Properties

Label 384.9.g.a.127.26
Level $384$
Weight $9$
Character 384.127
Analytic conductor $156.433$
Analytic rank $0$
Dimension $32$
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] = [384,9,Mod(127,384)]
 
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([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("384.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.26
Character \(\chi\) \(=\) 384.127
Dual form 384.9.g.a.127.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+46.7654i q^{3} +795.016 q^{5} +2625.43i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q+46.7654i q^{3} +795.016 q^{5} +2625.43i q^{7} -2187.00 q^{9} -10097.9i q^{11} -11030.0 q^{13} +37179.2i q^{15} +59630.3 q^{17} +135471. i q^{19} -122779. q^{21} -139801. i q^{23} +241425. q^{25} -102276. i q^{27} -621971. q^{29} -334997. i q^{31} +472231. q^{33} +2.08726e6i q^{35} -298771. q^{37} -515820. i q^{39} -4.49508e6 q^{41} +4.61305e6i q^{43} -1.73870e6 q^{45} +5.76388e6i q^{47} -1.12809e6 q^{49} +2.78863e6i q^{51} -1.06426e7 q^{53} -8.02798e6i q^{55} -6.33537e6 q^{57} +7.68703e6i q^{59} +8.82864e6 q^{61} -5.74182e6i q^{63} -8.76899e6 q^{65} -2.01447e7i q^{67} +6.53783e6 q^{69} -1.28859e7i q^{71} -2.72924e7 q^{73} +1.12903e7i q^{75} +2.65113e7 q^{77} +3.34620e7i q^{79} +4.78297e6 q^{81} -3.64077e6i q^{83} +4.74070e7 q^{85} -2.90867e7i q^{87} -2.49677e7 q^{89} -2.89584e7i q^{91} +1.56663e7 q^{93} +1.07702e8i q^{95} +6.99172e7 q^{97} +2.20841e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 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
\(3\) 46.7654i 0.577350i
\(4\) 0 0
\(5\) 795.016 1.27203 0.636013 0.771679i \(-0.280583\pi\)
0.636013 + 0.771679i \(0.280583\pi\)
\(6\) 0 0
\(7\) 2625.43i 1.09347i 0.837304 + 0.546737i \(0.184130\pi\)
−0.837304 + 0.546737i \(0.815870\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) − 10097.9i − 0.689699i −0.938658 0.344850i \(-0.887930\pi\)
0.938658 0.344850i \(-0.112070\pi\)
\(12\) 0 0
\(13\) −11030.0 −0.386189 −0.193095 0.981180i \(-0.561852\pi\)
−0.193095 + 0.981180i \(0.561852\pi\)
\(14\) 0 0
\(15\) 37179.2i 0.734404i
\(16\) 0 0
\(17\) 59630.3 0.713956 0.356978 0.934113i \(-0.383807\pi\)
0.356978 + 0.934113i \(0.383807\pi\)
\(18\) 0 0
\(19\) 135471.i 1.03952i 0.854312 + 0.519761i \(0.173979\pi\)
−0.854312 + 0.519761i \(0.826021\pi\)
\(20\) 0 0
\(21\) −122779. −0.631318
\(22\) 0 0
\(23\) − 139801.i − 0.499572i −0.968301 0.249786i \(-0.919640\pi\)
0.968301 0.249786i \(-0.0803602\pi\)
\(24\) 0 0
\(25\) 241425. 0.618049
\(26\) 0 0
\(27\) − 102276.i − 0.192450i
\(28\) 0 0
\(29\) −621971. −0.879383 −0.439691 0.898149i \(-0.644912\pi\)
−0.439691 + 0.898149i \(0.644912\pi\)
\(30\) 0 0
\(31\) − 334997.i − 0.362739i −0.983415 0.181370i \(-0.941947\pi\)
0.983415 0.181370i \(-0.0580530\pi\)
\(32\) 0 0
\(33\) 472231. 0.398198
\(34\) 0 0
\(35\) 2.08726e6i 1.39093i
\(36\) 0 0
\(37\) −298771. −0.159416 −0.0797079 0.996818i \(-0.525399\pi\)
−0.0797079 + 0.996818i \(0.525399\pi\)
\(38\) 0 0
\(39\) − 515820.i − 0.222967i
\(40\) 0 0
\(41\) −4.49508e6 −1.59075 −0.795375 0.606117i \(-0.792726\pi\)
−0.795375 + 0.606117i \(0.792726\pi\)
\(42\) 0 0
\(43\) 4.61305e6i 1.34932i 0.738129 + 0.674660i \(0.235710\pi\)
−0.738129 + 0.674660i \(0.764290\pi\)
\(44\) 0 0
\(45\) −1.73870e6 −0.424008
\(46\) 0 0
\(47\) 5.76388e6i 1.18120i 0.806964 + 0.590600i \(0.201109\pi\)
−0.806964 + 0.590600i \(0.798891\pi\)
\(48\) 0 0
\(49\) −1.12809e6 −0.195686
\(50\) 0 0
\(51\) 2.78863e6i 0.412203i
\(52\) 0 0
\(53\) −1.06426e7 −1.34879 −0.674393 0.738373i \(-0.735595\pi\)
−0.674393 + 0.738373i \(0.735595\pi\)
\(54\) 0 0
\(55\) − 8.02798e6i − 0.877315i
\(56\) 0 0
\(57\) −6.33537e6 −0.600168
\(58\) 0 0
\(59\) 7.68703e6i 0.634381i 0.948362 + 0.317191i \(0.102739\pi\)
−0.948362 + 0.317191i \(0.897261\pi\)
\(60\) 0 0
\(61\) 8.82864e6 0.637639 0.318819 0.947816i \(-0.396714\pi\)
0.318819 + 0.947816i \(0.396714\pi\)
\(62\) 0 0
\(63\) − 5.74182e6i − 0.364491i
\(64\) 0 0
\(65\) −8.76899e6 −0.491243
\(66\) 0 0
\(67\) − 2.01447e7i − 0.999680i −0.866118 0.499840i \(-0.833392\pi\)
0.866118 0.499840i \(-0.166608\pi\)
\(68\) 0 0
\(69\) 6.53783e6 0.288428
\(70\) 0 0
\(71\) − 1.28859e7i − 0.507088i −0.967324 0.253544i \(-0.918404\pi\)
0.967324 0.253544i \(-0.0815962\pi\)
\(72\) 0 0
\(73\) −2.72924e7 −0.961060 −0.480530 0.876978i \(-0.659556\pi\)
−0.480530 + 0.876978i \(0.659556\pi\)
\(74\) 0 0
\(75\) 1.12903e7i 0.356831i
\(76\) 0 0
