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Properties

Label 4235.2.a.a.1.1
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
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 \(\chi\) \(=\) 4235.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -4.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +4.00000 q^{39} -3.00000 q^{40} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} +6.00000 q^{47} +2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} +3.00000 q^{56} -8.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} -2.00000 q^{60} -4.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} +2.00000 q^{67} +2.00000 q^{68} -12.0000 q^{69} +1.00000 q^{70} +3.00000 q^{72} -10.0000 q^{73} +6.00000 q^{74} -2.00000 q^{75} -4.00000 q^{76} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +12.0000 q^{83} +2.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +8.00000 q^{87} -10.0000 q^{89} +1.00000 q^{90} -2.00000 q^{91} -6.00000 q^{92} -8.00000 q^{93} -6.00000 q^{94} -4.00000 q^{95} +10.0000 q^{96} -6.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −6.00000 −1.22474
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −2.00000 −0.365148
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) −1.00000 −0.166667
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) 4.00000 0.640513
\(40\) −3.00000 −0.474342
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −6.00000 −0.884652
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −8.00000 −1.05963
\(58\) 4.00000 0.525226
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −2.00000 −0.258199
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 2.00000 0.242536
\(69\) −12.0000 −1.44463
\(70\) 1.00000 0.119523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 6.00000 0.697486
\(75\) −2.00000 −0.230940
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 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
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) −2.00000 −0.209657
\(92\) −6.00000 −0.625543
\(93\) −8.00000 −0.829561
\(94\) −6.00000 −0.618853
\(95\) −4.00000 −0.410391
\(96\) 10.0000 1.02062
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −4.00000 −0.396059
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −6.00000 −0.588348
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −4.00000 −0.384900
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 8.00000 0.749269
\(115\) −6.00000 −0.559503
\(116\) 4.00000 0.371391
\(117\) −2.00000 −0.184900
\(118\) 8.00000 0.736460
\(119\) −2.00000 −0.183340
\(120\) 6.00000 0.547723
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) −8.00000 −0.704361
\(130\) −2.00000 −0.175412
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −2.00000 −0.172774
\(135\) −4.00000 −0.344265
\(136\) −6.00000 −0.514496
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 12.0000 1.02151
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.00000 0.0845154
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 4.00000 0.332182
\(146\) 10.0000 0.827606
\(147\) −2.00000 −0.164957
\(148\) 6.00000 0.493197
\(149\) −8.00000 −0.655386 −0.327693 − 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 2.00000 0.163299
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 12.0000 0.973329
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) −4.00000 −0.320256
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 8.00000 0.636446
\(159\) 12.0000 0.951662
\(160\) 5.00000 0.395285
\(161\) 6.00000 0.472866
\(162\) 11.0000 0.864242
\(163\) −14.0000 −1.09656 −0.548282 − 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −6.00000 −0.462910
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −8.00000 −0.606478
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 16.0000 1.20263
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) 18.0000 1.33793 0.668965 − 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 18.0000 1.32698
\(185\) 6.00000 0.441129
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 4.00000 0.290957
\(190\) 4.00000 0.290191
\(191\) −4.00000 −0.289430 −0.144715 − 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −14.0000 −1.01036
