Properties

Label 48.9.g.c.31.4
Level $48$
Weight $9$
Character 48.31
Analytic conductor $19.554$
Analytic rank $0$
Dimension $4$
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] = [48,9,Mod(31,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.g (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: \(19.5541732829\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{1801})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 451x^{2} + 450x + 202500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.4
Root \(-10.3595 + 17.9433i\) of defining polynomial
Character \(\chi\) \(=\) 48.31
Dual form 48.9.g.c.31.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+46.7654i q^{3} +952.517 q^{5} -3101.27i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q+46.7654i q^{3} +952.517 q^{5} -3101.27i q^{7} -2187.00 q^{9} -7175.83i q^{11} -26897.5 q^{13} +44544.8i q^{15} +146374. q^{17} -220215. i q^{19} +145032. q^{21} +96574.1i q^{23} +516663. q^{25} -102276. i q^{27} +169321. q^{29} -429668. i q^{31} +335580. q^{33} -2.95401e6i q^{35} +2.94894e6 q^{37} -1.25787e6i q^{39} +3.15721e6 q^{41} +4.89907e6i q^{43} -2.08315e6 q^{45} -808105. i q^{47} -3.85305e6 q^{49} +6.84524e6i q^{51} -1.21447e7 q^{53} -6.83510e6i q^{55} +1.02984e7 q^{57} +2.47774e6i q^{59} -6.21934e6 q^{61} +6.78247e6i q^{63} -2.56203e7 q^{65} +1.52139e7i q^{67} -4.51632e6 q^{69} -1.92087e7i q^{71} -3.17283e7 q^{73} +2.41619e7i q^{75} -2.22542e7 q^{77} +5.67932e7i q^{79} +4.78297e6 q^{81} -8.52789e7i q^{83} +1.39424e8 q^{85} +7.91835e6i q^{87} +3.06662e7 q^{89} +8.34163e7i q^{91} +2.00936e7 q^{93} -2.09758e8i q^{95} +5.15412e6 q^{97} +1.56935e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 264 q^{5} - 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 264 q^{5} - 8748 q^{9} + 14632 q^{13} + 332904 q^{17} + 250128 q^{21} + 2604428 q^{25} + 2343576 q^{29} + 2002320 q^{33} + 4315784 q^{37} + 9035496 q^{41} + 577368 q^{45} + 3458884 q^{49} - 42186600 q^{53} + 2253744 q^{57} - 48148408 q^{61} - 125450832 q^{65} + 23514624 q^{69} - 21215480 q^{73} - 32354496 q^{77} + 19131876 q^{81} + 235297584 q^{85} - 12675576 q^{89} + 193564080 q^{93} + 263153800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 46.7654i 0.577350i
\(4\) 0 0
\(5\) 952.517 1.52403 0.762013 0.647561i \(-0.224211\pi\)
0.762013 + 0.647561i \(0.224211\pi\)
\(6\) 0 0
\(7\) − 3101.27i − 1.29166i −0.763483 0.645828i \(-0.776512\pi\)
0.763483 0.645828i \(-0.223488\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) − 7175.83i − 0.490119i −0.969508 0.245059i \(-0.921193\pi\)
0.969508 0.245059i \(-0.0788075\pi\)
\(12\) 0 0
\(13\) −26897.5 −0.941756 −0.470878 0.882198i \(-0.656063\pi\)
−0.470878 + 0.882198i \(0.656063\pi\)
\(14\) 0 0
\(15\) 44544.8i 0.879897i
\(16\) 0 0
\(17\) 146374. 1.75254 0.876271 0.481819i \(-0.160024\pi\)
0.876271 + 0.481819i \(0.160024\pi\)
\(18\) 0 0
\(19\) − 220215.i − 1.68979i −0.534935 0.844893i \(-0.679664\pi\)
0.534935 0.844893i \(-0.320336\pi\)
\(20\) 0 0
\(21\) 145032. 0.745738
\(22\) 0 0
\(23\) 96574.1i 0.345104i 0.985000 + 0.172552i \(0.0552012\pi\)
−0.985000 + 0.172552i \(0.944799\pi\)
\(24\) 0 0
\(25\) 516663. 1.32266
\(26\) 0 0
\(27\) − 102276.i − 0.192450i
\(28\) 0 0
\(29\) 169321. 0.239397 0.119698 0.992810i \(-0.461807\pi\)
0.119698 + 0.992810i \(0.461807\pi\)
\(30\) 0 0
\(31\) − 429668.i − 0.465250i −0.972567 0.232625i \(-0.925269\pi\)
0.972567 0.232625i \(-0.0747314\pi\)
\(32\) 0 0
\(33\) 335580. 0.282970
\(34\) 0 0
\(35\) − 2.95401e6i − 1.96852i
\(36\) 0 0
\(37\) 2.94894e6 1.57347 0.786737 0.617289i \(-0.211769\pi\)
0.786737 + 0.617289i \(0.211769\pi\)
\(38\) 0 0
\(39\) − 1.25787e6i − 0.543723i
\(40\) 0 0
\(41\) 3.15721e6 1.11729 0.558647 0.829406i \(-0.311321\pi\)
0.558647 + 0.829406i \(0.311321\pi\)
\(42\) 0 0
\(43\) 4.89907e6i 1.43298i 0.697597 + 0.716490i \(0.254252\pi\)
−0.697597 + 0.716490i \(0.745748\pi\)
\(44\) 0 0
\(45\) −2.08315e6 −0.508009
\(46\) 0 0
\(47\) − 808105.i − 0.165606i −0.996566 0.0828031i \(-0.973613\pi\)
0.996566 0.0828031i \(-0.0263873\pi\)
\(48\) 0 0
\(49\) −3.85305e6 −0.668375
\(50\) 0 0
\(51\) 6.84524e6i 1.01183i
\(52\) 0 0
\(53\) −1.21447e7 −1.53916 −0.769579 0.638551i \(-0.779534\pi\)
−0.769579 + 0.638551i \(0.779534\pi\)
\(54\) 0 0
\(55\) − 6.83510e6i − 0.746954i
\(56\) 0 0
\(57\) 1.02984e7 0.975598
\(58\) 0 0
\(59\) 2.47774e6i 0.204478i 0.994760 + 0.102239i \(0.0326007\pi\)
−0.994760 + 0.102239i \(0.967399\pi\)
\(60\) 0 0
\(61\) −6.21934e6 −0.449184 −0.224592 0.974453i \(-0.572105\pi\)
