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Properties

Label 6045.2.a.g.1.1
Level $6045$
Weight $2$
Character 6045.1
Self dual yes
Analytic conductor $48.270$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
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 \(\chi\) \(=\) 6045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +9.00000 q^{29} +1.00000 q^{31} +6.00000 q^{33} +1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +1.00000 q^{39} -9.00000 q^{41} -1.00000 q^{43} -12.0000 q^{44} -1.00000 q^{45} -6.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} +6.00000 q^{51} -2.00000 q^{52} -12.0000 q^{53} -6.00000 q^{55} +2.00000 q^{57} +3.00000 q^{59} +2.00000 q^{60} +8.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} +5.00000 q^{67} -12.0000 q^{68} -6.00000 q^{69} -16.0000 q^{73} +1.00000 q^{75} -4.00000 q^{76} -6.00000 q^{77} +8.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +9.00000 q^{87} +6.00000 q^{89} -1.00000 q^{91} +12.0000 q^{92} +1.00000 q^{93} -2.00000 q^{95} +17.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) −2.00000 −0.577350
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −12.0000 −1.80907
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 4.00000 0.577350
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 2.00000 0.258199
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −12.0000 −1.45521
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 2.00000 0.218218
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 12.0000 1.25109
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 − 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) −2.00000 −0.200000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −9.00000 −0.870063 −0.435031 − 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) −2.00000 −0.192450
\(109\) 20.0000 1.91565 0.957826 − 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −4.00000 −0.377964
\(113\) 15.0000 1.41108 0.705541 − 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) −18.0000 −1.67126
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −12.0000 −1.04447
\(133\) −2.00000 −0.173422
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −2.00000 −0.169031
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 4.00000 0.333333
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) −4.00000 −0.328798
\(149\) 3.00000 0.245770 0.122885 − 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 − 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) −2.00000 −0.160128
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 18.0000 1.40556
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) −15.0000 −1.14043 −0.570214 − 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 24.0000 1.80907
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 − 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 2.00000 0.149071
\(181\) −16.0000 −1.18927 −0.594635 − 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 36.0000 2.63258
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 − 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −8.00000 −0.577350
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 12.0000 0.857143
\(197\) 3.00000 0.213741 0.106871 − 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) −12.0000 −0.840168
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 4.00000 0.277350
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 − 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 24.0000 1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 12.0000 0.809040
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.00000 −0.398234 −0.199117 − 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) −4.00000 −0.264906
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 − 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −6.00000 −0.390567
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −4.00000 −0.258199
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −16.0000 −1.02430
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 − 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 2.00000 0.125988
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 16.0000 1.00000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 2.00000 0.124035
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 − 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −10.0000 −0.610847
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 − 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 24.0000 1.45521
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 12.0000 0.722315
\(277\) −19.0000 −1.14160 −0.570800 − 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 − 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) 32.0000 1.87266
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) 0 0
\(297\) 6.00000 0.348155
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) −2.00000 −0.115470
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 12.0000 0.683763
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) −16.0000 −0.900070
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 54.0000 3.02342
\(320\) 8.00000 0.447214
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) −2.00000 −0.111111
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 20.0000 1.10600
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 − 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) −18.0000 −0.987878
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −5.00000 −0.273179
\(336\) −4.00000 −0.218218
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) 12.0000 0.650791
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −6.00000 −0.322097 −0.161048 − 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −18.0000 −0.964901
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −33.0000 −1.75641 −0.878206 − 0.478282i \(-0.841260\pi\)
−0.878206 + 0.478282i \(0.841260\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) −3.00000 −0.158334 −0.0791670 − 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 25.0000 1.31216
\(364\) 2.00000 0.104828
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −24.0000 −1.25109
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −2.00000 −0.103695
