Properties

Label 1792.2.a.x.1.1
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.9248.1
Defining polynomial: \(x^{4} - 5 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.662153\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.02045 q^{3} -1.69614 q^{5} +1.00000 q^{7} +6.12311 q^{9} +O(q^{10})\) \(q-3.02045 q^{3} -1.69614 q^{5} +1.00000 q^{7} +6.12311 q^{9} -1.32431 q^{11} -1.69614 q^{13} +5.12311 q^{15} +2.00000 q^{17} -3.02045 q^{19} -3.02045 q^{21} -5.12311 q^{23} -2.12311 q^{25} -9.43318 q^{27} +6.04090 q^{29} -10.2462 q^{31} +4.00000 q^{33} -1.69614 q^{35} +6.04090 q^{37} +5.12311 q^{39} -4.24621 q^{41} -1.32431 q^{43} -10.3857 q^{45} +1.00000 q^{49} -6.04090 q^{51} -2.64861 q^{53} +2.24621 q^{55} +9.12311 q^{57} -0.371834 q^{59} -1.69614 q^{61} +6.12311 q^{63} +2.87689 q^{65} +11.5012 q^{67} +15.4741 q^{69} +8.00000 q^{71} +6.00000 q^{73} +6.41273 q^{75} -1.32431 q^{77} +10.1231 q^{81} +5.66906 q^{83} -3.39228 q^{85} -18.2462 q^{87} +16.2462 q^{89} -1.69614 q^{91} +30.9481 q^{93} +5.12311 q^{95} +12.2462 q^{97} -8.10887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + 8q^{9} + O(q^{10}) \) \( 4q + 4q^{7} + 8q^{9} + 4q^{15} + 8q^{17} - 4q^{23} + 8q^{25} - 8q^{31} + 16q^{33} + 4q^{39} + 16q^{41} + 4q^{49} - 24q^{55} + 20q^{57} + 8q^{63} + 28q^{65} + 32q^{71} + 24q^{73} + 24q^{81} - 40q^{87} + 32q^{89} + 4q^{95} + 16q^{97} + O(q^{100}) \)

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
\(3\) −3.02045 −1.74386 −0.871928 0.489634i \(-0.837130\pi\)
−0.871928 + 0.489634i \(0.837130\pi\)
\(4\) 0 0
\(5\) −1.69614 −0.758537 −0.379269 0.925287i \(-0.623824\pi\)
−0.379269 + 0.925287i \(0.623824\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.12311 2.04104
\(10\) 0 0
\(11\) −1.32431 −0.399294 −0.199647 0.979868i \(-0.563979\pi\)
−0.199647 + 0.979868i \(0.563979\pi\)
\(12\) 0 0
\(13\) −1.69614 −0.470425 −0.235212 0.971944i \(-0.575579\pi\)
−0.235212 + 0.971944i \(0.575579\pi\)
\(14\) 0 0
\(15\) 5.12311 1.32278
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −3.02045 −0.692938 −0.346469 0.938061i \(-0.612619\pi\)
−0.346469 + 0.938061i \(0.612619\pi\)
\(20\) 0 0
\(21\) −3.02045 −0.659116
\(22\) 0 0
\(23\) −5.12311 −1.06824 −0.534121 0.845408i \(-0.679357\pi\)
−0.534121 + 0.845408i \(0.679357\pi\)
\(24\) 0 0
\(25\) −2.12311 −0.424621
\(26\) 0 0
\(27\) −9.43318 −1.81542
\(28\) 0 0
\(29\) 6.04090 1.12177 0.560883 0.827895i \(-0.310462\pi\)
0.560883 + 0.827895i \(0.310462\pi\)
\(30\) 0 0
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) −1.69614 −0.286700
\(36\) 0 0
\(37\) 6.04090 0.993117 0.496559 0.868003i \(-0.334597\pi\)
0.496559 + 0.868003i \(0.334597\pi\)
\(38\) 0 0
\(39\) 5.12311 0.820353
\(40\) 0 0
\(41\) −4.24621 −0.663147 −0.331573 0.943429i \(-0.607579\pi\)
−0.331573 + 0.943429i \(0.607579\pi\)
\(42\) 0 0
\(43\) −1.32431 −0.201955 −0.100977 0.994889i \(-0.532197\pi\)
−0.100977 + 0.994889i \(0.532197\pi\)
\(44\) 0 0
\(45\) −10.3857 −1.54820
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.04090 −0.845895
\(52\) 0 0
\(53\) −2.64861 −0.363815 −0.181908 0.983316i \(-0.558227\pi\)
−0.181908 + 0.983316i \(0.558227\pi\)
\(54\) 0 0
\(55\) 2.24621 0.302879
\(56\) 0 0
\(57\) 9.12311 1.20838
\(58\) 0 0
\(59\) −0.371834 −0.0484087 −0.0242043 0.999707i \(-0.507705\pi\)
−0.0242043 + 0.999707i \(0.507705\pi\)
\(60\) 0 0
\(61\) −1.69614 −0.217169 −0.108584 0.994087i \(-0.534632\pi\)
−0.108584 + 0.994087i \(0.534632\pi\)
\(62\) 0 0
\(63\) 6.12311 0.771439
\(64\) 0 0
\(65\) 2.87689 0.356835
\(66\) 0 0
\(67\) 11.5012 1.40509 0.702545 0.711640i \(-0.252047\pi\)
0.702545 + 0.711640i \(0.252047\pi\)
\(68\) 0 0
\(69\) 15.4741 1.86286
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 6.41273 0.740478
\(76\) 0 0
\(77\) −1.32431 −0.150919
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 10.1231 1.12479
\(82\) 0 0
\(83\) 5.66906 0.622260 0.311130 0.950367i \(-0.399292\pi\)
0.311130 + 0.950367i \(0.399292\pi\)
\(84\) 0 0
\(85\) −3.39228 −0.367945
\(86\) 0 0
\(87\) −18.2462 −1.95620
\(88\) 0 0
\(89\) 16.2462 1.72209 0.861047 0.508525i \(-0.169809\pi\)
0.861047 + 0.508525i \(0.169809\pi\)
\(90\) 0 0
\(91\) −1.69614 −0.177804
