Properties

Label 384.8.a.f.1.2
Level $384$
Weight $8$
Character 384.1
Self dual yes
Analytic conductor $119.956$
Analytic rank $1$
Dimension $2$
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] = [384,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(119.955849786\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{366}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 366 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(19.1311\) 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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} +241.049 q^{5} -1255.25 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} +241.049 q^{5} -1255.25 q^{7} +729.000 q^{9} +2415.27 q^{11} -13408.9 q^{13} -6508.32 q^{15} +25571.4 q^{17} +38780.3 q^{19} +33891.6 q^{21} +49641.1 q^{23} -20020.4 q^{25} -19683.0 q^{27} -231355. q^{29} +246856. q^{31} -65212.4 q^{33} -302576. q^{35} -297314. q^{37} +362041. q^{39} -616533. q^{41} +968744. q^{43} +175725. q^{45} -359217. q^{47} +752097. q^{49} -690426. q^{51} +1.39933e6 q^{53} +582199. q^{55} -1.04707e6 q^{57} -214279. q^{59} +1.75876e6 q^{61} -915074. q^{63} -3.23221e6 q^{65} -1.08452e6 q^{67} -1.34031e6 q^{69} -1.02995e6 q^{71} +822221. q^{73} +540550. q^{75} -3.03176e6 q^{77} -4.25755e6 q^{79} +531441. q^{81} +5.69750e6 q^{83} +6.16395e6 q^{85} +6.24659e6 q^{87} -6.57345e6 q^{89} +1.68315e7 q^{91} -6.66512e6 q^{93} +9.34796e6 q^{95} +7.55287e6 q^{97} +1.76073e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} + 176 q^{5} - 980 q^{7} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{3} + 176 q^{5} - 980 q^{7} + 1458 q^{9} - 3128 q^{11} - 8452 q^{13} - 4752 q^{15} + 5228 q^{17} + 87968 q^{19} + 26460 q^{21} + 104792 q^{23} - 93914 q^{25} - 39366 q^{27} - 98760 q^{29} - 1860 q^{31} + 84456 q^{33} - 320480 q^{35} - 455660 q^{37} + 228204 q^{39} - 205188 q^{41} + 803088 q^{43} + 128304 q^{45} + 522488 q^{47} + 4314 q^{49} - 141156 q^{51} + 2468376 q^{53} + 942784 q^{55} - 2375136 q^{57} + 989288 q^{59} + 2482292 q^{61} - 714420 q^{63} - 3554656 q^{65} + 434008 q^{67} - 2829384 q^{69} - 1554840 q^{71} + 279244 q^{73} + 2535678 q^{75} - 4557520 q^{77} - 1015396 q^{79} + 1062882 q^{81} + 2899560 q^{83} + 7487264 q^{85} + 2666520 q^{87} - 8115868 q^{89} + 18195880 q^{91} + 50220 q^{93} + 6148352 q^{95} - 2553676 q^{97} - 2280312 q^{99}+O(q^{100}) \) 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
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) 241.049 0.862403 0.431202 0.902256i \(-0.358090\pi\)
0.431202 + 0.902256i \(0.358090\pi\)
\(6\) 0 0
\(7\) −1255.25 −1.38320 −0.691601 0.722280i \(-0.743094\pi\)
−0.691601 + 0.722280i \(0.743094\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 2415.27 0.547132 0.273566 0.961853i \(-0.411797\pi\)
0.273566 + 0.961853i \(0.411797\pi\)
\(12\) 0 0
\(13\) −13408.9 −1.69275 −0.846375 0.532588i \(-0.821220\pi\)
−0.846375 + 0.532588i \(0.821220\pi\)
\(14\) 0 0
\(15\) −6508.32 −0.497909
\(16\) 0 0
\(17\) 25571.4 1.26236 0.631179 0.775637i \(-0.282571\pi\)
0.631179 + 0.775637i \(0.282571\pi\)
\(18\) 0 0
\(19\) 38780.3 1.29710 0.648551 0.761171i \(-0.275375\pi\)
0.648551 + 0.761171i \(0.275375\pi\)
\(20\) 0 0
\(21\) 33891.6 0.798592
\(22\) 0 0
\(23\) 49641.1 0.850735 0.425367 0.905021i \(-0.360145\pi\)
0.425367 + 0.905021i \(0.360145\pi\)
\(24\) 0 0
\(25\) −20020.4 −0.256261
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −231355. −1.76152 −0.880758 0.473567i \(-0.842966\pi\)
−0.880758 + 0.473567i \(0.842966\pi\)
\(30\) 0 0
\(31\) 246856. 1.48826 0.744129 0.668036i \(-0.232865\pi\)
0.744129 + 0.668036i \(0.232865\pi\)
\(32\) 0 0
\(33\) −65212.4 −0.315887
\(34\) 0 0
\(35\) −302576. −1.19288
\(36\) 0 0
\(37\) −297314. −0.964961 −0.482480 0.875907i \(-0.660264\pi\)
−0.482480 + 0.875907i \(0.660264\pi\)
\(38\) 0 0
\(39\) 362041. 0.977309
\(40\) 0 0
\(41\) −616533. −1.39705 −0.698526 0.715585i \(-0.746160\pi\)
−0.698526 + 0.715585i \(0.746160\pi\)
\(42\) 0 0
\(43\) 968744. 1.85810 0.929050 0.369953i \(-0.120626\pi\)
0.929050 + 0.369953i \(0.120626\pi\)
\(44\) 0 0
\(45\) 175725. 0.287468
\(46\) 0 0
\(47\) −359217. −0.504678 −0.252339 0.967639i \(-0.581200\pi\)
−0.252339 + 0.967639i \(0.581200\pi\)
\(48\) 0 0
\(49\) 752097. 0.913246
\(50\) 0 0
\(51\) −690426. −0.728823
\(52\) 0 0
\(53\) 1.39933e6 1.29108 0.645541 0.763725i \(-0.276632\pi\)
0.645541 + 0.763725i \(0.276632\pi\)
