Properties

Label 384.10.d.f.193.8
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,10,Mod(193,384)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (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: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{175}\cdot 3^{32} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.8
Root \(4.28001 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.f.193.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-81.0000i q^{3} +1655.59i q^{5} +11370.0 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q-81.0000i q^{3} +1655.59i q^{5} +11370.0 q^{7} -6561.00 q^{9} +94507.4i q^{11} -140234. i q^{13} +134103. q^{15} -531000. q^{17} +387485. i q^{19} -920969. i q^{21} +2.36100e6 q^{23} -787863. q^{25} +531441. i q^{27} +2.70715e6i q^{29} +5.18531e6 q^{31} +7.65510e6 q^{33} +1.88241e7i q^{35} -142841. i q^{37} -1.13589e7 q^{39} +1.92825e6 q^{41} +1.65243e7i q^{43} -1.08623e7i q^{45} +1.78851e7 q^{47} +8.89229e7 q^{49} +4.30110e7i q^{51} -2.01018e6i q^{53} -1.56466e8 q^{55} +3.13863e7 q^{57} +9.71099e7i q^{59} +7.23555e6i q^{61} -7.45985e7 q^{63} +2.32170e8 q^{65} -2.34701e8i q^{67} -1.91241e8i q^{69} +2.62515e7 q^{71} -3.59965e8 q^{73} +6.38169e7i q^{75} +1.07455e9i q^{77} +3.01072e8 q^{79} +4.30467e7 q^{81} -3.48363e8i q^{83} -8.79119e8i q^{85} +2.19279e8 q^{87} +9.32242e8 q^{89} -1.59446e9i q^{91} -4.20010e8i q^{93} -6.41517e8 q^{95} -9.09752e8 q^{97} -6.20063e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 131220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) 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
\(3\) − 81.0000i − 0.577350i
\(4\) 0 0
\(5\) 1655.59i 1.18465i 0.805701 + 0.592323i \(0.201789\pi\)
−0.805701 + 0.592323i \(0.798211\pi\)
\(6\) 0 0
\(7\) 11370.0 1.78986 0.894929 0.446208i \(-0.147226\pi\)
0.894929 + 0.446208i \(0.147226\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 94507.4i 1.94625i 0.230276 + 0.973125i \(0.426037\pi\)
−0.230276 + 0.973125i \(0.573963\pi\)
\(12\) 0 0
\(13\) − 140234.i − 1.36178i −0.732385 0.680891i \(-0.761593\pi\)
0.732385 0.680891i \(-0.238407\pi\)
\(14\) 0 0
\(15\) 134103. 0.683956
\(16\) 0 0
\(17\) −531000. −1.54196 −0.770982 0.636857i \(-0.780234\pi\)
−0.770982 + 0.636857i \(0.780234\pi\)
\(18\) 0 0
\(19\) 387485.i 0.682124i 0.940041 + 0.341062i \(0.110787\pi\)
−0.940041 + 0.341062i \(0.889213\pi\)
\(20\) 0 0
\(21\) − 920969.i − 1.03338i
\(22\) 0 0
\(23\) 2.36100e6 1.75922 0.879610 0.475695i \(-0.157803\pi\)
0.879610 + 0.475695i \(0.157803\pi\)
\(24\) 0 0
\(25\) −787863. −0.403386
\(26\) 0 0
\(27\) 531441.i 0.192450i
\(28\) 0 0
\(29\) 2.70715e6i 0.710756i 0.934723 + 0.355378i \(0.115648\pi\)
−0.934723 + 0.355378i \(0.884352\pi\)
\(30\) 0 0
\(31\) 5.18531e6 1.00843 0.504216 0.863577i \(-0.331781\pi\)
0.504216 + 0.863577i \(0.331781\pi\)
\(32\) 0 0
\(33\) 7.65510e6 1.12367
\(34\) 0 0
\(35\) 1.88241e7i 2.12035i
\(36\) 0 0
\(37\) − 142841.i − 0.0125298i −0.999980 0.00626492i \(-0.998006\pi\)
0.999980 0.00626492i \(-0.00199420\pi\)
\(38\) 0 0
\(39\) −1.13589e7 −0.786225
\(40\) 0 0
\(41\) 1.92825e6 0.106570 0.0532851 0.998579i \(-0.483031\pi\)
0.0532851 + 0.998579i \(0.483031\pi\)
\(42\) 0 0
\(43\) 1.65243e7i 0.737080i 0.929612 + 0.368540i \(0.120142\pi\)
−0.929612 + 0.368540i \(0.879858\pi\)
\(44\) 0 0
\(45\) − 1.08623e7i − 0.394882i
\(46\) 0 0
\(47\) 1.78851e7 0.534627 0.267314 0.963610i \(-0.413864\pi\)
0.267314 + 0.963610i \(0.413864\pi\)
\(48\) 0 0
\(49\) 8.89229e7 2.20359
\(50\) 0 0
\(51\) 4.30110e7i 0.890253i
\(52\) 0 0
\(53\) − 2.01018e6i − 0.0349940i −0.999847 0.0174970i \(-0.994430\pi\)
0.999847 0.0174970i \(-0.00556975\pi\)
\(54\) 0 0
\(55\) −1.56466e8 −2.30562
\(56\) 0 0
\(57\) 3.13863e7 0.393825
\(58\) 0 0
\(59\) 9.71099e7i 1.04335i 0.853145 + 0.521674i \(0.174692\pi\)
−0.853145 + 0.521674i \(0.825308\pi\)
\(60\) 0 0
\(61\) 7.23555e6i 0.0669094i 0.999440 + 0.0334547i \(0.0106509\pi\)
−0.999440 + 0.0334547i \(0.989349\pi\)
\(62\) 0 0
\(63\) −7.45985e7 −0.596619
\(64\) 0 0
\(65\) 2.32170e8 1.61323
\(66\) 0 0
\(67\) − 2.34701e8i − 1.42291i −0.702730 0.711457i \(-0.748036\pi\)
0.702730 0.711457i \(-0.251964\pi\)
\(68\) 0 0
\(69\) − 1.91241e8i − 1.01569i
\(70\) 0 0
\(71\) 2.62515e7 0.122600 0.0613002 0.998119i \(-0.480475\pi\)
0.0613002 + 0.998119i \(0.480475\pi\)
\(72\) 0 0
\(73\) −3.59965e8 −1.48357 −0.741785 0.670638i \(-0.766020\pi\)
−0.741785 + 0.670638i \(0.766020\pi\)
\(74\) 0 0
