Properties

Label 76.8.a.b.1.4
Level $76$
Weight $8$
Character 76.1
Self dual yes
Analytic conductor $23.741$
Analytic rank $0$
Dimension $6$
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] = [76,8,Mod(1,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("76.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.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: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-13.7370\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+20.7370 q^{3} -501.411 q^{5} -1008.02 q^{7} -1756.98 q^{9} +O(q^{10})\) \(q+20.7370 q^{3} -501.411 q^{5} -1008.02 q^{7} -1756.98 q^{9} +6783.78 q^{11} +10249.0 q^{13} -10397.8 q^{15} +27829.3 q^{17} +6859.00 q^{19} -20903.3 q^{21} +4350.81 q^{23} +173288. q^{25} -81786.3 q^{27} -171678. q^{29} -107058. q^{31} +140675. q^{33} +505430. q^{35} +489322. q^{37} +212534. q^{39} +25924.1 q^{41} +189649. q^{43} +880966. q^{45} -404901. q^{47} +192554. q^{49} +577098. q^{51} +1.48333e6 q^{53} -3.40146e6 q^{55} +142235. q^{57} +834818. q^{59} -344246. q^{61} +1.77106e6 q^{63} -5.13896e6 q^{65} +2.08156e6 q^{67} +90222.9 q^{69} +2.08187e6 q^{71} -1.99207e6 q^{73} +3.59347e6 q^{75} -6.83816e6 q^{77} -4.22247e6 q^{79} +2.14650e6 q^{81} +5.66320e6 q^{83} -1.39539e7 q^{85} -3.56010e6 q^{87} +1.04694e7 q^{89} -1.03312e7 q^{91} -2.22007e6 q^{93} -3.43918e6 q^{95} -1.45838e7 q^{97} -1.19189e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9} + 7983 q^{11} + 1250 q^{13} + 26790 q^{15} + 21735 q^{17} + 41154 q^{19} - 86346 q^{21} + 100920 q^{23} + 373305 q^{25} + 534790 q^{27} - 58656 q^{29} + 403808 q^{31} + 916430 q^{33} + 463497 q^{35} + 808780 q^{37} + 758704 q^{39} + 556944 q^{41} + 1220735 q^{43} + 3234843 q^{45} + 1915305 q^{47} + 2045883 q^{49} + 908816 q^{51} + 511650 q^{53} + 1813341 q^{55} + 274360 q^{57} + 1300572 q^{59} + 565335 q^{61} - 7170325 q^{63} - 6195012 q^{65} - 45010 q^{67} - 7381528 q^{69} - 1424106 q^{71} - 11153825 q^{73} + 3941974 q^{75} - 17515425 q^{77} - 6392144 q^{79} + 6187530 q^{81} - 3164160 q^{83} - 19479255 q^{85} - 25999500 q^{87} - 14502678 q^{89} - 9736226 q^{91} - 18344300 q^{93} + 1913661 q^{95} - 21377010 q^{97} + 24032935 q^{99}+O(q^{100}) \) 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
\(3\) 20.7370 0.443427 0.221713 0.975112i \(-0.428835\pi\)
0.221713 + 0.975112i \(0.428835\pi\)
\(4\) 0 0
\(5\) −501.411 −1.79390 −0.896951 0.442131i \(-0.854223\pi\)
−0.896951 + 0.442131i \(0.854223\pi\)
\(6\) 0 0
\(7\) −1008.02 −1.11077 −0.555385 0.831593i \(-0.687429\pi\)
−0.555385 + 0.831593i \(0.687429\pi\)
\(8\) 0 0
\(9\) −1756.98 −0.803373
\(10\) 0 0
\(11\) 6783.78 1.53673 0.768365 0.640012i \(-0.221071\pi\)
0.768365 + 0.640012i \(0.221071\pi\)
\(12\) 0 0
\(13\) 10249.0 1.29384 0.646919 0.762559i \(-0.276057\pi\)
0.646919 + 0.762559i \(0.276057\pi\)
\(14\) 0 0
\(15\) −10397.8 −0.795464
\(16\) 0 0
\(17\) 27829.3 1.37383 0.686913 0.726740i \(-0.258965\pi\)
0.686913 + 0.726740i \(0.258965\pi\)
\(18\) 0 0
\(19\) 6859.00 0.229416
\(20\) 0 0
\(21\) −20903.3 −0.492546
\(22\) 0 0
\(23\) 4350.81 0.0745629 0.0372815 0.999305i \(-0.488130\pi\)
0.0372815 + 0.999305i \(0.488130\pi\)
\(24\) 0 0
\(25\) 173288. 2.21808
\(26\) 0 0
\(27\) −81786.3 −0.799664
\(28\) 0 0
\(29\) −171678. −1.30714 −0.653571 0.756866i \(-0.726730\pi\)
−0.653571 + 0.756866i \(0.726730\pi\)
\(30\) 0 0
\(31\) −107058. −0.645438 −0.322719 0.946495i \(-0.604597\pi\)
−0.322719 + 0.946495i \(0.604597\pi\)
\(32\) 0 0
\(33\) 140675. 0.681427
\(34\) 0 0
\(35\) 505430. 1.99261
\(36\) 0 0
\(37\) 489322. 1.58814 0.794070 0.607826i \(-0.207958\pi\)
0.794070 + 0.607826i \(0.207958\pi\)
\(38\) 0 0
\(39\) 212534. 0.573723
\(40\) 0 0
\(41\) 25924.1 0.0587434 0.0293717 0.999569i \(-0.490649\pi\)
0.0293717 + 0.999569i \(0.490649\pi\)
\(42\) 0 0
\(43\) 189649. 0.363757 0.181878 0.983321i \(-0.441782\pi\)
0.181878 + 0.983321i \(0.441782\pi\)
\(44\) 0 0
\(45\) 880966. 1.44117
\(46\) 0 0
\(47\) −404901. −0.568862 −0.284431 0.958697i \(-0.591805\pi\)
−0.284431 + 0.958697i \(0.591805\pi\)
\(48\) 0 0
\(49\) 192554. 0.233811
\(50\) 0 0
\(51\) 577098. 0.609192
\(52\) 0 0
\(53\) 1.48333e6 1.36859 0.684295 0.729205i \(-0.260110\pi\)
0.684295 + 0.729205i \(0.260110\pi\)
\(54\) 0 0
\(55\) −3.40146e6 −2.75674
\(56\) 0 0
\(57\) 142235. 0.101729
\(58\) 0 0
\(59\) 834818. 0.529188 0.264594 0.964360i \(-0.414762\pi\)
