Properties

Label 6930.2.a.t.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{20} +1.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +10.0000 q^{37} -1.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -8.00000 q^{46} +12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} +8.00000 q^{71} +6.00000 q^{73} +10.0000 q^{74} -1.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +10.0000 q^{82} -4.00000 q^{83} -2.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} +6.00000 q^{91} -8.00000 q^{92} +12.0000 q^{94} +6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 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 \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 0 0
\(77\) −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
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −20.0000 −1.97066 −0.985329 0.170664i \(-0.945409\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −2.00000 −0.183340
\(120\) 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
\(130\) 6.00000 0.526235
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) −20.0000 −1.39347
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) −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
\(253\) −8.00000 −0.502956
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −8.00000 −0.479808
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 0 0
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −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
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) −10.0000 −0.519875
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −10.0000 −0.493865
\(411\) 0 0
\(412\) −20.0000 −0.985329
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −8.00000 −0.380091 −0.190046 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) 4.00000 0.182956
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 20.0000 0.881305
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 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
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) −10.0000 −0.417392
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 48.0000 1.96287
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −48.0000 −1.94826 −0.974130 0.225989i \(-0.927439\pi\)
−0.974130 + 0.225989i \(0.927439\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −72.0000 −2.91281
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 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
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) −44.0000 −1.72982 −0.864909 0.501928i \(-0.832624\pi\)
−0.864909 + 0.501928i \(0.832624\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 10.0000 0.390434
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −2.00000 −0.0752177
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 64.0000 2.39682
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) 2.00000 0.0734223
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 0 0
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) −72.0000 −2.59977
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −22.0000 −0.779280 −0.389640 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 48.0000 1.69073
\(807\) 0 0
\(808\) 2.00000 0.0703598
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −48.0000 −1.68551 −0.842754 0.538299i \(-0.819067\pi\)
−0.842754 + 0.538299i \(0.819067\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 10.0000 0.350500
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −20.0000 −0.696733
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) −80.0000 −2.74236
\(852\) 0 0
\(853\) 42.0000 1.43805 0.719026 0.694983i \(-0.244588\pi\)
0.719026 + 0.694983i \(0.244588\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −4.00000 −0.136241
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −32.0000 −1.07995
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −10.0000 −0.336909 −0.168454 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 0 0
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 10.0000 0.332964
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −6.00000 −0.198898
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −14.0000 −0.461065
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −80.0000 −2.60516
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −60.0000 −1.93448
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −52.0000 −1.65854 −0.829271 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −8.00000 −0.254000
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 36.0000 1.13956
\(999\) 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.t.1.1 1
3.2 odd 2 2310.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.j.1.1 1 3.2 odd 2
6930.2.a.t.1.1 1 1.1 even 1 trivial