Properties

Label 5520.2.a.cb.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.655762\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -4.46569 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -4.46569 q^{7} +1.00000 q^{9} -3.39413 q^{11} -4.70565 q^{13} -1.00000 q^{15} -6.54830 q^{17} +6.22574 q^{19} -4.46569 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.448521 q^{29} +5.15417 q^{31} -3.39413 q^{33} +4.46569 q^{35} -5.98578 q^{37} -4.70565 q^{39} +3.07157 q^{41} +4.47991 q^{43} -1.00000 q^{45} -6.22574 q^{47} +12.9424 q^{49} -6.54830 q^{51} -2.59166 q^{53} +3.39413 q^{55} +6.22574 q^{57} +1.55148 q^{59} -4.91421 q^{61} -4.46569 q^{63} +4.70565 q^{65} -6.08556 q^{67} -1.00000 q^{69} -1.07157 q^{71} -8.70565 q^{73} +1.00000 q^{75} +15.1571 q^{77} +13.1400 q^{79} +1.00000 q^{81} +13.0282 q^{83} +6.54830 q^{85} +0.448521 q^{87} +11.6199 q^{89} +21.0140 q^{91} +5.15417 q^{93} -6.22574 q^{95} +4.30834 q^{97} -3.39413 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 4 q^{37} - 2 q^{39} + 8 q^{41} + 20 q^{43} - 4 q^{45} - 6 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{53} + 6 q^{57} + 4 q^{59} - 4 q^{61} + 2 q^{65} + 26 q^{67} - 4 q^{69} - 18 q^{73} + 4 q^{75} + 6 q^{77} + 18 q^{79} + 4 q^{81} + 26 q^{83} - 2 q^{85} + 4 q^{87} + 14 q^{89} + 38 q^{91} + 6 q^{93} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) 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
\(7\) −4.46569 −1.68787 −0.843937 0.536443i \(-0.819768\pi\)
−0.843937 + 0.536443i \(0.819768\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.39413 −1.02337 −0.511684 0.859174i \(-0.670978\pi\)
−0.511684 + 0.859174i \(0.670978\pi\)
\(12\) 0 0
\(13\) −4.70565 −1.30511 −0.652556 0.757740i \(-0.726303\pi\)
−0.652556 + 0.757740i \(0.726303\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.54830 −1.58819 −0.794097 0.607790i \(-0.792056\pi\)
−0.794097 + 0.607790i \(0.792056\pi\)
\(18\) 0 0
\(19\) 6.22574 1.42828 0.714141 0.700002i \(-0.246817\pi\)
0.714141 + 0.700002i \(0.246817\pi\)
\(20\) 0 0
\(21\) −4.46569 −0.974494
\(22\) 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
\(29\) 0.448521 0.0832882 0.0416441 0.999133i \(-0.486740\pi\)
0.0416441 + 0.999133i \(0.486740\pi\)
\(30\) 0 0
\(31\) 5.15417 0.925716 0.462858 0.886432i \(-0.346824\pi\)
0.462858 + 0.886432i \(0.346824\pi\)
\(32\) 0 0
\(33\) −3.39413 −0.590841
\(34\) 0 0
\(35\) 4.46569 0.754840
\(36\) 0 0
\(37\) −5.98578 −0.984057 −0.492028 0.870579i \(-0.663744\pi\)
−0.492028 + 0.870579i \(0.663744\pi\)
\(38\) 0 0
\(39\) −4.70565 −0.753507
\(40\) 0 0
\(41\) 3.07157 0.479698 0.239849 0.970810i \(-0.422902\pi\)
0.239849 + 0.970810i \(0.422902\pi\)
\(42\) 0 0
\(43\) 4.47991 0.683180 0.341590 0.939849i \(-0.389035\pi\)
0.341590 + 0.939849i \(0.389035\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.22574 −0.908117 −0.454059 0.890972i \(-0.650024\pi\)
−0.454059 + 0.890972i \(0.650024\pi\)
\(48\) 0 0
\(49\) 12.9424 1.84892
\(50\) 0 0
\(51\) −6.54830 −0.916945
\(52\) 0 0
\(53\) −2.59166 −0.355991 −0.177996 0.984031i \(-0.556961\pi\)
−0.177996 + 0.984031i \(0.556961\pi\)
\(54\) 0 0
\(55\) 3.39413 0.457664
\(56\) 0 0
\(57\) 6.22574 0.824619
\(58\) 0 0
\(59\) 1.55148 0.201985 0.100993 0.994887i \(-0.467798\pi\)
0.100993 + 0.994887i \(0.467798\pi\)
\(60\) 0 0
\(61\) −4.91421 −0.629201 −0.314600 0.949224i \(-0.601870\pi\)
−0.314600 + 0.949224i \(0.601870\pi\)
\(62\) 0 0
\(63\) −4.46569 −0.562625
\(64\) 0 0
\(65\) 4.70565 0.583664
\(66\) 0 0
\(67\) −6.08556 −0.743469 −0.371735 0.928339i \(-0.621237\pi\)
−0.371735 + 0.928339i \(0.621237\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −1.07157 −0.127172 −0.0635859 0.997976i \(-0.520254\pi\)
−0.0635859 + 0.997976i \(0.520254\pi\)
\(72\) 0 0
\(73\) −8.70565 −1.01892 −0.509460 0.860495i \(-0.670155\pi\)
−0.509460 + 0.860495i \(0.670155\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 15.1571 1.72731
\(78\) 0 0
\(79\) 13.1400 1.47836 0.739180 0.673508i \(-0.235213\pi\)
0.739180 + 0.673508i \(0.235213\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.0282 1.43003 0.715016 0.699108i \(-0.246420\pi\)
