Properties

Label 6045.2.a.d.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} -1.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} -7.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{28} -8.00000 q^{29} +1.00000 q^{31} +3.00000 q^{33} +3.00000 q^{35} -2.00000 q^{36} -9.00000 q^{37} +1.00000 q^{39} -5.00000 q^{41} -2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} -8.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} +2.00000 q^{52} -9.00000 q^{53} +3.00000 q^{55} -6.00000 q^{57} -12.0000 q^{59} -2.00000 q^{60} -5.00000 q^{61} -3.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} +7.00000 q^{69} -5.00000 q^{71} -10.0000 q^{73} -1.00000 q^{75} -12.0000 q^{76} +9.00000 q^{77} +11.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{83} -6.00000 q^{84} +1.00000 q^{85} +8.00000 q^{87} +7.00000 q^{89} +3.00000 q^{91} +14.0000 q^{92} -1.00000 q^{93} -6.00000 q^{95} +11.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 2.00000 0.577350
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 2.00000 0.447214
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 6.00000 1.13389
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) −2.00000 −0.333333
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 6.00000 0.904534
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −4.00000 −0.577350
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −2.00000 −0.258199
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) −8.00000 −1.00000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) −12.0000 −1.37649
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −6.00000 −0.654654
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 14.0000 1.45960
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) −2.00000 −0.200000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 15.0000 1.45010 0.725052 0.688694i \(-0.241816\pi\)
0.725052 + 0.688694i \(0.241816\pi\)
\(108\) 2.00000 0.192450
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) −12.0000 −1.13389
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) 16.0000 1.48556
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −6.00000 −0.522233
\(133\) −18.0000 −1.56080
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) −23.0000 −1.95083 −0.975417 0.220366i \(-0.929275\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) −6.00000 −0.507093
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 4.00000 0.333333
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 18.0000 1.47959
\(149\) 1.00000 0.0819232 0.0409616 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) −2.00000 −0.160128
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 21.0000 1.65503
\(162\) 0 0
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 10.0000 0.780869
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) −12.0000 −0.904534
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 2.00000 0.149071
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 16.0000 1.16692
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 8.00000 0.577350
\(193\) −3.00000 −0.215945 −0.107972 0.994154i \(-0.534436\pi\)
−0.107972 + 0.994154i \(0.534436\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) −4.00000 −0.285714
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) −2.00000 −0.140028
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) −7.00000 −0.486534
\(208\) −4.00000 −0.277350
\(209\) −18.0000 −1.24509
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 18.0000 1.23625
\(213\) 5.00000 0.342594
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) −6.00000 −0.404520
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 12.0000 0.794719
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) −9.00000 −0.592157
\(232\) 0 0
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 24.0000 1.56227
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 4.00000 0.258199
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 6.00000 0.377964
\(253\) 21.0000 1.32026
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 16.0000 1.00000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) 27.0000 1.67770
\(260\) −2.00000 −0.124035
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −7.00000 −0.428393
\(268\) 8.00000 0.488678
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −4.00000 −0.242536
\(273\) −3.00000 −0.181568
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) −14.0000 −0.842701
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 10.0000 0.593391
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 15.0000 0.885422
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −11.0000 −0.644831
\(292\) 20.0000 1.17041
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) 7.00000 0.404820
\(300\) 2.00000 0.115470
