Properties

Label 4368.2.a.bk.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $0$
Dimension $2$
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] = [4368,2,Mod(1,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 4368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -1.56155 q^{15} +5.12311 q^{17} +2.43845 q^{19} -1.00000 q^{21} -4.68466 q^{23} -2.56155 q^{25} +1.00000 q^{27} -3.56155 q^{29} -1.56155 q^{31} +2.00000 q^{33} +1.56155 q^{35} +1.12311 q^{37} -1.00000 q^{39} +7.12311 q^{41} +9.56155 q^{43} -1.56155 q^{45} +6.68466 q^{47} +1.00000 q^{49} +5.12311 q^{51} +0.438447 q^{53} -3.12311 q^{55} +2.43845 q^{57} -5.12311 q^{59} -6.00000 q^{61} -1.00000 q^{63} +1.56155 q^{65} +13.3693 q^{67} -4.68466 q^{69} +2.87689 q^{71} -5.80776 q^{73} -2.56155 q^{75} -2.00000 q^{77} +11.8078 q^{79} +1.00000 q^{81} -9.80776 q^{83} -8.00000 q^{85} -3.56155 q^{87} +5.56155 q^{89} +1.00000 q^{91} -1.56155 q^{93} -3.80776 q^{95} -7.56155 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 9 q^{19} - 2 q^{21} + 3 q^{23} - q^{25} + 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} - 6 q^{37} - 2 q^{39} + 6 q^{41} + 15 q^{43} + q^{45} + q^{47} + 2 q^{49} + 2 q^{51} + 5 q^{53} + 2 q^{55} + 9 q^{57} - 2 q^{59} - 12 q^{61} - 2 q^{63} - q^{65} + 2 q^{67} + 3 q^{69} + 14 q^{71} + 9 q^{73} - q^{75} - 4 q^{77} + 3 q^{79} + 2 q^{81} + q^{83} - 16 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} + q^{93} + 13 q^{95} - 11 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.56155 −0.698348 −0.349174 0.937058i \(-0.613538\pi\)
−0.349174 + 0.937058i \(0.613538\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) 5.12311 1.24254 0.621268 0.783598i \(-0.286618\pi\)
0.621268 + 0.783598i \(0.286618\pi\)
\(18\) 0 0
\(19\) 2.43845 0.559418 0.279709 0.960085i \(-0.409762\pi\)
0.279709 + 0.960085i \(0.409762\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.68466 −0.976819 −0.488409 0.872615i \(-0.662423\pi\)
−0.488409 + 0.872615i \(0.662423\pi\)
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.56155 −0.661364 −0.330682 0.943742i \(-0.607279\pi\)
−0.330682 + 0.943742i \(0.607279\pi\)
\(30\) 0 0
\(31\) −1.56155 −0.280463 −0.140232 0.990119i \(-0.544785\pi\)
−0.140232 + 0.990119i \(0.544785\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 1.56155 0.263951
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) 0 0
\(43\) 9.56155 1.45812 0.729062 0.684448i \(-0.239957\pi\)
0.729062 + 0.684448i \(0.239957\pi\)
\(44\) 0 0
\(45\) −1.56155 −0.232783
\(46\) 0 0
\(47\) 6.68466 0.975058 0.487529 0.873107i \(-0.337898\pi\)
0.487529 + 0.873107i \(0.337898\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.12311 0.717378
\(52\) 0 0
\(53\) 0.438447 0.0602254 0.0301127 0.999547i \(-0.490413\pi\)
0.0301127 + 0.999547i \(0.490413\pi\)
\(54\) 0 0
\(55\) −3.12311 −0.421119
\(56\) 0 0
\(57\) 2.43845 0.322980
\(58\) 0 0
\(59\) −5.12311 −0.666972 −0.333486 0.942755i \(-0.608225\pi\)
−0.333486 + 0.942755i \(0.608225\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 1.56155 0.193687
\(66\) 0 0
\(67\) 13.3693 1.63332 0.816661 0.577118i \(-0.195823\pi\)
0.816661 + 0.577118i \(0.195823\pi\)
\(68\) 0 0
\(69\) −4.68466 −0.563967
\(70\) 0 0
\(71\) 2.87689 0.341425 0.170712 0.985321i \(-0.445393\pi\)
0.170712 + 0.985321i \(0.445393\pi\)
\(72\) 0 0
\(73\) −5.80776 −0.679747 −0.339874 0.940471i \(-0.610384\pi\)
−0.339874 + 0.940471i \(0.610384\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 11.8078 1.32848 0.664239 0.747521i \(-0.268756\pi\)
0.664239 + 0.747521i \(0.268756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.80776 −1.07654 −0.538271 0.842772i \(-0.680922\pi\)
−0.538271 + 0.842772i \(0.680922\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) −3.56155 −0.381839
\(88\) 0 0
\(89\) 5.56155 0.589523 0.294762 0.955571i \(-0.404760\pi\)