0.715016 + 0.699108i \(0.246420\pi\)
\(84\) 0 0
\(85\) 6.54830 0.710262
\(86\) 0 0
\(87\) 0.448521 0.0480865
\(88\) 0 0
\(89\) 11.6199 1.23170 0.615852 0.787862i \(-0.288812\pi\)
0.615852 + 0.787862i \(0.288812\pi\)
\(90\) 0 0
\(91\) 21.0140 2.20286
\(92\) 0 0
\(93\) 5.15417 0.534463
\(94\) 0 0
\(95\) −6.22574 −0.638747
\(96\) 0 0
\(97\) 4.30834 0.437446 0.218723 0.975787i \(-0.429811\pi\)
0.218723 + 0.975787i \(0.429811\pi\)
\(98\) 0 0
\(99\) −3.39413 −0.341122
\(100\) 0 0
\(101\) 10.3397 1.02884 0.514421 0.857538i \(-0.328007\pi\)
0.514421 + 0.857538i \(0.328007\pi\)
\(102\) 0 0
\(103\) 10.8316 1.06727 0.533635 0.845715i \(-0.320826\pi\)
0.533635 + 0.845715i \(0.320826\pi\)
\(104\) 0 0
\(105\) 4.46569 0.435807
\(106\) 0 0
\(107\) 1.76004 0.170150 0.0850750 0.996375i \(-0.472887\pi\)
0.0850750 + 0.996375i \(0.472887\pi\)
\(108\) 0 0
\(109\) −1.39413 −0.133533 −0.0667665 0.997769i \(-0.521268\pi\)
−0.0667665 + 0.997769i \(0.521268\pi\)
\(110\) 0 0
\(111\) −5.98578 −0.568145
\(112\) 0 0
\(113\) 1.17134 0.110191 0.0550954 0.998481i \(-0.482454\pi\)
0.0550954 + 0.998481i \(0.482454\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −4.70565 −0.435037
\(118\) 0 0
\(119\) 29.2427 2.68067
\(120\) 0 0
\(121\) 0.520089 0.0472808
\(122\) 0 0
\(123\) 3.07157 0.276954
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.9454 −1.23745 −0.618726 0.785607i \(-0.712351\pi\)
−0.618726 + 0.785607i \(0.712351\pi\)
\(128\) 0 0
\(129\) 4.47991 0.394434
\(130\) 0 0
\(131\) −13.8628 −1.21120 −0.605598 0.795771i \(-0.707066\pi\)
−0.605598 + 0.795771i \(0.707066\pi\)
\(132\) 0 0
\(133\) −27.8022 −2.41076
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −14.4081 −1.23097 −0.615484 0.788149i \(-0.711039\pi\)
−0.615484 + 0.788149i \(0.711039\pi\)
\(138\) 0 0
\(139\) 10.1140 0.857858 0.428929 0.903338i \(-0.358891\pi\)
0.428929 + 0.903338i \(0.358891\pi\)
\(140\) 0 0
\(141\) −6.22574 −0.524302
\(142\) 0 0
\(143\) 15.9716 1.33561
\(144\) 0 0
\(145\) −0.448521 −0.0372476
\(146\) 0 0
\(147\) 12.9424 1.06747
\(148\) 0 0
\(149\) 3.32870 0.272697 0.136349 0.990661i \(-0.456463\pi\)
0.136349 + 0.990661i \(0.456463\pi\)
\(150\) 0 0
\(151\) 22.3427 1.81822 0.909111 0.416554i \(-0.136762\pi\)
0.909111 + 0.416554i \(0.136762\pi\)
\(152\) 0 0
\(153\) −6.54830 −0.529398
\(154\) 0 0
\(155\) −5.15417 −0.413993
\(156\) 0 0
\(157\) 6.63090 0.529203 0.264602 0.964358i \(-0.414760\pi\)
0.264602 + 0.964358i \(0.414760\pi\)
\(158\) 0 0
\(159\) −2.59166 −0.205532
\(160\) 0 0
\(161\) 4.46569 0.351946
\(162\) 0 0
\(163\) 22.3083 1.74732 0.873662 0.486533i \(-0.161739\pi\)
0.873662 + 0.486533i \(0.161739\pi\)
\(164\) 0 0
\(165\) 3.39413 0.264232
\(166\) 0 0
\(167\) 24.0885 1.86403 0.932013 0.362426i \(-0.118051\pi\)
0.932013 + 0.362426i \(0.118051\pi\)
\(168\) 0 0
\(169\) 9.14314 0.703318
\(170\) 0 0
\(171\) 6.22574 0.476094
\(172\) 0 0
\(173\) −18.9314 −1.43933 −0.719663 0.694323i \(-0.755704\pi\)
−0.719663 + 0.694323i \(0.755704\pi\)
\(174\) 0 0
\(175\) −4.46569 −0.337575
\(176\) 0 0
\(177\) 1.55148 0.116616
\(178\) 0 0
\(179\) 18.7598 1.40217 0.701087 0.713075i \(-0.252698\pi\)
0.701087 + 0.713075i \(0.252698\pi\)
\(180\) 0 0
\(181\) −21.1964 −1.57551 −0.787757 0.615986i \(-0.788758\pi\)
−0.787757 + 0.615986i \(0.788758\pi\)
\(182\) 0 0
\(183\) −4.91421 −0.363269
\(184\) 0 0
\(185\) 5.98578 0.440083
\(186\) 0 0
\(187\) 22.2257 1.62531
\(188\) 0 0
\(189\) −4.46569 −0.324831
\(190\) 0 0
\(191\) −14.3909 −1.04129 −0.520646 0.853773i \(-0.674309\pi\)
−0.520646 + 0.853773i \(0.674309\pi\)
\(192\) 0 0
\(193\) 11.0966 0.798750 0.399375 0.916788i \(-0.369227\pi\)
0.399375 + 0.916788i \(0.369227\pi\)
\(194\) 0 0
\(195\) 4.70565 0.336979
\(196\) 0 0
\(197\) 14.1652 1.00923 0.504614 0.863345i \(-0.331635\pi\)
0.504614 + 0.863345i \(0.331635\pi\)
\(198\) 0 0
\(199\) −8.66004 −0.613894 −0.306947 0.951727i \(-0.599307\pi\)
−0.306947 + 0.951727i \(0.599307\pi\)
\(200\) 0 0
\(201\) −6.08556 −0.429242
\(202\) 0 0
\(203\) −2.00296 −0.140580
\(204\) 0 0
\(205\) −3.07157 −0.214528
