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Properties

Label 4002.2.a.c.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +3.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +1.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} -6.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +1.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} -3.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +4.00000 q^{50} -6.00000 q^{51} +1.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -4.00000 q^{56} -6.00000 q^{57} -1.00000 q^{58} -3.00000 q^{59} -1.00000 q^{60} +7.00000 q^{61} +1.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -3.00000 q^{66} +3.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -4.00000 q^{70} +9.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -3.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} -12.0000 q^{77} +1.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -10.0000 q^{83} -4.00000 q^{84} +6.00000 q^{85} -6.00000 q^{86} -1.00000 q^{87} +3.00000 q^{88} +12.0000 q^{89} -1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} +1.00000 q^{93} -4.00000 q^{94} +6.00000 q^{95} +1.00000 q^{96} -6.00000 q^{97} -9.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −4.00000 −1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.00000 −0.872872
\(22\) 3.00000 0.639602
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 1.00000 0.185695
\(30\) 1.00000 0.182574
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) −6.00000 −1.02899
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −6.00000 −0.973329
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 4.00000 0.617213
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −3.00000 −0.452267
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 4.00000 0.565685
\(51\) −6.00000 −0.840168
\(52\) 1.00000 0.138675
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) −4.00000 −0.534522
\(57\) −6.00000 −0.794719
\(58\) −1.00000 −0.131306
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −1.00000 −0.129099
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 1.00000 0.127000
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −3.00000 −0.369274
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) −4.00000 −0.478091
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −3.00000 −0.348743
\(75\) 4.00000 0.461880
\(76\) 6.00000 0.688247
\(77\) −12.0000 −1.36753
\(78\) 1.00000 0.113228
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −1.00000 −0.110432
\(83\) −10.0000 −1.09764 −0.548821 − 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) −4.00000 −0.436436
\(85\) 6.00000 0.650791
\(86\) −6.00000 −0.646997
\(87\) −1.00000 −0.107211
\(88\) 3.00000 0.319801
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −1.00000 −0.105409
\(91\) 4.00000 0.419314
\(92\) −1.00000 −0.104257
\(93\) 1.00000 0.103695
\(94\) −4.00000 −0.412568
\(95\) 6.00000 0.615587
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −9.00000 −0.909137
\(99\) −3.00000 −0.301511
\(100\) −4.00000 −0.400000
\(101\) −15.0000 −1.49256 −0.746278 − 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 6.00000 0.594089
\(103\) 11.0000 1.08386 0.541931 − 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −4.00000 −0.390360
\(106\) 10.0000 0.971286
\(107\) −10.0000 −0.966736 −0.483368 − 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 3.00000 0.286039
\(111\) −3.00000 −0.284747
\(112\) 4.00000 0.377964
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 6.00000 0.561951
\(115\) −1.00000 −0.0932505
\(116\) 1.00000 0.0928477
\(117\) 1.00000 0.0924500
\(118\) 3.00000 0.276172
\(119\) 24.0000 2.20008
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) −7.00000 −0.633750
\(123\) −1.00000 −0.0901670
\(124\) −1.00000 −0.0898027
\(125\) −9.00000 −0.804984
\(126\) −4.00000 −0.356348
\(127\) −21.0000 −1.86345 −0.931724 − 0.363166i \(-0.881696\pi\)
−0.931724 + 0.363166i \(0.881696\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) −1.00000 −0.0877058
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 3.00000 0.261116
\(133\) 24.0000 2.08106
