Properties

Label 4830.2.a.bc.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
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 \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} -6.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} +6.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -4.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} +1.00000 q^{56} -8.00000 q^{58} -1.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -6.00000 q^{66} -10.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -6.00000 q^{77} -4.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +1.00000 q^{84} -2.00000 q^{85} +6.00000 q^{86} -8.00000 q^{87} -6.00000 q^{88} +8.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} +1.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +4.00000 q^{97} +1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.00000 −1.04447
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 1.00000 0.154303
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) 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
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −4.00000 −0.554700
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 6.00000 0.809040
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −6.00000 −0.738549
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 2.00000 0.242536
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) −4.00000 −0.452911
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 0.109109
\(85\) −2.00000 −0.216930
\(86\) 6.00000 0.646997
\(87\) −8.00000 −0.857690
\(88\) −6.00000 −0.639602
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 −0.419314
\(92\) 1.00000 0.104257
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 1.00000 0.101015
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −4.00000 −0.392232
\(105\) −1.00000 −0.0975900
\(106\) −2.00000 −0.194257
\(107\) 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 \(0.733911\pi\)
\(110\) 6.00000 0.572078
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −8.00000 −0.742781
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) 6.00000 0.503509
\(143\) 24.0000 2.00698
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) −10.0000 −0.827606
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 1.00000 0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) −6.00000 −0.483494
\(155\) −4.00000 −0.321288
\(156\) −4.00000 −0.320256
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 8.00000 0.636446
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −10.0000 −0.780869
\(165\) 6.00000 0.467099
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) 3.00000 0.230769
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −8.00000 −0.606478
\(175\) 1.00000 0.0755929
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 8.00000 0.599625
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) −4.00000 −0.296500
\(183\) −2.00000 −0.147844
\(184\) 1.00000 0.0737210
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) −12.0000 −0.877527
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 4.00000 0.287183
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −6.00000 −0.426401
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.0000 −0.705346
\(202\) −12.0000 −0.844317
\(203\) −8.00000 −0.561490
\(204\) 2.00000 0.140028
\(205\) 10.0000 0.698430
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −2.00000 −0.137361
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 1.00000 0.0680414
\(217\) 4.00000 0.271538
\(218\) −14.0000 −0.948200
\(219\) −10.0000 −0.675737
\(220\) 6.00000 0.404520
\(221\) −8.00000 −0.538138
\(222\) −6.00000 −0.402694
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −6.00000 −0.394771
\(232\) −8.00000 −0.525226
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −4.00000 −0.261488
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 2.00000 0.129641
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) −10.0000 −0.637577
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 1.00000 0.0629941
\(253\) −6.00000 −0.377217
\(254\) −18.0000 −1.12942
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 6.00000 0.373544
\(259\) −6.00000 −0.372822
\(260\) 4.00000 0.248069
\(261\) −8.00000 −0.495188
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −6.00000 −0.369274
