Properties

Label 8045.2.a.a.1.1
Level $8045$
Weight $2$
Character 8045.1
Self dual yes
Analytic conductor $64.240$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8045,2,Mod(1,8045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8045.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: \(64.2396484261\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -3.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +2.00000 q^{28} +2.00000 q^{29} -4.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} +3.00000 q^{36} -10.0000 q^{37} -6.00000 q^{38} +3.00000 q^{40} -2.00000 q^{41} -6.00000 q^{43} -4.00000 q^{44} +3.00000 q^{45} -6.00000 q^{46} -6.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -4.00000 q^{55} +6.00000 q^{56} +2.00000 q^{58} +10.0000 q^{61} -4.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -6.00000 q^{68} +2.00000 q^{70} +14.0000 q^{71} +9.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} +6.00000 q^{76} -8.00000 q^{77} +2.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +10.0000 q^{83} -6.00000 q^{85} -6.00000 q^{86} -12.0000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -4.00000 q^{91} +6.00000 q^{92} -6.00000 q^{94} +6.00000 q^{95} -10.0000 q^{97} -3.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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 \(0.384973\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −3.00000 −1.06066
\(9\) −3.00000 −1.00000
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 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 \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −3.00000 −0.707107
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 3.00000 0.500000
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −4.00000 −0.603023
\(45\) 3.00000 0.447214
\(46\) −6.00000 −0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 6.00000 0.755929
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 9.00000 1.06066
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 3.00000 0.316228
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −3.00000 −0.303046
\(99\) −12.0000 −1.20605
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) −2.00000 −0.185695
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 6.00000 0.534522
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 14.0000 1.17485
\(143\) 8.00000 0.668994
\(144\) 3.00000 0.250000
\(145\) −2.00000 −0.166091
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 18.0000 1.45999
\(153\) −18.0000 −1.45521
\(154\) −8.00000 −0.644658
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) −5.00000 −0.395285
\(161\) 12.0000 0.945732
\(162\) 9.00000 0.707107
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 18.0000 1.37649
\(172\) 6.00000 0.457496
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −3.00000 −0.223607
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 18.0000 1.32698
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −12.0000 −0.852803
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 4.00000 0.278693
\(207\) 18.0000 1.25109
\(208\) −2.00000 −0.138675
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −10.0000 −0.668153
\(225\) −3.00000 −0.200000
\(226\) 14.0000 0.931266
\(227\) 26.0000 1.72568 0.862840 0.505477i \(-0.168683\pi\)
0.862840 + 0.505477i \(0.168683\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −6.00000 −0.392232
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 12.0000 0.762001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −6.00000 −0.377964
\(253\) −24.0000 −1.50887
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 20.0000 1.24274
\(260\) 2.00000 0.124035
\(261\) −6.00000 −0.371391
\(262\) 6.00000 0.370681
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −4.00000 −0.239904
\(279\) 12.0000 0.718421
\(280\) −6.00000 −0.358569
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 4.00000 0.236113
\(288\) −15.0000 −0.883883
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) −10.0000 −0.572598
\(306\) −18.0000 −1.02899
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 10.0000 0.564333
\(315\) −6.00000 −0.338062
\(316\) −2.00000 −0.112509
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) −36.0000 −2.00309
\(324\) −9.00000 −0.500000
\(325\) 2.00000 0.110940
\(326\) 18.0000 0.996928
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) −10.0000 −0.548821
\(333\) 30.0000 1.64399
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) −16.0000 −0.866449
\(342\) 18.0000 0.973329
\(343\) 20.0000 1.07990
\(344\) 18.0000 0.970495
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 20.0000 1.06600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −14.0000 −0.743043
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) −9.00000 −0.474342
\(361\) 17.0000 0.894737
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 6.00000 0.312772
\(369\) 6.00000 0.312348
\(370\) 10.0000 0.519875
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 18.0000 0.928279
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −2.00000 −0.101797
\(387\) 18.0000 0.914991
\(388\) 10.0000 0.507673
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) −2.00000 −0.100631
\(396\) 12.0000 0.603023
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 18.0000 0.902258
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −2.00000 −0.0995037
\(405\) −9.00000 −0.447214
\(406\) −4.00000 −0.198517
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 18.0000 0.884652
\(415\) −10.0000 −0.490881
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) −24.0000 −1.17388
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −6.00000 −0.292075
\(423\) 18.0000 0.875190
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 36.0000 1.72211
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 12.0000 0.572078
\(441\) 9.00000 0.428571
\(442\) 12.0000 0.570782
\(443\) 34.0000 1.61539 0.807694 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −10.0000 −0.473514
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −3.00000 −0.141421
\(451\) −8.00000 −0.376705
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 26.0000 1.22024
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 12.0000 0.550019
\(477\) 6.00000 0.274721
\(478\) 22.0000 1.00626
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −30.0000 −1.35804
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −12.0000 −0.539906
\(495\) 12.0000 0.539360
\(496\) 4.00000 0.179605
\(497\) −28.0000 −1.25597
\(498\) 0 0
\(499\) −2.00000 −0.0895323 −0.0447661 0.998997i \(-0.514254\pi\)
−0.0447661 + 0.998997i \(0.514254\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) −18.0000 −0.801784
\(505\) −2.00000 −0.0889988
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) −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
\(511\) −4.00000 −0.176950
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 20.0000 0.878750
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) −6.00000 −0.262613
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −10.0000 −0.429537
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 2.00000 0.0854358