\(134\) −3.00000 −0.259161
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 4.00000 0.338062
\(141\) −4.00000 −0.336861
\(142\) −9.00000 −0.755263
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 1.00000 0.0830455
\(146\) −2.00000 −0.165521
\(147\) −9.00000 −0.742307
\(148\) 3.00000 0.246598
\(149\) 3.00000 0.245770 0.122885 − 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) −4.00000 −0.326599
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −6.00000 −0.486664
\(153\) 6.00000 0.485071
\(154\) 12.0000 0.966988
\(155\) −1.00000 −0.0803219
\(156\) −1.00000 −0.0800641
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −12.0000 −0.954669
\(159\) 10.0000 0.793052
\(160\) −1.00000 −0.0790569
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) −3.00000 −0.234978 −0.117489 − 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(164\) 1.00000 0.0780869
\(165\) 3.00000 0.233550
\(166\) 10.0000 0.776151
\(167\) 11.0000 0.851206 0.425603 − 0.904910i \(-0.360062\pi\)
0.425603 + 0.904910i \(0.360062\pi\)
\(168\) 4.00000 0.308607
\(169\) −12.0000 −0.923077
\(170\) −6.00000 −0.460179
\(171\) 6.00000 0.458831
\(172\) 6.00000 0.457496
\(173\) 12.0000 0.912343 0.456172 − 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 1.00000 0.0758098
\(175\) −16.0000 −1.20949
\(176\) −3.00000 −0.226134
\(177\) 3.00000 0.225494
\(178\) −12.0000 −0.899438
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.00000 −0.296500
\(183\) −7.00000 −0.517455
\(184\) 1.00000 0.0737210
\(185\) 3.00000 0.220564
\(186\) −1.00000 −0.0733236
\(187\) −18.0000 −1.31629
\(188\) 4.00000 0.291730
\(189\) −4.00000 −0.290957
\(190\) −6.00000 −0.435286
\(191\) −9.00000 −0.651217 −0.325609 − 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 6.00000 0.430775
\(195\) −1.00000 −0.0716115
\(196\) 9.00000 0.642857
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 3.00000 0.213201
\(199\) −13.0000 −0.921546 −0.460773 − 0.887518i \(-0.652428\pi\)
−0.460773 + 0.887518i \(0.652428\pi\)
\(200\) 4.00000 0.282843
\(201\) −3.00000 −0.211604
\(202\) 15.0000 1.05540
\(203\) 4.00000 0.280745
\(204\) −6.00000 −0.420084
\(205\) 1.00000 0.0698430
\(206\) −11.0000 −0.766406
\(207\) −1.00000 −0.0695048
\(208\) 1.00000 0.0693375
\(209\) −18.0000 −1.24509
\(210\) 4.00000 0.276026
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −10.0000 −0.686803
\(213\) −9.00000 −0.616670
\(214\) 10.0000 0.683586
\(215\) 6.00000 0.409197
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) 10.0000 0.677285
\(219\) −2.00000 −0.135147
\(220\) −3.00000 −0.202260
\(221\) 6.00000 0.403604
\(222\) 3.00000 0.201347
\(223\) 18.0000 1.20537 0.602685 − 0.797980i \(-0.294098\pi\)
0.602685 + 0.797980i \(0.294098\pi\)
\(224\) −4.00000 −0.267261
\(225\) −4.00000 −0.266667
\(226\) 2.00000 0.133038
\(227\) −10.0000 −0.663723 −0.331862 − 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) −6.00000 −0.397360
\(229\) 3.00000 0.198246 0.0991228 − 0.995075i \(-0.468396\pi\)
0.0991228 + 0.995075i \(0.468396\pi\)
\(230\) 1.00000 0.0659380
\(231\) 12.0000 0.789542
\(232\) −1.00000 −0.0656532
\(233\) −8.00000 −0.524097 −0.262049 − 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 4.00000 0.260931
\(236\) −3.00000 −0.195283
\(237\) −12.0000 −0.779484
\(238\) −24.0000 −1.55569
\(239\) 29.0000 1.87585 0.937927 − 0.346833i \(-0.112743\pi\)
0.937927 + 0.346833i \(0.112743\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 6.00000 0.386494 0.193247 − 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) 7.00000 0.448129
\(245\) 9.00000 0.574989
\(246\) 1.00000 0.0637577
\(247\) 6.00000 0.381771
\(248\) 1.00000 0.0635001
\(249\) 10.0000 0.633724
\(250\) 9.00000 0.569210
\(251\) 9.00000 0.568075 0.284037 − 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 4.00000 0.251976
\(253\) 3.00000 0.188608
\(254\) 21.0000 1.31766
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 6.00000 0.373544
\(259\) 12.0000 0.745644
\(260\) 1.00000 0.0620174
\(261\) 1.00000 0.0618984
\(262\) −10.0000 −0.617802
\(263\) 28.0000 1.72655 0.863277 − 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −3.00000 −0.184637