0.264594 + 0.964360i \(0.414762\pi\)
\(60\) 0 0
\(61\) −344246. −0.194184 −0.0970922 0.995275i \(-0.530954\pi\)
−0.0970922 + 0.995275i \(0.530954\pi\)
\(62\) 0 0
\(63\) 1.77106e6 0.892363
\(64\) 0 0
\(65\) −5.13896e6 −2.32102
\(66\) 0 0
\(67\) 2.08156e6 0.845526 0.422763 0.906240i \(-0.361060\pi\)
0.422763 + 0.906240i \(0.361060\pi\)
\(68\) 0 0
\(69\) 90222.9 0.0330632
\(70\) 0 0
\(71\) 2.08187e6 0.690318 0.345159 0.938544i \(-0.387825\pi\)
0.345159 + 0.938544i \(0.387825\pi\)
\(72\) 0 0
\(73\) −1.99207e6 −0.599342 −0.299671 0.954043i \(-0.596877\pi\)
−0.299671 + 0.954043i \(0.596877\pi\)
\(74\) 0 0
\(75\) 3.59347e6 0.983557
\(76\) 0 0
\(77\) −6.83816e6 −1.70695
\(78\) 0 0
\(79\) −4.22247e6 −0.963544 −0.481772 0.876297i \(-0.660007\pi\)
−0.481772 + 0.876297i \(0.660007\pi\)
\(80\) 0 0
\(81\) 2.14650e6 0.448780
\(82\) 0 0
\(83\) 5.66320e6 1.08715 0.543574 0.839361i \(-0.317071\pi\)
0.543574 + 0.839361i \(0.317071\pi\)
\(84\) 0 0
\(85\) −1.39539e7 −2.46451
\(86\) 0 0
\(87\) −3.56010e6 −0.579622
\(88\) 0 0
\(89\) 1.04694e7 1.57418 0.787092 0.616835i \(-0.211586\pi\)
0.787092 + 0.616835i \(0.211586\pi\)
\(90\) 0 0
\(91\) −1.03312e7 −1.43716
\(92\) 0 0
\(93\) −2.22007e6 −0.286204
\(94\) 0 0
\(95\) −3.43918e6 −0.411549
\(96\) 0 0
\(97\) −1.45838e7 −1.62245 −0.811223 0.584738i \(-0.801198\pi\)
−0.811223 + 0.584738i \(0.801198\pi\)
\(98\) 0 0
\(99\) −1.19189e7 −1.23457
\(100\) 0 0
\(101\) 306934. 0.0296429 0.0148214 0.999890i \(-0.495282\pi\)
0.0148214 + 0.999890i \(0.495282\pi\)
\(102\) 0 0
\(103\) 1.57505e7 1.42024 0.710122 0.704078i \(-0.248640\pi\)
0.710122 + 0.704078i \(0.248640\pi\)
\(104\) 0 0
\(105\) 1.04811e7 0.883578
\(106\) 0 0
\(107\) 1.34511e7 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(108\) 0 0
\(109\) 1.39015e7 1.02818 0.514089 0.857737i \(-0.328130\pi\)
0.514089 + 0.857737i \(0.328130\pi\)
\(110\) 0 0
\(111\) 1.01471e7 0.704224
\(112\) 0 0
\(113\) 1.85824e7 1.21151 0.605755 0.795651i \(-0.292871\pi\)
0.605755 + 0.795651i \(0.292871\pi\)
\(114\) 0 0
\(115\) −2.18154e6 −0.133758
\(116\) 0 0
\(117\) −1.80073e7 −1.03943
\(118\) 0 0
\(119\) −2.80524e7 −1.52601
\(120\) 0 0
\(121\) 2.65325e7 1.36154
\(122\) 0 0
\(123\) 537588. 0.0260484
\(124\) 0 0
\(125\) −4.77155e7 −2.18512
\(126\) 0 0
\(127\) −3.05149e7 −1.32190 −0.660951 0.750429i \(-0.729847\pi\)
−0.660951 + 0.750429i \(0.729847\pi\)
\(128\) 0 0
\(129\) 3.93276e6 0.161300
\(130\) 0 0
\(131\) 4.28387e7 1.66489 0.832446 0.554105i \(-0.186940\pi\)
0.832446 + 0.554105i \(0.186940\pi\)
\(132\) 0 0
\(133\) −6.91398e6 −0.254828
\(134\) 0 0
\(135\) 4.10085e7 1.43452
\(136\) 0 0
\(137\) −3.68434e7 −1.22416 −0.612079 0.790797i \(-0.709667\pi\)
−0.612079 + 0.790797i \(0.709667\pi\)
\(138\) 0 0
\(139\) 5.18263e7 1.63681 0.818405 0.574643i \(-0.194859\pi\)
0.818405 + 0.574643i \(0.194859\pi\)
\(140\) 0 0
\(141\) −8.39645e6 −0.252249
\(142\) 0 0
\(143\) 6.95270e7 1.98828
\(144\) 0 0
\(145\) 8.60813e7 2.34488
\(146\) 0 0
\(147\) 3.99299e6 0.103678
\(148\) 0 0
\(149\) −4.36079e7 −1.07997 −0.539986 0.841674i \(-0.681571\pi\)
−0.539986 + 0.841674i \(0.681571\pi\)
\(150\) 0 0
\(151\) 4.26610e7 1.00835 0.504175 0.863601i \(-0.331797\pi\)
0.504175 + 0.863601i \(0.331797\pi\)
\(152\) 0 0
\(153\) −4.88955e7 −1.10369
\(154\) 0 0
\(155\) 5.36801e7 1.15785
\(156\) 0 0
\(157\) 7.75145e6 0.159858 0.0799290 0.996801i \(-0.474531\pi\)
0.0799290 + 0.996801i \(0.474531\pi\)
\(158\) 0 0
\(159\) 3.07599e7 0.606870
\(160\) 0 0
\(161\) −4.38569e6 −0.0828223
\(162\) 0 0
\(163\) −7.13287e7 −1.29005 −0.645027 0.764160i \(-0.723154\pi\)
−0.645027 + 0.764160i \(0.723154\pi\)
\(164\) 0 0
\(165\) −7.05361e7 −1.22241
\(166\) 0 0
\(167\) 7.82203e7 1.29961 0.649803 0.760102i \(-0.274851\pi\)
0.649803 + 0.760102i \(0.274851\pi\)
\(168\) 0 0
\(169\) 4.22936e7 0.674017
\(170\) 0 0
\(171\) −1.20511e7 −0.184306
\(172\) 0 0
\(173\) −5.48988e7 −0.806123 −0.403062 0.915173i \(-0.632054\pi\)
−0.403062 + 0.915173i \(0.632054\pi\)
\(174\) 0 0
\(175\) −1.74677e8 −2.46378
\(176\) 0 0
\(177\) 1.73116e7 0.234656
\(178\) 0 0
\(179\) −7.47688e7 −0.974395 −0.487197 0.873292i \(-0.661981\pi\)
−0.487197 + 0.873292i \(0.661981\pi\)
\(180\) 0 0
\(181\) −5.58486e7 −0.700063 −0.350031 0.936738i \(-0.613829\pi\)
−0.350031 + 0.936738i \(0.613829\pi\)
\(182\) 0 0
\(183\) −7.13864e6 −0.0861066
\(184\) 0 0
