Properties

Label 784.3.c.e.97.2
Level $784$
Weight $3$
Character 784.97
Analytic conductor $21.362$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,3,Mod(97,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 784.97
Dual form 784.3.c.e.97.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.717439i q^{3} +6.63103i q^{5} +8.48528 q^{9} +O(q^{10})\) \(q-0.717439i q^{3} +6.63103i q^{5} +8.48528 q^{9} +4.75736 q^{11} -15.2913i q^{13} +4.75736 q^{15} -3.76127i q^{17} +4.18154i q^{19} +27.7279 q^{23} -18.9706 q^{25} -12.5446i q^{27} +3.51472 q^{29} +48.8667i q^{31} -3.41311i q^{33} -2.94113 q^{37} -10.9706 q^{39} +27.9590i q^{41} +10.4853 q^{43} +56.2662i q^{45} +52.6790i q^{47} -2.69848 q^{51} +55.9706 q^{53} +31.5462i q^{55} +3.00000 q^{57} -38.7206i q^{59} +90.5080i q^{61} +101.397 q^{65} +34.6396 q^{67} -19.8931i q^{69} -36.4264 q^{71} +52.6069i q^{73} +13.6102i q^{75} +33.7868 q^{79} +67.3675 q^{81} -127.577i q^{83} +24.9411 q^{85} -2.52160i q^{87} +50.3314i q^{89} +35.0589 q^{93} -27.7279 q^{95} -101.792i q^{97} +40.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{11} + 36 q^{15} + 60 q^{23} - 8 q^{25} + 48 q^{29} + 124 q^{37} + 24 q^{39} + 8 q^{43} + 108 q^{51} + 156 q^{53} + 12 q^{57} + 168 q^{65} - 116 q^{67} + 24 q^{71} + 220 q^{79} - 36 q^{81} - 36 q^{85} + 276 q^{93} - 60 q^{95} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 0.717439i − 0.239146i −0.992825 0.119573i \(-0.961847\pi\)
0.992825 0.119573i \(-0.0381526\pi\)
\(4\) 0 0
\(5\) 6.63103i 1.32621i 0.748528 + 0.663103i \(0.230761\pi\)
−0.748528 + 0.663103i \(0.769239\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 8.48528 0.942809
\(10\) 0 0
\(11\) 4.75736 0.432487 0.216244 0.976339i \(-0.430619\pi\)
0.216244 + 0.976339i \(0.430619\pi\)
\(12\) 0 0
\(13\) − 15.2913i − 1.17625i −0.808769 0.588126i \(-0.799866\pi\)
0.808769 0.588126i \(-0.200134\pi\)
\(14\) 0 0
\(15\) 4.75736 0.317157
\(16\) 0 0
\(17\) − 3.76127i − 0.221251i −0.993862 0.110626i \(-0.964715\pi\)
0.993862 0.110626i \(-0.0352855\pi\)
\(18\) 0 0
\(19\) 4.18154i 0.220081i 0.993927 + 0.110041i \(0.0350981\pi\)
−0.993927 + 0.110041i \(0.964902\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 27.7279 1.20556 0.602781 0.797907i \(-0.294059\pi\)
0.602781 + 0.797907i \(0.294059\pi\)
\(24\) 0 0
\(25\) −18.9706 −0.758823
\(26\) 0 0
\(27\) − 12.5446i − 0.464616i
\(28\) 0 0
\(29\) 3.51472 0.121197 0.0605986 0.998162i \(-0.480699\pi\)
0.0605986 + 0.998162i \(0.480699\pi\)
\(30\) 0 0
\(31\) 48.8667i 1.57635i 0.615454 + 0.788173i \(0.288973\pi\)
−0.615454 + 0.788173i \(0.711027\pi\)
\(32\) 0 0
\(33\) − 3.41311i − 0.103428i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.94113 −0.0794899 −0.0397449 0.999210i \(-0.512655\pi\)
−0.0397449 + 0.999210i \(0.512655\pi\)
\(38\) 0 0
\(39\) −10.9706 −0.281296
\(40\) 0 0
\(41\) 27.9590i 0.681927i 0.940077 + 0.340963i \(0.110753\pi\)
−0.940077 + 0.340963i \(0.889247\pi\)
\(42\) 0 0
\(43\) 10.4853 0.243844 0.121922 0.992540i \(-0.461094\pi\)
0.121922 + 0.992540i \(0.461094\pi\)
\(44\) 0 0
\(45\) 56.2662i 1.25036i
\(46\) 0 0
\(47\) 52.6790i 1.12083i 0.828212 + 0.560415i \(0.189358\pi\)
−0.828212 + 0.560415i \(0.810642\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.69848 −0.0529115
\(52\) 0 0
\(53\) 55.9706 1.05605 0.528024 0.849229i \(-0.322933\pi\)
0.528024 + 0.849229i \(0.322933\pi\)
\(54\) 0 0
\(55\) 31.5462i 0.573567i
\(56\) 0 0
\(57\) 3.00000 0.0526316
\(58\) 0 0
\(59\) − 38.7206i − 0.656281i −0.944629 0.328141i \(-0.893578\pi\)
0.944629 0.328141i \(-0.106422\pi\)
\(60\) 0 0
\(61\) 90.5080i 1.48374i 0.670545 + 0.741869i \(0.266060\pi\)
−0.670545 + 0.741869i \(0.733940\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 101.397 1.55995
\(66\) 0 0
\(67\) 34.6396 0.517009 0.258505 0.966010i \(-0.416770\pi\)
0.258505 + 0.966010i \(0.416770\pi\)
\(68\) 0 0
\(69\) − 19.8931i − 0.288306i
\(70\) 0 0
\(71\) −36.4264 −0.513048 −0.256524 0.966538i \(-0.582577\pi\)
−0.256524 + 0.966538i \(0.582577\pi\)
\(72\) 0 0
\(73\) 52.6069i 0.720642i 0.932828 + 0.360321i \(0.117333\pi\)
−0.932828 + 0.360321i \(0.882667\pi\)
\(74\) 0 0
\(75\) 13.6102i 0.181470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 33.7868 0.427681 0.213840 0.976869i \(-0.431403\pi\)
0.213840 + 0.976869i \(0.431403\pi\)
\(80\) 0 0
\(81\) 67.3675 0.831698
\(82\) 0 0