\(265\) −10.0000 −0.614295
\(266\) −24.0000 −1.47153
\(267\) −12.0000 −0.734388
\(268\) 3.00000 0.183254
\(269\) 15.0000 0.914566 0.457283 − 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 1.00000 0.0608581
\(271\) 25.0000 1.51864 0.759321 − 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) 6.00000 0.363803
\(273\) −4.00000 −0.242091
\(274\) 6.00000 0.362473
\(275\) 12.0000 0.723627
\(276\) 1.00000 0.0601929
\(277\) 5.00000 0.300421 0.150210 − 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) −12.0000 −0.719712
\(279\) −1.00000 −0.0598684
\(280\) −4.00000 −0.239046
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 4.00000 0.238197
\(283\) 27.0000 1.60498 0.802492 − 0.596663i \(-0.203507\pi\)
0.802492 + 0.596663i \(0.203507\pi\)
\(284\) 9.00000 0.534052
\(285\) −6.00000 −0.355409
\(286\) 3.00000 0.177394
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −1.00000 −0.0587220
\(291\) 6.00000 0.351726
\(292\) 2.00000 0.117041
\(293\) −24.0000 −1.40209 −0.701047 − 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 9.00000 0.524891
\(295\) −3.00000 −0.174667
\(296\) −3.00000 −0.174371
\(297\) 3.00000 0.174078
\(298\) −3.00000 −0.173785
\(299\) −1.00000 −0.0578315
\(300\) 4.00000 0.230940
\(301\) 24.0000 1.38334
\(302\) 12.0000 0.690522
\(303\) 15.0000 0.861727
\(304\) 6.00000 0.344124
\(305\) 7.00000 0.400819
\(306\) −6.00000 −0.342997
\(307\) 19.0000 1.08439 0.542194 − 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(308\) −12.0000 −0.683763
\(309\) −11.0000 −0.625768
\(310\) 1.00000 0.0567962
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 1.00000 0.0566139
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −10.0000 −0.564333
\(315\) 4.00000 0.225374
\(316\) 12.0000 0.675053
\(317\) 1.00000 0.0561656 0.0280828 − 0.999606i \(-0.491060\pi\)
0.0280828 + 0.999606i \(0.491060\pi\)
\(318\) −10.0000 −0.560772
\(319\) −3.00000 −0.167968
\(320\) 1.00000 0.0559017
\(321\) 10.0000 0.558146
\(322\) 4.00000 0.222911
\(323\) 36.0000 2.00309
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 3.00000 0.166155
\(327\) 10.0000 0.553001
\(328\) −1.00000 −0.0552158
\(329\) 16.0000 0.882109
\(330\) −3.00000 −0.165145
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −10.0000 −0.548821
\(333\) 3.00000 0.164399
\(334\) −11.0000 −0.601893
\(335\) 3.00000 0.163908
\(336\) −4.00000 −0.218218
\(337\) 5.00000 0.272367 0.136184 − 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 12.0000 0.652714
\(339\) 2.00000 0.108625
\(340\) 6.00000 0.325396
\(341\) 3.00000 0.162459
\(342\) −6.00000 −0.324443
\(343\) 8.00000 0.431959
\(344\) −6.00000 −0.323498
\(345\) 1.00000 0.0538382
\(346\) −12.0000 −0.645124
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 23.0000 1.23116 0.615581 − 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 16.0000 0.855236
\(351\) −1.00000 −0.0533761
\(352\) 3.00000 0.159901
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) −3.00000 −0.159448
\(355\) 9.00000 0.477670
\(356\) 12.0000 0.635999
\(357\) −24.0000 −1.27021
\(358\) 4.00000 0.211407
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) −2.00000 −0.105118
\(363\) 2.00000 0.104973
\(364\) 4.00000 0.209657
\(365\) 2.00000 0.104685
\(366\) 7.00000 0.365896
\(367\) 10.0000 0.521996 0.260998 − 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 1.00000 0.0520579
\(370\) −3.00000 −0.155963
\(371\) −40.0000 −2.07670
\(372\) 1.00000 0.0518476
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 18.0000 0.930758
\(375\) 9.00000 0.464758
\(376\) −4.00000 −0.206284
\(377\) 1.00000 0.0515026
\(378\) 4.00000 0.205738
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 6.00000 0.307794
\(381\) 21.0000 1.07586
\(382\) 9.00000 0.460480
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 1.00000 0.0510310
\(385\) −12.0000 −0.611577
\(386\) −14.0000 −0.712581
\(387\) 6.00000 0.304997
\(388\) −6.00000 −0.304604
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 1.00000 0.0506370
\(391\) −6.00000 −0.303433
\(392\) −9.00000 −0.454569
\(393\) −10.0000 −0.504433
\(394\) 18.0000 0.906827
