Properties

Label 2352.3.m.g.1471.3
Level $2352$
Weight $3$
Character 2352.1471
Analytic conductor $64.087$
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] = [2352,3,Mod(1471,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), 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: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.3
Root \(2.13746 + 0.656712i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.g.1471.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} -4.27492 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} -4.27492 q^{5} -3.00000 q^{9} -3.10302i q^{11} +2.72508 q^{13} -7.40437i q^{15} +5.09967 q^{17} +25.7348i q^{19} +12.1819i q^{23} -6.72508 q^{25} -5.19615i q^{27} +41.0241 q^{29} -0.172632i q^{31} +5.37459 q^{33} -16.9244 q^{37} +4.71998i q^{39} +36.7492 q^{41} -53.3325i q^{43} +12.8248 q^{45} +24.2487i q^{47} +8.83289i q^{51} -103.375 q^{53} +13.2651i q^{55} -44.5739 q^{57} +29.2564i q^{59} -32.9003 q^{61} -11.6495 q^{65} +17.0170i q^{67} -21.0997 q^{69} +76.7857i q^{71} +99.2749 q^{73} -11.6482i q^{75} -87.7275i q^{79} +9.00000 q^{81} +151.452i q^{83} -21.8007 q^{85} +71.0558i q^{87} -68.5498 q^{89} +0.299007 q^{93} -110.014i q^{95} -104.522 q^{97} +9.30906i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 12 q^{9} + 26 q^{13} - 40 q^{17} - 42 q^{25} - 2 q^{29} - 54 q^{33} + 38 q^{37} - 4 q^{41} + 6 q^{45} - 338 q^{53} + 18 q^{57} - 192 q^{61} + 44 q^{65} - 24 q^{69} + 382 q^{73} + 36 q^{81} - 208 q^{85} - 244 q^{89} - 180 q^{93} + 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) −4.27492 −0.854983 −0.427492 0.904019i \(-0.640603\pi\)
−0.427492 + 0.904019i \(0.640603\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 3.10302i − 0.282093i −0.990003 0.141046i \(-0.954953\pi\)
0.990003 0.141046i \(-0.0450466\pi\)
\(12\) 0 0
\(13\) 2.72508 0.209622 0.104811 0.994492i \(-0.466576\pi\)
0.104811 + 0.994492i \(0.466576\pi\)
\(14\) 0 0
\(15\) − 7.40437i − 0.493625i
\(16\) 0 0
\(17\) 5.09967 0.299981 0.149990 0.988687i \(-0.452076\pi\)
0.149990 + 0.988687i \(0.452076\pi\)
\(18\) 0 0
\(19\) 25.7348i 1.35446i 0.735771 + 0.677231i \(0.236820\pi\)
−0.735771 + 0.677231i \(0.763180\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 12.1819i 0.529648i 0.964297 + 0.264824i \(0.0853138\pi\)
−0.964297 + 0.264824i \(0.914686\pi\)
\(24\) 0 0
\(25\) −6.72508 −0.269003
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) 41.0241 1.41462 0.707312 0.706902i \(-0.249908\pi\)
0.707312 + 0.706902i \(0.249908\pi\)
\(30\) 0 0
\(31\) − 0.172632i − 0.00556876i −0.999996 0.00278438i \(-0.999114\pi\)
0.999996 0.00278438i \(-0.000886297\pi\)
\(32\) 0 0
\(33\) 5.37459 0.162866
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −16.9244 −0.457417 −0.228708 0.973495i \(-0.573450\pi\)
−0.228708 + 0.973495i \(0.573450\pi\)
\(38\) 0 0
\(39\) 4.71998i 0.121025i
\(40\) 0 0
\(41\) 36.7492 0.896321 0.448161 0.893953i \(-0.352079\pi\)
0.448161 + 0.893953i \(0.352079\pi\)
\(42\) 0 0
\(43\) − 53.3325i − 1.24029i −0.784487 0.620145i \(-0.787074\pi\)
0.784487 0.620145i \(-0.212926\pi\)
\(44\) 0 0
\(45\) 12.8248 0.284994
\(46\) 0 0
\(47\) 24.2487i 0.515930i 0.966154 + 0.257965i \(0.0830519\pi\)
−0.966154 + 0.257965i \(0.916948\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 8.83289i 0.173194i
\(52\) 0 0
\(53\) −103.375 −1.95046 −0.975232 0.221185i \(-0.929008\pi\)
−0.975232 + 0.221185i \(0.929008\pi\)
\(54\) 0 0
\(55\) 13.2651i 0.241185i
\(56\) 0 0
\(57\) −44.5739 −0.781999
\(58\) 0 0
\(59\) 29.2564i 0.495871i 0.968776 + 0.247936i \(0.0797522\pi\)
−0.968776 + 0.247936i \(0.920248\pi\)
\(60\) 0 0
\(61\) −32.9003 −0.539350 −0.269675 0.962951i \(-0.586916\pi\)
−0.269675 + 0.962951i \(0.586916\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11.6495 −0.179223
\(66\) 0 0
\(67\) 17.0170i 0.253985i 0.991904 + 0.126992i \(0.0405324\pi\)
−0.991904 + 0.126992i \(0.959468\pi\)
\(68\) 0 0
\(69\) −21.0997 −0.305792
\(70\) 0 0
\(71\) 76.7857i 1.08149i 0.841187 + 0.540744i \(0.181857\pi\)
−0.841187 + 0.540744i \(0.818143\pi\)
\(72\) 0 0
\(73\) 99.2749 1.35993 0.679965 0.733244i \(-0.261995\pi\)
0.679965 + 0.733244i \(0.261995\pi\)
\(74\) 0 0
