Properties

Label 48.22.a.l.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
Analytic rank $0$
Dimension $3$
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] = [48,22,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(134.149125258\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2295485x - 828958533 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{4}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-386.305\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} -3.08294e7 q^{5} +1.10597e9 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q+59049.0 q^{3} -3.08294e7 q^{5} +1.10597e9 q^{7} +3.48678e9 q^{9} -1.54673e11 q^{11} +5.78668e11 q^{13} -1.82045e12 q^{15} +1.59754e12 q^{17} -7.29390e12 q^{19} +6.53065e13 q^{21} +1.56234e14 q^{23} +4.73615e14 q^{25} +2.05891e14 q^{27} -1.80995e15 q^{29} -5.98096e15 q^{31} -9.13328e15 q^{33} -3.40965e16 q^{35} +1.92628e16 q^{37} +3.41698e16 q^{39} -1.01900e17 q^{41} +2.34697e17 q^{43} -1.07495e17 q^{45} -1.31288e17 q^{47} +6.64628e17 q^{49} +9.43330e16 q^{51} +2.51169e18 q^{53} +4.76847e18 q^{55} -4.30697e17 q^{57} -4.37160e18 q^{59} -6.88287e18 q^{61} +3.85629e18 q^{63} -1.78400e19 q^{65} -1.11554e19 q^{67} +9.22545e18 q^{69} +4.02859e19 q^{71} +3.07636e19 q^{73} +2.79665e19 q^{75} -1.71064e20 q^{77} +1.48672e20 q^{79} +1.21577e19 q^{81} -9.43003e19 q^{83} -4.92511e19 q^{85} -1.06876e20 q^{87} -2.18162e20 q^{89} +6.39991e20 q^{91} -3.53169e20 q^{93} +2.24866e20 q^{95} +1.66723e20 q^{97} -5.39311e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9} - 13839247500 q^{11} + 718855551690 q^{13} + 122823455274 q^{15} + 2135189843046 q^{17} + 40122324686988 q^{19} + 71170700597136 q^{21} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 48\!\cdots\!00 q^{99}+O(q^{100}) \) 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
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) −3.08294e7 −1.41182 −0.705911 0.708300i \(-0.749462\pi\)
−0.705911 + 0.708300i \(0.749462\pi\)
\(6\) 0 0
\(7\) 1.10597e9 1.47984 0.739920 0.672695i \(-0.234863\pi\)
0.739920 + 0.672695i \(0.234863\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.54673e11 −1.79801 −0.899003 0.437943i \(-0.855707\pi\)
−0.899003 + 0.437943i \(0.855707\pi\)
\(12\) 0 0
\(13\) 5.78668e11 1.16419 0.582095 0.813120i \(-0.302233\pi\)
0.582095 + 0.813120i \(0.302233\pi\)
\(14\) 0 0
\(15\) −1.82045e12 −0.815116
\(16\) 0 0
\(17\) 1.59754e12 0.192193 0.0960964 0.995372i \(-0.469364\pi\)
0.0960964 + 0.995372i \(0.469364\pi\)
\(18\) 0 0
\(19\) −7.29390e12 −0.272927 −0.136464 0.990645i \(-0.543574\pi\)
−0.136464 + 0.990645i \(0.543574\pi\)
\(20\) 0 0
\(21\) 6.53065e13 0.854386
\(22\) 0 0
\(23\) 1.56234e14 0.786381 0.393190 0.919457i \(-0.371371\pi\)
0.393190 + 0.919457i \(0.371371\pi\)
\(24\) 0 0
\(25\) 4.73615e14 0.993242
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) −1.80995e15 −0.798890 −0.399445 0.916757i \(-0.630797\pi\)
−0.399445 + 0.916757i \(0.630797\pi\)
\(30\) 0 0
\(31\) −5.98096e15 −1.31061 −0.655304 0.755365i \(-0.727459\pi\)
−0.655304 + 0.755365i \(0.727459\pi\)
\(32\) 0 0
\(33\) −9.13328e15 −1.03808
\(34\) 0 0
\(35\) −3.40965e16 −2.08927
\(36\) 0 0
\(37\) 1.92628e16 0.658568 0.329284 0.944231i \(-0.393193\pi\)
0.329284 + 0.944231i \(0.393193\pi\)
\(38\) 0 0
\(39\) 3.41698e16 0.672146
\(40\) 0 0
\(41\) −1.01900e17 −1.18561 −0.592805 0.805346i \(-0.701980\pi\)
−0.592805 + 0.805346i \(0.701980\pi\)
\(42\) 0 0
\(43\) 2.34697e17 1.65611 0.828053 0.560650i \(-0.189449\pi\)
0.828053 + 0.560650i \(0.189449\pi\)
\(44\) 0 0
\(45\) −1.07495e17 −0.470607
\(46\) 0 0
\(47\) −1.31288e17 −0.364080 −0.182040 0.983291i \(-0.558270\pi\)
−0.182040 + 0.983291i \(0.558270\pi\)
\(48\) 0 0
\(49\) 6.64628e17 1.18993
\(50\) 0 0
\(51\) 9.43330e16 0.110963
\(52\) 0 0
\(53\) 2.51169e18 1.97274 0.986368 0.164554i \(-0.0526184\pi\)
0.986368 + 0.164554i \(0.0526184\pi\)
\(54\) 0 0
\(55\) 4.76847e18 2.53846
\(56\) 0 0
\(57\) −4.30697e17 −0.157575
\(58\) 0 0
\(59\) −4.37160e18 −1.11351 −0.556755 0.830677i \(-0.687954\pi\)
−0.556755 + 0.830677i \(0.687954\pi\)
\(60\) 0 0
\(61\) −6.88287e18 −1.23540 −0.617698 0.786415i \(-0.711935\pi\)
−0.617698 + 0.786415i \(0.711935\pi\)
\(62\) 0 0
\(63\) 3.85629e18 0.493280
\(64\) 0 0
\(65\) −1.78400e19 −1.64363
\(66\) 0 0
\(67\) −1.11554e19 −0.747650 −0.373825 0.927499i \(-0.621954\pi\)
−0.373825 + 0.927499i \(0.621954\pi\)
