Properties

Label 6592.2.a.t.1.2
Level $6592$
Weight $2$
Character 6592.1
Self dual yes
Analytic conductor $52.637$
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] = [6592,2,Mod(1,6592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6592, 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("6592.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6592 = 2^{6} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6592.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: \(52.6373850124\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 103)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 6592.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.61803 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.61803 q^{5} -1.00000 q^{7} -2.00000 q^{9} +0.381966 q^{11} -1.85410 q^{13} +2.61803 q^{15} -3.38197 q^{17} +0.854102 q^{19} -1.00000 q^{21} -4.47214 q^{23} +1.85410 q^{25} -5.00000 q^{27} +0.763932 q^{29} +6.70820 q^{31} +0.381966 q^{33} -2.61803 q^{35} +6.70820 q^{37} -1.85410 q^{39} -8.94427 q^{41} -4.70820 q^{43} -5.23607 q^{45} -7.09017 q^{47} -6.00000 q^{49} -3.38197 q^{51} +10.0902 q^{53} +1.00000 q^{55} +0.854102 q^{57} -8.61803 q^{59} -10.8541 q^{61} +2.00000 q^{63} -4.85410 q^{65} +12.4164 q^{67} -4.47214 q^{69} +7.09017 q^{71} -4.14590 q^{73} +1.85410 q^{75} -0.381966 q^{77} +13.5623 q^{79} +1.00000 q^{81} -9.32624 q^{83} -8.85410 q^{85} +0.763932 q^{87} -15.7082 q^{89} +1.85410 q^{91} +6.70820 q^{93} +2.23607 q^{95} +11.7082 q^{97} -0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} - 4 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{15} - 9 q^{17} - 5 q^{19} - 2 q^{21} - 3 q^{25} - 10 q^{27} + 6 q^{29} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 4 q^{43} - 6 q^{45} - 3 q^{47} - 12 q^{49} - 9 q^{51} + 9 q^{53} + 2 q^{55} - 5 q^{57} - 15 q^{59} - 15 q^{61} + 4 q^{63} - 3 q^{65} - 2 q^{67} + 3 q^{71} - 15 q^{73} - 3 q^{75} - 3 q^{77} + 7 q^{79} + 2 q^{81} - 3 q^{83} - 11 q^{85} + 6 q^{87} - 18 q^{89} - 3 q^{91} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 2.61803 1.17082 0.585410 0.810737i \(-0.300933\pi\)
0.585410 + 0.810737i \(0.300933\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0.381966 0.115167 0.0575835 0.998341i \(-0.481660\pi\)
0.0575835 + 0.998341i \(0.481660\pi\)
\(12\) 0 0
\(13\) −1.85410 −0.514235 −0.257118 0.966380i \(-0.582773\pi\)
−0.257118 + 0.966380i \(0.582773\pi\)
\(14\) 0 0
\(15\) 2.61803 0.675973
\(16\) 0 0
\(17\) −3.38197 −0.820247 −0.410124 0.912030i \(-0.634514\pi\)
−0.410124 + 0.912030i \(0.634514\pi\)
\(18\) 0 0
\(19\) 0.854102 0.195944 0.0979722 0.995189i \(-0.468764\pi\)
0.0979722 + 0.995189i \(0.468764\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 1.85410 0.370820
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 0.763932 0.141859 0.0709293 0.997481i \(-0.477404\pi\)
0.0709293 + 0.997481i \(0.477404\pi\)
\(30\) 0 0
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 0 0
\(33\) 0.381966 0.0664917
\(34\) 0 0
\(35\) −2.61803 −0.442529
\(36\) 0 0
\(37\) 6.70820 1.10282 0.551411 0.834234i \(-0.314090\pi\)
0.551411 + 0.834234i \(0.314090\pi\)
\(38\) 0 0
\(39\) −1.85410 −0.296894
\(40\) 0 0
\(41\) −8.94427 −1.39686 −0.698430 0.715678i \(-0.746118\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) 0 0
\(43\) −4.70820 −0.717994 −0.358997 0.933339i \(-0.616881\pi\)
−0.358997 + 0.933339i \(0.616881\pi\)
\(44\) 0 0
\(45\) −5.23607 −0.780547
\(46\) 0 0
\(47\) −7.09017 −1.03421 −0.517104 0.855923i \(-0.672990\pi\)
−0.517104 + 0.855923i \(0.672990\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.38197 −0.473570
\(52\) 0 0
\(53\) 10.0902 1.38599 0.692996 0.720942i \(-0.256290\pi\)
0.692996 + 0.720942i \(0.256290\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0.854102 0.113129
\(58\) 0 0
\(59\) −8.61803 −1.12197 −0.560986 0.827825i \(-0.689578\pi\)
−0.560986 + 0.827825i \(0.689578\pi\)
\(60\) 0 0
\(61\) −10.8541 −1.38973 −0.694863 0.719142i \(-0.744535\pi\)
−0.694863 + 0.719142i \(0.744535\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −4.85410 −0.602077