\(395\) 12.0000 0.603786
\(396\) −3.00000 −0.150756
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 13.0000 0.651631
\(399\) −24.0000 −1.20150
\(400\) −4.00000 −0.200000
\(401\) −11.0000 −0.549314 −0.274657 − 0.961542i \(-0.588564\pi\)
−0.274657 + 0.961542i \(0.588564\pi\)
\(402\) 3.00000 0.149626
\(403\) −1.00000 −0.0498135
\(404\) −15.0000 −0.746278
\(405\) 1.00000 0.0496904
\(406\) −4.00000 −0.198517
\(407\) −9.00000 −0.446113
\(408\) 6.00000 0.297044
\(409\) 20.0000 0.988936 0.494468 − 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) −1.00000 −0.0493865
\(411\) 6.00000 0.295958
\(412\) 11.0000 0.541931
\(413\) −12.0000 −0.590481
\(414\) 1.00000 0.0491473
\(415\) −10.0000 −0.490881
\(416\) −1.00000 −0.0490290
\(417\) −12.0000 −0.587643
\(418\) 18.0000 0.880409
\(419\) 10.0000 0.488532 0.244266 − 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) −4.00000 −0.195180
\(421\) 13.0000 0.633581 0.316791 − 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 13.0000 0.632830
\(423\) 4.00000 0.194487
\(424\) 10.0000 0.485643
\(425\) −24.0000 −1.16417
\(426\) 9.00000 0.436051
\(427\) 28.0000 1.35501
\(428\) −10.0000 −0.483368
\(429\) 3.00000 0.144841
\(430\) −6.00000 −0.289346
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 4.00000 0.192006
\(435\) −1.00000 −0.0479463
\(436\) −10.0000 −0.478913
\(437\) −6.00000 −0.287019
\(438\) 2.00000 0.0955637
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 3.00000 0.143019
\(441\) 9.00000 0.428571
\(442\) −6.00000 −0.285391
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −3.00000 −0.142374
\(445\) 12.0000 0.568855
\(446\) −18.0000 −0.852325
\(447\) −3.00000 −0.141895
\(448\) 4.00000 0.188982
\(449\) −19.0000 −0.896665 −0.448333 − 0.893867i \(-0.647982\pi\)
−0.448333 + 0.893867i \(0.647982\pi\)
\(450\) 4.00000 0.188562
\(451\) −3.00000 −0.141264
\(452\) −2.00000 −0.0940721
\(453\) 12.0000 0.563809
\(454\) 10.0000 0.469323
\(455\) 4.00000 0.187523
\(456\) 6.00000 0.280976
\(457\) −28.0000 −1.30978 −0.654892 − 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −3.00000 −0.140181
\(459\) −6.00000 −0.280056
\(460\) −1.00000 −0.0466252
\(461\) −27.0000 −1.25752 −0.628758 − 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) −12.0000 −0.558291
\(463\) 14.0000 0.650635 0.325318 − 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 1.00000 0.0464238
\(465\) 1.00000 0.0463739
\(466\) 8.00000 0.370593
\(467\) 21.0000 0.971764 0.485882 − 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) 1.00000 0.0462250
\(469\) 12.0000 0.554109
\(470\) −4.00000 −0.184506
\(471\) −10.0000 −0.460776
\(472\) 3.00000 0.138086
\(473\) −18.0000 −0.827641
\(474\) 12.0000 0.551178
\(475\) −24.0000 −1.10120
\(476\) 24.0000 1.10004
\(477\) −10.0000 −0.457869
\(478\) −29.0000 −1.32643
\(479\) −15.0000 −0.685367 −0.342684 − 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 1.00000 0.0456435
\(481\) 3.00000 0.136788
\(482\) −6.00000 −0.273293
\(483\) 4.00000 0.182006
\(484\) −2.00000 −0.0909091
\(485\) −6.00000 −0.272446
\(486\) 1.00000 0.0453609
\(487\) −22.0000 −0.996915 −0.498458 − 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −7.00000 −0.316875
\(489\) 3.00000 0.135665
\(490\) −9.00000 −0.406579
\(491\) −8.00000 −0.361035 −0.180517 − 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) −1.00000 −0.0450835
\(493\) 6.00000 0.270226
\(494\) −6.00000 −0.269953
\(495\) −3.00000 −0.134840
\(496\) −1.00000 −0.0449013
\(497\) 36.0000 1.61482
\(498\) −10.0000 −0.448111
\(499\) −26.0000 −1.16392 −0.581960 − 0.813217i \(-0.697714\pi\)
−0.581960 + 0.813217i \(0.697714\pi\)
\(500\) −9.00000 −0.402492
\(501\) −11.0000 −0.491444
\(502\) −9.00000 −0.401690
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −4.00000 −0.178174
\(505\) −15.0000 −0.667491
\(506\) −3.00000 −0.133366
\(507\) 12.0000 0.532939
\(508\) −21.0000 −0.931724
\(509\) −4.00000 −0.177297 −0.0886484 − 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 6.00000 0.265684
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) −12.0000 −0.529297
\(515\) 11.0000 0.484718
\(516\) −6.00000 −0.264135
\(517\) −12.0000 −0.527759
