Properties

Label 768.2.d.f.385.1
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: \(\Q(i)\)
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 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 385.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 768.385
Dual form 768.2.d.f.385.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -4.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -4.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} -4.00000 q^{15} -2.00000 q^{17} +8.00000i q^{19} -2.00000i q^{21} +4.00000 q^{23} -11.0000 q^{25} +1.00000i q^{27} +6.00000 q^{31} -4.00000 q^{33} -8.00000i q^{35} -2.00000i q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{45} -4.00000 q^{47} -3.00000 q^{49} +2.00000i q^{51} -16.0000 q^{55} +8.00000 q^{57} +4.00000i q^{59} -14.0000i q^{61} -2.00000 q^{63} -8.00000 q^{65} +4.00000i q^{67} -4.00000i q^{69} +12.0000 q^{71} +10.0000 q^{73} +11.0000i q^{75} -8.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +8.00000i q^{85} +14.0000 q^{89} -4.00000i q^{91} -6.00000i q^{93} +32.0000 q^{95} +10.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{7} - 2q^{9} - 8q^{15} - 4q^{17} + 8q^{23} - 22q^{25} + 12q^{31} - 8q^{33} - 4q^{39} - 12q^{41} - 8q^{47} - 6q^{49} - 32q^{55} + 16q^{57} - 4q^{63} - 16q^{65} + 24q^{71} + 20q^{73} - 20q^{79} + 2q^{81} + 28q^{89} + 64q^{95} + 20q^{97} + O(q^{100}) \)

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
\(5\) − 4.00000i − 1.78885i −0.447214 0.894427i \(-0.647584\pi\)
0.447214 0.894427i \(-0.352416\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 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
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 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 \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 4.00000i 0.596285i
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) − 14.0000i − 1.79252i −0.443533 0.896258i \(-0.646275\pi\)
0.443533 0.896258i \(-0.353725\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) − 4.00000i − 0.481543i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 11.0000i 1.27017i
\(76\) 0 0
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 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
\(85\) 8.00000i 0.867722i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) − 4.00000i − 0.419314i
\(92\) 0 0
\(93\) − 6.00000i − 0.622171i
\(94\) 0 0
\(95\) 32.0000 3.28313
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 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
\(115\) − 16.0000i − 1.49201i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 0 0
\(125\) 24.0000i 2.14663i
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) 12.0000i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) − 24.0000i − 1.92773i
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) 16.0000i 1.24560i
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 8.00000i − 0.611775i
\(172\) 0 0
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) 0 0
\(175\) −22.0000 −1.66304
\(176\) 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
\(181\) − 22.0000i − 1.63525i −0.575753 0.817624i \(-0.695291\pi\)
0.575753 0.817624i \(-0.304709\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 8.00000i 0.572892i
\(196\) 0 0
\(197\) − 8.00000i − 0.569976i −0.958531 0.284988i \(-0.908010\pi\)
0.958531 0.284988i \(-0.0919897\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 24.0000i 1.67623i
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 20.0000i 1.37686i 0.725304 + 0.688428i \(0.241699\pi\)
−0.725304 + 0.688428i \(0.758301\pi\)
\(212\) 0 0
\(213\) − 12.0000i − 0.822226i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) − 10.0000i − 0.675737i
\(220\) 0 0
\(221\) 4.00000i 0.269069i
\(222\) 0 0
\(223\) 18.0000 1.20537 0.602685 0.797980i \(-0.294098\pi\)
0.602685 + 0.797980i \(0.294098\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) − 14.0000i − 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 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
\(237\) 10.0000i 0.649570i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 12.0000i 0.766652i
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 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
\(253\) − 16.0000i − 1.00591i
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) − 4.00000i − 0.248548i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 14.0000i − 0.856786i
\(268\) 0 0
\(269\) − 16.0000i − 0.975537i −0.872973 0.487769i \(-0.837811\pi\)
0.872973 0.487769i \(-0.162189\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) 44.0000i 2.65330i
