Properties

Label 3840.2.k.r.1921.2
Level $3840$
Weight $2$
Character 3840.1921
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
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 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1921.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.1921
Dual form 3840.2.k.r.1921.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +1.00000 q^{15} +2.00000 q^{17} -4.00000i q^{19} -1.00000 q^{25} -1.00000i q^{27} -2.00000i q^{29} +4.00000 q^{33} +10.0000i q^{37} +2.00000 q^{39} -10.0000 q^{41} +4.00000i q^{43} +1.00000i q^{45} -8.00000 q^{47} -7.00000 q^{49} +2.00000i q^{51} +10.0000i q^{53} -4.00000 q^{55} +4.00000 q^{57} -4.00000i q^{59} -2.00000i q^{61} -2.00000 q^{65} -12.0000i q^{67} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{85} +2.00000 q^{87} +6.00000 q^{89} -4.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{15} + 4q^{17} - 2q^{25} + 8q^{33} + 4q^{39} - 20q^{41} - 16q^{47} - 14q^{49} - 8q^{55} + 8q^{57} - 4q^{65} - 16q^{71} - 20q^{73} + 2q^{81} + 4q^{87} + 12q^{89} - 8q^{95} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) − 6.00000i − 0.597022i −0.954406 0.298511i \(-0.903510\pi\)
0.954406 0.298511i \(-0.0964900\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 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
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 10.0000i − 0.901670i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) − 4.00000i − 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) − 8.00000i − 0.673722i
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) − 7.00000i − 0.577350i
\(148\) 0 0
\(149\) − 22.0000i − 1.80231i −0.433497 0.901155i \(-0.642720\pi\)
0.433497 0.901155i \(-0.357280\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) − 4.00000i − 0.311400i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) − 20.0000i − 1.49487i −0.664335 0.747435i \(-0.731285\pi\)
0.664335 0.747435i \(-0.268715\pi\)
\(180\) 0 0
\(181\) 10.0000i 0.743294i 0.928374 + 0.371647i \(0.121207\pi\)
−0.928374 + 0.371647i \(0.878793\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) − 2.00000i − 0.143223i
\(196\) 0 0
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000i 0.698430i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) − 20.0000i − 1.37686i −0.725304 0.688428i \(-0.758301\pi\)
0.725304 0.688428i \(-0.241699\pi\)
\(212\) 0 0
\(213\) − 8.00000i − 0.548151i
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 10.0000i − 0.675737i
\(220\) 0 0
\(221\) − 4.00000i − 0.269069i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) − 6.00000i − 0.396491i −0.980152 0.198246i \(-0.936476\pi\)
0.980152 0.198246i \(-0.0635244\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 8.00000i 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 7.00000i 0.447214i
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 12.0000i 0.757433i 0.925513 + 0.378717i \(0.123635\pi\)
−0.925513 + 0.378717i \(0.876365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 14.0000i 0.853595i 0.904347 + 0.426798i \(0.140358\pi\)
−0.904347 + 0.426798i \(0.859642\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000i 0.241209i
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 0 0
\(285\) − 4.00000i − 0.236940i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) − 16.0000i − 0.910208i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 2.00000i 0.110940i
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000i 0.659580i 0.944054 + 0.329790i \(0.106978\pi\)
−0.944054 + 0.329790i \(0.893022\pi\)
\(332\) 0 0
\(333\) − 10.0000i − 0.547997i
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 2.00000i 0.108625i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) 0 0
