Properties

Label 224.3.d.b.127.4
Level 224
Weight 3
Character 224.127
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1997017344.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.4
Root \(-0.277334i\) of \(x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4\)
Character \(\chi\) \(=\) 224.127
Dual form 224.3.d.b.127.5

$q$-expansion

\(f(q)\) \(=\) \(q-0.554669i q^{3} -4.57685 q^{5} +2.64575i q^{7} +8.69234 q^{9} +O(q^{10})\) \(q-0.554669i q^{3} -4.57685 q^{5} +2.64575i q^{7} +8.69234 q^{9} +15.7367i q^{11} +8.57685 q^{13} +2.53864i q^{15} +28.3197 q^{17} +6.33599i q^{19} +1.46752 q^{21} +31.0647i q^{23} -4.05242 q^{25} -9.81339i q^{27} -0.846294 q^{29} -21.6354i q^{31} +8.72866 q^{33} -12.1092i q^{35} -33.6637 q^{37} -4.75731i q^{39} +66.9757 q^{41} +44.8781i q^{43} -39.7836 q^{45} +38.4528i q^{47} -7.00000 q^{49} -15.7081i q^{51} -14.8174 q^{53} -72.0246i q^{55} +3.51438 q^{57} +5.80942i q^{59} -52.6015 q^{61} +22.9978i q^{63} -39.2550 q^{65} -117.397i q^{67} +17.2306 q^{69} -81.2543i q^{71} -47.8054 q^{73} +2.24775i q^{75} -41.6354 q^{77} +57.4900i q^{79} +72.7879 q^{81} -102.855i q^{83} -129.615 q^{85} +0.469413i q^{87} -89.2955 q^{89} +22.6922i q^{91} -12.0005 q^{93} -28.9989i q^{95} -3.44452 q^{97} +136.789i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 40q^{9} + O(q^{10}) \) \( 8q - 40q^{9} + 32q^{13} - 16q^{17} + 104q^{25} - 80q^{29} - 176q^{37} + 144q^{41} + 256q^{45} - 56q^{49} + 48q^{53} - 400q^{57} - 192q^{61} - 304q^{65} + 576q^{69} + 272q^{73} - 112q^{77} + 504q^{81} - 160q^{85} - 80q^{89} - 608q^{93} + 528q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 0.554669i − 0.184890i −0.995718 0.0924448i \(-0.970532\pi\)
0.995718 0.0924448i \(-0.0294682\pi\)
\(4\) 0 0
\(5\) −4.57685 −0.915371 −0.457685 0.889114i \(-0.651321\pi\)
−0.457685 + 0.889114i \(0.651321\pi\)
\(6\) 0 0
\(7\) 2.64575i 0.377964i
\(8\) 0 0
\(9\) 8.69234 0.965816
\(10\) 0 0
\(11\) 15.7367i 1.43061i 0.698812 + 0.715305i \(0.253712\pi\)
−0.698812 + 0.715305i \(0.746288\pi\)
\(12\) 0 0
\(13\) 8.57685 0.659758 0.329879 0.944023i \(-0.392992\pi\)
0.329879 + 0.944023i \(0.392992\pi\)
\(14\) 0 0
\(15\) 2.53864i 0.169242i
\(16\) 0 0
\(17\) 28.3197 1.66587 0.832933 0.553374i \(-0.186660\pi\)
0.832933 + 0.553374i \(0.186660\pi\)
\(18\) 0 0
\(19\) 6.33599i 0.333473i 0.986001 + 0.166737i \(0.0533230\pi\)
−0.986001 + 0.166737i \(0.946677\pi\)
\(20\) 0 0
\(21\) 1.46752 0.0698817
\(22\) 0 0
\(23\) 31.0647i 1.35064i 0.737525 + 0.675320i \(0.235995\pi\)
−0.737525 + 0.675320i \(0.764005\pi\)
\(24\) 0 0
\(25\) −4.05242 −0.162097
\(26\) 0 0
\(27\) − 9.81339i − 0.363459i
\(28\) 0 0
\(29\) −0.846294 −0.0291826 −0.0145913 0.999894i \(-0.504645\pi\)
−0.0145913 + 0.999894i \(0.504645\pi\)
\(30\) 0 0
\(31\) − 21.6354i − 0.697917i −0.937138 0.348958i \(-0.886535\pi\)
0.937138 0.348958i \(-0.113465\pi\)
\(32\) 0 0
\(33\) 8.72866 0.264505
\(34\) 0 0
\(35\) − 12.1092i − 0.345978i
\(36\) 0 0
\(37\) −33.6637 −0.909830 −0.454915 0.890535i \(-0.650330\pi\)
−0.454915 + 0.890535i \(0.650330\pi\)
\(38\) 0 0
\(39\) − 4.75731i − 0.121982i
\(40\) 0 0
\(41\) 66.9757 1.63355 0.816777 0.576953i \(-0.195758\pi\)
0.816777 + 0.576953i \(0.195758\pi\)
\(42\) 0 0
\(43\) 44.8781i 1.04368i 0.853044 + 0.521839i \(0.174754\pi\)
−0.853044 + 0.521839i \(0.825246\pi\)
\(44\) 0 0
\(45\) −39.7836 −0.884079
\(46\) 0 0
\(47\) 38.4528i 0.818145i 0.912502 + 0.409073i \(0.134148\pi\)
−0.912502 + 0.409073i \(0.865852\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) − 15.7081i − 0.308001i
\(52\) 0 0
\(53\) −14.8174 −0.279574 −0.139787 0.990182i \(-0.544642\pi\)
−0.139787 + 0.990182i \(0.544642\pi\)
\(54\) 0 0
\(55\) − 72.0246i − 1.30954i
\(56\) 0 0
\(57\) 3.51438 0.0616557
\(58\) 0 0
\(59\) 5.80942i 0.0984647i 0.998787 + 0.0492323i \(0.0156775\pi\)
−0.998787 + 0.0492323i \(0.984323\pi\)
\(60\) 0 0
\(61\) −52.6015 −0.862319 −0.431160 0.902276i \(-0.641895\pi\)
−0.431160 + 0.902276i \(0.641895\pi\)
\(62\) 0 0
\(63\) 22.9978i 0.365044i
\(64\) 0 0
\(65\) −39.2550 −0.603923
\(66\) 0 0
\(67\) − 117.397i − 1.75219i −0.482138 0.876095i \(-0.660140\pi\)
0.482138 0.876095i \(-0.339860\pi\)
\(68\) 0 0
\(69\) 17.2306 0.249719
\(70\) 0 0
\(71\) − 81.2543i − 1.14443i −0.820105 0.572213i \(-0.806085\pi\)
0.820105 0.572213i \(-0.193915\pi\)
\(72\) 0 0
\(73\) −47.8054 −0.654869 −0.327435 0.944874i \(-0.606184\pi\)
