Properties

Label 560.8.a.i.1.1
Level $560$
Weight $8$
Character 560.1
Self dual yes
Analytic conductor $174.936$
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] = [560,8,Mod(1,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("560.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 560.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: \(174.935614271\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31662\) of defining polynomial
Character \(\chi\) \(=\) 560.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-24.7995 q^{3} +125.000 q^{5} +343.000 q^{7} -1571.98 q^{9} +O(q^{10})\) \(q-24.7995 q^{3} +125.000 q^{5} +343.000 q^{7} -1571.98 q^{9} +1432.37 q^{11} -6136.30 q^{13} -3099.94 q^{15} -15858.5 q^{17} +38567.5 q^{19} -8506.23 q^{21} +63987.4 q^{23} +15625.0 q^{25} +93220.9 q^{27} +94236.6 q^{29} -275990. q^{31} -35521.9 q^{33} +42875.0 q^{35} +156532. q^{37} +152177. q^{39} -303738. q^{41} -636818. q^{43} -196498. q^{45} -512021. q^{47} +117649. q^{49} +393282. q^{51} -201249. q^{53} +179046. q^{55} -956454. q^{57} +1.81196e6 q^{59} -982021. q^{61} -539191. q^{63} -767038. q^{65} +4.45336e6 q^{67} -1.58686e6 q^{69} -725436. q^{71} +2.17602e6 q^{73} -387492. q^{75} +491301. q^{77} +5.21525e6 q^{79} +1.12610e6 q^{81} -6.07921e6 q^{83} -1.98231e6 q^{85} -2.33702e6 q^{87} -1.06137e7 q^{89} -2.10475e6 q^{91} +6.84442e6 q^{93} +4.82093e6 q^{95} +6.64483e6 q^{97} -2.25166e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 30 q^{3} + 250 q^{5} + 686 q^{7} - 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 30 q^{3} + 250 q^{5} + 686 q^{7} - 756 q^{9} + 7906 q^{11} - 17818 q^{13} + 3750 q^{15} - 2398 q^{17} + 3612 q^{19} + 10290 q^{21} - 13844 q^{23} + 31250 q^{25} + 18090 q^{27} - 126898 q^{29} - 252768 q^{31} + 319230 q^{33} + 85750 q^{35} - 265860 q^{37} - 487974 q^{39} - 111920 q^{41} - 947572 q^{43} - 94500 q^{45} - 271274 q^{47} + 235298 q^{49} + 1130910 q^{51} - 1267792 q^{53} + 988250 q^{55} - 2871996 q^{57} + 1360120 q^{59} - 1813680 q^{61} - 259308 q^{63} - 2227250 q^{65} + 2189312 q^{67} - 5851980 q^{69} + 1494928 q^{71} + 7169788 q^{73} + 468750 q^{75} + 2711758 q^{77} + 7942974 q^{79} - 4775598 q^{81} + 304712 q^{83} - 299750 q^{85} - 14455086 q^{87} - 17943528 q^{89} - 6111574 q^{91} + 8116992 q^{93} + 451500 q^{95} + 4258074 q^{97} + 3030732 q^{99}+O(q^{100}) \) 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
\(3\) −24.7995 −0.530296 −0.265148 0.964208i \(-0.585421\pi\)
−0.265148 + 0.964208i \(0.585421\pi\)
\(4\) 0 0
\(5\) 125.000 0.447214
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) −1571.98 −0.718786
\(10\) 0 0
\(11\) 1432.37 0.324474 0.162237 0.986752i \(-0.448129\pi\)
0.162237 + 0.986752i \(0.448129\pi\)
\(12\) 0 0
\(13\) −6136.30 −0.774649 −0.387325 0.921943i \(-0.626601\pi\)
−0.387325 + 0.921943i \(0.626601\pi\)
\(14\) 0 0
\(15\) −3099.94 −0.237156
\(16\) 0 0
\(17\) −15858.5 −0.782871 −0.391436 0.920205i \(-0.628021\pi\)
−0.391436 + 0.920205i \(0.628021\pi\)
\(18\) 0 0
\(19\) 38567.5 1.28998 0.644991 0.764190i \(-0.276861\pi\)
0.644991 + 0.764190i \(0.276861\pi\)
\(20\) 0 0
\(21\) −8506.23 −0.200433
\(22\) 0 0
\(23\) 63987.4 1.09660 0.548299 0.836282i \(-0.315276\pi\)
0.548299 + 0.836282i \(0.315276\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) 93220.9 0.911466
\(28\) 0 0
\(29\) 94236.6 0.717508 0.358754 0.933432i \(-0.383202\pi\)
0.358754 + 0.933432i \(0.383202\pi\)
\(30\) 0 0
\(31\) −275990. −1.66390 −0.831951 0.554849i \(-0.812776\pi\)
−0.831951 + 0.554849i \(0.812776\pi\)
\(32\) 0 0
\(33\) −35521.9 −0.172067
\(34\) 0 0
\(35\) 42875.0 0.169031
\(36\) 0 0
\(37\) 156532. 0.508038 0.254019 0.967199i \(-0.418247\pi\)
0.254019 + 0.967199i \(0.418247\pi\)
\(38\) 0 0
\(39\) 152177. 0.410793
\(40\) 0 0
\(41\) −303738. −0.688266 −0.344133 0.938921i \(-0.611827\pi\)
−0.344133 + 0.938921i \(0.611827\pi\)
\(42\) 0 0
\(43\) −636818. −1.22145 −0.610725 0.791843i \(-0.709122\pi\)
−0.610725 + 0.791843i \(0.709122\pi\)
\(44\) 0 0
\(45\) −196498. −0.321451
\(46\) 0 0
\(47\) −512021. −0.719358 −0.359679 0.933076i \(-0.617114\pi\)
−0.359679 + 0.933076i \(0.617114\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 393282. 0.415154
\(52\) 0 0
\(53\) −201249. −0.185681 −0.0928406 0.995681i \(-0.529595\pi\)
−0.0928406 + 0.995681i \(0.529595\pi\)
\(54\) 0 0
\(55\) 179046. 0.145109
\(56\) 0 0
\(57\) −956454. −0.684072
