Properties

Label 21.29.h.a.11.1
Level $21$
Weight $29$
Character 21.11
Analytic conductor $104.304$
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] = [21,29,Mod(2,21)]
 
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, 2]))
 
N = Newforms(chi, 29, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 29);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 29 \)
Character orbit: \([\chi]\) \(=\) 21.h (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: \(104.303795921\)
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 11.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.11
Dual form 21.29.h.a.2.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.39148e6 + 4.14217e6i) q^{3} +(-1.34218e8 + 2.32472e8i) q^{4} +(3.89159e11 - 5.55466e11i) q^{7} +(-1.14384e13 - 1.98119e13i) q^{9} +O(q^{10})\) \(q+(-2.39148e6 + 4.14217e6i) q^{3} +(-1.34218e8 + 2.32472e8i) q^{4} +(3.89159e11 - 5.55466e11i) q^{7} +(-1.14384e13 - 1.98119e13i) q^{9} +(-6.41959e14 - 1.11191e15i) q^{12} +6.87801e15 q^{13} +(-3.60288e16 - 6.24037e16i) q^{16} +(-4.51754e17 - 7.82461e17i) q^{19} +(1.37017e18 + 2.94035e18i) q^{21} +(-1.86265e19 + 3.22620e19i) q^{25} +1.09419e20 q^{27} +(7.68982e19 + 1.65022e20i) q^{28} +(-7.44144e20 + 1.28890e21i) q^{31} +6.14094e21 q^{36} +(6.03260e21 + 1.04488e22i) q^{37} +(-1.64487e22 + 2.84899e22i) q^{39} -6.43151e21 q^{43} +3.44649e23 q^{48} +(-1.57098e23 - 4.32329e23i) q^{49} +(-9.23151e23 + 1.59894e24i) q^{52} +4.32145e24 q^{57} +(-9.43379e24 - 1.63398e25i) q^{61} +(-1.54562e25 - 1.35633e24i) q^{63} +1.93428e25 q^{64} +(7.67328e24 - 1.32905e25i) q^{67} +(-8.94167e25 + 1.54874e26i) q^{73} +(-8.90897e25 - 1.54308e26i) q^{75} +2.42534e26 q^{76} +(2.06809e26 + 3.58204e26i) q^{79} +(-2.61674e26 + 4.53232e26i) q^{81} +(-8.67450e26 - 7.61216e25i) q^{84} +(2.67664e27 - 3.82050e27i) q^{91} +(-3.55922e27 - 6.16475e27i) q^{93} -1.29210e28 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4782969 q^{3} - 268435456 q^{4} + 778317387191 q^{7} - 22876792454961 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4782969 q^{3} - 268435456 q^{4} + 778317387191 q^{7} - 22876792454961 q^{9} - 12\!\cdots\!64 q^{12}+ \cdots - 25\!\cdots\!08 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) −2.39148e6 + 4.14217e6i −0.500000 + 0.866025i
\(4\) −1.34218e8 + 2.32472e8i −0.500000 + 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 3.89159e11 5.55466e11i 0.573792 0.819001i
\(8\) 0 0
\(9\) −1.14384e13 1.98119e13i −0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −6.41959e14 1.11191e15i −0.500000 0.866025i
\(13\) 6.87801e15 1.74685 0.873426 0.486958i \(-0.161893\pi\)
0.873426 + 0.486958i \(0.161893\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.60288e16 6.24037e16i −0.500000 0.866025i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −4.51754e17 7.82461e17i −0.565395 0.979293i −0.997013 0.0772361i \(-0.975390\pi\)
0.431618 0.902056i \(-0.357943\pi\)
\(20\) 0 0
\(21\) 1.37017e18 + 2.94035e18i 0.422380 + 0.906419i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −1.86265e19 + 3.22620e19i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 1.09419e20 1.00000
\(28\) 7.68982e19 + 1.65022e20i 0.422380 + 0.906419i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −7.44144e20 + 1.28890e21i −0.983090 + 1.70276i −0.332956 + 0.942942i \(0.608046\pi\)
−0.650134 + 0.759819i \(0.725287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.14094e21 1.00000
\(37\) 6.03260e21 + 1.04488e22i 0.669391 + 1.15942i 0.978075 + 0.208255i \(0.0667784\pi\)
−0.308683 + 0.951165i \(0.599888\pi\)
\(38\) 0 0
