Properties

Label 5929.2.a.r.1.1
Level $5929$
Weight $2$
Character 5929.1
Self dual yes
Analytic conductor $47.343$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5929,2,Mod(1,5929)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5929, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5929.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.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: \(47.3433033584\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 539)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5929.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} +1.41421 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} +1.41421 q^{12} -2.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} -5.65685 q^{19} -1.41421 q^{20} -2.00000 q^{23} +4.24264 q^{24} -3.00000 q^{25} +5.65685 q^{27} +6.00000 q^{29} -2.00000 q^{30} -4.24264 q^{31} +5.00000 q^{32} +1.00000 q^{36} -10.0000 q^{37} -5.65685 q^{38} -4.24264 q^{40} +5.65685 q^{41} +8.00000 q^{43} -1.41421 q^{45} -2.00000 q^{46} -9.89949 q^{47} +1.41421 q^{48} -3.00000 q^{50} +5.65685 q^{54} +8.00000 q^{57} +6.00000 q^{58} +9.89949 q^{59} +2.00000 q^{60} +8.48528 q^{61} -4.24264 q^{62} +7.00000 q^{64} +2.00000 q^{67} +2.82843 q^{69} +6.00000 q^{71} +3.00000 q^{72} -8.48528 q^{73} -10.0000 q^{74} +4.24264 q^{75} +5.65685 q^{76} +8.00000 q^{79} -1.41421 q^{80} -5.00000 q^{81} +5.65685 q^{82} -11.3137 q^{83} +8.00000 q^{86} -8.48528 q^{87} -15.5563 q^{89} -1.41421 q^{90} +2.00000 q^{92} +6.00000 q^{93} -9.89949 q^{94} -8.00000 q^{95} -7.07107 q^{96} -12.7279 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} - 4 q^{15} - 2 q^{16} - 2 q^{18} - 4 q^{23} - 6 q^{25} + 12 q^{29} - 4 q^{30} + 10 q^{32} + 2 q^{36} - 20 q^{37} + 16 q^{43} - 4 q^{46} - 6 q^{50} + 16 q^{57}+ \cdots - 16 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) −1.00000 −0.333333
\(10\) 1.41421 0.447214
\(11\) 0 0
\(12\) 1.41421 0.408248
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) −1.41421 −0.316228
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 4.24264 0.866025
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.24264 −0.762001 −0.381000 0.924575i \(-0.624420\pi\)
−0.381000 + 0.924575i \(0.624420\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −5.65685 −0.917663
\(39\) 0 0
\(40\) −4.24264 −0.670820
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −1.41421 −0.210819
\(46\) −2.00000 −0.294884
\(47\) −9.89949 −1.44399 −0.721995 0.691898i \(-0.756775\pi\)
−0.721995 + 0.691898i \(0.756775\pi\)
\(48\) 1.41421 0.204124
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 5.65685 0.769800
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 6.00000 0.787839
\(59\) 9.89949 1.28880 0.644402 0.764687i \(-0.277106\pi\)
0.644402 + 0.764687i \(0.277106\pi\)
\(60\) 2.00000 0.258199
\(61\) 8.48528 1.08643 0.543214 0.839594i \(-0.317207\pi\)
0.543214 + 0.839594i \(0.317207\pi\)
\(62\) −4.24264 −0.538816
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −10.0000 −1.16248
\(75\) 4.24264 0.489898
\(76\) 5.65685 0.648886
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.41421 −0.158114
\(81\) −5.00000 −0.555556
\(82\) 5.65685 0.624695
\(83\) −11.3137 −1.24184 −0.620920 0.783874i \(-0.713241\pi\)
−0.620920 + 0.783874i \(0.713241\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −8.48528 −0.909718
\(88\) 0 0
\(89\) −15.5563 −1.64897 −0.824485 0.565884i \(-0.808535\pi\)
−0.824485 + 0.565884i \(0.808535\pi\)
\(90\) −1.41421 −0.149071
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 6.00000 0.622171
\(94\) −9.89949 −1.02105
\(95\) −8.00000 −0.820783
\(96\) −7.07107 −0.721688
