Properties

Label 5520.2.a.bt.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +1.41421 q^{11} +0.585786 q^{13} +1.00000 q^{15} -2.41421 q^{17} +7.41421 q^{19} -1.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.58579 q^{29} +1.00000 q^{31} +1.41421 q^{33} -1.82843 q^{35} -7.48528 q^{37} +0.585786 q^{39} +7.24264 q^{41} +7.65685 q^{43} +1.00000 q^{45} +6.24264 q^{47} -3.65685 q^{49} -2.41421 q^{51} +5.24264 q^{53} +1.41421 q^{55} +7.41421 q^{57} -13.7279 q^{59} +1.41421 q^{61} -1.82843 q^{63} +0.585786 q^{65} +15.4853 q^{67} +1.00000 q^{69} +14.0711 q^{71} +2.24264 q^{73} +1.00000 q^{75} -2.58579 q^{77} -16.9706 q^{79} +1.00000 q^{81} -2.75736 q^{83} -2.41421 q^{85} -3.58579 q^{87} -9.17157 q^{89} -1.07107 q^{91} +1.00000 q^{93} +7.41421 q^{95} +18.1421 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} + 12 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} + 12 q^{57} - 2 q^{59} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 2 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 24 q^{89} + 12 q^{91} + 2 q^{93} + 12 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.82843 −0.691080 −0.345540 0.938404i \(-0.612304\pi\)
−0.345540 + 0.938404i \(0.612304\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.41421 −0.585533 −0.292766 0.956184i \(-0.594576\pi\)
−0.292766 + 0.956184i \(0.594576\pi\)
\(18\) 0 0
\(19\) 7.41421 1.70094 0.850469 0.526026i \(-0.176318\pi\)
0.850469 + 0.526026i \(0.176318\pi\)
\(20\) 0 0
\(21\) −1.82843 −0.398996
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.58579 −0.665864 −0.332932 0.942951i \(-0.608038\pi\)
−0.332932 + 0.942951i \(0.608038\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 1.41421 0.246183
\(34\) 0 0
\(35\) −1.82843 −0.309061
\(36\) 0 0
\(37\) −7.48528 −1.23057 −0.615286 0.788304i \(-0.710960\pi\)
−0.615286 + 0.788304i \(0.710960\pi\)
\(38\) 0 0
\(39\) 0.585786 0.0938009
\(40\) 0 0
\(41\) 7.24264 1.13111 0.565555 0.824710i \(-0.308662\pi\)
0.565555 + 0.824710i \(0.308662\pi\)
\(42\) 0 0
\(43\) 7.65685 1.16766 0.583830 0.811876i \(-0.301554\pi\)
0.583830 + 0.811876i \(0.301554\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.24264 0.910583 0.455291 0.890343i \(-0.349535\pi\)
0.455291 + 0.890343i \(0.349535\pi\)
\(48\) 0 0
\(49\) −3.65685 −0.522408
\(50\) 0 0
\(51\) −2.41421 −0.338058
\(52\) 0 0
\(53\) 5.24264 0.720132 0.360066 0.932927i \(-0.382754\pi\)
0.360066 + 0.932927i \(0.382754\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) 7.41421 0.982037
\(58\) 0 0
\(59\) −13.7279 −1.78722 −0.893612 0.448841i \(-0.851837\pi\)
−0.893612 + 0.448841i \(0.851837\pi\)
\(60\) 0 0
\(61\) 1.41421 0.181071 0.0905357 0.995893i \(-0.471142\pi\)
0.0905357 + 0.995893i \(0.471142\pi\)
\(62\) 0 0
\(63\) −1.82843 −0.230360
\(64\) 0 0
\(65\) 0.585786 0.0726579
\(66\) 0 0
\(67\) 15.4853 1.89183 0.945914 0.324417i \(-0.105168\pi\)
0.945914 + 0.324417i \(0.105168\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 14.0711 1.66993 0.834964 0.550304i \(-0.185488\pi\)
0.834964 + 0.550304i \(0.185488\pi\)
\(72\) 0 0
\(73\) 2.24264 0.262481 0.131241 0.991351i \(-0.458104\pi\)
0.131241 + 0.991351i \(0.458104\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.58579 −0.294678
\(78\) 0 0
\(79\) −16.9706 −1.90934 −0.954669 0.297670i \(-0.903790\pi\)
−0.954669 + 0.297670i \(0.903790\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.75736 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(84\) 0 0
\(85\) −2.41421 −0.261858
\(86\) 0 0
\(87\) −3.58579 −0.384437
\(88\) 0 0
\(89\) −9.17157 −0.972185 −0.486092 0.873907i \(-0.661578\pi\)
−0.486092 + 0.873907i \(0.661578\pi\)
\(90\) 0 0
\(91\) −1.07107 −0.112278
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 7.41421 0.760682
\(96\) 0 0
\(97\) 18.1421 1.84205 0.921027 0.389498i \(-0.127351\pi\)
0.921027 + 0.389498i \(0.127351\pi\)
\(98\) 0 0
\(99\) 1.41421 0.142134
\(100\) 0 0
\(101\) −10.0711 −1.00211 −0.501054 0.865416i \(-0.667054\pi\)
