Properties

Label 252.9.d.c.181.8
Level $252$
Weight $9$
Character 252.181
Analytic conductor $102.659$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,9,Mod(181,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.181");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), 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: \(102.659409735\)
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: \( x^{8} - 2 x^{7} + 229371 x^{6} - 1944986 x^{5} + 13911336469 x^{4} - 182147783748 x^{3} + 180227670127764 x^{2} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.8
Root \(-3.81971 - 274.634i\) of defining polynomial
Character \(\chi\) \(=\) 252.181
Dual form 252.9.d.c.181.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1071.65i q^{5} +(-436.993 + 2360.90i) q^{7} +O(q^{10})\) \(q+1071.65i q^{5} +(-436.993 + 2360.90i) q^{7} -10581.1 q^{11} -24672.2i q^{13} +63685.7i q^{17} +136002. i q^{19} -340275. q^{23} -757811. q^{25} -967918. q^{29} -94428.3i q^{31} +(-2.53006e6 - 468304. i) q^{35} +733395. q^{37} +2.00732e6i q^{41} +5.98286e6 q^{43} +1.51713e6i q^{47} +(-5.38287e6 - 2.06339e6i) q^{49} +3.99981e6 q^{53} -1.13393e7i q^{55} -9.18726e6i q^{59} +2.02054e7i q^{61} +2.64400e7 q^{65} -4.55149e6 q^{67} +2.39953e7 q^{71} -3.93925e7i q^{73} +(4.62388e6 - 2.49810e7i) q^{77} -7.44371e6 q^{79} +1.51689e7i q^{83} -6.82488e7 q^{85} +3.77542e7i q^{89} +(5.82485e7 + 1.07816e7i) q^{91} -1.45747e8 q^{95} -8.92820e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6076 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6076 q^{7} - 2154400 q^{25} + 2495096 q^{37} + 18340232 q^{43} - 13983424 q^{49} + 12320248 q^{67} + 53237896 q^{79} - 79347240 q^{85} + 257358192 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 1071.65i 1.71464i 0.514782 + 0.857321i \(0.327873\pi\)
−0.514782 + 0.857321i \(0.672127\pi\)
\(6\) 0 0
\(7\) −436.993 + 2360.90i −0.182005 + 0.983298i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −10581.1 −0.722706 −0.361353 0.932429i \(-0.617685\pi\)
−0.361353 + 0.932429i \(0.617685\pi\)
\(12\) 0 0
\(13\) 24672.2i 0.863842i −0.901911 0.431921i \(-0.857836\pi\)
0.901911 0.431921i \(-0.142164\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 63685.7i 0.762511i 0.924470 + 0.381255i \(0.124508\pi\)
−0.924470 + 0.381255i \(0.875492\pi\)
\(18\) 0 0
\(19\) 136002.i 1.04359i 0.853070 + 0.521796i \(0.174738\pi\)
−0.853070 + 0.521796i \(0.825262\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −340275. −1.21596 −0.607979 0.793953i \(-0.708019\pi\)
−0.607979 + 0.793953i \(0.708019\pi\)
\(24\) 0 0
\(25\) −757811. −1.94000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −967918. −1.36851 −0.684253 0.729245i \(-0.739872\pi\)
−0.684253 + 0.729245i \(0.739872\pi\)
\(30\) 0 0
\(31\) 94428.3i 0.102248i −0.998692 0.0511241i \(-0.983720\pi\)
0.998692 0.0511241i \(-0.0162804\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.53006e6 468304.i −1.68600 0.312073i
\(36\) 0 0
\(37\) 733395. 0.391319 0.195659 0.980672i \(-0.437315\pi\)
0.195659 + 0.980672i \(0.437315\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00732e6i 0.710365i 0.934797 + 0.355183i \(0.115581\pi\)
−0.934797 + 0.355183i \(0.884419\pi\)
\(42\) 0 0
\(43\) 5.98286e6 1.74999 0.874993 0.484135i \(-0.160866\pi\)
0.874993 + 0.484135i \(0.160866\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.51713e6i 0.310908i 0.987843 + 0.155454i \(0.0496841\pi\)
−0.987843 + 0.155454i \(0.950316\pi\)
\(48\) 0 0
\(49\) −5.38287e6 2.06339e6i −0.933749 0.357930i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.99981e6 0.506916 0.253458 0.967346i \(-0.418432\pi\)
0.253458 + 0.967346i \(0.418432\pi\)
\(54\) 0 0
\(55\) 1.13393e7i 1.23918i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.18726e6i 0.758190i −0.925358 0.379095i \(-0.876235\pi\)
0.925358 0.379095i \(-0.123765\pi\)
\(60\) 0 0
\(61\) 2.02054e7i 1.45931i 0.683813 + 0.729657i \(0.260320\pi\)
−0.683813 + 0.729657i \(0.739680\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.64400e7 1.48118
\(66\) 0 0
\(67\) −4.55149e6 −0.225868 −0.112934 0.993603i \(-0.536025\pi\)
−0.112934 + 0.993603i \(0.536025\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.39953e7 0.944263 0.472132 0.881528i \(-0.343485\pi\)
