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Properties

Label 8050.2.a.m.1.1
Level $8050$
Weight $2$
Character 8050.1
Self dual yes
Analytic conductor $64.280$
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] = [8050,2,Mod(1,8050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8050.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8050 = 2 \cdot 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8050.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: \(64.2795736271\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8050.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} -2.00000 q^{24} -4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +8.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} -2.00000 q^{44} +1.00000 q^{46} +2.00000 q^{48} +1.00000 q^{49} +12.0000 q^{51} +4.00000 q^{52} +12.0000 q^{53} +4.00000 q^{54} -1.00000 q^{56} +2.00000 q^{58} -10.0000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +2.00000 q^{67} +6.00000 q^{68} -2.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{77} -8.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} +16.0000 q^{83} +2.00000 q^{84} +6.00000 q^{86} -4.00000 q^{87} +2.00000 q^{88} +6.00000 q^{89} +4.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} -2.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 2.00000 0.577350
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 4.00000 0.554700
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) −8.00000 −0.905822
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −4.00000 −0.428845
\(88\) 2.00000 0.213201
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 − 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −12.0000 −1.18818
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −4.00000 −0.384900
\(109\) −20.0000 −1.91565 −0.957826 − 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 4.00000 0.369800
\(118\) 10.0000 0.920575
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 12.0000 1.08200
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 22.0000 1.92215 0.961074 − 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 2.00000 0.170251
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 8.00000 0.640513
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −8.00000 −0.636446
\(159\) 24.0000 1.90332
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 11.0000 0.864242
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) 24.0000 1.82469 0.912343 − 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −20.0000 −1.50329
\(178\) −6.00000 −0.449719
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −4.00000 −0.296500
\(183\) 4.00000 0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 2.00000 0.144338
\(193\) 18.0000 1.29567 0.647834 − 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 2.00000 0.142134
\(199\) 24.0000 1.70131 0.850657 − 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 8.00000 0.562878
\(203\) −2.00000 −0.140372
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 12.0000 0.824163
\(213\) 16.0000 1.09630
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 4.00000 0.271538
\(218\) 20.0000 1.35457
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 2.00000 0.131306
\(233\) −10.0000 −0.655122 −0.327561 − 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 16.0000 1.03931
\(238\) −6.00000 −0.388922
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 7.00000 0.449977
\(243\) −10.0000 −0.641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) 32.0000 2.02792
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) 2.00000 0.125739
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −22.0000 −1.35916
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 2.00000 0.122169
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000 0.363803
\(273\) 8.00000 0.484182
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 14.0000 0.839664
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 6.00000 0.354169
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) −2.00000 −0.117041
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 12.0000 0.695141
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) −16.0000 −0.919176
\(304\) 0 0
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −8.00000 −0.452911
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −24.0000 −1.34585
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −40.0000 −2.21201
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 − 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −3.00000 −0.163178
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −4.00000 −0.214423
\(349\) 24.0000 1.28469 0.642345 − 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 2.00000 0.106600
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 20.0000 1.06299
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 12.0000 0.635107
\(358\) −16.0000 −0.845626
\(359\) −4.00000 −0.211112 −0.105556 − 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) −14.0000 −0.734809
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000 0.414781
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 4.00000 0.205738
\(379\) 2.00000 0.102733 0.0513665 − 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) −32.0000 −1.63941
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −6.00000 −0.304997
\(388\) 2.00000 0.101535
\(389\) −28.0000 −1.41966 −0.709828 − 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −1.00000 −0.0505076
\(393\) 44.0000 2.21951
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 20.0000 1.00377 0.501886 − 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −4.00000 −0.199502
\(403\) 16.0000 0.797017
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 0 0
\(408\) −12.0000 −0.594089
\(409\) −34.0000 −1.68119 −0.840596 − 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 0 0