\(97\) −12.7279 −1.29232 −0.646162 0.763200i \(-0.723627\pi\)
−0.646162 + 0.763200i \(0.723627\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) −5.65685 −0.562878 −0.281439 0.959579i \(-0.590812\pi\)
−0.281439 + 0.959579i \(0.590812\pi\)
\(102\) 0 0
\(103\) 9.89949 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −5.65685 −0.544331
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 14.1421 1.34231
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 8.00000 0.749269
\(115\) −2.82843 −0.263752
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 9.89949 0.911322
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) 0 0
\(122\) 8.48528 0.768221
\(123\) −8.00000 −0.721336
\(124\) 4.24264 0.381000
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −3.00000 −0.265165
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) 19.7990 1.72985 0.864923 0.501905i \(-0.167367\pi\)
0.864923 + 0.501905i \(0.167367\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 2.82843 0.240772
\(139\) 16.9706 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(140\) 0 0
\(141\) 14.0000 1.17901
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.48528 0.704664
\(146\) −8.48528 −0.702247
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 4.24264 0.346410
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 16.9706 1.37649
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −9.89949 −0.790066 −0.395033 0.918667i \(-0.629267\pi\)
−0.395033 + 0.918667i \(0.629267\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 7.07107 0.559017
\(161\) 0 0
\(162\) −5.00000 −0.392837
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) −5.65685 −0.441726
\(165\) 0 0
\(166\) −11.3137 −0.878114
\(167\) −16.9706 −1.31322 −0.656611 0.754230i \(-0.728011\pi\)
−0.656611 + 0.754230i \(0.728011\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 5.65685 0.432590
\(172\) −8.00000 −0.609994
\(173\) −11.3137 −0.860165 −0.430083 0.902790i \(-0.641516\pi\)
−0.430083 + 0.902790i \(0.641516\pi\)
\(174\) −8.48528 −0.643268
\(175\) 0 0
\(176\) 0 0
\(177\) −14.0000 −1.05230
\(178\) −15.5563 −1.16600
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.41421 0.105409
\(181\) 1.41421 0.105118 0.0525588 0.998618i \(-0.483262\pi\)
0.0525588 + 0.998618i \(0.483262\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 6.00000 0.442326
\(185\) −14.1421 −1.03975
\(186\) 6.00000 0.439941
\(187\) 0 0
\(188\) 9.89949 0.721995
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −9.89949 −0.714435
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −12.7279 −0.913812
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −4.24264 −0.300753 −0.150376 0.988629i \(-0.548049\pi\)
−0.150376 + 0.988629i \(0.548049\pi\)
\(200\) 9.00000 0.636396
\(201\) −2.82843 −0.199502
\(202\) −5.65685 −0.398015
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 9.89949 0.689730
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) −8.48528 −0.581402
\(214\) −8.00000 −0.546869
\(215\) 11.3137 0.771589
\(216\) −16.9706 −1.15470
\(217\) 0 0
\(218\) 18.0000 1.21911
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 14.1421 0.949158
\(223\) 15.5563 1.04173 0.520865 0.853639i \(-0.325609\pi\)
0.520865 + 0.853639i \(0.325609\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 14.0000 0.931266
\(227\) 2.82843 0.187729 0.0938647 0.995585i \(-0.470078\pi\)
0.0938647 + 0.995585i \(0.470078\pi\)
\(228\) −8.00000 −0.529813
\(229\) 29.6985 1.96253 0.981266 0.192660i \(-0.0617115\pi\)
0.981266 + 0.192660i \(0.0617115\pi\)
\(230\) −2.82843 −0.186501
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −14.0000 −0.913259
