Properties

Label 2240.2.a.be.1.1
Level $2240$
Weight $2$
Character 2240.1
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2240.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.56155 q^{9} +2.56155 q^{11} +5.68466 q^{13} +2.56155 q^{15} +3.43845 q^{17} -1.12311 q^{19} -2.56155 q^{21} -5.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -4.56155 q^{29} -10.2462 q^{31} -6.56155 q^{33} -1.00000 q^{35} -8.24621 q^{37} -14.5616 q^{39} +7.12311 q^{41} -1.12311 q^{43} -3.56155 q^{45} +6.56155 q^{47} +1.00000 q^{49} -8.80776 q^{51} +4.87689 q^{53} -2.56155 q^{55} +2.87689 q^{57} +4.00000 q^{59} +15.1231 q^{61} +3.56155 q^{63} -5.68466 q^{65} +14.2462 q^{67} +13.1231 q^{69} +12.2462 q^{73} -2.56155 q^{75} +2.56155 q^{77} -11.6847 q^{79} -7.00000 q^{81} -12.0000 q^{83} -3.43845 q^{85} +11.6847 q^{87} -3.12311 q^{89} +5.68466 q^{91} +26.2462 q^{93} +1.12311 q^{95} +13.6847 q^{97} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + q^{11} - q^{13} + q^{15} + 11q^{17} + 6q^{19} - q^{21} - 2q^{23} + 2q^{25} - 7q^{27} - 5q^{29} - 4q^{31} - 9q^{33} - 2q^{35} - 25q^{39} + 6q^{41} + 6q^{43} - 3q^{45} + 9q^{47} + 2q^{49} + 3q^{51} + 18q^{53} - q^{55} + 14q^{57} + 8q^{59} + 22q^{61} + 3q^{63} + q^{65} + 12q^{67} + 18q^{69} + 8q^{73} - q^{75} + q^{77} - 11q^{79} - 14q^{81} - 24q^{83} - 11q^{85} + 11q^{87} + 2q^{89} - q^{91} + 36q^{93} - 6q^{95} + 15q^{97} + 10q^{99} + O(q^{100}) \)

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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 2.56155 0.772337 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) 0 0
\(13\) 5.68466 1.57664 0.788320 0.615265i \(-0.210951\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 0 0
\(17\) 3.43845 0.833946 0.416973 0.908919i \(-0.363091\pi\)
0.416973 + 0.908919i \(0.363091\pi\)
\(18\) 0 0
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 0 0
\(21\) −2.56155 −0.558977
\(22\) 0 0
\(23\) −5.12311 −1.06824 −0.534121 0.845408i \(-0.679357\pi\)
−0.534121 + 0.845408i \(0.679357\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −4.56155 −0.847059 −0.423530 0.905882i \(-0.639209\pi\)
−0.423530 + 0.905882i \(0.639209\pi\)
\(30\) 0 0
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) 0 0
\(33\) −6.56155 −1.14222
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −8.24621 −1.35567 −0.677834 0.735215i \(-0.737081\pi\)
−0.677834 + 0.735215i \(0.737081\pi\)
\(38\) 0 0
\(39\) −14.5616 −2.33171
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) 0 0
\(43\) −1.12311 −0.171272 −0.0856360 0.996326i \(-0.527292\pi\)
−0.0856360 + 0.996326i \(0.527292\pi\)
\(44\) 0 0
\(45\) −3.56155 −0.530925
\(46\) 0 0
\(47\) 6.56155 0.957101 0.478550 0.878060i \(-0.341162\pi\)
0.478550 + 0.878060i \(0.341162\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −8.80776 −1.23333
\(52\) 0 0
\(53\) 4.87689 0.669893 0.334946 0.942237i \(-0.391282\pi\)
0.334946 + 0.942237i \(0.391282\pi\)
\(54\) 0 0
\(55\) −2.56155 −0.345400
\(56\) 0 0
\(57\) 2.87689 0.381054
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 15.1231 1.93632 0.968158 0.250341i \(-0.0805429\pi\)
0.968158 + 0.250341i \(0.0805429\pi\)
\(62\) 0 0
\(63\) 3.56155 0.448713
\(64\) 0 0
\(65\) −5.68466 −0.705095
\(66\) 0 0
\(67\) 14.2462 1.74045 0.870226 0.492653i \(-0.163973\pi\)
0.870226 + 0.492653i \(0.163973\pi\)
\(68\) 0 0
\(69\) 13.1231 1.57984
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) 2.56155 0.291916
\(78\) 0 0
\(79\) −11.6847 −1.31463 −0.657313 0.753617i \(-0.728307\pi\)
−0.657313 + 0.753617i \(0.728307\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −3.43845 −0.372952
\(86\) 0 0
\(87\) 11.6847 1.25273
\(88\) 0 0
\(89\) −3.12311 −0.331049 −0.165524 0.986206i \(-0.552932\pi\)
−0.165524 + 0.986206i \(0.552932\pi\)
\(90\) 0 0
\(91\) 5.68466 0.595914
\(92\) 0 0
\(93\) 26.2462 2.72161
\(94\) 0 0
\(95\) 1.12311 0.115228
\(96\) 0 0
\(97\) 13.6847 1.38947 0.694733 0.719267i \(-0.255522\pi\)
