Properties

Label 1792.2.a.x.1.2
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
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] = [1792,2,Mod(1,1792)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.9248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.13578\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.936426 q^{3} +3.33513 q^{5} +1.00000 q^{7} -2.12311 q^{9} +O(q^{10})\) \(q-0.936426 q^{3} +3.33513 q^{5} +1.00000 q^{7} -2.12311 q^{9} -4.27156 q^{11} +3.33513 q^{13} -3.12311 q^{15} +2.00000 q^{17} -0.936426 q^{19} -0.936426 q^{21} +3.12311 q^{23} +6.12311 q^{25} +4.79741 q^{27} +1.87285 q^{29} +6.24621 q^{31} +4.00000 q^{33} +3.33513 q^{35} +1.87285 q^{37} -3.12311 q^{39} +12.2462 q^{41} -4.27156 q^{43} -7.08084 q^{45} +1.00000 q^{49} -1.87285 q^{51} -8.54312 q^{53} -14.2462 q^{55} +0.876894 q^{57} +7.60669 q^{59} +3.33513 q^{61} -2.12311 q^{63} +11.1231 q^{65} -15.7392 q^{67} -2.92456 q^{69} +8.00000 q^{71} +6.00000 q^{73} -5.73384 q^{75} -4.27156 q^{77} +1.87689 q^{81} +9.47954 q^{83} +6.67026 q^{85} -1.75379 q^{87} -0.246211 q^{89} +3.33513 q^{91} -5.84912 q^{93} -3.12311 q^{95} -4.24621 q^{97} +9.06897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 8 q^{9} + 4 q^{15} + 8 q^{17} - 4 q^{23} + 8 q^{25} - 8 q^{31} + 16 q^{33} + 4 q^{39} + 16 q^{41} + 4 q^{49} - 24 q^{55} + 20 q^{57} + 8 q^{63} + 28 q^{65} + 32 q^{71} + 24 q^{73} + 24 q^{81} - 40 q^{87} + 32 q^{89} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.936426 −0.540646 −0.270323 0.962770i \(-0.587130\pi\)
−0.270323 + 0.962770i \(0.587130\pi\)
\(4\) 0 0
\(5\) 3.33513 1.49152 0.745758 0.666217i \(-0.232087\pi\)
0.745758 + 0.666217i \(0.232087\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.12311 −0.707702
\(10\) 0 0
\(11\) −4.27156 −1.28792 −0.643962 0.765058i \(-0.722710\pi\)
−0.643962 + 0.765058i \(0.722710\pi\)
\(12\) 0 0
\(13\) 3.33513 0.924999 0.462500 0.886619i \(-0.346953\pi\)
0.462500 + 0.886619i \(0.346953\pi\)
\(14\) 0 0
\(15\) −3.12311 −0.806382
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −0.936426 −0.214831 −0.107415 0.994214i \(-0.534258\pi\)
−0.107415 + 0.994214i \(0.534258\pi\)
\(20\) 0 0
\(21\) −0.936426 −0.204345
\(22\) 0 0
\(23\) 3.12311 0.651213 0.325606 0.945505i \(-0.394432\pi\)
0.325606 + 0.945505i \(0.394432\pi\)
\(24\) 0 0
\(25\) 6.12311 1.22462
\(26\) 0 0
\(27\) 4.79741 0.923262
\(28\) 0 0
\(29\) 1.87285 0.347780 0.173890 0.984765i \(-0.444366\pi\)
0.173890 + 0.984765i \(0.444366\pi\)
\(30\) 0 0
\(31\) 6.24621 1.12185 0.560926 0.827866i \(-0.310445\pi\)
0.560926 + 0.827866i \(0.310445\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 3.33513 0.563740
\(36\) 0 0
\(37\) 1.87285 0.307895 0.153948 0.988079i \(-0.450801\pi\)
0.153948 + 0.988079i \(0.450801\pi\)
\(38\) 0 0
\(39\) −3.12311 −0.500097
\(40\) 0 0
\(41\) 12.2462 1.91254 0.956268 0.292490i \(-0.0944840\pi\)
0.956268 + 0.292490i \(0.0944840\pi\)
\(42\) 0 0
\(43\) −4.27156 −0.651407 −0.325703 0.945472i \(-0.605601\pi\)
−0.325703 + 0.945472i \(0.605601\pi\)
\(44\) 0 0
\(45\) −7.08084 −1.05555
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.87285 −0.262252
\(52\) 0 0
\(53\) −8.54312 −1.17349 −0.586744 0.809773i \(-0.699590\pi\)
−0.586744 + 0.809773i \(0.699590\pi\)
\(54\) 0 0
\(55\) −14.2462 −1.92096
\(56\) 0 0
\(57\) 0.876894 0.116147
\(58\) 0 0
\(59\) 7.60669 0.990307 0.495153 0.868806i \(-0.335112\pi\)
0.495153 + 0.868806i \(0.335112\pi\)
\(60\) 0 0
\(61\) 3.33513 0.427020 0.213510 0.976941i \(-0.431510\pi\)
0.213510 + 0.976941i \(0.431510\pi\)
\(62\) 0 0
\(63\) −2.12311 −0.267486
\(64\) 0 0
\(65\) 11.1231 1.37965
\(66\) 0 0
\(67\) −15.7392 −1.92285 −0.961427 0.275061i \(-0.911302\pi\)
−0.961427 + 0.275061i \(0.911302\pi\)
\(68\) 0 0
\(69\) −2.92456 −0.352075
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −5.73384 −0.662087
\(76\) 0 0
\(77\) −4.27156 −0.486789
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.87689 0.208544
\(82\) 0 0
\(83\) 9.47954 1.04052 0.520258 0.854009i \(-0.325836\pi\)
0.520258 + 0.854009i \(0.325836\pi\)
\(84\) 0 0
\(85\) 6.67026 0.723492
\(86\) 0 0
\(87\) −1.75379 −0.188026
\(88\) 0 0
\(89\) −0.246211 −0.0260983 −0.0130492 0.999915i \(-0.504154\pi\)
