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Properties

Label 4830.2.a.w.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +8.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -4.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +1.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +2.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -8.00000 q^{57} -1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} -4.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +16.0000 q^{73} +6.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +1.00000 q^{84} -4.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} +4.00000 q^{93} +12.0000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 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
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −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 \(0.335828\pi\)
\(38\) 8.00000 1.29777
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −4.00000 −0.485071
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) −2.00000 −0.209657
\(92\) 1.00000 0.104257
\(93\) 4.00000 0.414781
\(94\) 12.0000 1.23771
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 4.00000 0.396059
\(103\) 20.0000 1.97066 0.985329 − 0.170664i \(-0.0545913\pi\)
0.985329 + 0.170664i \(0.0545913\pi\)
\(104\) 2.00000 0.196116
\(105\) 1.00000 0.0975900
\(106\) −4.00000 −0.388514
\(107\) −14.0000 −1.35343 −0.676716 − 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.00000 −0.766261 −0.383131 − 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 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 \(0.470012\pi\)
\(114\) −8.00000 −0.749269
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) 4.00000 0.360668
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) −4.00000 −0.342997
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −12.0000 −1.01058
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 8.00000 0.648886
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) −2.00000 −0.160128
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) 4.00000 0.317221
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) −4.00000 −0.306786
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 6.00000 0.448461 0.224231 − 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −2.00000 −0.148250
\(183\) 2.00000 0.147844
\(184\) 1.00000 0.0737210
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 1.00000 0.0727393
\(190\) 8.00000 0.580381
\(191\) 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 \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 2.00000 0.143592
\(195\) −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
\(199\) −22.0000 −1.55954 −0.779769 − 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −4.00000 −0.279372
\(206\) 20.0000 1.39347
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −4.00000 −0.274721
\(213\) 8.00000 0.548151
\(214\) −14.0000 −0.957020
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) −8.00000 −0.541828
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −6.00000 −0.402694
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −8.00000 −0.529813
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 4.00000 0.259281
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) 4.00000 0.255031
\(247\) 16.0000 1.01806
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 20.0000 1.25491
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 4.00000 0.249029
\(259\) −6.00000 −0.372822
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) −8.00000 −0.490511
\(267\) −6.00000 −0.367194
\(268\) 12.0000 0.733017
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 28.0000 1.70088 0.850439 − 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −4.00000 −0.242536
\(273\) 2.00000 0.121046
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −24.0000 −1.44202 −0.721010 − 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 4.00000 0.239904
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −12.0000 −0.714590
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −8.00000 −0.474713
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 16.0000 0.936329
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) 2.00000 0.114897
\(304\) 8.00000 0.458831
\(305\) −2.00000 −0.114520
\(306\) −4.00000 −0.228665
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) −4.00000 −0.227185
\(311\) 22.0000 1.24751 0.623753 − 0.781622i \(-0.285607\pi\)
0.623753 + 0.781622i \(0.285607\pi\)
\(312\) −2.00000 −0.113228
\(313\) 30.0000 1.69570 0.847850 − 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 2.00000 0.112867
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 4.00000 0.224309
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 14.0000 0.781404
\(322\) −1.00000 −0.0557278
\(323\) −32.0000 −1.78053
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 2.00000 0.110770
\(327\) 8.00000 0.442401
\(328\) −4.00000 −0.220863
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 1.00000 0.0545545
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) −2.00000 −0.108625
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) −1.00000 −0.0538382
\(346\) 18.0000 0.967686