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 24.0000 1.37649
\(305\) 5.00000 0.286299
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) −18.0000 −1.02565
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) −22.0000 −1.23760
\(317\) −8.00000 −0.449325 −0.224662 0.974437i \(-0.572128\pi\)
−0.224662 + 0.974437i \(0.572128\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 8.00000 0.447214
\(321\) −15.0000 −0.837218
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) −2.00000 −0.111111
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 12.0000 0.658586
\(333\) −9.00000 −0.493197
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 12.0000 0.654654
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) −3.00000 −0.162459
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −7.00000 −0.376867
\(346\) 0 0
\(347\) −23.0000 −1.23470 −0.617352 0.786687i \(-0.711795\pi\)
−0.617352 + 0.786687i \(0.711795\pi\)
\(348\) −16.0000 −0.857690
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) 5.00000 0.265372
\(356\) −14.0000 −0.741999
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) −6.00000 −0.314485
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) −28.0000 −1.45960
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 27.0000 1.40177
\(372\) 2.00000 0.103695
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 12.0000 0.615587
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) −22.0000 −1.11688
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) 7.00000 0.354005
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) −11.0000 −0.553470
\(396\) 6.00000 0.301511
\(397\) 17.0000 0.853206 0.426603 0.904439i \(-0.359710\pi\)
0.426603 + 0.904439i \(0.359710\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 4.00000 0.200000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −1.00000 −0.0498135
\(404\) −4.00000 −0.199007
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 27.0000 1.33834
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 16.0000 0.788263
\(413\) 36.0000 1.77144
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 23.0000 1.12631
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 6.00000 0.292770
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 15.0000 0.725901
\(428\) −30.0000 −1.45010
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) −4.00000 −0.192450
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) −28.0000 −1.34096
\(437\) −42.0000 −2.00913
\(438\) 0 0
\(439\) 15.0000 0.715911 0.357955 0.933739i \(-0.383474\pi\)
0.357955 + 0.933739i \(0.383474\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 15.0000 0.712672 0.356336 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) −18.0000 −0.854242
\(445\) −7.00000 −0.331832
\(446\) 0 0
\(447\) −1.00000 −0.0472984
\(448\) 24.0000 1.13389
\(449\) −31.0000 −1.46298 −0.731490 0.681852i \(-0.761175\pi\)
−0.731490 + 0.681852i \(0.761175\pi\)
\(450\) 0 0
\(451\) 15.0000 0.706322
\(452\) 12.0000 0.564433
\(453\) −12.0000 −0.563809
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −35.0000 −1.63723 −0.818615 0.574342i \(-0.805258\pi\)
−0.818615 + 0.574342i \(0.805258\pi\)
\(458\) 0 0
\(459\) 1.00000 0.0466760
\(460\) −14.0000 −0.652753
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) −5.00000 −0.232370 −0.116185 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) −32.0000 −1.48556
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 2.00000 0.0924500
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) −6.00000 −0.275010
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 0 0
\(481\) 9.00000 0.410365
\(482\) 0 0
\(483\) −21.0000 −0.955533
\(484\) 4.00000 0.181818
\(485\) −11.0000 −0.499484
\(486\) 0 0
\(487\) 5.00000 0.226572 0.113286 0.993562i \(-0.463862\pi\)
0.113286 + 0.993562i \(0.463862\pi\)
\(488\) 0 0
\(489\) −17.0000 −0.768767
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) −10.0000 −0.450835
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 4.00000 0.179605
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 2.00000 0.0894427
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 32.0000 1.41977
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 30.0000 1.32712
\(512\) 0 0
\(513\) −6.00000 −0.264906
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) −4.00000 −0.176090
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) −12.0000 −0.524222
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −1.00000 −0.0435607
\(528\) 12.0000 0.522233
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 36.0000 1.56080
\(533\) 5.00000 0.216574
\(534\) 0 0
\(535\) −15.0000 −0.648507
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) −2.00000 −0.0860663
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) 1.00000 0.0429141
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 32.0000 1.36697
