Properties

Label 1792.3.d.g.1023.6
Level $1792$
Weight $3$
Character 1792.1023
Analytic conductor $48.828$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,3,Mod(1023,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1792.1023");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.d (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: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1023.6
Root \(-1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1023
Dual form 1792.3.d.g.1023.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.585786i q^{3} +9.03316 q^{5} -2.64575i q^{7} +8.65685 q^{9} +O(q^{10})\) \(q+0.585786i q^{3} +9.03316 q^{5} -2.64575i q^{7} +8.65685 q^{9} -12.4853i q^{11} +9.03316 q^{13} +5.29150i q^{15} +12.3431 q^{17} +28.8701i q^{19} +1.54985 q^{21} +24.6418i q^{23} +56.5980 q^{25} +10.3431i q^{27} -22.4499 q^{29} -16.7824i q^{31} +7.31371 q^{33} -23.8995i q^{35} +16.2506 q^{37} +5.29150i q^{39} -6.97056 q^{41} +22.8284i q^{43} +78.1987 q^{45} -6.19938i q^{47} -7.00000 q^{49} +7.23045i q^{51} +8.01514 q^{53} -112.782i q^{55} -16.9117 q^{57} -30.4437i q^{59} +15.2325 q^{61} -22.9039i q^{63} +81.5980 q^{65} -78.6274i q^{67} -14.4348 q^{69} +17.5345i q^{71} -46.6863 q^{73} +33.1543i q^{75} -33.0329 q^{77} -81.0325i q^{79} +71.8528 q^{81} +40.3848i q^{83} +111.498 q^{85} -13.1509i q^{87} -111.941 q^{89} -23.8995i q^{91} +9.83089 q^{93} +260.788i q^{95} -164.108 q^{97} -108.083i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 144 q^{17} + 136 q^{25} - 32 q^{33} + 80 q^{41} - 56 q^{49} + 272 q^{57} + 336 q^{65} - 464 q^{73} - 104 q^{81} - 624 q^{89} - 272 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0.585786i 0.195262i 0.995223 + 0.0976311i \(0.0311265\pi\)
−0.995223 + 0.0976311i \(0.968873\pi\)
\(4\) 0 0
\(5\) 9.03316 1.80663 0.903316 0.428976i \(-0.141125\pi\)
0.903316 + 0.428976i \(0.141125\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) 8.65685 0.961873
\(10\) 0 0
\(11\) − 12.4853i − 1.13503i −0.823365 0.567513i \(-0.807906\pi\)
0.823365 0.567513i \(-0.192094\pi\)
\(12\) 0 0
\(13\) 9.03316 0.694858 0.347429 0.937706i \(-0.387055\pi\)
0.347429 + 0.937706i \(0.387055\pi\)
\(14\) 0 0
\(15\) 5.29150i 0.352767i
\(16\) 0 0
\(17\) 12.3431 0.726067 0.363034 0.931776i \(-0.381741\pi\)
0.363034 + 0.931776i \(0.381741\pi\)
\(18\) 0 0
\(19\) 28.8701i 1.51948i 0.650229 + 0.759738i \(0.274673\pi\)
−0.650229 + 0.759738i \(0.725327\pi\)
\(20\) 0 0
\(21\) 1.54985 0.0738022
\(22\) 0 0
\(23\) 24.6418i 1.07138i 0.844414 + 0.535690i \(0.179949\pi\)
−0.844414 + 0.535690i \(0.820051\pi\)
\(24\) 0 0
\(25\) 56.5980 2.26392
\(26\) 0 0
\(27\) 10.3431i 0.383079i
\(28\) 0 0
\(29\) −22.4499 −0.774136 −0.387068 0.922051i \(-0.626512\pi\)
−0.387068 + 0.922051i \(0.626512\pi\)
\(30\) 0 0
\(31\) − 16.7824i − 0.541367i −0.962668 0.270684i \(-0.912750\pi\)
0.962668 0.270684i \(-0.0872497\pi\)
\(32\) 0 0
\(33\) 7.31371 0.221628
\(34\) 0 0
\(35\) − 23.8995i − 0.682843i
\(36\) 0 0
\(37\) 16.2506 0.439204 0.219602 0.975589i \(-0.429524\pi\)
0.219602 + 0.975589i \(0.429524\pi\)
\(38\) 0 0
\(39\) 5.29150i 0.135680i
\(40\) 0 0
\(41\) −6.97056 −0.170014 −0.0850069 0.996380i \(-0.527091\pi\)
−0.0850069 + 0.996380i \(0.527091\pi\)
\(42\) 0 0
\(43\) 22.8284i 0.530894i 0.964126 + 0.265447i \(0.0855195\pi\)
−0.964126 + 0.265447i \(0.914481\pi\)
\(44\) 0 0
\(45\) 78.1987 1.73775
\(46\) 0 0
\(47\) − 6.19938i − 0.131902i −0.997823 0.0659509i \(-0.978992\pi\)
0.997823 0.0659509i \(-0.0210081\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 7.23045i 0.141773i
\(52\) 0 0
\(53\) 8.01514 0.151229 0.0756145 0.997137i \(-0.475908\pi\)
0.0756145 + 0.997137i \(0.475908\pi\)
\(54\) 0 0
\(55\) − 112.782i − 2.05057i
\(56\) 0 0
\(57\) −16.9117 −0.296696
\(58\) 0 0
\(59\) − 30.4437i − 0.515994i −0.966146 0.257997i \(-0.916938\pi\)
0.966146 0.257997i \(-0.0830625\pi\)
\(60\) 0 0
\(61\) 15.2325 0.249714 0.124857 0.992175i \(-0.460153\pi\)
0.124857 + 0.992175i \(0.460153\pi\)
\(62\) 0 0
\(63\) − 22.9039i − 0.363554i
\(64\) 0 0
\(65\) 81.5980 1.25535
\(66\) 0 0
\(67\) − 78.6274i − 1.17354i −0.809752 0.586772i \(-0.800399\pi\)
0.809752 0.586772i \(-0.199601\pi\)
\(68\) 0 0
\(69\) −14.4348 −0.209200
\(70\) 0 0
\(71\) 17.5345i 0.246965i 0.992347 + 0.123482i \(0.0394062\pi\)
−0.992347 + 0.123482i \(0.960594\pi\)
\(72\) 0 0
\(73\) −46.6863 −0.639538 −0.319769 0.947495i \(-0.603605\pi\)
−0.319769 + 0.947495i \(0.603605\pi\)
\(74\) 0 0