\(92\) 0 0
\(93\) 30.9481 3.20917
\(94\) 0 0
\(95\) 5.12311 0.525620
\(96\) 0 0
\(97\) 12.2462 1.24341 0.621707 0.783250i \(-0.286439\pi\)
0.621707 + 0.783250i \(0.286439\pi\)
\(98\) 0 0
\(99\) −8.10887 −0.814972
\(100\) 0 0
\(101\) −10.3857 −1.03341 −0.516705 0.856163i \(-0.672842\pi\)
−0.516705 + 0.856163i \(0.672842\pi\)
\(102\) 0 0
\(103\) −2.24621 −0.221326 −0.110663 0.993858i \(-0.535297\pi\)
−0.110663 + 0.993858i \(0.535297\pi\)
\(104\) 0 0
\(105\) 5.12311 0.499964
\(106\) 0 0
\(107\) 14.1498 1.36791 0.683955 0.729524i \(-0.260259\pi\)
0.683955 + 0.729524i \(0.260259\pi\)
\(108\) 0 0
\(109\) −18.1227 −1.73584 −0.867919 0.496705i \(-0.834543\pi\)
−0.867919 + 0.496705i \(0.834543\pi\)
\(110\) 0 0
\(111\) −18.2462 −1.73185
\(112\) 0 0
\(113\) 4.87689 0.458780 0.229390 0.973335i \(-0.426327\pi\)
0.229390 + 0.973335i \(0.426327\pi\)
\(114\) 0 0
\(115\) 8.68951 0.810301
\(116\) 0 0
\(117\) −10.3857 −0.960154
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −9.24621 −0.840565
\(122\) 0 0
\(123\) 12.8255 1.15643
\(124\) 0 0
\(125\) 12.0818 1.08063
\(126\) 0 0
\(127\) 13.1231 1.16449 0.582244 0.813014i \(-0.302175\pi\)
0.582244 + 0.813014i \(0.302175\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.4536 1.08808 0.544039 0.839060i \(-0.316894\pi\)
0.544039 + 0.839060i \(0.316894\pi\)
\(132\) 0 0
\(133\) −3.02045 −0.261906
\(134\) 0 0
\(135\) 16.0000 1.37706
\(136\) 0 0
\(137\) 16.2462 1.38801 0.694004 0.719971i \(-0.255845\pi\)
0.694004 + 0.719971i \(0.255845\pi\)
\(138\) 0 0
\(139\) 8.31768 0.705496 0.352748 0.935718i \(-0.385247\pi\)
0.352748 + 0.935718i \(0.385247\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.24621 0.187838
\(144\) 0 0
\(145\) −10.2462 −0.850902
\(146\) 0 0
\(147\) −3.02045 −0.249122
\(148\) 0 0
\(149\) 14.7304 1.20676 0.603381 0.797453i \(-0.293820\pi\)
0.603381 + 0.797453i \(0.293820\pi\)
\(150\) 0 0
\(151\) −10.8769 −0.885149 −0.442575 0.896732i \(-0.645935\pi\)
−0.442575 + 0.896732i \(0.645935\pi\)
\(152\) 0 0
\(153\) 12.2462 0.990048
\(154\) 0 0
\(155\) 17.3790 1.39592
\(156\) 0 0
\(157\) −8.48071 −0.676834 −0.338417 0.940996i \(-0.609891\pi\)
−0.338417 + 0.940996i \(0.609891\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) 13.4061 1.05005 0.525023 0.851088i \(-0.324057\pi\)
0.525023 + 0.851088i \(0.324057\pi\)
\(164\) 0 0
\(165\) −6.78456 −0.528178
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −10.1231 −0.778700
\(170\) 0 0
\(171\) −18.4945 −1.41431
\(172\) 0 0
\(173\) −19.0752 −1.45026 −0.725129 0.688613i \(-0.758220\pi\)
−0.725129 + 0.688613i \(0.758220\pi\)
\(174\) 0 0
\(175\) −2.12311 −0.160492
\(176\) 0 0
\(177\) 1.12311 0.0844178
\(178\) 0 0
\(179\) 4.71659 0.352534 0.176267 0.984342i \(-0.443598\pi\)
0.176267 + 0.984342i \(0.443598\pi\)
\(180\) 0 0
\(181\) 6.99337 0.519813 0.259906 0.965634i \(-0.416308\pi\)
0.259906 + 0.965634i \(0.416308\pi\)
\(182\) 0 0
\(183\) 5.12311 0.378711
\(184\) 0 0
\(185\) −10.2462 −0.753316
\(186\) 0 0
\(187\) −2.64861 −0.193686
\(188\) 0 0
\(189\) −9.43318 −0.686163
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −11.1231 −0.800659 −0.400329 0.916371i \(-0.631104\pi\)
−0.400329 + 0.916371i \(0.631104\pi\)
\(194\) 0 0
\(195\) −8.68951 −0.622269
\(196\) 0 0
\(197\) 21.5150 1.53288 0.766439 0.642317i \(-0.222027\pi\)
0.766439 + 0.642317i \(0.222027\pi\)
\(198\) 0 0
\(199\) 18.2462 1.29344 0.646720 0.762728i \(-0.276140\pi\)
0.646720 + 0.762728i \(0.276140\pi\)
\(200\) 0 0
\(201\) −34.7386 −2.45027
\(202\) 0 0
\(203\) 6.04090 0.423988
\(204\) 0 0
\(205\) 7.20217 0.503022
\(206\) 0 0
\(207\) −31.3693 −2.18032
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.71659 0.324703 0.162352 0.986733i \(-0.448092\pi\)
0.162352 + 0.986733i \(0.448092\pi\)
\(212\) 0 0
\(213\) −24.1636 −1.65566
\(214\) 0 0
\(215\) 2.24621 0.153190
\(216\) 0 0
\(217\) −10.2462 −0.695558
\(218\) 0 0
\(219\) −18.1227 −1.22462
\(220\) 0 0
\(221\) −3.39228 −0.228190
\(222\) 0 0
\(223\) −5.75379 −0.385302 −0.192651 0.981267i \(-0.561709\pi\)
−0.192651 + 0.981267i \(0.561709\pi\)
\(224\) 0 0
\(225\) −13.0000 −0.866667
\(226\) 0 0
\(227\) −9.80501 −0.650782 −0.325391 0.945580i \(-0.605496\pi\)
