Properties

Label 4608.2.a.w.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
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] = [4608,2,Mod(1,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, 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("4608.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.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: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.46410 q^{5} +2.82843 q^{7} +O(q^{10})\) \(q-3.46410 q^{5} +2.82843 q^{7} +2.44949 q^{11} -3.46410 q^{13} -4.00000 q^{17} +2.44949 q^{19} -2.82843 q^{23} +7.00000 q^{25} -3.46410 q^{29} +5.65685 q^{31} -9.79796 q^{35} -3.46410 q^{37} +2.00000 q^{41} +12.2474 q^{43} +11.3137 q^{47} +1.00000 q^{49} -10.3923 q^{53} -8.48528 q^{55} -2.44949 q^{59} -3.46410 q^{61} +12.0000 q^{65} -7.34847 q^{67} +2.82843 q^{71} +8.00000 q^{73} +6.92820 q^{77} +5.65685 q^{79} -7.34847 q^{83} +13.8564 q^{85} +8.00000 q^{89} -9.79796 q^{91} -8.48528 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{17} + 28 q^{25} + 8 q^{41} + 4 q^{49} + 48 q^{65} + 32 q^{73} + 32 q^{89} + 48 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 2.44949 0.561951 0.280976 0.959715i \(-0.409342\pi\)
0.280976 + 0.959715i \(0.409342\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.79796 −1.65616
\(36\) 0 0
\(37\) −3.46410 −0.569495 −0.284747 0.958603i \(-0.591910\pi\)
−0.284747 + 0.958603i \(0.591910\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 12.2474 1.86772 0.933859 0.357641i \(-0.116419\pi\)
0.933859 + 0.357641i \(0.116419\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.3137 1.65027 0.825137 0.564933i \(-0.191098\pi\)
0.825137 + 0.564933i \(0.191098\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 0 0
\(55\) −8.48528 −1.14416
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.44949 −0.318896 −0.159448 0.987206i \(-0.550971\pi\)
−0.159448 + 0.987206i \(0.550971\pi\)
\(60\) 0 0
\(61\) −3.46410 −0.443533 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −7.34847 −0.897758 −0.448879 0.893592i \(-0.648177\pi\)
−0.448879 + 0.893592i \(0.648177\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.34847 −0.806599 −0.403300 0.915068i \(-0.632137\pi\)
−0.403300 + 0.915068i \(0.632137\pi\)
\(84\) 0 0
\(85\) 13.8564 1.50294
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −9.79796 −1.02711
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.48528 −0.870572
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46410 0.344691 0.172345 0.985037i \(-0.444865\pi\)
0.172345 + 0.985037i \(0.444865\pi\)
\(102\) 0 0
\(103\) −14.1421 −1.39347 −0.696733 0.717331i \(-0.745364\pi\)
−0.696733 + 0.717331i \(0.745364\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.44949 0.236801 0.118401 0.992966i \(-0.462223\pi\)
0.118401 + 0.992966i \(0.462223\pi\)
\(108\) 0 0
\(109\) −10.3923 −0.995402 −0.497701 0.867349i \(-0.665822\pi\)
−0.497701 + 0.867349i \(0.665822\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 9.79796 0.913664
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.3137 −1.03713
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.34847 0.642039 0.321019 0.947073i \(-0.395975\pi\)
0.321019 + 0.947073i \(0.395975\pi\)
\(132\) 0 0
\(133\) 6.92820 0.600751
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 12.2474 1.03882 0.519408 0.854527i \(-0.326153\pi\)
0.519408 + 0.854527i \(0.326153\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.48528 −0.709575
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.3923 0.851371 0.425685 0.904871i \(-0.360033\pi\)
0.425685 + 0.904871i \(0.360033\pi\)
\(150\) 0 0
\(151\) −19.7990 −1.61122 −0.805609 0.592447i \(-0.798162\pi\)
−0.805609 + 0.592447i \(0.798162\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −19.5959 −1.57398
\(156\) 0 0
\(157\) 24.2487 1.93526 0.967629 0.252377i \(-0.0812124\pi\)
0.967629 + 0.252377i \(0.0812124\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 17.1464 1.34301 0.671506 0.740999i \(-0.265648\pi\)
0.671506 + 0.740999i \(0.265648\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.48528 0.656611 0.328305 0.944572i \(-0.393522\pi\)
