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Properties

Label 4830.2.a.bi.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} +18.0000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −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
\(22\) 4.00000 0.852803
\(23\) −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
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 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
\(33\) 4.00000 0.696311
\(34\) 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
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −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 \(0.462659\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −4.00000 −0.455842
\(78\) −2.00000 −0.226455
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 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
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −4.00000 −0.374634
\(115\) −1.00000 −0.0932505
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) −2.00000 −0.183340
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) 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
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 2.00000 0.165521
\(147\) 1.00000 0.0824786
\(148\) 6.00000 0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −24.0000 −1.95309 −0.976546 − 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −4.00000 −0.324443
\(153\) 2.00000 0.161690
\(154\) −4.00000 −0.322329
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) 4.00000 0.311400
\(166\) 12.0000 0.931381
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) 4.00000 0.301511
\(177\) −4.00000 −0.300658
\(178\) −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
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 2.00000 0.148250
\(183\) 14.0000 1.03491
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) 8.00000 0.586588
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) −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
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 18.0000 1.29232
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) 2.00000 0.140028
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) −1.00000 −0.0690066
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −10.0000 −0.677285
\(219\) 2.00000 0.135147
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) 6.00000 0.402694
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −4.00000 −0.263181
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 1.00000 0.0645497
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 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
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 4.00000 0.249029
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 4.00000 0.246183
\(265\) −2.00000 −0.122859
\(266\) 4.00000 0.245256
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 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
\(275\) 4.00000 0.241209
\(276\) −1.00000 −0.0601929
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 12.0000 0.719712
\(279\) 8.00000 0.478947
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 8.00000 0.476393
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) −8.00000 −0.473050
\(287\) −2.00000 −0.118056
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 18.0000 1.05518
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 1.00000 0.0583212
\(295\) −4.00000 −0.232889
\(296\) 6.00000 0.348743
\(297\) 4.00000 0.232104
\(298\) 6.00000 0.347571
\(299\) 2.00000 0.115663
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) −24.0000 −1.38104
\(303\) −10.0000 −0.574485
\(304\) −4.00000 −0.229416
\(305\) 14.0000 0.801638
\(306\) 2.00000 0.114332
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −4.00000 −0.227921
\(309\) −8.00000 −0.455104
\(310\) 8.00000 0.454369
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.00000 −0.113228
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −2.00000 −0.112154
\(319\) −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
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) −10.0000 −0.553001
\(328\) 2.00000 0.110432
\(329\) −8.00000 −0.441054
\(330\) 4.00000 0.220193
\(331\) −36.0000 −1.97874 −0.989369 − 0.145424i \(-0.953545\pi\)
−0.989369 + 0.145424i \(0.953545\pi\)
\(332\) 12.0000 0.658586
\(333\) 6.00000 0.328798
\(334\) 16.0000 0.875481
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) −14.0000 −0.760376
\(340\) 2.00000 0.108465
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) −1.00000 −0.0538382
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −2.00000 −0.105851
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 5.00000 0.262432
\(364\) 2.00000 0.104828
\(365\) 2.00000 0.104685
\(366\) 14.0000 0.731792
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) 6.00000 0.311925
\(371\) 2.00000 0.103835
\(372\) 8.00000 0.414781
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 8.00000 0.413670
\(375\) 1.00000 0.0516398
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −4.00000 −0.205196
