Properties

Label 3822.2.a.s.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -3.00000 q^{20} +3.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -9.00000 q^{29} +3.00000 q^{30} -5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} -3.00000 q^{40} +12.0000 q^{41} -4.00000 q^{43} +3.00000 q^{44} -3.00000 q^{45} +6.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +6.00000 q^{51} -1.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -9.00000 q^{55} -4.00000 q^{57} -9.00000 q^{58} +9.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -3.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} -1.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -3.00000 q^{83} +18.0000 q^{85} -4.00000 q^{86} +9.00000 q^{87} +3.00000 q^{88} -3.00000 q^{90} +6.00000 q^{92} +5.00000 q^{93} +12.0000 q^{94} -12.0000 q^{95} -1.00000 q^{96} -5.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 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
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 3.00000 0.547723
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) 1.00000 0.160128
\(40\) −3.00000 −0.474342
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) −3.00000 −0.447214
\(46\) 6.00000 0.884652
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) −1.00000 −0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −9.00000 −1.18176
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 3.00000 0.387298
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) −3.00000 −0.369274
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −4.00000 −0.464991
\(75\) −4.00000 −0.461880
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) −4.00000 −0.431331
\(87\) 9.00000 0.964901
\(88\) 3.00000 0.319801
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 5.00000 0.518476
\(94\) 12.0000 1.23771
\(95\) −12.0000 −1.23117
\(96\) −1.00000 −0.102062
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 4.00000 0.400000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 6.00000 0.594089
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −9.00000 −0.858116
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −4.00000 −0.374634
\(115\) −18.0000 −1.67851
\(116\) −9.00000 −0.835629
\(117\) −1.00000 −0.0924500
\(118\) 9.00000 0.828517
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) −12.0000 −1.08200
\(124\) −5.00000 −0.449013
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 3.00000 0.263117
\(131\) −21.0000 −1.83478 −0.917389 0.397991i \(-0.869707\pi\)
−0.917389 + 0.397991i \(0.869707\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 3.00000 0.258199
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −6.00000 −0.510754
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 6.00000 0.503509
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 27.0000 2.24223
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −4.00000 −0.326599
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) 1.00000 0.0800641
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 9.00000 0.713746
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 12.0000 0.937043
\(165\) 9.00000 0.700649
\(166\) −3.00000 −0.232845
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 18.0000 1.38054
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −3.00000 −0.223607
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 6.00000 0.442326
\(185\) 12.0000 0.882258
\(186\) 5.00000 0.366618
\(187\) −18.0000 −1.31629
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) −5.00000 −0.358979
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 3.00000 0.213201
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 4.00000 0.282843
\(201\) 4.00000 0.282138
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −36.0000 −2.51435
\(206\) −8.00000 −0.557386
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −9.00000 −0.618123
\(213\) −6.00000 −0.411113
\(214\) 3.00000 0.205076
\(215\) 12.0000 0.818393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) 14.0000 0.946032
\(220\) −9.00000 −0.606780
\(221\) 6.00000 0.403604
\(222\) 4.00000 0.268462
\(223\) 1.00000 0.0669650 0.0334825 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −12.0000 −0.798228
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) −4.00000 −0.264906
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −18.0000 −1.18688
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −36.0000 −2.34838
\(236\) 9.00000 0.585850
\(237\) 1.00000 0.0649570
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 3.00000 0.193649
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) −2.00000 −0.128565
\(243\) −1.00000 −0.0641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) −4.00000 −0.254514
