Properties

Label 48.9.g.b.31.1
Level $48$
Weight $9$
Character 48.31
Analytic conductor $19.554$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 48.31
Dual form 48.9.g.b.31.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-46.7654i q^{3} +726.000 q^{5} -3055.34i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q-46.7654i q^{3} +726.000 q^{5} -3055.34i q^{7} -2187.00 q^{9} +13281.4i q^{11} +39034.0 q^{13} -33951.7i q^{15} -65814.0 q^{17} -130257. i q^{19} -142884. q^{21} -502073. i q^{23} +136451. q^{25} +102276. i q^{27} +202062. q^{29} -1.19563e6i q^{31} +621108. q^{33} -2.21818e6i q^{35} -1.87603e6 q^{37} -1.82544e6i q^{39} +3.09105e6 q^{41} -2.26388e6i q^{43} -1.58776e6 q^{45} +6.35672e6i q^{47} -3.57029e6 q^{49} +3.07782e6i q^{51} -1.06648e6 q^{53} +9.64227e6i q^{55} -6.09152e6 q^{57} -5.76355e6i q^{59} +1.71542e7 q^{61} +6.68202e6i q^{63} +2.83387e7 q^{65} +2.74275e7i q^{67} -2.34796e7 q^{69} +3.98336e7i q^{71} -5.32860e7 q^{73} -6.38118e6i q^{75} +4.05791e7 q^{77} -1.82696e7i q^{79} +4.78297e6 q^{81} -7.78905e6i q^{83} -4.77810e7 q^{85} -9.44950e6i q^{87} +8.66672e7 q^{89} -1.19262e8i q^{91} -5.59143e7 q^{93} -9.45667e7i q^{95} -7.39018e7 q^{97} -2.90463e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1452 q^{5} - 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1452 q^{5} - 4374 q^{9} + 78068 q^{13} - 131628 q^{17} - 285768 q^{21} + 272902 q^{25} + 404124 q^{29} + 1242216 q^{33} - 3752060 q^{37} + 6182100 q^{41} - 3175524 q^{45} - 7140574 q^{49} - 2132964 q^{53} - 12183048 q^{57} + 34308388 q^{61} + 56677368 q^{65} - 46959264 q^{69} - 106572028 q^{73} + 81158112 q^{77} + 9565938 q^{81} - 95561928 q^{85} + 173334468 q^{89} - 111828600 q^{93} - 147803644 q^{97}+O(q^{100}) \) 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
\(5\) 726.000 1.16160 0.580800 0.814046i \(-0.302740\pi\)
0.580800 + 0.814046i \(0.302740\pi\)
\(6\) 0 0
\(7\) − 3055.34i − 1.27253i −0.771472 0.636264i \(-0.780479\pi\)
0.771472 0.636264i \(-0.219521\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) 13281.4i 0.907135i 0.891222 + 0.453568i \(0.149849\pi\)
−0.891222 + 0.453568i \(0.850151\pi\)
\(12\) 0 0
\(13\) 39034.0 1.36669 0.683344 0.730096i \(-0.260525\pi\)
0.683344 + 0.730096i \(0.260525\pi\)
\(14\) 0 0
\(15\) − 33951.7i − 0.670650i
\(16\) 0 0
\(17\) −65814.0 −0.787993 −0.393997 0.919112i \(-0.628908\pi\)
−0.393997 + 0.919112i \(0.628908\pi\)
\(18\) 0 0
\(19\) − 130257.i − 0.999510i −0.866167 0.499755i \(-0.833423\pi\)
0.866167 0.499755i \(-0.166577\pi\)
\(20\) 0 0
\(21\) −142884. −0.734694
\(22\) 0 0
\(23\) − 502073.i − 1.79414i −0.441892 0.897068i \(-0.645692\pi\)
0.441892 0.897068i \(-0.354308\pi\)
\(24\) 0 0
\(25\) 136451. 0.349315
\(26\) 0 0
\(27\) 102276.i 0.192450i
\(28\) 0 0
\(29\) 202062. 0.285688 0.142844 0.989745i \(-0.454375\pi\)
0.142844 + 0.989745i \(0.454375\pi\)
\(30\) 0 0
\(31\) − 1.19563e6i − 1.29465i −0.762215 0.647324i \(-0.775888\pi\)
0.762215 0.647324i \(-0.224112\pi\)
\(32\) 0 0
\(33\) 621108. 0.523735
\(34\) 0 0
\(35\) − 2.21818e6i − 1.47817i
\(36\) 0 0
\(37\) −1.87603e6 −1.00100 −0.500499 0.865737i \(-0.666850\pi\)
−0.500499 + 0.865737i \(0.666850\pi\)
\(38\) 0 0
\(39\) − 1.82544e6i − 0.789058i
\(40\) 0 0
\(41\) 3.09105e6 1.09388 0.546941 0.837171i \(-0.315792\pi\)
0.546941 + 0.837171i \(0.315792\pi\)
\(42\) 0 0
\(43\) − 2.26388e6i − 0.662186i −0.943598 0.331093i \(-0.892583\pi\)
0.943598 0.331093i \(-0.107417\pi\)
\(44\) 0 0
\(45\) −1.58776e6 −0.387200
\(46\) 0 0
\(47\) 6.35672e6i 1.30269i 0.758781 + 0.651346i \(0.225795\pi\)
−0.758781 + 0.651346i \(0.774205\pi\)
\(48\) 0 0
\(49\) −3.57029e6 −0.619325
\(50\) 0 0
\(51\) 3.07782e6i 0.454948i
\(52\) 0 0
\(53\) −1.06648e6 −0.135161 −0.0675803 0.997714i \(-0.521528\pi\)
−0.0675803 + 0.997714i \(0.521528\pi\)
\(54\) 0 0
\(55\) 9.64227e6i 1.05373i
\(56\) 0 0
\(57\) −6.09152e6 −0.577067
\(58\) 0 0
\(59\) − 5.76355e6i − 0.475644i −0.971309 0.237822i \(-0.923566\pi\)
0.971309 0.237822i \(-0.0764335\pi\)
\(60\) 0 0
\(61\) 1.71542e7 1.23894 0.619471 0.785020i \(-0.287347\pi\)
0.619471 + 0.785020i \(0.287347\pi\)
\(62\) 0 0
\(63\) 6.68202e6i 0.424176i
\(64\) 0 0
\(65\) 2.83387e7 1.58755
\(66\) 0 0
\(67\) 2.74275e7i 1.36109i 0.732706 + 0.680546i \(0.238257\pi\)
−0.732706 + 0.680546i \(0.761743\pi\)
\(68\) 0 0
\(69\) −2.34796e7 −1.03585
\(70\) 0 0
\(71\) 3.98336e7i 1.56753i 0.621056 + 0.783766i \(0.286704\pi\)
−0.621056 + 0.783766i \(0.713296\pi\)
\(72\) 0 0