\(349\) − 2.00000i − 0.107058i −0.998566 0.0535288i \(-0.982953\pi\)
0.998566 0.0535288i \(-0.0170469\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 8.00000i 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 10.0000i 0.523424i
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) − 6.00000i − 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000i 0.295958i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) − 4.00000i − 0.195413i −0.995215 0.0977064i \(-0.968849\pi\)
0.995215 0.0977064i \(-0.0311506\pi\)
\(420\) 0 0
\(421\) 26.0000i 1.26716i 0.773676 + 0.633581i \(0.218416\pi\)
−0.773676 + 0.633581i \(0.781584\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 8.00000i − 0.386244i
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) − 2.00000i − 0.0958927i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) − 6.00000i − 0.284427i
\(446\) 0 0
\(447\) 22.0000 1.04056
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 40.0000i 1.88353i
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) − 18.0000i − 0.838344i −0.907907 0.419172i \(-0.862320\pi\)
0.907907 0.419172i \(-0.137680\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) − 10.0000i − 0.457869i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 2.00000i − 0.0908153i
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 28.0000i 1.26362i 0.775122 + 0.631811i \(0.217688\pi\)
−0.775122 + 0.631811i \(0.782312\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) − 34.0000i − 1.50702i −0.657434 0.753512i \(-0.728358\pi\)
0.657434 0.753512i \(-0.271642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 16.0000i 0.705044i
\(516\) 0 0
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) 20.0000i 0.866296i
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) 0 0
\(539\) 28.0000i 1.20605i
\(540\) 0 0
\(541\) 30.0000i 1.28980i 0.764267 + 0.644900i \(0.223101\pi\)
−0.764267 + 0.644900i \(0.776899\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) 2.00000i 0.0853579i
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.0000i 0.424476i
\(556\) 0 0
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) − 2.00000i − 0.0841406i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) − 4.00000i − 0.167395i −0.996491 0.0836974i \(-0.973327\pi\)
0.996491 0.0836974i \(-0.0266729\pi\)
\(572\) 0 0
\(573\) − 16.0000i − 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 2.00000i 0.0831172i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 8.00000i − 0.327418i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 5.00000i 0.203279i
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000i 0.647291i
\(612\) 0 0
\(613\) − 22.0000i − 0.888572i −0.895885 0.444286i \(-0.853457\pi\)
0.895885 0.444286i \(-0.146543\pi\)
\(614\) 0 0
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) − 16.0000i − 0.638978i
\(628\) 0 0
\(629\) 20.0000i 0.797452i
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) − 8.00000i − 0.317470i
\(636\) 0 0
\(637\) 14.0000i 0.554700i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 4.00000i 0.157500i
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.0000i 1.80012i 0.435767 + 0.900060i \(0.356477\pi\)
−0.435767 + 0.900060i \(0.643523\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) − 20.0000i − 0.779089i −0.921008 0.389545i \(-0.872632\pi\)
0.921008 0.389545i \(-0.127368\pi\)
\(660\) 0 0
\(661\) − 22.0000i − 0.855701i −0.903850 0.427850i \(-0.859271\pi\)
0.903850 0.427850i \(-0.140729\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 8.00000i − 0.309298i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 0 0
\(685\) − 6.00000i − 0.229248i