\(185\) −2.45351e8 −2.84897
\(186\) 0 0
\(187\) 1.88788e8 2.11120
\(188\) 0 0
\(189\) 8.24419e7 0.888243
\(190\) 0 0
\(191\) 6.71702e7 0.697525 0.348763 0.937211i \(-0.386602\pi\)
0.348763 + 0.937211i \(0.386602\pi\)
\(192\) 0 0
\(193\) −3.89356e7 −0.389849 −0.194925 0.980818i \(-0.562446\pi\)
−0.194925 + 0.980818i \(0.562446\pi\)
\(194\) 0 0
\(195\) −1.06567e8 −1.02920
\(196\) 0 0
\(197\) 3.94584e7 0.367712 0.183856 0.982953i \(-0.441142\pi\)
0.183856 + 0.982953i \(0.441142\pi\)
\(198\) 0 0
\(199\) −1.45572e8 −1.30946 −0.654731 0.755862i \(-0.727218\pi\)
−0.654731 + 0.755862i \(0.727218\pi\)
\(200\) 0 0
\(201\) 4.31654e7 0.374929
\(202\) 0 0
\(203\) 1.73055e8 1.45193
\(204\) 0 0
\(205\) −1.29986e7 −0.105380
\(206\) 0 0
\(207\) −7.64427e6 −0.0599018
\(208\) 0 0
\(209\) 4.65300e7 0.352550
\(210\) 0 0
\(211\) −1.82801e8 −1.33965 −0.669823 0.742521i \(-0.733630\pi\)
−0.669823 + 0.742521i \(0.733630\pi\)
\(212\) 0 0
\(213\) 4.31717e7 0.306105
\(214\) 0 0
\(215\) −9.50921e7 −0.652544
\(216\) 0 0
\(217\) 1.07916e8 0.716933
\(218\) 0 0
\(219\) −4.13096e7 −0.265765
\(220\) 0 0
\(221\) 2.85223e8 1.77751
\(222\) 0 0
\(223\) −4.04726e7 −0.244396 −0.122198 0.992506i \(-0.538994\pi\)
−0.122198 + 0.992506i \(0.538994\pi\)
\(224\) 0 0
\(225\) −3.04462e8 −1.78195
\(226\) 0 0
\(227\) −1.37641e8 −0.781013 −0.390507 0.920600i \(-0.627700\pi\)
−0.390507 + 0.920600i \(0.627700\pi\)
\(228\) 0 0
\(229\) −2.31082e8 −1.27157 −0.635787 0.771865i \(-0.719324\pi\)
−0.635787 + 0.771865i \(0.719324\pi\)
\(230\) 0 0
\(231\) −1.41803e8 −0.756909
\(232\) 0 0
\(233\) −4.37486e7 −0.226579 −0.113289 0.993562i \(-0.536139\pi\)
−0.113289 + 0.993562i \(0.536139\pi\)
\(234\) 0 0
\(235\) 2.03022e8 1.02048
\(236\) 0 0
\(237\) −8.75614e7 −0.427261
\(238\) 0 0
\(239\) 3.53555e8 1.67519 0.837597 0.546289i \(-0.183960\pi\)
0.837597 + 0.546289i \(0.183960\pi\)
\(240\) 0 0
\(241\) 2.02396e8 0.931411 0.465705 0.884940i \(-0.345801\pi\)
0.465705 + 0.884940i \(0.345801\pi\)
\(242\) 0 0
\(243\) 2.23379e8 0.998665
\(244\) 0 0
\(245\) −9.65484e7 −0.419434
\(246\) 0 0
\(247\) 7.02979e7 0.296827
\(248\) 0 0
\(249\) 1.17438e8 0.482070
\(250\) 0 0
\(251\) −5.33482e7 −0.212942 −0.106471 0.994316i \(-0.533955\pi\)
−0.106471 + 0.994316i \(0.533955\pi\)
\(252\) 0 0
\(253\) 2.95150e7 0.114583
\(254\) 0 0
\(255\) −2.89363e8 −1.09283
\(256\) 0 0
\(257\) 3.54687e8 1.30340 0.651702 0.758475i \(-0.274055\pi\)
0.651702 + 0.758475i \(0.274055\pi\)
\(258\) 0 0
\(259\) −4.93245e8 −1.76406
\(260\) 0 0
\(261\) 3.01635e8 1.05012
\(262\) 0 0
\(263\) 4.73491e8 1.60497 0.802484 0.596673i \(-0.203511\pi\)
0.802484 + 0.596673i \(0.203511\pi\)
\(264\) 0 0
\(265\) −7.43760e8 −2.45512
\(266\) 0 0
\(267\) 2.17104e8 0.698036
\(268\) 0 0
\(269\) 9.38349e7 0.293921 0.146961 0.989142i \(-0.453051\pi\)
0.146961 + 0.989142i \(0.453051\pi\)
\(270\) 0 0
\(271\) 3.34812e8 1.02190 0.510950 0.859610i \(-0.329294\pi\)
0.510950 + 0.859610i \(0.329294\pi\)
\(272\) 0 0
\(273\) −2.14238e8 −0.637274
\(274\) 0 0
\(275\) 1.17554e9 3.40859
\(276\) 0 0
\(277\) −1.19729e8 −0.338469 −0.169235 0.985576i \(-0.554130\pi\)
−0.169235 + 0.985576i \(0.554130\pi\)
\(278\) 0 0
\(279\) 1.88099e8 0.518527
\(280\) 0 0
\(281\) 2.81327e8 0.756378 0.378189 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(282\) 0 0
\(283\) 1.47945e7 0.0388016 0.0194008 0.999812i \(-0.493824\pi\)
0.0194008 + 0.999812i \(0.493824\pi\)
\(284\) 0 0
\(285\) −7.13183e7 −0.182492
\(286\) 0 0
\(287\) −2.61319e7 −0.0652505
\(288\) 0 0
\(289\) 3.64134e8 0.887399
\(290\) 0 0
\(291\) −3.02425e8 −0.719436
\(292\) 0 0
\(293\) 5.35486e8 1.24369 0.621845 0.783141i \(-0.286384\pi\)
0.621845 + 0.783141i \(0.286384\pi\)
\(294\) 0 0
\(295\) −4.18587e8 −0.949310
\(296\) 0 0
\(297\) −5.54820e8 −1.22887
\(298\) 0 0
\(299\) 4.45915e7 0.0964723
\(300\) 0 0
\(301\) −1.91169e8 −0.404050
\(302\) 0 0
\(303\) 6.36490e6 0.0131444
\(304\) 0 0
\(305\) 1.72609e8 0.348348
\(306\) 0 0
\(307\) −5.23353e8 −1.03231 −0.516155 0.856495i \(-0.672637\pi\)
−0.516155 + 0.856495i \(0.672637\pi\)
\(308\) 0 0
\(309\) 3.26618e8 0.629775
\(310\) 0 0
\(311\) −7.01475e8 −1.32236 −0.661182 0.750226i \(-0.729945\pi\)
−0.661182 + 0.750226i \(0.729945\pi\)
\(312\) 0 0
\(313\) −1.49136e8 −0.274902 −0.137451 0.990509i \(-0.543891\pi\)
−0.137451 + 0.990509i \(0.543891\pi\)
\(314\) 0 0
\(315\) −8.88028e8 −1.60081
