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Properties

Label 4015.2.a.a.1.1
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
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 \(\chi\) \(=\) 4015.1

$q$-expansion

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

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −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
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) 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
\(16\) 4.00000 1.00000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 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
\(27\) 5.00000 0.962250
\(28\) 6.00000 1.13389
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 4.00000 0.666667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −2.00000 −0.301511
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −4.00000 −0.577350
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) −8.00000 −1.09888 −0.549442 − 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −2.00000 −0.258199
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −9.00000 −1.09952 −0.549762 − 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 6.00000 0.727607
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) −12.0000 −1.37649
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −6.00000 −0.654654
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 14.0000 1.45960
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −2.00000 −0.200000
\(101\) 3.00000 0.298511 0.149256 − 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 − 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −10.0000 −0.962250
\(109\) −12.0000 −1.14939 −0.574696 − 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −12.0000 −1.13389
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) −12.0000 −1.11417
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 9.00000 0.825029
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 − 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 0 0
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 − 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 2.00000 0.174078
\(133\) −18.0000 −1.56080
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −6.00000 −0.507093
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) −8.00000 −0.666667
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) −4.00000 −0.328798
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 − 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 2.00000 0.160128
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 21.0000 1.65503
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −12.0000 −0.937043
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) −22.0000 −1.67748
\(173\) −4.00000 −0.304114 −0.152057 − 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 4.00000 0.301511
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −4.00000 −0.298142
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 8.00000 0.583460
\(189\) −15.0000 −1.09109
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 8.00000 0.577350
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) −4.00000 −0.285714
\(197\) −7.00000 −0.498729 −0.249365 − 0.968410i \(-0.580222\pi\)
−0.249365 + 0.968410i \(0.580222\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 − 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) −18.0000 −1.26335
\(204\) −6.00000 −0.420084
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 14.0000 0.973067
\(208\) 4.00000 0.277350
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 16.0000 1.09888
\(213\) 0 0
\(214\) 0 0
\(215\) −11.0000 −0.750194
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) −1.00000 −0.0675737
\(220\) 2.00000 0.134840
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 2.00000 0.132745 0.0663723 − 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 12.0000 0.794719
\(229\) 20.0000 1.32164 0.660819 − 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 7.00000 0.458585 0.229293 − 0.973358i \(-0.426359\pi\)
0.229293 + 0.973358i \(0.426359\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 16.0000 1.04151
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) −13.0000 −0.840900 −0.420450 − 0.907316i \(-0.638128\pi\)
−0.420450 + 0.907316i \(0.638128\pi\)
\(240\) 4.00000 0.258199
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) −20.0000 −1.28037
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 − 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) −12.0000 −0.755929
\(253\) −7.00000 −0.440086
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 16.0000 1.00000
\(257\) −7.00000 −0.436648 −0.218324 − 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 2.00000 0.124035
\(261\) −12.0000 −0.742781
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 18.0000 1.09952
\(269\) 5.00000 0.304855 0.152428 − 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 − 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) −12.0000 −0.727607
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) −14.0000 −0.842701
\(277\) −19.0000 −1.14160 −0.570800 − 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 21.0000 1.25275 0.626377 − 0.779520i \(-0.284537\pi\)
0.626377 + 0.779520i \(0.284537\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −2.00000 −0.117041
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −7.00000 −0.404820
