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Properties

Label 2070.2.a.o.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2070,2,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, 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("2070.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2070.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} +4.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} +4.00000 q^{28} -8.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -4.00000 q^{35} -10.0000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -6.00000 q^{41} +6.00000 q^{43} -2.00000 q^{44} +1.00000 q^{46} +4.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} +14.0000 q^{53} +2.00000 q^{55} +4.00000 q^{56} -8.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +14.0000 q^{67} +6.00000 q^{68} -4.00000 q^{70} -10.0000 q^{71} +14.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} -8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} +6.00000 q^{86} -2.00000 q^{88} +16.0000 q^{91} +1.00000 q^{92} +4.00000 q^{94} +4.00000 q^{95} -8.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 − 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 − 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −16.0000 −1.38738
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 16.0000 1.18600
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 8.00000 0.562878
\(203\) −32.0000 −2.24596
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 24.0000 1.55569
\(239\) −26.0000 −1.68180 −0.840900 − 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 2.00000 0.126239 0.0631194 − 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −40.0000 −2.48548
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −16.0000 −0.988483
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −12.0000 −0.731653 −0.365826 − 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 − 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) −4.00000 −0.239046
\(281\) −8.00000 −0.477240 −0.238620 − 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 − 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 8.00000 0.469776
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) −26.0000 −1.51894 −0.759468 − 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 14.0000 0.793867 0.396934 − 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 − 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 − 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) 0 0
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 16.0000 0.838628
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 56.0000 2.90738
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 − 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −32.0000 −1.64808
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) 14.0000 0.709828 0.354914 − 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −28.0000 −1.40528 −0.702640 − 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −20.0000 −0.998752 −0.499376 − 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) −32.0000 −1.58813
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) 30.0000 1.46560 0.732798 − 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 14.0000 0.679900
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 32.0000 1.53782 0.768911 − 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 − 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) 28.0000 1.33032 0.665160 − 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) 16.0000 0.748448 0.374224 − 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 − 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000 1.66588 0.832941 − 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 56.0000 2.58584
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 24.0000 1.10004
\(477\) 0 0
\(478\) −26.0000 −1.18921
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −10.0000 −0.453143 −0.226572 − 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) −48.0000 −2.16181
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −40.0000 −1.79425
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 2.00000 0.0892644
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) −40.0000 −1.75750
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −14.0000 −0.612177 −0.306089 − 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −14.0000 −0.608121
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) −12.0000 −0.517357
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 − 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) 44.0000 1.84458 0.922288 − 0.386503i \(-0.126317\pi\)
0.922288 + 0.386503i \(0.126317\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 − 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 8.00000 0.332182
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) −28.0000 −1.15964
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −24.0000 −0.983904
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 4.00000 0.163572
\(599\) −14.0000 −0.572024 −0.286012 − 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 24.0000 0.978167
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 30.0000 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 − 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 14.0000 0.561349
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.00000 −0.159872
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 36.0000 1.42637
\(638\) 16.0000 0.633446
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 24.0000 0.947943 0.473972 − 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 − 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −14.0000 −0.540867
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 56.0000 2.13343
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 − 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −30.0000 −1.13552
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 32.0000 1.20348
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 − 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 10.0000 0.375293
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −46.0000 −1.71551 −0.857755 − 0.514058i \(-0.828142\pi\)
−0.857755 + 0.514058i \(0.828142\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 16.0000 0.592999
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −28.0000 −1.03139
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) 56.0000 2.05582
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −18.0000 −0.659027
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) −32.0000 −1.16537
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −56.0000 −2.02734
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 50.0000 1.78231 0.891154 − 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 40.0000 1.42224
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −54.0000 −1.91278 −0.956389 − 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −20.0000 −0.706225
\(803\) −28.0000 −0.988099
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 32.0000 1.12715
\(807\) 0 0
\(808\) 8.00000 0.281439
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) −32.0000 −1.12298
\(813\) 0 0
\(814\) 20.0000 0.701000
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −12.0000 −0.418803 −0.209401 − 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) 42.0000 1.46403 0.732014 − 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) −10.0000 −0.342796
\(852\) 0 0
\(853\) −4.00000 −0.136957 −0.0684787 − 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) 36.0000 1.22545 0.612727 − 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) 32.0000 1.08740
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 56.0000 1.89749
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) −4.00000 −0.135225
\(876\) 0 0
\(877\) 20.0000 0.675352 0.337676 − 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 48.0000 1.61716 0.808581 − 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −64.0000 −2.13452
\(900\) 0 0
\(901\) 84.0000 2.79845
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) 46.0000 1.52740 0.763702 − 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −64.0000 −2.11347
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) 0 0
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 56.0000 1.82846
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −38.0000 −1.23876 −0.619382 − 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 20.0000 0.649913 0.324956 − 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) −38.0000 −1.23094 −0.615470 − 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) −26.0000 −0.840900
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −40.0000 −1.28965
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −6.00000 −0.192549 −0.0962746 − 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) −80.0000 −2.56468
\(974\) −10.0000 −0.320421
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −10.0000 −0.319928 −0.159964 − 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) −48.0000 −1.52863
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) −40.0000 −1.26872
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 4.00000 0.126681 0.0633406 − 0.997992i \(-0.479825\pi\)
0.0633406 + 0.997992i \(0.479825\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 2070.2.a.o.1.1 1
3.2 odd 2 690.2.a.d.1.1 ✓ 1
12.11 even 2 5520.2.a.bb.1.1 1
15.2 even 4 3450.2.d.s.2899.1 2
15.8 even 4 3450.2.d.s.2899.2 2
15.14 odd 2 3450.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.d.1.1 ✓ 1 3.2 odd 2
2070.2.a.o.1.1 1 1.1 even 1 trivial
3450.2.a.u.1.1 1 15.14 odd 2
3450.2.d.s.2899.1 2 15.2 even 4
3450.2.d.s.2899.2 2 15.8 even 4
5520.2.a.bb.1.1 1 12.11 even 2