Properties

Label 384.10.a.h.1.4
Level $384$
Weight $10$
Character 384.1
Self dual yes
Analytic conductor $197.774$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,10,Mod(1,384)]
 
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, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

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: yes
Analytic conductor: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 14124x^{2} - 170336x + 18391464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(119.418\) of defining polynomial
Character \(\chi\) \(=\) 384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+81.0000 q^{3} +1818.41 q^{5} -3817.40 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q+81.0000 q^{3} +1818.41 q^{5} -3817.40 q^{7} +6561.00 q^{9} +79487.2 q^{11} +154447. q^{13} +147291. q^{15} +343554. q^{17} +25500.6 q^{19} -309209. q^{21} +2.10072e6 q^{23} +1.35350e6 q^{25} +531441. q^{27} +1.13554e6 q^{29} +5.73826e6 q^{31} +6.43847e6 q^{33} -6.94160e6 q^{35} -9.67864e6 q^{37} +1.25102e7 q^{39} -1.08921e7 q^{41} -108899. q^{43} +1.19306e7 q^{45} -4.24010e7 q^{47} -2.57811e7 q^{49} +2.78279e7 q^{51} -7.58656e7 q^{53} +1.44541e8 q^{55} +2.06555e6 q^{57} +9.39466e7 q^{59} -1.38466e8 q^{61} -2.50459e7 q^{63} +2.80848e8 q^{65} +6.57461e7 q^{67} +1.70159e8 q^{69} +3.75829e7 q^{71} +2.83407e8 q^{73} +1.09633e8 q^{75} -3.03434e8 q^{77} +4.68780e8 q^{79} +4.30467e7 q^{81} -4.19210e8 q^{83} +6.24723e8 q^{85} +9.19785e7 q^{87} +8.32398e8 q^{89} -5.89586e8 q^{91} +4.64799e8 q^{93} +4.63705e7 q^{95} -1.37090e9 q^{97} +5.21516e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 324 q^{3} + 1728 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 324 q^{3} + 1728 q^{5} + 4840 q^{7} + 26244 q^{9} + 15824 q^{11} + 82440 q^{13} + 139968 q^{15} + 165912 q^{17} + 539904 q^{19} + 392040 q^{21} + 729680 q^{23} + 224812 q^{25} + 2125764 q^{27} + 1850864 q^{29} + 3197960 q^{31} + 1281744 q^{33} + 8574912 q^{35} + 4187992 q^{37} + 6677640 q^{39} + 227704 q^{41} + 1600352 q^{43} + 11337408 q^{45} + 18053904 q^{47} + 57728820 q^{49} + 13438872 q^{51} + 29418288 q^{53} + 45906816 q^{55} + 43732224 q^{57} + 38300048 q^{59} - 99764648 q^{61} + 31755240 q^{63} + 314120832 q^{65} - 183717008 q^{67} + 59104080 q^{69} + 181868080 q^{71} + 254539160 q^{73} + 18209772 q^{75} - 230564704 q^{77} + 831578184 q^{79} + 172186884 q^{81} - 687923952 q^{83} + 1041391104 q^{85} + 149919984 q^{87} + 627627272 q^{89} - 1018147632 q^{91} + 259034760 q^{93} + 2167118208 q^{95} - 889385880 q^{97} + 103821264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 81.0000 0.577350
\(4\) 0 0
\(5\) 1818.41 1.30115 0.650575 0.759442i \(-0.274528\pi\)
0.650575 + 0.759442i \(0.274528\pi\)
\(6\) 0 0
\(7\) −3817.40 −0.600933 −0.300467 0.953792i \(-0.597142\pi\)
−0.300467 + 0.953792i \(0.597142\pi\)
\(8\) 0 0
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 79487.2 1.63693 0.818465 0.574556i \(-0.194825\pi\)
0.818465 + 0.574556i \(0.194825\pi\)
\(12\) 0 0
\(13\) 154447. 1.49980 0.749902 0.661549i \(-0.230101\pi\)
0.749902 + 0.661549i \(0.230101\pi\)
\(14\) 0 0
\(15\) 147291. 0.751219
\(16\) 0 0
\(17\) 343554. 0.997643 0.498822 0.866705i \(-0.333766\pi\)
0.498822 + 0.866705i \(0.333766\pi\)
\(18\) 0 0
\(19\) 25500.6 0.0448909 0.0224455 0.999748i \(-0.492855\pi\)
0.0224455 + 0.999748i \(0.492855\pi\)
\(20\) 0 0
\(21\) −309209. −0.346949
\(22\) 0 0
\(23\) 2.10072e6 1.56529 0.782643 0.622470i \(-0.213871\pi\)
0.782643 + 0.622470i \(0.213871\pi\)
\(24\) 0 0
\(25\) 1.35350e6 0.692990
\(26\) 0 0
\(27\) 531441. 0.192450
\(28\) 0 0
\(29\) 1.13554e6 0.298133 0.149067 0.988827i \(-0.452373\pi\)
0.149067 + 0.988827i \(0.452373\pi\)
\(30\) 0 0
\(31\) 5.73826e6 1.11597 0.557985 0.829851i \(-0.311574\pi\)
0.557985 + 0.829851i \(0.311574\pi\)
\(32\) 0 0
\(33\) 6.43847e6 0.945082
\(34\) 0 0
\(35\) −6.94160e6 −0.781904
\(36\) 0 0
\(37\) −9.67864e6 −0.848998 −0.424499 0.905429i \(-0.639550\pi\)
−0.424499 + 0.905429i \(0.639550\pi\)
\(38\) 0 0
\(39\) 1.25102e7 0.865912
\(40\) 0 0
\(41\) −1.08921e7 −0.601982 −0.300991 0.953627i \(-0.597317\pi\)
−0.300991 + 0.953627i \(0.597317\pi\)
\(42\) 0 0
\(43\) −108899. −0.00485754 −0.00242877 0.999997i \(-0.500773\pi\)
−0.00242877 + 0.999997i \(0.500773\pi\)
\(44\) 0 0
\(45\) 1.19306e7 0.433717
\(46\) 0 0
\(47\) −4.24010e7 −1.26747 −0.633733 0.773552i \(-0.718478\pi\)
−0.633733 + 0.773552i \(0.718478\pi\)
\(48\) 0 0
\(49\) −2.57811e7 −0.638879
