Properties

Label 3332.2.a.g.1.1
Level $3332$
Weight $2$
Character 3332.1
Self dual yes
Analytic conductor $26.606$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,2,Mod(1,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.6061539535\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 3332.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61803 q^{3} -3.61803 q^{5} +3.85410 q^{9} +O(q^{10})\) \(q-2.61803 q^{3} -3.61803 q^{5} +3.85410 q^{9} -1.23607 q^{13} +9.47214 q^{15} -1.00000 q^{17} -6.47214 q^{19} +0.763932 q^{23} +8.09017 q^{25} -2.23607 q^{27} +7.23607 q^{29} +10.5623 q^{31} -9.70820 q^{37} +3.23607 q^{39} +7.85410 q^{41} -1.85410 q^{43} -13.9443 q^{45} +6.94427 q^{47} +2.61803 q^{51} -3.32624 q^{53} +16.9443 q^{57} -0.763932 q^{59} +7.56231 q^{61} +4.47214 q^{65} +6.09017 q^{67} -2.00000 q^{69} -9.23607 q^{71} -3.85410 q^{73} -21.1803 q^{75} +11.7082 q^{79} -5.70820 q^{81} +10.4721 q^{83} +3.61803 q^{85} -18.9443 q^{87} -8.76393 q^{89} -27.6525 q^{93} +23.4164 q^{95} +2.85410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 5 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 5 q^{5} + q^{9} + 2 q^{13} + 10 q^{15} - 2 q^{17} - 4 q^{19} + 6 q^{23} + 5 q^{25} + 10 q^{29} + q^{31} - 6 q^{37} + 2 q^{39} + 9 q^{41} + 3 q^{43} - 10 q^{45} - 4 q^{47} + 3 q^{51} + 9 q^{53} + 16 q^{57} - 6 q^{59} - 5 q^{61} + q^{67} - 4 q^{69} - 14 q^{71} - q^{73} - 20 q^{75} + 10 q^{79} + 2 q^{81} + 12 q^{83} + 5 q^{85} - 20 q^{87} - 22 q^{89} - 24 q^{93} + 20 q^{95} - q^{97}+O(q^{100}) \) 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
\(3\) −2.61803 −1.51152 −0.755761 0.654847i \(-0.772733\pi\)
−0.755761 + 0.654847i \(0.772733\pi\)
\(4\) 0 0
\(5\) −3.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.85410 1.28470
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) 0 0
\(15\) 9.47214 2.44569
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.763932 0.159291 0.0796454 0.996823i \(-0.474621\pi\)
0.0796454 + 0.996823i \(0.474621\pi\)
\(24\) 0 0
\(25\) 8.09017 1.61803
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) 7.23607 1.34370 0.671852 0.740685i \(-0.265499\pi\)
0.671852 + 0.740685i \(0.265499\pi\)
\(30\) 0 0
\(31\) 10.5623 1.89705 0.948523 0.316708i \(-0.102578\pi\)
0.948523 + 0.316708i \(0.102578\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.70820 −1.59602 −0.798009 0.602645i \(-0.794114\pi\)
−0.798009 + 0.602645i \(0.794114\pi\)
\(38\) 0 0
\(39\) 3.23607 0.518186
\(40\) 0 0
\(41\) 7.85410 1.22660 0.613302 0.789848i \(-0.289841\pi\)
0.613302 + 0.789848i \(0.289841\pi\)
\(42\) 0 0
\(43\) −1.85410 −0.282748 −0.141374 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(44\) 0 0
\(45\) −13.9443 −2.07869
\(46\) 0 0
\(47\) 6.94427 1.01293 0.506463 0.862262i \(-0.330953\pi\)
0.506463 + 0.862262i \(0.330953\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.61803 0.366598
\(52\) 0 0
\(53\) −3.32624 −0.456894 −0.228447 0.973556i \(-0.573365\pi\)
−0.228447 + 0.973556i \(0.573365\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 16.9443 2.24432
\(58\) 0 0
\(59\) −0.763932 −0.0994555 −0.0497277 0.998763i \(-0.515835\pi\)
−0.0497277 + 0.998763i \(0.515835\pi\)
\(60\) 0 0
\(61\) 7.56231 0.968254 0.484127 0.874998i \(-0.339137\pi\)
0.484127 + 0.874998i \(0.339137\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.47214 0.554700
\(66\) 0 0
\(67\) 6.09017 0.744033 0.372016 0.928226i \(-0.378667\pi\)
0.372016 + 0.928226i \(0.378667\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −9.23607 −1.09612 −0.548060 0.836439i \(-0.684633\pi\)
−0.548060 + 0.836439i \(0.684633\pi\)
\(72\) 0 0
\(73\) −3.85410 −0.451089 −0.225544 0.974233i \(-0.572416\pi\)
−0.225544 + 0.974233i \(0.572416\pi\)
\(74\) 0 0
\(75\) −21.1803 −2.44569
