Properties

Label 1296.5.e.g.161.3
Level $1296$
Weight $5$
Character 1296.161
Analytic conductor $133.967$
Analytic rank $0$
Dimension $8$
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] = [1296,5,Mod(161,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.161");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(133.967472157\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{20} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.3
Root \(4.23522 + 4.06612i\) of defining polynomial
Character \(\chi\) \(=\) 1296.161
Dual form 1296.5.e.g.161.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-12.2819i q^{5} +14.2840 q^{7} +O(q^{10})\) \(q-12.2819i q^{5} +14.2840 q^{7} -104.171i q^{11} +75.2346 q^{13} -341.998i q^{17} +706.329 q^{19} -596.312i q^{23} +474.155 q^{25} +1302.39i q^{29} -1029.02 q^{31} -175.435i q^{35} +563.132 q^{37} -99.1448i q^{41} +896.514 q^{43} -430.570i q^{47} -2196.97 q^{49} +5271.47i q^{53} -1279.41 q^{55} -5639.26i q^{59} +1131.25 q^{61} -924.023i q^{65} +1352.82 q^{67} -5681.42i q^{71} +4236.54 q^{73} -1487.98i q^{77} +6135.42 q^{79} -7509.88i q^{83} -4200.38 q^{85} +8721.70i q^{89} +1074.65 q^{91} -8675.05i q^{95} +5441.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 12.2819i − 0.491275i −0.969362 0.245638i \(-0.921003\pi\)
0.969362 0.245638i \(-0.0789974\pi\)
\(6\) 0 0
\(7\) 14.2840 0.291511 0.145756 0.989321i \(-0.453439\pi\)
0.145756 + 0.989321i \(0.453439\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 104.171i − 0.860916i −0.902611 0.430458i \(-0.858352\pi\)
0.902611 0.430458i \(-0.141648\pi\)
\(12\) 0 0
\(13\) 75.2346 0.445175 0.222588 0.974913i \(-0.428550\pi\)
0.222588 + 0.974913i \(0.428550\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 341.998i − 1.18338i −0.806164 0.591692i \(-0.798460\pi\)
0.806164 0.591692i \(-0.201540\pi\)
\(18\) 0 0
\(19\) 706.329 1.95659 0.978295 0.207218i \(-0.0664411\pi\)
0.978295 + 0.207218i \(0.0664411\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 596.312i − 1.12724i −0.826033 0.563622i \(-0.809407\pi\)
0.826033 0.563622i \(-0.190593\pi\)
\(24\) 0 0
\(25\) 474.155 0.758648
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1302.39i 1.54862i 0.632807 + 0.774309i \(0.281903\pi\)
−0.632807 + 0.774309i \(0.718097\pi\)
\(30\) 0 0
\(31\) −1029.02 −1.07078 −0.535391 0.844605i \(-0.679836\pi\)
−0.535391 + 0.844605i \(0.679836\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 175.435i − 0.143212i
\(36\) 0 0
\(37\) 563.132 0.411346 0.205673 0.978621i \(-0.434062\pi\)
0.205673 + 0.978621i \(0.434062\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 99.1448i − 0.0589797i −0.999565 0.0294898i \(-0.990612\pi\)
0.999565 0.0294898i \(-0.00938826\pi\)
\(42\) 0 0
\(43\) 896.514 0.484864 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 430.570i − 0.194916i −0.995240 0.0974581i \(-0.968929\pi\)
0.995240 0.0974581i \(-0.0310712\pi\)
\(48\) 0 0
\(49\) −2196.97 −0.915021
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5271.47i 1.87664i 0.345773 + 0.938318i \(0.387617\pi\)
−0.345773 + 0.938318i \(0.612383\pi\)
\(54\) 0 0
\(55\) −1279.41 −0.422947
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5639.26i − 1.62001i −0.586423 0.810005i \(-0.699464\pi\)
0.586423 0.810005i \(-0.300536\pi\)
\(60\) 0 0
\(61\) 1131.25 0.304018 0.152009 0.988379i \(-0.451426\pi\)
0.152009 + 0.988379i \(0.451426\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 924.023i − 0.218704i
\(66\) 0 0
\(67\) 1352.82 0.301364 0.150682 0.988582i \(-0.451853\pi\)
0.150682 + 0.988582i \(0.451853\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5681.42i − 1.12704i −0.826101 0.563522i \(-0.809446\pi\)
0.826101 0.563522i \(-0.190554\pi\)
\(72\) 0 0
\(73\) 4236.54 0.794996 0.397498 0.917603i \(-0.369878\pi\)
0.397498 + 0.917603i \(0.369878\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1487.98i − 0.250967i
\(78\) 0 0
\(79\) 6135.42 0.983083 0.491542 0.870854i \(-0.336434\pi\)
0.491542 + 0.870854i \(0.336434\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7509.88i − 1.09013i −0.838395 0.545063i \(-0.816506\pi\)