\(68\) 0 0
\(69\) 9.22545e18 0.454017
\(70\) 0 0
\(71\) 4.02859e19 1.46872 0.734362 0.678758i \(-0.237481\pi\)
0.734362 + 0.678758i \(0.237481\pi\)
\(72\) 0 0
\(73\) 3.07636e19 0.837812 0.418906 0.908030i \(-0.362414\pi\)
0.418906 + 0.908030i \(0.362414\pi\)
\(74\) 0 0
\(75\) 2.79665e19 0.573449
\(76\) 0 0
\(77\) −1.71064e20 −2.66076
\(78\) 0 0
\(79\) 1.48672e20 1.76662 0.883312 0.468785i \(-0.155308\pi\)
0.883312 + 0.468785i \(0.155308\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) −9.43003e19 −0.667104 −0.333552 0.942732i \(-0.608247\pi\)
−0.333552 + 0.942732i \(0.608247\pi\)
\(84\) 0 0
\(85\) −4.92511e19 −0.271342
\(86\) 0 0
\(87\) −1.06876e20 −0.461239
\(88\) 0 0
\(89\) −2.18162e20 −0.741626 −0.370813 0.928708i \(-0.620921\pi\)
−0.370813 + 0.928708i \(0.620921\pi\)
\(90\) 0 0
\(91\) 6.39991e20 1.72282
\(92\) 0 0
\(93\) −3.53169e20 −0.756680
\(94\) 0 0
\(95\) 2.24866e20 0.385325
\(96\) 0 0
\(97\) 1.66723e20 0.229557 0.114779 0.993391i \(-0.463384\pi\)
0.114779 + 0.993391i \(0.463384\pi\)
\(98\) 0 0
\(99\) −5.39311e20 −0.599335
\(100\) 0 0
\(101\) −1.30649e20 −0.117688 −0.0588440 0.998267i \(-0.518741\pi\)
−0.0588440 + 0.998267i \(0.518741\pi\)
\(102\) 0 0
\(103\) 1.79611e21 1.31687 0.658434 0.752639i \(-0.271219\pi\)
0.658434 + 0.752639i \(0.271219\pi\)
\(104\) 0 0
\(105\) −2.01336e21 −1.20624
\(106\) 0 0
\(107\) −1.52295e19 −0.00748437 −0.00374218 0.999993i \(-0.501191\pi\)
−0.00374218 + 0.999993i \(0.501191\pi\)
\(108\) 0 0
\(109\) −1.75681e20 −0.0710799 −0.0355400 0.999368i \(-0.511315\pi\)
−0.0355400 + 0.999368i \(0.511315\pi\)
\(110\) 0 0
\(111\) 1.13745e21 0.380224
\(112\) 0 0
\(113\) 5.54442e21 1.53650 0.768250 0.640150i \(-0.221128\pi\)
0.768250 + 0.640150i \(0.221128\pi\)
\(114\) 0 0
\(115\) −4.81660e21 −1.11023
\(116\) 0 0
\(117\) 2.01769e21 0.388064
\(118\) 0 0
\(119\) 1.76683e21 0.284415
\(120\) 0 0
\(121\) 1.65235e22 2.23282
\(122\) 0 0
\(123\) −6.01707e21 −0.684512
\(124\) 0 0
\(125\) 9.93400e19 0.00954045
\(126\) 0 0
\(127\) 7.59121e21 0.617123 0.308562 0.951204i \(-0.400152\pi\)
0.308562 + 0.951204i \(0.400152\pi\)
\(128\) 0 0
\(129\) 1.38586e22 0.956153
\(130\) 0 0
\(131\) 1.26195e22 0.740789 0.370395 0.928874i \(-0.379222\pi\)
0.370395 + 0.928874i \(0.379222\pi\)
\(132\) 0 0
\(133\) −8.06684e21 −0.403888
\(134\) 0 0
\(135\) −6.34750e21 −0.271705
\(136\) 0 0
\(137\) 2.04751e22 0.751036 0.375518 0.926815i \(-0.377465\pi\)
0.375518 + 0.926815i \(0.377465\pi\)
\(138\) 0 0
\(139\) 8.05555e21 0.253770 0.126885 0.991917i \(-0.459502\pi\)
0.126885 + 0.991917i \(0.459502\pi\)
\(140\) 0 0
\(141\) −7.75241e21 −0.210202
\(142\) 0 0
\(143\) −8.95042e22 −2.09322
\(144\) 0 0
\(145\) 5.57996e22 1.12789
\(146\) 0 0
\(147\) 3.92456e22 0.687004
\(148\) 0 0
\(149\) −1.91681e21 −0.0291155 −0.0145577 0.999894i \(-0.504634\pi\)
−0.0145577 + 0.999894i \(0.504634\pi\)
\(150\) 0 0
\(151\) −4.49230e22 −0.593213 −0.296606 0.955000i \(-0.595855\pi\)
−0.296606 + 0.955000i \(0.595855\pi\)
\(152\) 0 0
\(153\) 5.57027e21 0.0640643
\(154\) 0 0
\(155\) 1.84389e23 1.85035
\(156\) 0 0
\(157\) −1.35200e23 −1.18585 −0.592926 0.805257i \(-0.702027\pi\)
−0.592926 + 0.805257i \(0.702027\pi\)
\(158\) 0 0
\(159\) 1.48313e23 1.13896
\(160\) 0 0
\(161\) 1.72790e23 1.16372
\(162\) 0 0
\(163\) 2.95535e23 1.74839 0.874197 0.485572i \(-0.161389\pi\)
0.874197 + 0.485572i \(0.161389\pi\)
\(164\) 0 0
\(165\) 2.81574e23 1.46558
\(166\) 0 0
\(167\) 2.71414e23 1.24483 0.622413 0.782689i \(-0.286152\pi\)
0.622413 + 0.782689i \(0.286152\pi\)
\(168\) 0 0
\(169\) 8.77920e22 0.355340
\(170\) 0 0
\(171\) −2.54322e22 −0.0909757
\(172\) 0 0
\(173\) 3.84952e23 1.21877 0.609386 0.792873i \(-0.291416\pi\)
0.609386 + 0.792873i \(0.291416\pi\)
\(174\) 0 0
\(175\) 5.23805e23 1.46984
\(176\) 0 0
\(177\) −2.58138e23 −0.642885
\(178\) 0 0
\(179\) 5.60520e23 1.24061 0.620304 0.784362i \(-0.287009\pi\)
0.620304 + 0.784362i \(0.287009\pi\)
\(180\) 0 0
\(181\) −7.65962e23 −1.50863 −0.754314 0.656514i \(-0.772030\pi\)
−0.754314 + 0.656514i \(0.772030\pi\)
\(182\) 0 0
\(183\) −4.06427e23 −0.713256
\(184\) 0 0
\(185\) −5.93860e23 −0.929781
\(186\) 0 0
\(187\) −2.47096e23 −0.345564
\(188\) 0 0
\(189\) 2.27710e23 0.284795
\(190\) 0 0
\(191\) 4.28534e23 0.479883 0.239941 0.970787i \(-0.422872\pi\)
0.239941 + 0.970787i \(0.422872\pi\)
\(192\) 0 0
