Properties

Label 11.18.a.b.1.1
Level $11$
Weight $18$
Character 11.1
Self dual yes
Analytic conductor $20.154$
Analytic rank $0$
Dimension $8$
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] = [11,18,Mod(1,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 11.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: \(20.1544296079\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 715858 x^{6} - 57426812 x^{5} + 132277346400 x^{4} + 17831801296448 x^{3} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{5}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(713.940\) of defining polynomial
Character \(\chi\) \(=\) 11.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-681.940 q^{2} -16808.1 q^{3} +333970. q^{4} +1.17558e6 q^{5} +1.14621e7 q^{6} -2.07343e7 q^{7} -1.38364e8 q^{8} +1.53374e8 q^{9} +O(q^{10})\) \(q-681.940 q^{2} -16808.1 q^{3} +333970. q^{4} +1.17558e6 q^{5} +1.14621e7 q^{6} -2.07343e7 q^{7} -1.38364e8 q^{8} +1.53374e8 q^{9} -8.01675e8 q^{10} +2.14359e8 q^{11} -5.61341e9 q^{12} +5.57006e8 q^{13} +1.41395e10 q^{14} -1.97593e10 q^{15} +5.05819e10 q^{16} -4.13492e10 q^{17} -1.04592e11 q^{18} -7.69786e10 q^{19} +3.92608e11 q^{20} +3.48504e11 q^{21} -1.46180e11 q^{22} -1.06545e11 q^{23} +2.32564e12 q^{24} +6.19049e11 q^{25} -3.79844e11 q^{26} -4.07322e11 q^{27} -6.92461e12 q^{28} -2.21603e12 q^{29} +1.34747e13 q^{30} +1.88738e12 q^{31} -1.63581e13 q^{32} -3.60298e12 q^{33} +2.81977e13 q^{34} -2.43748e13 q^{35} +5.12222e13 q^{36} +1.71018e12 q^{37} +5.24947e13 q^{38} -9.36223e12 q^{39} -1.62658e14 q^{40} +4.62083e13 q^{41} -2.37659e14 q^{42} +3.90310e13 q^{43} +7.15894e13 q^{44} +1.80303e14 q^{45} +7.26573e13 q^{46} -1.67719e14 q^{47} -8.50188e14 q^{48} +1.97279e14 q^{49} -4.22154e14 q^{50} +6.95004e14 q^{51} +1.86023e14 q^{52} +2.26137e14 q^{53} +2.77769e14 q^{54} +2.51996e14 q^{55} +2.86888e15 q^{56} +1.29387e15 q^{57} +1.51120e15 q^{58} -1.73463e15 q^{59} -6.59902e15 q^{60} +1.80557e15 q^{61} -1.28708e15 q^{62} -3.18009e15 q^{63} +4.52539e15 q^{64} +6.54805e14 q^{65} +2.45701e15 q^{66} +3.06974e15 q^{67} -1.38094e16 q^{68} +1.79082e15 q^{69} +1.66221e16 q^{70} +1.54669e14 q^{71} -2.12214e16 q^{72} +1.32716e16 q^{73} -1.16624e15 q^{74} -1.04051e16 q^{75} -2.57085e16 q^{76} -4.44457e15 q^{77} +6.38448e15 q^{78} +5.63148e15 q^{79} +5.94630e16 q^{80} -1.29604e16 q^{81} -3.15113e16 q^{82} -2.96855e16 q^{83} +1.16390e17 q^{84} -4.86093e16 q^{85} -2.66168e16 q^{86} +3.72474e16 q^{87} -2.96596e16 q^{88} +4.62487e16 q^{89} -1.22956e17 q^{90} -1.15491e16 q^{91} -3.55828e16 q^{92} -3.17234e16 q^{93} +1.14375e17 q^{94} -9.04945e16 q^{95} +2.74950e17 q^{96} +1.10934e17 q^{97} -1.34532e17 q^{98} +3.28770e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 256 q^{2} + 3058 q^{3} + 391332 q^{4} + 1795234 q^{5} + 13170682 q^{6} - 896364 q^{7} - 101682420 q^{8} + 413432462 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 256 q^{2} + 3058 q^{3} + 391332 q^{4} + 1795234 q^{5} + 13170682 q^{6} - 896364 q^{7} - 101682420 q^{8} + 413432462 q^{9} - 717567014 q^{10} + 1714871048 q^{11} - 3285295504 q^{12} + 1804014468 q^{13} + 17794023508 q^{14} + 8875276234 q^{15} + 53793011976 q^{16} - 27416338904 q^{17} + 28258692878 q^{18} + 58429836440 q^{19} + 650385049400 q^{20} + 562668763708 q^{21} + 54875873536 q^{22} + 1231860549578 q^{23} + 3528111117084 q^{24} + 3225446670918 q^{25} + 3698352129748 q^{26} + 5544189136510 q^{27} + 553346903392 q^{28} + 4016405848668 q^{29} + 28329117219490 q^{30} + 21044142033258 q^{31} - 7034951233624 q^{32} + 655509458098 q^{33} + 10491977089288 q^{34} - 37564178328188 q^{35} + 18688387613044 q^{36} - 38179864040434 q^{37} - 32101053490680 q^{38} - 133370005047128 q^{39} - 229151934325836 q^{40} - 84601913468108 q^{41} - 374381853665348 q^{42} - 79795156805452 q^{43} + 83885489619492 q^{44} + 85333967988848 q^{45} - 63876340102558 q^{46} - 333992064138544 q^{47} - 921917930639032 q^{48} + 16663435022976 q^{49} - 203190218406730 q^{50} + 445337187172876 q^{51} + 383080290241336 q^{52} + 351380494472328 q^{53} + 21\!\cdots\!38 q^{54}+ \cdots + 88\!\cdots\!22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −681.940 −1.88361 −0.941805 0.336159i \(-0.890872\pi\)
−0.941805 + 0.336159i \(0.890872\pi\)
\(3\) −16808.1 −1.47907 −0.739536 0.673117i \(-0.764955\pi\)
−0.739536 + 0.673117i \(0.764955\pi\)
\(4\) 333970. 2.54799
\(5\) 1.17558e6 1.34588 0.672941 0.739696i \(-0.265031\pi\)
0.672941 + 0.739696i \(0.265031\pi\)
\(6\) 1.14621e7 2.78599
\(7\) −2.07343e7 −1.35942 −0.679712 0.733479i \(-0.737895\pi\)
−0.679712 + 0.733479i \(0.737895\pi\)
\(8\) −1.38364e8 −2.91581
\(9\) 1.53374e8 1.18765
\(10\) −8.01675e8 −2.53512
\(11\) 2.14359e8 0.301511
\(12\) −5.61341e9 −3.76866
\(13\) 5.57006e8 0.189383 0.0946915 0.995507i \(-0.469814\pi\)
0.0946915 + 0.995507i \(0.469814\pi\)
\(14\) 1.41395e10 2.56063
\(15\) −1.97593e10 −1.99066
\(16\) 5.05819e10 2.94425
\(17\) −4.13492e10 −1.43764 −0.718822 0.695194i \(-0.755318\pi\)
−0.718822 + 0.695194i \(0.755318\pi\)
\(18\) −1.04592e11 −2.23708
\(19\) −7.69786e10 −1.03983 −0.519917 0.854217i \(-0.674037\pi\)
−0.519917 + 0.854217i \(0.674037\pi\)
\(20\) 3.92608e11 3.42929
\(21\) 3.48504e11 2.01069
\(22\) −1.46180e11 −0.567930
\(23\) −1.06545e11 −0.283692 −0.141846 0.989889i \(-0.545304\pi\)
−0.141846 + 0.989889i \(0.545304\pi\)
\(24\) 2.32564e12 4.31268
\(25\) 6.19049e11 0.811400
\(26\) −3.79844e11 −0.356724
\(27\) −4.07322e11 −0.277553
\(28\) −6.92461e12 −3.46380
\(29\) −2.21603e12 −0.822609 −0.411304 0.911498i \(-0.634927\pi\)
−0.411304 + 0.911498i \(0.634927\pi\)
\(30\) 1.34747e13 3.74962
\(31\) 1.88738e12 0.397452 0.198726 0.980055i \(-0.436320\pi\)
