Properties

Label 256.8.b.c.129.1
Level $256$
Weight $8$
Character 256.129
Analytic conductor $79.971$
Analytic rank $0$
Dimension $2$
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] = [256,8,Mod(129,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.9705665239\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.8.b.c.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-84.0000i q^{3} -82.0000i q^{5} -456.000 q^{7} -4869.00 q^{9} +O(q^{10})\) \(q-84.0000i q^{3} -82.0000i q^{5} -456.000 q^{7} -4869.00 q^{9} +2524.00i q^{11} +10778.0i q^{13} -6888.00 q^{15} -11150.0 q^{17} +4124.00i q^{19} +38304.0i q^{21} +81704.0 q^{23} +71401.0 q^{25} +225288. i q^{27} -99798.0i q^{29} +40480.0 q^{31} +212016. q^{33} +37392.0i q^{35} -419442. i q^{37} +905352. q^{39} -141402. q^{41} +690428. i q^{43} +399258. i q^{45} +682032. q^{47} -615607. q^{49} +936600. i q^{51} +1.81312e6i q^{53} +206968. q^{55} +346416. q^{57} +966028. i q^{59} -1.88767e6i q^{61} +2.22026e6 q^{63} +883796. q^{65} +2.96587e6i q^{67} -6.86314e6i q^{69} -2.54823e6 q^{71} +1.68033e6 q^{73} -5.99768e6i q^{75} -1.15094e6i q^{77} -4.03806e6 q^{79} +8.27569e6 q^{81} -5.38576e6i q^{83} +914300. i q^{85} -8.38303e6 q^{87} +6.47305e6 q^{89} -4.91477e6i q^{91} -3.40032e6i q^{93} +338168. q^{95} -6.06576e6 q^{97} -1.22894e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 912 q^{7} - 9738 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 912 q^{7} - 9738 q^{9} - 13776 q^{15} - 22300 q^{17} + 163408 q^{23} + 142802 q^{25} + 80960 q^{31} + 424032 q^{33} + 1810704 q^{39} - 282804 q^{41} + 1364064 q^{47} - 1231214 q^{49} + 413936 q^{55} + 692832 q^{57} + 4440528 q^{63} + 1767592 q^{65} - 5096464 q^{71} + 3360652 q^{73} - 8076128 q^{79} + 16551378 q^{81} - 16766064 q^{87} + 12946092 q^{89} + 676336 q^{95} - 12131516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 84.0000i − 1.79620i −0.439790 0.898100i \(-0.644947\pi\)
0.439790 0.898100i \(-0.355053\pi\)
\(4\) 0 0
\(5\) − 82.0000i − 0.293372i −0.989183 0.146686i \(-0.953139\pi\)
0.989183 0.146686i \(-0.0468607\pi\)
\(6\) 0 0
\(7\) −456.000 −0.502483 −0.251242 0.967924i \(-0.580839\pi\)
−0.251242 + 0.967924i \(0.580839\pi\)
\(8\) 0 0
\(9\) −4869.00 −2.22634
\(10\) 0 0
\(11\) 2524.00i 0.571762i 0.958265 + 0.285881i \(0.0922861\pi\)
−0.958265 + 0.285881i \(0.907714\pi\)
\(12\) 0 0
\(13\) 10778.0i 1.36062i 0.732925 + 0.680309i \(0.238155\pi\)
−0.732925 + 0.680309i \(0.761845\pi\)
\(14\) 0 0
\(15\) −6888.00 −0.526955
\(16\) 0 0
\(17\) −11150.0 −0.550432 −0.275216 0.961382i \(-0.588749\pi\)
−0.275216 + 0.961382i \(0.588749\pi\)
\(18\) 0 0
\(19\) 4124.00i 0.137937i 0.997619 + 0.0689685i \(0.0219708\pi\)
−0.997619 + 0.0689685i \(0.978029\pi\)
\(20\) 0 0
\(21\) 38304.0i 0.902561i
\(22\) 0 0
\(23\) 81704.0 1.40022 0.700109 0.714036i \(-0.253135\pi\)
0.700109 + 0.714036i \(0.253135\pi\)
\(24\) 0 0
\(25\) 71401.0 0.913933
\(26\) 0 0
\(27\) 225288.i 2.20275i
\(28\) 0 0
\(29\) − 99798.0i − 0.759852i −0.925017 0.379926i \(-0.875949\pi\)
0.925017 0.379926i \(-0.124051\pi\)
\(30\) 0 0
\(31\) 40480.0 0.244048 0.122024 0.992527i \(-0.461062\pi\)
0.122024 + 0.992527i \(0.461062\pi\)
\(32\) 0 0
\(33\) 212016. 1.02700
\(34\) 0 0
\(35\) 37392.0i 0.147415i
\(36\) 0 0
\(37\) − 419442.i − 1.36134i −0.732591 0.680669i \(-0.761689\pi\)
0.732591 0.680669i \(-0.238311\pi\)
\(38\) 0 0
\(39\) 905352. 2.44394
\(40\) 0 0
\(41\) −141402. −0.320414 −0.160207 0.987083i \(-0.551216\pi\)
−0.160207 + 0.987083i \(0.551216\pi\)
\(42\) 0 0
\(43\) 690428.i 1.32428i 0.749382 + 0.662138i \(0.230351\pi\)
−0.749382 + 0.662138i \(0.769649\pi\)
\(44\) 0 0
\(45\) 399258.i 0.653145i
\(46\) 0 0
\(47\) 682032. 0.958213 0.479107 0.877757i \(-0.340961\pi\)
0.479107 + 0.877757i \(0.340961\pi\)
\(48\) 0 0
\(49\) −615607. −0.747510
\(50\) 0 0
\(51\) 936600.i 0.988686i
\(52\) 0 0
\(53\) 1.81312e6i 1.67286i 0.548071 + 0.836432i \(0.315362\pi\)
−0.548071 + 0.836432i \(0.684638\pi\)
\(54\) 0 0
\(55\) 206968. 0.167739
\(56\) 0 0
\(57\) 346416. 0.247763
\(58\) 0 0
\(59\) 966028.i 0.612361i 0.951973 + 0.306181i \(0.0990511\pi\)
−0.951973 + 0.306181i \(0.900949\pi\)
\(60\) 0 0
\(61\) − 1.88767e6i − 1.06481i −0.846490 0.532404i \(-0.821289\pi\)
0.846490 0.532404i \(-0.178711\pi\)
\(62\) 0 0
\(63\) 2.22026e6 1.11870
\(64\) 0 0
\(65\) 883796. 0.399168
\(66\) 0 0
\(67\) 2.96587e6i 1.20473i 0.798220 + 0.602365i \(0.205775\pi\)
−0.798220 + 0.602365i \(0.794225\pi\)
\(68\) 0 0
\(69\) − 6.86314e6i − 2.51507i
\(70\) 0 0
\(71\) −2.54823e6 −0.844957 −0.422479 0.906373i \(-0.638840\pi\)