−0.327435 + 0.944874i \(0.606184\pi\)
\(74\) 0 0
\(75\) 2.24775i 0.0299700i
\(76\) 0 0
\(77\) −41.6354 −0.540720
\(78\) 0 0
\(79\) 57.4900i 0.727722i 0.931453 + 0.363861i \(0.118542\pi\)
−0.931453 + 0.363861i \(0.881458\pi\)
\(80\) 0 0
\(81\) 72.7879 0.898616
\(82\) 0 0
\(83\) − 102.855i − 1.23921i −0.784913 0.619606i \(-0.787292\pi\)
0.784913 0.619606i \(-0.212708\pi\)
\(84\) 0 0
\(85\) −129.615 −1.52488
\(86\) 0 0
\(87\) 0.469413i 0.00539555i
\(88\) 0 0
\(89\) −89.2955 −1.00332 −0.501660 0.865065i \(-0.667277\pi\)
−0.501660 + 0.865065i \(0.667277\pi\)
\(90\) 0 0
\(91\) 22.6922i 0.249365i
\(92\) 0 0
\(93\) −12.0005 −0.129038
\(94\) 0 0
\(95\) − 28.9989i − 0.305252i
\(96\) 0 0
\(97\) −3.44452 −0.0355106 −0.0177553 0.999842i \(-0.505652\pi\)
−0.0177553 + 0.999842i \(0.505652\pi\)
\(98\) 0 0
\(99\) 136.789i 1.38171i
\(100\) 0 0
\(101\) 143.034 1.41618 0.708089 0.706123i \(-0.249557\pi\)
0.708089 + 0.706123i \(0.249557\pi\)
\(102\) 0 0
\(103\) − 173.424i − 1.68373i −0.539688 0.841865i \(-0.681458\pi\)
0.539688 0.841865i \(-0.318542\pi\)
\(104\) 0 0
\(105\) −6.71660 −0.0639676
\(106\) 0 0
\(107\) 95.5581i 0.893067i 0.894767 + 0.446533i \(0.147342\pi\)
−0.894767 + 0.446533i \(0.852658\pi\)
\(108\) 0 0
\(109\) 185.517 1.70199 0.850997 0.525170i \(-0.175998\pi\)
0.850997 + 0.525170i \(0.175998\pi\)
\(110\) 0 0
\(111\) 18.6722i 0.168218i
\(112\) 0 0
\(113\) −103.409 −0.915124 −0.457562 0.889178i \(-0.651277\pi\)
−0.457562 + 0.889178i \(0.651277\pi\)
\(114\) 0 0
\(115\) − 142.179i − 1.23634i
\(116\) 0 0
\(117\) 74.5529 0.637205
\(118\) 0 0
\(119\) 74.9269i 0.629638i
\(120\) 0 0
\(121\) −126.644 −1.04665
\(122\) 0 0
\(123\) − 37.1493i − 0.302027i
\(124\) 0 0
\(125\) 132.969 1.06375
\(126\) 0 0
\(127\) 147.970i 1.16512i 0.812787 + 0.582560i \(0.197949\pi\)
−0.812787 + 0.582560i \(0.802051\pi\)
\(128\) 0 0
\(129\) 24.8925 0.192965
\(130\) 0 0
\(131\) 259.412i 1.98025i 0.140200 + 0.990123i \(0.455225\pi\)
−0.140200 + 0.990123i \(0.544775\pi\)
\(132\) 0 0
\(133\) −16.7635 −0.126041
\(134\) 0 0
\(135\) 44.9144i 0.332700i
\(136\) 0 0
\(137\) 3.07777 0.0224654 0.0112327 0.999937i \(-0.496424\pi\)
0.0112327 + 0.999937i \(0.496424\pi\)
\(138\) 0 0
\(139\) − 90.3041i − 0.649670i −0.945771 0.324835i \(-0.894691\pi\)
0.945771 0.324835i \(-0.105309\pi\)
\(140\) 0 0
\(141\) 21.3286 0.151266
\(142\) 0 0
\(143\) 134.971i 0.943856i
\(144\) 0 0
\(145\) 3.87336 0.0267129
\(146\) 0 0
\(147\) 3.88268i 0.0264128i
\(148\) 0 0
\(149\) 41.3766 0.277695 0.138848 0.990314i \(-0.455660\pi\)
0.138848 + 0.990314i \(0.455660\pi\)
\(150\) 0 0
\(151\) 41.5182i 0.274955i 0.990505 + 0.137477i \(0.0438994\pi\)
−0.990505 + 0.137477i \(0.956101\pi\)
\(152\) 0 0
\(153\) 246.165 1.60892
\(154\) 0 0
\(155\) 99.0221i 0.638853i
\(156\) 0 0
\(157\) −26.9848 −0.171878 −0.0859389 0.996300i \(-0.527389\pi\)
−0.0859389 + 0.996300i \(0.527389\pi\)
\(158\) 0 0
\(159\) 8.21875i 0.0516902i
\(160\) 0 0
\(161\) −82.1895 −0.510494
\(162\) 0 0
\(163\) − 126.684i − 0.777200i −0.921407 0.388600i \(-0.872959\pi\)
0.921407 0.388600i \(-0.127041\pi\)
\(164\) 0 0
\(165\) −39.9498 −0.242120
\(166\) 0 0
\(167\) − 246.243i − 1.47451i −0.675615 0.737254i \(-0.736122\pi\)
0.675615 0.737254i \(-0.263878\pi\)
\(168\) 0 0
\(169\) −95.4376 −0.564720
\(170\) 0 0
\(171\) 55.0746i 0.322074i
\(172\) 0 0
\(173\) 74.5776 0.431084 0.215542 0.976495i \(-0.430848\pi\)
0.215542 + 0.976495i \(0.430848\pi\)
\(174\) 0 0
\(175\) − 10.7217i − 0.0612668i
\(176\) 0 0
\(177\) 3.22230 0.0182051
\(178\) 0 0
\(179\) − 201.073i − 1.12331i −0.827371 0.561656i \(-0.810165\pi\)
0.827371 0.561656i \(-0.189835\pi\)
\(180\) 0 0
\(181\) 252.696 1.39611 0.698054 0.716045i \(-0.254050\pi\)
0.698054 + 0.716045i \(0.254050\pi\)
\(182\) 0 0
\(183\) 29.1764i 0.159434i
\(184\) 0 0
\(185\) 154.074 0.832831
\(186\) 0 0
\(187\) 445.659i 2.38320i
\(188\) 0 0
\(189\) 25.9638 0.137375
\(190\) 0 0
\(191\) 332.333i 1.73996i 0.493085 + 0.869981i \(0.335869\pi\)
−0.493085 + 0.869981i \(0.664131\pi\)
\(192\) 0 0
\(193\) −53.2218 −0.275761 −0.137880 0.990449i \(-0.544029\pi\)
−0.137880 + 0.990449i \(0.544029\pi\)
\(194\) 0 0
\(195\) 21.7735i 0.111659i
\(196\) 0 0
\(197\) 276.248 1.40227 0.701137 0.713026i \(-0.252676\pi\)
0.701137 + 0.713026i \(0.252676\pi\)
\(198\) 0 0
\(199\) − 58.5094i − 0.294017i −0.989135 0.147008i \(-0.953036\pi\)