\(316\) 0 0
\(317\) 4.62409e8 0.815302 0.407651 0.913138i \(-0.366348\pi\)
0.407651 + 0.913138i \(0.366348\pi\)
\(318\) 0 0
\(319\) −1.16463e9 −2.00872
\(320\) 0 0
\(321\) 2.78935e8 0.470690
\(322\) 0 0
\(323\) 1.90882e8 0.315177
\(324\) 0 0
\(325\) 1.77602e9 2.86984
\(326\) 0 0
\(327\) 2.88275e8 0.455921
\(328\) 0 0
\(329\) 4.08147e8 0.631875
\(330\) 0 0
\(331\) 3.78961e8 0.574376 0.287188 0.957874i \(-0.407280\pi\)
0.287188 + 0.957874i \(0.407280\pi\)
\(332\) 0 0
\(333\) −8.59727e8 −1.27587
\(334\) 0 0
\(335\) −1.04372e9 −1.51679
\(336\) 0 0
\(337\) 4.67178e8 0.664933 0.332467 0.943115i \(-0.392119\pi\)
0.332467 + 0.943115i \(0.392119\pi\)
\(338\) 0 0
\(339\) 3.85343e8 0.537216
\(340\) 0 0
\(341\) −7.26260e8 −0.991863
\(342\) 0 0
\(343\) 6.36047e8 0.851060
\(344\) 0 0
\(345\) −4.52387e7 −0.0593121
\(346\) 0 0
\(347\) −8.93105e8 −1.14749 −0.573745 0.819034i \(-0.694510\pi\)
−0.573745 + 0.819034i \(0.694510\pi\)
\(348\) 0 0
\(349\) −5.93562e8 −0.747441 −0.373720 0.927541i \(-0.621918\pi\)
−0.373720 + 0.927541i \(0.621918\pi\)
\(350\) 0 0
\(351\) −8.38228e8 −1.03464
\(352\) 0 0
\(353\) −3.88892e8 −0.470563 −0.235282 0.971927i \(-0.575601\pi\)
−0.235282 + 0.971927i \(0.575601\pi\)
\(354\) 0 0
\(355\) −1.04387e9 −1.23836
\(356\) 0 0
\(357\) −5.81724e8 −0.676672
\(358\) 0 0
\(359\) 8.12699e7 0.0927042 0.0463521 0.998925i \(-0.485240\pi\)
0.0463521 + 0.998925i \(0.485240\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) 5.50205e8 0.603742
\(364\) 0 0
\(365\) 9.98846e8 1.07516
\(366\) 0 0
\(367\) 1.36670e8 0.144325 0.0721627 0.997393i \(-0.477010\pi\)
0.0721627 + 0.997393i \(0.477010\pi\)
\(368\) 0 0
\(369\) −4.55479e7 −0.0471929
\(370\) 0 0
\(371\) −1.49523e9 −1.52019
\(372\) 0 0
\(373\) 1.53262e8 0.152916 0.0764581 0.997073i \(-0.475639\pi\)
0.0764581 + 0.997073i \(0.475639\pi\)
\(374\) 0 0
\(375\) −9.89478e8 −0.968939
\(376\) 0 0
\(377\) −1.75953e9 −1.69123
\(378\) 0 0
\(379\) −2.04506e8 −0.192961 −0.0964803 0.995335i \(-0.530758\pi\)
−0.0964803 + 0.995335i \(0.530758\pi\)
\(380\) 0 0
\(381\) −6.32789e8 −0.586167
\(382\) 0 0
\(383\) 3.03716e8 0.276231 0.138115 0.990416i \(-0.455896\pi\)
0.138115 + 0.990416i \(0.455896\pi\)
\(384\) 0 0
\(385\) 3.42873e9 3.06211
\(386\) 0 0
\(387\) −3.33209e8 −0.292232
\(388\) 0 0
\(389\) −9.25621e8 −0.797278 −0.398639 0.917108i \(-0.630517\pi\)
−0.398639 + 0.917108i \(0.630517\pi\)
\(390\) 0 0
\(391\) 1.21080e8 0.102436
\(392\) 0 0
\(393\) 8.88346e8 0.738258
\(394\) 0 0
\(395\) 2.11719e9 1.72850
\(396\) 0 0
\(397\) −1.31806e9 −1.05723 −0.528615 0.848862i \(-0.677289\pi\)
−0.528615 + 0.848862i \(0.677289\pi\)
\(398\) 0 0
\(399\) −1.43375e8 −0.112998
\(400\) 0 0
\(401\) 1.42599e9 1.10436 0.552181 0.833724i \(-0.313796\pi\)
0.552181 + 0.833724i \(0.313796\pi\)
\(402\) 0 0
\(403\) −1.09724e9 −0.835092
\(404\) 0 0
\(405\) −1.07628e9 −0.805067
\(406\) 0 0
\(407\) 3.31945e9 2.44054
\(408\) 0 0
\(409\) 2.44675e9 1.76831 0.884154 0.467196i \(-0.154736\pi\)
0.884154 + 0.467196i \(0.154736\pi\)
\(410\) 0 0
\(411\) −7.64022e8 −0.542825
\(412\) 0 0
\(413\) −8.41510e8 −0.587806
\(414\) 0 0
\(415\) −2.83959e9 −1.95023
\(416\) 0 0
\(417\) 1.07472e9 0.725805
\(418\) 0 0
\(419\) 1.62114e9 1.07664 0.538320 0.842740i \(-0.319059\pi\)
0.538320 + 0.842740i \(0.319059\pi\)
\(420\) 0 0
\(421\) −7.97506e8 −0.520891 −0.260445 0.965489i \(-0.583869\pi\)
−0.260445 + 0.965489i \(0.583869\pi\)
\(422\) 0 0
\(423\) 7.11402e8 0.457008
\(424\) 0 0
\(425\) 4.82248e9 3.04726
\(426\) 0 0
\(427\) 3.47006e8 0.215694
\(428\) 0 0
\(429\) 1.44178e9 0.881656
\(430\) 0 0
\(431\) 1.33983e9 0.806085 0.403042 0.915181i \(-0.367953\pi\)
0.403042 + 0.915181i \(0.367953\pi\)
\(432\) 0 0
\(433\) −1.49611e9 −0.885639 −0.442819 0.896611i \(-0.646022\pi\)
−0.442819 + 0.896611i \(0.646022\pi\)
\(434\) 0 0
\(435\) 1.78507e9 1.03978
\(436\) 0 0
\(437\) 2.98422e7 0.0171059
\(438\) 0 0
\(439\) −2.18648e9 −1.23344 −0.616721 0.787182i \(-0.711539\pi\)
−0.616721 + 0.787182i \(0.711539\pi\)
\(440\) 0 0
\(441\) −3.38312e8 −0.187838
\(442\) 0 0
\(443\) 1.00352e9 0.548420 0.274210 0.961670i \(-0.411584\pi\)
0.274210 + 0.961670i \(0.411584\pi\)
\(444\) 0 0
\(445\) −5.24945e9 −2.82393
\(446\) 0 0
\(447\) −9.04297e8 −0.478889
\(448\) 0 0
\(449\) −2.07250e9 −1.08052 −0.540259 0.841499i \(-0.681674\pi\)
−0.540259 + 0.841499i \(0.681674\pi\)
\(450\) 0 0
