Properties

Label 3822.2.a.k.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: no (minimal twist has level 546)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} -3.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -5.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} -1.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} -10.0000 q^{43} +3.00000 q^{44} -2.00000 q^{45} -1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +5.00000 q^{51} -1.00000 q^{52} -3.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +1.00000 q^{57} +1.00000 q^{58} +3.00000 q^{59} -2.00000 q^{60} -5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -3.00000 q^{66} +5.00000 q^{67} +5.00000 q^{68} -1.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +1.00000 q^{78} -6.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +16.0000 q^{83} -10.0000 q^{85} +10.0000 q^{86} -1.00000 q^{87} -3.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} +4.00000 q^{93} +1.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} +4.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 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
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 2.00000 0.365148
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.00000 −0.160128
\(40\) 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
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 3.00000 0.452267
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −1.00000 −0.145865 −0.0729325 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 5.00000 0.700140
\(52\) −1.00000 −0.138675
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 1.00000 0.131306
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) −2.00000 −0.258199
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −3.00000 −0.369274
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 5.00000 0.606339
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 10.0000 1.07833
\(87\) −1.00000 −0.107211
\(88\) −3.00000 −0.319801
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 1.00000 0.103142
\(95\) −2.00000 −0.205196
\(96\) −1.00000 −0.102062
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) −1.00000 −0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −5.00000 −0.495074
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 6.00000 0.572078
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) −1.00000 −0.0924500
\(118\) −3.00000 −0.276172
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −2.00000 −0.181818
\(122\) 5.00000 0.452679
\(123\) 10.0000 0.901670
\(124\) 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
\(129\) −10.0000 −0.880451
\(130\) −2.00000 −0.175412
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) −5.00000 −0.431934
\(135\) −2.00000 −0.172133
\(136\) −5.00000 −0.428746
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 1.00000 0.0839181
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 1.00000 0.0816497
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.00000 0.404226
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −1.00000 −0.0800641
\(157\) 21.0000 1.67598 0.837991 0.545684i \(-0.183730\pi\)
0.837991 + 0.545684i \(0.183730\pi\)
\(158\) 6.00000 0.477334
\(159\) −3.00000 −0.237915
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 10.0000 0.780869
\(165\) −6.00000 −0.467099
\(166\) −16.0000 −1.24184
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 10.0000 0.766965
\(171\) 1.00000 0.0764719
\(172\) −10.0000 −0.762493
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 3.00000 0.225494
\(178\) 14.0000 1.04934
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) −2.00000 −0.149071
\(181\) 3.00000 0.222988 0.111494 0.993765i \(-0.464436\pi\)
0.111494 + 0.993765i \(0.464436\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) −4.00000 −0.293294
\(187\) 15.0000 1.09691
\(188\) −1.00000 −0.0729325
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −4.00000 −0.287183
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) −3.00000 −0.213201
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000 0.0707107
\(201\) 5.00000 0.352673
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 5.00000 0.350070
\(205\) −20.0000 −1.39686
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −3.00000 −0.206041
\(213\) −1.00000 −0.0685189
\(214\) −12.0000 −0.820303
\(215\) 20.0000 1.36399
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) 12.0000 0.810885
\(220\) −6.00000 −0.404520
\(221\) −5.00000 −0.336336
\(222\) 2.00000 0.134231
\(223\) 11.0000 0.736614 0.368307 0.929704i \(-0.379937\pi\)
0.368307 + 0.929704i \(0.379937\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −15.0000 −0.997785
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 1.00000 0.0662266