0.989135 0.147008i \(-0.0469644\pi\)
\(200\) 0 0
\(201\) −65.1163 −0.323962
\(202\) 0 0
\(203\) − 2.23908i − 0.0110300i
\(204\) 0 0
\(205\) −306.538 −1.49531
\(206\) 0 0
\(207\) 270.025i 1.30447i
\(208\) 0 0
\(209\) −99.7077 −0.477070
\(210\) 0 0
\(211\) − 146.237i − 0.693068i −0.938037 0.346534i \(-0.887359\pi\)
0.938037 0.346534i \(-0.112641\pi\)
\(212\) 0 0
\(213\) −45.0692 −0.211592
\(214\) 0 0
\(215\) − 205.401i − 0.955351i
\(216\) 0 0
\(217\) 57.2420 0.263788
\(218\) 0 0
\(219\) 26.5162i 0.121078i
\(220\) 0 0
\(221\) 242.894 1.09907
\(222\) 0 0
\(223\) − 422.290i − 1.89368i −0.321709 0.946839i \(-0.604257\pi\)
0.321709 0.946839i \(-0.395743\pi\)
\(224\) 0 0
\(225\) −35.2250 −0.156556
\(226\) 0 0
\(227\) − 271.103i − 1.19429i −0.802134 0.597144i \(-0.796302\pi\)
0.802134 0.597144i \(-0.203698\pi\)
\(228\) 0 0
\(229\) −400.572 −1.74922 −0.874611 0.484826i \(-0.838883\pi\)
−0.874611 + 0.484826i \(0.838883\pi\)
\(230\) 0 0
\(231\) 23.0939i 0.0999734i
\(232\) 0 0
\(233\) −257.611 −1.10563 −0.552813 0.833305i \(-0.686446\pi\)
−0.552813 + 0.833305i \(0.686446\pi\)
\(234\) 0 0
\(235\) − 175.993i − 0.748906i
\(236\) 0 0
\(237\) 31.8879 0.134548
\(238\) 0 0
\(239\) − 152.650i − 0.638704i −0.947636 0.319352i \(-0.896535\pi\)
0.947636 0.319352i \(-0.103465\pi\)
\(240\) 0 0
\(241\) −47.1218 −0.195526 −0.0977630 0.995210i \(-0.531169\pi\)
−0.0977630 + 0.995210i \(0.531169\pi\)
\(242\) 0 0
\(243\) − 128.694i − 0.529604i
\(244\) 0 0
\(245\) 32.0380 0.130767
\(246\) 0 0
\(247\) 54.3429i 0.220012i
\(248\) 0 0
\(249\) −57.0502 −0.229117
\(250\) 0 0
\(251\) 236.051i 0.940443i 0.882549 + 0.470221i \(0.155826\pi\)
−0.882549 + 0.470221i \(0.844174\pi\)
\(252\) 0 0
\(253\) −488.857 −1.93224
\(254\) 0 0
\(255\) 71.8935i 0.281935i
\(256\) 0 0
\(257\) −27.5524 −0.107208 −0.0536039 0.998562i \(-0.517071\pi\)
−0.0536039 + 0.998562i \(0.517071\pi\)
\(258\) 0 0
\(259\) − 89.0658i − 0.343883i
\(260\) 0 0
\(261\) −7.35628 −0.0281850
\(262\) 0 0
\(263\) 88.2675i 0.335618i 0.985820 + 0.167809i \(0.0536691\pi\)
−0.985820 + 0.167809i \(0.946331\pi\)
\(264\) 0 0
\(265\) 67.8170 0.255913
\(266\) 0 0
\(267\) 49.5294i 0.185503i
\(268\) 0 0
\(269\) 21.9135 0.0814629 0.0407314 0.999170i \(-0.487031\pi\)
0.0407314 + 0.999170i \(0.487031\pi\)
\(270\) 0 0
\(271\) − 428.897i − 1.58265i −0.611399 0.791323i \(-0.709393\pi\)
0.611399 0.791323i \(-0.290607\pi\)
\(272\) 0 0
\(273\) 12.5867 0.0461050
\(274\) 0 0
\(275\) − 63.7717i − 0.231897i
\(276\) 0 0
\(277\) −457.076 −1.65009 −0.825046 0.565065i \(-0.808851\pi\)
−0.825046 + 0.565065i \(0.808851\pi\)
\(278\) 0 0
\(279\) − 188.063i − 0.674059i
\(280\) 0 0
\(281\) 95.5032 0.339869 0.169934 0.985455i \(-0.445644\pi\)
0.169934 + 0.985455i \(0.445644\pi\)
\(282\) 0 0
\(283\) − 131.804i − 0.465737i −0.972508 0.232869i \(-0.925189\pi\)
0.972508 0.232869i \(-0.0748112\pi\)
\(284\) 0 0
\(285\) −16.0848 −0.0564379
\(286\) 0 0
\(287\) 177.201i 0.617426i
\(288\) 0 0
\(289\) 513.006 1.77511
\(290\) 0 0
\(291\) 1.91057i 0.00656553i
\(292\) 0 0
\(293\) −325.183 −1.10984 −0.554920 0.831904i \(-0.687251\pi\)
−0.554920 + 0.831904i \(0.687251\pi\)
\(294\) 0 0
\(295\) − 26.5888i − 0.0901317i
\(296\) 0 0
\(297\) 154.430 0.519968
\(298\) 0 0
\(299\) 266.438i 0.891096i
\(300\) 0 0
\(301\) −118.736 −0.394473
\(302\) 0 0
\(303\) − 79.3365i − 0.261837i
\(304\) 0 0
\(305\) 240.749 0.789341
\(306\) 0 0
\(307\) − 34.3658i − 0.111941i −0.998432 0.0559704i \(-0.982175\pi\)
0.998432 0.0559704i \(-0.0178252\pi\)
\(308\) 0 0
\(309\) −96.1930 −0.311304
\(310\) 0 0
\(311\) 195.190i 0.627620i 0.949486 + 0.313810i \(0.101606\pi\)
−0.949486 + 0.313810i \(0.898394\pi\)
\(312\) 0 0
\(313\) 19.8987 0.0635740 0.0317870 0.999495i \(-0.489880\pi\)
0.0317870 + 0.999495i \(0.489880\pi\)
\(314\) 0 0
\(315\) − 105.257i − 0.334151i
\(316\) 0 0
\(317\) 168.672 0.532088 0.266044 0.963961i \(-0.414283\pi\)
0.266044 + 0.963961i \(0.414283\pi\)
\(318\) 0 0
\(319\) − 13.3179i − 0.0417489i
\(320\) 0 0
\(321\) 53.0031 0.165119
\(322\) 0 0
\(323\) 179.434i 0.555522i
\(324\) 0 0
\(325\) −34.7570 −0.106945
\(326\) 0 0
\(327\) − 102.901i − 0.314681i
\(328\) 0 0
\(329\) −101.737 −0.309230
\(330\) 0 0
\(331\) − 603.602i − 1.82357i −0.410666 0.911786i \(-0.634704\pi\)
0.410666 0.911786i \(-0.365296\pi\)
\(332\) 0 0
\(333\) −292.616 −0.878728
\(334\) 0 0
\(335\) 537.308i 1.60390i
