Properties

Label 21.42.g.a.5.1
Level $21$
Weight $42$
Character 21.5
Analytic conductor $223.591$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,42,Mod(5,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 42, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 42);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), 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: \(223.590507958\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 5.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.5
Dual form 21.42.g.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.23018e9 - 3.01964e9i) q^{3} +(-1.09951e12 - 1.90441e12i) q^{4} +(2.10745e17 - 1.24129e16i) q^{7} +(1.82365e19 - 3.15865e19i) q^{9} +O(q^{10})\) \(q+(5.23018e9 - 3.01964e9i) q^{3} +(-1.09951e12 - 1.90441e12i) q^{4} +(2.10745e17 - 1.24129e16i) q^{7} +(1.82365e19 - 3.15865e19i) q^{9} +(-1.15013e22 - 6.64027e21i) q^{12} -6.81363e22i q^{13} +(-2.41785e24 + 4.18784e24i) q^{16} +(2.45919e26 + 1.41982e26i) q^{19} +(1.06475e27 - 7.01297e26i) q^{21} +(2.27374e28 + 3.93823e28i) q^{25} -2.20271e29i q^{27} +(-2.55356e29 - 3.87697e29i) q^{28} +(5.95816e30 - 3.43994e30i) q^{31} -8.02050e31 q^{36} +(3.23924e31 - 5.61053e31i) q^{37} +(-2.05747e32 - 3.56365e32i) q^{39} +1.97879e33 q^{43} +2.92042e34i q^{48} +(4.42595e34 - 5.23191e33i) q^{49} +(-1.29760e35 + 7.49167e34i) q^{52} +1.71494e36 q^{57} +(5.89124e34 + 3.40131e34i) q^{61} +(3.45118e36 - 6.88308e36i) q^{63} +1.06338e37 q^{64} +(2.54133e37 + 4.40171e37i) q^{67} +(7.06634e36 - 4.07976e36i) q^{73} +(2.37841e38 + 1.37318e38i) q^{75} -6.24442e38i q^{76} +(6.37702e38 - 1.10453e39i) q^{79} +(-6.65140e38 - 1.15206e39i) q^{81} +(-2.50626e39 - 1.25664e39i) q^{84} +(-8.45767e38 - 1.43594e40i) q^{91} +(2.07748e40 - 3.59830e40i) q^{93} +5.81679e40i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10460353203 q^{3} - 2199023255552 q^{4} + 42\!\cdots\!79 q^{7}+ \cdots + 36\!\cdots\!03 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10460353203 q^{3} - 2199023255552 q^{4} + 42\!\cdots\!79 q^{7}+ \cdots + 41\!\cdots\!85 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 5.23018e9 3.01964e9i 0.866025 0.500000i
\(4\) −1.09951e12 1.90441e12i −0.500000 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 2.10745e17 1.24129e16i 0.998270 0.0587980i
\(8\) 0 0
\(9\) 1.82365e19 3.15865e19i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −1.15013e22 6.64027e21i −0.866025 0.500000i
\(13\) 6.81363e22i 0.994352i −0.867650 0.497176i \(-0.834370\pi\)
0.867650 0.497176i \(-0.165630\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.41785e24 + 4.18784e24i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 2.45919e26 + 1.41982e26i 1.50087 + 0.866529i 0.999999 + 0.00100876i \(0.000321098\pi\)
0.500873 + 0.865521i \(0.333012\pi\)
\(20\) 0 0
\(21\) 1.06475e27 7.01297e26i 0.835128 0.550055i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 2.27374e28 + 3.93823e28i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 2.20271e29i 1.00000i
\(28\) −2.55356e29 3.87697e29i −0.550055 0.835128i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 5.95816e30 3.43994e30i 1.59293 0.919680i 0.600131 0.799902i \(-0.295115\pi\)
0.992801 0.119778i \(-0.0382183\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −8.02050e31 −1.00000
\(37\) 3.23924e31 5.61053e31i 0.230307 0.398904i −0.727591 0.686011i \(-0.759360\pi\)
0.957899 + 0.287107i \(0.0926935\pi\)
\(38\) 0 0
\(39\) −2.05747e32 3.56365e32i −0.497176 0.861134i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.97879e33 0.646095 0.323048 0.946383i \(-0.395293\pi\)
