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Properties

Label 4002.2.a.n.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +1.00000 q^{29} -2.00000 q^{30} +1.00000 q^{32} +6.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} +2.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} -2.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{56} +4.00000 q^{57} +1.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} +10.0000 q^{61} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} +8.00000 q^{70} +1.00000 q^{72} +10.0000 q^{73} +10.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +8.00000 q^{83} -4.00000 q^{84} -12.0000 q^{85} +4.00000 q^{86} +1.00000 q^{87} +14.0000 q^{89} -2.00000 q^{90} +8.00000 q^{91} -1.00000 q^{92} -8.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) −2.00000 −0.516398
\(16\) 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
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 1.00000 0.185695
\(30\) −2.00000 −0.365148
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −4.00000 −0.617213
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 4.00000 0.529813
\(58\) 1.00000 0.131306
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −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
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 8.00000 0.956183
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −4.00000 −0.436436
\(85\) −12.0000 −1.30158
\(86\) 4.00000 0.431331
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −2.00000 −0.210819
\(91\) 8.00000 0.838628
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 6.00000 0.594089
\(103\) 12.0000 1.18240 0.591198 − 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −2.00000 −0.196116
\(105\) 8.00000 0.780720
\(106\) −10.0000 −0.971286
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 2.00000 0.186501
\(116\) 1.00000 0.0928477
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) −24.0000 −2.20008
\(120\) −2.00000 −0.182574
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) −4.00000 −0.356348
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 4.00000 0.350823
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −16.0000 −1.38738
\(134\) −4.00000 −0.345547
\(135\) −2.00000 −0.172133
\(136\) 6.00000 0.514496
\(137\) 22.0000 1.87959 0.939793 − 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 8.00000 0.676123
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) 9.00000 0.742307
\(148\) 10.0000 0.821995
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 4.00000 0.318223
\(159\) −10.0000 −0.793052
\(160\) −2.00000 −0.158114
\(161\) 4.00000 0.315244
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 1.00000 0.0758098
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 14.0000 1.04934
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −2.00000 −0.149071
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 8.00000 0.592999
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) −4.00000 −0.290957
\(190\) −8.00000 −0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 10.0000 0.717958
\(195\) 4.00000 0.286446
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) −10.0000 −0.703598
\(203\) −4.00000 −0.280745
\(204\) 6.00000 0.420084
\(205\) −4.00000 −0.279372
\(206\) 12.0000 0.836080
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 8.00000 0.552052
\(211\) −28.0000 −1.92760 −0.963800 − 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 10.0000 0.671156
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −4.00000 −0.267261
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 4.00000 0.264906
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 26.0000 1.70332 0.851658 − 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) 12.0000 0.781133
\(237\) 4.00000 0.259828
\(238\) −24.0000 −1.55569
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.00000 −0.129099
\(241\) −30.0000 −1.93247 −0.966235 − 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −18.0000 −1.14998
\(246\) 2.00000 0.127515
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 12.0000 0.758947
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 − 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 4.00000 0.249029
\(259\) −40.0000 −2.48548
\(260\) 4.00000 0.248069
\(261\) 1.00000 0.0618984
\(262\) −4.00000 −0.247121
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) −16.0000 −0.981023
\(267\) 14.0000 0.856786
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 8.00000 0.484182
\(274\) 22.0000 1.32907
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 8.00000 0.478091
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −8.00000 −0.476393
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 10.0000 0.586210
\(292\) 10.0000 0.585206
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 9.00000 0.524891
\(295\) −24.0000 −1.39733
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) −16.0000 −0.922225
