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Properties

Label 4830.2.a.y.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -6.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -8.00000 q^{38} +1.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} -8.00000 q^{43} -6.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -4.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +1.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} +1.00000 q^{72} +4.00000 q^{73} +8.00000 q^{74} -1.00000 q^{75} -8.00000 q^{76} -6.00000 q^{77} -2.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} -1.00000 q^{84} +6.00000 q^{85} -8.00000 q^{86} +6.00000 q^{87} -6.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +1.00000 q^{92} +6.00000 q^{93} +8.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −6.00000 −0.904534
\(45\) 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
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.00000 −0.809040
\(56\) 1.00000 0.133631
\(57\) 8.00000 1.05963
\(58\) −6.00000 −0.787839
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −6.00000 −0.762001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 1.00000 0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) −1.00000 −0.115470
\(76\) −8.00000 −0.917663
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 6.00000 0.650791
\(86\) −8.00000 −0.862662
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 6.00000 0.622171
\(94\) 8.00000 0.825137
\(95\) −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
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −6.00000 −0.594089
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) −4.00000 −0.388514
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −6.00000 −0.572078
\(111\) −8.00000 −0.759326
\(112\) 1.00000 0.0944911
\(113\) 4.00000 0.376288 0.188144 − 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 8.00000 0.749269
\(115\) 1.00000 0.0932505
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) 10.0000 0.901670
\(124\) −6.00000 −0.538816
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 − 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 6.00000 0.522233
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 1.00000 0.0845154
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 4.00000 0.331042
\(147\) −1.00000 −0.0824786
\(148\) 8.00000 0.657596
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −8.00000 −0.648886
\(153\) 6.00000 0.485071
\(154\) −6.00000 −0.483494
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −2.00000 −0.159111
\(159\) 4.00000 0.317221
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −10.0000 −0.780869
\(165\) 6.00000 0.467099
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −13.0000 −1.00000
\(170\) 6.00000 0.460179
\(171\) −8.00000 −0.611775
\(172\) −8.00000 −0.609994
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) −6.00000 −0.452267
\(177\) 10.0000 0.751646
\(178\) −10.0000 −0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 1.00000 0.0737210
\(185\) 8.00000 0.588172
\(186\) 6.00000 0.439941
\(187\) −36.0000 −2.63258
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) −6.00000 −0.434145 −0.217072 − 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −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
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −22.0000 −1.56744 −0.783718 − 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −6.00000 −0.426401
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 2.00000 0.140720
\(203\) −6.00000 −0.421117
\(204\) −6.00000 −0.420084
\(205\) −10.0000 −0.698430
\(206\) −16.0000 −1.11477
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 48.0000 3.32023
\(210\) −1.00000 −0.0690066
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) −6.00000 −0.407307
\(218\) −14.0000 −0.948200
\(219\) −4.00000 −0.270295
\(220\) −6.00000 −0.404520
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 4.00000 0.266076
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 8.00000 0.529813
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 1.00000 0.0659380
\(231\) 6.00000 0.394771
\(232\) −6.00000 −0.393919
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −10.0000 −0.650945
\(237\) 2.00000 0.129914
\(238\) 6.00000 0.388922
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 25.0000 1.60706
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 1.00000 0.0638877
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 − 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 1.00000 0.0629941
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 14.0000 0.864923
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 6.00000 0.369274
\(265\) −4.00000 −0.245718
\(266\) −8.00000 −0.490511
\(267\) 10.0000 0.611990
