Properties

Label 252.9.d.b.181.4
Level $252$
Weight $9$
Character 252.181
Analytic conductor $102.659$
Analytic rank $0$
Dimension $6$
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] = [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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 2160x^{4} + 976392x^{2} + 85162752 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.4
Root \(-10.7169i\) of defining polynomial
Character \(\chi\) \(=\) 252.181
Dual form 252.9.d.b.181.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+333.657i q^{5} +(560.525 + 2334.65i) q^{7} +O(q^{10})\) \(q+333.657i q^{5} +(560.525 + 2334.65i) q^{7} +24127.6 q^{11} +43091.3i q^{13} -55998.7i q^{17} +208077. i q^{19} +159742. q^{23} +279298. q^{25} +326089. q^{29} -1.30508e6i q^{31} +(-778974. + 187023. i) q^{35} +1.59489e6 q^{37} +4.17224e6i q^{41} +1.46924e6 q^{43} -4.11221e6i q^{47} +(-5.13642e6 + 2.61726e6i) q^{49} -6.56996e6 q^{53} +8.05034e6i q^{55} -1.27843e7i q^{59} -1.65901e6i q^{61} -1.43777e7 q^{65} -1.08767e7 q^{67} +2.55338e7 q^{71} -1.79430e7i q^{73} +(1.35241e7 + 5.63296e7i) q^{77} +5.32302e7 q^{79} +2.08752e6i q^{83} +1.86844e7 q^{85} +9.88773e7i q^{89} +(-1.00603e8 + 2.41538e7i) q^{91} -6.94265e7 q^{95} +2.44104e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2166 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71} + 61390452 q^{77} + 186114540 q^{79} - 263210880 q^{85} - 179101056 q^{91} - 85912896 q^{95}+O(q^{100}) \) 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
\(5\) 333.657i 0.533851i 0.963717 + 0.266926i \(0.0860078\pi\)
−0.963717 + 0.266926i \(0.913992\pi\)
\(6\) 0 0
\(7\) 560.525 + 2334.65i 0.233455 + 0.972368i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 24127.6 1.64795 0.823973 0.566629i \(-0.191753\pi\)
0.823973 + 0.566629i \(0.191753\pi\)
\(12\) 0 0
\(13\) 43091.3i 1.50875i 0.656445 + 0.754374i \(0.272059\pi\)
−0.656445 + 0.754374i \(0.727941\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 55998.7i 0.670475i −0.942134 0.335237i \(-0.891183\pi\)
0.942134 0.335237i \(-0.108817\pi\)
\(18\) 0 0
\(19\) 208077.i 1.59665i 0.602225 + 0.798326i \(0.294281\pi\)
−0.602225 + 0.798326i \(0.705719\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 159742. 0.570831 0.285416 0.958404i \(-0.407868\pi\)
0.285416 + 0.958404i \(0.407868\pi\)
\(24\) 0 0
\(25\) 279298. 0.715003
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 326089. 0.461045 0.230523 0.973067i \(-0.425956\pi\)
0.230523 + 0.973067i \(0.425956\pi\)
\(30\) 0 0
\(31\) 1.30508e6i 1.41315i −0.707636 0.706577i \(-0.750238\pi\)
0.707636 0.706577i \(-0.249762\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −778974. + 187023.i −0.519100 + 0.124630i
\(36\) 0 0
\(37\) 1.59489e6 0.850989 0.425495 0.904961i \(-0.360100\pi\)
0.425495 + 0.904961i \(0.360100\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.17224e6i 1.47650i 0.674526 + 0.738251i \(0.264348\pi\)
−0.674526 + 0.738251i \(0.735652\pi\)
\(42\) 0 0
\(43\) 1.46924e6 0.429754 0.214877 0.976641i \(-0.431065\pi\)
0.214877 + 0.976641i \(0.431065\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.11221e6i 0.842720i −0.906893 0.421360i \(-0.861553\pi\)
0.906893 0.421360i \(-0.138447\pi\)
\(48\) 0 0
\(49\) −5.13642e6 + 2.61726e6i −0.890998 + 0.454008i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.56996e6 −0.832644 −0.416322 0.909217i \(-0.636681\pi\)
−0.416322 + 0.909217i \(0.636681\pi\)
\(54\) 0 0
\(55\) 8.05034e6i 0.879758i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.27843e7i 1.05504i −0.849544 0.527518i \(-0.823123\pi\)
0.849544 0.527518i \(-0.176877\pi\)
\(60\) 0 0
\(61\) 1.65901e6i 0.119820i −0.998204 0.0599100i \(-0.980919\pi\)
0.998204 0.0599100i \(-0.0190814\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.43777e7 −0.805446
\(66\) 0 0
\(67\) −1.08767e7 −0.539755 −0.269877 0.962895i \(-0.586983\pi\)
−0.269877 + 0.962895i \(0.586983\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.55338e7 1.00480 0.502402 0.864634i \(-0.332450\pi\)
0.502402 + 0.864634i \(0.332450\pi\)
\(72\) 0 0
\(73\) 1.79430e7i 0.631834i −0.948787 0.315917i \(-0.897688\pi\)
0.948787 0.315917i \(-0.102312\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.35241e7 + 5.63296e7i 0.384721 + 1.60241i
\(78\) 0 0