\(413\) −10.0000 −0.492068
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −28.0000 −1.37117
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 2.00000 0.0967868
\(428\) 18.0000 0.870063
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 − 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) −4.00000 −0.192450
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −24.0000 −1.14156
\(443\) 16.0000 0.760183 0.380091 − 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) −24.0000 −1.13516
\(448\) 1.00000 0.0472456
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −6.00000 −0.280362
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 − 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 4.00000 0.186097
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 40.0000 1.85098 0.925490 − 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) 4.00000 0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −36.0000 −1.65879
\(472\) 10.0000 0.460287
\(473\) 12.0000 0.551761
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 12.0000 0.549442
\(478\) 16.0000 0.731823
\(479\) −8.00000 −0.365529 −0.182765 − 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −14.0000 −0.637683
\(483\) −2.00000 −0.0910032
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 12.0000 0.541002
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 8.00000 0.358849
\(498\) −32.0000 −1.43395
\(499\) −24.0000 −1.07439 −0.537194 − 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 32.0000 1.42965
\(502\) 12.0000 0.535586
\(503\) 32.0000 1.42681 0.713405 − 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) 6.00000 0.266469
\(508\) −16.0000 −0.709885
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 48.0000 2.10697
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 2.00000 0.0875376
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 22.0000 0.961074
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 24.0000 1.04546
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 32.0000 1.38090
\(538\) −24.0000 −1.03471
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 − 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) −20.0000 −0.859074
\(543\) −4.00000 −0.171656
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 40.0000 1.71028 0.855138 − 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 2.00000 0.0854358
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 2.00000 0.0851257
\(553\) 8.00000 0.340195
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) −32.0000 −1.35588 −0.677942 − 0.735116i \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) −4.00000 −0.169334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 22.0000 0.928014
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) −11.0000 −0.461957
\(568\) −8.00000 −0.335673
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −38.0000 −1.59025 −0.795125 − 0.606445i \(-0.792595\pi\)
−0.795125 + 0.606445i \(0.792595\pi\)
\(572\) −8.00000 −0.334497
\(573\) −32.0000 −1.33682
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −19.0000 −0.790296
\(579\) 36.0000 1.49611
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) −4.00000 −0.165805
\(583\) −24.0000 −0.993978
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −18.0000 −0.742940 −0.371470 − 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −8.00000 −0.328244
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 48.0000 1.96451
\(598\) 4.00000 0.163572
\(599\) 40.0000 1.63436 0.817178 − 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 − 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 6.00000 0.244542
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 0 0
\(606\) 16.0000 0.649956
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −44.0000 −1.77714 −0.888572 − 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −24.0000 −0.962312
\(623\) 6.00000 0.240385
\(624\) 8.00000 0.320256
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 0 0
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 − 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −8.00000 −0.318223
\(633\) 8.00000 0.317971
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 24.0000 0.951662
\(637\) 4.00000 0.158486
\(638\) −4.00000 −0.158362
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −36.0000 −1.42081
\(643\) −24.0000 −0.946468 −0.473234 − 0.880937i \(-0.656913\pi\)
−0.473234 + 0.880937i \(0.656913\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 11.0000 0.432121
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 4.00000 0.156652
\(653\) −26.0000 −1.01746 −0.508729 − 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 40.0000 1.56412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 − 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) −16.0000 −0.621858
\(663\) 48.0000 1.86417
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 16.0000 0.619059
\(669\) 32.0000 1.23719
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) −2.00000 −0.0771517
\(673\) 50.0000 1.92736 0.963679 − 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −12.0000 −0.460857
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 8.00000 0.306336
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 12.0000 0.457829
\(688\) −6.00000 −0.228748
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) 46.0000 1.74992 0.874961 − 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) 24.0000 0.912343
\(693\) −2.00000 −0.0759737
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 36.0000 1.36360
\(698\) −24.0000 −0.908413
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 − 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 16.0000 0.603881
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) −8.00000 −0.300871
\(708\) −20.0000 −0.751646
\(709\) −24.0000 −0.901339 −0.450669 − 0.892691i \(-0.648815\pi\)