0.294762 + 0.955571i \(0.404760\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −1.56155 −0.161925
\(94\) 0 0
\(95\) −3.80776 −0.390668
\(96\) 0 0
\(97\) −7.56155 −0.767759 −0.383880 0.923383i \(-0.625412\pi\)
−0.383880 + 0.923383i \(0.625412\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −4.24621 −0.422514 −0.211257 0.977431i \(-0.567756\pi\)
−0.211257 + 0.977431i \(0.567756\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 1.56155 0.152392
\(106\) 0 0
\(107\) 14.2462 1.37723 0.688617 0.725126i \(-0.258218\pi\)
0.688617 + 0.725126i \(0.258218\pi\)
\(108\) 0 0
\(109\) 16.2462 1.55610 0.778052 0.628199i \(-0.216208\pi\)
0.778052 + 0.628199i \(0.216208\pi\)
\(110\) 0 0
\(111\) 1.12311 0.106600
\(112\) 0 0
\(113\) −10.6847 −1.00513 −0.502564 0.864540i \(-0.667610\pi\)
−0.502564 + 0.864540i \(0.667610\pi\)
\(114\) 0 0
\(115\) 7.31534 0.682159
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −5.12311 −0.469634
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 7.12311 0.642269
\(124\) 0 0
\(125\) 11.8078 1.05612
\(126\) 0 0
\(127\) 1.75379 0.155624 0.0778118 0.996968i \(-0.475207\pi\)
0.0778118 + 0.996968i \(0.475207\pi\)
\(128\) 0 0
\(129\) 9.56155 0.841848
\(130\) 0 0
\(131\) 20.4924 1.79043 0.895216 0.445633i \(-0.147021\pi\)
0.895216 + 0.445633i \(0.147021\pi\)
\(132\) 0 0
\(133\) −2.43845 −0.211440
\(134\) 0 0
\(135\) −1.56155 −0.134397
\(136\) 0 0
\(137\) −2.24621 −0.191907 −0.0959534 0.995386i \(-0.530590\pi\)
−0.0959534 + 0.995386i \(0.530590\pi\)
\(138\) 0 0
\(139\) 0.876894 0.0743772 0.0371886 0.999308i \(-0.488160\pi\)
0.0371886 + 0.999308i \(0.488160\pi\)
\(140\) 0 0
\(141\) 6.68466 0.562950
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 5.56155 0.461862
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −4.87689 −0.399531 −0.199765 0.979844i \(-0.564018\pi\)
−0.199765 + 0.979844i \(0.564018\pi\)
\(150\) 0 0
\(151\) 10.2462 0.833825 0.416912 0.908947i \(-0.363112\pi\)
0.416912 + 0.908947i \(0.363112\pi\)
\(152\) 0 0
\(153\) 5.12311 0.414179
\(154\) 0 0
\(155\) 2.43845 0.195861
\(156\) 0 0
\(157\) 15.3693 1.22661 0.613303 0.789848i \(-0.289841\pi\)
0.613303 + 0.789848i \(0.289841\pi\)
\(158\) 0 0
\(159\) 0.438447 0.0347711
\(160\) 0 0
\(161\) 4.68466 0.369203
\(162\) 0 0
\(163\) 22.2462 1.74246 0.871229 0.490877i \(-0.163324\pi\)
0.871229 + 0.490877i \(0.163324\pi\)
\(164\) 0 0
\(165\) −3.12311 −0.243133
\(166\) 0 0
\(167\) 9.80776 0.758948 0.379474 0.925202i \(-0.376105\pi\)
0.379474 + 0.925202i \(0.376105\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.43845 0.186473
\(172\) 0 0
\(173\) −7.36932 −0.560279 −0.280139 0.959959i \(-0.590381\pi\)
−0.280139 + 0.959959i \(0.590381\pi\)
\(174\) 0 0
\(175\) 2.56155 0.193635
\(176\) 0 0
\(177\) −5.12311 −0.385076
\(178\) 0 0
\(179\) 7.80776 0.583580 0.291790 0.956482i \(-0.405749\pi\)
0.291790 + 0.956482i \(0.405749\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −1.75379 −0.128941
\(186\) 0 0
\(187\) 10.2462 0.749277
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 1.56155 0.111825
\(196\) 0 0
\(197\) −8.87689 −0.632453 −0.316226 0.948684i \(-0.602416\pi\)
−0.316226 + 0.948684i \(0.602416\pi\)
\(198\) 0 0
\(199\) 20.4924 1.45267 0.726335 0.687341i \(-0.241222\pi\)
0.726335 + 0.687341i \(0.241222\pi\)
\(200\) 0 0
\(201\) 13.3693 0.942999
\(202\) 0 0
\(203\) 3.56155 0.249972
\(204\) 0 0
\(205\) −11.1231 −0.776871
\(206\) 0 0
\(207\) −4.68466 −0.325606
\(208\) 0 0
\(209\) 4.87689 0.337342
\(210\) 0 0
\(211\) −17.1771 −1.18252 −0.591260 0.806481i \(-0.701369\pi\)
−0.591260 + 0.806481i \(0.701369\pi\)
\(212\) 0 0
\(213\) 2.87689 0.197122
\(214\) 0 0
\(215\) −14.9309 −1.01828
\(216\) 0 0
\(217\) 1.56155 0.106005
\(218\) 0 0
\(219\) −5.80776 −0.392452
\(220\) 0 0
\(221\) −5.12311 −0.344617
\(222\) 0 0
\(223\) 3.31534 0.222012 0.111006 0.993820i \(-0.464593\pi\)
0.111006 + 0.993820i \(0.464593\pi\)