\(83\) − 127.577i − 1.53708i −0.639803 0.768539i \(-0.720984\pi\)
0.639803 0.768539i \(-0.279016\pi\)
\(84\) 0 0
\(85\) 24.9411 0.293425
\(86\) 0 0
\(87\) − 2.52160i − 0.0289839i
\(88\) 0 0
\(89\) 50.3314i 0.565522i 0.959190 + 0.282761i \(0.0912503\pi\)
−0.959190 + 0.282761i \(0.908750\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 35.0589 0.376977
\(94\) 0 0
\(95\) −27.7279 −0.291873
\(96\) 0 0
\(97\) − 101.792i − 1.04940i −0.851287 0.524700i \(-0.824177\pi\)
0.851287 0.524700i \(-0.175823\pi\)
\(98\) 0 0
\(99\) 40.3675 0.407753
\(100\) 0 0
\(101\) − 59.6793i − 0.590884i −0.955361 0.295442i \(-0.904533\pi\)
0.955361 0.295442i \(-0.0954669\pi\)
\(102\) 0 0
\(103\) 120.178i 1.16678i 0.812193 + 0.583388i \(0.198273\pi\)
−0.812193 + 0.583388i \(0.801727\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 113.610 1.06178 0.530889 0.847442i \(-0.321858\pi\)
0.530889 + 0.847442i \(0.321858\pi\)
\(108\) 0 0
\(109\) −145.309 −1.33311 −0.666553 0.745457i \(-0.732231\pi\)
−0.666553 + 0.745457i \(0.732231\pi\)
\(110\) 0 0
\(111\) 2.11008i 0.0190097i
\(112\) 0 0
\(113\) 34.5442 0.305700 0.152850 0.988249i \(-0.451155\pi\)
0.152850 + 0.988249i \(0.451155\pi\)
\(114\) 0 0
\(115\) 183.865i 1.59882i
\(116\) 0 0
\(117\) − 129.751i − 1.10898i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −98.3675 −0.812955
\(122\) 0 0
\(123\) 20.0589 0.163080
\(124\) 0 0
\(125\) 39.9814i 0.319851i
\(126\) 0 0
\(127\) 247.338 1.94754 0.973772 0.227526i \(-0.0730636\pi\)
0.973772 + 0.227526i \(0.0730636\pi\)
\(128\) 0 0
\(129\) − 7.52255i − 0.0583143i
\(130\) 0 0
\(131\) − 147.645i − 1.12706i −0.826096 0.563529i \(-0.809443\pi\)
0.826096 0.563529i \(-0.190557\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 83.1838 0.616176
\(136\) 0 0
\(137\) −32.5736 −0.237763 −0.118882 0.992908i \(-0.537931\pi\)
−0.118882 + 0.992908i \(0.537931\pi\)
\(138\) 0 0
\(139\) 68.5857i 0.493422i 0.969089 + 0.246711i \(0.0793499\pi\)
−0.969089 + 0.246711i \(0.920650\pi\)
\(140\) 0 0
\(141\) 37.7939 0.268042
\(142\) 0 0
\(143\) − 72.7461i − 0.508714i
\(144\) 0 0
\(145\) 23.3062i 0.160732i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 92.3970 0.620114 0.310057 0.950718i \(-0.399652\pi\)
0.310057 + 0.950718i \(0.399652\pi\)
\(150\) 0 0
\(151\) 91.7868 0.607860 0.303930 0.952694i \(-0.401701\pi\)
0.303930 + 0.952694i \(0.401701\pi\)
\(152\) 0 0
\(153\) − 31.9155i − 0.208598i
\(154\) 0 0
\(155\) −324.037 −2.09056
\(156\) 0 0
\(157\) 8.45631i 0.0538618i 0.999637 + 0.0269309i \(0.00857341\pi\)
−0.999637 + 0.0269309i \(0.991427\pi\)
\(158\) 0 0
\(159\) − 40.1555i − 0.252550i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −221.978 −1.36183 −0.680913 0.732364i \(-0.738417\pi\)
−0.680913 + 0.732364i \(0.738417\pi\)
\(164\) 0 0
\(165\) 22.6325 0.137166
\(166\) 0 0
\(167\) 168.841i 1.01102i 0.862820 + 0.505511i \(0.168696\pi\)
−0.862820 + 0.505511i \(0.831304\pi\)
\(168\) 0 0
\(169\) −64.8234 −0.383570
\(170\) 0 0
\(171\) 35.4815i 0.207494i
\(172\) 0 0
\(173\) 164.341i 0.949947i 0.880000 + 0.474974i \(0.157542\pi\)
−0.880000 + 0.474974i \(0.842458\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −27.7797 −0.156947
\(178\) 0 0
\(179\) −185.184 −1.03455 −0.517273 0.855820i \(-0.673053\pi\)
−0.517273 + 0.855820i \(0.673053\pi\)
\(180\) 0 0
\(181\) 155.086i 0.856830i 0.903582 + 0.428415i \(0.140928\pi\)
−0.903582 + 0.428415i \(0.859072\pi\)
\(182\) 0 0
\(183\) 64.9340 0.354831
\(184\) 0 0
\(185\) − 19.5027i − 0.105420i
\(186\) 0 0
\(187\) − 17.8937i − 0.0956884i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −248.095 −1.29893 −0.649465 0.760392i \(-0.725007\pi\)
−0.649465 + 0.760392i \(0.725007\pi\)
\(192\) 0 0
\(193\) 154.338 0.799679 0.399840 0.916585i \(-0.369066\pi\)
0.399840 + 0.916585i \(0.369066\pi\)
\(194\) 0 0
\(195\) − 72.7461i − 0.373057i
\(196\) 0 0
\(197\) −181.103 −0.919303 −0.459651 0.888099i \(-0.652026\pi\)
−0.459651 + 0.888099i \(0.652026\pi\)
\(198\) 0 0
\(199\) − 348.707i − 1.75229i −0.482043 0.876147i \(-0.660105\pi\)
0.482043 0.876147i \(-0.339895\pi\)
\(200\) 0 0
\(201\) − 24.8518i − 0.123641i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −185.397 −0.904375
\(206\) 0 0
\(207\) 235.279 1.13661
\(208\) 0 0
\(209\) 19.8931i 0.0951823i
\(210\) 0 0
\(211\) −364.073 −1.72547 −0.862733 0.505660i \(-0.831249\pi\)
−0.862733 + 0.505660i \(0.831249\pi\)
\(212\) 0 0
\(213\) 26.1337i 0.122694i