\(268\) −4.00000 −0.244339
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 −0.361814
\(276\) −1.00000 −0.0601929
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −14.0000 −0.839664
\(279\) −6.00000 −0.359211
\(280\) 1.00000 0.0597614
\(281\) −26.0000 −1.55103 −0.775515 − 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −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
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −2.00000 −0.117242
\(292\) 4.00000 0.234082
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −10.0000 −0.582223
\(296\) 8.00000 0.464991
\(297\) 6.00000 0.348155
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −8.00000 −0.461112
\(302\) −12.0000 −0.690522
\(303\) −2.00000 −0.114897
\(304\) −8.00000 −0.458831
\(305\) 2.00000 0.114520
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −6.00000 −0.341882
\(309\) 16.0000 0.910208
\(310\) −6.00000 −0.340777
\(311\) −2.00000 −0.113410 −0.0567048 − 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 18.0000 1.01580
\(315\) 1.00000 0.0563436
\(316\) −2.00000 −0.112509
\(317\) −14.0000 −0.786318 −0.393159 − 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 4.00000 0.224309
\(319\) 36.0000 2.01561
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) 1.00000 0.0557278
\(323\) −48.0000 −2.67079
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 14.0000 0.774202
\(328\) −10.0000 −0.552158
\(329\) 8.00000 0.441054
\(330\) 6.00000 0.330289
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) 8.00000 0.438397
\(334\) 16.0000 0.875481
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) −20.0000 −1.08947 −0.544735 − 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) −13.0000 −0.707107
\(339\) −4.00000 −0.217250
\(340\) 6.00000 0.325396
\(341\) 36.0000 1.94951
\(342\) −8.00000 −0.432590
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −6.00000 −0.317554
\(358\) −12.0000 −0.634220
\(359\) 14.0000 0.738892 0.369446 − 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) 6.00000 0.315353
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −2.00000 −0.104542
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) −10.0000 −0.520579
\(370\) 8.00000 0.415900
\(371\) −4.00000 −0.207670
\(372\) 6.00000 0.311086
\(373\) −16.0000 −0.828449 −0.414224 − 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −36.0000 −1.86152
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −18.0000 −0.924598 −0.462299 − 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) −6.00000 −0.306987
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.00000 −0.305788
\(386\) −18.0000 −0.916176
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 1.00000 0.0505076
\(393\) −14.0000 −0.706207
\(394\) −22.0000 −1.10834
\(395\) −2.00000 −0.100631
\(396\) −6.00000 −0.301511
\(397\) −4.00000 −0.200754 −0.100377 − 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 20.0000 1.00251
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) −6.00000 −0.297775
\(407\) −48.0000 −2.37927
\(408\) −6.00000 −0.297044
\(409\) 2.00000 0.0988936 0.0494468 − 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −10.0000 −0.493865
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) −10.0000 −0.492068
\(414\) 1.00000 0.0491473
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 48.0000 2.34776
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −18.0000 −0.877266 −0.438633 − 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 16.0000 0.778868
\(423\) 8.00000 0.388973
\(424\) −4.00000 −0.194257
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −2.00000 −0.0963366 −0.0481683 − 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −6.00000 −0.288009
\(435\) 6.00000 0.287678
\(436\) −14.0000 −0.670478
\(437\) −8.00000 −0.382692
\(438\) −4.00000 −0.191127
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) −6.00000 −0.286039
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −8.00000 −0.379663
\(445\) −10.0000 −0.474045
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 1.00000 0.0472456
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 1.00000 0.0471405
\(451\) 60.0000 2.82529
\(452\) 4.00000 0.188144
\(453\) 12.0000 0.563809
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −32.0000 −1.49690 −0.748448 − 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −14.0000 −0.654177
\(459\) −6.00000 −0.280056
\(460\) 1.00000 0.0466252
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 6.00000 0.279145
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −6.00000 −0.278543
\(465\) 6.00000 0.278243
\(466\) −18.0000 −0.833834
\(467\) 36.0000 1.66588 0.832941 − 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 8.00000 0.369012
\(471\) −18.0000 −0.829396
\(472\) −10.0000 −0.460287
\(473\) 48.0000 2.20704