\(58\) 0 0
\(59\) 1.81196e6 1.14859 0.574296 0.818648i \(-0.305276\pi\)
0.574296 + 0.818648i \(0.305276\pi\)
\(60\) 0 0
\(61\) −982021. −0.553945 −0.276972 0.960878i \(-0.589331\pi\)
−0.276972 + 0.960878i \(0.589331\pi\)
\(62\) 0 0
\(63\) −539191. −0.271676
\(64\) 0 0
\(65\) −767038. −0.346434
\(66\) 0 0
\(67\) 4.45336e6 1.80895 0.904474 0.426528i \(-0.140264\pi\)
0.904474 + 0.426528i \(0.140264\pi\)
\(68\) 0 0
\(69\) −1.58686e6 −0.581522
\(70\) 0 0
\(71\) −725436. −0.240544 −0.120272 0.992741i \(-0.538377\pi\)
−0.120272 + 0.992741i \(0.538377\pi\)
\(72\) 0 0
\(73\) 2.17602e6 0.654685 0.327343 0.944906i \(-0.393847\pi\)
0.327343 + 0.944906i \(0.393847\pi\)
\(74\) 0 0
\(75\) −387492. −0.106059
\(76\) 0 0
\(77\) 491301. 0.122639
\(78\) 0 0
\(79\) 5.21525e6 1.19009 0.595045 0.803692i \(-0.297134\pi\)
0.595045 + 0.803692i \(0.297134\pi\)
\(80\) 0 0
\(81\) 1.12610e6 0.235439
\(82\) 0 0
\(83\) −6.07921e6 −1.16701 −0.583504 0.812110i \(-0.698319\pi\)
−0.583504 + 0.812110i \(0.698319\pi\)
\(84\) 0 0
\(85\) −1.98231e6 −0.350111
\(86\) 0 0
\(87\) −2.33702e6 −0.380492
\(88\) 0 0
\(89\) −1.06137e7 −1.59589 −0.797946 0.602729i \(-0.794080\pi\)
−0.797946 + 0.602729i \(0.794080\pi\)
\(90\) 0 0
\(91\) −2.10475e6 −0.292790
\(92\) 0 0
\(93\) 6.84442e6 0.882361
\(94\) 0 0
\(95\) 4.82093e6 0.576897
\(96\) 0 0
\(97\) 6.64483e6 0.739236 0.369618 0.929184i \(-0.379489\pi\)
0.369618 + 0.929184i \(0.379489\pi\)
\(98\) 0 0
\(99\) −2.25166e6 −0.233227
\(100\) 0 0
\(101\) 1.07531e7 1.03851 0.519254 0.854620i \(-0.326210\pi\)
0.519254 + 0.854620i \(0.326210\pi\)
\(102\) 0 0
\(103\) 1.05886e7 0.954788 0.477394 0.878689i \(-0.341581\pi\)
0.477394 + 0.878689i \(0.341581\pi\)
\(104\) 0 0
\(105\) −1.06328e6 −0.0896364
\(106\) 0 0
\(107\) 8.37234e6 0.660699 0.330349 0.943859i \(-0.392833\pi\)
0.330349 + 0.943859i \(0.392833\pi\)
\(108\) 0 0
\(109\) −1.95948e7 −1.44926 −0.724632 0.689136i \(-0.757990\pi\)
−0.724632 + 0.689136i \(0.757990\pi\)
\(110\) 0 0
\(111\) −3.88191e6 −0.269411
\(112\) 0 0
\(113\) 1.36310e7 0.888694 0.444347 0.895855i \(-0.353436\pi\)
0.444347 + 0.895855i \(0.353436\pi\)
\(114\) 0 0
\(115\) 7.99843e6 0.490413
\(116\) 0 0
\(117\) 9.64617e6 0.556807
\(118\) 0 0
\(119\) −5.43946e6 −0.295898
\(120\) 0 0
\(121\) −1.74355e7 −0.894717
\(122\) 0 0
\(123\) 7.53256e6 0.364985
\(124\) 0 0
\(125\) 1.95312e6 0.0894427
\(126\) 0 0
\(127\) −2.23763e7 −0.969336 −0.484668 0.874698i \(-0.661060\pi\)
−0.484668 + 0.874698i \(0.661060\pi\)
\(128\) 0 0
\(129\) 1.57928e7 0.647730
\(130\) 0 0
\(131\) 4.53330e6 0.176183 0.0880917 0.996112i \(-0.471923\pi\)
0.0880917 + 0.996112i \(0.471923\pi\)
\(132\) 0 0
\(133\) 1.32286e7 0.487567
\(134\) 0 0
\(135\) 1.16526e7 0.407620
\(136\) 0 0
\(137\) −5.07657e7 −1.68674 −0.843371 0.537332i \(-0.819432\pi\)
−0.843371 + 0.537332i \(0.819432\pi\)
\(138\) 0 0
\(139\) −1.05183e7 −0.332195 −0.166097 0.986109i \(-0.553117\pi\)
−0.166097 + 0.986109i \(0.553117\pi\)
\(140\) 0 0
\(141\) 1.26979e7 0.381473
\(142\) 0 0
\(143\) −8.78942e6 −0.251353
\(144\) 0 0
\(145\) 1.17796e7 0.320879
\(146\) 0 0
\(147\) −2.91764e6 −0.0757566
\(148\) 0 0
\(149\) 5.43497e7 1.34600 0.673000 0.739642i \(-0.265005\pi\)
0.673000 + 0.739642i \(0.265005\pi\)
\(150\) 0 0
\(151\) 2.23258e7 0.527700 0.263850 0.964564i \(-0.415008\pi\)
0.263850 + 0.964564i \(0.415008\pi\)
\(152\) 0 0
\(153\) 2.49293e7 0.562717
\(154\) 0 0
\(155\) −3.44988e7 −0.744120
\(156\) 0 0
\(157\) −4.37788e7 −0.902848 −0.451424 0.892310i \(-0.649084\pi\)
−0.451424 + 0.892310i \(0.649084\pi\)
\(158\) 0 0
\(159\) 4.99087e6 0.0984661
\(160\) 0 0
\(161\) 2.19477e7 0.414475
\(162\) 0 0
\(163\) −4.05451e7 −0.733300 −0.366650 0.930359i \(-0.619495\pi\)
−0.366650 + 0.930359i \(0.619495\pi\)
\(164\) 0 0
\(165\) −4.44024e6 −0.0769507
\(166\) 0 0
\(167\) −9.73453e7 −1.61736 −0.808682 0.588247i \(-0.799818\pi\)
−0.808682 + 0.588247i \(0.799818\pi\)
\(168\) 0 0
\(169\) −2.50943e7 −0.399919
\(170\) 0 0
\(171\) −6.06275e7 −0.927221
\(172\) 0 0
\(173\) −5.10607e7 −0.749765 −0.374882 0.927072i \(-0.622317\pi\)
−0.374882 + 0.927072i \(0.622317\pi\)
\(174\) 0 0
\(175\) 5.35938e6 0.0755929
\(176\) 0 0
\(177\) −4.49356e7 −0.609094
\(178\) 0 0
\(179\) −1.45811e8 −1.90023 −0.950113 0.311907i \(-0.899032\pi\)
−0.950113 + 0.311907i \(0.899032\pi\)
\(180\) 0 0
\(181\) −6.09656e7 −0.764205 −0.382102 0.924120i \(-0.624800\pi\)
−0.382102 + 0.924120i \(0.624800\pi\)
