Properties

Label 6864.2.a.by.1.2
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
Dimension $4$
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] = [6864,2,Mod(1,6864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, 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("6864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.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: \(54.8093159474\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.22676.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.35449\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.16536 q^{5} +3.64193 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.16536 q^{5} +3.64193 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.16536 q^{15} +5.80729 q^{17} -2.70898 q^{19} +3.64193 q^{21} -9.50785 q^{23} -0.311211 q^{25} +1.00000 q^{27} -10.4095 q^{29} -10.5163 q^{31} +1.00000 q^{33} -7.88610 q^{35} +8.19664 q^{37} -1.00000 q^{39} -5.31121 q^{41} +8.74025 q^{43} -2.16536 q^{45} -5.41795 q^{47} +6.26368 q^{49} +5.80729 q^{51} -2.00000 q^{53} -2.16536 q^{55} -2.70898 q^{57} -6.35091 q^{59} -8.79887 q^{61} +3.64193 q^{63} +2.16536 q^{65} -1.70055 q^{67} -9.50785 q^{69} -2.91277 q^{71} -13.9727 q^{73} -0.311211 q^{75} +3.64193 q^{77} +3.80729 q^{79} +1.00000 q^{81} +16.7018 q^{83} -12.5749 q^{85} -10.4095 q^{87} -11.4290 q^{89} -3.64193 q^{91} -10.5163 q^{93} +5.86591 q^{95} +8.19664 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(5\) −2.16536 −0.968379 −0.484189 0.874963i \(-0.660885\pi\)
−0.484189 + 0.874963i \(0.660885\pi\)
\(6\) 0 0
\(7\) 3.64193 1.37652 0.688261 0.725463i \(-0.258375\pi\)
0.688261 + 0.725463i \(0.258375\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.16536 −0.559094
\(16\) 0 0
\(17\) 5.80729 1.40848 0.704238 0.709964i \(-0.251289\pi\)
0.704238 + 0.709964i \(0.251289\pi\)
\(18\) 0 0
\(19\) −2.70898 −0.621482 −0.310741 0.950495i \(-0.600577\pi\)
−0.310741 + 0.950495i \(0.600577\pi\)
\(20\) 0 0
\(21\) 3.64193 0.794735
\(22\) 0 0
\(23\) −9.50785 −1.98252 −0.991262 0.131911i \(-0.957889\pi\)
−0.991262 + 0.131911i \(0.957889\pi\)
\(24\) 0 0
\(25\) −0.311211 −0.0622422
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.4095 −1.93300 −0.966500 0.256665i \(-0.917376\pi\)
−0.966500 + 0.256665i \(0.917376\pi\)
\(30\) 0 0
\(31\) −10.5163 −1.88878 −0.944389 0.328830i \(-0.893346\pi\)
−0.944389 + 0.328830i \(0.893346\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −7.88610 −1.33299
\(36\) 0 0
\(37\) 8.19664 1.34752 0.673759 0.738951i \(-0.264678\pi\)
0.673759 + 0.738951i \(0.264678\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −5.31121 −0.829472 −0.414736 0.909942i \(-0.636126\pi\)
−0.414736 + 0.909942i \(0.636126\pi\)
\(42\) 0 0
\(43\) 8.74025 1.33288 0.666438 0.745561i \(-0.267818\pi\)
0.666438 + 0.745561i \(0.267818\pi\)
\(44\) 0 0
\(45\) −2.16536 −0.322793
\(46\) 0 0
\(47\) −5.41795 −0.790290 −0.395145 0.918619i \(-0.629306\pi\)
−0.395145 + 0.918619i \(0.629306\pi\)
\(48\) 0 0
\(49\) 6.26368 0.894811
\(50\) 0 0
\(51\) 5.80729 0.813184
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −2.16536 −0.291977
\(56\) 0 0
\(57\) −2.70898 −0.358813
\(58\) 0 0
\(59\) −6.35091 −0.826818 −0.413409 0.910545i \(-0.635662\pi\)
−0.413409 + 0.910545i \(0.635662\pi\)
\(60\) 0 0
\(61\) −8.79887 −1.12658 −0.563290 0.826259i \(-0.690465\pi\)
−0.563290 + 0.826259i \(0.690465\pi\)
\(62\) 0 0
\(63\) 3.64193 0.458840
\(64\) 0 0
\(65\) 2.16536 0.268580
\(66\) 0 0
\(67\) −1.70055 −0.207755 −0.103878 0.994590i \(-0.533125\pi\)
−0.103878 + 0.994590i \(0.533125\pi\)
\(68\) 0 0
\(69\) −9.50785 −1.14461
\(70\) 0 0
\(71\) −2.91277 −0.345682 −0.172841 0.984950i \(-0.555295\pi\)
−0.172841 + 0.984950i \(0.555295\pi\)
\(72\) 0 0
\(73\) −13.9727 −1.63538 −0.817688 0.575662i \(-0.804744\pi\)
−0.817688 + 0.575662i \(0.804744\pi\)
\(74\) 0 0
\(75\) −0.311211 −0.0359356
\(76\) 0 0
\(77\) 3.64193 0.415037
\(78\) 0 0
\(79\) 3.80729 0.428354 0.214177 0.976795i \(-0.431293\pi\)
0.214177 + 0.976795i \(0.431293\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.7018 1.83326 0.916631 0.399733i \(-0.130897\pi\)