\(75\) 33.1543i 0.442058i
\(76\) 0 0
\(77\) −33.0329 −0.428999
\(78\) 0 0
\(79\) − 81.0325i − 1.02573i −0.858470 0.512864i \(-0.828584\pi\)
0.858470 0.512864i \(-0.171416\pi\)
\(80\) 0 0
\(81\) 71.8528 0.887072
\(82\) 0 0
\(83\) 40.3848i 0.486564i 0.969956 + 0.243282i \(0.0782240\pi\)
−0.969956 + 0.243282i \(0.921776\pi\)
\(84\) 0 0
\(85\) 111.498 1.31174
\(86\) 0 0
\(87\) − 13.1509i − 0.151159i
\(88\) 0 0
\(89\) −111.941 −1.25777 −0.628883 0.777500i \(-0.716487\pi\)
−0.628883 + 0.777500i \(0.716487\pi\)
\(90\) 0 0
\(91\) − 23.8995i − 0.262632i
\(92\) 0 0
\(93\) 9.83089 0.105709
\(94\) 0 0
\(95\) 260.788i 2.74514i
\(96\) 0 0
\(97\) −164.108 −1.69183 −0.845916 0.533317i \(-0.820945\pi\)
−0.845916 + 0.533317i \(0.820945\pi\)
\(98\) 0 0
\(99\) − 108.083i − 1.09175i
\(100\) 0 0
\(101\) 12.1329 0.120127 0.0600636 0.998195i \(-0.480870\pi\)
0.0600636 + 0.998195i \(0.480870\pi\)
\(102\) 0 0
\(103\) 106.582i 1.03478i 0.855750 + 0.517389i \(0.173096\pi\)
−0.855750 + 0.517389i \(0.826904\pi\)
\(104\) 0 0
\(105\) 14.0000 0.133333
\(106\) 0 0
\(107\) 63.5980i 0.594374i 0.954819 + 0.297187i \(0.0960484\pi\)
−0.954819 + 0.297187i \(0.903952\pi\)
\(108\) 0 0
\(109\) 130.848 1.20044 0.600220 0.799835i \(-0.295080\pi\)
0.600220 + 0.799835i \(0.295080\pi\)
\(110\) 0 0
\(111\) 9.51936i 0.0857600i
\(112\) 0 0
\(113\) −138.225 −1.22323 −0.611617 0.791154i \(-0.709481\pi\)
−0.611617 + 0.791154i \(0.709481\pi\)
\(114\) 0 0
\(115\) 222.593i 1.93559i
\(116\) 0 0
\(117\) 78.1987 0.668365
\(118\) 0 0
\(119\) − 32.6569i − 0.274428i
\(120\) 0 0
\(121\) −34.8823 −0.288283
\(122\) 0 0
\(123\) − 4.08326i − 0.0331972i
\(124\) 0 0
\(125\) 285.430 2.28344
\(126\) 0 0
\(127\) − 114.442i − 0.901114i −0.892748 0.450557i \(-0.851225\pi\)
0.892748 0.450557i \(-0.148775\pi\)
\(128\) 0 0
\(129\) −13.3726 −0.103663
\(130\) 0 0
\(131\) − 168.350i − 1.28512i −0.766237 0.642558i \(-0.777873\pi\)
0.766237 0.642558i \(-0.222127\pi\)
\(132\) 0 0
\(133\) 76.3830 0.574308
\(134\) 0 0
\(135\) 93.4313i 0.692084i
\(136\) 0 0
\(137\) −34.6863 −0.253185 −0.126592 0.991955i \(-0.540404\pi\)
−0.126592 + 0.991955i \(0.540404\pi\)
\(138\) 0 0
\(139\) − 107.664i − 0.774561i −0.921962 0.387281i \(-0.873414\pi\)
0.921962 0.387281i \(-0.126586\pi\)
\(140\) 0 0
\(141\) 3.63151 0.0257554
\(142\) 0 0
\(143\) − 112.782i − 0.788682i
\(144\) 0 0
\(145\) −202.794 −1.39858
\(146\) 0 0
\(147\) − 4.10051i − 0.0278946i
\(148\) 0 0
\(149\) 252.176 1.69246 0.846229 0.532819i \(-0.178867\pi\)
0.846229 + 0.532819i \(0.178867\pi\)
\(150\) 0 0
\(151\) 234.486i 1.55289i 0.630186 + 0.776444i \(0.282979\pi\)
−0.630186 + 0.776444i \(0.717021\pi\)
\(152\) 0 0
\(153\) 106.853 0.698384
\(154\) 0 0
\(155\) − 151.598i − 0.978051i
\(156\) 0 0
\(157\) 10.0968 0.0643109 0.0321554 0.999483i \(-0.489763\pi\)
0.0321554 + 0.999483i \(0.489763\pi\)
\(158\) 0 0
\(159\) 4.69516i 0.0295293i
\(160\) 0 0
\(161\) 65.1960 0.404944
\(162\) 0 0
\(163\) 104.534i 0.641313i 0.947196 + 0.320657i \(0.103904\pi\)
−0.947196 + 0.320657i \(0.896096\pi\)
\(164\) 0 0
\(165\) 66.0659 0.400399
\(166\) 0 0
\(167\) − 296.765i − 1.77703i −0.458843 0.888517i \(-0.651736\pi\)
0.458843 0.888517i \(-0.348264\pi\)
\(168\) 0 0
\(169\) −87.4020 −0.517172
\(170\) 0 0
\(171\) 249.924i 1.46154i
\(172\) 0 0
\(173\) 40.0301 0.231388 0.115694 0.993285i \(-0.463091\pi\)
0.115694 + 0.993285i \(0.463091\pi\)
\(174\) 0 0
\(175\) − 149.744i − 0.855681i
\(176\) 0 0
\(177\) 17.8335 0.100754
\(178\) 0 0
\(179\) 294.794i 1.64689i 0.567394 + 0.823447i \(0.307952\pi\)
−0.567394 + 0.823447i \(0.692048\pi\)
\(180\) 0 0
\(181\) −40.4706 −0.223595 −0.111797 0.993731i \(-0.535661\pi\)
−0.111797 + 0.993731i \(0.535661\pi\)
\(182\) 0 0
\(183\) 8.92302i 0.0487596i
\(184\) 0 0
\(185\) 146.794 0.793481
\(186\) 0 0
\(187\) − 154.108i − 0.824105i
\(188\) 0 0
\(189\) 27.3654 0.144790
\(190\) 0 0
\(191\) 156.929i 0.821619i 0.911721 + 0.410810i \(0.134754\pi\)
−0.911721 + 0.410810i \(0.865246\pi\)
\(192\) 0 0
\(193\) −261.304 −1.35390 −0.676952 0.736027i \(-0.736700\pi\)
−0.676952 + 0.736027i \(0.736700\pi\)
\(194\) 0 0
\(195\) 47.7990i 0.245123i
\(196\) 0 0
\(197\) 145.283 0.737475 0.368738 0.929533i \(-0.379790\pi\)
0.368738 + 0.929533i \(0.379790\pi\)
\(198\) 0 0
\(199\) − 390.508i − 1.96235i −0.193122 0.981175i \(-0.561861\pi\)
0.193122 0.981175i \(-0.438139\pi\)
\(200\) 0 0
\(201\) 46.0589 0.229149
\(202\) 0 0
\(203\) 59.3970i 0.292596i
\(204\) 0 0
