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Properties

Label 4650.2.a.s.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
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 \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} +1.00000 q^{22} -1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -5.00000 q^{37} -1.00000 q^{39} -1.00000 q^{41} -1.00000 q^{42} +11.0000 q^{43} -1.00000 q^{44} -5.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +6.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -15.0000 q^{61} -1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -2.00000 q^{67} +6.00000 q^{68} +11.0000 q^{71} -1.00000 q^{72} +16.0000 q^{73} +5.00000 q^{74} -1.00000 q^{77} +1.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +1.00000 q^{82} +5.00000 q^{83} +1.00000 q^{84} -11.0000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -1.00000 q^{91} +1.00000 q^{93} +5.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 − 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) −1.00000 −0.154303
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) −5.00000 −0.729325 −0.364662 − 0.931140i \(-0.618816\pi\)
−0.364662 + 0.931140i \(0.618816\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −1.00000 −0.138675
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 − 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) −1.00000 −0.127000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 11.0000 1.30546 0.652730 − 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 5.00000 0.581238
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 1.00000 0.113228
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.00000 0.110432
\(83\) 5.00000 0.548821 0.274411 − 0.961613i \(-0.411517\pi\)
0.274411 + 0.961613i \(0.411517\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −11.0000 −1.18616
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 5.00000 0.515711
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 6.00000 0.606092
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −6.00000 −0.594089
\(103\) −3.00000 −0.295599 −0.147799 − 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 1.00000 0.0944911
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 15.0000 1.35804
\(123\) −1.00000 −0.0901670
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −6.00000 −0.532414 −0.266207 − 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 − 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) −5.00000 −0.421076
\(142\) −11.0000 −0.923099
\(143\) 1.00000 0.0836242
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) −6.00000 −0.494872
\(148\) −5.00000 −0.410997
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 2.00000 0.159111
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 24.0000 1.87983 0.939913 − 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) −5.00000 −0.388075
\(167\) 14.0000 1.08335 0.541676 − 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) 16.0000 1.21646 0.608229 − 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −4.00000 −0.300658
\(178\) 6.00000 0.449719
\(179\) 1.00000 0.0747435 0.0373718 − 0.999301i \(-0.488101\pi\)
0.0373718 + 0.999301i \(0.488101\pi\)
\(180\) 0 0
\(181\) −11.0000 −0.817624 −0.408812 − 0.912619i \(-0.634057\pi\)
−0.408812 + 0.912619i \(0.634057\pi\)
\(182\) 1.00000 0.0741249
\(183\) −15.0000 −1.10883
\(184\) 0 0
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −6.00000 −0.438763
\(188\) −5.00000 −0.364662
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −1.00000 −0.0719816 −0.0359908 − 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 1.00000 0.0710669
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −14.0000 −0.985037
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 3.00000 0.209020
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 − 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 9.00000 0.618123
\(213\) 11.0000 0.753708
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 1.00000 0.0678844
\(218\) 2.00000 0.135457
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 5.00000 0.335578
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 16.0000 1.06196 0.530979 − 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 − 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) −6.00000 −0.393919
\(233\) −1.00000 −0.0655122 −0.0327561 − 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −2.00000 −0.129914
\(238\) −6.00000 −0.388922
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 10.0000 0.642824
\(243\) 1.00000 0.0641500
\(244\) −15.0000 −0.960277
\(245\) 0 0
\(246\) 1.00000 0.0637577
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) −25.0000 −1.57799 −0.788993 − 0.614402i \(-0.789397\pi\)
