↑

Properties

Label 8004.2.a.c.1.1
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8004,2,Mod(1,8004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, 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("8004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.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: \(63.9122617778\)
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 \(\chi\) \(=\) 8004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} -5.00000 q^{17} -2.00000 q^{19} +3.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -6.00000 q^{31} +3.00000 q^{33} -6.00000 q^{35} -10.0000 q^{37} +1.00000 q^{39} -6.00000 q^{41} -2.00000 q^{43} -2.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} -5.00000 q^{51} +8.00000 q^{53} -6.00000 q^{55} -2.00000 q^{57} -4.00000 q^{59} -6.00000 q^{61} +3.00000 q^{63} -2.00000 q^{65} -7.00000 q^{67} -1.00000 q^{69} -12.0000 q^{71} -6.00000 q^{73} -1.00000 q^{75} +9.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +2.00000 q^{83} +10.0000 q^{85} -1.00000 q^{87} +9.00000 q^{89} +3.00000 q^{91} -6.00000 q^{93} +4.00000 q^{95} +6.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 − 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 1.00000 0.0995037 0.0497519 − 0.998762i \(-0.484157\pi\)
0.0497519 + 0.998762i \(0.484157\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 − 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 − 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 17.0000 1.48530 0.742648 − 0.669681i \(-0.233569\pi\)
0.742648 + 0.669681i \(0.233569\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 − 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 − 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 14.0000 1.08335 0.541676 − 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −2.00000 −0.149487 −0.0747435 − 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 − 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 20.0000 1.47043
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 13.0000 0.921546 0.460773 − 0.887518i \(-0.347572\pi\)
0.460773 + 0.887518i \(0.347572\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 − 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −18.0000 −1.22192
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) −13.0000 −0.870544 −0.435272 − 0.900299i \(-0.643348\pi\)
−0.435272 + 0.900299i \(0.643348\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 24.0000 1.58596 0.792982 − 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 0 0
\(231\) 9.00000 0.592157
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 − 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157 0.0322078 − 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.00000 −0.255551
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −13.0000 −0.820553 −0.410276 − 0.911961i \(-0.634568\pi\)
−0.410276 + 0.911961i \(0.634568\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) 10.0000 0.626224
\(256\) 0 0
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) −30.0000 −1.86411
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −16.0000 −0.982872
\(266\) 0 0
\(267\) 9.00000 0.550791
\(268\) 0 0
\(269\) −15.0000 −0.914566 −0.457283 − 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 23.0000 1.38194 0.690968 − 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) −27.0000 −1.57736 −0.788678 − 0.614806i \(-0.789234\pi\)
−0.788678 + 0.614806i \(0.789234\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 1.00000 0.0574485
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −24.0000 −1.36975 −0.684876 − 0.728659i \(-0.740144\pi\)
−0.684876 + 0.728659i \(0.740144\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 17.0000 0.963982 0.481991 − 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 − 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) 0 0
\(317\) 7.00000 0.393159 0.196580 − 0.980488i \(-0.437017\pi\)
0.196580 + 0.980488i \(0.437017\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) 27.0000 1.48856
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 14.0000 0.764902
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) −18.0000 −0.974755
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 2.00000 0.107676
\(346\) 0 0
\(347\) 10.0000 0.536828 0.268414 − 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 − 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 − 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 − 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 − 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 26.0000 1.32854 0.664269 − 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) 0 0
\(385\) −18.0000 −0.917365
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) 5.00000 0.253510 0.126755 − 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) 17.0000 0.857537
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 − 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 − 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −7.00000 −0.342791
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 − 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) −18.0000 −0.871081
\(428\) 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
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −21.0000 −1.00228 −0.501138 − 0.865368i \(-0.667085\pi\)
−0.501138 + 0.865368i \(0.667085\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −33.0000 −1.56788 −0.783939 − 0.620838i \(-0.786792\pi\)
−0.783939 + 0.620838i \(0.786792\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −31.0000 −1.46298 −0.731490 − 0.681852i \(-0.761175\pi\)
−0.731490 + 0.681852i \(0.761175\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 7.00000 0.327446 0.163723 − 0.986506i \(-0.447650\pi\)
0.163723 + 0.986506i \(0.447650\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −9.00000 −0.418265 −0.209133 − 0.977887i \(-0.567064\pi\)
−0.209133 + 0.977887i \(0.567064\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −21.0000 −0.969690
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 5.00000 0.225189
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) −36.0000 −1.61482
\(498\) 0 0
\(499\) −3.00000 −0.134298 −0.0671492 − 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(500\) 0 0
\(501\) 14.0000 0.625474
\(502\) 0 0
\(503\) 11.0000 0.490466 0.245233 − 0.969464i \(-0.421136\pi\)
0.245233 + 0.969464i \(0.421136\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) 27.0000 1.18746
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 19.0000 0.830812 0.415406 − 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) 0 0
\(525\) −3.00000 −0.130931
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2.00000 −0.0863064
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 11.0000 0.472055
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 − 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) 20.0000 0.848953
\(556\) 0 0
\(557\) 44.0000 1.86434 0.932170 − 0.362021i \(-0.117913\pi\)
0.932170 + 0.362021i \(0.117913\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 − 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 − 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −8.00000 −0.333044 −0.166522 − 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −38.0000 −1.56843 −0.784214 − 0.620491i \(-0.786934\pi\)
