Properties

Label 1648.2.a.f.1.1
Level $1648$
Weight $2$
Character 1648.1
Self dual yes
Analytic conductor $13.159$
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] = [1648,2,Mod(1,1648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1648, 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("1648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1648 = 2^{4} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1648.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: \(13.1593462531\)
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 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1648.1

$q$-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 \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 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 \(-0.417227\pi\)
0.257118 + 0.966380i \(0.417227\pi\)
\(14\) 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 \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 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 \(-0.522596\pi\)
−0.0709293 + 0.997481i \(0.522596\pi\)
\(30\) 0 0
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) 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 \(-0.685910\pi\)
−0.551411 + 0.834234i \(0.685910\pi\)
\(38\) 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 \(-0.327010\pi\)
0.517104 + 0.855923i \(0.327010\pi\)
\(48\) 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 \(-0.743710\pi\)
−0.692996 + 0.720942i \(0.743710\pi\)
\(54\) 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 \(-0.255465\pi\)
0.694863 + 0.719142i \(0.255465\pi\)
\(62\) 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 \(-0.638224\pi\)
−0.420724 + 0.907189i \(0.638224\pi\)
\(72\) 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 \(-0.776245\pi\)
−0.762939 + 0.646470i \(0.776245\pi\)
\(80\) 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 \(-0.274241\pi\)
0.651260 + 0.758854i \(0.274241\pi\)
\(102\) 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 \(-0.331182\pi\)
0.505843 + 0.862626i \(0.331182\pi\)
\(110\) 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 \(-0.199129\pi\)
0.810623 + 0.585569i \(0.199129\pi\)
\(128\) 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 \(-0.480794\pi\)
0.0603010 + 0.998180i \(0.480794\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 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 \(-0.458067\pi\)
0.131357 + 0.991335i \(0.458067\pi\)
\(158\) 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 \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 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 \(-0.665013\pi\)
−0.495495 + 0.868611i \(0.665013\pi\)
\(174\) 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 \(-0.533827\pi\)
−0.106072 + 0.994358i \(0.533827\pi\)
\(182\) 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 \(-0.539045\pi\)
−0.122355 + 0.992486i \(0.539045\pi\)
\(192\) 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 \(-0.378991\pi\)
0.371069 + 0.928605i \(0.378991\pi\)
\(198\) 0 0
\(199\) −3.41641 −0.242183 −0.121091 0.992641i \(-0.538639\pi\)
−0.121091 + 0.992641i \(0.538639\pi\)
\(200\) 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 \(-0.438786\pi\)
0.191125 + 0.981566i \(0.438786\pi\)
\(224\) 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 \(-0.571143\pi\)
−0.221645 + 0.975127i \(0.571143\pi\)
\(230\) 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 \(-0.788247\pi\)
−0.786767 + 0.617250i \(0.788247\pi\)
\(240\) 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 \(-0.308204\pi\)
0.566740 + 0.823896i \(0.308204\pi\)
\(264\) 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 \(-0.532333\pi\)
−0.101402 + 0.994846i \(0.532333\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 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 \(-0.584254\pi\)
−0.261613 + 0.965173i \(0.584254\pi\)
\(278\) 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 \(-0.409018\pi\)
0.281952 + 0.959429i \(0.409018\pi\)
\(294\) 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 \(-0.117650\pi\)
0.932469 + 0.361250i \(0.117650\pi\)
\(312\) 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 \(-0.485840\pi\)
0.0444717 + 0.999011i \(0.485840\pi\)
\(318\) 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 \(-0.598841\pi\)
−0.305553 + 0.952175i \(0.598841\pi\)
\(350\) 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 \(-0.795315\pi\)
−0.800279 + 0.599628i \(0.795315\pi\)
\(360\) 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 \(-0.903363\pi\)
−0.954268 + 0.298951i \(0.903363\pi\)
\(368\) 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 \(-0.0703485\pi\)
0.975677 + 0.219212i \(0.0703485\pi\)
\(374\) 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 \(-0.701752\pi\)
−0.592230 + 0.805769i \(0.701752\pi\)
\(384\) 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 \(-0.663816\pi\)
−0.492225 + 0.870468i \(0.663816\pi\)
\(390\) 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 \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 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 \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) 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 \(-0.419724\pi\)
0.249528 + 0.968368i \(0.419724\pi\)
