Properties

Label 384.10.d.f.193.17
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,10,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{175}\cdot 3^{32} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.17
Root \(-271.678 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.f.193.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+81.0000i q^{3} +573.512i q^{5} +2185.20 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q+81.0000i q^{3} +573.512i q^{5} +2185.20 q^{7} -6561.00 q^{9} -25190.5i q^{11} -170686. i q^{13} -46454.5 q^{15} +461173. q^{17} +165769. i q^{19} +177001. i q^{21} +953690. q^{23} +1.62421e6 q^{25} -531441. i q^{27} +1.12414e6i q^{29} -886392. q^{31} +2.04043e6 q^{33} +1.25324e6i q^{35} +1.04360e7i q^{37} +1.38256e7 q^{39} -2.74090e7 q^{41} -3.48442e7i q^{43} -3.76281e6i q^{45} -5.83629e7 q^{47} -3.55785e7 q^{49} +3.73550e7i q^{51} -4.24392e7i q^{53} +1.44471e7 q^{55} -1.34273e7 q^{57} +5.50758e7i q^{59} -3.22137e7i q^{61} -1.43371e7 q^{63} +9.78904e7 q^{65} -1.97628e8i q^{67} +7.72489e7i q^{69} -1.31495e8 q^{71} +1.88866e7 q^{73} +1.31561e8i q^{75} -5.50464e7i q^{77} -6.76946e7 q^{79} +4.30467e7 q^{81} +3.73246e8i q^{83} +2.64488e8i q^{85} -9.10553e7 q^{87} -2.50175e8 q^{89} -3.72983e8i q^{91} -7.17978e7i q^{93} -9.50706e7 q^{95} -7.73257e8 q^{97} +1.65275e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 131220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 81.0000i 0.577350i
\(4\) 0 0
\(5\) 573.512i 0.410372i 0.978723 + 0.205186i \(0.0657799\pi\)
−0.978723 + 0.205186i \(0.934220\pi\)
\(6\) 0 0
\(7\) 2185.20 0.343994 0.171997 0.985097i \(-0.444978\pi\)
0.171997 + 0.985097i \(0.444978\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) − 25190.5i − 0.518764i −0.965775 0.259382i \(-0.916481\pi\)
0.965775 0.259382i \(-0.0835189\pi\)
\(12\) 0 0
\(13\) − 170686.i − 1.65750i −0.559622 0.828748i \(-0.689054\pi\)
0.559622 0.828748i \(-0.310946\pi\)
\(14\) 0 0
\(15\) −46454.5 −0.236928
\(16\) 0 0
\(17\) 461173. 1.33919 0.669597 0.742725i \(-0.266467\pi\)
0.669597 + 0.742725i \(0.266467\pi\)
\(18\) 0 0
\(19\) 165769.i 0.291818i 0.989298 + 0.145909i \(0.0466107\pi\)
−0.989298 + 0.145909i \(0.953389\pi\)
\(20\) 0 0
\(21\) 177001.i 0.198605i
\(22\) 0 0
\(23\) 953690. 0.710611 0.355306 0.934750i \(-0.384377\pi\)
0.355306 + 0.934750i \(0.384377\pi\)
\(24\) 0 0
\(25\) 1.62421e6 0.831595
\(26\) 0 0
\(27\) − 531441.i − 0.192450i
\(28\) 0 0
\(29\) 1.12414e6i 0.295141i 0.989052 + 0.147570i \(0.0471453\pi\)
−0.989052 + 0.147570i \(0.952855\pi\)
\(30\) 0 0
\(31\) −886392. −0.172385 −0.0861923 0.996279i \(-0.527470\pi\)
−0.0861923 + 0.996279i \(0.527470\pi\)
\(32\) 0 0
\(33\) 2.04043e6 0.299509
\(34\) 0 0
\(35\) 1.25324e6i 0.141165i
\(36\) 0 0
\(37\) 1.04360e7i 0.915434i 0.889098 + 0.457717i \(0.151333\pi\)
−0.889098 + 0.457717i \(0.848667\pi\)
\(38\) 0 0
\(39\) 1.38256e7 0.956956
\(40\) 0 0
\(41\) −2.74090e7 −1.51484 −0.757420 0.652928i \(-0.773540\pi\)
−0.757420 + 0.652928i \(0.773540\pi\)
\(42\) 0 0
\(43\) − 3.48442e7i − 1.55426i −0.629342 0.777128i \(-0.716676\pi\)
0.629342 0.777128i \(-0.283324\pi\)
\(44\) 0 0
\(45\) − 3.76281e6i − 0.136791i
\(46\) 0 0
\(47\) −5.83629e7 −1.74460 −0.872301 0.488970i \(-0.837373\pi\)
−0.872301 + 0.488970i \(0.837373\pi\)
\(48\) 0 0
\(49\) −3.55785e7 −0.881668
\(50\) 0 0
\(51\) 3.73550e7i 0.773184i
\(52\) 0 0
\(53\) − 4.24392e7i − 0.738797i −0.929271 0.369399i \(-0.879564\pi\)
0.929271 0.369399i \(-0.120436\pi\)
\(54\) 0 0
\(55\) 1.44471e7 0.212886
\(56\) 0 0
\(57\) −1.34273e7 −0.168481
\(58\) 0 0
\(59\) 5.50758e7i 0.591734i 0.955229 + 0.295867i \(0.0956086\pi\)
−0.955229 + 0.295867i \(0.904391\pi\)
\(60\) 0 0
\(61\) − 3.22137e7i − 0.297890i −0.988845 0.148945i \(-0.952412\pi\)
0.988845 0.148945i \(-0.0475877\pi\)
\(62\) 0 0
\(63\) −1.43371e7 −0.114665
\(64\) 0 0
\(65\) 9.78904e7 0.680190
\(66\) 0 0
\(67\) − 1.97628e8i − 1.19815i −0.800692 0.599076i \(-0.795535\pi\)
0.800692 0.599076i \(-0.204465\pi\)
\(68\) 0 0
\(69\) 7.72489e7i 0.410272i
\(70\) 0 0
\(71\) −1.31495e8 −0.614111 −0.307056 0.951692i \(-0.599344\pi\)
−0.307056 + 0.951692i \(0.599344\pi\)
\(72\) 0 0
\(73\) 1.88866e7 0.0778398 0.0389199 0.999242i \(-0.487608\pi\)
0.0389199 + 0.999242i \(0.487608\pi\)
\(74\) 0 0
\(75\) 1.31561e8i 0.480122i
\(76\) 0 0
\(77\) − 5.50464e7i − 0.178452i
\(78\) 0 0
\(79\) −6.76946e7 −0.195538 −0.0977692 0.995209i \(-0.531171\pi\)