\(518\) −12.0000 −0.527250
\(519\) −12.0000 −0.526742
\(520\) −1.00000 −0.0438529
\(521\) −26.0000 −1.13908 −0.569540 − 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −25.0000 −1.09317 −0.546587 − 0.837402i \(-0.684073\pi\)
−0.546587 + 0.837402i \(0.684073\pi\)
\(524\) 10.0000 0.436852
\(525\) 16.0000 0.698297
\(526\) −28.0000 −1.22086
\(527\) −6.00000 −0.261364
\(528\) 3.00000 0.130558
\(529\) 1.00000 0.0434783
\(530\) 10.0000 0.434372
\(531\) −3.00000 −0.130189
\(532\) 24.0000 1.04053
\(533\) 1.00000 0.0433148
\(534\) 12.0000 0.519291
\(535\) −10.0000 −0.432338
\(536\) −3.00000 −0.129580
\(537\) 4.00000 0.172613
\(538\) −15.0000 −0.646696
\(539\) −27.0000 −1.16297
\(540\) −1.00000 −0.0430331
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −25.0000 −1.07384
\(543\) −2.00000 −0.0858282
\(544\) −6.00000 −0.257248
\(545\) −10.0000 −0.428353
\(546\) 4.00000 0.171184
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −6.00000 −0.256307
\(549\) 7.00000 0.298753
\(550\) −12.0000 −0.511682
\(551\) 6.00000 0.255609
\(552\) −1.00000 −0.0425628
\(553\) 48.0000 2.04117
\(554\) −5.00000 −0.212430
\(555\) −3.00000 −0.127343
\(556\) 12.0000 0.508913
\(557\) −45.0000 −1.90671 −0.953356 − 0.301849i \(-0.902396\pi\)
−0.953356 + 0.301849i \(0.902396\pi\)
\(558\) 1.00000 0.0423334
\(559\) 6.00000 0.253773
\(560\) 4.00000 0.169031
\(561\) 18.0000 0.759961
\(562\) −10.0000 −0.421825
\(563\) −13.0000 −0.547885 −0.273942 − 0.961746i \(-0.588328\pi\)
−0.273942 + 0.961746i \(0.588328\pi\)
\(564\) −4.00000 −0.168430
\(565\) −2.00000 −0.0841406
\(566\) −27.0000 −1.13489
\(567\) 4.00000 0.167984
\(568\) −9.00000 −0.377632
\(569\) −14.0000 −0.586911 −0.293455 − 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 6.00000 0.251312
\(571\) −23.0000 −0.962520 −0.481260 − 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) −3.00000 −0.125436
\(573\) 9.00000 0.375980
\(574\) −4.00000 −0.166957
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 − 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −19.0000 −0.790296
\(579\) −14.0000 −0.581820
\(580\) 1.00000 0.0415227
\(581\) −40.0000 −1.65948
\(582\) −6.00000 −0.248708
\(583\) 30.0000 1.24247
\(584\) −2.00000 −0.0827606
\(585\) 1.00000 0.0413449
\(586\) 24.0000 0.991431
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −9.00000 −0.371154
\(589\) −6.00000 −0.247226
\(590\) 3.00000 0.123508
\(591\) 18.0000 0.740421
\(592\) 3.00000 0.123299
\(593\) 36.0000 1.47834 0.739171 − 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) −3.00000 −0.123091
\(595\) 24.0000 0.983904
\(596\) 3.00000 0.122885
\(597\) 13.0000 0.532055
\(598\) 1.00000 0.0408930
\(599\) −26.0000 −1.06233 −0.531166 − 0.847268i \(-0.678246\pi\)
−0.531166 + 0.847268i \(0.678246\pi\)
\(600\) −4.00000 −0.163299
\(601\) −44.0000 −1.79480 −0.897399 − 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) −24.0000 −0.978167
\(603\) 3.00000 0.122169
\(604\) −12.0000 −0.488273
\(605\) −2.00000 −0.0813116
\(606\) −15.0000 −0.609333
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −6.00000 −0.243332
\(609\) −4.00000 −0.162088
\(610\) −7.00000 −0.283422
\(611\) 4.00000 0.161823
\(612\) 6.00000 0.242536
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −19.0000 −0.766778
\(615\) −1.00000 −0.0403239
\(616\) 12.0000 0.483494
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 11.0000 0.442485
\(619\) 36.0000 1.44696 0.723481 − 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 1.00000 0.0401286
\(622\) −4.00000 −0.160385
\(623\) 48.0000 1.92308
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) 8.00000 0.319744
\(627\) 18.0000 0.718851
\(628\) 10.0000 0.399043
\(629\) 18.0000 0.717707
\(630\) −4.00000 −0.159364
\(631\) 1.00000 0.0398094 0.0199047 − 0.999802i \(-0.493664\pi\)
0.0199047 + 0.999802i \(0.493664\pi\)
\(632\) −12.0000 −0.477334
\(633\) 13.0000 0.516704
\(634\) −1.00000 −0.0397151
\(635\) −21.0000 −0.833360
\(636\) 10.0000 0.396526
\(637\) 9.00000 0.356593
\(638\) 3.00000 0.118771
\(639\) 9.00000 0.356034
\(640\) −1.00000 −0.0395285
\(641\) 4.00000 0.157991 0.0789953 − 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) −10.0000 −0.394669
