Properties

Label 8624.2.a.bs.1.2
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{3} -4.24264 q^{5} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} -4.24264 q^{5} -1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{15} +5.65685 q^{17} -6.00000 q^{23} +13.0000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -1.41421 q^{31} +1.41421 q^{33} -10.0000 q^{37} +11.3137 q^{41} +8.00000 q^{43} +4.24264 q^{45} +4.24264 q^{47} +8.00000 q^{51} +8.00000 q^{53} -4.24264 q^{55} +1.41421 q^{59} +2.82843 q^{61} -2.00000 q^{67} -8.48528 q^{69} +2.00000 q^{71} -8.48528 q^{73} +18.3848 q^{75} -16.0000 q^{79} -5.00000 q^{81} -16.9706 q^{83} -24.0000 q^{85} +2.82843 q^{87} +7.07107 q^{89} -2.00000 q^{93} +9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 2 q^{11} - 12 q^{15} - 12 q^{23} + 26 q^{25} + 4 q^{29} - 20 q^{37} + 16 q^{43} + 16 q^{51} + 16 q^{53} - 4 q^{67} + 4 q^{71} - 32 q^{79} - 10 q^{81} - 48 q^{85} - 4 q^{93} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) −4.24264 −1.89737 −0.948683 0.316228i \(-0.897584\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −6.00000 −1.54919
\(16\) 0 0
\(17\) 5.65685 1.37199 0.685994 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 13.0000 2.60000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.41421 −0.254000 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) 0 0
\(33\) 1.41421 0.246183
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.3137 1.76690 0.883452 0.468521i \(-0.155213\pi\)
0.883452 + 0.468521i \(0.155213\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 4.24264 0.632456
\(46\) 0 0
\(47\) 4.24264 0.618853 0.309426 0.950923i \(-0.399863\pi\)
0.309426 + 0.950923i \(0.399863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) −4.24264 −0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) −8.48528 −1.02151
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 0 0
\(75\) 18.3848 2.12289
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −16.9706 −1.86276 −0.931381 0.364047i \(-0.881395\pi\)
−0.931381 + 0.364047i \(0.881395\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) 0 0
\(87\) 2.82843 0.303239
\(88\) 0 0
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −5.65685 −0.562878 −0.281439 0.959579i \(-0.590812\pi\)
−0.281439 + 0.959579i \(0.590812\pi\)
\(102\) 0 0
\(103\) 18.3848 1.81151 0.905753 0.423806i \(-0.139306\pi\)
0.905753 + 0.423806i \(0.139306\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −14.1421 −1.34231
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 25.4558 2.37377
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 16.0000 1.44267
\(124\) 0 0
\(125\) −33.9411 −3.03579
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 11.3137 0.996116
\(130\) 0 0
\(131\) −19.7990 −1.72985 −0.864923 0.501905i \(-0.832633\pi\)
−0.864923 + 0.501905i \(0.832633\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 24.0000 2.06559
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 11.3137 0.959616 0.479808 0.877373i \(-0.340706\pi\)
0.479808 + 0.877373i \(0.340706\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.48528 −0.704664
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −5.65685 −0.457330
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −4.24264 −0.338600 −0.169300 0.985565i \(-0.554151\pi\)
−0.169300 + 0.985565i \(0.554151\pi\)
\(158\) 0 0
\(159\) 11.3137 0.897235
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 5.65685 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.3137 0.860165 0.430083 0.902790i \(-0.358484\pi\)
0.430083 + 0.902790i \(0.358484\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 7.07107 0.525588 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) 42.4264 3.11925
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) −1.41421 −0.100251 −0.0501255 0.998743i \(-0.515962\pi\)
−0.0501255 + 0.998743i \(0.515962\pi\)
\(200\) 0 0
\(201\) −2.82843 −0.199502
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −48.0000 −3.35247
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 2.82843 0.193801