−0.0130492 + 0.999915i \(0.504154\pi\)
\(90\) 0 0
\(91\) 3.33513 0.349617
\(92\) 0 0
\(93\) −5.84912 −0.606525
\(94\) 0 0
\(95\) −3.12311 −0.320424
\(96\) 0 0
\(97\) −4.24621 −0.431137 −0.215569 0.976489i \(-0.569161\pi\)
−0.215569 + 0.976489i \(0.569161\pi\)
\(98\) 0 0
\(99\) 9.06897 0.911466
\(100\) 0 0
\(101\) −7.08084 −0.704570 −0.352285 0.935893i \(-0.614595\pi\)
−0.352285 + 0.935893i \(0.614595\pi\)
\(102\) 0 0
\(103\) 14.2462 1.40372 0.701860 0.712314i \(-0.252353\pi\)
0.701860 + 0.712314i \(0.252353\pi\)
\(104\) 0 0
\(105\) −3.12311 −0.304784
\(106\) 0 0
\(107\) −7.19612 −0.695675 −0.347837 0.937555i \(-0.613084\pi\)
−0.347837 + 0.937555i \(0.613084\pi\)
\(108\) 0 0
\(109\) −5.61856 −0.538160 −0.269080 0.963118i \(-0.586720\pi\)
−0.269080 + 0.963118i \(0.586720\pi\)
\(110\) 0 0
\(111\) −1.75379 −0.166462
\(112\) 0 0
\(113\) 13.1231 1.23452 0.617259 0.786760i \(-0.288243\pi\)
0.617259 + 0.786760i \(0.288243\pi\)
\(114\) 0 0
\(115\) 10.4160 0.971294
\(116\) 0 0
\(117\) −7.08084 −0.654624
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 7.24621 0.658746
\(122\) 0 0
\(123\) −11.4677 −1.03401
\(124\) 0 0
\(125\) 3.74571 0.335026
\(126\) 0 0
\(127\) 4.87689 0.432754 0.216377 0.976310i \(-0.430576\pi\)
0.216377 + 0.976310i \(0.430576\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −3.86098 −0.337336 −0.168668 0.985673i \(-0.553947\pi\)
−0.168668 + 0.985673i \(0.553947\pi\)
\(132\) 0 0
\(133\) −0.936426 −0.0811985
\(134\) 0 0
\(135\) 16.0000 1.37706
\(136\) 0 0
\(137\) −0.246211 −0.0210352 −0.0105176 0.999945i \(-0.503348\pi\)
−0.0105176 + 0.999945i \(0.503348\pi\)
\(138\) 0 0
\(139\) 18.0227 1.52866 0.764331 0.644824i \(-0.223069\pi\)
0.764331 + 0.644824i \(0.223069\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −14.2462 −1.19133
\(144\) 0 0
\(145\) 6.24621 0.518720
\(146\) 0 0
\(147\) −0.936426 −0.0772351
\(148\) 0 0
\(149\) 12.2888 1.00674 0.503370 0.864071i \(-0.332093\pi\)
0.503370 + 0.864071i \(0.332093\pi\)
\(150\) 0 0
\(151\) −19.1231 −1.55622 −0.778108 0.628130i \(-0.783820\pi\)
−0.778108 + 0.628130i \(0.783820\pi\)
\(152\) 0 0
\(153\) −4.24621 −0.343286
\(154\) 0 0
\(155\) 20.8319 1.67326
\(156\) 0 0
\(157\) 16.6757 1.33086 0.665431 0.746459i \(-0.268248\pi\)
0.665431 + 0.746459i \(0.268248\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 0 0
\(163\) 8.01726 0.627961 0.313980 0.949430i \(-0.398337\pi\)
0.313980 + 0.949430i \(0.398337\pi\)
\(164\) 0 0
\(165\) 13.3405 1.03856
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −1.87689 −0.144376
\(170\) 0 0
\(171\) 1.98813 0.152036
\(172\) 0 0
\(173\) −17.4968 −1.33026 −0.665129 0.746729i \(-0.731623\pi\)
−0.665129 + 0.746729i \(0.731623\pi\)
\(174\) 0 0
\(175\) 6.12311 0.462863
\(176\) 0 0
\(177\) −7.12311 −0.535405
\(178\) 0 0
\(179\) −2.39871 −0.179288 −0.0896438 0.995974i \(-0.528573\pi\)
−0.0896438 + 0.995974i \(0.528573\pi\)
\(180\) 0 0
\(181\) 13.7511 1.02211 0.511056 0.859548i \(-0.329255\pi\)
0.511056 + 0.859548i \(0.329255\pi\)
\(182\) 0 0
\(183\) −3.12311 −0.230867
\(184\) 0 0
\(185\) 6.24621 0.459231
\(186\) 0 0
\(187\) −8.54312 −0.624735
\(188\) 0 0
\(189\) 4.79741 0.348960
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −2.87689 −0.207083 −0.103542 0.994625i \(-0.533018\pi\)
−0.103542 + 0.994625i \(0.533018\pi\)
\(194\) 0 0
\(195\) −10.4160 −0.745903
\(196\) 0 0
\(197\) −1.05171 −0.0749309 −0.0374655 0.999298i \(-0.511928\pi\)
−0.0374655 + 0.999298i \(0.511928\pi\)
\(198\) 0 0
\(199\) 1.75379 0.124323 0.0621614 0.998066i \(-0.480201\pi\)
0.0621614 + 0.998066i \(0.480201\pi\)
\(200\) 0 0
\(201\) 14.7386 1.03958
\(202\) 0 0
\(203\) 1.87285 0.131448
\(204\) 0 0
\(205\) 40.8427 2.85258
\(206\) 0 0
\(207\) −6.63068 −0.460864
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −2.39871 −0.165134 −0.0825669 0.996586i \(-0.526312\pi\)
−0.0825669 + 0.996586i \(0.526312\pi\)
\(212\) 0 0
\(213\) −7.49141 −0.513303
\(214\) 0 0
\(215\) −14.2462 −0.971584
\(216\) 0 0
\(217\) 6.24621 0.424020
\(218\) 0 0
\(219\) −5.61856 −0.379667
\(220\) 0 0
\(221\) 6.67026 0.448691
\(222\) 0 0
\(223\) −22.2462 −1.48972 −0.744858 0.667223i \(-0.767483\pi\)
−0.744858 + 0.667223i \(0.767483\pi\)
