Properties

Label 4056.2.a.be.1.3
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $4$
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] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25488.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} - 6x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.28657\) of defining polynomial
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.55452 q^{5} +2.96046 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.55452 q^{5} +2.96046 q^{7} +1.00000 q^{9} -2.24703 q^{11} +1.55452 q^{15} -1.01862 q^{17} -5.35607 q^{19} +2.96046 q^{21} +0.782926 q^{23} -2.58347 q^{25} +1.00000 q^{27} +2.69251 q^{29} +7.28657 q^{31} -2.24703 q^{33} +4.60209 q^{35} +7.80155 q^{37} +5.34474 q^{41} +3.12766 q^{43} +1.55452 q^{45} +7.13799 q^{47} +1.76430 q^{49} -1.01862 q^{51} +13.8388 q^{53} -3.49305 q^{55} -5.35607 q^{57} +4.35506 q^{59} +5.14528 q^{61} +2.96046 q^{63} -12.9977 q^{67} +0.782926 q^{69} +13.8501 q^{71} +8.30519 q^{73} -2.58347 q^{75} -6.65223 q^{77} +1.23570 q^{79} +1.00000 q^{81} +7.67389 q^{83} -1.58347 q^{85} +2.69251 q^{87} -0.494055 q^{89} +7.28657 q^{93} -8.32611 q^{95} -11.1057 q^{97} -2.24703 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 10 q^{11} + 4 q^{15} + 12 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 4 q^{27} - 6 q^{29} + 20 q^{31} + 10 q^{33} - 10 q^{35} + 10 q^{37} + 6 q^{41} - 8 q^{43} + 4 q^{45} + 14 q^{47} + 18 q^{49} + 12 q^{51} + 2 q^{53} + 10 q^{55} + 2 q^{57} + 8 q^{59} - 2 q^{61} + 2 q^{63} - 10 q^{67} - 2 q^{69} - 6 q^{71} + 8 q^{73} + 2 q^{75} - 28 q^{77} - 6 q^{79} + 4 q^{81} + 30 q^{83} + 6 q^{85} - 6 q^{87} + 36 q^{89} + 20 q^{93} - 34 q^{95} - 12 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.55452 0.695202 0.347601 0.937642i \(-0.386996\pi\)
0.347601 + 0.937642i \(0.386996\pi\)
\(6\) 0 0
\(7\) 2.96046 1.11895 0.559474 0.828848i \(-0.311003\pi\)
0.559474 + 0.828848i \(0.311003\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.24703 −0.677504 −0.338752 0.940876i \(-0.610005\pi\)
−0.338752 + 0.940876i \(0.610005\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 1.55452 0.401375
\(16\) 0 0
\(17\) −1.01862 −0.247052 −0.123526 0.992341i \(-0.539420\pi\)
−0.123526 + 0.992341i \(0.539420\pi\)
\(18\) 0 0
\(19\) −5.35607 −1.22877 −0.614383 0.789008i \(-0.710595\pi\)
−0.614383 + 0.789008i \(0.710595\pi\)
\(20\) 0 0
\(21\) 2.96046 0.646025
\(22\) 0 0
\(23\) 0.782926 0.163251 0.0816257 0.996663i \(-0.473989\pi\)
0.0816257 + 0.996663i \(0.473989\pi\)
\(24\) 0 0
\(25\) −2.58347 −0.516694
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.69251 0.499986 0.249993 0.968248i \(-0.419572\pi\)
0.249993 + 0.968248i \(0.419572\pi\)
\(30\) 0 0
\(31\) 7.28657 1.30871 0.654353 0.756189i \(-0.272941\pi\)
0.654353 + 0.756189i \(0.272941\pi\)
\(32\) 0 0
\(33\) −2.24703 −0.391157
\(34\) 0 0
\(35\) 4.60209 0.777895
\(36\) 0 0
\(37\) 7.80155 1.28257 0.641283 0.767304i \(-0.278402\pi\)
0.641283 + 0.767304i \(0.278402\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.34474 0.834707 0.417354 0.908744i \(-0.362958\pi\)
0.417354 + 0.908744i \(0.362958\pi\)
\(42\) 0 0
\(43\) 3.12766 0.476964 0.238482 0.971147i \(-0.423350\pi\)
0.238482 + 0.971147i \(0.423350\pi\)
\(44\) 0 0
\(45\) 1.55452 0.231734
\(46\) 0 0
\(47\) 7.13799 1.04118 0.520591 0.853806i \(-0.325712\pi\)
0.520591 + 0.853806i \(0.325712\pi\)
\(48\) 0 0
\(49\) 1.76430 0.252043
\(50\) 0 0
\(51\) −1.01862 −0.142636
\(52\) 0 0
\(53\) 13.8388 1.90090 0.950452 0.310871i \(-0.100621\pi\)
0.950452 + 0.310871i \(0.100621\pi\)
\(54\) 0 0
\(55\) −3.49305 −0.471003
\(56\) 0 0
\(57\) −5.35607 −0.709428
\(58\) 0 0
\(59\) 4.35506 0.566981 0.283490 0.958975i \(-0.408508\pi\)
0.283490 + 0.958975i \(0.408508\pi\)
\(60\) 0 0
\(61\) 5.14528 0.658785 0.329393 0.944193i \(-0.393156\pi\)
0.329393 + 0.944193i \(0.393156\pi\)
\(62\) 0 0
\(63\) 2.96046 0.372982
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.9977 −1.58792 −0.793961 0.607969i \(-0.791985\pi\)
−0.793961 + 0.607969i \(0.791985\pi\)
\(68\) 0 0
\(69\) 0.782926 0.0942532
\(70\) 0 0
\(71\) 13.8501 1.64371 0.821854 0.569699i \(-0.192940\pi\)
0.821854 + 0.569699i \(0.192940\pi\)
\(72\) 0 0
\(73\) 8.30519 0.972049 0.486025 0.873945i \(-0.338447\pi\)