0.916631 + 0.399733i \(0.130897\pi\)
\(84\) 0 0
\(85\) −12.5749 −1.36394
\(86\) 0 0
\(87\) −10.4095 −1.11602
\(88\) 0 0
\(89\) −11.4290 −1.21148 −0.605738 0.795664i \(-0.707122\pi\)
−0.605738 + 0.795664i \(0.707122\pi\)
\(90\) 0 0
\(91\) −3.64193 −0.381778
\(92\) 0 0
\(93\) −10.5163 −1.09049
\(94\) 0 0
\(95\) 5.86591 0.601830
\(96\) 0 0
\(97\) 8.19664 0.832242 0.416121 0.909309i \(-0.363389\pi\)
0.416121 + 0.909309i \(0.363389\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 2.89453 0.288016 0.144008 0.989577i \(-0.454001\pi\)
0.144008 + 0.989577i \(0.454001\pi\)
\(102\) 0 0
\(103\) 2.35091 0.231642 0.115821 0.993270i \(-0.463050\pi\)
0.115821 + 0.993270i \(0.463050\pi\)
\(104\) 0 0
\(105\) −7.88610 −0.769605
\(106\) 0 0
\(107\) 3.06704 0.296502 0.148251 0.988950i \(-0.452636\pi\)
0.148251 + 0.988950i \(0.452636\pi\)
\(108\) 0 0
\(109\) 12.4408 1.19161 0.595806 0.803128i \(-0.296833\pi\)
0.595806 + 0.803128i \(0.296833\pi\)
\(110\) 0 0
\(111\) 8.19664 0.777990
\(112\) 0 0
\(113\) −17.3431 −1.63150 −0.815750 0.578405i \(-0.803675\pi\)
−0.815750 + 0.578405i \(0.803675\pi\)
\(114\) 0 0
\(115\) 20.5879 1.91983
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 21.1498 1.93880
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −5.31121 −0.478896
\(124\) 0 0
\(125\) 11.5007 1.02865
\(126\) 0 0
\(127\) −7.80729 −0.692785 −0.346393 0.938090i \(-0.612594\pi\)
−0.346393 + 0.938090i \(0.612594\pi\)
\(128\) 0 0
\(129\) 8.74025 0.769536
\(130\) 0 0
\(131\) 6.46815 0.565125 0.282562 0.959249i \(-0.408816\pi\)
0.282562 + 0.959249i \(0.408816\pi\)
\(132\) 0 0
\(133\) −9.86591 −0.855483
\(134\) 0 0
\(135\) −2.16536 −0.186365
\(136\) 0 0
\(137\) −18.7129 −1.59875 −0.799376 0.600831i \(-0.794836\pi\)
−0.799376 + 0.600831i \(0.794836\pi\)
\(138\) 0 0
\(139\) 8.13802 0.690258 0.345129 0.938555i \(-0.387835\pi\)
0.345129 + 0.938555i \(0.387835\pi\)
\(140\) 0 0
\(141\) −5.41795 −0.456274
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 22.5404 1.87188
\(146\) 0 0
\(147\) 6.26368 0.516620
\(148\) 0 0
\(149\) 20.0554 1.64300 0.821501 0.570207i \(-0.193137\pi\)
0.821501 + 0.570207i \(0.193137\pi\)
\(150\) 0 0
\(151\) −8.57489 −0.697815 −0.348907 0.937157i \(-0.613447\pi\)
−0.348907 + 0.937157i \(0.613447\pi\)
\(152\) 0 0
\(153\) 5.80729 0.469492
\(154\) 0 0
\(155\) 22.7715 1.82905
\(156\) 0 0
\(157\) −11.9928 −0.957133 −0.478567 0.878051i \(-0.658843\pi\)
−0.478567 + 0.878051i \(0.658843\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −34.6269 −2.72899
\(162\) 0 0
\(163\) −20.5990 −1.61344 −0.806719 0.590935i \(-0.798759\pi\)
−0.806719 + 0.590935i \(0.798759\pi\)
\(164\) 0 0
\(165\) −2.16536 −0.168573
\(166\) 0 0
\(167\) −0.154275 −0.0119381 −0.00596907 0.999982i \(-0.501900\pi\)
−0.00596907 + 0.999982i \(0.501900\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.70898 −0.207161
\(172\) 0 0
\(173\) 2.02412 0.153891 0.0769454 0.997035i \(-0.475483\pi\)
0.0769454 + 0.997035i \(0.475483\pi\)
\(174\) 0 0
\(175\) −1.13341 −0.0856778
\(176\) 0 0
\(177\) −6.35091 −0.477364
\(178\) 0 0
\(179\) 15.7011 1.17356 0.586779 0.809747i \(-0.300396\pi\)
0.586779 + 0.809747i \(0.300396\pi\)
\(180\) 0 0
\(181\) 4.26102 0.316719 0.158359 0.987382i \(-0.449380\pi\)
0.158359 + 0.987382i \(0.449380\pi\)
\(182\) 0 0
\(183\) −8.79887 −0.650431
\(184\) 0 0
\(185\) −17.7487 −1.30491
\(186\) 0 0
\(187\) 5.80729 0.424671
\(188\) 0 0
\(189\) 3.64193 0.264912
\(190\) 0 0
\(191\) 0.154275 0.0111629 0.00558147 0.999984i \(-0.498223\pi\)
0.00558147 + 0.999984i \(0.498223\pi\)
\(192\) 0 0
\(193\) 17.4108 1.25326 0.626628 0.779318i \(-0.284434\pi\)
0.626628 + 0.779318i \(0.284434\pi\)
\(194\) 0 0
\(195\) 2.16536 0.155065
\(196\) 0 0
\(197\) −19.6621 −1.40087 −0.700434 0.713717i \(-0.747010\pi\)
−0.700434 + 0.713717i \(0.747010\pi\)
\(198\) 0 0
\(199\) −5.42129 −0.384305 −0.192153 0.981365i \(-0.561547\pi\)
−0.192153 + 0.981365i \(0.561547\pi\)
\(200\) 0 0
\(201\) −1.70055 −0.119948
\(202\) 0 0
\(203\) −37.9108 −2.66082
\(204\) 0 0
\(205\) 11.5007 0.803243
\(206\) 0 0