0.323048 + 0.946383i \(0.395293\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 2.92042e34i 1.00000i
\(49\) 4.42595e34 5.23191e33i 0.993086 0.117392i
\(50\) 0 0
\(51\) 0 0
\(52\) −1.29760e35 + 7.49167e34i −0.861134 + 0.497176i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.71494e36 1.73306
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 5.89124e34 + 3.40131e34i 0.0148233 + 0.00855824i 0.507393 0.861715i \(-0.330609\pi\)
−0.492570 + 0.870273i \(0.663942\pi\)
\(62\) 0 0
\(63\) 3.45118e36 6.88308e36i 0.448214 0.893926i
\(64\) 1.06338e37 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.54133e37 + 4.40171e37i 0.934389 + 1.61841i 0.775719 + 0.631078i \(0.217387\pi\)
0.158670 + 0.987332i \(0.449279\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 7.06634e36 4.07976e36i 0.0447792 0.0258533i −0.477443 0.878663i \(-0.658436\pi\)
0.522222 + 0.852809i \(0.325103\pi\)
\(74\) 0 0
\(75\) 2.37841e38 + 1.37318e38i 0.866025 + 0.500000i
\(76\) 6.24442e38i 1.73306i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.37702e38 1.10453e39i 0.800317 1.38619i −0.119090 0.992883i \(-0.537998\pi\)
0.919407 0.393306i \(-0.128669\pi\)
\(80\) 0 0
\(81\) −6.65140e38 1.15206e39i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.50626e39 1.25664e39i −0.893926 0.448214i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −8.45767e38 1.43594e40i −0.0584659 0.992632i
\(92\) 0 0
\(93\) 2.07748e40 3.59830e40i 0.919680 1.59293i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.81679e40i 1.08608i 0.839706 + 0.543041i \(0.182727\pi\)
−0.839706 + 0.543041i \(0.817273\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000e40 8.66025e40i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 9.34402e40 + 5.39477e40i 0.509766 + 0.294314i 0.732737 0.680511i \(-0.238242\pi\)
−0.222971 + 0.974825i \(0.571576\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −4.19486e41 + 2.42190e41i −0.866025 + 0.500000i
\(109\) 5.64242e41 + 9.77296e41i 0.964322 + 1.67025i 0.711426 + 0.702761i \(0.248050\pi\)
0.252896 + 0.967493i \(0.418617\pi\)
\(110\) 0 0
\(111\) 3.91254e41i 0.460615i
\(112\) −4.57568e41 + 9.12580e41i −0.448214 + 0.893926i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.15219e42 1.24257e42i −0.861134 0.497176i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.48926e42 + 4.31152e42i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.31021e43 7.56452e42i −1.59293 0.919680i
\(125\) 0 0
\(126\) 0 0
\(127\) 7.28476e42 0.542549 0.271274 0.962502i \(-0.412555\pi\)
0.271274 + 0.962502i \(0.412555\pi\)
\(128\) 0 0
\(129\) 1.03494e43 5.97524e42i 0.559535 0.323048i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 5.35887e43 + 2.68694e43i 1.54923 + 0.776782i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 1.67231e44i 1.95665i −0.207086 0.978323i \(-0.566398\pi\)
0.207086 0.978323i \(-0.433602\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 8.81863e43 + 1.52743e44i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 2.15686e44 1.61012e44i 0.801341 0.598208i
\(148\) −1.42463e44 −0.460615
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −1.90169e44 3.29382e44i −0.407485 0.705785i 0.587122 0.809498i \(-0.300261\pi\)
−0.994607 + 0.103713i \(0.966928\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.52444e44 + 7.83655e44i −0.497176 + 0.861134i
\(157\) −1.21374e45 + 7.00754e44i −1.16999 + 0.675495i −0.953678 0.300828i \(-0.902737\pi\)
−0.216314 + 0.976324i \(0.569404\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.07118e45 + 1.85534e45i −0.478649 + 0.829045i −0.999700 0.0244805i \(-0.992207\pi\)
