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Properties

Label 4830.2.a.x.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} +1.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} +2.00000 q^{85} +8.00000 q^{86} +6.00000 q^{87} +6.00000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} -8.00000 q^{93} -4.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(17\) 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 \(-0.651732\pi\)
−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 \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 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
\(47\) 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
\(52\) 2.00000 0.277350
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 2.00000 0.242536
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) −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
\(82\) 2.00000 0.220863
\(83\) 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
\(86\) 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
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −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
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 1.00000 0.0975900
\(106\) −10.0000 −0.971286
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −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
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 4.00000 0.374634
\(115\) 1.00000 0.0932505
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) −2.00000 −0.183340
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) −2.00000 −0.180334
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 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
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −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
\(142\) 4.00000 0.335673
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −14.0000 −1.15865
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) 10.0000 0.793052
\(160\) 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 \(0.213555\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 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 \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) 2.00000 0.147844
\(184\) 1.00000 0.0737210
\(185\) 6.00000 0.441129
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 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
\(193\) 18.0000 1.29567 0.647834 − 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −10.0000 −0.717958
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) 2.00000 0.139686
\(206\) 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 \(0.543966\pi\)
\(212\) −10.0000 −0.686803
\(213\) −4.00000 −0.274075
\(214\) 4.00000 0.273434
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −6.00000 −0.402694
\(223\) 28.0000 1.87502 0.937509 − 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 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
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −8.00000 −0.519656
\(238\) −2.00000 −0.129641
\(239\) −4.00000 −0.258738 −0.129369 − 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) −2.00000 −0.127515
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) 16.0000 1.00991 0.504956 − 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −8.00000 −0.498058
\(259\) −6.00000 −0.372822
\(260\) 2.00000 0.124035
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 4.00000 0.245256
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 2.00000 0.121268
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 4.00000 0.239904
\(279\) 8.00000 0.478947
\(280\) −1.00000 −0.0597614
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 − 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 4.00000 0.237356
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 10.0000 0.586210
\(292\) −14.0000 −0.819288
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 12.0000 0.698667
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) −8.00000 −0.461112
\(302\) 16.0000 0.920697
\(303\) −10.0000 −0.574485
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 8.00000 0.454369
\(311\) −20.0000 −1.13410 −0.567048 − 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) −2.00000 −0.113228
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 14.0000 0.790066
\(315\) −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 \(0.446109\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) −1.00000 −0.0557278
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 20.0000 1.10770
\(327\) 2.00000 0.110600
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 12.0000 0.658586
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −9.00000 −0.489535
\(339\) −2.00000 −0.108625
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) −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 \(0.229302\pi\)
\(348\) 6.00000 0.321634
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 4.00000 0.212298
\(356\) 6.00000 0.317999
\(357\) 2.00000 0.105851
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 11.0000 0.577350
\(364\) −2.00000 −0.104828
\(365\) −14.0000 −0.732793
\(366\) 2.00000 0.104542
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000 0.104116
\(370\) 6.00000 0.311925
\(371\) 10.0000 0.519174
\(372\) −8.00000 −0.414781
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 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
\(381\) 4.00000 0.204926
\(382\) 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
\(386\) 18.0000 0.916176
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −2.00000 −0.101274
\(391\) 2.00000 0.101144
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 8.00000 0.401004
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 14.0000 0.699127 0.349563 − 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 10.0000 0.497519
\(405\) 1.00000 0.0496904
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 2.00000 0.0987730
\(411\) 6.00000 0.295958
\(412\) 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
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 1.00000 0.0487950
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 2.00000 0.0970143