\(77\) 2.65113e7 0.754169
\(78\) 0 0
\(79\) 3.34620e7i 0.859100i 0.903043 + 0.429550i \(0.141328\pi\)
−0.903043 + 0.429550i \(0.858672\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) − 3.64077e6i − 0.0767152i −0.999264 0.0383576i \(-0.987787\pi\)
0.999264 0.0383576i \(-0.0122126\pi\)
\(84\) 0 0
\(85\) 4.74070e7 0.908170
\(86\) 0 0
\(87\) − 2.90867e7i − 0.507712i
\(88\) 0 0
\(89\) −2.49677e7 −0.397941 −0.198970 0.980006i \(-0.563760\pi\)
−0.198970 + 0.980006i \(0.563760\pi\)
\(90\) 0 0
\(91\) − 2.89584e7i − 0.422288i
\(92\) 0 0
\(93\) 1.56663e7 0.209427
\(94\) 0 0
\(95\) 1.07702e8i 1.32230i
\(96\) 0 0
\(97\) 6.99172e7 0.789763 0.394882 0.918732i \(-0.370786\pi\)
0.394882 + 0.918732i \(0.370786\pi\)
\(98\) 0 0
\(99\) 2.20841e7i 0.229900i
\(100\) 0 0
\(101\) −1.54036e8 −1.48026 −0.740129 0.672464i \(-0.765236\pi\)
−0.740129 + 0.672464i \(0.765236\pi\)
\(102\) 0 0
\(103\) 1.81464e8i 1.61228i 0.591723 + 0.806141i \(0.298448\pi\)
−0.591723 + 0.806141i \(0.701552\pi\)
\(104\) 0 0
\(105\) −9.76115e7 −0.803052
\(106\) 0 0
\(107\) 7.04547e7i 0.537496i 0.963211 + 0.268748i \(0.0866098\pi\)
−0.963211 + 0.268748i \(0.913390\pi\)
\(108\) 0 0
\(109\) 2.56300e8 1.81569 0.907847 0.419301i \(-0.137725\pi\)
0.907847 + 0.419301i \(0.137725\pi\)
\(110\) 0 0
\(111\) − 1.39721e7i − 0.0920387i
\(112\) 0 0
\(113\) 2.31677e8 1.42092 0.710459 0.703739i \(-0.248488\pi\)
0.710459 + 0.703739i \(0.248488\pi\)
\(114\) 0 0
\(115\) − 1.11144e8i − 0.635468i
\(116\) 0 0
\(117\) 2.41225e7 0.128730
\(118\) 0 0
\(119\) 1.56555e8i 0.780692i
\(120\) 0 0
\(121\) 1.12392e8 0.524315
\(122\) 0 0
\(123\) − 2.10214e8i − 0.918420i
\(124\) 0 0
\(125\) −1.18616e8 −0.485852
\(126\) 0 0
\(127\) 1.08884e8i 0.418550i 0.977857 + 0.209275i \(0.0671104\pi\)
−0.977857 + 0.209275i \(0.932890\pi\)
\(128\) 0 0
\(129\) −2.15731e8 −0.779030
\(130\) 0 0
\(131\) 3.97482e8i 1.34969i 0.737961 + 0.674843i \(0.235789\pi\)
−0.737961 + 0.674843i \(0.764211\pi\)
\(132\) 0 0
\(133\) −3.55671e8 −1.13669
\(134\) 0 0
\(135\) − 8.13109e7i − 0.244801i
\(136\) 0 0
\(137\) −2.96120e8 −0.840592 −0.420296 0.907387i \(-0.638074\pi\)
−0.420296 + 0.907387i \(0.638074\pi\)
\(138\) 0 0
\(139\) − 5.70397e8i − 1.52798i −0.645227 0.763991i \(-0.723237\pi\)
0.645227 0.763991i \(-0.276763\pi\)
\(140\) 0 0
\(141\) −2.69550e8 −0.681966
\(142\) 0 0
\(143\) 1.11379e8i 0.266355i
\(144\) 0 0
\(145\) −4.94476e8 −1.11860
\(146\) 0 0
\(147\) − 5.27556e7i − 0.112979i
\(148\) 0 0
\(149\) −3.90505e8 −0.792284 −0.396142 0.918189i \(-0.629651\pi\)
−0.396142 + 0.918189i \(0.629651\pi\)
\(150\) 0 0
\(151\) − 8.29215e8i − 1.59500i −0.603322 0.797498i \(-0.706157\pi\)
0.603322 0.797498i \(-0.293843\pi\)
\(152\) 0 0
\(153\) −1.30411e8 −0.237985
\(154\) 0 0
\(155\) − 2.66328e8i − 0.461413i
\(156\) 0 0
\(157\) −4.15092e8 −0.683197 −0.341599 0.939846i \(-0.610968\pi\)
−0.341599 + 0.939846i \(0.610968\pi\)
\(158\) 0 0
\(159\) − 4.97704e8i − 0.778722i
\(160\) 0 0
\(161\) 3.67037e8 0.546269
\(162\) 0 0
\(163\) − 5.74012e7i − 0.0813150i −0.999173 0.0406575i \(-0.987055\pi\)
0.999173 0.0406575i \(-0.0129453\pi\)
\(164\) 0 0
\(165\) 3.75432e8 0.506518
\(166\) 0 0
\(167\) 1.10763e8i 0.142407i 0.997462 + 0.0712033i \(0.0226839\pi\)
−0.997462 + 0.0712033i \(0.977316\pi\)
\(168\) 0 0
\(169\) −6.94071e8 −0.850858
\(170\) 0 0
\(171\) − 2.96276e8i − 0.346507i
\(172\) 0 0
\(173\) −1.35188e9 −1.50922 −0.754612 0.656171i \(-0.772175\pi\)
−0.754612 + 0.656171i \(0.772175\pi\)
\(174\) 0 0
\(175\) 6.33846e8i 0.675820i
\(176\) 0 0
\(177\) −3.59487e8 −0.366260
\(178\) 0 0
\(179\) 2.03426e8i 0.198150i 0.995080 + 0.0990750i \(0.0315884\pi\)
−0.995080 + 0.0990750i \(0.968412\pi\)
\(180\) 0 0
\(181\) 7.31026e8 0.681112 0.340556 0.940224i \(-0.389385\pi\)
0.340556 + 0.940224i \(0.389385\pi\)
\(182\) 0 0
\(183\) 4.12875e8i 0.368141i
\(184\) 0 0
\(185\) −2.37528e8 −0.202781
\(186\) 0 0
\(187\) − 6.02140e8i − 0.492415i
\(188\) 0 0
\(189\) 2.68518e8 0.210439
\(190\) 0 0
\(191\) 8.41990e8i 0.632665i 0.948648 + 0.316332i \(0.102451\pi\)
−0.948648 + 0.316332i \(0.897549\pi\)
\(192\) 0 0
\(193\) −2.63204e9 −1.89698 −0.948489 0.316809i \(-0.897389\pi\)
−0.948489 + 0.316809i \(0.897389\pi\)
\(194\) 0 0
\(195\) − 4.10085e8i − 0.283619i
\(196\) 0 0
\(197\) −1.48020e9 −0.982780 −0.491390 0.870940i \(-0.663511\pi\)
−0.491390 + 0.870940i \(0.663511\pi\)
\(198\) 0 0
\(199\) − 1.37124e8i − 0.0874381i −0.999044 0.0437191i \(-0.986079\pi\)
