Properties

Label 4016.2.a.l.1.6
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} - 10845 x^{11} - 41921 x^{10} + 43551 x^{9} + 76260 x^{8} - 80907 x^{7} - 72526 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.31693\) of defining polynomial
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.31693 q^{3} -1.28443 q^{5} +1.02036 q^{7} -1.26569 q^{9} +O(q^{10})\) \(q-1.31693 q^{3} -1.28443 q^{5} +1.02036 q^{7} -1.26569 q^{9} -3.40771 q^{11} -6.32304 q^{13} +1.69151 q^{15} +5.36124 q^{17} +2.29090 q^{19} -1.34374 q^{21} -6.25279 q^{23} -3.35023 q^{25} +5.61763 q^{27} -8.07720 q^{29} -1.34002 q^{31} +4.48773 q^{33} -1.31058 q^{35} -7.74614 q^{37} +8.32702 q^{39} +9.95371 q^{41} +1.55335 q^{43} +1.62569 q^{45} +11.5709 q^{47} -5.95887 q^{49} -7.06039 q^{51} -8.43397 q^{53} +4.37697 q^{55} -3.01697 q^{57} +12.9358 q^{59} +0.212306 q^{61} -1.29145 q^{63} +8.12152 q^{65} -1.62764 q^{67} +8.23450 q^{69} -11.1685 q^{71} -10.7073 q^{73} +4.41203 q^{75} -3.47708 q^{77} +14.1646 q^{79} -3.60098 q^{81} -1.94879 q^{83} -6.88615 q^{85} +10.6371 q^{87} -4.10583 q^{89} -6.45176 q^{91} +1.76472 q^{93} -2.94251 q^{95} +13.6029 q^{97} +4.31309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.31693 −0.760332 −0.380166 0.924918i \(-0.624133\pi\)
−0.380166 + 0.924918i \(0.624133\pi\)
\(4\) 0 0
\(5\) −1.28443 −0.574416 −0.287208 0.957868i \(-0.592727\pi\)
−0.287208 + 0.957868i \(0.592727\pi\)
\(6\) 0 0
\(7\) 1.02036 0.385659 0.192829 0.981232i \(-0.438234\pi\)
0.192829 + 0.981232i \(0.438234\pi\)
\(8\) 0 0
\(9\) −1.26569 −0.421895
\(10\) 0 0
\(11\) −3.40771 −1.02746 −0.513731 0.857951i \(-0.671737\pi\)
−0.513731 + 0.857951i \(0.671737\pi\)
\(12\) 0 0
\(13\) −6.32304 −1.75370 −0.876848 0.480768i \(-0.840358\pi\)
−0.876848 + 0.480768i \(0.840358\pi\)
\(14\) 0 0
\(15\) 1.69151 0.436747
\(16\) 0 0
\(17\) 5.36124 1.30029 0.650145 0.759810i \(-0.274708\pi\)
0.650145 + 0.759810i \(0.274708\pi\)
\(18\) 0 0
\(19\) 2.29090 0.525569 0.262785 0.964855i \(-0.415359\pi\)
0.262785 + 0.964855i \(0.415359\pi\)
\(20\) 0 0
\(21\) −1.34374 −0.293229
\(22\) 0 0
\(23\) −6.25279 −1.30380 −0.651898 0.758307i \(-0.726027\pi\)
−0.651898 + 0.758307i \(0.726027\pi\)
\(24\) 0 0
\(25\) −3.35023 −0.670047
\(26\) 0 0
\(27\) 5.61763 1.08111
\(28\) 0 0
\(29\) −8.07720 −1.49990 −0.749949 0.661495i \(-0.769922\pi\)
−0.749949 + 0.661495i \(0.769922\pi\)
\(30\) 0 0
\(31\) −1.34002 −0.240675 −0.120337 0.992733i \(-0.538398\pi\)
−0.120337 + 0.992733i \(0.538398\pi\)
\(32\) 0 0
\(33\) 4.48773 0.781213
\(34\) 0 0
\(35\) −1.31058 −0.221528
\(36\) 0 0
\(37\) −7.74614 −1.27346 −0.636729 0.771088i \(-0.719713\pi\)
−0.636729 + 0.771088i \(0.719713\pi\)
\(38\) 0 0
\(39\) 8.32702 1.33339
\(40\) 0 0
\(41\) 9.95371 1.55451 0.777254 0.629187i \(-0.216612\pi\)
0.777254 + 0.629187i \(0.216612\pi\)
\(42\) 0 0
\(43\) 1.55335 0.236884 0.118442 0.992961i \(-0.462210\pi\)
0.118442 + 0.992961i \(0.462210\pi\)
\(44\) 0 0
\(45\) 1.62569 0.242343
\(46\) 0 0
\(47\) 11.5709 1.68780 0.843898 0.536503i \(-0.180255\pi\)
0.843898 + 0.536503i \(0.180255\pi\)
\(48\) 0 0
\(49\) −5.95887 −0.851267
\(50\) 0 0
\(51\) −7.06039 −0.988653
\(52\) 0 0
\(53\) −8.43397 −1.15850 −0.579248 0.815152i \(-0.696654\pi\)
−0.579248 + 0.815152i \(0.696654\pi\)
\(54\) 0 0
\(55\) 4.37697 0.590191
\(56\) 0 0
\(57\) −3.01697 −0.399607
\(58\) 0 0
\(59\) 12.9358 1.68410 0.842051 0.539397i \(-0.181348\pi\)
0.842051 + 0.539397i \(0.181348\pi\)
\(60\) 0 0
\(61\) 0.212306 0.0271830 0.0135915 0.999908i \(-0.495674\pi\)
0.0135915 + 0.999908i \(0.495674\pi\)
\(62\) 0 0
\(63\) −1.29145 −0.162708
\(64\) 0 0
\(65\) 8.12152 1.00735
\(66\) 0 0
\(67\) −1.62764 −0.198848 −0.0994240 0.995045i \(-0.531700\pi\)
−0.0994240 + 0.995045i \(0.531700\pi\)
\(68\) 0 0
\(69\) 8.23450 0.991318
\(70\) 0 0
\(71\) −11.1685 −1.32546 −0.662729 0.748860i \(-0.730602\pi\)
−0.662729 + 0.748860i \(0.730602\pi\)
\(72\) 0 0
\(73\) −10.7073 −1.25319 −0.626597 0.779343i \(-0.715553\pi\)
−0.626597 + 0.779343i \(0.715553\pi\)
\(74\) 0 0
\(75\) 4.41203 0.509458
\(76\) 0 0
\(77\) −3.47708 −0.396250
\(78\) 0 0