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) −33.0000 −1.40330
\(554\) 0 0
\(555\) −9.00000 −0.382029
\(556\) 46.0000 1.95083
\(557\) 20.0000 0.847427 0.423714 0.905796i \(-0.360726\pi\)
0.423714 + 0.905796i \(0.360726\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 12.0000 0.507093
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) −16.0000 −0.673722
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) −6.00000 −0.250873
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) −8.00000 −0.333333
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) 0 0
\(579\) 3.00000 0.124676
\(580\) −16.0000 −0.664364
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 32.0000 1.32078 0.660391 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 4.00000 0.164957
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) −36.0000 −1.47959
\(593\) −44.0000 −1.80686 −0.903432 0.428732i \(-0.858960\pi\)
−0.903432 + 0.428732i \(0.858960\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −2.00000 −0.0819232
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 33.0000 1.34610 0.673049 0.739598i \(-0.264984\pi\)
0.673049 + 0.739598i \(0.264984\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −24.0000 −0.976546
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 2.00000 0.0808452
\(613\) −43.0000 −1.73675 −0.868377 0.495905i \(-0.834836\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(614\) 0 0
\(615\) −5.00000 −0.201619
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 2.00000 0.0803219
\(621\) 7.00000 0.280900
\(622\) 0 0
\(623\) −21.0000 −0.841347
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 18.0000 0.718851
\(628\) −8.00000 −0.319235
\(629\) 9.00000 0.358854
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) −18.0000 −0.713746
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −5.00000 −0.197797
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) −42.0000 −1.65503
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) −34.0000 −1.33154
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −20.0000 −0.780869
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 50.0000 1.94772 0.973862 0.227142i \(-0.0729380\pi\)
0.973862 + 0.227142i \(0.0729380\pi\)
\(660\) 6.00000 0.233550
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) −1.00000 −0.0388368
\(664\) 0 0
\(665\) 18.0000 0.698010
\(666\) 0 0
\(667\) 56.0000 2.16833
\(668\) −16.0000 −0.619059
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 15.0000 0.579069
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) −2.00000 −0.0769231
\(677\) −21.0000 −0.807096 −0.403548 0.914959i \(-0.632223\pi\)
−0.403548 + 0.914959i \(0.632223\pi\)
\(678\) 0 0
\(679\) −33.0000 −1.26642
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −12.0000 −0.458831
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) −8.00000 −0.304997
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −28.0000 −1.06440
\(693\) 9.00000 0.341882
\(694\) 0 0
\(695\) 23.0000 0.872440
\(696\) 0 0
\(697\) 5.00000 0.189389
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 6.00000 0.226779
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −54.0000 −2.03665
\(704\) 24.0000 0.904534
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) −24.0000 −0.901975
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 11.0000 0.412532
\(712\) 0 0
\(713\) −7.00000 −0.262152
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 40.0000 1.49487
\(717\) 15.0000 0.560185
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) −4.00000 −0.149071
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 6.00000 0.223142
\(724\) 2.00000 0.0743294
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) −10.0000 −0.369611
\(733\) −3.00000 −0.110808 −0.0554038 0.998464i \(-0.517645\pi\)
−0.0554038 + 0.998464i \(0.517645\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −18.0000 −0.661693
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −1.00000 −0.0366372
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) −6.00000 −0.219382
\(749\) −45.0000 −1.64426
\(750\) 0 0
\(751\) −17.0000 −0.620339 −0.310169 0.950681i \(-0.600386\pi\)
−0.310169 + 0.950681i \(0.600386\pi\)
\(752\) −32.0000 −1.16692
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) −6.00000 −0.218218
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) −21.0000 −0.762252
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −42.0000 −1.52050
\(764\) −20.0000 −0.723575
\(765\) 1.00000 0.0361551
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) −16.0000 −0.577350
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) −10.0000 −0.360141
\(772\) 6.00000 0.215945
\(773\) 44.0000 1.58257 0.791285 0.611448i \(-0.209412\pi\)
0.791285 + 0.611448i \(0.209412\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) −27.0000 −0.968620
