Properties

Label 465.2.a.b.1.1
Level $465$
Weight $2$
Character 465.1
Self dual yes
Analytic conductor $3.713$
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] = [465,2,Mod(1,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, 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("465.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.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: \(3.71304369399\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 465.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} +1.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} +8.00000 q^{37} -8.00000 q^{38} -3.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} +4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +6.00000 q^{56} +8.00000 q^{57} +10.0000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} +2.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} -3.00000 q^{72} -16.0000 q^{73} +8.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} +8.00000 q^{77} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +2.00000 q^{85} +12.0000 q^{88} +4.00000 q^{89} +1.00000 q^{90} +8.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -8.00000 q^{95} -5.00000 q^{96} +6.00000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) −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 \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −4.00000 −0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.00000 0.179605
\(32\) 5.00000 0.883883
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −8.00000 −1.17954
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 6.00000 0.801784
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 1.00000 0.127000
\(63\) −2.00000 −0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 8.00000 0.963087
\(70\) −2.00000 −0.239046
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −3.00000 −0.353553
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 8.00000 0.929981
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 12.0000 1.27920
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) −1.00000 −0.103695
\(94\) 4.00000 0.412568
\(95\) −8.00000 −0.820783
\(96\) −5.00000 −0.510310
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −3.00000 −0.303046
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −2.00000 −0.198030
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −4.00000 −0.381385
\(111\) −8.00000 −0.759326
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) −4.00000 −0.366679
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) −14.0000 −1.26750
\(123\) 6.00000 0.541002
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) −4.00000 −0.348155
\(133\) 16.0000 1.38738
\(134\) 2.00000 0.172774
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 8.00000 0.681005
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 2.00000 0.169031
\(141\) −4.00000 −0.336861
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 24.0000 1.94666
\(153\) 2.00000 0.161690
\(154\) 8.00000 0.644658
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 5.00000 0.395285
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 6.00000 0.468521
\(165\) 4.00000 0.311400
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −6.00000 −0.462910
\(169\) −13.0000 −1.00000
\(170\) 2.00000 0.153393
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 4.00000 0.301511
\(177\) −10.0000 −0.751646
\(178\) 4.00000 0.299813
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 24.0000 1.76930
\(185\) 8.00000 0.588172
\(186\) −1.00000 −0.0733236
\(187\) −8.00000 −0.585018
\(188\) −4.00000 −0.291730
\(189\) 2.00000 0.145479
\(190\) −8.00000 −0.580381
\(191\) −14.0000 −1.01300 −0.506502 0.862239i \(-0.669062\pi\)
−0.506502 + 0.862239i \(0.669062\pi\)
\(192\) −7.00000 −0.505181
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −4.00000 −0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −3.00000 −0.212132
\(201\) −2.00000 −0.141069
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) −6.00000 −0.419058
\(206\) 2.00000 0.139347
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) 2.00000 0.138013
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) −2.00000 −0.135769
\(218\) −14.0000 −0.948200
\(219\) 16.0000 1.08118
\(220\) 4.00000 0.269680
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −10.0000 −0.668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −8.00000 −0.529813