0.521051 + 0.853526i \(0.325540\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 5.28914e43 0.0112644
\(170\) 0 0
\(171\) 8.96942e45 5.17849e45i 1.50087 0.866529i
\(172\) −2.17570e45 3.76843e45i −0.323048 0.559535i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 5.28064e45 + 8.01739e45i 0.550055 + 0.835128i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 7.80733e45i 0.407466i −0.979026 0.203733i \(-0.934693\pi\)
0.979026 0.203733i \(-0.0653075\pi\)
\(182\) 0 0
\(183\) 4.10830e44 0.0171165
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2.73419e45 4.64211e46i −0.0587980 0.998270i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 5.56168e46 3.21104e46i 0.866025 0.500000i
\(193\) −6.74472e46 1.16822e47i −0.944145 1.63531i −0.757454 0.652889i \(-0.773557\pi\)
−0.186692 0.982419i \(-0.559777\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.86275e46 7.85357e46i −0.598208 0.801341i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −2.18927e47 + 1.26398e47i −1.63611 + 0.944607i −0.653952 + 0.756536i \(0.726890\pi\)
−0.982155 + 0.188071i \(0.939776\pi\)
\(200\) 0 0
\(201\) 2.65832e47 + 1.53478e47i 1.61841 + 0.934389i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 2.85344e47 + 1.64744e47i 0.861134 + 0.497176i
\(209\) 0 0
\(210\) 0 0
\(211\) −8.34500e47 −1.87774 −0.938871 0.344270i \(-0.888127\pi\)
−0.938871 + 0.344270i \(0.888127\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.21295e48 7.98910e47i 1.53610 1.01175i
\(218\) 0 0
\(219\) 2.46388e46 4.26757e46i 0.0258533 0.0447792i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.50626e48i 1.81458i −0.420505 0.907290i \(-0.638147\pi\)
0.420505 0.907290i \(-0.361853\pi\)
\(224\) 0 0
\(225\) 1.65860e48 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) −1.88559e48 3.26594e48i −0.866529 1.50087i
\(229\) −4.07972e48 2.35543e48i −1.71397 0.989561i −0.929041 0.369976i \(-0.879366\pi\)
−0.784929 0.619585i \(-0.787301\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.70253e48i 1.60063i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.11143e49 6.41682e48i 1.63882 0.946173i 0.657579 0.753385i \(-0.271580\pi\)
0.981240 0.192788i \(-0.0617529\pi\)
\(242\) 0 0
\(243\) −6.95760e48 4.01697e48i −0.866025 0.500000i
\(244\) 1.49591e47i 0.0171165i
\(245\) 0 0
\(246\) 0 0
\(247\) 9.67411e48 1.67560e49i 0.861635 1.49240i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.69028e49 + 9.95574e47i −0.998270 + 0.0587980i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −1.16920e49 2.02512e49i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 6.13011e48 1.22260e49i 0.206454 0.411755i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 5.58845e49 9.67947e49i 0.934389 1.61841i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −8.60000e49 4.96521e49i −1.14453 0.660796i −0.196984 0.980407i \(-0.563115\pi\)
−0.947549 + 0.319611i \(0.896448\pi\)
\(272\) 0 0
\(273\) −4.77838e49 7.25483e49i −0.546949 0.830411i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.07368e50 1.85966e50i −0.912091 1.57979i −0.811105 0.584901i \(-0.801133\pi\)
−0.100987 0.994888i \(-0.532200\pi\)
\(278\) 0 0
\(279\) 2.50930e50i 1.83936i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 2.13079e50 1.23021e50i 1.16659 0.673533i 0.213718 0.976895i \(-0.431443\pi\)
0.952875 + 0.303362i \(0.0981092\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.40403e50 2.43185e50i 0.500000 0.866025i
\(290\) 0 0
\(291\) 1.75646e50 + 3.04228e50i 0.543041 + 0.940574i
\(292\) −1.55391e49 8.97148e48i −0.0447792 0.0258533i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 6.03929e50i 1.00000i
