Properties

Label 192.9.g.d.127.3
Level $192$
Weight $9$
Character 192.127
Analytic conductor $78.217$
Analytic rank $0$
Dimension $8$
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] = [192,9,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, 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("192.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3468738816.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.3
Root \(2.64070 + 2.64070i\) of defining polynomial
Character \(\chi\) \(=\) 192.127
Dual form 192.9.g.d.127.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-46.7654i q^{3} +149.264 q^{5} -3207.68i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q-46.7654i q^{3} +149.264 q^{5} -3207.68i q^{7} -2187.00 q^{9} -3235.02i q^{11} -176.686 q^{13} -6980.37i q^{15} +80443.0 q^{17} -239555. i q^{19} -150008. q^{21} -85338.9i q^{23} -368345. q^{25} +102276. i q^{27} +310329. q^{29} +494114. i q^{31} -151287. q^{33} -478790. i q^{35} +855708. q^{37} +8262.78i q^{39} -1.41237e6 q^{41} -2.15193e6i q^{43} -326439. q^{45} +1.10493e6i q^{47} -4.52442e6 q^{49} -3.76194e6i q^{51} +1.47339e7 q^{53} -482870. i q^{55} -1.12029e7 q^{57} -1.62594e7i q^{59} -9.65310e6 q^{61} +7.01520e6i q^{63} -26372.7 q^{65} -2.25759e7i q^{67} -3.99091e6 q^{69} +3.04163e7i q^{71} -4.01565e7 q^{73} +1.72258e7i q^{75} -1.03769e7 q^{77} +4.96729e7i q^{79} +4.78297e6 q^{81} +4.44132e7i q^{83} +1.20072e7 q^{85} -1.45126e7i q^{87} -1.11247e8 q^{89} +566752. i q^{91} +2.31074e7 q^{93} -3.57569e7i q^{95} +3.96005e6 q^{97} +7.07498e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 560 q^{5} - 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 560 q^{5} - 17496 q^{9} + 54256 q^{13} + 436176 q^{17} - 54432 q^{21} + 2456472 q^{25} - 195952 q^{29} - 1065312 q^{33} + 7023408 q^{37} - 12035120 q^{41} + 1224720 q^{45} - 9602040 q^{49} - 27342192 q^{53} + 2744928 q^{57} - 50803280 q^{61} + 34049888 q^{65} + 26687232 q^{69} + 59541648 q^{73} + 178489472 q^{77} + 38263752 q^{81} - 29428704 q^{85} - 170794992 q^{89} + 121951008 q^{93} + 272647184 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 149.264 0.238822 0.119411 0.992845i \(-0.461899\pi\)
0.119411 + 0.992845i \(0.461899\pi\)
\(6\) 0 0
\(7\) − 3207.68i − 1.33598i −0.744171 0.667989i \(-0.767155\pi\)
0.744171 0.667989i \(-0.232845\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) − 3235.02i − 0.220956i −0.993879 0.110478i \(-0.964762\pi\)
0.993879 0.110478i \(-0.0352382\pi\)
\(12\) 0 0
\(13\) −176.686 −0.00618626 −0.00309313 0.999995i \(-0.500985\pi\)
−0.00309313 + 0.999995i \(0.500985\pi\)
\(14\) 0 0
\(15\) − 6980.37i − 0.137884i
\(16\) 0 0
\(17\) 80443.0 0.963146 0.481573 0.876406i \(-0.340066\pi\)
0.481573 + 0.876406i \(0.340066\pi\)
\(18\) 0 0
\(19\) − 239555.i − 1.83819i −0.394030 0.919097i \(-0.628920\pi\)
0.394030 0.919097i \(-0.371080\pi\)
\(20\) 0 0
\(21\) −150008. −0.771327
\(22\) 0 0
\(23\) − 85338.9i − 0.304955i −0.988307 0.152478i \(-0.951275\pi\)
0.988307 0.152478i \(-0.0487252\pi\)
\(24\) 0 0
\(25\) −368345. −0.942964
\(26\) 0 0
\(27\) 102276.i 0.192450i
\(28\) 0 0
\(29\) 310329. 0.438763 0.219382 0.975639i \(-0.429596\pi\)
0.219382 + 0.975639i \(0.429596\pi\)
\(30\) 0 0
\(31\) 494114.i 0.535033i 0.963553 + 0.267516i \(0.0862029\pi\)
−0.963553 + 0.267516i \(0.913797\pi\)
\(32\) 0 0
\(33\) −151287. −0.127569
\(34\) 0 0
\(35\) − 478790.i − 0.319060i
\(36\) 0 0
\(37\) 855708. 0.456582 0.228291 0.973593i \(-0.426686\pi\)
0.228291 + 0.973593i \(0.426686\pi\)
\(38\) 0 0
\(39\) 8262.78i 0.00357164i
\(40\) 0 0
\(41\) −1.41237e6 −0.499818 −0.249909 0.968269i \(-0.580401\pi\)
−0.249909 + 0.968269i \(0.580401\pi\)
\(42\) 0 0
\(43\) − 2.15193e6i − 0.629439i −0.949185 0.314720i \(-0.898090\pi\)
0.949185 0.314720i \(-0.101910\pi\)
\(44\) 0 0
\(45\) −326439. −0.0796072
\(46\) 0 0
\(47\) 1.10493e6i 0.226434i 0.993570 + 0.113217i \(0.0361156\pi\)
−0.993570 + 0.113217i \(0.963884\pi\)
\(48\) 0 0
\(49\) −4.52442e6 −0.784835
\(50\) 0 0
\(51\) − 3.76194e6i − 0.556073i
\(52\) 0 0
\(53\) 1.47339e7 1.86730 0.933651 0.358184i \(-0.116604\pi\)
0.933651 + 0.358184i \(0.116604\pi\)
\(54\) 0 0
\(55\) − 482870.i − 0.0527691i
\(56\) 0 0
\(57\) −1.12029e7 −1.06128
\(58\) 0 0
\(59\) − 1.62594e7i − 1.34183i −0.741535 0.670914i \(-0.765902\pi\)
0.741535 0.670914i \(-0.234098\pi\)
\(60\) 0 0
\(61\) −9.65310e6 −0.697184 −0.348592 0.937275i \(-0.613340\pi\)
−0.348592 + 0.937275i \(0.613340\pi\)
\(62\) 0 0
\(63\) 7.01520e6i 0.445326i
\(64\) 0 0
\(65\) −26372.7 −0.00147741
