Properties

Label 176.8.a.e.1.1
Level $176$
Weight $8$
Character 176.1
Self dual yes
Analytic conductor $54.980$
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] = [176,8,Mod(1,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, 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("176.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 176.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: \(54.9797644852\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14881}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3720 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(61.4939\) of defining polynomial
Character \(\chi\) \(=\) 176.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-49.4939 q^{3} +104.506 q^{5} -43.0861 q^{7} +262.641 q^{9} +O(q^{10})\) \(q-49.4939 q^{3} +104.506 q^{5} -43.0861 q^{7} +262.641 q^{9} -1331.00 q^{11} +4494.27 q^{13} -5172.41 q^{15} -6878.55 q^{17} +21063.0 q^{19} +2132.50 q^{21} +60259.0 q^{23} -67203.5 q^{25} +95243.9 q^{27} -59039.5 q^{29} +88522.5 q^{31} +65876.3 q^{33} -4502.76 q^{35} +382131. q^{37} -222439. q^{39} +550122. q^{41} -693613. q^{43} +27447.6 q^{45} +126233. q^{47} -821687. q^{49} +340446. q^{51} -1.19815e6 q^{53} -139098. q^{55} -1.04249e6 q^{57} -1.31641e6 q^{59} +440165. q^{61} -11316.2 q^{63} +469679. q^{65} -3.56366e6 q^{67} -2.98245e6 q^{69} -3.13943e6 q^{71} -4.07747e6 q^{73} +3.32616e6 q^{75} +57347.6 q^{77} -1.89723e6 q^{79} -5.28839e6 q^{81} -6.49257e6 q^{83} -718851. q^{85} +2.92209e6 q^{87} +9.23033e6 q^{89} -193641. q^{91} -4.38132e6 q^{93} +2.20122e6 q^{95} -1.38062e7 q^{97} -349576. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 23 q^{3} + 331 q^{5} - 1794 q^{7} + 3331 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 23 q^{3} + 331 q^{5} - 1794 q^{7} + 3331 q^{9} - 2662 q^{11} - 5406 q^{13} + 11247 q^{15} + 15032 q^{17} - 16916 q^{19} - 124798 q^{21} + 51351 q^{23} - 94029 q^{25} + 159137 q^{27} - 207130 q^{29} + 19071 q^{31} - 30613 q^{33} - 401074 q^{35} + 351333 q^{37} - 940148 q^{39} + 123610 q^{41} + 159822 q^{43} + 722412 q^{45} - 451160 q^{47} + 1420470 q^{49} + 1928826 q^{51} - 1260832 q^{53} - 440561 q^{55} - 3795736 q^{57} - 887547 q^{59} - 597918 q^{61} - 5383748 q^{63} - 1772672 q^{65} - 2864711 q^{67} - 3628227 q^{69} - 1306267 q^{71} - 4577530 q^{73} + 1381472 q^{75} + 2387814 q^{77} + 2946342 q^{79} - 7367030 q^{81} - 9965450 q^{83} + 4243754 q^{85} - 7813560 q^{87} + 10185377 q^{89} + 17140888 q^{91} - 9416131 q^{93} - 6400800 q^{95} - 27765477 q^{97} - 4433561 q^{99}+O(q^{100}) \) 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
\(3\) −49.4939 −1.05834 −0.529172 0.848515i \(-0.677497\pi\)
−0.529172 + 0.848515i \(0.677497\pi\)
\(4\) 0 0
\(5\) 104.506 0.373893 0.186946 0.982370i \(-0.440141\pi\)
0.186946 + 0.982370i \(0.440141\pi\)
\(6\) 0 0
\(7\) −43.0861 −0.0474781 −0.0237391 0.999718i \(-0.507557\pi\)
−0.0237391 + 0.999718i \(0.507557\pi\)
\(8\) 0 0
\(9\) 262.641 0.120092
\(10\) 0 0
\(11\) −1331.00 −0.301511
\(12\) 0 0
\(13\) 4494.27 0.567359 0.283679 0.958919i \(-0.408445\pi\)
0.283679 + 0.958919i \(0.408445\pi\)
\(14\) 0 0
\(15\) −5172.41 −0.395707
\(16\) 0 0
\(17\) −6878.55 −0.339567 −0.169784 0.985481i \(-0.554307\pi\)
−0.169784 + 0.985481i \(0.554307\pi\)
\(18\) 0 0
\(19\) 21063.0 0.704503 0.352252 0.935905i \(-0.385416\pi\)
0.352252 + 0.935905i \(0.385416\pi\)
\(20\) 0 0
\(21\) 2132.50 0.0502482
\(22\) 0 0
\(23\) 60259.0 1.03270 0.516350 0.856377i \(-0.327290\pi\)
0.516350 + 0.856377i \(0.327290\pi\)
\(24\) 0 0
\(25\) −67203.5 −0.860204
\(26\) 0 0
\(27\) 95243.9 0.931245
\(28\) 0 0
\(29\) −59039.5 −0.449521 −0.224760 0.974414i \(-0.572160\pi\)
−0.224760 + 0.974414i \(0.572160\pi\)
\(30\) 0 0
\(31\) 88522.5 0.533689 0.266844 0.963740i \(-0.414019\pi\)
0.266844 + 0.963740i \(0.414019\pi\)
\(32\) 0 0
\(33\) 65876.3 0.319103
\(34\) 0 0
\(35\) −4502.76 −0.0177517
\(36\) 0 0
\(37\) 382131. 1.24024 0.620120 0.784507i \(-0.287084\pi\)
0.620120 + 0.784507i \(0.287084\pi\)
\(38\) 0 0
\(39\) −222439. −0.600461
\(40\) 0 0
\(41\) 550122. 1.24657 0.623283 0.781996i \(-0.285798\pi\)
0.623283 + 0.781996i \(0.285798\pi\)
\(42\) 0 0
\(43\) −693613. −1.33039 −0.665193 0.746671i \(-0.731651\pi\)
−0.665193 + 0.746671i \(0.731651\pi\)
\(44\) 0 0
\(45\) 27447.6 0.0449015
\(46\) 0 0
\(47\) 126233. 0.177349 0.0886745 0.996061i \(-0.471737\pi\)
0.0886745 + 0.996061i \(0.471737\pi\)
\(48\) 0 0
\(49\) −821687. −0.997746
