Properties

Label 384.2.d.c.193.2
Level $384$
Weight $2$
Character 384.193
Analytic conductor $3.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.2
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.2.d.c.193.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.82843i q^{5} -2.82843 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.82843i q^{5} -2.82843 q^{7} -1.00000 q^{9} +4.00000i q^{11} +5.65685i q^{13} +2.82843 q^{15} -2.00000 q^{17} -4.00000i q^{19} +2.82843i q^{21} +5.65685 q^{23} -3.00000 q^{25} +1.00000i q^{27} +2.82843i q^{29} -8.48528 q^{31} +4.00000 q^{33} -8.00000i q^{35} +5.65685 q^{39} +10.0000 q^{41} +12.0000i q^{43} -2.82843i q^{45} +5.65685 q^{47} +1.00000 q^{49} +2.00000i q^{51} -2.82843i q^{53} -11.3137 q^{55} -4.00000 q^{57} -4.00000i q^{59} -11.3137i q^{61} +2.82843 q^{63} -16.0000 q^{65} -4.00000i q^{67} -5.65685i q^{69} -5.65685 q^{71} -2.00000 q^{73} +3.00000i q^{75} -11.3137i q^{77} -8.48528 q^{79} +1.00000 q^{81} +4.00000i q^{83} -5.65685i q^{85} +2.82843 q^{87} +6.00000 q^{89} -16.0000i q^{91} +8.48528i q^{93} +11.3137 q^{95} +14.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 8q^{17} - 12q^{25} + 16q^{33} + 40q^{41} + 4q^{49} - 16q^{57} - 64q^{65} - 8q^{73} + 4q^{81} + 24q^{89} + 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 2.82843i 1.26491i 0.774597 + 0.632456i \(0.217953\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 5.65685i 1.56893i 0.620174 + 0.784465i \(0.287062\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 2.82843i 0.617213i
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) −8.48528 −1.52400 −0.762001 0.647576i \(-0.775783\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) − 8.00000i − 1.35225i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 0 0
\(45\) − 2.82843i − 0.421637i
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) − 2.82843i − 0.388514i −0.980951 0.194257i \(-0.937770\pi\)
0.980951 0.194257i \(-0.0622296\pi\)
\(54\) 0 0
\(55\) −11.3137 −1.52554
\(56\) 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
\(61\) − 11.3137i − 1.44857i −0.689500 0.724286i \(-0.742170\pi\)
0.689500 0.724286i \(-0.257830\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) −16.0000 −1.98456
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) − 5.65685i − 0.681005i
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 3.00000i 0.346410i
\(76\) 0 0
\(77\) − 11.3137i − 1.28932i
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 5.65685i − 0.613572i
\(86\) 0 0
\(87\) 2.82843 0.303239
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 16.0000i − 1.67726i
\(92\) 0 0
\(93\) 8.48528i 0.879883i
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
\(100\) 0 0
\(101\) − 14.1421i − 1.40720i −0.710599 0.703598i \(-0.751576\pi\)
0.710599 0.703598i \(-0.248424\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) − 5.65685i − 0.541828i −0.962604 0.270914i \(-0.912674\pi\)
0.962604 0.270914i \(-0.0873260\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 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
\(117\) − 5.65685i − 0.522976i
\(118\) 0 0
\(119\) 5.65685 0.518563
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 10.0000i − 0.901670i
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) 8.48528 0.752947 0.376473 0.926427i \(-0.377137\pi\)
0.376473 + 0.926427i \(0.377137\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 11.3137i 0.981023i
\(134\) 0 0
\(135\) −2.82843 −0.243432