0.328305 + 0.944572i \(0.393522\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.3205 1.31685 0.658427 0.752645i \(-0.271222\pi\)
0.658427 + 0.752645i \(0.271222\pi\)
\(174\) 0 0
\(175\) 19.7990 1.49666
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.2474 0.915417 0.457709 0.889102i \(-0.348670\pi\)
0.457709 + 0.889102i \(0.348670\pi\)
\(180\) 0 0
\(181\) −24.2487 −1.80239 −0.901196 0.433411i \(-0.857310\pi\)
−0.901196 + 0.433411i \(0.857310\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) −9.79796 −0.716498
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923 0.740421 0.370211 0.928948i \(-0.379286\pi\)
0.370211 + 0.928948i \(0.379286\pi\)
\(198\) 0 0
\(199\) 2.82843 0.200502 0.100251 0.994962i \(-0.468035\pi\)
0.100251 + 0.994962i \(0.468035\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.79796 −0.687682
\(204\) 0 0
\(205\) −6.92820 −0.483887
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −12.2474 −0.843149 −0.421575 0.906794i \(-0.638522\pi\)
−0.421575 + 0.906794i \(0.638522\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −42.4264 −2.89346
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.8564 0.932083
\(222\) 0 0
\(223\) −5.65685 −0.378811 −0.189405 0.981899i \(-0.560656\pi\)
−0.189405 + 0.981899i \(0.560656\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.0454 1.46321 0.731603 0.681731i \(-0.238773\pi\)
0.731603 + 0.681731i \(0.238773\pi\)
\(228\) 0 0
\(229\) 3.46410 0.228914 0.114457 0.993428i \(-0.463487\pi\)
0.114457 + 0.993428i \(0.463487\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) −39.1918 −2.55659
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.46410 −0.221313
\(246\) 0 0
\(247\) −8.48528 −0.539906
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.1464 1.08227 0.541136 0.840935i \(-0.317994\pi\)
0.541136 + 0.840935i \(0.317994\pi\)
\(252\) 0 0
\(253\) −6.92820 −0.435572
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) −9.79796 −0.608816
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 25.4558 1.56967 0.784837 0.619702i \(-0.212746\pi\)
0.784837 + 0.619702i \(0.212746\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.46410 −0.211210 −0.105605 0.994408i \(-0.533678\pi\)
−0.105605 + 0.994408i \(0.533678\pi\)
\(270\) 0 0
\(271\) 11.3137 0.687259 0.343629 0.939105i \(-0.388344\pi\)
0.343629 + 0.939105i \(0.388344\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.1464 1.03397
\(276\) 0 0
\(277\) 24.2487 1.45696 0.728482 0.685065i \(-0.240226\pi\)
0.728482 + 0.685065i \(0.240226\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) 26.9444 1.60168 0.800839 0.598880i \(-0.204387\pi\)
0.800839 + 0.598880i \(0.204387\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.65685 0.333914
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.3923 0.607125 0.303562 0.952812i \(-0.401824\pi\)
0.303562 + 0.952812i \(0.401824\pi\)
\(294\) 0 0
\(295\) 8.48528 0.494032
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.79796 0.566631
\(300\) 0 0
\(301\) 34.6410 1.99667
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 2.44949 0.139800 0.0698999 0.997554i \(-0.477732\pi\)
0.0698999 + 0.997554i \(0.477732\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.48528 −0.481156 −0.240578 0.970630i \(-0.577337\pi\)
−0.240578 + 0.970630i \(0.577337\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.3205 −0.972817 −0.486408 0.873732i \(-0.661693\pi\)
−0.486408 + 0.873732i \(0.661693\pi\)
\(318\) 0 0
\(319\) −8.48528 −0.475085
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.79796 −0.545173
\(324\) 0 0
\(325\) −24.2487 −1.34508
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) −2.44949 −0.134636 −0.0673181 0.997732i \(-0.521444\pi\)
−0.0673181 + 0.997732i \(0.521444\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 25.4558 1.39080
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 13.8564 0.750366
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.1464 0.920468 0.460234 0.887798i \(-0.347765\pi\)
0.460234 + 0.887798i \(0.347765\pi\)
\(348\) 0 0
\(349\) 17.3205 0.927146 0.463573 0.886059i \(-0.346567\pi\)