\(50\) 0 0
\(51\) 340446. 0.359379
\(52\) 0 0
\(53\) −1.19815e6 −1.10546 −0.552732 0.833359i \(-0.686415\pi\)
−0.552732 + 0.833359i \(0.686415\pi\)
\(54\) 0 0
\(55\) −139098. −0.112733
\(56\) 0 0
\(57\) −1.04249e6 −0.745607
\(58\) 0 0
\(59\) −1.31641e6 −0.834469 −0.417234 0.908799i \(-0.637001\pi\)
−0.417234 + 0.908799i \(0.637001\pi\)
\(60\) 0 0
\(61\) 440165. 0.248291 0.124145 0.992264i \(-0.460381\pi\)
0.124145 + 0.992264i \(0.460381\pi\)
\(62\) 0 0
\(63\) −11316.2 −0.00570175
\(64\) 0 0
\(65\) 469679. 0.212131
\(66\) 0 0
\(67\) −3.56366e6 −1.44755 −0.723777 0.690034i \(-0.757596\pi\)
−0.723777 + 0.690034i \(0.757596\pi\)
\(68\) 0 0
\(69\) −2.98245e6 −1.09295
\(70\) 0 0
\(71\) −3.13943e6 −1.04099 −0.520494 0.853865i \(-0.674252\pi\)
−0.520494 + 0.853865i \(0.674252\pi\)
\(72\) 0 0
\(73\) −4.07747e6 −1.22676 −0.613382 0.789787i \(-0.710191\pi\)
−0.613382 + 0.789787i \(0.710191\pi\)
\(74\) 0 0
\(75\) 3.32616e6 0.910392
\(76\) 0 0
\(77\) 57347.6 0.0143152
\(78\) 0 0
\(79\) −1.89723e6 −0.432937 −0.216468 0.976290i \(-0.569454\pi\)
−0.216468 + 0.976290i \(0.569454\pi\)
\(80\) 0 0
\(81\) −5.28839e6 −1.10567
\(82\) 0 0
\(83\) −6.49257e6 −1.24636 −0.623179 0.782079i \(-0.714159\pi\)
−0.623179 + 0.782079i \(0.714159\pi\)
\(84\) 0 0
\(85\) −718851. −0.126962
\(86\) 0 0
\(87\) 2.92209e6 0.475747
\(88\) 0 0
\(89\) 9.23033e6 1.38788 0.693940 0.720032i \(-0.255873\pi\)
0.693940 + 0.720032i \(0.255873\pi\)
\(90\) 0 0
\(91\) −193641. −0.0269371
\(92\) 0 0
\(93\) −4.38132e6 −0.564826
\(94\) 0 0
\(95\) 2.20122e6 0.263409
\(96\) 0 0
\(97\) −1.38062e7 −1.53593 −0.767967 0.640489i \(-0.778732\pi\)
−0.767967 + 0.640489i \(0.778732\pi\)
\(98\) 0 0
\(99\) −349576. −0.0362091
\(100\) 0 0
\(101\) 6.59478e6 0.636906 0.318453 0.947939i \(-0.396837\pi\)
0.318453 + 0.947939i \(0.396837\pi\)
\(102\) 0 0
\(103\) 1.98787e7 1.79250 0.896249 0.443551i \(-0.146282\pi\)
0.896249 + 0.443551i \(0.146282\pi\)
\(104\) 0 0
\(105\) 222859. 0.0187874
\(106\) 0 0
\(107\) −2.09533e7 −1.65352 −0.826758 0.562558i \(-0.809817\pi\)
−0.826758 + 0.562558i \(0.809817\pi\)
\(108\) 0 0
\(109\) −598244. −0.0442472 −0.0221236 0.999755i \(-0.507043\pi\)
−0.0221236 + 0.999755i \(0.507043\pi\)
\(110\) 0 0
\(111\) −1.89131e7 −1.31260
\(112\) 0 0
\(113\) −1.41807e6 −0.0924535 −0.0462267 0.998931i \(-0.514720\pi\)
−0.0462267 + 0.998931i \(0.514720\pi\)
\(114\) 0 0
\(115\) 6.29744e6 0.386119
\(116\) 0 0
\(117\) 1.18038e6 0.0681353
\(118\) 0 0
\(119\) 296370. 0.0161220
\(120\) 0 0
\(121\) 1.77156e6 0.0909091
\(122\) 0 0
\(123\) −2.72276e7 −1.31930
\(124\) 0 0
\(125\) −1.51877e7 −0.695517
\(126\) 0 0
\(127\) 1.92988e7 0.836023 0.418012 0.908442i \(-0.362727\pi\)
0.418012 + 0.908442i \(0.362727\pi\)
\(128\) 0 0
\(129\) 3.43296e7 1.40801
\(130\) 0 0
\(131\) −4.22451e7 −1.64182 −0.820912 0.571055i \(-0.806534\pi\)
−0.820912 + 0.571055i \(0.806534\pi\)
\(132\) 0 0
\(133\) −907523. −0.0334485
\(134\) 0 0
\(135\) 9.95358e6 0.348186
\(136\) 0 0
\(137\) −7.65888e6 −0.254474 −0.127237 0.991872i \(-0.540611\pi\)
−0.127237 + 0.991872i \(0.540611\pi\)
\(138\) 0 0
\(139\) 1.46018e7 0.461164 0.230582 0.973053i \(-0.425937\pi\)
0.230582 + 0.973053i \(0.425937\pi\)
\(140\) 0 0
\(141\) −6.24773e6 −0.187696
\(142\) 0 0
\(143\) −5.98188e6 −0.171065
\(144\) 0 0
\(145\) −6.16999e6 −0.168072
\(146\) 0 0
\(147\) 4.06684e7 1.05596
\(148\) 0 0
\(149\) −3.54730e7 −0.878507 −0.439254 0.898363i \(-0.644757\pi\)
−0.439254 + 0.898363i \(0.644757\pi\)
\(150\) 0 0
\(151\) −4.10281e7 −0.969754 −0.484877 0.874582i \(-0.661136\pi\)
−0.484877 + 0.874582i \(0.661136\pi\)
\(152\) 0 0
\(153\) −1.80659e6 −0.0407793
\(154\) 0 0
\(155\) 9.25115e6 0.199542
\(156\) 0 0
\(157\) −6.66977e7 −1.37550 −0.687752 0.725946i \(-0.741402\pi\)
−0.687752 + 0.725946i \(0.741402\pi\)
\(158\) 0 0
\(159\) 5.93009e7 1.16996
\(160\) 0 0
\(161\) −2.59632e6 −0.0490307
\(162\) 0 0
\(163\) −9.29744e6 −0.168154 −0.0840769 0.996459i \(-0.526794\pi\)
−0.0840769 + 0.996459i \(0.526794\pi\)
\(164\) 0 0
\(165\) 6.88448e6 0.119310
\(166\) 0 0
\(167\) 5.93268e7 0.985696 0.492848 0.870115i \(-0.335956\pi\)
0.492848 + 0.870115i \(0.335956\pi\)
\(168\) 0 0
\(169\) −4.25500e7 −0.678104
\(170\) 0 0
\(171\) 5.53202e6 0.0846053
\(172\) 0 0
\(173\) 1.19997e8 1.76201 0.881004 0.473108i \(-0.156868\pi\)
0.881004 + 0.473108i \(0.156868\pi\)
\(174\) 0 0
\(175\) 2.89553e6 0.0408409
\(176\) 0 0
\(177\) 6.51543e7 0.883155
\(178\) 0 0