\(451\) 1.75863e8 0.0902728
\(452\) 0 0
\(453\) 8.84662e8 0.447130
\(454\) 0 0
\(455\) 5.18015e9 2.57812
\(456\) 0 0
\(457\) −1.00176e9 −0.490974 −0.245487 0.969400i \(-0.578948\pi\)
−0.245487 + 0.969400i \(0.578948\pi\)
\(458\) 0 0
\(459\) −2.27606e9 −1.09860
\(460\) 0 0
\(461\) −2.48579e9 −1.18171 −0.590856 0.806777i \(-0.701210\pi\)
−0.590856 + 0.806777i \(0.701210\pi\)
\(462\) 0 0
\(463\) 1.38703e9 0.649459 0.324729 0.945807i \(-0.394727\pi\)
0.324729 + 0.945807i \(0.394727\pi\)
\(464\) 0 0
\(465\) 1.11317e9 0.513422
\(466\) 0 0
\(467\) −2.93622e9 −1.33407 −0.667036 0.745026i \(-0.732437\pi\)
−0.667036 + 0.745026i \(0.732437\pi\)
\(468\) 0 0
\(469\) −2.09825e9 −0.939186
\(470\) 0 0
\(471\) 1.60742e8 0.0708853
\(472\) 0 0
\(473\) 1.28654e9 0.558996
\(474\) 0 0
\(475\) 1.18858e9 0.508863
\(476\) 0 0
\(477\) −2.60618e9 −1.09949
\(478\) 0 0
\(479\) 1.12696e9 0.468526 0.234263 0.972173i \(-0.424732\pi\)
0.234263 + 0.972173i \(0.424732\pi\)
\(480\) 0 0
\(481\) 5.01506e9 2.05480
\(482\) 0 0
\(483\) −9.09461e7 −0.0367256
\(484\) 0 0
\(485\) 7.31248e9 2.91051
\(486\) 0 0
\(487\) 4.17540e9 1.63812 0.819062 0.573705i \(-0.194494\pi\)
0.819062 + 0.573705i \(0.194494\pi\)
\(488\) 0 0
\(489\) −1.47914e9 −0.572044
\(490\) 0 0
\(491\) 3.34194e9 1.27413 0.637064 0.770811i \(-0.280149\pi\)
0.637064 + 0.770811i \(0.280149\pi\)
\(492\) 0 0
\(493\) −4.77770e9 −1.79578
\(494\) 0 0
\(495\) 5.97628e9 2.21469
\(496\) 0 0
\(497\) −2.09856e9 −0.766784
\(498\) 0 0
\(499\) −3.98948e9 −1.43735 −0.718677 0.695344i \(-0.755252\pi\)
−0.718677 + 0.695344i \(0.755252\pi\)
\(500\) 0 0
\(501\) 1.62206e9 0.576281
\(502\) 0 0
\(503\) −4.47180e9 −1.56673 −0.783365 0.621562i \(-0.786498\pi\)
−0.783365 + 0.621562i \(0.786498\pi\)
\(504\) 0 0
\(505\) −1.53900e8 −0.0531764
\(506\) 0 0
\(507\) 8.77043e8 0.298877
\(508\) 0 0
\(509\) 2.36655e9 0.795433 0.397716 0.917508i \(-0.369803\pi\)
0.397716 + 0.917508i \(0.369803\pi\)
\(510\) 0 0
\(511\) 2.00804e9 0.665732
\(512\) 0 0
\(513\) −5.60972e8 −0.183455
\(514\) 0 0
\(515\) −7.89745e9 −2.54778
\(516\) 0 0
\(517\) −2.74676e9 −0.874186
\(518\) 0 0
\(519\) −1.13844e9 −0.357457
\(520\) 0 0
\(521\) −1.20799e9 −0.374222 −0.187111 0.982339i \(-0.559912\pi\)
−0.187111 + 0.982339i \(0.559912\pi\)
\(522\) 0 0
\(523\) 3.99004e9 1.21961 0.609806 0.792551i \(-0.291247\pi\)
0.609806 + 0.792551i \(0.291247\pi\)
\(524\) 0 0
\(525\) −3.62227e9 −1.09251
\(526\) 0 0
\(527\) −2.97936e9 −0.886719
\(528\) 0 0
\(529\) −3.38590e9 −0.994440
\(530\) 0 0
\(531\) −1.46675e9 −0.425135
\(532\) 0 0
\(533\) 2.65696e8 0.0760045
\(534\) 0 0
\(535\) −6.74450e9 −1.90420
\(536\) 0 0
\(537\) −1.55048e9 −0.432073
\(538\) 0 0
\(539\) 1.30624e9 0.359305
\(540\) 0 0
\(541\) 5.43642e8 0.147612 0.0738061 0.997273i \(-0.476485\pi\)
0.0738061 + 0.997273i \(0.476485\pi\)
\(542\) 0 0
\(543\) −1.15813e9 −0.310427
\(544\) 0 0
\(545\) −6.97034e9 −1.84445
\(546\) 0 0
\(547\) −1.39107e9 −0.363407 −0.181704 0.983353i \(-0.558161\pi\)
−0.181704 + 0.983353i \(0.558161\pi\)
\(548\) 0 0
\(549\) 6.04832e8 0.156002
\(550\) 0 0
\(551\) −1.17754e9 −0.299879
\(552\) 0 0
\(553\) 4.25632e9 1.07028
\(554\) 0 0
\(555\) −5.08786e9 −1.26331
\(556\) 0 0
\(557\) 3.95293e9 0.969229 0.484614 0.874728i \(-0.338960\pi\)
0.484614 + 0.874728i \(0.338960\pi\)
\(558\) 0 0
\(559\) 1.94371e9 0.470643
\(560\) 0 0
\(561\) 3.91491e9 0.936163
\(562\) 0 0
\(563\) 1.52515e9 0.360191 0.180096 0.983649i \(-0.442359\pi\)
0.180096 + 0.983649i \(0.442359\pi\)
\(564\) 0 0
\(565\) −9.31740e9 −2.17333
\(566\) 0 0
\(567\) −2.16371e9 −0.498492
\(568\) 0 0
\(569\) 5.48747e9 1.24876 0.624381 0.781120i \(-0.285351\pi\)
0.624381 + 0.781120i \(0.285351\pi\)
\(570\) 0 0
\(571\) 7.93275e9 1.78319 0.891595 0.452834i \(-0.149587\pi\)
0.891595 + 0.452834i \(0.149587\pi\)
\(572\) 0 0
\(573\) 1.39291e9 0.309302
\(574\) 0 0
\(575\) 7.53942e8 0.165387
\(576\) 0 0
\(577\) −3.64252e9 −0.789380 −0.394690 0.918814i \(-0.629148\pi\)
−0.394690 + 0.918814i \(0.629148\pi\)
\(578\) 0 0
\(579\) −8.07409e8 −0.172870
\(580\) 0 0
\(581\) −5.70860e9 −1.20757
\(582\) 0 0
\(583\) 1.00626e10 2.10315
\(584\) 0 0
\(585\) 9.02903e9 1.86464
\(586\) 0 0
\(587\) −4.40965e9 −0.899850 −0.449925 0.893066i \(-0.648549\pi\)
−0.449925 + 0.893066i \(0.648549\pi\)
\(588\) 0 0
\(589\) −7.34313e8 −0.148074
\(590\) 0 0
\(591\) 8.18250e8 0.163053
\(592\) 0 0
