Properties

Label 8.14.a.b.1.1
Level $8$
Weight $14$
Character 8.1
Self dual yes
Analytic conductor $8.578$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,14,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(8.57847431615\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{781}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 195 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(14.4732\) of defining polynomial
Character \(\chi\) \(=\) 8.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1352.57 q^{3} -12224.8 q^{5} +452526. q^{7} +235118. q^{9} +O(q^{10})\) \(q-1352.57 q^{3} -12224.8 q^{5} +452526. q^{7} +235118. q^{9} +7.28729e6 q^{11} +2.80664e7 q^{13} +1.65349e7 q^{15} -1.45191e8 q^{17} +1.08428e8 q^{19} -6.12072e8 q^{21} +1.59777e8 q^{23} -1.07126e9 q^{25} +1.83842e9 q^{27} +4.40214e9 q^{29} +5.53608e9 q^{31} -9.85656e9 q^{33} -5.53205e9 q^{35} +1.25151e10 q^{37} -3.79617e10 q^{39} +1.51058e10 q^{41} -3.15026e10 q^{43} -2.87427e9 q^{45} +3.50306e10 q^{47} +1.07891e11 q^{49} +1.96381e11 q^{51} -1.09912e11 q^{53} -8.90858e10 q^{55} -1.46656e11 q^{57} -1.18303e11 q^{59} -2.98314e11 q^{61} +1.06397e11 q^{63} -3.43107e11 q^{65} +7.44158e11 q^{67} -2.16109e11 q^{69} -1.15098e12 q^{71} +1.84863e12 q^{73} +1.44895e12 q^{75} +3.29769e12 q^{77} -2.20082e12 q^{79} -2.86144e12 q^{81} -1.92754e12 q^{83} +1.77494e12 q^{85} -5.95420e12 q^{87} -5.50546e12 q^{89} +1.27008e13 q^{91} -7.48793e12 q^{93} -1.32551e12 q^{95} +1.35322e13 q^{97} +1.71337e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 872 q^{3} + 18476 q^{5} + 110928 q^{7} + 3589498 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 872 q^{3} + 18476 q^{5} + 110928 q^{7} + 3589498 q^{9} + 16474040 q^{11} + 18744572 q^{13} + 84830960 q^{15} - 153793628 q^{17} - 118747640 q^{19} - 1371980736 q^{21} + 718268912 q^{23} - 1349419874 q^{25} + 5753785616 q^{27} + 309341340 q^{29} + 5767504192 q^{31} + 10579993952 q^{33} - 16019391264 q^{35} - 11621553300 q^{37} - 58698760912 q^{39} + 1311168276 q^{41} - 29595620104 q^{43} + 100107952252 q^{45} + 12313617888 q^{47} + 127691156146 q^{49} + 177245377360 q^{51} - 38006007028 q^{53} + 192954931664 q^{55} - 652023429728 q^{57} + 253345911704 q^{59} - 647244384292 q^{61} - 1039453222896 q^{63} - 629294375512 q^{65} + 1619993806312 q^{67} + 1026293771456 q^{69} - 1040270142512 q^{71} + 4005283908692 q^{73} + 830155668760 q^{75} + 159512776896 q^{77} - 2521777572064 q^{79} + 500597403058 q^{81} - 290486230904 q^{83} + 1510847254552 q^{85} - 15058906809936 q^{87} - 8723755657740 q^{89} + 15885098476896 q^{91} - 6973121519360 q^{93} - 8299984202576 q^{95} + 9601712299972 q^{97} + 32529223326424 q^{99}+O(q^{100}) \) 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
\(3\) −1352.57 −1.07120 −0.535601 0.844471i \(-0.679915\pi\)
−0.535601 + 0.844471i \(0.679915\pi\)
\(4\) 0 0
\(5\) −12224.8 −0.349895 −0.174947 0.984578i \(-0.555976\pi\)
−0.174947 + 0.984578i \(0.555976\pi\)
\(6\) 0 0
\(7\) 452526. 1.45381 0.726903 0.686740i \(-0.240959\pi\)
0.726903 + 0.686740i \(0.240959\pi\)
\(8\) 0 0
\(9\) 235118. 0.147472
\(10\) 0 0
\(11\) 7.28729e6 1.24026 0.620131 0.784498i \(-0.287079\pi\)
0.620131 + 0.784498i \(0.287079\pi\)
\(12\) 0 0
\(13\) 2.80664e7 1.61270 0.806352 0.591435i \(-0.201438\pi\)
0.806352 + 0.591435i \(0.201438\pi\)
\(14\) 0 0
\(15\) 1.65349e7 0.374808
\(16\) 0 0
\(17\) −1.45191e8 −1.45889 −0.729446 0.684038i \(-0.760222\pi\)
−0.729446 + 0.684038i \(0.760222\pi\)
\(18\) 0 0
\(19\) 1.08428e8 0.528740 0.264370 0.964421i \(-0.414836\pi\)
0.264370 + 0.964421i \(0.414836\pi\)
\(20\) 0 0
\(21\) −6.12072e8 −1.55732
\(22\) 0 0
\(23\) 1.59777e8 0.225052 0.112526 0.993649i \(-0.464106\pi\)
0.112526 + 0.993649i \(0.464106\pi\)
\(24\) 0 0
\(25\) −1.07126e9 −0.877574
\(26\) 0 0
\(27\) 1.83842e9 0.913229
\(28\) 0 0
\(29\) 4.40214e9 1.37429 0.687143 0.726522i \(-0.258865\pi\)
0.687143 + 0.726522i \(0.258865\pi\)
\(30\) 0 0
\(31\) 5.53608e9 1.12034 0.560172 0.828376i \(-0.310735\pi\)
0.560172 + 0.828376i \(0.310735\pi\)
\(32\) 0 0
\(33\) −9.85656e9 −1.32857
\(34\) 0 0
\(35\) −5.53205e9 −0.508679
\(36\) 0 0
\(37\) 1.25151e10 0.801905 0.400953 0.916099i \(-0.368679\pi\)
0.400953 + 0.916099i \(0.368679\pi\)
\(38\) 0 0
\(39\) −3.79617e10 −1.72753
\(40\) 0 0
\(41\) 1.51058e10 0.496648 0.248324 0.968677i \(-0.420120\pi\)
0.248324 + 0.968677i \(0.420120\pi\)
\(42\) 0 0
\(43\) −3.15026e10 −0.759979 −0.379990 0.924991i \(-0.624072\pi\)
−0.379990 + 0.924991i \(0.624072\pi\)
\(44\) 0 0
\(45\) −2.87427e9 −0.0515996