\(50\) 0 0
\(51\) 2.78279e7 0.575990
\(52\) 0 0
\(53\) −7.58656e7 −1.32070 −0.660349 0.750959i \(-0.729592\pi\)
−0.660349 + 0.750959i \(0.729592\pi\)
\(54\) 0 0
\(55\) 1.44541e8 2.12989
\(56\) 0 0
\(57\) 2.06555e6 0.0259178
\(58\) 0 0
\(59\) 9.39466e7 1.00936 0.504681 0.863306i \(-0.331610\pi\)
0.504681 + 0.863306i \(0.331610\pi\)
\(60\) 0 0
\(61\) −1.38466e8 −1.28044 −0.640221 0.768190i \(-0.721157\pi\)
−0.640221 + 0.768190i \(0.721157\pi\)
\(62\) 0 0
\(63\) −2.50459e7 −0.200311
\(64\) 0 0
\(65\) 2.80848e8 1.95147
\(66\) 0 0
\(67\) 6.57461e7 0.398597 0.199298 0.979939i \(-0.436134\pi\)
0.199298 + 0.979939i \(0.436134\pi\)
\(68\) 0 0
\(69\) 1.70159e8 0.903719
\(70\) 0 0
\(71\) 3.75829e7 0.175520 0.0877602 0.996142i \(-0.472029\pi\)
0.0877602 + 0.996142i \(0.472029\pi\)
\(72\) 0 0
\(73\) 2.83407e8 1.16804 0.584019 0.811740i \(-0.301479\pi\)
0.584019 + 0.811740i \(0.301479\pi\)
\(74\) 0 0
\(75\) 1.09633e8 0.400098
\(76\) 0 0
\(77\) −3.03434e8 −0.983686
\(78\) 0 0
\(79\) 4.68780e8 1.35409 0.677045 0.735942i \(-0.263260\pi\)
0.677045 + 0.735942i \(0.263260\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) −4.19210e8 −0.969573 −0.484786 0.874633i \(-0.661103\pi\)
−0.484786 + 0.874633i \(0.661103\pi\)
\(84\) 0 0
\(85\) 6.24723e8 1.29808
\(86\) 0 0
\(87\) 9.19785e7 0.172127
\(88\) 0 0
\(89\) 8.32398e8 1.40629 0.703147 0.711045i \(-0.251778\pi\)
0.703147 + 0.711045i \(0.251778\pi\)
\(90\) 0 0
\(91\) −5.89586e8 −0.901282
\(92\) 0 0
\(93\) 4.64799e8 0.644306
\(94\) 0 0
\(95\) 4.63705e7 0.0584098
\(96\) 0 0
\(97\) −1.37090e9 −1.57229 −0.786143 0.618044i \(-0.787925\pi\)
−0.786143 + 0.618044i \(0.787925\pi\)
\(98\) 0 0
\(99\) 5.21516e8 0.545643
\(100\) 0 0
\(101\) 1.29635e9 1.23958 0.619791 0.784767i \(-0.287218\pi\)
0.619791 + 0.784767i \(0.287218\pi\)
\(102\) 0 0
\(103\) −2.02804e8 −0.177545 −0.0887727 0.996052i \(-0.528294\pi\)
−0.0887727 + 0.996052i \(0.528294\pi\)
\(104\) 0 0
\(105\) −5.62270e8 −0.451433
\(106\) 0 0
\(107\) −2.25915e9 −1.66616 −0.833082 0.553150i \(-0.813426\pi\)
−0.833082 + 0.553150i \(0.813426\pi\)
\(108\) 0 0
\(109\) 2.66475e8 0.180816 0.0904082 0.995905i \(-0.471183\pi\)
0.0904082 + 0.995905i \(0.471183\pi\)
\(110\) 0 0
\(111\) −7.83970e8 −0.490169
\(112\) 0 0
\(113\) −2.42429e9 −1.39872 −0.699360 0.714770i \(-0.746532\pi\)
−0.699360 + 0.714770i \(0.746532\pi\)
\(114\) 0 0
\(115\) 3.81998e9 2.03667
\(116\) 0 0
\(117\) 1.01333e9 0.499935
\(118\) 0 0
\(119\) −1.31148e9 −0.599517
\(120\) 0 0
\(121\) 3.96027e9 1.67954
\(122\) 0 0
\(123\) −8.82259e8 −0.347555
\(124\) 0 0
\(125\) −1.09037e9 −0.399465
\(126\) 0 0
\(127\) −9.12647e8 −0.311305 −0.155652 0.987812i \(-0.549748\pi\)
−0.155652 + 0.987812i \(0.549748\pi\)
\(128\) 0 0
\(129\) −8.82084e6 −0.00280450
\(130\) 0 0
\(131\) −5.96692e9 −1.77023 −0.885114 0.465375i \(-0.845920\pi\)
−0.885114 + 0.465375i \(0.845920\pi\)
\(132\) 0 0
\(133\) −9.73458e7 −0.0269764
\(134\) 0 0
\(135\) 9.66379e8 0.250406
\(136\) 0 0
\(137\) −7.06316e9 −1.71300 −0.856498 0.516150i \(-0.827365\pi\)
−0.856498 + 0.516150i \(0.827365\pi\)
\(138\) 0 0
\(139\) −2.13879e9 −0.485961 −0.242980 0.970031i \(-0.578125\pi\)
−0.242980 + 0.970031i \(0.578125\pi\)
\(140\) 0 0
\(141\) −3.43448e9 −0.731772
\(142\) 0 0
\(143\) 1.22766e10 2.45508
\(144\) 0 0
\(145\) 2.06487e9 0.387916
\(146\) 0 0
\(147\) −2.08827e9 −0.368857
\(148\) 0 0
\(149\) 1.97318e9 0.327965 0.163983 0.986463i \(-0.447566\pi\)
0.163983 + 0.986463i \(0.447566\pi\)
\(150\) 0 0
\(151\) −2.03152e8 −0.0317998 −0.0158999 0.999874i \(-0.505061\pi\)
−0.0158999 + 0.999874i \(0.505061\pi\)
\(152\) 0 0
\(153\) 2.25406e9 0.332548
\(154\) 0 0
\(155\) 1.04345e10 1.45205
\(156\) 0 0
\(157\) 1.37720e10 1.80904 0.904520 0.426432i \(-0.140230\pi\)
0.904520 + 0.426432i \(0.140230\pi\)
\(158\) 0 0
\(159\) −6.14511e9 −0.762505
\(160\) 0 0
\(161\) −8.01930e9 −0.940633
\(162\) 0 0
\(163\) 3.01690e9 0.334747 0.167374 0.985894i \(-0.446471\pi\)
0.167374 + 0.985894i \(0.446471\pi\)
\(164\) 0 0
\(165\) 1.17078e10 1.22969
\(166\) 0 0
\(167\) 4.92894e9 0.490376 0.245188 0.969476i \(-0.421150\pi\)
0.245188 + 0.969476i \(0.421150\pi\)
\(168\) 0 0
\(169\) 1.32494e10 1.24941
\(170\) 0 0
\(171\) 1.67309e8 0.0149636
\(172\) 0 0
\(173\) −1.34775e10 −1.14393 −0.571967 0.820277i \(-0.693819\pi\)
−0.571967 + 0.820277i \(0.693819\pi\)
\(174\) 0 0
\(175\) −5.16684e9 −0.416441
\(176\) 0 0
\(177\) 7.60968e9 0.582756
\(178\) 0 0
\(179\) 1.28594e10 0.936231 0.468116 0.883667i \(-0.344933\pi\)