\(73\) −5.32860e7 −1.87638 −0.938192 0.346115i \(-0.887501\pi\)
−0.938192 + 0.346115i \(0.887501\pi\)
\(74\) 0 0
\(75\) − 6.38118e6i − 0.201677i
\(76\) 0 0
\(77\) 4.05791e7 1.15435
\(78\) 0 0
\(79\) − 1.82696e7i − 0.469052i −0.972110 0.234526i \(-0.924646\pi\)
0.972110 0.234526i \(-0.0753538\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) − 7.78905e6i − 0.164124i −0.996627 0.0820620i \(-0.973849\pi\)
0.996627 0.0820620i \(-0.0261506\pi\)
\(84\) 0 0
\(85\) −4.77810e7 −0.915333
\(86\) 0 0
\(87\) − 9.44950e6i − 0.164942i
\(88\) 0 0
\(89\) 8.66672e7 1.38132 0.690661 0.723179i \(-0.257320\pi\)
0.690661 + 0.723179i \(0.257320\pi\)
\(90\) 0 0
\(91\) − 1.19262e8i − 1.73915i
\(92\) 0 0
\(93\) −5.59143e7 −0.747465
\(94\) 0 0
\(95\) − 9.45667e7i − 1.16103i
\(96\) 0 0
\(97\) −7.39018e7 −0.834773 −0.417386 0.908729i \(-0.637054\pi\)
−0.417386 + 0.908729i \(0.637054\pi\)
\(98\) 0 0
\(99\) − 2.90463e7i − 0.302378i
\(100\) 0 0
\(101\) 1.91310e8 1.83845 0.919227 0.393727i \(-0.128815\pi\)
0.919227 + 0.393727i \(0.128815\pi\)
\(102\) 0 0
\(103\) 1.62781e8i 1.44629i 0.690699 + 0.723143i \(0.257303\pi\)
−0.690699 + 0.723143i \(0.742697\pi\)
\(104\) 0 0
\(105\) −1.03734e8 −0.853420
\(106\) 0 0
\(107\) − 2.00810e8i − 1.53197i −0.642857 0.765986i \(-0.722251\pi\)
0.642857 0.765986i \(-0.277749\pi\)
\(108\) 0 0
\(109\) 6.86083e7 0.486039 0.243019 0.970021i \(-0.421862\pi\)
0.243019 + 0.970021i \(0.421862\pi\)
\(110\) 0 0
\(111\) 8.77332e7i 0.577926i
\(112\) 0 0
\(113\) 3.30831e7 0.202905 0.101452 0.994840i \(-0.467651\pi\)
0.101452 + 0.994840i \(0.467651\pi\)
\(114\) 0 0
\(115\) − 3.64505e8i − 2.08407i
\(116\) 0 0
\(117\) −8.53674e7 −0.455563
\(118\) 0 0
\(119\) 2.01084e8i 1.00274i
\(120\) 0 0
\(121\) 3.79642e7 0.177106
\(122\) 0 0
\(123\) − 1.44554e8i − 0.631553i
\(124\) 0 0
\(125\) −1.84530e8 −0.755836
\(126\) 0 0
\(127\) 2.70471e8i 1.03970i 0.854259 + 0.519848i \(0.174011\pi\)
−0.854259 + 0.519848i \(0.825989\pi\)
\(128\) 0 0
\(129\) −1.05871e8 −0.382313
\(130\) 0 0
\(131\) 3.02851e8i 1.02836i 0.857683 + 0.514178i \(0.171903\pi\)
−0.857683 + 0.514178i \(0.828097\pi\)
\(132\) 0 0
\(133\) −3.97980e8 −1.27190
\(134\) 0 0
\(135\) 7.42523e7i 0.223550i
\(136\) 0 0
\(137\) 6.40316e8 1.81766 0.908828 0.417170i \(-0.136978\pi\)
0.908828 + 0.417170i \(0.136978\pi\)
\(138\) 0 0
\(139\) 4.90714e8i 1.31453i 0.753661 + 0.657263i \(0.228286\pi\)
−0.753661 + 0.657263i \(0.771714\pi\)
\(140\) 0 0
\(141\) 2.97275e8 0.752110
\(142\) 0 0
\(143\) 5.18425e8i 1.23977i
\(144\) 0 0
\(145\) 1.46697e8 0.331856
\(146\) 0 0
\(147\) 1.66966e8i 0.357568i
\(148\) 0 0
\(149\) −8.11121e7 −0.164566 −0.0822831 0.996609i \(-0.526221\pi\)
−0.0822831 + 0.996609i \(0.526221\pi\)
\(150\) 0 0
\(151\) 1.77325e8i 0.341086i 0.985350 + 0.170543i \(0.0545521\pi\)
−0.985350 + 0.170543i \(0.945448\pi\)
\(152\) 0 0
\(153\) 1.43935e8 0.262664
\(154\) 0 0
\(155\) − 8.68031e8i − 1.50386i
\(156\) 0 0
\(157\) −2.14784e7 −0.0353511 −0.0176755 0.999844i \(-0.505627\pi\)
−0.0176755 + 0.999844i \(0.505627\pi\)
\(158\) 0 0
\(159\) 4.98744e7i 0.0780350i
\(160\) 0 0
\(161\) −1.53400e9 −2.28309
\(162\) 0 0
\(163\) − 2.42230e8i − 0.343144i −0.985172 0.171572i \(-0.945115\pi\)
0.985172 0.171572i \(-0.0548847\pi\)
\(164\) 0 0
\(165\) 4.50924e8 0.608370
\(166\) 0 0
\(167\) 3.89012e8i 0.500146i 0.968227 + 0.250073i \(0.0804547\pi\)
−0.968227 + 0.250073i \(0.919545\pi\)
\(168\) 0 0
\(169\) 7.07922e8 0.867838
\(170\) 0 0
\(171\) 2.84872e8i 0.333170i
\(172\) 0 0
\(173\) −6.35072e7 −0.0708988 −0.0354494 0.999371i \(-0.511286\pi\)
−0.0354494 + 0.999371i \(0.511286\pi\)
\(174\) 0 0
\(175\) − 4.16904e8i − 0.444512i
\(176\) 0 0
\(177\) −2.69535e8 −0.274613
\(178\) 0 0
\(179\) − 5.33629e8i − 0.519789i −0.965637 0.259895i \(-0.916312\pi\)
0.965637 0.259895i \(-0.0836878\pi\)
\(180\) 0 0
\(181\) 8.56360e8 0.797888 0.398944 0.916975i \(-0.369377\pi\)
0.398944 + 0.916975i \(0.369377\pi\)
\(182\) 0 0
\(183\) − 8.02222e8i − 0.715303i
\(184\) 0 0
\(185\) −1.36200e9 −1.16276
\(186\) 0 0
\(187\) − 8.74100e8i − 0.714817i
\(188\) 0 0
\(189\) 3.12487e8 0.244898
\(190\) 0 0
\(191\) − 4.75759e8i − 0.357481i −0.983896 0.178741i \(-0.942798\pi\)
0.983896 0.178741i \(-0.0572023\pi\)
\(192\) 0 0
\(193\) 8.76708e8 0.631867 0.315933 0.948781i \(-0.397682\pi\)
0.315933 + 0.948781i \(0.397682\pi\)
\(194\) 0 0
\(195\) − 1.32527e9i − 0.916570i
\(196\) 0 0
\(197\) −2.76762e9 −1.83756 −0.918780 0.394771i \(-0.870824\pi\)
−0.918780 + 0.394771i \(0.870824\pi\)