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.7082 1.31728 0.658638 0.752460i \(-0.271133\pi\)
0.658638 + 0.752460i \(0.271133\pi\)
\(80\) 0 0
\(81\) −5.70820 −0.634245
\(82\) 0 0
\(83\) 10.4721 1.14947 0.574733 0.818341i \(-0.305106\pi\)
0.574733 + 0.818341i \(0.305106\pi\)
\(84\) 0 0
\(85\) 3.61803 0.392431
\(86\) 0 0
\(87\) −18.9443 −2.03104
\(88\) 0 0
\(89\) −8.76393 −0.928975 −0.464487 0.885580i \(-0.653761\pi\)
−0.464487 + 0.885580i \(0.653761\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −27.6525 −2.86743
\(94\) 0 0
\(95\) 23.4164 2.40247
\(96\) 0 0
\(97\) 2.85410 0.289790 0.144895 0.989447i \(-0.453716\pi\)
0.144895 + 0.989447i \(0.453716\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.52786 −0.749050 −0.374525 0.927217i \(-0.622194\pi\)
−0.374525 + 0.927217i \(0.622194\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 12.1803 1.16666 0.583332 0.812233i \(-0.301748\pi\)
0.583332 + 0.812233i \(0.301748\pi\)
\(110\) 0 0
\(111\) 25.4164 2.41242
\(112\) 0 0
\(113\) 12.9443 1.21769 0.608847 0.793287i \(-0.291632\pi\)
0.608847 + 0.793287i \(0.291632\pi\)
\(114\) 0 0
\(115\) −2.76393 −0.257738
\(116\) 0 0
\(117\) −4.76393 −0.440426
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −20.5623 −1.85404
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) −15.6180 −1.38588 −0.692938 0.720997i \(-0.743684\pi\)
−0.692938 + 0.720997i \(0.743684\pi\)
\(128\) 0 0
\(129\) 4.85410 0.427380
\(130\) 0 0
\(131\) −6.47214 −0.565473 −0.282737 0.959198i \(-0.591242\pi\)
−0.282737 + 0.959198i \(0.591242\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 8.09017 0.696291
\(136\) 0 0
\(137\) 11.0902 0.947497 0.473749 0.880660i \(-0.342901\pi\)
0.473749 + 0.880660i \(0.342901\pi\)
\(138\) 0 0
\(139\) 13.6180 1.15507 0.577533 0.816367i \(-0.304015\pi\)
0.577533 + 0.816367i \(0.304015\pi\)
\(140\) 0 0
\(141\) −18.1803 −1.53106
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −26.1803 −2.17416
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.38197 −0.277061 −0.138531 0.990358i \(-0.544238\pi\)
−0.138531 + 0.990358i \(0.544238\pi\)
\(150\) 0 0
\(151\) −21.2705 −1.73097 −0.865485 0.500935i \(-0.832989\pi\)
−0.865485 + 0.500935i \(0.832989\pi\)
\(152\) 0 0
\(153\) −3.85410 −0.311586
\(154\) 0 0
\(155\) −38.2148 −3.06949
\(156\) 0 0
\(157\) −23.7082 −1.89212 −0.946060 0.323991i \(-0.894975\pi\)
−0.946060 + 0.323991i \(0.894975\pi\)
\(158\) 0 0
\(159\) 8.70820 0.690605
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.437694 0.0338698 0.0169349 0.999857i \(-0.494609\pi\)
0.0169349 + 0.999857i \(0.494609\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 0 0
\(171\) −24.9443 −1.90754
\(172\) 0 0
\(173\) −22.3262 −1.69743 −0.848716 0.528849i \(-0.822624\pi\)
−0.848716 + 0.528849i \(0.822624\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −3.09017 −0.230970 −0.115485 0.993309i \(-0.536842\pi\)
−0.115485 + 0.993309i \(0.536842\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −19.7984 −1.46354
\(184\) 0 0
\(185\) 35.1246 2.58241
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.0902 1.59839 0.799194 0.601073i \(-0.205260\pi\)
0.799194 + 0.601073i \(0.205260\pi\)
\(192\) 0 0
\(193\) −5.52786 −0.397904 −0.198952 0.980009i \(-0.563754\pi\)
−0.198952 + 0.980009i \(0.563754\pi\)
\(194\) 0 0
\(195\) −11.7082 −0.838442
\(196\) 0 0
\(197\) −15.7082 −1.11916 −0.559582 0.828775i \(-0.689038\pi\)
−0.559582 + 0.828775i \(0.689038\pi\)
\(198\) 0 0
\(199\) −12.1459 −0.861000 −0.430500 0.902591i \(-0.641663\pi\)
−0.430500 + 0.902591i \(0.641663\pi\)
\(200\) 0 0
\(201\) −15.9443 −1.12462
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −28.4164 −1.98469
\(206\) 0 0
\(207\) 2.94427 0.204641