−0.325391 + 0.945580i \(0.605496\pi\)
\(228\) 0 0
\(229\) −25.8597 −1.70886 −0.854429 0.519568i \(-0.826093\pi\)
−0.854429 + 0.519568i \(0.826093\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 16.2462 1.06432 0.532162 0.846642i \(-0.321380\pi\)
0.532162 + 0.846642i \(0.321380\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −17.6155 −1.13945 −0.569727 0.821834i \(-0.692951\pi\)
−0.569727 + 0.821834i \(0.692951\pi\)
\(240\) 0 0
\(241\) −3.75379 −0.241803 −0.120901 0.992665i \(-0.538578\pi\)
−0.120901 + 0.992665i \(0.538578\pi\)
\(242\) 0 0
\(243\) −2.27678 −0.146055
\(244\) 0 0
\(245\) −1.69614 −0.108362
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) −17.1231 −1.08513
\(250\) 0 0
\(251\) −10.9663 −0.692186 −0.346093 0.938200i \(-0.612492\pi\)
−0.346093 + 0.938200i \(0.612492\pi\)
\(252\) 0 0
\(253\) 6.78456 0.426542
\(254\) 0 0
\(255\) 10.2462 0.641643
\(256\) 0 0
\(257\) 22.4924 1.40304 0.701519 0.712650i \(-0.252505\pi\)
0.701519 + 0.712650i \(0.252505\pi\)
\(258\) 0 0
\(259\) 6.04090 0.375363
\(260\) 0 0
\(261\) 36.9890 2.28956
\(262\) 0 0
\(263\) −12.4924 −0.770316 −0.385158 0.922851i \(-0.625853\pi\)
−0.385158 + 0.922851i \(0.625853\pi\)
\(264\) 0 0
\(265\) 4.49242 0.275967
\(266\) 0 0
\(267\) −49.0708 −3.00309
\(268\) 0 0
\(269\) 11.8730 0.723909 0.361954 0.932196i \(-0.382110\pi\)
0.361954 + 0.932196i \(0.382110\pi\)
\(270\) 0 0
\(271\) 10.2462 0.622413 0.311207 0.950342i \(-0.399267\pi\)
0.311207 + 0.950342i \(0.399267\pi\)
\(272\) 0 0
\(273\) 5.12311 0.310064
\(274\) 0 0
\(275\) 2.81164 0.169548
\(276\) 0 0
\(277\) −2.64861 −0.159140 −0.0795699 0.996829i \(-0.525355\pi\)
−0.0795699 + 0.996829i \(0.525355\pi\)
\(278\) 0 0
\(279\) −62.7386 −3.75606
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 21.8868 1.30104 0.650518 0.759491i \(-0.274552\pi\)
0.650518 + 0.759491i \(0.274552\pi\)
\(284\) 0 0
\(285\) −15.4741 −0.916605
\(286\) 0 0
\(287\) −4.24621 −0.250646
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −36.9890 −2.16834
\(292\) 0 0
\(293\) −10.3857 −0.606736 −0.303368 0.952873i \(-0.598111\pi\)
−0.303368 + 0.952873i \(0.598111\pi\)
\(294\) 0 0
\(295\) 0.630683 0.0367198
\(296\) 0 0
\(297\) 12.4924 0.724884
\(298\) 0 0
\(299\) 8.68951 0.502527
\(300\) 0 0
\(301\) −1.32431 −0.0763318
\(302\) 0 0
\(303\) 31.3693 1.80212
\(304\) 0 0
\(305\) 2.87689 0.164730
\(306\) 0 0
\(307\) −25.2791 −1.44275 −0.721377 0.692543i \(-0.756490\pi\)
−0.721377 + 0.692543i \(0.756490\pi\)
\(308\) 0 0
\(309\) 6.78456 0.385960
\(310\) 0 0
\(311\) 12.4924 0.708380 0.354190 0.935173i \(-0.384757\pi\)
0.354190 + 0.935173i \(0.384757\pi\)
\(312\) 0 0
\(313\) 20.7386 1.17222 0.586108 0.810233i \(-0.300659\pi\)
0.586108 + 0.810233i \(0.300659\pi\)
\(314\) 0 0
\(315\) −10.3857 −0.585165
\(316\) 0 0
\(317\) 4.13595 0.232298 0.116149 0.993232i \(-0.462945\pi\)
0.116149 + 0.993232i \(0.462945\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −42.7386 −2.38544
\(322\) 0 0
\(323\) −6.04090 −0.336124
\(324\) 0 0
\(325\) 3.60109 0.199752
\(326\) 0 0
\(327\) 54.7386 3.02705
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.8934 −0.818617 −0.409309 0.912396i \(-0.634230\pi\)
−0.409309 + 0.912396i \(0.634230\pi\)
\(332\) 0 0
\(333\) 36.9890 2.02699
\(334\) 0 0
\(335\) −19.5076 −1.06581
\(336\) 0 0
\(337\) −0.876894 −0.0477675 −0.0238837 0.999715i \(-0.507603\pi\)
−0.0238837 + 0.999715i \(0.507603\pi\)
\(338\) 0 0
\(339\) −14.7304 −0.800046
\(340\) 0 0
\(341\) 13.5691 0.734809
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −26.2462 −1.41305
\(346\) 0 0
\(347\) −8.10887 −0.435307 −0.217654 0.976026i \(-0.569840\pi\)
−0.217654 + 0.976026i \(0.569840\pi\)
\(348\) 0 0
\(349\) 27.3471 1.46385 0.731927 0.681383i \(-0.238621\pi\)
0.731927 + 0.681383i \(0.238621\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) 7.75379 0.412693 0.206346 0.978479i \(-0.433843\pi\)
0.206346 + 0.978479i \(0.433843\pi\)
\(354\) 0 0
\(355\) −13.5691 −0.720175
\(356\) 0 0
\(357\) −6.04090 −0.319718
\(358\) 0 0
\(359\) 31.3693 1.65561 0.827805 0.561017i \(-0.189590\pi\)
0.827805 + 0.561017i \(0.189590\pi\)
\(360\) 0 0
\(361\) −9.87689 −0.519837
\(362\) 0 0
\(363\) 27.9277 1.46582