\(301\) 4.17021e50 2.45625e49i 0.644977 0.0379891i
\(302\) 0 0
\(303\) 0 0
\(304\) −1.18919e51 + 6.86581e50i −1.50087 + 0.866529i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.73958e51i 1.79518i 0.440830 + 0.897590i \(0.354684\pi\)
−0.440830 + 0.897590i \(0.645316\pi\)
\(308\) 0 0
\(309\) 6.51612e50 0.588627
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −5.94246e50 3.43088e50i −0.412389 0.238093i 0.279427 0.960167i \(-0.409856\pi\)
−0.691816 + 0.722074i \(0.743189\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.80464e51 −1.60063
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.46266e51 + 2.53340e51i −0.500000 + 0.866025i
\(325\) 2.68336e51 1.54924e51i 0.861134 0.497176i
\(326\) 0 0
\(327\) 5.90217e51 + 3.40762e51i 1.67025 + 0.964322i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.09456e51 + 7.09199e51i −0.903095 + 1.56421i −0.0796418 + 0.996824i \(0.525378\pi\)
−0.823454 + 0.567384i \(0.807956\pi\)
\(332\) 0 0
\(333\) −1.18145e51 2.04633e51i −0.230307 0.398904i
\(334\) 0 0
\(335\) 0 0
\(336\) 3.62508e50 + 6.15465e51i 0.0587980 + 0.998270i
\(337\) 8.26335e51 1.26108 0.630541 0.776156i \(-0.282833\pi\)
0.630541 + 0.776156i \(0.282833\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 9.26253e51 1.65199e51i 0.984465 0.175581i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 2.68299e52i 1.99847i 0.0390895 + 0.999236i \(0.487554\pi\)
−0.0390895 + 0.999236i \(0.512446\pi\)
\(350\) 0 0
\(351\) −1.50085e52 −0.994352
\(352\) 0 0
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 2.68940e52 + 4.65818e52i 1.00175 + 1.73508i
\(362\) 0 0
\(363\) 3.00667e52i 1.00000i
\(364\) −2.64163e52 + 1.73990e52i −0.830411 + 0.546949i
\(365\) 0 0
\(366\) 0 0
\(367\) −6.48734e52 + 3.74547e52i −1.72350 + 0.995065i −0.812139 + 0.583464i \(0.801697\pi\)
−0.911364 + 0.411601i \(0.864970\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −9.13686e52 −1.83936
\(373\) 5.23785e52 9.07222e52i 0.997980 1.72855i 0.443974 0.896040i \(-0.353568\pi\)
0.554006 0.832512i \(-0.313098\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.33136e53 1.82891 0.914455 0.404688i \(-0.132620\pi\)
0.914455 + 0.404688i \(0.132620\pi\)
\(380\) 0 0
\(381\) 3.81006e52 2.19974e52i 0.469861 0.271274i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.60862e52 6.25032e52i 0.323048 0.559535i
\(388\) 1.10775e53 6.39562e52i 0.940574 0.543041i
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.26358e53 1.88423e53i −1.73176 0.999834i −0.874993 0.484135i \(-0.839134\pi\)
−0.856770 0.515699i \(-0.827532\pi\)
\(398\) 0 0
\(399\) 3.61415e53 2.12873e52i 1.73006 0.101900i
\(400\) −2.19902e53 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −2.34385e53 4.05967e53i −0.914485 1.58393i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3.97412e53 2.29446e53i 1.14528 0.661230i 0.197550 0.980293i \(-0.436701\pi\)
0.947734 + 0.319063i \(0.103368\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.37265e53i 0.588627i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.04977e53 8.74645e53i −0.978323 1.69450i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −9.73208e53 −1.55033 −0.775163 0.631762i \(-0.782332\pi\)
−0.775163 + 0.631762i \(0.782332\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.28377e52 + 6.43682e51i 0.0153009 + 0.00767185i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 9.22460e53 + 5.32582e53i 0.866025 + 0.500000i
\(433\) 2.21678e54i 1.98482i 0.122989 + 0.992408i \(0.460752\pi\)
−0.122989 + 0.992408i \(0.539248\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.24078e54 2.14910e54i 0.964322 1.67025i
