Properties

Label 27.11.b.b.26.1
Level $27$
Weight $11$
Character 27.26
Analytic conductor $17.155$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,11,Mod(26,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(17.1546458222\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 27.26
Dual form 27.11.b.b.26.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-51.9615i q^{2} -1676.00 q^{4} +831.384i q^{5} -16093.0 q^{7} +33878.9i q^{8} +O(q^{10})\) \(q-51.9615i q^{2} -1676.00 q^{4} +831.384i q^{5} -16093.0 q^{7} +33878.9i q^{8} +43200.0 q^{10} +293479. i q^{11} +608639. q^{13} +836217. i q^{14} +44176.0 q^{16} +17459.1i q^{17} -2.10455e6 q^{19} -1.39340e6i q^{20} +1.52496e7 q^{22} -4.58342e6i q^{23} +9.07442e6 q^{25} -3.16258e7i q^{26} +2.69719e7 q^{28} +3.14679e7i q^{29} -2.60141e7 q^{31} +3.23966e7i q^{32} +907200. q^{34} -1.33795e7i q^{35} +5.19466e7 q^{37} +1.09356e8i q^{38} -2.81664e7 q^{40} +1.79637e8i q^{41} -7.19650e7 q^{43} -4.91870e8i q^{44} -2.38162e8 q^{46} +1.49783e8i q^{47} -2.34906e7 q^{49} -4.71521e8i q^{50} -1.02008e9 q^{52} +1.05218e8i q^{53} -2.43994e8 q^{55} -5.45213e8i q^{56} +1.63512e9 q^{58} +6.06334e8i q^{59} -3.56794e8 q^{61} +1.35173e9i q^{62} +1.72861e9 q^{64} +5.06013e8i q^{65} -5.91549e8 q^{67} -2.92614e7i q^{68} -6.95218e8 q^{70} +3.09009e9i q^{71} +4.13274e8 q^{73} -2.69922e9i q^{74} +3.52722e9 q^{76} -4.72295e9i q^{77} -1.08052e9 q^{79} +3.67272e7i q^{80} +9.33422e9 q^{82} -5.70129e9i q^{83} -1.45152e7 q^{85} +3.73941e9i q^{86} -9.94274e9 q^{88} -2.84786e9i q^{89} -9.79483e9 q^{91} +7.68182e9i q^{92} +7.78296e9 q^{94} -1.74969e9i q^{95} -1.36942e10 q^{97} +1.22061e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3352 q^{4} - 32186 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3352 q^{4} - 32186 q^{7} + 86400 q^{10} + 1217278 q^{13} + 88352 q^{16} - 4209098 q^{19} + 30499200 q^{22} + 18148850 q^{25} + 53943736 q^{28} - 52028252 q^{31} + 1814400 q^{34} + 103893214 q^{37} - 56332800 q^{40} - 143929964 q^{43} - 476323200 q^{46} - 46981200 q^{49} - 2040157928 q^{52} - 487987200 q^{55} + 3270240000 q^{58} - 713588402 q^{61} + 3457221248 q^{64} - 1183097978 q^{67} - 1390435200 q^{70} + 826548286 q^{73} + 7054448248 q^{76} - 2161039898 q^{79} + 18668448000 q^{82} - 29030400 q^{85} - 19885478400 q^{88} - 19589654854 q^{91} + 15565910400 q^{94} - 27388355186 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 51.9615i − 1.62380i −0.583798 0.811899i \(-0.698434\pi\)
0.583798 0.811899i \(-0.301566\pi\)
\(3\) 0 0
\(4\) −1676.00 −1.63672
\(5\) 831.384i 0.266043i 0.991113 + 0.133022i \(0.0424679\pi\)
−0.991113 + 0.133022i \(0.957532\pi\)
\(6\) 0 0
\(7\) −16093.0 −0.957518 −0.478759 0.877946i \(-0.658913\pi\)
−0.478759 + 0.877946i \(0.658913\pi\)
\(8\) 33878.9i 1.03390i
\(9\) 0 0
\(10\) 43200.0 0.432000
\(11\) 293479.i 1.82227i 0.412106 + 0.911136i \(0.364793\pi\)
−0.412106 + 0.911136i \(0.635207\pi\)
\(12\) 0 0
\(13\) 608639. 1.63924 0.819621 0.572906i \(-0.194184\pi\)
0.819621 + 0.572906i \(0.194184\pi\)
\(14\) 836217.i 1.55481i
\(15\) 0 0
\(16\) 44176.0 0.0421295
\(17\) 17459.1i 0.0122964i 0.999981 + 0.00614818i \(0.00195704\pi\)
−0.999981 + 0.00614818i \(0.998043\pi\)
\(18\) 0 0
\(19\) −2.10455e6 −0.849945 −0.424973 0.905206i \(-0.639716\pi\)
−0.424973 + 0.905206i \(0.639716\pi\)
\(20\) − 1.39340e6i − 0.435438i
\(21\) 0 0
\(22\) 1.52496e7 2.95900
\(23\) − 4.58342e6i − 0.712116i −0.934464 0.356058i \(-0.884121\pi\)
0.934464 0.356058i \(-0.115879\pi\)
\(24\) 0 0
\(25\) 9.07442e6 0.929221
\(26\) − 3.16258e7i − 2.66180i
\(27\) 0 0
\(28\) 2.69719e7 1.56719
\(29\) 3.14679e7i 1.53419i 0.641536 + 0.767093i \(0.278297\pi\)
−0.641536 + 0.767093i \(0.721703\pi\)
\(30\) 0 0
\(31\) −2.60141e7 −0.908659 −0.454329 0.890834i \(-0.650121\pi\)
−0.454329 + 0.890834i \(0.650121\pi\)
\(32\) 3.23966e7i 0.965493i
\(33\) 0 0
\(34\) 907200. 0.0199668
\(35\) − 1.33795e7i − 0.254741i
\(36\) 0 0
\(37\) 5.19466e7 0.749115 0.374558 0.927204i \(-0.377795\pi\)
0.374558 + 0.927204i \(0.377795\pi\)
\(38\) 1.09356e8i 1.38014i
\(39\) 0 0
\(40\) −2.81664e7 −0.275062
\(41\) 1.79637e8i 1.55052i 0.631643 + 0.775259i \(0.282381\pi\)
−0.631643 + 0.775259i \(0.717619\pi\)
\(42\) 0 0
\(43\) −7.19650e7 −0.489530 −0.244765 0.969582i \(-0.578711\pi\)
−0.244765 + 0.969582i \(0.578711\pi\)
\(44\) − 4.91870e8i − 2.98255i
\(45\) 0 0
\(46\) −2.38162e8 −1.15633
\(47\) 1.49783e8i 0.653090i 0.945182 + 0.326545i \(0.105885\pi\)
−0.945182 + 0.326545i \(0.894115\pi\)
\(48\) 0 0
\(49\) −2.34906e7 −0.0831599
\(50\) − 4.71521e8i − 1.50887i
\(51\) 0 0
\(52\) −1.02008e9 −2.68298
\(53\) 1.05218e8i 0.251601i 0.992056 + 0.125800i \(0.0401499\pi\)
−0.992056 + 0.125800i \(0.959850\pi\)
\(54\) 0 0
\(55\) −2.43994e8 −0.484803
\(56\) − 5.45213e8i − 0.989980i
\(57\) 0 0
\(58\) 1.63512e9 2.49121
\(59\) 6.06334e8i 0.848110i 0.905636 + 0.424055i \(0.139394\pi\)
−0.905636 + 0.424055i \(0.860606\pi\)
\(60\) 0 0
\(61\) −3.56794e8 −0.422443 −0.211222 0.977438i \(-0.567744\pi\)