\(593\) −5.72019e9 −1.12647 −0.563235 0.826297i \(-0.690443\pi\)
−0.563235 + 0.826297i \(0.690443\pi\)
\(594\) 0 0
\(595\) 1.40658e10 2.73750
\(596\) 0 0
\(597\) −3.01874e9 −0.580651
\(598\) 0 0
\(599\) 9.46158e9 1.79875 0.899373 0.437182i \(-0.144023\pi\)
0.899373 + 0.437182i \(0.144023\pi\)
\(600\) 0 0
\(601\) −2.40265e9 −0.451470 −0.225735 0.974189i \(-0.572478\pi\)
−0.225735 + 0.974189i \(0.572478\pi\)
\(602\) 0 0
\(603\) −3.65725e9 −0.679273
\(604\) 0 0
\(605\) −1.33037e10 −2.44246
\(606\) 0 0
\(607\) −7.03151e9 −1.27611 −0.638055 0.769991i \(-0.720261\pi\)
−0.638055 + 0.769991i \(0.720261\pi\)
\(608\) 0 0
\(609\) 3.58864e9 0.643827
\(610\) 0 0
\(611\) −4.14983e9 −0.736015
\(612\) 0 0
\(613\) −3.30413e9 −0.579356 −0.289678 0.957124i \(-0.593548\pi\)
−0.289678 + 0.957124i \(0.593548\pi\)
\(614\) 0 0
\(615\) −2.69552e8 −0.0467283
\(616\) 0 0
\(617\) 1.23310e9 0.211349 0.105674 0.994401i \(-0.466300\pi\)
0.105674 + 0.994401i \(0.466300\pi\)
\(618\) 0 0
\(619\) 9.23102e9 1.56434 0.782172 0.623062i \(-0.214112\pi\)
0.782172 + 0.623062i \(0.214112\pi\)
\(620\) 0 0
\(621\) −3.55837e8 −0.0596253
\(622\) 0 0
\(623\) −1.05533e10 −1.74856
\(624\) 0 0
\(625\) 1.03870e10 1.70180
\(626\) 0 0
\(627\) 9.64893e8 0.156330
\(628\) 0 0
\(629\) 1.36175e10 2.18183
\(630\) 0 0
\(631\) 7.58497e9 1.20185 0.600926 0.799304i \(-0.294799\pi\)
0.600926 + 0.799304i \(0.294799\pi\)
\(632\) 0 0
\(633\) −3.79075e9 −0.594035
\(634\) 0 0
\(635\) 1.53005e10 2.37136
\(636\) 0 0
\(637\) 1.97348e9 0.302514
\(638\) 0 0
\(639\) −3.65779e9 −0.554582
\(640\) 0 0
\(641\) −1.00750e10 −1.51092 −0.755459 0.655196i \(-0.772586\pi\)
−0.755459 + 0.655196i \(0.772586\pi\)
\(642\) 0 0
\(643\) 3.40961e9 0.505785 0.252892 0.967494i \(-0.418618\pi\)
0.252892 + 0.967494i \(0.418618\pi\)
\(644\) 0 0
\(645\) −1.97193e9 −0.289356
\(646\) 0 0
\(647\) 5.78057e9 0.839085 0.419542 0.907736i \(-0.362191\pi\)
0.419542 + 0.907736i \(0.362191\pi\)
\(648\) 0 0
\(649\) 5.66322e9 0.813218
\(650\) 0 0
\(651\) 2.23787e9 0.317907
\(652\) 0 0
\(653\) 6.34746e9 0.892080 0.446040 0.895013i \(-0.352834\pi\)
0.446040 + 0.895013i \(0.352834\pi\)
\(654\) 0 0
\(655\) −2.14798e10 −2.98665
\(656\) 0 0
\(657\) 3.50002e9 0.481495
\(658\) 0 0
\(659\) 2.10542e9 0.286576 0.143288 0.989681i \(-0.454232\pi\)
0.143288 + 0.989681i \(0.454232\pi\)
\(660\) 0 0
\(661\) −3.50157e9 −0.471583 −0.235791 0.971804i \(-0.575768\pi\)
−0.235791 + 0.971804i \(0.575768\pi\)
\(662\) 0 0
\(663\) 5.91468e9 0.788195
\(664\) 0 0
\(665\) 3.46674e9 0.457137
\(666\) 0 0
\(667\) −7.46940e8 −0.0974642
\(668\) 0 0
\(669\) −8.39282e8 −0.108372
\(670\) 0 0
\(671\) −2.33529e9 −0.298409
\(672\) 0 0
\(673\) −3.06704e9 −0.387852 −0.193926 0.981016i \(-0.562122\pi\)
−0.193926 + 0.981016i \(0.562122\pi\)
\(674\) 0 0
\(675\) −1.41726e10 −1.77372
\(676\) 0 0
\(677\) −1.07375e10 −1.32997 −0.664984 0.746858i \(-0.731562\pi\)
−0.664984 + 0.746858i \(0.731562\pi\)
\(678\) 0 0
\(679\) 1.47007e10 1.80216
\(680\) 0 0
\(681\) −2.85427e9 −0.346322
\(682\) 0 0
\(683\) 1.34679e10 1.61743 0.808716 0.588200i \(-0.200163\pi\)
0.808716 + 0.588200i \(0.200163\pi\)
\(684\) 0 0
\(685\) 1.84737e10 2.19602
\(686\) 0 0
\(687\) −4.79195e9 −0.563850
\(688\) 0 0
\(689\) 1.52027e10 1.77073
\(690\) 0 0
\(691\) 1.06634e10 1.22948 0.614742 0.788728i \(-0.289260\pi\)
0.614742 + 0.788728i \(0.289260\pi\)
\(692\) 0 0
\(693\) 1.20145e10 1.37132
\(694\) 0 0
\(695\) −2.59862e10 −2.93627
\(696\) 0 0
\(697\) 7.21450e8 0.0807033
\(698\) 0 0
\(699\) −9.07217e8 −0.100471
\(700\) 0 0
\(701\) −1.21534e10 −1.33255 −0.666275 0.745707i \(-0.732112\pi\)
−0.666275 + 0.745707i \(0.732112\pi\)
\(702\) 0 0
\(703\) 3.35626e9 0.364344
\(704\) 0 0
\(705\) 4.21007e9 0.452509
\(706\) 0 0
\(707\) −3.09394e8 −0.0329264
\(708\) 0 0
\(709\) 7.57486e9 0.798202 0.399101 0.916907i \(-0.369322\pi\)
0.399101 + 0.916907i \(0.369322\pi\)
\(710\) 0 0
\(711\) 7.41877e9 0.774085
\(712\) 0 0
\(713\) −4.65790e8 −0.0481257
\(714\) 0 0
\(715\) −3.48616e10 −3.56678
\(716\) 0 0
\(717\) 7.33169e9 0.742826
\(718\) 0 0
\(719\) −8.12422e9 −0.815137 −0.407569 0.913175i \(-0.633623\pi\)
−0.407569 + 0.913175i \(0.633623\pi\)
\(720\) 0 0
\(721\) −1.58767e10 −1.57757
\(722\) 0 0
\(723\) 4.19708e9 0.413013
\(724\) 0 0
\(725\) −2.97497e10 −2.89934
\(726\) 0 0
\(727\) 1.10999e10 1.07140 0.535698 0.844410i \(-0.320049\pi\)
0.535698 + 0.844410i \(0.320049\pi\)