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −21.1309 −1.46166
\(210\) 0 0
\(211\) 11.1321 0.766366 0.383183 0.923673i \(-0.374828\pi\)
0.383183 + 0.923673i \(0.374828\pi\)
\(212\) 0 0
\(213\) −1.07157 −0.0734226
\(214\) 0 0
\(215\) −4.47991 −0.305527
\(216\) 0 0
\(217\) −23.0169 −1.56249
\(218\) 0 0
\(219\) −8.70565 −0.588273
\(220\) 0 0
\(221\) 30.8140 2.07277
\(222\) 0 0
\(223\) −20.1711 −1.35076 −0.675379 0.737471i \(-0.736020\pi\)
−0.675379 + 0.737471i \(0.736020\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0.523273 0.0347308 0.0173654 0.999849i \(-0.494472\pi\)
0.0173654 + 0.999849i \(0.494472\pi\)
\(228\) 0 0
\(229\) −3.33359 −0.220290 −0.110145 0.993916i \(-0.535132\pi\)
−0.110145 + 0.993916i \(0.535132\pi\)
\(230\) 0 0
\(231\) 15.1571 0.997266
\(232\) 0 0
\(233\) 15.5481 1.01859 0.509294 0.860593i \(-0.329907\pi\)
0.509294 + 0.860593i \(0.329907\pi\)
\(234\) 0 0
\(235\) 6.22574 0.406122
\(236\) 0 0
\(237\) 13.1400 0.853532
\(238\) 0 0
\(239\) −27.5451 −1.78175 −0.890873 0.454253i \(-0.849906\pi\)
−0.890873 + 0.454253i \(0.849906\pi\)
\(240\) 0 0
\(241\) 8.74000 0.562993 0.281496 0.959562i \(-0.409169\pi\)
0.281496 + 0.959562i \(0.409169\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −12.9424 −0.826861
\(246\) 0 0
\(247\) −29.2961 −1.86407
\(248\) 0 0
\(249\) 13.0282 0.825629
\(250\) 0 0
\(251\) −15.0312 −0.948759 −0.474379 0.880321i \(-0.657327\pi\)
−0.474379 + 0.880321i \(0.657327\pi\)
\(252\) 0 0
\(253\) 3.39413 0.213387
\(254\) 0 0
\(255\) 6.54830 0.410070
\(256\) 0 0
\(257\) −13.1571 −0.820719 −0.410359 0.911924i \(-0.634597\pi\)
−0.410359 + 0.911924i \(0.634597\pi\)
\(258\) 0 0
\(259\) 26.7307 1.66096
\(260\) 0 0
\(261\) 0.448521 0.0277627
\(262\) 0 0
\(263\) 23.5795 1.45397 0.726986 0.686652i \(-0.240921\pi\)
0.726986 + 0.686652i \(0.240921\pi\)
\(264\) 0 0
\(265\) 2.59166 0.159204
\(266\) 0 0
\(267\) 11.6199 0.711124
\(268\) 0 0
\(269\) −7.52304 −0.458688 −0.229344 0.973345i \(-0.573658\pi\)
−0.229344 + 0.973345i \(0.573658\pi\)
\(270\) 0 0
\(271\) −18.5370 −1.12604 −0.563022 0.826442i \(-0.690361\pi\)
−0.563022 + 0.826442i \(0.690361\pi\)
\(272\) 0 0
\(273\) 21.0140 1.27182
\(274\) 0 0
\(275\) −3.39413 −0.204673
\(276\) 0 0
\(277\) 23.8628 1.43378 0.716888 0.697189i \(-0.245566\pi\)
0.716888 + 0.697189i \(0.245566\pi\)
\(278\) 0 0
\(279\) 5.15417 0.308572
\(280\) 0 0
\(281\) −22.6118 −1.34891 −0.674453 0.738318i \(-0.735620\pi\)
−0.674453 + 0.738318i \(0.735620\pi\)
\(282\) 0 0
\(283\) −13.6684 −0.812504 −0.406252 0.913761i \(-0.633164\pi\)
−0.406252 + 0.913761i \(0.633164\pi\)
\(284\) 0 0
\(285\) −6.22574 −0.368781
\(286\) 0 0
\(287\) −13.7167 −0.809670
\(288\) 0 0
\(289\) 25.8802 1.52236
\(290\) 0 0
\(291\) 4.30834 0.252559
\(292\) 0 0
\(293\) −15.0431 −0.878829 −0.439415 0.898284i \(-0.644814\pi\)
−0.439415 + 0.898284i \(0.644814\pi\)
\(294\) 0 0
\(295\) −1.55148 −0.0903306
\(296\) 0 0
\(297\) −3.39413 −0.196947
\(298\) 0 0
\(299\) 4.70565 0.272135
\(300\) 0 0
\(301\) −20.0059 −1.15312
\(302\) 0 0
\(303\) 10.3397 0.594002
\(304\) 0 0
\(305\) 4.91421 0.281387
\(306\) 0 0
\(307\) 31.9674 1.82448 0.912239 0.409658i \(-0.134352\pi\)
0.912239 + 0.409658i \(0.134352\pi\)
\(308\) 0 0
\(309\) 10.8316 0.616189
\(310\) 0 0
\(311\) 34.1932 1.93892 0.969459 0.245254i \(-0.0788715\pi\)
0.969459 + 0.245254i \(0.0788715\pi\)
\(312\) 0 0
\(313\) −11.5713 −0.654049 −0.327024 0.945016i \(-0.606046\pi\)
−0.327024 + 0.945016i \(0.606046\pi\)
\(314\) 0 0
\(315\) 4.46569 0.251613
\(316\) 0 0
\(317\) 3.12915 0.175750 0.0878752 0.996131i \(-0.471992\pi\)
0.0878752 + 0.996131i \(0.471992\pi\)
\(318\) 0 0
\(319\) −1.52234 −0.0852344
\(320\) 0 0
\(321\) 1.76004 0.0982361
\(322\) 0 0
\(323\) −40.7680 −2.26839
\(324\) 0 0
\(325\) −4.70565 −0.261022
\(326\) 0 0
\(327\) −1.39413 −0.0770953
\(328\) 0 0
\(329\) 27.8022 1.53279
\(330\) 0 0
\(331\) 0.988966 0.0543585 0.0271793 0.999631i \(-0.491348\pi\)
0.0271793 + 0.999631i \(0.491348\pi\)
\(332\) 0 0
\(333\) −5.98578 −0.328019
\(334\) 0 0
\(335\) 6.08556 0.332490
\(336\) 0 0
\(337\) 21.2397 1.15700 0.578501 0.815682i \(-0.303638\pi\)