\(214\) 0 0
\(215\) −33.9411 −2.31477
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −21.2132 −1.42054 −0.710271 0.703929i \(-0.751427\pi\)
−0.710271 + 0.703929i \(0.751427\pi\)
\(224\) 0 0
\(225\) −13.0000 −0.866667
\(226\) 0 0
\(227\) −14.1421 −0.938647 −0.469323 0.883026i \(-0.655502\pi\)
−0.469323 + 0.883026i \(0.655502\pi\)
\(228\) 0 0
\(229\) −9.89949 −0.654177 −0.327089 0.944994i \(-0.606068\pi\)
−0.327089 + 0.944994i \(0.606068\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) −22.6274 −1.46981
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 18.3848 1.16044 0.580218 0.814461i \(-0.302967\pi\)
0.580218 + 0.814461i \(0.302967\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) −33.9411 −2.12548
\(256\) 0 0
\(257\) −1.41421 −0.0882162 −0.0441081 0.999027i \(-0.514045\pi\)
−0.0441081 + 0.999027i \(0.514045\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −33.9411 −2.08499
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) −18.3848 −1.12094 −0.560470 0.828175i \(-0.689379\pi\)
−0.560470 + 0.828175i \(0.689379\pi\)
\(270\) 0 0
\(271\) −8.48528 −0.515444 −0.257722 0.966219i \(-0.582972\pi\)
−0.257722 + 0.966219i \(0.582972\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.0000 0.783929
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 1.41421 0.0846668
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) −19.7990 −1.17693 −0.588464 0.808523i \(-0.700267\pi\)
−0.588464 + 0.808523i \(0.700267\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) −5.65685 −0.328244
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −25.4558 −1.45284 −0.726421 0.687250i \(-0.758818\pi\)
−0.726421 + 0.687250i \(0.758818\pi\)
\(308\) 0 0
\(309\) 26.0000 1.47909
\(310\) 0 0
\(311\) 18.3848 1.04251 0.521253 0.853402i \(-0.325465\pi\)
0.521253 + 0.853402i \(0.325465\pi\)
\(312\) 0 0
\(313\) 12.7279 0.719425 0.359712 0.933063i \(-0.382875\pi\)
0.359712 + 0.933063i \(0.382875\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −22.6274 −1.26294
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.82843 −0.156412
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 8.48528 0.463600
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −2.82843 −0.153619
\(340\) 0 0
\(341\) −1.41421 −0.0765840
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 36.0000 1.93817
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −14.1421 −0.757011 −0.378506 0.925599i \(-0.623562\pi\)
−0.378506 + 0.925599i \(0.623562\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.41421 −0.0752710 −0.0376355 0.999292i \(-0.511983\pi\)
−0.0376355 + 0.999292i \(0.511983\pi\)
\(354\) 0 0
\(355\) −8.48528 −0.450352
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 1.41421 0.0742270
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) 0 0
\(367\) −21.2132 −1.10732 −0.553660 0.832743i \(-0.686769\pi\)
−0.553660 + 0.832743i \(0.686769\pi\)
\(368\) 0 0
\(369\) −11.3137 −0.588968
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −48.0000 −2.47871
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) −22.6274 −1.15924
\(382\) 0 0
\(383\) 15.5563 0.794892 0.397446 0.917625i \(-0.369897\pi\)
0.397446 + 0.917625i \(0.369897\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) −33.9411 −1.71648
\(392\) 0 0
\(393\) −28.0000 −1.41241
\(394\) 0 0
\(395\) 67.8823 3.41553
\(396\) 0 0
\(397\) −12.7279 −0.638796 −0.319398 0.947621i \(-0.603481\pi\)
−0.319398 + 0.947621i \(0.603481\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 21.2132 1.05409
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 2.82843 0.139857 0.0699284 0.997552i \(-0.477723\pi\)
0.0699284 + 0.997552i \(0.477723\pi\)
\(410\) 0 0
\(411\) −25.4558 −1.25564
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 72.0000 3.53434
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 24.0416 1.17451 0.587255 0.809402i \(-0.300208\pi\)
0.587255 + 0.809402i \(0.300208\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) −4.24264 −0.206284
\(424\) 0 0
\(425\) 73.5391 3.56717