\(205\) −62.9662 −0.307152
\(206\) 0 0
\(207\) 213.320i 1.03053i
\(208\) 0 0
\(209\) 360.451 1.72464
\(210\) 0 0
\(211\) − 164.049i − 0.777482i −0.921347 0.388741i \(-0.872910\pi\)
0.921347 0.388741i \(-0.127090\pi\)
\(212\) 0 0
\(213\) −10.2715 −0.0482229
\(214\) 0 0
\(215\) 206.213i 0.959129i
\(216\) 0 0
\(217\) −44.4020 −0.204618
\(218\) 0 0
\(219\) − 27.3482i − 0.124878i
\(220\) 0 0
\(221\) 111.498 0.504514
\(222\) 0 0
\(223\) 10.5830i 0.0474574i 0.999718 + 0.0237287i \(0.00755379\pi\)
−0.999718 + 0.0237287i \(0.992446\pi\)
\(224\) 0 0
\(225\) 489.960 2.17760
\(226\) 0 0
\(227\) − 213.806i − 0.941877i −0.882166 0.470939i \(-0.843915\pi\)
0.882166 0.470939i \(-0.156085\pi\)
\(228\) 0 0
\(229\) 232.028 1.01322 0.506612 0.862174i \(-0.330898\pi\)
0.506612 + 0.862174i \(0.330898\pi\)
\(230\) 0 0
\(231\) − 19.3503i − 0.0837673i
\(232\) 0 0
\(233\) 192.863 0.827738 0.413869 0.910336i \(-0.364177\pi\)
0.413869 + 0.910336i \(0.364177\pi\)
\(234\) 0 0
\(235\) − 56.0000i − 0.238298i
\(236\) 0 0
\(237\) 47.4678 0.200286
\(238\) 0 0
\(239\) − 327.917i − 1.37204i −0.727583 0.686020i \(-0.759356\pi\)
0.727583 0.686020i \(-0.240644\pi\)
\(240\) 0 0
\(241\) 71.8721 0.298225 0.149112 0.988820i \(-0.452358\pi\)
0.149112 + 0.988820i \(0.452358\pi\)
\(242\) 0 0
\(243\) 135.179i 0.556291i
\(244\) 0 0
\(245\) −63.2321 −0.258090
\(246\) 0 0
\(247\) 260.788i 1.05582i
\(248\) 0 0
\(249\) −23.6569 −0.0950074
\(250\) 0 0
\(251\) 256.919i 1.02358i 0.859110 + 0.511790i \(0.171018\pi\)
−0.859110 + 0.511790i \(0.828982\pi\)
\(252\) 0 0
\(253\) 307.659 1.21604
\(254\) 0 0
\(255\) 65.3138i 0.256133i
\(256\) 0 0
\(257\) 319.352 1.24262 0.621308 0.783566i \(-0.286602\pi\)
0.621308 + 0.783566i \(0.286602\pi\)
\(258\) 0 0
\(259\) − 42.9949i − 0.166004i
\(260\) 0 0
\(261\) −194.346 −0.744620
\(262\) 0 0
\(263\) 377.357i 1.43482i 0.696653 + 0.717408i \(0.254672\pi\)
−0.696653 + 0.717408i \(0.745328\pi\)
\(264\) 0 0
\(265\) 72.4020 0.273215
\(266\) 0 0
\(267\) − 65.5736i − 0.245594i
\(268\) 0 0
\(269\) −28.1631 −0.104696 −0.0523478 0.998629i \(-0.516670\pi\)
−0.0523478 + 0.998629i \(0.516670\pi\)
\(270\) 0 0
\(271\) 399.715i 1.47496i 0.675367 + 0.737482i \(0.263985\pi\)
−0.675367 + 0.737482i \(0.736015\pi\)
\(272\) 0 0
\(273\) 14.0000 0.0512821
\(274\) 0 0
\(275\) − 706.642i − 2.56961i
\(276\) 0 0
\(277\) −102.951 −0.371663 −0.185831 0.982582i \(-0.559498\pi\)
−0.185831 + 0.982582i \(0.559498\pi\)
\(278\) 0 0
\(279\) − 145.283i − 0.520726i
\(280\) 0 0
\(281\) 150.235 0.534646 0.267323 0.963607i \(-0.413861\pi\)
0.267323 + 0.963607i \(0.413861\pi\)
\(282\) 0 0
\(283\) 178.561i 0.630959i 0.948932 + 0.315480i \(0.102165\pi\)
−0.948932 + 0.315480i \(0.897835\pi\)
\(284\) 0 0
\(285\) −152.766 −0.536021
\(286\) 0 0
\(287\) 18.4424i 0.0642591i
\(288\) 0 0
\(289\) −136.647 −0.472826
\(290\) 0 0
\(291\) − 96.1320i − 0.330351i
\(292\) 0 0
\(293\) −219.189 −0.748085 −0.374043 0.927411i \(-0.622029\pi\)
−0.374043 + 0.927411i \(0.622029\pi\)
\(294\) 0 0
\(295\) − 275.002i − 0.932211i
\(296\) 0 0
\(297\) 129.137 0.434805
\(298\) 0 0
\(299\) 222.593i 0.744458i
\(300\) 0 0
\(301\) 60.3983 0.200659
\(302\) 0 0
\(303\) 7.10726i 0.0234563i
\(304\) 0 0
\(305\) 137.598 0.451141
\(306\) 0 0
\(307\) 316.669i 1.03150i 0.856741 + 0.515748i \(0.172486\pi\)
−0.856741 + 0.515748i \(0.827514\pi\)
\(308\) 0 0
\(309\) −62.4344 −0.202053
\(310\) 0 0
\(311\) − 72.2653i − 0.232364i −0.993228 0.116182i \(-0.962934\pi\)
0.993228 0.116182i \(-0.0370656\pi\)
\(312\) 0 0
\(313\) −81.9512 −0.261825 −0.130913 0.991394i \(-0.541791\pi\)
−0.130913 + 0.991394i \(0.541791\pi\)
\(314\) 0 0
\(315\) − 206.894i − 0.656808i
\(316\) 0 0
\(317\) 109.150 0.344322 0.172161 0.985069i \(-0.444925\pi\)
0.172161 + 0.985069i \(0.444925\pi\)
\(318\) 0 0
\(319\) 280.294i 0.878664i
\(320\) 0 0
\(321\) −37.2548 −0.116059
\(322\) 0 0
\(323\) 356.347i 1.10324i
\(324\) 0 0
\(325\) 511.259 1.57310
\(326\) 0 0
\(327\) 76.6489i 0.234400i
\(328\) 0 0
\(329\) −16.4020 −0.0498542
\(330\) 0 0
\(331\) − 321.740i − 0.972025i −0.873952 0.486012i \(-0.838451\pi\)
0.873952 0.486012i \(-0.161549\pi\)
\(332\) 0 0
\(333\) 140.679 0.422459
\(334\) 0 0
\(335\) − 710.254i − 2.12016i
\(336\) 0 0
\(337\) −164.049 −0.486792 −0.243396 0.969927i \(-0.578261\pi\)
−0.243396 + 0.969927i \(0.578261\pi\)
\(338\) 0 0
\(339\) − 80.9706i − 0.238851i
\(340\) 0 0
\(341\) −209.533 −0.614466
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) −130.392 −0.377948