\(336\) 0 0
\(337\) 189.903 0.563511 0.281756 0.959486i \(-0.409083\pi\)
0.281756 + 0.959486i \(0.409083\pi\)
\(338\) 0 0
\(339\) 57.3578i 0.169197i
\(340\) 0 0
\(341\) 340.470 0.998447
\(342\) 0 0
\(343\) − 18.5203i − 0.0539949i
\(344\) 0 0
\(345\) −78.8621 −0.228586
\(346\) 0 0
\(347\) − 81.4988i − 0.234867i −0.993081 0.117433i \(-0.962533\pi\)
0.993081 0.117433i \(-0.0374667\pi\)
\(348\) 0 0
\(349\) −223.673 −0.640898 −0.320449 0.947266i \(-0.603834\pi\)
−0.320449 + 0.947266i \(0.603834\pi\)
\(350\) 0 0
\(351\) − 84.1680i − 0.239795i
\(352\) 0 0
\(353\) −613.342 −1.73751 −0.868756 0.495241i \(-0.835080\pi\)
−0.868756 + 0.495241i \(0.835080\pi\)
\(354\) 0 0
\(355\) 371.889i 1.04757i
\(356\) 0 0
\(357\) 41.5596 0.116414
\(358\) 0 0
\(359\) − 79.0559i − 0.220212i −0.993920 0.110106i \(-0.964881\pi\)
0.993920 0.110106i \(-0.0351190\pi\)
\(360\) 0 0
\(361\) 320.855 0.888796
\(362\) 0 0
\(363\) 70.2455i 0.193514i
\(364\) 0 0
\(365\) 218.798 0.599448
\(366\) 0 0
\(367\) − 0.705581i − 0.00192257i −1.00000 0.000961283i \(-0.999694\pi\)
1.00000 0.000961283i \(-0.000305986\pi\)
\(368\) 0 0
\(369\) 582.176 1.57771
\(370\) 0 0
\(371\) − 39.2031i − 0.105669i
\(372\) 0 0
\(373\) 250.574 0.671781 0.335890 0.941901i \(-0.390963\pi\)
0.335890 + 0.941901i \(0.390963\pi\)
\(374\) 0 0
\(375\) − 73.7535i − 0.196676i
\(376\) 0 0
\(377\) −7.25854 −0.0192534
\(378\) 0 0
\(379\) 532.859i 1.40596i 0.711209 + 0.702981i \(0.248148\pi\)
−0.711209 + 0.702981i \(0.751852\pi\)
\(380\) 0 0
\(381\) 82.0745 0.215419
\(382\) 0 0
\(383\) − 441.226i − 1.15203i −0.817441 0.576013i \(-0.804608\pi\)
0.817441 0.576013i \(-0.195392\pi\)
\(384\) 0 0
\(385\) 190.559 0.494959
\(386\) 0 0
\(387\) 390.096i 1.00800i
\(388\) 0 0
\(389\) 174.837 0.449452 0.224726 0.974422i \(-0.427851\pi\)
0.224726 + 0.974422i \(0.427851\pi\)
\(390\) 0 0
\(391\) 879.744i 2.24999i
\(392\) 0 0
\(393\) 143.888 0.366127
\(394\) 0 0
\(395\) − 263.123i − 0.666135i
\(396\) 0 0
\(397\) 129.750 0.326826 0.163413 0.986558i \(-0.447750\pi\)
0.163413 + 0.986558i \(0.447750\pi\)
\(398\) 0 0
\(399\) 9.29817i 0.0233037i
\(400\) 0 0
\(401\) −135.495 −0.337892 −0.168946 0.985625i \(-0.554036\pi\)
−0.168946 + 0.985625i \(0.554036\pi\)
\(402\) 0 0
\(403\) − 185.564i − 0.460456i
\(404\) 0 0
\(405\) −333.140 −0.822567
\(406\) 0 0
\(407\) − 529.756i − 1.30161i
\(408\) 0 0
\(409\) 243.575 0.595538 0.297769 0.954638i \(-0.403758\pi\)
0.297769 + 0.954638i \(0.403758\pi\)
\(410\) 0 0
\(411\) − 1.70714i − 0.00415363i
\(412\) 0 0
\(413\) −15.3703 −0.0372161
\(414\) 0 0
\(415\) 470.750i 1.13434i
\(416\) 0 0
\(417\) −50.0888 −0.120117
\(418\) 0 0
\(419\) 499.348i 1.19176i 0.803073 + 0.595881i \(0.203197\pi\)
−0.803073 + 0.595881i \(0.796803\pi\)
\(420\) 0 0
\(421\) 537.031 1.27561 0.637804 0.770199i \(-0.279843\pi\)
0.637804 + 0.770199i \(0.279843\pi\)
\(422\) 0 0
\(423\) 334.245i 0.790178i
\(424\) 0 0
\(425\) −114.763 −0.270031
\(426\) 0 0
\(427\) − 139.170i − 0.325926i
\(428\) 0 0
\(429\) 74.8644 0.174509
\(430\) 0 0
\(431\) 559.209i 1.29747i 0.761015 + 0.648735i \(0.224702\pi\)
−0.761015 + 0.648735i \(0.775298\pi\)
\(432\) 0 0
\(433\) −812.706 −1.87692 −0.938459 0.345389i \(-0.887747\pi\)
−0.938459 + 0.345389i \(0.887747\pi\)
\(434\) 0 0
\(435\) − 2.14843i − 0.00493893i
\(436\) 0 0
\(437\) −196.826 −0.450403
\(438\) 0 0
\(439\) 346.809i 0.789998i 0.918681 + 0.394999i \(0.129255\pi\)
−0.918681 + 0.394999i \(0.870745\pi\)
\(440\) 0 0
\(441\) −60.8464 −0.137974
\(442\) 0 0
\(443\) − 369.535i − 0.834166i −0.908868 0.417083i \(-0.863052\pi\)
0.908868 0.417083i \(-0.136948\pi\)
\(444\) 0 0
\(445\) 408.692 0.918409
\(446\) 0 0
\(447\) − 22.9503i − 0.0513430i
\(448\) 0 0
\(449\) 315.180 0.701961 0.350980 0.936383i \(-0.385848\pi\)
0.350980 + 0.936383i \(0.385848\pi\)
\(450\) 0 0
\(451\) 1053.98i 2.33698i
\(452\) 0 0
\(453\) 23.0288 0.0508363
\(454\) 0 0
\(455\) − 103.859i − 0.228261i
\(456\) 0 0
\(457\) 781.559 1.71019 0.855097 0.518468i \(-0.173497\pi\)
0.855097 + 0.518468i \(0.173497\pi\)
\(458\) 0 0
\(459\) − 277.912i − 0.605474i
\(460\) 0 0
\(461\) 526.503 1.14209 0.571044 0.820919i \(-0.306539\pi\)
0.571044 + 0.820919i \(0.306539\pi\)
\(462\) 0 0
\(463\) − 754.257i − 1.62906i −0.580118 0.814532i \(-0.696994\pi\)
0.580118 0.814532i \(-0.303006\pi\)
\(464\) 0 0
\(465\) 54.9245 0.118117
\(466\) 0 0
