Properties

Label 21.17.h.a.2.1
Level $21$
Weight $17$
Character 21.2
Analytic conductor $34.088$
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] = [21,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 17 \)
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: \(34.0881542099\)
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 2.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.2
Dual form 21.17.h.a.11.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3280.50 - 5681.99i) q^{3} +(-32768.0 - 56755.8i) q^{4} +(-5.73624e6 - 573128. i) q^{7} +(-2.15234e7 + 3.72796e7i) q^{9} +O(q^{10})\) \(q+(-3280.50 - 5681.99i) q^{3} +(-32768.0 - 56755.8i) q^{4} +(-5.73624e6 - 573128. i) q^{7} +(-2.15234e7 + 3.72796e7i) q^{9} +(-2.14991e8 + 3.72375e8i) q^{12} -1.20541e9 q^{13} +(-2.14748e9 + 3.71955e9i) q^{16} +(1.19275e10 - 2.06590e10i) q^{19} +(1.55612e10 + 3.44734e10i) q^{21} +(-7.62939e10 - 1.32145e11i) q^{25} +2.82430e11 q^{27} +(1.55437e11 + 3.44345e11i) q^{28} +(1.01393e11 + 1.75619e11i) q^{31} +2.82111e12 q^{36} +(-8.83011e11 + 1.52942e12i) q^{37} +(3.95435e12 + 6.84913e12i) q^{39} +2.33692e13 q^{43} +2.81793e13 q^{48} +(3.25760e13 + 6.57520e12i) q^{49} +(3.94989e13 + 6.84141e13i) q^{52} -1.56513e14 q^{57} +(-9.21444e13 + 1.59599e14i) q^{61} +(1.44829e14 - 2.01509e14i) q^{63} +2.81475e14 q^{64} +(-2.89437e14 - 5.01320e14i) q^{67} +(-7.19074e14 - 1.24547e15i) q^{73} +(-5.00565e14 + 8.67003e14i) q^{75} -1.56336e15 q^{76} +(-1.28746e15 + 2.22994e15i) q^{79} +(-9.26510e14 - 1.60476e15i) q^{81} +(1.44666e15 - 2.01282e15i) q^{84} +(6.91452e15 + 6.90854e14i) q^{91} +(6.65243e14 - 1.15223e15i) q^{93} +1.56216e16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6561 q^{3} - 65536 q^{4} - 11472481 q^{7} - 43046721 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6561 q^{3} - 65536 q^{4} - 11472481 q^{7} - 43046721 q^{9} - 429981696 q^{12} - 2410821122 q^{13} - 4294967296 q^{16} + 23855013601 q^{19} + 31122457794 q^{21} - 152587890625 q^{25} + 564859072962 q^{27} + 310873554944 q^{28} + 202786989793 q^{31} + 5642219814912 q^{36} - 1766022963839 q^{37} + 7908698690721 q^{39} + 46738333304254 q^{43} + 56358560858112 q^{48} + 65151959156159 q^{49} + 78997786525696 q^{52} - 313025488472322 q^{57} - 184288715234114 q^{61} + 289658243198367 q^{63} + 562949953421312 q^{64} - 578874088431839 q^{67} - 14\!\cdots\!99 q^{73}+ \cdots + 31\!\cdots\!68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −3280.50 5681.99i −0.500000 0.866025i
\(4\) −32768.0 56755.8i −0.500000 0.866025i
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −5.73624e6 573128.i −0.995046 0.0994185i
\(8\) 0 0
\(9\) −2.15234e7 + 3.72796e7i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −2.14991e8 + 3.72375e8i −0.500000 + 0.866025i
\(13\) −1.20541e9 −1.47771 −0.738853 0.673866i \(-0.764632\pi\)
−0.738853 + 0.673866i \(0.764632\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.14748e9 + 3.71955e9i −0.500000 + 0.866025i
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 1.19275e10 2.06590e10i 0.702297 1.21641i −0.265361 0.964149i \(-0.585491\pi\)
0.967658 0.252265i \(-0.0811755\pi\)
\(20\) 0 0
\(21\) 1.55612e10 + 3.44734e10i 0.411424 + 0.911444i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −7.62939e10 1.32145e11i −0.500000 0.866025i
\(26\) 0 0
\(27\) 2.82430e11 1.00000
\(28\) 1.55437e11 + 3.44345e11i 0.411424 + 0.911444i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.01393e11 + 1.75619e11i 0.118882 + 0.205910i 0.919325 0.393499i \(-0.128736\pi\)
−0.800443 + 0.599409i \(0.795402\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.82111e12 1.00000
\(37\) −8.83011e11 + 1.52942e12i −0.251393 + 0.435425i −0.963909 0.266230i \(-0.914222\pi\)