0.468116 + 0.883667i \(0.344933\pi\)
\(180\) 0 0
\(181\) −1.46209e10 −1.01256 −0.506281 0.862368i \(-0.668980\pi\)
−0.506281 + 0.862368i \(0.668980\pi\)
\(182\) 0 0
\(183\) −1.12158e10 −0.739264
\(184\) 0 0
\(185\) −1.75997e10 −1.10467
\(186\) 0 0
\(187\) 2.73082e10 1.63307
\(188\) 0 0
\(189\) −2.02872e9 −0.115650
\(190\) 0 0
\(191\) 2.49838e10 1.35834 0.679170 0.733981i \(-0.262340\pi\)
0.679170 + 0.733981i \(0.262340\pi\)
\(192\) 0 0
\(193\) −2.45447e10 −1.27336 −0.636678 0.771129i \(-0.719692\pi\)
−0.636678 + 0.771129i \(0.719692\pi\)
\(194\) 0 0
\(195\) 2.27487e10 1.12668
\(196\) 0 0
\(197\) −1.14198e10 −0.540209 −0.270104 0.962831i \(-0.587058\pi\)
−0.270104 + 0.962831i \(0.587058\pi\)
\(198\) 0 0
\(199\) 1.08567e10 0.490747 0.245373 0.969429i \(-0.421089\pi\)
0.245373 + 0.969429i \(0.421089\pi\)
\(200\) 0 0
\(201\) 5.32544e9 0.230130
\(202\) 0 0
\(203\) −4.33480e9 −0.179158
\(204\) 0 0
\(205\) −1.98063e10 −0.783269
\(206\) 0 0
\(207\) 1.37829e10 0.521762
\(208\) 0 0
\(209\) 2.02697e9 0.0734833
\(210\) 0 0
\(211\) 4.64028e10 1.61166 0.805830 0.592147i \(-0.201719\pi\)
0.805830 + 0.592147i \(0.201719\pi\)
\(212\) 0 0
\(213\) 3.04422e9 0.101337
\(214\) 0 0
\(215\) −1.98024e8 −0.00632039
\(216\) 0 0
\(217\) −2.19052e10 −0.670624
\(218\) 0 0
\(219\) 2.29559e10 0.674367
\(220\) 0 0
\(221\) 5.30610e10 1.49627
\(222\) 0 0
\(223\) 2.71348e10 0.734775 0.367388 0.930068i \(-0.380252\pi\)
0.367388 + 0.930068i \(0.380252\pi\)
\(224\) 0 0
\(225\) 8.88029e9 0.230997
\(226\) 0 0
\(227\) −7.40226e9 −0.185032 −0.0925162 0.995711i \(-0.529491\pi\)
−0.0925162 + 0.995711i \(0.529491\pi\)
\(228\) 0 0
\(229\) 4.21764e10 1.01347 0.506733 0.862103i \(-0.330853\pi\)
0.506733 + 0.862103i \(0.330853\pi\)
\(230\) 0 0
\(231\) −2.45782e10 −0.567931
\(232\) 0 0
\(233\) −7.28793e10 −1.61995 −0.809977 0.586462i \(-0.800520\pi\)
−0.809977 + 0.586462i \(0.800520\pi\)
\(234\) 0 0
\(235\) −7.71026e10 −1.64916
\(236\) 0 0
\(237\) 3.79712e10 0.781784
\(238\) 0 0
\(239\) 1.38185e10 0.273950 0.136975 0.990574i \(-0.456262\pi\)
0.136975 + 0.990574i \(0.456262\pi\)
\(240\) 0 0
\(241\) −4.86065e10 −0.928148 −0.464074 0.885796i \(-0.653613\pi\)
−0.464074 + 0.885796i \(0.653613\pi\)
\(242\) 0 0
\(243\) 3.48678e9 0.0641500
\(244\) 0 0
\(245\) −4.68806e10 −0.831277
\(246\) 0 0
\(247\) 3.93849e9 0.0673276
\(248\) 0 0
\(249\) −3.39560e10 −0.559783
\(250\) 0 0
\(251\) −9.95300e10 −1.58279 −0.791393 0.611308i \(-0.790644\pi\)
−0.791393 + 0.611308i \(0.790644\pi\)
\(252\) 0 0
\(253\) 1.66981e11 2.56227
\(254\) 0 0
\(255\) 5.06026e10 0.749449
\(256\) 0 0
\(257\) 4.82008e10 0.689216 0.344608 0.938747i \(-0.388012\pi\)
0.344608 + 0.938747i \(0.388012\pi\)
\(258\) 0 0
\(259\) 3.69472e10 0.510191
\(260\) 0 0
\(261\) 7.45026e9 0.0993777
\(262\) 0 0
\(263\) 3.11820e10 0.401886 0.200943 0.979603i \(-0.435599\pi\)
0.200943 + 0.979603i \(0.435599\pi\)
\(264\) 0 0
\(265\) −1.37955e11 −1.71843
\(266\) 0 0
\(267\) 6.74243e10 0.811924
\(268\) 0 0
\(269\) 4.79418e10 0.558251 0.279125 0.960255i \(-0.409955\pi\)
0.279125 + 0.960255i \(0.409955\pi\)
\(270\) 0 0
\(271\) −6.20036e10 −0.698321 −0.349160 0.937063i \(-0.613533\pi\)
−0.349160 + 0.937063i \(0.613533\pi\)
\(272\) 0 0
\(273\) −4.77565e10 −0.520356
\(274\) 0 0
\(275\) 1.07586e11 1.13438
\(276\) 0 0
\(277\) 7.51491e10 0.766946 0.383473 0.923552i \(-0.374728\pi\)
0.383473 + 0.923552i \(0.374728\pi\)
\(278\) 0 0
\(279\) 3.76487e10 0.371990
\(280\) 0 0
\(281\) 7.27701e9 0.0696265 0.0348133 0.999394i \(-0.488916\pi\)
0.0348133 + 0.999394i \(0.488916\pi\)
\(282\) 0 0
\(283\) 1.07526e11 0.996492 0.498246 0.867036i \(-0.333978\pi\)
0.498246 + 0.867036i \(0.333978\pi\)
\(284\) 0 0
\(285\) 3.75601e9 0.0337229
\(286\) 0 0
\(287\) 4.15794e10 0.361751
\(288\) 0 0
\(289\) −5.58306e8 −0.00470795
\(290\) 0 0
\(291\) −1.11043e11 −0.907760
\(292\) 0 0
\(293\) −2.31989e10 −0.183892 −0.0919462 0.995764i \(-0.529309\pi\)
−0.0919462 + 0.995764i \(0.529309\pi\)
\(294\) 0 0
\(295\) 1.70834e11 1.31333
\(296\) 0 0
\(297\) 4.22428e10 0.315027
\(298\) 0 0
\(299\) 3.24451e11 2.34762
\(300\) 0 0
\(301\) 4.15712e8 0.00291906
\(302\) 0 0
\(303\) 1.05004e11 0.715672
\(304\) 0 0
\(305\) −2.51789e11 −1.66605
\(306\) 0 0
\(307\) 1.84503e11 1.18544 0.592721 0.805408i \(-0.298054\pi\)
0.592721 + 0.805408i \(0.298054\pi\)
\(308\) 0 0
\(309\) −1.64271e10 −0.102506
\(310\) 0 0
\(311\) 1.83522e11 1.11241 0.556205 0.831045i \(-0.312257\pi\)
0.556205 + 0.831045i \(0.312257\pi\)