\(207\) −9.50785 −0.660841
\(208\) 0 0
\(209\) −2.70898 −0.187384
\(210\) 0 0
\(211\) 19.0912 1.31429 0.657145 0.753764i \(-0.271764\pi\)
0.657145 + 0.753764i \(0.271764\pi\)
\(212\) 0 0
\(213\) −2.91277 −0.199580
\(214\) 0 0
\(215\) −18.9258 −1.29073
\(216\) 0 0
\(217\) −38.2996 −2.59994
\(218\) 0 0
\(219\) −13.9727 −0.944185
\(220\) 0 0
\(221\) −5.80729 −0.390641
\(222\) 0 0
\(223\) −1.46942 −0.0983994 −0.0491997 0.998789i \(-0.515667\pi\)
−0.0491997 + 0.998789i \(0.515667\pi\)
\(224\) 0 0
\(225\) −0.311211 −0.0207474
\(226\) 0 0
\(227\) −5.62175 −0.373128 −0.186564 0.982443i \(-0.559735\pi\)
−0.186564 + 0.982443i \(0.559735\pi\)
\(228\) 0 0
\(229\) 20.0059 1.32203 0.661013 0.750375i \(-0.270127\pi\)
0.661013 + 0.750375i \(0.270127\pi\)
\(230\) 0 0
\(231\) 3.64193 0.239622
\(232\) 0 0
\(233\) 3.18487 0.208648 0.104324 0.994543i \(-0.466732\pi\)
0.104324 + 0.994543i \(0.466732\pi\)
\(234\) 0 0
\(235\) 11.7318 0.765300
\(236\) 0 0
\(237\) 3.80729 0.247310
\(238\) 0 0
\(239\) 9.60291 0.621161 0.310580 0.950547i \(-0.399477\pi\)
0.310580 + 0.950547i \(0.399477\pi\)
\(240\) 0 0
\(241\) −8.07940 −0.520440 −0.260220 0.965549i \(-0.583795\pi\)
−0.260220 + 0.965549i \(0.583795\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −13.5631 −0.866516
\(246\) 0 0
\(247\) 2.70898 0.172368
\(248\) 0 0
\(249\) 16.7018 1.05843
\(250\) 0 0
\(251\) −7.01569 −0.442827 −0.221413 0.975180i \(-0.571067\pi\)
−0.221413 + 0.975180i \(0.571067\pi\)
\(252\) 0 0
\(253\) −9.50785 −0.597753
\(254\) 0 0
\(255\) −12.5749 −0.787470
\(256\) 0 0
\(257\) −31.3262 −1.95408 −0.977038 0.213064i \(-0.931656\pi\)
−0.977038 + 0.213064i \(0.931656\pi\)
\(258\) 0 0
\(259\) 29.8516 1.85489
\(260\) 0 0
\(261\) −10.4095 −0.644334
\(262\) 0 0
\(263\) −6.95314 −0.428749 −0.214375 0.976752i \(-0.568771\pi\)
−0.214375 + 0.976752i \(0.568771\pi\)
\(264\) 0 0
\(265\) 4.33072 0.266034
\(266\) 0 0
\(267\) −11.4290 −0.699446
\(268\) 0 0
\(269\) −16.4101 −1.00054 −0.500271 0.865869i \(-0.666766\pi\)
−0.500271 + 0.865869i \(0.666766\pi\)
\(270\) 0 0
\(271\) 0.574890 0.0349221 0.0174610 0.999848i \(-0.494442\pi\)
0.0174610 + 0.999848i \(0.494442\pi\)
\(272\) 0 0
\(273\) −3.64193 −0.220420
\(274\) 0 0
\(275\) −0.311211 −0.0187667
\(276\) 0 0
\(277\) −10.8561 −0.652280 −0.326140 0.945322i \(-0.605748\pi\)
−0.326140 + 0.945322i \(0.605748\pi\)
\(278\) 0 0
\(279\) −10.5163 −0.629593
\(280\) 0 0
\(281\) −17.3516 −1.03511 −0.517554 0.855650i \(-0.673158\pi\)
−0.517554 + 0.855650i \(0.673158\pi\)
\(282\) 0 0
\(283\) 13.7871 0.819558 0.409779 0.912185i \(-0.365606\pi\)
0.409779 + 0.912185i \(0.365606\pi\)
\(284\) 0 0
\(285\) 5.86591 0.347467
\(286\) 0 0
\(287\) −19.3431 −1.14179
\(288\) 0 0
\(289\) 16.7247 0.983804
\(290\) 0 0
\(291\) 8.19664 0.480495
\(292\) 0 0
\(293\) −17.7962 −1.03967 −0.519833 0.854268i \(-0.674006\pi\)
−0.519833 + 0.854268i \(0.674006\pi\)
\(294\) 0 0
\(295\) 13.7520 0.800673
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 9.50785 0.549853
\(300\) 0 0
\(301\) 31.8314 1.83473
\(302\) 0 0
\(303\) 2.89453 0.166286
\(304\) 0 0
\(305\) 19.0527 1.09096
\(306\) 0 0
\(307\) 8.61527 0.491699 0.245850 0.969308i \(-0.420933\pi\)
0.245850 + 0.969308i \(0.420933\pi\)
\(308\) 0 0
\(309\) 2.35091 0.133739
\(310\) 0 0
\(311\) 3.03970 0.172366 0.0861828 0.996279i \(-0.472533\pi\)
0.0861828 + 0.996279i \(0.472533\pi\)
\(312\) 0 0
\(313\) −2.72916 −0.154262 −0.0771308 0.997021i \(-0.524576\pi\)
−0.0771308 + 0.997021i \(0.524576\pi\)
\(314\) 0 0
\(315\) −7.88610 −0.444331
\(316\) 0 0
\(317\) −27.7631 −1.55933 −0.779666 0.626196i \(-0.784611\pi\)
−0.779666 + 0.626196i \(0.784611\pi\)
\(318\) 0 0
\(319\) −10.4095 −0.582822
\(320\) 0 0
\(321\) 3.06704 0.171186
\(322\) 0 0
\(323\) −15.7318 −0.875342
\(324\) 0 0
\(325\) 0.311211 0.0172629
\(326\) 0 0
\(327\) 12.4408 0.687978
\(328\) 0 0
\(329\) −19.7318 −1.08785
\(330\) 0 0
\(331\) −4.16536 −0.228949 −0.114474 0.993426i \(-0.536518\pi\)
−0.114474 + 0.993426i \(0.536518\pi\)
\(332\) 0 0
\(333\) 8.19664 0.449173
\(334\) 0 0
\(335\) 3.68231 0.201186
\(336\) 0 0