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) − 4.00000i − 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) − 5.65685i − 0.476393i
\(142\) 0 0
\(143\) −22.6274 −1.89220
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) 14.1421i 1.15857i 0.815125 + 0.579284i \(0.196668\pi\)
−0.815125 + 0.579284i \(0.803332\pi\)
\(150\) 0 0
\(151\) 8.48528 0.690522 0.345261 0.938507i \(-0.387790\pi\)
0.345261 + 0.938507i \(0.387790\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) − 24.0000i − 1.92773i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −2.82843 −0.224309
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 11.3137i 0.880771i
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) −19.0000 −1.46154
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 19.7990i 1.50529i 0.658427 + 0.752645i \(0.271222\pi\)
−0.658427 + 0.752645i \(0.728778\pi\)
\(174\) 0 0
\(175\) 8.48528 0.641427
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) 5.65685i 0.420471i 0.977651 + 0.210235i \(0.0674230\pi\)
−0.977651 + 0.210235i \(0.932577\pi\)
\(182\) 0 0
\(183\) −11.3137 −0.836333
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) − 2.82843i − 0.205738i
\(190\) 0 0
\(191\) −11.3137 −0.818631 −0.409316 0.912393i \(-0.634232\pi\)
−0.409316 + 0.912393i \(0.634232\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 16.0000i 1.14578i
\(196\) 0 0
\(197\) 8.48528i 0.604551i 0.953221 + 0.302276i \(0.0977463\pi\)
−0.953221 + 0.302276i \(0.902254\pi\)
\(198\) 0 0
\(199\) 2.82843 0.200502 0.100251 0.994962i \(-0.468035\pi\)
0.100251 + 0.994962i \(0.468035\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) − 8.00000i − 0.561490i
\(204\) 0 0
\(205\) 28.2843i 1.97546i
\(206\) 0 0
\(207\) −5.65685 −0.393179
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) 5.65685i 0.387601i
\(214\) 0 0
\(215\) −33.9411 −2.31477
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) − 11.3137i − 0.761042i
\(222\) 0 0
\(223\) 19.7990 1.32584 0.662919 0.748691i \(-0.269317\pi\)
0.662919 + 0.748691i \(0.269317\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) − 5.65685i − 0.373815i −0.982377 0.186908i \(-0.940153\pi\)
0.982377 0.186908i \(-0.0598465\pi\)
\(230\) 0 0
\(231\) −11.3137 −0.744387
\(232\) 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
\(237\) 8.48528i 0.551178i
\(238\) 0 0
\(239\) 22.6274 1.46365 0.731823 0.681495i \(-0.238670\pi\)
0.731823 + 0.681495i \(0.238670\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 2.82843i 0.180702i
\(246\) 0 0
\(247\) 22.6274 1.43975
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) − 12.0000i − 0.757433i −0.925513 0.378717i \(-0.876365\pi\)
0.925513 0.378717i \(-0.123635\pi\)
\(252\) 0 0
\(253\) 22.6274i 1.42257i
\(254\) 0 0
\(255\) −5.65685 −0.354246
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) − 2.82843i − 0.175075i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) − 8.48528i − 0.517357i −0.965964 0.258678i \(-0.916713\pi\)
0.965964 0.258678i \(-0.0832870\pi\)
\(270\) 0 0
\(271\) 19.7990 1.20270 0.601351 0.798985i \(-0.294629\pi\)
0.601351 + 0.798985i \(0.294629\pi\)
\(272\) 0 0
\(273\) −16.0000 −0.968364
\(274\) 0 0
\(275\) − 12.0000i − 0.723627i
\(276\) 0 0
\(277\) − 16.9706i − 1.01966i −0.860274 0.509831i \(-0.829708\pi\)
0.860274 0.509831i \(-0.170292\pi\)
\(278\) 0 0
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) − 11.3137i − 0.670166i
\(286\) 0 0
\(287\) −28.2843 −1.66957
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) 19.7990i 1.15667i 0.815800 + 0.578335i \(0.196297\pi\)
−0.815800 + 0.578335i \(0.803703\pi\)