\(312\) 0 0
\(313\) 1.67363e11 0.985620 0.492810 0.870137i \(-0.335970\pi\)
0.492810 + 0.870137i \(0.335970\pi\)
\(314\) 0 0
\(315\) −4.55439e10 −0.260635
\(316\) 0 0
\(317\) 3.07406e11 1.70980 0.854899 0.518794i \(-0.173619\pi\)
0.854899 + 0.518794i \(0.173619\pi\)
\(318\) 0 0
\(319\) 9.02607e10 0.488023
\(320\) 0 0
\(321\) −1.82991e11 −0.961960
\(322\) 0 0
\(323\) 8.76083e9 0.0447851
\(324\) 0 0
\(325\) 2.09044e11 1.03935
\(326\) 0 0
\(327\) 2.15845e10 0.104394
\(328\) 0 0
\(329\) 1.61862e11 0.761662
\(330\) 0 0
\(331\) −2.22502e10 −0.101884 −0.0509421 0.998702i \(-0.516222\pi\)
−0.0509421 + 0.998702i \(0.516222\pi\)
\(332\) 0 0
\(333\) −6.35015e10 −0.282999
\(334\) 0 0
\(335\) 1.19554e11 0.518634
\(336\) 0 0
\(337\) −6.04892e10 −0.255472 −0.127736 0.991808i \(-0.540771\pi\)
−0.127736 + 0.991808i \(0.540771\pi\)
\(338\) 0 0
\(339\) −1.96367e11 −0.807551
\(340\) 0 0
\(341\) 4.56119e11 1.82677
\(342\) 0 0
\(343\) 2.52462e11 0.984857
\(344\) 0 0
\(345\) 3.09419e11 1.17587
\(346\) 0 0
\(347\) 1.97594e11 0.731628 0.365814 0.930688i \(-0.380791\pi\)
0.365814 + 0.930688i \(0.380791\pi\)
\(348\) 0 0
\(349\) −4.32441e11 −1.56032 −0.780159 0.625582i \(-0.784862\pi\)
−0.780159 + 0.625582i \(0.784862\pi\)
\(350\) 0 0
\(351\) 8.20795e10 0.288637
\(352\) 0 0
\(353\) 3.49455e10 0.119786 0.0598928 0.998205i \(-0.480924\pi\)
0.0598928 + 0.998205i \(0.480924\pi\)
\(354\) 0 0
\(355\) 6.83412e10 0.228378
\(356\) 0 0
\(357\) −1.06230e11 −0.346131
\(358\) 0 0
\(359\) −2.44648e10 −0.0777351 −0.0388676 0.999244i \(-0.512375\pi\)
−0.0388676 + 0.999244i \(0.512375\pi\)
\(360\) 0 0
\(361\) −3.22037e11 −0.997985
\(362\) 0 0
\(363\) 3.20782e11 0.969684
\(364\) 0 0
\(365\) 5.15350e11 1.51979
\(366\) 0 0
\(367\) 1.54134e11 0.443508 0.221754 0.975103i \(-0.428822\pi\)
0.221754 + 0.975103i \(0.428822\pi\)
\(368\) 0 0
\(369\) −7.14630e10 −0.200661
\(370\) 0 0
\(371\) 2.89609e11 0.793651
\(372\) 0 0
\(373\) −2.83594e11 −0.758590 −0.379295 0.925276i \(-0.623833\pi\)
−0.379295 + 0.925276i \(0.623833\pi\)
\(374\) 0 0
\(375\) −8.83200e10 −0.230631
\(376\) 0 0
\(377\) 1.75380e11 0.447141
\(378\) 0 0
\(379\) −4.98473e11 −1.24098 −0.620490 0.784214i \(-0.713066\pi\)
−0.620490 + 0.784214i \(0.713066\pi\)
\(380\) 0 0
\(381\) −7.39244e10 −0.179732
\(382\) 0 0
\(383\) 2.80570e11 0.666264 0.333132 0.942880i \(-0.391895\pi\)
0.333132 + 0.942880i \(0.391895\pi\)
\(384\) 0 0
\(385\) −5.51769e11 −1.27992
\(386\) 0 0
\(387\) −7.14488e8 −0.00161918
\(388\) 0 0
\(389\) 3.48132e10 0.0770853 0.0385426 0.999257i \(-0.487728\pi\)
0.0385426 + 0.999257i \(0.487728\pi\)
\(390\) 0 0
\(391\) 7.21713e11 1.56160
\(392\) 0 0
\(393\) −4.83320e11 −1.02204
\(394\) 0 0
\(395\) 8.52436e11 1.76187
\(396\) 0 0
\(397\) −2.60260e10 −0.0525835 −0.0262917 0.999654i \(-0.508370\pi\)
−0.0262917 + 0.999654i \(0.508370\pi\)
\(398\) 0 0
\(399\) −7.88501e9 −0.0155749
\(400\) 0 0
\(401\) −7.66429e11 −1.48021 −0.740103 0.672494i \(-0.765223\pi\)
−0.740103 + 0.672494i \(0.765223\pi\)
\(402\) 0 0
\(403\) 8.86258e11 1.67374
\(404\) 0 0
\(405\) 7.82767e10 0.144572
\(406\) 0 0
\(407\) −7.69328e11 −1.38975
\(408\) 0 0
\(409\) −3.76343e11 −0.665011 −0.332506 0.943101i \(-0.607894\pi\)
−0.332506 + 0.943101i \(0.607894\pi\)
\(410\) 0 0
\(411\) −5.72116e11 −0.988999
\(412\) 0 0
\(413\) −3.58632e11 −0.606559
\(414\) 0 0
\(415\) −7.62297e11 −1.26156
\(416\) 0 0
\(417\) −1.73242e11 −0.280569
\(418\) 0 0
\(419\) −9.24941e11 −1.46606 −0.733028 0.680198i \(-0.761894\pi\)
−0.733028 + 0.680198i \(0.761894\pi\)
\(420\) 0 0
\(421\) −5.98004e11 −0.927758 −0.463879 0.885899i \(-0.653543\pi\)
−0.463879 + 0.885899i \(0.653543\pi\)
\(422\) 0 0
\(423\) −2.78193e11 −0.422489
\(424\) 0 0
\(425\) 4.65000e11 0.691357
\(426\) 0 0
\(427\) 5.28581e11 0.769461
\(428\) 0 0
\(429\) 9.94402e11 1.41744
\(430\) 0 0
\(431\) 3.60269e10 0.0502897 0.0251448 0.999684i \(-0.491995\pi\)
0.0251448 + 0.999684i \(0.491995\pi\)
\(432\) 0 0
\(433\) −8.52251e11 −1.16512 −0.582562 0.812786i \(-0.697950\pi\)
−0.582562 + 0.812786i \(0.697950\pi\)
\(434\) 0 0
\(435\) 1.67255e11 0.223963
\(436\) 0 0
\(437\) 5.35697e10 0.0702672
\(438\) 0 0
\(439\) −1.94785e11 −0.250302 −0.125151 0.992138i \(-0.539942\pi\)
−0.125151 + 0.992138i \(0.539942\pi\)
\(440\) 0 0
\(441\) −1.69150e11 −0.212960
\(442\) 0 0
\(443\) 2.03960e11 0.251611 0.125805 0.992055i \(-0.459849\pi\)
0.125805 + 0.992055i \(0.459849\pi\)
\(444\) 0 0
\(445\) 1.51364e12 1.82980
\(446\) 0 0
\(447\) 1.59827e11 0.189351