0.486025 + 0.873945i \(0.338447\pi\)
\(74\) 0 0
\(75\) −2.58347 −0.298313
\(76\) 0 0
\(77\) −6.65223 −0.758092
\(78\) 0 0
\(79\) 1.23570 0.139027 0.0695133 0.997581i \(-0.477855\pi\)
0.0695133 + 0.997581i \(0.477855\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.67389 0.842318 0.421159 0.906987i \(-0.361623\pi\)
0.421159 + 0.906987i \(0.361623\pi\)
\(84\) 0 0
\(85\) −1.58347 −0.171751
\(86\) 0 0
\(87\) 2.69251 0.288667
\(88\) 0 0
\(89\) −0.494055 −0.0523697 −0.0261849 0.999657i \(-0.508336\pi\)
−0.0261849 + 0.999657i \(0.508336\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.28657 0.755582
\(94\) 0 0
\(95\) −8.32611 −0.854241
\(96\) 0 0
\(97\) −11.1057 −1.12762 −0.563808 0.825906i \(-0.690664\pi\)
−0.563808 + 0.825906i \(0.690664\pi\)
\(98\) 0 0
\(99\) −2.24703 −0.225835
\(100\) 0 0
\(101\) −15.1866 −1.51112 −0.755560 0.655080i \(-0.772635\pi\)
−0.755560 + 0.655080i \(0.772635\pi\)
\(102\) 0 0
\(103\) −1.12766 −0.111112 −0.0555559 0.998456i \(-0.517693\pi\)
−0.0555559 + 0.998456i \(0.517693\pi\)
\(104\) 0 0
\(105\) 4.60209 0.449118
\(106\) 0 0
\(107\) −17.0382 −1.64715 −0.823575 0.567208i \(-0.808024\pi\)
−0.823575 + 0.567208i \(0.808024\pi\)
\(108\) 0 0
\(109\) 15.2866 1.46419 0.732094 0.681204i \(-0.238543\pi\)
0.732094 + 0.681204i \(0.238543\pi\)
\(110\) 0 0
\(111\) 7.80155 0.740490
\(112\) 0 0
\(113\) 7.01862 0.660256 0.330128 0.943936i \(-0.392908\pi\)
0.330128 + 0.943936i \(0.392908\pi\)
\(114\) 0 0
\(115\) 1.21707 0.113493
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.01559 −0.276438
\(120\) 0 0
\(121\) −5.95087 −0.540988
\(122\) 0 0
\(123\) 5.34474 0.481919
\(124\) 0 0
\(125\) −11.7887 −1.05441
\(126\) 0 0
\(127\) −14.8015 −1.31342 −0.656712 0.754141i \(-0.728053\pi\)
−0.656712 + 0.754141i \(0.728053\pi\)
\(128\) 0 0
\(129\) 3.12766 0.275375
\(130\) 0 0
\(131\) 5.81916 0.508423 0.254211 0.967149i \(-0.418184\pi\)
0.254211 + 0.967149i \(0.418184\pi\)
\(132\) 0 0
\(133\) −15.8564 −1.37492
\(134\) 0 0
\(135\) 1.55452 0.133792
\(136\) 0 0
\(137\) 17.3674 1.48380 0.741899 0.670512i \(-0.233926\pi\)
0.741899 + 0.670512i \(0.233926\pi\)
\(138\) 0 0
\(139\) 5.88792 0.499407 0.249704 0.968322i \(-0.419667\pi\)
0.249704 + 0.968322i \(0.419667\pi\)
\(140\) 0 0
\(141\) 7.13799 0.601127
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.18556 0.347592
\(146\) 0 0
\(147\) 1.76430 0.145517
\(148\) 0 0
\(149\) −10.5845 −0.867114 −0.433557 0.901126i \(-0.642742\pi\)
−0.433557 + 0.901126i \(0.642742\pi\)
\(150\) 0 0
\(151\) 2.24703 0.182861 0.0914303 0.995811i \(-0.470856\pi\)
0.0914303 + 0.995811i \(0.470856\pi\)
\(152\) 0 0
\(153\) −1.01862 −0.0823507
\(154\) 0 0
\(155\) 11.3271 0.909816
\(156\) 0 0
\(157\) −8.45782 −0.675007 −0.337504 0.941324i \(-0.609583\pi\)
−0.337504 + 0.941324i \(0.609583\pi\)
\(158\) 0 0
\(159\) 13.8388 1.09749
\(160\) 0 0
\(161\) 2.31782 0.182670
\(162\) 0 0
\(163\) −4.20019 −0.328985 −0.164492 0.986378i \(-0.552599\pi\)
−0.164492 + 0.986378i \(0.552599\pi\)
\(164\) 0 0
\(165\) −3.49305 −0.271934
\(166\) 0 0
\(167\) 18.7121 1.44799 0.723994 0.689806i \(-0.242304\pi\)
0.723994 + 0.689806i \(0.242304\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −5.35607 −0.409589
\(172\) 0 0
\(173\) 2.89096 0.219796 0.109898 0.993943i \(-0.464948\pi\)
0.109898 + 0.993943i \(0.464948\pi\)
\(174\) 0 0
\(175\) −7.64824 −0.578153
\(176\) 0 0
\(177\) 4.35506 0.327346
\(178\) 0 0
\(179\) −22.8574 −1.70844 −0.854222 0.519909i \(-0.825966\pi\)
−0.854222 + 0.519909i \(0.825966\pi\)
\(180\) 0 0
\(181\) −8.09042 −0.601356 −0.300678 0.953726i \(-0.597213\pi\)
−0.300678 + 0.953726i \(0.597213\pi\)
\(182\) 0 0
\(183\) 5.14528 0.380350
\(184\) 0 0
\(185\) 12.1277 0.891643
\(186\) 0 0
\(187\) 2.28887 0.167379
\(188\) 0 0
\(189\) 2.96046 0.215342
\(190\) 0 0
\(191\) −15.3850 −1.11322 −0.556610 0.830774i \(-0.687898\pi\)
−0.556610 + 0.830774i \(0.687898\pi\)
\(192\) 0 0
\(193\) −0.347036 −0.0249802 −0.0124901 0.999922i \(-0.503976\pi\)
−0.0124901 + 0.999922i \(0.503976\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.8564 1.84219 0.921096 0.389335i \(-0.127295\pi\)
0.921096 + 0.389335i \(0.127295\pi\)
\(198\) 0 0
\(199\) −7.34574 −0.520726 −0.260363 0.965511i \(-0.583842\pi\)