0.578501 + 0.815682i \(0.303638\pi\)
\(338\) 0 0
\(339\) 1.17134 0.0636186
\(340\) 0 0
\(341\) −17.4939 −0.947348
\(342\) 0 0
\(343\) −26.5370 −1.43287
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 2.78825 0.149681 0.0748406 0.997196i \(-0.476155\pi\)
0.0748406 + 0.997196i \(0.476155\pi\)
\(348\) 0 0
\(349\) 30.3939 1.62695 0.813474 0.581601i \(-0.197574\pi\)
0.813474 + 0.581601i \(0.197574\pi\)
\(350\) 0 0
\(351\) −4.70565 −0.251169
\(352\) 0 0
\(353\) 10.6993 0.569465 0.284733 0.958607i \(-0.408095\pi\)
0.284733 + 0.958607i \(0.408095\pi\)
\(354\) 0 0
\(355\) 1.07157 0.0568729
\(356\) 0 0
\(357\) 29.2427 1.54769
\(358\) 0 0
\(359\) −18.5120 −0.977027 −0.488513 0.872556i \(-0.662461\pi\)
−0.488513 + 0.872556i \(0.662461\pi\)
\(360\) 0 0
\(361\) 19.7598 1.03999
\(362\) 0 0
\(363\) 0.520089 0.0272976
\(364\) 0 0
\(365\) 8.70565 0.455675
\(366\) 0 0
\(367\) 24.6937 1.28900 0.644500 0.764605i \(-0.277066\pi\)
0.644500 + 0.764605i \(0.277066\pi\)
\(368\) 0 0
\(369\) 3.07157 0.159899
\(370\) 0 0
\(371\) 11.5735 0.600869
\(372\) 0 0
\(373\) −24.9968 −1.29429 −0.647143 0.762369i \(-0.724036\pi\)
−0.647143 + 0.762369i \(0.724036\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −2.11058 −0.108700
\(378\) 0 0
\(379\) −20.6167 −1.05901 −0.529504 0.848308i \(-0.677622\pi\)
−0.529504 + 0.848308i \(0.677622\pi\)
\(380\) 0 0
\(381\) −13.9454 −0.714443
\(382\) 0 0
\(383\) −4.42645 −0.226181 −0.113091 0.993585i \(-0.536075\pi\)
−0.113091 + 0.993585i \(0.536075\pi\)
\(384\) 0 0
\(385\) −15.1571 −0.772479
\(386\) 0 0
\(387\) 4.47991 0.227727
\(388\) 0 0
\(389\) −19.7417 −1.00094 −0.500472 0.865753i \(-0.666840\pi\)
−0.500472 + 0.865753i \(0.666840\pi\)
\(390\) 0 0
\(391\) 6.54830 0.331162
\(392\) 0 0
\(393\) −13.8628 −0.699285
\(394\) 0 0
\(395\) −13.1400 −0.661143
\(396\) 0 0
\(397\) −5.68529 −0.285337 −0.142668 0.989771i \(-0.545568\pi\)
−0.142668 + 0.989771i \(0.545568\pi\)
\(398\) 0 0
\(399\) −27.8022 −1.39185
\(400\) 0 0
\(401\) 12.1341 0.605949 0.302975 0.952999i \(-0.402020\pi\)
0.302975 + 0.952999i \(0.402020\pi\)
\(402\) 0 0
\(403\) −24.2537 −1.20816
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 20.3165 1.00705
\(408\) 0 0
\(409\) 25.0454 1.23841 0.619207 0.785228i \(-0.287454\pi\)
0.619207 + 0.785228i \(0.287454\pi\)
\(410\) 0 0
\(411\) −14.4081 −0.710700
\(412\) 0 0
\(413\) −6.92843 −0.340926
\(414\) 0 0
\(415\) −13.0282 −0.639530
\(416\) 0 0
\(417\) 10.1140 0.495284
\(418\) 0 0
\(419\) −32.3314 −1.57949 −0.789747 0.613433i \(-0.789788\pi\)
−0.789747 + 0.613433i \(0.789788\pi\)
\(420\) 0 0
\(421\) 24.8858 1.21286 0.606429 0.795137i \(-0.292601\pi\)
0.606429 + 0.795137i \(0.292601\pi\)
\(422\) 0 0
\(423\) −6.22574 −0.302706
\(424\) 0 0
\(425\) −6.54830 −0.317639
\(426\) 0 0
\(427\) 21.9454 1.06201
\(428\) 0 0
\(429\) 15.9716 0.771114
\(430\) 0 0
\(431\) −28.0343 −1.35037 −0.675183 0.737650i \(-0.735936\pi\)
−0.675183 + 0.737650i \(0.735936\pi\)
\(432\) 0 0
\(433\) −18.7370 −0.900445 −0.450222 0.892916i \(-0.648655\pi\)
−0.450222 + 0.892916i \(0.648655\pi\)
\(434\) 0 0
\(435\) −0.448521 −0.0215049
\(436\) 0 0
\(437\) −6.22574 −0.297817
\(438\) 0 0
\(439\) 1.01811 0.0485918 0.0242959 0.999705i \(-0.492266\pi\)
0.0242959 + 0.999705i \(0.492266\pi\)
\(440\) 0 0
\(441\) 12.9424 0.616306
\(442\) 0 0
\(443\) −17.4311 −0.828177 −0.414089 0.910237i \(-0.635900\pi\)
−0.414089 + 0.910237i \(0.635900\pi\)
\(444\) 0 0
\(445\) −11.6199 −0.550834
\(446\) 0 0
\(447\) 3.32870 0.157442
\(448\) 0 0
\(449\) −33.6138 −1.58633 −0.793167 0.609005i \(-0.791569\pi\)
−0.793167 + 0.609005i \(0.791569\pi\)
\(450\) 0 0
\(451\) −10.4253 −0.490908
\(452\) 0 0
\(453\) 22.3427 1.04975
\(454\) 0 0
\(455\) −21.0140 −0.985151
\(456\) 0 0
\(457\) −5.01695 −0.234683 −0.117341 0.993092i \(-0.537437\pi\)
−0.117341 + 0.993092i \(0.537437\pi\)
\(458\) 0 0
\(459\) −6.54830 −0.305648
\(460\) 0 0
\(461\) −5.49165 −0.255772 −0.127886 0.991789i \(-0.540819\pi\)
−0.127886 + 0.991789i \(0.540819\pi\)
\(462\) 0 0
\(463\) 42.3684 1.96903 0.984514 0.175308i \(-0.0560920\pi\)