\(39\) −1.64487e22 + 2.84899e22i −0.873426 + 1.51282i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −6.43151e21 −0.0870471 −0.0435236 0.999052i \(-0.513858\pi\)
−0.0435236 + 0.999052i \(0.513858\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 3.44649e23 1.00000
\(49\) −1.57098e23 4.32329e23i −0.341526 0.939872i
\(50\) 0 0
\(51\) 0 0
\(52\) −9.23151e23 + 1.59894e24i −0.873426 + 1.51282i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.32145e24 1.13079
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −9.43379e24 1.63398e25i −0.955144 1.65436i −0.734039 0.679107i \(-0.762367\pi\)
−0.221104 0.975250i \(-0.570966\pi\)
\(62\) 0 0
\(63\) −1.54562e25 1.35633e24i −0.996172 0.0874174i
\(64\) 1.93428e25 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.67328e24 1.32905e25i 0.208898 0.361822i −0.742470 0.669880i \(-0.766346\pi\)
0.951368 + 0.308058i \(0.0996791\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −8.94167e25 + 1.54874e26i −0.732653 + 1.26899i 0.223092 + 0.974797i \(0.428385\pi\)
−0.955745 + 0.294195i \(0.904948\pi\)
\(74\) 0 0
\(75\) −8.90897e25 1.54308e26i −0.500000 0.866025i
\(76\) 2.42534e26 1.13079
\(77\) 0 0
\(78\) 0 0
\(79\) 2.06809e26 + 3.58204e26i 0.560777 + 0.971294i 0.997429 + 0.0716634i \(0.0228307\pi\)
−0.436652 + 0.899630i \(0.643836\pi\)
\(80\) 0 0
\(81\) −2.61674e26 + 4.53232e26i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −8.67450e26 7.61216e25i −0.996172 0.0874174i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 2.67664e27 3.82050e27i 1.00233 1.43067i
\(92\) 0 0
\(93\) −3.55922e27 6.16475e27i −0.983090 1.70276i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.29210e28 −1.97920 −0.989602 0.143832i \(-0.954058\pi\)
−0.989602 + 0.143832i \(0.954058\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000e27 8.66025e27i −0.500000 0.866025i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 1.37243e28 + 2.37712e28i 0.907338 + 1.57156i 0.817748 + 0.575577i \(0.195222\pi\)
0.0895902 + 0.995979i \(0.471444\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) −1.46860e28 + 2.54368e28i −0.500000 + 0.866025i
\(109\) −2.27187e27 + 3.93500e27i −0.0679850 + 0.117754i −0.898014 0.439966i \(-0.854990\pi\)
0.830029 + 0.557720i \(0.188324\pi\)
\(110\) 0 0
\(111\) −5.77074e28 −1.33878
\(112\) −4.86840e28 4.27219e27i −0.996172 0.0874174i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −7.86734e28 1.36266e29i −0.873426 1.51282i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.21050e28 1.24889e29i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.99755e29 3.45985e29i −0.983090 1.70276i
\(125\) 0 0
\(126\) 0 0
\(127\) −5.57094e29 −1.96190 −0.980949 0.194266i \(-0.937767\pi\)
−0.980949 + 0.194266i \(0.937767\pi\)
\(128\) 0 0
\(129\) 1.53809e28 2.66404e28i 0.0435236 0.0753850i
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) −6.10434e29 5.35677e28i −1.12646 0.0988507i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) −1.11082e30 −1.10519 −0.552595 0.833450i \(-0.686362\pi\)
−0.552595 + 0.833450i \(0.686362\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −8.24223e29 + 1.42760e30i −0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 2.16648e30 + 3.83182e29i 0.984716 + 0.174166i
\(148\) −3.23873e30 −1.33878
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 4.30608e29 7.45835e29i 0.134402 0.232791i −0.790967 0.611859i \(-0.790422\pi\)
0.925369 + 0.379068i \(0.123755\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.41540e30 7.64770e30i −0.873426 1.51282i
\(157\) 3.48131e30 6.02980e30i 0.629719 1.09071i −0.357889 0.933764i \(-0.616503\pi\)