\(294\) 0 0
\(295\) 11.3137 0.658710
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 32.0000i 1.85061i
\(300\) 0 0
\(301\) − 33.9411i − 1.95633i
\(302\) 0 0
\(303\) −14.1421 −0.812444
\(304\) 0 0
\(305\) 32.0000 1.83231
\(306\) 0 0
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) − 2.82843i − 0.160904i
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 8.00000i 0.450749i
\(316\) 0 0
\(317\) − 8.48528i − 0.476581i −0.971194 0.238290i \(-0.923413\pi\)
0.971194 0.238290i \(-0.0765870\pi\)
\(318\) 0 0
\(319\) −11.3137 −0.633446
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) − 16.9706i − 0.941357i
\(326\) 0 0
\(327\) −5.65685 −0.312825
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) − 2.00000i − 0.108625i
\(340\) 0 0
\(341\) − 33.9411i − 1.83801i
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) 36.0000i 1.93258i 0.257454 + 0.966291i \(0.417117\pi\)
−0.257454 + 0.966291i \(0.582883\pi\)
\(348\) 0 0
\(349\) − 11.3137i − 0.605609i −0.953053 0.302804i \(-0.902077\pi\)
0.953053 0.302804i \(-0.0979229\pi\)
\(350\) 0 0
\(351\) −5.65685 −0.301941
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) − 16.0000i − 0.849192i
\(356\) 0 0
\(357\) − 5.65685i − 0.299392i
\(358\) 0 0
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) − 5.65685i − 0.296093i
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 8.00000i 0.415339i
\(372\) 0 0
\(373\) 33.9411i 1.75740i 0.477370 + 0.878702i \(0.341590\pi\)
−0.477370 + 0.878702i \(0.658410\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) 0 0
\(381\) − 8.48528i − 0.434714i
\(382\) 0 0
\(383\) 11.3137 0.578103 0.289052 0.957313i \(-0.406660\pi\)
0.289052 + 0.957313i \(0.406660\pi\)
\(384\) 0 0
\(385\) 32.0000 1.63087
\(386\) 0 0
\(387\) − 12.0000i − 0.609994i
\(388\) 0 0
\(389\) − 31.1127i − 1.57748i −0.614729 0.788738i \(-0.710735\pi\)
0.614729 0.788738i \(-0.289265\pi\)
\(390\) 0 0
\(391\) −11.3137 −0.572159
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) − 24.0000i − 1.20757i
\(396\) 0 0
\(397\) − 33.9411i − 1.70346i −0.523984 0.851728i \(-0.675555\pi\)
0.523984 0.851728i \(-0.324445\pi\)
\(398\) 0 0
\(399\) 11.3137 0.566394
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) − 48.0000i − 2.39105i
\(404\) 0 0
\(405\) 2.82843i 0.140546i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) − 10.0000i − 0.493264i
\(412\) 0 0
\(413\) 11.3137i 0.556711i
\(414\) 0 0
\(415\) −11.3137 −0.555368
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) − 28.0000i − 1.36789i −0.729534 0.683945i \(-0.760263\pi\)
0.729534 0.683945i \(-0.239737\pi\)
\(420\) 0 0
\(421\) − 16.9706i − 0.827095i −0.910483 0.413547i \(-0.864290\pi\)
0.910483 0.413547i \(-0.135710\pi\)
\(422\) 0 0
\(423\) −5.65685 −0.275046
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 32.0000i 1.54859i
\(428\) 0 0
\(429\) 22.6274i 1.09246i
\(430\) 0 0
\(431\) −28.2843 −1.36241 −0.681203 0.732095i \(-0.738543\pi\)
−0.681203 + 0.732095i \(0.738543\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 8.00000i 0.383571i
\(436\) 0 0
\(437\) − 22.6274i − 1.08242i
\(438\) 0 0
\(439\) −31.1127 −1.48493 −0.742464 0.669886i \(-0.766343\pi\)
−0.742464 + 0.669886i \(0.766343\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 0 0
\(445\) 16.9706i 0.804482i
\(446\) 0 0
\(447\) 14.1421 0.668900
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 40.0000i 1.88353i
\(452\) 0 0
\(453\) − 8.48528i − 0.398673i
\(454\) 0 0
\(455\) 45.2548 2.12158
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) 8.48528i 0.395199i 0.980283 + 0.197599i \(0.0633145\pi\)