\(79\) 5.32302e7 1.36663 0.683313 0.730125i \(-0.260538\pi\)
0.683313 + 0.730125i \(0.260538\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.08752e6i 0.0439864i 0.999758 + 0.0219932i \(0.00700122\pi\)
−0.999758 + 0.0219932i \(0.992999\pi\)
\(84\) 0 0
\(85\) 1.86844e7 0.357934
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.88773e7i 1.57593i 0.615721 + 0.787964i \(0.288865\pi\)
−0.615721 + 0.787964i \(0.711135\pi\)
\(90\) 0 0
\(91\) −1.00603e8 + 2.41538e7i −1.46706 + 0.352224i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.94265e7 −0.852375
\(96\) 0 0
\(97\) 2.44104e7i 0.275733i 0.990451 + 0.137867i \(0.0440245\pi\)
−0.990451 + 0.137867i \(0.955976\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.28121e7i 0.795808i −0.917427 0.397904i \(-0.869738\pi\)
0.917427 0.397904i \(-0.130262\pi\)
\(102\) 0 0
\(103\) 8.72625e7i 0.775316i 0.921803 + 0.387658i \(0.126716\pi\)
−0.921803 + 0.387658i \(0.873284\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.14648e8 −1.63754 −0.818769 0.574123i \(-0.805343\pi\)
−0.818769 + 0.574123i \(0.805343\pi\)
\(108\) 0 0
\(109\) −8.44130e7 −0.598003 −0.299002 0.954253i \(-0.596654\pi\)
−0.299002 + 0.954253i \(0.596654\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.83510e8 1.73882 0.869411 0.494090i \(-0.164499\pi\)
0.869411 + 0.494090i \(0.164499\pi\)
\(114\) 0 0
\(115\) 5.32990e7i 0.304739i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.30738e8 3.13887e7i 0.651948 0.156526i
\(120\) 0 0
\(121\) 3.67782e8 1.71573
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.23525e8i 0.915556i
\(126\) 0 0
\(127\) −6.18914e7 −0.237911 −0.118956 0.992900i \(-0.537955\pi\)
−0.118956 + 0.992900i \(0.537955\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.29768e8i 1.45932i 0.683813 + 0.729658i \(0.260321\pi\)
−0.683813 + 0.729658i \(0.739679\pi\)
\(132\) 0 0
\(133\) −4.85789e8 + 1.16633e8i −1.55253 + 0.372746i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.92189e8 −0.829432 −0.414716 0.909951i \(-0.636119\pi\)
−0.414716 + 0.909951i \(0.636119\pi\)
\(138\) 0 0
\(139\) 9.34937e6i 0.0250451i −0.999922 0.0125226i \(-0.996014\pi\)
0.999922 0.0125226i \(-0.00398616\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.03969e9i 2.48633i
\(144\) 0 0
\(145\) 1.08802e8i 0.246130i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.12014e8 −1.64747 −0.823736 0.566973i \(-0.808114\pi\)
−0.823736 + 0.566973i \(0.808114\pi\)
\(150\) 0 0
\(151\) 1.07626e7 0.0207019 0.0103510 0.999946i \(-0.496705\pi\)
0.0103510 + 0.999946i \(0.496705\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.35448e8 0.754414
\(156\) 0 0
\(157\) 4.58487e8i 0.754620i 0.926087 + 0.377310i \(0.123151\pi\)
−0.926087 + 0.377310i \(0.876849\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.95394e7 + 3.72942e8i 0.133263 + 0.555058i
\(162\) 0 0
\(163\) 6.57211e8 0.931010 0.465505 0.885045i \(-0.345873\pi\)
0.465505 + 0.885045i \(0.345873\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.94686e8i 0.636010i 0.948089 + 0.318005i \(0.103013\pi\)
−0.948089 + 0.318005i \(0.896987\pi\)
\(168\) 0 0
\(169\) −1.04113e9 −1.27632
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.94129e8i 0.886557i −0.896384 0.443279i \(-0.853815\pi\)
0.896384 0.443279i \(-0.146185\pi\)
\(174\) 0 0
\(175\) 1.56553e8 + 6.52064e8i 0.166921 + 0.695246i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.85404e8 −0.180595 −0.0902976 0.995915i \(-0.528782\pi\)
−0.0902976 + 0.995915i \(0.528782\pi\)
\(180\) 0 0
\(181\) 1.44977e9i 1.35078i 0.737462 + 0.675389i \(0.236024\pi\)
−0.737462 + 0.675389i \(0.763976\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.32146e8i 0.454302i
\(186\) 0 0
\(187\) 1.35111e9i 1.10491i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.13964e8 0.0856319 0.0428159 0.999083i \(-0.486367\pi\)
0.0428159 + 0.999083i \(0.486367\pi\)
\(192\) 0 0
\(193\) 9.27036e8 0.668140 0.334070 0.942548i \(-0.391578\pi\)
0.334070 + 0.942548i \(0.391578\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.50846e9 −1.66549 −0.832747 0.553654i \(-0.813233\pi\)
−0.832747 + 0.553654i \(0.813233\pi\)
\(198\) 0 0
\(199\) 1.31010e9i 0.835394i 0.908586 + 0.417697i \(0.137163\pi\)
−0.908586 + 0.417697i \(0.862837\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.82781e8 + 7.61304e8i 0.107633 + 0.448305i