−0.788993 + 0.614402i \(0.789397\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000 0.935674 0.467837 − 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) −11.0000 −0.684830
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −4.00000 −0.247121
\(263\) −2.00000 −0.123325 −0.0616626 − 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −2.00000 −0.122169
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 − 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 6.00000 0.363803
\(273\) −1.00000 −0.0605228
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −9.00000 −0.539784
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 15.0000 0.894825 0.447412 − 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) 5.00000 0.297746
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 11.0000 0.652730
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) −1.00000 −0.0590281
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 16.0000 0.936329
\(293\) 16.0000 0.934730 0.467365 − 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 5.00000 0.290619
\(297\) −1.00000 −0.0580259
\(298\) −4.00000 −0.231714
\(299\) 0 0
\(300\) 0 0
\(301\) 11.0000 0.634029
\(302\) −10.0000 −0.575435
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 2.00000 0.114146 0.0570730 − 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −3.00000 −0.170664
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 − 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 1.00000 0.0566139
\(313\) 28.0000 1.58265 0.791327 − 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −9.00000 −0.504695
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −24.0000 −1.32924
\(327\) −2.00000 −0.110600
\(328\) 1.00000 0.0552158
\(329\) −5.00000 −0.275659
\(330\) 0 0
\(331\) −9.00000 −0.494685 −0.247342 − 0.968928i \(-0.579557\pi\)
−0.247342 + 0.968928i \(0.579557\pi\)
\(332\) 5.00000 0.274411
\(333\) −5.00000 −0.273998
\(334\) −14.0000 −0.766046
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 16.0000 0.871576 0.435788 − 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 12.0000 0.652714
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −1.00000 −0.0541530
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) −27.0000 −1.44944 −0.724718 − 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) 6.00000 0.321634
\(349\) 28.0000 1.49881 0.749403 − 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 1.00000 0.0533002
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 6.00000 0.317554
\(358\) −1.00000 −0.0528516
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 11.0000 0.578147
\(363\) −10.0000 −0.524864
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 15.0000 0.784063
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −1.00000 −0.0520579
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 1.00000 0.0518476
\(373\) −24.0000 −1.24267 −0.621336 − 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 5.00000 0.257855
\(377\) −6.00000 −0.309016
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 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
\(386\) 1.00000 0.0508987
\(387\) 11.0000 0.559161
\(388\) 2.00000 0.101535
\(389\) 14.0000 0.709828 0.354914 − 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 4.00000 0.201773
\(394\) 3.00000 0.151138
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 12.0000 0.602263 0.301131 − 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 − 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 2.00000 0.0997509
\(403\) −1.00000 −0.0498135
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 5.00000 0.247841
\(408\) −6.00000 −0.297044
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −3.00000 −0.147799
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 9.00000 0.440732
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 − 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 − 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) −2.00000 −0.0973585
\(423\) −5.00000 −0.243108
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −11.0000 −0.532952
\(427\) −15.0000 −0.725901
\(428\) 6.00000 0.290021
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 19.0000 0.915198 0.457599 − 0.889159i \(-0.348710\pi\)
0.457599 + 0.889159i \(0.348710\pi\)
\(432\) 1.00000 0.0481125
\(433\) 28.0000 1.34559 0.672797 − 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −16.0000 −0.764510
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 6.00000 0.285391
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −5.00000 −0.237289
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 4.00000 0.189194
\(448\) 1.00000 0.0472456
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 1.00000 0.0470882
\(452\) 14.0000 0.658505
\(453\) 10.0000 0.469841
\(454\) −16.0000 −0.750917
