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Properties

Label 9075.2.a.d.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
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] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +2.00000 q^{21} -4.00000 q^{23} -3.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} -6.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} +4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +6.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -6.00000 q^{56} -6.00000 q^{57} +6.00000 q^{58} -4.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} -12.0000 q^{67} +6.00000 q^{68} +4.00000 q^{69} +8.00000 q^{71} +3.00000 q^{72} +16.0000 q^{73} -6.00000 q^{74} -6.00000 q^{76} +4.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -2.00000 q^{84} +6.00000 q^{86} +6.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +5.00000 q^{96} +6.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −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 \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 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
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −6.00000 −0.973329
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) −6.00000 −0.794719
\(58\) 6.00000 0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 6.00000 0.727607
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000 0.353553
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 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
\(99\) 0 0
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 12.0000 1.17670
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 0 0
\(122\) 4.00000 0.362143
\(123\) −6.00000 −0.541002
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 3.00000 0.265165
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −4.00000 −0.340503
\(139\) 22.0000 1.86602 0.933008 − 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 3.00000 0.247436
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 − 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 18.0000 1.45999
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 2.00000 0.159111
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 6.00000 0.462910
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 6.00000 0.457496
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 − 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 8.00000 0.592999
\(183\) 4.00000 0.295689
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −7.00000 −0.505181
\(193\) 20.0000 1.43963 0.719816 − 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 14.0000 0.985037
\(203\) 12.0000 0.842235
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 − 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 6.00000 0.402694
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 16.0000 1.06196 0.530979 − 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 6.00000 0.397360
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) −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
\(241\) −20.0000 −1.28831 −0.644157 − 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 24.0000 1.52708
\(248\) 24.0000 1.52400
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −6.00000 −0.373544
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 16.0000 0.988483
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 − 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 6.00000 0.363803
\(273\) 8.00000 0.484182
\(274\) 14.0000 0.845771
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 4.00000 0.240337 0.120168 − 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) −22.0000 −1.31947
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 − 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) −8.00000 −0.476393
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) −16.0000 −0.936329
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 2.00000 0.115087
\(303\) 14.0000 0.804279
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 14.0000 0.799022 0.399511 − 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 − 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) −12.0000 −0.679366
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) −8.00000 −0.445823
\(323\) −36.0000 −2.00309
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −3.00000 −0.163178
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) 20.0000 1.07990
\(344\) −18.0000 −0.970495
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −8.00000 −0.429463 −0.214731 − 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) −6.00000 −0.321634
\(349\) −36.0000 −1.92704 −0.963518 − 0.267644i \(-0.913755\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −12.0000 −0.635107
\(358\) −12.0000 −0.634220
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 26.0000 1.36653
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) −24.0000 −1.25279 −0.626395 − 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000 0.414781
\(373\) 24.0000 1.24267 0.621336 − 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) −24.0000 −1.23606
\(378\) −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
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 − 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) −6.00000 −0.304997
\(388\) −6.00000 −0.304604
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) −9.00000 −0.454569
\(393\) 16.0000 0.807093
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −16.0000 −0.802008
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 − 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −12.0000 −0.598506
\(403\) 32.0000 1.59403
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 18.0000 0.891133
\(409\) 4.00000 0.197787 0.0988936 − 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 − 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −22.0000 −1.07094
\(423\) −8.00000 −0.388973
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 8.00000 0.387147
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −18.0000 −0.865025 −0.432512 − 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 16.0000 0.764510
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 24.0000 1.14156
\(443\) −16.0000 −0.760183 −0.380091 − 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) −6.00000 −0.283790
\(448\) −14.0000 −0.661438
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 2.00000 0.0939682
\(454\) −16.0000 −0.750917
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 22.0000 1.02799
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 − 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −16.0000 −0.740392 −0.370196 − 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) −4.00000 −0.184900
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) −2.00000 −0.0918630
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) −6.00000 −0.274721
\(478\) −12.0000 −0.548867
\(479\) −28.0000 −1.27935 −0.639676 − 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 20.0000 0.910975
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −12.0000 −0.543214
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 6.00000 0.270501
\(493\) 36.0000 1.62136
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 2.00000 0.0887357
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 11.0000 0.486136
\(513\) −6.00000 −0.264906
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 6.00000 0.262613
\(523\) −14.0000 −0.612177 −0.306089 − 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) 24.0000 1.03956
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −36.0000 −1.55496
\(537\) −12.0000 −0.517838
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 − 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 14.0000 0.601351
\(543\) 26.0000 1.11577
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 34.0000 1.45374 0.726868 − 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) 14.0000 0.598050
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 12.0000 0.510754
\(553\) 4.00000 0.170097
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −22.0000 −0.933008
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) −2.00000 −0.0839921
\(568\) 24.0000 1.00702
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 − 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −14.0000 −0.582828 −0.291414 − 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −19.0000 −0.790296
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) 0 0
\(582\) 6.00000 0.248708
\(583\) 0 0
\(584\) 48.0000 1.98625
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −3.00000 −0.123718