−0.0977692 + 0.995209i \(0.531171\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) 3.73246e8i 0.863264i 0.902050 + 0.431632i \(0.142062\pi\)
−0.902050 + 0.431632i \(0.857938\pi\)
\(84\) 0 0
\(85\) 2.64488e8i 0.549567i
\(86\) 0 0
\(87\) −9.10553e7 −0.170400
\(88\) 0 0
\(89\) −2.50175e8 −0.422658 −0.211329 0.977415i \(-0.567779\pi\)
−0.211329 + 0.977415i \(0.567779\pi\)
\(90\) 0 0
\(91\) − 3.72983e8i − 0.570168i
\(92\) 0 0
\(93\) − 7.17978e7i − 0.0995262i
\(94\) 0 0
\(95\) −9.50706e7 −0.119754
\(96\) 0 0
\(97\) −7.73257e8 −0.886852 −0.443426 0.896311i \(-0.646237\pi\)
−0.443426 + 0.896311i \(0.646237\pi\)
\(98\) 0 0
\(99\) 1.65275e8i 0.172921i
\(100\) 0 0
\(101\) − 8.34597e8i − 0.798051i −0.916940 0.399025i \(-0.869349\pi\)
0.916940 0.399025i \(-0.130651\pi\)
\(102\) 0 0
\(103\) −1.44701e9 −1.26678 −0.633392 0.773831i \(-0.718338\pi\)
−0.633392 + 0.773831i \(0.718338\pi\)
\(104\) 0 0
\(105\) −1.01513e8 −0.0815019
\(106\) 0 0
\(107\) 2.20932e8i 0.162942i 0.996676 + 0.0814708i \(0.0259617\pi\)
−0.996676 + 0.0814708i \(0.974038\pi\)
\(108\) 0 0
\(109\) − 8.53497e8i − 0.579139i −0.957157 0.289570i \(-0.906488\pi\)
0.957157 0.289570i \(-0.0935122\pi\)
\(110\) 0 0
\(111\) −8.45318e8 −0.528526
\(112\) 0 0
\(113\) 1.25755e9 0.725560 0.362780 0.931875i \(-0.381828\pi\)
0.362780 + 0.931875i \(0.381828\pi\)
\(114\) 0 0
\(115\) 5.46953e8i 0.291615i
\(116\) 0 0
\(117\) 1.11987e9i 0.552499i
\(118\) 0 0
\(119\) 1.00776e9 0.460674
\(120\) 0 0
\(121\) 1.72339e9 0.730884
\(122\) 0 0
\(123\) − 2.22013e9i − 0.874593i
\(124\) 0 0
\(125\) 2.05164e9i 0.751635i
\(126\) 0 0
\(127\) −2.98428e9 −1.01794 −0.508971 0.860783i \(-0.669974\pi\)
−0.508971 + 0.860783i \(0.669974\pi\)
\(128\) 0 0
\(129\) 2.82238e9 0.897351
\(130\) 0 0
\(131\) − 2.24225e9i − 0.665217i −0.943065 0.332609i \(-0.892071\pi\)
0.943065 0.332609i \(-0.107929\pi\)
\(132\) 0 0
\(133\) 3.62239e8i 0.100384i
\(134\) 0 0
\(135\) 3.04788e8 0.0789761
\(136\) 0 0
\(137\) −1.92870e8 −0.0467759 −0.0233879 0.999726i \(-0.507445\pi\)
−0.0233879 + 0.999726i \(0.507445\pi\)
\(138\) 0 0
\(139\) 5.70374e9i 1.29596i 0.761655 + 0.647982i \(0.224387\pi\)
−0.761655 + 0.647982i \(0.775613\pi\)
\(140\) 0 0
\(141\) − 4.72739e9i − 1.00725i
\(142\) 0 0
\(143\) −4.29966e9 −0.859849
\(144\) 0 0
\(145\) −6.44708e8 −0.121117
\(146\) 0 0
\(147\) − 2.88186e9i − 0.509031i
\(148\) 0 0
\(149\) − 8.87026e9i − 1.47434i −0.675707 0.737171i \(-0.736161\pi\)
0.675707 0.737171i \(-0.263839\pi\)
\(150\) 0 0
\(151\) 1.12257e10 1.75719 0.878595 0.477568i \(-0.158481\pi\)
0.878595 + 0.477568i \(0.158481\pi\)
\(152\) 0 0
\(153\) −3.02575e9 −0.446398
\(154\) 0 0
\(155\) − 5.08357e8i − 0.0707418i
\(156\) 0 0
\(157\) − 6.26221e9i − 0.822582i −0.911504 0.411291i \(-0.865078\pi\)
0.911504 0.411291i \(-0.134922\pi\)
\(158\) 0 0
\(159\) 3.43757e9 0.426545
\(160\) 0 0
\(161\) 2.08401e9 0.244446
\(162\) 0 0
\(163\) − 4.06846e9i − 0.451425i −0.974194 0.225712i \(-0.927529\pi\)
0.974194 0.225712i \(-0.0724710\pi\)
\(164\) 0 0
\(165\) 1.17021e9i 0.122910i
\(166\) 0 0
\(167\) −7.42771e9 −0.738976 −0.369488 0.929235i \(-0.620467\pi\)
−0.369488 + 0.929235i \(0.620467\pi\)
\(168\) 0 0
\(169\) −1.85292e10 −1.74729
\(170\) 0 0
\(171\) − 1.08761e9i − 0.0972727i
\(172\) 0 0
\(173\) 9.15808e9i 0.777315i 0.921382 + 0.388658i \(0.127061\pi\)
−0.921382 + 0.388658i \(0.872939\pi\)
\(174\) 0 0
\(175\) 3.54923e9 0.286064
\(176\) 0 0
\(177\) −4.46114e9 −0.341638
\(178\) 0 0
\(179\) 2.24462e10i 1.63420i 0.576499 + 0.817098i \(0.304418\pi\)
−0.576499 + 0.817098i \(0.695582\pi\)
\(180\) 0 0
\(181\) − 1.01662e10i − 0.704049i −0.935991 0.352025i \(-0.885493\pi\)
0.935991 0.352025i \(-0.114507\pi\)
\(182\) 0 0
\(183\) 2.60931e9 0.171987
\(184\) 0 0
\(185\) −5.98519e9 −0.375669
\(186\) 0 0
\(187\) − 1.16172e10i − 0.694726i
\(188\) 0 0
\(189\) − 1.16131e9i − 0.0662017i
\(190\) 0 0
\(191\) 2.71383e10 1.47548 0.737738 0.675088i \(-0.235894\pi\)
0.737738 + 0.675088i \(0.235894\pi\)
\(192\) 0 0
\(193\) 2.59970e10 1.34870 0.674350 0.738412i \(-0.264424\pi\)
0.674350 + 0.738412i \(0.264424\pi\)
\(194\) 0 0
\(195\) 7.92912e9i 0.392708i
\(196\) 0 0
\(197\) − 1.11903e10i − 0.529351i −0.964337 0.264676i \(-0.914735\pi\)
0.964337 0.264676i \(-0.0852649\pi\)
\(198\) 0 0
\(199\) −2.24980e10 −1.01696 −0.508482 0.861073i \(-0.669793\pi\)
−0.508482 + 0.861073i \(0.669793\pi\)
\(200\) 0 0
\(201\) 1.60079e10 0.691753
\(202\) 0 0
\(203\) 2.45647e9i 0.101527i
\(204\) 0 0