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 6.00000 0.430775
\(195\) −4.00000 −0.286446
\(196\) −1.00000 −0.0714286
\(197\) 10.0000 0.712470 0.356235 − 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 3.00000 0.212132
\(201\) −4.00000 −0.282138
\(202\) 12.0000 0.844317
\(203\) −4.00000 −0.280745
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) 6.00000 0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 12.0000 0.816497
\(217\) 4.00000 0.271538
\(218\) −16.0000 −1.08366
\(219\) 20.0000 1.35147
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −12.0000 −0.805387
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −5.00000 −0.334077
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 8.00000 0.529813
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) −12.0000 −0.787839
\(233\) 26.0000 1.70332 0.851658 − 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 2.00000 0.130744
\(235\) −6.00000 −0.391397
\(236\) 8.00000 0.520756
\(237\) 16.0000 1.03931
\(238\) 2.00000 0.129641
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −2.00000 −0.129099
\(241\) 16.0000 1.03065 0.515325 − 0.856995i \(-0.327671\pi\)
0.515325 + 0.856995i \(0.327671\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 12.0000 0.762001
\(249\) −24.0000 −1.52094
\(250\) 1.00000 0.0632456
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) −4.00000 −0.250490
\(256\) −17.0000 −1.06250
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 8.00000 0.498058
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) −4.00000 −0.247594
\(262\) −8.00000 −0.494242
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) 20.0000 1.22398
\(268\) −2.00000 −0.122169
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 4.00000 0.243432
\(271\) 32.0000 1.94386 0.971931 − 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 2.00000 0.121268
\(273\) 4.00000 0.242091
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 30.0000 1.80253 0.901263 − 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) −3.00000 −0.179284
\(281\) −20.0000 −1.19310 −0.596550 − 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 12.0000 0.714590
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 12.0000 0.703452
\(292\) 10.0000 0.585206
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 2.00000 0.116642
\(295\) 8.00000 0.465778
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) 8.00000 0.463428
\(299\) −12.0000 −0.693978
\(300\) 2.00000 0.115470
\(301\) 4.00000 0.230556
\(302\) 12.0000 0.690522
\(303\) 24.0000 1.37876
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 4.00000 0.227185
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 12.0000 0.679366
\(313\) 30.0000 1.69570 0.847850 − 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) −10.0000 −0.564333
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) −24.0000 −1.33955
\(322\) −6.00000 −0.334367
\(323\) −8.00000 −0.445132
\(324\) 11.0000 0.611111
\(325\) −2.00000 −0.110940
\(326\) 14.0000 0.775388
\(327\) −32.0000 −1.76960
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −12.0000 −0.658586
\(333\) −6.00000 −0.328798
\(334\) 24.0000 1.31322
\(335\) −2.00000 −0.109272
\(336\) 2.00000 0.109109
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 9.00000 0.489535
\(339\) 4.00000 0.217250
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) 12.0000 0.646058
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −8.00000 −0.428845
\(349\) −32.0000 −1.71292 −0.856460 − 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −8.00000 −0.427008
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −16.0000 −0.850390
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 4.00000 0.211702
\(358\) 12.0000 0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) −18.0000 −0.946059
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 − 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) −6.00000 −0.311504
\(372\) 8.00000 0.414781
\(373\) 26.0000 1.34623 0.673114 − 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 18.0000 0.928279
\(377\) 8.00000 0.412021
\(378\) −4.00000 −0.205738
\(379\) 32.0000 1.64373 0.821865 − 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 4.00000 0.205196
\(381\) 32.0000 1.63941