\(79\) 14.1646 1.59365 0.796824 0.604212i \(-0.206512\pi\)
0.796824 + 0.604212i \(0.206512\pi\)
\(80\) 0 0
\(81\) −3.60098 −0.400109
\(82\) 0 0
\(83\) −1.94879 −0.213907 −0.106954 0.994264i \(-0.534110\pi\)
−0.106954 + 0.994264i \(0.534110\pi\)
\(84\) 0 0
\(85\) −6.88615 −0.746907
\(86\) 0 0
\(87\) 10.6371 1.14042
\(88\) 0 0
\(89\) −4.10583 −0.435217 −0.217608 0.976036i \(-0.569826\pi\)
−0.217608 + 0.976036i \(0.569826\pi\)
\(90\) 0 0
\(91\) −6.45176 −0.676328
\(92\) 0 0
\(93\) 1.76472 0.182993
\(94\) 0 0
\(95\) −2.94251 −0.301895
\(96\) 0 0
\(97\) 13.6029 1.38117 0.690583 0.723253i \(-0.257354\pi\)
0.690583 + 0.723253i \(0.257354\pi\)
\(98\) 0 0
\(99\) 4.31309 0.433482
\(100\) 0 0
\(101\) −0.708875 −0.0705357 −0.0352679 0.999378i \(-0.511228\pi\)
−0.0352679 + 0.999378i \(0.511228\pi\)
\(102\) 0 0
\(103\) −0.913943 −0.0900535 −0.0450267 0.998986i \(-0.514337\pi\)
−0.0450267 + 0.998986i \(0.514337\pi\)
\(104\) 0 0
\(105\) 1.72595 0.168435
\(106\) 0 0
\(107\) −3.00141 −0.290157 −0.145078 0.989420i \(-0.546343\pi\)
−0.145078 + 0.989420i \(0.546343\pi\)
\(108\) 0 0
\(109\) 12.7818 1.22427 0.612137 0.790752i \(-0.290310\pi\)
0.612137 + 0.790752i \(0.290310\pi\)
\(110\) 0 0
\(111\) 10.2011 0.968250
\(112\) 0 0
\(113\) −3.91713 −0.368493 −0.184246 0.982880i \(-0.558984\pi\)
−0.184246 + 0.982880i \(0.558984\pi\)
\(114\) 0 0
\(115\) 8.03128 0.748921
\(116\) 0 0
\(117\) 8.00298 0.739876
\(118\) 0 0
\(119\) 5.47038 0.501468
\(120\) 0 0
\(121\) 0.612476 0.0556796
\(122\) 0 0
\(123\) −13.1084 −1.18194
\(124\) 0 0
\(125\) 10.7253 0.959301
\(126\) 0 0
\(127\) 8.76931 0.778150 0.389075 0.921206i \(-0.372795\pi\)
0.389075 + 0.921206i \(0.372795\pi\)
\(128\) 0 0
\(129\) −2.04566 −0.180110
\(130\) 0 0
\(131\) −3.48973 −0.304899 −0.152450 0.988311i \(-0.548716\pi\)
−0.152450 + 0.988311i \(0.548716\pi\)
\(132\) 0 0
\(133\) 2.33754 0.202690
\(134\) 0 0
\(135\) −7.21546 −0.621008
\(136\) 0 0
\(137\) 3.10500 0.265278 0.132639 0.991164i \(-0.457655\pi\)
0.132639 + 0.991164i \(0.457655\pi\)
\(138\) 0 0
\(139\) 3.59477 0.304904 0.152452 0.988311i \(-0.451283\pi\)
0.152452 + 0.988311i \(0.451283\pi\)
\(140\) 0 0
\(141\) −15.2382 −1.28329
\(142\) 0 0
\(143\) 21.5471 1.80186
\(144\) 0 0
\(145\) 10.3746 0.861565
\(146\) 0 0
\(147\) 7.84744 0.647246
\(148\) 0 0
\(149\) −14.2901 −1.17069 −0.585344 0.810785i \(-0.699041\pi\)
−0.585344 + 0.810785i \(0.699041\pi\)
\(150\) 0 0
\(151\) 4.83424 0.393405 0.196702 0.980463i \(-0.436977\pi\)
0.196702 + 0.980463i \(0.436977\pi\)
\(152\) 0 0
\(153\) −6.78564 −0.548587
\(154\) 0 0
\(155\) 1.72117 0.138247
\(156\) 0 0
\(157\) 10.3653 0.827240 0.413620 0.910450i \(-0.364264\pi\)
0.413620 + 0.910450i \(0.364264\pi\)
\(158\) 0 0
\(159\) 11.1070 0.880841
\(160\) 0 0
\(161\) −6.38007 −0.502820
\(162\) 0 0
\(163\) 5.74307 0.449832 0.224916 0.974378i \(-0.427789\pi\)
0.224916 + 0.974378i \(0.427789\pi\)
\(164\) 0 0
\(165\) −5.76418 −0.448741
\(166\) 0 0
\(167\) −7.01822 −0.543086 −0.271543 0.962426i \(-0.587534\pi\)
−0.271543 + 0.962426i \(0.587534\pi\)
\(168\) 0 0
\(169\) 26.9808 2.07545
\(170\) 0 0
\(171\) −2.89956 −0.221735
\(172\) 0 0
\(173\) 3.76197 0.286018 0.143009 0.989721i \(-0.454322\pi\)
0.143009 + 0.989721i \(0.454322\pi\)
\(174\) 0 0
\(175\) −3.41843 −0.258409
\(176\) 0 0
\(177\) −17.0356 −1.28048
\(178\) 0 0
\(179\) −19.7663 −1.47740 −0.738700 0.674035i \(-0.764560\pi\)
−0.738700 + 0.674035i \(0.764560\pi\)
\(180\) 0 0
\(181\) 14.9611 1.11205 0.556024 0.831167i \(-0.312326\pi\)
0.556024 + 0.831167i \(0.312326\pi\)
\(182\) 0 0
\(183\) −0.279593 −0.0206681
\(184\) 0 0
\(185\) 9.94939 0.731494
\(186\) 0 0
\(187\) −18.2695 −1.33600
\(188\) 0 0
\(189\) 5.73198 0.416940
\(190\) 0 0
\(191\) 10.6702 0.772071 0.386036 0.922484i \(-0.373844\pi\)
0.386036 + 0.922484i \(0.373844\pi\)
\(192\) 0 0
\(193\) 16.3570 1.17740 0.588702 0.808350i \(-0.299639\pi\)
0.588702 + 0.808350i \(0.299639\pi\)
\(194\) 0 0
\(195\) −10.6955 −0.765921
\(196\) 0 0
\(197\) −5.80928 −0.413894 −0.206947 0.978352i \(-0.566353\pi\)
−0.206947 + 0.978352i \(0.566353\pi\)
\(198\) 0 0
\(199\) 5.49860 0.389785 0.194893 0.980825i \(-0.437564\pi\)