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −4.00000 −0.157622
\(645\) −6.00000 −0.236250
\(646\) −36.0000 −1.41640
\(647\) 17.0000 0.668339 0.334169 − 0.942513i \(-0.391544\pi\)
0.334169 + 0.942513i \(0.391544\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 9.00000 0.353281
\(650\) 4.00000 0.156893
\(651\) 4.00000 0.156772
\(652\) −3.00000 −0.117489
\(653\) −1.00000 −0.0391330 −0.0195665 − 0.999809i \(-0.506229\pi\)
−0.0195665 + 0.999809i \(0.506229\pi\)
\(654\) −10.0000 −0.391031
\(655\) 10.0000 0.390732
\(656\) 1.00000 0.0390434
\(657\) 2.00000 0.0780274
\(658\) −16.0000 −0.623745
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 3.00000 0.116775
\(661\) 20.0000 0.777910 0.388955 − 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 8.00000 0.310929
\(663\) −6.00000 −0.233021
\(664\) 10.0000 0.388075
\(665\) 24.0000 0.930680
\(666\) −3.00000 −0.116248
\(667\) −1.00000 −0.0387202
\(668\) 11.0000 0.425603
\(669\) −18.0000 −0.695920
\(670\) −3.00000 −0.115900
\(671\) −21.0000 −0.810696
\(672\) 4.00000 0.154303
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −5.00000 −0.192593
\(675\) 4.00000 0.153960
\(676\) −12.0000 −0.461538
\(677\) 24.0000 0.922395 0.461197 − 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) −2.00000 −0.0768095
\(679\) −24.0000 −0.921035
\(680\) −6.00000 −0.230089
\(681\) 10.0000 0.383201
\(682\) −3.00000 −0.114876
\(683\) 9.00000 0.344375 0.172188 − 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 6.00000 0.229416
\(685\) −6.00000 −0.229248
\(686\) −8.00000 −0.305441
\(687\) −3.00000 −0.114457
\(688\) 6.00000 0.228748
\(689\) −10.0000 −0.380970
\(690\) −1.00000 −0.0380693
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 12.0000 0.456172
\(693\) −12.0000 −0.455842
\(694\) −4.00000 −0.151838
\(695\) 12.0000 0.455186
\(696\) 1.00000 0.0379049
\(697\) 6.00000 0.227266
\(698\) −23.0000 −0.870563
\(699\) 8.00000 0.302588
\(700\) −16.0000 −0.604743
\(701\) −13.0000 −0.491003 −0.245502 − 0.969396i \(-0.578953\pi\)
−0.245502 + 0.969396i \(0.578953\pi\)
\(702\) 1.00000 0.0377426
\(703\) 18.0000 0.678883
\(704\) −3.00000 −0.113067
\(705\) −4.00000 −0.150649
\(706\) −4.00000 −0.150542
\(707\) −60.0000 −2.25653
\(708\) 3.00000 0.112747
\(709\) −38.0000 −1.42712 −0.713560 − 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) −9.00000 −0.337764
\(711\) 12.0000 0.450035
\(712\) −12.0000 −0.449719
\(713\) 1.00000 0.0374503
\(714\) 24.0000 0.898177
\(715\) −3.00000 −0.112194
\(716\) −4.00000 −0.149487
\(717\) −29.0000 −1.08302
\(718\) 0 0
\(719\) −21.0000 −0.783168 −0.391584 − 0.920142i \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(720\) 1.00000 0.0372678
\(721\) 44.0000 1.63865
\(722\) −17.0000 −0.632674
\(723\) −6.00000 −0.223142
\(724\) 2.00000 0.0743294
\(725\) −4.00000 −0.148556
\(726\) −2.00000 −0.0742270
\(727\) 6.00000 0.222528 0.111264 − 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 36.0000 1.33151
\(732\) −7.00000 −0.258727
\(733\) 3.00000 0.110808 0.0554038 − 0.998464i \(-0.482355\pi\)
0.0554038 + 0.998464i \(0.482355\pi\)
\(734\) −10.0000 −0.369107
\(735\) −9.00000 −0.331970
\(736\) 1.00000 0.0368605
\(737\) −9.00000 −0.331519
\(738\) −1.00000 −0.0368105
\(739\) −29.0000 −1.06678 −0.533391 − 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 3.00000 0.110282
\(741\) −6.00000 −0.220416
\(742\) 40.0000 1.46845
\(743\) −5.00000 −0.183432 −0.0917161 − 0.995785i \(-0.529235\pi\)
−0.0917161 + 0.995785i \(0.529235\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 3.00000 0.109911
\(746\) −4.00000 −0.146450
\(747\) −10.0000 −0.365881
\(748\) −18.0000 −0.658145
\(749\) −40.0000 −1.46157
\(750\) −9.00000 −0.328634
\(751\) 22.0000 0.802791 0.401396 − 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 4.00000 0.145865
\(753\) −9.00000 −0.327978
\(754\) −1.00000 −0.0364179
\(755\) −12.0000 −0.436725
\(756\) −4.00000 −0.145479
\(757\) −17.0000 −0.617876 −0.308938 − 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) −6.00000 −0.217643
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −21.0000 −0.760750