\(224\) 0 0
\(225\) −13.0000 −0.866667
\(226\) 0 0
\(227\) 12.4041 0.823289 0.411645 0.911344i \(-0.364955\pi\)
0.411645 + 0.911344i \(0.364955\pi\)
\(228\) 0 0
\(229\) −4.15628 −0.274655 −0.137327 0.990526i \(-0.543851\pi\)
−0.137327 + 0.990526i \(0.543851\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −0.246211 −0.0161298 −0.00806492 0.999967i \(-0.502567\pi\)
−0.00806492 + 0.999967i \(0.502567\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.6155 1.52756 0.763781 0.645476i \(-0.223341\pi\)
0.763781 + 0.645476i \(0.223341\pi\)
\(240\) 0 0
\(241\) −20.2462 −1.30417 −0.652087 0.758145i \(-0.726106\pi\)
−0.652087 + 0.758145i \(0.726106\pi\)
\(242\) 0 0
\(243\) −16.1498 −1.03601
\(244\) 0 0
\(245\) 3.33513 0.213074
\(246\) 0 0
\(247\) −3.12311 −0.198718
\(248\) 0 0
\(249\) −8.87689 −0.562550
\(250\) 0 0
\(251\) −26.5658 −1.67682 −0.838408 0.545042i \(-0.816514\pi\)
−0.838408 + 0.545042i \(0.816514\pi\)
\(252\) 0 0
\(253\) −13.3405 −0.838712
\(254\) 0 0
\(255\) −6.24621 −0.391153
\(256\) 0 0
\(257\) −10.4924 −0.654499 −0.327250 0.944938i \(-0.606122\pi\)
−0.327250 + 0.944938i \(0.606122\pi\)
\(258\) 0 0
\(259\) 1.87285 0.116373
\(260\) 0 0
\(261\) −3.97626 −0.246125
\(262\) 0 0
\(263\) 20.4924 1.26362 0.631808 0.775125i \(-0.282313\pi\)
0.631808 + 0.775125i \(0.282313\pi\)
\(264\) 0 0
\(265\) −28.4924 −1.75028
\(266\) 0 0
\(267\) 0.230559 0.0141100
\(268\) 0 0
\(269\) −23.3459 −1.42343 −0.711713 0.702470i \(-0.752080\pi\)
−0.711713 + 0.702470i \(0.752080\pi\)
\(270\) 0 0
\(271\) −6.24621 −0.379430 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(272\) 0 0
\(273\) −3.12311 −0.189019
\(274\) 0 0
\(275\) −26.1552 −1.57722
\(276\) 0 0
\(277\) −8.54312 −0.513306 −0.256653 0.966504i \(-0.582620\pi\)
−0.256653 + 0.966504i \(0.582620\pi\)
\(278\) 0 0
\(279\) −13.2614 −0.793937
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −8.65840 −0.514688 −0.257344 0.966320i \(-0.582847\pi\)
−0.257344 + 0.966320i \(0.582847\pi\)
\(284\) 0 0
\(285\) 2.92456 0.173236
\(286\) 0 0
\(287\) 12.2462 0.722871
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 3.97626 0.233093
\(292\) 0 0
\(293\) −7.08084 −0.413667 −0.206833 0.978376i \(-0.566316\pi\)
−0.206833 + 0.978376i \(0.566316\pi\)
\(294\) 0 0
\(295\) 25.3693 1.47706
\(296\) 0 0
\(297\) −20.4924 −1.18909
\(298\) 0 0
\(299\) 10.4160 0.602371
\(300\) 0 0
\(301\) −4.27156 −0.246209
\(302\) 0 0
\(303\) 6.63068 0.380923
\(304\) 0 0
\(305\) 11.1231 0.636907
\(306\) 0 0
\(307\) 15.3287 0.874853 0.437426 0.899254i \(-0.355890\pi\)
0.437426 + 0.899254i \(0.355890\pi\)
\(308\) 0 0
\(309\) −13.3405 −0.758916
\(310\) 0 0
\(311\) −20.4924 −1.16202 −0.581009 0.813897i \(-0.697342\pi\)
−0.581009 + 0.813897i \(0.697342\pi\)
\(312\) 0 0
\(313\) −28.7386 −1.62440 −0.812202 0.583377i \(-0.801731\pi\)
−0.812202 + 0.583377i \(0.801731\pi\)
\(314\) 0 0
\(315\) −7.08084 −0.398960
\(316\) 0 0
\(317\) −21.8836 −1.22911 −0.614554 0.788875i \(-0.710664\pi\)
−0.614554 + 0.788875i \(0.710664\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 6.73863 0.376114
\(322\) 0 0
\(323\) −1.87285 −0.104208
\(324\) 0 0
\(325\) 20.4214 1.13277
\(326\) 0 0
\(327\) 5.26137 0.290954
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.4095 1.23174 0.615869 0.787849i \(-0.288805\pi\)
0.615869 + 0.787849i \(0.288805\pi\)
\(332\) 0 0
\(333\) −3.97626 −0.217898
\(334\) 0 0
\(335\) −52.4924 −2.86797
\(336\) 0 0
\(337\) −9.12311 −0.496967 −0.248484 0.968636i \(-0.579932\pi\)
−0.248484 + 0.968636i \(0.579932\pi\)
\(338\) 0 0
\(339\) −12.2888 −0.667437
\(340\) 0 0
\(341\) −26.6811 −1.44486
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −9.75379 −0.525126
\(346\) 0 0
\(347\) 9.06897 0.486848 0.243424 0.969920i \(-0.421729\pi\)
0.243424 + 0.969920i \(0.421729\pi\)
\(348\) 0 0
\(349\) −26.2705 −1.40623 −0.703113 0.711078i \(-0.748207\pi\)
−0.703113 + 0.711078i \(0.748207\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) 24.2462 1.29050 0.645248 0.763973i \(-0.276754\pi\)
0.645248 + 0.763973i \(0.276754\pi\)
\(354\) 0 0
\(355\) 26.6811 1.41608
\(356\) 0 0
\(357\) −1.87285 −0.0991219
\(358\) 0 0
\(359\) 6.63068 0.349954 0.174977 0.984573i \(-0.444015\pi\)