0.194893 + 0.980825i \(0.437564\pi\)
\(200\) 0 0
\(201\) 2.14350 0.151191
\(202\) 0 0
\(203\) −8.24163 −0.578449
\(204\) 0 0
\(205\) −12.7849 −0.892934
\(206\) 0 0
\(207\) 7.91406 0.550065
\(208\) 0 0
\(209\) −7.80673 −0.540003
\(210\) 0 0
\(211\) −9.32109 −0.641690 −0.320845 0.947132i \(-0.603967\pi\)
−0.320845 + 0.947132i \(0.603967\pi\)
\(212\) 0 0
\(213\) 14.7082 1.00779
\(214\) 0 0
\(215\) −1.99517 −0.136070
\(216\) 0 0
\(217\) −1.36730 −0.0928184
\(218\) 0 0
\(219\) 14.1008 0.952843
\(220\) 0 0
\(221\) −33.8993 −2.28031
\(222\) 0 0
\(223\) −22.9589 −1.53744 −0.768721 0.639584i \(-0.779106\pi\)
−0.768721 + 0.639584i \(0.779106\pi\)
\(224\) 0 0
\(225\) 4.24034 0.282690
\(226\) 0 0
\(227\) 26.8349 1.78109 0.890546 0.454893i \(-0.150323\pi\)
0.890546 + 0.454893i \(0.150323\pi\)
\(228\) 0 0
\(229\) 23.7770 1.57123 0.785613 0.618718i \(-0.212348\pi\)
0.785613 + 0.618718i \(0.212348\pi\)
\(230\) 0 0
\(231\) 4.57908 0.301282
\(232\) 0 0
\(233\) 6.34838 0.415896 0.207948 0.978140i \(-0.433321\pi\)
0.207948 + 0.978140i \(0.433321\pi\)
\(234\) 0 0
\(235\) −14.8621 −0.969497
\(236\) 0 0
\(237\) −18.6539 −1.21170
\(238\) 0 0
\(239\) −20.5726 −1.33073 −0.665366 0.746518i \(-0.731724\pi\)
−0.665366 + 0.746518i \(0.731724\pi\)
\(240\) 0 0
\(241\) −17.1473 −1.10455 −0.552277 0.833661i \(-0.686241\pi\)
−0.552277 + 0.833661i \(0.686241\pi\)
\(242\) 0 0
\(243\) −12.1106 −0.776897
\(244\) 0 0
\(245\) 7.65377 0.488981
\(246\) 0 0
\(247\) −14.4855 −0.921689
\(248\) 0 0
\(249\) 2.56643 0.162641
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 21.3077 1.33960
\(254\) 0 0
\(255\) 9.06860 0.567898
\(256\) 0 0
\(257\) 19.0889 1.19073 0.595367 0.803454i \(-0.297007\pi\)
0.595367 + 0.803454i \(0.297007\pi\)
\(258\) 0 0
\(259\) −7.90383 −0.491120
\(260\) 0 0
\(261\) 10.2232 0.632800
\(262\) 0 0
\(263\) 26.8459 1.65539 0.827695 0.561178i \(-0.189652\pi\)
0.827695 + 0.561178i \(0.189652\pi\)
\(264\) 0 0
\(265\) 10.8329 0.665458
\(266\) 0 0
\(267\) 5.40710 0.330909
\(268\) 0 0
\(269\) −17.8198 −1.08649 −0.543246 0.839573i \(-0.682805\pi\)
−0.543246 + 0.839573i \(0.682805\pi\)
\(270\) 0 0
\(271\) 25.9178 1.57440 0.787198 0.616700i \(-0.211531\pi\)
0.787198 + 0.616700i \(0.211531\pi\)
\(272\) 0 0
\(273\) 8.49654 0.514234
\(274\) 0 0
\(275\) 11.4166 0.688448
\(276\) 0 0
\(277\) 0.255098 0.0153273 0.00766367 0.999971i \(-0.497561\pi\)
0.00766367 + 0.999971i \(0.497561\pi\)
\(278\) 0 0
\(279\) 1.69605 0.101540
\(280\) 0 0
\(281\) 24.1427 1.44023 0.720115 0.693855i \(-0.244089\pi\)
0.720115 + 0.693855i \(0.244089\pi\)
\(282\) 0 0
\(283\) 30.9395 1.83916 0.919581 0.392899i \(-0.128528\pi\)
0.919581 + 0.392899i \(0.128528\pi\)
\(284\) 0 0
\(285\) 3.87509 0.229541
\(286\) 0 0
\(287\) 10.1563 0.599510
\(288\) 0 0
\(289\) 11.7429 0.690756
\(290\) 0 0
\(291\) −17.9141 −1.05014
\(292\) 0 0
\(293\) −23.8506 −1.39337 −0.696684 0.717378i \(-0.745342\pi\)
−0.696684 + 0.717378i \(0.745342\pi\)
\(294\) 0 0
\(295\) −16.6152 −0.967375
\(296\) 0 0
\(297\) −19.1432 −1.11080
\(298\) 0 0
\(299\) 39.5366 2.28646
\(300\) 0 0
\(301\) 1.58497 0.0913563
\(302\) 0 0
\(303\) 0.933542 0.0536306
\(304\) 0 0
\(305\) −0.272693 −0.0156143
\(306\) 0 0
\(307\) 16.2007 0.924622 0.462311 0.886718i \(-0.347020\pi\)
0.462311 + 0.886718i \(0.347020\pi\)
\(308\) 0 0
\(309\) 1.20360 0.0684705
\(310\) 0 0
\(311\) −6.09690 −0.345724 −0.172862 0.984946i \(-0.555301\pi\)
−0.172862 + 0.984946i \(0.555301\pi\)
\(312\) 0 0
\(313\) 4.43223 0.250525 0.125262 0.992124i \(-0.460023\pi\)
0.125262 + 0.992124i \(0.460023\pi\)
\(314\) 0 0
\(315\) 1.65878 0.0934618
\(316\) 0 0
\(317\) −8.00439 −0.449571 −0.224786 0.974408i \(-0.572168\pi\)
−0.224786 + 0.974408i \(0.572168\pi\)
\(318\) 0 0
\(319\) 27.5247 1.54109
\(320\) 0 0
\(321\) 3.95265 0.220616
\(322\) 0 0
\(323\) 12.2821 0.683393
\(324\) 0 0
\(325\) 21.1837 1.17506
\(326\) 0 0
\(327\) −16.8328 −0.930854
\(328\) 0 0
\(329\) 11.8065 0.650913
\(330\) 0 0
\(331\) 30.8762 1.69711 0.848554 0.529109i \(-0.177474\pi\)
0.848554 + 0.529109i \(0.177474\pi\)
\(332\) 0 0
\(333\) 9.80418 0.537266
\(334\) 0 0
\(335\) 2.09060 0.114221