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 11.0000 0.720634 0.360317 0.932830i \(-0.382669\pi\)
0.360317 + 0.932830i \(0.382669\pi\)
\(234\) 1.00000 0.0653720
\(235\) 2.00000 0.130466
\(236\) 3.00000 0.195283
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) 21.0000 1.35838 0.679189 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) −2.00000 −0.129099
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −1.00000 −0.0636285
\(248\) −4.00000 −0.254000
\(249\) 16.0000 1.01396
\(250\) −12.0000 −0.758947
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −10.0000 −0.626224
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 10.0000 0.622573
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −1.00000 −0.0618984
\(262\) 4.00000 0.247121
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −3.00000 −0.184637
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 5.00000 0.305424
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 2.00000 0.121716
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −27.0000 −1.62227 −0.811136 0.584857i \(-0.801151\pi\)
−0.811136 + 0.584857i \(0.801151\pi\)
\(278\) −2.00000 −0.119952
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 1.00000 0.0595491
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −1.00000 −0.0593391
\(285\) −2.00000 −0.118470
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) −2.00000 −0.117444
\(291\) 4.00000 0.234484
\(292\) 12.0000 0.702247
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 2.00000 0.116248
\(297\) 3.00000 0.174078
\(298\) −8.00000 −0.463428
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −19.0000 −1.09333
\(303\) 2.00000 0.114897
\(304\) 1.00000 0.0573539
\(305\) 10.0000 0.572598
\(306\) −5.00000 −0.285831
\(307\) −27.0000 −1.54097 −0.770486 0.637457i \(-0.779986\pi\)
−0.770486 + 0.637457i \(0.779986\pi\)
\(308\) 0 0
\(309\) −2.00000 −0.113776
\(310\) 8.00000 0.454369
\(311\) 14.0000 0.793867 0.396934 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 1.00000 0.0566139
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) −21.0000 −1.18510
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 3.00000 0.168232
\(319\) −3.00000 −0.167968
\(320\) −2.00000 −0.111803
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) −19.0000 −1.05231
\(327\) −16.0000 −0.884802
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 16.0000 0.878114
\(333\) −2.00000 −0.109599
\(334\) −21.0000 −1.14907
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 15.0000 0.814688
\(340\) −10.0000 −0.542326
\(341\) 12.0000 0.649836
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −3.00000 −0.161281
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 36.0000 1.92704 0.963518 0.267644i \(-0.0862451\pi\)
0.963518 + 0.267644i \(0.0862451\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −3.00000 −0.159901
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −3.00000 −0.159448
\(355\) 2.00000 0.106149
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 2.00000 0.105409
\(361\) −18.0000 −0.947368
\(362\) −3.00000 −0.157676
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 5.00000 0.261354
\(367\) 26.0000 1.35719 0.678594 0.734513i \(-0.262589\pi\)
0.678594 + 0.734513i \(0.262589\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) 13.0000 0.673114 0.336557 0.941663i \(-0.390737\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) −15.0000 −0.775632
\(375\) 12.0000 0.619677
\(376\) 1.00000 0.0515711
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −10.0000 −0.508329
\(388\) 4.00000 0.203069
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 24.0000 1.20910
\(395\) 12.0000 0.603786
\(396\) 3.00000 0.150756
\(397\) 36.0000 1.80679 0.903394 0.428811i \(-0.141067\pi\)
0.903394 + 0.428811i \(0.141067\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) −5.00000 −0.249377
\(403\) −4.00000 −0.199254
\(404\) 2.00000 0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) −5.00000 −0.247537
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 20.0000 0.987730
\(411\) 18.0000 0.887875
\(412\) −2.00000 −0.0985329
\(413\) 0 0
\(414\) 0 0
\(415\) −32.0000 −1.57082
\(416\) 1.00000 0.0490290
\(417\) 2.00000 0.0979404
\(418\) −3.00000 −0.146735
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 16.0000 0.778868
\(423\) −1.00000 −0.0486217
\(424\) 3.00000 0.145693
\(425\) −5.00000 −0.242536
\(426\) 1.00000 0.0484502
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −3.00000 −0.144841
\(430\) −20.0000 −0.964486
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) −12.0000 −0.573382
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 6.00000 0.286039
\(441\) 0 0
\(442\) 5.00000 0.237826