\(474\) 2.00000 0.0918630
\(475\) −8.00000 −0.367065
\(476\) 6.00000 0.275010
\(477\) −4.00000 −0.183147
\(478\) 24.0000 1.09773
\(479\) 36.0000 1.64488 0.822441 − 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −22.0000 −1.00207
\(483\) −1.00000 −0.0455016
\(484\) 25.0000 1.13636
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 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 \(0.717688\pi\)
\(492\) 10.0000 0.450835
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.0000 −0.714827
\(502\) −24.0000 −1.07117
\(503\) 40.0000 1.78351 0.891756 − 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 1.00000 0.0445435
\(505\) 2.00000 0.0889988
\(506\) −6.00000 −0.266733
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −6.00000 −0.265684
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 12.0000 0.529297
\(515\) −16.0000 −0.705044
\(516\) 8.00000 0.352180
\(517\) −48.0000 −2.11104
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 14.0000 0.611593
\(525\) −1.00000 −0.0436436
\(526\) 12.0000 0.523225
\(527\) −36.0000 −1.56818
\(528\) 6.00000 0.261116
\(529\) 1.00000 0.0434783
\(530\) −4.00000 −0.173749
\(531\) −10.0000 −0.433963
\(532\) −8.00000 −0.346844
\(533\) 0 0
\(534\) 10.0000 0.432742
\(535\) −4.00000 −0.172935
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) −6.00000 −0.258678
\(539\) −6.00000 −0.258438
\(540\) −1.00000 −0.0430331
\(541\) 34.0000 1.46177 0.730887 − 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) 2.00000 0.0859074
\(543\) −6.00000 −0.257485
\(544\) 6.00000 0.257248
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) −6.00000 −0.255841
\(551\) 48.0000 2.04487
\(552\) −1.00000 −0.0425628
\(553\) −2.00000 −0.0850487
\(554\) 18.0000 0.764747
\(555\) −8.00000 −0.339581
\(556\) −14.0000 −0.593732
\(557\) −12.0000 −0.508456 −0.254228 − 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 36.0000 1.51992
\(562\) −26.0000 −1.09674
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −8.00000 −0.336861
\(565\) 4.00000 0.168281
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 − 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 8.00000 0.335083
\(571\) 30.0000 1.25546 0.627730 − 0.778431i \(-0.283984\pi\)
0.627730 + 0.778431i \(0.283984\pi\)
\(572\) 0 0
\(573\) 6.00000 0.250654
\(574\) −10.0000 −0.417392
\(575\) 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 \(0.698054\pi\)
\(578\) 19.0000 0.790296
\(579\) 18.0000 0.748054
\(580\) −6.00000 −0.249136
\(581\) 4.00000 0.165948
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 28.0000 1.15568 0.577842 − 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 48.0000 1.97781
\(590\) −10.0000 −0.411693
\(591\) 22.0000 0.904959
\(592\) 8.00000 0.328798
\(593\) 16.0000 0.657041 0.328521 − 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 6.00000 0.246183
\(595\) 6.00000 0.245976
\(596\) 18.0000 0.737309
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −34.0000 −1.38689 −0.693444 − 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) −8.00000 −0.326056
\(603\) −4.00000 −0.162893
\(604\) −12.0000 −0.488273
\(605\) 25.0000 1.01639
\(606\) −2.00000 −0.0812444
\(607\) 12.0000 0.487065 0.243532 − 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) −8.00000 −0.324443
\(609\) 6.00000 0.243132
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 28.0000 1.13091 0.565455 − 0.824779i \(-0.308701\pi\)
0.565455 + 0.824779i \(0.308701\pi\)
\(614\) −12.0000 −0.484281
\(615\) 10.0000 0.403239
\(616\) −6.00000 −0.241747
\(617\) 4.00000 0.161034 0.0805170 − 0.996753i \(-0.474343\pi\)
0.0805170 + 0.996753i \(0.474343\pi\)
\(618\) 16.0000 0.643614
\(619\) 32.0000 1.28619 0.643094 − 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) −6.00000 −0.240966
\(621\) −1.00000 −0.0401286
\(622\) −2.00000 −0.0801927
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) −48.0000 −1.91694
\(628\) 18.0000 0.718278
\(629\) 48.0000 1.91389
\(630\) 1.00000 0.0398410
\(631\) −34.0000 −1.35352 −0.676759 − 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) −2.00000 −0.0795557
\(633\) −16.0000 −0.635943
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) 0 0
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) 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
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 1.00000 0.0394055
\(645\) 8.00000 0.315000
\(646\) −48.0000 −1.88853
\(647\) −4.00000 −0.157256 −0.0786281 − 0.996904i \(-0.525054\pi\)
−0.0786281 + 0.996904i \(0.525054\pi\)
\(648\) 1.00000 0.0392837
\(649\) 60.0000 2.35521
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −20.0000 −0.783260
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 14.0000 0.547443
\(655\) 14.0000 0.547025
\(656\) −10.0000 −0.390434
\(657\) 4.00000 0.156055
