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Properties

Label 6040.2.a.h.1.1
Level $6040$
Weight $2$
Character 6040.1
Self dual yes
Analytic conductor $48.230$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, 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("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.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: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.00000 q^{17} -5.00000 q^{19} +4.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -3.00000 q^{29} -7.00000 q^{31} -3.00000 q^{33} +4.00000 q^{35} -2.00000 q^{37} +1.00000 q^{39} +2.00000 q^{43} -2.00000 q^{45} -10.0000 q^{47} +9.00000 q^{49} -6.00000 q^{51} -2.00000 q^{53} -3.00000 q^{55} -5.00000 q^{57} +3.00000 q^{59} -10.0000 q^{61} -8.00000 q^{63} +1.00000 q^{65} +1.00000 q^{67} +3.00000 q^{69} -2.00000 q^{71} +1.00000 q^{73} +1.00000 q^{75} -12.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -11.0000 q^{83} -6.00000 q^{85} -3.00000 q^{87} +8.00000 q^{89} +4.00000 q^{91} -7.00000 q^{93} -5.00000 q^{95} -8.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −8.00000 −1.00791
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 1.00000 0.122169 0.0610847 − 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −7.00000 −0.725866
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 − 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 − 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −20.0000 −1.73422
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) 0 0
\(149\) 8.00000 0.655386 0.327693 − 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) −7.00000 −0.562254
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 − 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) −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
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 10.0000 0.764719
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) −26.0000 −1.94333 −0.971666 − 0.236360i \(-0.924046\pi\)
−0.971666 + 0.236360i \(0.924046\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 18.0000 1.31629
\(188\) 0 0
\(189\) −20.0000 −1.45479
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 − 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 1.00000 0.0705346
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) −2.00000 −0.137038
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −28.0000 −1.90076
\(218\) 0 0
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 − 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) −11.0000 −0.720634 −0.360317 − 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 − 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 21.0000 1.35273 0.676364 − 0.736567i \(-0.263554\pi\)
0.676364 + 0.736567i \(0.263554\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) 0 0
\(249\) −11.0000 −0.697097
\(250\) 0 0
\(251\) 9.00000 0.568075 0.284037 − 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) 11.0000 0.686161 0.343081 − 0.939306i \(-0.388530\pi\)
0.343081 + 0.939306i \(0.388530\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −21.0000 −1.29492 −0.647458 − 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) 0 0
\(269\) −1.00000 −0.0609711 −0.0304855 − 0.999535i \(-0.509705\pi\)
−0.0304855 + 0.999535i \(0.509705\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 − 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 − 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) 14.0000 0.838158
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 − 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 7.00000 0.416107 0.208053 − 0.978117i \(-0.433287\pi\)
0.208053 + 0.978117i \(0.433287\pi\)
\(284\) 0 0
\(285\) −5.00000 −0.296174
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 33.0000 1.87126 0.935629 − 0.352985i \(-0.114833\pi\)
0.935629 + 0.352985i \(0.114833\pi\)
\(312\) 0 0
\(313\) −20.0000 −1.13047 −0.565233 − 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 0 0
\(315\) −8.00000 −0.450749
\(316\) 0 0
\(317\) −27.0000 −1.51647 −0.758236 − 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 18.0000 0.995402
\(328\) 0 0
\(329\) −40.0000 −2.20527
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 1.00000 0.0546358
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 − 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 21.0000 1.13721
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 − 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −19.0000 −1.01127 −0.505634 − 0.862748i \(-0.668741\pi\)
−0.505634 + 0.862748i \(0.668741\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) 0 0
\(357\) −24.0000 −1.27021
\(358\) 0 0
\(359\) −18.0000 −0.950004 −0.475002 − 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 1.00000 0.0523424
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 − 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 21.0000 1.08734 0.543669 − 0.839299i \(-0.317035\pi\)
0.543669 + 0.839299i \(0.317035\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 18.0000 0.924598 0.462299 − 0.886724i \(-0.347025\pi\)