−0.260363 + 0.965511i \(0.583842\pi\)
\(200\) 0 0
\(201\) −12.9977 −0.916787
\(202\) 0 0
\(203\) 7.97105 0.559458
\(204\) 0 0
\(205\) 8.30850 0.580291
\(206\) 0 0
\(207\) 0.782926 0.0544171
\(208\) 0 0
\(209\) 12.0352 0.832494
\(210\) 0 0
\(211\) −28.6952 −1.97546 −0.987729 0.156175i \(-0.950084\pi\)
−0.987729 + 0.156175i \(0.950084\pi\)
\(212\) 0 0
\(213\) 13.8501 0.948995
\(214\) 0 0
\(215\) 4.86201 0.331586
\(216\) 0 0
\(217\) 21.5716 1.46437
\(218\) 0 0
\(219\) 8.30519 0.561213
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.13698 −0.143103 −0.0715514 0.997437i \(-0.522795\pi\)
−0.0715514 + 0.997437i \(0.522795\pi\)
\(224\) 0 0
\(225\) −2.58347 −0.172231
\(226\) 0 0
\(227\) −18.3860 −1.22032 −0.610162 0.792277i \(-0.708895\pi\)
−0.610162 + 0.792277i \(0.708895\pi\)
\(228\) 0 0
\(229\) 1.74669 0.115424 0.0577122 0.998333i \(-0.481619\pi\)
0.0577122 + 0.998333i \(0.481619\pi\)
\(230\) 0 0
\(231\) −6.65223 −0.437684
\(232\) 0 0
\(233\) −26.3525 −1.72641 −0.863204 0.504855i \(-0.831546\pi\)
−0.863204 + 0.504855i \(0.831546\pi\)
\(234\) 0 0
\(235\) 11.0961 0.723833
\(236\) 0 0
\(237\) 1.23570 0.0802671
\(238\) 0 0
\(239\) 6.36336 0.411611 0.205806 0.978593i \(-0.434019\pi\)
0.205806 + 0.978593i \(0.434019\pi\)
\(240\) 0 0
\(241\) −9.01862 −0.580940 −0.290470 0.956884i \(-0.593812\pi\)
−0.290470 + 0.956884i \(0.593812\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.74265 0.175221
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 7.67389 0.486313
\(250\) 0 0
\(251\) −18.2760 −1.15357 −0.576785 0.816896i \(-0.695693\pi\)
−0.576785 + 0.816896i \(0.695693\pi\)
\(252\) 0 0
\(253\) −1.75926 −0.110603
\(254\) 0 0
\(255\) −1.58347 −0.0991606
\(256\) 0 0
\(257\) 26.4263 1.64843 0.824214 0.566279i \(-0.191617\pi\)
0.824214 + 0.566279i \(0.191617\pi\)
\(258\) 0 0
\(259\) 23.0961 1.43512
\(260\) 0 0
\(261\) 2.69251 0.166662
\(262\) 0 0
\(263\) 4.82017 0.297224 0.148612 0.988896i \(-0.452519\pi\)
0.148612 + 0.988896i \(0.452519\pi\)
\(264\) 0 0
\(265\) 21.5127 1.32151
\(266\) 0 0
\(267\) −0.494055 −0.0302357
\(268\) 0 0
\(269\) −1.64034 −0.100013 −0.0500066 0.998749i \(-0.515924\pi\)
−0.0500066 + 0.998749i \(0.515924\pi\)
\(270\) 0 0
\(271\) −12.7300 −0.773294 −0.386647 0.922228i \(-0.626367\pi\)
−0.386647 + 0.922228i \(0.626367\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.80512 0.350062
\(276\) 0 0
\(277\) 24.5008 1.47211 0.736055 0.676922i \(-0.236687\pi\)
0.736055 + 0.676922i \(0.236687\pi\)
\(278\) 0 0
\(279\) 7.28657 0.436236
\(280\) 0 0
\(281\) 1.93853 0.115643 0.0578215 0.998327i \(-0.481585\pi\)
0.0578215 + 0.998327i \(0.481585\pi\)
\(282\) 0 0
\(283\) 8.07853 0.480219 0.240109 0.970746i \(-0.422817\pi\)
0.240109 + 0.970746i \(0.422817\pi\)
\(284\) 0 0
\(285\) −8.32611 −0.493196
\(286\) 0 0
\(287\) 15.8229 0.933994
\(288\) 0 0
\(289\) −15.9624 −0.938965
\(290\) 0 0
\(291\) −11.1057 −0.651030
\(292\) 0 0
\(293\) −28.3411 −1.65571 −0.827854 0.560944i \(-0.810438\pi\)
−0.827854 + 0.560944i \(0.810438\pi\)
\(294\) 0 0
\(295\) 6.77003 0.394166
\(296\) 0 0
\(297\) −2.24703 −0.130386
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 9.25931 0.533698
\(302\) 0 0
\(303\) −15.1866 −0.872445
\(304\) 0 0
\(305\) 7.99844 0.457989
\(306\) 0 0
\(307\) −30.7025 −1.75229 −0.876143 0.482051i \(-0.839892\pi\)
−0.876143 + 0.482051i \(0.839892\pi\)
\(308\) 0 0
\(309\) −1.12766 −0.0641504
\(310\) 0 0
\(311\) −13.7338 −0.778772 −0.389386 0.921075i \(-0.627313\pi\)
−0.389386 + 0.921075i \(0.627313\pi\)
\(312\) 0 0
\(313\) 32.1886 1.81941 0.909703 0.415260i \(-0.136309\pi\)
0.909703 + 0.415260i \(0.136309\pi\)
\(314\) 0 0
\(315\) 4.60209 0.259298
\(316\) 0 0
\(317\) −4.54723 −0.255398 −0.127699 0.991813i \(-0.540759\pi\)
−0.127699 + 0.991813i \(0.540759\pi\)
\(318\) 0 0
\(319\) −6.05014 −0.338743
\(320\) 0 0
\(321\) −17.0382 −0.950982
\(322\) 0 0
\(323\) 5.45581 0.303569
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 15.2866 0.845349
\(328\) 0 0
\(329\) 21.1317 1.16503
\(330\) 0 0
\(331\) 14.7899 0.812928 0.406464 0.913667i \(-0.366762\pi\)
0.406464 + 0.913667i \(0.366762\pi\)
\(332\) 0 0
\(333\) 7.80155 0.427522
\(334\) 0 0
\(335\) −20.2052 −1.10393
\(336\) 0 0