\(205\) − 1.57194e10i − 0.621648i
\(206\) 0 0
\(207\) −6.25716e9 −0.236870
\(208\) 0 0
\(209\) 4.17581e9 0.151385
\(210\) 0 0
\(211\) − 2.12635e10i − 0.738521i −0.929326 0.369261i \(-0.879611\pi\)
0.929326 0.369261i \(-0.120389\pi\)
\(212\) 0 0
\(213\) − 1.06511e10i − 0.354557i
\(214\) 0 0
\(215\) 1.99836e10 0.637823
\(216\) 0 0
\(217\) −1.93695e9 −0.0592992
\(218\) 0 0
\(219\) 1.52982e9i 0.0449408i
\(220\) 0 0
\(221\) − 7.87156e10i − 2.21971i
\(222\) 0 0
\(223\) 2.72002e10 0.736546 0.368273 0.929718i \(-0.379949\pi\)
0.368273 + 0.929718i \(0.379949\pi\)
\(224\) 0 0
\(225\) −1.06564e10 −0.277198
\(226\) 0 0
\(227\) 4.16425e9i 0.104093i 0.998645 + 0.0520464i \(0.0165744\pi\)
−0.998645 + 0.0520464i \(0.983426\pi\)
\(228\) 0 0
\(229\) − 5.12064e10i − 1.23045i −0.788351 0.615226i \(-0.789065\pi\)
0.788351 0.615226i \(-0.210935\pi\)
\(230\) 0 0
\(231\) 4.45876e9 0.103029
\(232\) 0 0
\(233\) −2.39479e10 −0.532311 −0.266155 0.963930i \(-0.585753\pi\)
−0.266155 + 0.963930i \(0.585753\pi\)
\(234\) 0 0
\(235\) − 3.34718e10i − 0.715935i
\(236\) 0 0
\(237\) − 5.48326e9i − 0.112894i
\(238\) 0 0
\(239\) 7.69723e9 0.152596 0.0762981 0.997085i \(-0.475690\pi\)
0.0762981 + 0.997085i \(0.475690\pi\)
\(240\) 0 0
\(241\) −9.87829e8 −0.0188628 −0.00943138 0.999956i \(-0.503002\pi\)
−0.00943138 + 0.999956i \(0.503002\pi\)
\(242\) 0 0
\(243\) 3.48678e9i 0.0641500i
\(244\) 0 0
\(245\) − 2.04047e10i − 0.361812i
\(246\) 0 0
\(247\) 2.82944e10 0.483687
\(248\) 0 0
\(249\) −3.02329e10 −0.498406
\(250\) 0 0
\(251\) 3.00907e10i 0.478521i 0.970955 + 0.239261i \(0.0769050\pi\)
−0.970955 + 0.239261i \(0.923095\pi\)
\(252\) 0 0
\(253\) − 2.40239e10i − 0.368640i
\(254\) 0 0
\(255\) −2.14235e10 −0.317293
\(256\) 0 0
\(257\) 1.68520e10 0.240963 0.120482 0.992716i \(-0.461556\pi\)
0.120482 + 0.992716i \(0.461556\pi\)
\(258\) 0 0
\(259\) 2.28048e10i 0.314904i
\(260\) 0 0
\(261\) − 7.37548e9i − 0.0983803i
\(262\) 0 0
\(263\) −2.47935e10 −0.319548 −0.159774 0.987154i \(-0.551077\pi\)
−0.159774 + 0.987154i \(0.551077\pi\)
\(264\) 0 0
\(265\) 2.43394e10 0.303182
\(266\) 0 0
\(267\) − 2.02642e10i − 0.244022i
\(268\) 0 0
\(269\) − 1.44586e11i − 1.68361i −0.539785 0.841803i \(-0.681494\pi\)
0.539785 0.841803i \(-0.318506\pi\)
\(270\) 0 0
\(271\) −9.33109e10 −1.05092 −0.525461 0.850818i \(-0.676107\pi\)
−0.525461 + 0.850818i \(0.676107\pi\)
\(272\) 0 0
\(273\) 3.02116e10 0.329187
\(274\) 0 0
\(275\) − 4.09146e10i − 0.431402i
\(276\) 0 0
\(277\) − 1.41736e11i − 1.44651i −0.690583 0.723253i \(-0.742646\pi\)
0.690583 0.723253i \(-0.257354\pi\)
\(278\) 0 0
\(279\) 5.81562e9 0.0574615
\(280\) 0 0
\(281\) 4.57492e10 0.437729 0.218864 0.975755i \(-0.429765\pi\)
0.218864 + 0.975755i \(0.429765\pi\)
\(282\) 0 0
\(283\) 8.57214e10i 0.794420i 0.917728 + 0.397210i \(0.130022\pi\)
−0.917728 + 0.397210i \(0.869978\pi\)
\(284\) 0 0
\(285\) − 7.70072e9i − 0.0691400i
\(286\) 0 0
\(287\) −5.98944e10 −0.521096
\(288\) 0 0
\(289\) 9.40923e10 0.793440
\(290\) 0 0
\(291\) − 6.26338e10i − 0.512024i
\(292\) 0 0
\(293\) − 7.92826e10i − 0.628454i −0.949348 0.314227i \(-0.898255\pi\)
0.949348 0.314227i \(-0.101745\pi\)
\(294\) 0 0
\(295\) −3.15866e10 −0.242831
\(296\) 0 0
\(297\) −1.33873e10 −0.0998362
\(298\) 0 0
\(299\) − 1.62781e11i − 1.17784i
\(300\) 0 0
\(301\) − 7.61417e10i − 0.534655i
\(302\) 0 0
\(303\) 6.76023e10 0.460755
\(304\) 0 0
\(305\) 1.84749e10 0.122246
\(306\) 0 0
\(307\) 2.70824e11i 1.74006i 0.492996 + 0.870031i \(0.335902\pi\)
−0.492996 + 0.870031i \(0.664098\pi\)
\(308\) 0 0
\(309\) − 1.17207e11i − 0.731378i
\(310\) 0 0
\(311\) −2.53125e11 −1.53431 −0.767155 0.641462i \(-0.778328\pi\)
−0.767155 + 0.641462i \(0.778328\pi\)
\(312\) 0 0
\(313\) −3.80905e10 −0.224320 −0.112160 0.993690i \(-0.535777\pi\)
−0.112160 + 0.993690i \(0.535777\pi\)
\(314\) 0 0
\(315\) − 8.22251e9i − 0.0470551i
\(316\) 0 0
\(317\) − 2.66537e11i − 1.48249i −0.671235 0.741245i \(-0.734236\pi\)
0.671235 0.741245i \(-0.265764\pi\)
\(318\) 0 0
\(319\) 2.83176e10 0.153108
\(320\) 0 0
\(321\) −1.78955e10 −0.0940744
\(322\) 0 0
\(323\) 7.64482e10i 0.390801i
\(324\) 0 0
\(325\) − 2.77229e11i − 1.37836i
\(326\) 0 0
\(327\) 6.91333e10 0.334366
\(328\) 0 0
\(329\) −1.27535e11 −0.600132
\(330\) 0 0
\(331\) − 1.32548e11i − 0.606942i −0.952841 0.303471i \(-0.901854\pi\)
0.952841 0.303471i \(-0.0981456\pi\)
\(332\) 0 0
\(333\) − 6.84707e10i − 0.305145i
\(334\) 0 0
\(335\) 1.13342e11 0.491688
\(336\) 0 0