−0.501054 + 0.865416i \(0.667054\pi\)
\(102\) 0 0
\(103\) 1.51472 0.149250 0.0746248 0.997212i \(-0.476224\pi\)
0.0746248 + 0.997212i \(0.476224\pi\)
\(104\) 0 0
\(105\) −1.82843 −0.178436
\(106\) 0 0
\(107\) −3.58579 −0.346651 −0.173326 0.984865i \(-0.555451\pi\)
−0.173326 + 0.984865i \(0.555451\pi\)
\(108\) 0 0
\(109\) −7.41421 −0.710153 −0.355076 0.934837i \(-0.615545\pi\)
−0.355076 + 0.934837i \(0.615545\pi\)
\(110\) 0 0
\(111\) −7.48528 −0.710471
\(112\) 0 0
\(113\) 18.8995 1.77791 0.888957 0.457990i \(-0.151430\pi\)
0.888957 + 0.457990i \(0.151430\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0.585786 0.0541560
\(118\) 0 0
\(119\) 4.41421 0.404650
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 7.24264 0.653047
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 19.5563 1.73535 0.867673 0.497136i \(-0.165615\pi\)
0.867673 + 0.497136i \(0.165615\pi\)
\(128\) 0 0
\(129\) 7.65685 0.674148
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) −13.5563 −1.17548
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 2.82843 0.241649 0.120824 0.992674i \(-0.461446\pi\)
0.120824 + 0.992674i \(0.461446\pi\)
\(138\) 0 0
\(139\) 17.4853 1.48308 0.741541 0.670907i \(-0.234095\pi\)
0.741541 + 0.670907i \(0.234095\pi\)
\(140\) 0 0
\(141\) 6.24264 0.525725
\(142\) 0 0
\(143\) 0.828427 0.0692766
\(144\) 0 0
\(145\) −3.58579 −0.297783
\(146\) 0 0
\(147\) −3.65685 −0.301612
\(148\) 0 0
\(149\) −2.24264 −0.183724 −0.0918621 0.995772i \(-0.529282\pi\)
−0.0918621 + 0.995772i \(0.529282\pi\)
\(150\) 0 0
\(151\) 21.3137 1.73448 0.867242 0.497886i \(-0.165890\pi\)
0.867242 + 0.497886i \(0.165890\pi\)
\(152\) 0 0
\(153\) −2.41421 −0.195178
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 0 0
\(159\) 5.24264 0.415768
\(160\) 0 0
\(161\) −1.82843 −0.144100
\(162\) 0 0
\(163\) −18.9706 −1.48589 −0.742945 0.669353i \(-0.766571\pi\)
−0.742945 + 0.669353i \(0.766571\pi\)
\(164\) 0 0
\(165\) 1.41421 0.110096
\(166\) 0 0
\(167\) 1.75736 0.135989 0.0679943 0.997686i \(-0.478340\pi\)
0.0679943 + 0.997686i \(0.478340\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) 7.41421 0.566979
\(172\) 0 0
\(173\) 21.6569 1.64654 0.823270 0.567650i \(-0.192147\pi\)
0.823270 + 0.567650i \(0.192147\pi\)
\(174\) 0 0
\(175\) −1.82843 −0.138216
\(176\) 0 0
\(177\) −13.7279 −1.03185
\(178\) 0 0
\(179\) 19.6569 1.46922 0.734611 0.678488i \(-0.237365\pi\)
0.734611 + 0.678488i \(0.237365\pi\)
\(180\) 0 0
\(181\) −3.51472 −0.261247 −0.130623 0.991432i \(-0.541698\pi\)
−0.130623 + 0.991432i \(0.541698\pi\)
\(182\) 0 0
\(183\) 1.41421 0.104542
\(184\) 0 0
\(185\) −7.48528 −0.550329
\(186\) 0 0
\(187\) −3.41421 −0.249672
\(188\) 0 0
\(189\) −1.82843 −0.132999
\(190\) 0 0
\(191\) −5.75736 −0.416588 −0.208294 0.978066i \(-0.566791\pi\)
−0.208294 + 0.978066i \(0.566791\pi\)
\(192\) 0 0
\(193\) 18.1421 1.30590 0.652950 0.757401i \(-0.273531\pi\)
0.652950 + 0.757401i \(0.273531\pi\)
\(194\) 0 0
\(195\) 0.585786 0.0419490
\(196\) 0 0
\(197\) −7.17157 −0.510953 −0.255477 0.966815i \(-0.582232\pi\)
−0.255477 + 0.966815i \(0.582232\pi\)
\(198\) 0 0
\(199\) 14.4853 1.02683 0.513417 0.858139i \(-0.328379\pi\)
0.513417 + 0.858139i \(0.328379\pi\)
\(200\) 0 0
\(201\) 15.4853 1.09225
\(202\) 0 0
\(203\) 6.55635 0.460166
\(204\) 0 0
\(205\) 7.24264 0.505848
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 10.4853 0.725282
\(210\) 0 0
\(211\) −20.7990 −1.43186 −0.715931 0.698171i \(-0.753997\pi\)
−0.715931 + 0.698171i \(0.753997\pi\)
\(212\) 0 0
\(213\) 14.0711 0.964134
\(214\) 0 0
\(215\) 7.65685 0.522193
\(216\) 0 0
\(217\) −1.82843 −0.124122
\(218\) 0 0
\(219\) 2.24264 0.151544
\(220\) 0 0
\(221\) −1.41421 −0.0951303
\(222\) 0 0
\(223\) 18.9706 1.27036 0.635181 0.772363i \(-0.280925\pi\)
0.635181 + 0.772363i \(0.280925\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0.686292 0.0455508 0.0227754 0.999741i \(-0.492750\pi\)
0.0227754 + 0.999741i \(0.492750\pi\)