\(364\) 0 0
\(365\) −10.1768 −0.532680
\(366\) 0 0
\(367\) 10.2462 0.534848 0.267424 0.963579i \(-0.413828\pi\)
0.267424 + 0.963579i \(0.413828\pi\)
\(368\) 0 0
\(369\) −26.0000 −1.35351
\(370\) 0 0
\(371\) −2.64861 −0.137509
\(372\) 0 0
\(373\) 14.7304 0.762711 0.381356 0.924428i \(-0.375457\pi\)
0.381356 + 0.924428i \(0.375457\pi\)
\(374\) 0 0
\(375\) −36.4924 −1.88446
\(376\) 0 0
\(377\) −10.2462 −0.527707
\(378\) 0 0
\(379\) 36.4084 1.87017 0.935087 0.354418i \(-0.115321\pi\)
0.935087 + 0.354418i \(0.115321\pi\)
\(380\) 0 0
\(381\) −39.6377 −2.03070
\(382\) 0 0
\(383\) −4.49242 −0.229552 −0.114776 0.993391i \(-0.536615\pi\)
−0.114776 + 0.993391i \(0.536615\pi\)
\(384\) 0 0
\(385\) 2.24621 0.114478
\(386\) 0 0
\(387\) −8.10887 −0.412197
\(388\) 0 0
\(389\) −24.9073 −1.26285 −0.631424 0.775438i \(-0.717529\pi\)
−0.631424 + 0.775438i \(0.717529\pi\)
\(390\) 0 0
\(391\) −10.2462 −0.518173
\(392\) 0 0
\(393\) −37.6155 −1.89745
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.5625 1.03200 0.516001 0.856588i \(-0.327420\pi\)
0.516001 + 0.856588i \(0.327420\pi\)
\(398\) 0 0
\(399\) 9.12311 0.456727
\(400\) 0 0
\(401\) −0.876894 −0.0437900 −0.0218950 0.999760i \(-0.506970\pi\)
−0.0218950 + 0.999760i \(0.506970\pi\)
\(402\) 0 0
\(403\) 17.3790 0.865711
\(404\) 0 0
\(405\) −17.1702 −0.853195
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −49.0708 −2.42049
\(412\) 0 0
\(413\) −0.371834 −0.0182968
\(414\) 0 0
\(415\) −9.61553 −0.472008
\(416\) 0 0
\(417\) −25.1231 −1.23028
\(418\) 0 0
\(419\) 27.9277 1.36436 0.682179 0.731185i \(-0.261033\pi\)
0.682179 + 0.731185i \(0.261033\pi\)
\(420\) 0 0
\(421\) −26.8122 −1.30675 −0.653373 0.757036i \(-0.726647\pi\)
−0.653373 + 0.757036i \(0.726647\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.24621 −0.205971
\(426\) 0 0
\(427\) −1.69614 −0.0820820
\(428\) 0 0
\(429\) −6.78456 −0.327562
\(430\) 0 0
\(431\) 17.6155 0.848510 0.424255 0.905543i \(-0.360536\pi\)
0.424255 + 0.905543i \(0.360536\pi\)
\(432\) 0 0
\(433\) −18.4924 −0.888689 −0.444345 0.895856i \(-0.646563\pi\)
−0.444345 + 0.895856i \(0.646563\pi\)
\(434\) 0 0
\(435\) 30.9481 1.48385
\(436\) 0 0
\(437\) 15.4741 0.740225
\(438\) 0 0
\(439\) −22.7386 −1.08526 −0.542628 0.839973i \(-0.682571\pi\)
−0.542628 + 0.839973i \(0.682571\pi\)
\(440\) 0 0
\(441\) 6.12311 0.291576
\(442\) 0 0
\(443\) 7.36520 0.349931 0.174966 0.984575i \(-0.444019\pi\)
0.174966 + 0.984575i \(0.444019\pi\)
\(444\) 0 0
\(445\) −27.5559 −1.30627
\(446\) 0 0
\(447\) −44.4924 −2.10442
\(448\) 0 0
\(449\) 16.7386 0.789945 0.394972 0.918693i \(-0.370754\pi\)
0.394972 + 0.918693i \(0.370754\pi\)
\(450\) 0 0
\(451\) 5.62329 0.264790
\(452\) 0 0
\(453\) 32.8531 1.54357
\(454\) 0 0
\(455\) 2.87689 0.134871
\(456\) 0 0
\(457\) −17.3693 −0.812502 −0.406251 0.913761i \(-0.633164\pi\)
−0.406251 + 0.913761i \(0.633164\pi\)
\(458\) 0 0
\(459\) −18.8664 −0.880606
\(460\) 0 0
\(461\) 24.3724 1.13514 0.567568 0.823327i \(-0.307885\pi\)
0.567568 + 0.823327i \(0.307885\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) −52.4924 −2.43428
\(466\) 0 0
\(467\) −3.02045 −0.139770 −0.0698848 0.997555i \(-0.522263\pi\)
−0.0698848 + 0.997555i \(0.522263\pi\)
\(468\) 0 0
\(469\) 11.5012 0.531074
\(470\) 0 0
\(471\) 25.6155 1.18030
\(472\) 0 0
\(473\) 1.75379 0.0806393
\(474\) 0 0
\(475\) 6.41273 0.294236
\(476\) 0 0
\(477\) −16.2177 −0.742559
\(478\) 0 0
\(479\) 10.2462 0.468161 0.234081 0.972217i \(-0.424792\pi\)
0.234081 + 0.972217i \(0.424792\pi\)
\(480\) 0 0
\(481\) −10.2462 −0.467187
\(482\) 0 0
\(483\) 15.4741 0.704095
\(484\) 0 0
\(485\) −20.7713 −0.943176
\(486\) 0 0
\(487\) −0.630683 −0.0285790 −0.0142895 0.999898i \(-0.504549\pi\)
−0.0142895 + 0.999898i \(0.504549\pi\)
\(488\) 0 0
\(489\) −40.4924 −1.83113
\(490\) 0 0
\(491\) −34.1774 −1.54240 −0.771202 0.636590i \(-0.780344\pi\)
−0.771202 + 0.636590i \(0.780344\pi\)
\(492\) 0 0
\(493\) 12.0818 0.544137
\(494\) 0 0
\(495\) 13.7538 0.618187
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −8.85254 −0.396294 −0.198147 0.980172i \(-0.563492\pi\)
−0.198147 + 0.980172i \(0.563492\pi\)
\(500\) 0 0
\(501\) 24.1636 1.07955
\(502\) 0 0