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −10.0000 −0.610847
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 2.00000 0.121268
\(273\) −4.00000 −0.242091
\(274\) −14.0000 −0.845771
\(275\) −6.00000 −0.361814
\(276\) 1.00000 0.0601929
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −4.00000 −0.239904
\(279\) 4.00000 0.239474
\(280\) −1.00000 −0.0597614
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) −8.00000 −0.476393
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) −10.0000 −0.590281
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 8.00000 0.469776
\(291\) 4.00000 0.234484
\(292\) −10.0000 −0.585206
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −6.00000 −0.348155
\(298\) −22.0000 −1.27443
\(299\) −4.00000 −0.231326
\(300\) 1.00000 0.0577350
\(301\) 6.00000 0.345834
\(302\) 4.00000 0.230174
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 2.00000 0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −6.00000 −0.341882
\(309\) −8.00000 −0.455104
\(310\) −4.00000 −0.227185
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) −4.00000 −0.226455
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 18.0000 1.01580
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −2.00000 −0.112154
\(319\) 48.0000 2.68748
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 8.00000 0.443079
\(327\) −14.0000 −0.774202
\(328\) −10.0000 −0.552158
\(329\) −8.00000 −0.441054
\(330\) 6.00000 0.330289
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) −12.0000 −0.656611
\(335\) 10.0000 0.546358
\(336\) 1.00000 0.0545545
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 3.00000 0.163178
\(339\) 2.00000 0.108625
\(340\) −2.00000 −0.108465
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 6.00000 0.323498
\(345\) −1.00000 −0.0538382
\(346\) 2.00000 0.107521
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −8.00000 −0.428845
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 1.00000 0.0534522
\(351\) −4.00000 −0.213504
\(352\) −6.00000 −0.319801
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 8.00000 0.423999
\(357\) 2.00000 0.105851
\(358\) 16.0000 0.845626
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) 22.0000 1.15629
\(363\) 25.0000 1.31216
\(364\) −4.00000 −0.209657
\(365\) 10.0000 0.523424
\(366\) −2.00000 −0.104542
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 1.00000 0.0521286
\(369\) −10.0000 −0.520579
\(370\) 6.00000 0.311925
\(371\) −2.00000 −0.103835
\(372\) 4.00000 0.207390
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −12.0000 −0.620505
\(375\) −1.00000 −0.0516398
\(376\) −8.00000 −0.412568
\(377\) 32.0000 1.64808
\(378\) 1.00000 0.0514344
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 8.00000 0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 6.00000 0.305788
\(386\) 10.0000 0.508987
\(387\) 6.00000 0.304997
\(388\) 4.00000 0.203069
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 4.00000 0.202548
\(391\) 2.00000 0.101144
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) −6.00000 −0.301511
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −10.0000 −0.498755
\(403\) −16.0000 −0.797017
\(404\) −12.0000 −0.597022
\(405\) −1.00000 −0.0496904
\(406\) −8.00000 −0.397033
\(407\) 36.0000 1.78445
\(408\) 2.00000 0.0990148
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 10.0000 0.493865
\(411\) −14.0000 −0.690569
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 6.00000 0.290701
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) −6.00000 −0.289346
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 4.00000 0.192006
\(435\) 8.00000 0.383571
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 6.00000 0.286039
\(441\) 1.00000 0.0476190
\(442\) −8.00000 −0.380521
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −6.00000 −0.284747
\(445\) −8.00000 −0.379236
\(446\) 6.00000 0.284108
\(447\) −22.0000 −1.04056
\(448\) 1.00000 0.0472456
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 1.00000 0.0471405
\(451\) 60.0000 2.82529
\(452\) 2.00000 0.0940721
\(453\) 4.00000 0.187936
\(454\) 20.0000 0.938647
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 26.0000 1.21490
\(459\) 2.00000 0.0933520
\(460\) −1.00000 −0.0466252
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) −6.00000 −0.279145
\(463\) −30.0000 −1.39422 −0.697109 0.716965i \(-0.745531\pi\)