\(75\) − 11.6482i − 0.155309i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 87.7275i − 1.11047i −0.831692 0.555237i \(-0.812627\pi\)
0.831692 0.555237i \(-0.187373\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 151.452i 1.82473i 0.409381 + 0.912363i \(0.365745\pi\)
−0.409381 + 0.912363i \(0.634255\pi\)
\(84\) 0 0
\(85\) −21.8007 −0.256478
\(86\) 0 0
\(87\) 71.0558i 0.816733i
\(88\) 0 0
\(89\) −68.5498 −0.770223 −0.385111 0.922870i \(-0.625837\pi\)
−0.385111 + 0.922870i \(0.625837\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.299007 0.00321512
\(94\) 0 0
\(95\) − 110.014i − 1.15804i
\(96\) 0 0
\(97\) −104.522 −1.07755 −0.538775 0.842449i \(-0.681113\pi\)
−0.538775 + 0.842449i \(0.681113\pi\)
\(98\) 0 0
\(99\) 9.30906i 0.0940309i
\(100\) 0 0
\(101\) −157.698 −1.56136 −0.780682 0.624929i \(-0.785128\pi\)
−0.780682 + 0.624929i \(0.785128\pi\)
\(102\) 0 0
\(103\) − 120.448i − 1.16940i −0.811250 0.584699i \(-0.801213\pi\)
0.811250 0.584699i \(-0.198787\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 104.022i − 0.972171i −0.873911 0.486086i \(-0.838424\pi\)
0.873911 0.486086i \(-0.161576\pi\)
\(108\) 0 0
\(109\) −103.625 −0.950692 −0.475346 0.879799i \(-0.657677\pi\)
−0.475346 + 0.879799i \(0.657677\pi\)
\(110\) 0 0
\(111\) − 29.3140i − 0.264090i
\(112\) 0 0
\(113\) −177.849 −1.57388 −0.786942 0.617027i \(-0.788337\pi\)
−0.786942 + 0.617027i \(0.788337\pi\)
\(114\) 0 0
\(115\) − 52.0766i − 0.452840i
\(116\) 0 0
\(117\) −8.17525 −0.0698739
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 111.371 0.920424
\(122\) 0 0
\(123\) 63.6514i 0.517491i
\(124\) 0 0
\(125\) 135.622 1.08498
\(126\) 0 0
\(127\) − 171.672i − 1.35174i −0.737019 0.675872i \(-0.763767\pi\)
0.737019 0.675872i \(-0.236233\pi\)
\(128\) 0 0
\(129\) 92.3746 0.716082
\(130\) 0 0
\(131\) − 125.152i − 0.955360i −0.878534 0.477680i \(-0.841478\pi\)
0.878534 0.477680i \(-0.158522\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 22.2131i 0.164542i
\(136\) 0 0
\(137\) −180.949 −1.32079 −0.660396 0.750918i \(-0.729612\pi\)
−0.660396 + 0.750918i \(0.729612\pi\)
\(138\) 0 0
\(139\) 155.204i 1.11658i 0.829647 + 0.558288i \(0.188542\pi\)
−0.829647 + 0.558288i \(0.811458\pi\)
\(140\) 0 0
\(141\) −42.0000 −0.297872
\(142\) 0 0
\(143\) − 8.45598i − 0.0591327i
\(144\) 0 0
\(145\) −175.375 −1.20948
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −39.1478 −0.262737 −0.131369 0.991334i \(-0.541937\pi\)
−0.131369 + 0.991334i \(0.541937\pi\)
\(150\) 0 0
\(151\) − 135.084i − 0.894597i −0.894385 0.447298i \(-0.852386\pi\)
0.894385 0.447298i \(-0.147614\pi\)
\(152\) 0 0
\(153\) −15.2990 −0.0999935
\(154\) 0 0
\(155\) 0.737986i 0.00476120i
\(156\) 0 0
\(157\) 56.0482 0.356995 0.178497 0.983940i \(-0.442876\pi\)
0.178497 + 0.983940i \(0.442876\pi\)
\(158\) 0 0
\(159\) − 179.050i − 1.12610i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 25.5780i 0.156920i 0.996917 + 0.0784600i \(0.0250003\pi\)
−0.996917 + 0.0784600i \(0.975000\pi\)
\(164\) 0 0
\(165\) −22.9759 −0.139248
\(166\) 0 0
\(167\) − 77.9998i − 0.467065i −0.972349 0.233532i \(-0.924971\pi\)
0.972349 0.233532i \(-0.0750285\pi\)
\(168\) 0 0
\(169\) −161.574 −0.956059
\(170\) 0 0
\(171\) − 77.2043i − 0.451487i
\(172\) 0 0
\(173\) −287.148 −1.65981 −0.829907 0.557902i \(-0.811607\pi\)
−0.829907 + 0.557902i \(0.811607\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −50.6736 −0.286291
\(178\) 0 0
\(179\) − 65.8496i − 0.367875i −0.982938 0.183937i \(-0.941116\pi\)
0.982938 0.183937i \(-0.0588844\pi\)
\(180\) 0 0
\(181\) 205.323 1.13438 0.567191 0.823586i \(-0.308030\pi\)
0.567191 + 0.823586i \(0.308030\pi\)
\(182\) 0 0
\(183\) − 56.9850i − 0.311394i
\(184\) 0 0
\(185\) 72.3505 0.391084
\(186\) 0 0
\(187\) − 15.8244i − 0.0846223i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 98.9830i 0.518235i 0.965846 + 0.259118i \(0.0834318\pi\)
−0.965846 + 0.259118i \(0.916568\pi\)
\(192\) 0 0
\(193\) −54.5531 −0.282659 −0.141329 0.989963i \(-0.545138\pi\)
−0.141329 + 0.989963i \(0.545138\pi\)
\(194\) 0 0
\(195\) − 20.1775i − 0.103475i
\(196\) 0 0
\(197\) −127.450 −0.646955 −0.323478 0.946236i \(-0.604852\pi\)
−0.323478 + 0.946236i \(0.604852\pi\)
\(198\) 0 0
\(199\) − 12.1819i − 0.0612156i −0.999531 0.0306078i \(-0.990256\pi\)
0.999531 0.0306078i \(-0.00974428\pi\)