0.472132 + 0.881528i \(0.343485\pi\)
\(72\) 0 0
\(73\) 3.93925e7i 1.38715i −0.720386 0.693574i \(-0.756035\pi\)
0.720386 0.693574i \(-0.243965\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.62388e6 2.49810e7i 0.131536 0.710635i
\(78\) 0 0
\(79\) −7.44371e6 −0.191109 −0.0955545 0.995424i \(-0.530462\pi\)
−0.0955545 + 0.995424i \(0.530462\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.51689e7i 0.319625i 0.987147 + 0.159813i \(0.0510890\pi\)
−0.987147 + 0.159813i \(0.948911\pi\)
\(84\) 0 0
\(85\) −6.82488e7 −1.30743
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.77542e7i 0.601735i 0.953666 + 0.300868i \(0.0972762\pi\)
−0.953666 + 0.300868i \(0.902724\pi\)
\(90\) 0 0
\(91\) 5.82485e7 + 1.07816e7i 0.849414 + 0.157223i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.45747e8 −1.78939
\(96\) 0 0
\(97\) 8.92820e7i 1.00850i −0.863557 0.504251i \(-0.831769\pi\)
0.863557 0.504251i \(-0.168231\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.02581e7i 0.386872i 0.981113 + 0.193436i \(0.0619632\pi\)
−0.981113 + 0.193436i \(0.938037\pi\)
\(102\) 0 0
\(103\) 1.16646e8i 1.03638i −0.855264 0.518192i \(-0.826605\pi\)
0.855264 0.518192i \(-0.173395\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.01941e8 −1.54059 −0.770297 0.637685i \(-0.779892\pi\)
−0.770297 + 0.637685i \(0.779892\pi\)
\(108\) 0 0
\(109\) 9.21391e7 0.652736 0.326368 0.945243i \(-0.394175\pi\)
0.326368 + 0.945243i \(0.394175\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.47904e8 0.907125 0.453563 0.891224i \(-0.350153\pi\)
0.453563 + 0.891224i \(0.350153\pi\)
\(114\) 0 0
\(115\) 3.64656e8i 2.08493i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.50355e8 2.78302e7i −0.749775 0.138781i
\(120\) 0 0
\(121\) −1.02399e8 −0.477697
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.93495e8i 1.61176i
\(126\) 0 0
\(127\) 9.15585e7 0.351952 0.175976 0.984394i \(-0.443692\pi\)
0.175976 + 0.984394i \(0.443692\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.92078e8i 1.33134i 0.746248 + 0.665668i \(0.231853\pi\)
−0.746248 + 0.665668i \(0.768147\pi\)
\(132\) 0 0
\(133\) −3.21087e8 5.94319e7i −1.02616 0.189939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.11204e8 1.16728 0.583639 0.812013i \(-0.301628\pi\)
0.583639 + 0.812013i \(0.301628\pi\)
\(138\) 0 0
\(139\) 3.29853e8i 0.883612i −0.897111 0.441806i \(-0.854338\pi\)
0.897111 0.441806i \(-0.145662\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.61060e8i 0.624304i
\(144\) 0 0
\(145\) 1.03727e9i 2.34650i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.82348e8 1.79017 0.895087 0.445892i \(-0.147114\pi\)
0.895087 + 0.445892i \(0.147114\pi\)
\(150\) 0 0
\(151\) −9.58744e8 −1.84414 −0.922072 0.387017i \(-0.873505\pi\)
−0.922072 + 0.387017i \(0.873505\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.01194e8 0.175319
\(156\) 0 0
\(157\) 3.02276e8i 0.497514i −0.968566 0.248757i \(-0.919978\pi\)
0.968566 0.248757i \(-0.0800220\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.48698e8 8.03354e8i 0.221310 1.19565i
\(162\) 0 0
\(163\) 3.20474e8 0.453986 0.226993 0.973896i \(-0.427111\pi\)
0.226993 + 0.973896i \(0.427111\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.22231e9i 1.57151i −0.618540 0.785753i \(-0.712276\pi\)
0.618540 0.785753i \(-0.287724\pi\)
\(168\) 0 0
\(169\) 2.07013e8 0.253776
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.69146e9i 1.88833i 0.329473 + 0.944165i \(0.393129\pi\)
−0.329473 + 0.944165i \(0.606871\pi\)
\(174\) 0 0
\(175\) 3.31158e8 1.78911e9i 0.353089 1.90759i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.28023e8 −0.903954 −0.451977 0.892030i \(-0.649281\pi\)
−0.451977 + 0.892030i \(0.649281\pi\)
\(180\) 0 0
\(181\) 1.94295e9i 1.81029i −0.425105 0.905144i \(-0.639763\pi\)
0.425105 0.905144i \(-0.360237\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.85943e8i 0.670972i
\(186\) 0 0
\(187\) 6.73867e8i 0.551071i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.11035e9 −0.834305 −0.417152 0.908837i \(-0.636972\pi\)
−0.417152 + 0.908837i \(0.636972\pi\)
\(192\) 0 0
\(193\) 1.88156e9 1.35609 0.678047 0.735019i \(-0.262827\pi\)
0.678047 + 0.735019i \(0.262827\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.85779e9 −1.89743 −0.948715 0.316133i \(-0.897616\pi\)