−0.211222 + 0.977438i \(0.567744\pi\)
\(62\) 1.35173e9i 1.47548i
\(63\) 0 0
\(64\) 1.72861e9 1.60989
\(65\) 5.06013e8i 0.436109i
\(66\) 0 0
\(67\) −5.91549e8 −0.438144 −0.219072 0.975709i \(-0.570303\pi\)
−0.219072 + 0.975709i \(0.570303\pi\)
\(68\) − 2.92614e7i − 0.0201257i
\(69\) 0 0
\(70\) −6.95218e8 −0.413648
\(71\) 3.09009e9i 1.71269i 0.516404 + 0.856345i \(0.327270\pi\)
−0.516404 + 0.856345i \(0.672730\pi\)
\(72\) 0 0
\(73\) 4.13274e8 0.199354 0.0996768 0.995020i \(-0.468219\pi\)
0.0996768 + 0.995020i \(0.468219\pi\)
\(74\) − 2.69922e9i − 1.21641i
\(75\) 0 0
\(76\) 3.52722e9 1.39112
\(77\) − 4.72295e9i − 1.74486i
\(78\) 0 0
\(79\) −1.08052e9 −0.351154 −0.175577 0.984466i \(-0.556179\pi\)
−0.175577 + 0.984466i \(0.556179\pi\)
\(80\) 3.67272e7i 0.0112083i
\(81\) 0 0
\(82\) 9.33422e9 2.51773
\(83\) − 5.70129e9i − 1.44738i −0.690125 0.723691i \(-0.742444\pi\)
0.690125 0.723691i \(-0.257556\pi\)
\(84\) 0 0
\(85\) −1.45152e7 −0.00327136
\(86\) 3.73941e9i 0.794897i
\(87\) 0 0
\(88\) −9.94274e9 −1.88405
\(89\) − 2.84786e9i − 0.509998i −0.966941 0.254999i \(-0.917925\pi\)
0.966941 0.254999i \(-0.0820751\pi\)
\(90\) 0 0
\(91\) −9.79483e9 −1.56960
\(92\) 7.68182e9i 1.16553i
\(93\) 0 0
\(94\) 7.78296e9 1.06049
\(95\) − 1.74969e9i − 0.226122i
\(96\) 0 0
\(97\) −1.36942e10 −1.59469 −0.797347 0.603521i \(-0.793764\pi\)
−0.797347 + 0.603521i \(0.793764\pi\)
\(98\) 1.22061e9i 0.135035i
\(99\) 0 0
\(100\) −1.52087e10 −1.52087
\(101\) 1.23670e10i 1.17667i 0.808616 + 0.588337i \(0.200217\pi\)
−0.808616 + 0.588337i \(0.799783\pi\)
\(102\) 0 0
\(103\) 2.71632e9 0.234312 0.117156 0.993114i \(-0.462622\pi\)
0.117156 + 0.993114i \(0.462622\pi\)
\(104\) 2.06200e10i 1.69482i
\(105\) 0 0
\(106\) 5.46731e9 0.408549
\(107\) − 1.23713e10i − 0.882056i −0.897493 0.441028i \(-0.854614\pi\)
0.897493 0.441028i \(-0.145386\pi\)
\(108\) 0 0
\(109\) 5.78399e9 0.375920 0.187960 0.982177i \(-0.439813\pi\)
0.187960 + 0.982177i \(0.439813\pi\)
\(110\) 1.26783e10i 0.787221i
\(111\) 0 0
\(112\) −7.10924e8 −0.0403398
\(113\) − 2.13143e10i − 1.15685i −0.815735 0.578426i \(-0.803667\pi\)
0.815735 0.578426i \(-0.196333\pi\)
\(114\) 0 0
\(115\) 3.81059e9 0.189453
\(116\) − 5.27402e10i − 2.51103i
\(117\) 0 0
\(118\) 3.15061e10 1.37716
\(119\) − 2.80969e8i − 0.0117740i
\(120\) 0 0
\(121\) −6.01923e10 −2.32067
\(122\) 1.85396e10i 0.685963i
\(123\) 0 0
\(124\) 4.35997e10 1.48722
\(125\) 1.56633e10i 0.513256i
\(126\) 0 0
\(127\) 1.59955e9 0.0484148 0.0242074 0.999707i \(-0.492294\pi\)
0.0242074 + 0.999707i \(0.492294\pi\)
\(128\) − 5.66472e10i − 1.64865i
\(129\) 0 0
\(130\) 2.62932e10 0.708152
\(131\) − 1.75475e10i − 0.454840i −0.973797 0.227420i \(-0.926971\pi\)
0.973797 0.227420i \(-0.0730290\pi\)
\(132\) 0 0
\(133\) 3.38685e10 0.813838
\(134\) 3.07378e10i 0.711457i
\(135\) 0 0
\(136\) −5.91494e8 −0.0127132
\(137\) 5.41009e10i 1.12099i 0.828158 + 0.560495i \(0.189389\pi\)
−0.828158 + 0.560495i \(0.810611\pi\)
\(138\) 0 0
\(139\) 2.58062e10 0.497336 0.248668 0.968589i \(-0.420007\pi\)
0.248668 + 0.968589i \(0.420007\pi\)
\(140\) 2.24240e10i 0.416939i
\(141\) 0 0
\(142\) 1.60566e11 2.78106
\(143\) 1.78623e11i 2.98714i
\(144\) 0 0
\(145\) −2.61619e10 −0.408159
\(146\) − 2.14744e10i − 0.323710i
\(147\) 0 0
\(148\) −8.70625e10 −1.22609
\(149\) − 8.88933e10i − 1.21042i −0.796064 0.605212i \(-0.793088\pi\)
0.796064 0.605212i \(-0.206912\pi\)
\(150\) 0 0
\(151\) −1.51414e11 −1.92878 −0.964389 0.264487i \(-0.914797\pi\)
−0.964389 + 0.264487i \(0.914797\pi\)
\(152\) − 7.12998e10i − 0.878761i
\(153\) 0 0
\(154\) −2.45412e11 −2.83330
\(155\) − 2.16277e10i − 0.241742i
\(156\) 0 0
\(157\) 1.15944e11 1.21549 0.607745 0.794132i \(-0.292074\pi\)
0.607745 + 0.794132i \(0.292074\pi\)
\(158\) 5.61455e10i 0.570203i
\(159\) 0 0
\(160\) −2.69340e10 −0.256863
\(161\) 7.37610e10i 0.681864i
\(162\) 0 0
\(163\) 9.85876e10 0.856809 0.428405 0.903587i \(-0.359076\pi\)
0.428405 + 0.903587i \(0.359076\pi\)
\(164\) − 3.01072e11i − 2.53776i
\(165\) 0 0
\(166\) −2.96248e11 −2.35025
\(167\) − 1.03461e11i − 0.796518i −0.917273 0.398259i \(-0.869614\pi\)
0.917273 0.398259i \(-0.130386\pi\)
\(168\) 0 0
\(169\) 2.32583e11 1.68711
\(170\) 7.54232e8i 0.00531203i
\(171\) 0 0
\(172\) 1.20613e11 0.801222
\(173\) 5.90424e10i 0.381008i 0.981686 + 0.190504i \(0.0610122\pi\)
−0.981686 + 0.190504i \(0.938988\pi\)
\(174\) 0 0
\(175\) −1.46035e11 −0.889746
\(176\) 1.29647e10i 0.0767714i
\(177\) 0 0
\(178\) −1.47979e11 −0.828133
\(179\) − 2.18650e11i − 1.18983i −0.803790 0.594913i \(-0.797187\pi\)
0.803790 0.594913i \(-0.202813\pi\)
\(180\) 0 0
\(181\) −8.39160e9 −0.0431968 −0.0215984 0.999767i \(-0.506876\pi\)
−0.0215984 + 0.999767i \(0.506876\pi\)
\(182\) 5.08954e11i 2.54872i
\(183\) 0 0
\(184\) 1.55281e11 0.736258
\(185\) 4.31876e10i 0.199297i
\(186\) 0 0
\(187\) −5.12387e9 −0.0224073
\(188\) − 2.51036e11i − 1.06893i
\(189\) 0 0
\(190\) −9.09165e10 −0.367176