\(381\) 8.00000 0.409852
\(382\) 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
\(385\) −4.00000 −0.203859
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 18.0000 0.913812
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) −2.00000 −0.101274
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) −12.0000 −0.605320
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) 2.00000 0.0992583
\(407\) 24.0000 1.18964
\(408\) 2.00000 0.0990148
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 2.00000 0.0987730
\(411\) −6.00000 −0.295958
\(412\) −8.00000 −0.394132
\(413\) 4.00000 0.196827
\(414\) −1.00000 −0.0491473
\(415\) 12.0000 0.589057
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) −16.0000 −0.782586
\(419\) −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 \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 4.00000 0.194717
\(423\) 8.00000 0.388973
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −14.0000 −0.677507
\(428\) −12.0000 −0.580042
\(429\) −8.00000 −0.386244
\(430\) 4.00000 0.192897
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −8.00000 −0.384012
\(435\) −2.00000 −0.0958927
\(436\) −10.0000 −0.478913
\(437\) 4.00000 0.191346
\(438\) 2.00000 0.0955637
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 4.00000 0.190693
\(441\) 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
\(446\) 8.00000 0.378811
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 8.00000 0.376705
\(452\) −14.0000 −0.658505
\(453\) −24.0000 −1.12762
\(454\) 12.0000 0.563188
\(455\) 2.00000 0.0937614
\(456\) −4.00000 −0.187317
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 6.00000 0.280362
\(459\) 2.00000 0.0933520
\(460\) −1.00000 −0.0466252
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −4.00000 −0.186097
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) 10.0000 0.463241
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 8.00000 0.369012
\(471\) −2.00000 −0.0921551
\(472\) −4.00000 −0.184115
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) −2.00000 −0.0915737
\(478\) −24.0000 −1.09773
\(479\) −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
\(482\) 2.00000 0.0910975
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) 18.0000 0.817338
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 14.0000 0.633750
\(489\) −4.00000 −0.180886
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) −4.00000 −0.180151
\(494\) 8.00000 0.359937
\(495\) 4.00000 0.179787
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 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
\(501\) 16.0000 0.714827
\(502\) −4.00000 −0.178529
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −10.0000 −0.444994
\(506\) −4.00000 −0.177822
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 2.00000 0.0885615
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 2.00000 0.0882162
\(515\) −8.00000 −0.352522
\(516\) 4.00000 0.176090
\(517\) 32.0000 1.40736
\(518\) −6.00000 −0.263625
\(519\) 6.00000 0.263371
\(520\) −2.00000 −0.0877058
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −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
\(526\) 8.00000 0.348817
\(527\) 16.0000 0.696971
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) −2.00000 −0.0868744
\(531\) −4.00000 −0.173585
\(532\) 4.00000 0.173422
\(533\) −4.00000 −0.173259
\(534\) −6.00000 −0.259645
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) −2.00000 −0.0862261
\(539\) 4.00000 0.172292
\(540\) 1.00000 0.0430331
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −8.00000 −0.343629
\(543\) −10.0000 −0.429141
\(544\) 2.00000 0.0857493
\(545\) −10.0000 −0.428353
\(546\) 2.00000 0.0855921
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −6.00000 −0.256307
\(549\) 14.0000 0.597505
\(550\) 4.00000 0.170561
\(551\) 8.00000 0.340811
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 6.00000 0.254686
\(556\) 12.0000 0.508913
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 8.00000 0.338667
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 8.00000 0.337760
\(562\) 18.0000 0.759284
\(563\) 28.0000 1.18006 0.590030 − 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 8.00000 0.336861
\(565\) −14.0000 −0.588984
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −4.00000 −0.167542
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) −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 \(0.651411\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) −2.00000 −0.0830455
\(581\) −12.0000 −0.497844
\(582\) 18.0000 0.746124
\(583\) −8.00000 −0.331326
\(584\) 2.00000 0.0827606
\(585\) −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
\(590\) −4.00000 −0.164677
\(591\) 22.0000 0.904959
\(592\) 6.00000 0.246598
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 4.00000 0.164122
\(595\) −2.00000 −0.0819920
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 2.00000 0.0817861
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −4.00000 −0.163028
\(603\) −4.00000 −0.162893