0.999044 0.0437191i \(-0.0139206\pi\)
\(200\) 0 0
\(201\) 9.42073e8 0.577166
\(202\) 0 0
\(203\) − 1.63294e9i − 0.961582i
\(204\) 0 0
\(205\) −3.57366e9 −2.02347
\(206\) 0 0
\(207\) 3.05744e8i 0.166524i
\(208\) 0 0
\(209\) 1.36798e9 0.716957
\(210\) 0 0
\(211\) − 2.83419e9i − 1.42988i −0.699186 0.714940i \(-0.746454\pi\)
0.699186 0.714940i \(-0.253546\pi\)
\(212\) 0 0
\(213\) 6.02616e8 0.292767
\(214\) 0 0
\(215\) 3.66745e9i 1.71637i
\(216\) 0 0
\(217\) 8.79512e8 0.396646
\(218\) 0 0
\(219\) − 1.27634e9i − 0.554868i
\(220\) 0 0
\(221\) −6.57720e8 −0.275722
\(222\) 0 0
\(223\) 2.95932e9i 1.19666i 0.801249 + 0.598331i \(0.204169\pi\)
−0.801249 + 0.598331i \(0.795831\pi\)
\(224\) 0 0
\(225\) −5.27997e8 −0.206016
\(226\) 0 0
\(227\) − 2.76691e9i − 1.04206i −0.853539 0.521029i \(-0.825549\pi\)
0.853539 0.521029i \(-0.174451\pi\)
\(228\) 0 0
\(229\) 1.33542e8 0.0485597 0.0242798 0.999705i \(-0.492271\pi\)
0.0242798 + 0.999705i \(0.492271\pi\)
\(230\) 0 0
\(231\) 1.23981e9i 0.435419i
\(232\) 0 0
\(233\) −1.60887e9 −0.545879 −0.272939 0.962031i \(-0.587996\pi\)
−0.272939 + 0.962031i \(0.587996\pi\)
\(234\) 0 0
\(235\) 4.58238e9i 1.50252i
\(236\) 0 0
\(237\) −1.56486e9 −0.496002
\(238\) 0 0
\(239\) 4.98543e9i 1.52796i 0.645241 + 0.763979i \(0.276757\pi\)
−0.645241 + 0.763979i \(0.723243\pi\)
\(240\) 0 0
\(241\) 8.11322e8 0.240506 0.120253 0.992743i \(-0.461629\pi\)
0.120253 + 0.992743i \(0.461629\pi\)
\(242\) 0 0
\(243\) 2.23677e8i 0.0641500i
\(244\) 0 0
\(245\) −8.96851e8 −0.248918
\(246\) 0 0
\(247\) − 1.49424e9i − 0.401452i
\(248\) 0 0
\(249\) 1.70262e8 0.0442915
\(250\) 0 0
\(251\) − 6.10021e9i − 1.53692i −0.639901 0.768458i \(-0.721025\pi\)
0.639901 0.768458i \(-0.278975\pi\)
\(252\) 0 0
\(253\) −1.41169e9 −0.344554
\(254\) 0 0
\(255\) 2.21701e9i 0.524332i
\(256\) 0 0
\(257\) −7.64150e9 −1.75164 −0.875822 0.482634i \(-0.839680\pi\)
−0.875822 + 0.482634i \(0.839680\pi\)
\(258\) 0 0
\(259\) − 7.84402e8i − 0.174317i
\(260\) 0 0
\(261\) 1.36025e9 0.293128
\(262\) 0 0
\(263\) − 4.62801e7i − 0.00967323i −0.999988 0.00483662i \(-0.998460\pi\)
0.999988 0.00483662i \(-0.00153955\pi\)
\(264\) 0 0
\(265\) −8.46101e9 −1.71569
\(266\) 0 0
\(267\) − 1.16762e9i − 0.229751i
\(268\) 0 0
\(269\) 6.33628e9 1.21011 0.605056 0.796183i \(-0.293151\pi\)
0.605056 + 0.796183i \(0.293151\pi\)
\(270\) 0 0
\(271\) 8.57083e9i 1.58908i 0.607212 + 0.794540i \(0.292288\pi\)
−0.607212 + 0.794540i \(0.707712\pi\)
\(272\) 0 0
\(273\) 1.35425e9 0.243808
\(274\) 0 0
\(275\) − 2.43789e9i − 0.426268i
\(276\) 0 0
\(277\) −3.18594e8 −0.0541152 −0.0270576 0.999634i \(-0.508614\pi\)
−0.0270576 + 0.999634i \(0.508614\pi\)
\(278\) 0 0
\(279\) 7.32639e8i 0.120913i
\(280\) 0 0
\(281\) −1.19967e10 −1.92414 −0.962070 0.272801i \(-0.912050\pi\)
−0.962070 + 0.272801i \(0.912050\pi\)
\(282\) 0 0
\(283\) − 2.24269e9i − 0.349643i −0.984600 0.174821i \(-0.944065\pi\)
0.984600 0.174821i \(-0.0559348\pi\)
\(284\) 0 0
\(285\) −5.03672e9 −0.763429
\(286\) 0 0
\(287\) − 1.18015e10i − 1.73944i
\(288\) 0 0
\(289\) −3.41998e9 −0.490267
\(290\) 0 0
\(291\) 3.26970e9i 0.455970i
\(292\) 0 0
\(293\) −1.22665e9 −0.166438 −0.0832189 0.996531i \(-0.526520\pi\)
−0.0832189 + 0.996531i \(0.526520\pi\)
\(294\) 0 0
\(295\) 6.11131e9i 0.806949i
\(296\) 0 0
\(297\) −1.03277e9 −0.132733
\(298\) 0 0
\(299\) 1.54200e9i 0.192929i
\(300\) 0 0
\(301\) −1.21113e10 −1.47545
\(302\) 0 0
\(303\) − 7.20357e9i − 0.854628i
\(304\) 0 0
\(305\) 7.01891e9 0.811092
\(306\) 0 0
\(307\) − 1.12906e10i − 1.27105i −0.772080 0.635526i \(-0.780783\pi\)
0.772080 0.635526i \(-0.219217\pi\)
\(308\) 0 0
\(309\) −8.48622e9 −0.930852
\(310\) 0 0
\(311\) 4.88837e9i 0.522544i 0.965265 + 0.261272i \(0.0841419\pi\)
−0.965265 + 0.261272i \(0.915858\pi\)
\(312\) 0 0
\(313\) −1.11505e10 −1.16176 −0.580880 0.813989i \(-0.697292\pi\)
−0.580880 + 0.813989i \(0.697292\pi\)
\(314\) 0 0
\(315\) − 4.56484e9i − 0.463642i
\(316\) 0 0
\(317\) −9.34931e9 −0.925854 −0.462927 0.886396i \(-0.653201\pi\)
−0.462927 + 0.886396i \(0.653201\pi\)
\(318\) 0 0
\(319\) 6.28059e9i 0.606510i
\(320\) 0 0
\(321\) −3.29484e9 −0.310323
\(322\) 0 0
\(323\) 8.07820e9i 0.742172i
\(324\) 0 0
\(325\) −2.66291e9 −0.238684
\(326\) 0 0
\(327\) 1.19860e10i 1.04829i
\(328\) 0 0
\(329\) −1.51327e10 −1.29161
\(330\) 0 0
\(331\) 1.66134e10i 1.38404i 0.721881 + 0.692018i \(0.243278\pi\)
−0.721881 + 0.692018i \(0.756722\pi\)
\(332\) 0 0
\(333\) 6.53412e8 0.0531386
\(334\) 0 0