\(337\) 0.291700 0.0158899 0.00794495 0.999968i \(-0.497471\pi\)
0.00794495 + 0.999968i \(0.497471\pi\)
\(338\) 0 0
\(339\) −17.3431 −0.941947
\(340\) 0 0
\(341\) −10.5163 −0.569488
\(342\) 0 0
\(343\) −2.68163 −0.144795
\(344\) 0 0
\(345\) 20.5879 1.10842
\(346\) 0 0
\(347\) −7.24349 −0.388851 −0.194425 0.980917i \(-0.562284\pi\)
−0.194425 + 0.980917i \(0.562284\pi\)
\(348\) 0 0
\(349\) 24.7475 1.32470 0.662352 0.749193i \(-0.269558\pi\)
0.662352 + 0.749193i \(0.269558\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 19.2012 1.02198 0.510990 0.859587i \(-0.329279\pi\)
0.510990 + 0.859587i \(0.329279\pi\)
\(354\) 0 0
\(355\) 6.30720 0.334751
\(356\) 0 0
\(357\) 21.1498 1.11937
\(358\) 0 0
\(359\) 10.2097 0.538846 0.269423 0.963022i \(-0.413167\pi\)
0.269423 + 0.963022i \(0.413167\pi\)
\(360\) 0 0
\(361\) −11.6614 −0.613760
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 30.2558 1.58366
\(366\) 0 0
\(367\) 12.3933 0.646923 0.323462 0.946241i \(-0.395153\pi\)
0.323462 + 0.946241i \(0.395153\pi\)
\(368\) 0 0
\(369\) −5.31121 −0.276491
\(370\) 0 0
\(371\) −7.28387 −0.378160
\(372\) 0 0
\(373\) −5.05273 −0.261620 −0.130810 0.991407i \(-0.541758\pi\)
−0.130810 + 0.991407i \(0.541758\pi\)
\(374\) 0 0
\(375\) 11.5007 0.593893
\(376\) 0 0
\(377\) 10.4095 0.536118
\(378\) 0 0
\(379\) −11.2728 −0.579044 −0.289522 0.957171i \(-0.593496\pi\)
−0.289522 + 0.957171i \(0.593496\pi\)
\(380\) 0 0
\(381\) −7.80729 −0.399980
\(382\) 0 0
\(383\) −3.84239 −0.196337 −0.0981684 0.995170i \(-0.531298\pi\)
−0.0981684 + 0.995170i \(0.531298\pi\)
\(384\) 0 0
\(385\) −7.88610 −0.401913
\(386\) 0 0
\(387\) 8.74025 0.444292
\(388\) 0 0
\(389\) 8.85808 0.449122 0.224561 0.974460i \(-0.427905\pi\)
0.224561 + 0.974460i \(0.427905\pi\)
\(390\) 0 0
\(391\) −55.2149 −2.79234
\(392\) 0 0
\(393\) 6.46815 0.326275
\(394\) 0 0
\(395\) −8.24417 −0.414809
\(396\) 0 0
\(397\) −16.9173 −0.849054 −0.424527 0.905415i \(-0.639560\pi\)
−0.424527 + 0.905415i \(0.639560\pi\)
\(398\) 0 0
\(399\) −9.86591 −0.493913
\(400\) 0 0
\(401\) 5.11517 0.255439 0.127720 0.991810i \(-0.459234\pi\)
0.127720 + 0.991810i \(0.459234\pi\)
\(402\) 0 0
\(403\) 10.5163 0.523853
\(404\) 0 0
\(405\) −2.16536 −0.107598
\(406\) 0 0
\(407\) 8.19664 0.406292
\(408\) 0 0
\(409\) 31.3906 1.55217 0.776083 0.630631i \(-0.217204\pi\)
0.776083 + 0.630631i \(0.217204\pi\)
\(410\) 0 0
\(411\) −18.7129 −0.923040
\(412\) 0 0
\(413\) −23.1296 −1.13813
\(414\) 0 0
\(415\) −36.1655 −1.77529
\(416\) 0 0
\(417\) 8.13802 0.398520
\(418\) 0 0
\(419\) −30.7572 −1.50259 −0.751294 0.659968i \(-0.770570\pi\)
−0.751294 + 0.659968i \(0.770570\pi\)
\(420\) 0 0
\(421\) −12.9565 −0.631460 −0.315730 0.948849i \(-0.602249\pi\)
−0.315730 + 0.948849i \(0.602249\pi\)
\(422\) 0 0
\(423\) −5.41795 −0.263430
\(424\) 0 0
\(425\) −1.80729 −0.0876667
\(426\) 0 0
\(427\) −32.0449 −1.55076
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) 11.9857 0.577330 0.288665 0.957430i \(-0.406789\pi\)
0.288665 + 0.957430i \(0.406789\pi\)
\(432\) 0 0
\(433\) −38.2247 −1.83696 −0.918481 0.395466i \(-0.870583\pi\)
−0.918481 + 0.395466i \(0.870583\pi\)
\(434\) 0 0
\(435\) 22.5404 1.08073
\(436\) 0 0
\(437\) 25.7565 1.23210
\(438\) 0 0
\(439\) −17.9414 −0.856295 −0.428148 0.903709i \(-0.640834\pi\)
−0.428148 + 0.903709i \(0.640834\pi\)
\(440\) 0 0
\(441\) 6.26368 0.298270
\(442\) 0 0
\(443\) 25.9063 1.23084 0.615422 0.788197i \(-0.288985\pi\)
0.615422 + 0.788197i \(0.288985\pi\)
\(444\) 0 0
\(445\) 24.7480 1.17317
\(446\) 0 0
\(447\) 20.0554 0.948588
\(448\) 0 0
\(449\) −36.4838 −1.72177 −0.860887 0.508795i \(-0.830091\pi\)
−0.860887 + 0.508795i \(0.830091\pi\)
\(450\) 0 0
\(451\) −5.31121 −0.250095
\(452\) 0 0
\(453\) −8.57489 −0.402883
\(454\) 0 0
\(455\) 7.88610 0.369706
\(456\) 0 0
\(457\) 1.80535 0.0844507 0.0422253 0.999108i \(-0.486555\pi\)
0.0422253 + 0.999108i \(0.486555\pi\)
\(458\) 0 0
\(459\) 5.80729 0.271061
\(460\) 0 0
\(461\) −35.4018 −1.64883 −0.824413 0.565988i \(-0.808495\pi\)
−0.824413 + 0.565988i \(0.808495\pi\)
\(462\) 0 0
\(463\) 28.7755 1.33731 0.668654 0.743573i \(-0.266871\pi\)
