Properties

Label 384.10.d.d.193.8
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,10,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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.8
Root \(-49.7838i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.d.193.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+81.0000i q^{3} -309.546i q^{5} -5272.51 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q+81.0000i q^{3} -309.546i q^{5} -5272.51 q^{7} -6561.00 q^{9} +67092.3i q^{11} -145625. i q^{13} +25073.2 q^{15} +280963. q^{17} +561917. i q^{19} -427073. i q^{21} -1.25321e6 q^{23} +1.85731e6 q^{25} -531441. i q^{27} -3.22372e6i q^{29} -4.89612e6 q^{31} -5.43448e6 q^{33} +1.63208e6i q^{35} -1.02493e7i q^{37} +1.17956e7 q^{39} -4.07376e6 q^{41} -2.29943e7i q^{43} +2.03093e6i q^{45} -9.57099e6 q^{47} -1.25542e7 q^{49} +2.27580e7i q^{51} -5.95368e7i q^{53} +2.07681e7 q^{55} -4.55153e7 q^{57} -8.97217e6i q^{59} +1.82954e8i q^{61} +3.45930e7 q^{63} -4.50777e7 q^{65} +2.19427e8i q^{67} -1.01510e8i q^{69} -2.11168e8 q^{71} +4.05164e8 q^{73} +1.50442e8i q^{75} -3.53745e8i q^{77} -1.72184e8 q^{79} +4.30467e7 q^{81} -7.57672e8i q^{83} -8.69710e7i q^{85} +2.61121e8 q^{87} +9.23467e8 q^{89} +7.67810e8i q^{91} -3.96586e8i q^{93} +1.73939e8 q^{95} +7.91926e8 q^{97} -4.40193e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5576 q^{7} - 65610 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5576 q^{7} - 65610 q^{9} - 90720 q^{15} + 195692 q^{17} - 135152 q^{23} - 1317134 q^{25} - 14887496 q^{31} + 3646296 q^{33} - 4703832 q^{39} + 32469956 q^{41} - 28692880 q^{47} + 80865242 q^{49} + 223324800 q^{55} + 40971096 q^{57} - 36584136 q^{63} + 511472000 q^{65} + 722817008 q^{71} + 642721212 q^{73} - 114892616 q^{79} + 430467210 q^{81} - 5955120 q^{87} + 1709981116 q^{89} - 657134976 q^{95} + 3919129836 q^{97}+O(q^{100}) \) 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
\(5\) − 309.546i − 0.221493i −0.993849 0.110747i \(-0.964676\pi\)
0.993849 0.110747i \(-0.0353242\pi\)
\(6\) 0 0
\(7\) −5272.51 −0.829997 −0.414998 0.909822i \(-0.636218\pi\)
−0.414998 + 0.909822i \(0.636218\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 67092.3i 1.38167i 0.723010 + 0.690837i \(0.242758\pi\)
−0.723010 + 0.690837i \(0.757242\pi\)
\(12\) 0 0
\(13\) − 145625.i − 1.41414i −0.707146 0.707068i \(-0.750017\pi\)
0.707146 0.707068i \(-0.249983\pi\)
\(14\) 0 0
\(15\) 25073.2 0.127879
\(16\) 0 0
\(17\) 280963. 0.815886 0.407943 0.913007i \(-0.366246\pi\)
0.407943 + 0.913007i \(0.366246\pi\)
\(18\) 0 0
\(19\) 561917.i 0.989194i 0.869123 + 0.494597i \(0.164684\pi\)
−0.869123 + 0.494597i \(0.835316\pi\)
\(20\) 0 0
\(21\) − 427073.i − 0.479199i
\(22\) 0 0
\(23\) −1.25321e6 −0.933790 −0.466895 0.884313i \(-0.654627\pi\)
−0.466895 + 0.884313i \(0.654627\pi\)
\(24\) 0 0
\(25\) 1.85731e6 0.950941
\(26\) 0 0
\(27\) − 531441.i − 0.192450i
\(28\) 0 0
\(29\) − 3.22372e6i − 0.846382i −0.906040 0.423191i \(-0.860910\pi\)
0.906040 0.423191i \(-0.139090\pi\)
\(30\) 0 0
\(31\) −4.89612e6 −0.952192 −0.476096 0.879393i \(-0.657948\pi\)
−0.476096 + 0.879393i \(0.657948\pi\)
\(32\) 0 0
\(33\) −5.43448e6 −0.797710
\(34\) 0 0
\(35\) 1.63208e6i 0.183838i
\(36\) 0 0
\(37\) − 1.02493e7i − 0.899058i −0.893266 0.449529i \(-0.851592\pi\)
0.893266 0.449529i \(-0.148408\pi\)
\(38\) 0 0
\(39\) 1.17956e7 0.816452
\(40\) 0 0
\(41\) −4.07376e6 −0.225148 −0.112574 0.993643i \(-0.535910\pi\)
−0.112574 + 0.993643i \(0.535910\pi\)
\(42\) 0 0
\(43\) − 2.29943e7i − 1.02568i −0.858484 0.512840i \(-0.828594\pi\)
0.858484 0.512840i \(-0.171406\pi\)
\(44\) 0 0
\(45\) 2.03093e6i 0.0738310i
\(46\) 0 0
\(47\) −9.57099e6 −0.286099 −0.143050 0.989716i \(-0.545691\pi\)
−0.143050 + 0.989716i \(0.545691\pi\)
\(48\) 0 0
\(49\) −1.25542e7 −0.311105
\(50\) 0 0
\(51\) 2.27580e7i 0.471052i
\(52\) 0 0
\(53\) − 5.95368e7i − 1.03644i −0.855247 0.518220i \(-0.826595\pi\)
0.855247 0.518220i \(-0.173405\pi\)
\(54\) 0 0
\(55\) 2.07681e7 0.306031
\(56\) 0 0
\(57\) −4.55153e7 −0.571111
\(58\) 0 0
\(59\) − 8.97217e6i − 0.0963969i −0.998838 0.0481985i \(-0.984652\pi\)
0.998838 0.0481985i \(-0.0153480\pi\)
\(60\) 0 0
\(61\) 1.82954e8i 1.69183i 0.533319 + 0.845914i \(0.320945\pi\)
−0.533319 + 0.845914i \(0.679055\pi\)
\(62\) 0 0
\(63\) 3.45930e7 0.276666
\(64\) 0 0
\(65\) −4.50777e7 −0.313221
\(66\) 0 0
\(67\) 2.19427e8i 1.33031i 0.746704 + 0.665156i \(0.231635\pi\)
−0.746704 + 0.665156i \(0.768365\pi\)
\(68\) 0 0