\(46\) 0 0
\(47\) 3.50306e10 0.474037 0.237018 0.971505i \(-0.423830\pi\)
0.237018 + 0.971505i \(0.423830\pi\)
\(48\) 0 0
\(49\) 1.07891e11 1.11355
\(50\) 0 0
\(51\) 1.96381e11 1.56277
\(52\) 0 0
\(53\) −1.09912e11 −0.681164 −0.340582 0.940215i \(-0.610624\pi\)
−0.340582 + 0.940215i \(0.610624\pi\)
\(54\) 0 0
\(55\) −8.90858e10 −0.433961
\(56\) 0 0
\(57\) −1.46656e11 −0.566387
\(58\) 0 0
\(59\) −1.18303e11 −0.365137 −0.182569 0.983193i \(-0.558441\pi\)
−0.182569 + 0.983193i \(0.558441\pi\)
\(60\) 0 0
\(61\) −2.98314e11 −0.741361 −0.370680 0.928760i \(-0.620875\pi\)
−0.370680 + 0.928760i \(0.620875\pi\)
\(62\) 0 0
\(63\) 1.06397e11 0.214395
\(64\) 0 0
\(65\) −3.43107e11 −0.564277
\(66\) 0 0
\(67\) 7.44158e11 1.00503 0.502515 0.864569i \(-0.332408\pi\)
0.502515 + 0.864569i \(0.332408\pi\)
\(68\) 0 0
\(69\) −2.16109e11 −0.241076
\(70\) 0 0
\(71\) −1.15098e12 −1.06632 −0.533162 0.846013i \(-0.678996\pi\)
−0.533162 + 0.846013i \(0.678996\pi\)
\(72\) 0 0
\(73\) 1.84863e12 1.42972 0.714861 0.699267i \(-0.246490\pi\)
0.714861 + 0.699267i \(0.246490\pi\)
\(74\) 0 0
\(75\) 1.44895e12 0.940058
\(76\) 0 0
\(77\) 3.29769e12 1.80310
\(78\) 0 0
\(79\) −2.20082e12 −1.01861 −0.509305 0.860586i \(-0.670098\pi\)
−0.509305 + 0.860586i \(0.670098\pi\)
\(80\) 0 0
\(81\) −2.86144e12 −1.12572
\(82\) 0 0
\(83\) −1.92754e12 −0.647136 −0.323568 0.946205i \(-0.604883\pi\)
−0.323568 + 0.946205i \(0.604883\pi\)
\(84\) 0 0
\(85\) 1.77494e12 0.510459
\(86\) 0 0
\(87\) −5.95420e12 −1.47214
\(88\) 0 0
\(89\) −5.50546e12 −1.17424 −0.587122 0.809498i \(-0.699739\pi\)
−0.587122 + 0.809498i \(0.699739\pi\)
\(90\) 0 0
\(91\) 1.27008e13 2.34456
\(92\) 0 0
\(93\) −7.48793e12 −1.20011
\(94\) 0 0
\(95\) −1.32551e12 −0.185003
\(96\) 0 0
\(97\) 1.35322e13 1.64950 0.824752 0.565495i \(-0.191315\pi\)
0.824752 + 0.565495i \(0.191315\pi\)
\(98\) 0 0
\(99\) 1.71337e12 0.182904
\(100\) 0 0
\(101\) −8.36824e12 −0.784414 −0.392207 0.919877i \(-0.628288\pi\)
−0.392207 + 0.919877i \(0.628288\pi\)
\(102\) 0 0
\(103\) 2.33937e12 0.193044 0.0965222 0.995331i \(-0.469228\pi\)
0.0965222 + 0.995331i \(0.469228\pi\)
\(104\) 0 0
\(105\) 7.48247e12 0.544897
\(106\) 0 0
\(107\) 2.43985e13 1.57170 0.785849 0.618418i \(-0.212226\pi\)
0.785849 + 0.618418i \(0.212226\pi\)
\(108\) 0 0
\(109\) 1.59545e11 0.00911196 0.00455598 0.999990i \(-0.498550\pi\)
0.00455598 + 0.999990i \(0.498550\pi\)
\(110\) 0 0
\(111\) −1.69275e13 −0.859002
\(112\) 0 0
\(113\) −2.90424e13 −1.31227 −0.656133 0.754645i \(-0.727809\pi\)
−0.656133 + 0.754645i \(0.727809\pi\)
\(114\) 0 0
\(115\) −1.95325e12 −0.0787446
\(116\) 0 0
\(117\) 6.59890e12 0.237828
\(118\) 0 0
\(119\) −6.57029e13 −2.12095
\(120\) 0 0
\(121\) 1.85819e13 0.538251
\(122\) 0 0
\(123\) −2.04316e13 −0.532010
\(124\) 0 0
\(125\) 2.80188e13 0.656953
\(126\) 0 0
\(127\) 5.45015e13 1.15261 0.576307 0.817233i \(-0.304493\pi\)
0.576307 + 0.817233i \(0.304493\pi\)
\(128\) 0 0
\(129\) 4.26094e13 0.814091
\(130\) 0 0
\(131\) −1.87712e12 −0.0324510 −0.0162255 0.999868i \(-0.505165\pi\)
−0.0162255 + 0.999868i \(0.505165\pi\)
\(132\) 0 0
\(133\) 4.90664e13 0.768686
\(134\) 0 0
\(135\) −2.24743e13 −0.319534
\(136\) 0 0
\(137\) −8.61421e13 −1.11309 −0.556547 0.830816i \(-0.687874\pi\)
−0.556547 + 0.830816i \(0.687874\pi\)
\(138\) 0 0
\(139\) 6.45595e13 0.759214 0.379607 0.925148i \(-0.376059\pi\)
0.379607 + 0.925148i \(0.376059\pi\)
\(140\) 0 0
\(141\) −4.73813e13 −0.507789
\(142\) 0 0
\(143\) 2.04528e14 2.00018
\(144\) 0 0
\(145\) −5.38154e13 −0.480855
\(146\) 0 0
\(147\) −1.45930e14 −1.19284
\(148\) 0 0
\(149\) −7.18629e13 −0.538015 −0.269007 0.963138i \(-0.586696\pi\)
−0.269007 + 0.963138i \(0.586696\pi\)
\(150\) 0 0
\(151\) −1.05962e14 −0.727445 −0.363723 0.931507i \(-0.618494\pi\)
−0.363723 + 0.931507i \(0.618494\pi\)
\(152\) 0 0
\(153\) −3.41371e13 −0.215145
\(154\) 0 0
\(155\) −6.76776e13 −0.392003
\(156\) 0 0
\(157\) −9.36766e13 −0.499210 −0.249605 0.968348i \(-0.580301\pi\)
−0.249605 + 0.968348i \(0.580301\pi\)
\(158\) 0 0
\(159\) 1.48663e14 0.729663
\(160\) 0 0
\(161\) 7.23033e13 0.327182
\(162\) 0 0
\(163\) −1.96048e14 −0.818734 −0.409367 0.912370i \(-0.634250\pi\)
−0.409367 + 0.912370i \(0.634250\pi\)
\(164\) 0 0
\(165\) 1.20495e14 0.464860
\(166\) 0 0
\(167\) 4.90223e14 1.74878 0.874392 0.485220i \(-0.161261\pi\)
0.874392 + 0.485220i \(0.161261\pi\)
\(168\) 0 0
\(169\) 4.84848e14 1.60082
\(170\) 0 0
\(171\) 2.54933e13 0.0779743
\(172\) 0 0
\(173\) −7.90181e13 −0.224092 −0.112046 0.993703i \(-0.535740\pi\)
−0.112046 + 0.993703i \(0.535740\pi\)