0.462299 + 0.886724i \(0.347025\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 − 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 0 0
\(399\) −20.0000 −1.00125
\(400\) 0 0
\(401\) −38.0000 −1.89763 −0.948815 − 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 0 0
\(403\) −7.00000 −0.348695
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −11.0000 −0.539969
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 − 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 − 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 0 0
\(423\) 20.0000 0.972433
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 0 0
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 0 0
\(437\) −15.0000 −0.717547
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 − 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 − 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) 8.00000 0.378387
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.00000 −0.0469841
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −16.0000 −0.748448 −0.374224 − 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 0 0
\(459\) 30.0000 1.40028
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) −7.00000 −0.324617
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 − 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) 2.00000 0.0913823 0.0456912 − 0.998956i \(-0.485451\pi\)
0.0456912 + 0.998956i \(0.485451\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 − 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) −2.00000 −0.0891756 −0.0445878 − 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) 12.0000 0.531891 0.265945 − 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) 25.0000 1.10378
\(514\) 0 0
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −23.0000 −1.00765 −0.503824 − 0.863806i \(-0.668074\pi\)
−0.503824 + 0.863806i \(0.668074\pi\)
\(522\) 0 0
\(523\) 27.0000 1.18063 0.590314 − 0.807174i \(-0.299004\pi\)
0.590314 + 0.807174i \(0.299004\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 42.0000 1.82955
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) 0 0
\(537\) −26.0000 −1.12198
\(538\) 0 0
\(539\) −27.0000 −1.16297
\(540\) 0 0
\(541\) −23.0000 −0.988847 −0.494424 − 0.869221i \(-0.664621\pi\)
−0.494424 + 0.869221i \(0.664621\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) −30.0000 −1.28271 −0.641354 − 0.767245i \(-0.721627\pi\)
−0.641354 + 0.767245i \(0.721627\pi\)
\(548\) 0 0
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) 15.0000 0.639021
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 25.0000 1.05928 0.529642 − 0.848221i \(-0.322326\pi\)
0.529642 + 0.848221i \(0.322326\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) −20.0000 −0.842900 −0.421450 − 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 − 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) −12.0000 −0.499567 −0.249783 − 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −44.0000 −1.82543
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 39.0000 1.60154 0.800769 − 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) −24.0000 −0.983904
\(596\) 0 0
\(597\) 18.0000 0.736691
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 − 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 − 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) −10.0000 −0.404557
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.00000 −0.120775 −0.0603877 − 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 0 0
\(623\) 32.0000 1.28205
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.0000 0.599042
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 − 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 0 0
\(633\) 10.0000 0.397464
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 9.00000 0.356593
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −3.00000 −0.118493 −0.0592464 − 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) −28.0000 −1.09741
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −43.0000 −1.67504 −0.837521 − 0.546405i \(-0.815996\pi\)
−0.837521 + 0.546405i \(0.815996\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 − 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 0 0
\(663\) −6.00000 −0.233021
\(664\) 0 0
\(665\) −20.0000 −0.775567
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) −34.0000 −1.30673 −0.653363 − 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 31.0000 1.18618 0.593091 − 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 15.0000 0.572286
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 − 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −11.0000 −0.416058
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 − 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) −10.0000 −0.376622
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000 0.0375558 0.0187779 − 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −21.0000 −0.786456
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 0 0
\(717\) 9.00000 0.336111
\(718\) 0 0