\(200\) 0 0
\(201\) −29.4743 −0.146638
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −157.100 −0.766340
\(206\) 0 0
\(207\) − 36.5457i − 0.176549i
\(208\) 0 0
\(209\) 79.8555 0.382084
\(210\) 0 0
\(211\) − 210.243i − 0.996411i −0.867059 0.498206i \(-0.833992\pi\)
0.867059 0.498206i \(-0.166008\pi\)
\(212\) 0 0
\(213\) −132.997 −0.624398
\(214\) 0 0
\(215\) 227.992i 1.06043i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 171.949i 0.785156i
\(220\) 0 0
\(221\) 13.8970 0.0628824
\(222\) 0 0
\(223\) − 264.323i − 1.18531i −0.805458 0.592653i \(-0.798080\pi\)
0.805458 0.592653i \(-0.201920\pi\)
\(224\) 0 0
\(225\) 20.1752 0.0896678
\(226\) 0 0
\(227\) 351.878i 1.55012i 0.631885 + 0.775062i \(0.282281\pi\)
−0.631885 + 0.775062i \(0.717719\pi\)
\(228\) 0 0
\(229\) −205.371 −0.896818 −0.448409 0.893829i \(-0.648009\pi\)
−0.448409 + 0.893829i \(0.648009\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 33.8488 0.145274 0.0726370 0.997358i \(-0.476859\pi\)
0.0726370 + 0.997358i \(0.476859\pi\)
\(234\) 0 0
\(235\) − 103.661i − 0.441112i
\(236\) 0 0
\(237\) 151.949 0.641133
\(238\) 0 0
\(239\) 78.4285i 0.328153i 0.986448 + 0.164076i \(0.0524644\pi\)
−0.986448 + 0.164076i \(0.947536\pi\)
\(240\) 0 0
\(241\) 174.474 0.723960 0.361980 0.932186i \(-0.382101\pi\)
0.361980 + 0.932186i \(0.382101\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 70.1294i 0.283925i
\(248\) 0 0
\(249\) −262.323 −1.05351
\(250\) 0 0
\(251\) − 128.124i − 0.510455i −0.966881 0.255228i \(-0.917850\pi\)
0.966881 0.255228i \(-0.0821504\pi\)
\(252\) 0 0
\(253\) 37.8007 0.149410
\(254\) 0 0
\(255\) − 37.7599i − 0.148078i
\(256\) 0 0
\(257\) −285.601 −1.11129 −0.555645 0.831420i \(-0.687528\pi\)
−0.555645 + 0.831420i \(0.687528\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −123.072 −0.471541
\(262\) 0 0
\(263\) − 510.227i − 1.94003i −0.243051 0.970013i \(-0.578148\pi\)
0.243051 0.970013i \(-0.421852\pi\)
\(264\) 0 0
\(265\) 441.918 1.66761
\(266\) 0 0
\(267\) − 118.732i − 0.444688i
\(268\) 0 0
\(269\) 233.471 0.867922 0.433961 0.900932i \(-0.357116\pi\)
0.433961 + 0.900932i \(0.357116\pi\)
\(270\) 0 0
\(271\) − 317.582i − 1.17189i −0.810350 0.585945i \(-0.800723\pi\)
0.810350 0.585945i \(-0.199277\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.8681i 0.0758838i
\(276\) 0 0
\(277\) −522.265 −1.88543 −0.942717 0.333595i \(-0.891738\pi\)
−0.942717 + 0.333595i \(0.891738\pi\)
\(278\) 0 0
\(279\) 0.517895i 0.00185625i
\(280\) 0 0
\(281\) −307.752 −1.09520 −0.547602 0.836739i \(-0.684459\pi\)
−0.547602 + 0.836739i \(0.684459\pi\)
\(282\) 0 0
\(283\) − 207.888i − 0.734586i −0.930105 0.367293i \(-0.880285\pi\)
0.930105 0.367293i \(-0.119715\pi\)
\(284\) 0 0
\(285\) 190.550 0.668596
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −262.993 −0.910012
\(290\) 0 0
\(291\) − 181.038i − 0.622124i
\(292\) 0 0
\(293\) −341.670 −1.16611 −0.583055 0.812433i \(-0.698143\pi\)
−0.583055 + 0.812433i \(0.698143\pi\)
\(294\) 0 0
\(295\) − 125.069i − 0.423962i
\(296\) 0 0
\(297\) −16.1238 −0.0542887
\(298\) 0 0
\(299\) 33.1967i 0.111026i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 273.140i − 0.901453i
\(304\) 0 0
\(305\) 140.646 0.461135
\(306\) 0 0
\(307\) 179.913i 0.586036i 0.956107 + 0.293018i \(0.0946597\pi\)
−0.956107 + 0.293018i \(0.905340\pi\)
\(308\) 0 0
\(309\) 208.622 0.675152
\(310\) 0 0
\(311\) 262.665i 0.844581i 0.906461 + 0.422290i \(0.138774\pi\)
−0.906461 + 0.422290i \(0.861226\pi\)
\(312\) 0 0
\(313\) 547.042 1.74774 0.873868 0.486163i \(-0.161604\pi\)
0.873868 + 0.486163i \(0.161604\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.2815 −0.0765979 −0.0382990 0.999266i \(-0.512194\pi\)
−0.0382990 + 0.999266i \(0.512194\pi\)
\(318\) 0 0
\(319\) − 127.299i − 0.399055i
\(320\) 0 0
\(321\) 180.172 0.561283
\(322\) 0 0
\(323\) 131.239i 0.406312i
\(324\) 0 0
\(325\) −18.3264 −0.0563889
\(326\) 0 0
\(327\) − 179.484i − 0.548882i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 306.819i 0.926946i 0.886111 + 0.463473i \(0.153397\pi\)
−0.886111 + 0.463473i \(0.846603\pi\)
\(332\) 0 0
\(333\) 50.7733 0.152472
\(334\) 0 0
\(335\) − 72.7461i − 0.217153i
\(336\) 0 0
\(337\) 312.093 0.926092 0.463046 0.886334i \(-0.346756\pi\)