−0.422479 + 0.906373i \(0.638840\pi\)
\(72\) 0 0
\(73\) 1.68033e6 0.505549 0.252775 0.967525i \(-0.418657\pi\)
0.252775 + 0.967525i \(0.418657\pi\)
\(74\) 0 0
\(75\) − 5.99768e6i − 1.64161i
\(76\) 0 0
\(77\) − 1.15094e6i − 0.287301i
\(78\) 0 0
\(79\) −4.03806e6 −0.921464 −0.460732 0.887539i \(-0.652413\pi\)
−0.460732 + 0.887539i \(0.652413\pi\)
\(80\) 0 0
\(81\) 8.27569e6 1.73024
\(82\) 0 0
\(83\) − 5.38576e6i − 1.03389i −0.856019 0.516945i \(-0.827069\pi\)
0.856019 0.516945i \(-0.172931\pi\)
\(84\) 0 0
\(85\) 914300.i 0.161481i
\(86\) 0 0
\(87\) −8.38303e6 −1.36485
\(88\) 0 0
\(89\) 6.47305e6 0.973293 0.486647 0.873599i \(-0.338220\pi\)
0.486647 + 0.873599i \(0.338220\pi\)
\(90\) 0 0
\(91\) − 4.91477e6i − 0.683688i
\(92\) 0 0
\(93\) − 3.40032e6i − 0.438359i
\(94\) 0 0
\(95\) 338168. 0.0404669
\(96\) 0 0
\(97\) −6.06576e6 −0.674814 −0.337407 0.941359i \(-0.609550\pi\)
−0.337407 + 0.941359i \(0.609550\pi\)
\(98\) 0 0
\(99\) − 1.22894e7i − 1.27293i
\(100\) 0 0
\(101\) 9.70069e6i 0.936866i 0.883499 + 0.468433i \(0.155181\pi\)
−0.883499 + 0.468433i \(0.844819\pi\)
\(102\) 0 0
\(103\) 4.10159e6 0.369847 0.184924 0.982753i \(-0.440796\pi\)
0.184924 + 0.982753i \(0.440796\pi\)
\(104\) 0 0
\(105\) 3.14093e6 0.264786
\(106\) 0 0
\(107\) − 72900.0i − 0.00575287i −0.999996 0.00287643i \(-0.999084\pi\)
0.999996 0.00287643i \(-0.000915598\pi\)
\(108\) 0 0
\(109\) − 9.55841e6i − 0.706957i −0.935443 0.353478i \(-0.884999\pi\)
0.935443 0.353478i \(-0.115001\pi\)
\(110\) 0 0
\(111\) −3.52331e7 −2.44524
\(112\) 0 0
\(113\) 9.33890e6 0.608865 0.304433 0.952534i \(-0.401533\pi\)
0.304433 + 0.952534i \(0.401533\pi\)
\(114\) 0 0
\(115\) − 6.69973e6i − 0.410785i
\(116\) 0 0
\(117\) − 5.24781e7i − 3.02920i
\(118\) 0 0
\(119\) 5.08440e6 0.276583
\(120\) 0 0
\(121\) 1.31166e7 0.673089
\(122\) 0 0
\(123\) 1.18778e7i 0.575529i
\(124\) 0 0
\(125\) − 1.22611e7i − 0.561495i
\(126\) 0 0
\(127\) 3.59794e7 1.55862 0.779311 0.626637i \(-0.215569\pi\)
0.779311 + 0.626637i \(0.215569\pi\)
\(128\) 0 0
\(129\) 5.79960e7 2.37867
\(130\) 0 0
\(131\) − 676052.i − 0.0262743i −0.999914 0.0131371i \(-0.995818\pi\)
0.999914 0.0131371i \(-0.00418180\pi\)
\(132\) 0 0
\(133\) − 1.88054e6i − 0.0693111i
\(134\) 0 0
\(135\) 1.84736e7 0.646225
\(136\) 0 0
\(137\) 2.95841e7 0.982962 0.491481 0.870888i \(-0.336456\pi\)
0.491481 + 0.870888i \(0.336456\pi\)
\(138\) 0 0
\(139\) 3.19084e7i 1.00775i 0.863776 + 0.503876i \(0.168093\pi\)
−0.863776 + 0.503876i \(0.831907\pi\)
\(140\) 0 0
\(141\) − 5.72907e7i − 1.72114i
\(142\) 0 0
\(143\) −2.72037e7 −0.777949
\(144\) 0 0
\(145\) −8.18344e6 −0.222919
\(146\) 0 0
\(147\) 5.17110e7i 1.34268i
\(148\) 0 0
\(149\) − 1.16603e7i − 0.288773i −0.989521 0.144386i \(-0.953879\pi\)
0.989521 0.144386i \(-0.0461208\pi\)
\(150\) 0 0
\(151\) −1.76295e7 −0.416698 −0.208349 0.978055i \(-0.566809\pi\)
−0.208349 + 0.978055i \(0.566809\pi\)
\(152\) 0 0
\(153\) 5.42894e7 1.22545
\(154\) 0 0
\(155\) − 3.31936e6i − 0.0715968i
\(156\) 0 0
\(157\) − 6.34658e6i − 0.130885i −0.997856 0.0654427i \(-0.979154\pi\)
0.997856 0.0654427i \(-0.0208460\pi\)
\(158\) 0 0
\(159\) 1.52302e8 3.00480
\(160\) 0 0
\(161\) −3.72570e7 −0.703587
\(162\) 0 0
\(163\) 8.04234e7i 1.45454i 0.686351 + 0.727271i \(0.259211\pi\)
−0.686351 + 0.727271i \(0.740789\pi\)
\(164\) 0 0
\(165\) − 1.73853e7i − 0.301293i
\(166\) 0 0
\(167\) 1.14767e8 1.90682 0.953411 0.301673i \(-0.0975451\pi\)
0.953411 + 0.301673i \(0.0975451\pi\)
\(168\) 0 0
\(169\) −5.34168e7 −0.851283
\(170\) 0 0
\(171\) − 2.00798e7i − 0.307095i
\(172\) 0 0
\(173\) 6.33755e7i 0.930594i 0.885155 + 0.465297i \(0.154053\pi\)
−0.885155 + 0.465297i \(0.845947\pi\)
\(174\) 0 0
\(175\) −3.25589e7 −0.459236
\(176\) 0 0
\(177\) 8.11464e7 1.09992
\(178\) 0 0
\(179\) − 1.13228e7i − 0.147559i −0.997275 0.0737796i \(-0.976494\pi\)
0.997275 0.0737796i \(-0.0235061\pi\)
\(180\) 0 0
\(181\) − 5.22650e6i − 0.0655143i −0.999463 0.0327571i \(-0.989571\pi\)
0.999463 0.0327571i \(-0.0104288\pi\)
\(182\) 0 0
\(183\) −1.58564e8 −1.91261
\(184\) 0 0
\(185\) −3.43942e7 −0.399379
\(186\) 0 0
\(187\) − 2.81426e7i − 0.314716i
\(188\) 0 0
\(189\) − 1.02731e8i − 1.10684i
\(190\) 0 0
\(191\) 8.50301e7 0.882990 0.441495 0.897264i \(-0.354448\pi\)
0.441495 + 0.897264i \(0.354448\pi\)
\(192\) 0 0
\(193\) 1.15092e8 1.15237 0.576186 0.817319i \(-0.304540\pi\)
0.576186 + 0.817319i \(0.304540\pi\)
\(194\) 0 0
\(195\) − 7.42389e7i − 0.716985i
\(196\) 0 0
\(197\) − 1.38522e8i − 1.29088i −0.763810 0.645441i \(-0.776674\pi\)
0.763810 0.645441i \(-0.223326\pi\)