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 23.5967 1.62447 0.812234 0.583332i \(-0.198251\pi\)
0.812234 + 0.583332i \(0.198251\pi\)
\(212\) 0 0
\(213\) 24.1803 1.65681
\(214\) 0 0
\(215\) 6.70820 0.457496
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 10.0902 0.681830
\(220\) 0 0
\(221\) 1.23607 0.0831469
\(222\) 0 0
\(223\) −15.8885 −1.06398 −0.531988 0.846752i \(-0.678555\pi\)
−0.531988 + 0.846752i \(0.678555\pi\)
\(224\) 0 0
\(225\) 31.1803 2.07869
\(226\) 0 0
\(227\) 7.14590 0.474290 0.237145 0.971474i \(-0.423788\pi\)
0.237145 + 0.971474i \(0.423788\pi\)
\(228\) 0 0
\(229\) 17.4164 1.15091 0.575454 0.817834i \(-0.304825\pi\)
0.575454 + 0.817834i \(0.304825\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.18034 −0.142839 −0.0714194 0.997446i \(-0.522753\pi\)
−0.0714194 + 0.997446i \(0.522753\pi\)
\(234\) 0 0
\(235\) −25.1246 −1.63895
\(236\) 0 0
\(237\) −30.6525 −1.99109
\(238\) 0 0
\(239\) −19.3820 −1.25372 −0.626858 0.779134i \(-0.715659\pi\)
−0.626858 + 0.779134i \(0.715659\pi\)
\(240\) 0 0
\(241\) −3.61803 −0.233058 −0.116529 0.993187i \(-0.537177\pi\)
−0.116529 + 0.993187i \(0.537177\pi\)
\(242\) 0 0
\(243\) 21.6525 1.38901
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) −27.4164 −1.73744
\(250\) 0 0
\(251\) 20.1803 1.27377 0.636886 0.770958i \(-0.280222\pi\)
0.636886 + 0.770958i \(0.280222\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −9.47214 −0.593168
\(256\) 0 0
\(257\) −1.52786 −0.0953055 −0.0476528 0.998864i \(-0.515174\pi\)
−0.0476528 + 0.998864i \(0.515174\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 27.8885 1.72626
\(262\) 0 0
\(263\) 19.4164 1.19727 0.598633 0.801023i \(-0.295711\pi\)
0.598633 + 0.801023i \(0.295711\pi\)
\(264\) 0 0
\(265\) 12.0344 0.739270
\(266\) 0 0
\(267\) 22.9443 1.40417
\(268\) 0 0
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 0 0
\(271\) 20.4721 1.24359 0.621797 0.783179i \(-0.286403\pi\)
0.621797 + 0.783179i \(0.286403\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −31.1246 −1.87010 −0.935048 0.354520i \(-0.884644\pi\)
−0.935048 + 0.354520i \(0.884644\pi\)
\(278\) 0 0
\(279\) 40.7082 2.43714
\(280\) 0 0
\(281\) −23.5066 −1.40228 −0.701142 0.713021i \(-0.747326\pi\)
−0.701142 + 0.713021i \(0.747326\pi\)
\(282\) 0 0
\(283\) −19.1459 −1.13811 −0.569053 0.822301i \(-0.692690\pi\)
−0.569053 + 0.822301i \(0.692690\pi\)
\(284\) 0 0
\(285\) −61.3050 −3.63139
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −7.47214 −0.438024
\(292\) 0 0
\(293\) 27.5967 1.61222 0.806110 0.591766i \(-0.201569\pi\)
0.806110 + 0.591766i \(0.201569\pi\)
\(294\) 0 0
\(295\) 2.76393 0.160922
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.944272 −0.0546087
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 19.7082 1.13221
\(304\) 0 0
\(305\) −27.3607 −1.56667
\(306\) 0 0
\(307\) 9.41641 0.537423 0.268711 0.963221i \(-0.413402\pi\)
0.268711 + 0.963221i \(0.413402\pi\)
\(308\) 0 0
\(309\) 15.7082 0.893609
\(310\) 0 0
\(311\) 30.7426 1.74326 0.871628 0.490168i \(-0.163065\pi\)
0.871628 + 0.490168i \(0.163065\pi\)
\(312\) 0 0
\(313\) −17.1459 −0.969143 −0.484572 0.874752i \(-0.661025\pi\)
−0.484572 + 0.874752i \(0.661025\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.8885 1.34171 0.670857 0.741587i \(-0.265926\pi\)
0.670857 + 0.741587i \(0.265926\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.47214 0.360119
\(324\) 0 0
\(325\) −10.0000 −0.554700
\(326\) 0 0
\(327\) −31.8885 −1.76344
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.3262 0.732476 0.366238 0.930521i \(-0.380646\pi\)
0.366238 + 0.930521i \(0.380646\pi\)
\(332\) 0 0
\(333\) −37.4164 −2.05041
\(334\) 0 0
\(335\) −22.0344 −1.20387
\(336\) 0 0
\(337\) −13.4164 −0.730838 −0.365419 0.930843i \(-0.619074\pi\)
−0.365419 + 0.930843i \(0.619074\pi\)