−0.697109 + 0.716965i \(0.745531\pi\)
\(464\) −8.00000 −0.371391
\(465\) −4.00000 −0.185496
\(466\) −10.0000 −0.463241
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) −4.00000 −0.184900
\(469\) −10.0000 −0.461757
\(470\) 8.00000 0.369012
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) −36.0000 −1.65528
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) −2.00000 −0.0915737
\(478\) 6.00000 0.274434
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 24.0000 1.09431
\(482\) 22.0000 1.00207
\(483\) 1.00000 0.0455016
\(484\) 25.0000 1.13636
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 8.00000 0.361773
\(490\) −1.00000 −0.0451754
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −10.0000 −0.450835
\(493\) −16.0000 −0.720604
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 4.00000 0.179605
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) 2.00000 0.0892644
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) 12.0000 0.533993
\(506\) −6.00000 −0.266733
\(507\) 3.00000 0.133235
\(508\) −18.0000 −0.798621
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 8.00000 0.352522
\(516\) 6.00000 0.264135
\(517\) 48.0000 2.11104
\(518\) −6.00000 −0.263625
\(519\) 2.00000 0.0877903
\(520\) 4.00000 0.175412
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) −8.00000 −0.350150
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) −12.0000 −0.524222
\(525\) 1.00000 0.0436436
\(526\) −16.0000 −0.697633
\(527\) 8.00000 0.348485
\(528\) −6.00000 −0.261116
\(529\) 1.00000 0.0434783
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) 40.0000 1.73259
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −10.0000 −0.431934
\(537\) 16.0000 0.690451
\(538\) 16.0000 0.689809
\(539\) −6.00000 −0.258438
\(540\) −1.00000 −0.0430331
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −12.0000 −0.515444
\(543\) 22.0000 0.944110
\(544\) 2.00000 0.0857493
\(545\) 14.0000 0.599694
\(546\) −4.00000 −0.171184
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −14.0000 −0.598050
\(549\) −2.00000 −0.0853579
\(550\) −6.00000 −0.255841
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 8.00000 0.340195
\(554\) −8.00000 −0.339887
\(555\) 6.00000 0.254686
\(556\) −4.00000 −0.169638
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 4.00000 0.169334
\(559\) −24.0000 −1.01509
\(560\) −1.00000 −0.0422577
\(561\) −12.0000 −0.506640
\(562\) 12.0000 0.506189
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −8.00000 −0.336861
\(565\) −2.00000 −0.0841406
\(566\) −2.00000 −0.0840663
\(567\) 1.00000 0.0419961
\(568\) 6.00000 0.251754
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 24.0000 1.00349
\(573\) 8.00000 0.334205
\(574\) −10.0000 −0.417392
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −13.0000 −0.540729
\(579\) 10.0000 0.415586
\(580\) 8.00000 0.332182
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 12.0000 0.496989
\(584\) −10.0000 −0.413803
\(585\) 4.00000 0.165380
\(586\) −2.00000 −0.0826192
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −6.00000 −0.246598
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) −6.00000 −0.246183
\(595\) −2.00000 −0.0819920
\(596\) −22.0000 −0.901155
\(597\) 8.00000 0.327418
\(598\) −4.00000 −0.163572
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 1.00000 0.0408248
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 6.00000 0.244542
\(603\) −10.0000 −0.407231
\(604\) 4.00000 0.162758
\(605\) −25.0000 −1.01639
\(606\) −12.0000 −0.487467
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 2.00000 0.0809776
\(611\) 32.0000 1.29458
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 12.0000 0.484281
\(615\) 10.0000 0.403239
\(616\) −6.00000 −0.241747
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) −8.00000 −0.321807
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) −4.00000 −0.160644
\(621\) 1.00000 0.0401286
\(622\) −6.00000 −0.240578
\(623\) 8.00000 0.320513
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 24.0000 0.959233
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −12.0000 −0.478471
\(630\) −1.00000 −0.0398410
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 8.00000 0.318223
\(633\) −12.0000 −0.476957
\(634\) 18.0000 0.714871
\(635\) 18.0000 0.714308
\(636\) −2.00000 −0.0793052
\(637\) −4.00000 −0.158486