−0.980283 + 0.197599i \(0.936685\pi\)
\(462\) 0 0
\(463\) 19.7990 0.920137 0.460069 0.887883i \(-0.347825\pi\)
0.460069 + 0.887883i \(0.347825\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) 0 0
\(469\) 11.3137i 0.522419i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) 2.82843i 0.129505i
\(478\) 0 0
\(479\) 28.2843 1.29234 0.646171 0.763193i \(-0.276369\pi\)
0.646171 + 0.763193i \(0.276369\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 16.0000i 0.728025i
\(484\) 0 0
\(485\) 39.5980i 1.79805i
\(486\) 0 0
\(487\) −14.1421 −0.640841 −0.320421 0.947275i \(-0.603824\pi\)
−0.320421 + 0.947275i \(0.603824\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) − 20.0000i − 0.902587i −0.892375 0.451294i \(-0.850963\pi\)
0.892375 0.451294i \(-0.149037\pi\)
\(492\) 0 0
\(493\) − 5.65685i − 0.254772i
\(494\) 0 0
\(495\) 11.3137 0.508513
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) − 11.3137i − 0.505459i
\(502\) 0 0
\(503\) −5.65685 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(504\) 0 0
\(505\) 40.0000 1.77998
\(506\) 0 0
\(507\) 19.0000i 0.843820i
\(508\) 0 0
\(509\) 2.82843i 0.125368i 0.998033 + 0.0626839i \(0.0199660\pi\)
−0.998033 + 0.0626839i \(0.980034\pi\)
\(510\) 0 0
\(511\) 5.65685 0.250244
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 8.00000i 0.352522i
\(516\) 0 0
\(517\) 22.6274i 0.995153i
\(518\) 0 0
\(519\) 19.7990 0.869079
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 0 0
\(525\) − 8.48528i − 0.370328i
\(526\) 0 0
\(527\) 16.9706 0.739249
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) 56.5685i 2.45026i
\(534\) 0 0
\(535\) −33.9411 −1.46740
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 4.00000i 0.172292i
\(540\) 0 0
\(541\) 16.9706i 0.729621i 0.931082 + 0.364811i \(0.118866\pi\)
−0.931082 + 0.364811i \(0.881134\pi\)
\(542\) 0 0
\(543\) 5.65685 0.242759
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 11.3137i 0.482857i
\(550\) 0 0
\(551\) 11.3137 0.481980
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.48528i 0.359533i 0.983709 + 0.179766i \(0.0575342\pi\)
−0.983709 + 0.179766i \(0.942466\pi\)
\(558\) 0 0
\(559\) −67.8823 −2.87111
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 5.65685i 0.237986i
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) − 4.00000i − 0.167395i −0.996491 0.0836974i \(-0.973327\pi\)
0.996491 0.0836974i \(-0.0266729\pi\)
\(572\) 0 0
\(573\) 11.3137i 0.472637i
\(574\) 0 0
\(575\) −16.9706 −0.707721
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) − 6.00000i − 0.249351i
\(580\) 0 0
\(581\) − 11.3137i − 0.469372i
\(582\) 0 0
\(583\) 11.3137 0.468566
\(584\) 0 0
\(585\) 16.0000 0.661519
\(586\) 0 0
\(587\) − 36.0000i − 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) 33.9411i 1.39852i
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 16.0000i 0.655936i
\(596\) 0 0
\(597\) − 2.82843i − 0.115760i
\(598\) 0 0
\(599\) 39.5980 1.61793 0.808965 0.587857i \(-0.200028\pi\)
0.808965 + 0.587857i \(0.200028\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) − 14.1421i − 0.574960i
\(606\) 0 0
\(607\) −14.1421 −0.574012 −0.287006 0.957929i \(-0.592660\pi\)
−0.287006 + 0.957929i \(0.592660\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 32.0000i 1.29458i
\(612\) 0 0
\(613\) 11.3137i 0.456956i 0.973549 + 0.228478i \(0.0733750\pi\)
−0.973549 + 0.228478i \(0.926625\pi\)
\(614\) 0 0
\(615\) 28.2843 1.14053
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) − 36.0000i − 1.44696i −0.690344 0.723481i \(-0.742541\pi\)