\(455\) 0 0
\(456\) 0 0
\(457\) −16.0000 −0.748448 −0.374224 − 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 26.0000 1.21490
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 − 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 1.00000 0.0465242
\(463\) −6.00000 −0.278844 −0.139422 − 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 1.00000 0.0463241
\(467\) 22.0000 1.01804 0.509019 − 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 4.00000 0.184115
\(473\) −11.0000 −0.505781
\(474\) 2.00000 0.0918630
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 9.00000 0.412082
\(478\) 20.0000 0.914779
\(479\) 40.0000 1.82765 0.913823 − 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 15.0000 0.679018
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −1.00000 −0.0450835
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 11.0000 0.493417
\(498\) −5.00000 −0.224055
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 14.0000 0.625474
\(502\) 25.0000 1.11580
\(503\) 7.00000 0.312115 0.156057 − 0.987748i \(-0.450122\pi\)
0.156057 + 0.987748i \(0.450122\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) −6.00000 −0.266207
\(509\) −1.00000 −0.0443242 −0.0221621 − 0.999754i \(-0.507055\pi\)
−0.0221621 + 0.999754i \(0.507055\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −15.0000 −0.661622
\(515\) 0 0
\(516\) 11.0000 0.484248
\(517\) 5.00000 0.219900
\(518\) 5.00000 0.219687
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 − 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 6.00000 0.261364
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 1.00000 0.0433148
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 1.00000 0.0431532
\(538\) −18.0000 −0.776035
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 12.0000 0.515444
\(543\) −11.0000 −0.472055
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 1.00000 0.0427960
\(547\) 6.00000 0.256541 0.128271 − 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) −6.00000 −0.256307
\(549\) −15.0000 −0.640184
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 9.00000 0.381685
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) −15.0000 −0.632737
\(563\) −30.0000 −1.26435 −0.632175 − 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) −5.00000 −0.210538
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 1.00000 0.0419961
\(568\) −11.0000 −0.461550
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 43.0000 1.79949 0.899747 − 0.436412i \(-0.143751\pi\)
0.899747 + 0.436412i \(0.143751\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 1.00000 0.0417392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 − 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −19.0000 −0.790296
\(579\) −1.00000 −0.0415586
\(580\) 0 0
\(581\) 5.00000 0.207435
\(582\) −2.00000 −0.0829027
\(583\) −9.00000 −0.372742
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −16.0000 −0.660954
\(587\) 27.0000 1.11441 0.557205 − 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) −6.00000 −0.247436
\(589\) 0 0
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) −5.00000 −0.205499
\(593\) −7.00000 −0.287456 −0.143728 − 0.989617i \(-0.545909\pi\)
−0.143728 + 0.989617i \(0.545909\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 21.0000 0.858037 0.429018 − 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 − 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) −11.0000 −0.448327
\(603\) −2.00000 −0.0814463
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) −31.0000 −1.25825 −0.629126 − 0.777304i \(-0.716587\pi\)
−0.629126 + 0.777304i \(0.716587\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 5.00000 0.202278
\(612\) 6.00000 0.242536
\(613\) 30.0000 1.21169 0.605844 − 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −3.00000 −0.120775 −0.0603877 − 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) 3.00000 0.120678
\(619\) 1.00000 0.0401934 0.0200967 − 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −15.0000 −0.601445
\(623\) −6.00000 −0.240385
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 − 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 2.00000 0.0795557
\(633\) 2.00000 0.0794929
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 6.00000 0.237729
\(638\) 6.00000 0.237542
\(639\) 11.0000 0.435153
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 − 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) −6.00000 −0.236801
\(643\) 29.0000 1.14365 0.571824 − 0.820376i \(-0.306236\pi\)
0.571824 + 0.820376i \(0.306236\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 − 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) 24.0000 0.939913
\(653\) 24.0000 0.939193 0.469596 − 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −1.00000 −0.0390434
