Properties

Label 384.8.d.e.193.6
Level $384$
Weight $8$
Character 384.193
Analytic conductor $119.956$
Analytic rank $0$
Dimension $12$
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] = [384,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.6
Root \(-19.8448 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.8.d.e.193.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000i q^{3} +467.267i q^{5} -31.2683 q^{7} -729.000 q^{9} +O(q^{10})\) \(q-27.0000i q^{3} +467.267i q^{5} -31.2683 q^{7} -729.000 q^{9} -5276.60i q^{11} -806.828i q^{13} +12616.2 q^{15} +18.4070 q^{17} +13657.0i q^{19} +844.244i q^{21} +58696.4 q^{23} -140214. q^{25} +19683.0i q^{27} +119183. i q^{29} -101880. q^{31} -142468. q^{33} -14610.7i q^{35} -259208. i q^{37} -21784.4 q^{39} -809267. q^{41} -15343.1i q^{43} -340638. i q^{45} -183648. q^{47} -822565. q^{49} -496.988i q^{51} +907643. i q^{53} +2.46558e6 q^{55} +368739. q^{57} -1.43398e6i q^{59} -884055. i q^{61} +22794.6 q^{63} +377004. q^{65} +2.48901e6i q^{67} -1.58480e6i q^{69} +4.68753e6 q^{71} +3.46295e6 q^{73} +3.78577e6i q^{75} +164990. i q^{77} -4.34533e6 q^{79} +531441. q^{81} -8.38216e6i q^{83} +8600.98i q^{85} +3.21793e6 q^{87} +1.29208e7 q^{89} +25228.1i q^{91} +2.75075e6i q^{93} -6.38146e6 q^{95} +4.60705e6 q^{97} +3.84664e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) \) 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
\(3\) − 27.0000i − 0.577350i
\(4\) 0 0
\(5\) 467.267i 1.67175i 0.548922 + 0.835873i \(0.315038\pi\)
−0.548922 + 0.835873i \(0.684962\pi\)
\(6\) 0 0
\(7\) −31.2683 −0.0344557 −0.0172279 0.999852i \(-0.505484\pi\)
−0.0172279 + 0.999852i \(0.505484\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) − 5276.60i − 1.19531i −0.801754 0.597654i \(-0.796100\pi\)
0.801754 0.597654i \(-0.203900\pi\)
\(12\) 0 0
\(13\) − 806.828i − 0.101854i −0.998702 0.0509271i \(-0.983782\pi\)
0.998702 0.0509271i \(-0.0162176\pi\)
\(14\) 0 0
\(15\) 12616.2 0.965183
\(16\) 0 0
\(17\) 18.4070 0.000908680 0 0.000454340 1.00000i \(-0.499855\pi\)
0.000454340 1.00000i \(0.499855\pi\)
\(18\) 0 0
\(19\) 13657.0i 0.456791i 0.973568 + 0.228395i \(0.0733479\pi\)
−0.973568 + 0.228395i \(0.926652\pi\)
\(20\) 0 0
\(21\) 844.244i 0.0198930i
\(22\) 0 0
\(23\) 58696.4 1.00592 0.502961 0.864309i \(-0.332244\pi\)
0.502961 + 0.864309i \(0.332244\pi\)
\(24\) 0 0
\(25\) −140214. −1.79474
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) 119183.i 0.907444i 0.891143 + 0.453722i \(0.149904\pi\)
−0.891143 + 0.453722i \(0.850096\pi\)
\(30\) 0 0
\(31\) −101880. −0.614216 −0.307108 0.951675i \(-0.599361\pi\)
−0.307108 + 0.951675i \(0.599361\pi\)
\(32\) 0 0
\(33\) −142468. −0.690111
\(34\) 0 0
\(35\) − 14610.7i − 0.0576012i
\(36\) 0 0
\(37\) − 259208.i − 0.841285i −0.907227 0.420642i \(-0.861805\pi\)
0.907227 0.420642i \(-0.138195\pi\)
\(38\) 0 0
\(39\) −21784.4 −0.0588056
\(40\) 0 0
\(41\) −809267. −1.83378 −0.916892 0.399135i \(-0.869311\pi\)
−0.916892 + 0.399135i \(0.869311\pi\)
\(42\) 0 0
\(43\) − 15343.1i − 0.0294289i −0.999892 0.0147145i \(-0.995316\pi\)
0.999892 0.0147145i \(-0.00468392\pi\)
\(44\) 0 0
\(45\) − 340638.i − 0.557249i
\(46\) 0 0
\(47\) −183648. −0.258015 −0.129007 0.991644i \(-0.541179\pi\)
−0.129007 + 0.991644i \(0.541179\pi\)
\(48\) 0 0
\(49\) −822565. −0.998813
\(50\) 0 0
\(51\) − 496.988i 0 0.000524627i
\(52\) 0 0
\(53\) 907643.i 0.837431i 0.908117 + 0.418716i \(0.137520\pi\)
−0.908117 + 0.418716i \(0.862480\pi\)
\(54\) 0 0
\(55\) 2.46558e6 1.99825
\(56\) 0 0
\(57\) 368739. 0.263728
\(58\) 0 0
\(59\) − 1.43398e6i − 0.908993i −0.890748 0.454497i \(-0.849819\pi\)
0.890748 0.454497i \(-0.150181\pi\)
\(60\) 0 0
\(61\) − 884055.i − 0.498683i −0.968416 0.249342i \(-0.919786\pi\)
0.968416 0.249342i \(-0.0802142\pi\)
\(62\) 0 0
\(63\) 22794.6 0.0114852
\(64\) 0 0
\(65\) 377004. 0.170274
\(66\) 0 0
\(67\) 2.48901e6i 1.01103i 0.862817 + 0.505516i \(0.168698\pi\)
−0.862817 + 0.505516i \(0.831302\pi\)
\(68\) 0 0
\(69\) − 1.58480e6i − 0.580769i
\(70\) 0 0
\(71\) 4.68753e6 1.55432 0.777158 0.629305i \(-0.216660\pi\)
0.777158 + 0.629305i \(0.216660\pi\)
\(72\) 0 0
\(73\) 3.46295e6 1.04188 0.520938 0.853594i \(-0.325582\pi\)
0.520938 + 0.853594i \(0.325582\pi\)
\(74\) 0 0
\(75\) 3.78577e6i 1.03619i
\(76\) 0 0
\(77\) 164990.i 0.0411852i
\(78\) 0 0
\(79\) −4.34533e6 −0.991580 −0.495790 0.868442i \(-0.665121\pi\)