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 12.0000i 0.713326i 0.934233 + 0.356663i \(0.116086\pi\)
−0.934233 + 0.356663i \(0.883914\pi\)
\(284\) 0 0
\(285\) − 32.0000i − 1.89552i
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 10.0000i − 0.586210i
\(292\) 0 0
\(293\) 16.0000i 0.934730i 0.884064 + 0.467365i \(0.154797\pi\)
−0.884064 + 0.467365i \(0.845203\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) − 8.00000i − 0.462652i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −56.0000 −3.20655
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 10.0000i 0.568880i
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) 8.00000i 0.450749i
\(316\) 0 0
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 16.0000i − 0.890264i
\(324\) 0 0
\(325\) 22.0000i 1.22034i
\(326\) 0 0
\(327\) 6.00000 0.331801
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) − 2.00000i − 0.108625i
\(340\) 0 0
\(341\) − 24.0000i − 1.29967i
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 2.00000i 0.107058i 0.998566 + 0.0535288i \(0.0170469\pi\)
−0.998566 + 0.0535288i \(0.982953\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) − 48.0000i − 2.54758i
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) − 40.0000i − 2.09370i
\(366\) 0 0
\(367\) 38.0000 1.98358 0.991792 0.127862i \(-0.0408116\pi\)
0.991792 + 0.127862i \(0.0408116\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 2.00000i − 0.103556i −0.998659 0.0517780i \(-0.983511\pi\)
0.998659 0.0517780i \(-0.0164888\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 32.0000i 1.64373i 0.569683 + 0.821865i \(0.307066\pi\)
−0.569683 + 0.821865i \(0.692934\pi\)
\(380\) 0 0
\(381\) − 2.00000i − 0.102463i
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −32.0000 −1.63087
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 20.0000i − 1.01404i −0.861934 0.507020i \(-0.830747\pi\)
0.861934 0.507020i \(-0.169253\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 40.0000i 2.01262i
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) − 12.0000i − 0.597763i
\(404\) 0 0
\(405\) − 4.00000i − 0.198762i
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) − 2.00000i − 0.0986527i
\(412\) 0 0
\(413\) 8.00000i 0.393654i
\(414\) 0 0
\(415\) −48.0000 −2.35623
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 20.0000i 0.977064i 0.872546 + 0.488532i \(0.162467\pi\)
−0.872546 + 0.488532i \(0.837533\pi\)
\(420\) 0 0
\(421\) − 38.0000i − 1.85201i −0.377515 0.926003i \(-0.623221\pi\)
0.377515 0.926003i \(-0.376779\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) − 28.0000i − 1.35501i
\(428\) 0 0
\(429\) 8.00000i 0.386244i
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 32.0000i 1.53077i
\(438\) 0 0
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) − 56.0000i − 2.65465i
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 24.0000i 1.13012i
\(452\) 0 0
\(453\) − 10.0000i − 0.469841i
\(454\) 0 0
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) − 28.0000i − 1.30409i −0.758180 0.652045i \(-0.773911\pi\)
0.758180 0.652045i \(-0.226089\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) 0 0
\(469\) 8.00000i 0.369406i
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 88.0000i − 4.03772i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) − 8.00000i − 0.364013i
\(484\) 0 0
\(485\) − 40.0000i − 1.81631i
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 4.00000i 0.180517i 0.995918 + 0.0902587i \(0.0287694\pi\)
−0.995918 + 0.0902587i \(0.971231\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) − 16.0000i − 0.714827i
\(502\) 0 0
\(503\) 44.0000 1.96186 0.980932 0.194354i \(-0.0622609\pi\)
0.980932 + 0.194354i \(0.0622609\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) 8.00000i 0.354594i 0.984157 + 0.177297i \(0.0567353\pi\)
−0.984157 + 0.177297i \(0.943265\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) 40.0000i 1.76261i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 0 0
\(525\) 22.0000i 0.960159i
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) − 4.00000i − 0.173585i
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 12.0000i 0.516877i
\(540\) 0 0
\(541\) 14.0000i 0.601907i 0.953639 + 0.300954i \(0.0973049\pi\)
−0.953639 + 0.300954i \(0.902695\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 16.0000i 0.684111i 0.939680 + 0.342055i \(0.111123\pi\)
−0.939680 + 0.342055i \(0.888877\pi\)
\(548\) 0 0
\(549\) 14.0000i 0.597505i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 0 0
\(555\) 8.00000i 0.339581i
\(556\) 0 0