\(214\) 0 0
\(215\) 69.5282i 0.323387i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 37.7422 0.172339
\(220\) 0 0
\(221\) −57.5147 −0.260248
\(222\) 0 0
\(223\) 123.231i 0.552603i 0.961071 + 0.276302i \(0.0891089\pi\)
−0.961071 + 0.276302i \(0.910891\pi\)
\(224\) 0 0
\(225\) −160.971 −0.715425
\(226\) 0 0
\(227\) 76.3756i 0.336456i 0.985748 + 0.168228i \(0.0538045\pi\)
−0.985748 + 0.168228i \(0.946195\pi\)
\(228\) 0 0
\(229\) − 357.286i − 1.56020i −0.625654 0.780101i \(-0.715168\pi\)
0.625654 0.780101i \(-0.284832\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −273.073 −1.17199 −0.585994 0.810315i \(-0.699296\pi\)
−0.585994 + 0.810315i \(0.699296\pi\)
\(234\) 0 0
\(235\) −349.316 −1.48645
\(236\) 0 0
\(237\) − 24.2400i − 0.102278i
\(238\) 0 0
\(239\) 265.103 1.10922 0.554608 0.832112i \(-0.312868\pi\)
0.554608 + 0.832112i \(0.312868\pi\)
\(240\) 0 0
\(241\) − 87.6383i − 0.363644i −0.983331 0.181822i \(-0.941800\pi\)
0.983331 0.181822i \(-0.0581995\pi\)
\(242\) 0 0
\(243\) − 161.234i − 0.663513i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 63.9411 0.258871
\(248\) 0 0
\(249\) −91.5290 −0.367586
\(250\) 0 0
\(251\) − 495.655i − 1.97472i −0.158491 0.987360i \(-0.550663\pi\)
0.158491 0.987360i \(-0.449337\pi\)
\(252\) 0 0
\(253\) 131.912 0.521390
\(254\) 0 0
\(255\) − 17.8937i − 0.0701715i
\(256\) 0 0
\(257\) − 400.536i − 1.55851i −0.626709 0.779254i \(-0.715598\pi\)
0.626709 0.779254i \(-0.284402\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 29.8234 0.114266
\(262\) 0 0
\(263\) 32.3452 0.122986 0.0614928 0.998108i \(-0.480414\pi\)
0.0614928 + 0.998108i \(0.480414\pi\)
\(264\) 0 0
\(265\) 371.142i 1.40054i
\(266\) 0 0
\(267\) 36.1097 0.135242
\(268\) 0 0
\(269\) − 306.963i − 1.14113i −0.821253 0.570564i \(-0.806725\pi\)
0.821253 0.570564i \(-0.193275\pi\)
\(270\) 0 0
\(271\) − 75.9852i − 0.280388i −0.990124 0.140194i \(-0.955227\pi\)
0.990124 0.140194i \(-0.0447726\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −90.2498 −0.328181
\(276\) 0 0
\(277\) 278.411 1.00509 0.502547 0.864550i \(-0.332396\pi\)
0.502547 + 0.864550i \(0.332396\pi\)
\(278\) 0 0
\(279\) 414.648i 1.48619i
\(280\) 0 0
\(281\) 394.690 1.40459 0.702296 0.711885i \(-0.252158\pi\)
0.702296 + 0.711885i \(0.252158\pi\)
\(282\) 0 0
\(283\) − 146.396i − 0.517301i −0.965971 0.258650i \(-0.916722\pi\)
0.965971 0.258650i \(-0.0832778\pi\)
\(284\) 0 0
\(285\) 19.8931i 0.0698003i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 274.853 0.951048
\(290\) 0 0
\(291\) −73.0294 −0.250960
\(292\) 0 0
\(293\) − 299.678i − 1.02279i −0.859345 0.511396i \(-0.829128\pi\)
0.859345 0.511396i \(-0.170872\pi\)
\(294\) 0 0
\(295\) 256.757 0.870364
\(296\) 0 0
\(297\) − 59.6793i − 0.200940i
\(298\) 0 0
\(299\) − 423.996i − 1.41805i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −42.8162 −0.141308
\(304\) 0 0
\(305\) −600.161 −1.96774
\(306\) 0 0
\(307\) − 20.9886i − 0.0683666i −0.999416 0.0341833i \(-0.989117\pi\)
0.999416 0.0341833i \(-0.0108830\pi\)
\(308\) 0 0
\(309\) 86.2203 0.279030
\(310\) 0 0
\(311\) − 182.039i − 0.585336i −0.956214 0.292668i \(-0.905457\pi\)
0.956214 0.292668i \(-0.0945430\pi\)
\(312\) 0 0
\(313\) 97.9286i 0.312871i 0.987688 + 0.156435i \(0.0500003\pi\)
−0.987688 + 0.156435i \(0.950000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −481.971 −1.52041 −0.760206 0.649682i \(-0.774902\pi\)
−0.760206 + 0.649682i \(0.774902\pi\)
\(318\) 0 0
\(319\) 16.7208 0.0524162
\(320\) 0 0
\(321\) − 81.5084i − 0.253920i
\(322\) 0 0
\(323\) 15.7279 0.0486933
\(324\) 0 0
\(325\) 290.084i 0.892567i
\(326\) 0 0
\(327\) 104.250i 0.318808i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −225.007 −0.679780 −0.339890 0.940465i \(-0.610390\pi\)
−0.339890 + 0.940465i \(0.610390\pi\)
\(332\) 0 0
\(333\) −24.9563 −0.0749438
\(334\) 0 0
\(335\) 229.696i 0.685661i
\(336\) 0 0
\(337\) −264.368 −0.784473 −0.392237 0.919864i \(-0.628299\pi\)
−0.392237 + 0.919864i \(0.628299\pi\)
\(338\) 0 0
\(339\) − 24.7833i − 0.0731071i
\(340\) 0 0
\(341\) 232.476i 0.681749i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 131.912 0.382353
\(346\) 0 0
\(347\) 191.257 0.551173 0.275586 0.961276i \(-0.411128\pi\)
0.275586 + 0.961276i \(0.411128\pi\)
\(348\) 0 0
\(349\) − 135.448i − 0.388104i −0.980991 0.194052i \(-0.937837\pi\)
0.980991 0.194052i \(-0.0621630\pi\)
\(350\) 0 0