\(346\) 0 0
\(347\) 330.309i 0.951898i 0.879473 + 0.475949i \(0.157895\pi\)
−0.879473 + 0.475949i \(0.842105\pi\)
\(348\) 0 0
\(349\) −262.402 −0.751869 −0.375934 0.926646i \(-0.622678\pi\)
−0.375934 + 0.926646i \(0.622678\pi\)
\(350\) 0 0
\(351\) 93.4313i 0.266186i
\(352\) 0 0
\(353\) −578.098 −1.63767 −0.818835 0.574029i \(-0.805380\pi\)
−0.818835 + 0.574029i \(0.805380\pi\)
\(354\) 0 0
\(355\) 158.392i 0.446174i
\(356\) 0 0
\(357\) 19.1300 0.0535853
\(358\) 0 0
\(359\) − 365.114i − 1.01703i −0.861053 0.508515i \(-0.830195\pi\)
0.861053 0.508515i \(-0.169805\pi\)
\(360\) 0 0
\(361\) −472.480 −1.30881
\(362\) 0 0
\(363\) − 20.4335i − 0.0562908i
\(364\) 0 0
\(365\) −421.725 −1.15541
\(366\) 0 0
\(367\) − 520.071i − 1.41709i −0.705666 0.708544i \(-0.749352\pi\)
0.705666 0.708544i \(-0.250648\pi\)
\(368\) 0 0
\(369\) −60.3431 −0.163532
\(370\) 0 0
\(371\) − 21.2061i − 0.0571592i
\(372\) 0 0
\(373\) 526.711 1.41210 0.706048 0.708164i \(-0.250476\pi\)
0.706048 + 0.708164i \(0.250476\pi\)
\(374\) 0 0
\(375\) 167.201i 0.445869i
\(376\) 0 0
\(377\) −202.794 −0.537915
\(378\) 0 0
\(379\) − 121.976i − 0.321835i −0.986968 0.160918i \(-0.948555\pi\)
0.986968 0.160918i \(-0.0514454\pi\)
\(380\) 0 0
\(381\) 67.0383 0.175954
\(382\) 0 0
\(383\) 316.427i 0.826179i 0.910690 + 0.413089i \(0.135550\pi\)
−0.910690 + 0.413089i \(0.864450\pi\)
\(384\) 0 0
\(385\) −298.392 −0.775044
\(386\) 0 0
\(387\) 197.622i 0.510652i
\(388\) 0 0
\(389\) −92.1474 −0.236883 −0.118441 0.992961i \(-0.537790\pi\)
−0.118441 + 0.992961i \(0.537790\pi\)
\(390\) 0 0
\(391\) 304.157i 0.777895i
\(392\) 0 0
\(393\) 98.6173 0.250935
\(394\) 0 0
\(395\) − 731.980i − 1.85311i
\(396\) 0 0
\(397\) 562.267 1.41629 0.708144 0.706068i \(-0.249533\pi\)
0.708144 + 0.706068i \(0.249533\pi\)
\(398\) 0 0
\(399\) 44.7441i 0.112141i
\(400\) 0 0
\(401\) 81.2061 0.202509 0.101254 0.994861i \(-0.467714\pi\)
0.101254 + 0.994861i \(0.467714\pi\)
\(402\) 0 0
\(403\) − 151.598i − 0.376174i
\(404\) 0 0
\(405\) 649.058 1.60261
\(406\) 0 0
\(407\) − 202.893i − 0.498508i
\(408\) 0 0
\(409\) −450.735 −1.10204 −0.551021 0.834491i \(-0.685762\pi\)
−0.551021 + 0.834491i \(0.685762\pi\)
\(410\) 0 0
\(411\) − 20.3188i − 0.0494374i
\(412\) 0 0
\(413\) −80.5463 −0.195027
\(414\) 0 0
\(415\) 364.802i 0.879041i
\(416\) 0 0
\(417\) 63.0681 0.151242
\(418\) 0 0
\(419\) 624.988i 1.49162i 0.666160 + 0.745809i \(0.267937\pi\)
−0.666160 + 0.745809i \(0.732063\pi\)
\(420\) 0 0
\(421\) −566.476 −1.34555 −0.672774 0.739848i \(-0.734897\pi\)
−0.672774 + 0.739848i \(0.734897\pi\)
\(422\) 0 0
\(423\) − 53.6671i − 0.126873i
\(424\) 0 0
\(425\) 698.597 1.64376
\(426\) 0 0
\(427\) − 40.3015i − 0.0943829i
\(428\) 0 0
\(429\) 66.0659 0.154000
\(430\) 0 0
\(431\) − 289.528i − 0.671760i −0.941905 0.335880i \(-0.890966\pi\)
0.941905 0.335880i \(-0.109034\pi\)
\(432\) 0 0
\(433\) −597.696 −1.38036 −0.690180 0.723638i \(-0.742468\pi\)
−0.690180 + 0.723638i \(0.742468\pi\)
\(434\) 0 0
\(435\) − 118.794i − 0.273090i
\(436\) 0 0
\(437\) −711.409 −1.62794
\(438\) 0 0
\(439\) − 38.3890i − 0.0874464i −0.999044 0.0437232i \(-0.986078\pi\)
0.999044 0.0437232i \(-0.0139220\pi\)
\(440\) 0 0
\(441\) −60.5980 −0.137410
\(442\) 0 0
\(443\) − 599.058i − 1.35228i −0.736775 0.676138i \(-0.763652\pi\)
0.736775 0.676138i \(-0.236348\pi\)
\(444\) 0 0
\(445\) −1011.18 −2.27232
\(446\) 0 0
\(447\) 147.721i 0.330473i
\(448\) 0 0
\(449\) −460.039 −1.02459 −0.512293 0.858811i \(-0.671204\pi\)
−0.512293 + 0.858811i \(0.671204\pi\)
\(450\) 0 0
\(451\) 87.0294i 0.192970i
\(452\) 0 0
\(453\) −137.359 −0.303220
\(454\) 0 0
\(455\) − 215.888i − 0.474479i
\(456\) 0 0
\(457\) −266.323 −0.582764 −0.291382 0.956607i \(-0.594115\pi\)
−0.291382 + 0.956607i \(0.594115\pi\)
\(458\) 0 0
\(459\) 127.667i 0.278142i
\(460\) 0 0
\(461\) −763.123 −1.65537 −0.827683 0.561196i \(-0.810341\pi\)
−0.827683 + 0.561196i \(0.810341\pi\)
\(462\) 0 0
\(463\) − 123.988i − 0.267792i −0.990995 0.133896i \(-0.957251\pi\)
0.990995 0.133896i \(-0.0427488\pi\)
\(464\) 0 0
\(465\) 88.8040 0.190976
\(466\) 0 0
\(467\) − 768.718i − 1.64608i −0.567986 0.823038i \(-0.692277\pi\)
0.567986 0.823038i \(-0.307723\pi\)
\(468\) 0 0
\(469\) −208.029 −0.443558
\(470\) 0 0
\(471\) 5.91457i 0.0125575i
\(472\) 0 0
\(473\) 285.019 0.602578
\(474\) 0 0
\(475\) 1633.99i 3.43997i
\(476\) 0 0
\(477\) 69.3859 0.145463
\(478\) 0 0
\(479\) 118.981i 0.248394i 0.992258 + 0.124197i \(0.0396355\pi\)