0.694733 + 0.719267i \(0.255522\pi\)
\(98\) 0 0
\(99\) 9.12311 0.916907
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.80776 0.867855 0.433927 0.900948i \(-0.357127\pi\)
0.433927 + 0.900948i \(0.357127\pi\)
\(104\) 0 0
\(105\) 2.56155 0.249982
\(106\) 0 0
\(107\) −6.87689 −0.664814 −0.332407 0.943136i \(-0.607861\pi\)
−0.332407 + 0.943136i \(0.607861\pi\)
\(108\) 0 0
\(109\) 8.56155 0.820048 0.410024 0.912075i \(-0.365520\pi\)
0.410024 + 0.912075i \(0.365520\pi\)
\(110\) 0 0
\(111\) 21.1231 2.00492
\(112\) 0 0
\(113\) −8.24621 −0.775738 −0.387869 0.921714i \(-0.626789\pi\)
−0.387869 + 0.921714i \(0.626789\pi\)
\(114\) 0 0
\(115\) 5.12311 0.477732
\(116\) 0 0
\(117\) 20.2462 1.87176
\(118\) 0 0
\(119\) 3.43845 0.315202
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 0 0
\(123\) −18.2462 −1.64521
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.2462 0.909204 0.454602 0.890695i \(-0.349781\pi\)
0.454602 + 0.890695i \(0.349781\pi\)
\(128\) 0 0
\(129\) 2.87689 0.253296
\(130\) 0 0
\(131\) −6.87689 −0.600837 −0.300419 0.953807i \(-0.597126\pi\)
−0.300419 + 0.953807i \(0.597126\pi\)
\(132\) 0 0
\(133\) −1.12311 −0.0973856
\(134\) 0 0
\(135\) 1.43845 0.123802
\(136\) 0 0
\(137\) 17.3693 1.48396 0.741980 0.670422i \(-0.233887\pi\)
0.741980 + 0.670422i \(0.233887\pi\)
\(138\) 0 0
\(139\) −3.36932 −0.285782 −0.142891 0.989738i \(-0.545640\pi\)
−0.142891 + 0.989738i \(0.545640\pi\)
\(140\) 0 0
\(141\) −16.8078 −1.41547
\(142\) 0 0
\(143\) 14.5616 1.21770
\(144\) 0 0
\(145\) 4.56155 0.378816
\(146\) 0 0
\(147\) −2.56155 −0.211273
\(148\) 0 0
\(149\) −16.2462 −1.33094 −0.665471 0.746424i \(-0.731769\pi\)
−0.665471 + 0.746424i \(0.731769\pi\)
\(150\) 0 0
\(151\) 0.807764 0.0657349 0.0328675 0.999460i \(-0.489536\pi\)
0.0328675 + 0.999460i \(0.489536\pi\)
\(152\) 0 0
\(153\) 12.2462 0.990048
\(154\) 0 0
\(155\) 10.2462 0.822995
\(156\) 0 0
\(157\) 12.2462 0.977354 0.488677 0.872465i \(-0.337480\pi\)
0.488677 + 0.872465i \(0.337480\pi\)
\(158\) 0 0
\(159\) −12.4924 −0.990714
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) 21.6155 1.69306 0.846529 0.532342i \(-0.178688\pi\)
0.846529 + 0.532342i \(0.178688\pi\)
\(164\) 0 0
\(165\) 6.56155 0.510816
\(166\) 0 0
\(167\) 24.8078 1.91968 0.959841 0.280544i \(-0.0905148\pi\)
0.959841 + 0.280544i \(0.0905148\pi\)
\(168\) 0 0
\(169\) 19.3153 1.48580
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −22.1771 −1.68609 −0.843046 0.537841i \(-0.819240\pi\)
−0.843046 + 0.537841i \(0.819240\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −10.2462 −0.770152
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −3.12311 −0.232139 −0.116069 0.993241i \(-0.537029\pi\)
−0.116069 + 0.993241i \(0.537029\pi\)
\(182\) 0 0
\(183\) −38.7386 −2.86364
\(184\) 0 0
\(185\) 8.24621 0.606274
\(186\) 0 0
\(187\) 8.80776 0.644087
\(188\) 0 0
\(189\) −1.43845 −0.104632
\(190\) 0 0
\(191\) 11.6847 0.845472 0.422736 0.906253i \(-0.361070\pi\)
0.422736 + 0.906253i \(0.361070\pi\)
\(192\) 0 0
\(193\) 4.87689 0.351047 0.175523 0.984475i \(-0.443838\pi\)
0.175523 + 0.984475i \(0.443838\pi\)
\(194\) 0 0
\(195\) 14.5616 1.04277
\(196\) 0 0
\(197\) 4.87689 0.347464 0.173732 0.984793i \(-0.444417\pi\)
0.173732 + 0.984793i \(0.444417\pi\)
\(198\) 0 0
\(199\) 2.24621 0.159230 0.0796148 0.996826i \(-0.474631\pi\)
0.0796148 + 0.996826i \(0.474631\pi\)
\(200\) 0 0
\(201\) −36.4924 −2.57398
\(202\) 0 0
\(203\) −4.56155 −0.320158
\(204\) 0 0
\(205\) −7.12311 −0.497499
\(206\) 0 0
\(207\) −18.2462 −1.26820
\(208\) 0 0
\(209\) −2.87689 −0.198999
\(210\) 0 0
\(211\) 13.4384 0.925141 0.462570 0.886583i \(-0.346927\pi\)
0.462570 + 0.886583i \(0.346927\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.12311 0.0765952
\(216\) 0 0
\(217\) −10.2462 −0.695558
\(218\) 0 0
\(219\) −31.3693 −2.11974
\(220\) 0 0
\(221\) 19.5464 1.31483
\(222\) 0 0
\(223\) 12.3153 0.824696 0.412348 0.911026i \(-0.364709\pi\)