0.712517 + 0.701655i \(0.247555\pi\)
\(38\) 0 0
\(39\) 3.95435e12 + 6.84913e12i 0.738853 + 1.27973i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 2.33692e13 1.99938 0.999691 0.0248759i \(-0.00791906\pi\)
0.999691 + 0.0248759i \(0.00791906\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 2.81793e13 1.00000
\(49\) 3.25760e13 + 6.57520e12i 0.980232 + 0.197852i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.94989e13 + 6.84141e13i 0.738853 + 1.27973i
\(53\) 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
\(57\) −1.56513e14 −1.40459
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −9.21444e13 + 1.59599e14i −0.480651 + 0.832512i −0.999754 0.0221997i \(-0.992933\pi\)
0.519102 + 0.854712i \(0.326266\pi\)
\(62\) 0 0
\(63\) 1.44829e14 2.01509e14i 0.583622 0.812026i
\(64\) 2.81475e14 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.89437e14 5.01320e14i −0.712780 1.23457i −0.963809 0.266592i \(-0.914102\pi\)
0.251029 0.967980i \(-0.419231\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −7.19074e14 1.24547e15i −0.891643 1.54437i −0.837905 0.545816i \(-0.816220\pi\)
−0.0537374 0.998555i \(-0.517113\pi\)
\(74\) 0 0
\(75\) −5.00565e14 + 8.67003e14i −0.500000 + 0.866025i
\(76\) −1.56336e15 −1.40459
\(77\) 0 0
\(78\) 0 0
\(79\) −1.28746e15 + 2.22994e15i −0.848626 + 1.46986i 0.0338077 + 0.999428i \(0.489237\pi\)
−0.882434 + 0.470436i \(0.844097\pi\)
\(80\) 0 0
\(81\) −9.26510e14 1.60476e15i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.44666e15 2.01282e15i 0.583622 0.812026i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 6.91452e15 + 6.90854e14i 1.47039 + 0.146911i
\(92\) 0 0
\(93\) 6.65243e14 1.15223e15i 0.118882 0.205910i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.56216e16 1.99320 0.996601 0.0823822i \(-0.0262528\pi\)
0.996601 + 0.0823822i \(0.0262528\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000e15 + 8.66025e15i −0.500000 + 0.866025i
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) −1.00871e16 + 1.74714e16i −0.796288 + 1.37921i 0.125729 + 0.992065i \(0.459873\pi\)
−0.922018 + 0.387147i \(0.873460\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −9.25465e15 1.60295e16i −0.500000 0.866025i
\(109\) 1.76028e16 + 3.04889e16i 0.883423 + 1.53013i 0.847510 + 0.530779i \(0.178101\pi\)
0.0359135 + 0.999355i \(0.488566\pi\)
\(110\) 0 0
\(111\) 1.15869e16 0.502785
\(112\) 1.44503e16 2.01055e16i 0.583622 0.812026i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.59445e16 4.49372e16i 0.738853 1.27973i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.29749e16 + 3.97936e16i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 6.64492e15 1.15093e16i 0.118882 0.205910i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.27734e17 −1.88745 −0.943725 0.330732i \(-0.892704\pi\)
−0.943725 + 0.330732i \(0.892704\pi\)
\(128\) 0 0
\(129\) −7.66626e16 1.32783e17i −0.999691 1.73151i
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) −8.02593e16 + 1.11669e17i −0.819752 + 1.14057i
\(134\) 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
\(139\) 3.06396e16 0.219870 0.109935 0.993939i \(-0.464936\pi\)
0.109935 + 0.993939i \(0.464936\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −9.24421e16 1.60114e17i −0.500000 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −6.95053e16 2.06666e17i −0.318771 0.947832i
\(148\) 1.15738e17 0.502785
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 2.33768e17 + 4.04898e17i 0.864906 + 1.49806i 0.867140 + 0.498064i \(0.165955\pi\)
−0.00223432 + 0.999998i \(0.500711\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 2.59152e17 4.48865e17i 0.738853 1.27973i