\(198\) 0 0
\(199\) −2.19614e7 −0.197548 −0.0987742 0.995110i \(-0.531492\pi\)
−0.0987742 + 0.995110i \(0.531492\pi\)
\(200\) 0 0
\(201\) 2.49133e8 2.16394
\(202\) 0 0
\(203\) 4.55079e7i 0.381813i
\(204\) 0 0
\(205\) 1.15950e7i 0.0940007i
\(206\) 0 0
\(207\) −3.97817e8 −3.11736
\(208\) 0 0
\(209\) −1.04090e7 −0.0788671
\(210\) 0 0
\(211\) − 6.10208e7i − 0.447187i −0.974682 0.223594i \(-0.928221\pi\)
0.974682 0.223594i \(-0.0717789\pi\)
\(212\) 0 0
\(213\) 2.14051e8i 1.51771i
\(214\) 0 0
\(215\) 5.66151e7 0.388506
\(216\) 0 0
\(217\) −1.84589e7 −0.122630
\(218\) 0 0
\(219\) − 1.41147e8i − 0.908068i
\(220\) 0 0
\(221\) − 1.20175e8i − 0.748928i
\(222\) 0 0
\(223\) 4.22448e7 0.255098 0.127549 0.991832i \(-0.459289\pi\)
0.127549 + 0.991832i \(0.459289\pi\)
\(224\) 0 0
\(225\) −3.47651e8 −2.03472
\(226\) 0 0
\(227\) − 2.39102e8i − 1.35673i −0.734726 0.678364i \(-0.762689\pi\)
0.734726 0.678364i \(-0.237311\pi\)
\(228\) 0 0
\(229\) − 4.67889e7i − 0.257465i −0.991679 0.128733i \(-0.958909\pi\)
0.991679 0.128733i \(-0.0410909\pi\)
\(230\) 0 0
\(231\) −9.66793e7 −0.516050
\(232\) 0 0
\(233\) −3.45225e8 −1.78795 −0.893977 0.448113i \(-0.852096\pi\)
−0.893977 + 0.448113i \(0.852096\pi\)
\(234\) 0 0
\(235\) − 5.59266e7i − 0.281113i
\(236\) 0 0
\(237\) 3.39197e8i 1.65513i
\(238\) 0 0
\(239\) −2.34413e8 −1.11068 −0.555340 0.831624i \(-0.687412\pi\)
−0.555340 + 0.831624i \(0.687412\pi\)
\(240\) 0 0
\(241\) −1.09557e8 −0.504175 −0.252087 0.967705i \(-0.581117\pi\)
−0.252087 + 0.967705i \(0.581117\pi\)
\(242\) 0 0
\(243\) − 2.02453e8i − 0.905112i
\(244\) 0 0
\(245\) 5.04798e7i 0.219299i
\(246\) 0 0
\(247\) −4.44485e7 −0.187680
\(248\) 0 0
\(249\) −4.52404e8 −1.85707
\(250\) 0 0
\(251\) 3.94031e8i 1.57280i 0.617720 + 0.786398i \(0.288057\pi\)
−0.617720 + 0.786398i \(0.711943\pi\)
\(252\) 0 0
\(253\) 2.06221e8i 0.800591i
\(254\) 0 0
\(255\) 7.68012e7 0.290053
\(256\) 0 0
\(257\) 3.19064e8 1.17250 0.586248 0.810131i \(-0.300604\pi\)
0.586248 + 0.810131i \(0.300604\pi\)
\(258\) 0 0
\(259\) 1.91266e8i 0.684050i
\(260\) 0 0
\(261\) 4.85916e8i 1.69169i
\(262\) 0 0
\(263\) 2.19359e8 0.743549 0.371774 0.928323i \(-0.378750\pi\)
0.371774 + 0.928323i \(0.378750\pi\)
\(264\) 0 0
\(265\) 1.48676e8 0.490772
\(266\) 0 0
\(267\) − 5.43736e8i − 1.74823i
\(268\) 0 0
\(269\) 1.48033e8i 0.463687i 0.972753 + 0.231844i \(0.0744757\pi\)
−0.972753 + 0.231844i \(0.925524\pi\)
\(270\) 0 0
\(271\) 3.69934e8 1.12910 0.564549 0.825399i \(-0.309050\pi\)
0.564549 + 0.825399i \(0.309050\pi\)
\(272\) 0 0
\(273\) −4.12841e8 −1.22804
\(274\) 0 0
\(275\) 1.80216e8i 0.522552i
\(276\) 0 0
\(277\) − 3.95860e8i − 1.11908i −0.828803 0.559541i \(-0.810977\pi\)
0.828803 0.559541i \(-0.189023\pi\)
\(278\) 0 0
\(279\) −1.97097e8 −0.543332
\(280\) 0 0
\(281\) 5.97760e8 1.60714 0.803572 0.595208i \(-0.202930\pi\)
0.803572 + 0.595208i \(0.202930\pi\)
\(282\) 0 0
\(283\) − 8.05797e7i − 0.211336i −0.994401 0.105668i \(-0.966302\pi\)
0.994401 0.105668i \(-0.0336981\pi\)
\(284\) 0 0
\(285\) − 2.84061e7i − 0.0726867i
\(286\) 0 0
\(287\) 6.44793e7 0.161003
\(288\) 0 0
\(289\) −2.86016e8 −0.697025
\(290\) 0 0
\(291\) 5.09524e8i 1.21210i
\(292\) 0 0
\(293\) 7.54530e8i 1.75243i 0.481924 + 0.876213i \(0.339938\pi\)
−0.481924 + 0.876213i \(0.660062\pi\)
\(294\) 0 0
\(295\) 7.92143e7 0.179650
\(296\) 0 0
\(297\) −5.68627e8 −1.25945
\(298\) 0 0
\(299\) 8.80606e8i 1.90516i
\(300\) 0 0
\(301\) − 3.14835e8i − 0.665427i
\(302\) 0 0
\(303\) 8.14858e8 1.68280
\(304\) 0 0
\(305\) −1.54789e8 −0.312385
\(306\) 0 0
\(307\) 8.20472e8i 1.61838i 0.587549 + 0.809188i \(0.300093\pi\)
−0.587549 + 0.809188i \(0.699907\pi\)
\(308\) 0 0
\(309\) − 3.44534e8i − 0.664320i
\(310\) 0 0
\(311\) 6.53503e8 1.23193 0.615965 0.787773i \(-0.288766\pi\)
0.615965 + 0.787773i \(0.288766\pi\)
\(312\) 0 0
\(313\) −6.63587e8 −1.22319 −0.611594 0.791172i \(-0.709471\pi\)
−0.611594 + 0.791172i \(0.709471\pi\)
\(314\) 0 0
\(315\) − 1.82062e8i − 0.328195i
\(316\) 0 0
\(317\) 3.54718e8i 0.625426i 0.949848 + 0.312713i \(0.101238\pi\)
−0.949848 + 0.312713i \(0.898762\pi\)
\(318\) 0 0
\(319\) 2.51890e8 0.434454
\(320\) 0 0
\(321\) −6.12360e6 −0.0103333
\(322\) 0 0
\(323\) − 4.59826e7i − 0.0759250i
\(324\) 0 0
\(325\) 7.69560e8i 1.24351i
\(326\) 0 0
\(327\) −8.02906e8 −1.26984
\(328\) 0 0
\(329\) −3.11007e8 −0.481486
\(330\) 0 0
\(331\) 3.05543e8i 0.463100i 0.972823 + 0.231550i \(0.0743797\pi\)
−0.972823 + 0.231550i \(0.925620\pi\)
\(332\) 0 0
\(333\) 2.04226e9i 3.03080i
\(334\) 0 0
\(335\) 2.43201e8 0.353434
\(336\) 0 0