0.412348 + 0.911026i \(0.364709\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) 23.6847 1.57201 0.786003 0.618223i \(-0.212147\pi\)
0.786003 + 0.618223i \(0.212147\pi\)
\(228\) 0 0
\(229\) −13.3693 −0.883469 −0.441735 0.897146i \(-0.645637\pi\)
−0.441735 + 0.897146i \(0.645637\pi\)
\(230\) 0 0
\(231\) −6.56155 −0.431718
\(232\) 0 0
\(233\) −19.1231 −1.25280 −0.626398 0.779503i \(-0.715472\pi\)
−0.626398 + 0.779503i \(0.715472\pi\)
\(234\) 0 0
\(235\) −6.56155 −0.428029
\(236\) 0 0
\(237\) 29.9309 1.94422
\(238\) 0 0
\(239\) 19.0540 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −6.38447 −0.406234
\(248\) 0 0
\(249\) 30.7386 1.94798
\(250\) 0 0
\(251\) 11.3693 0.717625 0.358812 0.933410i \(-0.383182\pi\)
0.358812 + 0.933410i \(0.383182\pi\)
\(252\) 0 0
\(253\) −13.1231 −0.825043
\(254\) 0 0
\(255\) 8.80776 0.551564
\(256\) 0 0
\(257\) 14.4924 0.904012 0.452006 0.892015i \(-0.350708\pi\)
0.452006 + 0.892015i \(0.350708\pi\)
\(258\) 0 0
\(259\) −8.24621 −0.512395
\(260\) 0 0
\(261\) −16.2462 −1.00562
\(262\) 0 0
\(263\) −15.3693 −0.947713 −0.473856 0.880602i \(-0.657138\pi\)
−0.473856 + 0.880602i \(0.657138\pi\)
\(264\) 0 0
\(265\) −4.87689 −0.299585
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) 0 0
\(269\) 12.2462 0.746665 0.373332 0.927698i \(-0.378215\pi\)
0.373332 + 0.927698i \(0.378215\pi\)
\(270\) 0 0
\(271\) 10.2462 0.622413 0.311207 0.950342i \(-0.399267\pi\)
0.311207 + 0.950342i \(0.399267\pi\)
\(272\) 0 0
\(273\) −14.5616 −0.881305
\(274\) 0 0
\(275\) 2.56155 0.154467
\(276\) 0 0
\(277\) −8.24621 −0.495467 −0.247733 0.968828i \(-0.579686\pi\)
−0.247733 + 0.968828i \(0.579686\pi\)
\(278\) 0 0
\(279\) −36.4924 −2.18474
\(280\) 0 0
\(281\) −15.4384 −0.920981 −0.460490 0.887665i \(-0.652326\pi\)
−0.460490 + 0.887665i \(0.652326\pi\)
\(282\) 0 0
\(283\) −7.68466 −0.456806 −0.228403 0.973567i \(-0.573350\pi\)
−0.228403 + 0.973567i \(0.573350\pi\)
\(284\) 0 0
\(285\) −2.87689 −0.170413
\(286\) 0 0
\(287\) 7.12311 0.420464
\(288\) 0 0
\(289\) −5.17708 −0.304534
\(290\) 0 0
\(291\) −35.0540 −2.05490
\(292\) 0 0
\(293\) 5.05398 0.295256 0.147628 0.989043i \(-0.452836\pi\)
0.147628 + 0.989043i \(0.452836\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) −3.68466 −0.213806
\(298\) 0 0
\(299\) −29.1231 −1.68423
\(300\) 0 0
\(301\) −1.12311 −0.0647347
\(302\) 0 0
\(303\) 15.3693 0.882944
\(304\) 0 0
\(305\) −15.1231 −0.865947
\(306\) 0 0
\(307\) 16.3153 0.931166 0.465583 0.885004i \(-0.345845\pi\)
0.465583 + 0.885004i \(0.345845\pi\)
\(308\) 0 0
\(309\) −22.5616 −1.28348
\(310\) 0 0
\(311\) −21.1231 −1.19778 −0.598891 0.800831i \(-0.704392\pi\)
−0.598891 + 0.800831i \(0.704392\pi\)
\(312\) 0 0
\(313\) 14.3153 0.809151 0.404575 0.914505i \(-0.367419\pi\)
0.404575 + 0.914505i \(0.367419\pi\)
\(314\) 0 0
\(315\) −3.56155 −0.200671
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −11.6847 −0.654215
\(320\) 0 0
\(321\) 17.6155 0.983203
\(322\) 0 0
\(323\) −3.86174 −0.214873
\(324\) 0 0
\(325\) 5.68466 0.315328
\(326\) 0 0
\(327\) −21.9309 −1.21278
\(328\) 0 0
\(329\) 6.56155 0.361750
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) −29.3693 −1.60943
\(334\) 0 0
\(335\) −14.2462 −0.778354
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 21.1231 1.14725
\(340\) 0 0
\(341\) −26.2462 −1.42131
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −13.1231 −0.706524
\(346\) 0 0
\(347\) −6.87689 −0.369171 −0.184586 0.982816i \(-0.559094\pi\)
−0.184586 + 0.982816i \(0.559094\pi\)
\(348\) 0 0
\(349\) 28.2462 1.51199 0.755993 0.654580i \(-0.227155\pi\)
0.755993 + 0.654580i \(0.227155\pi\)
\(350\) 0 0
\(351\) −8.17708 −0.436460
\(352\) 0 0
\(353\) −22.8078 −1.21393 −0.606967 0.794727i \(-0.707614\pi\)
−0.606967 + 0.794727i \(0.707614\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.80776 −0.466156
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) 11.3693 0.596734
\(364\) 0 0
\(365\) −12.2462 −0.640996
