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Properties

Label 2394.2.a.n.1.1
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
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] = [2394,2,Mod(1,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, 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("2394.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2394.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +2.00000 q^{35} -6.00000 q^{37} -1.00000 q^{38} +2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -10.0000 q^{53} +8.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} +2.00000 q^{70} +16.0000 q^{71} +10.0000 q^{73} -6.00000 q^{74} -1.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} +2.00000 q^{80} +10.0000 q^{82} +4.00000 q^{85} -4.00000 q^{86} +4.00000 q^{88} +10.0000 q^{89} -2.00000 q^{91} +4.00000 q^{92} +8.00000 q^{94} -2.00000 q^{95} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 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
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 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 \(0.363071\pi\)
\(24\) 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 \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 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 \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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
\(106\) −10.0000 −0.971286
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 14.0000 1.19610 0.598050 − 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 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
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 − 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 − 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −2.00000 −0.131024 −0.0655122 − 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) −20.0000 −1.22859
\(266\) −1.00000 −0.0613139
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 10.0000 0.590281
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 − 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 − 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 26.0000 1.38384 0.691920 − 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 32.0000 1.69838
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) −30.0000 −1.55334 −0.776671 − 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 − 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 8.00000 0.408781 0.204390 − 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 20.0000 0.987730
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −16.0000 −0.770693 −0.385346 − 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) −12.0000 −0.568216
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 − 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −2.00000 −0.0926482
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 16.0000 0.711287
\(507\) 0 0
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) −6.00000 −0.263625
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −20.0000 −0.868744
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 − 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 14.0000 0.598050
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 20.0000 0.841406
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 10.0000 0.417392
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 16.0000 0.660391 0.330195 − 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −24.0000 −0.988064
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 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
\(604\) 16.0000 0.651031
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 4.00000 0.162355 0.0811775 − 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) −8.00000 −0.316723
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) −44.0000 −1.73519 −0.867595 − 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) −48.0000 −1.88707 −0.943537 − 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 − 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 28.0000 1.06983
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 52.0000 1.97817 0.989087 − 0.147335i \(-0.0470696\pi\)
0.989087 + 0.147335i \(0.0470696\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 32.0000 1.20094
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) 16.0000 0.596699 0.298350 − 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 22.0000 0.817624
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 20.0000 0.740233
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −18.0000 −0.664845 −0.332423 − 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −32.0000 −1.17874
\(738\) 0 0
\(739\) 44.0000 1.61857 0.809283 − 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −30.0000 −1.09838
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −32.0000 −1.16229
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −10.0000 −0.354218 −0.177109 − 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 10.0000 0.353112
\(803\) 40.0000 1.41157
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −2.00000 −0.0703163 −0.0351581 − 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 20.0000 0.698430
\(821\) 46.0000 1.60541 0.802706 − 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 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
\(826\) −12.0000 −0.417533
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 − 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −24.0000 −0.829066
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 26.0000 0.888143 0.444072 − 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) 40.0000 1.36162 0.680808 − 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 − 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 8.00000 0.269680
\(881\) 10.0000 0.336909 0.168454 − 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −32.0000 −1.07445 −0.537227 − 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 44.0000 1.46261
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 − 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −4.00000 −0.132599
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −40.0000 −1.31448
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 58.0000 1.90292 0.951459 − 0.307775i \(-0.0995844\pi\)
0.951459 + 0.307775i \(0.0995844\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −2.00000 −0.0655122
\(933\) 0 0
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 − 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −44.0000 −1.41641
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 48.0000 1.53096 0.765481 − 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 6.00000 0.190022 0.0950110 − 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) −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 2394.2.a.n.1.1 1
3.2 odd 2 798.2.a.c.1.1 ✓ 1
12.11 even 2 6384.2.a.e.1.1 1
21.20 even 2 5586.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.c.1.1 ✓ 1 3.2 odd 2
2394.2.a.n.1.1 1 1.1 even 1 trivial
5586.2.a.i.1.1 1 21.20 even 2
6384.2.a.e.1.1 1 12.11 even 2