\(763\) −40.0000 −1.44810
\(764\) −9.00000 −0.325609
\(765\) 6.00000 0.216930
\(766\) 20.0000 0.722629
\(767\) −3.00000 −0.108324
\(768\) −1.00000 −0.0360844
\(769\) 49.0000 1.76699 0.883493 − 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 12.0000 0.432450
\(771\) −12.0000 −0.432169
\(772\) 14.0000 0.503871
\(773\) 36.0000 1.29483 0.647415 − 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) −6.00000 −0.215666
\(775\) 4.00000 0.143684
\(776\) 6.00000 0.215387
\(777\) −12.0000 −0.430498
\(778\) 6.00000 0.215110
\(779\) 6.00000 0.214972
\(780\) −1.00000 −0.0358057
\(781\) −27.0000 −0.966136
\(782\) 6.00000 0.214560
\(783\) −1.00000 −0.0357371
\(784\) 9.00000 0.321429
\(785\) 10.0000 0.356915
\(786\) 10.0000 0.356688
\(787\) −43.0000 −1.53278 −0.766392 − 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(788\) −18.0000 −0.641223
\(789\) −28.0000 −0.996826
\(790\) −12.0000 −0.426941
\(791\) −8.00000 −0.284447
\(792\) 3.00000 0.106600
\(793\) 7.00000 0.248577
\(794\) −30.0000 −1.06466
\(795\) 10.0000 0.354663
\(796\) −13.0000 −0.460773
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 24.0000 0.849591
\(799\) 24.0000 0.849059
\(800\) 4.00000 0.141421
\(801\) 12.0000 0.423999
\(802\) 11.0000 0.388424
\(803\) −6.00000 −0.211735
\(804\) −3.00000 −0.105802
\(805\) −4.00000 −0.140981
\(806\) 1.00000 0.0352235
\(807\) −15.0000 −0.528025
\(808\) 15.0000 0.527698
\(809\) 31.0000 1.08990 0.544951 − 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 22.0000 0.772524 0.386262 − 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 4.00000 0.140372
\(813\) −25.0000 −0.876788
\(814\) 9.00000 0.315450
\(815\) −3.00000 −0.105085
\(816\) −6.00000 −0.210042
\(817\) 36.0000 1.25948
\(818\) −20.0000 −0.699284
\(819\) 4.00000 0.139771
\(820\) 1.00000 0.0349215
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −6.00000 −0.209274
\(823\) −21.0000 −0.732014 −0.366007 − 0.930612i \(-0.619275\pi\)
−0.366007 + 0.930612i \(0.619275\pi\)
\(824\) −11.0000 −0.383203
\(825\) −12.0000 −0.417786
\(826\) 12.0000 0.417533
\(827\) −21.0000 −0.730242 −0.365121 − 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −30.0000 −1.04194 −0.520972 − 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 10.0000 0.347105
\(831\) −5.00000 −0.173448
\(832\) 1.00000 0.0346688
\(833\) 54.0000 1.87099
\(834\) 12.0000 0.415526
\(835\) 11.0000 0.380671
\(836\) −18.0000 −0.622543
\(837\) 1.00000 0.0345651
\(838\) −10.0000 −0.345444
\(839\) 33.0000 1.13929 0.569643 − 0.821892i \(-0.307081\pi\)
0.569643 + 0.821892i \(0.307081\pi\)
\(840\) 4.00000 0.138013
\(841\) 1.00000 0.0344828
\(842\) −13.0000 −0.448010
\(843\) −10.0000 −0.344418
\(844\) −13.0000 −0.447478
\(845\) −12.0000 −0.412813
\(846\) −4.00000 −0.137523
\(847\) −8.00000 −0.274883
\(848\) −10.0000 −0.343401
\(849\) −27.0000 −0.926638
\(850\) 24.0000 0.823193
\(851\) −3.00000 −0.102839
\(852\) −9.00000 −0.308335
\(853\) −54.0000 −1.84892 −0.924462 − 0.381273i \(-0.875486\pi\)
−0.924462 + 0.381273i \(0.875486\pi\)
\(854\) −28.0000 −0.958140
\(855\) 6.00000 0.205196
\(856\) 10.0000 0.341793
\(857\) −58.0000 −1.98124 −0.990621 − 0.136637i \(-0.956370\pi\)
−0.990621 + 0.136637i \(0.956370\pi\)
\(858\) −3.00000 −0.102418
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 6.00000 0.204598
\(861\) −4.00000 −0.136320
\(862\) −12.0000 −0.408722
\(863\) 11.0000 0.374444 0.187222 − 0.982318i \(-0.440052\pi\)
0.187222 + 0.982318i \(0.440052\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.0000 0.408012
\(866\) 2.00000 0.0679628
\(867\) −19.0000 −0.645274
\(868\) −4.00000 −0.135769
\(869\) −36.0000 −1.22122
\(870\) 1.00000 0.0339032
\(871\) 3.00000 0.101651
\(872\) 10.0000 0.338643
\(873\) −6.00000 −0.203069
\(874\) 6.00000 0.202953
\(875\) −36.0000 −1.21702
\(876\) −2.00000 −0.0675737
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 10.0000 0.337484
\(879\) 24.0000 0.809500
\(880\) −3.00000 −0.101130
\(881\) 28.0000 0.943344 0.471672 − 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) −9.00000 −0.303046
\(883\) 42.0000 1.41341 0.706706 − 0.707507i \(-0.250180\pi\)