\(66\) 0 0
\(67\) − 2.25759e7i − 1.12033i −0.828381 0.560164i \(-0.810738\pi\)
0.828381 0.560164i \(-0.189262\pi\)
\(68\) 0 0
\(69\) −3.99091e6 −0.176066
\(70\) 0 0
\(71\) 3.04163e7i 1.19694i 0.801144 + 0.598472i \(0.204225\pi\)
−0.801144 + 0.598472i \(0.795775\pi\)
\(72\) 0 0
\(73\) −4.01565e7 −1.41405 −0.707024 0.707190i \(-0.749963\pi\)
−0.707024 + 0.707190i \(0.749963\pi\)
\(74\) 0 0
\(75\) 1.72258e7i 0.544421i
\(76\) 0 0
\(77\) −1.03769e7 −0.295192
\(78\) 0 0
\(79\) 4.96729e7i 1.27530i 0.770328 + 0.637648i \(0.220092\pi\)
−0.770328 + 0.637648i \(0.779908\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) 4.44132e7i 0.935836i 0.883772 + 0.467918i \(0.154996\pi\)
−0.883772 + 0.467918i \(0.845004\pi\)
\(84\) 0 0
\(85\) 1.20072e7 0.230020
\(86\) 0 0
\(87\) − 1.45126e7i − 0.253320i
\(88\) 0 0
\(89\) −1.11247e8 −1.77307 −0.886537 0.462659i \(-0.846896\pi\)
−0.886537 + 0.462659i \(0.846896\pi\)
\(90\) 0 0
\(91\) 566752.i 0.00826470i
\(92\) 0 0
\(93\) 2.31074e7 0.308901
\(94\) 0 0
\(95\) − 3.57569e7i − 0.439001i
\(96\) 0 0
\(97\) 3.96005e6 0.0447316 0.0223658 0.999750i \(-0.492880\pi\)
0.0223658 + 0.999750i \(0.492880\pi\)
\(98\) 0 0
\(99\) 7.07498e6i 0.0736520i
\(100\) 0 0
\(101\) −1.74661e8 −1.67846 −0.839229 0.543778i \(-0.816993\pi\)
−0.839229 + 0.543778i \(0.816993\pi\)
\(102\) 0 0
\(103\) − 1.11312e7i − 0.0988997i −0.998777 0.0494498i \(-0.984253\pi\)
0.998777 0.0494498i \(-0.0157468\pi\)
\(104\) 0 0
\(105\) −2.23908e7 −0.184210
\(106\) 0 0
\(107\) − 2.58612e7i − 0.197294i −0.995122 0.0986471i \(-0.968548\pi\)
0.995122 0.0986471i \(-0.0314515\pi\)
\(108\) 0 0
\(109\) −2.23824e8 −1.58562 −0.792812 0.609466i \(-0.791384\pi\)
−0.792812 + 0.609466i \(0.791384\pi\)
\(110\) 0 0
\(111\) − 4.00175e7i − 0.263608i
\(112\) 0 0
\(113\) −1.13591e8 −0.696675 −0.348338 0.937369i \(-0.613254\pi\)
−0.348338 + 0.937369i \(0.613254\pi\)
\(114\) 0 0
\(115\) − 1.27380e7i − 0.0728299i
\(116\) 0 0
\(117\) 386412. 0.00206209
\(118\) 0 0
\(119\) − 2.58035e8i − 1.28674i
\(120\) 0 0
\(121\) 2.03894e8 0.951178
\(122\) 0 0
\(123\) 6.60499e7i 0.288570i
\(124\) 0 0
\(125\) −1.13287e8 −0.464022
\(126\) 0 0
\(127\) − 1.26585e8i − 0.486594i −0.969952 0.243297i \(-0.921771\pi\)
0.969952 0.243297i \(-0.0782290\pi\)
\(128\) 0 0
\(129\) −1.00636e8 −0.363407
\(130\) 0 0
\(131\) 2.37678e8i 0.807055i 0.914967 + 0.403528i \(0.132216\pi\)
−0.914967 + 0.403528i \(0.867784\pi\)
\(132\) 0 0
\(133\) −7.68417e8 −2.45579
\(134\) 0 0
\(135\) 1.52661e7i 0.0459613i
\(136\) 0 0
\(137\) −2.32073e8 −0.658783 −0.329392 0.944193i \(-0.606844\pi\)
−0.329392 + 0.944193i \(0.606844\pi\)
\(138\) 0 0
\(139\) − 1.04348e7i − 0.0279529i −0.999902 0.0139764i \(-0.995551\pi\)
0.999902 0.0139764i \(-0.00444898\pi\)
\(140\) 0 0
\(141\) 5.16724e7 0.130732
\(142\) 0 0
\(143\) 571581.i 0.00136689i
\(144\) 0 0
\(145\) 4.63208e7 0.104786
\(146\) 0 0
\(147\) 2.11586e8i 0.453125i
\(148\) 0 0
\(149\) −4.71028e8 −0.955656 −0.477828 0.878453i \(-0.658576\pi\)
−0.477828 + 0.878453i \(0.658576\pi\)
\(150\) 0 0
\(151\) 6.01838e8i 1.15764i 0.815457 + 0.578818i \(0.196486\pi\)
−0.815457 + 0.578818i \(0.803514\pi\)
\(152\) 0 0
\(153\) −1.75929e8 −0.321049
\(154\) 0 0
\(155\) 7.37532e7i 0.127777i
\(156\) 0 0
\(157\) −4.87702e7 −0.0802706 −0.0401353 0.999194i \(-0.512779\pi\)
−0.0401353 + 0.999194i \(0.512779\pi\)
\(158\) 0 0
\(159\) − 6.89037e8i − 1.07809i
\(160\) 0 0
\(161\) −2.73740e8 −0.407413
\(162\) 0 0
\(163\) − 7.37869e8i − 1.04527i −0.852556 0.522635i \(-0.824949\pi\)
0.852556 0.522635i \(-0.175051\pi\)
\(164\) 0 0
\(165\) −2.25816e7 −0.0304662
\(166\) 0 0
\(167\) − 1.45451e9i − 1.87004i −0.354600 0.935018i \(-0.615383\pi\)
0.354600 0.935018i \(-0.384617\pi\)
\(168\) 0 0
\(169\) −8.15700e8 −0.999962
\(170\) 0 0
\(171\) 5.23908e8i 0.612732i
\(172\) 0 0
\(173\) 1.13360e9 1.26554 0.632768 0.774341i \(-0.281919\pi\)
0.632768 + 0.774341i \(0.281919\pi\)
\(174\) 0 0
\(175\) 1.18153e9i 1.25978i
\(176\) 0 0
\(177\) −7.60378e8 −0.774705
\(178\) 0 0
\(179\) − 3.48018e8i − 0.338992i −0.985531 0.169496i \(-0.945786\pi\)
0.985531 0.169496i \(-0.0542140\pi\)
\(180\) 0 0
\(181\) −1.15025e9 −1.07172 −0.535858 0.844308i \(-0.680012\pi\)
−0.535858 + 0.844308i \(0.680012\pi\)
\(182\) 0 0
\(183\) 4.51431e8i 0.402519i
\(184\) 0 0
\(185\) 1.27726e8 0.109042
\(186\) 0 0
\(187\) − 2.60234e8i − 0.212813i
\(188\) 0 0
\(189\) 3.28068e8 0.257109
\(190\) 0 0
\(191\) 1.71289e9i 1.28705i 0.765424 + 0.643527i \(0.222529\pi\)
−0.765424 + 0.643527i \(0.777471\pi\)
\(192\) 0 0