\(224\) 0 0
\(225\) −2.56155 −0.170770
\(226\) 0 0
\(227\) −19.3693 −1.28559 −0.642793 0.766040i \(-0.722225\pi\)
−0.642793 + 0.766040i \(0.722225\pi\)
\(228\) 0 0
\(229\) 28.7386 1.89910 0.949551 0.313612i \(-0.101539\pi\)
0.949551 + 0.313612i \(0.101539\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) −21.8078 −1.42867 −0.714337 0.699802i \(-0.753272\pi\)
−0.714337 + 0.699802i \(0.753272\pi\)
\(234\) 0 0
\(235\) −10.4384 −0.680929
\(236\) 0 0
\(237\) 11.8078 0.766997
\(238\) 0 0
\(239\) −13.1231 −0.848863 −0.424432 0.905460i \(-0.639526\pi\)
−0.424432 + 0.905460i \(0.639526\pi\)
\(240\) 0 0
\(241\) 12.9309 0.832951 0.416475 0.909147i \(-0.363265\pi\)
0.416475 + 0.909147i \(0.363265\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.56155 −0.0997639
\(246\) 0 0
\(247\) −2.43845 −0.155155
\(248\) 0 0
\(249\) −9.80776 −0.621542
\(250\) 0 0
\(251\) 9.36932 0.591386 0.295693 0.955283i \(-0.404449\pi\)
0.295693 + 0.955283i \(0.404449\pi\)
\(252\) 0 0
\(253\) −9.36932 −0.589044
\(254\) 0 0
\(255\) −8.00000 −0.500979
\(256\) 0 0
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −1.12311 −0.0697864
\(260\) 0 0
\(261\) −3.56155 −0.220455
\(262\) 0 0
\(263\) −17.5616 −1.08289 −0.541446 0.840736i \(-0.682123\pi\)
−0.541446 + 0.840736i \(0.682123\pi\)
\(264\) 0 0
\(265\) −0.684658 −0.0420582
\(266\) 0 0
\(267\) 5.56155 0.340362
\(268\) 0 0
\(269\) −9.12311 −0.556246 −0.278123 0.960546i \(-0.589712\pi\)
−0.278123 + 0.960546i \(0.589712\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −5.12311 −0.308935
\(276\) 0 0
\(277\) 11.5616 0.694666 0.347333 0.937742i \(-0.387087\pi\)
0.347333 + 0.937742i \(0.387087\pi\)
\(278\) 0 0
\(279\) −1.56155 −0.0934877
\(280\) 0 0
\(281\) −7.12311 −0.424929 −0.212464 0.977169i \(-0.568149\pi\)
−0.212464 + 0.977169i \(0.568149\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) −3.80776 −0.225552
\(286\) 0 0
\(287\) −7.12311 −0.420464
\(288\) 0 0
\(289\) 9.24621 0.543895
\(290\) 0 0
\(291\) −7.56155 −0.443266
\(292\) 0 0
\(293\) 32.6847 1.90946 0.954729 0.297477i \(-0.0961451\pi\)
0.954729 + 0.297477i \(0.0961451\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 4.68466 0.270921
\(300\) 0 0
\(301\) −9.56155 −0.551119
\(302\) 0 0
\(303\) −4.24621 −0.243938
\(304\) 0 0
\(305\) 9.36932 0.536486
\(306\) 0 0
\(307\) 14.0540 0.802103 0.401051 0.916056i \(-0.368645\pi\)
0.401051 + 0.916056i \(0.368645\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.8769 −1.41064 −0.705320 0.708889i \(-0.749197\pi\)
−0.705320 + 0.708889i \(0.749197\pi\)
\(312\) 0 0
\(313\) −0.246211 −0.0139167 −0.00695834 0.999976i \(-0.502215\pi\)
−0.00695834 + 0.999976i \(0.502215\pi\)
\(314\) 0 0
\(315\) 1.56155 0.0879835
\(316\) 0 0
\(317\) 16.4924 0.926307 0.463153 0.886278i \(-0.346718\pi\)
0.463153 + 0.886278i \(0.346718\pi\)
\(318\) 0 0
\(319\) −7.12311 −0.398817
\(320\) 0 0
\(321\) 14.2462 0.795146
\(322\) 0 0
\(323\) 12.4924 0.695097
\(324\) 0 0
\(325\) 2.56155 0.142089
\(326\) 0 0
\(327\) 16.2462 0.898418
\(328\) 0 0
\(329\) −6.68466 −0.368537
\(330\) 0 0
\(331\) 2.63068 0.144595 0.0722977 0.997383i \(-0.476967\pi\)
0.0722977 + 0.997383i \(0.476967\pi\)
\(332\) 0 0
\(333\) 1.12311 0.0615458
\(334\) 0 0
\(335\) −20.8769 −1.14063
\(336\) 0 0
\(337\) −10.6847 −0.582030 −0.291015 0.956718i \(-0.593993\pi\)
−0.291015 + 0.956718i \(0.593993\pi\)
\(338\) 0 0
\(339\) −10.6847 −0.580311
\(340\) 0 0
\(341\) −3.12311 −0.169126
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 7.31534 0.393845
\(346\) 0 0
\(347\) −30.7386 −1.65014 −0.825068 0.565033i \(-0.808863\pi\)
−0.825068 + 0.565033i \(0.808863\pi\)
\(348\) 0 0
\(349\) 8.43845 0.451700 0.225850 0.974162i \(-0.427484\pi\)
0.225850 + 0.974162i \(0.427484\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −2.63068 −0.140017 −0.0700086 0.997546i \(-0.522303\pi\)