\(228\) 0 0
\(229\) −16.4853 −1.08938 −0.544689 0.838638i \(-0.683352\pi\)
−0.544689 + 0.838638i \(0.683352\pi\)
\(230\) 0 0
\(231\) −2.58579 −0.170132
\(232\) 0 0
\(233\) −5.65685 −0.370593 −0.185296 0.982683i \(-0.559325\pi\)
−0.185296 + 0.982683i \(0.559325\pi\)
\(234\) 0 0
\(235\) 6.24264 0.407225
\(236\) 0 0
\(237\) −16.9706 −1.10236
\(238\) 0 0
\(239\) −4.41421 −0.285532 −0.142766 0.989756i \(-0.545600\pi\)
−0.142766 + 0.989756i \(0.545600\pi\)
\(240\) 0 0
\(241\) −15.2132 −0.979969 −0.489984 0.871731i \(-0.662997\pi\)
−0.489984 + 0.871731i \(0.662997\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.65685 −0.233628
\(246\) 0 0
\(247\) 4.34315 0.276348
\(248\) 0 0
\(249\) −2.75736 −0.174741
\(250\) 0 0
\(251\) 14.9706 0.944934 0.472467 0.881348i \(-0.343364\pi\)
0.472467 + 0.881348i \(0.343364\pi\)
\(252\) 0 0
\(253\) 1.41421 0.0889108
\(254\) 0 0
\(255\) −2.41421 −0.151184
\(256\) 0 0
\(257\) −11.4142 −0.711999 −0.356000 0.934486i \(-0.615860\pi\)
−0.356000 + 0.934486i \(0.615860\pi\)
\(258\) 0 0
\(259\) 13.6863 0.850425
\(260\) 0 0
\(261\) −3.58579 −0.221955
\(262\) 0 0
\(263\) −12.4142 −0.765493 −0.382747 0.923853i \(-0.625022\pi\)
−0.382747 + 0.923853i \(0.625022\pi\)
\(264\) 0 0
\(265\) 5.24264 0.322053
\(266\) 0 0
\(267\) −9.17157 −0.561291
\(268\) 0 0
\(269\) 12.0711 0.735986 0.367993 0.929829i \(-0.380045\pi\)
0.367993 + 0.929829i \(0.380045\pi\)
\(270\) 0 0
\(271\) −18.3137 −1.11248 −0.556239 0.831022i \(-0.687756\pi\)
−0.556239 + 0.831022i \(0.687756\pi\)
\(272\) 0 0
\(273\) −1.07107 −0.0648240
\(274\) 0 0
\(275\) 1.41421 0.0852803
\(276\) 0 0
\(277\) 18.9706 1.13983 0.569915 0.821703i \(-0.306976\pi\)
0.569915 + 0.821703i \(0.306976\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 3.75736 0.224145 0.112073 0.993700i \(-0.464251\pi\)
0.112073 + 0.993700i \(0.464251\pi\)
\(282\) 0 0
\(283\) 25.6274 1.52339 0.761696 0.647935i \(-0.224367\pi\)
0.761696 + 0.647935i \(0.224367\pi\)
\(284\) 0 0
\(285\) 7.41421 0.439180
\(286\) 0 0
\(287\) −13.2426 −0.781688
\(288\) 0 0
\(289\) −11.1716 −0.657151
\(290\) 0 0
\(291\) 18.1421 1.06351
\(292\) 0 0
\(293\) 10.8995 0.636755 0.318378 0.947964i \(-0.396862\pi\)
0.318378 + 0.947964i \(0.396862\pi\)
\(294\) 0 0
\(295\) −13.7279 −0.799271
\(296\) 0 0
\(297\) 1.41421 0.0820610
\(298\) 0 0
\(299\) 0.585786 0.0338769
\(300\) 0 0
\(301\) −14.0000 −0.806947
\(302\) 0 0
\(303\) −10.0711 −0.578568
\(304\) 0 0
\(305\) 1.41421 0.0809776
\(306\) 0 0
\(307\) −3.89949 −0.222556 −0.111278 0.993789i \(-0.535494\pi\)
−0.111278 + 0.993789i \(0.535494\pi\)
\(308\) 0 0
\(309\) 1.51472 0.0861693
\(310\) 0 0
\(311\) −31.3137 −1.77564 −0.887819 0.460193i \(-0.847780\pi\)
−0.887819 + 0.460193i \(0.847780\pi\)
\(312\) 0 0
\(313\) 21.1421 1.19502 0.597512 0.801860i \(-0.296156\pi\)
0.597512 + 0.801860i \(0.296156\pi\)
\(314\) 0 0
\(315\) −1.82843 −0.103020
\(316\) 0 0
\(317\) −7.41421 −0.416424 −0.208212 0.978084i \(-0.566764\pi\)
−0.208212 + 0.978084i \(0.566764\pi\)
\(318\) 0 0
\(319\) −5.07107 −0.283925
\(320\) 0 0
\(321\) −3.58579 −0.200139
\(322\) 0 0
\(323\) −17.8995 −0.995955
\(324\) 0 0
\(325\) 0.585786 0.0324936
\(326\) 0 0
\(327\) −7.41421 −0.410007
\(328\) 0 0
\(329\) −11.4142 −0.629286
\(330\) 0 0
\(331\) 23.8284 1.30973 0.654864 0.755746i \(-0.272726\pi\)
0.654864 + 0.755746i \(0.272726\pi\)
\(332\) 0 0
\(333\) −7.48528 −0.410191
\(334\) 0 0
\(335\) 15.4853 0.846051
\(336\) 0 0
\(337\) −8.34315 −0.454480 −0.227240 0.973839i \(-0.572970\pi\)
−0.227240 + 0.973839i \(0.572970\pi\)
\(338\) 0 0
\(339\) 18.8995 1.02648
\(340\) 0 0
\(341\) 1.41421 0.0765840
\(342\) 0 0
\(343\) 19.4853 1.05211
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) −4.48528 −0.240783 −0.120391 0.992727i \(-0.538415\pi\)
−0.120391 + 0.992727i \(0.538415\pi\)
\(348\) 0 0
\(349\) −15.8284 −0.847276 −0.423638 0.905832i \(-0.639247\pi\)
−0.423638 + 0.905832i \(0.639247\pi\)
\(350\) 0 0
\(351\) 0.585786 0.0312670
