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Properties

Label 4830.2.a.be.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
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] = [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: \(0\)
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 \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000 q^{45} -1.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} +6.00000 q^{87} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} +12.0000 q^{94} +4.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) −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
\(11\) 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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 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
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 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
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) 8.00000 0.862662
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) 12.0000 1.23771
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) −1.00000 −0.0975900
\(106\) −6.00000 −0.582772
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −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
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 2.00000 0.165521
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −4.00000 −0.305888
\(172\) 8.00000 0.609994
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 2.00000 0.148250
\(183\) 2.00000 0.147844
\(184\) −1.00000 −0.0737210
\(185\) −2.00000 −0.147043
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) 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
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) 8.00000 0.557386
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) 2.00000 0.135457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) −12.0000 −0.782794
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 6.00000 0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) −4.00000 −0.254000
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 8.00000 0.498058
\(259\) 2.00000 0.124274
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −28.0000 −1.70088 −0.850439 − 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 6.00000 0.363803
\(273\) 2.00000 0.121046
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 12.0000 0.714590
\(283\) −28.0000 −1.66443 −0.832214 − 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −2.00000 −0.115663
\(300\) 1.00000 0.0577350
\(301\) 8.00000 0.461112
\(302\) 8.00000 0.460348
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 6.00000 0.342997
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 4.00000 0.227185
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 2.00000 0.112867
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 20.0000 1.10770
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000 0.658586
\(333\) 2.00000 0.109599
\(334\) 12.0000 0.656611
\(335\) −8.00000 −0.437087
\(336\) 1.00000 0.0545545
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 1.00000 0.0538382
\(346\) 18.0000 0.967686
\(347\) −36.0000 −1.93258 −0.966291 − 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 6.00000 0.321634
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 6.00000 0.317554
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) −11.0000 −0.577350
\(364\) 2.00000 0.104828
\(365\) −2.00000 −0.104685
\(366\) 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
\(369\) −6.00000 −0.312348
\(370\) −2.00000 −0.103975
\(371\) −6.00000 −0.311504
\(372\) −4.00000 −0.207390
\(373\) 2.00000 0.103556 0.0517780 − 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) 12.0000 0.618031
\(378\) 1.00000 0.0514344
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −2.00000 −0.101274
\(391\) −6.00000 −0.303433
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) −18.0000 −0.906827
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −16.0000 −0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 8.00000 0.399004
\(403\) −8.00000 −0.398508
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 6.00000 0.296319
\(411\) 18.0000 0.887875
\(412\) 8.00000 0.394132
\(413\) −12.0000 −0.590481
\(414\) −1.00000 −0.0491473
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.0000 −0.480569 −0.240285 − 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) −4.00000 −0.192006
\(435\) −6.00000 −0.287678
\(436\) 2.00000 0.0957826
\(437\) 4.00000 0.191346
\(438\) 2.00000 0.0955637
\(439\) −28.0000 −1.33637 −0.668184 − 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) −6.00000 −0.284427
\(446\) −4.00000 −0.189405
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) −2.00000 −0.0937614
\(456\) −4.00000 −0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 26.0000 1.21490
\(459\) 6.00000 0.280056
\(460\) 1.00000 0.0466252
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) 8.00000 0.369406
\(470\) −12.0000 −0.553519
\(471\) 2.00000 0.0921551
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) −4.00000 −0.183533
\(476\) 6.00000 0.275010