−0.495790 + 0.868442i \(0.665121\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) − 8.38216e6i − 1.60910i −0.593886 0.804549i \(-0.702407\pi\)
0.593886 0.804549i \(-0.297593\pi\)
\(84\) 0 0
\(85\) 8600.98i 0.00151908i
\(86\) 0 0
\(87\) 3.21793e6 0.523913
\(88\) 0 0
\(89\) 1.29208e7 1.94278 0.971390 0.237489i \(-0.0763245\pi\)
0.971390 + 0.237489i \(0.0763245\pi\)
\(90\) 0 0
\(91\) 25228.1i 0.00350946i
\(92\) 0 0
\(93\) 2.75075e6i 0.354618i
\(94\) 0 0
\(95\) −6.38146e6 −0.763638
\(96\) 0 0
\(97\) 4.60705e6 0.512533 0.256266 0.966606i \(-0.417508\pi\)
0.256266 + 0.966606i \(0.417508\pi\)
\(98\) 0 0
\(99\) 3.84664e6i 0.398436i
\(100\) 0 0
\(101\) − 1.50932e7i − 1.45766i −0.684696 0.728829i \(-0.740065\pi\)
0.684696 0.728829i \(-0.259935\pi\)
\(102\) 0 0
\(103\) 6.45820e6 0.582346 0.291173 0.956670i \(-0.405954\pi\)
0.291173 + 0.956670i \(0.405954\pi\)
\(104\) 0 0
\(105\) −394488. −0.0332561
\(106\) 0 0
\(107\) − 1.43047e7i − 1.12885i −0.825484 0.564425i \(-0.809098\pi\)
0.825484 0.564425i \(-0.190902\pi\)
\(108\) 0 0
\(109\) − 1.46215e7i − 1.08143i −0.841206 0.540715i \(-0.818154\pi\)
0.841206 0.540715i \(-0.181846\pi\)
\(110\) 0 0
\(111\) −6.99863e6 −0.485716
\(112\) 0 0
\(113\) −1.44884e7 −0.944596 −0.472298 0.881439i \(-0.656575\pi\)
−0.472298 + 0.881439i \(0.656575\pi\)
\(114\) 0 0
\(115\) 2.74269e7i 1.68165i
\(116\) 0 0
\(117\) 588177.i 0.0339514i
\(118\) 0 0
\(119\) −575.555 −3.13092e−5 0
\(120\) 0 0
\(121\) −8.35532e6 −0.428760
\(122\) 0 0
\(123\) 2.18502e7i 1.05874i
\(124\) 0 0
\(125\) − 2.90121e7i − 1.32860i
\(126\) 0 0
\(127\) 1.06200e7 0.460055 0.230028 0.973184i \(-0.426118\pi\)
0.230028 + 0.973184i \(0.426118\pi\)
\(128\) 0 0
\(129\) −414264. −0.0169908
\(130\) 0 0
\(131\) − 4.85018e7i − 1.88499i −0.334227 0.942493i \(-0.608475\pi\)
0.334227 0.942493i \(-0.391525\pi\)
\(132\) 0 0
\(133\) − 427031.i − 0.0157390i
\(134\) 0 0
\(135\) −9.19722e6 −0.321728
\(136\) 0 0
\(137\) −3.66799e7 −1.21873 −0.609363 0.792892i \(-0.708575\pi\)
−0.609363 + 0.792892i \(0.708575\pi\)
\(138\) 0 0
\(139\) − 5.99325e6i − 0.189282i −0.995511 0.0946412i \(-0.969830\pi\)
0.995511 0.0946412i \(-0.0301704\pi\)
\(140\) 0 0
\(141\) 4.95850e6i 0.148965i
\(142\) 0 0
\(143\) −4.25731e6 −0.121747
\(144\) 0 0
\(145\) −5.56902e7 −1.51702
\(146\) 0 0
\(147\) 2.22093e7i 0.576665i
\(148\) 0 0
\(149\) − 5.10670e7i − 1.26470i −0.774682 0.632351i \(-0.782090\pi\)
0.774682 0.632351i \(-0.217910\pi\)
\(150\) 0 0
\(151\) −6.49428e7 −1.53501 −0.767506 0.641042i \(-0.778502\pi\)
−0.767506 + 0.641042i \(0.778502\pi\)
\(152\) 0 0
\(153\) −13418.7 −0.000302893 0
\(154\) 0 0
\(155\) − 4.76050e7i − 1.02681i
\(156\) 0 0
\(157\) − 8.27200e7i − 1.70593i −0.521967 0.852966i \(-0.674802\pi\)
0.521967 0.852966i \(-0.325198\pi\)
\(158\) 0 0
\(159\) 2.45063e7 0.483491
\(160\) 0 0
\(161\) −1.83534e6 −0.0346598
\(162\) 0 0
\(163\) 4.55553e7i 0.823915i 0.911203 + 0.411958i \(0.135155\pi\)
−0.911203 + 0.411958i \(0.864845\pi\)
\(164\) 0 0
\(165\) − 6.65707e7i − 1.15369i
\(166\) 0 0
\(167\) −8.80939e7 −1.46365 −0.731827 0.681491i \(-0.761332\pi\)
−0.731827 + 0.681491i \(0.761332\pi\)
\(168\) 0 0
\(169\) 6.20975e7 0.989626
\(170\) 0 0
\(171\) − 9.95594e6i − 0.152264i
\(172\) 0 0
\(173\) 2.34223e6i 0.0343929i 0.999852 + 0.0171965i \(0.00547407\pi\)
−0.999852 + 0.0171965i \(0.994526\pi\)
\(174\) 0 0
\(175\) 4.38425e6 0.0618389
\(176\) 0 0
\(177\) −3.87174e7 −0.524807
\(178\) 0 0
\(179\) 5.96459e7i 0.777312i 0.921383 + 0.388656i \(0.127061\pi\)
−0.921383 + 0.388656i \(0.872939\pi\)
\(180\) 0 0
\(181\) − 4.32379e7i − 0.541987i −0.962581 0.270994i \(-0.912648\pi\)
0.962581 0.270994i \(-0.0873522\pi\)
\(182\) 0 0
\(183\) −2.38695e7 −0.287915
\(184\) 0 0
\(185\) 1.21120e8 1.40642
\(186\) 0 0
\(187\) − 97126.2i − 0.00108615i
\(188\) 0 0
\(189\) − 615454.i − 0.00663100i
\(190\) 0 0
\(191\) 1.53658e8 1.59565 0.797826 0.602887i \(-0.205983\pi\)
0.797826 + 0.602887i \(0.205983\pi\)
\(192\) 0 0
\(193\) 7.22178e7 0.723093 0.361546 0.932354i \(-0.382249\pi\)
0.361546 + 0.932354i \(0.382249\pi\)
\(194\) 0 0
\(195\) − 1.01791e7i − 0.0983080i
\(196\) 0 0
\(197\) − 1.48312e8i − 1.38212i −0.722799 0.691058i \(-0.757145\pi\)
0.722799 0.691058i \(-0.242855\pi\)
\(198\) 0 0
\(199\) 1.21013e8 1.08855 0.544274 0.838907i \(-0.316805\pi\)
0.544274 + 0.838907i \(0.316805\pi\)
\(200\) 0 0
\(201\) 6.72033e7 0.583720
\(202\) 0 0
\(203\) − 3.72664e6i − 0.0312666i
\(204\) 0 0