\(352\) 0 0
\(353\) −22.0416 −1.17316 −0.586579 0.809892i \(-0.699526\pi\)
−0.586579 + 0.809892i \(0.699526\pi\)
\(354\) 0 0
\(355\) 14.0711 0.746815
\(356\) 0 0
\(357\) 4.41421 0.233625
\(358\) 0 0
\(359\) 5.07107 0.267641 0.133820 0.991006i \(-0.457275\pi\)
0.133820 + 0.991006i \(0.457275\pi\)
\(360\) 0 0
\(361\) 35.9706 1.89319
\(362\) 0 0
\(363\) −9.00000 −0.472377
\(364\) 0 0
\(365\) 2.24264 0.117385
\(366\) 0 0
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) 0 0
\(369\) 7.24264 0.377037
\(370\) 0 0
\(371\) −9.58579 −0.497669
\(372\) 0 0
\(373\) 17.6569 0.914237 0.457119 0.889406i \(-0.348881\pi\)
0.457119 + 0.889406i \(0.348881\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −2.10051 −0.108182
\(378\) 0 0
\(379\) 14.3431 0.736758 0.368379 0.929676i \(-0.379913\pi\)
0.368379 + 0.929676i \(0.379913\pi\)
\(380\) 0 0
\(381\) 19.5563 1.00190
\(382\) 0 0
\(383\) −13.7279 −0.701464 −0.350732 0.936476i \(-0.614067\pi\)
−0.350732 + 0.936476i \(0.614067\pi\)
\(384\) 0 0
\(385\) −2.58579 −0.131784
\(386\) 0 0
\(387\) 7.65685 0.389220
\(388\) 0 0
\(389\) −19.4558 −0.986450 −0.493225 0.869902i \(-0.664182\pi\)
−0.493225 + 0.869902i \(0.664182\pi\)
\(390\) 0 0
\(391\) −2.41421 −0.122092
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) −16.9706 −0.853882
\(396\) 0 0
\(397\) −21.6569 −1.08693 −0.543463 0.839433i \(-0.682887\pi\)
−0.543463 + 0.839433i \(0.682887\pi\)
\(398\) 0 0
\(399\) −13.5563 −0.678666
\(400\) 0 0
\(401\) 38.4853 1.92186 0.960932 0.276786i \(-0.0892693\pi\)
0.960932 + 0.276786i \(0.0892693\pi\)
\(402\) 0 0
\(403\) 0.585786 0.0291801
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −10.5858 −0.524718
\(408\) 0 0
\(409\) 19.8284 0.980453 0.490226 0.871595i \(-0.336914\pi\)
0.490226 + 0.871595i \(0.336914\pi\)
\(410\) 0 0
\(411\) 2.82843 0.139516
\(412\) 0 0
\(413\) 25.1005 1.23512
\(414\) 0 0
\(415\) −2.75736 −0.135353
\(416\) 0 0
\(417\) 17.4853 0.856258
\(418\) 0 0
\(419\) −35.3553 −1.72722 −0.863611 0.504159i \(-0.831802\pi\)
−0.863611 + 0.504159i \(0.831802\pi\)
\(420\) 0 0
\(421\) 32.8701 1.60199 0.800994 0.598672i \(-0.204305\pi\)
0.800994 + 0.598672i \(0.204305\pi\)
\(422\) 0 0
\(423\) 6.24264 0.303528
\(424\) 0 0
\(425\) −2.41421 −0.117107
\(426\) 0 0
\(427\) −2.58579 −0.125135
\(428\) 0 0
\(429\) 0.828427 0.0399968
\(430\) 0 0
\(431\) −28.4853 −1.37209 −0.686044 0.727560i \(-0.740654\pi\)
−0.686044 + 0.727560i \(0.740654\pi\)
\(432\) 0 0
\(433\) 3.68629 0.177152 0.0885759 0.996069i \(-0.471768\pi\)
0.0885759 + 0.996069i \(0.471768\pi\)
\(434\) 0 0
\(435\) −3.58579 −0.171925
\(436\) 0 0
\(437\) 7.41421 0.354670
\(438\) 0 0
\(439\) 33.7990 1.61314 0.806569 0.591140i \(-0.201322\pi\)
0.806569 + 0.591140i \(0.201322\pi\)
\(440\) 0 0
\(441\) −3.65685 −0.174136
\(442\) 0 0
\(443\) −15.8995 −0.755408 −0.377704 0.925926i \(-0.623286\pi\)
−0.377704 + 0.925926i \(0.623286\pi\)
\(444\) 0 0
\(445\) −9.17157 −0.434774
\(446\) 0 0
\(447\) −2.24264 −0.106073
\(448\) 0 0
\(449\) 1.10051 0.0519360 0.0259680 0.999663i \(-0.491733\pi\)
0.0259680 + 0.999663i \(0.491733\pi\)
\(450\) 0 0
\(451\) 10.2426 0.482307
\(452\) 0 0
\(453\) 21.3137 1.00141
\(454\) 0 0
\(455\) −1.07107 −0.0502124
\(456\) 0 0
\(457\) −10.4558 −0.489104 −0.244552 0.969636i \(-0.578641\pi\)
−0.244552 + 0.969636i \(0.578641\pi\)
\(458\) 0 0
\(459\) −2.41421 −0.112686
\(460\) 0 0
\(461\) −27.4558 −1.27875 −0.639373 0.768897i \(-0.720806\pi\)
−0.639373 + 0.768897i \(0.720806\pi\)
\(462\) 0 0
\(463\) −22.0416 −1.02436 −0.512181 0.858878i \(-0.671162\pi\)
−0.512181 + 0.858878i \(0.671162\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −11.5858 −0.536126 −0.268063 0.963401i \(-0.586384\pi\)
−0.268063 + 0.963401i \(0.586384\pi\)
\(468\) 0 0
\(469\) −28.3137 −1.30741
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 0 0
\(473\) 10.8284 0.497892
\(474\) 0 0
\(475\) 7.41421 0.340187
\(476\) 0 0
\(477\) 5.24264 0.240044
\(478\) 0 0