\(174\) 0 0
\(175\) −4.84772e14 −1.27582
\(176\) 0 0
\(177\) 1.60012e14 0.391135
\(178\) 0 0
\(179\) −3.61625e14 −0.821700 −0.410850 0.911703i \(-0.634768\pi\)
−0.410850 + 0.911703i \(0.634768\pi\)
\(180\) 0 0
\(181\) −1.55078e14 −0.327822 −0.163911 0.986475i \(-0.552411\pi\)
−0.163911 + 0.986475i \(0.552411\pi\)
\(182\) 0 0
\(183\) 4.03490e14 0.794147
\(184\) 0 0
\(185\) −1.52995e14 −0.280582
\(186\) 0 0
\(187\) −1.05805e15 −1.80941
\(188\) 0 0
\(189\) 8.31932e14 1.32766
\(190\) 0 0
\(191\) 1.82528e14 0.272027 0.136013 0.990707i \(-0.456571\pi\)
0.136013 + 0.990707i \(0.456571\pi\)
\(192\) 0 0
\(193\) −1.28326e15 −1.78728 −0.893639 0.448786i \(-0.851857\pi\)
−0.893639 + 0.448786i \(0.851857\pi\)
\(194\) 0 0
\(195\) 4.64075e14 0.604454
\(196\) 0 0
\(197\) −3.04303e14 −0.370916 −0.185458 0.982652i \(-0.559377\pi\)
−0.185458 + 0.982652i \(0.559377\pi\)
\(198\) 0 0
\(199\) 6.78433e13 0.0774394 0.0387197 0.999250i \(-0.487672\pi\)
0.0387197 + 0.999250i \(0.487672\pi\)
\(200\) 0 0
\(201\) −1.00652e15 −1.07659
\(202\) 0 0
\(203\) 1.99208e15 1.99794
\(204\) 0 0
\(205\) −1.84666e14 −0.173775
\(206\) 0 0
\(207\) 3.75664e13 0.0331889
\(208\) 0 0
\(209\) 7.90145e14 0.655777
\(210\) 0 0
\(211\) 8.63768e14 0.673847 0.336924 0.941532i \(-0.390614\pi\)
0.336924 + 0.941532i \(0.390614\pi\)
\(212\) 0 0
\(213\) 1.55678e15 1.14225
\(214\) 0 0
\(215\) 3.85114e14 0.265913
\(216\) 0 0
\(217\) 2.50522e15 1.62876
\(218\) 0 0
\(219\) −2.50040e15 −1.53152
\(220\) 0 0
\(221\) −4.07500e15 −2.35276
\(222\) 0 0
\(223\) −2.22939e15 −1.21396 −0.606981 0.794717i \(-0.707619\pi\)
−0.606981 + 0.794717i \(0.707619\pi\)
\(224\) 0 0
\(225\) −2.51871e14 −0.129417
\(226\) 0 0
\(227\) 3.44059e15 1.66903 0.834516 0.550983i \(-0.185747\pi\)
0.834516 + 0.550983i \(0.185747\pi\)
\(228\) 0 0
\(229\) 7.09971e14 0.325320 0.162660 0.986682i \(-0.447993\pi\)
0.162660 + 0.986682i \(0.447993\pi\)
\(230\) 0 0
\(231\) −4.46035e15 −1.93148
\(232\) 0 0
\(233\) −8.27446e14 −0.338786 −0.169393 0.985549i \(-0.554181\pi\)
−0.169393 + 0.985549i \(0.554181\pi\)
\(234\) 0 0
\(235\) −4.28243e14 −0.165863
\(236\) 0 0
\(237\) 2.97676e15 1.09114
\(238\) 0 0
\(239\) −1.13283e15 −0.393167 −0.196584 0.980487i \(-0.562985\pi\)
−0.196584 + 0.980487i \(0.562985\pi\)
\(240\) 0 0
\(241\) 2.81149e13 0.00924329 0.00462164 0.999989i \(-0.498529\pi\)
0.00462164 + 0.999989i \(0.498529\pi\)
\(242\) 0 0
\(243\) 9.39259e14 0.292647
\(244\) 0 0
\(245\) −1.31895e15 −0.389626
\(246\) 0 0
\(247\) 3.04318e15 0.852702
\(248\) 0 0
\(249\) 2.60713e15 0.693213
\(250\) 0 0
\(251\) −5.45538e14 −0.137704 −0.0688519 0.997627i \(-0.521934\pi\)
−0.0688519 + 0.997627i \(0.521934\pi\)
\(252\) 0 0
\(253\) 1.16434e15 0.279124
\(254\) 0 0
\(255\) −2.40073e15 −0.546804
\(256\) 0 0
\(257\) 4.41616e14 0.0956048 0.0478024 0.998857i \(-0.484778\pi\)
0.0478024 + 0.998857i \(0.484778\pi\)
\(258\) 0 0
\(259\) 5.66341e15 1.16581
\(260\) 0 0
\(261\) 1.03502e15 0.202668
\(262\) 0 0
\(263\) 1.77399e15 0.330551 0.165276 0.986247i \(-0.447149\pi\)
0.165276 + 0.986247i \(0.447149\pi\)
\(264\) 0 0
\(265\) 1.34365e15 0.238336
\(266\) 0 0
\(267\) 7.44651e15 1.25785
\(268\) 0 0
\(269\) −1.01131e16 −1.62741 −0.813703 0.581281i \(-0.802552\pi\)
−0.813703 + 0.581281i \(0.802552\pi\)
\(270\) 0 0
\(271\) −3.06778e15 −0.470461 −0.235231 0.971940i \(-0.575585\pi\)
−0.235231 + 0.971940i \(0.575585\pi\)
\(272\) 0 0
\(273\) −1.71787e16 −2.51150
\(274\) 0 0
\(275\) −7.80656e15 −1.08842
\(276\) 0 0
\(277\) 7.55487e15 1.00487 0.502433 0.864616i \(-0.332438\pi\)
0.502433 + 0.864616i \(0.332438\pi\)
\(278\) 0 0
\(279\) 1.30163e15 0.165219
\(280\) 0 0
\(281\) −6.02256e15 −0.729777 −0.364888 0.931051i \(-0.618893\pi\)
−0.364888 + 0.931051i \(0.618893\pi\)
\(282\) 0 0
\(283\) −7.97392e14 −0.0922698 −0.0461349 0.998935i \(-0.514690\pi\)
−0.0461349 + 0.998935i \(0.514690\pi\)
\(284\) 0 0
\(285\) 1.79284e15 0.198176
\(286\) 0 0
\(287\) 6.83577e15 0.722030
\(288\) 0 0
\(289\) 1.11760e16 1.12837
\(290\) 0 0
\(291\) −1.83033e16 −1.76695
\(292\) 0 0
\(293\) 1.71127e16 1.58008 0.790039 0.613057i \(-0.210060\pi\)
0.790039 + 0.613057i \(0.210060\pi\)
\(294\) 0 0
\(295\) 1.44623e15 0.127760
\(296\) 0 0
\(297\) 1.33971e16 1.13264
\(298\) 0 0
\(299\) 4.48437e15 0.362943
\(300\) 0 0
\(301\) −1.42558e16 −1.10486
\(302\) 0 0
\(303\) 1.13186e16 0.840265
\(304\) 0 0
\(305\) 3.64683e15 0.259398
\(306\) 0 0
\(307\) −1.08209e16 −0.737676 −0.368838 0.929494i \(-0.620244\pi\)
−0.368838 + 0.929494i \(0.620244\pi\)
\(308\) 0 0
\(309\) −3.16416e15 −0.206789