\(248\) −5.00000 −0.317500
\(249\) 3.00000 0.190117
\(250\) 3.00000 0.189737
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) −7.00000 −0.439219
\(255\) −18.0000 −1.12720
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 3.00000 0.186052
\(261\) −9.00000 −0.557086
\(262\) −21.0000 −1.29738
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) −3.00000 −0.184637
\(265\) 27.0000 1.65860
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 3.00000 0.182574
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 12.0000 0.723627
\(276\) −6.00000 −0.361158
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 4.00000 0.239904
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −12.0000 −0.714590
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 6.00000 0.356034
\(285\) 12.0000 0.710819
\(286\) −3.00000 −0.177394
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 27.0000 1.58549
\(291\) 5.00000 0.293105
\(292\) −14.0000 −0.819288
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) −27.0000 −1.57200
\(296\) −4.00000 −0.232495
\(297\) −3.00000 −0.174078
\(298\) 6.00000 0.347571
\(299\) −6.00000 −0.346989
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −19.0000 −1.09333
\(303\) 18.0000 1.03407
\(304\) 4.00000 0.229416
\(305\) 24.0000 1.37424
\(306\) −6.00000 −0.342997
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 15.0000 0.851943
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 1.00000 0.0566139
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) 9.00000 0.504695
\(319\) −27.0000 −1.51171
\(320\) −3.00000 −0.167705
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 8.00000 0.443079
\(327\) 16.0000 0.884802
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 9.00000 0.495434
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −3.00000 −0.164646
\(333\) −4.00000 −0.219199
\(334\) −12.0000 −0.656611
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −31.0000 −1.68868 −0.844339 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) 1.00000 0.0543928
\(339\) 12.0000 0.651751
\(340\) 18.0000 0.976187
\(341\) −15.0000 −0.812296
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 18.0000 0.969087
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 9.00000 0.482451
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 3.00000 0.159901
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −9.00000 −0.478345
\(355\) −18.0000 −0.955341
\(356\) 0 0
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 16.0000 0.840941
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 42.0000 2.19838
\(366\) 8.00000 0.418167
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 6.00000 0.312772
\(369\) 12.0000 0.624695
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 5.00000 0.259238
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −18.0000 −0.930758
\(375\) −3.00000 −0.154919
\(376\) 12.0000 0.618853
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −12.0000 −0.615587
\(381\) 7.00000 0.358621
\(382\) 6.00000 0.306987
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −1.00000 −0.0508987
\(387\) −4.00000 −0.203331
\(388\) −5.00000 −0.253837
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −3.00000 −0.151911
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 21.0000 1.05931
\(394\) −18.0000 −0.906827
\(395\) 3.00000 0.150946
\(396\) 3.00000 0.150756
\(397\) −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
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) 5.00000 0.249068
\(404\) −18.0000 −0.895533
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 6.00000 0.297044
\(409\) 7.00000 0.346128 0.173064 0.984911i \(-0.444633\pi\)
0.173064 + 0.984911i \(0.444633\pi\)
\(410\) −36.0000 −1.77791
\(411\) 12.0000 0.591916
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 9.00000 0.441793
\(416\) −1.00000 −0.0490290
\(417\) −4.00000 −0.195881
\(418\) 12.0000 0.586939
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 8.00000 0.389434
\(423\) 12.0000 0.583460
\(424\) −9.00000 −0.437079
\(425\) −24.0000 −1.16417
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) 3.00000 0.144841
\(430\) 12.0000 0.578691
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) −27.0000 −1.29455
\(436\) −16.0000 −0.766261
\(437\) 24.0000 1.14808
\(438\) 14.0000 0.668946
\(439\) 1.00000 0.0477274 0.0238637 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) −9.00000 −0.429058
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −3.00000 −0.142534 −0.0712672 0.997457i \(-0.522704\pi\)
−0.0712672 + 0.997457i \(0.522704\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 1.00000 0.0473514
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 4.00000 0.188562