0.198726 + 0.980055i \(0.436320\pi\)
\(32\) −1.63581e13 −2.63002
\(33\) −3.60298e12 −0.445957
\(34\) 2.81977e13 2.70796
\(35\) −2.43748e13 −1.82963
\(36\) 5.12222e13 3.02613
\(37\) 1.71018e12 0.0800435 0.0400218 0.999199i \(-0.487257\pi\)
0.0400218 + 0.999199i \(0.487257\pi\)
\(38\) 5.24947e13 1.95864
\(39\) −9.36223e12 −0.280111
\(40\) −1.62658e14 −3.92433
\(41\) 4.62083e13 0.903768 0.451884 0.892077i \(-0.350752\pi\)
0.451884 + 0.892077i \(0.350752\pi\)
\(42\) −2.37659e14 −3.78735
\(43\) 3.90310e13 0.509246 0.254623 0.967040i \(-0.418049\pi\)
0.254623 + 0.967040i \(0.418049\pi\)
\(44\) 7.15894e13 0.768247
\(45\) 1.80303e14 1.59844
\(46\) 7.26573e13 0.534364
\(47\) −1.67719e14 −1.02743 −0.513714 0.857961i \(-0.671731\pi\)
−0.513714 + 0.857961i \(0.671731\pi\)
\(48\) −8.50188e14 −4.35476
\(49\) 1.97279e14 0.848034
\(50\) −4.22154e14 −1.52836
\(51\) 6.95004e14 2.12638
\(52\) 1.86023e14 0.482546
\(53\) 2.26137e14 0.498915 0.249458 0.968386i \(-0.419748\pi\)
0.249458 + 0.968386i \(0.419748\pi\)
\(54\) 2.77769e14 0.522801
\(55\) 2.51996e14 0.405799
\(56\) 2.86888e15 3.96382
\(57\) 1.29387e15 1.53799
\(58\) 1.51120e15 1.54947
\(59\) −1.73463e15 −1.53803 −0.769017 0.639229i \(-0.779254\pi\)
−0.769017 + 0.639229i \(0.779254\pi\)
\(60\) −6.59902e15 −5.07217
\(61\) 1.80557e15 1.20590 0.602949 0.797780i \(-0.293992\pi\)
0.602949 + 0.797780i \(0.293992\pi\)
\(62\) −1.28708e15 −0.748645
\(63\) −3.18009e15 −1.61452
\(64\) 4.52539e15 2.00968
\(65\) 6.54805e14 0.254887
\(66\) 2.45701e15 0.840009
\(67\) 3.06974e15 0.923563 0.461781 0.886994i \(-0.347210\pi\)
0.461781 + 0.886994i \(0.347210\pi\)
\(68\) −1.38094e16 −3.66310
\(69\) 1.79082e15 0.419600
\(70\) 1.66221e16 3.44630
\(71\) 1.54669e14 0.0284254 0.0142127 0.999899i \(-0.495476\pi\)
0.0142127 + 0.999899i \(0.495476\pi\)
\(72\) −2.12214e16 −3.46297
\(73\) 1.32716e16 1.92610 0.963052 0.269315i \(-0.0867973\pi\)
0.963052 + 0.269315i \(0.0867973\pi\)
\(74\) −1.16624e15 −0.150771
\(75\) −1.04051e16 −1.20012
\(76\) −2.57085e16 −2.64949
\(77\) −4.44457e15 −0.409882
\(78\) 6.38448e15 0.527620
\(79\) 5.63148e15 0.417631 0.208815 0.977955i \(-0.433039\pi\)
0.208815 + 0.977955i \(0.433039\pi\)
\(80\) 5.94630e16 3.96262
\(81\) −1.29604e16 −0.777133
\(82\) −3.15113e16 −1.70235
\(83\) −2.96855e16 −1.44671 −0.723353 0.690478i \(-0.757400\pi\)
−0.723353 + 0.690478i \(0.757400\pi\)
\(84\) 1.16390e17 5.12320
\(85\) −4.86093e16 −1.93490
\(86\) −2.66168e16 −0.959221
\(87\) 3.72474e16 1.21670
\(88\) −2.96596e16 −0.879148
\(89\) 4.62487e16 1.24533 0.622665 0.782488i \(-0.286050\pi\)
0.622665 + 0.782488i \(0.286050\pi\)
\(90\) −1.22956e17 −3.01084
\(91\) −1.15491e16 −0.257452
\(92\) −3.55828e16 −0.722843
\(93\) −3.17234e16 −0.587860
\(94\) 1.14375e17 1.93527
\(95\) −9.04945e16 −1.39950
\(96\) 2.74950e17 3.88999
\(97\) 1.10934e17 1.43716 0.718580 0.695444i \(-0.244792\pi\)
0.718580 + 0.695444i \(0.244792\pi\)
\(98\) −1.34532e17 −1.59737
\(99\) 3.28770e16 0.358091
\(100\) 2.06744e17 2.06744
\(101\) 1.99011e17 1.82872 0.914358 0.404906i \(-0.132696\pi\)
0.914358 + 0.404906i \(0.132696\pi\)
\(102\) −4.73951e17 −4.00527
\(103\) 2.73079e16 0.212409 0.106204 0.994344i \(-0.466130\pi\)
0.106204 + 0.994344i \(0.466130\pi\)
\(104\) −7.70696e16 −0.552204
\(105\) 4.09695e17 2.70615
\(106\) −1.54212e17 −0.939761
\(107\) 5.70109e16 0.320772 0.160386 0.987054i \(-0.448726\pi\)
0.160386 + 0.987054i \(0.448726\pi\)
\(108\) −1.36033e17 −0.707201
\(109\) 4.34668e16 0.208945 0.104473 0.994528i \(-0.466685\pi\)
0.104473 + 0.994528i \(0.466685\pi\)
\(110\) −1.71846e17 −0.764367
\(111\) −2.87449e16 −0.118390
\(112\) −1.04878e18 −4.00249
\(113\) −1.72825e17 −0.611561 −0.305781 0.952102i \(-0.598917\pi\)
−0.305781 + 0.952102i \(0.598917\pi\)
\(114\) −8.82340e17 −2.89697
\(115\) −1.25252e17 −0.381816
\(116\) −7.40088e17 −2.09600
\(117\) 8.54300e16 0.224921
\(118\) 1.18292e18 2.89706
\(119\) 8.57345e17 1.95437
\(120\) 2.73398e18 5.80437
\(121\) 4.59497e16 0.0909091
\(122\) −1.23129e18 −2.27144
\(123\) −7.76676e17 −1.33674
\(124\) 6.30328e17 1.01270
\(125\) −1.69155e17 −0.253833
\(126\) 2.16863e18 3.04114
\(127\) 6.39079e17 0.837959 0.418979 0.907996i \(-0.362388\pi\)
0.418979 + 0.907996i \(0.362388\pi\)
\(128\) −9.41953e17 −1.15543
\(129\) −6.56039e17 −0.753211
\(130\) −4.46537e17 −0.480108
\(131\) −5.08763e17 −0.512518 −0.256259 0.966608i \(-0.582490\pi\)
−0.256259 + 0.966608i \(0.582490\pi\)
\(132\) −1.20329e18 −1.13629
\(133\) 1.59609e18 1.41358
\(134\) −2.09338e18 −1.73963
\(135\) −4.78839e17 −0.373553
\(136\) 5.72124e18 4.19189
\(137\) 1.75027e18 1.20498 0.602492 0.798125i \(-0.294175\pi\)
0.602492 + 0.798125i \(0.294175\pi\)
\(138\) −1.22123e18 −0.790363
\(139\) −1.38318e18 −0.841883 −0.420942 0.907088i \(-0.638300\pi\)
−0.420942 + 0.907088i \(0.638300\pi\)
\(140\) −8.14044e18 −4.66186
\(141\) 2.81905e18 1.51964
\(142\) −1.05475e17 −0.0535423
\(143\) 1.19399e17 0.0571011
\(144\) 7.75793e18 3.49675
\(145\) −2.60512e18 −1.10713
\(146\) −9.05045e18 −3.62803
\(147\) −3.31589e18 −1.25430
\(148\) 5.71147e17 0.203950
\(149\) −7.01809e17 −0.236666 −0.118333 0.992974i \(-0.537755\pi\)
−0.118333 + 0.992974i \(0.537755\pi\)
\(150\) 7.09563e18 2.26056
\(151\) −2.47410e18 −0.744926 −0.372463 0.928047i \(-0.621487\pi\)
−0.372463 + 0.928047i \(0.621487\pi\)
\(152\) 1.06511e19 3.03195
\(153\) −6.34188e18 −1.70742
\(154\) 3.03093e18 0.772058
\(155\) 2.21877e18 0.534924
\(156\) −3.12670e18 −0.713719
\(157\) 6.25634e18 1.35261 0.676307 0.736620i \(-0.263579\pi\)
0.676307 + 0.736620i \(0.263579\pi\)
\(158\) −3.84033e18 −0.786653
\(159\) −3.80094e18 −0.737931
\(160\) −1.92303e19 −3.53970
\(161\) 2.20913e18 0.385657
\(162\) 8.83820e18 1.46382