0.463573 + 0.886059i \(0.346567\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −9.79796 −0.520022
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.82843 0.149279 0.0746393 0.997211i \(-0.476219\pi\)
0.0746393 + 0.997211i \(0.476219\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −27.7128 −1.45055
\(366\) 0 0
\(367\) 22.6274 1.18114 0.590571 0.806986i \(-0.298903\pi\)
0.590571 + 0.806986i \(0.298903\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −29.3939 −1.52605
\(372\) 0 0
\(373\) −3.46410 −0.179364 −0.0896822 0.995970i \(-0.528585\pi\)
−0.0896822 + 0.995970i \(0.528585\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 2.44949 0.125822 0.0629109 0.998019i \(-0.479962\pi\)
0.0629109 + 0.998019i \(0.479962\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −33.9411 −1.73431 −0.867155 0.498038i \(-0.834054\pi\)
−0.867155 + 0.498038i \(0.834054\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.46410 0.175637 0.0878185 0.996136i \(-0.472010\pi\)
0.0878185 + 0.996136i \(0.472010\pi\)
\(390\) 0 0
\(391\) 11.3137 0.572159
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −19.5959 −0.985978
\(396\) 0 0
\(397\) −24.2487 −1.21701 −0.608504 0.793551i \(-0.708230\pi\)
−0.608504 + 0.793551i \(0.708230\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 0 0
\(403\) −19.5959 −0.976142
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.48528 −0.420600
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 25.4558 1.24958
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −31.8434 −1.55565 −0.777825 0.628481i \(-0.783677\pi\)
−0.777825 + 0.628481i \(0.783677\pi\)
\(420\) 0 0
\(421\) −10.3923 −0.506490 −0.253245 0.967402i \(-0.581498\pi\)
−0.253245 + 0.967402i \(0.581498\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −28.0000 −1.35820
\(426\) 0 0
\(427\) −9.79796 −0.474156
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685 0.272481 0.136241 0.990676i \(-0.456498\pi\)
0.136241 + 0.990676i \(0.456498\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) −2.82843 −0.134993 −0.0674967 0.997719i \(-0.521501\pi\)
−0.0674967 + 0.997719i \(0.521501\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.8434 1.51292 0.756462 0.654038i \(-0.226926\pi\)
0.756462 + 0.654038i \(0.226926\pi\)
\(444\) 0 0
\(445\) −27.7128 −1.31371
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) 4.89898 0.230684
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 33.9411 1.59118
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17.3205 −0.806696 −0.403348 0.915047i \(-0.632154\pi\)
−0.403348 + 0.915047i \(0.632154\pi\)
\(462\) 0 0
\(463\) −5.65685 −0.262896 −0.131448 0.991323i \(-0.541963\pi\)
−0.131448 + 0.991323i \(0.541963\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.1464 0.793442 0.396721 0.917939i \(-0.370148\pi\)
0.396721 + 0.917939i \(0.370148\pi\)
\(468\) 0 0
\(469\) −20.7846 −0.959744
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) 17.1464 0.786732
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.65685 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −41.5692 −1.88756
\(486\) 0 0
\(487\) 14.1421 0.640841 0.320421 0.947275i \(-0.396176\pi\)
0.320421 + 0.947275i \(0.396176\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.34847 0.331632 0.165816 0.986157i \(-0.446974\pi\)
0.165816 + 0.986157i \(0.446974\pi\)
\(492\) 0 0
\(493\) 13.8564 0.624061
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −31.8434 −1.42550 −0.712752 0.701416i \(-0.752552\pi\)
−0.712752 + 0.701416i \(0.752552\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.1421 −0.630567 −0.315283 0.948998i \(-0.602100\pi\)
−0.315283 + 0.948998i \(0.602100\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.3923 −0.460631 −0.230315 0.973116i \(-0.573976\pi\)
−0.230315 + 0.973116i \(0.573976\pi\)
\(510\) 0 0
\(511\) 22.6274 1.00098
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 48.9898 2.15875
\(516\) 0 0
\(517\) 27.7128 1.21881
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) −22.0454 −0.963978 −0.481989 0.876177i \(-0.660086\pi\)
−0.481989 + 0.876177i \(0.660086\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −22.6274 −0.985666