\(310\) 0 0
\(311\) −1.10167e16 −0.690411 −0.345206 0.938527i \(-0.612191\pi\)
−0.345206 + 0.938527i \(0.612191\pi\)
\(312\) 0 0
\(313\) 2.64690e16 1.59111 0.795555 0.605882i \(-0.207180\pi\)
0.795555 + 0.605882i \(0.207180\pi\)
\(314\) 0 0
\(315\) −1.30068e15 −0.0750158
\(316\) 0 0
\(317\) −3.88517e15 −0.215043 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(318\) 0 0
\(319\) 3.20797e16 1.70447
\(320\) 0 0
\(321\) −3.30007e16 −1.68361
\(322\) 0 0
\(323\) −1.57428e16 −0.771375
\(324\) 0 0
\(325\) −3.00663e16 −1.41527
\(326\) 0 0
\(327\) −2.15796e14 −0.00976074
\(328\) 0 0
\(329\) 1.58523e16 0.689157
\(330\) 0 0
\(331\) −3.69659e16 −1.54497 −0.772483 0.635035i \(-0.780986\pi\)
−0.772483 + 0.635035i \(0.780986\pi\)
\(332\) 0 0
\(333\) 2.94252e15 0.118258
\(334\) 0 0
\(335\) −9.09720e15 −0.351655
\(336\) 0 0
\(337\) −3.50293e16 −1.30268 −0.651340 0.758786i \(-0.725793\pi\)
−0.651340 + 0.758786i \(0.725793\pi\)
\(338\) 0 0
\(339\) 3.92818e16 1.40570
\(340\) 0 0
\(341\) 4.03431e16 1.38952
\(342\) 0 0
\(343\) 4.97864e15 0.165082
\(344\) 0 0
\(345\) 2.64190e15 0.0843513
\(346\) 0 0
\(347\) 3.29030e16 1.01180 0.505899 0.862593i \(-0.331161\pi\)
0.505899 + 0.862593i \(0.331161\pi\)
\(348\) 0 0
\(349\) −4.00420e16 −1.18618 −0.593090 0.805136i \(-0.702092\pi\)
−0.593090 + 0.805136i \(0.702092\pi\)
\(350\) 0 0
\(351\) 5.15978e16 1.47277
\(352\) 0 0
\(353\) −1.37884e16 −0.379296 −0.189648 0.981852i \(-0.560735\pi\)
−0.189648 + 0.981852i \(0.560735\pi\)
\(354\) 0 0
\(355\) 1.40705e16 0.373101
\(356\) 0 0
\(357\) 8.88677e16 2.27196
\(358\) 0 0
\(359\) −4.30837e14 −0.0106218 −0.00531092 0.999986i \(-0.501691\pi\)
−0.00531092 + 0.999986i \(0.501691\pi\)
\(360\) 0 0
\(361\) −3.02964e16 −0.720434
\(362\) 0 0
\(363\) −2.51333e16 −0.576575
\(364\) 0 0
\(365\) −2.25992e16 −0.500252
\(366\) 0 0
\(367\) −6.94463e16 −1.48361 −0.741804 0.670616i \(-0.766030\pi\)
−0.741804 + 0.670616i \(0.766030\pi\)
\(368\) 0 0
\(369\) 3.55164e15 0.0732416
\(370\) 0 0
\(371\) −4.97380e16 −0.990280
\(372\) 0 0
\(373\) 2.24900e16 0.432396 0.216198 0.976350i \(-0.430634\pi\)
0.216198 + 0.976350i \(0.430634\pi\)
\(374\) 0 0
\(375\) −3.78973e16 −0.703729
\(376\) 0 0
\(377\) 1.23552e17 2.21632
\(378\) 0 0
\(379\) 4.69095e16 0.813029 0.406514 0.913644i \(-0.366744\pi\)
0.406514 + 0.913644i \(0.366744\pi\)
\(380\) 0 0
\(381\) −7.37169e16 −1.23468
\(382\) 0 0
\(383\) −6.37867e16 −1.03261 −0.516307 0.856403i \(-0.672694\pi\)
−0.516307 + 0.856403i \(0.672694\pi\)
\(384\) 0 0
\(385\) −4.03137e16 −0.630895
\(386\) 0 0
\(387\) −7.40682e15 −0.112075
\(388\) 0 0
\(389\) −1.51321e16 −0.221425 −0.110712 0.993852i \(-0.535313\pi\)
−0.110712 + 0.993852i \(0.535313\pi\)
\(390\) 0 0
\(391\) −2.31983e16 −0.328327
\(392\) 0 0
\(393\) 2.53893e15 0.0347616
\(394\) 0 0
\(395\) 2.69046e16 0.356406
\(396\) 0 0
\(397\) 1.79854e16 0.230558 0.115279 0.993333i \(-0.463224\pi\)
0.115279 + 0.993333i \(0.463224\pi\)
\(398\) 0 0
\(399\) −6.63657e16 −0.823417
\(400\) 0 0
\(401\) −8.16229e15 −0.0980332 −0.0490166 0.998798i \(-0.515609\pi\)
−0.0490166 + 0.998798i \(0.515609\pi\)
\(402\) 0 0
\(403\) 1.55378e17 1.80679
\(404\) 0 0
\(405\) 3.49806e16 0.393885
\(406\) 0 0
\(407\) 9.12012e16 0.994573
\(408\) 0 0
\(409\) 1.55109e16 0.163845 0.0819227 0.996639i \(-0.473894\pi\)
0.0819227 + 0.996639i \(0.473894\pi\)
\(410\) 0 0
\(411\) 1.16513e17 1.19235
\(412\) 0 0
\(413\) −5.35350e16 −0.530839
\(414\) 0 0
\(415\) 2.35638e16 0.226430
\(416\) 0 0
\(417\) −8.73212e16 −0.813271
\(418\) 0 0
\(419\) 1.34978e16 0.121863 0.0609314 0.998142i \(-0.480593\pi\)
0.0609314 + 0.998142i \(0.480593\pi\)
\(420\) 0 0
\(421\) 8.44210e16 0.738953 0.369477 0.929240i \(-0.379537\pi\)
0.369477 + 0.929240i \(0.379537\pi\)
\(422\) 0 0
\(423\) 8.23632e15 0.0699070
\(424\) 0 0
\(425\) 1.55537e17 1.28029
\(426\) 0 0
\(427\) −1.34995e17 −1.07779
\(428\) 0 0
\(429\) −2.76638e17 −2.14259
\(430\) 0 0
\(431\) −2.51699e17 −1.89139 −0.945693 0.325062i \(-0.894615\pi\)
−0.945693 + 0.325062i \(0.894615\pi\)
\(432\) 0 0
\(433\) −1.04387e17 −0.761162 −0.380581 0.924748i \(-0.624276\pi\)
−0.380581 + 0.924748i \(0.624276\pi\)
\(434\) 0 0
\(435\) 7.27890e16 0.515093
\(436\) 0 0
\(437\) 1.73243e16 0.118994
\(438\) 0 0
\(439\) 1.50949e17 1.00650 0.503248 0.864142i \(-0.332138\pi\)
0.503248 + 0.864142i \(0.332138\pi\)
\(440\) 0 0
\(441\) 2.53670e16 0.164217
\(442\) 0 0
\(443\) −1.21966e17 −0.766681 −0.383341 0.923607i \(-0.625226\pi\)
−0.383341 + 0.923607i \(0.625226\pi\)
\(444\) 0 0
\(445\) 6.73032e16 0.410862
\(446\) 0 0
