Properties

Label 2760.2.a.o.1.1
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -2.37228 q^{7} +1.00000 q^{9} +4.00000 q^{11} -6.74456 q^{13} -1.00000 q^{15} -0.372281 q^{17} +4.00000 q^{19} +2.37228 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.372281 q^{29} -2.37228 q^{31} -4.00000 q^{33} -2.37228 q^{35} +0.372281 q^{37} +6.74456 q^{39} +9.11684 q^{41} -4.00000 q^{43} +1.00000 q^{45} +4.74456 q^{47} -1.37228 q^{49} +0.372281 q^{51} -4.37228 q^{53} +4.00000 q^{55} -4.00000 q^{57} +14.3723 q^{59} -2.00000 q^{61} -2.37228 q^{63} -6.74456 q^{65} +9.62772 q^{67} -1.00000 q^{69} -2.37228 q^{71} -1.25544 q^{73} -1.00000 q^{75} -9.48913 q^{77} +9.48913 q^{79} +1.00000 q^{81} +15.8614 q^{83} -0.372281 q^{85} -0.372281 q^{87} +6.74456 q^{89} +16.0000 q^{91} +2.37228 q^{93} +4.00000 q^{95} -7.48913 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 8 q^{11} - 2 q^{13} - 2 q^{15} + 5 q^{17} + 8 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 5 q^{29} + q^{31} - 8 q^{33} + q^{35} - 5 q^{37} + 2 q^{39} + q^{41} - 8 q^{43} + 2 q^{45} - 2 q^{47} + 3 q^{49} - 5 q^{51} - 3 q^{53} + 8 q^{55} - 8 q^{57} + 23 q^{59} - 4 q^{61} + q^{63} - 2 q^{65} + 25 q^{67} - 2 q^{69} + q^{71} - 14 q^{73} - 2 q^{75} + 4 q^{77} - 4 q^{79} + 2 q^{81} + 3 q^{83} + 5 q^{85} + 5 q^{87} + 2 q^{89} + 32 q^{91} - q^{93} + 8 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(7\) −2.37228 −0.896638 −0.448319 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −6.74456 −1.87061 −0.935303 0.353849i \(-0.884873\pi\)
−0.935303 + 0.353849i \(0.884873\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.372281 −0.0902915 −0.0451457 0.998980i \(-0.514375\pi\)
−0.0451457 + 0.998980i \(0.514375\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 2.37228 0.517674
\(22\) 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
\(29\) 0.372281 0.0691309 0.0345655 0.999402i \(-0.488995\pi\)
0.0345655 + 0.999402i \(0.488995\pi\)
\(30\) 0 0
\(31\) −2.37228 −0.426074 −0.213037 0.977044i \(-0.568336\pi\)
−0.213037 + 0.977044i \(0.568336\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −2.37228 −0.400989
\(36\) 0 0
\(37\) 0.372281 0.0612027 0.0306013 0.999532i \(-0.490258\pi\)
0.0306013 + 0.999532i \(0.490258\pi\)
\(38\) 0 0
\(39\) 6.74456 1.07999
\(40\) 0 0
\(41\) 9.11684 1.42381 0.711906 0.702275i \(-0.247832\pi\)
0.711906 + 0.702275i \(0.247832\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 4.74456 0.692066 0.346033 0.938222i \(-0.387529\pi\)
0.346033 + 0.938222i \(0.387529\pi\)
\(48\) 0 0
\(49\) −1.37228 −0.196040
\(50\) 0 0
\(51\) 0.372281 0.0521298
\(52\) 0 0
\(53\) −4.37228 −0.600579 −0.300290 0.953848i \(-0.597083\pi\)
−0.300290 + 0.953848i \(0.597083\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 14.3723 1.87111 0.935556 0.353179i \(-0.114899\pi\)
0.935556 + 0.353179i \(0.114899\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −2.37228 −0.298879
\(64\) 0 0
\(65\) −6.74456 −0.836560
\(66\) 0 0
\(67\) 9.62772 1.17621 0.588107 0.808783i \(-0.299874\pi\)
0.588107 + 0.808783i \(0.299874\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −2.37228 −0.281538 −0.140769 0.990042i \(-0.544957\pi\)
−0.140769 + 0.990042i \(0.544957\pi\)
\(72\) 0 0
\(73\) −1.25544 −0.146938 −0.0734689 0.997298i \(-0.523407\pi\)
−0.0734689 + 0.997298i \(0.523407\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −9.48913 −1.08139
\(78\) 0 0
\(79\) 9.48913 1.06761 0.533805 0.845608i \(-0.320762\pi\)
0.533805 + 0.845608i \(0.320762\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.8614 1.74102 0.870508 0.492155i \(-0.163791\pi\)
0.870508 + 0.492155i \(0.163791\pi\)
\(84\) 0 0
\(85\) −0.372281 −0.0403796
\(86\) 0 0
\(87\) −0.372281 −0.0399127
\(88\) 0 0