\(503\) −13.7538 −0.613251 −0.306626 0.951830i \(-0.599200\pi\)
−0.306626 + 0.951830i \(0.599200\pi\)
\(504\) 0 0
\(505\) 17.6155 0.783881
\(506\) 0 0
\(507\) 30.5763 1.35794
\(508\) 0 0
\(509\) −30.7393 −1.36250 −0.681249 0.732052i \(-0.738563\pi\)
−0.681249 + 0.732052i \(0.738563\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 28.4924 1.25797
\(514\) 0 0
\(515\) 3.80989 0.167884
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 57.6155 2.52904
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 41.1708 1.80027 0.900136 0.435609i \(-0.143467\pi\)
0.900136 + 0.435609i \(0.143467\pi\)
\(524\) 0 0
\(525\) 6.41273 0.279874
\(526\) 0 0
\(527\) −20.4924 −0.892664
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 0 0
\(531\) −2.27678 −0.0988038
\(532\) 0 0
\(533\) 7.20217 0.311961
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) −14.2462 −0.614769
\(538\) 0 0
\(539\) −1.32431 −0.0570419
\(540\) 0 0
\(541\) −13.2431 −0.569364 −0.284682 0.958622i \(-0.591888\pi\)
−0.284682 + 0.958622i \(0.591888\pi\)
\(542\) 0 0
\(543\) −21.1231 −0.906479
\(544\) 0 0
\(545\) 30.7386 1.31670
\(546\) 0 0
\(547\) 9.59621 0.410304 0.205152 0.978730i \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(548\) 0 0
\(549\) −10.3857 −0.443249
\(550\) 0 0
\(551\) −18.2462 −0.777315
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 30.9481 1.31368
\(556\) 0 0
\(557\) −2.64861 −0.112225 −0.0561127 0.998424i \(-0.517871\pi\)
−0.0561127 + 0.998424i \(0.517871\pi\)
\(558\) 0 0
\(559\) 2.24621 0.0950046
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 29.8326 1.25730 0.628648 0.777690i \(-0.283609\pi\)
0.628648 + 0.777690i \(0.283609\pi\)
\(564\) 0 0
\(565\) −8.27190 −0.348001
\(566\) 0 0
\(567\) 10.1231 0.425130
\(568\) 0 0
\(569\) 13.3693 0.560471 0.280235 0.959931i \(-0.409587\pi\)
0.280235 + 0.959931i \(0.409587\pi\)
\(570\) 0 0
\(571\) 9.27015 0.387944 0.193972 0.981007i \(-0.437863\pi\)
0.193972 + 0.981007i \(0.437863\pi\)
\(572\) 0 0
\(573\) −48.3272 −2.01890
\(574\) 0 0
\(575\) 10.8769 0.453598
\(576\) 0 0
\(577\) 7.75379 0.322794 0.161397 0.986890i \(-0.448400\pi\)
0.161397 + 0.986890i \(0.448400\pi\)
\(578\) 0 0
\(579\) 33.5968 1.39623
\(580\) 0 0
\(581\) 5.66906 0.235192
\(582\) 0 0
\(583\) 3.50758 0.145269
\(584\) 0 0
\(585\) 17.6155 0.728312
\(586\) 0 0
\(587\) 21.8868 0.903365 0.451683 0.892179i \(-0.350824\pi\)
0.451683 + 0.892179i \(0.350824\pi\)
\(588\) 0 0
\(589\) 30.9481 1.27520
\(590\) 0 0
\(591\) −64.9848 −2.67312
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −3.39228 −0.139070
\(596\) 0 0
\(597\) −55.1117 −2.25557
\(598\) 0 0
\(599\) 12.4924 0.510427 0.255213 0.966885i \(-0.417854\pi\)
0.255213 + 0.966885i \(0.417854\pi\)
\(600\) 0 0
\(601\) 16.2462 0.662697 0.331348 0.943508i \(-0.392496\pi\)
0.331348 + 0.943508i \(0.392496\pi\)
\(602\) 0 0
\(603\) 70.4228 2.86784
\(604\) 0 0
\(605\) 15.6829 0.637600
\(606\) 0 0
\(607\) 40.9848 1.66352 0.831762 0.555133i \(-0.187333\pi\)
0.831762 + 0.555133i \(0.187333\pi\)
\(608\) 0 0
\(609\) −18.2462 −0.739374
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.23100 0.0901094 0.0450547 0.998985i \(-0.485654\pi\)
0.0450547 + 0.998985i \(0.485654\pi\)
\(614\) 0 0
\(615\) −21.7538 −0.877197
\(616\) 0 0
\(617\) 29.3693 1.18236 0.591182 0.806538i \(-0.298661\pi\)
0.591182 + 0.806538i \(0.298661\pi\)
\(618\) 0 0
\(619\) −19.6558 −0.790033 −0.395017 0.918674i \(-0.629261\pi\)
−0.395017 + 0.918674i \(0.629261\pi\)
\(620\) 0 0
\(621\) 48.3272 1.93930
\(622\) 0 0
\(623\) 16.2462 0.650891
\(624\) 0 0
\(625\) −9.87689 −0.395076
\(626\) 0 0
\(627\) −12.0818 −0.482500
\(628\) 0 0
\(629\) 12.0818 0.481733
\(630\) 0 0
\(631\) 3.50758 0.139634 0.0698172 0.997560i \(-0.477758\pi\)
0.0698172 + 0.997560i \(0.477758\pi\)
\(632\) 0 0
\(633\) −14.2462 −0.566236
\(634\) 0 0
\(635\) −22.2586 −0.883307
\(636\) 0 0
\(637\) −1.69614 −0.0672036
\(638\) 0 0
\(639\) 48.9848 1.93781
\(640\) 0 0
\(641\) −5.36932 −0.212075 −0.106038 0.994362i \(-0.533816\pi\)
−0.106038 + 0.994362i \(0.533816\pi\)
\(642\) 0 0
\(643\) 5.66906 0.223566 0.111783 0.993733i \(-0.464344\pi\)
0.111783 + 0.993733i \(0.464344\pi\)
\(644\) 0 0
\(645\) −6.78456 −0.267142