0.838395 0.545063i \(-0.183494\pi\)
\(84\) 0 0
\(85\) −4200.38 −0.581367
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8721.70i 1.10109i 0.834807 + 0.550543i \(0.185579\pi\)
−0.834807 + 0.550543i \(0.814421\pi\)
\(90\) 0 0
\(91\) 1074.65 0.129774
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 8675.05i − 0.961224i
\(96\) 0 0
\(97\) 5441.31 0.578309 0.289154 0.957282i \(-0.406626\pi\)
0.289154 + 0.957282i \(0.406626\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7482.63i 0.733519i 0.930316 + 0.366759i \(0.119533\pi\)
−0.930316 + 0.366759i \(0.880467\pi\)
\(102\) 0 0
\(103\) −13576.5 −1.27972 −0.639860 0.768492i \(-0.721008\pi\)
−0.639860 + 0.768492i \(0.721008\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16741.7i 1.46229i 0.682224 + 0.731143i \(0.261013\pi\)
−0.682224 + 0.731143i \(0.738987\pi\)
\(108\) 0 0
\(109\) −12068.7 −1.01579 −0.507897 0.861418i \(-0.669577\pi\)
−0.507897 + 0.861418i \(0.669577\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 704.601i − 0.0551806i −0.999619 0.0275903i \(-0.991217\pi\)
0.999619 0.0275903i \(-0.00878337\pi\)
\(114\) 0 0
\(115\) −7323.84 −0.553787
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 4885.11i − 0.344970i
\(120\) 0 0
\(121\) 3789.45 0.258824
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 13499.7i − 0.863981i
\(126\) 0 0
\(127\) 16050.0 0.995105 0.497552 0.867434i \(-0.334232\pi\)
0.497552 + 0.867434i \(0.334232\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 15504.3i − 0.903463i −0.892154 0.451732i \(-0.850806\pi\)
0.892154 0.451732i \(-0.149194\pi\)
\(132\) 0 0
\(133\) 10089.2 0.570368
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 890.615i − 0.0474514i −0.999719 0.0237257i \(-0.992447\pi\)
0.999719 0.0237257i \(-0.00755283\pi\)
\(138\) 0 0
\(139\) −9358.18 −0.484353 −0.242176 0.970232i \(-0.577861\pi\)
−0.242176 + 0.970232i \(0.577861\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 7837.25i − 0.383258i
\(144\) 0 0
\(145\) 15995.8 0.760798
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 40669.4i − 1.83187i −0.401323 0.915937i \(-0.631450\pi\)
0.401323 0.915937i \(-0.368550\pi\)
\(150\) 0 0
\(151\) 24974.5 1.09532 0.547661 0.836700i \(-0.315518\pi\)
0.547661 + 0.836700i \(0.315518\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12638.3i 0.526049i
\(156\) 0 0
\(157\) −32918.4 −1.33549 −0.667743 0.744392i \(-0.732739\pi\)
−0.667743 + 0.744392i \(0.732739\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 8517.75i − 0.328604i
\(162\) 0 0
\(163\) −13796.3 −0.519262 −0.259631 0.965708i \(-0.583601\pi\)
−0.259631 + 0.965708i \(0.583601\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11332.4i 0.406339i 0.979144 + 0.203170i \(0.0651243\pi\)
−0.979144 + 0.203170i \(0.934876\pi\)
\(168\) 0 0
\(169\) −22900.8 −0.801819
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 39170.4i − 1.30878i −0.756158 0.654389i \(-0.772926\pi\)
0.756158 0.654389i \(-0.227074\pi\)
\(174\) 0 0
\(175\) 6772.86 0.221155
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 24097.3i − 0.752078i −0.926604 0.376039i \(-0.877286\pi\)
0.926604 0.376039i \(-0.122714\pi\)
\(180\) 0 0
\(181\) 10277.1 0.313699 0.156850 0.987623i \(-0.449866\pi\)
0.156850 + 0.987623i \(0.449866\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 6916.33i − 0.202084i
\(186\) 0 0
\(187\) −35626.2 −1.01879
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 35030.4i 0.960237i 0.877204 + 0.480119i \(0.159406\pi\)
−0.877204 + 0.480119i \(0.840594\pi\)
\(192\) 0 0
\(193\) 5240.55 0.140690 0.0703448 0.997523i \(-0.477590\pi\)
0.0703448 + 0.997523i \(0.477590\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 42421.5i − 1.09308i −0.837431 0.546542i \(-0.815944\pi\)
0.837431 0.546542i \(-0.184056\pi\)
\(198\) 0 0
\(199\) −31270.0 −0.789627 −0.394814 0.918761i \(-0.629191\pi\)
−0.394814 + 0.918761i \(0.629191\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18603.4i 0.451440i
\(204\) 0 0
\(205\) −1217.69 −0.0289753
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 73578.8i − 1.68446i
\(210\) 0 0