\(193\) 1.58535e24 1.59138 0.795689 0.605705i \(-0.207109\pi\)
0.795689 + 0.605705i \(0.207109\pi\)
\(194\) 0 0
\(195\) −1.05343e24 −0.948951
\(196\) 0 0
\(197\) 1.58432e24 1.28217 0.641086 0.767469i \(-0.278484\pi\)
0.641086 + 0.767469i \(0.278484\pi\)
\(198\) 0 0
\(199\) 1.61590e24 1.17614 0.588068 0.808811i \(-0.299889\pi\)
0.588068 + 0.808811i \(0.299889\pi\)
\(200\) 0 0
\(201\) −6.58713e23 −0.431656
\(202\) 0 0
\(203\) −2.00175e24 −1.18223
\(204\) 0 0
\(205\) 3.14150e24 1.67387
\(206\) 0 0
\(207\) 5.44754e23 0.262127
\(208\) 0 0
\(209\) 1.12817e24 0.490724
\(210\) 0 0
\(211\) 4.40467e24 1.73359 0.866797 0.498661i \(-0.166175\pi\)
0.866797 + 0.498661i \(0.166175\pi\)
\(212\) 0 0
\(213\) 2.37884e24 0.847968
\(214\) 0 0
\(215\) −7.23556e24 −2.33813
\(216\) 0 0
\(217\) −6.61477e24 −1.93949
\(218\) 0 0
\(219\) 1.81656e24 0.483711
\(220\) 0 0
\(221\) 9.24444e23 0.223749
\(222\) 0 0
\(223\) −2.38725e24 −0.525650 −0.262825 0.964843i \(-0.584654\pi\)
−0.262825 + 0.964843i \(0.584654\pi\)
\(224\) 0 0
\(225\) 1.65139e24 0.331081
\(226\) 0 0
\(227\) −1.41267e24 −0.258089 −0.129045 0.991639i \(-0.541191\pi\)
−0.129045 + 0.991639i \(0.541191\pi\)
\(228\) 0 0
\(229\) −6.13078e24 −1.02151 −0.510755 0.859726i \(-0.670634\pi\)
−0.510755 + 0.859726i \(0.670634\pi\)
\(230\) 0 0
\(231\) −1.01012e25 −1.53619
\(232\) 0 0
\(233\) 3.43514e24 0.477208 0.238604 0.971117i \(-0.423310\pi\)
0.238604 + 0.971117i \(0.423310\pi\)
\(234\) 0 0
\(235\) 4.04753e24 0.514016
\(236\) 0 0
\(237\) 8.77892e24 1.01996
\(238\) 0 0
\(239\) 2.65401e23 0.0282309 0.0141154 0.999900i \(-0.495507\pi\)
0.0141154 + 0.999900i \(0.495507\pi\)
\(240\) 0 0
\(241\) −4.64377e24 −0.452577 −0.226288 0.974060i \(-0.572659\pi\)
−0.226288 + 0.974060i \(0.572659\pi\)
\(242\) 0 0
\(243\) 7.17898e23 0.0641500
\(244\) 0 0
\(245\) −2.04901e25 −1.67996
\(246\) 0 0
\(247\) −4.22074e24 −0.317739
\(248\) 0 0
\(249\) −5.56834e24 −0.385152
\(250\) 0 0
\(251\) 9.13351e24 0.580850 0.290425 0.956898i \(-0.406203\pi\)
0.290425 + 0.956898i \(0.406203\pi\)
\(252\) 0 0
\(253\) −2.41651e25 −1.41392
\(254\) 0 0
\(255\) −2.90823e24 −0.156660
\(256\) 0 0
\(257\) 2.42762e25 1.20471 0.602355 0.798228i \(-0.294229\pi\)
0.602355 + 0.798228i \(0.294229\pi\)
\(258\) 0 0
\(259\) 2.13041e25 0.974575
\(260\) 0 0
\(261\) −6.31089e24 −0.266297
\(262\) 0 0
\(263\) 5.72262e24 0.222874 0.111437 0.993772i \(-0.464455\pi\)
0.111437 + 0.993772i \(0.464455\pi\)
\(264\) 0 0
\(265\) −7.74338e25 −2.78515
\(266\) 0 0
\(267\) −1.28823e25 −0.428178
\(268\) 0 0
\(269\) −2.61502e25 −0.803668 −0.401834 0.915712i \(-0.631627\pi\)
−0.401834 + 0.915712i \(0.631627\pi\)
\(270\) 0 0
\(271\) 1.46407e25 0.416279 0.208139 0.978099i \(-0.433259\pi\)
0.208139 + 0.978099i \(0.433259\pi\)
\(272\) 0 0
\(273\) 3.77908e25 0.994668
\(274\) 0 0
\(275\) −7.32554e25 −1.78586
\(276\) 0 0
\(277\) −4.49242e25 −1.01495 −0.507473 0.861668i \(-0.669420\pi\)
−0.507473 + 0.861668i \(0.669420\pi\)
\(278\) 0 0
\(279\) −2.08543e25 −0.436869
\(280\) 0 0
\(281\) 5.52906e25 1.07457 0.537286 0.843400i \(-0.319450\pi\)
0.537286 + 0.843400i \(0.319450\pi\)
\(282\) 0 0
\(283\) −4.21386e25 −0.760192 −0.380096 0.924947i \(-0.624109\pi\)
−0.380096 + 0.924947i \(0.624109\pi\)
\(284\) 0 0
\(285\) 1.32781e25 0.222467
\(286\) 0 0
\(287\) −1.12698e26 −1.75451
\(288\) 0 0
\(289\) −6.65398e25 −0.963062
\(290\) 0 0
\(291\) 9.84481e24 0.132535
\(292\) 0 0
\(293\) −6.37530e25 −0.798713 −0.399356 0.916796i \(-0.630766\pi\)
−0.399356 + 0.916796i \(0.630766\pi\)
\(294\) 0 0
\(295\) 1.34774e26 1.57208
\(296\) 0 0
\(297\) −3.18458e25 −0.346026
\(298\) 0 0
\(299\) 9.04075e25 0.915497
\(300\) 0 0
\(301\) 2.59568e26 2.45077
\(302\) 0 0
\(303\) −7.71470e24 −0.0679472
\(304\) 0 0
\(305\) 2.12195e26 1.74416
\(306\) 0 0
\(307\) −7.93138e25 −0.608689 −0.304345 0.952562i \(-0.598437\pi\)
−0.304345 + 0.952562i \(0.598437\pi\)
\(308\) 0 0
\(309\) 1.06058e26 0.760294
\(310\) 0 0
\(311\) −1.28045e26 −0.857783 −0.428892 0.903356i \(-0.641096\pi\)
−0.428892 + 0.903356i \(0.641096\pi\)
\(312\) 0 0
\(313\) 1.71687e26 1.07528 0.537640 0.843174i \(-0.319316\pi\)
0.537640 + 0.843174i \(0.319316\pi\)
\(314\) 0 0
\(315\) −1.18887e26 −0.696424
\(316\) 0 0
\(317\) −3.54215e26 −1.94153 −0.970766 0.240028i \(-0.922843\pi\)
−0.970766 + 0.240028i \(0.922843\pi\)
\(318\) 0 0
\(319\) 2.79950e26 1.43641
\(320\) 0 0
\(321\) −8.99285e23 −0.00432110
\(322\) 0 0
