Properties

Label 4650.2.a.bm.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -2.00000 q^{21} -2.00000 q^{22} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{48} -3.00000 q^{49} -6.00000 q^{51} +2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} -8.00000 q^{57} +4.00000 q^{58} -14.0000 q^{59} -2.00000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} +12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{74} -8.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -2.00000 q^{84} +4.00000 q^{86} +4.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} -4.00000 q^{91} +1.00000 q^{93} +1.00000 q^{96} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −2.00000 −0.426401
\(23\) 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
\(28\) −2.00000 −0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −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 \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −8.00000 −1.05963
\(58\) 4.00000 0.525226
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.00000 0.127000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 4.00000 0.428845
\(88\) −2.00000 −0.213201
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −3.00000 −0.303046
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −16.0000 −1.59206 −0.796030 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) −6.00000 −0.594089
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) −2.00000 −0.188982
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 2.00000 0.184900
\(118\) −14.0000 −1.28880
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −2.00000 −0.174078
\(133\) 16.0000 1.38738
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) −10.0000 −0.821995
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −8.00000 −0.648886
\(153\) −6.00000 −0.485071
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −16.0000 −1.27289
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −14.0000 −1.05230
\(178\) 6.00000 0.449719
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) −2.00000 −0.142134
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −16.0000 −1.12576
\(203\) −8.00000 −0.561490
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 2.00000 0.137361
\(213\) 12.0000 0.822226
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −2.00000 −0.135769
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −10.0000 −0.671156
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −8.00000 −0.529813
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 4.00000 0.262613
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) −16.0000 −1.03931
\(238\) 12.0000 0.777844
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −16.0000 −1.01806
\(248\) 1.00000 0.0635001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −22.0000 −1.38863 −0.694314 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 4.00000 0.249029
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 6.00000 0.370681
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) −28.0000 −1.70719 −0.853595 0.520937i \(-0.825583\pi\)
−0.853595 + 0.520937i \(0.825583\pi\)
\(270\) 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
\(273\) −4.00000 −0.242091
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 16.0000 0.959616
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) −2.00000 −0.116052
\(298\) 4.00000 0.231714
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −16.0000 −0.919176
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 4.00000 0.227921
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 2.00000 0.113228
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 2.00000 0.112154
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) −9.00000 −0.489535
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) −8.00000 −0.432590
\(343\) 20.0000 1.07990
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 4.00000 0.214423
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −2.00000 −0.106600
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −14.0000 −0.744092
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 12.0000 0.635107
\(358\) 14.0000 0.739923
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) −7.00000 −0.367405
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 1.00000 0.0518476
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −4.00000 −0.204658
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 6.00000 0.302660
\(394\) 26.0000 1.30986
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −8.00000 −0.401004
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 8.00000 0.399004
\(403\) 2.00000 0.0996271
\(404\) −16.0000 −0.796030
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 20.0000 0.991363
\(408\) −6.00000 −0.297044
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 14.0000 0.689730
\(413\) 28.0000 1.37779
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 16.0000 0.783523
\(418\) 16.0000 0.782586
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 4.00000 0.193574
\(428\) −4.00000 −0.193347
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −12.0000 −0.570782
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 4.00000 0.189194
\(448\) −2.00000 −0.0944911
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 20.0000 0.935561 0.467780 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) −22.0000 −1.02799
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 4.00000 0.186097
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 2.00000 0.0924500
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −14.0000 −0.644402
\(473\) −8.00000 −0.367840
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 2.00000 0.0915737
\(478\) 12.0000 0.548867
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −42.0000 −1.90320 −0.951601 0.307337i \(-0.900562\pi\)
−0.951601 + 0.307337i \(0.900562\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) −6.00000 −0.270501
\(493\) −24.0000 −1.08091
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) −24.0000 −1.07655
\(498\) −12.0000 −0.537733
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) −22.0000 −0.981908
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 2.00000 0.0887357
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 20.0000 0.878750
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 4.00000 0.175075
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −6.00000 −0.261364
\(528\) −2.00000 −0.0870388
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −14.0000 −0.607548
\(532\) 16.0000 0.693688
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 14.0000 0.604145
\(538\) −28.0000 −1.20717
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 42.0000 1.80572 0.902861 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 8.00000 0.343629
\(543\) −10.0000 −0.429141
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −10.0000 −0.427179
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 1.00000 0.0423334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 6.00000 0.253095
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) −2.00000 −0.0839921
\(568\) 12.0000 0.503509
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −4.00000 −0.167248
\(573\) −4.00000 −0.167102
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 19.0000 0.790296
