Properties

Label 768.2.d.h.385.2
Level 768
Weight 2
Character 768.385
Analytic conductor 6.133
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 385.2
Root \(-1.00000i\) of \(x^{2} + 1\)
Character \(\chi\) \(=\) 768.385
Dual form 768.2.d.h.385.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} +1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{31} +4.00000 q^{33} -8.00000i q^{35} +2.00000i q^{37} +2.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} +2.00000i q^{45} +8.00000 q^{47} +9.00000 q^{49} -6.00000i q^{51} -10.0000i q^{53} -8.00000 q^{55} +4.00000 q^{57} +4.00000i q^{59} +6.00000i q^{61} -4.00000 q^{63} -4.00000 q^{65} +4.00000i q^{67} +16.0000 q^{71} +6.00000 q^{73} +1.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +12.0000i q^{85} -2.00000 q^{87} -10.0000 q^{89} -8.00000i q^{91} +4.00000i q^{93} -8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 8q^{7} - 2q^{9} + 4q^{15} - 12q^{17} + 2q^{25} + 8q^{31} + 8q^{33} + 4q^{39} - 4q^{41} + 16q^{47} + 18q^{49} - 16q^{55} + 8q^{57} - 8q^{63} - 8q^{65} + 32q^{71} + 12q^{73} + 8q^{79} + 2q^{81} - 4q^{87} - 20q^{89} - 16q^{95} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) − 2.00000i − 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 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
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 4.00000i 0.872872i
\(22\) 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 \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) − 8.00000i − 1.35225i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) − 6.00000i − 0.840168i
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i 0.923287 + 0.384111i \(0.125492\pi\)
−0.923287 + 0.384111i \(0.874508\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 12.0000i 1.30158i
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) − 8.00000i − 0.838628i
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 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
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) − 12.0000i − 1.07331i
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 4.00000i 0.349482i 0.984614 + 0.174741i \(0.0559088\pi\)
−0.984614 + 0.174741i \(0.944091\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 20.0000i 1.69638i 0.529694 + 0.848189i \(0.322307\pi\)
−0.529694 + 0.848189i \(0.677693\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) − 8.00000i − 0.642575i
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 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 \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) − 8.00000i − 0.622799i
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) − 12.0000i − 0.896922i −0.893802 0.448461i \(-0.851972\pi\)
0.893802 0.448461i \(-0.148028\pi\)
\(180\) 0 0
\(181\) 10.0000i 0.743294i 0.928374 + 0.371647i \(0.121207\pi\)
−0.928374 + 0.371647i \(0.878793\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) − 4.00000i − 0.290957i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) − 4.00000i − 0.286446i
\(196\) 0 0
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 8.00000i 0.561490i
\(204\) 0 0
\(205\) 4.00000i 0.279372i
\(206\) 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 \(0.241699\pi\)
−0.725304 + 0.688428i \(0.758301\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) − 16.0000i − 1.04372i
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 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
\(245\) − 18.0000i − 1.14998i
\(246\) 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
\(255\) −12.0000 −0.751469
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 8.00000i 0.497096i
\(260\) 0 0
\(261\) − 2.00000i − 0.123797i
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −20.0000 −1.22859
\(266\) 0 0
\(267\) − 10.0000i − 0.611990i
\(268\) 0 0
\(269\) 18.0000i 1.09748i 0.835993 + 0.548740i \(0.184892\pi\)
−0.835993 + 0.548740i \(0.815108\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 28.0000i − 1.66443i −0.554455 0.832214i \(-0.687073\pi\)
0.554455 0.832214i \(-0.312927\pi\)
\(284\) 0 0
\(285\) − 8.00000i − 0.473879i
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 16.0000i − 0.922225i
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 12.0000i 0.682656i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 8.00000i 0.450749i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) − 2.00000i − 0.110940i
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) − 28.0000i − 1.53902i −0.638635 0.769510i \(-0.720501\pi\)