\(337\) −3.91693e11 −1.65429 −0.827145 0.561988i \(-0.810037\pi\)
−0.827145 + 0.561988i \(0.810037\pi\)
\(338\) 0 0
\(339\) 1.01862e11i 0.418903i
\(340\) 0 0
\(341\) 2.23287e10i 0.0894269i
\(342\) 0 0
\(343\) −1.65927e11 −0.647282
\(344\) 0 0
\(345\) −4.43032e10 −0.168364
\(346\) 0 0
\(347\) − 5.17702e10i − 0.191689i −0.995396 0.0958445i \(-0.969445\pi\)
0.995396 0.0958445i \(-0.0305551\pi\)
\(348\) 0 0
\(349\) − 2.61383e11i − 0.943113i −0.881836 0.471556i \(-0.843692\pi\)
0.881836 0.471556i \(-0.156308\pi\)
\(350\) 0 0
\(351\) −9.07095e10 −0.318985
\(352\) 0 0
\(353\) −2.46204e11 −0.843934 −0.421967 0.906611i \(-0.638660\pi\)
−0.421967 + 0.906611i \(0.638660\pi\)
\(354\) 0 0
\(355\) − 7.54141e10i − 0.252014i
\(356\) 0 0
\(357\) 8.16282e10i 0.265971i
\(358\) 0 0
\(359\) 4.40472e11 1.39956 0.699782 0.714356i \(-0.253280\pi\)
0.699782 + 0.714356i \(0.253280\pi\)
\(360\) 0 0
\(361\) 2.95208e11 0.914842
\(362\) 0 0
\(363\) 1.39594e11i 0.421976i
\(364\) 0 0
\(365\) 1.08317e10i 0.0319433i
\(366\) 0 0
\(367\) −2.54837e11 −0.733271 −0.366636 0.930365i \(-0.619490\pi\)
−0.366636 + 0.930365i \(0.619490\pi\)
\(368\) 0 0
\(369\) 1.79831e11 0.504947
\(370\) 0 0
\(371\) − 9.27382e10i − 0.254142i
\(372\) 0 0
\(373\) − 4.92490e11i − 1.31737i −0.752419 0.658685i \(-0.771113\pi\)
0.752419 0.658685i \(-0.228887\pi\)
\(374\) 0 0
\(375\) −1.66183e11 −0.433957
\(376\) 0 0
\(377\) 1.91875e11 0.489195
\(378\) 0 0
\(379\) − 6.04503e11i − 1.50495i −0.658621 0.752475i \(-0.728860\pi\)
0.658621 0.752475i \(-0.271140\pi\)
\(380\) 0 0
\(381\) − 2.41727e11i − 0.587710i
\(382\) 0 0
\(383\) 9.93960e10 0.236034 0.118017 0.993012i \(-0.462346\pi\)
0.118017 + 0.993012i \(0.462346\pi\)
\(384\) 0 0
\(385\) 3.15698e10 0.0732315
\(386\) 0 0
\(387\) 2.28613e11i 0.518086i
\(388\) 0 0
\(389\) − 8.77921e11i − 1.94394i −0.235109 0.971969i \(-0.575545\pi\)
0.235109 0.971969i \(-0.424455\pi\)
\(390\) 0 0
\(391\) 4.39816e11 0.951646
\(392\) 0 0
\(393\) 1.81622e11 0.384063
\(394\) 0 0
\(395\) − 3.88237e10i − 0.0802435i
\(396\) 0 0
\(397\) 3.40922e9i 0.00688808i 0.999994 + 0.00344404i \(0.00109627\pi\)
−0.999994 + 0.00344404i \(0.998904\pi\)
\(398\) 0 0
\(399\) −2.93414e10 −0.0579565
\(400\) 0 0
\(401\) 4.36366e11 0.842755 0.421378 0.906885i \(-0.361547\pi\)
0.421378 + 0.906885i \(0.361547\pi\)
\(402\) 0 0
\(403\) 1.51295e11i 0.285727i
\(404\) 0 0
\(405\) 2.46878e10i 0.0455969i
\(406\) 0 0
\(407\) 2.62889e11 0.474894
\(408\) 0 0
\(409\) 1.53167e9 0.00270652 0.00135326 0.999999i \(-0.499569\pi\)
0.00135326 + 0.999999i \(0.499569\pi\)
\(410\) 0 0
\(411\) − 1.56225e10i − 0.0270061i
\(412\) 0 0
\(413\) 1.20352e11i 0.203553i
\(414\) 0 0
\(415\) −2.14061e11 −0.354259
\(416\) 0 0
\(417\) −4.62003e11 −0.748226
\(418\) 0 0
\(419\) 7.95996e11i 1.26168i 0.775915 + 0.630838i \(0.217289\pi\)
−0.775915 + 0.630838i \(0.782711\pi\)
\(420\) 0 0
\(421\) 7.16602e11i 1.11175i 0.831265 + 0.555877i \(0.187617\pi\)
−0.831265 + 0.555877i \(0.812383\pi\)
\(422\) 0 0
\(423\) 3.82919e11 0.581534
\(424\) 0 0
\(425\) 7.49041e11 1.11367
\(426\) 0 0
\(427\) − 7.03934e10i − 0.102472i
\(428\) 0 0
\(429\) − 3.48273e11i − 0.496434i
\(430\) 0 0
\(431\) −2.72042e11 −0.379741 −0.189871 0.981809i \(-0.560807\pi\)
−0.189871 + 0.981809i \(0.560807\pi\)
\(432\) 0 0
\(433\) −2.93461e11 −0.401195 −0.200598 0.979674i \(-0.564288\pi\)
−0.200598 + 0.979674i \(0.564288\pi\)
\(434\) 0 0
\(435\) − 5.22213e10i − 0.0699272i
\(436\) 0 0
\(437\) 1.58092e11i 0.207369i
\(438\) 0 0
\(439\) −2.95559e11 −0.379799 −0.189900 0.981804i \(-0.560816\pi\)
−0.189900 + 0.981804i \(0.560816\pi\)
\(440\) 0 0
\(441\) 2.33430e11 0.293889
\(442\) 0 0
\(443\) − 1.41221e12i − 1.74214i −0.491163 0.871068i \(-0.663428\pi\)
0.491163 0.871068i \(-0.336572\pi\)
\(444\) 0 0
\(445\) − 1.43479e11i − 0.173447i
\(446\) 0 0
\(447\) 7.18491e11 0.851211
\(448\) 0 0
\(449\) −8.27145e11 −0.960447 −0.480223 0.877146i \(-0.659444\pi\)
−0.480223 + 0.877146i \(0.659444\pi\)
\(450\) 0 0
\(451\) 6.90448e11i 0.785844i
\(452\) 0 0
\(453\) 9.09285e11i 1.01451i
\(454\) 0 0
\(455\) 2.13910e11 0.233981
\(456\) 0 0
\(457\) −8.57958e11 −0.920117 −0.460058 0.887889i \(-0.652172\pi\)
−0.460058 + 0.887889i \(0.652172\pi\)
\(458\) 0 0
\(459\) − 2.45086e11i − 0.257728i
\(460\) 0 0
\(461\) 4.68707e11i 0.483334i 0.970359 + 0.241667i \(0.0776941\pi\)
−0.970359 + 0.241667i \(0.922306\pi\)
\(462\) 0 0
\(463\) 8.90051e11 0.900120 0.450060 0.892998i \(-0.351403\pi\)