\(382\) 4.00000 0.204658
\(383\) 30.0000 1.53293 0.766464 − 0.642287i \(-0.222014\pi\)
0.766464 + 0.642287i \(0.222014\pi\)
\(384\) −6.00000 −0.306186
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 4.00000 0.203331
\(388\) 6.00000 0.304604
\(389\) −26.0000 −1.31825 −0.659126 − 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 4.00000 0.202548
\(391\) −12.0000 −0.606866
\(392\) 3.00000 0.151523
\(393\) −16.0000 −0.807093
\(394\) −10.0000 −0.503793
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 4.00000 0.200502
\(399\) −8.00000 −0.400501
\(400\) −1.00000 −0.0500000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000 0.199502
\(403\) −8.00000 −0.398508
\(404\) 12.0000 0.597022
\(405\) 11.0000 0.546594
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) 12.0000 0.594089
\(409\) −4.00000 −0.197787 −0.0988936 − 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) −6.00000 −0.295599
\(413\) −8.00000 −0.393654
\(414\) −6.00000 −0.294884
\(415\) −12.0000 −0.589057
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 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
\(421\) 34.0000 1.65706 0.828529 − 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 4.00000 0.194717
\(423\) 6.00000 0.291730
\(424\) −18.0000 −0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −4.00000 −0.192450
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −4.00000 −0.192006
\(435\) −8.00000 −0.383571
\(436\) −16.0000 −0.766261
\(437\) 24.0000 1.14808
\(438\) −20.0000 −0.955637
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −4.00000 −0.190261
\(443\) −14.0000 −0.665160 −0.332580 − 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) −12.0000 −0.569495
\(445\) 10.0000 0.474045
\(446\) −2.00000 −0.0947027
\(447\) 16.0000 0.756774
\(448\) 7.00000 0.330719
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) 24.0000 1.12762
\(454\) −20.0000 −0.938647
\(455\) 2.00000 0.0937614
\(456\) −24.0000 −1.12390
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −6.00000 −0.280362
\(459\) −8.00000 −0.373408
\(460\) 6.00000 0.279751
\(461\) −20.0000 −0.931493 −0.465746 − 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 4.00000 0.185695
\(465\) 8.00000 0.370991
\(466\) −26.0000 −1.20443
\(467\) −2.00000 −0.0925490 −0.0462745 − 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 2.00000 0.0924500
\(469\) 2.00000 0.0923514
\(470\) 6.00000 0.276759
\(471\) −20.0000 −0.921551
\(472\) −24.0000 −1.10469
\(473\) 0 0
\(474\) −16.0000 −0.734904
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) −6.00000 −0.274721
\(478\) 16.0000 0.731823
\(479\) −28.0000 −1.27935 −0.639676 − 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) −10.0000 −0.456435
\(481\) 12.0000 0.547153
\(482\) −16.0000 −0.728780
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) −10.0000 −0.453609
\(487\) 42.0000 1.90320 0.951601 − 0.307337i \(-0.0994378\pi\)
0.951601 + 0.307337i \(0.0994378\pi\)
\(488\) 0 0
\(489\) 28.0000 1.26620
\(490\) 1.00000 0.0451754
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 24.0000 1.07547
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 1.00000 0.0447214
\(501\) 48.0000 2.14448
\(502\) −28.0000 −1.24970
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 3.00000 0.133631
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) 16.0000 0.709885
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 4.00000 0.177123
\(511\) −10.0000 −0.442374
\(512\) 11.0000 0.486136
\(513\) 16.0000 0.706417
\(514\) −22.0000 −0.970378
\(515\) −6.00000 −0.264392
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) −36.0000 −1.58022
\(520\) 6.00000 0.263117
\(521\) 10.0000 0.438108 0.219054 − 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 4.00000 0.175075
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −8.00000 −0.349482
\(525\) −2.00000 −0.0872872
\(526\) −24.0000 −1.04645
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −6.00000 −0.260623
\(531\) −8.00000 −0.347170
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) −20.0000 −0.865485
\(535\) −12.0000 −0.518805
\(536\) 6.00000 0.259161
\(537\) 24.0000 1.03568
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −32.0000 −1.37452