\(157\) 3.68491e17 + 6.38245e17i 0.998228 + 1.72898i 0.550652 + 0.834735i \(0.314379\pi\)
0.447576 + 0.894246i \(0.352288\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.87763e17 + 3.25215e17i −0.376798 + 0.652634i −0.990594 0.136831i \(-0.956308\pi\)
0.613796 + 0.789465i \(0.289642\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 7.87598e17 1.18362
\(170\) 0 0
\(171\) 5.13440e17 + 8.89304e17i 0.702297 + 1.21641i
\(172\) −7.65761e17 1.32634e18i −0.999691 1.73151i
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 3.61904e17 + 8.01742e17i 0.411424 + 0.911444i
\(176\) 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
\(181\) −1.10757e18 −0.961482 −0.480741 0.876863i \(-0.659632\pi\)
−0.480741 + 0.876863i \(0.659632\pi\)
\(182\) 0 0
\(183\) 1.20912e18 0.961302
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.62008e18 1.61868e17i −0.995046 0.0994185i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −9.23379e17 1.59934e18i −0.500000 0.866025i
\(193\) −1.13278e18 1.96203e18i −0.588420 1.01917i −0.994440 0.105309i \(-0.966417\pi\)
0.406020 0.913864i \(-0.366916\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6.94269e17 2.06433e18i −0.318771 0.947832i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 9.54740e17 + 1.65366e18i 0.388205 + 0.672390i 0.992208 0.124592i \(-0.0397620\pi\)
−0.604003 + 0.796982i \(0.706429\pi\)
\(200\) 0 0
\(201\) −1.89900e18 + 3.28916e18i −0.712780 + 1.23457i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 2.58860e18 4.48359e18i 0.738853 1.27973i
\(209\) 0 0
\(210\) 0 0
\(211\) −7.82385e18 −1.99141 −0.995705 0.0925844i \(-0.970487\pi\)
−0.995705 + 0.0925844i \(0.970487\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.80966e17 1.06550e18i −0.0978219 0.216709i
\(218\) 0 0
\(219\) −4.71785e18 + 8.17155e18i −0.891643 + 1.54437i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.01164e19 1.65419 0.827096 0.562061i \(-0.189991\pi\)
0.827096 + 0.562061i \(0.189991\pi\)
\(224\) 0 0
\(225\) 6.56841e18 1.00000
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 5.12861e18 + 8.88301e18i 0.702297 + 1.21641i
\(229\) 6.27243e18 1.08642e19i 0.829377 1.43652i −0.0691508 0.997606i \(-0.522029\pi\)
0.898528 0.438917i \(-0.144638\pi\)
\(230\) 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
\(237\) 1.68940e19 1.69725
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −4.10913e18 7.11722e18i −0.361088 0.625423i 0.627052 0.778977i \(-0.284261\pi\)
−0.988140 + 0.153554i \(0.950928\pi\)
\(242\) 0 0
\(243\) −6.07883e18 + 1.05288e19i −0.500000 + 0.866025i
\(244\) 1.20775e19 0.961302
\(245\) 0 0
\(246\) 0 0
\(247\) −1.43775e19 + 2.49026e19i −1.03779 + 1.79750i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −1.61826e19 1.61686e18i −0.995046 0.0994185i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −9.22337e18 1.59753e19i −0.500000 0.866025i
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 5.94172e18 8.26705e18i 0.293436 0.408275i
\(260\) 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
\(268\) −1.89685e19 + 3.28545e19i −0.712780 + 1.23457i
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 2.46294e19 4.26593e19i 0.846640 1.46642i −0.0375498 0.999295i \(-0.511955\pi\)
0.884190 0.467128i \(-0.154711\pi\)
\(272\) 0 0
\(273\) −1.87577e19 4.15546e19i −0.607964 1.34685i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.29219e19 5.70223e19i −0.949830 1.64515i −0.745778 0.666195i \(-0.767922\pi\)
−0.204053 0.978960i \(-0.565411\pi\)
\(278\) 0 0
\(279\) −8.72931e18 −0.237764
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 4.02236e19 + 6.96693e19i 0.977663 + 1.69336i 0.670851 + 0.741592i \(0.265929\pi\)