\(448\) 0 0
\(449\) 3.72352e11 0.432360 0.216180 0.976354i \(-0.430640\pi\)
0.216180 + 0.976354i \(0.430640\pi\)
\(450\) 0 0
\(451\) −8.65782e11 −0.985403
\(452\) 0 0
\(453\) −1.64553e10 −0.0183596
\(454\) 0 0
\(455\) −1.07211e12 −1.17270
\(456\) 0 0
\(457\) 1.43400e12 1.53789 0.768947 0.639313i \(-0.220781\pi\)
0.768947 + 0.639313i \(0.220781\pi\)
\(458\) 0 0
\(459\) 1.82579e11 0.191997
\(460\) 0 0
\(461\) 1.21208e12 1.24991 0.624953 0.780662i \(-0.285118\pi\)
0.624953 + 0.780662i \(0.285118\pi\)
\(462\) 0 0
\(463\) −1.68275e12 −1.70179 −0.850895 0.525336i \(-0.823939\pi\)
−0.850895 + 0.525336i \(0.823939\pi\)
\(464\) 0 0
\(465\) 8.45197e11 0.838339
\(466\) 0 0
\(467\) −9.42499e11 −0.916969 −0.458485 0.888702i \(-0.651608\pi\)
−0.458485 + 0.888702i \(0.651608\pi\)
\(468\) 0 0
\(469\) −2.50979e11 −0.239530
\(470\) 0 0
\(471\) 1.11553e12 1.04445
\(472\) 0 0
\(473\) −8.65610e9 −0.00795146
\(474\) 0 0
\(475\) 3.45149e10 0.0311090
\(476\) 0 0
\(477\) −4.97754e11 −0.440233
\(478\) 0 0
\(479\) −9.15579e11 −0.794669 −0.397334 0.917674i \(-0.630065\pi\)
−0.397334 + 0.917674i \(0.630065\pi\)
\(480\) 0 0
\(481\) −1.49484e12 −1.27333
\(482\) 0 0
\(483\) −6.49564e11 −0.543075
\(484\) 0 0
\(485\) −2.49285e12 −2.04578
\(486\) 0 0
\(487\) 1.70542e12 1.37389 0.686943 0.726712i \(-0.258952\pi\)
0.686943 + 0.726712i \(0.258952\pi\)
\(488\) 0 0
\(489\) 2.44369e11 0.193266
\(490\) 0 0
\(491\) −1.95142e12 −1.51525 −0.757625 0.652690i \(-0.773640\pi\)
−0.757625 + 0.652690i \(0.773640\pi\)
\(492\) 0 0
\(493\) 3.90119e11 0.297431
\(494\) 0 0
\(495\) 9.48330e11 0.709964
\(496\) 0 0
\(497\) −1.43469e11 −0.105476
\(498\) 0 0
\(499\) 2.72335e12 1.96630 0.983152 0.182790i \(-0.0585127\pi\)
0.983152 + 0.182790i \(0.0585127\pi\)
\(500\) 0 0
\(501\) 3.99244e11 0.283119
\(502\) 0 0
\(503\) 2.70028e12 1.88084 0.940422 0.340011i \(-0.110431\pi\)
0.940422 + 0.340011i \(0.110431\pi\)
\(504\) 0 0
\(505\) 2.35729e12 1.61288
\(506\) 0 0
\(507\) 1.07320e12 0.721349
\(508\) 0 0
\(509\) −2.15879e12 −1.42554 −0.712770 0.701398i \(-0.752560\pi\)
−0.712770 + 0.701398i \(0.752560\pi\)
\(510\) 0 0
\(511\) −1.08188e12 −0.701913
\(512\) 0 0
\(513\) 1.35520e10 0.00863926
\(514\) 0 0
\(515\) −3.68782e11 −0.231013
\(516\) 0 0
\(517\) −3.37034e12 −2.07475
\(518\) 0 0
\(519\) −1.09167e12 −0.660450
\(520\) 0 0
\(521\) 6.59041e11 0.391871 0.195935 0.980617i \(-0.437226\pi\)
0.195935 + 0.980617i \(0.437226\pi\)
\(522\) 0 0
\(523\) 7.33748e11 0.428834 0.214417 0.976742i \(-0.431215\pi\)
0.214417 + 0.976742i \(0.431215\pi\)
\(524\) 0 0
\(525\) −4.18514e11 −0.240432
\(526\) 0 0
\(527\) 1.97141e12 1.11334
\(528\) 0 0
\(529\) 2.61189e12 1.45012
\(530\) 0 0
\(531\) 6.16384e11 0.336454
\(532\) 0 0
\(533\) −1.68225e12 −0.902856
\(534\) 0 0
\(535\) −4.10806e12 −2.16793
\(536\) 0 0
\(537\) 1.04161e12 0.540533
\(538\) 0 0
\(539\) −2.04927e12 −1.04580
\(540\) 0 0
\(541\) 2.56238e12 1.28604 0.643022 0.765848i \(-0.277680\pi\)
0.643022 + 0.765848i \(0.277680\pi\)
\(542\) 0 0
\(543\) −1.18430e12 −0.584603
\(544\) 0 0
\(545\) 4.84562e11 0.235269
\(546\) 0 0
\(547\) −2.42235e12 −1.15689 −0.578447 0.815720i \(-0.696341\pi\)
−0.578447 + 0.815720i \(0.696341\pi\)
\(548\) 0 0
\(549\) −9.08478e11 −0.426814
\(550\) 0 0
\(551\) 2.89568e10 0.0133835
\(552\) 0 0
\(553\) −1.78952e12 −0.813718
\(554\) 0 0
\(555\) −1.42558e12 −0.637783
\(556\) 0 0
\(557\) −2.99530e11 −0.131854 −0.0659269 0.997824i \(-0.521000\pi\)
−0.0659269 + 0.997824i \(0.521000\pi\)
\(558\) 0 0
\(559\) −1.68192e10 −0.00728537
\(560\) 0 0
\(561\) 2.21196e12 0.942855
\(562\) 0 0
\(563\) 2.58923e12 1.08613 0.543066 0.839690i \(-0.317263\pi\)
0.543066 + 0.839690i \(0.317263\pi\)
\(564\) 0 0
\(565\) −4.40835e12 −1.81994
\(566\) 0 0
\(567\) −1.64326e11 −0.0667704
\(568\) 0 0
\(569\) 3.09519e12 1.23789 0.618945 0.785434i \(-0.287560\pi\)
0.618945 + 0.785434i \(0.287560\pi\)
\(570\) 0 0
\(571\) −1.97701e12 −0.778298 −0.389149 0.921175i \(-0.627231\pi\)
−0.389149 + 0.921175i \(0.627231\pi\)
\(572\) 0 0
\(573\) 2.02369e12 0.784238
\(574\) 0 0
\(575\) 2.84332e12 1.08473
\(576\) 0 0
\(577\) −4.50291e11 −0.169123 −0.0845614 0.996418i \(-0.526949\pi\)
−0.0845614 + 0.996418i \(0.526949\pi\)
\(578\) 0 0
\(579\) −1.98812e12 −0.735173
\(580\) 0 0
\(581\) 1.60029e12 0.582648
\(582\) 0 0
\(583\) −6.03035e12 −2.16189
\(584\) 0 0
\(585\) 1.84265e12 0.650490
\(586\) 0 0
\(587\) −3.02724e12 −1.05239 −0.526193 0.850365i \(-0.676381\pi\)
−0.526193 + 0.850365i \(0.676381\pi\)