\(54\) 0 0
\(55\) 582199. 0.471848
\(56\) 0 0
\(57\) −1.04707e6 −0.748882
\(58\) 0 0
\(59\) −214279. −0.135831 −0.0679153 0.997691i \(-0.521635\pi\)
−0.0679153 + 0.997691i \(0.521635\pi\)
\(60\) 0 0
\(61\) 1.75876e6 0.992091 0.496046 0.868296i \(-0.334785\pi\)
0.496046 + 0.868296i \(0.334785\pi\)
\(62\) 0 0
\(63\) −915074. −0.461067
\(64\) 0 0
\(65\) −3.23221e6 −1.45983
\(66\) 0 0
\(67\) −1.08452e6 −0.440532 −0.220266 0.975440i \(-0.570693\pi\)
−0.220266 + 0.975440i \(0.570693\pi\)
\(68\) 0 0
\(69\) −1.34031e6 −0.491172
\(70\) 0 0
\(71\) −1.02995e6 −0.341517 −0.170758 0.985313i \(-0.554622\pi\)
−0.170758 + 0.985313i \(0.554622\pi\)
\(72\) 0 0
\(73\) 822221. 0.247376 0.123688 0.992321i \(-0.460528\pi\)
0.123688 + 0.992321i \(0.460528\pi\)
\(74\) 0 0
\(75\) 540550. 0.147952
\(76\) 0 0
\(77\) −3.03176e6 −0.756794
\(78\) 0 0
\(79\) −4.25755e6 −0.971550 −0.485775 0.874084i \(-0.661463\pi\)
−0.485775 + 0.874084i \(0.661463\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 5.69750e6 1.09373 0.546866 0.837220i \(-0.315821\pi\)
0.546866 + 0.837220i \(0.315821\pi\)
\(84\) 0 0
\(85\) 6.16395e6 1.08866
\(86\) 0 0
\(87\) 6.24659e6 1.01701
\(88\) 0 0
\(89\) −6.57345e6 −0.988390 −0.494195 0.869351i \(-0.664537\pi\)
−0.494195 + 0.869351i \(0.664537\pi\)
\(90\) 0 0
\(91\) 1.68315e7 2.34141
\(92\) 0 0
\(93\) −6.66512e6 −0.859246
\(94\) 0 0
\(95\) 9.34796e6 1.11862
\(96\) 0 0
\(97\) 7.55287e6 0.840254 0.420127 0.907465i \(-0.361985\pi\)
0.420127 + 0.907465i \(0.361985\pi\)
\(98\) 0 0
\(99\) 1.76073e6 0.182377
\(100\) 0 0
\(101\) 7.02082e6 0.678052 0.339026 0.940777i \(-0.389902\pi\)
0.339026 + 0.940777i \(0.389902\pi\)
\(102\) 0 0
\(103\) −1.83883e7 −1.65810 −0.829051 0.559172i \(-0.811119\pi\)
−0.829051 + 0.559172i \(0.811119\pi\)
\(104\) 0 0
\(105\) 8.16954e6 0.688708
\(106\) 0 0
\(107\) −1.41357e7 −1.11551 −0.557755 0.830005i \(-0.688337\pi\)
−0.557755 + 0.830005i \(0.688337\pi\)
\(108\) 0 0
\(109\) −4.63280e6 −0.342650 −0.171325 0.985215i \(-0.554805\pi\)
−0.171325 + 0.985215i \(0.554805\pi\)
\(110\) 0 0
\(111\) 8.02748e6 0.557120
\(112\) 0 0
\(113\) 1.05522e7 0.687970 0.343985 0.938975i \(-0.388223\pi\)
0.343985 + 0.938975i \(0.388223\pi\)
\(114\) 0 0
\(115\) 1.19659e7 0.733676
\(116\) 0 0
\(117\) −9.77512e6 −0.564250
\(118\) 0 0
\(119\) −3.20983e7 −1.74609
\(120\) 0 0
\(121\) −1.36536e7 −0.700647
\(122\) 0 0
\(123\) 1.66464e7 0.806588
\(124\) 0 0
\(125\) −2.36578e7 −1.08340
\(126\) 0 0
\(127\) 9.59436e6 0.415626 0.207813 0.978169i \(-0.433365\pi\)
0.207813 + 0.978169i \(0.433365\pi\)
\(128\) 0 0
\(129\) −2.61561e7 −1.07277
\(130\) 0 0
\(131\) −2.51622e7 −0.977910 −0.488955 0.872309i \(-0.662622\pi\)
−0.488955 + 0.872309i \(0.662622\pi\)
\(132\) 0 0
\(133\) −4.86788e7 −1.79415
\(134\) 0 0
\(135\) −4.74457e6 −0.165970
\(136\) 0 0
\(137\) −4.08875e7 −1.35853 −0.679263 0.733895i \(-0.737701\pi\)
−0.679263 + 0.733895i \(0.737701\pi\)
\(138\) 0 0
\(139\) −3.93884e7 −1.24399 −0.621994 0.783022i \(-0.713677\pi\)
−0.621994 + 0.783022i \(0.713677\pi\)
\(140\) 0 0
\(141\) 9.69885e6 0.291376
\(142\) 0 0
\(143\) −3.23863e7 −0.926157
\(144\) 0 0
\(145\) −5.57680e7 −1.51914
\(146\) 0 0
\(147\) −2.03066e7 −0.527263
\(148\) 0 0
\(149\) 3.70043e6 0.0916431 0.0458216 0.998950i \(-0.485409\pi\)
0.0458216 + 0.998950i \(0.485409\pi\)
\(150\) 0 0
\(151\) −1.24292e7 −0.293781 −0.146891 0.989153i \(-0.546927\pi\)
−0.146891 + 0.989153i \(0.546927\pi\)
\(152\) 0 0
\(153\) 1.86415e7 0.420786
\(154\) 0 0
\(155\) 5.95045e7 1.28348
\(156\) 0 0
\(157\) −2.82244e7 −0.582072 −0.291036 0.956712i \(-0.594000\pi\)
−0.291036 + 0.956712i \(0.594000\pi\)
\(158\) 0 0
\(159\) −3.77819e7 −0.745407
\(160\) 0 0
\(161\) −6.23118e7 −1.17674
\(162\) 0 0
\(163\) 6.47825e7 1.17166 0.585830 0.810434i \(-0.300769\pi\)
0.585830 + 0.810434i \(0.300769\pi\)
\(164\) 0 0
\(165\) −1.57194e7 −0.272422
\(166\) 0 0
\(167\) −5.65214e7 −0.939086 −0.469543 0.882910i \(-0.655581\pi\)
−0.469543 + 0.882910i \(0.655581\pi\)
\(168\) 0 0
\(169\) 1.17051e8 1.86540
\(170\) 0 0
\(171\) 2.82709e7 0.432367
\(172\) 0 0
\(173\) −1.07852e7 −0.158367 −0.0791836 0.996860i \(-0.525231\pi\)
−0.0791836 + 0.996860i \(0.525231\pi\)
\(174\) 0 0
\(175\) 2.51305e7 0.354460
\(176\) 0 0
\(177\) 5.78553e6 0.0784218
\(178\) 0 0
\(179\) 1.53198e6 0.0199649 0.00998247 0.999950i \(-0.496822\pi\)
0.00998247 + 0.999950i \(0.496822\pi\)
\(180\) 0 0