\(236\) −9.89949 −0.644402
\(237\) −11.3137 −0.734904
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 2.00000 0.129099
\(241\) −16.9706 −1.09317 −0.546585 0.837404i \(-0.684072\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) −8.48528 −0.543214
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 0 0
\(248\) 12.7279 0.808224
\(249\) 16.0000 1.01396
\(250\) −11.3137 −0.715542
\(251\) 15.5563 0.981908 0.490954 0.871185i \(-0.336648\pi\)
0.490954 + 0.871185i \(0.336648\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −1.41421 −0.0882162 −0.0441081 0.999027i \(-0.514045\pi\)
−0.0441081 + 0.999027i \(0.514045\pi\)
\(258\) −11.3137 −0.704361
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 19.7990 1.22319
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 22.0000 1.34638
\(268\) −2.00000 −0.122169
\(269\) 9.89949 0.603583 0.301791 0.953374i \(-0.402415\pi\)
0.301791 + 0.953374i \(0.402415\pi\)
\(270\) 8.00000 0.486864
\(271\) 14.1421 0.859074 0.429537 0.903049i \(-0.358677\pi\)
0.429537 + 0.903049i \(0.358677\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 0 0
\(276\) −2.82843 −0.170251
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 16.9706 1.01783
\(279\) 4.24264 0.254000
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 14.0000 0.833688
\(283\) 31.1127 1.84946 0.924729 0.380626i \(-0.124292\pi\)
0.924729 + 0.380626i \(0.124292\pi\)
\(284\) −6.00000 −0.356034
\(285\) 11.3137 0.670166
\(286\) 0 0
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −17.0000 −1.00000
\(290\) 8.48528 0.498273
\(291\) 18.0000 1.05518
\(292\) 8.48528 0.496564
\(293\) 14.1421 0.826192 0.413096 0.910687i \(-0.364447\pi\)
0.413096 + 0.910687i \(0.364447\pi\)
\(294\) 0 0
\(295\) 14.0000 0.815112
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −4.24264 −0.244949
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) 8.00000 0.459588
\(304\) 5.65685 0.324443
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −19.7990 −1.12999 −0.564994 0.825095i \(-0.691122\pi\)
−0.564994 + 0.825095i \(0.691122\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) −6.00000 −0.340777
\(311\) −1.41421 −0.0801927 −0.0400963 0.999196i \(-0.512766\pi\)
−0.0400963 + 0.999196i \(0.512766\pi\)
\(312\) 0 0
\(313\) 24.0416 1.35891 0.679457 0.733716i \(-0.262216\pi\)
0.679457 + 0.733716i \(0.262216\pi\)
\(314\) −9.89949 −0.558661
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 9.89949 0.553399
\(321\) 11.3137 0.631470
\(322\) 0 0
\(323\) 0 0
\(324\) 5.00000 0.277778
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) −25.4558 −1.40771
\(328\) −16.9706 −0.937043
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 11.3137 0.620920
\(333\) 10.0000 0.547997
\(334\) −16.9706 −0.928588
\(335\) 2.82843 0.154533
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −13.0000 −0.707107
\(339\) −19.7990 −1.07533
\(340\) 0 0
\(341\) 0 0
\(342\) 5.65685 0.305888
\(343\) 0 0
\(344\) −24.0000 −1.29399
\(345\) 4.00000 0.215353
\(346\) −11.3137 −0.608229
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 8.48528 0.454859
\(349\) 25.4558 1.36262 0.681310 0.731995i \(-0.261411\pi\)
0.681310 + 0.731995i \(0.261411\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.41421 −0.0752710 −0.0376355 0.999292i \(-0.511983\pi\)
−0.0376355 + 0.999292i \(0.511983\pi\)
\(354\) −14.0000 −0.744092
\(355\) 8.48528 0.450352
\(356\) 15.5563 0.824485
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 4.24264 0.223607
\(361\) 13.0000 0.684211
\(362\) 1.41421 0.0743294
\(363\) 0 0
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) −12.0000 −0.627250