\(66\) 0 0
\(67\) 12.4164 1.51691 0.758453 0.651728i \(-0.225956\pi\)
0.758453 + 0.651728i \(0.225956\pi\)
\(68\) 0 0
\(69\) −4.47214 −0.538382
\(70\) 0 0
\(71\) 7.09017 0.841448 0.420724 0.907189i \(-0.361776\pi\)
0.420724 + 0.907189i \(0.361776\pi\)
\(72\) 0 0
\(73\) −4.14590 −0.485241 −0.242620 0.970121i \(-0.578007\pi\)
−0.242620 + 0.970121i \(0.578007\pi\)
\(74\) 0 0
\(75\) 1.85410 0.214093
\(76\) 0 0
\(77\) −0.381966 −0.0435291
\(78\) 0 0
\(79\) 13.5623 1.52588 0.762939 0.646470i \(-0.223755\pi\)
0.762939 + 0.646470i \(0.223755\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.32624 −1.02369 −0.511844 0.859079i \(-0.671037\pi\)
−0.511844 + 0.859079i \(0.671037\pi\)
\(84\) 0 0
\(85\) −8.85410 −0.960362
\(86\) 0 0
\(87\) 0.763932 0.0819021
\(88\) 0 0
\(89\) −15.7082 −1.66507 −0.832533 0.553975i \(-0.813110\pi\)
−0.832533 + 0.553975i \(0.813110\pi\)
\(90\) 0 0
\(91\) 1.85410 0.194363
\(92\) 0 0
\(93\) 6.70820 0.695608
\(94\) 0 0
\(95\) 2.23607 0.229416
\(96\) 0 0
\(97\) 11.7082 1.18879 0.594394 0.804174i \(-0.297392\pi\)
0.594394 + 0.804174i \(0.297392\pi\)
\(98\) 0 0
\(99\) −0.763932 −0.0767781
\(100\) 0 0
\(101\) −13.0902 −1.30252 −0.651260 0.758854i \(-0.725759\pi\)
−0.651260 + 0.758854i \(0.725759\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329
\(104\) 0 0
\(105\) −2.61803 −0.255494
\(106\) 0 0
\(107\) −4.09017 −0.395412 −0.197706 0.980261i \(-0.563349\pi\)
−0.197706 + 0.980261i \(0.563349\pi\)
\(108\) 0 0
\(109\) −10.5623 −1.01169 −0.505843 0.862626i \(-0.668818\pi\)
−0.505843 + 0.862626i \(0.668818\pi\)
\(110\) 0 0
\(111\) 6.70820 0.636715
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) −11.7082 −1.09180
\(116\) 0 0
\(117\) 3.70820 0.342824
\(118\) 0 0
\(119\) 3.38197 0.310024
\(120\) 0 0
\(121\) −10.8541 −0.986737
\(122\) 0 0
\(123\) −8.94427 −0.806478
\(124\) 0 0
\(125\) −8.23607 −0.736656
\(126\) 0 0
\(127\) −18.2705 −1.62125 −0.810623 0.585569i \(-0.800871\pi\)
−0.810623 + 0.585569i \(0.800871\pi\)
\(128\) 0 0
\(129\) −4.70820 −0.414534
\(130\) 0 0
\(131\) −2.23607 −0.195366 −0.0976831 0.995218i \(-0.531143\pi\)
−0.0976831 + 0.995218i \(0.531143\pi\)
\(132\) 0 0
\(133\) −0.854102 −0.0740600
\(134\) 0 0
\(135\) −13.0902 −1.12662
\(136\) 0 0
\(137\) 0.708204 0.0605059 0.0302530 0.999542i \(-0.490369\pi\)
0.0302530 + 0.999542i \(0.490369\pi\)
\(138\) 0 0
\(139\) 17.8541 1.51437 0.757183 0.653203i \(-0.226575\pi\)
0.757183 + 0.653203i \(0.226575\pi\)
\(140\) 0 0
\(141\) −7.09017 −0.597100
\(142\) 0 0
\(143\) −0.708204 −0.0592230
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) −1.47214 −0.120602 −0.0603010 0.998180i \(-0.519206\pi\)
−0.0603010 + 0.998180i \(0.519206\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 6.76393 0.546831
\(154\) 0 0
\(155\) 17.5623 1.41064
\(156\) 0 0
\(157\) −3.29180 −0.262714 −0.131357 0.991335i \(-0.541933\pi\)
−0.131357 + 0.991335i \(0.541933\pi\)
\(158\) 0 0
\(159\) 10.0902 0.800203
\(160\) 0 0
\(161\) 4.47214 0.352454
\(162\) 0 0
\(163\) 10.7082 0.838731 0.419366 0.907817i \(-0.362253\pi\)
0.419366 + 0.907817i \(0.362253\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) −9.56231 −0.735562
\(170\) 0 0
\(171\) −1.70820 −0.130630
\(172\) 0 0
\(173\) 13.0344 0.990990 0.495495 0.868611i \(-0.334987\pi\)
0.495495 + 0.868611i \(0.334987\pi\)
\(174\) 0 0
\(175\) −1.85410 −0.140157
\(176\) 0 0
\(177\) −8.61803 −0.647771
\(178\) 0 0
\(179\) 1.14590 0.0856484 0.0428242 0.999083i \(-0.486364\pi\)
0.0428242 + 0.999083i \(0.486364\pi\)
\(180\) 0 0
\(181\) 2.85410 0.212144 0.106072 0.994358i \(-0.466173\pi\)
0.106072 + 0.994358i \(0.466173\pi\)
\(182\) 0 0
\(183\) −10.8541 −0.802358
\(184\) 0 0
\(185\) 17.5623 1.29121
\(186\) 0 0
\(187\) −1.29180 −0.0944655
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 3.38197 0.244710 0.122355 0.992486i \(-0.460955\pi\)
0.122355 + 0.992486i \(0.460955\pi\)
\(192\) 0 0
\(193\) −20.1246 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(194\) 0 0
\(195\) −4.85410 −0.347609
\(196\) 0 0
\(197\) −10.4164 −0.742138 −0.371069 0.928605i \(-0.621009\pi\)