\(163\) −6.44391e18 −1.01287 −0.506436 0.862277i \(-0.669037\pi\)
−0.506436 + 0.862277i \(0.669037\pi\)
\(164\) 1.54322e19 2.30279
\(165\) −4.23559e18 −0.600206
\(166\) 2.02437e19 2.72503
\(167\) 1.11330e19 1.42404 0.712021 0.702158i \(-0.247780\pi\)
0.712021 + 0.702158i \(0.247780\pi\)
\(168\) −4.82205e19 −5.86277
\(169\) −8.34016e18 −0.964134
\(170\) 3.31486e19 3.64460
\(171\) −1.18065e19 −1.23496
\(172\) 1.30352e19 1.29755
\(173\) −4.68185e18 −0.443635 −0.221817 0.975088i \(-0.571199\pi\)
−0.221817 + 0.975088i \(0.571199\pi\)
\(174\) −2.54005e19 −2.29178
\(175\) −1.28355e19 −1.10304
\(176\) 1.08427e19 0.887726
\(177\) 2.91560e19 2.27486
\(178\) −3.15388e19 −2.34572
\(179\) 2.31874e19 1.64438 0.822190 0.569214i \(-0.192752\pi\)
0.822190 + 0.569214i \(0.192752\pi\)
\(180\) 6.02158e19 4.07281
\(181\) 2.69496e18 0.173894 0.0869469 0.996213i \(-0.472289\pi\)
0.0869469 + 0.996213i \(0.472289\pi\)
\(182\) 7.87579e18 0.484939
\(183\) −3.03483e19 −1.78361
\(184\) 1.47420e19 0.827189
\(185\) 2.01045e18 0.107729
\(186\) 2.16334e19 1.10730
\(187\) −8.86357e18 −0.433466
\(188\) −5.60132e19 −2.61787
\(189\) 8.44551e18 0.377312
\(190\) 6.17118e19 2.63610
\(191\) 1.07776e19 0.440291 0.220146 0.975467i \(-0.429347\pi\)
0.220146 + 0.975467i \(0.429347\pi\)
\(192\) −7.60635e19 −2.97246
\(193\) 3.71281e18 0.138824 0.0694122 0.997588i \(-0.477888\pi\)
0.0694122 + 0.997588i \(0.477888\pi\)
\(194\) −7.56504e19 −2.70705
\(195\) −1.10061e19 −0.376997
\(196\) 6.58851e19 2.16078
\(197\) −3.64967e18 −0.114628 −0.0573140 0.998356i \(-0.518254\pi\)
−0.0573140 + 0.998356i \(0.518254\pi\)
\(198\) −2.24201e19 −0.674504
\(199\) 2.71668e19 0.783046 0.391523 0.920168i \(-0.371948\pi\)
0.391523 + 0.920168i \(0.371948\pi\)
\(200\) −8.56542e19 −2.36588
\(201\) −5.15967e19 −1.36602
\(202\) −1.35714e20 −3.44459
\(203\) 4.59478e19 1.11827
\(204\) 2.32110e20 5.41799
\(205\) 5.43215e19 1.21637
\(206\) −1.86224e19 −0.400095
\(207\) −1.63412e19 −0.336927
\(208\) 2.81744e19 0.557591
\(209\) −1.65010e19 −0.313522
\(210\) −2.79387e20 −5.09733
\(211\) 7.26013e19 1.27217 0.636083 0.771621i \(-0.280554\pi\)
0.636083 + 0.771621i \(0.280554\pi\)
\(212\) 7.55229e19 1.27123
\(213\) −2.59969e18 −0.0420431
\(214\) −3.88780e19 −0.604209
\(215\) 4.58841e19 0.685385
\(216\) 5.63587e19 0.809290
\(217\) −3.91334e19 −0.540306
\(218\) −2.96417e19 −0.393571
\(219\) −2.23071e20 −2.84885
\(220\) 8.41591e19 1.03397
\(221\) −2.30317e19 −0.272265
\(222\) 1.96023e19 0.223001
\(223\) 1.52861e20 1.67381 0.836904 0.547349i \(-0.184363\pi\)
0.836904 + 0.547349i \(0.184363\pi\)
\(224\) 3.39174e20 3.57531
\(225\) 9.49459e19 0.963662
\(226\) 1.17856e20 1.15194
\(227\) −1.10733e19 −0.104246 −0.0521229 0.998641i \(-0.516599\pi\)
−0.0521229 + 0.998641i \(0.516599\pi\)
\(228\) 4.32113e20 3.91878
\(229\) −1.29979e20 −1.13572 −0.567861 0.823124i \(-0.692229\pi\)
−0.567861 + 0.823124i \(0.692229\pi\)
\(230\) 8.54144e19 0.719192
\(231\) 7.47050e19 0.606245
\(232\) 3.06619e20 2.39857
\(233\) −2.35354e20 −1.77499 −0.887495 0.460817i \(-0.847556\pi\)
−0.887495 + 0.460817i \(0.847556\pi\)
\(234\) −5.82581e19 −0.423664
\(235\) −1.97168e20 −1.38280
\(236\) −5.79315e20 −3.91889
\(237\) −9.46548e19 −0.617706
\(238\) −5.84658e20 −3.68127
\(239\) 1.09974e19 0.0668204 0.0334102 0.999442i \(-0.489363\pi\)
0.0334102 + 0.999442i \(0.489363\pi\)
\(240\) −9.99464e20 −5.86100
\(241\) −7.17676e19 −0.406241 −0.203120 0.979154i \(-0.565108\pi\)
−0.203120 + 0.979154i \(0.565108\pi\)
\(242\) −3.13349e19 −0.171237
\(243\) 2.70442e20 1.42699
\(244\) 6.03005e20 3.07261
\(245\) 2.31917e20 1.14135
\(246\) 5.29646e20 2.51789
\(247\) −4.28775e19 −0.196927
\(248\) −2.61145e20 −1.15889
\(249\) 4.98958e20 2.13978
\(250\) 1.15353e20 0.478123
\(251\) 3.00496e20 1.20396 0.601980 0.798511i \(-0.294379\pi\)
0.601980 + 0.798511i \(0.294379\pi\)
\(252\) −1.06205e21 −4.11379
\(253\) −2.28389e19 −0.0855362
\(254\) −4.35813e20 −1.57839
\(255\) 8.17032e20 2.86186
\(256\) 4.92026e19 0.166705
\(257\) −2.18500e20 −0.716177 −0.358088 0.933688i \(-0.616571\pi\)
−0.358088 + 0.933688i \(0.616571\pi\)
\(258\) 4.47379e20 1.41876
\(259\) −3.54592e19 −0.108813
\(260\) 2.18685e20 0.649450
\(261\) −3.39881e20 −0.976974
\(262\) 3.46946e20 0.965384
\(263\) 3.76045e20 1.01301 0.506507 0.862236i \(-0.330936\pi\)
0.506507 + 0.862236i \(0.330936\pi\)
\(264\) 4.98523e20 1.30032
\(265\) 2.65842e20 0.671481
\(266\) −1.08844e21 −2.66263
\(267\) −7.77355e20 −1.84193
\(268\) 1.02520e21 2.35323
\(269\) −7.45654e20 −1.65822 −0.829111 0.559083i \(-0.811153\pi\)
−0.829111 + 0.559083i \(0.811153\pi\)
\(270\) 3.26539e20 0.703629
\(271\) 2.17881e20 0.454967 0.227484 0.973782i \(-0.426950\pi\)
0.227484 + 0.973782i \(0.426950\pi\)
\(272\) −2.09152e21 −4.23279
\(273\) 1.94119e20 0.380790
\(274\) −1.19358e21 −2.26972
\(275\) 1.32699e20 0.244646
\(276\) 5.98081e20 1.06914
\(277\) −4.87360e20 −0.844835 −0.422417 0.906401i \(-0.638818\pi\)
−0.422417 + 0.906401i \(0.638818\pi\)
\(278\) 9.43243e20 1.58578
\(279\) 2.89474e20 0.472035
\(280\) 3.37259e21 5.33483
\(281\) 7.81004e20 1.19853 0.599266 0.800550i \(-0.295459\pi\)
0.599266 + 0.800550i \(0.295459\pi\)
\(282\) −1.92242e21 −2.86241
\(283\) −1.05855e21 −1.52943 −0.764713 0.644371i \(-0.777119\pi\)
−0.764713 + 0.644371i \(0.777119\pi\)
\(284\) 5.16546e19 0.0724275
\(285\) 1.52104e21 2.06995
\(286\) −8.14230e19 −0.107556
\(287\) −9.58094e20 −1.22860
\(288\) −2.50891e21 −3.12355
\(289\) 8.82516e20 1.06682
\(290\) 1.77654e21 2.08541
\(291\) −1.86460e21 −2.12566
\(292\) 4.43232e21 4.90769
\(293\) 5.06688e20 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(294\) 2.26124e21 2.36262
\(295\) −2.03920e21 −2.07001
\(296\) −2.36627e20 −0.233391
\(297\) −8.73130e19 −0.0836853