\(205\) − 3.78144e8i − 3.06562i
\(206\) 0 0
\(207\) −4.27897e7 −0.335307
\(208\) 0 0
\(209\) 7.20624e7 0.546005
\(210\) 0 0
\(211\) − 1.37134e8i − 1.00498i −0.864584 0.502489i \(-0.832418\pi\)
0.864584 0.502489i \(-0.167582\pi\)
\(212\) 0 0
\(213\) − 1.26563e8i − 0.897385i
\(214\) 0 0
\(215\) 7.16934e6 0.0491977
\(216\) 0 0
\(217\) 3.18560e6 0.0211632
\(218\) 0 0
\(219\) − 9.34996e7i − 0.601528i
\(220\) 0 0
\(221\) − 14851.3i 0 9.25529e-5i
\(222\) 0 0
\(223\) 2.14479e8 1.29514 0.647572 0.762004i \(-0.275785\pi\)
0.647572 + 0.762004i \(0.275785\pi\)
\(224\) 0 0
\(225\) 1.02216e8 0.598246
\(226\) 0 0
\(227\) 2.91532e8i 1.65423i 0.562035 + 0.827113i \(0.310019\pi\)
−0.562035 + 0.827113i \(0.689981\pi\)
\(228\) 0 0
\(229\) 1.31691e8i 0.724657i 0.932050 + 0.362329i \(0.118018\pi\)
−0.932050 + 0.362329i \(0.881982\pi\)
\(230\) 0 0
\(231\) 4.45474e6 0.0237783
\(232\) 0 0
\(233\) −1.40022e8 −0.725189 −0.362595 0.931947i \(-0.618109\pi\)
−0.362595 + 0.931947i \(0.618109\pi\)
\(234\) 0 0
\(235\) − 8.58129e7i − 0.431335i
\(236\) 0 0
\(237\) 1.17324e8i 0.572489i
\(238\) 0 0
\(239\) −3.32407e8 −1.57499 −0.787495 0.616321i \(-0.788622\pi\)
−0.787495 + 0.616321i \(0.788622\pi\)
\(240\) 0 0
\(241\) 1.15264e8 0.530436 0.265218 0.964188i \(-0.414556\pi\)
0.265218 + 0.964188i \(0.414556\pi\)
\(242\) 0 0
\(243\) − 1.43489e7i − 0.0641500i
\(244\) 0 0
\(245\) − 3.84358e8i − 1.66976i
\(246\) 0 0
\(247\) 1.10188e7 0.0465261
\(248\) 0 0
\(249\) −2.26318e8 −0.929014
\(250\) 0 0
\(251\) 6.42315e7i 0.256384i 0.991749 + 0.128192i \(0.0409173\pi\)
−0.991749 + 0.128192i \(0.959083\pi\)
\(252\) 0 0
\(253\) − 3.09718e8i − 1.20239i
\(254\) 0 0
\(255\) 232226. 0.000877043 0
\(256\) 0 0
\(257\) −1.87951e8 −0.690682 −0.345341 0.938477i \(-0.612237\pi\)
−0.345341 + 0.938477i \(0.612237\pi\)
\(258\) 0 0
\(259\) 8.10501e6i 0.0289871i
\(260\) 0 0
\(261\) − 8.68841e7i − 0.302481i
\(262\) 0 0
\(263\) 4.53334e8 1.53664 0.768322 0.640064i \(-0.221092\pi\)
0.768322 + 0.640064i \(0.221092\pi\)
\(264\) 0 0
\(265\) −4.24112e8 −1.39997
\(266\) 0 0
\(267\) − 3.48861e8i − 1.12166i
\(268\) 0 0
\(269\) − 3.11428e8i − 0.975495i −0.872985 0.487747i \(-0.837819\pi\)
0.872985 0.487747i \(-0.162181\pi\)
\(270\) 0 0
\(271\) −3.59086e8 −1.09599 −0.547994 0.836483i \(-0.684608\pi\)
−0.547994 + 0.836483i \(0.684608\pi\)
\(272\) 0 0
\(273\) 681160. 0.00202619
\(274\) 0 0
\(275\) 7.39852e8i 2.14526i
\(276\) 0 0
\(277\) 2.12866e8i 0.601767i 0.953661 + 0.300883i \(0.0972815\pi\)
−0.953661 + 0.300883i \(0.902719\pi\)
\(278\) 0 0
\(279\) 7.42702e7 0.204739
\(280\) 0 0
\(281\) −2.34768e8 −0.631200 −0.315600 0.948892i \(-0.602206\pi\)
−0.315600 + 0.948892i \(0.602206\pi\)
\(282\) 0 0
\(283\) − 4.61638e8i − 1.21073i −0.795947 0.605367i \(-0.793026\pi\)
0.795947 0.605367i \(-0.206974\pi\)
\(284\) 0 0
\(285\) 1.72299e8i 0.440887i
\(286\) 0 0
\(287\) 2.53044e7 0.0631843
\(288\) 0 0
\(289\) −4.10338e8 −0.999999
\(290\) 0 0
\(291\) − 1.24390e8i − 0.295911i
\(292\) 0 0
\(293\) − 2.81937e8i − 0.654811i −0.944884 0.327406i \(-0.893826\pi\)
0.944884 0.327406i \(-0.106174\pi\)
\(294\) 0 0
\(295\) 6.70051e8 1.51961
\(296\) 0 0
\(297\) 1.03859e8 0.230037
\(298\) 0 0
\(299\) − 4.73579e7i − 0.102457i
\(300\) 0 0
\(301\) 479753.i 0.00101399i
\(302\) 0 0
\(303\) −4.07515e8 −0.841579
\(304\) 0 0
\(305\) 4.13090e8 0.833672
\(306\) 0 0
\(307\) 2.31592e8i 0.456813i 0.973566 + 0.228407i \(0.0733515\pi\)
−0.973566 + 0.228407i \(0.926648\pi\)
\(308\) 0 0
\(309\) − 1.74371e8i − 0.336218i
\(310\) 0 0
\(311\) −6.15818e8 −1.16089 −0.580445 0.814299i \(-0.697121\pi\)
−0.580445 + 0.814299i \(0.697121\pi\)
\(312\) 0 0
\(313\) 2.86280e8 0.527699 0.263849 0.964564i \(-0.415008\pi\)
0.263849 + 0.964564i \(0.415008\pi\)
\(314\) 0 0
\(315\) 1.06512e7i 0.0192004i
\(316\) 0 0
\(317\) 3.20478e8i 0.565055i 0.959259 + 0.282527i \(0.0911728\pi\)
−0.959259 + 0.282527i \(0.908827\pi\)
\(318\) 0 0
\(319\) 6.28879e8 1.08467
\(320\) 0 0
\(321\) −3.86228e8 −0.651742
\(322\) 0 0
\(323\) 251384.i 0 0.000415077i
\(324\) 0 0
\(325\) 1.13128e8i 0.182802i
\(326\) 0 0
\(327\) −3.94780e8 −0.624364
\(328\) 0 0
\(329\) 5.74237e6 0.00889008
\(330\) 0 0
\(331\) − 6.95072e8i − 1.05349i −0.850022 0.526746i \(-0.823412\pi\)
0.850022 0.526746i \(-0.176588\pi\)
\(332\) 0 0
\(333\) 1.88963e8i 0.280428i
\(334\) 0 0
\(335\) −1.16303e9 −1.69019
\(336\) 0 0
\(337\) 7.90458e8 1.12506 0.562528 0.826778i \(-0.309829\pi\)