\(181\) −1.26121e8 −1.58093 −0.790467 0.612505i \(-0.790162\pi\)
−0.790467 + 0.612505i \(0.790162\pi\)
\(182\) 0 0
\(183\) −4.74865e7 −0.572784
\(184\) 0 0
\(185\) −7.16673e7 −0.832185
\(186\) 0 0
\(187\) 6.17618e7 0.690676
\(188\) 0 0
\(189\) 2.47070e7 0.266197
\(190\) 0 0
\(191\) −1.41139e8 −1.46565 −0.732826 0.680417i \(-0.761799\pi\)
−0.732826 + 0.680417i \(0.761799\pi\)
\(192\) 0 0
\(193\) −1.15274e8 −1.15420 −0.577100 0.816674i \(-0.695816\pi\)
−0.577100 + 0.816674i \(0.695816\pi\)
\(194\) 0 0
\(195\) 8.72697e7 0.842835
\(196\) 0 0
\(197\) 3.22603e7 0.300633 0.150316 0.988638i \(-0.451971\pi\)
0.150316 + 0.988638i \(0.451971\pi\)
\(198\) 0 0
\(199\) −5.49688e7 −0.494459 −0.247230 0.968957i \(-0.579520\pi\)
−0.247230 + 0.968957i \(0.579520\pi\)
\(200\) 0 0
\(201\) 2.92822e7 0.254341
\(202\) 0 0
\(203\) 2.90408e8 2.43653
\(204\) 0 0
\(205\) −1.48615e8 −1.20482
\(206\) 0 0
\(207\) 3.61884e7 0.283578
\(208\) 0 0
\(209\) 9.36651e7 0.709686
\(210\) 0 0
\(211\) −5.55364e7 −0.406995 −0.203498 0.979075i \(-0.565231\pi\)
−0.203498 + 0.979075i \(0.565231\pi\)
\(212\) 0 0
\(213\) 2.78087e7 0.197175
\(214\) 0 0
\(215\) 2.33515e8 1.60243
\(216\) 0 0
\(217\) −3.09865e8 −2.05856
\(218\) 0 0
\(219\) −2.22000e7 −0.142823
\(220\) 0 0
\(221\) −3.42885e8 −2.13686
\(222\) 0 0
\(223\) −1.26355e8 −0.763004 −0.381502 0.924368i \(-0.624593\pi\)
−0.381502 + 0.924368i \(0.624593\pi\)
\(224\) 0 0
\(225\) −1.45949e7 −0.0854203
\(226\) 0 0
\(227\) 1.81110e8 1.02767 0.513834 0.857890i \(-0.328225\pi\)
0.513834 + 0.857890i \(0.328225\pi\)
\(228\) 0 0
\(229\) 1.46997e8 0.808880 0.404440 0.914564i \(-0.367466\pi\)
0.404440 + 0.914564i \(0.367466\pi\)
\(230\) 0 0
\(231\) 8.18576e7 0.436935
\(232\) 0 0
\(233\) 1.68176e8 0.870998 0.435499 0.900189i \(-0.356572\pi\)
0.435499 + 0.900189i \(0.356572\pi\)
\(234\) 0 0
\(235\) −8.65888e7 −0.435236
\(236\) 0 0
\(237\) 1.14954e8 0.560925
\(238\) 0 0
\(239\) −2.06268e8 −0.977326 −0.488663 0.872473i \(-0.662515\pi\)
−0.488663 + 0.872473i \(0.662515\pi\)
\(240\) 0 0
\(241\) −3.39136e8 −1.56068 −0.780341 0.625354i \(-0.784954\pi\)
−0.780341 + 0.625354i \(0.784954\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) 1.81292e8 0.787586
\(246\) 0 0
\(247\) −5.20003e8 −2.19567
\(248\) 0 0
\(249\) −1.53833e8 −0.631467
\(250\) 0 0
\(251\) 3.62781e7 0.144806 0.0724030 0.997375i \(-0.476933\pi\)
0.0724030 + 0.997375i \(0.476933\pi\)
\(252\) 0 0
\(253\) 1.19897e8 0.465464
\(254\) 0 0
\(255\) −1.66427e8 −0.628539
\(256\) 0 0
\(257\) −5.14951e8 −1.89234 −0.946172 0.323665i \(-0.895085\pi\)
−0.946172 + 0.323665i \(0.895085\pi\)
\(258\) 0 0
\(259\) 3.73202e8 1.33474
\(260\) 0 0
\(261\) −1.68658e8 −0.587172
\(262\) 0 0
\(263\) −9.03571e7 −0.306279 −0.153139 0.988205i \(-0.548938\pi\)
−0.153139 + 0.988205i \(0.548938\pi\)
\(264\) 0 0
\(265\) 3.37307e8 1.11343
\(266\) 0 0
\(267\) 1.77483e8 0.570647
\(268\) 0 0
\(269\) −3.05304e8 −0.956311 −0.478156 0.878275i \(-0.658694\pi\)
−0.478156 + 0.878275i \(0.658694\pi\)
\(270\) 0 0
\(271\) 1.67428e8 0.511019 0.255509 0.966807i \(-0.417757\pi\)
0.255509 + 0.966807i \(0.417757\pi\)
\(272\) 0 0
\(273\) −4.54451e8 −1.35182
\(274\) 0 0
\(275\) −4.83547e7 −0.140208
\(276\) 0 0
\(277\) 2.46391e8 0.696538 0.348269 0.937395i \(-0.386770\pi\)
0.348269 + 0.937395i \(0.386770\pi\)
\(278\) 0 0
\(279\) 1.79958e8 0.496086
\(280\) 0 0
\(281\) −3.35835e8 −0.902929 −0.451465 0.892289i \(-0.649098\pi\)
−0.451465 + 0.892289i \(0.649098\pi\)
\(282\) 0 0
\(283\) −3.73786e8 −0.980325 −0.490162 0.871631i \(-0.663063\pi\)
−0.490162 + 0.871631i \(0.663063\pi\)
\(284\) 0 0
\(285\) −2.52395e8 −0.645838
\(286\) 0 0
\(287\) 7.73899e8 1.93240
\(288\) 0 0
\(289\) 2.43555e8 0.593547
\(290\) 0 0
\(291\) −2.03927e8 −0.485121
\(292\) 0 0
\(293\) −4.15803e8 −0.965720 −0.482860 0.875697i \(-0.660402\pi\)
−0.482860 + 0.875697i \(0.660402\pi\)
\(294\) 0 0
\(295\) −5.16517e7 −0.117141
\(296\) 0 0
\(297\) −4.75398e7 −0.105296
\(298\) 0 0
\(299\) −6.65635e8 −1.44008
\(300\) 0 0
\(301\) −1.21601e9 −2.57013
\(302\) 0 0
\(303\) −1.89562e8 −0.391474
\(304\) 0 0
\(305\) 4.23947e8 0.855583
\(306\) 0 0
\(307\) 2.19257e8 0.432484 0.216242 0.976340i \(-0.430620\pi\)
0.216242 + 0.976340i \(0.430620\pi\)
\(308\) 0 0
\(309\) 4.96484e8 0.957306
\(310\) 0 0
\(311\) 2.13624e8 0.402707 0.201354 0.979519i \(-0.435466\pi\)
0.201354 + 0.979519i \(0.435466\pi\)
\(312\) 0 0
\(313\) 8.19563e8 1.51070 0.755348 0.655324i \(-0.227468\pi\)