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −8.00000 −0.527504
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) −3.00000 −0.191663
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 2.00000 0.125988
\(253\) 32.0000 2.01182
\(254\) −4.00000 −0.250982
\(255\) −2.00000 −0.125245
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) −14.0000 −0.864923
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −12.0000 −0.738549
\(265\) 6.00000 0.368577
\(266\) 16.0000 0.981023
\(267\) −4.00000 −0.244796
\(268\) −2.00000 −0.122169
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −4.00000 −0.241209
\(276\) −8.00000 −0.481543
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) −12.0000 −0.719712
\(279\) 1.00000 0.0598684
\(280\) 6.00000 0.358569
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −4.00000 −0.238197
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) −6.00000 −0.356034
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 16.0000 0.936329
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 3.00000 0.174964
\(295\) 10.0000 0.582223
\(296\) −24.0000 −1.39497
\(297\) 4.00000 0.232104
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −24.0000 −1.38104
\(303\) −14.0000 −0.804279
\(304\) 8.00000 0.458831
\(305\) −14.0000 −0.801638
\(306\) 2.00000 0.114332
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) −8.00000 −0.455842
\(309\) −2.00000 −0.113776
\(310\) 1.00000 0.0567962
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 2.00000 0.112867
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 7.00000 0.391312
\(321\) −16.0000 −0.893033
\(322\) 16.0000 0.891645
\(323\) −16.0000 −0.890264
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 2.00000 0.110770
\(327\) 14.0000 0.774202
\(328\) 18.0000 0.993884
\(329\) −8.00000 −0.441054
\(330\) 4.00000 0.220193
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −4.00000 −0.219529
\(333\) 8.00000 0.438397
\(334\) 16.0000 0.875481
\(335\) 2.00000 0.109272
\(336\) −2.00000 −0.109109
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) −13.0000 −0.707107
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) −4.00000 −0.216612
\(342\) −8.00000 −0.432590
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) −10.0000 −0.531494
\(355\) 6.00000 0.318447
\(356\) −4.00000 −0.212000
\(357\) 4.00000 0.211702
\(358\) −4.00000 −0.211407
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) −3.00000 −0.158114
\(361\) 45.0000 2.36842
\(362\) −6.00000 −0.315353
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 14.0000 0.731792
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) −6.00000 −0.312348
\(370\) 8.00000 0.415900
\(371\) −12.0000 −0.623009
\(372\) 1.00000 0.0518476
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −8.00000 −0.413670
\(375\) −1.00000 −0.0516398
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 8.00000 0.410391
\(381\) 4.00000 0.204926
\(382\) −14.0000 −0.716302
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 3.00000 0.153093
\(385\) 8.00000 0.407718
\(386\) 6.00000 0.305392
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 9.00000 0.454569
\(393\) 14.0000 0.706207
\(394\) 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 \(0.385733\pi\)
\(398\) −8.00000 −0.401004
\(399\) −16.0000 −0.801002
\(400\) −1.00000 −0.0500000
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −32.0000 −1.58618
\(408\) 6.00000 0.297044
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) −6.00000 −0.296319
\(411\) 18.0000 0.887875
\(412\) −2.00000 −0.0985329
\(413\) −20.0000 −0.984136
\(414\) −8.00000 −0.393179
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 32.0000 1.56517
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 8.00000 0.389434
\(423\) 4.00000 0.194487
\(424\) −18.0000 −0.874157
\(425\) 2.00000 0.0970143
\(426\) −6.00000 −0.290701
\(427\) 28.0000 1.35501
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 1.00000 0.0481125
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 64.0000 3.06154
\(438\) 16.0000 0.764510
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 12.0000 0.572078
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 8.00000 0.379663
\(445\) 4.00000 0.189618
\(446\) 16.0000 0.757622
\(447\) 18.0000 0.851371
\(448\) −14.0000 −0.661438
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 1.00000 0.0471405
\(451\) 24.0000 1.13012
\(452\) 6.00000 0.282216
\(453\) 24.0000 1.12762