\(646\) 0 0
\(647\) −43.2311 −1.69959 −0.849794 0.527115i \(-0.823274\pi\)
−0.849794 + 0.527115i \(0.823274\pi\)
\(648\) 0 0
\(649\) 0.492423 0.0193293
\(650\) 0 0
\(651\) 30.9481 1.21295
\(652\) 0 0
\(653\) −31.6918 −1.24020 −0.620098 0.784524i \(-0.712907\pi\)
−0.620098 + 0.784524i \(0.712907\pi\)
\(654\) 0 0
\(655\) −21.1231 −0.825348
\(656\) 0 0
\(657\) 36.7386 1.43331
\(658\) 0 0
\(659\) 6.62153 0.257938 0.128969 0.991649i \(-0.458833\pi\)
0.128969 + 0.991649i \(0.458833\pi\)
\(660\) 0 0
\(661\) 44.7261 1.73964 0.869821 0.493367i \(-0.164234\pi\)
0.869821 + 0.493367i \(0.164234\pi\)
\(662\) 0 0
\(663\) 10.2462 0.397930
\(664\) 0 0
\(665\) 5.12311 0.198666
\(666\) 0 0
\(667\) −30.9481 −1.19832
\(668\) 0 0
\(669\) 17.3790 0.671912
\(670\) 0 0
\(671\) 2.24621 0.0867140
\(672\) 0 0
\(673\) −38.9848 −1.50276 −0.751378 0.659872i \(-0.770610\pi\)
−0.751378 + 0.659872i \(0.770610\pi\)
\(674\) 0 0
\(675\) 20.0276 0.770864
\(676\) 0 0
\(677\) −17.1702 −0.659905 −0.329952 0.943998i \(-0.607033\pi\)
−0.329952 + 0.943998i \(0.607033\pi\)
\(678\) 0 0
\(679\) 12.2462 0.469966
\(680\) 0 0
\(681\) 29.6155 1.13487
\(682\) 0 0
\(683\) 0.580639 0.0222175 0.0111088 0.999938i \(-0.496464\pi\)
0.0111088 + 0.999938i \(0.496464\pi\)
\(684\) 0 0
\(685\) −27.5559 −1.05286
\(686\) 0 0
\(687\) 78.1080 2.98000
\(688\) 0 0
\(689\) 4.49242 0.171148
\(690\) 0 0
\(691\) −38.8482 −1.47786 −0.738928 0.673785i \(-0.764668\pi\)
−0.738928 + 0.673785i \(0.764668\pi\)
\(692\) 0 0
\(693\) −8.10887 −0.308031
\(694\) 0 0
\(695\) −14.1080 −0.535145
\(696\) 0 0
\(697\) −8.49242 −0.321673
\(698\) 0 0
\(699\) −49.0708 −1.85603
\(700\) 0 0
\(701\) 2.23100 0.0842639 0.0421319 0.999112i \(-0.486585\pi\)
0.0421319 + 0.999112i \(0.486585\pi\)
\(702\) 0 0
\(703\) −18.2462 −0.688169
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.3857 −0.390593
\(708\) 0 0
\(709\) −28.7171 −1.07849 −0.539247 0.842147i \(-0.681291\pi\)
−0.539247 + 0.842147i \(0.681291\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 52.4924 1.96586
\(714\) 0 0
\(715\) −3.80989 −0.142482
\(716\) 0 0
\(717\) 53.2068 1.98704
\(718\) 0 0
\(719\) −52.4924 −1.95764 −0.978819 0.204730i \(-0.934368\pi\)
−0.978819 + 0.204730i \(0.934368\pi\)
\(720\) 0 0
\(721\) −2.24621 −0.0836533
\(722\) 0 0
\(723\) 11.3381 0.421669
\(724\) 0 0
\(725\) −12.8255 −0.476326
\(726\) 0 0
\(727\) 16.9848 0.629933 0.314967 0.949103i \(-0.398007\pi\)
0.314967 + 0.949103i \(0.398007\pi\)
\(728\) 0 0
\(729\) −23.4924 −0.870090
\(730\) 0 0
\(731\) −2.64861 −0.0979625
\(732\) 0 0
\(733\) 29.2520 1.08045 0.540224 0.841521i \(-0.318340\pi\)
0.540224 + 0.841521i \(0.318340\pi\)
\(734\) 0 0
\(735\) 5.12311 0.188969
\(736\) 0 0
\(737\) −15.2311 −0.561043
\(738\) 0 0
\(739\) −21.3519 −0.785444 −0.392722 0.919657i \(-0.628467\pi\)
−0.392722 + 0.919657i \(0.628467\pi\)
\(740\) 0 0
\(741\) −15.4741 −0.568454
\(742\) 0 0
\(743\) −0.630683 −0.0231375 −0.0115688 0.999933i \(-0.503683\pi\)
−0.0115688 + 0.999933i \(0.503683\pi\)
\(744\) 0 0
\(745\) −24.9848 −0.915374
\(746\) 0 0
\(747\) 34.7123 1.27006
\(748\) 0 0
\(749\) 14.1498 0.517021
\(750\) 0 0
\(751\) 8.63068 0.314938 0.157469 0.987524i \(-0.449667\pi\)
0.157469 + 0.987524i \(0.449667\pi\)
\(752\) 0 0
\(753\) 33.1231 1.20707
\(754\) 0 0
\(755\) 18.4487 0.671419
\(756\) 0 0
\(757\) 26.3946 0.959328 0.479664 0.877452i \(-0.340759\pi\)
0.479664 + 0.877452i \(0.340759\pi\)
\(758\) 0 0
\(759\) −20.4924 −0.743828
\(760\) 0 0
\(761\) −8.73863 −0.316775 −0.158388 0.987377i \(-0.550630\pi\)
−0.158388 + 0.987377i \(0.550630\pi\)
\(762\) 0 0
\(763\) −18.1227 −0.656085
\(764\) 0 0
\(765\) −20.7713 −0.750988
\(766\) 0 0
\(767\) 0.630683 0.0227726
\(768\) 0 0
\(769\) −40.2462 −1.45132 −0.725658 0.688056i \(-0.758464\pi\)
−0.725658 + 0.688056i \(0.758464\pi\)
\(770\) 0 0
\(771\) −67.9372 −2.44670
\(772\) 0 0
\(773\) −1.69614 −0.0610060 −0.0305030 0.999535i \(-0.509711\pi\)
−0.0305030 + 0.999535i \(0.509711\pi\)
\(774\) 0 0
\(775\) 21.7538 0.781419
\(776\) 0 0
\(777\) −18.2462 −0.654579
\(778\) 0 0
\(779\) 12.8255 0.459520
\(780\) 0 0
\(781\) −10.5945 −0.379099
\(782\) 0 0
\(783\) −56.9848 −2.03647
\(784\) 0 0
\(785\) 14.3845 0.513404