\(89\) 6.74456 0.714922 0.357461 0.933928i \(-0.383642\pi\)
0.357461 + 0.933928i \(0.383642\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) 2.37228 0.245994
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −7.48913 −0.760405 −0.380203 0.924903i \(-0.624146\pi\)
−0.380203 + 0.924903i \(0.624146\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −7.62772 −0.758986 −0.379493 0.925195i \(-0.623902\pi\)
−0.379493 + 0.925195i \(0.623902\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 2.37228 0.231511
\(106\) 0 0
\(107\) −1.62772 −0.157358 −0.0786788 0.996900i \(-0.525070\pi\)
−0.0786788 + 0.996900i \(0.525070\pi\)
\(108\) 0 0
\(109\) −6.74456 −0.646012 −0.323006 0.946397i \(-0.604693\pi\)
−0.323006 + 0.946397i \(0.604693\pi\)
\(110\) 0 0
\(111\) −0.372281 −0.0353354
\(112\) 0 0
\(113\) 9.11684 0.857641 0.428820 0.903390i \(-0.358929\pi\)
0.428820 + 0.903390i \(0.358929\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −6.74456 −0.623535
\(118\) 0 0
\(119\) 0.883156 0.0809588
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −9.11684 −0.822038
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.7446 1.13090 0.565449 0.824784i \(-0.308703\pi\)
0.565449 + 0.824784i \(0.308703\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −9.48913 −0.822812
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 11.4891 0.981582 0.490791 0.871277i \(-0.336708\pi\)
0.490791 + 0.871277i \(0.336708\pi\)
\(138\) 0 0
\(139\) 14.3723 1.21904 0.609520 0.792770i \(-0.291362\pi\)
0.609520 + 0.792770i \(0.291362\pi\)
\(140\) 0 0
\(141\) −4.74456 −0.399564
\(142\) 0 0
\(143\) −26.9783 −2.25603
\(144\) 0 0
\(145\) 0.372281 0.0309163
\(146\) 0 0
\(147\) 1.37228 0.113184
\(148\) 0 0
\(149\) −19.4891 −1.59661 −0.798306 0.602252i \(-0.794270\pi\)
−0.798306 + 0.602252i \(0.794270\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −0.372281 −0.0300972
\(154\) 0 0
\(155\) −2.37228 −0.190546
\(156\) 0 0
\(157\) 22.6060 1.80415 0.902076 0.431576i \(-0.142042\pi\)
0.902076 + 0.431576i \(0.142042\pi\)
\(158\) 0 0
\(159\) 4.37228 0.346744
\(160\) 0 0
\(161\) −2.37228 −0.186962
\(162\) 0 0
\(163\) 16.7446 1.31154 0.655768 0.754963i \(-0.272345\pi\)
0.655768 + 0.754963i \(0.272345\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) 12.7446 0.986204 0.493102 0.869972i \(-0.335863\pi\)
0.493102 + 0.869972i \(0.335863\pi\)
\(168\) 0 0
\(169\) 32.4891 2.49916
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 20.2337 1.53834 0.769169 0.639045i \(-0.220670\pi\)
0.769169 + 0.639045i \(0.220670\pi\)
\(174\) 0 0
\(175\) −2.37228 −0.179328
\(176\) 0 0
\(177\) −14.3723 −1.08029
\(178\) 0 0
\(179\) −13.4891 −1.00822 −0.504112 0.863638i \(-0.668180\pi\)
−0.504112 + 0.863638i \(0.668180\pi\)
\(180\) 0 0
\(181\) 20.2337 1.50396 0.751979 0.659187i \(-0.229099\pi\)
0.751979 + 0.659187i \(0.229099\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0.372281 0.0273707
\(186\) 0 0
\(187\) −1.48913 −0.108896
\(188\) 0 0
\(189\) 2.37228 0.172558
\(190\) 0 0
\(191\) −14.2337 −1.02991 −0.514957 0.857216i \(-0.672192\pi\)
−0.514957 + 0.857216i \(0.672192\pi\)
\(192\) 0 0
\(193\) −9.25544 −0.666221 −0.333110 0.942888i \(-0.608098\pi\)
−0.333110 + 0.942888i \(0.608098\pi\)
\(194\) 0 0
\(195\) 6.74456 0.482988
\(196\) 0 0
\(197\) −11.4891 −0.818566 −0.409283 0.912407i \(-0.634221\pi\)
−0.409283 + 0.912407i \(0.634221\pi\)
\(198\) 0 0
\(199\) 12.7446 0.903438 0.451719 0.892160i \(-0.350811\pi\)
0.451719 + 0.892160i \(0.350811\pi\)
\(200\) 0 0
\(201\) −9.62772 −0.679087
\(202\) 0 0
\(203\) −0.883156 −0.0619854
\(204\) 0 0
\(205\) 9.11684 0.636748
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −7.86141 −0.541202 −0.270601 0.962692i \(-0.587222\pi\)
−0.270601 + 0.962692i \(0.587222\pi\)