0.987608 0.156942i \(-0.0501634\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.19289e30 + 1.41905e31i 0.876619 + 1.51835i 0.855028 + 0.518581i \(0.173540\pi\)
0.0215904 + 0.999767i \(0.493127\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.18041e31 2.05149
\(170\) 0 0
\(171\) −1.03347e31 + 1.79002e31i −0.565395 + 0.979293i
\(172\) 8.63223e29 1.49515e30i 0.0435236 0.0753850i
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 1.06718e31 + 2.29014e31i 0.422380 + 0.906419i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 3.77614e31 0.932283 0.466142 0.884710i \(-0.345644\pi\)
0.466142 + 0.884710i \(0.345644\pi\)
\(182\) 0 0
\(183\) 9.02431e31 1.91029
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.25814e31 6.07785e31i 0.573792 0.819001i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −4.62580e31 + 8.01213e31i −0.500000 + 0.866025i
\(193\) −9.91992e31 + 1.71818e32i −0.997024 + 1.72690i −0.431750 + 0.901993i \(0.642104\pi\)
−0.565274 + 0.824903i \(0.691230\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.21590e32 + 2.15054e31i 0.984716 + 0.174166i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.45688e32 2.52338e32i 0.953848 1.65211i 0.216866 0.976201i \(-0.430416\pi\)
0.736982 0.675912i \(-0.236250\pi\)
\(200\) 0 0
\(201\) 3.67011e31 + 6.35681e31i 0.208898 + 0.361822i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −2.47806e32 4.29213e32i −0.873426 1.51282i
\(209\) 0 0
\(210\) 0 0
\(211\) −5.63075e32 −1.62410 −0.812048 0.583590i \(-0.801647\pi\)
−0.812048 + 0.583590i \(0.801647\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.26347e32 + 9.14931e32i 0.830475 + 1.78218i
\(218\) 0 0
\(219\) −4.27677e32 7.40759e32i −0.732653 1.26899i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.30064e33 −1.72938 −0.864690 0.502306i \(-0.832485\pi\)
−0.864690 + 0.502306i \(0.832485\pi\)
\(224\) 0 0
\(225\) 8.52227e32 1.00000
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) −5.80016e32 + 1.00462e33i −0.565395 + 0.979293i
\(229\) −1.05623e33 1.82944e33i −0.968413 1.67734i −0.700151 0.713995i \(-0.746884\pi\)
−0.268262 0.963346i \(-0.586449\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.97832e33 −1.12155
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.92197e33 + 3.32896e33i −0.862004 + 1.49303i 0.00798813 + 0.999968i \(0.497457\pi\)
−0.869992 + 0.493066i \(0.835876\pi\)
\(242\) 0 0
\(243\) −1.25158e33 2.16780e33i −0.500000 0.866025i
\(244\) 5.06473e33 1.91029
\(245\) 0 0
\(246\) 0 0
\(247\) −3.10717e33 5.38178e33i −0.987661 1.71068i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 2.38980e33 3.41108e33i 0.573792 0.819001i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −2.59615e33 + 4.49666e33i −0.500000 + 0.866025i
\(257\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) 8.15157e33 + 7.15327e32i 1.33366 + 0.117033i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2.05978e33 + 3.56764e33i 0.208898 + 0.361822i
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 1.13011e34 + 1.95740e34i 0.980731 + 1.69868i 0.659555 + 0.751657i \(0.270745\pi\)
0.321176 + 0.947019i \(0.395922\pi\)
\(272\) 0 0
\(273\) 9.42403e33 + 2.02238e34i 0.737835 + 1.58338i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.27285e33 1.43290e34i 0.528370 0.915165i −0.471082 0.882089i \(-0.656137\pi\)
0.999453 0.0330753i \(-0.0105301\pi\)
\(278\) 0 0
\(279\) 3.40473e34 1.96618
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.93428e34 + 3.35028e34i −0.915188 + 1.58515i −0.108562 + 0.994090i \(0.534625\pi\)
−0.806626 + 0.591062i \(0.798709\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41755e34 2.45528e34i −0.500000 0.866025i
\(290\) 0 0
\(291\) 3.09003e34 5.35209e34i 0.989602 1.71404i