−0.371069 + 0.928605i \(0.621009\pi\)
\(198\) 0 0
\(199\) 3.41641 0.242183 0.121091 0.992641i \(-0.461361\pi\)
0.121091 + 0.992641i \(0.461361\pi\)
\(200\) 0 0
\(201\) 12.4164 0.875786
\(202\) 0 0
\(203\) −0.763932 −0.0536175
\(204\) 0 0
\(205\) −23.4164 −1.63547
\(206\) 0 0
\(207\) 8.94427 0.621670
\(208\) 0 0
\(209\) 0.326238 0.0225663
\(210\) 0 0
\(211\) −8.14590 −0.560787 −0.280393 0.959885i \(-0.590465\pi\)
−0.280393 + 0.959885i \(0.590465\pi\)
\(212\) 0 0
\(213\) 7.09017 0.485810
\(214\) 0 0
\(215\) −12.3262 −0.840642
\(216\) 0 0
\(217\) −6.70820 −0.455383
\(218\) 0 0
\(219\) −4.14590 −0.280154
\(220\) 0 0
\(221\) 6.27051 0.421800
\(222\) 0 0
\(223\) −5.70820 −0.382250 −0.191125 0.981566i \(-0.561214\pi\)
−0.191125 + 0.981566i \(0.561214\pi\)
\(224\) 0 0
\(225\) −3.70820 −0.247214
\(226\) 0 0
\(227\) 14.9443 0.991886 0.495943 0.868355i \(-0.334822\pi\)
0.495943 + 0.868355i \(0.334822\pi\)
\(228\) 0 0
\(229\) 6.70820 0.443291 0.221645 0.975127i \(-0.428857\pi\)
0.221645 + 0.975127i \(0.428857\pi\)
\(230\) 0 0
\(231\) −0.381966 −0.0251315
\(232\) 0 0
\(233\) 8.88854 0.582308 0.291154 0.956676i \(-0.405961\pi\)
0.291154 + 0.956676i \(0.405961\pi\)
\(234\) 0 0
\(235\) −18.5623 −1.21087
\(236\) 0 0
\(237\) 13.5623 0.880966
\(238\) 0 0
\(239\) 24.3262 1.57353 0.786767 0.617250i \(-0.211753\pi\)
0.786767 + 0.617250i \(0.211753\pi\)
\(240\) 0 0
\(241\) 25.2705 1.62782 0.813908 0.580993i \(-0.197336\pi\)
0.813908 + 0.580993i \(0.197336\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −15.7082 −1.00356
\(246\) 0 0
\(247\) −1.58359 −0.100762
\(248\) 0 0
\(249\) −9.32624 −0.591026
\(250\) 0 0
\(251\) −6.76393 −0.426936 −0.213468 0.976950i \(-0.568476\pi\)
−0.213468 + 0.976950i \(0.568476\pi\)
\(252\) 0 0
\(253\) −1.70820 −0.107394
\(254\) 0 0
\(255\) −8.85410 −0.554465
\(256\) 0 0
\(257\) −4.52786 −0.282440 −0.141220 0.989978i \(-0.545103\pi\)
−0.141220 + 0.989978i \(0.545103\pi\)
\(258\) 0 0
\(259\) −6.70820 −0.416828
\(260\) 0 0
\(261\) −1.52786 −0.0945724
\(262\) 0 0
\(263\) −18.3820 −1.13348 −0.566740 0.823896i \(-0.691796\pi\)
−0.566740 + 0.823896i \(0.691796\pi\)
\(264\) 0 0
\(265\) 26.4164 1.62275
\(266\) 0 0
\(267\) −15.7082 −0.961326
\(268\) 0 0
\(269\) 3.32624 0.202804 0.101402 0.994846i \(-0.467667\pi\)
0.101402 + 0.994846i \(0.467667\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) 0 0
\(273\) 1.85410 0.112215
\(274\) 0 0
\(275\) 0.708204 0.0427063
\(276\) 0 0
\(277\) 8.70820 0.523225 0.261613 0.965173i \(-0.415746\pi\)
0.261613 + 0.965173i \(0.415746\pi\)
\(278\) 0 0
\(279\) −13.4164 −0.803219
\(280\) 0 0
\(281\) −22.5279 −1.34390 −0.671950 0.740597i \(-0.734543\pi\)
−0.671950 + 0.740597i \(0.734543\pi\)
\(282\) 0 0
\(283\) 6.29180 0.374008 0.187004 0.982359i \(-0.440122\pi\)
0.187004 + 0.982359i \(0.440122\pi\)
\(284\) 0 0
\(285\) 2.23607 0.132453
\(286\) 0 0
\(287\) 8.94427 0.527964
\(288\) 0 0
\(289\) −5.56231 −0.327194
\(290\) 0 0
\(291\) 11.7082 0.686347
\(292\) 0 0
\(293\) −9.65248 −0.563904 −0.281952 0.959429i \(-0.590982\pi\)
−0.281952 + 0.959429i \(0.590982\pi\)
\(294\) 0 0
\(295\) −22.5623 −1.31363
\(296\) 0 0
\(297\) −1.90983 −0.110820
\(298\) 0 0
\(299\) 8.29180 0.479527
\(300\) 0 0
\(301\) 4.70820 0.271376
\(302\) 0 0
\(303\) −13.0902 −0.752011
\(304\) 0 0
\(305\) −28.4164 −1.62712
\(306\) 0 0
\(307\) −3.85410 −0.219965 −0.109983 0.993934i \(-0.535080\pi\)
−0.109983 + 0.993934i \(0.535080\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −32.8885 −1.86494 −0.932469 0.361250i \(-0.882350\pi\)
−0.932469 + 0.361250i \(0.882350\pi\)
\(312\) 0 0
\(313\) 29.7082 1.67921 0.839603 0.543200i \(-0.182787\pi\)
0.839603 + 0.543200i \(0.182787\pi\)
\(314\) 0 0
\(315\) 5.23607 0.295019
\(316\) 0 0
\(317\) −1.58359 −0.0889434 −0.0444717 0.999011i \(-0.514160\pi\)
−0.0444717 + 0.999011i \(0.514160\pi\)
\(318\) 0 0
\(319\) 0.291796 0.0163374
\(320\) 0 0
\(321\) −4.09017 −0.228291
\(322\) 0 0
\(323\) −2.88854 −0.160723
\(324\) 0 0
\(325\) −3.43769 −0.190689
\(326\) 0 0
\(327\) −10.5623 −0.584097
\(328\) 0 0
\(329\) 7.09017 0.390894
