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Properties

Label 35.2.a.a.1.1
Level $35$
Weight $2$
Character 35.1
Self dual yes
Analytic conductor $0.279$
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] = [35,2,Mod(1,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("35.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 35.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: \(0.279476407074\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 35.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} +2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -2.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} -3.00000 q^{33} -1.00000 q^{35} +4.00000 q^{36} +2.00000 q^{37} +5.00000 q^{39} -12.0000 q^{41} -10.0000 q^{43} +6.00000 q^{44} +2.00000 q^{45} +9.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} -10.0000 q^{52} +12.0000 q^{53} +3.00000 q^{55} +2.00000 q^{57} +2.00000 q^{60} +8.00000 q^{61} -2.00000 q^{63} -8.00000 q^{64} -5.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +2.00000 q^{73} +1.00000 q^{75} -4.00000 q^{76} -3.00000 q^{77} -1.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{84} -3.00000 q^{85} +3.00000 q^{87} -12.0000 q^{89} +5.00000 q^{91} +12.0000 q^{92} -4.00000 q^{93} -2.00000 q^{95} -1.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) −2.00000 −0.377964
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 4.00000 0.666667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 6.00000 0.904534
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 4.00000 0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −10.0000 −1.38675
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) −5.00000 −0.620174
\(66\) 0 0
\(67\) −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
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 12.0000 1.25109
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) −2.00000 −0.200000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 − 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 10.0000 0.962250
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) −6.00000 −0.557086
\(117\) −10.0000 −0.924500
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 6.00000 0.522233
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 − 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 2.00000 0.169031
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) −15.0000 −1.25436
\(144\) −8.00000 −0.666667
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) −4.00000 −0.328798
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 − 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −10.0000 −0.800641
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 24.0000 1.87409
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 20.0000 1.52499
\(173\) −9.00000 −0.684257 −0.342129 − 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −4.00000 −0.298142
\(181\) 20.0000 1.48659 0.743294 − 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) −18.0000 −1.31278
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 − 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) −8.00000 −0.577350
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −5.00000 −0.358057
\(196\) −2.00000 −0.142857
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) −6.00000 −0.420084
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 20.0000 1.38675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −24.0000 −1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) −6.00000 −0.404520
\(221\) 15.0000 1.00901
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 − 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 − 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −4.00000 −0.264906
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) −4.00000 −0.258199
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) −16.0000 −1.02430
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 4.00000 0.251976
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 16.0000 1.00000
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 10.0000 0.620174
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 8.00000 0.488678
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 12.0000 0.727607
\(273\) 5.00000 0.302614
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 12.0000 0.722315
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 − 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 − 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) −4.00000 −0.234082
\(293\) −21.0000 −1.22683 −0.613417 − 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) −30.0000 −1.73494
\(300\) −2.00000 −0.115470
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 8.00000 0.458831
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 − 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 6.00000 0.341882
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 − 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 2.00000 0.112509
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 8.00000 0.447214
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) −2.00000 −0.111111
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) −7.00000 −0.387101
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −24.0000 −1.31717
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 4.00000 0.218218
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 6.00000 0.325396
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) −6.00000 −0.321634
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −25.0000 −1.33440
\(352\) 0 0
\(353\) 15.0000 0.798369 0.399185 − 0.916871i \(-0.369293\pi\)
0.399185 + 0.916871i \(0.369293\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 24.0000 1.27200
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) −10.0000 −0.524142
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 − 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) −24.0000 −1.25109
\(369\) 24.0000 1.24939
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 8.00000 0.414781
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 15.0000 0.772539
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) 20.0000 1.01666
\(388\) 2.00000 0.101535