\(335\) − 1.60153e10i − 1.27162i
\(336\) 0 0
\(337\) 1.93404e10 1.49950 0.749749 0.661722i \(-0.230174\pi\)
0.749749 + 0.661722i \(0.230174\pi\)
\(338\) 0 0
\(339\) 1.08345e10i 0.820367i
\(340\) 0 0
\(341\) −3.38276e9 −0.250181
\(342\) 0 0
\(343\) 1.21734e10i 0.879497i
\(344\) 0 0
\(345\) 5.19768e9 0.366888
\(346\) 0 0
\(347\) − 2.33924e10i − 1.61346i −0.590923 0.806728i \(-0.701236\pi\)
0.590923 0.806728i \(-0.298764\pi\)
\(348\) 0 0
\(349\) 8.26158e9 0.556880 0.278440 0.960454i \(-0.410183\pi\)
0.278440 + 0.960454i \(0.410183\pi\)
\(350\) 0 0
\(351\) 1.12810e9i 0.0743222i
\(352\) 0 0
\(353\) 5.37397e9 0.346096 0.173048 0.984913i \(-0.444638\pi\)
0.173048 + 0.984913i \(0.444638\pi\)
\(354\) 0 0
\(355\) − 1.02445e10i − 0.645028i
\(356\) 0 0
\(357\) −7.32137e9 −0.450733
\(358\) 0 0
\(359\) − 1.94858e9i − 0.117312i −0.998278 0.0586559i \(-0.981319\pi\)
0.998278 0.0586559i \(-0.0186815\pi\)
\(360\) 0 0
\(361\) −1.36895e9 −0.0806043
\(362\) 0 0
\(363\) 5.25603e9i 0.302713i
\(364\) 0 0
\(365\) −2.16979e10 −1.22249
\(366\) 0 0
\(367\) 1.05117e10i 0.579442i 0.957111 + 0.289721i \(0.0935626\pi\)
−0.957111 + 0.289721i \(0.906437\pi\)
\(368\) 0 0
\(369\) 9.83074e9 0.530250
\(370\) 0 0
\(371\) − 2.79413e10i − 1.47486i
\(372\) 0 0
\(373\) 1.52843e10 0.789606 0.394803 0.918766i \(-0.370813\pi\)
0.394803 + 0.918766i \(0.370813\pi\)
\(374\) 0 0
\(375\) − 5.54713e9i − 0.280507i
\(376\) 0 0
\(377\) 6.86031e9 0.339608
\(378\) 0 0
\(379\) − 9.70465e9i − 0.470352i −0.971953 0.235176i \(-0.924433\pi\)
0.971953 0.235176i \(-0.0755667\pi\)
\(380\) 0 0
\(381\) −5.09198e9 −0.241650
\(382\) 0 0
\(383\) 1.56132e10i 0.725599i 0.931867 + 0.362800i \(0.118179\pi\)
−0.931867 + 0.362800i \(0.881821\pi\)
\(384\) 0 0
\(385\) 2.10769e10 0.959322
\(386\) 0 0
\(387\) − 1.00887e10i − 0.449773i
\(388\) 0 0
\(389\) 4.20022e10 1.83432 0.917158 0.398525i \(-0.130478\pi\)
0.917158 + 0.398525i \(0.130478\pi\)
\(390\) 0 0
\(391\) − 8.33636e9i − 0.356672i
\(392\) 0 0
\(393\) −1.85884e10 −0.779242
\(394\) 0 0
\(395\) 2.66028e10i 1.09280i
\(396\) 0 0
\(397\) 1.82342e10 0.734047 0.367024 0.930212i \(-0.380377\pi\)
0.367024 + 0.930212i \(0.380377\pi\)
\(398\) 0 0
\(399\) − 1.66331e10i − 0.656268i
\(400\) 0 0
\(401\) 2.08304e10 0.805601 0.402801 0.915288i \(-0.368037\pi\)
0.402801 + 0.915288i \(0.368037\pi\)
\(402\) 0 0
\(403\) 3.69500e9i 0.140086i
\(404\) 0 0
\(405\) 3.80254e9 0.141336
\(406\) 0 0
\(407\) 3.01695e9i 0.109949i
\(408\) 0 0
\(409\) 1.99212e9 0.0711904 0.0355952 0.999366i \(-0.488667\pi\)
0.0355952 + 0.999366i \(0.488667\pi\)
\(410\) 0 0
\(411\) − 1.38482e10i − 0.485316i
\(412\) 0 0
\(413\) −2.01818e10 −0.693680
\(414\) 0 0
\(415\) − 2.89447e9i − 0.0975837i
\(416\) 0 0
\(417\) 2.66748e10 0.882181
\(418\) 0 0
\(419\) 2.20707e9i 0.0716078i 0.999359 + 0.0358039i \(0.0113992\pi\)
−0.999359 + 0.0358039i \(0.988601\pi\)
\(420\) 0 0
\(421\) 3.87341e10 1.23300 0.616502 0.787353i \(-0.288549\pi\)
0.616502 + 0.787353i \(0.288549\pi\)
\(422\) 0 0
\(423\) − 1.26056e10i − 0.393733i
\(424\) 0 0
\(425\) 1.43963e10 0.441259
\(426\) 0 0
\(427\) 2.31790e10i 0.697241i
\(428\) 0 0
\(429\) −5.20869e9 −0.153780
\(430\) 0 0
\(431\) − 1.79774e10i − 0.520975i −0.965477 0.260488i \(-0.916117\pi\)
0.965477 0.260488i \(-0.0838833\pi\)
\(432\) 0 0
\(433\) −3.58344e10 −1.01941 −0.509705 0.860349i \(-0.670245\pi\)
−0.509705 + 0.860349i \(0.670245\pi\)
\(434\) 0 0
\(435\) − 2.31244e10i − 0.645822i
\(436\) 0 0
\(437\) 1.89390e10 0.519315
\(438\) 0 0
\(439\) − 5.47851e10i − 1.47504i −0.675324 0.737521i \(-0.735996\pi\)
0.675324 0.737521i \(-0.264004\pi\)
\(440\) 0 0
\(441\) 2.46714e9 0.0652287
\(442\) 0 0
\(443\) 4.90120e9i 0.127259i 0.997974 + 0.0636294i \(0.0202676\pi\)
−0.997974 + 0.0636294i \(0.979732\pi\)
\(444\) 0 0
\(445\) −1.98497e10 −0.506191
\(446\) 0 0
\(447\) − 1.82621e10i − 0.457425i
\(448\) 0 0
\(449\) 1.43247e10 0.352452 0.176226 0.984350i \(-0.443611\pi\)
0.176226 + 0.984350i \(0.443611\pi\)
\(450\) 0 0
\(451\) 4.53908e10i 1.09714i
\(452\) 0 0
\(453\) 3.87786e10 0.920871
\(454\) 0 0
\(455\) − 2.30224e10i − 0.537161i
\(456\) 0 0
\(457\) −3.28090e10 −0.752191 −0.376096 0.926581i \(-0.622734\pi\)
−0.376096 + 0.926581i \(0.622734\pi\)
\(458\) 0 0
\(459\) − 6.09874e9i − 0.137401i
\(460\) 0 0
\(461\) −8.80132e10 −1.94870 −0.974348 0.225045i \(-0.927747\pi\)
−0.974348 + 0.225045i \(0.927747\pi\)
\(462\) 0 0
\(463\) 2.36196e10i 0.513983i 0.966414 + 0.256992i \(0.0827313\pi\)