0.668654 + 0.743573i \(0.266871\pi\)
\(464\) 0 0
\(465\) 22.7715 1.05600
\(466\) 0 0
\(467\) 11.2599 0.521044 0.260522 0.965468i \(-0.416105\pi\)
0.260522 + 0.965468i \(0.416105\pi\)
\(468\) 0 0
\(469\) −6.19330 −0.285980
\(470\) 0 0
\(471\) −11.9928 −0.552601
\(472\) 0 0
\(473\) 8.74025 0.401877
\(474\) 0 0
\(475\) 0.843064 0.0386824
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −10.2644 −0.468990 −0.234495 0.972117i \(-0.575344\pi\)
−0.234495 + 0.972117i \(0.575344\pi\)
\(480\) 0 0
\(481\) −8.19664 −0.373734
\(482\) 0 0
\(483\) −34.6269 −1.57558
\(484\) 0 0
\(485\) −17.7487 −0.805926
\(486\) 0 0
\(487\) −11.6257 −0.526810 −0.263405 0.964685i \(-0.584845\pi\)
−0.263405 + 0.964685i \(0.584845\pi\)
\(488\) 0 0
\(489\) −20.5990 −0.931519
\(490\) 0 0
\(491\) 10.9565 0.494459 0.247230 0.968957i \(-0.420480\pi\)
0.247230 + 0.968957i \(0.420480\pi\)
\(492\) 0 0
\(493\) −60.4512 −2.72259
\(494\) 0 0
\(495\) −2.16536 −0.0973257
\(496\) 0 0
\(497\) −10.6081 −0.475839
\(498\) 0 0
\(499\) 29.3008 1.31169 0.655843 0.754898i \(-0.272313\pi\)
0.655843 + 0.754898i \(0.272313\pi\)
\(500\) 0 0
\(501\) −0.154275 −0.00689249
\(502\) 0 0
\(503\) 30.4168 1.35622 0.678109 0.734961i \(-0.262800\pi\)
0.678109 + 0.734961i \(0.262800\pi\)
\(504\) 0 0
\(505\) −6.26769 −0.278909
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −41.4929 −1.83914 −0.919571 0.392924i \(-0.871464\pi\)
−0.919571 + 0.392924i \(0.871464\pi\)
\(510\) 0 0
\(511\) −50.8875 −2.25113
\(512\) 0 0
\(513\) −2.70898 −0.119604
\(514\) 0 0
\(515\) −5.09057 −0.224317
\(516\) 0 0
\(517\) −5.41795 −0.238281
\(518\) 0 0
\(519\) 2.02412 0.0888489
\(520\) 0 0
\(521\) 9.67262 0.423765 0.211883 0.977295i \(-0.432041\pi\)
0.211883 + 0.977295i \(0.432041\pi\)
\(522\) 0 0
\(523\) 8.61849 0.376860 0.188430 0.982087i \(-0.439660\pi\)
0.188430 + 0.982087i \(0.439660\pi\)
\(524\) 0 0
\(525\) −1.13341 −0.0494661
\(526\) 0 0
\(527\) −61.0711 −2.66030
\(528\) 0 0
\(529\) 67.3991 2.93040
\(530\) 0 0
\(531\) −6.35091 −0.275606
\(532\) 0 0
\(533\) 5.31121 0.230054
\(534\) 0 0
\(535\) −6.64126 −0.287127
\(536\) 0 0
\(537\) 15.7011 0.677554
\(538\) 0 0
\(539\) 6.26368 0.269796
\(540\) 0 0
\(541\) 3.51302 0.151036 0.0755182 0.997144i \(-0.475939\pi\)
0.0755182 + 0.997144i \(0.475939\pi\)
\(542\) 0 0
\(543\) 4.26102 0.182858
\(544\) 0 0
\(545\) −26.9388 −1.15393
\(546\) 0 0
\(547\) 36.1204 1.54440 0.772198 0.635382i \(-0.219158\pi\)
0.772198 + 0.635382i \(0.219158\pi\)
\(548\) 0 0
\(549\) −8.79887 −0.375527
\(550\) 0 0
\(551\) 28.1992 1.20133
\(552\) 0 0
\(553\) 13.8659 0.589639
\(554\) 0 0
\(555\) −17.7487 −0.753389
\(556\) 0 0
\(557\) −9.54235 −0.404322 −0.202161 0.979352i \(-0.564796\pi\)
−0.202161 + 0.979352i \(0.564796\pi\)
\(558\) 0 0
\(559\) −8.74025 −0.369673
\(560\) 0 0
\(561\) 5.80729 0.245184
\(562\) 0 0
\(563\) 19.2435 0.811016 0.405508 0.914091i \(-0.367095\pi\)
0.405508 + 0.914091i \(0.367095\pi\)
\(564\) 0 0
\(565\) 37.5540 1.57991
\(566\) 0 0
\(567\) 3.64193 0.152947
\(568\) 0 0
\(569\) 36.1694 1.51630 0.758150 0.652080i \(-0.226103\pi\)
0.758150 + 0.652080i \(0.226103\pi\)
\(570\) 0 0
\(571\) −0.133495 −0.00558660 −0.00279330 0.999996i \(-0.500889\pi\)
−0.00279330 + 0.999996i \(0.500889\pi\)
\(572\) 0 0
\(573\) 0.154275 0.00644492
\(574\) 0 0
\(575\) 2.95895 0.123397
\(576\) 0 0
\(577\) 15.5977 0.649342 0.324671 0.945827i \(-0.394746\pi\)
0.324671 + 0.945827i \(0.394746\pi\)
\(578\) 0 0
\(579\) 17.4108 0.723568
\(580\) 0 0
\(581\) 60.8269 2.52353
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) 2.16536 0.0895267
\(586\) 0 0
\(587\) −13.8693 −0.572445 −0.286223 0.958163i \(-0.592400\pi\)
−0.286223 + 0.958163i \(0.592400\pi\)
\(588\) 0 0
\(589\) 28.4883 1.17384
\(590\) 0 0
\(591\) −19.6621 −0.808792
\(592\) 0 0
\(593\) −16.5514 −0.679683 −0.339842 0.940483i \(-0.610373\pi\)
−0.339842 + 0.940483i \(0.610373\pi\)
\(594\) 0 0
\(595\) −45.7969 −1.87749
\(596\) 0 0
\(597\) −5.42129 −0.221879
\(598\) 0 0
\(599\) 12.8561 0.525286 0.262643 0.964893i \(-0.415406\pi\)
0.262643 + 0.964893i \(0.415406\pi\)
\(600\) 0 0