\(389\) −3.00000 −0.152106 −0.0760530 − 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) 1.00000 0.0503155
\(396\) −12.0000 −0.603023
\(397\) −25.0000 −1.25471 −0.627357 − 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 4.00000 0.200000
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −12.0000 −0.597022
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −10.0000 −0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) 17.0000 0.828529 0.414265 − 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 0 0
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) −12.0000 −0.580042
\(429\) −15.0000 −0.724207
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 − 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) −20.0000 −0.962250
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 14.0000 0.670478
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 − 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) −4.00000 −0.189832
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) −8.00000 −0.377964
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) −12.0000 −0.564433
\(453\) −1.00000 −0.0469841
\(454\) 0 0
\(455\) −5.00000 −0.234404
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) −12.0000 −0.559503
\(461\) −24.0000 −1.11779 −0.558896 − 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 12.0000 0.557086
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 − 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 20.0000 0.924500
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −6.00000 −0.275010
\(477\) −24.0000 −1.09888
\(478\) 0 0
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 4.00000 0.181818
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 − 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 − 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 24.0000 1.08200
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) −16.0000 −0.718421
\(497\) 0 0
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 − 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 2.00000 0.0894427
\(501\) −3.00000 −0.134030
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 − 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 32.0000 1.41977
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) −10.0000 −0.441511
\(514\) 0 0
\(515\) −5.00000 −0.220326
\(516\) 20.0000 0.880451
\(517\) −27.0000 −1.18746
\(518\) 0 0
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) −12.0000 −0.522233
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −60.0000 −2.59889
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) −10.0000 −0.430331
\(541\) 11.0000 0.472927 0.236463 − 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 24.0000 1.02523
\(549\) −16.0000 −0.682863
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −1.00000 −0.0425243
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) −28.0000 −1.18746
\(557\) −24.0000 −1.01691 −0.508456 − 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −50.0000 −2.11477
\(560\) −4.00000 −0.169031
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −18.0000 −0.757937
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 30.0000 1.25436
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 16.0000 0.666667
\(577\) −7.00000 −0.291414 −0.145707 − 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 6.00000 0.249136
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 10.0000 0.413449
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −39.0000 −1.60154 −0.800769 − 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 12.0000 0.491539
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 45.0000 1.83865 0.919325 − 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 2.00000 0.0813788
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 − 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 45.0000 1.82051
\(612\) 12.0000 0.485071
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) −8.00000 −0.321288
\(621\) 30.0000 1.20386
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 20.0000 0.800641
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) −28.0000 −1.11732
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 − 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) −13.0000 −0.516704
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) −24.0000 −0.951662
\(637\) 5.00000 0.198107
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 41.0000 1.61688 0.808441 − 0.588577i \(-0.200312\pi\)
0.808441 + 0.588577i \(0.200312\pi\)
\(644\) 12.0000 0.472866
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −48.0000 −1.87409
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 − 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) −6.00000 −0.233550
\(661\) 32.0000 1.24466 0.622328 − 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) 15.0000 0.582552
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 6.00000 0.232147
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −28.0000 −1.07932 −0.539660 − 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) −24.0000 −0.923077
\(677\) 45.0000 1.72949 0.864745 − 0.502211i \(-0.167480\pi\)
0.864745 + 0.502211i \(0.167480\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 − 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 8.00000 0.305888
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) −40.0000 −1.52499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000 0.684257
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) −2.00000 −0.0755929
\(701\) −9.00000 −0.339925 −0.169963 − 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 24.0000 0.904534
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 − 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 15.0000 0.560968
\(716\) −24.0000 −0.896922
\(717\) −21.0000 −0.784259
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 8.00000 0.298142
\(721\) 5.00000 0.186210
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) −40.0000 −1.48659
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) −16.0000 −0.591377
\(733\) −31.0000 −1.14501 −0.572506 − 0.819901i \(-0.694029\pi\)
−0.572506 + 0.819901i \(0.694029\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −43.0000 −1.58178 −0.790890 − 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) 4.00000 0.147043