\(447\) 9.71995e16 0.576322
\(448\) 0 0
\(449\) −2.40890e17 −1.38745 −0.693724 0.720241i \(-0.744031\pi\)
−0.693724 + 0.720241i \(0.744031\pi\)
\(450\) 0 0
\(451\) 1.10080e17 0.615974
\(452\) 0 0
\(453\) 1.43321e17 0.779240
\(454\) 0 0
\(455\) −1.55265e17 −0.820349
\(456\) 0 0
\(457\) 1.12095e17 0.575615 0.287807 0.957688i \(-0.407074\pi\)
0.287807 + 0.957688i \(0.407074\pi\)
\(458\) 0 0
\(459\) −2.66923e17 −1.33230
\(460\) 0 0
\(461\) 2.60788e17 1.26541 0.632706 0.774392i \(-0.281944\pi\)
0.632706 + 0.774392i \(0.281944\pi\)
\(462\) 0 0
\(463\) 1.50149e17 0.708348 0.354174 0.935180i \(-0.384762\pi\)
0.354174 + 0.935180i \(0.384762\pi\)
\(464\) 0 0
\(465\) 9.15386e16 0.419914
\(466\) 0 0
\(467\) −1.57516e17 −0.702691 −0.351346 0.936246i \(-0.614276\pi\)
−0.351346 + 0.936246i \(0.614276\pi\)
\(468\) 0 0
\(469\) 3.36751e17 1.46112
\(470\) 0 0
\(471\) 1.26704e17 0.534754
\(472\) 0 0
\(473\) −2.29569e17 −0.942574
\(474\) 0 0
\(475\) −1.16154e17 −0.464009
\(476\) 0 0
\(477\) −2.58422e16 −0.100452
\(478\) 0 0
\(479\) 3.75827e17 1.42170 0.710848 0.703345i \(-0.248311\pi\)
0.710848 + 0.703345i \(0.248311\pi\)
\(480\) 0 0
\(481\) 3.51254e17 1.29324
\(482\) 0 0
\(483\) −9.77951e16 −0.350478
\(484\) 0 0
\(485\) −1.65429e17 −0.577153
\(486\) 0 0
\(487\) −4.53837e17 −1.54156 −0.770782 0.637099i \(-0.780134\pi\)
−0.770782 + 0.637099i \(0.780134\pi\)
\(488\) 0 0
\(489\) 2.65168e17 0.877029
\(490\) 0 0
\(491\) 5.98005e17 1.92608 0.963042 0.269351i \(-0.0868091\pi\)
0.963042 + 0.269351i \(0.0868091\pi\)
\(492\) 0 0
\(493\) −6.39153e17 −2.00493
\(494\) 0 0
\(495\) −2.09456e16 −0.0639970
\(496\) 0 0
\(497\) −5.20849e17 −1.55023
\(498\) 0 0
\(499\) −4.80727e17 −1.39394 −0.696971 0.717099i \(-0.745470\pi\)
−0.696971 + 0.717099i \(0.745470\pi\)
\(500\) 0 0
\(501\) −6.63059e17 −1.87330
\(502\) 0 0
\(503\) 5.47213e17 1.50649 0.753243 0.657743i \(-0.228489\pi\)
0.753243 + 0.657743i \(0.228489\pi\)
\(504\) 0 0
\(505\) 1.02300e17 0.274462
\(506\) 0 0
\(507\) −6.55790e17 −1.71480
\(508\) 0 0
\(509\) −8.68173e16 −0.221279 −0.110640 0.993861i \(-0.535290\pi\)
−0.110640 + 0.993861i \(0.535290\pi\)
\(510\) 0 0
\(511\) 8.36553e17 2.07854
\(512\) 0 0
\(513\) 1.99336e17 0.482861
\(514\) 0 0
\(515\) −2.85984e16 −0.0675452
\(516\) 0 0
\(517\) 2.55278e17 0.587930
\(518\) 0 0
\(519\) 1.06877e17 0.240048
\(520\) 0 0
\(521\) 1.21497e17 0.266147 0.133074 0.991106i \(-0.457515\pi\)
0.133074 + 0.991106i \(0.457515\pi\)
\(522\) 0 0
\(523\) 4.79553e17 1.02465 0.512325 0.858792i \(-0.328784\pi\)
0.512325 + 0.858792i \(0.328784\pi\)
\(524\) 0 0
\(525\) 6.55687e17 1.36666
\(526\) 0 0
\(527\) −8.03792e17 −1.63446
\(528\) 0 0
\(529\) −4.78508e17 −0.949351
\(530\) 0 0
\(531\) −2.78150e16 −0.0538474
\(532\) 0 0
\(533\) 4.23966e17 0.800947
\(534\) 0 0
\(535\) −2.98268e17 −0.549929
\(536\) 0 0
\(537\) 4.89122e17 0.880206
\(538\) 0 0
\(539\) 7.86232e17 1.38110
\(540\) 0 0
\(541\) −5.38054e16 −0.0922664 −0.0461332 0.998935i \(-0.514690\pi\)
−0.0461332 + 0.998935i \(0.514690\pi\)
\(542\) 0 0
\(543\) 2.09753e17 0.351164
\(544\) 0 0
\(545\) −1.95041e15 −0.00318823
\(546\) 0 0
\(547\) −9.62528e17 −1.53637 −0.768185 0.640228i \(-0.778840\pi\)
−0.768185 + 0.640228i \(0.778840\pi\)
\(548\) 0 0
\(549\) −7.01389e16 −0.109330
\(550\) 0 0
\(551\) 4.77315e17 0.726640
\(552\) 0 0
\(553\) −9.95928e17 −1.48086
\(554\) 0 0
\(555\) 2.06936e17 0.300560
\(556\) 0 0
\(557\) 3.40799e17 0.483548 0.241774 0.970333i \(-0.422271\pi\)
0.241774 + 0.970333i \(0.422271\pi\)
\(558\) 0 0
\(559\) −8.84165e17 −1.22562
\(560\) 0 0
\(561\) 1.43109e18 1.93824
\(562\) 0 0
\(563\) −2.05601e15 −0.00272095 −0.00136048 0.999999i \(-0.500433\pi\)
−0.00136048 + 0.999999i \(0.500433\pi\)
\(564\) 0 0
\(565\) 3.55038e17 0.459155
\(566\) 0 0
\(567\) −1.29488e18 −1.63658
\(568\) 0 0
\(569\) −1.51190e18 −1.86764 −0.933821 0.357741i \(-0.883547\pi\)
−0.933821 + 0.357741i \(0.883547\pi\)
\(570\) 0 0
\(571\) −8.11054e17 −0.979298 −0.489649 0.871920i \(-0.662875\pi\)
−0.489649 + 0.871920i \(0.662875\pi\)
\(572\) 0 0
\(573\) −2.46881e17 −0.291395
\(574\) 0 0
\(575\) −1.71162e17 −0.197500
\(576\) 0 0
\(577\) 1.32718e17 0.149723 0.0748614 0.997194i \(-0.476149\pi\)
0.0748614 + 0.997194i \(0.476149\pi\)
\(578\) 0 0
\(579\) 1.73570e18 1.91453
\(580\) 0 0
\(581\) −8.72262e17 −0.940811
\(582\) 0 0
\(583\) −8.00960e17 −0.844822
\(584\) 0 0
\(585\) −8.06704e16 −0.0832149
\(586\) 0 0
\(587\) 1.10453e18 1.11437 0.557187 0.830387i \(-0.311881\pi\)
0.557187 + 0.830387i \(0.311881\pi\)
\(588\) 0 0
\(589\) 6.00266e17 0.592371