\(193\) 1.78226e9 1.28452 0.642261 0.766486i \(-0.277996\pi\)
0.642261 + 0.766486i \(0.277996\pi\)
\(194\) 0 0
\(195\) 1.23333e6i 0 0.000852985i
\(196\) 0 0
\(197\) 2.49946e9 1.65951 0.829756 0.558126i \(-0.188479\pi\)
0.829756 + 0.558126i \(0.188479\pi\)
\(198\) 0 0
\(199\) 2.17140e9i 1.38461i 0.721604 + 0.692306i \(0.243405\pi\)
−0.721604 + 0.692306i \(0.756595\pi\)
\(200\) 0 0
\(201\) −1.05577e9 −0.646822
\(202\) 0 0
\(203\) − 9.95436e8i − 0.586177i
\(204\) 0 0
\(205\) −2.10815e8 −0.119367
\(206\) 0 0
\(207\) 1.86636e8i 0.101652i
\(208\) 0 0
\(209\) −7.74965e8 −0.406160
\(210\) 0 0
\(211\) − 1.98390e9i − 1.00090i −0.865766 0.500449i \(-0.833168\pi\)
0.865766 0.500449i \(-0.166832\pi\)
\(212\) 0 0
\(213\) 1.42243e9 0.691056
\(214\) 0 0
\(215\) − 3.21204e8i − 0.150324i
\(216\) 0 0
\(217\) 1.58496e9 0.714791
\(218\) 0 0
\(219\) 1.87793e9i 0.816401i
\(220\) 0 0
\(221\) −1.42131e7 −0.00595827
\(222\) 0 0
\(223\) 9.04205e8i 0.365635i 0.983147 + 0.182817i \(0.0585217\pi\)
−0.983147 + 0.182817i \(0.941478\pi\)
\(224\) 0 0
\(225\) 8.05571e8 0.314321
\(226\) 0 0
\(227\) 1.18796e9i 0.447402i 0.974658 + 0.223701i \(0.0718138\pi\)
−0.974658 + 0.223701i \(0.928186\pi\)
\(228\) 0 0
\(229\) 2.73614e9 0.994940 0.497470 0.867481i \(-0.334262\pi\)
0.497470 + 0.867481i \(0.334262\pi\)
\(230\) 0 0
\(231\) 4.85280e8i 0.170429i
\(232\) 0 0
\(233\) −3.83321e9 −1.30059 −0.650293 0.759683i \(-0.725354\pi\)
−0.650293 + 0.759683i \(0.725354\pi\)
\(234\) 0 0
\(235\) 1.64925e8i 0.0540775i
\(236\) 0 0
\(237\) 2.32297e9 0.736292
\(238\) 0 0
\(239\) − 5.01203e9i − 1.53611i −0.640385 0.768054i \(-0.721225\pi\)
0.640385 0.768054i \(-0.278775\pi\)
\(240\) 0 0
\(241\) 4.52278e9 1.34072 0.670359 0.742037i \(-0.266140\pi\)
0.670359 + 0.742037i \(0.266140\pi\)
\(242\) 0 0
\(243\) − 2.23677e8i − 0.0641500i
\(244\) 0 0
\(245\) −6.75331e8 −0.187436
\(246\) 0 0
\(247\) 4.23260e7i 0.0113716i
\(248\) 0 0
\(249\) 2.07700e9 0.540305
\(250\) 0 0
\(251\) − 4.93482e9i − 1.24330i −0.783294 0.621651i \(-0.786462\pi\)
0.783294 0.621651i \(-0.213538\pi\)
\(252\) 0 0
\(253\) −2.76073e8 −0.0673816
\(254\) 0 0
\(255\) − 5.61521e8i − 0.132802i
\(256\) 0 0
\(257\) 5.31230e9 1.21773 0.608864 0.793275i \(-0.291626\pi\)
0.608864 + 0.793275i \(0.291626\pi\)
\(258\) 0 0
\(259\) − 2.74484e9i − 0.609983i
\(260\) 0 0
\(261\) −6.78689e8 −0.146254
\(262\) 0 0
\(263\) − 9.22967e9i − 1.92914i −0.263830 0.964569i \(-0.584986\pi\)
0.263830 0.964569i \(-0.415014\pi\)
\(264\) 0 0
\(265\) 2.19924e9 0.445952
\(266\) 0 0
\(267\) 5.20249e9i 1.02368i
\(268\) 0 0
\(269\) 2.92523e9 0.558664 0.279332 0.960195i \(-0.409887\pi\)
0.279332 + 0.960195i \(0.409887\pi\)
\(270\) 0 0
\(271\) 2.04474e9i 0.379106i 0.981870 + 0.189553i \(0.0607039\pi\)
−0.981870 + 0.189553i \(0.939296\pi\)
\(272\) 0 0
\(273\) 2.65044e7 0.00477163
\(274\) 0 0
\(275\) 1.19160e9i 0.208354i
\(276\) 0 0
\(277\) 3.15752e9 0.536323 0.268162 0.963374i \(-0.413584\pi\)
0.268162 + 0.963374i \(0.413584\pi\)
\(278\) 0 0
\(279\) − 1.08063e9i − 0.178344i
\(280\) 0 0
\(281\) 1.16146e10 1.86286 0.931431 0.363917i \(-0.118561\pi\)
0.931431 + 0.363917i \(0.118561\pi\)
\(282\) 0 0
\(283\) 9.59477e9i 1.49585i 0.663782 + 0.747926i \(0.268950\pi\)
−0.663782 + 0.747926i \(0.731050\pi\)
\(284\) 0 0
\(285\) −1.67218e9 −0.253457
\(286\) 0 0
\(287\) 4.53042e9i 0.667746i
\(288\) 0 0
\(289\) −5.04689e8 −0.0723490
\(290\) 0 0
\(291\) − 1.85193e8i − 0.0258258i
\(292\) 0 0
\(293\) −1.10305e9 −0.149667 −0.0748335 0.997196i \(-0.523843\pi\)
−0.0748335 + 0.997196i \(0.523843\pi\)
\(294\) 0 0
\(295\) − 2.42694e9i − 0.320458i
\(296\) 0 0
\(297\) 3.30864e8 0.0425230
\(298\) 0 0
\(299\) 1.50782e7i 0.00188653i
\(300\) 0 0
\(301\) −6.90270e9 −0.840916
\(302\) 0 0
\(303\) 8.16809e9i 0.969058i
\(304\) 0 0
\(305\) −1.44086e9 −0.166503
\(306\) 0 0
\(307\) 9.19662e9i 1.03532i 0.855586 + 0.517660i \(0.173197\pi\)
−0.855586 + 0.517660i \(0.826803\pi\)
\(308\) 0 0
\(309\) −5.20557e8 −0.0570997
\(310\) 0 0
\(311\) − 7.75359e9i − 0.828822i −0.910090 0.414411i \(-0.863988\pi\)
0.910090 0.414411i \(-0.136012\pi\)
\(312\) 0 0
\(313\) 1.32776e10 1.38339 0.691693 0.722192i \(-0.256865\pi\)
0.691693 + 0.722192i \(0.256865\pi\)
\(314\) 0 0
\(315\) 1.04711e9i 0.106353i
\(316\) 0 0
\(317\) −3.22186e8 −0.0319058 −0.0159529 0.999873i \(-0.505078\pi\)
−0.0159529 + 0.999873i \(0.505078\pi\)
\(318\) 0 0
\(319\) − 1.00392e9i − 0.0969473i
\(320\) 0 0
\(321\) −1.20941e9 −0.113908
\(322\) 0 0
\(323\) − 1.92705e10i − 1.77045i
\(324\) 0 0
\(325\) 6.50814e7 0.00583342