\(479\) −20.1005 −0.918416 −0.459208 0.888329i \(-0.651867\pi\)
−0.459208 + 0.888329i \(0.651867\pi\)
\(480\) 0 0
\(481\) −4.38478 −0.199929
\(482\) 0 0
\(483\) −1.82843 −0.0831963
\(484\) 0 0
\(485\) 18.1421 0.823792
\(486\) 0 0
\(487\) −31.0711 −1.40796 −0.703982 0.710218i \(-0.748597\pi\)
−0.703982 + 0.710218i \(0.748597\pi\)
\(488\) 0 0
\(489\) −18.9706 −0.857879
\(490\) 0 0
\(491\) −23.1005 −1.04251 −0.521256 0.853401i \(-0.674536\pi\)
−0.521256 + 0.853401i \(0.674536\pi\)
\(492\) 0 0
\(493\) 8.65685 0.389885
\(494\) 0 0
\(495\) 1.41421 0.0635642
\(496\) 0 0
\(497\) −25.7279 −1.15406
\(498\) 0 0
\(499\) 24.4558 1.09479 0.547397 0.836873i \(-0.315619\pi\)
0.547397 + 0.836873i \(0.315619\pi\)
\(500\) 0 0
\(501\) 1.75736 0.0785130
\(502\) 0 0
\(503\) −20.7574 −0.925525 −0.462762 0.886482i \(-0.653142\pi\)
−0.462762 + 0.886482i \(0.653142\pi\)
\(504\) 0 0
\(505\) −10.0711 −0.448157
\(506\) 0 0
\(507\) −12.6569 −0.562111
\(508\) 0 0
\(509\) −20.4853 −0.907994 −0.453997 0.891003i \(-0.650002\pi\)
−0.453997 + 0.891003i \(0.650002\pi\)
\(510\) 0 0
\(511\) −4.10051 −0.181396
\(512\) 0 0
\(513\) 7.41421 0.327346
\(514\) 0 0
\(515\) 1.51472 0.0667465
\(516\) 0 0
\(517\) 8.82843 0.388274
\(518\) 0 0
\(519\) 21.6569 0.950630
\(520\) 0 0
\(521\) −24.7279 −1.08335 −0.541675 0.840588i \(-0.682210\pi\)
−0.541675 + 0.840588i \(0.682210\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) −1.82843 −0.0797991
\(526\) 0 0
\(527\) −2.41421 −0.105165
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −13.7279 −0.595741
\(532\) 0 0
\(533\) 4.24264 0.183769
\(534\) 0 0
\(535\) −3.58579 −0.155027
\(536\) 0 0
\(537\) 19.6569 0.848256
\(538\) 0 0
\(539\) −5.17157 −0.222755
\(540\) 0 0
\(541\) −43.1127 −1.85356 −0.926780 0.375605i \(-0.877435\pi\)
−0.926780 + 0.375605i \(0.877435\pi\)
\(542\) 0 0
\(543\) −3.51472 −0.150831
\(544\) 0 0
\(545\) −7.41421 −0.317590
\(546\) 0 0
\(547\) −6.34315 −0.271213 −0.135607 0.990763i \(-0.543298\pi\)
−0.135607 + 0.990763i \(0.543298\pi\)
\(548\) 0 0
\(549\) 1.41421 0.0603572
\(550\) 0 0
\(551\) −26.5858 −1.13259
\(552\) 0 0
\(553\) 31.0294 1.31951
\(554\) 0 0
\(555\) −7.48528 −0.317732
\(556\) 0 0
\(557\) 33.5269 1.42058 0.710290 0.703909i \(-0.248564\pi\)
0.710290 + 0.703909i \(0.248564\pi\)
\(558\) 0 0
\(559\) 4.48528 0.189707
\(560\) 0 0
\(561\) −3.41421 −0.144148
\(562\) 0 0
\(563\) 43.5858 1.83692 0.918461 0.395512i \(-0.129433\pi\)
0.918461 + 0.395512i \(0.129433\pi\)
\(564\) 0 0
\(565\) 18.8995 0.795108
\(566\) 0 0
\(567\) −1.82843 −0.0767867
\(568\) 0 0
\(569\) 18.4853 0.774943 0.387472 0.921882i \(-0.373349\pi\)
0.387472 + 0.921882i \(0.373349\pi\)
\(570\) 0 0
\(571\) 1.89949 0.0794914 0.0397457 0.999210i \(-0.487345\pi\)
0.0397457 + 0.999210i \(0.487345\pi\)
\(572\) 0 0
\(573\) −5.75736 −0.240517
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 24.2843 1.01097 0.505484 0.862836i \(-0.331314\pi\)
0.505484 + 0.862836i \(0.331314\pi\)
\(578\) 0 0
\(579\) 18.1421 0.753961
\(580\) 0 0
\(581\) 5.04163 0.209162
\(582\) 0 0
\(583\) 7.41421 0.307065
\(584\) 0 0
\(585\) 0.585786 0.0242193
\(586\) 0 0
\(587\) −36.6274 −1.51178 −0.755888 0.654701i \(-0.772794\pi\)
−0.755888 + 0.654701i \(0.772794\pi\)
\(588\) 0 0
\(589\) 7.41421 0.305497
\(590\) 0 0
\(591\) −7.17157 −0.294999
\(592\) 0 0
\(593\) −6.92893 −0.284537 −0.142269 0.989828i \(-0.545440\pi\)
−0.142269 + 0.989828i \(0.545440\pi\)
\(594\) 0 0
\(595\) 4.41421 0.180965
\(596\) 0 0
\(597\) 14.4853 0.592843
\(598\) 0 0
\(599\) 18.1421 0.741268 0.370634 0.928779i \(-0.379141\pi\)
0.370634 + 0.928779i \(0.379141\pi\)
\(600\) 0 0
\(601\) 22.6569 0.924192 0.462096 0.886830i \(-0.347097\pi\)
0.462096 + 0.886830i \(0.347097\pi\)
\(602\) 0 0
\(603\) 15.4853 0.630609
\(604\) 0 0
\(605\) −9.00000 −0.365902
\(606\) 0 0
\(607\) 12.7279 0.516610 0.258305 0.966063i \(-0.416836\pi\)
0.258305 + 0.966063i \(0.416836\pi\)
\(608\) 0 0
\(609\) 6.55635 0.265677
\(610\) 0 0
\(611\) 3.65685 0.147940