\(211\) 31124.4 0.699096 0.349548 0.936918i \(-0.386335\pi\)
0.349548 + 0.936918i \(0.386335\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 11010.9i − 0.238202i
\(216\) 0 0
\(217\) −14698.6 −0.312145
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 25730.1i − 0.526813i
\(222\) 0 0
\(223\) −16027.9 −0.322305 −0.161153 0.986930i \(-0.551521\pi\)
−0.161153 + 0.986930i \(0.551521\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 67528.9i − 1.31050i −0.755411 0.655251i \(-0.772563\pi\)
0.755411 0.655251i \(-0.227437\pi\)
\(228\) 0 0
\(229\) 78065.8 1.48864 0.744321 0.667823i \(-0.232773\pi\)
0.744321 + 0.667823i \(0.232773\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 61204.9i − 1.12739i −0.825983 0.563696i \(-0.809379\pi\)
0.825983 0.563696i \(-0.190621\pi\)
\(234\) 0 0
\(235\) −5288.21 −0.0957576
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 43154.9i 0.755500i 0.925908 + 0.377750i \(0.123302\pi\)
−0.925908 + 0.377750i \(0.876698\pi\)
\(240\) 0 0
\(241\) −56207.6 −0.967745 −0.483873 0.875138i \(-0.660770\pi\)
−0.483873 + 0.875138i \(0.660770\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 26982.9i 0.449527i
\(246\) 0 0
\(247\) 53140.4 0.871025
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 15739.4i − 0.249828i −0.992168 0.124914i \(-0.960135\pi\)
0.992168 0.124914i \(-0.0398655\pi\)
\(252\) 0 0
\(253\) −62118.3 −0.970462
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 58760.1i 0.889643i 0.895619 + 0.444822i \(0.146733\pi\)
−0.895619 + 0.444822i \(0.853267\pi\)
\(258\) 0 0
\(259\) 8043.81 0.119912
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 94338.8i − 1.36389i −0.731404 0.681944i \(-0.761135\pi\)
0.731404 0.681944i \(-0.238865\pi\)
\(264\) 0 0
\(265\) 64743.6 0.921945
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 79782.6i − 1.10256i −0.834319 0.551282i \(-0.814139\pi\)
0.834319 0.551282i \(-0.185861\pi\)
\(270\) 0 0
\(271\) 76677.1 1.04406 0.522032 0.852926i \(-0.325174\pi\)
0.522032 + 0.852926i \(0.325174\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 49393.1i − 0.653132i
\(276\) 0 0
\(277\) 113156. 1.47475 0.737374 0.675485i \(-0.236066\pi\)
0.737374 + 0.675485i \(0.236066\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 96549.7i − 1.22275i −0.791341 0.611376i \(-0.790616\pi\)
0.791341 0.611376i \(-0.209384\pi\)
\(282\) 0 0
\(283\) −30537.1 −0.381289 −0.190645 0.981659i \(-0.561058\pi\)
−0.190645 + 0.981659i \(0.561058\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1416.19i − 0.0171932i
\(288\) 0 0
\(289\) −33441.6 −0.400397
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 4054.67i − 0.0472303i −0.999721 0.0236151i \(-0.992482\pi\)
0.999721 0.0236151i \(-0.00751763\pi\)
\(294\) 0 0
\(295\) −69260.7 −0.795872
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 44863.3i − 0.501821i
\(300\) 0 0
\(301\) 12805.8 0.141343
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 13893.9i − 0.149357i
\(306\) 0 0
\(307\) −44297.5 −0.470005 −0.235002 0.971995i \(-0.575510\pi\)
−0.235002 + 0.971995i \(0.575510\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 95089.7i − 0.983134i −0.870840 0.491567i \(-0.836424\pi\)
0.870840 0.491567i \(-0.163576\pi\)
\(312\) 0 0
\(313\) −170572. −1.74108 −0.870541 0.492097i \(-0.836231\pi\)
−0.870541 + 0.492097i \(0.836231\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 121998.i 1.21405i 0.794684 + 0.607024i \(0.207637\pi\)
−0.794684 + 0.607024i \(0.792363\pi\)
\(318\) 0 0
\(319\) 135671. 1.33323
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 241563.i − 2.31540i
\(324\) 0 0
\(325\) 35672.9 0.337731
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 6150.28i − 0.0568203i
\(330\) 0 0
\(331\) 95214.2 0.869052 0.434526 0.900659i \(-0.356916\pi\)
0.434526 + 0.900659i \(0.356916\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 16615.2i − 0.148053i
\(336\) 0 0
\(337\) 8409.77 0.0740499 0.0370249 0.999314i \(-0.488212\pi\)
0.0370249 + 0.999314i \(0.488212\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 107194.i 0.921852i
\(342\) 0 0