−0.966414 + 0.256992i \(0.917269\pi\)
\(464\) 0 0
\(465\) 1.24549e10 0.266397
\(466\) 0 0
\(467\) 2.39463e10i 0.503466i 0.967797 + 0.251733i \(0.0810005\pi\)
−0.967797 + 0.251733i \(0.919000\pi\)
\(468\) 0 0
\(469\) 5.28885e10 1.09312
\(470\) 0 0
\(471\) − 1.94120e10i − 0.394444i
\(472\) 0 0
\(473\) 4.65821e10 0.930625
\(474\) 0 0
\(475\) 3.27062e10i 0.642475i
\(476\) 0 0
\(477\) 2.32753e10 0.449595
\(478\) 0 0
\(479\) 4.18896e9i 0.0795727i 0.999208 + 0.0397864i \(0.0126677\pi\)
−0.999208 + 0.0397864i \(0.987332\pi\)
\(480\) 0 0
\(481\) 3.29543e9 0.0615647
\(482\) 0 0
\(483\) 1.71646e10i 0.315389i
\(484\) 0 0
\(485\) 5.55853e10 1.00460
\(486\) 0 0
\(487\) 7.65755e9i 0.136136i 0.997681 + 0.0680681i \(0.0216835\pi\)
−0.997681 + 0.0680681i \(0.978316\pi\)
\(488\) 0 0
\(489\) 2.68439e9 0.0469472
\(490\) 0 0
\(491\) 5.78263e10i 0.994946i 0.867479 + 0.497473i \(0.165739\pi\)
−0.867479 + 0.497473i \(0.834261\pi\)
\(492\) 0 0
\(493\) −3.70883e10 −0.627840
\(494\) 0 0
\(495\) 1.75572e10i 0.292438i
\(496\) 0 0
\(497\) 3.38312e10 0.554487
\(498\) 0 0
\(499\) 8.98539e10i 1.44922i 0.689158 + 0.724611i \(0.257981\pi\)
−0.689158 + 0.724611i \(0.742019\pi\)
\(500\) 0 0
\(501\) −5.17989e9 −0.0822184
\(502\) 0 0
\(503\) 8.89949e9i 0.139025i 0.997581 + 0.0695126i \(0.0221444\pi\)
−0.997581 + 0.0695126i \(0.977856\pi\)
\(504\) 0 0
\(505\) −1.22461e11 −1.88293
\(506\) 0 0
\(507\) − 3.24585e10i − 0.491243i
\(508\) 0 0
\(509\) 2.83459e10 0.422298 0.211149 0.977454i \(-0.432279\pi\)
0.211149 + 0.977454i \(0.432279\pi\)
\(510\) 0 0
\(511\) − 7.16543e10i − 1.05089i
\(512\) 0 0
\(513\) 1.38555e10 0.200056
\(514\) 0 0
\(515\) 1.44267e11i 2.05086i
\(516\) 0 0
\(517\) 5.82030e10 0.814673
\(518\) 0 0
\(519\) − 6.32212e10i − 0.871351i
\(520\) 0 0
\(521\) 8.48484e10 1.15158 0.575788 0.817599i \(-0.304695\pi\)
0.575788 + 0.817599i \(0.304695\pi\)
\(522\) 0 0
\(523\) 1.22214e11i 1.63348i 0.577003 + 0.816742i \(0.304222\pi\)
−0.577003 + 0.816742i \(0.695778\pi\)
\(524\) 0 0
\(525\) −2.96420e10 −0.390185
\(526\) 0 0
\(527\) − 1.99760e10i − 0.258980i
\(528\) 0 0
\(529\) 5.87668e10 0.750428
\(530\) 0 0
\(531\) − 1.68115e10i − 0.211460i
\(532\) 0 0
\(533\) 4.95805e10 0.614331
\(534\) 0 0
\(535\) 5.60126e10i 0.683708i
\(536\) 0 0
\(537\) −9.51329e9 −0.114402
\(538\) 0 0
\(539\) 1.13913e10i 0.134965i
\(540\) 0 0
\(541\) 1.26688e11 1.47892 0.739462 0.673198i \(-0.235080\pi\)
0.739462 + 0.673198i \(0.235080\pi\)
\(542\) 0 0
\(543\) 3.41867e10i 0.393240i
\(544\) 0 0
\(545\) 2.03763e11 2.30961
\(546\) 0 0
\(547\) 9.79694e10i 1.09431i 0.837031 + 0.547156i \(0.184290\pi\)
−0.837031 + 0.547156i \(0.815710\pi\)
\(548\) 0 0
\(549\) −1.93082e10 −0.212546
\(550\) 0 0
\(551\) − 8.42593e10i − 0.914137i
\(552\) 0 0
\(553\) −8.78522e10 −0.939404
\(554\) 0 0
\(555\) − 1.11081e10i − 0.117076i
\(556\) 0 0
\(557\) −8.25333e10 −0.857449 −0.428725 0.903435i \(-0.641037\pi\)
−0.428725 + 0.903435i \(0.641037\pi\)
\(558\) 0 0
\(559\) − 5.08818e10i − 0.521093i
\(560\) 0 0
\(561\) 2.81593e10 0.284296
\(562\) 0 0
\(563\) − 1.35561e11i − 1.34928i −0.738147 0.674640i \(-0.764299\pi\)
0.738147 0.674640i \(-0.235701\pi\)
\(564\) 0 0
\(565\) 1.84187e11 1.80744
\(566\) 0 0
\(567\) 1.25574e10i 0.121497i
\(568\) 0 0
\(569\) −1.20885e11 −1.15325 −0.576624 0.817010i \(-0.695630\pi\)
−0.576624 + 0.817010i \(0.695630\pi\)
\(570\) 0 0
\(571\) − 1.61019e11i − 1.51472i −0.652998 0.757360i \(-0.726489\pi\)
0.652998 0.757360i \(-0.273511\pi\)
\(572\) 0 0
\(573\) −3.93760e10 −0.365269
\(574\) 0 0
\(575\) − 3.37514e10i − 0.308760i
\(576\) 0 0
\(577\) 1.39635e11 1.25977 0.629884 0.776689i \(-0.283102\pi\)
0.629884 + 0.776689i \(0.283102\pi\)
\(578\) 0 0
\(579\) − 1.23088e11i − 1.09522i
\(580\) 0 0
\(581\) 9.55860e9 0.0838861
\(582\) 0 0
\(583\) 1.07467e11i 0.930257i
\(584\) 0 0
\(585\) 1.91778e10 0.163748
\(586\) 0 0
\(587\) − 1.23680e11i − 1.04171i −0.853646 0.520854i \(-0.825614\pi\)
0.853646 0.520854i \(-0.174386\pi\)
\(588\) 0 0
\(589\) 4.53825e10 0.377075
\(590\) 0 0
\(591\) − 6.92222e10i − 0.567408i
\(592\) 0 0
\(593\) 1.29499e11 1.04725 0.523623 0.851950i \(-0.324580\pi\)
0.523623 + 0.851950i \(0.324580\pi\)
\(594\) 0 0
\(595\) 1.24464e11i 0.993061i
\(596\) 0 0
\(597\) 6.41265e9 0.0504824
\(598\) 0 0
\(599\) − 7.24974e10i − 0.563139i −0.959541 0.281569i \(-0.909145\pi\)
0.959541 0.281569i \(-0.0908549\pi\)
\(600\) 0 0
\(601\) −1.03270e10 −0.0791545 −0.0395773 0.999217i \(-0.512601\pi\)