\(292\) −2.40026e34 4.15737e34i −0.732653 1.26899i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 4.78297e34 1.00000
\(301\) −2.50288e33 + 3.57248e33i −0.0499469 + 0.0712917i
\(302\) 0 0
\(303\) 0 0
\(304\) −3.25523e34 + 5.63823e34i −0.565395 + 0.979293i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.44698e34 0.370419 0.185210 0.982699i \(-0.440704\pi\)
0.185210 + 0.982699i \(0.440704\pi\)
\(308\) 0 0
\(309\) −1.31286e35 −1.81468
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 4.78758e34 + 8.29234e34i 0.552708 + 0.957318i 0.998078 + 0.0619715i \(0.0197388\pi\)
−0.445370 + 0.895347i \(0.646928\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.11030e35 −1.12155
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −7.02425e34 1.21664e35i −0.500000 0.866025i
\(325\) −1.28113e35 + 2.21898e35i −0.873426 + 1.51282i
\(326\) 0 0
\(327\) −1.08663e34 1.88210e34i −0.0679850 0.117754i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.36091e35 2.35716e35i −0.718185 1.24393i −0.961718 0.274040i \(-0.911640\pi\)
0.243534 0.969892i \(-0.421693\pi\)
\(332\) 0 0
\(333\) 1.38006e35 2.39034e35i 0.669391 1.15942i
\(334\) 0 0
\(335\) 0 0
\(336\) 1.34123e35 1.91441e35i 0.573792 0.819001i
\(337\) −3.77901e35 −1.55081 −0.775407 0.631461i \(-0.782455\pi\)
−0.775407 + 0.631461i \(0.782455\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −3.01280e35 8.09821e34i −0.965722 0.259580i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) −1.90214e35 −0.478285 −0.239143 0.970984i \(-0.576866\pi\)
−0.239143 + 0.970984i \(0.576866\pi\)
\(350\) 0 0
\(351\) 7.52585e35 1.74685
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) −8.89580e34 + 1.54080e35i −0.139343 + 0.241349i
\(362\) 0 0
\(363\) 6.89752e35 1.00000
\(364\) 5.28906e35 + 1.13502e36i 0.737835 + 1.58338i
\(365\) 0 0
\(366\) 0 0
\(367\) −5.40238e35 + 9.35720e35i −0.671832 + 1.16365i 0.305552 + 0.952175i \(0.401159\pi\)
−0.977384 + 0.211471i \(0.932174\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.91084e36 1.96618
\(373\) 9.91585e35 + 1.71748e36i 0.982667 + 1.70203i 0.651877 + 0.758324i \(0.273982\pi\)
0.330789 + 0.943705i \(0.392685\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.14016e36 −1.69628 −0.848139 0.529774i \(-0.822277\pi\)
−0.848139 + 0.529774i \(0.822277\pi\)
\(380\) 0 0
\(381\) 1.33228e36 2.30758e36i 0.980949 1.69905i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.35662e34 + 1.27420e35i 0.0435236 + 0.0753850i
\(388\) 1.73422e36 3.00376e36i 0.989602 1.71404i
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −4.30402e35 7.45478e35i −0.178159 0.308580i 0.763091 0.646291i \(-0.223681\pi\)
−0.941250 + 0.337711i \(0.890347\pi\)
\(398\) 0 0
\(399\) 1.68173e36 2.40042e36i 0.648838 0.926118i
\(400\) 2.68435e36 1.00000
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −5.11823e36 + 8.86503e36i −1.71731 + 2.97447i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.76446e35 + 9.98433e35i −0.157265 + 0.272391i −0.933881 0.357583i \(-0.883601\pi\)
0.776616 + 0.629974i \(0.216934\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −7.36818e36 −1.81468
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.65651e36 4.60121e36i 0.552595 0.957123i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 6.72538e36 1.22396 0.611980 0.790873i \(-0.290373\pi\)
0.611980 + 0.790873i \(0.290373\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.27474e37 1.11863e36i −1.90297 0.166992i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) −3.94223e36 6.82815e36i −0.500000 0.866025i
\(433\) −1.55655e37 −1.91131 −0.955657 0.294484i \(-0.904852\pi\)