\(330\) 0 0
\(331\) 22.8541 1.25618 0.628088 0.778143i \(-0.283838\pi\)
0.628088 + 0.778143i \(0.283838\pi\)
\(332\) 0 0
\(333\) −13.4164 −0.735215
\(334\) 0 0
\(335\) 32.5066 1.77602
\(336\) 0 0
\(337\) −22.5623 −1.22905 −0.614524 0.788898i \(-0.710652\pi\)
−0.614524 + 0.788898i \(0.710652\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) 2.56231 0.138757
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) −11.7082 −0.630349
\(346\) 0 0
\(347\) −7.47214 −0.401125 −0.200563 0.979681i \(-0.564277\pi\)
−0.200563 + 0.979681i \(0.564277\pi\)
\(348\) 0 0
\(349\) 11.4164 0.611106 0.305553 0.952175i \(-0.401159\pi\)
0.305553 + 0.952175i \(0.401159\pi\)
\(350\) 0 0
\(351\) 9.27051 0.494823
\(352\) 0 0
\(353\) 4.03444 0.214732 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(354\) 0 0
\(355\) 18.5623 0.985185
\(356\) 0 0
\(357\) 3.38197 0.178993
\(358\) 0 0
\(359\) 30.3262 1.60056 0.800279 0.599628i \(-0.204685\pi\)
0.800279 + 0.599628i \(0.204685\pi\)
\(360\) 0 0
\(361\) −18.2705 −0.961606
\(362\) 0 0
\(363\) −10.8541 −0.569693
\(364\) 0 0
\(365\) −10.8541 −0.568130
\(366\) 0 0
\(367\) 36.5623 1.90854 0.954268 0.298951i \(-0.0966368\pi\)
0.954268 + 0.298951i \(0.0966368\pi\)
\(368\) 0 0
\(369\) 17.8885 0.931240
\(370\) 0 0
\(371\) −10.0902 −0.523856
\(372\) 0 0
\(373\) −37.6869 −1.95135 −0.975677 0.219212i \(-0.929651\pi\)
−0.975677 + 0.219212i \(0.929651\pi\)
\(374\) 0 0
\(375\) −8.23607 −0.425309
\(376\) 0 0
\(377\) −1.41641 −0.0729487
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −18.2705 −0.936027
\(382\) 0 0
\(383\) 23.1803 1.18446 0.592230 0.805769i \(-0.298248\pi\)
0.592230 + 0.805769i \(0.298248\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) 9.41641 0.478663
\(388\) 0 0
\(389\) 19.4164 0.984451 0.492225 0.870468i \(-0.336184\pi\)
0.492225 + 0.870468i \(0.336184\pi\)
\(390\) 0 0
\(391\) 15.1246 0.764884
\(392\) 0 0
\(393\) −2.23607 −0.112795
\(394\) 0 0
\(395\) 35.5066 1.78653
\(396\) 0 0
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 0 0
\(399\) −0.854102 −0.0427586
\(400\) 0 0
\(401\) 11.8885 0.593686 0.296843 0.954926i \(-0.404066\pi\)
0.296843 + 0.954926i \(0.404066\pi\)
\(402\) 0 0
\(403\) −12.4377 −0.619566
\(404\) 0 0
\(405\) 2.61803 0.130091
\(406\) 0 0
\(407\) 2.56231 0.127009
\(408\) 0 0
\(409\) 23.2918 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(410\) 0 0
\(411\) 0.708204 0.0349331
\(412\) 0 0
\(413\) 8.61803 0.424066
\(414\) 0 0
\(415\) −24.4164 −1.19855
\(416\) 0 0
\(417\) 17.8541 0.874319
\(418\) 0 0
\(419\) −7.09017 −0.346377 −0.173189 0.984889i \(-0.555407\pi\)
−0.173189 + 0.984889i \(0.555407\pi\)
\(420\) 0 0
\(421\) −3.00000 −0.146211 −0.0731055 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) 0 0
\(423\) 14.1803 0.689472
\(424\) 0 0
\(425\) −6.27051 −0.304164
\(426\) 0 0
\(427\) 10.8541 0.525267
\(428\) 0 0
\(429\) −0.708204 −0.0341924
\(430\) 0 0
\(431\) −10.3607 −0.499056 −0.249528 0.968368i \(-0.580276\pi\)
−0.249528 + 0.968368i \(0.580276\pi\)
\(432\) 0 0
\(433\) −12.4164 −0.596694 −0.298347 0.954457i \(-0.596435\pi\)
−0.298347 + 0.954457i \(0.596435\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) −3.81966 −0.182719
\(438\) 0 0
\(439\) 9.43769 0.450437 0.225218 0.974308i \(-0.427690\pi\)
0.225218 + 0.974308i \(0.427690\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 35.5623 1.68962 0.844808 0.535069i \(-0.179715\pi\)
0.844808 + 0.535069i \(0.179715\pi\)
\(444\) 0 0
\(445\) −41.1246 −1.94949
\(446\) 0 0
\(447\) −1.47214 −0.0696296
\(448\) 0 0
\(449\) 31.3607 1.48000 0.740001 0.672606i \(-0.234825\pi\)
0.740001 + 0.672606i \(0.234825\pi\)
\(450\) 0 0
\(451\) −3.41641 −0.160872
\(452\) 0 0
\(453\) −19.0000 −0.892698
\(454\) 0 0
\(455\) 4.85410 0.227564
\(456\) 0 0
\(457\) −3.14590 −0.147159 −0.0735795 0.997289i \(-0.523442\pi\)
−0.0735795 + 0.997289i \(0.523442\pi\)
\(458\) 0 0
\(459\) 16.9098 0.789283
\(460\) 0 0
\(461\) 39.2148 1.82641 0.913207 0.407495i \(-0.133598\pi\)
0.913207 + 0.407495i \(0.133598\pi\)
\(462\) 0 0
\(463\) 5.41641 0.251722 0.125861 0.992048i \(-0.459831\pi\)