0.984514 + 0.175308i \(0.0560920\pi\)
\(464\) 0 0
\(465\) −5.15417 −0.239019
\(466\) 0 0
\(467\) 5.95959 0.275777 0.137889 0.990448i \(-0.455968\pi\)
0.137889 + 0.990448i \(0.455968\pi\)
\(468\) 0 0
\(469\) 27.1762 1.25488
\(470\) 0 0
\(471\) 6.63090 0.305536
\(472\) 0 0
\(473\) −15.2054 −0.699144
\(474\) 0 0
\(475\) 6.22574 0.285656
\(476\) 0 0
\(477\) −2.59166 −0.118664
\(478\) 0 0
\(479\) −33.2135 −1.51757 −0.758783 0.651344i \(-0.774206\pi\)
−0.758783 + 0.651344i \(0.774206\pi\)
\(480\) 0 0
\(481\) 28.1670 1.28430
\(482\) 0 0
\(483\) 4.46569 0.203196
\(484\) 0 0
\(485\) −4.30834 −0.195632
\(486\) 0 0
\(487\) 28.7336 1.30204 0.651022 0.759058i \(-0.274340\pi\)
0.651022 + 0.759058i \(0.274340\pi\)
\(488\) 0 0
\(489\) 22.3083 1.00882
\(490\) 0 0
\(491\) −19.4427 −0.877436 −0.438718 0.898625i \(-0.644567\pi\)
−0.438718 + 0.898625i \(0.644567\pi\)
\(492\) 0 0
\(493\) −2.93705 −0.132278
\(494\) 0 0
\(495\) 3.39413 0.152555
\(496\) 0 0
\(497\) 4.78530 0.214650
\(498\) 0 0
\(499\) 5.21695 0.233543 0.116771 0.993159i \(-0.462746\pi\)
0.116771 + 0.993159i \(0.462746\pi\)
\(500\) 0 0
\(501\) 24.0885 1.07620
\(502\) 0 0
\(503\) 6.06574 0.270458 0.135229 0.990814i \(-0.456823\pi\)
0.135229 + 0.990814i \(0.456823\pi\)
\(504\) 0 0
\(505\) −10.3397 −0.460112
\(506\) 0 0
\(507\) 9.14314 0.406061
\(508\) 0 0
\(509\) 12.5079 0.554402 0.277201 0.960812i \(-0.410593\pi\)
0.277201 + 0.960812i \(0.410593\pi\)
\(510\) 0 0
\(511\) 38.8768 1.71981
\(512\) 0 0
\(513\) 6.22574 0.274873
\(514\) 0 0
\(515\) −10.8316 −0.477298
\(516\) 0 0
\(517\) 21.1309 0.929338
\(518\) 0 0
\(519\) −18.9314 −0.830996
\(520\) 0 0
\(521\) −24.1539 −1.05820 −0.529102 0.848558i \(-0.677471\pi\)
−0.529102 + 0.848558i \(0.677471\pi\)
\(522\) 0 0
\(523\) 28.1427 1.23059 0.615297 0.788296i \(-0.289036\pi\)
0.615297 + 0.788296i \(0.289036\pi\)
\(524\) 0 0
\(525\) −4.46569 −0.194899
\(526\) 0 0
\(527\) −33.7510 −1.47022
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.55148 0.0673285
\(532\) 0 0
\(533\) −14.4537 −0.626060
\(534\) 0 0
\(535\) −1.76004 −0.0760934
\(536\) 0 0
\(537\) 18.7598 0.809546
\(538\) 0 0
\(539\) −43.9282 −1.89212
\(540\) 0 0
\(541\) −17.0117 −0.731392 −0.365696 0.930734i \(-0.619169\pi\)
−0.365696 + 0.930734i \(0.619169\pi\)
\(542\) 0 0
\(543\) −21.1964 −0.909623
\(544\) 0 0
\(545\) 1.39413 0.0597178
\(546\) 0 0
\(547\) 19.0402 0.814099 0.407050 0.913406i \(-0.366558\pi\)
0.407050 + 0.913406i \(0.366558\pi\)
\(548\) 0 0
\(549\) −4.91421 −0.209734
\(550\) 0 0
\(551\) 2.79237 0.118959
\(552\) 0 0
\(553\) −58.6790 −2.49529
\(554\) 0 0
\(555\) 5.98578 0.254082
\(556\) 0 0
\(557\) 23.6882 1.00370 0.501852 0.864954i \(-0.332652\pi\)
0.501852 + 0.864954i \(0.332652\pi\)
\(558\) 0 0
\(559\) −21.0809 −0.891627
\(560\) 0 0
\(561\) 22.2257 0.938371
\(562\) 0 0
\(563\) 12.8630 0.542111 0.271055 0.962564i \(-0.412627\pi\)
0.271055 + 0.962564i \(0.412627\pi\)
\(564\) 0 0
\(565\) −1.17134 −0.0492788
\(566\) 0 0
\(567\) −4.46569 −0.187542
\(568\) 0 0
\(569\) −10.4709 −0.438963 −0.219481 0.975617i \(-0.570437\pi\)
−0.219481 + 0.975617i \(0.570437\pi\)
\(570\) 0 0
\(571\) 11.0361 0.461845 0.230922 0.972972i \(-0.425826\pi\)
0.230922 + 0.972972i \(0.425826\pi\)
\(572\) 0 0
\(573\) −14.3909 −0.601190
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −45.9535 −1.91307 −0.956535 0.291616i \(-0.905807\pi\)
−0.956535 + 0.291616i \(0.905807\pi\)
\(578\) 0 0
\(579\) 11.0966 0.461159
\(580\) 0 0
\(581\) −58.1800 −2.41371
\(582\) 0 0
\(583\) 8.79641 0.364310
\(584\) 0 0
\(585\) 4.70565 0.194555
\(586\) 0 0
\(587\) 24.2578 1.00123 0.500614 0.865671i \(-0.333108\pi\)
0.500614 + 0.865671i \(0.333108\pi\)
\(588\) 0 0
\(589\) 32.0885 1.32218
\(590\) 0 0
\(591\) 14.1652 0.582679
\(592\) 0 0
\(593\) −27.5282 −1.13045 −0.565225 0.824937i \(-0.691211\pi\)
−0.565225 + 0.824937i \(0.691211\pi\)
\(594\) 0 0
\(595\) −29.2427 −1.19883
\(596\) 0 0
\(597\) −8.66004 −0.354432
\(598\) 0 0
\(599\) −31.2741 −1.27782 −0.638912 0.769280i \(-0.720615\pi\)
−0.638912 + 0.769280i \(0.720615\pi\)
\(600\) 0 0
\(601\) 43.3253 1.76728 0.883638 0.468171i \(-0.155087\pi\)