−0.955657 + 0.294484i \(0.904852\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.09852e35 1.05629e36i −0.0679850 0.117754i
\(437\) 0 0
\(438\) 0 0
\(439\) 6.81343e35 + 1.18012e36i 0.0690022 + 0.119515i 0.898462 0.439051i \(-0.144685\pi\)
−0.829460 + 0.558566i \(0.811352\pi\)
\(440\) 0 0
\(441\) −6.76830e36 + 8.05754e36i −0.643190 + 0.765707i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 7.74536e36 1.34154e37i 0.669391 1.15942i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 7.52742e36 1.07443e37i 0.573792 0.819001i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.05959e36 + 3.56731e36i 0.134402 + 0.232791i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.38593e36 + 5.86459e36i 0.195367 + 0.338385i 0.947021 0.321173i \(-0.104077\pi\)
−0.751654 + 0.659558i \(0.770744\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −4.14736e37 −1.99360 −0.996799 0.0799547i \(-0.974522\pi\)
−0.996799 + 0.0799547i \(0.974522\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 4.22375e37 1.74685
\(469\) −4.39630e36 9.43436e36i −0.176469 0.378698i
\(470\) 0 0
\(471\) 1.66510e37 + 2.88403e37i 0.629719 + 1.09071i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.36583e37 1.13079
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 4.14923e37 + 7.18667e37i 1.16933 + 2.02533i
\(482\) 0 0
\(483\) 0 0
\(484\) 3.87111e37 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.71380e37 2.96840e37i 0.406028 0.703261i −0.588413 0.808561i \(-0.700247\pi\)
0.994441 + 0.105300i \(0.0335802\pi\)
\(488\) 0 0
\(489\) −7.83727e37 −1.75324
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.07242e38 1.96618
\(497\) 0 0
\(498\) 0 0
\(499\) −9.38658e36 1.62580e37i −0.158161 0.273943i 0.776044 0.630678i \(-0.217223\pi\)
−0.934206 + 0.356735i \(0.883890\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.60590e37 + 1.31738e38i −1.02574 + 1.77664i
\(508\) 7.47719e37 1.29509e38i 0.980949 1.69905i
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 5.12300e37 + 1.09939e38i 0.618916 + 1.32818i
\(512\) 0 0
\(513\) −4.94305e37 8.56161e37i −0.565395 0.979293i
\(514\) 0 0
\(515\) 0 0
\(516\) 4.12877e36 + 7.15123e36i 0.0435236 + 0.0753850i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) −8.50630e37 1.47333e38i −0.742535 1.28611i −0.951338 0.308151i \(-0.900290\pi\)
0.208802 0.977958i \(-0.433043\pi\)
\(524\) 0 0
\(525\) −1.20383e38 1.05640e37i −0.996172 0.0874174i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.71969e37 + 1.16388e38i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 9.43841e37 1.34719e38i 0.648838 0.926118i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.83975e38 3.18655e38i −0.999995 1.73204i −0.502668 0.864479i \(-0.667648\pi\)
−0.497327 0.867563i \(-0.665685\pi\)
\(542\) 0 0
\(543\) −9.03058e37 + 1.56414e38i −0.466142 + 0.807381i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.86913e38 −1.80215 −0.901075 0.433664i \(-0.857220\pi\)
−0.901075 + 0.433664i \(0.857220\pi\)
\(548\) 0 0
\(549\) −2.15815e38 + 3.73802e38i −0.955144 + 1.65436i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.79451e38 + 2.45228e37i 1.11726 + 0.0980433i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.49092e38 2.58235e38i 0.552595 0.957123i
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) −4.42360e37 −0.152058
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.49922e38 + 3.21730e38i 0.422380 + 0.906419i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) −2.39362e38 + 4.14586e38i −0.611151 + 1.05855i 0.379895 + 0.925029i \(0.375960\pi\)
−0.991047 + 0.133516i \(0.957373\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −2.21251e38 3.83218e38i −0.500000 0.866025i