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 28.0000 1.32733
\(446\) −11.0000 −0.520865
\(447\) 8.00000 0.378387
\(448\) 0 0
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 1.00000 0.0471405
\(451\) 30.0000 1.41264
\(452\) 15.0000 0.705541
\(453\) 19.0000 0.892698
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) −6.00000 −0.280362
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −8.00000 −0.370991
\(466\) −11.0000 −0.509565
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −2.00000 −0.0922531
\(471\) 21.0000 0.967629
\(472\) −3.00000 −0.138086
\(473\) −30.0000 −1.37940
\(474\) 6.00000 0.275589
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) −21.0000 −0.960518
\(479\) −33.0000 −1.50781 −0.753904 0.656984i \(-0.771832\pi\)
−0.753904 + 0.656984i \(0.771832\pi\)
\(480\) 2.00000 0.0912871
\(481\) 2.00000 0.0911922
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 5.00000 0.226339
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 10.0000 0.450835
\(493\) −5.00000 −0.225189
\(494\) 1.00000 0.0449921
\(495\) −6.00000 −0.269680
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 12.0000 0.536656
\(501\) 21.0000 0.938211
\(502\) 18.0000 0.803379
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 10.0000 0.442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −6.00000 −0.264649
\(515\) 4.00000 0.176261
\(516\) −10.0000 −0.440225
\(517\) −3.00000 −0.131940
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) −2.00000 −0.0877058
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 1.00000 0.0437688
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 20.0000 0.871214
\(528\) 3.00000 0.130558
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 3.00000 0.130189
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 14.0000 0.605839
\(535\) −24.0000 −1.03761
\(536\) −5.00000 −0.215967
\(537\) 2.00000 0.0863064
\(538\) 9.00000 0.388018
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −11.0000 −0.472490
\(543\) 3.00000 0.128742
\(544\) −5.00000 −0.214373
\(545\) 32.0000 1.37073
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 18.0000 0.768922
\(549\) −5.00000 −0.213395
\(550\) 3.00000 0.127920
\(551\) −1.00000 −0.0426014
\(552\) 0 0
\(553\) 0 0
\(554\) 27.0000 1.14712
\(555\) 4.00000 0.169791
\(556\) 2.00000 0.0848189
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −4.00000 −0.169334
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) −16.0000 −0.674919
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −1.00000 −0.0421076
\(565\) −30.0000 −1.26211
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 1.00000 0.0419591
\(569\) 19.0000 0.796521 0.398261 0.917272i \(-0.369614\pi\)
0.398261 + 0.917272i \(0.369614\pi\)
\(570\) 2.00000 0.0837708
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) −3.00000 −0.125436
\(573\) 20.0000 0.835512
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −8.00000 −0.332756
\(579\) −14.0000 −0.581820
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) −4.00000 −0.165805
\(583\) −9.00000 −0.372742
\(584\) −12.0000 −0.496564
\(585\) 2.00000 0.0826898
\(586\) 12.0000 0.495715
\(587\) 31.0000 1.27951 0.639753 0.768580i \(-0.279036\pi\)
0.639753 + 0.768580i \(0.279036\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 6.00000 0.247016
\(591\) −24.0000 −0.987228
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) 8.00000 0.327693
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −38.0000 −1.55264 −0.776319 0.630340i \(-0.782915\pi\)
−0.776319 + 0.630340i \(0.782915\pi\)
\(600\) 1.00000 0.0408248
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) 19.0000 0.773099
\(605\) 4.00000 0.162623
\(606\) −2.00000 −0.0812444
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 1.00000 0.0404557
\(612\) 5.00000 0.202113
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 27.0000 1.08963
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 2.00000 0.0804518
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −14.0000 −0.561349
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) −34.0000 −1.35891
\(627\) 3.00000 0.119808
\(628\) 21.0000 0.837991
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 6.00000 0.238667
\(633\) −16.0000 −0.635943
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) 3.00000 0.118771
\(639\) −1.00000 −0.0395594
\(640\) 2.00000 0.0790569
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −12.0000 −0.473602
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 0 0
\(645\) 20.0000 0.787499
\(646\) −5.00000 −0.196722
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 9.00000 0.353281
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) 19.0000 0.744097
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 16.0000 0.625650
\(655\) 8.00000 0.312586