\(179\) −3.18894e7 −0.415586 −0.207793 0.978173i \(-0.566628\pi\)
−0.207793 + 0.978173i \(0.566628\pi\)
\(180\) 0 0
\(181\) 2.14661e7 0.269079 0.134539 0.990908i \(-0.457045\pi\)
0.134539 + 0.990908i \(0.457045\pi\)
\(182\) 0 0
\(183\) −2.17854e7 −0.262777
\(184\) 0 0
\(185\) 3.99350e7 0.463717
\(186\) 0 0
\(187\) 9.15535e6 0.102383
\(188\) 0 0
\(189\) −4.10369e6 −0.0442138
\(190\) 0 0
\(191\) 5.37382e7 0.558042 0.279021 0.960285i \(-0.409990\pi\)
0.279021 + 0.960285i \(0.409990\pi\)
\(192\) 0 0
\(193\) 1.72101e8 1.72319 0.861593 0.507600i \(-0.169467\pi\)
0.861593 + 0.507600i \(0.169467\pi\)
\(194\) 0 0
\(195\) −2.32462e7 −0.224508
\(196\) 0 0
\(197\) −1.14526e8 −1.06726 −0.533631 0.845718i \(-0.679173\pi\)
−0.533631 + 0.845718i \(0.679173\pi\)
\(198\) 0 0
\(199\) 3.40644e7 0.306418 0.153209 0.988194i \(-0.451039\pi\)
0.153209 + 0.988194i \(0.451039\pi\)
\(200\) 0 0
\(201\) 1.76379e8 1.53201
\(202\) 0 0
\(203\) 2.54378e6 0.0213424
\(204\) 0 0
\(205\) 5.74911e7 0.466082
\(206\) 0 0
\(207\) 1.58265e7 0.124019
\(208\) 0 0
\(209\) −2.80349e7 −0.212416
\(210\) 0 0
\(211\) 1.74288e8 1.27726 0.638628 0.769516i \(-0.279502\pi\)
0.638628 + 0.769516i \(0.279502\pi\)
\(212\) 0 0
\(213\) 1.55382e8 1.10172
\(214\) 0 0
\(215\) −7.24868e7 −0.497421
\(216\) 0 0
\(217\) −3.81409e6 −0.0253385
\(218\) 0 0
\(219\) 2.01810e8 1.29834
\(220\) 0 0
\(221\) −3.09141e7 −0.192656
\(222\) 0 0
\(223\) −1.45730e8 −0.879998 −0.439999 0.897998i \(-0.645021\pi\)
−0.439999 + 0.897998i \(0.645021\pi\)
\(224\) 0 0
\(225\) −1.76504e7 −0.103304
\(226\) 0 0
\(227\) 1.27102e8 0.721211 0.360606 0.932718i \(-0.382570\pi\)
0.360606 + 0.932718i \(0.382570\pi\)
\(228\) 0 0
\(229\) −1.29750e8 −0.713975 −0.356987 0.934109i \(-0.616196\pi\)
−0.356987 + 0.934109i \(0.616196\pi\)
\(230\) 0 0
\(231\) −2.83835e6 −0.0151504
\(232\) 0 0
\(233\) −1.52210e8 −0.788310 −0.394155 0.919044i \(-0.628963\pi\)
−0.394155 + 0.919044i \(0.628963\pi\)
\(234\) 0 0
\(235\) 1.31921e7 0.0663095
\(236\) 0 0
\(237\) 9.39011e7 0.458196
\(238\) 0 0
\(239\) 2.43061e8 1.15165 0.575827 0.817571i \(-0.304680\pi\)
0.575827 + 0.817571i \(0.304680\pi\)
\(240\) 0 0
\(241\) −6.84592e7 −0.315045 −0.157522 0.987515i \(-0.550351\pi\)
−0.157522 + 0.987515i \(0.550351\pi\)
\(242\) 0 0
\(243\) 5.34441e7 0.238934
\(244\) 0 0
\(245\) −8.58713e7 −0.373050
\(246\) 0 0
\(247\) 9.46630e7 0.399706
\(248\) 0 0
\(249\) 3.21342e8 1.31908
\(250\) 0 0
\(251\) −1.67292e8 −0.667755 −0.333878 0.942616i \(-0.608357\pi\)
−0.333878 + 0.942616i \(0.608357\pi\)
\(252\) 0 0
\(253\) −8.02047e7 −0.311371
\(254\) 0 0
\(255\) 3.55787e7 0.134369
\(256\) 0 0
\(257\) 3.84789e8 1.41402 0.707012 0.707202i \(-0.250043\pi\)
0.707012 + 0.707202i \(0.250043\pi\)
\(258\) 0 0
\(259\) −1.64645e7 −0.0588843
\(260\) 0 0
\(261\) −1.55062e7 −0.0539839
\(262\) 0 0
\(263\) −5.09459e8 −1.72689 −0.863444 0.504445i \(-0.831697\pi\)
−0.863444 + 0.504445i \(0.831697\pi\)
\(264\) 0 0
\(265\) −1.25214e8 −0.413325
\(266\) 0 0
\(267\) −4.56845e8 −1.46886
\(268\) 0 0
\(269\) −5.51297e8 −1.72684 −0.863421 0.504484i \(-0.831683\pi\)
−0.863421 + 0.504484i \(0.831683\pi\)
\(270\) 0 0
\(271\) −3.42271e8 −1.04467 −0.522333 0.852742i \(-0.674938\pi\)
−0.522333 + 0.852742i \(0.674938\pi\)
\(272\) 0 0
\(273\) 9.58402e6 0.0285088
\(274\) 0 0
\(275\) 8.94478e7 0.259361
\(276\) 0 0
\(277\) −5.10974e8 −1.44451 −0.722253 0.691629i \(-0.756893\pi\)
−0.722253 + 0.691629i \(0.756893\pi\)
\(278\) 0 0
\(279\) 2.32497e7 0.0640918
\(280\) 0 0
\(281\) −1.51750e7 −0.0407998 −0.0203999 0.999792i \(-0.506494\pi\)
−0.0203999 + 0.999792i \(0.506494\pi\)
\(282\) 0 0
\(283\) 2.25341e8 0.590999 0.295500 0.955343i \(-0.404514\pi\)
0.295500 + 0.955343i \(0.404514\pi\)
\(284\) 0 0
\(285\) −1.08947e8 −0.278777
\(286\) 0 0
\(287\) −2.37026e7 −0.0591846
\(288\) 0 0
\(289\) −3.63024e8 −0.884694
\(290\) 0 0
\(291\) 6.83322e8 1.62555
\(292\) 0 0
\(293\) 3.84004e8 0.891864 0.445932 0.895067i \(-0.352872\pi\)
0.445932 + 0.895067i \(0.352872\pi\)
\(294\) 0 0
\(295\) −1.37573e8 −0.312002
\(296\) 0 0
\(297\) −1.26770e8 −0.280781
\(298\) 0 0
\(299\) 2.70821e8 0.585912
\(300\) 0 0
\(301\) 2.98851e7 0.0631643
\(302\) 0 0
\(303\) −3.26401e8 −0.674066
\(304\) 0 0
\(305\) 4.59999e7 0.0928341
\(306\) 0 0
\(307\) −7.46289e8 −1.47205 −0.736026 0.676954i \(-0.763300\pi\)
−0.736026 + 0.676954i \(0.763300\pi\)
\(308\) 0 0
\(309\) −9.83876e8 −1.89708
\(310\) 0 0
\(311\) −6.40000e8 −1.20648 −0.603238 0.797561i \(-0.706123\pi\)