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) −10.0000 −0.410997
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 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
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 0 0
\(606\) −16.0000 −0.649956
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −8.00000 −0.324443
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 14.0000 0.563163
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −20.0000 −0.801927
\(623\) −12.0000 −0.480770
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 10.0000 0.399043
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −16.0000 −0.636446
\(633\) −4.00000 −0.158986
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) −6.00000 −0.237729
\(638\) −8.00000 −0.316723
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) −4.00000 −0.157867
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 1.00000 0.0392837
\(649\) 28.0000 1.09910
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 16.0000 0.626608
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 12.0000 0.466393
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) −2.00000 −0.0771517
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −2.00000 −0.0765840
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 24.0000 0.913003 0.456502 0.889723i \(-0.349102\pi\)
0.456502 + 0.889723i \(0.349102\pi\)
\(692\) −18.0000 −0.684257
\(693\) 4.00000 0.151947
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 36.0000 1.36360
\(698\) −14.0000 −0.529908
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −16.0000 −0.604312 −0.302156 0.953259i \(-0.597706\pi\)
−0.302156 + 0.953259i \(0.597706\pi\)
\(702\) 2.00000 0.0754851
\(703\) 80.0000 3.01726
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) 32.0000 1.20348
\(708\) −14.0000 −0.526152
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 14.0000 0.523205
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 45.0000 1.67473
\(723\) −18.0000 −0.669427
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 10.0000 0.370879 0.185440 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) −2.00000 −0.0739221
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) −4.00000 −0.146845
\(743\) −28.0000 −1.02722 −0.513610 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 34.0000 1.24483
\(747\) −12.0000 −0.439057
\(748\) 12.0000 0.438763
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −22.0000 −0.801725
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 2.00000 0.0724524
\(763\) −20.0000 −0.724049
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −28.0000 −1.01102
\(768\) 1.00000 0.0360844
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) −4.00000 −0.143963
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 0 0
\(777\) 20.0000 0.717496
\(778\) −16.0000 −0.573628
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 0 0
\(783\) 4.00000 0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 26.0000 0.926212
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 28.0000 0.995565
\(792\) −2.00000 −0.0710669
\(793\) −4.00000 −0.142044
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 16.0000 0.566394
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) −28.0000 −0.985647
\(808\) −16.0000 −0.562878
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −8.00000 −0.280745
\(813\) 8.00000 0.280572
\(814\) 20.0000 0.701000
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −32.0000 −1.11954
\(818\) 38.0000 1.32864
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −20.0000 −0.698005 −0.349002 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) −10.0000 −0.348790
\(823\) −2.00000 −0.0697156 −0.0348578 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 28.0000 0.974245
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) 18.0000 0.623663
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 1.00000 0.0345651
\(838\) −18.0000 −0.621800
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −2.00000 −0.0689246
\(843\) 6.00000 0.206651
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 2.00000 0.0686803
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 12.0000 0.411113
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) −4.00000 −0.136558
\(859\) −48.0000 −1.63774 −0.818869 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 16.0000 0.544962
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 19.0000 0.645274
\(868\) −2.00000 −0.0678844
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 8.00000 0.269987
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −3.00000 −0.101015
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) −4.00000 −0.134307 −0.0671534 0.997743i \(-0.521392\pi\)
−0.0671534 + 0.997743i \(0.521392\pi\)
\(888\) −10.0000 −0.335578
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 4.00000 0.133780
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 12.0000 0.399556
\(903\) −8.00000 −0.266223
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 12.0000 0.398234
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −8.00000 −0.264906
\(913\) 24.0000 0.794284
\(914\) 20.0000 0.661541
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −12.0000 −0.396275
\(918\) −6.00000 −0.198030
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) 6.00000 0.197172
\(927\) 14.0000 0.459820
\(928\) 4.00000 0.131306
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) −10.0000 −0.327561
\(933\) −20.0000 −0.654771
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −48.0000 −1.56809 −0.784046 0.620703i \(-0.786847\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) −16.0000 −0.522419
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 10.0000 0.325818
\(943\) 0 0
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 12.0000 0.388922
\(953\) −58.0000 −1.87880 −0.939402 0.342817i \(-0.888619\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) −8.00000 −0.258603
\(958\) −12.0000 −0.387702
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −20.0000 −0.644826
\(963\) −4.00000 −0.128898
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) 10.0000 0.321578 0.160789 0.986989i \(-0.448596\pi\)
0.160789 + 0.986989i \(0.448596\pi\)
\(968\) −7.00000 −0.224989
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) −34.0000 −1.09111 −0.545556 0.838074i \(-0.683681\pi\)
−0.545556 + 0.838074i \(0.683681\pi\)
\(972\) 1.00000 0.0320750
\(973\) −32.0000 −1.02587
\(974\) −42.0000 −1.34577
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 16.0000 0.511624
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 18.0000 0.574403
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 1.00000 0.0317500
\(993\) 12.0000 0.380808
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −36.0000 −1.13956
\(999\) −10.0000 −0.316386
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 4650.2.a.bm.1.1 1
5.2 odd 4 930.2.d.f.559.2 yes 2
5.3 odd 4 930.2.d.f.559.1 2
5.4 even 2 4650.2.a.j.1.1 1
15.2 even 4 2790.2.d.b.559.1 2
15.8 even 4 2790.2.d.b.559.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.f.559.1 2 5.3 odd 4
930.2.d.f.559.2 yes 2 5.2 odd 4
2790.2.d.b.559.1 2 15.2 even 4
2790.2.d.b.559.2 2 15.8 even 4
4650.2.a.j.1.1 1 5.4 even 2
4650.2.a.bm.1.1 1 1.1 even 1 trivial