\(198\) 0 0
\(199\) − 1.42932e9i − 0.911420i −0.890128 0.455710i \(-0.849385\pi\)
0.890128 0.455710i \(-0.150615\pi\)
\(200\) 0 0
\(201\) 1.28266e9 0.785826
\(202\) 0 0
\(203\) − 6.17368e8i − 0.363546i
\(204\) 0 0
\(205\) 2.24410e9 1.27065
\(206\) 0 0
\(207\) 1.09803e9i 0.598046i
\(208\) 0 0
\(209\) 1.72999e9 0.906691
\(210\) 0 0
\(211\) − 4.61738e8i − 0.232952i −0.993194 0.116476i \(-0.962840\pi\)
0.993194 0.116476i \(-0.0371598\pi\)
\(212\) 0 0
\(213\) 1.86283e9 0.905015
\(214\) 0 0
\(215\) − 1.64358e9i − 0.769195i
\(216\) 0 0
\(217\) −3.65307e9 −1.64747
\(218\) 0 0
\(219\) 2.49194e9i 1.08333i
\(220\) 0 0
\(221\) −2.56898e9 −1.07694
\(222\) 0 0
\(223\) 3.40037e9i 1.37501i 0.726179 + 0.687506i \(0.241295\pi\)
−0.726179 + 0.687506i \(0.758705\pi\)
\(224\) 0 0
\(225\) −2.98418e8 −0.116438
\(226\) 0 0
\(227\) 4.52697e9i 1.70492i 0.522792 + 0.852460i \(0.324891\pi\)
−0.522792 + 0.852460i \(0.675109\pi\)
\(228\) 0 0
\(229\) −9.90176e8 −0.360056 −0.180028 0.983661i \(-0.557619\pi\)
−0.180028 + 0.983661i \(0.557619\pi\)
\(230\) 0 0
\(231\) − 1.89769e9i − 0.666467i
\(232\) 0 0
\(233\) 2.23709e9 0.759032 0.379516 0.925185i \(-0.376091\pi\)
0.379516 + 0.925185i \(0.376091\pi\)
\(234\) 0 0
\(235\) 4.61498e9i 1.51321i
\(236\) 0 0
\(237\) −8.54385e8 −0.270807
\(238\) 0 0
\(239\) − 2.63524e9i − 0.807659i −0.914834 0.403830i \(-0.867679\pi\)
0.914834 0.403830i \(-0.132321\pi\)
\(240\) 0 0
\(241\) −6.19651e8 −0.183687 −0.0918436 0.995773i \(-0.529276\pi\)
−0.0918436 + 0.995773i \(0.529276\pi\)
\(242\) 0 0
\(243\) − 2.23677e8i − 0.0641500i
\(244\) 0 0
\(245\) −2.59203e9 −0.719408
\(246\) 0 0
\(247\) − 5.08446e9i − 1.36602i
\(248\) 0 0
\(249\) −3.64258e8 −0.0947571
\(250\) 0 0
\(251\) 6.81334e9i 1.71659i 0.513161 + 0.858293i \(0.328474\pi\)
−0.513161 + 0.858293i \(0.671526\pi\)
\(252\) 0 0
\(253\) 6.66822e9 1.62752
\(254\) 0 0
\(255\) 2.23449e9i 0.528468i
\(256\) 0 0
\(257\) 3.95756e9 0.907184 0.453592 0.891209i \(-0.350142\pi\)
0.453592 + 0.891209i \(0.350142\pi\)
\(258\) 0 0
\(259\) 5.73191e9i 1.27380i
\(260\) 0 0
\(261\) −4.41910e8 −0.0952295
\(262\) 0 0
\(263\) 1.86129e9i 0.389037i 0.980899 + 0.194518i \(0.0623144\pi\)
−0.980899 + 0.194518i \(0.937686\pi\)
\(264\) 0 0
\(265\) −7.74266e8 −0.157003
\(266\) 0 0
\(267\) − 4.05303e9i − 0.797507i
\(268\) 0 0
\(269\) 1.17367e9 0.224149 0.112074 0.993700i \(-0.464250\pi\)
0.112074 + 0.993700i \(0.464250\pi\)
\(270\) 0 0
\(271\) − 1.90505e9i − 0.353207i −0.984282 0.176604i \(-0.943489\pi\)
0.984282 0.176604i \(-0.0565110\pi\)
\(272\) 0 0
\(273\) −5.57733e9 −1.00410
\(274\) 0 0
\(275\) 1.81226e9i 0.316876i
\(276\) 0 0
\(277\) −5.03752e9 −0.855654 −0.427827 0.903861i \(-0.640721\pi\)
−0.427827 + 0.903861i \(0.640721\pi\)
\(278\) 0 0
\(279\) 2.61485e9i 0.431549i
\(280\) 0 0
\(281\) 6.66317e9 1.06870 0.534350 0.845264i \(-0.320557\pi\)
0.534350 + 0.845264i \(0.320557\pi\)
\(282\) 0 0
\(283\) − 5.54295e9i − 0.864162i −0.901835 0.432081i \(-0.857779\pi\)
0.901835 0.432081i \(-0.142221\pi\)
\(284\) 0 0
\(285\) −4.42245e9 −0.670321
\(286\) 0 0
\(287\) − 9.44420e9i − 1.39199i
\(288\) 0 0
\(289\) −2.64427e9 −0.379066
\(290\) 0 0
\(291\) 3.45605e9i 0.481956i
\(292\) 0 0
\(293\) 6.67390e9 0.905543 0.452772 0.891627i \(-0.350435\pi\)
0.452772 + 0.891627i \(0.350435\pi\)
\(294\) 0 0
\(295\) − 4.18434e9i − 0.552508i
\(296\) 0 0
\(297\) −1.35836e9 −0.174578
\(298\) 0 0
\(299\) − 1.95979e10i − 2.45203i
\(300\) 0 0
\(301\) −6.91692e9 −0.842649
\(302\) 0 0
\(303\) − 8.94670e9i − 1.06143i
\(304\) 0 0
\(305\) 1.24539e10 1.43916
\(306\) 0 0
\(307\) 6.49752e8i 0.0731466i 0.999331 + 0.0365733i \(0.0116442\pi\)
−0.999331 + 0.0365733i \(0.988356\pi\)
\(308\) 0 0
\(309\) 7.61250e9 0.835013
\(310\) 0 0
\(311\) 3.97832e8i 0.0425264i 0.999774 + 0.0212632i \(0.00676879\pi\)
−0.999774 + 0.0212632i \(0.993231\pi\)
\(312\) 0 0
\(313\) −1.58217e10 −1.64845 −0.824223 0.566266i \(-0.808388\pi\)
−0.824223 + 0.566266i \(0.808388\pi\)
\(314\) 0 0
\(315\) 4.85115e9i 0.492723i
\(316\) 0 0
\(317\) −9.60836e9 −0.951507 −0.475754 0.879579i \(-0.657825\pi\)
−0.475754 + 0.879579i \(0.657825\pi\)
\(318\) 0 0
\(319\) 2.68366e9i 0.259158i
\(320\) 0 0
\(321\) −9.39097e9 −0.884485
\(322\) 0 0
\(323\) 8.57274e9i 0.787607i
\(324\) 0 0
\(325\) 5.32623e9 0.477404
\(326\) 0 0
\(327\) − 3.20849e9i − 0.280615i
\(328\) 0 0
\(329\) 1.94219e10 1.65771
\(330\) 0 0
\(331\) − 1.41991e10i − 1.18290i −0.806341 0.591451i \(-0.798555\pi\)
0.806341 0.591451i \(-0.201445\pi\)