−0.948715 + 0.316133i \(0.897616\pi\)
\(198\) 0 0
\(199\) 8.58663e8i 0.547533i −0.961796 0.273767i \(-0.911730\pi\)
0.961796 0.273767i \(-0.0882696\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.22974e8 2.28516e9i 0.249075 1.34565i
\(204\) 0 0
\(205\) −2.15115e9 −1.21802
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.43905e9i 0.754209i
\(210\) 0 0
\(211\) −3.18684e9 −1.60779 −0.803897 0.594769i \(-0.797243\pi\)
−0.803897 + 0.594769i \(0.797243\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.41153e9i 3.00060i
\(216\) 0 0
\(217\) 2.22936e8 + 4.12645e7i 0.100540 + 0.0186096i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.57127e9 0.658689
\(222\) 0 0
\(223\) 7.41102e8i 0.299681i −0.988710 0.149840i \(-0.952124\pi\)
0.988710 0.149840i \(-0.0478760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.75636e8i 0.292116i −0.989276 0.146058i \(-0.953341\pi\)
0.989276 0.146058i \(-0.0466585\pi\)
\(228\) 0 0
\(229\) 1.44246e9i 0.524521i −0.964997 0.262261i \(-0.915532\pi\)
0.964997 0.262261i \(-0.0844680\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.18218e9 −1.07969 −0.539847 0.841763i \(-0.681518\pi\)
−0.539847 + 0.841763i \(0.681518\pi\)
\(234\) 0 0
\(235\) −1.62584e9 −0.533096
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.01977e9 1.53848 0.769240 0.638960i \(-0.220635\pi\)
0.769240 + 0.638960i \(0.220635\pi\)
\(240\) 0 0
\(241\) 1.09821e8i 0.0325549i 0.999868 + 0.0162774i \(0.00518150\pi\)
−0.999868 + 0.0162774i \(0.994819\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.21124e9 5.76856e9i 0.613721 1.60104i
\(246\) 0 0
\(247\) 3.35547e9 0.901499
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.07598e9i 1.78276i −0.453261 0.891378i \(-0.649739\pi\)
0.453261 0.891378i \(-0.350261\pi\)
\(252\) 0 0
\(253\) 3.60049e9 0.878779
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.92292e9i 1.35770i 0.734278 + 0.678849i \(0.237521\pi\)
−0.734278 + 0.678849i \(0.762479\pi\)
\(258\) 0 0
\(259\) −3.20488e8 + 1.73147e9i −0.0712219 + 0.384783i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.58926e9 1.37725 0.688626 0.725116i \(-0.258214\pi\)
0.688626 + 0.725116i \(0.258214\pi\)
\(264\) 0 0
\(265\) 4.28641e9i 0.869180i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.23482e9i 1.19073i 0.803454 + 0.595367i \(0.202993\pi\)
−0.803454 + 0.595367i \(0.797007\pi\)
\(270\) 0 0
\(271\) 4.05326e9i 0.751498i 0.926722 + 0.375749i \(0.122614\pi\)
−0.926722 + 0.375749i \(0.877386\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.01850e9 1.40205
\(276\) 0 0
\(277\) −4.65609e9 −0.790865 −0.395433 0.918495i \(-0.629405\pi\)
−0.395433 + 0.918495i \(0.629405\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.83530e9 −0.615141 −0.307570 0.951525i \(-0.599516\pi\)
−0.307570 + 0.951525i \(0.599516\pi\)
\(282\) 0 0
\(283\) 4.32516e9i 0.674306i −0.941450 0.337153i \(-0.890536\pi\)
0.941450 0.337153i \(-0.109464\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.73908e9 8.77186e8i −0.698500 0.129290i
\(288\) 0 0
\(289\) 2.91989e9 0.418577
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.99484e9i 0.949089i −0.880231 0.474545i \(-0.842613\pi\)
0.880231 0.474545i \(-0.157387\pi\)
\(294\) 0 0
\(295\) 9.84554e9 1.30002
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.39533e9i 1.05040i
\(300\) 0 0
\(301\) −2.61447e9 + 1.41249e10i −0.318506 + 1.72076i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.16532e10 −2.50220
\(306\) 0 0
\(307\) 8.11196e9i 0.913214i 0.889669 + 0.456607i \(0.150935\pi\)
−0.889669 + 0.456607i \(0.849065\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.49867e9i 0.160201i 0.996787 + 0.0801006i \(0.0255242\pi\)
−0.996787 + 0.0801006i \(0.974476\pi\)
\(312\) 0 0
\(313\) 9.72355e9i 1.01309i 0.862214 + 0.506545i \(0.169077\pi\)
−0.862214 + 0.506545i \(0.830923\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.16325e8 0.0511312 0.0255656 0.999673i \(-0.491861\pi\)
0.0255656 + 0.999673i \(0.491861\pi\)
\(318\) 0 0
\(319\) 1.02417e10 0.989027
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8.66137e9 −0.795750
\(324\) 0 0
\(325\) 1.86969e10i 1.67585i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.58180e9 6.62977e8i −0.305715 0.0565868i
\(330\) 0 0