0.755348 + 0.655324i \(0.227468\pi\)
\(314\) 0 0
\(315\) −2.20578e8 −0.397626
\(316\) 0 0
\(317\) 4.31470e8 0.760753 0.380376 0.924832i \(-0.375794\pi\)
0.380376 + 0.924832i \(0.375794\pi\)
\(318\) 0 0
\(319\) −5.58786e8 −0.963781
\(320\) 0 0
\(321\) 3.81664e8 0.644040
\(322\) 0 0
\(323\) 9.91666e8 1.63741
\(324\) 0 0
\(325\) 2.68452e8 0.433785
\(326\) 0 0
\(327\) 1.25086e8 0.197829
\(328\) 0 0
\(329\) 4.50905e8 0.698071
\(330\) 0 0
\(331\) 8.22158e7 0.124611 0.0623056 0.998057i \(-0.480155\pi\)
0.0623056 + 0.998057i \(0.480155\pi\)
\(332\) 0 0
\(333\) −2.16742e8 −0.321654
\(334\) 0 0
\(335\) −2.61424e8 −0.379916
\(336\) 0 0
\(337\) 8.93277e8 1.27140 0.635699 0.771937i \(-0.280712\pi\)
0.635699 + 0.771937i \(0.280712\pi\)
\(338\) 0 0
\(339\) −2.84910e8 −0.397199
\(340\) 0 0
\(341\) 5.96226e8 0.814274
\(342\) 0 0
\(343\) 8.96820e7 0.119999
\(344\) 0 0
\(345\) −3.23080e8 −0.423588
\(346\) 0 0
\(347\) 7.08416e8 0.910197 0.455099 0.890441i \(-0.349604\pi\)
0.455099 + 0.890441i \(0.349604\pi\)
\(348\) 0 0
\(349\) −1.85138e8 −0.233134 −0.116567 0.993183i \(-0.537189\pi\)
−0.116567 + 0.993183i \(0.537189\pi\)
\(350\) 0 0
\(351\) 2.63928e8 0.325770
\(352\) 0 0
\(353\) −1.62041e9 −1.96071 −0.980355 0.197240i \(-0.936802\pi\)
−0.980355 + 0.197240i \(0.936802\pi\)
\(354\) 0 0
\(355\) −2.48269e8 −0.294525
\(356\) 0 0
\(357\) 8.66654e8 1.00811
\(358\) 0 0
\(359\) −2.30456e8 −0.262880 −0.131440 0.991324i \(-0.541960\pi\)
−0.131440 + 0.991324i \(0.541960\pi\)
\(360\) 0 0
\(361\) 6.10043e8 0.682472
\(362\) 0 0
\(363\) 3.68648e8 0.404519
\(364\) 0 0
\(365\) 1.98195e8 0.213338
\(366\) 0 0
\(367\) −1.30643e9 −1.37960 −0.689801 0.723999i \(-0.742302\pi\)
−0.689801 + 0.723999i \(0.742302\pi\)
\(368\) 0 0
\(369\) −4.49452e8 −0.465684
\(370\) 0 0
\(371\) −1.75650e9 −1.78583
\(372\) 0 0
\(373\) 6.36011e8 0.634576 0.317288 0.948329i \(-0.397228\pi\)
0.317288 + 0.948329i \(0.397228\pi\)
\(374\) 0 0
\(375\) 6.38762e8 0.625503
\(376\) 0 0
\(377\) 3.10223e9 2.98180
\(378\) 0 0
\(379\) −7.80291e8 −0.736240 −0.368120 0.929778i \(-0.619998\pi\)
−0.368120 + 0.929778i \(0.619998\pi\)
\(380\) 0 0
\(381\) −2.59048e8 −0.239962
\(382\) 0 0
\(383\) −3.06788e6 −0.00279024 −0.00139512 0.999999i \(-0.500444\pi\)
−0.00139512 + 0.999999i \(0.500444\pi\)
\(384\) 0 0
\(385\) −7.30803e8 −0.652661
\(386\) 0 0
\(387\) 7.06214e8 0.619367
\(388\) 0 0
\(389\) 1.50816e9 1.29905 0.649524 0.760341i \(-0.274968\pi\)
0.649524 + 0.760341i \(0.274968\pi\)
\(390\) 0 0
\(391\) 1.26939e9 1.07393
\(392\) 0 0
\(393\) 6.79379e8 0.564597
\(394\) 0 0
\(395\) −1.02628e9 −0.837868
\(396\) 0 0
\(397\) 2.76390e8 0.221694 0.110847 0.993837i \(-0.464644\pi\)
0.110847 + 0.993837i \(0.464644\pi\)
\(398\) 0 0
\(399\) 1.31433e9 1.03585
\(400\) 0 0
\(401\) −1.86311e9 −1.44289 −0.721444 0.692473i \(-0.756521\pi\)
−0.721444 + 0.692473i \(0.756521\pi\)
\(402\) 0 0
\(403\) −3.31008e9 −2.51925
\(404\) 0 0
\(405\) 1.28103e8 0.0958226
\(406\) 0 0
\(407\) −7.18095e8 −0.527961
\(408\) 0 0
\(409\) 1.11458e9 0.805524 0.402762 0.915305i \(-0.368050\pi\)
0.402762 + 0.915305i \(0.368050\pi\)
\(410\) 0 0
\(411\) 1.10396e9 0.784346
\(412\) 0 0
\(413\) 2.68973e8 0.187881
\(414\) 0 0
\(415\) 1.37338e9 0.943239
\(416\) 0 0
\(417\) 1.06349e9 0.718217
\(418\) 0 0
\(419\) −1.75638e9 −1.16646 −0.583228 0.812308i \(-0.698211\pi\)
−0.583228 + 0.812308i \(0.698211\pi\)
\(420\) 0 0
\(421\) −8.24327e8 −0.538409 −0.269204 0.963083i \(-0.586761\pi\)
−0.269204 + 0.963083i \(0.586761\pi\)
\(422\) 0 0
\(423\) −2.61869e8 −0.168226
\(424\) 0 0
\(425\) −5.11948e8 −0.323493
\(426\) 0 0
\(427\) −2.20767e9 −1.37226
\(428\) 0 0
\(429\) 8.74429e8 0.534717
\(430\) 0 0
\(431\) 2.21141e9 1.33045 0.665225 0.746643i \(-0.268335\pi\)
0.665225 + 0.746643i \(0.268335\pi\)
\(432\) 0 0
\(433\) −8.91107e8 −0.527499 −0.263750 0.964591i \(-0.584959\pi\)
−0.263750 + 0.964591i \(0.584959\pi\)
\(434\) 0 0
\(435\) 1.50573e9 0.877074
\(436\) 0 0
\(437\) 1.92510e9 1.10349
\(438\) 0 0
\(439\) −1.22630e8 −0.0691785 −0.0345893 0.999402i \(-0.511012\pi\)
−0.0345893 + 0.999402i \(0.511012\pi\)
\(440\) 0 0
\(441\) 5.48279e8 0.304415
\(442\) 0 0
\(443\) 3.32516e9 1.81719 0.908594 0.417681i \(-0.137157\pi\)
0.908594 + 0.417681i \(0.137157\pi\)
\(444\) 0 0
\(445\) −1.58452e9 −0.852390
\(446\) 0 0
\(447\) −9.99115e7 −0.0529102
\(448\) 0 0
\(449\) −9.94013e8 −0.518239 −0.259119 0.965845i \(-0.583432\pi\)