\(786\) 0 0
\(787\) 10.5487 0.376020 0.188010 0.982167i \(-0.439796\pi\)
0.188010 + 0.982167i \(0.439796\pi\)
\(788\) 0 0
\(789\) 37.7327 1.34332
\(790\) 0 0
\(791\) 4.87689 0.173402
\(792\) 0 0
\(793\) 2.87689 0.102162
\(794\) 0 0
\(795\) −13.5691 −0.481247
\(796\) 0 0
\(797\) −25.8597 −0.915998 −0.457999 0.888953i \(-0.651434\pi\)
−0.457999 + 0.888953i \(0.651434\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 99.4773 3.51486
\(802\) 0 0
\(803\) −7.94584 −0.280403
\(804\) 0 0
\(805\) 8.68951 0.306265
\(806\) 0 0
\(807\) −35.8617 −1.26239
\(808\) 0 0
\(809\) −37.8617 −1.33115 −0.665574 0.746332i \(-0.731813\pi\)
−0.665574 + 0.746332i \(0.731813\pi\)
\(810\) 0 0
\(811\) −15.8459 −0.556425 −0.278213 0.960520i \(-0.589742\pi\)
−0.278213 + 0.960520i \(0.589742\pi\)
\(812\) 0 0
\(813\) −30.9481 −1.08540
\(814\) 0 0
\(815\) −22.7386 −0.796500
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) −10.3857 −0.362904
\(820\) 0 0
\(821\) −33.5968 −1.17254 −0.586268 0.810118i \(-0.699403\pi\)
−0.586268 + 0.810118i \(0.699403\pi\)
\(822\) 0 0
\(823\) 32.9848 1.14978 0.574890 0.818231i \(-0.305045\pi\)
0.574890 + 0.818231i \(0.305045\pi\)
\(824\) 0 0
\(825\) −8.49242 −0.295668
\(826\) 0 0
\(827\) −6.20393 −0.215732 −0.107866 0.994165i \(-0.534402\pi\)
−0.107866 + 0.994165i \(0.534402\pi\)
\(828\) 0 0
\(829\) 46.6310 1.61956 0.809781 0.586732i \(-0.199586\pi\)
0.809781 + 0.586732i \(0.199586\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 13.5691 0.469579
\(836\) 0 0
\(837\) 96.6543 3.34086
\(838\) 0 0
\(839\) 6.73863 0.232643 0.116322 0.993212i \(-0.462890\pi\)
0.116322 + 0.993212i \(0.462890\pi\)
\(840\) 0 0
\(841\) 7.49242 0.258359
\(842\) 0 0
\(843\) −18.1227 −0.624179
\(844\) 0 0
\(845\) 17.1702 0.590673
\(846\) 0 0
\(847\) −9.24621 −0.317704
\(848\) 0 0
\(849\) −66.1080 −2.26882
\(850\) 0 0
\(851\) −30.9481 −1.06089
\(852\) 0 0
\(853\) 25.4421 0.871121 0.435561 0.900159i \(-0.356550\pi\)
0.435561 + 0.900159i \(0.356550\pi\)
\(854\) 0 0
\(855\) 31.3693 1.07281
\(856\) 0 0
\(857\) 46.9848 1.60497 0.802486 0.596671i \(-0.203510\pi\)
0.802486 + 0.596671i \(0.203510\pi\)
\(858\) 0 0
\(859\) −28.3453 −0.967129 −0.483565 0.875309i \(-0.660658\pi\)
−0.483565 + 0.875309i \(0.660658\pi\)
\(860\) 0 0
\(861\) 12.8255 0.437091
\(862\) 0 0
\(863\) 36.4924 1.24222 0.621108 0.783725i \(-0.286683\pi\)
0.621108 + 0.783725i \(0.286683\pi\)
\(864\) 0 0
\(865\) 32.3542 1.10007
\(866\) 0 0
\(867\) 39.2658 1.33354
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −19.5076 −0.660989
\(872\) 0 0
\(873\) 74.9848 2.53785
\(874\) 0 0
\(875\) 12.0818 0.408439
\(876\) 0 0
\(877\) −35.5017 −1.19881 −0.599404 0.800447i \(-0.704596\pi\)
−0.599404 + 0.800447i \(0.704596\pi\)
\(878\) 0 0
\(879\) 31.3693 1.05806
\(880\) 0 0
\(881\) 44.2462 1.49069 0.745346 0.666677i \(-0.232284\pi\)
0.745346 + 0.666677i \(0.232284\pi\)
\(882\) 0 0
\(883\) 4.71659 0.158726 0.0793629 0.996846i \(-0.474711\pi\)
0.0793629 + 0.996846i \(0.474711\pi\)
\(884\) 0 0
\(885\) −1.90495 −0.0640340
\(886\) 0 0
\(887\) 28.4924 0.956682 0.478341 0.878174i \(-0.341238\pi\)
0.478341 + 0.878174i \(0.341238\pi\)
\(888\) 0 0
\(889\) 13.1231 0.440135
\(890\) 0 0
\(891\) −13.4061 −0.449121
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −26.2462 −0.876335
\(898\) 0 0
\(899\) −61.8963 −2.06436
\(900\) 0 0
\(901\) −5.29723 −0.176476
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −11.8617 −0.394298
\(906\) 0 0
\(907\) −51.5564 −1.71190 −0.855951 0.517056i \(-0.827028\pi\)
−0.855951 + 0.517056i \(0.827028\pi\)
\(908\) 0 0
\(909\) −63.5924 −2.10923
\(910\) 0 0
\(911\) 11.8617 0.392997 0.196498 0.980504i \(-0.437043\pi\)
0.196498 + 0.980504i \(0.437043\pi\)
\(912\) 0 0
\(913\) −7.50758 −0.248465
\(914\) 0 0
\(915\) −8.68951 −0.287266
\(916\) 0 0
\(917\) 12.4536 0.411255
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 76.3542 2.51596
\(922\) 0 0
\(923\) −13.5691 −0.446633
\(924\) 0 0
\(925\) −12.8255 −0.421699
\(926\) 0 0
\(927\) −13.7538 −0.451734
\(928\) 0 0
\(929\) −2.49242 −0.0817737 −0.0408869 0.999164i \(-0.513018\pi\)
−0.0408869 + 0.999164i \(0.513018\pi\)
\(930\) 0 0
\(931\) −3.02045 −0.0989912
\(932\) 0 0