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 44.0000i 1.67384i 0.547326 + 0.836919i \(0.315646\pi\)
−0.547326 + 0.836919i \(0.684354\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) − 2.00000i − 0.0755390i −0.999286 0.0377695i \(-0.987975\pi\)
0.999286 0.0377695i \(-0.0120253\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26.0000i 0.976450i 0.872718 + 0.488225i \(0.162356\pi\)
−0.872718 + 0.488225i \(0.837644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000i 0.299183i
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 14.0000i − 0.520666i
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000i 0.295891i
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) −7.00000 −0.258199
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 44.0000i 1.61857i 0.587419 + 0.809283i \(0.300144\pi\)
−0.587419 + 0.809283i \(0.699856\pi\)
\(740\) 0 0
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −22.0000 −0.806018
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 8.00000i 0.291150i
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000i 0.0723102i
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 18.0000i 0.648254i
\(772\) 0 0
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 40.0000i 1.43315i
\(780\) 0 0
\(781\) 32.0000i 1.14505i
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 16.0000i 0.569615i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 10.0000i 0.354663i
\(796\) 0 0
\(797\) − 2.00000i − 0.0708436i −0.999372 0.0354218i \(-0.988723\pi\)
0.999372 0.0354218i \(-0.0112775\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 40.0000i 1.41157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 12.0000i 0.421377i 0.977553 + 0.210688i \(0.0675706\pi\)
−0.977553 + 0.210688i \(0.932429\pi\)
\(812\) 0 0
\(813\) − 16.0000i − 0.561144i
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 54.0000i − 1.88461i −0.334751 0.942306i \(-0.608652\pi\)
0.334751 0.942306i \(-0.391348\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 0 0
\(829\) 30.0000i 1.04194i 0.853574 + 0.520972i \(0.174430\pi\)
−0.853574 + 0.520972i \(0.825570\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 0 0
\(845\) − 9.00000i − 0.309609i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 6.00000i − 0.205436i −0.994711 0.102718i \(-0.967246\pi\)
0.994711 0.102718i \(-0.0327539\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) − 20.0000i − 0.682391i −0.939992 0.341196i \(-0.889168\pi\)
0.939992 0.341196i \(-0.110832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 56.0000 1.90626 0.953131 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 30.0000i 1.01303i 0.862232 + 0.506514i \(0.169066\pi\)
−0.862232 + 0.506514i \(0.830934\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) 0 0
\(885\) − 4.00000i − 0.134459i
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) 32.0000i 1.07084i
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 20.0000i 0.666297i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 0 0
\(909\) 6.00000i 0.199007i
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) − 2.00000i − 0.0661180i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) − 10.0000i − 0.328798i
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 28.0000i 0.917663i
\(932\) 0 0
\(933\) − 24.0000i − 0.785725i
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) 0 0
\(939\) − 26.0000i − 0.848478i
\(940\) 0 0
\(941\) − 50.0000i − 1.62995i −0.579494 0.814977i \(-0.696750\pi\)
0.579494 0.814977i \(-0.303250\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) 20.0000i 0.649227i
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 0 0
\(955\) 16.0000i 0.517748i
\(956\) 0 0