\(778\) 0 0
\(779\) −30.0000 −1.07486
\(780\) 2.00000 0.0716115
\(781\) 15.0000 0.536742
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 8.00000 0.285714
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 5.00000 0.177555
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) −32.0000 −1.13421
\(797\) −55.0000 −1.94820 −0.974100 0.226118i \(-0.927397\pi\)
−0.974100 + 0.226118i \(0.927397\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 7.00000 0.247333
\(802\) 0 0
\(803\) 30.0000 1.05868
\(804\) −8.00000 −0.282138
\(805\) −21.0000 −0.740153
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) 0 0
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) −48.0000 −1.68447
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −17.0000 −0.595484
\(816\) 4.00000 0.140028
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 3.00000 0.104828
\(820\) −10.0000 −0.349215
\(821\) −35.0000 −1.22151 −0.610754 0.791820i \(-0.709134\pi\)
−0.610754 + 0.791820i \(0.709134\pi\)
\(822\) 0 0
\(823\) −12.0000 −0.418294 −0.209147 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 14.0000 0.486534
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 8.00000 0.277350
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 36.0000 1.24509
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −21.0000 −0.725001 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −22.0000 −0.757720
\(844\) −40.0000 −1.37686
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) −36.0000 −1.23625
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) 63.0000 2.15961
\(852\) −10.0000 −0.342594
\(853\) −41.0000 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) 41.0000 1.39890 0.699451 0.714681i \(-0.253428\pi\)
0.699451 + 0.714681i \(0.253428\pi\)
\(860\) −4.00000 −0.136399
\(861\) −15.0000 −0.511199
\(862\) 0 0
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 6.00000 0.203653
\(869\) −33.0000 −1.11945
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 11.0000 0.372294
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) −20.0000 −0.675737
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) 12.0000 0.404520
\(881\) 44.0000 1.48240 0.741199 0.671286i \(-0.234258\pi\)
0.741199 + 0.671286i \(0.234258\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) −7.00000 −0.235037 −0.117518 0.993071i \(-0.537494\pi\)
−0.117518 + 0.993071i \(0.537494\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 48.0000 1.60716
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) −7.00000 −0.233723
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) 1.00000 0.0332411
\(906\) 0 0
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) −36.0000 −1.19470
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 50.0000 1.65657 0.828287 0.560304i \(-0.189316\pi\)
0.828287 + 0.560304i \(0.189316\pi\)
\(912\) −24.0000 −0.794719
\(913\) 18.0000 0.595713
\(914\) 0 0
\(915\) −5.00000 −0.165295
\(916\) 52.0000 1.71813
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 25.0000 0.824674 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) 0 0
\(923\) 5.00000 0.164577
\(924\) 18.0000 0.592157
\(925\) −9.00000 −0.295918
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 18.0000 0.589610
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) −16.0000 −0.521862
\(941\) −41.0000 −1.33656 −0.668281 0.743909i \(-0.732970\pi\)
−0.668281 + 0.743909i \(0.732970\pi\)
\(942\) 0 0
\(943\) 35.0000 1.13976
\(944\) −48.0000 −1.56227
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 22.0000 0.714527
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 8.00000 0.259418
\(952\) 0 0
\(953\) −25.0000 −0.809829 −0.404915 0.914354i \(-0.632699\pi\)
−0.404915 + 0.914354i \(0.632699\pi\)
\(954\) 0 0
\(955\) −10.0000 −0.323592
\(956\) 30.0000 0.970269
\(957\) −24.0000 −0.775810
\(958\) 0 0
\(959\) 48.0000 1.55000
\(960\) −8.00000 −0.258199
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 15.0000 0.483368
\(964\) 12.0000 0.386494
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 2.00000 0.0641500
\(973\) 69.0000 2.21204
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) −20.0000 −0.640184
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 0 0
\(979\) −21.0000 −0.671163
\(980\) 4.00000 0.127775
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −26.0000 −0.829271 −0.414636 0.909988i \(-0.636091\pi\)
−0.414636 + 0.909988i \(0.636091\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 12.0000 0.381771
\(989\) 14.0000 0.445174
\(990\) 0 0
\(991\) 29.0000 0.921215 0.460608 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) 0 0
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −12.0000 −0.380235
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6045.2.a.d.1.1 1 1.1 even 1 trivial