\(338\) 0 0
\(339\) −33.8885 −1.84057
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 7.23607 0.389577
\(346\) 0 0
\(347\) 25.8885 1.38977 0.694885 0.719121i \(-0.255455\pi\)
0.694885 + 0.719121i \(0.255455\pi\)
\(348\) 0 0
\(349\) −8.47214 −0.453503 −0.226752 0.973953i \(-0.572811\pi\)
−0.226752 + 0.973953i \(0.572811\pi\)
\(350\) 0 0
\(351\) 2.76393 0.147528
\(352\) 0 0
\(353\) −34.0689 −1.81330 −0.906652 0.421880i \(-0.861370\pi\)
−0.906652 + 0.421880i \(0.861370\pi\)
\(354\) 0 0
\(355\) 33.4164 1.77356
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.8541 0.572858 0.286429 0.958102i \(-0.407532\pi\)
0.286429 + 0.958102i \(0.407532\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 0 0
\(363\) 28.7984 1.51152
\(364\) 0 0
\(365\) 13.9443 0.729877
\(366\) 0 0
\(367\) 13.3820 0.698533 0.349266 0.937023i \(-0.386431\pi\)
0.349266 + 0.937023i \(0.386431\pi\)
\(368\) 0 0
\(369\) 30.2705 1.57582
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −30.5066 −1.57957 −0.789785 0.613383i \(-0.789808\pi\)
−0.789785 + 0.613383i \(0.789808\pi\)
\(374\) 0 0
\(375\) 29.2705 1.51152
\(376\) 0 0
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) −9.12461 −0.468700 −0.234350 0.972152i \(-0.575296\pi\)
−0.234350 + 0.972152i \(0.575296\pi\)
\(380\) 0 0
\(381\) 40.8885 2.09478
\(382\) 0 0
\(383\) 9.88854 0.505281 0.252640 0.967560i \(-0.418701\pi\)
0.252640 + 0.967560i \(0.418701\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.14590 −0.363246
\(388\) 0 0
\(389\) 27.5066 1.39464 0.697319 0.716761i \(-0.254376\pi\)
0.697319 + 0.716761i \(0.254376\pi\)
\(390\) 0 0
\(391\) −0.763932 −0.0386337
\(392\) 0 0
\(393\) 16.9443 0.854725
\(394\) 0 0
\(395\) −42.3607 −2.13140
\(396\) 0 0
\(397\) 17.3262 0.869579 0.434789 0.900532i \(-0.356823\pi\)
0.434789 + 0.900532i \(0.356823\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.0557 0.552097 0.276048 0.961144i \(-0.410975\pi\)
0.276048 + 0.961144i \(0.410975\pi\)
\(402\) 0 0
\(403\) −13.0557 −0.650352
\(404\) 0 0
\(405\) 20.6525 1.02623
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) −29.0344 −1.43216
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −37.8885 −1.85988
\(416\) 0 0
\(417\) −35.6525 −1.74591
\(418\) 0 0
\(419\) −35.5066 −1.73461 −0.867305 0.497777i \(-0.834150\pi\)
−0.867305 + 0.497777i \(0.834150\pi\)
\(420\) 0 0
\(421\) −10.1459 −0.494481 −0.247240 0.968954i \(-0.579524\pi\)
−0.247240 + 0.968954i \(0.579524\pi\)
\(422\) 0 0
\(423\) 26.7639 1.30131
\(424\) 0 0
\(425\) −8.09017 −0.392431
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.94427 −0.141821 −0.0709103 0.997483i \(-0.522590\pi\)
−0.0709103 + 0.997483i \(0.522590\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 68.5410 3.28629
\(436\) 0 0
\(437\) −4.94427 −0.236517
\(438\) 0 0
\(439\) 16.1459 0.770602 0.385301 0.922791i \(-0.374098\pi\)
0.385301 + 0.922791i \(0.374098\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.05573 −0.335228 −0.167614 0.985853i \(-0.553606\pi\)
−0.167614 + 0.985853i \(0.553606\pi\)
\(444\) 0 0
\(445\) 31.7082 1.50311
\(446\) 0 0
\(447\) 8.85410 0.418785
\(448\) 0 0
\(449\) 0.944272 0.0445629 0.0222815 0.999752i \(-0.492907\pi\)
0.0222815 + 0.999752i \(0.492907\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 55.6869 2.61640
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.2705 0.573990 0.286995 0.957932i \(-0.407344\pi\)
0.286995 + 0.957932i \(0.407344\pi\)
\(458\) 0 0
\(459\) 2.23607 0.104371
\(460\) 0 0
\(461\) 26.8328 1.24973 0.624864 0.780733i \(-0.285154\pi\)
0.624864 + 0.780733i \(0.285154\pi\)
\(462\) 0 0
\(463\) −2.79837 −0.130051 −0.0650257 0.997884i \(-0.520713\pi\)
−0.0650257 + 0.997884i \(0.520713\pi\)
\(464\) 0 0
\(465\) 100.048 4.63960
\(466\) 0 0