\(75\) 6.38169e7i 0.232895i
\(76\) 0 0
\(77\) 1.07455e9i 3.48351i
\(78\) 0 0
\(79\) 3.01072e8 0.869658 0.434829 0.900513i \(-0.356809\pi\)
0.434829 + 0.900513i \(0.356809\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) − 3.48363e8i − 0.805714i −0.915263 0.402857i \(-0.868017\pi\)
0.915263 0.402857i \(-0.131983\pi\)
\(84\) 0 0
\(85\) − 8.79119e8i − 1.82668i
\(86\) 0 0
\(87\) 2.19279e8 0.410355
\(88\) 0 0
\(89\) 9.32242e8 1.57497 0.787487 0.616331i \(-0.211382\pi\)
0.787487 + 0.616331i \(0.211382\pi\)
\(90\) 0 0
\(91\) − 1.59446e9i − 2.43740i
\(92\) 0 0
\(93\) − 4.20010e8i − 0.582219i
\(94\) 0 0
\(95\) −6.41517e8 −0.808075
\(96\) 0 0
\(97\) −9.09752e8 −1.04340 −0.521699 0.853129i \(-0.674702\pi\)
−0.521699 + 0.853129i \(0.674702\pi\)
\(98\) 0 0
\(99\) − 6.20063e8i − 0.648750i
\(100\) 0 0
\(101\) − 3.56743e8i − 0.341121i −0.985347 0.170561i \(-0.945442\pi\)
0.985347 0.170561i \(-0.0545579\pi\)
\(102\) 0 0
\(103\) −1.27419e9 −1.11550 −0.557748 0.830010i \(-0.688334\pi\)
−0.557748 + 0.830010i \(0.688334\pi\)
\(104\) 0 0
\(105\) 1.52475e9 1.22418
\(106\) 0 0
\(107\) 1.90069e9i 1.40179i 0.713264 + 0.700896i \(0.247216\pi\)
−0.713264 + 0.700896i \(0.752784\pi\)
\(108\) 0 0
\(109\) 1.92409e8i 0.130559i 0.997867 + 0.0652793i \(0.0207938\pi\)
−0.997867 + 0.0652793i \(0.979206\pi\)
\(110\) 0 0
\(111\) −1.15701e7 −0.00723411
\(112\) 0 0
\(113\) −2.29060e9 −1.32159 −0.660793 0.750568i \(-0.729780\pi\)
−0.660793 + 0.750568i \(0.729780\pi\)
\(114\) 0 0
\(115\) 3.90885e9i 2.08405i
\(116\) 0 0
\(117\) 9.20074e8i 0.453927i
\(118\) 0 0
\(119\) −6.03746e9 −2.75990
\(120\) 0 0
\(121\) −6.57370e9 −2.78789
\(122\) 0 0
\(123\) − 1.56188e8i − 0.0615283i
\(124\) 0 0
\(125\) 1.92920e9i 0.706776i
\(126\) 0 0
\(127\) 9.60539e7 0.0327641 0.0163821 0.999866i \(-0.494785\pi\)
0.0163821 + 0.999866i \(0.494785\pi\)
\(128\) 0 0
\(129\) 1.33847e9 0.425554
\(130\) 0 0
\(131\) 2.10259e9i 0.623783i 0.950118 + 0.311892i \(0.100963\pi\)
−0.950118 + 0.311892i \(0.899037\pi\)
\(132\) 0 0
\(133\) 4.40569e9i 1.22091i
\(134\) 0 0
\(135\) −8.79850e8 −0.227985
\(136\) 0 0
\(137\) −8.18137e8 −0.198419 −0.0992095 0.995067i \(-0.531631\pi\)
−0.0992095 + 0.995067i \(0.531631\pi\)
\(138\) 0 0
\(139\) 7.00619e8i 0.159190i 0.996827 + 0.0795950i \(0.0253627\pi\)
−0.996827 + 0.0795950i \(0.974637\pi\)
\(140\) 0 0
\(141\) − 1.44869e9i − 0.308667i
\(142\) 0 0
\(143\) 1.32531e10 2.65037
\(144\) 0 0
\(145\) −4.48193e9 −0.841994
\(146\) 0 0
\(147\) − 7.20276e9i − 1.27224i
\(148\) 0 0
\(149\) − 4.80548e9i − 0.798728i −0.916792 0.399364i \(-0.869231\pi\)
0.916792 0.399364i \(-0.130769\pi\)
\(150\) 0 0
\(151\) 1.05075e9 0.164477 0.0822383 0.996613i \(-0.473793\pi\)
0.0822383 + 0.996613i \(0.473793\pi\)
\(152\) 0 0
\(153\) 3.48389e9 0.513988
\(154\) 0 0
\(155\) 8.58476e9i 1.19464i
\(156\) 0 0
\(157\) 2.84280e9i 0.373421i 0.982415 + 0.186710i \(0.0597826\pi\)
−0.982415 + 0.186710i \(0.940217\pi\)
\(158\) 0 0
\(159\) −1.62825e8 −0.0202038
\(160\) 0 0
\(161\) 2.68445e10 3.14876
\(162\) 0 0
\(163\) − 1.36183e9i − 0.151105i −0.997142 0.0755527i \(-0.975928\pi\)
0.997142 0.0755527i \(-0.0240721\pi\)
\(164\) 0 0
\(165\) 1.26737e10i 1.33115i
\(166\) 0 0
\(167\) 3.81425e9 0.379477 0.189738 0.981835i \(-0.439236\pi\)
0.189738 + 0.981835i \(0.439236\pi\)
\(168\) 0 0
\(169\) −9.06101e9 −0.854450
\(170\) 0 0
\(171\) − 2.54229e9i − 0.227375i
\(172\) 0 0
\(173\) − 1.35181e10i − 1.14738i −0.819071 0.573692i \(-0.805511\pi\)
0.819071 0.573692i \(-0.194489\pi\)
\(174\) 0 0
\(175\) −8.95799e9 −0.722004
\(176\) 0 0
\(177\) 7.86590e9 0.602378
\(178\) 0 0
\(179\) 2.34665e9i 0.170848i 0.996345 + 0.0854240i \(0.0272245\pi\)
−0.996345 + 0.0854240i \(0.972776\pi\)
\(180\) 0 0
\(181\) 2.69581e10i 1.86696i 0.358627 + 0.933481i \(0.383245\pi\)
−0.358627 + 0.933481i \(0.616755\pi\)
\(182\) 0 0
\(183\) 5.86079e8 0.0386302
\(184\) 0 0
\(185\) 2.36487e8 0.0148434
\(186\) 0 0
\(187\) − 5.01834e10i − 3.00105i
\(188\) 0 0
\(189\) 6.04248e9i 0.344458i
\(190\) 0 0
\(191\) −3.44999e9 −0.187572 −0.0937859 0.995592i \(-0.529897\pi\)
−0.0937859 + 0.995592i \(0.529897\pi\)
\(192\) 0 0
\(193\) 1.93617e10 1.00447 0.502234 0.864732i \(-0.332512\pi\)
0.502234 + 0.864732i \(0.332512\pi\)
\(194\) 0 0
\(195\) − 1.88058e10i − 0.931398i
\(196\) 0 0
\(197\) 2.11248e10i 0.999298i 0.866228 + 0.499649i \(0.166538\pi\)
−0.866228 + 0.499649i \(0.833462\pi\)
\(198\) 0 0
\(199\) 3.06345e10 1.38475 0.692376 0.721536i \(-0.256564\pi\)