0.450060 + 0.892998i \(0.351403\pi\)
\(464\) 0 0
\(465\) 4.11769e10 0.0408428
\(466\) 0 0
\(467\) 1.17567e12i 1.14383i 0.820314 + 0.571913i \(0.193799\pi\)
−0.820314 + 0.571913i \(0.806201\pi\)
\(468\) 0 0
\(469\) − 4.31857e11i − 0.412157i
\(470\) 0 0
\(471\) 5.07239e11 0.474918
\(472\) 0 0
\(473\) −8.77744e11 −0.806293
\(474\) 0 0
\(475\) 2.69244e11i 0.242674i
\(476\) 0 0
\(477\) 2.78443e11i 0.246266i
\(478\) 0 0
\(479\) 7.61386e11 0.660838 0.330419 0.943834i \(-0.392810\pi\)
0.330419 + 0.943834i \(0.392810\pi\)
\(480\) 0 0
\(481\) 1.78128e12 1.51733
\(482\) 0 0
\(483\) 1.68805e11i 0.141131i
\(484\) 0 0
\(485\) − 4.43473e11i − 0.363939i
\(486\) 0 0
\(487\) −2.08842e12 −1.68243 −0.841217 0.540698i \(-0.818160\pi\)
−0.841217 + 0.540698i \(0.818160\pi\)
\(488\) 0 0
\(489\) 3.29545e11 0.260630
\(490\) 0 0
\(491\) − 5.35372e11i − 0.415709i −0.978160 0.207854i \(-0.933352\pi\)
0.978160 0.207854i \(-0.0666480\pi\)
\(492\) 0 0
\(493\) 5.18422e11i 0.395251i
\(494\) 0 0
\(495\) −9.47872e10 −0.0709621
\(496\) 0 0
\(497\) −2.87344e11 −0.211251
\(498\) 0 0
\(499\) 1.24709e12i 0.900422i 0.892922 + 0.450211i \(0.148651\pi\)
−0.892922 + 0.450211i \(0.851349\pi\)
\(500\) 0 0
\(501\) − 6.01644e11i − 0.426648i
\(502\) 0 0
\(503\) −4.38151e11 −0.305188 −0.152594 0.988289i \(-0.548763\pi\)
−0.152594 + 0.988289i \(0.548763\pi\)
\(504\) 0 0
\(505\) 4.78651e11 0.327498
\(506\) 0 0
\(507\) − 1.50086e12i − 1.00880i
\(508\) 0 0
\(509\) − 8.18520e11i − 0.540505i −0.962790 0.270252i \(-0.912893\pi\)
0.962790 0.270252i \(-0.0871071\pi\)
\(510\) 0 0
\(511\) 4.12712e10 0.0267764
\(512\) 0 0
\(513\) 8.80965e10 0.0561604
\(514\) 0 0
\(515\) − 8.29875e11i − 0.519853i
\(516\) 0 0
\(517\) 1.47019e12i 0.905037i
\(518\) 0 0
\(519\) −7.41805e11 −0.448783
\(520\) 0 0
\(521\) 1.40738e12 0.836835 0.418418 0.908255i \(-0.362585\pi\)
0.418418 + 0.908255i \(0.362585\pi\)
\(522\) 0 0
\(523\) − 1.18942e12i − 0.695147i −0.937653 0.347574i \(-0.887006\pi\)
0.937653 0.347574i \(-0.112994\pi\)
\(524\) 0 0
\(525\) 2.87487e11i 0.165159i
\(526\) 0 0
\(527\) −4.08780e11 −0.230856
\(528\) 0 0
\(529\) −8.91627e11 −0.495032
\(530\) 0 0
\(531\) − 3.61352e11i − 0.197245i
\(532\) 0 0
\(533\) 4.67834e12i 2.51084i
\(534\) 0 0
\(535\) −1.26707e11 −0.0668667
\(536\) 0 0
\(537\) −1.81814e12 −0.943503
\(538\) 0 0
\(539\) 8.96240e11i 0.457378i
\(540\) 0 0
\(541\) 2.99696e12i 1.50416i 0.659072 + 0.752080i \(0.270949\pi\)
−0.659072 + 0.752080i \(0.729051\pi\)
\(542\) 0 0
\(543\) 8.23459e11 0.406483
\(544\) 0 0
\(545\) 4.89491e11 0.237662
\(546\) 0 0
\(547\) 1.19878e12i 0.572527i 0.958151 + 0.286264i \(0.0924133\pi\)
−0.958151 + 0.286264i \(0.907587\pi\)
\(548\) 0 0
\(549\) 2.11354e11i 0.0992966i
\(550\) 0 0
\(551\) −1.86348e11 −0.0861274
\(552\) 0 0
\(553\) −1.47927e11 −0.0672640
\(554\) 0 0
\(555\) − 4.84800e11i − 0.216892i
\(556\) 0 0
\(557\) 2.94419e12i 1.29604i 0.761625 + 0.648018i \(0.224402\pi\)
−0.761625 + 0.648018i \(0.775598\pi\)
\(558\) 0 0
\(559\) −5.94742e12 −2.57617
\(560\) 0 0
\(561\) 9.40991e11 0.401100
\(562\) 0 0
\(563\) 2.99061e12i 1.25450i 0.778816 + 0.627252i \(0.215820\pi\)
−0.778816 + 0.627252i \(0.784180\pi\)
\(564\) 0 0
\(565\) 7.21222e11i 0.297750i
\(566\) 0 0
\(567\) 9.40658e10 0.0382215
\(568\) 0 0
\(569\) −3.33223e12 −1.33269 −0.666346 0.745643i \(-0.732142\pi\)
−0.666346 + 0.745643i \(0.732142\pi\)
\(570\) 0 0
\(571\) − 3.69698e12i − 1.45541i −0.685891 0.727704i \(-0.740587\pi\)
0.685891 0.727704i \(-0.259413\pi\)
\(572\) 0 0
\(573\) 2.19820e12i 0.851866i
\(574\) 0 0
\(575\) 1.54899e12 0.590941
\(576\) 0 0
\(577\) −1.68443e12 −0.632647 −0.316323 0.948651i \(-0.602448\pi\)
−0.316323 + 0.948651i \(0.602448\pi\)
\(578\) 0 0
\(579\) 2.10576e12i 0.778672i
\(580\) 0 0
\(581\) 8.15618e11i 0.296958i
\(582\) 0 0
\(583\) −1.06906e12 −0.383262
\(584\) 0 0
\(585\) −6.42259e11 −0.226730
\(586\) 0 0
\(587\) − 5.18518e12i − 1.80257i −0.433224 0.901286i \(-0.642624\pi\)
0.433224 0.901286i \(-0.357376\pi\)
\(588\) 0 0
\(589\) − 1.46936e11i − 0.0503049i
\(590\) 0 0
\(591\) 9.06415e11 0.305621
\(592\) 0 0
\(593\) 5.75208e10 0.0191020 0.00955100 0.999954i \(-0.496960\pi\)
0.00955100 + 0.999954i \(0.496960\pi\)
\(594\) 0 0
\(595\) 5.77960e11i 0.189048i
\(596\) 0 0
\(597\) − 1.82234e12i − 0.587144i
\(598\) 0 0
\(599\) 5.13376e12 1.62935 0.814677 0.579916i \(-0.196914\pi\)
0.814677 + 0.579916i \(0.196914\pi\)
\(600\) 0 0
\(601\) 1.69179e12 0.528947 0.264474 0.964393i \(-0.414802\pi\)