\(204\) 0 0
\(205\) −1.39210e9 −0.788232
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.02040e9i 2.63120i
\(210\) 0 0
\(211\) −9.73665e8 −0.491224 −0.245612 0.969368i \(-0.578989\pi\)
−0.245612 + 0.969368i \(0.578989\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.90223e8i 0.229425i
\(216\) 0 0
\(217\) 3.04691e9 7.31529e8i 1.37411 0.329908i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.41306e9 1.01158
\(222\) 0 0
\(223\) 1.64534e9i 0.665329i 0.943045 + 0.332665i \(0.107948\pi\)
−0.943045 + 0.332665i \(0.892052\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.98469e8i 0.225392i 0.993630 + 0.112696i \(0.0359486\pi\)
−0.993630 + 0.112696i \(0.964051\pi\)
\(228\) 0 0
\(229\) 5.33903e8i 0.194142i −0.995277 0.0970712i \(-0.969053\pi\)
0.995277 0.0970712i \(-0.0309475\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.41592e9 0.819708 0.409854 0.912151i \(-0.365580\pi\)
0.409854 + 0.912151i \(0.365580\pi\)
\(234\) 0 0
\(235\) 1.37207e9 0.449887
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.20541e9 −0.369440 −0.184720 0.982791i \(-0.559138\pi\)
−0.184720 + 0.982791i \(0.559138\pi\)
\(240\) 0 0
\(241\) 2.76959e9i 0.821007i −0.911859 0.410504i \(-0.865353\pi\)
0.911859 0.410504i \(-0.134647\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8.73268e8 1.71380e9i −0.242373 0.475660i
\(246\) 0 0
\(247\) −8.96633e9 −2.40894
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.72275e9i 1.44182i −0.693031 0.720908i \(-0.743725\pi\)
0.693031 0.720908i \(-0.256275\pi\)
\(252\) 0 0
\(253\) 3.85419e9 0.940700
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.33650e8i 0.0535591i 0.999641 + 0.0267796i \(0.00852522\pi\)
−0.999641 + 0.0267796i \(0.991475\pi\)
\(258\) 0 0
\(259\) 8.93976e8 + 3.72352e9i 0.198667 + 0.827474i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.04999e8 −0.0637494 −0.0318747 0.999492i \(-0.510148\pi\)
−0.0318747 + 0.999492i \(0.510148\pi\)
\(264\) 0 0
\(265\) 2.19211e9i 0.444508i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.13017e9i 1.17075i −0.810763 0.585374i \(-0.800948\pi\)
0.810763 0.585374i \(-0.199052\pi\)
\(270\) 0 0
\(271\) 3.43372e9i 0.636632i 0.947985 + 0.318316i \(0.103117\pi\)
−0.947985 + 0.318316i \(0.896883\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.73879e9 1.17829
\(276\) 0 0
\(277\) −7.30404e9 −1.24064 −0.620318 0.784351i \(-0.712996\pi\)
−0.620318 + 0.784351i \(0.712996\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.46382e9 −0.715947 −0.357974 0.933732i \(-0.616532\pi\)
−0.357974 + 0.933732i \(0.616532\pi\)
\(282\) 0 0
\(283\) 5.02284e9i 0.783075i −0.920162 0.391537i \(-0.871943\pi\)
0.920162 0.391537i \(-0.128057\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.74074e9 + 2.33864e9i −1.43570 + 0.344696i
\(288\) 0 0
\(289\) 3.83990e9 0.550463
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.99782e9i 0.678125i −0.940764 0.339063i \(-0.889890\pi\)
0.940764 0.339063i \(-0.110110\pi\)
\(294\) 0 0
\(295\) 4.26556e9 0.563232
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.88350e9i 0.861240i
\(300\) 0 0
\(301\) 8.23547e8 + 3.43017e9i 0.100328 + 0.417879i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.53540e8 0.0639660
\(306\) 0 0
\(307\) 1.40127e10i 1.57750i −0.614714 0.788750i \(-0.710729\pi\)
0.614714 0.788750i \(-0.289271\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.19762e9i 0.983182i −0.870826 0.491591i \(-0.836415\pi\)
0.870826 0.491591i \(-0.163585\pi\)
\(312\) 0 0
\(313\) 1.63979e10i 1.70848i −0.519876 0.854242i \(-0.674022\pi\)
0.519876 0.854242i \(-0.325978\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.51375e9 0.942138 0.471069 0.882096i \(-0.343868\pi\)
0.471069 + 0.882096i \(0.343868\pi\)
\(318\) 0 0
\(319\) 7.86773e9 0.759778
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.16521e10 1.07052
\(324\) 0 0
\(325\) 1.20353e10i 1.07876i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.60058e9 2.30499e9i 0.819434 0.196737i
\(330\) 0 0
\(331\) 4.91615e9 0.409556 0.204778 0.978808i \(-0.434353\pi\)
0.204778 + 0.978808i \(0.434353\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.62907e9i 0.288149i
\(336\) 0 0
\(337\) 2.07250e10 1.60685 0.803425 0.595406i \(-0.203009\pi\)
