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Properties

Label 3450.2.a.f.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
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] = [3450,2,Mod(1,3450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, 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("3450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.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: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} -10.0000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} -2.00000 q^{38} +2.00000 q^{39} +12.0000 q^{41} +1.00000 q^{42} +10.0000 q^{43} -1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +1.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} -9.00000 q^{58} -6.00000 q^{59} +14.0000 q^{61} +10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -14.0000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +3.00000 q^{71} -1.00000 q^{72} -11.0000 q^{73} +11.0000 q^{74} +2.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} +9.00000 q^{83} -1.00000 q^{84} -10.0000 q^{86} -9.00000 q^{87} +3.00000 q^{89} -2.00000 q^{91} +1.00000 q^{92} +10.0000 q^{93} -9.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −11.0000 −1.80839 −0.904194 − 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 − 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 1.00000 0.154303
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.00000 −0.264906
\(58\) −9.00000 −1.18176
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 10.0000 1.27000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) −3.00000 −0.363803
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 11.0000 1.27872
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 1.00000 0.104257
\(93\) 10.0000 1.03695
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 −0.895533 −0.447767 − 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) −3.00000 −0.297044
\(103\) 7.00000 0.689730 0.344865 − 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.0000 1.05361 0.526804 − 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 1.00000 0.0944911
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) −12.0000 −1.08200
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −9.00000 −0.768922 −0.384461 − 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 1.00000 0.0851257
\(139\) 5.00000 0.424094 0.212047 − 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) −3.00000 −0.251754
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 11.0000 0.910366
\(147\) 6.00000 0.494872
\(148\) −11.0000 −0.904194
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 3.00000 0.232147 0.116073 − 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 10.0000 0.762493
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) −3.00000 −0.224860
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −1.00000 −0.0743294 −0.0371647 − 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 2.00000 0.148250
\(183\) −14.0000 −1.03491
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −10.0000 −0.733236
\(187\) 0 0
\(188\) 9.00000 0.656392
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 − 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 25.0000 1.79954 0.899770 − 0.436365i \(-0.143734\pi\)
0.899770 + 0.436365i \(0.143734\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 15.0000 1.06871 0.534353 − 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 − 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) 9.00000 0.633238
\(203\) 9.00000 0.631676
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −7.00000 −0.487713
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 − 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 0 0
\(213\) −3.00000 −0.205557
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −10.0000 −0.678844
\(218\) −11.0000 −0.745014
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −11.0000 −0.738272
\(223\) 22.0000 1.47323 0.736614 − 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 21.0000 1.39382 0.696909 − 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) −2.00000 −0.132453
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) −8.00000 −0.519656
\(238\) 3.00000 0.194461
\(239\) 27.0000 1.74648 0.873242 − 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 − 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) −4.00000 −0.254514
\(248\) 10.0000 0.635001
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 − 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 10.0000 0.622573
\(259\) −11.0000 −0.683507
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) −12.0000 −0.741362
\(263\) 18.0000 1.10993 0.554964 − 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) −3.00000 −0.183597
\(268\) −14.0000 −0.855186
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 − 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −3.00000 −0.181902
\(273\) 2.00000 0.121046
\(274\) 9.00000 0.543710
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −5.00000 −0.299880
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 − 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 9.00000 0.535942
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −11.0000 −0.643726
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 11.0000 0.639362
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) −8.00000 −0.460348
\(303\) 9.00000 0.517036
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) −5.00000 −0.285365 −0.142683 − 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) −15.0000 −0.850572 −0.425286 − 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) −2.00000 −0.113228
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 27.0000 1.51647 0.758236 − 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) −1.00000 −0.0557278
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) −11.0000 −0.608301
\(328\) −12.0000 −0.662589
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 − 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 9.00000 0.493939
\(333\) −11.0000 −0.602796
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 28.0000 1.52526 0.762629 − 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 9.00000 0.489535
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) −13.0000 −0.701934
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 30.0000 1.61048 0.805242 − 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) −9.00000 −0.482451
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 3.00000 0.159000
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 − 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 1.00000 0.0525588
\(363\) 11.0000 0.577350
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 1.00000 0.0521286
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 10.0000 0.518476