−0.603238 + 0.797561i \(0.706123\pi\)
\(312\) 0 0
\(313\) 1.35452e8 0.249677 0.124839 0.992177i \(-0.460159\pi\)
0.124839 + 0.992177i \(0.460159\pi\)
\(314\) 0 0
\(315\) −1.18261e6 −0.00213184
\(316\) 0 0
\(317\) 9.75198e7 0.171943 0.0859717 0.996298i \(-0.472601\pi\)
0.0859717 + 0.996298i \(0.472601\pi\)
\(318\) 0 0
\(319\) 7.85816e7 0.135536
\(320\) 0 0
\(321\) 1.03706e9 1.74999
\(322\) 0 0
\(323\) −1.44883e8 −0.239226
\(324\) 0 0
\(325\) −3.02031e8 −0.488045
\(326\) 0 0
\(327\) 2.96094e7 0.0468288
\(328\) 0 0
\(329\) −5.43886e6 −0.00842020
\(330\) 0 0
\(331\) −3.49842e8 −0.530242 −0.265121 0.964215i \(-0.585412\pi\)
−0.265121 + 0.964215i \(0.585412\pi\)
\(332\) 0 0
\(333\) 1.00363e8 0.148943
\(334\) 0 0
\(335\) −3.72425e8 −0.541230
\(336\) 0 0
\(337\) 7.20870e8 1.02601 0.513006 0.858385i \(-0.328532\pi\)
0.513006 + 0.858385i \(0.328532\pi\)
\(338\) 0 0
\(339\) 7.01858e7 0.0978476
\(340\) 0 0
\(341\) −1.17823e8 −0.160913
\(342\) 0 0
\(343\) 7.08865e7 0.0948493
\(344\) 0 0
\(345\) −3.11684e8 −0.408647
\(346\) 0 0
\(347\) −1.28079e8 −0.164560 −0.0822798 0.996609i \(-0.526220\pi\)
−0.0822798 + 0.996609i \(0.526220\pi\)
\(348\) 0 0
\(349\) −1.00379e8 −0.126402 −0.0632012 0.998001i \(-0.520131\pi\)
−0.0632012 + 0.998001i \(0.520131\pi\)
\(350\) 0 0
\(351\) 4.28052e8 0.528350
\(352\) 0 0
\(353\) 7.51413e8 0.909216 0.454608 0.890692i \(-0.349779\pi\)
0.454608 + 0.890692i \(0.349779\pi\)
\(354\) 0 0
\(355\) −3.28089e8 −0.389218
\(356\) 0 0
\(357\) −1.46685e7 −0.0170626
\(358\) 0 0
\(359\) 9.17064e8 1.04609 0.523045 0.852305i \(-0.324796\pi\)
0.523045 + 0.852305i \(0.324796\pi\)
\(360\) 0 0
\(361\) −4.50221e8 −0.503675
\(362\) 0 0
\(363\) −8.76814e7 −0.0962131
\(364\) 0 0
\(365\) −4.26121e8 −0.458678
\(366\) 0 0
\(367\) 9.16435e8 0.967766 0.483883 0.875133i \(-0.339226\pi\)
0.483883 + 0.875133i \(0.339226\pi\)
\(368\) 0 0
\(369\) 1.44485e8 0.149703
\(370\) 0 0
\(371\) 5.16234e7 0.0524854
\(372\) 0 0
\(373\) −1.56099e9 −1.55747 −0.778736 0.627352i \(-0.784139\pi\)
−0.778736 + 0.627352i \(0.784139\pi\)
\(374\) 0 0
\(375\) 7.51699e8 0.736096
\(376\) 0 0
\(377\) −2.65340e8 −0.255040
\(378\) 0 0
\(379\) −4.34986e7 −0.0410429 −0.0205214 0.999789i \(-0.506533\pi\)
−0.0205214 + 0.999789i \(0.506533\pi\)
\(380\) 0 0
\(381\) −9.55174e8 −0.884800
\(382\) 0 0
\(383\) −1.95422e8 −0.177737 −0.0888685 0.996043i \(-0.528325\pi\)
−0.0888685 + 0.996043i \(0.528325\pi\)
\(384\) 0 0
\(385\) 5.99317e6 0.00535235
\(386\) 0 0
\(387\) −1.82171e8 −0.159769
\(388\) 0 0
\(389\) −1.60846e9 −1.38543 −0.692716 0.721210i \(-0.743586\pi\)
−0.692716 + 0.721210i \(0.743586\pi\)
\(390\) 0 0
\(391\) −4.14495e8 −0.350671
\(392\) 0 0
\(393\) 2.09087e9 1.73761
\(394\) 0 0
\(395\) −1.98272e8 −0.161872
\(396\) 0 0
\(397\) 1.20591e9 0.967268 0.483634 0.875270i \(-0.339317\pi\)
0.483634 + 0.875270i \(0.339317\pi\)
\(398\) 0 0
\(399\) 4.49168e7 0.0354000
\(400\) 0 0
\(401\) 7.40689e8 0.573628 0.286814 0.957986i \(-0.407404\pi\)
0.286814 + 0.957986i \(0.407404\pi\)
\(402\) 0 0
\(403\) 3.97845e8 0.302793
\(404\) 0 0
\(405\) −5.52669e8 −0.413402
\(406\) 0 0
\(407\) −5.08616e8 −0.373947
\(408\) 0 0
\(409\) 1.37187e7 0.00991476 0.00495738 0.999988i \(-0.498422\pi\)
0.00495738 + 0.999988i \(0.498422\pi\)
\(410\) 0 0
\(411\) 3.79067e8 0.269321
\(412\) 0 0
\(413\) 5.67190e7 0.0396190
\(414\) 0 0
\(415\) −6.78513e8 −0.466004
\(416\) 0 0
\(417\) −7.22700e8 −0.488070
\(418\) 0 0
\(419\) −1.95596e9 −1.29901 −0.649503 0.760359i \(-0.725023\pi\)
−0.649503 + 0.760359i \(0.725023\pi\)
\(420\) 0 0
\(421\) −2.31110e9 −1.50950 −0.754748 0.656015i \(-0.772241\pi\)
−0.754748 + 0.656015i \(0.772241\pi\)
\(422\) 0 0
\(423\) 3.31539e7 0.0212982
\(424\) 0 0
\(425\) 4.62262e8 0.292097
\(426\) 0 0
\(427\) −1.89650e7 −0.0117884
\(428\) 0 0
\(429\) 2.96066e8 0.181046
\(430\) 0 0
\(431\) 2.03521e9 1.22444 0.612222 0.790686i \(-0.290276\pi\)
0.612222 + 0.790686i \(0.290276\pi\)
\(432\) 0 0
\(433\) 2.81028e9 1.66357 0.831786 0.555096i \(-0.187318\pi\)
0.831786 + 0.555096i \(0.187318\pi\)
\(434\) 0 0
\(435\) 3.05377e8 0.177878
\(436\) 0 0
\(437\) 1.26924e9 0.727541
\(438\) 0 0
\(439\) −1.58225e9 −0.892587 −0.446293 0.894887i \(-0.647256\pi\)
−0.446293 + 0.894887i \(0.647256\pi\)
\(440\) 0 0
\(441\) −2.15809e8 −0.119821
\(442\) 0 0
\(443\) 4.88989e8 0.267231 0.133615 0.991033i \(-0.457341\pi\)
0.133615 + 0.991033i \(0.457341\pi\)
\(444\) 0 0
\(445\) 9.64626e8 0.518918
\(446\) 0 0
\(447\) 1.75569e9 0.929763