\(332\) 0 0
\(333\) 4.10288e9 0.333666
\(334\) 0 0
\(335\) 1.99124e10i 1.58104i
\(336\) 0 0
\(337\) 3.39383e9 0.263130 0.131565 0.991308i \(-0.458000\pi\)
0.131565 + 0.991308i \(0.458000\pi\)
\(338\) 0 0
\(339\) − 1.54714e9i − 0.117147i
\(340\) 0 0
\(341\) 1.58797e10 1.17442
\(342\) 0 0
\(343\) − 6.70498e9i − 0.484419i
\(344\) 0 0
\(345\) −1.70462e10 −1.20324
\(346\) 0 0
\(347\) − 1.35188e8i − 0.00932439i −0.999989 0.00466219i \(-0.998516\pi\)
0.999989 0.00466219i \(-0.00148403\pi\)
\(348\) 0 0
\(349\) 1.13213e10 0.763122 0.381561 0.924344i \(-0.375387\pi\)
0.381561 + 0.924344i \(0.375387\pi\)
\(350\) 0 0
\(351\) 3.99224e9i 0.263019i
\(352\) 0 0
\(353\) −1.42650e10 −0.918697 −0.459348 0.888256i \(-0.651917\pi\)
−0.459348 + 0.888256i \(0.651917\pi\)
\(354\) 0 0
\(355\) 2.89192e10i 1.82085i
\(356\) 0 0
\(357\) 9.40377e9 0.578934
\(358\) 0 0
\(359\) − 8.15636e9i − 0.491042i −0.969391 0.245521i \(-0.921041\pi\)
0.969391 0.245521i \(-0.0789590\pi\)
\(360\) 0 0
\(361\) 1.66382e7 0.000979664 0
\(362\) 0 0
\(363\) − 1.77541e9i − 0.102252i
\(364\) 0 0
\(365\) −3.86856e10 −2.17961
\(366\) 0 0
\(367\) 2.06760e10i 1.13973i 0.821738 + 0.569865i \(0.193005\pi\)
−0.821738 + 0.569865i \(0.806995\pi\)
\(368\) 0 0
\(369\) −6.76013e9 −0.364627
\(370\) 0 0
\(371\) 3.25846e9i 0.171996i
\(372\) 0 0
\(373\) −7.71358e9 −0.398493 −0.199247 0.979949i \(-0.563849\pi\)
−0.199247 + 0.979949i \(0.563849\pi\)
\(374\) 0 0
\(375\) 8.62963e9i 0.436382i
\(376\) 0 0
\(377\) 7.88729e9 0.390447
\(378\) 0 0
\(379\) − 1.53767e9i − 0.0745256i −0.999306 0.0372628i \(-0.988136\pi\)
0.999306 0.0372628i \(-0.0118639\pi\)
\(380\) 0 0
\(381\) 1.26487e10 0.600268
\(382\) 0 0
\(383\) 3.20555e10i 1.48973i 0.667216 + 0.744864i \(0.267486\pi\)
−0.667216 + 0.744864i \(0.732514\pi\)
\(384\) 0 0
\(385\) 2.94604e10 1.34090
\(386\) 0 0
\(387\) 4.95111e9i 0.220729i
\(388\) 0 0
\(389\) −2.99296e10 −1.30708 −0.653541 0.756891i \(-0.726717\pi\)
−0.653541 + 0.756891i \(0.726717\pi\)
\(390\) 0 0
\(391\) 3.30434e10i 1.41377i
\(392\) 0 0
\(393\) 1.41629e10 0.593722
\(394\) 0 0
\(395\) − 1.32637e10i − 0.544851i
\(396\) 0 0
\(397\) 1.32156e10 0.532016 0.266008 0.963971i \(-0.414295\pi\)
0.266008 + 0.963971i \(0.414295\pi\)
\(398\) 0 0
\(399\) 1.86117e10i 0.734334i
\(400\) 0 0
\(401\) 2.51637e10 0.973190 0.486595 0.873628i \(-0.338239\pi\)
0.486595 + 0.873628i \(0.338239\pi\)
\(402\) 0 0
\(403\) − 4.66704e10i − 1.76938i
\(404\) 0 0
\(405\) 3.47244e9 0.129067
\(406\) 0 0
\(407\) − 2.49162e10i − 0.908040i
\(408\) 0 0
\(409\) −3.78473e10 −1.35251 −0.676257 0.736666i \(-0.736399\pi\)
−0.676257 + 0.736666i \(0.736399\pi\)
\(410\) 0 0
\(411\) − 2.99446e10i − 1.04942i
\(412\) 0 0
\(413\) −1.76096e10 −0.605270
\(414\) 0 0
\(415\) − 5.65485e9i − 0.190647i
\(416\) 0 0
\(417\) 2.29484e10 0.758942
\(418\) 0 0
\(419\) − 1.88088e10i − 0.610246i −0.952313 0.305123i \(-0.901302\pi\)
0.952313 0.305123i \(-0.0986976\pi\)
\(420\) 0 0
\(421\) 6.04555e9 0.192445 0.0962227 0.995360i \(-0.469324\pi\)
0.0962227 + 0.995360i \(0.469324\pi\)
\(422\) 0 0
\(423\) − 1.39022e10i − 0.434231i
\(424\) 0 0
\(425\) −8.98039e9 −0.275258
\(426\) 0 0
\(427\) − 5.24119e10i − 1.57659i
\(428\) 0 0
\(429\) 2.42443e10 0.715782
\(430\) 0 0
\(431\) 4.60486e10i 1.33447i 0.744849 + 0.667233i \(0.232521\pi\)
−0.744849 + 0.667233i \(0.767479\pi\)
\(432\) 0 0
\(433\) 1.85654e9 0.0528145 0.0264072 0.999651i \(-0.491593\pi\)
0.0264072 + 0.999651i \(0.491593\pi\)
\(434\) 0 0
\(435\) − 6.86034e9i − 0.191597i
\(436\) 0 0
\(437\) −6.53986e10 −1.79326
\(438\) 0 0
\(439\) − 1.24165e10i − 0.334303i −0.985931 0.167152i \(-0.946543\pi\)
0.985931 0.167152i \(-0.0534569\pi\)
\(440\) 0 0
\(441\) 7.80822e9 0.206442
\(442\) 0 0
\(443\) − 7.57779e9i − 0.196756i −0.995149 0.0983779i \(-0.968635\pi\)
0.995149 0.0983779i \(-0.0313654\pi\)
\(444\) 0 0
\(445\) 6.29204e10 1.60454
\(446\) 0 0
\(447\) 3.79324e9i 0.0950123i
\(448\) 0 0
\(449\) −3.37970e10 −0.831559 −0.415780 0.909465i \(-0.636491\pi\)
−0.415780 + 0.909465i \(0.636491\pi\)
\(450\) 0 0
\(451\) 4.10534e10i 0.992299i
\(452\) 0 0
\(453\) 8.29269e9 0.196926
\(454\) 0 0
\(455\) − 8.65842e10i − 2.02020i
\(456\) 0 0
\(457\) −2.01366e10 −0.461659 −0.230829 0.972994i \(-0.574144\pi\)
−0.230829 + 0.972994i \(0.574144\pi\)
\(458\) 0 0
\(459\) − 6.73118e9i − 0.151649i
\(460\) 0 0
\(461\) −2.54155e10 −0.562724 −0.281362 0.959602i \(-0.590786\pi\)
−0.281362 + 0.959602i \(0.590786\pi\)
\(462\) 0 0