0.562528 + 0.826778i \(0.309829\pi\)
\(338\) 0 0
\(339\) 3.91187e8i 0.545363i
\(340\) 0 0
\(341\) 5.37577e8i 0.734177i
\(342\) 0 0
\(343\) 5.14710e7 0.0688705
\(344\) 0 0
\(345\) 7.40527e8 0.970899
\(346\) 0 0
\(347\) − 8.49503e8i − 1.09147i −0.837958 0.545735i \(-0.816251\pi\)
0.837958 0.545735i \(-0.183749\pi\)
\(348\) 0 0
\(349\) − 3.96115e8i − 0.498806i −0.968400 0.249403i \(-0.919766\pi\)
0.968400 0.249403i \(-0.0802345\pi\)
\(350\) 0 0
\(351\) 1.58808e7 0.0196019
\(352\) 0 0
\(353\) 6.57428e8 0.795494 0.397747 0.917495i \(-0.369792\pi\)
0.397747 + 0.917495i \(0.369792\pi\)
\(354\) 0 0
\(355\) 2.19033e9i 2.59842i
\(356\) 0 0
\(357\) 15540.0i 0 1.80764e-5i
\(358\) 0 0
\(359\) −5.97068e8 −0.681072 −0.340536 0.940231i \(-0.610609\pi\)
−0.340536 + 0.940231i \(0.610609\pi\)
\(360\) 0 0
\(361\) 7.07359e8 0.791342
\(362\) 0 0
\(363\) 2.25594e8i 0.247545i
\(364\) 0 0
\(365\) 1.61812e9i 1.74175i
\(366\) 0 0
\(367\) −2.31319e8 −0.244275 −0.122138 0.992513i \(-0.538975\pi\)
−0.122138 + 0.992513i \(0.538975\pi\)
\(368\) 0 0
\(369\) 5.89956e8 0.611262
\(370\) 0 0
\(371\) − 2.83804e7i − 0.0288543i
\(372\) 0 0
\(373\) 1.47548e9i 1.47215i 0.676897 + 0.736077i \(0.263324\pi\)
−0.676897 + 0.736077i \(0.736676\pi\)
\(374\) 0 0
\(375\) −7.83326e8 −0.767067
\(376\) 0 0
\(377\) 9.61599e7 0.0924270
\(378\) 0 0
\(379\) − 9.58315e8i − 0.904214i −0.891964 0.452107i \(-0.850673\pi\)
0.891964 0.452107i \(-0.149327\pi\)
\(380\) 0 0
\(381\) − 2.86739e8i − 0.265613i
\(382\) 0 0
\(383\) 1.90346e9 1.73121 0.865603 0.500731i \(-0.166935\pi\)
0.865603 + 0.500731i \(0.166935\pi\)
\(384\) 0 0
\(385\) −7.70946e7 −0.0688512
\(386\) 0 0
\(387\) 1.11851e7i 0.00980963i
\(388\) 0 0
\(389\) 1.68380e9i 1.45033i 0.688575 + 0.725165i \(0.258236\pi\)
−0.688575 + 0.725165i \(0.741764\pi\)
\(390\) 0 0
\(391\) 1.08042e6 0.000914061 0
\(392\) 0 0
\(393\) −1.30955e9 −1.08830
\(394\) 0 0
\(395\) − 2.03043e9i − 1.65767i
\(396\) 0 0
\(397\) 9.50529e8i 0.762428i 0.924487 + 0.381214i \(0.124494\pi\)
−0.924487 + 0.381214i \(0.875506\pi\)
\(398\) 0 0
\(399\) −1.15298e7 −0.00908694
\(400\) 0 0
\(401\) −5.52085e8 −0.427564 −0.213782 0.976881i \(-0.568578\pi\)
−0.213782 + 0.976881i \(0.568578\pi\)
\(402\) 0 0
\(403\) 8.21992e7i 0.0625605i
\(404\) 0 0
\(405\) 2.48325e8i 0.185750i
\(406\) 0 0
\(407\) −1.36774e9 −1.00559
\(408\) 0 0
\(409\) −4.84172e8 −0.349919 −0.174960 0.984576i \(-0.555980\pi\)
−0.174960 + 0.984576i \(0.555980\pi\)
\(410\) 0 0
\(411\) 9.90357e8i 0.703631i
\(412\) 0 0
\(413\) 4.48381e7i 0.0313200i
\(414\) 0 0
\(415\) 3.91671e9 2.69001
\(416\) 0 0
\(417\) −1.61818e8 −0.109282
\(418\) 0 0
\(419\) − 4.02693e8i − 0.267439i −0.991019 0.133719i \(-0.957308\pi\)
0.991019 0.133719i \(-0.0426921\pi\)
\(420\) 0 0
\(421\) − 1.16230e9i − 0.759159i −0.925159 0.379579i \(-0.876069\pi\)
0.925159 0.379579i \(-0.123931\pi\)
\(422\) 0 0
\(423\) 1.33880e8 0.0860049
\(424\) 0 0
\(425\) −2.58091e6 −0.00163084
\(426\) 0 0
\(427\) 2.76429e7i 0.0171825i
\(428\) 0 0
\(429\) 1.14947e8i 0.0702907i
\(430\) 0 0
\(431\) −8.65230e8 −0.520548 −0.260274 0.965535i \(-0.583813\pi\)
−0.260274 + 0.965535i \(0.583813\pi\)
\(432\) 0 0
\(433\) −1.88729e9 −1.11720 −0.558599 0.829438i \(-0.688661\pi\)
−0.558599 + 0.829438i \(0.688661\pi\)
\(434\) 0 0
\(435\) 1.50363e9i 0.875850i
\(436\) 0 0
\(437\) 8.01616e8i 0.459496i
\(438\) 0 0
\(439\) 1.78047e9 1.00440 0.502202 0.864751i \(-0.332524\pi\)
0.502202 + 0.864751i \(0.332524\pi\)
\(440\) 0 0
\(441\) 5.99650e8 0.332938
\(442\) 0 0
\(443\) − 2.21282e9i − 1.20930i −0.796492 0.604649i \(-0.793314\pi\)
0.796492 0.604649i \(-0.206686\pi\)
\(444\) 0 0
\(445\) 6.03746e9i 3.24784i
\(446\) 0 0
\(447\) −1.37881e9 −0.730176
\(448\) 0 0
\(449\) −1.81035e9 −0.943844 −0.471922 0.881640i \(-0.656440\pi\)
−0.471922 + 0.881640i \(0.656440\pi\)
\(450\) 0 0
\(451\) 4.27018e9i 2.19194i
\(452\) 0 0
\(453\) 1.75346e9i 0.886239i
\(454\) 0 0
\(455\) −1.17883e7 −0.00586693
\(456\) 0 0
\(457\) −2.26015e9 −1.10772 −0.553861 0.832609i \(-0.686846\pi\)
−0.553861 + 0.832609i \(0.686846\pi\)
\(458\) 0 0
\(459\) 362304.i 0 0.000174876i
\(460\) 0 0
\(461\) − 2.29944e8i − 0.109312i −0.998505 0.0546561i \(-0.982594\pi\)
0.998505 0.0546561i \(-0.0174062\pi\)
\(462\) 0 0
\(463\) 2.87071e9 1.34418 0.672088 0.740472i \(-0.265398\pi\)
0.672088 + 0.740472i \(0.265398\pi\)
\(464\) 0 0
\(465\) −1.28533e9 −0.592831
\(466\) 0 0