\(366\) 0 0
\(367\) −13.9309 −0.727185 −0.363593 0.931558i \(-0.618450\pi\)
−0.363593 + 0.931558i \(0.618450\pi\)
\(368\) 0 0
\(369\) 25.3693 1.32067
\(370\) 0 0
\(371\) 4.87689 0.253196
\(372\) 0 0
\(373\) 9.36932 0.485125 0.242562 0.970136i \(-0.422012\pi\)
0.242562 + 0.970136i \(0.422012\pi\)
\(374\) 0 0
\(375\) 2.56155 0.132278
\(376\) 0 0
\(377\) −25.9309 −1.33551
\(378\) 0 0
\(379\) 0.492423 0.0252940 0.0126470 0.999920i \(-0.495974\pi\)
0.0126470 + 0.999920i \(0.495974\pi\)
\(380\) 0 0
\(381\) −26.2462 −1.34463
\(382\) 0 0
\(383\) −26.2462 −1.34112 −0.670559 0.741856i \(-0.733946\pi\)
−0.670559 + 0.741856i \(0.733946\pi\)
\(384\) 0 0
\(385\) −2.56155 −0.130549
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 31.3002 1.58698 0.793491 0.608582i \(-0.208261\pi\)
0.793491 + 0.608582i \(0.208261\pi\)
\(390\) 0 0
\(391\) −17.6155 −0.890856
\(392\) 0 0
\(393\) 17.6155 0.888586
\(394\) 0 0
\(395\) 11.6847 0.587919
\(396\) 0 0
\(397\) −22.1771 −1.11304 −0.556518 0.830836i \(-0.687863\pi\)
−0.556518 + 0.830836i \(0.687863\pi\)
\(398\) 0 0
\(399\) 2.87689 0.144025
\(400\) 0 0
\(401\) 15.9309 0.795550 0.397775 0.917483i \(-0.369783\pi\)
0.397775 + 0.917483i \(0.369783\pi\)
\(402\) 0 0
\(403\) −58.2462 −2.90145
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) −21.1231 −1.04703
\(408\) 0 0
\(409\) 14.4924 0.716604 0.358302 0.933606i \(-0.383356\pi\)
0.358302 + 0.933606i \(0.383356\pi\)
\(410\) 0 0
\(411\) −44.4924 −2.19465
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 8.63068 0.422646
\(418\) 0 0
\(419\) 1.75379 0.0856782 0.0428391 0.999082i \(-0.486360\pi\)
0.0428391 + 0.999082i \(0.486360\pi\)
\(420\) 0 0
\(421\) 0.561553 0.0273684 0.0136842 0.999906i \(-0.495644\pi\)
0.0136842 + 0.999906i \(0.495644\pi\)
\(422\) 0 0
\(423\) 23.3693 1.13626
\(424\) 0 0
\(425\) 3.43845 0.166789
\(426\) 0 0
\(427\) 15.1231 0.731858
\(428\) 0 0
\(429\) −37.3002 −1.80087
\(430\) 0 0
\(431\) 5.93087 0.285680 0.142840 0.989746i \(-0.454377\pi\)
0.142840 + 0.989746i \(0.454377\pi\)
\(432\) 0 0
\(433\) 36.2462 1.74188 0.870941 0.491388i \(-0.163510\pi\)
0.870941 + 0.491388i \(0.163510\pi\)
\(434\) 0 0
\(435\) −11.6847 −0.560236
\(436\) 0 0
\(437\) 5.75379 0.275241
\(438\) 0 0
\(439\) 10.8769 0.519126 0.259563 0.965726i \(-0.416422\pi\)
0.259563 + 0.965726i \(0.416422\pi\)
\(440\) 0 0
\(441\) 3.56155 0.169598
\(442\) 0 0
\(443\) −21.6155 −1.02698 −0.513492 0.858094i \(-0.671649\pi\)
−0.513492 + 0.858094i \(0.671649\pi\)
\(444\) 0 0
\(445\) 3.12311 0.148049
\(446\) 0 0
\(447\) 41.6155 1.96835
\(448\) 0 0
\(449\) −33.6847 −1.58968 −0.794839 0.606821i \(-0.792445\pi\)
−0.794839 + 0.606821i \(0.792445\pi\)
\(450\) 0 0
\(451\) 18.2462 0.859181
\(452\) 0 0
\(453\) −2.06913 −0.0972162
\(454\) 0 0
\(455\) −5.68466 −0.266501
\(456\) 0 0
\(457\) 33.3693 1.56095 0.780475 0.625186i \(-0.214977\pi\)
0.780475 + 0.625186i \(0.214977\pi\)
\(458\) 0 0
\(459\) −4.94602 −0.230861
\(460\) 0 0
\(461\) −27.1231 −1.26325 −0.631624 0.775274i \(-0.717612\pi\)
−0.631624 + 0.775274i \(0.717612\pi\)
\(462\) 0 0
\(463\) 10.2462 0.476182 0.238091 0.971243i \(-0.423478\pi\)
0.238091 + 0.971243i \(0.423478\pi\)
\(464\) 0 0
\(465\) −26.2462 −1.21714
\(466\) 0 0
\(467\) −21.4384 −0.992053 −0.496027 0.868307i \(-0.665208\pi\)
−0.496027 + 0.868307i \(0.665208\pi\)
\(468\) 0 0
\(469\) 14.2462 0.657829
\(470\) 0 0
\(471\) −31.3693 −1.44542
\(472\) 0 0
\(473\) −2.87689 −0.132280
\(474\) 0 0
\(475\) −1.12311 −0.0515316
\(476\) 0 0
\(477\) 17.3693 0.795286
\(478\) 0 0
\(479\) −7.36932 −0.336713 −0.168356 0.985726i \(-0.553846\pi\)
−0.168356 + 0.985726i \(0.553846\pi\)
\(480\) 0 0
\(481\) −46.8769 −2.13740
\(482\) 0 0
\(483\) 13.1231 0.597122
\(484\) 0 0
\(485\) −13.6847 −0.621388
\(486\) 0 0
\(487\) −5.12311 −0.232150 −0.116075 0.993240i \(-0.537031\pi\)
−0.116075 + 0.993240i \(0.537031\pi\)
\(488\) 0 0
\(489\) −55.3693 −2.50389
\(490\) 0 0
\(491\) 15.6847 0.707839 0.353919 0.935276i \(-0.384849\pi\)