\(331\) −1.13625e10 −0.946590 −0.473295 0.880904i \(-0.656936\pi\)
−0.473295 + 0.880904i \(0.656936\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.87761e9i 0.387282i
\(336\) 0 0
\(337\) −1.90839e10 −1.47961 −0.739805 0.672822i \(-0.765082\pi\)
−0.739805 + 0.672822i \(0.765082\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.99158e8i 0.0738953i
\(342\) 0 0
\(343\) 7.22374e9 1.18067e10i 0.521898 0.853008i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.18230e9 0.426415 0.213207 0.977007i \(-0.431609\pi\)
0.213207 + 0.977007i \(0.431609\pi\)
\(348\) 0 0
\(349\) 1.57447e10i 1.06129i 0.847595 + 0.530643i \(0.178050\pi\)
−0.847595 + 0.530643i \(0.821950\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.96364e9i 0.190865i −0.995436 0.0954327i \(-0.969577\pi\)
0.995436 0.0954327i \(-0.0304235\pi\)
\(354\) 0 0
\(355\) 2.57146e10i 1.61907i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.13412e10 −0.682782 −0.341391 0.939921i \(-0.610898\pi\)
−0.341391 + 0.939921i \(0.610898\pi\)
\(360\) 0 0
\(361\) −1.51295e9 −0.0890833
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.22151e10 2.37846
\(366\) 0 0
\(367\) 2.34457e10i 1.29241i 0.763164 + 0.646204i \(0.223645\pi\)
−0.763164 + 0.646204i \(0.776355\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.74789e9 + 9.44315e9i −0.0922612 + 0.498450i
\(372\) 0 0
\(373\) 3.30201e10 1.70586 0.852930 0.522026i \(-0.174824\pi\)
0.852930 + 0.522026i \(0.174824\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.38807e10i 1.18217i
\(378\) 0 0
\(379\) −6.57661e9 −0.318746 −0.159373 0.987218i \(-0.550947\pi\)
−0.159373 + 0.987218i \(0.550947\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.31609e10i 1.54110i −0.637379 0.770550i \(-0.719982\pi\)
0.637379 0.770550i \(-0.280018\pi\)
\(384\) 0 0
\(385\) 2.67709e10 + 4.95519e9i 1.21848 + 0.225537i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.63950e10 1.15272 0.576359 0.817197i \(-0.304473\pi\)
0.576359 + 0.817197i \(0.304473\pi\)
\(390\) 0 0
\(391\) 2.16706e10i 0.927181i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.97707e9i 0.327684i
\(396\) 0 0
\(397\) 4.49632e9i 0.181007i −0.995896 0.0905034i \(-0.971152\pi\)
0.995896 0.0905034i \(-0.0288476\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.56656e10 0.605855 0.302928 0.953014i \(-0.402036\pi\)
0.302928 + 0.953014i \(0.402036\pi\)
\(402\) 0 0
\(403\) −2.32975e9 −0.0883263
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.76014e9 −0.282808
\(408\) 0 0
\(409\) 8.48977e9i 0.303391i 0.988427 + 0.151695i \(0.0484733\pi\)
−0.988427 + 0.151695i \(0.951527\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.16902e10 + 4.01477e9i 0.745527 + 0.137994i
\(414\) 0 0
\(415\) −1.62558e10 −0.548043
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.39173e10i 1.42488i 0.701731 + 0.712442i \(0.252411\pi\)
−0.701731 + 0.712442i \(0.747589\pi\)
\(420\) 0 0
\(421\) 4.09583e10 1.30381 0.651903 0.758302i \(-0.273971\pi\)
0.651903 + 0.758302i \(0.273971\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.82617e10i 1.47927i
\(426\) 0 0
\(427\) −4.77030e10 8.82964e9i −1.43494 0.265602i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.31374e10 −0.380716 −0.190358 0.981715i \(-0.560965\pi\)
−0.190358 + 0.981715i \(0.560965\pi\)
\(432\) 0 0
\(433\) 6.11013e10i 1.73820i 0.494638 + 0.869099i \(0.335301\pi\)
−0.494638 + 0.869099i \(0.664699\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.62780e10i 1.26896i
\(438\) 0 0
\(439\) 5.25752e10i 1.41554i −0.706442 0.707771i \(-0.749701\pi\)
0.706442 0.707771i \(-0.250299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.52730e10 −1.43515 −0.717576 0.696480i \(-0.754748\pi\)
−0.717576 + 0.696480i \(0.754748\pi\)
\(444\) 0 0
\(445\) −4.04594e10 −1.03176
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.08377e10 −1.49688 −0.748440 0.663202i \(-0.769197\pi\)
−0.748440 + 0.663202i \(0.769197\pi\)
\(450\) 0 0
\(451\) 2.12397e10i 0.513385i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.15541e10 + 6.24221e10i −0.269582 + 1.45644i
\(456\) 0 0
\(457\) −1.75283e10 −0.401860 −0.200930 0.979606i \(-0.564396\pi\)
−0.200930 + 0.979606i \(0.564396\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.14631e10i 1.58226i 0.611648 + 0.791130i \(0.290507\pi\)