\(182\) 0 0
\(183\) 2.43536e7 0.293755
\(184\) 0 0
\(185\) 1.95665e7 0.227202
\(186\) 0 0
\(187\) −2.27151e7 −0.254021
\(188\) 0 0
\(189\) 3.19748e7 0.344502
\(190\) 0 0
\(191\) 1.52578e8 1.58444 0.792219 0.610237i \(-0.208926\pi\)
0.792219 + 0.610237i \(0.208926\pi\)
\(192\) 0 0
\(193\) −1.39277e8 −1.39453 −0.697267 0.716812i \(-0.745601\pi\)
−0.697267 + 0.716812i \(0.745601\pi\)
\(194\) 0 0
\(195\) 1.90221e7 0.183712
\(196\) 0 0
\(197\) 6.52480e7 0.608044 0.304022 0.952665i \(-0.401670\pi\)
0.304022 + 0.952665i \(0.401670\pi\)
\(198\) 0 0
\(199\) −1.93503e6 −0.0174061 −0.00870307 0.999962i \(-0.502770\pi\)
−0.00870307 + 0.999962i \(0.502770\pi\)
\(200\) 0 0
\(201\) −1.10441e8 −0.959278
\(202\) 0 0
\(203\) 3.23232e7 0.271192
\(204\) 0 0
\(205\) −3.79673e7 −0.307802
\(206\) 0 0
\(207\) −1.00587e8 −0.788219
\(208\) 0 0
\(209\) 5.52427e7 0.418565
\(210\) 0 0
\(211\) 5.17848e7 0.379502 0.189751 0.981832i \(-0.439232\pi\)
0.189751 + 0.981832i \(0.439232\pi\)
\(212\) 0 0
\(213\) 1.79904e7 0.127560
\(214\) 0 0
\(215\) −7.96023e7 −0.546249
\(216\) 0 0
\(217\) −9.46647e7 −0.628896
\(218\) 0 0
\(219\) −5.39642e7 −0.347177
\(220\) 0 0
\(221\) 9.73124e7 0.606450
\(222\) 0 0
\(223\) 1.25065e8 0.755209 0.377605 0.925967i \(-0.376748\pi\)
0.377605 + 0.925967i \(0.376748\pi\)
\(224\) 0 0
\(225\) −2.45623e7 −0.143757
\(226\) 0 0
\(227\) −1.92108e7 −0.109007 −0.0545036 0.998514i \(-0.517358\pi\)
−0.0545036 + 0.998514i \(0.517358\pi\)
\(228\) 0 0
\(229\) −1.05650e8 −0.581360 −0.290680 0.956820i \(-0.593882\pi\)
−0.290680 + 0.956820i \(0.593882\pi\)
\(230\) 0 0
\(231\) −1.21840e7 −0.0650353
\(232\) 0 0
\(233\) −2.31646e8 −1.19972 −0.599859 0.800106i \(-0.704776\pi\)
−0.599859 + 0.800106i \(0.704776\pi\)
\(234\) 0 0
\(235\) −6.40026e7 −0.321707
\(236\) 0 0
\(237\) −1.29336e8 −0.631101
\(238\) 0 0
\(239\) −1.09174e8 −0.517281 −0.258641 0.965974i \(-0.583275\pi\)
−0.258641 + 0.965974i \(0.583275\pi\)
\(240\) 0 0
\(241\) −8.25277e7 −0.379787 −0.189893 0.981805i \(-0.560814\pi\)
−0.189893 + 0.981805i \(0.560814\pi\)
\(242\) 0 0
\(243\) −2.31801e8 −1.03632
\(244\) 0 0
\(245\) 1.47061e7 0.0638877
\(246\) 0 0
\(247\) −2.36662e8 −0.999283
\(248\) 0 0
\(249\) 1.50761e8 0.618860
\(250\) 0 0
\(251\) 2.40987e7 0.0961912 0.0480956 0.998843i \(-0.484685\pi\)
0.0480956 + 0.998843i \(0.484685\pi\)
\(252\) 0 0
\(253\) 9.16534e7 0.355817
\(254\) 0 0
\(255\) 4.91603e7 0.185662
\(256\) 0 0
\(257\) 9.75049e7 0.358311 0.179156 0.983821i \(-0.442663\pi\)
0.179156 + 0.983821i \(0.442663\pi\)
\(258\) 0 0
\(259\) 5.36904e7 0.192020
\(260\) 0 0
\(261\) −1.48139e8 −0.515735
\(262\) 0 0
\(263\) −2.98637e8 −1.01228 −0.506138 0.862452i \(-0.668927\pi\)
−0.506138 + 0.862452i \(0.668927\pi\)
\(264\) 0 0
\(265\) −2.51561e7 −0.0830392
\(266\) 0 0
\(267\) 2.63216e8 0.846296
\(268\) 0 0
\(269\) −3.90722e8 −1.22387 −0.611934 0.790909i \(-0.709608\pi\)
−0.611934 + 0.790909i \(0.709608\pi\)
\(270\) 0 0
\(271\) −2.12098e8 −0.647357 −0.323678 0.946167i \(-0.604920\pi\)
−0.323678 + 0.946167i \(0.604920\pi\)
\(272\) 0 0
\(273\) 5.21968e7 0.155265
\(274\) 0 0
\(275\) 2.23807e7 0.0648947
\(276\) 0 0
\(277\) 1.86723e8 0.527861 0.263930 0.964542i \(-0.414981\pi\)
0.263930 + 0.964542i \(0.414981\pi\)
\(278\) 0 0
\(279\) 4.33853e8 1.19599
\(280\) 0 0
\(281\) −7.38791e8 −1.98632 −0.993161 0.116756i \(-0.962750\pi\)
−0.993161 + 0.116756i \(0.962750\pi\)
\(282\) 0 0
\(283\) 3.11903e8 0.818026 0.409013 0.912529i \(-0.365873\pi\)
0.409013 + 0.912529i \(0.365873\pi\)
\(284\) 0 0
\(285\) −1.19557e8 −0.305926
\(286\) 0 0
\(287\) −1.04182e8 −0.260140
\(288\) 0 0
\(289\) −1.58847e8 −0.387113
\(290\) 0 0
\(291\) −1.64789e8 −0.392014
\(292\) 0 0
\(293\) −5.05466e8 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(294\) 0 0
\(295\) 2.26495e8 0.513666
\(296\) 0 0
\(297\) 1.33526e8 0.295747
\(298\) 0 0
\(299\) −3.92646e8 −0.849478
\(300\) 0 0
\(301\) −2.18429e8 −0.461665
\(302\) 0 0
\(303\) −2.66672e8 −0.550717
\(304\) 0 0
\(305\) −1.22753e8 −0.247732
\(306\) 0 0
\(307\) −4.67463e8 −0.922067 −0.461034 0.887383i \(-0.652521\pi\)
−0.461034 + 0.887383i \(0.652521\pi\)
\(308\) 0 0
\(309\) −2.62591e8 −0.506321
\(310\) 0 0
\(311\) −1.16022e7 −0.0218714 −0.0109357 0.999940i \(-0.503481\pi\)
−0.0109357 + 0.999940i \(0.503481\pi\)
\(312\) 0 0
\(313\) 8.23197e8 1.51740 0.758698 0.651443i \(-0.225836\pi\)
0.758698 + 0.651443i \(0.225836\pi\)
\(314\) 0 0
\(315\) −6.73989e7 −0.121497