0.353919 + 0.935276i \(0.384849\pi\)
\(492\) 0 0
\(493\) −15.6847 −0.706401
\(494\) 0 0
\(495\) −9.12311 −0.410053
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 20.8078 0.931483 0.465742 0.884921i \(-0.345788\pi\)
0.465742 + 0.884921i \(0.345788\pi\)
\(500\) 0 0
\(501\) −63.5464 −2.83904
\(502\) 0 0
\(503\) 16.1771 0.721300 0.360650 0.932701i \(-0.382555\pi\)
0.360650 + 0.932701i \(0.382555\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −49.4773 −2.19736
\(508\) 0 0
\(509\) 16.7386 0.741927 0.370963 0.928647i \(-0.379028\pi\)
0.370963 + 0.928647i \(0.379028\pi\)
\(510\) 0 0
\(511\) 12.2462 0.541740
\(512\) 0 0
\(513\) 1.61553 0.0713273
\(514\) 0 0
\(515\) −8.80776 −0.388116
\(516\) 0 0
\(517\) 16.8078 0.739205
\(518\) 0 0
\(519\) 56.8078 2.49358
\(520\) 0 0
\(521\) 4.24621 0.186030 0.0930149 0.995665i \(-0.470350\pi\)
0.0930149 + 0.995665i \(0.470350\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) −2.56155 −0.111795
\(526\) 0 0
\(527\) −35.2311 −1.53469
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 0 0
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) 40.4924 1.75392
\(534\) 0 0
\(535\) 6.87689 0.297314
\(536\) 0 0
\(537\) −10.2462 −0.442157
\(538\) 0 0
\(539\) 2.56155 0.110334
\(540\) 0 0
\(541\) 20.4233 0.878066 0.439033 0.898471i \(-0.355321\pi\)
0.439033 + 0.898471i \(0.355321\pi\)
\(542\) 0 0
\(543\) 8.00000 0.343313
\(544\) 0 0
\(545\) −8.56155 −0.366737
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 53.8617 2.29876
\(550\) 0 0
\(551\) 5.12311 0.218252
\(552\) 0 0
\(553\) −11.6847 −0.496882
\(554\) 0 0
\(555\) −21.1231 −0.896626
\(556\) 0 0
\(557\) −23.6155 −1.00062 −0.500311 0.865846i \(-0.666781\pi\)
−0.500311 + 0.865846i \(0.666781\pi\)
\(558\) 0 0
\(559\) −6.38447 −0.270034
\(560\) 0 0
\(561\) −22.5616 −0.952550
\(562\) 0 0
\(563\) 40.4924 1.70655 0.853276 0.521459i \(-0.174612\pi\)
0.853276 + 0.521459i \(0.174612\pi\)
\(564\) 0 0
\(565\) 8.24621 0.346921
\(566\) 0 0
\(567\) −7.00000 −0.293972
\(568\) 0 0
\(569\) −46.9848 −1.96971 −0.984854 0.173388i \(-0.944529\pi\)
−0.984854 + 0.173388i \(0.944529\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −29.9309 −1.25038
\(574\) 0 0
\(575\) −5.12311 −0.213648
\(576\) 0 0
\(577\) 0.561553 0.0233777 0.0116889 0.999932i \(-0.496279\pi\)
0.0116889 + 0.999932i \(0.496279\pi\)
\(578\) 0 0
\(579\) −12.4924 −0.519167
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 12.4924 0.517383
\(584\) 0 0
\(585\) −20.2462 −0.837078
\(586\) 0 0
\(587\) 6.24621 0.257809 0.128904 0.991657i \(-0.458854\pi\)
0.128904 + 0.991657i \(0.458854\pi\)
\(588\) 0 0
\(589\) 11.5076 0.474161
\(590\) 0 0
\(591\) −12.4924 −0.513870
\(592\) 0 0
\(593\) −24.4233 −1.00294 −0.501472 0.865174i \(-0.667208\pi\)
−0.501472 + 0.865174i \(0.667208\pi\)
\(594\) 0 0
\(595\) −3.43845 −0.140963
\(596\) 0 0
\(597\) −5.75379 −0.235487
\(598\) 0 0
\(599\) −18.0691 −0.738285 −0.369142 0.929373i \(-0.620349\pi\)
−0.369142 + 0.929373i \(0.620349\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 50.7386 2.06624
\(604\) 0 0
\(605\) 4.43845 0.180449
\(606\) 0 0
\(607\) 3.68466 0.149556 0.0747778 0.997200i \(-0.476175\pi\)
0.0747778 + 0.997200i \(0.476175\pi\)
\(608\) 0 0
\(609\) 11.6847 0.473486
\(610\) 0 0
\(611\) 37.3002 1.50900
\(612\) 0 0
\(613\) 32.7386 1.32230 0.661150 0.750253i \(-0.270068\pi\)
0.661150 + 0.750253i \(0.270068\pi\)
\(614\) 0 0
\(615\) 18.2462 0.735758
\(616\) 0 0
\(617\) −32.2462 −1.29818 −0.649092 0.760710i \(-0.724851\pi\)
−0.649092 + 0.760710i \(0.724851\pi\)
\(618\) 0 0
\(619\) −35.3693 −1.42161 −0.710806 0.703388i \(-0.751670\pi\)
−0.710806 + 0.703388i \(0.751670\pi\)
\(620\) 0 0
\(621\) 7.36932 0.295720
\(622\) 0 0
\(623\) −3.12311 −0.125125
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.36932 0.294302
\(628\) 0 0
\(629\) −28.3542 −1.13055
\(630\) 0 0
\(631\) 34.4233 1.37037 0.685185 0.728369i \(-0.259721\pi\)