\(467\) 28.7639 1.33104 0.665518 0.746382i \(-0.268211\pi\)
0.665518 + 0.746382i \(0.268211\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 62.0689 2.85998
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −52.3607 −2.40247
\(476\) 0 0
\(477\) −12.8197 −0.586972
\(478\) 0 0
\(479\) 10.6180 0.485150 0.242575 0.970133i \(-0.422008\pi\)
0.242575 + 0.970133i \(0.422008\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.3262 −0.468890
\(486\) 0 0
\(487\) 19.0557 0.863497 0.431749 0.901994i \(-0.357897\pi\)
0.431749 + 0.901994i \(0.357897\pi\)
\(488\) 0 0
\(489\) 15.7082 0.710350
\(490\) 0 0
\(491\) −39.7984 −1.79608 −0.898038 0.439918i \(-0.855007\pi\)
−0.898038 + 0.439918i \(0.855007\pi\)
\(492\) 0 0
\(493\) −7.23607 −0.325896
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −35.3050 −1.58047 −0.790233 0.612806i \(-0.790041\pi\)
−0.790233 + 0.612806i \(0.790041\pi\)
\(500\) 0 0
\(501\) −1.14590 −0.0511949
\(502\) 0 0
\(503\) −12.2705 −0.547115 −0.273557 0.961856i \(-0.588200\pi\)
−0.273557 + 0.961856i \(0.588200\pi\)
\(504\) 0 0
\(505\) 27.2361 1.21199
\(506\) 0 0
\(507\) 30.0344 1.33388
\(508\) 0 0
\(509\) 23.2361 1.02992 0.514960 0.857214i \(-0.327807\pi\)
0.514960 + 0.857214i \(0.327807\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 14.4721 0.638960
\(514\) 0 0
\(515\) 21.7082 0.956578
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 58.4508 2.56571
\(520\) 0 0
\(521\) 13.0902 0.573491 0.286745 0.958007i \(-0.407427\pi\)
0.286745 + 0.958007i \(0.407427\pi\)
\(522\) 0 0
\(523\) 7.34752 0.321285 0.160642 0.987013i \(-0.448643\pi\)
0.160642 + 0.987013i \(0.448643\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.5623 −0.460101
\(528\) 0 0
\(529\) −22.4164 −0.974626
\(530\) 0 0
\(531\) −2.94427 −0.127771
\(532\) 0 0
\(533\) −9.70820 −0.420509
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 8.09017 0.349117
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 36.6525 1.57291
\(544\) 0 0
\(545\) −44.0689 −1.88770
\(546\) 0 0
\(547\) −13.1246 −0.561168 −0.280584 0.959829i \(-0.590528\pi\)
−0.280584 + 0.959829i \(0.590528\pi\)
\(548\) 0 0
\(549\) 29.1459 1.24392
\(550\) 0 0
\(551\) −46.8328 −1.99515
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −91.9574 −3.90338
\(556\) 0 0
\(557\) 24.4721 1.03692 0.518459 0.855103i \(-0.326506\pi\)
0.518459 + 0.855103i \(0.326506\pi\)
\(558\) 0 0
\(559\) 2.29180 0.0969326
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 17.1246 0.721716 0.360858 0.932621i \(-0.382484\pi\)
0.360858 + 0.932621i \(0.382484\pi\)
\(564\) 0 0
\(565\) −46.8328 −1.97027
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.09017 0.297235 0.148618 0.988895i \(-0.452518\pi\)
0.148618 + 0.988895i \(0.452518\pi\)
\(570\) 0 0
\(571\) 28.6525 1.19907 0.599534 0.800349i \(-0.295352\pi\)
0.599534 + 0.800349i \(0.295352\pi\)
\(572\) 0 0
\(573\) −57.8328 −2.41600
\(574\) 0 0
\(575\) 6.18034 0.257738
\(576\) 0 0
\(577\) −45.4164 −1.89071 −0.945355 0.326043i \(-0.894285\pi\)
−0.945355 + 0.326043i \(0.894285\pi\)
\(578\) 0 0
\(579\) 14.4721 0.601441
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 17.2361 0.712624
\(586\) 0 0
\(587\) 30.0000 1.23823 0.619116 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 0 0
\(589\) −68.3607 −2.81675
\(590\) 0 0
\(591\) 41.1246 1.69164
\(592\) 0 0
\(593\) −29.5967 −1.21539 −0.607696 0.794169i \(-0.707906\pi\)
−0.607696 + 0.794169i \(0.707906\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 31.7984 1.30142
\(598\) 0 0
\(599\) −12.0344 −0.491714 −0.245857 0.969306i \(-0.579069\pi\)
−0.245857 + 0.969306i \(0.579069\pi\)
\(600\) 0 0
\(601\) −23.5279 −0.959722 −0.479861 0.877345i \(-0.659313\pi\)
−0.479861 + 0.877345i \(0.659313\pi\)
\(602\) 0 0
\(603\) 23.4721 0.955859
\(604\) 0 0