0.883638 + 0.468171i \(0.155087\pi\)
\(602\) 0 0
\(603\) −6.08556 −0.247823
\(604\) 0 0
\(605\) −0.520089 −0.0211446
\(606\) 0 0
\(607\) −29.7802 −1.20874 −0.604370 0.796704i \(-0.706575\pi\)
−0.604370 + 0.796704i \(0.706575\pi\)
\(608\) 0 0
\(609\) −2.00296 −0.0811639
\(610\) 0 0
\(611\) 29.2961 1.18520
\(612\) 0 0
\(613\) 13.6763 0.552380 0.276190 0.961103i \(-0.410928\pi\)
0.276190 + 0.961103i \(0.410928\pi\)
\(614\) 0 0
\(615\) −3.07157 −0.123858
\(616\) 0 0
\(617\) 14.9130 0.600377 0.300188 0.953880i \(-0.402950\pi\)
0.300188 + 0.953880i \(0.402950\pi\)
\(618\) 0 0
\(619\) 41.3021 1.66007 0.830035 0.557712i \(-0.188320\pi\)
0.830035 + 0.557712i \(0.188320\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −51.8908 −2.07896
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −21.1309 −0.843888
\(628\) 0 0
\(629\) 39.1967 1.56287
\(630\) 0 0
\(631\) −44.2104 −1.75999 −0.879993 0.474986i \(-0.842453\pi\)
−0.879993 + 0.474986i \(0.842453\pi\)
\(632\) 0 0
\(633\) 11.1321 0.442461
\(634\) 0 0
\(635\) 13.9454 0.553405
\(636\) 0 0
\(637\) −60.9025 −2.41304
\(638\) 0 0
\(639\) −1.07157 −0.0423906
\(640\) 0 0
\(641\) 32.2750 1.27479 0.637393 0.770539i \(-0.280013\pi\)
0.637393 + 0.770539i \(0.280013\pi\)
\(642\) 0 0
\(643\) 23.2689 0.917635 0.458817 0.888531i \(-0.348273\pi\)
0.458817 + 0.888531i \(0.348273\pi\)
\(644\) 0 0
\(645\) −4.47991 −0.176396
\(646\) 0 0
\(647\) 1.49390 0.0587313 0.0293656 0.999569i \(-0.490651\pi\)
0.0293656 + 0.999569i \(0.490651\pi\)
\(648\) 0 0
\(649\) −5.26592 −0.206705
\(650\) 0 0
\(651\) −23.0169 −0.902105
\(652\) 0 0
\(653\) 9.30663 0.364197 0.182098 0.983280i \(-0.441711\pi\)
0.182098 + 0.983280i \(0.441711\pi\)
\(654\) 0 0
\(655\) 13.8628 0.541663
\(656\) 0 0
\(657\) −8.70565 −0.339640
\(658\) 0 0
\(659\) −34.8903 −1.35913 −0.679566 0.733614i \(-0.737832\pi\)
−0.679566 + 0.733614i \(0.737832\pi\)
\(660\) 0 0
\(661\) 2.14578 0.0834613 0.0417307 0.999129i \(-0.486713\pi\)
0.0417307 + 0.999129i \(0.486713\pi\)
\(662\) 0 0
\(663\) 30.8140 1.19672
\(664\) 0 0
\(665\) 27.8022 1.07812
\(666\) 0 0
\(667\) −0.448521 −0.0173668
\(668\) 0 0
\(669\) −20.1711 −0.779860
\(670\) 0 0
\(671\) 16.6795 0.643903
\(672\) 0 0
\(673\) 27.0483 1.04264 0.521318 0.853362i \(-0.325440\pi\)
0.521318 + 0.853362i \(0.325440\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 45.9442 1.76578 0.882890 0.469580i \(-0.155595\pi\)
0.882890 + 0.469580i \(0.155595\pi\)
\(678\) 0 0
\(679\) −19.2397 −0.738353
\(680\) 0 0
\(681\) 0.523273 0.0200518
\(682\) 0 0
\(683\) 24.9571 0.954958 0.477479 0.878643i \(-0.341551\pi\)
0.477479 + 0.878643i \(0.341551\pi\)
\(684\) 0 0
\(685\) 14.4081 0.550506
\(686\) 0 0
\(687\) −3.33359 −0.127184
\(688\) 0 0
\(689\) 12.1954 0.464609
\(690\) 0 0
\(691\) 18.8167 0.715820 0.357910 0.933756i \(-0.383489\pi\)
0.357910 + 0.933756i \(0.383489\pi\)
\(692\) 0 0
\(693\) 15.1571 0.575772
\(694\) 0 0
\(695\) −10.1140 −0.383646
\(696\) 0 0
\(697\) −20.1135 −0.761855
\(698\) 0 0
\(699\) 15.5481 0.588082
\(700\) 0 0
\(701\) 42.0601 1.58859 0.794294 0.607534i \(-0.207841\pi\)
0.794294 + 0.607534i \(0.207841\pi\)
\(702\) 0 0
\(703\) −37.2659 −1.40551
\(704\) 0 0
\(705\) 6.22574 0.234475
\(706\) 0 0
\(707\) −46.1741 −1.73655
\(708\) 0 0
\(709\) −29.3133 −1.10088 −0.550442 0.834873i \(-0.685541\pi\)
−0.550442 + 0.834873i \(0.685541\pi\)
\(710\) 0 0
\(711\) 13.1400 0.492787
\(712\) 0 0
\(713\) −5.15417 −0.193025
\(714\) 0 0
\(715\) −15.9716 −0.597303
\(716\) 0 0
\(717\) −27.5451 −1.02869
\(718\) 0 0
\(719\) 8.95641 0.334018 0.167009 0.985955i \(-0.446589\pi\)
0.167009 + 0.985955i \(0.446589\pi\)
\(720\) 0 0
\(721\) −48.3707 −1.80142
\(722\) 0 0
\(723\) 8.74000 0.325044
\(724\) 0 0
\(725\) 0.448521 0.0166576
\(726\) 0 0
\(727\) 16.4313 0.609405 0.304702 0.952448i \(-0.401443\pi\)
0.304702 + 0.952448i \(0.401443\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −29.3358 −1.08502
\(732\) 0 0
\(733\) 8.41519 0.310822 0.155411 0.987850i \(-0.450330\pi\)
0.155411 + 0.987850i \(0.450330\pi\)
\(734\) 0 0
\(735\) −12.9424 −0.477388
\(736\) 0 0