\(351\) −191.823 −0.546505
\(352\) 0 0
\(353\) − 348.490i − 0.987225i −0.869682 0.493612i \(-0.835676\pi\)
0.869682 0.493612i \(-0.164324\pi\)
\(354\) 0 0
\(355\) − 241.545i − 0.680407i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −304.831 −0.849110 −0.424555 0.905402i \(-0.639569\pi\)
−0.424555 + 0.905402i \(0.639569\pi\)
\(360\) 0 0
\(361\) 343.515 0.951564
\(362\) 0 0
\(363\) 70.5727i 0.194415i
\(364\) 0 0
\(365\) −348.838 −0.955720
\(366\) 0 0
\(367\) 95.0042i 0.258867i 0.991588 + 0.129434i \(0.0413159\pi\)
−0.991588 + 0.129434i \(0.958684\pi\)
\(368\) 0 0
\(369\) 237.240i 0.642927i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 253.558 0.679781 0.339891 0.940465i \(-0.389610\pi\)
0.339891 + 0.940465i \(0.389610\pi\)
\(374\) 0 0
\(375\) 28.6842 0.0764912
\(376\) 0 0
\(377\) − 53.7446i − 0.142559i
\(378\) 0 0
\(379\) −508.250 −1.34103 −0.670514 0.741897i \(-0.733926\pi\)
−0.670514 + 0.741897i \(0.733926\pi\)
\(380\) 0 0
\(381\) − 177.450i − 0.465748i
\(382\) 0 0
\(383\) 477.761i 1.24742i 0.781656 + 0.623709i \(0.214375\pi\)
−0.781656 + 0.623709i \(0.785625\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 88.9706 0.229898
\(388\) 0 0
\(389\) 170.220 0.437584 0.218792 0.975771i \(-0.429788\pi\)
0.218792 + 0.975771i \(0.429788\pi\)
\(390\) 0 0
\(391\) − 104.292i − 0.266732i
\(392\) 0 0
\(393\) −105.926 −0.269532
\(394\) 0 0
\(395\) 224.041i 0.567193i
\(396\) 0 0
\(397\) − 244.550i − 0.615995i −0.951387 0.307997i \(-0.900341\pi\)
0.951387 0.307997i \(-0.0996588\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −417.573 −1.04133 −0.520664 0.853762i \(-0.674316\pi\)
−0.520664 + 0.853762i \(0.674316\pi\)
\(402\) 0 0
\(403\) 747.235 1.85418
\(404\) 0 0
\(405\) 446.716i 1.10300i
\(406\) 0 0
\(407\) −13.9920 −0.0343784
\(408\) 0 0
\(409\) 308.212i 0.753574i 0.926300 + 0.376787i \(0.122971\pi\)
−0.926300 + 0.376787i \(0.877029\pi\)
\(410\) 0 0
\(411\) 23.3696i 0.0568603i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 845.970 2.03848
\(416\) 0 0
\(417\) 49.2061 0.118000
\(418\) 0 0
\(419\) − 103.142i − 0.246163i −0.992397 0.123081i \(-0.960722\pi\)
0.992397 0.123081i \(-0.0392776\pi\)
\(420\) 0 0
\(421\) −165.220 −0.392447 −0.196224 0.980559i \(-0.562868\pi\)
−0.196224 + 0.980559i \(0.562868\pi\)
\(422\) 0 0
\(423\) 446.996i 1.05673i
\(424\) 0 0
\(425\) 71.3535i 0.167891i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −52.1909 −0.121657
\(430\) 0 0
\(431\) −594.536 −1.37943 −0.689717 0.724079i \(-0.742265\pi\)
−0.689717 + 0.724079i \(0.742265\pi\)
\(432\) 0 0
\(433\) 40.6267i 0.0938261i 0.998899 + 0.0469131i \(0.0149384\pi\)
−0.998899 + 0.0469131i \(0.985062\pi\)
\(434\) 0 0
\(435\) 16.7208 0.0384386
\(436\) 0 0
\(437\) 115.945i 0.265321i
\(438\) 0 0
\(439\) 146.600i 0.333941i 0.985962 + 0.166971i \(0.0533985\pi\)
−0.985962 + 0.166971i \(0.946602\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −107.360 −0.242349 −0.121174 0.992631i \(-0.538666\pi\)
−0.121174 + 0.992631i \(0.538666\pi\)
\(444\) 0 0
\(445\) −333.749 −0.749999
\(446\) 0 0
\(447\) − 66.2892i − 0.148298i
\(448\) 0 0
\(449\) 135.161 0.301028 0.150514 0.988608i \(-0.451907\pi\)
0.150514 + 0.988608i \(0.451907\pi\)
\(450\) 0 0
\(451\) 133.011i 0.294925i
\(452\) 0 0
\(453\) − 65.8514i − 0.145367i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 159.735 0.349530 0.174765 0.984610i \(-0.444083\pi\)
0.174765 + 0.984610i \(0.444083\pi\)
\(458\) 0 0
\(459\) −47.1838 −0.102797
\(460\) 0 0
\(461\) − 310.250i − 0.672993i −0.941685 0.336497i \(-0.890758\pi\)
0.941685 0.336497i \(-0.109242\pi\)
\(462\) 0 0
\(463\) 326.014 0.704135 0.352067 0.935975i \(-0.385479\pi\)
0.352067 + 0.935975i \(0.385479\pi\)
\(464\) 0 0
\(465\) 232.476i 0.499949i
\(466\) 0 0
\(467\) 595.558i 1.27529i 0.770332 + 0.637643i \(0.220090\pi\)
−0.770332 + 0.637643i \(0.779910\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.06688 0.0128809
\(472\) 0 0
\(473\) 49.8823 0.105459
\(474\) 0 0
\(475\) − 79.3262i − 0.167002i
\(476\) 0 0
\(477\) 474.926 0.995652
\(478\) 0 0
\(479\) − 506.680i − 1.05779i −0.848688 0.528894i \(-0.822607\pi\)
0.848688 0.528894i \(-0.177393\pi\)
\(480\) 0 0
\(481\) 44.9736i 0.0935002i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 674.985 1.39172
\(486\) 0 0
\(487\) −211.302 −0.433884 −0.216942 0.976184i \(-0.569608\pi\)
−0.216942 + 0.976184i \(0.569608\pi\)