\(337\) 3.54965e7 0.0505220 0.0252610 0.999681i \(-0.491958\pi\)
0.0252610 + 0.999681i \(0.491958\pi\)
\(338\) 0 0
\(339\) − 7.84467e8i − 1.09364i
\(340\) 0 0
\(341\) 1.02172e8i 0.139537i
\(342\) 0 0
\(343\) 6.56252e8 0.878095
\(344\) 0 0
\(345\) −5.62777e8 −0.737853
\(346\) 0 0
\(347\) 1.90594e8i 0.244882i 0.992476 + 0.122441i \(0.0390723\pi\)
−0.992476 + 0.122441i \(0.960928\pi\)
\(348\) 0 0
\(349\) − 8.60864e8i − 1.08404i −0.840366 0.542020i \(-0.817660\pi\)
0.840366 0.542020i \(-0.182340\pi\)
\(350\) 0 0
\(351\) −2.42815e9 −2.99710
\(352\) 0 0
\(353\) −1.04544e9 −1.26500 −0.632498 0.774562i \(-0.717970\pi\)
−0.632498 + 0.774562i \(0.717970\pi\)
\(354\) 0 0
\(355\) 2.08955e8i 0.247887i
\(356\) 0 0
\(357\) − 4.27090e8i − 0.496798i
\(358\) 0 0
\(359\) 7.63303e8 0.870696 0.435348 0.900262i \(-0.356625\pi\)
0.435348 + 0.900262i \(0.356625\pi\)
\(360\) 0 0
\(361\) 8.76864e8 0.980973
\(362\) 0 0
\(363\) − 1.10179e9i − 1.20900i
\(364\) 0 0
\(365\) − 1.37787e8i − 0.148314i
\(366\) 0 0
\(367\) 1.38692e9 1.46460 0.732302 0.680980i \(-0.238446\pi\)
0.732302 + 0.680980i \(0.238446\pi\)
\(368\) 0 0
\(369\) 6.88486e8 0.713351
\(370\) 0 0
\(371\) − 8.26782e8i − 0.840586i
\(372\) 0 0
\(373\) 4.77105e8i 0.476029i 0.971262 + 0.238015i \(0.0764966\pi\)
−0.971262 + 0.238015i \(0.923503\pi\)
\(374\) 0 0
\(375\) −1.02994e9 −1.00856
\(376\) 0 0
\(377\) 1.07562e9 1.03387
\(378\) 0 0
\(379\) 3.92468e8i 0.370311i 0.982709 + 0.185156i \(0.0592789\pi\)
−0.982709 + 0.185156i \(0.940721\pi\)
\(380\) 0 0
\(381\) − 3.02227e9i − 2.79960i
\(382\) 0 0
\(383\) −2.10409e9 −1.91368 −0.956839 0.290617i \(-0.906139\pi\)
−0.956839 + 0.290617i \(0.906139\pi\)
\(384\) 0 0
\(385\) −9.43774e7 −0.0842860
\(386\) 0 0
\(387\) − 3.36169e9i − 2.94829i
\(388\) 0 0
\(389\) − 1.26019e9i − 1.08546i −0.839907 0.542730i \(-0.817391\pi\)
0.839907 0.542730i \(-0.182609\pi\)
\(390\) 0 0
\(391\) −9.11000e8 −0.770725
\(392\) 0 0
\(393\) −5.67884e7 −0.0471939
\(394\) 0 0
\(395\) 3.31121e8i 0.270332i
\(396\) 0 0
\(397\) 9.81298e8i 0.787107i 0.919302 + 0.393554i \(0.128754\pi\)
−0.919302 + 0.393554i \(0.871246\pi\)
\(398\) 0 0
\(399\) −1.57966e8 −0.124497
\(400\) 0 0
\(401\) 9.09981e8 0.704737 0.352369 0.935861i \(-0.385376\pi\)
0.352369 + 0.935861i \(0.385376\pi\)
\(402\) 0 0
\(403\) 4.36293e8i 0.332056i
\(404\) 0 0
\(405\) − 6.78606e8i − 0.507604i
\(406\) 0 0
\(407\) 1.05867e9 0.778361
\(408\) 0 0
\(409\) 3.55609e7 0.0257004 0.0128502 0.999917i \(-0.495910\pi\)
0.0128502 + 0.999917i \(0.495910\pi\)
\(410\) 0 0
\(411\) − 2.48507e9i − 1.76560i
\(412\) 0 0
\(413\) − 4.40509e8i − 0.307701i
\(414\) 0 0
\(415\) −4.41633e8 −0.303314
\(416\) 0 0
\(417\) 2.68031e9 1.81013
\(418\) 0 0
\(419\) 2.65360e9i 1.76233i 0.472813 + 0.881163i \(0.343239\pi\)
−0.472813 + 0.881163i \(0.656761\pi\)
\(420\) 0 0
\(421\) − 1.12113e9i − 0.732264i −0.930563 0.366132i \(-0.880682\pi\)
0.930563 0.366132i \(-0.119318\pi\)
\(422\) 0 0
\(423\) −3.32081e9 −2.13331
\(424\) 0 0
\(425\) −7.96121e8 −0.503058
\(426\) 0 0
\(427\) 8.60778e8i 0.535049i
\(428\) 0 0
\(429\) 2.28511e9i 1.39735i
\(430\) 0 0
\(431\) 1.06344e9 0.639799 0.319900 0.947451i \(-0.396351\pi\)
0.319900 + 0.947451i \(0.396351\pi\)
\(432\) 0 0
\(433\) −7.05962e8 −0.417901 −0.208951 0.977926i \(-0.567005\pi\)
−0.208951 + 0.977926i \(0.567005\pi\)
\(434\) 0 0
\(435\) 6.87409e8i 0.400408i
\(436\) 0 0
\(437\) 3.36947e8i 0.193142i
\(438\) 0 0
\(439\) 1.48506e9 0.837760 0.418880 0.908042i \(-0.362423\pi\)
0.418880 + 0.908042i \(0.362423\pi\)
\(440\) 0 0
\(441\) 2.99739e9 1.66421
\(442\) 0 0
\(443\) 7.22153e8i 0.394654i 0.980338 + 0.197327i \(0.0632260\pi\)
−0.980338 + 0.197327i \(0.936774\pi\)
\(444\) 0 0
\(445\) − 5.30790e8i − 0.285537i
\(446\) 0 0
\(447\) −9.79462e8 −0.518694
\(448\) 0 0
\(449\) −1.22968e9 −0.641109 −0.320554 0.947230i \(-0.603869\pi\)
−0.320554 + 0.947230i \(0.603869\pi\)
\(450\) 0 0
\(451\) − 3.56899e8i − 0.183201i
\(452\) 0 0
\(453\) 1.48088e9i 0.748473i
\(454\) 0 0
\(455\) −4.03011e8 −0.200575
\(456\) 0 0
\(457\) 8.85551e7 0.0434017 0.0217009 0.999765i \(-0.493092\pi\)
0.0217009 + 0.999765i \(0.493092\pi\)
\(458\) 0 0
\(459\) − 2.51196e9i − 1.21246i
\(460\) 0 0
\(461\) − 2.10937e8i − 0.100277i −0.998742 0.0501384i \(-0.984034\pi\)
0.998742 0.0501384i \(-0.0159662\pi\)
\(462\) 0 0
\(463\) 3.29775e9 1.54413 0.772066 0.635543i \(-0.219224\pi\)
0.772066 + 0.635543i \(0.219224\pi\)
\(464\) 0 0
\(465\) −2.78826e8 −0.128602
\(466\) 0 0
\(467\) 8.82873e7i 0.0401134i 0.999799 + 0.0200567i \(0.00638467\pi\)
−0.999799 + 0.0200567i \(0.993615\pi\)