\(737\) 20.6551 0.760842
\(738\) 0 0
\(739\) −10.3375 −0.380270 −0.190135 0.981758i \(-0.560893\pi\)
−0.190135 + 0.981758i \(0.560893\pi\)
\(740\) 0 0
\(741\) −29.2961 −1.07622
\(742\) 0 0
\(743\) 35.6388 1.30746 0.653731 0.756727i \(-0.273203\pi\)
0.653731 + 0.756727i \(0.273203\pi\)
\(744\) 0 0
\(745\) −3.32870 −0.121954
\(746\) 0 0
\(747\) 13.0282 0.476677
\(748\) 0 0
\(749\) −7.85982 −0.287192
\(750\) 0 0
\(751\) −30.4402 −1.11078 −0.555390 0.831590i \(-0.687431\pi\)
−0.555390 + 0.831590i \(0.687431\pi\)
\(752\) 0 0
\(753\) −15.0312 −0.547766
\(754\) 0 0
\(755\) −22.3427 −0.813134
\(756\) 0 0
\(757\) −10.4941 −0.381416 −0.190708 0.981647i \(-0.561078\pi\)
−0.190708 + 0.981647i \(0.561078\pi\)
\(758\) 0 0
\(759\) 3.39413 0.123199
\(760\) 0 0
\(761\) −3.99704 −0.144893 −0.0724464 0.997372i \(-0.523081\pi\)
−0.0724464 + 0.997372i \(0.523081\pi\)
\(762\) 0 0
\(763\) 6.22574 0.225387
\(764\) 0 0
\(765\) 6.54830 0.236754
\(766\) 0 0
\(767\) −7.30072 −0.263614
\(768\) 0 0
\(769\) −15.6307 −0.563656 −0.281828 0.959465i \(-0.590941\pi\)
−0.281828 + 0.959465i \(0.590941\pi\)
\(770\) 0 0
\(771\) −13.1571 −0.473842
\(772\) 0 0
\(773\) 33.6912 1.21179 0.605894 0.795545i \(-0.292815\pi\)
0.605894 + 0.795545i \(0.292815\pi\)
\(774\) 0 0
\(775\) 5.15417 0.185143
\(776\) 0 0
\(777\) 26.7307 0.958958
\(778\) 0 0
\(779\) 19.1228 0.685145
\(780\) 0 0
\(781\) 3.63704 0.130143
\(782\) 0 0
\(783\) 0.448521 0.0160288
\(784\) 0 0
\(785\) −6.63090 −0.236667
\(786\) 0 0
\(787\) −1.57721 −0.0562215 −0.0281108 0.999605i \(-0.508949\pi\)
−0.0281108 + 0.999605i \(0.508949\pi\)
\(788\) 0 0
\(789\) 23.5795 0.839451
\(790\) 0 0
\(791\) −5.23086 −0.185988
\(792\) 0 0
\(793\) 23.1246 0.821178
\(794\) 0 0
\(795\) 2.59166 0.0919166
\(796\) 0 0
\(797\) −14.5677 −0.516015 −0.258007 0.966143i \(-0.583066\pi\)
−0.258007 + 0.966143i \(0.583066\pi\)
\(798\) 0 0
\(799\) 40.7680 1.44227
\(800\) 0 0
\(801\) 11.6199 0.410568
\(802\) 0 0
\(803\) 29.5481 1.04273
\(804\) 0 0
\(805\) −4.46569 −0.157395
\(806\) 0 0
\(807\) −7.52304 −0.264824
\(808\) 0 0
\(809\) 29.1280 1.02409 0.512043 0.858960i \(-0.328889\pi\)
0.512043 + 0.858960i \(0.328889\pi\)
\(810\) 0 0
\(811\) 28.0351 0.984444 0.492222 0.870470i \(-0.336185\pi\)
0.492222 + 0.870470i \(0.336185\pi\)
\(812\) 0 0
\(813\) −18.5370 −0.650122
\(814\) 0 0
\(815\) −22.3083 −0.781427
\(816\) 0 0
\(817\) 27.8908 0.975774
\(818\) 0 0
\(819\) 21.0140 0.734288
\(820\) 0 0
\(821\) 42.5015 1.48331 0.741657 0.670780i \(-0.234040\pi\)
0.741657 + 0.670780i \(0.234040\pi\)
\(822\) 0 0
\(823\) 14.5662 0.507745 0.253873 0.967238i \(-0.418296\pi\)
0.253873 + 0.967238i \(0.418296\pi\)
\(824\) 0 0
\(825\) −3.39413 −0.118168
\(826\) 0 0
\(827\) −11.7880 −0.409910 −0.204955 0.978771i \(-0.565705\pi\)
−0.204955 + 0.978771i \(0.565705\pi\)
\(828\) 0 0
\(829\) −2.34385 −0.0814053 −0.0407027 0.999171i \(-0.512960\pi\)
−0.0407027 + 0.999171i \(0.512960\pi\)
\(830\) 0 0
\(831\) 23.8628 0.827791
\(832\) 0 0
\(833\) −84.7508 −2.93644
\(834\) 0 0
\(835\) −24.0885 −0.833617
\(836\) 0 0
\(837\) 5.15417 0.178154
\(838\) 0 0
\(839\) −7.97157 −0.275209 −0.137604 0.990487i \(-0.543940\pi\)
−0.137604 + 0.990487i \(0.543940\pi\)
\(840\) 0 0
\(841\) −28.7988 −0.993063
\(842\) 0 0
\(843\) −22.6118 −0.778791
\(844\) 0 0
\(845\) −9.14314 −0.314533
\(846\) 0 0
\(847\) −2.32256 −0.0798040
\(848\) 0 0
\(849\) −13.6684 −0.469099
\(850\) 0 0
\(851\) 5.98578 0.205190
\(852\) 0 0
\(853\) 49.4736 1.69394 0.846972 0.531637i \(-0.178423\pi\)
0.846972 + 0.531637i \(0.178423\pi\)
\(854\) 0 0
\(855\) −6.22574 −0.212916
\(856\) 0 0
\(857\) 36.7359 1.25487 0.627437 0.778667i \(-0.284104\pi\)
0.627437 + 0.778667i \(0.284104\pi\)
\(858\) 0 0
\(859\) −13.8277 −0.471796 −0.235898 0.971778i \(-0.575803\pi\)
−0.235898 + 0.971778i \(0.575803\pi\)
\(860\) 0 0
\(861\) −13.7167 −0.467463
\(862\) 0 0
\(863\) −3.43337 −0.116873 −0.0584366 0.998291i \(-0.518612\pi\)
−0.0584366 + 0.998291i \(0.518612\pi\)
\(864\) 0 0
\(865\) 18.9314 0.643686
\(866\) 0 0
\(867\) 25.8802 0.878937
\(868\) 0 0
\(869\) −44.5987 −1.51291
\(870\) 0 0