\(326\) 0 0
\(327\) 1.04672e10i 0.915460i
\(328\) 0 0
\(329\) 3.54426e9 0.302511
\(330\) 0 0
\(331\) 1.16547e10i 0.970931i 0.874256 + 0.485466i \(0.161350\pi\)
−0.874256 + 0.485466i \(0.838650\pi\)
\(332\) 0 0
\(333\) −1.87143e9 −0.152194
\(334\) 0 0
\(335\) − 3.36976e9i − 0.267559i
\(336\) 0 0
\(337\) −2.36011e10 −1.82983 −0.914917 0.403641i \(-0.867744\pi\)
−0.914917 + 0.403641i \(0.867744\pi\)
\(338\) 0 0
\(339\) 5.31213e9i 0.402226i
\(340\) 0 0
\(341\) 1.59847e9 0.118219
\(342\) 0 0
\(343\) − 3.97875e9i − 0.287455i
\(344\) 0 0
\(345\) −5.95697e8 −0.0420484
\(346\) 0 0
\(347\) − 2.40544e9i − 0.165912i −0.996553 0.0829559i \(-0.973564\pi\)
0.996553 0.0829559i \(-0.0264361\pi\)
\(348\) 0 0
\(349\) 2.05149e10 1.38283 0.691415 0.722458i \(-0.256988\pi\)
0.691415 + 0.722458i \(0.256988\pi\)
\(350\) 0 0
\(351\) − 1.80707e7i − 0.00119055i
\(352\) 0 0
\(353\) −1.10340e10 −0.710612 −0.355306 0.934750i \(-0.615623\pi\)
−0.355306 + 0.934750i \(0.615623\pi\)
\(354\) 0 0
\(355\) 4.54005e9i 0.285856i
\(356\) 0 0
\(357\) −1.20671e10 −0.742901
\(358\) 0 0
\(359\) − 1.81015e10i − 1.08978i −0.838509 0.544888i \(-0.816572\pi\)
0.838509 0.544888i \(-0.183428\pi\)
\(360\) 0 0
\(361\) −4.04032e10 −2.37896
\(362\) 0 0
\(363\) − 9.53516e9i − 0.549163i
\(364\) 0 0
\(365\) −5.99390e9 −0.337705
\(366\) 0 0
\(367\) 2.78190e10i 1.53348i 0.641959 + 0.766739i \(0.278122\pi\)
−0.641959 + 0.766739i \(0.721878\pi\)
\(368\) 0 0
\(369\) 3.08885e9 0.166606
\(370\) 0 0
\(371\) − 4.72617e10i − 2.49467i
\(372\) 0 0
\(373\) 8.10899e8 0.0418920 0.0209460 0.999781i \(-0.493332\pi\)
0.0209460 + 0.999781i \(0.493332\pi\)
\(374\) 0 0
\(375\) 5.29789e9i 0.267903i
\(376\) 0 0
\(377\) −5.48307e7 −0.00271430
\(378\) 0 0
\(379\) 6.25965e9i 0.303384i 0.988428 + 0.151692i \(0.0484722\pi\)
−0.988428 + 0.151692i \(0.951528\pi\)
\(380\) 0 0
\(381\) −5.91979e9 −0.280935
\(382\) 0 0
\(383\) 1.50020e10i 0.697195i 0.937272 + 0.348598i \(0.113342\pi\)
−0.937272 + 0.348598i \(0.886658\pi\)
\(384\) 0 0
\(385\) −1.54889e9 −0.0704983
\(386\) 0 0
\(387\) 4.70627e9i 0.209813i
\(388\) 0 0
\(389\) −4.77166e9 −0.208387 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(390\) 0 0
\(391\) − 6.86492e9i − 0.293716i
\(392\) 0 0
\(393\) 1.11151e10 0.465954
\(394\) 0 0
\(395\) 7.41435e9i 0.304568i
\(396\) 0 0
\(397\) −1.06262e10 −0.427777 −0.213889 0.976858i \(-0.568613\pi\)
−0.213889 + 0.976858i \(0.568613\pi\)
\(398\) 0 0
\(399\) 3.59353e10i 1.41785i
\(400\) 0 0
\(401\) −2.34131e10 −0.905485 −0.452743 0.891641i \(-0.649554\pi\)
−0.452743 + 0.891641i \(0.649554\pi\)
\(402\) 0 0
\(403\) − 8.73029e7i − 0.00330985i
\(404\) 0 0
\(405\) 7.13923e8 0.0265357
\(406\) 0 0
\(407\) − 2.76823e9i − 0.100884i
\(408\) 0 0
\(409\) −3.53006e10 −1.26150 −0.630752 0.775985i \(-0.717253\pi\)
−0.630752 + 0.775985i \(0.717253\pi\)
\(410\) 0 0
\(411\) 1.08530e10i 0.380349i
\(412\) 0 0
\(413\) −5.21551e10 −1.79265
\(414\) 0 0
\(415\) 6.62927e9i 0.223498i
\(416\) 0 0
\(417\) −4.87989e8 −0.0161386
\(418\) 0 0
\(419\) − 5.73589e10i − 1.86099i −0.366299 0.930497i \(-0.619375\pi\)
0.366299 0.930497i \(-0.380625\pi\)
\(420\) 0 0
\(421\) 4.22283e10 1.34423 0.672117 0.740445i \(-0.265385\pi\)
0.672117 + 0.740445i \(0.265385\pi\)
\(422\) 0 0
\(423\) − 2.41648e9i − 0.0754782i
\(424\) 0 0
\(425\) −2.96308e10 −0.908213
\(426\) 0 0
\(427\) 3.09641e10i 0.931422i
\(428\) 0 0
\(429\) 2.67302e7 0.000789175 0
\(430\) 0 0
\(431\) − 4.42681e10i − 1.28287i −0.767179 0.641433i \(-0.778340\pi\)
0.767179 0.641433i \(-0.221660\pi\)
\(432\) 0 0
\(433\) 1.90037e8 0.00540614 0.00270307 0.999996i \(-0.499140\pi\)
0.00270307 + 0.999996i \(0.499140\pi\)
\(434\) 0 0
\(435\) − 2.16621e9i − 0.0604983i
\(436\) 0 0
\(437\) −2.04434e10 −0.560567
\(438\) 0 0
\(439\) − 6.76040e10i − 1.82018i −0.414409 0.910091i \(-0.636012\pi\)
0.414409 0.910091i \(-0.363988\pi\)
\(440\) 0 0
\(441\) 9.89490e9 0.261612
\(442\) 0 0
\(443\) − 1.35521e10i − 0.351877i −0.984401 0.175939i \(-0.943704\pi\)
0.984401 0.175939i \(-0.0562960\pi\)
\(444\) 0 0
\(445\) −1.66051e10 −0.423448
\(446\) 0 0
\(447\) 2.20278e10i 0.551748i
\(448\) 0 0
\(449\) 1.25951e10 0.309897 0.154948 0.987923i \(-0.450479\pi\)
0.154948 + 0.987923i \(0.450479\pi\)
\(450\) 0 0
\(451\) 4.56903e9i 0.110438i
\(452\) 0 0
\(453\) 2.81452e10 0.668361
\(454\) 0 0
\(455\) 8.45954e7i 0.00197379i
\(456\) 0 0
\(457\) 3.90759e10 0.895869 0.447935 0.894066i \(-0.352160\pi\)
0.447935 + 0.894066i \(0.352160\pi\)
\(458\) 0 0
\(459\) 8.22737e9i 0.185358i
\(460\) 0 0