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) 4.00000 0.229416
\(305\) −20.0000 −1.14520
\(306\) 6.00000 0.342997
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −2.00000 −0.113228
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 18.0000 1.01580
\(315\) 8.00000 0.450749
\(316\) 4.00000 0.225018
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −10.0000 −0.560772
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) −14.0000 −0.774202
\(328\) 2.00000 0.110432
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 8.00000 0.439057
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) −4.00000 −0.218218
\(337\) −30.0000 −1.63420 −0.817102 − 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −8.00000 −0.431959
\(344\) 4.00000 0.215666
\(345\) 2.00000 0.107676
\(346\) −2.00000 −0.107521
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 1.00000 0.0536056
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 4.00000 0.213809
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −24.0000 −1.27021
\(358\) 12.0000 0.634220
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) −11.0000 −0.577350
\(364\) 8.00000 0.419314
\(365\) −20.0000 −1.04685
\(366\) 10.0000 0.522708
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) −20.0000 −1.03975
\(371\) 40.0000 2.07670
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 − 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) −8.00000 −0.412568
\(377\) −2.00000 −0.103005
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.00000 −0.410391
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 4.00000 0.202548
\(391\) −6.00000 −0.303433
\(392\) 9.00000 0.454569
\(393\) −4.00000 −0.201773
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 4.00000 0.200502
\(399\) −16.0000 −0.801002
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 − 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) −2.00000 −0.0993808
\(406\) −4.00000 −0.198517
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −4.00000 −0.197546
\(411\) 22.0000 1.08518
\(412\) 12.0000 0.591198
\(413\) −48.0000 −2.36193
\(414\) −1.00000 −0.0491473
\(415\) −16.0000 −0.785409
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 − 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 8.00000 0.390360
\(421\) −14.0000 −0.682318 −0.341159 − 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −28.0000 −1.36302
\(423\) −8.00000 −0.388973
\(424\) −10.0000 −0.485643
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) −14.0000 −0.670478
\(437\) −4.00000 −0.191346
\(438\) 10.0000 0.477818
\(439\) −24.0000 −1.14546 −0.572729 − 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −12.0000 −0.570782
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 10.0000 0.474579
\(445\) −28.0000 −1.32733
\(446\) −8.00000 −0.378811
\(447\) 14.0000 0.662177
\(448\) −4.00000 −0.188982
\(449\) 10.0000 0.471929 0.235965 − 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −16.0000 −0.751746
\(454\) −24.0000 −1.12638
\(455\) −16.0000 −0.750092
\(456\) 4.00000 0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −22.0000 −1.02799
\(459\) 6.00000 0.280056
\(460\) 2.00000 0.0932505
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) 16.0000 0.738025
\(471\) 18.0000 0.829396
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) −24.0000 −1.10004
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −20.0000 −0.911922
\(482\) −30.0000 −1.36646
\(483\) 4.00000 0.182006
\(484\) −11.0000 −0.500000
\(485\) −20.0000 −0.908153
\(486\) 1.00000 0.0453609
\(487\) 40.0000 1.81257 0.906287 − 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 10.0000 0.452679
\(489\) −4.00000 −0.180886
\(490\) −18.0000 −0.813157
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000 0.0901670
\(493\) 6.00000 0.270226
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −4.00000 −0.178174
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) −12.0000 −0.531369
\(511\) −40.0000 −1.76950
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −30.0000 −1.32324
\(515\) −24.0000 −1.05757
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −40.0000 −1.75750
\(519\) −2.00000 −0.0877903
\(520\) 4.00000 0.175412
\(521\) 22.0000 0.963837 0.481919 − 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 1.00000 0.0437688
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −4.00000 −0.174741
\(525\) 4.00000 0.174574
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 20.0000 0.868744
\(531\) 12.0000 0.520756
\(532\) −16.0000 −0.693688
\(533\) −4.00000 −0.173259
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 8.00000 0.343629
\(543\) −6.00000 −0.257485
\(544\) 6.00000 0.257248
\(545\) 28.0000 1.19939
\(546\) 8.00000 0.342368
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 22.0000 0.939793
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) −1.00000 −0.0425628
\(553\) −16.0000 −0.680389
\(554\) −2.00000 −0.0849719
\(555\) −20.0000 −0.848953
\(556\) 12.0000 0.508913