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −24.0000 −1.12390
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 6.00000 0.280362
\(459\) −2.00000 −0.0933520
\(460\) 8.00000 0.373002
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) −8.00000 −0.372194
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) −1.00000 −0.0463739
\(466\) 6.00000 0.277945
\(467\) 40.0000 1.85098 0.925490 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 4.00000 0.184506
\(471\) −2.00000 −0.0921551
\(472\) −30.0000 −1.38086
\(473\) 0 0
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 4.00000 0.183340
\(477\) 6.00000 0.274721
\(478\) −20.0000 −0.914779
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) −5.00000 −0.228218
\(481\) 0 0
\(482\) −22.0000 −1.00207
\(483\) −16.0000 −0.728025
\(484\) −5.00000 −0.227273
\(485\) 6.00000 0.272446
\(486\) −1.00000 −0.0453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 42.0000 1.90125
\(489\) −2.00000 −0.0904431
\(490\) −3.00000 −0.135526
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) −1.00000 −0.0449013
\(497\) −12.0000 −0.538274
\(498\) −4.00000 −0.179244
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −16.0000 −0.714827
\(502\) 20.0000 0.892644
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 6.00000 0.267261
\(505\) 14.0000 0.622992
\(506\) 32.0000 1.42257
\(507\) 13.0000 0.577350
\(508\) 4.00000 0.177471
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 32.0000 1.41560
\(512\) −11.0000 −0.486136
\(513\) 8.00000 0.353209
\(514\) −18.0000 −0.793946
\(515\) 2.00000 0.0881305
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) −16.0000 −0.703000
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 14.0000 0.611593
\(525\) 2.00000 0.0872872
\(526\) −24.0000 −1.04645
\(527\) 2.00000 0.0871214
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 10.0000 0.433963
\(532\) −16.0000 −0.693688
\(533\) 0 0
\(534\) −4.00000 −0.173097
\(535\) 16.0000 0.691740
\(536\) −6.00000 −0.259161
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 1.00000 0.0430331
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) 8.00000 0.343629
\(543\) 6.00000 0.257485
\(544\) 10.0000 0.428746
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 18.0000 0.768922
\(549\) −14.0000 −0.597505
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) −24.0000 −1.02151
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) −8.00000 −0.339581
\(556\) 12.0000 0.508913
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 1.00000 0.0423334
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 8.00000 0.337760
\(562\) 30.0000 1.26547
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 4.00000 0.168430
\(565\) −6.00000 −0.252422
\(566\) 10.0000 0.420331
\(567\) −2.00000 −0.0839921
\(568\) −18.0000 −0.755263
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 8.00000 0.335083
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 14.0000 0.584858
\(574\) 12.0000 0.500870
\(575\) −8.00000 −0.333623
\(576\) 7.00000 0.291667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −13.0000 −0.540729
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −6.00000 −0.248708
\(583\) −24.0000 −0.993978
\(584\) 48.0000 1.98625
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.00000 −0.329634
\(590\) 10.0000 0.411693
\(591\) −2.00000 −0.0822690
\(592\) −8.00000 −0.328798
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 4.00000 0.164122
\(595\) −4.00000 −0.163984
\(596\) 18.0000 0.737309
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 3.00000 0.122474
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 24.0000 0.976546
\(605\) 5.00000 0.203279
\(606\) −14.0000 −0.568711
\(607\) −46.0000 −1.86708 −0.933541 0.358470i \(-0.883298\pi\)
−0.933541 + 0.358470i \(0.883298\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 22.0000 0.887848
\(615\) 6.00000 0.241943
\(616\) −24.0000 −0.966988
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −2.00000 −0.0804518
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 8.00000 0.321029
\(622\) 18.0000 0.721734
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) −32.0000 −1.27796
\(628\) −2.00000 −0.0798087
\(629\) 16.0000 0.637962
\(630\) −2.00000 −0.0796819
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) −6.00000 −0.238290
\(635\) −4.00000 −0.158735
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) −3.00000 −0.118585