\(461\) −2.22395e9 −0.0492404 −0.0246202 0.999697i \(-0.507838\pi\)
−0.0246202 + 0.999697i \(0.507838\pi\)
\(462\) 0 0
\(463\) 1.65068e10i 0.359203i 0.983739 + 0.179602i \(0.0574809\pi\)
−0.983739 + 0.179602i \(0.942519\pi\)
\(464\) 0 0
\(465\) 3.44910e9 0.0737723
\(466\) 0 0
\(467\) − 7.20146e9i − 0.151409i −0.997130 0.0757047i \(-0.975879\pi\)
0.997130 0.0757047i \(-0.0241206\pi\)
\(468\) 0 0
\(469\) −7.24162e10 −1.49673
\(470\) 0 0
\(471\) 2.28076e9i 0.0463442i
\(472\) 0 0
\(473\) −6.96152e9 −0.139078
\(474\) 0 0
\(475\) 8.82391e10i 1.73335i
\(476\) 0 0
\(477\) −3.22231e10 −0.622434
\(478\) 0 0
\(479\) − 2.84033e10i − 0.539543i −0.962924 0.269772i \(-0.913052\pi\)
0.962924 0.269772i \(-0.0869482\pi\)
\(480\) 0 0
\(481\) −1.51191e8 −0.00282453
\(482\) 0 0
\(483\) 1.28016e10i 0.235220i
\(484\) 0 0
\(485\) 5.91092e8 0.0106829
\(486\) 0 0
\(487\) − 6.74024e9i − 0.119828i −0.998204 0.0599142i \(-0.980917\pi\)
0.998204 0.0599142i \(-0.0190827\pi\)
\(488\) 0 0
\(489\) −3.45067e10 −0.603487
\(490\) 0 0
\(491\) 7.72506e10i 1.32916i 0.747219 + 0.664578i \(0.231389\pi\)
−0.747219 + 0.664578i \(0.768611\pi\)
\(492\) 0 0
\(493\) 2.49638e10 0.422593
\(494\) 0 0
\(495\) 1.05604e9i 0.0175897i
\(496\) 0 0
\(497\) 9.75659e10 1.59909
\(498\) 0 0
\(499\) − 2.02155e10i − 0.326049i −0.986622 0.163024i \(-0.947875\pi\)
0.986622 0.163024i \(-0.0521249\pi\)
\(500\) 0 0
\(501\) −6.80206e10 −1.07967
\(502\) 0 0
\(503\) − 6.27908e10i − 0.980899i −0.871470 0.490449i \(-0.836833\pi\)
0.871470 0.490449i \(-0.163167\pi\)
\(504\) 0 0
\(505\) −2.60705e10 −0.400852
\(506\) 0 0
\(507\) 3.81465e10i 0.577328i
\(508\) 0 0
\(509\) 7.85179e10 1.16976 0.584881 0.811119i \(-0.301141\pi\)
0.584881 + 0.811119i \(0.301141\pi\)
\(510\) 0 0
\(511\) 1.28809e11i 1.88913i
\(512\) 0 0
\(513\) 2.45007e10 0.353761
\(514\) 0 0
\(515\) − 1.66149e9i − 0.0236194i
\(516\) 0 0
\(517\) 3.57446e9 0.0500320
\(518\) 0 0
\(519\) − 5.30131e10i − 0.730658i
\(520\) 0 0
\(521\) 1.08134e11 1.46761 0.733806 0.679359i \(-0.237742\pi\)
0.733806 + 0.679359i \(0.237742\pi\)
\(522\) 0 0
\(523\) − 1.90750e8i − 0.00254952i −0.999999 0.00127476i \(-0.999594\pi\)
0.999999 0.00127476i \(-0.000405769\pi\)
\(524\) 0 0
\(525\) 5.52549e10 0.727334
\(526\) 0 0
\(527\) 3.97480e10i 0.515315i
\(528\) 0 0
\(529\) 7.10282e10 0.907002
\(530\) 0 0
\(531\) 3.55594e10i 0.447276i
\(532\) 0 0
\(533\) 2.49545e8 0.00309200
\(534\) 0 0
\(535\) − 3.86014e9i − 0.0471181i
\(536\) 0 0
\(537\) −1.62752e10 −0.195717
\(538\) 0 0
\(539\) 1.46366e10i 0.173414i
\(540\) 0 0
\(541\) −8.80067e9 −0.102737 −0.0513685 0.998680i \(-0.516358\pi\)
−0.0513685 + 0.998680i \(0.516358\pi\)
\(542\) 0 0
\(543\) 5.37921e10i 0.618755i
\(544\) 0 0
\(545\) −3.34087e10 −0.378681
\(546\) 0 0
\(547\) − 1.38634e11i − 1.54853i −0.632862 0.774265i \(-0.718120\pi\)
0.632862 0.774265i \(-0.281880\pi\)
\(548\) 0 0
\(549\) 2.11113e10 0.232395
\(550\) 0 0
\(551\) − 7.43409e10i − 0.806532i
\(552\) 0 0
\(553\) 1.59335e11 1.70377
\(554\) 0 0
\(555\) − 5.97315e9i − 0.0629552i
\(556\) 0 0
\(557\) 9.91532e10 1.03012 0.515058 0.857156i \(-0.327771\pi\)
0.515058 + 0.857156i \(0.327771\pi\)
\(558\) 0 0
\(559\) 3.80215e8i 0.00389387i
\(560\) 0 0
\(561\) −1.21699e10 −0.122868
\(562\) 0 0
\(563\) − 1.30106e11i − 1.29498i −0.762073 0.647492i \(-0.775818\pi\)
0.762073 0.647492i \(-0.224182\pi\)
\(564\) 0 0
\(565\) −1.69550e10 −0.166381
\(566\) 0 0
\(567\) − 1.53422e10i − 0.148442i
\(568\) 0 0
\(569\) −7.48058e10 −0.713651 −0.356826 0.934171i \(-0.616141\pi\)
−0.356826 + 0.934171i \(0.616141\pi\)
\(570\) 0 0
\(571\) − 4.89520e10i − 0.460496i −0.973132 0.230248i \(-0.926046\pi\)
0.973132 0.230248i \(-0.0739537\pi\)
\(572\) 0 0
\(573\) 8.01040e10 0.743081
\(574\) 0 0
\(575\) 3.14342e10i 0.287562i
\(576\) 0 0
\(577\) −8.33708e9 −0.0752161 −0.0376080 0.999293i \(-0.511974\pi\)
−0.0376080 + 0.999293i \(0.511974\pi\)
\(578\) 0 0
\(579\) − 8.33480e10i − 0.741619i
\(580\) 0 0
\(581\) 1.42463e11 1.25025
\(582\) 0 0
\(583\) − 4.76644e10i − 0.412591i
\(584\) 0 0
\(585\) 5.76772e7 0.000492471 0
\(586\) 0 0
\(587\) 1.05583e11i 0.889289i 0.895707 + 0.444644i \(0.146670\pi\)
−0.895707 + 0.444644i \(0.853330\pi\)
\(588\) 0 0
\(589\) 1.18368e11 0.983494
\(590\) 0 0
\(591\) − 1.16888e11i − 0.958120i
\(592\) 0 0
\(593\) −1.10652e11 −0.894832 −0.447416 0.894326i \(-0.647656\pi\)
−0.447416 + 0.894326i \(0.647656\pi\)
\(594\) 0 0
\(595\) − 3.85153e10i − 0.307302i
\(596\) 0 0
\(597\) 1.01547e11 0.799407
\(598\) 0 0
\(599\) − 8.69285e10i − 0.675235i −0.941283 0.337617i \(-0.890379\pi\)