\(605\) 39.7984 1.61803
\(606\) 0 0
\(607\) 24.8541 1.00880 0.504398 0.863471i \(-0.331714\pi\)
0.504398 + 0.863471i \(0.331714\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.58359 −0.347255
\(612\) 0 0
\(613\) −0.854102 −0.0344969 −0.0172484 0.999851i \(-0.505491\pi\)
−0.0172484 + 0.999851i \(0.505491\pi\)
\(614\) 0 0
\(615\) 74.3951 2.99990
\(616\) 0 0
\(617\) 45.5967 1.83566 0.917828 0.396978i \(-0.129941\pi\)
0.917828 + 0.396978i \(0.129941\pi\)
\(618\) 0 0
\(619\) −46.8328 −1.88237 −0.941185 0.337892i \(-0.890286\pi\)
−0.941185 + 0.337892i \(0.890286\pi\)
\(620\) 0 0
\(621\) −1.70820 −0.0685479
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.70820 0.387091
\(630\) 0 0
\(631\) 15.9787 0.636103 0.318051 0.948074i \(-0.396972\pi\)
0.318051 + 0.948074i \(0.396972\pi\)
\(632\) 0 0
\(633\) −61.7771 −2.45542
\(634\) 0 0
\(635\) 56.5066 2.24240
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −35.5967 −1.40819
\(640\) 0 0
\(641\) 41.7771 1.65010 0.825048 0.565063i \(-0.191148\pi\)
0.825048 + 0.565063i \(0.191148\pi\)
\(642\) 0 0
\(643\) −29.3820 −1.15871 −0.579356 0.815075i \(-0.696696\pi\)
−0.579356 + 0.815075i \(0.696696\pi\)
\(644\) 0 0
\(645\) −17.5623 −0.691515
\(646\) 0 0
\(647\) −27.5967 −1.08494 −0.542470 0.840075i \(-0.682511\pi\)
−0.542470 + 0.840075i \(0.682511\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.2361 −0.987564 −0.493782 0.869586i \(-0.664386\pi\)
−0.493782 + 0.869586i \(0.664386\pi\)
\(654\) 0 0
\(655\) 23.4164 0.914955
\(656\) 0 0
\(657\) −14.8541 −0.579514
\(658\) 0 0
\(659\) −9.90983 −0.386032 −0.193016 0.981196i \(-0.561827\pi\)
−0.193016 + 0.981196i \(0.561827\pi\)
\(660\) 0 0
\(661\) 4.58359 0.178281 0.0891405 0.996019i \(-0.471588\pi\)
0.0891405 + 0.996019i \(0.471588\pi\)
\(662\) 0 0
\(663\) −3.23607 −0.125678
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.52786 0.214040
\(668\) 0 0
\(669\) 41.5967 1.60822
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.944272 0.0363990 0.0181995 0.999834i \(-0.494207\pi\)
0.0181995 + 0.999834i \(0.494207\pi\)
\(674\) 0 0
\(675\) −18.0902 −0.696291
\(676\) 0 0
\(677\) 20.8328 0.800670 0.400335 0.916369i \(-0.368894\pi\)
0.400335 + 0.916369i \(0.368894\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −18.7082 −0.716900
\(682\) 0 0
\(683\) −45.8885 −1.75588 −0.877938 0.478774i \(-0.841081\pi\)
−0.877938 + 0.478774i \(0.841081\pi\)
\(684\) 0 0
\(685\) −40.1246 −1.53308
\(686\) 0 0
\(687\) −45.5967 −1.73962
\(688\) 0 0
\(689\) 4.11146 0.156634
\(690\) 0 0
\(691\) −10.9787 −0.417650 −0.208825 0.977953i \(-0.566964\pi\)
−0.208825 + 0.977953i \(0.566964\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −49.2705 −1.86894
\(696\) 0 0
\(697\) −7.85410 −0.297495
\(698\) 0 0
\(699\) 5.70820 0.215904
\(700\) 0 0
\(701\) −23.3050 −0.880216 −0.440108 0.897945i \(-0.645060\pi\)
−0.440108 + 0.897945i \(0.645060\pi\)
\(702\) 0 0
\(703\) 62.8328 2.36978
\(704\) 0 0
\(705\) 65.7771 2.47731
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.65248 −0.0996158 −0.0498079 0.998759i \(-0.515861\pi\)
−0.0498079 + 0.998759i \(0.515861\pi\)
\(710\) 0 0
\(711\) 45.1246 1.69231
\(712\) 0 0
\(713\) 8.06888 0.302182
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 50.7426 1.89502
\(718\) 0 0
\(719\) 10.9098 0.406868 0.203434 0.979089i \(-0.434790\pi\)
0.203434 + 0.979089i \(0.434790\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9.47214 0.352273
\(724\) 0 0
\(725\) 58.5410 2.17416
\(726\) 0 0
\(727\) 13.7082 0.508409 0.254205 0.967150i \(-0.418186\pi\)
0.254205 + 0.967150i \(0.418186\pi\)
\(728\) 0 0
\(729\) −39.5623 −1.46527
\(730\) 0 0
\(731\) 1.85410 0.0685764
\(732\) 0 0
\(733\) −46.2492 −1.70825 −0.854127 0.520064i \(-0.825908\pi\)