0.174977 + 0.984573i \(0.444015\pi\)
\(360\) 0 0
\(361\) −18.1231 −0.953848
\(362\) 0 0
\(363\) −6.78554 −0.356149
\(364\) 0 0
\(365\) 20.0108 1.04741
\(366\) 0 0
\(367\) −6.24621 −0.326050 −0.163025 0.986622i \(-0.552125\pi\)
−0.163025 + 0.986622i \(0.552125\pi\)
\(368\) 0 0
\(369\) −26.0000 −1.35351
\(370\) 0 0
\(371\) −8.54312 −0.443537
\(372\) 0 0
\(373\) 12.2888 0.636291 0.318146 0.948042i \(-0.396940\pi\)
0.318146 + 0.948042i \(0.396940\pi\)
\(374\) 0 0
\(375\) −3.50758 −0.181131
\(376\) 0 0
\(377\) 6.24621 0.321696
\(378\) 0 0
\(379\) −23.4612 −1.20512 −0.602561 0.798073i \(-0.705853\pi\)
−0.602561 + 0.798073i \(0.705853\pi\)
\(380\) 0 0
\(381\) −4.56685 −0.233967
\(382\) 0 0
\(383\) 28.4924 1.45589 0.727947 0.685633i \(-0.240474\pi\)
0.727947 + 0.685633i \(0.240474\pi\)
\(384\) 0 0
\(385\) −14.2462 −0.726054
\(386\) 0 0
\(387\) 9.06897 0.461002
\(388\) 0 0
\(389\) 7.72197 0.391519 0.195760 0.980652i \(-0.437283\pi\)
0.195760 + 0.980652i \(0.437283\pi\)
\(390\) 0 0
\(391\) 6.24621 0.315884
\(392\) 0 0
\(393\) 3.61553 0.182379
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12.9300 −0.648936 −0.324468 0.945897i \(-0.605185\pi\)
−0.324468 + 0.945897i \(0.605185\pi\)
\(398\) 0 0
\(399\) 0.876894 0.0438996
\(400\) 0 0
\(401\) −9.12311 −0.455586 −0.227793 0.973710i \(-0.573151\pi\)
−0.227793 + 0.973710i \(0.573151\pi\)
\(402\) 0 0
\(403\) 20.8319 1.03771
\(404\) 0 0
\(405\) 6.25969 0.311047
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0.230559 0.0113726
\(412\) 0 0
\(413\) 7.60669 0.374301
\(414\) 0 0
\(415\) 31.6155 1.55195
\(416\) 0 0
\(417\) −16.8769 −0.826465
\(418\) 0 0
\(419\) −6.78554 −0.331495 −0.165748 0.986168i \(-0.553004\pi\)
−0.165748 + 0.986168i \(0.553004\pi\)
\(420\) 0 0
\(421\) −16.0345 −0.781475 −0.390738 0.920502i \(-0.627780\pi\)
−0.390738 + 0.920502i \(0.627780\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.2462 0.594029
\(426\) 0 0
\(427\) 3.33513 0.161398
\(428\) 0 0
\(429\) 13.3405 0.644087
\(430\) 0 0
\(431\) −23.6155 −1.13752 −0.568760 0.822504i \(-0.692577\pi\)
−0.568760 + 0.822504i \(0.692577\pi\)
\(432\) 0 0
\(433\) 14.4924 0.696461 0.348231 0.937409i \(-0.386783\pi\)
0.348231 + 0.937409i \(0.386783\pi\)
\(434\) 0 0
\(435\) −5.84912 −0.280444
\(436\) 0 0
\(437\) −2.92456 −0.139901
\(438\) 0 0
\(439\) 26.7386 1.27617 0.638083 0.769968i \(-0.279728\pi\)
0.638083 + 0.769968i \(0.279728\pi\)
\(440\) 0 0
\(441\) −2.12311 −0.101100
\(442\) 0 0
\(443\) 6.14441 0.291930 0.145965 0.989290i \(-0.453371\pi\)
0.145965 + 0.989290i \(0.453371\pi\)
\(444\) 0 0
\(445\) −0.821147 −0.0389261
\(446\) 0 0
\(447\) −11.5076 −0.544290
\(448\) 0 0
\(449\) −32.7386 −1.54503 −0.772516 0.634996i \(-0.781002\pi\)
−0.772516 + 0.634996i \(0.781002\pi\)
\(450\) 0 0
\(451\) −52.3104 −2.46320
\(452\) 0 0
\(453\) 17.9074 0.841362
\(454\) 0 0
\(455\) 11.1231 0.521459
\(456\) 0 0
\(457\) 7.36932 0.344722 0.172361 0.985034i \(-0.444860\pi\)
0.172361 + 0.985034i \(0.444860\pi\)
\(458\) 0 0
\(459\) 9.59482 0.447848
\(460\) 0 0
\(461\) 34.5830 1.61069 0.805346 0.592805i \(-0.201979\pi\)
0.805346 + 0.592805i \(0.201979\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) −19.5076 −0.904642
\(466\) 0 0
\(467\) −0.936426 −0.0433326 −0.0216663 0.999765i \(-0.506897\pi\)
−0.0216663 + 0.999765i \(0.506897\pi\)
\(468\) 0 0
\(469\) −15.7392 −0.726770
\(470\) 0 0
\(471\) −15.6155 −0.719526
\(472\) 0 0
\(473\) 18.2462 0.838962
\(474\) 0 0
\(475\) −5.73384 −0.263087
\(476\) 0 0
\(477\) 18.1379 0.830479
\(478\) 0 0
\(479\) −6.24621 −0.285397 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(480\) 0 0
\(481\) 6.24621 0.284803
\(482\) 0 0
\(483\) −2.92456 −0.133072
\(484\) 0 0
\(485\) −14.1617 −0.643049
\(486\) 0 0
\(487\) −25.3693 −1.14959 −0.574797 0.818296i \(-0.694919\pi\)
−0.574797 + 0.818296i \(0.694919\pi\)
\(488\) 0 0
\(489\) −7.50758 −0.339504
\(490\) 0 0
\(491\) −22.1789 −1.00092 −0.500461 0.865759i \(-0.666836\pi\)
−0.500461 + 0.865759i \(0.666836\pi\)
\(492\) 0 0
\(493\) 3.74571 0.168698
\(494\) 0 0
\(495\) 30.2462 1.35947
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 24.2824 1.08703 0.543514 0.839400i \(-0.317094\pi\)
0.543514 + 0.839400i \(0.317094\pi\)
\(500\) 0 0