0.685185 + 0.728369i \(0.259721\pi\)
\(632\) 0 0
\(633\) −34.4233 −1.36820
\(634\) 0 0
\(635\) −10.2462 −0.406608
\(636\) 0 0
\(637\) 5.68466 0.225234
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.9848 1.69780 0.848900 0.528554i \(-0.177266\pi\)
0.848900 + 0.528554i \(0.177266\pi\)
\(642\) 0 0
\(643\) −7.05398 −0.278182 −0.139091 0.990280i \(-0.544418\pi\)
−0.139091 + 0.990280i \(0.544418\pi\)
\(644\) 0 0
\(645\) −2.87689 −0.113278
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 10.2462 0.402199
\(650\) 0 0
\(651\) 26.2462 1.02867
\(652\) 0 0
\(653\) 5.50758 0.215528 0.107764 0.994176i \(-0.465631\pi\)
0.107764 + 0.994176i \(0.465631\pi\)
\(654\) 0 0
\(655\) 6.87689 0.268702
\(656\) 0 0
\(657\) 43.6155 1.70160
\(658\) 0 0
\(659\) 20.8078 0.810555 0.405278 0.914194i \(-0.367175\pi\)
0.405278 + 0.914194i \(0.367175\pi\)
\(660\) 0 0
\(661\) −23.6155 −0.918538 −0.459269 0.888297i \(-0.651889\pi\)
−0.459269 + 0.888297i \(0.651889\pi\)
\(662\) 0 0
\(663\) −50.0691 −1.94452
\(664\) 0 0
\(665\) 1.12311 0.0435522
\(666\) 0 0
\(667\) 23.3693 0.904864
\(668\) 0 0
\(669\) −31.5464 −1.21965
\(670\) 0 0
\(671\) 38.7386 1.49549
\(672\) 0 0
\(673\) 0.384472 0.0148203 0.00741015 0.999973i \(-0.497641\pi\)
0.00741015 + 0.999973i \(0.497641\pi\)
\(674\) 0 0
\(675\) −1.43845 −0.0553659
\(676\) 0 0
\(677\) 13.6847 0.525944 0.262972 0.964803i \(-0.415297\pi\)
0.262972 + 0.964803i \(0.415297\pi\)
\(678\) 0 0
\(679\) 13.6847 0.525169
\(680\) 0 0
\(681\) −60.6695 −2.32486
\(682\) 0 0
\(683\) 20.9848 0.802963 0.401481 0.915867i \(-0.368495\pi\)
0.401481 + 0.915867i \(0.368495\pi\)
\(684\) 0 0
\(685\) −17.3693 −0.663647
\(686\) 0 0
\(687\) 34.2462 1.30657
\(688\) 0 0
\(689\) 27.7235 1.05618
\(690\) 0 0
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) 0 0
\(693\) 9.12311 0.346558
\(694\) 0 0
\(695\) 3.36932 0.127805
\(696\) 0 0
\(697\) 24.4924 0.927717
\(698\) 0 0
\(699\) 48.9848 1.85278
\(700\) 0 0
\(701\) 21.6847 0.819018 0.409509 0.912306i \(-0.365700\pi\)
0.409509 + 0.912306i \(0.365700\pi\)
\(702\) 0 0
\(703\) 9.26137 0.349299
\(704\) 0 0
\(705\) 16.8078 0.633017
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −33.0540 −1.24137 −0.620684 0.784061i \(-0.713145\pi\)
−0.620684 + 0.784061i \(0.713145\pi\)
\(710\) 0 0
\(711\) −41.6155 −1.56070
\(712\) 0 0
\(713\) 52.4924 1.96586
\(714\) 0 0
\(715\) −14.5616 −0.544571
\(716\) 0 0
\(717\) −48.8078 −1.82276
\(718\) 0 0
\(719\) 17.6155 0.656948 0.328474 0.944513i \(-0.393466\pi\)
0.328474 + 0.944513i \(0.393466\pi\)
\(720\) 0 0
\(721\) 8.80776 0.328018
\(722\) 0 0
\(723\) −5.12311 −0.190530
\(724\) 0 0
\(725\) −4.56155 −0.169412
\(726\) 0 0
\(727\) −16.9848 −0.629933 −0.314967 0.949103i \(-0.601993\pi\)
−0.314967 + 0.949103i \(0.601993\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −3.86174 −0.142832
\(732\) 0 0
\(733\) −20.5616 −0.759458 −0.379729 0.925098i \(-0.623983\pi\)
−0.379729 + 0.925098i \(0.623983\pi\)
\(734\) 0 0
\(735\) 2.56155 0.0944843
\(736\) 0 0
\(737\) 36.4924 1.34422
\(738\) 0 0
\(739\) 10.5616 0.388513 0.194257 0.980951i \(-0.437771\pi\)
0.194257 + 0.980951i \(0.437771\pi\)
\(740\) 0 0
\(741\) 16.3542 0.600785
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 16.2462 0.595215
\(746\) 0 0
\(747\) −42.7386 −1.56372
\(748\) 0 0
\(749\) −6.87689 −0.251276
\(750\) 0 0
\(751\) −43.6847 −1.59408 −0.797038 0.603929i \(-0.793601\pi\)
−0.797038 + 0.603929i \(0.793601\pi\)
\(752\) 0 0
\(753\) −29.1231 −1.06130
\(754\) 0 0
\(755\) −0.807764 −0.0293975
\(756\) 0 0
\(757\) 9.36932 0.340534 0.170267 0.985398i \(-0.445537\pi\)
0.170267 + 0.985398i \(0.445537\pi\)
\(758\) 0 0
\(759\) 33.6155 1.22017
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 8.56155 0.309949
\(764\) 0 0
\(765\) −12.2462 −0.442763
\(766\) 0 0
\(767\) 22.7386 0.821044
\(768\) 0 0
\(769\) −22.9848 −0.828855 −0.414427 0.910082i \(-0.636018\pi\)
−0.414427 + 0.910082i \(0.636018\pi\)
\(770\) 0 0