\(588\) 0 0
\(589\) 1.46329e11 0.0500970
\(590\) 0 0
\(591\) −9.25007e11 −0.311890
\(592\) 0 0
\(593\) −4.55472e12 −1.51257 −0.756284 0.654243i \(-0.772987\pi\)
−0.756284 + 0.654243i \(0.772987\pi\)
\(594\) 0 0
\(595\) −2.38482e12 −0.780061
\(596\) 0 0
\(597\) 8.79390e11 0.283333
\(598\) 0 0
\(599\) 2.02438e12 0.642497 0.321249 0.946995i \(-0.395897\pi\)
0.321249 + 0.946995i \(0.395897\pi\)
\(600\) 0 0
\(601\) 5.89526e11 0.184318 0.0921591 0.995744i \(-0.470623\pi\)
0.0921591 + 0.995744i \(0.470623\pi\)
\(602\) 0 0
\(603\) 4.31360e11 0.132866
\(604\) 0 0
\(605\) 7.20140e12 2.18533
\(606\) 0 0
\(607\) 5.99382e12 1.79207 0.896035 0.443984i \(-0.146435\pi\)
0.896035 + 0.443984i \(0.146435\pi\)
\(608\) 0 0
\(609\) −3.51119e11 −0.103437
\(610\) 0 0
\(611\) −6.54872e12 −1.90095
\(612\) 0 0
\(613\) −3.98091e12 −1.13870 −0.569351 0.822094i \(-0.692806\pi\)
−0.569351 + 0.822094i \(0.692806\pi\)
\(614\) 0 0
\(615\) −1.60431e12 −0.452221
\(616\) 0 0
\(617\) 6.04076e12 1.67806 0.839032 0.544082i \(-0.183122\pi\)
0.839032 + 0.544082i \(0.183122\pi\)
\(618\) 0 0
\(619\) −1.19083e12 −0.326019 −0.163010 0.986624i \(-0.552120\pi\)
−0.163010 + 0.986624i \(0.552120\pi\)
\(620\) 0 0
\(621\) 1.11641e12 0.301240
\(622\) 0 0
\(623\) −3.17760e12 −0.845089
\(624\) 0 0
\(625\) −4.62629e12 −1.21275
\(626\) 0 0
\(627\) 1.64184e11 0.0424256
\(628\) 0 0
\(629\) −3.32514e12 −0.846997
\(630\) 0 0
\(631\) −7.09497e12 −1.78163 −0.890816 0.454363i \(-0.849867\pi\)
−0.890816 + 0.454363i \(0.849867\pi\)
\(632\) 0 0
\(633\) 3.75863e12 0.930493
\(634\) 0 0
\(635\) −1.65957e12 −0.405054
\(636\) 0 0
\(637\) −3.98181e12 −0.958194
\(638\) 0 0
\(639\) 2.46581e11 0.0585068
\(640\) 0 0
\(641\) 4.21353e12 0.985792 0.492896 0.870088i \(-0.335938\pi\)
0.492896 + 0.870088i \(0.335938\pi\)
\(642\) 0 0
\(643\) 2.10169e12 0.484864 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(644\) 0 0
\(645\) −1.60399e10 −0.00364908
\(646\) 0 0
\(647\) −3.45447e12 −0.775018 −0.387509 0.921866i \(-0.626664\pi\)
−0.387509 + 0.921866i \(0.626664\pi\)
\(648\) 0 0
\(649\) 7.46755e12 1.65226
\(650\) 0 0
\(651\) −1.77432e12 −0.387185
\(652\) 0 0
\(653\) −5.38398e12 −1.15876 −0.579380 0.815057i \(-0.696705\pi\)
−0.579380 + 0.815057i \(0.696705\pi\)
\(654\) 0 0
\(655\) −1.08503e13 −2.30333
\(656\) 0 0
\(657\) 1.85943e12 0.389346
\(658\) 0 0
\(659\) 6.08428e12 1.25668 0.628340 0.777939i \(-0.283735\pi\)
0.628340 + 0.777939i \(0.283735\pi\)
\(660\) 0 0
\(661\) 4.81235e12 0.980506 0.490253 0.871580i \(-0.336904\pi\)
0.490253 + 0.871580i \(0.336904\pi\)
\(662\) 0 0
\(663\) 4.29794e12 0.863872
\(664\) 0 0
\(665\) −1.77015e11 −0.0351004
\(666\) 0 0
\(667\) 2.38545e12 0.466664
\(668\) 0 0
\(669\) 2.19792e12 0.424223
\(670\) 0 0
\(671\) −1.10063e13 −2.09600
\(672\) 0 0
\(673\) −8.82624e12 −1.65847 −0.829235 0.558899i \(-0.811224\pi\)
−0.829235 + 0.558899i \(0.811224\pi\)
\(674\) 0 0
\(675\) 7.19304e11 0.133366
\(676\) 0 0
\(677\) −3.95703e12 −0.723969 −0.361984 0.932184i \(-0.617901\pi\)
−0.361984 + 0.932184i \(0.617901\pi\)
\(678\) 0 0
\(679\) 5.23326e12 0.944839
\(680\) 0 0
\(681\) −5.99583e11 −0.106829
\(682\) 0 0
\(683\) −5.76613e12 −1.01389 −0.506946 0.861978i \(-0.669226\pi\)
−0.506946 + 0.861978i \(0.669226\pi\)
\(684\) 0 0
\(685\) −1.28437e13 −2.22887
\(686\) 0 0
\(687\) 3.41629e12 0.585125
\(688\) 0 0
\(689\) −1.17172e13 −1.98079
\(690\) 0 0
\(691\) −4.26637e12 −0.711881 −0.355941 0.934509i \(-0.615839\pi\)
−0.355941 + 0.934509i \(0.615839\pi\)
\(692\) 0 0
\(693\) −1.99083e12 −0.327895
\(694\) 0 0
\(695\) −3.88920e12 −0.632307
\(696\) 0 0
\(697\) −3.74202e12 −0.600564
\(698\) 0 0
\(699\) −5.90322e12 −0.935281
\(700\) 0 0
\(701\) 5.62492e12 0.879802 0.439901 0.898046i \(-0.355013\pi\)
0.439901 + 0.898046i \(0.355013\pi\)
\(702\) 0 0
\(703\) −2.46811e11 −0.0381123
\(704\) 0 0
\(705\) −6.24531e12 −0.952144
\(706\) 0 0
\(707\) −4.94867e12 −0.744906
\(708\) 0 0
\(709\) −2.74160e12 −0.407470 −0.203735 0.979026i \(-0.565308\pi\)
−0.203735 + 0.979026i \(0.565308\pi\)
\(710\) 0 0
\(711\) 3.07567e12 0.451363
\(712\) 0 0
\(713\) 1.20545e13 1.74681
\(714\) 0 0
\(715\) 2.23239e13 3.19442
\(716\) 0 0
\(717\) 1.11930e12 0.158165
\(718\) 0 0
\(719\) 3.15437e12 0.440182 0.220091 0.975479i \(-0.429365\pi\)
0.220091 + 0.975479i \(0.429365\pi\)
\(720\) 0 0
\(721\) 7.74184e11 0.106693
\(722\) 0 0
\(723\) −3.93712e12 −0.535866
\(724\) 0 0
\(725\) 1.53695e12 0.206603
\(726\) 0 0
\(727\) −8.74710e12 −1.16134 −0.580670 0.814139i \(-0.697209\pi\)