\(337\) 3.21808 0.175300 0.0876500 0.996151i \(-0.472064\pi\)
0.0876500 + 0.996151i \(0.472064\pi\)
\(338\) 0 0
\(339\) 7.01862 0.381199
\(340\) 0 0
\(341\) −16.3731 −0.886654
\(342\) 0 0
\(343\) −15.5001 −0.836924
\(344\) 0 0
\(345\) 1.21707 0.0655251
\(346\) 0 0
\(347\) −4.30345 −0.231021 −0.115511 0.993306i \(-0.536850\pi\)
−0.115511 + 0.993306i \(0.536850\pi\)
\(348\) 0 0
\(349\) −11.1284 −0.595689 −0.297845 0.954614i \(-0.596268\pi\)
−0.297845 + 0.954614i \(0.596268\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.8295 0.576396 0.288198 0.957571i \(-0.406944\pi\)
0.288198 + 0.957571i \(0.406944\pi\)
\(354\) 0 0
\(355\) 21.5303 1.14271
\(356\) 0 0
\(357\) −3.01559 −0.159602
\(358\) 0 0
\(359\) 32.0616 1.69215 0.846073 0.533067i \(-0.178961\pi\)
0.846073 + 0.533067i \(0.178961\pi\)
\(360\) 0 0
\(361\) 9.68746 0.509866
\(362\) 0 0
\(363\) −5.95087 −0.312340
\(364\) 0 0
\(365\) 12.9106 0.675771
\(366\) 0 0
\(367\) 0.222122 0.0115947 0.00579734 0.999983i \(-0.498155\pi\)
0.00579734 + 0.999983i \(0.498155\pi\)
\(368\) 0 0
\(369\) 5.34474 0.278236
\(370\) 0 0
\(371\) 40.9691 2.12701
\(372\) 0 0
\(373\) 24.0961 1.24765 0.623826 0.781564i \(-0.285578\pi\)
0.623826 + 0.781564i \(0.285578\pi\)
\(374\) 0 0
\(375\) −11.7887 −0.608763
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.3518 −0.839933 −0.419967 0.907540i \(-0.637958\pi\)
−0.419967 + 0.907540i \(0.637958\pi\)
\(380\) 0 0
\(381\) −14.8015 −0.758306
\(382\) 0 0
\(383\) −19.4967 −0.996237 −0.498119 0.867109i \(-0.665976\pi\)
−0.498119 + 0.867109i \(0.665976\pi\)
\(384\) 0 0
\(385\) −10.3410 −0.527027
\(386\) 0 0
\(387\) 3.12766 0.158988
\(388\) 0 0
\(389\) 29.6972 1.50571 0.752854 0.658187i \(-0.228677\pi\)
0.752854 + 0.658187i \(0.228677\pi\)
\(390\) 0 0
\(391\) −0.797505 −0.0403316
\(392\) 0 0
\(393\) 5.81916 0.293538
\(394\) 0 0
\(395\) 1.92091 0.0966517
\(396\) 0 0
\(397\) −15.8743 −0.796708 −0.398354 0.917232i \(-0.630418\pi\)
−0.398354 + 0.917232i \(0.630418\pi\)
\(398\) 0 0
\(399\) −15.8564 −0.793813
\(400\) 0 0
\(401\) −11.5402 −0.576288 −0.288144 0.957587i \(-0.593038\pi\)
−0.288144 + 0.957587i \(0.593038\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.55452 0.0772447
\(406\) 0 0
\(407\) −17.5303 −0.868944
\(408\) 0 0
\(409\) −35.0979 −1.73548 −0.867739 0.497019i \(-0.834428\pi\)
−0.867739 + 0.497019i \(0.834428\pi\)
\(410\) 0 0
\(411\) 17.3674 0.856671
\(412\) 0 0
\(413\) 12.8930 0.634422
\(414\) 0 0
\(415\) 11.9292 0.585582
\(416\) 0 0
\(417\) 5.88792 0.288333
\(418\) 0 0
\(419\) 22.6284 1.10547 0.552736 0.833356i \(-0.313584\pi\)
0.552736 + 0.833356i \(0.313584\pi\)
\(420\) 0 0
\(421\) 13.3833 0.652261 0.326130 0.945325i \(-0.394255\pi\)
0.326130 + 0.945325i \(0.394255\pi\)
\(422\) 0 0
\(423\) 7.13799 0.347061
\(424\) 0 0
\(425\) 2.63158 0.127650
\(426\) 0 0
\(427\) 15.2324 0.737146
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −20.0616 −0.966333 −0.483166 0.875529i \(-0.660513\pi\)
−0.483166 + 0.875529i \(0.660513\pi\)
\(432\) 0 0
\(433\) 37.8445 1.81869 0.909346 0.416041i \(-0.136583\pi\)
0.909346 + 0.416041i \(0.136583\pi\)
\(434\) 0 0
\(435\) 4.18556 0.200682
\(436\) 0 0
\(437\) −4.19340 −0.200598
\(438\) 0 0
\(439\) −25.6796 −1.22562 −0.612811 0.790230i \(-0.709961\pi\)
−0.612811 + 0.790230i \(0.709961\pi\)
\(440\) 0 0
\(441\) 1.76430 0.0840145
\(442\) 0 0
\(443\) −37.4822 −1.78083 −0.890416 0.455148i \(-0.849586\pi\)
−0.890416 + 0.455148i \(0.849586\pi\)
\(444\) 0 0
\(445\) −0.768019 −0.0364076
\(446\) 0 0
\(447\) −10.5845 −0.500628
\(448\) 0 0
\(449\) −5.92091 −0.279425 −0.139713 0.990192i \(-0.544618\pi\)
−0.139713 + 0.990192i \(0.544618\pi\)
\(450\) 0 0
\(451\) −12.0098 −0.565518
\(452\) 0 0
\(453\) 2.24703 0.105575
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −40.6184 −1.90005 −0.950025 0.312175i \(-0.898942\pi\)
−0.950025 + 0.312175i \(0.898942\pi\)
\(458\) 0 0
\(459\) −1.01862 −0.0475452
\(460\) 0 0
\(461\) 7.86627 0.366369 0.183184 0.983079i \(-0.441360\pi\)
0.183184 + 0.983079i \(0.441360\pi\)
\(462\) 0 0
\(463\) −3.96775 −0.184397 −0.0921984 0.995741i \(-0.529389\pi\)
−0.0921984 + 0.995741i \(0.529389\pi\)
\(464\) 0 0
\(465\) 11.3271 0.525283
\(466\) 0 0
\(467\) −1.41249 −0.0653622 −0.0326811 0.999466i \(-0.510405\pi\)