−0.0395773 + 0.999217i \(0.512601\pi\)
\(602\) 0 0
\(603\) 4.40564e10i 0.333227i
\(604\) 0 0
\(605\) 8.93531e10 0.666942
\(606\) 0 0
\(607\) 5.17177e10i 0.380964i 0.981691 + 0.190482i \(0.0610052\pi\)
−0.981691 + 0.190482i \(0.938995\pi\)
\(608\) 0 0
\(609\) 7.63651e10 0.555170
\(610\) 0 0
\(611\) − 6.35753e10i − 0.456167i
\(612\) 0 0
\(613\) 1.79758e11 1.27305 0.636527 0.771254i \(-0.280370\pi\)
0.636527 + 0.771254i \(0.280370\pi\)
\(614\) 0 0
\(615\) − 1.67124e11i − 1.16825i
\(616\) 0 0
\(617\) −9.69245e10 −0.668795 −0.334397 0.942432i \(-0.608533\pi\)
−0.334397 + 0.942432i \(0.608533\pi\)
\(618\) 0 0
\(619\) − 7.61781e10i − 0.518881i −0.965759 0.259440i \(-0.916462\pi\)
0.965759 0.259440i \(-0.0835381\pi\)
\(620\) 0 0
\(621\) −1.42982e10 −0.0961426
\(622\) 0 0
\(623\) − 6.55510e10i − 0.435138i
\(624\) 0 0
\(625\) −1.88608e11 −1.23606
\(626\) 0 0
\(627\) 6.39739e10i 0.413935i
\(628\) 0 0
\(629\) −1.78158e10 −0.113816
\(630\) 0 0
\(631\) 1.18736e11i 0.748972i 0.927233 + 0.374486i \(0.122181\pi\)
−0.927233 + 0.374486i \(0.877819\pi\)
\(632\) 0 0
\(633\) 1.32542e11 0.825542
\(634\) 0 0
\(635\) 8.65642e10i 0.532407i
\(636\) 0 0
\(637\) 1.24428e10 0.0755719
\(638\) 0 0
\(639\) 2.81816e10i 0.169029i
\(640\) 0 0
\(641\) −1.38064e10 −0.0817805 −0.0408903 0.999164i \(-0.513019\pi\)
−0.0408903 + 0.999164i \(0.513019\pi\)
\(642\) 0 0
\(643\) 6.32942e10i 0.370272i 0.982713 + 0.185136i \(0.0592725\pi\)
−0.982713 + 0.185136i \(0.940728\pi\)
\(644\) 0 0
\(645\) −1.71510e11 −0.990946
\(646\) 0 0
\(647\) − 3.83819e10i − 0.219033i −0.993985 0.109516i \(-0.965070\pi\)
0.993985 0.109516i \(-0.0349302\pi\)
\(648\) 0 0
\(649\) 7.76227e10 0.437532
\(650\) 0 0
\(651\) 4.11307e10i 0.229004i
\(652\) 0 0
\(653\) 4.77321e10 0.262517 0.131259 0.991348i \(-0.458098\pi\)
0.131259 + 0.991348i \(0.458098\pi\)
\(654\) 0 0
\(655\) 3.16005e11i 1.71683i
\(656\) 0 0
\(657\) 5.96885e10 0.320353
\(658\) 0 0
\(659\) − 1.78691e11i − 0.947461i −0.880670 0.473730i \(-0.842907\pi\)
0.880670 0.473730i \(-0.157093\pi\)
\(660\) 0 0
\(661\) −1.62385e11 −0.850630 −0.425315 0.905045i \(-0.639837\pi\)
−0.425315 + 0.905045i \(0.639837\pi\)
\(662\) 0 0
\(663\) − 3.07585e10i − 0.159188i
\(664\) 0 0
\(665\) −2.82764e11 −1.44590
\(666\) 0 0
\(667\) 8.69519e10i 0.439315i
\(668\) 0 0
\(669\) −1.38393e11 −0.690894
\(670\) 0 0
\(671\) − 8.91506e10i − 0.439779i
\(672\) 0 0
\(673\) 3.25895e11 1.58861 0.794304 0.607520i \(-0.207836\pi\)
0.794304 + 0.607520i \(0.207836\pi\)
\(674\) 0 0
\(675\) − 2.46920e10i − 0.118944i
\(676\) 0 0
\(677\) −2.56252e11 −1.21987 −0.609934 0.792452i \(-0.708804\pi\)
−0.609934 + 0.792452i \(0.708804\pi\)
\(678\) 0 0
\(679\) 1.83563e11i 0.863586i
\(680\) 0 0
\(681\) 1.29396e11 0.601632
\(682\) 0 0
\(683\) − 3.96209e10i − 0.182071i −0.995848 0.0910357i \(-0.970982\pi\)
0.995848 0.0910357i \(-0.0290177\pi\)
\(684\) 0 0
\(685\) −2.35420e11 −1.06925
\(686\) 0 0
\(687\) 6.24514e9i 0.0280360i
\(688\) 0 0
\(689\) 1.17387e11 0.520887
\(690\) 0 0
\(691\) 3.75065e11i 1.64511i 0.568687 + 0.822554i \(0.307452\pi\)
−0.568687 + 0.822554i \(0.692548\pi\)
\(692\) 0 0
\(693\) −5.79803e10 −0.251390
\(694\) 0 0
\(695\) − 4.53475e11i − 1.94363i
\(696\) 0 0
\(697\) −2.68043e11 −1.13573
\(698\) 0 0
\(699\) − 7.52392e10i − 0.315163i
\(700\) 0 0
\(701\) −2.96001e11 −1.22580 −0.612901 0.790160i \(-0.709998\pi\)
−0.612901 + 0.790160i \(0.709998\pi\)
\(702\) 0 0
\(703\) − 4.04749e10i − 0.165716i
\(704\) 0 0
\(705\) −2.14296e11 −0.867478
\(706\) 0 0
\(707\) − 4.04412e11i − 1.61863i
\(708\) 0 0
\(709\) 2.94063e11 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(710\) 0 0
\(711\) − 7.31814e10i − 0.286367i
\(712\) 0 0
\(713\) −4.68328e10 −0.181214
\(714\) 0 0
\(715\) 8.85483e10i 0.338810i
\(716\) 0 0
\(717\) −2.33146e11 −0.882167
\(718\) 0 0
\(719\) − 1.88372e11i − 0.704855i −0.935839 0.352428i \(-0.885356\pi\)
0.935839 0.352428i \(-0.114644\pi\)
\(720\) 0 0
\(721\) −4.76421e11 −1.76299
\(722\) 0 0
\(723\) 3.79418e10i 0.138856i
\(724\) 0 0
\(725\) −1.50159e11 −0.543501
\(726\) 0 0
\(727\) 3.55301e11i 1.27192i 0.771723 + 0.635959i \(0.219395\pi\)
−0.771723 + 0.635959i \(0.780605\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) 2.75078e11i 0.963354i
\(732\) 0 0
\(733\) 1.06552e11 0.369103 0.184551 0.982823i \(-0.440917\pi\)
0.184551 + 0.982823i \(0.440917\pi\)
\(734\) 0 0
\(735\) − 4.19416e10i − 0.143713i
\(736\) 0 0
\(737\) −2.03419e11 −0.689479
