Properties

Label 9702.2.a.df.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +2.82843 q^{10} -1.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} -1.41421 q^{19} +2.82843 q^{20} -1.00000 q^{22} -6.00000 q^{23} +3.00000 q^{25} -4.24264 q^{26} -8.00000 q^{29} +1.41421 q^{31} +1.00000 q^{32} -2.82843 q^{34} -6.00000 q^{37} -1.41421 q^{38} +2.82843 q^{40} +8.48528 q^{41} +10.0000 q^{43} -1.00000 q^{44} -6.00000 q^{46} -7.07107 q^{47} +3.00000 q^{50} -4.24264 q^{52} -6.00000 q^{53} -2.82843 q^{55} -8.00000 q^{58} -14.1421 q^{59} +4.24264 q^{61} +1.41421 q^{62} +1.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} -2.82843 q^{68} -8.48528 q^{73} -6.00000 q^{74} -1.41421 q^{76} +2.82843 q^{80} +8.48528 q^{82} -7.07107 q^{83} -8.00000 q^{85} +10.0000 q^{86} -1.00000 q^{88} +18.3848 q^{89} -6.00000 q^{92} -7.07107 q^{94} -4.00000 q^{95} +1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} - 12 q^{23} + 6 q^{25} - 16 q^{29} + 2 q^{32} - 12 q^{37} + 20 q^{43} - 2 q^{44} - 12 q^{46} + 6 q^{50} - 12 q^{53} - 16 q^{58} + 2 q^{64} - 24 q^{65} - 8 q^{67} - 12 q^{74} - 16 q^{85} + 20 q^{86} - 2 q^{88} - 12 q^{92} - 8 q^{95}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.82843 0.894427
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) −1.41421 −0.324443 −0.162221 0.986754i \(-0.551866\pi\)
−0.162221 + 0.986754i \(0.551866\pi\)
\(20\) 2.82843 0.632456
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) −4.24264 −0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −1.41421 −0.229416
\(39\) 0 0
\(40\) 2.82843 0.447214
\(41\) 8.48528 1.32518 0.662589 0.748983i \(-0.269458\pi\)
0.662589 + 0.748983i \(0.269458\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −7.07107 −1.03142 −0.515711 0.856763i \(-0.672472\pi\)
−0.515711 + 0.856763i \(0.672472\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 3.00000 0.424264
\(51\) 0 0
\(52\) −4.24264 −0.588348
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −2.82843 −0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) −14.1421 −1.84115 −0.920575 0.390567i \(-0.872279\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 4.24264 0.543214 0.271607 0.962408i \(-0.412445\pi\)
0.271607 + 0.962408i \(0.412445\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.82843 −0.342997
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −1.41421 −0.162221
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.82843 0.316228
\(81\) 0 0
\(82\) 8.48528 0.937043
\(83\) −7.07107 −0.776151 −0.388075 0.921628i \(-0.626860\pi\)
−0.388075 + 0.921628i \(0.626860\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 18.3848 1.94878 0.974391 0.224860i \(-0.0721923\pi\)
0.974391 + 0.224860i \(0.0721923\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −7.07107 −0.729325
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 1.41421 0.143592 0.0717958 0.997419i \(-0.477127\pi\)
0.0717958 + 0.997419i \(0.477127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) −1.41421 −0.140720 −0.0703598 0.997522i \(-0.522415\pi\)
−0.0703598 + 0.997522i \(0.522415\pi\)
\(102\) 0 0
\(103\) −4.24264 −0.418040 −0.209020 0.977911i \(-0.567027\pi\)
−0.209020 + 0.977911i \(0.567027\pi\)
\(104\) −4.24264 −0.416025
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −2.82843 −0.269680
\(111\) 0 0
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) −16.9706 −1.58251
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) −14.1421 −1.30189
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 4.24264 0.384111
\(123\) 0 0
\(124\) 1.41421 0.127000
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) 7.07107 0.617802 0.308901 0.951094i \(-0.400039\pi\)
0.308901 + 0.951094i \(0.400039\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −2.82843 −0.242536
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 21.2132 1.79928 0.899640 0.436632i \(-0.143829\pi\)
0.899640 + 0.436632i \(0.143829\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.24264 0.354787
\(144\) 0 0
\(145\) −22.6274 −1.87910
\(146\) −8.48528 −0.702247
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) −1.41421 −0.114708