\(336\) 0 0
\(337\) 25.1121 1.36794 0.683971 0.729509i \(-0.260252\pi\)
0.683971 + 0.729509i \(0.260252\pi\)
\(338\) 0 0
\(339\) 5.15860 0.280177
\(340\) 0 0
\(341\) 4.56640 0.247285
\(342\) 0 0
\(343\) −13.2227 −0.713957
\(344\) 0 0
\(345\) −10.5767 −0.569428
\(346\) 0 0
\(347\) 21.6440 1.16191 0.580956 0.813935i \(-0.302679\pi\)
0.580956 + 0.813935i \(0.302679\pi\)
\(348\) 0 0
\(349\) 31.8202 1.70330 0.851648 0.524113i \(-0.175603\pi\)
0.851648 + 0.524113i \(0.175603\pi\)
\(350\) 0 0
\(351\) −35.5205 −1.89594
\(352\) 0 0
\(353\) −18.9369 −1.00791 −0.503954 0.863730i \(-0.668122\pi\)
−0.503954 + 0.863730i \(0.668122\pi\)
\(354\) 0 0
\(355\) 14.3452 0.761364
\(356\) 0 0
\(357\) −7.20412 −0.381282
\(358\) 0 0
\(359\) −16.4551 −0.868468 −0.434234 0.900800i \(-0.642981\pi\)
−0.434234 + 0.900800i \(0.642981\pi\)
\(360\) 0 0
\(361\) −13.7518 −0.723777
\(362\) 0 0
\(363\) −0.806590 −0.0423350
\(364\) 0 0
\(365\) 13.7528 0.719854
\(366\) 0 0
\(367\) −7.04730 −0.367866 −0.183933 0.982939i \(-0.558883\pi\)
−0.183933 + 0.982939i \(0.558883\pi\)
\(368\) 0 0
\(369\) −12.5983 −0.655840
\(370\) 0 0
\(371\) −8.60566 −0.446784
\(372\) 0 0
\(373\) −10.9354 −0.566211 −0.283106 0.959089i \(-0.591365\pi\)
−0.283106 + 0.959089i \(0.591365\pi\)
\(374\) 0 0
\(375\) −14.1245 −0.729387
\(376\) 0 0
\(377\) 51.0725 2.63037
\(378\) 0 0
\(379\) −3.20641 −0.164702 −0.0823511 0.996603i \(-0.526243\pi\)
−0.0823511 + 0.996603i \(0.526243\pi\)
\(380\) 0 0
\(381\) −11.5486 −0.591652
\(382\) 0 0
\(383\) 25.5125 1.30363 0.651814 0.758379i \(-0.274008\pi\)
0.651814 + 0.758379i \(0.274008\pi\)
\(384\) 0 0
\(385\) 4.46607 0.227612
\(386\) 0 0
\(387\) −1.96605 −0.0999402
\(388\) 0 0
\(389\) 19.7967 1.00373 0.501866 0.864945i \(-0.332647\pi\)
0.501866 + 0.864945i \(0.332647\pi\)
\(390\) 0 0
\(391\) −33.5227 −1.69531
\(392\) 0 0
\(393\) 4.59574 0.231824
\(394\) 0 0
\(395\) −18.1935 −0.915416
\(396\) 0 0
\(397\) −20.8522 −1.04654 −0.523270 0.852167i \(-0.675288\pi\)
−0.523270 + 0.852167i \(0.675288\pi\)
\(398\) 0 0
\(399\) −3.07838 −0.154112
\(400\) 0 0
\(401\) −30.4625 −1.52122 −0.760612 0.649207i \(-0.775101\pi\)
−0.760612 + 0.649207i \(0.775101\pi\)
\(402\) 0 0
\(403\) 8.47301 0.422071
\(404\) 0 0
\(405\) 4.62522 0.229829
\(406\) 0 0
\(407\) 26.3966 1.30843
\(408\) 0 0
\(409\) 21.9664 1.08617 0.543085 0.839678i \(-0.317256\pi\)
0.543085 + 0.839678i \(0.317256\pi\)
\(410\) 0 0
\(411\) −4.08907 −0.201699
\(412\) 0 0
\(413\) 13.1992 0.649489
\(414\) 0 0
\(415\) 2.50309 0.122872
\(416\) 0 0
\(417\) −4.73407 −0.231828
\(418\) 0 0
\(419\) −6.11485 −0.298730 −0.149365 0.988782i \(-0.547723\pi\)
−0.149365 + 0.988782i \(0.547723\pi\)
\(420\) 0 0
\(421\) −7.04861 −0.343528 −0.171764 0.985138i \(-0.554947\pi\)
−0.171764 + 0.985138i \(0.554947\pi\)
\(422\) 0 0
\(423\) −14.6452 −0.712073
\(424\) 0 0
\(425\) −17.9614 −0.871255
\(426\) 0 0
\(427\) 0.216628 0.0104834
\(428\) 0 0
\(429\) −28.3761 −1.37001
\(430\) 0 0
\(431\) 1.56669 0.0754646 0.0377323 0.999288i \(-0.487987\pi\)
0.0377323 + 0.999288i \(0.487987\pi\)
\(432\) 0 0
\(433\) 22.5996 1.08607 0.543034 0.839711i \(-0.317276\pi\)
0.543034 + 0.839711i \(0.317276\pi\)
\(434\) 0 0
\(435\) −13.6627 −0.655076
\(436\) 0 0
\(437\) −14.3245 −0.685235
\(438\) 0 0
\(439\) −26.3805 −1.25907 −0.629537 0.776970i \(-0.716756\pi\)
−0.629537 + 0.776970i \(0.716756\pi\)
\(440\) 0 0
\(441\) 7.54206 0.359146
\(442\) 0 0
\(443\) −31.1676 −1.48082 −0.740409 0.672157i \(-0.765368\pi\)
−0.740409 + 0.672157i \(0.765368\pi\)
\(444\) 0 0
\(445\) 5.27366 0.249995
\(446\) 0 0
\(447\) 18.8191 0.890111
\(448\) 0 0
\(449\) −10.6847 −0.504242 −0.252121 0.967696i \(-0.581128\pi\)
−0.252121 + 0.967696i \(0.581128\pi\)
\(450\) 0 0
\(451\) −33.9193 −1.59720
\(452\) 0 0
\(453\) −6.36637 −0.299118
\(454\) 0 0
\(455\) 8.28685 0.388494
\(456\) 0 0
\(457\) −2.95521 −0.138239 −0.0691194 0.997608i \(-0.522019\pi\)
−0.0691194 + 0.997608i \(0.522019\pi\)
\(458\) 0 0
\(459\) 30.1174 1.40576
\(460\) 0 0
\(461\) 40.6933 1.89528 0.947639 0.319344i \(-0.103463\pi\)
0.947639 + 0.319344i \(0.103463\pi\)
\(462\) 0 0