−0.854127 + 0.520064i \(0.825908\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 7.38197 0.271550 0.135775 0.990740i \(-0.456648\pi\)
0.135775 + 0.990740i \(0.456648\pi\)
\(740\) 0 0
\(741\) −20.9443 −0.769407
\(742\) 0 0
\(743\) 0.111456 0.00408893 0.00204447 0.999998i \(-0.499349\pi\)
0.00204447 + 0.999998i \(0.499349\pi\)
\(744\) 0 0
\(745\) 12.2361 0.448295
\(746\) 0 0
\(747\) 40.3607 1.47672
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 50.8328 1.85492 0.927458 0.373928i \(-0.121989\pi\)
0.927458 + 0.373928i \(0.121989\pi\)
\(752\) 0 0
\(753\) −52.8328 −1.92533
\(754\) 0 0
\(755\) 76.9574 2.80077
\(756\) 0 0
\(757\) −5.56231 −0.202165 −0.101083 0.994878i \(-0.532231\pi\)
−0.101083 + 0.994878i \(0.532231\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.1115 −0.656540 −0.328270 0.944584i \(-0.606466\pi\)
−0.328270 + 0.944584i \(0.606466\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 13.9443 0.504156
\(766\) 0 0
\(767\) 0.944272 0.0340957
\(768\) 0 0
\(769\) −35.5967 −1.28365 −0.641826 0.766850i \(-0.721823\pi\)
−0.641826 + 0.766850i \(0.721823\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 0 0
\(773\) 11.5279 0.414628 0.207314 0.978274i \(-0.433528\pi\)
0.207314 + 0.978274i \(0.433528\pi\)
\(774\) 0 0
\(775\) 85.4508 3.06949
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −50.8328 −1.82127
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −16.1803 −0.578238
\(784\) 0 0
\(785\) 85.7771 3.06152
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) −50.8328 −1.80970
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.34752 −0.331940
\(794\) 0 0
\(795\) −31.5066 −1.11742
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) −6.94427 −0.245671
\(800\) 0 0
\(801\) −33.7771 −1.19345
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 35.1246 1.23644
\(808\) 0 0
\(809\) −41.1246 −1.44586 −0.722932 0.690919i \(-0.757206\pi\)
−0.722932 + 0.690919i \(0.757206\pi\)
\(810\) 0 0
\(811\) 14.9787 0.525974 0.262987 0.964799i \(-0.415292\pi\)
0.262987 + 0.964799i \(0.415292\pi\)
\(812\) 0 0
\(813\) −53.5967 −1.87972
\(814\) 0 0
\(815\) 21.7082 0.760405
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 37.5279 1.30973 0.654866 0.755745i \(-0.272725\pi\)
0.654866 + 0.755745i \(0.272725\pi\)
\(822\) 0 0
\(823\) −42.5410 −1.48289 −0.741443 0.671015i \(-0.765858\pi\)
−0.741443 + 0.671015i \(0.765858\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.7639 −1.27841 −0.639204 0.769038i \(-0.720736\pi\)
−0.639204 + 0.769038i \(0.720736\pi\)
\(828\) 0 0
\(829\) 11.5967 0.402772 0.201386 0.979512i \(-0.435455\pi\)
0.201386 + 0.979512i \(0.435455\pi\)
\(830\) 0 0
\(831\) 81.4853 2.82669
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.58359 −0.0548025
\(836\) 0 0
\(837\) −23.6180 −0.816359
\(838\) 0 0
\(839\) 32.9443 1.13736 0.568681 0.822558i \(-0.307454\pi\)
0.568681 + 0.822558i \(0.307454\pi\)
\(840\) 0 0
\(841\) 23.3607 0.805541
\(842\) 0 0
\(843\) 61.5410 2.11959
\(844\) 0 0
\(845\) 41.5066 1.42787
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 50.1246 1.72027
\(850\) 0 0
\(851\) −7.41641 −0.254231
\(852\) 0 0
\(853\) 25.7771 0.882591 0.441295 0.897362i \(-0.354519\pi\)
0.441295 + 0.897362i \(0.354519\pi\)
\(854\) 0 0
\(855\) 90.2492 3.08646
\(856\) 0 0
\(857\) 20.2016 0.690074 0.345037 0.938589i \(-0.387866\pi\)
0.345037 + 0.938589i \(0.387866\pi\)
\(858\) 0 0
\(859\) −16.8328 −0.574328 −0.287164 0.957881i \(-0.592713\pi\)
−0.287164 + 0.957881i \(0.592713\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.6738 −0.363339 −0.181670 0.983360i \(-0.558150\pi\)
−0.181670 + 0.983360i \(0.558150\pi\)
\(864\) 0 0
\(865\) 80.7771 2.74650
\(866\) 0 0
\(867\) −2.61803 −0.0889131
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −7.52786 −0.255072