\(367\) −7.07107 −0.369107 −0.184553 0.982822i \(-0.559084\pi\)
−0.184553 + 0.982822i \(0.559084\pi\)
\(368\) 2.00000 0.104257
\(369\) −5.65685 −0.294484
\(370\) −14.1421 −0.735215
\(371\) 0 0
\(372\) −6.00000 −0.311086
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 16.0000 0.826236
\(376\) 29.6985 1.53158
\(377\) 0 0
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 8.00000 0.410391
\(381\) −11.3137 −0.579619
\(382\) 0 0
\(383\) 24.0416 1.22847 0.614235 0.789123i \(-0.289465\pi\)
0.614235 + 0.789123i \(0.289465\pi\)
\(384\) 4.24264 0.216506
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −8.00000 −0.406663
\(388\) 12.7279 0.646162
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −28.0000 −1.41241
\(394\) 18.0000 0.906827
\(395\) 11.3137 0.569254
\(396\) 0 0
\(397\) 15.5563 0.780751 0.390375 0.920656i \(-0.372345\pi\)
0.390375 + 0.920656i \(0.372345\pi\)
\(398\) −4.24264 −0.212664
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) −2.82843 −0.141069
\(403\) 0 0
\(404\) 5.65685 0.281439
\(405\) −7.07107 −0.351364
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −19.7990 −0.978997 −0.489499 0.872004i \(-0.662820\pi\)
−0.489499 + 0.872004i \(0.662820\pi\)
\(410\) 8.00000 0.395092
\(411\) −19.7990 −0.976612
\(412\) −9.89949 −0.487713
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) −16.0000 −0.785409
\(416\) 0 0
\(417\) −24.0000 −1.17529
\(418\) 0 0
\(419\) −24.0416 −1.17451 −0.587255 0.809402i \(-0.699792\pi\)
−0.587255 + 0.809402i \(0.699792\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 16.0000 0.778868
\(423\) 9.89949 0.481330
\(424\) 0 0
\(425\) 0 0
\(426\) −8.48528 −0.411113
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 11.3137 0.545595
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −5.65685 −0.272166
\(433\) −9.89949 −0.475739 −0.237870 0.971297i \(-0.576449\pi\)
−0.237870 + 0.971297i \(0.576449\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) −18.0000 −0.862044
\(437\) 11.3137 0.541208
\(438\) 12.0000 0.573382
\(439\) −2.82843 −0.134993 −0.0674967 0.997719i \(-0.521501\pi\)
−0.0674967 + 0.997719i \(0.521501\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −14.1421 −0.671156
\(445\) −22.0000 −1.04290
\(446\) 15.5563 0.736614
\(447\) 8.48528 0.401340
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 3.00000 0.141421
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) −16.9706 −0.797347
\(454\) 2.82843 0.132745
\(455\) 0 0
\(456\) −24.0000 −1.12390
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 29.6985 1.38772
\(459\) 0 0
\(460\) 2.82843 0.131876
\(461\) −42.4264 −1.97599 −0.987997 0.154471i \(-0.950633\pi\)
−0.987997 + 0.154471i \(0.950633\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) −6.00000 −0.278543
\(465\) 8.48528 0.393496
\(466\) 10.0000 0.463241
\(467\) 4.24264 0.196326 0.0981630 0.995170i \(-0.468703\pi\)
0.0981630 + 0.995170i \(0.468703\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −14.0000 −0.645772
\(471\) 14.0000 0.645086
\(472\) −29.6985 −1.36698
\(473\) 0 0
\(474\) −11.3137 −0.519656
\(475\) 16.9706 0.778663
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −19.7990 −0.904639 −0.452319 0.891856i \(-0.649403\pi\)
−0.452319 + 0.891856i \(0.649403\pi\)
\(480\) −10.0000 −0.456435
\(481\) 0 0
\(482\) −16.9706 −0.772988
\(483\) 0 0
\(484\) 0 0
\(485\) −18.0000 −0.817338
\(486\) −9.89949 −0.449050
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −25.4558 −1.15233
\(489\) −31.1127 −1.40696
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 8.00000 0.360668
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 4.24264 0.190500