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) −7.00000 −0.358621
\(382\) 0 0
\(383\) −3.00000 −0.153293 −0.0766464 − 0.997058i \(-0.524421\pi\)
−0.0766464 + 0.997058i \(0.524421\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) −34.0000 −1.72609
\(389\) −21.0000 −1.06474 −0.532371 − 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) −12.0000 −0.603023
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 4.00000 0.200000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 1.00000 0.0498135
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 8.00000 0.394132
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 − 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −28.0000 −1.36464 −0.682318 − 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 18.0000 0.870063
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −27.0000 −1.30054 −0.650272 − 0.759701i \(-0.725345\pi\)
−0.650272 + 0.759701i \(0.725345\pi\)
\(432\) 4.00000 0.192450
\(433\) 11.0000 0.528626 0.264313 − 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) −40.0000 −1.91565
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −19.0000 −0.906821 −0.453410 − 0.891302i \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 − 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) −4.00000 −0.189832
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 3.00000 0.141895
\(448\) 8.00000 0.377964
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −54.0000 −2.54276
\(452\) −30.0000 −1.41108
\(453\) 17.0000 0.798730
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) −12.0000 −0.559503
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 36.0000 1.67126
\(465\) −1.00000 −0.0463739
\(466\) 0 0
\(467\) 27.0000 1.24941 0.624705 − 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −5.00000 −0.230879
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 12.0000 0.550019
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 39.0000 1.78196 0.890978 − 0.454047i \(-0.150020\pi\)
0.890978 + 0.454047i \(0.150020\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 6.00000 0.273009
\(484\) −50.0000 −2.27273
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 − 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 18.0000 0.811503
\(493\) 54.0000 2.43204
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 − 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 2.00000 0.0894427
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 33.0000 1.47140 0.735699 − 0.677309i \(-0.236854\pi\)
0.735699 + 0.677309i \(0.236854\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 14.0000 0.621150
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 4.00000 0.176261
\(516\) 2.00000 0.0880451
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) −24.0000 −1.05146 −0.525730 − 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 − 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 24.0000 1.04447
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 4.00000 0.173422
\(533\) −9.00000 −0.389833
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) 0 0
\(537\) 9.00000 0.388379
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 2.00000 0.0860663
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 12.0000 0.512615
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) −28.0000 −1.18746
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 4.00000 0.169031
\(561\) 36.0000 1.51992
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 12.0000 0.505291
\(565\) −15.0000 −0.631055
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −12.0000 −0.501745
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) −8.00000 −0.333333
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) −13.0000 −0.540262
\(580\) 18.0000 0.747409
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) −72.0000 −2.98194
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −9.00000 −0.371470 −0.185735 − 0.982600i \(-0.559467\pi\)
−0.185735 + 0.982600i \(0.559467\pi\)
\(588\) 12.0000 0.494872
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 3.00000 0.123404
\(592\) 8.00000 0.328798
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −6.00000 −0.245770
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) −34.0000 −1.38344
\(605\) −25.0000 −1.01639
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −12.0000 −0.485071
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 − 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 0 0
\(619\) −1.00000 −0.0401934 −0.0200967 − 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 2.00000 0.0803219
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) −28.0000 −1.11732
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 7.00000 0.277787
\(636\) 24.0000 0.951662
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 − 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 − 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) −12.0000 −0.472866
\(645\) 1.00000 0.0393750
\(646\) 0 0
\(647\) −30.0000 −1.17942 −0.589711 − 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) 32.0000 1.25322
\(653\) −15.0000 −0.586995 −0.293498 − 0.955960i \(-0.594819\pi\)
−0.293498 + 0.955960i \(0.594819\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36.0000 −1.40556
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 12.0000 0.467099
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) −54.0000 −2.09089
\(668\) −48.0000 −1.85718
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) 0 0
\(673\) 35.0000 1.34915 0.674575 − 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) −2.00000 −0.0769231
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) −17.0000 −0.652400
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 − 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) −4.00000 −0.152944
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 30.0000 1.14043
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) −54.0000 −2.04540
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 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
\(703\) 4.00000 0.150863
\(704\) −48.0000 −1.80907
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) 26.0000 0.976450 0.488225 − 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) −18.0000 −0.672692