\(316\) 0 0
\(317\) 3.89154e8 0.686142 0.343071 0.939309i \(-0.388533\pi\)
0.343071 + 0.939309i \(0.388533\pi\)
\(318\) 0 0
\(319\) 1.34981e8 0.232812
\(320\) 0 0
\(321\) −2.07630e8 −0.350366
\(322\) 0 0
\(323\) −6.11621e8 −1.00989
\(324\) 0 0
\(325\) −9.58797e7 −0.154930
\(326\) 0 0
\(327\) 4.85940e8 0.768539
\(328\) 0 0
\(329\) −1.75623e8 −0.271892
\(330\) 0 0
\(331\) −1.48582e8 −0.225199 −0.112600 0.993640i \(-0.535918\pi\)
−0.112600 + 0.993640i \(0.535918\pi\)
\(332\) 0 0
\(333\) −2.46066e8 −0.365171
\(334\) 0 0
\(335\) 5.56670e8 0.808986
\(336\) 0 0
\(337\) 1.23379e8 0.175605 0.0878023 0.996138i \(-0.472016\pi\)
0.0878023 + 0.996138i \(0.472016\pi\)
\(338\) 0 0
\(339\) −3.38041e8 −0.471271
\(340\) 0 0
\(341\) −3.95319e8 −0.539892
\(342\) 0 0
\(343\) 4.03536e7 0.0539949
\(344\) 0 0
\(345\) −1.98357e8 −0.260064
\(346\) 0 0
\(347\) −1.31658e9 −1.69159 −0.845793 0.533511i \(-0.820872\pi\)
−0.845793 + 0.533511i \(0.820872\pi\)
\(348\) 0 0
\(349\) 2.64521e8 0.333097 0.166549 0.986033i \(-0.446738\pi\)
0.166549 + 0.986033i \(0.446738\pi\)
\(350\) 0 0
\(351\) −5.72032e8 −0.706066
\(352\) 0 0
\(353\) 1.30271e9 1.57629 0.788144 0.615490i \(-0.211042\pi\)
0.788144 + 0.615490i \(0.211042\pi\)
\(354\) 0 0
\(355\) −9.06795e7 −0.107575
\(356\) 0 0
\(357\) 1.34896e8 0.156913
\(358\) 0 0
\(359\) 1.03262e9 1.17790 0.588952 0.808168i \(-0.299541\pi\)
0.588952 + 0.808168i \(0.299541\pi\)
\(360\) 0 0
\(361\) 5.93578e8 0.664053
\(362\) 0 0
\(363\) 4.32392e8 0.474465
\(364\) 0 0
\(365\) 2.72002e8 0.292784
\(366\) 0 0
\(367\) −1.13124e9 −1.19460 −0.597302 0.802017i \(-0.703760\pi\)
−0.597302 + 0.802017i \(0.703760\pi\)
\(368\) 0 0
\(369\) 4.77472e8 0.494716
\(370\) 0 0
\(371\) −6.90284e7 −0.0701809
\(372\) 0 0
\(373\) 5.38130e8 0.536916 0.268458 0.963291i \(-0.413486\pi\)
0.268458 + 0.963291i \(0.413486\pi\)
\(374\) 0 0
\(375\) −4.84365e7 −0.0474311
\(376\) 0 0
\(377\) −5.78264e8 −0.555817
\(378\) 0 0
\(379\) −7.83114e8 −0.738904 −0.369452 0.929250i \(-0.620455\pi\)
−0.369452 + 0.929250i \(0.620455\pi\)
\(380\) 0 0
\(381\) 5.54920e8 0.514035
\(382\) 0 0
\(383\) −8.22468e8 −0.748038 −0.374019 0.927421i \(-0.622020\pi\)
−0.374019 + 0.927421i \(0.622020\pi\)
\(384\) 0 0
\(385\) 6.14127e7 0.0548460
\(386\) 0 0
\(387\) 1.00107e9 0.877961
\(388\) 0 0
\(389\) −1.07007e9 −0.921696 −0.460848 0.887479i \(-0.652455\pi\)
−0.460848 + 0.887479i \(0.652455\pi\)
\(390\) 0 0
\(391\) −1.01474e9 −0.858495
\(392\) 0 0
\(393\) −1.12424e8 −0.0934293
\(394\) 0 0
\(395\) 6.51906e8 0.532225
\(396\) 0 0
\(397\) −9.64552e8 −0.773676 −0.386838 0.922148i \(-0.626433\pi\)
−0.386838 + 0.922148i \(0.626433\pi\)
\(398\) 0 0
\(399\) −3.28064e8 −0.258555
\(400\) 0 0
\(401\) −1.94810e9 −1.50871 −0.754357 0.656465i \(-0.772051\pi\)
−0.754357 + 0.656465i \(0.772051\pi\)
\(402\) 0 0
\(403\) 1.69356e9 1.28894
\(404\) 0 0
\(405\) 1.40762e8 0.105292
\(406\) 0 0
\(407\) 2.24211e8 0.164845
\(408\) 0 0
\(409\) 8.63865e8 0.624330 0.312165 0.950028i \(-0.398946\pi\)
0.312165 + 0.950028i \(0.398946\pi\)
\(410\) 0 0
\(411\) 1.25896e9 0.894473
\(412\) 0 0
\(413\) 6.21501e8 0.434127
\(414\) 0 0
\(415\) −7.59901e8 −0.521902
\(416\) 0 0
\(417\) 2.60848e8 0.176162
\(418\) 0 0
\(419\) −2.21337e9 −1.46996 −0.734978 0.678091i \(-0.762808\pi\)
−0.734978 + 0.678091i \(0.762808\pi\)
\(420\) 0 0
\(421\) −2.89866e9 −1.89326 −0.946631 0.322321i \(-0.895537\pi\)
−0.946631 + 0.322321i \(0.895537\pi\)
\(422\) 0 0
\(423\) 8.04889e8 0.517065
\(424\) 0 0
\(425\) −2.47789e8 −0.156574
\(426\) 0 0
\(427\) −3.36833e8 −0.209371
\(428\) 0 0
\(429\) 2.17973e8 0.133292
\(430\) 0 0
\(431\) 2.42056e9 1.45628 0.728142 0.685426i \(-0.240384\pi\)
0.728142 + 0.685426i \(0.240384\pi\)
\(432\) 0 0
\(433\) −2.26686e9 −1.34189 −0.670946 0.741506i \(-0.734112\pi\)
−0.670946 + 0.741506i \(0.734112\pi\)
\(434\) 0 0
\(435\) −2.92128e8 −0.170161
\(436\) 0 0
\(437\) 2.46783e9 1.41459
\(438\) 0 0
\(439\) −1.98911e9 −1.12210 −0.561052 0.827780i \(-0.689603\pi\)
−0.561052 + 0.827780i \(0.689603\pi\)
\(440\) 0 0
\(441\) −1.84942e8 −0.102684
\(442\) 0 0
\(443\) 8.78038e8 0.479844 0.239922 0.970792i \(-0.422878\pi\)
0.239922 + 0.970792i \(0.422878\pi\)
\(444\) 0 0
\(445\) −1.32672e9 −0.713705
\(446\) 0 0
\(447\) −1.34785e9 −0.713779
\(448\) 0 0
\(449\) 1.53113e8 0.0798270 0.0399135 0.999203i \(-0.487292\pi\)
0.0399135 + 0.999203i \(0.487292\pi\)
\(450\) 0 0
\(451\) −4.35064e8 −0.223324