\(437\) 0 0
\(438\) 0 0
\(439\) −4.01604e53 2.31866e53i −0.271191 0.156572i 0.358238 0.933630i \(-0.383378\pi\)
−0.629429 + 0.777058i \(0.716711\pi\)
\(440\) 0 0
\(441\) 6.41880e53 1.49342e54i 0.394878 0.918734i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) −7.45108e53 + 4.30188e53i −0.398904 + 0.230307i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 2.24103e54 1.31996e53i 0.998270 0.0587980i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.98923e54 1.14848e54i −0.705785 0.407485i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.40419e54 4.16417e54i 0.712338 1.23380i −0.251640 0.967821i \(-0.580970\pi\)
0.963977 0.265984i \(-0.0856968\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −7.96507e54 −1.80626 −0.903128 0.429372i \(-0.858735\pi\)
−0.903128 + 0.429372i \(0.858735\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 5.46487e54i 0.994352i
\(469\) 5.90211e54 + 8.96095e54i 1.02793 + 1.56067i
\(470\) 0 0
\(471\) −4.23205e54 + 7.33013e54i −0.675495 + 1.16999i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.29132e55i 1.73306i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) −3.82281e54 2.20710e54i −0.396651 0.229006i
\(482\) 0 0
\(483\) 0 0
\(484\) 1.09479e55 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 5.33694e54 + 9.24385e54i 0.429486 + 0.743891i 0.996828 0.0795914i \(-0.0253616\pi\)
−0.567342 + 0.823482i \(0.692028\pi\)
\(488\) 0 0
\(489\) 1.29384e55i 0.957299i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 3.32691e55i 1.83936i
\(497\) 0 0
\(498\) 0 0
\(499\) −5.50521e54 + 9.53530e54i −0.268976 + 0.465879i −0.968598 0.248634i \(-0.920018\pi\)
0.699622 + 0.714513i \(0.253352\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.76631e53 1.59713e53i 0.00975524 0.00563219i
\(508\) −8.00968e54 1.38732e55i −0.271274 0.469861i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 1.43856e54 9.47503e53i 0.0431816 0.0284415i
\(512\) 0 0
\(513\) 3.12744e55 5.41689e55i 0.866529 1.50087i
\(514\) 0 0
\(515\) 0 0
\(516\) −2.27586e55 1.31397e55i −0.559535 0.323048i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) −3.62921e55 2.09532e55i −0.676904 0.390811i 0.121783 0.992557i \(-0.461139\pi\)
−0.798688 + 0.601746i \(0.794472\pi\)
\(524\) 0 0
\(525\) 5.18283e55 + 2.59867e55i 0.893926 + 0.448214i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.38697e55 5.86640e55i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −7.75111e54 1.31598e56i −0.101900 1.73006i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6.72699e55 + 1.16515e56i −0.627014 + 1.08602i 0.361134 + 0.932514i \(0.382390\pi\)
−0.988148 + 0.153506i \(0.950944\pi\)
\(542\) 0 0
\(543\) −2.35753e55 4.08337e55i −0.203733 0.352876i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.52425e56 1.87669 0.938345 0.345700i \(-0.112359\pi\)
0.938345 + 0.345700i \(0.112359\pi\)
\(548\) 0 0
\(549\) 2.14871e54 1.24056e54i 0.0148233 0.00855824i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.20682e56 2.40690e56i 0.717427 1.43085i
\(554\) 0 0
\(555\) 0 0
\(556\) −3.18475e56 + 1.83872e56i −1.69450 + 0.978323i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 1.34828e56i 0.642446i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.54475e56 2.34534e56i −0.550055 0.835128i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −1.99927e56 3.46284e56i −0.616356 1.06756i −0.990145 0.140047i \(-0.955275\pi\)
0.373789 0.927514i \(-0.378059\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.93924e56 3.35886e56i 0.500000 0.866025i