\(543\) −36.0000 −1.54491
\(544\) 10.0000 0.428746
\(545\) −16.0000 −0.685365
\(546\) −4.00000 −0.171184
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) −36.0000 −1.53226
\(553\) −8.00000 −0.340195
\(554\) −30.0000 −1.27458
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 − 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −4.00000 −0.169334
\(559\) −8.00000 −0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) −28.0000 −1.18006 −0.590030 − 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 12.0000 0.505291
\(565\) 2.00000 0.0841406
\(566\) −12.0000 −0.504398
\(567\) −11.0000 −0.461957
\(568\) 0 0
\(569\) −8.00000 −0.335377 −0.167689 − 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) −8.00000 −0.335083
\(571\) −8.00000 −0.334790 −0.167395 − 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 7.00000 0.291667
\(577\) −14.0000 −0.582828 −0.291414 − 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 13.0000 0.540729
\(579\) −44.0000 −1.82858
\(580\) −4.00000 −0.166091
\(581\) 12.0000 0.497844
\(582\) −12.0000 −0.497416
\(583\) 0 0
\(584\) −30.0000 −1.24141
\(585\) 2.00000 0.0826898
\(586\) −14.0000 −0.578335
\(587\) 38.0000 1.56843 0.784214 − 0.620491i \(-0.213066\pi\)
0.784214 + 0.620491i \(0.213066\pi\)
\(588\) 2.00000 0.0824786
\(589\) 16.0000 0.659269
\(590\) −8.00000 −0.329355
\(591\) −20.0000 −0.822690
\(592\) 6.00000 0.246598
\(593\) −26.0000 −1.06769 −0.533846 − 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 8.00000 0.327693
\(597\) 8.00000 0.327418
\(598\) 12.0000 0.490716
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) −6.00000 −0.244949
\(601\) 8.00000 0.326327 0.163163 − 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −4.00000 −0.163028
\(603\) 2.00000 0.0814463
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −24.0000 −0.974933
\(607\) 16.0000 0.649420 0.324710 − 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −20.0000 −0.811107
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 2.00000 0.0808452
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 − 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 12.0000 0.482711
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 4.00000 0.160644
\(621\) 24.0000 0.963087
\(622\) −8.00000 −0.320771
\(623\) −10.0000 −0.400642
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −30.0000 −1.19904
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 12.0000 0.478471
\(630\) 1.00000 0.0398410
\(631\) 16.0000 0.636950 0.318475 − 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −24.0000 −0.954669
\(633\) 8.00000 0.317971
\(634\) −22.0000 −0.873732
\(635\) 16.0000 0.634941
\(636\) −12.0000 −0.475831
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 24.0000 0.947204
\(643\) 22.0000 0.867595 0.433798 − 0.901010i \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) −6.00000 −0.236433
\(645\) 8.00000 0.315000
\(646\) 8.00000 0.314756
\(647\) 2.00000 0.0786281 0.0393141 − 0.999227i \(-0.487483\pi\)
0.0393141 + 0.999227i \(0.487483\pi\)
\(648\) −33.0000 −1.29636
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) −8.00000 −0.313545
\(652\) 14.0000 0.548282
\(653\) 2.00000 0.0782660 0.0391330 − 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 32.0000 1.25130
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) −6.00000 −0.233904
\(659\) −4.00000 −0.155818 −0.0779089 − 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −8.00000 −0.310929
\(663\) −8.00000 −0.310694
\(664\) 36.0000 1.39707
\(665\) −4.00000 −0.155113
\(666\) 6.00000 0.232495
\(667\) −24.0000 −0.929284
\(668\) 24.0000 0.928588
\(669\) −4.00000 −0.154649
\(670\) 2.00000 0.0772667
\(671\) 0 0
\(672\) 10.0000 0.385758
\(673\) 46.0000 1.77317 0.886585 − 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) −6.00000 −0.231111
\(675\) 4.00000 0.153960
\(676\) 9.00000 0.346154
\(677\) 30.0000 1.15299 0.576497 − 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) −4.00000 −0.153619
\(679\) −6.00000 −0.230259
\(680\) 6.00000 0.230089
\(681\) −40.0000 −1.53280
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 − 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) −4.00000 −0.152944
\(685\) 18.0000 0.687745