\(448\) 0 0
\(449\) 1.81072e9 0.944036 0.472018 0.881589i \(-0.343526\pi\)
0.472018 + 0.881589i \(0.343526\pi\)
\(450\) 0 0
\(451\) −7.32212e8 −0.375854
\(452\) 0 0
\(453\) 2.03064e9 1.02633
\(454\) 0 0
\(455\) −2.02366e7 −0.0100716
\(456\) 0 0
\(457\) −8.64311e8 −0.423607 −0.211804 0.977312i \(-0.567934\pi\)
−0.211804 + 0.977312i \(0.567934\pi\)
\(458\) 0 0
\(459\) −6.55140e8 −0.316220
\(460\) 0 0
\(461\) 4.85430e8 0.230767 0.115383 0.993321i \(-0.463190\pi\)
0.115383 + 0.993321i \(0.463190\pi\)
\(462\) 0 0
\(463\) −1.85376e9 −0.867999 −0.434000 0.900913i \(-0.642898\pi\)
−0.434000 + 0.900913i \(0.642898\pi\)
\(464\) 0 0
\(465\) −4.57875e8 −0.211184
\(466\) 0 0
\(467\) −3.16192e7 −0.0143662 −0.00718310 0.999974i \(-0.502286\pi\)
−0.00718310 + 0.999974i \(0.502286\pi\)
\(468\) 0 0
\(469\) 1.53544e8 0.0687272
\(470\) 0 0
\(471\) 3.30112e9 1.45576
\(472\) 0 0
\(473\) 9.23199e8 0.401126
\(474\) 0 0
\(475\) −1.41551e9 −0.606017
\(476\) 0 0
\(477\) −3.14683e8 −0.132757
\(478\) 0 0
\(479\) 3.11838e9 1.29645 0.648223 0.761451i \(-0.275513\pi\)
0.648223 + 0.761451i \(0.275513\pi\)
\(480\) 0 0
\(481\) 1.71740e9 0.703661
\(482\) 0 0
\(483\) 1.28502e8 0.0518914
\(484\) 0 0
\(485\) −1.44283e9 −0.574275
\(486\) 0 0
\(487\) 3.37608e9 1.32453 0.662265 0.749270i \(-0.269596\pi\)
0.662265 + 0.749270i \(0.269596\pi\)
\(488\) 0 0
\(489\) 4.60166e8 0.177965
\(490\) 0 0
\(491\) 4.55043e9 1.73487 0.867436 0.497548i \(-0.165766\pi\)
0.867436 + 0.497548i \(0.165766\pi\)
\(492\) 0 0
\(493\) 4.06106e8 0.152642
\(494\) 0 0
\(495\) −3.65328e7 −0.0135383
\(496\) 0 0
\(497\) 1.35266e8 0.0494242
\(498\) 0 0
\(499\) 6.25733e8 0.225443 0.112722 0.993627i \(-0.464043\pi\)
0.112722 + 0.993627i \(0.464043\pi\)
\(500\) 0 0
\(501\) −2.93631e9 −1.04321
\(502\) 0 0
\(503\) 3.43240e9 1.20257 0.601285 0.799035i \(-0.294656\pi\)
0.601285 + 0.799035i \(0.294656\pi\)
\(504\) 0 0
\(505\) 6.89195e8 0.238135
\(506\) 0 0
\(507\) 2.10596e9 0.717667
\(508\) 0 0
\(509\) 1.36035e9 0.457234 0.228617 0.973516i \(-0.426580\pi\)
0.228617 + 0.973516i \(0.426580\pi\)
\(510\) 0 0
\(511\) 1.75682e8 0.0582445
\(512\) 0 0
\(513\) 2.00613e9 0.656066
\(514\) 0 0
\(515\) 2.07745e9 0.670202
\(516\) 0 0
\(517\) −1.68016e8 −0.0534727
\(518\) 0 0
\(519\) −5.93910e9 −1.86481
\(520\) 0 0
\(521\) −2.51109e9 −0.777912 −0.388956 0.921256i \(-0.627164\pi\)
−0.388956 + 0.921256i \(0.627164\pi\)
\(522\) 0 0
\(523\) −4.25045e9 −1.29921 −0.649605 0.760272i \(-0.725066\pi\)
−0.649605 + 0.760272i \(0.725066\pi\)
\(524\) 0 0
\(525\) −1.43311e8 −0.0432237
\(526\) 0 0
\(527\) −6.08907e8 −0.181223
\(528\) 0 0
\(529\) 2.26323e8 0.0664713
\(530\) 0 0
\(531\) −3.45744e8 −0.100213
\(532\) 0 0
\(533\) 2.47240e9 0.707250
\(534\) 0 0
\(535\) −2.18974e9 −0.618237
\(536\) 0 0
\(537\) 1.57833e9 0.439833
\(538\) 0 0
\(539\) 1.09366e9 0.300832
\(540\) 0 0
\(541\) 1.06279e9 0.288574 0.144287 0.989536i \(-0.453911\pi\)
0.144287 + 0.989536i \(0.453911\pi\)
\(542\) 0 0
\(543\) −1.06244e9 −0.284778
\(544\) 0 0
\(545\) −6.25202e7 −0.0165437
\(546\) 0 0
\(547\) 3.28564e9 0.858350 0.429175 0.903221i \(-0.358804\pi\)
0.429175 + 0.903221i \(0.358804\pi\)
\(548\) 0 0
\(549\) 1.15605e8 0.0298178
\(550\) 0 0
\(551\) −1.24355e9 −0.316689
\(552\) 0 0
\(553\) 8.17441e7 0.0205550
\(554\) 0 0
\(555\) −1.97654e9 −0.490772
\(556\) 0 0
\(557\) −4.04436e9 −0.991645 −0.495823 0.868424i \(-0.665133\pi\)
−0.495823 + 0.868424i \(0.665133\pi\)
\(558\) 0 0
\(559\) −3.11729e9 −0.754806
\(560\) 0 0
\(561\) −4.53133e8 −0.108357
\(562\) 0 0
\(563\) 1.60197e9 0.378333 0.189167 0.981945i \(-0.439421\pi\)
0.189167 + 0.981945i \(0.439421\pi\)
\(564\) 0 0
\(565\) −1.48197e8 −0.0345677
\(566\) 0 0
\(567\) 2.27856e8 0.0524952
\(568\) 0 0
\(569\) −5.94461e9 −1.35279 −0.676396 0.736539i \(-0.736459\pi\)
−0.676396 + 0.736539i \(0.736459\pi\)
\(570\) 0 0
\(571\) 1.01942e9 0.229154 0.114577 0.993414i \(-0.463449\pi\)
0.114577 + 0.993414i \(0.463449\pi\)
\(572\) 0 0
\(573\) −2.65971e9 −0.590600
\(574\) 0 0
\(575\) −4.04961e9 −0.888334
\(576\) 0 0
\(577\) −5.05194e8 −0.109482 −0.0547410 0.998501i \(-0.517433\pi\)
−0.0547410 + 0.998501i \(0.517433\pi\)
\(578\) 0 0
\(579\) −8.51792e9 −1.82372
\(580\) 0 0
\(581\) 2.79739e8 0.0591748
\(582\) 0 0
\(583\) 1.59473e9 0.333310
\(584\) 0 0
\(585\) 1.23357e8 0.0254753
\(586\) 0 0
\(587\) −6.05441e8 −0.123549 −0.0617744 0.998090i \(-0.519676\pi\)
−0.0617744 + 0.998090i \(0.519676\pi\)
\(588\) 0 0
\(589\) 1.86455e9 0.375985