0.306812 + 0.951770i \(0.400738\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.43306e19 + 4.21418e19i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) −5.12466e19 8.87617e19i −0.996601 1.72616i
\(292\) −4.71253e19 + 8.16233e19i −0.891643 + 1.54437i
\(293\) 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
\(300\) 6.56100e19 1.00000
\(301\) −1.34051e20 1.33935e19i −1.98948 0.198775i
\(302\) 0 0
\(303\) 0 0
\(304\) 5.12283e19 + 8.87299e19i 0.702297 + 1.21641i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.10946e20 1.40606 0.703028 0.711162i \(-0.251831\pi\)
0.703028 + 0.711162i \(0.251831\pi\)
\(308\) 0 0
\(309\) 1.32364e20 1.59258
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 6.52667e19 1.13045e20i 0.708495 1.22715i −0.256921 0.966433i \(-0.582708\pi\)
0.965415 0.260716i \(-0.0839588\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.68750e20 1.69725
\(317\) 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
\(324\) −6.07198e19 + 1.05170e20i −0.500000 + 0.866025i
\(325\) 9.19655e19 + 1.59289e20i 0.738853 + 1.27973i
\(326\) 0 0
\(327\) 1.15492e20 2.00038e20i 0.883423 1.53013i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.30376e20 + 2.25819e20i −0.904847 + 1.56724i −0.0837255 + 0.996489i \(0.526682\pi\)
−0.821122 + 0.570753i \(0.806651\pi\)
\(332\) 0 0
\(333\) −3.80107e19 6.58365e19i −0.251393 0.435425i
\(334\) 0 0
\(335\) 0 0
\(336\) −1.61643e20 1.61503e19i −0.995046 0.0994185i
\(337\) −1.91944e20 −1.15381 −0.576905 0.816811i \(-0.695740\pi\)
−0.576905 + 0.816811i \(0.695740\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.83095e20 5.63871e19i −0.955705 0.294325i
\(344\) 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
\(349\) 4.35999e20 1.98099 0.990494 0.137553i \(-0.0439236\pi\)
0.990494 + 0.137553i \(0.0439236\pi\)
\(350\) 0 0
\(351\) −3.40444e20 −1.47771
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 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
\(361\) −1.40310e20 2.43024e20i −0.486442 0.842543i
\(362\) 0 0
\(363\) 3.01476e20 1.00000
\(364\) −1.87365e20 4.15078e20i −0.607964 1.34685i
\(365\) 0 0
\(366\) 0 0
\(367\) −3.00369e20 5.20255e20i −0.912697 1.58084i −0.810238 0.586101i \(-0.800662\pi\)
−0.102459 0.994737i \(-0.532671\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −8.71947e19 −0.237764
\(373\) −2.63443e20 + 4.56297e20i −0.703099 + 1.21780i 0.264274 + 0.964448i \(0.414868\pi\)
−0.967373 + 0.253356i \(0.918466\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.95267e20 0.693588 0.346794 0.937941i \(-0.387270\pi\)
0.346794 + 0.937941i \(0.387270\pi\)
\(380\) 0 0
\(381\) 4.19030e20 + 7.25781e20i 0.943725 + 1.63458i
\(382\) 0 0
\(383\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.02983e20 + 8.71192e20i −0.999691 + 1.73151i
\(388\) −5.11888e20 8.86616e20i −0.996601 1.72616i
\(389\) 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
\(397\) 7.51236e19 1.30118e20i 0.121745 0.210869i −0.798711 0.601715i \(-0.794484\pi\)
0.920456 + 0.390846i \(0.127818\pi\)
\(398\) 0 0
\(399\) 8.97795e20 + 8.97018e19i 1.39764 + 0.139643i
\(400\) 6.55360e20 1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −1.22221e20 2.11693e20i −0.175673 0.304274i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.33246e20 + 1.27002e21i 0.936404 + 1.62190i 0.772112 + 0.635487i \(0.219201\pi\)
0.164292 + 0.986412i \(0.447466\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.32214e21 1.59258
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.00513e20 1.74094e20i −0.109935 0.190413i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 2.98700e20 0.302676 0.151338 0.988482i \(-0.451642\pi\)
0.151338 + 0.988482i \(0.451642\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.20033e20 8.62686e20i 0.561037 0.780602i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) −6.06513e20 + 1.05051e21i −0.500000 + 0.866025i