\(467\) 8.25672e8i 0.375145i 0.982251 + 0.187572i \(0.0600619\pi\)
−0.982251 + 0.187572i \(0.939938\pi\)
\(468\) 0 0
\(469\) − 7.78272e7i − 0.0348359i
\(470\) 0 0
\(471\) −2.23344e9 −0.984920
\(472\) 0 0
\(473\) −8.09595e7 −0.0351766
\(474\) 0 0
\(475\) − 1.91490e9i − 0.819819i
\(476\) 0 0
\(477\) − 6.61671e8i − 0.279144i
\(478\) 0 0
\(479\) −3.09397e9 −1.28630 −0.643150 0.765740i \(-0.722373\pi\)
−0.643150 + 0.765740i \(0.722373\pi\)
\(480\) 0 0
\(481\) −2.09137e8 −0.0856884
\(482\) 0 0
\(483\) 4.95541e7i 0.0200108i
\(484\) 0 0
\(485\) 2.15272e9i 0.856825i
\(486\) 0 0
\(487\) 2.88844e9 1.13321 0.566607 0.823988i \(-0.308256\pi\)
0.566607 + 0.823988i \(0.308256\pi\)
\(488\) 0 0
\(489\) 1.22999e9 0.475688
\(490\) 0 0
\(491\) − 3.69018e9i − 1.40690i −0.710746 0.703449i \(-0.751643\pi\)
0.710746 0.703449i \(-0.248357\pi\)
\(492\) 0 0
\(493\) 2.19379e6i 0 0.000824577i
\(494\) 0 0
\(495\) −1.79741e9 −0.666084
\(496\) 0 0
\(497\) −1.46571e8 −0.0535551
\(498\) 0 0
\(499\) − 1.03673e9i − 0.373519i −0.982406 0.186759i \(-0.940202\pi\)
0.982406 0.186759i \(-0.0597985\pi\)
\(500\) 0 0
\(501\) 2.37854e9i 0.845040i
\(502\) 0 0
\(503\) 6.16186e8 0.215886 0.107943 0.994157i \(-0.465574\pi\)
0.107943 + 0.994157i \(0.465574\pi\)
\(504\) 0 0
\(505\) 7.05254e9 2.43683
\(506\) 0 0
\(507\) − 1.67663e9i − 0.571361i
\(508\) 0 0
\(509\) − 1.70431e9i − 0.572844i −0.958103 0.286422i \(-0.907534\pi\)
0.958103 0.286422i \(-0.0924660\pi\)
\(510\) 0 0
\(511\) −1.08281e8 −0.0358986
\(512\) 0 0
\(513\) −2.68810e8 −0.0879094
\(514\) 0 0
\(515\) 3.01771e9i 0.973535i
\(516\) 0 0
\(517\) 9.69038e8i 0.308407i
\(518\) 0 0
\(519\) 6.32403e7 0.0198568
\(520\) 0 0
\(521\) −1.12737e9 −0.349249 −0.174624 0.984635i \(-0.555871\pi\)
−0.174624 + 0.984635i \(0.555871\pi\)
\(522\) 0 0
\(523\) 7.00952e8i 0.214256i 0.994245 + 0.107128i \(0.0341654\pi\)
−0.994245 + 0.107128i \(0.965835\pi\)
\(524\) 0 0
\(525\) − 1.18375e8i − 0.0357027i
\(526\) 0 0
\(527\) −1.87529e6 −0.000558126 0
\(528\) 0 0
\(529\) 4.04465e7 0.0118792
\(530\) 0 0
\(531\) 1.04537e9i 0.302998i
\(532\) 0 0
\(533\) 6.52939e8i 0.186779i
\(534\) 0 0
\(535\) 6.68413e9 1.88715
\(536\) 0 0
\(537\) 1.61044e9 0.448781
\(538\) 0 0
\(539\) 4.34035e9i 1.19389i
\(540\) 0 0
\(541\) 2.93472e9i 0.796851i 0.917201 + 0.398425i \(0.130443\pi\)
−0.917201 + 0.398425i \(0.869557\pi\)
\(542\) 0 0
\(543\) −1.16742e9 −0.312916
\(544\) 0 0
\(545\) 6.83214e9 1.80788
\(546\) 0 0
\(547\) − 3.77357e9i − 0.985819i −0.870080 0.492910i \(-0.835933\pi\)
0.870080 0.492910i \(-0.164067\pi\)
\(548\) 0 0
\(549\) 6.44476e8i 0.166228i
\(550\) 0 0
\(551\) −1.62767e9 −0.414512
\(552\) 0 0
\(553\) 1.35871e8 0.0341656
\(554\) 0 0
\(555\) − 3.27023e9i − 0.811994i
\(556\) 0 0
\(557\) 1.41429e9i 0.346774i 0.984854 + 0.173387i \(0.0554711\pi\)
−0.984854 + 0.173387i \(0.944529\pi\)
\(558\) 0 0
\(559\) −1.23793e7 −0.00299746
\(560\) 0 0
\(561\) −2.62241e6 −0.000627090 0
\(562\) 0 0
\(563\) 6.13087e9i 1.44791i 0.689845 + 0.723957i \(0.257679\pi\)
−0.689845 + 0.723957i \(0.742321\pi\)
\(564\) 0 0
\(565\) − 6.76996e9i − 1.57913i
\(566\) 0 0
\(567\) −1.66173e7 −0.00382841
\(568\) 0 0
\(569\) 1.93798e9 0.441017 0.220509 0.975385i \(-0.429228\pi\)
0.220509 + 0.975385i \(0.429228\pi\)
\(570\) 0 0
\(571\) − 8.05170e8i − 0.180993i −0.995897 0.0904964i \(-0.971155\pi\)
0.995897 0.0904964i \(-0.0288453\pi\)
\(572\) 0 0
\(573\) − 4.14877e9i − 0.921250i
\(574\) 0 0
\(575\) −8.23005e9 −1.80537
\(576\) 0 0
\(577\) −1.00273e9 −0.217305 −0.108653 0.994080i \(-0.534654\pi\)
−0.108653 + 0.994080i \(0.534654\pi\)
\(578\) 0 0
\(579\) − 1.94988e9i − 0.417478i
\(580\) 0 0
\(581\) 2.62096e8i 0.0554426i
\(582\) 0 0
\(583\) 4.78927e9 1.00099
\(584\) 0 0
\(585\) −2.74836e8 −0.0567582
\(586\) 0 0
\(587\) 1.87262e9i 0.382135i 0.981577 + 0.191067i \(0.0611949\pi\)
−0.981577 + 0.191067i \(0.938805\pi\)
\(588\) 0 0
\(589\) − 1.39137e9i − 0.280568i
\(590\) 0 0
\(591\) −4.00442e9 −0.797965
\(592\) 0 0
\(593\) 5.41357e9 1.06609 0.533043 0.846088i \(-0.321048\pi\)
0.533043 + 0.846088i \(0.321048\pi\)
\(594\) 0 0
\(595\) − 268938.i 0 5.23411e-5i
\(596\) 0 0
\(597\) − 3.26736e9i − 0.628474i
\(598\) 0 0
\(599\) −4.71914e9 −0.897159 −0.448579 0.893743i \(-0.648070\pi\)
−0.448579 + 0.893743i \(0.648070\pi\)
\(600\) 0 0
\(601\) −2.69993e9 −0.507331 −0.253666 0.967292i \(-0.581636\pi\)
−0.253666 + 0.967292i \(0.581636\pi\)
\(602\) 0 0
\(603\) − 1.81449e9i − 0.337011i