\(590\) 0 0
\(591\) 5.66831e9 1.12953
\(592\) 0 0
\(593\) −2.44359e8 −0.0481212 −0.0240606 0.999711i \(-0.507659\pi\)
−0.0240606 + 0.999711i \(0.507659\pi\)
\(594\) 0 0
\(595\) 3.09725e7 0.00602790
\(596\) 0 0
\(597\) −1.68598e9 −0.324296
\(598\) 0 0
\(599\) −4.76209e9 −0.905324 −0.452662 0.891682i \(-0.649526\pi\)
−0.452662 + 0.891682i \(0.649526\pi\)
\(600\) 0 0
\(601\) −4.27387e9 −0.803083 −0.401542 0.915841i \(-0.631525\pi\)
−0.401542 + 0.915841i \(0.631525\pi\)
\(602\) 0 0
\(603\) −9.35966e8 −0.173840
\(604\) 0 0
\(605\) 1.85139e8 0.0339902
\(606\) 0 0
\(607\) −6.40384e9 −1.16220 −0.581099 0.813833i \(-0.697377\pi\)
−0.581099 + 0.813833i \(0.697377\pi\)
\(608\) 0 0
\(609\) −1.25901e8 −0.0225876
\(610\) 0 0
\(611\) 5.67324e8 0.100621
\(612\) 0 0
\(613\) −8.98899e9 −1.57616 −0.788078 0.615575i \(-0.788924\pi\)
−0.788078 + 0.615575i \(0.788924\pi\)
\(614\) 0 0
\(615\) −2.84546e9 −0.493275
\(616\) 0 0
\(617\) −2.11391e9 −0.362316 −0.181158 0.983454i \(-0.557985\pi\)
−0.181158 + 0.983454i \(0.557985\pi\)
\(618\) 0 0
\(619\) 4.25875e9 0.721713 0.360856 0.932621i \(-0.382484\pi\)
0.360856 + 0.932621i \(0.382484\pi\)
\(620\) 0 0
\(621\) 5.73930e9 0.961698
\(622\) 0 0
\(623\) −3.97699e8 −0.0658940
\(624\) 0 0
\(625\) 3.66306e9 0.600156
\(626\) 0 0
\(627\) 1.38755e9 0.224809
\(628\) 0 0
\(629\) −2.62850e9 −0.421145
\(630\) 0 0
\(631\) −9.55885e9 −1.51462 −0.757309 0.653057i \(-0.773486\pi\)
−0.757309 + 0.653057i \(0.773486\pi\)
\(632\) 0 0
\(633\) −8.62617e9 −1.35178
\(634\) 0 0
\(635\) 2.01685e9 0.312583
\(636\) 0 0
\(637\) −3.69289e9 −0.566080
\(638\) 0 0
\(639\) −8.24543e8 −0.125015
\(640\) 0 0
\(641\) 7.93190e9 1.18953 0.594763 0.803901i \(-0.297246\pi\)
0.594763 + 0.803901i \(0.297246\pi\)
\(642\) 0 0
\(643\) 3.67790e9 0.545583 0.272792 0.962073i \(-0.412053\pi\)
0.272792 + 0.962073i \(0.412053\pi\)
\(644\) 0 0
\(645\) 3.58765e9 0.526443
\(646\) 0 0
\(647\) 3.34066e9 0.484917 0.242459 0.970162i \(-0.422046\pi\)
0.242459 + 0.970162i \(0.422046\pi\)
\(648\) 0 0
\(649\) 1.75215e9 0.251602
\(650\) 0 0
\(651\) 1.88774e8 0.0268169
\(652\) 0 0
\(653\) 1.23649e10 1.73778 0.868891 0.495004i \(-0.164833\pi\)
0.868891 + 0.495004i \(0.164833\pi\)
\(654\) 0 0
\(655\) −4.41487e9 −0.613865
\(656\) 0 0
\(657\) −1.07091e9 −0.147325
\(658\) 0 0
\(659\) −8.58087e9 −1.16797 −0.583986 0.811764i \(-0.698508\pi\)
−0.583986 + 0.811764i \(0.698508\pi\)
\(660\) 0 0
\(661\) −8.09007e9 −1.08955 −0.544775 0.838582i \(-0.683385\pi\)
−0.544775 + 0.838582i \(0.683385\pi\)
\(662\) 0 0
\(663\) 1.53006e9 0.203897
\(664\) 0 0
\(665\) −9.48417e7 −0.0125062
\(666\) 0 0
\(667\) −3.55766e9 −0.464220
\(668\) 0 0
\(669\) 7.21274e9 0.931341
\(670\) 0 0
\(671\) −5.85859e8 −0.0748625
\(672\) 0 0
\(673\) −1.38929e10 −1.75687 −0.878436 0.477860i \(-0.841413\pi\)
−0.878436 + 0.477860i \(0.841413\pi\)
\(674\) 0 0
\(675\) −6.40072e9 −0.801061
\(676\) 0 0
\(677\) 6.36018e9 0.787788 0.393894 0.919156i \(-0.371128\pi\)
0.393894 + 0.919156i \(0.371128\pi\)
\(678\) 0 0
\(679\) 5.94855e8 0.0729233
\(680\) 0 0
\(681\) −6.29077e9 −0.763289
\(682\) 0 0
\(683\) 7.46187e9 0.896139 0.448069 0.893999i \(-0.352112\pi\)
0.448069 + 0.893999i \(0.352112\pi\)
\(684\) 0 0
\(685\) −8.00400e8 −0.0951459
\(686\) 0 0
\(687\) 6.42182e9 0.755631
\(688\) 0 0
\(689\) −5.38480e9 −0.627195
\(690\) 0 0
\(691\) 1.01338e10 1.16842 0.584212 0.811601i \(-0.301404\pi\)
0.584212 + 0.811601i \(0.301404\pi\)
\(692\) 0 0
\(693\) 1.50618e7 0.00171914
\(694\) 0 0
\(695\) 1.52598e9 0.172426
\(696\) 0 0
\(697\) −3.78404e9 −0.423293
\(698\) 0 0
\(699\) 7.53345e9 0.834303
\(700\) 0 0
\(701\) 1.31006e10 1.43641 0.718206 0.695831i \(-0.244964\pi\)
0.718206 + 0.695831i \(0.244964\pi\)
\(702\) 0 0
\(703\) 8.04883e9 0.873754
\(704\) 0 0
\(705\) −6.52927e8 −0.0701782
\(706\) 0 0
\(707\) −2.84143e8 −0.0302391
\(708\) 0 0
\(709\) −1.04601e10 −1.10223 −0.551116 0.834429i \(-0.685798\pi\)
−0.551116 + 0.834429i \(0.685798\pi\)
\(710\) 0 0
\(711\) −4.98290e8 −0.0519923
\(712\) 0 0
\(713\) 5.33428e9 0.551141
\(714\) 0 0
\(715\) −6.25143e8 −0.0639600
\(716\) 0 0
\(717\) −1.20300e10 −1.21885
\(718\) 0 0
\(719\) −6.14452e9 −0.616505 −0.308253 0.951305i \(-0.599744\pi\)
−0.308253 + 0.951305i \(0.599744\pi\)
\(720\) 0 0
\(721\) −8.56497e8 −0.0851045
\(722\) 0 0
\(723\) 3.38831e9 0.333426
\(724\) 0 0
\(725\) 3.96766e9 0.386680
\(726\) 0 0
\(727\) 1.47022e9 0.141910 0.0709550 0.997480i \(-0.477395\pi\)
