Properties

Label 1568.3.g.j.687.6
Level $1568$
Weight $3$
Character 1568.687
Analytic conductor $42.725$
Analytic rank $0$
Dimension $6$
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] = [1568,3,Mod(687,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.687");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.g (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: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.15582448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 687.6
Root \(0.500000 - 0.148124i\) of defining polynomial
Character \(\chi\) \(=\) 1568.687
Dual form 1568.3.g.j.687.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.98103 q^{3} +1.88252i q^{5} +6.84860 q^{9} +O(q^{10})\) \(q+3.98103 q^{3} +1.88252i q^{5} +6.84860 q^{9} +7.87946 q^{11} +11.4863i q^{13} +7.49437i q^{15} -2.89843 q^{17} +30.0447 q^{19} -38.5483i q^{23} +21.4561 q^{25} -8.56478 q^{27} +27.8701i q^{29} -22.4831i q^{31} +31.3684 q^{33} +45.5552i q^{37} +45.7274i q^{39} -40.6313 q^{41} +47.2806 q^{43} +12.8926i q^{45} +82.5809i q^{47} -11.5387 q^{51} -26.8841i q^{53} +14.8332i q^{55} +119.609 q^{57} +10.4111 q^{59} +22.1624i q^{61} -21.6233 q^{65} +59.3099 q^{67} -153.462i q^{69} +38.2541i q^{71} +13.9778 q^{73} +85.4175 q^{75} +51.1895i q^{79} -95.7341 q^{81} +89.4458 q^{83} -5.45635i q^{85} +110.952i q^{87} +105.258 q^{89} -89.5059i q^{93} +56.5597i q^{95} +55.3301 q^{97} +53.9633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 40 q^{9} + 30 q^{11} - 30 q^{17} + 78 q^{19} + 92 q^{25} - 78 q^{27} + 78 q^{33} - 116 q^{41} + 100 q^{43} + 10 q^{51} + 166 q^{57} - 110 q^{59} + 32 q^{65} + 434 q^{67} - 102 q^{73} - 60 q^{75} + 82 q^{81} + 268 q^{83} - 214 q^{89} - 76 q^{97} - 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 3.98103 1.32701 0.663505 0.748172i \(-0.269068\pi\)
0.663505 + 0.748172i \(0.269068\pi\)
\(4\) 0 0
\(5\) 1.88252i 0.376504i 0.982121 + 0.188252i \(0.0602822\pi\)
−0.982121 + 0.188252i \(0.939718\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.84860 0.760956
\(10\) 0 0
\(11\) 7.87946 0.716314 0.358157 0.933661i \(-0.383405\pi\)
0.358157 + 0.933661i \(0.383405\pi\)
\(12\) 0 0
\(13\) 11.4863i 0.883564i 0.897122 + 0.441782i \(0.145654\pi\)
−0.897122 + 0.441782i \(0.854346\pi\)
\(14\) 0 0
\(15\) 7.49437i 0.499625i
\(16\) 0 0
\(17\) −2.89843 −0.170496 −0.0852478 0.996360i \(-0.527168\pi\)
−0.0852478 + 0.996360i \(0.527168\pi\)
\(18\) 0 0
\(19\) 30.0447 1.58130 0.790649 0.612270i \(-0.209743\pi\)
0.790649 + 0.612270i \(0.209743\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 38.5483i − 1.67601i −0.545661 0.838006i \(-0.683721\pi\)
0.545661 0.838006i \(-0.316279\pi\)
\(24\) 0 0
\(25\) 21.4561 0.858245
\(26\) 0 0
\(27\) −8.56478 −0.317214
\(28\) 0 0
\(29\) 27.8701i 0.961039i 0.876984 + 0.480519i \(0.159552\pi\)
−0.876984 + 0.480519i \(0.840448\pi\)
\(30\) 0 0
\(31\) − 22.4831i − 0.725262i −0.931933 0.362631i \(-0.881879\pi\)
0.931933 0.362631i \(-0.118121\pi\)
\(32\) 0 0
\(33\) 31.3684 0.950556
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 45.5552i 1.23122i 0.788050 + 0.615611i \(0.211091\pi\)
−0.788050 + 0.615611i \(0.788909\pi\)
\(38\) 0 0
\(39\) 45.7274i 1.17250i
\(40\) 0 0
\(41\) −40.6313 −0.991007 −0.495503 0.868606i \(-0.665016\pi\)
−0.495503 + 0.868606i \(0.665016\pi\)
\(42\) 0 0
\(43\) 47.2806 1.09955 0.549774 0.835313i \(-0.314714\pi\)
0.549774 + 0.835313i \(0.314714\pi\)
\(44\) 0 0
\(45\) 12.8926i 0.286503i
\(46\) 0 0
\(47\) 82.5809i 1.75704i 0.477705 + 0.878520i \(0.341469\pi\)
−0.477705 + 0.878520i \(0.658531\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −11.5387 −0.226249
\(52\) 0 0
\(53\) − 26.8841i − 0.507247i −0.967303 0.253623i \(-0.918378\pi\)
0.967303 0.253623i \(-0.0816224\pi\)
\(54\) 0 0
\(55\) 14.8332i 0.269695i
\(56\) 0 0
\(57\) 119.609 2.09840
\(58\) 0 0
\(59\) 10.4111 0.176459 0.0882296 0.996100i \(-0.471879\pi\)
0.0882296 + 0.996100i \(0.471879\pi\)
\(60\) 0 0
\(61\) 22.1624i 0.363318i 0.983362 + 0.181659i \(0.0581467\pi\)
−0.983362 + 0.181659i \(0.941853\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −21.6233 −0.332665
\(66\) 0 0
\(67\) 59.3099 0.885222 0.442611 0.896714i \(-0.354052\pi\)
0.442611 + 0.896714i \(0.354052\pi\)
\(68\) 0 0
\(69\) − 153.462i − 2.22409i
\(70\) 0 0
\(71\) 38.2541i 0.538791i 0.963030 + 0.269395i \(0.0868238\pi\)