−0.259119 + 0.965845i \(0.583432\pi\)
\(450\) 0 0
\(451\) −1.48910e9 −0.764372
\(452\) 0 0
\(453\) 3.35589e8 0.169615
\(454\) 0 0
\(455\) 4.05722e9 2.01924
\(456\) 0 0
\(457\) 2.67555e9 1.31131 0.655657 0.755059i \(-0.272392\pi\)
0.655657 + 0.755059i \(0.272392\pi\)
\(458\) 0 0
\(459\) −5.03321e8 −0.242941
\(460\) 0 0
\(461\) 2.02753e9 0.963861 0.481931 0.876209i \(-0.339936\pi\)
0.481931 + 0.876209i \(0.339936\pi\)
\(462\) 0 0
\(463\) 3.02693e9 1.41732 0.708661 0.705549i \(-0.249300\pi\)
0.708661 + 0.705549i \(0.249300\pi\)
\(464\) 0 0
\(465\) −1.60662e9 −0.741017
\(466\) 0 0
\(467\) 3.30446e9 1.50138 0.750690 0.660654i \(-0.229721\pi\)
0.750690 + 0.660654i \(0.229721\pi\)
\(468\) 0 0
\(469\) 1.36134e9 0.609345
\(470\) 0 0
\(471\) 7.62060e8 0.336059
\(472\) 0 0
\(473\) 2.33978e9 1.01663
\(474\) 0 0
\(475\) −7.76397e8 −0.332396
\(476\) 0 0
\(477\) 1.02011e9 0.430361
\(478\) 0 0
\(479\) −6.88113e8 −0.286079 −0.143039 0.989717i \(-0.545688\pi\)
−0.143039 + 0.989717i \(0.545688\pi\)
\(480\) 0 0
\(481\) 3.98667e9 1.63344
\(482\) 0 0
\(483\) 1.68242e9 0.679390
\(484\) 0 0
\(485\) 1.82061e9 0.724638
\(486\) 0 0
\(487\) −1.85856e9 −0.729163 −0.364581 0.931172i \(-0.618788\pi\)
−0.364581 + 0.931172i \(0.618788\pi\)
\(488\) 0 0
\(489\) −1.74913e9 −0.676458
\(490\) 0 0
\(491\) −3.10855e9 −1.18515 −0.592574 0.805516i \(-0.701888\pi\)
−0.592574 + 0.805516i \(0.701888\pi\)
\(492\) 0 0
\(493\) −5.91607e9 −2.22366
\(494\) 0 0
\(495\) 4.24423e8 0.157283
\(496\) 0 0
\(497\) 1.29284e9 0.472387
\(498\) 0 0
\(499\) −2.95742e6 −0.00106552 −0.000532759 1.00000i \(-0.500170\pi\)
−0.000532759 1.00000i \(0.500170\pi\)
\(500\) 0 0
\(501\) 1.52608e9 0.542182
\(502\) 0 0
\(503\) 7.89330e8 0.276548 0.138274 0.990394i \(-0.455845\pi\)
0.138274 + 0.990394i \(0.455845\pi\)
\(504\) 0 0
\(505\) 1.69236e9 0.584755
\(506\) 0 0
\(507\) −3.16038e9 −1.07699
\(508\) 0 0
\(509\) 4.84341e8 0.162794 0.0813971 0.996682i \(-0.474062\pi\)
0.0813971 + 0.996682i \(0.474062\pi\)
\(510\) 0 0
\(511\) −1.03209e9 −0.342171
\(512\) 0 0
\(513\) −7.63313e8 −0.249627
\(514\) 0 0
\(515\) −4.43248e9 −1.42995
\(516\) 0 0
\(517\) −8.67607e8 −0.276125
\(518\) 0 0
\(519\) 2.91199e8 0.0914334
\(520\) 0 0
\(521\) −4.48726e9 −1.39011 −0.695056 0.718956i \(-0.744620\pi\)
−0.695056 + 0.718956i \(0.744620\pi\)
\(522\) 0 0
\(523\) 2.86000e9 0.874200 0.437100 0.899413i \(-0.356006\pi\)
0.437100 + 0.899413i \(0.356006\pi\)
\(524\) 0 0
\(525\) −6.78523e8 −0.204648
\(526\) 0 0
\(527\) 6.31245e9 1.87871
\(528\) 0 0
\(529\) −9.40585e8 −0.276251
\(530\) 0 0
\(531\) −1.56209e8 −0.0452769
\(532\) 0 0
\(533\) 8.26705e9 2.36486
\(534\) 0 0
\(535\) −3.40739e9 −0.962020
\(536\) 0 0
\(537\) −4.13635e7 −0.0115268
\(538\) 0 0
\(539\) 1.81652e9 0.499666
\(540\) 0 0
\(541\) −2.29923e9 −0.624297 −0.312148 0.950033i \(-0.601049\pi\)
−0.312148 + 0.950033i \(0.601049\pi\)
\(542\) 0 0
\(543\) 3.40528e9 0.912752
\(544\) 0 0
\(545\) −1.11673e9 −0.295503
\(546\) 0 0
\(547\) −8.06117e8 −0.210592 −0.105296 0.994441i \(-0.533579\pi\)
−0.105296 + 0.994441i \(0.533579\pi\)
\(548\) 0 0
\(549\) 1.28213e9 0.330697
\(550\) 0 0
\(551\) −8.97203e9 −2.28486
\(552\) 0 0
\(553\) 5.34427e9 1.34385
\(554\) 0 0
\(555\) 1.93502e9 0.480462
\(556\) 0 0
\(557\) −5.67124e9 −1.39054 −0.695272 0.718747i \(-0.744716\pi\)
−0.695272 + 0.718747i \(0.744716\pi\)
\(558\) 0 0
\(559\) −1.29898e10 −3.14530
\(560\) 0 0
\(561\) −1.66757e9 −0.398762
\(562\) 0 0
\(563\) 2.83103e9 0.668598 0.334299 0.942467i \(-0.391500\pi\)
0.334299 + 0.942467i \(0.391500\pi\)
\(564\) 0 0
\(565\) 2.54360e9 0.593307
\(566\) 0 0
\(567\) −6.67089e8 −0.153689
\(568\) 0 0
\(569\) 1.35162e8 0.0307583 0.0153792 0.999882i \(-0.495104\pi\)
0.0153792 + 0.999882i \(0.495104\pi\)
\(570\) 0 0
\(571\) −6.87175e9 −1.54469 −0.772344 0.635204i \(-0.780916\pi\)
−0.772344 + 0.635204i \(0.780916\pi\)
\(572\) 0 0
\(573\) 3.81076e9 0.846194
\(574\) 0 0
\(575\) −9.93834e8 −0.218010
\(576\) 0 0
\(577\) 2.78800e9 0.604195 0.302098 0.953277i \(-0.402313\pi\)
0.302098 + 0.953277i \(0.402313\pi\)
\(578\) 0 0
\(579\) 3.11240e9 0.666377
\(580\) 0 0
\(581\) −7.15176e9 −1.51285
\(582\) 0 0
\(583\) 3.37976e9 0.706392
\(584\) 0 0
\(585\) −2.35628e9 −0.486611
\(586\) 0 0
\(587\) 3.87741e9 0.791240 0.395620 0.918414i \(-0.370530\pi\)
0.395620 + 0.918414i \(0.370530\pi\)
\(588\) 0 0
\(589\) 9.57317e9 1.93042
\(590\) 0 0
\(591\) −8.71027e8 −0.173570
\(592\) 0 0