\(468\) 0 0
\(469\) − 1.35244e9i − 0.605357i
\(470\) 0 0
\(471\) −5.33113e8 −0.235096
\(472\) 0 0
\(473\) −1.74264e9 −0.757171
\(474\) 0 0
\(475\) 2.94458e8i 0.126065i
\(476\) 0 0
\(477\) − 8.82807e9i − 3.72436i
\(478\) 0 0
\(479\) 4.51507e9 1.87711 0.938557 0.345125i \(-0.112164\pi\)
0.938557 + 0.345125i \(0.112164\pi\)
\(480\) 0 0
\(481\) 4.52075e9 1.85226
\(482\) 0 0
\(483\) 3.12959e9i 1.26378i
\(484\) 0 0
\(485\) 4.97392e8i 0.197972i
\(486\) 0 0
\(487\) 3.31338e9 1.29993 0.649964 0.759965i \(-0.274784\pi\)
0.649964 + 0.759965i \(0.274784\pi\)
\(488\) 0 0
\(489\) 6.75557e9 2.61265
\(490\) 0 0
\(491\) 4.01694e9i 1.53147i 0.643154 + 0.765737i \(0.277626\pi\)
−0.643154 + 0.765737i \(0.722374\pi\)
\(492\) 0 0
\(493\) 1.11275e9i 0.418247i
\(494\) 0 0
\(495\) −1.00773e9 −0.373443
\(496\) 0 0
\(497\) 1.16199e9 0.424577
\(498\) 0 0
\(499\) 2.70976e9i 0.976290i 0.872763 + 0.488145i \(0.162326\pi\)
−0.872763 + 0.488145i \(0.837674\pi\)
\(500\) 0 0
\(501\) − 9.64045e9i − 3.42504i
\(502\) 0 0
\(503\) 3.04579e8 0.106712 0.0533558 0.998576i \(-0.483008\pi\)
0.0533558 + 0.998576i \(0.483008\pi\)
\(504\) 0 0
\(505\) 7.95456e8 0.274850
\(506\) 0 0
\(507\) 4.48701e9i 1.52908i
\(508\) 0 0
\(509\) 1.88202e8i 0.0632575i 0.999500 + 0.0316287i \(0.0100694\pi\)
−0.999500 + 0.0316287i \(0.989931\pi\)
\(510\) 0 0
\(511\) −7.66229e8 −0.254030
\(512\) 0 0
\(513\) −9.29088e8 −0.303841
\(514\) 0 0
\(515\) − 3.36331e8i − 0.108503i
\(516\) 0 0
\(517\) 1.72145e9i 0.547870i
\(518\) 0 0
\(519\) 5.32355e9 1.67153
\(520\) 0 0
\(521\) −4.14963e9 −1.28552 −0.642758 0.766069i \(-0.722210\pi\)
−0.642758 + 0.766069i \(0.722210\pi\)
\(522\) 0 0
\(523\) − 2.51360e9i − 0.768318i −0.923267 0.384159i \(-0.874491\pi\)
0.923267 0.384159i \(-0.125509\pi\)
\(524\) 0 0
\(525\) 2.73494e9i 0.824880i
\(526\) 0 0
\(527\) −4.51352e8 −0.134332
\(528\) 0 0
\(529\) 3.27072e9 0.960613
\(530\) 0 0
\(531\) − 4.70359e9i − 1.36332i
\(532\) 0 0
\(533\) − 1.52403e9i − 0.435962i
\(534\) 0 0
\(535\) −5.97780e6 −0.00168773
\(536\) 0 0
\(537\) −9.51112e8 −0.265046
\(538\) 0 0
\(539\) − 1.55379e9i − 0.427398i
\(540\) 0 0
\(541\) 1.32416e9i 0.359543i 0.983708 + 0.179772i \(0.0575358\pi\)
−0.983708 + 0.179772i \(0.942464\pi\)
\(542\) 0 0
\(543\) −4.39026e8 −0.117677
\(544\) 0 0
\(545\) −7.83789e8 −0.207401
\(546\) 0 0
\(547\) 5.58047e8i 0.145786i 0.997340 + 0.0728929i \(0.0232231\pi\)
−0.997340 + 0.0728929i \(0.976777\pi\)
\(548\) 0 0
\(549\) 9.19107e9i 2.37062i
\(550\) 0 0
\(551\) 4.11567e8 0.104812
\(552\) 0 0
\(553\) 1.84136e9 0.463020
\(554\) 0 0
\(555\) 2.88912e9i 0.717364i
\(556\) 0 0
\(557\) 3.30331e9i 0.809946i 0.914329 + 0.404973i \(0.132719\pi\)
−0.914329 + 0.404973i \(0.867281\pi\)
\(558\) 0 0
\(559\) −7.44143e9 −1.80184
\(560\) 0 0
\(561\) −2.36398e9 −0.565293
\(562\) 0 0
\(563\) − 1.22011e8i − 0.0288152i −0.999896 0.0144076i \(-0.995414\pi\)
0.999896 0.0144076i \(-0.00458623\pi\)
\(564\) 0 0
\(565\) − 7.65790e8i − 0.178624i
\(566\) 0 0
\(567\) −3.77371e9 −0.869417
\(568\) 0 0
\(569\) −5.00925e8 −0.113993 −0.0569967 0.998374i \(-0.518152\pi\)
−0.0569967 + 0.998374i \(0.518152\pi\)
\(570\) 0 0
\(571\) 6.98702e9i 1.57060i 0.619116 + 0.785300i \(0.287491\pi\)
−0.619116 + 0.785300i \(0.712509\pi\)
\(572\) 0 0
\(573\) − 7.14253e9i − 1.58603i
\(574\) 0 0
\(575\) 5.83375e9 1.27971
\(576\) 0 0
\(577\) −8.16573e9 −1.76962 −0.884809 0.465954i \(-0.845711\pi\)
−0.884809 + 0.465954i \(0.845711\pi\)
\(578\) 0 0
\(579\) − 9.66769e9i − 2.06989i
\(580\) 0 0
\(581\) 2.45591e9i 0.519512i
\(582\) 0 0
\(583\) −4.57631e9 −0.956479
\(584\) 0 0
\(585\) −4.30320e9 −0.888682
\(586\) 0 0
\(587\) − 8.53182e9i − 1.74104i −0.492135 0.870519i \(-0.663783\pi\)
0.492135 0.870519i \(-0.336217\pi\)
\(588\) 0 0
\(589\) 1.66940e8i 0.0336632i
\(590\) 0 0
\(591\) −1.16358e10 −2.31868
\(592\) 0 0
\(593\) −1.71175e9 −0.337092 −0.168546 0.985694i \(-0.553907\pi\)
−0.168546 + 0.985694i \(0.553907\pi\)
\(594\) 0 0
\(595\) − 4.16921e8i − 0.0811417i
\(596\) 0 0
\(597\) 1.84475e9i 0.354836i
\(598\) 0 0
\(599\) −4.77362e9 −0.907516 −0.453758 0.891125i \(-0.649917\pi\)
−0.453758 + 0.891125i \(0.649917\pi\)
\(600\) 0 0
\(601\) −7.89998e8 −0.148445 −0.0742224 0.997242i \(-0.523647\pi\)
−0.0742224 + 0.997242i \(0.523647\pi\)
\(602\) 0 0
\(603\) − 1.44408e10i − 2.68214i
\(604\) 0 0
\(605\) − 1.07556e9i − 0.197465i
\(606\) 0 0
\(607\) 1.82652e9 0.331485 0.165743 0.986169i \(-0.446998\pi\)
0.165743 + 0.986169i \(0.446998\pi\)
\(608\) 0 0
\(609\) 3.82266e9 0.685813
\(610\) 0 0
\(611\) 7.35094e9i 1.30376i
\(612\) 0 0