0.803425 + 0.595406i \(0.203009\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.14884e10i 2.32880i
\(342\) 0 0
\(343\) −8.98950e9 1.05247e10i −0.649470 0.760387i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.68352e9 0.185092 0.0925459 0.995708i \(-0.470500\pi\)
0.0925459 + 0.995708i \(0.470500\pi\)
\(348\) 0 0
\(349\) 1.30154e10i 0.877312i 0.898655 + 0.438656i \(0.144545\pi\)
−0.898655 + 0.438656i \(0.855455\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.69314e10i 1.09042i −0.838299 0.545211i \(-0.816450\pi\)
0.838299 0.545211i \(-0.183550\pi\)
\(354\) 0 0
\(355\) 8.51952e9i 0.536416i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.74172e10 −1.65061 −0.825307 0.564685i \(-0.808998\pi\)
−0.825307 + 0.564685i \(0.808998\pi\)
\(360\) 0 0
\(361\) −2.63126e10 −1.54930
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.98680e9 0.337305
\(366\) 0 0
\(367\) 9.59511e9i 0.528915i −0.964397 0.264457i \(-0.914807\pi\)
0.964397 0.264457i \(-0.0851929\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.68263e9 1.53386e10i −0.194385 0.809636i
\(372\) 0 0
\(373\) 1.78953e10 0.924492 0.462246 0.886752i \(-0.347044\pi\)
0.462246 + 0.886752i \(0.347044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.40516e10i 0.695601i
\(378\) 0 0
\(379\) −2.75716e10 −1.33630 −0.668152 0.744025i \(-0.732914\pi\)
−0.668152 + 0.744025i \(0.732914\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.23133e9i 0.150171i −0.997177 0.0750855i \(-0.976077\pi\)
0.997177 0.0750855i \(-0.0239230\pi\)
\(384\) 0 0
\(385\) −1.87948e10 + 4.51241e9i −0.855449 + 0.205384i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.43217e9 −0.280905 −0.140452 0.990087i \(-0.544856\pi\)
−0.140452 + 0.990087i \(0.544856\pi\)
\(390\) 0 0
\(391\) 8.94535e9i 0.382728i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.77606e10i 0.729575i
\(396\) 0 0
\(397\) 2.57203e10i 1.03541i −0.855558 0.517707i \(-0.826786\pi\)
0.855558 0.517707i \(-0.173214\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.25006e10 −1.25694 −0.628469 0.777835i \(-0.716318\pi\)
−0.628469 + 0.777835i \(0.716318\pi\)
\(402\) 0 0
\(403\) 5.62375e10 2.13209
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.84809e10 1.40238
\(408\) 0 0
\(409\) 2.57138e10i 0.918909i −0.888201 0.459454i \(-0.848045\pi\)
0.888201 0.459454i \(-0.151955\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.98468e10 7.16589e9i 1.02588 0.246303i
\(414\) 0 0
\(415\) −6.96516e8 −0.0234822
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.24082e10i 0.402581i 0.979532 + 0.201291i \(0.0645136\pi\)
−0.979532 + 0.201291i \(0.935486\pi\)
\(420\) 0 0
\(421\) −2.05827e10 −0.655201 −0.327600 0.944816i \(-0.606240\pi\)
−0.327600 + 0.944816i \(0.606240\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.56403e10i 0.479392i
\(426\) 0 0
\(427\) 3.87321e9 9.29915e8i 0.116509 0.0279725i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.23479e9 0.122722 0.0613611 0.998116i \(-0.480456\pi\)
0.0613611 + 0.998116i \(0.480456\pi\)
\(432\) 0 0
\(433\) 4.21039e10i 1.19776i −0.800838 0.598882i \(-0.795612\pi\)
0.800838 0.598882i \(-0.204388\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.32387e10i 0.911419i
\(438\) 0 0
\(439\) 4.07786e10i 1.09793i 0.835845 + 0.548965i \(0.184978\pi\)
−0.835845 + 0.548965i \(0.815022\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.88976e8 0.0100997 0.00504984 0.999987i \(-0.498393\pi\)
0.00504984 + 0.999987i \(0.498393\pi\)
\(444\) 0 0
\(445\) −3.29911e10 −0.841312
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.69794e10 −0.663814 −0.331907 0.943312i \(-0.607692\pi\)
−0.331907 + 0.943312i \(0.607692\pi\)
\(450\) 0 0
\(451\) 1.00666e11i 2.43320i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.05907e9 3.35670e10i −0.188035 0.783190i
\(456\) 0 0
\(457\) 6.34464e10 1.45460 0.727298 0.686322i \(-0.240776\pi\)
0.727298 + 0.686322i \(0.240776\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.53281e10i 1.22502i −0.790464 0.612508i \(-0.790161\pi\)
0.790464 0.612508i \(-0.209839\pi\)
\(462\) 0 0
\(463\) −6.29238e10 −1.36928 −0.684638 0.728883i \(-0.740040\pi\)
−0.684638 + 0.728883i \(0.740040\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.29439e10i 1.95413i 0.212943 + 0.977065i \(0.431695\pi\)