0.0709550 + 0.997480i \(0.477395\pi\)
\(728\) 0 0
\(729\) 8.92054e9 0.852796
\(730\) 0 0
\(731\) 4.77105e9 0.451755
\(732\) 0 0
\(733\) 6.30989e8 0.0591777 0.0295888 0.999562i \(-0.490580\pi\)
0.0295888 + 0.999562i \(0.490580\pi\)
\(734\) 0 0
\(735\) 4.25010e9 0.394815
\(736\) 0 0
\(737\) 4.74324e9 0.436454
\(738\) 0 0
\(739\) −7.22368e9 −0.658420 −0.329210 0.944257i \(-0.606782\pi\)
−0.329210 + 0.944257i \(0.606782\pi\)
\(740\) 0 0
\(741\) −4.68524e9 −0.423027
\(742\) 0 0
\(743\) −7.72789e9 −0.691194 −0.345597 0.938383i \(-0.612324\pi\)
−0.345597 + 0.938383i \(0.612324\pi\)
\(744\) 0 0
\(745\) −3.70714e9 −0.328467
\(746\) 0 0
\(747\) −1.70522e9 −0.149678
\(748\) 0 0
\(749\) 9.02793e8 0.0785059
\(750\) 0 0
\(751\) 1.54417e10 1.33032 0.665158 0.746702i \(-0.268364\pi\)
0.665158 + 0.746702i \(0.268364\pi\)
\(752\) 0 0
\(753\) 8.27993e9 0.706715
\(754\) 0 0
\(755\) −4.28769e9 −0.362584
\(756\) 0 0
\(757\) 1.94159e10 1.62676 0.813378 0.581735i \(-0.197626\pi\)
0.813378 + 0.581735i \(0.197626\pi\)
\(758\) 0 0
\(759\) 3.96964e9 0.329538
\(760\) 0 0
\(761\) 5.03230e8 0.0413924 0.0206962 0.999786i \(-0.493412\pi\)
0.0206962 + 0.999786i \(0.493412\pi\)
\(762\) 0 0
\(763\) 2.57760e7 0.00210078
\(764\) 0 0
\(765\) −1.88800e8 −0.0152471
\(766\) 0 0
\(767\) −5.91632e9 −0.473443
\(768\) 0 0
\(769\) −1.87701e10 −1.48842 −0.744208 0.667948i \(-0.767173\pi\)
−0.744208 + 0.667948i \(0.767173\pi\)
\(770\) 0 0
\(771\) −1.90447e10 −1.49652
\(772\) 0 0
\(773\) 2.39166e10 1.86239 0.931196 0.364518i \(-0.118766\pi\)
0.931196 + 0.364518i \(0.118766\pi\)
\(774\) 0 0
\(775\) −5.94902e9 −0.459081
\(776\) 0 0
\(777\) 8.14892e8 0.0623199
\(778\) 0 0
\(779\) 1.15872e10 0.878210
\(780\) 0 0
\(781\) 4.17858e9 0.313870
\(782\) 0 0
\(783\) −5.62315e9 −0.418614
\(784\) 0 0
\(785\) −6.97032e9 −0.514291
\(786\) 0 0
\(787\) −2.09777e10 −1.53408 −0.767038 0.641602i \(-0.778270\pi\)
−0.767038 + 0.641602i \(0.778270\pi\)
\(788\) 0 0
\(789\) 2.52151e10 1.82764
\(790\) 0 0
\(791\) 6.10991e7 0.00438952
\(792\) 0 0
\(793\) 1.97822e9 0.140870
\(794\) 0 0
\(795\) 6.19731e9 0.437440
\(796\) 0 0
\(797\) 1.55513e10 1.08809 0.544043 0.839057i \(-0.316893\pi\)
0.544043 + 0.839057i \(0.316893\pi\)
\(798\) 0 0
\(799\) −8.68297e8 −0.0602219
\(800\) 0 0
\(801\) 2.42427e9 0.166673
\(802\) 0 0
\(803\) 5.42711e9 0.369883
\(804\) 0 0
\(805\) −2.71332e8 −0.0183322
\(806\) 0 0
\(807\) 2.72858e10 1.82759
\(808\) 0 0
\(809\) −1.19959e10 −0.796548 −0.398274 0.917266i \(-0.630391\pi\)
−0.398274 + 0.917266i \(0.630391\pi\)
\(810\) 0 0
\(811\) −1.18093e10 −0.777412 −0.388706 0.921362i \(-0.627078\pi\)
−0.388706 + 0.921362i \(0.627078\pi\)
\(812\) 0 0
\(813\) 1.69403e10 1.10562
\(814\) 0 0
\(815\) −9.71640e8 −0.0628715
\(816\) 0 0
\(817\) −1.46096e10 −0.937262
\(818\) 0 0
\(819\) −5.08580e7 −0.00323494
\(820\) 0 0
\(821\) −1.96308e10 −1.23805 −0.619024 0.785372i \(-0.712472\pi\)
−0.619024 + 0.785372i \(0.712472\pi\)
\(822\) 0 0
\(823\) 2.67601e9 0.167336 0.0836679 0.996494i \(-0.473337\pi\)
0.0836679 + 0.996494i \(0.473337\pi\)
\(824\) 0 0
\(825\) −4.42712e9 −0.274494
\(826\) 0 0
\(827\) −9.26722e8 −0.0569745 −0.0284872 0.999594i \(-0.509069\pi\)
−0.0284872 + 0.999594i \(0.509069\pi\)
\(828\) 0 0
\(829\) −2.94467e10 −1.79513 −0.897565 0.440882i \(-0.854666\pi\)
−0.897565 + 0.440882i \(0.854666\pi\)
\(830\) 0 0
\(831\) 2.52900e10 1.52878
\(832\) 0 0
\(833\) 5.65201e9 0.338802
\(834\) 0 0
\(835\) 6.20001e9 0.368544
\(836\) 0 0
\(837\) 8.43123e9 0.496995
\(838\) 0 0
\(839\) 8.93988e9 0.522595 0.261297 0.965258i \(-0.415850\pi\)
0.261297 + 0.965258i \(0.415850\pi\)
\(840\) 0 0
\(841\) −1.37642e10 −0.797931
\(842\) 0 0
\(843\) 7.51071e8 0.0431802
\(844\) 0 0
\(845\) −4.44674e9 −0.253538
\(846\) 0 0
\(847\) −7.63296e7 −0.00431619
\(848\) 0 0
\(849\) −1.11530e10 −0.625480
\(850\) 0 0
\(851\) 2.30268e10 1.28080
\(852\) 0 0
\(853\) 1.71708e10 0.947257 0.473628 0.880725i \(-0.342944\pi\)
0.473628 + 0.880725i \(0.342944\pi\)
\(854\) 0 0
\(855\) 5.78130e8 0.0316333
\(856\) 0 0
\(857\) −7.71458e9 −0.418677 −0.209339 0.977843i \(-0.567131\pi\)
−0.209339 + 0.977843i \(0.567131\pi\)
\(858\) 0 0
\(859\) −1.68606e10 −0.907606 −0.453803 0.891102i \(-0.649933\pi\)
−0.453803 + 0.891102i \(0.649933\pi\)
\(860\) 0 0
\(861\) 1.17313e9 0.0626377
\(862\) 0 0
\(863\) 2.42922e10 1.28656 0.643278 0.765633i \(-0.277574\pi\)
0.643278 + 0.765633i \(0.277574\pi\)
\(864\) 0 0
\(865\) 1.25404e10 0.658802
\(866\) 0 0