−0.963030 + 0.269395i \(0.913176\pi\)
\(72\) 0 0
\(73\) 13.9778 0.191477 0.0957383 0.995407i \(-0.469479\pi\)
0.0957383 + 0.995407i \(0.469479\pi\)
\(74\) 0 0
\(75\) 85.4175 1.13890
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 51.1895i 0.647969i 0.946062 + 0.323984i \(0.105022\pi\)
−0.946062 + 0.323984i \(0.894978\pi\)
\(80\) 0 0
\(81\) −95.7341 −1.18190
\(82\) 0 0
\(83\) 89.4458 1.07766 0.538830 0.842414i \(-0.318866\pi\)
0.538830 + 0.842414i \(0.318866\pi\)
\(84\) 0 0
\(85\) − 5.45635i − 0.0641923i
\(86\) 0 0
\(87\) 110.952i 1.27531i
\(88\) 0 0
\(89\) 105.258 1.18267 0.591335 0.806426i \(-0.298601\pi\)
0.591335 + 0.806426i \(0.298601\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 89.5059i − 0.962429i
\(94\) 0 0
\(95\) 56.5597i 0.595365i
\(96\) 0 0
\(97\) 55.3301 0.570413 0.285206 0.958466i \(-0.407938\pi\)
0.285206 + 0.958466i \(0.407938\pi\)
\(98\) 0 0
\(99\) 53.9633 0.545084
\(100\) 0 0
\(101\) 31.4328i 0.311216i 0.987819 + 0.155608i \(0.0497337\pi\)
−0.987819 + 0.155608i \(0.950266\pi\)
\(102\) 0 0
\(103\) 79.9862i 0.776565i 0.921540 + 0.388283i \(0.126932\pi\)
−0.921540 + 0.388283i \(0.873068\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −48.7634 −0.455732 −0.227866 0.973692i \(-0.573175\pi\)
−0.227866 + 0.973692i \(0.573175\pi\)
\(108\) 0 0
\(109\) − 115.069i − 1.05568i −0.849344 0.527840i \(-0.823002\pi\)
0.849344 0.527840i \(-0.176998\pi\)
\(110\) 0 0
\(111\) 181.357i 1.63384i
\(112\) 0 0
\(113\) −55.7570 −0.493425 −0.246712 0.969089i \(-0.579350\pi\)
−0.246712 + 0.969089i \(0.579350\pi\)
\(114\) 0 0
\(115\) 72.5679 0.631025
\(116\) 0 0
\(117\) 78.6653i 0.672353i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −58.9142 −0.486894
\(122\) 0 0
\(123\) −161.754 −1.31508
\(124\) 0 0
\(125\) 87.4546i 0.699637i
\(126\) 0 0
\(127\) − 35.6964i − 0.281074i −0.990075 0.140537i \(-0.955117\pi\)
0.990075 0.140537i \(-0.0448828\pi\)
\(128\) 0 0
\(129\) 188.225 1.45911
\(130\) 0 0
\(131\) 121.292 0.925896 0.462948 0.886385i \(-0.346792\pi\)
0.462948 + 0.886385i \(0.346792\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 16.1234i − 0.119432i
\(136\) 0 0
\(137\) −8.49670 −0.0620197 −0.0310099 0.999519i \(-0.509872\pi\)
−0.0310099 + 0.999519i \(0.509872\pi\)
\(138\) 0 0
\(139\) 3.05942 0.0220102 0.0110051 0.999939i \(-0.496497\pi\)
0.0110051 + 0.999939i \(0.496497\pi\)
\(140\) 0 0
\(141\) 328.757i 2.33161i
\(142\) 0 0
\(143\) 90.5061i 0.632910i
\(144\) 0 0
\(145\) −52.4661 −0.361835
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 31.7242i − 0.212914i −0.994317 0.106457i \(-0.966049\pi\)
0.994317 0.106457i \(-0.0339507\pi\)
\(150\) 0 0
\(151\) − 253.596i − 1.67945i −0.543016 0.839723i \(-0.682718\pi\)
0.543016 0.839723i \(-0.317282\pi\)
\(152\) 0 0
\(153\) −19.8502 −0.129740
\(154\) 0 0
\(155\) 42.3249 0.273064
\(156\) 0 0
\(157\) − 49.3273i − 0.314186i −0.987584 0.157093i \(-0.949788\pi\)
0.987584 0.157093i \(-0.0502123\pi\)
\(158\) 0 0
\(159\) − 107.026i − 0.673121i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −115.719 −0.709934 −0.354967 0.934879i \(-0.615508\pi\)
−0.354967 + 0.934879i \(0.615508\pi\)
\(164\) 0 0
\(165\) 59.0516i 0.357888i
\(166\) 0 0
\(167\) − 32.3859i − 0.193928i −0.995288 0.0969639i \(-0.969087\pi\)
0.995288 0.0969639i \(-0.0309131\pi\)
\(168\) 0 0
\(169\) 37.0641 0.219314
\(170\) 0 0
\(171\) 205.764 1.20330
\(172\) 0 0
\(173\) − 207.245i − 1.19795i −0.800769 0.598973i \(-0.795576\pi\)
0.800769 0.598973i \(-0.204424\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 41.4469 0.234163
\(178\) 0 0
\(179\) −174.167 −0.973002 −0.486501 0.873680i \(-0.661727\pi\)
−0.486501 + 0.873680i \(0.661727\pi\)
\(180\) 0 0
\(181\) 204.244i 1.12842i 0.825632 + 0.564209i \(0.190819\pi\)
−0.825632 + 0.564209i \(0.809181\pi\)
\(182\) 0 0
\(183\) 88.2291i 0.482126i
\(184\) 0 0
\(185\) −85.7586 −0.463560
\(186\) 0 0
\(187\) −22.8380 −0.122128
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 298.511i − 1.56288i −0.623978 0.781442i \(-0.714485\pi\)
0.623978 0.781442i \(-0.285515\pi\)
\(192\) 0 0
\(193\) −330.762 −1.71379 −0.856896 0.515490i \(-0.827610\pi\)
−0.856896 + 0.515490i \(0.827610\pi\)
\(194\) 0 0
\(195\) −86.0828 −0.441450
\(196\) 0 0
\(197\) − 327.309i − 1.66146i −0.556672 0.830732i \(-0.687922\pi\)
0.556672 0.830732i \(-0.312078\pi\)
\(198\) 0 0