\(69\) − 1.01510e8i − 0.539124i
\(70\) 0 0
\(71\) −2.11168e8 −0.986203 −0.493101 0.869972i \(-0.664137\pi\)
−0.493101 + 0.869972i \(0.664137\pi\)
\(72\) 0 0
\(73\) 4.05164e8 1.66985 0.834927 0.550361i \(-0.185510\pi\)
0.834927 + 0.550361i \(0.185510\pi\)
\(74\) 0 0
\(75\) 1.50442e8i 0.549026i
\(76\) 0 0
\(77\) − 3.53745e8i − 1.14679i
\(78\) 0 0
\(79\) −1.72184e8 −0.497359 −0.248680 0.968586i \(-0.579997\pi\)
−0.248680 + 0.968586i \(0.579997\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) − 7.57672e8i − 1.75239i −0.481960 0.876193i \(-0.660075\pi\)
0.481960 0.876193i \(-0.339925\pi\)
\(84\) 0 0
\(85\) − 8.69710e7i − 0.180713i
\(86\) 0 0
\(87\) 2.61121e8 0.488659
\(88\) 0 0
\(89\) 9.23467e8 1.56015 0.780075 0.625686i \(-0.215181\pi\)
0.780075 + 0.625686i \(0.215181\pi\)
\(90\) 0 0
\(91\) 7.67810e8i 1.17373i
\(92\) 0 0
\(93\) − 3.96586e8i − 0.549748i
\(94\) 0 0
\(95\) 1.73939e8 0.219099
\(96\) 0 0
\(97\) 7.91926e8 0.908264 0.454132 0.890935i \(-0.349950\pi\)
0.454132 + 0.890935i \(0.349950\pi\)
\(98\) 0 0
\(99\) − 4.40193e8i − 0.460558i
\(100\) 0 0
\(101\) 1.08355e9i 1.03611i 0.855349 + 0.518053i \(0.173343\pi\)
−0.855349 + 0.518053i \(0.826657\pi\)
\(102\) 0 0
\(103\) 1.17446e9 1.02819 0.514093 0.857735i \(-0.328129\pi\)
0.514093 + 0.857735i \(0.328129\pi\)
\(104\) 0 0
\(105\) −1.32199e8 −0.106139
\(106\) 0 0
\(107\) − 8.23598e8i − 0.607419i −0.952765 0.303710i \(-0.901775\pi\)
0.952765 0.303710i \(-0.0982252\pi\)
\(108\) 0 0
\(109\) − 1.44977e9i − 0.983736i −0.870670 0.491868i \(-0.836314\pi\)
0.870670 0.491868i \(-0.163686\pi\)
\(110\) 0 0
\(111\) 8.30196e8 0.519071
\(112\) 0 0
\(113\) −2.16643e9 −1.24995 −0.624974 0.780646i \(-0.714890\pi\)
−0.624974 + 0.780646i \(0.714890\pi\)
\(114\) 0 0
\(115\) 3.87927e8i 0.206828i
\(116\) 0 0
\(117\) 9.55446e8i 0.471379i
\(118\) 0 0
\(119\) −1.48138e9 −0.677182
\(120\) 0 0
\(121\) −2.14343e9 −0.909023
\(122\) 0 0
\(123\) − 3.29975e8i − 0.129989i
\(124\) 0 0
\(125\) − 1.17950e9i − 0.432120i
\(126\) 0 0
\(127\) −3.66249e9 −1.24928 −0.624640 0.780912i \(-0.714754\pi\)
−0.624640 + 0.780912i \(0.714754\pi\)
\(128\) 0 0
\(129\) 1.86254e9 0.592176
\(130\) 0 0
\(131\) 3.73034e9i 1.10669i 0.832951 + 0.553346i \(0.186649\pi\)
−0.832951 + 0.553346i \(0.813351\pi\)
\(132\) 0 0
\(133\) − 2.96272e9i − 0.821027i
\(134\) 0 0
\(135\) −1.64505e8 −0.0426264
\(136\) 0 0
\(137\) 4.65957e9 1.13006 0.565032 0.825069i \(-0.308864\pi\)
0.565032 + 0.825069i \(0.308864\pi\)
\(138\) 0 0
\(139\) 2.13924e9i 0.486064i 0.970018 + 0.243032i \(0.0781420\pi\)
−0.970018 + 0.243032i \(0.921858\pi\)
\(140\) 0 0
\(141\) − 7.75251e8i − 0.165179i
\(142\) 0 0
\(143\) 9.77032e9 1.95388
\(144\) 0 0
\(145\) −9.97890e8 −0.187468
\(146\) 0 0
\(147\) − 1.01689e9i − 0.179617i
\(148\) 0 0
\(149\) − 2.49237e9i − 0.414261i −0.978313 0.207130i \(-0.933588\pi\)
0.978313 0.207130i \(-0.0664124\pi\)
\(150\) 0 0
\(151\) −8.88631e8 −0.139099 −0.0695497 0.997578i \(-0.522156\pi\)
−0.0695497 + 0.997578i \(0.522156\pi\)
\(152\) 0 0
\(153\) −1.84340e9 −0.271962
\(154\) 0 0
\(155\) 1.51557e9i 0.210904i
\(156\) 0 0
\(157\) − 2.27297e9i − 0.298569i −0.988794 0.149285i \(-0.952303\pi\)
0.988794 0.149285i \(-0.0476971\pi\)
\(158\) 0 0
\(159\) 4.82248e9 0.598389
\(160\) 0 0
\(161\) 6.60757e9 0.775043
\(162\) 0 0
\(163\) 8.57075e7i 0.00950987i 0.999989 + 0.00475494i \(0.00151355\pi\)
−0.999989 + 0.00475494i \(0.998486\pi\)
\(164\) 0 0
\(165\) 1.68222e9i 0.176687i
\(166\) 0 0
\(167\) 1.54333e10 1.53545 0.767724 0.640781i \(-0.221389\pi\)
0.767724 + 0.640781i \(0.221389\pi\)
\(168\) 0 0
\(169\) −1.06022e10 −0.999781
\(170\) 0 0
\(171\) − 3.68674e9i − 0.329731i
\(172\) 0 0
\(173\) 8.71262e9i 0.739505i 0.929130 + 0.369753i \(0.120558\pi\)
−0.929130 + 0.369753i \(0.879442\pi\)
\(174\) 0 0
\(175\) −9.79267e9 −0.789278
\(176\) 0 0
\(177\) 7.26745e8 0.0556548
\(178\) 0 0
\(179\) 7.50026e9i 0.546056i 0.962006 + 0.273028i \(0.0880252\pi\)
−0.962006 + 0.273028i \(0.911975\pi\)
\(180\) 0 0
\(181\) 1.92534e10i 1.33338i 0.745336 + 0.666689i \(0.232289\pi\)
−0.745336 + 0.666689i \(0.767711\pi\)
\(182\) 0 0
\(183\) −1.48192e10 −0.976778
\(184\) 0 0
\(185\) −3.17264e9 −0.199135
\(186\) 0 0
\(187\) 1.88505e10i 1.12729i
\(188\) 0 0
\(189\) 2.80203e9i 0.159733i
\(190\) 0 0
\(191\) −7.84333e9 −0.426433 −0.213216 0.977005i \(-0.568394\pi\)
−0.213216 + 0.977005i \(0.568394\pi\)
\(192\) 0 0
\(193\) 1.46224e10 0.758595 0.379297 0.925275i \(-0.376166\pi\)
0.379297 + 0.925275i \(0.376166\pi\)
\(194\) 0 0