\(298\) 4.78592e20 0.445786
\(299\) −5.93462e19 −0.0537264
\(300\) −3.47498e21 −3.05789
\(301\) −8.09278e20 −0.692281
\(302\) 1.68719e21 1.40315
\(303\) −3.34501e21 −2.70480
\(304\) −3.89372e21 −3.06154
\(305\) 2.12259e21 1.62300
\(306\) 4.32478e21 3.21612
\(307\) 2.00355e21 1.44919 0.724594 0.689176i \(-0.242027\pi\)
0.724594 + 0.689176i \(0.242027\pi\)
\(308\) −1.48435e21 −1.04437
\(309\) −4.58996e20 −0.314168
\(310\) −1.51306e21 −1.00759
\(311\) −2.52349e21 −1.63508 −0.817538 0.575874i \(-0.804662\pi\)
−0.817538 + 0.575874i \(0.804662\pi\)
\(312\) 1.29540e21 0.816749
\(313\) 2.05053e21 1.25817 0.629084 0.777337i \(-0.283430\pi\)
0.629084 + 0.777337i \(0.283430\pi\)
\(314\) −4.26645e21 −2.54780
\(315\) −3.73845e21 −2.17296
\(316\) 1.88074e21 1.06412
\(317\) −2.75915e21 −1.51975 −0.759873 0.650072i \(-0.774739\pi\)
−0.759873 + 0.650072i \(0.774739\pi\)
\(318\) 2.59201e21 1.38997
\(319\) −4.75026e20 −0.248026
\(320\) 5.31996e21 2.70479
\(321\) −9.58248e20 −0.474444
\(322\) −1.50649e21 −0.726428
\(323\) 3.18300e21 1.49491
\(324\) −4.32838e21 −1.98012
\(325\) 3.44814e20 0.153665
\(326\) 4.39436e21 1.90786
\(327\) −7.30596e20 −0.309045
\(328\) −6.39357e21 −2.63521
\(329\) 3.47754e21 1.39671
\(330\) 2.88842e21 1.13055
\(331\) 2.54943e21 0.972535 0.486268 0.873810i \(-0.338358\pi\)
0.486268 + 0.873810i \(0.338358\pi\)
\(332\) −9.91406e21 −3.68619
\(333\) 2.62296e20 0.0950639
\(334\) −7.59204e21 −2.68234
\(335\) 3.60873e21 1.24301
\(336\) 1.76280e22 5.91997
\(337\) 2.19535e21 0.718867 0.359434 0.933171i \(-0.382970\pi\)
0.359434 + 0.933171i \(0.382970\pi\)
\(338\) 5.68749e21 1.81605
\(339\) 2.90487e21 0.904543
\(340\) −1.62340e22 −4.93010
\(341\) 4.04576e20 0.119836
\(342\) 8.05132e21 2.32619
\(343\) 7.32994e20 0.206586
\(344\) −5.40049e21 −1.48486
\(345\) 2.10526e21 0.564733
\(346\) 3.19274e21 0.835635
\(347\) −4.75493e20 −0.121435 −0.0607174 0.998155i \(-0.519339\pi\)
−0.0607174 + 0.998155i \(0.519339\pi\)
\(348\) 1.24395e22 3.10013
\(349\) 3.91716e21 0.952697 0.476349 0.879257i \(-0.341960\pi\)
0.476349 + 0.879257i \(0.341960\pi\)
\(350\) 8.75305e21 2.07769
\(351\) −2.26880e20 −0.0525638
\(352\) −3.50651e21 −0.792981
\(353\) −1.24417e21 −0.274661 −0.137330 0.990525i \(-0.543852\pi\)
−0.137330 + 0.990525i \(0.543852\pi\)
\(354\) −1.98826e22 −4.28495
\(355\) 1.81825e20 0.0382572
\(356\) 1.54457e22 3.17309
\(357\) −1.44104e22 −2.89065
\(358\) −1.58124e22 −3.09737
\(359\) 3.07812e20 0.0588820 0.0294410 0.999567i \(-0.490627\pi\)
0.0294410 + 0.999567i \(0.490627\pi\)
\(360\) −2.49475e22 −4.66075
\(361\) 4.45314e20 0.0812560
\(362\) −1.83780e21 −0.327548
\(363\) −7.72330e20 −0.134461
\(364\) −3.85705e21 −0.655984
\(365\) 1.56019e22 2.59231
\(366\) 2.06957e22 3.35962
\(367\) −1.70106e21 −0.269811 −0.134905 0.990858i \(-0.543073\pi\)
−0.134905 + 0.990858i \(0.543073\pi\)
\(368\) −5.38925e21 −0.835260
\(369\) 7.08714e21 1.07336
\(370\) −1.37100e21 −0.202920
\(371\) −4.68878e21 −0.678237
\(372\) −1.05946e22 −1.49786
\(373\) −1.88241e21 −0.260129 −0.130065 0.991506i \(-0.541518\pi\)
−0.130065 + 0.991506i \(0.541518\pi\)
\(374\) 6.04442e21 0.816481
\(375\) 2.84318e21 0.375438
\(376\) 2.32063e22 2.99578
\(377\) −1.23434e21 −0.155788
\(378\) −5.75933e21 −0.710709
\(379\) 2.24229e21 0.270557 0.135278 0.990808i \(-0.456807\pi\)
0.135278 + 0.990808i \(0.456807\pi\)
\(380\) −3.02224e22 −3.56590
\(381\) −1.07417e22 −1.23940
\(382\) −7.34970e21 −0.829337
\(383\) −3.53424e21 −0.390038 −0.195019 0.980799i \(-0.562477\pi\)
−0.195019 + 0.980799i \(0.562477\pi\)
\(384\) 1.58325e22 1.70897
\(385\) −5.22495e21 −0.551653
\(386\) −2.53191e21 −0.261491
\(387\) 5.98633e21 0.604808
\(388\) 3.70486e22 3.66187
\(389\) −3.71170e21 −0.358923 −0.179461 0.983765i \(-0.557435\pi\)
−0.179461 + 0.983765i \(0.557435\pi\)
\(390\) 7.50547e21 0.710115
\(391\) 4.40555e21 0.407848
\(392\) −2.72963e22 −2.47270
\(393\) 8.55136e21 0.758051
\(394\) 2.48886e21 0.215915
\(395\) 6.62026e21 0.562082
\(396\) 1.09799e22 0.912411
\(397\) 6.48398e21 0.527379 0.263689 0.964608i \(-0.415061\pi\)
0.263689 + 0.964608i \(0.415061\pi\)
\(398\) −1.85261e22 −1.47495
\(399\) −2.68274e22 −2.09078
\(400\) 3.13127e22 2.38897
\(401\) 1.54416e22 1.15336 0.576680 0.816970i \(-0.304348\pi\)
0.576680 + 0.816970i \(0.304348\pi\)
\(402\) 3.51859e22 2.57304
\(403\) 1.05128e21 0.0752707
\(404\) 6.64638e22 4.65955
\(405\) −1.52360e22 −1.04593
\(406\) −3.13336e22 −2.10639
\(407\) 3.66591e20 0.0241340
\(408\) −9.61635e22 −6.20011
\(409\) 2.99028e21 0.188827 0.0944134 0.995533i \(-0.469902\pi\)
0.0944134 + 0.995533i \(0.469902\pi\)
\(410\) −3.70440e22 −2.29116
\(411\) −2.94189e22 −1.78226
\(412\) 9.12002e21 0.541214
\(413\) 3.59663e22 2.09084
\(414\) 1.11437e22 0.634640
\(415\) −3.48977e22 −1.94710
\(416\) −9.11157e21 −0.498081
\(417\) 2.32486e22 1.24521
\(418\) 1.12527e22 0.590553
\(419\) 1.94485e22 1.00015 0.500076 0.865981i \(-0.333305\pi\)
0.500076 + 0.865981i \(0.333305\pi\)
\(420\) 1.36826e23 6.89523
\(421\) 3.85665e21 0.190464 0.0952320 0.995455i \(-0.469641\pi\)
0.0952320 + 0.995455i \(0.469641\pi\)
\(422\) −4.95097e22 −2.39626
\(423\) −2.57238e22 −1.22023
\(424\) −3.12892e22 −1.45474
\(425\) −2.55972e22 −1.16650
\(426\) 1.77283e21 0.0791929
\(427\) −3.74371e22 −1.63933
\(428\) 1.90399e22 0.817322
\(429\) −2.00688e21 −0.0844567
\(430\) −3.12902e22 −1.29100
\(431\) −1.17839e22 −0.476686 −0.238343 0.971181i \(-0.576604\pi\)
−0.238343 + 0.971181i \(0.576604\pi\)
\(432\) −2.06031e22 −0.817185
\(433\) −2.89668e22 −1.12656 −0.563280 0.826266i \(-0.690461\pi\)
−0.563280 + 0.826266i \(0.690461\pi\)
\(434\) 2.66866e22 1.01773
\(435\) 4.37873e22 1.63753
\(436\) 1.45166e22 0.532389
\(437\) 8.20168e21 0.294992
