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Properties

Label 3822.2.a.bd.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
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] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} +8.00000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} +4.00000 q^{57} -8.00000 q^{59} -2.00000 q^{60} +14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} -14.0000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +4.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -4.00000 q^{83} +4.00000 q^{85} +8.00000 q^{86} +4.00000 q^{88} -2.00000 q^{90} +4.00000 q^{92} -4.00000 q^{93} +6.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −2.00000 −0.365148
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 4.00000 0.648886
\(39\) −1.00000 −0.160128
\(40\) −2.00000 −0.316228
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 4.00000 0.603023
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) −1.00000 −0.138675
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −2.00000 −0.258199
\(61\) 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
\(66\) 4.00000 0.492366
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 4.00000 0.464991
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −4.00000 −0.414781
\(94\) 6.00000 0.618853
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −2.00000 −0.198030
\(103\) −6.00000 −0.591198 −0.295599 − 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 2.00000 0.193347 0.0966736 − 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.00000 0.766261 0.383131 − 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) −8.00000 −0.762770
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 4.00000 0.374634
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 2.00000 0.175412
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 4.00000 0.340503
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 16.0000 1.34269
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −4.00000 −0.325515 −0.162758 − 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 4.00000 0.324443
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) −1.00000 −0.0800641
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −8.00000 −0.636446
\(159\) 4.00000 0.317221
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 6.00000 0.469956 0.234978 − 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) −8.00000 −0.622799
\(166\) −4.00000 −0.310460
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) 4.00000 0.305888
\(172\) 8.00000 0.609994
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) 2.00000 0.149487 0.0747435 − 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) −2.00000 −0.149071
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 4.00000 0.294884
\(185\) −8.00000 −0.588172
\(186\) −4.00000 −0.293294
\(187\) −8.00000 −0.585018
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 2.00000 0.143592
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 4.00000 0.284268
\(199\) 2.00000 0.141776 0.0708881 − 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.0000 −0.987484
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) 4.00000 0.278019
\(208\) −1.00000 −0.0693375
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 4.00000 0.274721
\(213\) 16.0000 1.09630
\(214\) 2.00000 0.136717
\(215\) −16.0000 −1.09119
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 8.00000 0.541828
\(219\) 10.0000 0.675737
\(220\) −8.00000 −0.539360
\(221\) 2.00000 0.134535
\(222\) 4.00000 0.268462
\(223\) −12.0000 −0.803579 −0.401790 − 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 2.00000 0.133038
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 − 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −12.0000 −0.782794
\(236\) −8.00000 −0.520756
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −2.00000 −0.129099
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) −4.00000 −0.254000
\(249\) −4.00000 −0.253490
\(250\) 12.0000 0.758947
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 16.0000 1.00393
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −2.00000 −0.121716
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) 4.00000 0.240772
\(277\) −18.0000 −1.08152 −0.540758 − 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −16.0000 −0.959616
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 − 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 6.00000 0.357295
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 16.0000 0.949425
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 10.0000 0.585206
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 4.00000 0.232495
\(297\) 4.00000 0.232104
\(298\) −6.00000 −0.347571
\(299\) −4.00000 −0.231326
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) −28.0000 −1.60328
\(306\) −2.00000 −0.114332
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 8.00000 0.454369