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) −8.48528 −0.366851
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.44949 0.105507
\(540\) 0 0
\(541\) −31.1769 −1.34040 −0.670200 0.742180i \(-0.733792\pi\)
−0.670200 + 0.742180i \(0.733792\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.0000 1.54207
\(546\) 0 0
\(547\) 12.2474 0.523663 0.261832 0.965114i \(-0.415674\pi\)
0.261832 + 0.965114i \(0.415674\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.48528 −0.361485
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.3205 0.733893 0.366947 0.930242i \(-0.380403\pi\)
0.366947 + 0.930242i \(0.380403\pi\)
\(558\) 0 0
\(559\) −42.4264 −1.79445
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.0454 −0.929103 −0.464552 0.885546i \(-0.653784\pi\)
−0.464552 + 0.885546i \(0.653784\pi\)
\(564\) 0 0
\(565\) −34.6410 −1.45736
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) 7.34847 0.307524 0.153762 0.988108i \(-0.450861\pi\)
0.153762 + 0.988108i \(0.450861\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −19.7990 −0.825675
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −20.7846 −0.862291
\(582\) 0 0
\(583\) −25.4558 −1.05427
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.0454 0.909911 0.454956 0.890514i \(-0.349655\pi\)
0.454956 + 0.890514i \(0.349655\pi\)
\(588\) 0 0
\(589\) 13.8564 0.570943
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 39.1918 1.60671
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.82843 0.115566 0.0577832 0.998329i \(-0.481597\pi\)
0.0577832 + 0.998329i \(0.481597\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17.3205 0.704179
\(606\) 0 0
\(607\) −45.2548 −1.83684 −0.918419 0.395610i \(-0.870533\pi\)
−0.918419 + 0.395610i \(0.870533\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −39.1918 −1.58553
\(612\) 0 0
\(613\) −10.3923 −0.419741 −0.209871 0.977729i \(-0.567304\pi\)
−0.209871 + 0.977729i \(0.567304\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 0 0
\(619\) 17.1464 0.689173 0.344587 0.938755i \(-0.388019\pi\)
0.344587 + 0.938755i \(0.388019\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 22.6274 0.906548
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) −19.7990 −0.788185 −0.394093 0.919071i \(-0.628941\pi\)
−0.394093 + 0.919071i \(0.628941\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −19.5959 −0.777640
\(636\) 0 0
\(637\) −3.46410 −0.137253
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) −7.34847 −0.289795 −0.144898 0.989447i \(-0.546285\pi\)
−0.144898 + 0.989447i \(0.546285\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 0 0
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.1769 −1.22005 −0.610023 0.792383i \(-0.708840\pi\)
−0.610023 + 0.792383i \(0.708840\pi\)
\(654\) 0 0
\(655\) −25.4558 −0.994642
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −46.5403 −1.81295 −0.906476 0.422256i \(-0.861238\pi\)
−0.906476 + 0.422256i \(0.861238\pi\)
\(660\) 0 0
\(661\) 31.1769 1.21264 0.606321 0.795220i \(-0.292645\pi\)
0.606321 + 0.795220i \(0.292645\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 9.79796 0.379378
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.48528 −0.327571
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.46410 0.133136 0.0665681 0.997782i \(-0.478795\pi\)
0.0665681 + 0.997782i \(0.478795\pi\)
\(678\) 0 0
\(679\) 33.9411 1.30254
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.2474 −0.468636 −0.234318 0.972160i \(-0.575286\pi\)
−0.234318 + 0.972160i \(0.575286\pi\)
\(684\) 0 0
\(685\) 6.92820 0.264713
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 7.34847 0.279549 0.139774 0.990183i \(-0.455362\pi\)
0.139774 + 0.990183i \(0.455362\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −42.4264 −1.60933
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 45.0333 1.70089 0.850443 0.526068i \(-0.176334\pi\)
0.850443 + 0.526068i \(0.176334\pi\)
\(702\) 0 0
\(703\) −8.48528 −0.320028
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.79796 0.368490
\(708\) 0 0
\(709\) 10.3923 0.390291 0.195146 0.980774i \(-0.437482\pi\)
0.195146 + 0.980774i \(0.437482\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 29.3939 1.09927
