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Properties

Label 6930.2.a.h.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +8.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} -8.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -14.0000 q^{53} -1.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} +14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +12.0000 q^{67} -6.00000 q^{68} +1.00000 q^{70} +8.00000 q^{71} -2.00000 q^{73} +2.00000 q^{74} +8.00000 q^{76} +1.00000 q^{77} -8.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} -16.0000 q^{83} -6.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} -2.00000 q^{89} +2.00000 q^{91} -4.00000 q^{94} +8.00000 q^{95} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 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
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 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
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −6.00000 −0.727607
\(69\) 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
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 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
\(106\) 14.0000 1.35980
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 − 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 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
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −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
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 − 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −22.0000 −1.67263 −0.836315 − 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 26.0000 1.85242 0.926212 − 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −14.0000 −0.961524
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 18.0000 1.21911
\(219\) 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
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −26.0000 −1.55103 −0.775515 − 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 − 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 14.0000 0.801638
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 34.0000 1.92179 0.960897 − 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −48.0000 −2.67079
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −16.0000 −0.878114
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 − 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 12.0000 0.626395 0.313197 − 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 − 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 24.0000 1.22795
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −26.0000 −1.30986
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 − 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 14.0000 0.699127 0.349563 − 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) −20.0000 −0.977064 −0.488532 − 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 − 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 14.0000 0.679900
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −14.0000 −0.677507
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 − 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) −28.0000 −1.32584
\(447\) 0 0
\(448\) −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
\(451\) −2.00000 −0.0941763
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 0 0
\(479\) −32.0000 −1.46212 −0.731059 − 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −14.0000 −0.633750
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 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
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 14.0000 0.613351 0.306676 − 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 − 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 14.0000 0.608121
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 26.0000 1.09674
\(563\) −8.00000 −0.337160 −0.168580 − 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 38.0000 1.59304 0.796521 − 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 − 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 0 0
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 − 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 14.0000 0.579821
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −20.0000 −0.825488 −0.412744 − 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −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
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 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
\(604\) −8.00000 −0.325515
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −26.0000 −1.04672 −0.523360 − 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) 12.0000 0.482321 0.241160 − 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 − 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 2.00000 0.0791808
\(639\) 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
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 2.00000 0.0782660 0.0391330 − 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 4.00000 0.155936
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 − 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −30.0000 −1.15299 −0.576497 − 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 28.0000 1.06672
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −50.0000 −1.87779 −0.938895 − 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 − 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −14.0000 −0.513956
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 − 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 18.0000 0.651644
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 26.0000 0.926212
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −14.0000 −0.494357
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 14.0000 0.492214 0.246107 − 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 − 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 − 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 4.00000 0.139094 0.0695468 − 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 16.0000 0.555368
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) −8.00000 −0.276686
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −6.00000 −0.206774
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −14.0000 −0.480762
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 10.0000 0.341593 0.170797 − 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 − 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −56.0000 −1.90626 −0.953131 − 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) 0 0
\(865\) −22.0000 −0.748022
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 18.0000 0.609557
\(873\) 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 \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 14.0000 0.471672 0.235836 − 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 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
\(886\) −12.0000 −0.403148
\(887\) −56.0000 −1.88030 −0.940148 − 0.340766i \(-0.889313\pi\)
−0.940148 + 0.340766i \(0.889313\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 2.00000 0.0670402
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 84.0000 2.79845
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 − 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.00000 −0.0658665
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) 58.0000 1.89075 0.945373 − 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −20.0000 −0.649913 −0.324956 − 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −34.0000 −1.10137 −0.550684 − 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 2.00000 0.0639203
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) −52.0000 −1.65854 −0.829271 − 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 0 0
\(985\) 26.0000 0.828429
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −34.0000 −1.07679 −0.538395 − 0.842692i \(-0.680969\pi\)
−0.538395 + 0.842692i \(0.680969\pi\)
\(998\) 20.0000 0.633089
\(999\) 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.h.1.1 1
3.2 odd 2 2310.2.a.r.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.r.1.1 ✓ 1 3.2 odd 2
6930.2.a.h.1.1 1 1.1 even 1 trivial