0.706706 + 0.707507i \(0.250180\pi\)
\(884\) 6.00000 0.201802
\(885\) 3.00000 0.100844
\(886\) −36.0000 −1.20944
\(887\) 20.0000 0.671534 0.335767 − 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 3.00000 0.100673
\(889\) −84.0000 −2.81727
\(890\) −12.0000 −0.402241
\(891\) −3.00000 −0.100504
\(892\) 18.0000 0.602685
\(893\) 24.0000 0.803129
\(894\) 3.00000 0.100335
\(895\) −4.00000 −0.133705
\(896\) −4.00000 −0.133631
\(897\) 1.00000 0.0333890
\(898\) 19.0000 0.634038
\(899\) −1.00000 −0.0333519
\(900\) −4.00000 −0.133333
\(901\) −60.0000 −1.99889
\(902\) 3.00000 0.0998891
\(903\) −24.0000 −0.798670
\(904\) 2.00000 0.0665190
\(905\) 2.00000 0.0664822
\(906\) −12.0000 −0.398673
\(907\) 10.0000 0.332045 0.166022 − 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −10.0000 −0.331862
\(909\) −15.0000 −0.497519
\(910\) −4.00000 −0.132599
\(911\) −29.0000 −0.960813 −0.480406 − 0.877046i \(-0.659511\pi\)
−0.480406 + 0.877046i \(0.659511\pi\)
\(912\) −6.00000 −0.198680
\(913\) 30.0000 0.992855
\(914\) 28.0000 0.926158
\(915\) −7.00000 −0.231413
\(916\) 3.00000 0.0991228
\(917\) 40.0000 1.32092
\(918\) 6.00000 0.198030
\(919\) −29.0000 −0.956622 −0.478311 − 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 1.00000 0.0329690
\(921\) −19.0000 −0.626071
\(922\) 27.0000 0.889198
\(923\) 9.00000 0.296239
\(924\) 12.0000 0.394771
\(925\) −12.0000 −0.394558
\(926\) −14.0000 −0.460069
\(927\) 11.0000 0.361287
\(928\) −1.00000 −0.0328266
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) −1.00000 −0.0327913
\(931\) 54.0000 1.76978
\(932\) −8.00000 −0.262049
\(933\) −4.00000 −0.130954
\(934\) −21.0000 −0.687141
\(935\) −18.0000 −0.588663
\(936\) −1.00000 −0.0326860
\(937\) 20.0000 0.653372 0.326686 − 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) −12.0000 −0.391814
\(939\) 8.00000 0.261070
\(940\) 4.00000 0.130466
\(941\) −15.0000 −0.488986 −0.244493 − 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 10.0000 0.325818
\(943\) −1.00000 −0.0325645
\(944\) −3.00000 −0.0976417
\(945\) −4.00000 −0.130120
\(946\) 18.0000 0.585230
\(947\) 38.0000 1.23483 0.617417 − 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) −12.0000 −0.389742
\(949\) 2.00000 0.0649227
\(950\) 24.0000 0.778663
\(951\) −1.00000 −0.0324272
\(952\) −24.0000 −0.777844
\(953\) 18.0000 0.583077 0.291539 − 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 10.0000 0.323762
\(955\) −9.00000 −0.291233
\(956\) 29.0000 0.937927
\(957\) 3.00000 0.0969762
\(958\) 15.0000 0.484628
\(959\) −24.0000 −0.775000
\(960\) −1.00000 −0.0322749
\(961\) −30.0000 −0.967742
\(962\) −3.00000 −0.0967239
\(963\) −10.0000 −0.322245
\(964\) 6.00000 0.193247
\(965\) 14.0000 0.450676
\(966\) −4.00000 −0.128698
\(967\) −20.0000 −0.643157 −0.321578 − 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 2.00000 0.0642824
\(969\) −36.0000 −1.15649
\(970\) 6.00000 0.192648
\(971\) 35.0000 1.12320 0.561602 − 0.827408i \(-0.310185\pi\)
0.561602 + 0.827408i \(0.310185\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 48.0000 1.53881
\(974\) 22.0000 0.704925
\(975\) 4.00000 0.128103
\(976\) 7.00000 0.224065
\(977\) −3.00000 −0.0959785 −0.0479893 − 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) −3.00000 −0.0959294
\(979\) −36.0000 −1.15056
\(980\) 9.00000 0.287494
\(981\) −10.0000 −0.319275
\(982\) 8.00000 0.255290
\(983\) −29.0000 −0.924956 −0.462478 − 0.886631i \(-0.653040\pi\)
−0.462478 + 0.886631i \(0.653040\pi\)
\(984\) 1.00000 0.0318788
\(985\) −18.0000 −0.573528
\(986\) −6.00000 −0.191079
\(987\) −16.0000 −0.509286
\(988\) 6.00000 0.190885
\(989\) −6.00000 −0.190789
\(990\) 3.00000 0.0953463
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 1.00000 0.0317500
\(993\) 8.00000 0.253872
\(994\) −36.0000 −1.14185
\(995\) −13.0000 −0.412128
\(996\) 10.0000 0.316862
\(997\) −32.0000 −1.01345 −0.506725 − 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) 26.0000 0.823016
\(999\) −3.00000 −0.0949158
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 4002.2.a.c.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.c.1.1 ✓ 1 1.1 even 1 trivial