−0.224592 + 0.974453i \(0.572105\pi\)
\(62\) 0 0
\(63\) 6.78247e6i 0.430552i
\(64\) 0 0
\(65\) −2.56203e7 −1.43526
\(66\) 0 0
\(67\) 1.52139e7i 0.754989i 0.926012 + 0.377494i \(0.123214\pi\)
−0.926012 + 0.377494i \(0.876786\pi\)
\(68\) 0 0
\(69\) −4.51632e6 −0.199246
\(70\) 0 0
\(71\) − 1.92087e7i − 0.755899i −0.925826 0.377949i \(-0.876629\pi\)
0.925826 0.377949i \(-0.123371\pi\)
\(72\) 0 0
\(73\) −3.17283e7 −1.11726 −0.558631 0.829416i \(-0.688673\pi\)
−0.558631 + 0.829416i \(0.688673\pi\)
\(74\) 0 0
\(75\) 2.41619e7i 0.763636i
\(76\) 0 0
\(77\) −2.22542e7 −0.633065
\(78\) 0 0
\(79\) 5.67932e7i 1.45810i 0.684460 + 0.729051i \(0.260038\pi\)
−0.684460 + 0.729051i \(0.739962\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) − 8.52789e7i − 1.79692i −0.439053 0.898461i \(-0.644686\pi\)
0.439053 0.898461i \(-0.355314\pi\)
\(84\) 0 0
\(85\) 1.39424e8 2.67092
\(86\) 0 0
\(87\) 7.91835e6i 0.138216i
\(88\) 0 0
\(89\) 3.06662e7 0.488765 0.244383 0.969679i \(-0.421415\pi\)
0.244383 + 0.969679i \(0.421415\pi\)
\(90\) 0 0
\(91\) 8.34163e7i 1.21642i
\(92\) 0 0
\(93\) 2.00936e7 0.268612
\(94\) 0 0
\(95\) − 2.09758e8i − 2.57528i
\(96\) 0 0
\(97\) 5.15412e6 0.0582194 0.0291097 0.999576i \(-0.490733\pi\)
0.0291097 + 0.999576i \(0.490733\pi\)
\(98\) 0 0
\(99\) 1.56935e7i 0.163373i
\(100\) 0 0
\(101\) −1.01284e8 −0.973316 −0.486658 0.873593i \(-0.661784\pi\)
−0.486658 + 0.873593i \(0.661784\pi\)
\(102\) 0 0
\(103\) − 2.04461e8i − 1.81661i −0.418310 0.908304i \(-0.637377\pi\)
0.418310 0.908304i \(-0.362623\pi\)
\(104\) 0 0
\(105\) 1.38145e8 1.13652
\(106\) 0 0
\(107\) 1.61055e8i 1.22868i 0.789041 + 0.614340i \(0.210578\pi\)
−0.789041 + 0.614340i \(0.789422\pi\)
\(108\) 0 0
\(109\) 5.73094e7 0.405994 0.202997 0.979179i \(-0.434932\pi\)
0.202997 + 0.979179i \(0.434932\pi\)
\(110\) 0 0
\(111\) 1.37908e8i 0.908445i
\(112\) 0 0
\(113\) 1.49424e7 0.0916446 0.0458223 0.998950i \(-0.485409\pi\)
0.0458223 + 0.998950i \(0.485409\pi\)
\(114\) 0 0
\(115\) 9.19884e7i 0.525947i
\(116\) 0 0
\(117\) 5.88248e7 0.313919
\(118\) 0 0
\(119\) − 4.53945e8i − 2.26368i
\(120\) 0 0
\(121\) 1.62866e8 0.759784
\(122\) 0 0
\(123\) 1.47648e8i 0.645070i
\(124\) 0 0
\(125\) 1.20053e8 0.491738
\(126\) 0 0
\(127\) 8.68934e7i 0.334019i 0.985955 + 0.167010i \(0.0534111\pi\)
−0.985955 + 0.167010i \(0.946589\pi\)
\(128\) 0 0
\(129\) −2.29107e8 −0.827331
\(130\) 0 0
\(131\) 2.93118e8i 0.995308i 0.867376 + 0.497654i \(0.165805\pi\)
−0.867376 + 0.497654i \(0.834195\pi\)
\(132\) 0 0
\(133\) −6.82944e8 −2.18262
\(134\) 0 0
\(135\) − 9.74195e7i − 0.293299i
\(136\) 0 0
\(137\) −4.93337e8 −1.40043 −0.700215 0.713932i \(-0.746913\pi\)
−0.700215 + 0.713932i \(0.746913\pi\)
\(138\) 0 0
\(139\) 5.21073e8i 1.39585i 0.716170 + 0.697926i \(0.245894\pi\)
−0.716170 + 0.697926i \(0.754106\pi\)
\(140\) 0 0
\(141\) 3.77913e7 0.0956128
\(142\) 0 0
\(143\) 1.93012e8i 0.461572i
\(144\) 0 0
\(145\) 1.61281e8 0.364847
\(146\) 0 0
\(147\) − 1.80189e8i − 0.385886i
\(148\) 0 0
\(149\) −4.09228e8 −0.830271 −0.415135 0.909760i \(-0.636266\pi\)
−0.415135 + 0.909760i \(0.636266\pi\)
\(150\) 0 0
\(151\) − 8.92557e6i − 0.0171683i −0.999963 0.00858417i \(-0.997268\pi\)
0.999963 0.00858417i \(-0.00273246\pi\)
\(152\) 0 0
\(153\) −3.20120e8 −0.584181
\(154\) 0 0
\(155\) − 4.09266e8i − 0.709053i
\(156\) 0 0
\(157\) 2.58929e8 0.426169 0.213085 0.977034i \(-0.431649\pi\)
0.213085 + 0.977034i \(0.431649\pi\)
\(158\) 0 0
\(159\) − 5.67952e8i − 0.888634i
\(160\) 0 0
\(161\) 2.99502e8 0.445755
\(162\) 0 0
\(163\) − 2.01640e8i − 0.285645i −0.989748 0.142823i \(-0.954382\pi\)
0.989748 0.142823i \(-0.0456179\pi\)
\(164\) 0 0
\(165\) 3.19646e8 0.431254
\(166\) 0 0
\(167\) 1.45117e9i 1.86575i 0.360202 + 0.932874i \(0.382708\pi\)
−0.360202 + 0.932874i \(0.617292\pi\)
\(168\) 0 0
\(169\) −9.22554e7 −0.113095
\(170\) 0 0
\(171\) 4.81609e8i 0.563262i
\(172\) 0 0
\(173\) 6.36420e8 0.710492 0.355246 0.934773i \(-0.384397\pi\)
0.355246 + 0.934773i \(0.384397\pi\)
\(174\) 0 0
\(175\) − 1.60231e9i − 1.70842i
\(176\) 0 0
\(177\) −1.15872e8 −0.118056
\(178\) 0 0
\(179\) 1.84104e9i 1.79329i 0.442751 + 0.896644i \(0.354002\pi\)
−0.442751 + 0.896644i \(0.645998\pi\)
\(180\) 0 0
\(181\) −7.25192e8 −0.675676 −0.337838 0.941204i \(-0.609696\pi\)
−0.337838 + 0.941204i \(0.609696\pi\)
\(182\) 0 0
\(183\) − 2.90850e8i − 0.259337i
\(184\) 0 0
\(185\) 2.80892e9 2.39802
\(186\) 0 0
\(187\) − 1.05036e9i − 0.858954i
\(188\) 0 0
\(189\) −3.17185e8 −0.248579
\(190\) 0 0
\(191\) − 1.59853e9i − 1.20112i −0.799579 0.600561i \(-0.794944\pi\)