\(557\) 38.0000 1.61011 0.805056 − 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −8.00000 −0.336861
\(565\) −12.0000 −0.504844
\(566\) −20.0000 −0.840663
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) −2.00000 −0.0838444 −0.0419222 − 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) −8.00000 −0.335083
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 19.0000 0.790296
\(579\) 2.00000 0.0831172
\(580\) −2.00000 −0.0830455
\(581\) −32.0000 −1.32758
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) 10.0000 0.413803
\(585\) 4.00000 0.165380
\(586\) 14.0000 0.578335
\(587\) 20.0000 0.825488 0.412744 − 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) 6.00000 0.246807
\(592\) 10.0000 0.410997
\(593\) −46.0000 −1.88899 −0.944497 − 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) 48.0000 1.96781
\(596\) 14.0000 0.573462
\(597\) 4.00000 0.163709
\(598\) 2.00000 0.0817861
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −16.0000 −0.652111
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) 22.0000 0.894427
\(606\) −10.0000 −0.406222
\(607\) −24.0000 −0.974130 −0.487065 − 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 4.00000 0.162221
\(609\) −4.00000 −0.162088
\(610\) −20.0000 −0.809776
\(611\) 16.0000 0.647291
\(612\) 6.00000 0.242536
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −20.0000 −0.807134
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 12.0000 0.482711
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 24.0000 0.962312
\(623\) −56.0000 −2.24359
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 60.0000 2.39236
\(630\) 8.00000 0.318728
\(631\) 12.0000 0.477712 0.238856 − 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 4.00000 0.159111
\(633\) −28.0000 −1.11290
\(634\) 30.0000 1.19145
\(635\) −16.0000 −0.634941
\(636\) −10.0000 −0.396526
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 22.0000 0.868948 0.434474 − 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 − 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 4.00000 0.157622
\(645\) −8.00000 −0.315000
\(646\) 24.0000 0.944267
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −14.0000 −0.547443
\(655\) 8.00000 0.312586
\(656\) 2.00000 0.0780869
\(657\) 10.0000 0.390137
\(658\) 32.0000 1.24749
\(659\) −24.0000 −0.934907 −0.467454 − 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) 8.00000 0.310460
\(665\) 32.0000 1.24091
\(666\) 10.0000 0.387492
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −30.0000 −1.15556
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) −40.0000 −1.53506
\(680\) −12.0000 −0.460179
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) −44.0000 −1.68115
\(686\) −8.00000 −0.305441
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) 2.00000 0.0761387
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −24.0000 −0.910372
\(696\) 1.00000 0.0379049
\(697\) 12.0000 0.454532
\(698\) −10.0000 −0.378506
\(699\) 26.0000 0.983410
\(700\) 4.00000 0.151186
\(701\) 46.0000 1.73740 0.868698 − 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) −6.00000 −0.225813
\(707\) 40.0000 1.50435
\(708\) 12.0000 0.450988
\(709\) −30.0000 −1.12667 −0.563337 − 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −48.0000 −1.78761
\(722\) −3.00000 −0.111648
\(723\) −30.0000 −1.11571
\(724\) −6.00000 −0.222988
\(725\) −1.00000 −0.0371391
\(726\) −11.0000 −0.408248
\(727\) 4.00000 0.148352 0.0741759 − 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 24.0000 0.887672
\(732\) 10.0000 0.369611
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −12.0000 −0.442928
\(735\) −18.0000 −0.663940
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −20.0000 −0.735215
\(741\) −8.00000 −0.293887
\(742\) 40.0000 1.46845
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −28.0000 −1.02584
\(746\) 26.0000 0.951928
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 4.00000 0.145962 0.0729810 − 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) −8.00000 −0.291730
\(753\) −16.0000 −0.583072
\(754\) −2.00000 −0.0728357
\(755\) 32.0000 1.16460
\(756\) −4.00000 −0.145479
\(757\) 42.0000 1.52652 0.763258 − 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 8.00000 0.289809
\(763\) 56.0000 2.02734
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 2.00000 0.0719816
\(773\) 46.0000 1.65451 0.827253 − 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) −40.0000 −1.43499
\(778\) 6.00000 0.215110
\(779\) 8.00000 0.286630
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) −6.00000 −0.214560
\(783\) 1.00000 0.0357371
\(784\) 9.00000 0.321429
\(785\) −36.0000 −1.28490
\(786\) −4.00000 −0.142675
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 6.00000 0.213741
\(789\) 16.0000 0.569615
\(790\) −8.00000 −0.284627
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 30.0000 1.06466
\(795\) 20.0000 0.709327
\(796\) 4.00000 0.141776
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −16.0000 −0.566394