\(452\) 0 0
\(453\) −5.53668e8 −0.279837
\(454\) 0 0
\(455\) −2.63094e8 −0.130940
\(456\) 0 0
\(457\) −2.39624e9 −1.17442 −0.587210 0.809435i \(-0.699774\pi\)
−0.587210 + 0.809435i \(0.699774\pi\)
\(458\) 0 0
\(459\) −1.47834e9 −0.713560
\(460\) 0 0
\(461\) 1.61913e9 0.769713 0.384856 0.922977i \(-0.374251\pi\)
0.384856 + 0.922977i \(0.374251\pi\)
\(462\) 0 0
\(463\) −1.16133e9 −0.543778 −0.271889 0.962329i \(-0.587648\pi\)
−0.271889 + 0.962329i \(0.587648\pi\)
\(464\) 0 0
\(465\) 8.55553e8 0.394604
\(466\) 0 0
\(467\) −2.83969e9 −1.29021 −0.645107 0.764092i \(-0.723187\pi\)
−0.645107 + 0.764092i \(0.723187\pi\)
\(468\) 0 0
\(469\) 1.52750e9 0.683718
\(470\) 0 0
\(471\) 1.08569e9 0.478777
\(472\) 0 0
\(473\) −9.12156e8 −0.396328
\(474\) 0 0
\(475\) 6.02617e8 0.257996
\(476\) 0 0
\(477\) 3.16360e8 0.133465
\(478\) 0 0
\(479\) −2.38771e9 −0.992676 −0.496338 0.868129i \(-0.665322\pi\)
−0.496338 + 0.868129i \(0.665322\pi\)
\(480\) 0 0
\(481\) −9.60526e8 −0.393551
\(482\) 0 0
\(483\) −5.44292e8 −0.219794
\(484\) 0 0
\(485\) 8.30604e8 0.330596
\(486\) 0 0
\(487\) 2.20508e9 0.865113 0.432556 0.901607i \(-0.357612\pi\)
0.432556 + 0.901607i \(0.357612\pi\)
\(488\) 0 0
\(489\) 1.00550e9 0.388866
\(490\) 0 0
\(491\) −4.28064e8 −0.163201 −0.0816006 0.996665i \(-0.526003\pi\)
−0.0816006 + 0.996665i \(0.526003\pi\)
\(492\) 0 0
\(493\) −1.49445e9 −0.561716
\(494\) 0 0
\(495\) −2.81457e8 −0.104302
\(496\) 0 0
\(497\) −2.48824e8 −0.0909171
\(498\) 0 0
\(499\) 2.95178e9 1.06349 0.531743 0.846906i \(-0.321537\pi\)
0.531743 + 0.846906i \(0.321537\pi\)
\(500\) 0 0
\(501\) 2.41412e9 0.857681
\(502\) 0 0
\(503\) 5.22380e9 1.83020 0.915099 0.403229i \(-0.132112\pi\)
0.915099 + 0.403229i \(0.132112\pi\)
\(504\) 0 0
\(505\) 1.34414e9 0.464435
\(506\) 0 0
\(507\) 6.22326e8 0.212075
\(508\) 0 0
\(509\) −2.80532e9 −0.942911 −0.471455 0.881890i \(-0.656271\pi\)
−0.471455 + 0.881890i \(0.656271\pi\)
\(510\) 0 0
\(511\) 7.46374e8 0.247448
\(512\) 0 0
\(513\) 3.59530e9 1.17577
\(514\) 0 0
\(515\) 1.32357e9 0.426994
\(516\) 0 0
\(517\) −7.33401e8 −0.233413
\(518\) 0 0
\(519\) 1.26628e9 0.397597
\(520\) 0 0
\(521\) 1.40563e8 0.0435450 0.0217725 0.999763i \(-0.493069\pi\)
0.0217725 + 0.999763i \(0.493069\pi\)
\(522\) 0 0
\(523\) 1.81127e9 0.553638 0.276819 0.960922i \(-0.410720\pi\)
0.276819 + 0.960922i \(0.410720\pi\)
\(524\) 0 0
\(525\) −1.32910e8 −0.0400866
\(526\) 0 0
\(527\) 4.37679e9 1.30262
\(528\) 0 0
\(529\) 6.89567e8 0.202526
\(530\) 0 0
\(531\) −2.84837e9 −0.825592
\(532\) 0 0
\(533\) 1.86383e9 0.533164
\(534\) 0 0
\(535\) 1.04654e9 0.295474
\(536\) 0 0
\(537\) 3.61604e9 1.00768
\(538\) 0 0
\(539\) 1.68516e8 0.0463534
\(540\) 0 0
\(541\) −7.11633e9 −1.93226 −0.966130 0.258058i \(-0.916918\pi\)
−0.966130 + 0.258058i \(0.916918\pi\)
\(542\) 0 0
\(543\) 1.51192e9 0.405255
\(544\) 0 0
\(545\) −2.44935e9 −0.648130
\(546\) 0 0
\(547\) 6.02390e9 1.57370 0.786850 0.617144i \(-0.211710\pi\)
0.786850 + 0.617144i \(0.211710\pi\)
\(548\) 0 0
\(549\) 1.54372e9 0.398168
\(550\) 0 0
\(551\) 3.63447e9 0.925572
\(552\) 0 0
\(553\) 1.78883e9 0.449812
\(554\) 0 0
\(555\) −4.85239e8 −0.120484
\(556\) 0 0
\(557\) 3.55726e9 0.872214 0.436107 0.899895i \(-0.356357\pi\)
0.436107 + 0.899895i \(0.356357\pi\)
\(558\) 0 0
\(559\) 3.90771e9 0.946195
\(560\) 0 0
\(561\) 5.63324e8 0.134706
\(562\) 0 0
\(563\) −2.51240e9 −0.593347 −0.296673 0.954979i \(-0.595877\pi\)
−0.296673 + 0.954979i \(0.595877\pi\)
\(564\) 0 0
\(565\) 1.70387e9 0.397436
\(566\) 0 0
\(567\) 3.86252e8 0.0889877
\(568\) 0 0
\(569\) 3.02191e9 0.687683 0.343841 0.939028i \(-0.388272\pi\)
0.343841 + 0.939028i \(0.388272\pi\)
\(570\) 0 0
\(571\) −4.13151e9 −0.928716 −0.464358 0.885648i \(-0.653715\pi\)
−0.464358 + 0.885648i \(0.653715\pi\)
\(572\) 0 0
\(573\) −3.78386e9 −0.840221
\(574\) 0 0
\(575\) 9.99804e8 0.219320
\(576\) 0 0
\(577\) −3.66048e9 −0.793274 −0.396637 0.917976i \(-0.629823\pi\)
−0.396637 + 0.917976i \(0.629823\pi\)
\(578\) 0 0
\(579\) 3.45400e9 0.739516
\(580\) 0 0
\(581\) −2.08517e9 −0.441088
\(582\) 0 0
\(583\) −2.88262e8 −0.0602487
\(584\) 0 0
\(585\) 1.20577e9 0.249012
\(586\) 0 0
\(587\) −8.93156e9 −1.82261 −0.911305 0.411731i \(-0.864924\pi\)
−0.911305 + 0.411731i \(0.864924\pi\)
\(588\) 0 0
\(589\) −1.06442e10 −2.14640
\(590\) 0 0
\(591\) −1.61812e9 −0.322444
\(592\) 0 0
\(593\) 8.00218e9 1.57586 0.787929 0.615766i \(-0.211153\pi\)