\(467\) − 480.381i − 1.02865i −0.857594 0.514326i \(-0.828042\pi\)
0.857594 0.514326i \(-0.171958\pi\)
\(468\) 0 0
\(469\) 310.603 0.662266
\(470\) 0 0
\(471\) 14.9676i 0.0317784i
\(472\) 0 0
\(473\) −706.234 −1.49309
\(474\) 0 0
\(475\) − 25.6761i − 0.0540549i
\(476\) 0 0
\(477\) −128.798 −0.270017
\(478\) 0 0
\(479\) − 23.0908i − 0.0482062i −0.999709 0.0241031i \(-0.992327\pi\)
0.999709 0.0241031i \(-0.00767300\pi\)
\(480\) 0 0
\(481\) −288.729 −0.600267
\(482\) 0 0
\(483\) 45.5880i 0.0943850i
\(484\) 0 0
\(485\) 15.7651 0.0325053
\(486\) 0 0
\(487\) 517.867i 1.06338i 0.846939 + 0.531691i \(0.178443\pi\)
−0.846939 + 0.531691i \(0.821557\pi\)
\(488\) 0 0
\(489\) −70.2674 −0.143696
\(490\) 0 0
\(491\) − 385.683i − 0.785505i −0.919644 0.392753i \(-0.871523\pi\)
0.919644 0.392753i \(-0.128477\pi\)
\(492\) 0 0
\(493\) −23.9668 −0.0486142
\(494\) 0 0
\(495\) − 626.063i − 1.26477i
\(496\) 0 0
\(497\) 214.979 0.432552
\(498\) 0 0
\(499\) − 726.518i − 1.45595i −0.685605 0.727974i \(-0.740462\pi\)
0.685605 0.727974i \(-0.259538\pi\)
\(500\) 0 0
\(501\) −136.583 −0.272621
\(502\) 0 0
\(503\) − 253.383i − 0.503743i −0.967761 0.251872i \(-0.918954\pi\)
0.967761 0.251872i \(-0.0810461\pi\)
\(504\) 0 0
\(505\) −654.646 −1.29633
\(506\) 0 0
\(507\) 52.9362i 0.104411i
\(508\) 0 0
\(509\) −338.344 −0.664723 −0.332361 0.943152i \(-0.607845\pi\)
−0.332361 + 0.943152i \(0.607845\pi\)
\(510\) 0 0
\(511\) − 126.481i − 0.247517i
\(512\) 0 0
\(513\) 62.1776 0.121204
\(514\) 0 0
\(515\) 793.737i 1.54124i
\(516\) 0 0
\(517\) −605.121 −1.17045
\(518\) 0 0
\(519\) − 41.3658i − 0.0797030i
\(520\) 0 0
\(521\) −333.599 −0.640305 −0.320152 0.947366i \(-0.603734\pi\)
−0.320152 + 0.947366i \(0.603734\pi\)
\(522\) 0 0
\(523\) 38.7942i 0.0741763i 0.999312 + 0.0370882i \(0.0118082\pi\)
−0.999312 + 0.0370882i \(0.988192\pi\)
\(524\) 0 0
\(525\) −5.94699 −0.0113276
\(526\) 0 0
\(527\) − 612.709i − 1.16264i
\(528\) 0 0
\(529\) −436.017 −0.824229
\(530\) 0 0
\(531\) 50.4974i 0.0950987i
\(532\) 0 0
\(533\) 574.441 1.07775
\(534\) 0 0
\(535\) − 437.355i − 0.817487i
\(536\) 0 0
\(537\) −111.529 −0.207689
\(538\) 0 0
\(539\) − 110.157i − 0.204373i
\(540\) 0 0
\(541\) 543.111 1.00390 0.501951 0.864896i \(-0.332616\pi\)
0.501951 + 0.864896i \(0.332616\pi\)
\(542\) 0 0
\(543\) − 140.162i − 0.258126i
\(544\) 0 0
\(545\) −849.086 −1.55796
\(546\) 0 0
\(547\) 262.532i 0.479949i 0.970779 + 0.239974i \(0.0771391\pi\)
−0.970779 + 0.239974i \(0.922861\pi\)
\(548\) 0 0
\(549\) −457.230 −0.832841
\(550\) 0 0
\(551\) − 5.36212i − 0.00973161i
\(552\) 0 0
\(553\) −152.104 −0.275053
\(554\) 0 0
\(555\) − 85.4599i − 0.153982i
\(556\) 0 0
\(557\) −652.276 −1.17105 −0.585526 0.810654i \(-0.699112\pi\)
−0.585526 + 0.810654i \(0.699112\pi\)
\(558\) 0 0
\(559\) 384.913i 0.688574i
\(560\) 0 0
\(561\) 247.193 0.440630
\(562\) 0 0
\(563\) − 993.435i − 1.76454i −0.470746 0.882269i \(-0.656015\pi\)
0.470746 0.882269i \(-0.343985\pi\)
\(564\) 0 0
\(565\) 473.288 0.837678
\(566\) 0 0
\(567\) 192.579i 0.339645i
\(568\) 0 0
\(569\) −450.590 −0.791898 −0.395949 0.918272i \(-0.629584\pi\)
−0.395949 + 0.918272i \(0.629584\pi\)
\(570\) 0 0
\(571\) 373.493i 0.654103i 0.945007 + 0.327051i \(0.106055\pi\)
−0.945007 + 0.327051i \(0.893945\pi\)
\(572\) 0 0
\(573\) 184.335 0.321701
\(574\) 0 0
\(575\) − 125.887i − 0.218934i
\(576\) 0 0
\(577\) 370.687 0.642439 0.321219 0.947005i \(-0.395907\pi\)
0.321219 + 0.947005i \(0.395907\pi\)
\(578\) 0 0
\(579\) 29.5205i 0.0509853i
\(580\) 0 0
\(581\) 272.128 0.468378
\(582\) 0 0
\(583\) − 233.177i − 0.399961i
\(584\) 0 0
\(585\) −341.218 −0.583278
\(586\) 0 0
\(587\) 182.976i 0.311714i 0.987780 + 0.155857i \(0.0498139\pi\)
−0.987780 + 0.155857i \(0.950186\pi\)
\(588\) 0 0
\(589\) 137.082 0.232737
\(590\) 0 0
\(591\) − 153.226i − 0.259266i
\(592\) 0 0
\(593\) 333.716 0.562759 0.281379 0.959597i \(-0.409208\pi\)
0.281379 + 0.959597i \(0.409208\pi\)
\(594\) 0 0
\(595\) − 342.930i − 0.576352i
\(596\) 0 0
\(597\) −32.4533 −0.0543607
\(598\) 0 0
\(599\) 219.331i 0.366163i 0.983098 + 0.183081i \(0.0586072\pi\)
−0.983098 + 0.183081i \(0.941393\pi\)
\(600\) 0 0
\(601\) 268.193 0.446245 0.223122 0.974790i \(-0.428375\pi\)
0.223122 + 0.974790i \(0.428375\pi\)
\(602\) 0 0
\(603\) − 1020.45i − 1.69229i
\(604\) 0 0
\(605\) 579.631 0.958068
\(606\) 0 0