\(195\) − 3.65129e9i − 0.180838i
\(196\) 0 0
\(197\) 2.90067e10i 1.37215i 0.727532 + 0.686074i \(0.240667\pi\)
−0.727532 + 0.686074i \(0.759333\pi\)
\(198\) 0 0
\(199\) 3.70653e10 1.67544 0.837720 0.546100i \(-0.183888\pi\)
0.837720 + 0.546100i \(0.183888\pi\)
\(200\) 0 0
\(201\) −1.77736e10 −0.768056
\(202\) 0 0
\(203\) 1.69971e10i 0.702495i
\(204\) 0 0
\(205\) 1.26102e9i 0.0498688i
\(206\) 0 0
\(207\) 8.22232e9 0.311263
\(208\) 0 0
\(209\) −3.77003e10 −1.36674
\(210\) 0 0
\(211\) 1.06794e10i 0.370917i 0.982652 + 0.185459i \(0.0593770\pi\)
−0.982652 + 0.185459i \(0.940623\pi\)
\(212\) 0 0
\(213\) − 1.71046e10i − 0.569384i
\(214\) 0 0
\(215\) −7.11778e9 −0.227181
\(216\) 0 0
\(217\) 2.58148e10 0.790316
\(218\) 0 0
\(219\) 3.28183e10i 0.964090i
\(220\) 0 0
\(221\) − 4.09153e10i − 1.15377i
\(222\) 0 0
\(223\) 4.10565e10 1.11176 0.555879 0.831263i \(-0.312382\pi\)
0.555879 + 0.831263i \(0.312382\pi\)
\(224\) 0 0
\(225\) −1.21858e10 −0.316980
\(226\) 0 0
\(227\) 5.35803e10i 1.33933i 0.742662 + 0.669666i \(0.233563\pi\)
−0.742662 + 0.669666i \(0.766437\pi\)
\(228\) 0 0
\(229\) 2.94863e10i 0.708534i 0.935144 + 0.354267i \(0.115270\pi\)
−0.935144 + 0.354267i \(0.884730\pi\)
\(230\) 0 0
\(231\) 2.86533e10 0.662097
\(232\) 0 0
\(233\) −2.35264e10 −0.522941 −0.261471 0.965211i \(-0.584207\pi\)
−0.261471 + 0.965211i \(0.584207\pi\)
\(234\) 0 0
\(235\) 2.96266e9i 0.0633690i
\(236\) 0 0
\(237\) − 1.39469e10i − 0.287150i
\(238\) 0 0
\(239\) 3.57950e10 0.709629 0.354815 0.934937i \(-0.384544\pi\)
0.354815 + 0.934937i \(0.384544\pi\)
\(240\) 0 0
\(241\) −3.63821e10 −0.694722 −0.347361 0.937731i \(-0.612922\pi\)
−0.347361 + 0.937731i \(0.612922\pi\)
\(242\) 0 0
\(243\) 3.48678e9i 0.0641500i
\(244\) 0 0
\(245\) 3.88611e9i 0.0689077i
\(246\) 0 0
\(247\) 8.18293e10 1.39885
\(248\) 0 0
\(249\) 6.13714e10 1.01174
\(250\) 0 0
\(251\) − 7.59465e10i − 1.20775i −0.797080 0.603873i \(-0.793623\pi\)
0.797080 0.603873i \(-0.206377\pi\)
\(252\) 0 0
\(253\) − 8.40809e10i − 1.29019i
\(254\) 0 0
\(255\) 7.04465e9 0.104335
\(256\) 0 0
\(257\) 8.54224e10 1.22144 0.610721 0.791846i \(-0.290880\pi\)
0.610721 + 0.791846i \(0.290880\pi\)
\(258\) 0 0
\(259\) 5.40397e10i 0.746215i
\(260\) 0 0
\(261\) 2.11508e10i 0.282127i
\(262\) 0 0
\(263\) 4.34744e10 0.560316 0.280158 0.959954i \(-0.409613\pi\)
0.280158 + 0.959954i \(0.409613\pi\)
\(264\) 0 0
\(265\) −1.84294e10 −0.229564
\(266\) 0 0
\(267\) 7.48009e10i 0.900753i
\(268\) 0 0
\(269\) 2.19478e10i 0.255568i 0.991802 + 0.127784i \(0.0407864\pi\)
−0.991802 + 0.127784i \(0.959214\pi\)
\(270\) 0 0
\(271\) 8.41640e10 0.947904 0.473952 0.880551i \(-0.342827\pi\)
0.473952 + 0.880551i \(0.342827\pi\)
\(272\) 0 0
\(273\) −6.21926e10 −0.677652
\(274\) 0 0
\(275\) 1.24611e11i 1.31389i
\(276\) 0 0
\(277\) − 4.30614e10i − 0.439470i −0.975560 0.219735i \(-0.929481\pi\)
0.975560 0.219735i \(-0.0705193\pi\)
\(278\) 0 0
\(279\) 3.21234e10 0.317397
\(280\) 0 0
\(281\) 1.94411e10 0.186012 0.0930062 0.995666i \(-0.470352\pi\)
0.0930062 + 0.995666i \(0.470352\pi\)
\(282\) 0 0
\(283\) 7.38936e10i 0.684807i 0.939553 + 0.342403i \(0.111241\pi\)
−0.939553 + 0.342403i \(0.888759\pi\)
\(284\) 0 0
\(285\) 1.40891e10i 0.126497i
\(286\) 0 0
\(287\) 2.14790e10 0.186872
\(288\) 0 0
\(289\) −3.96476e10 −0.334331
\(290\) 0 0
\(291\) 6.41460e10i 0.524386i
\(292\) 0 0
\(293\) − 3.07308e9i − 0.0243595i −0.999926 0.0121798i \(-0.996123\pi\)
0.999926 0.0121798i \(-0.00387704\pi\)
\(294\) 0 0
\(295\) −2.77730e9 −0.0213513
\(296\) 0 0
\(297\) 3.56556e10 0.265903
\(298\) 0 0
\(299\) 1.82499e11i 1.32051i
\(300\) 0 0
\(301\) 1.21238e11i 0.851311i
\(302\) 0 0
\(303\) −8.77678e10 −0.598196
\(304\) 0 0
\(305\) 5.66325e10 0.374728
\(306\) 0 0
\(307\) − 7.68581e9i − 0.0493818i −0.999695 0.0246909i \(-0.992140\pi\)
0.999695 0.0246909i \(-0.00786016\pi\)
\(308\) 0 0
\(309\) 9.51315e10i 0.593623i
\(310\) 0 0
\(311\) 2.31483e11 1.40313 0.701564 0.712607i \(-0.252486\pi\)
0.701564 + 0.712607i \(0.252486\pi\)
\(312\) 0 0
\(313\) 7.14841e10 0.420979 0.210489 0.977596i \(-0.432494\pi\)
0.210489 + 0.977596i \(0.432494\pi\)
\(314\) 0 0
\(315\) − 1.07081e10i − 0.0612795i
\(316\) 0 0
\(317\) 6.96471e10i 0.387380i 0.981063 + 0.193690i \(0.0620455\pi\)
−0.981063 + 0.193690i \(0.937954\pi\)
\(318\) 0 0
\(319\) 2.16287e11 1.16942
\(320\) 0 0
\(321\) 6.67115e10 0.350694
\(322\) 0 0
\(323\) 1.57878e11i 0.807069i
\(324\) 0 0
\(325\) − 2.70470e11i − 1.34476i
\(326\) 0 0
\(327\) 1.17431e11 0.567960
\(328\) 0 0