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 − 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) −4.00000 −0.211702
\(358\) 6.00000 0.317110
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) 2.00000 0.105118
\(363\) 11.0000 0.577350
\(364\) −2.00000 −0.104828
\(365\) 16.0000 0.837478
\(366\) 2.00000 0.104542
\(367\) 24.0000 1.25279 0.626395 − 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 1.00000 0.0521286
\(369\) −4.00000 −0.208232
\(370\) 6.00000 0.311925
\(371\) 4.00000 0.207670
\(372\) 4.00000 0.207390
\(373\) −30.0000 −1.55334 −0.776671 − 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −22.0000 −1.13006 −0.565032 − 0.825069i \(-0.691136\pi\)
−0.565032 + 0.825069i \(0.691136\pi\)
\(380\) 8.00000 0.410391
\(381\) −20.0000 −1.02463
\(382\) 0 0
\(383\) 18.0000 0.919757 0.459879 − 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) −2.00000 −0.101274
\(391\) −4.00000 −0.202289
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 − 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −22.0000 −1.10276
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −34.0000 −1.69788 −0.848939 − 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) −12.0000 −0.598506
\(403\) −8.00000 −0.398508
\(404\) −2.00000 −0.0995037
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 4.00000 0.198030
\(409\) 34.0000 1.68119 0.840596 − 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) −4.00000 −0.197546
\(411\) −18.0000 −0.887875
\(412\) 20.0000 0.985329
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 − 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 1.00000 0.0487950
\(421\) −36.0000 −1.75453 −0.877266 − 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 8.00000 0.389434
\(423\) 12.0000 0.583460
\(424\) −4.00000 −0.194257
\(425\) −4.00000 −0.194029
\(426\) 8.00000 0.387601
\(427\) 2.00000 0.0967868
\(428\) −14.0000 −0.676716
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 − 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 8.00000 0.382692
\(438\) −16.0000 −0.764510
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −8.00000 −0.380521
\(443\) 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
\(446\) −2.00000 −0.0947027
\(447\) −14.0000 −0.662177
\(448\) −1.00000 −0.0472456
\(449\) −26.0000 −1.22702 −0.613508 − 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 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
\(456\) −8.00000 −0.374634
\(457\) −6.00000 −0.280668 −0.140334 − 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) −22.0000 −1.02799
\(459\) 4.00000 0.186704
\(460\) 1.00000 0.0466252
\(461\) 10.0000 0.465746 0.232873 − 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 − 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 6.00000 0.277945
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 2.00000 0.0924500
\(469\) −12.0000 −0.554109
\(470\) 12.0000 0.553519
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 4.00000 0.183726
\(475\) 8.00000 0.367065
\(476\) 4.00000 0.183340
\(477\) −4.00000 −0.183147
\(478\) −16.0000 −0.731823
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 12.0000 0.547153
\(482\) −4.00000 −0.182195
\(483\) 1.00000 0.0455016
\(484\) −11.0000 −0.500000
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −2.00000 −0.0904431
\(490\) 1.00000 0.0451754
\(491\) 2.00000 0.0902587 0.0451294 − 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 4.00000 0.180334
\(493\) 0 0
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 − 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) 26.0000 1.15928 0.579641 − 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 20.0000 0.887357
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 4.00000 0.177123
\(511\) −16.0000 −0.707798
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −18.0000 −0.793946
\(515\) 20.0000 0.881305
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) −18.0000 −0.790112
\(520\) 2.00000 0.0877058
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 − 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −4.00000 −0.173749
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) −8.00000 −0.346518
\(534\) −6.00000 −0.259645
\(535\) −14.0000 −0.605273
\(536\) 12.0000 0.518321
\(537\) −6.00000 −0.258919
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −18.0000 −0.773880 −0.386940 − 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 28.0000 1.20270
\(543\) −2.00000 −0.0858282
\(544\) −4.00000 −0.171499
\(545\) −8.00000 −0.342682
\(546\) 2.00000 0.0855921
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 18.0000 0.768922
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 4.00000 0.170097
\(554\) −24.0000 −1.01966
\(555\) −6.00000 −0.254686
\(556\) 4.00000 0.169638
\(557\) −40.0000 −1.69485 −0.847427 − 0.530912i \(-0.821850\pi\)
−0.847427 + 0.530912i \(0.821850\pi\)
\(558\) −4.00000 −0.169334
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −12.0000 −0.505291
\(565\) 2.00000 0.0841406
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) −8.00000 −0.335673
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) −8.00000 −0.335083
\(571\) 10.0000 0.418487 0.209243 − 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −28.0000 −1.16566 −0.582828 − 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 16.0000 0.662085