\(593\) 3.58123e9 0.705245 0.352623 0.935766i \(-0.385290\pi\)
0.352623 + 0.935766i \(0.385290\pi\)
\(594\) 0 0
\(595\) −7.73727e9 −1.50584
\(596\) 0 0
\(597\) 1.48416e9 0.285476
\(598\) 0 0
\(599\) 2.81275e9 0.534734 0.267367 0.963595i \(-0.413846\pi\)
0.267367 + 0.963595i \(0.413846\pi\)
\(600\) 0 0
\(601\) 7.21845e9 1.35639 0.678193 0.734884i \(-0.262764\pi\)
0.678193 + 0.734884i \(0.262764\pi\)
\(602\) 0 0
\(603\) −7.90619e8 −0.146844
\(604\) 0 0
\(605\) −3.29119e9 −0.604240
\(606\) 0 0
\(607\) −9.43874e9 −1.71298 −0.856492 0.516160i \(-0.827361\pi\)
−0.856492 + 0.516160i \(0.827361\pi\)
\(608\) 0 0
\(609\) −7.84100e9 −1.40673
\(610\) 0 0
\(611\) 4.81672e9 0.854293
\(612\) 0 0
\(613\) −6.64596e9 −1.16532 −0.582661 0.812715i \(-0.697989\pi\)
−0.582661 + 0.812715i \(0.697989\pi\)
\(614\) 0 0
\(615\) 4.01259e9 0.695604
\(616\) 0 0
\(617\) −6.57521e9 −1.12697 −0.563484 0.826127i \(-0.690539\pi\)
−0.563484 + 0.826127i \(0.690539\pi\)
\(618\) 0 0
\(619\) 5.34093e9 0.905107 0.452553 0.891737i \(-0.350513\pi\)
0.452553 + 0.891737i \(0.350513\pi\)
\(620\) 0 0
\(621\) −9.77086e8 −0.163724
\(622\) 0 0
\(623\) 8.25129e9 1.36714
\(624\) 0 0
\(625\) −4.13861e9 −0.678070
\(626\) 0 0
\(627\) −2.52896e9 −0.409737
\(628\) 0 0
\(629\) −7.60273e9 −1.21813
\(630\) 0 0
\(631\) −2.30915e9 −0.365889 −0.182945 0.983123i \(-0.558563\pi\)
−0.182945 + 0.983123i \(0.558563\pi\)
\(632\) 0 0
\(633\) 1.49948e9 0.234979
\(634\) 0 0
\(635\) 2.31271e9 0.358437
\(636\) 0 0
\(637\) −1.00848e10 −1.54590
\(638\) 0 0
\(639\) −7.50834e8 −0.113839
\(640\) 0 0
\(641\) 4.73538e9 0.710152 0.355076 0.934837i \(-0.384455\pi\)
0.355076 + 0.934837i \(0.384455\pi\)
\(642\) 0 0
\(643\) −4.22576e8 −0.0626854 −0.0313427 0.999509i \(-0.509978\pi\)
−0.0313427 + 0.999509i \(0.509978\pi\)
\(644\) 0 0
\(645\) −6.30490e9 −0.925165
\(646\) 0 0
\(647\) 7.01310e9 1.01799 0.508997 0.860769i \(-0.330017\pi\)
0.508997 + 0.860769i \(0.330017\pi\)
\(648\) 0 0
\(649\) −5.17543e8 −0.0743173
\(650\) 0 0
\(651\) 8.36636e9 1.18851
\(652\) 0 0
\(653\) −1.04168e10 −1.46399 −0.731995 0.681310i \(-0.761410\pi\)
−0.731995 + 0.681310i \(0.761410\pi\)
\(654\) 0 0
\(655\) −6.06532e9 −0.843353
\(656\) 0 0
\(657\) 5.99399e8 0.0824588
\(658\) 0 0
\(659\) 1.07019e10 1.45667 0.728337 0.685219i \(-0.240294\pi\)
0.728337 + 0.685219i \(0.240294\pi\)
\(660\) 0 0
\(661\) −7.65225e9 −1.03059 −0.515293 0.857014i \(-0.672317\pi\)
−0.515293 + 0.857014i \(0.672317\pi\)
\(662\) 0 0
\(663\) 9.25789e9 1.23371
\(664\) 0 0
\(665\) −1.17340e10 −1.54728
\(666\) 0 0
\(667\) −1.14847e10 −1.49858
\(668\) 0 0
\(669\) 3.41159e9 0.440520
\(670\) 0 0
\(671\) 4.24788e9 0.542805
\(672\) 0 0
\(673\) −3.47157e9 −0.439009 −0.219505 0.975611i \(-0.570444\pi\)
−0.219505 + 0.975611i \(0.570444\pi\)
\(674\) 0 0
\(675\) 3.94061e8 0.0493174
\(676\) 0 0
\(677\) 1.01392e10 1.25587 0.627936 0.778265i \(-0.283900\pi\)
0.627936 + 0.778265i \(0.283900\pi\)
\(678\) 0 0
\(679\) −9.48070e9 −1.16224
\(680\) 0 0
\(681\) −4.88998e9 −0.593324
\(682\) 0 0
\(683\) 6.29238e9 0.755689 0.377844 0.925869i \(-0.376665\pi\)
0.377844 + 0.925869i \(0.376665\pi\)
\(684\) 0 0
\(685\) −9.85588e9 −1.17160
\(686\) 0 0
\(687\) −3.96892e9 −0.467007
\(688\) 0 0
\(689\) −1.87635e10 −2.18548
\(690\) 0 0
\(691\) 1.09543e10 1.26303 0.631514 0.775365i \(-0.282434\pi\)
0.631514 + 0.775365i \(0.282434\pi\)
\(692\) 0 0
\(693\) −2.21015e9 −0.252265
\(694\) 0 0
\(695\) −9.49453e9 −1.07282
\(696\) 0 0
\(697\) −1.57656e10 −1.76358
\(698\) 0 0
\(699\) −4.54074e9 −0.502871
\(700\) 0 0
\(701\) 1.41208e10 1.54827 0.774137 0.633018i \(-0.218184\pi\)
0.774137 + 0.633018i \(0.218184\pi\)
\(702\) 0 0
\(703\) −1.15299e10 −1.25165
\(704\) 0 0
\(705\) 2.33790e9 0.251283
\(706\) 0 0
\(707\) −8.81285e9 −0.937883
\(708\) 0 0
\(709\) −1.66122e10 −1.75052 −0.875258 0.483656i \(-0.839309\pi\)
−0.875258 + 0.483656i \(0.839309\pi\)
\(710\) 0 0
\(711\) −3.10376e9 −0.323850
\(712\) 0 0
\(713\) 1.22542e10 1.26611
\(714\) 0 0
\(715\) −7.80668e9 −0.798721
\(716\) 0 0
\(717\) 5.56924e9 0.564259
\(718\) 0 0
\(719\) −1.53237e10 −1.53749 −0.768747 0.639553i \(-0.779119\pi\)
−0.768747 + 0.639553i \(0.779119\pi\)
\(720\) 0 0
\(721\) 2.30818e10 2.29349
\(722\) 0 0
\(723\) 9.15667e9 0.901060
\(724\) 0 0
\(725\) 4.63182e9 0.451407
\(726\) 0 0
\(727\) 3.65903e9 0.353180 0.176590 0.984285i \(-0.443493\pi\)