0.638635 0.769510i \(-0.279499\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 2.00000i 0.108625i
\(340\) 0 0
\(341\) − 16.0000i − 0.866449i
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 20.0000i − 1.07366i −0.843692 0.536828i \(-0.819622\pi\)
0.843692 0.536828i \(-0.180378\pi\)
\(348\) 0 0
\(349\) − 26.0000i − 1.39175i −0.718164 0.695874i \(-0.755017\pi\)
0.718164 0.695874i \(-0.244983\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) − 32.0000i − 1.69838i
\(356\) 0 0
\(357\) − 24.0000i − 1.27021i
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) − 12.0000i − 0.628109i
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) − 40.0000i − 2.07670i
\(372\) 0 0
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) 0 0
\(381\) − 20.0000i − 1.02463i
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) −32.0000 −1.63087
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 30.0000i 1.52106i 0.649303 + 0.760530i \(0.275061\pi\)
−0.649303 + 0.760530i \(0.724939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) − 8.00000i − 0.402524i
\(396\) 0 0
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) − 8.00000i − 0.398508i
\(404\) 0 0
\(405\) − 2.00000i − 0.0993808i
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) − 18.0000i − 0.887875i
\(412\) 0 0
\(413\) 16.0000i 0.787309i
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) − 20.0000i − 0.977064i −0.872546 0.488532i \(-0.837533\pi\)
0.872546 0.488532i \(-0.162467\pi\)
\(420\) 0 0
\(421\) − 38.0000i − 1.85201i −0.377515 0.926003i \(-0.623221\pi\)
0.377515 0.926003i \(-0.376779\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) − 8.00000i − 0.386244i
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 4.00000i 0.191785i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 20.0000i 0.948091i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 8.00000i 0.376705i
\(452\) 0 0
\(453\) − 12.0000i − 0.563809i
\(454\) 0 0
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) 6.00000i 0.280056i
\(460\) 0 0
\(461\) − 6.00000i − 0.279448i −0.990190 0.139724i \(-0.955378\pi\)
0.990190 0.139724i \(-0.0446215\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) 0 0
\(469\) 16.0000i 0.738811i
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 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
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.0000i 1.27141i
\(486\) 0 0
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) − 28.0000i − 1.26362i −0.775122 0.631811i \(-0.782312\pi\)
0.775122 0.631811i \(-0.217688\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 64.0000 2.87079
\(498\) 0 0
\(499\) − 12.0000i − 0.537194i −0.963253 0.268597i \(-0.913440\pi\)
0.963253 0.268597i \(-0.0865599\pi\)
\(500\) 0 0
\(501\) − 8.00000i − 0.357414i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 18.0000i 0.797836i 0.916987 + 0.398918i \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) − 24.0000i − 1.05757i
\(516\) 0 0
\(517\) − 32.0000i − 1.40736i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 12.0000i 0.524723i 0.964970 + 0.262362i \(0.0845013\pi\)
−0.964970 + 0.262362i \(0.915499\pi\)
\(524\) 0 0
\(525\) 4.00000i 0.174574i
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 4.00000i − 0.173585i
\(532\) 0 0
\(533\) 4.00000i 0.173259i
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) − 36.0000i − 1.55063i
\(540\) 0 0
\(541\) − 2.00000i − 0.0859867i −0.999075 0.0429934i \(-0.986311\pi\)
0.999075 0.0429934i \(-0.0136894\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 0 0
\(549\) − 6.00000i − 0.256074i
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) 4.00000i 0.169791i
\(556\) 0 0
\(557\) − 22.0000i − 0.932170i −0.884740 0.466085i \(-0.845664\pi\)
0.884740 0.466085i \(-0.154336\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) − 4.00000i − 0.168281i
\(566\) 0 0
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 46.0000 1.92842 0.964210 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 0 0
\(571\) − 44.0000i − 1.84134i −0.390339 0.920671i \(-0.627642\pi\)
0.390339 0.920671i \(-0.372358\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 18.0000i 0.748054i
\(580\) 0 0
\(581\) 48.0000i 1.99138i
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) − 16.0000i − 0.659269i
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 48.0000i 1.96781i
\(596\) 0 0
\(597\) − 4.00000i − 0.163709i
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 10.0000i 0.406558i