\(463\) −14.1764 −0.658834 −0.329417 0.944185i \(-0.606852\pi\)
−0.329417 + 0.944185i \(0.606852\pi\)
\(464\) 0 0
\(465\) −2.26666 −0.105114
\(466\) 0 0
\(467\) 6.85700 0.317304 0.158652 0.987335i \(-0.449285\pi\)
0.158652 + 0.987335i \(0.449285\pi\)
\(468\) 0 0
\(469\) −1.66078 −0.0766875
\(470\) 0 0
\(471\) −13.6504 −0.628977
\(472\) 0 0
\(473\) −5.29337 −0.243389
\(474\) 0 0
\(475\) −7.67506 −0.352156
\(476\) 0 0
\(477\) 10.6748 0.488764
\(478\) 0 0
\(479\) −20.6396 −0.943048 −0.471524 0.881853i \(-0.656296\pi\)
−0.471524 + 0.881853i \(0.656296\pi\)
\(480\) 0 0
\(481\) 48.9791 2.23326
\(482\) 0 0
\(483\) 8.40213 0.382310
\(484\) 0 0
\(485\) −17.4720 −0.793363
\(486\) 0 0
\(487\) 33.9244 1.53726 0.768631 0.639693i \(-0.220938\pi\)
0.768631 + 0.639693i \(0.220938\pi\)
\(488\) 0 0
\(489\) −7.56325 −0.342022
\(490\) 0 0
\(491\) −29.3239 −1.32337 −0.661685 0.749782i \(-0.730158\pi\)
−0.661685 + 0.749782i \(0.730158\pi\)
\(492\) 0 0
\(493\) −43.3038 −1.95030
\(494\) 0 0
\(495\) −5.53987 −0.248999
\(496\) 0 0
\(497\) −11.3959 −0.511174
\(498\) 0 0
\(499\) 8.74198 0.391345 0.195672 0.980669i \(-0.437311\pi\)
0.195672 + 0.980669i \(0.437311\pi\)
\(500\) 0 0
\(501\) 9.24253 0.412926
\(502\) 0 0
\(503\) −6.56731 −0.292822 −0.146411 0.989224i \(-0.546772\pi\)
−0.146411 + 0.989224i \(0.546772\pi\)
\(504\) 0 0
\(505\) 0.910503 0.0405168
\(506\) 0 0
\(507\) −35.5320 −1.57803
\(508\) 0 0
\(509\) 34.1486 1.51361 0.756804 0.653642i \(-0.226760\pi\)
0.756804 + 0.653642i \(0.226760\pi\)
\(510\) 0 0
\(511\) −10.9253 −0.483305
\(512\) 0 0
\(513\) 12.8694 0.568199
\(514\) 0 0
\(515\) 1.17390 0.0517281
\(516\) 0 0
\(517\) −39.4304 −1.73415
\(518\) 0 0
\(519\) −4.95427 −0.217468
\(520\) 0 0
\(521\) −18.7659 −0.822150 −0.411075 0.911601i \(-0.634847\pi\)
−0.411075 + 0.911601i \(0.634847\pi\)
\(522\) 0 0
\(523\) −25.5983 −1.11934 −0.559669 0.828716i \(-0.689072\pi\)
−0.559669 + 0.828716i \(0.689072\pi\)
\(524\) 0 0
\(525\) 4.50185 0.196477
\(526\) 0 0
\(527\) −7.18417 −0.312947
\(528\) 0 0
\(529\) 16.0973 0.699884
\(530\) 0 0
\(531\) −16.3727 −0.710515
\(532\) 0 0
\(533\) −62.9377 −2.72613
\(534\) 0 0
\(535\) 3.85510 0.166671
\(536\) 0 0
\(537\) 26.0308 1.12331
\(538\) 0 0
\(539\) 20.3061 0.874645
\(540\) 0 0
\(541\) −25.6658 −1.10346 −0.551729 0.834023i \(-0.686032\pi\)
−0.551729 + 0.834023i \(0.686032\pi\)
\(542\) 0 0
\(543\) −19.7027 −0.845525
\(544\) 0 0
\(545\) −16.4174 −0.703242
\(546\) 0 0
\(547\) −20.9323 −0.895000 −0.447500 0.894284i \(-0.647686\pi\)
−0.447500 + 0.894284i \(0.647686\pi\)
\(548\) 0 0
\(549\) −0.268713 −0.0114684
\(550\) 0 0
\(551\) −18.5041 −0.788301
\(552\) 0 0
\(553\) 14.4530 0.614604
\(554\) 0 0
\(555\) −13.1027 −0.556178
\(556\) 0 0
\(557\) −36.1674 −1.53246 −0.766230 0.642566i \(-0.777870\pi\)
−0.766230 + 0.642566i \(0.777870\pi\)
\(558\) 0 0
\(559\) −9.82190 −0.415422
\(560\) 0 0
\(561\) 24.0598 1.01580
\(562\) 0 0
\(563\) −41.5029 −1.74914 −0.874569 0.484901i \(-0.838856\pi\)
−0.874569 + 0.484901i \(0.838856\pi\)
\(564\) 0 0
\(565\) 5.03129 0.211668
\(566\) 0 0
\(567\) −3.67429 −0.154306
\(568\) 0 0
\(569\) −19.7499 −0.827958 −0.413979 0.910286i \(-0.635861\pi\)
−0.413979 + 0.910286i \(0.635861\pi\)
\(570\) 0 0
\(571\) 1.83271 0.0766964 0.0383482 0.999264i \(-0.487790\pi\)
0.0383482 + 0.999264i \(0.487790\pi\)
\(572\) 0 0
\(573\) −14.0520 −0.587030
\(574\) 0 0
\(575\) 20.9483 0.873604
\(576\) 0 0
\(577\) 39.2668 1.63470 0.817349 0.576143i \(-0.195443\pi\)
0.817349 + 0.576143i \(0.195443\pi\)
\(578\) 0 0
\(579\) −21.5411 −0.895217
\(580\) 0 0
\(581\) −1.98846 −0.0824952
\(582\) 0 0
\(583\) 28.7405 1.19031
\(584\) 0 0
\(585\) −10.2793 −0.424996
\(586\) 0 0
\(587\) −33.6019 −1.38690 −0.693449 0.720506i \(-0.743910\pi\)
−0.693449 + 0.720506i \(0.743910\pi\)
\(588\) 0 0
\(589\) −3.06986 −0.126491
\(590\) 0 0
\(591\) 7.65044 0.314697
\(592\) 0 0
\(593\) −26.0943 −1.07157 −0.535783 0.844356i \(-0.679984\pi\)
−0.535783 + 0.844356i \(0.679984\pi\)
\(594\) 0 0
\(595\) −7.02633 −0.288051
\(596\) 0 0
\(597\) −7.24129 −0.296366
\(598\) 0 0
\(599\) 35.7789 1.46189 0.730944 0.682438i \(-0.239080\pi\)