−0.212943 + 0.977065i \(0.568305\pi\)
\(468\) 0 0
\(469\) −6.09664e9 2.53932e10i −0.126008 0.524840i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.54493e10 0.708211
\(474\) 0 0
\(475\) 5.81156e10i 1.14161i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.92837e10i 0.556268i 0.960542 + 0.278134i \(0.0897159\pi\)
−0.960542 + 0.278134i \(0.910284\pi\)
\(480\) 0 0
\(481\) 6.87260e10i 1.28393i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.14472e9 −0.147200
\(486\) 0 0
\(487\) 9.87418e10 1.75544 0.877718 0.479178i \(-0.159065\pi\)
0.877718 + 0.479178i \(0.159065\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.84236e10 0.316992 0.158496 0.987360i \(-0.449335\pi\)
0.158496 + 0.987360i \(0.449335\pi\)
\(492\) 0 0
\(493\) 1.82605e10i 0.309119i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.43123e10 + 5.96125e10i 0.234576 + 0.977039i
\(498\) 0 0
\(499\) −9.28072e10 −1.49685 −0.748427 0.663217i \(-0.769191\pi\)
−0.748427 + 0.663217i \(0.769191\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.06679e11i 1.66651i 0.552886 + 0.833257i \(0.313527\pi\)
−0.552886 + 0.833257i \(0.686473\pi\)
\(504\) 0 0
\(505\) 2.76308e10 0.424843
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.90061e10i 1.32602i 0.748612 + 0.663008i \(0.230720\pi\)
−0.748612 + 0.663008i \(0.769280\pi\)
\(510\) 0 0
\(511\) 4.18907e10 1.00575e10i 0.614375 0.147505i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.91157e10 −0.413903
\(516\) 0 0
\(517\) 9.92176e10i 1.38876i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.08443e11i 1.47181i −0.677086 0.735904i \(-0.736758\pi\)
0.677086 0.735904i \(-0.263242\pi\)
\(522\) 0 0
\(523\) 3.49280e10i 0.466839i −0.972376 0.233420i \(-0.925008\pi\)
0.972376 0.233420i \(-0.0749916\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.30827e10 −0.947485
\(528\) 0 0
\(529\) −5.27935e10 −0.674152
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.79787e11 −2.22767
\(534\) 0 0
\(535\) 7.16188e10i 0.874202i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.23930e11 + 6.31483e10i −1.46832 + 0.748180i
\(540\) 0 0
\(541\) −8.47596e10 −0.989464 −0.494732 0.869046i \(-0.664734\pi\)
−0.494732 + 0.869046i \(0.664734\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.81650e10i 0.319245i
\(546\) 0 0
\(547\) −1.64069e10 −0.183264 −0.0916320 0.995793i \(-0.529208\pi\)
−0.0916320 + 0.995793i \(0.529208\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.78516e10i 0.736129i
\(552\) 0 0
\(553\) 2.98369e10 + 1.24274e11i 0.319045 + 1.32886i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.61798e11 1.68094 0.840471 0.541857i \(-0.182279\pi\)
0.840471 + 0.541857i \(0.182279\pi\)
\(558\) 0 0
\(559\) 6.33116e10i 0.648390i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.20561e10i 0.418597i 0.977852 + 0.209298i \(0.0671180\pi\)
−0.977852 + 0.209298i \(0.932882\pi\)
\(564\) 0 0
\(565\) 9.45952e10i 0.928272i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.48271e11 1.41451 0.707256 0.706957i \(-0.249933\pi\)
0.707256 + 0.706957i \(0.249933\pi\)
\(570\) 0 0
\(571\) −8.03731e9 −0.0756078 −0.0378039 0.999285i \(-0.512036\pi\)
−0.0378039 + 0.999285i \(0.512036\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.46156e10 0.408146
\(576\) 0 0
\(577\) 1.50045e11i 1.35369i 0.736126 + 0.676845i \(0.236653\pi\)
−0.736126 + 0.676845i \(0.763347\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4.87364e9 + 1.17011e9i −0.0427710 + 0.0102688i
\(582\) 0 0
\(583\) −1.58517e11 −1.37215
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.31985e11i 1.11166i 0.831296 + 0.555830i \(0.187599\pi\)
−0.831296 + 0.555830i \(0.812401\pi\)
\(588\) 0 0
\(589\) 2.71557e11 2.25632
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.77348e11i 1.43419i −0.696976 0.717094i \(-0.745472\pi\)
0.696976 0.717094i \(-0.254528\pi\)
\(594\) 0 0
\(595\) 1.04731e10 + 4.36216e10i 0.0835613 + 0.348043i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.94212e10 −0.694597 −0.347299 0.937755i \(-0.612901\pi\)
−0.347299 + 0.937755i \(0.612901\pi\)
\(600\) 0 0
\(601\) 6.85319e10i 0.525285i 0.964893 + 0.262642i \(0.0845939\pi\)
−0.964893 + 0.262642i \(0.915406\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.22713e11i 0.915944i
\(606\) 0 0