\(799\) −48.0000 −1.69812
\(800\) −1.00000 −0.0353553
\(801\) 14.0000 0.494666
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) −10.0000 −0.351799
\(809\) −22.0000 −0.773479 −0.386739 − 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) −4.00000 −0.140372
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) −6.00000 −0.209785
\(819\) 8.00000 0.279543
\(820\) −4.00000 −0.139686
\(821\) 22.0000 0.767805 0.383903 − 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 22.0000 0.767338
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −16.0000 −0.555368
\(831\) −2.00000 −0.0693792
\(832\) −2.00000 −0.0693375
\(833\) 54.0000 1.87099
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −8.00000 −0.276355
\(839\) −32.0000 −1.10476 −0.552381 − 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 8.00000 0.276026
\(841\) 1.00000 0.0344828
\(842\) −14.0000 −0.482472
\(843\) 30.0000 1.03325
\(844\) −28.0000 −0.963800
\(845\) 18.0000 0.619219
\(846\) −8.00000 −0.275046
\(847\) 44.0000 1.51186
\(848\) −10.0000 −0.343401
\(849\) −20.0000 −0.686398
\(850\) −6.00000 −0.205798
\(851\) −10.0000 −0.342796
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 − 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −40.0000 −1.36877
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −8.00000 −0.272798
\(861\) −8.00000 −0.272639
\(862\) 24.0000 0.817443
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 1.00000 0.0340207
\(865\) 4.00000 0.136004
\(866\) 2.00000 0.0679628
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 8.00000 0.271070
\(872\) −14.0000 −0.474100
\(873\) 10.0000 0.338449
\(874\) −4.00000 −0.135302
\(875\) −48.0000 −1.62270
\(876\) 10.0000 0.337869
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −24.0000 −0.809961
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −10.0000 −0.336909 −0.168454 − 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 9.00000 0.303046
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −12.0000 −0.403604
\(885\) −24.0000 −0.806751
\(886\) −4.00000 −0.134383
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 10.0000 0.335578
\(889\) −32.0000 −1.07325
\(890\) −28.0000 −0.938562
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) 14.0000 0.468230
\(895\) −24.0000 −0.802232
\(896\) −4.00000 −0.133631
\(897\) 2.00000 0.0667781
\(898\) 10.0000 0.333704
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −60.0000 −1.99889
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 6.00000 0.199557
\(905\) 12.0000 0.398893
\(906\) −16.0000 −0.531564
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −24.0000 −0.796468
\(909\) −10.0000 −0.331679
\(910\) −16.0000 −0.530395
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −20.0000 −0.661180
\(916\) −22.0000 −0.726900
\(917\) 16.0000 0.528367
\(918\) 6.00000 0.198030
\(919\) 4.00000 0.131948 0.0659739 − 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 2.00000 0.0659380
\(921\) −20.0000 −0.659022
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 16.0000 0.525793
\(927\) 12.0000 0.394132
\(928\) 1.00000 0.0328266
\(929\) 50.0000 1.64045 0.820223 − 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 26.0000 0.851658
\(933\) 24.0000 0.785725
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 16.0000 0.522419
\(939\) −14.0000 −0.456873
\(940\) 16.0000 0.521862
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 18.0000 0.586472
\(943\) −2.00000 −0.0651290
\(944\) 12.0000 0.390567
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 4.00000 0.129914
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) 30.0000 0.972817
\(952\) −24.0000 −0.777844
\(953\) −34.0000 −1.10137 −0.550684 − 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −88.0000 −2.84167
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) −30.0000 −0.966235
\(965\) −4.00000 −0.128765
\(966\) 4.00000 0.128698
\(967\) −56.0000 −1.80084 −0.900419 − 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) −11.0000 −0.353553
\(969\) 24.0000 0.770991
\(970\) −20.0000 −0.642161
\(971\) −56.0000 −1.79713 −0.898563 − 0.438845i \(-0.855388\pi\)
−0.898563 + 0.438845i \(0.855388\pi\)
\(972\) 1.00000 0.0320750
\(973\) −48.0000 −1.53881
\(974\) 40.0000 1.28168
\(975\) 2.00000 0.0640513
\(976\) 10.0000 0.320092
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) −18.0000 −0.574989
\(981\) −14.0000 −0.446986
\(982\) 20.0000 0.638226
\(983\) 48.0000 1.53096 0.765481 − 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 2.00000 0.0637577
\(985\) −12.0000 −0.382352
\(986\) 6.00000 0.191079
\(987\) 32.0000 1.01857
\(988\) −8.00000 −0.254514
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 8.00000 0.253490
\(997\) 6.00000 0.190022 0.0950110 − 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 36.0000 1.13956
\(999\) 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 4002.2.a.n.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.n.1.1 ✓ 1 1.1 even 1 trivial