\(577\) 2.95149e38 5.11214e38i 0.650999 1.12756i −0.331881 0.943321i \(-0.607683\pi\)
0.982881 0.184243i \(-0.0589833\pi\)
\(578\) 0 0
\(579\) −4.74467e38 8.21800e38i −0.997024 1.72690i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −3.79858e38 + 4.52215e38i −0.643190 + 0.765707i
\(589\) 1.34468e39 2.22334
\(590\) 0 0
\(591\) 0 0
\(592\) 4.34694e38 7.52913e38i 0.669391 1.15942i
\(593\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.96819e38 + 1.20693e39i 0.953848 + 1.65211i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 8.81133e38 1.09850 0.549249 0.835659i \(-0.314914\pi\)
0.549249 + 0.835659i \(0.314914\pi\)
\(602\) 0 0
\(603\) −3.51080e38 −0.417796
\(604\) 1.15591e38 + 2.00209e38i 0.134402 + 0.232791i
\(605\) 0 0
\(606\) 0 0
\(607\) 9.06348e38 + 1.56984e39i 0.983226 + 1.70300i 0.649569 + 0.760303i \(0.274950\pi\)
0.333657 + 0.942694i \(0.391717\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8.79362e38 + 1.52310e39i −0.831230 + 1.43973i 0.0658342 + 0.997831i \(0.479029\pi\)
−0.897064 + 0.441901i \(0.854304\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −2.23916e38 + 3.87834e38i −0.184678 + 0.319872i −0.943468 0.331463i \(-0.892458\pi\)
0.758790 + 0.651336i \(0.225791\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 2.37050e39 1.74685
\(625\) −6.93889e38 1.20185e39i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 9.34506e38 + 1.61861e39i 0.629719 + 1.09071i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.07122e38 0.0675256 0.0337628 0.999430i \(-0.489251\pi\)
0.0337628 + 0.999430i \(0.489251\pi\)
\(632\) 0 0
\(633\) 1.34658e39 2.33235e39i 0.812048 1.40651i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.08052e39 2.97356e39i −0.596596 1.64182i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) −3.06254e39 −1.48295 −0.741477 0.670979i \(-0.765874\pi\)
−0.741477 + 0.670979i \(0.765874\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.80941e39 4.22041e38i −1.95865 0.171878i
\(652\) −4.39853e39 −1.75324
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.09113e39 1.46531
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 2.11648e39 3.66585e39i 0.696296 1.20602i −0.273446 0.961887i \(-0.588163\pi\)
0.969742 0.244133i \(-0.0785032\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.11047e39 5.38749e39i 0.864690 1.49769i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7.59787e39 −1.94304 −0.971519 0.236963i \(-0.923848\pi\)
−0.971519 + 0.236963i \(0.923848\pi\)
\(674\) 0 0
\(675\) −2.03809e39 + 3.53007e39i −0.500000 + 0.866025i
\(676\) −4.26867e39 + 7.39356e39i −1.02574 + 1.77664i
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) −5.02831e39 + 7.17715e39i −1.13565 + 1.62097i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) −2.77420e39 4.80505e39i −0.565395 0.979293i
\(685\) 0 0
\(686\) 0 0
\(687\) 1.01038e40 1.93683
\(688\) 2.31720e38 + 4.01350e38i 0.0435236 + 0.0753850i
\(689\) 0 0
\(690\) 0 0
\(691\) 3.25126e39 + 5.63136e39i 0.574592 + 0.995223i 0.996086 + 0.0883914i \(0.0281726\pi\)
−0.421494 + 0.906831i \(0.638494\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −6.75627e39 5.92885e38i −0.996172 0.0874174i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 5.45050e39 9.44055e39i 0.756941 1.31106i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.49571e39 1.12509e40i −0.800902 1.38720i −0.919023 0.394203i \(-0.871021\pi\)
0.118122 0.992999i \(-0.462313\pi\)
\(710\) 0 0
\(711\) 4.73113e39 8.19455e39i 0.560777 0.971294i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) 1.85450e40 + 1.62739e39i 1.80773 + 0.158634i
\(722\) 0 0