\(501\) 7.49141 0.334692
\(502\) 0 0
\(503\) −30.2462 −1.34861 −0.674306 0.738452i \(-0.735557\pi\)
−0.674306 + 0.738452i \(0.735557\pi\)
\(504\) 0 0
\(505\) −23.6155 −1.05088
\(506\) 0 0
\(507\) 1.75757 0.0780566
\(508\) 0 0
\(509\) 32.9407 1.46007 0.730036 0.683408i \(-0.239503\pi\)
0.730036 + 0.683408i \(0.239503\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) −4.49242 −0.198345
\(514\) 0 0
\(515\) 47.5130 2.09367
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 16.3845 0.719198
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 35.9300 1.57111 0.785555 0.618791i \(-0.212377\pi\)
0.785555 + 0.618791i \(0.212377\pi\)
\(524\) 0 0
\(525\) −5.73384 −0.250245
\(526\) 0 0
\(527\) 12.4924 0.544178
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) −16.1498 −0.700842
\(532\) 0 0
\(533\) 40.8427 1.76910
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) 2.24621 0.0969312
\(538\) 0 0
\(539\) −4.27156 −0.183989
\(540\) 0 0
\(541\) −42.7156 −1.83649 −0.918243 0.396017i \(-0.870392\pi\)
−0.918243 + 0.396017i \(0.870392\pi\)
\(542\) 0 0
\(543\) −12.8769 −0.552600
\(544\) 0 0
\(545\) −18.7386 −0.802675
\(546\) 0 0
\(547\) −39.4957 −1.68872 −0.844358 0.535780i \(-0.820018\pi\)
−0.844358 + 0.535780i \(0.820018\pi\)
\(548\) 0 0
\(549\) −7.08084 −0.302203
\(550\) 0 0
\(551\) −1.75379 −0.0747139
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −5.84912 −0.248281
\(556\) 0 0
\(557\) −8.54312 −0.361983 −0.180992 0.983485i \(-0.557931\pi\)
−0.180992 + 0.983485i \(0.557931\pi\)
\(558\) 0 0
\(559\) −14.2462 −0.602551
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 16.9710 0.715240 0.357620 0.933867i \(-0.383588\pi\)
0.357620 + 0.933867i \(0.383588\pi\)
\(564\) 0 0
\(565\) 43.7673 1.84130
\(566\) 0 0
\(567\) 1.87689 0.0788222
\(568\) 0 0
\(569\) −11.3693 −0.476627 −0.238313 0.971188i \(-0.576595\pi\)
−0.238313 + 0.971188i \(0.576595\pi\)
\(570\) 0 0
\(571\) 29.9009 1.25131 0.625657 0.780098i \(-0.284831\pi\)
0.625657 + 0.780098i \(0.284831\pi\)
\(572\) 0 0
\(573\) −14.9828 −0.625916
\(574\) 0 0
\(575\) 19.1231 0.797489
\(576\) 0 0
\(577\) 24.2462 1.00938 0.504691 0.863300i \(-0.331606\pi\)
0.504691 + 0.863300i \(0.331606\pi\)
\(578\) 0 0
\(579\) 2.69400 0.111959
\(580\) 0 0
\(581\) 9.47954 0.393278
\(582\) 0 0
\(583\) 36.4924 1.51136
\(584\) 0 0
\(585\) −23.6155 −0.976382
\(586\) 0 0
\(587\) −8.65840 −0.357370 −0.178685 0.983906i \(-0.557184\pi\)
−0.178685 + 0.983906i \(0.557184\pi\)
\(588\) 0 0
\(589\) −5.84912 −0.241009
\(590\) 0 0
\(591\) 0.984845 0.0405111
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 6.67026 0.273454
\(596\) 0 0
\(597\) −1.64229 −0.0672146
\(598\) 0 0
\(599\) −20.4924 −0.837298 −0.418649 0.908148i \(-0.637496\pi\)
−0.418649 + 0.908148i \(0.637496\pi\)
\(600\) 0 0
\(601\) −0.246211 −0.0100432 −0.00502158 0.999987i \(-0.501598\pi\)
−0.00502158 + 0.999987i \(0.501598\pi\)
\(602\) 0 0
\(603\) 33.4161 1.36081
\(604\) 0 0
\(605\) 24.1671 0.982531
\(606\) 0 0
\(607\) −24.9848 −1.01410 −0.507052 0.861916i \(-0.669265\pi\)
−0.507052 + 0.861916i \(0.669265\pi\)
\(608\) 0 0
\(609\) −1.75379 −0.0710671
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −45.6401 −1.84339 −0.921694 0.387918i \(-0.873194\pi\)
−0.921694 + 0.387918i \(0.873194\pi\)
\(614\) 0 0
\(615\) −38.2462 −1.54224
\(616\) 0 0
\(617\) 4.63068 0.186424 0.0932121 0.995646i \(-0.470287\pi\)
0.0932121 + 0.995646i \(0.470287\pi\)
\(618\) 0 0
\(619\) −36.9817 −1.48642 −0.743211 0.669057i \(-0.766698\pi\)
−0.743211 + 0.669057i \(0.766698\pi\)
\(620\) 0 0
\(621\) 14.9828 0.601240
\(622\) 0 0
\(623\) −0.246211 −0.00986425
\(624\) 0 0
\(625\) −18.1231 −0.724924
\(626\) 0 0
\(627\) −3.74571 −0.149589
\(628\) 0 0
\(629\) 3.74571 0.149351
\(630\) 0 0
\(631\) 36.4924 1.45274 0.726370 0.687304i \(-0.241206\pi\)
0.726370 + 0.687304i \(0.241206\pi\)
\(632\) 0 0
\(633\) 2.24621 0.0892789
\(634\) 0 0
\(635\) 16.2651 0.645460
\(636\) 0 0
\(637\) 3.33513 0.132143
\(638\) 0 0
\(639\) −16.9848 −0.671910
\(640\) 0 0
\(641\) 19.3693 0.765042 0.382521 0.923947i \(-0.375056\pi\)
0.382521 + 0.923947i \(0.375056\pi\)
\(642\) 0 0
\(643\) 9.47954 0.373837 0.186918 0.982375i \(-0.440150\pi\)