0.941283 0.337617i \(-0.109621\pi\)
\(600\) 0 0
\(601\) 3.34335e10 0.256262 0.128131 0.991757i \(-0.459102\pi\)
0.128131 + 0.991757i \(0.459102\pi\)
\(602\) 0 0
\(603\) 4.93734e10i 0.373443i
\(604\) 0 0
\(605\) 3.04339e10 0.227162
\(606\) 0 0
\(607\) 1.12699e11i 0.830167i 0.909783 + 0.415084i \(0.136248\pi\)
−0.909783 + 0.415084i \(0.863752\pi\)
\(608\) 0 0
\(609\) −4.65519e10 −0.338430
\(610\) 0 0
\(611\) − 1.95225e8i − 0.00140078i
\(612\) 0 0
\(613\) −4.91478e10 −0.348067 −0.174033 0.984740i \(-0.555680\pi\)
−0.174033 + 0.984740i \(0.555680\pi\)
\(614\) 0 0
\(615\) 9.85884e9i 0.0689168i
\(616\) 0 0
\(617\) 2.19370e10 0.151369 0.0756845 0.997132i \(-0.475886\pi\)
0.0756845 + 0.997132i \(0.475886\pi\)
\(618\) 0 0
\(619\) − 6.88531e10i − 0.468987i −0.972118 0.234493i \(-0.924657\pi\)
0.972118 0.234493i \(-0.0753432\pi\)
\(620\) 0 0
\(621\) 8.72811e9 0.0586886
\(622\) 0 0
\(623\) 3.56844e11i 2.36879i
\(624\) 0 0
\(625\) 1.26975e11 0.832146
\(626\) 0 0
\(627\) 3.62415e10i 0.234497i
\(628\) 0 0
\(629\) 6.88357e10 0.439755
\(630\) 0 0
\(631\) 2.01724e11i 1.27245i 0.771505 + 0.636223i \(0.219504\pi\)
−0.771505 + 0.636223i \(0.780496\pi\)
\(632\) 0 0
\(633\) −9.27777e10 −0.577868
\(634\) 0 0
\(635\) − 1.88945e10i − 0.116209i
\(636\) 0 0
\(637\) 7.99400e8 0.00485519
\(638\) 0 0
\(639\) − 6.65205e10i − 0.398981i
\(640\) 0 0
\(641\) −3.11594e10 −0.184568 −0.0922841 0.995733i \(-0.529417\pi\)
−0.0922841 + 0.995733i \(0.529417\pi\)
\(642\) 0 0
\(643\) − 1.11577e11i − 0.652728i −0.945244 0.326364i \(-0.894177\pi\)
0.945244 0.326364i \(-0.105823\pi\)
\(644\) 0 0
\(645\) −1.50212e10 −0.0867895
\(646\) 0 0
\(647\) − 9.54210e10i − 0.544536i −0.962221 0.272268i \(-0.912226\pi\)
0.962221 0.272268i \(-0.0877737\pi\)
\(648\) 0 0
\(649\) −5.25995e10 −0.296485
\(650\) 0 0
\(651\) − 7.41212e10i − 0.412685i
\(652\) 0 0
\(653\) −1.57875e11 −0.868281 −0.434140 0.900845i \(-0.642948\pi\)
−0.434140 + 0.900845i \(0.642948\pi\)
\(654\) 0 0
\(655\) 3.54766e10i 0.192742i
\(656\) 0 0
\(657\) 8.78222e10 0.471349
\(658\) 0 0
\(659\) 1.05320e11i 0.558433i 0.960228 + 0.279216i \(0.0900747\pi\)
−0.960228 + 0.279216i \(0.909925\pi\)
\(660\) 0 0
\(661\) −1.99905e10 −0.104717 −0.0523585 0.998628i \(-0.516674\pi\)
−0.0523585 + 0.998628i \(0.516674\pi\)
\(662\) 0 0
\(663\) 6.64682e8i 0.00344001i
\(664\) 0 0
\(665\) −1.14697e11 −0.586495
\(666\) 0 0
\(667\) − 2.64831e10i − 0.133803i
\(668\) 0 0
\(669\) 4.22855e10 0.211099
\(670\) 0 0
\(671\) 3.12279e10i 0.154047i
\(672\) 0 0
\(673\) 1.01495e11 0.494750 0.247375 0.968920i \(-0.420432\pi\)
0.247375 + 0.968920i \(0.420432\pi\)
\(674\) 0 0
\(675\) − 3.76728e10i − 0.181474i
\(676\) 0 0
\(677\) −1.34737e10 −0.0641406 −0.0320703 0.999486i \(-0.510210\pi\)
−0.0320703 + 0.999486i \(0.510210\pi\)
\(678\) 0 0
\(679\) − 1.27026e10i − 0.0597604i
\(680\) 0 0
\(681\) 5.55553e10 0.258307
\(682\) 0 0
\(683\) 2.42663e11i 1.11512i 0.830138 + 0.557558i \(0.188262\pi\)
−0.830138 + 0.557558i \(0.811738\pi\)
\(684\) 0 0
\(685\) −3.46400e10 −0.157332
\(686\) 0 0
\(687\) − 1.27957e11i − 0.574429i
\(688\) 0 0
\(689\) −2.60327e9 −0.0115516
\(690\) 0 0
\(691\) − 2.15881e11i − 0.946896i −0.880822 0.473448i \(-0.843009\pi\)
0.880822 0.473448i \(-0.156991\pi\)
\(692\) 0 0
\(693\) 2.26943e10 0.0983974
\(694\) 0 0
\(695\) − 1.55754e9i − 0.00667575i
\(696\) 0 0
\(697\) −1.13615e11 −0.481398
\(698\) 0 0
\(699\) 1.79262e11i 0.750894i
\(700\) 0 0
\(701\) −2.45336e11 −1.01599 −0.507994 0.861361i \(-0.669613\pi\)
−0.507994 + 0.861361i \(0.669613\pi\)
\(702\) 0 0
\(703\) − 2.04989e11i − 0.839286i
\(704\) 0 0
\(705\) 7.71280e9 0.0312216
\(706\) 0 0
\(707\) 5.60257e11i 2.24238i
\(708\) 0 0
\(709\) 1.40515e10 0.0556080 0.0278040 0.999613i \(-0.491149\pi\)
0.0278040 + 0.999613i \(0.491149\pi\)
\(710\) 0 0
\(711\) − 1.08635e11i − 0.425099i
\(712\) 0 0
\(713\) 4.21672e10 0.163161
\(714\) 0 0
\(715\) 8.53162e7i 0 0.000326443i
\(716\) 0 0
\(717\) −2.34389e11 −0.886873
\(718\) 0 0
\(719\) 1.91174e11i 0.715343i 0.933848 + 0.357671i \(0.116429\pi\)
−0.933848 + 0.357671i \(0.883571\pi\)
\(720\) 0 0
\(721\) −3.57055e10 −0.132128
\(722\) 0 0
\(723\) − 2.11510e11i − 0.774064i
\(724\) 0 0
\(725\) −1.14308e11 −0.413738
\(726\) 0 0
\(727\) − 3.91555e11i − 1.40170i −0.713308 0.700850i \(-0.752804\pi\)
0.713308 0.700850i \(-0.247196\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) − 1.73107e11i − 0.606242i
\(732\) 0 0
\(733\) −5.52391e11 −1.91351 −0.956754 0.290898i \(-0.906046\pi\)
−0.956754 + 0.290898i \(0.906046\pi\)