\(433\) −2.37647e21 −1.92322 −0.961609 0.274424i \(-0.911513\pi\)
−0.961609 + 0.274424i \(0.911513\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.15361e21 1.99812e21i 0.883423 1.53013i
\(437\) 0 0
\(438\) 0 0
\(439\) −9.01972e20 + 1.56226e21i −0.653848 + 1.13250i 0.328333 + 0.944562i \(0.393513\pi\)
−0.982181 + 0.187937i \(0.939820\pi\)
\(440\) 0 0
\(441\) −9.46265e20 + 1.07290e21i −0.661461 + 0.749980i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) −3.79679e20 6.57623e20i −0.251393 0.435425i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.61461e21 1.61321e20i −0.995046 0.0994185i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.53375e21 2.65653e21i 0.864906 1.49806i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.12710e20 3.68425e20i 0.111804 0.193651i −0.804693 0.593691i \(-0.797670\pi\)
0.916498 + 0.400040i \(0.131004\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −3.88510e21 −1.83973 −0.919864 0.392237i \(-0.871701\pi\)
−0.919864 + 0.392237i \(0.871701\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) −3.40060e21 −1.47771
\(469\) 1.37296e21 + 3.04157e21i 0.586510 + 1.29932i
\(470\) 0 0
\(471\) 2.41767e21 4.18753e21i 0.998228 1.72898i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.63999e21 −1.40459
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 1.06439e21 1.84358e21i 0.371485 0.643430i
\(482\) 0 0
\(483\) 0 0
\(484\) 3.01136e21 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 3.09453e21 + 5.35989e21i 0.978055 + 1.69404i 0.669460 + 0.742848i \(0.266526\pi\)
0.308595 + 0.951193i \(0.400141\pi\)
\(488\) 0 0
\(489\) 2.46382e21 0.753597
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −8.70963e20 −0.237764
\(497\) 0 0
\(498\) 0 0
\(499\) 1.19157e21 2.06386e21i 0.309967 0.536879i −0.668388 0.743813i \(-0.733015\pi\)
0.978355 + 0.206934i \(0.0663486\pi\)
\(500\) 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
\(507\) −2.58372e21 4.47513e21i −0.591808 1.02504i
\(508\) 4.18557e21 + 7.24963e21i 0.943725 + 1.63458i
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 3.41097e21 + 7.55646e21i 0.733686 + 1.62537i
\(512\) 0 0
\(513\) 3.36868e21 5.83473e21i 0.702297 1.21641i
\(514\) 0 0
\(515\) 0 0
\(516\) −5.02416e21 + 8.70210e21i −0.999691 + 1.73151i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) −1.08466e21 + 1.87868e21i −0.193766 + 0.335613i −0.946495 0.322717i \(-0.895404\pi\)
0.752729 + 0.658331i \(0.228737\pi\)
\(524\) 0 0
\(525\) 3.36826e21 4.68645e21i 0.583622 0.812026i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.06631e21 5.31100e21i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 8.96782e21 + 8.96006e20i 1.39764 + 0.139643i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.64546e21 2.85003e21i 0.224239 0.388393i −0.731852 0.681463i \(-0.761344\pi\)
0.956091 + 0.293071i \(0.0946772\pi\)
\(542\) 0 0
\(543\) 3.63337e21 + 6.29318e21i 0.480741 + 0.832667i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.22073e22 −1.52307 −0.761536 0.648122i \(-0.775555\pi\)
−0.761536 + 0.648122i \(0.775555\pi\)
\(548\) 0 0
\(549\) −3.96651e21 6.87020e21i −0.480651 0.832512i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.66322e21 1.20536e22i 0.990554 1.37821i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.00400e21 1.73898e21i −0.109935 0.190413i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) −2.81694e22 −2.95450
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.39495e21 + 9.73631e21i 0.411424 + 0.911444i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −3.31220e21 5.73690e21i −0.293108 0.507678i 0.681435 0.731879i \(-0.261356\pi\)