−0.0700086 + 0.997546i \(0.522303\pi\)
\(354\) 0 0
\(355\) −4.49242 −0.238433
\(356\) 0 0
\(357\) −5.12311 −0.271144
\(358\) 0 0
\(359\) −0.630683 −0.0332862 −0.0166431 0.999861i \(-0.505298\pi\)
−0.0166431 + 0.999861i \(0.505298\pi\)
\(360\) 0 0
\(361\) −13.0540 −0.687051
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 9.06913 0.474700
\(366\) 0 0
\(367\) −19.1231 −0.998218 −0.499109 0.866539i \(-0.666339\pi\)
−0.499109 + 0.866539i \(0.666339\pi\)
\(368\) 0 0
\(369\) 7.12311 0.370814
\(370\) 0 0
\(371\) −0.438447 −0.0227630
\(372\) 0 0
\(373\) −15.7538 −0.815700 −0.407850 0.913049i \(-0.633721\pi\)
−0.407850 + 0.913049i \(0.633721\pi\)
\(374\) 0 0
\(375\) 11.8078 0.609750
\(376\) 0 0
\(377\) 3.56155 0.183429
\(378\) 0 0
\(379\) 31.6155 1.62398 0.811990 0.583671i \(-0.198384\pi\)
0.811990 + 0.583671i \(0.198384\pi\)
\(380\) 0 0
\(381\) 1.75379 0.0898493
\(382\) 0 0
\(383\) 11.3693 0.580945 0.290472 0.956883i \(-0.406188\pi\)
0.290472 + 0.956883i \(0.406188\pi\)
\(384\) 0 0
\(385\) 3.12311 0.159168
\(386\) 0 0
\(387\) 9.56155 0.486041
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) 20.4924 1.03371
\(394\) 0 0
\(395\) −18.4384 −0.927739
\(396\) 0 0
\(397\) −4.05398 −0.203463 −0.101732 0.994812i \(-0.532438\pi\)
−0.101732 + 0.994812i \(0.532438\pi\)
\(398\) 0 0
\(399\) −2.43845 −0.122075
\(400\) 0 0
\(401\) −22.2462 −1.11092 −0.555461 0.831542i \(-0.687458\pi\)
−0.555461 + 0.831542i \(0.687458\pi\)
\(402\) 0 0
\(403\) 1.56155 0.0777865
\(404\) 0 0
\(405\) −1.56155 −0.0775942
\(406\) 0 0
\(407\) 2.24621 0.111341
\(408\) 0 0
\(409\) −13.8078 −0.682750 −0.341375 0.939927i \(-0.610893\pi\)
−0.341375 + 0.939927i \(0.610893\pi\)
\(410\) 0 0
\(411\) −2.24621 −0.110797
\(412\) 0 0
\(413\) 5.12311 0.252092
\(414\) 0 0
\(415\) 15.3153 0.751801
\(416\) 0 0
\(417\) 0.876894 0.0429417
\(418\) 0 0
\(419\) −12.8769 −0.629077 −0.314539 0.949245i \(-0.601850\pi\)
−0.314539 + 0.949245i \(0.601850\pi\)
\(420\) 0 0
\(421\) −10.8769 −0.530107 −0.265054 0.964234i \(-0.585390\pi\)
−0.265054 + 0.964234i \(0.585390\pi\)
\(422\) 0 0
\(423\) 6.68466 0.325019
\(424\) 0 0
\(425\) −13.1231 −0.636564
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 8.24621 0.397206 0.198603 0.980080i \(-0.436360\pi\)
0.198603 + 0.980080i \(0.436360\pi\)
\(432\) 0 0
\(433\) 31.3693 1.50751 0.753757 0.657154i \(-0.228240\pi\)
0.753757 + 0.657154i \(0.228240\pi\)
\(434\) 0 0
\(435\) 5.56155 0.266656
\(436\) 0 0
\(437\) −11.4233 −0.546450
\(438\) 0 0
\(439\) −23.6155 −1.12711 −0.563554 0.826079i \(-0.690566\pi\)
−0.563554 + 0.826079i \(0.690566\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −21.1771 −1.00615 −0.503077 0.864242i \(-0.667799\pi\)
−0.503077 + 0.864242i \(0.667799\pi\)
\(444\) 0 0
\(445\) −8.68466 −0.411692
\(446\) 0 0
\(447\) −4.87689 −0.230669
\(448\) 0 0
\(449\) −1.75379 −0.0827664 −0.0413832 0.999143i \(-0.513176\pi\)
−0.0413832 + 0.999143i \(0.513176\pi\)
\(450\) 0 0
\(451\) 14.2462 0.670828
\(452\) 0 0
\(453\) 10.2462 0.481409
\(454\) 0 0
\(455\) −1.56155 −0.0732067
\(456\) 0 0
\(457\) −5.12311 −0.239649 −0.119824 0.992795i \(-0.538233\pi\)
−0.119824 + 0.992795i \(0.538233\pi\)
\(458\) 0 0
\(459\) 5.12311 0.239126
\(460\) 0 0
\(461\) 2.63068 0.122523 0.0612616 0.998122i \(-0.480488\pi\)
0.0612616 + 0.998122i \(0.480488\pi\)
\(462\) 0 0
\(463\) −21.8617 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(464\) 0 0
\(465\) 2.43845 0.113080
\(466\) 0 0
\(467\) 17.3693 0.803756 0.401878 0.915693i \(-0.368358\pi\)
0.401878 + 0.915693i \(0.368358\pi\)
\(468\) 0 0
\(469\) −13.3693 −0.617338
\(470\) 0 0
\(471\) 15.3693 0.708181
\(472\) 0 0
\(473\) 19.1231 0.879281
\(474\) 0 0
\(475\) −6.24621 −0.286596
\(476\) 0 0
\(477\) 0.438447 0.0200751
\(478\) 0 0
\(479\) −8.43845 −0.385562 −0.192781 0.981242i \(-0.561751\pi\)
−0.192781 + 0.981242i \(0.561751\pi\)
\(480\) 0 0
\(481\) −1.12311 −0.0512092