\(212\) 0 0
\(213\) 2.37228 0.162546
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 5.62772 0.382034
\(218\) 0 0
\(219\) 1.25544 0.0848346
\(220\) 0 0
\(221\) 2.51087 0.168900
\(222\) 0 0
\(223\) 28.7446 1.92488 0.962439 0.271498i \(-0.0875189\pi\)
0.962439 + 0.271498i \(0.0875189\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 17.2554 1.14027 0.570136 0.821551i \(-0.306891\pi\)
0.570136 + 0.821551i \(0.306891\pi\)
\(230\) 0 0
\(231\) 9.48913 0.624339
\(232\) 0 0
\(233\) 8.23369 0.539407 0.269703 0.962943i \(-0.413074\pi\)
0.269703 + 0.962943i \(0.413074\pi\)
\(234\) 0 0
\(235\) 4.74456 0.309501
\(236\) 0 0
\(237\) −9.48913 −0.616385
\(238\) 0 0
\(239\) −5.62772 −0.364027 −0.182013 0.983296i \(-0.558261\pi\)
−0.182013 + 0.983296i \(0.558261\pi\)
\(240\) 0 0
\(241\) −18.7446 −1.20744 −0.603722 0.797195i \(-0.706316\pi\)
−0.603722 + 0.797195i \(0.706316\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.37228 −0.0876718
\(246\) 0 0
\(247\) −26.9783 −1.71658
\(248\) 0 0
\(249\) −15.8614 −1.00518
\(250\) 0 0
\(251\) −7.25544 −0.457959 −0.228980 0.973431i \(-0.573539\pi\)
−0.228980 + 0.973431i \(0.573539\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0.372281 0.0233132
\(256\) 0 0
\(257\) 11.4891 0.716672 0.358336 0.933593i \(-0.383344\pi\)
0.358336 + 0.933593i \(0.383344\pi\)
\(258\) 0 0
\(259\) −0.883156 −0.0548766
\(260\) 0 0
\(261\) 0.372281 0.0230436
\(262\) 0 0
\(263\) −27.8614 −1.71801 −0.859004 0.511969i \(-0.828916\pi\)
−0.859004 + 0.511969i \(0.828916\pi\)
\(264\) 0 0
\(265\) −4.37228 −0.268587
\(266\) 0 0
\(267\) −6.74456 −0.412761
\(268\) 0 0
\(269\) −13.8614 −0.845145 −0.422572 0.906329i \(-0.638873\pi\)
−0.422572 + 0.906329i \(0.638873\pi\)
\(270\) 0 0
\(271\) −23.1168 −1.40425 −0.702124 0.712055i \(-0.747765\pi\)
−0.702124 + 0.712055i \(0.747765\pi\)
\(272\) 0 0
\(273\) −16.0000 −0.968364
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) −2.37228 −0.142025
\(280\) 0 0
\(281\) −12.2337 −0.729801 −0.364900 0.931047i \(-0.618897\pi\)
−0.364900 + 0.931047i \(0.618897\pi\)
\(282\) 0 0
\(283\) 6.37228 0.378793 0.189396 0.981901i \(-0.439347\pi\)
0.189396 + 0.981901i \(0.439347\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −21.6277 −1.27664
\(288\) 0 0
\(289\) −16.8614 −0.991847
\(290\) 0 0
\(291\) 7.48913 0.439020
\(292\) 0 0
\(293\) 9.86141 0.576110 0.288055 0.957614i \(-0.406991\pi\)
0.288055 + 0.957614i \(0.406991\pi\)
\(294\) 0 0
\(295\) 14.3723 0.836787
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) −6.74456 −0.390048
\(300\) 0 0
\(301\) 9.48913 0.546944
\(302\) 0 0
\(303\) 7.62772 0.438201
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −25.8614 −1.46177 −0.730887 0.682499i \(-0.760893\pi\)
−0.730887 + 0.682499i \(0.760893\pi\)
\(314\) 0 0
\(315\) −2.37228 −0.133663
\(316\) 0 0
\(317\) 31.4891 1.76861 0.884303 0.466914i \(-0.154634\pi\)
0.884303 + 0.466914i \(0.154634\pi\)
\(318\) 0 0
\(319\) 1.48913 0.0833750
\(320\) 0 0
\(321\) 1.62772 0.0908504
\(322\) 0 0
\(323\) −1.48913 −0.0828571
\(324\) 0 0
\(325\) −6.74456 −0.374121
\(326\) 0 0
\(327\) 6.74456 0.372975
\(328\) 0 0
\(329\) −11.2554 −0.620532
\(330\) 0 0
\(331\) 19.1168 1.05076 0.525378 0.850869i \(-0.323924\pi\)
0.525378 + 0.850869i \(0.323924\pi\)
\(332\) 0 0
\(333\) 0.372281 0.0204009
\(334\) 0 0
\(335\) 9.62772 0.526018
\(336\) 0 0
\(337\) −4.51087 −0.245723 −0.122862 0.992424i \(-0.539207\pi\)
−0.122862 + 0.992424i \(0.539207\pi\)
\(338\) 0 0
\(339\) −9.11684 −0.495159
\(340\) 0 0
\(341\) −9.48913 −0.513865
\(342\) 0 0
\(343\) 19.8614 1.07242
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 13.1168 0.702129 0.351064 0.936351i \(-0.385820\pi\)
0.351064 + 0.936351i \(0.385820\pi\)
\(350\) 0 0
\(351\) 6.74456 0.359998