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 39.5980 1.47676 0.738378 0.674387i \(-0.235592\pi\)
0.738378 + 0.674387i \(0.235592\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −24.2487 −0.900575
\(726\) 0 0
\(727\) 19.7990 0.734304 0.367152 0.930161i \(-0.380333\pi\)
0.367152 + 0.930161i \(0.380333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −48.9898 −1.81195
\(732\) 0 0
\(733\) 17.3205 0.639748 0.319874 0.947460i \(-0.396359\pi\)
0.319874 + 0.947460i \(0.396359\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.0000 −0.663039
\(738\) 0 0
\(739\) 36.7423 1.35159 0.675795 0.737090i \(-0.263801\pi\)
0.675795 + 0.737090i \(0.263801\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 42.4264 1.55647 0.778237 0.627971i \(-0.216114\pi\)
0.778237 + 0.627971i \(0.216114\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.92820 0.253151
\(750\) 0 0
\(751\) 28.2843 1.03211 0.516054 0.856556i \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 68.5857 2.49609
\(756\) 0 0
\(757\) 45.0333 1.63676 0.818382 0.574675i \(-0.194871\pi\)
0.818382 + 0.574675i \(0.194871\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) −29.3939 −1.06413
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.48528 0.306386
\(768\) 0 0
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 45.0333 1.61974 0.809868 0.586612i \(-0.199539\pi\)
0.809868 + 0.586612i \(0.199539\pi\)
\(774\) 0 0
\(775\) 39.5980 1.42240
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.89898 0.175524
\(780\) 0 0
\(781\) 6.92820 0.247911
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −84.0000 −2.99809
\(786\) 0 0
\(787\) 7.34847 0.261945 0.130972 0.991386i \(-0.458190\pi\)
0.130972 + 0.991386i \(0.458190\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 28.2843 1.00567
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 45.0333 1.59516 0.797581 0.603212i \(-0.206113\pi\)
0.797581 + 0.603212i \(0.206113\pi\)
\(798\) 0 0
\(799\) −45.2548 −1.60100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.5959 0.691525
\(804\) 0 0
\(805\) 27.7128 0.976748
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −46.0000 −1.61727 −0.808637 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 36.7423 1.29020 0.645099 0.764099i \(-0.276816\pi\)
0.645099 + 0.764099i \(0.276816\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −59.3970 −2.08059
\(816\) 0 0
\(817\) 30.0000 1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.46410 0.120898 0.0604490 0.998171i \(-0.480747\pi\)
0.0604490 + 0.998171i \(0.480747\pi\)
\(822\) 0 0
\(823\) 19.7990 0.690149 0.345075 0.938575i \(-0.387854\pi\)
0.345075 + 0.938575i \(0.387854\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.34847 −0.255531 −0.127766 0.991804i \(-0.540781\pi\)
−0.127766 + 0.991804i \(0.540781\pi\)
\(828\) 0 0
\(829\) −17.3205 −0.601566 −0.300783 0.953693i \(-0.597248\pi\)
−0.300783 + 0.953693i \(0.597248\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) −29.3939 −1.01722
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −36.7696 −1.26943 −0.634713 0.772748i \(-0.718882\pi\)
−0.634713 + 0.772748i \(0.718882\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.46410 0.119169
\(846\) 0 0
\(847\) −14.1421 −0.485930
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.79796 0.335870
\(852\) 0 0
\(853\) −17.3205 −0.593043 −0.296521 0.955026i \(-0.595827\pi\)
−0.296521 + 0.955026i \(0.595827\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) −17.1464 −0.585029 −0.292514 0.956261i \(-0.594492\pi\)
−0.292514 + 0.956261i \(0.594492\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −16.9706 −0.577685 −0.288842 0.957377i \(-0.593270\pi\)
−0.288842 + 0.957377i \(0.593270\pi\)
\(864\) 0 0
\(865\) −60.0000 −2.04006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.8564 0.470046
\(870\) 0 0
\(871\) 25.4558 0.862538
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −19.5959 −0.662463
\(876\) 0 0
\(877\) −38.1051 −1.28672 −0.643359 0.765564i \(-0.722460\pi\)
−0.643359 + 0.765564i \(0.722460\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 2.44949 0.0824319 0.0412159 0.999150i \(-0.486877\pi\)