\(199\) − 12.7358i − 0.0639989i −0.999488 0.0319994i \(-0.989813\pi\)
0.999488 0.0319994i \(-0.0101875\pi\)
\(200\) 0 0
\(201\) 236.114 1.17470
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) − 76.4892i − 0.373118i
\(206\) 0 0
\(207\) − 264.002i − 1.27537i
\(208\) 0 0
\(209\) 236.736 1.13271
\(210\) 0 0
\(211\) −120.455 −0.570875 −0.285437 0.958397i \(-0.592139\pi\)
−0.285437 + 0.958397i \(0.592139\pi\)
\(212\) 0 0
\(213\) 152.291i 0.714981i
\(214\) 0 0
\(215\) 89.0067i 0.413984i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 55.6460 0.254092
\(220\) 0 0
\(221\) − 33.2923i − 0.150644i
\(222\) 0 0
\(223\) − 372.958i − 1.67246i −0.548382 0.836228i \(-0.684756\pi\)
0.548382 0.836228i \(-0.315244\pi\)
\(224\) 0 0
\(225\) 146.944 0.653086
\(226\) 0 0
\(227\) −73.4256 −0.323461 −0.161730 0.986835i \(-0.551708\pi\)
−0.161730 + 0.986835i \(0.551708\pi\)
\(228\) 0 0
\(229\) 424.453i 1.85350i 0.375673 + 0.926752i \(0.377412\pi\)
−0.375673 + 0.926752i \(0.622588\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −83.4140 −0.358000 −0.179000 0.983849i \(-0.557286\pi\)
−0.179000 + 0.983849i \(0.557286\pi\)
\(234\) 0 0
\(235\) −155.460 −0.661533
\(236\) 0 0
\(237\) 203.787i 0.859861i
\(238\) 0 0
\(239\) 112.561i 0.470967i 0.971878 + 0.235484i \(0.0756674\pi\)
−0.971878 + 0.235484i \(0.924333\pi\)
\(240\) 0 0
\(241\) 280.432 1.16362 0.581809 0.813325i \(-0.302345\pi\)
0.581809 + 0.813325i \(0.302345\pi\)
\(242\) 0 0
\(243\) −304.037 −1.25118
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 345.103i 1.39718i
\(248\) 0 0
\(249\) 356.086 1.43007
\(250\) 0 0
\(251\) 32.9560 0.131299 0.0656493 0.997843i \(-0.479088\pi\)
0.0656493 + 0.997843i \(0.479088\pi\)
\(252\) 0 0
\(253\) − 303.740i − 1.20055i
\(254\) 0 0
\(255\) − 21.7219i − 0.0851838i
\(256\) 0 0
\(257\) −224.219 −0.872446 −0.436223 0.899839i \(-0.643684\pi\)
−0.436223 + 0.899839i \(0.643684\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 190.871i 0.731308i
\(262\) 0 0
\(263\) 169.961i 0.646239i 0.946358 + 0.323119i \(0.104732\pi\)
−0.946358 + 0.323119i \(0.895268\pi\)
\(264\) 0 0
\(265\) 50.6098 0.190980
\(266\) 0 0
\(267\) 419.034 1.56942
\(268\) 0 0
\(269\) 108.526i 0.403442i 0.979443 + 0.201721i \(0.0646535\pi\)
−0.979443 + 0.201721i \(0.935347\pi\)
\(270\) 0 0
\(271\) 19.3631i 0.0714507i 0.999362 + 0.0357253i \(0.0113741\pi\)
−0.999362 + 0.0357253i \(0.988626\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 169.063 0.614773
\(276\) 0 0
\(277\) 129.446i 0.467315i 0.972319 + 0.233658i \(0.0750695\pi\)
−0.972319 + 0.233658i \(0.924931\pi\)
\(278\) 0 0
\(279\) − 153.978i − 0.551892i
\(280\) 0 0
\(281\) 83.3608 0.296658 0.148329 0.988938i \(-0.452611\pi\)
0.148329 + 0.988938i \(0.452611\pi\)
\(282\) 0 0
\(283\) −28.9293 −0.102224 −0.0511118 0.998693i \(-0.516276\pi\)
−0.0511118 + 0.998693i \(0.516276\pi\)
\(284\) 0 0
\(285\) 225.166i 0.790055i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −280.599 −0.970931
\(290\) 0 0
\(291\) 220.271 0.756944
\(292\) 0 0
\(293\) − 214.613i − 0.732468i −0.930523 0.366234i \(-0.880647\pi\)
0.930523 0.366234i \(-0.119353\pi\)
\(294\) 0 0
\(295\) 19.5991i 0.0664376i
\(296\) 0 0
\(297\) −67.4858 −0.227225
\(298\) 0 0
\(299\) 442.778 1.48086
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 125.135i 0.412987i
\(304\) 0 0
\(305\) −41.7211 −0.136791
\(306\) 0 0
\(307\) 120.542 0.392644 0.196322 0.980539i \(-0.437100\pi\)
0.196322 + 0.980539i \(0.437100\pi\)
\(308\) 0 0
\(309\) 318.427i 1.03051i
\(310\) 0 0
\(311\) − 325.361i − 1.04618i −0.852278 0.523089i \(-0.824780\pi\)
0.852278 0.523089i \(-0.175220\pi\)
\(312\) 0 0
\(313\) 456.756 1.45928 0.729642 0.683829i \(-0.239687\pi\)
0.729642 + 0.683829i \(0.239687\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 121.220i − 0.382399i −0.981551 0.191199i \(-0.938762\pi\)
0.981551 0.191199i \(-0.0612377\pi\)
\(318\) 0 0
\(319\) 219.601i 0.688406i
\(320\) 0 0
\(321\) −194.128 −0.604761
\(322\) 0 0
\(323\) −87.0822 −0.269604
\(324\) 0 0
\(325\) 246.452i 0.758314i
\(326\) 0 0
\(327\) − 458.094i − 1.40090i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −121.168 −0.366068 −0.183034 0.983107i \(-0.558592\pi\)
−0.183034 + 0.983107i \(0.558592\pi\)
\(332\) 0 0
\(333\) 311.989i 0.936905i
\(334\) 0 0
\(335\) 111.652i 0.333290i
\(336\) 0 0
\(337\) −464.021 −1.37692 −0.688459 0.725275i \(-0.741713\pi\)