\(933\) −37.7327 −1.23531
\(934\) 0 0
\(935\) 4.49242 0.146918
\(936\) 0 0
\(937\) −34.9848 −1.14291 −0.571453 0.820635i \(-0.693620\pi\)
−0.571453 + 0.820635i \(0.693620\pi\)
\(938\) 0 0
\(939\) −62.6400 −2.04418
\(940\) 0 0
\(941\) −30.7393 −1.00207 −0.501037 0.865426i \(-0.667048\pi\)
−0.501037 + 0.865426i \(0.667048\pi\)
\(942\) 0 0
\(943\) 21.7538 0.708401
\(944\) 0 0
\(945\) 16.0000 0.520480
\(946\) 0 0
\(947\) −41.7056 −1.35525 −0.677625 0.735407i \(-0.736991\pi\)
−0.677625 + 0.735407i \(0.736991\pi\)
\(948\) 0 0
\(949\) −10.1768 −0.330354
\(950\) 0 0
\(951\) −12.4924 −0.405095
\(952\) 0 0
\(953\) 17.5076 0.567126 0.283563 0.958954i \(-0.408483\pi\)
0.283563 + 0.958954i \(0.408483\pi\)
\(954\) 0 0
\(955\) −27.1383 −0.878173
\(956\) 0 0
\(957\) 24.1636 0.781098
\(958\) 0 0
\(959\) 16.2462 0.524618
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) 86.6405 2.79195
\(964\) 0 0
\(965\) 18.8664 0.607329
\(966\) 0 0
\(967\) 10.8769 0.349777 0.174889 0.984588i \(-0.444043\pi\)
0.174889 + 0.984588i \(0.444043\pi\)
\(968\) 0 0
\(969\) 18.2462 0.586153
\(970\) 0 0
\(971\) −10.9663 −0.351925 −0.175962 0.984397i \(-0.556304\pi\)
−0.175962 + 0.984397i \(0.556304\pi\)
\(972\) 0 0
\(973\) 8.31768 0.266652
\(974\) 0 0
\(975\) −10.8769 −0.348339
\(976\) 0 0
\(977\) 23.7538 0.759951 0.379976 0.924997i \(-0.375932\pi\)
0.379976 + 0.924997i \(0.375932\pi\)
\(978\) 0 0
\(979\) −21.5150 −0.687621
\(980\) 0 0
\(981\) −110.967 −3.54291
\(982\) 0 0
\(983\) 34.2462 1.09228 0.546142 0.837692i \(-0.316096\pi\)
0.546142 + 0.837692i \(0.316096\pi\)
\(984\) 0 0
\(985\) −36.4924 −1.16275
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.78456 0.215737
\(990\) 0 0
\(991\) −36.4924 −1.15922 −0.579610 0.814894i \(-0.696795\pi\)
−0.579610 + 0.814894i \(0.696795\pi\)
\(992\) 0 0
\(993\) 44.9848 1.42755
\(994\) 0 0
\(995\) −30.9481 −0.981122
\(996\) 0 0
\(997\) 33.0619 1.04708 0.523540 0.852001i \(-0.324611\pi\)
0.523540 + 0.852001i \(0.324611\pi\)
\(998\) 0 0
\(999\) −56.9848 −1.80292
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 1792.2.a.x.1.1 4
4.3 odd 2 1792.2.a.v.1.4 4
8.3 odd 2 1792.2.a.v.1.1 4
8.5 even 2 inner 1792.2.a.x.1.4 4
16.3 odd 4 224.2.b.b.113.4 4
16.5 even 4 56.2.b.b.29.3 4
16.11 odd 4 224.2.b.b.113.1 4
16.13 even 4 56.2.b.b.29.4 yes 4
48.5 odd 4 504.2.c.d.253.2 4
48.11 even 4 2016.2.c.c.1009.3 4
48.29 odd 4 504.2.c.d.253.1 4
48.35 even 4 2016.2.c.c.1009.2 4
112.3 even 12 1568.2.t.e.177.1 8
112.5 odd 12 392.2.p.e.165.1 8
112.11 odd 12 1568.2.t.d.177.1 8
112.13 odd 4 392.2.b.c.197.4 4
112.19 even 12 1568.2.t.e.753.4 8
112.27 even 4 1568.2.b.d.785.4 4
112.37 even 12 392.2.p.f.165.1 8
112.45 odd 12 392.2.p.e.373.1 8
112.51 odd 12 1568.2.t.d.753.1 8
112.53 even 12 392.2.p.f.373.3 8
112.59 even 12 1568.2.t.e.177.4 8
112.61 odd 12 392.2.p.e.165.3 8
112.67 odd 12 1568.2.t.d.177.4 8
112.69 odd 4 392.2.b.c.197.3 4
112.75 even 12 1568.2.t.e.753.1 8
112.83 even 4 1568.2.b.d.785.1 4
112.93 even 12 392.2.p.f.165.3 8
112.101 odd 12 392.2.p.e.373.3 8
112.107 odd 12 1568.2.t.d.753.4 8
112.109 even 12 392.2.p.f.373.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.b.b.29.3 4 16.5 even 4
56.2.b.b.29.4 yes 4 16.13 even 4
224.2.b.b.113.1 4 16.11 odd 4
224.2.b.b.113.4 4 16.3 odd 4
392.2.b.c.197.3 4 112.69 odd 4
392.2.b.c.197.4 4 112.13 odd 4
392.2.p.e.165.1 8 112.5 odd 12
392.2.p.e.165.3 8 112.61 odd 12
392.2.p.e.373.1 8 112.45 odd 12
392.2.p.e.373.3 8 112.101 odd 12
392.2.p.f.165.1 8 112.37 even 12
392.2.p.f.165.3 8 112.93 even 12
392.2.p.f.373.1 8 112.109 even 12
392.2.p.f.373.3 8 112.53 even 12
504.2.c.d.253.1 4 48.29 odd 4
504.2.c.d.253.2 4 48.5 odd 4
1568.2.b.d.785.1 4 112.83 even 4
1568.2.b.d.785.4 4 112.27 even 4
1568.2.t.d.177.1 8 112.11 odd 12
1568.2.t.d.177.4 8 112.67 odd 12
1568.2.t.d.753.1 8 112.51 odd 12
1568.2.t.d.753.4 8 112.107 odd 12
1568.2.t.e.177.1 8 112.3 even 12
1568.2.t.e.177.4 8 112.59 even 12
1568.2.t.e.753.1 8 112.75 even 12
1568.2.t.e.753.4 8 112.19 even 12
1792.2.a.v.1.1 4 8.3 odd 2
1792.2.a.v.1.4 4 4.3 odd 2
1792.2.a.x.1.1 4 1.1 even 1 trivial
1792.2.a.x.1.4 4 8.5 even 2 inner
2016.2.c.c.1009.2 4 48.35 even 4
2016.2.c.c.1009.3 4 48.11 even 4