\(373\) 13.0000 0.673114 0.336557 − 0.941663i \(-0.390737\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −18.0000 −0.927047
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 18.0000 0.920960
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −25.0000 −1.27247
\(387\) 10.0000 0.508329
\(388\) 10.0000 0.507673
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 6.00000 0.303046
\(393\) −12.0000 −0.605320
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 − 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 1.00000 0.0501255
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −14.0000 −0.698257
\(403\) 20.0000 0.996271
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) 0 0
\(408\) −3.00000 −0.148522
\(409\) 17.0000 0.840596 0.420298 − 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 7.00000 0.344865
\(413\) −6.00000 −0.295241
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 − 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 − 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −23.0000 −1.11962
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) 0 0
\(426\) 3.00000 0.145350
\(427\) 14.0000 0.677507
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 − 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 2.00000 0.0956730
\(438\) −11.0000 −0.525600
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −6.00000 −0.285391
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 11.0000 0.522037
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 9.00000 0.423324
\(453\) −8.00000 −0.375873
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) −32.0000 −1.49690 −0.748448 − 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 − 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) 33.0000 1.52706 0.763529 − 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) −27.0000 −1.23495
\(479\) −18.0000 −0.822441 −0.411220 − 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 22.0000 1.00311
\(482\) −8.00000 −0.364390
\(483\) −1.00000 −0.0455016
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 − 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −14.0000 −0.633750
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 − 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −12.0000 −0.541002
\(493\) −27.0000 −1.21602
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 3.00000 0.134568
\(498\) 9.00000 0.403300
\(499\) −19.0000 −0.850557 −0.425278 − 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) −3.00000 −0.133897
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) −27.0000 −1.19675 −0.598377 − 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) −1.00000 −0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 0 0
\(518\) 11.0000 0.483312
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 − 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) −9.00000 −0.393919
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 2.00000 0.0867110
\(533\) −24.0000 −1.03956
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 44.0000 1.89171 0.945854 − 0.324593i \(-0.105227\pi\)
0.945854 + 0.324593i \(0.105227\pi\)
\(542\) 28.0000 1.20270
\(543\) 1.00000 0.0429141
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) 1.00000 0.0427569 0.0213785 − 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) −9.00000 −0.384461
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 1.00000 0.0425628
\(553\) 8.00000 0.340195
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) 42.0000 1.77960 0.889799 − 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 10.0000 0.423334
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) −3.00000 −0.126547
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −9.00000 −0.378968
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 1.00000 0.0419961
\(568\) −3.00000 −0.125877
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 − 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 8.00000 0.332756
\(579\) −25.0000 −1.03896
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 6.00000 0.247436
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) −11.0000 −0.452097
\(593\) −36.0000 −1.47834 −0.739171 − 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.00000 0.0409273
\(598\) 2.00000 0.0817861
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 − 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) −10.0000 −0.407570
\(603\) −14.0000 −0.570124
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −9.00000 −0.365600
\(607\) −14.0000 −0.568242 −0.284121 − 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) −3.00000 −0.121268
\(613\) 19.0000 0.767403 0.383701 − 0.923457i \(-0.374649\pi\)
0.383701 + 0.923457i \(0.374649\pi\)
\(614\) 5.00000 0.201784
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 7.00000 0.281581
\(619\) 44.0000 1.76851 0.884255 − 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 15.0000 0.601445
\(623\) 3.00000 0.120192
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 33.0000 1.31580
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 − 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) −8.00000 −0.318223
\(633\) −23.0000 −0.914168
\(634\) −27.0000 −1.07231
\(635\) 0 0
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 − 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) −12.0000 −0.473602
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) −21.0000 −0.825595 −0.412798 − 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 16.0000 0.626608
\(653\) −9.00000 −0.352197 −0.176099 − 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) 11.0000 0.430134
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) −11.0000 −0.429151
\(658\) −9.00000 −0.350857
\(659\) 15.0000 0.584317 0.292159 − 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) −31.0000 −1.20576 −0.602880 − 0.797832i \(-0.705980\pi\)
−0.602880 + 0.797832i \(0.705980\pi\)
\(662\) −5.00000 −0.194331
\(663\) −6.00000 −0.233021
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 11.0000 0.426241
\(667\) 9.00000 0.348481
\(668\) 3.00000 0.116073
\(669\) −22.0000 −0.850569
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 7.00000 0.269830 0.134915 − 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −12.0000 −0.461197 −0.230599 − 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 9.00000 0.345643
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) −2.00000 −0.0763048
\(688\) 10.0000 0.381246
\(689\) 0 0
\(690\) 0 0
\(691\) 11.0000 0.418460 0.209230 − 0.977866i \(-0.432904\pi\)
0.209230 + 0.977866i \(0.432904\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) −36.0000 −1.36360
\(698\) 10.0000 0.378506
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −22.0000 −0.829746
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −9.00000 −0.338480