\(463\) 1.19712e9i 0.0260504i 0.999915 + 0.0130252i \(0.00414617\pi\)
−0.999915 + 0.0130252i \(0.995854\pi\)
\(464\) 0 0
\(465\) −4.05938e10 −0.868256
\(466\) 0 0
\(467\) − 2.92676e10i − 0.615347i −0.951492 0.307673i \(-0.900450\pi\)
0.951492 0.307673i \(-0.0995504\pi\)
\(468\) 0 0
\(469\) 8.38003e10 1.73203
\(470\) 0 0
\(471\) 1.00444e9i 0.0204100i
\(472\) 0 0
\(473\) 3.00674e10 0.600692
\(474\) 0 0
\(475\) − 1.77737e10i − 0.349143i
\(476\) 0 0
\(477\) 2.33240e9 0.0450535
\(478\) 0 0
\(479\) 2.51066e10i 0.476919i 0.971152 + 0.238460i \(0.0766425\pi\)
−0.971152 + 0.238460i \(0.923357\pi\)
\(480\) 0 0
\(481\) −7.32290e10 −1.36805
\(482\) 0 0
\(483\) 7.17382e10i 1.31814i
\(484\) 0 0
\(485\) −5.36527e10 −0.969672
\(486\) 0 0
\(487\) − 5.82581e10i − 1.03571i −0.855467 0.517857i \(-0.826730\pi\)
0.855467 0.517857i \(-0.173270\pi\)
\(488\) 0 0
\(489\) −1.13280e10 −0.198115
\(490\) 0 0
\(491\) 3.61816e10i 0.622532i 0.950323 + 0.311266i \(0.100753\pi\)
−0.950323 + 0.311266i \(0.899247\pi\)
\(492\) 0 0
\(493\) −1.32985e10 −0.225121
\(494\) 0 0
\(495\) − 2.10876e10i − 0.351243i
\(496\) 0 0
\(497\) 1.21705e11 1.99473
\(498\) 0 0
\(499\) 5.58440e10i 0.900687i 0.892855 + 0.450344i \(0.148699\pi\)
−0.892855 + 0.450344i \(0.851301\pi\)
\(500\) 0 0
\(501\) 1.81923e10 0.288760
\(502\) 0 0
\(503\) 1.84340e10i 0.287971i 0.989580 + 0.143985i \(0.0459918\pi\)
−0.989580 + 0.143985i \(0.954008\pi\)
\(504\) 0 0
\(505\) 1.38891e11 2.13555
\(506\) 0 0
\(507\) − 3.31063e10i − 0.501047i
\(508\) 0 0
\(509\) 1.42165e10 0.211798 0.105899 0.994377i \(-0.466228\pi\)
0.105899 + 0.994377i \(0.466228\pi\)
\(510\) 0 0
\(511\) 1.62807e11i 2.38775i
\(512\) 0 0
\(513\) 1.33222e10 0.192356
\(514\) 0 0
\(515\) 1.18179e11i 1.68001i
\(516\) 0 0
\(517\) −8.44260e10 −1.18172
\(518\) 0 0
\(519\) 2.96994e9i 0.0409334i
\(520\) 0 0
\(521\) −6.81614e10 −0.925098 −0.462549 0.886594i \(-0.653065\pi\)
−0.462549 + 0.886594i \(0.653065\pi\)
\(522\) 0 0
\(523\) 5.63922e10i 0.753724i 0.926269 + 0.376862i \(0.122997\pi\)
−0.926269 + 0.376862i \(0.877003\pi\)
\(524\) 0 0
\(525\) −1.94967e10 −0.256639
\(526\) 0 0
\(527\) 7.86895e10i 1.02017i
\(528\) 0 0
\(529\) −1.73766e11 −2.21893
\(530\) 0 0
\(531\) 1.26049e10i 0.158548i
\(532\) 0 0
\(533\) 1.20656e11 1.49500
\(534\) 0 0
\(535\) − 1.45788e11i − 1.77954i
\(536\) 0 0
\(537\) −2.49554e10 −0.300100
\(538\) 0 0
\(539\) − 4.74183e10i − 0.561812i
\(540\) 0 0
\(541\) −7.61478e10 −0.888932 −0.444466 0.895796i \(-0.646607\pi\)
−0.444466 + 0.895796i \(0.646607\pi\)
\(542\) 0 0
\(543\) − 4.00480e10i − 0.460661i
\(544\) 0 0
\(545\) 4.98096e10 0.564583
\(546\) 0 0
\(547\) 5.84939e10i 0.653373i 0.945133 + 0.326686i \(0.105932\pi\)
−0.945133 + 0.326686i \(0.894068\pi\)
\(548\) 0 0
\(549\) −3.75162e10 −0.412981
\(550\) 0 0
\(551\) − 2.63200e10i − 0.285548i
\(552\) 0 0
\(553\) −5.58198e10 −0.596881
\(554\) 0 0
\(555\) 6.36943e10i 0.671319i
\(556\) 0 0
\(557\) 1.61301e11 1.67577 0.837887 0.545844i \(-0.183791\pi\)
0.837887 + 0.545844i \(0.183791\pi\)
\(558\) 0 0
\(559\) − 8.83683e10i − 0.905002i
\(560\) 0 0
\(561\) −4.08776e10 −0.412700
\(562\) 0 0
\(563\) 6.88172e9i 0.0684957i 0.999413 + 0.0342479i \(0.0109036\pi\)
−0.999413 + 0.0342479i \(0.989096\pi\)
\(564\) 0 0
\(565\) 2.40183e10 0.235694
\(566\) 0 0
\(567\) − 1.46136e10i − 0.141392i
\(568\) 0 0
\(569\) 9.38382e10 0.895221 0.447611 0.894229i \(-0.352275\pi\)
0.447611 + 0.894229i \(0.352275\pi\)
\(570\) 0 0
\(571\) − 1.92744e10i − 0.181316i −0.995882 0.0906582i \(-0.971103\pi\)
0.995882 0.0906582i \(-0.0288971\pi\)
\(572\) 0 0
\(573\) −2.22490e10 −0.206392
\(574\) 0 0
\(575\) − 6.85084e10i − 0.626718i
\(576\) 0 0
\(577\) 1.65488e11 1.49301 0.746507 0.665378i \(-0.231730\pi\)
0.746507 + 0.665378i \(0.231730\pi\)
\(578\) 0 0
\(579\) − 4.09996e10i − 0.364809i
\(580\) 0 0
\(581\) −2.37982e10 −0.208852
\(582\) 0 0
\(583\) − 1.41643e10i − 0.122609i
\(584\) 0 0
\(585\) −6.19767e10 −0.529182
\(586\) 0 0
\(587\) − 2.09633e11i − 1.76566i −0.469695 0.882829i \(-0.655636\pi\)
0.469695 0.882829i \(-0.344364\pi\)
\(588\) 0 0
\(589\) −1.55740e11 −1.29401
\(590\) 0 0
\(591\) 1.29429e11i 1.06092i
\(592\) 0 0
\(593\) −7.33746e10 −0.593372 −0.296686 0.954975i \(-0.595881\pi\)
−0.296686 + 0.954975i \(0.595881\pi\)
\(594\) 0 0
\(595\) 1.45987e11i 1.16479i
\(596\) 0 0
\(597\) −6.68429e10 −0.526209
\(598\) 0 0
\(599\) − 1.33326e11i − 1.03563i −0.855491 0.517817i \(-0.826745\pi\)
0.855491 0.517817i \(-0.173255\pi\)
\(600\) 0 0