−0.992258 + 0.124197i \(0.960364\pi\)
\(480\) 0 0
\(481\) 146.794 0.305185
\(482\) 0 0
\(483\) 38.1909i 0.0790702i
\(484\) 0 0
\(485\) −1482.41 −3.05652
\(486\) 0 0
\(487\) − 282.577i − 0.580240i −0.956990 0.290120i \(-0.906305\pi\)
0.956990 0.290120i \(-0.0936952\pi\)
\(488\) 0 0
\(489\) −61.2346 −0.125224
\(490\) 0 0
\(491\) 388.049i 0.790323i 0.918612 + 0.395162i \(0.129311\pi\)
−0.918612 + 0.395162i \(0.870689\pi\)
\(492\) 0 0
\(493\) −277.103 −0.562075
\(494\) 0 0
\(495\) − 976.333i − 1.97239i
\(496\) 0 0
\(497\) 46.3919 0.0933439
\(498\) 0 0
\(499\) − 27.7157i − 0.0555425i −0.999614 0.0277713i \(-0.991159\pi\)
0.999614 0.0277713i \(-0.00884101\pi\)
\(500\) 0 0
\(501\) 173.841 0.346988
\(502\) 0 0
\(503\) 727.477i 1.44628i 0.690703 + 0.723138i \(0.257301\pi\)
−0.690703 + 0.723138i \(0.742699\pi\)
\(504\) 0 0
\(505\) 109.598 0.217026
\(506\) 0 0
\(507\) − 51.1989i − 0.100984i
\(508\) 0 0
\(509\) −634.183 −1.24594 −0.622969 0.782246i \(-0.714074\pi\)
−0.622969 + 0.782246i \(0.714074\pi\)
\(510\) 0 0
\(511\) 123.520i 0.241723i
\(512\) 0 0
\(513\) −298.607 −0.582080
\(514\) 0 0
\(515\) 962.774i 1.86946i
\(516\) 0 0
\(517\) −77.4010 −0.149712
\(518\) 0 0
\(519\) 23.4491i 0.0451813i
\(520\) 0 0
\(521\) −833.127 −1.59909 −0.799546 0.600605i \(-0.794927\pi\)
−0.799546 + 0.600605i \(0.794927\pi\)
\(522\) 0 0
\(523\) − 876.434i − 1.67578i −0.545838 0.837891i \(-0.683789\pi\)
0.545838 0.837891i \(-0.316211\pi\)
\(524\) 0 0
\(525\) 87.7181 0.167082
\(526\) 0 0
\(527\) − 207.147i − 0.393069i
\(528\) 0 0
\(529\) −78.2162 −0.147857
\(530\) 0 0
\(531\) − 263.546i − 0.496321i
\(532\) 0 0
\(533\) −62.9662 −0.118135
\(534\) 0 0
\(535\) 574.491i 1.07381i
\(536\) 0 0
\(537\) −172.686 −0.321576
\(538\) 0 0
\(539\) 87.3970i 0.162147i
\(540\) 0 0
\(541\) 405.915 0.750305 0.375152 0.926963i \(-0.377590\pi\)
0.375152 + 0.926963i \(0.377590\pi\)
\(542\) 0 0
\(543\) − 23.7072i − 0.0436596i
\(544\) 0 0
\(545\) 1181.97 2.16875
\(546\) 0 0
\(547\) − 606.024i − 1.10791i −0.832548 0.553953i \(-0.813119\pi\)
0.832548 0.553953i \(-0.186881\pi\)
\(548\) 0 0
\(549\) 131.866 0.240193
\(550\) 0 0
\(551\) − 648.131i − 1.17628i
\(552\) 0 0
\(553\) −214.392 −0.387689
\(554\) 0 0
\(555\) 85.9899i 0.154937i
\(556\) 0 0
\(557\) −36.4442 −0.0654294 −0.0327147 0.999465i \(-0.510415\pi\)
−0.0327147 + 0.999465i \(0.510415\pi\)
\(558\) 0 0
\(559\) 206.213i 0.368896i
\(560\) 0 0
\(561\) 90.2742 0.160917
\(562\) 0 0
\(563\) − 186.389i − 0.331064i −0.986204 0.165532i \(-0.947066\pi\)
0.986204 0.165532i \(-0.0529341\pi\)
\(564\) 0 0
\(565\) −1248.61 −2.20993
\(566\) 0 0
\(567\) − 190.105i − 0.335282i
\(568\) 0 0
\(569\) 670.891 1.17907 0.589536 0.807742i \(-0.299311\pi\)
0.589536 + 0.807742i \(0.299311\pi\)
\(570\) 0 0
\(571\) 677.082i 1.18578i 0.805282 + 0.592892i \(0.202014\pi\)
−0.805282 + 0.592892i \(0.797986\pi\)
\(572\) 0 0
\(573\) −91.9271 −0.160431
\(574\) 0 0
\(575\) 1394.67i 2.42552i
\(576\) 0 0
\(577\) 927.901 1.60815 0.804073 0.594530i \(-0.202662\pi\)
0.804073 + 0.594530i \(0.202662\pi\)
\(578\) 0 0
\(579\) − 153.068i − 0.264366i
\(580\) 0 0
\(581\) 106.848 0.183904
\(582\) 0 0
\(583\) − 100.071i − 0.171649i
\(584\) 0 0
\(585\) 706.382 1.20749
\(586\) 0 0
\(587\) 321.120i 0.547053i 0.961865 + 0.273526i \(0.0881900\pi\)
−0.961865 + 0.273526i \(0.911810\pi\)
\(588\) 0 0
\(589\) 484.508 0.822595
\(590\) 0 0
\(591\) 85.1046i 0.144001i
\(592\) 0 0
\(593\) −219.255 −0.369738 −0.184869 0.982763i \(-0.559186\pi\)
−0.184869 + 0.982763i \(0.559186\pi\)
\(594\) 0 0
\(595\) − 294.995i − 0.495790i
\(596\) 0 0
\(597\) 228.754 0.383173
\(598\) 0 0
\(599\) − 154.802i − 0.258434i −0.991616 0.129217i \(-0.958754\pi\)
0.991616 0.129217i \(-0.0412464\pi\)
\(600\) 0 0
\(601\) −205.862 −0.342533 −0.171266 0.985225i \(-0.554786\pi\)
−0.171266 + 0.985225i \(0.554786\pi\)
\(602\) 0 0
\(603\) − 680.666i − 1.12880i
\(604\) 0 0
\(605\) −315.097 −0.520821
\(606\) 0 0
\(607\) 790.663i 1.30258i 0.758831 + 0.651288i \(0.225771\pi\)
−0.758831 + 0.651288i \(0.774229\pi\)
\(608\) 0 0
\(609\) −34.7939 −0.0571329
\(610\) 0 0
\(611\) − 56.0000i − 0.0916530i
\(612\) 0 0
\(613\) 741.471 1.20958 0.604789 0.796386i \(-0.293257\pi\)
0.604789 + 0.796386i \(0.293257\pi\)
\(614\) 0 0
\(615\) − 36.8848i − 0.0599752i
\(616\) 0 0
\(617\) −171.578 −0.278084 −0.139042 0.990286i \(-0.544402\pi\)
−0.139042 + 0.990286i \(0.544402\pi\)
\(618\) 0 0
\(619\) 540.198i 0.872695i 0.899778 + 0.436347i \(0.143728\pi\)