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) − 16.0000i − 0.647291i
\(612\) 0 0
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) 0 0
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) − 12.0000i − 0.482321i −0.970485 0.241160i \(-0.922472\pi\)
0.970485 0.241160i \(-0.0775280\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −40.0000 −1.60257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) − 16.0000i − 0.638978i
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) 40.0000i 1.58735i
\(636\) 0 0
\(637\) − 18.0000i − 0.713186i
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) − 8.00000i − 0.315000i
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 0 0
\(653\) 10.0000i 0.391330i 0.980671 + 0.195665i \(0.0626866\pi\)
−0.980671 + 0.195665i \(0.937313\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 36.0000i 1.40236i 0.712984 + 0.701180i \(0.247343\pi\)
−0.712984 + 0.701180i \(0.752657\pi\)
\(660\) 0 0
\(661\) − 14.0000i − 0.544537i −0.962221 0.272268i \(-0.912226\pi\)
0.962221 0.272268i \(-0.0877739\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −32.0000 −1.24091
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 4.00000i − 0.154649i
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) − 1.00000i − 0.0384900i
\(676\) 0 0
\(677\) 14.0000i 0.538064i 0.963131 + 0.269032i \(0.0867037\pi\)
−0.963131 + 0.269032i \(0.913296\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 36.0000i 1.37549i
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) − 52.0000i − 1.97817i −0.147335 0.989087i \(-0.547070\pi\)
0.147335 0.989087i \(-0.452930\pi\)
\(692\) 0 0
\(693\) 16.0000i 0.607790i
\(694\) 0 0
\(695\) 40.0000 1.51729
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 18.0000i 0.679851i 0.940452 + 0.339925i \(0.110402\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) − 22.0000i − 0.826227i −0.910679 0.413114i \(-0.864441\pi\)
0.910679 0.413114i \(-0.135559\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 16.0000i 0.598366i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) − 14.0000i − 0.520666i
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000i 0.887672i
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 16.0000i 0.584627i
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 24.0000i 0.873449i
\(756\) 0 0
\(757\) − 6.00000i − 0.218074i −0.994038 0.109037i \(-0.965223\pi\)
0.994038 0.109037i \(-0.0347767\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 56.0000i 2.02734i
\(764\) 0 0
\(765\) − 12.0000i − 0.433861i
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 0 0
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) 8.00000i 0.286630i
\(780\) 0 0
\(781\) − 64.0000i − 2.29010i
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) − 20.0000i − 0.709327i
\(796\) 0 0
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) − 24.0000i − 0.846942i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) 28.0000i 0.983213i 0.870817 + 0.491606i \(0.163590\pi\)
−0.870817 + 0.491606i \(0.836410\pi\)
\(812\) 0 0
\(813\) 12.0000i 0.420858i
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 8.00000i 0.279543i
\(820\) 0 0
\(821\) − 26.0000i − 0.907406i −0.891153 0.453703i \(-0.850103\pi\)
0.891153 0.453703i \(-0.149897\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 20.0000i 0.695468i 0.937593 + 0.347734i \(0.113049\pi\)
−0.937593 + 0.347734i \(0.886951\pi\)
\(828\) 0 0
\(829\) − 34.0000i − 1.18087i −0.807086 0.590434i \(-0.798956\pi\)
0.807086 0.590434i \(-0.201044\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −54.0000 −1.87099
\(834\) 0 0
\(835\) 16.0000i 0.553703i
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 0 0
\(845\) − 18.0000i − 0.619219i
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) 0 0
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 18.0000i 0.616308i 0.951336 + 0.308154i \(0.0997113\pi\)
−0.951336 + 0.308154i \(0.900289\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 0 0
\(861\) − 8.00000i − 0.272639i
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 0 0
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) − 16.0000i − 0.542763i
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) − 48.0000i − 1.62270i
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 0 0
\(885\) 8.00000i 0.268917i
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −80.0000 −2.68311
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.00000i 0.266815i
\(900\) 0 0
\(901\) 60.0000i 1.99889i
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) − 20.0000i − 0.664089i −0.943264 0.332045i \(-0.892262\pi\)