−0.580670 + 0.814139i \(0.697209\pi\)
\(728\) 0 0
\(729\) 2.82430e11 0.0370370
\(730\) 0 0
\(731\) −3.74128e10 −0.00484610
\(732\) 0 0
\(733\) −9.44871e10 −0.0120894 −0.00604470 0.999982i \(-0.501924\pi\)
−0.00604470 + 0.999982i \(0.501924\pi\)
\(734\) 0 0
\(735\) −3.79733e12 −0.479938
\(736\) 0 0
\(737\) 5.22598e12 0.652475
\(738\) 0 0
\(739\) 7.12664e12 0.878992 0.439496 0.898244i \(-0.355157\pi\)
0.439496 + 0.898244i \(0.355157\pi\)
\(740\) 0 0
\(741\) 3.19017e11 0.0388716
\(742\) 0 0
\(743\) 7.11604e11 0.0856621 0.0428310 0.999082i \(-0.486362\pi\)
0.0428310 + 0.999082i \(0.486362\pi\)
\(744\) 0 0
\(745\) 3.58805e12 0.426732
\(746\) 0 0
\(747\) −2.75044e12 −0.323191
\(748\) 0 0
\(749\) 8.62407e12 1.00125
\(750\) 0 0
\(751\) 1.09650e13 1.25785 0.628927 0.777464i \(-0.283494\pi\)
0.628927 + 0.777464i \(0.283494\pi\)
\(752\) 0 0
\(753\) −8.06193e12 −0.913822
\(754\) 0 0
\(755\) −3.69414e11 −0.0413763
\(756\) 0 0
\(757\) −1.22871e13 −1.35993 −0.679966 0.733243i \(-0.738006\pi\)
−0.679966 + 0.733243i \(0.738006\pi\)
\(758\) 0 0
\(759\) 1.35254e13 1.47932
\(760\) 0 0
\(761\) 1.02506e13 1.10794 0.553971 0.832536i \(-0.313112\pi\)
0.553971 + 0.832536i \(0.313112\pi\)
\(762\) 0 0
\(763\) −1.01724e12 −0.108659
\(764\) 0 0
\(765\) 4.09881e12 0.432694
\(766\) 0 0
\(767\) 1.45098e13 1.51385
\(768\) 0 0
\(769\) −9.06562e12 −0.934822 −0.467411 0.884040i \(-0.654813\pi\)
−0.467411 + 0.884040i \(0.654813\pi\)
\(770\) 0 0
\(771\) 3.90427e12 0.397919
\(772\) 0 0
\(773\) −1.84509e13 −1.85870 −0.929350 0.369201i \(-0.879632\pi\)
−0.929350 + 0.369201i \(0.879632\pi\)
\(774\) 0 0
\(775\) 7.76672e12 0.773357
\(776\) 0 0
\(777\) 2.99272e12 0.294559
\(778\) 0 0
\(779\) −2.77754e11 −0.0270235
\(780\) 0 0
\(781\) 2.98736e12 0.287315
\(782\) 0 0
\(783\) 6.03471e11 0.0573758
\(784\) 0 0
\(785\) 2.50431e13 2.35383
\(786\) 0 0
\(787\) 3.03454e12 0.281972 0.140986 0.990012i \(-0.454973\pi\)
0.140986 + 0.990012i \(0.454973\pi\)
\(788\) 0 0
\(789\) 2.52574e12 0.232029
\(790\) 0 0
\(791\) 9.25446e12 0.840537
\(792\) 0 0
\(793\) −2.13857e13 −1.92041
\(794\) 0 0
\(795\) −1.11743e13 −0.992134
\(796\) 0 0
\(797\) 1.71637e13 1.50677 0.753386 0.657579i \(-0.228419\pi\)
0.753386 + 0.657579i \(0.228419\pi\)
\(798\) 0 0
\(799\) −1.45671e13 −1.26448
\(800\) 0 0
\(801\) 5.46136e12 0.468765
\(802\) 0 0
\(803\) 2.25272e13 1.91200
\(804\) 0 0
\(805\) −1.45824e13 −1.22390
\(806\) 0 0
\(807\) 3.88329e12 0.322306
\(808\) 0 0
\(809\) −3.84376e12 −0.315491 −0.157746 0.987480i \(-0.550423\pi\)
−0.157746 + 0.987480i \(0.550423\pi\)
\(810\) 0 0
\(811\) 5.81504e12 0.472019 0.236009 0.971751i \(-0.424160\pi\)
0.236009 + 0.971751i \(0.424160\pi\)
\(812\) 0 0
\(813\) −5.02229e12 −0.403176
\(814\) 0 0
\(815\) 5.48597e12 0.435556
\(816\) 0 0
\(817\) −2.77699e9 −0.000218060 0
\(818\) 0 0
\(819\) −3.86827e12 −0.300427
\(820\) 0 0
\(821\) 1.98531e13 1.52505 0.762526 0.646957i \(-0.223959\pi\)
0.762526 + 0.646957i \(0.223959\pi\)
\(822\) 0 0
\(823\) −1.43301e13 −1.08881 −0.544403 0.838824i \(-0.683244\pi\)
−0.544403 + 0.838824i \(0.683244\pi\)
\(824\) 0 0
\(825\) 8.71444e12 0.654933
\(826\) 0 0
\(827\) 8.25152e12 0.613422 0.306711 0.951803i \(-0.400772\pi\)
0.306711 + 0.951803i \(0.400772\pi\)
\(828\) 0 0
\(829\) −3.31194e12 −0.243549 −0.121775 0.992558i \(-0.538859\pi\)
−0.121775 + 0.992558i \(0.538859\pi\)
\(830\) 0 0
\(831\) 6.08708e12 0.442797
\(832\) 0 0
\(833\) −8.85720e12 −0.637373
\(834\) 0 0
\(835\) 8.96284e12 0.638053
\(836\) 0 0
\(837\) 3.04955e12 0.214769
\(838\) 0 0
\(839\) −1.20859e13 −0.842072 −0.421036 0.907044i \(-0.638333\pi\)
−0.421036 + 0.907044i \(0.638333\pi\)
\(840\) 0 0
\(841\) −1.32177e13 −0.911117
\(842\) 0 0
\(843\) 5.89438e11 0.0401989
\(844\) 0 0
\(845\) 2.40929e13 1.62567
\(846\) 0 0
\(847\) −1.51179e13 −1.00929
\(848\) 0 0
\(849\) 8.70959e12 0.575325
\(850\) 0 0
\(851\) −2.03322e13 −1.32892
\(852\) 0 0
\(853\) −3.65968e12 −0.236686 −0.118343 0.992973i \(-0.537758\pi\)
−0.118343 + 0.992973i \(0.537758\pi\)
\(854\) 0 0
\(855\) 3.04237e11 0.0194699
\(856\) 0 0
\(857\) −1.91677e13 −1.21383 −0.606913 0.794768i \(-0.707592\pi\)
−0.606913 + 0.794768i \(0.707592\pi\)
\(858\) 0 0
\(859\) 2.06228e13 1.29234 0.646172 0.763192i \(-0.276369\pi\)
0.646172 + 0.763192i \(0.276369\pi\)
\(860\) 0 0
\(861\) 3.36793e12 0.208857
\(862\) 0 0
\(863\) −3.08452e12 −0.189295 −0.0946475 0.995511i \(-0.530172\pi\)
−0.0946475 + 0.995511i \(0.530172\pi\)
\(864\) 0 0
\(865\) −2.45076e13 −1.48843