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 12.7279 0.611665 0.305832 0.952085i \(-0.401065\pi\)
0.305832 + 0.952085i \(0.401065\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −25.4558 −1.21494 −0.607471 0.794342i \(-0.707816\pi\)
−0.607471 + 0.794342i \(0.707816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) −30.0000 −1.42214
\(446\) 0 0
\(447\) −14.1421 −0.668900
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 11.3137 0.532742
\(452\) 0 0
\(453\) −5.65685 −0.265782
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) −32.0000 −1.49363
\(460\) 0 0
\(461\) −2.82843 −0.131733 −0.0658665 0.997828i \(-0.520981\pi\)
−0.0658665 + 0.997828i \(0.520981\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 0 0
\(465\) 8.48528 0.393496
\(466\) 0 0
\(467\) −4.24264 −0.196326 −0.0981630 0.995170i \(-0.531297\pi\)
−0.0981630 + 0.995170i \(0.531297\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.00000 −0.366295
\(478\) 0 0
\(479\) 2.82843 0.129234 0.0646171 0.997910i \(-0.479417\pi\)
0.0646171 + 0.997910i \(0.479417\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −42.0000 −1.90712
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 14.1421 0.639529
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 11.3137 0.509544
\(494\) 0 0
\(495\) 4.24264 0.190693
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) 31.1127 1.38725 0.693623 0.720338i \(-0.256013\pi\)
0.693623 + 0.720338i \(0.256013\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) −18.3848 −0.816497
\(508\) 0 0
\(509\) −12.7279 −0.564155 −0.282078 0.959392i \(-0.591024\pi\)
−0.282078 + 0.959392i \(0.591024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −78.0000 −3.43709
\(516\) 0 0
\(517\) 4.24264 0.186591
\(518\) 0 0
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) −1.41421 −0.0619578 −0.0309789 0.999520i \(-0.509862\pi\)
−0.0309789 + 0.999520i \(0.509862\pi\)
\(522\) 0 0
\(523\) 8.48528 0.371035 0.185518 0.982641i \(-0.440604\pi\)
0.185518 + 0.982641i \(0.440604\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −1.41421 −0.0613716
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 67.8823 2.93481
\(536\) 0 0
\(537\) −16.9706 −0.732334
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 8.48528 0.363470
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 0 0
\(549\) −2.82843 −0.120714
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 60.0000 2.54686
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 22.6274 0.953632 0.476816 0.879003i \(-0.341791\pi\)
0.476816 + 0.879003i \(0.341791\pi\)
\(564\) 0 0
\(565\) 8.48528 0.356978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) −22.6274 −0.945274
\(574\) 0 0
\(575\) −78.0000 −3.25282
\(576\) 0 0
\(577\) 7.07107 0.294372 0.147186 0.989109i \(-0.452978\pi\)
0.147186 + 0.989109i \(0.452978\pi\)
\(578\) 0 0
\(579\) −8.48528 −0.352636
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −26.8701 −1.10905 −0.554523 0.832168i \(-0.687099\pi\)
−0.554523 + 0.832168i \(0.687099\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 31.1127 1.27981
\(592\) 0 0
\(593\) −42.4264 −1.74224 −0.871122 0.491067i \(-0.836607\pi\)
−0.871122 + 0.491067i \(0.836607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.00000 −0.0818546
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −5.65685 −0.230748 −0.115374 0.993322i \(-0.536807\pi\)
−0.115374 + 0.993322i \(0.536807\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −4.24264 −0.172488
\(606\) 0 0
\(607\) 16.9706 0.688814 0.344407 0.938820i \(-0.388080\pi\)
0.344407 + 0.938820i \(0.388080\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) −67.8823 −2.73728
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 9.89949 0.397894 0.198947 0.980010i \(-0.436248\pi\)
0.198947 + 0.980010i \(0.436248\pi\)
\(620\) 0 0
\(621\) 33.9411 1.36201
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 79.0000 3.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −56.5685 −2.25554
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) 0 0