\(607\) 3.66702e9i 0.0270121i −0.999909 0.0135061i \(-0.995701\pi\)
0.999909 0.0135061i \(-0.00429924\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.77200e11 1.27145
\(612\) 0 0
\(613\) −6.49244e10 −0.459797 −0.229898 0.973215i \(-0.573839\pi\)
−0.229898 + 0.973215i \(0.573839\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.79839e10 0.676105 0.338052 0.941127i \(-0.390232\pi\)
0.338052 + 0.941127i \(0.390232\pi\)
\(618\) 0 0
\(619\) 1.16922e11i 0.796403i −0.917298 0.398202i \(-0.869634\pi\)
0.917298 0.398202i \(-0.130366\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.30844e11 + 5.54232e10i −1.53238 + 0.367908i
\(624\) 0 0
\(625\) 3.45203e10 0.226232
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.93119e10i 0.570567i
\(630\) 0 0
\(631\) 1.32605e11 0.836457 0.418229 0.908342i \(-0.362651\pi\)
0.418229 + 0.908342i \(0.362651\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.06505e10i 0.127009i
\(636\) 0 0
\(637\) −1.12781e11 2.21335e11i −0.684983 1.34429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.82510e11 1.08107 0.540536 0.841321i \(-0.318221\pi\)
0.540536 + 0.841321i \(0.318221\pi\)
\(642\) 0 0
\(643\) 4.16413e10i 0.243602i 0.992555 + 0.121801i \(0.0388669\pi\)
−0.992555 + 0.121801i \(0.961133\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.19404e11i 0.681402i 0.940172 + 0.340701i \(0.110664\pi\)
−0.940172 + 0.340701i \(0.889336\pi\)
\(648\) 0 0
\(649\) 3.08453e11i 1.73864i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.16240e11 −1.73926 −0.869629 0.493705i \(-0.835642\pi\)
−0.869629 + 0.493705i \(0.835642\pi\)
\(654\) 0 0
\(655\) −1.43395e11 −0.779057
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.31924e10 0.175994 0.0879969 0.996121i \(-0.471953\pi\)
0.0879969 + 0.996121i \(0.471953\pi\)
\(660\) 0 0
\(661\) 1.67799e11i 0.878990i −0.898245 0.439495i \(-0.855157\pi\)
0.898245 0.439495i \(-0.144843\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.89153e10 1.62087e11i −0.198991 0.828822i
\(666\) 0 0
\(667\) 5.20900e10 0.263179
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.00279e10i 0.197457i
\(672\) 0 0
\(673\) 9.56218e10 0.466119 0.233059 0.972462i \(-0.425126\pi\)
0.233059 + 0.972462i \(0.425126\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.73296e10i 0.368122i −0.982915 0.184061i \(-0.941076\pi\)
0.982915 0.184061i \(-0.0589243\pi\)
\(678\) 0 0
\(679\) −5.69900e10 + 1.36827e10i −0.268114 + 0.0643712i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.15222e10 0.190808 0.0954042 0.995439i \(-0.469586\pi\)
0.0954042 + 0.995439i \(0.469586\pi\)
\(684\) 0 0
\(685\) 9.74908e10i 0.442793i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.83108e11i 1.25625i
\(690\) 0 0
\(691\) 2.32923e11i 1.02164i 0.859687 + 0.510822i \(0.170659\pi\)
−0.859687 + 0.510822i \(0.829341\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.11948e9 0.0133704
\(696\) 0 0
\(697\) 2.33640e11 0.989957
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.07467e11 0.445044 0.222522 0.974928i \(-0.428571\pi\)
0.222522 + 0.974928i \(0.428571\pi\)
\(702\) 0 0
\(703\) 3.31861e11i 1.35873i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.93338e11 4.64183e10i 0.773818 0.185785i
\(708\) 0 0
\(709\) −7.41741e10 −0.293540 −0.146770 0.989171i \(-0.546888\pi\)
−0.146770 + 0.989171i \(0.546888\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.08476e11i 0.806673i
\(714\) 0 0
\(715\) −3.46900e11 −1.32733
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.87359e11i 0.701065i −0.936550 0.350533i \(-0.886001\pi\)
0.936550 0.350533i \(-0.113999\pi\)
\(720\) 0 0
\(721\) −2.03728e11 + 4.89128e10i −0.753892 + 0.181001i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.10759e10 0.329649
\(726\) 0 0
\(727\) 3.65939e11i 1.31000i 0.755629 + 0.655000i \(0.227331\pi\)
−0.755629 + 0.655000i \(0.772669\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.22757e10i 0.288139i
\(732\) 0 0
\(733\) 3.01274e11i 1.04363i −0.853060 0.521813i \(-0.825256\pi\)
0.853060 0.521813i \(-0.174744\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.62428e11 −0.889487
\(738\) 0 0
\(739\) 2.65721e11 0.890940 0.445470 0.895297i \(-0.353037\pi\)
0.445470 + 0.895297i \(0.353037\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.14572e11 1.36033 0.680166 0.733058i \(-0.261908\pi\)
