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Properties

Label 4830.2.a.bh.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} +2.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +10.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +10.0000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +2.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{77} +16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +2.00000 q^{86} +4.00000 q^{87} -2.00000 q^{88} +1.00000 q^{90} -1.00000 q^{92} +4.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} -4.00000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −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
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 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
\(33\) −2.00000 −0.348155
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −2.00000 −0.301511
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) 4.00000 0.525226
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 2.00000 0.242536
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 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
\(85\) 2.00000 0.216930
\(86\) 2.00000 0.215666
\(87\) 4.00000 0.428845
\(88\) −2.00000 −0.213201
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 1.00000 0.101015
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −2.00000 −0.190693
\(111\) −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
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −2.00000 −0.183340
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −14.0000 −1.26750
\(123\) 6.00000 0.541002
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −14.0000 −1.24230 −0.621150 − 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −2.00000 −0.174078
\(133\) −4.00000 −0.346844
\(134\) 10.0000 0.863868
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −2.00000 −0.165521
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 2.00000 0.161165
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 16.0000 1.27289
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −13.0000 −1.00000
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) 2.00000 0.152499
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.00000 0.303239
\(175\) −1.00000 −0.0755929
\(176\) −2.00000 −0.150756
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 1.00000 0.0745356
\(181\) 26.0000 1.93256 0.966282 − 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) −1.00000 −0.0737210
\(185\) −2.00000 −0.147043
\(186\) 4.00000 0.293294
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 10.0000 0.705346
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 2.00000 0.140028
\(205\) 6.00000 0.419058
\(206\) 8.00000 0.557386
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) −1.00000 −0.0690066
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 2.00000 0.137038
\(214\) 12.0000 0.820303
\(215\) 2.00000 0.136399
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) −10.0000 −0.677285
\(219\) −2.00000 −0.135147
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −14.0000 −0.937509 −0.468755 − 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 4.00000 0.264906
\(229\) −18.0000 −1.18947 −0.594737 − 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 2.00000 0.131590
\(232\) 4.00000 0.262613
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 16.0000 1.03931
\(238\) −2.00000 −0.129641
\(239\) 26.0000 1.68180 0.840900 − 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 1.00000 0.0638877
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) −18.0000 −1.13615 −0.568075 − 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 2.00000 0.125739
\(254\) −14.0000 −0.878438
\(255\) 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 \(0.356166\pi\)
\(258\) 2.00000 0.124515
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −2.00000 −0.123091
\(265\) 10.0000 0.614295
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 12.0000 0.731653 0.365826 − 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 1.00000 0.0608581
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −2.00000 −0.120605
\(276\) −1.00000 −0.0601929
\(277\) −20.0000 −1.20168 −0.600842 − 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 4.00000 0.239904
\(279\) 4.00000 0.239474
\(280\) −1.00000 −0.0597614
\(281\) −28.0000 −1.67034 −0.835170 − 0.549992i \(-0.814631\pi\)
−0.835170 + 0.549992i \(0.814631\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 − 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 2.00000 0.118678
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) −4.00000 −0.234484
\(292\) −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
\(295\) −8.00000 −0.465778
\(296\) −2.00000 −0.116248
\(297\) −2.00000 −0.116052
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −2.00000 −0.115278
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −14.0000 −0.801638
\(306\) 2.00000 0.114332
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 4.00000 0.227185
\(311\) 6.00000 0.340229 0.170114 − 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 14.0000 0.790066
\(315\) −1.00000 −0.0563436
\(316\) 16.0000 0.900070
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 10.0000 0.560772
\(319\) −8.00000 −0.447914