\(426\) −4.00000 −0.193801
\(427\) 2.00000 0.0967868
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −16.0000 −0.770693 −0.385346 − 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −8.00000 −0.384012
\(435\) 6.00000 0.287678
\(436\) −2.00000 −0.0957826
\(437\) −4.00000 −0.191346
\(438\) 14.0000 0.668946
\(439\) −24.0000 −1.14546 −0.572729 − 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −6.00000 −0.284747
\(445\) 6.00000 0.284427
\(446\) 28.0000 1.32584
\(447\) 10.0000 0.472984
\(448\) −1.00000 −0.0472456
\(449\) −38.0000 −1.79333 −0.896665 − 0.442709i \(-0.854018\pi\)
−0.896665 + 0.442709i \(0.854018\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −16.0000 −0.751746
\(454\) −12.0000 −0.563188
\(455\) −2.00000 −0.0937614
\(456\) 4.00000 0.187317
\(457\) 6.00000 0.280668 0.140334 − 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 1.00000 0.0466252
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 − 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −6.00000 −0.278543
\(465\) −8.00000 −0.370991
\(466\) −6.00000 −0.277945
\(467\) −4.00000 −0.185098 −0.0925490 − 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) −10.0000 −0.457869
\(478\) −4.00000 −0.182956
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 12.0000 0.547153
\(482\) 26.0000 1.18427
\(483\) 1.00000 0.0455016
\(484\) −11.0000 −0.500000
\(485\) −10.0000 −0.454077
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −20.0000 −0.904431
\(490\) 1.00000 0.0451754
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −12.0000 −0.540453
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −4.00000 −0.179425
\(498\) −12.0000 −0.537733
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 16.0000 0.714115
\(503\) −40.0000 −1.78351 −0.891756 − 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −4.00000 −0.177471
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 18.0000 0.793946
\(515\) 8.00000 0.352522
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) −6.00000 −0.263371
\(520\) 2.00000 0.0877058
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) 40.0000 1.74908 0.874539 − 0.484955i \(-0.161164\pi\)
0.874539 + 0.484955i \(0.161164\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −10.0000 −0.434372
\(531\) 12.0000 0.520756
\(532\) 4.00000 0.173422
\(533\) 4.00000 0.173259
\(534\) −6.00000 −0.259645
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 6.00000 0.257960 0.128980 − 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) −8.00000 −0.343629
\(543\) 10.0000 0.429141
\(544\) 2.00000 0.0857493
\(545\) −2.00000 −0.0856706
\(546\) 2.00000 0.0855921
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) −1.00000 −0.0425628
\(553\) −8.00000 −0.340195
\(554\) 18.0000 0.764747
\(555\) −6.00000 −0.254686
\(556\) 4.00000 0.169638
\(557\) −34.0000 −1.44063 −0.720313 − 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 8.00000 0.338667
\(559\) 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 \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 8.00000 0.336265
\(567\) −1.00000 −0.0419961
\(568\) 4.00000 0.167836
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 26.0000 1.08239 0.541197 − 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) −13.0000 −0.540729
\(579\) −18.0000 −0.748054
\(580\) −6.00000 −0.249136
\(581\) −12.0000 −0.497844
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) −14.0000 −0.579324
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) −4.00000 −0.165098 −0.0825488 − 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) 12.0000 0.494032
\(591\) −6.00000 −0.246807
\(592\) 6.00000 0.246598
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −10.0000 −0.409616
\(597\) −8.00000 −0.327418
\(598\) 2.00000 0.0817861
\(599\) −20.0000 −0.817178 −0.408589 − 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −11.0000 −0.447214
\(606\) −10.0000 −0.406222
\(607\) 12.0000 0.487065 0.243532 − 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) −4.00000 −0.162221
\(609\) −6.00000 −0.243132
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 28.0000 1.12999
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −8.00000 −0.321807
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 8.00000 0.321288
\(621\) −1.00000 −0.0401286
\(622\) −20.0000 −0.801927
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 12.0000 0.478471
\(630\) −1.00000 −0.0398410
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 8.00000 0.318223
\(633\) 4.00000 0.158986
\(634\) 6.00000 0.238290
\(635\) −4.00000 −0.158735
\(636\) 10.0000 0.396526
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 1.00000 0.0395285
\(641\) 38.0000 1.50091 0.750455 − 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) −4.00000 −0.157867
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −8.00000 −0.315000
\(646\) −8.00000 −0.314756
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) 20.0000 0.783260
\(653\) −26.0000 −1.01746 −0.508729 − 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 2.00000 0.0782062
\(655\) 12.0000 0.468879
\(656\) 2.00000 0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 − 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −28.0000 −1.08825
\(663\) −4.00000 −0.155347
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) 6.00000 0.232495
\(667\) −6.00000 −0.232321
\(668\) 0 0
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −26.0000 −1.00148
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 10.0000 0.383765
\(680\) 2.00000 0.0766965
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) −1.00000 −0.0381802
\(687\) 10.0000 0.381524
\(688\) 8.00000 0.304997
\(689\) −20.0000 −0.761939
\(690\) −1.00000 −0.0380693
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) 4.00000 0.151511
\(698\) −2.00000 −0.0757011
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) −10.0000 −0.376089
\(708\) −12.0000 −0.450988
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 4.00000 0.150117
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 8.00000 0.299602
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 4.00000 0.149383
\(718\) −24.0000 −0.895672