\(451\) 36.0000 1.69517
\(452\) −12.0000 −0.564433
\(453\) 19.0000 0.892698
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −19.0000 −0.888783 −0.444391 0.895833i \(-0.646580\pi\)
−0.444391 + 0.895833i \(0.646580\pi\)
\(458\) −26.0000 −1.21490
\(459\) 6.00000 0.280056
\(460\) −18.0000 −0.839254
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −9.00000 −0.417815
\(465\) −15.0000 −0.695608
\(466\) −6.00000 −0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −36.0000 −1.66056
\(471\) 2.00000 0.0921551
\(472\) 9.00000 0.414259
\(473\) −12.0000 −0.551761
\(474\) 1.00000 0.0459315
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 6.00000 0.274434
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 3.00000 0.136931
\(481\) 4.00000 0.182384
\(482\) −17.0000 −0.774329
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 15.0000 0.681115
\(486\) −1.00000 −0.0453609
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) −8.00000 −0.362143
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) −12.0000 −0.541002
\(493\) 54.0000 2.43204
\(494\) −4.00000 −0.179969
\(495\) −9.00000 −0.404520
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) 3.00000 0.134433
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 3.00000 0.134164
\(501\) 12.0000 0.536120
\(502\) 3.00000 0.133897
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 54.0000 2.40297
\(506\) 18.0000 0.800198
\(507\) −1.00000 −0.0444116
\(508\) −7.00000 −0.310575
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) −18.0000 −0.797053
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 30.0000 1.32324
\(515\) 24.0000 1.05757
\(516\) 4.00000 0.176090
\(517\) 36.0000 1.58328
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 3.00000 0.131559
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −9.00000 −0.393919
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) −21.0000 −0.917389
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 30.0000 1.30682
\(528\) −3.00000 −0.130558
\(529\) 13.0000 0.565217
\(530\) 27.0000 1.17281
\(531\) 9.00000 0.390567
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) −4.00000 −0.172774
\(537\) −12.0000 −0.517838
\(538\) −3.00000 −0.129339
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 7.00000 0.300676
\(543\) −16.0000 −0.686626
\(544\) −6.00000 −0.257248
\(545\) 48.0000 2.05609
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −12.0000 −0.512615
\(549\) −8.00000 −0.341432
\(550\) 12.0000 0.511682
\(551\) −36.0000 −1.53365
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) −12.0000 −0.509372
\(556\) 4.00000 0.169638
\(557\) −21.0000 −0.889799 −0.444899 0.895581i \(-0.646761\pi\)
−0.444899 + 0.895581i \(0.646761\pi\)
\(558\) −5.00000 −0.211667
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) −33.0000 −1.39078 −0.695392 0.718631i \(-0.744769\pi\)
−0.695392 + 0.718631i \(0.744769\pi\)
\(564\) −12.0000 −0.505291
\(565\) 36.0000 1.51453
\(566\) −2.00000 −0.0840663
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 12.0000 0.502625
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) −3.00000 −0.125436
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 1.00000 0.0416667
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 19.0000 0.790296
\(579\) 1.00000 0.0415586
\(580\) 27.0000 1.12111
\(581\) 0 0
\(582\) 5.00000 0.207257
\(583\) −27.0000 −1.11823
\(584\) −14.0000 −0.579324
\(585\) 3.00000 0.124035
\(586\) −21.0000 −0.867502
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) −27.0000 −1.11157
\(591\) 18.0000 0.740421
\(592\) −4.00000 −0.164399
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 8.00000 0.327418
\(598\) −6.00000 −0.245358
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) −4.00000 −0.163299
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −19.0000 −0.773099
\(605\) 6.00000 0.243935
\(606\) 18.0000 0.731200
\(607\) −5.00000 −0.202944 −0.101472 0.994838i \(-0.532355\pi\)
−0.101472 + 0.994838i \(0.532355\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) −12.0000 −0.485468
\(612\) −6.00000 −0.242536
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 36.0000 1.45166
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 8.00000 0.321807
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 15.0000 0.602414
\(621\) −6.00000 −0.240772
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −29.0000 −1.16000
\(626\) 19.0000 0.759393
\(627\) −12.0000 −0.479234
\(628\) −2.00000 −0.0798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −8.00000 −0.317971
\(634\) −15.0000 −0.595726
\(635\) 21.0000 0.833360
\(636\) 9.00000 0.356873
\(637\) 0 0
\(638\) −27.0000 −1.06894
\(639\) 6.00000 0.237356