\(191\) 1.44581e11i 0.568780i 0.958709 + 0.284390i \(0.0917911\pi\)
−0.958709 + 0.284390i \(0.908209\pi\)
\(192\) 0 0
\(193\) 3.99338e10 0.149126 0.0745631 0.997216i \(-0.476244\pi\)
0.0745631 + 0.997216i \(0.476244\pi\)
\(194\) 7.11570e11i 2.58946i
\(195\) 0 0
\(196\) 3.93702e10 0.136109
\(197\) − 2.03053e9i − 0.00684351i −0.999994 0.00342175i \(-0.998911\pi\)
0.999994 0.00342175i \(-0.00108918\pi\)
\(198\) 0 0
\(199\) 4.87714e11 1.56279 0.781394 0.624038i \(-0.214509\pi\)
0.781394 + 0.624038i \(0.214509\pi\)
\(200\) 3.07432e11i 0.960724i
\(201\) 0 0
\(202\) 6.42606e11 1.91068
\(203\) − 5.06413e11i − 1.46901i
\(204\) 0 0
\(205\) −1.49348e11 −0.412505
\(206\) − 1.41144e11i − 0.380475i
\(207\) 0 0
\(208\) 2.68872e10 0.0690605
\(209\) − 6.17640e11i − 1.54883i
\(210\) 0 0
\(211\) 5.65646e11 1.35248 0.676242 0.736679i \(-0.263607\pi\)
0.676242 + 0.736679i \(0.263607\pi\)
\(212\) − 1.76346e11i − 0.411800i
\(213\) 0 0
\(214\) −6.42831e11 −1.43228
\(215\) − 5.98306e10i − 0.130236i
\(216\) 0 0
\(217\) 4.18645e11 0.870057
\(218\) − 3.00545e11i − 0.610418i
\(219\) 0 0
\(220\) 4.08933e11 0.793486
\(221\) 1.06263e10i 0.0201567i
\(222\) 0 0
\(223\) −5.10500e11 −0.925703 −0.462852 0.886436i \(-0.653174\pi\)
−0.462852 + 0.886436i \(0.653174\pi\)
\(224\) − 5.21358e11i − 0.924476i
\(225\) 0 0
\(226\) −1.10752e12 −1.87849
\(227\) 7.40075e11i 1.22785i 0.789363 + 0.613927i \(0.210411\pi\)
−0.789363 + 0.613927i \(0.789589\pi\)
\(228\) 0 0
\(229\) −4.13154e11 −0.656047 −0.328023 0.944670i \(-0.606382\pi\)
−0.328023 + 0.944670i \(0.606382\pi\)
\(230\) − 1.98004e11i − 0.307634i
\(231\) 0 0
\(232\) −1.06610e12 −1.58620
\(233\) 6.00400e10i 0.0874301i 0.999044 + 0.0437150i \(0.0139194\pi\)
−0.999044 + 0.0437150i \(0.986081\pi\)
\(234\) 0 0
\(235\) −1.24527e11 −0.173750
\(236\) − 1.01622e12i − 1.38812i
\(237\) 0 0
\(238\) −1.45996e10 −0.0191186
\(239\) 3.72788e11i 0.478049i 0.971014 + 0.239024i \(0.0768276\pi\)
−0.971014 + 0.239024i \(0.923172\pi\)
\(240\) 0 0
\(241\) 5.07345e11 0.624049 0.312024 0.950074i \(-0.398993\pi\)
0.312024 + 0.950074i \(0.398993\pi\)
\(242\) 3.12768e12i 3.76831i
\(243\) 0 0
\(244\) 5.97987e11 0.691421
\(245\) − 1.95297e10i − 0.0221241i
\(246\) 0 0
\(247\) −1.28091e12 −1.39327
\(248\) − 8.81330e11i − 0.939464i
\(249\) 0 0
\(250\) 8.13890e11 0.833424
\(251\) 1.29052e12i 1.29538i 0.761905 + 0.647689i \(0.224264\pi\)
−0.761905 + 0.647689i \(0.775736\pi\)
\(252\) 0 0
\(253\) 1.34514e12 1.29767
\(254\) − 8.31149e10i − 0.0786159i
\(255\) 0 0
\(256\) −1.17338e12 −1.06718
\(257\) 1.10308e12i 0.983876i 0.870630 + 0.491938i \(0.163711\pi\)
−0.870630 + 0.491938i \(0.836289\pi\)
\(258\) 0 0
\(259\) −8.35977e11 −0.717291
\(260\) − 8.48078e11i − 0.713787i
\(261\) 0 0
\(262\) −9.11794e11 −0.738568
\(263\) − 1.32992e12i − 1.05693i −0.848955 0.528466i \(-0.822768\pi\)
0.848955 0.528466i \(-0.177232\pi\)
\(264\) 0 0
\(265\) −8.74769e10 −0.0669366
\(266\) − 1.75986e12i − 1.32151i
\(267\) 0 0
\(268\) 9.91436e11 0.717118
\(269\) − 1.38777e12i − 0.985275i −0.870235 0.492638i \(-0.836033\pi\)
0.870235 0.492638i \(-0.163967\pi\)
\(270\) 0 0
\(271\) −1.14179e12 −0.781159 −0.390579 0.920569i \(-0.627725\pi\)
−0.390579 + 0.920569i \(0.627725\pi\)
\(272\) 7.71272e8i 0 0.000518040i
\(273\) 0 0
\(274\) 2.81116e12 1.82026
\(275\) 2.66315e12i 1.69329i
\(276\) 0 0
\(277\) −2.29508e12 −1.40734 −0.703670 0.710527i \(-0.748457\pi\)
−0.703670 + 0.710527i \(0.748457\pi\)
\(278\) − 1.34093e12i − 0.807574i
\(279\) 0 0
\(280\) 4.53282e11 0.263377
\(281\) 1.85770e12i 1.06034i 0.847893 + 0.530168i \(0.177871\pi\)
−0.847893 + 0.530168i \(0.822129\pi\)
\(282\) 0 0
\(283\) 3.74722e11 0.206432 0.103216 0.994659i \(-0.467087\pi\)
0.103216 + 0.994659i \(0.467087\pi\)
\(284\) − 5.17898e12i − 2.80319i
\(285\) 0 0
\(286\) 9.28150e12 4.85052
\(287\) − 2.89090e12i − 1.48465i
\(288\) 0 0
\(289\) 2.01569e12 0.999849
\(290\) 1.35941e12i 0.662768i
\(291\) 0 0
\(292\) −6.92647e11 −0.326286
\(293\) 1.36489e12i 0.632062i 0.948749 + 0.316031i \(0.102350\pi\)
−0.948749 + 0.316031i \(0.897650\pi\)
\(294\) 0 0
\(295\) −5.04097e11 −0.225634
\(296\) 1.75989e12i 0.774512i
\(297\) 0 0
\(298\) −4.61903e12 −1.96548
\(299\) − 2.78965e12i − 1.16733i
\(300\) 0 0
\(301\) 1.15813e12 0.468733
\(302\) 7.86772e12i 3.13195i
\(303\) 0 0
\(304\) −9.29706e10 −0.0358078
\(305\) − 2.96633e11i − 0.112388i
\(306\) 0 0
\(307\) −1.96662e12 −0.721153 −0.360577 0.932730i \(-0.617420\pi\)
−0.360577 + 0.932730i \(0.617420\pi\)
\(308\) 7.91567e12i 2.85584i
\(309\) 0 0
\(310\) −1.12381e12 −0.392541
\(311\) − 5.77004e11i − 0.198325i −0.995071 0.0991624i \(-0.968384\pi\)
0.995071 0.0991624i \(-0.0316163\pi\)
\(312\) 0 0
\(313\) −1.55124e12 −0.516366 −0.258183 0.966096i \(-0.583124\pi\)
−0.258183 + 0.966096i \(0.583124\pi\)
\(314\) − 6.02465e12i − 1.97371i
\(315\) 0 0
\(316\) 1.81095e12 0.574740
\(317\) 1.30028e12i 0.406201i 0.979158 + 0.203100i \(0.0651017\pi\)
−0.979158 + 0.203100i \(0.934898\pi\)
\(318\) 0 0