\(329\) 5.04632e10 0.237461
\(330\) 0 0
\(331\) 3.26115e11i 1.49329i 0.665221 + 0.746646i \(0.268337\pi\)
−0.665221 + 0.746646i \(0.731663\pi\)
\(332\) 0 0
\(333\) 6.72458e10i 0.299686i
\(334\) 0 0
\(335\) 6.79228e10 0.294655
\(336\) 0 0
\(337\) 2.97450e11 1.25626 0.628130 0.778108i \(-0.283821\pi\)
0.628130 + 0.778108i \(0.283821\pi\)
\(338\) 0 0
\(339\) − 1.75481e11i − 0.721657i
\(340\) 0 0
\(341\) − 3.28492e11i − 1.31562i
\(342\) 0 0
\(343\) 2.78957e11 1.08821
\(344\) 0 0
\(345\) −3.14221e10 −0.119412
\(346\) 0 0
\(347\) 2.03340e11i 0.752905i 0.926436 + 0.376452i \(0.122856\pi\)
−0.926436 + 0.376452i \(0.877144\pi\)
\(348\) 0 0
\(349\) − 1.09580e11i − 0.395384i −0.980264 0.197692i \(-0.936655\pi\)
0.980264 0.197692i \(-0.0633445\pi\)
\(350\) 0 0
\(351\) −7.73912e10 −0.272151
\(352\) 0 0
\(353\) −7.46299e10 −0.255815 −0.127908 0.991786i \(-0.540826\pi\)
−0.127908 + 0.991786i \(0.540826\pi\)
\(354\) 0 0
\(355\) 6.53663e10i 0.218437i
\(356\) 0 0
\(357\) − 1.19992e11i − 0.390971i
\(358\) 0 0
\(359\) 4.98306e11 1.58333 0.791664 0.610957i \(-0.209215\pi\)
0.791664 + 0.610957i \(0.209215\pi\)
\(360\) 0 0
\(361\) 6.93654e9 0.0214961
\(362\) 0 0
\(363\) − 1.73618e11i − 0.524825i
\(364\) 0 0
\(365\) − 1.25417e11i − 0.369861i
\(366\) 0 0
\(367\) −2.00929e11 −0.578158 −0.289079 0.957305i \(-0.593349\pi\)
−0.289079 + 0.957305i \(0.593349\pi\)
\(368\) 0 0
\(369\) 2.67280e10 0.0750494
\(370\) 0 0
\(371\) 3.13909e11i 0.860242i
\(372\) 0 0
\(373\) 2.20541e11i 0.589930i 0.955508 + 0.294965i \(0.0953079\pi\)
−0.955508 + 0.294965i \(0.904692\pi\)
\(374\) 0 0
\(375\) 9.55398e10 0.249484
\(376\) 0 0
\(377\) −4.69455e11 −1.19690
\(378\) 0 0
\(379\) 3.61774e11i 0.900660i 0.892862 + 0.450330i \(0.148694\pi\)
−0.892862 + 0.450330i \(0.851306\pi\)
\(380\) 0 0
\(381\) − 2.96662e11i − 0.721273i
\(382\) 0 0
\(383\) 9.73049e10 0.231068 0.115534 0.993304i \(-0.463142\pi\)
0.115534 + 0.993304i \(0.463142\pi\)
\(384\) 0 0
\(385\) −1.09500e11 −0.254005
\(386\) 0 0
\(387\) 1.50865e11i 0.341893i
\(388\) 0 0
\(389\) 6.77438e11i 1.50002i 0.661428 + 0.750009i \(0.269951\pi\)
−0.661428 + 0.750009i \(0.730049\pi\)
\(390\) 0 0
\(391\) −3.52106e11 −0.761866
\(392\) 0 0
\(393\) −3.02157e11 −0.638949
\(394\) 0 0
\(395\) 5.32988e10i 0.110162i
\(396\) 0 0
\(397\) 4.50301e11i 0.909799i 0.890543 + 0.454899i \(0.150325\pi\)
−0.890543 + 0.454899i \(0.849675\pi\)
\(398\) 0 0
\(399\) 2.39980e11 0.474020
\(400\) 0 0
\(401\) −6.04222e11 −1.16694 −0.583468 0.812136i \(-0.698305\pi\)
−0.583468 + 0.812136i \(0.698305\pi\)
\(402\) 0 0
\(403\) 7.12998e11i 1.34653i
\(404\) 0 0
\(405\) − 1.33249e10i − 0.0246103i
\(406\) 0 0
\(407\) 6.87651e11 1.24220
\(408\) 0 0
\(409\) −4.46829e11 −0.789563 −0.394782 0.918775i \(-0.629180\pi\)
−0.394782 + 0.918775i \(0.629180\pi\)
\(410\) 0 0
\(411\) 3.77425e11i 0.652443i
\(412\) 0 0
\(413\) 4.73059e10i 0.0800092i
\(414\) 0 0
\(415\) −2.34534e11 −0.388141
\(416\) 0 0
\(417\) −1.73279e11 −0.280629
\(418\) 0 0
\(419\) − 2.15985e11i − 0.342342i −0.985241 0.171171i \(-0.945245\pi\)
0.985241 0.171171i \(-0.0547551\pi\)
\(420\) 0 0
\(421\) 2.01623e11i 0.312803i 0.987694 + 0.156402i \(0.0499894\pi\)
−0.987694 + 0.156402i \(0.950011\pi\)
\(422\) 0 0
\(423\) 6.27953e10 0.0953664
\(424\) 0 0
\(425\) 5.21835e11 0.775859
\(426\) 0 0
\(427\) − 9.64625e11i − 1.40421i
\(428\) 0 0
\(429\) 7.91396e11i 1.12807i
\(430\) 0 0
\(431\) −1.02060e12 −1.42466 −0.712328 0.701847i \(-0.752359\pi\)
−0.712328 + 0.701847i \(0.752359\pi\)
\(432\) 0 0
\(433\) 1.31704e12 1.80054 0.900269 0.435335i \(-0.143370\pi\)
0.900269 + 0.435335i \(0.143370\pi\)
\(434\) 0 0
\(435\) − 8.08291e10i − 0.108235i
\(436\) 0 0
\(437\) − 7.04201e11i − 0.923699i
\(438\) 0 0
\(439\) 1.17939e12 1.51554 0.757769 0.652523i \(-0.226289\pi\)
0.757769 + 0.652523i \(0.226289\pi\)
\(440\) 0 0
\(441\) 8.23683e10 0.103702
\(442\) 0 0
\(443\) − 7.50044e11i − 0.925273i −0.886548 0.462636i \(-0.846904\pi\)
0.886548 0.462636i \(-0.153096\pi\)
\(444\) 0 0
\(445\) − 2.85856e11i − 0.345563i
\(446\) 0 0
\(447\) 2.01882e11 0.239174
\(448\) 0 0
\(449\) −8.15176e11 −0.946548 −0.473274 0.880915i \(-0.656928\pi\)
−0.473274 + 0.880915i \(0.656928\pi\)
\(450\) 0 0
\(451\) − 2.73318e11i − 0.311082i
\(452\) 0 0
\(453\) − 7.19791e10i − 0.0803091i
\(454\) 0 0
\(455\) 2.37672e11 0.259973
\(456\) 0 0
\(457\) 3.63242e11 0.389559 0.194779 0.980847i \(-0.437601\pi\)
0.194779 + 0.980847i \(0.437601\pi\)
\(458\) 0 0
\(459\) − 1.49315e11i − 0.157017i
\(460\) 0 0