−0.688459 + 0.725275i \(0.741713\pi\)
\(338\) 0 0
\(339\) −221.970 −0.654780
\(340\) 0 0
\(341\) − 177.155i − 0.519515i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 288.895 0.837377
\(346\) 0 0
\(347\) 371.186 1.06970 0.534851 0.844947i \(-0.320368\pi\)
0.534851 + 0.844947i \(0.320368\pi\)
\(348\) 0 0
\(349\) 207.871i 0.595619i 0.954625 + 0.297809i \(0.0962560\pi\)
−0.954625 + 0.297809i \(0.903744\pi\)
\(350\) 0 0
\(351\) − 98.3779i − 0.280279i
\(352\) 0 0
\(353\) −614.014 −1.73942 −0.869708 0.493567i \(-0.835693\pi\)
−0.869708 + 0.493567i \(0.835693\pi\)
\(354\) 0 0
\(355\) −72.0142 −0.202857
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 108.072i − 0.301036i −0.988607 0.150518i \(-0.951906\pi\)
0.988607 0.150518i \(-0.0480941\pi\)
\(360\) 0 0
\(361\) 541.682 1.50050
\(362\) 0 0
\(363\) −234.539 −0.646113
\(364\) 0 0
\(365\) 26.3135i 0.0720917i
\(366\) 0 0
\(367\) − 454.815i − 1.23928i −0.784887 0.619639i \(-0.787279\pi\)
0.784887 0.619639i \(-0.212721\pi\)
\(368\) 0 0
\(369\) −278.268 −0.754113
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 271.752i − 0.728557i −0.931290 0.364279i \(-0.881316\pi\)
0.931290 0.364279i \(-0.118684\pi\)
\(374\) 0 0
\(375\) 348.159i 0.928425i
\(376\) 0 0
\(377\) −320.126 −0.849140
\(378\) 0 0
\(379\) −268.351 −0.708051 −0.354026 0.935236i \(-0.615187\pi\)
−0.354026 + 0.935236i \(0.615187\pi\)
\(380\) 0 0
\(381\) − 142.108i − 0.372988i
\(382\) 0 0
\(383\) 327.610i 0.855377i 0.903926 + 0.427689i \(0.140672\pi\)
−0.903926 + 0.427689i \(0.859328\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 323.806 0.836708
\(388\) 0 0
\(389\) − 126.301i − 0.324682i −0.986735 0.162341i \(-0.948096\pi\)
0.986735 0.162341i \(-0.0519044\pi\)
\(390\) 0 0
\(391\) 111.729i 0.285753i
\(392\) 0 0
\(393\) 482.869 1.22867
\(394\) 0 0
\(395\) −96.3653 −0.243963
\(396\) 0 0
\(397\) 127.144i 0.320262i 0.987096 + 0.160131i \(0.0511917\pi\)
−0.987096 + 0.160131i \(0.948808\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 60.1502 0.150001 0.0750003 0.997184i \(-0.476104\pi\)
0.0750003 + 0.997184i \(0.476104\pi\)
\(402\) 0 0
\(403\) 258.248 0.640815
\(404\) 0 0
\(405\) − 180.221i − 0.444991i
\(406\) 0 0
\(407\) 358.950i 0.881941i
\(408\) 0 0
\(409\) −69.7747 −0.170598 −0.0852991 0.996355i \(-0.527185\pi\)
−0.0852991 + 0.996355i \(0.527185\pi\)
\(410\) 0 0
\(411\) −33.8256 −0.0823008
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 168.384i 0.405743i
\(416\) 0 0
\(417\) 12.1796 0.0292078
\(418\) 0 0
\(419\) −714.794 −1.70595 −0.852976 0.521950i \(-0.825205\pi\)
−0.852976 + 0.521950i \(0.825205\pi\)
\(420\) 0 0
\(421\) 303.440i 0.720759i 0.932806 + 0.360380i \(0.117353\pi\)
−0.932806 + 0.360380i \(0.882647\pi\)
\(422\) 0 0
\(423\) 565.564i 1.33703i
\(424\) 0 0
\(425\) −62.1890 −0.146327
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 360.307i 0.839877i
\(430\) 0 0
\(431\) − 431.495i − 1.00115i −0.865694 0.500574i \(-0.833122\pi\)
0.865694 0.500574i \(-0.166878\pi\)
\(432\) 0 0
\(433\) 194.875 0.450057 0.225029 0.974352i \(-0.427752\pi\)
0.225029 + 0.974352i \(0.427752\pi\)
\(434\) 0 0
\(435\) −208.869 −0.480159
\(436\) 0 0
\(437\) − 1158.17i − 2.65028i
\(438\) 0 0
\(439\) − 305.970i − 0.696969i −0.937314 0.348485i \(-0.886696\pi\)
0.937314 0.348485i \(-0.113304\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −250.197 −0.564779 −0.282390 0.959300i \(-0.591127\pi\)
−0.282390 + 0.959300i \(0.591127\pi\)
\(444\) 0 0
\(445\) 198.150i 0.445280i
\(446\) 0 0
\(447\) − 126.295i − 0.282539i
\(448\) 0 0
\(449\) −688.681 −1.53381 −0.766905 0.641761i \(-0.778204\pi\)
−0.766905 + 0.641761i \(0.778204\pi\)
\(450\) 0 0
\(451\) −320.152 −0.709872
\(452\) 0 0
\(453\) − 1009.57i − 2.22864i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 804.518 1.76043 0.880217 0.474572i \(-0.157397\pi\)
0.880217 + 0.474572i \(0.157397\pi\)
\(458\) 0 0
\(459\) 24.8244 0.0540836
\(460\) 0 0
\(461\) − 693.657i − 1.50468i −0.658776 0.752339i \(-0.728926\pi\)
0.658776 0.752339i \(-0.271074\pi\)
\(462\) 0 0
\(463\) − 321.194i − 0.693724i −0.937916 0.346862i \(-0.887247\pi\)
0.937916 0.346862i \(-0.112753\pi\)
\(464\) 0 0
\(465\) 168.497 0.362359
\(466\) 0 0
\(467\) −751.875 −1.61001 −0.805005 0.593268i \(-0.797837\pi\)
−0.805005 + 0.593268i \(0.797837\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 196.373i − 0.416929i