\(604\) −24.0000 −0.976546
\(605\) 5.00000 0.203279
\(606\) −10.0000 −0.406222
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −4.00000 −0.162221
\(609\) 2.00000 0.0810441
\(610\) 14.0000 0.566843
\(611\) −16.0000 −0.647291
\(612\) 2.00000 0.0808452
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −12.0000 −0.484281
\(615\) 2.00000 0.0806478
\(616\) −4.00000 −0.161165
\(617\) −38.0000 −1.52982 −0.764911 − 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −8.00000 −0.321807
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 8.00000 0.321288
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) −16.0000 −0.638978
\(628\) −2.00000 −0.0798087
\(629\) 12.0000 0.478471
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) −18.0000 −0.714871
\(635\) 8.00000 0.317470
\(636\) −2.00000 −0.0793052
\(637\) −2.00000 −0.0792429
\(638\) −8.00000 −0.316723
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −12.0000 −0.473602
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 1.00000 0.0394055
\(645\) 4.00000 0.157500
\(646\) −8.00000 −0.314756
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) −2.00000 −0.0784465
\(651\) −8.00000 −0.313545
\(652\) −4.00000 −0.156652
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −10.0000 −0.391031
\(655\) −12.0000 −0.468879
\(656\) 2.00000 0.0780869
\(657\) 2.00000 0.0780274
\(658\) −8.00000 −0.311872
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 4.00000 0.155700
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −36.0000 −1.39918
\(663\) −4.00000 −0.155347
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) 6.00000 0.232495
\(667\) 2.00000 0.0774403
\(668\) 16.0000 0.619059
\(669\) 8.00000 0.309298
\(670\) −4.00000 −0.154533
\(671\) 56.0000 2.16186
\(672\) −1.00000 −0.0385758
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −14.0000 −0.537667
\(679\) −18.0000 −0.690777
\(680\) 2.00000 0.0766965
\(681\) 12.0000 0.459841
\(682\) 32.0000 1.22534
\(683\) 28.0000 1.07139 0.535695 − 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) −1.00000 −0.0381802
\(687\) 6.00000 0.228914
\(688\) 4.00000 0.152499
\(689\) 4.00000 0.152388
\(690\) −1.00000 −0.0380693
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000 0.228086
\(693\) −4.00000 −0.151947
\(694\) 12.0000 0.455514
\(695\) 12.0000 0.455186
\(696\) −2.00000 −0.0758098
\(697\) 4.00000 0.151511
\(698\) −26.0000 −0.984115
\(699\) 10.0000 0.378235
\(700\) −1.00000 −0.0377964
\(701\) 14.0000 0.528773 0.264386 − 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −24.0000 −0.905177
\(704\) 4.00000 0.150756
\(705\) 8.00000 0.301297
\(706\) −14.0000 −0.526897
\(707\) 10.0000 0.376089
\(708\) −4.00000 −0.150329
\(709\) 14.0000 0.525781 0.262891 − 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −8.00000 −0.299602
\(714\) −2.00000 −0.0748481
\(715\) −8.00000 −0.299183
\(716\) 12.0000 0.448461
\(717\) −24.0000 −0.896296
\(718\) 24.0000 0.895672
\(719\) −8.00000 −0.298350 −0.149175 − 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) 2.00000 0.0743808
\(724\) −10.0000 −0.371647
\(725\) −2.00000 −0.0742781
\(726\) 5.00000 0.185567
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 8.00000 0.295891
\(732\) 14.0000 0.517455
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 16.0000 0.590571
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −16.0000 −0.589368
\(738\) 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
\(742\) 2.00000 0.0734223
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 8.00000 0.293294
\(745\) 6.00000 0.219823
\(746\) 6.00000 0.219676
\(747\) 12.0000 0.439057
\(748\) 8.00000 0.292509
\(749\) 12.0000 0.438470
\(750\) 1.00000 0.0365148
\(751\) 48.0000 1.75154 0.875772 − 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 8.00000 0.291730
\(753\) −4.00000 −0.145768
\(754\) 4.00000 0.145671
\(755\) −24.0000 −0.873449
\(756\) −1.00000 −0.0363696
\(757\) −26.0000 −0.944986 −0.472493 − 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 20.0000 0.726433
\(759\) −4.00000 −0.145191
\(760\) −4.00000 −0.145095
\(761\) −14.0000 −0.507500 −0.253750 − 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 8.00000 0.289809
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −4.00000 −0.144150
\(771\) 2.00000 0.0720282
\(772\) −14.0000 −0.503871
\(773\) −10.0000 −0.359675 −0.179838 − 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) 18.0000 0.646162
\(777\) −6.00000 −0.215249
\(778\) −10.0000 −0.358517
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) −2.00000 −0.0715199
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) −12.0000 −0.428026
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 22.0000 0.783718
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 4.00000 0.142134
\(793\) −28.0000 −0.994309
\(794\) −34.0000 −1.20661
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 4.00000 0.141598
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −6.00000 −0.211867
\(803\) 8.00000 0.282314
\(804\) −4.00000 −0.141069
\(805\) 1.00000 0.0352454
\(806\) −16.0000 −0.563576