\(461\) 1.30313e12i 1.34380i 0.740643 + 0.671899i \(0.234521\pi\)
−0.740643 + 0.671899i \(0.765479\pi\)
\(462\) 0 0
\(463\) −1.12709e12 −1.13984 −0.569920 0.821700i \(-0.693026\pi\)
−0.569920 + 0.821700i \(0.693026\pi\)
\(464\) 0 0
\(465\) −1.22761e11 −0.121765
\(466\) 0 0
\(467\) − 1.62119e12i − 1.57727i −0.614859 0.788637i \(-0.710787\pi\)
0.614859 0.788637i \(-0.289213\pi\)
\(468\) 0 0
\(469\) − 1.15693e12i − 1.10416i
\(470\) 0 0
\(471\) 1.84111e11 0.172379
\(472\) 0 0
\(473\) 1.54274e12 1.41715
\(474\) 0 0
\(475\) 1.04365e12i 0.940665i
\(476\) 0 0
\(477\) 3.90621e11i 0.345480i
\(478\) 0 0
\(479\) −6.14450e11 −0.533306 −0.266653 0.963793i \(-0.585918\pi\)
−0.266653 + 0.963793i \(0.585918\pi\)
\(480\) 0 0
\(481\) −1.49256e12 −1.27139
\(482\) 0 0
\(483\) 5.35213e11i 0.447471i
\(484\) 0 0
\(485\) − 2.45138e11i − 0.201174i
\(486\) 0 0
\(487\) 2.12192e12 1.70942 0.854709 0.519107i \(-0.173735\pi\)
0.854709 + 0.519107i \(0.173735\pi\)
\(488\) 0 0
\(489\) −6.94231e9 −0.00549053
\(490\) 0 0
\(491\) 1.80606e12i 1.40238i 0.712974 + 0.701191i \(0.247348\pi\)
−0.712974 + 0.701191i \(0.752652\pi\)
\(492\) 0 0
\(493\) − 9.05747e11i − 0.690551i
\(494\) 0 0
\(495\) −1.36260e11 −0.102010
\(496\) 0 0
\(497\) 1.11339e12 0.818545
\(498\) 0 0
\(499\) 5.53469e11i 0.399614i 0.979835 + 0.199807i \(0.0640315\pi\)
−0.979835 + 0.199807i \(0.935968\pi\)
\(500\) 0 0
\(501\) 1.25010e12i 0.886491i
\(502\) 0 0
\(503\) −2.40841e12 −1.67754 −0.838772 0.544482i \(-0.816726\pi\)
−0.838772 + 0.544482i \(0.816726\pi\)
\(504\) 0 0
\(505\) 3.35409e11 0.229490
\(506\) 0 0
\(507\) − 8.58776e11i − 0.577224i
\(508\) 0 0
\(509\) 1.01973e12i 0.673375i 0.941616 + 0.336687i \(0.109307\pi\)
−0.941616 + 0.336687i \(0.890693\pi\)
\(510\) 0 0
\(511\) −2.13623e12 −1.38597
\(512\) 0 0
\(513\) 2.98626e11 0.190370
\(514\) 0 0
\(515\) − 3.63550e11i − 0.227736i
\(516\) 0 0
\(517\) − 6.42140e11i − 0.395296i
\(518\) 0 0
\(519\) −7.05722e11 −0.426954
\(520\) 0 0
\(521\) −2.56824e12 −1.52709 −0.763547 0.645753i \(-0.776544\pi\)
−0.763547 + 0.645753i \(0.776544\pi\)
\(522\) 0 0
\(523\) − 1.89472e12i − 1.10736i −0.832731 0.553678i \(-0.813224\pi\)
0.832731 0.553678i \(-0.186776\pi\)
\(524\) 0 0
\(525\) − 7.93206e11i − 0.455690i
\(526\) 0 0
\(527\) −1.37563e12 −0.776879
\(528\) 0 0
\(529\) −2.30613e11 −0.128036
\(530\) 0 0
\(531\) 5.88664e10i 0.0321323i
\(532\) 0 0
\(533\) 5.93242e11i 0.318390i
\(534\) 0 0
\(535\) −2.54941e11 −0.134539
\(536\) 0 0
\(537\) −6.07521e11 −0.315266
\(538\) 0 0
\(539\) − 8.42292e11i − 0.429846i
\(540\) 0 0
\(541\) − 1.87444e12i − 0.940772i −0.882461 0.470386i \(-0.844115\pi\)
0.882461 0.470386i \(-0.155885\pi\)
\(542\) 0 0
\(543\) −1.55952e12 −0.769826
\(544\) 0 0
\(545\) −4.48769e11 −0.217891
\(546\) 0 0
\(547\) − 3.54576e12i − 1.69343i −0.532049 0.846713i \(-0.678578\pi\)
0.532049 0.846713i \(-0.321422\pi\)
\(548\) 0 0
\(549\) − 1.20036e12i − 0.563943i
\(550\) 0 0
\(551\) 1.81147e12 0.837236
\(552\) 0 0
\(553\) 9.07841e11 0.412807
\(554\) 0 0
\(555\) − 2.56984e11i − 0.114971i
\(556\) 0 0
\(557\) − 3.10406e12i − 1.36641i −0.730226 0.683206i \(-0.760585\pi\)
0.730226 0.683206i \(-0.239415\pi\)
\(558\) 0 0
\(559\) −3.34854e12 −1.45045
\(560\) 0 0
\(561\) −1.52689e12 −0.650840
\(562\) 0 0
\(563\) − 5.85117e11i − 0.245445i −0.992441 0.122723i \(-0.960837\pi\)
0.992441 0.122723i \(-0.0391625\pi\)
\(564\) 0 0
\(565\) 6.70610e11i 0.276855i
\(566\) 0 0
\(567\) −2.26964e11 −0.0922219
\(568\) 0 0
\(569\) 7.06739e11 0.282653 0.141327 0.989963i \(-0.454863\pi\)
0.141327 + 0.989963i \(0.454863\pi\)
\(570\) 0 0
\(571\) − 4.79486e12i − 1.88762i −0.330495 0.943808i \(-0.607216\pi\)
0.330495 0.943808i \(-0.392784\pi\)
\(572\) 0 0
\(573\) − 6.35310e11i − 0.246201i
\(574\) 0 0
\(575\) −2.32760e12 −0.887979
\(576\) 0 0
\(577\) 3.75415e12 1.41000 0.705001 0.709206i \(-0.250946\pi\)
0.705001 + 0.709206i \(0.250946\pi\)
\(578\) 0 0
\(579\) 1.18441e12i 0.437975i
\(580\) 0 0
\(581\) 3.99483e12i 1.45447i
\(582\) 0 0
\(583\) 3.99446e12 1.43202
\(584\) 0 0
\(585\) 2.95755e11 0.104407
\(586\) 0 0
\(587\) − 2.48518e12i − 0.863946i −0.901887 0.431973i \(-0.857818\pi\)
0.901887 0.431973i \(-0.142182\pi\)
\(588\) 0 0
\(589\) − 2.75121e12i − 0.941902i
\(590\) 0 0
\(591\) −2.34955e12 −0.792210
\(592\) 0 0
\(593\) 1.26542e12 0.420230 0.210115 0.977677i \(-0.432616\pi\)
0.210115 + 0.977677i \(0.432616\pi\)
\(594\) 0 0
\(595\) 4.58556e11i 0.149991i
\(596\) 0 0
\(597\) 3.00229e12i 0.967315i
\(598\) 0 0