\(741\) 10.0000 0.367359
\(742\) 0 0
\(743\) −12.0000 −0.440237 −0.220119 − 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 18.0000 0.658145
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 − 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 36.0000 1.31278
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 1.00000 0.0363937
\(756\) 10.0000 0.363696
\(757\) −16.0000 −0.581530 −0.290765 − 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) −7.00000 −0.253417
\(764\) −18.0000 −0.651217
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 0 0
\(768\) 16.0000 0.577350
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 8.00000 0.287926
\(773\) 21.0000 0.755318 0.377659 − 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 10.0000 0.358057
\(781\) 0 0
\(782\) 0 0
\(783\) −15.0000 −0.536056
\(784\) 4.00000 0.142857
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) 5.00000 0.178231 0.0891154 − 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) 32.0000 1.13421
\(797\) 15.0000 0.531327 0.265664 − 0.964066i \(-0.414409\pi\)
0.265664 + 0.964066i \(0.414409\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 8.00000 0.282138
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 − 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) −6.00000 −0.210559
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 12.0000 0.420084
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) −10.0000 −0.349428
\(820\) −24.0000 −0.838116
\(821\) −27.0000 −0.942306 −0.471153 − 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) 0 0
\(827\) 54.0000 1.87776 0.938882 − 0.344239i \(-0.111863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) −24.0000 −0.834058
\(829\) −52.0000 −1.80603 −0.903017 − 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) −40.0000 −1.38675
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 12.0000 0.415029
\(837\) 20.0000 0.691301
\(838\) 0 0
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 3.00000 0.103325
\(844\) 26.0000 0.894957
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 48.0000 1.64833
\(849\) −13.0000 −0.446159
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 − 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −20.0000 −0.681994
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 8.00000 0.271538
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −4.00000 −0.135147
\(877\) 50.0000 1.68838 0.844190 − 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 0 0
\(879\) −21.0000 −0.708312
\(880\) 12.0000 0.404520
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 38.0000 1.27233
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −30.0000 −1.00167
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 4.00000 0.133333
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 − 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 6.00000 0.199117
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 8.00000 0.264906
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 8.00000 0.264327
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) 0 0
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −10.0000 −0.328443
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −48.0000 −1.57229
\(933\) 18.0000 0.589294
\(934\) 0 0
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 − 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 18.0000 0.587095
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 72.0000 2.34464
\(944\) 0 0
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 2.00000 0.0649570
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −9.00000 −0.291233
\(956\) 42.0000 1.35838
\(957\) −9.00000 −0.290929
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 8.00000 0.258199
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 20.0000 0.644157
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 − 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) −32.0000 −1.02640
\(973\) 14.0000 0.448819
\(974\) 0 0
\(975\) 5.00000 0.160128
\(976\) 32.0000 1.02430
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) 2.00000 0.0638877
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −21.0000 −0.669796 −0.334898 − 0.942254i \(-0.608702\pi\)
−0.334898 + 0.942254i \(0.608702\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.00000 0.286473
\(988\) −20.0000 −0.636285
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 − 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) −24.0000 −0.760469
\(997\) −37.0000 −1.17180 −0.585901 − 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 35.2.a.a.1.1 ✓ 1
3.2 odd 2 315.2.a.b.1.1 1
4.3 odd 2 560.2.a.b.1.1 1
5.2 odd 4 175.2.b.a.99.1 2
5.3 odd 4 175.2.b.a.99.2 2
5.4 even 2 175.2.a.b.1.1 1
7.2 even 3 245.2.e.a.116.1 2
7.3 odd 6 245.2.e.b.226.1 2
7.4 even 3 245.2.e.a.226.1 2
7.5 odd 6 245.2.e.b.116.1 2
7.6 odd 2 245.2.a.c.1.1 1
8.3 odd 2 2240.2.a.u.1.1 1
8.5 even 2 2240.2.a.k.1.1 1
11.10 odd 2 4235.2.a.c.1.1 1
12.11 even 2 5040.2.a.v.1.1 1
13.12 even 2 5915.2.a.f.1.1 1
15.2 even 4 1575.2.d.c.1324.2 2
15.8 even 4 1575.2.d.c.1324.1 2
15.14 odd 2 1575.2.a.f.1.1 1
20.3 even 4 2800.2.g.l.449.1 2
20.7 even 4 2800.2.g.l.449.2 2
20.19 odd 2 2800.2.a.z.1.1 1
21.20 even 2 2205.2.a.e.1.1 1
28.27 even 2 3920.2.a.ba.1.1 1
35.13 even 4 1225.2.b.d.99.1 2
35.27 even 4 1225.2.b.d.99.2 2
35.34 odd 2 1225.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
175.2.a.b.1.1 1 5.4 even 2
175.2.b.a.99.1 2 5.2 odd 4
175.2.b.a.99.2 2 5.3 odd 4
245.2.a.c.1.1 1 7.6 odd 2
245.2.e.a.116.1 2 7.2 even 3
245.2.e.a.226.1 2 7.4 even 3
245.2.e.b.116.1 2 7.5 odd 6
245.2.e.b.226.1 2 7.3 odd 6
315.2.a.b.1.1 1 3.2 odd 2
560.2.a.b.1.1 1 4.3 odd 2
1225.2.a.e.1.1 1 35.34 odd 2
1225.2.b.d.99.1 2 35.13 even 4
1225.2.b.d.99.2 2 35.27 even 4
1575.2.a.f.1.1 1 15.14 odd 2
1575.2.d.c.1324.1 2 15.8 even 4
1575.2.d.c.1324.2 2 15.2 even 4
2205.2.a.e.1.1 1 21.20 even 2
2240.2.a.k.1.1 1 8.5 even 2
2240.2.a.u.1.1 1 8.3 odd 2
2800.2.a.z.1.1 1 20.19 odd 2
2800.2.g.l.449.1 2 20.3 even 4
2800.2.g.l.449.2 2 20.7 even 4
3920.2.a.ba.1.1 1 28.27 even 2
4235.2.a.c.1.1 1 11.10 odd 2
5040.2.a.v.1.1 1 12.11 even 2
5915.2.a.f.1.1 1 13.12 even 2