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 4.00000 0.224309
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 6.00000 0.332309
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −4.00000 −0.219529
\(333\) 4.00000 0.219199
\(334\) 18.0000 0.984916
\(335\) 28.0000 1.52980
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 1.00000 0.0543928
\(339\) 2.00000 0.108625
\(340\) 4.00000 0.216930
\(341\) −16.0000 −0.866449
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) −8.00000 −0.430706
\(346\) −14.0000 −0.752645
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 4.00000 0.213201
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) −8.00000 −0.425195
\(355\) −32.0000 −1.69838
\(356\) 0 0
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −14.0000 −0.735824
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) 14.0000 0.731792
\(367\) −26.0000 −1.35719 −0.678594 − 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) 18.0000 0.932005 0.466002 − 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −8.00000 −0.413670
\(375\) 12.0000 0.619677
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) −38.0000 −1.95193 −0.975964 − 0.217930i \(-0.930070\pi\)
−0.975964 + 0.217930i \(0.930070\pi\)
\(380\) −8.00000 −0.410391
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 22.0000 1.12415 0.562074 − 0.827087i \(-0.310004\pi\)
0.562074 + 0.827087i \(0.310004\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 8.00000 0.406663
\(388\) 2.00000 0.101535
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 2.00000 0.101274
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) −2.00000 −0.100759
\(395\) 16.0000 0.805047
\(396\) 4.00000 0.201008
\(397\) 38.0000 1.90717 0.953583 − 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −14.0000 −0.698257
\(403\) 4.00000 0.199254
\(404\) 6.00000 0.298511
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) −2.00000 −0.0990148
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 8.00000 0.392705
\(416\) −1.00000 −0.0490290
\(417\) −16.0000 −0.783523
\(418\) 16.0000 0.782586
\(419\) −4.00000 −0.195413 −0.0977064 − 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 4.00000 0.194257
\(425\) 2.00000 0.0970143
\(426\) 16.0000 0.775203
\(427\) 0 0
\(428\) 2.00000 0.0966736
\(429\) −4.00000 −0.193122
\(430\) −16.0000 −0.771589
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 16.0000 0.765384
\(438\) 10.0000 0.477818
\(439\) 14.0000 0.668184 0.334092 − 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −18.0000 −0.855206 −0.427603 − 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −4.00000 −0.187936
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −2.00000 −0.0935561 −0.0467780 − 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) −8.00000 −0.373002
\(461\) −22.0000 −1.02464 −0.512321 − 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 22.0000 1.01913
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) 2.00000 0.0921551
\(472\) −8.00000 −0.368230
\(473\) 32.0000 1.47136
\(474\) −8.00000 −0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 24.0000 1.09773
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −4.00000 −0.182384
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 14.0000 0.633750
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) −22.0000 −0.992846 −0.496423 − 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) −8.00000 −0.359573
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 12.0000 0.536656
\(501\) 18.0000 0.804181
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 16.0000 0.711287
\(507\) 1.00000 0.0444116
\(508\) 16.0000 0.709885
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 14.0000 0.617514
\(515\) 12.0000 0.528783
\(516\) 8.00000 0.352180
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 2.00000 0.0877058
\(521\) 2.00000 0.0876216 0.0438108 − 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 8.00000 0.348485
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) −8.00000 −0.347498
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) −14.0000 −0.604708
\(537\) 2.00000 0.0863064
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −20.0000 −0.859867 −0.429934 − 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 12.0000 0.515444
\(543\) −14.0000 −0.600798
\(544\) −2.00000 −0.0857493
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 − 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 10.0000 0.427179
\(549\) 14.0000 0.597505
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −18.0000 −0.764747
\(555\) −8.00000 −0.339581
\(556\) −16.0000 −0.678551
\(557\) 10.0000 0.423714 0.211857 − 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −4.00000 −0.169334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 22.0000 0.928014
\(563\) 44.0000 1.85438 0.927189 − 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 6.00000 0.252646
\(565\) −4.00000 −0.168281
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −8.00000 −0.335083