−0.0326811 + 0.999466i \(0.510405\pi\)
\(468\) 0 0
\(469\) −38.4791 −1.77680
\(470\) 0 0
\(471\) −8.45782 −0.389716
\(472\) 0 0
\(473\) −7.02794 −0.323145
\(474\) 0 0
\(475\) 13.8372 0.634896
\(476\) 0 0
\(477\) 13.8388 0.633635
\(478\) 0 0
\(479\) 21.4368 0.979474 0.489737 0.871870i \(-0.337093\pi\)
0.489737 + 0.871870i \(0.337093\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 2.31782 0.105464
\(484\) 0 0
\(485\) −17.2641 −0.783922
\(486\) 0 0
\(487\) 11.1898 0.507059 0.253529 0.967328i \(-0.418409\pi\)
0.253529 + 0.967328i \(0.418409\pi\)
\(488\) 0 0
\(489\) −4.20019 −0.189939
\(490\) 0 0
\(491\) 19.9665 0.901073 0.450537 0.892758i \(-0.351233\pi\)
0.450537 + 0.892758i \(0.351233\pi\)
\(492\) 0 0
\(493\) −2.74265 −0.123523
\(494\) 0 0
\(495\) −3.49305 −0.157001
\(496\) 0 0
\(497\) 41.0027 1.83922
\(498\) 0 0
\(499\) −4.15156 −0.185849 −0.0929247 0.995673i \(-0.529622\pi\)
−0.0929247 + 0.995673i \(0.529622\pi\)
\(500\) 0 0
\(501\) 18.7121 0.835997
\(502\) 0 0
\(503\) −22.3488 −0.996483 −0.498241 0.867038i \(-0.666021\pi\)
−0.498241 + 0.867038i \(0.666021\pi\)
\(504\) 0 0
\(505\) −23.6078 −1.05053
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 34.6523 1.53594 0.767969 0.640487i \(-0.221267\pi\)
0.767969 + 0.640487i \(0.221267\pi\)
\(510\) 0 0
\(511\) 24.5872 1.08767
\(512\) 0 0
\(513\) −5.35607 −0.236476
\(514\) 0 0
\(515\) −1.75297 −0.0772452
\(516\) 0 0
\(517\) −16.0393 −0.705406
\(518\) 0 0
\(519\) 2.89096 0.126899
\(520\) 0 0
\(521\) 25.0186 1.09609 0.548043 0.836450i \(-0.315373\pi\)
0.548043 + 0.836450i \(0.315373\pi\)
\(522\) 0 0
\(523\) −27.7484 −1.21335 −0.606676 0.794949i \(-0.707497\pi\)
−0.606676 + 0.794949i \(0.707497\pi\)
\(524\) 0 0
\(525\) −7.64824 −0.333797
\(526\) 0 0
\(527\) −7.42226 −0.323319
\(528\) 0 0
\(529\) −22.3870 −0.973349
\(530\) 0 0
\(531\) 4.35506 0.188994
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −26.4863 −1.14510
\(536\) 0 0
\(537\) −22.8574 −0.986370
\(538\) 0 0
\(539\) −3.96444 −0.170761
\(540\) 0 0
\(541\) −31.4592 −1.35254 −0.676269 0.736655i \(-0.736404\pi\)
−0.676269 + 0.736655i \(0.736404\pi\)
\(542\) 0 0
\(543\) −8.09042 −0.347193
\(544\) 0 0
\(545\) 23.7633 1.01791
\(546\) 0 0
\(547\) −18.1510 −0.776081 −0.388040 0.921642i \(-0.626848\pi\)
−0.388040 + 0.921642i \(0.626848\pi\)
\(548\) 0 0
\(549\) 5.14528 0.219595
\(550\) 0 0
\(551\) −14.4213 −0.614366
\(552\) 0 0
\(553\) 3.65822 0.155563
\(554\) 0 0
\(555\) 12.1277 0.514791
\(556\) 0 0
\(557\) 14.8977 0.631235 0.315618 0.948886i \(-0.397788\pi\)
0.315618 + 0.948886i \(0.397788\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2.28887 0.0966362
\(562\) 0 0
\(563\) −6.23873 −0.262931 −0.131466 0.991321i \(-0.541968\pi\)
−0.131466 + 0.991321i \(0.541968\pi\)
\(564\) 0 0
\(565\) 10.9106 0.459012
\(566\) 0 0
\(567\) 2.96046 0.124327
\(568\) 0 0
\(569\) −15.2268 −0.638342 −0.319171 0.947697i \(-0.603405\pi\)
−0.319171 + 0.947697i \(0.603405\pi\)
\(570\) 0 0
\(571\) −14.8801 −0.622712 −0.311356 0.950293i \(-0.600783\pi\)
−0.311356 + 0.950293i \(0.600783\pi\)
\(572\) 0 0
\(573\) −15.3850 −0.642718
\(574\) 0 0
\(575\) −2.02266 −0.0843509
\(576\) 0 0
\(577\) 15.6078 0.649762 0.324881 0.945755i \(-0.394676\pi\)
0.324881 + 0.945755i \(0.394676\pi\)
\(578\) 0 0
\(579\) −0.347036 −0.0144223
\(580\) 0 0
\(581\) 22.7182 0.942510
\(582\) 0 0
\(583\) −31.0961 −1.28787
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −40.2301 −1.66047 −0.830236 0.557411i \(-0.811795\pi\)
−0.830236 + 0.557411i \(0.811795\pi\)
\(588\) 0 0
\(589\) −39.0274 −1.60809
\(590\) 0 0
\(591\) 25.8564 1.06359
\(592\) 0 0
\(593\) 37.7225 1.54908 0.774538 0.632527i \(-0.217982\pi\)
0.774538 + 0.632527i \(0.217982\pi\)
\(594\) 0 0
\(595\) −4.68779 −0.192181
\(596\) 0 0
\(597\) −7.34574 −0.300641
\(598\) 0 0
\(599\) −30.0518 −1.22788 −0.613942 0.789351i \(-0.710417\pi\)
−0.613942 + 0.789351i \(0.710417\pi\)
\(600\) 0 0
\(601\) −3.58347 −0.146173 −0.0730863 0.997326i \(-0.523285\pi\)
−0.0730863 + 0.997326i \(0.523285\pi\)
\(602\) 0 0
\(603\) −12.9977 −0.529307
\(604\) 0 0
\(605\) −9.25074 −0.376096
\(606\) 0 0
\(607\) −23.5679 −0.956590 −0.478295 0.878199i \(-0.658745\pi\)
−0.478295 + 0.878199i \(0.658745\pi\)
\(608\) 0 0