\(323\) −1.16523e25 −0.0524547
\(324\) 0 0
\(325\) 2.74066e26 1.15632
\(326\) 0 0
\(327\) −1.03738e25 −0.0410380
\(328\) 0 0
\(329\) −1.45201e26 −0.538780
\(330\) 0 0
\(331\) 2.50034e26 0.870573 0.435286 0.900292i \(-0.356647\pi\)
0.435286 + 0.900292i \(0.356647\pi\)
\(332\) 0 0
\(333\) 6.71651e25 0.219523
\(334\) 0 0
\(335\) 3.43913e26 1.05555
\(336\) 0 0
\(337\) −1.04705e26 −0.301893 −0.150947 0.988542i \(-0.548232\pi\)
−0.150947 + 0.988542i \(0.548232\pi\)
\(338\) 0 0
\(339\) 3.27393e26 0.887099
\(340\) 0 0
\(341\) 9.25092e26 2.35648
\(342\) 0 0
\(343\) 1.17324e26 0.281061
\(344\) 0 0
\(345\) −2.84415e26 −0.640992
\(346\) 0 0
\(347\) −7.57106e25 −0.160582 −0.0802910 0.996771i \(-0.525585\pi\)
−0.0802910 + 0.996771i \(0.525585\pi\)
\(348\) 0 0
\(349\) −6.14974e26 −1.22798 −0.613988 0.789315i \(-0.710436\pi\)
−0.613988 + 0.789315i \(0.710436\pi\)
\(350\) 0 0
\(351\) 1.19143e26 0.224049
\(352\) 0 0
\(353\) −3.59357e25 −0.0636637 −0.0318319 0.999493i \(-0.510134\pi\)
−0.0318319 + 0.999493i \(0.510134\pi\)
\(354\) 0 0
\(355\) −1.24199e27 −2.07358
\(356\) 0 0
\(357\) 1.04330e26 0.164207
\(358\) 0 0
\(359\) 2.77822e26 0.412359 0.206180 0.978514i \(-0.433897\pi\)
0.206180 + 0.978514i \(0.433897\pi\)
\(360\) 0 0
\(361\) −6.61009e26 −0.925511
\(362\) 0 0
\(363\) 9.75693e26 1.28912
\(364\) 0 0
\(365\) −9.48422e26 −1.18284
\(366\) 0 0
\(367\) 2.81108e26 0.331039 0.165520 0.986206i \(-0.447070\pi\)
0.165520 + 0.986206i \(0.447070\pi\)
\(368\) 0 0
\(369\) −3.55302e26 −0.395203
\(370\) 0 0
\(371\) 2.77786e27 2.91933
\(372\) 0 0
\(373\) 1.46468e27 1.45479 0.727396 0.686218i \(-0.240730\pi\)
0.727396 + 0.686218i \(0.240730\pi\)
\(374\) 0 0
\(375\) 5.86593e24 0.00550818
\(376\) 0 0
\(377\) −1.04736e27 −0.930060
\(378\) 0 0
\(379\) −4.24665e26 −0.356726 −0.178363 0.983965i \(-0.557080\pi\)
−0.178363 + 0.983965i \(0.557080\pi\)
\(380\) 0 0
\(381\) 4.48253e26 0.356296
\(382\) 0 0
\(383\) −9.06730e26 −0.682167 −0.341083 0.940033i \(-0.610794\pi\)
−0.341083 + 0.940033i \(0.610794\pi\)
\(384\) 0 0
\(385\) 5.27380e27 3.75652
\(386\) 0 0
\(387\) 8.18337e26 0.552035
\(388\) 0 0
\(389\) 1.38481e26 0.0884951 0.0442476 0.999021i \(-0.485911\pi\)
0.0442476 + 0.999021i \(0.485911\pi\)
\(390\) 0 0
\(391\) 2.49589e26 0.151137
\(392\) 0 0
\(393\) 7.45171e26 0.427695
\(394\) 0 0
\(395\) −4.58346e27 −2.49416
\(396\) 0 0
\(397\) −3.07714e27 −1.58799 −0.793993 0.607927i \(-0.792001\pi\)
−0.793993 + 0.607927i \(0.792001\pi\)
\(398\) 0 0
\(399\) −4.76339e26 −0.233185
\(400\) 0 0
\(401\) −2.92618e27 −1.35920 −0.679602 0.733581i \(-0.737847\pi\)
−0.679602 + 0.733581i \(0.737847\pi\)
\(402\) 0 0
\(403\) −3.46099e27 −1.52580
\(404\) 0 0
\(405\) −3.74814e26 −0.156869
\(406\) 0 0
\(407\) −2.97943e27 −1.18411
\(408\) 0 0
\(409\) 3.30155e27 1.24630 0.623151 0.782102i \(-0.285852\pi\)
0.623151 + 0.782102i \(0.285852\pi\)
\(410\) 0 0
\(411\) 1.20904e27 0.433611
\(412\) 0 0
\(413\) −4.83487e27 −1.64782
\(414\) 0 0
\(415\) 2.90722e27 0.941832
\(416\) 0 0
\(417\) 4.75672e26 0.146514
\(418\) 0 0
\(419\) −3.60698e27 −1.05656 −0.528282 0.849069i \(-0.677164\pi\)
−0.528282 + 0.849069i \(0.677164\pi\)
\(420\) 0 0
\(421\) 1.13735e27 0.316908 0.158454 0.987366i \(-0.449349\pi\)
0.158454 + 0.987366i \(0.449349\pi\)
\(422\) 0 0
\(423\) −4.57772e26 −0.121360
\(424\) 0 0
\(425\) 7.56617e26 0.190894
\(426\) 0 0
\(427\) −7.61226e27 −1.82819
\(428\) 0 0
\(429\) −5.28514e27 −1.20852
\(430\) 0 0
\(431\) 1.65432e27 0.360254 0.180127 0.983643i \(-0.442349\pi\)
0.180127 + 0.983643i \(0.442349\pi\)
\(432\) 0 0
\(433\) −3.17892e27 −0.659412 −0.329706 0.944084i \(-0.606950\pi\)
−0.329706 + 0.944084i \(0.606950\pi\)
\(434\) 0 0
\(435\) 3.29491e27 0.651188
\(436\) 0 0
\(437\) −1.13955e27 −0.214625
\(438\) 0 0
\(439\) −2.89492e27 −0.519707 −0.259853 0.965648i \(-0.583674\pi\)
−0.259853 + 0.965648i \(0.583674\pi\)
\(440\) 0 0
\(441\) 2.31742e27 0.396642
\(442\) 0 0
\(443\) 3.00624e27 0.490665 0.245332 0.969439i \(-0.421103\pi\)
0.245332 + 0.969439i \(0.421103\pi\)
\(444\) 0 0
\(445\) 6.72582e27 1.04704
\(446\) 0 0
\(447\) −1.13186e26 −0.0168098
\(448\) 0 0
\(449\) −2.37879e27 −0.337109 −0.168554 0.985692i \(-0.553910\pi\)
−0.168554 + 0.985692i \(0.553910\pi\)
\(450\) 0 0
\(451\) 1.57611e28 2.13173
\(452\) 0 0
\(453\) −2.65266e27 −0.342491
\(454\) 0 0
\(455\) −1.97305e28 −2.43231
\(456\) 0 0