\(590\) 0 0
\(591\) 4.11591e17 0.397326
\(592\) 0 0
\(593\) 1.44365e18 1.36335 0.681675 0.731656i \(-0.261252\pi\)
0.681675 + 0.731656i \(0.261252\pi\)
\(594\) 0 0
\(595\) 8.03207e17 0.742108
\(596\) 0 0
\(597\) −9.17627e16 −0.0829532
\(598\) 0 0
\(599\) 1.84559e17 0.163253 0.0816266 0.996663i \(-0.473989\pi\)
0.0816266 + 0.996663i \(0.473989\pi\)
\(600\) 0 0
\(601\) 5.19182e17 0.449403 0.224701 0.974428i \(-0.427859\pi\)
0.224701 + 0.974428i \(0.427859\pi\)
\(602\) 0 0
\(603\) 1.74965e17 0.148213
\(604\) 0 0
\(605\) −2.27160e17 −0.188331
\(606\) 0 0
\(607\) 1.80021e18 1.46082 0.730410 0.683009i \(-0.239329\pi\)
0.730410 + 0.683009i \(0.239329\pi\)
\(608\) 0 0
\(609\) −2.69443e18 −2.14020
\(610\) 0 0
\(611\) 9.83184e17 0.764481
\(612\) 0 0
\(613\) −1.41838e18 −1.07969 −0.539847 0.841763i \(-0.681518\pi\)
−0.539847 + 0.841763i \(0.681518\pi\)
\(614\) 0 0
\(615\) 2.49773e17 0.186148
\(616\) 0 0
\(617\) 8.74021e17 0.637776 0.318888 0.947792i \(-0.396691\pi\)
0.318888 + 0.947792i \(0.396691\pi\)
\(618\) 0 0
\(619\) −8.56739e16 −0.0612152 −0.0306076 0.999531i \(-0.509744\pi\)
−0.0306076 + 0.999531i \(0.509744\pi\)
\(620\) 0 0
\(621\) 2.93737e17 0.205524
\(622\) 0 0
\(623\) −2.49136e18 −1.70712
\(624\) 0 0
\(625\) 9.65162e17 0.647709
\(626\) 0 0
\(627\) −1.06873e18 −0.702469
\(628\) 0 0
\(629\) −1.81709e18 −1.16989
\(630\) 0 0
\(631\) −2.34850e17 −0.148115 −0.0740575 0.997254i \(-0.523595\pi\)
−0.0740575 + 0.997254i \(0.523595\pi\)
\(632\) 0 0
\(633\) −1.16831e18 −0.721826
\(634\) 0 0
\(635\) −6.66270e17 −0.403294
\(636\) 0 0
\(637\) 3.02811e18 1.79583
\(638\) 0 0
\(639\) −2.70616e17 −0.157253
\(640\) 0 0
\(641\) 2.86283e18 1.63011 0.815057 0.579380i \(-0.196705\pi\)
0.815057 + 0.579380i \(0.196705\pi\)
\(642\) 0 0
\(643\) 4.45824e17 0.248766 0.124383 0.992234i \(-0.460305\pi\)
0.124383 + 0.992234i \(0.460305\pi\)
\(644\) 0 0
\(645\) −5.20893e17 −0.284846
\(646\) 0 0
\(647\) −2.70439e18 −1.44941 −0.724705 0.689059i \(-0.758024\pi\)
−0.724705 + 0.689059i \(0.758024\pi\)
\(648\) 0 0
\(649\) −8.62105e17 −0.452866
\(650\) 0 0
\(651\) −3.38848e18 −1.74473
\(652\) 0 0
\(653\) −2.96211e17 −0.149508 −0.0747541 0.997202i \(-0.523817\pi\)
−0.0747541 + 0.997202i \(0.523817\pi\)
\(654\) 0 0
\(655\) 2.29475e16 0.0113544
\(656\) 0 0
\(657\) 4.34645e17 0.210843
\(658\) 0 0
\(659\) 1.93623e18 0.920875 0.460438 0.887692i \(-0.347692\pi\)
0.460438 + 0.887692i \(0.347692\pi\)
\(660\) 0 0
\(661\) −2.14190e18 −0.998825 −0.499412 0.866364i \(-0.666451\pi\)
−0.499412 + 0.866364i \(0.666451\pi\)
\(662\) 0 0
\(663\) 5.51172e18 2.52028
\(664\) 0 0
\(665\) −5.99828e17 −0.268959
\(666\) 0 0
\(667\) 7.03361e17 0.309286
\(668\) 0 0
\(669\) 3.01540e18 1.30040
\(670\) 0 0
\(671\) −2.17390e18 −0.919482
\(672\) 0 0
\(673\) 3.09007e18 1.28195 0.640974 0.767563i \(-0.278531\pi\)
0.640974 + 0.767563i \(0.278531\pi\)
\(674\) 0 0
\(675\) −1.96942e18 −0.801426
\(676\) 0 0
\(677\) 4.35504e18 1.73847 0.869233 0.494403i \(-0.164613\pi\)
0.869233 + 0.494403i \(0.164613\pi\)
\(678\) 0 0
\(679\) 6.12369e18 2.39806
\(680\) 0 0
\(681\) −4.65363e18 −1.78787
\(682\) 0 0
\(683\) −1.73059e18 −0.652319 −0.326159 0.945315i \(-0.605755\pi\)
−0.326159 + 0.945315i \(0.605755\pi\)
\(684\) 0 0
\(685\) 1.05307e18 0.389466
\(686\) 0 0
\(687\) −9.60284e17 −0.348483
\(688\) 0 0
\(689\) −3.08483e18 −1.09852
\(690\) 0 0
\(691\) −3.55737e17 −0.124315 −0.0621573 0.998066i \(-0.519798\pi\)
−0.0621573 + 0.998066i \(0.519798\pi\)
\(692\) 0 0
\(693\) 7.75345e17 0.265906
\(694\) 0 0
\(695\) −7.89229e17 −0.265645
\(696\) 0 0
\(697\) −2.19324e18 −0.724556
\(698\) 0 0
\(699\) 1.11918e18 0.362908
\(700\) 0 0
\(701\) −2.63200e18 −0.837759 −0.418880 0.908042i \(-0.637577\pi\)
−0.418880 + 0.908042i \(0.637577\pi\)
\(702\) 0 0
\(703\) 1.35699e18 0.424000
\(704\) 0 0
\(705\) 5.79228e17 0.177673
\(706\) 0 0
\(707\) −3.78685e18 −1.14039
\(708\) 0 0
\(709\) −6.27661e18 −1.85577 −0.927886 0.372863i \(-0.878376\pi\)
−0.927886 + 0.372863i \(0.878376\pi\)
\(710\) 0 0
\(711\) −5.17451e17 −0.150216
\(712\) 0 0
\(713\) 8.84539e17 0.252136
\(714\) 0 0
\(715\) −2.50032e18 −0.699852
\(716\) 0 0
\(717\) 1.53223e18 0.421161
\(718\) 0 0
\(719\) 5.58024e18 1.50631 0.753156 0.657842i \(-0.228531\pi\)
0.753156 + 0.657842i \(0.228531\pi\)
\(720\) 0 0
\(721\) 1.05863e18 0.280649
\(722\) 0 0
\(723\) −3.80274e16 −0.00990142
\(724\) 0 0
\(725\) −4.71582e18 −1.20604
\(726\) 0 0
\(727\) 2.15542e18 0.541449 0.270725 0.962657i \(-0.412737\pi\)
0.270725 + 0.962657i \(0.412737\pi\)
\(728\) 0 0
\(729\) 3.29165e18 0.812240