0.125861 + 0.992048i \(0.459831\pi\)
\(464\) 0 0
\(465\) 17.5623 0.814432
\(466\) 0 0
\(467\) −27.6525 −1.27960 −0.639802 0.768540i \(-0.720984\pi\)
−0.639802 + 0.768540i \(0.720984\pi\)
\(468\) 0 0
\(469\) −12.4164 −0.573336
\(470\) 0 0
\(471\) −3.29180 −0.151678
\(472\) 0 0
\(473\) −1.79837 −0.0826893
\(474\) 0 0
\(475\) 1.58359 0.0726602
\(476\) 0 0
\(477\) −20.1803 −0.923994
\(478\) 0 0
\(479\) −14.1803 −0.647916 −0.323958 0.946071i \(-0.605014\pi\)
−0.323958 + 0.946071i \(0.605014\pi\)
\(480\) 0 0
\(481\) −12.4377 −0.567110
\(482\) 0 0
\(483\) 4.47214 0.203489
\(484\) 0 0
\(485\) 30.6525 1.39186
\(486\) 0 0
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) 0 0
\(489\) 10.7082 0.484242
\(490\) 0 0
\(491\) 35.7771 1.61460 0.807299 0.590143i \(-0.200929\pi\)
0.807299 + 0.590143i \(0.200929\pi\)
\(492\) 0 0
\(493\) −2.58359 −0.116359
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) −7.09017 −0.318038
\(498\) 0 0
\(499\) 20.2705 0.907433 0.453716 0.891146i \(-0.350098\pi\)
0.453716 + 0.891146i \(0.350098\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) −31.3607 −1.39830 −0.699152 0.714973i \(-0.746439\pi\)
−0.699152 + 0.714973i \(0.746439\pi\)
\(504\) 0 0
\(505\) −34.2705 −1.52502
\(506\) 0 0
\(507\) −9.56231 −0.424677
\(508\) 0 0
\(509\) −24.3820 −1.08071 −0.540356 0.841437i \(-0.681710\pi\)
−0.540356 + 0.841437i \(0.681710\pi\)
\(510\) 0 0
\(511\) 4.14590 0.183404
\(512\) 0 0
\(513\) −4.27051 −0.188548
\(514\) 0 0
\(515\) −2.61803 −0.115364
\(516\) 0 0
\(517\) −2.70820 −0.119107
\(518\) 0 0
\(519\) 13.0344 0.572148
\(520\) 0 0
\(521\) −5.18034 −0.226955 −0.113477 0.993541i \(-0.536199\pi\)
−0.113477 + 0.993541i \(0.536199\pi\)
\(522\) 0 0
\(523\) 37.4164 1.63611 0.818053 0.575143i \(-0.195054\pi\)
0.818053 + 0.575143i \(0.195054\pi\)
\(524\) 0 0
\(525\) −1.85410 −0.0809196
\(526\) 0 0
\(527\) −22.6869 −0.988258
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 0 0
\(531\) 17.2361 0.747982
\(532\) 0 0
\(533\) 16.5836 0.718315
\(534\) 0 0
\(535\) −10.7082 −0.462956
\(536\) 0 0
\(537\) 1.14590 0.0494492
\(538\) 0 0
\(539\) −2.29180 −0.0987146
\(540\) 0 0
\(541\) −15.8541 −0.681621 −0.340811 0.940132i \(-0.610701\pi\)
−0.340811 + 0.940132i \(0.610701\pi\)
\(542\) 0 0
\(543\) 2.85410 0.122481
\(544\) 0 0
\(545\) −27.6525 −1.18450
\(546\) 0 0
\(547\) −31.2705 −1.33703 −0.668515 0.743698i \(-0.733070\pi\)
−0.668515 + 0.743698i \(0.733070\pi\)
\(548\) 0 0
\(549\) 21.7082 0.926484
\(550\) 0 0
\(551\) 0.652476 0.0277964
\(552\) 0 0
\(553\) −13.5623 −0.576728
\(554\) 0 0
\(555\) 17.5623 0.745478
\(556\) 0 0
\(557\) −28.6869 −1.21550 −0.607752 0.794127i \(-0.707928\pi\)
−0.607752 + 0.794127i \(0.707928\pi\)
\(558\) 0 0
\(559\) 8.72949 0.369218
\(560\) 0 0
\(561\) −1.29180 −0.0545397
\(562\) 0 0
\(563\) 25.7984 1.08727 0.543636 0.839321i \(-0.317047\pi\)
0.543636 + 0.839321i \(0.317047\pi\)
\(564\) 0 0
\(565\) −39.2705 −1.65212
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 42.1591 1.76740 0.883700 0.468054i \(-0.155045\pi\)
0.883700 + 0.468054i \(0.155045\pi\)
\(570\) 0 0
\(571\) 2.43769 0.102014 0.0510072 0.998698i \(-0.483757\pi\)
0.0510072 + 0.998698i \(0.483757\pi\)
\(572\) 0 0
\(573\) 3.38197 0.141284
\(574\) 0 0
\(575\) −8.29180 −0.345792
\(576\) 0 0
\(577\) 16.8328 0.700759 0.350380 0.936608i \(-0.386053\pi\)
0.350380 + 0.936608i \(0.386053\pi\)
\(578\) 0 0
\(579\) −20.1246 −0.836350
\(580\) 0 0
\(581\) 9.32624 0.386918
\(582\) 0 0
\(583\) 3.85410 0.159621
\(584\) 0 0
\(585\) 9.70820 0.401385
\(586\) 0 0
\(587\) −11.0689 −0.456862 −0.228431 0.973560i \(-0.573359\pi\)
−0.228431 + 0.973560i \(0.573359\pi\)
\(588\) 0 0
\(589\) 5.72949 0.236080
\(590\) 0 0
\(591\) −10.4164 −0.428474
\(592\) 0 0
\(593\) −20.1803 −0.828707 −0.414354 0.910116i \(-0.635992\pi\)
−0.414354 + 0.910116i \(0.635992\pi\)
\(594\) 0 0
\(595\) 8.85410 0.362983
\(596\) 0 0
\(597\) 3.41641 0.139824
\(598\) 0 0
\(599\) 20.4508 0.835599 0.417800 0.908539i \(-0.362801\pi\)
0.417800 + 0.908539i \(0.362801\pi\)
\(600\) 0 0
\(601\) −16.5623 −0.675591 −0.337795 0.941220i \(-0.609681\pi\)