\(557\) 28.0000i 1.18640i 0.805056 + 0.593199i \(0.202135\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) − 36.0000i − 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) − 8.00000i − 0.336563i
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 4.00000i 0.167395i 0.996491 + 0.0836974i \(0.0266729\pi\)
−0.996491 + 0.0836974i \(0.973327\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) −44.0000 −1.83493
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) 2.00000i 0.0831172i
\(580\) 0 0
\(581\) − 24.0000i − 0.995688i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.00000 0.330759
\(586\) 0 0
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) 0 0
\(589\) 48.0000i 1.97781i
\(590\) 0 0
\(591\) −8.00000 −0.329076
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 16.0000i 0.655936i
\(596\) 0 0
\(597\) − 14.0000i − 0.572982i
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 20.0000i 0.813116i
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000i 0.323645i
\(612\) 0 0
\(613\) 46.0000i 1.85792i 0.370177 + 0.928961i \(0.379297\pi\)
−0.370177 + 0.928961i \(0.620703\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) − 32.0000i − 1.27796i
\(628\) 0 0
\(629\) 4.00000i 0.159490i
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) − 8.00000i − 0.317470i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 48.0000i 1.89294i 0.322799 + 0.946468i \(0.395376\pi\)
−0.322799 + 0.946468i \(0.604624\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) − 12.0000i − 0.470317i
\(652\) 0 0
\(653\) 12.0000i 0.469596i 0.972044 + 0.234798i \(0.0754429\pi\)
−0.972044 + 0.234798i \(0.924557\pi\)
\(654\) 0 0
\(655\) 48.0000 1.87552
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 36.0000i 1.40236i 0.712984 + 0.701180i \(0.247343\pi\)
−0.712984 + 0.701180i \(0.752657\pi\)
\(660\) 0 0
\(661\) 22.0000i 0.855701i 0.903850 + 0.427850i \(0.140729\pi\)
−0.903850 + 0.427850i \(0.859271\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 64.0000 2.48181
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 18.0000i − 0.695920i
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) − 11.0000i − 0.423390i
\(676\) 0 0
\(677\) 20.0000i 0.768662i 0.923195 + 0.384331i \(0.125568\pi\)
−0.923195 + 0.384331i \(0.874432\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) − 8.00000i − 0.305664i
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 24.0000i − 0.913003i −0.889723 0.456502i \(-0.849102\pi\)
0.889723 0.456502i \(-0.150898\pi\)
\(692\) 0 0
\(693\) 8.00000i 0.303895i
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) − 6.00000i − 0.226941i
\(700\) 0 0
\(701\) 40.0000i 1.51078i 0.655276 + 0.755390i \(0.272552\pi\)
−0.655276 + 0.755390i \(0.727448\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.00000i 0.0751116i 0.999295 + 0.0375558i \(0.0119572\pi\)
−0.999295 + 0.0375558i \(0.988043\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 32.0000i 1.19673i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 42.0000 1.55769 0.778847 0.627214i \(-0.215805\pi\)
0.778847 + 0.627214i \(0.215805\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 42.0000i − 1.55131i −0.631160 0.775653i \(-0.717421\pi\)
0.631160 0.775653i \(-0.282579\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) − 12.0000i − 0.441427i −0.975339 0.220714i \(-0.929161\pi\)
0.975339 0.220714i \(-0.0708386\pi\)
\(740\) 0 0
\(741\) − 16.0000i − 0.587775i
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 48.0000 1.75858
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) − 8.00000i − 0.292314i
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) − 40.0000i − 1.45575i
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 0 0
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 12.0000i 0.434429i
\(764\) 0 0
\(765\) − 8.00000i − 0.289241i
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) − 18.0000i − 0.648254i
\(772\) 0 0
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 0 0
\(775\) −66.0000 −2.37079
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) − 48.0000i − 1.71978i
\(780\) 0 0
\(781\) − 48.0000i − 1.71758i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 72.0000 2.56979
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 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
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) − 40.0000i − 1.41157i
\(804\) 0 0
\(805\) − 32.0000i − 1.12785i
\(806\) 0 0
\(807\) −16.0000 −0.563227
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) − 24.0000i − 0.842754i −0.906886 0.421377i \(-0.861547\pi\)
0.906886 0.421377i \(-0.138453\pi\)
\(812\) 0 0
\(813\) 14.0000i 0.491001i
\(814\) 0 0
\(815\) 32.0000 1.12091