\(807\) −2.00000 −0.0704033
\(808\) −10.0000 −0.351799
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 2.00000 0.0701862
\(813\) −8.00000 −0.280572
\(814\) 24.0000 0.841200
\(815\) −4.00000 −0.140114
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) −6.00000 −0.209785
\(819\) 2.00000 0.0698857
\(820\) 2.00000 0.0698430
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −6.00000 −0.209274
\(823\) −32.0000 −1.11545 −0.557725 − 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) −8.00000 −0.278693
\(825\) 4.00000 0.139262
\(826\) 4.00000 0.139178
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 12.0000 0.416526
\(831\) 22.0000 0.763172
\(832\) −2.00000 −0.0693375
\(833\) 2.00000 0.0692959
\(834\) 12.0000 0.415526
\(835\) 16.0000 0.553703
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) −12.0000 −0.414533
\(839\) 40.0000 1.38095 0.690477 − 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −25.0000 −0.862069
\(842\) −2.00000 −0.0689246
\(843\) 18.0000 0.619953
\(844\) 4.00000 0.137686
\(845\) −9.00000 −0.309609
\(846\) 8.00000 0.275046
\(847\) −5.00000 −0.171802
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) 2.00000 0.0685994
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 − 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −14.0000 −0.479070
\(855\) −4.00000 −0.136797
\(856\) −12.0000 −0.410152
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −8.00000 −0.273115
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 4.00000 0.136399
\(861\) −2.00000 −0.0681598
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) 2.00000 0.0679628
\(867\) −13.0000 −0.441503
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) 18.0000 0.609208
\(874\) 4.00000 0.135302
\(875\) −1.00000 −0.0338062
\(876\) 2.00000 0.0675737
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 32.0000 1.07995
\(879\) 6.00000 0.202375
\(880\) 4.00000 0.134840
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 1.00000 0.0336718
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −4.00000 −0.134535
\(885\) −4.00000 −0.134459
\(886\) 12.0000 0.403148
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 6.00000 0.201347
\(889\) −8.00000 −0.268311
\(890\) −6.00000 −0.201120
\(891\) 4.00000 0.134005
\(892\) 8.00000 0.267860
\(893\) −32.0000 −1.07084
\(894\) 6.00000 0.200670
\(895\) 12.0000 0.401116
\(896\) −1.00000 −0.0334077
\(897\) 2.00000 0.0667781
\(898\) −30.0000 −1.00111
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) 8.00000 0.266371
\(903\) −4.00000 −0.133112
\(904\) −14.0000 −0.465633
\(905\) −10.0000 −0.332411
\(906\) −24.0000 −0.797347
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 12.0000 0.398234
\(909\) −10.0000 −0.331679
\(910\) 2.00000 0.0662994
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) 2.00000 0.0661541
\(915\) 14.0000 0.462826
\(916\) 6.00000 0.198246
\(917\) 12.0000 0.396275
\(918\) 2.00000 0.0660098
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −12.0000 −0.395413
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 6.00000 0.197279
\(926\) −40.0000 −1.31448
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) −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
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 4.00000 0.130605
\(939\) −6.00000 −0.195803
\(940\) 8.00000 0.260931
\(941\) 14.0000 0.456387 0.228193 − 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −2.00000 −0.0651290
\(944\) −4.00000 −0.130189
\(945\) −1.00000 −0.0325300
\(946\) 16.0000 0.520205
\(947\) −60.0000 −1.94974 −0.974869 − 0.222779i \(-0.928487\pi\)
−0.974869 + 0.222779i \(0.928487\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) −4.00000 −0.129777
\(951\) −18.0000 −0.583690
\(952\) −2.00000 −0.0648204
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −8.00000 −0.258603
\(958\) −16.0000 −0.516937
\(959\) 6.00000 0.193750
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) −14.0000 −0.450676
\(966\) 1.00000 0.0321745
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 5.00000 0.160706
\(969\) −8.00000 −0.256997
\(970\) 18.0000 0.577945
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) −12.0000 −0.384702
\(974\) −32.0000 −1.02535
\(975\) −2.00000 −0.0640513
\(976\) 14.0000 0.448129
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −4.00000 −0.127906
\(979\) −24.0000 −0.767043
\(980\) 1.00000 0.0319438
\(981\) −10.0000 −0.319275
\(982\) −12.0000 −0.382935
\(983\) 32.0000 1.02064 0.510321 − 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 2.00000 0.0637577
\(985\) 22.0000 0.700978
\(986\) −4.00000 −0.127386
\(987\) −8.00000 −0.254643
\(988\) 8.00000 0.254514
\(989\) −4.00000 −0.127193
\(990\) 4.00000 0.127128
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000 0.254000
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 6.00000 0.190022 0.0950110 − 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 4.00000 0.126618
\(999\) 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.bi.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bi.1.1 ✓ 1 1.1 even 1 trivial