\(319\) −9.23516e12 −2.79570
\(320\) 1.43714e12i 0.428301i
\(321\) 0 0
\(322\) 3.83273e12 1.10721
\(323\) − 3.67435e10i − 0.0104512i
\(324\) 0 0
\(325\) 5.52305e12 1.52322
\(326\) − 5.12276e12i − 1.39128i
\(327\) 0 0
\(328\) −6.08591e12 −1.60309
\(329\) − 2.41046e12i − 0.625346i
\(330\) 0 0
\(331\) −3.14492e12 −0.791533 −0.395767 0.918351i \(-0.629521\pi\)
−0.395767 + 0.918351i \(0.629521\pi\)
\(332\) 9.55537e12i 2.36896i
\(333\) 0 0
\(334\) −5.37601e12 −1.29338
\(335\) − 4.91805e11i − 0.116565i
\(336\) 0 0
\(337\) 3.16283e12 0.727657 0.363828 0.931466i \(-0.381469\pi\)
0.363828 + 0.931466i \(0.381469\pi\)
\(338\) − 1.20854e13i − 2.73953i
\(339\) 0 0
\(340\) 2.43275e10 0.00535430
\(341\) − 7.63459e12i − 1.65582i
\(342\) 0 0
\(343\) 4.92391e12 1.03714
\(344\) − 2.43810e12i − 0.506126i
\(345\) 0 0
\(346\) 3.06794e12 0.618679
\(347\) 1.57357e12i 0.312781i 0.987695 + 0.156390i \(0.0499858\pi\)
−0.987695 + 0.156390i \(0.950014\pi\)
\(348\) 0 0
\(349\) 3.57543e12 0.690559 0.345279 0.938500i \(-0.387784\pi\)
0.345279 + 0.938500i \(0.387784\pi\)
\(350\) 7.58819e12i 1.44477i
\(351\) 0 0
\(352\) −9.50770e12 −1.75939
\(353\) − 8.03200e12i − 1.46538i −0.680562 0.732690i \(-0.738264\pi\)
0.680562 0.732690i \(-0.261736\pi\)
\(354\) 0 0
\(355\) −2.56905e12 −0.455649
\(356\) 4.77301e12i 0.834723i
\(357\) 0 0
\(358\) −1.13614e13 −1.93204
\(359\) 3.65454e12i 0.612859i 0.951893 + 0.306430i \(0.0991344\pi\)
−0.951893 + 0.306430i \(0.900866\pi\)
\(360\) 0 0
\(361\) −1.70194e12 −0.277593
\(362\) 4.36040e11i 0.0701429i
\(363\) 0 0
\(364\) 1.64161e13 2.56900
\(365\) 3.43590e11i 0.0530366i
\(366\) 0 0
\(367\) −1.70763e12 −0.256487 −0.128243 0.991743i \(-0.540934\pi\)
−0.128243 + 0.991743i \(0.540934\pi\)
\(368\) − 2.02477e11i − 0.0300011i
\(369\) 0 0
\(370\) 2.24409e12 0.323618
\(371\) − 1.69328e12i − 0.240912i
\(372\) 0 0
\(373\) 8.49123e12 1.17605 0.588026 0.808842i \(-0.299905\pi\)
0.588026 + 0.808842i \(0.299905\pi\)
\(374\) 2.66244e11i 0.0363849i
\(375\) 0 0
\(376\) −5.07449e12 −0.675232
\(377\) 1.91526e13i 2.51490i
\(378\) 0 0
\(379\) 3.92340e12 0.501726 0.250863 0.968023i \(-0.419286\pi\)
0.250863 + 0.968023i \(0.419286\pi\)
\(380\) 2.93248e12i 0.370098i
\(381\) 0 0
\(382\) 7.51265e12 0.923584
\(383\) 3.50870e12i 0.425748i 0.977080 + 0.212874i \(0.0682823\pi\)
−0.977080 + 0.212874i \(0.931718\pi\)
\(384\) 0 0
\(385\) 3.92659e12 0.464207
\(386\) − 2.07502e12i − 0.242151i
\(387\) 0 0
\(388\) 2.29514e13 2.61007
\(389\) 1.28204e13i 1.43931i 0.694334 + 0.719653i \(0.255699\pi\)
−0.694334 + 0.719653i \(0.744301\pi\)
\(390\) 0 0
\(391\) 8.00223e10 0.00875643
\(392\) − 7.95836e11i − 0.0859792i
\(393\) 0 0
\(394\) −1.05510e11 −0.0111125
\(395\) − 8.98327e11i − 0.0934220i
\(396\) 0 0
\(397\) 9.98220e12 1.01222 0.506108 0.862470i \(-0.331084\pi\)
0.506108 + 0.862470i \(0.331084\pi\)
\(398\) − 2.53424e13i − 2.53765i
\(399\) 0 0
\(400\) 4.00872e11 0.0391476
\(401\) − 6.13719e12i − 0.591900i −0.955204 0.295950i \(-0.904364\pi\)
0.955204 0.295950i \(-0.0956361\pi\)
\(402\) 0 0
\(403\) −1.58332e13 −1.48951
\(404\) − 2.07270e13i − 1.92588i
\(405\) 0 0
\(406\) −2.63140e13 −2.38537
\(407\) 1.52452e13i 1.36509i
\(408\) 0 0
\(409\) 2.93317e12 0.256283 0.128142 0.991756i \(-0.459099\pi\)
0.128142 + 0.991756i \(0.459099\pi\)
\(410\) 7.76033e12i 0.669824i
\(411\) 0 0
\(412\) −4.55255e12 −0.383503
\(413\) − 9.75774e12i − 0.812080i
\(414\) 0 0
\(415\) 4.73997e12 0.385066
\(416\) 1.97178e13i 1.58268i
\(417\) 0 0
\(418\) −3.20935e13 −2.51499
\(419\) − 2.51641e13i − 1.94855i −0.225371 0.974273i \(-0.572360\pi\)
0.225371 0.974273i \(-0.427640\pi\)
\(420\) 0 0
\(421\) 6.35524e12 0.480531 0.240266 0.970707i \(-0.422765\pi\)
0.240266 + 0.970707i \(0.422765\pi\)
\(422\) − 2.93918e13i − 2.19616i
\(423\) 0 0
\(424\) −3.56468e12 −0.260131
\(425\) 1.58431e11i 0.0114260i
\(426\) 0 0
\(427\) 5.74189e12 0.404497
\(428\) 2.07343e13i 1.44368i
\(429\) 0 0
\(430\) −3.10889e12 −0.211477
\(431\) 6.51142e12i 0.437814i 0.975746 + 0.218907i \(0.0702491\pi\)
−0.975746 + 0.218907i \(0.929751\pi\)
\(432\) 0 0
\(433\) 1.82238e13 1.19729 0.598646 0.801014i \(-0.295706\pi\)
0.598646 + 0.801014i \(0.295706\pi\)
\(434\) − 2.17534e13i − 1.41280i
\(435\) 0 0
\(436\) −9.69397e12 −0.615275
\(437\) 9.64604e12i 0.605260i
\(438\) 0 0
\(439\) 7.24114e12 0.444103 0.222052 0.975035i \(-0.428725\pi\)
0.222052 + 0.975035i \(0.428725\pi\)
\(440\) − 8.26624e12i − 0.501239i
\(441\) 0 0
\(442\) 5.52157e11 0.0327304
\(443\) 1.61604e13i 0.947185i 0.880744 + 0.473592i \(0.157043\pi\)
−0.880744 + 0.473592i \(0.842957\pi\)
\(444\) 0 0
\(445\) 2.36766e12 0.135681
\(446\) 2.65264e13i 1.50315i
\(447\) 0 0
\(448\) −2.78185e13 −1.54150
\(449\) 1.89232e13i 1.03696i 0.855088 + 0.518482i \(0.173503\pi\)
−0.855088 + 0.518482i \(0.826497\pi\)
\(450\) 0 0
\(451\) −5.27197e13 −2.82547
\(452\) 3.57227e13i 1.89344i
\(453\) 0 0
\(454\) 3.84554e13 1.99379
\(455\) − 8.14327e12i − 0.417582i
\(456\) 0 0