\(343\) −65677.6 −0.558250
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6032.75i 0.0501022i 0.999686 + 0.0250511i \(0.00797484\pi\)
−0.999686 + 0.0250511i \(0.992025\pi\)
\(348\) 0 0
\(349\) 139600. 1.14613 0.573065 0.819510i \(-0.305754\pi\)
0.573065 + 0.819510i \(0.305754\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 85395.3i − 0.685306i −0.939462 0.342653i \(-0.888674\pi\)
0.939462 0.342653i \(-0.111326\pi\)
\(354\) 0 0
\(355\) −69778.6 −0.553689
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 90712.4i − 0.703846i −0.936029 0.351923i \(-0.885528\pi\)
0.936029 0.351923i \(-0.114472\pi\)
\(360\) 0 0
\(361\) 368579. 2.82824
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 52032.6i − 0.390562i
\(366\) 0 0
\(367\) 25541.3 0.189632 0.0948160 0.995495i \(-0.469774\pi\)
0.0948160 + 0.995495i \(0.469774\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 75297.9i 0.547060i
\(372\) 0 0
\(373\) −234528. −1.68569 −0.842844 0.538158i \(-0.819120\pi\)
−0.842844 + 0.538158i \(0.819120\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 97984.7i 0.689407i
\(378\) 0 0
\(379\) 107483. 0.748278 0.374139 0.927373i \(-0.377938\pi\)
0.374139 + 0.927373i \(0.377938\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 164427.i − 1.12093i −0.828180 0.560463i \(-0.810623\pi\)
0.828180 0.560463i \(-0.189377\pi\)
\(384\) 0 0
\(385\) −18275.2 −0.123294
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 182900.i 1.20869i 0.796724 + 0.604343i \(0.206565\pi\)
−0.796724 + 0.604343i \(0.793435\pi\)
\(390\) 0 0
\(391\) −203937. −1.33396
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 75354.6i − 0.482965i
\(396\) 0 0
\(397\) −277717. −1.76206 −0.881031 0.473059i \(-0.843150\pi\)
−0.881031 + 0.473059i \(0.843150\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 168062.i 1.04515i 0.852593 + 0.522576i \(0.175029\pi\)
−0.852593 + 0.522576i \(0.824971\pi\)
\(402\) 0 0
\(403\) −77418.0 −0.476685
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 58661.9i − 0.354134i
\(408\) 0 0
\(409\) −88381.1 −0.528339 −0.264170 0.964476i \(-0.585098\pi\)
−0.264170 + 0.964476i \(0.585098\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 80551.4i − 0.472251i
\(414\) 0 0
\(415\) −92235.5 −0.535552
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 151429.i − 0.862542i −0.902222 0.431271i \(-0.858065\pi\)
0.902222 0.431271i \(-0.141935\pi\)
\(420\) 0 0
\(421\) −132220. −0.745992 −0.372996 0.927833i \(-0.621669\pi\)
−0.372996 + 0.927833i \(0.621669\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 162160.i − 0.897772i
\(426\) 0 0
\(427\) 16158.9 0.0886247
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 231482.i 1.24613i 0.782170 + 0.623065i \(0.214113\pi\)
−0.782170 + 0.623065i \(0.785887\pi\)
\(432\) 0 0
\(433\) −218090. −1.16321 −0.581607 0.813470i \(-0.697576\pi\)
−0.581607 + 0.813470i \(0.697576\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 421192.i − 2.20555i
\(438\) 0 0
\(439\) −89557.0 −0.464698 −0.232349 0.972633i \(-0.574641\pi\)
−0.232349 + 0.972633i \(0.574641\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 264286.i − 1.34669i −0.739330 0.673343i \(-0.764858\pi\)
0.739330 0.673343i \(-0.235142\pi\)
\(444\) 0 0
\(445\) 107119. 0.540936
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 33967.2i − 0.168487i −0.996445 0.0842435i \(-0.973153\pi\)
0.996445 0.0842435i \(-0.0268474\pi\)
\(450\) 0 0
\(451\) −10328.0 −0.0507765
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 13198.8i − 0.0637546i
\(456\) 0 0
\(457\) 83273.7 0.398727 0.199363 0.979926i \(-0.436113\pi\)
0.199363 + 0.979926i \(0.436113\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 57510.4i − 0.270610i −0.990804 0.135305i \(-0.956799\pi\)
0.990804 0.135305i \(-0.0432015\pi\)
\(462\) 0 0
\(463\) −81793.9 −0.381557 −0.190778 0.981633i \(-0.561101\pi\)
−0.190778 + 0.981633i \(0.561101\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 161762.i 0.741726i 0.928688 + 0.370863i \(0.120938\pi\)
−0.928688 + 0.370863i \(0.879062\pi\)
\(468\) 0 0
\(469\) 19323.8 0.0878511