0.0412159 + 0.999150i \(0.486877\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.4558 0.854724 0.427362 0.904081i \(-0.359443\pi\)
0.427362 + 0.904081i \(0.359443\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.7128 0.927374
\(894\) 0 0
\(895\) −42.4264 −1.41816
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −19.5959 −0.653560
\(900\) 0 0
\(901\) 41.5692 1.38487
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 84.0000 2.79225
\(906\) 0 0
\(907\) −7.34847 −0.244002 −0.122001 0.992530i \(-0.538931\pi\)
−0.122001 + 0.992530i \(0.538931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.7846 0.686368
\(918\) 0 0
\(919\) 36.7696 1.21292 0.606458 0.795116i \(-0.292590\pi\)
0.606458 + 0.795116i \(0.292590\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −9.79796 −0.322504
\(924\) 0 0
\(925\) −24.2487 −0.797293
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 0 0
\(931\) 2.44949 0.0802788
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 33.9411 1.10999
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 38.1051 1.24219 0.621096 0.783735i \(-0.286688\pi\)
0.621096 + 0.783735i \(0.286688\pi\)
\(942\) 0 0
\(943\) −5.65685 −0.184213
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41.6413 −1.35316 −0.676581 0.736369i \(-0.736539\pi\)
−0.676581 + 0.736369i \(0.736539\pi\)
\(948\) 0 0
\(949\) −27.7128 −0.899596
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −58.7878 −1.90233
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5.65685 −0.182669
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −41.5692 −1.33816
\(966\) 0 0
\(967\) 14.1421 0.454780 0.227390 0.973804i \(-0.426981\pi\)
0.227390 + 0.973804i \(0.426981\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −51.4393 −1.65077 −0.825383 0.564574i \(-0.809041\pi\)
−0.825383 + 0.564574i \(0.809041\pi\)
\(972\) 0 0
\(973\) 34.6410 1.11054
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 0 0
\(979\) 19.5959 0.626288
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −42.4264 −1.35319 −0.676596 0.736354i \(-0.736546\pi\)
−0.676596 + 0.736354i \(0.736546\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −34.6410 −1.10152
\(990\) 0 0
\(991\) 45.2548 1.43757 0.718784 0.695234i \(-0.244699\pi\)
0.718784 + 0.695234i \(0.244699\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.79796 −0.310616
\(996\) 0 0
\(997\) 10.3923 0.329128 0.164564 0.986366i \(-0.447378\pi\)
0.164564 + 0.986366i \(0.447378\pi\)
\(998\) 0 0
\(999\) 0 0
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 4608.2.a.w.1.2 4
3.2 odd 2 512.2.a.g.1.2 yes 4
4.3 odd 2 inner 4608.2.a.w.1.1 4
8.3 odd 2 inner 4608.2.a.w.1.3 4
8.5 even 2 inner 4608.2.a.w.1.4 4
12.11 even 2 512.2.a.g.1.4 yes 4
16.3 odd 4 4608.2.d.d.2305.4 4
16.5 even 4 4608.2.d.d.2305.1 4
16.11 odd 4 4608.2.d.d.2305.2 4
16.13 even 4 4608.2.d.d.2305.3 4
24.5 odd 2 512.2.a.g.1.3 yes 4
24.11 even 2 512.2.a.g.1.1 4
48.5 odd 4 512.2.b.d.257.4 4
48.11 even 4 512.2.b.d.257.2 4
48.29 odd 4 512.2.b.d.257.1 4
48.35 even 4 512.2.b.d.257.3 4
96.5 odd 8 1024.2.e.p.769.4 8
96.11 even 8 1024.2.e.p.769.3 8
96.29 odd 8 1024.2.e.p.257.1 8
96.35 even 8 1024.2.e.p.257.3 8
96.53 odd 8 1024.2.e.p.769.1 8
96.59 even 8 1024.2.e.p.769.2 8
96.77 odd 8 1024.2.e.p.257.4 8
96.83 even 8 1024.2.e.p.257.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.g.1.1 4 24.11 even 2
512.2.a.g.1.2 yes 4 3.2 odd 2
512.2.a.g.1.3 yes 4 24.5 odd 2
512.2.a.g.1.4 yes 4 12.11 even 2
512.2.b.d.257.1 4 48.29 odd 4
512.2.b.d.257.2 4 48.11 even 4
512.2.b.d.257.3 4 48.35 even 4
512.2.b.d.257.4 4 48.5 odd 4
1024.2.e.p.257.1 8 96.29 odd 8
1024.2.e.p.257.2 8 96.83 even 8
1024.2.e.p.257.3 8 96.35 even 8
1024.2.e.p.257.4 8 96.77 odd 8
1024.2.e.p.769.1 8 96.53 odd 8
1024.2.e.p.769.2 8 96.59 even 8
1024.2.e.p.769.3 8 96.11 even 8
1024.2.e.p.769.4 8 96.5 odd 8
4608.2.a.w.1.1 4 4.3 odd 2 inner
4608.2.a.w.1.2 4 1.1 even 1 trivial
4608.2.a.w.1.3 4 8.3 odd 2 inner
4608.2.a.w.1.4 4 8.5 even 2 inner
4608.2.d.d.2305.1 4 16.5 even 4
4608.2.d.d.2305.2 4 16.11 odd 4
4608.2.d.d.2305.3 4 16.13 even 4
4608.2.d.d.2305.4 4 16.3 odd 4