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) −26.0000 −1.16392 −0.581960 0.813217i \(-0.697714\pi\)
−0.581960 + 0.813217i \(0.697714\pi\)
\(500\) 11.3137 0.505964
\(501\) 24.0000 1.07224
\(502\) 15.5563 0.694314
\(503\) −2.82843 −0.126113 −0.0630567 0.998010i \(-0.520085\pi\)
−0.0630567 + 0.998010i \(0.520085\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 18.3848 0.816497
\(508\) −8.00000 −0.354943
\(509\) 15.5563 0.689523 0.344762 0.938690i \(-0.387960\pi\)
0.344762 + 0.938690i \(0.387960\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) −32.0000 −1.41283
\(514\) −1.41421 −0.0623783
\(515\) 14.0000 0.616914
\(516\) 11.3137 0.498058
\(517\) 0 0
\(518\) 0 0
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) 21.2132 0.929367 0.464684 0.885477i \(-0.346168\pi\)
0.464684 + 0.885477i \(0.346168\pi\)
\(522\) −6.00000 −0.262613
\(523\) −8.48528 −0.371035 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(524\) −19.7990 −0.864923
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −9.89949 −0.429601
\(532\) 0 0
\(533\) 0 0
\(534\) 22.0000 0.952033
\(535\) −11.3137 −0.489134
\(536\) −6.00000 −0.259161
\(537\) −16.9706 −0.732334
\(538\) 9.89949 0.426798
\(539\) 0 0
\(540\) −8.00000 −0.344265
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 14.1421 0.607457
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 25.4558 1.09041
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −14.0000 −0.598050
\(549\) −8.48528 −0.362143
\(550\) 0 0
\(551\) −33.9411 −1.44594
\(552\) −8.48528 −0.361158
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) 20.0000 0.848953
\(556\) −16.9706 −0.719712
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 4.24264 0.179605
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) 28.2843 1.19204 0.596020 0.802970i \(-0.296748\pi\)
0.596020 + 0.802970i \(0.296748\pi\)
\(564\) −14.0000 −0.589506
\(565\) 19.7990 0.832950
\(566\) 31.1127 1.30776
\(567\) 0 0
\(568\) −18.0000 −0.755263
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 11.3137 0.473879
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) −7.00000 −0.291667
\(577\) 18.3848 0.765368 0.382684 0.923879i \(-0.375000\pi\)
0.382684 + 0.923879i \(0.375000\pi\)
\(578\) −17.0000 −0.707107
\(579\) 2.82843 0.117545
\(580\) −8.48528 −0.352332
\(581\) 0 0
\(582\) 18.0000 0.746124
\(583\) 0 0
\(584\) 25.4558 1.05337
\(585\) 0 0
\(586\) 14.1421 0.584206
\(587\) 15.5563 0.642079 0.321040 0.947066i \(-0.395968\pi\)
0.321040 + 0.947066i \(0.395968\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 14.0000 0.576371
\(591\) −25.4558 −1.04711
\(592\) 10.0000 0.410997
\(593\) 2.82843 0.116150 0.0580748 0.998312i \(-0.481504\pi\)
0.0580748 + 0.998312i \(0.481504\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 6.00000 0.245564
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) −12.7279 −0.519615
\(601\) 22.6274 0.922992 0.461496 0.887142i \(-0.347313\pi\)
0.461496 + 0.887142i \(0.347313\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) 39.5980 1.60723 0.803616 0.595148i \(-0.202907\pi\)
0.803616 + 0.595148i \(0.202907\pi\)
\(608\) −28.2843 −1.14708
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 0 0
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −19.7990 −0.799022
\(615\) −11.3137 −0.456213
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −14.0000 −0.563163
\(619\) −9.89949 −0.397894 −0.198947 0.980010i \(-0.563752\pi\)
−0.198947 + 0.980010i \(0.563752\pi\)
\(620\) 6.00000 0.240966
\(621\) −11.3137 −0.454003
\(622\) −1.41421 −0.0567048
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 24.0416 0.960897