\(601\) 2.01691e11 1.54593 0.772965 0.634449i \(-0.218773\pi\)
0.772965 + 0.634449i \(0.218773\pi\)
\(602\) 0 0
\(603\) − 5.99840e10i − 0.453697i
\(604\) 0 0
\(605\) 2.75620e10 0.205726
\(606\) 0 0
\(607\) − 1.55515e11i − 1.14556i −0.819710 0.572779i \(-0.805866\pi\)
0.819710 0.572779i \(-0.194134\pi\)
\(608\) 0 0
\(609\) −2.88714e10 −0.209894
\(610\) 0 0
\(611\) 2.48128e11i 1.78038i
\(612\) 0 0
\(613\) 3.19775e10 0.226466 0.113233 0.993568i \(-0.463879\pi\)
0.113233 + 0.993568i \(0.463879\pi\)
\(614\) 0 0
\(615\) − 1.04946e11i − 0.733612i
\(616\) 0 0
\(617\) 5.63108e10 0.388553 0.194277 0.980947i \(-0.437764\pi\)
0.194277 + 0.980947i \(0.437764\pi\)
\(618\) 0 0
\(619\) − 2.66432e11i − 1.81478i −0.420287 0.907391i \(-0.638071\pi\)
0.420287 0.907391i \(-0.361929\pi\)
\(620\) 0 0
\(621\) 5.13500e10 0.345282
\(622\) 0 0
\(623\) − 2.64798e11i − 1.75777i
\(624\) 0 0
\(625\) −1.87270e11 −1.22729
\(626\) 0 0
\(627\) − 8.09038e10i − 0.523478i
\(628\) 0 0
\(629\) 1.23469e11 0.788779
\(630\) 0 0
\(631\) − 8.55727e9i − 0.0539781i −0.999636 0.0269891i \(-0.991408\pi\)
0.999636 0.0269891i \(-0.00859193\pi\)
\(632\) 0 0
\(633\) −2.15934e10 −0.134495
\(634\) 0 0
\(635\) 1.96362e11i 1.20771i
\(636\) 0 0
\(637\) −1.39363e11 −0.846425
\(638\) 0 0
\(639\) − 8.71161e10i − 0.522511i
\(640\) 0 0
\(641\) 2.90248e10 0.171924 0.0859620 0.996298i \(-0.472604\pi\)
0.0859620 + 0.996298i \(0.472604\pi\)
\(642\) 0 0
\(643\) 5.13563e10i 0.300435i 0.988653 + 0.150217i \(0.0479973\pi\)
−0.988653 + 0.150217i \(0.952003\pi\)
\(644\) 0 0
\(645\) −7.68625e10 −0.444095
\(646\) 0 0
\(647\) − 5.58175e10i − 0.318532i −0.987236 0.159266i \(-0.949087\pi\)
0.987236 0.159266i \(-0.0509128\pi\)
\(648\) 0 0
\(649\) 7.65478e10 0.431473
\(650\) 0 0
\(651\) 1.70837e11i 0.951170i
\(652\) 0 0
\(653\) −6.40717e10 −0.352382 −0.176191 0.984356i \(-0.556378\pi\)
−0.176191 + 0.984356i \(0.556378\pi\)
\(654\) 0 0
\(655\) 2.19870e11i 1.19454i
\(656\) 0 0
\(657\) 1.16537e11 0.625461
\(658\) 0 0
\(659\) 3.17581e11i 1.68389i 0.539567 + 0.841943i \(0.318588\pi\)
−0.539567 + 0.841943i \(0.681412\pi\)
\(660\) 0 0
\(661\) −1.33716e11 −0.700449 −0.350224 0.936666i \(-0.613895\pi\)
−0.350224 + 0.936666i \(0.613895\pi\)
\(662\) 0 0
\(663\) 1.20139e11i 0.621773i
\(664\) 0 0
\(665\) −2.88933e11 −1.47744
\(666\) 0 0
\(667\) − 1.01450e11i − 0.512564i
\(668\) 0 0
\(669\) 1.59019e11 0.793864
\(670\) 0 0
\(671\) 2.27831e11i 1.12389i
\(672\) 0 0
\(673\) −4.12429e10 −0.201043 −0.100521 0.994935i \(-0.532051\pi\)
−0.100521 + 0.994935i \(0.532051\pi\)
\(674\) 0 0
\(675\) 1.39556e10i 0.0672256i
\(676\) 0 0
\(677\) −5.06159e10 −0.240953 −0.120476 0.992716i \(-0.538442\pi\)
−0.120476 + 0.992716i \(0.538442\pi\)
\(678\) 0 0
\(679\) 2.25795e11i 1.06227i
\(680\) 0 0
\(681\) 2.11705e11 0.984337
\(682\) 0 0
\(683\) − 3.59716e11i − 1.65302i −0.562924 0.826508i \(-0.690324\pi\)
0.562924 0.826508i \(-0.309676\pi\)
\(684\) 0 0
\(685\) 4.64869e11 2.11139
\(686\) 0 0
\(687\) 4.63059e10i 0.207879i
\(688\) 0 0
\(689\) −4.16291e10 −0.184722
\(690\) 0 0
\(691\) 1.38563e11i 0.607763i 0.952710 + 0.303882i \(0.0982827\pi\)
−0.952710 + 0.303882i \(0.901717\pi\)
\(692\) 0 0
\(693\) −8.87464e10 −0.384785
\(694\) 0 0
\(695\) 3.56258e11i 1.52695i
\(696\) 0 0
\(697\) −2.03434e11 −0.861972
\(698\) 0 0
\(699\) − 1.04618e11i − 0.438227i
\(700\) 0 0
\(701\) 2.39409e11 0.991445 0.495722 0.868481i \(-0.334903\pi\)
0.495722 + 0.868481i \(0.334903\pi\)
\(702\) 0 0
\(703\) 2.44366e11i 1.00051i
\(704\) 0 0
\(705\) 2.15821e11 0.873651
\(706\) 0 0
\(707\) − 5.84518e11i − 2.33948i
\(708\) 0 0
\(709\) 1.08904e11 0.430981 0.215490 0.976506i \(-0.430865\pi\)
0.215490 + 0.976506i \(0.430865\pi\)
\(710\) 0 0
\(711\) 3.99556e10i 0.156351i
\(712\) 0 0
\(713\) −6.00296e11 −2.32278
\(714\) 0 0
\(715\) 3.76376e11i 1.44012i
\(716\) 0 0
\(717\) −1.23238e11 −0.466302
\(718\) 0 0
\(719\) 2.60327e11i 0.974101i 0.873374 + 0.487051i \(0.161927\pi\)
−0.873374 + 0.487051i \(0.838073\pi\)
\(720\) 0 0
\(721\) 4.97350e11 1.84044
\(722\) 0 0
\(723\) 2.89782e10i 0.106052i
\(724\) 0 0
\(725\) 2.75716e10 0.0997951
\(726\) 0 0
\(727\) − 4.81879e11i − 1.72504i −0.506020 0.862522i \(-0.668884\pi\)
0.506020 0.862522i \(-0.331116\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) 1.48995e11i 0.521798i
\(732\) 0 0
\(733\) −1.82274e11 −0.631405 −0.315702 0.948858i \(-0.602240\pi\)
−0.315702 + 0.948858i \(0.602240\pi\)
\(734\) 0 0
\(735\) 1.21217e11i 0.415351i
\(736\) 0 0
\(737\) −3.64275e11 −1.23469