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −13.0000 −0.540729
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 16.0000 0.662652
\(584\) 10.0000 0.413803
\(585\) 2.00000 0.0826898
\(586\) 14.0000 0.578335
\(587\) 48.0000 1.98117 0.990586 − 0.136892i \(-0.0437113\pi\)
0.990586 + 0.136892i \(0.0437113\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 16.0000 0.658710
\(591\) −2.00000 −0.0822690
\(592\) 4.00000 0.164399
\(593\) −16.0000 −0.657041 −0.328521 − 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 2.00000 0.0818546
\(598\) −4.00000 −0.163572
\(599\) 16.0000 0.653742 0.326871 − 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 28.0000 1.14214 0.571072 − 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) −14.0000 −0.570124
\(604\) −4.00000 −0.162758
\(605\) −10.0000 −0.406558
\(606\) 6.00000 0.243733
\(607\) −10.0000 −0.405887 −0.202944 − 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −28.0000 −1.13369
\(611\) −6.00000 −0.242734
\(612\) −2.00000 −0.0808452
\(613\) 40.0000 1.61558 0.807792 − 0.589467i \(-0.200662\pi\)
0.807792 + 0.589467i \(0.200662\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −6.00000 −0.241355
\(619\) −12.0000 −0.482321 −0.241160 − 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 8.00000 0.321288
\(621\) 4.00000 0.160514
\(622\) 4.00000 0.160385
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) −16.0000 −0.639489
\(627\) 16.0000 0.638978
\(628\) 2.00000 0.0798087
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −32.0000 −1.26988
\(636\) 4.00000 0.158610
\(637\) 0 0
\(638\) 0 0
\(639\) 16.0000 0.632950
\(640\) −2.00000 −0.0790569
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 2.00000 0.0789337
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) −8.00000 −0.314756
\(647\) 48.0000 1.88707 0.943537 − 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000 0.0392837
\(649\) −32.0000 −1.25611
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 8.00000 0.312825
\(655\) 8.00000 0.312586
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −34.0000 −1.32445 −0.662226 − 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) −8.00000 −0.311400
\(661\) −6.00000 −0.233373 −0.116686 − 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) −10.0000 −0.388661
\(663\) 2.00000 0.0776736
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) 18.0000 0.696441
\(669\) −12.0000 −0.463947
\(670\) 28.0000 1.08173
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −34.0000 −1.30963
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) −12.0000 −0.459841
\(682\) −16.0000 −0.612672
\(683\) 16.0000 0.612223 0.306111 − 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 4.00000 0.152944
\(685\) −20.0000 −0.764161
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 8.00000 0.304997
\(689\) −4.00000 −0.152388
\(690\) −8.00000 −0.304555
\(691\) 52.0000 1.97817 0.989087 − 0.147335i \(-0.0470696\pi\)
0.989087 + 0.147335i \(0.0470696\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 32.0000 1.21383
\(696\) 0 0
\(697\) 0 0
\(698\) −26.0000 −0.984115
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 4.00000 0.151078 0.0755390 − 0.997143i \(-0.475932\pi\)
0.0755390 + 0.997143i \(0.475932\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 16.0000 0.603451
\(704\) 4.00000 0.150756
\(705\) −12.0000 −0.451946
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) −8.00000 −0.300658
\(709\) 12.0000 0.450669 0.225335 − 0.974281i \(-0.427652\pi\)
0.225335 + 0.974281i \(0.427652\pi\)
\(710\) −32.0000 −1.20094
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 2.00000 0.0747435
\(717\) 24.0000 0.896296
\(718\) 8.00000 0.298557
\(719\) 40.0000 1.49175 0.745874 − 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −14.0000 −0.520666
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −26.0000 −0.964287 −0.482143 − 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) −16.0000 −0.591781
\(732\) 14.0000 0.517455
\(733\) 26.0000 0.960332 0.480166 − 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −56.0000 −2.06279
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 − 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) −8.00000 −0.294086
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −4.00000 −0.146647
\(745\) 12.0000 0.439646
\(746\) 18.0000 0.659027
\(747\) −4.00000 −0.146352
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 6.00000 0.218797
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −38.0000 −1.38022
\(759\) 16.0000 0.580763
\(760\) −8.00000 −0.290191
\(761\) 40.0000 1.45000 0.724999 − 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 22.0000 0.794892
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −18.0000 −0.647834
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 8.00000 0.287554
\(775\) 4.00000 0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) 64.0000 2.29010