\(604\) 0 0
\(605\) − 3.90417e9i − 0.716778i
\(606\) 0 0
\(607\) 3.86996e8 0.0702337 0.0351169 0.999383i \(-0.488820\pi\)
0.0351169 + 0.999383i \(0.488820\pi\)
\(608\) 0 0
\(609\) −1.00619e8 −0.0180518
\(610\) 0 0
\(611\) 1.48173e8i 0.0262799i
\(612\) 0 0
\(613\) − 7.20217e9i − 1.26285i −0.775437 0.631425i \(-0.782470\pi\)
0.775437 0.631425i \(-0.217530\pi\)
\(614\) 0 0
\(615\) −1.02099e10 −1.76994
\(616\) 0 0
\(617\) −1.11331e10 −1.90817 −0.954087 0.299531i \(-0.903170\pi\)
−0.954087 + 0.299531i \(0.903170\pi\)
\(618\) 0 0
\(619\) − 5.50625e9i − 0.933123i −0.884489 0.466561i \(-0.845493\pi\)
0.884489 0.466561i \(-0.154507\pi\)
\(620\) 0 0
\(621\) 1.15532e9i 0.193590i
\(622\) 0 0
\(623\) −4.04011e8 −0.0669399
\(624\) 0 0
\(625\) 2.60219e9 0.426343
\(626\) 0 0
\(627\) − 1.94569e9i − 0.315236i
\(628\) 0 0
\(629\) − 4.77124e6i 0 0.000764459i
\(630\) 0 0
\(631\) 4.99750e9 0.791863 0.395931 0.918280i \(-0.370422\pi\)
0.395931 + 0.918280i \(0.370422\pi\)
\(632\) 0 0
\(633\) −3.70262e9 −0.580224
\(634\) 0 0
\(635\) 4.96236e9i 0.769096i
\(636\) 0 0
\(637\) 6.63669e8i 0.101733i
\(638\) 0 0
\(639\) −3.41721e9 −0.518106
\(640\) 0 0
\(641\) −8.92704e9 −1.33877 −0.669383 0.742918i \(-0.733441\pi\)
−0.669383 + 0.742918i \(0.733441\pi\)
\(642\) 0 0
\(643\) − 1.87437e9i − 0.278046i −0.990289 0.139023i \(-0.955604\pi\)
0.990289 0.139023i \(-0.0443963\pi\)
\(644\) 0 0
\(645\) − 1.93572e8i − 0.0284043i
\(646\) 0 0
\(647\) −6.12131e9 −0.888545 −0.444273 0.895892i \(-0.646538\pi\)
−0.444273 + 0.895892i \(0.646538\pi\)
\(648\) 0 0
\(649\) −7.56653e9 −1.08653
\(650\) 0 0
\(651\) − 8.60112e7i − 0.0122186i
\(652\) 0 0
\(653\) − 3.55775e9i − 0.500010i −0.968245 0.250005i \(-0.919568\pi\)
0.968245 0.250005i \(-0.0804323\pi\)
\(654\) 0 0
\(655\) 2.26633e10 3.15122
\(656\) 0 0
\(657\) −2.52449e9 −0.347292
\(658\) 0 0
\(659\) 2.14258e9i 0.291634i 0.989312 + 0.145817i \(0.0465811\pi\)
−0.989312 + 0.145817i \(0.953419\pi\)
\(660\) 0 0
\(661\) 1.02631e9i 0.138221i 0.997609 + 0.0691105i \(0.0220161\pi\)
−0.997609 + 0.0691105i \(0.977984\pi\)
\(662\) 0 0
\(663\) −400984. −5.34355e−5 0
\(664\) 0 0
\(665\) 1.99537e8 0.0263117
\(666\) 0 0
\(667\) 6.99560e9i 0.912818i
\(668\) 0 0
\(669\) − 5.79094e9i − 0.747752i
\(670\) 0 0
\(671\) −4.66480e9 −0.596080
\(672\) 0 0
\(673\) −4.26856e9 −0.539794 −0.269897 0.962889i \(-0.586990\pi\)
−0.269897 + 0.962889i \(0.586990\pi\)
\(674\) 0 0
\(675\) − 2.75983e9i − 0.345397i
\(676\) 0 0
\(677\) − 5.61660e9i − 0.695686i −0.937553 0.347843i \(-0.886914\pi\)
0.937553 0.347843i \(-0.113086\pi\)
\(678\) 0 0
\(679\) −1.44055e8 −0.0176597
\(680\) 0 0
\(681\) 7.87135e9 0.955068
\(682\) 0 0
\(683\) 5.51352e9i 0.662150i 0.943604 + 0.331075i \(0.107411\pi\)
−0.943604 + 0.331075i \(0.892589\pi\)
\(684\) 0 0
\(685\) − 1.71393e10i − 2.03740i
\(686\) 0 0
\(687\) 3.55566e9 0.418381
\(688\) 0 0
\(689\) 7.32311e8 0.0852959
\(690\) 0 0
\(691\) − 1.03033e10i − 1.18796i −0.804480 0.593980i \(-0.797556\pi\)
0.804480 0.593980i \(-0.202444\pi\)
\(692\) 0 0
\(693\) − 1.20278e8i − 0.0137284i
\(694\) 0 0
\(695\) 2.80045e9 0.316432
\(696\) 0 0
\(697\) −1.48962e7 −0.00166632
\(698\) 0 0
\(699\) 3.78060e9i 0.418688i
\(700\) 0 0
\(701\) − 1.05192e10i − 1.15337i −0.816966 0.576686i \(-0.804345\pi\)
0.816966 0.576686i \(-0.195655\pi\)
\(702\) 0 0
\(703\) 3.54000e9 0.384291
\(704\) 0 0
\(705\) −2.31695e9 −0.249031
\(706\) 0 0
\(707\) 4.71938e8i 0.0502246i
\(708\) 0 0
\(709\) − 7.67229e9i − 0.808468i −0.914656 0.404234i \(-0.867538\pi\)
0.914656 0.404234i \(-0.132462\pi\)
\(710\) 0 0
\(711\) 3.16774e9 0.330527
\(712\) 0 0
\(713\) −5.97996e9 −0.617853
\(714\) 0 0
\(715\) − 1.98930e9i − 0.203530i
\(716\) 0 0
\(717\) 8.97500e9i 0.909321i
\(718\) 0 0
\(719\) 1.54882e10 1.55400 0.776999 0.629502i \(-0.216741\pi\)
0.776999 + 0.629502i \(0.216741\pi\)
\(720\) 0 0
\(721\) −2.01937e8 −0.0200652
\(722\) 0 0
\(723\) − 3.11212e9i − 0.306247i
\(724\) 0 0
\(725\) − 1.67110e10i − 1.62862i
\(726\) 0 0
\(727\) −1.36057e9 −0.131326 −0.0656631 0.997842i \(-0.520916\pi\)
−0.0656631 + 0.997842i \(0.520916\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) − 282420.i 0 2.67415e-5i
\(732\) 0 0
\(733\) 9.50290e8i 0.0891235i 0.999007 + 0.0445617i \(0.0141891\pi\)
−0.999007 + 0.0445617i \(0.985811\pi\)
\(734\) 0 0
\(735\) −1.03777e10 −0.964037
\(736\) 0 0
\(737\) 1.31335e10 1.20850
\(738\) 0 0