−0.337795 + 0.941220i \(0.609681\pi\)
\(602\) 0 0
\(603\) −24.8328 −1.01127
\(604\) 0 0
\(605\) −28.4164 −1.15529
\(606\) 0 0
\(607\) −7.70820 −0.312866 −0.156433 0.987689i \(-0.550000\pi\)
−0.156433 + 0.987689i \(0.550000\pi\)
\(608\) 0 0
\(609\) −0.763932 −0.0309561
\(610\) 0 0
\(611\) 13.1459 0.531826
\(612\) 0 0
\(613\) 2.41641 0.0975978 0.0487989 0.998809i \(-0.484461\pi\)
0.0487989 + 0.998809i \(0.484461\pi\)
\(614\) 0 0
\(615\) −23.4164 −0.944241
\(616\) 0 0
\(617\) −30.2705 −1.21864 −0.609322 0.792923i \(-0.708558\pi\)
−0.609322 + 0.792923i \(0.708558\pi\)
\(618\) 0 0
\(619\) −31.6869 −1.27360 −0.636802 0.771027i \(-0.719743\pi\)
−0.636802 + 0.771027i \(0.719743\pi\)
\(620\) 0 0
\(621\) 22.3607 0.897303
\(622\) 0 0
\(623\) 15.7082 0.629336
\(624\) 0 0
\(625\) −30.8328 −1.23331
\(626\) 0 0
\(627\) 0.326238 0.0130287
\(628\) 0 0
\(629\) −22.6869 −0.904587
\(630\) 0 0
\(631\) 8.72949 0.347516 0.173758 0.984788i \(-0.444409\pi\)
0.173758 + 0.984788i \(0.444409\pi\)
\(632\) 0 0
\(633\) −8.14590 −0.323770
\(634\) 0 0
\(635\) −47.8328 −1.89819
\(636\) 0 0
\(637\) 11.1246 0.440773
\(638\) 0 0
\(639\) −14.1803 −0.560966
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) −12.3262 −0.485345
\(646\) 0 0
\(647\) 43.7426 1.71970 0.859850 0.510546i \(-0.170557\pi\)
0.859850 + 0.510546i \(0.170557\pi\)
\(648\) 0 0
\(649\) −3.29180 −0.129214
\(650\) 0 0
\(651\) −6.70820 −0.262915
\(652\) 0 0
\(653\) 0.763932 0.0298950 0.0149475 0.999888i \(-0.495242\pi\)
0.0149475 + 0.999888i \(0.495242\pi\)
\(654\) 0 0
\(655\) −5.85410 −0.228739
\(656\) 0 0
\(657\) 8.29180 0.323494
\(658\) 0 0
\(659\) 12.5967 0.490700 0.245350 0.969435i \(-0.421097\pi\)
0.245350 + 0.969435i \(0.421097\pi\)
\(660\) 0 0
\(661\) 11.4377 0.444875 0.222437 0.974947i \(-0.428599\pi\)
0.222437 + 0.974947i \(0.428599\pi\)
\(662\) 0 0
\(663\) 6.27051 0.243526
\(664\) 0 0
\(665\) −2.23607 −0.0867110
\(666\) 0 0
\(667\) −3.41641 −0.132284
\(668\) 0 0
\(669\) −5.70820 −0.220692
\(670\) 0 0
\(671\) −4.14590 −0.160051
\(672\) 0 0
\(673\) −37.7082 −1.45354 −0.726772 0.686879i \(-0.758980\pi\)
−0.726772 + 0.686879i \(0.758980\pi\)
\(674\) 0 0
\(675\) −9.27051 −0.356822
\(676\) 0 0
\(677\) −10.9656 −0.421441 −0.210720 0.977546i \(-0.567581\pi\)
−0.210720 + 0.977546i \(0.567581\pi\)
\(678\) 0 0
\(679\) −11.7082 −0.449320
\(680\) 0 0
\(681\) 14.9443 0.572666
\(682\) 0 0
\(683\) −42.7639 −1.63632 −0.818158 0.574993i \(-0.805005\pi\)
−0.818158 + 0.574993i \(0.805005\pi\)
\(684\) 0 0
\(685\) 1.85410 0.0708416
\(686\) 0 0
\(687\) 6.70820 0.255934
\(688\) 0 0
\(689\) −18.7082 −0.712726
\(690\) 0 0
\(691\) −1.14590 −0.0435920 −0.0217960 0.999762i \(-0.506938\pi\)
−0.0217960 + 0.999762i \(0.506938\pi\)
\(692\) 0 0
\(693\) 0.763932 0.0290194
\(694\) 0 0
\(695\) 46.7426 1.77305
\(696\) 0 0
\(697\) 30.2492 1.14577
\(698\) 0 0
\(699\) 8.88854 0.336196
\(700\) 0 0
\(701\) 1.20163 0.0453848 0.0226924 0.999742i \(-0.492776\pi\)
0.0226924 + 0.999742i \(0.492776\pi\)
\(702\) 0 0
\(703\) 5.72949 0.216092
\(704\) 0 0
\(705\) −18.5623 −0.699097
\(706\) 0 0
\(707\) 13.0902 0.492307
\(708\) 0 0
\(709\) 47.9787 1.80188 0.900939 0.433945i \(-0.142879\pi\)
0.900939 + 0.433945i \(0.142879\pi\)
\(710\) 0 0
\(711\) −27.1246 −1.01725
\(712\) 0 0
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) −1.85410 −0.0693395
\(716\) 0 0
\(717\) 24.3262 0.908480
\(718\) 0 0
\(719\) 23.6738 0.882882 0.441441 0.897290i \(-0.354467\pi\)
0.441441 + 0.897290i \(0.354467\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) 0 0
\(723\) 25.2705 0.939820
\(724\) 0 0
\(725\) 1.41641 0.0526041
\(726\) 0 0
\(727\) −38.2705 −1.41937 −0.709687 0.704517i \(-0.751164\pi\)
−0.709687 + 0.704517i \(0.751164\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 15.9230 0.588933
\(732\) 0 0
\(733\) −1.29180 −0.0477136 −0.0238568 0.999715i \(-0.507595\pi\)
−0.0238568 + 0.999715i \(0.507595\pi\)
\(734\) 0 0
\(735\) −15.7082 −0.579406
\(736\) 0 0
\(737\) 4.74265 0.174698
\(738\) 0 0
\(739\) −36.8328 −1.35492 −0.677459 0.735561i \(-0.736919\pi\)