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) −2.37228 −0.125908
\(356\) 0 0
\(357\) −0.883156 −0.0467416
\(358\) 0 0
\(359\) 31.7228 1.67427 0.837133 0.546999i \(-0.184230\pi\)
0.837133 + 0.546999i \(0.184230\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −1.25544 −0.0657126
\(366\) 0 0
\(367\) 3.86141 0.201564 0.100782 0.994909i \(-0.467866\pi\)
0.100782 + 0.994909i \(0.467866\pi\)
\(368\) 0 0
\(369\) 9.11684 0.474604
\(370\) 0 0
\(371\) 10.3723 0.538502
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −2.51087 −0.129317
\(378\) 0 0
\(379\) −13.4891 −0.692890 −0.346445 0.938070i \(-0.612611\pi\)
−0.346445 + 0.938070i \(0.612611\pi\)
\(380\) 0 0
\(381\) −12.7446 −0.652924
\(382\) 0 0
\(383\) −0.883156 −0.0451272 −0.0225636 0.999745i \(-0.507183\pi\)
−0.0225636 + 0.999745i \(0.507183\pi\)
\(384\) 0 0
\(385\) −9.48913 −0.483611
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −19.4891 −0.988138 −0.494069 0.869423i \(-0.664491\pi\)
−0.494069 + 0.869423i \(0.664491\pi\)
\(390\) 0 0
\(391\) −0.372281 −0.0188271
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) 9.48913 0.477450
\(396\) 0 0
\(397\) −0.233688 −0.0117285 −0.00586423 0.999983i \(-0.501867\pi\)
−0.00586423 + 0.999983i \(0.501867\pi\)
\(398\) 0 0
\(399\) 9.48913 0.475050
\(400\) 0 0
\(401\) 0.510875 0.0255119 0.0127559 0.999919i \(-0.495940\pi\)
0.0127559 + 0.999919i \(0.495940\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 1.48913 0.0738132
\(408\) 0 0
\(409\) 20.3723 1.00734 0.503672 0.863895i \(-0.331982\pi\)
0.503672 + 0.863895i \(0.331982\pi\)
\(410\) 0 0
\(411\) −11.4891 −0.566717
\(412\) 0 0
\(413\) −34.0951 −1.67771
\(414\) 0 0
\(415\) 15.8614 0.778606
\(416\) 0 0
\(417\) −14.3723 −0.703814
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −14.7446 −0.718606 −0.359303 0.933221i \(-0.616986\pi\)
−0.359303 + 0.933221i \(0.616986\pi\)
\(422\) 0 0
\(423\) 4.74456 0.230689
\(424\) 0 0
\(425\) −0.372281 −0.0180583
\(426\) 0 0
\(427\) 4.74456 0.229605
\(428\) 0 0
\(429\) 26.9783 1.30252
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) −32.0951 −1.54239 −0.771196 0.636598i \(-0.780341\pi\)
−0.771196 + 0.636598i \(0.780341\pi\)
\(434\) 0 0
\(435\) −0.372281 −0.0178495
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −18.9783 −0.905782 −0.452891 0.891566i \(-0.649607\pi\)
−0.452891 + 0.891566i \(0.649607\pi\)
\(440\) 0 0
\(441\) −1.37228 −0.0653467
\(442\) 0 0
\(443\) −32.7446 −1.55574 −0.777871 0.628425i \(-0.783700\pi\)
−0.777871 + 0.628425i \(0.783700\pi\)
\(444\) 0 0
\(445\) 6.74456 0.319723
\(446\) 0 0
\(447\) 19.4891 0.921804
\(448\) 0 0
\(449\) 21.8614 1.03170 0.515852 0.856678i \(-0.327476\pi\)
0.515852 + 0.856678i \(0.327476\pi\)
\(450\) 0 0
\(451\) 36.4674 1.71718
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 2.60597 0.121902 0.0609511 0.998141i \(-0.480587\pi\)
0.0609511 + 0.998141i \(0.480587\pi\)
\(458\) 0 0
\(459\) 0.372281 0.0173766
\(460\) 0 0
\(461\) 34.4674 1.60531 0.802653 0.596446i \(-0.203421\pi\)
0.802653 + 0.596446i \(0.203421\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 2.37228 0.110012
\(466\) 0 0
\(467\) 9.35053 0.432691 0.216346 0.976317i \(-0.430586\pi\)
0.216346 + 0.976317i \(0.430586\pi\)
\(468\) 0 0
\(469\) −22.8397 −1.05464
\(470\) 0 0
\(471\) −22.6060 −1.04163
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −4.37228 −0.200193
\(478\) 0 0
\(479\) 23.7228 1.08392 0.541962 0.840403i \(-0.317682\pi\)
0.541962 + 0.840403i \(0.317682\pi\)
\(480\) 0 0
\(481\) −2.51087 −0.114486
\(482\) 0 0
\(483\) 2.37228 0.107943
\(484\) 0 0
\(485\) −7.48913 −0.340064
\(486\) 0 0
\(487\) −23.7228 −1.07498 −0.537492 0.843269i \(-0.680628\pi\)
−0.537492 + 0.843269i \(0.680628\pi\)
\(488\) 0 0
\(489\) −16.7446 −0.757215