\(601\) 0.622422 0.0253891 0.0126946 0.999919i \(-0.495959\pi\)
0.0126946 + 0.999919i \(0.495959\pi\)
\(602\) 0 0
\(603\) −1.70055 −0.0692518
\(604\) 0 0
\(605\) −2.16536 −0.0880344
\(606\) 0 0
\(607\) 2.17704 0.0883633 0.0441816 0.999024i \(-0.485932\pi\)
0.0441816 + 0.999024i \(0.485932\pi\)
\(608\) 0 0
\(609\) −37.9108 −1.53622
\(610\) 0 0
\(611\) 5.41795 0.219187
\(612\) 0 0
\(613\) −23.0085 −0.929306 −0.464653 0.885493i \(-0.653821\pi\)
−0.464653 + 0.885493i \(0.653821\pi\)
\(614\) 0 0
\(615\) 11.5007 0.463753
\(616\) 0 0
\(617\) 6.53194 0.262966 0.131483 0.991318i \(-0.458026\pi\)
0.131483 + 0.991318i \(0.458026\pi\)
\(618\) 0 0
\(619\) 22.0313 0.885512 0.442756 0.896642i \(-0.354001\pi\)
0.442756 + 0.896642i \(0.354001\pi\)
\(620\) 0 0
\(621\) −9.50785 −0.381537
\(622\) 0 0
\(623\) −41.6238 −1.66762
\(624\) 0 0
\(625\) −23.3471 −0.933884
\(626\) 0 0
\(627\) −2.70898 −0.108186
\(628\) 0 0
\(629\) 47.6003 1.89795
\(630\) 0 0
\(631\) −42.4057 −1.68814 −0.844072 0.536229i \(-0.819848\pi\)
−0.844072 + 0.536229i \(0.819848\pi\)
\(632\) 0 0
\(633\) 19.0912 0.758806
\(634\) 0 0
\(635\) 16.9056 0.670879
\(636\) 0 0
\(637\) −6.26368 −0.248176
\(638\) 0 0
\(639\) −2.91277 −0.115227
\(640\) 0 0
\(641\) 18.5879 0.734179 0.367089 0.930186i \(-0.380354\pi\)
0.367089 + 0.930186i \(0.380354\pi\)
\(642\) 0 0
\(643\) −45.4551 −1.79257 −0.896287 0.443474i \(-0.853746\pi\)
−0.896287 + 0.443474i \(0.853746\pi\)
\(644\) 0 0
\(645\) −18.9258 −0.745203
\(646\) 0 0
\(647\) 18.4727 0.726236 0.363118 0.931743i \(-0.381712\pi\)
0.363118 + 0.931743i \(0.381712\pi\)
\(648\) 0 0
\(649\) −6.35091 −0.249295
\(650\) 0 0
\(651\) −38.2996 −1.50108
\(652\) 0 0
\(653\) −29.1733 −1.14164 −0.570820 0.821075i \(-0.693375\pi\)
−0.570820 + 0.821075i \(0.693375\pi\)
\(654\) 0 0
\(655\) −14.0059 −0.547255
\(656\) 0 0
\(657\) −13.9727 −0.545125
\(658\) 0 0
\(659\) −44.3568 −1.72790 −0.863948 0.503582i \(-0.832015\pi\)
−0.863948 + 0.503582i \(0.832015\pi\)
\(660\) 0 0
\(661\) −18.0079 −0.700425 −0.350212 0.936670i \(-0.613891\pi\)
−0.350212 + 0.936670i \(0.613891\pi\)
\(662\) 0 0
\(663\) −5.80729 −0.225537
\(664\) 0 0
\(665\) 21.3633 0.828432
\(666\) 0 0
\(667\) 98.9722 3.83222
\(668\) 0 0
\(669\) −1.46942 −0.0568109
\(670\) 0 0
\(671\) −8.79887 −0.339677
\(672\) 0 0
\(673\) 38.6875 1.49129 0.745647 0.666341i \(-0.232141\pi\)
0.745647 + 0.666341i \(0.232141\pi\)
\(674\) 0 0
\(675\) −0.311211 −0.0119785
\(676\) 0 0
\(677\) 27.9271 1.07332 0.536662 0.843797i \(-0.319685\pi\)
0.536662 + 0.843797i \(0.319685\pi\)
\(678\) 0 0
\(679\) 29.8516 1.14560
\(680\) 0 0
\(681\) −5.62175 −0.215426
\(682\) 0 0
\(683\) −1.28188 −0.0490499 −0.0245249 0.999699i \(-0.507807\pi\)
−0.0245249 + 0.999699i \(0.507807\pi\)
\(684\) 0 0
\(685\) 40.5202 1.54820
\(686\) 0 0
\(687\) 20.0059 0.763272
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 0.436873 0.0166194 0.00830972 0.999965i \(-0.497355\pi\)
0.00830972 + 0.999965i \(0.497355\pi\)
\(692\) 0 0
\(693\) 3.64193 0.138346
\(694\) 0 0
\(695\) −17.6217 −0.668431
\(696\) 0 0
\(697\) −30.8438 −1.16829
\(698\) 0 0
\(699\) 3.18487 0.120463
\(700\) 0 0
\(701\) −20.7234 −0.782712 −0.391356 0.920239i \(-0.627994\pi\)
−0.391356 + 0.920239i \(0.627994\pi\)
\(702\) 0 0
\(703\) −22.2045 −0.837458
\(704\) 0 0
\(705\) 11.7318 0.441846
\(706\) 0 0
\(707\) 10.5417 0.396460
\(708\) 0 0
\(709\) 40.9355 1.53736 0.768682 0.639631i \(-0.220913\pi\)
0.768682 + 0.639631i \(0.220913\pi\)
\(710\) 0 0
\(711\) 3.80729 0.142785
\(712\) 0 0
\(713\) 99.9871 3.74455
\(714\) 0 0
\(715\) 2.16536 0.0809799
\(716\) 0 0
\(717\) 9.60291 0.358627
\(718\) 0 0
\(719\) 9.53853 0.355727 0.177864 0.984055i \(-0.443081\pi\)
0.177864 + 0.984055i \(0.443081\pi\)
\(720\) 0 0
\(721\) 8.56186 0.318860
\(722\) 0 0
\(723\) −8.07940 −0.300476
\(724\) 0 0
\(725\) 3.23956 0.120314
\(726\) 0 0
\(727\) 9.17195 0.340169 0.170084 0.985429i \(-0.445596\pi\)
0.170084 + 0.985429i \(0.445596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 50.7572 1.87732
\(732\) 0 0
\(733\) −39.2363 −1.44923 −0.724614 0.689155i \(-0.757982\pi\)