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 4.00000i 0.139771i
\(820\) 0 0
\(821\) − 8.00000i − 0.279202i −0.990208 0.139601i \(-0.955418\pi\)
0.990208 0.139601i \(-0.0445820\pi\)
\(822\) 0 0
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 0 0
\(825\) 44.0000 1.53188
\(826\) 0 0
\(827\) 44.0000i 1.53003i 0.644013 + 0.765015i \(0.277268\pi\)
−0.644013 + 0.765015i \(0.722732\pi\)
\(828\) 0 0
\(829\) 38.0000i 1.31979i 0.751356 + 0.659897i \(0.229400\pi\)
−0.751356 + 0.659897i \(0.770600\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) − 64.0000i − 2.21481i
\(836\) 0 0
\(837\) 6.00000i 0.207390i
\(838\) 0 0
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) − 36.0000i − 1.23844i
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) − 8.00000i − 0.274236i
\(852\) 0 0
\(853\) 38.0000i 1.30110i 0.759465 + 0.650548i \(0.225461\pi\)
−0.759465 + 0.650548i \(0.774539\pi\)
\(854\) 0 0
\(855\) −32.0000 −1.09438
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 8.00000i 0.272956i 0.990643 + 0.136478i \(0.0435784\pi\)
−0.990643 + 0.136478i \(0.956422\pi\)
\(860\) 0 0
\(861\) 12.0000i 0.408959i
\(862\) 0 0
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 40.0000i 1.35691i
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 48.0000i 1.62270i
\(876\) 0 0
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 0 0
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) 40.0000i 1.34611i 0.739594 + 0.673054i \(0.235018\pi\)
−0.739594 + 0.673054i \(0.764982\pi\)
\(884\) 0 0
\(885\) − 16.0000i − 0.537834i
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −88.0000 −2.92522
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(915\) 56.0000i 1.85130i
\(916\) 0 0
\(917\) 24.0000i 0.792550i
\(918\) 0 0
\(919\) −6.00000 −0.197922 −0.0989609 0.995091i \(-0.531552\pi\)
−0.0989609 + 0.995091i \(0.531552\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) − 24.0000i − 0.789970i
\(924\) 0 0
\(925\) 22.0000i 0.723356i
\(926\) 0 0
\(927\) 10.0000 0.328443
\(928\) 0 0
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) − 24.0000i − 0.786568i
\(932\) 0 0
\(933\) 16.0000i 0.523816i
\(934\) 0 0
\(935\) 32.0000 1.04651
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) − 26.0000i − 0.848478i
\(940\) 0 0
\(941\) − 28.0000i − 0.912774i −0.889781 0.456387i \(-0.849143\pi\)
0.889781 0.456387i \(-0.150857\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 0 0
\(949\) − 20.0000i − 0.649227i
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) 32.0000i 1.03550i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) 8.00000i 0.257529i
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 0 0
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) 4.00000i 0.128366i 0.997938 + 0.0641831i \(0.0204442\pi\)
−0.997938 + 0.0641831i \(0.979556\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 0 0
\(975\) 22.0000 0.704564
\(976\) 0 0
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) − 56.0000i − 1.78977i
\(980\) 0 0
\(981\) − 6.00000i − 0.191565i
\(982\) 0 0
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 0 0
\(987\) 8.00000i 0.254643i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) − 56.0000i − 1.77532i
\(996\) 0 0
\(997\) 54.0000i 1.71020i 0.518465 + 0.855099i \(0.326503\pi\)
−0.518465 + 0.855099i \(0.673497\pi\)
\(998\) 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.f.385.1 2
3.2 odd 2 2304.2.d.o.1153.2 2
4.3 odd 2 768.2.d.c.385.2 2
8.3 odd 2 768.2.d.c.385.1 2
8.5 even 2 inner 768.2.d.f.385.2 2
12.11 even 2 2304.2.d.f.1153.2 2
16.3 odd 4 384.2.a.a.1.1 1
16.5 even 4 384.2.a.d.1.1 yes 1
16.11 odd 4 384.2.a.h.1.1 yes 1
16.13 even 4 384.2.a.e.1.1 yes 1
24.5 odd 2 2304.2.d.o.1153.1 2
24.11 even 2 2304.2.d.f.1153.1 2
48.5 odd 4 1152.2.a.a.1.1 1
48.11 even 4 1152.2.a.b.1.1 1
48.29 odd 4 1152.2.a.s.1.1 1
48.35 even 4 1152.2.a.t.1.1 1
80.19 odd 4 9600.2.a.bk.1.1 1
80.29 even 4 9600.2.a.t.1.1 1
80.59 odd 4 9600.2.a.e.1.1 1
80.69 even 4 9600.2.a.bz.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.a.a.1.1 1 16.3 odd 4
384.2.a.d.1.1 yes 1 16.5 even 4
384.2.a.e.1.1 yes 1 16.13 even 4
384.2.a.h.1.1 yes 1 16.11 odd 4
768.2.d.c.385.1 2 8.3 odd 2
768.2.d.c.385.2 2 4.3 odd 2
768.2.d.f.385.1 2 1.1 even 1 trivial
768.2.d.f.385.2 2 8.5 even 2 inner
1152.2.a.a.1.1 1 48.5 odd 4
1152.2.a.b.1.1 1 48.11 even 4
1152.2.a.s.1.1 1 48.29 odd 4
1152.2.a.t.1.1 1 48.35 even 4
2304.2.d.f.1153.1 2 24.11 even 2
2304.2.d.f.1153.2 2 12.11 even 2
2304.2.d.o.1153.1 2 24.5 odd 2
2304.2.d.o.1153.2 2 3.2 odd 2
9600.2.a.e.1.1 1 80.59 odd 4
9600.2.a.t.1.1 1 80.29 even 4
9600.2.a.bk.1.1 1 80.19 odd 4
9600.2.a.bz.1.1 1 80.69 even 4