\(638\) 48.0000 1.90034
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 1.00000 0.0394055
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 4.00000 0.156772
\(652\) 8.00000 0.313304
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −14.0000 −0.547443
\(655\) 12.0000 0.468879
\(656\) −10.0000 −0.390434
\(657\) −10.0000 −0.390137
\(658\) −8.00000 −0.311872
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 6.00000 0.233550
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 20.0000 0.777322
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −8.00000 −0.309761
\(668\) −12.0000 −0.464294
\(669\) 6.00000 0.231973
\(670\) 10.0000 0.386334
\(671\) 12.0000 0.463255
\(672\) 1.00000 0.0385758
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) −20.0000 −0.770371
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 2.00000 0.0768095
\(679\) 4.00000 0.153506
\(680\) −2.00000 −0.0766965
\(681\) 20.0000 0.766402
\(682\) −24.0000 −0.919007
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 1.00000 0.0381802
\(687\) 26.0000 0.991962
\(688\) 6.00000 0.228748
\(689\) 8.00000 0.304776
\(690\) −1.00000 −0.0380693
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 2.00000 0.0760286
\(693\) −6.00000 −0.227921
\(694\) −12.0000 −0.455514
\(695\) 4.00000 0.151729
\(696\) −8.00000 −0.303239
\(697\) −20.0000 −0.757554
\(698\) −18.0000 −0.681310
\(699\) −10.0000 −0.378235
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −4.00000 −0.150970
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) 8.00000 0.301297
\(706\) −14.0000 −0.526897
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −6.00000 −0.225176
\(711\) 8.00000 0.300023
\(712\) 8.00000 0.299813
\(713\) 4.00000 0.149801
\(714\) 2.00000 0.0748481
\(715\) −24.0000 −0.897549
\(716\) 16.0000 0.597948
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) −19.0000 −0.707107
\(723\) 22.0000 0.818189
\(724\) 22.0000 0.817624
\(725\) −8.00000 −0.297113
\(726\) 25.0000 0.927837
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 12.0000 0.443836
\(732\) −2.00000 −0.0739221
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 8.00000 0.295285
\(735\) −1.00000 −0.0368856
\(736\) 1.00000 0.0368605
\(737\) 60.0000 2.21013
\(738\) −10.0000 −0.368105
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 4.00000 0.146647
\(745\) 22.0000 0.806018
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) −8.00000 −0.291730
\(753\) 2.00000 0.0728841
\(754\) 32.0000 1.16537
\(755\) −4.00000 −0.145575
\(756\) 1.00000 0.0363696
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −32.0000 −1.16229
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) −18.0000 −0.652071
\(763\) −14.0000 −0.506834
\(764\) 8.00000 0.289430
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 6.00000 0.216225
\(771\) 14.0000 0.504198
\(772\) 10.0000 0.359908
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 6.00000 0.215666
\(775\) 4.00000 0.143684
\(776\) 4.00000 0.143592
\(777\) −6.00000 −0.215249
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) −36.0000 −1.28818
\(782\) 2.00000 0.0715199
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) −18.0000 −0.642448
\(786\) −12.0000 −0.428026
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 6.00000 0.213741
\(789\) −16.0000 −0.569615
\(790\) −8.00000 −0.284627
\(791\) 2.00000 0.0711118
\(792\) −6.00000 −0.213201
\(793\) 8.00000 0.284088
\(794\) 20.0000 0.709773
\(795\) 2.00000 0.0709327
\(796\) 8.00000 0.283552
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 1.00000 0.0353553
\(801\) 8.00000 0.282666
\(802\) 0 0
\(803\) 60.0000 2.11735
\(804\) −10.0000 −0.352673
\(805\) −1.00000 −0.0352454
\(806\) −16.0000 −0.563576
\(807\) 16.0000 0.563227
\(808\) −12.0000 −0.422159
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −8.00000 −0.280745
\(813\) −12.0000 −0.420858
\(814\) 36.0000 1.26180
\(815\) −8.00000 −0.280228
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) −18.0000 −0.629355
\(819\) −4.00000 −0.139771
\(820\) 10.0000 0.349215
\(821\) −52.0000 −1.81481 −0.907406 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) −14.0000 −0.488306
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) −8.00000 −0.278693
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 1.00000 0.0347524
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) −4.00000 −0.138675