\(300\) 2.00000 0.115470
\(301\) −33.0000 −1.90209
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 24.0000 1.37649
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 6.00000 0.341882
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 19.0000 1.07739 0.538696 − 0.842500i \(-0.318917\pi\)
0.538696 + 0.842500i \(0.318917\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) −28.0000 −1.57512
\(317\) 21.0000 1.17948 0.589739 − 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 8.00000 0.447214
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −18.0000 −1.00155
\(324\) −2.00000 −0.111111
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 12.0000 0.663602
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −30.0000 −1.64895 −0.824475 − 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 18.0000 0.987878
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 9.00000 0.491723
\(336\) 12.0000 0.654654
\(337\) 10.0000 0.544735 0.272367 − 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) −6.00000 −0.325396
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −7.00000 −0.376867
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 12.0000 0.643268
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −35.0000 −1.86286 −0.931431 − 0.363918i \(-0.881439\pi\)
−0.931431 + 0.363918i \(0.881439\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) −9.00000 −0.476331
\(358\) 0 0
\(359\) 34.0000 1.79445 0.897226 − 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 6.00000 0.314485
\(365\) −1.00000 −0.0523424
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 − 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) −28.0000 −1.45960
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 4.00000 0.207390
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 − 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 12.0000 0.615587
\(381\) −10.0000 −0.512316
\(382\) 0 0
\(383\) −1.00000 −0.0510976 −0.0255488 − 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) −22.0000 −1.11832
\(388\) −4.00000 −0.203069
\(389\) 33.0000 1.67317 0.836583 − 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) 21.0000 1.06202
\(392\) 0 0
\(393\) −3.00000 −0.151330
\(394\) 0 0
\(395\) −14.0000 −0.704416
\(396\) 4.00000 0.201008
\(397\) −33.0000 −1.65622 −0.828111 − 0.560564i \(-0.810584\pi\)
−0.828111 + 0.560564i \(0.810584\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 4.00000 0.200000
\(401\) −37.0000 −1.84769 −0.923846 − 0.382765i \(-0.874972\pi\)
−0.923846 + 0.382765i \(0.874972\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) −24.0000 −1.18240
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 6.00000 0.292770
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) −30.0000 −1.45180
\(428\) −8.00000 −0.386695
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 − 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 20.0000 0.962250
\(433\) −30.0000 −1.44171 −0.720854 − 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 24.0000 1.14939
\(437\) −42.0000 −2.00913
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 14.0000 0.665160 0.332580 − 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) 4.00000 0.189832
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 24.0000 1.13389
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −12.0000 −0.564433
\(453\) 3.00000 0.140952
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) −14.0000 −0.652753
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −41.0000 −1.90543 −0.952716 − 0.303863i \(-0.901724\pi\)
−0.952716 + 0.303863i \(0.901724\pi\)
\(464\) 24.0000 1.11417
\(465\) 2.00000 0.0927478
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 − 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 4.00000 0.184900
\(469\) 27.0000 1.24674
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 11.0000 0.505781
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) −18.0000 −0.825029
\(477\) 16.0000 0.732590
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 − 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) −21.0000 −0.955533
\(484\) −2.00000 −0.0909091
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 29.0000 1.31412 0.657058 − 0.753840i \(-0.271801\pi\)
0.657058 + 0.753840i \(0.271801\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 − 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 12.0000 0.541002
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 2.00000 0.0894427
\(501\) 3.00000 0.134030
\(502\) 0 0
\(503\) 26.0000 1.15928 0.579641 − 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) −20.0000 −0.887357
\(509\) −19.0000 −0.842160 −0.421080 − 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) 30.0000 1.32453
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 22.0000 0.968496
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 − 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) −6.00000 −0.262111
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) −4.00000 −0.174078
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 16.0000 0.694341
\(532\) 36.0000 1.56080
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 10.0000 0.430331
\(541\) −11.0000 −0.472927 −0.236463 − 0.971640i \(-0.575988\pi\)
−0.236463 + 0.971640i \(0.575988\pi\)
\(542\) 0 0
\(543\) 5.00000 0.214571
\(544\) 0 0
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 − 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 36.0000 1.53784