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) −4.00000 −0.142675
\(787\) −44.0000 −1.56843 −0.784215 − 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −24.0000 −0.854423
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −14.0000 −0.497155
\(794\) 38.0000 1.34857
\(795\) −8.00000 −0.283731
\(796\) 2.00000 0.0708881
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 40.0000 1.41157
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 14.0000 0.492823
\(808\) 6.00000 0.211079
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 16.0000 0.560800
\(815\) −12.0000 −0.420342
\(816\) −2.00000 −0.0700140
\(817\) 32.0000 1.11954
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 − 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 10.0000 0.348790
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −6.00000 −0.209020
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 4.00000 0.139010
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 8.00000 0.277684
\(831\) −18.0000 −0.624413
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) −36.0000 −1.24583
\(836\) 16.0000 0.553372
\(837\) −4.00000 −0.138260
\(838\) −4.00000 −0.138178
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 22.0000 0.757720
\(844\) 0 0
\(845\) −2.00000 −0.0688021
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) −20.0000 −0.686398
\(850\) 2.00000 0.0685994
\(851\) 16.0000 0.548473
\(852\) 16.0000 0.548151
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 2.00000 0.0683586
\(857\) 14.0000 0.478231 0.239115 − 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) −4.00000 −0.136558
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 32.0000 1.08929 0.544646 − 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 28.0000 0.952029
\(866\) 16.0000 0.543702
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 8.00000 0.270914
\(873\) 2.00000 0.0676897
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −56.0000 −1.89099 −0.945493 − 0.325643i \(-0.894419\pi\)
−0.945493 + 0.325643i \(0.894419\pi\)
\(878\) 14.0000 0.472477
\(879\) 14.0000 0.472208
\(880\) −8.00000 −0.269680
\(881\) 50.0000 1.68454 0.842271 − 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 − 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 2.00000 0.0672673
\(885\) 16.0000 0.537834
\(886\) −18.0000 −0.604722
\(887\) −44.0000 −1.47738 −0.738688 − 0.674048i \(-0.764554\pi\)
−0.738688 + 0.674048i \(0.764554\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) −12.0000 −0.401790
\(893\) 24.0000 0.803129
\(894\) −6.00000 −0.200670
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 28.0000 0.930751
\(906\) −4.00000 −0.132891
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −12.0000 −0.398234
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 4.00000 0.132453
\(913\) −16.0000 −0.529523
\(914\) −2.00000 −0.0661541
\(915\) −28.0000 −0.925651
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −8.00000 −0.263752
\(921\) −20.0000 −0.659022
\(922\) −22.0000 −0.724531
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 4.00000 0.131448
\(927\) −6.00000 −0.197066
\(928\) 0 0
\(929\) 56.0000 1.83730 0.918650 − 0.395072i \(-0.129280\pi\)
0.918650 + 0.395072i \(0.129280\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) 22.0000 0.720634
\(933\) 4.00000 0.130954
\(934\) −28.0000 −0.916188
\(935\) 16.0000 0.523256
\(936\) −1.00000 −0.0326860
\(937\) −32.0000 −1.04539 −0.522697 − 0.852518i \(-0.675074\pi\)
−0.522697 + 0.852518i \(0.675074\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) −12.0000 −0.391397
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) 32.0000 1.03986 0.519930 − 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) −8.00000 −0.259828
\(949\) −10.0000 −0.324614
\(950\) −4.00000 −0.129777
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) 2.00000 0.0644491
\(964\) −14.0000 −0.450910
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 5.00000 0.160706
\(969\) −8.00000 −0.256997
\(970\) −4.00000 −0.128432
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 1.00000 0.0320256
\(976\) 14.0000 0.448129
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 6.00000 0.191859
\(979\) 0 0
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) −22.0000 −0.702048
\(983\) −46.0000 −1.46717 −0.733586 − 0.679597i \(-0.762155\pi\)
−0.733586 + 0.679597i \(0.762155\pi\)
\(984\) 0 0
\(985\) 4.00000 0.127451
\(986\) 0 0
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 32.0000 1.01754
\(990\) −8.00000 −0.254257
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −4.00000 −0.127000
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) −4.00000 −0.126745
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 14.0000 0.443162
\(999\) 4.00000 0.126554
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 3822.2.a.bd.1.1 yes 1
7.6 odd 2 3822.2.a.x.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.x.1.1 ✓ 1 7.6 odd 2
3822.2.a.bd.1.1 yes 1 1.1 even 1 trivial