0.680166 + 0.733058i \(0.261908\pi\)
\(744\) 0 0
\(745\) 2.70934e11i 0.879505i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.20315e11 5.01129e11i −0.382291 1.59229i
\(750\) 0 0
\(751\) 4.44651e11 1.39785 0.698924 0.715196i \(-0.253663\pi\)
0.698924 + 0.715196i \(0.253663\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.59103e9i 0.0110518i
\(756\) 0 0
\(757\) −2.60251e11 −0.792517 −0.396258 0.918139i \(-0.629692\pi\)
−0.396258 + 0.918139i \(0.629692\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.68056e10i 0.139559i 0.997562 + 0.0697797i \(0.0222296\pi\)
−0.997562 + 0.0697797i \(0.977770\pi\)
\(762\) 0 0
\(763\) −4.73156e10 1.97075e11i −0.139607 0.581479i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.50890e11 1.59178
\(768\) 0 0
\(769\) 3.82049e11i 1.09248i −0.837628 0.546240i \(-0.816058\pi\)
0.837628 0.546240i \(-0.183942\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.60161e10i 0.100874i 0.998727 + 0.0504369i \(0.0160614\pi\)
−0.998727 + 0.0504369i \(0.983939\pi\)
\(774\) 0 0
\(775\) 3.64506e11i 1.01041i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.68149e11 −2.35746
\(780\) 0 0
\(781\) 6.16068e11 1.65586
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.52977e11 −0.402855
\(786\) 0 0
\(787\) 3.27297e11i 0.853184i −0.904444 0.426592i \(-0.859714\pi\)
0.904444 0.426592i \(-0.140286\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.58914e11 + 6.61898e11i 0.405936 + 1.69077i
\(792\) 0 0
\(793\) 7.14889e10 0.180778
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.29152e11i 1.55927i −0.626232 0.779636i \(-0.715404\pi\)
0.626232 0.779636i \(-0.284596\pi\)
\(798\) 0 0
\(799\) −2.30278e11 −0.565023
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.32921e11i 1.04123i
\(804\) 0 0
\(805\) −1.24435e11 + 2.98754e10i −0.296318 + 0.0711428i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.75487e11 0.643142 0.321571 0.946885i \(-0.395789\pi\)
0.321571 + 0.946885i \(0.395789\pi\)
\(810\) 0 0
\(811\) 3.31344e11i 0.765942i −0.923760 0.382971i \(-0.874901\pi\)
0.923760 0.382971i \(-0.125099\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.19283e11i 0.497021i
\(816\) 0 0
\(817\) 3.05716e11i 0.686167i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.05208e11 −0.231566 −0.115783 0.993275i \(-0.536938\pi\)
−0.115783 + 0.993275i \(0.536938\pi\)
\(822\) 0 0
\(823\) 2.65593e11 0.578918 0.289459 0.957190i \(-0.406525\pi\)
0.289459 + 0.957190i \(0.406525\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.81754e11 −0.602349 −0.301175 0.953569i \(-0.597379\pi\)
−0.301175 + 0.953569i \(0.597379\pi\)
\(828\) 0 0
\(829\) 1.61031e11i 0.340950i 0.985362 + 0.170475i \(0.0545303\pi\)
−0.985362 + 0.170475i \(0.945470\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.46563e11 + 2.87633e11i 0.304401 + 0.597392i
\(834\) 0 0
\(835\) −1.65056e11 −0.339535
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.23152e11i 0.248539i 0.992248 + 0.124270i \(0.0396588\pi\)
−0.992248 + 0.124270i \(0.960341\pi\)
\(840\) 0 0
\(841\) −3.93913e11 −0.787437
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.47381e11i 0.681364i
\(846\) 0 0
\(847\) 2.06151e11 + 8.58643e11i 0.400545 + 1.66832i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.54771e11 0.485771
\(852\) 0 0
\(853\) 9.07583e11i 1.71431i 0.515056 + 0.857156i \(0.327771\pi\)
−0.515056 + 0.857156i \(0.672229\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.59324e11i 1.40768i −0.710359 0.703840i \(-0.751467\pi\)
0.710359 0.703840i \(-0.248533\pi\)
\(858\) 0 0
\(859\) 1.59744e11i 0.293394i −0.989181 0.146697i \(-0.953136\pi\)
0.989181 0.146697i \(-0.0468643\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.96333e11 −0.894807 −0.447404 0.894332i \(-0.647651\pi\)
−0.447404 + 0.894332i \(0.647651\pi\)
\(864\) 0 0
\(865\) 2.64967e11 0.473290
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.28432e12 2.25213
\(870\) 0 0
\(871\) 4.68690e11i 0.814353i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.21853e11 + 1.25291e11i −0.890257 + 0.213741i
\(876\) 0 0
\(877\) −5.74134e11 −0.970543 −0.485272 0.874363i \(-0.661279\pi\)
−0.485272 + 0.874363i \(0.661279\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.34834e11i 0.223819i 0.993718 + 0.111909i \(0.0356967\pi\)