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 2.82843 0.223607
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 8.48528 0.662589
\(165\) 0 0
\(166\) −7.07107 −0.548821
\(167\) −14.1421 −1.09435 −0.547176 0.837018i \(-0.684297\pi\)
−0.547176 + 0.837018i \(0.684297\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) 10.0000 0.762493
\(173\) −15.5563 −1.18273 −0.591364 0.806405i \(-0.701410\pi\)
−0.591364 + 0.806405i \(0.701410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 18.3848 1.37800
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) −16.9706 −1.24770
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) −7.07107 −0.515711
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 22.0000 1.59186 0.795932 0.605386i \(-0.206981\pi\)
0.795932 + 0.605386i \(0.206981\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 1.41421 0.101535
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −4.24264 −0.300753 −0.150376 0.988629i \(-0.548049\pi\)
−0.150376 + 0.988629i \(0.548049\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) −1.41421 −0.0995037
\(203\) 0 0
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) −4.24264 −0.295599
\(207\) 0 0
\(208\) −4.24264 −0.294174
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 28.2843 1.92897
\(216\) 0 0
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) −2.82843 −0.190693
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 12.7279 0.852325 0.426162 0.904647i \(-0.359865\pi\)
0.426162 + 0.904647i \(0.359865\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) 26.8701 1.78343 0.891714 0.452599i \(-0.149503\pi\)
0.891714 + 0.452599i \(0.149503\pi\)
\(228\) 0 0
\(229\) 25.4558 1.68217 0.841085 0.540903i \(-0.181918\pi\)
0.841085 + 0.540903i \(0.181918\pi\)
\(230\) −16.9706 −1.11901
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) −20.0000 −1.30466
\(236\) −14.1421 −0.920575
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 2.82843 0.182195 0.0910975 0.995842i \(-0.470963\pi\)
0.0910975 + 0.995842i \(0.470963\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 4.24264 0.271607
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 1.41421 0.0898027
\(249\) 0 0
\(250\) −5.65685 −0.357771
\(251\) 25.4558 1.60676 0.803379 0.595468i \(-0.203033\pi\)
0.803379 + 0.595468i \(0.203033\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.3848 1.14681 0.573405 0.819272i \(-0.305622\pi\)
0.573405 + 0.819272i \(0.305622\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 7.07107 0.436852
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −16.9706 −1.04249
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 25.4558 1.55207 0.776035 0.630690i \(-0.217228\pi\)
0.776035 + 0.630690i \(0.217228\pi\)
\(270\) 0 0
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) −2.82843 −0.171499
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 21.2132 1.27228
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.24264 0.252199 0.126099 0.992018i \(-0.459754\pi\)
0.126099 + 0.992018i \(0.459754\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.24264 0.250873
\(287\) 0 0
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) −22.6274 −1.32873
\(291\) 0 0
\(292\) −8.48528 −0.496564
\(293\) −15.5563 −0.908812 −0.454406 0.890795i \(-0.650148\pi\)
−0.454406 + 0.890795i \(0.650148\pi\)
\(294\) 0 0
\(295\) −40.0000 −2.32889
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 25.4558 1.47215
\(300\) 0 0
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) −1.41421 −0.0811107
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 24.0416 1.37213 0.686064 0.727541i \(-0.259337\pi\)
0.686064 + 0.727541i \(0.259337\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −24.0416 −1.36328 −0.681638 0.731690i \(-0.738732\pi\)
−0.681638 + 0.731690i \(0.738732\pi\)
\(312\) 0 0
\(313\) 12.7279 0.719425 0.359712 0.933063i \(-0.382875\pi\)
0.359712 + 0.933063i \(0.382875\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 2.82843 0.158114
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) −12.7279 −0.706018
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 8.48528 0.468521
\(329\) 0 0