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) −16.0000 −0.631470
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) −3.00000 −0.117851
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) −2.00000 −0.0783260
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 14.0000 0.547443
\(655\) −14.0000 −0.547025
\(656\) 6.00000 0.234261
\(657\) −16.0000 −0.624219
\(658\) −8.00000 −0.311872
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) −4.00000 −0.155700
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 16.0000 0.620453
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) −16.0000 −0.618596
\(670\) 2.00000 0.0772667
\(671\) 56.0000 2.16186
\(672\) 10.0000 0.385758
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 28.0000 1.07852
\(675\) −1.00000 −0.0384900
\(676\) 13.0000 0.500000
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 6.00000 0.230429
\(679\) −12.0000 −0.460518
\(680\) −6.00000 −0.230089
\(681\) 20.0000 0.766402
\(682\) −4.00000 −0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 8.00000 0.305888
\(685\) −18.0000 −0.687745
\(686\) 20.0000 0.763604
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) 0 0
\(690\) 8.00000 0.304555
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −6.00000 −0.228086
\(693\) 8.00000 0.303895
\(694\) −12.0000 −0.455514
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −14.0000 −0.529908
\(699\) −6.00000 −0.226941
\(700\) 2.00000 0.0755929
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −64.0000 −2.41381
\(704\) −28.0000 −1.05529
\(705\) −4.00000 −0.150649
\(706\) −14.0000 −0.526897
\(707\) −28.0000 −1.05305
\(708\) 10.0000 0.375823
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) −8.00000 −0.299602
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 20.0000 0.746914
\(718\) 26.0000 0.970311
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −4.00000 −0.148968
\(722\) 45.0000 1.67473
\(723\) 22.0000 0.818189
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −16.0000 −0.592187
\(731\) 0 0
\(732\) −14.0000 −0.517455
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −32.0000 −1.18114
\(735\) 3.00000 0.110657
\(736\) −40.0000 −1.47442
\(737\) −8.00000 −0.294684
\(738\) −6.00000 −0.220863
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 3.00000 0.109985
\(745\) −18.0000 −0.659469
\(746\) −22.0000 −0.805477
\(747\) 4.00000 0.146352
\(748\) 8.00000 0.292509
\(749\) −32.0000 −1.16925
\(750\) −1.00000 −0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −4.00000 −0.145865
\(753\) −20.0000 −0.728841
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) −2.00000 −0.0727393
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 16.0000 0.581146
\(759\) −32.0000 −1.16153
\(760\) 24.0000 0.870572
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 4.00000 0.144905
\(763\) 28.0000 1.01367
\(764\) 14.0000 0.506502
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 8.00000 0.288300
\(771\) 18.0000 0.648254
\(772\) −6.00000 −0.215945
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) −18.0000 −0.646162
\(777\) 16.0000 0.573997
\(778\) 16.0000 0.573628
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 2.00000 0.0713831
\(786\) 14.0000 0.499363
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 12.0000 0.426401
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) −6.00000 −0.212798
\(796\) 8.00000 0.283552
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −16.0000 −0.566394
\(799\) 8.00000 0.283020
\(800\) 5.00000 0.176777
\(801\) 4.00000 0.141333
\(802\) 16.0000 0.564980
\(803\) 64.0000 2.25851
\(804\) 2.00000 0.0705346
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) −42.0000 −1.47755
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 1.00000 0.0351364
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −32.0000 −1.12160
\(815\) 2.00000 0.0700569
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 18.0000 0.627822
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −6.00000 −0.209020
\(825\) 4.00000 0.139262
\(826\) −20.0000 −0.695889
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 8.00000 0.278019
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 4.00000 0.138842
\(831\) 28.0000 0.971309
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 12.0000 0.415526
\(835\) 16.0000 0.553703
\(836\) −32.0000 −1.10674
\(837\) −1.00000 −0.0345651
\(838\) 14.0000 0.483622
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) −6.00000 −0.207020