\(771\) −37.1231 −1.33696
\(772\) 0 0
\(773\) 9.19224 0.330622 0.165311 0.986242i \(-0.447137\pi\)
0.165311 + 0.986242i \(0.447137\pi\)
\(774\) 0 0
\(775\) −10.2462 −0.368055
\(776\) 0 0
\(777\) 21.1231 0.757787
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.56155 0.234491
\(784\) 0 0
\(785\) −12.2462 −0.437086
\(786\) 0 0
\(787\) 12.1771 0.434066 0.217033 0.976164i \(-0.430362\pi\)
0.217033 + 0.976164i \(0.430362\pi\)
\(788\) 0 0
\(789\) 39.3693 1.40158
\(790\) 0 0
\(791\) −8.24621 −0.293202
\(792\) 0 0
\(793\) 85.9697 3.05287
\(794\) 0 0
\(795\) 12.4924 0.443061
\(796\) 0 0
\(797\) 2.80776 0.0994561 0.0497281 0.998763i \(-0.484165\pi\)
0.0497281 + 0.998763i \(0.484165\pi\)
\(798\) 0 0
\(799\) 22.5616 0.798170
\(800\) 0 0
\(801\) −11.1231 −0.393016
\(802\) 0 0
\(803\) 31.3693 1.10700
\(804\) 0 0
\(805\) 5.12311 0.180566
\(806\) 0 0
\(807\) −31.3693 −1.10425
\(808\) 0 0
\(809\) −33.0540 −1.16212 −0.581058 0.813862i \(-0.697361\pi\)
−0.581058 + 0.813862i \(0.697361\pi\)
\(810\) 0 0
\(811\) −13.6155 −0.478106 −0.239053 0.971007i \(-0.576837\pi\)
−0.239053 + 0.971007i \(0.576837\pi\)
\(812\) 0 0
\(813\) −26.2462 −0.920495
\(814\) 0 0
\(815\) −21.6155 −0.757159
\(816\) 0 0
\(817\) 1.26137 0.0441296
\(818\) 0 0
\(819\) 20.2462 0.707460
\(820\) 0 0
\(821\) −0.699813 −0.0244237 −0.0122118 0.999925i \(-0.503887\pi\)
−0.0122118 + 0.999925i \(0.503887\pi\)
\(822\) 0 0
\(823\) −34.2462 −1.19375 −0.596874 0.802335i \(-0.703591\pi\)
−0.596874 + 0.802335i \(0.703591\pi\)
\(824\) 0 0
\(825\) −6.56155 −0.228444
\(826\) 0 0
\(827\) −33.1231 −1.15180 −0.575902 0.817519i \(-0.695349\pi\)
−0.575902 + 0.817519i \(0.695349\pi\)
\(828\) 0 0
\(829\) 35.6155 1.23698 0.618489 0.785793i \(-0.287745\pi\)
0.618489 + 0.785793i \(0.287745\pi\)
\(830\) 0 0
\(831\) 21.1231 0.732752
\(832\) 0 0
\(833\) 3.43845 0.119135
\(834\) 0 0
\(835\) −24.8078 −0.858508
\(836\) 0 0
\(837\) 14.7386 0.509442
\(838\) 0 0
\(839\) 10.8769 0.375512 0.187756 0.982216i \(-0.439879\pi\)
0.187756 + 0.982216i \(0.439879\pi\)
\(840\) 0 0
\(841\) −8.19224 −0.282491
\(842\) 0 0
\(843\) 39.5464 1.36205
\(844\) 0 0
\(845\) −19.3153 −0.664468
\(846\) 0 0
\(847\) −4.43845 −0.152507
\(848\) 0 0
\(849\) 19.6847 0.675576
\(850\) 0 0
\(851\) 42.2462 1.44818
\(852\) 0 0
\(853\) −32.2462 −1.10409 −0.552045 0.833815i \(-0.686152\pi\)
−0.552045 + 0.833815i \(0.686152\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 3.26137 0.111406 0.0557031 0.998447i \(-0.482260\pi\)
0.0557031 + 0.998447i \(0.482260\pi\)
\(858\) 0 0
\(859\) 28.9848 0.988950 0.494475 0.869192i \(-0.335360\pi\)
0.494475 + 0.869192i \(0.335360\pi\)
\(860\) 0 0
\(861\) −18.2462 −0.621829
\(862\) 0 0
\(863\) −4.49242 −0.152924 −0.0764619 0.997073i \(-0.524362\pi\)
−0.0764619 + 0.997073i \(0.524362\pi\)
\(864\) 0 0
\(865\) 22.1771 0.754043
\(866\) 0 0
\(867\) 13.2614 0.450380
\(868\) 0 0
\(869\) −29.9309 −1.01534
\(870\) 0 0
\(871\) 80.9848 2.74407
\(872\) 0 0
\(873\) 48.7386 1.64955
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 46.4924 1.56994 0.784969 0.619535i \(-0.212679\pi\)
0.784969 + 0.619535i \(0.212679\pi\)
\(878\) 0 0
\(879\) −12.9460 −0.436659
\(880\) 0 0
\(881\) −20.1080 −0.677454 −0.338727 0.940885i \(-0.609996\pi\)
−0.338727 + 0.940885i \(0.609996\pi\)
\(882\) 0 0
\(883\) −38.2462 −1.28709 −0.643544 0.765409i \(-0.722537\pi\)
−0.643544 + 0.765409i \(0.722537\pi\)
\(884\) 0 0
\(885\) 10.2462 0.344423
\(886\) 0 0
\(887\) −3.50758 −0.117773 −0.0588865 0.998265i \(-0.518755\pi\)
−0.0588865 + 0.998265i \(0.518755\pi\)
\(888\) 0 0
\(889\) 10.2462 0.343647
\(890\) 0 0
\(891\) −17.9309 −0.600707
\(892\) 0 0
\(893\) −7.36932 −0.246605
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 74.6004 2.49083
\(898\) 0 0
\(899\) 46.7386 1.55882
\(900\) 0 0
\(901\) 16.7689 0.558655
\(902\) 0 0
\(903\) 2.87689 0.0957371
\(904\) 0 0
\(905\) 3.12311 0.103816
\(906\) 0 0
\(907\) −26.1080 −0.866900 −0.433450 0.901178i \(-0.642704\pi\)