\(871\) 28.6365 0.970311
\(872\) 0 0
\(873\) 4.30834 0.145815
\(874\) 0 0
\(875\) 4.46569 0.150968
\(876\) 0 0
\(877\) 27.9838 0.944947 0.472474 0.881345i \(-0.343361\pi\)
0.472474 + 0.881345i \(0.343361\pi\)
\(878\) 0 0
\(879\) −15.0431 −0.507392
\(880\) 0 0
\(881\) −2.77108 −0.0933600 −0.0466800 0.998910i \(-0.514864\pi\)
−0.0466800 + 0.998910i \(0.514864\pi\)
\(882\) 0 0
\(883\) −42.9052 −1.44387 −0.721937 0.691958i \(-0.756748\pi\)
−0.721937 + 0.691958i \(0.756748\pi\)
\(884\) 0 0
\(885\) −1.55148 −0.0521524
\(886\) 0 0
\(887\) −54.1991 −1.81983 −0.909914 0.414798i \(-0.863852\pi\)
−0.909914 + 0.414798i \(0.863852\pi\)
\(888\) 0 0
\(889\) 62.2758 2.08866
\(890\) 0 0
\(891\) −3.39413 −0.113707
\(892\) 0 0
\(893\) −38.7598 −1.29705
\(894\) 0 0
\(895\) −18.7598 −0.627072
\(896\) 0 0
\(897\) 4.70565 0.157117
\(898\) 0 0
\(899\) 2.31175 0.0771012
\(900\) 0 0
\(901\) 16.9709 0.565384
\(902\) 0 0
\(903\) −20.0059 −0.665755
\(904\) 0 0
\(905\) 21.1964 0.704591
\(906\) 0 0
\(907\) 26.4003 0.876606 0.438303 0.898827i \(-0.355580\pi\)
0.438303 + 0.898827i \(0.355580\pi\)
\(908\) 0 0
\(909\) 10.3397 0.342947
\(910\) 0 0
\(911\) 38.5662 1.27775 0.638877 0.769309i \(-0.279399\pi\)
0.638877 + 0.769309i \(0.279399\pi\)
\(912\) 0 0
\(913\) −44.2194 −1.46345
\(914\) 0 0
\(915\) 4.91421 0.162459
\(916\) 0 0
\(917\) 61.9069 2.04435
\(918\) 0 0
\(919\) 7.56345 0.249495 0.124748 0.992189i \(-0.460188\pi\)
0.124748 + 0.992189i \(0.460188\pi\)
\(920\) 0 0
\(921\) 31.9674 1.05336
\(922\) 0 0
\(923\) 5.04242 0.165973
\(924\) 0 0
\(925\) −5.98578 −0.196811
\(926\) 0 0
\(927\) 10.8316 0.355757
\(928\) 0 0
\(929\) −28.5329 −0.936135 −0.468067 0.883693i \(-0.655050\pi\)
−0.468067 + 0.883693i \(0.655050\pi\)
\(930\) 0 0
\(931\) 80.5761 2.64078
\(932\) 0 0
\(933\) 34.1932 1.11943
\(934\) 0 0
\(935\) −22.2257 −0.726859
\(936\) 0 0
\(937\) −37.9535 −1.23989 −0.619944 0.784646i \(-0.712845\pi\)
−0.619944 + 0.784646i \(0.712845\pi\)
\(938\) 0 0
\(939\) −11.5713 −0.377615
\(940\) 0 0
\(941\) −5.79633 −0.188955 −0.0944775 0.995527i \(-0.530118\pi\)
−0.0944775 + 0.995527i \(0.530118\pi\)
\(942\) 0 0
\(943\) −3.07157 −0.100024
\(944\) 0 0
\(945\) 4.46569 0.145269
\(946\) 0 0
\(947\) 34.1588 1.11001 0.555007 0.831846i \(-0.312716\pi\)
0.555007 + 0.831846i \(0.312716\pi\)
\(948\) 0 0
\(949\) 40.9657 1.32980
\(950\) 0 0
\(951\) 3.12915 0.101470
\(952\) 0 0
\(953\) −22.3110 −0.722724 −0.361362 0.932426i \(-0.617688\pi\)
−0.361362 + 0.932426i \(0.617688\pi\)
\(954\) 0 0
\(955\) 14.3909 0.465680
\(956\) 0 0
\(957\) −1.52234 −0.0492101
\(958\) 0 0
\(959\) 64.3422 2.07772
\(960\) 0 0
\(961\) −4.43453 −0.143049
\(962\) 0 0
\(963\) 1.76004 0.0567166
\(964\) 0 0
\(965\) −11.0966 −0.357212
\(966\) 0 0
\(967\) −21.5282 −0.692302 −0.346151 0.938179i \(-0.612511\pi\)
−0.346151 + 0.938179i \(0.612511\pi\)
\(968\) 0 0
\(969\) −40.7680 −1.30966
\(970\) 0 0
\(971\) 27.0939 0.869486 0.434743 0.900555i \(-0.356839\pi\)
0.434743 + 0.900555i \(0.356839\pi\)
\(972\) 0 0
\(973\) −45.1660 −1.44796
\(974\) 0 0
\(975\) −4.70565 −0.150701
\(976\) 0 0
\(977\) −27.9880 −0.895416 −0.447708 0.894180i \(-0.647760\pi\)
−0.447708 + 0.894180i \(0.647760\pi\)
\(978\) 0 0
\(979\) −39.4393 −1.26048
\(980\) 0 0
\(981\) −1.39413 −0.0445110
\(982\) 0 0
\(983\) −21.2711 −0.678443 −0.339222 0.940706i \(-0.610164\pi\)
−0.339222 + 0.940706i \(0.610164\pi\)
\(984\) 0 0
\(985\) −14.1652 −0.451341
\(986\) 0 0
\(987\) 27.8022 0.884955
\(988\) 0 0
\(989\) −4.47991 −0.142453
\(990\) 0 0
\(991\) −51.7023 −1.64238 −0.821189 0.570656i \(-0.806689\pi\)
−0.821189 + 0.570656i \(0.806689\pi\)
\(992\) 0 0
\(993\) 0.988966 0.0313839
\(994\) 0 0
\(995\) 8.66004 0.274542
\(996\) 0 0
\(997\) 27.8687 0.882610 0.441305 0.897357i \(-0.354516\pi\)
0.441305 + 0.897357i \(0.354516\pi\)
\(998\) 0 0
\(999\) −5.98578 −0.189382
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.cb.1.1 4
4.3 odd 2 2760.2.a.v.1.4 4
12.11 even 2 8280.2.a.bq.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.v.1.4 4 4.3 odd 2
5520.2.a.cb.1.1 4 1.1 even 1 trivial
8280.2.a.bq.1.4 4 12.11 even 2