\(457\) −8.43469e27 −0.992999 −0.496500 0.868037i \(-0.665382\pi\)
−0.496500 + 0.868037i \(0.665382\pi\)
\(458\) 0 0
\(459\) 3.28919e26 0.0369875
\(460\) 0 0
\(461\) −1.82405e27 −0.195965 −0.0979823 0.995188i \(-0.531239\pi\)
−0.0979823 + 0.995188i \(0.531239\pi\)
\(462\) 0 0
\(463\) 1.00432e28 1.03103 0.515516 0.856880i \(-0.327600\pi\)
0.515516 + 0.856880i \(0.327600\pi\)
\(464\) 0 0
\(465\) 1.08880e28 1.06830
\(466\) 0 0
\(467\) 1.94652e28 1.82571 0.912856 0.408282i \(-0.133872\pi\)
0.912856 + 0.408282i \(0.133872\pi\)
\(468\) 0 0
\(469\) −1.23375e28 −1.10640
\(470\) 0 0
\(471\) −7.98343e27 −0.684652
\(472\) 0 0
\(473\) −3.63012e28 −2.97769
\(474\) 0 0
\(475\) −3.45450e27 −0.271083
\(476\) 0 0
\(477\) 8.75771e27 0.657579
\(478\) 0 0
\(479\) −1.75628e28 −1.26204 −0.631018 0.775768i \(-0.717362\pi\)
−0.631018 + 0.775768i \(0.717362\pi\)
\(480\) 0 0
\(481\) 1.11467e28 0.766699
\(482\) 0 0
\(483\) 1.02031e28 0.671873
\(484\) 0 0
\(485\) −5.13996e27 −0.324094
\(486\) 0 0
\(487\) 1.86000e28 1.12320 0.561602 0.827407i \(-0.310185\pi\)
0.561602 + 0.827407i \(0.310185\pi\)
\(488\) 0 0
\(489\) 1.74510e28 1.00944
\(490\) 0 0
\(491\) −3.81388e27 −0.211354 −0.105677 0.994400i \(-0.533701\pi\)
−0.105677 + 0.994400i \(0.533701\pi\)
\(492\) 0 0
\(493\) −2.89146e27 −0.153541
\(494\) 0 0
\(495\) 1.66266e28 0.846155
\(496\) 0 0
\(497\) 4.45551e28 2.17348
\(498\) 0 0
\(499\) 9.64495e27 0.451070 0.225535 0.974235i \(-0.427587\pi\)
0.225535 + 0.974235i \(0.427587\pi\)
\(500\) 0 0
\(501\) 1.60267e28 0.718701
\(502\) 0 0
\(503\) −1.64738e27 −0.0708485 −0.0354243 0.999372i \(-0.511278\pi\)
−0.0354243 + 0.999372i \(0.511278\pi\)
\(504\) 0 0
\(505\) 4.02784e27 0.166155
\(506\) 0 0
\(507\) 5.18403e27 0.205156
\(508\) 0 0
\(509\) 2.39315e28 0.908725 0.454363 0.890817i \(-0.349867\pi\)
0.454363 + 0.890817i \(0.349867\pi\)
\(510\) 0 0
\(511\) 3.40236e28 1.23983
\(512\) 0 0
\(513\) −1.50175e27 −0.0525249
\(514\) 0 0
\(515\) −5.53730e28 −1.85918
\(516\) 0 0
\(517\) 2.03067e28 0.654618
\(518\) 0 0
\(519\) 2.27310e28 0.703659
\(520\) 0 0
\(521\) 1.46945e28 0.436876 0.218438 0.975851i \(-0.429904\pi\)
0.218438 + 0.975851i \(0.429904\pi\)
\(522\) 0 0
\(523\) −2.10247e27 −0.0600430 −0.0300215 0.999549i \(-0.509558\pi\)
−0.0300215 + 0.999549i \(0.509558\pi\)
\(524\) 0 0
\(525\) 3.09302e28 0.848612
\(526\) 0 0
\(527\) −9.55480e27 −0.251890
\(528\) 0 0
\(529\) −1.50626e28 −0.381605
\(530\) 0 0
\(531\) −1.52428e28 −0.371170
\(532\) 0 0
\(533\) −5.89660e28 −1.38028
\(534\) 0 0
\(535\) 4.69515e26 0.0105666
\(536\) 0 0
\(537\) 3.30982e28 0.716265
\(538\) 0 0
\(539\) −1.02800e29 −2.13949
\(540\) 0 0
\(541\) −7.08289e27 −0.141788 −0.0708939 0.997484i \(-0.522585\pi\)
−0.0708939 + 0.997484i \(0.522585\pi\)
\(542\) 0 0
\(543\) −4.52293e28 −0.871007
\(544\) 0 0
\(545\) 5.41614e27 0.100352
\(546\) 0 0
\(547\) 6.08725e28 1.08531 0.542655 0.839955i \(-0.317419\pi\)
0.542655 + 0.839955i \(0.317419\pi\)
\(548\) 0 0
\(549\) −2.39991e28 −0.411799
\(550\) 0 0
\(551\) 1.32016e28 0.218039
\(552\) 0 0
\(553\) 1.64427e29 2.61432
\(554\) 0 0
\(555\) −3.50668e28 −0.536809
\(556\) 0 0
\(557\) −1.08630e29 −1.60129 −0.800643 0.599142i \(-0.795508\pi\)
−0.800643 + 0.599142i \(0.795508\pi\)
\(558\) 0 0
\(559\) 1.35811e29 1.92802
\(560\) 0 0
\(561\) −1.45908e28 −0.199511
\(562\) 0 0
\(563\) 7.40541e28 0.975463 0.487732 0.872994i \(-0.337824\pi\)
0.487732 + 0.872994i \(0.337824\pi\)
\(564\) 0 0
\(565\) −1.70931e29 −2.16927
\(566\) 0 0
\(567\) 1.34460e28 0.164427
\(568\) 0 0
\(569\) 2.96450e28 0.349359 0.174680 0.984625i \(-0.444111\pi\)
0.174680 + 0.984625i \(0.444111\pi\)
\(570\) 0 0
\(571\) 1.55764e29 1.76924 0.884622 0.466309i \(-0.154416\pi\)
0.884622 + 0.466309i \(0.154416\pi\)
\(572\) 0 0
\(573\) 2.53045e28 0.277060
\(574\) 0 0
\(575\) 7.39947e28 0.781067
\(576\) 0 0
\(577\) 1.06756e29 1.08654 0.543270 0.839558i \(-0.317186\pi\)
0.543270 + 0.839558i \(0.317186\pi\)
\(578\) 0 0
\(579\) 9.36133e28 0.918783
\(580\) 0 0
\(581\) −1.04294e29 −0.987207
\(582\) 0 0
\(583\) −3.88490e29 −3.54699
\(584\) 0 0
\(585\) −6.22042e28 −0.547877
\(586\) 0 0
\(587\) −1.34711e29 −1.14473 −0.572365 0.819999i \(-0.693974\pi\)
−0.572365 + 0.819999i \(0.693974\pi\)
\(588\) 0 0
\(589\) 4.36245e28 0.357700
\(590\) 0 0
\(591\) 9.35523e28 0.740263
\(592\) 0 0
\(593\) 1.79961e28 0.137437 0.0687185 0.997636i \(-0.478109\pi\)
0.0687185 + 0.997636i \(0.478109\pi\)