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) −6.00000 −0.246598
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −16.0000 −0.654836
\(598\) 16.0000 0.654289
\(599\) −4.00000 −0.163436 −0.0817178 − 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 − 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) −12.0000 −0.489083
\(603\) −12.0000 −0.488678
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) −42.0000 −1.70473 −0.852364 − 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) −30.0000 −1.21666
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 6.00000 0.242536
\(613\) 8.00000 0.323117 0.161558 − 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −14.0000 −0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 8.00000 0.321807
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 28.0000 1.12270
\(623\) −20.0000 −0.801283
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 − 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −6.00000 −0.238667
\(633\) −22.0000 −0.874421
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 − 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 8.00000 0.315735
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) 20.0000 0.786281 0.393141 − 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 16.0000 0.624219
\(658\) −16.0000 −0.623745
\(659\) −40.0000 −1.55818 −0.779089 − 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 28.0000 1.08825
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) −10.0000 −0.385758
\(673\) 4.00000 0.154189 0.0770943 − 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 2.00000 0.0768662 0.0384331 − 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) −6.00000 −0.230429
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 − 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 22.0000 0.839352
\(688\) 6.00000 0.228748
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) 18.0000 0.682288
\(697\) −36.0000 −1.36360
\(698\) 36.0000 1.36262
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 − 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 4.00000 0.150970
\(703\) 36.0000 1.35777
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 28.0000 1.05305
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 30.0000 1.12430
\(713\) −32.0000 −1.19841
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −12.0000 −0.448148
\(718\) 8.00000 0.298557
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −17.0000 −0.632674
\(723\) 20.0000 0.743808
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −24.0000 −0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) −4.00000 −0.147844
\(733\) −44.0000 −1.62518 −0.812589 − 0.582838i \(-0.801942\pi\)
−0.812589 + 0.582838i \(0.801942\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 34.0000 1.25071 0.625355 − 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) −12.0000 −0.440534
\(743\) −20.0000 −0.733729 −0.366864 − 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) −24.0000 −0.879883
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) 0 0
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 8.00000 0.291730
\(753\) −20.0000 −0.728841
\(754\) 24.0000 0.874028
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 50.0000 1.81250 0.906249 − 0.422744i \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) −40.0000 −1.44244 −0.721218 − 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −20.0000 −0.719816
\(773\) −54.0000 −1.94225 −0.971123 − 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 12.0000 0.430498
\(778\) 6.00000 0.215110
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) −24.0000 −0.858238
\(783\) 6.00000 0.214423
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −16.0000 −0.570701
\(787\) −22.0000 −0.784215 −0.392108 − 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −18.0000 −0.641223
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −26.0000 −0.920967 −0.460484 − 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) −12.0000 −0.424795
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) −6.00000 −0.211210
\(808\) −42.0000 −1.47755
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 − 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) −12.0000 −0.421117
\(813\) 14.0000 0.491001
\(814\) 0 0
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) −36.0000 −1.25948
\(818\) −4.00000 −0.139857
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −14.0000 −0.488306
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 − 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 4.00000 0.139010
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) 28.0000 0.970725
\(833\) 18.0000 0.623663
\(834\) 22.0000 0.761798
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 28.0000 0.967244
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) −22.0000 −0.757720
\(844\) −22.0000 −0.757271
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 8.00000 0.274075
\(853\) 24.0000 0.821744 0.410872 − 0.911693i \(-0.365224\pi\)
0.410872 + 0.911693i \(0.365224\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 24.0000 0.820303
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −12.0000 −0.408722
\(863\) 36.0000 1.22545 0.612727 − 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 18.0000 0.611665
\(867\) −19.0000 −0.645274
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −32.0000 −1.08056 −0.540282 − 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) −26.0000 −0.877457
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 46.0000 1.54978 0.774890 − 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) 3.00000 0.101015
\(883\) 52.0000 1.74994 0.874970 − 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −18.0000 −0.604040
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −48.0000 −1.60626
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 16.0000 0.534224
\(898\) 30.0000 1.00111
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) −20.0000 −0.664089 −0.332045 − 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −16.0000 −0.530979
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 − 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 6.00000 0.198680
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 32.0000 1.05673
\(918\) −6.00000 −0.198030
\(919\) −22.0000 −0.725713 −0.362857 − 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) −14.0000 −0.461316
\(922\) −22.0000 −0.724531
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) 30.0000 0.984798
\(929\) 14.0000 0.459325 0.229663 − 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) −6.00000 −0.196537
\(933\) 28.0000 0.916679
\(934\) 16.0000 0.523536
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) −20.0000 −0.653372 −0.326686 − 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(938\) −24.0000 −0.783628
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 − 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) −18.0000 −0.586472
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 36.0000 1.16677
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 28.0000 0.904639
\(959\) 28.0000 0.904167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −24.0000 −0.773791
\(963\) 8.00000 0.257796
\(964\) 20.0000 0.644157
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 46.0000 1.47926 0.739630 − 0.673014i \(-0.235000\pi\)
0.739630 + 0.673014i \(0.235000\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 1.00000 0.0320750
\(973\) −44.0000 −1.41058
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −30.0000 −0.959785 −0.479893 − 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 4.00000 0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) 4.00000 0.127580 0.0637901 − 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) −16.0000 −0.509286
\(988\) −24.0000 −0.763542
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −40.0000 −1.27000
\(993\) 28.0000 0.888553
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 24.0000 0.760088 0.380044 − 0.924968i \(-0.375909\pi\)
0.380044 + 0.924968i \(0.375909\pi\)
\(998\) 4.00000 0.126618
\(999\) −6.00000 −0.189832
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 9075.2.a.d.1.1 1
5.4 even 2 1815.2.a.e.1.1 yes 1
11.10 odd 2 9075.2.a.n.1.1 1
15.14 odd 2 5445.2.a.e.1.1 1
55.54 odd 2 1815.2.a.a.1.1 ✓ 1
165.164 even 2 5445.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.a.1.1 ✓ 1 55.54 odd 2
1815.2.a.e.1.1 yes 1 5.4 even 2
5445.2.a.e.1.1 1 15.14 odd 2
5445.2.a.j.1.1 1 165.164 even 2
9075.2.a.d.1.1 1 1.1 even 1 trivial
9075.2.a.n.1.1 1 11.10 odd 2