\(657\) 16.0000 0.624219
\(658\) 5.00000 0.194920
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 9.00000 0.349795
\(663\) −6.00000 −0.233021
\(664\) −5.00000 −0.194038
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) 0 0
\(668\) 14.0000 0.541676
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 15.0000 0.579069
\(672\) −1.00000 −0.0385758
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −3.00000 −0.115299 −0.0576497 − 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) −14.0000 −0.537667
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 1.00000 0.0382920
\(683\) −2.00000 −0.0765279 −0.0382639 − 0.999268i \(-0.512183\pi\)
−0.0382639 + 0.999268i \(0.512183\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) −26.0000 −0.991962
\(688\) 11.0000 0.419371
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 − 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 16.0000 0.608229
\(693\) −1.00000 −0.0379869
\(694\) 27.0000 1.02491
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −6.00000 −0.227266
\(698\) −28.0000 −1.05982
\(699\) −1.00000 −0.0378235
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 − 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) 1.00000 0.0377426
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 14.0000 0.526524
\(708\) −4.00000 −0.150329
\(709\) −2.00000 −0.0751116 −0.0375558 − 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 1.00000 0.0373718
\(717\) −20.0000 −0.746914
\(718\) 24.0000 0.895672
\(719\) 6.00000 0.223762 0.111881 − 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 19.0000 0.707107
\(723\) −4.00000 −0.148762
\(724\) −11.0000 −0.408812
\(725\) 0 0
\(726\) 10.0000 0.371135
\(727\) −9.00000 −0.333792 −0.166896 − 0.985975i \(-0.553374\pi\)
−0.166896 + 0.985975i \(0.553374\pi\)
\(728\) 1.00000 0.0370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 66.0000 2.44110
\(732\) −15.0000 −0.554416
\(733\) −8.00000 −0.295487 −0.147743 − 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 2.00000 0.0736709
\(738\) 1.00000 0.0368105
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −9.00000 −0.330400
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) 5.00000 0.182940
\(748\) −6.00000 −0.219382
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −5.00000 −0.182453 −0.0912263 − 0.995830i \(-0.529079\pi\)
−0.0912263 + 0.995830i \(0.529079\pi\)
\(752\) −5.00000 −0.182331
\(753\) −25.0000 −0.911051
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −3.00000 −0.109037 −0.0545184 − 0.998513i \(-0.517362\pi\)
−0.0545184 + 0.998513i \(0.517362\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −32.0000 −1.16000 −0.580000 − 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) 6.00000 0.217357
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 4.00000 0.144432
\(768\) 1.00000 0.0360844
\(769\) −13.0000 −0.468792 −0.234396 − 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) −1.00000 −0.0359908
\(773\) −14.0000 −0.503545 −0.251773 − 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −11.0000 −0.395387
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −5.00000 −0.179374
\(778\) −14.0000 −0.501924
\(779\) 0 0
\(780\) 0 0
\(781\) −11.0000 −0.393611
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −19.0000 −0.677277 −0.338638 − 0.940917i \(-0.609966\pi\)
−0.338638 + 0.940917i \(0.609966\pi\)
\(788\) −3.00000 −0.106871
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 1.00000 0.0355335
\(793\) 15.0000 0.532666
\(794\) −12.0000 −0.425864
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 38.0000 1.34603 0.673015 − 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) −30.0000 −1.06132
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −12.0000 −0.423735
\(803\) −16.0000 −0.564628
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 1.00000 0.0352235
\(807\) 18.0000 0.633630
\(808\) −14.0000 −0.492518
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 − 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 6.00000 0.210559
\(813\) −12.0000 −0.420858
\(814\) −5.00000 −0.175250
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 18.0000 0.629355
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 − 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 6.00000 0.209274
\(823\) 24.0000 0.836587 0.418294 − 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 3.00000 0.104510
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 52.0000 1.80822 0.904109 − 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) 0 0
\(829\) 1.00000 0.0347314 0.0173657 − 0.999849i \(-0.494472\pi\)
0.0173657 + 0.999849i \(0.494472\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) −1.00000 −0.0346688
\(833\) −36.0000 −1.24733