\(738\) 0 0
\(739\) − 2.71902e11i − 0.911664i −0.890066 0.455832i \(-0.849342\pi\)
0.890066 0.455832i \(-0.150658\pi\)
\(740\) 0 0
\(741\) 6.98789e10 0.231778
\(742\) 0 0
\(743\) 4.45168e11i 1.46073i 0.683060 + 0.730363i \(0.260649\pi\)
−0.683060 + 0.730363i \(0.739351\pi\)
\(744\) 0 0
\(745\) −3.10457e11 −1.00781
\(746\) 0 0
\(747\) 7.96237e9i 0.0255717i
\(748\) 0 0
\(749\) −1.84974e11 −0.587738
\(750\) 0 0
\(751\) 3.05844e11i 0.961479i 0.876863 + 0.480740i \(0.159632\pi\)
−0.876863 + 0.480740i \(0.840368\pi\)
\(752\) 0 0
\(753\) 2.85279e11 0.887338
\(754\) 0 0
\(755\) − 6.59239e11i − 2.02887i
\(756\) 0 0
\(757\) 1.31463e11 0.400330 0.200165 0.979762i \(-0.435852\pi\)
0.200165 + 0.979762i \(0.435852\pi\)
\(758\) 0 0
\(759\) − 6.60183e10i − 0.198929i
\(760\) 0 0
\(761\) 4.23715e11 1.26338 0.631692 0.775219i \(-0.282361\pi\)
0.631692 + 0.775219i \(0.282361\pi\)
\(762\) 0 0
\(763\) 6.72899e11i 1.98542i
\(764\) 0 0
\(765\) −1.03679e11 −0.302723
\(766\) 0 0
\(767\) − 8.47875e10i − 0.244991i
\(768\) 0 0
\(769\) −5.39328e11 −1.54223 −0.771113 0.636699i \(-0.780300\pi\)
−0.771113 + 0.636699i \(0.780300\pi\)
\(770\) 0 0
\(771\) − 3.57357e11i − 1.01131i
\(772\) 0 0
\(773\) 2.77304e10 0.0776673 0.0388336 0.999246i \(-0.487636\pi\)
0.0388336 + 0.999246i \(0.487636\pi\)
\(774\) 0 0
\(775\) − 8.08767e10i − 0.224190i
\(776\) 0 0
\(777\) 3.66829e10 0.100642
\(778\) 0 0
\(779\) − 6.08955e11i − 1.65362i
\(780\) 0 0
\(781\) −1.30121e11 −0.349738
\(782\) 0 0
\(783\) 6.36126e10i 0.169237i
\(784\) 0 0
\(785\) −3.30005e11 −0.869044
\(786\) 0 0
\(787\) 1.92830e11i 0.502662i 0.967901 + 0.251331i \(0.0808683\pi\)
−0.967901 + 0.251331i \(0.919132\pi\)
\(788\) 0 0
\(789\) 2.16431e9 0.00558484
\(790\) 0 0
\(791\) 6.08252e11i 1.55374i
\(792\) 0 0
\(793\) −9.73795e10 −0.246249
\(794\) 0 0
\(795\) − 3.95682e11i − 0.990554i
\(796\) 0 0
\(797\) 4.96593e11 1.23074 0.615371 0.788238i \(-0.289006\pi\)
0.615371 + 0.788238i \(0.289006\pi\)
\(798\) 0 0
\(799\) 3.43702e11i 0.843325i
\(800\) 0 0
\(801\) 5.46043e10 0.132647
\(802\) 0 0
\(803\) 2.75596e11i 0.662842i
\(804\) 0 0
\(805\) 2.91800e11 0.694868
\(806\) 0 0
\(807\) 2.96319e11i 0.698658i
\(808\) 0 0
\(809\) −3.18435e11 −0.743407 −0.371704 0.928351i \(-0.621226\pi\)
−0.371704 + 0.928351i \(0.621226\pi\)
\(810\) 0 0
\(811\) − 7.47780e11i − 1.72858i −0.502991 0.864292i \(-0.667767\pi\)
0.502991 0.864292i \(-0.332233\pi\)
\(812\) 0 0
\(813\) −4.00818e11 −0.917456
\(814\) 0 0
\(815\) − 4.56349e10i − 0.103435i
\(816\) 0 0
\(817\) −6.24937e11 −1.40265
\(818\) 0 0
\(819\) 6.33320e10i 0.140763i
\(820\) 0 0
\(821\) −2.17778e11 −0.479338 −0.239669 0.970855i \(-0.577039\pi\)
−0.239669 + 0.970855i \(0.577039\pi\)
\(822\) 0 0
\(823\) 6.07755e11i 1.32474i 0.749179 + 0.662368i \(0.230448\pi\)
−0.749179 + 0.662368i \(0.769552\pi\)
\(824\) 0 0
\(825\) 1.14009e11 0.246106
\(826\) 0 0
\(827\) 1.03109e11i 0.220431i 0.993908 + 0.110215i \(0.0351541\pi\)
−0.993908 + 0.110215i \(0.964846\pi\)
\(828\) 0 0
\(829\) 4.78415e11 1.01295 0.506474 0.862255i \(-0.330949\pi\)
0.506474 + 0.862255i \(0.330949\pi\)
\(830\) 0 0
\(831\) − 1.48992e10i − 0.0312434i
\(832\) 0 0
\(833\) −6.72685e10 −0.139711
\(834\) 0 0
\(835\) 8.80586e10i 0.181145i
\(836\) 0 0
\(837\) −3.42621e10 −0.0698092
\(838\) 0 0
\(839\) − 9.56688e10i − 0.193073i −0.995329 0.0965367i \(-0.969223\pi\)
0.995329 0.0965367i \(-0.0307765\pi\)
\(840\) 0 0
\(841\) −1.13399e11 −0.226686
\(842\) 0 0
\(843\) − 5.61031e11i − 1.11090i
\(844\) 0 0
\(845\) −5.51797e11 −1.08231
\(846\) 0 0
\(847\) 2.95076e11i 0.573325i
\(848\) 0 0
\(849\) 1.04880e11 0.201866
\(850\) 0 0
\(851\) 4.17684e10i 0.0796396i
\(852\) 0 0
\(853\) −7.23586e11 −1.36677 −0.683383 0.730060i \(-0.739492\pi\)
−0.683383 + 0.730060i \(0.739492\pi\)
\(854\) 0 0
\(855\) − 2.35544e11i − 0.440766i
\(856\) 0 0
\(857\) 7.92851e11 1.46983 0.734917 0.678157i \(-0.237221\pi\)
0.734917 + 0.678157i \(0.237221\pi\)
\(858\) 0 0
\(859\) − 2.17760e10i − 0.0399949i −0.999800 0.0199975i \(-0.993634\pi\)
0.999800 0.0199975i \(-0.00636581\pi\)
\(860\) 0 0
\(861\) 5.51903e11 1.00427
\(862\) 0 0
\(863\) 9.20359e10i 0.165926i 0.996553 + 0.0829629i \(0.0264383\pi\)
−0.996553 + 0.0829629i \(0.973562\pi\)
\(864\) 0 0
\(865\) −1.07477e12 −1.91977
\(866\) 0 0
\(867\) − 1.59937e11i − 0.283056i
\(868\) 0 0
\(869\) 3.37896e11 0.592521
\(870\) 0 0
\(871\) 2.22195e11i 0.386066i
\(872\) 0 0
\(873\) −1.52909e11 −0.263254
\(874\) 0 0