\(658\) 8.00000 0.311872
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 6.00000 0.233550
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) −8.00000 −0.310227
\(666\) 8.00000 0.309994
\(667\) −6.00000 −0.232321
\(668\) 16.0000 0.619059
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −12.0000 −0.463255
\(672\) −1.00000 −0.0385758
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −20.0000 −0.770371
\(675\) −1.00000 −0.0384900
\(676\) −13.0000 −0.500000
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −4.00000 −0.153619
\(679\) 2.00000 0.0767530
\(680\) 6.00000 0.230089
\(681\) −20.0000 −0.766402
\(682\) 36.0000 1.37851
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 14.0000 0.534133
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) −38.0000 −1.44559 −0.722794 − 0.691063i \(-0.757142\pi\)
−0.722794 + 0.691063i \(0.757142\pi\)
\(692\) 0 0
\(693\) −6.00000 −0.227921
\(694\) −12.0000 −0.455514
\(695\) −14.0000 −0.531050
\(696\) 6.00000 0.227429
\(697\) −60.0000 −2.27266
\(698\) 14.0000 0.529908
\(699\) 18.0000 0.680823
\(700\) 1.00000 0.0377964
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −64.0000 −2.41381
\(704\) −6.00000 −0.226134
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) 2.00000 0.0752177
\(708\) 10.0000 0.375823
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) −10.0000 −0.374766
\(713\) −6.00000 −0.224702
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −24.0000 −0.896296
\(718\) 14.0000 0.522475
\(719\) 42.0000 1.56634 0.783168 − 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.0000 −0.595871
\(722\) 45.0000 1.67473
\(723\) 22.0000 0.818189
\(724\) 6.00000 0.222988
\(725\) −6.00000 −0.222834
\(726\) −25.0000 −0.927837
\(727\) 48.0000 1.78022 0.890111 − 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) −48.0000 −1.77534
\(732\) −2.00000 −0.0739221
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 16.0000 0.590571
\(735\) −1.00000 −0.0368856
\(736\) 1.00000 0.0368605
\(737\) 24.0000 0.884051
\(738\) −10.0000 −0.368105
\(739\) 24.0000 0.882854 0.441427 − 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 6.00000 0.219971
\(745\) 18.0000 0.659469
\(746\) −16.0000 −0.585802
\(747\) 4.00000 0.146352
\(748\) −36.0000 −1.31629
\(749\) −4.00000 −0.146157
\(750\) −1.00000 −0.0365148
\(751\) −26.0000 −0.948753 −0.474377 − 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 8.00000 0.291730
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) −1.00000 −0.0363696
\(757\) −44.0000 −1.59921 −0.799604 − 0.600528i \(-0.794957\pi\)
−0.799604 + 0.600528i \(0.794957\pi\)
\(758\) −18.0000 −0.653789
\(759\) 6.00000 0.217786
\(760\) −8.00000 −0.290191
\(761\) 26.0000 0.942499 0.471250 − 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) −14.0000 −0.506834
\(764\) −6.00000 −0.217072
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −6.00000 −0.216225
\(771\) −12.0000 −0.432169
\(772\) −18.0000 −0.647834
\(773\) 34.0000 1.22290 0.611448 − 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) −8.00000 −0.287554
\(775\) −6.00000 −0.215526
\(776\) 2.00000 0.0717958
\(777\) −8.00000 −0.286998
\(778\) −6.00000 −0.215110
\(779\) 80.0000 2.86630
\(780\) 0 0
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 18.0000 0.642448
\(786\) −14.0000 −0.499363
\(787\) 20.0000 0.712923 0.356462 − 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −22.0000 −0.783718
\(789\) −12.0000 −0.427211
\(790\) −2.00000 −0.0711568
\(791\) 4.00000 0.142224
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −4.00000 −0.141955
\(795\) 4.00000 0.141865
\(796\) 20.0000 0.708881
\(797\) 46.0000 1.62940 0.814702 − 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 8.00000 0.283197
\(799\) 48.0000 1.69812
\(800\) 1.00000 0.0353553
\(801\) −10.0000 −0.353333
\(802\) −14.0000 −0.494357
\(803\) −24.0000 −0.846942
\(804\) 4.00000 0.141069
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 2.00000 0.0703598
\(809\) −14.0000 −0.492214 −0.246107 − 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 1.00000 0.0351364
\(811\) 26.0000 0.912983 0.456492 − 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) −6.00000 −0.210559
\(813\) −2.00000 −0.0701431
\(814\) −48.0000 −1.68240
\(815\) −20.0000 −0.700569
\(816\) −6.00000 −0.210042
\(817\) 64.0000 2.23908
\(818\) 2.00000 0.0699284
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) 54.0000 1.88461 0.942306 − 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −16.0000 −0.557386
\(825\) 6.00000 0.208893
\(826\) −10.0000 −0.347945
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 1.00000 0.0347524
\(829\) 18.0000 0.625166 0.312583 − 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 4.00000 0.138842
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 14.0000 0.484780