−0.611648 + 0.791130i \(0.709493\pi\)
\(462\) 0 0
\(463\) −4.51271e9 −0.0982004 −0.0491002 0.998794i \(-0.515635\pi\)
−0.0491002 + 0.998794i \(0.515635\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.62414e10i 1.18247i −0.806501 0.591233i \(-0.798641\pi\)
0.806501 0.591233i \(-0.201359\pi\)
\(468\) 0 0
\(469\) 1.98897e9 1.07456e10i 0.0411090 0.222095i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.33054e10 −1.26473
\(474\) 0 0
\(475\) 1.03064e11i 2.02456i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.56246e10i 0.676719i 0.941017 + 0.338359i \(0.109872\pi\)
−0.941017 + 0.338359i \(0.890128\pi\)
\(480\) 0 0
\(481\) 1.80945e10i 0.338038i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.56792e10 1.72922
\(486\) 0 0
\(487\) −5.09256e10 −0.905359 −0.452679 0.891673i \(-0.649532\pi\)
−0.452679 + 0.891673i \(0.649532\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.24005e10 −1.07365 −0.536825 0.843694i \(-0.680376\pi\)
−0.536825 + 0.843694i \(0.680376\pi\)
\(492\) 0 0
\(493\) 6.16425e10i 1.04350i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.04858e10 + 5.66505e10i −0.171860 + 0.928492i
\(498\) 0 0
\(499\) 9.04114e10 1.45821 0.729106 0.684401i \(-0.239936\pi\)
0.729106 + 0.684401i \(0.239936\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.50934e10i 0.548219i −0.961698 0.274109i \(-0.911617\pi\)
0.961698 0.274109i \(-0.0883831\pi\)
\(504\) 0 0
\(505\) −4.31426e10 −0.663347
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.94040e10i 1.48092i −0.672098 0.740462i \(-0.734607\pi\)
0.672098 0.740462i \(-0.265393\pi\)
\(510\) 0 0
\(511\) 9.30018e10 + 1.72143e10i 1.36398 + 0.252467i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.25004e11 1.77703
\(516\) 0 0
\(517\) 1.60530e10i 0.224695i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.21734e10i 0.708106i −0.935225 0.354053i \(-0.884803\pi\)
0.935225 0.354053i \(-0.115197\pi\)
\(522\) 0 0
\(523\) 1.15036e11i 1.53755i −0.639522 0.768773i \(-0.720868\pi\)
0.639522 0.768773i \(-0.279132\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.01373e9 0.0779653
\(528\) 0 0
\(529\) 3.74759e10 0.478552
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.95250e10 0.613643
\(534\) 0 0
\(535\) 2.16410e11i 2.64157i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.69569e10 + 2.18330e10i 0.674825 + 0.258678i
\(540\) 0 0
\(541\) −3.48673e10 −0.407033 −0.203517 0.979072i \(-0.565237\pi\)
−0.203517 + 0.979072i \(0.565237\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.87409e10i 1.11921i
\(546\) 0 0
\(547\) −1.32416e11 −1.47908 −0.739540 0.673113i \(-0.764957\pi\)
−0.739540 + 0.673113i \(0.764957\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.31639e11i 1.42816i
\(552\) 0 0
\(553\) 3.25285e9 1.75738e10i 0.0347828 0.187917i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.05214e10 0.732656 0.366328 0.930486i \(-0.380615\pi\)
0.366328 + 0.930486i \(0.380615\pi\)
\(558\) 0 0
\(559\) 1.47610e11i 1.51171i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.67216e10i 0.763632i 0.924238 + 0.381816i \(0.124701\pi\)
−0.924238 + 0.381816i \(0.875299\pi\)
\(564\) 0 0
\(565\) 1.58502e11i 1.55540i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.53682e11 −1.46613 −0.733067 0.680156i \(-0.761912\pi\)
−0.733067 + 0.680156i \(0.761912\pi\)
\(570\) 0 0
\(571\) −9.69767e10 −0.912269 −0.456135 0.889911i \(-0.650766\pi\)
−0.456135 + 0.889911i \(0.650766\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.57864e11 2.35895
\(576\) 0 0
\(577\) 1.00096e11i 0.903049i 0.892259 + 0.451525i \(0.149120\pi\)
−0.892259 + 0.451525i \(0.850880\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.58122e10 6.62870e9i −0.314287 0.0581733i
\(582\) 0 0
\(583\) −4.23226e10 −0.366351
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.00055e11i 0.842728i −0.906892 0.421364i \(-0.861551\pi\)
0.906892 0.421364i \(-0.138449\pi\)
\(588\) 0 0
\(589\) 1.28424e10 0.106705
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.05395e11i 1.66100i 0.557017 + 0.830501i \(0.311946\pi\)
−0.557017 + 0.830501i \(0.688054\pi\)
\(594\) 0 0
\(595\) 2.98243e10 1.61128e11i 0.237959 1.28560i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.46446e11 −1.13755 −0.568776 0.822492i \(-0.692583\pi\)