\(656\) 10.0000 0.390434
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −6.00000 −0.233550
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 20.0000 0.777322
\(663\) −5.00000 −0.194184
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 21.0000 0.812514
\(669\) 11.0000 0.425285
\(670\) 10.0000 0.386334
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 17.0000 0.654816
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) −15.0000 −0.576072
\(679\) 0 0
\(680\) 10.0000 0.383482
\(681\) 8.00000 0.306561
\(682\) −12.0000 −0.459504
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 1.00000 0.0382360
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) −10.0000 −0.381246
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) −17.0000 −0.646710 −0.323355 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) 3.00000 0.114043
\(693\) 0 0
\(694\) −2.00000 −0.0759190
\(695\) −4.00000 −0.151729
\(696\) 1.00000 0.0379049
\(697\) 50.0000 1.89389
\(698\) −36.0000 −1.36262
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 1.00000 0.0377426
\(703\) −2.00000 −0.0754314
\(704\) 3.00000 0.113067
\(705\) 2.00000 0.0753244
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 3.00000 0.112747
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) −2.00000 −0.0750587
\(711\) −6.00000 −0.225018
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 2.00000 0.0747435
\(717\) 21.0000 0.784259
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 18.0000 0.669891
\(723\) 4.00000 0.148762
\(724\) 3.00000 0.111494
\(725\) 1.00000 0.0371391
\(726\) 2.00000 0.0742270
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 24.0000 0.888280
\(731\) −50.0000 −1.84932
\(732\) −5.00000 −0.184805
\(733\) −40.0000 −1.47743 −0.738717 0.674016i \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) −10.0000 −0.368105
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 4.00000 0.147043
\(741\) −1.00000 −0.0367359
\(742\) 0 0
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) −4.00000 −0.146647
\(745\) −16.0000 −0.586195
\(746\) −13.0000 −0.475964
\(747\) 16.0000 0.585409
\(748\) 15.0000 0.548454
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −1.00000 −0.0364662
\(753\) −18.0000 −0.655956
\(754\) −1.00000 −0.0364179
\(755\) −38.0000 −1.38296
\(756\) 0 0
\(757\) −37.0000 −1.34479 −0.672394 0.740193i \(-0.734734\pi\)
−0.672394 + 0.740193i \(0.734734\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 20.0000 0.723575
\(765\) −10.0000 −0.361551
\(766\) 32.0000 1.15621
\(767\) −3.00000 −0.108324
\(768\) 1.00000 0.0360844
\(769\) −44.0000 −1.58668 −0.793340 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −14.0000 −0.503871
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 10.0000 0.359443
\(775\) −4.00000 −0.143684
\(776\) −4.00000 −0.143592
\(777\) 0 0
\(778\) −9.00000 −0.322666
\(779\) 10.0000 0.358287
\(780\) 2.00000 0.0716115
\(781\) −3.00000 −0.107348
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) −42.0000 −1.49904
\(786\) 4.00000 0.142675
\(787\) −5.00000 −0.178231 −0.0891154 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) −24.0000 −0.854965
\(789\) −6.00000 −0.213606
\(790\) −12.0000 −0.426941
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) 5.00000 0.177555
\(794\) −36.0000 −1.27759
\(795\) 6.00000 0.212798
\(796\) −20.0000 −0.708881
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −5.00000 −0.176887
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) 36.0000 1.27120
\(803\) 36.0000 1.27041
\(804\) 5.00000 0.176336
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −9.00000 −0.316815
\(808\) −2.00000 −0.0703598
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 2.00000 0.0702728
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 11.0000 0.385787
\(814\) 6.00000 0.210300
\(815\) −38.0000 −1.33108
\(816\) 5.00000 0.175035
\(817\) −10.0000 −0.349856
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −18.0000 −0.627822
\(823\) −30.0000 −1.04573 −0.522867 0.852414i \(-0.675138\pi\)
−0.522867 + 0.852414i \(0.675138\pi\)
\(824\) 2.00000 0.0696733
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) −35.0000 −1.21707 −0.608535 0.793527i \(-0.708242\pi\)
−0.608535 + 0.793527i \(0.708242\pi\)
\(828\) 0 0
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) 32.0000 1.11074
\(831\) −27.0000 −0.936620
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −2.00000 −0.0692543
\(835\) −42.0000 −1.45347
\(836\) 3.00000 0.103757
\(837\) 4.00000 0.138260
\(838\) −10.0000 −0.345444
\(839\) −9.00000 −0.310715 −0.155357 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −8.00000 −0.275698
\(843\) 16.0000 0.551069
\(844\) −16.0000 −0.550743
\(845\) −2.00000 −0.0688021
\(846\) 1.00000 0.0343807