\(835\) 16.0000 0.553703
\(836\) 48.0000 1.66011
\(837\) 6.00000 0.207390
\(838\) 4.00000 0.138178
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) −18.0000 −0.620321
\(843\) 26.0000 0.895488
\(844\) 16.0000 0.550743
\(845\) −13.0000 −0.447214
\(846\) 8.00000 0.275046
\(847\) 25.0000 0.859010
\(848\) −4.00000 −0.137361
\(849\) 4.00000 0.137280
\(850\) 6.00000 0.205798
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −36.0000 −1.23262 −0.616308 − 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 2.00000 0.0684386
\(855\) −8.00000 −0.273594
\(856\) −4.00000 −0.136717
\(857\) −24.0000 −0.819824 −0.409912 − 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 − 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) −8.00000 −0.272798
\(861\) 10.0000 0.340799
\(862\) −2.00000 −0.0681203
\(863\) 16.0000 0.544646 0.272323 − 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −19.0000 −0.645274
\(868\) −6.00000 −0.203653
\(869\) 12.0000 0.407072
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 2.00000 0.0676897
\(874\) −8.00000 −0.270604
\(875\) 1.00000 0.0338062
\(876\) −4.00000 −0.135147
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −26.0000 −0.877457
\(879\) −6.00000 −0.202375
\(880\) −6.00000 −0.202260
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 1.00000 0.0336718
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 10.0000 0.336146
\(886\) −4.00000 −0.134383
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −8.00000 −0.268462
\(889\) 0 0
\(890\) −10.0000 −0.335201
\(891\) −6.00000 −0.201008
\(892\) 0 0
\(893\) −64.0000 −2.14168
\(894\) −18.0000 −0.602010
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) 36.0000 1.20067
\(900\) 1.00000 0.0333333
\(901\) −24.0000 −0.799556
\(902\) 60.0000 1.99778
\(903\) 8.00000 0.266223
\(904\) 4.00000 0.133038
\(905\) 6.00000 0.199447
\(906\) 12.0000 0.398673
\(907\) 24.0000 0.796907 0.398453 − 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 20.0000 0.663723
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −50.0000 −1.65657 −0.828287 − 0.560304i \(-0.810684\pi\)
−0.828287 + 0.560304i \(0.810684\pi\)
\(912\) 8.00000 0.264906
\(913\) −24.0000 −0.794284
\(914\) −32.0000 −1.05847
\(915\) −2.00000 −0.0661180
\(916\) −14.0000 −0.462573
\(917\) 14.0000 0.462321
\(918\) −6.00000 −0.198030
\(919\) −30.0000 −0.989609 −0.494804 − 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 1.00000 0.0329690
\(921\) 12.0000 0.395413
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 8.00000 0.263038
\(926\) 32.0000 1.05159
\(927\) −16.0000 −0.525509
\(928\) −6.00000 −0.196960
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 6.00000 0.196748
\(931\) −8.00000 −0.262189
\(932\) −18.0000 −0.589610
\(933\) 2.00000 0.0654771
\(934\) 36.0000 1.17796
\(935\) −36.0000 −1.17733
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) −4.00000 −0.130605
\(939\) −14.0000 −0.456873
\(940\) 8.00000 0.260931
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −18.0000 −0.586472
\(943\) −10.0000 −0.325645
\(944\) −10.0000 −0.325472
\(945\) −1.00000 −0.0325300
\(946\) 48.0000 1.56061
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 2.00000 0.0649570
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) 14.0000 0.453981
\(952\) 6.00000 0.194461
\(953\) 24.0000 0.777436 0.388718 − 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −4.00000 −0.129505
\(955\) −6.00000 −0.194155
\(956\) 24.0000 0.776215
\(957\) −36.0000 −1.16371
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) −22.0000 −0.708572
\(965\) −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 \(0.541058\pi\)
\(968\) 25.0000 0.803530
\(969\) 48.0000 1.54198
\(970\) 2.00000 0.0642161
\(971\) −48.0000 −1.54039 −0.770197 − 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −14.0000 −0.448819
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 44.0000 1.40768 0.703842 − 0.710356i \(-0.251466\pi\)
0.703842 + 0.710356i \(0.251466\pi\)
\(978\) 20.0000 0.639529
\(979\) 60.0000 1.91761
\(980\) 1.00000 0.0319438
\(981\) −14.0000 −0.446986
\(982\) −28.0000 −0.893516
\(983\) −8.00000 −0.255160 −0.127580 − 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 10.0000 0.318788
\(985\) −22.0000 −0.700978
\(986\) −36.0000 −1.14647
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) −6.00000 −0.190693
\(991\) −4.00000 −0.127064 −0.0635321 − 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −6.00000 −0.190500
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) −4.00000 −0.126745
\(997\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) −32.0000 −1.01294
\(999\) −8.00000 −0.253109
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.y.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.y.1.1 ✓ 1 1.1 even 1 trivial