0.264474 + 0.964393i \(0.414802\pi\)
\(602\) 0 0
\(603\) 1.29664e12i 0.399384i
\(604\) 0 0
\(605\) 9.88383e11i 0.299934i
\(606\) 0 0
\(607\) −1.63728e12 −0.489524 −0.244762 0.969583i \(-0.578710\pi\)
−0.244762 + 0.969583i \(0.578710\pi\)
\(608\) 0 0
\(609\) −1.98974e11 −0.0586164
\(610\) 0 0
\(611\) 9.96171e12i 2.89167i
\(612\) 0 0
\(613\) 2.61028e12i 0.746646i 0.927701 + 0.373323i \(0.121782\pi\)
−0.927701 + 0.373323i \(0.878218\pi\)
\(614\) 0 0
\(615\) 1.27327e12 0.358908
\(616\) 0 0
\(617\) 6.80946e11 0.189160 0.0945800 0.995517i \(-0.469849\pi\)
0.0945800 + 0.995517i \(0.469849\pi\)
\(618\) 0 0
\(619\) − 5.97226e12i − 1.63505i −0.575894 0.817524i \(-0.695346\pi\)
0.575894 0.817524i \(-0.304654\pi\)
\(620\) 0 0
\(621\) − 5.06830e11i − 0.136757i
\(622\) 0 0
\(623\) −5.46684e11 −0.145392
\(624\) 0 0
\(625\) 1.99564e12 0.523145
\(626\) 0 0
\(627\) 3.38240e11i 0.0874020i
\(628\) 0 0
\(629\) 4.81281e12i 1.22594i
\(630\) 0 0
\(631\) −5.38581e12 −1.35244 −0.676221 0.736698i \(-0.736384\pi\)
−0.676221 + 0.736698i \(0.736384\pi\)
\(632\) 0 0
\(633\) 1.72234e12 0.426386
\(634\) 0 0
\(635\) − 1.71152e12i − 0.417735i
\(636\) 0 0
\(637\) 6.07275e12i 1.46136i
\(638\) 0 0
\(639\) 8.62740e11 0.204704
\(640\) 0 0
\(641\) 6.98854e12 1.63503 0.817514 0.575909i \(-0.195352\pi\)
0.817514 + 0.575909i \(0.195352\pi\)
\(642\) 0 0
\(643\) 1.21879e12i 0.281176i 0.990068 + 0.140588i \(0.0448994\pi\)
−0.990068 + 0.140588i \(0.955101\pi\)
\(644\) 0 0
\(645\) 1.61867e12i 0.368247i
\(646\) 0 0
\(647\) −8.11593e11 −0.182083 −0.0910414 0.995847i \(-0.529020\pi\)
−0.0910414 + 0.995847i \(0.529020\pi\)
\(648\) 0 0
\(649\) 1.38739e12 0.306970
\(650\) 0 0
\(651\) − 1.56893e11i − 0.0342364i
\(652\) 0 0
\(653\) 2.33692e12i 0.502962i 0.967862 + 0.251481i \(0.0809175\pi\)
−0.967862 + 0.251481i \(0.919082\pi\)
\(654\) 0 0
\(655\) 1.28596e12 0.272986
\(656\) 0 0
\(657\) −1.23915e11 −0.0259466
\(658\) 0 0
\(659\) − 5.06294e12i − 1.04573i −0.852416 0.522863i \(-0.824864\pi\)
0.852416 0.522863i \(-0.175136\pi\)
\(660\) 0 0
\(661\) 1.34073e12i 0.273170i 0.990628 + 0.136585i \(0.0436127\pi\)
−0.990628 + 0.136585i \(0.956387\pi\)
\(662\) 0 0
\(663\) 6.37597e12 1.28155
\(664\) 0 0
\(665\) −2.07749e11 −0.0411946
\(666\) 0 0
\(667\) 1.07208e12i 0.209730i
\(668\) 0 0
\(669\) 2.20321e12i 0.425245i
\(670\) 0 0
\(671\) −8.11478e11 −0.154535
\(672\) 0 0
\(673\) 4.40610e12 0.827916 0.413958 0.910296i \(-0.364146\pi\)
0.413958 + 0.910296i \(0.364146\pi\)
\(674\) 0 0
\(675\) − 8.63171e11i − 0.160041i
\(676\) 0 0
\(677\) − 2.41655e12i − 0.442126i −0.975259 0.221063i \(-0.929047\pi\)
0.975259 0.221063i \(-0.0709527\pi\)
\(678\) 0 0
\(679\) −1.68972e12 −0.305072
\(680\) 0 0
\(681\) −3.37304e11 −0.0600980
\(682\) 0 0
\(683\) 3.19393e12i 0.561606i 0.959765 + 0.280803i \(0.0906008\pi\)
−0.959765 + 0.280803i \(0.909399\pi\)
\(684\) 0 0
\(685\) − 1.10613e11i − 0.0191955i
\(686\) 0 0
\(687\) 4.14772e12 0.710401
\(688\) 0 0
\(689\) −7.24376e12 −1.22455
\(690\) 0 0
\(691\) 1.11051e13i 1.85298i 0.376318 + 0.926491i \(0.377190\pi\)
−0.376318 + 0.926491i \(0.622810\pi\)
\(692\) 0 0
\(693\) 3.61159e11i 0.0594839i
\(694\) 0 0
\(695\) −3.27116e12 −0.531828
\(696\) 0 0
\(697\) −1.26403e13 −2.02866
\(698\) 0 0
\(699\) − 1.93978e12i − 0.307330i
\(700\) 0 0
\(701\) 1.08592e12i 0.169850i 0.996387 + 0.0849249i \(0.0270650\pi\)
−0.996387 + 0.0849249i \(0.972935\pi\)
\(702\) 0 0
\(703\) −1.72997e12 −0.267140
\(704\) 0 0
\(705\) 2.71122e12 0.413346
\(706\) 0 0
\(707\) − 1.82376e12i − 0.274525i
\(708\) 0 0
\(709\) 4.38234e12i 0.651325i 0.945486 + 0.325662i \(0.105587\pi\)
−0.945486 + 0.325662i \(0.894413\pi\)
\(710\) 0 0
\(711\) 4.44144e11 0.0651795
\(712\) 0 0
\(713\) −8.45344e11 −0.122498
\(714\) 0 0
\(715\) − 2.46591e12i − 0.352858i
\(716\) 0 0
\(717\) 6.23475e11i 0.0881015i
\(718\) 0 0
\(719\) 1.08321e13 1.51158 0.755789 0.654815i \(-0.227253\pi\)
0.755789 + 0.654815i \(0.227253\pi\)
\(720\) 0 0
\(721\) −3.16200e12 −0.435766
\(722\) 0 0
\(723\) − 8.00142e10i − 0.0108904i
\(724\) 0 0
\(725\) 1.82584e12i 0.245438i
\(726\) 0 0
\(727\) −4.52246e12 −0.600440 −0.300220 0.953870i \(-0.597060\pi\)
−0.300220 + 0.953870i \(0.597060\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) − 1.60692e13i − 2.08145i
\(732\) 0 0
\(733\) − 1.16220e13i − 1.48701i −0.668729 0.743506i \(-0.733161\pi\)
0.668729 0.743506i \(-0.266839\pi\)
\(734\) 0 0
\(735\) 1.65278e12 0.208892
\(736\) 0 0