0.730944 + 0.682438i \(0.239080\pi\)
\(600\) 0 0
\(601\) 4.13530 0.168682 0.0843411 0.996437i \(-0.473121\pi\)
0.0843411 + 0.996437i \(0.473121\pi\)
\(602\) 0 0
\(603\) 2.06008 0.0838931
\(604\) 0 0
\(605\) −0.786684 −0.0319833
\(606\) 0 0
\(607\) −35.4600 −1.43928 −0.719638 0.694349i \(-0.755692\pi\)
−0.719638 + 0.694349i \(0.755692\pi\)
\(608\) 0 0
\(609\) 10.8537 0.439813
\(610\) 0 0
\(611\) −73.1636 −2.95988
\(612\) 0 0
\(613\) −12.4080 −0.501156 −0.250578 0.968096i \(-0.580621\pi\)
−0.250578 + 0.968096i \(0.580621\pi\)
\(614\) 0 0
\(615\) 16.8368 0.678926
\(616\) 0 0
\(617\) −10.4736 −0.421653 −0.210826 0.977524i \(-0.567615\pi\)
−0.210826 + 0.977524i \(0.567615\pi\)
\(618\) 0 0
\(619\) 31.7351 1.27554 0.637770 0.770227i \(-0.279857\pi\)
0.637770 + 0.770227i \(0.279857\pi\)
\(620\) 0 0
\(621\) −35.1258 −1.40955
\(622\) 0 0
\(623\) −4.18941 −0.167845
\(624\) 0 0
\(625\) 2.97522 0.119009
\(626\) 0 0
\(627\) 10.2809 0.410581
\(628\) 0 0
\(629\) −41.5289 −1.65586
\(630\) 0 0
\(631\) 47.7008 1.89894 0.949470 0.313857i \(-0.101621\pi\)
0.949470 + 0.313857i \(0.101621\pi\)
\(632\) 0 0
\(633\) 12.2753 0.487898
\(634\) 0 0
\(635\) −11.2636 −0.446982
\(636\) 0 0
\(637\) 37.6782 1.49286
\(638\) 0 0
\(639\) 14.1358 0.559204
\(640\) 0 0
\(641\) 29.2583 1.15563 0.577817 0.816166i \(-0.303905\pi\)
0.577817 + 0.816166i \(0.303905\pi\)
\(642\) 0 0
\(643\) 21.3885 0.843481 0.421740 0.906717i \(-0.361419\pi\)
0.421740 + 0.906717i \(0.361419\pi\)
\(644\) 0 0
\(645\) 2.62751 0.103458
\(646\) 0 0
\(647\) −21.9187 −0.861715 −0.430857 0.902420i \(-0.641789\pi\)
−0.430857 + 0.902420i \(0.641789\pi\)
\(648\) 0 0
\(649\) −44.0816 −1.73035
\(650\) 0 0
\(651\) 1.80064 0.0705728
\(652\) 0 0
\(653\) 13.0864 0.512110 0.256055 0.966662i \(-0.417577\pi\)
0.256055 + 0.966662i \(0.417577\pi\)
\(654\) 0 0
\(655\) 4.48232 0.175139
\(656\) 0 0
\(657\) 13.5521 0.528717
\(658\) 0 0
\(659\) −1.04943 −0.0408800 −0.0204400 0.999791i \(-0.506507\pi\)
−0.0204400 + 0.999791i \(0.506507\pi\)
\(660\) 0 0
\(661\) 7.93829 0.308764 0.154382 0.988011i \(-0.450661\pi\)
0.154382 + 0.988011i \(0.450661\pi\)
\(662\) 0 0
\(663\) 44.6431 1.73380
\(664\) 0 0
\(665\) −3.00241 −0.116429
\(666\) 0 0
\(667\) 50.5050 1.95556
\(668\) 0 0
\(669\) 30.2353 1.16897
\(670\) 0 0
\(671\) −0.723477 −0.0279295
\(672\) 0 0
\(673\) −31.2158 −1.20328 −0.601640 0.798767i \(-0.705486\pi\)
−0.601640 + 0.798767i \(0.705486\pi\)
\(674\) 0 0
\(675\) −18.8204 −0.724396
\(676\) 0 0
\(677\) 21.1051 0.811135 0.405568 0.914065i \(-0.367074\pi\)
0.405568 + 0.914065i \(0.367074\pi\)
\(678\) 0 0
\(679\) 13.8798 0.532659
\(680\) 0 0
\(681\) −35.3397 −1.35422
\(682\) 0 0
\(683\) 15.8340 0.605872 0.302936 0.953011i \(-0.402033\pi\)
0.302936 + 0.953011i \(0.402033\pi\)
\(684\) 0 0
\(685\) −3.98816 −0.152380
\(686\) 0 0
\(687\) −31.3127 −1.19465
\(688\) 0 0
\(689\) 53.3284 2.03165
\(690\) 0 0
\(691\) 36.7200 1.39689 0.698447 0.715662i \(-0.253875\pi\)
0.698447 + 0.715662i \(0.253875\pi\)
\(692\) 0 0
\(693\) 4.40089 0.167176
\(694\) 0 0
\(695\) −4.61724 −0.175142
\(696\) 0 0
\(697\) 53.3642 2.02131
\(698\) 0 0
\(699\) −8.36039 −0.316219
\(700\) 0 0
\(701\) 17.9385 0.677528 0.338764 0.940871i \(-0.389991\pi\)
0.338764 + 0.940871i \(0.389991\pi\)
\(702\) 0 0
\(703\) −17.7457 −0.669290
\(704\) 0 0
\(705\) 19.5724 0.737139
\(706\) 0 0
\(707\) −0.723306 −0.0272027
\(708\) 0 0
\(709\) −32.2917 −1.21274 −0.606371 0.795182i \(-0.707375\pi\)
−0.606371 + 0.795182i \(0.707375\pi\)
\(710\) 0 0
\(711\) −17.9280 −0.672352
\(712\) 0 0
\(713\) 8.37887 0.313791
\(714\) 0 0
\(715\) −27.6758 −1.03502
\(716\) 0 0
\(717\) 27.0928 1.01180
\(718\) 0 0
\(719\) 11.5719 0.431560 0.215780 0.976442i \(-0.430771\pi\)
0.215780 + 0.976442i \(0.430771\pi\)
\(720\) 0 0
\(721\) −0.932548 −0.0347299
\(722\) 0 0
\(723\) 22.5819 0.839828
\(724\) 0 0
\(725\) 27.0605 1.00500
\(726\) 0 0
\(727\) −6.59189 −0.244480 −0.122240 0.992501i \(-0.539008\pi\)
−0.122240 + 0.992501i \(0.539008\pi\)
\(728\) 0 0
\(729\) 26.7518 0.990808
\(730\) 0 0
\(731\) 8.32788 0.308018
\(732\) 0 0
\(733\) −6.34073 −0.234200 −0.117100 0.993120i \(-0.537360\pi\)