\(488\) 0 0
\(489\) 159.255i 0.325676i
\(490\) 0 0
\(491\) 784.161 1.59707 0.798534 0.601949i \(-0.205609\pi\)
0.798534 + 0.601949i \(0.205609\pi\)
\(492\) 0 0
\(493\) − 13.2198i − 0.0268151i
\(494\) 0 0
\(495\) 267.678i 0.540764i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 171.492 0.343672 0.171836 0.985126i \(-0.445030\pi\)
0.171836 + 0.985126i \(0.445030\pi\)
\(500\) 0 0
\(501\) 121.133 0.241782
\(502\) 0 0
\(503\) − 20.0883i − 0.0399370i −0.999801 0.0199685i \(-0.993643\pi\)
0.999801 0.0199685i \(-0.00635659\pi\)
\(504\) 0 0
\(505\) 395.735 0.783634
\(506\) 0 0
\(507\) 46.5068i 0.0917294i
\(508\) 0 0
\(509\) 476.764i 0.936668i 0.883551 + 0.468334i \(0.155146\pi\)
−0.883551 + 0.468334i \(0.844854\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 52.4558 0.102253
\(514\) 0 0
\(515\) −796.904 −1.54739
\(516\) 0 0
\(517\) 250.613i 0.484744i
\(518\) 0 0
\(519\) 117.905 0.227176
\(520\) 0 0
\(521\) 854.274i 1.63968i 0.572592 + 0.819841i \(0.305938\pi\)
−0.572592 + 0.819841i \(0.694062\pi\)
\(522\) 0 0
\(523\) − 593.002i − 1.13385i −0.823771 0.566923i \(-0.808134\pi\)
0.823771 0.566923i \(-0.191866\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 183.801 0.348769
\(528\) 0 0
\(529\) 239.838 0.453379
\(530\) 0 0
\(531\) − 328.555i − 0.618748i
\(532\) 0 0
\(533\) 427.529 0.802118
\(534\) 0 0
\(535\) 753.352i 1.40814i
\(536\) 0 0
\(537\) 132.858i 0.247408i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 855.191 1.58076 0.790380 0.612617i \(-0.209883\pi\)
0.790380 + 0.612617i \(0.209883\pi\)
\(542\) 0 0
\(543\) 111.265 0.204908
\(544\) 0 0
\(545\) − 963.546i − 1.76797i
\(546\) 0 0
\(547\) −415.897 −0.760323 −0.380161 0.924920i \(-0.624132\pi\)
−0.380161 + 0.924920i \(0.624132\pi\)
\(548\) 0 0
\(549\) 767.986i 1.39888i
\(550\) 0 0
\(551\) 14.6969i 0.0266732i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −13.9920 −0.0252108
\(556\) 0 0
\(557\) −584.220 −1.04887 −0.524435 0.851451i \(-0.675723\pi\)
−0.524435 + 0.851451i \(0.675723\pi\)
\(558\) 0 0
\(559\) − 160.333i − 0.286822i
\(560\) 0 0
\(561\) −12.8377 −0.0228835
\(562\) 0 0
\(563\) − 911.147i − 1.61838i −0.587548 0.809189i \(-0.699907\pi\)
0.587548 0.809189i \(-0.300093\pi\)
\(564\) 0 0
\(565\) 229.063i 0.405422i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 699.999 1.23023 0.615113 0.788439i \(-0.289110\pi\)
0.615113 + 0.788439i \(0.289110\pi\)
\(570\) 0 0
\(571\) 562.463 0.985049 0.492525 0.870299i \(-0.336074\pi\)
0.492525 + 0.870299i \(0.336074\pi\)
\(572\) 0 0
\(573\) 177.993i 0.310634i
\(574\) 0 0
\(575\) −526.014 −0.914807
\(576\) 0 0
\(577\) − 661.659i − 1.14672i −0.819302 0.573362i \(-0.805639\pi\)
0.819302 0.573362i \(-0.194361\pi\)
\(578\) 0 0
\(579\) − 110.728i − 0.191240i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 266.272 0.456727
\(584\) 0 0
\(585\) 860.382 1.47074
\(586\) 0 0
\(587\) − 823.029i − 1.40209i −0.713116 0.701046i \(-0.752717\pi\)
0.713116 0.701046i \(-0.247283\pi\)
\(588\) 0 0
\(589\) −204.338 −0.346924
\(590\) 0 0
\(591\) 129.930i 0.219848i
\(592\) 0 0
\(593\) 622.256i 1.04934i 0.851307 + 0.524668i \(0.175811\pi\)
−0.851307 + 0.524668i \(0.824189\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −250.176 −0.419055
\(598\) 0 0
\(599\) 512.845 0.856168 0.428084 0.903739i \(-0.359189\pi\)
0.428084 + 0.903739i \(0.359189\pi\)
\(600\) 0 0
\(601\) − 680.160i − 1.13171i −0.824504 0.565857i \(-0.808546\pi\)
0.824504 0.565857i \(-0.191454\pi\)
\(602\) 0 0
\(603\) 293.927 0.487441
\(604\) 0 0
\(605\) − 652.278i − 1.07815i
\(606\) 0 0
\(607\) 38.7381i 0.0638189i 0.999491 + 0.0319095i \(0.0101588\pi\)
−0.999491 + 0.0319095i \(0.989841\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 805.529 1.31838
\(612\) 0 0
\(613\) 401.103 0.654329 0.327164 0.944967i \(-0.393907\pi\)
0.327164 + 0.944967i \(0.393907\pi\)
\(614\) 0 0
\(615\) 133.011i 0.216278i
\(616\) 0 0
\(617\) −959.044 −1.55437 −0.777183 0.629275i \(-0.783352\pi\)
−0.777183 + 0.629275i \(0.783352\pi\)
\(618\) 0 0
\(619\) 1004.53i 1.62283i 0.584469 + 0.811416i \(0.301303\pi\)
−0.584469 + 0.811416i \(0.698697\pi\)
\(620\) 0 0
\(621\) − 347.836i − 0.560123i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −739.382 −1.18301
\(626\) 0 0
\(627\) 14.2721 0.0227625
\(628\) 0 0
\(629\) 11.0624i 0.0175873i
\(630\) 0 0
\(631\) 386.514 0.612542 0.306271 0.951944i \(-0.400919\pi\)