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 93390.6i − 0.417427i
\(474\) 0 0
\(475\) 334910. 1.48436
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 128459.i − 0.559878i −0.960018 0.279939i \(-0.909686\pi\)
0.960018 0.279939i \(-0.0903142\pi\)
\(480\) 0 0
\(481\) 42367.0 0.183121
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 66829.5i − 0.284109i
\(486\) 0 0
\(487\) 255021. 1.07527 0.537636 0.843177i \(-0.319318\pi\)
0.537636 + 0.843177i \(0.319318\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 174548.i − 0.724023i −0.932174 0.362012i \(-0.882090\pi\)
0.932174 0.362012i \(-0.117910\pi\)
\(492\) 0 0
\(493\) 445414. 1.83261
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 81153.7i − 0.328546i
\(498\) 0 0
\(499\) −209729. −0.842282 −0.421141 0.906995i \(-0.638370\pi\)
−0.421141 + 0.906995i \(0.638370\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 288148.i 1.13888i 0.822031 + 0.569442i \(0.192841\pi\)
−0.822031 + 0.569442i \(0.807159\pi\)
\(504\) 0 0
\(505\) 91900.8 0.360360
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 84680.9i − 0.326851i −0.986556 0.163426i \(-0.947746\pi\)
0.986556 0.163426i \(-0.0522544\pi\)
\(510\) 0 0
\(511\) 60514.9 0.231750
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 166746.i 0.628695i
\(516\) 0 0
\(517\) −44852.8 −0.167806
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 80148.3i 0.295270i 0.989042 + 0.147635i \(0.0471660\pi\)
−0.989042 + 0.147635i \(0.952834\pi\)
\(522\) 0 0
\(523\) −127783. −0.467163 −0.233581 0.972337i \(-0.575045\pi\)
−0.233581 + 0.972337i \(0.575045\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 351923.i 1.26715i
\(528\) 0 0
\(529\) −75747.1 −0.270679
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 7459.12i − 0.0262563i
\(534\) 0 0
\(535\) 205620. 0.718385
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 228860.i 0.787756i
\(540\) 0 0
\(541\) 1167.97 0.00399060 0.00199530 0.999998i \(-0.499365\pi\)
0.00199530 + 0.999998i \(0.499365\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 148226.i 0.499035i
\(546\) 0 0
\(547\) −277805. −0.928465 −0.464232 0.885713i \(-0.653670\pi\)
−0.464232 + 0.885713i \(0.653670\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 919914.i 3.03001i
\(552\) 0 0
\(553\) 87638.7 0.286580
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 127126.i 0.409755i 0.978788 + 0.204877i \(0.0656795\pi\)
−0.978788 + 0.204877i \(0.934320\pi\)
\(558\) 0 0
\(559\) 67448.9 0.215850
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 123792.i 0.390550i 0.980749 + 0.195275i \(0.0625599\pi\)
−0.980749 + 0.195275i \(0.937440\pi\)
\(564\) 0 0
\(565\) −8653.83 −0.0271089
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 61812.5i 0.190920i 0.995433 + 0.0954601i \(0.0304322\pi\)
−0.995433 + 0.0954601i \(0.969568\pi\)
\(570\) 0 0
\(571\) 204135. 0.626101 0.313050 0.949737i \(-0.398649\pi\)
0.313050 + 0.949737i \(0.398649\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 282745.i − 0.855182i
\(576\) 0 0
\(577\) 555504. 1.66854 0.834269 0.551358i \(-0.185890\pi\)
0.834269 + 0.551358i \(0.185890\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 107272.i − 0.317784i
\(582\) 0 0
\(583\) 549133. 1.61563
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7016.77i − 0.0203639i −0.999948 0.0101820i \(-0.996759\pi\)
0.999948 0.0101820i \(-0.00324107\pi\)
\(588\) 0 0
\(589\) −726827. −2.09508
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 77064.9i 0.219153i 0.993978 + 0.109576i \(0.0349494\pi\)
−0.993978 + 0.109576i \(0.965051\pi\)
\(594\) 0 0
\(595\) −59998.4 −0.169475
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 546962.i 1.52442i 0.647333 + 0.762208i \(0.275884\pi\)
−0.647333 + 0.762208i \(0.724116\pi\)
\(600\) 0 0
\(601\) −539976. −1.49495 −0.747473 0.664292i \(-0.768733\pi\)
−0.747473 + 0.664292i \(0.768733\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 46541.6i − 0.127154i
\(606\) 0 0
\(607\) −499326. −1.35521 −0.677606 0.735425i \(-0.736982\pi\)
−0.677606 + 0.735425i \(0.736982\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 32393.8i − 0.0867719i