0.799579 0.600561i \(-0.205056\pi\)
\(192\) 0 0
\(193\) −5.05071e8 −0.364018 −0.182009 0.983297i \(-0.558260\pi\)
−0.182009 + 0.983297i \(0.558260\pi\)
\(194\) 0 0
\(195\) − 1.19814e9i − 0.828649i
\(196\) 0 0
\(197\) 2.23833e9 1.48614 0.743070 0.669214i \(-0.233369\pi\)
0.743070 + 0.669214i \(0.233369\pi\)
\(198\) 0 0
\(199\) − 1.11582e9i − 0.711510i −0.934579 0.355755i \(-0.884224\pi\)
0.934579 0.355755i \(-0.115776\pi\)
\(200\) 0 0
\(201\) −7.11482e8 −0.435893
\(202\) 0 0
\(203\) − 5.25109e8i − 0.309218i
\(204\) 0 0
\(205\) 3.00729e9 1.70279
\(206\) 0 0
\(207\) − 2.11208e8i − 0.115035i
\(208\) 0 0
\(209\) −1.58022e9 −0.828196
\(210\) 0 0
\(211\) 2.94087e7i 0.0148370i 0.999972 + 0.00741850i \(0.00236140\pi\)
−0.999972 + 0.00741850i \(0.997639\pi\)
\(212\) 0 0
\(213\) 8.98300e8 0.436418
\(214\) 0 0
\(215\) 4.66645e9i 2.18390i
\(216\) 0 0
\(217\) −1.33251e9 −0.600942
\(218\) 0 0
\(219\) − 1.48378e9i − 0.645051i
\(220\) 0 0
\(221\) −3.93709e9 −1.65047
\(222\) 0 0
\(223\) 3.32442e9i 1.34430i 0.740414 + 0.672151i \(0.234629\pi\)
−0.740414 + 0.672151i \(0.765371\pi\)
\(224\) 0 0
\(225\) −1.12994e9 −0.440886
\(226\) 0 0
\(227\) 3.68532e8i 0.138794i 0.997589 + 0.0693971i \(0.0221076\pi\)
−0.997589 + 0.0693971i \(0.977892\pi\)
\(228\) 0 0
\(229\) −1.32990e9 −0.483591 −0.241795 0.970327i \(-0.577736\pi\)
−0.241795 + 0.970327i \(0.577736\pi\)
\(230\) 0 0
\(231\) − 1.04072e9i − 0.365500i
\(232\) 0 0
\(233\) 2.49591e9 0.846846 0.423423 0.905932i \(-0.360828\pi\)
0.423423 + 0.905932i \(0.360828\pi\)
\(234\) 0 0
\(235\) − 7.69734e8i − 0.252388i
\(236\) 0 0
\(237\) −2.65595e9 −0.841835
\(238\) 0 0
\(239\) − 5.77198e7i − 0.0176902i −0.999961 0.00884511i \(-0.997184\pi\)
0.999961 0.00884511i \(-0.00281552\pi\)
\(240\) 0 0
\(241\) −1.57199e9 −0.465996 −0.232998 0.972477i \(-0.574854\pi\)
−0.232998 + 0.972477i \(0.574854\pi\)
\(242\) 0 0
\(243\) 2.23677e8i 0.0641500i
\(244\) 0 0
\(245\) −3.67009e9 −1.01862
\(246\) 0 0
\(247\) 5.92322e9i 1.59137i
\(248\) 0 0
\(249\) 3.98810e9 1.03745
\(250\) 0 0
\(251\) − 2.63479e9i − 0.663822i −0.943311 0.331911i \(-0.892307\pi\)
0.943311 0.331911i \(-0.107693\pi\)
\(252\) 0 0
\(253\) 6.92999e8 0.169142
\(254\) 0 0
\(255\) 6.52020e9i 1.54206i
\(256\) 0 0
\(257\) 3.67876e9 0.843276 0.421638 0.906764i \(-0.361455\pi\)
0.421638 + 0.906764i \(0.361455\pi\)
\(258\) 0 0
\(259\) − 9.14545e9i − 2.03239i
\(260\) 0 0
\(261\) −3.70304e8 −0.0797989
\(262\) 0 0
\(263\) 1.21643e9i 0.254252i 0.991887 + 0.127126i \(0.0405753\pi\)
−0.991887 + 0.127126i \(0.959425\pi\)
\(264\) 0 0
\(265\) −1.15680e10 −2.34572
\(266\) 0 0
\(267\) 1.43412e9i 0.282189i
\(268\) 0 0
\(269\) −9.26695e9 −1.76981 −0.884907 0.465768i \(-0.845778\pi\)
−0.884907 + 0.465768i \(0.845778\pi\)
\(270\) 0 0
\(271\) 3.54288e7i 0.00656871i 0.999995 + 0.00328435i \(0.00104544\pi\)
−0.999995 + 0.00328435i \(0.998955\pi\)
\(272\) 0 0
\(273\) −3.90099e9 −0.702303
\(274\) 0 0
\(275\) − 3.70748e9i − 0.648259i
\(276\) 0 0
\(277\) 5.73085e9 0.973419 0.486709 0.873564i \(-0.338197\pi\)
0.486709 + 0.873564i \(0.338197\pi\)
\(278\) 0 0
\(279\) 9.39683e8i 0.155083i
\(280\) 0 0
\(281\) 6.79380e9 1.08965 0.544825 0.838550i \(-0.316596\pi\)
0.544825 + 0.838550i \(0.316596\pi\)
\(282\) 0 0
\(283\) 5.67140e9i 0.884188i 0.896969 + 0.442094i \(0.145764\pi\)
−0.896969 + 0.442094i \(0.854236\pi\)
\(284\) 0 0
\(285\) 9.80941e9 1.48684
\(286\) 0 0
\(287\) − 9.79133e9i − 1.44316i
\(288\) 0 0
\(289\) 1.44496e10 2.07140
\(290\) 0 0
\(291\) 2.41034e8i 0.0336130i
\(292\) 0 0
\(293\) −1.52415e9 −0.206803 −0.103402 0.994640i \(-0.532973\pi\)
−0.103402 + 0.994640i \(0.532973\pi\)
\(294\) 0 0
\(295\) 2.36009e9i 0.311631i
\(296\) 0 0
\(297\) −7.33914e8 −0.0943234
\(298\) 0 0
\(299\) − 2.59760e9i − 0.325003i
\(300\) 0 0
\(301\) 1.51933e10 1.85092
\(302\) 0 0
\(303\) − 4.73657e9i − 0.561944i
\(304\) 0 0
\(305\) −5.92402e9 −0.684569
\(306\) 0 0
\(307\) 5.36906e9i 0.604428i 0.953240 + 0.302214i \(0.0977257\pi\)
−0.953240 + 0.302214i \(0.902274\pi\)
\(308\) 0 0
\(309\) 9.56169e9 1.04882
\(310\) 0 0
\(311\) − 6.21354e9i − 0.664198i −0.943245 0.332099i \(-0.892243\pi\)
0.943245 0.332099i \(-0.107757\pi\)
\(312\) 0 0
\(313\) 2.95026e8 0.0307385 0.0153693 0.999882i \(-0.495108\pi\)
0.0153693 + 0.999882i \(0.495108\pi\)
\(314\) 0 0
\(315\) 6.46041e9i 0.656173i
\(316\) 0 0
\(317\) 1.22463e9 0.121274 0.0606369 0.998160i \(-0.480687\pi\)
0.0606369 + 0.998160i \(0.480687\pi\)
\(318\) 0 0
\(319\) − 1.21502e9i − 0.117333i
\(320\) 0 0