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) 4.00000 0.137280
\(850\) 5.00000 0.171499
\(851\) 0 0
\(852\) −1.00000 −0.0342594
\(853\) −40.0000 −1.36957 −0.684787 0.728743i \(-0.740105\pi\)
−0.684787 + 0.728743i \(0.740105\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) −12.0000 −0.410152
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 3.00000 0.102418
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 20.0000 0.681994
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −6.00000 −0.204006
\(866\) 19.0000 0.645646
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −18.0000 −0.610608
\(870\) −2.00000 −0.0678064
\(871\) −5.00000 −0.169419
\(872\) 16.0000 0.541828
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −40.0000 −1.34993
\(879\) −12.0000 −0.404750
\(880\) −6.00000 −0.202260
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) −38.0000 −1.27880 −0.639401 0.768874i \(-0.720818\pi\)
−0.639401 + 0.768874i \(0.720818\pi\)
\(884\) −5.00000 −0.168168
\(885\) −6.00000 −0.201688
\(886\) 18.0000 0.604722
\(887\) −14.0000 −0.470074 −0.235037 0.971986i \(-0.575521\pi\)
−0.235037 + 0.971986i \(0.575521\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −28.0000 −0.938562
\(891\) 3.00000 0.100504
\(892\) 11.0000 0.368307
\(893\) −1.00000 −0.0334637
\(894\) −8.00000 −0.267560
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 8.00000 0.266963
\(899\) −4.00000 −0.133407
\(900\) −1.00000 −0.0333333
\(901\) −15.0000 −0.499722
\(902\) −30.0000 −0.998891
\(903\) 0 0
\(904\) −15.0000 −0.498893
\(905\) −6.00000 −0.199447
\(906\) −19.0000 −0.631233
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 8.00000 0.265489
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 26.0000 0.861418 0.430709 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(912\) 1.00000 0.0331133
\(913\) 48.0000 1.58857
\(914\) 16.0000 0.529233
\(915\) 10.0000 0.330590
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) −5.00000 −0.165025
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 0 0
\(921\) −27.0000 −0.889680
\(922\) −20.0000 −0.658665
\(923\) 1.00000 0.0329154
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −8.00000 −0.262896
\(927\) −2.00000 −0.0656886
\(928\) 1.00000 0.0328266
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) 11.0000 0.360317
\(933\) 14.0000 0.458339
\(934\) 24.0000 0.785304
\(935\) −30.0000 −0.981105
\(936\) 1.00000 0.0326860
\(937\) −5.00000 −0.163343 −0.0816714 0.996659i \(-0.526026\pi\)
−0.0816714 + 0.996659i \(0.526026\pi\)
\(938\) 0 0
\(939\) 34.0000 1.10955
\(940\) 2.00000 0.0652328
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) −21.0000 −0.684217
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) −25.0000 −0.812391 −0.406195 0.913786i \(-0.633145\pi\)
−0.406195 + 0.913786i \(0.633145\pi\)
\(948\) −6.00000 −0.194871
\(949\) −12.0000 −0.389536
\(950\) 1.00000 0.0324443
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 19.0000 0.615470 0.307735 0.951472i \(-0.400429\pi\)
0.307735 + 0.951472i \(0.400429\pi\)
\(954\) 3.00000 0.0971286
\(955\) −40.0000 −1.29437
\(956\) 21.0000 0.679189
\(957\) −3.00000 −0.0969762
\(958\) 33.0000 1.06618
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) −2.00000 −0.0644826
\(963\) 12.0000 0.386695
\(964\) 4.00000 0.128831
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 2.00000 0.0642824
\(969\) 5.00000 0.160623
\(970\) 8.00000 0.256865
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −11.0000 −0.352463
\(975\) 1.00000 0.0320256
\(976\) −5.00000 −0.160046
\(977\) 62.0000 1.98356 0.991778 0.127971i \(-0.0408466\pi\)
0.991778 + 0.127971i \(0.0408466\pi\)
\(978\) −19.0000 −0.607553
\(979\) −42.0000 −1.34233
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −22.0000 −0.702048
\(983\) −59.0000 −1.88181 −0.940904 0.338674i \(-0.890022\pi\)
−0.940904 + 0.338674i \(0.890022\pi\)
\(984\) −10.0000 −0.318788
\(985\) 48.0000 1.52941
\(986\) 5.00000 0.159232
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) 0 0
\(990\) 6.00000 0.190693
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) −4.00000 −0.127000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) 16.0000 0.506979
\(997\) 19.0000 0.601736 0.300868 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(998\) 28.0000 0.886325
\(999\) −2.00000 −0.0632772
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.k.1.1 1
7.2 even 3 546.2.i.e.235.1 yes 2
7.4 even 3 546.2.i.e.79.1 2
7.6 odd 2 3822.2.a.i.1.1 1
21.2 odd 6 1638.2.j.a.235.1 2
21.11 odd 6 1638.2.j.a.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.e.79.1 2 7.4 even 3
546.2.i.e.235.1 yes 2 7.2 even 3
1638.2.j.a.235.1 2 21.2 odd 6
1638.2.j.a.1171.1 2 21.11 odd 6
3822.2.a.i.1.1 1 7.6 odd 2
3822.2.a.k.1.1 1 1.1 even 1 trivial