−0.993718 + 0.111909i \(0.964303\pi\)
\(882\) 0 0
\(883\) 2.79034e11 0.459002 0.229501 0.973308i \(-0.426291\pi\)
0.229501 + 0.973308i \(0.426291\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.83305e11i 0.457677i 0.973464 + 0.228838i \(0.0734928\pi\)
−0.973464 + 0.228838i \(0.926507\pi\)
\(888\) 0 0
\(889\) −3.46916e10 1.44495e11i −0.0555415 0.231337i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.55657e11 1.34553
\(894\) 0 0
\(895\) 6.18612e10i 0.0964109i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.25571e11i 0.651528i
\(900\) 0 0
\(901\) 3.67910e11i 0.558267i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.83725e11 −0.721115
\(906\) 0 0
\(907\) −4.97993e11 −0.735858 −0.367929 0.929854i \(-0.619933\pi\)
−0.367929 + 0.929854i \(0.619933\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.98004e11 −1.15859 −0.579297 0.815117i \(-0.696673\pi\)
−0.579297 + 0.815117i \(0.696673\pi\)
\(912\) 0 0
\(913\) 5.03669e10i 0.0724873i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.00336e12 + 2.40896e11i −1.41899 + 0.340684i
\(918\) 0 0
\(919\) −3.12582e11 −0.438230 −0.219115 0.975699i \(-0.570317\pi\)
−0.219115 + 0.975699i \(0.570317\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.10028e12i 1.51600i
\(924\) 0 0
\(925\) 4.45450e11 0.608460
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.16004e12i 1.55743i 0.627377 + 0.778716i \(0.284129\pi\)
−0.627377 + 0.778716i \(0.715871\pi\)
\(930\) 0 0
\(931\) −5.44593e11 1.06877e12i −0.724892 1.42261i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.50809e11 0.589856
\(936\) 0 0
\(937\) 9.95746e11i 1.29179i −0.763428 0.645893i \(-0.776485\pi\)
0.763428 0.645893i \(-0.223515\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.68630e11i 0.725222i 0.931941 + 0.362611i \(0.118115\pi\)
−0.931941 + 0.362611i \(0.881885\pi\)
\(942\) 0 0
\(943\) 6.66482e11i 0.842833i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.13162e12 1.40702 0.703512 0.710684i \(-0.251614\pi\)
0.703512 + 0.710684i \(0.251614\pi\)
\(948\) 0 0
\(949\) 7.73187e11 0.953278
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.59922e11 −0.193882 −0.0969411 0.995290i \(-0.530906\pi\)
−0.0969411 + 0.995290i \(0.530906\pi\)
\(954\) 0 0
\(955\) 3.80250e10i 0.0457147i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.63779e11 6.82159e11i −0.193635 0.806513i
\(960\) 0 0
\(961\) −8.50338e11 −0.997006
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.09312e11i 0.356687i
\(966\) 0 0
\(967\) 1.40368e12 1.60532 0.802660 0.596436i \(-0.203417\pi\)
0.802660 + 0.596436i \(0.203417\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.57502e12i 1.77178i 0.463896 + 0.885890i \(0.346451\pi\)
−0.463896 + 0.885890i \(0.653549\pi\)
\(972\) 0 0
\(973\) 2.18276e10 5.24055e9i 0.0243531 0.00584690i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.31282e11 0.692859 0.346429 0.938076i \(-0.387394\pi\)
0.346429 + 0.938076i \(0.387394\pi\)
\(978\) 0 0
\(979\) 2.38567e12i 2.59705i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.48559e12i 1.59106i 0.605916 + 0.795528i \(0.292807\pi\)
−0.605916 + 0.795528i \(0.707193\pi\)
\(984\) 0 0
\(985\) 8.36966e11i 0.889126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.34700e11 0.245317
\(990\) 0 0
\(991\) −1.22790e12 −1.27312 −0.636558 0.771228i \(-0.719643\pi\)
−0.636558 + 0.771228i \(0.719643\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.37123e11 −0.445976
\(996\) 0 0
\(997\) 1.74247e12i 1.76354i 0.471680 + 0.881770i \(0.343648\pi\)
−0.471680 + 0.881770i \(0.656352\pi\)
\(998\) 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.b.181.4 6
3.2 odd 2 28.9.b.a.13.4 yes 6
7.6 odd 2 inner 252.9.d.b.181.3 6
12.11 even 2 112.9.c.d.97.3 6
21.2 odd 6 196.9.h.b.129.3 12
21.5 even 6 196.9.h.b.129.4 12
21.11 odd 6 196.9.h.b.117.4 12
21.17 even 6 196.9.h.b.117.3 12
21.20 even 2 28.9.b.a.13.3 6
84.83 odd 2 112.9.c.d.97.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.9.b.a.13.3 6 21.20 even 2
28.9.b.a.13.4 yes 6 3.2 odd 2
112.9.c.d.97.3 6 12.11 even 2
112.9.c.d.97.4 6 84.83 odd 2
196.9.h.b.117.3 12 21.17 even 6
196.9.h.b.117.4 12 21.11 odd 6
196.9.h.b.129.3 12 21.2 odd 6
196.9.h.b.129.4 12 21.5 even 6
252.9.d.b.181.3 6 7.6 odd 2 inner
252.9.d.b.181.4 6 1.1 even 1 trivial