−0.899778 + 0.436347i \(0.856272\pi\)
\(620\) 0 0
\(621\) −254.873 −0.410424
\(622\) 0 0
\(623\) 296.168i 0.475391i
\(624\) 0 0
\(625\) 1163.38 1.86141
\(626\) 0 0
\(627\) 211.147i 0.336758i
\(628\) 0 0
\(629\) 200.583 0.318892
\(630\) 0 0
\(631\) − 269.399i − 0.426940i −0.976950 0.213470i \(-0.931523\pi\)
0.976950 0.213470i \(-0.0684766\pi\)
\(632\) 0 0
\(633\) 96.0975 0.151813
\(634\) 0 0
\(635\) − 1033.77i − 1.62798i
\(636\) 0 0
\(637\) −63.2321 −0.0992655
\(638\) 0 0
\(639\) 151.794i 0.237549i
\(640\) 0 0
\(641\) 36.1867 0.0564535 0.0282268 0.999602i \(-0.491014\pi\)
0.0282268 + 0.999602i \(0.491014\pi\)
\(642\) 0 0
\(643\) 266.297i 0.414148i 0.978325 + 0.207074i \(0.0663941\pi\)
−0.978325 + 0.207074i \(0.933606\pi\)
\(644\) 0 0
\(645\) −120.797 −0.187282
\(646\) 0 0
\(647\) 1086.24i 1.67888i 0.543452 + 0.839440i \(0.317117\pi\)
−0.543452 + 0.839440i \(0.682883\pi\)
\(648\) 0 0
\(649\) −380.098 −0.585666
\(650\) 0 0
\(651\) − 26.0101i − 0.0399541i
\(652\) 0 0
\(653\) −1195.35 −1.83055 −0.915274 0.402832i \(-0.868026\pi\)
−0.915274 + 0.402832i \(0.868026\pi\)
\(654\) 0 0
\(655\) − 1520.74i − 2.32173i
\(656\) 0 0
\(657\) −404.156 −0.615154
\(658\) 0 0
\(659\) − 685.220i − 1.03979i −0.854231 0.519894i \(-0.825971\pi\)
0.854231 0.519894i \(-0.174029\pi\)
\(660\) 0 0
\(661\) −993.382 −1.50285 −0.751423 0.659820i \(-0.770632\pi\)
−0.751423 + 0.659820i \(0.770632\pi\)
\(662\) 0 0
\(663\) 65.3138i 0.0985125i
\(664\) 0 0
\(665\) 689.980 1.03756
\(666\) 0 0
\(667\) − 553.206i − 0.829394i
\(668\) 0 0
\(669\) −6.19938 −0.00926664
\(670\) 0 0
\(671\) − 190.183i − 0.283432i
\(672\) 0 0
\(673\) 106.569 0.158349 0.0791743 0.996861i \(-0.474772\pi\)
0.0791743 + 0.996861i \(0.474772\pi\)
\(674\) 0 0
\(675\) 585.401i 0.867261i
\(676\) 0 0
\(677\) 1004.18 1.48329 0.741643 0.670795i \(-0.234047\pi\)
0.741643 + 0.670795i \(0.234047\pi\)
\(678\) 0 0
\(679\) 434.188i 0.639452i
\(680\) 0 0
\(681\) 125.245 0.183913
\(682\) 0 0
\(683\) 678.225i 0.993009i 0.868034 + 0.496505i \(0.165383\pi\)
−0.868034 + 0.496505i \(0.834617\pi\)
\(684\) 0 0
\(685\) −313.327 −0.457411
\(686\) 0 0
\(687\) 135.919i 0.197844i
\(688\) 0 0
\(689\) 72.4020 0.105083
\(690\) 0 0
\(691\) − 365.175i − 0.528473i −0.964458 0.264236i \(-0.914880\pi\)
0.964458 0.264236i \(-0.0851199\pi\)
\(692\) 0 0
\(693\) −285.961 −0.412643
\(694\) 0 0
\(695\) − 972.546i − 1.39935i
\(696\) 0 0
\(697\) −86.0387 −0.123441
\(698\) 0 0
\(699\) 112.976i 0.161626i
\(700\) 0 0
\(701\) −940.292 −1.34136 −0.670679 0.741748i \(-0.733997\pi\)
−0.670679 + 0.741748i \(0.733997\pi\)
\(702\) 0 0
\(703\) 469.155i 0.667361i
\(704\) 0 0
\(705\) 32.8040 0.0465306
\(706\) 0 0
\(707\) − 32.1005i − 0.0454038i
\(708\) 0 0
\(709\) −1057.46 −1.49148 −0.745738 0.666239i \(-0.767903\pi\)
−0.745738 + 0.666239i \(0.767903\pi\)
\(710\) 0 0
\(711\) − 701.487i − 0.986620i
\(712\) 0 0
\(713\) 413.547 0.580010
\(714\) 0 0
\(715\) − 1018.77i − 1.42486i
\(716\) 0 0
\(717\) 192.090 0.267907
\(718\) 0 0
\(719\) − 1034.82i − 1.43926i −0.694360 0.719628i \(-0.744312\pi\)
0.694360 0.719628i \(-0.255688\pi\)
\(720\) 0 0
\(721\) 281.990 0.391109
\(722\) 0 0
\(723\) 42.1017i 0.0582320i
\(724\) 0 0
\(725\) −1270.62 −1.75258
\(726\) 0 0
\(727\) 495.145i 0.681080i 0.940230 + 0.340540i \(0.110610\pi\)
−0.940230 + 0.340540i \(0.889390\pi\)
\(728\) 0 0
\(729\) 567.489 0.778449
\(730\) 0 0
\(731\) 281.775i 0.385465i
\(732\) 0 0
\(733\) 567.494 0.774207 0.387103 0.922036i \(-0.373476\pi\)
0.387103 + 0.922036i \(0.373476\pi\)
\(734\) 0 0
\(735\) − 37.0405i − 0.0503953i
\(736\) 0 0
\(737\) −981.685 −1.33200
\(738\) 0 0
\(739\) − 544.701i − 0.737078i −0.929612 0.368539i \(-0.879858\pi\)
0.929612 0.368539i \(-0.120142\pi\)
\(740\) 0 0
\(741\) −152.766 −0.206162
\(742\) 0 0
\(743\) 731.264i 0.984205i 0.870537 + 0.492102i \(0.163771\pi\)
−0.870537 + 0.492102i \(0.836229\pi\)
\(744\) 0 0
\(745\) 2277.95 3.05765
\(746\) 0 0
\(747\) 349.605i 0.468012i
\(748\) 0 0
\(749\) 168.264 0.224652
\(750\) 0 0
\(751\) − 666.262i − 0.887166i −0.896233 0.443583i \(-0.853707\pi\)
0.896233 0.443583i \(-0.146293\pi\)
\(752\) 0 0
\(753\) −150.500 −0.199867
\(754\) 0 0
\(755\) 2118.15i 2.80550i
\(756\) 0 0
\(757\) 238.623 0.315222 0.157611 0.987501i \(-0.449621\pi\)
0.157611 + 0.987501i \(0.449621\pi\)
\(758\) 0 0
\(759\) 180.223i 0.237447i
\(760\) 0 0
\(761\) −614.930 −0.808055 −0.404028 0.914747i \(-0.632390\pi\)
−0.404028 + 0.914747i \(0.632390\pi\)
