.p.1.1 1
20.19 odd 2 4800.2.f.bi.3649.2 2
24.5 odd 2 900.2.d.a.649.1 2
24.11 even 2 3600.2.f.v.2449.2 2
40.3 even 4 1200.2.a.f.1.1 1
40.13 odd 4 300.2.a.c.1.1 yes 1
40.19 odd 2 1200.2.f.a.49.1 2
40.27 even 4 1200.2.a.n.1.1 1
40.29 even 2 300.2.d.a.49.2 2
40.37 odd 4 300.2.a.b.1.1 1
120.29 odd 2 900.2.d.a.649.2 2
120.53 even 4 900.2.a.c.1.1 1
120.59 even 2 3600.2.f.v.2449.1 2
120.77 even 4 900.2.a.e.1.1 1
120.83 odd 4 3600.2.a.z.1.1 1
120.107 odd 4 3600.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.a.b.1.1 1 40.37 odd 4
300.2.a.c.1.1 yes 1 40.13 odd 4
300.2.d.a.49.1 2 8.5 even 2
300.2.d.a.49.2 2 40.29 even 2
900.2.a.c.1.1 1 120.53 even 4
900.2.a.e.1.1 1 120.77 even 4
900.2.d.a.649.1 2 24.5 odd 2
900.2.d.a.649.2 2 120.29 odd 2
1200.2.a.f.1.1 1 40.3 even 4
1200.2.a.n.1.1 1 40.27 even 4
1200.2.f.a.49.1 2 40.19 odd 2
1200.2.f.a.49.2 2 8.3 odd 2
3600.2.a.s.1.1 1 120.107 odd 4
3600.2.a.z.1.1 1 120.83 odd 4
3600.2.f.v.2449.1 2 120.59 even 2
3600.2.f.v.2449.2 2 24.11 even 2
4800.2.a.o.1.1 1 5.3 odd 4
4800.2.a.p.1.1 1 20.7 even 4
4800.2.a.ce.1.1 1 5.2 odd 4
4800.2.a.cf.1.1 1 20.3 even 4
4800.2.f.b.3649.1 2 5.4 even 2 inner
4800.2.f.b.3649.2 2 1.1 even 1 trivial
4800.2.f.bi.3649.1 2 4.3 odd 2
4800.2.f.bi.3649.2 2 20.19 odd 2