0.692376 + 0.721536i \(0.256564\pi\)
\(200\) 0 0
\(201\) −1.90108e10 −0.821519
\(202\) 0 0
\(203\) 3.07802e10i 1.27215i
\(204\) 0 0
\(205\) 3.19239e9i 0.126248i
\(206\) 0 0
\(207\) −1.54905e10 −0.586407
\(208\) 0 0
\(209\) −3.66202e10 −1.32758
\(210\) 0 0
\(211\) − 3.75426e9i − 0.130393i −0.997872 0.0651964i \(-0.979233\pi\)
0.997872 0.0651964i \(-0.0207674\pi\)
\(212\) 0 0
\(213\) − 2.12637e9i − 0.0707834i
\(214\) 0 0
\(215\) −2.73575e10 −0.873179
\(216\) 0 0
\(217\) 5.89569e10 1.80495
\(218\) 0 0
\(219\) 2.91572e10i 0.856539i
\(220\) 0 0
\(221\) 7.44641e10i 2.09982i
\(222\) 0 0
\(223\) −5.80461e9 −0.157181 −0.0785907 0.996907i \(-0.525042\pi\)
−0.0785907 + 0.996907i \(0.525042\pi\)
\(224\) 0 0
\(225\) 5.16917e9 0.134462
\(226\) 0 0
\(227\) 4.37575e10i 1.09379i 0.837200 + 0.546897i \(0.184191\pi\)
−0.837200 + 0.546897i \(0.815809\pi\)
\(228\) 0 0
\(229\) − 5.00629e10i − 1.20298i −0.798882 0.601488i \(-0.794575\pi\)
0.798882 0.601488i \(-0.205425\pi\)
\(230\) 0 0
\(231\) 8.70384e10 2.01121
\(232\) 0 0
\(233\) −5.78995e10 −1.28698 −0.643492 0.765453i \(-0.722515\pi\)
−0.643492 + 0.765453i \(0.722515\pi\)
\(234\) 0 0
\(235\) 2.96104e10i 0.633344i
\(236\) 0 0
\(237\) − 2.43868e10i − 0.502097i
\(238\) 0 0
\(239\) 6.59252e10 1.30696 0.653478 0.756945i \(-0.273309\pi\)
0.653478 + 0.756945i \(0.273309\pi\)
\(240\) 0 0
\(241\) −1.03616e10 −0.197857 −0.0989284 0.995095i \(-0.531541\pi\)
−0.0989284 + 0.995095i \(0.531541\pi\)
\(242\) 0 0
\(243\) − 3.48678e9i − 0.0641500i
\(244\) 0 0
\(245\) 1.47220e11i 2.61048i
\(246\) 0 0
\(247\) 5.43384e10 0.928904
\(248\) 0 0
\(249\) −2.82174e10 −0.465179
\(250\) 0 0
\(251\) 7.88299e10i 1.25360i 0.779180 + 0.626800i \(0.215636\pi\)
−0.779180 + 0.626800i \(0.784364\pi\)
\(252\) 0 0
\(253\) 2.23132e11i 3.42388i
\(254\) 0 0
\(255\) −7.12087e10 −1.05463
\(256\) 0 0
\(257\) 4.14179e9 0.0592228 0.0296114 0.999561i \(-0.490573\pi\)
0.0296114 + 0.999561i \(0.490573\pi\)
\(258\) 0 0
\(259\) − 1.62410e9i − 0.0224267i
\(260\) 0 0
\(261\) − 1.77616e10i − 0.236919i
\(262\) 0 0
\(263\) −6.51316e10 −0.839442 −0.419721 0.907653i \(-0.637872\pi\)
−0.419721 + 0.907653i \(0.637872\pi\)
\(264\) 0 0
\(265\) 3.32804e9 0.0414555
\(266\) 0 0
\(267\) − 7.55116e10i − 0.909312i
\(268\) 0 0
\(269\) 4.43073e10i 0.515930i 0.966154 + 0.257965i \(0.0830519\pi\)
−0.966154 + 0.257965i \(0.916948\pi\)
\(270\) 0 0
\(271\) −9.19147e10 −1.03520 −0.517598 0.855624i \(-0.673174\pi\)
−0.517598 + 0.855624i \(0.673174\pi\)
\(272\) 0 0
\(273\) −1.29151e11 −1.40723
\(274\) 0 0
\(275\) − 7.44589e10i − 0.785090i
\(276\) 0 0
\(277\) − 3.23505e10i − 0.330158i −0.986280 0.165079i \(-0.947212\pi\)
0.986280 0.165079i \(-0.0527880\pi\)
\(278\) 0 0
\(279\) −3.40208e10 −0.336144
\(280\) 0 0
\(281\) −5.98617e10 −0.572757 −0.286379 0.958117i \(-0.592451\pi\)
−0.286379 + 0.958117i \(0.592451\pi\)
\(282\) 0 0
\(283\) 1.24438e11i 1.15323i 0.817017 + 0.576614i \(0.195626\pi\)
−0.817017 + 0.576614i \(0.804374\pi\)
\(284\) 0 0
\(285\) 5.19629e10i 0.466543i
\(286\) 0 0
\(287\) 2.19241e10 0.190745
\(288\) 0 0
\(289\) 1.63373e11 1.37765
\(290\) 0 0
\(291\) 7.36899e10i 0.602407i
\(292\) 0 0
\(293\) 9.22003e10i 0.730850i 0.930841 + 0.365425i \(0.119076\pi\)
−0.930841 + 0.365425i \(0.880924\pi\)
\(294\) 0 0
\(295\) −1.60774e11 −1.23600
\(296\) 0 0
\(297\) −5.02251e10 −0.374556
\(298\) 0 0
\(299\) − 3.31092e11i − 2.39567i
\(300\) 0 0
\(301\) 1.87881e11i 1.31927i
\(302\) 0 0
\(303\) −2.88961e10 −0.196946
\(304\) 0 0
\(305\) −1.19791e10 −0.0792639
\(306\) 0 0
\(307\) − 2.08578e11i − 1.34013i −0.742303 0.670064i \(-0.766267\pi\)
0.742303 0.670064i \(-0.233733\pi\)
\(308\) 0 0
\(309\) 1.03210e11i 0.644032i
\(310\) 0 0
\(311\) −1.15604e11 −0.700728 −0.350364 0.936614i \(-0.613942\pi\)
−0.350364 + 0.936614i \(0.613942\pi\)
\(312\) 0 0
\(313\) −1.38691e11 −0.816769 −0.408384 0.912810i \(-0.633908\pi\)
−0.408384 + 0.912810i \(0.633908\pi\)
\(314\) 0 0
\(315\) − 1.23505e11i − 0.706783i
\(316\) 0 0
\(317\) 5.75006e10i 0.319820i 0.987132 + 0.159910i \(0.0511204\pi\)
−0.987132 + 0.159910i \(0.948880\pi\)
\(318\) 0 0
\(319\) −2.55845e11 −1.38331
\(320\) 0 0
\(321\) 1.53956e11 0.809325
\(322\) 0 0
\(323\) − 2.05754e11i − 1.05181i
\(324\) 0 0
\(325\) 1.10485e11i 0.549324i
\(326\) 0 0
\(327\) 1.55851e10 0.0753780
\(328\) 0 0
\(329\) 2.03353e11 0.956907
\(330\) 0 0
\(331\) 3.43458e11i 1.57271i 0.617776 + 0.786354i \(0.288034\pi\)
−0.617776 + 0.786354i \(0.711966\pi\)
\(332\) 0 0