\(737\) −4.97835e12 −0.621558
\(738\) 0 0
\(739\) 2.43315e12i 0.300102i 0.988678 + 0.150051i \(0.0479438\pi\)
−0.988678 + 0.150051i \(0.952056\pi\)
\(740\) 0 0
\(741\) 2.29185e12i 0.279257i
\(742\) 0 0
\(743\) 1.63463e13 1.96774 0.983872 0.178875i \(-0.0572457\pi\)
0.983872 + 0.178875i \(0.0572457\pi\)
\(744\) 0 0
\(745\) 5.08720e12 0.605028
\(746\) 0 0
\(747\) − 2.44887e12i − 0.287755i
\(748\) 0 0
\(749\) 4.82782e11i 0.0560509i
\(750\) 0 0
\(751\) −1.35795e13 −1.55778 −0.778888 0.627164i \(-0.784216\pi\)
−0.778888 + 0.627164i \(0.784216\pi\)
\(752\) 0 0
\(753\) −2.43735e12 −0.276274
\(754\) 0 0
\(755\) 6.43810e12i 0.721101i
\(756\) 0 0
\(757\) 5.13921e12i 0.568807i 0.958705 + 0.284403i \(0.0917955\pi\)
−0.958705 + 0.284403i \(0.908205\pi\)
\(758\) 0 0
\(759\) 1.94594e12 0.212834
\(760\) 0 0
\(761\) 1.01011e13 1.09178 0.545892 0.837855i \(-0.316191\pi\)
0.545892 + 0.837855i \(0.316191\pi\)
\(762\) 0 0
\(763\) − 1.86507e12i − 0.199220i
\(764\) 0 0
\(765\) − 1.73531e12i − 0.183189i
\(766\) 0 0
\(767\) 9.40066e12 0.980797
\(768\) 0 0
\(769\) −2.55912e12 −0.263889 −0.131945 0.991257i \(-0.542122\pi\)
−0.131945 + 0.991257i \(0.542122\pi\)
\(770\) 0 0
\(771\) 1.36501e12i 0.139120i
\(772\) 0 0
\(773\) − 1.50470e13i − 1.51580i −0.652372 0.757899i \(-0.726226\pi\)
0.652372 0.757899i \(-0.273774\pi\)
\(774\) 0 0
\(775\) −1.43969e12 −0.143354
\(776\) 0 0
\(777\) −1.84719e12 −0.181810
\(778\) 0 0
\(779\) − 4.54357e12i − 0.442058i
\(780\) 0 0
\(781\) 3.31243e12i 0.318579i
\(782\) 0 0
\(783\) 5.97414e11 0.0567999
\(784\) 0 0
\(785\) 3.59145e12 0.337564
\(786\) 0 0
\(787\) 1.23403e13i 1.14667i 0.819320 + 0.573337i \(0.194351\pi\)
−0.819320 + 0.573337i \(0.805649\pi\)
\(788\) 0 0
\(789\) − 2.00827e12i − 0.184491i
\(790\) 0 0
\(791\) 2.74801e12 0.249588
\(792\) 0 0
\(793\) −5.49841e12 −0.493751
\(794\) 0 0
\(795\) 1.97149e12i 0.175042i
\(796\) 0 0
\(797\) 7.96003e12i 0.698799i 0.936974 + 0.349399i \(0.113614\pi\)
−0.936974 + 0.349399i \(0.886386\pi\)
\(798\) 0 0
\(799\) −2.69154e13 −2.33636
\(800\) 0 0
\(801\) 1.64140e12 0.140886
\(802\) 0 0
\(803\) − 4.75764e11i − 0.0403805i
\(804\) 0 0
\(805\) 1.19520e12i 0.100314i
\(806\) 0 0
\(807\) 1.17115e13 0.972030
\(808\) 0 0
\(809\) 1.33262e13 1.09380 0.546902 0.837197i \(-0.315807\pi\)
0.546902 + 0.837197i \(0.315807\pi\)
\(810\) 0 0
\(811\) − 3.23877e12i − 0.262897i −0.991323 0.131449i \(-0.958037\pi\)
0.991323 0.131449i \(-0.0419628\pi\)
\(812\) 0 0
\(813\) − 7.55818e12i − 0.606750i
\(814\) 0 0
\(815\) 2.33331e12 0.185252
\(816\) 0 0
\(817\) 5.77609e12 0.453560
\(818\) 0 0
\(819\) 2.44714e12i 0.190056i
\(820\) 0 0
\(821\) − 1.53482e13i − 1.17900i −0.807768 0.589501i \(-0.799324\pi\)
0.807768 0.589501i \(-0.200676\pi\)
\(822\) 0 0
\(823\) −2.18794e13 −1.66240 −0.831202 0.555971i \(-0.812347\pi\)
−0.831202 + 0.555971i \(0.812347\pi\)
\(824\) 0 0
\(825\) 3.31409e12 0.249070
\(826\) 0 0
\(827\) 1.40288e12i 0.104291i 0.998640 + 0.0521454i \(0.0166059\pi\)
−0.998640 + 0.0521454i \(0.983394\pi\)
\(828\) 0 0
\(829\) − 1.59294e13i − 1.17139i −0.810530 0.585697i \(-0.800821\pi\)
0.810530 0.585697i \(-0.199179\pi\)
\(830\) 0 0
\(831\) 1.14806e13 0.835140
\(832\) 0 0
\(833\) −1.64078e13 −1.18072
\(834\) 0 0
\(835\) − 4.25988e12i − 0.303255i
\(836\) 0 0
\(837\) 4.71065e11i 0.0331754i
\(838\) 0 0
\(839\) 1.81369e12 0.126367 0.0631836 0.998002i \(-0.479875\pi\)
0.0631836 + 0.998002i \(0.479875\pi\)
\(840\) 0 0
\(841\) 1.32435e13 0.912892
\(842\) 0 0
\(843\) 3.70569e12i 0.252723i
\(844\) 0 0
\(845\) − 1.06267e13i − 0.717040i
\(846\) 0 0
\(847\) 3.76595e12 0.251420
\(848\) 0 0
\(849\) −6.94343e12 −0.458659
\(850\) 0 0
\(851\) 9.95273e12i 0.650518i
\(852\) 0 0
\(853\) − 2.21558e13i − 1.43290i −0.697637 0.716451i \(-0.745765\pi\)
0.697637 0.716451i \(-0.254235\pi\)
\(854\) 0 0
\(855\) 6.23758e11 0.0399180
\(856\) 0 0
\(857\) −1.33185e13 −0.843416 −0.421708 0.906732i \(-0.638569\pi\)
−0.421708 + 0.906732i \(0.638569\pi\)
\(858\) 0 0
\(859\) − 4.69713e12i − 0.294349i −0.989111 0.147175i \(-0.952982\pi\)
0.989111 0.147175i \(-0.0470179\pi\)
\(860\) 0 0
\(861\) − 4.85144e12i − 0.300855i
\(862\) 0 0
\(863\) −1.20688e12 −0.0740653 −0.0370326 0.999314i \(-0.511791\pi\)
−0.0370326 + 0.999314i \(0.511791\pi\)
\(864\) 0 0
\(865\) −5.25227e12 −0.318988
\(866\) 0 0
\(867\) 7.62148e12i 0.458093i
\(868\) 0 0
\(869\) 1.70526e12i 0.101438i
\(870\) 0 0
\(871\) −3.37323e13 −1.98593
\(872\) 0 0
\(873\) 5.07334e12 0.295617