\(640\) −3.00000 −0.118585
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) −3.00000 −0.118401
\(643\) −8.00000 −0.315489 −0.157745 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) −24.0000 −0.944267
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) 27.0000 1.05984
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 16.0000 0.625650
\(655\) 63.0000 2.46161
\(656\) 12.0000 0.468521
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 9.00000 0.350325
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 32.0000 1.24372
\(663\) −6.00000 −0.233021
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −54.0000 −2.09089
\(668\) −12.0000 −0.464294
\(669\) −1.00000 −0.0386622
\(670\) 12.0000 0.463600
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −31.0000 −1.19496 −0.597481 0.801883i \(-0.703832\pi\)
−0.597481 + 0.801883i \(0.703832\pi\)
\(674\) −31.0000 −1.19408
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 18.0000 0.690268
\(681\) −3.00000 −0.114960
\(682\) −15.0000 −0.574380
\(683\) −39.0000 −1.49229 −0.746147 0.665782i \(-0.768098\pi\)
−0.746147 + 0.665782i \(0.768098\pi\)
\(684\) 4.00000 0.152944
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) −4.00000 −0.152499
\(689\) 9.00000 0.342873
\(690\) 18.0000 0.685248
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −12.0000 −0.455186
\(696\) 9.00000 0.341144
\(697\) −72.0000 −2.72719
\(698\) 34.0000 1.28692
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 1.00000 0.0377426
\(703\) −16.0000 −0.603451
\(704\) 3.00000 0.113067
\(705\) 36.0000 1.35584
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) −9.00000 −0.338241
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −18.0000 −0.675528
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) 12.0000 0.448461
\(717\) −6.00000 −0.224074
\(718\) −6.00000 −0.223918
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 17.0000 0.632237
\(724\) 16.0000 0.594635
\(725\) −36.0000 −1.33701
\(726\) 2.00000 0.0742270
\(727\) 37.0000 1.37225 0.686127 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 42.0000 1.55449
\(731\) 24.0000 0.887672
\(732\) 8.00000 0.295689
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 7.00000 0.258375
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −12.0000 −0.442026
\(738\) 12.0000 0.441726
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 12.0000 0.441129
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 5.00000 0.183309
\(745\) −18.0000 −0.659469
\(746\) 14.0000 0.512576
\(747\) −3.00000 −0.109764
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 12.0000 0.437595
\(753\) −3.00000 −0.109326
\(754\) 9.00000 0.327761
\(755\) 57.0000 2.07444
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −16.0000 −0.581146
\(759\) −18.0000 −0.653359
\(760\) −12.0000 −0.435286
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 7.00000 0.253583
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 18.0000 0.650791
\(766\) 6.00000 0.216789
\(767\) −9.00000 −0.324971
\(768\) −1.00000 −0.0360844
\(769\) 1.00000 0.0360609 0.0180305 0.999837i \(-0.494260\pi\)
0.0180305 + 0.999837i \(0.494260\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) −1.00000 −0.0359908
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −4.00000 −0.143777
\(775\) −20.0000 −0.718421
\(776\) −5.00000 −0.179490
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 48.0000 1.71978
\(780\) −3.00000 −0.107417
\(781\) 18.0000 0.644091
\(782\) −36.0000 −1.28736
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 21.0000 0.749045
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −18.0000 −0.641223
\(789\) −6.00000 −0.213606
\(790\) 3.00000 0.106735
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) 8.00000 0.284088
\(794\) −2.00000 −0.0709773
\(795\) −27.0000 −0.957591
\(796\) −8.00000 −0.283552
\(797\) −9.00000 −0.318796 −0.159398 0.987214i \(-0.550955\pi\)
−0.159398 + 0.987214i \(0.550955\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −42.0000 −1.48215
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 5.00000 0.176117
\(807\) 3.00000 0.105605
\(808\) −18.0000 −0.633238
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) −3.00000 −0.105409
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −7.00000 −0.245501
\(814\) −12.0000 −0.420600
\(815\) −24.0000 −0.840683
\(816\) 6.00000 0.210042
\(817\) −16.0000 −0.559769
\(818\) 7.00000 0.244749
\(819\) 0 0
\(820\) −36.0000 −1.25717
\(821\) −39.0000 −1.36111 −0.680555 0.732697i \(-0.738261\pi\)
−0.680555 + 0.732697i \(0.738261\pi\)
\(822\) 12.0000 0.418548
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −8.00000 −0.278693
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 6.00000 0.208514