\(472\) 0 0
\(473\) 372.545 0.787622
\(474\) 0 0
\(475\) 644.642 1.35714
\(476\) 0 0
\(477\) − 184.118i − 0.385992i
\(478\) 0 0
\(479\) 722.140i 1.50760i 0.657104 + 0.753800i \(0.271781\pi\)
−0.657104 + 0.753800i \(0.728219\pi\)
\(480\) 0 0
\(481\) −523.262 −1.08786
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 104.160i 0.214763i
\(486\) 0 0
\(487\) 47.8290i 0.0982114i 0.998794 + 0.0491057i \(0.0156371\pi\)
−0.998794 + 0.0491057i \(0.984363\pi\)
\(488\) 0 0
\(489\) −460.682 −0.942090
\(490\) 0 0
\(491\) −44.4724 −0.0905752 −0.0452876 0.998974i \(-0.514420\pi\)
−0.0452876 + 0.998974i \(0.514420\pi\)
\(492\) 0 0
\(493\) − 80.7795i − 0.163853i
\(494\) 0 0
\(495\) 101.587i 0.205226i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 501.572 1.00515 0.502577 0.864532i \(-0.332385\pi\)
0.502577 + 0.864532i \(0.332385\pi\)
\(500\) 0 0
\(501\) − 128.929i − 0.257344i
\(502\) 0 0
\(503\) 462.733i 0.919946i 0.887933 + 0.459973i \(0.152141\pi\)
−0.887933 + 0.459973i \(0.847859\pi\)
\(504\) 0 0
\(505\) −59.1729 −0.117174
\(506\) 0 0
\(507\) 147.553 0.291032
\(508\) 0 0
\(509\) − 471.985i − 0.927278i −0.886024 0.463639i \(-0.846543\pi\)
0.886024 0.463639i \(-0.153457\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −257.326 −0.501610
\(514\) 0 0
\(515\) −150.576 −0.292380
\(516\) 0 0
\(517\) 650.693i 1.25859i
\(518\) 0 0
\(519\) − 825.047i − 1.58969i
\(520\) 0 0
\(521\) −90.7419 −0.174169 −0.0870843 0.996201i \(-0.527755\pi\)
−0.0870843 + 0.996201i \(0.527755\pi\)
\(522\) 0 0
\(523\) 360.512 0.689316 0.344658 0.938728i \(-0.387995\pi\)
0.344658 + 0.938728i \(0.387995\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 65.1656i 0.123654i
\(528\) 0 0
\(529\) −956.970 −1.80902
\(530\) 0 0
\(531\) 71.3014 0.134278
\(532\) 0 0
\(533\) − 466.705i − 0.875618i
\(534\) 0 0
\(535\) − 91.7980i − 0.171585i
\(536\) 0 0
\(537\) −693.366 −1.29118
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 561.148i 1.03724i 0.855004 + 0.518621i \(0.173555\pi\)
−0.855004 + 0.518621i \(0.826445\pi\)
\(542\) 0 0
\(543\) 813.101i 1.49742i
\(544\) 0 0
\(545\) 216.620 0.397468
\(546\) 0 0
\(547\) −1043.62 −1.90790 −0.953952 0.299960i \(-0.903027\pi\)
−0.953952 + 0.299960i \(0.903027\pi\)
\(548\) 0 0
\(549\) 151.781i 0.276469i
\(550\) 0 0
\(551\) 837.349i 1.51969i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −341.407 −0.615148
\(556\) 0 0
\(557\) 48.6444i 0.0873328i 0.999046 + 0.0436664i \(0.0139039\pi\)
−0.999046 + 0.0436664i \(0.986096\pi\)
\(558\) 0 0
\(559\) 543.081i 0.971522i
\(560\) 0 0
\(561\) −90.9189 −0.162066
\(562\) 0 0
\(563\) −726.031 −1.28957 −0.644787 0.764362i \(-0.723054\pi\)
−0.644787 + 0.764362i \(0.723054\pi\)
\(564\) 0 0
\(565\) − 104.964i − 0.185776i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −481.392 −0.846032 −0.423016 0.906122i \(-0.639029\pi\)
−0.423016 + 0.906122i \(0.639029\pi\)
\(570\) 0 0
\(571\) −110.127 −0.192866 −0.0964331 0.995339i \(-0.530743\pi\)
−0.0964331 + 0.995339i \(0.530743\pi\)
\(572\) 0 0
\(573\) − 1188.38i − 2.07396i
\(574\) 0 0
\(575\) − 827.097i − 1.43843i
\(576\) 0 0
\(577\) −202.336 −0.350669 −0.175335 0.984509i \(-0.556101\pi\)
−0.175335 + 0.984509i \(0.556101\pi\)
\(578\) 0 0
\(579\) −1316.77 −2.27422
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 211.832i − 0.363348i
\(584\) 0 0
\(585\) −148.089 −0.253144
\(586\) 0 0
\(587\) 568.689 0.968805 0.484403 0.874845i \(-0.339037\pi\)
0.484403 + 0.874845i \(0.339037\pi\)
\(588\) 0 0
\(589\) − 675.497i − 1.14685i
\(590\) 0 0
\(591\) − 1303.03i − 2.20478i
\(592\) 0 0
\(593\) −685.373 −1.15577 −0.577886 0.816117i \(-0.696122\pi\)
−0.577886 + 0.816117i \(0.696122\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 50.7015i − 0.0849271i
\(598\) 0 0
\(599\) − 367.283i − 0.613161i −0.951845 0.306580i \(-0.900815\pi\)
0.951845 0.306580i \(-0.0991848\pi\)
\(600\) 0 0
\(601\) 412.344 0.686097 0.343049 0.939318i \(-0.388540\pi\)
0.343049 + 0.939318i \(0.388540\pi\)
\(602\) 0 0
\(603\) 406.190 0.673615
\(604\) 0 0
\(605\) − 110.907i − 0.183318i
\(606\) 0 0
\(607\) − 1166.78i − 1.92220i −0.276195 0.961102i \(-0.589074\pi\)
0.276195 0.961102i \(-0.410926\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −948.552 −1.55246
\(612\) 0 0
\(613\) − 431.743i − 0.704311i −0.935942 0.352155i \(-0.885449\pi\)