\(734\) 0 0
\(735\) 3.15821e10i 0.108216i
\(736\) 0 0
\(737\) −7.30333e10 −0.247543
\(738\) 0 0
\(739\) 1.83932e10i 0.0616709i 0.999524 + 0.0308354i \(0.00981678\pi\)
−0.999524 + 0.0308354i \(0.990183\pi\)
\(740\) 0 0
\(741\) 1.97939e9 0.00656537
\(742\) 0 0
\(743\) 1.79242e11i 0.588145i 0.955783 + 0.294072i \(0.0950106\pi\)
−0.955783 + 0.294072i \(0.904989\pi\)
\(744\) 0 0
\(745\) −7.03073e10 −0.228231
\(746\) 0 0
\(747\) − 9.71316e10i − 0.311945i
\(748\) 0 0
\(749\) −8.29546e10 −0.263581
\(750\) 0 0
\(751\) − 3.38227e11i − 1.06328i −0.846970 0.531641i \(-0.821576\pi\)
0.846970 0.531641i \(-0.178424\pi\)
\(752\) 0 0
\(753\) −2.30779e11 −0.717821
\(754\) 0 0
\(755\) 8.98325e10i 0.276468i
\(756\) 0 0
\(757\) −3.36576e11 −1.02494 −0.512471 0.858705i \(-0.671270\pi\)
−0.512471 + 0.858705i \(0.671270\pi\)
\(758\) 0 0
\(759\) 1.29106e10i 0.0389028i
\(760\) 0 0
\(761\) 3.50280e11 1.04442 0.522212 0.852816i \(-0.325107\pi\)
0.522212 + 0.852816i \(0.325107\pi\)
\(762\) 0 0
\(763\) 7.17955e11i 2.11836i
\(764\) 0 0
\(765\) −2.62597e10 −0.0766734
\(766\) 0 0
\(767\) 2.87281e9i 0.00830090i
\(768\) 0 0
\(769\) −3.89703e11 −1.11437 −0.557184 0.830389i \(-0.688118\pi\)
−0.557184 + 0.830389i \(0.688118\pi\)
\(770\) 0 0
\(771\) − 2.48432e11i − 0.703055i
\(772\) 0 0
\(773\) 5.62526e11 1.57552 0.787761 0.615981i \(-0.211240\pi\)
0.787761 + 0.615981i \(0.211240\pi\)
\(774\) 0 0
\(775\) − 1.82005e11i − 0.504517i
\(776\) 0 0
\(777\) −1.28363e11 −0.352174
\(778\) 0 0
\(779\) 3.38340e11i 0.918763i
\(780\) 0 0
\(781\) 9.83973e10 0.264472
\(782\) 0 0
\(783\) 3.17391e10i 0.0844400i
\(784\) 0 0
\(785\) −7.27962e9 −0.0191704
\(786\) 0 0
\(787\) 7.83441e10i 0.204224i 0.994773 + 0.102112i \(0.0325600\pi\)
−0.994773 + 0.102112i \(0.967440\pi\)
\(788\) 0 0
\(789\) −4.31629e11 −1.11379
\(790\) 0 0
\(791\) 3.64364e11i 0.930742i
\(792\) 0 0
\(793\) 1.70556e9 0.00431296
\(794\) 0 0
\(795\) − 1.02848e11i − 0.257471i
\(796\) 0 0
\(797\) −4.07970e11 −1.01110 −0.505551 0.862797i \(-0.668711\pi\)
−0.505551 + 0.862797i \(0.668711\pi\)
\(798\) 0 0
\(799\) 8.88837e10i 0.218090i
\(800\) 0 0
\(801\) 2.43296e11 0.591024
\(802\) 0 0
\(803\) 1.29907e11i 0.312442i
\(804\) 0 0
\(805\) −4.08594e10 −0.0972991
\(806\) 0 0
\(807\) − 1.36799e11i − 0.322545i
\(808\) 0 0
\(809\) −6.48839e10 −0.151476 −0.0757378 0.997128i \(-0.524131\pi\)
−0.0757378 + 0.997128i \(0.524131\pi\)
\(810\) 0 0
\(811\) − 5.68236e11i − 1.31355i −0.754088 0.656773i \(-0.771921\pi\)
0.754088 0.656773i \(-0.228079\pi\)
\(812\) 0 0
\(813\) 9.56230e10 0.218877
\(814\) 0 0
\(815\) − 1.10137e11i − 0.249633i
\(816\) 0 0
\(817\) −5.15506e11 −1.15703
\(818\) 0 0
\(819\) − 1.23949e9i − 0.00275490i
\(820\) 0 0
\(821\) −8.62816e11 −1.89909 −0.949545 0.313631i \(-0.898454\pi\)
−0.949545 + 0.313631i \(0.898454\pi\)
\(822\) 0 0
\(823\) − 5.20649e10i − 0.113487i −0.998389 0.0567435i \(-0.981928\pi\)
0.998389 0.0567435i \(-0.0180717\pi\)
\(824\) 0 0
\(825\) 5.57258e10 0.120293
\(826\) 0 0
\(827\) − 6.71579e11i − 1.43574i −0.696179 0.717869i \(-0.745118\pi\)
0.696179 0.717869i \(-0.254882\pi\)
\(828\) 0 0
\(829\) 7.34136e11 1.55439 0.777193 0.629263i \(-0.216643\pi\)
0.777193 + 0.629263i \(0.216643\pi\)
\(830\) 0 0
\(831\) − 1.47662e11i − 0.309646i
\(832\) 0 0
\(833\) −3.63958e11 −0.755911
\(834\) 0 0
\(835\) − 2.17105e11i − 0.446605i
\(836\) 0 0
\(837\) −5.05359e10 −0.102967
\(838\) 0 0
\(839\) 6.60308e11i 1.33260i 0.745685 + 0.666299i \(0.232122\pi\)
−0.745685 + 0.666299i \(0.767878\pi\)
\(840\) 0 0
\(841\) −4.03942e11 −0.807487
\(842\) 0 0
\(843\) − 5.43163e11i − 1.07552i
\(844\) 0 0
\(845\) −1.21754e11 −0.238813
\(846\) 0 0
\(847\) − 6.54026e11i − 1.27075i
\(848\) 0 0
\(849\) 4.48703e11 0.863631
\(850\) 0 0
\(851\) − 7.30252e10i − 0.139237i
\(852\) 0 0
\(853\) 3.67260e11 0.693709 0.346854 0.937919i \(-0.387250\pi\)
0.346854 + 0.937919i \(0.387250\pi\)
\(854\) 0 0
\(855\) 7.82003e10i 0.146334i
\(856\) 0 0
\(857\) 5.83979e11 1.08261 0.541307 0.840825i \(-0.317930\pi\)
0.541307 + 0.840825i \(0.317930\pi\)
\(858\) 0 0
\(859\) − 4.73945e11i − 0.870473i −0.900316 0.435236i \(-0.856665\pi\)
0.900316 0.435236i \(-0.143335\pi\)
\(860\) 0 0
\(861\) 2.11867e11 0.385523
\(862\) 0 0
\(863\) 4.46554e11i 0.805065i 0.915406 + 0.402533i \(0.131870\pi\)
−0.915406 + 0.402533i \(0.868130\pi\)
\(864\) 0 0
\(865\) 1.69205e11 0.302238
\(866\) 0 0
\(867\) 2.36020e10i 0.0417707i
\(868\) 0 0
\(869\) 1.60693e11 0.281784
\(870\) 0 0
\(871\) 3.98884e9i 0.00693064i
\(872\) 0 0