\(739\) 1.61058e10i 1.46800i 0.679150 + 0.733999i \(0.262349\pi\)
−0.679150 + 0.733999i \(0.737651\pi\)
\(740\) 0 0
\(741\) − 2.97508e8i − 0.0268618i
\(742\) 0 0
\(743\) −1.62026e10 −1.44918 −0.724592 0.689178i \(-0.757972\pi\)
−0.724592 + 0.689178i \(0.757972\pi\)
\(744\) 0 0
\(745\) 2.38619e10 2.11426
\(746\) 0 0
\(747\) 6.11060e9i 0.536366i
\(748\) 0 0
\(749\) 4.47284e8i 0.0388953i
\(750\) 0 0
\(751\) 1.60043e10 1.37879 0.689393 0.724388i \(-0.257878\pi\)
0.689393 + 0.724388i \(0.257878\pi\)
\(752\) 0 0
\(753\) 1.73425e9 0.148023
\(754\) 0 0
\(755\) − 3.03457e10i − 2.56615i
\(756\) 0 0
\(757\) 1.44950e10i 1.21446i 0.794527 + 0.607229i \(0.207719\pi\)
−0.794527 + 0.607229i \(0.792281\pi\)
\(758\) 0 0
\(759\) −8.36237e9 −0.694198
\(760\) 0 0
\(761\) 3.48731e8 0.0286843 0.0143421 0.999897i \(-0.495435\pi\)
0.0143421 + 0.999897i \(0.495435\pi\)
\(762\) 0 0
\(763\) 4.57189e8i 0.0372615i
\(764\) 0 0
\(765\) − 6.27011e6i 0 0.000506361i
\(766\) 0 0
\(767\) −1.15697e9 −0.0925848
\(768\) 0 0
\(769\) −4.31946e9 −0.342521 −0.171260 0.985226i \(-0.554784\pi\)
−0.171260 + 0.985226i \(0.554784\pi\)
\(770\) 0 0
\(771\) 5.07467e9i 0.398765i
\(772\) 0 0
\(773\) − 1.47386e10i − 1.14770i −0.818961 0.573849i \(-0.805449\pi\)
0.818961 0.573849i \(-0.194551\pi\)
\(774\) 0 0
\(775\) 1.42849e10 1.10236
\(776\) 0 0
\(777\) 2.18835e8 0.0167357
\(778\) 0 0
\(779\) − 1.10521e10i − 0.837655i
\(780\) 0 0
\(781\) − 2.47342e10i − 1.85789i
\(782\) 0 0
\(783\) −2.34587e9 −0.174638
\(784\) 0 0
\(785\) 3.86523e10 2.85189
\(786\) 0 0
\(787\) 8.74976e9i 0.639859i 0.947441 + 0.319930i \(0.103659\pi\)
−0.947441 + 0.319930i \(0.896341\pi\)
\(788\) 0 0
\(789\) − 1.22400e10i − 0.887182i
\(790\) 0 0
\(791\) 4.53028e8 0.0325467
\(792\) 0 0
\(793\) −7.13280e8 −0.0507930
\(794\) 0 0
\(795\) 1.14510e10i 0.808275i
\(796\) 0 0
\(797\) − 1.95855e9i − 0.137035i −0.997650 0.0685173i \(-0.978173\pi\)
0.997650 0.0685173i \(-0.0218268\pi\)
\(798\) 0 0
\(799\) −3.38041e6 −0.000234453 0
\(800\) 0 0
\(801\) −9.41925e9 −0.647593
\(802\) 0 0
\(803\) − 1.82726e10i − 1.24536i
\(804\) 0 0
\(805\) − 8.57594e8i − 0.0579423i
\(806\) 0 0
\(807\) −8.40856e9 −0.563202
\(808\) 0 0
\(809\) −1.64242e10 −1.09060 −0.545299 0.838242i \(-0.683584\pi\)
−0.545299 + 0.838242i \(0.683584\pi\)
\(810\) 0 0
\(811\) 2.27404e10i 1.49701i 0.663127 + 0.748507i \(0.269229\pi\)
−0.663127 + 0.748507i \(0.730771\pi\)
\(812\) 0 0
\(813\) 9.69531e9i 0.632768i
\(814\) 0 0
\(815\) −2.12865e10 −1.37738
\(816\) 0 0
\(817\) 2.09541e8 0.0134428
\(818\) 0 0
\(819\) − 1.83913e7i − 0.00116982i
\(820\) 0 0
\(821\) − 2.18898e9i − 0.138052i −0.997615 0.0690258i \(-0.978011\pi\)
0.997615 0.0690258i \(-0.0219891\pi\)
\(822\) 0 0
\(823\) 2.09467e10 1.30983 0.654916 0.755702i \(-0.272704\pi\)
0.654916 + 0.755702i \(0.272704\pi\)
\(824\) 0 0
\(825\) 1.99760e10 1.23857
\(826\) 0 0
\(827\) 1.60221e10i 0.985033i 0.870303 + 0.492517i \(0.163923\pi\)
−0.870303 + 0.492517i \(0.836077\pi\)
\(828\) 0 0
\(829\) 8.26352e9i 0.503761i 0.967758 + 0.251880i \(0.0810490\pi\)
−0.967758 + 0.251880i \(0.918951\pi\)
\(830\) 0 0
\(831\) 5.74739e9 0.347430
\(832\) 0 0
\(833\) −1.51409e7 −0.000907601 0
\(834\) 0 0
\(835\) − 4.11634e10i − 2.44686i
\(836\) 0 0
\(837\) − 2.00529e9i − 0.118206i
\(838\) 0 0
\(839\) 9.38387e9 0.548549 0.274274 0.961651i \(-0.411562\pi\)
0.274274 + 0.961651i \(0.411562\pi\)
\(840\) 0 0
\(841\) 3.04538e9 0.176545
\(842\) 0 0
\(843\) 6.33874e9i 0.364424i
\(844\) 0 0
\(845\) 2.90162e10i 1.65440i
\(846\) 0 0
\(847\) 2.61257e8 0.0147732
\(848\) 0 0
\(849\) −1.24642e10 −0.699017
\(850\) 0 0
\(851\) − 1.52146e10i − 0.846267i
\(852\) 0 0
\(853\) 3.49929e10i 1.93045i 0.261424 + 0.965224i \(0.415808\pi\)
−0.261424 + 0.965224i \(0.584192\pi\)
\(854\) 0 0
\(855\) 4.65209e9 0.254546
\(856\) 0 0
\(857\) −2.73150e10 −1.48241 −0.741205 0.671279i \(-0.765745\pi\)
−0.741205 + 0.671279i \(0.765745\pi\)
\(858\) 0 0
\(859\) 3.65662e10i 1.96836i 0.177181 + 0.984178i \(0.443302\pi\)
−0.177181 + 0.984178i \(0.556698\pi\)
\(860\) 0 0
\(861\) − 6.83219e8i − 0.0364795i
\(862\) 0 0
\(863\) −1.34559e10 −0.712646 −0.356323 0.934363i \(-0.615970\pi\)
−0.356323 + 0.934363i \(0.615970\pi\)
\(864\) 0 0
\(865\) −1.09445e9 −0.0574962
\(866\) 0 0
\(867\) 1.10791e10i 0.577350i
\(868\) 0 0
\(869\) 2.29286e10i 1.18524i
\(870\) 0 0
\(871\) 2.00820e9 0.102978
\(872\) 0 0
\(873\) −3.35854e9 −0.170844
\(874\) 0 0
\(875\) 9.07158e8i 0.0457778i