−0.117100 + 0.993120i \(0.537360\pi\)
\(734\) 0 0
\(735\) −10.0795 −0.371788
\(736\) 0 0
\(737\) 5.54653 0.204309
\(738\) 0 0
\(739\) −23.9608 −0.881411 −0.440705 0.897652i \(-0.645272\pi\)
−0.440705 + 0.897652i \(0.645272\pi\)
\(740\) 0 0
\(741\) 19.0764 0.700789
\(742\) 0 0
\(743\) −40.3846 −1.48157 −0.740784 0.671743i \(-0.765546\pi\)
−0.740784 + 0.671743i \(0.765546\pi\)
\(744\) 0 0
\(745\) 18.3546 0.672462
\(746\) 0 0
\(747\) 2.46655 0.0902465
\(748\) 0 0
\(749\) −3.06251 −0.111902
\(750\) 0 0
\(751\) −14.5929 −0.532503 −0.266251 0.963904i \(-0.585785\pi\)
−0.266251 + 0.963904i \(0.585785\pi\)
\(752\) 0 0
\(753\) −1.31693 −0.0479917
\(754\) 0 0
\(755\) −6.20925 −0.225978
\(756\) 0 0
\(757\) −15.6202 −0.567727 −0.283863 0.958865i \(-0.591616\pi\)
−0.283863 + 0.958865i \(0.591616\pi\)
\(758\) 0 0
\(759\) −28.0608 −1.01854
\(760\) 0 0
\(761\) 40.6289 1.47280 0.736399 0.676548i \(-0.236525\pi\)
0.736399 + 0.676548i \(0.236525\pi\)
\(762\) 0 0
\(763\) 13.0420 0.472152
\(764\) 0 0
\(765\) 8.71570 0.315117
\(766\) 0 0
\(767\) −81.7938 −2.95340
\(768\) 0 0
\(769\) −13.1395 −0.473824 −0.236912 0.971531i \(-0.576135\pi\)
−0.236912 + 0.971531i \(0.576135\pi\)
\(770\) 0 0
\(771\) −25.1388 −0.905353
\(772\) 0 0
\(773\) −40.3078 −1.44977 −0.724886 0.688869i \(-0.758108\pi\)
−0.724886 + 0.688869i \(0.758108\pi\)
\(774\) 0 0
\(775\) 4.48938 0.161263
\(776\) 0 0
\(777\) 10.4088 0.373414
\(778\) 0 0
\(779\) 22.8030 0.817002
\(780\) 0 0
\(781\) 38.0590 1.36186
\(782\) 0 0
\(783\) −45.3747 −1.62156
\(784\) 0 0
\(785\) −13.3135 −0.475179
\(786\) 0 0
\(787\) 12.4714 0.444557 0.222279 0.974983i \(-0.428651\pi\)
0.222279 + 0.974983i \(0.428651\pi\)
\(788\) 0 0
\(789\) −35.3543 −1.25865
\(790\) 0 0
\(791\) −3.99687 −0.142112
\(792\) 0 0
\(793\) −1.34242 −0.0476707
\(794\) 0 0
\(795\) −14.2662 −0.505969
\(796\) 0 0
\(797\) −14.5834 −0.516570 −0.258285 0.966069i \(-0.583157\pi\)
−0.258285 + 0.966069i \(0.583157\pi\)
\(798\) 0 0
\(799\) 62.0346 2.19463
\(800\) 0 0
\(801\) 5.19669 0.183616
\(802\) 0 0
\(803\) 36.4873 1.28761
\(804\) 0 0
\(805\) 8.19478 0.288828
\(806\) 0 0
\(807\) 23.4675 0.826095
\(808\) 0 0
\(809\) −36.7020 −1.29037 −0.645186 0.764025i \(-0.723220\pi\)
−0.645186 + 0.764025i \(0.723220\pi\)
\(810\) 0 0
\(811\) −8.80062 −0.309032 −0.154516 0.987990i \(-0.549382\pi\)
−0.154516 + 0.987990i \(0.549382\pi\)
\(812\) 0 0
\(813\) −34.1321 −1.19706
\(814\) 0 0
\(815\) −7.37659 −0.258391
\(816\) 0 0
\(817\) 3.55858 0.124499
\(818\) 0 0
\(819\) 8.16590 0.285340
\(820\) 0 0
\(821\) −14.9979 −0.523432 −0.261716 0.965145i \(-0.584288\pi\)
−0.261716 + 0.965145i \(0.584288\pi\)
\(822\) 0 0
\(823\) 8.68961 0.302901 0.151450 0.988465i \(-0.451606\pi\)
0.151450 + 0.988465i \(0.451606\pi\)
\(824\) 0 0
\(825\) −15.0349 −0.523449
\(826\) 0 0
\(827\) 12.9115 0.448976 0.224488 0.974477i \(-0.427929\pi\)
0.224488 + 0.974477i \(0.427929\pi\)
\(828\) 0 0
\(829\) −51.2578 −1.78026 −0.890128 0.455711i \(-0.849385\pi\)
−0.890128 + 0.455711i \(0.849385\pi\)
\(830\) 0 0
\(831\) −0.335947 −0.0116539
\(832\) 0 0
\(833\) −31.9469 −1.10690
\(834\) 0 0
\(835\) 9.01443 0.311957
\(836\) 0 0
\(837\) −7.52774 −0.260197
\(838\) 0 0
\(839\) 8.57114 0.295909 0.147954 0.988994i \(-0.452731\pi\)
0.147954 + 0.988994i \(0.452731\pi\)
\(840\) 0 0
\(841\) 36.2412 1.24970
\(842\) 0 0
\(843\) −31.7943 −1.09505
\(844\) 0 0
\(845\) −34.6551 −1.19217
\(846\) 0 0
\(847\) 0.624944 0.0214733
\(848\) 0 0
\(849\) −40.7453 −1.39837
\(850\) 0 0
\(851\) 48.4349 1.66033
\(852\) 0 0
\(853\) −45.0161 −1.54132 −0.770661 0.637245i \(-0.780074\pi\)
−0.770661 + 0.637245i \(0.780074\pi\)
\(854\) 0 0
\(855\) 3.72430 0.127368
\(856\) 0 0
\(857\) −44.6140 −1.52398 −0.761992 0.647587i \(-0.775778\pi\)
−0.761992 + 0.647587i \(0.775778\pi\)
\(858\) 0 0
\(859\) −37.0606 −1.26449 −0.632245 0.774768i \(-0.717866\pi\)
−0.632245 + 0.774768i \(0.717866\pi\)
\(860\) 0 0
\(861\) −13.3752 −0.455826
\(862\) 0 0
\(863\) 15.7527 0.536229 0.268114 0.963387i \(-0.413599\pi\)
0.268114 + 0.963387i \(0.413599\pi\)
\(864\) 0 0
\(865\) −4.83200 −0.164293
\(866\) 0 0