\(438\) 1.52121e23 5.36612
\(439\) −2.18950e22 −0.757524 −0.378762 0.925494i \(-0.623650\pi\)
−0.378762 + 0.925494i \(0.623650\pi\)
\(440\) −3.48672e22 −1.18323
\(441\) 3.02574e22 1.00717
\(442\) 1.57063e22 0.512842
\(443\) 6.03618e22 1.93344 0.966720 0.255839i \(-0.0823516\pi\)
0.966720 + 0.255839i \(0.0823516\pi\)
\(444\) −9.59993e21 −0.301656
\(445\) 5.43691e22 1.67607
\(446\) −1.04242e23 −3.15280
\(447\) 1.17961e22 0.350046
\(448\) −9.38307e22 −2.73201
\(449\) −1.87858e22 −0.536704 −0.268352 0.963321i \(-0.586479\pi\)
−0.268352 + 0.963321i \(0.586479\pi\)
\(450\) −6.47474e22 −1.81516
\(451\) 9.90515e21 0.272496
\(452\) −5.77184e22 −1.55825
\(453\) 4.15851e22 1.10180
\(454\) 7.55134e21 0.196358
\(455\) −1.35769e22 −0.346500
\(456\) −1.79025e23 −4.48448
\(457\) −1.04123e22 −0.256010 −0.128005 0.991774i \(-0.540857\pi\)
−0.128005 + 0.991774i \(0.540857\pi\)
\(458\) 8.86379e22 2.13926
\(459\) 1.68424e22 0.399022
\(460\) −4.18305e22 −0.972861
\(461\) −4.41492e22 −1.00801 −0.504005 0.863701i \(-0.668141\pi\)
−0.504005 + 0.863701i \(0.668141\pi\)
\(462\) −5.09443e22 −1.14193
\(463\) −1.91173e21 −0.0420716 −0.0210358 0.999779i \(-0.506696\pi\)
−0.0210358 + 0.999779i \(0.506696\pi\)
\(464\) −1.12091e23 −2.42197
\(465\) −3.72933e22 −0.791191
\(466\) 1.60497e23 3.34339
\(467\) −1.96144e21 −0.0401220 −0.0200610 0.999799i \(-0.506386\pi\)
−0.0200610 + 0.999799i \(0.506386\pi\)
\(468\) 2.85310e22 0.573097
\(469\) −6.36488e22 −1.25551
\(470\) 1.34456e23 2.60465
\(471\) −1.05158e23 −2.00061
\(472\) 2.40011e23 4.48461
\(473\) 8.36664e21 0.153543
\(474\) 6.45488e22 1.16352
\(475\) −4.76535e22 −0.843722
\(476\) 2.86327e23 4.97971
\(477\) 3.46835e22 0.592538
\(478\) −7.49959e21 −0.125864
\(479\) −9.08054e22 −1.49713 −0.748565 0.663061i \(-0.769257\pi\)
−0.748565 + 0.663061i \(0.769257\pi\)
\(480\) 3.23226e23 5.23547
\(481\) 9.52577e20 0.0151589
\(482\) 4.89412e22 0.765199
\(483\) −3.71314e22 −0.570415
\(484\) 1.53458e22 0.231635
\(485\) 1.30412e23 1.93425
\(486\) −1.84425e23 −2.68789
\(487\) −3.00468e22 −0.430331 −0.215165 0.976578i \(-0.569029\pi\)
−0.215165 + 0.976578i \(0.569029\pi\)
\(488\) −2.49826e23 −3.51616
\(489\) 1.08310e23 1.49811
\(490\) −1.58153e23 −2.14987
\(491\) 3.39667e22 0.453796 0.226898 0.973918i \(-0.427142\pi\)
0.226898 + 0.973918i \(0.427142\pi\)
\(492\) −2.59386e23 −3.40599
\(493\) 9.16312e22 1.18262
\(494\) 2.92399e22 0.370934
\(495\) 3.86496e22 0.481948
\(496\) 9.54672e22 1.17020
\(497\) −3.20694e21 −0.0386421
\(498\) −3.40259e23 −4.03052
\(499\) −1.22594e23 −1.42762 −0.713811 0.700338i \(-0.753033\pi\)
−0.713811 + 0.700338i \(0.753033\pi\)
\(500\) −5.64925e22 −0.646764
\(501\) −1.87125e23 −2.10626
\(502\) −2.04920e23 −2.26779
\(503\) 1.04506e23 1.13714 0.568569 0.822636i \(-0.307497\pi\)
0.568569 + 0.822636i \(0.307497\pi\)
\(504\) 4.40010e23 4.70764
\(505\) 2.33954e23 2.46124
\(506\) 1.55747e22 0.161117
\(507\) 1.40183e23 1.42602
\(508\) 2.13433e23 2.13511
\(509\) 5.57313e21 0.0548275 0.0274137 0.999624i \(-0.491273\pi\)
0.0274137 + 0.999624i \(0.491273\pi\)
\(510\) −5.57167e23 −5.39062
\(511\) −2.75177e23 −2.61839
\(512\) 8.99104e22 0.841425
\(513\) 3.13550e22 0.288609
\(514\) 1.49004e23 1.34900
\(515\) 3.21026e22 0.285877
\(516\) −2.19097e23 −1.91917
\(517\) −3.59521e22 −0.309781
\(518\) 2.41810e22 0.204961
\(519\) 7.86932e22 0.656168
\(520\) −9.06014e22 −0.743202
\(521\) 1.52614e23 1.23161 0.615804 0.787899i \(-0.288831\pi\)
0.615804 + 0.787899i \(0.288831\pi\)
\(522\) 2.31779e23 1.84024
\(523\) −3.85396e22 −0.301053 −0.150526 0.988606i \(-0.548097\pi\)
−0.150526 + 0.988606i \(0.548097\pi\)
\(524\) −1.69911e23 −1.30589
\(525\) 2.15741e23 1.63147
\(526\) −2.56440e23 −1.90813
\(527\) −7.80416e22 −0.571395
\(528\) −1.82245e23 −1.31301
\(529\) −1.29698e23 −0.919519
\(530\) −1.81288e23 −1.26481
\(531\) −2.66047e23 −1.82665
\(532\) 5.33047e23 3.60177
\(533\) 2.57383e22 0.171158
\(534\) 5.30109e23 3.46948
\(535\) 6.70209e22 0.431721
\(536\) −4.24742e23 −2.69293
\(537\) −3.89738e23 −2.43215
\(538\) 5.08491e23 3.12345
\(539\) 4.22884e22 0.255692
\(540\) −1.59918e23 −0.951809
\(541\) 7.66558e22 0.449126 0.224563 0.974460i \(-0.427905\pi\)
0.224563 + 0.974460i \(0.427905\pi\)
\(542\) −1.48582e23 −0.856981
\(543\) −4.52973e22 −0.257201
\(544\) 6.76396e23 3.78103
\(545\) 5.10987e22 0.281215
\(546\) −1.32377e23 −0.717260
\(547\) 2.41503e23 1.28834 0.644171 0.764882i \(-0.277203\pi\)
0.644171 + 0.764882i \(0.277203\pi\)
\(548\) 5.84538e23 3.07028
\(549\) 2.76927e23 1.43219
\(550\) −9.04925e22 −0.460818
\(551\) 1.70587e23 0.855377
\(552\) −2.47786e23 −1.22347
\(553\) −1.16765e23 −0.567737
\(554\) 3.32350e23 1.59134
\(555\) −3.37919e22 −0.159339
\(556\) −4.61939e23 −2.14511
\(557\) −2.40237e23 −1.09868 −0.549340 0.835599i \(-0.685121\pi\)
−0.549340 + 0.835599i \(0.685121\pi\)
\(558\) −1.97404e23 −0.889130
\(559\) 2.17405e22 0.0964425
\(560\) −1.23292e24 −5.38688
\(561\) 1.48980e23 0.641127
\(562\) −5.32598e23 −2.25757
\(563\) 1.12536e23 0.469862 0.234931 0.972012i \(-0.424514\pi\)
0.234931 + 0.972012i \(0.424514\pi\)
\(564\) 9.41479e23 3.87202
\(565\) −2.03170e23 −0.823090
\(566\) 7.21870e23 2.88084
\(567\) 2.68724e23 1.05645
\(568\) −2.14006e22 −0.0828828
\(569\) 1.53285e23 0.584853 0.292426 0.956288i \(-0.405537\pi\)
0.292426 + 0.956288i \(0.405537\pi\)
\(570\) −1.03726e24 −3.89899
\(571\) 1.14055e23 0.422383 0.211191 0.977445i \(-0.432266\pi\)
0.211191 + 0.977445i \(0.432266\pi\)
\(572\) 3.98757e22 0.145493
\(573\) −1.81152e23 −0.651222
\(574\) 6.53362e23 2.31421
\(575\) −6.59566e22 −0.230187
\(576\) 6.94077e23 2.38680
\(577\) −9.95652e22 −0.337375 −0.168688 0.985670i \(-0.553953\pi\)
−0.168688 + 0.985670i \(0.553953\pi\)