0.787929 + 0.615766i \(0.211153\pi\)
\(594\) 0 0
\(595\) −6.79932e8 −0.132329
\(596\) 0 0
\(597\) 4.79879e7 0.00923041
\(598\) 0 0
\(599\) −6.37081e9 −1.21116 −0.605579 0.795785i \(-0.707059\pi\)
−0.605579 + 0.795785i \(0.707059\pi\)
\(600\) 0 0
\(601\) 7.97677e9 1.49888 0.749439 0.662073i \(-0.230323\pi\)
0.749439 + 0.662073i \(0.230323\pi\)
\(602\) 0 0
\(603\) −7.00062e9 −1.30025
\(604\) 0 0
\(605\) −2.17944e9 −0.400130
\(606\) 0 0
\(607\) −5.42119e9 −0.983863 −0.491931 0.870634i \(-0.663709\pi\)
−0.491931 + 0.870634i \(0.663709\pi\)
\(608\) 0 0
\(609\) −8.01598e8 −0.143812
\(610\) 0 0
\(611\) 3.14191e9 0.557250
\(612\) 0 0
\(613\) 8.21824e9 1.44101 0.720505 0.693450i \(-0.243910\pi\)
0.720505 + 0.693450i \(0.243910\pi\)
\(614\) 0 0
\(615\) 9.41570e8 0.163226
\(616\) 0 0
\(617\) 8.15621e9 1.39795 0.698973 0.715148i \(-0.253641\pi\)
0.698973 + 0.715148i \(0.253641\pi\)
\(618\) 0 0
\(619\) 6.46052e9 1.09484 0.547420 0.836858i \(-0.315610\pi\)
0.547420 + 0.836858i \(0.315610\pi\)
\(620\) 0 0
\(621\) 5.96497e9 0.999511
\(622\) 0 0
\(623\) −3.64051e9 −0.603191
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) 0 0
\(627\) −1.36999e9 −0.221963
\(628\) 0 0
\(629\) −2.48236e9 −0.397729
\(630\) 0 0
\(631\) 8.82660e9 1.39859 0.699295 0.714833i \(-0.253497\pi\)
0.699295 + 0.714833i \(0.253497\pi\)
\(632\) 0 0
\(633\) −1.28424e9 −0.201248
\(634\) 0 0
\(635\) −2.79703e9 −0.433500
\(636\) 0 0
\(637\) −7.21930e8 −0.110664
\(638\) 0 0
\(639\) 1.14037e9 0.172900
\(640\) 0 0
\(641\) 8.54151e9 1.28095 0.640474 0.767980i \(-0.278738\pi\)
0.640474 + 0.767980i \(0.278738\pi\)
\(642\) 0 0
\(643\) 1.20342e10 1.78517 0.892585 0.450878i \(-0.148889\pi\)
0.892585 + 0.450878i \(0.148889\pi\)
\(644\) 0 0
\(645\) 1.97410e9 0.289674
\(646\) 0 0
\(647\) −1.89174e8 −0.0274598 −0.0137299 0.999906i \(-0.504370\pi\)
−0.0137299 + 0.999906i \(0.504370\pi\)
\(648\) 0 0
\(649\) 2.59539e9 0.372688
\(650\) 0 0
\(651\) 2.34764e9 0.333501
\(652\) 0 0
\(653\) 8.70977e9 1.22408 0.612041 0.790826i \(-0.290349\pi\)
0.612041 + 0.790826i \(0.290349\pi\)
\(654\) 0 0
\(655\) 5.66662e8 0.0787916
\(656\) 0 0
\(657\) −3.42067e9 −0.470579
\(658\) 0 0
\(659\) 7.48288e8 0.101852 0.0509260 0.998702i \(-0.483783\pi\)
0.0509260 + 0.998702i \(0.483783\pi\)
\(660\) 0 0
\(661\) 8.45586e9 1.13881 0.569407 0.822056i \(-0.307173\pi\)
0.569407 + 0.822056i \(0.307173\pi\)
\(662\) 0 0
\(663\) −2.41330e9 −0.321598
\(664\) 0 0
\(665\) 1.65358e9 0.218047
\(666\) 0 0
\(667\) 6.02996e9 0.786817
\(668\) 0 0
\(669\) −3.10154e9 −0.400485
\(670\) 0 0
\(671\) −1.40661e9 −0.179740
\(672\) 0 0
\(673\) 4.78543e9 0.605157 0.302578 0.953124i \(-0.402153\pi\)
0.302578 + 0.953124i \(0.402153\pi\)
\(674\) 0 0
\(675\) 1.45658e9 0.182293
\(676\) 0 0
\(677\) −1.29662e10 −1.60603 −0.803015 0.595958i \(-0.796772\pi\)
−0.803015 + 0.595958i \(0.796772\pi\)
\(678\) 0 0
\(679\) 2.27918e9 0.279405
\(680\) 0 0
\(681\) 4.76419e8 0.0578061
\(682\) 0 0
\(683\) 9.15988e9 1.10006 0.550031 0.835144i \(-0.314616\pi\)
0.550031 + 0.835144i \(0.314616\pi\)
\(684\) 0 0
\(685\) −6.34572e9 −0.754334
\(686\) 0 0
\(687\) 2.62007e9 0.308293
\(688\) 0 0
\(689\) 1.23492e9 0.143838
\(690\) 0 0
\(691\) −1.05298e10 −1.21407 −0.607037 0.794673i \(-0.707642\pi\)
−0.607037 + 0.794673i \(0.707642\pi\)
\(692\) 0 0
\(693\) −7.72318e8 −0.0881515
\(694\) 0 0
\(695\) −1.31478e9 −0.148562
\(696\) 0 0
\(697\) 4.81683e9 0.538824
\(698\) 0 0
\(699\) 5.74470e9 0.636205
\(700\) 0 0
\(701\) 1.27411e9 0.139699 0.0698497 0.997558i \(-0.477748\pi\)
0.0698497 + 0.997558i \(0.477748\pi\)
\(702\) 0 0
\(703\) 6.03703e9 0.655360
\(704\) 0 0
\(705\) 1.58723e9 0.170600
\(706\) 0 0
\(707\) 3.68832e9 0.392519
\(708\) 0 0
\(709\) −7.17795e9 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(710\) 0 0
\(711\) −8.19829e9 −0.855421
\(712\) 0 0
\(713\) −1.76599e10 −1.82463
\(714\) 0 0
\(715\) −1.09868e9 −0.112409
\(716\) 0 0
\(717\) 2.70746e9 0.274312
\(718\) 0 0
\(719\) −1.18502e10 −1.18898 −0.594488 0.804104i \(-0.702645\pi\)
−0.594488 + 0.804104i \(0.702645\pi\)
\(720\) 0 0
\(721\) 3.63188e9 0.360876
\(722\) 0 0
\(723\) 2.04665e9 0.201400
\(724\) 0 0
\(725\) 1.47245e9 0.143502
\(726\) 0 0
\(727\) 4.67874e9 0.451605 0.225802 0.974173i \(-0.427500\pi\)
0.225802 + 0.974173i \(0.427500\pi\)
\(728\) 0 0
\(729\) 3.28577e9 0.314116
\(730\) 0 0
\(731\) 1.00990e10 0.956238
\(732\) 0 0
\(733\) 1.28552e9 0.120563 0.0602817 0.998181i \(-0.480800\pi\)