−0.677459 + 0.735561i \(0.736919\pi\)
\(740\) 0 0
\(741\) −1.58359 −0.0581747
\(742\) 0 0
\(743\) 17.2918 0.634374 0.317187 0.948363i \(-0.397262\pi\)
0.317187 + 0.948363i \(0.397262\pi\)
\(744\) 0 0
\(745\) −3.85410 −0.141203
\(746\) 0 0
\(747\) 18.6525 0.682458
\(748\) 0 0
\(749\) 4.09017 0.149452
\(750\) 0 0
\(751\) −3.87539 −0.141415 −0.0707075 0.997497i \(-0.522526\pi\)
−0.0707075 + 0.997497i \(0.522526\pi\)
\(752\) 0 0
\(753\) −6.76393 −0.246491
\(754\) 0 0
\(755\) −49.7426 −1.81032
\(756\) 0 0
\(757\) 34.7082 1.26149 0.630746 0.775990i \(-0.282749\pi\)
0.630746 + 0.775990i \(0.282749\pi\)
\(758\) 0 0
\(759\) −1.70820 −0.0620039
\(760\) 0 0
\(761\) 1.47214 0.0533649 0.0266824 0.999644i \(-0.491506\pi\)
0.0266824 + 0.999644i \(0.491506\pi\)
\(762\) 0 0
\(763\) 10.5623 0.382381
\(764\) 0 0
\(765\) 17.7082 0.640241
\(766\) 0 0
\(767\) 15.9787 0.576958
\(768\) 0 0
\(769\) −42.3951 −1.52881 −0.764404 0.644738i \(-0.776966\pi\)
−0.764404 + 0.644738i \(0.776966\pi\)
\(770\) 0 0
\(771\) −4.52786 −0.163067
\(772\) 0 0
\(773\) 25.4721 0.916169 0.458085 0.888909i \(-0.348536\pi\)
0.458085 + 0.888909i \(0.348536\pi\)
\(774\) 0 0
\(775\) 12.4377 0.446775
\(776\) 0 0
\(777\) −6.70820 −0.240655
\(778\) 0 0
\(779\) −7.63932 −0.273707
\(780\) 0 0
\(781\) 2.70820 0.0969072
\(782\) 0 0
\(783\) −3.81966 −0.136504
\(784\) 0 0
\(785\) −8.61803 −0.307591
\(786\) 0 0
\(787\) −16.5836 −0.591141 −0.295571 0.955321i \(-0.595510\pi\)
−0.295571 + 0.955321i \(0.595510\pi\)
\(788\) 0 0
\(789\) −18.3820 −0.654415
\(790\) 0 0
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) 20.1246 0.714646
\(794\) 0 0
\(795\) 26.4164 0.936893
\(796\) 0 0
\(797\) 9.87539 0.349804 0.174902 0.984586i \(-0.444039\pi\)
0.174902 + 0.984586i \(0.444039\pi\)
\(798\) 0 0
\(799\) 23.9787 0.848306
\(800\) 0 0
\(801\) 31.4164 1.11004
\(802\) 0 0
\(803\) −1.58359 −0.0558838
\(804\) 0 0
\(805\) 11.7082 0.412660
\(806\) 0 0
\(807\) 3.32624 0.117089
\(808\) 0 0
\(809\) −14.9443 −0.525413 −0.262706 0.964876i \(-0.584615\pi\)
−0.262706 + 0.964876i \(0.584615\pi\)
\(810\) 0 0
\(811\) 45.5410 1.59916 0.799581 0.600559i \(-0.205055\pi\)
0.799581 + 0.600559i \(0.205055\pi\)
\(812\) 0 0
\(813\) 1.00000 0.0350715
\(814\) 0 0
\(815\) 28.0344 0.982004
\(816\) 0 0
\(817\) −4.02129 −0.140687
\(818\) 0 0
\(819\) −3.70820 −0.129575
\(820\) 0 0
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 0 0
\(823\) 23.5623 0.821330 0.410665 0.911786i \(-0.365297\pi\)
0.410665 + 0.911786i \(0.365297\pi\)
\(824\) 0 0
\(825\) 0.708204 0.0246565
\(826\) 0 0
\(827\) 6.70820 0.233267 0.116634 0.993175i \(-0.462790\pi\)
0.116634 + 0.993175i \(0.462790\pi\)
\(828\) 0 0
\(829\) 19.2705 0.669292 0.334646 0.942344i \(-0.391383\pi\)
0.334646 + 0.942344i \(0.391383\pi\)
\(830\) 0 0
\(831\) 8.70820 0.302084
\(832\) 0 0
\(833\) 20.2918 0.703069
\(834\) 0 0
\(835\) 23.5623 0.815407
\(836\) 0 0
\(837\) −33.5410 −1.15935
\(838\) 0 0
\(839\) 21.3820 0.738187 0.369094 0.929392i \(-0.379668\pi\)
0.369094 + 0.929392i \(0.379668\pi\)
\(840\) 0 0
\(841\) −28.4164 −0.979876
\(842\) 0 0
\(843\) −22.5279 −0.775901
\(844\) 0 0
\(845\) −25.0344 −0.861211
\(846\) 0 0
\(847\) 10.8541 0.372951
\(848\) 0 0
\(849\) 6.29180 0.215934
\(850\) 0 0
\(851\) −30.0000 −1.02839
\(852\) 0 0
\(853\) 10.7295 0.367371 0.183685 0.982985i \(-0.441197\pi\)
0.183685 + 0.982985i \(0.441197\pi\)
\(854\) 0 0
\(855\) −4.47214 −0.152944
\(856\) 0 0
\(857\) 3.76393 0.128573 0.0642867 0.997931i \(-0.479523\pi\)
0.0642867 + 0.997931i \(0.479523\pi\)
\(858\) 0 0
\(859\) 9.56231 0.326262 0.163131 0.986604i \(-0.447841\pi\)
0.163131 + 0.986604i \(0.447841\pi\)
\(860\) 0 0
\(861\) 8.94427 0.304820
\(862\) 0 0
\(863\) −8.29180 −0.282256 −0.141128 0.989991i \(-0.545073\pi\)
−0.141128 + 0.989991i \(0.545073\pi\)
\(864\) 0 0
\(865\) 34.1246 1.16027
\(866\) 0 0
\(867\) −5.56231 −0.188906
\(868\) 0 0
\(869\) 5.18034 0.175731
\(870\) 0 0
\(871\) −23.0213 −0.780047
\(872\) 0 0
\(873\) −23.4164 −0.792525
\(874\) 0 0
\(875\) 8.23607 0.278430
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 0 0