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) −21.0000 −0.783168 −0.391584 − 0.920142i \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(720\) −4.00000 −0.149071
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 17.0000 0.632237
\(724\) 32.0000 1.18927
\(725\) 9.00000 0.334252
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) −16.0000 −0.591377
\(733\) −13.0000 −0.480166 −0.240083 − 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 4.00000 0.147043
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 39.0000 1.43077 0.715386 − 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 0 0
\(747\) 9.00000 0.329293
\(748\) −72.0000 −2.63258
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 − 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) −24.0000 −0.875190
\(753\) 15.0000 0.546630
\(754\) 0 0
\(755\) −17.0000 −0.618693
\(756\) 2.00000 0.0727393
\(757\) −7.00000 −0.254419 −0.127210 − 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(758\) 0 0
\(759\) −36.0000 −1.30672
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 12.0000 0.434145
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 3.00000 0.108324
\(768\) 16.0000 0.577350
\(769\) −4.00000 −0.144244 −0.0721218 − 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 26.0000 0.935760
\(773\) 21.0000 0.755318 0.377659 − 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) −24.0000 −0.857143
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 − 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −6.00000 −0.213741
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 32.0000 1.13421
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −96.0000 −3.38777
\(804\) −10.0000 −0.352673
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) −27.0000 −0.949269 −0.474635 − 0.880183i \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 − 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 18.0000 0.631676
\(813\) 11.0000 0.385787
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 24.0000 0.840168
\(817\) −2.00000 −0.0699711
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) −18.0000 −0.628587
\(821\) 54.0000 1.88461 0.942306 − 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 − 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 − 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 12.0000 0.417029
\(829\) −52.0000 −1.80603 −0.903017 − 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) −8.00000 −0.277350
\(833\) −36.0000 −1.24733
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) −24.0000 −0.830057
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) −39.0000 −1.34643 −0.673215 − 0.739447i \(-0.735087\pi\)
−0.673215 + 0.739447i \(0.735087\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) −15.0000 −0.516627
\(844\) −10.0000 −0.344214
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) −48.0000 −1.64833
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −19.0000 −0.650548 −0.325274 − 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 9.00000 0.306719
\(862\) 0 0
\(863\) 33.0000 1.12333 0.561667 − 0.827364i \(-0.310160\pi\)
0.561667 + 0.827364i \(0.310160\pi\)
\(864\) 0 0
\(865\) 15.0000 0.510015
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 2.00000 0.0678844
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 5.00000 0.169419
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 32.0000 1.08118
\(877\) 50.0000 1.68838 0.844190 − 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) −24.0000 −0.809040
\(881\) 39.0000 1.31394 0.656972 − 0.753915i \(-0.271837\pi\)
0.656972 + 0.753915i \(0.271837\pi\)
\(882\) 0 0
\(883\) 11.0000 0.370179 0.185090 − 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) −12.0000 −0.403604
\(885\) −3.00000 −0.100844
\(886\) 0 0
\(887\) −21.0000 −0.705111 −0.352555 − 0.935791i \(-0.614687\pi\)
−0.352555 + 0.935791i \(0.614687\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) −16.0000 −0.535720
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −9.00000 −0.300837
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) −2.00000 −0.0666667
\(901\) −72.0000 −2.39867
\(902\) 0 0
\(903\) 1.00000 0.0332779
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 − 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) −57.0000 −1.88849 −0.944247 − 0.329238i \(-0.893208\pi\)
−0.944247 + 0.329238i \(0.893208\pi\)
\(912\) 8.00000 0.264906
\(913\) 54.0000 1.78714
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 44.0000 1.45380
\(917\) 0 0
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) −18.0000 −0.589610
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) −36.0000 −1.17733
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) −1.00000 −0.0326338
\(940\) −12.0000 −0.391397
\(941\) 48.0000 1.56476 0.782378 − 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 54.0000 1.75848
\(944\) 12.0000 0.390567
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −16.0000 −0.519656
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 − 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) −24.0000 −0.776215
\(957\) 54.0000 1.74557
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 8.00000 0.258199
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −9.00000 −0.290021
\(964\) −34.0000 −1.09507
\(965\) 13.0000 0.418485
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 − 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −14.0000 −0.448819
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 32.0000 1.02430
\(977\) 12.0000 0.383914 0.191957 − 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) −12.0000 −0.383326
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) 45.0000 1.43528 0.717639 − 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 0 0
\(985\) −3.00000 −0.0955879
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) −4.00000 −0.127257
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 − 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) −19.0000 −0.602947
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) −18.0000 −0.570352
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 6045.2.a.g.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6045.2.a.g.1.1 ✓ 1 1.1 even 1 trivial