0.935942 0.352155i \(-0.114551\pi\)
\(614\) 0 0
\(615\) − 304.506i − 0.495131i
\(616\) 0 0
\(617\) −333.751 −0.540926 −0.270463 0.962730i \(-0.587177\pi\)
−0.270463 + 0.962730i \(0.587177\pi\)
\(618\) 0 0
\(619\) 976.316 1.57725 0.788624 0.614876i \(-0.210794\pi\)
0.788624 + 0.614876i \(0.210794\pi\)
\(620\) 0 0
\(621\) 330.157i 0.531655i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 371.768 0.594829
\(626\) 0 0
\(627\) 942.452 1.50311
\(628\) 0 0
\(629\) − 132.038i − 0.209918i
\(630\) 0 0
\(631\) − 639.885i − 1.01408i −0.861922 0.507040i \(-0.830739\pi\)
0.861922 0.507040i \(-0.169261\pi\)
\(632\) 0 0
\(633\) −479.533 −0.757556
\(634\) 0 0
\(635\) 67.1991 0.105825
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 261.987i 0.409996i
\(640\) 0 0
\(641\) 20.8591 0.0325415 0.0162707 0.999868i \(-0.494821\pi\)
0.0162707 + 0.999868i \(0.494821\pi\)
\(642\) 0 0
\(643\) 69.1348 0.107519 0.0537596 0.998554i \(-0.482880\pi\)
0.0537596 + 0.998554i \(0.482880\pi\)
\(644\) 0 0
\(645\) 354.338i 0.549362i
\(646\) 0 0
\(647\) − 358.959i − 0.554806i −0.960754 0.277403i \(-0.910526\pi\)
0.960754 0.277403i \(-0.0894737\pi\)
\(648\) 0 0
\(649\) 82.0338 0.126400
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 37.0921i − 0.0568026i −0.999597 0.0284013i \(-0.990958\pi\)
0.999597 0.0284013i \(-0.00904163\pi\)
\(654\) 0 0
\(655\) 228.335i 0.348604i
\(656\) 0 0
\(657\) 95.7284 0.145705
\(658\) 0 0
\(659\) −197.302 −0.299396 −0.149698 0.988732i \(-0.547830\pi\)
−0.149698 + 0.988732i \(0.547830\pi\)
\(660\) 0 0
\(661\) 1083.83i 1.63969i 0.572589 + 0.819843i \(0.305939\pi\)
−0.572589 + 0.819843i \(0.694061\pi\)
\(662\) 0 0
\(663\) − 132.538i − 0.199906i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1074.35 1.61071
\(668\) 0 0
\(669\) − 1484.76i − 2.21937i
\(670\) 0 0
\(671\) 174.628i 0.260250i
\(672\) 0 0
\(673\) −286.066 −0.425061 −0.212530 0.977154i \(-0.568170\pi\)
−0.212530 + 0.977154i \(0.568170\pi\)
\(674\) 0 0
\(675\) −183.767 −0.272247
\(676\) 0 0
\(677\) − 500.360i − 0.739083i −0.929214 0.369542i \(-0.879515\pi\)
0.929214 0.369542i \(-0.120485\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −292.310 −0.429236
\(682\) 0 0
\(683\) −946.450 −1.38572 −0.692862 0.721070i \(-0.743651\pi\)
−0.692862 + 0.721070i \(0.743651\pi\)
\(684\) 0 0
\(685\) − 15.9952i − 0.0233507i
\(686\) 0 0
\(687\) 1689.76i 2.45962i
\(688\) 0 0
\(689\) 308.799 0.448185
\(690\) 0 0
\(691\) −303.264 −0.438877 −0.219439 0.975626i \(-0.570423\pi\)
−0.219439 + 0.975626i \(0.570423\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.75942i 0.00828694i
\(696\) 0 0
\(697\) 117.767 0.168962
\(698\) 0 0
\(699\) −332.074 −0.475069
\(700\) 0 0
\(701\) − 390.864i − 0.557580i −0.960352 0.278790i \(-0.910067\pi\)
0.960352 0.278790i \(-0.0899333\pi\)
\(702\) 0 0
\(703\) 1368.69i 1.94693i
\(704\) 0 0
\(705\) −618.892 −0.877861
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 961.165i − 1.35566i −0.735217 0.677831i \(-0.762920\pi\)
0.735217 0.677831i \(-0.237080\pi\)
\(710\) 0 0
\(711\) 350.577i 0.493075i
\(712\) 0 0
\(713\) −866.685 −1.21555
\(714\) 0 0
\(715\) −170.379 −0.238293
\(716\) 0 0
\(717\) 448.109i 0.624978i
\(718\) 0 0
\(719\) − 525.126i − 0.730356i −0.930938 0.365178i \(-0.881008\pi\)
0.930938 0.365178i \(-0.118992\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1116.41 1.54413
\(724\) 0 0
\(725\) 597.985i 0.824807i
\(726\) 0 0
\(727\) 108.633i 0.149426i 0.997205 + 0.0747131i \(0.0238041\pi\)
−0.997205 + 0.0747131i \(0.976196\pi\)
\(728\) 0 0
\(729\) −348.775 −0.478429
\(730\) 0 0
\(731\) −137.039 −0.187468
\(732\) 0 0
\(733\) − 40.2540i − 0.0549167i −0.999623 0.0274584i \(-0.991259\pi\)
0.999623 0.0274584i \(-0.00874137\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 467.330 0.634097
\(738\) 0 0
\(739\) −997.203 −1.34940 −0.674698 0.738094i \(-0.735726\pi\)
−0.674698 + 0.738094i \(0.735726\pi\)
\(740\) 0 0
\(741\) 1373.87i 1.85407i
\(742\) 0 0
\(743\) 476.575i 0.641420i 0.947177 + 0.320710i \(0.103922\pi\)
−0.947177 + 0.320710i \(0.896078\pi\)
\(744\) 0 0
\(745\) 59.7214 0.0801630
\(746\) 0 0
\(747\) 612.579 0.820052
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 528.496i − 0.703723i −0.936052 0.351862i \(-0.885549\pi\)
0.936052 0.351862i \(-0.114451\pi\)
\(752\) 0 0
\(753\) 131.199 0.174235