\(723\) −9.19275e39 1.59223e40i −0.862004 1.49303i
\(724\) −5.06825e39 + 8.77846e39i −0.466142 + 0.807381i
\(725\) 0 0
\(726\) 0 0
\(727\) 2.01599e40 1.74987 0.874935 0.484241i \(-0.160904\pi\)
0.874935 + 0.484241i \(0.160904\pi\)
\(728\) 0 0
\(729\) 1.19725e40 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −1.21122e40 + 2.09790e40i −0.955144 + 1.65436i
\(733\) 1.14153e40 + 1.97718e40i 0.883142 + 1.52965i 0.847829 + 0.530270i \(0.177909\pi\)
0.0353127 + 0.999376i \(0.488757\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.44871e40 + 2.50924e40i −0.999905 + 1.73189i −0.488016 + 0.872834i \(0.662279\pi\)
−0.511889 + 0.859052i \(0.671054\pi\)
\(740\) 0 0
\(741\) 2.97230e40 1.97532
\(742\) 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
\(751\) −1.84159e39 3.18973e39i −0.101446 0.175709i 0.810835 0.585275i \(-0.199013\pi\)
−0.912281 + 0.409566i \(0.865680\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 8.41412e39 + 1.80565e40i 0.422380 + 0.906419i
\(757\) 1.42168e40 0.700581 0.350290 0.936641i \(-0.386083\pi\)
0.350290 + 0.936641i \(0.386083\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) 1.30164e39 + 2.79329e39i 0.0574311 + 0.123246i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.24173e40 2.15074e40i −0.500000 0.866025i
\(769\) 2.36412e40 0.934762 0.467381 0.884056i \(-0.345198\pi\)
0.467381 + 0.884056i \(0.345198\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.66286e40 4.61220e40i −0.997024 1.72690i
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) −2.77215e40 4.80151e40i −0.983090 1.70276i
\(776\) 0 0
\(777\) −2.24574e40 + 3.20545e40i −0.768182 + 1.09646i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −2.13189e40 + 2.53797e40i −0.643190 + 0.765707i
\(785\) 0 0
\(786\) 0 0
\(787\) 5.53794e39 9.59199e39i 0.158381 0.274323i −0.775904 0.630851i \(-0.782706\pi\)
0.934285 + 0.356527i \(0.116039\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.48857e40 1.12385e41i −1.66849 2.88992i
\(794\) 0 0
\(795\) 0 0
\(796\) 3.91077e40 + 6.77365e40i 0.953848 + 1.65211i
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −1.97037e40 −0.417796
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 4.39401e40 0.825219 0.412609 0.910908i \(-0.364617\pi\)
0.412609 + 0.910908i \(0.364617\pi\)
\(812\) 0 0
\(813\) −1.08105e41 −1.96146
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.90546e39 + 5.03241e39i 0.0492160 + 0.0852446i
\(818\) 0 0
\(819\) −1.06308e41 9.32886e39i −1.74016 0.152705i
\(820\) 0 0
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) −6.14658e40 + 1.06462e41i −0.939800 + 1.62778i −0.173959 + 0.984753i \(0.555656\pi\)
−0.765842 + 0.643029i \(0.777677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 6.97708e40 1.20847e41i 0.963630 1.66906i 0.250379 0.968148i \(-0.419445\pi\)
0.713251 0.700909i \(-0.247222\pi\)
\(830\) 0 0
\(831\) 3.95688e40 + 6.85352e40i 0.528370 + 0.915165i
\(832\) 1.33040e41 1.74685
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.14235e40 + 1.41030e41i −0.983090 + 1.70276i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 8.85409e40 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 7.55746e40 1.30899e41i 0.812048 1.40651i
\(845\) 0 0
\(846\) 0 0
\(847\) −9.74321e40 8.54999e39i −0.996172 0.0874174i
\(848\) 0 0
\(849\) −9.25162e40 1.60243e41i −0.915188 1.58515i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.40896e41 1.30501 0.652503 0.757786i \(-0.273719\pi\)
0.652503 + 0.757786i \(0.273719\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(858\) 0 0
\(859\) 6.17383e40 + 1.06934e41i 0.518383 + 0.897865i 0.999772 + 0.0213584i \(0.00679910\pi\)