0.690344 0.723481i \(-0.257459\pi\)
\(620\) 0 0
\(621\) 5.65685i 0.227002i
\(622\) 0 0
\(623\) −16.9706 −0.679911
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) − 16.0000i − 0.638978i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 8.48528 0.337794 0.168897 0.985634i \(-0.445980\pi\)
0.168897 + 0.985634i \(0.445980\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 24.0000i 0.952411i
\(636\) 0 0
\(637\) 5.65685i 0.224133i
\(638\) 0 0
\(639\) 5.65685 0.223782
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) 33.9411i 1.33643i
\(646\) 0 0
\(647\) −39.5980 −1.55676 −0.778379 0.627795i \(-0.783958\pi\)
−0.778379 + 0.627795i \(0.783958\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) − 24.0000i − 0.940634i
\(652\) 0 0
\(653\) 42.4264i 1.66027i 0.557560 + 0.830137i \(0.311738\pi\)
−0.557560 + 0.830137i \(0.688262\pi\)
\(654\) 0 0
\(655\) −33.9411 −1.32619
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) − 4.00000i − 0.155818i −0.996960 0.0779089i \(-0.975176\pi\)
0.996960 0.0779089i \(-0.0248243\pi\)
\(660\) 0 0
\(661\) − 45.2548i − 1.76021i −0.474780 0.880105i \(-0.657472\pi\)
0.474780 0.880105i \(-0.342528\pi\)
\(662\) 0 0
\(663\) −11.3137 −0.439388
\(664\) 0 0
\(665\) −32.0000 −1.24091
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) − 19.7990i − 0.765473i
\(670\) 0 0
\(671\) 45.2548 1.74704
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) − 3.00000i − 0.115470i
\(676\) 0 0
\(677\) − 19.7990i − 0.760937i −0.924794 0.380468i \(-0.875763\pi\)
0.924794 0.380468i \(-0.124237\pi\)
\(678\) 0 0
\(679\) −39.5980 −1.51963
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 0 0
\(685\) 28.2843i 1.08069i
\(686\) 0 0
\(687\) −5.65685 −0.215822
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 28.0000i 1.06517i 0.846376 + 0.532585i \(0.178779\pi\)
−0.846376 + 0.532585i \(0.821221\pi\)
\(692\) 0 0
\(693\) 11.3137i 0.429772i
\(694\) 0 0
\(695\) 11.3137 0.429153
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 0 0
\(699\) − 6.00000i − 0.226941i
\(700\) 0 0
\(701\) − 19.7990i − 0.747798i −0.927470 0.373899i \(-0.878021\pi\)
0.927470 0.373899i \(-0.121979\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 40.0000i 1.50435i
\(708\) 0 0
\(709\) 16.9706i 0.637343i 0.947865 + 0.318671i \(0.103237\pi\)
−0.947865 + 0.318671i \(0.896763\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) − 64.0000i − 2.39346i
\(716\) 0 0
\(717\) − 22.6274i − 0.845036i
\(718\) 0 0
\(719\) 16.9706 0.632895 0.316448 0.948610i \(-0.397510\pi\)
0.316448 + 0.948610i \(0.397510\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) − 6.00000i − 0.223142i
\(724\) 0 0
\(725\) − 8.48528i − 0.315135i
\(726\) 0 0
\(727\) −25.4558 −0.944105 −0.472052 0.881570i \(-0.656487\pi\)
−0.472052 + 0.881570i \(0.656487\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 24.0000i − 0.887672i
\(732\) 0 0
\(733\) − 16.9706i − 0.626822i −0.949618 0.313411i \(-0.898528\pi\)
0.949618 0.313411i \(-0.101472\pi\)
\(734\) 0 0
\(735\) 2.82843 0.104328
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) − 36.0000i − 1.32428i −0.749380 0.662141i \(-0.769648\pi\)
0.749380 0.662141i \(-0.230352\pi\)
\(740\) 0 0
\(741\) − 22.6274i − 0.831239i
\(742\) 0 0
\(743\) −22.6274 −0.830119 −0.415060 0.909794i \(-0.636239\pi\)
−0.415060 + 0.909794i \(0.636239\pi\)
\(744\) 0 0
\(745\) −40.0000 −1.46549
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) − 33.9411i − 1.24018i
\(750\) 0 0
\(751\) 48.0833 1.75458 0.877292 0.479958i \(-0.159348\pi\)