\(457\) 2.27666e13 1.14214 0.571068 0.820903i \(-0.306529\pi\)
0.571068 + 0.820903i \(0.306529\pi\)
\(458\) 2.14681e13i 1.06529i
\(459\) 0 0
\(460\) −6.38654e12 −0.310082
\(461\) − 2.54886e13i − 1.22417i −0.790793 0.612084i \(-0.790331\pi\)
0.790793 0.612084i \(-0.209669\pi\)
\(462\) 0 0
\(463\) −2.89243e13 −1.35943 −0.679716 0.733475i \(-0.737897\pi\)
−0.679716 + 0.733475i \(0.737897\pi\)
\(464\) 1.39013e12i 0.0646345i
\(465\) 0 0
\(466\) 3.11977e12 0.141969
\(467\) − 1.23497e13i − 0.555996i −0.960582 0.277998i \(-0.910329\pi\)
0.960582 0.277998i \(-0.0896708\pi\)
\(468\) 0 0
\(469\) 9.51980e12 0.419530
\(470\) 6.47063e12i 0.282135i
\(471\) 0 0
\(472\) −2.05420e13 −0.876863
\(473\) − 2.11202e13i − 0.892056i
\(474\) 0 0
\(475\) −1.90976e13 −0.789787
\(476\) 4.70904e11i 0.0192707i
\(477\) 0 0
\(478\) 1.93706e13 0.776254
\(479\) − 8.53563e12i − 0.338499i −0.985573 0.169250i \(-0.945866\pi\)
0.985573 0.169250i \(-0.0541344\pi\)
\(480\) 0 0
\(481\) 3.16167e13 1.22798
\(482\) − 2.63624e13i − 1.01333i
\(483\) 0 0
\(484\) 1.00882e14 3.79829
\(485\) − 1.13851e13i − 0.424257i
\(486\) 0 0
\(487\) −2.84369e12 −0.103810 −0.0519048 0.998652i \(-0.516529\pi\)
−0.0519048 + 0.998652i \(0.516529\pi\)
\(488\) − 1.20878e13i − 0.436765i
\(489\) 0 0
\(490\) −1.01479e12 −0.0359251
\(491\) 3.03433e12i 0.106330i 0.998586 + 0.0531650i \(0.0169309\pi\)
−0.998586 + 0.0531650i \(0.983069\pi\)
\(492\) 0 0
\(493\) −5.49400e11 −0.0188649
\(494\) 6.65581e13i 2.26238i
\(495\) 0 0
\(496\) −1.14920e12 −0.0382814
\(497\) − 4.97288e13i − 1.63993i
\(498\) 0 0
\(499\) −1.92796e13 −0.623155 −0.311578 0.950221i \(-0.600857\pi\)
−0.311578 + 0.950221i \(0.600857\pi\)
\(500\) − 2.62517e13i − 0.840055i
\(501\) 0 0
\(502\) 6.70574e13 2.10343
\(503\) 7.96962e12i 0.247513i 0.992313 + 0.123756i \(0.0394942\pi\)
−0.992313 + 0.123756i \(0.960506\pi\)
\(504\) 0 0
\(505\) −1.02817e13 −0.313046
\(506\) − 6.98954e13i − 2.10715i
\(507\) 0 0
\(508\) −2.68084e12 −0.0792414
\(509\) − 1.37925e13i − 0.403695i −0.979417 0.201847i \(-0.935306\pi\)
0.979417 0.201847i \(-0.0646945\pi\)
\(510\) 0 0
\(511\) −6.65082e12 −0.190885
\(512\) 2.96370e12i 0.0842335i
\(513\) 0 0
\(514\) 5.73176e13 1.59761
\(515\) 2.25830e12i 0.0623370i
\(516\) 0 0
\(517\) −4.39581e13 −1.19011
\(518\) 4.34386e13i 1.16474i
\(519\) 0 0
\(520\) −1.71432e13 −0.450894
\(521\) 4.32531e13i 1.12675i 0.826201 + 0.563376i \(0.190498\pi\)
−0.826201 + 0.563376i \(0.809502\pi\)
\(522\) 0 0
\(523\) 4.88631e12 0.124874 0.0624371 0.998049i \(-0.480113\pi\)
0.0624371 + 0.998049i \(0.480113\pi\)
\(524\) 2.94096e13i 0.744445i
\(525\) 0 0
\(526\) −6.91047e13 −1.71624
\(527\) − 4.54183e11i − 0.0111732i
\(528\) 0 0
\(529\) 2.04188e13 0.492891
\(530\) 4.54543e12i 0.108692i
\(531\) 0 0
\(532\) −5.67636e13 −1.33202
\(533\) 1.09334e14i 2.54168i
\(534\) 0 0
\(535\) 1.02853e13 0.234665
\(536\) − 2.00410e13i − 0.452998i
\(537\) 0 0
\(538\) −7.21109e13 −1.59989
\(539\) − 6.89399e12i − 0.151540i
\(540\) 0 0
\(541\) −6.70080e13 −1.44591 −0.722953 0.690897i \(-0.757216\pi\)
−0.722953 + 0.690897i \(0.757216\pi\)
\(542\) 5.93291e13i 1.26844i
\(543\) 0 0
\(544\) −5.65614e11 −0.0118720
\(545\) 4.80872e12i 0.100011i
\(546\) 0 0
\(547\) −1.75593e11 −0.00358567 −0.00179284 0.999998i \(-0.500571\pi\)
−0.00179284 + 0.999998i \(0.500571\pi\)
\(548\) − 9.06730e13i − 1.83474i
\(549\) 0 0
\(550\) 1.38381e14 2.74957
\(551\) − 6.62257e13i − 1.30397i
\(552\) 0 0
\(553\) 1.73888e13 0.336236
\(554\) 1.19256e14i 2.28523i
\(555\) 0 0
\(556\) −4.32512e13 −0.814000
\(557\) − 5.28769e13i − 0.986256i −0.869957 0.493128i \(-0.835853\pi\)
0.869957 0.493128i \(-0.164147\pi\)
\(558\) 0 0
\(559\) −4.38007e13 −0.802457
\(560\) − 5.91051e11i − 0.0107321i
\(561\) 0 0
\(562\) 9.65288e13 1.72177
\(563\) 9.20469e13i 1.62730i 0.581356 + 0.813650i \(0.302522\pi\)
−0.581356 + 0.813650i \(0.697478\pi\)
\(564\) 0 0
\(565\) 1.77203e13 0.307773
\(566\) − 1.94711e13i − 0.335204i
\(567\) 0 0
\(568\) −1.04689e14 −1.77075
\(569\) 6.27692e13i 1.05241i 0.850357 + 0.526206i \(0.176386\pi\)
−0.850357 + 0.526206i \(0.823614\pi\)
\(570\) 0 0
\(571\) 5.95299e13 0.980741 0.490371 0.871514i \(-0.336861\pi\)
0.490371 + 0.871514i \(0.336861\pi\)
\(572\) − 2.99371e14i − 4.88911i
\(573\) 0 0
\(574\) −1.50216e14 −2.41077
\(575\) − 4.15919e13i − 0.661713i
\(576\) 0 0
\(577\) −1.13204e14 −1.77003 −0.885016 0.465561i \(-0.845853\pi\)
−0.885016 + 0.465561i \(0.845853\pi\)
\(578\) − 1.04738e14i − 1.62355i
\(579\) 0 0
\(580\) 4.38474e13 0.668042
\(581\) 9.17509e13i 1.38589i
\(582\) 0 0
\(583\) −3.08793e13 −0.458485
\(584\) 1.40013e13i 0.206112i
\(585\) 0 0
\(586\) 7.09218e13 1.02634
\(587\) − 2.71880e13i − 0.390110i −0.980792 0.195055i \(-0.937511\pi\)
0.980792 0.195055i \(-0.0624885\pi\)
\(588\) 0 0
\(589\) 5.47480e13 0.772310
\(590\) 2.61936e13i 0.366384i
\(591\) 0 0
\(592\) 2.29479e12 0.0315599
\(593\) − 9.56909e13i − 1.30496i −0.757806 0.652480i \(-0.773729\pi\)
0.757806 0.652480i \(-0.226271\pi\)
\(594\) 0 0