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 1.00000 0.0557278
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 12.0000 0.658586
\(333\) −2.00000 −0.109599
\(334\) −12.0000 −0.656611
\(335\) 10.0000 0.546358
\(336\) −1.00000 −0.0545545
\(337\) 12.0000 0.653682 0.326841 − 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) −13.0000 −0.707107
\(339\) −6.00000 −0.325875
\(340\) 2.00000 0.108465
\(341\) −8.00000 −0.433224
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 2.00000 0.107833
\(345\) −1.00000 −0.0538382
\(346\) −6.00000 −0.322562
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 4.00000 0.214423
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −8.00000 −0.425195
\(355\) 2.00000 0.106149
\(356\) 0 0
\(357\) −2.00000 −0.105851
\(358\) −24.0000 −1.26844
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 26.0000 1.36653
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) −14.0000 −0.731792
\(367\) 24.0000 1.25279 0.626395 − 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) −2.00000 −0.103975
\(371\) −10.0000 −0.519174
\(372\) 4.00000 0.207390
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −4.00000 −0.206835
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 4.00000 0.205196
\(381\) −14.0000 −0.717242
\(382\) −8.00000 −0.409316
\(383\) 8.00000 0.408781 0.204390 − 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.00000 0.101929
\(386\) −14.0000 −0.712581
\(387\) 2.00000 0.101666
\(388\) −4.00000 −0.203069
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) −10.0000 −0.503793
\(395\) 16.0000 0.805047
\(396\) −2.00000 −0.100504
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 16.0000 0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 10.0000 0.498755
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) −4.00000 −0.198517
\(407\) 4.00000 0.198273
\(408\) 2.00000 0.0990148
\(409\) −2.00000 −0.0988936 −0.0494468 − 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 6.00000 0.296319
\(411\) 2.00000 0.0986527
\(412\) 8.00000 0.394132
\(413\) 8.00000 0.393654
\(414\) −1.00000 −0.0491473
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) −8.00000 −0.391293
\(419\) 26.0000 1.27018 0.635092 − 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 2.00000 0.0970143
\(426\) 2.00000 0.0969003
\(427\) 14.0000 0.677507
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) −20.0000 −0.961139 −0.480569 − 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) −4.00000 −0.192006
\(435\) 4.00000 0.191785
\(436\) −10.0000 −0.478913
\(437\) −4.00000 −0.191346
\(438\) −2.00000 −0.0955637
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 14.0000 0.662177
\(448\) −1.00000 −0.0472456
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 1.00000 0.0471405
\(451\) −12.0000 −0.565058
\(452\) −6.00000 −0.282216
\(453\) 20.0000 0.939682
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −18.0000 −0.841085
\(459\) 2.00000 0.0933520
\(460\) −1.00000 −0.0466252
\(461\) 24.0000 1.11779 0.558896 − 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 2.00000 0.0930484
\(463\) 14.0000 0.650635 0.325318 − 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 4.00000 0.185695
\(465\) 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 \(0.589555\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −8.00000 −0.368230
\(473\) −4.00000 −0.183920
\(474\) 16.0000 0.734904
\(475\) 4.00000 0.183533
\(476\) −2.00000 −0.0916698
\(477\) 10.0000 0.457869
\(478\) 26.0000 1.18921
\(479\) −32.0000 −1.46212 −0.731059 − 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) 1.00000 0.0455016
\(484\) −7.00000 −0.318182
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) −14.0000 −0.633750
\(489\) 0 0
\(490\) 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
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 4.00000 0.179605
\(497\) −2.00000 −0.0897123
\(498\) 12.0000 0.537733
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) −12.0000 −0.536120
\(502\) −18.0000 −0.803379
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −13.0000 −0.577350
\(508\) −14.0000 −0.621150
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 2.00000 0.0885615
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 14.0000 0.617514
\(515\) 8.00000 0.352522
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 40.0000 1.75243 0.876216 − 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) 4.00000 0.175075
\(523\) −38.0000 −1.66162 −0.830812 − 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) −16.0000 −0.697633
\(527\) 8.00000 0.348485
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 10.0000 0.434372
\(531\) −8.00000 −0.347170
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 10.0000 0.431934
\(537\) −24.0000 −1.03568
\(538\) 12.0000 0.517357
\(539\) −2.00000 −0.0861461
\(540\) 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
\(543\) 26.0000 1.11577
\(544\) 2.00000 0.0857493
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 2.00000 0.0854358
\(549\) −14.0000 −0.597505
\(550\) −2.00000 −0.0852803
\(551\) 16.0000 0.681623
\(552\) −1.00000 −0.0425628
\(553\) −16.0000 −0.680389
\(554\) −20.0000 −0.849719
\(555\) −2.00000 −0.0848953
\(556\) 4.00000 0.169638
\(557\) 22.0000 0.932170 0.466085 − 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) −1.00000 −0.0422577
\(561\) −4.00000 −0.168880