\(633\) −11.3137 −0.449680
\(634\) 0 0
\(635\) 67.8823 2.69382
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) −38.1838 −1.50582 −0.752910 0.658123i \(-0.771351\pi\)
−0.752910 + 0.658123i \(0.771351\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) −15.5563 −0.611583 −0.305792 0.952098i \(-0.598921\pi\)
−0.305792 + 0.952098i \(0.598921\pi\)
\(648\) 0 0
\(649\) 1.41421 0.0555127
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 0 0
\(655\) 84.0000 3.28215
\(656\) 0 0
\(657\) 8.48528 0.331042
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −12.7279 −0.495059 −0.247529 0.968880i \(-0.579619\pi\)
−0.247529 + 0.968880i \(0.579619\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) −30.0000 −1.15987
\(670\) 0 0
\(671\) 2.82843 0.109190
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) −73.5391 −2.83052
\(676\) 0 0
\(677\) 39.5980 1.52187 0.760937 0.648826i \(-0.224740\pi\)
0.760937 + 0.648826i \(0.224740\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 0 0
\(685\) 76.3675 2.91785
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 15.5563 0.591791 0.295896 0.955220i \(-0.404382\pi\)
0.295896 + 0.955220i \(0.404382\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −48.0000 −1.82074
\(696\) 0 0
\(697\) 64.0000 2.42417
\(698\) 0 0
\(699\) 19.7990 0.748867
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −25.4558 −0.958723
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) 8.48528 0.317776
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.9706 0.633777
\(718\) 0 0
\(719\) 18.3848 0.685636 0.342818 0.939402i \(-0.388619\pi\)
0.342818 + 0.939402i \(0.388619\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 26.0000 0.965616
\(726\) 0 0
\(727\) 12.7279 0.472052 0.236026 0.971747i \(-0.424155\pi\)
0.236026 + 0.971747i \(0.424155\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 45.2548 1.67381
\(732\) 0 0
\(733\) −14.1421 −0.522352 −0.261176 0.965291i \(-0.584110\pi\)
−0.261176 + 0.965291i \(0.584110\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.0000 1.61420 0.807102 0.590412i \(-0.201035\pi\)
0.807102 + 0.590412i \(0.201035\pi\)
\(744\) 0 0
\(745\) 42.4264 1.55438
\(746\) 0 0
\(747\) 16.9706 0.620920
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 0 0
\(753\) 26.0000 0.947493
\(754\) 0 0
\(755\) 16.9706 0.617622
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) −8.48528 −0.307996
\(760\) 0 0
\(761\) 42.4264 1.53796 0.768978 0.639275i \(-0.220766\pi\)
0.768978 + 0.639275i \(0.220766\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 24.0000 0.867722
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −19.7990 −0.713970 −0.356985 0.934110i \(-0.616195\pi\)
−0.356985 + 0.934110i \(0.616195\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) −1.41421 −0.0508657 −0.0254329 0.999677i \(-0.508096\pi\)
−0.0254329 + 0.999677i \(0.508096\pi\)
\(774\) 0 0
\(775\) −18.3848 −0.660401
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 0 0
\(783\) −11.3137 −0.404319
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −5.65685 −0.201645 −0.100823 0.994904i \(-0.532147\pi\)
−0.100823 + 0.994904i \(0.532147\pi\)
\(788\) 0 0
\(789\) 11.3137 0.402779
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −48.0000 −1.70238
\(796\) 0 0
\(797\) −7.07107 −0.250470 −0.125235 0.992127i \(-0.539968\pi\)
−0.125235 + 0.992127i \(0.539968\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −7.07107 −0.249844
\(802\) 0 0
\(803\) −8.48528 −0.299439
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −26.0000 −0.915243
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) 36.7696 1.29115 0.645577 0.763695i \(-0.276617\pi\)
0.645577 + 0.763695i \(0.276617\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) −42.4264 −1.48613
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 18.3848 0.640076
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) 4.24264 0.147353 0.0736765 0.997282i \(-0.476527\pi\)
0.0736765 + 0.997282i \(0.476527\pi\)
\(830\) 0 0
\(831\) −2.82843 −0.0981170