\(607\) 585.050i 0.963839i 0.876216 + 0.481919i \(0.160060\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(608\) 0 0
\(609\) −1.24195 −0.00203933
\(610\) 0 0
\(611\) 329.804i 0.539778i
\(612\) 0 0
\(613\) 29.3420 0.0478662 0.0239331 0.999714i \(-0.492381\pi\)
0.0239331 + 0.999714i \(0.492381\pi\)
\(614\) 0 0
\(615\) 170.027i 0.276467i
\(616\) 0 0
\(617\) −413.079 −0.669497 −0.334748 0.942308i \(-0.608651\pi\)
−0.334748 + 0.942308i \(0.608651\pi\)
\(618\) 0 0
\(619\) − 225.596i − 0.364453i −0.983257 0.182226i \(-0.941670\pi\)
0.983257 0.182226i \(-0.0583304\pi\)
\(620\) 0 0
\(621\) 304.850 0.490902
\(622\) 0 0
\(623\) − 236.254i − 0.379219i
\(624\) 0 0
\(625\) −507.267 −0.811628
\(626\) 0 0
\(627\) 55.3047i 0.0882053i
\(628\) 0 0
\(629\) −953.346 −1.51565
\(630\) 0 0
\(631\) 914.619i 1.44948i 0.689025 + 0.724738i \(0.258039\pi\)
−0.689025 + 0.724738i \(0.741961\pi\)
\(632\) 0 0
\(633\) −81.1132 −0.128141
\(634\) 0 0
\(635\) − 677.239i − 1.06652i
\(636\) 0 0
\(637\) −60.0380 −0.0942511
\(638\) 0 0
\(639\) − 706.290i − 1.10530i
\(640\) 0 0
\(641\) −1025.61 −1.60002 −0.800009 0.599987i \(-0.795172\pi\)
−0.800009 + 0.599987i \(0.795172\pi\)
\(642\) 0 0
\(643\) 864.377i 1.34429i 0.740421 + 0.672144i \(0.234626\pi\)
−0.740421 + 0.672144i \(0.765374\pi\)
\(644\) 0 0
\(645\) −113.929 −0.176634
\(646\) 0 0
\(647\) 689.240i 1.06529i 0.846340 + 0.532643i \(0.178801\pi\)
−0.846340 + 0.532643i \(0.821199\pi\)
\(648\) 0 0
\(649\) −91.4211 −0.140865
\(650\) 0 0
\(651\) − 31.7503i − 0.0487716i
\(652\) 0 0
\(653\) −565.975 −0.866730 −0.433365 0.901218i \(-0.642674\pi\)
−0.433365 + 0.901218i \(0.642674\pi\)
\(654\) 0 0
\(655\) − 1187.29i − 1.81266i
\(656\) 0 0
\(657\) −415.541 −0.632483
\(658\) 0 0
\(659\) − 289.064i − 0.438640i −0.975653 0.219320i \(-0.929616\pi\)
0.975653 0.219320i \(-0.0703838\pi\)
\(660\) 0 0
\(661\) 283.787 0.429329 0.214665 0.976688i \(-0.431134\pi\)
0.214665 + 0.976688i \(0.431134\pi\)
\(662\) 0 0
\(663\) − 134.726i − 0.203206i
\(664\) 0 0
\(665\) 76.7239 0.115374
\(666\) 0 0
\(667\) − 26.2899i − 0.0394151i
\(668\) 0 0
\(669\) −234.231 −0.350121
\(670\) 0 0
\(671\) − 827.774i − 1.23364i
\(672\) 0 0
\(673\) 987.489 1.46729 0.733647 0.679531i \(-0.237817\pi\)
0.733647 + 0.679531i \(0.237817\pi\)
\(674\) 0 0
\(675\) 39.7680i 0.0589155i
\(676\) 0 0
\(677\) 787.301 1.16293 0.581463 0.813573i \(-0.302481\pi\)
0.581463 + 0.813573i \(0.302481\pi\)
\(678\) 0 0
\(679\) − 9.11336i − 0.0134217i
\(680\) 0 0
\(681\) −150.373 −0.220811
\(682\) 0 0
\(683\) 193.168i 0.282823i 0.989951 + 0.141411i \(0.0451640\pi\)
−0.989951 + 0.141411i \(0.954836\pi\)
\(684\) 0 0
\(685\) −14.0865 −0.0205642
\(686\) 0 0
\(687\) 222.185i 0.323413i
\(688\) 0 0
\(689\) −127.087 −0.184451
\(690\) 0 0
\(691\) 967.147i 1.39963i 0.714322 + 0.699817i \(0.246735\pi\)
−0.714322 + 0.699817i \(0.753265\pi\)
\(692\) 0 0
\(693\) −361.909 −0.522236
\(694\) 0 0
\(695\) 413.308i 0.594688i
\(696\) 0 0
\(697\) 1896.73 2.72128
\(698\) 0 0
\(699\) 142.889i 0.204419i
\(700\) 0 0
\(701\) −246.322 −0.351386 −0.175693 0.984445i \(-0.556217\pi\)
−0.175693 + 0.984445i \(0.556217\pi\)
\(702\) 0 0
\(703\) − 213.293i − 0.303404i
\(704\) 0 0
\(705\) −97.6178 −0.138465
\(706\) 0 0
\(707\) 378.432i 0.535265i
\(708\) 0 0
\(709\) 1191.27 1.68021 0.840103 0.542428i \(-0.182495\pi\)
0.840103 + 0.542428i \(0.182495\pi\)
\(710\) 0 0
\(711\) 499.723i 0.702845i
\(712\) 0 0
\(713\) 672.098 0.942635
\(714\) 0 0
\(715\) − 617.744i − 0.863978i
\(716\) 0 0
\(717\) −84.6703 −0.118090
\(718\) 0 0
\(719\) 434.163i 0.603843i 0.953333 + 0.301922i \(0.0976281\pi\)
−0.953333 + 0.301922i \(0.902372\pi\)
\(720\) 0 0
\(721\) 458.838 0.636390
\(722\) 0 0
\(723\) 26.1370i 0.0361507i
\(724\) 0 0
\(725\) 3.42954 0.00473040
\(726\) 0 0
\(727\) 931.815i 1.28173i 0.767655 + 0.640863i \(0.221423\pi\)
−0.767655 + 0.640863i \(0.778577\pi\)
\(728\) 0 0
\(729\) 583.709 0.800698
\(730\) 0 0
\(731\) 1270.94i 1.73863i
\(732\) 0 0
\(733\) −1426.30 −1.94584 −0.972918 0.231150i \(-0.925751\pi\)
−0.972918 + 0.231150i \(0.925751\pi\)
\(734\) 0 0
\(735\) − 17.7705i − 0.0241775i
\(736\) 0 0
\(737\) 1847.44 2.50670
\(738\) 0 0
\(739\) 216.382i 0.292804i 0.989225 + 0.146402i \(0.0467693\pi\)
−0.989225 + 0.146402i \(0.953231\pi\)
\(740\) 0 0
\(741\) 30.1423 0.0406779
\(742\) 0 0
\(743\) 81.5074i 0.109700i 0.998495 + 0.0548502i \(0.0174681\pi\)
−0.998495 + 0.0548502i \(0.982532\pi\)