\(876\) 0 0
\(877\) − 3.73698e10i − 1.87078i −0.353624 0.935388i \(-0.615051\pi\)
0.353624 0.935388i \(-0.384949\pi\)
\(878\) 0 0
\(879\) −7.61231e9 −0.378055
\(880\) 0 0
\(881\) −3.50627e10 −1.72754 −0.863772 0.503883i \(-0.831904\pi\)
−0.863772 + 0.503883i \(0.831904\pi\)
\(882\) 0 0
\(883\) 1.59555e10i 0.779918i 0.920832 + 0.389959i \(0.127511\pi\)
−0.920832 + 0.389959i \(0.872489\pi\)
\(884\) 0 0
\(885\) − 1.80914e10i − 0.877345i
\(886\) 0 0
\(887\) 1.32602e10 0.637993 0.318996 0.947756i \(-0.396654\pi\)
0.318996 + 0.947756i \(0.396654\pi\)
\(888\) 0 0
\(889\) −3.32068e8 −0.0158515
\(890\) 0 0
\(891\) − 2.80420e9i − 0.132812i
\(892\) 0 0
\(893\) − 2.50808e9i − 0.117859i
\(894\) 0 0
\(895\) −2.78706e10 −1.29947
\(896\) 0 0
\(897\) −1.27866e9 −0.0591538
\(898\) 0 0
\(899\) − 1.21423e10i − 0.557366i
\(900\) 0 0
\(901\) 1.67069e7i 0 0.000760957i
\(902\) 0 0
\(903\) 1.29533e7 0.000585429 0
\(904\) 0 0
\(905\) 2.02036e10 0.906065
\(906\) 0 0
\(907\) − 4.01584e10i − 1.78711i −0.448953 0.893555i \(-0.648203\pi\)
0.448953 0.893555i \(-0.351797\pi\)
\(908\) 0 0
\(909\) 1.10029e10i 0.485886i
\(910\) 0 0
\(911\) −9.17715e8 −0.0402155 −0.0201078 0.999798i \(-0.506401\pi\)
−0.0201078 + 0.999798i \(0.506401\pi\)
\(912\) 0 0
\(913\) −4.42293e10 −1.92337
\(914\) 0 0
\(915\) − 1.11534e10i − 0.481321i
\(916\) 0 0
\(917\) 1.51657e9i 0.0649485i
\(918\) 0 0
\(919\) −1.08713e10 −0.462039 −0.231019 0.972949i \(-0.574206\pi\)
−0.231019 + 0.972949i \(0.574206\pi\)
\(920\) 0 0
\(921\) 6.25297e9 0.263741
\(922\) 0 0
\(923\) − 3.78203e9i − 0.158314i
\(924\) 0 0
\(925\) 3.63446e10i 1.50988i
\(926\) 0 0
\(927\) −4.70803e9 −0.194115
\(928\) 0 0
\(929\) 5.19109e9 0.212424 0.106212 0.994344i \(-0.466128\pi\)
0.106212 + 0.994344i \(0.466128\pi\)
\(930\) 0 0
\(931\) − 1.12338e10i − 0.456248i
\(932\) 0 0
\(933\) 1.66271e10i 0.670240i
\(934\) 0 0
\(935\) 4.53839e7 0.00181577
\(936\) 0 0
\(937\) −1.42896e10 −0.567453 −0.283727 0.958905i \(-0.591571\pi\)
−0.283727 + 0.958905i \(0.591571\pi\)
\(938\) 0 0
\(939\) − 7.72957e9i − 0.304667i
\(940\) 0 0
\(941\) 4.37976e10i 1.71351i 0.515722 + 0.856756i \(0.327524\pi\)
−0.515722 + 0.856756i \(0.672476\pi\)
\(942\) 0 0
\(943\) −4.75011e10 −1.84464
\(944\) 0 0
\(945\) 2.87582e8 0.0110854
\(946\) 0 0
\(947\) − 1.19780e10i − 0.458309i −0.973390 0.229154i \(-0.926404\pi\)
0.973390 0.229154i \(-0.0735961\pi\)
\(948\) 0 0
\(949\) − 2.79400e9i − 0.106120i
\(950\) 0 0
\(951\) 8.65290e9 0.326234
\(952\) 0 0
\(953\) −1.67815e10 −0.628068 −0.314034 0.949412i \(-0.601681\pi\)
−0.314034 + 0.949412i \(0.601681\pi\)
\(954\) 0 0
\(955\) 7.17994e10i 2.66753i
\(956\) 0 0
\(957\) − 1.69797e10i − 0.626237i
\(958\) 0 0
\(959\) 1.14692e9 0.0419920
\(960\) 0 0
\(961\) −1.71332e10 −0.622739
\(962\) 0 0
\(963\) 1.04281e10i 0.376283i
\(964\) 0 0
\(965\) 3.37450e10i 1.20883i
\(966\) 0 0
\(967\) −7.39750e9 −0.263083 −0.131541 0.991311i \(-0.541993\pi\)
−0.131541 + 0.991311i \(0.541993\pi\)
\(968\) 0 0
\(969\) 6.78736e6 0.000239645 0
\(970\) 0 0
\(971\) 3.13191e10i 1.09785i 0.835872 + 0.548924i \(0.184962\pi\)
−0.835872 + 0.548924i \(0.815038\pi\)
\(972\) 0 0
\(973\) 1.87399e8i 0.00652186i
\(974\) 0 0
\(975\) 3.05447e9 0.105541
\(976\) 0 0
\(977\) 1.00101e10 0.343405 0.171703 0.985149i \(-0.445073\pi\)
0.171703 + 0.985149i \(0.445073\pi\)
\(978\) 0 0
\(979\) − 6.81778e10i − 2.32222i
\(980\) 0 0
\(981\) 1.06591e10i 0.360477i
\(982\) 0 0
\(983\) −4.04299e10 −1.35758 −0.678789 0.734333i \(-0.737495\pi\)
−0.678789 + 0.734333i \(0.737495\pi\)
\(984\) 0 0
\(985\) 6.93014e10 2.31055
\(986\) 0 0
\(987\) − 1.55044e8i − 0.00513269i
\(988\) 0 0
\(989\) − 9.00586e8i − 0.0296032i
\(990\) 0 0
\(991\) 6.02049e10 1.96505 0.982526 0.186125i \(-0.0595930\pi\)
0.982526 + 0.186125i \(0.0595930\pi\)
\(992\) 0 0
\(993\) −1.87669e10 −0.608234
\(994\) 0 0
\(995\) 5.65456e10i 1.81978i
\(996\) 0 0
\(997\) 4.18147e10i 1.33628i 0.744037 + 0.668138i \(0.232909\pi\)
−0.744037 + 0.668138i \(0.767091\pi\)
\(998\) 0 0
\(999\) 5.10200e9 0.161905
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 384.8.d.e.193.6 yes 12
4.3 odd 2 inner 384.8.d.e.193.12 yes 12
8.3 odd 2 inner 384.8.d.e.193.1 12
8.5 even 2 inner 384.8.d.e.193.7 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.e.193.1 12 8.3 odd 2 inner
384.8.d.e.193.6 yes 12 1.1 even 1 trivial
384.8.d.e.193.7 yes 12 8.5 even 2 inner
384.8.d.e.193.12 yes 12 4.3 odd 2 inner