\(609\) 7.97105 0.323003
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.2390 −0.413549 −0.206775 0.978389i \(-0.566297\pi\)
−0.206775 + 0.978389i \(0.566297\pi\)
\(614\) 0 0
\(615\) 8.30850 0.335031
\(616\) 0 0
\(617\) 37.7804 1.52098 0.760490 0.649350i \(-0.224959\pi\)
0.760490 + 0.649350i \(0.224959\pi\)
\(618\) 0 0
\(619\) 20.1165 0.808551 0.404275 0.914637i \(-0.367524\pi\)
0.404275 + 0.914637i \(0.367524\pi\)
\(620\) 0 0
\(621\) 0.782926 0.0314177
\(622\) 0 0
\(623\) −1.46263 −0.0585990
\(624\) 0 0
\(625\) −5.40836 −0.216334
\(626\) 0 0
\(627\) 12.0352 0.480641
\(628\) 0 0
\(629\) −7.94682 −0.316861
\(630\) 0 0
\(631\) −23.6225 −0.940395 −0.470198 0.882561i \(-0.655817\pi\)
−0.470198 + 0.882561i \(0.655817\pi\)
\(632\) 0 0
\(633\) −28.6952 −1.14053
\(634\) 0 0
\(635\) −23.0093 −0.913096
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 13.8501 0.547902
\(640\) 0 0
\(641\) 3.57370 0.141153 0.0705763 0.997506i \(-0.477516\pi\)
0.0705763 + 0.997506i \(0.477516\pi\)
\(642\) 0 0
\(643\) 32.6876 1.28907 0.644537 0.764573i \(-0.277050\pi\)
0.644537 + 0.764573i \(0.277050\pi\)
\(644\) 0 0
\(645\) 4.86201 0.191442
\(646\) 0 0
\(647\) 37.0998 1.45855 0.729273 0.684223i \(-0.239859\pi\)
0.729273 + 0.684223i \(0.239859\pi\)
\(648\) 0 0
\(649\) −9.78594 −0.384132
\(650\) 0 0
\(651\) 21.5716 0.845457
\(652\) 0 0
\(653\) 25.9143 1.01410 0.507052 0.861915i \(-0.330735\pi\)
0.507052 + 0.861915i \(0.330735\pi\)
\(654\) 0 0
\(655\) 9.04601 0.353457
\(656\) 0 0
\(657\) 8.30519 0.324016
\(658\) 0 0
\(659\) −14.9861 −0.583776 −0.291888 0.956453i \(-0.594283\pi\)
−0.291888 + 0.956453i \(0.594283\pi\)
\(660\) 0 0
\(661\) −12.1765 −0.473611 −0.236806 0.971557i \(-0.576100\pi\)
−0.236806 + 0.971557i \(0.576100\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.6491 −0.955851
\(666\) 0 0
\(667\) 2.10803 0.0816234
\(668\) 0 0
\(669\) −2.13698 −0.0826205
\(670\) 0 0
\(671\) −11.5616 −0.446330
\(672\) 0 0
\(673\) 38.8564 1.49780 0.748902 0.662681i \(-0.230581\pi\)
0.748902 + 0.662681i \(0.230581\pi\)
\(674\) 0 0
\(675\) −2.58347 −0.0994377
\(676\) 0 0
\(677\) −25.3624 −0.974754 −0.487377 0.873192i \(-0.662046\pi\)
−0.487377 + 0.873192i \(0.662046\pi\)
\(678\) 0 0
\(679\) −32.8780 −1.26174
\(680\) 0 0
\(681\) −18.3860 −0.704554
\(682\) 0 0
\(683\) 15.9189 0.609120 0.304560 0.952493i \(-0.401491\pi\)
0.304560 + 0.952493i \(0.401491\pi\)
\(684\) 0 0
\(685\) 26.9980 1.03154
\(686\) 0 0
\(687\) 1.74669 0.0666403
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 3.57084 0.135841 0.0679206 0.997691i \(-0.478364\pi\)
0.0679206 + 0.997691i \(0.478364\pi\)
\(692\) 0 0
\(693\) −6.65223 −0.252697
\(694\) 0 0
\(695\) 9.15289 0.347189
\(696\) 0 0
\(697\) −5.44426 −0.206216
\(698\) 0 0
\(699\) −26.3525 −0.996742
\(700\) 0 0
\(701\) −1.56585 −0.0591414 −0.0295707 0.999563i \(-0.509414\pi\)
−0.0295707 + 0.999563i \(0.509414\pi\)
\(702\) 0 0
\(703\) −41.7856 −1.57597
\(704\) 0 0
\(705\) 11.0961 0.417905
\(706\) 0 0
\(707\) −44.9592 −1.69086
\(708\) 0 0
\(709\) −40.7930 −1.53201 −0.766006 0.642833i \(-0.777759\pi\)
−0.766006 + 0.642833i \(0.777759\pi\)
\(710\) 0 0
\(711\) 1.23570 0.0463422
\(712\) 0 0
\(713\) 5.70485 0.213648
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.36336 0.237644
\(718\) 0 0
\(719\) −0.831053 −0.0309930 −0.0154965 0.999880i \(-0.504933\pi\)
−0.0154965 + 0.999880i \(0.504933\pi\)
\(720\) 0 0
\(721\) −3.33839 −0.124328
\(722\) 0 0
\(723\) −9.01862 −0.335406
\(724\) 0 0
\(725\) −6.95601 −0.258340
\(726\) 0 0
\(727\) −9.85844 −0.365629 −0.182815 0.983147i \(-0.558521\pi\)
−0.182815 + 0.983147i \(0.558521\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.18590 −0.117835
\(732\) 0 0
\(733\) −11.6467 −0.430180 −0.215090 0.976594i \(-0.569004\pi\)
−0.215090 + 0.976594i \(0.569004\pi\)
\(734\) 0 0
\(735\) 2.74265 0.101164
\(736\) 0 0
\(737\) 29.2062 1.07582
\(738\) 0 0
\(739\) 28.1132 1.03416 0.517080 0.855937i \(-0.327019\pi\)
0.517080 + 0.855937i \(0.327019\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.8078 0.616619 0.308309 0.951286i \(-0.400237\pi\)
0.308309 + 0.951286i \(0.400237\pi\)
\(744\) 0 0
\(745\) −16.4538 −0.602820
\(746\) 0 0
\(747\) 7.67389 0.280773
\(748\) 0 0
\(749\) −50.4410 −1.84307