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 4.00000 0.182384
\(482\) −10.0000 −0.455488
\(483\) −1.00000 −0.0455016
\(484\) −11.0000 −0.500000
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 2.00000 0.0905357
\(489\) 20.0000 0.904431
\(490\) −1.00000 −0.0451754
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) 36.0000 1.62136
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 12.0000 0.535586
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −6.00000 −0.265684
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) −8.00000 −0.352522
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 18.0000 0.790112
\(520\) −2.00000 −0.0877058
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −12.0000 −0.520756
\(532\) −4.00000 −0.173422
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 12.0000 0.517838
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −28.0000 −1.20270
\(543\) −22.0000 −0.944110
\(544\) 6.00000 0.257248
\(545\) −2.00000 −0.0856706
\(546\) 2.00000 0.0855921
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) −1.00000 −0.0425628
\(553\) 8.00000 0.340195
\(554\) −10.0000 −0.424859
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) −6.00000 −0.254228 −0.127114 − 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −4.00000 −0.169334
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 12.0000 0.505291
\(565\) 6.00000 0.252422
\(566\) −28.0000 −1.17693
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 4.00000 0.167542
\(571\) 8.00000 0.334790 0.167395 − 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 − 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 19.0000 0.790296
\(579\) 2.00000 0.0831172
\(580\) −6.00000 −0.249136
\(581\) 12.0000 0.497844
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) −2.00000 −0.0826898
\(586\) −6.00000 −0.247858
\(587\) −36.0000 −1.48588 −0.742940 − 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 1.00000 0.0412393
\(589\) 16.0000 0.659269
\(590\) 12.0000 0.494032
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) −6.00000 −0.245770
\(597\) −16.0000 −0.654836
\(598\) −2.00000 −0.0817861
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) −46.0000 −1.87638 −0.938190 − 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 8.00000 0.326056
\(603\) 8.00000 0.325785
\(604\) 8.00000 0.325515
\(605\) 11.0000 0.447214
\(606\) −6.00000 −0.243733
\(607\) −28.0000 −1.13648 −0.568242 − 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −4.00000 −0.162221
\(609\) 6.00000 0.243132
\(610\) −2.00000 −0.0809776
\(611\) 24.0000 0.970936
\(612\) 6.00000 0.242536
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −4.00000 −0.161427
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 8.00000 0.321807
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 4.00000 0.160644
\(621\) −1.00000 −0.0401286
\(622\) −12.0000 −0.481156
\(623\) 6.00000 0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 12.0000 0.478471
\(630\) −1.00000 −0.0398410
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) 6.00000 0.238290
\(635\) 16.0000 0.634941
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −8.00000 −0.315000
\(646\) −24.0000 −0.944267
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) −4.00000 −0.156772
\(652\) 20.0000 0.783260
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 2.00000 0.0782062
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 12.0000 0.467809
\(659\) −24.0000 −0.934907 −0.467454 − 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 − 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −4.00000 −0.155464
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) 2.00000 0.0774984
\(667\) −6.00000 −0.232321
\(668\) 12.0000 0.464294
\(669\) −4.00000 −0.154649
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 2.00000 0.0770371
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −6.00000 −0.230429
\(679\) −10.0000 −0.383765
\(680\) −6.00000 −0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −4.00000 −0.152944
\(685\) −18.0000 −0.687745
\(686\) 1.00000 0.0381802
\(687\) 26.0000 0.991962
\(688\) 8.00000 0.304997
\(689\) −12.0000 −0.457164
\(690\) 1.00000 0.0380693
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −36.0000 −1.36654
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) −36.0000 −1.36360
\(698\) −22.0000 −0.832712
\(699\) −6.00000 −0.226941
\(700\) 1.00000 0.0377964
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) −30.0000 −1.12906
\(707\) −6.00000 −0.225653
\(708\) −12.0000 −0.450988
\(709\) −46.0000 −1.72757 −0.863783 − 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 4.00000 0.149801
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) −22.0000 −0.817624
\(725\) 6.00000 0.222834
\(726\) −11.0000 −0.408248
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 48.0000 1.77534
\(732\) 2.00000 0.0739221
\(733\) 50.0000 1.84679 0.923396 − 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 8.00000 0.295285
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −8.00000 −0.293887
\(742\) −6.00000 −0.220267
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −4.00000 −0.146647
\(745\) 6.00000 0.219823