\(867\) −15.4646 −0.525204
\(868\) 0 0
\(869\) −48.2690 −1.63741
\(870\) 0 0
\(871\) 10.2916 0.348719
\(872\) 0 0
\(873\) −17.2170 −0.582707
\(874\) 0 0
\(875\) 10.9436 0.369963
\(876\) 0 0
\(877\) 6.05982 0.204626 0.102313 0.994752i \(-0.467376\pi\)
0.102313 + 0.994752i \(0.467376\pi\)
\(878\) 0 0
\(879\) 31.4097 1.05942
\(880\) 0 0
\(881\) −37.4775 −1.26265 −0.631324 0.775519i \(-0.717488\pi\)
−0.631324 + 0.775519i \(0.717488\pi\)
\(882\) 0 0
\(883\) −9.74089 −0.327807 −0.163903 0.986476i \(-0.552409\pi\)
−0.163903 + 0.986476i \(0.552409\pi\)
\(884\) 0 0
\(885\) 21.8811 0.735526
\(886\) 0 0
\(887\) 27.8151 0.933940 0.466970 0.884273i \(-0.345346\pi\)
0.466970 + 0.884273i \(0.345346\pi\)
\(888\) 0 0
\(889\) 8.94783 0.300100
\(890\) 0 0
\(891\) 12.2711 0.411097
\(892\) 0 0
\(893\) 26.5079 0.887054
\(894\) 0 0
\(895\) 25.3884 0.848641
\(896\) 0 0
\(897\) −52.0671 −1.73847
\(898\) 0 0
\(899\) 10.8236 0.360988
\(900\) 0 0
\(901\) −45.2165 −1.50638
\(902\) 0 0
\(903\) −2.08730 −0.0694611
\(904\) 0 0
\(905\) −19.2165 −0.638777
\(906\) 0 0
\(907\) −18.9267 −0.628452 −0.314226 0.949348i \(-0.601745\pi\)
−0.314226 + 0.949348i \(0.601745\pi\)
\(908\) 0 0
\(909\) 0.897214 0.0297587
\(910\) 0 0
\(911\) 7.37167 0.244234 0.122117 0.992516i \(-0.461032\pi\)
0.122117 + 0.992516i \(0.461032\pi\)
\(912\) 0 0
\(913\) 6.64090 0.219782
\(914\) 0 0
\(915\) 0.359118 0.0118721
\(916\) 0 0
\(917\) −3.56077 −0.117587
\(918\) 0 0
\(919\) 29.9676 0.988540 0.494270 0.869308i \(-0.335435\pi\)
0.494270 + 0.869308i \(0.335435\pi\)
\(920\) 0 0
\(921\) −21.3352 −0.703020
\(922\) 0 0
\(923\) 70.6189 2.32445
\(924\) 0 0
\(925\) 25.9514 0.853276
\(926\) 0 0
\(927\) 1.15677 0.0379932
\(928\) 0 0
\(929\) 9.27346 0.304252 0.152126 0.988361i \(-0.451388\pi\)
0.152126 + 0.988361i \(0.451388\pi\)
\(930\) 0 0
\(931\) −13.6512 −0.447400
\(932\) 0 0
\(933\) 8.02922 0.262865
\(934\) 0 0
\(935\) 23.4660 0.767420
\(936\) 0 0
\(937\) 9.16311 0.299346 0.149673 0.988736i \(-0.452178\pi\)
0.149673 + 0.988736i \(0.452178\pi\)
\(938\) 0 0
\(939\) −5.83696 −0.190482
\(940\) 0 0
\(941\) 48.4655 1.57993 0.789965 0.613152i \(-0.210099\pi\)
0.789965 + 0.613152i \(0.210099\pi\)
\(942\) 0 0
\(943\) −62.2384 −2.02676
\(944\) 0 0
\(945\) −7.36235 −0.239497
\(946\) 0 0
\(947\) −16.2316 −0.527455 −0.263727 0.964597i \(-0.584952\pi\)
−0.263727 + 0.964597i \(0.584952\pi\)
\(948\) 0 0
\(949\) 67.7026 2.19772
\(950\) 0 0
\(951\) 10.5412 0.341823
\(952\) 0 0
\(953\) −37.7780 −1.22375 −0.611874 0.790955i \(-0.709584\pi\)
−0.611874 + 0.790955i \(0.709584\pi\)
\(954\) 0 0
\(955\) −13.7052 −0.443490
\(956\) 0 0
\(957\) −36.2483 −1.17174
\(958\) 0 0
\(959\) 3.16821 0.102307
\(960\) 0 0
\(961\) −29.2043 −0.942076
\(962\) 0 0
\(963\) 3.79884 0.122416
\(964\) 0 0
\(965\) −21.0095 −0.676319
\(966\) 0 0
\(967\) −11.6664 −0.375167 −0.187583 0.982249i \(-0.560066\pi\)
−0.187583 + 0.982249i \(0.560066\pi\)
\(968\) 0 0
\(969\) −16.1747 −0.519605
\(970\) 0 0
\(971\) 56.7485 1.82115 0.910574 0.413347i \(-0.135640\pi\)
0.910574 + 0.413347i \(0.135640\pi\)
\(972\) 0 0
\(973\) 3.66795 0.117589
\(974\) 0 0
\(975\) −27.8975 −0.893434
\(976\) 0 0
\(977\) −15.8276 −0.506371 −0.253185 0.967418i \(-0.581478\pi\)
−0.253185 + 0.967418i \(0.581478\pi\)
\(978\) 0 0
\(979\) 13.9915 0.447169
\(980\) 0 0
\(981\) −16.1777 −0.516515
\(982\) 0 0
\(983\) −22.0625 −0.703685 −0.351842 0.936059i \(-0.614445\pi\)
−0.351842 + 0.936059i \(0.614445\pi\)
\(984\) 0 0
\(985\) 7.46163 0.237747
\(986\) 0 0
\(987\) −15.5484 −0.494910
\(988\) 0 0
\(989\) −9.71277 −0.308848
\(990\) 0 0
\(991\) 12.2127 0.387949 0.193974 0.981007i \(-0.437862\pi\)
0.193974 + 0.981007i \(0.437862\pi\)
\(992\) 0 0
\(993\) −40.6618 −1.29036
\(994\) 0 0
\(995\) −7.06258 −0.223899
\(996\) 0 0
\(997\) −36.0107 −1.14047 −0.570236 0.821481i \(-0.693148\pi\)
−0.570236 + 0.821481i \(0.693148\pi\)
\(998\) 0 0
\(999\) −43.5149 −1.37675
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 4016.2.a.l.1.6 19
4.3 odd 2 2008.2.a.c.1.14 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.c.1.14 19 4.3 odd 2
4016.2.a.l.1.6 19 1.1 even 1 trivial