\(432\) 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 \(-0.572310\pi\)
−0.225218 + 0.974308i \(0.572310\pi\)
\(440\) 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 \(-0.866402\pi\)
−0.913207 + 0.407495i \(0.866402\pi\)
\(462\) 0 0
\(463\) −5.41641 −0.251722 −0.125861 0.992048i \(-0.540169\pi\)
−0.125861 + 0.992048i \(0.540169\pi\)
\(464\) 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 \(-0.394986\pi\)
0.323958 + 0.946071i \(0.394986\pi\)
\(480\) 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 \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 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 \(-0.253561\pi\)
0.699152 + 0.714973i \(0.253561\pi\)
\(504\) 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 \(-0.318290\pi\)
0.540356 + 0.841437i \(0.318290\pi\)
\(510\) 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 \(-0.389299\pi\)
0.340811 + 0.940132i \(0.389299\pi\)
\(542\) 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 \(-0.292072\pi\)
0.607752 + 0.794127i \(0.292072\pi\)
\(558\) 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 \(-0.637199\pi\)
−0.417800 + 0.908539i \(0.637199\pi\)
\(600\) 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 \(-0.450000\pi\)
0.156433 + 0.987689i \(0.450000\pi\)
\(608\) 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 \(-0.515539\pi\)
−0.0487989 + 0.998809i \(0.515539\pi\)
\(614\) 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 \(-0.555591\pi\)
−0.173758 + 0.984788i \(0.555591\pi\)
\(632\) 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 \(-0.829443\pi\)
−0.859850 + 0.510546i \(0.829443\pi\)
\(648\) 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 \(-0.504758\pi\)
−0.0149475 + 0.999888i \(0.504758\pi\)
\(654\) 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 \(-0.571401\pi\)
−0.222437 + 0.974947i \(0.571401\pi\)
\(662\) 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 \(-0.432419\pi\)
0.210720 + 0.977546i \(0.432419\pi\)
\(678\) 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 \(-0.507224\pi\)
−0.0226924 + 0.999742i \(0.507224\pi\)
\(702\) 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 \(-0.857121\pi\)
−0.900939 + 0.433945i \(0.857121\pi\)
\(710\) 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 \(-0.645533\pi\)
−0.441441 + 0.897290i \(0.645533\pi\)
\(720\) 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 \(-0.248836\pi\)
0.709687 + 0.704517i \(0.248836\pi\)
\(728\) 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 \(-0.492405\pi\)
0.0238568 + 0.999715i \(0.492405\pi\)
\(734\) 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 \(-0.602738\pi\)
−0.317187 + 0.948363i \(0.602738\pi\)
\(744\) 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 \(-0.477474\pi\)
0.0707075 + 0.997497i \(0.477474\pi\)
\(752\) 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 \(-0.717251\pi\)
−0.630746 + 0.775990i \(0.717251\pi\)
\(758\) 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 \(-0.651464\pi\)
−0.458085 + 0.888909i \(0.651464\pi\)
\(774\) 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 \(-0.555961\pi\)
−0.174902 + 0.984586i \(0.555961\pi\)
\(798\) 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 \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) −23.5623 −0.821330 −0.410665 0.911786i \(-0.634703\pi\)
−0.410665 + 0.911786i \(0.634703\pi\)
\(824\) 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 \(-0.608617\pi\)
−0.334646 + 0.942344i \(0.608617\pi\)
\(830\) 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 \(-0.620332\pi\)
−0.369094 + 0.929392i \(0.620332\pi\)
\(840\) 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 \(-0.558803\pi\)
−0.183685 + 0.982985i \(0.558803\pi\)
\(854\) 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 \(-0.454927\pi\)
0.141128 + 0.989991i \(0.454927\pi\)
\(864\) 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 \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 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 \(-0.0684414\pi\)
0.976973 + 0.213362i \(0.0684414\pi\)
\(888\) 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 \(-0.602112\pi\)
−0.315320 + 0.948986i \(0.602112\pi\)
\(912\) 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 \(-0.601342\pi\)
−0.313025 + 0.949745i \(0.601342\pi\)
\(920\) 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 \(-0.193184\pi\)
0.821417 + 0.570328i \(0.193184\pi\)
\(942\) 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 \(-0.563978\pi\)
−0.199642 + 0.979869i \(0.563978\pi\)
\(968\) 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 \(-0.435329\pi\)
0.201776 + 0.979432i \(0.435329\pi\)
\(984\) 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 \(-0.452960\pi\)
0.147244 + 0.989100i \(0.452960\pi\)
\(992\) 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 \(-0.814326\pi\)
−0.834643 + 0.550792i \(0.814326\pi\)
\(998\) 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 1648.2.a.f.1.1 2
4.3 odd 2 103.2.a.a.1.2 2
8.3 odd 2 6592.2.a.t.1.2 2
8.5 even 2 6592.2.a.h.1.2 2
12.11 even 2 927.2.a.b.1.1 2
20.19 odd 2 2575.2.a.g.1.1 2
28.27 even 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 4.3 odd 2
927.2.a.b.1.1 2 12.11 even 2
1648.2.a.f.1.1 2 1.1 even 1 trivial
2575.2.a.g.1.1 2 20.19 odd 2
5047.2.a.a.1.2 2 28.27 even 2
6592.2.a.h.1.2 2 8.5 even 2
6592.2.a.t.1.2 2 8.3 odd 2