\(879\) −9.65248 −0.325570
\(880\) 0 0
\(881\) −5.88854 −0.198390 −0.0991950 0.995068i \(-0.531627\pi\)
−0.0991950 + 0.995068i \(0.531627\pi\)
\(882\) 0 0
\(883\) −46.1246 −1.55222 −0.776108 0.630600i \(-0.782809\pi\)
−0.776108 + 0.630600i \(0.782809\pi\)
\(884\) 0 0
\(885\) −22.5623 −0.758424
\(886\) 0 0
\(887\) −58.1935 −1.95395 −0.976973 0.213362i \(-0.931559\pi\)
−0.976973 + 0.213362i \(0.931559\pi\)
\(888\) 0 0
\(889\) 18.2705 0.612773
\(890\) 0 0
\(891\) 0.381966 0.0127963
\(892\) 0 0
\(893\) −6.05573 −0.202647
\(894\) 0 0
\(895\) 3.00000 0.100279
\(896\) 0 0
\(897\) 8.29180 0.276855
\(898\) 0 0
\(899\) 5.12461 0.170915
\(900\) 0 0
\(901\) −34.1246 −1.13686
\(902\) 0 0
\(903\) 4.70820 0.156679
\(904\) 0 0
\(905\) 7.47214 0.248382
\(906\) 0 0
\(907\) −33.1246 −1.09988 −0.549942 0.835203i \(-0.685350\pi\)
−0.549942 + 0.835203i \(0.685350\pi\)
\(908\) 0 0
\(909\) 26.1803 0.868347
\(910\) 0 0
\(911\) 19.0344 0.630639 0.315320 0.948986i \(-0.397888\pi\)
0.315320 + 0.948986i \(0.397888\pi\)
\(912\) 0 0
\(913\) −3.56231 −0.117895
\(914\) 0 0
\(915\) −28.4164 −0.939417
\(916\) 0 0
\(917\) 2.23607 0.0738415
\(918\) 0 0
\(919\) 18.9787 0.626050 0.313025 0.949745i \(-0.398658\pi\)
0.313025 + 0.949745i \(0.398658\pi\)
\(920\) 0 0
\(921\) −3.85410 −0.126997
\(922\) 0 0
\(923\) −13.1459 −0.432703
\(924\) 0 0
\(925\) 12.4377 0.408949
\(926\) 0 0
\(927\) 2.00000 0.0656886
\(928\) 0 0
\(929\) −2.94427 −0.0965984 −0.0482992 0.998833i \(-0.515380\pi\)
−0.0482992 + 0.998833i \(0.515380\pi\)
\(930\) 0 0
\(931\) −5.12461 −0.167952
\(932\) 0 0
\(933\) −32.8885 −1.07672
\(934\) 0 0
\(935\) −3.38197 −0.110602
\(936\) 0 0
\(937\) −11.0000 −0.359354 −0.179677 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) 0 0
\(939\) 29.7082 0.969491
\(940\) 0 0
\(941\) −50.3951 −1.64283 −0.821417 0.570328i \(-0.806816\pi\)
−0.821417 + 0.570328i \(0.806816\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) 0 0
\(945\) 13.0902 0.425823
\(946\) 0 0
\(947\) −35.0132 −1.13777 −0.568887 0.822415i \(-0.692626\pi\)
−0.568887 + 0.822415i \(0.692626\pi\)
\(948\) 0 0
\(949\) 7.68692 0.249528
\(950\) 0 0
\(951\) −1.58359 −0.0513515
\(952\) 0 0
\(953\) −31.3607 −1.01587 −0.507936 0.861395i \(-0.669591\pi\)
−0.507936 + 0.861395i \(0.669591\pi\)
\(954\) 0 0
\(955\) 8.85410 0.286512
\(956\) 0 0
\(957\) 0.291796 0.00943243
\(958\) 0 0
\(959\) −0.708204 −0.0228691
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 0 0
\(963\) 8.18034 0.263608
\(964\) 0 0
\(965\) −52.6869 −1.69605
\(966\) 0 0
\(967\) 12.4164 0.399285 0.199642 0.979869i \(-0.436022\pi\)
0.199642 + 0.979869i \(0.436022\pi\)
\(968\) 0 0
\(969\) −2.88854 −0.0927934
\(970\) 0 0
\(971\) −23.0132 −0.738527 −0.369264 0.929325i \(-0.620390\pi\)
−0.369264 + 0.929325i \(0.620390\pi\)
\(972\) 0 0
\(973\) −17.8541 −0.572376
\(974\) 0 0
\(975\) −3.43769 −0.110094
\(976\) 0 0
\(977\) 4.25735 0.136205 0.0681024 0.997678i \(-0.478306\pi\)
0.0681024 + 0.997678i \(0.478306\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 21.1246 0.674457
\(982\) 0 0
\(983\) −12.6525 −0.403551 −0.201776 0.979432i \(-0.564671\pi\)
−0.201776 + 0.979432i \(0.564671\pi\)
\(984\) 0 0
\(985\) −27.2705 −0.868911
\(986\) 0 0
\(987\) 7.09017 0.225683
\(988\) 0 0
\(989\) 21.0557 0.669533
\(990\) 0 0
\(991\) −9.27051 −0.294487 −0.147244 0.989100i \(-0.547040\pi\)
−0.147244 + 0.989100i \(0.547040\pi\)
\(992\) 0 0
\(993\) 22.8541 0.725253
\(994\) 0 0
\(995\) 8.94427 0.283552
\(996\) 0 0
\(997\) 52.7082 1.66929 0.834643 0.550792i \(-0.185674\pi\)
0.834643 + 0.550792i \(0.185674\pi\)
\(998\) 0 0
\(999\) −33.5410 −1.06119
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 6592.2.a.t.1.2 2
4.3 odd 2 6592.2.a.h.1.2 2
8.3 odd 2 1648.2.a.f.1.1 2
8.5 even 2 103.2.a.a.1.2 2
24.5 odd 2 927.2.a.b.1.1 2
40.29 even 2 2575.2.a.g.1.1 2
56.13 odd 2 5047.2.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.2 2 8.5 even 2
927.2.a.b.1.1 2 24.5 odd 2
1648.2.a.f.1.1 2 8.3 odd 2
2575.2.a.g.1.1 2 40.29 even 2
5047.2.a.a.1.2 2 56.13 odd 2
6592.2.a.h.1.2 2 4.3 odd 2
6592.2.a.t.1.2 2 1.1 even 1 trivial