\(612\) 0 0
\(613\) 80551.4 0.214364 0.107182 0.994239i \(-0.465817\pi\)
0.107182 + 0.994239i \(0.465817\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 444845.i − 1.16853i −0.811564 0.584263i \(-0.801383\pi\)
0.811564 0.584263i \(-0.198617\pi\)
\(618\) 0 0
\(619\) −560941. −1.46398 −0.731992 0.681314i \(-0.761409\pi\)
−0.731992 + 0.681314i \(0.761409\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 124581.i 0.320979i
\(624\) 0 0
\(625\) 130545. 0.334196
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 192590.i − 0.486780i
\(630\) 0 0
\(631\) −621772. −1.56161 −0.780805 0.624775i \(-0.785191\pi\)
−0.780805 + 0.624775i \(0.785191\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 197125.i − 0.488871i
\(636\) 0 0
\(637\) −165288. −0.407345
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 322644.i − 0.785248i −0.919699 0.392624i \(-0.871567\pi\)
0.919699 0.392624i \(-0.128433\pi\)
\(642\) 0 0
\(643\) 489813. 1.18470 0.592350 0.805681i \(-0.298200\pi\)
0.592350 + 0.805681i \(0.298200\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26885.7i 0.0642263i 0.999484 + 0.0321132i \(0.0102237\pi\)
−0.999484 + 0.0321132i \(0.989776\pi\)
\(648\) 0 0
\(649\) −587446. −1.39469
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 714229.i 1.67498i 0.546449 + 0.837492i \(0.315979\pi\)
−0.546449 + 0.837492i \(0.684021\pi\)
\(654\) 0 0
\(655\) −190422. −0.443849
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 234697.i − 0.540428i −0.962800 0.270214i \(-0.912906\pi\)
0.962800 0.270214i \(-0.0870944\pi\)
\(660\) 0 0
\(661\) 626562. 1.43404 0.717020 0.697053i \(-0.245506\pi\)
0.717020 + 0.697053i \(0.245506\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 123915.i − 0.280208i
\(666\) 0 0
\(667\) 776630. 1.74567
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 117843.i − 0.261734i
\(672\) 0 0
\(673\) 641998. 1.41744 0.708718 0.705491i \(-0.249274\pi\)
0.708718 + 0.705491i \(0.249274\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 656946.i 1.43335i 0.697407 + 0.716675i \(0.254337\pi\)
−0.697407 + 0.716675i \(0.745663\pi\)
\(678\) 0 0
\(679\) 77723.9 0.168583
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 76710.6i − 0.164442i −0.996614 0.0822212i \(-0.973799\pi\)
0.996614 0.0822212i \(-0.0262014\pi\)
\(684\) 0 0
\(685\) −10938.4 −0.0233117
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 396597.i 0.835432i
\(690\) 0 0
\(691\) 570869. 1.19559 0.597793 0.801651i \(-0.296045\pi\)
0.597793 + 0.801651i \(0.296045\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 114936.i 0.237951i
\(696\) 0 0
\(697\) −33907.3 −0.0697956
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4271.71i 0.00869292i 0.999991 + 0.00434646i \(0.00138353\pi\)
−0.999991 + 0.00434646i \(0.998616\pi\)
\(702\) 0 0
\(703\) 397757. 0.804835
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 106882.i 0.213829i
\(708\) 0 0
\(709\) 148443. 0.295303 0.147652 0.989039i \(-0.452829\pi\)
0.147652 + 0.989039i \(0.452829\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 613618.i 1.20703i
\(714\) 0 0
\(715\) −96256.2 −0.188285
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 549881.i 1.06368i 0.846845 + 0.531840i \(0.178499\pi\)
−0.846845 + 0.531840i \(0.821501\pi\)
\(720\) 0 0
\(721\) −193928. −0.373052
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 617534.i 1.17486i
\(726\) 0 0
\(727\) −644007. −1.21849 −0.609244 0.792982i \(-0.708527\pi\)
−0.609244 + 0.792982i \(0.708527\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 306606.i − 0.573780i
\(732\) 0 0
\(733\) 832831. 1.55006 0.775031 0.631924i \(-0.217734\pi\)
0.775031 + 0.631924i \(0.217734\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 140925.i − 0.259449i
\(738\) 0 0
\(739\) −472487. −0.865170 −0.432585 0.901593i \(-0.642398\pi\)
−0.432585 + 0.901593i \(0.642398\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 978422.i 1.77235i 0.463354 + 0.886173i \(0.346646\pi\)
−0.463354 + 0.886173i \(0.653354\pi\)
\(744\) 0 0
\(745\) −499497. −0.899954