0.943264 0.332045i \(-0.107738\pi\)
\(908\) 0 0
\(909\) − 6.00000i − 0.199007i
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 0 0
\(915\) 12.0000i 0.396708i
\(916\) 0 0
\(917\) 16.0000i 0.528367i
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) − 32.0000i − 1.05329i
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) − 36.0000i − 1.17985i
\(932\) 0 0
\(933\) 24.0000i 0.785725i
\(934\) 0 0
\(935\) 48.0000 1.56977
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) − 10.0000i − 0.326338i
\(940\) 0 0
\(941\) 58.0000i 1.89075i 0.325991 + 0.945373i \(0.394302\pi\)
−0.325991 + 0.945373i \(0.605698\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 0 0
\(949\) − 12.0000i − 0.389536i
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000i 0.258603i
\(958\) 0 0
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 0 0
\(965\) − 36.0000i − 1.15888i
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) − 52.0000i − 1.66876i −0.551190 0.834380i \(-0.685826\pi\)
0.551190 0.834380i \(-0.314174\pi\)
\(972\) 0 0
\(973\) 80.0000i 2.56468i
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 0 0
\(979\) 40.0000i 1.27841i
\(980\) 0 0
\(981\) − 14.0000i − 0.446986i
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 44.0000 1.40196
\(986\) 0 0
\(987\) 32.0000i 1.01857i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 8.00000i 0.253617i
\(996\) 0 0
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 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 768.2.d.h.385.2 2
3.2 odd 2 2304.2.d.s.1153.2 2
4.3 odd 2 768.2.d.a.385.1 2
8.3 odd 2 768.2.d.a.385.2 2
8.5 even 2 inner 768.2.d.h.385.1 2
12.11 even 2 2304.2.d.c.1153.2 2
16.3 odd 4 192.2.a.c.1.1 1
16.5 even 4 96.2.a.b.1.1 yes 1
16.11 odd 4 96.2.a.a.1.1 1
16.13 even 4 192.2.a.a.1.1 1
24.5 odd 2 2304.2.d.s.1153.1 2
24.11 even 2 2304.2.d.c.1153.1 2
48.5 odd 4 288.2.a.b.1.1 1
48.11 even 4 288.2.a.c.1.1 1
48.29 odd 4 576.2.a.g.1.1 1
48.35 even 4 576.2.a.h.1.1 1
80.3 even 4 4800.2.f.bh.3649.2 2
80.13 odd 4 4800.2.f.e.3649.1 2
80.19 odd 4 4800.2.a.f.1.1 1
80.27 even 4 2400.2.f.a.1249.2 2
80.29 even 4 4800.2.a.co.1.1 1
80.37 odd 4 2400.2.f.r.1249.1 2
80.43 even 4 2400.2.f.a.1249.1 2
80.53 odd 4 2400.2.f.r.1249.2 2
80.59 odd 4 2400.2.a.r.1.1 1
80.67 even 4 4800.2.f.bh.3649.1 2
80.69 even 4 2400.2.a.q.1.1 1
80.77 odd 4 4800.2.f.e.3649.2 2
112.13 odd 4 9408.2.a.ct.1.1 1
112.27 even 4 4704.2.a.t.1.1 1
112.69 odd 4 4704.2.a.e.1.1 1
112.83 even 4 9408.2.a.bj.1.1 1
144.5 odd 12 2592.2.i.w.865.1 2
144.11 even 12 2592.2.i.q.1729.1 2
144.43 odd 12 2592.2.i.b.1729.1 2
144.59 even 12 2592.2.i.q.865.1 2
144.85 even 12 2592.2.i.h.865.1 2
144.101 odd 12 2592.2.i.w.1729.1 2
144.133 even 12 2592.2.i.h.1729.1 2
144.139 odd 12 2592.2.i.b.865.1 2
240.53 even 4 7200.2.f.f.6049.2 2
240.59 even 4 7200.2.a.e.1.1 1
240.107 odd 4 7200.2.f.x.6049.2 2
240.149 odd 4 7200.2.a.bx.1.1 1
240.197 even 4 7200.2.f.f.6049.1 2
240.203 odd 4 7200.2.f.x.6049.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.a.a.1.1 1 16.11 odd 4
96.2.a.b.1.1 yes 1 16.5 even 4
192.2.a.a.1.1 1 16.13 even 4
192.2.a.c.1.1 1 16.3 odd 4
288.2.a.b.1.1 1 48.5 odd 4
288.2.a.c.1.1 1 48.11 even 4
576.2.a.g.1.1 1 48.29 odd 4
576.2.a.h.1.1 1 48.35 even 4
768.2.d.a.385.1 2 4.3 odd 2
768.2.d.a.385.2 2 8.3 odd 2
768.2.d.h.385.1 2 8.5 even 2 inner
768.2.d.h.385.2 2 1.1 even 1 trivial
2304.2.d.c.1153.1 2 24.11 even 2
2304.2.d.c.1153.2 2 12.11 even 2
2304.2.d.s.1153.1 2 24.5 odd 2
2304.2.d.s.1153.2 2 3.2 odd 2
2400.2.a.q.1.1 1 80.69 even 4
2400.2.a.r.1.1 1 80.59 odd 4
2400.2.f.a.1249.1 2 80.43 even 4
2400.2.f.a.1249.2 2 80.27 even 4
2400.2.f.r.1249.1 2 80.37 odd 4
2400.2.f.r.1249.2 2 80.53 odd 4
2592.2.i.b.865.1 2 144.139 odd 12
2592.2.i.b.1729.1 2 144.43 odd 12
2592.2.i.h.865.1 2 144.85 even 12
2592.2.i.h.1729.1 2 144.133 even 12
2592.2.i.q.865.1 2 144.59 even 12
2592.2.i.q.1729.1 2 144.11 even 12
2592.2.i.w.865.1 2 144.5 odd 12
2592.2.i.w.1729.1 2 144.101 odd 12
4704.2.a.e.1.1 1 112.69 odd 4
4704.2.a.t.1.1 1 112.27 even 4
4800.2.a.f.1.1 1 80.19 odd 4
4800.2.a.co.1.1 1 80.29 even 4
4800.2.f.e.3649.1 2 80.13 odd 4
4800.2.f.e.3649.2 2 80.77 odd 4
4800.2.f.bh.3649.1 2 80.67 even 4
4800.2.f.bh.3649.2 2 80.3 even 4
7200.2.a.e.1.1 1 240.59 even 4
7200.2.a.bx.1.1 1 240.149 odd 4
7200.2.f.f.6049.1 2 240.197 even 4
7200.2.f.f.6049.2 2 240.53 even 4
7200.2.f.x.6049.1 2 240.203 odd 4
7200.2.f.x.6049.2 2 240.107 odd 4
9408.2.a.bj.1.1 1 112.83 even 4
9408.2.a.ct.1.1 1 112.13 odd 4