0.463046 + 0.886334i \(0.346756\pi\)
\(338\) 0 0
\(339\) − 308.043i − 0.908682i
\(340\) 0 0
\(341\) −0.535679 −0.00157091
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 90.1993 0.261447
\(346\) 0 0
\(347\) 380.004i 1.09511i 0.836769 + 0.547556i \(0.184442\pi\)
−0.836769 + 0.547556i \(0.815558\pi\)
\(348\) 0 0
\(349\) −201.395 −0.577064 −0.288532 0.957470i \(-0.593167\pi\)
−0.288532 + 0.957470i \(0.593167\pi\)
\(350\) 0 0
\(351\) − 14.1599i − 0.0403417i
\(352\) 0 0
\(353\) 319.746 0.905796 0.452898 0.891562i \(-0.350390\pi\)
0.452898 + 0.891562i \(0.350390\pi\)
\(354\) 0 0
\(355\) − 328.252i − 0.924655i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 547.463i − 1.52497i −0.647007 0.762484i \(-0.723980\pi\)
0.647007 0.762484i \(-0.276020\pi\)
\(360\) 0 0
\(361\) −301.278 −0.834566
\(362\) 0 0
\(363\) 192.901i 0.531407i
\(364\) 0 0
\(365\) −424.392 −1.16272
\(366\) 0 0
\(367\) 186.250i 0.507494i 0.967271 + 0.253747i \(0.0816630\pi\)
−0.967271 + 0.253747i \(0.918337\pi\)
\(368\) 0 0
\(369\) −110.248 −0.298774
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 31.8729 0.0854502 0.0427251 0.999087i \(-0.486396\pi\)
0.0427251 + 0.999087i \(0.486396\pi\)
\(374\) 0 0
\(375\) 234.904i 0.626412i
\(376\) 0 0
\(377\) 111.794 0.296536
\(378\) 0 0
\(379\) 644.960i 1.70174i 0.525375 + 0.850871i \(0.323925\pi\)
−0.525375 + 0.850871i \(0.676075\pi\)
\(380\) 0 0
\(381\) 297.344 0.780430
\(382\) 0 0
\(383\) 539.678i 1.40908i 0.709664 + 0.704540i \(0.248847\pi\)
−0.709664 + 0.704540i \(0.751153\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 159.997i 0.413430i
\(388\) 0 0
\(389\) 255.704 0.657338 0.328669 0.944445i \(-0.393400\pi\)
0.328669 + 0.944445i \(0.393400\pi\)
\(390\) 0 0
\(391\) 62.1237i 0.158884i
\(392\) 0 0
\(393\) 216.770 0.551577
\(394\) 0 0
\(395\) 375.028i 0.949438i
\(396\) 0 0
\(397\) −193.419 −0.487203 −0.243601 0.969875i \(-0.578329\pi\)
−0.243601 + 0.969875i \(0.578329\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.0000 0.0798005 0.0399002 0.999204i \(-0.487296\pi\)
0.0399002 + 0.999204i \(0.487296\pi\)
\(402\) 0 0
\(403\) − 0.470435i − 0.00116733i
\(404\) 0 0
\(405\) −38.4743 −0.0949982
\(406\) 0 0
\(407\) 52.5168i 0.129034i
\(408\) 0 0
\(409\) −370.739 −0.906453 −0.453226 0.891395i \(-0.649727\pi\)
−0.453226 + 0.891395i \(0.649727\pi\)
\(410\) 0 0
\(411\) − 313.412i − 0.762560i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 647.446i − 1.56011i
\(416\) 0 0
\(417\) −268.821 −0.644656
\(418\) 0 0
\(419\) 735.457i 1.75527i 0.479332 + 0.877634i \(0.340879\pi\)
−0.479332 + 0.877634i \(0.659121\pi\)
\(420\) 0 0
\(421\) −483.021 −1.14732 −0.573659 0.819094i \(-0.694476\pi\)
−0.573659 + 0.819094i \(0.694476\pi\)
\(422\) 0 0
\(423\) − 72.7461i − 0.171977i
\(424\) 0 0
\(425\) −34.2957 −0.0806958
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 14.6462 0.0341403
\(430\) 0 0
\(431\) − 569.117i − 1.32046i −0.751065 0.660228i \(-0.770460\pi\)
0.751065 0.660228i \(-0.229540\pi\)
\(432\) 0 0
\(433\) 45.3297 0.104688 0.0523438 0.998629i \(-0.483331\pi\)
0.0523438 + 0.998629i \(0.483331\pi\)
\(434\) 0 0
\(435\) − 303.758i − 0.698294i
\(436\) 0 0
\(437\) −313.498 −0.717388
\(438\) 0 0
\(439\) − 59.1041i − 0.134633i −0.997732 0.0673167i \(-0.978556\pi\)
0.997732 0.0673167i \(-0.0214438\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 218.191i − 0.492530i −0.969202 0.246265i \(-0.920797\pi\)
0.969202 0.246265i \(-0.0792034\pi\)
\(444\) 0 0
\(445\) 293.045 0.658528
\(446\) 0 0
\(447\) − 67.8061i − 0.151691i
\(448\) 0 0
\(449\) −369.643 −0.823258 −0.411629 0.911351i \(-0.635040\pi\)
−0.411629 + 0.911351i \(0.635040\pi\)
\(450\) 0 0
\(451\) − 114.033i − 0.252846i
\(452\) 0 0
\(453\) 233.973 0.516496
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 514.650 1.12615 0.563074 0.826407i \(-0.309619\pi\)
0.563074 + 0.826407i \(0.309619\pi\)
\(458\) 0 0
\(459\) − 26.4987i − 0.0577313i
\(460\) 0 0
\(461\) −656.440 −1.42395 −0.711974 0.702206i \(-0.752199\pi\)
−0.711974 + 0.702206i \(0.752199\pi\)
\(462\) 0 0
\(463\) 90.7673i 0.196042i 0.995184 + 0.0980208i \(0.0312512\pi\)
−0.995184 + 0.0980208i \(0.968749\pi\)
\(464\) 0 0
\(465\) −1.27823 −0.00274888
\(466\) 0 0
\(467\) 341.010i 0.730214i 0.930966 + 0.365107i \(0.118968\pi\)