−0.974543 + 0.224201i \(0.928023\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −6.05829e21 + 1.04933e22i −0.500000 + 0.866025i
\(577\) −1.08257e22 1.87506e22i −0.881148 1.52619i −0.850066 0.526676i \(-0.823438\pi\)
−0.0310817 0.999517i \(-0.509895\pi\)
\(578\) 0 0
\(579\) −7.43217e21 + 1.28729e22i −0.588420 + 1.01917i
\(580\) 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 \(\pi\)
\(588\) −9.45198e21 + 1.07169e22i −0.661461 + 0.749980i
\(589\) 4.83749e21 0.333962
\(590\) 0 0
\(591\) 0 0
\(592\) −3.79251e21 6.56881e21i −0.251393 0.435425i
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.26405e21 1.08497e22i 0.388205 0.672390i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 2.87461e22 1.68882 0.844411 0.535696i \(-0.179951\pi\)
0.844411 + 0.535696i \(0.179951\pi\)
\(602\) 0 0
\(603\) 2.49186e22 1.42556
\(604\) 1.53202e22 2.65354e22i 0.864906 1.49806i
\(605\) 0 0
\(606\) 0 0
\(607\) 2.19870e21 3.80826e21i 0.119304 0.206641i −0.800188 0.599749i \(-0.795267\pi\)
0.919492 + 0.393108i \(0.128600\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.98133e22 + 3.43176e22i 0.993740 + 1.72121i 0.593622 + 0.804744i \(0.297698\pi\)
0.400118 + 0.916464i \(0.368969\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.03008e22 + 3.51621e22i 0.941865 + 1.63136i 0.761911 + 0.647682i \(0.224262\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −3.39676e22 −1.47771
\(625\) −1.16415e22 + 2.01637e22i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 2.41494e22 4.18280e22i 0.998228 1.72898i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.02373e22 −0.407336 −0.203668 0.979040i \(-0.565286\pi\)
−0.203668 + 0.979040i \(0.565286\pi\)
\(632\) 0 0
\(633\) 2.56661e22 + 4.44550e22i 0.995705 + 1.72461i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.92674e22 7.92581e21i −1.44850 0.292367i
\(638\) 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
\(643\) −3.51207e22 −1.20192 −0.600960 0.799279i \(-0.705215\pi\)
−0.600960 + 0.799279i \(0.705215\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.47637e21 + 6.22822e21i −0.138764 + 0.193071i
\(652\) 2.46105e22 0.753597
\(653\) 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
\(657\) 6.19076e22 1.78329
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −7.80091e21 1.35116e22i −0.214059 0.370761i 0.738922 0.673791i \(-0.235335\pi\)
−0.952981 + 0.303030i \(0.902002\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.31867e22 5.74811e22i −0.827096 1.43257i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.33834e22 1.98134 0.990670 0.136283i \(-0.0435155\pi\)
0.990670 + 0.136283i \(0.0435155\pi\)
\(674\) 0 0
\(675\) −2.15477e22 3.73216e22i −0.500000 0.866025i
\(676\) −2.58080e22 4.47008e22i −0.591808 1.02504i
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) −8.96092e22 8.95317e21i −1.98333 0.198161i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 3.36488e22 5.82814e22i 0.702297 1.21641i
\(685\) 0 0
\(686\) 0 0
\(687\) −8.23068e22 −1.65875
\(688\) −5.01849e22 + 8.69228e22i −0.999691 + 1.73151i
\(689\) 0 0
\(690\) 0 0
\(691\) −4.88313e22 + 8.45782e22i −0.939450 + 1.62718i −0.172950 + 0.984931i \(0.555330\pi\)
−0.766500 + 0.642244i \(0.778003\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 3.36446e22 4.68117e22i 0.583622 0.812026i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 2.10643e22 + 3.64844e22i 0.353105 + 0.611595i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −5.50674e22 + 9.53795e22i −0.862431 + 1.49378i 0.00714362 + 0.999974i \(0.497726\pi\)
−0.869575 + 0.493801i \(0.835607\pi\)
\(710\) 0 0
\(711\) −5.54209e22 9.59918e22i −0.848626 1.46986i
\(712\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 6.78757e22 9.44392e22i 0.929463 1.29321i