\(738\) 0 0
\(739\) − 5.49813e11i − 1.84347i −0.387815 0.921737i \(-0.626770\pi\)
0.387815 0.921737i \(-0.373230\pi\)
\(740\) 0 0
\(741\) −2.37777e11 −0.788672
\(742\) 0 0
\(743\) 2.24817e11i 0.737690i 0.929491 + 0.368845i \(0.120247\pi\)
−0.929491 + 0.368845i \(0.879753\pi\)
\(744\) 0 0
\(745\) −5.88874e10 −0.191160
\(746\) 0 0
\(747\) 1.70347e10i 0.0547080i
\(748\) 0 0
\(749\) −6.13543e11 −1.94948
\(750\) 0 0
\(751\) 4.17556e11i 1.31267i 0.754470 + 0.656334i \(0.227894\pi\)
−0.754470 + 0.656334i \(0.772106\pi\)
\(752\) 0 0
\(753\) 3.18629e11 0.991071
\(754\) 0 0
\(755\) 1.28738e11i 0.396205i
\(756\) 0 0
\(757\) 6.29371e11 1.91656 0.958282 0.285826i \(-0.0922679\pi\)
0.958282 + 0.285826i \(0.0922679\pi\)
\(758\) 0 0
\(759\) − 3.11842e11i − 0.939652i
\(760\) 0 0
\(761\) −1.72289e11 −0.513710 −0.256855 0.966450i \(-0.582686\pi\)
−0.256855 + 0.966450i \(0.582686\pi\)
\(762\) 0 0
\(763\) − 2.09622e11i − 0.618497i
\(764\) 0 0
\(765\) 1.04497e11 0.305111
\(766\) 0 0
\(767\) − 2.24974e11i − 0.650058i
\(768\) 0 0
\(769\) 9.21192e10 0.263418 0.131709 0.991288i \(-0.457954\pi\)
0.131709 + 0.991288i \(0.457954\pi\)
\(770\) 0 0
\(771\) − 1.85077e11i − 0.523763i
\(772\) 0 0
\(773\) 4.70219e11 1.31699 0.658495 0.752585i \(-0.271194\pi\)
0.658495 + 0.752585i \(0.271194\pi\)
\(774\) 0 0
\(775\) − 1.63146e11i − 0.452239i
\(776\) 0 0
\(777\) 2.68055e11 0.735427
\(778\) 0 0
\(779\) − 4.02631e11i − 1.09335i
\(780\) 0 0
\(781\) −5.29045e11 −1.42196
\(782\) 0 0
\(783\) 2.06661e10i 0.0549808i
\(784\) 0 0
\(785\) −1.55933e10 −0.0410638
\(786\) 0 0
\(787\) − 5.86015e11i − 1.52760i −0.645452 0.763801i \(-0.723331\pi\)
0.645452 0.763801i \(-0.276669\pi\)
\(788\) 0 0
\(789\) 8.70439e10 0.224611
\(790\) 0 0
\(791\) − 1.01080e11i − 0.258202i
\(792\) 0 0
\(793\) 6.69597e11 1.69325
\(794\) 0 0
\(795\) 3.62088e10i 0.0906455i
\(796\) 0 0
\(797\) −2.08165e11 −0.515912 −0.257956 0.966157i \(-0.583049\pi\)
−0.257956 + 0.966157i \(0.583049\pi\)
\(798\) 0 0
\(799\) − 4.18361e11i − 1.02651i
\(800\) 0 0
\(801\) −1.89541e11 −0.460441
\(802\) 0 0
\(803\) − 7.07711e11i − 1.70213i
\(804\) 0 0
\(805\) −1.11369e12 −2.65203
\(806\) 0 0
\(807\) − 5.48871e10i − 0.129412i
\(808\) 0 0
\(809\) 3.24105e11 0.756645 0.378322 0.925674i \(-0.376501\pi\)
0.378322 + 0.925674i \(0.376501\pi\)
\(810\) 0 0
\(811\) 6.94202e11i 1.60473i 0.596833 + 0.802366i \(0.296426\pi\)
−0.596833 + 0.802366i \(0.703574\pi\)
\(812\) 0 0
\(813\) −8.90904e10 −0.203924
\(814\) 0 0
\(815\) − 1.75859e11i − 0.398597i
\(816\) 0 0
\(817\) −2.94887e11 −0.661861
\(818\) 0 0
\(819\) 2.60826e11i 0.579716i
\(820\) 0 0
\(821\) 4.79835e11 1.05614 0.528068 0.849202i \(-0.322917\pi\)
0.528068 + 0.849202i \(0.322917\pi\)
\(822\) 0 0
\(823\) 3.34155e11i 0.728365i 0.931328 + 0.364183i \(0.118652\pi\)
−0.931328 + 0.364183i \(0.881348\pi\)
\(824\) 0 0
\(825\) 8.47508e10 0.182948
\(826\) 0 0
\(827\) − 6.22860e9i − 0.0133158i −0.999978 0.00665792i \(-0.997881\pi\)
0.999978 0.00665792i \(-0.00211930\pi\)
\(828\) 0 0
\(829\) 6.97808e11 1.47747 0.738733 0.673998i \(-0.235424\pi\)
0.738733 + 0.673998i \(0.235424\pi\)
\(830\) 0 0
\(831\) 2.35582e11i 0.494012i
\(832\) 0 0
\(833\) 2.34975e11 0.488024
\(834\) 0 0
\(835\) 2.82423e11i 0.580970i
\(836\) 0 0
\(837\) 1.22285e11 0.249155
\(838\) 0 0
\(839\) 1.01762e11i 0.205370i 0.994714 + 0.102685i \(0.0327433\pi\)
−0.994714 + 0.102685i \(0.967257\pi\)
\(840\) 0 0
\(841\) −4.59417e11 −0.918382
\(842\) 0 0
\(843\) − 3.11606e11i − 0.617014i
\(844\) 0 0
\(845\) 5.13952e11 1.00808
\(846\) 0 0
\(847\) − 1.15993e11i − 0.225372i
\(848\) 0 0
\(849\) −2.59218e11 −0.498924
\(850\) 0 0
\(851\) 9.41904e11i 1.79593i
\(852\) 0 0
\(853\) −8.81799e10 −0.166561 −0.0832805 0.996526i \(-0.526540\pi\)
−0.0832805 + 0.996526i \(0.526540\pi\)
\(854\) 0 0
\(855\) 2.06817e11i 0.387010i
\(856\) 0 0
\(857\) −7.95365e11 −1.47449 −0.737247 0.675623i \(-0.763875\pi\)
−0.737247 + 0.675623i \(0.763875\pi\)
\(858\) 0 0
\(859\) − 7.01767e11i − 1.28890i −0.764645 0.644452i \(-0.777086\pi\)
0.764645 0.644452i \(-0.222914\pi\)
\(860\) 0 0
\(861\) −4.41662e11 −0.803669
\(862\) 0 0
\(863\) 2.80009e11i 0.504811i 0.967622 + 0.252405i \(0.0812217\pi\)
−0.967622 + 0.252405i \(0.918778\pi\)
\(864\) 0 0
\(865\) −4.61062e10 −0.0823560
\(866\) 0 0
\(867\) 1.23660e11i 0.218854i
\(868\) 0 0
\(869\) 2.42645e11 0.425493
\(870\) 0 0
\(871\) 1.07061e12i 1.86019i
\(872\) 0 0
\(873\) 1.61623e11 0.278258
\(874\) 0 0
\(875\) 5.63802e11i 0.961822i