\(957\) − 8.00000i − 0.258603i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) − 2.00000i − 0.0643823i
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 60.0000i 1.92549i 0.270408 + 0.962746i \(0.412841\pi\)
−0.270408 + 0.962746i \(0.587159\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) − 24.0000i − 0.767043i
\(980\) 0 0
\(981\) − 14.0000i − 0.446986i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 8.00000i 0.253617i
\(996\) 0 0
\(997\) − 54.0000i − 1.71020i −0.518465 0.855099i \(-0.673497\pi\)
0.518465 0.855099i \(-0.326503\pi\)
\(998\) 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 3840.2.k.r.1921.2 2
4.3 odd 2 3840.2.k.m.1921.1 2
8.3 odd 2 3840.2.k.m.1921.2 2
8.5 even 2 inner 3840.2.k.r.1921.1 2
16.3 odd 4 960.2.a.l.1.1 1
16.5 even 4 240.2.a.d.1.1 1
16.11 odd 4 15.2.a.a.1.1 1
16.13 even 4 960.2.a.a.1.1 1
48.5 odd 4 720.2.a.c.1.1 1
48.11 even 4 45.2.a.a.1.1 1
48.29 odd 4 2880.2.a.bc.1.1 1
48.35 even 4 2880.2.a.y.1.1 1
80.3 even 4 4800.2.f.bf.3649.2 2
80.13 odd 4 4800.2.f.c.3649.1 2
80.19 odd 4 4800.2.a.t.1.1 1
80.27 even 4 75.2.b.b.49.1 2
80.29 even 4 4800.2.a.bz.1.1 1
80.37 odd 4 1200.2.f.h.49.1 2
80.43 even 4 75.2.b.b.49.2 2
80.53 odd 4 1200.2.f.h.49.2 2
80.59 odd 4 75.2.a.b.1.1 1
80.67 even 4 4800.2.f.bf.3649.1 2
80.69 even 4 1200.2.a.e.1.1 1
80.77 odd 4 4800.2.f.c.3649.2 2
112.11 odd 12 735.2.i.e.226.1 2
112.27 even 4 735.2.a.c.1.1 1
112.59 even 12 735.2.i.d.226.1 2
112.75 even 12 735.2.i.d.361.1 2
112.107 odd 12 735.2.i.e.361.1 2
144.11 even 12 405.2.e.c.271.1 2
144.43 odd 12 405.2.e.f.271.1 2
144.59 even 12 405.2.e.c.136.1 2
144.139 odd 12 405.2.e.f.136.1 2
176.43 even 4 1815.2.a.d.1.1 1
208.155 odd 4 2535.2.a.j.1.1 1
240.53 even 4 3600.2.f.e.2449.1 2
240.59 even 4 225.2.a.b.1.1 1
240.107 odd 4 225.2.b.b.199.2 2
240.149 odd 4 3600.2.a.u.1.1 1
240.197 even 4 3600.2.f.e.2449.2 2
240.203 odd 4 225.2.b.b.199.1 2
272.203 odd 4 4335.2.a.c.1.1 1
304.75 even 4 5415.2.a.j.1.1 1
336.251 odd 4 2205.2.a.i.1.1 1
368.91 even 4 7935.2.a.d.1.1 1
528.395 odd 4 5445.2.a.c.1.1 1
560.139 even 4 3675.2.a.j.1.1 1
624.155 even 4 7605.2.a.g.1.1 1
880.219 even 4 9075.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 16.11 odd 4
45.2.a.a.1.1 1 48.11 even 4
75.2.a.b.1.1 1 80.59 odd 4
75.2.b.b.49.1 2 80.27 even 4
75.2.b.b.49.2 2 80.43 even 4
225.2.a.b.1.1 1 240.59 even 4
225.2.b.b.199.1 2 240.203 odd 4
225.2.b.b.199.2 2 240.107 odd 4
240.2.a.d.1.1 1 16.5 even 4
405.2.e.c.136.1 2 144.59 even 12
405.2.e.c.271.1 2 144.11 even 12
405.2.e.f.136.1 2 144.139 odd 12
405.2.e.f.271.1 2 144.43 odd 12
720.2.a.c.1.1 1 48.5 odd 4
735.2.a.c.1.1 1 112.27 even 4
735.2.i.d.226.1 2 112.59 even 12
735.2.i.d.361.1 2 112.75 even 12
735.2.i.e.226.1 2 112.11 odd 12
735.2.i.e.361.1 2 112.107 odd 12
960.2.a.a.1.1 1 16.13 even 4
960.2.a.l.1.1 1 16.3 odd 4
1200.2.a.e.1.1 1 80.69 even 4
1200.2.f.h.49.1 2 80.37 odd 4
1200.2.f.h.49.2 2 80.53 odd 4
1815.2.a.d.1.1 1 176.43 even 4
2205.2.a.i.1.1 1 336.251 odd 4
2535.2.a.j.1.1 1 208.155 odd 4
2880.2.a.y.1.1 1 48.35 even 4
2880.2.a.bc.1.1 1 48.29 odd 4
3600.2.a.u.1.1 1 240.149 odd 4
3600.2.f.e.2449.1 2 240.53 even 4
3600.2.f.e.2449.2 2 240.197 even 4
3675.2.a.j.1.1 1 560.139 even 4
3840.2.k.m.1921.1 2 4.3 odd 2
3840.2.k.m.1921.2 2 8.3 odd 2
3840.2.k.r.1921.1 2 8.5 even 2 inner
3840.2.k.r.1921.2 2 1.1 even 1 trivial
4335.2.a.c.1.1 1 272.203 odd 4
4800.2.a.t.1.1 1 80.19 odd 4
4800.2.a.bz.1.1 1 80.29 even 4
4800.2.f.c.3649.1 2 80.13 odd 4
4800.2.f.c.3649.2 2 80.77 odd 4
4800.2.f.bf.3649.1 2 80.67 even 4
4800.2.f.bf.3649.2 2 80.3 even 4
5415.2.a.j.1.1 1 304.75 even 4
5445.2.a.c.1.1 1 528.395 odd 4
7605.2.a.g.1.1 1 624.155 even 4
7935.2.a.d.1.1 1 368.91 even 4
9075.2.a.g.1.1 1 880.219 even 4