\(612\) 0 0
\(613\) −4.62742 −0.186900 −0.0934498 0.995624i \(-0.529789\pi\)
−0.0934498 + 0.995624i \(0.529789\pi\)
\(614\) 0 0
\(615\) 7.24264 0.292051
\(616\) 0 0
\(617\) 17.2426 0.694163 0.347081 0.937835i \(-0.387173\pi\)
0.347081 + 0.937835i \(0.387173\pi\)
\(618\) 0 0
\(619\) −16.2843 −0.654520 −0.327260 0.944934i \(-0.606125\pi\)
−0.327260 + 0.944934i \(0.606125\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 16.7696 0.671858
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 10.4853 0.418742
\(628\) 0 0
\(629\) 18.0711 0.720541
\(630\) 0 0
\(631\) −1.41421 −0.0562990 −0.0281495 0.999604i \(-0.508961\pi\)
−0.0281495 + 0.999604i \(0.508961\pi\)
\(632\) 0 0
\(633\) −20.7990 −0.826686
\(634\) 0 0
\(635\) 19.5563 0.776070
\(636\) 0 0
\(637\) −2.14214 −0.0848745
\(638\) 0 0
\(639\) 14.0711 0.556643
\(640\) 0 0
\(641\) −21.4142 −0.845811 −0.422905 0.906174i \(-0.638990\pi\)
−0.422905 + 0.906174i \(0.638990\pi\)
\(642\) 0 0
\(643\) −31.3431 −1.23605 −0.618027 0.786157i \(-0.712068\pi\)
−0.618027 + 0.786157i \(0.712068\pi\)
\(644\) 0 0
\(645\) 7.65685 0.301488
\(646\) 0 0
\(647\) 2.38478 0.0937552 0.0468776 0.998901i \(-0.485073\pi\)
0.0468776 + 0.998901i \(0.485073\pi\)
\(648\) 0 0
\(649\) −19.4142 −0.762075
\(650\) 0 0
\(651\) −1.82843 −0.0716617
\(652\) 0 0
\(653\) −16.7279 −0.654614 −0.327307 0.944918i \(-0.606141\pi\)
−0.327307 + 0.944918i \(0.606141\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 2.24264 0.0874937
\(658\) 0 0
\(659\) 6.38478 0.248716 0.124358 0.992237i \(-0.460313\pi\)
0.124358 + 0.992237i \(0.460313\pi\)
\(660\) 0 0
\(661\) 13.1127 0.510025 0.255012 0.966938i \(-0.417920\pi\)
0.255012 + 0.966938i \(0.417920\pi\)
\(662\) 0 0
\(663\) −1.41421 −0.0549235
\(664\) 0 0
\(665\) −13.5563 −0.525693
\(666\) 0 0
\(667\) −3.58579 −0.138842
\(668\) 0 0
\(669\) 18.9706 0.733444
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 7.27208 0.280318 0.140159 0.990129i \(-0.455239\pi\)
0.140159 + 0.990129i \(0.455239\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −43.1838 −1.65969 −0.829843 0.557996i \(-0.811570\pi\)
−0.829843 + 0.557996i \(0.811570\pi\)
\(678\) 0 0
\(679\) −33.1716 −1.27301
\(680\) 0 0
\(681\) 0.686292 0.0262987
\(682\) 0 0
\(683\) −43.0122 −1.64582 −0.822908 0.568175i \(-0.807650\pi\)
−0.822908 + 0.568175i \(0.807650\pi\)
\(684\) 0 0
\(685\) 2.82843 0.108069
\(686\) 0 0
\(687\) −16.4853 −0.628953
\(688\) 0 0
\(689\) 3.07107 0.116998
\(690\) 0 0
\(691\) 14.9706 0.569507 0.284754 0.958601i \(-0.408088\pi\)
0.284754 + 0.958601i \(0.408088\pi\)
\(692\) 0 0
\(693\) −2.58579 −0.0982259
\(694\) 0 0
\(695\) 17.4853 0.663255
\(696\) 0 0
\(697\) −17.4853 −0.662302
\(698\) 0 0
\(699\) −5.65685 −0.213962
\(700\) 0 0
\(701\) 49.5563 1.87172 0.935859 0.352375i \(-0.114626\pi\)
0.935859 + 0.352375i \(0.114626\pi\)
\(702\) 0 0
\(703\) −55.4975 −2.09313
\(704\) 0 0
\(705\) 6.24264 0.235111
\(706\) 0 0
\(707\) 18.4142 0.692538
\(708\) 0 0
\(709\) 17.6985 0.664681 0.332340 0.943160i \(-0.392162\pi\)
0.332340 + 0.943160i \(0.392162\pi\)
\(710\) 0 0
\(711\) −16.9706 −0.636446
\(712\) 0 0
\(713\) 1.00000 0.0374503
\(714\) 0 0
\(715\) 0.828427 0.0309814
\(716\) 0 0
\(717\) −4.41421 −0.164852
\(718\) 0 0
\(719\) −32.5563 −1.21415 −0.607073 0.794646i \(-0.707657\pi\)
−0.607073 + 0.794646i \(0.707657\pi\)
\(720\) 0 0
\(721\) −2.76955 −0.103144
\(722\) 0 0
\(723\) −15.2132 −0.565785
\(724\) 0 0
\(725\) −3.58579 −0.133173
\(726\) 0 0
\(727\) −9.00000 −0.333792 −0.166896 0.985975i \(-0.553374\pi\)
−0.166896 + 0.985975i \(0.553374\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.4853 −0.683703
\(732\) 0 0
\(733\) 16.3137 0.602561 0.301280 0.953536i \(-0.402586\pi\)
0.301280 + 0.953536i \(0.402586\pi\)
\(734\) 0 0
\(735\) −3.65685 −0.134885
\(736\) 0 0
\(737\) 21.8995 0.806678
\(738\) 0 0
\(739\) −17.1421 −0.630584 −0.315292 0.948995i \(-0.602102\pi\)
−0.315292 + 0.948995i \(0.602102\pi\)
\(740\) 0 0
\(741\) 4.34315 0.159549