−0.784214 + 0.620491i \(0.786934\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 − 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 30.0000 1.22988
\(596\) 0 0
\(597\) 13.0000 0.532055
\(598\) 0 0
\(599\) 33.0000 1.34834 0.674172 − 0.738575i \(-0.264501\pi\)
0.674172 + 0.738575i \(0.264501\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 − 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 0 0
\(603\) −7.00000 −0.285062
\(604\) 0 0
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 − 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) −1.00000 −0.0403896 −0.0201948 − 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 27.0000 1.08173
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) 0 0
\(629\) 50.0000 1.99363
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 − 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) 0 0
\(633\) 24.0000 0.953914
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 − 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) 37.0000 1.45914 0.729569 − 0.683907i \(-0.239721\pi\)
0.729569 + 0.683907i \(0.239721\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −18.0000 −0.705476
\(652\) 0 0
\(653\) 15.0000 0.586995 0.293498 − 0.955960i \(-0.405181\pi\)
0.293498 + 0.955960i \(0.405181\pi\)
\(654\) 0 0
\(655\) −34.0000 −1.32849
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 33.0000 1.28550 0.642749 − 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 − 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 0 0
\(663\) −5.00000 −0.194184
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −13.0000 −0.502609
\(670\) 0 0
\(671\) −18.0000 −0.694882
\(672\) 0 0
\(673\) 45.0000 1.73462 0.867311 − 0.497766i \(-0.165846\pi\)
0.867311 + 0.497766i \(0.165846\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −13.0000 −0.499631 −0.249815 − 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) 0 0
\(679\) 18.0000 0.690777
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 24.0000 0.915657
\(688\) 0 0
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 − 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) 0 0
\(693\) 9.00000 0.341882
\(694\) 0 0
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) 30.0000 1.13633
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 − 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) −18.0000 −0.677919
\(706\) 0 0
\(707\) 3.00000 0.112827
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 − 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.00000 −0.0745874 −0.0372937 − 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) 1.00000 0.0371904
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 − 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.0000 0.369863
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) −21.0000 −0.773545
\(738\) 0 0
\(739\) 42.0000 1.54499 0.772497 − 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 51.0000 1.87101 0.935504 − 0.353315i \(-0.114946\pi\)
0.935504 + 0.353315i \(0.114946\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) −13.0000 −0.473746
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 − 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 15.0000 0.543036
\(764\) 0 0
\(765\) 10.0000 0.361551
\(766\) 0 0
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 − 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) −30.0000 −1.07624
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) −6.00000 −0.213066
\(794\) 0 0
\(795\) −16.0000 −0.567462
\(796\) 0 0
\(797\) 26.0000 0.920967 0.460484 − 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) −45.0000 −1.59199
\(800\) 0 0
\(801\) 9.00000 0.317999
\(802\) 0 0
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) −15.0000 −0.528025
\(808\) 0 0
\(809\) −35.0000 −1.23053 −0.615267 − 0.788319i \(-0.710952\pi\)
−0.615267 + 0.788319i \(0.710952\pi\)
\(810\) 0 0
\(811\) 11.0000 0.386262 0.193131 − 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 − 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 22.0000 0.764092 0.382046 − 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(830\) 0 0
\(831\) 23.0000 0.797861
\(832\) 0 0
\(833\) −10.0000 −0.346479
\(834\) 0 0
\(835\) −28.0000 −0.968980
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) 35.0000 1.20833 0.604167 − 0.796858i \(-0.293506\pi\)
0.604167 + 0.796858i \(0.293506\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) 24.0000 0.825625
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 − 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 10.0000 0.341593 0.170797 − 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) 10.0000 0.340404 0.170202 − 0.985409i \(-0.445558\pi\)
0.170202 + 0.985409i \(0.445558\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 36.0000 1.21702
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) −27.0000 −0.910687
\(880\) 0 0
\(881\) −23.0000 −0.774890 −0.387445 − 0.921893i \(-0.626642\pi\)
−0.387445 + 0.921893i \(0.626642\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 15.0000 0.503651 0.251825 − 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 0 0
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 0 0
\(909\) 1.00000 0.0331679
\(910\) 0 0
\(911\) −25.0000 −0.828287 −0.414143 − 0.910212i \(-0.635919\pi\)
−0.414143 + 0.910212i \(0.635919\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 0 0
\(915\) 12.0000 0.396708
\(916\) 0 0
\(917\) 51.0000 1.68417
\(918\) 0 0
\(919\) 13.0000 0.428830 0.214415 − 0.976743i \(-0.431215\pi\)
0.214415 + 0.976743i \(0.431215\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) 17.0000 0.556555
\(934\) 0 0
\(935\) 30.0000 0.981105
\(936\) 0 0
\(937\) 33.0000 1.07806 0.539032 − 0.842286i \(-0.318790\pi\)
0.539032 + 0.842286i \(0.318790\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) −6.00000 −0.195180
\(946\) 0 0
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 7.00000 0.226991
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −32.0000 −1.03012
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 − 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 0 0
\(969\) 10.0000 0.321246
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 − 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −21.0000 −0.673229
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) −4.00000 −0.127971 −0.0639857 − 0.997951i \(-0.520381\pi\)
−0.0639857 + 0.997951i \(0.520381\pi\)
\(978\) 0 0
\(979\) 27.0000 0.862924
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) 0 0
\(983\) 56.0000 1.78612 0.893061 − 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 27.0000 0.859419
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 37.0000 1.17534 0.587672 − 0.809099i \(-0.300045\pi\)
0.587672 + 0.809099i \(0.300045\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −26.0000 −0.824255
\(996\) 0 0
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) 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 8004.2.a.c.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.c.1.1 ✓ 1 1.1 even 1 trivial