\(482\) 0 0
\(483\) 4.68466 0.213159
\(484\) 0 0
\(485\) 11.8078 0.536163
\(486\) 0 0
\(487\) −33.8617 −1.53442 −0.767211 0.641395i \(-0.778356\pi\)
−0.767211 + 0.641395i \(0.778356\pi\)
\(488\) 0 0
\(489\) 22.2462 1.00601
\(490\) 0 0
\(491\) −1.75379 −0.0791474 −0.0395737 0.999217i \(-0.512600\pi\)
−0.0395737 + 0.999217i \(0.512600\pi\)
\(492\) 0 0
\(493\) −18.2462 −0.821768
\(494\) 0 0
\(495\) −3.12311 −0.140373
\(496\) 0 0
\(497\) −2.87689 −0.129046
\(498\) 0 0
\(499\) −21.7538 −0.973833 −0.486917 0.873448i \(-0.661878\pi\)
−0.486917 + 0.873448i \(0.661878\pi\)
\(500\) 0 0
\(501\) 9.80776 0.438179
\(502\) 0 0
\(503\) −2.63068 −0.117296 −0.0586482 0.998279i \(-0.518679\pi\)
−0.0586482 + 0.998279i \(0.518679\pi\)
\(504\) 0 0
\(505\) 6.63068 0.295062
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −13.0691 −0.579279 −0.289640 0.957136i \(-0.593535\pi\)
−0.289640 + 0.957136i \(0.593535\pi\)
\(510\) 0 0
\(511\) 5.80776 0.256920
\(512\) 0 0
\(513\) 2.43845 0.107660
\(514\) 0 0
\(515\) −12.4924 −0.550482
\(516\) 0 0
\(517\) 13.3693 0.587982
\(518\) 0 0
\(519\) −7.36932 −0.323477
\(520\) 0 0
\(521\) −6.87689 −0.301282 −0.150641 0.988589i \(-0.548134\pi\)
−0.150641 + 0.988589i \(0.548134\pi\)
\(522\) 0 0
\(523\) 11.6155 0.507912 0.253956 0.967216i \(-0.418268\pi\)
0.253956 + 0.967216i \(0.418268\pi\)
\(524\) 0 0
\(525\) 2.56155 0.111795
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −1.05398 −0.0458250
\(530\) 0 0
\(531\) −5.12311 −0.222324
\(532\) 0 0
\(533\) −7.12311 −0.308536
\(534\) 0 0
\(535\) −22.2462 −0.961788
\(536\) 0 0
\(537\) 7.80776 0.336930
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −13.6155 −0.585377 −0.292689 0.956208i \(-0.594550\pi\)
−0.292689 + 0.956208i \(0.594550\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −25.3693 −1.08670
\(546\) 0 0
\(547\) −22.0540 −0.942960 −0.471480 0.881877i \(-0.656280\pi\)
−0.471480 + 0.881877i \(0.656280\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −8.68466 −0.369979
\(552\) 0 0
\(553\) −11.8078 −0.502117
\(554\) 0 0
\(555\) −1.75379 −0.0744442
\(556\) 0 0
\(557\) 20.4924 0.868292 0.434146 0.900843i \(-0.357050\pi\)
0.434146 + 0.900843i \(0.357050\pi\)
\(558\) 0 0
\(559\) −9.56155 −0.404411
\(560\) 0 0
\(561\) 10.2462 0.432595
\(562\) 0 0
\(563\) −22.7386 −0.958319 −0.479160 0.877728i \(-0.659058\pi\)
−0.479160 + 0.877728i \(0.659058\pi\)
\(564\) 0 0
\(565\) 16.6847 0.701929
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −34.3002 −1.43794 −0.718969 0.695042i \(-0.755386\pi\)
−0.718969 + 0.695042i \(0.755386\pi\)
\(570\) 0 0
\(571\) 5.06913 0.212137 0.106068 0.994359i \(-0.466174\pi\)
0.106068 + 0.994359i \(0.466174\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 9.80776 0.406895
\(582\) 0 0
\(583\) 0.876894 0.0363173
\(584\) 0 0
\(585\) 1.56155 0.0645623
\(586\) 0 0
\(587\) −1.80776 −0.0746144 −0.0373072 0.999304i \(-0.511878\pi\)
−0.0373072 + 0.999304i \(0.511878\pi\)
\(588\) 0 0
\(589\) −3.80776 −0.156896
\(590\) 0 0
\(591\) −8.87689 −0.365147
\(592\) 0 0
\(593\) 18.4384 0.757176 0.378588 0.925565i \(-0.376410\pi\)
0.378588 + 0.925565i \(0.376410\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) 0 0
\(597\) 20.4924 0.838699
\(598\) 0 0
\(599\) −36.6847 −1.49889 −0.749447 0.662064i \(-0.769681\pi\)
−0.749447 + 0.662064i \(0.769681\pi\)
\(600\) 0 0
\(601\) −3.75379 −0.153120 −0.0765601 0.997065i \(-0.524394\pi\)
−0.0765601 + 0.997065i \(0.524394\pi\)
\(602\) 0 0
\(603\) 13.3693 0.544441
\(604\) 0 0
\(605\) 10.9309 0.444403
\(606\) 0 0
\(607\) 15.6155 0.633815 0.316907 0.948456i \(-0.397356\pi\)
0.316907 + 0.948456i \(0.397356\pi\)
\(608\) 0 0
\(609\) 3.56155 0.144321
\(610\) 0 0
\(611\) −6.68466 −0.270432
\(612\) 0 0
\(613\) 15.3693 0.620761 0.310380 0.950612i \(-0.399544\pi\)
0.310380 + 0.950612i \(0.399544\pi\)
\(614\) 0 0
\(615\) −11.1231 −0.448527