\(321\) −7.53180e9 −0.709379
\(322\) 0 0
\(323\) − 3.22337e10i − 2.96142i
\(324\) 0 0
\(325\) −1.38969e10 −1.24562
\(326\) 0 0
\(327\) 2.68010e9i 0.234401i
\(328\) 0 0
\(329\) −2.50615e9 −0.213906
\(330\) 0 0
\(331\) 1.49808e10i 1.24802i 0.781415 + 0.624012i \(0.214498\pi\)
−0.781415 + 0.624012i \(0.785502\pi\)
\(332\) 0 0
\(333\) −6.44934e9 −0.524491
\(334\) 0 0
\(335\) 1.44915e10i 1.15062i
\(336\) 0 0
\(337\) 1.22500e10 0.949768 0.474884 0.880049i \(-0.342490\pi\)
0.474884 + 0.880049i \(0.342490\pi\)
\(338\) 0 0
\(339\) 6.98787e8i 0.0529110i
\(340\) 0 0
\(341\) −3.08322e9 −0.228028
\(342\) 0 0
\(343\) − 5.92885e9i − 0.428346i
\(344\) 0 0
\(345\) −4.30187e9 −0.303656
\(346\) 0 0
\(347\) 9.66765e9i 0.666811i 0.942783 + 0.333406i \(0.108198\pi\)
−0.942783 + 0.333406i \(0.891802\pi\)
\(348\) 0 0
\(349\) −8.41057e9 −0.566923 −0.283461 0.958984i \(-0.591483\pi\)
−0.283461 + 0.958984i \(0.591483\pi\)
\(350\) 0 0
\(351\) 2.75096e9i 0.181241i
\(352\) 0 0
\(353\) 1.96120e10 1.26305 0.631527 0.775354i \(-0.282428\pi\)
0.631527 + 0.775354i \(0.282428\pi\)
\(354\) 0 0
\(355\) − 1.82966e10i − 1.15201i
\(356\) 0 0
\(357\) 2.12289e10 1.30694
\(358\) 0 0
\(359\) 3.86050e9i 0.232416i 0.993225 + 0.116208i \(0.0370739\pi\)
−0.993225 + 0.116208i \(0.962926\pi\)
\(360\) 0 0
\(361\) −3.15109e10 −1.85538
\(362\) 0 0
\(363\) 7.61651e9i 0.438661i
\(364\) 0 0
\(365\) −3.02217e10 −1.70274
\(366\) 0 0
\(367\) − 4.07167e9i − 0.224444i −0.993683 0.112222i \(-0.964203\pi\)
0.993683 0.112222i \(-0.0357968\pi\)
\(368\) 0 0
\(369\) −6.90481e9 −0.372431
\(370\) 0 0
\(371\) 3.76639e10i 1.98806i
\(372\) 0 0
\(373\) 9.73315e8 0.0502827 0.0251413 0.999684i \(-0.491996\pi\)
0.0251413 + 0.999684i \(0.491996\pi\)
\(374\) 0 0
\(375\) 5.61433e9i 0.283905i
\(376\) 0 0
\(377\) −4.55430e9 −0.225453
\(378\) 0 0
\(379\) 1.03474e10i 0.501506i 0.968051 + 0.250753i \(0.0806781\pi\)
−0.968051 + 0.250753i \(0.919322\pi\)
\(380\) 0 0
\(381\) −4.06360e9 −0.192846
\(382\) 0 0
\(383\) − 1.47254e10i − 0.684342i −0.939638 0.342171i \(-0.888838\pi\)
0.939638 0.342171i \(-0.111162\pi\)
\(384\) 0 0
\(385\) −2.11974e10 −0.964808
\(386\) 0 0
\(387\) − 1.07143e10i − 0.477660i
\(388\) 0 0
\(389\) −2.12190e10 −0.926673 −0.463336 0.886183i \(-0.653348\pi\)
−0.463336 + 0.886183i \(0.653348\pi\)
\(390\) 0 0
\(391\) 1.41359e10i 0.604808i
\(392\) 0 0
\(393\) −1.37078e10 −0.574641
\(394\) 0 0
\(395\) 5.40964e10i 2.22219i
\(396\) 0 0
\(397\) 2.05017e10 0.825331 0.412665 0.910883i \(-0.364598\pi\)
0.412665 + 0.910883i \(0.364598\pi\)
\(398\) 0 0
\(399\) − 3.19381e10i − 1.26014i
\(400\) 0 0
\(401\) 3.20594e10 1.23987 0.619937 0.784652i \(-0.287158\pi\)
0.619937 + 0.784652i \(0.287158\pi\)
\(402\) 0 0
\(403\) 1.15570e10i 0.438152i
\(404\) 0 0
\(405\) 4.55586e9 0.169336
\(406\) 0 0
\(407\) − 2.11611e10i − 0.771189i
\(408\) 0 0
\(409\) 2.67550e10 0.956117 0.478058 0.878328i \(-0.341341\pi\)
0.478058 + 0.878328i \(0.341341\pi\)
\(410\) 0 0
\(411\) − 2.30711e10i − 0.808539i
\(412\) 0 0
\(413\) 7.68413e9 0.264116
\(414\) 0 0
\(415\) − 8.12296e10i − 2.73856i
\(416\) 0 0
\(417\) −2.43682e10 −0.805895
\(418\) 0 0
\(419\) − 1.08653e10i − 0.352521i −0.984344 0.176261i \(-0.943600\pi\)
0.984344 0.176261i \(-0.0564002\pi\)
\(420\) 0 0
\(421\) 2.03721e9 0.0648495 0.0324247 0.999474i \(-0.489677\pi\)
0.0324247 + 0.999474i \(0.489677\pi\)
\(422\) 0 0
\(423\) 1.76733e9i 0.0552021i
\(424\) 0 0
\(425\) 7.56260e10 2.31801
\(426\) 0 0
\(427\) 1.92878e10i 0.580192i
\(428\) 0 0
\(429\) −9.02627e9 −0.266489
\(430\) 0 0
\(431\) − 3.19963e10i − 0.927236i −0.886035 0.463618i \(-0.846551\pi\)
0.886035 0.463618i \(-0.153449\pi\)
\(432\) 0 0
\(433\) −2.78313e10 −0.791738 −0.395869 0.918307i \(-0.629557\pi\)
−0.395869 + 0.918307i \(0.629557\pi\)
\(434\) 0 0
\(435\) 7.54236e9i 0.210644i
\(436\) 0 0
\(437\) 2.12670e10 0.583151
\(438\) 0 0
\(439\) 3.88247e10i 1.04532i 0.852540 + 0.522661i \(0.175061\pi\)
−0.852540 + 0.522661i \(0.824939\pi\)
\(440\) 0 0
\(441\) 8.42662e9 0.222792
\(442\) 0 0
\(443\) 4.68689e10i 1.21694i 0.793576 + 0.608471i \(0.208217\pi\)
−0.793576 + 0.608471i \(0.791783\pi\)
\(444\) 0 0
\(445\) 2.92101e10 0.744891
\(446\) 0 0
\(447\) − 1.91377e10i − 0.479357i
\(448\) 0 0
\(449\) 3.48851e10 0.858331 0.429166 0.903226i \(-0.358808\pi\)
0.429166 + 0.903226i \(0.358808\pi\)
\(450\) 0 0
\(451\) − 2.26556e10i − 0.547607i
\(452\) 0 0
\(453\) 4.17408e8 0.00991214
\(454\) 0 0
\(455\) 7.94554e10i 1.85386i
\(456\) 0 0
\(457\) −7.46508e9 −0.171147 −0.0855736 0.996332i \(-0.527272\pi\)