0.306271 + 0.951944i \(0.400919\pi\)
\(632\) 0 0
\(633\) 261.200i 0.412639i
\(634\) 0 0
\(635\) 1640.11i 2.58284i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −309.088 −0.483706
\(640\) 0 0
\(641\) −992.147 −1.54781 −0.773906 0.633301i \(-0.781700\pi\)
−0.773906 + 0.633301i \(0.781700\pi\)
\(642\) 0 0
\(643\) 944.986i 1.46965i 0.678256 + 0.734826i \(0.262736\pi\)
−0.678256 + 0.734826i \(0.737264\pi\)
\(644\) 0 0
\(645\) 49.8823 0.0773368
\(646\) 0 0
\(647\) − 2.89088i − 0.00446812i −0.999998 0.00223406i \(-0.999289\pi\)
0.999998 0.00223406i \(-0.000711124\pi\)
\(648\) 0 0
\(649\) − 184.208i − 0.283833i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −323.059 −0.494730 −0.247365 0.968922i \(-0.579565\pi\)
−0.247365 + 0.968922i \(0.579565\pi\)
\(654\) 0 0
\(655\) 979.036 1.49471
\(656\) 0 0
\(657\) 446.384i 0.679428i
\(658\) 0 0
\(659\) −295.955 −0.449098 −0.224549 0.974463i \(-0.572091\pi\)
−0.224549 + 0.974463i \(0.572091\pi\)
\(660\) 0 0
\(661\) 20.7511i 0.0313935i 0.999877 + 0.0156968i \(0.00499664\pi\)
−0.999877 + 0.0156968i \(0.995003\pi\)
\(662\) 0 0
\(663\) 41.2633i 0.0622373i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 97.4558 0.146111
\(668\) 0 0
\(669\) 88.4104 0.132153
\(670\) 0 0
\(671\) 430.579i 0.641698i
\(672\) 0 0
\(673\) 627.044 0.931714 0.465857 0.884860i \(-0.345746\pi\)
0.465857 + 0.884860i \(0.345746\pi\)
\(674\) 0 0
\(675\) 237.979i 0.352561i
\(676\) 0 0
\(677\) 109.246i 0.161368i 0.996740 + 0.0806838i \(0.0257104\pi\)
−0.996740 + 0.0806838i \(0.974290\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 54.7948 0.0804623
\(682\) 0 0
\(683\) −793.566 −1.16188 −0.580941 0.813946i \(-0.697315\pi\)
−0.580941 + 0.813946i \(0.697315\pi\)
\(684\) 0 0
\(685\) − 215.996i − 0.315323i
\(686\) 0 0
\(687\) −256.331 −0.373116
\(688\) 0 0
\(689\) − 855.862i − 1.24218i
\(690\) 0 0
\(691\) 183.889i 0.266121i 0.991108 + 0.133060i \(0.0424804\pi\)
−0.991108 + 0.133060i \(0.957520\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −454.794 −0.654380
\(696\) 0 0
\(697\) 105.161 0.150877
\(698\) 0 0
\(699\) 195.913i 0.280277i
\(700\) 0 0
\(701\) −1043.82 −1.48905 −0.744525 0.667595i \(-0.767324\pi\)
−0.744525 + 0.667595i \(0.767324\pi\)
\(702\) 0 0
\(703\) − 12.2984i − 0.0174942i
\(704\) 0 0
\(705\) 250.613i 0.355479i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −980.558 −1.38301 −0.691507 0.722369i \(-0.743053\pi\)
−0.691507 + 0.722369i \(0.743053\pi\)
\(710\) 0 0
\(711\) 286.690 0.403221
\(712\) 0 0
\(713\) 1354.97i 1.90038i
\(714\) 0 0
\(715\) 482.382 0.674660
\(716\) 0 0
\(717\) − 190.195i − 0.265265i
\(718\) 0 0
\(719\) − 778.484i − 1.08273i −0.840787 0.541366i \(-0.817907\pi\)
0.840787 0.541366i \(-0.182093\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −62.8751 −0.0869642
\(724\) 0 0
\(725\) −66.6762 −0.0919672
\(726\) 0 0
\(727\) − 735.255i − 1.01135i −0.862723 0.505677i \(-0.831243\pi\)
0.862723 0.505677i \(-0.168757\pi\)
\(728\) 0 0
\(729\) 490.632 0.673021
\(730\) 0 0
\(731\) − 39.4380i − 0.0539508i
\(732\) 0 0
\(733\) 478.860i 0.653288i 0.945147 + 0.326644i \(0.105918\pi\)
−0.945147 + 0.326644i \(0.894082\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 164.793 0.223600
\(738\) 0 0
\(739\) 19.9045 0.0269344 0.0134672 0.999909i \(-0.495713\pi\)
0.0134672 + 0.999909i \(0.495713\pi\)
\(740\) 0 0
\(741\) − 45.8739i − 0.0619080i
\(742\) 0 0
\(743\) −43.3095 −0.0582901 −0.0291450 0.999575i \(-0.509278\pi\)
−0.0291450 + 0.999575i \(0.509278\pi\)
\(744\) 0 0
\(745\) 612.687i 0.822399i
\(746\) 0 0
\(747\) − 1082.53i − 1.44917i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −225.330 −0.300040 −0.150020 0.988683i \(-0.547934\pi\)
−0.150020 + 0.988683i \(0.547934\pi\)
\(752\) 0 0
\(753\) −355.602 −0.472247
\(754\) 0 0
\(755\) 608.641i 0.806147i
\(756\) 0 0
\(757\) 935.779 1.23617 0.618084 0.786112i \(-0.287909\pi\)
0.618084 + 0.786112i \(0.287909\pi\)
\(758\) 0 0
\(759\) − 94.6386i − 0.124689i
\(760\) 0 0
\(761\) − 1402.71i − 1.84325i −0.388081 0.921625i \(-0.626862\pi\)
0.388081 0.921625i \(-0.373138\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 211.632 0.276644
\(766\) 0 0
\(767\) −592.087 −0.771952
\(768\) 0 0
\(769\) − 1.72330i − 0.00224097i −0.999999 0.00112048i \(-0.999643\pi\)
0.999999 0.00112048i \(-0.000356661\pi\)
\(770\) 0 0
\(771\) −287.360 −0.372711
\(772\) 0 0