\(866\) 0 0
\(867\) −4.52228e10 −0.00271814
\(868\) 0 0
\(869\) 3.72621e13 2.21655
\(870\) 0 0
\(871\) 1.01543e13 0.597817
\(872\) 0 0
\(873\) −8.99445e12 −0.524096
\(874\) 0 0
\(875\) 4.16238e12 0.240052
\(876\) 0 0
\(877\) 2.90264e13 1.65689 0.828446 0.560069i \(-0.189225\pi\)
0.828446 + 0.560069i \(0.189225\pi\)
\(878\) 0 0
\(879\) −1.87911e12 −0.106170
\(880\) 0 0
\(881\) −1.69337e13 −0.947021 −0.473511 0.880788i \(-0.657013\pi\)
−0.473511 + 0.880788i \(0.657013\pi\)
\(882\) 0 0
\(883\) −1.73642e13 −0.961242 −0.480621 0.876928i \(-0.659589\pi\)
−0.480621 + 0.876928i \(0.659589\pi\)
\(884\) 0 0
\(885\) 1.38375e13 0.758252
\(886\) 0 0
\(887\) 1.69498e13 0.919406 0.459703 0.888073i \(-0.347956\pi\)
0.459703 + 0.888073i \(0.347956\pi\)
\(888\) 0 0
\(889\) 3.48394e12 0.187073
\(890\) 0 0
\(891\) 3.42166e12 0.181881
\(892\) 0 0
\(893\) −1.08125e12 −0.0568977
\(894\) 0 0
\(895\) 2.33837e13 1.21818
\(896\) 0 0
\(897\) 2.62805e13 1.35540
\(898\) 0 0
\(899\) 6.51601e12 0.332708
\(900\) 0 0
\(901\) −2.60640e13 −1.31759
\(902\) 0 0
\(903\) 3.36726e10 0.00168532
\(904\) 0 0
\(905\) −2.65869e13 −1.31750
\(906\) 0 0
\(907\) −1.95992e13 −0.961622 −0.480811 0.876824i \(-0.659658\pi\)
−0.480811 + 0.876824i \(0.659658\pi\)
\(908\) 0 0
\(909\) 8.50533e12 0.413194
\(910\) 0 0
\(911\) 1.77979e13 0.856124 0.428062 0.903749i \(-0.359197\pi\)
0.428062 + 0.903749i \(0.359197\pi\)
\(912\) 0 0
\(913\) −3.33218e13 −1.58712
\(914\) 0 0
\(915\) −2.03949e13 −0.961893
\(916\) 0 0
\(917\) 2.27781e13 1.06379
\(918\) 0 0
\(919\) 6.61520e11 0.0305931 0.0152965 0.999883i \(-0.495131\pi\)
0.0152965 + 0.999883i \(0.495131\pi\)
\(920\) 0 0
\(921\) 1.49447e13 0.684415
\(922\) 0 0
\(923\) 5.80457e12 0.263246
\(924\) 0 0
\(925\) −1.31000e13 −0.588347
\(926\) 0 0
\(927\) −1.33060e12 −0.0591818
\(928\) 0 0
\(929\) 1.85586e12 0.0817475 0.0408737 0.999164i \(-0.486986\pi\)
0.0408737 + 0.999164i \(0.486986\pi\)
\(930\) 0 0
\(931\) −6.57432e11 −0.0286799
\(932\) 0 0
\(933\) 1.48652e13 0.642251
\(934\) 0 0
\(935\) 4.96575e13 2.12487
\(936\) 0 0
\(937\) −1.97151e13 −0.835548 −0.417774 0.908551i \(-0.637190\pi\)
−0.417774 + 0.908551i \(0.637190\pi\)
\(938\) 0 0
\(939\) 1.35564e13 0.569048
\(940\) 0 0
\(941\) −1.18755e13 −0.493739 −0.246869 0.969049i \(-0.579402\pi\)
−0.246869 + 0.969049i \(0.579402\pi\)
\(942\) 0 0
\(943\) −2.28813e13 −0.942275
\(944\) 0 0
\(945\) −3.68905e12 −0.150478
\(946\) 0 0
\(947\) −1.15724e13 −0.467572 −0.233786 0.972288i \(-0.575112\pi\)
−0.233786 + 0.972288i \(0.575112\pi\)
\(948\) 0 0
\(949\) 4.37713e13 1.75183
\(950\) 0 0
\(951\) 2.48998e13 0.987153
\(952\) 0 0
\(953\) 4.16575e13 1.63597 0.817985 0.575239i \(-0.195091\pi\)
0.817985 + 0.575239i \(0.195091\pi\)
\(954\) 0 0
\(955\) 4.54309e13 1.76740
\(956\) 0 0
\(957\) 7.31112e12 0.281760
\(958\) 0 0
\(959\) 2.69629e13 1.02940
\(960\) 0 0
\(961\) 6.48804e12 0.245391
\(962\) 0 0
\(963\) −1.48223e13 −0.555388
\(964\) 0 0
\(965\) −4.46324e13 −1.65683
\(966\) 0 0
\(967\) −2.31308e13 −0.850692 −0.425346 0.905031i \(-0.639848\pi\)
−0.425346 + 0.905031i \(0.639848\pi\)
\(968\) 0 0
\(969\) 7.09627e11 0.0258567
\(970\) 0 0
\(971\) −2.35529e13 −0.850272 −0.425136 0.905129i \(-0.639774\pi\)
−0.425136 + 0.905129i \(0.639774\pi\)
\(972\) 0 0
\(973\) 8.16460e12 0.292030
\(974\) 0 0
\(975\) 1.69325e13 0.600069
\(976\) 0 0
\(977\) 3.82311e13 1.34243 0.671215 0.741263i \(-0.265773\pi\)
0.671215 + 0.741263i \(0.265773\pi\)
\(978\) 0 0
\(979\) 6.61650e13 2.30201
\(980\) 0 0
\(981\) 1.74834e12 0.0602721
\(982\) 0 0
\(983\) 2.76306e13 0.943843 0.471922 0.881641i \(-0.343561\pi\)
0.471922 + 0.881641i \(0.343561\pi\)
\(984\) 0 0
\(985\) −2.07660e13 −0.702893
\(986\) 0 0
\(987\) 1.31108e13 0.439746
\(988\) 0 0
\(989\) −2.28767e11 −0.00760345
\(990\) 0 0
\(991\) −2.13754e13 −0.704017 −0.352008 0.935997i \(-0.614501\pi\)
−0.352008 + 0.935997i \(0.614501\pi\)
\(992\) 0 0
\(993\) −1.80226e12 −0.0588229
\(994\) 0 0
\(995\) 1.97419e13 0.638535
\(996\) 0 0
\(997\) −5.63793e12 −0.180714 −0.0903569 0.995909i \(-0.528801\pi\)
−0.0903569 + 0.995909i \(0.528801\pi\)
\(998\) 0 0
\(999\) −5.14362e12 −0.163390
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.a.h.1.4 yes 4
4.3 odd 2 384.10.a.d.1.4 yes 4
8.3 odd 2 384.10.a.e.1.1 yes 4
8.5 even 2 384.10.a.a.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.a.a.1.1 4 8.5 even 2
384.10.a.d.1.4 yes 4 4.3 odd 2
384.10.a.e.1.1 yes 4 8.3 odd 2
384.10.a.h.1.4 yes 4 1.1 even 1 trivial