−0.481389 + 0.876507i \(0.659868\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.35602e41 1.00000
\(868\) −2.69919e41 2.36863e40i −1.95865 0.171878i
\(869\) 0 0
\(870\) 0 0
\(871\) 5.27769e40 9.14123e40i 0.364913 0.632048i
\(872\) 0 0
\(873\) 1.47795e41 + 2.55989e41i 0.989602 + 1.71404i
\(874\) 0 0
\(875\) 0 0
\(876\) 2.29607e41 1.46531
\(877\) 2.51101e40 + 4.34920e40i 0.157708 + 0.273159i 0.934042 0.357164i \(-0.116256\pi\)
−0.776334 + 0.630322i \(0.782923\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 3.21546e41 1.83566 0.917831 0.396970i \(-0.129938\pi\)
0.917831 + 0.396970i \(0.129938\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) −2.16798e41 + 3.09447e41i −1.12572 + 1.60680i
\(890\) 0 0
\(891\) 0 0
\(892\) 1.74569e41 3.02363e41i 0.864690 1.49769i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.14384e41 + 1.98119e41i −0.500000 + 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) −8.81225e39 1.89109e40i −0.0367670 0.0789012i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −2.09246e41 + 3.62425e41i −0.820646 + 1.42140i 0.0845564 + 0.996419i \(0.473053\pi\)
−0.905202 + 0.424981i \(0.860281\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −1.55697e41 2.69675e41i −0.565395 0.979293i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 5.67058e41 1.93683
\(917\) 0 0
\(918\) 0 0
\(919\) −2.74381e41 4.75242e41i −0.895235 1.55059i −0.833514 0.552499i \(-0.813674\pi\)
−0.0617212 0.998093i \(-0.519659\pi\)
\(920\) 0 0
\(921\) −5.85192e40 + 1.01358e41i −0.185210 + 0.320792i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −4.49463e41 −1.33878
\(926\) 0 0
\(927\) 3.13968e41 5.43809e41i 0.907338 1.57156i
\(928\) 0 0
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) −2.67311e41 + 3.18229e41i −0.727313 + 0.865853i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.90090e41 −0.472721 −0.236361 0.971665i \(-0.575955\pi\)
−0.236361 + 0.971665i \(0.575955\pi\)
\(938\) 0 0
\(939\) −4.57977e41 −1.10542
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 2.65526e41 4.59904e41i 0.560777 0.971294i
\(949\) −6.15009e41 + 1.06523e42i −1.27984 + 2.21674i
\(950\) 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
\(961\) −8.21019e41 1.42205e42i −1.43293 2.48191i
\(962\) 0 0
\(963\) 0 0
\(964\) −5.15926e41 8.93610e41i −0.862004 1.49303i
\(965\) 0 0
\(966\) 0 0
\(967\) 8.80431e41 1.40839 0.704197 0.710004i \(-0.251307\pi\)
0.704197 + 0.710004i \(0.251307\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 6.71936e41 1.00000
\(973\) −4.32286e41 + 6.17023e41i −0.634149 + 0.905152i
\(974\) 0 0
\(975\) −6.12760e41 1.06133e42i −0.873426 1.51282i
\(976\) −6.79776e41 + 1.17741e42i −0.955144 + 1.65436i
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.03946e41 0.135970
\(982\) 0 0
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.66815e42 1.97532
\(989\) 0 0
\(990\) 0 0
\(991\) −1.73880e41 + 3.01169e41i −0.197341 + 0.341805i −0.947666 0.319265i \(-0.896564\pi\)
0.750324 + 0.661070i \(0.229897\pi\)
\(992\) 0 0
\(993\) 1.30183e42 1.43637
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.25641e41 + 1.60326e42i −0.965407 + 1.67213i −0.256889 + 0.966441i \(0.582698\pi\)
−0.708518 + 0.705693i \(0.750636\pi\)
\(998\) 0 0
\(999\) 6.60081e41 + 1.14329e42i 0.669391 + 1.15942i
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 21.29.h.a.11.1 yes 2
3.2 odd 2 CM 21.29.h.a.11.1 yes 2
7.2 even 3 inner 21.29.h.a.2.1 2
21.2 odd 6 inner 21.29.h.a.2.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.29.h.a.2.1 2 7.2 even 3 inner
21.29.h.a.2.1 2 21.2 odd 6 inner
21.29.h.a.11.1 yes 2 1.1 even 1 trivial
21.29.h.a.11.1 yes 2 3.2 odd 2 CM