\(585\) 2.00000 0.0826898
\(586\) −18.0000 −0.743573
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 6.00000 0.246598
\(593\) −26.0000 −1.06769 −0.533846 − 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 14.0000 0.573462
\(597\) 22.0000 0.900400
\(598\) 2.00000 0.0817861
\(599\) 4.00000 0.163436 0.0817178 − 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −14.0000 −0.571072 −0.285536 − 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 4.00000 0.163028
\(603\) 12.0000 0.488678
\(604\) 16.0000 0.651031
\(605\) −11.0000 −0.447214
\(606\) 2.00000 0.0812444
\(607\) 18.0000 0.730597 0.365299 − 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 24.0000 0.970936
\(612\) −4.00000 −0.161690
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −8.00000 −0.322854
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −20.0000 −0.804518
\(619\) 40.0000 1.60774 0.803868 − 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) −4.00000 −0.160644
\(621\) −1.00000 −0.0401286
\(622\) 22.0000 0.882120
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −24.0000 −0.956943
\(630\) −1.00000 −0.0398410
\(631\) −36.0000 −1.43314 −0.716569 − 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) −4.00000 −0.159111
\(633\) −8.00000 −0.317971
\(634\) −6.00000 −0.238290
\(635\) 20.0000 0.793676
\(636\) 4.00000 0.158610
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −46.0000 −1.81689 −0.908445 − 0.418004i \(-0.862730\pi\)
−0.908445 + 0.418004i \(0.862730\pi\)
\(642\) 14.0000 0.552536
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 4.00000 0.157500
\(646\) −32.0000 −1.25902
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) −4.00000 −0.156772
\(652\) 2.00000 0.0783260
\(653\) −26.0000 −1.01746 −0.508729 − 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 16.0000 0.624219
\(658\) −12.0000 −0.467809
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 20.0000 0.777322
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) 0 0
\(669\) 2.00000 0.0773245
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 34.0000 1.30673 0.653363 − 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −2.00000 −0.0768095
\(679\) −2.00000 −0.0767530
\(680\) −4.00000 −0.153393
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 8.00000 0.305888
\(685\) 18.0000 0.687745
\(686\) −1.00000 −0.0381802
\(687\) 22.0000 0.839352
\(688\) −4.00000 −0.152499
\(689\) −8.00000 −0.304776
\(690\) −1.00000 −0.0380693
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 34.0000 1.28692
\(699\) −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 \(0.389598\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 48.0000 1.81035
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) −10.0000 −0.376355
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) −24.0000 −0.901339 −0.450669 − 0.892691i \(-0.648815\pi\)
−0.450669 + 0.892691i \(0.648815\pi\)
\(710\) −8.00000 −0.300235
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) −4.00000 −0.149801
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 16.0000 0.597531
\(718\) 24.0000 0.895672
\(719\) −34.0000 −1.26799 −0.633993 − 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 1.00000 0.0372678
\(721\) −20.0000 −0.744839
\(722\) 45.0000 1.67473
\(723\) 4.00000 0.148762
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) −12.0000 −0.445055 −0.222528 − 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 16.0000 0.592187
\(731\) 16.0000 0.591781
\(732\) 2.00000 0.0739221
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 24.0000 0.885856
\(735\) −1.00000 −0.0368856
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −4.00000 −0.147242
\(739\) −8.00000 −0.294285 −0.147142 − 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 6.00000 0.220564
\(741\) −16.0000 −0.587775
\(742\) 4.00000 0.146845
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 4.00000 0.146647
\(745\) 14.0000 0.512920
\(746\) −30.0000 −1.09838
\(747\) 0 0
\(748\) 0 0
\(749\) 14.0000 0.511549
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 12.0000 0.437595
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 1.00000 0.0363696
\(757\) 6.00000 0.218074 0.109037 − 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) −22.0000 −0.799076
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 20.0000 0.724999 0.362500 − 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) −20.0000 −0.724524
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 18.0000 0.650366
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −18.0000 −0.647834
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −4.00000 −0.143777
\(775\) −4.00000 −0.143684
\(776\) 2.00000 0.0717958
\(777\) 6.00000 0.215249
\(778\) −30.0000 −1.07555
\(779\) −32.0000 −1.14652
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) −4.00000 −0.143040
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 − 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) −6.00000 −0.212932
\(795\) 4.00000 0.141865
\(796\) −22.0000 −0.779769
\(797\) 26.0000 0.920967 0.460484 − 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 8.00000 0.283197
\(799\) −48.0000 −1.69812