\(613\) − 6.90339e9i − 1.21046i −0.796050 0.605231i \(-0.793081\pi\)
0.796050 0.605231i \(-0.206919\pi\)
\(614\) 0 0
\(615\) 9.73977e8 0.168844
\(616\) 0 0
\(617\) 5.69235e9 0.975649 0.487825 0.872942i \(-0.337791\pi\)
0.487825 + 0.872942i \(0.337791\pi\)
\(618\) 0 0
\(619\) 4.28594e9i 0.726321i 0.931727 + 0.363161i \(0.118302\pi\)
−0.931727 + 0.363161i \(0.881698\pi\)
\(620\) 0 0
\(621\) 1.84069e10i 3.08433i
\(622\) 0 0
\(623\) −2.95171e9 −0.489064
\(624\) 0 0
\(625\) 4.57279e9 0.749206
\(626\) 0 0
\(627\) 8.74354e8i 0.141661i
\(628\) 0 0
\(629\) 4.67678e9i 0.749324i
\(630\) 0 0
\(631\) −5.61602e8 −0.0889869 −0.0444935 0.999010i \(-0.514167\pi\)
−0.0444935 + 0.999010i \(0.514167\pi\)
\(632\) 0 0
\(633\) −5.12575e9 −0.803238
\(634\) 0 0
\(635\) − 2.95031e9i − 0.457256i
\(636\) 0 0
\(637\) − 6.63501e9i − 1.01708i
\(638\) 0 0
\(639\) 1.24073e10 1.88116
\(640\) 0 0
\(641\) 5.17445e9 0.775998 0.387999 0.921660i \(-0.373166\pi\)
0.387999 + 0.921660i \(0.373166\pi\)
\(642\) 0 0
\(643\) − 1.04374e10i − 1.54830i −0.633004 0.774148i \(-0.718178\pi\)
0.633004 0.774148i \(-0.281822\pi\)
\(644\) 0 0
\(645\) − 4.75567e9i − 0.697835i
\(646\) 0 0
\(647\) −9.71623e8 −0.141037 −0.0705185 0.997510i \(-0.522465\pi\)
−0.0705185 + 0.997510i \(0.522465\pi\)
\(648\) 0 0
\(649\) −2.43825e9 −0.350125
\(650\) 0 0
\(651\) 1.55055e9i 0.220268i
\(652\) 0 0
\(653\) 7.25223e9i 1.01924i 0.860400 + 0.509619i \(0.170214\pi\)
−0.860400 + 0.509619i \(0.829786\pi\)
\(654\) 0 0
\(655\) −5.54363e7 −0.00770814
\(656\) 0 0
\(657\) −8.18151e9 −1.12552
\(658\) 0 0
\(659\) 3.81924e9i 0.519851i 0.965629 + 0.259925i \(0.0836979\pi\)
−0.965629 + 0.259925i \(0.916302\pi\)
\(660\) 0 0
\(661\) 1.07881e10i 1.45292i 0.687210 + 0.726459i \(0.258835\pi\)
−0.687210 + 0.726459i \(0.741165\pi\)
\(662\) 0 0
\(663\) −1.00947e10 −1.34523
\(664\) 0 0
\(665\) −1.54205e8 −0.0203339
\(666\) 0 0
\(667\) − 8.15390e9i − 1.06396i
\(668\) 0 0
\(669\) − 3.54857e9i − 0.458207i
\(670\) 0 0
\(671\) 4.76448e9 0.608817
\(672\) 0 0
\(673\) −6.34833e9 −0.802798 −0.401399 0.915903i \(-0.631476\pi\)
−0.401399 + 0.915903i \(0.631476\pi\)
\(674\) 0 0
\(675\) 1.60858e10i 2.01316i
\(676\) 0 0
\(677\) 8.82566e9i 1.09317i 0.837404 + 0.546584i \(0.184072\pi\)
−0.837404 + 0.546584i \(0.815928\pi\)
\(678\) 0 0
\(679\) 2.76599e9 0.339083
\(680\) 0 0
\(681\) −2.00846e10 −2.43696
\(682\) 0 0
\(683\) − 4.92331e9i − 0.591268i −0.955301 0.295634i \(-0.904469\pi\)
0.955301 0.295634i \(-0.0955309\pi\)
\(684\) 0 0
\(685\) − 2.42590e9i − 0.288374i
\(686\) 0 0
\(687\) −3.93027e9 −0.462459
\(688\) 0 0
\(689\) −1.95418e10 −2.27613
\(690\) 0 0
\(691\) − 5.68449e9i − 0.655418i −0.944779 0.327709i \(-0.893723\pi\)
0.944779 0.327709i \(-0.106277\pi\)
\(692\) 0 0
\(693\) 5.60395e9i 0.639628i
\(694\) 0 0
\(695\) 2.61649e9 0.295646
\(696\) 0 0
\(697\) 1.57663e9 0.176366
\(698\) 0 0
\(699\) 2.89989e10i 3.21152i
\(700\) 0 0
\(701\) 1.70567e9i 0.187017i 0.995618 + 0.0935085i \(0.0298082\pi\)
−0.995618 + 0.0935085i \(0.970192\pi\)
\(702\) 0 0
\(703\) 1.72978e9 0.187779
\(704\) 0 0
\(705\) −4.69784e9 −0.504936
\(706\) 0 0
\(707\) − 4.42351e9i − 0.470760i
\(708\) 0 0
\(709\) 4.52189e9i 0.476495i 0.971204 + 0.238248i \(0.0765730\pi\)
−0.971204 + 0.238248i \(0.923427\pi\)
\(710\) 0 0
\(711\) 1.96613e10 2.05149
\(712\) 0 0
\(713\) 3.30738e9 0.341720
\(714\) 0 0
\(715\) 2.23070e9i 0.228229i
\(716\) 0 0
\(717\) 1.96907e10i 1.99500i
\(718\) 0 0
\(719\) 3.09206e9 0.310239 0.155120 0.987896i \(-0.450424\pi\)
0.155120 + 0.987896i \(0.450424\pi\)
\(720\) 0 0
\(721\) −1.87033e9 −0.185842
\(722\) 0 0
\(723\) 9.20280e9i 0.905599i
\(724\) 0 0
\(725\) − 7.12568e9i − 0.694453i
\(726\) 0 0
\(727\) 1.44622e10 1.39593 0.697965 0.716132i \(-0.254089\pi\)
0.697965 + 0.716132i \(0.254089\pi\)
\(728\) 0 0
\(729\) 1.09288e9 0.104478
\(730\) 0 0
\(731\) − 7.69827e9i − 0.728924i
\(732\) 0 0
\(733\) − 3.15415e9i − 0.295814i −0.989001 0.147907i \(-0.952746\pi\)
0.989001 0.147907i \(-0.0472536\pi\)
\(734\) 0 0
\(735\) 4.24030e9 0.393905
\(736\) 0 0
\(737\) −7.48585e9 −0.688819
\(738\) 0 0
\(739\) 1.54236e10i 1.40582i 0.711277 + 0.702912i \(0.248117\pi\)
−0.711277 + 0.702912i \(0.751883\pi\)
\(740\) 0 0
\(741\) 3.73367e9i 0.337111i
\(742\) 0 0
\(743\) −1.59520e10 −1.42677 −0.713385 0.700772i \(-0.752839\pi\)
−0.713385 + 0.700772i \(0.752839\pi\)
\(744\) 0 0
\(745\) −9.56141e8 −0.0847179
\(746\) 0 0
\(747\) 2.62233e10i 2.30179i
\(748\) 0 0
\(749\) 3.32424e7i 0.00289072i
\(750\) 0 0
\(751\) −6.13964e9 −0.528936 −0.264468 0.964395i \(-0.585196\pi\)
−0.264468 + 0.964395i \(0.585196\pi\)