−0.0855736 + 0.996332i \(0.527272\pi\)
\(458\) 0 0
\(459\) − 1.49705e10i − 0.337277i
\(460\) 0 0
\(461\) −5.60087e10 −1.24008 −0.620042 0.784568i \(-0.712885\pi\)
−0.620042 + 0.784568i \(0.712885\pi\)
\(462\) 0 0
\(463\) − 4.51649e10i − 0.982826i −0.870927 0.491413i \(-0.836481\pi\)
0.870927 0.491413i \(-0.163519\pi\)
\(464\) 0 0
\(465\) 1.91395e10 0.409372
\(466\) 0 0
\(467\) − 1.30108e10i − 0.273550i −0.990602 0.136775i \(-0.956326\pi\)
0.990602 0.136775i \(-0.0436737\pi\)
\(468\) 0 0
\(469\) 4.71822e10 0.975186
\(470\) 0 0
\(471\) 1.21089e10i 0.246049i
\(472\) 0 0
\(473\) 3.51549e10 0.702330
\(474\) 0 0
\(475\) − 1.13777e11i − 2.23501i
\(476\) 0 0
\(477\) 2.65605e10 0.513053
\(478\) 0 0
\(479\) − 5.63024e9i − 0.106951i −0.998569 0.0534755i \(-0.982970\pi\)
0.998569 0.0534755i \(-0.0170299\pi\)
\(480\) 0 0
\(481\) −7.93192e10 −1.48183
\(482\) 0 0
\(483\) 1.40063e10i 0.257357i
\(484\) 0 0
\(485\) 4.90939e9 0.0887279
\(486\) 0 0
\(487\) − 2.21903e10i − 0.394501i −0.980353 0.197250i \(-0.936799\pi\)
0.980353 0.197250i \(-0.0632012\pi\)
\(488\) 0 0
\(489\) 9.42979e9 0.164917
\(490\) 0 0
\(491\) 2.38353e10i 0.410104i 0.978751 + 0.205052i \(0.0657364\pi\)
−0.978751 + 0.205052i \(0.934264\pi\)
\(492\) 0 0
\(493\) 2.47842e10 0.419553
\(494\) 0 0
\(495\) 1.49484e10i 0.248985i
\(496\) 0 0
\(497\) −5.95712e10 −0.976361
\(498\) 0 0
\(499\) 8.49679e10i 1.37042i 0.728347 + 0.685209i \(0.240289\pi\)
−0.728347 + 0.685209i \(0.759711\pi\)
\(500\) 0 0
\(501\) −6.78646e10 −1.07719
\(502\) 0 0
\(503\) 8.35912e10i 1.30584i 0.757428 + 0.652918i \(0.226456\pi\)
−0.757428 + 0.652918i \(0.773544\pi\)
\(504\) 0 0
\(505\) −9.64744e10 −1.48336
\(506\) 0 0
\(507\) − 4.31436e9i − 0.0652956i
\(508\) 0 0
\(509\) −4.04693e10 −0.602912 −0.301456 0.953480i \(-0.597473\pi\)
−0.301456 + 0.953480i \(0.597473\pi\)
\(510\) 0 0
\(511\) 9.83978e10i 1.44312i
\(512\) 0 0
\(513\) −2.25226e10 −0.325199
\(514\) 0 0
\(515\) − 1.94752e11i − 2.76856i
\(516\) 0 0
\(517\) −5.79883e9 −0.0811667
\(518\) 0 0
\(519\) 2.97624e10i 0.410203i
\(520\) 0 0
\(521\) −8.68115e10 −1.17822 −0.589110 0.808053i \(-0.700522\pi\)
−0.589110 + 0.808053i \(0.700522\pi\)
\(522\) 0 0
\(523\) 1.28187e11i 1.71332i 0.515881 + 0.856660i \(0.327465\pi\)
−0.515881 + 0.856660i \(0.672535\pi\)
\(524\) 0 0
\(525\) 7.49326e10 0.986355
\(526\) 0 0
\(527\) − 6.28922e10i − 0.815369i
\(528\) 0 0
\(529\) 6.89844e10 0.880904
\(530\) 0 0
\(531\) − 5.41882e9i − 0.0681595i
\(532\) 0 0
\(533\) −8.49209e10 −1.05222
\(534\) 0 0
\(535\) 1.53408e11i 1.87254i
\(536\) 0 0
\(537\) −8.60968e10 −1.03536
\(538\) 0 0
\(539\) 2.76488e10i 0.327583i
\(540\) 0 0
\(541\) −9.63677e10 −1.12497 −0.562487 0.826806i \(-0.690155\pi\)
−0.562487 + 0.826806i \(0.690155\pi\)
\(542\) 0 0
\(543\) − 3.39139e10i − 0.390102i
\(544\) 0 0
\(545\) 5.45882e10 0.618746
\(546\) 0 0
\(547\) − 1.13025e11i − 1.26248i −0.775587 0.631241i \(-0.782546\pi\)
0.775587 0.631241i \(-0.217454\pi\)
\(548\) 0 0
\(549\) 1.36017e10 0.149728
\(550\) 0 0
\(551\) − 3.72869e10i − 0.404529i
\(552\) 0 0
\(553\) 1.76131e11 1.88337
\(554\) 0 0
\(555\) 1.31360e11i 1.38449i
\(556\) 0 0
\(557\) 1.42898e11 1.48458 0.742291 0.670078i \(-0.233739\pi\)
0.742291 + 0.670078i \(0.233739\pi\)
\(558\) 0 0
\(559\) − 1.31773e11i − 1.34952i
\(560\) 0 0
\(561\) 4.91202e10 0.495917
\(562\) 0 0
\(563\) − 8.62444e10i − 0.858415i −0.903206 0.429207i \(-0.858793\pi\)
0.903206 0.429207i \(-0.141207\pi\)
\(564\) 0 0
\(565\) 1.42329e10 0.139669
\(566\) 0 0
\(567\) − 1.48333e10i − 0.143517i
\(568\) 0 0
\(569\) −7.08653e10 −0.676059 −0.338029 0.941136i \(-0.609760\pi\)
−0.338029 + 0.941136i \(0.609760\pi\)
\(570\) 0 0
\(571\) 1.62595e11i 1.52955i 0.644298 + 0.764774i \(0.277150\pi\)
−0.644298 + 0.764774i \(0.722850\pi\)
\(572\) 0 0
\(573\) 7.47558e10 0.693468
\(574\) 0 0
\(575\) 4.98963e10i 0.456453i
\(576\) 0 0
\(577\) 7.57766e10 0.683647 0.341824 0.939764i \(-0.388955\pi\)
0.341824 + 0.939764i \(0.388955\pi\)
\(578\) 0 0
\(579\) − 2.36198e10i − 0.210166i
\(580\) 0 0
\(581\) −2.64473e11 −2.32101
\(582\) 0 0
\(583\) 8.71483e10i 0.754371i
\(584\) 0 0
\(585\) 5.60316e10 0.478420
\(586\) 0 0
\(587\) − 1.92106e11i − 1.61804i −0.587784 0.809018i \(-0.700000\pi\)
0.587784 0.809018i \(-0.300000\pi\)
\(588\) 0 0
\(589\) −9.46191e10 −0.786172
\(590\) 0 0
\(591\) 1.04676e11i 0.858023i
\(592\) 0 0
\(593\) −5.12974e10 −0.414836 −0.207418 0.978252i \(-0.566506\pi\)
−0.207418 + 0.978252i \(0.566506\pi\)
\(594\) 0 0
\(595\) − 4.32390e11i − 3.44991i
\(596\) 0 0
\(597\) 5.21817e10 0.410791
\(598\) 0 0