\(333\) 9.37181e8i 0.00417662i
\(334\) 0 0
\(335\) 3.88569e11 1.68565
\(336\) 0 0
\(337\) 3.16926e11 1.33851 0.669257 0.743031i \(-0.266612\pi\)
0.669257 + 0.743031i \(0.266612\pi\)
\(338\) 0 0
\(339\) 1.85538e11i 0.763019i
\(340\) 0 0
\(341\) 4.90050e11i 1.96266i
\(342\) 0 0
\(343\) 5.52232e11 2.15426
\(344\) 0 0
\(345\) 3.16617e11 1.20323
\(346\) 0 0
\(347\) 6.34668e10i 0.234998i 0.993073 + 0.117499i \(0.0374877\pi\)
−0.993073 + 0.117499i \(0.962512\pi\)
\(348\) 0 0
\(349\) 4.80264e11i 1.73287i 0.499291 + 0.866434i \(0.333594\pi\)
−0.499291 + 0.866434i \(0.666406\pi\)
\(350\) 0 0
\(351\) 7.45260e10 0.262075
\(352\) 0 0
\(353\) 5.25576e10 0.180156 0.0900781 0.995935i \(-0.471288\pi\)
0.0900781 + 0.995935i \(0.471288\pi\)
\(354\) 0 0
\(355\) 4.34618e10i 0.145238i
\(356\) 0 0
\(357\) 4.89034e11i 1.59343i
\(358\) 0 0
\(359\) −3.68799e11 −1.17183 −0.585916 0.810372i \(-0.699265\pi\)
−0.585916 + 0.810372i \(0.699265\pi\)
\(360\) 0 0
\(361\) 1.72543e11 0.534707
\(362\) 0 0
\(363\) 5.32470e11i 1.60959i
\(364\) 0 0
\(365\) − 5.95956e11i − 1.75750i
\(366\) 0 0
\(367\) 2.50239e11 0.720043 0.360022 0.932944i \(-0.382769\pi\)
0.360022 + 0.932944i \(0.382769\pi\)
\(368\) 0 0
\(369\) −1.26512e10 −0.0355234
\(370\) 0 0
\(371\) − 2.28557e10i − 0.0626343i
\(372\) 0 0
\(373\) 6.47052e11i 1.73081i 0.501073 + 0.865405i \(0.332939\pi\)
−0.501073 + 0.865405i \(0.667061\pi\)
\(374\) 0 0
\(375\) 1.56265e11 0.408058
\(376\) 0 0
\(377\) 3.79633e11 0.967895
\(378\) 0 0
\(379\) 3.44133e11i 0.856742i 0.903603 + 0.428371i \(0.140912\pi\)
−0.903603 + 0.428371i \(0.859088\pi\)
\(380\) 0 0
\(381\) − 7.78037e9i − 0.0189164i
\(382\) 0 0
\(383\) −2.36072e11 −0.560595 −0.280298 0.959913i \(-0.590433\pi\)
−0.280298 + 0.959913i \(0.590433\pi\)
\(384\) 0 0
\(385\) −1.77901e12 −4.12673
\(386\) 0 0
\(387\) − 1.08416e11i − 0.245693i
\(388\) 0 0
\(389\) − 2.69565e11i − 0.596885i −0.954428 0.298442i \(-0.903533\pi\)
0.954428 0.298442i \(-0.0964671\pi\)
\(390\) 0 0
\(391\) −1.25369e12 −2.71265
\(392\) 0 0
\(393\) 1.70310e11 0.360141
\(394\) 0 0
\(395\) 4.98453e11i 1.03024i
\(396\) 0 0
\(397\) − 2.24972e11i − 0.454540i −0.973832 0.227270i \(-0.927020\pi\)
0.973832 0.227270i \(-0.0729800\pi\)
\(398\) 0 0
\(399\) 3.56861e11 0.704890
\(400\) 0 0
\(401\) −2.94935e11 −0.569609 −0.284805 0.958586i \(-0.591929\pi\)
−0.284805 + 0.958586i \(0.591929\pi\)
\(402\) 0 0
\(403\) − 7.27155e11i − 1.37327i
\(404\) 0 0
\(405\) 7.12678e10i 0.131627i
\(406\) 0 0
\(407\) 1.34996e10 0.0243862
\(408\) 0 0
\(409\) −7.05914e11 −1.24737 −0.623687 0.781674i \(-0.714366\pi\)
−0.623687 + 0.781674i \(0.714366\pi\)
\(410\) 0 0
\(411\) 6.62691e10i 0.114557i
\(412\) 0 0
\(413\) 1.10414e12i 1.86745i
\(414\) 0 0
\(415\) 5.76748e11 0.954486
\(416\) 0 0
\(417\) 5.67502e10 0.0919083
\(418\) 0 0
\(419\) − 6.73207e11i − 1.06705i −0.845784 0.533526i \(-0.820867\pi\)
0.845784 0.533526i \(-0.179133\pi\)
\(420\) 0 0
\(421\) − 7.27739e11i − 1.12903i −0.825422 0.564516i \(-0.809063\pi\)
0.825422 0.564516i \(-0.190937\pi\)
\(422\) 0 0
\(423\) −1.17344e11 −0.178209
\(424\) 0 0
\(425\) 4.18355e11 0.622006
\(426\) 0 0
\(427\) 8.22680e10i 0.119758i
\(428\) 0 0
\(429\) − 1.07350e12i − 1.53019i
\(430\) 0 0
\(431\) 5.32994e11 0.744003 0.372001 0.928232i \(-0.378672\pi\)
0.372001 + 0.928232i \(0.378672\pi\)
\(432\) 0 0
\(433\) 2.23588e11 0.305670 0.152835 0.988252i \(-0.451160\pi\)
0.152835 + 0.988252i \(0.451160\pi\)
\(434\) 0 0
\(435\) 3.63036e11i 0.486126i
\(436\) 0 0
\(437\) 9.14851e11i 1.20001i
\(438\) 0 0
\(439\) 9.15099e10 0.117592 0.0587960 0.998270i \(-0.481274\pi\)
0.0587960 + 0.998270i \(0.481274\pi\)
\(440\) 0 0
\(441\) −5.83423e11 −0.734531
\(442\) 0 0
\(443\) − 6.99027e11i − 0.862337i −0.902271 0.431169i \(-0.858101\pi\)
0.902271 0.431169i \(-0.141899\pi\)
\(444\) 0 0
\(445\) 1.54341e12i 1.86579i
\(446\) 0 0
\(447\) −3.89244e11 −0.461146
\(448\) 0 0
\(449\) 2.69460e11 0.312886 0.156443 0.987687i \(-0.449997\pi\)
0.156443 + 0.987687i \(0.449997\pi\)
\(450\) 0 0
\(451\) 1.82234e11i 0.207412i
\(452\) 0 0
\(453\) − 8.51109e10i − 0.0949606i
\(454\) 0 0
\(455\) 2.63977e12 2.88745
\(456\) 0 0
\(457\) 7.28944e11 0.781756 0.390878 0.920443i \(-0.372171\pi\)
0.390878 + 0.920443i \(0.372171\pi\)
\(458\) 0 0
\(459\) − 2.82195e11i − 0.296751i
\(460\) 0 0
\(461\) 1.09764e12i 1.13190i 0.824440 + 0.565949i \(0.191490\pi\)
−0.824440 + 0.565949i \(0.808510\pi\)
\(462\) 0 0
\(463\) 2.32754e11 0.235387 0.117693 0.993050i \(-0.462450\pi\)