\(841\) −29.0000 −1.00000
\(842\) 18.0000 0.620321
\(843\) −30.0000 −1.03325
\(844\) −8.00000 −0.275371
\(845\) −13.0000 −0.447214
\(846\) 4.00000 0.137523
\(847\) −10.0000 −0.343604
\(848\) −6.00000 −0.206041
\(849\) −10.0000 −0.343199
\(850\) 2.00000 0.0685994
\(851\) −64.0000 −2.19389
\(852\) 6.00000 0.205557
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 28.0000 0.958140
\(855\) −8.00000 −0.273594
\(856\) −48.0000 −1.64061
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 2.00000 0.0681203
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −5.00000 −0.170103
\(865\) 6.00000 0.204006
\(866\) 24.0000 0.815553
\(867\) 13.0000 0.441503
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 42.0000 1.42230
\(873\) 6.00000 0.203069
\(874\) 64.0000 2.16483
\(875\) −2.00000 −0.0676123
\(876\) −16.0000 −0.540590
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 24.0000 0.809961
\(879\) −26.0000 −0.876958
\(880\) 4.00000 0.134840
\(881\) −28.0000 −0.943344 −0.471672 0.881774i \(-0.656349\pi\)
−0.471672 + 0.881774i \(0.656349\pi\)
\(882\) −3.00000 −0.101015
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) −10.0000 −0.336146
\(886\) −20.0000 −0.671913
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 24.0000 0.805387
\(889\) 8.00000 0.268311
\(890\) 4.00000 0.134080
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) −32.0000 −1.07084
\(894\) 18.0000 0.602010
\(895\) −4.00000 −0.133705
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) −6.00000 −0.199447
\(906\) 24.0000 0.797347
\(907\) −18.0000 −0.597680 −0.298840 0.954303i \(-0.596600\pi\)
−0.298840 + 0.954303i \(0.596600\pi\)
\(908\) 20.0000 0.663723
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) −8.00000 −0.264906
\(913\) −16.0000 −0.529523
\(914\) −8.00000 −0.264616
\(915\) 14.0000 0.462826
\(916\) −6.00000 −0.198246
\(917\) 28.0000 0.924641
\(918\) −2.00000 −0.0660098
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 24.0000 0.791257
\(921\) −22.0000 −0.724925
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 8.00000 0.263038
\(926\) 20.0000 0.657241
\(927\) 2.00000 0.0656886
\(928\) 0 0
\(929\) −44.0000 −1.44359 −0.721797 0.692105i \(-0.756683\pi\)
−0.721797 + 0.692105i \(0.756683\pi\)
\(930\) −1.00000 −0.0327913
\(931\) 24.0000 0.786568
\(932\) −6.00000 −0.196537
\(933\) −18.0000 −0.589294
\(934\) 40.0000 1.30884
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) −4.00000 −0.130605
\(939\) 8.00000 0.261070
\(940\) −4.00000 −0.130466
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 48.0000 1.56310
\(944\) −10.0000 −0.325472
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) 6.00000 0.194563
\(952\) 12.0000 0.388922
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 6.00000 0.194257
\(955\) −14.0000 −0.453029
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) −10.0000 −0.323085
\(959\) 36.0000 1.16250
\(960\) −7.00000 −0.225924
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 22.0000 0.708572
\(965\) 6.00000 0.193147
\(966\) −16.0000 −0.514792
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −15.0000 −0.482118
\(969\) 16.0000 0.513994
\(970\) 6.00000 0.192648
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 1.00000 0.0320750
\(973\) 24.0000 0.769405
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −2.00000 −0.0639529
\(979\) −16.0000 −0.511362
\(980\) 3.00000 0.0958315
\(981\) −14.0000 −0.446986
\(982\) −24.0000 −0.765871
\(983\) −8.00000 −0.255160 −0.127580 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) −18.0000 −0.573819
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 0 0
\(990\) −4.00000 −0.127128
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 5.00000 0.158750
\(993\) 4.00000 0.126936
\(994\) −12.0000 −0.380617
\(995\) −8.00000 −0.253617
\(996\) 4.00000 0.126745
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −20.0000 −0.633089
\(999\) −8.00000 −0.253109
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 465.2.a.b.1.1 1
3.2 odd 2 1395.2.a.a.1.1 1
4.3 odd 2 7440.2.a.ba.1.1 1
5.2 odd 4 2325.2.c.d.1024.2 2
5.3 odd 4 2325.2.c.d.1024.1 2
5.4 even 2 2325.2.a.d.1.1 1
15.14 odd 2 6975.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.b.1.1 1 1.1 even 1 trivial
1395.2.a.a.1.1 1 3.2 odd 2
2325.2.a.d.1.1 1 5.4 even 2
2325.2.c.d.1024.1 2 5.3 odd 4
2325.2.c.d.1024.2 2 5.2 odd 4
6975.2.a.q.1.1 1 15.14 odd 2
7440.2.a.ba.1.1 1 4.3 odd 2