\(730\) 0 0
\(731\) 4.57391e18 1.10873
\(732\) 0 0
\(733\) −3.43698e18 −0.818467 −0.409234 0.912430i \(-0.634204\pi\)
−0.409234 + 0.912430i \(0.634204\pi\)
\(734\) 0 0
\(735\) 1.78396e18 0.417367
\(736\) 0 0
\(737\) 5.42289e18 1.24650
\(738\) 0 0
\(739\) −2.34366e18 −0.529305 −0.264653 0.964344i \(-0.585257\pi\)
−0.264653 + 0.964344i \(0.585257\pi\)
\(740\) 0 0
\(741\) −4.11611e18 −0.913416
\(742\) 0 0
\(743\) −6.88793e18 −1.50197 −0.750985 0.660319i \(-0.770421\pi\)
−0.750985 + 0.660319i \(0.770421\pi\)
\(744\) 0 0
\(745\) 8.78511e17 0.188248
\(746\) 0 0
\(747\) −4.53199e17 −0.0954343
\(748\) 0 0
\(749\) 1.10410e19 2.28494
\(750\) 0 0
\(751\) −1.85684e17 −0.0377671 −0.0188836 0.999822i \(-0.506011\pi\)
−0.0188836 + 0.999822i \(0.506011\pi\)
\(752\) 0 0
\(753\) 7.37878e17 0.147508
\(754\) 0 0
\(755\) 1.29537e18 0.254529
\(756\) 0 0
\(757\) 8.78485e18 1.69672 0.848362 0.529417i \(-0.177589\pi\)
0.848362 + 0.529417i \(0.177589\pi\)
\(758\) 0 0
\(759\) −1.57485e18 −0.298998
\(760\) 0 0
\(761\) 1.88495e17 0.0351803 0.0175901 0.999845i \(-0.494401\pi\)
0.0175901 + 0.999845i \(0.494401\pi\)
\(762\) 0 0
\(763\) 7.21984e16 0.0132470
\(764\) 0 0
\(765\) 4.17320e17 0.0752782
\(766\) 0 0
\(767\) −3.32033e18 −0.588859
\(768\) 0 0
\(769\) 5.16573e17 0.0900763 0.0450381 0.998985i \(-0.485659\pi\)
0.0450381 + 0.998985i \(0.485659\pi\)
\(770\) 0 0
\(771\) −5.97316e17 −0.102412
\(772\) 0 0
\(773\) 2.41255e18 0.406733 0.203366 0.979103i \(-0.434812\pi\)
0.203366 + 0.979103i \(0.434812\pi\)
\(774\) 0 0
\(775\) −5.93057e18 −0.983185
\(776\) 0 0
\(777\) −7.66015e18 −1.24882
\(778\) 0 0
\(779\) 1.63789e18 0.262598
\(780\) 0 0
\(781\) −8.38753e18 −1.32252
\(782\) 0 0
\(783\) 8.09297e18 1.25504
\(784\) 0 0
\(785\) 1.14518e18 0.174671
\(786\) 0 0
\(787\) −1.24281e18 −0.186453 −0.0932265 0.995645i \(-0.529718\pi\)
−0.0932265 + 0.995645i \(0.529718\pi\)
\(788\) 0 0
\(789\) −2.39944e18 −0.354087
\(790\) 0 0
\(791\) −1.31424e19 −1.90778
\(792\) 0 0
\(793\) −8.37260e18 −1.19560
\(794\) 0 0
\(795\) −1.81738e18 −0.255305
\(796\) 0 0
\(797\) −5.76552e17 −0.0796818 −0.0398409 0.999206i \(-0.512685\pi\)
−0.0398409 + 0.999206i \(0.512685\pi\)
\(798\) 0 0
\(799\) −5.08615e18 −0.691568
\(800\) 0 0
\(801\) −1.29443e18 −0.173168
\(802\) 0 0
\(803\) 1.34715e19 1.77323
\(804\) 0 0
\(805\) −8.83895e17 −0.114479
\(806\) 0 0
\(807\) 1.36787e19 1.74328
\(808\) 0 0
\(809\) 4.34559e18 0.544983 0.272492 0.962158i \(-0.412152\pi\)
0.272492 + 0.962158i \(0.412152\pi\)
\(810\) 0 0
\(811\) 1.69949e18 0.209741 0.104871 0.994486i \(-0.466557\pi\)
0.104871 + 0.994486i \(0.466557\pi\)
\(812\) 0 0
\(813\) 4.14938e18 0.503958
\(814\) 0 0
\(815\) 2.39665e18 0.286471
\(816\) 0 0
\(817\) −3.41576e18 −0.401832
\(818\) 0 0
\(819\) 2.98618e18 0.345756
\(820\) 0 0
\(821\) −3.06171e17 −0.0348926 −0.0174463 0.999848i \(-0.505554\pi\)
−0.0174463 + 0.999848i \(0.505554\pi\)
\(822\) 0 0
\(823\) −4.19904e18 −0.471033 −0.235516 0.971870i \(-0.575678\pi\)
−0.235516 + 0.971870i \(0.575678\pi\)
\(824\) 0 0
\(825\) 1.05589e19 1.16592
\(826\) 0 0
\(827\) 1.76597e19 1.91954 0.959772 0.280780i \(-0.0905931\pi\)
0.959772 + 0.280780i \(0.0905931\pi\)
\(828\) 0 0
\(829\) −1.95676e18 −0.209379 −0.104689 0.994505i \(-0.533385\pi\)
−0.104689 + 0.994505i \(0.533385\pi\)
\(830\) 0 0
\(831\) −1.02185e19 −1.07641
\(832\) 0 0
\(833\) −1.56648e19 −1.62455
\(834\) 0 0
\(835\) −5.99288e18 −0.611890
\(836\) 0 0
\(837\) 1.01776e19 1.02313
\(838\) 0 0
\(839\) −8.20221e18 −0.811855 −0.405927 0.913905i \(-0.633051\pi\)
−0.405927 + 0.913905i \(0.633051\pi\)
\(840\) 0 0
\(841\) 9.11822e18 0.888660
\(842\) 0 0
\(843\) 8.14592e18 0.781737
\(844\) 0 0
\(845\) −5.92717e18 −0.560117
\(846\) 0 0
\(847\) 8.40879e18 0.782513
\(848\) 0 0
\(849\) 1.07853e18 0.0988396
\(850\) 0 0
\(851\) 1.99963e18 0.180471
\(852\) 0 0
\(853\) 6.81541e18 0.605792 0.302896 0.953024i \(-0.402047\pi\)
0.302896 + 0.953024i \(0.402047\pi\)
\(854\) 0 0
\(855\) −3.11651e17 −0.0272828
\(856\) 0 0
\(857\) −4.95407e18 −0.427156 −0.213578 0.976926i \(-0.568512\pi\)
−0.213578 + 0.976926i \(0.568512\pi\)
\(858\) 0 0
\(859\) −1.25946e19 −1.06962 −0.534809 0.844973i \(-0.679616\pi\)
−0.534809 + 0.844973i \(0.679616\pi\)
\(860\) 0 0
\(861\) −9.24585e18 −0.773440
\(862\) 0 0
\(863\) −9.02789e18 −0.743903 −0.371951 0.928252i \(-0.621311\pi\)
−0.371951 + 0.928252i \(0.621311\pi\)
\(864\) 0 0
\(865\) 9.65982e17 0.0784087
\(866\) 0 0
\(867\) −1.51163e19 −1.20871
\(868\) 0 0
\(869\) −1.60380e19 −1.26334
\(870\) 0 0
\(871\) 2.08858e19 1.62082