\(867\) 1.79675e10 0.936311
\(868\) 0 0
\(869\) 2.52521e9 0.130535
\(870\) 0 0
\(871\) −1.60161e10 −0.821283
\(872\) 0 0
\(873\) −3.62608e9 −0.184454
\(874\) 0 0
\(875\) 6.54379e8 0.0330218
\(876\) 0 0
\(877\) 2.67740e10 1.34034 0.670169 0.742209i \(-0.266222\pi\)
0.670169 + 0.742209i \(0.266222\pi\)
\(878\) 0 0
\(879\) −1.90058e10 −0.943899
\(880\) 0 0
\(881\) 1.10654e10 0.545197 0.272598 0.962128i \(-0.412117\pi\)
0.272598 + 0.962128i \(0.412117\pi\)
\(882\) 0 0
\(883\) −2.06264e10 −1.00824 −0.504118 0.863635i \(-0.668182\pi\)
−0.504118 + 0.863635i \(0.668182\pi\)
\(884\) 0 0
\(885\) 6.80903e9 0.330205
\(886\) 0 0
\(887\) 7.62436e9 0.366835 0.183417 0.983035i \(-0.441284\pi\)
0.183417 + 0.983035i \(0.441284\pi\)
\(888\) 0 0
\(889\) −8.31512e8 −0.0396928
\(890\) 0 0
\(891\) 7.03884e9 0.333372
\(892\) 0 0
\(893\) 2.65884e9 0.124943
\(894\) 0 0
\(895\) −3.33264e9 −0.155385
\(896\) 0 0
\(897\) −1.34040e10 −0.620097
\(898\) 0 0
\(899\) −5.22633e9 −0.239904
\(900\) 0 0
\(901\) 8.24151e9 0.375379
\(902\) 0 0
\(903\) −1.47913e9 −0.0668495
\(904\) 0 0
\(905\) 2.24334e9 0.100606
\(906\) 0 0
\(907\) 3.36091e10 1.49566 0.747828 0.663893i \(-0.231097\pi\)
0.747828 + 0.663893i \(0.231097\pi\)
\(908\) 0 0
\(909\) 1.73206e9 0.0764874
\(910\) 0 0
\(911\) 3.89170e10 1.70540 0.852698 0.522404i \(-0.174965\pi\)
0.852698 + 0.522404i \(0.174965\pi\)
\(912\) 0 0
\(913\) 8.64161e9 0.375791
\(914\) 0 0
\(915\) −2.27671e9 −0.0982504
\(916\) 0 0
\(917\) 1.82017e9 0.0779507
\(918\) 0 0
\(919\) −4.35993e10 −1.85300 −0.926499 0.376296i \(-0.877197\pi\)
−0.926499 + 0.376296i \(0.877197\pi\)
\(920\) 0 0
\(921\) 3.69367e10 1.55794
\(922\) 0 0
\(923\) −1.41094e10 −0.590614
\(924\) 0 0
\(925\) −2.56805e10 −1.06686
\(926\) 0 0
\(927\) 5.22098e9 0.215265
\(928\) 0 0
\(929\) −3.06911e10 −1.25591 −0.627954 0.778250i \(-0.716108\pi\)
−0.627954 + 0.778250i \(0.716108\pi\)
\(930\) 0 0
\(931\) −1.73072e10 −0.702915
\(932\) 0 0
\(933\) 3.16761e10 1.27687
\(934\) 0 0
\(935\) 9.56790e8 0.0382804
\(936\) 0 0
\(937\) 3.51153e9 0.139446 0.0697232 0.997566i \(-0.477788\pi\)
0.0697232 + 0.997566i \(0.477788\pi\)
\(938\) 0 0
\(939\) −6.70402e9 −0.264244
\(940\) 0 0
\(941\) −1.84266e10 −0.720911 −0.360455 0.932776i \(-0.617379\pi\)
−0.360455 + 0.932776i \(0.617379\pi\)
\(942\) 0 0
\(943\) 3.31498e10 1.28733
\(944\) 0 0
\(945\) −4.28860e8 −0.0165312
\(946\) 0 0
\(947\) −8.78798e9 −0.336251 −0.168126 0.985766i \(-0.553771\pi\)
−0.168126 + 0.985766i \(0.553771\pi\)
\(948\) 0 0
\(949\) −1.83253e10 −0.696015
\(950\) 0 0
\(951\) −4.82663e9 −0.181975
\(952\) 0 0
\(953\) 1.85304e10 0.693521 0.346761 0.937954i \(-0.387282\pi\)
0.346761 + 0.937954i \(0.387282\pi\)
\(954\) 0 0
\(955\) 5.61598e9 0.208648
\(956\) 0 0
\(957\) −3.88930e9 −0.143443
\(958\) 0 0
\(959\) 3.29991e8 0.0120819
\(960\) 0 0
\(961\) −1.96764e10 −0.715176
\(962\) 0 0
\(963\) −5.50319e9 −0.198574
\(964\) 0 0
\(965\) 1.79856e10 0.644286
\(966\) 0 0
\(967\) −4.71498e10 −1.67682 −0.838412 0.545037i \(-0.816516\pi\)
−0.838412 + 0.545037i \(0.816516\pi\)
\(968\) 0 0
\(969\) 7.17082e9 0.253184
\(970\) 0 0
\(971\) 8.76215e9 0.307145 0.153572 0.988137i \(-0.450922\pi\)
0.153572 + 0.988137i \(0.450922\pi\)
\(972\) 0 0
\(973\) −6.29135e8 −0.0218952
\(974\) 0 0
\(975\) 1.49487e10 0.516519
\(976\) 0 0
\(977\) −6.33661e9 −0.217383 −0.108692 0.994076i \(-0.534666\pi\)
−0.108692 + 0.994076i \(0.534666\pi\)
\(978\) 0 0
\(979\) −1.22856e10 −0.418462
\(980\) 0 0
\(981\) −1.57124e8 −0.00531374
\(982\) 0 0
\(983\) −4.47312e10 −1.50201 −0.751006 0.660295i \(-0.770431\pi\)
−0.751006 + 0.660295i \(0.770431\pi\)
\(984\) 0 0
\(985\) −1.19686e10 −0.399041
\(986\) 0 0
\(987\) 2.69190e8 0.00891147
\(988\) 0 0
\(989\) −4.17964e10 −1.37389
\(990\) 0 0
\(991\) −4.62109e10 −1.50830 −0.754148 0.656705i \(-0.771950\pi\)
−0.754148 + 0.656705i \(0.771950\pi\)
\(992\) 0 0
\(993\) 1.73150e10 0.561178
\(994\) 0 0
\(995\) 3.55994e9 0.114567
\(996\) 0 0
\(997\) −1.52392e10 −0.487001 −0.243501 0.969901i \(-0.578296\pi\)
−0.243501 + 0.969901i \(0.578296\pi\)
\(998\) 0 0
\(999\) 3.63956e10 1.15497
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 176.8.a.e.1.1 2
4.3 odd 2 22.8.a.d.1.2 2
12.11 even 2 198.8.a.f.1.2 2
20.19 odd 2 550.8.a.d.1.1 2
44.43 even 2 242.8.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.8.a.d.1.2 2 4.3 odd 2
176.8.a.e.1.1 2 1.1 even 1 trivial
198.8.a.f.1.2 2 12.11 even 2
242.8.a.h.1.2 2 44.43 even 2
550.8.a.d.1.1 2 20.19 odd 2