\(686\) −1.00000 −0.0381802
\(687\) −12.0000 −0.457829
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) −12.0000 −0.456832
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 24.0000 0.909718
\(697\) 0 0
\(698\) 32.0000 1.21122
\(699\) −52.0000 −1.96682
\(700\) −1.00000 −0.0377964
\(701\) 4.00000 0.151078 0.0755390 − 0.997143i \(-0.475932\pi\)
0.0755390 + 0.997143i \(0.475932\pi\)
\(702\) 8.00000 0.301941
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 18.0000 0.677439
\(707\) −12.0000 −0.451306
\(708\) −16.0000 −0.601317
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −30.0000 −1.12430
\(713\) 24.0000 0.898807
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 32.0000 1.19506
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 1.00000 0.0372678
\(721\) 6.00000 0.223452
\(722\) 3.00000 0.111648
\(723\) −32.0000 −1.19009
\(724\) −18.0000 −0.668965
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) −6.00000 −0.222375
\(729\) 13.0000 0.481481
\(730\) −10.0000 −0.370117
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 18.0000 0.664392
\(735\) 2.00000 0.0737711
\(736\) −30.0000 −1.10581
\(737\) 0 0
\(738\) 0 0
\(739\) 52.0000 1.91285 0.956425 − 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) −6.00000 −0.220564
\(741\) 16.0000 0.587775
\(742\) 6.00000 0.220267
\(743\) 32.0000 1.17397 0.586983 − 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) −24.0000 −0.879883
\(745\) 8.00000 0.293097
\(746\) −26.0000 −0.951928
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −2.00000 −0.0730297
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −6.00000 −0.218797
\(753\) −56.0000 −2.04075
\(754\) −8.00000 −0.291343
\(755\) 12.0000 0.436725
\(756\) −4.00000 −0.145479
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −32.0000 −1.16229
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) 36.0000 1.30500 0.652499 − 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) −32.0000 −1.15924
\(763\) 16.0000 0.579239
\(764\) 4.00000 0.144715
\(765\) 2.00000 0.0723102
\(766\) −30.0000 −1.08394
\(767\) 16.0000 0.577727
\(768\) 34.0000 1.22687
\(769\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) −22.0000 −0.791797
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) −18.0000 −0.646162
\(777\) 12.0000 0.430498
\(778\) 26.0000 0.932145
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 12.0000 0.429119
\(783\) −16.0000 −0.571793
\(784\) −1.00000 −0.0357143
\(785\) −10.0000 −0.356915
\(786\) 16.0000 0.570701
\(787\) −36.0000 −1.28326 −0.641631 − 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) −10.0000 −0.356235
\(789\) −48.0000 −1.70885
\(790\) −8.00000 −0.284627
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 0 0
\(794\) 22.0000 0.780751
\(795\) −12.0000 −0.425596
\(796\) 4.00000 0.141776
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 8.00000 0.283197
\(799\) −12.0000 −0.424529
\(800\) −5.00000 −0.176777
\(801\) −10.0000 −0.353333
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) −6.00000 −0.211472
\(806\) 8.00000 0.281788
\(807\) 12.0000 0.422420
\(808\) −36.0000 −1.26648
\(809\) −20.0000 −0.703163 −0.351581 − 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) −11.0000 −0.386501
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 4.00000 0.140372
\(813\) −64.0000 −2.24458
\(814\) 0 0
\(815\) 14.0000 0.490399
\(816\) −4.00000 −0.140028
\(817\) 16.0000 0.559769
\(818\) 4.00000 0.139857
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −52.0000 −1.81481 −0.907406 − 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) −36.0000 −1.25564
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 18.0000 0.627060
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 12.0000 0.416526
\(831\) −60.0000 −2.08138
\(832\) −14.0000 −0.485363
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 12.0000 0.414533
\(839\) −4.00000 −0.138095 −0.0690477 − 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 6.00000 0.207020
\(841\) −13.0000 −0.448276
\(842\) −34.0000 −1.17172
\(843\) 40.0000 1.37767
\(844\) 4.00000 0.137686
\(845\) 9.00000 0.309609
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −24.0000 −0.823678
\(850\) 2.00000 0.0685994