0.117693 + 0.993050i \(0.462450\pi\)
\(464\) 0 0
\(465\) 6.95365e11 0.689723
\(466\) 0 0
\(467\) − 6.68871e11i − 0.650754i −0.945584 0.325377i \(-0.894509\pi\)
0.945584 0.325377i \(-0.105491\pi\)
\(468\) 0 0
\(469\) − 2.66855e12i − 2.54681i
\(470\) 0 0
\(471\) 2.30267e11 0.215595
\(472\) 0 0
\(473\) −1.56167e12 −1.43454
\(474\) 0 0
\(475\) − 3.05285e11i − 0.275159i
\(476\) 0 0
\(477\) 1.31888e10i 0.0116647i
\(478\) 0 0
\(479\) −1.61992e12 −1.40599 −0.702996 0.711194i \(-0.748155\pi\)
−0.702996 + 0.711194i \(0.748155\pi\)
\(480\) 0 0
\(481\) −2.00312e10 −0.0170629
\(482\) 0 0
\(483\) − 2.17441e12i − 1.81794i
\(484\) 0 0
\(485\) − 1.50618e12i − 1.23606i
\(486\) 0 0
\(487\) 2.49759e11 0.201206 0.100603 0.994927i \(-0.467923\pi\)
0.100603 + 0.994927i \(0.467923\pi\)
\(488\) 0 0
\(489\) −1.10309e11 −0.0872407
\(490\) 0 0
\(491\) 1.89367e12i 1.47040i 0.677848 + 0.735202i \(0.262913\pi\)
−0.677848 + 0.735202i \(0.737087\pi\)
\(492\) 0 0
\(493\) − 1.43749e12i − 1.09596i
\(494\) 0 0
\(495\) 1.02657e12 0.768539
\(496\) 0 0
\(497\) 2.98479e11 0.219437
\(498\) 0 0
\(499\) − 4.18338e11i − 0.302047i −0.988530 0.151024i \(-0.951743\pi\)
0.988530 0.151024i \(-0.0482569\pi\)
\(500\) 0 0
\(501\) − 3.08954e11i − 0.219091i
\(502\) 0 0
\(503\) 2.09661e12 1.46036 0.730182 0.683253i \(-0.239435\pi\)
0.730182 + 0.683253i \(0.239435\pi\)
\(504\) 0 0
\(505\) 5.90620e11 0.404108
\(506\) 0 0
\(507\) 7.33942e11i 0.493317i
\(508\) 0 0
\(509\) − 1.81715e12i − 1.19994i −0.800022 0.599970i \(-0.795179\pi\)
0.800022 0.599970i \(-0.204821\pi\)
\(510\) 0 0
\(511\) −4.09280e12 −2.65538
\(512\) 0 0
\(513\) −2.05925e11 −0.131275
\(514\) 0 0
\(515\) − 2.10955e12i − 1.32147i
\(516\) 0 0
\(517\) 1.69027e12i 1.04052i
\(518\) 0 0
\(519\) −1.09497e12 −0.662443
\(520\) 0 0
\(521\) −8.17238e11 −0.485936 −0.242968 0.970034i \(-0.578121\pi\)
−0.242968 + 0.970034i \(0.578121\pi\)
\(522\) 0 0
\(523\) 1.36193e12i 0.795969i 0.917392 + 0.397984i \(0.130290\pi\)
−0.917392 + 0.397984i \(0.869710\pi\)
\(524\) 0 0
\(525\) 7.25597e11i 0.416849i
\(526\) 0 0
\(527\) −2.75340e12 −1.55497
\(528\) 0 0
\(529\) 3.77316e12 2.09486
\(530\) 0 0
\(531\) − 6.37138e11i − 0.347783i
\(532\) 0 0
\(533\) − 2.70406e11i − 0.145125i
\(534\) 0 0
\(535\) −3.14676e12 −1.66063
\(536\) 0 0
\(537\) 1.90079e11 0.0986392
\(538\) 0 0
\(539\) 8.40387e12i 4.28874i
\(540\) 0 0
\(541\) − 2.08586e12i − 1.04688i −0.852063 0.523440i \(-0.824649\pi\)
0.852063 0.523440i \(-0.175351\pi\)
\(542\) 0 0
\(543\) 2.18360e12 1.07789
\(544\) 0 0
\(545\) −3.18550e11 −0.154666
\(546\) 0 0
\(547\) 2.54074e12i 1.21344i 0.794917 + 0.606719i \(0.207515\pi\)
−0.794917 + 0.606719i \(0.792485\pi\)
\(548\) 0 0
\(549\) − 4.74724e10i − 0.0223031i
\(550\) 0 0
\(551\) −1.04898e12 −0.484824
\(552\) 0 0
\(553\) 3.42319e12 1.55657
\(554\) 0 0
\(555\) − 1.91554e10i − 0.00856986i
\(556\) 0 0
\(557\) − 2.62379e12i − 1.15500i −0.816391 0.577499i \(-0.804029\pi\)
0.816391 0.577499i \(-0.195971\pi\)
\(558\) 0 0
\(559\) 2.31726e12 1.00374
\(560\) 0 0
\(561\) −4.06485e12 −1.73266
\(562\) 0 0
\(563\) 1.67317e12i 0.701863i 0.936401 + 0.350932i \(0.114135\pi\)
−0.936401 + 0.350932i \(0.885865\pi\)
\(564\) 0 0
\(565\) − 3.79230e12i − 1.56561i
\(566\) 0 0
\(567\) 4.89441e11 0.198873
\(568\) 0 0
\(569\) 3.63787e12 1.45493 0.727466 0.686144i \(-0.240698\pi\)
0.727466 + 0.686144i \(0.240698\pi\)
\(570\) 0 0
\(571\) − 2.12669e12i − 0.837223i −0.908165 0.418612i \(-0.862517\pi\)
0.908165 0.418612i \(-0.137483\pi\)
\(572\) 0 0
\(573\) 2.79449e11i 0.108295i
\(574\) 0 0
\(575\) −1.86014e12 −0.709645
\(576\) 0 0
\(577\) 2.52863e12 0.949716 0.474858 0.880062i \(-0.342499\pi\)
0.474858 + 0.880062i \(0.342499\pi\)
\(578\) 0 0
\(579\) − 1.56830e12i − 0.579930i
\(580\) 0 0
\(581\) − 3.96088e12i − 1.44211i
\(582\) 0 0
\(583\) 1.89977e11 0.0681071
\(584\) 0 0
\(585\) −1.52327e12 −0.537743
\(586\) 0 0
\(587\) − 1.94162e12i − 0.674981i −0.941329 0.337491i \(-0.890422\pi\)
0.941329 0.337491i \(-0.109578\pi\)
\(588\) 0 0
\(589\) 2.00923e12i 0.687876i
\(590\) 0 0
\(591\) 1.71111e12 0.576945
\(592\) 0 0
\(593\) −2.39070e12 −0.793926 −0.396963 0.917835i \(-0.629936\pi\)
−0.396963 + 0.917835i \(0.629936\pi\)
\(594\) 0 0
\(595\) − 9.99557e12i − 3.26950i
\(596\) 0 0
\(597\) − 2.48140e12i − 0.799488i
\(598\) 0 0
\(599\) 1.02268e11 0.0324576 0.0162288 0.999868i \(-0.494834\pi\)
0.0162288 + 0.999868i \(0.494834\pi\)
\(600\) 0 0