\(578\) −6.01823e23 −2.00947
\(579\) −6.24055e22 −0.205331
\(580\) −8.70033e23 −2.82097
\(581\) 6.15506e23 1.96669
\(582\) 1.27154e24 4.00392
\(583\) 4.84745e22 0.150429
\(584\) −1.83632e24 −5.61614
\(585\) 1.00430e23 0.302718
\(586\) −3.45531e23 −1.02650
\(587\) −1.09270e22 −0.0319946 −0.0159973 0.999872i \(-0.505092\pi\)
−0.0159973 + 0.999872i \(0.505092\pi\)
\(588\) −1.10741e24 −3.19595
\(589\) −1.45288e23 −0.413284
\(590\) 1.39061e24 3.89910
\(591\) 6.13442e22 0.169543
\(592\) 8.65039e22 0.235668
\(593\) −7.16856e23 −1.92516 −0.962581 0.270994i \(-0.912648\pi\)
−0.962581 + 0.270994i \(0.912648\pi\)
\(594\) 5.95422e22 0.157630
\(595\) 1.00788e24 2.63035
\(596\) −2.34383e23 −0.603022
\(597\) −4.56623e23 −1.15818
\(598\) 4.04705e22 0.101200
\(599\) −4.47207e23 −1.10250 −0.551252 0.834339i \(-0.685850\pi\)
−0.551252 + 0.834339i \(0.685850\pi\)
\(600\) 1.43969e24 3.49931
\(601\) −2.21734e23 −0.531374 −0.265687 0.964059i \(-0.585599\pi\)
−0.265687 + 0.964059i \(0.585599\pi\)
\(602\) 5.51879e23 1.30399
\(603\) 4.70818e23 1.09687
\(604\) −8.26275e23 −1.89806
\(605\) 5.40176e22 0.122353
\(606\) 2.28110e24 5.09479
\(607\) −8.29393e21 −0.0182666 −0.00913328 0.999958i \(-0.502907\pi\)
−0.00913328 + 0.999958i \(0.502907\pi\)
\(608\) 1.25923e24 2.73479
\(609\) −7.72297e23 −1.65401
\(610\) −1.44748e24 −3.05709
\(611\) −9.34206e22 −0.194577
\(612\) −2.11800e24 −4.35049
\(613\) 7.73224e23 1.56636 0.783179 0.621796i \(-0.213597\pi\)
0.783179 + 0.621796i \(0.213597\pi\)
\(614\) −1.36630e24 −2.72971
\(615\) −9.13044e23 −1.79909
\(616\) 6.14969e23 1.19514
\(617\) −9.48149e23 −1.81741 −0.908705 0.417439i \(-0.862928\pi\)
−0.908705 + 0.417439i \(0.862928\pi\)
\(618\) 3.13007e23 0.591769
\(619\) −5.21614e23 −0.972699 −0.486350 0.873764i \(-0.661672\pi\)
−0.486350 + 0.873764i \(0.661672\pi\)
\(620\) 7.41001e23 1.36298
\(621\) 4.33981e22 0.0787394
\(622\) 1.72087e24 3.07985
\(623\) −9.58932e23 −1.69293
\(624\) −4.73559e23 −0.824718
\(625\) −6.71152e23 −1.15303
\(626\) −1.39834e24 −2.36990
\(627\) 2.77352e23 0.463721
\(628\) 2.08943e24 3.44644
\(629\) −7.07144e22 −0.115074
\(630\) 2.54940e24 4.09301
\(631\) −5.47428e23 −0.867116 −0.433558 0.901126i \(-0.642742\pi\)
−0.433558 + 0.901126i \(0.642742\pi\)
\(632\) −7.79194e23 −1.21773
\(633\) −1.22029e24 −1.88162
\(634\) 1.88157e24 2.86261
\(635\) 7.51288e23 1.12779
\(636\) −1.26940e24 −1.88024
\(637\) 1.09885e23 0.160603
\(638\) 3.23939e23 0.467184
\(639\) 2.37221e22 0.0337595
\(640\) −1.10734e24 −1.55508
\(641\) 1.04352e24 1.44614 0.723068 0.690776i \(-0.242731\pi\)
0.723068 + 0.690776i \(0.242731\pi\)
\(642\) 6.53468e23 0.893668
\(643\) 1.48086e24 1.99857 0.999287 0.0377577i \(-0.0120215\pi\)
0.999287 + 0.0377577i \(0.0120215\pi\)
\(644\) 7.37783e23 0.982650
\(645\) −7.71226e23 −1.01373
\(646\) −2.17062e24 −2.81583
\(647\) 1.06979e24 1.36965 0.684827 0.728705i \(-0.259878\pi\)
0.684827 + 0.728705i \(0.259878\pi\)
\(648\) 1.79325e24 2.26597
\(649\) −3.71834e23 −0.463735
\(650\) −2.35142e23 −0.289446
\(651\) 6.57760e23 0.799151
\(652\) −2.15207e24 −2.58079
\(653\) −1.36660e24 −1.61763 −0.808815 0.588064i \(-0.799890\pi\)
−0.808815 + 0.588064i \(0.799890\pi\)
\(654\) 4.98222e23 0.582120
\(655\) −5.98091e23 −0.689789
\(656\) 2.33730e24 2.66092
\(657\) 2.03552e24 2.28754
\(658\) −2.37147e24 −2.63086
\(659\) 1.19225e24 1.30570 0.652849 0.757488i \(-0.273574\pi\)
0.652849 + 0.757488i \(0.273574\pi\)
\(660\) −1.41456e24 −1.52932
\(661\) 1.29055e23 0.137741 0.0688703 0.997626i \(-0.478061\pi\)
0.0688703 + 0.997626i \(0.478061\pi\)
\(662\) −1.73856e24 −1.83188
\(663\) 3.87121e23 0.402700
\(664\) 4.10741e24 4.21831
\(665\) 1.87634e24 1.90251
\(666\) −1.78870e23 −0.179063
\(667\) 2.36107e23 0.233367
\(668\) 3.71809e24 3.62844
\(669\) −2.56931e24 −2.47568
\(670\) −2.46094e24 −2.34134
\(671\) 3.87040e23 0.363592
\(672\) −5.70088e24 −5.28814
\(673\) 7.96311e23 0.729382 0.364691 0.931129i \(-0.381175\pi\)
0.364691 + 0.931129i \(0.381175\pi\)
\(674\) −1.49709e24 −1.35407
\(675\) −2.52152e23 −0.225206
\(676\) −2.78536e24 −2.45660
\(677\) 2.36216e23 0.205734 0.102867 0.994695i \(-0.467198\pi\)
0.102867 + 0.994695i \(0.467198\pi\)
\(678\) −1.98095e24 −1.70381
\(679\) −2.30014e24 −1.95371
\(680\) 6.72578e24 5.64179
\(681\) 1.86122e23 0.154187
\(682\) −2.75897e23 −0.225725
\(683\) −7.98251e23 −0.645006 −0.322503 0.946568i \(-0.604524\pi\)
−0.322503 + 0.946568i \(0.604524\pi\)
\(684\) −3.94301e24 −3.14667
\(685\) 2.05759e24 1.62177
\(686\) −4.99858e23 −0.389127
\(687\) 2.18471e24 1.67981
\(688\) 1.97426e24 1.49935
\(689\) 1.25960e23 0.0944860
\(690\) −1.43566e24 −1.06374
\(691\) 1.84393e24 1.34953 0.674764 0.738034i \(-0.264245\pi\)
0.674764 + 0.738034i \(0.264245\pi\)
\(692\) −1.56360e24 −1.13038
\(693\) −6.81680e23 −0.486798
\(694\) 3.24257e23 0.228736
\(695\) −1.62604e24 −1.13308
\(696\) −5.15371e24 −3.54765
\(697\) −1.91068e24 −1.29930
\(698\) −2.67126e24 −1.79451
\(699\) 3.95586e24 2.62534
\(700\) −4.28668e24 −2.81052
\(701\) 1.27061e24 0.823021 0.411510 0.911405i \(-0.365001\pi\)
0.411510 + 0.911405i \(0.365001\pi\)
\(702\) 1.54719e23 0.0990097
\(703\) −1.31647e23 −0.0832320
\(704\) 9.70059e23 0.605941
\(705\) 3.31402e24 2.04526
\(706\) 8.48452e23 0.517354
\(707\) −4.12635e24 −2.48600
\(708\) 9.73722e24 5.79632
\(709\) 9.86619e23 0.580305 0.290153 0.956980i \(-0.406294\pi\)
0.290153 + 0.956980i \(0.406294\pi\)
\(710\) −1.23994e23 −0.0720616
\(711\) 8.63721e23 0.496000
\(712\) −6.39916e24 −3.63114
\(713\) −2.01091e23 −0.112754
\(714\) 9.82701e24 5.44486
\(715\) 1.40363e23 0.0768514
\(716\) 7.74390e24 4.18986
\(717\) −1.84846e23 −0.0988322
\(718\) −2.09909e23 −0.110911
\(719\) 3.10705e23 0.162238 0.0811189 0.996704i \(-0.474151\pi\)