0.186918 + 0.982375i \(0.440150\pi\)
\(644\) 0 0
\(645\) 13.3405 0.525283
\(646\) 0 0
\(647\) 39.2311 1.54233 0.771166 0.636634i \(-0.219674\pi\)
0.771166 + 0.636634i \(0.219674\pi\)
\(648\) 0 0
\(649\) −32.4924 −1.27544
\(650\) 0 0
\(651\) −5.84912 −0.229245
\(652\) 0 0
\(653\) 21.0625 0.824239 0.412120 0.911130i \(-0.364789\pi\)
0.412120 + 0.911130i \(0.364789\pi\)
\(654\) 0 0
\(655\) −12.8769 −0.503142
\(656\) 0 0
\(657\) −12.7386 −0.496981
\(658\) 0 0
\(659\) 21.3578 0.831981 0.415991 0.909369i \(-0.363435\pi\)
0.415991 + 0.909369i \(0.363435\pi\)
\(660\) 0 0
\(661\) −5.43854 −0.211535 −0.105767 0.994391i \(-0.533730\pi\)
−0.105767 + 0.994391i \(0.533730\pi\)
\(662\) 0 0
\(663\) −6.24621 −0.242583
\(664\) 0 0
\(665\) −3.12311 −0.121109
\(666\) 0 0
\(667\) 5.84912 0.226479
\(668\) 0 0
\(669\) 20.8319 0.805409
\(670\) 0 0
\(671\) −14.2462 −0.549969
\(672\) 0 0
\(673\) 26.9848 1.04019 0.520095 0.854109i \(-0.325897\pi\)
0.520095 + 0.854109i \(0.325897\pi\)
\(674\) 0 0
\(675\) 29.3751 1.13065
\(676\) 0 0
\(677\) 6.25969 0.240579 0.120290 0.992739i \(-0.461618\pi\)
0.120290 + 0.992739i \(0.461618\pi\)
\(678\) 0 0
\(679\) −4.24621 −0.162955
\(680\) 0 0
\(681\) −11.6155 −0.445108
\(682\) 0 0
\(683\) 19.4849 0.745570 0.372785 0.927918i \(-0.378403\pi\)
0.372785 + 0.927918i \(0.378403\pi\)
\(684\) 0 0
\(685\) −0.821147 −0.0313744
\(686\) 0 0
\(687\) 3.89205 0.148491
\(688\) 0 0
\(689\) −28.4924 −1.08547
\(690\) 0 0
\(691\) 42.0097 1.59812 0.799062 0.601248i \(-0.205330\pi\)
0.799062 + 0.601248i \(0.205330\pi\)
\(692\) 0 0
\(693\) 9.06897 0.344502
\(694\) 0 0
\(695\) 60.1080 2.28002
\(696\) 0 0
\(697\) 24.4924 0.927717
\(698\) 0 0
\(699\) 0.230559 0.00872053
\(700\) 0 0
\(701\) −45.6401 −1.72380 −0.861902 0.507075i \(-0.830727\pi\)
−0.861902 + 0.507075i \(0.830727\pi\)
\(702\) 0 0
\(703\) −1.75379 −0.0661454
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.08084 −0.266302
\(708\) 0 0
\(709\) −39.7910 −1.49438 −0.747192 0.664609i \(-0.768598\pi\)
−0.747192 + 0.664609i \(0.768598\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 19.5076 0.730565
\(714\) 0 0
\(715\) −47.5130 −1.77689
\(716\) 0 0
\(717\) −22.1142 −0.825870
\(718\) 0 0
\(719\) −19.5076 −0.727510 −0.363755 0.931495i \(-0.618505\pi\)
−0.363755 + 0.931495i \(0.618505\pi\)
\(720\) 0 0
\(721\) 14.2462 0.530557
\(722\) 0 0
\(723\) 18.9591 0.705096
\(724\) 0 0
\(725\) 11.4677 0.425899
\(726\) 0 0
\(727\) −48.9848 −1.81675 −0.908374 0.418159i \(-0.862675\pi\)
−0.908374 + 0.418159i \(0.862675\pi\)
\(728\) 0 0
\(729\) 9.49242 0.351571
\(730\) 0 0
\(731\) −8.54312 −0.315979
\(732\) 0 0
\(733\) −2.51398 −0.0928562 −0.0464281 0.998922i \(-0.514784\pi\)
−0.0464281 + 0.998922i \(0.514784\pi\)
\(734\) 0 0
\(735\) −3.12311 −0.115197
\(736\) 0 0
\(737\) 67.2311 2.47649
\(738\) 0 0
\(739\) −33.6466 −1.23771 −0.618855 0.785505i \(-0.712403\pi\)
−0.618855 + 0.785505i \(0.712403\pi\)
\(740\) 0 0
\(741\) 2.92456 0.107436
\(742\) 0 0
\(743\) −25.3693 −0.930710 −0.465355 0.885124i \(-0.654073\pi\)
−0.465355 + 0.885124i \(0.654073\pi\)
\(744\) 0 0
\(745\) 40.9848 1.50157
\(746\) 0 0
\(747\) −20.1261 −0.736374
\(748\) 0 0
\(749\) −7.19612 −0.262940
\(750\) 0 0
\(751\) 33.3693 1.21766 0.608832 0.793299i \(-0.291638\pi\)
0.608832 + 0.793299i \(0.291638\pi\)
\(752\) 0 0
\(753\) 24.8769 0.906564
\(754\) 0 0
\(755\) −63.7781 −2.32112
\(756\) 0 0
\(757\) −38.1487 −1.38654 −0.693270 0.720678i \(-0.743830\pi\)
−0.693270 + 0.720678i \(0.743830\pi\)
\(758\) 0 0
\(759\) 12.4924 0.453446
\(760\) 0 0
\(761\) 40.7386 1.47677 0.738387 0.674377i \(-0.235588\pi\)
0.738387 + 0.674377i \(0.235588\pi\)
\(762\) 0 0
\(763\) −5.61856 −0.203405
\(764\) 0 0
\(765\) −14.1617 −0.512016
\(766\) 0 0
\(767\) 25.3693 0.916033
\(768\) 0 0
\(769\) −23.7538 −0.856584 −0.428292 0.903641i \(-0.640884\pi\)
−0.428292 + 0.903641i \(0.640884\pi\)
\(770\) 0 0
\(771\) 9.82538 0.353852
\(772\) 0 0
\(773\) 3.33513 0.119956 0.0599782 0.998200i \(-0.480897\pi\)
0.0599782 + 0.998200i \(0.480897\pi\)
\(774\) 0 0
\(775\) 38.2462 1.37384
\(776\) 0 0
\(777\) −1.75379 −0.0629168
\(778\) 0 0
\(779\) −11.4677 −0.410872
\(780\) 0 0
\(781\) −34.1725 −1.22279
\(782\) 0 0
\(783\) 8.98485 0.321092
\(784\) 0 0
