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Properties

Label 5550.2.a.br.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +4.00000 q^{21} +4.00000 q^{22} +1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +8.00000 q^{38} -2.00000 q^{39} -6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} +12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} -6.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{72} +6.00000 q^{73} +1.00000 q^{74} +8.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} -14.0000 q^{89} -8.00000 q^{91} -8.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000 0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 8.00000 1.29777
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 8.00000 1.05963
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 16.0000 1.82337
\(78\) −2.00000 −0.226455
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 4.00000 0.377964
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 4.00000 0.348155
\(133\) 32.0000 2.77475
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 9.00000 0.742307
\(148\) 1.00000 0.0821995
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 8.00000 0.648886
\(153\) 2.00000 0.161690
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 22.0000 1.75579 0.877896 − 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 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 \(0.399829\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 24.0000 1.79384 0.896922 − 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −8.00000 −0.592999
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 8.00000 0.585018
\(188\) 12.0000 0.875190
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) −10.0000 −0.703598
\(203\) −24.0000 −1.68447
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −32.0000 −2.17230
\(218\) 2.00000 0.135457
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 1.00000 0.0671156
\(223\) 12.0000 0.803579 0.401790 − 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 8.00000 0.529813
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 8.00000 0.518563
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −16.0000 −1.01806
\(248\) −8.00000 −0.508001
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) −16.0000 −0.988483
\(263\) 4.00000 0.246651 0.123325 − 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 32.0000 1.96205
\(267\) −14.0000 −0.856786
\(268\) −12.0000 −0.733017
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) −8.00000 −0.484182
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 − 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 12.0000 0.714590
\(283\) −12.0000 −0.713326 −0.356663 − 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 6.00000 0.351123
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 4.00000 0.232104
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 16.0000 0.911685
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −2.00000 −0.113228
\(313\) −22.0000 −1.24351 −0.621757 − 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 6.00000 0.336463
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −4.00000 −0.219529
\(333\) 1.00000 0.0547997
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 8.00000 0.432590
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) 26.0000 1.38384 0.691920 − 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 8.00000 0.423405
\(358\) 24.0000 1.26844
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 14.0000 0.735824
\(363\) 5.00000 0.262432
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) −8.00000 −0.414781
\(373\) −34.0000 −1.76045 −0.880227 − 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 12.0000 0.618031
\(378\) 4.00000 0.205738
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 24.0000 1.22795
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −4.00000 −0.203331
\(388\) −6.00000 −0.304604
\(389\) 34.0000 1.72387 0.861934 − 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) −16.0000 −0.807093
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 32.0000 1.60200
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −12.0000 −0.598506
\(403\) 16.0000 0.797017
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) 4.00000 0.198273
\(408\) 2.00000 0.0990148
\(409\) 2.00000 0.0988936 0.0494468 − 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) 32.0000 1.56517
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −20.0000 −0.973585
\(423\) 12.0000 0.583460
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) −4.00000 −0.193347
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −4.00000 −0.190261
\(443\) −20.0000 −0.950229 −0.475114 − 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 12.0000 0.568216
\(447\) 6.00000 0.283790
\(448\) 4.00000 0.188982
\(449\) −22.0000 −1.03824 −0.519122 − 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 18.0000 0.846649
\(453\) −16.0000 −0.751746
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −14.0000 −0.654892 −0.327446 − 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 14.0000 0.654177
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 − 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 16.0000 0.744387
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −48.0000 −2.21643
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −6.00000 −0.271607
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) −12.0000 −0.540453
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −16.0000 −0.714115
\(503\) −8.00000 −0.356702 −0.178351 − 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −4.00000 −0.177471
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 48.0000 2.11104
\(518\) 4.00000 0.175750
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −6.00000 −0.262613
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) −16.0000 −0.696971
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 32.0000 1.38738
\(533\) 12.0000 0.519778
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 24.0000 1.03568
\(538\) 30.0000 1.29339
\(539\) 36.0000 1.55063
\(540\) 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
\(543\) 14.0000 0.600798
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −4.00000 −0.171028 −0.0855138 − 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.00000 0.256307
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −14.0000 −0.590554
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 24.0000 1.00261
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) −6.00000 −0.248708
\(583\) 24.0000 0.993978
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 9.00000 0.371154
\(589\) −64.0000 −2.63707
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 1.00000 0.0410997
\(593\) −2.00000 −0.0821302 −0.0410651 − 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 − 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) −16.0000 −0.652111
\(603\) −12.0000 −0.488678
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −40.0000 −1.62355 −0.811775 − 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 8.00000 0.324443
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 2.00000 0.0808452
\(613\) 22.0000 0.888572 0.444286 − 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −56.0000 −2.24359