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) −42.0000 −1.78602
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 8.00000 0.339276
\(557\) −26.0000 −1.10166 −0.550828 − 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) 0 0
\(559\) 11.0000 0.465250
\(560\) 12.0000 0.507093
\(561\) 3.00000 0.126660
\(562\) 0 0
\(563\) −5.00000 −0.210725 −0.105362 − 0.994434i \(-0.533600\pi\)
−0.105362 + 0.994434i \(0.533600\pi\)
\(564\) −8.00000 −0.336861
\(565\) −6.00000 −0.252422
\(566\) 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
\(571\) −27.0000 −1.12991 −0.564957 − 0.825120i \(-0.691107\pi\)
−0.564957 + 0.825120i \(0.691107\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 16.0000 0.666667
\(577\) −32.0000 −1.33218 −0.666089 − 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 12.0000 0.498273
\(581\) 27.0000 1.12015
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 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 \(0.696105\pi\)
\(588\) 4.00000 0.164957
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 7.00000 0.287942
\(592\) 8.00000 0.328798
\(593\) −4.00000 −0.164260 −0.0821302 − 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 20.0000 0.819232
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 − 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) 18.0000 0.733017
\(604\) 6.00000 0.244137
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 18.0000 0.730597 0.365299 − 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 0 0
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) −12.0000 −0.485071
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 − 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 − 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 4.00000 0.160644
\(621\) −35.0000 −1.40450
\(622\) 0 0
\(623\) 18.0000 0.721155
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) 20.0000 0.798087
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) −10.0000 −0.397464
\(634\) 0 0
\(635\) −10.0000 −0.396838
\(636\) −16.0000 −0.634441
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.00000 −0.276483 −0.138242 − 0.990399i \(-0.544145\pi\)
−0.138242 + 0.990399i \(0.544145\pi\)
\(642\) 0 0
\(643\) 6.00000 0.236617 0.118308 − 0.992977i \(-0.462253\pi\)
0.118308 + 0.992977i \(0.462253\pi\)
\(644\) −42.0000 −1.65503
\(645\) 11.0000 0.433125
\(646\) 0 0
\(647\) 26.0000 1.02217 0.511083 − 0.859532i \(-0.329245\pi\)
0.511083 + 0.859532i \(0.329245\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 8.00000 0.313304
\(653\) 17.0000 0.665261 0.332631 − 0.943057i \(-0.392064\pi\)
0.332631 + 0.943057i \(0.392064\pi\)
\(654\) 0 0
\(655\) −3.00000 −0.117220
\(656\) 24.0000 0.937043
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 5.00000 0.194477 0.0972387 − 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 0 0
\(663\) 3.00000 0.116510
\(664\) 0 0
\(665\) 18.0000 0.698010
\(666\) 0 0
\(667\) −42.0000 −1.62625
\(668\) 6.00000 0.232147
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 − 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 24.0000 0.923077
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 0 0
\(683\) −48.0000 −1.83667 −0.918334 − 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 24.0000 0.917663
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −20.0000 −0.763048
\(688\) 44.0000 1.67748
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 − 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 8.00000 0.304114
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) −7.00000 −0.264764
\(700\) 6.00000 0.226779
\(701\) −14.0000 −0.528773 −0.264386 − 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) −8.00000 −0.301511
\(705\) −4.00000 −0.150649
\(706\) 0 0
\(707\) −9.00000 −0.338480
\(708\) −16.0000 −0.601317
\(709\) 24.0000 0.901339 0.450669 − 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 0 0
\(711\) −28.0000 −1.05008
\(712\) 0 0
\(713\) −14.0000 −0.524304
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) 36.0000 1.34538
\(717\) 13.0000 0.485494
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 8.00000 0.298142
\(721\) −36.0000 −1.34071
\(722\) 0 0
\(723\) −17.0000 −0.632237
\(724\) 10.0000 0.371647
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) 20.0000 0.739221
\(733\) −8.00000 −0.295487 −0.147743 − 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) −9.00000 −0.331519
\(738\) 0 0
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 4.00000 0.147043
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) 19.0000 0.697042 0.348521 − 0.937301i \(-0.386684\pi\)
0.348521 + 0.937301i \(0.386684\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 18.0000 0.658586
\(748\) 6.00000 0.219382
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 − 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −16.0000 −0.583460
\(753\) −5.00000 −0.182210
\(754\) 0 0
\(755\) 3.00000 0.109181
\(756\) 30.0000 1.09109
\(757\) 22.0000 0.799604 0.399802 − 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 7.00000 0.254084
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 36.0000 1.30329
\(764\) −12.0000 −0.434145
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) −16.0000 −0.577350
\(769\) 23.0000 0.829401 0.414701 − 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) −12.0000 −0.431889
\(773\) 2.00000 0.0719350 0.0359675 − 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) 36.0000 1.28983