\(577\) 3.19378e56 1.84393e56i 0.794696 0.458818i −0.0469175 0.998899i \(-0.514940\pi\)
0.841613 + 0.540081i \(0.181606\pi\)
\(578\) 0 0
\(579\) −7.05521e56 4.07333e56i −1.63531 0.944145i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −5.43782e56 2.33721e56i −0.918734 0.394878i
\(589\) 1.95363e57 3.18772
\(590\) 0 0
\(591\) 0 0
\(592\) 1.56640e56 + 2.71308e56i 0.230307 + 0.398904i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.63352e56 + 1.32216e57i −0.944607 + 1.63611i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 1.83954e57i 1.98510i 0.121829 + 0.992551i \(0.461124\pi\)
−0.121829 + 0.992551i \(0.538876\pi\)
\(602\) 0 0
\(603\) 1.85380e57 1.86878
\(604\) −4.18186e56 + 7.24319e56i −0.407485 + 0.705785i
\(605\) 0 0
\(606\) 0 0
\(607\) −1.82664e57 1.05461e57i −1.60800 0.928376i −0.989818 0.142337i \(-0.954538\pi\)
−0.618177 0.786039i \(-0.712128\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.23063e57 + 2.13152e57i 0.885499 + 1.53373i 0.845140 + 0.534545i \(0.179517\pi\)
0.0403591 + 0.999185i \(0.487150\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −2.28222e57 + 1.31764e57i −1.34493 + 0.776493i −0.987526 0.157458i \(-0.949670\pi\)
−0.357400 + 0.933951i \(0.616337\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.98987e57 0.994352
\(625\) −1.03398e57 + 1.79090e57i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 2.66904e57 + 1.54097e57i 1.16999 + 0.675495i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.15205e57 −0.458002 −0.229001 0.973426i \(-0.573546\pi\)
−0.229001 + 0.973426i \(0.573546\pi\)
\(632\) 0 0
\(633\) −4.36458e57 + 2.51989e57i −1.62617 + 0.938871i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.56483e56 3.01568e57i −0.116729 0.987477i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 6.08414e57i 1.64390i 0.569562 + 0.821948i \(0.307113\pi\)
−0.569562 + 0.821948i \(0.692887\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 3.93154e57 7.84113e57i 0.824427 1.64425i
\(652\) 4.71111e57 0.957299
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.97602e56i 0.0517065i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 6.98743e56 4.03419e56i 0.107198 0.0618909i −0.445442 0.895311i \(-0.646954\pi\)
0.552641 + 0.833420i \(0.313620\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −7.56801e57 1.31082e58i −0.907290 1.57147i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.70949e58 1.81367 0.906835 0.421487i \(-0.138492\pi\)
0.906835 + 0.421487i \(0.138492\pi\)
\(674\) 0 0
\(675\) 8.67477e57 5.00838e57i 0.866025 0.500000i
\(676\) −5.81547e55 1.00727e56i −0.00563219 0.00975524i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 7.22030e56 + 1.22586e58i 0.0638594 + 1.08420i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) −1.97240e58 1.13876e58i −1.50087 0.866529i
\(685\) 0 0
\(686\) 0 0
\(687\) −2.84502e58 −1.97912
\(688\) −4.78442e57 + 8.28686e57i −0.323048 + 0.559535i
\(689\) 0 0
\(690\) 0 0
\(691\) −1.35341e58 7.81393e57i −0.835854 0.482580i 0.0199989 0.999800i \(-0.493634\pi\)
−0.855853 + 0.517220i \(0.826967\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 9.46228e57 1.88717e58i 0.448214 0.893926i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.59318e58 9.19825e57i 0.691324 0.399136i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.69787e58 4.67284e58i 0.983496 1.70347i 0.335060 0.942197i \(-0.391244\pi\)
0.648437 0.761269i \(-0.275423\pi\)
\(710\) 0 0
\(711\) −2.32589e58 4.02856e58i −0.800317 1.38619i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 2.03617e58 + 1.02094e58i 0.526189 + 0.263831i
\(722\) 0 0
\(723\) 3.87530e58 6.71222e58i 0.946173 1.63882i