0.0811189 + 0.996704i \(0.474151\pi\)
\(720\) 9.12007e24 4.70622
\(721\) −5.66209e23 −0.288753
\(722\) −3.03677e23 −0.153055
\(723\) 1.20628e24 0.600859
\(724\) 9.00035e23 0.443079
\(725\) −1.37183e24 −0.667465
\(726\) 5.26682e23 0.253272
\(727\) −9.91632e22 −0.0471312 −0.0235656 0.999722i \(-0.507502\pi\)
−0.0235656 + 0.999722i \(0.507502\pi\)
\(728\) 1.59798e24 0.750679
\(729\) −2.87192e24 −1.33348
\(730\) −1.06395e25 −4.88290
\(731\) −1.61390e24 −0.732114
\(732\) −1.01354e25 −4.54461
\(733\) 9.22202e22 0.0408736 0.0204368 0.999791i \(-0.493494\pi\)
0.0204368 + 0.999791i \(0.493494\pi\)
\(734\) 1.16002e24 0.508218
\(735\) −3.89809e24 −1.68815
\(736\) 1.74288e24 0.746114
\(737\) 6.58027e23 0.278465
\(738\) −4.83300e24 −2.02180
\(739\) 3.12320e24 1.29158 0.645790 0.763515i \(-0.276528\pi\)
0.645790 + 0.763515i \(0.276528\pi\)
\(740\) 6.71429e23 0.274493
\(741\) 7.20691e23 0.291269
\(742\) 3.19747e24 1.27753
\(743\) 9.03411e23 0.356846 0.178423 0.983954i \(-0.442900\pi\)
0.178423 + 0.983954i \(0.442900\pi\)
\(744\) 4.38937e24 1.71409
\(745\) −8.25033e23 −0.318525
\(746\) 1.28369e24 0.489982
\(747\) −4.55297e24 −1.71819
\(748\) −2.96016e24 −1.10447
\(749\) −1.18208e24 −0.436065
\(750\) −1.93888e24 −0.707179
\(751\) 2.52046e24 0.908951 0.454476 0.890759i \(-0.349827\pi\)
0.454476 + 0.890759i \(0.349827\pi\)
\(752\) −8.48356e24 −3.02501
\(753\) −5.05078e24 −1.78074
\(754\) 8.41747e23 0.293444
\(755\) −2.90850e24 −1.00258
\(756\) 2.82055e24 0.961386
\(757\) 3.50173e24 1.18023 0.590117 0.807318i \(-0.299082\pi\)
0.590117 + 0.807318i \(0.299082\pi\)
\(758\) −1.52911e24 −0.509624
\(759\) 3.83879e23 0.126514
\(760\) 1.25212e25 4.08066
\(761\) −1.67830e24 −0.540878 −0.270439 0.962737i \(-0.587169\pi\)
−0.270439 + 0.962737i \(0.587169\pi\)
\(762\) 7.32521e24 2.33455
\(763\) −9.01251e23 −0.284045
\(764\) 3.59940e24 1.12186
\(765\) −7.45539e24 −2.29799
\(766\) 2.41014e24 0.734680
\(767\) −9.66201e23 −0.291277
\(768\) −8.27005e23 −0.246569
\(769\) 4.76727e24 1.40571 0.702855 0.711333i \(-0.251908\pi\)
0.702855 + 0.711333i \(0.251908\pi\)
\(770\) 3.56310e24 1.03910
\(771\) 3.67258e24 1.05928
\(772\) 1.23997e24 0.353723
\(773\) −9.44535e23 −0.266497 −0.133249 0.991083i \(-0.542541\pi\)
−0.133249 + 0.991083i \(0.542541\pi\)
\(774\) −4.08232e24 −1.13922
\(775\) 1.16838e24 0.322493
\(776\) −1.53493e25 −4.19048
\(777\) 5.96004e23 0.160942
\(778\) 2.53115e24 0.676070
\(779\) −3.55705e24 −0.939769
\(780\) −3.67569e24 −0.960583
\(781\) 3.31546e22 0.00857057
\(782\) −3.00432e24 −0.768226
\(783\) 9.02638e23 0.228317
\(784\) 9.97873e24 2.49683
\(785\) 7.35483e24 1.82046
\(786\) −5.83151e24 −1.42787
\(787\) 3.45699e24 0.837360 0.418680 0.908134i \(-0.362493\pi\)
0.418680 + 0.908134i \(0.362493\pi\)
\(788\) −1.21888e24 −0.292071
\(789\) −6.32062e24 −1.49832
\(790\) −4.51462e24 −1.05874
\(791\) 3.58340e24 0.831371
\(792\) −4.54900e24 −1.04412
\(793\) 1.00571e24 0.228377
\(794\) −4.42169e24 −0.993376
\(795\) −4.46831e24 −0.993169
\(796\) 9.07289e24 1.99519
\(797\) −4.45740e24 −0.969808 −0.484904 0.874567i \(-0.661146\pi\)
−0.484904 + 0.874567i \(0.661146\pi\)
\(798\) 1.82947e25 3.93822
\(799\) 6.93506e24 1.47708
\(800\) −1.01265e25 −2.13400
\(801\) 7.09334e24 1.47902
\(802\) −1.05302e25 −2.17248
\(803\) 2.84489e24 0.580742
\(804\) −1.72317e25 −3.48059
\(805\) 2.59701e24 0.519049
\(806\) −7.16910e23 −0.141781
\(807\) 1.25331e25 2.45263
\(808\) −2.75360e25 −5.33218
\(809\) −9.65318e24 −1.84973 −0.924864 0.380297i \(-0.875822\pi\)
−0.924864 + 0.380297i \(0.875822\pi\)
\(810\) 1.03900e25 1.97012
\(811\) 2.41658e24 0.453444 0.226722 0.973960i \(-0.427199\pi\)
0.226722 + 0.973960i \(0.427199\pi\)
\(812\) 1.53452e25 2.84935
\(813\) −3.66217e24 −0.672929
\(814\) −2.49993e23 −0.0454591
\(815\) −7.57533e24 −1.36321
\(816\) 3.51546e25 6.26060
\(817\) −3.00455e24 −0.529532
\(818\) −2.03919e24 −0.355676
\(819\) −1.77133e24 −0.305764
\(820\) 1.81418e25 3.09928
\(821\) 4.17116e24 0.705245 0.352622 0.935766i \(-0.385290\pi\)
0.352622 + 0.935766i \(0.385290\pi\)
\(822\) 2.00619e25 3.35708
\(823\) 5.89474e24 0.976261 0.488131 0.872771i \(-0.337679\pi\)
0.488131 + 0.872771i \(0.337679\pi\)
\(824\) −3.77843e24 −0.619342
\(825\) −2.23042e24 −0.361849
\(826\) −2.45269e25 −3.93833
\(827\) −9.04242e24 −1.43710 −0.718552 0.695474i \(-0.755195\pi\)
−0.718552 + 0.695474i \(0.755195\pi\)
\(828\) −5.45747e24 −0.858486
\(829\) −8.31631e24 −1.29484 −0.647422 0.762132i \(-0.724153\pi\)
−0.647422 + 0.762132i \(0.724153\pi\)
\(830\) 2.37981e25 3.66757
\(831\) 8.19162e24 1.24957
\(832\) 2.52067e24 0.380599
\(833\) −8.15732e24 −1.21917
\(834\) −1.58542e25 −2.34548
\(835\) 1.30878e25 1.91659
\(836\) −5.51085e24 −0.798850
\(837\) −7.68770e23 −0.110314
\(838\) −1.32627e25 −1.88390
\(839\) 4.62141e24 0.649827 0.324913 0.945744i \(-0.394665\pi\)
0.324913 + 0.945744i \(0.394665\pi\)
\(840\) −5.66871e25 −7.89060
\(841\) −2.34634e24 −0.323315
\(842\) −2.63001e24 −0.358760
\(843\) −1.31272e25 −1.77272
\(844\) 2.42466e25 3.24146
\(845\) −9.80453e24 −1.29761
\(846\) 1.75421e25 2.29844
\(847\) −9.52733e23 −0.123584
\(848\) 1.14384e25 1.46893
\(849\) 1.77923e25 2.26213
\(850\) 1.74557e25 2.19724
\(851\) −1.82211e23 −0.0227077
\(852\) −8.68219e23 −0.107125
\(853\) −4.20936e24 −0.514221 −0.257110 0.966382i \(-0.582770\pi\)
−0.257110 + 0.966382i \(0.582770\pi\)
\(854\) 2.55299e25 3.08785
\(855\) −1.38795e25 −1.66212
\(856\) −7.88826e24 −0.935308
\(857\) 1.04305e25 1.22452 0.612261 0.790656i \(-0.290260\pi\)
0.612261 + 0.790656i \(0.290260\pi\)
\(858\) 1.36857e24 0.159083
\(859\) −4.56105e24 −0.524956 −0.262478 0.964938i \(-0.584540\pi\)
−0.262478 + 0.964938i \(0.584540\pi\)
\(860\) 1.53239e25 1.74635