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) −34.0000 −1.20058
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) −1.00000 −0.0352454
\(806\) −8.00000 −0.281788
\(807\) 6.00000 0.211210
\(808\) −2.00000 −0.0703598
\(809\) −26.0000 −0.914111 −0.457056 − 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 1.00000 0.0351364
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 4.00000 0.140028
\(817\) −32.0000 −1.11954
\(818\) 34.0000 1.18878
\(819\) −2.00000 −0.0698857
\(820\) −4.00000 −0.139686
\(821\) 16.0000 0.558404 0.279202 − 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) −18.0000 −0.627822
\(823\) 36.0000 1.25488 0.627441 − 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) 20.0000 0.696733
\(825\) 0 0
\(826\) 0 0
\(827\) −34.0000 −1.18230 −0.591148 − 0.806563i \(-0.701325\pi\)
−0.591148 + 0.806563i \(0.701325\pi\)
\(828\) 1.00000 0.0347524
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 2.00000 0.0693375
\(833\) −4.00000 −0.138592
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −36.0000 −1.24360
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 1.00000 0.0345033
\(841\) −29.0000 −1.00000
\(842\) −36.0000 −1.24064
\(843\) −18.0000 −0.619953
\(844\) 8.00000 0.275371
\(845\) −9.00000 −0.309609
\(846\) 12.0000 0.412568
\(847\) 11.0000 0.377964
\(848\) −4.00000 −0.137361
\(849\) 4.00000 0.137280
\(850\) −4.00000 −0.137199
\(851\) 6.00000 0.205677
\(852\) 8.00000 0.274075
\(853\) −22.0000 −0.753266 −0.376633 − 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 2.00000 0.0684386
\(855\) 8.00000 0.273594
\(856\) −14.0000 −0.478510
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −4.00000 −0.136399
\(861\) −4.00000 −0.136320
\(862\) 8.00000 0.272481
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 18.0000 0.612018
\(866\) 34.0000 1.15537
\(867\) 1.00000 0.0339618
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) −8.00000 −0.270914
\(873\) 2.00000 0.0676897
\(874\) 8.00000 0.270604
\(875\) −1.00000 −0.0338062
\(876\) −16.0000 −0.540590
\(877\) 28.0000 0.945493 0.472746 − 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) −12.0000 −0.404980
\(879\) 18.0000 0.607125
\(880\) 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
\(883\) −14.0000 −0.471138 −0.235569 − 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −20.0000 −0.671534 −0.335767 − 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) −6.00000 −0.201347
\(889\) −20.0000 −0.670778
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 96.0000 3.21252
\(894\) −14.0000 −0.468230
\(895\) 6.00000 0.200558
\(896\) −1.00000 −0.0334077
\(897\) −2.00000 −0.0667781
\(898\) −26.0000 −0.867631
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 16.0000 0.533037
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 2.00000 0.0665190
\(905\) 2.00000 0.0664822
\(906\) −16.0000 −0.531564
\(907\) 44.0000 1.46100 0.730498 − 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 12.0000 0.398234
\(909\) −2.00000 −0.0663358
\(910\) −2.00000 −0.0662994
\(911\) −40.0000 −1.32526 −0.662630 − 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 2.00000 0.0661180
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 1.00000 0.0329690
\(921\) 8.00000 0.263609
\(922\) 10.0000 0.329332
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 20.0000 0.657241
\(927\) 20.0000 0.656886
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 4.00000 0.131165
\(931\) 8.00000 0.262189
\(932\) 6.00000 0.196537
\(933\) −22.0000 −0.720248
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −50.0000 −1.63343 −0.816714 − 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) −12.0000 −0.391814
\(939\) −30.0000 −0.979013
\(940\) 12.0000 0.391397
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −4.00000 −0.130258
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −8.00000 −0.259965 −0.129983 − 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 4.00000 0.129914
\(949\) 32.0000 1.03876
\(950\) 8.00000 0.259554
\(951\) 6.00000 0.194563
\(952\) 4.00000 0.129641
\(953\) −42.0000 −1.36051 −0.680257 − 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) −4.00000 −0.129505
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −40.0000 −1.29234
\(959\) −18.0000 −0.581250
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) −14.0000 −0.451144
\(964\) −4.00000 −0.128831
\(965\) −18.0000 −0.579441
\(966\) 1.00000 0.0321745
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −11.0000 −0.353553
\(969\) 32.0000 1.02799
\(970\) 2.00000 0.0642161
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −8.00000 −0.255420
\(982\) 2.00000 0.0638226
\(983\) −38.0000 −1.21201 −0.606006 − 0.795460i \(-0.707229\pi\)
−0.606006 + 0.795460i \(0.707229\pi\)
\(984\) 4.00000 0.127515
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 16.0000 0.509028
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −4.00000 −0.127000
\(993\) −20.0000 −0.634681
\(994\) 8.00000 0.253745
\(995\) −22.0000 −0.697447
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −8.00000 −0.253236
\(999\) −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.w.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.w.1.1 ✓ 1 1.1 even 1 trivial