\(746\) 2.00000 0.0732252
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 12.0000 0.437595
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) −8.00000 −0.291150
\(756\) 1.00000 0.0363696
\(757\) 26.0000 0.944986 0.472493 − 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −16.0000 −0.579619
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 2.00000 0.0719816
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 8.00000 0.287554
\(775\) −4.00000 −0.143684
\(776\) −10.0000 −0.358979
\(777\) 2.00000 0.0717496
\(778\) 18.0000 0.645331
\(779\) 24.0000 0.859889
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) −6.00000 −0.214560
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) 12.0000 0.428026
\(787\) 20.0000 0.712923 0.356462 − 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −18.0000 −0.641223
\(789\) 24.0000 0.854423
\(790\) −8.00000 −0.284627
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 2.00000 0.0709773
\(795\) 6.00000 0.212798
\(796\) −16.0000 −0.567105
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −4.00000 −0.141598
\(799\) 72.0000 2.54718
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 1.00000 0.0352454
\(806\) −8.00000 −0.281788
\(807\) −6.00000 −0.211210
\(808\) −6.00000 −0.211079
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 6.00000 0.210042
\(817\) −32.0000 −1.11954
\(818\) −22.0000 −0.769212
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 18.0000 0.627822
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) −12.0000 −0.416526
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) 6.00000 0.207888
\(834\) −4.00000 −0.138509
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) −12.0000 −0.414533
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) 18.0000 0.619953
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) 12.0000 0.412568
\(847\) −11.0000 −0.377964
\(848\) −6.00000 −0.206041
\(849\) −28.0000 −0.960958
\(850\) 6.00000 0.205798
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 2.00000 0.0684386
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −8.00000 −0.272798
\(861\) −6.00000 −0.204479
\(862\) −24.0000 −0.817443
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) −10.0000 −0.339814
\(867\) 19.0000 0.645274
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 4.00000 0.135302
\(875\) −1.00000 −0.0338062
\(876\) 2.00000 0.0675737
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −28.0000 −0.944954
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 1.00000 0.0336718
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 12.0000 0.403604
\(885\) 12.0000 0.403376
\(886\) 12.0000 0.403148
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 2.00000 0.0671156
\(889\) −16.0000 −0.536623
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −48.0000 −1.60626
\(894\) −6.00000 −0.200670
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) −2.00000 −0.0667781
\(898\) 18.0000 0.600668
\(899\) −24.0000 −0.800445
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) −6.00000 −0.199557
\(905\) 22.0000 0.731305
\(906\) 8.00000 0.265782
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) −2.00000 −0.0662994
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −2.00000 −0.0661180
\(916\) 26.0000 0.859064
\(917\) 12.0000 0.396275
\(918\) 6.00000 0.198030
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 1.00000 0.0329690
\(921\) −4.00000 −0.131804
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 8.00000 0.262896
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 4.00000 0.131165
\(931\) −4.00000 −0.131095
\(932\) −6.00000 −0.196537
\(933\) −12.0000 −0.392862
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −34.0000 −1.11073 −0.555366 − 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 8.00000 0.261209
\(939\) −10.0000 −0.326338
\(940\) −12.0000 −0.391397
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 2.00000 0.0651635
\(943\) 6.00000 0.195387
\(944\) −12.0000 −0.390567
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 8.00000 0.259828
\(949\) 4.00000 0.129845
\(950\) −4.00000 −0.129777
\(951\) 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 \(0.530982\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −2.00000 −0.0643823
\(966\) −1.00000 −0.0321745
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −11.0000 −0.353553
\(969\) −24.0000 −0.770991
\(970\) 10.0000 0.321081
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) 32.0000 1.02535
\(975\) 2.00000 0.0640513
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) 48.0000 1.53096 0.765481 − 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) −6.00000 −0.191273
\(985\) 18.0000 0.573528
\(986\) 36.0000 1.14647
\(987\) 12.0000 0.381964
\(988\) −8.00000 −0.254514
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −4.00000 −0.127000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 12.0000 0.380235
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 20.0000 0.633089
\(999\) 2.00000 0.0632772
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.be.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.be.1.1 ✓ 1 1.1 even 1 trivial