\(872\) 0 0
\(873\) 11.0000 0.372294
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 45.3050 1.52984 0.764920 0.644126i \(-0.222779\pi\)
0.764920 + 0.644126i \(0.222779\pi\)
\(878\) 0 0
\(879\) −72.2492 −2.43691
\(880\) 0 0
\(881\) −48.1591 −1.62252 −0.811260 0.584686i \(-0.801218\pi\)
−0.811260 + 0.584686i \(0.801218\pi\)
\(882\) 0 0
\(883\) 45.6869 1.53749 0.768744 0.639557i \(-0.220882\pi\)
0.768744 + 0.639557i \(0.220882\pi\)
\(884\) 0 0
\(885\) −7.23607 −0.243238
\(886\) 0 0
\(887\) −30.7426 −1.03224 −0.516119 0.856517i \(-0.672624\pi\)
−0.516119 + 0.856517i \(0.672624\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −44.9443 −1.50400
\(894\) 0 0
\(895\) 11.1803 0.373718
\(896\) 0 0
\(897\) 2.47214 0.0825422
\(898\) 0 0
\(899\) 76.4296 2.54907
\(900\) 0 0
\(901\) 3.32624 0.110813
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 50.6525 1.68375
\(906\) 0 0
\(907\) −13.8197 −0.458874 −0.229437 0.973323i \(-0.573689\pi\)
−0.229437 + 0.973323i \(0.573689\pi\)
\(908\) 0 0
\(909\) −29.0132 −0.962306
\(910\) 0 0
\(911\) −38.7214 −1.28290 −0.641448 0.767167i \(-0.721666\pi\)
−0.641448 + 0.767167i \(0.721666\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 71.6312 2.36805
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 36.6869 1.21019 0.605095 0.796153i \(-0.293135\pi\)
0.605095 + 0.796153i \(0.293135\pi\)
\(920\) 0 0
\(921\) −24.6525 −0.812327
\(922\) 0 0
\(923\) 11.4164 0.375776
\(924\) 0 0
\(925\) −78.5410 −2.58241
\(926\) 0 0
\(927\) −23.1246 −0.759512
\(928\) 0 0
\(929\) 20.2016 0.662794 0.331397 0.943491i \(-0.392480\pi\)
0.331397 + 0.943491i \(0.392480\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −80.4853 −2.63497
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.58359 −0.149739 −0.0748697 0.997193i \(-0.523854\pi\)
−0.0748697 + 0.997193i \(0.523854\pi\)
\(938\) 0 0
\(939\) 44.8885 1.46488
\(940\) 0 0
\(941\) 19.2016 0.625955 0.312978 0.949761i \(-0.398674\pi\)
0.312978 + 0.949761i \(0.398674\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41.1935 −1.33861 −0.669304 0.742988i \(-0.733408\pi\)
−0.669304 + 0.742988i \(0.733408\pi\)
\(948\) 0 0
\(949\) 4.76393 0.154644
\(950\) 0 0
\(951\) −62.5410 −2.02803
\(952\) 0 0
\(953\) −19.9098 −0.644943 −0.322471 0.946579i \(-0.604514\pi\)
−0.322471 + 0.946579i \(0.604514\pi\)
\(954\) 0 0
\(955\) −79.9230 −2.58625
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 80.5623 2.59878
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −24.9098 −0.801046 −0.400523 0.916287i \(-0.631172\pi\)
−0.400523 + 0.916287i \(0.631172\pi\)
\(968\) 0 0
\(969\) −16.9443 −0.544328
\(970\) 0 0
\(971\) −22.7639 −0.730529 −0.365265 0.930904i \(-0.619022\pi\)
−0.365265 + 0.930904i \(0.619022\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 26.1803 0.838442
\(976\) 0 0
\(977\) −36.4508 −1.16617 −0.583083 0.812413i \(-0.698154\pi\)
−0.583083 + 0.812413i \(0.698154\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 46.9443 1.49882
\(982\) 0 0
\(983\) −0.270510 −0.00862792 −0.00431396 0.999991i \(-0.501373\pi\)
−0.00431396 + 0.999991i \(0.501373\pi\)
\(984\) 0 0
\(985\) 56.8328 1.81084
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.41641 −0.0450391
\(990\) 0 0
\(991\) 8.54102 0.271314 0.135657 0.990756i \(-0.456685\pi\)
0.135657 + 0.990756i \(0.456685\pi\)
\(992\) 0 0
\(993\) −34.8885 −1.10715
\(994\) 0 0
\(995\) 43.9443 1.39313
\(996\) 0 0
\(997\) −29.7426 −0.941959 −0.470980 0.882144i \(-0.656099\pi\)
−0.470980 + 0.882144i \(0.656099\pi\)
\(998\) 0 0
\(999\) 21.7082 0.686817
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 3332.2.a.g.1.1 2
7.6 odd 2 3332.2.a.o.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.2.a.g.1.1 2 1.1 even 1 trivial
3332.2.a.o.1.2 yes 2 7.6 odd 2