\(754\) 0 0
\(755\) 477.400 0.632318
\(756\) 0 0
\(757\) 455.964i 0.602331i 0.953572 + 0.301165i \(0.0973756\pi\)
−0.953572 + 0.301165i \(0.902624\pi\)
\(758\) 0 0
\(759\) − 1209.20i − 1.59314i
\(760\) 0 0
\(761\) 476.650 0.626347 0.313174 0.949696i \(-0.398608\pi\)
0.313174 + 0.949696i \(0.398608\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) − 37.3683i − 0.0488475i
\(766\) 0 0
\(767\) 119.585i 0.155913i
\(768\) 0 0
\(769\) −568.246 −0.738941 −0.369471 0.929242i \(-0.620461\pi\)
−0.369471 + 0.929242i \(0.620461\pi\)
\(770\) 0 0
\(771\) −892.621 −1.15774
\(772\) 0 0
\(773\) − 1197.03i − 1.54855i −0.632849 0.774275i \(-0.718115\pi\)
0.632849 0.774275i \(-0.281885\pi\)
\(774\) 0 0
\(775\) − 482.400i − 0.622452i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1220.75 −1.56708
\(780\) 0 0
\(781\) 301.422i 0.385943i
\(782\) 0 0
\(783\) − 238.701i − 0.304855i
\(784\) 0 0
\(785\) 92.8596 0.118292
\(786\) 0 0
\(787\) −452.269 −0.574674 −0.287337 0.957830i \(-0.592770\pi\)
−0.287337 + 0.957830i \(0.592770\pi\)
\(788\) 0 0
\(789\) 676.619i 0.857565i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −254.565 −0.321015
\(794\) 0 0
\(795\) 201.479 0.253433
\(796\) 0 0
\(797\) 1047.47i 1.31426i 0.753777 + 0.657130i \(0.228230\pi\)
−0.753777 + 0.657130i \(0.771770\pi\)
\(798\) 0 0
\(799\) − 239.355i − 0.299568i
\(800\) 0 0
\(801\) 720.868 0.899960
\(802\) 0 0
\(803\) 110.137 0.137157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 432.045i 0.535372i
\(808\) 0 0
\(809\) −298.665 −0.369178 −0.184589 0.982816i \(-0.559095\pi\)
−0.184589 + 0.982816i \(0.559095\pi\)
\(810\) 0 0
\(811\) −633.054 −0.780584 −0.390292 0.920691i \(-0.627626\pi\)
−0.390292 + 0.920691i \(0.627626\pi\)
\(812\) 0 0
\(813\) 77.0852i 0.0948158i
\(814\) 0 0
\(815\) − 217.844i − 0.267293i
\(816\) 0 0
\(817\) 1420.53 1.73871
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1251.50i 1.52436i 0.647367 + 0.762178i \(0.275870\pi\)
−0.647367 + 0.762178i \(0.724130\pi\)
\(822\) 0 0
\(823\) − 1226.82i − 1.49066i −0.666693 0.745332i \(-0.732291\pi\)
0.666693 0.745332i \(-0.267709\pi\)
\(824\) 0 0
\(825\) 673.043 0.815810
\(826\) 0 0
\(827\) 1438.26 1.73913 0.869566 0.493816i \(-0.164398\pi\)
0.869566 + 0.493816i \(0.164398\pi\)
\(828\) 0 0
\(829\) 626.488i 0.755715i 0.925864 + 0.377858i \(0.123339\pi\)
−0.925864 + 0.377858i \(0.876661\pi\)
\(830\) 0 0
\(831\) 515.330i 0.620132i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 60.9672 0.0730146
\(836\) 0 0
\(837\) 192.563i 0.230063i
\(838\) 0 0
\(839\) − 1062.37i − 1.26624i −0.774055 0.633118i \(-0.781775\pi\)
0.774055 0.633118i \(-0.218225\pi\)
\(840\) 0 0
\(841\) 64.2558 0.0764041
\(842\) 0 0
\(843\) 331.862 0.393668
\(844\) 0 0
\(845\) 69.7739i 0.0825727i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −115.168 −0.135652
\(850\) 0 0
\(851\) 1756.07 2.06354
\(852\) 0 0
\(853\) − 698.388i − 0.818743i −0.912368 0.409372i \(-0.865748\pi\)
0.912368 0.409372i \(-0.134252\pi\)
\(854\) 0 0
\(855\) 387.355i 0.453047i
\(856\) 0 0
\(857\) −730.324 −0.852187 −0.426094 0.904679i \(-0.640111\pi\)
−0.426094 + 0.904679i \(0.640111\pi\)
\(858\) 0 0
\(859\) 966.259 1.12487 0.562433 0.826843i \(-0.309866\pi\)
0.562433 + 0.826843i \(0.309866\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 294.475i − 0.341222i −0.985338 0.170611i \(-0.945426\pi\)
0.985338 0.170611i \(-0.0545742\pi\)
\(864\) 0 0
\(865\) 390.142 0.451032
\(866\) 0 0
\(867\) −1117.07 −1.28844
\(868\) 0 0
\(869\) 403.346i 0.464149i
\(870\) 0 0
\(871\) 681.253i 0.782151i
\(872\) 0 0
\(873\) 378.934 0.434059
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 944.891i 1.07741i 0.842494 + 0.538706i \(0.181087\pi\)
−0.842494 + 0.538706i \(0.818913\pi\)
\(878\) 0 0
\(879\) − 854.382i − 0.971993i
\(880\) 0 0
\(881\) 744.098 0.844606 0.422303 0.906455i \(-0.361222\pi\)
0.422303 + 0.906455i \(0.361222\pi\)
\(882\) 0 0
\(883\) −23.9032 −0.0270704 −0.0135352 0.999908i \(-0.504309\pi\)
−0.0135352 + 0.999908i \(0.504309\pi\)
\(884\) 0 0
\(885\) 78.0246i 0.0881634i
\(886\) 0 0
\(887\) 947.312i 1.06800i 0.845486 + 0.533998i \(0.179311\pi\)
−0.845486 + 0.533998i \(0.820689\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −754.332 −0.846613
\(892\) 0 0
\(893\) 2481.12i 2.77840i
\(894\) 0 0
\(895\) − 327.874i − 0.366339i
\(896\) 0 0
\(897\) 1762.71 1.96512