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 8.00000 0.285714
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 − 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) −8.00000 −0.283731
\(796\) 28.0000 0.992434
\(797\) −34.0000 −1.20434 −0.602171 − 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) 1.00000 0.0352892
\(804\) −18.0000 −0.634811
\(805\) −21.0000 −0.740153
\(806\) 0 0
\(807\) −5.00000 −0.176008
\(808\) 0 0
\(809\) −14.0000 −0.492214 −0.246107 − 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 − 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 36.0000 1.26335
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 12.0000 0.420084
\(817\) 66.0000 2.30905
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 12.0000 0.419058
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 − 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 − 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) −28.0000 −0.973067
\(829\) 12.0000 0.416777 0.208389 − 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 0 0
\(831\) 19.0000 0.659103
\(832\) −8.00000 −0.277350
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) −12.0000 −0.415029
\(837\) 10.0000 0.345651
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 − 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −21.0000 −0.723278
\(844\) −20.0000 −0.688428
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −3.00000 −0.103081
\(848\) −32.0000 −1.09888
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −14.0000 −0.479914
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 − 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 − 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 22.0000 0.750194
\(861\) 18.0000 0.613438
\(862\) 0 0
\(863\) 14.0000 0.476566 0.238283 − 0.971196i \(-0.423415\pi\)
0.238283 + 0.971196i \(0.423415\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 12.0000 0.407307
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 2.00000 0.0675737
\(877\) 44.0000 1.48577 0.742887 − 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(878\) 0 0
\(879\) −24.0000 −0.809500
\(880\) −4.00000 −0.134840
\(881\) −8.00000 −0.269527 −0.134763 − 0.990878i \(-0.543027\pi\)
−0.134763 + 0.990878i \(0.543027\pi\)
\(882\) 0 0
\(883\) 26.0000 0.874970 0.437485 − 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 6.00000 0.201802
\(885\) −8.00000 −0.268917
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) −30.0000 −1.00617
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 8.00000 0.267860
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) 7.00000 0.233723
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 4.00000 0.133333
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 33.0000 1.09817
\(904\) 0 0
\(905\) 5.00000 0.166206
\(906\) 0 0
\(907\) −36.0000 −1.19536 −0.597680 − 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −4.00000 −0.132745
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −13.0000 −0.430709 −0.215355 − 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) −24.0000 −0.794719
\(913\) −9.00000 −0.297857
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) −40.0000 −1.32164
\(917\) −9.00000 −0.297206
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 − 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −24.0000 −0.788263
\(928\) 0 0
\(929\) −20.0000 −0.656179 −0.328089 − 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −14.0000 −0.458585
\(933\) −19.0000 −0.622032
\(934\) 0 0
\(935\) 3.00000 0.0981105
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) −8.00000 −0.260931
\(941\) −4.00000 −0.130396 −0.0651981 − 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) −42.0000 −1.36771
\(944\) −32.0000 −1.04151
\(945\) 15.0000 0.487950
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 28.0000 0.909398
\(949\) 1.00000 0.0324614
\(950\) 0 0
\(951\) −21.0000 −0.680972
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 − 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 26.0000 0.840900
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) 54.0000 1.74375
\(960\) −8.00000 −0.258199
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) −34.0000 −1.09507
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 30.0000 0.964735 0.482367 − 0.875969i \(-0.339777\pi\)
0.482367 + 0.875969i \(0.339777\pi\)
\(968\) 0 0
\(969\) 18.0000 0.578243
\(970\) 0 0
\(971\) 10.0000 0.320915 0.160458 − 0.987043i \(-0.448703\pi\)
0.160458 + 0.987043i \(0.448703\pi\)
\(972\) 32.0000 1.02640
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 40.0000 1.28037
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 4.00000 0.127775
\(981\) 24.0000 0.766261
\(982\) 0 0
\(983\) −28.0000 −0.893061 −0.446531 − 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) 0 0
\(985\) 7.00000 0.223039
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) −12.0000 −0.381771
\(989\) −77.0000 −2.44846
\(990\) 0 0
\(991\) 38.0000 1.20711 0.603555 − 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 0 0
\(993\) 30.0000 0.952021
\(994\) 0 0
\(995\) 14.0000 0.443830
\(996\) −18.0000 −0.570352
\(997\) −24.0000 −0.760088 −0.380044 − 0.924968i \(-0.624091\pi\)
−0.380044 + 0.924968i \(0.624091\pi\)
\(998\) 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 4015.2.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.a.1.1 ✓ 1 1.1 even 1 trivial