\(549\) −30.0000 −1.28037
\(550\) 4.00000 0.170561
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 12.0000 0.508001
\(559\) −12.0000 −0.507546
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −14.0000 −0.588464
\(567\) −18.0000 −0.755929
\(568\) −42.0000 −1.76228
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) −6.00000 −0.250217
\(576\) −21.0000 −0.875000
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) −20.0000 −0.829740
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) −6.00000 −0.248282
\(585\) 6.00000 0.248069
\(586\) 10.0000 0.413096
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) 10.0000 0.408589 0.204294 0.978909i \(-0.434510\pi\)
0.204294 + 0.978909i \(0.434510\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −12.0000 −0.485468
\(612\) 18.0000 0.727607
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 24.0000 0.966988
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −60.0000 −2.39236
\(630\) −6.00000 −0.239046
\(631\) 10.0000 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) 26.0000 1.03259
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 8.00000 0.316723
\(639\) −42.0000 −1.66149
\(640\) 3.00000 0.118585
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −36.0000 −1.41640
\(647\) −34.0000 −1.33668 −0.668339 0.743857i \(-0.732994\pi\)
−0.668339 + 0.743857i \(0.732994\pi\)
\(648\) −27.0000 −1.06066
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −18.0000 −0.704934
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 2.00000 0.0780869
\(657\) −6.00000 −0.234082
\(658\) 12.0000 0.467809
\(659\) −46.0000 −1.79191 −0.895953 0.444149i \(-0.853506\pi\)
−0.895953 + 0.444149i \(0.853506\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 14.0000 0.544125
\(663\) 0 0
\(664\) −30.0000 −1.16423
\(665\) −12.0000 −0.465340
\(666\) 30.0000 1.16248
\(667\) −12.0000 −0.464642
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 18.0000 0.690268
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) −18.0000 −0.688247
\(685\) 2.00000 0.0764161
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −14.0000 −0.532200
\(693\) 24.0000 0.911685
\(694\) 6.00000 0.227757
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 60.0000 2.26294
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) −14.0000 −0.525411
\(711\) −6.00000 −0.225018
\(712\) 18.0000 0.674579
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) −3.00000 −0.111803
\(721\) −8.00000 −0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 12.0000 0.444750
\(729\) −27.0000 −1.00000
\(730\) −2.00000 −0.0740233
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) −30.0000 −1.10581
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) −6.00000 −0.219676
\(747\) −30.0000 −1.09764
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −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
\(760\) −18.0000 −0.652929
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) 18.0000 0.650791
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 18.0000 0.646997
\(775\) −4.00000 −0.143684
\(776\) 30.0000 1.07694
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 56.0000 2.00384
\(782\) −36.0000 −1.28736
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) −28.0000 −0.995565
\(792\) 36.0000 1.27920
\(793\) 20.0000 0.710221
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −18.0000 −0.637993
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 5.00000 0.176777
\(801\) 18.0000 0.635999
\(802\) 18.0000 0.635602
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) −12.0000 −0.422944
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) −9.00000 −0.316228
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) −6.00000 −0.209785
\(819\) 12.0000 0.419314
\(820\) −2.00000 −0.0698430
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −50.0000 −1.74289 −0.871445 0.490493i \(-0.836817\pi\)
−0.871445 + 0.490493i \(0.836817\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) −18.0000 −0.625543
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −10.0000 −0.347105
\(831\) 0 0
\(832\) 14.0000 0.485363
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) −16.0000 −0.552711
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −34.0000 −1.17172
\(843\) 0 0
\(844\) 6.00000 0.206529
\(845\) 9.00000 0.309609
\(846\) 18.0000 0.618853
\(847\) −10.0000 −0.343604
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −20.0000 −0.684386
\(855\) −18.0000 −0.615587
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −56.0000 −1.91070 −0.955348 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 14.0000 0.476842
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) −30.0000 −1.01944
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) 30.0000 1.01593
\(873\) 30.0000 1.01535
\(874\) 36.0000 1.21772
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −26.0000 −0.877958 −0.438979 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 16.0000 0.539974
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 9.00000 0.303046
\(883\) −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
\(886\) 34.0000 1.14225
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 6.00000 0.201120
\(891\) 36.0000 1.20605
\(892\) 10.0000 0.334825
\(893\) 36.0000 1.20469
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) 3.00000 0.100000
\(901\) −12.0000 −0.399778
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) −26.0000 −0.862840
\(909\) −6.00000 −0.199007
\(910\) 4.00000 0.132599
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 40.0000 1.32381
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) −18.0000 −0.593442
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 28.0000 0.921631
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −28.0000 −0.920137
\(927\) −12.0000 −0.394132
\(928\) 10.0000 0.328266
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) −24.0000 −0.784884
\(936\) 18.0000 0.588348
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) 0 0
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 50.0000 1.62478 0.812391 0.583113i \(-0.198166\pi\)
0.812391 + 0.583113i \(0.198166\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 36.0000 1.16677
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −22.0000 −0.711531
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −15.0000 −0.482118
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 26.0000 0.834380 0.417190 0.908819i \(-0.363015\pi\)
0.417190 + 0.908819i \(0.363015\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) −3.00000 −0.0958315
\(981\) 30.0000 0.957826
\(982\) 22.0000 0.702048
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 36.0000 1.14473
\(990\) 12.0000 0.381385
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −20.0000 −0.635001
\(993\) 0 0
\(994\) −28.0000 −0.888106
\(995\) −18.0000 −0.570638
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −2.00000 −0.0633089
\(999\) 0 0
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 8045.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8045.2.a.a.1.1 1 1.1 even 1 trivial