\(744\) 0 0
\(745\) −189.375 −0.254194
\(746\) 0 0
\(747\) − 894.048i − 1.19685i
\(748\) 0 0
\(749\) −252.823 −0.337547
\(750\) 0 0
\(751\) − 142.603i − 0.189884i −0.995483 0.0949422i \(-0.969733\pi\)
0.995483 0.0949422i \(-0.0302666\pi\)
\(752\) 0 0
\(753\) 130.930 0.173878
\(754\) 0 0
\(755\) − 190.023i − 0.251686i
\(756\) 0 0
\(757\) −372.503 −0.492078 −0.246039 0.969260i \(-0.579129\pi\)
−0.246039 + 0.969260i \(0.579129\pi\)
\(758\) 0 0
\(759\) 271.153i 0.357251i
\(760\) 0 0
\(761\) 663.881 0.872380 0.436190 0.899855i \(-0.356328\pi\)
0.436190 + 0.899855i \(0.356328\pi\)
\(762\) 0 0
\(763\) 490.833i 0.643294i
\(764\) 0 0
\(765\) −1126.66 −1.47276
\(766\) 0 0
\(767\) 49.8265i 0.0649628i
\(768\) 0 0
\(769\) 732.653 0.952734 0.476367 0.879247i \(-0.341953\pi\)
0.476367 + 0.879247i \(0.341953\pi\)
\(770\) 0 0
\(771\) 15.2824i 0.0198216i
\(772\) 0 0
\(773\) −928.973 −1.20178 −0.600888 0.799333i \(-0.705186\pi\)
−0.600888 + 0.799333i \(0.705186\pi\)
\(774\) 0 0
\(775\) 87.6758i 0.113130i
\(776\) 0 0
\(777\) −49.4020 −0.0635804
\(778\) 0 0
\(779\) 424.358i 0.544747i
\(780\) 0 0
\(781\) 1278.67 1.63723
\(782\) 0 0
\(783\) 8.30502i 0.0106067i
\(784\) 0 0
\(785\) 123.506 0.157332
\(786\) 0 0
\(787\) 470.202i 0.597461i 0.954338 + 0.298730i \(0.0965631\pi\)
−0.954338 + 0.298730i \(0.903437\pi\)
\(788\) 0 0
\(789\) 48.9592 0.0620522
\(790\) 0 0
\(791\) − 273.595i − 0.345884i
\(792\) 0 0
\(793\) −451.155 −0.568922
\(794\) 0 0
\(795\) − 37.6160i − 0.0473157i
\(796\) 0 0
\(797\) 832.694 1.04479 0.522393 0.852705i \(-0.325040\pi\)
0.522393 + 0.852705i \(0.325040\pi\)
\(798\) 0 0
\(799\) 1088.97i 1.36292i
\(800\) 0 0
\(801\) −776.187 −0.969022
\(802\) 0 0
\(803\) − 752.300i − 0.936862i
\(804\) 0 0
\(805\) 376.169 0.467291
\(806\) 0 0
\(807\) − 12.1547i − 0.0150616i
\(808\) 0 0
\(809\) 976.293 1.20679 0.603395 0.797442i \(-0.293814\pi\)
0.603395 + 0.797442i \(0.293814\pi\)
\(810\) 0 0
\(811\) − 564.470i − 0.696018i −0.937491 0.348009i \(-0.886858\pi\)
0.937491 0.348009i \(-0.113142\pi\)
\(812\) 0 0
\(813\) −237.896 −0.292615
\(814\) 0 0
\(815\) 579.812i 0.711426i
\(816\) 0 0
\(817\) −284.347 −0.348038
\(818\) 0 0
\(819\) 197.249i 0.240841i
\(820\) 0 0
\(821\) 779.033 0.948883 0.474441 0.880287i \(-0.342650\pi\)
0.474441 + 0.880287i \(0.342650\pi\)
\(822\) 0 0
\(823\) − 363.394i − 0.441548i −0.975325 0.220774i \(-0.929142\pi\)
0.975325 0.220774i \(-0.0708582\pi\)
\(824\) 0 0
\(825\) −35.3722 −0.0428754
\(826\) 0 0
\(827\) 620.477i 0.750275i 0.926969 + 0.375137i \(0.122404\pi\)
−0.926969 + 0.375137i \(0.877596\pi\)
\(828\) 0 0
\(829\) 737.842 0.890038 0.445019 0.895521i \(-0.353197\pi\)
0.445019 + 0.895521i \(0.353197\pi\)
\(830\) 0 0
\(831\) 253.525i 0.305085i
\(832\) 0 0
\(833\) −198.238 −0.237981
\(834\) 0 0
\(835\) 1127.02i 1.34972i
\(836\) 0 0
\(837\) −212.317 −0.253664
\(838\) 0 0
\(839\) − 897.093i − 1.06924i −0.845092 0.534620i \(-0.820455\pi\)
0.845092 0.534620i \(-0.179545\pi\)
\(840\) 0 0
\(841\) −840.284 −0.999148
\(842\) 0 0
\(843\) − 52.9726i − 0.0628382i
\(844\) 0 0
\(845\) 436.804 0.516928
\(846\) 0 0
\(847\) − 335.069i − 0.395595i
\(848\) 0 0
\(849\) −73.1074 −0.0861100
\(850\) 0 0
\(851\) − 1045.75i − 1.22885i
\(852\) 0 0
\(853\) −1470.98 −1.72448 −0.862239 0.506501i \(-0.830939\pi\)
−0.862239 + 0.506501i \(0.830939\pi\)
\(854\) 0 0
\(855\) − 252.068i − 0.294817i
\(856\) 0 0
\(857\) 851.862 0.994005 0.497002 0.867749i \(-0.334434\pi\)
0.497002 + 0.867749i \(0.334434\pi\)
\(858\) 0 0
\(859\) − 140.400i − 0.163446i −0.996655 0.0817229i \(-0.973958\pi\)
0.996655 0.0817229i \(-0.0260422\pi\)
\(860\) 0 0
\(861\) 98.2879 0.114156
\(862\) 0 0
\(863\) − 32.1489i − 0.0372525i −0.999827 0.0186263i \(-0.994071\pi\)
0.999827 0.0186263i \(-0.00592927\pi\)
\(864\) 0 0
\(865\) −341.330 −0.394602
\(866\) 0 0
\(867\) − 284.549i − 0.328199i
\(868\) 0 0
\(869\) −904.704 −1.04109
\(870\) 0 0
\(871\) − 1006.89i − 1.15602i
\(872\) 0 0
\(873\) −29.9410 −0.0342967
\(874\) 0 0
\(875\) 351.802i 0.402059i
\(876\) 0 0
\(877\) −1039.93 −1.18578 −0.592888 0.805285i \(-0.702012\pi\)
−0.592888 + 0.805285i \(0.702012\pi\)
\(878\) 0 0
\(879\) 180.369i 0.205198i
\(880\) 0 0
\(881\) −416.344 −0.472581 −0.236291 0.971682i \(-0.575932\pi\)
−0.236291 + 0.971682i \(0.575932\pi\)
\(882\) 0 0
\(883\) − 231.281i − 0.261926i −0.991387 0.130963i \(-0.958193\pi\)