\(599\) − 6.95617e10i − 0.540335i −0.962813 0.270167i \(-0.912921\pi\)
0.962813 0.270167i \(-0.0870790\pi\)
\(600\) 0 0
\(601\) 1.46315e11 1.12148 0.560739 0.827993i \(-0.310517\pi\)
0.560739 + 0.827993i \(0.310517\pi\)
\(602\) 0 0
\(603\) − 3.32727e10i − 0.251663i
\(604\) 0 0
\(605\) 1.55133e11 1.15793
\(606\) 0 0
\(607\) 6.48265e10i 0.477527i 0.971078 + 0.238763i \(0.0767420\pi\)
−0.971078 + 0.238763i \(0.923258\pi\)
\(608\) 0 0
\(609\) 2.45569e10 0.178527
\(610\) 0 0
\(611\) 2.17360e10i 0.155961i
\(612\) 0 0
\(613\) 5.62833e9 0.0398600 0.0199300 0.999801i \(-0.493656\pi\)
0.0199300 + 0.999801i \(0.493656\pi\)
\(614\) 0 0
\(615\) 1.40637e11i 0.983104i
\(616\) 0 0
\(617\) 1.70885e11 1.17913 0.589567 0.807719i \(-0.299298\pi\)
0.589567 + 0.807719i \(0.299298\pi\)
\(618\) 0 0
\(619\) 2.59388e10i 0.176680i 0.996090 + 0.0883401i \(0.0281562\pi\)
−0.996090 + 0.0883401i \(0.971844\pi\)
\(620\) 0 0
\(621\) 9.87720e9 0.0664152
\(622\) 0 0
\(623\) − 9.51041e10i − 0.631316i
\(624\) 0 0
\(625\) −8.74688e10 −0.573236
\(626\) 0 0
\(627\) − 7.38997e10i − 0.478159i
\(628\) 0 0
\(629\) 4.31649e11 2.75758
\(630\) 0 0
\(631\) − 3.47218e9i − 0.0219021i −0.999940 0.0109510i \(-0.996514\pi\)
0.999940 0.0109510i \(-0.00348589\pi\)
\(632\) 0 0
\(633\) −1.37531e9 −0.00856615
\(634\) 0 0
\(635\) 8.27674e10i 0.509055i
\(636\) 0 0
\(637\) 1.03637e11 0.629446
\(638\) 0 0
\(639\) 4.20093e10i 0.251966i
\(640\) 0 0
\(641\) −1.20527e11 −0.713927 −0.356963 0.934118i \(-0.616188\pi\)
−0.356963 + 0.934118i \(0.616188\pi\)
\(642\) 0 0
\(643\) 1.04525e11i 0.611470i 0.952117 + 0.305735i \(0.0989021\pi\)
−0.952117 + 0.305735i \(0.901098\pi\)
\(644\) 0 0
\(645\) −2.18228e11 −1.26087
\(646\) 0 0
\(647\) − 4.75874e10i − 0.271566i −0.990739 0.135783i \(-0.956645\pi\)
0.990739 0.135783i \(-0.0433550\pi\)
\(648\) 0 0
\(649\) 1.77798e10 0.100219
\(650\) 0 0
\(651\) − 6.23155e10i − 0.346954i
\(652\) 0 0
\(653\) −1.53529e11 −0.844381 −0.422190 0.906507i \(-0.638739\pi\)
−0.422190 + 0.906507i \(0.638739\pi\)
\(654\) 0 0
\(655\) 2.79200e11i 1.51688i
\(656\) 0 0
\(657\) 6.93897e10 0.372421
\(658\) 0 0
\(659\) 1.08583e11i 0.575731i 0.957671 + 0.287865i \(0.0929456\pi\)
−0.957671 + 0.287865i \(0.907054\pi\)
\(660\) 0 0
\(661\) 1.09736e11 0.574834 0.287417 0.957806i \(-0.407204\pi\)
0.287417 + 0.957806i \(0.407204\pi\)
\(662\) 0 0
\(663\) − 1.84120e11i − 0.952897i
\(664\) 0 0
\(665\) −6.50515e11 −3.32637
\(666\) 0 0
\(667\) 1.63520e10i 0.0826166i
\(668\) 0 0
\(669\) −1.55468e11 −0.776133
\(670\) 0 0
\(671\) 4.46289e10i 0.220154i
\(672\) 0 0
\(673\) −2.53333e11 −1.23490 −0.617449 0.786611i \(-0.711834\pi\)
−0.617449 + 0.786611i \(0.711834\pi\)
\(674\) 0 0
\(675\) − 5.28421e10i − 0.254545i
\(676\) 0 0
\(677\) 8.20339e10 0.390516 0.195258 0.980752i \(-0.437446\pi\)
0.195258 + 0.980752i \(0.437446\pi\)
\(678\) 0 0
\(679\) − 1.59843e10i − 0.0751994i
\(680\) 0 0
\(681\) −1.72345e10 −0.0801329
\(682\) 0 0
\(683\) − 3.43876e10i − 0.158022i −0.996874 0.0790112i \(-0.974824\pi\)
0.996874 0.0790112i \(-0.0251763\pi\)
\(684\) 0 0
\(685\) −4.69912e11 −2.13429
\(686\) 0 0
\(687\) − 6.21934e10i − 0.279201i
\(688\) 0 0
\(689\) 3.26662e11 1.44951
\(690\) 0 0
\(691\) 1.89939e11i 0.833111i 0.909110 + 0.416556i \(0.136763\pi\)
−0.909110 + 0.416556i \(0.863237\pi\)
\(692\) 0 0
\(693\) 4.86698e10 0.211022
\(694\) 0 0
\(695\) 4.96331e11i 2.12732i
\(696\) 0 0
\(697\) 4.62133e11 1.95810
\(698\) 0 0
\(699\) 1.16722e11i 0.488927i
\(700\) 0 0
\(701\) −1.49808e11 −0.620388 −0.310194 0.950673i \(-0.600394\pi\)
−0.310194 + 0.950673i \(0.600394\pi\)
\(702\) 0 0
\(703\) − 6.49400e11i − 2.65883i
\(704\) 0 0
\(705\) 3.59969e10 0.145716
\(706\) 0 0
\(707\) 3.14107e11i 1.25719i
\(708\) 0 0
\(709\) −1.29962e11 −0.514317 −0.257158 0.966369i \(-0.582786\pi\)
−0.257158 + 0.966369i \(0.582786\pi\)
\(710\) 0 0
\(711\) − 1.24207e11i − 0.486034i
\(712\) 0 0
\(713\) 4.14948e10 0.160559
\(714\) 0 0
\(715\) 1.83847e11i 0.703449i
\(716\) 0 0
\(717\) 2.69929e9 0.0102135
\(718\) 0 0
\(719\) − 3.22515e11i − 1.20680i −0.797440 0.603398i \(-0.793813\pi\)
0.797440 0.603398i \(-0.206187\pi\)
\(720\) 0 0
\(721\) −6.34088e11 −2.34643
\(722\) 0 0
\(723\) − 7.35148e10i − 0.269043i
\(724\) 0 0
\(725\) 8.74817e10 0.316640
\(726\) 0 0
\(727\) − 4.13304e11i − 1.47956i −0.672850 0.739779i \(-0.734930\pi\)
0.672850 0.739779i \(-0.265070\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) 7.17097e11i 2.51136i
\(732\) 0 0
\(733\) 3.98880e11 1.38174 0.690871 0.722978i \(-0.257227\pi\)
0.690871 + 0.722978i \(0.257227\pi\)