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 32.0000 1.27796
\(628\) 22.0000 0.877896
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −8.00000 −0.318223
\(633\) −20.0000 −0.794929
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −18.0000 −0.713186
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −4.00000 −0.157867
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) 8.00000 0.314512 0.157256 − 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 12.0000 0.469956
\(653\) −34.0000 −1.33052 −0.665261 − 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000 0.234082
\(658\) 48.0000 1.87123
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 − 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −8.00000 −0.310929
\(663\) −4.00000 −0.155347
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 4.00000 0.154303
\(673\) 6.00000 0.231283 0.115642 − 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −34.0000 −1.30673 −0.653363 − 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 18.0000 0.691286
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) −32.0000 −1.22534
\(683\) −20.0000 −0.765279 −0.382639 − 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 16.0000 0.607790
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −12.0000 −0.454532
\(698\) −2.00000 −0.0757011
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 − 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 8.00000 0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −40.0000 −1.50435
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 − 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −14.0000 −0.524672
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) −8.00000 −0.298765
\(718\) 24.0000 0.895672
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 45.0000 1.67473
\(723\) −14.0000 −0.520666
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) −6.00000 −0.221766
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) −6.00000 −0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 24.0000 0.881068
\(743\) −36.0000 −1.32071 −0.660356 − 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) −4.00000 −0.146352
\(748\) 8.00000 0.292509
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 − 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 12.0000 0.437595
\(753\) −16.0000 −0.583072
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −36.0000 −1.30758
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −4.00000 −0.144905
\(763\) 8.00000 0.289619
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 18.0000 0.649097 0.324548 − 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −14.0000 −0.503871
\(773\) 54.0000 1.94225 0.971123 − 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 4.00000 0.143499
\(778\) 34.0000 1.21896
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −16.0000 −0.570701
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 4.00000 0.142134
\(793\) 12.0000 0.426132
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 32.0000 1.13279
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) −6.00000 −0.211867
\(803\) 24.0000 0.846942
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 30.0000 1.05605
\(808\) −10.0000 −0.351799
\(809\) 26.0000 0.914111 0.457056 − 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −24.0000 −0.842235
\(813\) 8.00000 0.280572
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −32.0000 −1.11954
\(818\) 2.00000 0.0699284
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 6.00000 0.209274
\(823\) 20.0000 0.697156 0.348578 − 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 44.0000 1.53003 0.765015 − 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 32.0000 1.10674
\(837\) −8.00000 −0.276520
\(838\) −12.0000 −0.414533
\(839\) −56.0000 −1.93333 −0.966667 − 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −22.0000 −0.758170
\(843\) −14.0000 −0.482186
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 20.0000 0.687208
\(848\) 6.00000 0.206041
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −46.0000 −1.57133 −0.785665 − 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) −8.00000 −0.273115
\(859\) −32.0000 −1.09183 −0.545913 − 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 24.0000 0.817443
\(863\) −12.0000 −0.408485 −0.204242 − 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 30.0000 1.01944
\(867\) −13.0000 −0.441503
\(868\) −32.0000 −1.08615
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 2.00000 0.0677285
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −58.0000 −1.95852 −0.979260 − 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 16.0000 0.539974
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 9.00000 0.303046
\(883\) −44.0000 −1.48072 −0.740359 − 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 52.0000 1.74599 0.872995 − 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) 1.00000 0.0335578
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 12.0000 0.401790
\(893\) 96.0000 3.21252
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) −16.0000 −0.532447
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) −20.0000 −0.664089 −0.332045 − 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −12.0000 −0.398234
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 8.00000 0.264906
\(913\) −16.0000 −0.529523
\(914\) −14.0000 −0.463079
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −64.0000 −2.11347
\(918\) 2.00000 0.0660098
\(919\) 56.0000 1.84727 0.923635 − 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) −38.0000 −1.25146
\(923\) 0 0
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) −8.00000 −0.262754
\(928\) −6.00000 −0.196960
\(929\) −46.0000 −1.50921 −0.754606 − 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) 6.00000 0.196537
\(933\) −24.0000 −0.785725
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −48.0000 −1.56726
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 − 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −8.00000 −0.259828
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 8.00000 0.259281
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −24.0000 −0.775810
\(958\) 24.0000 0.775405
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −2.00000 −0.0644826
\(963\) −4.00000 −0.128898
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 5.00000 0.160706
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) 4.00000 0.128366 0.0641831 − 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 50.0000 1.59964 0.799821 − 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) 12.0000 0.383718
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) −36.0000 −1.14822 −0.574111 − 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 48.0000 1.52786
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 − 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) −8.00000 −0.254000
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 32.0000 1.01294
\(999\) 1.00000 0.0316386
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 5550.2.a.br.1.1 1
5.4 even 2 1110.2.a.a.1.1 ✓ 1
15.14 odd 2 3330.2.a.s.1.1 1
20.19 odd 2 8880.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.a.1.1 ✓ 1 5.4 even 2
3330.2.a.s.1.1 1 15.14 odd 2
5550.2.a.br.1.1 1 1.1 even 1 trivial
8880.2.a.w.1.1 1 20.19 odd 2