\(562\) −28.0000 −1.18111
\(563\) 28.0000 1.18006 0.590030 − 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −22.0000 −0.924729
\(567\) −1.00000 −0.0419961
\(568\) 2.00000 0.0839181
\(569\) 16.0000 0.670755 0.335377 − 0.942084i \(-0.391136\pi\)
0.335377 + 0.942084i \(0.391136\pi\)
\(570\) 4.00000 0.167542
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) −6.00000 −0.250435
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) 4.00000 0.166091
\(581\) −12.0000 −0.497844
\(582\) −4.00000 −0.165805
\(583\) −20.0000 −0.828315
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 1.00000 0.0412393
\(589\) 16.0000 0.659269
\(590\) −8.00000 −0.329355
\(591\) −10.0000 −0.411345
\(592\) −2.00000 −0.0821995
\(593\) −2.00000 −0.0821302 −0.0410651 − 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) −2.00000 −0.0820610
\(595\) −2.00000 −0.0819920
\(596\) 14.0000 0.573462
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 1.00000 0.0408248
\(601\) 6.00000 0.244745 0.122373 − 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 10.0000 0.407231
\(604\) 20.0000 0.813788
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 − 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 4.00000 0.162221
\(609\) −4.00000 −0.162088
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 12.0000 0.484281
\(615\) 6.00000 0.241943
\(616\) 2.00000 0.0805823
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 8.00000 0.321807
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 4.00000 0.160644
\(621\) −1.00000 −0.0401286
\(622\) 6.00000 0.240578
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −16.0000 −0.639489
\(627\) −8.00000 −0.319489
\(628\) 14.0000 0.558661
\(629\) −4.00000 −0.159490
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 16.0000 0.636446
\(633\) −20.0000 −0.794929
\(634\) −22.0000 −0.873732
\(635\) −14.0000 −0.555573
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) 2.00000 0.0791188
\(640\) 1.00000 0.0395285
\(641\) −12.0000 −0.473972 −0.236986 − 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 12.0000 0.473602
\(643\) −6.00000 −0.236617 −0.118308 − 0.992977i \(-0.537747\pi\)
−0.118308 + 0.992977i \(0.537747\pi\)
\(644\) 1.00000 0.0394055
\(645\) 2.00000 0.0787499
\(646\) 8.00000 0.314756
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −10.0000 −0.391031
\(655\) −12.0000 −0.468879
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 − 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −4.00000 −0.155113
\(666\) −2.00000 −0.0774984
\(667\) −4.00000 −0.154881
\(668\) −12.0000 −0.464294
\(669\) −14.0000 −0.541271
\(670\) 10.0000 0.386334
\(671\) 28.0000 1.08093
\(672\) −1.00000 −0.0385758
\(673\) −50.0000 −1.92736 −0.963679 − 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 12.0000 0.462223
\(675\) 1.00000 0.0384900
\(676\) −13.0000 −0.500000
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −6.00000 −0.230429
\(679\) 4.00000 0.153506
\(680\) 2.00000 0.0766965
\(681\) 24.0000 0.919682
\(682\) −8.00000 −0.306336
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 4.00000 0.152944
\(685\) 2.00000 0.0764161
\(686\) −1.00000 −0.0381802
\(687\) −18.0000 −0.686743
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −6.00000 −0.228086
\(693\) 2.00000 0.0759737
\(694\) 28.0000 1.06287
\(695\) 4.00000 0.151729
\(696\) 4.00000 0.151620
\(697\) 12.0000 0.454532
\(698\) −10.0000 −0.378506
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 2.00000 0.0750587
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) −2.00000 −0.0748481
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 26.0000 0.970988
\(718\) −32.0000 −1.19423
\(719\) −14.0000 −0.522112 −0.261056 − 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) 26.0000 0.966282
\(725\) 4.00000 0.148556
\(726\) −7.00000 −0.259794
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 4.00000 0.147945
\(732\) −14.0000 −0.517455
\(733\) 38.0000 1.40356 0.701781 − 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 24.0000 0.885856
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −20.0000 −0.736709
\(738\) 6.00000 0.220863
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 4.00000 0.146647
\(745\) 14.0000 0.512920
\(746\) −10.0000 −0.366126
\(747\) 12.0000 0.439057
\(748\) −4.00000 −0.146254
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) −1.00000 −0.0363696
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −28.0000 −1.01701
\(759\) 2.00000 0.0725954
\(760\) 4.00000 0.145095
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) −14.0000 −0.507166
\(763\) 10.0000 0.362024
\(764\) −8.00000 −0.289430
\(765\) 2.00000 0.0723102
\(766\) 8.00000 0.289052
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 2.00000 0.0720750
\(771\) 14.0000 0.504198
\(772\) −14.0000 −0.503871
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 2.00000 0.0718885
\(775\) 4.00000 0.143684
\(776\) −4.00000 −0.143592
\(777\) 2.00000 0.0717496
\(778\) −30.0000 −1.07555
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −2.00000 −0.0715199
\(783\) 4.00000 0.142948
\(784\) 1.00000 0.0357143
\(785\) 14.0000 0.499681
\(786\) −12.0000 −0.428026
\(787\) −6.00000 −0.213877 −0.106938 − 0.994266i \(-0.534105\pi\)
−0.106938 + 0.994266i \(0.534105\pi\)
\(788\) −10.0000 −0.356235
\(789\) −16.0000 −0.569615