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) −7.07107 −0.388075
\(333\) 0 0
\(334\) −14.1421 −0.773823
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 5.00000 0.271964
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) −1.41421 −0.0765840
\(342\) 0 0
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −15.5563 −0.836315
\(347\) −10.0000 −0.536828 −0.268414 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) 0 0
\(349\) 9.89949 0.529908 0.264954 0.964261i \(-0.414643\pi\)
0.264954 + 0.964261i \(0.414643\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −29.6985 −1.58069 −0.790345 0.612661i \(-0.790099\pi\)
−0.790345 + 0.612661i \(0.790099\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 18.3848 0.974391
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 8.48528 0.445976
\(363\) 0 0
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 0 0
\(367\) −24.0416 −1.25496 −0.627481 0.778632i \(-0.715914\pi\)
−0.627481 + 0.778632i \(0.715914\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −16.9706 −0.882258
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) −7.07107 −0.364662
\(377\) 33.9411 1.74806
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 22.0000 1.12562
\(383\) 35.3553 1.80657 0.903287 0.429037i \(-0.141147\pi\)
0.903287 + 0.429037i \(0.141147\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 1.41421 0.0717958
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 16.9706 0.858238
\(392\) 0 0
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) 11.3137 0.567819 0.283909 0.958851i \(-0.408369\pi\)
0.283909 + 0.958851i \(0.408369\pi\)
\(398\) −4.24264 −0.212664
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) −1.41421 −0.0703598
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 28.2843 1.39857 0.699284 0.714844i \(-0.253502\pi\)
0.699284 + 0.714844i \(0.253502\pi\)
\(410\) 24.0000 1.18528
\(411\) 0 0
\(412\) −4.24264 −0.209020
\(413\) 0 0
\(414\) 0 0
\(415\) −20.0000 −0.981761
\(416\) −4.24264 −0.208013
\(417\) 0 0
\(418\) 1.41421 0.0691714
\(419\) −22.6274 −1.10542 −0.552711 0.833373i \(-0.686407\pi\)
−0.552711 + 0.833373i \(0.686407\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −8.48528 −0.411597
\(426\) 0 0
\(427\) 0 0
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 28.2843 1.36399
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) −24.0416 −1.15537 −0.577684 0.816261i \(-0.696043\pi\)
−0.577684 + 0.816261i \(0.696043\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 8.48528 0.405906
\(438\) 0 0
\(439\) 31.1127 1.48493 0.742464 0.669886i \(-0.233657\pi\)
0.742464 + 0.669886i \(0.233657\pi\)
\(440\) −2.82843 −0.134840
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 52.0000 2.46504
\(446\) 12.7279 0.602685
\(447\) 0 0
\(448\) 0 0
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) −8.48528 −0.399556
\(452\) −8.00000 −0.376288
\(453\) 0 0
\(454\) 26.8701 1.26107
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 25.4558 1.18947
\(459\) 0 0
\(460\) −16.9706 −0.791257
\(461\) 18.3848 0.856264 0.428132 0.903716i \(-0.359172\pi\)
0.428132 + 0.903716i \(0.359172\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −2.00000 −0.0926482
\(467\) −33.9411 −1.57061 −0.785304 0.619110i \(-0.787493\pi\)
−0.785304 + 0.619110i \(0.787493\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −20.0000 −0.922531
\(471\) 0 0
\(472\) −14.1421 −0.650945
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) −4.24264 −0.194666
\(476\) 0 0
\(477\) 0 0
\(478\) 4.00000 0.182956
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 0 0
\(481\) 25.4558 1.16069
\(482\) 2.82843 0.128831
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 4.24264 0.192055
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 22.6274 1.01909
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 1.41421 0.0635001
\(497\) 0 0
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) −5.65685 −0.252982
\(501\) 0 0
\(502\) 25.4558 1.13615
\(503\) −14.1421 −0.630567 −0.315283 0.948998i \(-0.602100\pi\)
−0.315283 + 0.948998i \(0.602100\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 6.00000 0.266733