\(595\) 2.33593e11 0.00313239
\(596\) 1.48985e14i 1.98112i
\(597\) 0 0
\(598\) −1.44954e14 −1.89551
\(599\) 6.04515e13i 0.783922i 0.919982 + 0.391961i \(0.128203\pi\)
−0.919982 + 0.391961i \(0.871797\pi\)
\(600\) 0 0
\(601\) 4.25151e13 0.542214 0.271107 0.962549i \(-0.412610\pi\)
0.271107 + 0.962549i \(0.412610\pi\)
\(602\) − 6.01783e13i − 0.761128i
\(603\) 0 0
\(604\) 2.53771e14 3.15687
\(605\) − 5.00430e13i − 0.617399i
\(606\) 0 0
\(607\) −3.83936e13 −0.465924 −0.232962 0.972486i \(-0.574842\pi\)
−0.232962 + 0.972486i \(0.574842\pi\)
\(608\) − 6.81801e13i − 0.820616i
\(609\) 0 0
\(610\) −1.54135e13 −0.182496
\(611\) 9.11638e13i 1.07057i
\(612\) 0 0
\(613\) 1.50166e14 1.73488 0.867439 0.497543i \(-0.165764\pi\)
0.867439 + 0.497543i \(0.165764\pi\)
\(614\) 1.02188e14i 1.17101i
\(615\) 0 0
\(616\) 1.60009e14 1.80401
\(617\) 4.36949e13i 0.488658i 0.969692 + 0.244329i \(0.0785677\pi\)
−0.969692 + 0.244329i \(0.921432\pi\)
\(618\) 0 0
\(619\) 1.31355e14 1.44541 0.722706 0.691155i \(-0.242898\pi\)
0.722706 + 0.691155i \(0.242898\pi\)
\(620\) 3.62481e13i 0.395664i
\(621\) 0 0
\(622\) −2.99820e13 −0.322039
\(623\) 4.58306e13i 0.488332i
\(624\) 0 0
\(625\) 7.55952e13 0.792673
\(626\) 8.06049e13i 0.838474i
\(627\) 0 0
\(628\) −1.94323e14 −1.98942
\(629\) 9.06940e11i 0.00921139i
\(630\) 0 0
\(631\) 3.88509e13 0.388377 0.194189 0.980964i \(-0.437793\pi\)
0.194189 + 0.980964i \(0.437793\pi\)
\(632\) − 3.66068e13i − 0.363059i
\(633\) 0 0
\(634\) 6.75645e13 0.659587
\(635\) 1.32984e12i 0.0128804i
\(636\) 0 0
\(637\) −1.42973e13 −0.136319
\(638\) 4.79873e14i 4.53965i
\(639\) 0 0
\(640\) 4.70956e13 0.438612
\(641\) − 1.40133e14i − 1.29494i −0.762090 0.647471i \(-0.775827\pi\)
0.762090 0.647471i \(-0.224173\pi\)
\(642\) 0 0
\(643\) −1.41701e14 −1.28920 −0.644598 0.764522i \(-0.722975\pi\)
−0.644598 + 0.764522i \(0.722975\pi\)
\(644\) − 1.23623e14i − 1.11602i
\(645\) 0 0
\(646\) −1.90925e12 −0.0169707
\(647\) − 1.07999e14i − 0.952570i −0.879291 0.476285i \(-0.841983\pi\)
0.879291 0.476285i \(-0.158017\pi\)
\(648\) 0 0
\(649\) −1.77946e14 −1.54549
\(650\) − 2.86986e14i − 2.47340i
\(651\) 0 0
\(652\) −1.65233e14 −1.40236
\(653\) − 1.71739e14i − 1.44645i −0.690613 0.723224i \(-0.742659\pi\)
0.690613 0.723224i \(-0.257341\pi\)
\(654\) 0 0
\(655\) 1.45887e13 0.121007
\(656\) 7.93565e12i 0.0653226i
\(657\) 0 0
\(658\) −1.25251e14 −1.01543
\(659\) − 1.09924e13i − 0.0884434i −0.999022 0.0442217i \(-0.985919\pi\)
0.999022 0.0442217i \(-0.0140808\pi\)
\(660\) 0 0
\(661\) 8.24850e13 0.653684 0.326842 0.945079i \(-0.394016\pi\)
0.326842 + 0.945079i \(0.394016\pi\)
\(662\) 1.63415e14i 1.28529i
\(663\) 0 0
\(664\) 1.93154e14 1.49645
\(665\) 2.81577e13i 0.216516i
\(666\) 0 0
\(667\) 1.44231e14 1.09252
\(668\) 1.73401e14i 1.30368i
\(669\) 0 0
\(670\) −2.55549e13 −0.189278
\(671\) − 1.04711e14i − 0.769807i
\(672\) 0 0
\(673\) −5.31597e13 −0.385041 −0.192520 0.981293i \(-0.561666\pi\)
−0.192520 + 0.981293i \(0.561666\pi\)
\(674\) − 1.64346e14i − 1.18157i
\(675\) 0 0
\(676\) −3.89809e14 −2.76133
\(677\) − 1.08669e14i − 0.764122i −0.924137 0.382061i \(-0.875214\pi\)
0.924137 0.382061i \(-0.124786\pi\)
\(678\) 0 0
\(679\) 2.20380e14 1.52695
\(680\) − 4.91759e11i − 0.00338227i
\(681\) 0 0
\(682\) −3.96705e14 −2.68872
\(683\) − 9.47666e12i − 0.0637605i −0.999492 0.0318803i \(-0.989850\pi\)
0.999492 0.0318803i \(-0.0101495\pi\)
\(684\) 0 0
\(685\) −4.49786e13 −0.298231
\(686\) − 2.55854e14i − 1.68411i
\(687\) 0 0
\(688\) −3.17913e12 −0.0206236
\(689\) 6.40400e13i 0.412435i
\(690\) 0 0
\(691\) 2.82614e14 1.79392 0.896961 0.442109i \(-0.145770\pi\)
0.896961 + 0.442109i \(0.145770\pi\)
\(692\) − 9.89551e13i − 0.623603i
\(693\) 0 0
\(694\) 8.17653e13 0.507892
\(695\) 2.14549e13i 0.132313i
\(696\) 0 0
\(697\) −3.13630e12 −0.0190657
\(698\) − 1.85785e14i − 1.12133i
\(699\) 0 0
\(700\) 2.44754e14 1.45626
\(701\) 1.40290e14i 0.828778i 0.910100 + 0.414389i \(0.136005\pi\)
−0.910100 + 0.414389i \(0.863995\pi\)
\(702\) 0 0
\(703\) −1.09324e14 −0.636707
\(704\) 5.07310e14i 2.93366i
\(705\) 0 0
\(706\) −4.17355e14 −2.37948
\(707\) − 1.99021e14i − 1.12669i
\(708\) 0 0
\(709\) −2.77750e14 −1.55032 −0.775162 0.631762i \(-0.782332\pi\)
−0.775162 + 0.631762i \(0.782332\pi\)
\(710\) 1.33492e14i 0.739882i
\(711\) 0 0
\(712\) 9.64823e13 0.527288
\(713\) 1.19234e14i 0.647070i
\(714\) 0 0
\(715\) −1.48504e14 −0.794709
\(716\) 3.66457e14i 1.94741i
\(717\) 0 0
\(718\) 1.89896e14 0.995159
\(719\) 9.14397e13i 0.475872i 0.971281 + 0.237936i \(0.0764709\pi\)
−0.971281 + 0.237936i \(0.923529\pi\)
\(720\) 0 0
\(721\) −4.37137e13 −0.224358
\(722\) 8.84354e13i 0.450755i
\(723\) 0 0
\(724\) 1.40643e13 0.0707011
\(725\) 2.85553e14i 1.42560i
\(726\) 0 0
\(727\) 2.33069e14 1.14766 0.573829 0.818975i \(-0.305457\pi\)
0.573829 + 0.818975i \(0.305457\pi\)
\(728\) − 3.31838e14i − 1.62282i
\(729\) 0 0
\(730\) 1.78534e13 0.0861207
\(731\) − 1.25644e12i − 0.00601943i
\(732\) 0 0