\(834\) −9.00000 −0.311645
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) −30.0000 −1.03633
\(839\) −29.0000 −1.00119 −0.500596 − 0.865681i \(-0.666886\pi\)
−0.500596 + 0.865681i \(0.666886\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 12.0000 0.413547
\(843\) 15.0000 0.516627
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 5.00000 0.171904
\(847\) −10.0000 −0.343604
\(848\) 9.00000 0.309061
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 11.0000 0.376854
\(853\) −12.0000 −0.410872 −0.205436 − 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) 15.0000 0.513289
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) −43.0000 −1.46885 −0.734426 − 0.678689i \(-0.762549\pi\)
−0.734426 + 0.678689i \(0.762549\pi\)
\(858\) −1.00000 −0.0341394
\(859\) 13.0000 0.443554 0.221777 − 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) −1.00000 −0.0340799
\(862\) −19.0000 −0.647143
\(863\) 30.0000 1.02121 0.510606 − 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −28.0000 −0.951479
\(867\) 19.0000 0.645274
\(868\) 1.00000 0.0339422
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 2.00000 0.0677285
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −36.0000 −1.21563 −0.607817 − 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(878\) 0 0
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) 4.00000 0.134763 0.0673817 − 0.997727i \(-0.478535\pi\)
0.0673817 + 0.997727i \(0.478535\pi\)
\(882\) 6.00000 0.202031
\(883\) 9.00000 0.302874 0.151437 − 0.988467i \(-0.451610\pi\)
0.151437 + 0.988467i \(0.451610\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −25.0000 −0.839418 −0.419709 − 0.907659i \(-0.637868\pi\)
−0.419709 + 0.907659i \(0.637868\pi\)
\(888\) 5.00000 0.167789
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) −4.00000 −0.133780
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 54.0000 1.79900
\(902\) −1.00000 −0.0332964
\(903\) 11.0000 0.366057
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) −10.0000 −0.332228
\(907\) −8.00000 −0.265636 −0.132818 − 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 16.0000 0.530979
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) −5.00000 −0.165476
\(914\) 16.0000 0.529233
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 4.00000 0.132092
\(918\) −6.00000 −0.198030
\(919\) −11.0000 −0.362857 −0.181428 − 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) −3.00000 −0.0987997
\(923\) −11.0000 −0.362069
\(924\) −1.00000 −0.0328976
\(925\) 0 0
\(926\) 6.00000 0.197172
\(927\) −3.00000 −0.0985329
\(928\) −6.00000 −0.196960
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.00000 −0.0327561
\(933\) 15.0000 0.491078
\(934\) −22.0000 −0.719862
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −54.0000 −1.76410 −0.882052 − 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) 2.00000 0.0653023
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 − 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 6.00000 0.195491
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 11.0000 0.357641
\(947\) −9.00000 −0.292461 −0.146230 − 0.989251i \(-0.546714\pi\)
−0.146230 + 0.989251i \(0.546714\pi\)
\(948\) −2.00000 −0.0649570
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) −6.00000 −0.194461
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) −20.0000 −0.646846
\(957\) −6.00000 −0.193952
\(958\) −40.0000 −1.29234
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −5.00000 −0.161206
\(963\) 6.00000 0.193347
\(964\) −4.00000 −0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 − 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 9.00000 0.288527
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −15.0000 −0.480138
\(977\) −21.0000 −0.671850 −0.335925 − 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) −24.0000 −0.767435
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −12.0000 −0.382935
\(983\) −32.0000 −1.02064 −0.510321 − 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) 1.00000 0.0318788
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) −5.00000 −0.159152
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 − 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −9.00000 −0.285606
\(994\) −11.0000 −0.348899
\(995\) 0 0
\(996\) 5.00000 0.158431
\(997\) −18.0000 −0.570066 −0.285033 − 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 20.0000 0.633089
\(999\) −5.00000 −0.158193
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 4650.2.a.s.1.1 ✓ 1
5.2 odd 4 4650.2.d.f.3349.1 2
5.3 odd 4 4650.2.d.f.3349.2 2
5.4 even 2 4650.2.a.ba.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4650.2.a.s.1.1 ✓ 1 1.1 even 1 trivial
4650.2.a.ba.1.1 yes 1 5.4 even 2
4650.2.d.f.3349.1 2 5.2 odd 4
4650.2.d.f.3349.2 2 5.3 odd 4