−0.930966 + 0.365107i \(0.881032\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 97.0783i 0.206111i
\(472\) 0 0
\(473\) −165.492 −0.349877
\(474\) 0 0
\(475\) − 173.068i − 0.364355i
\(476\) 0 0
\(477\) 310.124 0.650155
\(478\) 0 0
\(479\) − 252.376i − 0.526881i −0.964676 0.263440i \(-0.915143\pi\)
0.964676 0.263440i \(-0.0848573\pi\)
\(480\) 0 0
\(481\) −46.1204 −0.0958845
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 446.825 0.921288
\(486\) 0 0
\(487\) 586.411i 1.20413i 0.798447 + 0.602065i \(0.205655\pi\)
−0.798447 + 0.602065i \(0.794345\pi\)
\(488\) 0 0
\(489\) −44.3023 −0.0905978
\(490\) 0 0
\(491\) 462.384i 0.941719i 0.882208 + 0.470860i \(0.156056\pi\)
−0.882208 + 0.470860i \(0.843944\pi\)
\(492\) 0 0
\(493\) 209.209 0.424360
\(494\) 0 0
\(495\) − 39.7954i − 0.0803948i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 882.142i 1.76782i 0.467658 + 0.883910i \(0.345098\pi\)
−0.467658 + 0.883910i \(0.654902\pi\)
\(500\) 0 0
\(501\) 135.100 0.269660
\(502\) 0 0
\(503\) 19.1303i 0.0380323i 0.999819 + 0.0190162i \(0.00605340\pi\)
−0.999819 + 0.0190162i \(0.993947\pi\)
\(504\) 0 0
\(505\) 674.145 1.33494
\(506\) 0 0
\(507\) − 279.854i − 0.551981i
\(508\) 0 0
\(509\) −427.725 −0.840324 −0.420162 0.907449i \(-0.638027\pi\)
−0.420162 + 0.907449i \(0.638027\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 133.722 0.260666
\(514\) 0 0
\(515\) 514.905i 0.999816i
\(516\) 0 0
\(517\) 75.2442 0.145540
\(518\) 0 0
\(519\) − 497.355i − 0.958294i
\(520\) 0 0
\(521\) −787.244 −1.51103 −0.755513 0.655134i \(-0.772612\pi\)
−0.755513 + 0.655134i \(0.772612\pi\)
\(522\) 0 0
\(523\) − 603.935i − 1.15475i −0.816479 0.577376i \(-0.804077\pi\)
0.816479 0.577376i \(-0.195923\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 0.880364i − 0.00167052i
\(528\) 0 0
\(529\) 380.601 0.719473
\(530\) 0 0
\(531\) − 87.7692i − 0.165290i
\(532\) 0 0
\(533\) 100.145 0.187888
\(534\) 0 0
\(535\) 444.687i 0.831190i
\(536\) 0 0
\(537\) 114.055 0.212393
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −851.914 −1.57470 −0.787352 0.616504i \(-0.788548\pi\)
−0.787352 + 0.616504i \(0.788548\pi\)
\(542\) 0 0
\(543\) 355.630i 0.654936i
\(544\) 0 0
\(545\) 442.990 0.812826
\(546\) 0 0
\(547\) − 887.898i − 1.62321i −0.584204 0.811607i \(-0.698593\pi\)
0.584204 0.811607i \(-0.301407\pi\)
\(548\) 0 0
\(549\) 98.7010 0.179783
\(550\) 0 0
\(551\) 1055.75i 1.91605i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 125.315i 0.225792i
\(556\) 0 0
\(557\) 592.908 1.06447 0.532233 0.846598i \(-0.321353\pi\)
0.532233 + 0.846598i \(0.321353\pi\)
\(558\) 0 0
\(559\) − 145.335i − 0.259992i
\(560\) 0 0
\(561\) 27.4086 0.0488567
\(562\) 0 0
\(563\) − 153.849i − 0.273266i −0.990622 0.136633i \(-0.956372\pi\)
0.990622 0.136633i \(-0.0436282\pi\)
\(564\) 0 0
\(565\) 760.289 1.34564
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 289.450 0.508700 0.254350 0.967112i \(-0.418139\pi\)
0.254350 + 0.967112i \(0.418139\pi\)
\(570\) 0 0
\(571\) 640.702i 1.12207i 0.827792 + 0.561035i \(0.189597\pi\)
−0.827792 + 0.561035i \(0.810403\pi\)
\(572\) 0 0
\(573\) −171.444 −0.299203
\(574\) 0 0
\(575\) − 81.9243i − 0.142477i
\(576\) 0 0
\(577\) −190.209 −0.329652 −0.164826 0.986323i \(-0.552706\pi\)
−0.164826 + 0.986323i \(0.552706\pi\)
\(578\) 0 0
\(579\) − 94.4888i − 0.163193i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 320.773i 0.550211i
\(584\) 0 0
\(585\) 34.9485 0.0597410
\(586\) 0 0
\(587\) − 338.241i − 0.576219i −0.957597 0.288109i \(-0.906973\pi\)
0.957597 0.288109i \(-0.0930267\pi\)
\(588\) 0 0
\(589\) 4.44263 0.00754267
\(590\) 0 0
\(591\) − 220.750i − 0.373520i
\(592\) 0 0
\(593\) −736.736 −1.24239 −0.621194 0.783657i \(-0.713352\pi\)
−0.621194 + 0.783657i \(0.713352\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21.0997 0.0353428
\(598\) 0 0
\(599\) − 371.266i − 0.619810i −0.950768 0.309905i \(-0.899703\pi\)
0.950768 0.309905i \(-0.100297\pi\)
\(600\) 0 0
\(601\) −965.894 −1.60714 −0.803572 0.595207i \(-0.797070\pi\)
−0.803572 + 0.595207i \(0.797070\pi\)
\(602\) 0 0
\(603\) − 51.0509i − 0.0846615i
\(604\) 0 0
\(605\) −476.103 −0.786947
\(606\) 0 0
\(607\) 304.887i 0.502285i 0.967950 + 0.251143i \(0.0808063\pi\)