\(790\) 16.0000 0.569254
\(791\) 6.00000 0.213335
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 10.0000 0.354663
\(796\) 16.0000 0.567105
\(797\) 22.0000 0.779280 0.389640 − 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 10.0000 0.352673
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −4.00000 −0.140372
\(813\) 12.0000 0.420858
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 8.00000 0.279885
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 8.00000 0.279202 0.139601 − 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 2.00000 0.0697580
\(823\) −6.00000 −0.209147 −0.104573 − 0.994517i \(-0.533348\pi\)
−0.104573 + 0.994517i \(0.533348\pi\)
\(824\) 8.00000 0.278693
\(825\) −2.00000 −0.0696311
\(826\) 8.00000 0.278356
\(827\) 4.00000 0.139094 0.0695468 − 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 12.0000 0.416526
\(831\) −20.0000 −0.693792
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 4.00000 0.138509
\(835\) −12.0000 −0.415277
\(836\) −8.00000 −0.276686
\(837\) 4.00000 0.138260
\(838\) 26.0000 0.898155
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −13.0000 −0.448276
\(842\) 2.00000 0.0689246
\(843\) −28.0000 −0.964371
\(844\) −20.0000 −0.688428
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 10.0000 0.343401
\(849\) −22.0000 −0.755038
\(850\) 2.00000 0.0685994
\(851\) 2.00000 0.0685591
\(852\) 2.00000 0.0685189
\(853\) 32.0000 1.09566 0.547830 − 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) 14.0000 0.479070
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) 26.0000 0.888143 0.444072 − 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 2.00000 0.0681994
\(861\) −6.00000 −0.204479
\(862\) 24.0000 0.817443
\(863\) 20.0000 0.680808 0.340404 − 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −20.0000 −0.679628
\(867\) −13.0000 −0.441503
\(868\) −4.00000 −0.135769
\(869\) −32.0000 −1.08553
\(870\) 4.00000 0.135613
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) −4.00000 −0.135379
\(874\) −4.00000 −0.135302
\(875\) −1.00000 −0.0338062
\(876\) −2.00000 −0.0675737
\(877\) −4.00000 −0.135070 −0.0675352 − 0.997717i \(-0.521513\pi\)
−0.0675352 + 0.997717i \(0.521513\pi\)
\(878\) −8.00000 −0.269987
\(879\) −6.00000 −0.202375
\(880\) −2.00000 −0.0674200
\(881\) −56.0000 −1.88669 −0.943344 − 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) 1.00000 0.0336718
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) 36.0000 1.20944
\(887\) −8.00000 −0.268614 −0.134307 − 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 14.0000 0.469545
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) −24.0000 −0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) 20.0000 0.666297
\(902\) −12.0000 −0.399556
\(903\) −2.00000 −0.0665558
\(904\) −6.00000 −0.199557
\(905\) 26.0000 0.864269
\(906\) 20.0000 0.664455
\(907\) −30.0000 −0.996134 −0.498067 − 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 4.00000 0.132453
\(913\) −24.0000 −0.794284
\(914\) 8.00000 0.264616
\(915\) −14.0000 −0.462826
\(916\) −18.0000 −0.594737
\(917\) 12.0000 0.396275
\(918\) 2.00000 0.0660098
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 12.0000 0.395413
\(922\) 24.0000 0.790398
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) −2.00000 −0.0657596
\(926\) 14.0000 0.460069
\(927\) 8.00000 0.262754
\(928\) 4.00000 0.131306
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 4.00000 0.131165
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) 6.00000 0.196431
\(934\) −12.0000 −0.392652
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 36.0000 1.17607 0.588034 − 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) −10.0000 −0.326512
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 14.0000 0.456145
\(943\) −6.00000 −0.195387
\(944\) −8.00000 −0.260378
\(945\) −1.00000 −0.0325300
\(946\) −4.00000 −0.130051
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) −22.0000 −0.713399
\(952\) −2.00000 −0.0648204
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 10.0000 0.323762
\(955\) −8.00000 −0.258874
\(956\) 26.0000 0.840900
\(957\) −8.00000 −0.258603
\(958\) −32.0000 −1.03387
\(959\) −2.00000 −0.0645834
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) −18.0000 −0.579741
\(965\) −14.0000 −0.450676
\(966\) 1.00000 0.0321745
\(967\) −2.00000 −0.0643157 −0.0321578 − 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) −7.00000 −0.224989
\(969\) 8.00000 0.256997
\(970\) −4.00000 −0.128432
\(971\) 30.0000 0.962746 0.481373 − 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 38.0000 1.21573 0.607864 − 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −10.0000 −0.319275
\(982\) 12.0000 0.382935
\(983\) 40.0000 1.27580 0.637901 − 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 6.00000 0.191273
\(985\) −10.0000 −0.318626
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) −2.00000 −0.0635642
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000 0.127000
\(993\) 28.0000 0.888553
\(994\) −2.00000 −0.0634361
\(995\) 16.0000 0.507234
\(996\) 12.0000 0.380235
\(997\) 8.00000 0.253363 0.126681 − 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 4.00000 0.126618
\(999\) −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.bh.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bh.1.1 ✓ 1 1.1 even 1 trivial