\(594\) 0 0
\(595\) −5.44704e28 −0.401543
\(596\) 0 0
\(597\) 9.54174e28 0.679043
\(598\) 0 0
\(599\) −4.20344e28 −0.288817 −0.144409 0.989518i \(-0.546128\pi\)
−0.144409 + 0.989518i \(0.546128\pi\)
\(600\) 0 0
\(601\) −2.69124e28 −0.178554 −0.0892772 0.996007i \(-0.528456\pi\)
−0.0892772 + 0.996007i \(0.528456\pi\)
\(602\) 0 0
\(603\) −3.88964e28 −0.249217
\(604\) 0 0
\(605\) −5.09408e29 −3.15235
\(606\) 0 0
\(607\) 1.40422e29 0.839373 0.419687 0.907669i \(-0.362140\pi\)
0.419687 + 0.907669i \(0.362140\pi\)
\(608\) 0 0
\(609\) −1.18201e29 −0.682560
\(610\) 0 0
\(611\) −7.59720e28 −0.423859
\(612\) 0 0
\(613\) −2.71738e29 −1.46493 −0.732464 0.680806i \(-0.761630\pi\)
−0.732464 + 0.680806i \(0.761630\pi\)
\(614\) 0 0
\(615\) 1.85503e29 0.966410
\(616\) 0 0
\(617\) 2.15284e29 1.08397 0.541984 0.840389i \(-0.317673\pi\)
0.541984 + 0.840389i \(0.317673\pi\)
\(618\) 0 0
\(619\) −1.12642e29 −0.548212 −0.274106 0.961699i \(-0.588382\pi\)
−0.274106 + 0.961699i \(0.588382\pi\)
\(620\) 0 0
\(621\) 3.21672e28 0.151339
\(622\) 0 0
\(623\) −2.41282e29 −1.09749
\(624\) 0 0
\(625\) −2.28900e29 −1.00671
\(626\) 0 0
\(627\) 6.66172e28 0.283320
\(628\) 0 0
\(629\) 3.07730e28 0.126572
\(630\) 0 0
\(631\) −4.46959e29 −1.77811 −0.889056 0.457798i \(-0.848638\pi\)
−0.889056 + 0.457798i \(0.848638\pi\)
\(632\) 0 0
\(633\) 2.60091e29 1.00089
\(634\) 0 0
\(635\) −2.34033e29 −0.871269
\(636\) 0 0
\(637\) 3.84599e29 1.38530
\(638\) 0 0
\(639\) 1.40468e29 0.489575
\(640\) 0 0
\(641\) −2.05893e29 −0.694435 −0.347218 0.937785i \(-0.612873\pi\)
−0.347218 + 0.937785i \(0.612873\pi\)
\(642\) 0 0
\(643\) −1.28958e29 −0.420951 −0.210476 0.977599i \(-0.567501\pi\)
−0.210476 + 0.977599i \(0.567501\pi\)
\(644\) 0 0
\(645\) −4.27253e29 −1.34992
\(646\) 0 0
\(647\) −2.06845e28 −0.0632629 −0.0316315 0.999500i \(-0.510070\pi\)
−0.0316315 + 0.999500i \(0.510070\pi\)
\(648\) 0 0
\(649\) 6.76168e29 2.00210
\(650\) 0 0
\(651\) −3.90596e29 −1.11977
\(652\) 0 0
\(653\) −1.59343e28 −0.0442328 −0.0221164 0.999755i \(-0.507040\pi\)
−0.0221164 + 0.999755i \(0.507040\pi\)
\(654\) 0 0
\(655\) −3.89053e29 −1.04586
\(656\) 0 0
\(657\) 1.07266e29 0.279271
\(658\) 0 0
\(659\) 5.85081e29 1.47543 0.737716 0.675111i \(-0.235904\pi\)
0.737716 + 0.675111i \(0.235904\pi\)
\(660\) 0 0
\(661\) 6.08260e29 1.48585 0.742923 0.669377i \(-0.233439\pi\)
0.742923 + 0.669377i \(0.233439\pi\)
\(662\) 0 0
\(663\) 5.45875e28 0.129182
\(664\) 0 0
\(665\) 2.48696e29 0.570219
\(666\) 0 0
\(667\) −2.82775e29 −0.628231
\(668\) 0 0
\(669\) −1.40965e29 −0.303484
\(670\) 0 0
\(671\) 1.06459e30 2.22125
\(672\) 0 0
\(673\) 1.17199e29 0.237009 0.118505 0.992953i \(-0.462190\pi\)
0.118505 + 0.992953i \(0.462190\pi\)
\(674\) 0 0
\(675\) 9.75131e28 0.191150
\(676\) 0 0
\(677\) −1.66070e29 −0.315581 −0.157790 0.987473i \(-0.550437\pi\)
−0.157790 + 0.987473i \(0.550437\pi\)
\(678\) 0 0
\(679\) 1.84391e29 0.339708
\(680\) 0 0
\(681\) −8.34169e28 −0.149008
\(682\) 0 0
\(683\) 3.97118e29 0.687863 0.343931 0.938995i \(-0.388241\pi\)
0.343931 + 0.938995i \(0.388241\pi\)
\(684\) 0 0
\(685\) −6.31236e29 −1.06033
\(686\) 0 0
\(687\) −3.62016e29 −0.589770
\(688\) 0 0
\(689\) 1.45343e30 2.29664
\(690\) 0 0
\(691\) 5.43975e29 0.833795 0.416897 0.908954i \(-0.363117\pi\)
0.416897 + 0.908954i \(0.363117\pi\)
\(692\) 0 0
\(693\) −5.96463e29 −0.886920
\(694\) 0 0
\(695\) −2.48348e29 −0.358278
\(696\) 0 0
\(697\) −1.62788e29 −0.227866
\(698\) 0 0
\(699\) 2.02842e29 0.275516
\(700\) 0 0
\(701\) 5.32602e29 0.702043 0.351022 0.936367i \(-0.385834\pi\)
0.351022 + 0.936367i \(0.385834\pi\)
\(702\) 0 0
\(703\) −1.40501e29 −0.179741
\(704\) 0 0
\(705\) 2.39002e29 0.296768
\(706\) 0 0
\(707\) −1.44494e29 −0.174159
\(708\) 0 0
\(709\) −1.45514e30 −1.70262 −0.851312 0.524660i \(-0.824193\pi\)
−0.851312 + 0.524660i \(0.824193\pi\)
\(710\) 0 0
\(711\) 5.18387e29 0.588875
\(712\) 0 0
\(713\) −9.34428e29 −1.03064
\(714\) 0 0
\(715\) 2.75936e30 2.95526
\(716\) 0 0
\(717\) 1.56717e28 0.0162991
\(718\) 0 0
\(719\) −5.61730e29 −0.567381 −0.283690 0.958916i \(-0.591559\pi\)
−0.283690 + 0.958916i \(0.591559\pi\)
\(720\) 0 0
\(721\) 1.98645e30 1.94875
\(722\) 0 0
\(723\) −2.74210e29 −0.261295
\(724\) 0 0
\(725\) −8.57218e29 −0.793491
\(726\) 0 0
\(727\) 9.24645e29 0.831503 0.415751 0.909478i \(-0.363519\pi\)
0.415751 + 0.909478i \(0.363519\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 3.74937e29 0.318292