\(490\) 0 0
\(491\) −22.3723 −1.00965 −0.504823 0.863223i \(-0.668442\pi\)
−0.504823 + 0.863223i \(0.668442\pi\)
\(492\) 0 0
\(493\) −0.138593 −0.00624193
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 5.62772 0.252438
\(498\) 0 0
\(499\) 3.39403 0.151938 0.0759688 0.997110i \(-0.475795\pi\)
0.0759688 + 0.997110i \(0.475795\pi\)
\(500\) 0 0
\(501\) −12.7446 −0.569385
\(502\) 0 0
\(503\) −34.3723 −1.53258 −0.766292 0.642492i \(-0.777900\pi\)
−0.766292 + 0.642492i \(0.777900\pi\)
\(504\) 0 0
\(505\) −7.62772 −0.339429
\(506\) 0 0
\(507\) −32.4891 −1.44289
\(508\) 0 0
\(509\) 24.9783 1.10714 0.553571 0.832802i \(-0.313265\pi\)
0.553571 + 0.832802i \(0.313265\pi\)
\(510\) 0 0
\(511\) 2.97825 0.131750
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 18.9783 0.834663
\(518\) 0 0
\(519\) −20.2337 −0.888160
\(520\) 0 0
\(521\) −12.2337 −0.535968 −0.267984 0.963423i \(-0.586357\pi\)
−0.267984 + 0.963423i \(0.586357\pi\)
\(522\) 0 0
\(523\) −6.97825 −0.305138 −0.152569 0.988293i \(-0.548755\pi\)
−0.152569 + 0.988293i \(0.548755\pi\)
\(524\) 0 0
\(525\) 2.37228 0.103535
\(526\) 0 0
\(527\) 0.883156 0.0384709
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 14.3723 0.623704
\(532\) 0 0
\(533\) −61.4891 −2.66339
\(534\) 0 0
\(535\) −1.62772 −0.0703724
\(536\) 0 0
\(537\) 13.4891 0.582099
\(538\) 0 0
\(539\) −5.48913 −0.236433
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) −20.2337 −0.868311
\(544\) 0 0
\(545\) −6.74456 −0.288905
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 1.48913 0.0634389
\(552\) 0 0
\(553\) −22.5109 −0.957260
\(554\) 0 0
\(555\) −0.372281 −0.0158025
\(556\) 0 0
\(557\) −15.3505 −0.650423 −0.325211 0.945641i \(-0.605435\pi\)
−0.325211 + 0.945641i \(0.605435\pi\)
\(558\) 0 0
\(559\) 26.9783 1.14106
\(560\) 0 0
\(561\) 1.48913 0.0628709
\(562\) 0 0
\(563\) 1.62772 0.0686002 0.0343001 0.999412i \(-0.489080\pi\)
0.0343001 + 0.999412i \(0.489080\pi\)
\(564\) 0 0
\(565\) 9.11684 0.383549
\(566\) 0 0
\(567\) −2.37228 −0.0996265
\(568\) 0 0
\(569\) −23.4891 −0.984715 −0.492358 0.870393i \(-0.663865\pi\)
−0.492358 + 0.870393i \(0.663865\pi\)
\(570\) 0 0
\(571\) −34.2337 −1.43264 −0.716318 0.697774i \(-0.754174\pi\)
−0.716318 + 0.697774i \(0.754174\pi\)
\(572\) 0 0
\(573\) 14.2337 0.594621
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −18.7446 −0.780346 −0.390173 0.920741i \(-0.627585\pi\)
−0.390173 + 0.920741i \(0.627585\pi\)
\(578\) 0 0
\(579\) 9.25544 0.384643
\(580\) 0 0
\(581\) −37.6277 −1.56106
\(582\) 0 0
\(583\) −17.4891 −0.724326
\(584\) 0 0
\(585\) −6.74456 −0.278853
\(586\) 0 0
\(587\) −10.2337 −0.422390 −0.211195 0.977444i \(-0.567735\pi\)
−0.211195 + 0.977444i \(0.567735\pi\)
\(588\) 0 0
\(589\) −9.48913 −0.390993
\(590\) 0 0
\(591\) 11.4891 0.472599
\(592\) 0 0
\(593\) −32.9783 −1.35425 −0.677127 0.735866i \(-0.736775\pi\)
−0.677127 + 0.735866i \(0.736775\pi\)
\(594\) 0 0
\(595\) 0.883156 0.0362059
\(596\) 0 0
\(597\) −12.7446 −0.521600
\(598\) 0 0
\(599\) −33.4891 −1.36833 −0.684164 0.729328i \(-0.739833\pi\)
−0.684164 + 0.729328i \(0.739833\pi\)
\(600\) 0 0
\(601\) −9.86141 −0.402255 −0.201128 0.979565i \(-0.564461\pi\)
−0.201128 + 0.979565i \(0.564461\pi\)
\(602\) 0 0
\(603\) 9.62772 0.392071
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −44.7446 −1.81613 −0.908063 0.418834i \(-0.862439\pi\)
−0.908063 + 0.418834i \(0.862439\pi\)
\(608\) 0 0
\(609\) 0.883156 0.0357873
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) −16.5109 −0.666868 −0.333434 0.942773i \(-0.608207\pi\)
−0.333434 + 0.942773i \(0.608207\pi\)
\(614\) 0 0
\(615\) −9.11684 −0.367627
\(616\) 0 0
\(617\) 31.3505 1.26212 0.631062 0.775732i \(-0.282619\pi\)
0.631062 + 0.775732i \(0.282619\pi\)
\(618\) 0 0