−0.724614 + 0.689155i \(0.757982\pi\)
\(734\) 0 0
\(735\) −13.5631 −0.500283
\(736\) 0 0
\(737\) −1.70055 −0.0626406
\(738\) 0 0
\(739\) 35.4187 1.30290 0.651448 0.758693i \(-0.274162\pi\)
0.651448 + 0.758693i \(0.274162\pi\)
\(740\) 0 0
\(741\) 2.70898 0.0995168
\(742\) 0 0
\(743\) 42.1453 1.54616 0.773080 0.634308i \(-0.218715\pi\)
0.773080 + 0.634308i \(0.218715\pi\)
\(744\) 0 0
\(745\) −43.4272 −1.59105
\(746\) 0 0
\(747\) 16.7018 0.611088
\(748\) 0 0
\(749\) 11.1700 0.408142
\(750\) 0 0
\(751\) 26.3184 0.960372 0.480186 0.877167i \(-0.340569\pi\)
0.480186 + 0.877167i \(0.340569\pi\)
\(752\) 0 0
\(753\) −7.01569 −0.255666
\(754\) 0 0
\(755\) 18.5677 0.675749
\(756\) 0 0
\(757\) 7.73898 0.281278 0.140639 0.990061i \(-0.455084\pi\)
0.140639 + 0.990061i \(0.455084\pi\)
\(758\) 0 0
\(759\) −9.50785 −0.345113
\(760\) 0 0
\(761\) 11.9154 0.431934 0.215967 0.976401i \(-0.430710\pi\)
0.215967 + 0.976401i \(0.430710\pi\)
\(762\) 0 0
\(763\) 45.3086 1.64028
\(764\) 0 0
\(765\) −12.5749 −0.454646
\(766\) 0 0
\(767\) 6.35091 0.229318
\(768\) 0 0
\(769\) −5.63224 −0.203104 −0.101552 0.994830i \(-0.532381\pi\)
−0.101552 + 0.994830i \(0.532381\pi\)
\(770\) 0 0
\(771\) −31.3262 −1.12819
\(772\) 0 0
\(773\) −34.7937 −1.25144 −0.625720 0.780047i \(-0.715195\pi\)
−0.625720 + 0.780047i \(0.715195\pi\)
\(774\) 0 0
\(775\) 3.27278 0.117562
\(776\) 0 0
\(777\) 29.8516 1.07092
\(778\) 0 0
\(779\) 14.3879 0.515502
\(780\) 0 0
\(781\) −2.91277 −0.104227
\(782\) 0 0
\(783\) −10.4095 −0.372006
\(784\) 0 0
\(785\) 25.9688 0.926868
\(786\) 0 0
\(787\) −18.1438 −0.646756 −0.323378 0.946270i \(-0.604819\pi\)
−0.323378 + 0.946270i \(0.604819\pi\)
\(788\) 0 0
\(789\) −6.95314 −0.247538
\(790\) 0 0
\(791\) −63.1623 −2.24579
\(792\) 0 0
\(793\) 8.79887 0.312457
\(794\) 0 0
\(795\) 4.33072 0.153595
\(796\) 0 0
\(797\) 10.4701 0.370871 0.185436 0.982656i \(-0.440630\pi\)
0.185436 + 0.982656i \(0.440630\pi\)
\(798\) 0 0
\(799\) −31.4637 −1.11310
\(800\) 0 0
\(801\) −11.4290 −0.403825
\(802\) 0 0
\(803\) −13.9727 −0.493084
\(804\) 0 0
\(805\) 74.9798 2.64269
\(806\) 0 0
\(807\) −16.4101 −0.577664
\(808\) 0 0
\(809\) 10.4869 0.368701 0.184351 0.982861i \(-0.440982\pi\)
0.184351 + 0.982861i \(0.440982\pi\)
\(810\) 0 0
\(811\) −17.8027 −0.625138 −0.312569 0.949895i \(-0.601190\pi\)
−0.312569 + 0.949895i \(0.601190\pi\)
\(812\) 0 0
\(813\) 0.574890 0.0201623
\(814\) 0 0
\(815\) 44.6043 1.56242
\(816\) 0 0
\(817\) −23.6771 −0.828358
\(818\) 0 0
\(819\) −3.64193 −0.127259
\(820\) 0 0
\(821\) 34.6386 1.20890 0.604448 0.796645i \(-0.293394\pi\)
0.604448 + 0.796645i \(0.293394\pi\)
\(822\) 0 0
\(823\) 32.8002 1.14334 0.571672 0.820482i \(-0.306295\pi\)
0.571672 + 0.820482i \(0.306295\pi\)
\(824\) 0 0
\(825\) −0.311211 −0.0108350
\(826\) 0 0
\(827\) −50.6765 −1.76219 −0.881097 0.472936i \(-0.843194\pi\)
−0.881097 + 0.472936i \(0.843194\pi\)
\(828\) 0 0
\(829\) 14.9767 0.520161 0.260081 0.965587i \(-0.416251\pi\)
0.260081 + 0.965587i \(0.416251\pi\)
\(830\) 0 0
\(831\) −10.8561 −0.376594
\(832\) 0 0
\(833\) 36.3750 1.26032
\(834\) 0 0
\(835\) 0.334061 0.0115606
\(836\) 0 0
\(837\) −10.5163 −0.363496
\(838\) 0 0
\(839\) 6.60811 0.228137 0.114069 0.993473i \(-0.463612\pi\)
0.114069 + 0.993473i \(0.463612\pi\)
\(840\) 0 0
\(841\) 79.3583 2.73649
\(842\) 0 0
\(843\) −17.3516 −0.597620
\(844\) 0 0
\(845\) −2.16536 −0.0744907
\(846\) 0 0
\(847\) 3.64193 0.125138
\(848\) 0 0
\(849\) 13.7871 0.473172
\(850\) 0 0
\(851\) −77.9324 −2.67149
\(852\) 0 0
\(853\) 1.51834 0.0519870 0.0259935 0.999662i \(-0.491725\pi\)
0.0259935 + 0.999662i \(0.491725\pi\)
\(854\) 0 0
\(855\) 5.86591 0.200610
\(856\) 0 0
\(857\) 20.9806 0.716684 0.358342 0.933590i \(-0.383342\pi\)
0.358342 + 0.933590i \(0.383342\pi\)
\(858\) 0 0
\(859\) −44.4011 −1.51495 −0.757474 0.652866i \(-0.773567\pi\)
−0.757474 + 0.652866i \(0.773567\pi\)
\(860\) 0 0
\(861\) −19.3431 −0.659210
\(862\) 0 0
\(863\) −11.6953 −0.398114 −0.199057 0.979988i \(-0.563788\pi\)
−0.199057 + 0.979988i \(0.563788\pi\)
\(864\) 0 0
\(865\) −4.38295 −0.149025
\(866\) 0 0
\(867\) 16.7247 0.568000