\(599\) 3.21302e12 1.01975 0.509873 0.860249i \(-0.329692\pi\)
0.509873 + 0.860249i \(0.329692\pi\)
\(600\) 0 0
\(601\) 5.89466e12 1.84299 0.921496 0.388387i \(-0.126968\pi\)
0.921496 + 0.388387i \(0.126968\pi\)
\(602\) 0 0
\(603\) − 1.43966e12i − 0.443438i
\(604\) 0 0
\(605\) 6.63490e11i 0.201342i
\(606\) 0 0
\(607\) −6.40258e12 −1.91428 −0.957141 0.289622i \(-0.906470\pi\)
−0.957141 + 0.289622i \(0.906470\pi\)
\(608\) 0 0
\(609\) −1.37677e12 −0.405585
\(610\) 0 0
\(611\) 1.39378e12i 0.404583i
\(612\) 0 0
\(613\) − 4.04612e12i − 1.15735i −0.815557 0.578677i \(-0.803569\pi\)
0.815557 0.578677i \(-0.196431\pi\)
\(614\) 0 0
\(615\) −1.02142e11 −0.0287917
\(616\) 0 0
\(617\) −6.75659e12 −1.87691 −0.938456 0.345398i \(-0.887744\pi\)
−0.938456 + 0.345398i \(0.887744\pi\)
\(618\) 0 0
\(619\) 2.17914e10i 0.00596592i 0.999996 + 0.00298296i \(0.000949507\pi\)
−0.999996 + 0.00298296i \(0.999050\pi\)
\(620\) 0 0
\(621\) 6.66008e11i 0.179708i
\(622\) 0 0
\(623\) −4.86899e12 −1.29492
\(624\) 0 0
\(625\) 3.26244e12 0.855229
\(626\) 0 0
\(627\) − 3.05373e12i − 0.789089i
\(628\) 0 0
\(629\) − 2.87968e12i − 0.733528i
\(630\) 0 0
\(631\) −5.02427e12 −1.26166 −0.630828 0.775922i \(-0.717285\pi\)
−0.630828 + 0.775922i \(0.717285\pi\)
\(632\) 0 0
\(633\) −8.65033e11 −0.214149
\(634\) 0 0
\(635\) 1.13371e12i 0.276707i
\(636\) 0 0
\(637\) 1.82821e12i 0.439945i
\(638\) 0 0
\(639\) 1.38548e12 0.328734
\(640\) 0 0
\(641\) 4.91095e12 1.14896 0.574479 0.818519i \(-0.305205\pi\)
0.574479 + 0.818519i \(0.305205\pi\)
\(642\) 0 0
\(643\) − 7.32211e12i − 1.68922i −0.535381 0.844611i \(-0.679832\pi\)
0.535381 0.844611i \(-0.320168\pi\)
\(644\) 0 0
\(645\) − 5.76540e11i − 0.131163i
\(646\) 0 0
\(647\) 3.54218e12 0.794697 0.397349 0.917668i \(-0.369930\pi\)
0.397349 + 0.917668i \(0.369930\pi\)
\(648\) 0 0
\(649\) 6.01963e11 0.133189
\(650\) 0 0
\(651\) 2.09100e12i 0.456289i
\(652\) 0 0
\(653\) − 8.23728e11i − 0.177286i −0.996063 0.0886430i \(-0.971747\pi\)
0.996063 0.0886430i \(-0.0282530\pi\)
\(654\) 0 0
\(655\) 1.15471e12 0.245125
\(656\) 0 0
\(657\) −2.65828e12 −0.556618
\(658\) 0 0
\(659\) − 5.70546e12i − 1.17844i −0.807974 0.589218i \(-0.799436\pi\)
0.807974 0.589218i \(-0.200564\pi\)
\(660\) 0 0
\(661\) 4.16492e12i 0.848593i 0.905523 + 0.424297i \(0.139479\pi\)
−0.905523 + 0.424297i \(0.860521\pi\)
\(662\) 0 0
\(663\) 3.31414e12 0.666131
\(664\) 0 0
\(665\) −9.17097e11 −0.181852
\(666\) 0 0
\(667\) 4.04001e12i 0.790343i
\(668\) 0 0
\(669\) 3.32558e12i 0.641873i
\(670\) 0 0
\(671\) −1.22748e13 −2.33756
\(672\) 0 0
\(673\) 4.31368e11 0.0810551 0.0405276 0.999178i \(-0.487096\pi\)
0.0405276 + 0.999178i \(0.487096\pi\)
\(674\) 0 0
\(675\) − 9.87049e11i − 0.183009i
\(676\) 0 0
\(677\) 8.06501e12i 1.47556i 0.675043 + 0.737778i \(0.264125\pi\)
−0.675043 + 0.737778i \(0.735875\pi\)
\(678\) 0 0
\(679\) −4.17544e12 −0.753856
\(680\) 0 0
\(681\) −4.34000e12 −0.773264
\(682\) 0 0
\(683\) − 3.19549e12i − 0.561882i −0.959725 0.280941i \(-0.909354\pi\)
0.959725 0.280941i \(-0.0906465\pi\)
\(684\) 0 0
\(685\) − 1.44235e12i − 0.250301i
\(686\) 0 0
\(687\) −2.38839e12 −0.409072
\(688\) 0 0
\(689\) −8.67006e12 −1.46567
\(690\) 0 0
\(691\) 8.23728e12i 1.37446i 0.726439 + 0.687231i \(0.241174\pi\)
−0.726439 + 0.687231i \(0.758826\pi\)
\(692\) 0 0
\(693\) 2.32092e12i 0.382262i
\(694\) 0 0
\(695\) 6.62194e11 0.107660
\(696\) 0 0
\(697\) −1.14458e12 −0.183695
\(698\) 0 0
\(699\) − 1.90564e12i − 0.301920i
\(700\) 0 0
\(701\) 9.25378e12i 1.44740i 0.690116 + 0.723699i \(0.257560\pi\)
−0.690116 + 0.723699i \(0.742440\pi\)
\(702\) 0 0
\(703\) 5.75928e12 0.889342
\(704\) 0 0
\(705\) −2.39976e11 −0.0365861
\(706\) 0 0
\(707\) − 5.71305e12i − 0.859964i
\(708\) 0 0
\(709\) − 8.53630e12i − 1.26871i −0.773043 0.634354i \(-0.781266\pi\)
0.773043 0.634354i \(-0.218734\pi\)
\(710\) 0 0
\(711\) 1.12970e12 0.165786
\(712\) 0 0
\(713\) 6.13587e12 0.889147
\(714\) 0 0
\(715\) − 3.02436e12i − 0.432770i
\(716\) 0 0
\(717\) 2.89939e12i 0.409705i
\(718\) 0 0
\(719\) −4.29268e11 −0.0599031 −0.0299515 0.999551i \(-0.509535\pi\)
−0.0299515 + 0.999551i \(0.509535\pi\)
\(720\) 0 0
\(721\) −6.19237e12 −0.853391
\(722\) 0 0
\(723\) − 2.94695e12i − 0.401098i
\(724\) 0 0
\(725\) − 5.98744e12i − 0.804859i
\(726\) 0 0
\(727\) 1.30507e13 1.73272 0.866360 0.499420i \(-0.166454\pi\)
0.866360 + 0.499420i \(0.166454\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) − 6.46055e12i − 0.836837i
\(732\) 0 0
\(733\) − 1.38098e13i − 1.76693i −0.468500 0.883463i \(-0.655206\pi\)