\(898\) 0 0
\(899\) 626.607 0.697005
\(900\) 0 0
\(901\) 77.9215i 0.0864834i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −384.493 −0.424854
\(906\) 0 0
\(907\) −551.023 −0.607523 −0.303761 0.952748i \(-0.598243\pi\)
−0.303761 + 0.952748i \(0.598243\pi\)
\(908\) 0 0
\(909\) 215.271i 0.236822i
\(910\) 0 0
\(911\) 827.652i 0.908509i 0.890872 + 0.454254i \(0.150094\pi\)
−0.890872 + 0.454254i \(0.849906\pi\)
\(912\) 0 0
\(913\) 704.784 0.771943
\(914\) 0 0
\(915\) −166.093 −0.181523
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 356.993i − 0.388458i −0.980956 0.194229i \(-0.937779\pi\)
0.980956 0.194229i \(-0.0622205\pi\)
\(920\) 0 0
\(921\) 479.880 0.521043
\(922\) 0 0
\(923\) −439.400 −0.476056
\(924\) 0 0
\(925\) 977.438i 1.05669i
\(926\) 0 0
\(927\) 547.794i 0.590932i
\(928\) 0 0
\(929\) 969.745 1.04386 0.521930 0.852988i \(-0.325212\pi\)
0.521930 + 0.852988i \(0.325212\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 1295.27i − 1.38829i
\(934\) 0 0
\(935\) − 42.9930i − 0.0459819i
\(936\) 0 0
\(937\) 1287.55 1.37412 0.687061 0.726600i \(-0.258901\pi\)
0.687061 + 0.726600i \(0.258901\pi\)
\(938\) 0 0
\(939\) 1818.36 1.93649
\(940\) 0 0
\(941\) − 456.796i − 0.485436i −0.970097 0.242718i \(-0.921961\pi\)
0.970097 0.242718i \(-0.0780390\pi\)
\(942\) 0 0
\(943\) 1566.27i 1.66094i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1147.85 1.21209 0.606045 0.795430i \(-0.292755\pi\)
0.606045 + 0.795430i \(0.292755\pi\)
\(948\) 0 0
\(949\) 160.554i 0.169182i
\(950\) 0 0
\(951\) − 482.582i − 0.507447i
\(952\) 0 0
\(953\) 873.170 0.916232 0.458116 0.888892i \(-0.348524\pi\)
0.458116 + 0.888892i \(0.348524\pi\)
\(954\) 0 0
\(955\) 561.952 0.588432
\(956\) 0 0
\(957\) 874.240i 0.913522i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 455.510 0.473996
\(962\) 0 0
\(963\) −333.961 −0.346792
\(964\) 0 0
\(965\) − 622.666i − 0.645249i
\(966\) 0 0
\(967\) − 449.047i − 0.464372i −0.972671 0.232186i \(-0.925412\pi\)
0.972671 0.232186i \(-0.0745878\pi\)
\(968\) 0 0
\(969\) −346.677 −0.357768
\(970\) 0 0
\(971\) 1237.86 1.27483 0.637414 0.770521i \(-0.280004\pi\)
0.637414 + 0.770521i \(0.280004\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 981.134i 1.00629i
\(976\) 0 0
\(977\) 1706.89 1.74707 0.873536 0.486760i \(-0.161821\pi\)
0.873536 + 0.486760i \(0.161821\pi\)
\(978\) 0 0
\(979\) 829.373 0.847164
\(980\) 0 0
\(981\) − 788.063i − 0.803326i
\(982\) 0 0
\(983\) 1152.15i 1.17208i 0.810283 + 0.586039i \(0.199313\pi\)
−0.810283 + 0.586039i \(0.800687\pi\)
\(984\) 0 0
\(985\) 616.165 0.625548
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1822.59i − 1.84286i
\(990\) 0 0
\(991\) 412.818i 0.416567i 0.978068 + 0.208283i \(0.0667877\pi\)
−0.978068 + 0.208283i \(0.933212\pi\)
\(992\) 0 0
\(993\) −482.375 −0.485776
\(994\) 0 0
\(995\) 23.9753 0.0240958
\(996\) 0 0
\(997\) 301.152i 0.302058i 0.988529 + 0.151029i \(0.0482587\pi\)
−0.988529 + 0.151029i \(0.951741\pi\)
\(998\) 0 0
\(999\) − 390.170i − 0.390561i
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 1568.3.g.j.687.6 6
4.3 odd 2 392.3.g.j.99.5 6
7.2 even 3 224.3.o.d.207.2 12
7.4 even 3 224.3.o.d.79.1 12
7.6 odd 2 1568.3.g.l.687.1 6
8.3 odd 2 inner 1568.3.g.j.687.5 6
8.5 even 2 392.3.g.j.99.6 6
28.3 even 6 392.3.k.l.275.3 12
28.11 odd 6 56.3.k.d.51.3 yes 12
28.19 even 6 392.3.k.l.67.1 12
28.23 odd 6 56.3.k.d.11.1 12
28.27 even 2 392.3.g.i.99.5 6
56.5 odd 6 392.3.k.l.67.3 12
56.11 odd 6 224.3.o.d.79.2 12
56.13 odd 2 392.3.g.i.99.6 6
56.27 even 2 1568.3.g.l.687.2 6
56.37 even 6 56.3.k.d.11.3 yes 12
56.45 odd 6 392.3.k.l.275.1 12
56.51 odd 6 224.3.o.d.207.1 12
56.53 even 6 56.3.k.d.51.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.k.d.11.1 12 28.23 odd 6
56.3.k.d.11.3 yes 12 56.37 even 6
56.3.k.d.51.1 yes 12 56.53 even 6
56.3.k.d.51.3 yes 12 28.11 odd 6
224.3.o.d.79.1 12 7.4 even 3
224.3.o.d.79.2 12 56.11 odd 6
224.3.o.d.207.1 12 56.51 odd 6
224.3.o.d.207.2 12 7.2 even 3
392.3.g.i.99.5 6 28.27 even 2
392.3.g.i.99.6 6 56.13 odd 2
392.3.g.j.99.5 6 4.3 odd 2
392.3.g.j.99.6 6 8.5 even 2
392.3.k.l.67.1 12 28.19 even 6
392.3.k.l.67.3 12 56.5 odd 6
392.3.k.l.275.1 12 56.45 odd 6
392.3.k.l.275.3 12 28.3 even 6
1568.3.g.j.687.5 6 8.3 odd 2 inner
1568.3.g.j.687.6 6 1.1 even 1 trivial
1568.3.g.l.687.1 6 7.6 odd 2
1568.3.g.l.687.2 6 56.27 even 2