\(601\) −3.63671e12 −1.13704 −0.568518 0.822671i \(-0.692483\pi\)
−0.568518 + 0.822671i \(0.692483\pi\)
\(602\) 0 0
\(603\) 1.53987e12i 0.474304i
\(604\) 0 0
\(605\) − 1.08834e13i − 3.30266i
\(606\) 0 0
\(607\) −3.46914e12 −1.03723 −0.518613 0.855009i \(-0.673551\pi\)
−0.518613 + 0.855009i \(0.673551\pi\)
\(608\) 0 0
\(609\) 2.49320e12 0.734478
\(610\) 0 0
\(611\) − 2.50810e12i − 0.728046i
\(612\) 0 0
\(613\) 3.39770e12i 0.971881i 0.873992 + 0.485941i \(0.161523\pi\)
−0.873992 + 0.485941i \(0.838477\pi\)
\(614\) 0 0
\(615\) 2.58584e11 0.0728893
\(616\) 0 0
\(617\) −2.17197e12 −0.603351 −0.301676 0.953411i \(-0.597546\pi\)
−0.301676 + 0.953411i \(0.597546\pi\)
\(618\) 0 0
\(619\) 4.30312e12i 1.17808i 0.808103 + 0.589041i \(0.200494\pi\)
−0.808103 + 0.589041i \(0.799506\pi\)
\(620\) 0 0
\(621\) 1.25473e12i 0.338562i
\(622\) 0 0
\(623\) 1.05996e13 2.81898
\(624\) 0 0
\(625\) −4.73276e12 −1.24067
\(626\) 0 0
\(627\) 2.96623e12i 0.766481i
\(628\) 0 0
\(629\) 7.58486e10i 0.0193206i
\(630\) 0 0
\(631\) −4.20938e12 −1.05703 −0.528513 0.848925i \(-0.677250\pi\)
−0.528513 + 0.848925i \(0.677250\pi\)
\(632\) 0 0
\(633\) −3.04095e11 −0.0752823
\(634\) 0 0
\(635\) 1.59026e11i 0.0388139i
\(636\) 0 0
\(637\) − 1.24700e13i − 3.00081i
\(638\) 0 0
\(639\) −1.72236e11 −0.0408668
\(640\) 0 0
\(641\) −7.60479e11 −0.177920 −0.0889602 0.996035i \(-0.528354\pi\)
−0.0889602 + 0.996035i \(0.528354\pi\)
\(642\) 0 0
\(643\) − 7.42913e12i − 1.71391i −0.515389 0.856957i \(-0.672352\pi\)
0.515389 0.856957i \(-0.327648\pi\)
\(644\) 0 0
\(645\) 2.21596e12i 0.504130i
\(646\) 0 0
\(647\) 2.60122e12 0.583590 0.291795 0.956481i \(-0.405748\pi\)
0.291795 + 0.956481i \(0.405748\pi\)
\(648\) 0 0
\(649\) −9.17760e12 −2.03062
\(650\) 0 0
\(651\) − 4.77551e12i − 1.04209i
\(652\) 0 0
\(653\) 3.06819e12i 0.660348i 0.943920 + 0.330174i \(0.107107\pi\)
−0.943920 + 0.330174i \(0.892893\pi\)
\(654\) 0 0
\(655\) −3.48103e12 −0.738962
\(656\) 0 0
\(657\) 2.36173e12 0.494523
\(658\) 0 0
\(659\) − 2.46719e12i − 0.509586i −0.966996 0.254793i \(-0.917993\pi\)
0.966996 0.254793i \(-0.0820073\pi\)
\(660\) 0 0
\(661\) 4.01622e12i 0.818297i 0.912468 + 0.409148i \(0.134174\pi\)
−0.912468 + 0.409148i \(0.865826\pi\)
\(662\) 0 0
\(663\) 6.03159e12 1.21233
\(664\) 0 0
\(665\) −7.29404e12 −1.44634
\(666\) 0 0
\(667\) 6.39157e12i 1.25038i
\(668\) 0 0
\(669\) 4.70173e11i 0.0907487i
\(670\) 0 0
\(671\) −6.83813e11 −0.130222
\(672\) 0 0
\(673\) 7.90823e12 1.48598 0.742988 0.669305i \(-0.233408\pi\)
0.742988 + 0.669305i \(0.233408\pi\)
\(674\) 0 0
\(675\) − 4.18703e11i − 0.0776316i
\(676\) 0 0
\(677\) 2.08763e11i 0.0381949i 0.999818 + 0.0190974i \(0.00607927\pi\)
−0.999818 + 0.0190974i \(0.993921\pi\)
\(678\) 0 0
\(679\) −1.03439e13 −1.86754
\(680\) 0 0
\(681\) 3.54435e12 0.631503
\(682\) 0 0
\(683\) 3.71560e12i 0.653335i 0.945139 + 0.326667i \(0.105926\pi\)
−0.945139 + 0.326667i \(0.894074\pi\)
\(684\) 0 0
\(685\) − 1.35450e12i − 0.235056i
\(686\) 0 0
\(687\) −4.05510e12 −0.694538
\(688\) 0 0
\(689\) −2.81895e11 −0.0476542
\(690\) 0 0
\(691\) − 6.47722e12i − 1.08078i −0.841415 0.540390i \(-0.818277\pi\)
0.841415 0.540390i \(-0.181723\pi\)
\(692\) 0 0
\(693\) − 7.05011e12i − 1.16117i
\(694\) 0 0
\(695\) −1.15994e12 −0.188584
\(696\) 0 0
\(697\) −1.02390e12 −0.164327
\(698\) 0 0
\(699\) 4.68986e12i 0.743040i
\(700\) 0 0
\(701\) 7.68628e12i 1.20222i 0.799165 + 0.601112i \(0.205275\pi\)
−0.799165 + 0.601112i \(0.794725\pi\)
\(702\) 0 0
\(703\) 5.53488e10 0.00854691
\(704\) 0 0
\(705\) 2.39845e12 0.365661
\(706\) 0 0
\(707\) − 4.05616e12i − 0.610559i
\(708\) 0 0
\(709\) − 1.27745e13i − 1.89861i −0.314358 0.949304i \(-0.601789\pi\)
0.314358 0.949304i \(-0.398211\pi\)
\(710\) 0 0
\(711\) −1.97533e12 −0.289886
\(712\) 0 0
\(713\) 1.22425e13 1.77406
\(714\) 0 0
\(715\) 2.19418e13i 3.13975i
\(716\) 0 0
\(717\) − 5.33994e12i − 0.754571i
\(718\) 0 0
\(719\) 9.64154e12 1.34545 0.672723 0.739894i \(-0.265124\pi\)
0.672723 + 0.739894i \(0.265124\pi\)
\(720\) 0 0
\(721\) −1.44876e13 −1.99658
\(722\) 0 0
\(723\) 8.39292e11i 0.114233i
\(724\) 0 0
\(725\) − 2.13286e12i − 0.286709i
\(726\) 0 0
\(727\) −1.93455e12 −0.256848 −0.128424 0.991719i \(-0.540992\pi\)
−0.128424 + 0.991719i \(0.540992\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) − 8.77439e12i − 1.13655i
\(732\) 0 0
\(733\) − 4.64764e12i − 0.594654i −0.954776 0.297327i \(-0.903905\pi\)
0.954776 0.297327i \(-0.0960951\pi\)
\(734\) 0 0
\(735\) 1.19248e13 1.50716