\(619\) −6.97825 −0.280480 −0.140240 0.990118i \(-0.544787\pi\)
−0.140240 + 0.990118i \(0.544787\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −16.0000 −0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −16.0000 −0.638978
\(628\) 0 0
\(629\) −0.138593 −0.00552608
\(630\) 0 0
\(631\) −28.7446 −1.14430 −0.572151 0.820148i \(-0.693891\pi\)
−0.572151 + 0.820148i \(0.693891\pi\)
\(632\) 0 0
\(633\) 7.86141 0.312463
\(634\) 0 0
\(635\) 12.7446 0.505753
\(636\) 0 0
\(637\) 9.25544 0.366714
\(638\) 0 0
\(639\) −2.37228 −0.0938460
\(640\) 0 0
\(641\) −13.7228 −0.542019 −0.271009 0.962577i \(-0.587357\pi\)
−0.271009 + 0.962577i \(0.587357\pi\)
\(642\) 0 0
\(643\) 41.6277 1.64164 0.820818 0.571189i \(-0.193518\pi\)
0.820818 + 0.571189i \(0.193518\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) 1.48913 0.0585436 0.0292718 0.999571i \(-0.490681\pi\)
0.0292718 + 0.999571i \(0.490681\pi\)
\(648\) 0 0
\(649\) 57.4891 2.25665
\(650\) 0 0
\(651\) −5.62772 −0.220568
\(652\) 0 0
\(653\) −24.2337 −0.948337 −0.474169 0.880434i \(-0.657251\pi\)
−0.474169 + 0.880434i \(0.657251\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) −1.25544 −0.0489793
\(658\) 0 0
\(659\) −22.9783 −0.895106 −0.447553 0.894258i \(-0.647704\pi\)
−0.447553 + 0.894258i \(0.647704\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) −2.51087 −0.0975143
\(664\) 0 0
\(665\) −9.48913 −0.367972
\(666\) 0 0
\(667\) 0.372281 0.0144148
\(668\) 0 0
\(669\) −28.7446 −1.11133
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −39.4891 −1.52219 −0.761097 0.648638i \(-0.775339\pi\)
−0.761097 + 0.648638i \(0.775339\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 22.8832 0.879471 0.439736 0.898127i \(-0.355072\pi\)
0.439736 + 0.898127i \(0.355072\pi\)
\(678\) 0 0
\(679\) 17.7663 0.681808
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 2.23369 0.0854697 0.0427348 0.999086i \(-0.486393\pi\)
0.0427348 + 0.999086i \(0.486393\pi\)
\(684\) 0 0
\(685\) 11.4891 0.438977
\(686\) 0 0
\(687\) −17.2554 −0.658336
\(688\) 0 0
\(689\) 29.4891 1.12345
\(690\) 0 0
\(691\) 21.4891 0.817485 0.408742 0.912650i \(-0.365967\pi\)
0.408742 + 0.912650i \(0.365967\pi\)
\(692\) 0 0
\(693\) −9.48913 −0.360462
\(694\) 0 0
\(695\) 14.3723 0.545172
\(696\) 0 0
\(697\) −3.39403 −0.128558
\(698\) 0 0
\(699\) −8.23369 −0.311427
\(700\) 0 0
\(701\) 51.9565 1.96237 0.981185 0.193070i \(-0.0618445\pi\)
0.981185 + 0.193070i \(0.0618445\pi\)
\(702\) 0 0
\(703\) 1.48913 0.0561634
\(704\) 0 0
\(705\) −4.74456 −0.178691
\(706\) 0 0
\(707\) 18.0951 0.680536
\(708\) 0 0
\(709\) 24.9783 0.938078 0.469039 0.883177i \(-0.344600\pi\)
0.469039 + 0.883177i \(0.344600\pi\)
\(710\) 0 0
\(711\) 9.48913 0.355870
\(712\) 0 0
\(713\) −2.37228 −0.0888426
\(714\) 0 0
\(715\) −26.9783 −1.00893
\(716\) 0 0
\(717\) 5.62772 0.210171
\(718\) 0 0
\(719\) −34.0951 −1.27153 −0.635766 0.771882i \(-0.719316\pi\)
−0.635766 + 0.771882i \(0.719316\pi\)
\(720\) 0 0
\(721\) 18.9783 0.706787
\(722\) 0 0
\(723\) 18.7446 0.697118
\(724\) 0 0
\(725\) 0.372281 0.0138262
\(726\) 0 0
\(727\) 2.37228 0.0879830 0.0439915 0.999032i \(-0.485993\pi\)
0.0439915 + 0.999032i \(0.485993\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.48913 0.0550773
\(732\) 0 0
\(733\) −17.1168 −0.632225 −0.316113 0.948722i \(-0.602378\pi\)
−0.316113 + 0.948722i \(0.602378\pi\)
\(734\) 0 0
\(735\) 1.37228 0.0506174
\(736\) 0 0
\(737\) 38.5109 1.41857
\(738\) 0 0
\(739\) 33.6277 1.23702 0.618508 0.785779i \(-0.287738\pi\)
0.618508 + 0.785779i \(0.287738\pi\)
\(740\) 0 0
\(741\) 26.9783 0.991071
\(742\) 0 0
\(743\) −10.9783 −0.402753 −0.201376 0.979514i \(-0.564541\pi\)
−0.201376 + 0.979514i \(0.564541\pi\)
\(744\) 0 0
\(745\) −19.4891 −0.714026
\(746\) 0 0