\(616\) 0 0
\(617\) 29.3693 1.18236 0.591182 0.806538i \(-0.298661\pi\)
0.591182 + 0.806538i \(0.298661\pi\)
\(618\) 0 0
\(619\) 36.9848 1.48655 0.743273 0.668988i \(-0.233272\pi\)
0.743273 + 0.668988i \(0.233272\pi\)
\(620\) 0 0
\(621\) −4.68466 −0.187989
\(622\) 0 0
\(623\) −5.56155 −0.222819
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) 4.87689 0.194764
\(628\) 0 0
\(629\) 5.75379 0.229419
\(630\) 0 0
\(631\) 23.1231 0.920516 0.460258 0.887785i \(-0.347757\pi\)
0.460258 + 0.887785i \(0.347757\pi\)
\(632\) 0 0
\(633\) −17.1771 −0.682728
\(634\) 0 0
\(635\) −2.73863 −0.108679
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 2.87689 0.113808
\(640\) 0 0
\(641\) −10.1922 −0.402569 −0.201285 0.979533i \(-0.564512\pi\)
−0.201285 + 0.979533i \(0.564512\pi\)
\(642\) 0 0
\(643\) 5.75379 0.226907 0.113454 0.993543i \(-0.463809\pi\)
0.113454 + 0.993543i \(0.463809\pi\)
\(644\) 0 0
\(645\) −14.9309 −0.587902
\(646\) 0 0
\(647\) −33.3693 −1.31188 −0.655942 0.754812i \(-0.727728\pi\)
−0.655942 + 0.754812i \(0.727728\pi\)
\(648\) 0 0
\(649\) −10.2462 −0.402199
\(650\) 0 0
\(651\) 1.56155 0.0612021
\(652\) 0 0
\(653\) 22.4924 0.880197 0.440098 0.897950i \(-0.354944\pi\)
0.440098 + 0.897950i \(0.354944\pi\)
\(654\) 0 0
\(655\) −32.0000 −1.25034
\(656\) 0 0
\(657\) −5.80776 −0.226582
\(658\) 0 0
\(659\) 16.1922 0.630760 0.315380 0.948965i \(-0.397868\pi\)
0.315380 + 0.948965i \(0.397868\pi\)
\(660\) 0 0
\(661\) −33.3153 −1.29582 −0.647908 0.761718i \(-0.724356\pi\)
−0.647908 + 0.761718i \(0.724356\pi\)
\(662\) 0 0
\(663\) −5.12311 −0.198965
\(664\) 0 0
\(665\) 3.80776 0.147659
\(666\) 0 0
\(667\) 16.6847 0.646033
\(668\) 0 0
\(669\) 3.31534 0.128179
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −0.438447 −0.0169009 −0.00845045 0.999964i \(-0.502690\pi\)
−0.00845045 + 0.999964i \(0.502690\pi\)
\(674\) 0 0
\(675\) −2.56155 −0.0985942
\(676\) 0 0
\(677\) 10.4924 0.403257 0.201628 0.979462i \(-0.435377\pi\)
0.201628 + 0.979462i \(0.435377\pi\)
\(678\) 0 0
\(679\) 7.56155 0.290186
\(680\) 0 0
\(681\) −19.3693 −0.742234
\(682\) 0 0
\(683\) 25.6155 0.980151 0.490075 0.871680i \(-0.336969\pi\)
0.490075 + 0.871680i \(0.336969\pi\)
\(684\) 0 0
\(685\) 3.50758 0.134018
\(686\) 0 0
\(687\) 28.7386 1.09645
\(688\) 0 0
\(689\) −0.438447 −0.0167035
\(690\) 0 0
\(691\) 27.3153 1.03912 0.519562 0.854433i \(-0.326095\pi\)
0.519562 + 0.854433i \(0.326095\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 0 0
\(695\) −1.36932 −0.0519411
\(696\) 0 0
\(697\) 36.4924 1.38225
\(698\) 0 0
\(699\) −21.8078 −0.824845
\(700\) 0 0
\(701\) −28.4384 −1.07411 −0.537053 0.843549i \(-0.680462\pi\)
−0.537053 + 0.843549i \(0.680462\pi\)
\(702\) 0 0
\(703\) 2.73863 0.103290
\(704\) 0 0
\(705\) −10.4384 −0.393135
\(706\) 0 0
\(707\) 4.24621 0.159695
\(708\) 0 0
\(709\) 20.7386 0.778856 0.389428 0.921057i \(-0.372673\pi\)
0.389428 + 0.921057i \(0.372673\pi\)
\(710\) 0 0
\(711\) 11.8078 0.442826
\(712\) 0 0
\(713\) 7.31534 0.273962
\(714\) 0 0
\(715\) 3.12311 0.116798
\(716\) 0 0
\(717\) −13.1231 −0.490091
\(718\) 0 0
\(719\) 43.6155 1.62658 0.813292 0.581855i \(-0.197673\pi\)
0.813292 + 0.581855i \(0.197673\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 12.9309 0.480904
\(724\) 0 0
\(725\) 9.12311 0.338824
\(726\) 0 0
\(727\) 3.12311 0.115830 0.0579148 0.998322i \(-0.481555\pi\)
0.0579148 + 0.998322i \(0.481555\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.9848 1.81177
\(732\) 0 0
\(733\) 39.1771 1.44704 0.723519 0.690304i \(-0.242523\pi\)
0.723519 + 0.690304i \(0.242523\pi\)
\(734\) 0 0
\(735\) −1.56155 −0.0575987
\(736\) 0 0
\(737\) 26.7386 0.984930
\(738\) 0 0
\(739\) −38.3542 −1.41088 −0.705440 0.708769i \(-0.749251\pi\)
−0.705440 + 0.708769i \(0.749251\pi\)
\(740\) 0 0
\(741\) −2.43845 −0.0895786
\(742\) 0 0
\(743\) 5.61553 0.206014 0.103007 0.994681i \(-0.467154\pi\)