0.0602817 + 0.998181i \(0.480800\pi\)
\(734\) 0 0
\(735\) −3.64705e8 −0.0338794
\(736\) 0 0
\(737\) 6.37884e9 0.586956
\(738\) 0 0
\(739\) 5.26720e9 0.480091 0.240046 0.970762i \(-0.422838\pi\)
0.240046 + 0.970762i \(0.422838\pi\)
\(740\) 0 0
\(741\) 5.86909e9 0.529916
\(742\) 0 0
\(743\) −4.15012e9 −0.371193 −0.185596 0.982626i \(-0.559422\pi\)
−0.185596 + 0.982626i \(0.559422\pi\)
\(744\) 0 0
\(745\) 6.79371e9 0.601950
\(746\) 0 0
\(747\) 9.55643e9 0.838829
\(748\) 0 0
\(749\) 2.87171e9 0.249721
\(750\) 0 0
\(751\) 6.37970e9 0.549618 0.274809 0.961499i \(-0.411385\pi\)
0.274809 + 0.961499i \(0.411385\pi\)
\(752\) 0 0
\(753\) −5.97635e8 −0.0510098
\(754\) 0 0
\(755\) 2.79072e9 0.235995
\(756\) 0 0
\(757\) −1.19658e10 −1.00255 −0.501274 0.865289i \(-0.667135\pi\)
−0.501274 + 0.865289i \(0.667135\pi\)
\(758\) 0 0
\(759\) −2.27296e9 −0.188688
\(760\) 0 0
\(761\) −2.00959e10 −1.65296 −0.826479 0.562967i \(-0.809660\pi\)
−0.826479 + 0.562967i \(0.809660\pi\)
\(762\) 0 0
\(763\) −6.72100e9 −0.547770
\(764\) 0 0
\(765\) 3.11616e9 0.251655
\(766\) 0 0
\(767\) −1.11187e10 −0.889756
\(768\) 0 0
\(769\) 2.46683e10 1.95613 0.978064 0.208304i \(-0.0667944\pi\)
0.978064 + 0.208304i \(0.0667944\pi\)
\(770\) 0 0
\(771\) −2.41807e9 −0.190011
\(772\) 0 0
\(773\) 8.88824e9 0.692130 0.346065 0.938211i \(-0.387518\pi\)
0.346065 + 0.938211i \(0.387518\pi\)
\(774\) 0 0
\(775\) −4.31235e9 −0.332781
\(776\) 0 0
\(777\) −1.33149e9 −0.101828
\(778\) 0 0
\(779\) −1.17144e10 −0.887850
\(780\) 0 0
\(781\) −1.03909e9 −0.0780502
\(782\) 0 0
\(783\) 8.78483e9 0.653984
\(784\) 0 0
\(785\) −5.47235e9 −0.403766
\(786\) 0 0
\(787\) 4.65006e9 0.340053 0.170027 0.985439i \(-0.445615\pi\)
0.170027 + 0.985439i \(0.445615\pi\)
\(788\) 0 0
\(789\) 7.40606e9 0.536806
\(790\) 0 0
\(791\) 4.67542e9 0.335895
\(792\) 0 0
\(793\) 6.02598e9 0.429113
\(794\) 0 0
\(795\) 6.23859e8 0.0440354
\(796\) 0 0
\(797\) 1.42890e10 0.999762 0.499881 0.866094i \(-0.333377\pi\)
0.499881 + 0.866094i \(0.333377\pi\)
\(798\) 0 0
\(799\) 8.11987e9 0.563165
\(800\) 0 0
\(801\) 1.66846e10 1.14711
\(802\) 0 0
\(803\) 3.11685e9 0.212428
\(804\) 0 0
\(805\) 2.74346e9 0.185359
\(806\) 0 0
\(807\) 9.68971e9 0.649013
\(808\) 0 0
\(809\) −4.92320e9 −0.326909 −0.163455 0.986551i \(-0.552264\pi\)
−0.163455 + 0.986551i \(0.552264\pi\)
\(810\) 0 0
\(811\) −2.35801e10 −1.55229 −0.776145 0.630555i \(-0.782827\pi\)
−0.776145 + 0.630555i \(0.782827\pi\)
\(812\) 0 0
\(813\) 5.25992e9 0.343291
\(814\) 0 0
\(815\) −5.06813e9 −0.327942
\(816\) 0 0
\(817\) −2.45605e10 −1.57565
\(818\) 0 0
\(819\) 3.30864e9 0.210453
\(820\) 0 0
\(821\) −2.86630e10 −1.80768 −0.903838 0.427875i \(-0.859262\pi\)
−0.903838 + 0.427875i \(0.859262\pi\)
\(822\) 0 0
\(823\) 2.76897e10 1.73148 0.865742 0.500490i \(-0.166847\pi\)
0.865742 + 0.500490i \(0.166847\pi\)
\(824\) 0 0
\(825\) −5.55030e8 −0.0344134
\(826\) 0 0
\(827\) 1.27176e10 0.781873 0.390936 0.920418i \(-0.372151\pi\)
0.390936 + 0.920418i \(0.372151\pi\)
\(828\) 0 0
\(829\) −1.50770e10 −0.919127 −0.459563 0.888145i \(-0.651994\pi\)
−0.459563 + 0.888145i \(0.651994\pi\)
\(830\) 0 0
\(831\) −4.63064e9 −0.279923
\(832\) 0 0
\(833\) −1.86573e9 −0.111839
\(834\) 0 0
\(835\) −1.21682e10 −0.723307
\(836\) 0 0
\(837\) −2.57281e10 −1.51659
\(838\) 0 0
\(839\) −4.59511e9 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(840\) 0 0
\(841\) −8.36934e9 −0.485182
\(842\) 0 0
\(843\) 1.83216e10 1.05334
\(844\) 0 0
\(845\) −3.13679e9 −0.178849
\(846\) 0 0
\(847\) −5.98038e9 −0.338171
\(848\) 0 0
\(849\) −7.73504e9 −0.433796
\(850\) 0 0
\(851\) 1.00161e10 0.557114
\(852\) 0 0
\(853\) −1.13971e9 −0.0628740 −0.0314370 0.999506i \(-0.510008\pi\)
−0.0314370 + 0.999506i \(0.510008\pi\)
\(854\) 0 0
\(855\) −7.57844e9 −0.414666
\(856\) 0 0
\(857\) 7.79419e9 0.422998 0.211499 0.977378i \(-0.432166\pi\)
0.211499 + 0.977378i \(0.432166\pi\)
\(858\) 0 0
\(859\) −1.27280e10 −0.685147 −0.342573 0.939491i \(-0.611299\pi\)
−0.342573 + 0.939491i \(0.611299\pi\)
\(860\) 0 0
\(861\) 2.58367e9 0.137951
\(862\) 0 0
\(863\) −2.53204e9 −0.134101 −0.0670507 0.997750i \(-0.521359\pi\)
−0.0670507 + 0.997750i \(0.521359\pi\)
\(864\) 0 0
\(865\) −6.38258e9 −0.335305
\(866\) 0 0
\(867\) 3.93933e9 0.205284
\(868\) 0 0
\(869\) 7.47014e9 0.386153
\(870\) 0 0
\(871\) −2.73272e10 −1.40130
\(872\) 0 0
\(873\) −1.04456e10 −0.531352
\(874\) 0 0