−0.433450 + 0.901178i \(0.642704\pi\)
\(908\) 0 0
\(909\) −21.3693 −0.708776
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −30.7386 −1.01730
\(914\) 0 0
\(915\) 38.7386 1.28066
\(916\) 0 0
\(917\) −6.87689 −0.227095
\(918\) 0 0
\(919\) 6.56155 0.216446 0.108223 0.994127i \(-0.465484\pi\)
0.108223 + 0.994127i \(0.465484\pi\)
\(920\) 0 0
\(921\) −41.7926 −1.37711
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.24621 −0.271134
\(926\) 0 0
\(927\) 31.3693 1.03030
\(928\) 0 0
\(929\) −11.1231 −0.364937 −0.182469 0.983212i \(-0.558409\pi\)
−0.182469 + 0.983212i \(0.558409\pi\)
\(930\) 0 0
\(931\) −1.12311 −0.0368083
\(932\) 0 0
\(933\) 54.1080 1.77141
\(934\) 0 0
\(935\) −8.80776 −0.288045
\(936\) 0 0
\(937\) 17.1922 0.561646 0.280823 0.959760i \(-0.409393\pi\)
0.280823 + 0.959760i \(0.409393\pi\)
\(938\) 0 0
\(939\) −36.6695 −1.19666
\(940\) 0 0
\(941\) −53.3693 −1.73979 −0.869895 0.493237i \(-0.835814\pi\)
−0.869895 + 0.493237i \(0.835814\pi\)
\(942\) 0 0
\(943\) −36.4924 −1.18836
\(944\) 0 0
\(945\) 1.43845 0.0467927
\(946\) 0 0
\(947\) −47.2311 −1.53480 −0.767402 0.641167i \(-0.778451\pi\)
−0.767402 + 0.641167i \(0.778451\pi\)
\(948\) 0 0
\(949\) 69.6155 2.25982
\(950\) 0 0
\(951\) −25.6155 −0.830640
\(952\) 0 0
\(953\) 5.86174 0.189880 0.0949402 0.995483i \(-0.469734\pi\)
0.0949402 + 0.995483i \(0.469734\pi\)
\(954\) 0 0
\(955\) −11.6847 −0.378107
\(956\) 0 0
\(957\) 29.9309 0.967528
\(958\) 0 0
\(959\) 17.3693 0.560884
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) −24.4924 −0.789257
\(964\) 0 0
\(965\) −4.87689 −0.156993
\(966\) 0 0
\(967\) −30.1080 −0.968206 −0.484103 0.875011i \(-0.660854\pi\)
−0.484103 + 0.875011i \(0.660854\pi\)
\(968\) 0 0
\(969\) 9.89205 0.317778
\(970\) 0 0
\(971\) −26.7386 −0.858084 −0.429042 0.903285i \(-0.641149\pi\)
−0.429042 + 0.903285i \(0.641149\pi\)
\(972\) 0 0
\(973\) −3.36932 −0.108015
\(974\) 0 0
\(975\) −14.5616 −0.466343
\(976\) 0 0
\(977\) −54.9848 −1.75912 −0.879561 0.475787i \(-0.842163\pi\)
−0.879561 + 0.475787i \(0.842163\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 30.4924 0.973548
\(982\) 0 0
\(983\) 16.1771 0.515969 0.257984 0.966149i \(-0.416942\pi\)
0.257984 + 0.966149i \(0.416942\pi\)
\(984\) 0 0
\(985\) −4.87689 −0.155391
\(986\) 0 0
\(987\) −16.8078 −0.534997
\(988\) 0 0
\(989\) 5.75379 0.182960
\(990\) 0 0
\(991\) 12.4924 0.396835 0.198417 0.980118i \(-0.436420\pi\)
0.198417 + 0.980118i \(0.436420\pi\)
\(992\) 0 0
\(993\) −30.7386 −0.975461
\(994\) 0 0
\(995\) −2.24621 −0.0712097
\(996\) 0 0
\(997\) 28.0691 0.888958 0.444479 0.895789i \(-0.353389\pi\)
0.444479 + 0.895789i \(0.353389\pi\)
\(998\) 0 0
\(999\) 11.8617 0.375289
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 2240.2.a.be.1.1 2
4.3 odd 2 2240.2.a.bi.1.2 2
8.3 odd 2 560.2.a.g.1.1 2
8.5 even 2 280.2.a.d.1.2 2
24.5 odd 2 2520.2.a.w.1.2 2
24.11 even 2 5040.2.a.bq.1.1 2
40.3 even 4 2800.2.g.u.449.1 4
40.13 odd 4 1400.2.g.k.449.4 4
40.19 odd 2 2800.2.a.bn.1.2 2
40.27 even 4 2800.2.g.u.449.4 4
40.29 even 2 1400.2.a.p.1.1 2
40.37 odd 4 1400.2.g.k.449.1 4
56.5 odd 6 1960.2.q.u.361.2 4
56.13 odd 2 1960.2.a.r.1.1 2
56.27 even 2 3920.2.a.bu.1.2 2
56.37 even 6 1960.2.q.s.361.1 4
56.45 odd 6 1960.2.q.u.961.2 4
56.53 even 6 1960.2.q.s.961.1 4
280.69 odd 2 9800.2.a.by.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.2 2 8.5 even 2
560.2.a.g.1.1 2 8.3 odd 2
1400.2.a.p.1.1 2 40.29 even 2
1400.2.g.k.449.1 4 40.37 odd 4
1400.2.g.k.449.4 4 40.13 odd 4
1960.2.a.r.1.1 2 56.13 odd 2
1960.2.q.s.361.1 4 56.37 even 6
1960.2.q.s.961.1 4 56.53 even 6
1960.2.q.u.361.2 4 56.5 odd 6
1960.2.q.u.961.2 4 56.45 odd 6
2240.2.a.be.1.1 2 1.1 even 1 trivial
2240.2.a.bi.1.2 2 4.3 odd 2
2520.2.a.w.1.2 2 24.5 odd 2
2800.2.a.bn.1.2 2 40.19 odd 2
2800.2.g.u.449.1 4 40.3 even 4
2800.2.g.u.449.4 4 40.27 even 4
3920.2.a.bu.1.2 2 56.27 even 2
5040.2.a.bq.1.1 2 24.11 even 2
9800.2.a.by.1.2 2 280.69 odd 2