0.103007 + 0.994681i \(0.467154\pi\)
\(744\) 0 0
\(745\) 7.61553 0.279011
\(746\) 0 0
\(747\) −9.80776 −0.358847
\(748\) 0 0
\(749\) −14.2462 −0.520545
\(750\) 0 0
\(751\) 10.0540 0.366875 0.183437 0.983031i \(-0.441278\pi\)
0.183437 + 0.983031i \(0.441278\pi\)
\(752\) 0 0
\(753\) 9.36932 0.341437
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −30.6847 −1.11525 −0.557626 0.830092i \(-0.688288\pi\)
−0.557626 + 0.830092i \(0.688288\pi\)
\(758\) 0 0
\(759\) −9.36932 −0.340085
\(760\) 0 0
\(761\) 10.9309 0.396244 0.198122 0.980177i \(-0.436516\pi\)
0.198122 + 0.980177i \(0.436516\pi\)
\(762\) 0 0
\(763\) −16.2462 −0.588152
\(764\) 0 0
\(765\) −8.00000 −0.289241
\(766\) 0 0
\(767\) 5.12311 0.184985
\(768\) 0 0
\(769\) −14.6847 −0.529542 −0.264771 0.964311i \(-0.585296\pi\)
−0.264771 + 0.964311i \(0.585296\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) 0 0
\(773\) 22.6307 0.813969 0.406985 0.913435i \(-0.366580\pi\)
0.406985 + 0.913435i \(0.366580\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −1.12311 −0.0402912
\(778\) 0 0
\(779\) 17.3693 0.622320
\(780\) 0 0
\(781\) 5.75379 0.205887
\(782\) 0 0
\(783\) −3.56155 −0.127280
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) −3.31534 −0.118179 −0.0590896 0.998253i \(-0.518820\pi\)
−0.0590896 + 0.998253i \(0.518820\pi\)
\(788\) 0 0
\(789\) −17.5616 −0.625208
\(790\) 0 0
\(791\) 10.6847 0.379903
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) −0.684658 −0.0242823
\(796\) 0 0
\(797\) −18.9848 −0.672478 −0.336239 0.941777i \(-0.609155\pi\)
−0.336239 + 0.941777i \(0.609155\pi\)
\(798\) 0 0
\(799\) 34.2462 1.21154
\(800\) 0 0
\(801\) 5.56155 0.196508
\(802\) 0 0
\(803\) −11.6155 −0.409903
\(804\) 0 0
\(805\) −7.31534 −0.257832
\(806\) 0 0
\(807\) −9.12311 −0.321149
\(808\) 0 0
\(809\) −44.4384 −1.56237 −0.781186 0.624298i \(-0.785385\pi\)
−0.781186 + 0.624298i \(0.785385\pi\)
\(810\) 0 0
\(811\) −22.2462 −0.781170 −0.390585 0.920567i \(-0.627727\pi\)
−0.390585 + 0.920567i \(0.627727\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) −34.7386 −1.21684
\(816\) 0 0
\(817\) 23.3153 0.815701
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −51.1231 −1.78421 −0.892104 0.451829i \(-0.850772\pi\)
−0.892104 + 0.451829i \(0.850772\pi\)
\(822\) 0 0
\(823\) −19.5076 −0.679991 −0.339996 0.940427i \(-0.610426\pi\)
−0.339996 + 0.940427i \(0.610426\pi\)
\(824\) 0 0
\(825\) −5.12311 −0.178364
\(826\) 0 0
\(827\) −42.1080 −1.46424 −0.732118 0.681177i \(-0.761468\pi\)
−0.732118 + 0.681177i \(0.761468\pi\)
\(828\) 0 0
\(829\) 14.4924 0.503343 0.251671 0.967813i \(-0.419020\pi\)
0.251671 + 0.967813i \(0.419020\pi\)
\(830\) 0 0
\(831\) 11.5616 0.401066
\(832\) 0 0
\(833\) 5.12311 0.177505
\(834\) 0 0
\(835\) −15.3153 −0.530009
\(836\) 0 0
\(837\) −1.56155 −0.0539752
\(838\) 0 0
\(839\) −10.8769 −0.375512 −0.187756 0.982216i \(-0.560121\pi\)
−0.187756 + 0.982216i \(0.560121\pi\)
\(840\) 0 0
\(841\) −16.3153 −0.562598
\(842\) 0 0
\(843\) −7.12311 −0.245333
\(844\) 0 0
\(845\) −1.56155 −0.0537190
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −5.26137 −0.180357
\(852\) 0 0
\(853\) 41.8078 1.43147 0.715735 0.698372i \(-0.246092\pi\)
0.715735 + 0.698372i \(0.246092\pi\)
\(854\) 0 0
\(855\) −3.80776 −0.130223
\(856\) 0 0
\(857\) −48.2462 −1.64806 −0.824030 0.566547i \(-0.808279\pi\)
−0.824030 + 0.566547i \(0.808279\pi\)
\(858\) 0 0
\(859\) 15.1231 0.515994 0.257997 0.966146i \(-0.416938\pi\)
0.257997 + 0.966146i \(0.416938\pi\)
\(860\) 0 0
\(861\) −7.12311 −0.242755
\(862\) 0 0
\(863\) −53.6155 −1.82509 −0.912547 0.408972i \(-0.865887\pi\)
−0.912547 + 0.408972i \(0.865887\pi\)
\(864\) 0 0
\(865\) 11.5076 0.391269
\(866\) 0 0
\(867\) 9.24621 0.314018
\(868\) 0 0
\(869\) 23.6155 0.801102
\(870\) 0 0
\(871\) −13.3693 −0.453002
\(872\) 0 0
\(873\) −7.56155 −0.255920
\(874\) 0 0
\(875\) −11.8078 −0.399175