\(773\) 224.258i 0.290113i 0.989423 + 0.145057i \(0.0463364\pi\)
−0.989423 + 0.145057i \(0.953664\pi\)
\(774\) 0 0
\(775\) − 927.029i − 1.19617i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −116.912 −0.150079
\(780\) 0 0
\(781\) −173.294 −0.221887
\(782\) 0 0
\(783\) − 44.0908i − 0.0563101i
\(784\) 0 0
\(785\) −56.0740 −0.0714319
\(786\) 0 0
\(787\) − 70.2034i − 0.0892038i −0.999005 0.0446019i \(-0.985798\pi\)
0.999005 0.0446019i \(-0.0142019\pi\)
\(788\) 0 0
\(789\) − 23.2057i − 0.0294116i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1383.98 1.74525
\(794\) 0 0
\(795\) 266.272 0.334933
\(796\) 0 0
\(797\) − 1305.38i − 1.63787i −0.573889 0.818933i \(-0.694566\pi\)
0.573889 0.818933i \(-0.305434\pi\)
\(798\) 0 0
\(799\) 198.140 0.247985
\(800\) 0 0
\(801\) 427.076i 0.533179i
\(802\) 0 0
\(803\) 250.270i 0.311668i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −220.227 −0.272897
\(808\) 0 0
\(809\) 762.765 0.942849 0.471424 0.881907i \(-0.343740\pi\)
0.471424 + 0.881907i \(0.343740\pi\)
\(810\) 0 0
\(811\) 1214.98i 1.49813i 0.662498 + 0.749064i \(0.269496\pi\)
−0.662498 + 0.749064i \(0.730504\pi\)
\(812\) 0 0
\(813\) −54.5147 −0.0670538
\(814\) 0 0
\(815\) − 1471.94i − 1.80606i
\(816\) 0 0
\(817\) 43.8446i 0.0536654i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −583.368 −0.710557 −0.355279 0.934760i \(-0.615614\pi\)
−0.355279 + 0.934760i \(0.615614\pi\)
\(822\) 0 0
\(823\) −1030.74 −1.25242 −0.626210 0.779655i \(-0.715395\pi\)
−0.626210 + 0.779655i \(0.715395\pi\)
\(824\) 0 0
\(825\) 64.7487i 0.0784833i
\(826\) 0 0
\(827\) −152.102 −0.183920 −0.0919599 0.995763i \(-0.529313\pi\)
−0.0919599 + 0.995763i \(0.529313\pi\)
\(828\) 0 0
\(829\) 614.410i 0.741146i 0.928803 + 0.370573i \(0.120839\pi\)
−0.928803 + 0.370573i \(0.879161\pi\)
\(830\) 0 0
\(831\) − 199.743i − 0.240365i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1119.59 −1.34082
\(836\) 0 0
\(837\) 613.014 0.732395
\(838\) 0 0
\(839\) − 1546.14i − 1.84284i −0.388568 0.921420i \(-0.627030\pi\)
0.388568 0.921420i \(-0.372970\pi\)
\(840\) 0 0
\(841\) −828.647 −0.985311
\(842\) 0 0
\(843\) − 283.166i − 0.335903i
\(844\) 0 0
\(845\) − 429.846i − 0.508693i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −105.030 −0.123711
\(850\) 0 0
\(851\) −81.5513 −0.0958300
\(852\) 0 0
\(853\) 1235.15i 1.44800i 0.689798 + 0.724002i \(0.257699\pi\)
−0.689798 + 0.724002i \(0.742301\pi\)
\(854\) 0 0
\(855\) −235.279 −0.275180
\(856\) 0 0
\(857\) − 1100.68i − 1.28434i −0.766561 0.642172i \(-0.778034\pi\)
0.766561 0.642172i \(-0.221966\pi\)
\(858\) 0 0
\(859\) 591.771i 0.688906i 0.938803 + 0.344453i \(0.111936\pi\)
−0.938803 + 0.344453i \(0.888064\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 64.7271 0.0750024 0.0375012 0.999297i \(-0.488060\pi\)
0.0375012 + 0.999297i \(0.488060\pi\)
\(864\) 0 0
\(865\) −1089.75 −1.25983
\(866\) 0 0
\(867\) − 197.190i − 0.227440i
\(868\) 0 0
\(869\) 160.736 0.184967
\(870\) 0 0
\(871\) − 529.684i − 0.608133i
\(872\) 0 0
\(873\) − 863.732i − 0.989384i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −304.192 −0.346855 −0.173427 0.984847i \(-0.555484\pi\)
−0.173427 + 0.984847i \(0.555484\pi\)
\(878\) 0 0
\(879\) −215.001 −0.244597
\(880\) 0 0
\(881\) − 863.732i − 0.980400i −0.871610 0.490200i \(-0.836924\pi\)
0.871610 0.490200i \(-0.163076\pi\)
\(882\) 0 0
\(883\) 567.456 0.642645 0.321323 0.946970i \(-0.395873\pi\)
0.321323 + 0.946970i \(0.395873\pi\)
\(884\) 0 0
\(885\) − 184.208i − 0.208144i
\(886\) 0 0
\(887\) − 889.761i − 1.00311i −0.865125 0.501556i \(-0.832761\pi\)
0.865125 0.501556i \(-0.167239\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 320.492 0.359699
\(892\) 0 0
\(893\) −220.279 −0.246673
\(894\) 0 0
\(895\) − 1227.96i − 1.37202i
\(896\) 0 0
\(897\) −304.191 −0.339120
\(898\) 0 0
\(899\) 171.753i 0.191049i
\(900\) 0 0
\(901\) − 210.521i − 0.233652i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1028.38 −1.13633
\(906\) 0 0
\(907\) −373.978 −0.412324 −0.206162 0.978518i \(-0.566097\pi\)
−0.206162 + 0.978518i \(0.566097\pi\)
\(908\) 0 0
\(909\) − 506.395i − 0.557091i
\(910\) 0 0
\(911\) −1133.75 −1.24451 −0.622256 0.782814i \(-0.713784\pi\)
−0.622256 + 0.782814i \(0.713784\pi\)
\(912\) 0 0
\(913\) − 606.932i − 0.664766i
\(914\) 0 0
\(915\) 430.579i 0.470578i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −456.302 −0.496521 −0.248260 0.968693i \(-0.579859\pi\)