\(762\) 0 0
\(763\) − 346.191i − 0.453723i
\(764\) 0 0
\(765\) 965.219 1.26172
\(766\) 0 0
\(767\) − 275.002i − 0.358543i
\(768\) 0 0
\(769\) 178.950 0.232705 0.116353 0.993208i \(-0.462880\pi\)
0.116353 + 0.993208i \(0.462880\pi\)
\(770\) 0 0
\(771\) 187.072i 0.242636i
\(772\) 0 0
\(773\) 631.615 0.817095 0.408548 0.912737i \(-0.366035\pi\)
0.408548 + 0.912737i \(0.366035\pi\)
\(774\) 0 0
\(775\) − 949.849i − 1.22561i
\(776\) 0 0
\(777\) 25.1859 0.0324142
\(778\) 0 0
\(779\) − 201.241i − 0.258332i
\(780\) 0 0
\(781\) 218.923 0.280311
\(782\) 0 0
\(783\) − 232.203i − 0.296556i
\(784\) 0 0
\(785\) 91.2061 0.116186
\(786\) 0 0
\(787\) 456.655i 0.580247i 0.956989 + 0.290124i \(0.0936965\pi\)
−0.956989 + 0.290124i \(0.906304\pi\)
\(788\) 0 0
\(789\) −221.050 −0.280165
\(790\) 0 0
\(791\) 365.710i 0.462339i
\(792\) 0 0
\(793\) 137.598 0.173516
\(794\) 0 0
\(795\) 42.4121i 0.0533486i
\(796\) 0 0
\(797\) 218.566 0.274236 0.137118 0.990555i \(-0.456216\pi\)
0.137118 + 0.990555i \(0.456216\pi\)
\(798\) 0 0
\(799\) − 76.5199i − 0.0957695i
\(800\) 0 0
\(801\) −969.058 −1.20981
\(802\) 0 0
\(803\) 582.891i 0.725892i
\(804\) 0 0
\(805\) 588.926 0.731585
\(806\) 0 0
\(807\) − 16.4976i − 0.0204431i
\(808\) 0 0
\(809\) −1347.46 −1.66559 −0.832794 0.553584i \(-0.813260\pi\)
−0.832794 + 0.553584i \(0.813260\pi\)
\(810\) 0 0
\(811\) 672.620i 0.829371i 0.909965 + 0.414686i \(0.136108\pi\)
−0.909965 + 0.414686i \(0.863892\pi\)
\(812\) 0 0
\(813\) −234.148 −0.288005
\(814\) 0 0
\(815\) 944.273i 1.15862i
\(816\) 0 0
\(817\) −659.058 −0.806681
\(818\) 0 0
\(819\) − 206.894i − 0.252618i
\(820\) 0 0
\(821\) 1162.57 1.41604 0.708022 0.706190i \(-0.249588\pi\)
0.708022 + 0.706190i \(0.249588\pi\)
\(822\) 0 0
\(823\) − 1041.65i − 1.26567i −0.774286 0.632835i \(-0.781891\pi\)
0.774286 0.632835i \(-0.218109\pi\)
\(824\) 0 0
\(825\) 413.941 0.501747
\(826\) 0 0
\(827\) − 278.432i − 0.336678i −0.985729 0.168339i \(-0.946160\pi\)
0.985729 0.168339i \(-0.0538403\pi\)
\(828\) 0 0
\(829\) −1065.74 −1.28557 −0.642785 0.766046i \(-0.722221\pi\)
−0.642785 + 0.766046i \(0.722221\pi\)
\(830\) 0 0
\(831\) − 60.3071i − 0.0725717i
\(832\) 0 0
\(833\) −86.4020 −0.103724
\(834\) 0 0
\(835\) − 2680.72i − 3.21045i
\(836\) 0 0
\(837\) 173.583 0.207387
\(838\) 0 0
\(839\) 305.844i 0.364533i 0.983249 + 0.182267i \(0.0583434\pi\)
−0.983249 + 0.182267i \(0.941657\pi\)
\(840\) 0 0
\(841\) −337.000 −0.400713
\(842\) 0 0
\(843\) 88.0059i 0.104396i
\(844\) 0 0
\(845\) −789.516 −0.934339
\(846\) 0 0
\(847\) 92.2898i 0.108961i
\(848\) 0 0
\(849\) −104.599 −0.123202
\(850\) 0 0
\(851\) 400.442i 0.470555i
\(852\) 0 0
\(853\) 164.018 0.192283 0.0961417 0.995368i \(-0.469350\pi\)
0.0961417 + 0.995368i \(0.469350\pi\)
\(854\) 0 0
\(855\) 2257.60i 2.64047i
\(856\) 0 0
\(857\) −851.068 −0.993078 −0.496539 0.868014i \(-0.665396\pi\)
−0.496539 + 0.868014i \(0.665396\pi\)
\(858\) 0 0
\(859\) 1179.69i 1.37333i 0.726973 + 0.686666i \(0.240927\pi\)
−0.726973 + 0.686666i \(0.759073\pi\)
\(860\) 0 0
\(861\) −10.8033 −0.0125474
\(862\) 0 0
\(863\) − 279.048i − 0.323346i −0.986844 0.161673i \(-0.948311\pi\)
0.986844 0.161673i \(-0.0516890\pi\)
\(864\) 0 0
\(865\) 361.598 0.418032
\(866\) 0 0
\(867\) − 80.0458i − 0.0923250i
\(868\) 0 0
\(869\) −1011.71 −1.16423
\(870\) 0 0
\(871\) − 710.254i − 0.815447i
\(872\) 0 0
\(873\) −1420.66 −1.62733
\(874\) 0 0
\(875\) − 755.176i − 0.863058i
\(876\) 0 0
\(877\) −674.159 −0.768711 −0.384355 0.923185i \(-0.625576\pi\)
−0.384355 + 0.923185i \(0.625576\pi\)
\(878\) 0 0
\(879\) − 128.398i − 0.146073i
\(880\) 0 0
\(881\) −1001.29 −1.13654 −0.568271 0.822841i \(-0.692387\pi\)
−0.568271 + 0.822841i \(0.692387\pi\)
\(882\) 0 0
\(883\) 882.010i 0.998879i 0.866349 + 0.499439i \(0.166461\pi\)
−0.866349 + 0.499439i \(0.833539\pi\)
\(884\) 0 0
\(885\) 161.093 0.182026
\(886\) 0 0
\(887\) 7.08053i 0.00798256i 0.999992 + 0.00399128i \(0.00127047\pi\)
−0.999992 + 0.00399128i \(0.998730\pi\)
\(888\) 0 0
\(889\) −302.784 −0.340589
\(890\) 0 0
\(891\) − 897.103i − 1.00685i
\(892\) 0 0
\(893\) 178.976 0.200422
\(894\) 0 0
\(895\) 2662.92i 2.97533i
\(896\) 0 0
\(897\) −130.392 −0.145364
\(898\) 0 0
\(899\) 376.764i 0.419092i
\(900\) 0 0
\(901\) 98.9320 0.109802
\(902\) 0 0
\(903\) 35.3805i 0.0391811i
\(904\) 0 0
\(905\) −365.578 −0.403953
\(906\) 0 0
\(907\) − 450.372i − 0.496551i −0.968689 0.248275i \(-0.920136\pi\)
0.968689 0.248275i \(-0.0798638\pi\)
\(908\) 0 0