−0.568776 + 0.822492i \(0.692583\pi\)
\(600\) 0 0
\(601\) 1.44783e10i 0.110974i −0.998459 0.0554869i \(-0.982329\pi\)
0.998459 0.0554869i \(-0.0176711\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.09735e11i 0.819079i
\(606\) 0 0
\(607\) 9.26418e10i 0.682421i −0.939987 0.341211i \(-0.889163\pi\)
0.939987 0.341211i \(-0.110837\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.74310e10 0.268576
\(612\) 0 0
\(613\) −5.36935e10 −0.380259 −0.190130 0.981759i \(-0.560891\pi\)
−0.190130 + 0.981759i \(0.560891\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.61826e11 −1.80665 −0.903323 0.428962i \(-0.858880\pi\)
−0.903323 + 0.428962i \(0.858880\pi\)
\(618\) 0 0
\(619\) 1.54542e10i 0.105265i 0.998614 + 0.0526323i \(0.0167611\pi\)
−0.998614 + 0.0526323i \(0.983239\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.91338e10 1.64983e10i −0.591685 0.109519i
\(624\) 0 0
\(625\) 1.25670e11 0.823590
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.67067e10i 0.298385i
\(630\) 0 0
\(631\) 1.70767e11 1.07718 0.538588 0.842569i \(-0.318958\pi\)
0.538588 + 0.842569i \(0.318958\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.81188e10i 0.603472i
\(636\) 0 0
\(637\) −5.09084e10 + 1.32807e11i −0.309195 + 0.806612i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.09185e11 −1.83142 −0.915708 0.401845i \(-0.868369\pi\)
−0.915708 + 0.401845i \(0.868369\pi\)
\(642\) 0 0
\(643\) 1.68796e10i 0.0987456i 0.998780 + 0.0493728i \(0.0157222\pi\)
−0.998780 + 0.0493728i \(0.984278\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.14934e11i 0.655893i 0.944696 + 0.327947i \(0.106357\pi\)
−0.944696 + 0.327947i \(0.893643\pi\)
\(648\) 0 0
\(649\) 9.72116e10i 0.547948i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.00660e10 −0.110359 −0.0551797 0.998476i \(-0.517573\pi\)
−0.0551797 + 0.998476i \(0.517573\pi\)
\(654\) 0 0
\(655\) −4.20171e11 −2.28276
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.04521e10 0.320531 0.160265 0.987074i \(-0.448765\pi\)
0.160265 + 0.987074i \(0.448765\pi\)
\(660\) 0 0
\(661\) 1.36602e11i 0.715567i 0.933805 + 0.357783i \(0.116467\pi\)
−0.933805 + 0.357783i \(0.883533\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.36903e10 3.44093e11i 0.325677 1.75950i
\(666\) 0 0
\(667\) 3.29358e11 1.66404
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.13796e11i 1.05465i
\(672\) 0 0
\(673\) 8.00609e10 0.390266 0.195133 0.980777i \(-0.437486\pi\)
0.195133 + 0.980777i \(0.437486\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.07291e11i 1.46284i 0.681930 + 0.731418i \(0.261141\pi\)
−0.681930 + 0.731418i \(0.738859\pi\)
\(678\) 0 0
\(679\) 2.10786e11 + 3.90156e10i 0.991658 + 0.183552i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.86033e10 0.0854882 0.0427441 0.999086i \(-0.486390\pi\)
0.0427441 + 0.999086i \(0.486390\pi\)
\(684\) 0 0
\(685\) 4.40667e11i 2.00147i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.86842e10i 0.437896i
\(690\) 0 0
\(691\) 2.32933e11i 1.02169i −0.859673 0.510845i \(-0.829332\pi\)
0.859673 0.510845i \(-0.170668\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.53488e11 1.51508
\(696\) 0 0
\(697\) −1.27838e11 −0.541661
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.24988e11 0.931722 0.465861 0.884858i \(-0.345745\pi\)
0.465861 + 0.884858i \(0.345745\pi\)
\(702\) 0 0
\(703\) 9.97430e10i 0.408377i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.50452e10 1.75925e10i −0.380411 0.0704126i
\(708\) 0 0
\(709\) 3.33098e11 1.31822 0.659109 0.752047i \(-0.270933\pi\)
0.659109 + 0.752047i \(0.270933\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.21316e10i 0.124329i
\(714\) 0 0
\(715\) −2.79765e11 −1.07046
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.44582e11i 1.28937i 0.764449 + 0.644684i \(0.223011\pi\)
−0.764449 + 0.644684i \(0.776989\pi\)
\(720\) 0 0
\(721\) 2.75389e11 + 5.09735e10i 1.01907 + 0.188627i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.33499e11 2.65490
\(726\) 0 0
\(727\) 2.17564e11i 0.778841i 0.921060 + 0.389421i \(0.127325\pi\)
−0.921060 + 0.389421i \(0.872675\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.81022e11i 1.33438i
\(732\) 0 0
\(733\) 3.55371e11i 1.23102i 0.788128 + 0.615511i \(0.211050\pi\)
