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Properties

Label 6930.2.a.q.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
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] = [6930,2,Mod(1,6930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6930, 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("6930.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.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: \(55.3363286007\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6930.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} -10.0000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} +1.00000 q^{44} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} -12.0000 q^{71} -10.0000 q^{73} +10.0000 q^{74} -4.00000 q^{76} +1.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} +12.0000 q^{83} -6.00000 q^{85} -8.00000 q^{86} -1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} -4.00000 q^{95} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 − 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −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
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 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
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 4.00000 0.230174
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 − 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 − 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 − 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 − 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −42.0000 −1.65890 −0.829450 − 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 − 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −48.0000 −1.83667 −0.918334 − 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 − 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 48.0000 1.76095 0.880475 − 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 − 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 54.0000 1.91278 0.956389 − 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 10.0000 0.350500
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −54.0000 −1.88461 −0.942306 − 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −2.00000 −0.0677285
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 − 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −32.0000 −1.05159
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −48.0000 −1.55979 −0.779895 − 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 20.0000 0.644826
\(963\) 0 0
\(964\) 14.0000 0.450910
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −54.0000 −1.72761 −0.863807 − 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) −8.00000 −0.254000
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 50.0000 1.58352 0.791758 − 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 6930.2.a.q.1.1 1
3.2 odd 2 2310.2.a.u.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.u.1.1 ✓ 1 3.2 odd 2
6930.2.a.q.1.1 1 1.1 even 1 trivial