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 239139.i 0.426273i
\(750\) 0 0
\(751\) 166281. 0.294824 0.147412 0.989075i \(-0.452906\pi\)
0.147412 + 0.989075i \(0.452906\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 306733.i − 0.538105i
\(756\) 0 0
\(757\) −238453. −0.416114 −0.208057 0.978117i \(-0.566714\pi\)
−0.208057 + 0.978117i \(0.566714\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 598335.i 1.03318i 0.856233 + 0.516589i \(0.172799\pi\)
−0.856233 + 0.516589i \(0.827201\pi\)
\(762\) 0 0
\(763\) −172389. −0.296116
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 424267.i − 0.721189i
\(768\) 0 0
\(769\) 667323. 1.12845 0.564227 0.825620i \(-0.309174\pi\)
0.564227 + 0.825620i \(0.309174\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 848153.i 1.41943i 0.704487 + 0.709717i \(0.251177\pi\)
−0.704487 + 0.709717i \(0.748823\pi\)
\(774\) 0 0
\(775\) −487916. −0.812347
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 70028.8i − 0.115399i
\(780\) 0 0
\(781\) −591838. −0.970289
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 404300.i 0.656091i
\(786\) 0 0
\(787\) 967733. 1.56245 0.781225 0.624250i \(-0.214595\pi\)
0.781225 + 0.624250i \(0.214595\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 10064.6i − 0.0160858i
\(792\) 0 0
\(793\) 85109.3 0.135341
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 674146.i 1.06130i 0.847592 + 0.530649i \(0.178052\pi\)
−0.847592 + 0.530649i \(0.821948\pi\)
\(798\) 0 0
\(799\) −147254. −0.230661
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 441323.i − 0.684425i
\(804\) 0 0
\(805\) −104614. −0.161435
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 570317.i 0.871403i 0.900091 + 0.435702i \(0.143500\pi\)
−0.900091 + 0.435702i \(0.856500\pi\)
\(810\) 0 0
\(811\) 111731. 0.169876 0.0849379 0.996386i \(-0.472931\pi\)
0.0849379 + 0.996386i \(0.472931\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 169444.i 0.255100i
\(816\) 0 0
\(817\) 633234. 0.948680
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 1.04162e6i − 1.54534i −0.634811 0.772668i \(-0.718922\pi\)
0.634811 0.772668i \(-0.281078\pi\)
\(822\) 0 0
\(823\) 108973. 0.160886 0.0804432 0.996759i \(-0.474366\pi\)
0.0804432 + 0.996759i \(0.474366\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 550908.i − 0.805505i −0.915309 0.402753i \(-0.868053\pi\)
0.915309 0.402753i \(-0.131947\pi\)
\(828\) 0 0
\(829\) −57256.0 −0.0833128 −0.0416564 0.999132i \(-0.513263\pi\)
−0.0416564 + 0.999132i \(0.513263\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 751358.i 1.08282i
\(834\) 0 0
\(835\) 139183. 0.199624
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 951705.i 1.35201i 0.736899 + 0.676003i \(0.236289\pi\)
−0.736899 + 0.676003i \(0.763711\pi\)
\(840\) 0 0
\(841\) −988934. −1.39822
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 281264.i 0.393914i
\(846\) 0 0
\(847\) 54128.7 0.0754502
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 335803.i − 0.463687i
\(852\) 0 0
\(853\) −1.02287e6 −1.40579 −0.702896 0.711293i \(-0.748110\pi\)
−0.702896 + 0.711293i \(0.748110\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 661433.i 0.900584i 0.892881 + 0.450292i \(0.148680\pi\)
−0.892881 + 0.450292i \(0.851320\pi\)
\(858\) 0 0
\(859\) −102749. −0.139249 −0.0696246 0.997573i \(-0.522180\pi\)
−0.0696246 + 0.997573i \(0.522180\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 168806.i 0.226655i 0.993558 + 0.113328i \(0.0361509\pi\)
−0.993558 + 0.113328i \(0.963849\pi\)
\(864\) 0 0
\(865\) −481087. −0.642971
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 639132.i − 0.846352i
\(870\) 0 0
\(871\) 101779. 0.134160
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 192830.i − 0.251860i
\(876\) 0 0
\(877\) −190156. −0.247235 −0.123618 0.992330i \(-0.539450\pi\)
−0.123618 + 0.992330i \(0.539450\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 171490.i 0.220946i 0.993879 + 0.110473i \(0.0352366\pi\)
−0.993879 + 0.110473i \(0.964763\pi\)
\(882\) 0 0
\(883\) −869284. −1.11491 −0.557456 0.830207i \(-0.688222\pi\)
