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Properties

Label 1386.2.a.g.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
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] = [1386,2,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, 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("1386.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.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: \(11.0672657201\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1386.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -8.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +2.00000 q^{35} +6.00000 q^{37} -8.00000 q^{38} -2.00000 q^{40} -6.00000 q^{41} +8.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -4.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -10.0000 q^{53} +2.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +2.00000 q^{70} +4.00000 q^{71} -14.0000 q^{73} +6.00000 q^{74} -8.00000 q^{76} +1.00000 q^{77} -8.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -4.00000 q^{83} +12.0000 q^{85} +8.00000 q^{86} -1.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -4.00000 q^{94} +16.0000 q^{95} +18.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 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
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 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
\(130\) −4.00000 −0.350823
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 4.00000 0.335673
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −16.0000 −1.28515
\(156\) 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
\(160\) −2.00000 −0.158114
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −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
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 16.0000 1.16076
\(191\) 4.00000 0.289430 0.144715 − 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −14.0000 −0.896258
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 − 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 − 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 28.0000 1.60328
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −16.0000 −0.908739
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 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
\(319\) 2.00000 0.111979
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 12.0000 0.650791
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 36.0000 1.93258 0.966291 − 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −26.0000 −1.38384 −0.691920 − 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 14.0000 0.741999
\(357\) 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
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 28.0000 1.46559
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 16.0000 0.820783
\(381\) 0 0
\(382\) 4.00000 0.204658
\(383\) 28.0000 1.43073 0.715367 − 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) 18.0000 0.913812
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −34.0000 −1.69788 −0.848939 − 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) −40.0000 −1.90910 −0.954548 − 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −28.0000 −1.32733
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −2.00000 −0.0931493 −0.0465746 − 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −36.0000 −1.63468
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 − 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) −14.0000 −0.633750
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) −6.00000 −0.263625
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 20.0000 0.868744
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 − 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 − 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 20.0000 0.841406
\(566\) 0 0
\(567\) 0 0
\(568\) 4.00000 0.167836
\(569\) −38.0000 −1.59304 −0.796521 − 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −64.0000 −2.63707
\(590\) 8.00000 0.329355
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 − 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 46.0000 1.85189 0.925945 − 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 6.00000 0.236986 0.118493 − 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 − 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 4.00000 0.155936
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 − 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 4.00000 0.155464
\(663\) 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
\(670\) 8.00000 0.309067
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −18.0000 −0.690777
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) 32.0000 1.21383
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −48.0000 −1.81035
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −12.0000 −0.447524 −0.223762 − 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 45.0000 1.67473
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 28.0000 1.03633
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) −40.0000 −1.46746 −0.733729 − 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 − 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 − 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 16.0000 0.580381
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 − 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) 22.0000 0.791285 0.395643 − 0.918405i \(-0.370522\pi\)
0.395643 + 0.918405i \(0.370522\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 − 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 16.0000 0.569254
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 22.0000 0.779280 0.389640 − 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −34.0000 −1.20058
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 56.0000 1.96643 0.983213 − 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −64.0000 −2.23908
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 − 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) 44.0000 1.49778 0.748889 − 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) 0 0
\(865\) −28.0000 −0.952029
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −40.0000 −1.34993
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −18.0000 −0.606435 −0.303218 − 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 − 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −32.0000 −1.07445 −0.537227 − 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −28.0000 −0.938562
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) −20.0000 −0.662630 −0.331315 − 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −2.00000 −0.0658665
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 22.0000 0.721797 0.360898 − 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) −22.0000 −0.712650 −0.356325 − 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −36.0000 −1.15589
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) −14.0000 −0.447442
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) −20.0000 −0.638226
\(983\) 4.00000 0.127580 0.0637901 − 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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 1386.2.a.g.1.1 1
3.2 odd 2 462.2.a.c.1.1 ✓ 1
7.6 odd 2 9702.2.a.by.1.1 1
12.11 even 2 3696.2.a.bb.1.1 1
21.20 even 2 3234.2.a.i.1.1 1
33.32 even 2 5082.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.c.1.1 ✓ 1 3.2 odd 2
1386.2.a.g.1.1 1 1.1 even 1 trivial
3234.2.a.i.1.1 1 21.20 even 2
3696.2.a.bb.1.1 1 12.11 even 2
5082.2.a.v.1.1 1 33.32 even 2
9702.2.a.by.1.1 1 7.6 odd 2