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 0 0
\(721\) −56.0000 −2.08555
\(722\) 0 0
\(723\) 21.0000 0.780998
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 9.00000 0.332423 0.166211 − 0.986090i \(-0.446847\pi\)
0.166211 + 0.986090i \(0.446847\pi\)
\(734\) 0 0
\(735\) 9.00000 0.331970
\(736\) 0 0
\(737\) −3.00000 −0.110506
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 − 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) −5.00000 −0.183680
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 − 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 0 0
\(747\) 22.0000 0.804938
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 − 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) −1.00000 −0.0363937
\(756\) 0 0
\(757\) 4.00000 0.145382 0.0726912 − 0.997354i \(-0.476841\pi\)
0.0726912 + 0.997354i \(0.476841\pi\)
\(758\) 0 0
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 72.0000 2.60658
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 0 0
\(767\) 3.00000 0.108324
\(768\) 0 0
\(769\) −26.0000 −0.937584 −0.468792 − 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 11.0000 0.396155
\(772\) 0 0
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) −7.00000 −0.251447
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) −13.0000 −0.463990
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) −21.0000 −0.747620
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 60.0000 2.12265
\(800\) 0 0
\(801\) −16.0000 −0.565332
\(802\) 0 0
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −1.00000 −0.0352017
\(808\) 0 0
\(809\) 44.0000 1.54696 0.773479 − 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) 1.00000 0.0350285
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) 0 0
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −22.0000 −0.766872 −0.383436 − 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) 18.0000 0.625921 0.312961 − 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 − 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) −54.0000 −1.87099
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 35.0000 1.20978
\(838\) 0 0
\(839\) −4.00000 −0.138095 −0.0690477 − 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −12.0000 −0.413302
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −8.00000 −0.274883
\(848\) 0 0
\(849\) 7.00000 0.240239
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 10.0000 0.341993
\(856\) 0 0
\(857\) −1.00000 −0.0341593 −0.0170797 − 0.999854i \(-0.505437\pi\)
−0.0170797 + 0.999854i \(0.505437\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 1.00000 0.0338837
\(872\) 0 0
\(873\) 16.0000 0.541518
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) 25.0000 0.844190 0.422095 − 0.906552i \(-0.361295\pi\)
0.422095 + 0.906552i \(0.361295\pi\)
\(878\) 0 0
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 − 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) 3.00000 0.100844
\(886\) 0 0
\(887\) 11.0000 0.369344 0.184672 − 0.982800i \(-0.440878\pi\)
0.184672 + 0.982800i \(0.440878\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 50.0000 1.67319
\(894\) 0 0
\(895\) −26.0000 −0.869084
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) 0 0
\(899\) 21.0000 0.700389
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −35.0000 −1.15960 −0.579801 − 0.814758i \(-0.696870\pi\)
−0.579801 + 0.814758i \(0.696870\pi\)
\(912\) 0 0
\(913\) 33.0000 1.09214
\(914\) 0 0
\(915\) −10.0000 −0.330590
\(916\) 0 0
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 14.0000 0.461817 0.230909 − 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) −2.00000 −0.0658308
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) 16.0000 0.524943 0.262471 − 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 0 0
\(931\) −45.0000 −1.47482
\(932\) 0 0
\(933\) 33.0000 1.08037
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 − 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 − 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −20.0000 −0.650600
\(946\) 0 0
\(947\) 55.0000 1.78726 0.893630 − 0.448805i \(-0.148150\pi\)
0.893630 + 0.448805i \(0.148150\pi\)
\(948\) 0 0
\(949\) 1.00000 0.0324614
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −23.0000 −0.739630 −0.369815 − 0.929105i \(-0.620579\pi\)
−0.369815 + 0.929105i \(0.620579\pi\)
\(968\) 0 0
\(969\) 30.0000 0.963739
\(970\) 0 0
\(971\) 44.0000 1.41203 0.706014 − 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) 0 0
\(973\) 80.0000 2.56468
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) 0 0
\(983\) 56.0000 1.78612 0.893061 − 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) −40.0000 −1.27321
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −55.0000 −1.74713 −0.873566 − 0.486705i \(-0.838199\pi\)
−0.873566 + 0.486705i \(0.838199\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 18.0000 0.570638
\(996\) 0 0
\(997\) 44.0000 1.39349 0.696747 − 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 6040.2.a.h.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6040.2.a.h.1.1 ✓ 1 1.1 even 1 trivial