\(627\) 0 0
\(628\) 9.89949 0.395033
\(629\) 0 0
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −24.0000 −0.954669
\(633\) −22.6274 −0.899359
\(634\) −18.0000 −0.714871
\(635\) 11.3137 0.448971
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) −4.24264 −0.167705
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 11.3137 0.446516
\(643\) −18.3848 −0.725025 −0.362512 0.931979i \(-0.618081\pi\)
−0.362512 + 0.931979i \(0.618081\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) −24.0416 −0.945174 −0.472587 0.881284i \(-0.656680\pi\)
−0.472587 + 0.881284i \(0.656680\pi\)
\(648\) 15.0000 0.589256
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) −25.4558 −0.995402
\(655\) 28.0000 1.09405
\(656\) −5.65685 −0.220863
\(657\) 8.48528 0.331042
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −41.0122 −1.59519 −0.797595 0.603194i \(-0.793895\pi\)
−0.797595 + 0.603194i \(0.793895\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 33.9411 1.31717
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) −12.0000 −0.464642
\(668\) 16.9706 0.656611
\(669\) −22.0000 −0.850569
\(670\) 2.82843 0.109272
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −2.00000 −0.0770371
\(675\) −16.9706 −0.653197
\(676\) 13.0000 0.500000
\(677\) −5.65685 −0.217411 −0.108705 0.994074i \(-0.534670\pi\)
−0.108705 + 0.994074i \(0.534670\pi\)
\(678\) −19.7990 −0.760376
\(679\) 0 0
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −5.65685 −0.216295
\(685\) 19.7990 0.756481
\(686\) 0 0
\(687\) −42.0000 −1.60240
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 4.00000 0.152277
\(691\) 18.3848 0.699390 0.349695 0.936864i \(-0.386285\pi\)
0.349695 + 0.936864i \(0.386285\pi\)
\(692\) 11.3137 0.430083
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 24.0000 0.910372
\(696\) 25.4558 0.964901
\(697\) 0 0
\(698\) 25.4558 0.963518
\(699\) −14.1421 −0.534905
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 0 0
\(703\) 56.5685 2.13352
\(704\) 0 0
\(705\) 19.7990 0.745673
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) 14.0000 0.526152
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 8.48528 0.318447
\(711\) −8.00000 −0.300023
\(712\) 46.6690 1.74900
\(713\) 8.48528 0.317776
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 16.9706 0.633777
\(718\) −4.00000 −0.149279
\(719\) 9.89949 0.369189 0.184594 0.982815i \(-0.440903\pi\)
0.184594 + 0.982815i \(0.440903\pi\)
\(720\) 1.41421 0.0527046
\(721\) 0 0
\(722\) 13.0000 0.483810
\(723\) 24.0000 0.892570
\(724\) −1.41421 −0.0525588
\(725\) −18.0000 −0.668503
\(726\) 0 0
\(727\) −7.07107 −0.262251 −0.131126 0.991366i \(-0.541859\pi\)
−0.131126 + 0.991366i \(0.541859\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) −12.0000 −0.444140
\(731\) 0 0
\(732\) 12.0000 0.443533
\(733\) −19.7990 −0.731292 −0.365646 0.930754i \(-0.619152\pi\)
−0.365646 + 0.930754i \(0.619152\pi\)
\(734\) −7.07107 −0.260998
\(735\) 0 0
\(736\) −10.0000 −0.368605
\(737\) 0 0
\(738\) −5.65685 −0.208232
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 14.1421 0.519875
\(741\) 0 0
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) −18.0000 −0.659912
\(745\) −8.48528 −0.310877
\(746\) −14.0000 −0.512576
\(747\) 11.3137 0.413947
\(748\) 0 0
\(749\) 0 0
\(750\) 16.0000 0.584237
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 9.89949 0.360997
\(753\) −22.0000 −0.801725
\(754\) 0 0
\(755\) 16.9706 0.617622
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −48.0833 −1.74302 −0.871508 0.490381i \(-0.836858\pi\)
−0.871508 + 0.490381i \(0.836858\pi\)
\(762\) −11.3137 −0.409852