\(872\) 0 0
\(873\) 3.18167e18 0.243255
\(874\) 0 0
\(875\) 1.26792e19 0.955082
\(876\) 0 0
\(877\) 4.25224e18 0.315588 0.157794 0.987472i \(-0.449562\pi\)
0.157794 + 0.987472i \(0.449562\pi\)
\(878\) 0 0
\(879\) −2.31461e19 −1.69258
\(880\) 0 0
\(881\) 1.10187e19 0.793937 0.396968 0.917832i \(-0.370062\pi\)
0.396968 + 0.917832i \(0.370062\pi\)
\(882\) 0 0
\(883\) −1.99194e19 −1.41427 −0.707135 0.707079i \(-0.750013\pi\)
−0.707135 + 0.707079i \(0.750013\pi\)
\(884\) 0 0
\(885\) −1.95612e18 −0.136856
\(886\) 0 0
\(887\) −2.18455e19 −1.50612 −0.753058 0.657954i \(-0.771422\pi\)
−0.753058 + 0.657954i \(0.771422\pi\)
\(888\) 0 0
\(889\) 2.46633e19 1.67568
\(890\) 0 0
\(891\) −2.08521e19 −1.39619
\(892\) 0 0
\(893\) 3.79830e18 0.250642
\(894\) 0 0
\(895\) 4.42080e18 0.287509
\(896\) 0 0
\(897\) −6.06541e18 −0.388785
\(898\) 0 0
\(899\) 2.43706e19 1.53967
\(900\) 0 0
\(901\) 1.59583e19 0.993744
\(902\) 0 0
\(903\) 1.92819e19 1.18353
\(904\) 0 0
\(905\) 1.89580e18 0.114703
\(906\) 0 0
\(907\) 1.13902e19 0.679336 0.339668 0.940545i \(-0.389685\pi\)
0.339668 + 0.940545i \(0.389685\pi\)
\(908\) 0 0
\(909\) −1.96752e18 −0.115679
\(910\) 0 0
\(911\) 3.34308e19 1.93766 0.968829 0.247732i \(-0.0796851\pi\)
0.968829 + 0.247732i \(0.0796851\pi\)
\(912\) 0 0
\(913\) −1.40465e19 −0.802619
\(914\) 0 0
\(915\) −4.93259e18 −0.277868
\(916\) 0 0
\(917\) −8.49446e17 −0.0471775
\(918\) 0 0
\(919\) 2.89154e19 1.58336 0.791678 0.610939i \(-0.209208\pi\)
0.791678 + 0.610939i \(0.209208\pi\)
\(920\) 0 0
\(921\) 1.46361e19 0.790199
\(922\) 0 0
\(923\) −3.23039e19 −1.71966
\(924\) 0 0
\(925\) −1.34069e19 −0.703731
\(926\) 0 0
\(927\) 5.50028e17 0.0284686
\(928\) 0 0
\(929\) −2.80750e19 −1.43291 −0.716453 0.697635i \(-0.754236\pi\)
−0.716453 + 0.697635i \(0.754236\pi\)
\(930\) 0 0
\(931\) 1.16984e19 0.588780
\(932\) 0 0
\(933\) 1.49008e19 0.739569
\(934\) 0 0
\(935\) 1.29345e19 0.633103
\(936\) 0 0
\(937\) −2.72300e19 −1.31444 −0.657220 0.753699i \(-0.728268\pi\)
−0.657220 + 0.753699i \(0.728268\pi\)
\(938\) 0 0
\(939\) −3.58012e19 −1.70440
\(940\) 0 0
\(941\) −2.67460e19 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(942\) 0 0
\(943\) 2.41356e18 0.111772
\(944\) 0 0
\(945\) −1.01702e19 −0.464540
\(946\) 0 0
\(947\) 3.01628e19 1.35893 0.679464 0.733709i \(-0.262212\pi\)
0.679464 + 0.733709i \(0.262212\pi\)
\(948\) 0 0
\(949\) 5.18844e19 2.30572
\(950\) 0 0
\(951\) 5.25496e18 0.230354
\(952\) 0 0
\(953\) −1.66980e19 −0.722037 −0.361019 0.932559i \(-0.617571\pi\)
−0.361019 + 0.932559i \(0.617571\pi\)
\(954\) 0 0
\(955\) −2.23137e18 −0.0951807
\(956\) 0 0
\(957\) −4.33900e19 −1.82584
\(958\) 0 0
\(959\) −3.89815e19 −1.61822
\(960\) 0 0
\(961\) 6.23068e18 0.255172
\(962\) 0 0
\(963\) 5.73653e18 0.231781
\(964\) 0 0
\(965\) 1.56876e19 0.625359
\(966\) 0 0
\(967\) −2.79699e19 −1.10007 −0.550033 0.835143i \(-0.685385\pi\)
−0.550033 + 0.835143i \(0.685385\pi\)
\(968\) 0 0
\(969\) 2.12932e19 0.826298
\(970\) 0 0
\(971\) −5.42017e18 −0.207534 −0.103767 0.994602i \(-0.533090\pi\)
−0.103767 + 0.994602i \(0.533090\pi\)
\(972\) 0 0
\(973\) 2.92149e19 1.10375
\(974\) 0 0
\(975\) 4.06668e19 1.51604
\(976\) 0 0
\(977\) −2.85933e19 −1.05184 −0.525921 0.850534i \(-0.676279\pi\)
−0.525921 + 0.850534i \(0.676279\pi\)
\(978\) 0 0
\(979\) −4.01199e19 −1.45637
\(980\) 0 0
\(981\) 3.75119e16 0.00134376
\(982\) 0 0
\(983\) 7.34661e18 0.259710 0.129855 0.991533i \(-0.458549\pi\)
0.129855 + 0.991533i \(0.458549\pi\)
\(984\) 0 0
\(985\) 3.72005e18 0.129782
\(986\) 0 0
\(987\) −2.14413e19 −0.738226
\(988\) 0 0
\(989\) −5.03340e18 −0.171035
\(990\) 0 0
\(991\) 1.78440e19 0.598431 0.299215 0.954186i \(-0.403275\pi\)
0.299215 + 0.954186i \(0.403275\pi\)
\(992\) 0 0
\(993\) 4.99989e19 1.65497
\(994\) 0 0
\(995\) −8.29372e17 −0.0270956
\(996\) 0 0
\(997\) −5.53089e19 −1.78351 −0.891757 0.452514i \(-0.850527\pi\)
−0.891757 + 0.452514i \(0.850527\pi\)
\(998\) 0 0
\(999\) 2.30080e19 0.732323
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 8.14.a.b.1.1 2
3.2 odd 2 72.14.a.c.1.2 2
4.3 odd 2 16.14.a.e.1.2 2
5.2 odd 4 200.14.c.b.49.3 4
5.3 odd 4 200.14.c.b.49.2 4
5.4 even 2 200.14.a.b.1.2 2
8.3 odd 2 64.14.a.l.1.1 2
8.5 even 2 64.14.a.j.1.2 2
12.11 even 2 144.14.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.14.a.b.1.1 2 1.1 even 1 trivial
16.14.a.e.1.2 2 4.3 odd 2
64.14.a.j.1.2 2 8.5 even 2
64.14.a.l.1.1 2 8.3 odd 2
72.14.a.c.1.2 2 3.2 odd 2
144.14.a.n.1.2 2 12.11 even 2
200.14.a.b.1.2 2 5.4 even 2
200.14.c.b.49.2 4 5.3 odd 4
200.14.c.b.49.3 4 5.2 odd 4