\(747\) 15.8614 0.580339
\(748\) 0 0
\(749\) 3.86141 0.141093
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 7.25544 0.264403
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 14.6060 0.530863 0.265431 0.964130i \(-0.414486\pi\)
0.265431 + 0.964130i \(0.414486\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −28.8397 −1.04544 −0.522718 0.852505i \(-0.675082\pi\)
−0.522718 + 0.852505i \(0.675082\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) −0.372281 −0.0134599
\(766\) 0 0
\(767\) −96.9348 −3.50011
\(768\) 0 0
\(769\) 43.4891 1.56826 0.784129 0.620598i \(-0.213110\pi\)
0.784129 + 0.620598i \(0.213110\pi\)
\(770\) 0 0
\(771\) −11.4891 −0.413771
\(772\) 0 0
\(773\) 40.9783 1.47389 0.736943 0.675955i \(-0.236269\pi\)
0.736943 + 0.675955i \(0.236269\pi\)
\(774\) 0 0
\(775\) −2.37228 −0.0852149
\(776\) 0 0
\(777\) 0.883156 0.0316830
\(778\) 0 0
\(779\) 36.4674 1.30658
\(780\) 0 0
\(781\) −9.48913 −0.339548
\(782\) 0 0
\(783\) −0.372281 −0.0133042
\(784\) 0 0
\(785\) 22.6060 0.806842
\(786\) 0 0
\(787\) −6.37228 −0.227147 −0.113574 0.993530i \(-0.536230\pi\)
−0.113574 + 0.993530i \(0.536230\pi\)
\(788\) 0 0
\(789\) 27.8614 0.991892
\(790\) 0 0
\(791\) −21.6277 −0.768993
\(792\) 0 0
\(793\) 13.4891 0.479013
\(794\) 0 0
\(795\) 4.37228 0.155069
\(796\) 0 0
\(797\) −1.11684 −0.0395606 −0.0197803 0.999804i \(-0.506297\pi\)
−0.0197803 + 0.999804i \(0.506297\pi\)
\(798\) 0 0
\(799\) −1.76631 −0.0624876
\(800\) 0 0
\(801\) 6.74456 0.238307
\(802\) 0 0
\(803\) −5.02175 −0.177214
\(804\) 0 0
\(805\) −2.37228 −0.0836119
\(806\) 0 0
\(807\) 13.8614 0.487945
\(808\) 0 0
\(809\) 36.3723 1.27878 0.639391 0.768882i \(-0.279187\pi\)
0.639391 + 0.768882i \(0.279187\pi\)
\(810\) 0 0
\(811\) 35.1168 1.23312 0.616560 0.787308i \(-0.288526\pi\)
0.616560 + 0.787308i \(0.288526\pi\)
\(812\) 0 0
\(813\) 23.1168 0.810743
\(814\) 0 0
\(815\) 16.7446 0.586536
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) −36.9783 −1.29055 −0.645275 0.763950i \(-0.723257\pi\)
−0.645275 + 0.763950i \(0.723257\pi\)
\(822\) 0 0
\(823\) 34.9783 1.21927 0.609633 0.792684i \(-0.291317\pi\)
0.609633 + 0.792684i \(0.291317\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −3.39403 −0.118022 −0.0590110 0.998257i \(-0.518795\pi\)
−0.0590110 + 0.998257i \(0.518795\pi\)
\(828\) 0 0
\(829\) −20.0951 −0.697931 −0.348966 0.937135i \(-0.613467\pi\)
−0.348966 + 0.937135i \(0.613467\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) 0.510875 0.0177008
\(834\) 0 0
\(835\) 12.7446 0.441044
\(836\) 0 0
\(837\) 2.37228 0.0819980
\(838\) 0 0
\(839\) 3.25544 0.112390 0.0561951 0.998420i \(-0.482103\pi\)
0.0561951 + 0.998420i \(0.482103\pi\)
\(840\) 0 0
\(841\) −28.8614 −0.995221
\(842\) 0 0
\(843\) 12.2337 0.421351
\(844\) 0 0
\(845\) 32.4891 1.11766
\(846\) 0 0
\(847\) −11.8614 −0.407563
\(848\) 0 0
\(849\) −6.37228 −0.218696
\(850\) 0 0
\(851\) 0.372281 0.0127616
\(852\) 0 0
\(853\) −12.9783 −0.444367 −0.222183 0.975005i \(-0.571318\pi\)
−0.222183 + 0.975005i \(0.571318\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) −18.4674 −0.630834 −0.315417 0.948953i \(-0.602144\pi\)
−0.315417 + 0.948953i \(0.602144\pi\)
\(858\) 0 0
\(859\) 12.6060 0.430110 0.215055 0.976602i \(-0.431007\pi\)
0.215055 + 0.976602i \(0.431007\pi\)
\(860\) 0 0
\(861\) 21.6277 0.737071
\(862\) 0 0
\(863\) 4.74456 0.161507 0.0807534 0.996734i \(-0.474267\pi\)
0.0807534 + 0.996734i \(0.474267\pi\)
\(864\) 0 0
\(865\) 20.2337 0.687966
\(866\) 0 0
\(867\) 16.8614 0.572643
\(868\) 0 0
\(869\) 37.9565 1.28759
\(870\) 0 0
\(871\) −64.9348 −2.20023
\(872\) 0 0
\(873\) −7.48913 −0.253468
\(874\) 0 0
\(875\) −2.37228 −0.0801977
\(876\) 0 0
\(877\) −43.4891 −1.46852 −0.734262 0.678867i \(-0.762471\pi\)