\(785\) 55.6155 1.98500
\(786\) 0 0
\(787\) −27.6175 −0.984457 −0.492228 0.870466i \(-0.663818\pi\)
−0.492228 + 0.870466i \(0.663818\pi\)
\(788\) 0 0
\(789\) −19.1896 −0.683169
\(790\) 0 0
\(791\) 13.1231 0.466604
\(792\) 0 0
\(793\) 11.1231 0.394993
\(794\) 0 0
\(795\) 26.6811 0.946280
\(796\) 0 0
\(797\) −4.15628 −0.147223 −0.0736115 0.997287i \(-0.523452\pi\)
−0.0736115 + 0.997287i \(0.523452\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.522732 0.0184698
\(802\) 0 0
\(803\) −25.6294 −0.904440
\(804\) 0 0
\(805\) 10.4160 0.367115
\(806\) 0 0
\(807\) 21.8617 0.769570
\(808\) 0 0
\(809\) 19.8617 0.698302 0.349151 0.937067i \(-0.386470\pi\)
0.349151 + 0.937067i \(0.386470\pi\)
\(810\) 0 0
\(811\) 10.5312 0.369802 0.184901 0.982757i \(-0.440804\pi\)
0.184901 + 0.982757i \(0.440804\pi\)
\(812\) 0 0
\(813\) 5.84912 0.205137
\(814\) 0 0
\(815\) 26.7386 0.936613
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) −7.08084 −0.247424
\(820\) 0 0
\(821\) −2.69400 −0.0940212 −0.0470106 0.998894i \(-0.514969\pi\)
−0.0470106 + 0.998894i \(0.514969\pi\)
\(822\) 0 0
\(823\) −32.9848 −1.14978 −0.574890 0.818231i \(-0.694955\pi\)
−0.574890 + 0.818231i \(0.694955\pi\)
\(824\) 0 0
\(825\) 24.4924 0.852717
\(826\) 0 0
\(827\) 32.8255 1.14145 0.570727 0.821140i \(-0.306662\pi\)
0.570727 + 0.821140i \(0.306662\pi\)
\(828\) 0 0
\(829\) 18.3180 0.636209 0.318104 0.948056i \(-0.396954\pi\)
0.318104 + 0.948056i \(0.396954\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −26.6811 −0.923336
\(836\) 0 0
\(837\) 29.9656 1.03576
\(838\) 0 0
\(839\) −42.7386 −1.47550 −0.737751 0.675073i \(-0.764112\pi\)
−0.737751 + 0.675073i \(0.764112\pi\)
\(840\) 0 0
\(841\) −25.4924 −0.879049
\(842\) 0 0
\(843\) −5.61856 −0.193513
\(844\) 0 0
\(845\) −6.25969 −0.215340
\(846\) 0 0
\(847\) 7.24621 0.248983
\(848\) 0 0
\(849\) 8.10795 0.278264
\(850\) 0 0
\(851\) 5.84912 0.200505
\(852\) 0 0
\(853\) −50.0270 −1.71289 −0.856446 0.516237i \(-0.827332\pi\)
−0.856446 + 0.516237i \(0.827332\pi\)
\(854\) 0 0
\(855\) 6.63068 0.226765
\(856\) 0 0
\(857\) −18.9848 −0.648510 −0.324255 0.945970i \(-0.605114\pi\)
−0.324255 + 0.945970i \(0.605114\pi\)
\(858\) 0 0
\(859\) −47.3977 −1.61719 −0.808595 0.588366i \(-0.799771\pi\)
−0.808595 + 0.588366i \(0.799771\pi\)
\(860\) 0 0
\(861\) −11.4677 −0.390817
\(862\) 0 0
\(863\) 3.50758 0.119399 0.0596997 0.998216i \(-0.480986\pi\)
0.0596997 + 0.998216i \(0.480986\pi\)
\(864\) 0 0
\(865\) −58.3542 −1.98410
\(866\) 0 0
\(867\) 12.1735 0.413435
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −52.4924 −1.77864
\(872\) 0 0
\(873\) 9.01515 0.305117
\(874\) 0 0
\(875\) 3.74571 0.126628
\(876\) 0 0
\(877\) −26.4505 −0.893170 −0.446585 0.894741i \(-0.647360\pi\)
−0.446585 + 0.894741i \(0.647360\pi\)
\(878\) 0 0
\(879\) 6.63068 0.223647
\(880\) 0 0
\(881\) 27.7538 0.935049 0.467524 0.883980i \(-0.345146\pi\)
0.467524 + 0.883980i \(0.345146\pi\)
\(882\) 0 0
\(883\) −2.39871 −0.0807229 −0.0403614 0.999185i \(-0.512851\pi\)
−0.0403614 + 0.999185i \(0.512851\pi\)
\(884\) 0 0
\(885\) −23.7565 −0.798566
\(886\) 0 0
\(887\) −4.49242 −0.150841 −0.0754204 0.997152i \(-0.524030\pi\)
−0.0754204 + 0.997152i \(0.524030\pi\)
\(888\) 0 0
\(889\) 4.87689 0.163566
\(890\) 0 0
\(891\) −8.01726 −0.268588
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −9.75379 −0.325670
\(898\) 0 0
\(899\) 11.6982 0.390158
\(900\) 0 0
\(901\) −17.0862 −0.569225
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 45.8617 1.52450
\(906\) 0 0
\(907\) −43.0109 −1.42815 −0.714076 0.700068i \(-0.753153\pi\)
−0.714076 + 0.700068i \(0.753153\pi\)
\(908\) 0 0
\(909\) 15.0334 0.498625
\(910\) 0 0
\(911\) −45.8617 −1.51947 −0.759734 0.650234i \(-0.774671\pi\)
−0.759734 + 0.650234i \(0.774671\pi\)
\(912\) 0 0
\(913\) −40.4924 −1.34010
\(914\) 0 0
\(915\) −10.4160 −0.344341
\(916\) 0 0
\(917\) −3.86098 −0.127501
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −14.3542 −0.472986
\(922\) 0 0
\(923\) 26.6811 0.878218
\(924\) 0 0
\(925\) 11.4677 0.377055
\(926\) 0 0
\(927\) −30.2462 −0.993416
\(928\) 0 0
\(929\) 30.4924 1.00042 0.500212 0.865903i \(-0.333255\pi\)
0.500212 + 0.865903i \(0.333255\pi\)
\(930\) 0 0