\(732\) 0 0
\(733\) −1.10815e30 −0.914124 −0.457062 0.889435i \(-0.651098\pi\)
−0.457062 + 0.889435i \(0.651098\pi\)
\(734\) 0 0
\(735\) −1.20992e30 −0.969928
\(736\) 0 0
\(737\) 1.72543e30 1.34428
\(738\) 0 0
\(739\) −2.39445e30 −1.81317 −0.906585 0.422023i \(-0.861320\pi\)
−0.906585 + 0.422023i \(0.861320\pi\)
\(740\) 0 0
\(741\) −2.49231e29 −0.183447
\(742\) 0 0
\(743\) 1.95615e30 1.39965 0.699825 0.714314i \(-0.253261\pi\)
0.699825 + 0.714314i \(0.253261\pi\)
\(744\) 0 0
\(745\) 5.90942e28 0.0411059
\(746\) 0 0
\(747\) −3.28805e29 −0.222368
\(748\) 0 0
\(749\) −1.68434e28 −0.0110757
\(750\) 0 0
\(751\) 1.69999e30 1.08699 0.543497 0.839411i \(-0.317100\pi\)
0.543497 + 0.839411i \(0.317100\pi\)
\(752\) 0 0
\(753\) 5.39325e29 0.335354
\(754\) 0 0
\(755\) 1.38495e30 0.837511
\(756\) 0 0
\(757\) −1.95617e30 −1.15054 −0.575269 0.817964i \(-0.695103\pi\)
−0.575269 + 0.817964i \(0.695103\pi\)
\(758\) 0 0
\(759\) −1.42693e30 −0.816325
\(760\) 0 0
\(761\) 6.47879e29 0.360541 0.180271 0.983617i \(-0.442303\pi\)
0.180271 + 0.983617i \(0.442303\pi\)
\(762\) 0 0
\(763\) −1.94298e29 −0.105187
\(764\) 0 0
\(765\) −1.71728e29 −0.0904474
\(766\) 0 0
\(767\) −2.52970e30 −1.29634
\(768\) 0 0
\(769\) 3.06878e30 1.53017 0.765083 0.643931i \(-0.222698\pi\)
0.765083 + 0.643931i \(0.222698\pi\)
\(770\) 0 0
\(771\) 1.43348e30 0.695540
\(772\) 0 0
\(773\) 3.32643e30 1.57070 0.785351 0.619050i \(-0.212482\pi\)
0.785351 + 0.619050i \(0.212482\pi\)
\(774\) 0 0
\(775\) −2.83267e30 −1.30175
\(776\) 0 0
\(777\) 1.25798e30 0.562671
\(778\) 0 0
\(779\) 7.43245e29 0.323585
\(780\) 0 0
\(781\) −6.23113e30 −2.64077
\(782\) 0 0
\(783\) −3.72652e29 −0.153746
\(784\) 0 0
\(785\) 4.16814e30 1.67421
\(786\) 0 0
\(787\) −3.66843e30 −1.43465 −0.717324 0.696740i \(-0.754633\pi\)
−0.717324 + 0.696740i \(0.754633\pi\)
\(788\) 0 0
\(789\) 3.37915e29 0.128676
\(790\) 0 0
\(791\) 6.13198e30 2.27378
\(792\) 0 0
\(793\) −3.98290e30 −1.43824
\(794\) 0 0
\(795\) −4.57239e30 −1.60801
\(796\) 0 0
\(797\) 9.14635e29 0.313282 0.156641 0.987656i \(-0.449933\pi\)
0.156641 + 0.987656i \(0.449933\pi\)
\(798\) 0 0
\(799\) −2.09737e29 −0.0699736
\(800\) 0 0
\(801\) −7.60685e29 −0.247209
\(802\) 0 0
\(803\) −4.75829e30 −1.50639
\(804\) 0 0
\(805\) −5.32702e30 −1.64296
\(806\) 0 0
\(807\) −1.54415e30 −0.463998
\(808\) 0 0
\(809\) −1.04306e30 −0.305385 −0.152693 0.988274i \(-0.548794\pi\)
−0.152693 + 0.988274i \(0.548794\pi\)
\(810\) 0 0
\(811\) −4.59742e28 −0.0131158 −0.00655791 0.999978i \(-0.502087\pi\)
−0.00655791 + 0.999978i \(0.502087\pi\)
\(812\) 0 0
\(813\) 8.64519e29 0.240339
\(814\) 0 0
\(815\) −9.11117e30 −2.46842
\(816\) 0 0
\(817\) −1.71185e30 −0.451996
\(818\) 0 0
\(819\) 2.23151e30 0.574272
\(820\) 0 0
\(821\) 2.75888e30 0.692037 0.346018 0.938228i \(-0.387533\pi\)
0.346018 + 0.938228i \(0.387533\pi\)
\(822\) 0 0
\(823\) 2.37654e30 0.581094 0.290547 0.956861i \(-0.406163\pi\)
0.290547 + 0.956861i \(0.406163\pi\)
\(824\) 0 0
\(825\) −4.32566e30 −1.03106
\(826\) 0 0
\(827\) 3.99308e30 0.927897 0.463949 0.885862i \(-0.346432\pi\)
0.463949 + 0.885862i \(0.346432\pi\)
\(828\) 0 0
\(829\) 7.15523e30 1.62107 0.810534 0.585691i \(-0.199177\pi\)
0.810534 + 0.585691i \(0.199177\pi\)
\(830\) 0 0
\(831\) −2.65273e30 −0.585979
\(832\) 0 0
\(833\) 1.06177e30 0.228695
\(834\) 0 0
\(835\) −8.36752e30 −1.75747
\(836\) 0 0
\(837\) −1.23143e30 −0.252227
\(838\) 0 0
\(839\) 5.19244e30 1.03722 0.518610 0.855011i \(-0.326450\pi\)
0.518610 + 0.855011i \(0.326450\pi\)
\(840\) 0 0
\(841\) −1.85694e30 −0.361775
\(842\) 0 0
\(843\) 3.26486e30 0.620404
\(844\) 0 0
\(845\) −2.70657e30 −0.501677
\(846\) 0 0
\(847\) 1.82745e31 3.30422
\(848\) 0 0
\(849\) −2.48825e30 −0.438897
\(850\) 0 0
\(851\) 3.00950e30 0.517885
\(852\) 0 0
\(853\) 9.61096e30 1.61362 0.806811 0.590809i \(-0.201191\pi\)
0.806811 + 0.590809i \(0.201191\pi\)
\(854\) 0 0
\(855\) 7.84061e29 0.128442
\(856\) 0 0
\(857\) −4.96150e30 −0.793075 −0.396537 0.918019i \(-0.629788\pi\)
−0.396537 + 0.918019i \(0.629788\pi\)
\(858\) 0 0
\(859\) −4.43332e30 −0.691513 −0.345757 0.938324i \(-0.612378\pi\)
−0.345757 + 0.938324i \(0.612378\pi\)
\(860\) 0 0
\(861\) −6.65471e30 −1.01297
\(862\) 0 0
\(863\) 3.32513e30 0.493965 0.246982 0.969020i \(-0.420561\pi\)
0.246982 + 0.969020i \(0.420561\pi\)
\(864\) 0 0
\(865\) −1.18678e31 −1.72069