−0.967950 + 0.251143i \(0.919194\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 66.0797i 0.108150i
\(612\) 0 0
\(613\) 425.341 0.693867 0.346934 0.937890i \(-0.387223\pi\)
0.346934 + 0.937890i \(0.387223\pi\)
\(614\) 0 0
\(615\) − 272.105i − 0.442447i
\(616\) 0 0
\(617\) 821.189 1.33094 0.665470 0.746425i \(-0.268231\pi\)
0.665470 + 0.746425i \(0.268231\pi\)
\(618\) 0 0
\(619\) − 897.086i − 1.44925i −0.689143 0.724625i \(-0.742013\pi\)
0.689143 0.724625i \(-0.257987\pi\)
\(620\) 0 0
\(621\) 63.2990 0.101931
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −411.646 −0.658634
\(626\) 0 0
\(627\) 138.314i 0.220596i
\(628\) 0 0
\(629\) −86.3089 −0.137216
\(630\) 0 0
\(631\) − 347.398i − 0.550552i −0.961365 0.275276i \(-0.911231\pi\)
0.961365 0.275276i \(-0.0887693\pi\)
\(632\) 0 0
\(633\) 364.151 0.575278
\(634\) 0 0
\(635\) 733.882i 1.15572i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 230.357i − 0.360496i
\(640\) 0 0
\(641\) 405.855 0.633160 0.316580 0.948566i \(-0.397465\pi\)
0.316580 + 0.948566i \(0.397465\pi\)
\(642\) 0 0
\(643\) − 52.5787i − 0.0817709i −0.999164 0.0408854i \(-0.986982\pi\)
0.999164 0.0408854i \(-0.0130179\pi\)
\(644\) 0 0
\(645\) −394.894 −0.612238
\(646\) 0 0
\(647\) 1121.44i 1.73329i 0.498928 + 0.866644i \(0.333727\pi\)
−0.498928 + 0.866644i \(0.666273\pi\)
\(648\) 0 0
\(649\) 90.7832 0.139882
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −41.9660 −0.0642664 −0.0321332 0.999484i \(-0.510230\pi\)
−0.0321332 + 0.999484i \(0.510230\pi\)
\(654\) 0 0
\(655\) 535.015i 0.816817i
\(656\) 0 0
\(657\) −297.825 −0.453310
\(658\) 0 0
\(659\) 1224.43i 1.85801i 0.370069 + 0.929004i \(0.379334\pi\)
−0.370069 + 0.929004i \(0.620666\pi\)
\(660\) 0 0
\(661\) 27.1171 0.0410244 0.0205122 0.999790i \(-0.493470\pi\)
0.0205122 + 0.999790i \(0.493470\pi\)
\(662\) 0 0
\(663\) 24.0703i 0.0363052i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 499.751i 0.749252i
\(668\) 0 0
\(669\) 457.821 0.684337
\(670\) 0 0
\(671\) 102.090i 0.152147i
\(672\) 0 0
\(673\) −418.238 −0.621453 −0.310726 0.950499i \(-0.600572\pi\)
−0.310726 + 0.950499i \(0.600572\pi\)
\(674\) 0 0
\(675\) 34.9446i 0.0517697i
\(676\) 0 0
\(677\) −233.410 −0.344770 −0.172385 0.985030i \(-0.555147\pi\)
−0.172385 + 0.985030i \(0.555147\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −609.471 −0.894965
\(682\) 0 0
\(683\) − 414.221i − 0.606472i −0.952915 0.303236i \(-0.901933\pi\)
0.952915 0.303236i \(-0.0980671\pi\)
\(684\) 0 0
\(685\) 773.540 1.12926
\(686\) 0 0
\(687\) − 355.713i − 0.517778i
\(688\) 0 0
\(689\) −281.704 −0.408860
\(690\) 0 0
\(691\) 471.232i 0.681956i 0.940071 + 0.340978i \(0.110758\pi\)
−0.940071 + 0.340978i \(0.889242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 663.485i − 0.954654i
\(696\) 0 0
\(697\) 187.409 0.268879
\(698\) 0 0
\(699\) 58.6279i 0.0838740i
\(700\) 0 0
\(701\) 1150.00 1.64051 0.820257 0.571995i \(-0.193830\pi\)
0.820257 + 0.571995i \(0.193830\pi\)
\(702\) 0 0
\(703\) − 435.546i − 0.619553i
\(704\) 0 0
\(705\) 179.547 0.254676
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1388.97 1.95906 0.979528 0.201310i \(-0.0645198\pi\)
0.979528 + 0.201310i \(0.0645198\pi\)
\(710\) 0 0
\(711\) 263.183i 0.370158i
\(712\) 0 0
\(713\) 2.10298 0.00294948
\(714\) 0 0
\(715\) 36.1486i 0.0505575i
\(716\) 0 0
\(717\) −135.842 −0.189459
\(718\) 0 0
\(719\) 1179.12i 1.63995i 0.572400 + 0.819975i \(0.306013\pi\)
−0.572400 + 0.819975i \(0.693987\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 302.198i 0.417978i
\(724\) 0 0
\(725\) −275.890 −0.380538
\(726\) 0 0
\(727\) − 1168.99i − 1.60796i −0.594656 0.803980i \(-0.702712\pi\)
0.594656 0.803980i \(-0.297288\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 271.978i − 0.372063i
\(732\) 0 0
\(733\) 768.375 1.04826 0.524130 0.851638i \(-0.324391\pi\)
0.524130 + 0.851638i \(0.324391\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 52.8040 0.0716472
\(738\) 0 0
\(739\) 378.288i 0.511891i 0.966691 + 0.255946i \(0.0823868\pi\)
−0.966691 + 0.255946i \(0.917613\pi\)
\(740\) 0 0
\(741\) −121.468 −0.163924
\(742\) 0 0
\(743\) 1461.21i 1.96664i 0.181889 + 0.983319i \(0.441779\pi\)
−0.181889 + 0.983319i \(0.558221\pi\)
\(744\) 0 0
\(745\) 167.354 0.224636