\(876\) 0 0
\(877\) −6.13288e11 −1.03673 −0.518365 0.855159i \(-0.673459\pi\)
−0.518365 + 0.855159i \(0.673459\pi\)
\(878\) 0 0
\(879\) − 3.12107e11i − 0.522816i
\(880\) 0 0
\(881\) −2.48326e11 −0.412210 −0.206105 0.978530i \(-0.566079\pi\)
−0.206105 + 0.978530i \(0.566079\pi\)
\(882\) 0 0
\(883\) − 3.43124e11i − 0.564428i −0.959351 0.282214i \(-0.908931\pi\)
0.959351 0.282214i \(-0.0910689\pi\)
\(884\) 0 0
\(885\) −1.95682e11 −0.318991
\(886\) 0 0
\(887\) − 1.46020e11i − 0.235895i −0.993020 0.117947i \(-0.962369\pi\)
0.993020 0.117947i \(-0.0376314\pi\)
\(888\) 0 0
\(889\) 8.26381e11 1.32304
\(890\) 0 0
\(891\) 6.35244e10i 0.100793i
\(892\) 0 0
\(893\) 8.28009e11 1.30205
\(894\) 0 0
\(895\) − 3.87415e11i − 0.603787i
\(896\) 0 0
\(897\) −9.16504e11 −1.41568
\(898\) 0 0
\(899\) − 2.41592e11i − 0.369866i
\(900\) 0 0
\(901\) 7.01894e10 0.106506
\(902\) 0 0
\(903\) 3.23472e11i 0.486504i
\(904\) 0 0
\(905\) 6.21717e11 0.926827
\(906\) 0 0
\(907\) 9.39725e11i 1.38858i 0.719695 + 0.694291i \(0.244282\pi\)
−0.719695 + 0.694291i \(0.755718\pi\)
\(908\) 0 0
\(909\) −4.18396e11 −0.612818
\(910\) 0 0
\(911\) 4.25662e11i 0.618004i 0.951061 + 0.309002i \(0.0999950\pi\)
−0.951061 + 0.309002i \(0.900005\pi\)
\(912\) 0 0
\(913\) 1.03449e11 0.148883
\(914\) 0 0
\(915\) − 5.82413e11i − 0.830897i
\(916\) 0 0
\(917\) 9.25312e11 1.30861
\(918\) 0 0
\(919\) 1.21782e12i 1.70734i 0.520815 + 0.853670i \(0.325628\pi\)
−0.520815 + 0.853670i \(0.674372\pi\)
\(920\) 0 0
\(921\) 3.03859e10 0.0422312
\(922\) 0 0
\(923\) 1.55487e12i 2.14233i
\(924\) 0 0
\(925\) −2.55986e11 −0.349663
\(926\) 0 0
\(927\) − 3.56001e11i − 0.482095i
\(928\) 0 0
\(929\) 9.97179e11 1.33878 0.669392 0.742910i \(-0.266555\pi\)
0.669392 + 0.742910i \(0.266555\pi\)
\(930\) 0 0
\(931\) 4.65055e11i 0.619022i
\(932\) 0 0
\(933\) 1.86048e10 0.0245526
\(934\) 0 0
\(935\) − 6.34596e11i − 0.830331i
\(936\) 0 0
\(937\) 3.43206e11 0.445243 0.222621 0.974905i \(-0.428539\pi\)
0.222621 + 0.974905i \(0.428539\pi\)
\(938\) 0 0
\(939\) 7.39906e11i 0.951730i
\(940\) 0 0
\(941\) 1.72310e11 0.219762 0.109881 0.993945i \(-0.464953\pi\)
0.109881 + 0.993945i \(0.464953\pi\)
\(942\) 0 0
\(943\) − 1.55193e12i − 1.96257i
\(944\) 0 0
\(945\) 2.26866e11 0.284473
\(946\) 0 0
\(947\) − 8.75107e11i − 1.08808i −0.839059 0.544041i \(-0.816894\pi\)
0.839059 0.544041i \(-0.183106\pi\)
\(948\) 0 0
\(949\) −2.07997e12 −2.56443
\(950\) 0 0
\(951\) 4.49339e11i 0.549353i
\(952\) 0 0
\(953\) −1.43349e12 −1.73789 −0.868946 0.494907i \(-0.835202\pi\)
−0.868946 + 0.494907i \(0.835202\pi\)
\(954\) 0 0
\(955\) − 3.45401e11i − 0.415250i
\(956\) 0 0
\(957\) 1.25502e11 0.149625
\(958\) 0 0
\(959\) − 1.95638e12i − 2.31302i
\(960\) 0 0
\(961\) −5.76651e11 −0.676114
\(962\) 0 0
\(963\) 4.39172e11i 0.510657i
\(964\) 0 0
\(965\) 6.36490e11 0.733977
\(966\) 0 0
\(967\) − 1.53627e12i − 1.75696i −0.477777 0.878481i \(-0.658557\pi\)
0.477777 0.878481i \(-0.341443\pi\)
\(968\) 0 0
\(969\) 4.00908e11 0.454725
\(970\) 0 0
\(971\) 2.88911e11i 0.325003i 0.986708 + 0.162501i \(0.0519562\pi\)
−0.986708 + 0.162501i \(0.948044\pi\)
\(972\) 0 0
\(973\) 1.49930e12 1.67277
\(974\) 0 0
\(975\) − 2.49083e11i − 0.275630i
\(976\) 0 0
\(977\) 4.59815e11 0.504667 0.252333 0.967640i \(-0.418802\pi\)
0.252333 + 0.967640i \(0.418802\pi\)
\(978\) 0 0
\(979\) 1.15106e12i 1.25305i
\(980\) 0 0
\(981\) −1.50046e11 −0.162013
\(982\) 0 0
\(983\) − 3.07463e11i − 0.329291i −0.986353 0.164645i \(-0.947352\pi\)
0.986353 0.164645i \(-0.0526479\pi\)
\(984\) 0 0
\(985\) −2.00929e12 −2.13451
\(986\) 0 0
\(987\) − 9.08274e11i − 0.957080i
\(988\) 0 0
\(989\) −1.13663e12 −1.18805
\(990\) 0 0
\(991\) − 1.10256e12i − 1.14316i −0.820547 0.571579i \(-0.806331\pi\)
0.820547 0.571579i \(-0.193669\pi\)
\(992\) 0 0
\(993\) −6.64026e11 −0.682949
\(994\) 0 0
\(995\) − 1.03769e12i − 1.05871i
\(996\) 0 0
\(997\) 5.02913e10 0.0508993 0.0254497 0.999676i \(-0.491898\pi\)
0.0254497 + 0.999676i \(0.491898\pi\)
\(998\) 0 0
\(999\) − 1.91873e11i − 0.192642i
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.b.31.1 2
3.2 odd 2 144.9.g.c.127.1 2
4.3 odd 2 inner 48.9.g.b.31.2 yes 2
8.3 odd 2 192.9.g.a.127.1 2
8.5 even 2 192.9.g.a.127.2 2
12.11 even 2 144.9.g.c.127.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.b.31.1 2 1.1 even 1 trivial
48.9.g.b.31.2 yes 2 4.3 odd 2 inner
144.9.g.c.127.1 2 3.2 odd 2
144.9.g.c.127.2 2 12.11 even 2
192.9.g.a.127.1 2 8.3 odd 2
192.9.g.a.127.2 2 8.5 even 2