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −31.1127 −1.37905 −0.689523 0.724264i \(-0.742180\pi\)
−0.689523 + 0.724264i \(0.742180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.3848 0.810918
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) 7.07107 0.310985
\(518\) 0 0
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) −41.0122 −1.79678 −0.898388 0.439202i \(-0.855261\pi\)
−0.898388 + 0.439202i \(0.855261\pi\)
\(522\) 0 0
\(523\) 18.3848 0.803910 0.401955 0.915659i \(-0.368331\pi\)
0.401955 + 0.915659i \(0.368331\pi\)
\(524\) 7.07107 0.308901
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −16.9706 −0.737154
\(531\) 0 0
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) −50.9117 −2.20110
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 25.4558 1.09748
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.48528 0.364474
\(543\) 0 0
\(544\) −2.82843 −0.121268
\(545\) 39.5980 1.69619
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) 11.3137 0.481980
\(552\) 0 0
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 21.2132 0.899640
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) −42.4264 −1.79445
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −9.89949 −0.417214 −0.208607 0.978000i \(-0.566893\pi\)
−0.208607 + 0.978000i \(0.566893\pi\)
\(564\) 0 0
\(565\) −22.6274 −0.951943
\(566\) 4.24264 0.178331
\(567\) 0 0
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 4.24264 0.177394
\(573\) 0 0
\(574\) 0 0
\(575\) −18.0000 −0.750652
\(576\) 0 0
\(577\) −12.7279 −0.529870 −0.264935 0.964266i \(-0.585351\pi\)
−0.264935 + 0.964266i \(0.585351\pi\)
\(578\) −9.00000 −0.374351
\(579\) 0 0
\(580\) −22.6274 −0.939552
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −8.48528 −0.351123
\(585\) 0 0
\(586\) −15.5563 −0.642627
\(587\) −19.7990 −0.817192 −0.408596 0.912715i \(-0.633981\pi\)
−0.408596 + 0.912715i \(0.633981\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) −40.0000 −1.64677
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 2.82843 0.116150 0.0580748 0.998312i \(-0.481504\pi\)
0.0580748 + 0.998312i \(0.481504\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) 25.4558 1.04097
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) 2.82843 0.114992
\(606\) 0 0
\(607\) 33.9411 1.37763 0.688814 0.724938i \(-0.258132\pi\)
0.688814 + 0.724938i \(0.258132\pi\)
\(608\) −1.41421 −0.0573539
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 30.0000 1.21367
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 24.0416 0.970241
\(615\) 0 0
\(616\) 0 0
\(617\) 4.00000 0.161034 0.0805170 0.996753i \(-0.474343\pi\)
0.0805170 + 0.996753i \(0.474343\pi\)
\(618\) 0 0
\(619\) −28.2843 −1.13684 −0.568420 0.822738i \(-0.692445\pi\)
−0.568420 + 0.822738i \(0.692445\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −24.0416 −0.963982
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 12.7279 0.508710
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9706 0.676661
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) 22.6274 0.897942
\(636\) 0 0
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 2.82843 0.111803
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) −25.4558 −1.00388 −0.501940 0.864902i \(-0.667380\pi\)
−0.501940 + 0.864902i \(0.667380\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) −7.07107 −0.277992 −0.138996 0.990293i \(-0.544388\pi\)
−0.138996 + 0.990293i \(0.544388\pi\)
\(648\) 0 0
\(649\) 14.1421 0.555127
\(650\) −12.7279 −0.499230
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 8.48528 0.331295
\(657\) 0 0
\(658\) 0 0
\(659\) −46.0000 −1.79191 −0.895953 0.444149i \(-0.853506\pi\)
−0.895953 + 0.444149i \(0.853506\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) −7.07107 −0.274411
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) −14.1421 −0.547176
\(669\) 0 0
\(670\) −11.3137 −0.437087
\(671\) −4.24264 −0.163785
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 5.00000 0.192308
\(677\) −1.41421 −0.0543526 −0.0271763 0.999631i \(-0.508652\pi\)
−0.0271763 + 0.999631i \(0.508652\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) −1.41421 −0.0541530