\(750\) 0 0
\(751\) 18.4378 0.672806 0.336403 0.941718i \(-0.390789\pi\)
0.336403 + 0.941718i \(0.390789\pi\)
\(752\) 0 0
\(753\) −18.2760 −0.666014
\(754\) 0 0
\(755\) 3.49305 0.127125
\(756\) 0 0
\(757\) −1.44805 −0.0526303 −0.0263151 0.999654i \(-0.508377\pi\)
−0.0263151 + 0.999654i \(0.508377\pi\)
\(758\) 0 0
\(759\) −1.75926 −0.0638570
\(760\) 0 0
\(761\) −14.7121 −0.533314 −0.266657 0.963791i \(-0.585919\pi\)
−0.266657 + 0.963791i \(0.585919\pi\)
\(762\) 0 0
\(763\) 45.2552 1.63835
\(764\) 0 0
\(765\) −1.58347 −0.0572504
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −9.10703 −0.328408 −0.164204 0.986426i \(-0.552506\pi\)
−0.164204 + 0.986426i \(0.552506\pi\)
\(770\) 0 0
\(771\) 26.4263 0.951720
\(772\) 0 0
\(773\) 23.2027 0.834543 0.417272 0.908782i \(-0.362986\pi\)
0.417272 + 0.908782i \(0.362986\pi\)
\(774\) 0 0
\(775\) −18.8246 −0.676200
\(776\) 0 0
\(777\) 23.0961 0.828570
\(778\) 0 0
\(779\) −28.6268 −1.02566
\(780\) 0 0
\(781\) −31.1216 −1.11362
\(782\) 0 0
\(783\) 2.69251 0.0962224
\(784\) 0 0
\(785\) −13.1478 −0.469267
\(786\) 0 0
\(787\) −12.7879 −0.455840 −0.227920 0.973680i \(-0.573192\pi\)
−0.227920 + 0.973680i \(0.573192\pi\)
\(788\) 0 0
\(789\) 4.82017 0.171603
\(790\) 0 0
\(791\) 20.7783 0.738792
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 21.5127 0.762976
\(796\) 0 0
\(797\) 40.4330 1.43221 0.716106 0.697992i \(-0.245923\pi\)
0.716106 + 0.697992i \(0.245923\pi\)
\(798\) 0 0
\(799\) −7.27091 −0.257226
\(800\) 0 0
\(801\) −0.494055 −0.0174566
\(802\) 0 0
\(803\) −18.6620 −0.658568
\(804\) 0 0
\(805\) 3.60310 0.126992
\(806\) 0 0
\(807\) −1.64034 −0.0577426
\(808\) 0 0
\(809\) 44.5607 1.56667 0.783335 0.621599i \(-0.213517\pi\)
0.783335 + 0.621599i \(0.213517\pi\)
\(810\) 0 0
\(811\) −1.90102 −0.0667537 −0.0333769 0.999443i \(-0.510626\pi\)
−0.0333769 + 0.999443i \(0.510626\pi\)
\(812\) 0 0
\(813\) −12.7300 −0.446461
\(814\) 0 0
\(815\) −6.52929 −0.228711
\(816\) 0 0
\(817\) −16.7520 −0.586077
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 43.0062 1.50093 0.750463 0.660913i \(-0.229831\pi\)
0.750463 + 0.660913i \(0.229831\pi\)
\(822\) 0 0
\(823\) 32.1490 1.12064 0.560321 0.828275i \(-0.310678\pi\)
0.560321 + 0.828275i \(0.310678\pi\)
\(824\) 0 0
\(825\) 5.80512 0.202108
\(826\) 0 0
\(827\) −2.20149 −0.0765533 −0.0382766 0.999267i \(-0.512187\pi\)
−0.0382766 + 0.999267i \(0.512187\pi\)
\(828\) 0 0
\(829\) 37.7013 1.30942 0.654709 0.755881i \(-0.272791\pi\)
0.654709 + 0.755881i \(0.272791\pi\)
\(830\) 0 0
\(831\) 24.5008 0.849923
\(832\) 0 0
\(833\) −1.79716 −0.0622679
\(834\) 0 0
\(835\) 29.0884 1.00665
\(836\) 0 0
\(837\) 7.28657 0.251861
\(838\) 0 0
\(839\) −27.8776 −0.962442 −0.481221 0.876599i \(-0.659806\pi\)
−0.481221 + 0.876599i \(0.659806\pi\)
\(840\) 0 0
\(841\) −21.7504 −0.750014
\(842\) 0 0
\(843\) 1.93853 0.0667665
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −17.6173 −0.605337
\(848\) 0 0
\(849\) 8.07853 0.277254
\(850\) 0 0
\(851\) 6.10803 0.209381
\(852\) 0 0
\(853\) 3.38475 0.115891 0.0579457 0.998320i \(-0.481545\pi\)
0.0579457 + 0.998320i \(0.481545\pi\)
\(854\) 0 0
\(855\) −8.32611 −0.284747
\(856\) 0 0
\(857\) 0.567882 0.0193985 0.00969925 0.999953i \(-0.496913\pi\)
0.00969925 + 0.999953i \(0.496913\pi\)
\(858\) 0 0
\(859\) −43.1391 −1.47189 −0.735944 0.677043i \(-0.763261\pi\)
−0.735944 + 0.677043i \(0.763261\pi\)
\(860\) 0 0
\(861\) 15.8229 0.539242
\(862\) 0 0
\(863\) 21.6466 0.736860 0.368430 0.929656i \(-0.379895\pi\)
0.368430 + 0.929656i \(0.379895\pi\)
\(864\) 0 0
\(865\) 4.49406 0.152802
\(866\) 0 0
\(867\) −15.9624 −0.542112
\(868\) 0 0
\(869\) −2.77664 −0.0941911
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −11.1057 −0.375872
\(874\) 0 0
\(875\) −34.8998 −1.17983
\(876\) 0 0
\(877\) −42.3494 −1.43004 −0.715019 0.699105i \(-0.753582\pi\)
−0.715019 + 0.699105i \(0.753582\pi\)
\(878\) 0 0
\(879\) −28.3411 −0.955923
\(880\) 0 0
\(881\) −29.1903 −0.983445 −0.491722 0.870752i \(-0.663633\pi\)
−0.491722 + 0.870752i \(0.663633\pi\)
\(882\) 0 0
\(883\) 41.6461 1.40150 0.700751 0.713406i \(-0.252848\pi\)
0.700751 + 0.713406i \(0.252848\pi\)
\(884\) 0 0
\(885\) 6.77003 0.227572
\(886\) 0 0
\(887\) 40.6471 1.36480 0.682398 0.730981i \(-0.260937\pi\)