\(708\) 6.00000 0.225494
\(709\) −19.0000 −0.713560 −0.356780 − 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −3.00000 −0.112430
\(713\) −10.0000 −0.374503
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 0 0
\(717\) −27.0000 −1.00833
\(718\) −36.0000 −1.34351
\(719\) −15.0000 −0.559406 −0.279703 − 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 15.0000 0.558242
\(723\) −8.00000 −0.297523
\(724\) −1.00000 −0.0371647
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) −32.0000 −1.18681 −0.593407 − 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) −14.0000 −0.517455
\(733\) −29.0000 −1.07114 −0.535570 − 0.844491i \(-0.679903\pi\)
−0.535570 + 0.844491i \(0.679903\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −12.0000 −0.441726
\(739\) −1.00000 −0.0367856 −0.0183928 − 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 − 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) −10.0000 −0.366618
\(745\) 0 0
\(746\) −13.0000 −0.475964
\(747\) 9.00000 0.329293
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 − 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 9.00000 0.328196
\(753\) −3.00000 −0.109326
\(754\) 18.0000 0.655521
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 13.0000 0.472493 0.236247 − 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −8.00000 −0.289809
\(763\) 11.0000 0.398227
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 12.0000 0.433295
\(768\) −1.00000 −0.0360844
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 25.0000 0.899770
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −10.0000 −0.359443
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 11.0000 0.394623
\(778\) 30.0000 1.07555
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 3.00000 0.107280
\(783\) −9.00000 −0.321634
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 15.0000 0.534353
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −1.00000 −0.0354441
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 2.00000 0.0707992
\(799\) −27.0000 −0.955191
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) −6.00000 −0.211210
\(808\) 9.00000 0.316619
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 − 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 9.00000 0.315838
\(813\) 28.0000 0.982003
\(814\) 0 0
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 20.0000 0.699711
\(818\) −17.0000 −0.594391
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) −9.00000 −0.313911
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −7.00000 −0.243857
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 33.0000 1.14752 0.573761 − 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) 1.00000 0.0347524
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) 5.00000 0.173136
\(835\) 0 0
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) −15.0000 −0.518166
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −26.0000 −0.896019
\(843\) −3.00000 −0.103325
\(844\) 23.0000 0.791693
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −11.0000 −0.377075
\(852\) −3.00000 −0.102778
\(853\) −44.0000 −1.50653 −0.753266 − 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) −14.0000 −0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) −30.0000 −1.02180
\(863\) −27.0000 −0.919091 −0.459545 − 0.888154i \(-0.651988\pi\)
−0.459545 + 0.888154i \(0.651988\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 8.00000 0.271694
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) −11.0000 −0.372507
\(873\) 10.0000 0.338449
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 11.0000 0.371656
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −20.0000 −0.674967
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 6.00000 0.202031
\(883\) −11.0000 −0.370179 −0.185090 − 0.982722i \(-0.559258\pi\)
−0.185090 + 0.982722i \(0.559258\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 0 0
\(887\) −39.0000 −1.30949 −0.654746 − 0.755849i \(-0.727224\pi\)
−0.654746 + 0.755849i \(0.727224\pi\)
\(888\) −11.0000 −0.369136
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 22.0000 0.736614
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 2.00000 0.0667781
\(898\) 6.00000 0.200223
\(899\) −90.0000 −3.00167
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −38.0000 −1.26177 −0.630885 − 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) 21.0000 0.696909
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 12.0000 0.396275
\(918\) −3.00000 −0.0990148
\(919\) −31.0000 −1.02260 −0.511298 − 0.859404i \(-0.670835\pi\)
−0.511298 + 0.859404i \(0.670835\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) −3.00000 −0.0987997
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) 7.00000 0.229910
\(928\) −9.00000 −0.295439
\(929\) −42.0000 −1.37798 −0.688988 − 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 24.0000 0.786146
\(933\) 15.0000 0.491078
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −50.0000 −1.63343 −0.816714 − 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) 14.0000 0.457116
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −14.0000 −0.456145
\(943\) 12.0000 0.390774
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −8.00000 −0.259828
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 3.00000 0.0972306
\(953\) −15.0000 −0.485898 −0.242949 − 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 27.0000 0.873242
\(957\) 0 0
\(958\) 18.0000 0.581554
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −22.0000 −0.709308
\(963\) −12.0000 −0.386695
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) −2.00000 −0.0643157 −0.0321578 − 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 11.0000 0.353553
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 − 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 5.00000 0.160293
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) −21.0000 −0.671850 −0.335925 − 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) 30.0000 0.957338
\(983\) −36.0000 −1.14822 −0.574111 − 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 27.0000 0.859855
\(987\) −9.00000 −0.286473
\(988\) −4.00000 −0.127257
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 10.0000 0.317500
\(993\) −5.00000 −0.158670
\(994\) −3.00000 −0.0951542
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) −26.0000 −0.823428 −0.411714 − 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 19.0000 0.601434
\(999\) 11.0000 0.348025
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 3450.2.a.f.1.1 ✓ 1
5.2 odd 4 3450.2.d.p.2899.1 2
5.3 odd 4 3450.2.d.p.2899.2 2
5.4 even 2 3450.2.a.x.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.f.1.1 ✓ 1 1.1 even 1 trivial
3450.2.a.x.1.1 yes 1 5.4 even 2
3450.2.d.p.2899.1 2 5.2 odd 4
3450.2.d.p.2899.2 2 5.3 odd 4