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0416 0.868659
\(767\) 0 0
\(768\) 24.0416 0.867528
\(769\) −19.7990 −0.713970 −0.356985 0.934110i \(-0.616195\pi\)
−0.356985 + 0.934110i \(0.616195\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 2.00000 0.0719816
\(773\) −29.6985 −1.06818 −0.534090 0.845428i \(-0.679346\pi\)
−0.534090 + 0.845428i \(0.679346\pi\)
\(774\) −8.00000 −0.287554
\(775\) 12.7279 0.457200
\(776\) 38.1838 1.37072
\(777\) 0 0
\(778\) 0 0
\(779\) −32.0000 −1.14652
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 33.9411 1.21296
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) −28.0000 −0.998727
\(787\) 22.6274 0.806580 0.403290 0.915072i \(-0.367867\pi\)
0.403290 + 0.915072i \(0.367867\pi\)
\(788\) −18.0000 −0.641223
\(789\) 11.3137 0.402779
\(790\) 11.3137 0.402524
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 15.5563 0.552074
\(795\) 0 0
\(796\) 4.24264 0.150376
\(797\) −24.0416 −0.851598 −0.425799 0.904818i \(-0.640007\pi\)
−0.425799 + 0.904818i \(0.640007\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −15.0000 −0.530330
\(801\) 15.5563 0.549657
\(802\) 28.0000 0.988714
\(803\) 0 0
\(804\) 2.82843 0.0997509
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 16.9706 0.597022
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −7.07107 −0.248452
\(811\) −25.4558 −0.893876 −0.446938 0.894565i \(-0.647485\pi\)
−0.446938 + 0.894565i \(0.647485\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 31.1127 1.08983
\(816\) 0 0
\(817\) −45.2548 −1.58327
\(818\) −19.7990 −0.692255
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) −19.7990 −0.690569
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −29.6985 −1.03460
\(825\) 0 0
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −46.6690 −1.62088 −0.810442 0.585820i \(-0.800773\pi\)
−0.810442 + 0.585820i \(0.800773\pi\)
\(830\) −16.0000 −0.555368
\(831\) −25.4558 −0.883053
\(832\) 0 0
\(833\) 0 0
\(834\) −24.0000 −0.831052
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) −24.0000 −0.829561
\(838\) −24.0416 −0.830504
\(839\) 4.24264 0.146472 0.0732361 0.997315i \(-0.476667\pi\)
0.0732361 + 0.997315i \(0.476667\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −28.0000 −0.964944
\(843\) −31.1127 −1.07158
\(844\) −16.0000 −0.550743
\(845\) −18.3848 −0.632456
\(846\) 9.89949 0.340352
\(847\) 0 0
\(848\) 0 0
\(849\) −44.0000 −1.51008
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) 8.48528 0.290701
\(853\) −5.65685 −0.193687 −0.0968435 0.995300i \(-0.530875\pi\)
−0.0968435 + 0.995300i \(0.530875\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 24.0000 0.820303
\(857\) 2.82843 0.0966172 0.0483086 0.998832i \(-0.484617\pi\)
0.0483086 + 0.998832i \(0.484617\pi\)
\(858\) 0 0
\(859\) 21.2132 0.723785 0.361893 0.932220i \(-0.382131\pi\)
0.361893 + 0.932220i \(0.382131\pi\)
\(860\) −11.3137 −0.385794
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 28.2843 0.962250
\(865\) −16.0000 −0.544016
\(866\) −9.89949 −0.336399
\(867\) 24.0416 0.816497
\(868\) 0 0
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) 0 0
\(872\) −54.0000 −1.82867
\(873\) 12.7279 0.430775
\(874\) 11.3137 0.382692
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) −2.82843 −0.0954548
\(879\) −20.0000 −0.674583
\(880\) 0 0
\(881\) −4.24264 −0.142938 −0.0714691 0.997443i \(-0.522769\pi\)
−0.0714691 + 0.997443i \(0.522769\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) −19.7990 −0.665536
\(886\) 4.00000 0.134383
\(887\) −8.48528 −0.284908 −0.142454 0.989801i \(-0.545499\pi\)
−0.142454 + 0.989801i \(0.545499\pi\)
\(888\) −42.4264 −1.42374
\(889\) 0 0