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) −2.24264 −0.0821640
\(746\) 0 0
\(747\) −2.75736 −0.100887
\(748\) 0 0
\(749\) 6.55635 0.239564
\(750\) 0 0
\(751\) 45.0711 1.64467 0.822333 0.569006i \(-0.192672\pi\)
0.822333 + 0.569006i \(0.192672\pi\)
\(752\) 0 0
\(753\) 14.9706 0.545558
\(754\) 0 0
\(755\) 21.3137 0.775685
\(756\) 0 0
\(757\) −1.82843 −0.0664553 −0.0332277 0.999448i \(-0.510579\pi\)
−0.0332277 + 0.999448i \(0.510579\pi\)
\(758\) 0 0
\(759\) 1.41421 0.0513327
\(760\) 0 0
\(761\) −23.9289 −0.867423 −0.433712 0.901052i \(-0.642796\pi\)
−0.433712 + 0.901052i \(0.642796\pi\)
\(762\) 0 0
\(763\) 13.5563 0.490773
\(764\) 0 0
\(765\) −2.41421 −0.0872861
\(766\) 0 0
\(767\) −8.04163 −0.290366
\(768\) 0 0
\(769\) −15.2721 −0.550725 −0.275363 0.961340i \(-0.588798\pi\)
−0.275363 + 0.961340i \(0.588798\pi\)
\(770\) 0 0
\(771\) −11.4142 −0.411073
\(772\) 0 0
\(773\) −13.7990 −0.496315 −0.248158 0.968720i \(-0.579825\pi\)
−0.248158 + 0.968720i \(0.579825\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) 13.6863 0.490993
\(778\) 0 0
\(779\) 53.6985 1.92395
\(780\) 0 0
\(781\) 19.8995 0.712060
\(782\) 0 0
\(783\) −3.58579 −0.128146
\(784\) 0 0
\(785\) −3.00000 −0.107075
\(786\) 0 0
\(787\) 19.2843 0.687410 0.343705 0.939078i \(-0.388318\pi\)
0.343705 + 0.939078i \(0.388318\pi\)
\(788\) 0 0
\(789\) −12.4142 −0.441958
\(790\) 0 0
\(791\) −34.5563 −1.22868
\(792\) 0 0
\(793\) 0.828427 0.0294183
\(794\) 0 0
\(795\) 5.24264 0.185937
\(796\) 0 0
\(797\) −34.8995 −1.23620 −0.618102 0.786098i \(-0.712098\pi\)
−0.618102 + 0.786098i \(0.712098\pi\)
\(798\) 0 0
\(799\) −15.0711 −0.533176
\(800\) 0 0
\(801\) −9.17157 −0.324062
\(802\) 0 0
\(803\) 3.17157 0.111922
\(804\) 0 0
\(805\) −1.82843 −0.0644436
\(806\) 0 0
\(807\) 12.0711 0.424922
\(808\) 0 0
\(809\) 29.7279 1.04518 0.522589 0.852585i \(-0.324966\pi\)
0.522589 + 0.852585i \(0.324966\pi\)
\(810\) 0 0
\(811\) 5.14214 0.180565 0.0902824 0.995916i \(-0.471223\pi\)
0.0902824 + 0.995916i \(0.471223\pi\)
\(812\) 0 0
\(813\) −18.3137 −0.642290
\(814\) 0 0
\(815\) −18.9706 −0.664510
\(816\) 0 0
\(817\) 56.7696 1.98612
\(818\) 0 0
\(819\) −1.07107 −0.0374261
\(820\) 0 0
\(821\) 33.4558 1.16762 0.583809 0.811891i \(-0.301562\pi\)
0.583809 + 0.811891i \(0.301562\pi\)
\(822\) 0 0
\(823\) 53.6569 1.87036 0.935180 0.354172i \(-0.115237\pi\)
0.935180 + 0.354172i \(0.115237\pi\)
\(824\) 0 0
\(825\) 1.41421 0.0492366
\(826\) 0 0
\(827\) −2.27208 −0.0790079 −0.0395039 0.999219i \(-0.512578\pi\)
−0.0395039 + 0.999219i \(0.512578\pi\)
\(828\) 0 0
\(829\) −7.82843 −0.271893 −0.135946 0.990716i \(-0.543407\pi\)
−0.135946 + 0.990716i \(0.543407\pi\)
\(830\) 0 0
\(831\) 18.9706 0.658082
\(832\) 0 0
\(833\) 8.82843 0.305887
\(834\) 0 0
\(835\) 1.75736 0.0608159
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) −28.3431 −0.978514 −0.489257 0.872140i \(-0.662732\pi\)
−0.489257 + 0.872140i \(0.662732\pi\)
\(840\) 0 0
\(841\) −16.1421 −0.556625
\(842\) 0 0
\(843\) 3.75736 0.129410
\(844\) 0 0
\(845\) −12.6569 −0.435409
\(846\) 0 0
\(847\) 16.4558 0.565429
\(848\) 0 0
\(849\) 25.6274 0.879531
\(850\) 0 0
\(851\) −7.48528 −0.256592
\(852\) 0 0
\(853\) 6.48528 0.222052 0.111026 0.993818i \(-0.464586\pi\)
0.111026 + 0.993818i \(0.464586\pi\)
\(854\) 0 0
\(855\) 7.41421 0.253561
\(856\) 0 0
\(857\) −30.6863 −1.04822 −0.524112 0.851649i \(-0.675603\pi\)
−0.524112 + 0.851649i \(0.675603\pi\)
\(858\) 0 0
\(859\) 53.8284 1.83660 0.918301 0.395883i \(-0.129561\pi\)
0.918301 + 0.395883i \(0.129561\pi\)
\(860\) 0 0
\(861\) −13.2426 −0.451308
\(862\) 0 0
\(863\) −46.4853 −1.58238 −0.791189 0.611572i \(-0.790537\pi\)
−0.791189 + 0.611572i \(0.790537\pi\)
\(864\) 0 0
\(865\) 21.6569 0.736355
\(866\) 0 0
\(867\) −11.1716 −0.379407
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 9.07107 0.307361
\(872\) 0 0
\(873\) 18.1421 0.614018
\(874\) 0 0
\(875\) −1.82843 −0.0618121
\(876\) 0 0