\(719\) −4.00000 −0.149175 −0.0745874 − 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) −26.0000 −0.966950
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 11.0000 0.408248
\(727\) −48.0000 −1.78022 −0.890111 − 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 16.0000 0.591781
\(732\) 2.00000 0.0739221
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 6.00000 0.220564
\(741\) 8.00000 0.293887
\(742\) 10.0000 0.367112
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) −8.00000 −0.293294
\(745\) −10.0000 −0.366372
\(746\) 6.00000 0.219676
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) −1.00000 −0.0365148
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) −16.0000 −0.583072
\(754\) −12.0000 −0.437014
\(755\) 16.0000 0.582300
\(756\) 1.00000 0.0363696
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −22.0000 −0.797499 −0.398750 − 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 4.00000 0.144905
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 18.0000 0.647834
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 8.00000 0.287554
\(775\) 8.00000 0.287368
\(776\) −10.0000 −0.358979
\(777\) 6.00000 0.215249
\(778\) 30.0000 1.07555
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 2.00000 0.0715199
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 14.0000 0.499681
\(786\) −12.0000 −0.428026
\(787\) 48.0000 1.71102 0.855508 − 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 18.0000 0.638796
\(795\) 10.0000 0.354663
\(796\) 8.00000 0.283552
\(797\) 14.0000 0.495905 0.247953 − 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 14.0000 0.494357
\(803\) 0 0
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) 16.0000 0.563576
\(807\) 6.00000 0.211210
\(808\) 10.0000 0.351799
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 6.00000 0.210559
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) 10.0000 0.349642
\(819\) −2.00000 −0.0698857
\(820\) 2.00000 0.0698430
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 6.00000 0.209274
\(823\) −12.0000 −0.418294 −0.209147 − 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 1.00000 0.0347524
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 12.0000 0.416526
\(831\) −18.0000 −0.624413
\(832\) 2.00000 0.0693375
\(833\) 2.00000 0.0692959
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 1.00000 0.0345033
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 18.0000 0.619953
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) −10.0000 −0.343401
\(849\) −8.00000 −0.274559
\(850\) 2.00000 0.0685994
\(851\) 6.00000 0.205677
\(852\) −4.00000 −0.137038
\(853\) 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
\(856\) 4.00000 0.136717
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 8.00000 0.272798
\(861\) 2.00000 0.0681598
\(862\) −16.0000 −0.544962
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −26.0000 −0.883516
\(867\) 13.0000 0.441503
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) −2.00000 −0.0677285
\(873\) −10.0000 −0.338449
\(874\) −4.00000 −0.135302
\(875\) −1.00000 −0.0338062
\(876\) 14.0000 0.473016
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −24.0000 −0.809961
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 − 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 1.00000 0.0336718
\(883\) −44.0000 −1.48072 −0.740359 − 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 4.00000 0.134535
\(885\) −12.0000 −0.403376
\(886\) 12.0000 0.403148
\(887\) −8.00000 −0.268614 −0.134307 − 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −6.00000 −0.201347
\(889\) 4.00000 0.134156
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) −12.0000 −0.401116
\(896\) −1.00000 −0.0334077
\(897\) −2.00000 −0.0667781
\(898\) −38.0000 −1.26808
\(899\) −48.0000 −1.60089
\(900\) 1.00000 0.0333333
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 2.00000 0.0665190
\(905\) −10.0000 −0.332411
\(906\) −16.0000 −0.531564
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −12.0000 −0.398234
\(909\) 10.0000 0.331679
\(910\) −2.00000 −0.0662994
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 2.00000 0.0661180
\(916\) −10.0000 −0.330409
\(917\) −12.0000 −0.396275
\(918\) −2.00000 −0.0660098
\(919\) 8.00000 0.263896 0.131948 − 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 1.00000 0.0329690
\(921\) −28.0000 −0.922631
\(922\) −14.0000 −0.461065
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 20.0000 0.657241
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) −54.0000 −1.77168 −0.885841 − 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) −8.00000 −0.262330
\(931\) −4.00000 −0.131095
\(932\) −6.00000 −0.196537
\(933\) 20.0000 0.654771
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −14.0000 −0.456145
\(943\) 2.00000 0.0651290
\(944\) 12.0000 0.390567
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −8.00000 −0.259828
\(949\) −28.0000 −0.908918
\(950\) −4.00000 −0.129777
\(951\) −6.00000 −0.194563
\(952\) −2.00000 −0.0648204
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) 6.00000 0.193750
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) 4.00000 0.128898
\(964\) 26.0000 0.837404
\(965\) 18.0000 0.579441
\(966\) 1.00000 0.0321745
\(967\) 28.0000 0.900419 0.450210 − 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −11.0000 −0.353553
\(969\) 8.00000 0.256997
\(970\) −10.0000 −0.321081
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 12.0000 0.384505
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) −20.0000 −0.639529
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −2.00000 −0.0638551
\(982\) −28.0000 −0.893516
\(983\) −32.0000 −1.02064 −0.510321 − 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 6.00000 0.191176
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 8.00000 0.254000
\(993\) 28.0000 0.888553
\(994\) −4.00000 −0.126872
\(995\) 8.00000 0.253617
\(996\) −12.0000 −0.380235
\(997\) −38.0000 −1.20347 −0.601736 − 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −20.0000 −0.633089
\(999\) −6.00000 −0.189832
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4830.2.a.x.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.x.1.1 ✓ 1 1.1 even 1 trivial