0.877292 + 0.479958i \(0.159348\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 24.0000i 0.873449i
\(756\) 0 0
\(757\) − 28.2843i − 1.02801i −0.857787 0.514005i \(-0.828161\pi\)
0.857787 0.514005i \(-0.171839\pi\)
\(758\) 0 0
\(759\) 22.6274 0.821323
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 16.0000i 0.579239i
\(764\) 0 0
\(765\) 5.65685i 0.204524i
\(766\) 0 0
\(767\) 22.6274 0.817029
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) 30.0000i 1.08042i
\(772\) 0 0
\(773\) − 2.82843i − 0.101731i −0.998706 0.0508657i \(-0.983802\pi\)
0.998706 0.0508657i \(-0.0161981\pi\)
\(774\) 0 0
\(775\) 25.4558 0.914401
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 40.0000i − 1.43315i
\(780\) 0 0
\(781\) − 22.6274i − 0.809673i
\(782\) 0 0
\(783\) −2.82843 −0.101080
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 20.0000i − 0.712923i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.65685 −0.201135
\(792\) 0 0
\(793\) 64.0000 2.27271
\(794\) 0 0
\(795\) − 8.00000i − 0.283731i
\(796\) 0 0
\(797\) 19.7990i 0.701316i 0.936504 + 0.350658i \(0.114042\pi\)
−0.936504 + 0.350658i \(0.885958\pi\)
\(798\) 0 0
\(799\) −11.3137 −0.400250
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) − 8.00000i − 0.282314i
\(804\) 0 0
\(805\) − 45.2548i − 1.59502i
\(806\) 0 0
\(807\) −8.48528 −0.298696
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) − 20.0000i − 0.702295i −0.936320 0.351147i \(-0.885792\pi\)
0.936320 0.351147i \(-0.114208\pi\)
\(812\) 0 0
\(813\) − 19.7990i − 0.694381i
\(814\) 0 0
\(815\) 11.3137 0.396302
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) 0 0
\(819\) 16.0000i 0.559085i
\(820\) 0 0
\(821\) 31.1127i 1.08584i 0.839784 + 0.542920i \(0.182681\pi\)
−0.839784 + 0.542920i \(0.817319\pi\)
\(822\) 0 0
\(823\) 19.7990 0.690149 0.345075 0.938575i \(-0.387854\pi\)
0.345075 + 0.938575i \(0.387854\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) − 50.9117i − 1.76824i −0.467264 0.884118i \(-0.654760\pi\)
0.467264 0.884118i \(-0.345240\pi\)
\(830\) 0 0
\(831\) −16.9706 −0.588702
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 32.0000i 1.10741i
\(836\) 0 0
\(837\) − 8.48528i − 0.293294i
\(838\) 0 0
\(839\) 5.65685 0.195296 0.0976481 0.995221i \(-0.468868\pi\)
0.0976481 + 0.995221i \(0.468868\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 10.0000i 0.344418i
\(844\) 0 0
\(845\) − 53.7401i − 1.84872i
\(846\) 0 0
\(847\) 14.1421 0.485930
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 22.6274i 0.774748i 0.921923 + 0.387374i \(0.126618\pi\)
−0.921923 + 0.387374i \(0.873382\pi\)
\(854\) 0 0
\(855\) −11.3137 −0.386921
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 44.0000i 1.50126i 0.660722 + 0.750630i \(0.270250\pi\)
−0.660722 + 0.750630i \(0.729750\pi\)
\(860\) 0 0
\(861\) 28.2843i 0.963925i
\(862\) 0 0
\(863\) −56.5685 −1.92562 −0.962808 0.270187i \(-0.912914\pi\)
−0.962808 + 0.270187i \(0.912914\pi\)
\(864\) 0 0
\(865\) −56.0000 −1.90406
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) − 33.9411i − 1.15137i
\(870\) 0 0
\(871\) 22.6274 0.766701
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) − 16.0000i − 0.540899i
\(876\) 0 0
\(877\) 11.3137i 0.382037i 0.981586 + 0.191018i \(0.0611790\pi\)
−0.981586 + 0.191018i \(0.938821\pi\)
\(878\) 0 0
\(879\) 19.7990 0.667803
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) 12.0000i 0.403832i 0.979403 + 0.201916i \(0.0647168\pi\)
−0.979403 + 0.201916i \(0.935283\pi\)
\(884\) 0 0