\(861\) 1.61038e25 1.81719
\(862\) 8.03591e24 0.897890
\(863\) −1.21177e25 −1.34069 −0.670343 0.742052i \(-0.733853\pi\)
−0.670343 + 0.742052i \(0.733853\pi\)
\(864\) 6.66302e24 0.729969
\(865\) −5.50389e24 −0.597080
\(866\) 1.97536e25 2.12200
\(867\) −1.48335e25 −1.57790
\(868\) −1.30694e25 −1.37669
\(869\) 1.20716e24 0.125920
\(870\) −2.98603e25 −3.08447
\(871\) 1.70986e24 0.174907
\(872\) −6.01424e24 −0.609243
\(873\) 1.70144e25 1.70685
\(874\) −5.59305e24 −0.555651
\(875\) 3.50729e24 0.345067
\(876\) −7.44991e25 −7.25883
\(877\) −6.90979e24 −0.666758 −0.333379 0.942793i \(-0.608189\pi\)
−0.333379 + 0.942793i \(0.608189\pi\)
\(878\) 1.49311e25 1.42688
\(879\) −8.51649e24 −0.806037
\(880\) 1.27464e25 1.19477
\(881\) −3.14171e24 −0.291656 −0.145828 0.989310i \(-0.546585\pi\)
−0.145828 + 0.989310i \(0.546585\pi\)
\(882\) −2.06337e25 −1.89712
\(883\) −1.31965e24 −0.120169 −0.0600845 0.998193i \(-0.519137\pi\)
−0.0600845 + 0.998193i \(0.519137\pi\)
\(884\) −7.69190e24 −0.693729
\(885\) 3.42752e25 3.06170
\(886\) −4.11631e25 −3.64185
\(887\) −9.68039e24 −0.848286 −0.424143 0.905595i \(-0.639425\pi\)
−0.424143 + 0.905595i \(0.639425\pi\)
\(888\) 3.97726e24 0.345202
\(889\) −1.32508e25 −1.13914
\(890\) −3.70764e25 −3.15706
\(891\) −2.77817e24 −0.234314
\(892\) 5.10510e25 4.26484
\(893\) 1.29108e25 1.06836
\(894\) −8.04424e24 −0.659350
\(895\) 2.72587e25 2.21314
\(896\) 1.95307e25 1.57072
\(897\) 9.97499e23 0.0794652
\(898\) 1.28108e25 1.01094
\(899\) −4.18250e24 −0.326948
\(900\) 3.17091e25 2.45540
\(901\) −9.35058e24 −0.717262
\(902\) −6.75472e24 −0.513277
\(903\) 1.36025e25 1.02393
\(904\) 2.39128e25 1.78319
\(905\) 3.16814e24 0.234041
\(906\) −2.83585e25 −2.07536
\(907\) −5.92079e24 −0.429257 −0.214629 0.976696i \(-0.568854\pi\)
−0.214629 + 0.976696i \(0.568854\pi\)
\(908\) −3.69816e24 −0.265617
\(909\) 3.05231e25 2.17188
\(910\) 9.25862e24 0.652671
\(911\) 6.95449e23 0.0485690 0.0242845 0.999705i \(-0.492269\pi\)
0.0242845 + 0.999705i \(0.492269\pi\)
\(912\) 6.54462e25 4.52823
\(913\) −6.36335e24 −0.436198
\(914\) 7.10053e24 0.482223
\(915\) −3.56768e25 −2.40053
\(916\) −4.34091e25 −2.89381
\(917\) 1.05488e25 0.696730
\(918\) −1.14855e25 −0.751602
\(919\) 1.22171e24 0.0792110 0.0396055 0.999215i \(-0.487390\pi\)
0.0396055 + 0.999215i \(0.487390\pi\)
\(920\) 1.73304e25 1.11330
\(921\) −3.36760e25 −2.14345
\(922\) 3.01071e25 1.89870
\(923\) 8.61512e22 0.00538328
\(924\) 2.49492e25 1.54470
\(925\) 1.05868e24 0.0649473
\(926\) 1.30369e24 0.0792465
\(927\) 4.18832e24 0.252268
\(928\) 3.62502e25 2.16348
\(929\) −3.60079e24 −0.212944 −0.106472 0.994316i \(-0.533955\pi\)
−0.106472 + 0.994316i \(0.533955\pi\)
\(930\) 2.54318e25 1.49029
\(931\) −1.51862e25 −0.881815
\(932\) −7.86011e25 −4.52265
\(933\) 4.24152e25 2.41840
\(934\) 1.33759e24 0.0755742
\(935\) −1.04198e25 −0.583394
\(936\) −1.18204e25 −0.655827
\(937\) 2.05869e25 1.13189 0.565946 0.824442i \(-0.308511\pi\)
0.565946 + 0.824442i \(0.308511\pi\)
\(938\) 4.34047e25 2.36490
\(939\) −3.44656e25 −1.86092
\(940\) −6.58480e25 −3.52335
\(941\) 1.57986e25 0.837735 0.418867 0.908047i \(-0.362427\pi\)
0.418867 + 0.908047i \(0.362427\pi\)
\(942\) 7.17111e25 3.76837
\(943\) −4.92326e24 −0.256391
\(944\) −8.77410e25 −4.52836
\(945\) 9.92837e24 0.507818
\(946\) −5.70554e24 −0.289216
\(947\) 6.49895e24 0.326489 0.163244 0.986586i \(-0.447804\pi\)
0.163244 + 0.986586i \(0.447804\pi\)
\(948\) −3.16118e25 −1.57391
\(949\) 7.39237e24 0.364771
\(950\) 3.24968e25 1.58924
\(951\) 4.63761e25 2.24781
\(952\) −1.18626e26 −5.69856
\(953\) 3.84574e25 1.83101 0.915504 0.402309i \(-0.131792\pi\)
0.915504 + 0.402309i \(0.131792\pi\)
\(954\) −2.36520e25 −1.11611
\(955\) 1.26700e25 0.592580
\(956\) 3.67281e24 0.170258
\(957\) 7.98432e24 0.366848
\(958\) 6.19238e25 2.82001
\(959\) −3.62906e25 −1.63808
\(960\) −8.94187e25 −4.00058
\(961\) −1.89879e25 −0.842032
\(962\) −6.49600e23 −0.0285534
\(963\) 8.74398e24 0.380966
\(964\) −2.39682e25 −1.03510
\(965\) 4.36471e24 0.186841
\(966\) 2.53214e25 1.07444
\(967\) −2.88586e25 −1.21381 −0.606905 0.794774i \(-0.707589\pi\)
−0.606905 + 0.794774i \(0.707589\pi\)
\(968\) −6.35779e24 −0.265073
\(969\) −5.35004e25 −2.21108
\(970\) −8.89331e25 −3.64337
\(971\) −2.30275e25 −0.935153 −0.467576 0.883953i \(-0.654873\pi\)
−0.467576 + 0.883953i \(0.654873\pi\)
\(972\) 9.03194e25 3.63595
\(973\) 2.86791e25 1.14448
\(974\) 2.04901e25 0.810576
\(975\) −5.79568e24 −0.227282
\(976\) 9.13290e25 3.55047
\(977\) 1.73913e24 0.0670238 0.0335119 0.999438i \(-0.489331\pi\)
0.0335119 + 0.999438i \(0.489331\pi\)
\(978\) −7.38610e25 −2.82186
\(979\) 9.91382e24 0.375481
\(980\) 7.74532e25 2.90816
\(981\) 6.66666e24 0.248154
\(982\) −2.31633e25 −0.854776
\(983\) 5.37513e25 1.96645 0.983227 0.182387i \(-0.0583824\pi\)
0.983227 + 0.182387i \(0.0583824\pi\)
\(984\) 1.07464e26 3.89767
\(985\) −4.29048e24 −0.154276
\(986\) −6.24870e25 −2.22759
\(987\) −5.84510e25 −2.06584
\(988\) −1.43198e25 −0.501768
\(989\) −4.15856e24 −0.144469
\(990\) −2.63567e25 −0.907803
\(991\) −4.37853e25 −1.49521 −0.747605 0.664144i \(-0.768796\pi\)
−0.747605 + 0.664144i \(0.768796\pi\)
\(992\) −3.08740e25 −1.04531
\(993\) −4.28512e25 −1.43845
\(994\) 2.18694e24 0.0727867
\(995\) 3.19367e25 1.05389
\(996\) 1.66637e26 5.45214
\(997\) 1.18208e25 0.383474 0.191737 0.981446i \(-0.438588\pi\)
0.191737 + 0.981446i \(0.438588\pi\)
\(998\) 8.36015e25 2.68908
\(999\) −6.96592e23 −0.0222163
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 11.18.a.b.1.1 8
3.2 odd 2 99.18.a.e.1.8 8
11.10 odd 2 121.18.a.d.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.18.a.b.1.1 8 1.1 even 1 trivial
99.18.a.e.1.8 8 3.2 odd 2
121.18.a.d.1.8 8 11.10 odd 2