\(722\) 0 0
\(723\) −2.69600e22 + 4.66961e22i −0.361088 + 0.625423i
\(724\) 3.62927e22 + 6.28608e22i 0.480741 + 0.832667i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.28031e23 1.64073 0.820367 0.571837i \(-0.193769\pi\)
0.820367 + 0.571837i \(0.193769\pi\)
\(728\) 0 0
\(729\) 7.97664e22 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −3.96204e22 6.86245e22i −0.480651 0.832512i
\(733\) 4.02029e22 6.96335e22i 0.482420 0.835577i −0.517376 0.855758i \(-0.673091\pi\)
0.999796 + 0.0201815i \(0.00642441\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 7.98365e22 + 1.38281e23i 0.897524 + 1.55456i 0.830650 + 0.556795i \(0.187969\pi\)
0.0668739 + 0.997761i \(0.478697\pi\)
\(740\) 0 0
\(741\) 1.88662e23 2.07558
\(742\) 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
\(751\) −5.83907e22 + 1.01136e23i −0.577064 + 0.999505i 0.418750 + 0.908102i \(0.362468\pi\)
−0.995814 + 0.0914030i \(0.970865\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 4.38999e22 + 9.72533e22i 0.411424 + 0.911444i
\(757\) −4.29074e21 −0.0397892 −0.0198946 0.999802i \(-0.506333\pi\)
−0.0198946 + 0.999802i \(0.506333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) −8.34997e22 1.84980e23i −0.726923 1.61038i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −6.05145e22 + 1.04814e23i −0.500000 + 0.866025i
\(769\) 1.30797e22 0.106951 0.0534756 0.998569i \(-0.482970\pi\)
0.0534756 + 0.998569i \(0.482970\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.42379e22 + 1.28584e23i −0.588420 + 1.01917i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 1.54714e22 2.67973e22i 0.118882 0.205910i
\(776\) 0 0
\(777\) −6.64651e22 6.64076e21i −0.500294 0.0499862i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −9.44132e22 + 1.07048e23i −0.661461 + 0.749980i
\(785\) 0 0
\(786\) 0 0
\(787\) −8.09332e22 1.40180e23i −0.549957 0.952553i −0.998277 0.0586800i \(-0.981311\pi\)
0.448320 0.893873i \(-0.352023\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.11072e23 1.92382e23i 0.710261 1.23021i
\(794\) 0 0
\(795\) 0 0
\(796\) 6.25699e22 1.08374e23i 0.388205 0.672390i
\(797\) 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
\(804\) 2.48905e23 1.42556
\(805\) 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
\(811\) −3.66385e23 −1.95781 −0.978906 0.204312i \(-0.934504\pi\)
−0.978906 + 0.204312i \(0.934504\pi\)
\(812\) 0 0
\(813\) −3.23186e23 −1.69328
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.78736e23 4.82785e23i 1.40416 2.43208i
\(818\) 0 0
\(819\) −1.74579e23 + 2.42901e23i −0.862422 + 1.19994i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) −1.61208e23 2.79221e23i −0.765929 1.32663i −0.939754 0.341851i \(-0.888946\pi\)
0.173826 0.984776i \(-0.444387\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 8.25302e22 + 1.42947e23i 0.369979 + 0.640822i 0.989562 0.144109i \(-0.0460316\pi\)
−0.619583 + 0.784931i \(0.712698\pi\)
\(830\) 0 0
\(831\) −2.16000e23 + 3.74123e23i −0.949830 + 1.64515i
\(832\) −3.39293e23 −1.47771
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.86365e22 + 4.95999e22i 0.118882 + 0.205910i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.50246e23 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.56372e23 + 4.44049e23i 0.995705 + 1.72461i
\(845\) 0 0
\(846\) 0 0
\(847\) 1.54596e23 2.15098e23i 0.583622 0.812026i
\(848\) 0 0
\(849\) 2.63907e23 4.57100e23i 0.977663 1.69336i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −3.67842e23 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) −2.57599e22 + 4.46175e22i −0.0868958 + 0.150508i −0.906197 0.422855i \(-0.861028\pi\)
0.819302 + 0.573363i \(0.194361\pi\)