\(877\) −7.17157 −0.242167 −0.121083 0.992642i \(-0.538637\pi\)
−0.121083 + 0.992642i \(0.538637\pi\)
\(878\) 0 0
\(879\) 10.8995 0.367631
\(880\) 0 0
\(881\) −10.9289 −0.368205 −0.184103 0.982907i \(-0.558938\pi\)
−0.184103 + 0.982907i \(0.558938\pi\)
\(882\) 0 0
\(883\) 32.9289 1.10815 0.554073 0.832468i \(-0.313073\pi\)
0.554073 + 0.832468i \(0.313073\pi\)
\(884\) 0 0
\(885\) −13.7279 −0.461459
\(886\) 0 0
\(887\) 4.82843 0.162123 0.0810614 0.996709i \(-0.474169\pi\)
0.0810614 + 0.996709i \(0.474169\pi\)
\(888\) 0 0
\(889\) −35.7574 −1.19926
\(890\) 0 0
\(891\) 1.41421 0.0473779
\(892\) 0 0
\(893\) 46.2843 1.54884
\(894\) 0 0
\(895\) 19.6569 0.657056
\(896\) 0 0
\(897\) 0.585786 0.0195588
\(898\) 0 0
\(899\) −3.58579 −0.119593
\(900\) 0 0
\(901\) −12.6569 −0.421661
\(902\) 0 0
\(903\) −14.0000 −0.465891
\(904\) 0 0
\(905\) −3.51472 −0.116833
\(906\) 0 0
\(907\) −18.1716 −0.603377 −0.301689 0.953407i \(-0.597550\pi\)
−0.301689 + 0.953407i \(0.597550\pi\)
\(908\) 0 0
\(909\) −10.0711 −0.334036
\(910\) 0 0
\(911\) −28.9706 −0.959838 −0.479919 0.877313i \(-0.659334\pi\)
−0.479919 + 0.877313i \(0.659334\pi\)
\(912\) 0 0
\(913\) −3.89949 −0.129054
\(914\) 0 0
\(915\) 1.41421 0.0467525
\(916\) 0 0
\(917\) −29.2548 −0.966080
\(918\) 0 0
\(919\) −24.2843 −0.801064 −0.400532 0.916283i \(-0.631175\pi\)
−0.400532 + 0.916283i \(0.631175\pi\)
\(920\) 0 0
\(921\) −3.89949 −0.128493
\(922\) 0 0
\(923\) 8.24264 0.271310
\(924\) 0 0
\(925\) −7.48528 −0.246115
\(926\) 0 0
\(927\) 1.51472 0.0497499
\(928\) 0 0
\(929\) −32.7574 −1.07473 −0.537367 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(930\) 0 0
\(931\) −27.1127 −0.888583
\(932\) 0 0
\(933\) −31.3137 −1.02516
\(934\) 0 0
\(935\) −3.41421 −0.111657
\(936\) 0 0
\(937\) 16.4853 0.538551 0.269275 0.963063i \(-0.413216\pi\)
0.269275 + 0.963063i \(0.413216\pi\)
\(938\) 0 0
\(939\) 21.1421 0.689948
\(940\) 0 0
\(941\) 6.52691 0.212771 0.106386 0.994325i \(-0.466072\pi\)
0.106386 + 0.994325i \(0.466072\pi\)
\(942\) 0 0
\(943\) 7.24264 0.235853
\(944\) 0 0
\(945\) −1.82843 −0.0594787
\(946\) 0 0
\(947\) 28.4853 0.925647 0.462824 0.886450i \(-0.346836\pi\)
0.462824 + 0.886450i \(0.346836\pi\)
\(948\) 0 0
\(949\) 1.31371 0.0426448
\(950\) 0 0
\(951\) −7.41421 −0.240422
\(952\) 0 0
\(953\) −4.62742 −0.149897 −0.0749484 0.997187i \(-0.523879\pi\)
−0.0749484 + 0.997187i \(0.523879\pi\)
\(954\) 0 0
\(955\) −5.75736 −0.186304
\(956\) 0 0
\(957\) −5.07107 −0.163924
\(958\) 0 0
\(959\) −5.17157 −0.166999
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −3.58579 −0.115550
\(964\) 0 0
\(965\) 18.1421 0.584016
\(966\) 0 0
\(967\) −42.3848 −1.36300 −0.681501 0.731817i \(-0.738673\pi\)
−0.681501 + 0.731817i \(0.738673\pi\)
\(968\) 0 0
\(969\) −17.8995 −0.575015
\(970\) 0 0
\(971\) 34.4264 1.10480 0.552398 0.833581i \(-0.313713\pi\)
0.552398 + 0.833581i \(0.313713\pi\)
\(972\) 0 0
\(973\) −31.9706 −1.02493
\(974\) 0 0
\(975\) 0.585786 0.0187602
\(976\) 0 0
\(977\) −49.8701 −1.59548 −0.797742 0.602999i \(-0.793972\pi\)
−0.797742 + 0.602999i \(0.793972\pi\)
\(978\) 0 0
\(979\) −12.9706 −0.414541
\(980\) 0 0
\(981\) −7.41421 −0.236718
\(982\) 0 0
\(983\) 37.5269 1.19692 0.598461 0.801152i \(-0.295779\pi\)
0.598461 + 0.801152i \(0.295779\pi\)
\(984\) 0 0
\(985\) −7.17157 −0.228505
\(986\) 0 0
\(987\) −11.4142 −0.363318
\(988\) 0 0
\(989\) 7.65685 0.243474
\(990\) 0 0
\(991\) −50.4558 −1.60278 −0.801391 0.598140i \(-0.795907\pi\)
−0.801391 + 0.598140i \(0.795907\pi\)
\(992\) 0 0
\(993\) 23.8284 0.756172
\(994\) 0 0
\(995\) 14.4853 0.459214
\(996\) 0 0
\(997\) 19.5147 0.618037 0.309019 0.951056i \(-0.399999\pi\)
0.309019 + 0.951056i \(0.399999\pi\)
\(998\) 0 0
\(999\) −7.48528 −0.236824
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 5520.2.a.bt.1.1 2
4.3 odd 2 2760.2.a.n.1.2 2
12.11 even 2 8280.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.n.1.2 2 4.3 odd 2
5520.2.a.bt.1.1 2 1.1 even 1 trivial
8280.2.a.x.1.2 2 12.11 even 2