0.468500 0.883463i \(-0.344794\pi\)
\(734\) 0 0
\(735\) −3.14775e11 −0.0397839
\(736\) 0 0
\(737\) −1.47219e13 −1.83806
\(738\) 0 0
\(739\) 2.03935e12i 0.251532i 0.992060 + 0.125766i \(0.0401388\pi\)
−0.992060 + 0.125766i \(0.959861\pi\)
\(740\) 0 0
\(741\) 6.62817e12i 0.807629i
\(742\) 0 0
\(743\) 8.47512e11 0.102023 0.0510113 0.998698i \(-0.483756\pi\)
0.0510113 + 0.998698i \(0.483756\pi\)
\(744\) 0 0
\(745\) −7.71502e11 −0.0917558
\(746\) 0 0
\(747\) 4.97109e12i 0.584129i
\(748\) 0 0
\(749\) 4.34243e12i 0.504156i
\(750\) 0 0
\(751\) −1.58689e13 −1.82040 −0.910198 0.414173i \(-0.864071\pi\)
−0.910198 + 0.414173i \(0.864071\pi\)
\(752\) 0 0
\(753\) 6.15166e12 0.697293
\(754\) 0 0
\(755\) 2.75072e11i 0.0308095i
\(756\) 0 0
\(757\) − 5.67467e12i − 0.628072i −0.949411 0.314036i \(-0.898319\pi\)
0.949411 0.314036i \(-0.101681\pi\)
\(758\) 0 0
\(759\) 6.81055e12 0.744893
\(760\) 0 0
\(761\) 8.49613e11 0.0918312 0.0459156 0.998945i \(-0.485379\pi\)
0.0459156 + 0.998945i \(0.485379\pi\)
\(762\) 0 0
\(763\) 7.64390e12i 0.816497i
\(764\) 0 0
\(765\) 5.70617e11i 0.0602377i
\(766\) 0 0
\(767\) −1.30657e12 −0.136318
\(768\) 0 0
\(769\) 1.44948e13 1.49467 0.747333 0.664450i \(-0.231334\pi\)
0.747333 + 0.664450i \(0.231334\pi\)
\(770\) 0 0
\(771\) 6.91921e12i 0.705200i
\(772\) 0 0
\(773\) − 7.06395e12i − 0.711607i −0.934561 0.355803i \(-0.884207\pi\)
0.934561 0.355803i \(-0.115793\pi\)
\(774\) 0 0
\(775\) −9.09359e12 −0.905478
\(776\) 0 0
\(777\) −4.37722e12 −0.430827
\(778\) 0 0
\(779\) − 2.28912e12i − 0.222715i
\(780\) 0 0
\(781\) − 1.41678e13i − 1.36261i
\(782\) 0 0
\(783\) −1.71322e12 −0.162886
\(784\) 0 0
\(785\) −7.03589e11 −0.0661310
\(786\) 0 0
\(787\) 4.72071e11i 0.0438653i 0.999759 + 0.0219326i \(0.00698193\pi\)
−0.999759 + 0.0219326i \(0.993018\pi\)
\(788\) 0 0
\(789\) 3.52143e12i 0.323499i
\(790\) 0 0
\(791\) 1.14225e13 1.03745
\(792\) 0 0
\(793\) 2.66426e13 2.39248
\(794\) 0 0
\(795\) − 1.49278e12i − 0.132539i
\(796\) 0 0
\(797\) 1.16384e13i 1.02171i 0.859666 + 0.510857i \(0.170672\pi\)
−0.859666 + 0.510857i \(0.829328\pi\)
\(798\) 0 0
\(799\) −2.68910e12 −0.233424
\(800\) 0 0
\(801\) −6.05887e12 −0.520050
\(802\) 0 0
\(803\) 2.71834e13i 2.30719i
\(804\) 0 0
\(805\) − 2.04535e12i − 0.171667i
\(806\) 0 0
\(807\) −1.77777e12 −0.147552
\(808\) 0 0
\(809\) −1.50266e13 −1.23337 −0.616684 0.787210i \(-0.711525\pi\)
−0.616684 + 0.787210i \(0.711525\pi\)
\(810\) 0 0
\(811\) 7.77282e12i 0.630935i 0.948936 + 0.315468i \(0.102161\pi\)
−0.948936 + 0.315468i \(0.897839\pi\)
\(812\) 0 0
\(813\) 6.81728e12i 0.547273i
\(814\) 0 0
\(815\) 2.65304e10 0.00210637
\(816\) 0 0
\(817\) 1.29209e13 1.01460
\(818\) 0 0
\(819\) − 5.03760e12i − 0.391243i
\(820\) 0 0
\(821\) 1.88473e13i 1.44779i 0.689911 + 0.723894i \(0.257649\pi\)
−0.689911 + 0.723894i \(0.742351\pi\)
\(822\) 0 0
\(823\) 1.03328e13 0.785089 0.392545 0.919733i \(-0.371595\pi\)
0.392545 + 0.919733i \(0.371595\pi\)
\(824\) 0 0
\(825\) −1.00935e13 −0.758575
\(826\) 0 0
\(827\) 4.38883e12i 0.326268i 0.986604 + 0.163134i \(0.0521603\pi\)
−0.986604 + 0.163134i \(0.947840\pi\)
\(828\) 0 0
\(829\) 2.55336e12i 0.187766i 0.995583 + 0.0938828i \(0.0299279\pi\)
−0.995583 + 0.0938828i \(0.970072\pi\)
\(830\) 0 0
\(831\) 3.48798e12 0.253728
\(832\) 0 0
\(833\) −3.52728e12 −0.253826
\(834\) 0 0
\(835\) − 4.77732e12i − 0.340091i
\(836\) 0 0
\(837\) 2.60200e12i 0.183249i
\(838\) 0 0
\(839\) 1.82154e13 1.26914 0.634570 0.772866i \(-0.281177\pi\)
0.634570 + 0.772866i \(0.281177\pi\)
\(840\) 0 0
\(841\) 4.11476e12 0.283637
\(842\) 0 0
\(843\) 1.57473e12i 0.107394i
\(844\) 0 0
\(845\) 3.28186e12i 0.221444i
\(846\) 0 0
\(847\) 1.13013e13 0.754486
\(848\) 0 0
\(849\) −5.98538e12 −0.395373
\(850\) 0 0
\(851\) 1.28446e13i 0.839531i
\(852\) 0 0
\(853\) − 7.23877e12i − 0.468160i −0.972217 0.234080i \(-0.924792\pi\)
0.972217 0.234080i \(-0.0752077\pi\)
\(854\) 0 0
\(855\) −1.14122e12 −0.0730332
\(856\) 0 0
\(857\) −6.12329e11 −0.0387768 −0.0193884 0.999812i \(-0.506172\pi\)
−0.0193884 + 0.999812i \(0.506172\pi\)
\(858\) 0 0
\(859\) − 5.23751e12i − 0.328213i −0.986443 0.164106i \(-0.947526\pi\)
0.986443 0.164106i \(-0.0524741\pi\)
\(860\) 0 0
\(861\) 1.73980e12i 0.107891i
\(862\) 0 0
\(863\) 2.53764e13 1.55733 0.778667 0.627437i \(-0.215896\pi\)
0.778667 + 0.627437i \(0.215896\pi\)
\(864\) 0 0
\(865\) 2.69696e12 0.163795
\(866\) 0 0
\(867\) − 3.21145e12i − 0.193026i
\(868\) 0 0
\(869\) − 1.15522e13i − 0.687188i