\(909\) 105.032 0.115547
\(910\) 0 0
\(911\) − 202.426i − 0.222201i −0.993809 0.111101i \(-0.964562\pi\)
0.993809 0.111101i \(-0.0354376\pi\)
\(912\) 0 0
\(913\) 504.215 0.552262
\(914\) 0 0
\(915\) 80.6030i 0.0880907i
\(916\) 0 0
\(917\) −445.413 −0.485728
\(918\) 0 0
\(919\) 1593.73i 1.73420i 0.498138 + 0.867098i \(0.334017\pi\)
−0.498138 + 0.867098i \(0.665983\pi\)
\(920\) 0 0
\(921\) −185.500 −0.201412
\(922\) 0 0
\(923\) 158.392i 0.171606i
\(924\) 0 0
\(925\) 919.749 0.994323
\(926\) 0 0
\(927\) 922.666i 0.995325i
\(928\) 0 0
\(929\) −1039.40 −1.11884 −0.559419 0.828885i \(-0.688976\pi\)
−0.559419 + 0.828885i \(0.688976\pi\)
\(930\) 0 0
\(931\) − 202.090i − 0.217068i
\(932\) 0 0
\(933\) 42.3320 0.0453719
\(934\) 0 0
\(935\) − 1392.08i − 1.48885i
\(936\) 0 0
\(937\) −881.765 −0.941051 −0.470525 0.882386i \(-0.655936\pi\)
−0.470525 + 0.882386i \(0.655936\pi\)
\(938\) 0 0
\(939\) − 48.0059i − 0.0511245i
\(940\) 0 0
\(941\) 953.344 1.01312 0.506559 0.862205i \(-0.330917\pi\)
0.506559 + 0.862205i \(0.330917\pi\)
\(942\) 0 0
\(943\) − 171.767i − 0.182149i
\(944\) 0 0
\(945\) 247.196 0.261583
\(946\) 0 0
\(947\) 16.8957i 0.0178413i 0.999960 + 0.00892063i \(0.00283956\pi\)
−0.999960 + 0.00892063i \(0.997160\pi\)
\(948\) 0 0
\(949\) −421.725 −0.444389
\(950\) 0 0
\(951\) 63.9386i 0.0672330i
\(952\) 0 0
\(953\) −1526.31 −1.60159 −0.800794 0.598940i \(-0.795589\pi\)
−0.800794 + 0.598940i \(0.795589\pi\)
\(954\) 0 0
\(955\) 1417.57i 1.48436i
\(956\) 0 0
\(957\) −164.192 −0.171570
\(958\) 0 0
\(959\) 91.7713i 0.0956948i
\(960\) 0 0
\(961\) 679.352 0.706921
\(962\) 0 0
\(963\) 550.558i 0.571712i
\(964\) 0 0
\(965\) −2360.40 −2.44601
\(966\) 0 0
\(967\) − 1410.39i − 1.45852i −0.684235 0.729262i \(-0.739864\pi\)
0.684235 0.729262i \(-0.260136\pi\)
\(968\) 0 0
\(969\) −208.743 −0.215421
\(970\) 0 0
\(971\) 596.497i 0.614312i 0.951659 + 0.307156i \(0.0993773\pi\)
−0.951659 + 0.307156i \(0.900623\pi\)
\(972\) 0 0
\(973\) −284.852 −0.292757
\(974\) 0 0
\(975\) 299.488i 0.307168i
\(976\) 0 0
\(977\) 146.686 0.150140 0.0750698 0.997178i \(-0.476082\pi\)
0.0750698 + 0.997178i \(0.476082\pi\)
\(978\) 0 0
\(979\) 1397.62i 1.42760i
\(980\) 0 0
\(981\) 1132.73 1.15467
\(982\) 0 0
\(983\) − 169.457i − 0.172388i −0.996278 0.0861939i \(-0.972530\pi\)
0.996278 0.0861939i \(-0.0274704\pi\)
\(984\) 0 0
\(985\) 1312.36 1.33235
\(986\) 0 0
\(987\) − 9.60808i − 0.00973463i
\(988\) 0 0
\(989\) −562.533 −0.568789
\(990\) 0 0
\(991\) − 1686.90i − 1.70222i −0.524988 0.851109i \(-0.675930\pi\)
0.524988 0.851109i \(-0.324070\pi\)
\(992\) 0 0
\(993\) 188.471 0.189800
\(994\) 0 0
\(995\) − 3527.52i − 3.54524i
\(996\) 0 0
\(997\) 1736.14 1.74136 0.870680 0.491849i \(-0.163679\pi\)
0.870680 + 0.491849i \(0.163679\pi\)
\(998\) 0 0
\(999\) 168.082i 0.168250i
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 1792.3.d.g.1023.6 8
4.3 odd 2 inner 1792.3.d.g.1023.4 8
8.3 odd 2 inner 1792.3.d.g.1023.5 8
8.5 even 2 inner 1792.3.d.g.1023.3 8
16.3 odd 4 56.3.g.a.43.1 4
16.5 even 4 56.3.g.a.43.2 yes 4
16.11 odd 4 224.3.g.a.15.4 4
16.13 even 4 224.3.g.a.15.3 4
48.5 odd 4 504.3.g.a.379.3 4
48.11 even 4 2016.3.g.a.1135.1 4
48.29 odd 4 2016.3.g.a.1135.4 4
48.35 even 4 504.3.g.a.379.4 4
112.3 even 12 392.3.k.j.275.4 8
112.5 odd 12 392.3.k.j.67.4 8
112.13 odd 4 1568.3.g.h.687.2 4
112.19 even 12 392.3.k.j.67.2 8
112.27 even 4 1568.3.g.h.687.1 4
112.37 even 12 392.3.k.i.67.4 8
112.51 odd 12 392.3.k.i.67.2 8
112.53 even 12 392.3.k.i.275.2 8
112.67 odd 12 392.3.k.i.275.4 8
112.69 odd 4 392.3.g.h.99.2 4
112.83 even 4 392.3.g.h.99.1 4
112.101 odd 12 392.3.k.j.275.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.a.43.1 4 16.3 odd 4
56.3.g.a.43.2 yes 4 16.5 even 4
224.3.g.a.15.3 4 16.13 even 4
224.3.g.a.15.4 4 16.11 odd 4
392.3.g.h.99.1 4 112.83 even 4
392.3.g.h.99.2 4 112.69 odd 4
392.3.k.i.67.2 8 112.51 odd 12
392.3.k.i.67.4 8 112.37 even 12
392.3.k.i.275.2 8 112.53 even 12
392.3.k.i.275.4 8 112.67 odd 12
392.3.k.j.67.2 8 112.19 even 12
392.3.k.j.67.4 8 112.5 odd 12
392.3.k.j.275.2 8 112.101 odd 12
392.3.k.j.275.4 8 112.3 even 12
504.3.g.a.379.3 4 48.5 odd 4
504.3.g.a.379.4 4 48.35 even 4
1568.3.g.h.687.1 4 112.27 even 4
1568.3.g.h.687.2 4 112.13 odd 4
1792.3.d.g.1023.3 8 8.5 even 2 inner
1792.3.d.g.1023.4 8 4.3 odd 2 inner
1792.3.d.g.1023.5 8 8.3 odd 2 inner
1792.3.d.g.1023.6 8 1.1 even 1 trivial
2016.3.g.a.1135.1 4 48.11 even 4
2016.3.g.a.1135.4 4 48.29 odd 4