\(875\) − 3.11419e11i − 0.531267i
\(876\) 0 0
\(877\) 7.09630e11 1.19959 0.599796 0.800153i \(-0.295248\pi\)
0.599796 + 0.800153i \(0.295248\pi\)
\(878\) 0 0
\(879\) − 5.73650e10i − 0.0960929i
\(880\) 0 0
\(881\) 9.65687e11 1.60300 0.801499 0.597997i \(-0.204036\pi\)
0.801499 + 0.597997i \(0.204036\pi\)
\(882\) 0 0
\(883\) 9.29054e9i 0.0152826i 0.999971 + 0.00764131i \(0.00243233\pi\)
−0.999971 + 0.00764131i \(0.997568\pi\)
\(884\) 0 0
\(885\) −2.85798e11 −0.465892
\(886\) 0 0
\(887\) 1.20513e12i 1.94689i 0.228926 + 0.973444i \(0.426479\pi\)
−0.228926 + 0.973444i \(0.573521\pi\)
\(888\) 0 0
\(889\) −2.85867e11 −0.457674
\(890\) 0 0
\(891\) − 4.82979e10i − 0.0766333i
\(892\) 0 0
\(893\) −7.80841e11 −1.22788
\(894\) 0 0
\(895\) 1.61727e11i 0.252052i
\(896\) 0 0
\(897\) −7.21120e10 −0.111388
\(898\) 0 0
\(899\) 2.08358e11i 0.318986i
\(900\) 0 0
\(901\) −6.34620e11 −0.962973
\(902\) 0 0
\(903\) − 5.66387e11i − 0.851849i
\(904\) 0 0
\(905\) 5.81177e11 0.866391
\(906\) 0 0
\(907\) 8.50561e11i 1.25683i 0.777878 + 0.628415i \(0.216296\pi\)
−0.777878 + 0.628415i \(0.783704\pi\)
\(908\) 0 0
\(909\) 3.36877e11 0.493420
\(910\) 0 0
\(911\) 2.04501e11i 0.296908i 0.988919 + 0.148454i \(0.0474296\pi\)
−0.988919 + 0.148454i \(0.952570\pi\)
\(912\) 0 0
\(913\) −3.67641e10 −0.0529104
\(914\) 0 0
\(915\) 3.28242e11i 0.468284i
\(916\) 0 0
\(917\) −1.04356e12 −1.47585
\(918\) 0 0
\(919\) − 5.59082e11i − 0.783815i −0.920005 0.391907i \(-0.871815\pi\)
0.920005 0.391907i \(-0.128185\pi\)
\(920\) 0 0
\(921\) 5.28009e11 0.733842
\(922\) 0 0
\(923\) 1.42131e11i 0.195832i
\(924\) 0 0
\(925\) −7.21308e10 −0.0985267
\(926\) 0 0
\(927\) − 3.96861e11i − 0.537427i
\(928\) 0 0
\(929\) −8.45154e11 −1.13468 −0.567340 0.823484i \(-0.692027\pi\)
−0.567340 + 0.823484i \(0.692027\pi\)
\(930\) 0 0
\(931\) − 1.52824e11i − 0.203420i
\(932\) 0 0
\(933\) −2.28607e11 −0.301691
\(934\) 0 0
\(935\) − 4.78711e11i − 0.626364i
\(936\) 0 0
\(937\) 3.92001e10 0.0508544 0.0254272 0.999677i \(-0.491905\pi\)
0.0254272 + 0.999677i \(0.491905\pi\)
\(938\) 0 0
\(939\) − 5.21457e11i − 0.670743i
\(940\) 0 0
\(941\) −3.60504e10 −0.0459782 −0.0229891 0.999736i \(-0.507318\pi\)
−0.0229891 + 0.999736i \(0.507318\pi\)
\(942\) 0 0
\(943\) 6.28415e11i 0.794694i
\(944\) 0 0
\(945\) 2.13476e11 0.267684
\(946\) 0 0
\(947\) − 5.80923e11i − 0.722301i −0.932508 0.361151i \(-0.882384\pi\)
0.932508 0.361151i \(-0.117616\pi\)
\(948\) 0 0
\(949\) 3.01034e11 0.371151
\(950\) 0 0
\(951\) − 4.37224e11i − 0.534542i
\(952\) 0 0
\(953\) 1.45669e11 0.176602 0.0883009 0.996094i \(-0.471856\pi\)
0.0883009 + 0.996094i \(0.471856\pi\)
\(954\) 0 0
\(955\) 6.69395e11i 0.804765i
\(956\) 0 0
\(957\) −2.93714e11 −0.350168
\(958\) 0 0
\(959\) − 7.77443e11i − 0.919166i
\(960\) 0 0
\(961\) 7.40668e11 0.868420
\(962\) 0 0
\(963\) − 1.54084e11i − 0.179165i
\(964\) 0 0
\(965\) −2.09251e12 −2.41301
\(966\) 0 0
\(967\) − 1.27650e12i − 1.45988i −0.683514 0.729938i \(-0.739549\pi\)
0.683514 0.729938i \(-0.260451\pi\)
\(968\) 0 0
\(969\) −3.77780e11 −0.428493
\(970\) 0 0
\(971\) 3.14012e11i 0.353239i 0.984279 + 0.176620i \(0.0565162\pi\)
−0.984279 + 0.176620i \(0.943484\pi\)
\(972\) 0 0
\(973\) 1.49754e12 1.67081
\(974\) 0 0
\(975\) − 1.24532e11i − 0.137804i
\(976\) 0 0
\(977\) 1.03341e10 0.0113421 0.00567107 0.999984i \(-0.498195\pi\)
0.00567107 + 0.999984i \(0.498195\pi\)
\(978\) 0 0
\(979\) 2.52121e11i 0.274459i
\(980\) 0 0
\(981\) −5.60528e11 −0.605232
\(982\) 0 0
\(983\) 4.49178e11i 0.481066i 0.970641 + 0.240533i \(0.0773222\pi\)
−0.970641 + 0.240533i \(0.922678\pi\)
\(984\) 0 0
\(985\) −1.17678e12 −1.25012
\(986\) 0 0
\(987\) − 7.07685e11i − 0.745712i
\(988\) 0 0
\(989\) 6.44908e11 0.674082
\(990\) 0 0
\(991\) 1.52082e12i 1.57682i 0.615149 + 0.788411i \(0.289096\pi\)
−0.615149 + 0.788411i \(0.710904\pi\)
\(992\) 0 0
\(993\) −7.76933e11 −0.799073
\(994\) 0 0
\(995\) − 1.09016e11i − 0.111223i
\(996\) 0 0
\(997\) −2.35823e11 −0.238675 −0.119337 0.992854i \(-0.538077\pi\)
−0.119337 + 0.992854i \(0.538077\pi\)
\(998\) 0 0
\(999\) 3.05570e10i 0.0306796i
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 384.9.g.a.127.26 yes 32
4.3 odd 2 inner 384.9.g.a.127.25 32
8.3 odd 2 384.9.g.b.127.8 yes 32
8.5 even 2 384.9.g.b.127.7 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.9.g.a.127.25 32 4.3 odd 2 inner
384.9.g.a.127.26 yes 32 1.1 even 1 trivial
384.9.g.b.127.7 yes 32 8.5 even 2
384.9.g.b.127.8 yes 32 8.3 odd 2