−0.734262 + 0.678867i \(0.762471\pi\)
\(878\) 0 0
\(879\) −9.86141 −0.332617
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 29.2119 0.983060 0.491530 0.870861i \(-0.336438\pi\)
0.491530 + 0.870861i \(0.336438\pi\)
\(884\) 0 0
\(885\) −14.3723 −0.483119
\(886\) 0 0
\(887\) 31.7228 1.06515 0.532574 0.846383i \(-0.321225\pi\)
0.532574 + 0.846383i \(0.321225\pi\)
\(888\) 0 0
\(889\) −30.2337 −1.01401
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 18.9783 0.635083
\(894\) 0 0
\(895\) −13.4891 −0.450892
\(896\) 0 0
\(897\) 6.74456 0.225194
\(898\) 0 0
\(899\) −0.883156 −0.0294549
\(900\) 0 0
\(901\) 1.62772 0.0542272
\(902\) 0 0
\(903\) −9.48913 −0.315778
\(904\) 0 0
\(905\) 20.2337 0.672591
\(906\) 0 0
\(907\) −14.3723 −0.477224 −0.238612 0.971115i \(-0.576692\pi\)
−0.238612 + 0.971115i \(0.576692\pi\)
\(908\) 0 0
\(909\) −7.62772 −0.252995
\(910\) 0 0
\(911\) 12.4674 0.413063 0.206531 0.978440i \(-0.433782\pi\)
0.206531 + 0.978440i \(0.433782\pi\)
\(912\) 0 0
\(913\) 63.4456 2.09974
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) 9.48913 0.313359
\(918\) 0 0
\(919\) 14.5109 0.478670 0.239335 0.970937i \(-0.423071\pi\)
0.239335 + 0.970937i \(0.423071\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 0.372281 0.0122405
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −6.60597 −0.216735 −0.108367 0.994111i \(-0.534562\pi\)
−0.108367 + 0.994111i \(0.534562\pi\)
\(930\) 0 0
\(931\) −5.48913 −0.179899
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) −1.48913 −0.0486996
\(936\) 0 0
\(937\) −34.4674 −1.12600 −0.563000 0.826457i \(-0.690353\pi\)
−0.563000 + 0.826457i \(0.690353\pi\)
\(938\) 0 0
\(939\) 25.8614 0.843955
\(940\) 0 0
\(941\) 32.9783 1.07506 0.537530 0.843245i \(-0.319357\pi\)
0.537530 + 0.843245i \(0.319357\pi\)
\(942\) 0 0
\(943\) 9.11684 0.296885
\(944\) 0 0
\(945\) 2.37228 0.0771703
\(946\) 0 0
\(947\) 9.02175 0.293167 0.146584 0.989198i \(-0.453172\pi\)
0.146584 + 0.989198i \(0.453172\pi\)
\(948\) 0 0
\(949\) 8.46738 0.274863
\(950\) 0 0
\(951\) −31.4891 −1.02110
\(952\) 0 0
\(953\) −7.48913 −0.242597 −0.121298 0.992616i \(-0.538706\pi\)
−0.121298 + 0.992616i \(0.538706\pi\)
\(954\) 0 0
\(955\) −14.2337 −0.460591
\(956\) 0 0
\(957\) −1.48913 −0.0481366
\(958\) 0 0
\(959\) −27.2554 −0.880124
\(960\) 0 0
\(961\) −25.3723 −0.818461
\(962\) 0 0
\(963\) −1.62772 −0.0524525
\(964\) 0 0
\(965\) −9.25544 −0.297943
\(966\) 0 0
\(967\) 46.2337 1.48678 0.743388 0.668861i \(-0.233218\pi\)
0.743388 + 0.668861i \(0.233218\pi\)
\(968\) 0 0
\(969\) 1.48913 0.0478376
\(970\) 0 0
\(971\) −21.4891 −0.689619 −0.344809 0.938673i \(-0.612056\pi\)
−0.344809 + 0.938673i \(0.612056\pi\)
\(972\) 0 0
\(973\) −34.0951 −1.09304
\(974\) 0 0
\(975\) 6.74456 0.215999
\(976\) 0 0
\(977\) 36.3723 1.16365 0.581826 0.813313i \(-0.302338\pi\)
0.581826 + 0.813313i \(0.302338\pi\)
\(978\) 0 0
\(979\) 26.9783 0.862229
\(980\) 0 0
\(981\) −6.74456 −0.215337
\(982\) 0 0
\(983\) −20.1386 −0.642321 −0.321161 0.947025i \(-0.604073\pi\)
−0.321161 + 0.947025i \(0.604073\pi\)
\(984\) 0 0
\(985\) −11.4891 −0.366074
\(986\) 0 0
\(987\) 11.2554 0.358265
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −24.8832 −0.790440 −0.395220 0.918587i \(-0.629332\pi\)
−0.395220 + 0.918587i \(0.629332\pi\)
\(992\) 0 0
\(993\) −19.1168 −0.606655
\(994\) 0 0
\(995\) 12.7446 0.404030
\(996\) 0 0
\(997\) −11.7663 −0.372643 −0.186321 0.982489i \(-0.559657\pi\)
−0.186321 + 0.982489i \(0.559657\pi\)
\(998\) 0 0
\(999\) −0.372281 −0.0117785
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 2760.2.a.o.1.1 2
3.2 odd 2 8280.2.a.z.1.1 2
4.3 odd 2 5520.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.o.1.1 2 1.1 even 1 trivial
5520.2.a.bq.1.2 2 4.3 odd 2
8280.2.a.z.1.1 2 3.2 odd 2