−0.788128 + 0.615511i \(0.788950\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.81599e10 0.163236
\(738\) 0 0
\(739\) 1.02474e11 0.343588 0.171794 0.985133i \(-0.445044\pi\)
0.171794 + 0.985133i \(0.445044\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.25856e11 −1.06923 −0.534614 0.845096i \(-0.679543\pi\)
−0.534614 + 0.845096i \(0.679543\pi\)
\(744\) 0 0
\(745\) 9.45570e11i 3.06951i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.82467e10 4.76761e11i 0.280395 1.51486i
\(750\) 0 0
\(751\) −4.85707e11 −1.52691 −0.763457 0.645858i \(-0.776500\pi\)
−0.763457 + 0.645858i \(0.776500\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.02744e12i 3.16205i
\(756\) 0 0
\(757\) −3.42261e11 −1.04226 −0.521128 0.853479i \(-0.674489\pi\)
−0.521128 + 0.853479i \(0.674489\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.32098e11i 0.990210i 0.868833 + 0.495105i \(0.164870\pi\)
−0.868833 + 0.495105i \(0.835130\pi\)
\(762\) 0 0
\(763\) −4.02642e10 + 2.17531e11i −0.118801 + 0.641834i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.26670e11 −0.654957
\(768\) 0 0
\(769\) 3.52075e10i 0.100677i 0.998732 + 0.0503384i \(0.0160300\pi\)
−0.998732 + 0.0503384i \(0.983970\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.04549e11i 0.292820i −0.989224 0.146410i \(-0.953228\pi\)
0.989224 0.146410i \(-0.0467719\pi\)
\(774\) 0 0
\(775\) 7.15588e10i 0.198361i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.73000e11 −0.741331
\(780\) 0 0
\(781\) −2.53898e11 −0.682424
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.23935e11 0.853058
\(786\) 0 0
\(787\) 6.08147e11i 1.58529i −0.609682 0.792646i \(-0.708703\pi\)
0.609682 0.792646i \(-0.291297\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.46332e10 + 3.49187e11i −0.165101 + 0.891974i
\(792\) 0 0
\(793\) 4.98513e11 1.26062
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.76664e11i 0.437841i 0.975743 + 0.218920i \(0.0702535\pi\)
−0.975743 + 0.218920i \(0.929747\pi\)
\(798\) 0 0
\(799\) −9.66197e10 −0.237071
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.16818e11i 1.00250i
\(804\) 0 0
\(805\) 8.60915e11 + 1.59352e11i 2.05011 + 0.379467i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.14399e11 −0.267071 −0.133536 0.991044i \(-0.542633\pi\)
−0.133536 + 0.991044i \(0.542633\pi\)
\(810\) 0 0
\(811\) 4.13662e11i 0.956230i 0.878297 + 0.478115i \(0.158680\pi\)
−0.878297 + 0.478115i \(0.841320\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.43436e11i 0.778423i
\(816\) 0 0
\(817\) 8.13680e11i 1.82627i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.20987e11 −1.58692 −0.793459 0.608623i \(-0.791722\pi\)
−0.793459 + 0.608623i \(0.791722\pi\)
\(822\) 0 0
\(823\) −5.28868e11 −1.15278 −0.576392 0.817174i \(-0.695540\pi\)
−0.576392 + 0.817174i \(0.695540\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.45686e11 −1.80795 −0.903977 0.427582i \(-0.859365\pi\)
−0.903977 + 0.427582i \(0.859365\pi\)
\(828\) 0 0
\(829\) 2.55650e11i 0.541288i −0.962680 0.270644i \(-0.912763\pi\)
0.962680 0.270644i \(-0.0872366\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.31409e11 3.42812e11i 0.272925 0.711993i
\(834\) 0 0
\(835\) 1.30989e12 2.69457
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.54287e11i 0.916817i −0.888742 0.458409i \(-0.848420\pi\)
0.888742 0.458409i \(-0.151580\pi\)
\(840\) 0 0
\(841\) 4.36620e11 0.872809
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.21846e11i 0.435136i
\(846\) 0 0
\(847\) 4.47475e10 2.41752e11i 0.0869430 0.469718i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.49556e11 −0.475827
\(852\) 0 0
\(853\) 6.04456e11i 1.14174i −0.821039 0.570871i \(-0.806605\pi\)
0.821039 0.570871i \(-0.193395\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.79880e10i 0.144579i −0.997384 0.0722894i \(-0.976969\pi\)
0.997384 0.0722894i \(-0.0230305\pi\)
\(858\) 0 0
\(859\) 1.42565e11i 0.261843i −0.991393 0.130922i \(-0.958206\pi\)
0.991393 0.130922i \(-0.0417936\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.37615e11 −0.969233 −0.484616 0.874727i \(-0.661041\pi\)
−0.484616 + 0.874727i \(0.661041\pi\)
\(864\) 0 0
\(865\) −1.81266e12 −3.23781
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7.87629e10 0.138116
\(870\) 0 0