\(868\) 0 0
\(869\) 3.80729 0.129154
\(870\) 0 0
\(871\) 1.70055 0.0576210
\(872\) 0 0
\(873\) 8.19664 0.277414
\(874\) 0 0
\(875\) 41.8847 1.41596
\(876\) 0 0
\(877\) 4.89060 0.165144 0.0825718 0.996585i \(-0.473687\pi\)
0.0825718 + 0.996585i \(0.473687\pi\)
\(878\) 0 0
\(879\) −17.7962 −0.600251
\(880\) 0 0
\(881\) −47.5032 −1.60042 −0.800212 0.599717i \(-0.795280\pi\)
−0.800212 + 0.599717i \(0.795280\pi\)
\(882\) 0 0
\(883\) −40.9779 −1.37901 −0.689507 0.724279i \(-0.742173\pi\)
−0.689507 + 0.724279i \(0.742173\pi\)
\(884\) 0 0
\(885\) 13.7520 0.462269
\(886\) 0 0
\(887\) −9.09391 −0.305344 −0.152672 0.988277i \(-0.548788\pi\)
−0.152672 + 0.988277i \(0.548788\pi\)
\(888\) 0 0
\(889\) −28.4336 −0.953634
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 14.6771 0.491151
\(894\) 0 0
\(895\) −33.9986 −1.13645
\(896\) 0 0
\(897\) 9.50785 0.317458
\(898\) 0 0
\(899\) 109.469 3.65101
\(900\) 0 0
\(901\) −11.6146 −0.386938
\(902\) 0 0
\(903\) 31.8314 1.05928
\(904\) 0 0
\(905\) −9.22664 −0.306704
\(906\) 0 0
\(907\) 21.4036 0.710696 0.355348 0.934734i \(-0.384362\pi\)
0.355348 + 0.934734i \(0.384362\pi\)
\(908\) 0 0
\(909\) 2.89453 0.0960054
\(910\) 0 0
\(911\) −3.77984 −0.125232 −0.0626158 0.998038i \(-0.519944\pi\)
−0.0626158 + 0.998038i \(0.519944\pi\)
\(912\) 0 0
\(913\) 16.7018 0.552750
\(914\) 0 0
\(915\) 19.0527 0.629864
\(916\) 0 0
\(917\) 23.5566 0.777906
\(918\) 0 0
\(919\) −42.8230 −1.41260 −0.706300 0.707913i \(-0.749637\pi\)
−0.706300 + 0.707913i \(0.749637\pi\)
\(920\) 0 0
\(921\) 8.61527 0.283883
\(922\) 0 0
\(923\) 2.91277 0.0958749
\(924\) 0 0
\(925\) −2.55088 −0.0838725
\(926\) 0 0
\(927\) 2.35091 0.0772140
\(928\) 0 0
\(929\) 56.6817 1.85967 0.929834 0.367980i \(-0.119950\pi\)
0.929834 + 0.367980i \(0.119950\pi\)
\(930\) 0 0
\(931\) −16.9682 −0.556109
\(932\) 0 0
\(933\) 3.03970 0.0995153
\(934\) 0 0
\(935\) −12.5749 −0.411243
\(936\) 0 0
\(937\) −33.0937 −1.08113 −0.540563 0.841304i \(-0.681789\pi\)
−0.540563 + 0.841304i \(0.681789\pi\)
\(938\) 0 0
\(939\) −2.72916 −0.0890629
\(940\) 0 0
\(941\) −49.1830 −1.60332 −0.801660 0.597780i \(-0.796050\pi\)
−0.801660 + 0.597780i \(0.796050\pi\)
\(942\) 0 0
\(943\) 50.4982 1.64445
\(944\) 0 0
\(945\) −7.88610 −0.256535
\(946\) 0 0
\(947\) 16.3152 0.530174 0.265087 0.964225i \(-0.414599\pi\)
0.265087 + 0.964225i \(0.414599\pi\)
\(948\) 0 0
\(949\) 13.9727 0.453572
\(950\) 0 0
\(951\) −27.7631 −0.900280
\(952\) 0 0
\(953\) 56.3686 1.82596 0.912979 0.408007i \(-0.133776\pi\)
0.912979 + 0.408007i \(0.133776\pi\)
\(954\) 0 0
\(955\) −0.334061 −0.0108099
\(956\) 0 0
\(957\) −10.4095 −0.336492
\(958\) 0 0
\(959\) −68.1512 −2.20072
\(960\) 0 0
\(961\) 79.5920 2.56748
\(962\) 0 0
\(963\) 3.06704 0.0988341
\(964\) 0 0
\(965\) −37.7007 −1.21363
\(966\) 0 0
\(967\) −1.89444 −0.0609211 −0.0304606 0.999536i \(-0.509697\pi\)
−0.0304606 + 0.999536i \(0.509697\pi\)
\(968\) 0 0
\(969\) −15.7318 −0.505379
\(970\) 0 0
\(971\) 44.4030 1.42496 0.712479 0.701693i \(-0.247572\pi\)
0.712479 + 0.701693i \(0.247572\pi\)
\(972\) 0 0
\(973\) 29.6381 0.950154
\(974\) 0 0
\(975\) 0.311211 0.00996673
\(976\) 0 0
\(977\) 1.34964 0.0431789 0.0215894 0.999767i \(-0.493127\pi\)
0.0215894 + 0.999767i \(0.493127\pi\)
\(978\) 0 0
\(979\) −11.4290 −0.365274
\(980\) 0 0
\(981\) 12.4408 0.397204
\(982\) 0 0
\(983\) 53.2202 1.69746 0.848730 0.528826i \(-0.177368\pi\)
0.848730 + 0.528826i \(0.177368\pi\)
\(984\) 0 0
\(985\) 42.5756 1.35657
\(986\) 0 0
\(987\) −19.7318 −0.628071
\(988\) 0 0
\(989\) −83.1010 −2.64246
\(990\) 0 0
\(991\) −24.7832 −0.787264 −0.393632 0.919268i \(-0.628782\pi\)
−0.393632 + 0.919268i \(0.628782\pi\)
\(992\) 0 0
\(993\) −4.16536 −0.132184
\(994\) 0 0
\(995\) 11.7391 0.372153
\(996\) 0 0
\(997\) 7.00703 0.221915 0.110957 0.993825i \(-0.464608\pi\)
0.110957 + 0.993825i \(0.464608\pi\)
\(998\) 0 0
\(999\) 8.19664 0.259330
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 6864.2.a.by.1.2 4
4.3 odd 2 3432.2.a.q.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.q.1.2 4 4.3 odd 2
6864.2.a.by.1.2 4 1.1 even 1 trivial