\(866\) 0 0
\(867\) −3.92911e30 −0.556024
\(868\) 0 0
\(869\) −2.29955e31 −3.17640
\(870\) 0 0
\(871\) −6.45525e30 −0.870408
\(872\) 0 0
\(873\) 5.81326e29 0.0765192
\(874\) 0 0
\(875\) 1.09867e29 0.0141183
\(876\) 0 0
\(877\) 4.83563e30 0.606677 0.303339 0.952883i \(-0.401899\pi\)
0.303339 + 0.952883i \(0.401899\pi\)
\(878\) 0 0
\(879\) −3.76455e30 −0.461137
\(880\) 0 0
\(881\) 1.40504e30 0.168051 0.0840257 0.996464i \(-0.473222\pi\)
0.0840257 + 0.996464i \(0.473222\pi\)
\(882\) 0 0
\(883\) 3.49210e30 0.407849 0.203924 0.978987i \(-0.434630\pi\)
0.203924 + 0.978987i \(0.434630\pi\)
\(884\) 0 0
\(885\) 7.95825e30 0.907640
\(886\) 0 0
\(887\) −6.00845e30 −0.669213 −0.334607 0.942358i \(-0.608603\pi\)
−0.334607 + 0.942358i \(0.608603\pi\)
\(888\) 0 0
\(889\) 8.39567e30 0.913244
\(890\) 0 0
\(891\) −1.88046e30 −0.199778
\(892\) 0 0
\(893\) 9.57600e29 0.0993673
\(894\) 0 0
\(895\) −1.72805e31 −1.75152
\(896\) 0 0
\(897\) 5.33847e30 0.528563
\(898\) 0 0
\(899\) 1.08252e31 1.04703
\(900\) 0 0
\(901\) 4.01251e30 0.379146
\(902\) 0 0
\(903\) 1.53272e31 1.41495
\(904\) 0 0
\(905\) 2.36141e31 2.12991
\(906\) 0 0
\(907\) 1.46441e31 1.29058 0.645290 0.763938i \(-0.276737\pi\)
0.645290 + 0.763938i \(0.276737\pi\)
\(908\) 0 0
\(909\) −4.55545e29 −0.0392293
\(910\) 0 0
\(911\) −8.68894e30 −0.731179 −0.365590 0.930776i \(-0.619133\pi\)
−0.365590 + 0.930776i \(0.619133\pi\)
\(912\) 0 0
\(913\) 1.45857e31 1.19946
\(914\) 0 0
\(915\) 1.25299e31 1.00699
\(916\) 0 0
\(917\) 1.39569e31 1.09625
\(918\) 0 0
\(919\) −1.84101e30 −0.141333 −0.0706664 0.997500i \(-0.522513\pi\)
−0.0706664 + 0.997500i \(0.522513\pi\)
\(920\) 0 0
\(921\) −4.68340e30 −0.351427
\(922\) 0 0
\(923\) 2.33122e31 1.70988
\(924\) 0 0
\(925\) 9.12313e30 0.654118
\(926\) 0 0
\(927\) 6.26265e30 0.438956
\(928\) 0 0
\(929\) −1.19812e31 −0.820983 −0.410491 0.911864i \(-0.634643\pi\)
−0.410491 + 0.911864i \(0.634643\pi\)
\(930\) 0 0
\(931\) −4.84773e30 −0.324763
\(932\) 0 0
\(933\) −7.56091e30 −0.495242
\(934\) 0 0
\(935\) 7.61781e30 0.487875
\(936\) 0 0
\(937\) −3.39787e30 −0.212785 −0.106392 0.994324i \(-0.533930\pi\)
−0.106392 + 0.994324i \(0.533930\pi\)
\(938\) 0 0
\(939\) 1.01379e31 0.620813
\(940\) 0 0
\(941\) 2.36473e31 1.41609 0.708045 0.706167i \(-0.249577\pi\)
0.708045 + 0.706167i \(0.249577\pi\)
\(942\) 0 0
\(943\) −1.59202e31 −0.932341
\(944\) 0 0
\(945\) −7.02016e30 −0.402080
\(946\) 0 0
\(947\) −3.07342e31 −1.72166 −0.860829 0.508894i \(-0.830055\pi\)
−0.860829 + 0.508894i \(0.830055\pi\)
\(948\) 0 0
\(949\) 1.78019e31 0.975373
\(950\) 0 0
\(951\) −2.09160e31 −1.12094
\(952\) 0 0
\(953\) 2.23214e31 1.17016 0.585082 0.810974i \(-0.301062\pi\)
0.585082 + 0.810974i \(0.301062\pi\)
\(954\) 0 0
\(955\) −1.32115e31 −0.677509
\(956\) 0 0
\(957\) 1.65307e31 0.829310
\(958\) 0 0
\(959\) 2.26449e31 1.11141
\(960\) 0 0
\(961\) 1.49463e31 0.717693
\(962\) 0 0
\(963\) −5.31019e28 −0.00249479
\(964\) 0 0
\(965\) −4.88754e31 −2.24674
\(966\) 0 0
\(967\) −1.97944e31 −0.890356 −0.445178 0.895442i \(-0.646860\pi\)
−0.445178 + 0.895442i \(0.646860\pi\)
\(968\) 0 0
\(969\) −6.88055e29 −0.0302847
\(970\) 0 0
\(971\) −2.03370e31 −0.875961 −0.437980 0.898985i \(-0.644306\pi\)
−0.437980 + 0.898985i \(0.644306\pi\)
\(972\) 0 0
\(973\) 8.90921e30 0.375538
\(974\) 0 0
\(975\) 1.61833e31 0.667604
\(976\) 0 0
\(977\) −4.80480e31 −1.93991 −0.969956 0.243280i \(-0.921777\pi\)
−0.969956 + 0.243280i \(0.921777\pi\)
\(978\) 0 0
\(979\) 3.37438e31 1.33345
\(980\) 0 0
\(981\) −6.12562e29 −0.0236933
\(982\) 0 0
\(983\) 1.07092e31 0.405458 0.202729 0.979235i \(-0.435019\pi\)
0.202729 + 0.979235i \(0.435019\pi\)
\(984\) 0 0
\(985\) −4.88435e31 −1.81020
\(986\) 0 0
\(987\) −8.57395e30 −0.311065
\(988\) 0 0
\(989\) 3.66676e31 1.30233
\(990\) 0 0
\(991\) 2.08415e31 0.724694 0.362347 0.932043i \(-0.381976\pi\)
0.362347 + 0.932043i \(0.381976\pi\)
\(992\) 0 0
\(993\) 1.47643e31 0.502625
\(994\) 0 0
\(995\) −4.98173e31 −1.66050
\(996\) 0 0
\(997\) −1.63103e31 −0.532306 −0.266153 0.963931i \(-0.585753\pi\)
−0.266153 + 0.963931i \(0.585753\pi\)
\(998\) 0 0
\(999\) 3.96603e30 0.126741
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 48.22.a.l.1.1 3
4.3 odd 2 24.22.a.c.1.1 3
12.11 even 2 72.22.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.c.1.1 3 4.3 odd 2
48.22.a.l.1.1 3 1.1 even 1 trivial
72.22.a.d.1.3 3 12.11 even 2