\(860\) 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
\(867\) 3.19266e23 1.00000
\(868\) −4.47132e22 + 6.22120e22i −0.138764 + 0.193071i
\(869\) 0 0
\(870\) 0 0
\(871\) 3.48890e23 + 6.04296e23i 1.05328 + 1.82433i
\(872\) 0 0
\(873\) −3.36229e23 + 5.82366e23i −0.996601 + 1.72616i
\(874\) 0 0
\(875\) 0 0
\(876\) 6.18378e23 1.78329
\(877\) 3.27719e23 5.67626e23i 0.936494 1.62205i 0.164545 0.986370i \(-0.447384\pi\)
0.771949 0.635685i \(-0.219282\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −3.26584e23 −0.883708 −0.441854 0.897087i \(-0.645679\pi\)
−0.441854 + 0.897087i \(0.645679\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 7.32711e23 + 7.32077e22i 1.87810 + 0.187647i
\(890\) 0 0
\(891\) 0 0
\(892\) −3.31493e23 5.74163e23i −0.827096 1.43257i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −2.15234e23 3.72796e23i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 3.63653e23 + 8.05615e23i 0.822593 + 1.82232i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.55648e23 + 7.89206e23i 0.994881 + 1.72318i 0.584953 + 0.811067i \(0.301113\pi\)
0.409928 + 0.912118i \(0.365554\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 3.36109e23 5.82157e23i 0.702297 1.21641i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −8.22140e23 −1.65875
\(917\) 0 0
\(918\) 0 0
\(919\) −4.91158e23 + 8.50711e23i −0.965378 + 1.67208i −0.256782 + 0.966469i \(0.582662\pi\)
−0.708596 + 0.705614i \(0.750671\pi\)
\(920\) 0 0
\(921\) −3.63957e23 6.30392e23i −0.703028 1.21768i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.69474e23 0.502785
\(926\) 0 0
\(927\) −4.34218e23 7.52088e23i −0.796288 1.37921i
\(928\) 0 0
\(929\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 5.24388e23 5.94563e23i 0.929084 1.05342i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.13082e24 −1.90317 −0.951585 0.307387i \(-0.900546\pi\)
−0.951585 + 0.307387i \(0.900546\pi\)
\(938\) 0 0
\(939\) −8.56429e23 −1.41699
\(940\) 0 0
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 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 \(0.666667\pi\)
\(948\) −5.53584e23 9.58835e23i −0.848626 1.46986i
\(949\) 8.66780e23 + 1.50131e24i 1.31759 + 2.28213i
\(950\) 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
\(961\) 3.43150e23 5.94354e23i 0.471734 0.817067i
\(962\) 0 0
\(963\) 0 0
\(964\) −2.69296e23 + 4.66434e23i −0.361088 + 0.625423i
\(965\) 0 0
\(966\) 0 0
\(967\) −1.52912e24 −1.99999 −0.999997 0.00234625i \(-0.999253\pi\)
−0.999997 + 0.00234625i \(0.999253\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 7.96765e23 1.00000
\(973\) −1.75756e23 1.75604e22i −0.218780 0.0218591i
\(974\) 0 0
\(975\) 6.03386e23 1.04509e24i 0.738853 1.27973i
\(976\) −3.95757e23 6.85471e23i −0.480651 0.832512i
\(977\) 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
\(981\) −1.51548e24 −1.76685
\(982\) 0 0
\(983\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.88449e24 2.07558
\(989\) 0 0
\(990\) 0 0
\(991\) 3.08386e23 + 5.34141e23i 0.331517 + 0.574204i 0.982810 0.184622i \(-0.0591062\pi\)
−0.651293 + 0.758827i \(0.725773\pi\)
\(992\) 0 0
\(993\) 1.71080e24 1.80969
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 7.18443e22 + 1.24438e23i 0.0735920 + 0.127465i 0.900473 0.434912i \(-0.143220\pi\)
−0.826881 + 0.562377i \(0.809887\pi\)
\(998\) 0 0
\(999\) −2.49389e23 + 4.31954e23i −0.251393 + 0.435425i
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 21.17.h.a.2.1 2
3.2 odd 2 CM 21.17.h.a.2.1 2
7.4 even 3 inner 21.17.h.a.11.1 yes 2
21.11 odd 6 inner 21.17.h.a.11.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.17.h.a.2.1 2 1.1 even 1 trivial
21.17.h.a.2.1 2 3.2 odd 2 CM
21.17.h.a.11.1 yes 2 7.4 even 3 inner
21.17.h.a.11.1 yes 2 21.11 odd 6 inner