−0.557456 + 0.830207i \(0.688222\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 144353.i 0.183476i 0.995783 + 0.0917378i \(0.0292422\pi\)
−0.995783 + 0.0917378i \(0.970758\pi\)
\(888\) 0 0
\(889\) 229260. 0.290084
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 304124.i − 0.381371i
\(894\) 0 0
\(895\) −295961. −0.369477
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1.34018e6i − 1.65823i
\(900\) 0 0
\(901\) 1.80283e6 2.22078
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 126222.i − 0.154113i
\(906\) 0 0
\(907\) −107697. −0.130915 −0.0654574 0.997855i \(-0.520851\pi\)
−0.0654574 + 0.997855i \(0.520851\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 712977.i 0.859091i 0.903045 + 0.429545i \(0.141326\pi\)
−0.903045 + 0.429545i \(0.858674\pi\)
\(912\) 0 0
\(913\) −782310. −0.938507
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 221465.i − 0.263370i
\(918\) 0 0
\(919\) −239512. −0.283594 −0.141797 0.989896i \(-0.545288\pi\)
−0.141797 + 0.989896i \(0.545288\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 427440.i − 0.501732i
\(924\) 0 0
\(925\) 267012. 0.312067
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.28407e6i 1.48785i 0.668266 + 0.743923i \(0.267037\pi\)
−0.668266 + 0.743923i \(0.732963\pi\)
\(930\) 0 0
\(931\) −1.55178e6 −1.79032
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 437557.i 0.500508i
\(936\) 0 0
\(937\) −471000. −0.536466 −0.268233 0.963354i \(-0.586440\pi\)
−0.268233 + 0.963354i \(0.586440\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 721460.i − 0.814766i −0.913257 0.407383i \(-0.866441\pi\)
0.913257 0.407383i \(-0.133559\pi\)
\(942\) 0 0
\(943\) −59121.2 −0.0664845
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 605552.i 0.675229i 0.941284 + 0.337615i \(0.109620\pi\)
−0.941284 + 0.337615i \(0.890380\pi\)
\(948\) 0 0
\(949\) 318734. 0.353913
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 527043.i 0.580310i 0.956980 + 0.290155i \(0.0937069\pi\)
−0.956980 + 0.290155i \(0.906293\pi\)
\(954\) 0 0
\(955\) 430240. 0.471741
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 12721.6i − 0.0138326i
\(960\) 0 0
\(961\) 135363. 0.146573
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 64363.8i − 0.0691174i
\(966\) 0 0
\(967\) 1.24031e6 1.32641 0.663203 0.748439i \(-0.269196\pi\)
0.663203 + 0.748439i \(0.269196\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.27298e6i 1.35015i 0.737748 + 0.675076i \(0.235889\pi\)
−0.737748 + 0.675076i \(0.764111\pi\)
\(972\) 0 0
\(973\) −133673. −0.141194
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1.38469e6i − 1.45065i −0.688406 0.725325i \(-0.741689\pi\)
0.688406 0.725325i \(-0.258311\pi\)
\(978\) 0 0
\(979\) 908546. 0.947942
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 169030.i 0.174927i 0.996168 + 0.0874633i \(0.0278761\pi\)
−0.996168 + 0.0874633i \(0.972124\pi\)
\(984\) 0 0
\(985\) −521016. −0.537006
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 534602.i − 0.546560i
\(990\) 0 0
\(991\) 1.57481e6 1.60354 0.801772 0.597630i \(-0.203891\pi\)
0.801772 + 0.597630i \(0.203891\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 384055.i 0.387924i
\(996\) 0 0
\(997\) 570747. 0.574187 0.287094 0.957903i \(-0.407311\pi\)
0.287094 + 0.957903i \(0.407311\pi\)
\(998\) 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 1296.5.e.g.161.3 8
3.2 odd 2 inner 1296.5.e.g.161.6 8
4.3 odd 2 324.5.c.a.161.3 8
9.2 odd 6 432.5.q.c.17.2 8
9.4 even 3 432.5.q.c.305.2 8
9.5 odd 6 144.5.q.c.65.1 8
9.7 even 3 144.5.q.c.113.1 8
12.11 even 2 324.5.c.a.161.6 8
36.7 odd 6 36.5.g.a.5.4 8
36.11 even 6 108.5.g.a.17.2 8
36.23 even 6 36.5.g.a.29.4 yes 8
36.31 odd 6 108.5.g.a.89.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.5.g.a.5.4 8 36.7 odd 6
36.5.g.a.29.4 yes 8 36.23 even 6
108.5.g.a.17.2 8 36.11 even 6
108.5.g.a.89.2 8 36.31 odd 6
144.5.q.c.65.1 8 9.5 odd 6
144.5.q.c.113.1 8 9.7 even 3
324.5.c.a.161.3 8 4.3 odd 2
324.5.c.a.161.6 8 12.11 even 2
432.5.q.c.17.2 8 9.2 odd 6
432.5.q.c.305.2 8 9.4 even 3
1296.5.e.g.161.3 8 1.1 even 1 trivial
1296.5.e.g.161.6 8 3.2 odd 2 inner