\(728\) 0 0
\(729\) −6.21877e7 −0.00594508
\(730\) 0 0
\(731\) 5.27781e9 0.499739
\(732\) 0 0
\(733\) −1.18891e10 −1.11502 −0.557511 0.830170i \(-0.688243\pi\)
−0.557511 + 0.830170i \(0.688243\pi\)
\(734\) 0 0
\(735\) −2.00213e9 −0.185988
\(736\) 0 0
\(737\) 1.41208e10 1.29935
\(738\) 0 0
\(739\) 5.97595e9 0.544693 0.272346 0.962199i \(-0.412200\pi\)
0.272346 + 0.962199i \(0.412200\pi\)
\(740\) 0 0
\(741\) 1.45777e9 0.131621
\(742\) 0 0
\(743\) −5.33399e8 −0.0477080 −0.0238540 0.999715i \(-0.507594\pi\)
−0.0238540 + 0.999715i \(0.507594\pi\)
\(744\) 0 0
\(745\) 2.18654e10 1.93736
\(746\) 0 0
\(747\) −9.95010e9 −0.873384
\(748\) 0 0
\(749\) −1.35589e10 −1.17906
\(750\) 0 0
\(751\) −4.50180e9 −0.387834 −0.193917 0.981018i \(-0.562119\pi\)
−0.193917 + 0.981018i \(0.562119\pi\)
\(752\) 0 0
\(753\) −1.10628e9 −0.0944244
\(754\) 0 0
\(755\) −2.13907e10 −1.80888
\(756\) 0 0
\(757\) −1.89821e10 −1.59041 −0.795204 0.606342i \(-0.792636\pi\)
−0.795204 + 0.606342i \(0.792636\pi\)
\(758\) 0 0
\(759\) 6.12052e8 0.0508092
\(760\) 0 0
\(761\) −8.94728e9 −0.735944 −0.367972 0.929837i \(-0.619948\pi\)
−0.367972 + 0.929837i \(0.619948\pi\)
\(762\) 0 0
\(763\) −1.40129e10 −1.14207
\(764\) 0 0
\(765\) 2.45167e10 1.97992
\(766\) 0 0
\(767\) 8.55605e9 0.684683
\(768\) 0 0
\(769\) 1.75298e10 1.39006 0.695031 0.718979i \(-0.255390\pi\)
0.695031 + 0.718979i \(0.255390\pi\)
\(770\) 0 0
\(771\) 7.35515e9 0.577965
\(772\) 0 0
\(773\) 1.24048e10 0.965963 0.482982 0.875631i \(-0.339554\pi\)
0.482982 + 0.875631i \(0.339554\pi\)
\(774\) 0 0
\(775\) −1.85519e10 −1.43163
\(776\) 0 0
\(777\) −1.02284e10 −0.782231
\(778\) 0 0
\(779\) 1.77813e8 0.0134767
\(780\) 0 0
\(781\) 1.41229e10 1.06083
\(782\) 0 0
\(783\) 1.40409e10 1.04527
\(784\) 0 0
\(785\) −3.88666e9 −0.286769
\(786\) 0 0
\(787\) −3.69738e9 −0.270385 −0.135193 0.990819i \(-0.543165\pi\)
−0.135193 + 0.990819i \(0.543165\pi\)
\(788\) 0 0
\(789\) 9.81879e9 0.711686
\(790\) 0 0
\(791\) −1.87313e10 −1.34571
\(792\) 0 0
\(793\) −3.52818e9 −0.251243
\(794\) 0 0
\(795\) −1.54234e10 −1.08866
\(796\) 0 0
\(797\) −1.67626e10 −1.17284 −0.586419 0.810008i \(-0.699463\pi\)
−0.586419 + 0.810008i \(0.699463\pi\)
\(798\) 0 0
\(799\) −1.12681e10 −0.781517
\(800\) 0 0
\(801\) −1.83944e10 −1.26466
\(802\) 0 0
\(803\) −1.35138e10 −0.921027
\(804\) 0 0
\(805\) 2.19903e9 0.148575
\(806\) 0 0
\(807\) 1.94586e9 0.130333
\(808\) 0 0
\(809\) 8.30672e9 0.551582 0.275791 0.961218i \(-0.411060\pi\)
0.275791 + 0.961218i \(0.411060\pi\)
\(810\) 0 0
\(811\) 1.35615e10 0.892757 0.446378 0.894844i \(-0.352714\pi\)
0.446378 + 0.894844i \(0.352714\pi\)
\(812\) 0 0
\(813\) 6.94300e9 0.453138
\(814\) 0 0
\(815\) 3.57650e10 2.31423
\(816\) 0 0
\(817\) 1.30080e9 0.0834516
\(818\) 0 0
\(819\) 1.81516e10 1.15457
\(820\) 0 0
\(821\) 4.43086e9 0.279439 0.139719 0.990191i \(-0.455380\pi\)
0.139719 + 0.990191i \(0.455380\pi\)
\(822\) 0 0
\(823\) 1.75941e10 1.10019 0.550095 0.835102i \(-0.314591\pi\)
0.550095 + 0.835102i \(0.314591\pi\)
\(824\) 0 0
\(825\) 2.43773e10 1.51146
\(826\) 0 0
\(827\) −9.61056e9 −0.590853 −0.295427 0.955365i \(-0.595462\pi\)
−0.295427 + 0.955365i \(0.595462\pi\)
\(828\) 0 0
\(829\) −2.63344e10 −1.60540 −0.802699 0.596384i \(-0.796604\pi\)
−0.802699 + 0.596384i \(0.796604\pi\)
\(830\) 0 0
\(831\) −2.48282e9 −0.150086
\(832\) 0 0
\(833\) 5.35864e9 0.321216
\(834\) 0 0
\(835\) −3.92205e10 −2.33137
\(836\) 0 0
\(837\) 8.75590e9 0.516133
\(838\) 0 0
\(839\) −8.52726e9 −0.498474 −0.249237 0.968443i \(-0.580180\pi\)
−0.249237 + 0.968443i \(0.580180\pi\)
\(840\) 0 0
\(841\) 1.22236e10 0.708618
\(842\) 0 0
\(843\) 5.83388e9 0.335398
\(844\) 0 0
\(845\) −2.12064e10 −1.20912
\(846\) 0 0
\(847\) −2.67452e10 −1.51236
\(848\) 0 0
\(849\) 3.06795e8 0.0172057
\(850\) 0 0
\(851\) 2.12895e9 0.118416
\(852\) 0 0
\(853\) −1.87090e10 −1.03212 −0.516058 0.856554i \(-0.672601\pi\)
−0.516058 + 0.856554i \(0.672601\pi\)
\(854\) 0 0
\(855\) 6.04255e9 0.330627
\(856\) 0 0
\(857\) −1.07860e10 −0.585367 −0.292684 0.956209i \(-0.594548\pi\)
−0.292684 + 0.956209i \(0.594548\pi\)
\(858\) 0 0
\(859\) −5.42048e9 −0.291784 −0.145892 0.989301i \(-0.546605\pi\)
−0.145892 + 0.989301i \(0.546605\pi\)
\(860\) 0 0
\(861\) −5.41897e8 −0.0289338
\(862\) 0 0
\(863\) 6.53103e8 0.0345895 0.0172947 0.999850i \(-0.494495\pi\)
0.0172947 + 0.999850i \(0.494495\pi\)