\(870\) 0 0
\(871\) 3.19541e13 1.88124
\(872\) 0 0
\(873\) −5.19583e12 −0.302755
\(874\) 0 0
\(875\) 6.21895e12i 0.358658i
\(876\) 0 0
\(877\) 3.79628e12i 0.216701i 0.994113 + 0.108350i \(0.0345568\pi\)
−0.994113 + 0.108350i \(0.965443\pi\)
\(878\) 0 0
\(879\) 2.48919e11 0.0140640
\(880\) 0 0
\(881\) 1.09066e13 0.609955 0.304977 0.952360i \(-0.401351\pi\)
0.304977 + 0.952360i \(0.401351\pi\)
\(882\) 0 0
\(883\) − 2.44991e12i − 0.135621i −0.997698 0.0678105i \(-0.978399\pi\)
0.997698 0.0678105i \(-0.0216013\pi\)
\(884\) 0 0
\(885\) − 2.24961e11i − 0.0123272i
\(886\) 0 0
\(887\) −2.15847e13 −1.17082 −0.585410 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(888\) 0 0
\(889\) 1.93105e13 1.03690
\(890\) 0 0
\(891\) 2.88810e12i 0.153519i
\(892\) 0 0
\(893\) − 5.37811e12i − 0.283008i
\(894\) 0 0
\(895\) 2.32167e12 0.120948
\(896\) 0 0
\(897\) −1.47824e13 −0.762394
\(898\) 0 0
\(899\) 1.57837e13i 0.805918i
\(900\) 0 0
\(901\) − 1.67277e13i − 0.845617i
\(902\) 0 0
\(903\) −9.82024e12 −0.491504
\(904\) 0 0
\(905\) 5.95980e12 0.295334
\(906\) 0 0
\(907\) 1.44358e13i 0.708287i 0.935191 + 0.354144i \(0.115228\pi\)
−0.935191 + 0.354144i \(0.884772\pi\)
\(908\) 0 0
\(909\) − 7.10919e12i − 0.345368i
\(910\) 0 0
\(911\) 2.86787e13 1.37952 0.689759 0.724039i \(-0.257716\pi\)
0.689759 + 0.724039i \(0.257716\pi\)
\(912\) 0 0
\(913\) 5.08340e13 2.42123
\(914\) 0 0
\(915\) 4.58723e12i 0.216349i
\(916\) 0 0
\(917\) − 1.96682e13i − 0.918551i
\(918\) 0 0
\(919\) 2.98333e13 1.37969 0.689845 0.723957i \(-0.257678\pi\)
0.689845 + 0.723957i \(0.257678\pi\)
\(920\) 0 0
\(921\) 6.22551e11 0.0285106
\(922\) 0 0
\(923\) 3.07514e13i 1.39462i
\(924\) 0 0
\(925\) − 1.90361e13i − 0.854951i
\(926\) 0 0
\(927\) −7.70565e12 −0.342729
\(928\) 0 0
\(929\) −1.98679e13 −0.875148 −0.437574 0.899182i \(-0.644162\pi\)
−0.437574 + 0.899182i \(0.644162\pi\)
\(930\) 0 0
\(931\) − 7.05444e12i − 0.307743i
\(932\) 0 0
\(933\) 1.87501e13i 0.810096i
\(934\) 0 0
\(935\) 5.83509e12 0.249686
\(936\) 0 0
\(937\) −9.96309e12 −0.422246 −0.211123 0.977459i \(-0.567712\pi\)
−0.211123 + 0.977459i \(0.567712\pi\)
\(938\) 0 0
\(939\) 5.79021e12i 0.243052i
\(940\) 0 0
\(941\) 2.05934e13i 0.856197i 0.903732 + 0.428099i \(0.140816\pi\)
−0.903732 + 0.428099i \(0.859184\pi\)
\(942\) 0 0
\(943\) 5.10529e12 0.210241
\(944\) 0 0
\(945\) 8.67357e11 0.0353797
\(946\) 0 0
\(947\) − 1.73914e13i − 0.702681i −0.936248 0.351341i \(-0.885726\pi\)
0.936248 0.351341i \(-0.114274\pi\)
\(948\) 0 0
\(949\) − 5.90021e13i − 2.36140i
\(950\) 0 0
\(951\) −5.64142e12 −0.223654
\(952\) 0 0
\(953\) −1.85939e13 −0.730219 −0.365109 0.930965i \(-0.618968\pi\)
−0.365109 + 0.930965i \(0.618968\pi\)
\(954\) 0 0
\(955\) 2.42787e12i 0.0944518i
\(956\) 0 0
\(957\) 1.75192e13i 0.675168i
\(958\) 0 0
\(959\) −2.45676e13 −0.937949
\(960\) 0 0
\(961\) −2.46764e12 −0.0933312
\(962\) 0 0
\(963\) 5.40363e12i 0.202473i
\(964\) 0 0
\(965\) − 4.52629e12i − 0.168023i
\(966\) 0 0
\(967\) −2.68504e13 −0.987489 −0.493744 0.869607i \(-0.664372\pi\)
−0.493744 + 0.869607i \(0.664372\pi\)
\(968\) 0 0
\(969\) −1.27881e13 −0.465961
\(970\) 0 0
\(971\) 2.06066e13i 0.743908i 0.928251 + 0.371954i \(0.121312\pi\)
−0.928251 + 0.371954i \(0.878688\pi\)
\(972\) 0 0
\(973\) − 1.12792e13i − 0.403432i
\(974\) 0 0
\(975\) 2.19081e13 0.776397
\(976\) 0 0
\(977\) 2.31007e13 0.811146 0.405573 0.914063i \(-0.367072\pi\)
0.405573 + 0.914063i \(0.367072\pi\)
\(978\) 0 0
\(979\) 6.19576e13i 2.15562i
\(980\) 0 0
\(981\) 9.51191e12i 0.327912i
\(982\) 0 0
\(983\) −8.68661e12 −0.296729 −0.148364 0.988933i \(-0.547401\pi\)
−0.148364 + 0.988933i \(0.547401\pi\)
\(984\) 0 0
\(985\) 8.97892e12 0.303921
\(986\) 0 0
\(987\) 4.08752e12i 0.137098i
\(988\) 0 0
\(989\) 2.88167e13i 0.957769i
\(990\) 0 0
\(991\) 2.74347e13 0.903585 0.451793 0.892123i \(-0.350785\pi\)
0.451793 + 0.892123i \(0.350785\pi\)
\(992\) 0 0
\(993\) −2.64153e13 −0.862153
\(994\) 0 0
\(995\) − 1.14734e13i − 0.371098i
\(996\) 0 0
\(997\) 6.62173e12i 0.212248i 0.994353 + 0.106124i \(0.0338440\pi\)
−0.994353 + 0.106124i \(0.966156\pi\)
\(998\) 0 0
\(999\) −5.44691e12 −0.173024
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.d.193.8 yes 10
4.3 odd 2 384.10.d.c.193.3 10
8.3 odd 2 384.10.d.c.193.8 yes 10
8.5 even 2 inner 384.10.d.d.193.3 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.c.193.3 10 4.3 odd 2
384.10.d.c.193.8 yes 10 8.3 odd 2
384.10.d.d.193.3 yes 10 8.5 even 2 inner
384.10.d.d.193.8 yes 10 1.1 even 1 trivial