\(851\) −36.0000 −1.23406
\(852\) 0 0
\(853\) 18.0000 0.616308 0.308154 − 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 36.0000 1.23045
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −20.0000 −0.680414
\(865\) −18.0000 −0.612018
\(866\) 14.0000 0.475739
\(867\) 26.0000 0.883006
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) 8.00000 0.271225
\(871\) −4.00000 −0.135535
\(872\) 48.0000 1.62549
\(873\) −6.00000 −0.203069
\(874\) −24.0000 −0.811812
\(875\) −1.00000 −0.0338062
\(876\) −20.0000 −0.675737
\(877\) 42.0000 1.41824 0.709120 − 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) −32.0000 −1.07995
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 34.0000 1.14419 0.572096 − 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) −4.00000 −0.134535
\(885\) −16.0000 −0.537834
\(886\) 14.0000 0.470339
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 36.0000 1.20808
\(889\) −16.0000 −0.536623
\(890\) −10.0000 −0.335201
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 24.0000 0.803129
\(894\) −16.0000 −0.535120
\(895\) 12.0000 0.401116
\(896\) 3.00000 0.100223
\(897\) 24.0000 0.801337
\(898\) 30.0000 1.00111
\(899\) −16.0000 −0.533630
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −6.00000 −0.199557
\(905\) −18.0000 −0.598340
\(906\) −24.0000 −0.797347
\(907\) −54.0000 −1.79304 −0.896520 − 0.443003i \(-0.853913\pi\)
−0.896520 + 0.443003i \(0.853913\pi\)
\(908\) −20.0000 −0.663723
\(909\) −12.0000 −0.398015
\(910\) −2.00000 −0.0662994
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 8.00000 0.264183
\(918\) 8.00000 0.264039
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) −18.0000 −0.593442
\(921\) −8.00000 −0.263609
\(922\) 20.0000 0.658665
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 26.0000 0.854413
\(927\) 6.00000 0.197066
\(928\) 20.0000 0.656532
\(929\) −14.0000 −0.459325 −0.229663 − 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) −8.00000 −0.262330
\(931\) 4.00000 0.131095
\(932\) −26.0000 −0.851658
\(933\) −16.0000 −0.523816
\(934\) 2.00000 0.0654420
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 58.0000 1.89478 0.947389 − 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) −2.00000 −0.0653023
\(939\) −60.0000 −1.95803
\(940\) 6.00000 0.195698
\(941\) 28.0000 0.912774 0.456387 − 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −14.0000 −0.454939 −0.227469 − 0.973785i \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(948\) −16.0000 −0.519656
\(949\) 20.0000 0.649227
\(950\) −4.00000 −0.129777
\(951\) −44.0000 −1.42680
\(952\) −6.00000 −0.194461
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 6.00000 0.194257
\(955\) 4.00000 0.129437
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 28.0000 0.904639
\(959\) −18.0000 −0.581250
\(960\) 14.0000 0.451848
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) 12.0000 0.386695
\(964\) −16.0000 −0.515325
\(965\) −22.0000 −0.708205
\(966\) 12.0000 0.386094
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 0 0
\(969\) 16.0000 0.513994
\(970\) −6.00000 −0.192648
\(971\) −56.0000 −1.79713 −0.898563 − 0.438845i \(-0.855388\pi\)
−0.898563 + 0.438845i \(0.855388\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) −42.0000 −1.34577
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) 38.0000 1.21573 0.607864 − 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −28.0000 −0.895341
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 16.0000 0.510841
\(982\) −24.0000 −0.765871
\(983\) −50.0000 −1.59475 −0.797376 − 0.603483i \(-0.793779\pi\)
−0.797376 + 0.603483i \(0.793779\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) −8.00000 −0.254772
\(987\) −12.0000 −0.381964
\(988\) 8.00000 0.254514
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 − 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −20.0000 −0.635001
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 24.0000 0.760469
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 0 0
\(999\) −24.0000 −0.759326
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 4235.2.a.a.1.1 ✓ 1
11.10 odd 2 4235.2.a.d.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
4235.2.a.d.1.1 yes 1 11.10 odd 2