\(931\) −0.936426 −0.0306901
\(932\) 0 0
\(933\) 19.1896 0.628241
\(934\) 0 0
\(935\) −28.4924 −0.931802
\(936\) 0 0
\(937\) 30.9848 1.01223 0.506115 0.862466i \(-0.331081\pi\)
0.506115 + 0.862466i \(0.331081\pi\)
\(938\) 0 0
\(939\) 26.9116 0.878227
\(940\) 0 0
\(941\) 32.9407 1.07384 0.536919 0.843634i \(-0.319588\pi\)
0.536919 + 0.843634i \(0.319588\pi\)
\(942\) 0 0
\(943\) 38.2462 1.24547
\(944\) 0 0
\(945\) 16.0000 0.520480
\(946\) 0 0
\(947\) 6.37497 0.207159 0.103579 0.994621i \(-0.466970\pi\)
0.103579 + 0.994621i \(0.466970\pi\)
\(948\) 0 0
\(949\) 20.0108 0.649578
\(950\) 0 0
\(951\) 20.4924 0.664512
\(952\) 0 0
\(953\) 50.4924 1.63561 0.817805 0.575495i \(-0.195191\pi\)
0.817805 + 0.575495i \(0.195191\pi\)
\(954\) 0 0
\(955\) 53.3621 1.72676
\(956\) 0 0
\(957\) 7.49141 0.242163
\(958\) 0 0
\(959\) −0.246211 −0.00795058
\(960\) 0 0
\(961\) 8.01515 0.258553
\(962\) 0 0
\(963\) 15.2781 0.492330
\(964\) 0 0
\(965\) −9.59482 −0.308868
\(966\) 0 0
\(967\) 19.1231 0.614958 0.307479 0.951555i \(-0.400515\pi\)
0.307479 + 0.951555i \(0.400515\pi\)
\(968\) 0 0
\(969\) 1.75379 0.0563398
\(970\) 0 0
\(971\) −26.5658 −0.852536 −0.426268 0.904597i \(-0.640172\pi\)
−0.426268 + 0.904597i \(0.640172\pi\)
\(972\) 0 0
\(973\) 18.0227 0.577780
\(974\) 0 0
\(975\) −19.1231 −0.612430
\(976\) 0 0
\(977\) 40.2462 1.28759 0.643795 0.765198i \(-0.277359\pi\)
0.643795 + 0.765198i \(0.277359\pi\)
\(978\) 0 0
\(979\) 1.05171 0.0336127
\(980\) 0 0
\(981\) 11.9288 0.380857
\(982\) 0 0
\(983\) 17.7538 0.566258 0.283129 0.959082i \(-0.408628\pi\)
0.283129 + 0.959082i \(0.408628\pi\)
\(984\) 0 0
\(985\) −3.50758 −0.111761
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13.3405 −0.424204
\(990\) 0 0
\(991\) −3.50758 −0.111422 −0.0557109 0.998447i \(-0.517743\pi\)
−0.0557109 + 0.998447i \(0.517743\pi\)
\(992\) 0 0
\(993\) −20.9848 −0.665934
\(994\) 0 0
\(995\) 5.84912 0.185429
\(996\) 0 0
\(997\) 44.9990 1.42513 0.712566 0.701605i \(-0.247533\pi\)
0.712566 + 0.701605i \(0.247533\pi\)
\(998\) 0 0
\(999\) 8.98485 0.284268
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 1792.2.a.x.1.2 4
4.3 odd 2 1792.2.a.v.1.3 4
8.3 odd 2 1792.2.a.v.1.2 4
8.5 even 2 inner 1792.2.a.x.1.3 4
16.3 odd 4 224.2.b.b.113.3 4
16.5 even 4 56.2.b.b.29.2 yes 4
16.11 odd 4 224.2.b.b.113.2 4
16.13 even 4 56.2.b.b.29.1 4
48.5 odd 4 504.2.c.d.253.3 4
48.11 even 4 2016.2.c.c.1009.1 4
48.29 odd 4 504.2.c.d.253.4 4
48.35 even 4 2016.2.c.c.1009.4 4
112.3 even 12 1568.2.t.e.177.2 8
112.5 odd 12 392.2.p.e.165.4 8
112.11 odd 12 1568.2.t.d.177.2 8
112.13 odd 4 392.2.b.c.197.1 4
112.19 even 12 1568.2.t.e.753.3 8
112.27 even 4 1568.2.b.d.785.3 4
112.37 even 12 392.2.p.f.165.4 8
112.45 odd 12 392.2.p.e.373.4 8
112.51 odd 12 1568.2.t.d.753.2 8
112.53 even 12 392.2.p.f.373.2 8
112.59 even 12 1568.2.t.e.177.3 8
112.61 odd 12 392.2.p.e.165.2 8
112.67 odd 12 1568.2.t.d.177.3 8
112.69 odd 4 392.2.b.c.197.2 4
112.75 even 12 1568.2.t.e.753.2 8
112.83 even 4 1568.2.b.d.785.2 4
112.93 even 12 392.2.p.f.165.2 8
112.101 odd 12 392.2.p.e.373.2 8
112.107 odd 12 1568.2.t.d.753.3 8
112.109 even 12 392.2.p.f.373.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.b.b.29.1 4 16.13 even 4
56.2.b.b.29.2 yes 4 16.5 even 4
224.2.b.b.113.2 4 16.11 odd 4
224.2.b.b.113.3 4 16.3 odd 4
392.2.b.c.197.1 4 112.13 odd 4
392.2.b.c.197.2 4 112.69 odd 4
392.2.p.e.165.2 8 112.61 odd 12
392.2.p.e.165.4 8 112.5 odd 12
392.2.p.e.373.2 8 112.101 odd 12
392.2.p.e.373.4 8 112.45 odd 12
392.2.p.f.165.2 8 112.93 even 12
392.2.p.f.165.4 8 112.37 even 12
392.2.p.f.373.2 8 112.53 even 12
392.2.p.f.373.4 8 112.109 even 12
504.2.c.d.253.3 4 48.5 odd 4
504.2.c.d.253.4 4 48.29 odd 4
1568.2.b.d.785.2 4 112.83 even 4
1568.2.b.d.785.3 4 112.27 even 4
1568.2.t.d.177.2 8 112.11 odd 12
1568.2.t.d.177.3 8 112.67 odd 12
1568.2.t.d.753.2 8 112.51 odd 12
1568.2.t.d.753.3 8 112.107 odd 12
1568.2.t.e.177.2 8 112.3 even 12
1568.2.t.e.177.3 8 112.59 even 12
1568.2.t.e.753.2 8 112.75 even 12
1568.2.t.e.753.3 8 112.19 even 12
1792.2.a.v.1.2 4 8.3 odd 2
1792.2.a.v.1.3 4 4.3 odd 2
1792.2.a.x.1.2 4 1.1 even 1 trivial
1792.2.a.x.1.3 4 8.5 even 2 inner
2016.2.c.c.1009.1 4 48.11 even 4
2016.2.c.c.1009.4 4 48.35 even 4