\(736\) 0 0
\(737\) 2.21810e13 2.76935
\(738\) 0 0
\(739\) 1.50955e13i 1.86186i 0.365194 + 0.930932i \(0.381003\pi\)
−0.365194 + 0.930932i \(0.618997\pi\)
\(740\) 0 0
\(741\) − 4.40141e12i − 0.536303i
\(742\) 0 0
\(743\) 1.13339e13 1.36437 0.682183 0.731182i \(-0.261031\pi\)
0.682183 + 0.731182i \(0.261031\pi\)
\(744\) 0 0
\(745\) 7.95593e12 0.946210
\(746\) 0 0
\(747\) 2.28561e12i 0.268571i
\(748\) 0 0
\(749\) 2.16108e13i 2.50901i
\(750\) 0 0
\(751\) −6.43395e12 −0.738071 −0.369035 0.929415i \(-0.620312\pi\)
−0.369035 + 0.929415i \(0.620312\pi\)
\(752\) 0 0
\(753\) 6.38522e12 0.723766
\(754\) 0 0
\(755\) 1.73962e12i 0.194847i
\(756\) 0 0
\(757\) 1.59289e11i 0.0176301i 0.999961 + 0.00881507i \(0.00280596\pi\)
−0.999961 + 0.00881507i \(0.997194\pi\)
\(758\) 0 0
\(759\) 1.80737e13 1.97678
\(760\) 0 0
\(761\) −1.38855e12 −0.150083 −0.0750413 0.997180i \(-0.523909\pi\)
−0.0750413 + 0.997180i \(0.523909\pi\)
\(762\) 0 0
\(763\) 2.18768e12i 0.233681i
\(764\) 0 0
\(765\) 5.76790e12i 0.608894i
\(766\) 0 0
\(767\) 1.36181e13 1.42081
\(768\) 0 0
\(769\) −7.69105e12 −0.793080 −0.396540 0.918017i \(-0.629789\pi\)
−0.396540 + 0.918017i \(0.629789\pi\)
\(770\) 0 0
\(771\) − 3.35485e11i − 0.0341923i
\(772\) 0 0
\(773\) − 1.48855e13i − 1.49954i −0.661701 0.749768i \(-0.730165\pi\)
0.661701 0.749768i \(-0.269835\pi\)
\(774\) 0 0
\(775\) −4.08531e12 −0.406787
\(776\) 0 0
\(777\) −1.31552e11 −0.0129480
\(778\) 0 0
\(779\) 7.47167e11i 0.0726941i
\(780\) 0 0
\(781\) 2.48096e12i 0.238611i
\(782\) 0 0
\(783\) −1.43869e12 −0.136785
\(784\) 0 0
\(785\) −4.70653e12 −0.442372
\(786\) 0 0
\(787\) − 8.96061e12i − 0.832629i −0.909221 0.416314i \(-0.863322\pi\)
0.909221 0.416314i \(-0.136678\pi\)
\(788\) 0 0
\(789\) 5.27566e12i 0.484652i
\(790\) 0 0
\(791\) −2.60441e13 −2.36545
\(792\) 0 0
\(793\) 1.01467e12 0.0911160
\(794\) 0 0
\(795\) − 2.69571e11i − 0.0239343i
\(796\) 0 0
\(797\) 6.13567e12i 0.538641i 0.963051 + 0.269320i \(0.0867991\pi\)
−0.963051 + 0.269320i \(0.913201\pi\)
\(798\) 0 0
\(799\) −9.49698e12 −0.824375
\(800\) 0 0
\(801\) −6.11644e12 −0.524992
\(802\) 0 0
\(803\) − 3.40194e13i − 2.88740i
\(804\) 0 0
\(805\) 4.44436e13i 3.73016i
\(806\) 0 0
\(807\) 3.58890e12 0.297872
\(808\) 0 0
\(809\) 1.68903e13 1.38634 0.693170 0.720774i \(-0.256214\pi\)
0.693170 + 0.720774i \(0.256214\pi\)
\(810\) 0 0
\(811\) − 1.31863e13i − 1.07036i −0.844739 0.535179i \(-0.820244\pi\)
0.844739 0.535179i \(-0.179756\pi\)
\(812\) 0 0
\(813\) 7.44509e12i 0.597671i
\(814\) 0 0
\(815\) 2.25464e12 0.179006
\(816\) 0 0
\(817\) −6.40291e12 −0.502780
\(818\) 0 0
\(819\) 1.04612e13i 0.812466i
\(820\) 0 0
\(821\) − 6.47323e12i − 0.497252i −0.968600 0.248626i \(-0.920021\pi\)
0.968600 0.248626i \(-0.0799790\pi\)
\(822\) 0 0
\(823\) 1.72880e13 1.31355 0.656773 0.754088i \(-0.271921\pi\)
0.656773 + 0.754088i \(0.271921\pi\)
\(824\) 0 0
\(825\) −6.03117e12 −0.453272
\(826\) 0 0
\(827\) 1.84398e12i 0.137082i 0.997648 + 0.0685410i \(0.0218344\pi\)
−0.997648 + 0.0685410i \(0.978166\pi\)
\(828\) 0 0
\(829\) 1.85743e12i 0.136589i 0.997665 + 0.0682947i \(0.0217558\pi\)
−0.997665 + 0.0682947i \(0.978244\pi\)
\(830\) 0 0
\(831\) −2.62039e12 −0.190617
\(832\) 0 0
\(833\) −4.72180e13 −3.39786
\(834\) 0 0
\(835\) 6.31485e12i 0.449546i
\(836\) 0 0
\(837\) 2.75568e12i 0.194073i
\(838\) 0 0
\(839\) −1.04502e12 −0.0728106 −0.0364053 0.999337i \(-0.511591\pi\)
−0.0364053 + 0.999337i \(0.511591\pi\)
\(840\) 0 0
\(841\) 7.17851e12 0.494826
\(842\) 0 0
\(843\) 4.84880e12i 0.330681i
\(844\) 0 0
\(845\) − 1.50013e13i − 1.01222i
\(846\) 0 0
\(847\) −7.47429e13 −4.98993
\(848\) 0 0
\(849\) 1.00795e13 0.665816
\(850\) 0 0
\(851\) − 3.37248e11i − 0.0220428i
\(852\) 0 0
\(853\) − 4.28041e12i − 0.276831i −0.990374 0.138415i \(-0.955799\pi\)
0.990374 0.138415i \(-0.0442009\pi\)
\(854\) 0 0
\(855\) 4.20899e12 0.269358
\(856\) 0 0
\(857\) 9.25941e12 0.586367 0.293184 0.956056i \(-0.405285\pi\)
0.293184 + 0.956056i \(0.405285\pi\)
\(858\) 0 0
\(859\) − 9.57537e12i − 0.600049i −0.953932 0.300024i \(-0.903005\pi\)
0.953932 0.300024i \(-0.0969948\pi\)
\(860\) 0 0
\(861\) − 1.77586e12i − 0.110127i
\(862\) 0 0
\(863\) 1.56057e13 0.957713 0.478856 0.877893i \(-0.341051\pi\)
0.478856 + 0.877893i \(0.341051\pi\)
\(864\) 0 0
\(865\) 2.23805e13 1.35924
\(866\) 0 0
\(867\) − 1.32332e13i − 0.795387i
\(868\) 0 0
\(869\) 2.84535e13i 1.69257i
\(870\) 0 0
\(871\) −3.29130e13 −1.93770
\(872\) 0 0