\(871\) 1.12295e11i 0.195114i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.29002e11 + 1.71955e11i 1.58484 + 0.293347i
\(876\) 0 0
\(877\) 1.81440e11 0.306715 0.153357 0.988171i \(-0.450991\pi\)
0.153357 + 0.988171i \(0.450991\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.93727e11i 1.64954i 0.565467 + 0.824771i \(0.308696\pi\)
−0.565467 + 0.824771i \(0.691304\pi\)
\(882\) 0 0
\(883\) −1.60514e10 −0.0264040 −0.0132020 0.999913i \(-0.504202\pi\)
−0.0132020 + 0.999913i \(0.504202\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.10193e11i 0.985763i −0.870096 0.492882i \(-0.835944\pi\)
0.870096 0.492882i \(-0.164056\pi\)
\(888\) 0 0
\(889\) −4.00105e10 + 2.16160e11i −0.0640570 + 0.346074i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.06333e11 −0.324461
\(894\) 0 0
\(895\) 9.94516e11i 1.54996i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.13989e10i 0.139927i
\(900\) 0 0
\(901\) 2.54731e11i 0.386529i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.08217e12 3.10400
\(906\) 0 0
\(907\) −5.22879e11 −0.772630 −0.386315 0.922367i \(-0.626252\pi\)
−0.386315 + 0.922367i \(0.626252\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.98988e11 0.869650 0.434825 0.900515i \(-0.356810\pi\)
0.434825 + 0.900515i \(0.356810\pi\)
\(912\) 0 0
\(913\) 1.60504e11i 0.230995i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.25657e11 1.71336e11i −1.30910 0.242309i
\(918\) 0 0
\(919\) 2.21058e10 0.0309916 0.0154958 0.999880i \(-0.495067\pi\)
0.0154958 + 0.999880i \(0.495067\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.92017e11i 0.815694i
\(924\) 0 0
\(925\) −5.55775e11 −0.759157
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.56340e11i 1.28395i −0.766724 0.641977i \(-0.778115\pi\)
0.766724 0.641977i \(-0.221885\pi\)
\(930\) 0 0
\(931\) 2.80625e11 7.32081e11i 0.373532 0.974452i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.22150e11 0.944889
\(936\) 0 0
\(937\) 6.08996e11i 0.790052i 0.918670 + 0.395026i \(0.129264\pi\)
−0.918670 + 0.395026i \(0.870736\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.11301e12i 1.41952i −0.704445 0.709759i \(-0.748804\pi\)
0.704445 0.709759i \(-0.251196\pi\)
\(942\) 0 0
\(943\) 6.83041e11i 0.863774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.71686e11 0.959491 0.479745 0.877408i \(-0.340729\pi\)
0.479745 + 0.877408i \(0.340729\pi\)
\(948\) 0 0
\(949\) −9.71901e11 −1.19828
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.10736e10 −0.0255485 −0.0127743 0.999918i \(-0.504066\pi\)
−0.0127743 + 0.999918i \(0.504066\pi\)
\(954\) 0 0
\(955\) 1.18990e12i 1.43053i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.79693e11 + 9.70810e11i −0.212450 + 1.14778i
\(960\) 0 0
\(961\) 8.43974e11 0.989545
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.01638e12i 2.32521i
\(966\) 0 0
\(967\) −2.86699e11 −0.327884 −0.163942 0.986470i \(-0.552421\pi\)
−0.163942 + 0.986470i \(0.552421\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.09523e11i 0.573175i −0.958054 0.286587i \(-0.907479\pi\)
0.958054 0.286587i \(-0.0925209\pi\)
\(972\) 0 0
\(973\) 7.78750e11 + 1.44144e11i 0.868854 + 0.160822i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.89697e11 0.647218 0.323609 0.946191i \(-0.395104\pi\)
0.323609 + 0.946191i \(0.395104\pi\)
\(978\) 0 0
\(979\) 3.99482e11i 0.434877i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.49357e11i 0.374159i 0.982345 + 0.187079i \(0.0599021\pi\)
−0.982345 + 0.187079i \(0.940098\pi\)
\(984\) 0 0
\(985\) 3.06256e12i 3.25341i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.03581e12 −2.12791
\(990\) 0 0
\(991\) −9.86530e11 −1.02286 −0.511429 0.859325i \(-0.670884\pi\)
−0.511429 + 0.859325i \(0.670884\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.20188e11 0.938824
\(996\) 0 0
\(997\) 7.46139e11i 0.755160i −0.925977 0.377580i \(-0.876756\pi\)
0.925977 0.377580i \(-0.123244\pi\)
\(998\) 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 252.9.d.c.181.8 yes 8
3.2 odd 2 inner 252.9.d.c.181.2 yes 8
7.6 odd 2 inner 252.9.d.c.181.1 8
21.20 even 2 inner 252.9.d.c.181.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.d.c.181.1 8 7.6 odd 2 inner
252.9.d.c.181.2 yes 8 3.2 odd 2 inner
252.9.d.c.181.7 yes 8 21.20 even 2 inner
252.9.d.c.181.8 yes 8 1.1 even 1 trivial