\(752\) 0 0
\(753\) 3.30986e10 2.82506
\(754\) 0 0
\(755\) 1.44562e9i 0.122248i
\(756\) 0 0
\(757\) − 1.42818e10i − 1.19660i −0.801273 0.598299i \(-0.795843\pi\)
0.801273 0.598299i \(-0.204157\pi\)
\(758\) 0 0
\(759\) 1.73226e10 1.43802
\(760\) 0 0
\(761\) −1.47536e10 −1.21353 −0.606767 0.794880i \(-0.707534\pi\)
−0.606767 + 0.794880i \(0.707534\pi\)
\(762\) 0 0
\(763\) 4.35863e9i 0.355234i
\(764\) 0 0
\(765\) − 4.45173e9i − 0.359512i
\(766\) 0 0
\(767\) −1.04118e10 −0.833190
\(768\) 0 0
\(769\) 1.97592e10 1.56685 0.783424 0.621487i \(-0.213471\pi\)
0.783424 + 0.621487i \(0.213471\pi\)
\(770\) 0 0
\(771\) − 2.68014e10i − 2.10604i
\(772\) 0 0
\(773\) 1.01370e10i 0.789374i 0.918816 + 0.394687i \(0.129147\pi\)
−0.918816 + 0.394687i \(0.870853\pi\)
\(774\) 0 0
\(775\) 2.89031e9 0.223043
\(776\) 0 0
\(777\) 1.60663e10 1.22869
\(778\) 0 0
\(779\) − 5.83142e8i − 0.0441970i
\(780\) 0 0
\(781\) − 6.43174e9i − 0.483114i
\(782\) 0 0
\(783\) 2.24833e10 1.67376
\(784\) 0 0
\(785\) −5.20420e8 −0.0383981
\(786\) 0 0
\(787\) − 1.27882e10i − 0.935188i −0.883943 0.467594i \(-0.845121\pi\)
0.883943 0.467594i \(-0.154879\pi\)
\(788\) 0 0
\(789\) − 1.84261e10i − 1.33556i
\(790\) 0 0
\(791\) −4.25854e9 −0.305945
\(792\) 0 0
\(793\) 2.03453e10 1.44880
\(794\) 0 0
\(795\) − 1.24888e10i − 0.881524i
\(796\) 0 0
\(797\) 7.38617e9i 0.516791i 0.966039 + 0.258396i \(0.0831938\pi\)
−0.966039 + 0.258396i \(0.916806\pi\)
\(798\) 0 0
\(799\) −7.60466e9 −0.527431
\(800\) 0 0
\(801\) −3.15173e10 −2.16688
\(802\) 0 0
\(803\) 4.24114e9i 0.289054i
\(804\) 0 0
\(805\) 3.05508e9i 0.206413i
\(806\) 0 0
\(807\) 1.24348e10 0.832875
\(808\) 0 0
\(809\) −1.53742e10 −1.02087 −0.510437 0.859915i \(-0.670516\pi\)
−0.510437 + 0.859915i \(0.670516\pi\)
\(810\) 0 0
\(811\) − 9.77882e9i − 0.643744i −0.946783 0.321872i \(-0.895688\pi\)
0.946783 0.321872i \(-0.104312\pi\)
\(812\) 0 0
\(813\) − 3.10745e10i − 2.02809i
\(814\) 0 0
\(815\) 6.59472e9 0.426722
\(816\) 0 0
\(817\) −2.84733e9 −0.182667
\(818\) 0 0
\(819\) 2.39300e10i 1.52212i
\(820\) 0 0
\(821\) 1.83470e10i 1.15708i 0.815654 + 0.578540i \(0.196377\pi\)
−0.815654 + 0.578540i \(0.803623\pi\)
\(822\) 0 0
\(823\) −3.16960e10 −1.98201 −0.991004 0.133829i \(-0.957273\pi\)
−0.991004 + 0.133829i \(0.957273\pi\)
\(824\) 0 0
\(825\) 1.51382e10 0.938608
\(826\) 0 0
\(827\) 6.12845e9i 0.376774i 0.982095 + 0.188387i \(0.0603260\pi\)
−0.982095 + 0.188387i \(0.939674\pi\)
\(828\) 0 0
\(829\) 1.24652e10i 0.759904i 0.925006 + 0.379952i \(0.124060\pi\)
−0.925006 + 0.379952i \(0.875940\pi\)
\(830\) 0 0
\(831\) −3.32522e10 −2.01010
\(832\) 0 0
\(833\) 6.86402e9 0.411454
\(834\) 0 0
\(835\) − 9.41091e9i − 0.559409i
\(836\) 0 0
\(837\) 9.11966e9i 0.537575i
\(838\) 0 0
\(839\) −1.82237e10 −1.06530 −0.532648 0.846337i \(-0.678803\pi\)
−0.532648 + 0.846337i \(0.678803\pi\)
\(840\) 0 0
\(841\) 7.29024e9 0.422625
\(842\) 0 0
\(843\) − 5.02118e10i − 2.88675i
\(844\) 0 0
\(845\) 4.38017e9i 0.249743i
\(846\) 0 0
\(847\) −5.98117e9 −0.338216
\(848\) 0 0
\(849\) −6.76870e9 −0.379602
\(850\) 0 0
\(851\) − 3.42701e10i − 1.90617i
\(852\) 0 0
\(853\) − 2.48619e10i − 1.37155i −0.727812 0.685777i \(-0.759463\pi\)
0.727812 0.685777i \(-0.240537\pi\)
\(854\) 0 0
\(855\) −1.64654e9 −0.0900930
\(856\) 0 0
\(857\) −2.96761e10 −1.61055 −0.805275 0.592902i \(-0.797982\pi\)
−0.805275 + 0.592902i \(0.797982\pi\)
\(858\) 0 0
\(859\) − 1.14772e10i − 0.617819i −0.951091 0.308910i \(-0.900036\pi\)
0.951091 0.308910i \(-0.0999640\pi\)
\(860\) 0 0
\(861\) − 5.41626e9i − 0.289194i
\(862\) 0 0
\(863\) −2.13485e10 −1.13066 −0.565328 0.824866i \(-0.691250\pi\)
−0.565328 + 0.824866i \(0.691250\pi\)
\(864\) 0 0
\(865\) 5.19679e9 0.273010
\(866\) 0 0
\(867\) 2.40254e10i 1.25200i
\(868\) 0 0
\(869\) − 1.01921e10i − 0.526858i
\(870\) 0 0
\(871\) −3.19661e10 −1.63918
\(872\) 0 0
\(873\) 2.95342e10 1.50236
\(874\) 0 0
\(875\) 5.59108e9i 0.282142i
\(876\) 0 0
\(877\) 7.92753e9i 0.396862i 0.980115 + 0.198431i \(0.0635846\pi\)
−0.980115 + 0.198431i \(0.936415\pi\)
\(878\) 0 0
\(879\) 6.33805e10 3.14771
\(880\) 0 0
\(881\) 7.32045e9 0.360680 0.180340 0.983604i \(-0.442280\pi\)
0.180340 + 0.983604i \(0.442280\pi\)
\(882\) 0 0
\(883\) − 3.54988e9i − 0.173521i −0.996229 0.0867604i \(-0.972349\pi\)
0.996229 0.0867604i \(-0.0276514\pi\)
\(884\) 0 0
\(885\) − 6.65400e9i − 0.322687i
\(886\) 0 0
\(887\) 5.80634e9 0.279364 0.139682 0.990196i \(-0.455392\pi\)
0.139682 + 0.990196i \(0.455392\pi\)
\(888\) 0 0
\(889\) −1.64066e10 −0.783182
\(890\) 0 0
\(891\) 2.08878e10i 0.989285i