0.991387 0.130963i \(-0.0418069\pi\)
\(884\) 0 0
\(885\) −14.7480 −0.0166644
\(886\) 0 0
\(887\) − 6.35223i − 0.00716148i −0.999994 0.00358074i \(-0.998860\pi\)
0.999994 0.00358074i \(-0.00113979\pi\)
\(888\) 0 0
\(889\) −391.493 −0.440374
\(890\) 0 0
\(891\) 1145.44i 1.28557i
\(892\) 0 0
\(893\) −243.637 −0.272830
\(894\) 0 0
\(895\) 920.280i 1.02825i
\(896\) 0 0
\(897\) 147.785 0.164754
\(898\) 0 0
\(899\) 18.3099i 0.0203670i
\(900\) 0 0
\(901\) −419.624 −0.465732
\(902\) 0 0
\(903\) 65.8593i 0.0729339i
\(904\) 0 0
\(905\) −1156.55 −1.27796
\(906\) 0 0
\(907\) − 201.716i − 0.222399i −0.993798 0.111200i \(-0.964531\pi\)
0.993798 0.111200i \(-0.0354693\pi\)
\(908\) 0 0
\(909\) 1243.30 1.36777
\(910\) 0 0
\(911\) 484.675i 0.532025i 0.963970 + 0.266013i \(0.0857063\pi\)
−0.963970 + 0.266013i \(0.914294\pi\)
\(912\) 0 0
\(913\) 1618.59 1.77283
\(914\) 0 0
\(915\) − 133.536i − 0.145941i
\(916\) 0 0
\(917\) −686.340 −0.748463
\(918\) 0 0
\(919\) − 650.067i − 0.707363i −0.935366 0.353681i \(-0.884930\pi\)
0.935366 0.353681i \(-0.115070\pi\)
\(920\) 0 0
\(921\) −19.0616 −0.0206967
\(922\) 0 0
\(923\) − 696.906i − 0.755044i
\(924\) 0 0
\(925\) 136.419 0.147480
\(926\) 0 0
\(927\) − 1507.46i − 1.62617i
\(928\) 0 0
\(929\) 522.394 0.562318 0.281159 0.959661i \(-0.409281\pi\)
0.281159 + 0.959661i \(0.409281\pi\)
\(930\) 0 0
\(931\) − 44.3520i − 0.0476391i
\(932\) 0 0
\(933\) 108.266 0.116040
\(934\) 0 0
\(935\) − 2039.72i − 2.18151i
\(936\) 0 0
\(937\) 1579.17 1.68535 0.842676 0.538422i \(-0.180979\pi\)
0.842676 + 0.538422i \(0.180979\pi\)
\(938\) 0 0
\(939\) − 11.0372i − 0.0117542i
\(940\) 0 0
\(941\) −1080.68 −1.14844 −0.574218 0.818703i \(-0.694694\pi\)
−0.574218 + 0.818703i \(0.694694\pi\)
\(942\) 0 0
\(943\) 2080.58i 2.20634i
\(944\) 0 0
\(945\) −118.832 −0.125749
\(946\) 0 0
\(947\) − 538.713i − 0.568863i −0.958696 0.284431i \(-0.908195\pi\)
0.958696 0.284431i \(-0.0918048\pi\)
\(948\) 0 0
\(949\) −410.020 −0.432055
\(950\) 0 0
\(951\) − 93.5570i − 0.0983775i
\(952\) 0 0
\(953\) −1099.01 −1.15322 −0.576608 0.817021i \(-0.695624\pi\)
−0.576608 + 0.817021i \(0.695624\pi\)
\(954\) 0 0
\(955\) − 1521.04i − 1.59271i
\(956\) 0 0
\(957\) −7.38702 −0.00771893
\(958\) 0 0
\(959\) 8.14300i 0.00849114i
\(960\) 0 0
\(961\) 492.908 0.512912
\(962\) 0 0
\(963\) 830.624i 0.862538i
\(964\) 0 0
\(965\) 243.588 0.252423
\(966\) 0 0
\(967\) 847.122i 0.876031i 0.898967 + 0.438016i \(0.144319\pi\)
−0.898967 + 0.438016i \(0.855681\pi\)
\(968\) 0 0
\(969\) 99.5262 0.102710
\(970\) 0 0
\(971\) − 1059.01i − 1.09064i −0.838227 0.545322i \(-0.816408\pi\)
0.838227 0.545322i \(-0.183592\pi\)
\(972\) 0 0
\(973\) 238.922 0.245552
\(974\) 0 0
\(975\) 19.2786i 0.0197729i
\(976\) 0 0
\(977\) −1223.46 −1.25226 −0.626131 0.779718i \(-0.715362\pi\)
−0.626131 + 0.779718i \(0.715362\pi\)
\(978\) 0 0
\(979\) − 1405.22i − 1.43536i
\(980\) 0 0
\(981\) 1612.58 1.64381
\(982\) 0 0
\(983\) 256.802i 0.261244i 0.991432 + 0.130622i \(0.0416974\pi\)
−0.991432 + 0.130622i \(0.958303\pi\)
\(984\) 0 0
\(985\) −1264.35 −1.28360
\(986\) 0 0
\(987\) 56.4301i 0.0571734i
\(988\) 0 0
\(989\) −1394.13 −1.40963
\(990\) 0 0
\(991\) − 988.781i − 0.997761i −0.866671 0.498881i \(-0.833745\pi\)
0.866671 0.498881i \(-0.166255\pi\)
\(992\) 0 0
\(993\) −334.799 −0.337159
\(994\) 0 0
\(995\) 267.789i 0.269134i
\(996\) 0 0
\(997\) −581.918 −0.583669 −0.291835 0.956469i \(-0.594266\pi\)
−0.291835 + 0.956469i \(0.594266\pi\)
\(998\) 0 0
\(999\) 330.355i 0.330686i
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 224.3.d.b.127.4 8
3.2 odd 2 2016.3.m.c.127.6 8
4.3 odd 2 inner 224.3.d.b.127.5 yes 8
7.6 odd 2 1568.3.d.n.1471.5 8
8.3 odd 2 448.3.d.e.127.4 8
8.5 even 2 448.3.d.e.127.5 8
12.11 even 2 2016.3.m.c.127.5 8
16.3 odd 4 1792.3.g.f.127.4 8
16.5 even 4 1792.3.g.f.127.3 8
16.11 odd 4 1792.3.g.d.127.5 8
16.13 even 4 1792.3.g.d.127.6 8
28.27 even 2 1568.3.d.n.1471.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.d.b.127.4 8 1.1 even 1 trivial
224.3.d.b.127.5 yes 8 4.3 odd 2 inner
448.3.d.e.127.4 8 8.3 odd 2
448.3.d.e.127.5 8 8.5 even 2
1568.3.d.n.1471.4 8 28.27 even 2
1568.3.d.n.1471.5 8 7.6 odd 2
1792.3.g.d.127.5 8 16.11 odd 4
1792.3.g.d.127.6 8 16.13 even 4
1792.3.g.f.127.3 8 16.5 even 4
1792.3.g.f.127.4 8 16.3 odd 4
2016.3.m.c.127.5 8 12.11 even 2
2016.3.m.c.127.6 8 3.2 odd 2