−0.450669 + 0.892691i \(0.648815\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) −4.00000 −0.149801
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 16.0000 0.597948
\(717\) −32.0000 −1.19506
\(718\) 4.00000 0.149279
\(719\) −20.0000 −0.745874 −0.372937 − 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 28.0000 1.04133
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −4.00000 −0.148250
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 4.00000 0.147844
\(733\) −38.0000 −1.40356 −0.701781 − 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −4.00000 −0.147342
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 16.0000 0.585409
\(748\) −12.0000 −0.438763
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −44.0000 −1.59921 −0.799604 − 0.600528i \(-0.794957\pi\)
−0.799604 + 0.600528i \(0.794957\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 32.0000 1.15924
\(763\) −20.0000 −0.724049
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −40.0000 −1.44432
\(768\) 2.00000 0.0721688
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 18.0000 0.647834
\(773\) −10.0000 −0.359675 −0.179838 − 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 28.0000 1.00385
\(779\) 0 0
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 6.00000 0.214560
\(783\) 8.00000 0.285897
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −44.0000 −1.56943
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −18.0000 −0.641223
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 2.00000 0.0710669
\(793\) 8.00000 0.284088
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) −14.0000 −0.495905 −0.247953 − 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −6.00000 −0.211867
\(803\) 4.00000 0.141157
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 48.0000 1.68968
\(808\) 8.00000 0.281439
\(809\) −42.0000 −1.47664 −0.738321 − 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 40.0000 1.40286
\(814\) 0 0
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 − 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) −4.00000 −0.139516
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) −14.0000 −0.486828 −0.243414 − 0.969923i \(-0.578267\pi\)
−0.243414 + 0.969923i \(0.578267\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 12.0000 0.416777 0.208389 − 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 0 0
\(831\) −52.0000 −1.80386
\(832\) 4.00000 0.138675
\(833\) 6.00000 0.207888
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) −4.00000 −0.138178
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 20.0000 0.689246
\(843\) −44.0000 −1.51544
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 12.0000 0.412082
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 16.0000 0.548151
\(853\) 44.0000 1.50653 0.753266 − 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −38.0000 −1.29806 −0.649028 − 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 16.0000 0.546231
\(859\) −42.0000 −1.43302 −0.716511 − 0.697576i \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −28.0000 −0.953684
\(863\) 8.00000 0.272323 0.136162 − 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 38.0000 1.29055
\(868\) 4.00000 0.135769
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 20.0000 0.677285
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −20.0000 −0.674967
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −12.0000 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 22.0000 0.737028
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 24.0000 0.802680
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −8.00000 −0.267112
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 12.0000 0.399556
\(903\) −12.0000 −0.399335
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 46.0000 1.52740 0.763702 − 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) 12.0000 0.398234
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 22.0000 0.726504
\(918\) 24.0000 0.792118
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) −8.00000 −0.263466
\(923\) 32.0000 1.05329
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 48.0000 1.57145
\(934\) −40.0000 −1.30884
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −30.0000 −0.980057 −0.490029 − 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) −2.00000 −0.0653023
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 36.0000 1.17294
\(943\) −6.00000 −0.195387
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −60.0000 −1.94974 −0.974869 − 0.222779i \(-0.928487\pi\)
−0.974869 + 0.222779i \(0.928487\pi\)
\(948\) 16.0000 0.519656
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) −6.00000 −0.194461
\(953\) 26.0000 0.842223 0.421111 − 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 8.00000 0.258603
\(958\) 8.00000 0.258468
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −10.0000 −0.320750
\(973\) −14.0000 −0.448819
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −30.0000 −0.959785 −0.479893 − 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −8.00000 −0.255812
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −4.00000 −0.127000
\(993\) 32.0000 1.01549
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 32.0000 1.01396
\(997\) 36.0000 1.14013 0.570066 − 0.821599i \(-0.306918\pi\)
0.570066 + 0.821599i \(0.306918\pi\)
\(998\) 24.0000 0.759707
\(999\) 0 0
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 8050.2.a.m.1.1 1
5.4 even 2 322.2.a.c.1.1 ✓ 1
15.14 odd 2 2898.2.a.g.1.1 1
20.19 odd 2 2576.2.a.m.1.1 1
35.34 odd 2 2254.2.a.g.1.1 1
115.114 odd 2 7406.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.a.c.1.1 ✓ 1 5.4 even 2
2254.2.a.g.1.1 1 35.34 odd 2
2576.2.a.m.1.1 1 20.19 odd 2
2898.2.a.g.1.1 1 15.14 odd 2
7406.2.a.f.1.1 1 115.114 odd 2
8050.2.a.m.1.1 1 1.1 even 1 trivial