\(864\) 0 0
\(865\) 2.75268e10 1.44611
\(866\) 0 0
\(867\) 7.55106e9 0.393496
\(868\) 0 0
\(869\) −2.86443e10 −1.48071
\(870\) 0 0
\(871\) 2.13339e10 1.09397
\(872\) 0 0
\(873\) 2.56234e10 1.30343
\(874\) 0 0
\(875\) 4.80980e10 2.42716
\(876\) 0 0
\(877\) −1.04507e10 −0.523174 −0.261587 0.965180i \(-0.584246\pi\)
−0.261587 + 0.965180i \(0.584246\pi\)
\(878\) 0 0
\(879\) 1.11044e10 0.551485
\(880\) 0 0
\(881\) −8.20111e8 −0.0404070 −0.0202035 0.999796i \(-0.506431\pi\)
−0.0202035 + 0.999796i \(0.506431\pi\)
\(882\) 0 0
\(883\) −2.27244e10 −1.11079 −0.555393 0.831588i \(-0.687432\pi\)
−0.555393 + 0.831588i \(0.687432\pi\)
\(884\) 0 0
\(885\) −8.68024e9 −0.420950
\(886\) 0 0
\(887\) −6.09428e9 −0.293217 −0.146609 0.989195i \(-0.546836\pi\)
−0.146609 + 0.989195i \(0.546836\pi\)
\(888\) 0 0
\(889\) 3.07596e10 1.46833
\(890\) 0 0
\(891\) 1.45614e10 0.689654
\(892\) 0 0
\(893\) −2.77722e9 −0.130506
\(894\) 0 0
\(895\) 3.74899e10 1.74797
\(896\) 0 0
\(897\) 9.24695e8 0.0427784
\(898\) 0 0
\(899\) 1.83796e10 0.843678
\(900\) 0 0
\(901\) 4.12802e10 1.88021
\(902\) 0 0
\(903\) −3.96428e9 −0.179167
\(904\) 0 0
\(905\) 2.80031e10 1.25584
\(906\) 0 0
\(907\) 4.41819e9 0.196616 0.0983080 0.995156i \(-0.468657\pi\)
0.0983080 + 0.995156i \(0.468657\pi\)
\(908\) 0 0
\(909\) −5.39276e8 −0.0238143
\(910\) 0 0
\(911\) −9.45414e8 −0.0414293 −0.0207146 0.999785i \(-0.506594\pi\)
−0.0207146 + 0.999785i \(0.506594\pi\)
\(912\) 0 0
\(913\) 3.84179e10 1.67065
\(914\) 0 0
\(915\) 3.57939e9 0.154467
\(916\) 0 0
\(917\) −4.31821e10 −1.84931
\(918\) 0 0
\(919\) −1.00356e10 −0.426520 −0.213260 0.976995i \(-0.568408\pi\)
−0.213260 + 0.976995i \(0.568408\pi\)
\(920\) 0 0
\(921\) −1.08528e10 −0.457754
\(922\) 0 0
\(923\) 2.13371e10 0.893159
\(924\) 0 0
\(925\) 8.47934e10 3.52262
\(926\) 0 0
\(927\) −2.76732e10 −1.14099
\(928\) 0 0
\(929\) 3.51917e10 1.44008 0.720038 0.693935i \(-0.244125\pi\)
0.720038 + 0.693935i \(0.244125\pi\)
\(930\) 0 0
\(931\) 1.32073e9 0.0536400
\(932\) 0 0
\(933\) −1.45465e10 −0.586371
\(934\) 0 0
\(935\) −9.46604e10 −3.78728
\(936\) 0 0
\(937\) 1.89212e10 0.751381 0.375690 0.926745i \(-0.377406\pi\)
0.375690 + 0.926745i \(0.377406\pi\)
\(938\) 0 0
\(939\) −3.09264e9 −0.121899
\(940\) 0 0
\(941\) 1.63328e10 0.638995 0.319497 0.947587i \(-0.396486\pi\)
0.319497 + 0.947587i \(0.396486\pi\)
\(942\) 0 0
\(943\) 1.12791e8 0.00438008
\(944\) 0 0
\(945\) −4.13373e10 −1.59342
\(946\) 0 0
\(947\) −3.04581e10 −1.16541 −0.582704 0.812684i \(-0.698005\pi\)
−0.582704 + 0.812684i \(0.698005\pi\)
\(948\) 0 0
\(949\) −2.04168e10 −0.775452
\(950\) 0 0
\(951\) 9.58898e9 0.361527
\(952\) 0 0
\(953\) −2.31953e9 −0.0868110 −0.0434055 0.999058i \(-0.513821\pi\)
−0.0434055 + 0.999058i \(0.513821\pi\)
\(954\) 0 0
\(955\) −3.36799e10 −1.25129
\(956\) 0 0
\(957\) −2.41509e10 −0.890722
\(958\) 0 0
\(959\) 3.71387e10 1.35976
\(960\) 0 0
\(961\) −1.60511e10 −0.583410
\(962\) 0 0
\(963\) −2.36332e10 −0.852767
\(964\) 0 0
\(965\) 1.95227e10 0.699351
\(966\) 0 0
\(967\) −1.94483e10 −0.691653 −0.345827 0.938298i \(-0.612401\pi\)
−0.345827 + 0.938298i \(0.612401\pi\)
\(968\) 0 0
\(969\) 3.95831e9 0.139758
\(970\) 0 0
\(971\) −4.73991e10 −1.66151 −0.830755 0.556638i \(-0.812091\pi\)
−0.830755 + 0.556638i \(0.812091\pi\)
\(972\) 0 0
\(973\) −5.22417e10 −1.81812
\(974\) 0 0
\(975\) 3.68295e10 1.27256
\(976\) 0 0
\(977\) −2.15960e10 −0.740871 −0.370436 0.928858i \(-0.620792\pi\)
−0.370436 + 0.928858i \(0.620792\pi\)
\(978\) 0 0
\(979\) 7.10219e10 2.41910
\(980\) 0 0
\(981\) −2.44245e10 −0.826009
\(982\) 0 0
\(983\) −3.36016e10 −1.12829 −0.564147 0.825674i \(-0.690795\pi\)
−0.564147 + 0.825674i \(0.690795\pi\)
\(984\) 0 0
\(985\) −1.97849e10 −0.659639
\(986\) 0 0
\(987\) 8.46376e9 0.280190
\(988\) 0 0
\(989\) 8.25128e8 0.0271228
\(990\) 0 0
\(991\) 3.87751e10 1.26560 0.632799 0.774316i \(-0.281906\pi\)
0.632799 + 0.774316i \(0.281906\pi\)
\(992\) 0 0
\(993\) 7.85852e9 0.254694
\(994\) 0 0
\(995\) 7.29915e10 2.34905
\(996\) 0 0
\(997\) −6.57638e9 −0.210162 −0.105081 0.994464i \(-0.533510\pi\)
−0.105081 + 0.994464i \(0.533510\pi\)
\(998\) 0 0
\(999\) −4.00199e10 −1.26998
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 76.8.a.b.1.4 6
4.3 odd 2 304.8.a.i.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.a.b.1.4 6 1.1 even 1 trivial
304.8.a.i.1.3 6 4.3 odd 2