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 1.41421 0.0488241 0.0244120 0.999702i \(-0.492229\pi\)
0.0244120 + 0.999702i \(0.492229\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −19.7990 −0.681913
\(844\) 0 0
\(845\) 55.1543 1.89737
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) −28.2843 −0.968435 −0.484218 0.874948i \(-0.660896\pi\)
−0.484218 + 0.874948i \(0.660896\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.4558 0.869555 0.434778 0.900538i \(-0.356827\pi\)
0.434778 + 0.900538i \(0.356827\pi\)
\(858\) 0 0
\(859\) 12.7279 0.434271 0.217136 0.976141i \(-0.430329\pi\)
0.217136 + 0.976141i \(0.430329\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 0 0
\(867\) 21.2132 0.720438
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −9.89949 −0.335047
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 29.6985 1.00057 0.500284 0.865862i \(-0.333229\pi\)
0.500284 + 0.865862i \(0.333229\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) −8.48528 −0.285230
\(886\) 0 0
\(887\) 36.7696 1.23460 0.617300 0.786728i \(-0.288226\pi\)
0.617300 + 0.786728i \(0.288226\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 50.9117 1.70179
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.82843 −0.0943333
\(900\) 0 0
\(901\) 45.2548 1.50766
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.0000 −0.997234
\(906\) 0 0
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 0 0
\(909\) 5.65685 0.187626
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) −16.9706 −0.561644
\(914\) 0 0
\(915\) −16.9706 −0.561029
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −130.000 −4.27437
\(926\) 0 0
\(927\) −18.3848 −0.603835
\(928\) 0 0
\(929\) 32.5269 1.06717 0.533587 0.845745i \(-0.320844\pi\)
0.533587 + 0.845745i \(0.320844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 26.0000 0.851202
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −25.4558 −0.831606 −0.415803 0.909455i \(-0.636499\pi\)
−0.415803 + 0.909455i \(0.636499\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −31.1127 −1.01424 −0.507122 0.861874i \(-0.669291\pi\)
−0.507122 + 0.861874i \(0.669291\pi\)
\(942\) 0 0
\(943\) −67.8823 −2.21055
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 42.4264 1.37577
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) 67.8823 2.19662
\(956\) 0 0
\(957\) 2.82843 0.0914301
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) 25.4558 0.819453
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 52.3259 1.67922 0.839609 0.543191i \(-0.182784\pi\)
0.839609 + 0.543191i \(0.182784\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.0000 1.66363 0.831814 0.555055i \(-0.187303\pi\)
0.831814 + 0.555055i \(0.187303\pi\)
\(978\) 0 0
\(979\) 7.07107 0.225992
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 1.41421 0.0451064 0.0225532 0.999746i \(-0.492820\pi\)
0.0225532 + 0.999746i \(0.492820\pi\)
\(984\) 0 0
\(985\) −93.3381 −2.97400
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 46.0000 1.46124 0.730619 0.682785i \(-0.239232\pi\)
0.730619 + 0.682785i \(0.239232\pi\)
\(992\) 0 0
\(993\) −28.2843 −0.897574
\(994\) 0 0
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) −16.9706 −0.537463 −0.268732 0.963215i \(-0.586604\pi\)
−0.268732 + 0.963215i \(0.586604\pi\)
\(998\) 0 0
\(999\) 56.5685 1.78975
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 8624.2.a.bs.1.2 2
4.3 odd 2 1078.2.a.u.1.1 2
7.6 odd 2 inner 8624.2.a.bs.1.1 2
12.11 even 2 9702.2.a.cp.1.2 2
28.3 even 6 1078.2.e.p.177.1 4
28.11 odd 6 1078.2.e.p.177.2 4
28.19 even 6 1078.2.e.p.67.1 4
28.23 odd 6 1078.2.e.p.67.2 4
28.27 even 2 1078.2.a.u.1.2 yes 2
84.83 odd 2 9702.2.a.cp.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.u.1.1 2 4.3 odd 2
1078.2.a.u.1.2 yes 2 28.27 even 2
1078.2.e.p.67.1 4 28.19 even 6
1078.2.e.p.67.2 4 28.23 odd 6
1078.2.e.p.177.1 4 28.3 even 6
1078.2.e.p.177.2 4 28.11 odd 6
8624.2.a.bs.1.1 2 7.6 odd 2 inner
8624.2.a.bs.1.2 2 1.1 even 1 trivial
9702.2.a.cp.1.1 2 84.83 odd 2
9702.2.a.cp.1.2 2 12.11 even 2