\(746\) 0 0
\(747\) − 454.357i − 0.608242i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 1109.38i − 1.47721i −0.674139 0.738605i \(-0.735485\pi\)
0.674139 0.738605i \(-0.264515\pi\)
\(752\) 0 0
\(753\) 221.918 0.294712
\(754\) 0 0
\(755\) 577.474i 0.764866i
\(756\) 0 0
\(757\) 925.286 1.22231 0.611153 0.791512i \(-0.290706\pi\)
0.611153 + 0.791512i \(0.290706\pi\)
\(758\) 0 0
\(759\) 65.4727i 0.0862617i
\(760\) 0 0
\(761\) 1373.79 1.80524 0.902620 0.430439i \(-0.141641\pi\)
0.902620 + 0.430439i \(0.141641\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 65.4020 0.0854928
\(766\) 0 0
\(767\) 79.7261i 0.103945i
\(768\) 0 0
\(769\) −1212.84 −1.57716 −0.788580 0.614933i \(-0.789183\pi\)
−0.788580 + 0.614933i \(0.789183\pi\)
\(770\) 0 0
\(771\) − 494.676i − 0.641603i
\(772\) 0 0
\(773\) −796.392 −1.03026 −0.515131 0.857112i \(-0.672257\pi\)
−0.515131 + 0.857112i \(0.672257\pi\)
\(774\) 0 0
\(775\) 1.16096i 0.00149801i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 945.731i 1.21403i
\(780\) 0 0
\(781\) 238.267 0.305080
\(782\) 0 0
\(783\) − 213.167i − 0.272244i
\(784\) 0 0
\(785\) −239.601 −0.305225
\(786\) 0 0
\(787\) 169.301i 0.215122i 0.994199 + 0.107561i \(0.0343040\pi\)
−0.994199 + 0.107561i \(0.965696\pi\)
\(788\) 0 0
\(789\) 883.739 1.12008
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −89.6561 −0.113059
\(794\) 0 0
\(795\) 765.424i 0.962798i
\(796\) 0 0
\(797\) −181.725 −0.228011 −0.114006 0.993480i \(-0.536368\pi\)
−0.114006 + 0.993480i \(0.536368\pi\)
\(798\) 0 0
\(799\) 123.660i 0.154769i
\(800\) 0 0
\(801\) 205.650 0.256741
\(802\) 0 0
\(803\) − 308.052i − 0.383626i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 404.384i 0.501095i
\(808\) 0 0
\(809\) 1284.05 1.58720 0.793602 0.608437i \(-0.208203\pi\)
0.793602 + 0.608437i \(0.208203\pi\)
\(810\) 0 0
\(811\) 627.609i 0.773871i 0.922107 + 0.386936i \(0.126466\pi\)
−0.922107 + 0.386936i \(0.873534\pi\)
\(812\) 0 0
\(813\) 550.069 0.676592
\(814\) 0 0
\(815\) − 109.344i − 0.134164i
\(816\) 0 0
\(817\) 1372.50 1.67993
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 644.577 0.785112 0.392556 0.919728i \(-0.371591\pi\)
0.392556 + 0.919728i \(0.371591\pi\)
\(822\) 0 0
\(823\) − 698.701i − 0.848969i −0.905435 0.424484i \(-0.860455\pi\)
0.905435 0.424484i \(-0.139545\pi\)
\(824\) 0 0
\(825\) −36.1445 −0.0438116
\(826\) 0 0
\(827\) 736.603i 0.890693i 0.895358 + 0.445347i \(0.146920\pi\)
−0.895358 + 0.445347i \(0.853080\pi\)
\(828\) 0 0
\(829\) 854.107 1.03029 0.515143 0.857104i \(-0.327739\pi\)
0.515143 + 0.857104i \(0.327739\pi\)
\(830\) 0 0
\(831\) − 904.589i − 1.08856i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 333.443i 0.399333i
\(836\) 0 0
\(837\) −0.897020 −0.00107171
\(838\) 0 0
\(839\) 261.597i 0.311796i 0.987773 + 0.155898i \(0.0498272\pi\)
−0.987773 + 0.155898i \(0.950173\pi\)
\(840\) 0 0
\(841\) 841.976 1.00116
\(842\) 0 0
\(843\) − 533.043i − 0.632317i
\(844\) 0 0
\(845\) 690.715 0.817414
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 360.072 0.424113
\(850\) 0 0
\(851\) − 206.172i − 0.242270i
\(852\) 0 0
\(853\) −300.278 −0.352026 −0.176013 0.984388i \(-0.556320\pi\)
−0.176013 + 0.984388i \(0.556320\pi\)
\(854\) 0 0
\(855\) 330.042i 0.386014i
\(856\) 0 0
\(857\) −256.344 −0.299118 −0.149559 0.988753i \(-0.547785\pi\)
−0.149559 + 0.988753i \(0.547785\pi\)
\(858\) 0 0
\(859\) − 887.342i − 1.03299i −0.856289 0.516497i \(-0.827236\pi\)
0.856289 0.516497i \(-0.172764\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1543.01i − 1.78796i −0.448106 0.893980i \(-0.647901\pi\)
0.448106 0.893980i \(-0.352099\pi\)
\(864\) 0 0
\(865\) 1227.53 1.41911
\(866\) 0 0
\(867\) − 455.518i − 0.525395i
\(868\) 0 0
\(869\) −272.220 −0.313257
\(870\) 0 0
\(871\) 46.3726i 0.0532407i
\(872\) 0 0
\(873\) 313.567 0.359184
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −574.028 −0.654536 −0.327268 0.944932i \(-0.606128\pi\)
−0.327268 + 0.944932i \(0.606128\pi\)
\(878\) 0 0
\(879\) − 591.790i − 0.673254i
\(880\) 0 0
\(881\) 201.842 0.229106 0.114553 0.993417i \(-0.463456\pi\)
0.114553 + 0.993417i \(0.463456\pi\)
\(882\) 0 0
\(883\) 661.078i 0.748673i 0.927293 + 0.374336i \(0.122129\pi\)
−0.927293 + 0.374336i \(0.877871\pi\)