\(873\) −8.66064e9 −0.0149105
\(874\) 0 0
\(875\) 3.63387e11i 0.619923i
\(876\) 0 0
\(877\) 6.58064e11 1.11242 0.556211 0.831041i \(-0.312254\pi\)
0.556211 + 0.831041i \(0.312254\pi\)
\(878\) 0 0
\(879\) 5.15847e10i 0.0864103i
\(880\) 0 0
\(881\) −1.09401e12 −1.81600 −0.908001 0.418968i \(-0.862392\pi\)
−0.908001 + 0.418968i \(0.862392\pi\)
\(882\) 0 0
\(883\) 7.41168e11i 1.21920i 0.792710 + 0.609599i \(0.208669\pi\)
−0.792710 + 0.609599i \(0.791331\pi\)
\(884\) 0 0
\(885\) −1.13497e11 −0.185016
\(886\) 0 0
\(887\) 5.47892e11i 0.885116i 0.896740 + 0.442558i \(0.145929\pi\)
−0.896740 + 0.442558i \(0.854071\pi\)
\(888\) 0 0
\(889\) −4.06044e11 −0.650079
\(890\) 0 0
\(891\) − 1.54730e10i − 0.0245507i
\(892\) 0 0
\(893\) 2.64691e11 0.416231
\(894\) 0 0
\(895\) − 5.19464e10i − 0.0809587i
\(896\) 0 0
\(897\) 7.05137e8 0.00108919
\(898\) 0 0
\(899\) 1.53338e11i 0.234753i
\(900\) 0 0
\(901\) 1.18524e12 1.79849
\(902\) 0 0
\(903\) 3.22807e11i 0.485503i
\(904\) 0 0
\(905\) −1.71691e11 −0.255949
\(906\) 0 0
\(907\) − 2.33352e11i − 0.344813i −0.985026 0.172406i \(-0.944846\pi\)
0.985026 0.172406i \(-0.0551542\pi\)
\(908\) 0 0
\(909\) 3.81984e11 0.559486
\(910\) 0 0
\(911\) − 2.97901e11i − 0.432513i −0.976337 0.216256i \(-0.930615\pi\)
0.976337 0.216256i \(-0.0693847\pi\)
\(912\) 0 0
\(913\) 1.43677e11 0.206778
\(914\) 0 0
\(915\) 6.73821e10i 0.0961303i
\(916\) 0 0
\(917\) 7.62394e11 1.07821
\(918\) 0 0
\(919\) 8.26039e11i 1.15808i 0.815299 + 0.579040i \(0.196572\pi\)
−0.815299 + 0.579040i \(0.803428\pi\)
\(920\) 0 0
\(921\) 4.30083e11 0.597742
\(922\) 0 0
\(923\) − 5.37414e9i − 0.00740460i
\(924\) 0 0
\(925\) −3.15196e11 −0.430540
\(926\) 0 0
\(927\) 2.43440e10i 0.0329666i
\(928\) 0 0
\(929\) 7.14762e11 0.959619 0.479809 0.877373i \(-0.340706\pi\)
0.479809 + 0.877373i \(0.340706\pi\)
\(930\) 0 0
\(931\) 1.08385e12i 1.44268i
\(932\) 0 0
\(933\) −3.62600e11 −0.478521
\(934\) 0 0
\(935\) − 3.88435e10i − 0.0508243i
\(936\) 0 0
\(937\) 3.24075e11 0.420424 0.210212 0.977656i \(-0.432585\pi\)
0.210212 + 0.977656i \(0.432585\pi\)
\(938\) 0 0
\(939\) − 6.20933e11i − 0.798698i
\(940\) 0 0
\(941\) 6.78533e10 0.0865391 0.0432696 0.999063i \(-0.486223\pi\)
0.0432696 + 0.999063i \(0.486223\pi\)
\(942\) 0 0
\(943\) 1.20530e11i 0.152422i
\(944\) 0 0
\(945\) 4.89687e10 0.0614032
\(946\) 0 0
\(947\) − 8.07653e11i − 1.00421i −0.864806 0.502105i \(-0.832559\pi\)
0.864806 0.502105i \(-0.167441\pi\)
\(948\) 0 0
\(949\) 7.09507e9 0.00874766
\(950\) 0 0
\(951\) 1.50671e10i 0.0184208i
\(952\) 0 0
\(953\) 7.72345e11 0.936353 0.468176 0.883635i \(-0.344911\pi\)
0.468176 + 0.883635i \(0.344911\pi\)
\(954\) 0 0
\(955\) 2.55672e11i 0.307376i
\(956\) 0 0
\(957\) −4.69486e10 −0.0559725
\(958\) 0 0
\(959\) 7.44416e11i 0.880119i
\(960\) 0 0
\(961\) 6.08743e11 0.713740
\(962\) 0 0
\(963\) 5.65585e10i 0.0657647i
\(964\) 0 0
\(965\) 2.66026e11 0.306772
\(966\) 0 0
\(967\) − 5.00769e11i − 0.572706i −0.958124 0.286353i \(-0.907557\pi\)
0.958124 0.286353i \(-0.0924430\pi\)
\(968\) 0 0
\(969\) −9.01194e11 −1.02217
\(970\) 0 0
\(971\) 8.51918e11i 0.958343i 0.877721 + 0.479171i \(0.159063\pi\)
−0.877721 + 0.479171i \(0.840937\pi\)
\(972\) 0 0
\(973\) −3.34716e10 −0.0373444
\(974\) 0 0
\(975\) − 3.04356e9i − 0.00336793i
\(976\) 0 0
\(977\) 9.46264e11 1.03857 0.519283 0.854603i \(-0.326199\pi\)
0.519283 + 0.854603i \(0.326199\pi\)
\(978\) 0 0
\(979\) 3.59884e11i 0.391771i
\(980\) 0 0
\(981\) 4.89503e11 0.528541
\(982\) 0 0
\(983\) 9.35217e11i 1.00161i 0.865560 + 0.500804i \(0.166962\pi\)
−0.865560 + 0.500804i \(0.833038\pi\)
\(984\) 0 0
\(985\) 3.73078e11 0.396328
\(986\) 0 0
\(987\) − 1.65748e11i − 0.174655i
\(988\) 0 0
\(989\) −1.83643e11 −0.191951
\(990\) 0 0
\(991\) − 9.50357e11i − 0.985354i −0.870212 0.492677i \(-0.836019\pi\)
0.870212 0.492677i \(-0.163981\pi\)
\(992\) 0 0
\(993\) 5.45036e11 0.560568
\(994\) 0 0
\(995\) 3.24112e11i 0.330676i
\(996\) 0 0
\(997\) −1.84651e12 −1.86884 −0.934419 0.356175i \(-0.884081\pi\)
−0.934419 + 0.356175i \(0.884081\pi\)
\(998\) 0 0
\(999\) 8.75183e10i 0.0878692i
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 192.9.g.d.127.3 8
4.3 odd 2 inner 192.9.g.d.127.7 8
8.3 odd 2 96.9.g.b.31.2 8
8.5 even 2 96.9.g.b.31.6 yes 8
24.5 odd 2 288.9.g.c.127.5 8
24.11 even 2 288.9.g.c.127.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.9.g.b.31.2 8 8.3 odd 2
96.9.g.b.31.6 yes 8 8.5 even 2
192.9.g.d.127.3 8 1.1 even 1 trivial
192.9.g.d.127.7 8 4.3 odd 2 inner
288.9.g.c.127.5 8 24.5 odd 2
288.9.g.c.127.6 8 24.11 even 2