\(724\) −1.48684e58 + 8.58425e57i −0.352876 + 0.203733i
\(725\) 0 0
\(726\) 0 0
\(727\) 8.51197e58i 1.85599i 0.372598 + 0.927993i \(0.378467\pi\)
−0.372598 + 0.927993i \(0.621533\pi\)
\(728\) 0 0
\(729\) −4.85193e58 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −4.51712e56 7.82388e56i −0.00855824 0.0148233i
\(733\) 5.32997e58 + 3.07726e58i 0.981959 + 0.566935i 0.902861 0.429932i \(-0.141463\pi\)
0.0790983 + 0.996867i \(0.474796\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 6.28190e58 + 1.08806e59i 0.979220 + 1.69606i 0.665241 + 0.746629i \(0.268329\pi\)
0.313979 + 0.949430i \(0.398338\pi\)
\(740\) 0 0
\(741\) 1.16849e59i 1.72327i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −8.26198e57 + 1.43102e58i −0.0925690 + 0.160334i −0.908591 0.417686i \(-0.862841\pi\)
0.816022 + 0.578020i \(0.196175\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −8.53984e58 + 5.62475e58i −0.835128 + 0.550055i
\(757\) −2.03933e59 −1.94098 −0.970490 0.241141i \(-0.922478\pi\)
−0.970490 + 0.241141i \(0.922478\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 1.31042e59 + 1.98957e59i 1.06086 + 1.61066i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.22303e59 7.06114e58i −0.866025 0.500000i
\(769\) 2.63478e59i 1.81658i 0.418344 + 0.908289i \(0.362611\pi\)
−0.418344 + 0.908289i \(0.637389\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.48318e59 + 2.56894e59i −0.944145 + 1.63531i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 2.70946e59 + 1.56431e59i 1.59293 + 0.919680i
\(776\) 0 0
\(777\) −4.85658e57 8.24549e58i −0.0270832 0.459818i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −8.51025e58 + 1.98002e59i −0.394878 + 0.918734i
\(785\) 0 0
\(786\) 0 0
\(787\) 2.30796e59 1.33250e59i 0.990251 0.571722i 0.0849019 0.996389i \(-0.472942\pi\)
0.905349 + 0.424667i \(0.139609\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.31753e57 4.01407e57i 0.00850990 0.0147396i
\(794\) 0 0
\(795\) 0 0
\(796\) 4.81426e59 + 2.77952e59i 1.63611 + 0.944607i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 6.75005e59i 1.86878i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) 8.16079e59i 1.89149i −0.324906 0.945746i \(-0.605333\pi\)
0.324906 0.945746i \(-0.394667\pi\)
\(812\) 0 0
\(813\) −5.99727e59 −1.32159
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.86623e59 + 2.80952e59i 0.969707 + 0.559860i
\(818\) 0 0
\(819\) −4.68988e59 2.35150e59i −0.888877 0.445683i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 3.34637e59 + 5.79609e59i 0.573955 + 0.994119i 0.996154 + 0.0876158i \(0.0279248\pi\)
−0.422200 + 0.906503i \(0.638742\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −3.28312e59 + 1.89551e59i −0.485198 + 0.280129i −0.722580 0.691287i \(-0.757044\pi\)
0.237382 + 0.971416i \(0.423711\pi\)
\(830\) 0 0
\(831\) −1.12310e60 6.48424e59i −1.57979 0.912091i
\(832\) 7.24550e59i 0.994352i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −7.57720e59 1.31241e60i −0.919680 1.59293i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 9.08485e59 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 9.17542e59 + 1.58923e60i 0.938871 + 1.62617i
\(845\) 0 0
\(846\) 0 0
\(847\) −4.71081e59 + 9.39532e59i −0.448214 + 0.893926i
\(848\) 0 0
\(849\) 7.42962e59 1.28685e60i 0.673533 1.16659i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 8.59336e59i 0.707470i −0.935346 0.353735i \(-0.884912\pi\)
0.935346 0.353735i \(-0.115088\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) −6.43015e59 3.71245e59i −0.458523 0.264728i 0.252900 0.967492i \(-0.418616\pi\)