\(734\) 0 0
\(735\) − 1.71633e11i − 0.588101i
\(736\) 0 0
\(737\) 1.09172e11 0.370034
\(738\) 0 0
\(739\) − 1.18702e11i − 0.397998i −0.980000 0.198999i \(-0.936231\pi\)
0.980000 0.198999i \(-0.0637691\pi\)
\(740\) 0 0
\(741\) −2.77002e11 −0.918776
\(742\) 0 0
\(743\) 2.31061e11i 0.758179i 0.925360 + 0.379089i \(0.123763\pi\)
−0.925360 + 0.379089i \(0.876237\pi\)
\(744\) 0 0
\(745\) −3.89796e11 −1.26535
\(746\) 0 0
\(747\) 1.86505e11i 0.598974i
\(748\) 0 0
\(749\) 4.99474e11 1.58703
\(750\) 0 0
\(751\) 1.94612e11i 0.611801i 0.952064 + 0.305900i \(0.0989574\pi\)
−0.952064 + 0.305900i \(0.901043\pi\)
\(752\) 0 0
\(753\) 1.23217e11 0.383258
\(754\) 0 0
\(755\) − 8.50175e9i − 0.0261650i
\(756\) 0 0
\(757\) 3.63936e10 0.110826 0.0554130 0.998464i \(-0.482352\pi\)
0.0554130 + 0.998464i \(0.482352\pi\)
\(758\) 0 0
\(759\) 3.24084e10i 0.0976540i
\(760\) 0 0
\(761\) −3.62354e10 −0.108043 −0.0540213 0.998540i \(-0.517204\pi\)
−0.0540213 + 0.998540i \(0.517204\pi\)
\(762\) 0 0
\(763\) − 1.77732e11i − 0.524405i
\(764\) 0 0
\(765\) −3.04920e11 −0.890307
\(766\) 0 0
\(767\) − 6.66450e10i − 0.192569i
\(768\) 0 0
\(769\) 4.96871e11 1.42082 0.710409 0.703789i \(-0.248510\pi\)
0.710409 + 0.703789i \(0.248510\pi\)
\(770\) 0 0
\(771\) 1.72039e11i 0.486865i
\(772\) 0 0
\(773\) 6.63957e11 1.85961 0.929806 0.368051i \(-0.119975\pi\)
0.929806 + 0.368051i \(0.119975\pi\)
\(774\) 0 0
\(775\) − 2.21993e11i − 0.615366i
\(776\) 0 0
\(777\) 4.27691e11 1.17340
\(778\) 0 0
\(779\) − 6.95263e11i − 1.88799i
\(780\) 0 0
\(781\) −1.37838e11 −0.370480
\(782\) 0 0
\(783\) − 1.73174e10i − 0.0460719i
\(784\) 0 0
\(785\) 2.46634e11 0.649493
\(786\) 0 0
\(787\) 6.50846e10i 0.169660i 0.996395 + 0.0848299i \(0.0270347\pi\)
−0.996395 + 0.0848299i \(0.972965\pi\)
\(788\) 0 0
\(789\) −5.68869e10 −0.146793
\(790\) 0 0
\(791\) − 4.63404e10i − 0.118373i
\(792\) 0 0
\(793\) 1.67285e11 0.423022
\(794\) 0 0
\(795\) − 5.40983e11i − 1.35430i
\(796\) 0 0
\(797\) 5.57879e10 0.138263 0.0691316 0.997608i \(-0.477977\pi\)
0.0691316 + 0.997608i \(0.477977\pi\)
\(798\) 0 0
\(799\) − 1.18286e11i − 0.290232i
\(800\) 0 0
\(801\) −6.70670e10 −0.162922
\(802\) 0 0
\(803\) 2.27677e11i 0.547591i
\(804\) 0 0
\(805\) 2.85281e11 0.679342
\(806\) 0 0
\(807\) − 4.33372e11i − 1.02180i
\(808\) 0 0
\(809\) −6.06402e11 −1.41568 −0.707842 0.706370i \(-0.750331\pi\)
−0.707842 + 0.706370i \(0.750331\pi\)
\(810\) 0 0
\(811\) − 4.98163e11i − 1.15156i −0.817603 0.575782i \(-0.804698\pi\)
0.817603 0.575782i \(-0.195302\pi\)
\(812\) 0 0
\(813\) −1.65684e9 −0.00379244
\(814\) 0 0
\(815\) − 1.92066e11i − 0.435331i
\(816\) 0 0
\(817\) 1.07885e12 2.42143
\(818\) 0 0
\(819\) − 1.82431e11i − 0.405475i
\(820\) 0 0
\(821\) −3.16956e11 −0.697632 −0.348816 0.937191i \(-0.613416\pi\)
−0.348816 + 0.937191i \(0.613416\pi\)
\(822\) 0 0
\(823\) 6.55634e11i 1.42910i 0.699585 + 0.714549i \(0.253368\pi\)
−0.699585 + 0.714549i \(0.746632\pi\)
\(824\) 0 0
\(825\) 1.73382e11 0.374272
\(826\) 0 0
\(827\) − 8.69774e10i − 0.185945i −0.995669 0.0929724i \(-0.970363\pi\)
0.995669 0.0929724i \(-0.0296368\pi\)
\(828\) 0 0
\(829\) −3.79641e11 −0.803813 −0.401906 0.915681i \(-0.631652\pi\)
−0.401906 + 0.915681i \(0.631652\pi\)
\(830\) 0 0
\(831\) 2.68005e11i 0.562004i
\(832\) 0 0
\(833\) −5.63986e11 −1.17135
\(834\) 0 0
\(835\) 1.38227e12i 2.84345i
\(836\) 0 0
\(837\) −4.39446e10 −0.0895373
\(838\) 0 0
\(839\) − 2.07825e11i − 0.419421i −0.977764 0.209710i \(-0.932748\pi\)
0.977764 0.209710i \(-0.0672521\pi\)
\(840\) 0 0
\(841\) −4.71577e11 −0.942689
\(842\) 0 0
\(843\) 3.17714e11i 0.629110i
\(844\) 0 0
\(845\) −8.78748e10 −0.172360
\(846\) 0 0
\(847\) − 5.05092e11i − 0.981379i
\(848\) 0 0
\(849\) −2.65225e11 −0.510486
\(850\) 0 0
\(851\) 2.84791e11i 0.543011i
\(852\) 0 0
\(853\) −7.32516e11 −1.38363 −0.691817 0.722073i \(-0.743189\pi\)
−0.691817 + 0.722073i \(0.743189\pi\)
\(854\) 0 0
\(855\) 4.58741e11i 0.858426i
\(856\) 0 0
\(857\) 2.23294e11 0.413955 0.206978 0.978346i \(-0.433637\pi\)
0.206978 + 0.978346i \(0.433637\pi\)
\(858\) 0 0
\(859\) 2.41807e11i 0.444116i 0.975034 + 0.222058i \(0.0712774\pi\)
−0.975034 + 0.222058i \(0.928723\pi\)
\(860\) 0 0
\(861\) 4.57895e11 0.833208
\(862\) 0 0
\(863\) 1.52874e11i 0.275607i 0.990460 + 0.137804i \(0.0440043\pi\)
−0.990460 + 0.137804i \(0.955996\pi\)
\(864\) 0 0
\(865\) 6.06200e11 1.08281
\(866\) 0 0
\(867\) 6.75741e11i 1.19592i
\(868\) 0 0
\(869\) 4.07538e11 0.714643
\(870\) 0 0
\(871\) − 4.09215e11i − 0.711015i
\(872\) 0 0
\(873\) −1.12721e10 −0.0194065