\(873\) 5.96888e12 0.347800
\(874\) 0 0
\(875\) 2.19350e13i 1.26503i
\(876\) 0 0
\(877\) 2.51400e13i 1.43505i 0.696532 + 0.717526i \(0.254726\pi\)
−0.696532 + 0.717526i \(0.745274\pi\)
\(878\) 0 0
\(879\) 7.46822e12 0.421956
\(880\) 0 0
\(881\) 1.06427e12 0.0595198 0.0297599 0.999557i \(-0.490526\pi\)
0.0297599 + 0.999557i \(0.490526\pi\)
\(882\) 0 0
\(883\) − 3.43027e13i − 1.89891i −0.313902 0.949455i \(-0.601636\pi\)
0.313902 0.949455i \(-0.398364\pi\)
\(884\) 0 0
\(885\) 1.30227e13i 0.713604i
\(886\) 0 0
\(887\) 1.38759e13 0.752670 0.376335 0.926484i \(-0.377184\pi\)
0.376335 + 0.926484i \(0.377184\pi\)
\(888\) 0 0
\(889\) 1.09213e12 0.0586431
\(890\) 0 0
\(891\) 4.06823e12i 0.216250i
\(892\) 0 0
\(893\) 6.93020e12i 0.364682i
\(894\) 0 0
\(895\) −3.88510e12 −0.202394
\(896\) 0 0
\(897\) −2.68184e13 −1.38314
\(898\) 0 0
\(899\) 1.40374e13i 0.716750i
\(900\) 0 0
\(901\) 1.06740e12i 0.0539595i
\(902\) 0 0
\(903\) 1.52184e13 0.761681
\(904\) 0 0
\(905\) −4.46316e13 −2.21169
\(906\) 0 0
\(907\) − 7.19852e12i − 0.353192i −0.984283 0.176596i \(-0.943491\pi\)
0.984283 0.176596i \(-0.0565086\pi\)
\(908\) 0 0
\(909\) 2.34059e12i 0.113707i
\(910\) 0 0
\(911\) 2.48977e12 0.119764 0.0598820 0.998205i \(-0.480928\pi\)
0.0598820 + 0.998205i \(0.480928\pi\)
\(912\) 0 0
\(913\) 3.29229e13 1.56812
\(914\) 0 0
\(915\) 9.70309e11i 0.0457631i
\(916\) 0 0
\(917\) 2.39064e13i 1.11648i
\(918\) 0 0
\(919\) −1.73700e13 −0.803303 −0.401652 0.915793i \(-0.631564\pi\)
−0.401652 + 0.915793i \(0.631564\pi\)
\(920\) 0 0
\(921\) −1.68948e13 −0.773723
\(922\) 0 0
\(923\) − 3.68135e12i − 0.166955i
\(924\) 0 0
\(925\) 1.12539e11i 0.00505436i
\(926\) 0 0
\(927\) 8.35999e12 0.371832
\(928\) 0 0
\(929\) −4.98860e12 −0.219739 −0.109870 0.993946i \(-0.535043\pi\)
−0.109870 + 0.993946i \(0.535043\pi\)
\(930\) 0 0
\(931\) 3.44563e13i 1.50312i
\(932\) 0 0
\(933\) 9.36389e12i 0.404565i
\(934\) 0 0
\(935\) 8.30833e13 3.55518
\(936\) 0 0
\(937\) 2.01453e12 0.0853781 0.0426890 0.999088i \(-0.486408\pi\)
0.0426890 + 0.999088i \(0.486408\pi\)
\(938\) 0 0
\(939\) 1.12340e13i 0.471562i
\(940\) 0 0
\(941\) − 2.75127e13i − 1.14388i −0.820296 0.571939i \(-0.806191\pi\)
0.820296 0.571939i \(-0.193809\pi\)
\(942\) 0 0
\(943\) 4.55259e12 0.187480
\(944\) 0 0
\(945\) −1.00039e13 −0.408061
\(946\) 0 0
\(947\) 2.57827e13i 1.04173i 0.853640 + 0.520863i \(0.174390\pi\)
−0.853640 + 0.520863i \(0.825610\pi\)
\(948\) 0 0
\(949\) 5.04793e13i 2.02030i
\(950\) 0 0
\(951\) 4.65755e12 0.184648
\(952\) 0 0
\(953\) −2.06001e13 −0.809004 −0.404502 0.914537i \(-0.632555\pi\)
−0.404502 + 0.914537i \(0.632555\pi\)
\(954\) 0 0
\(955\) − 5.71178e12i − 0.222206i
\(956\) 0 0
\(957\) 2.07235e13i 0.798654i
\(958\) 0 0
\(959\) −9.30220e12 −0.355142
\(960\) 0 0
\(961\) 4.47786e11 0.0169362
\(962\) 0 0
\(963\) − 1.24704e13i − 0.467264i
\(964\) 0 0
\(965\) 3.20551e13i 1.18994i
\(966\) 0 0
\(967\) −3.94563e13 −1.45110 −0.725550 0.688170i \(-0.758414\pi\)
−0.725550 + 0.688170i \(0.758414\pi\)
\(968\) 0 0
\(969\) −1.66661e13 −0.607263
\(970\) 0 0
\(971\) 8.55423e12i 0.308812i 0.988007 + 0.154406i \(0.0493464\pi\)
−0.988007 + 0.154406i \(0.950654\pi\)
\(972\) 0 0
\(973\) 7.96603e12i 0.284927i
\(974\) 0 0
\(975\) 8.94929e12 0.317152
\(976\) 0 0
\(977\) −1.68374e12 −0.0591221 −0.0295611 0.999563i \(-0.509411\pi\)
−0.0295611 + 0.999563i \(0.509411\pi\)
\(978\) 0 0
\(979\) 8.81038e13i 3.06530i
\(980\) 0 0
\(981\) − 1.26239e12i − 0.0435195i
\(982\) 0 0
\(983\) −5.66001e13 −1.93342 −0.966711 0.255872i \(-0.917637\pi\)
−0.966711 + 0.255872i \(0.917637\pi\)
\(984\) 0 0
\(985\) −3.49741e13 −1.18381
\(986\) 0 0
\(987\) − 1.64716e13i − 0.552470i
\(988\) 0 0
\(989\) 3.90138e13i 1.29669i
\(990\) 0 0
\(991\) 7.49565e12 0.246875 0.123438 0.992352i \(-0.460608\pi\)
0.123438 + 0.992352i \(0.460608\pi\)
\(992\) 0 0
\(993\) 2.78201e13 0.908003
\(994\) 0 0
\(995\) 5.07183e13i 1.64044i
\(996\) 0 0
\(997\) − 5.52784e13i − 1.77185i −0.463828 0.885925i \(-0.653524\pi\)
0.463828 0.885925i \(-0.346476\pi\)
\(998\) 0 0
\(999\) 7.59117e10 0.00241137
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 384.10.d.f.193.8 yes 20
4.3 odd 2 inner 384.10.d.f.193.18 yes 20
8.3 odd 2 inner 384.10.d.f.193.3 20
8.5 even 2 inner 384.10.d.f.193.13 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.f.193.3 20 8.3 odd 2 inner
384.10.d.f.193.8 yes 20 1.1 even 1 trivial
384.10.d.f.193.13 yes 20 8.5 even 2 inner
384.10.d.f.193.18 yes 20 4.3 odd 2 inner