\(892\) 0 0
\(893\) 2.81270e9i 0.132173i
\(894\) 0 0
\(895\) −9.28466e8 −0.0432898
\(896\) 0 0
\(897\) 7.39709e10 3.42206
\(898\) 0 0
\(899\) − 4.03982e9i − 0.185440i
\(900\) 0 0
\(901\) − 2.02163e10i − 0.920798i
\(902\) 0 0
\(903\) −2.64462e10 −1.19524
\(904\) 0 0
\(905\) −4.28573e8 −0.0192201
\(906\) 0 0
\(907\) − 1.78240e10i − 0.793196i −0.917992 0.396598i \(-0.870191\pi\)
0.917992 0.396598i \(-0.129809\pi\)
\(908\) 0 0
\(909\) − 4.72326e10i − 2.08578i
\(910\) 0 0
\(911\) −1.87703e10 −0.822538 −0.411269 0.911514i \(-0.634914\pi\)
−0.411269 + 0.911514i \(0.634914\pi\)
\(912\) 0 0
\(913\) 1.35937e10 0.591138
\(914\) 0 0
\(915\) 1.30023e10i 0.561107i
\(916\) 0 0
\(917\) 3.08280e8i 0.0132024i
\(918\) 0 0
\(919\) 3.75844e10 1.59736 0.798681 0.601754i \(-0.205531\pi\)
0.798681 + 0.601754i \(0.205531\pi\)
\(920\) 0 0
\(921\) 6.89197e10 2.90693
\(922\) 0 0
\(923\) − 2.74648e10i − 1.14966i
\(924\) 0 0
\(925\) − 2.99486e10i − 1.24417i
\(926\) 0 0
\(927\) −1.99707e10 −0.823404
\(928\) 0 0
\(929\) −1.92372e10 −0.787205 −0.393602 0.919281i \(-0.628771\pi\)
−0.393602 + 0.919281i \(0.628771\pi\)
\(930\) 0 0
\(931\) − 2.53876e9i − 0.103109i
\(932\) 0 0
\(933\) − 5.48943e10i − 2.21279i
\(934\) 0 0
\(935\) −2.30769e9 −0.0923289
\(936\) 0 0
\(937\) −1.04732e9 −0.0415900 −0.0207950 0.999784i \(-0.506620\pi\)
−0.0207950 + 0.999784i \(0.506620\pi\)
\(938\) 0 0
\(939\) 5.57413e10i 2.19709i
\(940\) 0 0
\(941\) 7.97861e9i 0.312150i 0.987745 + 0.156075i \(0.0498842\pi\)
−0.987745 + 0.156075i \(0.950116\pi\)
\(942\) 0 0
\(943\) −1.15531e10 −0.448650
\(944\) 0 0
\(945\) −8.42397e9 −0.324717
\(946\) 0 0
\(947\) − 4.26943e9i − 0.163360i −0.996659 0.0816799i \(-0.973971\pi\)
0.996659 0.0816799i \(-0.0260285\pi\)
\(948\) 0 0
\(949\) 1.81106e10i 0.687860i
\(950\) 0 0
\(951\) 2.97963e10 1.12339
\(952\) 0 0
\(953\) 1.06048e10 0.396897 0.198449 0.980111i \(-0.436410\pi\)
0.198449 + 0.980111i \(0.436410\pi\)
\(954\) 0 0
\(955\) − 6.97247e9i − 0.259045i
\(956\) 0 0
\(957\) − 2.11588e10i − 0.780367i
\(958\) 0 0
\(959\) −1.34904e10 −0.493922
\(960\) 0 0
\(961\) −2.58740e10 −0.940441
\(962\) 0 0
\(963\) 3.54950e8i 0.0128078i
\(964\) 0 0
\(965\) − 9.43750e9i − 0.338074i
\(966\) 0 0
\(967\) −1.65090e10 −0.587123 −0.293562 0.955940i \(-0.594841\pi\)
−0.293562 + 0.955940i \(0.594841\pi\)
\(968\) 0 0
\(969\) −3.86254e9 −0.136377
\(970\) 0 0
\(971\) 2.46094e10i 0.862649i 0.902197 + 0.431324i \(0.141954\pi\)
−0.902197 + 0.431324i \(0.858046\pi\)
\(972\) 0 0
\(973\) − 1.45503e10i − 0.506379i
\(974\) 0 0
\(975\) 6.46430e10 2.23360
\(976\) 0 0
\(977\) −2.53886e10 −0.870979 −0.435489 0.900194i \(-0.643425\pi\)
−0.435489 + 0.900194i \(0.643425\pi\)
\(978\) 0 0
\(979\) 1.63380e10i 0.556492i
\(980\) 0 0
\(981\) 4.65399e10i 1.57392i
\(982\) 0 0
\(983\) 1.87585e10 0.629884 0.314942 0.949111i \(-0.398015\pi\)
0.314942 + 0.949111i \(0.398015\pi\)
\(984\) 0 0
\(985\) −1.13588e10 −0.378709
\(986\) 0 0
\(987\) 2.61246e10i 0.864846i
\(988\) 0 0
\(989\) 5.64107e10i 1.85428i
\(990\) 0 0
\(991\) 3.59792e9 0.117434 0.0587170 0.998275i \(-0.481299\pi\)
0.0587170 + 0.998275i \(0.481299\pi\)
\(992\) 0 0
\(993\) 2.56656e10 0.831821
\(994\) 0 0
\(995\) 1.80083e9i 0.0579552i
\(996\) 0 0
\(997\) − 1.34287e10i − 0.429143i −0.976708 0.214571i \(-0.931165\pi\)
0.976708 0.214571i \(-0.0688354\pi\)
\(998\) 0 0
\(999\) 9.44952e10 2.99868
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 256.8.b.c.129.1 2
4.3 odd 2 256.8.b.e.129.2 2
8.3 odd 2 256.8.b.e.129.1 2
8.5 even 2 inner 256.8.b.c.129.2 2
16.3 odd 4 8.8.a.a.1.1 1
16.5 even 4 64.8.a.a.1.1 1
16.11 odd 4 64.8.a.g.1.1 1
16.13 even 4 16.8.a.c.1.1 1
48.5 odd 4 576.8.a.k.1.1 1
48.11 even 4 576.8.a.j.1.1 1
48.29 odd 4 144.8.a.g.1.1 1
48.35 even 4 72.8.a.d.1.1 1
80.3 even 4 200.8.c.a.49.1 2
80.13 odd 4 400.8.c.b.49.2 2
80.19 odd 4 200.8.a.i.1.1 1
80.29 even 4 400.8.a.b.1.1 1
80.67 even 4 200.8.c.a.49.2 2
80.77 odd 4 400.8.c.b.49.1 2
112.83 even 4 392.8.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 16.3 odd 4
16.8.a.c.1.1 1 16.13 even 4
64.8.a.a.1.1 1 16.5 even 4
64.8.a.g.1.1 1 16.11 odd 4
72.8.a.d.1.1 1 48.35 even 4
144.8.a.g.1.1 1 48.29 odd 4
200.8.a.i.1.1 1 80.19 odd 4
200.8.c.a.49.1 2 80.3 even 4
200.8.c.a.49.2 2 80.67 even 4
256.8.b.c.129.1 2 1.1 even 1 trivial
256.8.b.c.129.2 2 8.5 even 2 inner
256.8.b.e.129.1 2 8.3 odd 2
256.8.b.e.129.2 2 4.3 odd 2
392.8.a.d.1.1 1 112.83 even 4
400.8.a.b.1.1 1 80.29 even 4
400.8.c.b.49.1 2 80.77 odd 4
400.8.c.b.49.2 2 80.13 odd 4
576.8.a.j.1.1 1 48.11 even 4
576.8.a.k.1.1 1 48.5 odd 4