\(884\) 0 0
\(885\) 216.625 0.244774
\(886\) 0 0
\(887\) − 430.783i − 0.485663i −0.970069 0.242831i \(-0.921924\pi\)
0.970069 0.242831i \(-0.0780762\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 27.9272i − 0.0313436i
\(892\) 0 0
\(893\) −624.035 −0.698807
\(894\) 0 0
\(895\) 281.501i 0.314527i
\(896\) 0 0
\(897\) −57.4983 −0.0641007
\(898\) 0 0
\(899\) − 7.08205i − 0.00787770i
\(900\) 0 0
\(901\) −527.176 −0.585101
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −877.739 −0.969878
\(906\) 0 0
\(907\) 1012.97i 1.11684i 0.829559 + 0.558419i \(0.188592\pi\)
−0.829559 + 0.558419i \(0.811408\pi\)
\(908\) 0 0
\(909\) 473.093 0.520454
\(910\) 0 0
\(911\) 507.799i 0.557408i 0.960377 + 0.278704i \(0.0899049\pi\)
−0.960377 + 0.278704i \(0.910095\pi\)
\(912\) 0 0
\(913\) 469.959 0.514742
\(914\) 0 0
\(915\) 243.606i 0.266236i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 179.536i − 0.195360i −0.995218 0.0976802i \(-0.968858\pi\)
0.995218 0.0976802i \(-0.0311422\pi\)
\(920\) 0 0
\(921\) −311.619 −0.338348
\(922\) 0 0
\(923\) 209.247i 0.226703i
\(924\) 0 0
\(925\) 113.818 0.123047
\(926\) 0 0
\(927\) 361.344i 0.389799i
\(928\) 0 0
\(929\) 298.578 0.321397 0.160699 0.987004i \(-0.448625\pi\)
0.160699 + 0.987004i \(0.448625\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −454.949 −0.487619
\(934\) 0 0
\(935\) 67.6479i 0.0723507i
\(936\) 0 0
\(937\) 568.691 0.606927 0.303464 0.952843i \(-0.401857\pi\)
0.303464 + 0.952843i \(0.401857\pi\)
\(938\) 0 0
\(939\) 947.504i 1.00906i
\(940\) 0 0
\(941\) −570.715 −0.606499 −0.303249 0.952911i \(-0.598071\pi\)
−0.303249 + 0.952911i \(0.598071\pi\)
\(942\) 0 0
\(943\) 447.675i 0.474735i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 435.346i 0.459711i 0.973225 + 0.229855i \(0.0738253\pi\)
−0.973225 + 0.229855i \(0.926175\pi\)
\(948\) 0 0
\(949\) 270.532 0.285071
\(950\) 0 0
\(951\) − 42.0569i − 0.0442238i
\(952\) 0 0
\(953\) −1282.02 −1.34525 −0.672624 0.739984i \(-0.734833\pi\)
−0.672624 + 0.739984i \(0.734833\pi\)
\(954\) 0 0
\(955\) − 423.144i − 0.443083i
\(956\) 0 0
\(957\) 220.488 0.230394
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 960.970 0.999969
\(962\) 0 0
\(963\) 312.067i 0.324057i
\(964\) 0 0
\(965\) 233.210 0.241669
\(966\) 0 0
\(967\) − 625.993i − 0.647355i −0.946167 0.323678i \(-0.895081\pi\)
0.946167 0.323678i \(-0.104919\pi\)
\(968\) 0 0
\(969\) −227.312 −0.234584
\(970\) 0 0
\(971\) 1260.98i 1.29864i 0.760516 + 0.649319i \(0.224946\pi\)
−0.760516 + 0.649319i \(0.775054\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 31.7423i − 0.0325562i
\(976\) 0 0
\(977\) 1575.49 1.61257 0.806287 0.591524i \(-0.201474\pi\)
0.806287 + 0.591524i \(0.201474\pi\)
\(978\) 0 0
\(979\) 212.711i 0.217274i
\(980\) 0 0
\(981\) 310.876 0.316897
\(982\) 0 0
\(983\) 1047.12i 1.06523i 0.846358 + 0.532614i \(0.178790\pi\)
−0.846358 + 0.532614i \(0.821210\pi\)
\(984\) 0 0
\(985\) 544.839 0.553136
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 649.691 0.656917
\(990\) 0 0
\(991\) − 781.385i − 0.788481i −0.919007 0.394241i \(-0.871008\pi\)
0.919007 0.394241i \(-0.128992\pi\)
\(992\) 0 0
\(993\) −531.426 −0.535172
\(994\) 0 0
\(995\) 52.0766i 0.0523383i
\(996\) 0 0
\(997\) −700.155 −0.702262 −0.351131 0.936326i \(-0.614203\pi\)
−0.351131 + 0.936326i \(0.614203\pi\)
\(998\) 0 0
\(999\) 87.9419i 0.0880299i
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 2352.3.m.g.1471.3 4
4.3 odd 2 inner 2352.3.m.g.1471.1 4
7.2 even 3 336.3.be.a.319.2 yes 4
7.4 even 3 336.3.be.b.79.2 yes 4
7.6 odd 2 2352.3.m.h.1471.2 4
21.2 odd 6 1008.3.cd.g.991.1 4
21.11 odd 6 1008.3.cd.f.415.1 4
28.11 odd 6 336.3.be.a.79.2 4
28.23 odd 6 336.3.be.b.319.2 yes 4
28.27 even 2 2352.3.m.h.1471.4 4
84.11 even 6 1008.3.cd.g.415.1 4
84.23 even 6 1008.3.cd.f.991.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.be.a.79.2 4 28.11 odd 6
336.3.be.a.319.2 yes 4 7.2 even 3
336.3.be.b.79.2 yes 4 7.4 even 3
336.3.be.b.319.2 yes 4 28.23 odd 6
1008.3.cd.f.415.1 4 21.11 odd 6
1008.3.cd.f.991.1 4 84.23 even 6
1008.3.cd.g.415.1 4 84.11 even 6
1008.3.cd.g.991.1 4 21.2 odd 6
2352.3.m.g.1471.1 4 4.3 odd 2 inner
2352.3.m.g.1471.3 4 1.1 even 1 trivial
2352.3.m.h.1471.2 4 7.6 odd 2
2352.3.m.h.1471.4 4 28.27 even 2