−0.248260 + 0.968693i \(0.579859\pi\)
\(920\) 0 0
\(921\) −15.0580 −0.0163496
\(922\) 0 0
\(923\) 557.007i 0.603474i
\(924\) 0 0
\(925\) 55.7948 0.0603187
\(926\) 0 0
\(927\) 1019.74i 1.10005i
\(928\) 0 0
\(929\) 951.540i 1.02426i 0.858907 + 0.512131i \(0.171144\pi\)
−0.858907 + 0.512131i \(0.828856\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −130.602 −0.139981
\(934\) 0 0
\(935\) 118.654 0.126903
\(936\) 0 0
\(937\) 1295.71i 1.38283i 0.722460 + 0.691413i \(0.243011\pi\)
−0.722460 + 0.691413i \(0.756989\pi\)
\(938\) 0 0
\(939\) 70.2578 0.0748219
\(940\) 0 0
\(941\) − 1356.89i − 1.44197i −0.692952 0.720984i \(-0.743690\pi\)
0.692952 0.720984i \(-0.256310\pi\)
\(942\) 0 0
\(943\) 775.245i 0.822105i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 709.462 0.749168 0.374584 0.927193i \(-0.377786\pi\)
0.374584 + 0.927193i \(0.377786\pi\)
\(948\) 0 0
\(949\) 804.426 0.847657
\(950\) 0 0
\(951\) 345.784i 0.363601i
\(952\) 0 0
\(953\) 936.603 0.982794 0.491397 0.870936i \(-0.336486\pi\)
0.491397 + 0.870936i \(0.336486\pi\)
\(954\) 0 0
\(955\) − 1645.13i − 1.72265i
\(956\) 0 0
\(957\) − 11.9961i − 0.0125351i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1426.95 −1.48486
\(962\) 0 0
\(963\) 964.014 1.00105
\(964\) 0 0
\(965\) 1023.42i 1.06054i
\(966\) 0 0
\(967\) −1374.37 −1.42127 −0.710635 0.703561i \(-0.751592\pi\)
−0.710635 + 0.703561i \(0.751592\pi\)
\(968\) 0 0
\(969\) − 11.2838i − 0.0116448i
\(970\) 0 0
\(971\) − 31.4617i − 0.0324014i −0.999869 0.0162007i \(-0.994843\pi\)
0.999869 0.0162007i \(-0.00515706\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 208.118 0.213454
\(976\) 0 0
\(977\) −541.897 −0.554654 −0.277327 0.960776i \(-0.589449\pi\)
−0.277327 + 0.960776i \(0.589449\pi\)
\(978\) 0 0
\(979\) 239.445i 0.244581i
\(980\) 0 0
\(981\) −1232.98 −1.25687
\(982\) 0 0
\(983\) − 22.4198i − 0.0228075i −0.999935 0.0114038i \(-0.996370\pi\)
0.999935 0.0114038i \(-0.00363001\pi\)
\(984\) 0 0
\(985\) − 1200.90i − 1.21918i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 290.735 0.293969
\(990\) 0 0
\(991\) −678.035 −0.684193 −0.342096 0.939665i \(-0.611137\pi\)
−0.342096 + 0.939665i \(0.611137\pi\)
\(992\) 0 0
\(993\) 161.429i 0.162567i
\(994\) 0 0
\(995\) 2312.28 2.32390
\(996\) 0 0
\(997\) 876.163i 0.878799i 0.898292 + 0.439400i \(0.144809\pi\)
−0.898292 + 0.439400i \(0.855191\pi\)
\(998\) 0 0
\(999\) 36.8953i 0.0369322i
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 784.3.c.e.97.2 4
4.3 odd 2 98.3.b.b.97.4 4
7.2 even 3 784.3.s.c.129.2 4
7.3 odd 6 784.3.s.c.705.2 4
7.4 even 3 112.3.s.b.33.1 4
7.5 odd 6 112.3.s.b.17.1 4
7.6 odd 2 inner 784.3.c.e.97.3 4
12.11 even 2 882.3.c.f.685.1 4
21.5 even 6 1008.3.cg.l.577.2 4
21.11 odd 6 1008.3.cg.l.145.2 4
28.3 even 6 98.3.d.a.19.1 4
28.11 odd 6 14.3.d.a.5.1 yes 4
28.19 even 6 14.3.d.a.3.1 4
28.23 odd 6 98.3.d.a.31.1 4
28.27 even 2 98.3.b.b.97.3 4
56.5 odd 6 448.3.s.c.129.2 4
56.11 odd 6 448.3.s.d.257.1 4
56.19 even 6 448.3.s.d.129.1 4
56.53 even 6 448.3.s.c.257.2 4
84.11 even 6 126.3.n.c.19.2 4
84.23 even 6 882.3.n.b.325.2 4
84.47 odd 6 126.3.n.c.73.2 4
84.59 odd 6 882.3.n.b.19.2 4
84.83 odd 2 882.3.c.f.685.2 4
140.19 even 6 350.3.k.a.101.2 4
140.39 odd 6 350.3.k.a.201.2 4
140.47 odd 12 350.3.i.a.199.3 8
140.67 even 12 350.3.i.a.299.2 8
140.103 odd 12 350.3.i.a.199.2 8
140.123 even 12 350.3.i.a.299.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.3.d.a.3.1 4 28.19 even 6
14.3.d.a.5.1 yes 4 28.11 odd 6
98.3.b.b.97.3 4 28.27 even 2
98.3.b.b.97.4 4 4.3 odd 2
98.3.d.a.19.1 4 28.3 even 6
98.3.d.a.31.1 4 28.23 odd 6
112.3.s.b.17.1 4 7.5 odd 6
112.3.s.b.33.1 4 7.4 even 3
126.3.n.c.19.2 4 84.11 even 6
126.3.n.c.73.2 4 84.47 odd 6
350.3.i.a.199.2 8 140.103 odd 12
350.3.i.a.199.3 8 140.47 odd 12
350.3.i.a.299.2 8 140.67 even 12
350.3.i.a.299.3 8 140.123 even 12
350.3.k.a.101.2 4 140.19 even 6
350.3.k.a.201.2 4 140.39 odd 6
448.3.s.c.129.2 4 56.5 odd 6
448.3.s.c.257.2 4 56.53 even 6
448.3.s.d.129.1 4 56.19 even 6
448.3.s.d.257.1 4 56.11 odd 6
784.3.c.e.97.2 4 1.1 even 1 trivial
784.3.c.e.97.3 4 7.6 odd 2 inner
784.3.s.c.129.2 4 7.2 even 3
784.3.s.c.705.2 4 7.3 odd 6
882.3.c.f.685.1 4 12.11 even 2
882.3.c.f.685.2 4 84.83 odd 2
882.3.n.b.19.2 4 84.59 odd 6
882.3.n.b.325.2 4 84.23 even 6
1008.3.cg.l.145.2 4 21.11 odd 6
1008.3.cg.l.577.2 4 21.5 even 6