−0.711423 + 0.702764i \(0.751949\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.69587e60i 1.00000i
\(868\) −2.85511e60 1.43155e60i −1.64425 0.824427i
\(869\) 0 0
\(870\) 0 0
\(871\) 2.99917e60 1.73157e60i 1.60927 0.929112i
\(872\) 0 0
\(873\) 1.83732e60 + 1.06078e60i 0.940574 + 0.543041i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.08363e59 −0.0517065
\(877\) −1.36347e59 + 2.36160e59i −0.0635556 + 0.110082i −0.896052 0.443948i \(-0.853577\pi\)
0.832497 + 0.554030i \(0.186911\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 1.45751e60 0.590766 0.295383 0.955379i \(-0.404553\pi\)
0.295383 + 0.955379i \(0.404553\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 1.53523e60 9.04248e58i 0.541610 0.0319008i
\(890\) 0 0
\(891\) 0 0
\(892\) −4.77294e60 + 2.75566e60i −1.57147 + 0.907290i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.82365e60 3.15865e60i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 2.10692e60 1.38772e60i 0.539572 0.355388i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.88158e60 3.25900e60i −0.440123 0.762316i 0.557575 0.830127i \(-0.311732\pi\)
−0.997698 + 0.0678108i \(0.978399\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −4.14646e60 + 7.18188e60i −0.866529 + 1.50087i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.03593e61i 1.97912i
\(917\) 0 0
\(918\) 0 0
\(919\) −3.50317e60 + 6.06767e60i −0.625883 + 1.08406i 0.362486 + 0.931989i \(0.381928\pi\)
−0.988369 + 0.152072i \(0.951405\pi\)
\(920\) 0 0
\(921\) 5.25292e60 + 9.09832e60i 0.897590 + 1.55467i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.94607e60 0.460615
\(926\) 0 0
\(927\) 3.40805e60 1.96764e60i 0.509766 0.294314i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) 1.16271e61 + 4.99741e60i 1.59222 + 0.684347i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.63882e61i 1.96729i 0.180118 + 0.983645i \(0.442352\pi\)
−0.180118 + 0.983645i \(0.557648\pi\)
\(938\) 0 0
\(939\) −4.14401e60 −0.476186
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) −1.46688e61 + 8.46902e60i −1.38619 + 0.800317i
\(949\) −2.77980e59 4.81475e59i −0.0257073 0.0445263i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.66712e61 2.88754e61i 1.19162 2.06395i
\(962\) 0 0
\(963\) 0 0
\(964\) −2.44405e61 1.41107e61i −1.63882 0.946173i
\(965\) 0 0
\(966\) 0 0
\(967\) 6.01411e60 0.378380 0.189190 0.981940i \(-0.439414\pi\)
0.189190 + 0.981940i \(0.439414\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 1.76668e61i 1.00000i
\(973\) −2.07581e60 3.52430e61i −0.115047 1.95326i
\(974\) 0 0
\(975\) 9.35631e60 1.62056e61i 0.497176 0.861134i
\(976\) −2.84883e59 + 1.64477e59i −0.0148233 + 0.00855824i
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.11592e61 1.92864
\(982\) 0 0
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −4.25472e61 −1.72327
\(989\) 0 0
\(990\) 0 0
\(991\) −2.56736e61 4.44679e61i −0.977184 1.69253i −0.672532 0.740068i \(-0.734793\pi\)
−0.304652 0.952464i \(-0.598540\pi\)
\(992\) 0 0
\(993\) 4.94565e61i 1.80619i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.45749e61 8.41480e60i 0.490178 0.283004i −0.234470 0.972123i \(-0.575336\pi\)
0.724648 + 0.689119i \(0.242002\pi\)
\(998\) 0 0
\(999\) −1.23584e61 7.13510e60i −0.398904 0.230307i
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 21.42.g.a.5.1 2
3.2 odd 2 CM 21.42.g.a.5.1 2
7.3 odd 6 inner 21.42.g.a.17.1 yes 2
21.17 even 6 inner 21.42.g.a.17.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.42.g.a.5.1 2 1.1 even 1 trivial
21.42.g.a.5.1 2 3.2 odd 2 CM
21.42.g.a.17.1 yes 2 7.3 odd 6 inner
21.42.g.a.17.1 yes 2 21.17 even 6 inner