\(733\) −2.37866e14 −1.12412 −0.562059 0.827097i \(-0.689991\pi\)
−0.562059 + 0.827097i \(0.689991\pi\)
\(734\) 8.87313e13i 0.416482i
\(735\) 0 0
\(736\) 1.48487e14 0.687543
\(737\) − 1.73607e14i − 0.798417i
\(738\) 0 0
\(739\) 2.85116e14 1.29360 0.646799 0.762660i \(-0.276107\pi\)
0.646799 + 0.762660i \(0.276107\pi\)
\(740\) − 7.23824e13i − 0.326193i
\(741\) 0 0
\(742\) −8.79853e13 −0.391193
\(743\) 1.78676e14i 0.789083i 0.918878 + 0.394541i \(0.129097\pi\)
−0.918878 + 0.394541i \(0.870903\pi\)
\(744\) 0 0
\(745\) 7.39045e13 0.322025
\(746\) − 4.41217e14i − 1.90967i
\(747\) 0 0
\(748\) 8.58760e12 0.0366745
\(749\) 1.99091e14i 0.844585i
\(750\) 0 0
\(751\) 3.98917e13 0.166987 0.0834935 0.996508i \(-0.473392\pi\)
0.0834935 + 0.996508i \(0.473392\pi\)
\(752\) 6.61682e12i 0.0275144i
\(753\) 0 0
\(754\) 9.95198e14 4.08369
\(755\) − 1.25884e14i − 0.513138i
\(756\) 0 0
\(757\) 3.34577e13 0.134591 0.0672956 0.997733i \(-0.478563\pi\)
0.0672956 + 0.997733i \(0.478563\pi\)
\(758\) − 2.03866e14i − 0.814701i
\(759\) 0 0
\(760\) 5.92776e13 0.233788
\(761\) − 1.40098e14i − 0.548920i −0.961599 0.274460i \(-0.911501\pi\)
0.961599 0.274460i \(-0.0884990\pi\)
\(762\) 0 0
\(763\) −9.30818e13 −0.359950
\(764\) − 2.42318e14i − 0.930933i
\(765\) 0 0
\(766\) 1.82317e14 0.691328
\(767\) 3.69039e14i 1.39026i
\(768\) 0 0
\(769\) 1.53854e14 0.572106 0.286053 0.958214i \(-0.407657\pi\)
0.286053 + 0.958214i \(0.407657\pi\)
\(770\) − 2.04032e14i − 0.753778i
\(771\) 0 0
\(772\) −6.69290e13 −0.244078
\(773\) 2.43492e14i 0.882243i 0.897447 + 0.441121i \(0.145419\pi\)
−0.897447 + 0.441121i \(0.854581\pi\)
\(774\) 0 0
\(775\) −2.36063e14 −0.844345
\(776\) − 4.63944e14i − 1.64876i
\(777\) 0 0
\(778\) 6.66167e14 2.33714
\(779\) − 3.78055e14i − 1.31786i
\(780\) 0 0
\(781\) −9.06874e14 −3.12099
\(782\) − 4.15808e12i − 0.0142187i
\(783\) 0 0
\(784\) −1.03772e12 −0.00350348
\(785\) 9.63944e13i 0.323373i
\(786\) 0 0
\(787\) 3.47268e13 0.115025 0.0575123 0.998345i \(-0.481683\pi\)
0.0575123 + 0.998345i \(0.481683\pi\)
\(788\) 3.40317e12i 0.0112009i
\(789\) 0 0
\(790\) −4.66785e13 −0.151698
\(791\) 3.43010e14i 1.10771i
\(792\) 0 0
\(793\) −2.17159e14 −0.692487
\(794\) − 5.18690e14i − 1.64364i
\(795\) 0 0
\(796\) −8.17409e14 −2.55785
\(797\) − 4.86090e14i − 1.51156i −0.654826 0.755780i \(-0.727258\pi\)
0.654826 0.755780i \(-0.272742\pi\)
\(798\) 0 0
\(799\) −2.61507e12 −0.00803064
\(800\) 2.93980e14i 0.897156i
\(801\) 0 0
\(802\) −3.18898e14 −0.961125
\(803\) 1.21287e14i 0.363276i
\(804\) 0 0
\(805\) −6.13238e13 −0.181405
\(806\) 8.22718e14i 2.41866i
\(807\) 0 0
\(808\) −4.18979e14 −1.21657
\(809\) 1.50911e14i 0.435489i 0.976006 + 0.217744i \(0.0698699\pi\)
−0.976006 + 0.217744i \(0.930130\pi\)
\(810\) 0 0
\(811\) 5.05292e14 1.44025 0.720125 0.693844i \(-0.244084\pi\)
0.720125 + 0.693844i \(0.244084\pi\)
\(812\) 8.48748e14i 2.40436i
\(813\) 0 0
\(814\) 7.92165e14 2.21663
\(815\) 8.19642e13i 0.227948i
\(816\) 0 0
\(817\) 1.51454e14 0.416073
\(818\) − 1.52412e14i − 0.416152i
\(819\) 0 0
\(820\) 2.50307e14 0.675154
\(821\) − 5.32014e14i − 1.42629i −0.701017 0.713145i \(-0.747270\pi\)
0.701017 0.713145i \(-0.252730\pi\)
\(822\) 0 0
\(823\) −1.35362e14 −0.358506 −0.179253 0.983803i \(-0.557368\pi\)
−0.179253 + 0.983803i \(0.557368\pi\)
\(824\) 9.20259e13i 0.242256i
\(825\) 0 0
\(826\) −5.07027e14 −1.31865
\(827\) 4.80302e14i 1.24162i 0.783963 + 0.620808i \(0.213195\pi\)
−0.783963 + 0.620808i \(0.786805\pi\)
\(828\) 0 0
\(829\) −3.67003e14 −0.937340 −0.468670 0.883373i \(-0.655267\pi\)
−0.468670 + 0.883373i \(0.655267\pi\)
\(830\) − 2.46296e14i − 0.625269i
\(831\) 0 0
\(832\) 1.05210e15 2.63901
\(833\) − 4.10124e11i − 0.00102256i
\(834\) 0 0
\(835\) 8.60162e13 0.211908
\(836\) 1.03517e15i 2.53500i
\(837\) 0 0
\(838\) −1.30756e15 −3.16404
\(839\) − 5.69381e14i − 1.36960i −0.728731 0.684800i \(-0.759890\pi\)
0.728731 0.684800i \(-0.240110\pi\)
\(840\) 0 0
\(841\) −5.69521e14 −1.35372
\(842\) − 3.30228e14i − 0.780285i
\(843\) 0 0
\(844\) −9.48022e14 −2.21364
\(845\) 1.93366e14i 0.448845i
\(846\) 0 0
\(847\) 9.68675e14 2.22209
\(848\) 4.64813e12i 0.0105998i
\(849\) 0 0
\(850\) 8.23232e12 0.0185536
\(851\) − 2.38093e14i − 0.533457i
\(852\) 0 0
\(853\) 1.83927e13 0.0407286 0.0203643 0.999793i \(-0.493517\pi\)
0.0203643 + 0.999793i \(0.493517\pi\)
\(854\) − 2.98357e14i − 0.656821i
\(855\) 0 0
\(856\) 4.19126e14 0.911960
\(857\) 7.22916e14i 1.56381i 0.623398 + 0.781905i \(0.285752\pi\)
−0.623398 + 0.781905i \(0.714248\pi\)
\(858\) 0 0
\(859\) −5.83296e14 −1.24716 −0.623582 0.781758i \(-0.714323\pi\)
−0.623582 + 0.781758i \(0.714323\pi\)
\(860\) 1.00276e14i 0.213160i
\(861\) 0 0
\(862\) 3.38343e14 0.710921
\(863\) − 3.20743e14i − 0.670044i −0.942210 0.335022i \(-0.891256\pi\)
0.942210 0.335022i \(-0.108744\pi\)
\(864\) 0 0
\(865\) −4.90870e13 −0.101364
\(866\) − 9.46937e14i − 1.94416i
\(867\) 0 0
\(868\) −7.01650e14 −1.42404
\(869\) − 3.17110e14i − 0.639898i
\(870\) 0 0