0.176590 + 0.984285i \(0.443493\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 2.47721e10 2.34559
\(732\) 0 0
\(733\) 5.09932e9 0.478243 0.239121 0.970990i \(-0.423141\pi\)
0.239121 + 0.970990i \(0.423141\pi\)
\(734\) 0 0
\(735\) −4.89489e9 −0.454713
\(736\) 0 0
\(737\) −2.61942e9 −0.241029
\(738\) 0 0
\(739\) −1.78005e9 −0.162247 −0.0811236 0.996704i \(-0.525851\pi\)
−0.0811236 + 0.996704i \(0.525851\pi\)
\(740\) 0 0
\(741\) 1.40401e10 1.26767
\(742\) 0 0
\(743\) −3.36627e9 −0.301084 −0.150542 0.988604i \(-0.548102\pi\)
−0.150542 + 0.988604i \(0.548102\pi\)
\(744\) 0 0
\(745\) 8.91984e8 0.0790333
\(746\) 0 0
\(747\) 4.15348e9 0.364578
\(748\) 0 0
\(749\) 1.77438e10 1.54298
\(750\) 0 0
\(751\) 2.21469e10 1.90798 0.953989 0.299842i \(-0.0969341\pi\)
0.953989 + 0.299842i \(0.0969341\pi\)
\(752\) 0 0
\(753\) −9.79509e8 −0.0836038
\(754\) 0 0
\(755\) −2.99605e9 −0.253358
\(756\) 0 0
\(757\) −1.43073e10 −1.19873 −0.599365 0.800476i \(-0.704580\pi\)
−0.599365 + 0.800476i \(0.704580\pi\)
\(758\) 0 0
\(759\) −3.23722e9 −0.268736
\(760\) 0 0
\(761\) −1.23905e10 −1.01916 −0.509579 0.860424i \(-0.670199\pi\)
−0.509579 + 0.860424i \(0.670199\pi\)
\(762\) 0 0
\(763\) 5.81530e9 0.473954
\(764\) 0 0
\(765\) 4.49352e9 0.362887
\(766\) 0 0
\(767\) 2.87325e9 0.229927
\(768\) 0 0
\(769\) −1.76200e10 −1.39722 −0.698608 0.715505i \(-0.746197\pi\)
−0.698608 + 0.715505i \(0.746197\pi\)
\(770\) 0 0
\(771\) 1.39037e10 1.09255
\(772\) 0 0
\(773\) −9.38497e9 −0.730811 −0.365405 0.930849i \(-0.619070\pi\)
−0.365405 + 0.930849i \(0.619070\pi\)
\(774\) 0 0
\(775\) −4.94216e9 −0.381382
\(776\) 0 0
\(777\) −1.00765e10 −0.770610
\(778\) 0 0
\(779\) −2.39093e10 −1.81212
\(780\) 0 0
\(781\) −2.48761e9 −0.186855
\(782\) 0 0
\(783\) 4.55377e9 0.339004
\(784\) 0 0
\(785\) −6.80347e9 −0.501980
\(786\) 0 0
\(787\) 2.16968e10 1.58666 0.793330 0.608791i \(-0.208345\pi\)
0.793330 + 0.608791i \(0.208345\pi\)
\(788\) 0 0
\(789\) 2.43964e9 0.176830
\(790\) 0 0
\(791\) −1.32456e10 −0.951600
\(792\) 0 0
\(793\) −2.35831e10 −1.67936
\(794\) 0 0
\(795\) −9.10728e9 −0.642841
\(796\) 0 0
\(797\) 6.39497e9 0.447439 0.223720 0.974654i \(-0.428180\pi\)
0.223720 + 0.974654i \(0.428180\pi\)
\(798\) 0 0
\(799\) −9.18566e9 −0.637084
\(800\) 0 0
\(801\) −4.79204e9 −0.329463
\(802\) 0 0
\(803\) 1.98589e9 0.135348
\(804\) 0 0
\(805\) −1.50202e10 −1.01482
\(806\) 0 0
\(807\) 8.24320e9 0.552126
\(808\) 0 0
\(809\) −4.80487e9 −0.319052 −0.159526 0.987194i \(-0.550997\pi\)
−0.159526 + 0.987194i \(0.550997\pi\)
\(810\) 0 0
\(811\) −1.20422e10 −0.792745 −0.396372 0.918090i \(-0.629731\pi\)
−0.396372 + 0.918090i \(0.629731\pi\)
\(812\) 0 0
\(813\) −4.52057e9 −0.295037
\(814\) 0 0
\(815\) 1.56158e10 1.01044
\(816\) 0 0
\(817\) 3.75682e10 2.41015
\(818\) 0 0
\(819\) 1.22702e10 0.780471
\(820\) 0 0
\(821\) −2.71353e10 −1.71133 −0.855666 0.517528i \(-0.826852\pi\)
−0.855666 + 0.517528i \(0.826852\pi\)
\(822\) 0 0
\(823\) −2.01740e10 −1.26152 −0.630758 0.775980i \(-0.717256\pi\)
−0.630758 + 0.775980i \(0.717256\pi\)
\(824\) 0 0
\(825\) 1.30558e9 0.0809494
\(826\) 0 0
\(827\) 2.66064e10 1.63575 0.817875 0.575396i \(-0.195152\pi\)
0.817875 + 0.575396i \(0.195152\pi\)
\(828\) 0 0
\(829\) 8.75290e9 0.533594 0.266797 0.963753i \(-0.414035\pi\)
0.266797 + 0.963753i \(0.414035\pi\)
\(830\) 0 0
\(831\) −6.65255e9 −0.402147
\(832\) 0 0
\(833\) 1.92321e10 1.15284
\(834\) 0 0
\(835\) −1.36244e10 −0.809871
\(836\) 0 0
\(837\) −4.85887e9 −0.286415
\(838\) 0 0
\(839\) −1.75179e9 −0.102404 −0.0512018 0.998688i \(-0.516305\pi\)
−0.0512018 + 0.998688i \(0.516305\pi\)
\(840\) 0 0
\(841\) 3.62754e10 2.10294
\(842\) 0 0
\(843\) 9.06754e9 0.521306
\(844\) 0 0
\(845\) 2.82151e10 1.60873
\(846\) 0 0
\(847\) 1.71386e10 0.969135
\(848\) 0 0
\(849\) 1.00922e10 0.565991
\(850\) 0 0
\(851\) −1.47590e10 −0.820926
\(852\) 0 0
\(853\) 5.65193e9 0.311799 0.155900 0.987773i \(-0.450172\pi\)
0.155900 + 0.987773i \(0.450172\pi\)
\(854\) 0 0
\(855\) 6.81466e9 0.372875
\(856\) 0 0
\(857\) −7.32497e9 −0.397533 −0.198766 0.980047i \(-0.563694\pi\)
−0.198766 + 0.980047i \(0.563694\pi\)
\(858\) 0 0
\(859\) −1.93541e10 −1.04183 −0.520914 0.853609i \(-0.674409\pi\)
−0.520914 + 0.853609i \(0.674409\pi\)
\(860\) 0 0
\(861\) −2.08953e10 −1.11567
\(862\) 0 0
\(863\) −1.55088e10 −0.821371 −0.410685 0.911777i \(-0.634711\pi\)
−0.410685 + 0.911777i \(0.634711\pi\)
\(864\) 0 0
\(865\) −2.59975e9 −0.136576
\(866\) 0 0
\(867\) −6.57599e9 −0.342685