\(833\) 2.00000 0.0692959
\(834\) −4.00000 −0.138509
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −18.0000 −0.621800
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 35.0000 1.20690
\(842\) −10.0000 −0.344623
\(843\) 12.0000 0.413302
\(844\) −12.0000 −0.413057
\(845\) −3.00000 −0.103203
\(846\) −8.00000 −0.275046
\(847\) 25.0000 0.859010
\(848\) −2.00000 −0.0686803
\(849\) −2.00000 −0.0686398
\(850\) 2.00000 0.0685994
\(851\) −6.00000 −0.205677
\(852\) 6.00000 0.205557
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 24.0000 0.819346
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) −6.00000 −0.204598
\(861\) −10.0000 −0.340799
\(862\) 8.00000 0.272481
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.00000 −0.0680020
\(866\) 4.00000 0.135926
\(867\) −13.0000 −0.441503
\(868\) 4.00000 0.135769
\(869\) −48.0000 −1.62829
\(870\) 8.00000 0.271225
\(871\) 40.0000 1.35535
\(872\) −14.0000 −0.474100
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −10.0000 −0.337869
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) −8.00000 −0.269987
\(879\) −2.00000 −0.0674583
\(880\) 6.00000 0.202260
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 1.00000 0.0336718
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −6.00000 −0.201347
\(889\) −18.0000 −0.603701
\(890\) −8.00000 −0.268161
\(891\) −6.00000 −0.201008
\(892\) 6.00000 0.200895
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) −16.0000 −0.534821
\(896\) 1.00000 0.0334077
\(897\) −4.00000 −0.133556
\(898\) −26.0000 −0.867631
\(899\) −32.0000 −1.06726
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) 60.0000 1.99778
\(903\) 6.00000 0.199667
\(904\) 2.00000 0.0665190
\(905\) −22.0000 −0.731305
\(906\) 4.00000 0.132891
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) 20.0000 0.663723
\(909\) −12.0000 −0.398015
\(910\) 4.00000 0.132599
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 32.0000 1.05847
\(915\) 2.00000 0.0661180
\(916\) 26.0000 0.859064
\(917\) −12.0000 −0.396275
\(918\) 2.00000 0.0660098
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 12.0000 0.395413
\(922\) −28.0000 −0.922131
\(923\) −24.0000 −0.789970
\(924\) −6.00000 −0.197386
\(925\) −6.00000 −0.197279
\(926\) −30.0000 −0.985861
\(927\) −8.00000 −0.262754
\(928\) −8.00000 −0.262613
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) −4.00000 −0.131165
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) −6.00000 −0.196431
\(934\) −24.0000 −0.785304
\(935\) 12.0000 0.392442
\(936\) −4.00000 −0.130744
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) −10.0000 −0.326512
\(939\) 24.0000 0.783210
\(940\) 8.00000 0.260931
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 18.0000 0.586472
\(943\) −10.0000 −0.325645
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) −36.0000 −1.17046
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 8.00000 0.259828
\(949\) 40.0000 1.29845
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 2.00000 0.0648204
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −8.00000 −0.258874
\(956\) 6.00000 0.194054
\(957\) 48.0000 1.55162
\(958\) 16.0000 0.516937
\(959\) −14.0000 −0.452084
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 24.0000 0.773791
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) −10.0000 −0.321911
\(966\) 1.00000 0.0321745
\(967\) −38.0000 −1.22200 −0.610999 0.791632i \(-0.709232\pi\)
−0.610999 + 0.791632i \(0.709232\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) 18.0000 0.576757
\(975\) −4.00000 −0.128103
\(976\) −2.00000 −0.0640184
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 8.00000 0.255812
\(979\) −48.0000 −1.53409
\(980\) −1.00000 −0.0319438
\(981\) −14.0000 −0.446986
\(982\) 28.0000 0.893516
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) −10.0000 −0.318788
\(985\) −6.00000 −0.191176
\(986\) −16.0000 −0.509544
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 6.00000 0.190693
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 4.00000 0.127000
\(993\) 20.0000 0.634681
\(994\) 6.00000 0.190308
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −28.0000 −0.886325
\(999\) −6.00000 −0.189832
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 4830.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bc.1.1 1 1.1 even 1 trivial