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) −28.2843 −1.08069
\(686\) 0 0
\(687\) 0 0
\(688\) 10.0000 0.381246
\(689\) 25.4558 0.969790
\(690\) 0 0
\(691\) −8.48528 −0.322795 −0.161398 0.986889i \(-0.551600\pi\)
−0.161398 + 0.986889i \(0.551600\pi\)
\(692\) −15.5563 −0.591364
\(693\) 0 0
\(694\) −10.0000 −0.379595
\(695\) 60.0000 2.27593
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 9.89949 0.374701
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −29.6985 −1.11772
\(707\) 0 0
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.3848 0.688999
\(713\) −8.48528 −0.317776
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 4.24264 0.158224 0.0791119 0.996866i \(-0.474792\pi\)
0.0791119 + 0.996866i \(0.474792\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 8.48528 0.315353
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) −15.5563 −0.576953 −0.288477 0.957487i \(-0.593149\pi\)
−0.288477 + 0.957487i \(0.593149\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −24.0000 −0.888280
\(731\) −28.2843 −1.04613
\(732\) 0 0
\(733\) −12.7279 −0.470117 −0.235058 0.971981i \(-0.575528\pi\)
−0.235058 + 0.971981i \(0.575528\pi\)
\(734\) −24.0416 −0.887393
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −16.9706 −0.623850
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −11.3137 −0.414502
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) 2.82843 0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) −7.07107 −0.257855
\(753\) 0 0
\(754\) 33.9411 1.23606
\(755\) −56.5685 −2.05874
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 22.6274 0.820243 0.410122 0.912031i \(-0.365486\pi\)
0.410122 + 0.912031i \(0.365486\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 22.0000 0.795932
\(765\) 0 0
\(766\) 35.3553 1.27744
\(767\) 60.0000 2.16647
\(768\) 0 0
\(769\) −16.9706 −0.611974 −0.305987 0.952036i \(-0.598986\pi\)
−0.305987 + 0.952036i \(0.598986\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −22.6274 −0.813852 −0.406926 0.913461i \(-0.633399\pi\)
−0.406926 + 0.913461i \(0.633399\pi\)
\(774\) 0 0
\(775\) 4.24264 0.152400
\(776\) 1.41421 0.0507673
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 16.9706 0.606866
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −15.5563 −0.554524 −0.277262 0.960794i \(-0.589427\pi\)
−0.277262 + 0.960794i \(0.589427\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 11.3137 0.401508
\(795\) 0 0
\(796\) −4.24264 −0.150376
\(797\) −39.5980 −1.40263 −0.701316 0.712850i \(-0.747404\pi\)
−0.701316 + 0.712850i \(0.747404\pi\)
\(798\) 0 0
\(799\) 20.0000 0.707549
\(800\) 3.00000 0.106066
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 8.48528 0.299439
\(804\) 0 0
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) 0 0
\(808\) −1.41421 −0.0497519
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −18.3848 −0.645577 −0.322788 0.946471i \(-0.604620\pi\)
−0.322788 + 0.946471i \(0.604620\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) −45.2548 −1.58521
\(816\) 0 0
\(817\) −14.1421 −0.494771
\(818\) 28.2843 0.988936
\(819\) 0 0
\(820\) 24.0000 0.838116
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) 0 0
\(823\) −42.0000 −1.46403 −0.732014 0.681290i \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(824\) −4.24264 −0.147799
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −25.4558 −0.884118 −0.442059 0.896986i \(-0.645752\pi\)
−0.442059 + 0.896986i \(0.645752\pi\)
\(830\) −20.0000 −0.694210
\(831\) 0 0
\(832\) −4.24264 −0.147087
\(833\) 0 0
\(834\) 0 0
\(835\) −40.0000 −1.38426
\(836\) 1.41421 0.0489116
\(837\) 0 0
\(838\) −22.6274 −0.781651
\(839\) −15.5563 −0.537065 −0.268532 0.963271i \(-0.586539\pi\)
−0.268532 + 0.963271i \(0.586539\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) −14.0000 −0.481900
\(845\) 14.1421 0.486504
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −8.48528 −0.291043
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) 38.1838 1.30739 0.653694 0.756759i \(-0.273219\pi\)
0.653694 + 0.756759i \(0.273219\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 22.6274 0.772938 0.386469 0.922302i \(-0.373695\pi\)