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 9.00000 0.312395
\(831\) −2.00000 −0.0693792
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 36.0000 1.24583
\(836\) 12.0000 0.415029
\(837\) 5.00000 0.172825
\(838\) −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
\(841\) 52.0000 1.79310
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −3.00000 −0.103203
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) 2.00000 0.0686398
\(850\) −24.0000 −0.823193
\(851\) −24.0000 −0.822709
\(852\) −6.00000 −0.205557
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 3.00000 0.102538
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 3.00000 0.102418
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 30.0000 1.02180
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 18.0000 0.612018
\(866\) −2.00000 −0.0679628
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −3.00000 −0.101768
\(870\) −27.0000 −0.915386
\(871\) 4.00000 0.135535
\(872\) −16.0000 −0.541828
\(873\) −5.00000 −0.169224
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 1.00000 0.0337484
\(879\) 21.0000 0.708312
\(880\) −9.00000 −0.303390
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 6.00000 0.201802
\(885\) 27.0000 0.907595
\(886\) −3.00000 −0.100787
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 1.00000 0.0334825
\(893\) 48.0000 1.60626
\(894\) −6.00000 −0.200670
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −12.0000 −0.400445
\(899\) 45.0000 1.50083
\(900\) 4.00000 0.133333
\(901\) 54.0000 1.79900
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) −48.0000 −1.59557
\(906\) 19.0000 0.631233
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 3.00000 0.0995585
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) −4.00000 −0.132453
\(913\) −9.00000 −0.297857
\(914\) −19.0000 −0.628464
\(915\) −24.0000 −0.793416
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −18.0000 −0.593442
\(921\) 2.00000 0.0659022
\(922\) −18.0000 −0.592798
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) −16.0000 −0.525793
\(927\) −8.00000 −0.262754
\(928\) −9.00000 −0.295439
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −15.0000 −0.491869
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −18.0000 −0.589294
\(934\) 36.0000 1.17796
\(935\) 54.0000 1.76599
\(936\) −1.00000 −0.0326860
\(937\) 13.0000 0.424691 0.212346 0.977195i \(-0.431890\pi\)
0.212346 + 0.977195i \(0.431890\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) −36.0000 −1.17419
\(941\) 45.0000 1.46696 0.733479 0.679712i \(-0.237895\pi\)
0.733479 + 0.679712i \(0.237895\pi\)
\(942\) 2.00000 0.0651635
\(943\) 72.0000 2.34464
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 1.00000 0.0324785
\(949\) 14.0000 0.454459
\(950\) 16.0000 0.519109
\(951\) 15.0000 0.486408
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) −9.00000 −0.291386
\(955\) −18.0000 −0.582466
\(956\) 6.00000 0.194054
\(957\) 27.0000 0.872786
\(958\) −6.00000 −0.193851
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) −6.00000 −0.193548
\(962\) 4.00000 0.128965
\(963\) 3.00000 0.0966736
\(964\) −17.0000 −0.547533
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) 29.0000 0.932577 0.466289 0.884633i \(-0.345591\pi\)
0.466289 + 0.884633i \(0.345591\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 24.0000 0.770991
\(970\) 15.0000 0.481621
\(971\) 33.0000 1.05902 0.529510 0.848304i \(-0.322376\pi\)
0.529510 + 0.848304i \(0.322376\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 23.0000 0.736968
\(975\) 4.00000 0.128103
\(976\) −8.00000 −0.256074
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −15.0000 −0.478669
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) −12.0000 −0.382546
\(985\) 54.0000 1.72058
\(986\) 54.0000 1.71971
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −24.0000 −0.763156
\(990\) −9.00000 −0.286039
\(991\) 35.0000 1.11181 0.555906 0.831245i \(-0.312372\pi\)
0.555906 + 0.831245i \(0.312372\pi\)
\(992\) −5.00000 −0.158750
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 3.00000 0.0950586
\(997\) 16.0000 0.506725 0.253363 0.967371i \(-0.418463\pi\)
0.253363 + 0.967371i \(0.418463\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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.s.1.1 1
7.3 odd 6 546.2.i.a.79.1 2
7.5 odd 6 546.2.i.a.235.1 yes 2
7.6 odd 2 3822.2.a.bh.1.1 1
21.5 even 6 1638.2.j.j.235.1 2
21.17 even 6 1638.2.j.j.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.a.79.1 2 7.3 odd 6
546.2.i.a.235.1 yes 2 7.5 odd 6
1638.2.j.j.235.1 2 21.5 even 6
1638.2.j.j.1171.1 2 21.17 even 6
3822.2.a.s.1.1 1 1.1 even 1 trivial
3822.2.a.bh.1.1 1 7.6 odd 2