\(868\) 0 0
\(869\) −1.02832e10 −0.531566
\(870\) 0 0
\(871\) 1.45423e10 0.745711
\(872\) 0 0
\(873\) 5.50604e9 0.280085
\(874\) 0 0
\(875\) 2.96964e10 1.49856
\(876\) 0 0
\(877\) −2.13613e10 −1.06937 −0.534686 0.845051i \(-0.679570\pi\)
−0.534686 + 0.845051i \(0.679570\pi\)
\(878\) 0 0
\(879\) 1.12267e10 0.557559
\(880\) 0 0
\(881\) 4.16334e9 0.205129 0.102564 0.994726i \(-0.467295\pi\)
0.102564 + 0.994726i \(0.467295\pi\)
\(882\) 0 0
\(883\) −1.68226e10 −0.822301 −0.411151 0.911567i \(-0.634873\pi\)
−0.411151 + 0.911567i \(0.634873\pi\)
\(884\) 0 0
\(885\) 1.39460e9 0.0676312
\(886\) 0 0
\(887\) −2.68120e9 −0.129002 −0.0645010 0.997918i \(-0.520546\pi\)
−0.0645010 + 0.997918i \(0.520546\pi\)
\(888\) 0 0
\(889\) −1.20433e10 −0.574895
\(890\) 0 0
\(891\) 1.28358e9 0.0607924
\(892\) 0 0
\(893\) −1.39305e10 −0.654618
\(894\) 0 0
\(895\) 3.69283e8 0.0172178
\(896\) 0 0
\(897\) 1.79721e10 0.831431
\(898\) 0 0
\(899\) −5.71115e10 −2.62159
\(900\) 0 0
\(901\) 3.57827e10 1.62981
\(902\) 0 0
\(903\) 3.28323e10 1.48386
\(904\) 0 0
\(905\) −3.04014e10 −1.36340
\(906\) 0 0
\(907\) 2.89904e9 0.129011 0.0645057 0.997917i \(-0.479453\pi\)
0.0645057 + 0.997917i \(0.479453\pi\)
\(908\) 0 0
\(909\) 5.11818e9 0.226017
\(910\) 0 0
\(911\) 1.65643e10 0.725868 0.362934 0.931815i \(-0.381775\pi\)
0.362934 + 0.931815i \(0.381775\pi\)
\(912\) 0 0
\(913\) 1.37610e10 0.598416
\(914\) 0 0
\(915\) −1.14466e10 −0.493971
\(916\) 0 0
\(917\) 3.15847e10 1.35265
\(918\) 0 0
\(919\) −6.44040e9 −0.273721 −0.136861 0.990590i \(-0.543701\pi\)
−0.136861 + 0.990590i \(0.543701\pi\)
\(920\) 0 0
\(921\) −5.91995e9 −0.249695
\(922\) 0 0
\(923\) 1.38106e10 0.578103
\(924\) 0 0
\(925\) 5.95234e9 0.247282
\(926\) 0 0
\(927\) −1.34051e10 −0.552701
\(928\) 0 0
\(929\) 3.45986e10 1.41580 0.707902 0.706311i \(-0.249642\pi\)
0.707902 + 0.706311i \(0.249642\pi\)
\(930\) 0 0
\(931\) 2.91666e10 1.18457
\(932\) 0 0
\(933\) −5.76786e9 −0.232503
\(934\) 0 0
\(935\) 1.48876e10 0.595641
\(936\) 0 0
\(937\) −3.81853e10 −1.51638 −0.758190 0.652034i \(-0.773916\pi\)
−0.758190 + 0.652034i \(0.773916\pi\)
\(938\) 0 0
\(939\) −2.21282e10 −0.872201
\(940\) 0 0
\(941\) −6.28907e9 −0.246049 −0.123025 0.992404i \(-0.539259\pi\)
−0.123025 + 0.992404i \(0.539259\pi\)
\(942\) 0 0
\(943\) −3.06054e10 −1.18852
\(944\) 0 0
\(945\) 5.95560e9 0.229569
\(946\) 0 0
\(947\) −2.30506e10 −0.881977 −0.440988 0.897513i \(-0.645372\pi\)
−0.440988 + 0.897513i \(0.645372\pi\)
\(948\) 0 0
\(949\) −1.10251e10 −0.418746
\(950\) 0 0
\(951\) −1.16497e10 −0.439221
\(952\) 0 0
\(953\) 1.46553e10 0.548493 0.274246 0.961659i \(-0.411572\pi\)
0.274246 + 0.961659i \(0.411572\pi\)
\(954\) 0 0
\(955\) −3.40214e10 −1.26398
\(956\) 0 0
\(957\) 1.50872e10 0.556439
\(958\) 0 0
\(959\) 5.13238e10 1.87912
\(960\) 0 0
\(961\) 3.34254e10 1.21491
\(962\) 0 0
\(963\) −1.03049e10 −0.371837
\(964\) 0 0
\(965\) −2.77867e10 −0.995385
\(966\) 0 0
\(967\) −2.23044e9 −0.0793226 −0.0396613 0.999213i \(-0.512628\pi\)
−0.0396613 + 0.999213i \(0.512628\pi\)
\(968\) 0 0
\(969\) −2.67750e10 −0.945357
\(970\) 0 0
\(971\) −3.25778e10 −1.14197 −0.570984 0.820961i \(-0.693438\pi\)
−0.570984 + 0.820961i \(0.693438\pi\)
\(972\) 0 0
\(973\) 4.94421e10 1.72069
\(974\) 0 0
\(975\) −7.24820e9 −0.250446
\(976\) 0 0
\(977\) −3.10973e10 −1.06682 −0.533411 0.845856i \(-0.679090\pi\)
−0.533411 + 0.845856i \(0.679090\pi\)
\(978\) 0 0
\(979\) −1.58767e10 −0.540780
\(980\) 0 0
\(981\) −3.37731e9 −0.114217
\(982\) 0 0
\(983\) 3.12824e9 0.105042 0.0525210 0.998620i \(-0.483274\pi\)
0.0525210 + 0.998620i \(0.483274\pi\)
\(984\) 0 0
\(985\) 7.77630e9 0.259266
\(986\) 0 0
\(987\) −1.21744e10 −0.403031
\(988\) 0 0
\(989\) 4.80895e10 1.58075
\(990\) 0 0
\(991\) −4.75891e10 −1.55328 −0.776639 0.629946i \(-0.783077\pi\)
−0.776639 + 0.629946i \(0.783077\pi\)
\(992\) 0 0
\(993\) −2.21983e9 −0.0719444
\(994\) 0 0
\(995\) −1.32502e10 −0.426423
\(996\) 0 0
\(997\) 1.10819e10 0.354144 0.177072 0.984198i \(-0.443338\pi\)
0.177072 + 0.984198i \(0.443338\pi\)
\(998\) 0 0
\(999\) 5.85204e9 0.185707
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 384.8.a.f.1.2 yes 2
4.3 odd 2 384.8.a.h.1.2 yes 2
8.3 odd 2 384.8.a.e.1.1 2
8.5 even 2 384.8.a.g.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.a.e.1.1 2 8.3 odd 2
384.8.a.f.1.2 yes 2 1.1 even 1 trivial
384.8.a.g.1.1 yes 2 8.5 even 2
384.8.a.h.1.2 yes 2 4.3 odd 2