0.386469 + 0.922302i \(0.373695\pi\)
\(858\) 0 0
\(859\) 19.7990 0.675533 0.337766 0.941230i \(-0.390329\pi\)
0.337766 + 0.941230i \(0.390329\pi\)
\(860\) 28.2843 0.964486
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −46.0000 −1.56586 −0.782929 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 0 0
\(865\) −44.0000 −1.49604
\(866\) −24.0416 −0.816968
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.9706 0.575026
\(872\) 14.0000 0.474100
\(873\) 0 0
\(874\) 8.48528 0.287019
\(875\) 0 0
\(876\) 0 0
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 31.1127 1.05000
\(879\) 0 0
\(880\) −2.82843 −0.0953463
\(881\) 43.8406 1.47703 0.738514 0.674238i \(-0.235528\pi\)
0.738514 + 0.674238i \(0.235528\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 5.65685 0.189939 0.0949693 0.995480i \(-0.469725\pi\)
0.0949693 + 0.995480i \(0.469725\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 52.0000 1.74304
\(891\) 0 0
\(892\) 12.7279 0.426162
\(893\) 10.0000 0.334637
\(894\) 0 0
\(895\) 33.9411 1.13453
\(896\) 0 0
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) 16.9706 0.565371
\(902\) −8.48528 −0.282529
\(903\) 0 0
\(904\) −8.00000 −0.266076
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 26.8701 0.891714
\(909\) 0 0
\(910\) 0 0
\(911\) 10.0000 0.331315 0.165657 0.986183i \(-0.447025\pi\)
0.165657 + 0.986183i \(0.447025\pi\)
\(912\) 0 0
\(913\) 7.07107 0.234018
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 25.4558 0.841085
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −16.9706 −0.559503
\(921\) 0 0
\(922\) 18.3848 0.605470
\(923\) 0 0
\(924\) 0 0
\(925\) −18.0000 −0.591836
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) 1.41421 0.0463988 0.0231994 0.999731i \(-0.492615\pi\)
0.0231994 + 0.999731i \(0.492615\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) 0 0
\(934\) −33.9411 −1.11059
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 19.7990 0.646805 0.323402 0.946262i \(-0.395173\pi\)
0.323402 + 0.946262i \(0.395173\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −20.0000 −0.652328
\(941\) 38.1838 1.24476 0.622378 0.782717i \(-0.286167\pi\)
0.622378 + 0.782717i \(0.286167\pi\)
\(942\) 0 0
\(943\) −50.9117 −1.65791
\(944\) −14.1421 −0.460287
\(945\) 0 0
\(946\) −10.0000 −0.325128
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) −4.24264 −0.137649
\(951\) 0 0
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) 62.2254 2.01357
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) −11.3137 −0.365529
\(959\) 0 0
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 25.4558 0.820729
\(963\) 0 0
\(964\) 2.82843 0.0910975
\(965\) −39.5980 −1.27470
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −11.3137 −0.363074 −0.181537 0.983384i \(-0.558107\pi\)
−0.181537 + 0.983384i \(0.558107\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 4.24264 0.135804
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) 0 0
\(979\) −18.3848 −0.587580
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −21.2132 −0.676596 −0.338298 0.941039i \(-0.609851\pi\)
−0.338298 + 0.941039i \(0.609851\pi\)
\(984\) 0 0
\(985\) −28.2843 −0.901212
\(986\) 22.6274 0.720604
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 1.41421 0.0449013
\(993\) 0 0
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) −57.9828 −1.83633 −0.918166 0.396196i \(-0.870330\pi\)
−0.918166 + 0.396196i \(0.870330\pi\)
\(998\) 16.0000 0.506471
\(999\) 0 0
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 9702.2.a.df.1.2 2
3.2 odd 2 1078.2.a.p.1.1 2
7.6 odd 2 inner 9702.2.a.df.1.1 2
12.11 even 2 8624.2.a.bl.1.1 2
21.2 odd 6 1078.2.e.t.67.2 4
21.5 even 6 1078.2.e.t.67.1 4
21.11 odd 6 1078.2.e.t.177.2 4
21.17 even 6 1078.2.e.t.177.1 4
21.20 even 2 1078.2.a.p.1.2 yes 2
84.83 odd 2 8624.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.p.1.1 2 3.2 odd 2
1078.2.a.p.1.2 yes 2 21.20 even 2
1078.2.e.t.67.1 4 21.5 even 6
1078.2.e.t.67.2 4 21.2 odd 6
1078.2.e.t.177.1 4 21.17 even 6
1078.2.e.t.177.2 4 21.11 odd 6
8624.2.a.bl.1.1 2 12.11 even 2
8624.2.a.bl.1.2 2 84.83 odd 2
9702.2.a.df.1.1 2 7.6 odd 2 inner
9702.2.a.df.1.2 2 1.1 even 1 trivial