0.682398 + 0.730981i \(0.260937\pi\)
\(888\) 0 0
\(889\) −43.8193 −1.46965
\(890\) 0 0
\(891\) −2.24703 −0.0752783
\(892\) 0 0
\(893\) −38.2315 −1.27937
\(894\) 0 0
\(895\) −35.5323 −1.18771
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19.6191 0.654335
\(900\) 0 0
\(901\) −14.0965 −0.469622
\(902\) 0 0
\(903\) 9.25931 0.308130
\(904\) 0 0
\(905\) −12.5767 −0.418064
\(906\) 0 0
\(907\) 37.7561 1.25367 0.626836 0.779151i \(-0.284350\pi\)
0.626836 + 0.779151i \(0.284350\pi\)
\(908\) 0 0
\(909\) −15.1866 −0.503706
\(910\) 0 0
\(911\) −42.4249 −1.40560 −0.702801 0.711387i \(-0.748067\pi\)
−0.702801 + 0.711387i \(0.748067\pi\)
\(912\) 0 0
\(913\) −17.2434 −0.570674
\(914\) 0 0
\(915\) 7.99844 0.264420
\(916\) 0 0
\(917\) 17.2274 0.568898
\(918\) 0 0
\(919\) 49.8346 1.64389 0.821947 0.569565i \(-0.192888\pi\)
0.821947 + 0.569565i \(0.192888\pi\)
\(920\) 0 0
\(921\) −30.7025 −1.01168
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.1550 −0.662694
\(926\) 0 0
\(927\) −1.12766 −0.0370373
\(928\) 0 0
\(929\) 44.1893 1.44980 0.724901 0.688853i \(-0.241885\pi\)
0.724901 + 0.688853i \(0.241885\pi\)
\(930\) 0 0
\(931\) −9.44973 −0.309703
\(932\) 0 0
\(933\) −13.7338 −0.449624
\(934\) 0 0
\(935\) 3.55810 0.116362
\(936\) 0 0
\(937\) −22.9644 −0.750215 −0.375107 0.926981i \(-0.622394\pi\)
−0.375107 + 0.926981i \(0.622394\pi\)
\(938\) 0 0
\(939\) 32.1886 1.05043
\(940\) 0 0
\(941\) 6.09255 0.198611 0.0993057 0.995057i \(-0.468338\pi\)
0.0993057 + 0.995057i \(0.468338\pi\)
\(942\) 0 0
\(943\) 4.18453 0.136267
\(944\) 0 0
\(945\) 4.60209 0.149706
\(946\) 0 0
\(947\) −41.6982 −1.35501 −0.677505 0.735518i \(-0.736939\pi\)
−0.677505 + 0.735518i \(0.736939\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −4.54723 −0.147454
\(952\) 0 0
\(953\) 35.3850 1.14623 0.573117 0.819474i \(-0.305734\pi\)
0.573117 + 0.819474i \(0.305734\pi\)
\(954\) 0 0
\(955\) −23.9163 −0.773914
\(956\) 0 0
\(957\) −6.05014 −0.195573
\(958\) 0 0
\(959\) 51.4154 1.66029
\(960\) 0 0
\(961\) 22.0941 0.712713
\(962\) 0 0
\(963\) −17.0382 −0.549050
\(964\) 0 0
\(965\) −0.539474 −0.0173663
\(966\) 0 0
\(967\) −4.08684 −0.131424 −0.0657120 0.997839i \(-0.520932\pi\)
−0.0657120 + 0.997839i \(0.520932\pi\)
\(968\) 0 0
\(969\) 5.45581 0.175266
\(970\) 0 0
\(971\) −5.72402 −0.183693 −0.0918463 0.995773i \(-0.529277\pi\)
−0.0918463 + 0.995773i \(0.529277\pi\)
\(972\) 0 0
\(973\) 17.4309 0.558810
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.04498 0.289375 0.144687 0.989477i \(-0.453782\pi\)
0.144687 + 0.989477i \(0.453782\pi\)
\(978\) 0 0
\(979\) 1.11016 0.0354807
\(980\) 0 0
\(981\) 15.2866 0.488063
\(982\) 0 0
\(983\) −7.05061 −0.224879 −0.112440 0.993659i \(-0.535867\pi\)
−0.112440 + 0.993659i \(0.535867\pi\)
\(984\) 0 0
\(985\) 40.1943 1.28070
\(986\) 0 0
\(987\) 21.1317 0.672630
\(988\) 0 0
\(989\) 2.44873 0.0778650
\(990\) 0 0
\(991\) −19.9292 −0.633072 −0.316536 0.948580i \(-0.602520\pi\)
−0.316536 + 0.948580i \(0.602520\pi\)
\(992\) 0 0
\(993\) 14.7899 0.469344
\(994\) 0 0
\(995\) −11.4191 −0.362010
\(996\) 0 0
\(997\) −22.1395 −0.701164 −0.350582 0.936532i \(-0.614016\pi\)
−0.350582 + 0.936532i \(0.614016\pi\)
\(998\) 0 0
\(999\) 7.80155 0.246830
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 4056.2.a.be.1.3 4
4.3 odd 2 8112.2.a.cs.1.3 4
13.5 odd 4 4056.2.c.p.337.2 8
13.6 odd 12 312.2.bf.b.49.1 8
13.8 odd 4 4056.2.c.p.337.7 8
13.11 odd 12 312.2.bf.b.121.4 yes 8
13.12 even 2 4056.2.a.bd.1.2 4
39.11 even 12 936.2.bi.c.433.1 8
39.32 even 12 936.2.bi.c.361.4 8
52.11 even 12 624.2.bv.g.433.4 8
52.19 even 12 624.2.bv.g.49.1 8
52.51 odd 2 8112.2.a.cq.1.2 4
156.11 odd 12 1872.2.by.m.433.1 8
156.71 odd 12 1872.2.by.m.1297.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.bf.b.49.1 8 13.6 odd 12
312.2.bf.b.121.4 yes 8 13.11 odd 12
624.2.bv.g.49.1 8 52.19 even 12
624.2.bv.g.433.4 8 52.11 even 12
936.2.bi.c.361.4 8 39.32 even 12
936.2.bi.c.433.1 8 39.11 even 12
1872.2.by.m.433.1 8 156.11 odd 12
1872.2.by.m.1297.4 8 156.71 odd 12
4056.2.a.bd.1.2 4 13.12 even 2
4056.2.a.be.1.3 4 1.1 even 1 trivial
4056.2.c.p.337.2 8 13.5 odd 4
4056.2.c.p.337.7 8 13.8 odd 4
8112.2.a.cq.1.2 4 52.51 odd 2
8112.2.a.cs.1.3 4 4.3 odd 2