\(876\) 0 0
\(877\) −33.6155 −1.13512 −0.567558 0.823334i \(-0.692112\pi\)
−0.567558 + 0.823334i \(0.692112\pi\)
\(878\) 0 0
\(879\) 32.6847 1.10243
\(880\) 0 0
\(881\) 0.738634 0.0248852 0.0124426 0.999923i \(-0.496039\pi\)
0.0124426 + 0.999923i \(0.496039\pi\)
\(882\) 0 0
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 40.9848 1.37614 0.688068 0.725646i \(-0.258459\pi\)
0.688068 + 0.725646i \(0.258459\pi\)
\(888\) 0 0
\(889\) −1.75379 −0.0588202
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 16.3002 0.545465
\(894\) 0 0
\(895\) −12.1922 −0.407542
\(896\) 0 0
\(897\) 4.68466 0.156416
\(898\) 0 0
\(899\) 5.56155 0.185488
\(900\) 0 0
\(901\) 2.24621 0.0748321
\(902\) 0 0
\(903\) −9.56155 −0.318189
\(904\) 0 0
\(905\) 15.6155 0.519078
\(906\) 0 0
\(907\) 14.0540 0.466655 0.233327 0.972398i \(-0.425039\pi\)
0.233327 + 0.972398i \(0.425039\pi\)
\(908\) 0 0
\(909\) −4.24621 −0.140838
\(910\) 0 0
\(911\) −31.3153 −1.03752 −0.518762 0.854919i \(-0.673607\pi\)
−0.518762 + 0.854919i \(0.673607\pi\)
\(912\) 0 0
\(913\) −19.6155 −0.649179
\(914\) 0 0
\(915\) 9.36932 0.309740
\(916\) 0 0
\(917\) −20.4924 −0.676719
\(918\) 0 0
\(919\) −19.5076 −0.643496 −0.321748 0.946825i \(-0.604270\pi\)
−0.321748 + 0.946825i \(0.604270\pi\)
\(920\) 0 0
\(921\) 14.0540 0.463094
\(922\) 0 0
\(923\) −2.87689 −0.0946941
\(924\) 0 0
\(925\) −2.87689 −0.0945917
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −48.7926 −1.60083 −0.800417 0.599444i \(-0.795388\pi\)
−0.800417 + 0.599444i \(0.795388\pi\)
\(930\) 0 0
\(931\) 2.43845 0.0799169
\(932\) 0 0
\(933\) −24.8769 −0.814433
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) −51.3693 −1.67816 −0.839081 0.544006i \(-0.816907\pi\)
−0.839081 + 0.544006i \(0.816907\pi\)
\(938\) 0 0
\(939\) −0.246211 −0.00803480
\(940\) 0 0
\(941\) −6.93087 −0.225940 −0.112970 0.993598i \(-0.536036\pi\)
−0.112970 + 0.993598i \(0.536036\pi\)
\(942\) 0 0
\(943\) −33.3693 −1.08665
\(944\) 0 0
\(945\) 1.56155 0.0507973
\(946\) 0 0
\(947\) 34.4924 1.12085 0.560427 0.828204i \(-0.310637\pi\)
0.560427 + 0.828204i \(0.310637\pi\)
\(948\) 0 0
\(949\) 5.80776 0.188528
\(950\) 0 0
\(951\) 16.4924 0.534803
\(952\) 0 0
\(953\) −12.4384 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(954\) 0 0
\(955\) −12.4924 −0.404245
\(956\) 0 0
\(957\) −7.12311 −0.230257
\(958\) 0 0
\(959\) 2.24621 0.0725339
\(960\) 0 0
\(961\) −28.5616 −0.921340
\(962\) 0 0
\(963\) 14.2462 0.459078
\(964\) 0 0
\(965\) −15.6155 −0.502682
\(966\) 0 0
\(967\) 52.6004 1.69151 0.845757 0.533568i \(-0.179149\pi\)
0.845757 + 0.533568i \(0.179149\pi\)
\(968\) 0 0
\(969\) 12.4924 0.401314
\(970\) 0 0
\(971\) 10.6307 0.341155 0.170577 0.985344i \(-0.445437\pi\)
0.170577 + 0.985344i \(0.445437\pi\)
\(972\) 0 0
\(973\) −0.876894 −0.0281119
\(974\) 0 0
\(975\) 2.56155 0.0820353
\(976\) 0 0
\(977\) −49.3693 −1.57946 −0.789732 0.613452i \(-0.789781\pi\)
−0.789732 + 0.613452i \(0.789781\pi\)
\(978\) 0 0
\(979\) 11.1231 0.355496
\(980\) 0 0
\(981\) 16.2462 0.518702
\(982\) 0 0
\(983\) −18.1922 −0.580242 −0.290121 0.956990i \(-0.593696\pi\)
−0.290121 + 0.956990i \(0.593696\pi\)
\(984\) 0 0
\(985\) 13.8617 0.441672
\(986\) 0 0
\(987\) −6.68466 −0.212775
\(988\) 0 0
\(989\) −44.7926 −1.42432
\(990\) 0 0
\(991\) −38.2462 −1.21493 −0.607465 0.794346i \(-0.707814\pi\)
−0.607465 + 0.794346i \(0.707814\pi\)
\(992\) 0 0
\(993\) 2.63068 0.0834822
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 0 0
\(997\) −33.6155 −1.06461 −0.532307 0.846551i \(-0.678675\pi\)
−0.532307 + 0.846551i \(0.678675\pi\)
\(998\) 0 0
\(999\) 1.12311 0.0355335
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 4368.2.a.bk.1.1 2
4.3 odd 2 2184.2.a.p.1.1 2
12.11 even 2 6552.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.p.1.1 2 4.3 odd 2
4368.2.a.bk.1.1 2 1.1 even 1 trivial
6552.2.a.be.1.2 2 12.11 even 2