\(890\) −22.0000 −0.737442
\(891\) 0 0
\(892\) −15.5563 −0.520865
\(893\) 56.0000 1.87397
\(894\) 8.48528 0.283790
\(895\) 16.9706 0.567263
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) −25.4558 −0.849000
\(900\) −3.00000 −0.100000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) 2.00000 0.0664822
\(906\) −16.9706 −0.563809
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −2.82843 −0.0938647
\(909\) 5.65685 0.187626
\(910\) 0 0
\(911\) −58.0000 −1.92163 −0.960813 0.277198i \(-0.910594\pi\)
−0.960813 + 0.277198i \(0.910594\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −16.9706 −0.561029
\(916\) −29.6985 −0.981266
\(917\) 0 0
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 8.48528 0.279751
\(921\) 28.0000 0.922631
\(922\) −42.4264 −1.39724
\(923\) 0 0
\(924\) 0 0
\(925\) 30.0000 0.986394
\(926\) 34.0000 1.11731
\(927\) −9.89949 −0.325142
\(928\) 30.0000 0.984798
\(929\) −35.3553 −1.15997 −0.579986 0.814627i \(-0.696942\pi\)
−0.579986 + 0.814627i \(0.696942\pi\)
\(930\) 8.48528 0.278243
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 2.00000 0.0654771
\(934\) 4.24264 0.138823
\(935\) 0 0
\(936\) 0 0
\(937\) −59.3970 −1.94041 −0.970207 0.242277i \(-0.922106\pi\)
−0.970207 + 0.242277i \(0.922106\pi\)
\(938\) 0 0
\(939\) −34.0000 −1.10955
\(940\) 14.0000 0.456630
\(941\) −36.7696 −1.19865 −0.599327 0.800505i \(-0.704565\pi\)
−0.599327 + 0.800505i \(0.704565\pi\)
\(942\) 14.0000 0.456145
\(943\) −11.3137 −0.368425
\(944\) −9.89949 −0.322201
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 11.3137 0.367452
\(949\) 0 0
\(950\) 16.9706 0.550598
\(951\) 25.4558 0.825462
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −19.7990 −0.639676
\(959\) 0 0
\(960\) −14.0000 −0.451848
\(961\) −13.0000 −0.419355
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 16.9706 0.546585
\(965\) −2.82843 −0.0910503
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −18.0000 −0.577945
\(971\) −29.6985 −0.953070 −0.476535 0.879156i \(-0.658107\pi\)
−0.476535 + 0.879156i \(0.658107\pi\)
\(972\) 9.89949 0.317526
\(973\) 0 0
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −8.48528 −0.271607
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) −31.1127 −0.994874
\(979\) 0 0
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 20.0000 0.638226
\(983\) 49.4975 1.57872 0.789362 0.613928i \(-0.210411\pi\)
0.789362 + 0.613928i \(0.210411\pi\)
\(984\) 24.0000 0.765092
\(985\) 25.4558 0.811091
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) −21.2132 −0.673520
\(993\) 16.9706 0.538545
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) −16.0000 −0.506979
\(997\) −16.9706 −0.537463 −0.268732 0.963215i \(-0.586604\pi\)
−0.268732 + 0.963215i \(0.586604\pi\)
\(998\) −26.0000 −0.823016
\(999\) −56.5685 −1.78975
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 5929.2.a.r.1.1 2
7.6 odd 2 inner 5929.2.a.r.1.2 2
11.10 odd 2 539.2.a.e.1.1 2
33.32 even 2 4851.2.a.bc.1.1 2
44.43 even 2 8624.2.a.bv.1.2 2
77.10 even 6 539.2.e.k.177.1 4
77.32 odd 6 539.2.e.k.177.2 4
77.54 even 6 539.2.e.k.67.1 4
77.65 odd 6 539.2.e.k.67.2 4
77.76 even 2 539.2.a.e.1.2 yes 2
231.230 odd 2 4851.2.a.bc.1.2 2
308.307 odd 2 8624.2.a.bv.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
539.2.a.e.1.1 2 11.10 odd 2
539.2.a.e.1.2 yes 2 77.76 even 2
539.2.e.k.67.1 4 77.54 even 6
539.2.e.k.67.2 4 77.65 odd 6
539.2.e.k.177.1 4 77.10 even 6
539.2.e.k.177.2 4 77.32 odd 6
4851.2.a.bc.1.1 2 33.32 even 2
4851.2.a.bc.1.2 2 231.230 odd 2
5929.2.a.r.1.1 2 1.1 even 1 trivial
5929.2.a.r.1.2 2 7.6 odd 2 inner
8624.2.a.bv.1.1 2 308.307 odd 2
8624.2.a.bv.1.2 2 44.43 even 2