\(871\) −3.60040e14 −0.718224
\(872\) 1.95955e14i 0.388664i
\(873\) 0 0
\(874\) 5.01223e14 0.982819
\(875\) − 2.52070e14i − 0.491451i
\(876\) 0 0
\(877\) −8.65432e13 −0.166815 −0.0834075 0.996516i \(-0.526580\pi\)
−0.0834075 + 0.996516i \(0.526580\pi\)
\(878\) − 3.76261e14i − 0.721134i
\(879\) 0 0
\(880\) −1.07787e13 −0.0204245
\(881\) − 8.97308e14i − 1.69068i −0.534227 0.845341i \(-0.679397\pi\)
0.534227 0.845341i \(-0.320603\pi\)
\(882\) 0 0
\(883\) 2.43623e14 0.453853 0.226927 0.973912i \(-0.427132\pi\)
0.226927 + 0.973912i \(0.427132\pi\)
\(884\) − 1.78096e13i − 0.0329909i
\(885\) 0 0
\(886\) 8.39721e14 1.53804
\(887\) 3.92471e13i 0.0714808i 0.999361 + 0.0357404i \(0.0113789\pi\)
−0.999361 + 0.0357404i \(0.988621\pi\)
\(888\) 0 0
\(889\) −2.57415e13 −0.0463580
\(890\) − 1.23027e14i − 0.220319i
\(891\) 0 0
\(892\) 8.55599e14 1.51512
\(893\) − 3.15226e14i − 0.555091i
\(894\) 0 0
\(895\) 1.81782e14 0.316545
\(896\) 9.11623e14i 1.57861i
\(897\) 0 0
\(898\) 9.83281e14 1.68382
\(899\) − 8.18610e14i − 1.39405i
\(900\) 0 0
\(901\) −1.83701e12 −0.00309377
\(902\) 2.73940e15i 4.58799i
\(903\) 0 0
\(904\) 7.22104e14 1.19607
\(905\) − 6.97664e12i − 0.0114922i
\(906\) 0 0
\(907\) −1.09694e15 −1.78709 −0.893544 0.448975i \(-0.851789\pi\)
−0.893544 + 0.448975i \(0.851789\pi\)
\(908\) − 1.24037e15i − 2.00965i
\(909\) 0 0
\(910\) −4.23137e14 −0.678069
\(911\) − 8.26823e14i − 1.31771i −0.752269 0.658856i \(-0.771041\pi\)
0.752269 0.658856i \(-0.228959\pi\)
\(912\) 0 0
\(913\) 1.67321e15 2.63752
\(914\) − 1.18299e15i − 1.85460i
\(915\) 0 0
\(916\) 6.92447e14 1.07376
\(917\) 2.82392e14i 0.435517i
\(918\) 0 0
\(919\) 8.42155e14 1.28474 0.642368 0.766396i \(-0.277952\pi\)
0.642368 + 0.766396i \(0.277952\pi\)
\(920\) 1.29099e14i 0.195876i
\(921\) 0 0
\(922\) −1.32443e15 −1.98780
\(923\) 1.88075e15i 2.80751i
\(924\) 0 0
\(925\) 4.71386e14 0.696094
\(926\) 1.50295e15i 2.20744i
\(927\) 0 0
\(928\) −1.01945e15 −1.48124
\(929\) 8.33534e14i 1.20460i 0.798268 + 0.602302i \(0.205750\pi\)
−0.798268 + 0.602302i \(0.794250\pi\)
\(930\) 0 0
\(931\) 4.94371e13 0.0706813
\(932\) − 1.00627e14i − 0.143098i
\(933\) 0 0
\(934\) −6.41708e14 −0.902824
\(935\) − 4.25990e12i − 0.00596131i
\(936\) 0 0
\(937\) −2.78438e14 −0.385506 −0.192753 0.981247i \(-0.561742\pi\)
−0.192753 + 0.981247i \(0.561742\pi\)
\(938\) − 4.94663e14i − 0.681233i
\(939\) 0 0
\(940\) 2.08708e14 0.284380
\(941\) 7.38535e14i 1.00097i 0.865744 + 0.500487i \(0.166846\pi\)
−0.865744 + 0.500487i \(0.833154\pi\)
\(942\) 0 0
\(943\) 8.23353e14 1.10415
\(944\) 2.67854e13i 0.0357305i
\(945\) 0 0
\(946\) −1.09744e15 −1.44852
\(947\) 7.85947e14i 1.03191i 0.856615 + 0.515957i \(0.172563\pi\)
−0.856615 + 0.515957i \(0.827437\pi\)
\(948\) 0 0
\(949\) 2.51535e14 0.326789
\(950\) 9.92339e14i 1.28245i
\(951\) 0 0
\(952\) 9.51892e12 0.0121731
\(953\) − 5.37814e14i − 0.684175i −0.939668 0.342088i \(-0.888866\pi\)
0.939668 0.342088i \(-0.111134\pi\)
\(954\) 0 0
\(955\) −1.20202e14 −0.151320
\(956\) − 6.24792e14i − 0.782431i
\(957\) 0 0
\(958\) −4.43524e14 −0.549654
\(959\) − 8.70645e14i − 1.07337i
\(960\) 0 0
\(961\) −1.42894e14 −0.174339
\(962\) − 1.64285e15i − 1.99399i
\(963\) 0 0
\(964\) −8.50311e14 −1.02139
\(965\) 3.32003e13i 0.0396740i
\(966\) 0 0
\(967\) −1.07681e15 −1.27353 −0.636763 0.771059i \(-0.719727\pi\)
−0.636763 + 0.771059i \(0.719727\pi\)
\(968\) − 2.03925e15i − 2.39935i
\(969\) 0 0
\(970\) −5.91588e14 −0.688908
\(971\) 1.33870e14i 0.155091i 0.996989 + 0.0775454i \(0.0247083\pi\)
−0.996989 + 0.0775454i \(0.975292\pi\)
\(972\) 0 0
\(973\) −4.15299e14 −0.476208
\(974\) 1.47762e14i 0.168566i
\(975\) 0 0
\(976\) −1.57617e13 −0.0177973
\(977\) 6.28694e14i 0.706263i 0.935574 + 0.353132i \(0.114883\pi\)
−0.935574 + 0.353132i \(0.885117\pi\)
\(978\) 0 0
\(979\) 8.35785e14 0.929354
\(980\) 3.27318e13i 0.0362109i
\(981\) 0 0
\(982\) 1.57668e14 0.172658
\(983\) 9.03733e14i 0.984629i 0.870417 + 0.492314i \(0.163849\pi\)
−0.870417 + 0.492314i \(0.836151\pi\)
\(984\) 0 0
\(985\) 1.68815e12 0.00182067
\(986\) 2.85477e13i 0.0306328i
\(987\) 0 0
\(988\) 2.14681e15 2.28038
\(989\) 3.29846e14i 0.348602i
\(990\) 0 0
\(991\) 1.70337e15 1.78213 0.891067 0.453871i \(-0.149957\pi\)
0.891067 + 0.453871i \(0.149957\pi\)
\(992\) − 8.42768e14i − 0.877303i
\(993\) 0 0
\(994\) −2.58398e15 −2.66292
\(995\) 4.05478e14i 0.415769i
\(996\) 0 0
\(997\) −5.98425e14 −0.607482 −0.303741 0.952755i \(-0.598236\pi\)
−0.303741 + 0.952755i \(0.598236\pi\)
\(998\) 1.00180e15i 1.01188i
\(999\) 0 0
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 27.11.b.b.26.1 2
3.2 odd 2 inner 27.11.b.b.26.2 yes 2
9.2 odd 6 81.11.d.c.53.1 2
9.4 even 3 81.11.d.c.26.1 2
9.5 odd 6 81.11.d.a.26.1 2
9.7 even 3 81.11.d.a.53.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.11.b.b.26.1 2 1.1 even 1 trivial
27.11.b.b.26.2 yes 2 3.2 odd 2 inner
81.11.d.a.26.1 2 9.5 odd 6
81.11.d.a.53.1 2 9.7 even 3
81.11.d.c.26.1 2 9.4 even 3
81.11.d.c.53.1 2 9.2 odd 6