\(874\) 0 0
\(875\) − 3.72317e11i − 0.635156i
\(876\) 0 0
\(877\) −1.94466e11 −0.328735 −0.164367 0.986399i \(-0.552558\pi\)
−0.164367 + 0.986399i \(0.552558\pi\)
\(878\) 0 0
\(879\) − 7.12774e10i − 0.119398i
\(880\) 0 0
\(881\) −9.23735e11 −1.53336 −0.766679 0.642030i \(-0.778092\pi\)
−0.766679 + 0.642030i \(0.778092\pi\)
\(882\) 0 0
\(883\) − 1.15008e12i − 1.89184i −0.324405 0.945918i \(-0.605164\pi\)
0.324405 0.945918i \(-0.394836\pi\)
\(884\) 0 0
\(885\) −1.10370e11 −0.179920
\(886\) 0 0
\(887\) − 1.00865e12i − 1.62947i −0.579831 0.814737i \(-0.696881\pi\)
0.579831 0.814737i \(-0.303119\pi\)
\(888\) 0 0
\(889\) 2.69479e11 0.431438
\(890\) 0 0
\(891\) − 3.43218e10i − 0.0544576i
\(892\) 0 0
\(893\) −1.77957e11 −0.279839
\(894\) 0 0
\(895\) 1.75362e12i 2.73302i
\(896\) 0 0
\(897\) 1.21478e11 0.187641
\(898\) 0 0
\(899\) − 7.27517e10i − 0.111379i
\(900\) 0 0
\(901\) −1.77767e12 −2.69744
\(902\) 0 0
\(903\) 7.10522e11i 1.06863i
\(904\) 0 0
\(905\) −6.90758e11 −1.02975
\(906\) 0 0
\(907\) 1.63003e11i 0.240861i 0.992722 + 0.120430i \(0.0384274\pi\)
−0.992722 + 0.120430i \(0.961573\pi\)
\(908\) 0 0
\(909\) 2.21507e11 0.324439
\(910\) 0 0
\(911\) 2.86326e11i 0.415707i 0.978160 + 0.207854i \(0.0666478\pi\)
−0.978160 + 0.207854i \(0.933352\pi\)
\(912\) 0 0
\(913\) −6.11947e11 −0.880705
\(914\) 0 0
\(915\) − 2.77039e11i − 0.395236i
\(916\) 0 0
\(917\) 9.09037e11 1.28560
\(918\) 0 0
\(919\) − 4.28536e11i − 0.600793i −0.953814 0.300396i \(-0.902881\pi\)
0.953814 0.300396i \(-0.0971190\pi\)
\(920\) 0 0
\(921\) −2.51086e11 −0.348967
\(922\) 0 0
\(923\) 5.16665e11i 0.711872i
\(924\) 0 0
\(925\) 1.52361e12 2.08117
\(926\) 0 0
\(927\) 4.47156e11i 0.605536i
\(928\) 0 0
\(929\) −7.91477e11 −1.06261 −0.531307 0.847180i \(-0.678299\pi\)
−0.531307 + 0.847180i \(0.678299\pi\)
\(930\) 0 0
\(931\) 8.48497e11i 1.12941i
\(932\) 0 0
\(933\) 2.90578e11 0.383475
\(934\) 0 0
\(935\) − 1.00048e12i − 1.30907i
\(936\) 0 0
\(937\) −8.10675e11 −1.05169 −0.525846 0.850580i \(-0.676251\pi\)
−0.525846 + 0.850580i \(0.676251\pi\)
\(938\) 0 0
\(939\) 1.37970e10i 0.0177469i
\(940\) 0 0
\(941\) −1.14478e12 −1.46004 −0.730021 0.683425i \(-0.760490\pi\)
−0.730021 + 0.683425i \(0.760490\pi\)
\(942\) 0 0
\(943\) 3.04904e11i 0.385582i
\(944\) 0 0
\(945\) −3.02124e11 −0.378841
\(946\) 0 0
\(947\) − 1.36874e12i − 1.70185i −0.525284 0.850927i \(-0.676041\pi\)
0.525284 0.850927i \(-0.323959\pi\)
\(948\) 0 0
\(949\) 8.53411e11 1.05219
\(950\) 0 0
\(951\) 5.72702e10i 0.0700175i
\(952\) 0 0
\(953\) 1.59249e11 0.193065 0.0965327 0.995330i \(-0.469225\pi\)
0.0965327 + 0.995330i \(0.469225\pi\)
\(954\) 0 0
\(955\) − 1.52263e12i − 1.83054i
\(956\) 0 0
\(957\) 5.68207e10 0.0677421
\(958\) 0 0
\(959\) 1.52997e12i 1.80887i
\(960\) 0 0
\(961\) 6.68277e11 0.783543
\(962\) 0 0
\(963\) − 3.52227e11i − 0.409560i
\(964\) 0 0
\(965\) −4.81088e11 −0.554773
\(966\) 0 0
\(967\) 9.07631e11i 1.03801i 0.854770 + 0.519007i \(0.173698\pi\)
−0.854770 + 0.519007i \(0.826302\pi\)
\(968\) 0 0
\(969\) 1.50742e12 1.70978
\(970\) 0 0
\(971\) − 1.07988e12i − 1.21478i −0.794405 0.607389i \(-0.792217\pi\)
0.794405 0.607389i \(-0.207783\pi\)
\(972\) 0 0
\(973\) 1.61599e12 1.80296
\(974\) 0 0
\(975\) − 6.49895e11i − 0.719159i
\(976\) 0 0
\(977\) 1.39047e12 1.52610 0.763049 0.646341i \(-0.223702\pi\)
0.763049 + 0.646341i \(0.223702\pi\)
\(978\) 0 0
\(979\) − 2.20056e11i − 0.239553i
\(980\) 0 0
\(981\) −1.25336e11 −0.135331
\(982\) 0 0
\(983\) − 1.23832e12i − 1.32623i −0.748516 0.663117i \(-0.769233\pi\)
0.748516 0.663117i \(-0.230767\pi\)
\(984\) 0 0
\(985\) 2.13205e12 2.26492
\(986\) 0 0
\(987\) − 1.17201e11i − 0.123499i
\(988\) 0 0
\(989\) −4.73124e11 −0.494526
\(990\) 0 0
\(991\) 1.64879e12i 1.70951i 0.519033 + 0.854754i \(0.326292\pi\)
−0.519033 + 0.854754i \(0.673708\pi\)
\(992\) 0 0
\(993\) −7.00582e11 −0.720547
\(994\) 0 0
\(995\) − 1.06284e12i − 1.08436i
\(996\) 0 0
\(997\) 1.01638e10 0.0102867 0.00514335 0.999987i \(-0.498363\pi\)
0.00514335 + 0.999987i \(0.498363\pi\)
\(998\) 0 0
\(999\) − 3.01606e11i − 0.302815i
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 48.9.g.c.31.4 yes 4
3.2 odd 2 144.9.g.i.127.1 4
4.3 odd 2 inner 48.9.g.c.31.2 4
8.3 odd 2 192.9.g.c.127.3 4
8.5 even 2 192.9.g.c.127.1 4
12.11 even 2 144.9.g.i.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.c.31.2 4 4.3 odd 2 inner
48.9.g.c.31.4 yes 4 1.1 even 1 trivial
144.9.g.i.127.1 4 3.2 odd 2
144.9.g.i.127.2 4 12.11 even 2
192.9.g.c.127.1 4 8.5 even 2
192.9.g.c.127.3 4 8.3 odd 2