\(874\) 0 0
\(875\) 4.48326e12i 0.258558i
\(876\) 0 0
\(877\) − 2.40216e13i − 1.37121i −0.727975 0.685604i \(-0.759538\pi\)
0.727975 0.685604i \(-0.240462\pi\)
\(878\) 0 0
\(879\) 6.42189e12 0.362838
\(880\) 0 0
\(881\) −1.51231e13 −0.845765 −0.422883 0.906184i \(-0.638982\pi\)
−0.422883 + 0.906184i \(0.638982\pi\)
\(882\) 0 0
\(883\) − 2.63940e13i − 1.46111i −0.682854 0.730555i \(-0.739262\pi\)
0.682854 0.730555i \(-0.260738\pi\)
\(884\) 0 0
\(885\) − 2.55852e12i − 0.140199i
\(886\) 0 0
\(887\) −2.20527e13 −1.19621 −0.598103 0.801419i \(-0.704079\pi\)
−0.598103 + 0.801419i \(0.704079\pi\)
\(888\) 0 0
\(889\) −6.52127e12 −0.350166
\(890\) 0 0
\(891\) − 1.08437e12i − 0.0576405i
\(892\) 0 0
\(893\) − 9.67476e12i − 0.509106i
\(894\) 0 0
\(895\) −1.28732e13 −0.670628
\(896\) 0 0
\(897\) 1.31853e13 0.680023
\(898\) 0 0
\(899\) − 9.96428e11i − 0.0508777i
\(900\) 0 0
\(901\) − 1.95718e13i − 0.989393i
\(902\) 0 0
\(903\) 6.16748e12 0.308683
\(904\) 0 0
\(905\) 5.83041e12 0.288922
\(906\) 0 0
\(907\) − 5.06988e12i − 0.248751i −0.992235 0.124376i \(-0.960307\pi\)
0.992235 0.124376i \(-0.0396927\pi\)
\(908\) 0 0
\(909\) 5.47579e12i 0.266017i
\(910\) 0 0
\(911\) 2.16345e13 1.04067 0.520337 0.853961i \(-0.325806\pi\)
0.520337 + 0.853961i \(0.325806\pi\)
\(912\) 0 0
\(913\) 9.40225e12 0.447830
\(914\) 0 0
\(915\) 1.49647e12i 0.0705785i
\(916\) 0 0
\(917\) − 4.89978e12i − 0.228831i
\(918\) 0 0
\(919\) 8.37636e12 0.387378 0.193689 0.981063i \(-0.437955\pi\)
0.193689 + 0.981063i \(0.437955\pi\)
\(920\) 0 0
\(921\) −2.19368e13 −1.00463
\(922\) 0 0
\(923\) 2.24444e13i 1.01789i
\(924\) 0 0
\(925\) 1.69503e13i 0.761271i
\(926\) 0 0
\(927\) 9.49380e12 0.422261
\(928\) 0 0
\(929\) 3.81132e13 1.67882 0.839412 0.543496i \(-0.182900\pi\)
0.839412 + 0.543496i \(0.182900\pi\)
\(930\) 0 0
\(931\) − 5.89781e12i − 0.257287i
\(932\) 0 0
\(933\) − 2.05031e13i − 0.885834i
\(934\) 0 0
\(935\) 6.66259e12 0.285096
\(936\) 0 0
\(937\) 2.86158e13 1.21277 0.606383 0.795172i \(-0.292620\pi\)
0.606383 + 0.795172i \(0.292620\pi\)
\(938\) 0 0
\(939\) − 3.08533e12i − 0.129511i
\(940\) 0 0
\(941\) − 3.31159e13i − 1.37684i −0.725312 0.688421i \(-0.758304\pi\)
0.725312 0.688421i \(-0.241696\pi\)
\(942\) 0 0
\(943\) −2.61397e13 −1.07646
\(944\) 0 0
\(945\) 6.66024e11 0.0271673
\(946\) 0 0
\(947\) − 6.14318e12i − 0.248209i −0.992269 0.124105i \(-0.960394\pi\)
0.992269 0.124105i \(-0.0396059\pi\)
\(948\) 0 0
\(949\) − 3.22368e12i − 0.129019i
\(950\) 0 0
\(951\) 2.15895e13 0.855915
\(952\) 0 0
\(953\) 1.53935e13 0.604531 0.302266 0.953224i \(-0.402257\pi\)
0.302266 + 0.953224i \(0.402257\pi\)
\(954\) 0 0
\(955\) 1.55641e13i 0.605494i
\(956\) 0 0
\(957\) 2.29373e12i 0.0883972i
\(958\) 0 0
\(959\) −4.21460e11 −0.0160906
\(960\) 0 0
\(961\) −2.56539e13 −0.970284
\(962\) 0 0
\(963\) − 1.44954e12i − 0.0543139i
\(964\) 0 0
\(965\) 1.49096e13i 0.553468i
\(966\) 0 0
\(967\) −4.90791e13 −1.80500 −0.902500 0.430691i \(-0.858270\pi\)
−0.902500 + 0.430691i \(0.858270\pi\)
\(968\) 0 0
\(969\) −6.19230e12 −0.225629
\(970\) 0 0
\(971\) − 2.88291e13i − 1.04075i −0.853939 0.520373i \(-0.825793\pi\)
0.853939 0.520373i \(-0.174207\pi\)
\(972\) 0 0
\(973\) 1.24638e13i 0.445804i
\(974\) 0 0
\(975\) 2.24556e13 0.795799
\(976\) 0 0
\(977\) −1.64010e13 −0.575899 −0.287949 0.957646i \(-0.592973\pi\)
−0.287949 + 0.957646i \(0.592973\pi\)
\(978\) 0 0
\(979\) 6.30204e12i 0.219260i
\(980\) 0 0
\(981\) 5.59980e12i 0.193046i
\(982\) 0 0
\(983\) 5.06192e13 1.72912 0.864559 0.502531i \(-0.167598\pi\)
0.864559 + 0.502531i \(0.167598\pi\)
\(984\) 0 0
\(985\) 6.41778e12 0.217231
\(986\) 0 0
\(987\) − 1.03303e13i − 0.346486i
\(988\) 0 0
\(989\) − 3.32306e13i − 1.10447i
\(990\) 0 0
\(991\) −2.53288e13 −0.834225 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(992\) 0 0
\(993\) 1.07364e13 0.350418
\(994\) 0 0
\(995\) − 1.29029e13i − 0.417333i
\(996\) 0 0
\(997\) − 5.28104e13i − 1.69275i −0.532591 0.846373i \(-0.678782\pi\)
0.532591 0.846373i \(-0.321218\pi\)
\(998\) 0 0
\(999\) 5.54613e12 0.176175
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 384.10.d.f.193.17 yes 20
4.3 odd 2 inner 384.10.d.f.193.7 yes 20
8.3 odd 2 inner 384.10.d.f.193.14 yes 20
8.5 even 2 inner 384.10.d.f.193.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.f.193.4 20 8.5 even 2 inner
384.10.d.f.193.7 yes 20 4.3 odd 2 inner
384.10.d.f.193.14 yes 20 8.3 odd 2 inner
384.10.d.f.193.17 yes 20 1.1 even 1 trivial