\(875\) 6.69922e8 0.0338062
\(876\) 0 0
\(877\) 5.00988e9 0.250800 0.125400 0.992106i \(-0.459979\pi\)
0.125400 + 0.992106i \(0.459979\pi\)
\(878\) 0 0
\(879\) 1.25353e10 0.622550
\(880\) 0 0
\(881\) 9.46900e9 0.466539 0.233270 0.972412i \(-0.425058\pi\)
0.233270 + 0.972412i \(0.425058\pi\)
\(882\) 0 0
\(883\) −1.11146e10 −0.543289 −0.271644 0.962398i \(-0.587567\pi\)
−0.271644 + 0.962398i \(0.587567\pi\)
\(884\) 0 0
\(885\) −5.61696e9 −0.272395
\(886\) 0 0
\(887\) 7.27986e9 0.350260 0.175130 0.984545i \(-0.443965\pi\)
0.175130 + 0.984545i \(0.443965\pi\)
\(888\) 0 0
\(889\) −7.67506e9 −0.366375
\(890\) 0 0
\(891\) 1.61298e9 0.0763938
\(892\) 0 0
\(893\) −1.97473e10 −0.927959
\(894\) 0 0
\(895\) −1.82264e10 −0.849807
\(896\) 0 0
\(897\) 9.73743e9 0.450475
\(898\) 0 0
\(899\) −2.60084e10 −1.19386
\(900\) 0 0
\(901\) 3.19150e9 0.145365
\(902\) 0 0
\(903\) 5.41692e9 0.244819
\(904\) 0 0
\(905\) −7.62070e9 −0.341763
\(906\) 0 0
\(907\) 1.39503e10 0.620809 0.310405 0.950605i \(-0.399535\pi\)
0.310405 + 0.950605i \(0.399535\pi\)
\(908\) 0 0
\(909\) −1.69038e10 −0.746465
\(910\) 0 0
\(911\) −2.98148e8 −0.0130653 −0.00653263 0.999979i \(-0.502079\pi\)
−0.00653263 + 0.999979i \(0.502079\pi\)
\(912\) 0 0
\(913\) −8.70765e9 −0.378663
\(914\) 0 0
\(915\) 3.04420e9 0.131371
\(916\) 0 0
\(917\) 1.55492e9 0.0665910
\(918\) 0 0
\(919\) −2.67202e10 −1.13563 −0.567814 0.823157i \(-0.692211\pi\)
−0.567814 + 0.823157i \(0.692211\pi\)
\(920\) 0 0
\(921\) 1.15928e10 0.488969
\(922\) 0 0
\(923\) 4.45149e9 0.186337
\(924\) 0 0
\(925\) 2.44581e9 0.101608
\(926\) 0 0
\(927\) −1.66451e10 −0.686288
\(928\) 0 0
\(929\) −3.66336e10 −1.49908 −0.749540 0.661959i \(-0.769725\pi\)
−0.749540 + 0.661959i \(0.769725\pi\)
\(930\) 0 0
\(931\) 4.53742e9 0.184283
\(932\) 0 0
\(933\) 2.87728e8 0.0115983
\(934\) 0 0
\(935\) −2.83939e9 −0.113602
\(936\) 0 0
\(937\) 1.28088e10 0.508649 0.254325 0.967119i \(-0.418147\pi\)
0.254325 + 0.967119i \(0.418147\pi\)
\(938\) 0 0
\(939\) −2.04149e10 −0.804669
\(940\) 0 0
\(941\) −1.20663e10 −0.472073 −0.236037 0.971744i \(-0.575849\pi\)
−0.236037 + 0.971744i \(0.575849\pi\)
\(942\) 0 0
\(943\) −1.94354e10 −0.754751
\(944\) 0 0
\(945\) 3.99685e9 0.154066
\(946\) 0 0
\(947\) −8.36023e9 −0.319885 −0.159942 0.987126i \(-0.551131\pi\)
−0.159942 + 0.987126i \(0.551131\pi\)
\(948\) 0 0
\(949\) −1.33527e10 −0.507151
\(950\) 0 0
\(951\) −9.65082e9 −0.363859
\(952\) 0 0
\(953\) 4.49530e10 1.68242 0.841209 0.540710i \(-0.181844\pi\)
0.841209 + 0.540710i \(0.181844\pi\)
\(954\) 0 0
\(955\) 1.90722e10 0.708582
\(956\) 0 0
\(957\) −3.34747e9 −0.123459
\(958\) 0 0
\(959\) −1.74126e10 −0.637528
\(960\) 0 0
\(961\) 4.86580e10 1.76857
\(962\) 0 0
\(963\) −1.31612e10 −0.474901
\(964\) 0 0
\(965\) −1.74096e10 −0.623654
\(966\) 0 0
\(967\) −1.34247e8 −0.00477432 −0.00238716 0.999997i \(-0.500760\pi\)
−0.00238716 + 0.999997i \(0.500760\pi\)
\(968\) 0 0
\(969\) 1.51679e10 0.535541
\(970\) 0 0
\(971\) 3.00377e10 1.05293 0.526465 0.850197i \(-0.323517\pi\)
0.526465 + 0.850197i \(0.323517\pi\)
\(972\) 0 0
\(973\) −3.60777e9 −0.125558
\(974\) 0 0
\(975\) 2.37777e9 0.0821587
\(976\) 0 0
\(977\) 4.52860e10 1.55358 0.776789 0.629761i \(-0.216847\pi\)
0.776789 + 0.629761i \(0.216847\pi\)
\(978\) 0 0
\(979\) −1.52028e10 −0.517825
\(980\) 0 0
\(981\) 3.08027e10 1.04171
\(982\) 0 0
\(983\) −4.61443e10 −1.54946 −0.774731 0.632290i \(-0.782115\pi\)
−0.774731 + 0.632290i \(0.782115\pi\)
\(984\) 0 0
\(985\) 8.15600e9 0.271926
\(986\) 0 0
\(987\) 4.35537e9 0.144183
\(988\) 0 0
\(989\) −4.07484e10 −1.33944
\(990\) 0 0
\(991\) 1.05400e10 0.344018 0.172009 0.985095i \(-0.444974\pi\)
0.172009 + 0.985095i \(0.444974\pi\)
\(992\) 0 0
\(993\) 3.68475e9 0.119422
\(994\) 0 0
\(995\) −2.41879e8 −0.00778427
\(996\) 0 0
\(997\) −5.00734e10 −1.60020 −0.800099 0.599868i \(-0.795220\pi\)
−0.800099 + 0.599868i \(0.795220\pi\)
\(998\) 0 0
\(999\) 1.45920e10 0.463059
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 560.8.a.i.1.1 2
4.3 odd 2 35.8.a.a.1.1 2
12.11 even 2 315.8.a.c.1.2 2
20.3 even 4 175.8.b.c.99.2 4
20.7 even 4 175.8.b.c.99.3 4
20.19 odd 2 175.8.a.b.1.2 2
28.27 even 2 245.8.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.1 2 4.3 odd 2
175.8.a.b.1.2 2 20.19 odd 2
175.8.b.c.99.2 4 20.3 even 4
175.8.b.c.99.3 4 20.7 even 4
245.8.a.b.1.1 2 28.27 even 2
315.8.a.c.1.2 2 12.11 even 2
560.8.a.i.1.1 2 1.1 even 1 trivial