\(885\) − 11.3137i − 0.380306i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) − 22.6274i − 0.757198i
\(894\) 0 0
\(895\) −33.9411 −1.13453
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) − 24.0000i − 0.800445i
\(900\) 0 0
\(901\) 5.65685i 0.188457i
\(902\) 0 0
\(903\) −33.9411 −1.12949
\(904\) 0 0
\(905\) −16.0000 −0.531858
\(906\) 0 0
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 0 0
\(909\) 14.1421i 0.469065i
\(910\) 0 0
\(911\) −22.6274 −0.749680 −0.374840 0.927090i \(-0.622302\pi\)
−0.374840 + 0.927090i \(0.622302\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) − 32.0000i − 1.05789i
\(916\) 0 0
\(917\) − 33.9411i − 1.12083i
\(918\) 0 0
\(919\) −2.82843 −0.0933012 −0.0466506 0.998911i \(-0.514855\pi\)
−0.0466506 + 0.998911i \(0.514855\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) − 32.0000i − 1.05329i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.82843 −0.0928977
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) − 11.3137i − 0.370394i
\(934\) 0 0
\(935\) 22.6274 0.739996
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 14.0000i 0.456873i
\(940\) 0 0
\(941\) − 14.1421i − 0.461020i −0.973070 0.230510i \(-0.925960\pi\)
0.973070 0.230510i \(-0.0740395\pi\)
\(942\) 0 0
\(943\) 56.5685 1.84213
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) − 11.3137i − 0.367259i
\(950\) 0 0
\(951\) −8.48528 −0.275154
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) − 32.0000i − 1.03550i
\(956\) 0 0
\(957\) 11.3137i 0.365720i
\(958\) 0 0
\(959\) −28.2843 −0.913347
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) 16.9706i 0.546302i
\(966\) 0 0
\(967\) 14.1421 0.454780 0.227390 0.973804i \(-0.426981\pi\)
0.227390 + 0.973804i \(0.426981\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) − 12.0000i − 0.385098i −0.981287 0.192549i \(-0.938325\pi\)
0.981287 0.192549i \(-0.0616755\pi\)
\(972\) 0 0
\(973\) 11.3137i 0.362701i
\(974\) 0 0
\(975\) −16.9706 −0.543493
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767043i
\(980\) 0 0
\(981\) 5.65685i 0.180609i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 16.0000i 0.509286i
\(988\) 0 0
\(989\) 67.8823i 2.15853i
\(990\) 0 0
\(991\) 25.4558 0.808632 0.404316 0.914619i \(-0.367510\pi\)
0.404316 + 0.914619i \(0.367510\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 8.00000i 0.253617i
\(996\) 0 0
\(997\) 33.9411i 1.07493i 0.843287 + 0.537463i \(0.180617\pi\)
−0.843287 + 0.537463i \(0.819383\pi\)
\(998\) 0 0
\(999\) 0 0
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 384.2.d.c.193.2 yes 4
3.2 odd 2 1152.2.d.h.577.1 4
4.3 odd 2 inner 384.2.d.c.193.4 yes 4
8.3 odd 2 inner 384.2.d.c.193.1 4
8.5 even 2 inner 384.2.d.c.193.3 yes 4
12.11 even 2 1152.2.d.h.577.2 4
16.3 odd 4 768.2.a.i.1.2 2
16.5 even 4 768.2.a.i.1.1 2
16.11 odd 4 768.2.a.l.1.1 2
16.13 even 4 768.2.a.l.1.2 2
24.5 odd 2 1152.2.d.h.577.3 4
24.11 even 2 1152.2.d.h.577.4 4
48.5 odd 4 2304.2.a.x.1.2 2
48.11 even 4 2304.2.a.r.1.2 2
48.29 odd 4 2304.2.a.r.1.1 2
48.35 even 4 2304.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.c.193.1 4 8.3 odd 2 inner
384.2.d.c.193.2 yes 4 1.1 even 1 trivial
384.2.d.c.193.3 yes 4 8.5 even 2 inner
384.2.d.c.193.4 yes 4 4.3 odd 2 inner
768.2.a.i.1.1 2 16.5 even 4
768.2.a.i.1.2 2 16.3 odd 4
768.2.a.l.1.1 2 16.11 odd 4
768.2.a.l.1.2 2 16.13 even 4
1152.2.d.h.577.1 4 3.2 odd 2
1152.2.d.h.577.2 4 12.11 even 2
1152.2.d.h.577.3 4 24.5 odd 2
1152.2.d.h.577.4 4 24.11 even 2
2304.2.a.r.1.1 2 48.29 odd 4
2304.2.a.r.1.2 2 48.11 even 4
2304.2.a.x.1.1 2 48.35 even 4
2304.2.a.x.1.2 2 48.5 odd 4